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1210.4607
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# Event anisotropy $v_{2}$ in Au+Au collisions at $\sqrt{s_{NN}}=$ 7.7 - 62.4
GeV with STAR
Shusu Shia,b (for the STAR Collaboration) Institute of Particle Physics,
Huazhong Normal University, Wuhan, Hubei, 430079, China Key Laboratory of
Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central
China Normal University, Wuhan 430079, China
###### Abstract
We present the $v_{2}$ measurement at midrapidity from Au+Au collisions at
$\sqrt{s_{NN}}=$ 7.7, 11.5, 19.6, 27, 39 and 62.4 GeV for inclusive charged
hadrons and identified hadrons ($\pi^{\pm}$, $K^{\pm}$, $K_{S}^{0}$, $p$,
$\bar{p}$, $\phi$, $\Lambda$, $\bar{\Lambda}$, $\Xi^{-}$, $\bar{\Xi}^{+}$,
$\Omega^{-}$, $\bar{\Omega}^{+}$) up to 4 GeV/$c$ in $p_{T}$. The beam energy
and centrality dependence of charged hadron $v_{2}$ are presented with
comparison to higher energies at RHIC and LHC. The identified hadron $v_{2}$
are used to discuss the NCQ scaling for different beam energies. Significant
difference in $v_{2}(p_{T})$ is observed between particles and corresponding
anti-particles for $\sqrt{s_{NN}}<$ 39 GeV. These differences are more
pronounced for baryons compared to mesons and they increase with decreasing
energy.
††journal: Nuclear Physics A
## 1 Introduction
Searching for the phase boundary in the Quantum ChromoDynamics (QCD) phase
diagram is one of the main motivations of the Beam Energy Scan (BES) program
at RHIC. The elliptic flow ($v_{2}$) could be used as a powerful tool [1],
because of the sensitivity of underlying dynamics in the early stage of the
collisions. The Number of Constituent Quark (NCQ) scaling in the top energy
collisions at RHIC ($\sqrt{\mathrm{\it s_{NN}}}$ = 200 GeV) indicates the
collectivity has been built up in the partonic level [2, 3]. Recently, the
similar NCQ scaling of multi-strange hadrons, $\phi$ and $\Omega$, which are
less sensitive to the late hadronic interactions provides the clear evidence
of partonic collectivity [4]. An energy dependence study based on A Multi-
Phase Transport (AMPT) model indicates the NCQ scaling is related to the
degrees of freedom in the system [5]. If the partonic degree of freedom is
included in the AMPT model, the NCQ scaling (including multi-strange hadrons)
could be observed; whereas the NCQ scaling is broken in the case of including
only hadronic degree of freedom. The BES data offer us the opportunity to
investigate the QCD phase boundary with $v_{2}$ measurements. In this paper,
we present the $v_{2}$ results from the STAR experiment in Au+Au collisions at
$\sqrt{\mathrm{\it s_{NN}}}$ = 7.7 - 62.4 GeV. The particle identification for
$\pi^{\pm}$, $K^{\pm}$ and $p~{}(\overline{p})$ is achieved via the energy
loss in the Time Projection Chamber (TPC) [6] and the time of flight
information from the multi-gap resistive plate chamber detector [7]. Strange
and multi-strange hadrons are reconstructed with the decay channels:
${K}^{0}_{S}$ $\rightarrow\pi^{+}+\pi^{-}$, $\phi\rightarrow K^{+}+K^{-}$,
$\Lambda$ $\rightarrow p+\pi^{-}$
($\overline{\Lambda}\rightarrow\overline{p}+\pi^{+}$), $\Xi^{-}\rightarrow$
$\Lambda$ $+\ \pi^{-}$ ($\overline{\Xi}^{+}\rightarrow$ $\overline{\Lambda}$\+
$\pi^{+}$) and $\Omega^{-}\rightarrow$ $\Lambda$ $+\ K^{-}$
($\overline{\Omega}^{+}\rightarrow$ $\overline{\Lambda}$\+ $K^{+}$). The event
plane method [8] and cumulant method [9, 10] are used for the $v_{2}$
measurement.
## 2 Results and Discussions
Figure 1: (Color online) The top panels show $v_{2}\\{4\\}$ vs. $p_{T}$ at
midrapidity for various collision energies ($\sqrt{\mathrm{\it s_{NN}}}$ = 7.7
GeV to 2.76 TeV). The results for $\sqrt{\mathrm{\it s_{NN}}}$ = 7.7 to 200
GeV are for Au+Au collisions and those for 2.76 TeV are for Pb + Pb
collisions. The dashed red curves show the fifth order polynomial fits to the
results from Au+Au collisions at $\sqrt{\mathrm{\it s_{NN}}}$ = 200 GeV. The
bottom panels show the ratio of $v_{2}\\{4\\}$ vs. $p_{T}$ for all
$\sqrt{\mathrm{\it s_{NN}}}$ with respect to the fit curve. The results are
shown for three collision centrality classes: $10-20\%$ (a1), $20-30\%$ (b1)
and $30-40\%$ (c1) [11]. Error bars are shown only for the statistical
uncertainties respectively. Figure 2: (Color online) The difference in $v_{2}$
between particles and their corrsponding anti-particles
($v_{2}(X)-v_{2}(\bar{X})$) as a function of beam energy in Au+Au collisions
(0-80%). The statistical and systematic uncertainties are shown by vertical
line and cap respectively. The dashed lines in the plot are fits with the
equation described in the text. Figure 3: (Color online) The elliptic flow
($v_{2}$) as a function of transverse mass ($m_{T}-m_{0}$) for the selected
particles in Au+Au collisions (0-80%) at $\sqrt{\mathrm{\it s_{NN}}}$ = 7.7,
11.5, 19.6, 27, 39 and 62.4 GeV. Error bars are shown only for the statistical
uncertainties.
Figure 1 [11] shows the transverse momentrum ($p_{T}$) dependence of
$v_{2}\\{4\\}$ from $\sqrt{\mathrm{\it s_{NN}}}$ = 7.7 GeV to 2.76 TeV in
$10-20\%$ (a1), $20-30\%$ (b1) and $30-40\%$ (c1) centrality bins. The ALICE
results in Pb + Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV are taken from
Ref. [12]. The 200 GeV data is empirically fit by a fifth order polynomial
function. For comparison, the $v_{2}$ from other energies are divided by the
fit and shown in the lower panels of Fig. 1. For $p_{T}$ below 2
$\mathrm{GeV}/c$, the $v_{2}$ values rise with increasing collision energy.
Beyond $p_{T}=2~{}\mbox{$\mathrm{GeV}/c$}$ the $v_{2}$ results show comparable
values within statistical errors. The increase of $v_{2}(p_{T})$ as a function
of energy could be due to the change of chemical composition from low to high
energies and/or larger collectivity at higher collision energy.
Figure 2 shows the $p_{T}$ independent difference in $v_{2}$ between particles
and their corresponding anti-particles. The data of collisions at
$\sqrt{\mathrm{\it s_{NN}}}$ = 19.6 and 27 GeV is new since Quark Matter 2011
[13]. The $\eta$-sub event plane method is used for the measurement. In this
method, one defines the flow vector for each particle based on particles
measured in the opposite hemisphere in pseudorapidity ($\eta$). An $\eta$ gap
of $|\eta|<0.05$ is used between negative/positive $\eta$ sub-event to reduce
the non-flow effects by enlarging the separation between the correlated
particles. $v_{2}(X)-v_{2}(\bar{X})$ ($\Delta v_{2}$) represents the average
values of the difference in $v_{2}$ between particles and corresponding anti-
particles over measured $p_{T}$ range. The dashed lines in the Fig. 2 are fits
with the function: $f_{\Delta v_{2}}(\sqrt{s_{NN}})=a\sqrt{s_{NN}}^{-b}+c$. A
monotonic increase of $\Delta v_{2}$ with decreasing collision energy is
observed and the slope of the difference increases towards lowers energies.
The difference is more pronounced for baryons compared to mesons. The observed
difference in $v_{2}$ reflects that the particles vs. anti-particles could not
fit into the NCQ scaling. The breaking of NCQ scaling between particles and
anti-particles indicates the contributions from hadronic interactions increase
in the system evolution with decreasing collision energy. The energy
dependence of $\Delta v_{2}$ could be qualitatively reproduced by the baryon
transport effect [14] or hadronic potential effect [15]. So far theoretical
calculations fail to quantitatively reproduce the measured $v_{2}$ difference
and none of the calculations could explain the correct order of particles.
Figure 3 shows the $v_{2}$ as a function of transverse mass ($m_{T}-m_{0}$)
for the selected particles for all six collision energies. In the top energy
($\sqrt{\mathrm{\it s_{NN}}}$ = 200 GeV) collisions, a clear splitting in
$v_{2}$ between baryons and mesons is observed for $m_{T}-m_{0}>1~{}{\rm
GeV}/c^{2}$. The splitting between baryons and mesons suggest the system
created in the collisions is sensitive to the quark degree of freedom. The
selected particles show a similar splitting for collision energy $\geq$ 39
GeV. The bayron and meson groups become closer to each other at all lower
energies. At $\sqrt{\mathrm{\it s_{NN}}}$ = 11.5 GeV, the splitting between
baryons and mesons is almost gone. The clear trend, a decreasing baryon-meson
splitting of $v_{2}(m_{T}-m_{0})$ beyond $m_{T}-m_{0}>1~{}{\rm GeV}/c^{2}$
indicates the hadronic interactions become more important in the lower
collision energies.
## 3 Summary
In summary, we present the $v_{2}$ measurements for charged hadrons and
identified hadrons in Au+Au collisions at $\sqrt{\mathrm{\it s_{NN}}}$ = 7.7 -
62.4 GeV. The comparison with Au+Au collisions at higher energies at RHIC
($\sqrt{\mathrm{\it s_{NN}}}$ = 62.4 and 200 GeV) and at LHC (Pb + Pb
collisions at $\sqrt{\mathrm{\it s_{NN}}}$ = 2.76 TeV) shows the
$v_{2}\\{4\\}$ values at low $p_{T}$ ($p_{T}<$ 2.0 GeV/$c$) increase with
increase in collision energy. The baryon and anti-baryon $v_{2}$ show
significant difference for $\sqrt{\mathrm{\it s_{NN}}}$ $<~{}$ 39 GeV. The
difference of $v_{2}$ between particles and corresponding anti-particles
(pions, kaons, protons, $\Lambda$s and $\Xi$s) increases with decreasing the
beam energy. The baryon-meson splitting of $v_{2}(m_{T}-m_{0})$ beyond
$m_{T}-m_{0}>1~{}{\rm GeV}/c^{2}$ becomes smaller in the lower collisions
energy and is almost gone in collisions at $\sqrt{\mathrm{\it s_{NN}}}$ = 11.5
GeV. Experimental data indicate that the hadronic interactions become more
important at the lower beam energy.
## 4 Acknowledgments
This work was supported in part by the National Natural Science Foundation of
China under grant No. 11105060, 10775060, 11135011, 11221504 and self
determined research funds of CCNU from the colleges’ basic research and
operation of MOE.
## References
* [1] S. A. Voloshin, A. M. Poskanzer and R. Snellings, arXiv:0809.2949.
* [2] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 92, 052302 (2004).
* [3] B. I. Abelev et al., (STAR Collaboration), Phys. Rev. C 81, 044902 (2010).
* [4] S. S. Shi (for the STAR collaboration), Nucl. Phys. A 830, 187c (2009); Nucl. Phys. A 862-863, 263 (2011).
* [5] F. Liu, K.J. Wu, and N. Xu, J. Phys. G 37 094029(2010).
* [6] K. H. Ackermann et al. (STAR Collaboration), Nucl. Instrum. Methods A 499, 624 (2003).
* [7] W. J. Llope (STAR TOF Group), Nucl. Instr. and Meth. B 241, 306 (2005).
* [8] A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58 1671 (1998).
* [9] N. Borghini, P. M. Dinh, and J.-Y. Ollitrault, Phys. Rev. C 63, 054906 (2001).
* [10] A. Bilandzic, R. Snellings and S.Voloshin, Phys. Rev. C 83, 044913 (2011).
* [11] L. Adamczyk et al., (STAR Collaboration), arXiv:1206.5528.
* [12] K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 105, 252302 (2010).
* [13] A. Schmah (for the STAR collaboration), J. Phys. G 38 124049 (2011).
* [14] J. Dunlop, M.A. Lisa and P. Sorensen, Phys. Rev. C 84, 044914 (2011).
* [15] J. Xu et al., Phys. Rev. C 85, 041901 (2012).
|
arxiv-papers
| 2012-10-17T01:21:13 |
2024-09-04T02:49:36.687732
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shusu Shi (for the STAR collaboration)",
"submitter": "Shusu Shi",
"url": "https://arxiv.org/abs/1210.4607"
}
|
1210.4637
|
Interacting Two-Fluid Viscous Dark Energy Models In Non-Flat Universe
Hassan Amirhashchi1, Anirudh Pradhan2, H. Zainuddin3
1Young Researchers Club, Mahshahr Branch, Islamic Azad University, Mahshahr,
Iran
E-mail: h.amirhashchi@mahshahriau.ac.ir
2Department of Mathematics, Hindu Post-graduate College, Zamania-232 331,
Ghazipur, India
E-mail: pradhan@iucaa.ernet.in
2,3Laboratory of Computational Sciences and Mathematical Physics, Institute
for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang,
Selangor D.E., Malaysia
E-mail: hisham@putra.upm.edu.my
###### Abstract
We study the evolution of the dark energy parameter within the scope of a
spatially non-flat and isotropic Friedmann-Robertson-Walker (FRW) model filled
with barotropic fluid and bulk viscous stresses. We have obtained cosmological
solutions which exhibit without a big rip singularity. It is concluded that in
both non-interacting and interacting cases non-flat open universe crosses the
phantom region. We find that during the evolution of the universe, the
equation of state (EoS) for dark energy $\omega_{D}$ changes from
$\omega^{eff}_{D}<-1$ to $\omega^{eff}_{D}>-1$, which is consistent with
recent observations.
Keywords: FRW universe, Dark energy, Viscous Fluid
PACS number: 98.80.Es, 98.80-k, 95.36.+x
## 1 INTRODUCTION
Observations of distant Supernovae (SNe Ia) (Perlmutter et al. 1997, 1998,
1999; Riess et al. 1998, 2000; Garnavich et al. 1998a,b; Schmidt et al. 1998;
Tonry et al. 2003; Clocchiatti et al. 2006), fluctuation of cosmic microwave
background radiation (CMBR) (de Bernardis et al. 1998; Hanany et al. 2000),
large scale structure (LSS) (Spergel et al. 2003; Tegmark et al. 2004), sloan
digital sky survey (SDSS) (Seljak et al. 2005; Adelman-McCarthy et al. 2006),
Wilkinson microwave anisotropy probe (WMAP) (Bennett. et al 2003) and Chandra
x-ray observatory (Allen et al. 2004) by means of ground and altitudinal
experiments have established that our Universe is undergoing a late-time
accelerating expansion, and we live in a priviledged spatially flat Universe
composed of approximately $4\%$ baryonic matter, $22\%$ dark matter and $74\%$
dark energy. The simplest candidate for dark energy is the cosmological
constant. Recently, a great number of theme have been proposed to explain the
current accelerating Universe, partly such as scalar field model, exotic
equation of state (EoS), modified gravity, and the inhomogeneous cosmology
model. There are several dark energy models which can be distinguished by, for
instance, their EoS ($\omega=\frac{p_{de}}{\rho_{de}}$) during the evolution
of the universe.
The introduction of viscosity into cosmology has been investigated from
different view points (Gr$\o$n 1990; Padmanabhan $\&$ Chitre 1987; Barrow
1986; Zimdahl 1996; Farzin et al. 2012). Misner (1966, 1967;) noted that the
“measurement of the isotropy of the cosmic background radiation represents the
most accurate observational datum in cosmology”. An explanation of this
isotropy was provided by showing that in large class of homogeneous but
anisotropic universe, the anisotropy dies away rapidly. It was found that the
most important mechanism in reducing the anisotropy is neutrino viscosity at
temperatures just above $10^{10}K$ (when the Universe was about 1 s old: cf.
Zel’dovich and Novikov (Zel’dovich $\&$ Novikov 1971)). The astrophysical
observations also indicate some evidences that cosmic media is not a perfect
fluid (Jaffe et al. 2005), and the viscosity effect could be concerned in the
evolution of the universe (Brevik $\&$ Gorbunova, 2005; Brevik et al. 2005;
Cataldo et al. 2005). On the other hand, in the standard cosmological model,
if the EoS parameter $\omega$ is less than $-1$, so-called phantom, the
universe shows the future finite time singularity called the Big Rip (Caldwell
et al. 2003; Nojiri et al. 2005) or Cosmic Doomsday. Several mechanisms are
proposed to prevent the future big rip, like by considering quantum effects
terms in the action (Nojiri $\&$ Odintsov 2004; Elizalde et al. 2004), or by
including viscosity effects for the Universe evolution (Meng et al. 2007). A
well known result of the FRW cosmological solutions, corresponding to
universes filled with perfect fluid and bulk viscous stresses, is the
possibility of violating dominant energy condition (Barrow 1987, 1988;
Folomeev $\&$ Gurovich 2008; Ren $\&$ Meng 2006; Brevikc $\&$ Gorbunovac 2005;
Nojiri $\&$ Odintsov 2005). Setare (Setare 2007a,b,c) and Setare and Saridakis
(Setare $\&$ Saridakis 2000) have studied the interacting models of dark
energy in different context. Interacting new agegraphic viscous dark energy
with varying $G$ has been studied by Sheykhi and Setare (Sheykhi $\&$ Setare
2010).
Recently, Amirhashchi et al. (2011a,b); Pradhan et al. (2011); Saha et al.
(2012) have studied the two-fluid scenario for dark energy in FRW universe in
different context. Very recently Singh and Chaubey (2012) have studied
interacting dark energy in Bianchi type I space-time. Some experimental data
implied that our universe is not a perfectly flat universe and recent papers
(Spergel et al. 2003; Bennett et al. 2003; Ichikawa et al. 2006) favoured a
universe with spatial curvature. Setare et al. (2009) have studied the tachyon
cosmology in non-interacting and interacting cases in non-flat FRW universe.
Due to these considerations and motivations, in this Letter, we study the
evolution of the dark energy parameter within the framework of a FRW open
cosmological model filled with two fluids (i.e., barotropic fluid and bulk
viscous stresses). In doing so we consider both interacting and non-
interacting cases.
## 2 THE METRIC AND FIELD EQUATIONS
We consider the spherically symmetric Friedmann-Robertson-Walker (FRW) metric
as
$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (1)
where $a(t)$ is the scale factor and the curvature constant $k$ is $-1,0,+1$
respectively for open, flat and close models of the universe.
The Einstein’s field equations (with $8\pi G=1$ and $c=1$) read as
$R^{j}_{i}-\frac{1}{2}R\delta^{j}_{i}=-T^{j}_{i},$ (2)
where the symbols have their usual meaning and $T^{j}_{i}$ is the two-fluid
energy-momentum tensor due to bulk viscous dark and barotropic fluids written
in the form.
$T^{j}_{i}=(\rho+\bar{p})u^{j}_{i}+\bar{p}g^{j}_{i},$ (3)
where
$\bar{p}=p-\xi u^{i}_{;i}$ (4)
and
$u^{i}u_{i}=-1,$ (5)
where $\rho$ is the energy density; $p$, the pressure; $\xi$, the bulk-viscous
coefficient; and $u^{i}$, the four-velocity vector of the distribution. Here
after the semi-colon denotes covariant differentiation.
The expansion factor $\theta$ is defined by
$\theta=u^{i}_{;i}=3\frac{\dot{a}}{a}$. Hence Eq. (4) leads to
$\bar{p}=p-3\xi H,$ (6)
where $H$ is Hubble’s constant defined by
$H=\frac{\dot{a}}{a}.$ (7)
Now with the aid of Equations (3)-(5) and metric (1), the surviving field
equations (2) take the explicit forms
$\rho=3\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right),$ (8)
and
$\bar{p}=-\left(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a}+\frac{k}{a^{2}}\right).$
(9)
Also in space-time (1) the Bianchi identity for the bulk-viscous fluid
distribution $G^{;j}_{ij}=0$ leads to $T^{;j}_{ij}=0$ which yields
$\rho u^{i}+(\rho+\bar{p})u^{i}_{;i}$ (10)
which leads to
$\dot{\rho}+3H(\rho+\bar{p})=0.$ (11)
Using Eq. (7) in Eqs. (8) and (9) we get
$\rho=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right),$ (12)
and
$\bar{p}=-\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right),$ (13)
where $\bar{p}=p_{m}+\bar{p}_{D}$ and $\rho=\rho_{m}+\rho_{D}$. Here $p_{m}$
and $\rho_{m}$ are pressure and energy density of barotropic fluid and $p_{D}$
and $\rho_{D}$ are pressure and energy density of dark fluid respectively.
The equation of state (EoS) for the barotropic fluid $\omega_{m}$ and dark
field $\omega_{D}$ are given by
$\omega_{m}=\frac{p_{m}}{\rho_{m}},$ (14)
and
$\omega_{D}=\frac{\bar{p}_{D}}{\rho_{D}},$ (15)
respectively.
From Eqs. (11)-(13) we obtain
$\frac{\dot{\rho}}{3H}=\frac{2k}{a^{2}}e^{-2Ht}.$ (16)
Now we assume
$\rho=\alpha\theta^{2}~{}\mbox{or}~{}\rho=9\alpha H^{2},$ (17)
where $\alpha$ is an arbitrary constant. Eq. (17) ensure us that our universe
approaches homogeneity (Collins 1977). This condition has also been used by
Banerjee et al. (1986) for deriving a viscous-fluid cosmological model with
Bianchi type II space time.
Putting Eq. (17) in Eq. (16) and after integrating we get
$e^{-2Ht}=-\frac{3\alpha A^{2}}{2kt^{2}},$ (18)
which yields
$H=\frac{1}{2t}\ln\left(-\frac{2kt^{2}}{3\alpha A^{2}}\right),$ (19)
where $A$ is an arbitrary constant. From Eq. (19), we observe that the
condition given by (17) restrict our study to the case when $k=-1$ (i.e. only
for open universe). In the following sections we deal with two cases, (i) non-
interacting two-fluid model and (ii) interacting two-fluid model.
## 3 NON-INTERACTING TWO-FLUID MODEL
In this section we assume that two-fluid do not interact with each other.
Therefor, the general form of conservation equation (11) leads us to write the
conservation equation for the dark and barotropic fluid separately as,
$\dot{\rho}_{m}+3\frac{\dot{a}}{a}\left(\rho_{m}+p_{m}\right)=0,$ (20)
and
$\dot{\rho}_{D}+3\frac{\dot{a}}{a}\left(\rho_{D}+\bar{p}_{D}\right)=0.$ (21)
Integration Eq. (20) and using (7) leads to
$\rho_{m}=\rho_{0}a^{-3(1+\omega_{m})}~{}\mbox{or}~{}\rho_{m}=\rho_{0}Be^{-3H(1+\omega_{m})t},$
(22)
where $\rho_{0}$ is an integrating constant and $B=A^{-3(1+\omega_{m})}$. By
using Eq. (22) in Eqs. (12) and (13), we first obtain the $\rho_{D}$ and
$p_{D}$ in term of Hubble’s constant $H$ as
$\rho_{D}=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t},$
(23)
and
$\bar{p}_{D}=\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}.$
(24)
Figure 1: The plot of $\rho_{D}$ vs $t$ for $\alpha=0.1,A=100,\omega_{m}=0.5$
in both non-interacting and interacting two-fluid model Figure 2: The plot of
EoS parameter $\omega^{eff}_{D}$ vs $t$ for
$\rho_{0}=10,\omega_{m}=0.5,\alpha=0.01,B=1$ in non-interacting two-fluid
model
respectively. By using Eqs. (23) and (24) in Eq. (15), we can find the EoS of
dark energy in term of time as
$\omega_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t}}.$
(25)
Therefore the effective EoS parameter for viscous DE can be written as
$\omega^{eff}_{D}=\omega_{D}-\frac{3\xi
H}{\rho_{D}}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+3\xi
H+\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t}}.$
(26)
The expressions for the matter-energy density $\Omega_{m}$ and dark-energy
density $\Omega_{D}$ are given by
$\Omega_{m}=\frac{\rho_{m}}{3H^{2}}=\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m})}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (27)
and
$\Omega_{D}=\frac{\rho_{D}}{3H^{2}}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1-\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m})}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (28)
Figure 3: The plot of density parameter ($\Omega$) vs $t$ for
$A=1,\alpha=0.01$ in non-interacting two-fluid model
respectively. Adding Eqs. (27) and (28), we obtain
$\Omega=\Omega_{m}+\Omega_{D}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1.$ (29)
From the right hand side of Eq. (29), it is clear that for open universe,
$\Omega<1$ but at late time we see that $\Omega\to 1$ i.e. the flat universe
scenario. This result is also compatible with the observational results. Since
our model predicts a flat universe for large times and the present-day
universe is very close to flat, so being flat, the derived model is thus
compatible with the observational results.
Fig. $1$ depicts the energy density of DE ($\rho_{D}$) versus $t$. From this
figure, we observe that ($\rho_{D}$), in both non-interacting and interacting
cases, is a decreasing function of time and approaches a small positive value
at late time and never go to infinity. Thus, in both cases the universe is
free from big rip.
The behavior of EoS for DE in term of cosmic time $t$ is shown in Fig. $2$. It
is observed that for open universe, the $\omega^{eff}_{D}$ is an decreasing
function of time, the rapidity of its decrease at the early stage depends on
the larger value of bulk viscous coefficient. The EoS parameter of the DE
begins in non-dark ($\omega_{D}>-\frac{1}{3}$) region at early stage and cross
the phantom divide or cosmological constant ($\omega_{D}=-1$) region and then
pass over into phantom ($\omega_{D}<-1$) region. The property of DE is a
violation of the null energy condition (NEC) since the DE crosses the Phantom
Divide Line (PDL), in particular depending on the direction (Rodrigues 2008;
Kumar $\&$ Yadav 2011; Pradhan $\&$ Amirhashchi 2011). In theory, despite the
observational constraints, extensions of general relativity are the prime
candidate class of theories consistent with PDL crossing (Nesseris $\&$
Perivolaropoulos 2007). On the other hand, while the current cosmological data
from SN Ia (Supernova Legacy Survey, Gold Sample of Hubble Space Telescope)
(Riess et al. 2004; Astier et al. 2006). CMB (WMAP, BOOMERANG) (Komatsu et al.
2009; MacTavish et al. 2006) and large scale structure (SDSS) (Eisenstein et
al. 2005) data rule out that $\omega_{D}\ll-1$, they mildly favour dynamically
evolving DE crossing the PDL (see Rodrigues 2008; Kumar $\&$ Yadav 2011;
Pradhan $\&$ Amirhashchi 2011; Nesseris $\&$ Perivolaropoulos 2007; Zhao et
al. 2007; Coperland et al. 2006) for theoretical and observational status of
crossing the PDL). Thus our DE model is in good agreement with well
established theoretical result as well as the recent observations. From Fig.
$2$, it is observed that in absence of viscosity (i.e. for $\xi=0$), the
universe does not cross the PDL but approaches to cosmological constant
($\omega_{D}=-1$) scenario. Thus, it clearly indicates the impact of viscosity
on the evolution of the universe.
The variation of density parameter ($\Omega$) with cosmic time $t$ for open
universe has been shown in Fig. $3$. From the figure, it can be seen that in
an open universe, $\Omega$ is an increasing function of time and at late time,
it approaches to the flat universe’s scenario.
## 4 INTERACTING TWO-FLUID MODEL
In this section we consider the interaction between dark viscous and
barotropic fluids. For this purpose we can write the continuity equations for
barotropic and dark viscous fluids as
$\dot{\rho}_{m}+3\frac{\dot{a}}{a}(\rho_{m}+p_{m})=Q,$ (30)
and
$\dot{\rho}_{D}+3\frac{\dot{a}}{a}(\rho_{D}+\bar{p}_{D})=-Q,$ (31)
where the quantity $Q$ expresses the interaction between the dark components.
Since we are interested in an energy transfer from the dark energy to dark
matter, we consider $Q>0$ which ensures that the second law of thermodynamics
is fulfilled (Pavon $\&$ Wang 2009). Here we emphasize that the continuity
Eqs. (11) and (30) imply that the interaction term ($Q$) should be
proportional to a quantity with units of inverse of time i.e
$Q\propto\frac{1}{t}$. Therefor, a first and natural candidate can be the
Hubble factor $H$ multiplied with the energy density. Following Amendola et
al. (2007) and Gou et al. (2007), we consider
$Q=3H\sigma\rho_{m},$ (32)
where $\sigma$ is a coupling constant. Using Eq. (32) in Eq. (30) and after
integrating, we obtain
$\rho_{m}=\rho_{0}a^{-3(1+\omega_{m}-\sigma)}~{}\mbox{or}~{}\rho_{m}=\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}.$
(33)
By using Eq. (33) in Eqs. (12) and (13), we again obtain the $\rho_{D}$ and
$p_{D}$ in term of Hubble’s constant $H$ as
$\rho_{D}=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t},$
(34)
and
$\bar{p}_{D}=\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t},$
(35)
respectively. By using Eqs. (34) and (35) in Eq. (15), we can find the EoS of
dark energy in term of time as
$\omega_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}.$
(36)
Again we can write the effective EoS parameter of viscous DE as
$\omega^{eff}_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-3\xi
H+(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}.$
(37)
Figure 4: The plot of EoS parameter $\omega^{eff}_{D}$ vs $t$ for
$\rho_{0}=10,\omega_{m}=0.5,\alpha=0.01,B=1,\sigma=0.3$ in interacting two-
fluid model
The expressions for the matter-energy density $\Omega_{m}$ and dark-energy
density $\Omega_{D}$ are given by
$\Omega_{m}=\frac{\rho_{m}}{3H^{2}}=\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m}-\sigma)}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (38)
and
$\Omega_{D}=\frac{\rho_{D}}{3H^{2}}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1-\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m}-\sigma)}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (39)
respectively. Adding Eqs. (38) and (39), we obtain
$\Omega=\Omega_{m}+\Omega_{D}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1,$ (40)
which is the same expression as in previous case of non-interacting two-fluid.
Fig. $4$ shows a plot of EoS parameter ($\omega^{eff}_{D}$) versus $t$. The
characteristic of $\omega^{eff}_{D}$ in this case is the same as in the
previous case.
## 5 CONCLUSION
In this Letter, we have studied the evolution of dark energy parameter within
the frame work of an open FRW space-time filled with barotropic and bulk
viscous dark fluid. In both non-interacting and interacting cases, we have
observed that for all values of bulk viscous coefficient, the universe has
transition from non-dark region ($\omega^{eff}_{D}>-\frac{1}{3}$) to phantom
region ($\omega^{eff}_{D}<-1$). In summary, we have investigated the
possibility of constructing a two-fluid dark energy models which have the
equation of state ($\omega^{eff}_{D}$) crossing - 1 by using the two-fluid
(barotropic and bulk viscous dark fluid) naturally. Therefore, the two-fluid
scenario discussed in the present paper is a viable candidate for dark energy.
It is also worth mentioned here that in both interacting and non-interacting
cases, our models are free from big rip.
## ACKNOWLEDGMENT
This work has been supported by the FRGS Grant by the Ministry of Higher
Education, Malaysia under the Project Number 02-10-10-969 FR. H. Amirhashchi &
A. Pradhan also thank the Laboratory of Computational Sciences and
Mathematical Physics, Institute for Mathematical Research, Universiti Putra
Malaysia for providing facility where this work was done.
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|
arxiv-papers
| 2012-10-17T06:41:13 |
2024-09-04T02:49:36.695194
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hassan Amirhashchi, Anirudh Pradhan, Hishamuddin Zainuddin",
"submitter": "Hassan Amirhashchi",
"url": "https://arxiv.org/abs/1210.4637"
}
|
1210.4670
|
# Completeness of $n$–tuple of projections in $C^{*}$–algebras
Shanwen Hu and Yifeng Xue
Department of mathematics and Research Center for Operator Algebras
East China Normal University, Shanghai 200241, P.R. China E-mail:
swhu@math.ecnu.edu.cnE-mail: yfxue@math.ecnu.edu.cn
###### Abstract
Let $(P_{1},\cdots,P_{n})$ be an $n$–tuple of projections in a unital
$C^{*}$–algebra $\mathcal{A}$. We say $(P_{1},\cdots,P_{n})$ is complete in
$\mathcal{A}$ if $\mathcal{A}$ is the linear direct sum of the closed
subspaces $P_{1}\mathcal{A},\cdots,P_{n}\mathcal{A}$. In this paper, we give
some necessary and sufficient conditions for the completeness of
$(P_{1},\cdots,P_{n})$ and discuss the perturbation problem and topology of
the set of all complete $n$–tuple of projections in $\mathcal{A}$. Some
interesting and significant results are obtained in this paper.
2010 Mathematics Subject Classification: 46L05, 47B65
Key words: projection, idempotent, complete $n$–tuple of projections
## 0 Introduction
Throughout the paper, we always assume that $H$ is a complex Hilbert space
with inner product $<\cdot,\cdot>$, $B(H)$ is the $C^{*}$–algebra of all
bounded linear operators on $H$ and $\mathcal{A}$ is a $C^{*}$–algebra with
the unit $1$. Let $\mathcal{A}_{+}$ denote the set of all positive elements in
$\mathcal{A}$. It is well–known that $\mathcal{A}$ has a faithful
representation $(\psi,H_{\psi})$ with $\psi(1)=I$ (cf. [8, Theorem 1.6.17] or
[17, Theorem 1.5.36]). For $T\in B(H)$, let $\mathrm{Ran}(T)$ (resp.
$\mathrm{Ker}(T)$) denote the range (resp. kernel) of $T$.
Let $V_{1},V_{2}$ be closed subspaces in $H$ such that $H=V_{1}\dotplus
V_{2}=V_{1}^{\perp}\dotplus V_{2}$, that is, $V_{1}$ and $V_{2}$ is in generic
position (cf. [6]). Let $P_{i}$ be projection of $H$ onto $V_{i}$, $i=1,2$.
Then
$H=\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})=\mathrm{Ran}(I-P_{1})\dotplus\mathrm{Ran}(P_{2})$.
In this case, Halmos gave very useful matrix representations of $P_{1}$ and
$P_{2}$ in [6]. Following Halmos’ work, Sunder investigated in [14] the
$n$–tuple closed subspaces $(V_{1},\cdots,V_{n})$ in $H$ which satisfying the
condition $H=V_{1}\dotplus\cdots\dotplus V_{n}$. If let $P_{i}$ be the
projection of $H$ onto $V_{i}$, $i=1,\cdots,n$, then the condition
$H=V_{1}\dotplus\cdots\dotplus V_{n}$ is equivalent to
$H=\mathrm{Ran}(P_{1})\dotplus\cdots\dotplus\mathrm{Ran}(P_{n})$.
Now the question yields: when does the relation
$H=\mathrm{Ran}(P_{1})\dotplus\cdots\dotplus\mathrm{Ran}(P_{n})$ hold for an
$n$–tuple of projections $(P_{1},\cdots,P_{n})$? When $n=2$, Buckholdtz proved
in [3] that $\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})=H$ iff
$P_{1}-P_{2}$ is invertible in $B(H)$ iff $I-P_{1}P_{2}$ is invertible in
$B(H)$ and iff $P_{1}+P_{2}-P_{1}P_{2}$ is invertible in $B(H)$. More
information about two projections can be found in [2]. Koliha and Rakočević
generalized Buckholdtz’s work to the set of $C^{*}$–algebras and rings. They
gave some equivalent conditions for decomposition
$\mathfrak{R}=P\,\mathfrak{R}\dotplus Q\,\mathfrak{R}$ or
$\mathfrak{R}=\mathfrak{R}\,P\dotplus\mathfrak{R}\,Q$ in [9] and [10] for
idempotent elements $P$ and $Q$ in a unital ring $\mathfrak{R}$. They also
characterized the Fredhomness of the difference of projections on $H$ in [11].
For $n\geq 3$, the question remains unknown so far. But there are some works
concerning with this problem. For example, the estimation of the spectrum of
the finite sum of projections on $H$ is given in [1] and the $C^{*}$–algebra
generated by certain projections is investigated in [13] and [15], etc..
Let $\mathbf{P}\\!_{n}(\mathcal{A})$ denote the set of $n$–tuple ($n\geq 2$)
of non–trivial projections in $\mathcal{A}$ and put
$\mathbf{PC}_{n}(\mathcal{A})=\\{(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})\,|\,P_{1}\mathcal{A}\dotplus\cdots\dotplus
P_{n}\mathcal{A}=\mathcal{A}\\}.$
It is worth to note that if $\mathcal{A}=B(H)$ and
$(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(B(H))$, then $(P_{1},\cdots,P_{n})$
$\in\mathbf{PC}_{n}(B(H))$ if and only if
$\mathrm{Ran}(P_{1})\dotplus\cdots\dotplus\mathrm{Ran}(P_{n})=H$ (see Theorem
1.2 below).
In this paper, we will investigate the set $\mathbf{PC}_{n}(\mathcal{A})$ for
$n\geq 3$. The paper consists of four sections. In Section 1, we give some
necessary and sufficient conditions that make
$(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ be in
$\mathbf{PC}_{n}(\mathcal{A})$. In Section 2, using some equivalent conditions
for $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ obtained in §1, we
obtain an explicit expression of $P_{i_{1}}\vee\cdots\vee P_{i_{k}}$ for
$\\{i_{1},\cdots,i_{k}\\}\subset\\{1,\cdots,n\\}$. We discuss the perturbation
problems for $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ in Section
3. We find an interesting result: if
$(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ with
$A=\sum\limits_{i=1}^{n}P_{i}$ invertible in $\mathcal{A}$, then
$\|P_{i}A^{-1}P_{j}\|<\big{[}(n-1)\|A^{-1}\|\|A\|^{2}\big{]}^{-1}$, $i\not=j$
implies $P_{i}A^{-1}P_{j}=0$, $i\not=j$, $i,j=1,\cdots,n$ in this section. We
show in this section that for given $\epsilon\in(0,1)$, if
$(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ satisfies condition
$\|P_{i}P_{j}\|<\epsilon$, then there exists an $n$–tuple of mutually
orthogonal projections
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ such
that $\|P_{i}-P^{\prime}_{i}\|<2(n-1)\epsilon$, $i=1,\cdots,n,$ which improves
a conventional estimate: $\|P_{i}-P^{\prime}_{i}\|<(12)^{n-1}n!\epsilon$,
$i=1,\cdots,n$ (cf. [8]). In the final section, we will study the topological
properties and equivalent relations on $\mathbf{PC}_{n}(\mathcal{A})$.
## 1 Equivalent conditions for complete $n$–tuples of projections in
$C^{*}$–algebras
Let $GL(\mathcal{A})$ (resp. $U(\mathcal{A})$) denote the group of all
invertible (resp. unitary) elements in $\mathcal{A}$. Let
$\mathrm{M}_{k}(\mathcal{A})$ denote matrix algebra of all $k\times k$
matrices over $\mathcal{A}$. For any $a\in\mathcal{A}$, we set
$a\,\mathcal{A}=\\{ax|\,x\in\mathcal{A}\\}\subset\mathcal{A}$.
###### Definition 1.1.
An $n$–tuple of projections $(P_{1},\cdots,P_{n})$ in $\mathcal{A}$ is called
complete in $\mathcal{A}$, if
$(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$.
###### Theorem 1.2.
Let $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$. Then the
following statements are equivalent:
1. $(1)$
$(P_{1},\cdots,P_{n})$ is complete in $\mathcal{A}$.
2. $(2)$
$H_{\psi}=\mathrm{Ran}(\psi(P_{1}))\dotplus\cdots\dotplus\mathrm{Ran}(\psi(P_{n}))$
for any faithful representation $(\psi,H_{\psi})$ of $\mathcal{A}$ with
$\psi(1)=I$.
3. $(3)$
$H_{\psi}=\mathrm{Ran}(\psi(P_{1}))\dotplus\cdots\dotplus\mathrm{Ran}(\psi(P_{n}))$
for some faithful representation $(\psi,H_{\psi})$ of $\mathcal{A}$ with
$\psi(1)=I$.
4. $(4)$
$\sum\limits_{j\not=i}P_{j}+\lambda P_{i}\in GL(\mathcal{A})$,
$i=1,2,\cdots,n$ and $\forall\,\lambda\in[1-n,0)$.
5. $(5)$
$\lambda\big{(}\sum\limits_{j\not=i}P_{j}\big{)}+P_{i}\in GL(\mathcal{A})$ for
$1\leq i\leq n$ and all $\lambda\in\mathbb{C}\backslash\\{0\\}$.
6. $(6)$
$A=\sum\limits_{i=1}^{n}P_{i}\in GL(\mathcal{A})$ and
$P_{i}A^{-1}P_{i}=P_{i}$, $i=1,\cdots,n$.
7. $(7)$
$A=\sum\limits_{i=1}^{n}P_{i}\in GL(\mathcal{A})$ and $P_{i}A^{-1}P_{j}=0$,
$i\not=j$, $i,j=1,\cdots,n$.
8. $(8)$
there is an $n$-tuple of idempotent operators ${(E_{1},\cdots,E_{n})}$ in
$\mathcal{A}$ such that $E_{i}P_{i}=E_{i},\,P_{i}E_{i}=P_{i}$, $i=1,\cdots,n$
and $E_{i}E_{j}=0,\ i\not=j,\ i,j=1,\cdots,n,\ \sum\limits_{i=1}^{n}E_{i}=1.$
In order to show Theorem 1.2, we need following lemmas.
###### Lemma 1.3.
Let $B,\,C\in\mathcal{A}_{+}\backslash\\{0\\}$ and suppose that $\lambda B+C$
is invertible in $\mathcal{A}$ for every
$\lambda\in\mathbb{R}\backslash\\{0\\}$. Then there is a non–trivial
orthogonal projection $P\in\mathcal{A}$ such that
$B=(B+C)^{1/2}P(B+C)^{1/2},\quad C=(B+C)^{1/2}(1-P)(B+C)^{1/2}.$
###### Proof.
Put $D=B+C$ and $D_{\lambda}=\lambda B+C$, $\forall\,\lambda\in
R\backslash\\{0\\}$. Then $D\geq 0$, $D$ and $D_{\lambda}$ are all invertible
in $\mathcal{A}$, $\forall\,\lambda\in\mathbb{R}\backslash\\{0\\}$.
Put $B_{1}=D^{-1/2}BD^{-1/2}$, $C_{1}=D^{-1/2}CD^{-1/2}$. Then $B_{1}+C_{1}=1$
and
$D^{-1/2}D_{\lambda}D^{-1/2}=\lambda
B_{1}+C_{1}=\lambda+(1-\lambda)C_{1}=(1-\lambda)(\lambda(1-\lambda)^{-1}+C_{1})$
is invertible in $\mathcal{A}$ for any
$\lambda\in\mathbb{R}\backslash\\{0,1\\}$. Since
$\lambda\mapsto\dfrac{\lambda}{1-\lambda}$ is a homeomorphism from
$\mathbb{R}\backslash\\{0,1\\}$ onto $\mathbb{R}\backslash\\{-1,0\\}$, it
follows that $\sigma(C_{1})\subset\\{0,1\\}$. Note that $B_{1}$ and $C_{1}$
are all non–zero. So $\sigma(C_{1})=\\{0,1\\}=\sigma(B_{1})$ and hence
$P=B_{1}$ is a non–zero projection in $\mathcal{A}$ and $B=D^{1/2}PD^{1/2}$,
$C=D^{1/2}(1-P)D^{1/2}$. ∎
###### Lemma 1.4.
Let $B,\,C\in\mathcal{A}_{+}\backslash\\{0\\}$. Then the following statements
are equivalent:
1. $(1)$
for any non–zero real number $\lambda$, $\lambda B+C$ is invertible in
$\mathcal{A}$.
2. $(2)$
$B+C$ is invertible in $\mathcal{A}$ and $B(B+C)^{-1}B=B$.
3. $(3)$
$B+C$ is invertible in $\mathcal{A}$ and $B(B+C)^{-1}C=0$.
4. $(4)$
$B+C$ is invertible in $\mathcal{A}$ and for any
$B^{\prime},\,C^{\prime}\in\mathcal{A}_{+}$ with $B^{\prime}\leq B$ and
$C^{\prime}\leq C$, $B^{\prime}(B+C)^{-1}C^{\prime}=0$.
###### Proof.
(1)$\Rightarrow$(2) By Lemma 1.3, there is a non–zero projection $P$ in
$\mathcal{A}$ such that $B=D^{1/2}PD^{1/2}$, $C=D^{1/2}(1-P)D^{1/2}$, where
$D=B+C\in GL(\mathcal{A})$. So
$B(B+C)^{-1}B=D^{1/2}PD^{1/2}D^{-1}D^{1/2}PD^{1/2}=B.$
The assertion (2) $\Leftrightarrow$ (3) follows from
$B(B+C)^{-1}B=B(B+C)^{-1}(B+C-C)=B-B(B+C)^{-1}C.$
(3) $\Rightarrow$ (4) For any $C^{\prime}$ with $0\leq C^{\prime}\leq C$,
$0\leq B(B+C)^{-1}C^{\prime}(B+C)^{-1}B\leq B(B+C)^{-1}C(B+C)^{-1}B=0,$
we have $B(B+C)^{-1}C^{\prime}=B(B+C)^{-1}C^{\prime 1/2}C^{\prime 1/2}=0.$
This implies $C^{\prime}(B+C)^{-1}B=0$. In the same way, we get that for any
$B^{\prime}$ with $0\leq B^{\prime}\leq B$,
$C^{\prime}(B+C)^{-1}B^{\prime}=0$.
(4)$\Rightarrow$(3) is obvious.
(2)$\Rightarrow$(1) Set $X=(B+C)^{-1/2}B$ and $Y=(B+C)^{-1/2}C$. Then
$X,\,Y\in\mathcal{A}$ and $X^{*}X=B$, $X+Y=(B+C)^{1/2}$. Thus, for any
$\lambda\in\mathbb{R}\backslash\\{0\\}$,
$\displaystyle X+\lambda Y$ $\displaystyle=(B+C)^{-1/2}(B+\lambda C)$
$\displaystyle(X+\lambda Y)^{*}(X+\lambda Y)$
$\displaystyle=((1-\lambda)X+\lambda(B+C)^{1/2})^{*}((1-\lambda)X+\lambda(B+C)^{1/2})$
$\displaystyle=(1-\lambda)^{2}B+2\lambda(1-\lambda)B+\lambda^{2}(B+C)$
$\displaystyle=B+\lambda^{2}C$
and consequently, $(X+\lambda Y)^{*}(X+\lambda Y)\geq B+C$ if $|\lambda|>1$
and $(X+\lambda Y)^{*}(X+\lambda Y)\geq\lambda^{2}(B+C)$ when $|\lambda|<1$.
This indicates that $(X+\lambda Y)^{*}(X+\lambda Y)$ is invertible in
$\mathcal{A}$. Noting that $B+C\geq\|(B+C)^{-1}\|^{-1}\cdot 1$, we have, for
any $\lambda\in\mathbb{R}\backslash\\{0\\}$,
$\displaystyle(B+\lambda C)^{2}$ $\displaystyle=(X+\lambda
Y)^{*}(B+C)(X+\lambda Y)$ $\displaystyle\geq\|(B+C)^{-1}\|^{-1}(X+\lambda
Y)^{*}(X+\lambda Y).$
Therefore, $B+\lambda C$ is invertible in $\mathcal{A}$,
$\forall\,\lambda\in\mathbb{R}\backslash\\{0\\}$. ∎
Now we begin to prove Theorem 1.2.
(1)$\Rightarrow$(6) Statement (1) implies that there are
$b_{1},\cdots,b_{n}\in\mathcal{A}$ such that
$1=\sum\limits^{n}_{i=1}P_{i}b_{i}$. Put $\hat{I}=\begin{bmatrix}1\\\ \ &0\\\
\ &\ &\ddots\\\ \ &\ &\ \ &0\end{bmatrix}$,
$X=\begin{bmatrix}P_{1}&\cdots&P_{n}\\\ 0&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&0\end{bmatrix}$ and $Y=\begin{bmatrix}b_{1}&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ b_{n}&0&\cdots&0\end{bmatrix}$. Then
$\hat{I}=XY=XYY^{*}X^{*}\leq\|Y\|^{2}XX^{*}=\|Y\|^{2}\begin{bmatrix}\sum\limits^{n}_{i=1}P_{i}\\\
\ &0\\\ \ &\ &\ddots\\\ \ &\ &\ \ &0\end{bmatrix}$
and so that $A=\sum\limits^{n}_{i=1}P_{i}$ is invertible in $\mathcal{A}$.
Therefore, from $\mathcal{A}=P_{1}\mathcal{A}\dotplus\cdots\dotplus
P_{n}\mathcal{A}$ and
$P_{i}=P_{1}A^{-1}P_{i}+\cdots+P_{i}A^{-1}P_{i}+\cdots+P_{n}A^{-1}P_{i}=\underbrace{0+\cdots+0}_{i-1}+P_{i}+\underbrace{0+\cdots+0}_{n-i},$
$i=1,\cdots,n$, we get that $P_{i}=P_{i}A^{-1}P_{i}$, $i=1,\cdots,n$.
(2)$\Rightarrow$(3) is obvious.
(3)$\Rightarrow$(4) Set $Q_{i}=\psi(P_{i})$, $i=1,\cdots,n$. From
$H_{\psi}=\mathrm{Ran}(Q_{1})\dotplus\cdots\dotplus\mathrm{Ran}(Q_{n})$, we
obtain idempotent operators $F_{1},\cdots,F_{n}$ in $B(H_{\psi})$ such that
$\sum\limits^{n}_{i=1}F_{i}=I$, $F_{i}F_{j}=0$, $i\not=j$ and
$F_{i}H_{\psi}=Q_{i}H_{\psi}$, $i,j=1,\cdots,n$. So $F_{i}Q_{i}=Q_{i}$,
$Q_{i}F_{i}=F_{i}$ and $F_{j}Q_{i}=0$, $i\not=j$, $1\leq i,j\leq n$. Using
these relations, it is easy to check that
$\displaystyle\big{(}\sum_{i=1}^{n}\lambda_{i}Q_{i}\big{)}\big{(}\sum_{i=1}^{n}\lambda_{i}^{-1}F_{i}^{*}F_{i}\big{)}$
$\displaystyle=\sum_{i=1}^{n}F_{i}=I,$
$\displaystyle\big{(}\sum_{i=1}^{n}\lambda_{i}^{-1}F_{i}^{*}F_{i}\big{)}\big{(}\sum_{i=1}^{n}\lambda_{i}Q_{i}\big{)}$
$\displaystyle=\sum_{i=1}^{n}F_{i}^{*}=I,$
for any non–zero complex number $\lambda_{i}$, $i=1,\cdots,n$. Particularly,
for any $\lambda\in[1-n,0)$,
$\big{(}\lambda\big{(}\sum\limits_{j\not=i}Q_{j}\big{)}+Q_{i}\big{)}^{-1}=\lambda^{-1}\sum_{j\not=i}F_{j}^{*}F_{j}+F_{i}^{*}F_{i}$
in $B(H_{\psi})$. Thus,
$\lambda\big{(}\sum\limits_{j\not=i}Q_{j}\big{)}+Q_{i}$ is invertible
$\psi(\mathcal{A})$, $1\leq i\leq n$ by [17, Corollary 1.5.8] and so that
$\lambda\big{(}\sum\limits_{j\not=i}P_{j}\big{)}+P_{i}\in GL(\mathcal{A})$
since $\psi$ is faithful and $\psi(1)=I$.
(4)$\Rightarrow$(5) Put $A_{i}(\lambda)=\sum\limits_{j\not=i}P_{j}+\lambda
P_{i}$, $i=1,\cdots,n$, $\lambda\in\mathbb{R}\backslash\\{0\\}$, then
$\displaystyle(A_{i}(\lambda))^{2}$ $\displaystyle\leq
2\big{(}\sum\limits_{j\not=i}P_{j}\big{)}^{2}+2\lambda^{2}P_{i}\leq
2(n-1)\sum\limits_{j\not=i}P_{j}+2\lambda^{2}P_{i}$ $\displaystyle\leq
2\max\\{n-1,\lambda^{2}\\}(P_{1}+\cdots+P_{n}).$
So $A_{i}(\lambda)$ is invertible in $\mathcal{A}$,
$\forall\,\lambda\in[1-n,0)$ means that $A=P_{1}+\cdots+P_{n}$ is invertible
in $\mathcal{A}$. Consequently, $A_{i}(\lambda)$ is invertible in
$\mathcal{A}$ when $\lambda>0$, $\forall\,1\leq i\leq n$.
Now we show that $A_{i}(\lambda)$ is invertible in $\mathcal{A}$ for
$i=1,\cdots,n$ and $\lambda<1-n$. Put
$A_{1i}=P_{i}AP_{i},\ A_{2i}=P_{i}A(1-P_{i}),\ A_{4i}=(1-P_{i})A(1-P_{i}),\
i=1,\cdots,n.$
Express $A_{i}(\lambda)$ as the form
$A_{i}(\lambda)=\begin{bmatrix}A_{1i}+(\lambda-1)P_{i}&A_{2i}\\\
A_{2i}^{*}&A_{4i}\end{bmatrix}$, $i=1,\cdots,n$. Noting that $A_{4i}$ is
invertible in $(1-P_{i})\mathcal{A}(1-P_{i})$ ($A\geq\|A^{-1}\|^{-1}\cdot 1$,
$A_{4i}\geq\|A^{-1}\|^{-1}(1-P_{i})$) and
$A_{i}(\lambda)\begin{bmatrix}P_{i}&0\\\
-A_{4i}^{-1}A_{2i}^{*}&1-P_{i}\end{bmatrix}=\begin{bmatrix}A_{1i}-A_{2i}A_{4i}^{-1}A_{2i}^{*}+(\lambda-1)P_{i}&A_{2i}\\\
0&A_{4i}\end{bmatrix},$
we get that $A_{i}(\lambda)$ is invertible iff
$A_{1i}-A_{2i}A_{4i}^{-1}A_{2i}^{*}+(\lambda-1)P_{i}$ is invertible in
$P_{i}\mathcal{A}P_{i}$, $i=1,\cdots,n$. Since $A_{1i}\leq nP_{i}$, it follows
that
$-A_{1i}+A_{2i}A_{4i}^{-1}A_{2i}^{*}-(\lambda-1)P_{i}\geq(1-n-\lambda)P_{i}+A_{2i}A_{4i}^{-1}A_{2i}^{*}\geq(1-n-\lambda)P_{i}$
when $\lambda<1-n$, $i=1,\cdots,n$. Therefore, $A_{i}(\lambda)$ is invertible
in $\mathcal{A}$ for $\lambda<1-n$ and $i=1,\cdots,n$.
Applying Lemma 1.4 to $\sum\limits_{j\not=i}P_{j}$ and $P_{i}$, $1\leq i\leq
n$, we can get the implications (5)$\Rightarrow$(6) and (6)$\Rightarrow$ (7)
easily.
(7)$\Rightarrow$(8) Set $E_{i}=P_{i}A^{-1}$, $i=1\cdots,n$. Then $E_{i}$ is an
idempotent elements in $\mathcal{A}$ and $E_{i}E_{j}=0$, $i\not=j$,
$i,j=1,\cdots,n$. It is obvious that $\sum\limits_{i=1}^{n}E_{i}=1$ and
$P_{i}E_{i}=E_{i}$, $E_{i}P_{i}=P_{i}$, $i=1,\cdots,n$.
(8)$\Rightarrow$(1) Let $E_{1},\cdots,E_{n}$ be idempotent elements in
$\mathcal{A}$ such that $E_{i}E_{j}=\delta_{ij}E_{i}$,
$\sum\limits^{n}_{i=1}E_{i}=1$ and $E_{i}P_{i}=P_{i}$, $P_{i}E_{i}=E_{i}$,
$i,j=1,\cdots,n$. Then $E_{i}\mathcal{A}=P_{i}\mathcal{A}$, $i=1,\cdots,n$ and
$\mathcal{A}=E_{1}\mathcal{A}\dotplus\cdots
E_{n}\mathcal{A}=P_{1}\mathcal{A}\dotplus\cdots\dotplus P_{n}\mathcal{A}$.
(8)$\Rightarrow$(2) Let $E_{1},\cdots,E_{n}$ be idempotent elements in
$\mathcal{A}$ such that $E_{i}E_{j}=\delta_{ij}E_{i}$,
$\sum\limits^{n}_{i=1}E_{i}=1$ and $E_{i}P_{i}=P_{i}$, $P_{i}E_{i}=E_{i}$,
$i,j=1,\cdots,n$. Let $(\psi,H_{\psi})$ be any faithful representation of
$\mathcal{A}$ with $\psi(1)=I$. Put $E_{i}^{\prime}=\psi(E_{i})$ and
$Q_{i}=\psi(P_{i})$, $i=1,\cdots,n$. Then
$E_{i}^{\prime}E_{j}^{\prime}=\delta_{ij}E_{i}^{\prime}$,
$\sum\limits^{n}_{i=1}E_{i}^{\prime}=I$ and
$\mathrm{Ran}(E_{i}^{\prime})=\mathrm{Ran}(Q_{i})$, $i,j=1,\cdots,n$.
Consequently,
$H_{\psi}=\mathrm{Ran}(Q_{1})\dotplus\cdots\dotplus\mathrm{Ran}(Q_{n})$. ∎
###### Remark 1.5.
(1) Statement (3) in Theorem 1.2 can not be replaced by “for any $1\leq i\leq
n$, $P_{i}-\sum\limits_{j\not=i}P_{j}$ is invertible”.
For example, let $H^{(4)}=\bigoplus\limits^{4}_{i=1}H$ and put
$\mathcal{A}=B(H^{(4)})$,
$P_{1}=\begin{bmatrix}I\\\ \ &I\\\ \ &\ &0\\\ \ &\ &\ &0\end{bmatrix},\quad
P_{2}=\begin{bmatrix}I\\\ \ &0\\\ \ &\ &I\\\ \ &\ &\ &0\end{bmatrix},\quad
P_{3}=\begin{bmatrix}I\\\ \ &0\\\ \ &\ &0\\\ \ &\ &\ &I\end{bmatrix}.$
Clearly, $P_{i}-\sum\limits_{j\not=i}{P_{j}}$ is invertible, $1\leq i\leq 3$,
but $P_{2}+P_{3}-2P_{1}$ is not invertible, that is, $(P_{1},P_{2},P_{3})$ is
not complete in $\mathcal{A}$.
(2) According to the proof of (3)$\Rightarrow$(4) of Theorem 1.2, we see that
for $(P_{1},\cdots,P_{n})$ $\in\mathbf{P}\\!_{n}(\mathcal{A})$, if
$\sum\limits^{n}_{i=1}P_{i}\in GL(\mathcal{A})$, then
$\sum\limits_{i\not=j}P_{i}-\lambda P_{i}\in GL(\mathcal{A})$, $\forall\,1\leq
i\leq n$ and $\lambda>n-1$.
###### Corollary 1.6 ([3, Theorem 1]).
Let $P_{1},P_{2}$ be non–trivial projections in $B(H)$. Then
$H=\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})$ iff $P_{1}-P_{2}$ is
invertible in $B(H)$.
###### Proof.
By Theorem 1.2, $H=\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})$ implies
that $P_{1}-P_{2}\in GL(B(H))$.
Conversely, if $P_{1}-P_{2}\in GL(B(H))$, then from
$2(P_{1}+P_{2})\geq(P_{1}-P_{2})^{2}$, we get that $P_{1}+P_{2}\in GL(B(H))$
and so that $P_{1}-\lambda P_{2},P_{2}-\lambda P_{1}\in GL(B(H))$,
$\forall\,\lambda>1$ by Remark 1.5 (2). Thus, for any $\lambda\in(0,1]$,
$P_{1}-\lambda P_{2}$ and $P_{2}-\lambda P_{1}$ are all invertible in $B(H)$.
Consequently, $H=\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})$ by Theorem
1.2. ∎
## 2 Some representations concerning the complete $n$–tuple of projections
We first statement two lemmas which are frequently used in this section and
the later sections.
###### Lemma 2.1.
Let $B\in\mathcal{A}_{+}$ such that $0\in\sigma(B)$ is an isolated point. Then
there is a unique element $B^{\dagger}\in\mathcal{A}_{+}$ such that
$BB^{\dagger}B=B,\ B^{\dagger}BB^{\dagger}=B^{\dagger},\
BB^{\dagger}=B^{\dagger}B.$
###### Proof.
The assertion follows from Proposition 3.5.8, Proposition 3.5.3 and Lemma
3.5.1 of [17]. ∎
###### Remark 2.2.
The element $B^{\dagger}$ in Lemma 2.1 is called the Moore–Penrose inverse of
$B$. When $0\not\in\sigma(B)$, $B^{\dagger}\triangleq B^{-1}$. The detailed
information can be found in [17].
The following lemma comes from [17, Lemma 3.5.5] and [4, Lemma 1]:
###### Lemma 2.3.
Let $P\in\mathcal{A}$ be an idempotent element. Then
1. $(1)$
$P+P^{*}-1\in GL(\mathcal{A})$.
2. $(2)$
$R=P(P+P^{*}-1)^{-1}$ is a projection in $\mathcal{A}$ satisfying $PR=R$ and
$RP=P$.
Moreover, if $R^{\prime}\in\mathcal{A}$ is a projection such that
$PR^{\prime}=R^{\prime}$ and $R^{\prime}P=P$, then $R^{\prime}=R$.
Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ and put
$A=\sum\limits^{n}_{i=1}P_{i}$. By Theorem 1.2, $A\in GL(\mathcal{A})$ and
$E_{i}=P_{i}A^{-1}$, $1\leq i\leq n$ are idempotent elements satisfying
conditions
$E_{i}E_{j}=0,\ i\not=j,\ E_{i}P_{i}=P_{i},\ P_{i}E_{i}=E_{i},\ i=1,\cdots,n,\
\text{and}\ \sum\limits^{n}_{i=1}E_{i}=1.$
By Lemma 2.3, $P_{i}=E_{i}(E_{i}^{*}+E_{i}-1)^{-1}$, $1\leq i\leq n$. So the
$C^{*}$–algebra $C^{*}(P_{1},\cdots,P_{n})$ generated by $P_{1},\cdots,P_{n}$
is equal to the $C^{*}$–algebra $C^{*}(E_{1},\cdots,E_{n})$ generated by
$E_{1},\cdots,E_{n}$.
Put $Q_{i}=A^{-1/2}P_{i}A^{-1/2}$, $i=1,\cdots,n$. Then
$Q_{i}Q_{j}=\delta_{ij}Q_{i}$ by Theorem 1.2, $i,j=1,\cdots,n$ and
$\sum\limits_{i=1}^{n}Q_{i}=1$. Thus,
$P_{i}=A^{1/2}Q_{i}A^{1/2}\ \text{and}\
E_{i}=P_{i}A^{-1}=A^{1/2}Q_{i}A^{-1/2},\ i=1,\cdots,n.$ (2.1)
###### Proposition 2.4.
Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ with
$A=\sum\limits^{n}_{i=1}P_{i}$. Then for any $\lambda_{i}\not=0$,
$i=1,\cdots,n$,
$\big{(}\sum\limits_{i=1}^{n}\lambda_{i}P_{i}\big{)}^{-1}=A^{-1}\big{(}\sum\limits_{i=1}^{n}\lambda_{i}^{-1}P_{i}\big{)}A^{-1}.$
Proof. Keeping the symbols as above. We have
$\sum\limits_{i=1}^{n}\lambda_{i}P_{i}=A^{1/2}\big{(}\sum\limits^{n}_{i=1}\lambda_{i}Q_{i}\big{)}A^{1/2}$.
Thus,
$\hskip
48.36958pt\big{(}\sum\limits_{i=1}^{n}\lambda_{i}P_{i}\big{)}^{-1}=A^{-1/2}\big{(}\sum\limits^{n}_{i=1}\lambda_{i}^{-1}Q_{i}\big{)}A^{-1/2}=A^{-1}\big{(}\sum_{i=1}^{n}\lambda_{i}^{-1}P_{i}\big{)}A^{-1}.\hskip
28.45274pt\qed$
Now for $i_{1},i_{2},\cdots,i_{k}\in\\{1,2,\cdots,n\\}$ with
$i_{1}<i_{2}<\cdots<i_{k}$, put $A_{0}=\sum\limits^{k}_{r=1}P_{i_{r}}$ and
$Q_{0}=\sum\limits^{k}_{r=1}Q_{i_{r}}$. Then $A_{0},Q_{0}\in\mathcal{A}$ and
$Q_{0}$ is a projection. From (2.1), $A_{0}=A^{1/2}Q_{0}A^{1/2}$. Thus,
$\sigma(A_{0})\backslash\\{0\\}=\sigma(Q_{0}AQ_{0})\backslash\\{0\\}$ (cf.
[17, Proposition 1.4.14]). Since $Q_{0}AQ_{0}$ is invertible in
$Q_{0}\mathcal{A}Q_{0}$, it follows that $0\in\sigma(Q_{0}AQ_{0})$ is an
isolated point and so that $0\in\sigma(A_{0})$ is also an isolated point. So
we can define $P_{i_{1}}\vee\cdots\vee P_{i_{k}}$ to be the projection
$A_{0}^{\dagger}A_{0}\in\mathcal{A}$ by Lemma 2.1. This definition is
reasonable:
if $P\in\mathcal{A}$ is a projection such that $P\geq P_{i_{r}}$,
$r=1,\cdots,k$, then $PA_{0}=A_{0}$ and hence
$PA_{0}A_{0}^{\dagger}=A_{0}A_{0}^{\dagger}$, i.e., $P\geq
P_{i_{1}}\vee\cdots\vee P_{i_{k}}$; Since $A_{0}\geq P_{i_{r}}$, we have
$0=(1-A_{0}^{\dagger}A_{0})A_{0}(1-A_{0}^{\dagger}A_{0})\geq(1-A_{0}^{\dagger}A_{0})P_{i_{r}}(1-A_{0}^{\dagger}A_{0})$
and consequently, $P_{i_{r}}(1-A_{0}^{\dagger}A_{0})=0$, that is,
$P_{i_{r}}\leq P_{i_{1}}\vee\cdots\vee P_{i_{k}}$, $i=1,\cdots,k$.
###### Proposition 2.5.
Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ with
$A=\sum\limits^{n}_{i=1}P_{i}$. Let $i_{1},\cdots,i_{k}$ be as above and
$\\{j_{1},\cdots,j_{l}\\}=\\{1,\cdots,n\\}\backslash\\{i_{1},\cdots,i_{k}\\}$
with $j_{1}<\cdots<j_{l}$. Then
$\displaystyle P_{i_{1}}\vee\cdots\vee P_{i_{k}}$
$\displaystyle=A^{1/2}\big{[}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\big{]}^{-1}A^{1/2}$
(2.2)
$\displaystyle=\big{(}\sum_{r=1}^{k}P_{i_{r}}\big{)}\big{[}\big{(}\sum_{r=1}^{k}P_{i_{r}}\big{)}^{2}+\sum_{t=1}^{l}P_{j_{t}}\big{]}^{-1}\big{(}\sum_{r=1}^{k}P_{i_{r}}\big{)}.$
(2.3)
###### Proof.
Using the symbols $P_{i},Q_{i},E_{i}$ as above. According to (2.1),
$\sum\limits_{r=1}^{k}P_{i_{r}}=A^{1/2}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A^{1/2},\
\sum\limits_{r=1}^{k}E_{i_{r}}=A^{1/2}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A^{-1/2}.$
Thus
$\big{(}\sum\limits_{r=1}^{k}E_{i_{r}}\big{)}\big{(}\sum\limits_{r=1}^{k}P_{i_{r}}\big{)}=\sum\limits_{r=1}^{k}P_{i_{r}}$
and
$\sum\limits_{r=1}^{k}E_{i_{r}}=\big{(}\sum\limits_{r=1}^{k}P_{i_{r}}\big{)}A^{-1}$.
Then we have
$\big{(}\sum\limits_{r=1}^{k}E_{i_{r}}\big{)}P_{i_{1}}\vee\cdots\vee
P_{i_{k}}=P_{i_{1}}\vee\cdots\vee P_{i_{k}},\quad P_{i_{1}}\vee\cdots\vee
P_{i_{k}}\big{(}\sum\limits_{r=1}^{k}E_{i_{r}}\big{)}=\sum\limits_{r=1}^{k}E_{i_{r}},$
according to the definition of $P_{i_{1}}\vee\cdots\vee P_{i_{k}}$.
Since $\sum\limits_{r=1}^{k}E_{i_{r}}$ is an idempotent element in
$\mathcal{A}$, it follows from Lemma 2.3 that
$P_{i_{1}}\vee\cdots\vee
P_{i_{k}}=\big{(}\sum\limits_{r=1}^{k}E_{i_{r}}\big{)}\big{[}\sum\limits_{r=1}^{k}(E_{i_{r}}^{*}+E_{i_{r}})-1\big{]}^{-1}\in\mathcal{A}.$
(2.4)
Noting that
$\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\in
GL\big{(}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\mathcal{A}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\big{)}$;
$\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}A\big{(}\sum\limits_{t=1}^{k}Q_{j_{t}}\big{)}$
is invertible in
$\big{(}\sum\limits_{t=1}^{k}Q_{j_{t}}\big{)}\mathcal{A}\big{(}\sum\limits_{t=1}^{k}Q_{j_{t}}\big{)}$
and
$\displaystyle\sum\limits_{r=1}^{k}(E_{i_{r}}^{*}+E_{i_{r}})-1$
$\displaystyle=A^{-1/2}\big{[}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A+A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}-A\big{]}A^{-1/2}$
$\displaystyle=A^{-1/2}\big{[}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\\!-\\!\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}A\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}\big{]}A^{-1/2},$
we obtain that
$\displaystyle\big{[}\sum\limits_{r=1}^{k}(E_{i_{r}}^{*}$
$\displaystyle+E_{i_{r}})-1\big{]}^{-1}$
$\displaystyle=A^{1/2}\big{[}\big{[}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\big{]}^{-1}\\!-\\!\big{[}\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}A\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}\big{]}^{-1}\big{]}A^{1/2}.$
Combining this with (2.4), we can get (2.2).
Note that
$\sum\limits_{r=1}^{k}P_{i_{r}}=A^{1/2}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A^{1/2}$,
$\sum\limits_{t=1}^{l}P_{j_{t}}=A^{1/2}\big{(}\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}A^{1/2}$
and $\big{(}\sum\limits_{r=1}^{k}P_{i_{r}}\big{)}^{2}$
$=A^{1/2}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A^{1/2}.$
Therefore,
$\displaystyle\big{(}$
$\displaystyle\sum_{r=1}^{k}P_{i_{r}}\big{)}\big{[}\big{(}\sum_{r=1}^{k}P_{i_{r}}\big{)}^{2}+\sum_{t=1}^{l}P_{j_{t}}\big{]}^{-1}\big{(}\sum_{r=1}^{k}P_{i_{r}}\big{)}$
$\displaystyle=A^{1/2}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\big{(}\big{[}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}\big{]}^{-1}+\sum\limits_{t=1}^{l}Q_{j_{t}}\big{)}\big{(}\sum\limits_{r=1}^{k}Q_{i_{r}}\big{)}A^{1/2}$
$\displaystyle=P_{i_{1}}\vee\cdots\vee P_{i_{k}}$
by (2.2). ∎
## 3 Perturbations of a complete $n$–tuple of projections
Recall that for any non–zero operator $C\in B(H)$, the reduced minimum modulus
$\gamma(C)$ is given by
$\gamma(C)=\\{\|Cx\|\,|\,x\in(\mathrm{Ker}(C))^{\perp},\,\|x\|=1\\}$ (cf. [17,
Remark 1.2.10]). We list some properties of the reduced minimum modulus as our
lemma as follows.
###### Lemma 3.1 (cf. [17]).
Let $C$ be in $B(H)\backslash\\{0\\}$, Then
1. $(1)$
$\|Cx\|\geq\gamma(C)\|x\|$, $\forall\,x\in(\mathrm{Ker}(C))^{\perp}$.
2. $(2)$
$\gamma(C)=\inf\\{\lambda\,|\,\lambda\in\sigma(|C|)\backslash\\{0\\}\\}$,
where $|C|=(C^{*}C)^{1/2}$.
3. $(3)$
$\gamma(C)>0$ iff $\mathrm{Ran}(C)$ is closed iff $0$ is an isolated point of
$\sigma(|C|)$ if $0\in\sigma(|C|)$.
4. $(4)$
$\gamma(C)=\|C^{-1}\|^{-1}$ when $C$ is invertible.
5. $(5)$
$\gamma(C)\geq\|B\|^{-1}$ when $CBC=C$ for $B\in B(H)\backslash\\{0\\}$.
For $a\in\mathcal{A}_{+}$, put
$\beta(a)=\inf\\{\lambda|\,\lambda\in\sigma(a)\backslash\\{0\\}\\}$. Combining
Lemma 3.1 with the faithful representation of $\mathcal{A}$, we can obtain
###### Corollary 3.2.
Let $a\in\mathcal{A}_{+}$. Then
1. $(1)$
$\beta(a)>0$ if and only if $0\in\sigma(a)$ is isolated when $a\not\in
GL(\mathcal{A})$.
2. $(2)$
$\beta(c)\geq\|c\|^{-1}$ when $aca=a$ for some
$c\in\mathcal{A}_{+}\backslash\\{0\\}$.
Let $\mathcal{E}$ be a $C^{*}$–subalgebra of $B(H)$ with the unit $I$. Let
$(T_{1},\cdots,T_{n})$ be an $n$–tuple of positive operators in $\mathcal{E}$
with $\mathrm{Ran}(T_{i})$ closed, $i=1,\cdots,n$. Put
$H_{0}=\bigoplus\limits^{n}_{i=1}\mathrm{Ran}(T_{i})\subset\bigoplus\limits^{n}_{i=1}H\triangleq\hat{H}$
and $H_{1}=\bigoplus\limits^{n}_{i=1}\mathrm{Ker}(T_{i})\subset\hat{H}$. Since
$H=\mathrm{Ran}(T_{i})\oplus\mathrm{Ker}(T_{i})$, $i=1,\cdots,n$, it follows
that $H_{0}\oplus H_{1}=\hat{H}$. Put
$T_{ij}=T_{i}T_{j}\big{|}_{\mathrm{Ran}(T_{j})}$, $i,j=1,\cdots,n$ and set
$T=\begin{bmatrix}T_{1}^{2}&T_{1}T_{2}&\cdots&T_{1}T_{n}\\\
T_{2}T_{1}&T_{2}^{2}&\cdots&T_{2}T_{n}\\\ \cdots&\cdots&\cdots&\cdots\\\
T_{n}T_{1}&T_{2}T_{2}&\cdots&T_{n}^{2}\end{bmatrix}\in\mathrm{M}_{n}(\mathcal{E}),\
\hat{T}=\begin{bmatrix}T_{11}&T_{12}&\cdots&T_{1n}\\\
T_{21}&T_{22}&\cdots&T_{2n}\\\ \cdots&\cdots&\cdots&\cdots\\\
T_{n1}&T_{n2}&\cdots&T_{nn}\end{bmatrix}\in B(H_{0}),$ (3.1)
Clearly, $H_{1}\subset\mathrm{Ker}(T)$ and it is easy to check that
$\mathrm{Ker}(T)=H_{1}$ when $\mathrm{Ker}(\hat{T})=\\{0\\}$. Thus, in this
case, $T$ can be expressed as $T=\begin{bmatrix}\hat{T}&0\\\ 0&0\end{bmatrix}$
with respect to the orthogonal decomposition $\hat{H}=H_{0}\oplus H_{1}$ and
consequently, $\sigma(T)=\sigma(\hat{T})\cup\\{0\\}$.
###### Lemma 3.3.
Let $(T_{1},\cdots,T_{n})$ be an $n$–tuple of positive operators in
$\mathcal{E}$ with $\mathrm{Ran}(T_{i})$ closed, $i=1,\cdots,n$. Let
$H_{0},H_{1},\hat{H}$ be as above and $T,\,\hat{T}$ be given in (3.1). Suppose
that $\hat{T}$ is invertible in $B(H_{0})$. Then
1. $(1)$
$\sigma(\hat{T})=\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}\backslash\\{0\\}$.
2. $(2)$
$0$ is an isolated point in $\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$
if $0\in\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$.
3. $(3)$
$\\{T_{1}a_{1},\cdots,T_{n}a_{n}\\}$ is linearly independent for any
$a_{1},\cdots,a_{n}\in\mathcal{E}$ with $T_{i}a_{i}\not=0$, $i=1,\cdots,n$.
###### Proof.
(1) Put $Z=\begin{bmatrix}T_{1}&\cdots&T_{n}\\\ 0&\cdots&0\\\
\vdots&\ddots&\vdots\\\
0&\cdots&0\end{bmatrix}\in\mathrm{M}_{n}(\mathcal{E})$. Then
$ZZ^{*}=\begin{bmatrix}\sum\limits^{n}_{i=1}T_{i}^{2}&\\\ \ &0\\\ \ &\
&\ddots\\\ \ &\ &\ &0\end{bmatrix}$ and $Z^{*}Z=T$. Thus,
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}\backslash\\{0\\}=\sigma(T)\backslash\\{0\\}=\sigma(\hat{T})$.
(2) According to (1), $0$ is an isolated point of
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}$ if
$\sum\limits^{n}_{i=1}T_{i}^{2}$ is not invertible in $\mathcal{E}$. So by
Lemma 2.1, there is $G\in\mathcal{E}_{+}$ such that
$\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}G\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}=\sum\limits^{n}_{i=1}T_{i}^{2},\
G\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}G=G,\
\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}G=G\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}.$
Put $P_{0}=I-\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}G\in\mathcal{E}$.
Then $P_{0}$ is a projection with
$\mathrm{Ran}(P_{0})=\mathrm{Ker}\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}$.
Noting that
$\mathrm{Ker}\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}=\mathrm{Ker}\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}=\bigcap\limits^{n}_{i=1}\mathrm{Ker}(T_{i})$,
$\sum\limits^{n}_{i=1}T_{i}^{2}\in GL((I-P_{0})\mathcal{E}(I-P_{0}))$ with the
inverse $G$ and $\sum\limits^{n}_{i=1}T^{2}_{i}\leq(\max\limits_{1\leq i\leq
n}\|T_{i}\|)\sum\limits^{n}_{i=1}T_{i}$, we get that
$\sum\limits^{n}_{i=1}T_{i}$ is invertible in $(I-P_{0})\mathcal{E}(I-P_{0})$.
Thus, $0$ is an isolated point of
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$ when
$0\in\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$.
(3) By Lemma 3.1 (3) and Lemma 2.1, there is
$T_{i}^{\dagger}\in\mathcal{E}_{+}$ such that
$T_{i}T^{\dagger}_{i}T_{i}=T_{i}$,
$T^{\dagger}_{i}T_{i}T^{\dagger}_{i}=T_{i}^{\dagger}$,
$T^{\dagger}_{i}T_{i}=T_{i}T_{i}^{\dagger}$, $i=1,\cdots,n$. Thus,
$\mathrm{Ran}(T_{i})=\mathrm{Ran}(T_{i}T_{i}^{\dagger})$, $i=1,\cdots,n$.
Let $a_{1},\cdots,a_{n}\in\mathcal{E}$ with $T_{i}a_{i}\not=0$, $i=1,\cdots,n$
such that $\sum\limits^{n}_{i=1}\lambda_{i}T_{i}a_{i}=0$ for some
$\lambda_{1},\cdots,\lambda_{n}\in\mathbb{C}$. For any $\xi\in H$, put
$x=\bigoplus\limits^{n}_{i=1}\lambda_{i}T_{i}T_{i}^{\dagger}a_{i}\xi\in
H_{0}$. Then $\hat{T}x=0$ and $x=0$ since $\hat{T}$ is invertible. Thus,
$\lambda_{i}T_{i}T_{i}^{\dagger}a_{i}\xi=0$, $\forall\,\xi\in H$ and hence
$\lambda_{i}=0$, $i=1,\cdots,n$. ∎
The following result duo to Levy and Dedplanques is very useful in Matrix
Theory:
###### Lemma 3.4 (cf. [7]).
Suppose complex $n\times n$ self–adjoint matrix $C=[c_{ij}]_{n\times n}$ is
strictly diagonally dominant, that is, $\sum\limits_{j\not=i}|c_{ij}|<c_{ii}$,
$i=1,\cdots,n$. Then $C$ is invertible and positive.
###### Proposition 3.5.
Let $T_{1},\cdots,T_{n}\in\mathcal{A}_{+}$. Assume that
1. $(1)$
$\gamma=\min\\{\beta(T_{1}),\cdots,\beta(T_{n})\\}>0$ and
2. $(2)$
there exists $\rho\in(0,\gamma]$ such that
$\eta=\max\\{\|T_{i}T_{j}\|\,|\,i\not=j,i,j=1,\cdots,n\\}<$
$(n-1)^{-1}\rho^{2}$.
Then for any $\delta\in[\eta,(n-1)^{-1}\rho^{2})$, we have
1. $(1)$
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}\backslash\\{0\\}\subset[\rho^{2}-(n-1)\delta,\rho^{2}+(n-1)\delta]$.
2. $(2)$
$0$ is an isolated point of $\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$
if $0\in\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}$.
3. $(3)$
$\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}\mathcal{A}=T_{1}\mathcal{A}\dotplus\cdots\dotplus
T_{n}\mathcal{A}$.
###### Proof.
(1) Let $(\psi,H_{\psi})$ be a faithful representation of $\mathcal{A}$ with
$\psi(1)=I$. We may assume that $H=H_{\psi}$ and
$\mathcal{E}=\psi(\mathcal{A})$. Put $S_{i}=\psi(T_{i})$,
$S_{ij}=S_{i}S_{j}\big{|}_{\mathrm{Ran}(S_{j})}$, $i,j=1,\cdots,n$. Then
$\max\\{\|S_{i}S_{j}\|\,|\,1\leq i\not=j\leq n\\}=\eta$ and
$\gamma(S_{i})=\beta(T_{i})$ by Lemma 3.1, $1\leq i\leq n$. Set
$H_{0}=\bigoplus\limits^{n}_{i=1}\mathrm{Ran}(S_{i})$ and
$\hat{S}=\begin{bmatrix}S_{11}&S_{12}&\cdots&S_{1n}\\\
S_{21}&S_{22}&\cdots&S_{2n}\\\ \cdots&\cdots&\cdots&\cdots\\\
S_{n1}&S_{n2}&\cdots&S_{nn}\end{bmatrix}\in B(H_{0}),\quad
S_{0}=\begin{bmatrix}\rho^{2}-\lambda&-\|S_{12}\|&\cdots&-\|S_{1n}\|\\\
-\|S_{21}\|&\rho^{2}-\lambda&\cdots&-\|S_{2n}\|\\\
\cdots&\cdots&\cdots&\cdots\\\
-\|S_{n1}\|&-\|S_{n2}\|&\cdots&\rho^{2}-\lambda\end{bmatrix},$
Then for any $\lambda<\rho^{2}-(n-1)\delta$, we have
$\sum\limits_{j\not=i}\|S_{ij}\|\leq(n-1)\eta<\rho^{2}-\lambda.$ It follows
from Lemma 3.4 that $S_{0}$ is positive and invertible. Therefore the
quadratic form
$f(x_{1},x_{2},\cdots,x_{n})=\sum_{i=1}^{n}(\rho^{2}-\lambda)x_{i}^{2}-2\sum_{1\leq
i<j\leq n}\|S_{ij}\|x_{i}x_{j}$
is positive definite and hence there exists $\alpha>0$ such that for any
$(x_{1},\cdots,x_{n})\in{\mathbb{R}^{n}}$,
$f(x_{1},\cdots,x_{n})\geq\alpha(x_{1}^{2}+\cdots+x_{n}^{2}).$
So for any $\xi=\bigoplus\limits^{n}_{i=1}\xi_{i}\in H_{0}$,
$\|S_{i}\xi_{i}\|\geq\gamma(S_{i})\|\xi_{i}\|\geq\rho\|\xi_{i}\|$,
$\xi_{i}\in\mathrm{Ran}(S_{i})=(\mathrm{Ker}(S_{i}))^{\perp}$, $i=1,\cdots,n$,
by Lemma 3.1 and
$\displaystyle<(\hat{S}-\lambda I)\xi,\xi>$
$\displaystyle=\sum_{i=1}^{n}\|S_{i}\xi_{i}\|^{2}-\sum_{i}^{n}\lambda\|\xi_{i}\|^{2}+\sum_{1\leq
i<j\leq n}(<S_{ij}\xi_{j},\xi_{i}>+<S_{ij}^{*}\xi_{i},\xi_{j}>)$
$\displaystyle\geq\sum_{i=1}^{n}(\rho^{2}-\lambda)\|\xi_{i}\|^{2}-2\sum_{1\leq
i<j\leq n}\|S_{ij}\|\|\xi_{i}\|\|\xi_{j}\|$
$\displaystyle=f(\|\xi_{1}\|,\cdots,\|\xi_{k}\|)\geq\alpha\sum_{i=1}^{k}\|\xi_{i}\|^{2}.$
Therefore, $\hat{S}-\lambda I$ is invertible.
Similarly, for any $\lambda>\rho^{2}+(n-1)\delta$, we can obtain that $\lambda
I-\hat{S}$ is invertible. So
$\sigma(\hat{S})\subset[\rho^{2}-(n-1)\delta,\rho^{2}+(n-1)\delta]\subset(0,\rho^{2}+(n-1)\delta]$
and consequently,
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}^{2}\big{)}\backslash\\{0\\}=\sigma\big{(}\sum\limits^{n}_{i=1}S_{i}^{2}\big{)}\backslash\\{0\\}\subset[\rho^{2}-(n-1)\delta,\rho^{2}+(n-1)\delta]$
by Lemma 3.3.
(2) Since
$\sigma\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}=\sigma\big{(}\sum\limits^{n}_{i=1}S_{i}\big{)}$,
the assertion follows from Lemma 3.3 (2).
(3) By (2) and Lemma 2.1,
$\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}^{\dagger}\in\mathcal{A}$ exists. Set
$E=\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}^{\dagger}$.
Obviously,
$E\mathcal{A}=\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}\mathcal{A}\subset
T_{1}\mathcal{A}+\cdots+T_{n}\mathcal{A}$ for
$E\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}=\sum\limits^{n}_{i=1}T_{i}$.
From $T_{i}\leq\sum\limits^{n}_{i=1}T_{i}$, we get that
$(1-E)T_{i}(1-E)\leq(1-E)\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}(1-E)=0$,
i.e., $T_{i}=ET_{i}$, $i=1,\cdots,n$. So $T_{i}\mathcal{A}\subset
E\mathcal{A}$, $i=1,\cdots,n$ and hence
$T_{1}\mathcal{A}+\cdots+T_{n}\mathcal{A}\subset
E\mathcal{A}=\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}\mathcal{A}\subset
T_{1}\mathcal{A}+\cdots+T_{n}\mathcal{A}.$
Since for any $a_{1},\cdots,a_{n}\in\mathcal{A}$ with $T_{i}a_{i}\not=0$,
$i=1,\cdots,n$, $\\{S_{1}\psi(a_{i}),\cdots,S_{n}\psi(a_{n})\\}$ is linearly
independent in $\mathcal{E}$, we have $\\{T_{1}a_{1},\cdots,T_{n}a_{n}\\}$ is
linearly independent in $\mathcal{A}$. Therefore,
$\big{(}\sum\limits^{n}_{i=1}T_{i}\big{)}\mathcal{A}=E\mathcal{A}=T_{1}\mathcal{A}\dotplus\cdots\dotplus
T_{n}\mathcal{A}$. ∎
Let $P_{1},P_{2}$ be projections on $H$. Buckholtz shows in [3] that
$\mathrm{Ran}(P_{1})\dotplus\mathrm{Ran}(P_{2})=H$ iff $\|P_{1}+P_{2}-I\|<1$.
For $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$, we have
###### Corollary 3.6.
Let $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ with
$\big{\|}\sum\limits^{n}_{i=1}P_{i}-1\big{\|}<(n-1)^{-2}$. Then
$(P_{1},\cdots,P_{n})$ is complete in $\mathcal{A}$.
###### Proof.
For any $i\not=j$,
$\displaystyle\|P_{i}P_{j}\|^{2}$
$\displaystyle=\|P_{i}P_{j}P_{i}\|\leq\big{\|}P_{i}\big{(}\sum_{k\not=i}P_{k}\big{)}P_{i}\big{\|}$
$\displaystyle=\big{\|}P_{i}\big{(}\sum_{k=1}^{n}P_{k}-1\big{)}P_{i}\big{\|}\leq\big{\|}\sum_{k=1}^{n}P_{k}-1\big{\|}<\frac{1}{(n-1)^{2}}.$
Thus $\|P_{i}P_{j}\|<(n-1)^{-1}$. Noting that
$\rho=\min\\{\beta(P_{1}),\cdots,\beta(P_{n})\\}=1,\
\eta=\max\\{\|P_{i}P_{j}\|\,|\,1\leq i<j\leq n\\}<\frac{1}{n-1},$
we have
$\big{(}\sum\limits^{n}_{i=1}P_{i}\big{)}\mathcal{A}=P_{1}\mathcal{A}\dotplus\cdots\dotplus
P_{n}\mathcal{A}$ by Proposition 3.5.
From $\big{\|}\sum\limits^{n}_{i=1}P_{i}-1\big{\|}<(n-1)^{-2}$, we have
$\sum\limits_{i=1}^{n}P_{i}$ is invertible in $\mathcal{A}$ and so that
$\mathcal{A}=P_{1}\mathcal{A}\dotplus\cdots\dotplus P_{n}\mathcal{A}$. Thus,
$(P_{1},\cdots,P_{n})$ is complete in $\mathcal{A}$. ∎
Combing Corollary 3.6 with Theorem 1.2 (3), we have
###### Corollary 3.7.
Let $P_{1},\cdots,P_{n}$ be non–trivial projections in $B(H)$ with
$\big{\|}\sum\limits^{n}_{i=1}P_{i}-I\big{\|}<(n-1)^{-2}$. Then
$H=\mathrm{Ran}(P_{1})\dotplus\cdots\dotplus\mathrm{Ran}(P_{n})$.
Let $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$. A well–known
statement says: “for any $\epsilon>0$, there is $\delta>0$ such that if
$\|P_{i}P_{j}\|<\delta$, $i\not=j$, $i,j=1,\cdots,n$, then there are mutually
orthogonal projections $P^{\prime}_{1},\cdots,P^{\prime}_{n}\in\mathcal{A}$
with $\|P_{i}-P_{i}^{\prime}\|<\epsilon$, $i=1,\cdots,n$”. It may be the first
time appeared in Glimm’s paper [5]. By using the induction on $n$, he gave its
proof. But how $\delta$ depends on $\epsilon$ is not given. Lemma 2.5.6 of [8]
states this statement and the author gives a slightly different proof. We can
find from the proof of [8, Lemma 2.5.6] that the relation between $\delta$ and
$\epsilon$ is $\delta\leq\dfrac{\epsilon}{(12)^{(n-1)}n!}$.
The next corollary will give a new proof of this statement with the relation
$\delta=\dfrac{\epsilon}{2(n-1)}$ for $\epsilon\in(0,1)$.
###### Corollary 3.8.
Let $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$. Given
$\epsilon\in(0,1)$. If $P_{1},\cdots,P_{n}$ satisfy condition:
$\|P_{i}P_{j}\|<\delta=\dfrac{\epsilon}{2(n-1)}$, $1\leq i<j\leq n$, then
there are mutually orthogonal projections
$P^{\prime}_{1},\cdots,P^{\prime}_{n}\in\mathcal{A}$ such that
$\|P_{i}-P_{i}^{\prime}\|<\epsilon$, $i=1,\cdots,n$.
###### Proof.
Set $A=\sum\limits^{n}_{i=1}P_{i}$. Noting that
$\gamma=\min\\{\beta(P_{1}),\cdots,\beta(P_{n})\\}=1$,
$\|P_{i}P_{j}\|<\dfrac{1}{n-1}$, $1\leq i<j\leq n$ and taking $\rho=1$, we
have $\sigma(A)\backslash\\{0\\}\subset[1-(n-1)\delta,1+(n-1)\delta]$ by
Proposition 3.5 (1). So the positive element $A^{\dagger}$ exists by Lemma
2.1. Set $P=A^{\dagger}A=AA^{\dagger}\in\mathcal{A}$. From $AA^{\dagger}A=A$
and $A^{\dagger}AA^{\dagger}=A^{\dagger}$, we get that $P_{i}\leq P$,
$i=1,\cdots,n$ and $AP=PA=A$, $A^{\dagger}P=PA^{\dagger}=A^{\dagger}$. So
$A\in GL(P\mathcal{A}P)$ with the inverse $A^{\dagger}\in P\mathcal{A}P$. Let
$\sigma_{P\mathcal{A}P}(A^{\dagger})$ stand for the spectrum of $A^{\dagger}$
in $P\mathcal{A}P$. Then
$\displaystyle\sigma_{P\mathcal{A}P}(A^{\dagger})$
$\displaystyle=\sigma(A^{\dagger})\backslash\\{0\\}=\\{\lambda^{-1}|\,\lambda\in\sigma(A)\backslash\\{0\\}\\}$
$\displaystyle\subset[(1+(n-1)\delta)^{-1},(1-(n-1)\delta)^{-1}],$ (3.2)
Now by Proposition 3.5,
$P\mathcal{A}=A\mathcal{A}=P_{1}\mathcal{A}\dotplus\cdots\dotplus
P_{n}\mathcal{A}$. Thus, by using $P_{i}\leq P$, $i=1,\cdots,n$, we have
$P\mathcal{A}P=P_{1}(P\mathcal{A}P)\dotplus\cdots\dotplus
P_{n}(P\mathcal{A}P)$ and then $P_{i}A^{\dagger}P_{j}=\delta_{ij}P_{i}$,
$i,j=1,\cdots,n$ by Theorem 1.2. Put
$P_{i}^{\prime}=(A^{\dagger})^{1/2}P_{i}(A^{\dagger})^{1/2}\in\mathcal{A}$,
$i=1,\cdots,n$. Then $P_{1}^{\prime},\cdots,P_{n}^{\prime}$ are mutually
orthogonal projections and moreover, for $1\leq i\leq n$,
$\displaystyle\|P^{\prime}_{i}-P_{i}\|$
$\displaystyle\leq\|(A^{\dagger})^{1/2}P_{i}(A^{\dagger})^{1/2}-P_{i}(A^{\dagger})^{1/2}\|+\|P_{i}(A^{\dagger})^{1/2}-P_{i}\|$
$\displaystyle\leq(\|(A^{\dagger})^{1/2}\|+1)\|(A^{\dagger})^{1/2}-P\|.$ (3.3)
Note that $0<(n-1)\delta<1/2$. Applying Spectrum Mapping Theorem to (3.2), we
get that $\|(A^{\dagger})^{1/2}\|)\leq(1-(n-1)\delta)^{-1/2}<\sqrt{2}$ and
$\|P-(A^{\dagger})^{1/2}\|\leq(1-(n-1)\delta)^{-1/2}-1<\frac{2}{1+\sqrt{2}}\,(n-1)\delta.$
Thus $\|P^{\prime}_{i}-P_{i}\|<2(n-1)\delta=\epsilon$ by (3.3). ∎
Applying Theorem 1.2 and Corollary 3.8 to an $n$–tuple of linear independent
unit vectors, we have:
###### Corollary 3.9.
Let $(\alpha_{1},\cdots,\alpha_{n})$ be an $n$–tuple of linear independent
unit vectors in Hilbert space $H$.
1. $(1)$
There is an invertible, positive operator $K$ in $B(H)$ and an $n$–tuple of
mutually orthogonal unit vectors $(\gamma_{1},\cdots,\gamma_{n})$ in $H$ such
that $\gamma_{i}=K\alpha_{i}$, $i=1,\cdots,n$.
2. $(2)$
Given $\epsilon\in(0,1)$. If
$|<\alpha_{i},\alpha_{j}>|<\dfrac{\epsilon}{2(n-1)}$, $1\leq i<j\leq n$, then
there exists an $n$–tuple of mutually orthogonal unit vectors
$(\beta_{1},\cdots,\beta_{n})$ in $H$ such that
$\|\alpha_{i}-\beta_{j}\|<2\epsilon$, $i=1,\cdots,n.$
###### Proof.
Set $H_{1}=\mathrm{span}\\{\alpha_{1},\cdots,\alpha_{n}\\}$ and
$P_{i}\xi=<\xi,\alpha_{i}>\alpha_{i}$, $\forall\,\xi\in H_{1}$,
$i=1,\cdots,n$. Then $(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(B(H_{1}))$ and
$\mathrm{Ran}(P_{1})\dotplus\cdots\dotplus\mathrm{Ran}(P_{n})=H_{1}$.
By Theorem 1.2, $A_{0}=\sum\limits^{n}_{i=1}P_{i}$ is invertible in $B(H_{1})$
and $P_{i}A_{0}^{-1}P_{j}=\delta_{ij}P_{i}$, $i,j=1,\cdots,n$. Put
$K=A_{0}^{-1/2}+P_{0}$ and $\gamma_{i}=A_{0}^{-1/2}\alpha_{i}$,
$i=1,\cdots,n$, where $P_{0}$ is the projection of $H$ onto $H_{1}^{\perp}$.
It is easy to check that $K$ is invertible and positive in $B(H)$ with
$\gamma_{i}=K\alpha_{i}$, $i=1,\cdots,n$ and $(\gamma_{1},\cdots,\gamma_{n})$
is an $n$–tuple of mutually orthogonal unit vectors. This proves (1).
(2) Note that
$\|P_{i}P_{j}\|=|<\alpha_{i},\alpha_{j}>|<\dfrac{\epsilon}{2(n-1)}$, $1\leq
i<j\leq n$. Thus, by Corollary 3.8, there are mutually orthogonal projections
$P^{\prime}_{1},\cdots,P^{\prime}_{n}\in\mathcal{A}$ such that
$\|P_{i}-P_{i}^{\prime}\|<\epsilon$, $i=1,\cdots,n$. Put
$\beta_{i}^{\prime}=P^{\prime}_{i}\alpha_{i}$, $i=1,\cdots,n$. Then
$\beta_{1}^{\prime},\cdots,\beta_{n}^{\prime}$ are mutually orthogonal and
$\|\alpha_{i}-\beta^{\prime}_{i}\|<\epsilon$, $i=1,\cdots,n$. Set
$\beta_{i}=\|\beta_{i}^{\prime}\|^{-1}\beta^{\prime}_{i}$, $i=1,\cdots,n$.
Then $<\beta_{i},\beta_{j}>=\delta_{ij}\beta_{i}$, $i,j=1,\cdots,n$ and
$\|\alpha_{i}-\beta_{i}\|\leq\|\alpha_{i}-\beta^{\prime}_{i}\|+|1-\|\beta^{\prime}_{i}\||<2\epsilon,$
for $i=1,\cdots,n$. ∎
Now we give a simple characterization of the completeness of a given $n$–tuple
of projections in $C^{*}$–algebra $\mathcal{A}$ as follows.
###### Theorem 3.10.
Let $P_{1},\cdots,P_{n}$ be projections in $\mathcal{A}$. Then
$(P_{1},\cdots,P_{n})$ is complete if and only if
$A=\sum\limits_{i=1}^{n}P_{i}$ is invertible in $\mathcal{A}$ and
$\|P_{i}A^{-1}P_{j}\|<\big{[}(n-1)\|A^{-1}\|\|A\|^{2}\big{]}^{-1}$,
$\forall\,i\not=j$, $i,j=1,\cdots,n$.
###### Proof.
If $(P_{1},\cdots,P_{n})$ is complete, then by Theorem 1.2, $A$ is invertible
in $\mathcal{A}$ and $P_{i}A^{-1}P_{j}=0$, $\forall\,i\not=j$,
$i,j=1,\cdots,n$.
Now we prove the converse.
Put $T_{i}=A^{-1/2}P_{i}A^{-1/2}$, $1\leq i\leq n$. Then
$\sum\limits^{n}_{i=1}T_{i}=1$. Since $T_{i}(A^{1/2}P_{i}A^{1/2})T_{i}=T_{i}$,
we have $\beta(T_{i})\geq\|A^{1/2}P_{i}A^{1/2}\|^{-1}\geq\|A\|^{-1}$,
$i=1,\cdots,n$ by Corollary 3.2. Put $\rho=\|A\|^{-1}$. Then
$\|T_{i}T_{j}\|\leq\|A^{-1}\|\|P_{i}A^{-1}P_{j}\|<\big{[}(n-1)\|A\|^{2}\big{]}^{-1}=\frac{\rho^{2}}{n-1},\
i\not=j,\ i,j=1,\cdots,n.$
Thus by Proposition 3.5 (3),
$\mathcal{A}=T_{1}\mathcal{A}\dotplus\cdots\dotplus T_{n}\mathcal{A}$. Note
that $T_{i}\mathcal{A}=A^{-1/2}(P_{i}\mathcal{A})$, $i=1,\cdots,n$. So
$P_{1}\mathcal{A}\dotplus\cdots\dotplus
P_{n}\mathcal{A}=A^{1/2}\mathcal{A}=\mathcal{A}$, i.e.,
$(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$. ∎
###### Corollary 3.11.
Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ and let
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$.
Assume that
$\|P_{i}-P^{\prime}_{i}\|<\big{[}4n^{2}(n-1)\|A^{-1}\|^{2}(n\|A^{-1}\|+1)\big{]}^{-1}$,
$i=1,\cdots,n$, where $A=\sum\limits^{n}_{i=1}P_{i}$, then
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})\in\mathbf{PC}_{n}(\mathcal{A})$.
###### Proof.
Set $B=\sum\limits^{n}_{i=1}P^{\prime}_{i}$. Since
$n\|A^{-1}\|\geq\|A\|\|A^{-1}\|\geq 1$, it follows that
$\|A-B\|<\dfrac{1}{2\|A^{-1}\|}$. Thus $B$ is invertible in $\mathcal{A}$ with
$\|B^{-1}\|\leq\dfrac{\|A^{-1}\|}{1-\|A^{-1}\|\|A-B\|}<2\|A^{-1}\|,\
\|B^{-1}-A^{-1}\|<2\|A^{-1}\|^{2}\|A-B\|.$
Note that $P_{i}A^{-1}P_{j}=0$, $i\not=j$, $i,j=1,\cdots,n$, we have
$\displaystyle\|P^{\prime}_{i}B^{-1}P^{\prime}_{j}\|$
$\displaystyle\leq\|P^{\prime}_{i}(B^{-1}-A^{-1})P^{\prime}_{j}\|+\|(P^{\prime}_{i}-P_{i})A^{-1}P^{\prime}_{j}\|+\|P_{i}A^{-1}(P_{j}-P^{\prime}_{j})\|$
$\displaystyle\leq
2\|A^{-1}\|^{2}\|A-B\|+\|A^{-1}\|\|P_{i}-P^{\prime}_{i}\|+\|A^{-1}\|\|P_{j}-P^{\prime}_{j}\|$
$\displaystyle<\frac{1}{2n^{2}(n-1)\|A^{-1}\|}<\frac{1}{(n-1)\|B^{-1}\|\|B\|^{2}}.$
So $(P_{1}^{\prime},\cdots,P_{n}^{\prime})$ is complete in $\mathcal{A}$ by
Theorem 3.10. ∎
## 4 Some equivalent relations and topological properties on
$\mathbf{PC}_{n}(\mathcal{A})$
Let $\mathcal{A}$ be a $C^{*}$–algebra with the unit $1$ and let
$GL_{0}(\mathcal{A})$ (resp. $U_{0}(\mathcal{A})$) be the connected component
of $1$ in $GL(\mathcal{A})$ (resp. in $U(\mathcal{A})$). Set
$\displaystyle\mathbf{PI}_{n}(\mathcal{A})$
$\displaystyle=\big{\\{}(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})\,|\,\sum\limits^{n}_{i=1}P_{i}\in
GL(\mathcal{A})\big{\\}}$ $\displaystyle\mathbf{PO}_{n}(\mathcal{A})$
$\displaystyle=\big{\\{}(P_{1},\cdots,P_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})\,|\,\sum\limits^{n}_{i=1}P_{i}=1,\
P_{i}P_{j}=0,\ i\not=j,\ i,j=1,\cdots,n\big{\\}}.$
###### Definition 4.1.
Let $(P_{1},\cdots,P_{n})$ and $(P_{1}^{\prime},\cdots,P_{n}^{\prime})$ be in
$\mathbf{PC}_{n}(\mathcal{A})$.
1. $(1)$
We say $(P_{1},\cdots,P_{n})$ is equivalent to
$(P_{1}^{\prime},\cdots,P_{n}^{\prime})$, denoted by
$(P_{1},\cdots,P_{n})\sim(P_{1}^{\prime},\cdots,P_{n}^{\prime})$, if there are
$U_{1},\cdots,U_{n}\in\mathcal{A}$ such that $P_{i}=U_{i}^{*}U_{i}$,
$P_{i}^{\prime}=U_{i}U_{i}^{*}$.
2. $(2)$
$(P_{1},\cdots,P_{n})$ and $(P_{1}^{\prime},\cdots,P_{n}^{\prime})$ are called
to be unitarily equivalent, denoted by
$(P_{1},\cdots,P_{n})\sim_{u}(P_{1}^{\prime},\cdots,P_{n}^{\prime})$, if there
is $U\in U(\mathcal{A})$ such that $UP_{i}U^{*}=P_{i}^{\prime}$,
$i=1,\cdots,n$.
3. $(3)$
$(P_{1},\cdots,P_{n})$ and $(P_{1}^{\prime},\cdots,P_{n}^{\prime})$ are called
homotopically equivalent, denoted by
$(P_{1},\cdots,P_{n})\sim_{h}(P_{1}^{\prime},\cdots,P_{n}^{\prime})$, if there
exists a continuous mapping
$F\colon[0,1]\rightarrow\mathbf{PC}_{n}(\mathcal{A})$ such that
$F(0)=(P_{1},\cdots,P_{n})$ and $F(1)=(P_{1}^{\prime},\cdots,P_{n}^{\prime})$.
It is well–know that
$(P_{1},\cdots,P_{n})\sim_{h}(P_{1}^{\prime},\cdots,P_{n}^{\prime})\Rightarrow(P_{1},\cdots,P_{n})\sim(P_{1}^{\prime},\cdots,P_{n}^{\prime})$
and if $U(\mathcal{A})$ is path–connected,
$(P_{1},\cdots,P_{n})\sim_{u}(P_{1}^{\prime},\cdots,P_{n}^{\prime})\Rightarrow(P_{1},\cdots,P_{n})\sim_{h}(P_{1}^{\prime},\cdots,P_{n}^{\prime}).$
###### Lemma 4.2.
Let $(P_{1},\cdots,P_{n})$ be in $\mathbf{PC}_{n}(\mathcal{A})$ and $C$ be a
positive and invertible element in $\mathcal{A}$ with $P_{i}C^{2}P_{i}=P_{i}$,
$i=1,\cdots,n$. Then $(CP_{1}C,\cdots,CP_{n}C)\in\mathbf{PC}_{n}(\mathcal{A})$
and $(P_{1},\cdots,P_{n})\sim_{h}(CP_{1}C,\cdots,CP_{n}C)$ in
$\mathbf{PC}_{n}(\mathcal{A})$.
###### Proof.
From $(CP_{i}C)^{2}=CP_{i}C^{2}P_{i}C=CP_{i}C$, $1\leq i\leq n$, we have
$(CP_{1}C,\cdots,CP_{n}C)$ $\in\mathbf{P}\\!_{n}(\mathcal{A})$.
$(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ implies that
$A=\sum\limits^{n}_{i=1}P_{i}\in GL(\mathcal{A})$ and
$P_{i}A^{-1}P_{i}=P_{i}$, $1\leq i\leq n$ by Theorem 1.2. So
$(CP_{i}C)\Big{(}\sum\limits_{i=1}^{n}(CP_{i}C)\Big{)}^{-1}(CP_{i}C)=CP_{i}A^{-1}P_{i}C$
and hence $(CP_{1}C,\cdots,CP_{n}C)\in\mathbf{PC}_{n}(\mathcal{A})$ by Theorem
1.2.
Put $A_{i}(t)=C^{t}P_{i}C^{t}$ and $B_{i}(t)=C^{-t}P_{i}C^{-t}$,
$\forall\,t\in[0,1]$, $i=1,\cdots,n$. Then $Q_{i}(t)\triangleq
A_{i}(t)B_{i}(t)=C^{t}P_{i}C^{-t}$ is idempotent and
$A_{i}(t)B_{i}(t)A_{i}(t)=A_{i}(t)$, $\forall\,t\in[0,1]$, $i=1,\cdots,n$.
Thus $A_{i}(t)\mathcal{A}=Q_{i}(t)\mathcal{A}$, $\forall\,t\in[0,1]$,
$i=1,\cdots,n$.
By Lemma 2.3, $P_{i}(t)\triangleq Q_{i}(t)(Q_{i}(t)+(Q_{i}(t))^{*}-1)^{-1}$ is
a projection in $\mathcal{A}$ satisfying $Q_{i}(t)P_{i}(t)=P_{i}(t)$ and
$P_{i}(t)Q_{i}(t)=Q_{i}(t)$, $\forall\,t\in[0,1]$, $i=1,\cdots,n$. Clearly,
$A_{i}(t)\mathcal{A}=Q_{i}(t)\mathcal{A}=P_{i}(t)\mathcal{A}$,
$\forall\,t\in[0,1]$ and $t\mapsto P_{i}(t)$ is a continuous mapping from
$[0,1]$ into $\mathcal{A}$, $i=1,\cdots,n$. Thus, from
$(C^{t}P_{1}C^{t})\mathcal{A}\dotplus\cdots\dotplus(C^{t}P_{n}C^{t})\mathcal{A}=\mathcal{A},\quad\forall\,t\in[0,1],$
we get that $F(t)=(P_{1}(t),\cdots,P_{n}(t))\in\mathbf{PC}_{n}(\mathcal{A})$,
$\forall\,t\in[0,1]$. Note that
$F\colon[0,1]\rightarrow\mathbf{PC}_{n}(\mathcal{A})$ is continuous with
$F(0)=(P_{1},\cdots,P_{n})$. Note that $A_{i}(1)=CP_{i}C$ is a projection with
$A_{i}(1)Q_{i}(1)=CP_{i}CCP_{i}C^{-1}=Q_{i}(1)$ and
$Q_{i}(1)A_{i}(1)=A_{i}(1)$, $i=1,\cdots,n$. So $P_{i}(1)=A_{i}(1)$,
$i=1,\cdots,n$ and $F(1)=(CP_{1}C,\cdots,CP_{n}C)$. The assertion follows. ∎
For $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$,
$A=\sum\limits^{n}_{i=1}P_{i}\in GL(\mathcal{A})$ and
$Q_{i}=A^{-1/2}P_{i}A^{-1/2}$ is a projection with $Q_{i}Q_{j}=0$, $i\not=j$,
$i,j=1,\cdots,n$ (see Theorem 1.2), that is,
$(Q_{1},\cdots,Q_{n})\in\mathbf{PO}_{n}(\mathcal{A})$. Since $C=A^{-1/2}$
satisfies the condition given in Lemma 4.2, we have the following:
###### Corollary 4.3.
Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ and let
$(Q_{1},\cdots,Q_{n})$ be as above. Then
$(P_{1},\cdots,P_{n})\sim_{h}(Q_{1},\cdots,Q_{n})$ in
$\mathbf{PC}_{n}(\mathcal{A})$.
###### Theorem 4.4.
Let $(P_{1},\cdots,P_{n})$ and
$(P_{1}^{\prime},\cdots,P_{n}^{\prime})\in\mathbf{PC}_{n}(\mathcal{A})$. Then
the following statements are equivalent:
1. $(1)$
$(P_{1},\cdots,P_{n})\sim(P_{1}^{\prime},\cdots,P_{n}^{\prime})$.
2. $(2)$
there is $D\in GL(\mathcal{A})$ such that for $1\leq i\leq n$,
$P_{i}DD^{*}P_{i}=P_{i}$ and $P^{\prime}_{i}=D^{*}P_{i}D$.
3. $(3)$
there is $(S_{1},\cdots,S_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ such that
$(P_{1},\cdots,P_{n})\sim_{u}(S_{1},\cdots,S_{n})\sim_{h}(P_{1}^{\prime},\cdots,P_{n}^{\prime}).$
###### Proof.
The implication (3)$\Rightarrow$(1) is obvious. We now prove the implications
(1)$\Rightarrow$(2) and (2)$\Rightarrow$(3) as follows.
(1)$\Rightarrow$ (2) Let $U_{i}\in\mathcal{A}$ be partial isometries such that
$U_{i}^{*}U_{i}=P_{i}$, $U_{i}U^{*}_{i}=P^{\prime}_{i}$, $i=1,\cdots,n$. Put
$A=\sum\limits^{n}_{i=1}P_{i}$,
$A^{\prime}=\sum\limits^{n}_{i=1}P_{i}^{\prime}$ and
$W=A^{-1/2}\big{(}\sum\limits_{i=1}^{n}P_{i}U^{*}_{i}P^{\prime}_{i}\big{)}A^{\prime-1/2}$.
Then
$\displaystyle W^{*}W$
$\displaystyle=A^{\prime-1/2}\big{(}\sum_{i=1}^{n}P^{\prime}_{i}U_{i}P_{i}\big{)}A^{-1}\big{(}\sum_{i=1}^{n}P_{i}U^{*}_{i}P^{\prime}_{i}\big{)}A^{\prime-1/2}$
$\displaystyle=A^{\prime-1/2}\big{(}\sum_{i=1}^{n}P^{\prime}_{i}U_{i}P_{i}U^{*}_{i}P^{\prime}_{i}\big{)}A^{\prime-1/2}=A^{\prime-1/2}\big{(}\sum_{i=1}^{n}P^{\prime}_{i}\big{)}A^{\prime-1/2}=1.$
Similarly, $WW^{*}=1$. Thus, $W\in U(\mathcal{A})$. Set $D=A^{-1/2}WA^{\prime
1/2}\in GL(\mathcal{A})$. Then, for $1\leq i\leq n$,
$D^{*}P_{i}D=\big{(}\sum_{i=1}^{n}P^{\prime}_{i}U_{i}P_{i}\big{)}A^{-1}P_{i}A^{-1}\big{(}\sum_{i=1}^{n}P_{i}U^{*}_{i}P^{\prime}_{i}\big{)}=P^{\prime}_{i}U_{i}P_{i}U^{*}_{i}P^{\prime}_{i}=P^{\prime}_{i}$
and $P_{i}DD^{*}P_{i}=P_{i}$ follows from $(D^{*}P_{i}D)^{2}=D^{*}P_{i}D$.
(2)$\Rightarrow$(3) Put $U=(DD^{*})^{-1/2}D$. Then $U\in U(\mathcal{A})$. Set
$C=U^{*}(DD^{*})^{1/2}U$ and $S_{i}=U^{*}P_{i}U$, $1\leq i\leq n$. Then
$(S_{1},\cdots,S_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ with
$(S_{1},\cdots,S_{n})$$\sim_{u}(P_{1},\cdots,P_{n})$ and
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})=(CS_{1}C,\cdots,CS_{n}C)$.
Since $S_{i}C^{2}S_{i}=U^{*}P_{i}DD^{*}P_{i}U=S_{i}$, $i=1,\cdots,n$, it
follows from Lemma 4.2 that
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})\sim_{h}(S_{1},\cdots,S_{n})$ in
$\mathbf{PC}_{n}(\mathcal{A})$. ∎
###### Proposition 4.5.
For $\mathbf{P}\\!_{n}(\mathcal{A})$, $\mathbf{PC}_{n}(\mathcal{A})$,
$\mathbf{PI}_{n}(\mathcal{A})$ and $\mathbf{PO}_{n}(\mathcal{A})$, we have
1. $(1)$
$\mathbf{PI}_{n}(\mathcal{A})$ is open in $\mathbf{P}\\!_{n}(\mathcal{A})$.
2. $(2)$
$\mathbf{PC}_{n}(\mathcal{A})$ is a clopen subset of
$\mathbf{PI}_{n}(\mathcal{A})$.
3. $(3)$
$\mathbf{PO}_{n}(\mathcal{A})$ is a strong deformation retract of
$\mathbf{PC}_{n}(\mathcal{A})$.
4. $(4)$
$\mathbf{PC}_{n}(\mathcal{A})$ is locally connected. Thus every connected
component of $\mathbf{PC}_{n}(\mathcal{A})$ is path–connected.
5. $(5)$
$(P_{1},\cdots,P_{n}),\,(P_{1}^{\prime},\cdots,P_{n}^{\prime})\in\mathbf{PC}_{n}(\mathcal{A})$
are in the same connected component iff there is $D\in GL_{0}(\mathcal{A})$
such that $P_{i}=D^{*}P_{i}^{\prime}D$, $i=1,\cdots,n$.
###### Proof.
(1) Since $h(P_{1},\cdots,P_{n})=\sum\limits^{n}_{i=1}P_{i}$ is a continuous
mapping from $\mathbf{P}\\!_{n}(\mathcal{A})$ into $\mathcal{A}$ and
$GL(\mathcal{A})$ is open in $\mathcal{A}$, it follows that
$\mathbf{PI}_{n}(\mathcal{A})=h^{-1}(GL(\mathcal{A}))$ is open in
$\mathbf{P}\\!_{n}(\mathcal{A})$.
(2) Define $F\colon\mathbf{PI}_{n}(\mathcal{A})\rightarrow\mathbb{R}$ by
$F(P_{1},\cdots,P_{n})=\sum_{1\leq i<j\leq
n}(n-1)\big{\|}\sum^{n}_{i=1}P_{i}\big{\|}^{2}\big{\|}\big{(}\sum^{n}_{i=1}P_{i}\big{)}^{-1}\big{\|}\big{\|}P_{i}\big{(}\sum^{n}_{i=1}P_{i}\big{)}^{-1}P_{j}\|.$
Clearly, $F$ is continuous on $\mathbf{PI}_{n}(\mathcal{A})$. By means of
Theorem 3.10, we get that $\mathbf{PC}_{n}(\mathcal{A})=F^{-1}((-1,1))$ is
open in $\mathbf{PI}_{n}(\mathcal{A})$ and
$\mathbf{PC}_{n}(\mathcal{A})=F^{-1}(\\{0\\})$ is closed in
$\mathbf{PI}_{n}(\mathcal{A})$.
(3) Define the continuous mapping
$r\colon\mathbf{PC}_{n}(\mathcal{A})\rightarrow\mathbf{PO}_{n}(\mathcal{A})$
by
$r(P_{1},\cdots,P_{n})=\big{(}A^{-1/2}P_{1}A^{-1/2},\cdots,A^{-1/2}P_{n}A^{-1/2}\big{)},\quad
A=\sum^{n}_{i=1}P_{i}.$
by Theorem 1.2. Clearly, $r(P_{1},\cdots,P_{n})=(P_{1},\cdots,P_{n})$ when
$(P_{1},\cdots,P_{n})\in\mathbf{PO}_{n}(\mathcal{A})$. This means that
$\mathbf{PO}_{n}(\mathcal{A})$ is a retract of $\mathbf{PC}_{n}(\mathcal{A})$.
For any $t\in[0,1]$ and $i=1,\cdots,n$, put
$H_{i}(P_{1},\cdots,P_{n},t)=A^{-t/2}P_{i}A^{t/2}(A^{-t/2}P_{i}A^{t/2}+A^{t/2}P_{i}A^{-t/2}-1)^{-1}.$
Similar to the proof of Lemma 4.2, we have
$H(P_{1},\cdots,P_{n},t)=(H_{1}(P_{1},\cdots,P_{n},t),\cdots,H_{n}(P_{1},\cdots,P_{n},t))$
is a continuous mapping from $\mathbf{PC}_{n}(\mathcal{A})\times[0,1]$ to
$\mathbf{PC}_{n}(\mathcal{A})$ with
$H(P_{1},\cdots,P_{n},0)=(P_{1},\cdots,P_{n})$ and
$H(P_{1},\cdots,P_{n},1)=r(P_{1},\cdots,P_{n})$. Furthermore, when
$(P_{1},\cdots,P_{n})$ $\in\mathbf{PO}_{n}(\mathcal{A})$, $A=1$. In this case,
$H(P_{1},\cdots,P_{n},t)=(P_{1},\cdots,P_{n})$, $\forall\,t\in[0,1]$.
Therefore, $\mathbf{PO}_{n}(\mathcal{A})$ is a strong deformation retract of
$\mathbf{PC}_{n}(\mathcal{A})$.
(4) Let $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$. Then by
Corollary 3.11, there is $\delta\in(0,1/2)$ such that for any
$(R_{1},\cdots,R_{n})\in\mathbf{P}\\!_{n}(\mathcal{A})$ with
$\|P_{i}-R_{i}\|<\delta$, $1\leq i\leq n$, we have
$(R_{1},\cdots,R_{n})\in\mathbf{PC}_{n}(\mathcal{A})$.
Let $(R_{1},\cdots,R_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ with
$\|P_{j}-R_{j}\|<\delta$, $i=1,\cdots,n$. put $P_{i}(t)=P_{i}$,
$R_{i}(t)=R_{i}$ and $a_{i}(t)=(1-t)P_{i}+tR_{i}$, $\forall\,t\in[0,1]$,
$i=1,\cdots,n$. Then $P_{i},R_{i},a_{i}$ are self–adjoint elements in
$C([0,1],\mathcal{A})=\mathcal{B}$ and
$\|P_{i}-a_{i}\|=\max\limits_{t\in[0,1]}\|P_{i}-a_{i}(t)\|<\delta$,
$i=1,\cdots,n$. It follows from [17, Lemm 6.5.9 (1)] that there exists a
projection $f_{i}\in C^{*}(a_{i})$ (the $C^{*}$–subalgebra of $\mathcal{B}$
generated by $a_{i}$) such that $\|P_{i}-f_{i}\|\leq\|P_{i}-a_{i}\|<\delta$,
$i=1,\cdots,n$. Thus, $\|P_{i}-f_{i}(t)\|<\delta$, $i=1,\cdots,n$ and
consequently, $F(t)=(f_{1}(t),\cdots,f_{n}(t))$ is a continuous mapping of
$[0,1]$ into $\mathbf{PC}_{n}(\mathcal{A})$. Since $a_{i}(0)=P_{i}$,
$a_{i}(1)=R_{i}$ and $f_{i}(t)\in C^{*}(a_{i}(t))$, $\forall\,t\in[0,1]$, we
have $f(0)=(P_{1},\cdots,P_{n})$ and $f(1)=(R_{1},\cdots,R_{n})$. This means
that $\mathbf{PC}_{n}(\mathcal{A})$ is locally path–connected.
(5) There is a continuous path
$P(t)=(P_{1}(t),\cdots,P_{n}(t))\in\mathbf{PC}_{n}(\mathcal{A})$,
$\forall\,t\in[0,1]$ such that $P(0)=(P_{1},\cdots,P_{n})$ and
$P(1)=(P_{1}^{\prime},\cdots,P_{n}^{\prime})$. By [12, Corollary 5.2.9.],
there is a continuous mapping $t\mapsto U_{i}(t)$ of $[0,1]$ into
$U(\mathcal{A})$ with $U_{i}(0)=1$ such that
$P_{i}(t)=U_{i}(t)P_{1}U_{i}^{*}(t)$, $\forall\,t\in[0,1]$ and $i=1,\cdots,n$.
Set
$W(t)=\Big{(}\sum\limits^{n}_{i=1}P_{i}\Big{)}^{-1/2}\Big{(}\sum\limits^{n}_{i=1}P_{i}U_{i}^{*}(t)P_{i}(t)\Big{)}\Big{(}\sum\limits^{n}_{i=1}U_{i}(t)P_{i}U_{i}^{*}(t)\Big{)}^{-1/2}$
and
$D(t)=\Big{(}\sum\limits^{n}_{i=1}P_{i}\Big{)}^{-1/2}W(t)\Big{(}\sum\limits^{n}_{i=1}U_{i}(t)P_{i}U_{i}^{*}(t)\Big{)}^{1/2}$,
$\forall\,t\in[0,1]$. Then $W(t)\in U(\mathcal{A})$ with $W(0)=1$, $D(t)\in
GL(\mathcal{A})$ with $D(0)=1$ and $W(t)$, $D(t)$ are all continuous on
$[0,1]$ with $D^{*}(t)P_{i}D(t)=P_{i}(t)$ (see the proof of
(1)$\Rightarrow$(2) in Theorem 4.4), $\forall\,t\in[0,1]$ and $i=1,\cdots,n$.
Put $D=D(1)$. Then $D\in GL_{0}(\mathcal{A})$ and
$D^{*}P_{i}D=P_{i}^{\prime}$, $i=1,\cdots,n$.
Conversely, if there is $D\in GL_{0}(\mathcal{A})$ such that
$D^{*}P_{i}D=P_{i}^{\prime}$, $i=1,\cdots,n$. Then $U=(DD^{*})^{-1/2}D\in
U_{0}(\mathcal{A})$ and $P_{i}DD^{*}P_{i}=P_{i}$,
$UP_{i}^{\prime}U^{*}=(DD^{*})^{1/2}P_{i}(DD^{*})^{1/2}$, $i=1,\cdots,n$.
Thus,
$(P_{1}^{\prime},\cdots,P_{n}^{\prime})\sim_{h}(UP_{1}^{\prime}U^{*},\cdots,UP_{n}^{\prime}U^{*})$
and
$((DD^{*})^{1/2}P_{1}(DD^{*})^{1/2},\cdots,(DD^{*})^{1/2}P_{n}(DD^{*})^{1/2})\sim_{h}(P_{1},\cdots,P_{n})$
by Lemma 4.2. Consequently,
$(P_{1}^{\prime},\cdots,P_{n}^{\prime})\sim_{h}(P_{1},\cdots,P_{n})$. ∎
As ending of this section, we consider following examples:
###### Example 4.6.
Let $\mathcal{A}=\mathrm{M}_{k}(\mathbb{C})$, $k\geq 2$. Define a mapping
$\rho\colon\mathbf{PC}_{n}(\mathcal{A})\rightarrow\mathbb{N}^{n-1}$ by
$\rho(P_{1},\cdots,P_{n})=(\mathrm{Tr}(P_{1}),\cdots,\mathrm{Tr}(P_{n-1}))$,
where $2\leq n\leq k$ and $\mathrm{Tr}(\cdot)$ is the canonical trace on
$\mathcal{A}$.
By Theorem 1.2, $(P_{1},\cdots,P_{n})\in\mathbf{PC}_{n}(\mathcal{A})$ means
that $A=\sum\limits^{n}_{i=1}P_{i}\in GL(\mathcal{A})$ and
$(A^{-1/2}P_{1}A^{-1/2},\cdots,A^{-1/2}P_{n}A^{-1/2})\in\mathbf{PO}_{n}(\mathcal{A})$.
Put $Q_{i}=A^{-1/2}P_{i}A^{-1/2}$, $i=1,\cdots,n$. Since
$(P_{1},\cdots,P_{n})\sim_{h}(Q_{1},\cdots,Q_{n})$ by Corollary 4.3, it
follows that $\mathrm{Tr}(P_{i})=\mathrm{Tr}(Q_{i})$, $i=1,\cdots,n$ and
$\mathrm{Tr}(A)=k$. Thus $\mathrm{Tr}(P_{n})=k-\sum\limits^{n-1}_{i=1}P_{i}$.
Note that $U(\mathcal{A})$ is path–connected. So, for
$(P_{1},\cdots,P_{n}),\,(P_{1}^{\prime},\cdots,P_{n}^{\prime})\in\mathbf{PC}_{n}(\mathcal{A})$,
$(P_{1},\cdots,P_{n})$ and $(P_{1}^{\prime},\cdots,P_{n}^{\prime})$ are in the
same connected component if and only if
$\rho(P_{1},\cdots,P_{n})=\rho(P_{1}^{\prime},\cdots,P_{n}^{\prime})$.
The above shows that $\mathbf{PC}_{k}(\mathcal{A})$ is connected and
$\mathbf{PC}_{n}(\mathcal{A})$ is not connected when $k\geq 3$ and $2\leq
n\leq k-1$.
###### Example 4.7.
Let $H$ be a separable complex Hilbert space and $\mathcal{K}(H)$ be the
$C^{*}$–algebra of all compact operators in $B(H)$. Let
$\mathcal{A}=B(H)/\mathcal{K}(H)$ be the Calkin algebra and $\pi\colon
B(H)\rightarrow\mathcal{A}$ be the quotient mapping. Then
$\mathbf{PC}_{n}(\mathcal{A})$ is path–connected.
In fact, if
$(P_{1},\cdots,P_{n}),(P_{1}^{\prime},\cdots,P_{n}^{\prime})\in\mathbf{PC}_{n}(\mathcal{A})$,
then we can find $(Q_{1},\cdots,Q_{n})$,
$(Q_{1}^{\prime},\cdots,Q_{n}^{\prime})\in\mathbf{PO}_{n}(\mathcal{A})$ such
that $(P_{1},\cdots,P_{n})\sim_{h}(Q_{1},\cdots,Q_{n})$ and
$(P^{\prime}_{1},\cdots,P^{\prime}_{n})\sim_{h}(Q^{\prime}_{1},\cdots,Q^{\prime}_{n})$
by Corollary 4.3. Since $B(H)$ is of real rank zero, it follows from [17,
Corollary B.2.2] or [16, Lemma 3.2] that there are projections
$R_{1},\cdots,R_{n}$ and $R_{1}^{\prime},\cdots,R_{n}^{\prime}$ in $B(H)$ such
that $\pi(R_{i})=Q_{i}$, $\pi(R_{i}^{\prime})=Q_{i}^{\prime}$, $i=1,\cdots,n$
and
$R_{i}R_{j}=\delta_{ij}R_{i},\
R_{i}^{\prime}R_{j}^{\prime}=\delta_{ij}R_{i}^{\prime},\ i,j=1,\cdots,n,\
\sum\limits^{n}_{i=1}R_{i}=\sum\limits^{n}_{i=1}R_{i}^{\prime}=I.$
Note that
$R_{1},\cdots,R_{n},R_{1}^{\prime},\cdots,R_{n}^{\prime}\not\in\mathcal{K}(H)$.
So there are partial isometry operators $V_{1},\cdots,V_{n}$ in $B(H)$ such
that $V_{i}^{*}V_{i}=R_{i}$, $V_{i}V_{i}^{*}=R_{i}^{\prime}$, $i=1,\cdots,n$.
Put $V=\sum\limits^{n}_{i=1}V_{i}$. Then $V\in U(B(H))$ and
$VR_{i}V^{*}=R_{i}^{\prime}$, $i=1,\cdots,n$. Put $U=\pi(V)\in
U(\mathcal{A})$. Then
$(UQ_{1}U^{*},\cdots,UQ_{n}U^{*})=(Q_{1}^{\prime},\cdots,Q_{n})$ in
$\mathbf{PO}_{n}(\mathcal{A})$. Since $U(B(H))$ is path–connected, we have
$(Q_{1},\cdots,Q_{n})\sim_{h}(Q_{1}^{\prime},\cdots,Q_{n}^{\prime})$ in
$\mathbf{PC}_{n}(\mathcal{A})$. Finally,
$(P_{1},\cdots,P_{n})\sim_{h}(P_{1}^{\prime},\cdots,P_{n}^{\prime})$. This
means that $\mathbf{PC}_{n}(\mathcal{A})$ is path–connected.
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* [6] P. Halmos, Two subspace, Tran. Amer. Math. Soc., 144 (1969), 381–389.
* [7] R.A. Horn and R. Johnson, Matrix Analysis, Cambridge University Press, 1986.
* [8] H. Lin, An Intruduction to the Classification of Amenable $C^{*}$-Algebras, World Scintific, 2001.
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* [10] J.J. Koliha and V. Rakočević, Invertibility of the difference of idempotents, Linear and Multilinear Algebra 51 (2003), 97–110.
* [11] J.J. Koliha and V. Rakočević, Fredholm properties of the difference of orthogonal projections in a Hilbert space, Integr. Equ. Oper. Theory, 52 (2005), 125–134.
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|
arxiv-papers
| 2012-10-17T08:55:44 |
2024-09-04T02:49:36.704738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shanwen Hu and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1210.4670"
}
|
1210.4714
|
arxiv-papers
| 2012-10-17T12:32:06 |
2024-09-04T02:49:36.717867
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ramazan Akg\\\"un",
"submitter": "Ramazan Akg\\\"un",
"url": "https://arxiv.org/abs/1210.4714"
}
|
|
1210.4773
|
The spectrum of Fe II]The spectrum of Fe II
Gillian Nave^1, Sveneric Johansson^2
1. National Institute of Standards and Technology,
Gaithersburg, MD, USA.
2. Lund Observatory, University of Lund, Sweden.
The spectrum of singly-ionized iron (Fe II) has been recorded using
high-resolution Fourier transform and grating spectroscopy over the
wavelength range 900 Å to 5.5 $\mu$m. The spectra were observed in
high-current continuous and pulsed hollow cathode discharges using
Fourier transform (FT) spectrometers at the Kitt Peak National
Observatory, Tucson, AZ and Imperial College, London and with the
10.7 m Normal Incidence Spectrograph at the National Institute of
Standards and Technology. Roughly 12 900 lines were classified
using 1027 energy levels of Fe II that were optimized to measured
wavenumbers. The wavenumber uncertainties of lines in the FT
spectra range from 10$^{-4}$ cm$^{-1}$ for strong lines around
4 $\mu$m to 0.05 cm$^{-1}$ for weaker lines around 1500 Å. The
wavelength uncertainty of lines in the grating spectra is 0.005 Å.
The ionization energy of (130 655.4$\pm$0.4) cm$^{-1}$ was
estimated from the $\rm 3d^6(^5D)5g$ and $\rm 3d^6(^5D)6h$ levels.
§ PREFACE
The title of Sveneric Johansson's last paper, “A half-life of Fe II”
[Johansson, 2009], accurately describes his association with the
spectrum during his career. He began work on the spectrum as a graduate
student of Edlén, with papers on new energy levels
[Johansson & Litzén, 1974] and a comprehensive analysis of the spectrum
[Johansson, 1978]. Much of his subsequent work focussed on the
discovery of new energy levels, identification of Fe II lines in
astrophysical spectra, and their use for diagnostics, including the
discovery of stimulated emission from Fe II in $\eta$ Carinae
[S. Johansson & V. S. Letokhov, 2004]. This work continued for the rest of his
life, with his final paper on new energy levels in Fe II observable
solely in stellar spectra [Johansson, 2009], completed just
days before his death in 2008.
In 1987, I began working on the spectrum of Fe I at Imperial College,
London, UK (IC), using Fourier transform (FT) spectra of iron-neon and
iron-argon hollow cathode lamps recorded at Kitt Peak National Observatory
(KPNO) and IC. This work resulted in two papers on precise wavelengths in
Fe I and Fe II [Nave et al., 1991, Nave et al., 1992]. In 1992, I went to Lund University
and began work with Sveneric to combine the FT spectra from IC with grating
spectra recorded by him in 1988 on the 10.7-m normal incidence spectrograph
at the National Institute of Standards and Technology (NIST). The
combined data were used to produce a linelist of about 34 000 spectral
lines covering wavelengths from 833 Å to 5 $\mu$m. Roughly 9500 of these
lines are due to Fe I, and these were used to produce optimized energy
levels and a new Fe I multiplet table [Nave et al., 1994]. About 15 000 of the
remaining lines belong to Fe II. After completion of the work on Fe I,
I continued to collaborate with Sveneric on the spectrum of Fe II, with the
aim of preparing a comprehensive analysis and linelist for Fe II. However,
progress was slow due to competing work and his subsequent illness.
Consequently, this work remained uncompleted when Sveneric died in
October, 2008.
This paper is my best attempt to complete the analysis of Fe II in the
way that Sveneric was unable to do. It has benefitted from
unpublished line lists and energy levels that he gave me
while I was working in Lund and from the insights into
the spectra of iron-group elements that I gained while there.
However, the loss of Sveneric's knowledge and lack
of access to some of the original data behind the unpublished
linelists have resulted in some inconsistencies in the
analysis. In particular, it has proven to be impossible to put the
intensities on a consistent scale over the whole wavelength
region. Since I do not have the original data behind the unpublished
linelists, I was also unable to verify all of the lines in these
lists in a way that I would have done otherwise. However, these
lists appear to be reliable, and I have no reason to doubt that Sveneric prepared them
well. Although Sveneric's considerable contributions of lines and
energy levels make him a co-author of this paper, the final selection
of lines and energy levels in the tables is solely mine.
Gillian Nave
§ INTRODUCTION
The iron-group elements have complex spectra with thousands of lines
from the VUV to the infrared [Johansson, 1978]. They are important
in the absorption spectra of many astrophysical objects including the ISM,
hot and cool stars, gaseous nebulae, galaxies and
quasi-stellar object absorption line systems. Lines from
Fe II in particular account for as many as half of the absorption
features in A and late B-type stars in all wavelength regions
[Brandt et al., 1999]. They also form prominent emission lines from a wide
variety of objects including chromospheres of cool stars
[Dupree et al., 2005], active galactic nuclei [Hamann & Ferland, 1999], and nebular
regions around massive stars.
Problems in modeling stellar spectra in the UV and VUV often can be traced to
the lack of adequate data for iron group elements, particularly for
very highly-excited levels of these elements. The importance of these
highly-excited levels has been demonstrated in observations of the
Bp star HR6000 made with the HIRES spectrograph on the Very Large
Telescope. Castelli et al., 2009 found it possible to identify strong
lines in the region around 5178 Å as transitions of the
3d$^6$($^3$H) 4d-4f array due to unpublished
energy levels of Fe II. The upper levels of this transition array are
located around 128,000 cm$^{-1}$ (15 eV) and are the highest known levels of Fe II.
Previous papers on Fe II containing substantial numbers of lines or
energy levels are summarized in Table <ref>. The last
comprehensive publication on Fe II was in 1978
[Johansson, 1978]. It contained classification of 3272
lines from 576 energy levels, covering the wavelength range 900 Åto 11 200 Å. The wavelength uncertainty of the best lines was
0.02 Å. Since then, precise wavelengths of Fe II in the UV
[Nave et al., 1991, Nave & Sansonetti, 2004] and VUV [Nave et al., 1997] have been
published. Additional Fe II wavelengths have been reported by
Adam et al., 1987, Rosberg & Johansson, 1992, Johansson et al., 1995,
Biémont et al., 1997, Aldenius et al., 2006, Aldenius, 2009,
Castelli et al., 2009, and Castelli & Kurucz, 2010. These publications
contain approximately 500 new spectral lines. Studies by
Adam et al., 1987, Johansson & Baschek, 1988, Rosberg & Johansson, 1992,
Biémont et al., 1997 and Castelli & Kurucz, 2010 account for an additional
301 new energy levels.
Previous work on Fe II
Reference No. energy levels No. lines Principal configurations
Johansson, 1978 576 3272 3d$^6$ns, 3d$^6$np, 3d$^6$nd, 3d$^6$4f
3d$^5$ 4sns,(3d$^5$4s)np, (3d$^5$4s)4d, 3d$^5$4p$^2$
Adam et al., 1987 20 120 3d$^5$ 4s($^7$S)4d, 3d$^5$($^6$S)4p$^2$
Johansson & Baschek, 1988 73 - 3d$^6$($^3$L)4d, where L=P, F, G, H
Nave et al., 1991 - 221
Rosberg & Johansson, 1992 49 220 3d$^6$($^5$D)5g
Nave et al., 1997 - 473
Biémont et al., 1997 50 74 3d$^6$($^5$D)6h
Castelli & Kurucz, 2010 109 (58 used) 137$^a$ 3d$^6$($^3$L)4f,3d$^6$($^3$L)nd where L=P, F, G, H
This work 102 12 909
$^a$ Lines listed as laboratory measurements that are not included in
Johansson, 1978. 51 of the energy levels in Castelli & Kurucz, 2010
could not be confirmed using our spectra.
§ EXPERIMENTAL WORK
The laboratory spectra used here are the same as those
in our previous studies of Fe I and Fe II
[Nave et al., 1991, Nave et al., 1992, Nave et al., 1994, Nave et al., 1997]. The spectra were obtained on four
different instruments: the f/55 IR-visible-UV FT spectrometer at the Kitt Peak
National Observatory (KPNO), Tucson, Arizona for the region 2000 to
35 000 (5 $\mu$m to 2900 Å ); the f/25 vacuum UV FT spectrometer at
Imperial College, London (IC) for the region
33 000 to 67 000 (3000 Å to 1500 Å); the f/25 UV FT
spectrometer at Lund University, Sweden for the region 31 900 to
55 000 (3135 Å to 1820 Å); and the 10.7m Normal
Incidence Grating Spectrograph at NIST for high-dispersion grating
spectra above 30 770 ($<$3250 Å).
§.§ Fourier transform spectra
The light source used for the FT investigations was a hollow cathode
lamp, run in either neon or argon. This source emits lines of Fe I,
Fe II, the neutral and singly-ionized spectra of the carrier gas used,
and a small number of impurities. The cathode was a 35 mm long cylinder of
pure iron with a 8 mm bore. A water cooled cathode was used
for some of the spectrograms. The metal case of the lamp formed the
anode. The gas pressures were about 500 Pa of Ne or 400 Pa of Ar for
the visible and IR observations made at KPNO, and 300 Pa to 800 Pa of Ne for the UV
observations made at IC. The currents ranged from 320 mA to 1.1 A. Argon-iron
spectra were recorded in the region 17 500 to 35 000 to provide an
absolute wavelength calibration based on Ar II lines. One infrared
spectrum was recorded with a water-cooled cathode as source and a
higher current of 1.4 A. The spectra are summarized in Table
Spectra used in this analysis$^a$
Name$^b$ lower
Resolution Gas Pressure Current Notes
wavenumber wavenumber
(cm$^{-1}$) Pa (Torr) A
5lKPNO FT spectrometer. CaF$_2$ beamsplitter
820216R0.001 2006 8996 0.012 Ne 373 (2.8) 1.4 Water cooled cathode;
N6 in Nave et al., 1992
801211R0.002 5007 5544 0.0095 Ne 533 (4.0) 0.85 N1 in Nave et al., 1992
5lKPNO FT spectrometer. Visible-UV beamsplitter
800321R0.001 5000 9436 0.012 Ne 533 (4.0) 0.86 N2 in Nave et al., 1992
810731R0.007 5007 11 743 0.015 Ne 546 (4.1) 1.04
810731R0.009 10 343 13 350 0.020 Ne 560 (4.2) 1.05 N3 in Nave et al., 1992
810731R0.005 10 505 13 820 0.018 Ne 546 (4.1) 1.04
810622R0.021 18 024 35 984 0.043 Ne 533 (4.0) 0.75 k11 in Nave et al., 1991;
N2 in Learner & Thorne, 1988
810622R0.009 17 198 34 040 0.043 Ar 400 (3.0) 0.4 k19 in Nave et al., 1991;
A1 in Learner & Thorne, 1988
810724R0.003 16 013 17 950 0.025 Ar 413 (3.1) 1.0
810724R0.002 20 801 26 000 0.036 Ar 400 (3.0) 1.0
800325R0.007 14 220 17 060 0.025 Ar 400 (3.0) 0.81
810622R0.012 17 006 25 000 0.033 Ne 533 (4.0) 0.75 N1 in Learner & Thorne, 1988
5lIC FT spectrometer.
fe6 33 000 44 500 0.07 Ne 533 (4.0) 0.75
i74 35 000 46 000 0.105 Ne 400 (3.0) 0.35
i20 41 500 48 000 0.17 Ne 333 (2.5) 0.95
i22 41 500 48 000 0.17 Ne 333 (2.5) 0.95
i24 38 000 44 000 0.17 Ne 333 (2.5) 0.95
fen180 44 000 59 000 0.08 Ne 800 (6.0) 0.5
fen200 44 000 59 000 0.08 Ne 800 (6.0) 0.5
fe7 56 000 75 000 0.07 Ne $^c$ 0.5
fe8 50 300 67 000 0.08 Ne $^c$ 0.5
5lLund FT spectrometer.
fe04193 31 600 47 000 0.07 Ar/Ne $^c$ 1.0
fe04201 31 600 47 000 0.07 Ne 67 (0.5) 1.0
5lNIST normal incidence grating spectrograph.
lower Upper Gas Pressure Current Notes
Pa (Torr) A
(Å) (Å)
plate 8,6$^\circ$ 836 1523 Ne 130 (1.0) 0.66 Pulsed, 100 A peak,
pulse width 70 $\mu$s, 100 Hz
plate 8,6$^\circ$ 836 1523 Ar 40 (0.3) 0.61 Pulsed, 100 A peak,
pulse width 70 $\mu$s, 100 Hz
plate 4,8$^\circ$ 1400 2107 Ne 130 (1.0) 0.64 Pulsed, 100 A peak,
pulse width 60 $\mu$s, 100 Hz
plate 4,8$^\circ$ 1400 2107 Ar 40 (0.3) 0.64 Pulsed, 100 A peak,
pulse width 60 $\mu$s, 100 Hz
$^a$ Additional grating spectra were taken between 2107 Å and 3249 Åunder similar conditions to the above spectra. Linelists for these
spectra were given to G. Nave by S. Johansson in the early 1990's but
full details are not known. Roughly 2000 Fe II lines have been taken
from these spectra and are included in Table <ref>.
$^b$ Name of spectrum in the National Solar Observatory digital Library [Hill & Suarez, 2012].
$^c$ The pressure was not recorded for these archival spectra.
The wavenumber, integrated intensity, and width of all lines
in the FT spectra were obtained using the Decomp program
of Brault & Abrams, 1989 and its modification Xgremlin
[Nave et al., 1997], which fit Voigt profiles to the spectral lines. Each line
was measured in up to 9 different spectra. The
full width at half maximum (FWHM) of the lines varied from about
0.024 at 5000 (0.1 Å at 2 ) to about 0.24 at
50 000 (10 mÅ at 2000 Å).
The spectra were calibrated with lines of Ar II between
19 429 cm$^{-1}$ and 22 826 cm$^{-1}$ taken from
Whaling et al., 1995. The calibration was carried into the UV and
infra-red using wide-range spectra (Nave 1991, 1992). Details of
the calibration of the visible and UV spectra are given in
Nave & Sansonetti, 2011, and of the IR spectra in Nave et al., 1992. The
calibration of the UV spectra based on Ar II agrees to better than
1:10$^8$ with a calibration based on Mg I and Mg II lines measured with a frequency comb
[Salumbides et al., 2006, Hannemann et al., 2006, Batteiger et al., 2009].
The uncertainty of the wavenumber of a line measured in a single spectrum
is a sum in quadrature of the statistical uncertainty in the measurement of its position and of the calibration uncertainty for the whole spectrum. The
statistical uncertainty was estimated from the FWHM of the line
divided by twice the signal-to-noise ratio (SNR). This was derived
from equation 9.3 of Davis et al., 2001, assuming 4
statistically independent points in a line width. The SNR was
estimated using a global noise level for each spectrum, but was
limited to 100 for strong lines for the purposes of calculating the
uncertainty. This accounts for an increased noise level around the
strongest lines in the KPNO spectra due to low frequency ghosts and
also ensures that strong lines
measured in more than one spectrum receive similar weighting in the
calculation of uncertainty of the weighted mean wavenumber. The
statistical uncertainty of strong lines (SNR$>$100) varies from
0.0001 (0.8 mÅ at 2 ) in the infra-red to 0.001 (0.01 mÅ at 3000 Å) in the ultraviolet. The weakest lines in the
spectra have a SNR of about 3, and their uncertainty varies from about
0.005 (0.02 Å at 2 ) in the infrared to 0.05 (5 mÅat 3000 Å) in the ultraviolet. For lines measured in several
spectra, a weighted average wavenumber and uncertainty were calculated
using the squared reciprocal of the statistical uncertainty as a
weight. Since the statistical uncertainties of the individual lines
are uncorrelated, their squared reciprocal was also used as a weighting
factor in the optimization of the energy levels.
The calibration uncertainty of a spectrum consists of two parts: the
uncertainty of the original standards and the uncertainty of the calibration
constant derived from those standards. The uncertainty of the original
Ar II standards is 2x10$^{-4}$
cm$^{-1}$. These Ar II standards are used to calibrate a `master spectrum',
covering the wavenumber region 17 200 cm$^{-1}$ to 34 040 cm$^{-1}$
(810622R0.009 in Table <ref>), with a one standard
uncertainty of 1.8 parts in 10$^8$. This calibration is propagated
to the UV and IR regions using overlapping spectra, each of which
increases the calibration uncertainty. The shortest UV spectrum (fe7 in Table
<ref>) thus has the largest calibration uncertainty of 4 parts
in 10$^8$. The calibration uncertainty for a particular line can thus vary, depending on
how the spectra in which it was measured were calibrated. For simplicity, we
adopted a global calibration uncertainty of 4 parts in 10$^8$ for
all spectral lines. Since this calibration uncertainty is common to
all lines, it is not included in the optimization of the energy
levels, but is added in quadrature to the uncertainties of the
optimized energy level values after the optimization.
§.§ Grating spectra
Grating spectra were recorded in the region 30 770 to 119 617 (3250 Å to 836 Å) using iron-neon and iron-argon hollow cathode lamps.
The iron-neon hollow cathode lamp was run in pulsed mode with a peak current
of 100 A, pulse width of 70 $\mu$s, pulse frequency of 100 Hz,
and a gas pressure of 130 Pa. The iron-argon hollow cathode lamp was run in
pulsed mode under similar conditions with a gas pressure of about
40 Pa. Similar spectra were recorded with a continuous hollow cathode
lamp for comparison, but the results are not presented here. Details
of the grating spectra are given in Table <ref>.
Spectra of the pulsed iron-neon hollow cathode between 850 Å and
2107 Å were read from the photographic plates using an
automatic comparator at Lund University that produced a
signal proportional to the optical density of the plate integrated over the
full height of the slit image. This produced a file of optical density
of the recorded spectrum as a function of position along the direction
of dispersion. This file was read in to Xgremlin and
Gaussian profiles were fitted to the spectral lines to obtain the
wavelength and peak intensity. Strong spectral lines saturate the
image on the photographic plates and give profiles with a flat top
that cannot be fitted with a Gaussian profile. The centroid of these
lines was estimated by integrating the profile between two points
taken on either side of the line profile. These wavelengths are,
however, less reliable than those obtained by fitting the line profile
with a Gaussian.
Additional lines in the spectra listed in Table <ref> are
present in linelists given to G. Nave by S. Johansson in 1992. These are all either
weak lines that are visible on the photographic plates but not easily
fit by the Xgremlin software, or lines blended with much
stronger lines. Roughly 360 lines of this type between 850 Å and 1600 Å have
been included in Table <ref>. Linelists for additional
spectra between 1600 Å and 3250 Å were given to G. Nave by S. Johansson in the
mid 1990's. These were obtained from spectra recorded under similar
conditions to the spectra in Table <ref>, but full details
of those spectra are not known. Roughly 2200 lines from these spectra are included in
Table <ref>.
All grating spectra were calibrated from Ritz wavelengths of Fe II
lines derived from energy levels determined from the FT spectra, with
particular care being taken to avoid lines that were weak, were saturated
on the photographic plate, or were significantly asymmetric. The wavelength
uncertainty of these calibration lines is approximately 2 mÅ. Details of the
calibration procedure are given in Nave et al., 1997.
§ LINE IDENTIFICATIONS
The initial identification of the lines was performed solely in the FT
spectra. These line identifications were carefully examined to eliminate spurious
coincidences of lines and energy level differences and were then used to obtain
optimized values for many of the energy levels. The remaining
levels were found using unidentified lines in the FT and
grating spectra and the identifications were again examined to
eliminate spurious coincidences. Finally, both the FT and grating
spectra were used to obtain optimized energy level values and Ritz
wavelengths and wavenumbers. The full procedure is as follows.
About 28 000 lines were measured in the FT spectra. Known lines that
belong to species other than Fe II were identified by comparison to
previously published wavelengths in Fe I [Nave et al., 1994], Ar I-II
[Whaling et al., 1995, Whaling et al., 2002], Ne I-III
[Sansonetti et al., 2004, Saloman & Sansonetti, 2004, Kramida & Nave, 2006],
and various impurities present in the spectra [Ralchenko et al., 2012]. An initial
identification of Fe II lines was made by comparison with Ritz
wavelengths calculated from Fe II energy levels taken from the
references in Table <ref>. Many of these levels
had large uncertainties as
they were obtained from grating spectra of lower resolution and
accuracy than our FT spectra. Setting a large tolerance window for
the agreement between the Ritz and experimental wavelengths leads to
many spurious identifications of lines from these levels. A better
approach is to use a few lines known to combine with these levels to
obtain a better value for the energy level. The energy levels then have
low uncertainties and the tolerance window
can be set lower, leading to fewer spurious identifications.
Roughly 10 000 lines in the FT spectra matched energy level
differences in Fe II. About 1/3 of these lines had more than one
possible identification. The identifications of all lines were checked
to ensure that very few spurious identifications contributed to the
energy level optimization, as even a small number of mis-identified
lines can have a significant effect on the optimized energy
levels. This task was aided by the small uncertainty of the
wavenumbers obtained from a FT spectrometer, and by predicted
intensities from the atomic structure calculations of
Kurucz, 2010. Although the accuracy of these calculations is
limited by strong configuration interaction in Fe II, they were useful
for locating lines in FT spectra from levels that had been found using
less accurate data from grating spectrographs. After eliminating
spurious identifications, the total number of Fe II lines in the FT
spectra was 8930, of which 798 had more than one plausible
The lines measured in FT spectra were used to derive optimized energy levels and
Ritz wavenumbers using the computer program lopt
[Kramida, 2011]. Values for 942 energy levels of Fe II were derived
from 8930 lines covering wavenumbers from 2008 cm$^{-1}$ to
67 851 cm$^{-1}$. The line uncertainties assigned for use in the level
optimization omit the calibration uncertainty (see section
<ref>). The statistical
uncertainty, estimated as described in section <ref>, was
added in quadrature to a minimum estimated uncertainty of
0.001 cm$^{-1}$. This value was chosen to ensure that the level
optimization was not dominated by the infrared region, where narrow
linewidths give very precise wavenumbers, but the calibration uncertainty of the
spectra is higher as many overlapping spectra are required to reach the Ar II
standards. Weights were then assigned proportional to the
squared reciprocal of the estimated uncertainty of the wavenumber of
the line. Lines with more than one possible classification, lines that
were blended, and lines with a large difference between the observed
and Ritz wavenumber were assigned a low weight so that they did
not significantly affect the values of the optimized energy levels.
The optimization was performed in three different steps. The first step
was designed to obtain accurate values and uncertainties for the ground
term, a$^6$D. The values for the a$^6$D intervals can be
determined from differences between lines close to one another in the
same spectrum sharing the same calibration, hence the calibration
uncertainty does not contribute to the uncertainty in the relative
values of these energy levels. An optimization was thus performed with
a set of lines connecting the lowest a$^6$D term to higher
$\rm 3d^6\,(^5D)4p$ levels. These lines were assigned a
weight proportional to the squared reciprocal of the statistical
uncertainty of the wavenumber. The minimum estimated
uncertainty of 0.001 cm$^{-1}$ was omitted as all the lines are in the
same spectral region. In the second step, the a$^6$D
levels were fixed to the values and uncertainties determined from the
first step. The weights of all the lines in the FT spectra were assigned by
combining in quadrature the statistical uncertainty and the minimum
estimated uncertainty of 0.001 cm$^{-1}$ in order to obtain accurate
uncertainties for the $\rm 3d^6\,(^5D)4p$ and higher levels.
After the second step of the level optimization, a further 86
energy levels that had not been optimized with the FT spectra were
added. Most of these were from Biémont et al., 1997 and
Castelli & Kurucz, 2010. Roughly half of the levels in Castelli & Kurucz, 2010
could not be matched definitively to lines in the FT spectra. Levels
were adopted from Castelli & Kurucz, 2010 if they had strong transitions
that matched at least two lines in the FT spectra. All the energy
levels were then used to identify additional lines in both the FT and
grating spectra. The final optimization step derived values for 1027 energy
levels from 13 653 transitions in both FT and grating spectra, again fixing
the values of the a$^6$D levels to the values obtained in the first step.
An uncertainty of 0.005 Å was assigned to all grating lines. The energy
level uncertainties from this iteration were added in quadrature to a global
calibration uncertainty of 4x10$^{-8}$ times the value of the energy level.
§ ENERGY LEVELS AND LINES
The full table of 1027 energy levels is available online and a small
section is given in Table <ref>.
The configurations and terms in columns 1 and 2 are taken from the
NIST Atomic Spectra Database [Ralchenko et al., 2012], the papers in Table
<ref>, or the calculations of
Kurucz, 2010. Many of the levels between 114$\,$212 cm$^{-1}$ and
114$\,$673 cm$^{-1}$ have not been assigned to configurations. The levels
in this region are from the two overlapping configurations
3d$^6$($^5$D)6d and 3d$^6$($^3$D)4d. The energy level values are given
in column 4. The uncertainties with respect to the ground term are
given in column 5 and represent one standard uncertainty. They were obtained
from the lopt program by combining the uncertainties derived
from the level optimization in quadrature with a global calibration
uncertainty of 4$\times$10$^{-8}$ times the level value. The last
column lists the number of observed lines that combine with the level.
Energy levels of Fe II (full table available online)
$\mathrm{3d^6(^5D)4s }$ $\mathrm{a^6D}$ 9/2 0.0000 0.0000 42
7/2 384.7872 0.0003 60
5/2 667.6829 0.0003 64
3/2 862.6118 0.0004 52
1/2 977.0498 0.0004 30
$\mathrm{3d^7 }$ $\mathrm{ a^4F }$ 9/2 1872.5998 0.0006 63
7/2 2430.1369 0.0006 77
5/2 2837.9807 0.0007 76
3/2 3117.4877 0.0008 54
$\mathrm{3d^6(^5D)4s }$ $\mathrm{a^4D}$ 7/2 7955.3186 0.0007 65
5/2 8391.9554 0.0007 72
3/2 8680.4706 0.0007 58
1/2 8846.7837 0.0008 32
$\mathrm{3d^7 }$ $\mathrm{ a^4P }$ 5/2 13474.4474 0.0009 67
3/2 13673.2045 0.0010 63
1/2 13904.8604 0.0012 38
$\mathrm{ a^2G }$ 9/2 15844.6485 0.0012 55
7/2 16369.4098 0.0013 59
$\mathrm{ a^2P }$ 3/2 18360.6399 0.0016 42
1/2 18886.773 0.002 29
$\mathrm{ a^2H }$ 11/2 20340.2461 0.0013 43
9/2 20805.7632 0.0014 57
$\mathrm{ a^2D2 }$ 5/2 20516.9534 0.0016 62
3/2 21307.999 0.002 47
$\mathrm{3d^6(^3P2)4s }$ $\mathrm{b^4P }$ 5/2 20830.5534 0.0011 75
3/2 21812.0454 0.0012 61
1/2 22409.8178 0.0013 47
$^a$ One standard uncertainty
A small section of the table of observed Fe II lines is given in Table
<ref>. The full table of 13 653 transitions is available
online. The intensities in column 1 depend on the source conditions
and method of measurement. The wavenumbers and intensities of all of
the lines above 3250 Å were taken from a weighted average of up to
nine individual measurements in the FT spectra. Since these
intensities were measured using several different source conditions,
they are useful only as a guide to the approximate strength of the
line and should not be relied upon for accurate intensity
measurements. Wavenumbers and intensities of lines below 3250 Åthat are marked with an `F' in column 11 were also taken from FT
spectra. The wavelengths and intensities of lines marked `G' in column
11 are from the grating spectra of the pulsed iron-neon hollow cathode
lamp listed in Table <ref>. The intensity scale of the
grating lines is different from the intensity scale of the FT spectra,
and are given as decimal numbers in Table <ref>. Strong
lines that saturate the photographic plate are indicated with a 'S' in
column 11. Wavelengths and intensities of lines taken from the
additional linelists are indicated with `N' in column 11 if taken from
the pulsed iron-neon hollow cathode, `A' if taken from the pulsed
iron-argon hollow cathode lamp. The intensities are visual estimates
of the photographic density and are also given as decimal numbers,
ranging from 0 to 5 for most lines. The intensity scale is different
from both the lines in the FT spectra and the grating spectra lines
marked with a `G'. Lines marked `b' are broad, those marked `0d' are
faint and diffuse and those marked `0d?' are hardly detectable from
the background. Lines below 1600 Å taken from the linelists given
to G. Nave by S. Johansson have uncertain intensities as they are all
weak or blended. They have been assigned an intensity of “00” and
indicated with `L' in column 11.
The wavelength in column 2 was derived from the wavenumber in column 4 for
lines taken from the FT spectra. Air wavelengths, given for lines
between 2000 Å and 2 $\mu$m, were derived from the 5 parameter formula in
equation 3 of Peck & Reeder, 1972. The wavelengths were measured directly in the
grating spectra and the wavenumbers derived from them. The corresponding one
standard uncertainties in columns 3 and 5 include contributions from
the statistical uncertainty in the measurement of the position of the
line and the calibration uncertainty for the spectrum. The one standard
uncertainty of lines taken from grating spectra has been estimated at
0.005 Å. Ritz wavelengths and their statistical uncertainties were obtained
from lopt and are given in columns 6 and 7 respectively. The
uncertainties were derived by adding the uncertainties from lopt to a
global calibration uncertainty of 4$\times$10$^{-8}$. The classification of
the line is given in column 10, with the values of the lower and upper energy
levels in columns 8 and 9 respectively. In addition to the codes indicating the
source of the data, column 11 also indicates if there are any other
identifications for the line.
§ IONIZATION ENERGY
Johansson, 1978 obtained an estimate of 130 563$\pm$10 cm$^{-1}$
for the ionization energy of Fe II by using a two parameter fit to the highest
J-value levels of the lowest three $\rm 3d^6(^5D)ns$ configurations
and comparing the value to similar terms in other singly-ionized
iron-group elements. A better estimate can be obtained by
using highly-excited levels that are well-described by the J$_c$K
coupling scheme.
We have used the quadrupole-polarization model of
Schoenfeld et al., 1995 to obtain the ionization energy from the $\rm
3d^6(^5D)4f$, $\rm 3d^6(^5D)5g$, and $\rm 3d^6(^5D)6h$ levels. The outer
electron in these levels is weakly bound to the core and the
levels form five groups, separated by the fine structure intervals of the $\rm
3d^6\,^5D$ term of Fe III. Within each group the levels form pairs and the energy of the
center of gravity of the pairs, $E(nlJ_cK)$, can be described by:
\begin{equation}
E(nlJ_cK) = IE +E(J_c)- R_{Fe}Z_c^2(1/n^2+\alpha <r^{-4}>_{nl}-\frac{AB}{D} Q<r^{-3}>_{nl})
\end{equation}
where $IE$ is the ionization energy, $E(J_c)$ is the energy of the $\rm
3d^6\,^5D_{J_c}$ level in Fe III, $R_{Fe}$ is the Rydberg constant for
Fe II (109 736.248 cm$^{-1}$), $Z_c$ is the effective charge of the core
(2 for Fe II), $<r^{-3}>_{nl}$ and $<r^{-4}>_{nl}$ are hydrogenic
radial expectation values in atomic units given by equations 3 and 4 of
Schoenfeld et al., 1995, and $A$, $B$, and $D$ are given by equation 5 of
Schoenfeld et al., 1995. The dipole polarizability $\alpha$ of the core
and its quadrupole moment Q in atomic units are obtained from the
experimental energy levels. By plotting
$E(nlJ_cK)+ R_{Fe}Z_c^2/n^2-E(J_c)$ for each configuration against
$\mathrm R_{Fe}Z_c^2\frac{AB}{D} <r^{-3}>_{nl}$, a straight line of slope Q and
intercept $\mathrm IE-\alpha R_{Fe}Z_c^2<r^{-4}>_{nl}$ is obtained.
This is shown in Fig. <ref>. The
ionization energy and $\alpha$ can then be obtained by simultaneously solving
the equations for the intercepts for pairs of configurations.
Calculation of the ionization energy for Fe II from the
$\rm 3d^6(^5D)4f$, $\rm 3d^6(^5D)5g$, and $\rm 3d^6(^5D)6h$
levels. The intercept for each configuration is $\mathrm IE-\alpha
The values of Q obtained from the slopes in Figure <ref> are
0.99$\pm$0.08 ea$_o^2$ (where e is the elementary charge and a$_o$
the Bohr radius) for the $\rm 3d^6(^5D)4f$ levels, 1.05$\pm$0.02 ea$_o^2$ for the
$\rm 3d^6(^5D)5g$ levels, and 1.098$\pm$0.008 ea$_o^2$ for the $\rm 3d^6(^5D)6h$
levels. By solving the two equations for the intercepts for the 4f and
5g levels, an ionization energy of 130 660$\pm 5$ cm$^{-1}$ is obtained with
$\alpha$ = 17.0$\pm$0.3. For the 5g and 6h levels, the ionization energy
obtained is 130 655.4$\pm$0.4 cm$^{-1}$ with $\alpha$ = 15.0$\pm$0.2 a$_o^3$.
The 4f levels are
not well described by the quadrupole-polarization model as the centers
of gravity of the pairs show a large scatter around a straight line as
shown in Fig. <ref>. We thus recommend the value
130 655.4$\pm$0.4 cm$^{-1}$ obtained from the 5g and 6h levels.
§ SUMMARY
Table <ref> contains 12 909 spectral lines from 13 653
transitions in Fe II, over a factor of three more than the number
of previously published lines in Fe II. About 900 of these lines have
alternate plausible identifications due to Fe I, Fe III, Ne,
Ar, or other Fe II transitions. These lines come from a total list of about 37 000
lines. Roughly 11 500 of the lines in this total list remain
unidentified, and about 460 of these lines are present in the FT spectra
with a SNR $>$ 50. About half of these strong unidentified lines are
in the infrared region and are probably due to
highly-excited levels in Fe I or Fe II that have yet to be identified.
The uncertainties of the strongest lines in Table <ref>
vary from 0.0001 in the infrared to 0.001 in the
ultraviolet, and are more than an order of magnitude lower than the
best lines in [Johansson, 1978].
Observed lines of Fe II (full table available online)
1c Vacuum
1cRitz Vacuum
1cLower level
1cUpper level
1cwavelength 1cwavelength
1c(Å )
1c(Å )
1c($ \mathrm{cm^{-1}}$)
1c($ \mathrm{cm^{-1}}$)
1c($ \mathrm{\AA\ }$)
1c($ \mathrm{\AA\ }$)
1c($ \mathrm{cm^{-1}}$)
1c($ \mathrm{cm^{-1}}$)
0.9 1824.980 0.005 54 795.13 0.06 1824.979 49 0.000 11 20 805.7632 75 600.900 $\mathrm{3d^7 }$ $\mathrm{ a^2H }$$_{9/2 }$ $\mathrm{3d^6(^3D)4p }$ $\mathrm{ w^2F^{\circ}}$$_{7/2 }$ G
0 1824.860 0.005 54 798.73 0.15 1824.860 52 0.000 08 67 516.328 122 315.037 $\mathrm{3d^6(^3G)4p }$ $\mathrm{ y^2H^{\circ} }$$_{11/2 }$ $\mathrm{3d^6(^3H)6s }$ $\mathrm{^2H }$ $_{11/2 }$ N
0 1823.929 0.005 54 826.70 0.15 1823.929 60 0.000 12 18 360.6399 73 187.318 $\mathrm{3d^7 }$ $\mathrm{ a^2P }$$_{3/2 }$ $\mathrm{3d^6(^3D)4p }$ $\mathrm{ y^2P^{\circ}}$$_{1/2 }$ N
4 1823.8711 0.0008 54 828.44 0.02 1823.869 85 0.000 10 18 360.6399 73 189.114 $\mathrm{3d^7 }$ $\mathrm{ a^2P }$$_{3/2 }$ $\mathrm{3d^6(^3D)4p }$ $\mathrm{ y^2P^{\circ}}$$_{3/2 }$ F
1.0 1822.196 0.005 54 878.83 0.06 1822.189 74 0.000 08 8 680.4706 63 559.498 $\mathrm{3d^6(^5D)4s }$ $\mathrm{a^4D}$$_{3/2 }$ $\mathrm{3d^6(^3F2)4p }$ $\mathrm{x^4D^{\circ} }$$_{1/2 }$ G II
1.0 1822.196 0.005 54 878.83 0.06 1822.190 10 0.000 19 44 929.532 99 808.549 $\mathrm{3d^6(^1F)4s }$ $\mathrm{ c^2F}$$_{5/2 }$ $\mathrm{3d^6(^1G1)4p }$ $\mathrm{^2G^{\circ} }$$_{7/2 }$ G II
1.3 1822.135 0.005 54 880.67 0.15 1822.123 38 0.000 08 8 391.9554 63 272.981 $\mathrm{3d^6(^5D)4s }$ $\mathrm{a^4D}$$_{5/2 }$ $\mathrm{3d^6(^3F2)4p }$ $\mathrm{x^4D^{\circ} }$$_{5/2 }$ S
4 1820.9161 0.0008 54 917.41 0.03 1820.916 18 0.000 08 54 232.201 109 149.611 $\mathrm{3d^5\,4s^2 }$ $\mathrm{b^4G }$$_{11/2 }$ $\mathrm{3d^5(^2I)\,4s4p(^3P) }$ $\mathrm{^2I^{\circ}}$$_{13/2 }$ F
0.9 1820.480 0.005 54 930.58 0.06 1820.478 44 0.000 17 31 999.049 86 929.664 $\mathrm{3d^7 }$ $\mathrm{ b^2F }$$_{7/2 }$ $\mathrm{3d^6(^3P1)4p }$ $\mathrm{v^4D^{\circ} }$$_{7/2 }$ G
1.0 1819.643 0.005 54 955.82 0.06 1819.644 13 0.000 14 31 811.814 86 767.614 $\mathrm{3d^7 }$ $\mathrm{ b^2F }$$_{5/2 }$ $\mathrm{3d^6(^3P1)4p }$ $\mathrm{v^4D^{\circ} }$$_{5/2 }$ G
9 1818.5202 0.0003 54 989.767 0.010 1818.521 51 0.000 08 7 955.3186 62 945.045 $\mathrm{3d^6(^5D)4s }$ $\mathrm{a^4D}$$_{7/2 }$ $\mathrm{3d^6(^3F2)4p }$ $\mathrm{x^4D^{\circ} }$$_{7/2 }$ F *
0 1816.086 0.005 55 063.48 0.15 1816.085 38 0.000 08 46 967.4751 102 030.965 $\mathrm{3d^6(^5D)4p }$ $\mathrm{z^4P^{\circ} }$$_{5/2 }$ $\mathrm{3d^6(^5D)6s }$ $\mathrm{^6D }$$_{7/2 }$ N
1.3 1815.766 0.005 55 073.17 0.15 1815.765 92 0.000 08 8 391.9554 63 465.134 $\mathrm{3d^6(^5D)4s }$ $\mathrm{a^4D}$$_{5/2 }$ $\mathrm{3d^6(^3F2)4p }$ $\mathrm{x^4D^{\circ} }$$_{3/2 }$ S
4 1815.4116 0.0005 55 083.927 0.016 1815.410 97 0.000 11 20 516.9534 75 600.900 $\mathrm{3d^7 }$ $\mathrm{ a^2D2 }$$_{5/2 }$ $\mathrm{3d^6(^3D)4p }$ $\mathrm{ w^2F^{\circ}}$$_{7/2 }$ F
0d? 1810.261 0.005 55 240.66 0.15 1810.262 75 0.000 11 54 904.241 110 144.841 $\mathrm{3d^6(^3F1)4s }$ $\mathrm{d^2F }$$_{7/2 }$ $\mathrm{3d^6(^3F)5p }$ $\mathrm{ ^4F^{\circ}}$$_{7/2 }$ N
0 1810.117 0.005 55 245.06 0.15 1810.110 58 0.000 11 61 093.406 116 338.649 $\mathrm{3d^6(^3P2)4p }$ $\mathrm{z^2D^{\circ} }$$_{5/2 }$ $\mathrm{3d^5\,4s(^7S)4d }$ $\mathrm{ ^6D }$$_{5/2 }$ N Ne
0d? 1809.945 0.005 55 250.31 0.15 1809.944 8 0.001 0 50 157.475 105 407.78 $\mathrm{3d^6(^3F1)4s }$ $\mathrm{c^4F }$$_{9/2 }$ $\mathrm{3d^5(^2D)\,4s4p(^3P) }$ $\mathrm{^4F^{\circ}}$$_{9/2 }$ N
6 1809.3168 0.0008 55 269.48 0.02 1809.318 25 0.000 11 21 307.999 76 577.436 $\mathrm{3d^7 }$ $\mathrm{ a^2D2 }$$_{3/2 }$ $\mathrm{3d^6(^1S2)4p }$ $\mathrm{x^2P^{\circ} }$$_{1/2 }$ F
2. 1809.291 0.005 55 270.28 0.15 1809.288 12 0.000 08 42 237.0575 97 507.414 $\mathrm{3d^6(^5D)4p }$ $\mathrm{z^6F^{\circ}}$$_{7/2 }$ $\mathrm{3d^6(^3P2)5s }$ $\mathrm{ ^4P }$$_{5/2 }$ N
14l$^a$ Intensities of the lines with different codes in `Notes' column
are on different scale and should not be relied on for accurate intensity measurements.
14lDecimal numbers are from grating spectra. Integer numbers are from FT spectra, which are in general the
stronger lines.
14lIntensities labeled `00' below 1600 Å are from unpublished linelists.
Intensities labeled `d' are diffuse; those marked `d0?' are faint and diffuse; those marked `b'
are broad.
14l$^b$ One standard uncertainty of preceding column.
5l$^c$ *: Line given low weight in level optimization;
5l F: Line measured in FT spectra;
5l G: Line measured in grating spectra;
5l N: Line is from unpublished iron-neon linelists (see section 3.2).
5l A: Line is from unpublished iron-argon linelists (see section 3.2).
5l L: Line is from unpublished linelists below 1600 AA (see section 3.2).
5l S: Line measured in grating spectra and is saturated on plate.
5l I, II, III : Line blended with Fe I, Fe II, or Fe III.
5l Ne, Ar, Si, Ti, Cr, Co, Ni : Line blended with species indicated.
5l gh : Line possibly blended with ghost.
§ ACKNOWLEGMENTS
G. Nave thanks Craig J. Sansonetti for many invaluable discussions on
wavelength calibration, spectral analysis and on the best way to
approach and present a project of this size. She also thanks Robert L. Kurucz and
Alexander E. Kramida for their assistance in finding errors in the tables. James W. Brault, Richard
C. M. Learner, Victor Kaufman, and Anne P. Thorne took or assisted
with taking many of the spectra used in this paper. Some of these
spectra are available in the National Solar Observatory digital
Library [Hill & Suarez, 2012]. This work was partially supported by the National
Aeronautics and Space Administration under the inter-agency agreement
§ REFERENCES
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spectrometry (Academic Press)
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[Kramida & Nave, 2006]
Kramida, A. E., & Nave, G. 2006, Eur. Phys. J. D, 39, 331
[Kurucz, 2010]
Kurucz, R. 2010, Atomic line data for Fe II: file gf2601.pos created on
23rd July, 2010, http://kurucz.harvard.edu/atoms/2601/gf2601.pos
[Learner & Thorne, 1988]
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Transform Spectroscopy:New Methods and Applications, Vol. 3, 38–40
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|
arxiv-papers
| 2012-10-17T15:35:29 |
2024-09-04T02:49:36.729457
|
{
"license": "Public Domain",
"authors": "Gillian Nave, Sveneric Johansson",
"submitter": "Gillian Nave",
"url": "https://arxiv.org/abs/1210.4773"
}
|
1210.4779
|
# Free unitary groups are (almost) simple
Alexandru Chirvasitu111University of California at Berkeley,
chirvasitua@math.berkeley.edu
###### Abstract
We show that the quotients of Wang and Van Daele’s universal quantum groups by
their centers are simple in the sense that they have no normal quantum
subgroups, thus providing the first examples of simple compact quantum groups
with non-commutative fusion rings.
Keywords: free unitary group, simple compact quantum group, CQG algebra
###### Contents
1. 1 Preliminaries
1. 1.1 Free unitary groups
2. 1.2 Normal quantum subgroups
2. 2 Statements and proofs
## Introduction
A short note such as this one can hardly do justice to the richness of the
subject of quantum groups, so we will simply refer the reader to the papers
mentioned below, as well as to the vast literature cited by those papers, for
a comprehensive view of the history and intricacies of the subject. Let us
just mention that the quantum groups in this paper are of function algebra
type, rather than of the quantized universal enveloping algebra kind
introduced in [13, 9]. The former are non-commutative analogues of algebras of
continuous functions on a (here compact) group, and were formalized in
essentially their present form in [22]. To be more specific, we only work with
the algebraic counterparts of the objects described by Woronowicz. These are
the so-called CQG algebras of [8], and mimic the algebras of representative
functions on a compact group, minus the commutativity. They are complex Hopf
$*$-algebras satisfying an additional condition whih ensures, among other
things, the semisimplicity of their categories of comodules.
It makes sense, in view of the importance and ubiquity of simple compact Lie
groups, to study analogous notions in the quantum setting. This program was
initiated in [21], where simple compact quantum groups are defined (along with
the notion of normal compact quantum subgroup) and examples are provided. It
is observed there that these examples are all almost classical, in the sense
that their so-called fusion algebras (meaning Grothendieck rings of their
categories of finite-dimensional comodules) are isomorphic to fusion algebras
of compact Lie groups.
One source of compact quantum groups is provided by ordinary compact Lie
groups deformed in some sense ([15, 18] and references therein), but more
exotic examples can be obtained as quantum automorphism groups of various
structures, such as, say, bilinear forms [10], finite-dimensional
$C^{*}$-algebras endowed with a trace [20, 3], finite graphs [5], finite
metric spaces [4], etc. In the context of simplicity, it turns out [21, $\S$5]
that deforming simple compact Lie groups produces, as expected, simple compact
quantum groups. On the other hand, it is shown in the same paper ($\S$4) that
quantum automorphism groups of traced finite-dimensional $C^{*}$-algebras are
simple, while quantum automorphism groups of non-degenerate bilinear forms
have a single non-trivial normal compact quantum subgroup of order $2$.
The universal quantum groups $A_{u}(Q)$ (the free unitary groups of the title)
parametrized by invertible positive matrices $Q$ were introduced in [19].
Since they jointly play the same role as that of the family of unitary groups
in the theory of compact Lie groups, they are arguably the first examples one
should test compact quantum group notions or tentative results against. Wang
notes in [21, Proposition 4.5] that $A_{u}(Q)$ always has a central one-
dimensional torus (just like classical unitary groups), and hence cannot
possibly be simple. The main result of this paper is that the quotient by this
central circle group is nevertheless simple, again as in the classical case
(see Section 1 for an explanation of the terminology):
###### Theorem.
Let $Q\in GL_{n}$ be a positive invertible matrix. The quotient of the quantum
group $A_{u}(Q)$ by its central circle subgroup $A_{u}(Q)\to
C({\mathbb{S}}^{1})$ is simple.
The outline of the paper: We recall the relevant terminology, conventions and
results in Section 1, and prove the main results (one of which is stated
above) in Section 2.
### Acknowledgement
I am grateful to Shuzhou Wang for fruitful conversations on the contents of
[21], and for his many suggestions on the improvement of this paper.
## 1 Preliminaries
We assume the basics of Hopf algebra and coalgebra theory, as covered in, say,
[16, 7]. Comultiplications, counits, and antipodes are denoted as usual by
$\Delta,\varepsilon$ and $S$ perhaps adorned with the name of the coalgebra or
Hopf algebra as in $\Delta_{H}$, and we use Sweedler notation with suppressed
summation symbol for comultiplication, as in
$\displaystyle\Delta(x)=x_{1}\otimes x_{2}$. Comodules are always right and
finite-dimensional, and everything in sight is complex.
For CQG algebras, unitary and unitarizable comodules, and all other notions
that go into formalizing compact quantum groups in a purely algebraic setting
we refer to [8] or [14, $\S$11]. Here, we only remind the reader that a CQG
algebra is a Hopf $*$-algebra (Hopf algebra endowed with a conjugate linear,
involutive algebra anti-automorphism ‘$*$’ making both $\Delta$ and
$\varepsilon$ morphisms of $*$-algebras) satisfying an additional condition
which ensures that all finite-dimensional comodules admit an inner product
invariant under the coaction in some sense ([14, 11.1.5]). This accords with
the point of view that representations of the compact quantum group are
comodules over the Hopf algebra which is supposed to behave like the algebra
of representative functions on the otherwise fictitious group. Denote by
$\operatorname{\mathrm{CQG}}$ the category of CQG algebras and Hopf
$*$-algebra morphisms.
We also take for granted the correspondence between comodules over a coalgebra
$C$ and subcoalgebras of $C$, sending a $C$-comodule $V$ to the smallest
subcoalgebra $C_{V}\subseteq C$ such that the structure map $V\mapsto V\otimes
C$ factors through $V\otimes C_{V}$. This correspondence induces a bijection
between isomorphism classes of simple comodules and simple subcoalgebras of
$C$, and in a Hopf algebra $H$ it behaves well with respect to multiplication
and the antipode: The coalgebra associated to the tensor product $V\otimes W$
is exactly the product $C_{V}C_{W}$ in $H$ (that is, the linear span of
products of elements from $C_{V}$ and $C_{W}$), while the coalgebra
corresponding to the dual $V^{*}$ is $S(C_{V})$.
The fusion semiring of a Hopf algebra is the Grothendieck semiring of its
category of (finite-dimensional, right) comodules. Note that it admits a
natural anti-endomorphism $\alpha\mapsto\alpha^{*}$, sending (the class of) a
comodule to (the class of) its dual. When the Hopf algebra is cosemisimple
(e.g. a CQG algebra), this anti-endomorphism is involutive. By a slight abuse
of notation, if $x$ is the class of $V$ in the fusion semiring, we write
$C_{x}$ for $C_{V}$.
### 1.1 Free unitary groups
The main characters in this paper are the objects $A_{u}(Q)$ mentioned in the
introduction. Here, $Q$ is a positive invertible $n\times n$ matrix, and by
definition, $A=A_{u}(Q)$ is the $*$-algebra freely generated by $n^{2}$
elements $u_{ij}$, $i,j=\overline{1,n}$ subject to the conditions that both
$u=(u_{ij})_{i,j}$ and $Q^{\frac{1}{2}}\overline{u}Q^{-\frac{1}{2}}$ be
unitary $n\times n$ matrices in $A$ (where $\overline{u}=(u_{ij}^{*})_{i,j}$);
cf. [14, 11.3.1, Example 6], or [19], where these objects were first
introduced in their $C^{*}$-algebraic versions.
It turns out that $A$ can be made into a CQG algebra by demanding that
$u_{ij}$ be the $n^{2}$ “matrix units” of an $n\times n$ matrix coalgebra:
$\Delta(u_{ij})=\sum_{k}u_{ik}\otimes
u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij}.$
The antipode is then defined by $S(u)=u^{*}=(u_{ji}^{*})_{i,j}$, and one
checks the CQG condition by observing that both the comodule $V$ with basis
$e_{i}$, $i=\overline{1,n}$ whose comodule structure is defined by
$e_{j}\mapsto\sum_{i}e_{i}\otimes u_{ij}$ and its dual are unitarizable; we
refer to the cited sources for details on how this works.
The reason why these are called ‘free unitary’ is that every CQG algebra
finitely generated as an algebra is a quotient of some $A_{u}(Q)$; keeping in
mind the arrow reversal implicit in passing from groups to functions on them,
this means that every “compact quantum Lie group” embeds in the compact
quantum group associated to some $A_{u}(Q)$. In other words, collectively, the
$A_{u}(Q)$’s play the same role in the world of compact quantum groups as the
family of all unitary groups does in the theory of ordinary compact groups.
It will be important to recall the structure of the fusion semiring $R_{+}$ of
$A=A_{u}(Q)$, as worked out in [2]. One of the main results of that paper is
that there is a bijection ${\mathbb{N}}*{\mathbb{N}}\ni x\mapsto r_{x}\in
R_{+}$ between the free monoid on two elements and the set of simple
comodules, with multiplication in $R_{+}$ given by
$r_{x}r_{y}=\sum_{x=ag,y=g^{*}b}r_{ab}.$ (1)
Here, ‘$*$’ is the involutive anti-automorphim of ${\mathbb{N}}*{\mathbb{N}}$
interchanging the two copies of ${\mathbb{N}}$, and the generators $\alpha$
and $\alpha^{*}$ of the two ${\mathbb{N}}$’s correspond respectively to the
fundamental comodule $V$ from the next-to-last paragraph and its dual.
### 1.2 Normal quantum subgroups
Always keeping in mind arrow reversal, a (closed) quantum subgroup of the
(compact quantum group with) CQG algebra $A$ should be a quotient CQG algebra
$A\to B$. This is indeed the standard definition in the literature, and the
one we employ here. We regard the arrow itself as the quantum subgroup, and
identify $A\to B$ and $A\to B^{\prime}$ as quantum subgroups provided the two
are isomorphic in the category of arrows in $\operatorname{\mathrm{CQG}}$
sourced at $A$ (in the terminology of [21, 2.7], we identify quantum subgroups
whenever they have the same imbedding).
Following [1, 1.1.5] and [21, 2.2], a quantum subgroup $\pi:A\to B$ of a
compact quantum group is said to be normal if the corresponding right and left
quantum coset spaces
$\operatorname{\textsc{LKer}}(\pi)=\\{a\in A\ |\ \pi(a_{1})\otimes
a_{2}=1_{B}\otimes a\\}$
and respectively
$\operatorname{\textsc{RKer}}(\pi)=\\{a\in A\ |\
a_{1}\otimes\pi(a_{2})=a\otimes 1_{B}\\}$
coincide. The resulting linear subspace
$\operatorname{\textsc{LKer}}=\operatorname{\textsc{RKer}}=C\leq A$ then turns
out to be a CQG subalgebra of $A$ and can be interpreted as functions on the
quotient of the compact quantum group corresponding to $A$ by the normal
quantum subgroup $\pi:A\to B$.
It is perhaps worth pointing out that $\pi$ and $\iota$ determine each other:
We have just seen how to get $\iota$ from $\pi$, and conversely, it can be
shown (e.g. [21, 4.4]) that $\pi:A\to B$ is precisely the quotient of $A$ by
the ideal $AC^{+}=C^{+}A$, where $C^{+}=\ker(\varepsilon_{C})$. What this
means, in other words, is that the inclusion $iota:C\to A$ and the surjection
$\pi:A\to B$ fit into an exact sequence $0\to C\to A\to B\to 0$ of Hopf (in
our case also $*$-) algebras in the sense of [1, 1.2.3].
###### 1.2.1 Remark.
If $\iota:C\to P$ is to be half of an exact sequence, then $C$ must be
invariant under the right (and also left) adjoint action of $P$ on itself
given by $q\triangleleft p=S(p_{1})qp_{2}$ (respectively $p\triangleright
q=p_{1}qS(p_{2})$). Indeed, it is observed in the proof of [1, 1.1.12] that
for any Hopf algebra morphism $f$, $\operatorname{\textsc{LKer}}(f)$ as
defined in § 1.2 is invariant under the right adjoint action. It is for this
reason that we refer to the CQG subalgebras $\iota:C\to P$ of interest, i.e.
those giving rise to exact sequences, as ad-invariant.
In fact, invariance of a Hopf $*$-subalgebra $C\to P$ under either the left or
right adjoint action is also sufficient in order that it be part of an exact
sequence. This is proven in [1, 1.2.5] provided $P$ is faithfuly flat over
$C$; the latter condition always holds (that is, a CQG algebra is always
faithfully flat over a CQG subalgebra) by [6]. $\blacklozenge$
In view of the above discussion, the following is very reasonable:
###### 1.2.2 Definition.
A compact quantum group $A\in\operatorname{\mathrm{CQG}}$ is simple if there
are no normal quantum subgroups $\pi:A\to B$ apart from $\varepsilon_{A}$ and
$\operatorname{id}_{A}$. $\blacklozenge$
###### 1.2.3 Example.
As shown in [21, 4.5], sending $u_{ij}$ to $\delta_{ij}t$ implements a normal
embedding of the one-dimensional torus ${\mathbb{S}}^{1}$ with algebra of
representative functions $C({\mathbb{S}}^{1})=\mathbb{C}[t,t^{-1}]$ into any
of the free unitary groups $A_{u}(Q)$. The aim of this paper is to prove that
the resulting quotient is simple in the sense of 1.2.2.
We denote the CQG algebra associated to this quotient by $P_{u}(Q)$, standing
for ‘projective’. This is motivated by the fact that $P_{u}(Q)$ is a kind of
“projectivized” version of the free unitary group. $\blacklozenge$
###### 1.2.4 Remark.
The terminology conflicts slightly with that of [21, 3.3]: On the one hand,
Wang’s definition of simplicity only demands that there be no non-trivial
connected normal quantum subgroups, and refers to the stronger form of
simplicity from 1.2.2 as absolute. I prefer the shorter term because there is
no need for that distinction in this paper. In other ways though, 1.2.2 might
seem weaker than [21, 3.3], because it makes no mention of the other three
conditions of the latter (numbered as in that paper):
(1) In order to be simple, Wang requires that a CQG algebra be finitely
generated. This is indeed not the case for $P_{u}(Q)$ (2.0.3), but we remedy
the problem in Proposition 2.0.5, where we provide a smaller, “less canonical”
example of a simple CQG satisfying this additional condition.
(2) The simple CQG’s of [21] are required to be connected, in the sense that
the $*$-subalgebra generated by any simple subcoalgebra is infinite-
dimensional. $A_{u}(Q)$ and hence its CQG subalgebras are easily seen to
satisfy this condition, so we need not worry about it any longer.
(4) Simple CQG’s are not supposed to have any non-trivial group-like elements,
or equivalently, one-dimensional comodules, or again, one-dimensional
subcoalgebras. Once more, this is automatically satisfied by $A_{u}(Q)$ and
its CQG subalgebras, for example because the fusion ring is freely generated
(as a ring) by the fundamental representation and its dual ([2, Théorème 1
(ii)]), which immediately shows that its only invertible elements are $\pm 1$.
$\blacklozenge$
## 2 Statements and proofs
Let us restate the result announced in the introduction:
###### 2.0.1 Theorem.
For any positive invertible matrix $Q$, the compact quantum group $P_{u}(Q)$
is simple in the sense of 1.2.2.
The proof consists of showing that whenever a CQG subalgebra $\iota:C\to
P=P_{u}(Q)$ fits into an exact sequence as explained in § 1.2, $C$ is either
the scalars or the entire $P$. Since inclusions of CQG subalgebras induce
inclusions of fusion semirings, we can attack the problem by limiting the
possibilities for a fusion semiring of $R_{+}=R_{+}(P)$ if it is to correspond
to such an embedding $\iota$.
In working with fusion semirings, all of which are embedded in that of
$A=A_{u}(Q)$ described in § 1.1, we will identify the free monoid
${\mathbb{N}}*{\mathbb{N}}$ (and hence the set of simple comodules of $A$)
with words in the generators $\alpha$ and $\alpha^{*}$. It will be less
cumbersome, notationally, to substitute $0$ and $1$ respectively for $\alpha$
and $\alpha^{*}$, and write the elements of the free monoid as binary words;
note that the ‘$*$’ involution changes $0$’s into $1$’s and vice versa. First,
we determine precisely which binary words correspond to simple comodules of
$P_{u}(Q)$.
###### 2.0.2 Lemma.
The simple comodules of $P$ are parametrized by the binary words with equal
numbers of $0$’s and $1$’s.
###### Proof.
Since by definition $P$ is the quotient of the free unitary group by the
circle subgroup $A\to C({\mathbb{S}}^{1})$, its simple comodules are precisely
those which, when regarded as comodules over $C({\mathbb{S}}^{1})$, break up
as direct sums of the trivial comodule. The conclusion follows from the
observations that (a) simple representations over the circle group are
parametrized by ${\mathbb{Z}}$, and (b) under this identification, scalar
corestriction via $A\to C({\mathbb{S}}^{1})$ turns the simple corresponding to
a binary word $w$ into a direct sum of copies of the simple
$C({\mathbb{S}}^{1})$-comodule corresponding to the integer $(\sharp\text{ of
}0\text{'s in }w)-(\sharp\text{ of }1\text{'s in }w)$. $\blacksquare$
We refer to binary words with equal numbers of $0$’s and $1$’s as balanced.
###### 2.0.3 Remark.
Using the fusion rules 1, it can be shown that given any finite set $S$ of
binary words, no simple in the semiring generated by $S$ can start with a
longer contiguous segment of $0$’s than the longer such segment in a member of
$S$. Together with Lemma 2.0.2, this shows that as noted in 1.2.4, $P$ is not
finitely generated. $\blacklozenge$
We now start working towards proving Theorem 2.0.1.
###### 2.0.4 Lemma.
If $\iota:C\to P$ is ad-invariant and strictly larger than the scalars, then
both $r_{01}$ and $r_{10}$ are in the fusion semiring $R_{+}(C)\subseteq
R_{+}$ of $C$.
###### Proof.
By assumption, $C$ has some non-trivial simple comodule, whose corresponding
binary word we may as well assume starts with a zero: $r_{0x}$. But then its
dual will be $r_{x^{*}1}$, and it follows from the fusion rules 1 that
$r_{01}$ is a summand in the product $r_{0x}r_{x^{*}1}$. In conclusion, the
simple $r_{01}$ must be in $R_{+}(C)$.
We now have to prove the same about $r_{10}$, and this is where ad-invariance
comes into the picture. First, the argument in the previous paragraph solves
the problem as soon as we can show that some $r_{1\cdots}$ is in $R_{+}(C)$
(then multiply it with its dual $r_{\cdots 0}$, etc.). To this end, fix some
non-zero $p\in P$ in the coalgebra $C(r_{10})$, and let it act on a non-zero
$c\in C$ in the coalgebra $C(r_{01})$ via the left adjoint action. On the one
hand, the result must be in $C$ by ad-invariance. On the other, the fusion
rules 1 say that $r_{10}r_{01}r_{10}$ equals $r_{100110}$, and hence the
multiplication map $C_{r_{10}}\otimes C_{r_{01}}\otimes C_{r_{10}}\to
C_{r_{100110}}$ is an isomorphism. It follows from this that $p\triangleright
c=p_{1}cS(p_{2})$ is a non-zero element of $C_{r_{100110}}$ (note that
$C_{r_{10}}$ is fixed by the antipode, so it contains both $p_{1}$ and
$S(p_{2})$). All in all, we get $r_{100110}\in R_{+}(C)$, and as observed at
the beginning of this paragraph, this will do to finish the proof.
$\blacksquare$
In the above proof and elsewhere we are tacitly using the correspondence
between CQG subalgebras of a CQG algebra $A$ and sub-semirings of its fusion
semiring $R_{+}(A)$. In one direction, an inclusion $\iota:C\to A$ of a CQG
subalgebra induces a fully faithful monoidal functor between categories of
comodules, and hence a fusion semiring inclusion. Moreover, $R_{+}(C)$ is the
${\mathbb{N}}$-span of precisely those simples in $R_{+}(A)$ which correspond
to simple subcoalgebras of $C$. In the other direction, given a sub-semiring
$R_{+}$ of $R_{+}(A)$ which is an ${\mathbb{N}}$-span of simples and is closed
under the involution, the direct sum of simple subcoalgebras of $A$
corresponding to the simples in $R_{+}$ is a CQG subalgebra. These two
constructions are inverse to one another, and implement the correspondence.
Let us stop for a moment to record the fact that we now already have an
example of simple compact quantum group with non-commutative fusion semiring:
The proof of Lemma 2.0.4 only uses $r_{01}$ and $r_{10}$ and the fusion rules
of $A_{u}(Q)$, so it provides a proof for
###### 2.0.5 Proposition.
For any positive invertible matrix $Q$, the compact quantum group whose
underlying CQG algebra is the subalgebra of $A_{u}(Q)$ generated by
$u_{ij}u_{kl}^{*}$ and $u_{ij}^{*}u_{kl}$ is simple.
As promised in 1.2.4, this example, unlike $P_{u}(Q)$, is finitely generated
as an algebra and hence conforms to all of the conditions for simplicity in
[21, 3.3]. Apart from commutativity of the fusion semiring, this example
felicitously lacks another property. It is observed in [21, $\S$5] that the
examples of simple compact quantum groups mentioned in the introduction have
what in that paper is called property F. It means that every embedding of a
CQG subalgebra is part of an exact sequence, or, in view of 1.2.1, every CQG
subalgebra is invariant under the left adjoint action. The CQG algebra of
Proposition 2.0.5 clearly does not have property F, as its CQG subalgebra
generated by $u_{ij}u_{kl}^{*}$, whose fusion semiring is generated by
$r_{01}$, is not ad-invariant by the proof of Lemma 2.0.4.
###### 2.0.6 Lemma.
Keeping the hypotheses and notation of Lemma 2.0.4, all simples of the form
$r_{0\cdots 01\cdots 1}$ and $r_{1\cdots 10\cdots 0}$ belong to $R_{+}(C)$.
###### Proof.
Let us prove the statement for $r_{0\cdots 01\cdots 1}$ ($n$ $0$’s and also
$n$ $1$’s, since the binary word must be balanced). First, note that it is
enough to show that some simple of the form $r_{0\cdots 0x}$ ($n$ $0$’s) is in
$R_{+}(C)$. Indeed, its dual would correspond to the word $x^{*}$ followed by
$n$ $1$’s, and their product would contain the desired simple as a summand. We
know from Lemma 2.0.4 that $r_{10}$ is in $R_{+}(C)$, and now we can argue as
in the proof of that lemma, acting on a non-zero element of $C_{r_{10}}$ by a
non-zero element of $C_{r_{0\cdots 01\cdots 1}}$ via the left adjoint action
to conclude. $\blacksquare$
###### Proof of Theorem 2.0.1.
Keeping the notations and assumptions of Lemma 2.0.4, we show by induction on
the length of a balanced word $x\in{\mathbb{N}}*{\mathbb{N}}$ that $r_{x}$
belongs to $R_{+}(C)$. Having taken care of the base step of the induction in
Lemma 2.0.4, we can assume $x$ is at least four symbols long. There are two
possibilities:
(1) $x$ consists of only two contiguous blocks of symbols, one of $0$’s and
one of $1$’s, as in, say, $x=0\cdots 01\cdots 1$. This case is covered by
Lemma 2.0.6.
(2) $x$ consists of more than two contiguous blocks of symbols. Write it
without loss of generality as, say, $x=0\cdots 01y$, starting with $n\geq 1$
$0$’s. Then, by the fusion rules 1, $x$ is a summand in the product of
$r_{0\cdots 01\cdots 1}$ ($n$ $0$’s) and $r_{0\cdots 0y}$ ($n-1$ $0$’s). These
two words are both strictly shorter than $x$, so the induction hypothesis
implies that both simples are in $R_{+}(C)$. This finishes the proof.
$\blacksquare$
While these results partially address [21, Problem 4.6 (2)], which asks for
examples of simple compact quantum groups with non-commutative fusion
semirings, it will still be interesting to investigate how much further
Proposition 2.0.5 can be pushed, and hence make some progress towards the
classification of simple quotients of free unitary groups. Note that here
‘quotient quantum group’ is used in the weak sense, meaning simply ‘CQG
subalgebra’. Quotients in the stronger sense of left hand halves of exact
sequences are taken care of by the following consequence of Theorem 2.0.1 (or
rather of its proof):
###### 2.0.7 Corollary.
Any proper normal quantum subgroup of $A=A_{u}(Q)$ is contained in the circle
subgroup $A\to C({\mathbb{S}}^{1})$ from 1.2.3.
###### Proof.
The previous results go through practically verbatim for $A$ (instead of
$P=P_{u}(Q)$), and show that any non-trivial ad-invariant $\iota:C\to A$
contains the CQG subalgebra $P\subset A$. $\blacksquare$
We saw in 1.2.1 that CQG subalgebras $\iota:C\to H$ corresponding to quotients
by normal quantum subgroups have a simple characterization as precisely those
which are ad-invariant. It would be interesting though, as well as convenient,
to have a purely combinatorial characterization in terms of fusion semirings:
###### 2.0.8 Question.
Let $\iota:C\to H$ be an inclusion of CQG algebras. Can the ad-invariance of
$C$ be characterized solely in terms of the fusion ring inclusion
$R_{+}(C)\subseteq R_{+}(H)$?
Alternatively, and also probably more tractably,
###### 2.0.9 Question.
Does simplicity for a compact quantum group depend only on its fusion
semiring?
Positive answers would provide an alternative approach to the invariance of
simplicity under deformation proved in [21, $\S$5], and would be natural
companions to such results as the possibility of lifting isomorphisms of
fusion semirings to honest isomorphisms for compact connected Lie groups
([12]) and the fusion semiring characterization of the center for a compact
group (as in [17] or [11, 3.9]).
## References
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* [2] Teodor Banica. Le groupe quantique compact libre ${\rm U}(n)$. Comm. Math. Phys., 190(1):143–172, 1997.
* [3] Teodor Banica. Symmetries of a generic coaction. Math. Ann., 314(4):763–780, 1999.
* [4] Teodor Banica. Quantum automorphism groups of small metric spaces. Pacific J. Math., 219(1):27–51, 2005.
* [5] Julien Bichon. Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc., 131(3):665–673 (electronic), 2003.
* [6] A. Chirvasitu. Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. ArXiv e-prints, October 2011.
* [7] Sorin Dăscălescu, Constantin Năstăsescu, and Şerban Raianu. Hopf algebras, volume 235 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 2001. An introduction.
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* [9] V. G. Drinfeld. Quantum groups. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 798–820, Providence, RI, 1987. Amer. Math. Soc.
* [10] Michel Dubois-Violette and Guy Launer. The quantum group of a nondegenerate bilinear form. Phys. Lett. B, 245(2):175–177, 1990.
* [11] Shlomo Gelaki and Dmitri Nikshych. Nilpotent fusion categories. Adv. Math., 217(3):1053–1071, 2008.
* [12] David Handelman. Representation rings as invariants for compact groups and limit ratio theorems for them. Internat. J. Math., 4(1):59–88, 1993.
* [13] Michio Jimbo. A $q$-difference analogue of $U({\mathfrak{g}})$ and the Yang-Baxter equation. Lett. Math. Phys., 10(1):63–69, 1985.
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|
arxiv-papers
| 2012-10-17T16:01:49 |
2024-09-04T02:49:36.741124
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandru Chirvasitu",
"submitter": "Alexandru Chirv\\u{a}situ L.",
"url": "https://arxiv.org/abs/1210.4779"
}
|
1210.4793
|
# Essential norm of Toeplitz operators and Hankel operators on the weighted
Bergman space
Fengying Li111Supported by China Scholarship Council
Abstract. In this paper, we show that on the weighted Bergman space of the
unit disk the essential norm of a noncompact Hankel operator equals its
distance to the set of compact Hankel operators and is realized by infinitely
many compact Hankel operators, which is analogous to the theorem of Axler,
Berg, Jewell and Shields on the Hardy space in [4]; moreover, the distance is
realized by infinitely many compact Hankel operators with symbols continuous
on the closure of the unit disk and vanishing on the unit circle.
Mathematics Subject Classification(2010). 11K70, 46-XX, 47S10, 97I80.
Keywords. Hankel operators, Toeplitz operators, essential norm, weighted
Bergman space.
## 1 Introduction
Let $D$ be the open unit disk in the complex plane $\mathbb{C}$. Let
$L^{\infty}(D)$ denote the space of bounded measurable functions on the unit
disk $D$, and let $H^{\infty}(D)$ denote its subalgebra of bounded analytic
functions. We write $dA$ to denote the normalized Lebesgue area measure on the
unit disk $D$. For $\alpha>-1$, $L^{2}(D,dA_{\alpha})$ consists of all
function $f$ on $D$ such that
$\|f\|_{2,\alpha}^{2}=\int_{D}|f(z)|^{2}dA_{\alpha}(z)<\infty$
where $dA_{\alpha}(z)=(1+\alpha)(1-|z|^{2})^{\alpha}dA(z)$, it easy to see
that $\int_{D}dA_{\alpha}(z)=1$.
For $\alpha>-1$ we define the weighted Bergman space
$A_{\alpha}^{2}(D)=H(D)\cap L^{2}(D,dA_{\alpha})$
where $H(D)$ is the space of analytic functions on $D$. When $\alpha=0$ we
recover the standard definition of the Bergman space.
The Toeplitz operator $T_{\varphi}$ with symbol $\varphi\in L^{\infty}(D)$ on
the weighted Bergman spaces is defined by
$T_{\varphi}f(z)=P_{\alpha}(\varphi
f)(z)=\int_{D}K_{\alpha}(z,w)f(w)\varphi(w)dA_{\alpha}(w)\ \ \forall f\in
A_{\alpha}^{2}(D)$
where $K_{\alpha}(z,w)=\frac{1}{(1-z\overline{w})^{2+\alpha}}$ is the
reproducing kernel and $P_{\alpha}$ is the orthogonal projection of
$L^{2}(D,dA_{\alpha})$ onto $A^{2}_{\alpha}(D)$. Theorem 4.24 in [16]
guarantees $P_{\alpha}$ is bounded operator on the weighted Bergman space.
The Hankel operator $H_{\varphi}$ with symbol $\varphi\in L^{\infty}(D)$ on
the weighted Bergman space is defined by
$H_{\varphi}f=(I-P_{\alpha})(\varphi f)\ \ \forall f\in A^{2}_{\alpha}(D)$
where $I-P_{\alpha}$ is the orthogonal projection from $L^{2}(D,dA_{\alpha})$
onto $(A_{\alpha}^{2}(D))^{\perp}$.
Let $\mathcal{K}(D)$ denote the space of compact operators on $D$. The
essential norm of an operator $T$ is defined by
$\|T\|_{e}=inf\\{\|T-K\|:K\in\mathcal{K}(D)\\};$
i.e., the distance $T$ to the space of compact operators.
In order to address an approximation problem in $L^{\infty}(\partial D)$ where
$\partial D$ is the boundary of unit disk $D$ ([1], [12]), Axler, Berg, Jewell
and Shields [4] obtained the following beautiful result for the Hardy space
$H^{2}(\partial D)$.
###### Theorem 1.1
Let $H_{f}$ be a noncompact Hankel operator on Hardy space $H^{2}(\partial
D)$. Then there exist infinitely many different compact Hankel operators
$H_{\varphi}$ such that $\|H_{f}-H_{\varphi}\|=\|H_{f}\|_{e}$.
In other words, for a noncompact Hankel operator $H_{f}$ with symbol $f\in
L^{\infty}(\partial D)$ on Hardy space $H^{2}(\partial D)$, its distance to
the space of compact operators is realized by infinitely many compact Hankel
operators. On the Hardy space, a theorem of Nehari [3] states
$\|H_{f}\|=dist(f,H^{\infty}).$
It is also true that $\|H_{f}\|_{e}=dist(f,H^{\infty}+C(\partial D))$.
For Hankel (Toeplitz) operators on unweighted Bergman space, it is known the
essential norm is realized by some compact operators([7],[8]), and in [15],
the essential norm was estimated by
$\overline{\lim_{|z|\rightarrow
1}}\|f\circ\varphi_{z}-P(f\circ\varphi_{z})\|_{2}\leq\|H_{f}\|_{e}\leq
C\overline{\lim_{|z|\rightarrow
1}}\|f\circ\varphi_{z}-P(f\circ\varphi_{z})\|_{2}^{1/10}$
where $\varphi_{z}(w)=\frac{z-w}{1-\overline{z}w}$ and $C$ is a constant. The
estimation of the Teoplitz operator $T_{f}$ is
$\overline{\lim_{|z|\rightarrow
1}}\|P(f\circ\varphi_{z})\|_{2}\leq\|T_{f}\|_{e}\leq
C\overline{\lim_{|z|\rightarrow 1}}\|P(f\circ\varphi_{z})\|_{2}^{1/10}.$
A natural question is whether Theorem 1.1 can be extended to the case of
Hankel (Toeplitz) operators on the weighted Bergman space.
Indeed, we shall prove that the conclusion of Theorem 1.1 holds for noncompact
Hankel operators $H_{f}$ and noncompact Toeplitz operators $T_{f}$, with
symbol $f\in L^{\infty}(D)$; moreover, the symbols $\varphi$ of the
approximation operators $H_{\varphi}$ ($T_{\varphi}$) will actually reside in
a better behaved function space, which will guarantee that $H_{\varphi}$,
$H_{\overline{\varphi}}$, $T_{\varphi}$, and $T_{\overline{\varphi}}$ are
compact.
Let $C_{\partial}(\overline{D})$ denote the space of continuous functions on
the closure $\overline{D}$ of the unit disk and vanishing on the unit circle
$\partial D.$ By results in [2], [10], [13], [14], [15], [16], [17], we easily
see that for each $\varphi$ in $C_{\partial}(\overline{D})$, $H_{\varphi}$,
$H_{\overline{\varphi}}$, $T_{\varphi}$, and $T_{\overline{\varphi}}$ are
compact. The first two theorems are inspired by and analogous to Theorem 1.1.
###### Theorem 1.2
Let $f\in L^{\infty}(D)$, and $H_{f}$ the associated noncompact Hankel
operator on $A^{2}_{\alpha}(D)$. There exist infinitely many distinct compact
Hankel operators $H_{\varphi}$ with symbol $\varphi$ in
$C_{\partial}(\overline{D})$ such that
$\|H_{f}-H_{\varphi}\|=\|H_{f}\|_{e}.$
###### Theorem 1.3
Let $f\in L^{\infty}(D)$, and $T_{f}$ the associated noncompact Teoplitz
operator on $A^{2}_{\alpha}(D)$. There exist infinitely many distinct compact
Toeplitz operators $T_{\varphi}$ with symbol $\varphi$ in
$C_{\partial}(\overline{D})$ such that $\|T_{f}-T_{\varphi}\|=\|T_{f}\|_{e}$.
If $f$ is harmonic on the unit disk, we have the following result.
###### Theorem 1.4
Let $f$ be a bounded harmonic function on the unit disk, and $H_{f}$ the
associated noncompact Hankel operator on $A^{2}_{a}(D)$. There exist
infinitely many distinct harmonic functions $\varphi$ on the unit disk and
continuous on the closure of the unit disk such that
$\|H_{f}-H_{\varphi}\|=\|H_{f}\|_{e}.$
For the Hardy space, the reproducing kernel is given by
$K_{z}(\xi)=\frac{1}{1-z\overline{\xi}}$ for $z\in D$ and $\xi\in\partial D$
(see Corollary 2.11 in [9]); i.e., for any $f\in H^{2}(\partial D)$,
$f(z)=<f,K_{z}>$. Since the reproducing kernel for $A_{\alpha}^{2}(D)$ is
$K_{\alpha}(z,w)=\frac{1}{(1-z\overline{w})^{2+\alpha}}$ with $\alpha>-1$, it
is tempting to consider the Hardy space as a limiting situation for the
weighted Bergman spaces as $\alpha\rightarrow-1$; for this reason, the Hardy
space is often denoted as $A^{2}_{-1}$. As a result, we view the theorems in
this paper as a generalization in which the theorem of Axler, Berg, Jewell and
Shields’s appears as a special limiting case.
Recall that a sequence $\\{A_{n}\\}$ of bounded linear operators on a Banach
space $H$ converges to an operator $A$ in the strong operator topology if
$\|(A_{n}-A)f\|\rightarrow 0$ for every $f\in H$.
Our proof will make use of the following result which we will record here, and
whose proof can be found in [4].
###### Theorem 1.5
Let $H_{1}$ and $H_{2}$ be two Hilbert spaces, and $T:H_{1}\rightarrow H_{2}$
a noncompact bounded operator. Let $\\{T_{n}\\}_{n\geq 1}$ be a sequence of
compact operators from $H_{1}$ to $H_{2}$ such that $T_{n}\rightarrow T$ and
$T_{n}^{*}\rightarrow T^{*}$ in the strong operator topology. Then there exist
sequences $\\{a_{n}\\}_{n\geq 1}$ and $\\{b_{n}\\}_{n\geq 1}$ of non-negative
real numbers such that $\sum_{n\geq 1}a_{n}=\sum_{n\geq 1}b_{n}=1$ and
$\|T-K_{1}\|=\|T-K_{2}\|=\|T\|_{e}$, where $K_{1}=\sum_{n\geq 1}a_{n}T_{n}$
and $K_{2}=\sum_{n\geq 1}b_{n}T_{n}$; moreover, $K_{1}\neq K_{2}$.
## 2 Proof of Theorems
In order to use Theorem 1.5 to prove Theorems 1.2 and 1.3, we need to
establish all the conditions in the premise; i.e., there exists a sequence of
functions $\psi_{n}\in C_{\partial}(\overline{D})$ such that the sequence of
compact Hankel(Toeplitz) operators $\\{H_{\psi_{n}}\\}(\\{T_{\psi_{n}}\\})$
and $\\{H_{\psi_{n}}^{*}\\}(\\{T_{\psi_{n}}^{*}\\})$ respectively converge to
$H_{f}(T_{f})$ and $H_{f}^{*}(T_{f}^{*})$ in the strong operator topology. We
start by approximating $f\in L^{\infty}(D)$ by continuous functions.
Suppose $\delta$ is a positive smooth function on the complex plane
$\mathbb{C}$ such that
(a) $\delta$ is compactly supported and identically zero outside of $D$,
(b) $\int_{\mathbb{C}}\delta(z)dA_{\alpha}(z)=1$,
(c) For $\varepsilon>0$, $\lim_{\varepsilon\rightarrow
0}\delta_{\varepsilon}(z)$ is a Dirac delta function where
$\delta_{\varepsilon}(z)=\frac{1}{\epsilon^{2}}\delta(\frac{z}{\epsilon})$,
(d) $\int_{|z|>\varepsilon}\delta_{\varepsilon}(z)dA_{\alpha}(z)=0$.
Then $\delta$ is called a mollifier and
$\int_{\mathbb{C}}\delta_{\varepsilon}(z)dA_{\alpha}(z)=1$.
Any $f\in L^{\infty}(D)$ can be extended to the whole complex plane
$\mathbb{C}$ by taking it to be zero outside of $D$. For convenience, we will
denote it by the same function and so we can assume $f\in
L^{\infty}_{loc}(\mathbb{C})$; thus, $f\in
L^{1}_{loc}(\mathbb{C},dA_{\alpha})$ since
$\int_{\mathbb{C}}|f(z)|dA_{\alpha}(z)=\int_{D}|f(z)|dA_{\alpha}(z)\leq\|f\|_{\infty}<\infty.$
We can define the convolution
$\delta_{\varepsilon}\ast
f(z)=\int_{\mathbb{C}}\delta_{\varepsilon}(z-w)f(w)dA_{\alpha}(w)=\int_{\mathbb{C}}\delta_{\varepsilon}(w)f(z-w)dA_{\alpha}(w).$
For each fixed $z\in D$, the non-trivial domain of integration for
$\int_{\mathbb{C}}\delta_{\varepsilon}(z-w)f(w)dA_{\alpha}(w)$ is the disk
centered at $z$ and of radius $\epsilon$. Note the convolution is still
defined for $z\in\partial D$; hence $\delta_{\varepsilon}\ast f$ is a
mollification of $f$.
It is well known that
(a) $\delta_{\varepsilon}\ast f\in C^{\infty}(\mathbb{C},dA_{\alpha})$,
(b) $\delta_{\varepsilon}\ast f\in L^{2}(\mathbb{C},dA_{\alpha})$ and
$\|\delta_{\varepsilon}\ast f-f\|_{2,\alpha}\rightarrow 0$ as
$\varepsilon\rightarrow 0$.
The reader may wish to consult [5], [6] and [11] for more information.
Note, even if the function $f\in L^{\infty}(D)$ is zero on the boundary
$\partial D$, the convolution $\delta_{\varepsilon}\ast f$ may not be
identically zero on $\partial D$. We will need to modify the convolution to
make sure that does not happen.
For a sequence $\\{r_{n}\\}$ such that $0<r_{n}<1$ and $z\in D$, we define
$f_{r_{n}}(z)=\begin{cases}f(z)\ \ \ &|z|<r_{n},\\\ 0\ \ \ &|z|\geq
r_{n}.\end{cases}$ (2.1)
We claim that for any $\varepsilon>0$ the convolution
$\delta_{\varepsilon}\ast f_{r_{n}}$ has the following properties:
* (a)
$\delta_{\varepsilon}\ast f_{r_{n}}(z)$ converges to $\delta_{\varepsilon}\ast
f(z)$ pointwise as $r_{n}\rightarrow 1$;
* (b)
If $dist(r_{n}D,\partial D)>\varepsilon$, the convolution
$\delta_{\varepsilon}\ast f_{r_{n}}$ is equal to zero on $\partial D$.
Proof of this claim:
(a) To see this, for every $z\in D$
$\displaystyle|\delta_{\varepsilon}\ast f(z)-\delta_{\varepsilon}\ast
f_{r_{n}}(z)|$ (2.2)
$\displaystyle\leq\int_{\mathbb{C}}\delta_{\varepsilon}(z-w)|f(w)-f_{r_{n}}(w)|dA_{\alpha}(w)$
$\displaystyle=\int_{|z-w|\leq\epsilon}\delta_{\varepsilon}(z-w)|f(w)-f_{r_{n}}(w)|dA_{\alpha}(w)$
$\displaystyle+2\|f\|_{\infty}\int_{|z-w|>\epsilon}\delta_{\varepsilon}(z-w)dA_{\alpha}(w)$
Since $f_{r_{n}}(z)\rightarrow f(z)$ pointwise as $r_{n}\rightarrow 1$ and
$\|f_{r_{n}}\|_{\infty}\leq\|f\|_{\infty}$, for any $z\in\mathbb{C}$ we have
$\delta_{\varepsilon}(z-w)f_{r_{n}}(w)\rightarrow\delta_{\varepsilon}(z-w)f(w)$
pointwise as $r_{n}\rightarrow 1$ and
$|\delta_{\varepsilon}(z-w)f_{r_{n}}(w)|\leq\delta_{\varepsilon}(z-w)\|f\|_{\infty}$.
By the dominated convergence theorem, the last equality of (2.2) goes to 0 as
$r_{n}\rightarrow 1$.
(b) We’ll show that $\delta_{\varepsilon}\ast f_{r_{n}}|\partial D=0$. In
fact,
$\displaystyle\delta_{\varepsilon}\ast f_{r_{n}}(z)|\partial D$ (2.3)
$\displaystyle=\int_{\mathbb{C}}\delta_{\varepsilon}(z-w)f_{r_{n}}(w)dA(w)|\partial
D$
$\displaystyle=\int_{|z-w|<\epsilon}\delta_{\varepsilon}(z-w)f(w)\chi_{r_{n}D}(w)dA(w)|\partial
D$
where $\chi_{A}$ is the characteristic function of the set $A$. For
$z\in\partial D$, from the assumptions $dist(r_{n}D,\partial D)>\epsilon$ and
$|z-w|<\epsilon$, the domain of the last integration is empty. Thus
$\delta_{\varepsilon}\ast f_{r_{n}}(z)|\partial D=0$ for all $0<r_{n}<1$.
Hence $\delta_{\varepsilon}\ast f_{r_{n}}\in C_{\partial}(\overline{D})$. This
finishes the claim.
For any $f\in L^{\infty}(D)$, we have a sequence of functions
$\delta_{\varepsilon}\ast f_{r_{n}}(z)\in C_{\partial}(\overline{D})$. In [15]
and [16], it was established that $H_{\delta_{\varepsilon}\ast
f_{r_{n}}}$($H_{\delta_{\varepsilon}\ast f_{r_{n}}}^{*}$) and
$T_{\delta_{\varepsilon}\ast f_{r_{n}}}$ ($T_{\delta_{\varepsilon}\ast
f_{r_{n}}}^{*}$) are compact on Bergman space for $0<r_{n}<1$. The next step
is to prove convergence in the strong operator topology of these sequences of
operators on the weighted Bergman space.
Proof of theorem 1.2. First, we show that $H_{\delta_{\varepsilon}\ast
f_{r_{n}}}$ converges to $H_{f}$ in the strong operator topology.
It is well known that the subalgebra $H^{\infty}(D)$ is dense in
$A_{\alpha}^{2}(D)$, i.e. $\forall\varepsilon_{1}>0$ and for any $g\in
A_{\alpha}^{2}(D)$, there exists a $g_{1}\in H^{\infty}(D)$ such that
$\|g-g_{1}\|_{2,\alpha}<\varepsilon_{1}$. Using the Holder inequality, the
bounds $\|\delta_{\varepsilon}\ast f_{r_{n}}\|_{2,\alpha}\leq\|f\|_{\infty}$,
and the fact that the orthogonal projection $I-P_{\alpha}$ is a bounded
operator on the weighted Bergman space $A^{2}_{\alpha}(D)$, we see that
$\displaystyle\|(H_{f}-H_{\delta_{\varepsilon}\ast f_{r_{n}}})g\|_{2,\alpha}$
(2.4) $\displaystyle\leq\|(H_{f}-H_{\delta_{\varepsilon}\ast
f_{r_{n}}})(g-g_{1})\|_{2,\alpha}+\|(H_{f}-H_{\delta_{\varepsilon}\ast
f_{r_{n}}})g_{1}\|_{2,\alpha}$
$\displaystyle=\|(I-P_{\alpha})(f-\delta_{\varepsilon}\ast
f_{r_{n}})(g-g_{1})\|_{2,\alpha}+\|(I-P_{\alpha})(f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq\|(f-\delta_{\varepsilon}\ast
f_{r_{n}})(g-g_{1})\|_{2,\alpha}+\|(f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq\|f-\delta_{\varepsilon}\ast
f_{r_{n}}\|_{2,\alpha}\|g-g_{1}\|_{2,\alpha}+\|(f-\delta_{\varepsilon}\ast
f)g_{1}\|_{2,\alpha}$ $\displaystyle+\|(\delta_{\varepsilon}\ast
f-\delta_{\varepsilon}\ast f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq
2\varepsilon_{1}\|f\|_{\infty}+\|f-\delta_{\varepsilon}\ast
f\|_{2,\alpha}\|g_{1}\|_{\infty}+\|(\delta_{\varepsilon}\ast
f-\delta_{\varepsilon}\ast f_{r_{n}})g_{1}\|_{2,\alpha}$
From claim (a), we see that $\delta_{\varepsilon}\ast
f_{r_{n}}(z)\rightarrow\delta_{\varepsilon}\ast f(z)$ pointwise as
$r_{n}\rightarrow 1$. For $g_{1}\in H^{\infty}(D)$, we also have
$\delta_{\varepsilon}\ast
f_{r_{n}}(z)g_{1}(z)\rightarrow\delta_{\varepsilon}\ast f(z)g_{1}(z)$
pointwise. It is easy to see that $|\delta_{\varepsilon}\ast
f_{r_{n}}(z)g_{1}(z)|\leq\|f\|_{\infty}|g_{1}(z)|$. By the dominated
convergence theorem $\|(\delta_{\varepsilon}\ast f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}\rightarrow 0$ as $r_{n}\rightarrow 1$; thus,
$\|(H_{f}-H_{\delta_{\varepsilon}\ast f_{r_{n}}})g\|_{2}\rightarrow 0$ as
$r_{n}\rightarrow 1$ and $\varepsilon\rightarrow 0$ for all
$\varepsilon_{1}>0$. We have shown that $H_{\delta_{\varepsilon}\ast
f_{r_{n}}}$ converges to $H_{f}$ in the strong operator topology.
Next we show that $H_{\delta_{\varepsilon}\ast f_{r_{n}}}^{*}$ converges to
$H_{f}^{*}$ in the strong operator topology.
For any $g\in A_{\alpha}^{2}(D)$ and $\forall\varepsilon_{1}>0$, there exists
a $g_{1}\in H^{\infty}(D)$ such that $\|g-g_{1}\|_{2,\alpha}<\varepsilon_{1}$.
Similar to the previous argument,
$\displaystyle\|(H_{f}^{*}-H_{\delta_{\varepsilon}\ast
f_{r_{n}}}^{*})g\|_{2,\alpha}$ (2.5)
$\displaystyle\leq\|(H_{f}^{*}-H_{\delta_{\varepsilon}\ast
f_{r_{n}}}^{*})(g-g_{1})\|_{2,\alpha}+\|(H_{f}^{*}-H_{\delta_{\varepsilon}\ast
f_{r_{n}}}^{*})g_{1}\|_{2,\alpha}$
$\displaystyle=\|(I-P_{\alpha})^{*}(f-\delta_{\varepsilon}\ast
f_{r_{n}})^{*}(g-g_{1})\|_{2,\alpha}$
$\displaystyle+\|(I-P_{\alpha})^{*}(f-\delta_{\varepsilon}\ast
f_{r_{n}})^{*}g_{1}\|_{,\alpha 2}$
$\displaystyle\leq\|(\overline{f-\delta_{\varepsilon}\ast
f_{r_{n}}})(g-g_{1})\|_{2,\alpha}+\|(\overline{f-\delta_{\varepsilon}\ast
f_{r_{n}}})g_{1}\|_{2,\alpha}$
$\displaystyle\leq\|\overline{f-\delta_{\varepsilon}\ast
f_{r_{n}}}\|_{2,\alpha}\|g-g_{1}\|_{2,\alpha}+\|(\overline{f-\delta_{\varepsilon}\ast
f})g_{1}\|_{2,\alpha}$ $\displaystyle+\|\overline{(\delta_{\varepsilon}\ast
f-\delta_{\varepsilon}\ast f_{r_{n}}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq
2\varepsilon_{1}\|f\|_{\infty}+\|(\overline{f-\delta_{\varepsilon}\ast
f})\|_{2,\alpha}\|g_{1}\|_{\infty}+\|(\overline{\delta_{\varepsilon}\ast
f-\delta_{\varepsilon}\ast f_{r_{n}}})g_{1}\|_{2,\alpha}$
For $g_{1}\in H^{\infty}(D)$, we have $\overline{\delta_{\varepsilon}\ast
f_{r_{n}}}(z)g_{1}(z)\rightarrow\overline{\delta_{\varepsilon}\ast
f}(z)g_{1}(z)$ pointwise and $|\overline{\delta_{\varepsilon}\ast
f_{r_{n}}}(z)g_{1}(z)|\leq\|f\|_{\infty}|g_{1}(z)|$. By the dominated
convergence theorem, we have $\|(\overline{f-\delta_{\varepsilon}\ast
f_{r_{n}}})g_{1}\|_{2,\alpha}\rightarrow 0$ as $r_{n}\rightarrow 1$. Using the
fact that $\|\overline{f-\delta_{\varepsilon}\ast f}\|_{2,\alpha}\rightarrow
0$ as $\varepsilon\rightarrow 0$, and for all $\varepsilon_{1}>0$, we see that
the last line in (2.5) goes to 0.
From Theorem 1.5, there exist two sequences $\\{a_{n}\\}_{n\geq 1}$ and
$\\{b_{n}\\}_{n\geq 1}$ of non-negative real numbers such that $\sum_{n\geq
1}a_{n}=\sum_{n\geq 1}b_{n}=1$. Let
$\varphi_{1}=\sum^{\infty}_{n=1}a_{n}\psi_{n}$ and
$\varphi_{2}=\sum^{\infty}_{n=1}b_{n}\psi_{n}$ where
$\psi_{n}=\delta_{\varepsilon}\ast f_{r_{n}}$. Since each $\psi_{n}$ is
continuous on $\overline{D}$ and $\psi_{n}|_{\partial D}=0$, $\varphi_{1}$ and
$\varphi_{2}$ are continuous on $\overline{D}$ and equal to zero on $\partial
D$. This implies $H_{\psi_{n}}$ is compact for $n\geq 1$. From the formula,
$H_{\varphi_{1}}=H_{\sum^{\infty}_{n=1}a_{n}\psi_{n}}=\sum^{\infty}_{n=1}a_{n}H_{\psi_{n}}$
it follows that $H_{\varphi_{1}}$ is compact; similarly for $H_{\varphi_{2}}$.
Theorem 1.5 guarantees that $H_{\varphi_{1}}\neq H_{\varphi_{2}}$.
The two distinct compact Hankel operators $H_{\varphi_{1}}$ and
$H_{\varphi_{2}}$ satisfy
$\|H_{f}-H_{\varphi_{1}}\|=\|H_{f}-H_{\varphi_{2}}\|=\|H_{f}\|_{e}$. Let
$\varphi=s\varphi_{1}+(1-s)\varphi_{2}$, for $s\in(0,1)$. Hence there exist
infinitely many compact Hankel operators $H_{\varphi}$ such that
$\|H_{f}-H_{\varphi}\|=\|H_{f}\|_{e}$. This finishes the proof of Theorem 1.2.
Proof of Theorem 1.3. We show $T_{\delta_{\varepsilon}\ast f_{r_{n}}}$
converges to $T_{f}$ in the strong operator topology.
For any $g\in A^{2}_{\alpha}(D)$, and $\varepsilon_{1}>0$, there exists a
$g_{1}\in H^{\infty}(D)$ such that $\|g-g_{1}\|_{2,\alpha}<\varepsilon_{1}$.
Then using the Holder inequality, $\|\delta_{\varepsilon}\ast
f_{r_{n}}\|_{2,\alpha}\leq\|f\|_{\infty}$ and the boundedness of the
orthogonal projection $P_{\alpha}$ on the weighted Bergman space, we have
$\displaystyle\|(T_{f}-T_{\delta_{\varepsilon}\ast f_{r_{n}}})g\|_{2,\alpha}$
(2.6) $\displaystyle\leq\|(T_{f}-T_{\delta_{\varepsilon}\ast
f_{r_{n}}})(g-g_{1})\|_{2,\alpha}+\|(T_{f}-T_{\delta_{\varepsilon}\ast
f_{r_{n}}})g_{1}\|_{2,\alpha}$
$\displaystyle=\|P_{\alpha}(f-\delta_{\varepsilon}\ast
f_{r_{n}})(g-g_{1})\|_{2,\alpha}+\|P_{\alpha}(f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq\|f-\delta_{\varepsilon}\ast
f_{r_{n}}\|_{2,\alpha}\|(g-g_{1})\|_{2,\alpha}+\|(f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq
2\varepsilon_{1}\|f\|_{\infty}+\|(f-\delta_{\varepsilon}\ast
f)g_{1}\|_{2,\alpha}+\|(\delta_{\varepsilon}\ast f-\delta_{\varepsilon}\ast
f_{r_{n}})g_{1}\|_{2,\alpha}$ $\displaystyle\leq
2\varepsilon_{1}\|f\|_{\infty}+\|(f-\delta_{\varepsilon}\ast
f)\|_{2,\alpha}\|g_{1}\|_{\infty}+\|(\delta_{\varepsilon}\ast
f-\delta_{\varepsilon}\ast f_{r_{n}})g_{1}\|_{2,\alpha}$
Similar to the argument of theorem 1.2, it can be seen that the last line of
equation (2.6) goes to 0. Hence $\|(T_{f}-T_{\delta_{\varepsilon}\ast
f_{r_{n}}})g\|_{2,\alpha}\rightarrow 0$ as $r_{n}\rightarrow 1$,
$\varepsilon\rightarrow 0$ and for all $\varepsilon_{1}>0$. This finishes the
demonstration that $T_{\delta_{\varepsilon}\ast f_{r_{n}}}\rightarrow T_{f}$
in the strong operator topology.
The reader may show that $T_{\delta_{\varepsilon}\ast
f_{r_{n}}}^{*}\rightarrow T_{f}^{*}$ in the strong operator topology by an
argument like the ones previously given.
From Theorem 1.5, there are two sequences $\\{a_{n}\\}_{n\geq 1}$ and
$\\{b_{n}\\}_{n\geq 1}$ of non-negative real numbers such that $\sum_{n\geq
1}a_{n}=\sum_{n\geq 1}b_{n}=1$. Set
$\varphi_{1}=\sum^{\infty}_{n=1}a_{n}\psi_{n}$ and
$\varphi_{2}=\sum^{\infty}_{n=1}b_{n}\psi_{n}$. Since every $\psi_{n}\in
C_{\partial}(\overline{D})$, we have $\varphi_{1},\varphi_{2}\in
C_{\partial}(\overline{D})$.
Hence $T_{\varphi_{1}}$ and $T_{\varphi_{2}}$ are compact and they satisfy
$\|T_{f}-T_{\varphi_{1}}\|=\|T_{f}-T_{\varphi_{2}}\|=\|T_{g}\|_{e}$. Theorem
1.5 guarantees $T_{\varphi_{1}}\neq T_{\varphi_{2}}$. By setting
$\varphi=s\varphi_{1}+(1-s)\varphi_{2}$, for $s\in(0,1)$, we produce
infinitely many compact Toeplitz operators $T_{\varphi}$ such that
$\|T_{f}-T_{\varphi}\|=\|T_{f}\|_{e}$.
Proof of Theorem 1.4. Let $\hbar(D)$ denote the collection of all bounded
harmonic functions on the unit disk $D$. For $f\in\hbar(D)$, we’ll show that
there exists a sequence $\\{f_{n}\\}$ of functions harmonic on $D$ and
continuous on $\overline{D}$ such that the compact Hankel operators
$H_{f_{n}}$($H_{f_{n}}^{*}$) converge to $H_{f}$($H_{f}^{*}$) in the strong
operator topology.
For $f\in\hbar(D)$ and $z\in D$, let
$f_{n}(z)=f(r_{n}z)$
where $0<r_{n}<1$ and $r_{n}\rightarrow 1$ as $n\rightarrow\infty$. We have
$\triangle f_{n}(z)=r_{n}^{2}\triangle f(r_{n}z)=0$ where $\triangle$ is the
Laplace operator; thus $f_{n}$ is harmonic on $D$ and continuous on
$\overline{D}$, and by [16, Sec 8.4], $H_{f_{n}}$ is compact on the weighted
Bergman space. It’s not hard to see that $f_{n}(z)$ converges to $f(z)$
pointwise as $n\rightarrow\infty$. By the maximum modulus principle, we have
$\|f_{n}\|_{\infty}\leq\|f\|_{\infty}$. For $g\in A^{2}_{\alpha}(D)$,
$|f_{n}(z)g(z)|\leq\|f\|_{\infty}|g(z)|$ and $f_{n}(z)g(z)$ pointwise
converges to $f(z)g(z)$. We can now apply the dominated convergence theorem to
see that
$\lim_{n\rightarrow\infty}\int_{D}|(f_{n}(z)-f(z))g(z)|^{2}dA_{\alpha}(z)=0.$
This yields
$\displaystyle\|(H_{f_{n}}-H_{f})g\|_{2,\alpha}^{2}=\|(I-P)(f_{n}-f)g\|_{2,\alpha}^{2}$
(2.7)
$\displaystyle\leq\|(f_{n}-f)g\|^{2}_{2,\alpha}=\int_{D}|(f_{n}(w)-f(w))g(w)|^{2}dA_{\alpha}(w)\rightarrow
0$
which implies $H_{f_{n}}\rightarrow H_{f}$ in the strong operator topology.
Similarly, $H_{f_{n}}^{*}\rightarrow H_{f}^{*}$ in the strong operator
topology.
Let $\\{a_{n}\\}_{n\geq 1}$ and $\\{b_{n}\\}_{n\geq 1}$ be two non-negative
real valued sequences such that $\sum_{n\geq 1}a_{n}=\sum_{n\geq 1}b_{n}=1$,
and set $\varphi_{1}=\sum^{\infty}_{n=1}a_{n}f_{n}$ and
$\varphi_{2}=\sum^{\infty}_{n=1}b_{n}f_{n}$. Since each $f_{n}$ is harmonic on
$D$ and continuous on $\overline{D}$, we have $\varphi_{1}$ and $\varphi_{2}$
are harmonic on $D$ and continuous on $\overline{D}$; thus, $H_{\varphi_{1}}$
and $H_{\varphi_{2}}$ are compact operators which satisfy
$\|H_{f}-H_{\varphi_{1}}\|=\|H_{f}-H_{\varphi_{2}}\|=\|H_{f}\|_{e}$, and are
distinct by Theorem 1.5. The proof can be finished in the now standard
fashion.
$\mathbf{Acknowledgement}$: I would like to thank Prof. Dechao Zheng for
introducing the subject of Hankel and Toplitz operators to me, and for his
guidance in all aspects of this project. This research was conducted while I
was a visiting student at Vanderbilt University. I would like to thank
Vanderbilt University for their hospitality.
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* [12] D. Sarason, Spaces of analytic functions (Sem. Functional Anal. Function Theory, Kristiansand, 1975), pp. 117–130, Lecture Notes in Math., Vol. 512, Springer, Berlin, 1976.
* [13] Karel Stroethoff , Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in ${\bf C}^{n}$, J. Operator Theory 23 (1990), no. 1, 153 C170.
* [14] Karel Stroethoff and Dechao Zheng, Algebraic and spectral properties of dual Toeplitz operators, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2495 C2520
* [15] Dechao Zheng, Toeplitz operators and Hankel operators, Integral equations and operator theory 12 (1989), 280–299.
* [16] Kehe Zhu, Operator Theory in Function Spaces, Second edition, Mathematical Surveys and Monographs 138(2007).
* [17] Kehe Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, T.A.M.S. V302. N2. August 1987.
Fengying Li e-mail: lify0308@163.com
Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064, China
|
arxiv-papers
| 2012-10-17T16:58:32 |
2024-09-04T02:49:36.750782
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fengying Li",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1210.4793"
}
|
1210.5079
|
# Efficient Raman Frequency Conversion by Feedbacks of Pump and Stokes Fields
Bing Chen1, Kai Zhang1, Chun-Hua Yuan1, Chengling Bian1,
Cheng Qiu1, L. Q. Chen1,a) 00footnotetext: a) Electronic mail:
lqchen@phy.ecnu.edu.cn., Z. Y. Ou1,2 , and Weiping Zhang1 1Quantum Institute
for Light and Atoms, State Key Laboratory of Precision Spectroscopy,
Department of Physics, East China Normal University, Shanghai 200062, P. R.
China
2Department of Physics, Indiana University-Purdue University Indianapolis, 402
North Blackford Street, Indianapolis, Indiana 46202, USA
###### Abstract
We experimentally demonstrate efficient Raman conversion to respective Stokes
and anti-Stokes fields in both pulsed and continuous modes with a Rb-87 atomic
vapor cell. The conversion efficiency is about 40-50% for the Stokes field and
20-30% for the anti-Stokes field, respectively. This conversion process is
realized with feedback of both the Raman pump and the frequency-converted
fields (Stokes or anti-Stokes). The experimental setup is very simple and can
be applied easily to produce the light source with larger frequency difference
using other Raman media. They may have wide applications in nonlinear optics,
atomic physics, quantum optics and precise measurement.
###### pacs:
42.65.Ky,42.65.Dr,42.65.-k,42.50.-p
Efficient nonlinear interaction and frequency conversion at low light
intensity is of interest in many areas of nonlinear and quantum optics because
of its potential applications to high-precision spectroscopy and quantum
information processing and storage. However, efficient conversion is almost
always required to have high-power pumping because nonlinear coefficients are
usually small in a nonlinear medium. In particular for Raman scattering, the
conversion efficiency from the Raman pump field to the Stokes field is quite
low. Traditionally, people could increase the conversion efficient by high-
finesse optical cavity kuh ; bra ; mat or stimulated Raman process ray ; mis
. But these methods are complicated to some degree.
On the other hand, in the past two decades, it was discovered that nonlinear
conversion can be greatly enhanced in coherent atomic ensembles. One approach
is to prepare atomic spin wave before the Raman conversion process, the atomic
spin wave acts as a seed to the Raman amplification process for enhanced Raman
conversion. Jain et al jai and Merriam et al mer achieved high frequency
conversion efficiencies with the help of an atomic coherence prepared via
electromagnetically induced transparency boi ; harris . The conversion
efficiency has reached near 40% when the Raman write lasers have an intensity
as high as several MW/cm2. Recently, we demonstrated a high Raman conversion
of 40% with a low pump field intensity of 0.1 W/cm2. This is achieved by first
preparing a spatially distributed atomic spin wave in Rb-87 vapor with another
Raman laser chen ; yuan . Nonlinear conversion efficiency can be enhanced with
coherent medium prepared by counter-propagating fields and efficient intrinsic
feedback fle ; zib ; berre ; berre2 . Zibrov et al zib observed a 4%
conversion efficiency with laser power of 300 $\mu$W and a spot size of 0.3mm.
However, these schemes need other fields to prepare the atomic spin waves and
can only operate in pulse mode.
In this paper, we experimentally demonstrate a simple and efficient Raman
conversion scheme with coherent feedback. After the first Raman process in the
forward direction, we reflect back both the original Raman pump and the
forward generated fields to the atomic medium. We obtain a conversion
efficiency as high as 50% for the Stokes field and 30% for the anti-Stokes
field with pump field power as low as 200$\mu$W. The scheme works in both
pulsed and continuous wave (CW) modes with the same conversion efficiency. By
beating two Stokes fields generated from a common write field, we observe a
narrow line width of 10 kHz, which is determined by the decoherence time of
the atomic spin wave in the medium.
Figure 1: (color online) (a) experimental arrangement. PBS: polarization beam
splitter; $CF_{i,j}$: frequency-converted fields, $i=F,B$ ($F$: forward
direction; $B$: backward direction), and $j=S,AS$ ($S$: Stokes field; $AS$:
anti-Stokes field); OP: optical pumping laser. (b) and (c) Energy levels of
87Rb for Stokes generation (b) and anti-Stokes generation (c); $g$ and $m$
states are the hyperfine split ground states $|5^{2}S_{1/2},F=1,2\rangle$;
$|e_{1}\rangle=|5^{2}P_{1/2},F=2\rangle$ and
$|e_{2}\rangle=|5^{2}P_{3/2},F=3\rangle$. (d) Timing sequence.
The experimental setup is shown in the Fig.1(a). The protocol is based on the
feedback Raman process in a pure 87Rb atomic ensemble. The 87Rb atoms are
contained in a 50mm long glass cell with paraffin coating. The cell is placed
inside a four-layer $\mu$-magnetic shielding to reduce stray magnetic fields
and is heated up to 70oC using a bifilar resistive heater. The energy levels
of 87Rb atom are given in Figs.1(b)-(c) together with laser frequencies. The
lower two energy states $|g\rangle$ and $|m\rangle$ are the hyperfine split
ground states $5S_{1/2}$ $(F=1,2)$ with a frequency difference of 6.87GHz and
the two higher energy states $|e_{1}\rangle$ and $|e_{2}\rangle$ are the
excited states ($5P_{1/2},{5P_{3/2}}$). An optical pumping field (OP) is used
to prepare the atoms in either $|g\rangle$ or $|m\rangle$ state. $W$ is the
Raman pump field with a diameter of 1.0 mm. Fig.1(b) is for Stokes generation
while Fig.1(c) is for anti-Stokes generation. If we tune all laser frequency
($W$ and OP) as shown in Fig.1(b), $W$ field couples the states
$|e_{1}\rangle$ and $|g\rangle$, the frequency of the generated converted
field is equal to the frequency of $W$ minus 6.87GHz, corresponding to Stokes
field generation. If we tune the laser frequency as in Fig.1(c), the frequency
of the converted field is equal to the frequency of $W$ plus 6.87GHz,
corresponding to anti-Stokes field generation. A mirror is placed behind the
atomic cell to reflect both $W$ and the generated fields back into the atomic
cell for subsequent efficient conversion. This arrangement of the mirror is a
crucial part in the setup. The generated fields are separated from $W$ field
by a polarization beam splitter (PBS1) because their polarizations are
orthogonal to each other. Photo-detector D1 is used to measure the generated
fields.
Figure 2: (color online) (a) and (c) The temporal behavior of the converted
field when $W$ field is in (a) pulsed mode and (c) CW mode; the inset is the
frequency analysis of the converted field by a FP cavity. (b) and (d)
Conversion efficiency from $W$ to the generated fields in (b) pulsed mode and
(d) CW mode; black solid square is for Stokes field and red hollow square is
for anti-Stokes field. Figure 3: (color online) (a) Intensity of the converted
field as the frequency of $W$ field is scanned; the left red curve is for
anti-Stokes and the right black curve is for Stokes. (b) Absorption spectrum
of Rb (85 and 87) for frequency calibration in (a).
Firstly, we perform the experiment in pulsed mode with a timing sequence shown
in Fig.1(d). W and OP lasers are chopped into pulse by acoustic-optic
modulators (AOM). The OP pulse lasts 200$\mu s$ to prepare all atoms in the
ground state $|g\rangle$ or the state $|m\rangle$. Then the $W$ laser turns on
for 1000 $\mu s$ and interacts with the atomic ensemble to generate the Stokes
or anti-Stokes light. The temporal behavior of the converted fields is shown
in Fig. 2(a). The intensity peaks quickly and decreases with the time due to
the decay of the atomic coherence. The coherence time of the paraffin cell is
500$\mu s$. The inset in Fig.2(a) is the frequency analysis of the generated
field by a Fabry-Perot cavity (FP). Almost all part are the generated field
with a small leaked $W$ also shown. We measure the conversion efficiency from
$W$ laser to the generated field and the results are given in Fig.2(b). The
efficiency ranges around 40-50% for Stokes and 20-30% for anti-Stokes,
depending on the power of $W$.
Next, we perform the experiment in continuous wave (CW) mode by applying
continuously the OP field and $W$ field. A steady frequency-converted field is
generated as shown in Fig.2(c). The conversion efficiency is almost the same
as the pulsed case, as shown in Fig.2(d) where we plot the efficiency as a
function of the power of $W$. In the CW mode, we can check the tuning range of
the generated field by scanning the frequency of $W$. The result is shown in
Fig.3 together with the absorption spectrum of 87Rb for frequency calibration.
The right black and left red curves are for the Stokes and anti-Stokes fields,
respectively. The red and black curves each consist of three peaks, which
match well the Raman gain profile. The two large side peaks correspond to blue
and red detuned Raman process, respectively. The small middle peak is due to
the crossover of the two hyperfine lines of $5^{2}S_{1/2},F\rightarrow
5^{2}P_{1/2},F^{\prime}=1,2$ transitions. The frequency difference between
$5^{2}P_{1/2},F^{\prime}=1,2$ energy levels is 800MHz, while the Doppler
broadening at cell temperature of 70 degree is about 600-700MHz. From this
figure, we obtain a tuning range of 3.0 and 4.0GHz for anti-Stokes and Stokes,
respectively.
In the CW mode, we are able to look at the coherence property of the converted
field. To do this, we split $W$ into two beams and convert each beam to Stokes
field. We then superimpose the two generated fields for interference. AC Stark
effect leads to a slight difference between the frequencies of the two
generated fields because of the difference in power and geometry in the
interaction of the two beams with atoms. So we observe a beat signal shown in
the inset of Fig.4. Fourier transformation of the beat signal is recorded by a
spectrum analyzer and shown in Fig.4. The line width of the beat signal is
about 10kHz, corresponding to a coherence time of 500 $\mu s$. This is in the
same order as the decoherence time of atoms in a paraffin-coated cell.
Figure 4: (color online) Demonstration of coherence of the generated field:
beating signal (inset) and its Fourier transformation between two similarly
generated fields. Figure 5: (color online) experimental sketch of Raman
conversion process without feedback. The conversion efficiency in (b) pulsed
mode and (c) CW mode.
Finally, to show the enhancement effect of the feedback, we add a PBS between
the flat mirror and the cell to separate the pump field $W$ and the forward
generated field. We reflect back only the pump field $W$. The experimental
arrangement is shown in Fig.5(a). The efficiency is given in Fig.5(b) and (c)
for the pulsed and CW cases, respectively. The efficiency is around a few
percent, an order of magnitude smaller than the scheme with feedback of the
forward generated field. This clearly demonstrates the advantage of the scheme
with feedback.
Let us understand the feedback mechanism here. When the pump field $W$ in the
forward direction interacts with the atomic ensemble in the ground state by
Raman scattering, a converted field in forward direction and an atomic spin
wave are generated. The atomic spin wave stays in the cell. When the forward
pump and the converted fields are both reflected back to the cell by the flat
mirror, the Raman process by the pump field in the backward direction will be
stimulated by the reflected forward converted field and enhanced by the
previously produced atomic spin wave at the same time. An interference effect
occurs between backward converted fields produced by the two mechanisms
because of the phases correlation between the forward converted field and the
atomic spin wave mis ; bian . The co-propagation of the reflected fields will
lead to in-phase constructive interference and thus enhanced conversion
efficiency.
In conclusion, we have demonstrated an efficient way to make Raman conversion
with feed-back. The conversion efficiency is about 40% for Stokes field and
20% for anti-Stokes field with as little as a few hundreds of $\mu$W of Raman
pump. Such a scheme can replace traditional techniques such as EOM and AOM to
obtain a large frequency shift for studying light interaction with atoms such
as the EIT effect for manipulation of atomic spin waves harris and Raman
atomic interferometer chu .
This work was supported by the National Basic Research Program of China (973
Program grant no. 2011CB921604), the National Natural Science Foundation of
China (grant numbers 11004058, 11004059, 11129402, J1030309, 11274118,
10828408 and 10588402) and the Program of Shanghai Subject Chief Scientist
(grant number 08XD14017).
## References
* (1) A. Kuhn, M. Hennrich, and G. Rempe, Phys. Rev. Lett. 89, 067901(2002).
* (2) Simon Brattke1,2, Benjamin T. H. Varcoe1, and Herbert Walther, Phys. Rev. Lett. 86, 3534 (2001).
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* (6) Maneesh Jain, Hui Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, Phys. Rev. Lett. 77, 4326 (1996).
* (7) A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G.Y. Yin, and S. E. Harris, Phys. Rev. Lett. 84, 5308 (2000).
* (8) K. J. Boiler, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
* (9) S. E. Harris, Phys. Today 50, 36 (1997).
* (10) L. Q. Chen, G. W. Zhang, C. H. Yuan, J. Jing, Z. Y. Ou, and W. P. Zhang, Applied Physics Letters 95, 041115 (2009).
* (11) C. H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. P. Zhang, Phys. Rev. A 82, 013817 (2010).
* (12) M. Fleischhauer, M. D. Lukin, A. B. Matsko, and M. O. Scully, Phys. Rev. Lett. 84, 3558 (2000).
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* (14) M. L. Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 43, 6345 (1991).
* (15) M. L. Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 44, 5958 (1991).
* (16) C. L. Bian, L. Q. Chen, G. W. Zhang, Z. Y. Ou and W. P. Zhang, EPL, 97, 34005 (2012).
* (17) M. Kasevich and S. Chu, Phys. Rev. Lett. 67, 181 (1991).
|
arxiv-papers
| 2012-10-18T10:11:24 |
2024-09-04T02:49:36.776631
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bing Chen (1), Kai Zhang (1), Chun-Hua Yuan (1), Chengling Bian (1),\n Cheng Qiu (1), L. Q. Chen (1), Z. Y. Ou (1 and 2), and Weiping Zhang (1) ((1)\n Quantum Institute for Light and Atoms, State Key Laboratory of Precision\n Spectroscopy, P. R. China (2) Department of Physics, Indiana\n University-Purdue University Indianapolis, Indianapolis)",
"submitter": "Liqing Chen",
"url": "https://arxiv.org/abs/1210.5079"
}
|
1210.5253
|
# Design and Simulation of Molecular Nonvolatile Single-Electron Resistive
Switches
Nikita Simonian simonian@grad.physics.sunysb.edu Konstantin K. Likharev
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794 Andreas Mayr Department of Chemistry, Stony Brook University, Stony
Brook, NY 11794
###### Abstract
We have carried out a preliminary design and simulation of a single-electron
resistive switch based on a system of two linear, parallel, electrostatically-
coupled molecules: one implementing a single-electron transistor and another
serving as a single-electron trap. To verify our design, we have performed a
theoretical analysis of this “memristive” device, based on a combination of
ab-initio calculations of the electronic structures of the molecules and the
general theory of single-electron tunneling in systems with discrete energy
spectra. Our results show that such molecular assemblies, with a length below
10 nm and a footprint area of about 5 nm2, may combine sub-second switching
times with multi-year retention times and high ($>10^{3}$) ON/OFF current
ratios, at room temperature. Moreover, Monte Carlo simulations of self-
assembled monolayers (SAM) based on such molecular assemblies have shown that
such monolayers may also be used as resistive switches, with comparable
characteristics and, in addition, be highly tolerant to defects and stray
offset charges.
## I Introduction
Recently, a substantial progress was made in the fabrication of two-terminal
“memristive” devices (including bistable “resistive” or “latching” switches)
based on metal oxide thin-films, whose bistability is apparently based on the
reversible formation/dissolution of conducting filaments — see, e.g., recent
reviews likharev08 ; waser09 ; StrukovKohlstedt12 . However, scaling of
resistive memories and hybrid CMOS/nano-crossbar integrated circuits
likharev08 , based on such switches, beyond the 10 nm frontier may still
require more reproducible devices based on other physical principles. One
possibility here is to use a molecular version of single-electron switches
folling01 .
Such a switch, schematically shown in Fig. 1a, is a combination of two
electrostatically-coupled devices: a “single-electron trap” and a “single-
electron transistor” likharev99 placed in parallel between two electrodes.
(It will be convenient for us to call these electrodes the “drain/control” and
the “source” — see Fig. 1a.) When the charge state of the trap island is
electroneutral ($Q=0$), the Coulomb blockade threshold $V_{C}$ of the
transistor is large (Fig. 1b), so that at all applied voltages with
$|V|<V_{C}$ the transistor carries virtually no current — the so-called OFF
state of the switch. As soon as the voltage exceeds a certain threshold value
$V_{\leftarrow}<V_{C}$, the rate of tunneling into the single-electron trap
island increases sharply (Fig. 1c), and an additional elementary charge $q$
(either a hole or an electron) enters the trap island from the source
electrode, charging it to $Q=q=\pm e$. The electrostatic field of this charge
shifts the background electrostatic potential of transistor’s island and as a
result reduces the Coulomb blockade threshold of the transistor to a lower
value $V_{C}^{\prime}$. This is the ON state of the switch, with a substantial
average current flowing through the transistor at $V>V_{C}^{\prime}$. The
device may be switched back into the OFF state by applying a reverse voltage
in excess of the trap-discharging threshold $|V_{\rightarrow}|$. As was
experimentally demonstrated for metallic, low-temperature prototypes of the
single-electron switch dresselhaus94 , its retention time may be very long
111A resistive switch with a sufficiently long (a-few-year) retention time at
$V=0$ may be classified as a nonvolatile memory cell. . However, for that the
scale $e^{2}/2C$ of the single-electron charging energy of the trap island,
with effective capacitance $C$, has to be much higher than the scale of
thermal fluctuations, $k_{\mbox{\scriptsize B}}T$. For room temperature, this
means the need for few-nm-sized islands likharev99 , and so far the only way
of reproducible fabrication of features so small has been the chemical
synthesis of suitable molecules — see, e.g., tour03 ; cuniberti05 .
Transport properties of single molecules, captured between two metallic
electrodes, have been repeatedly studied by several research groups for more
than a decade — see, e.g., BummArnoldCygan96 ; Reed97 ;
KerguerisBourgoinPalacin99 ; ParkPasupathyGoldsmith02 ; KubatkinDanilovHjort03
; DanilovKubatkinKafanov08 ; WangBatsanovBryce09 . The practical use of such
devices in VLSI circuits is still impeded by the unacceptably low yield of
their fabrication and large device-to-device variability. The main reason of
this problem is apparently the lack of atomic-scale control of the contacts
between the molecules and the metallic electrodes. In addition, in three-
terminal single-molecule devices that can work as transistors
ParkPasupathyGoldsmith02 ; KubatkinDanilovHjort03 , there is an additional
huge challenge of reproducible patterning of three-electrode geometries with
the required sub-nm precision. These two challenges make single-molecule
three-terminal stand-alone devices rather unlikely candidates for post-CMOS
VLSI circuit technology. However, we believe that for resistive memories and
CMOS/nano hybrids based on nano-crossbars (Fig. 1d likharev08 ), these
challenges may be met. Indeed, such crossbars use two-terminal crosspoint
devices, so that their only critical dimension (the distance between the two
electrodes) may be precisely controlled by layer thickness. In addition, if
such devices are based on self-assembled monolayers (SAMs), the large number
$N$ of molecules in a single device may mitigate the negative effects of
interfacial and other uncertainties of single molecules akkerman07 . (Since
the electrode footprint of a quasi-linear functional molecule stretched
between the electrodes may be very small, $N$ may be as high as $\sim 10^{2}$
even for sub-10-nm-scale devices.)
The goal of this paper is to describe the results of the design and ab-initio
calculations of basic properties of a molecular resistive switch and SAMs
based on such molecules. Our basic design is described, and its physics is
discussed in the next section (Sec. II). In Sec. III we formulate a
theoretical model which allows an approximate but reasonably accurate
numerical simulation of electron transport properties of this device. The
results of simulation of our most promising switch version are described in
Sec. IV. In Sec. V we describe our approach to simulation of SAM layers
consisting of resistive switch assemblies, and the results of these
simulations. Finally, in Sec. VI we summarize our results and discuss the
necessary further work towards the practical implementation of reproducible
resistive switches.
Figure 1: (a) The traditional version of the single-electron resistive switch,
(b) its $I-V$ curve (schematically), and (c) the ON/OFF switching rates of the
device, calculated using the orthodox theory of single-electron tunneling
averin91 , for $e^{2}/C=20k_{\mbox{\scriptsize B}}T$. The inset in panel (c)
schematically shows the switching between charge states of the trap resulting
from repeated voltage sweeps with a rate
$\Gamma_{0}=\left|dV/dt\right|/(e/C)\ll\Gamma_{r}$. (d) Nano-crossbar with
resistive switches as crosspoint devices.
## II Resistive switch design
Our initial design likharev03 ; mayr07 of the molecular resistive switch was
based on oligophenyleneethynylene (OPE) chains as tunnel barriers and diimide
(namely, pyromellitdiimide, naphthalenediimide, and perylenediimide) groups as
trap and transistor islands. However, already the first quantitative
simulations have shown that the relatively narrow HOMO-LUMO bandgap of the OPE
chains (of the order of 1.5 eV ke07 ) cannot provide a tunnel barrier high
enough to ensure sufficiently long electron retention times in traps with
acceptable lengths. As a result, we have concluded that alkane chains
(CH2-CH2-…), with a bandgap of $\sim 9$ eV wangreed03 , are much better
candidates for resistive switches. There is also substantial experience in the
chemical synthesis of molecular electronic devices and SAMs using such chains
as tunnel barriers tour03 ; cuniberti05 .
Figure 2 shows a possible realization of such a device, based on benzene-
benzobisoxazole and naphthalenediimide acceptor groups (playing the role of
single-electron islands), and alkane chains. In contrast with the usual
(“orthodox”) design of the trap folling01 ; dresselhaus94 ; likharev89 , where
a long charge retention time is achieved by incorporation of several
additional single-electron islands into the trap charging path (Fig. 1a), the
long alkane chain used in the molecular trap has a band structure which
enables its use simultaneously in two roles: as a tunnel barrier as well as a
replacement of intermediate islands.
Figure 2: Our final version of a molecular resistive switch featuring an
alkane-naphthalenediimide single-electron trap electrostatically coupled to an
alkane-benzobisoxazole-benzene single-electron transistor.
In order to explain this novel approach, let us first review the role of
intermediate islands in the conventional design of the trap (Fig. 1a). If a
single-electron island is so large that the electron motion quantization
inside it is negligible, its energy spectrum, at a fixed net charge $Q$, may
be treated as a continuum. Elementary charging of the island with either an
additional electron or an additional hole raises all energies in the spectrum
by $e^{2}/2C$, where $C$ is the effective capacitance of the island likharev89
; likharev99 . As a result, the continua of the effective single-particle
energies of the system for electrons and holes are separated by an effective
energy gap $e^{2}/C$ — essentially, the “Coulomb gap” EfrosShklovskii75 . If
this gap is much larger than $k_{\mbox{\scriptsize B}}T$ at applied voltages
$V$ close to the “energy-equilibrating” voltage $V_{e}$ (see the middle panel
of Fig. 3a), it may ensure a very low rate $\Gamma_{r}$ of single-charge
tunneling in either direction and hence a sufficiently long retention time
$t_{r}=1/\Gamma_{r}$ of the trap. The energy gap may be suppressed by applying
sufficiently high voltages $V\sim e/C$ of the proper polarity, enabling fast
switching of the device into the counterpart charge state — see the left and
right panels of Fig. 3a, and also Fig. 1c.
Figure 3: Schematic single-particle energy diagrams of (a) the usual single-
electron trap shown in Fig. 1a (for the sake of simplicity, with just one
intermediate island) and (b) the molecular trap analyzed in this work (Fig.
2), each for three values of the applied voltage $V$. Occupied/unoccupied
energy levels are shown in black/green. (The dotted green/black line denotes
the energy level of the “working” orbital that is either empty or occupied
during the device operation, defining its ON/OFF state.) Horizontal arrows
show (elastic) tunneling transitions, while vertical arrows indicate inelastic
relaxation transitions within an island, a molecule, or an electrode.
In the molecular single-electron trap shown in Fig. 2, the “energy-
equilibrating” voltage $V_{e}$ aligns the Fermi energy of the source electrode
with the lowest unoccupied level of the acceptor group that is, by design,
located in the middle of the HOMO-LUMO gap of the alkane chain. As a result,
an electron from the source electrode may elastically tunnel into the group
only with a very low rate $\Gamma_{r}$ — see the middle panel in Fig. 3b. The
reciprocal process (at the same voltage) may be viewed as electron tunneling
from the highest occupied molecular orbital of the singly-negatively charged
molecule. (To simplify the terminology, in the reminder of the paper we call
this molecular orbital the “working orbital” (indexed $W$), instead of HOMO or
LUMO of the counterpart molecular ions, to make the name independent of the
charge state of the device.) The energy-balance condition of both processes is
similar, and may be expressed via the effective single-particle energy
$\varepsilon_{W}$ of the working orbital simonian07 :
$\varepsilon_{W}=\Delta E(n)\equiv E_{gr}(n)-E_{gr}(n-1),$ (1)
where $E_{gr}(n)$ is the ground-state energy of the molecular ion with $n$
electrons. (In the case of singly-negative ion we are discussing, $n=n_{0}+1$
222We use the notation in which the fundamental electric charge unit $e$ is
positive, so that the electric charge of an ion with $n$ electrons is
$Q(n)=-e(n-n_{0})$., where $n_{0}$ is the number of protons in the molecule.)
In this notation, the energy-balance (level-alignment) condition, which
defines the voltage $V_{e}$, is
$\varepsilon_{W}=W-e\gamma V_{e},$ (2)
where $W$ is the workfunction of the source electrode material, and $\gamma$
is a constant factor imposed by the geometry of the junction; $0<\gamma<1$.
(Its physical meaning is the fraction of the applied voltage, which drops
between the trapping island and the source electrode.) At the charging
threshold voltage $V_{\leftarrow}$, energy $\varepsilon_{W}$ becomes aligned
with the valence band edge of the chain, allowing for a fast charging of the
molecule — see the left panel in Fig. 3b. Similarly, as shown on the right
panel in Fig. 3b, at $V_{\rightarrow}$ this energy becomes aligned with the
conduction band edge of the chain, allowing for a fast discharging of the
molecule.
As an example, Fig. 4a shows the atomic self-interaction corrected (ASIC)
pemmaraju07 Kohn-Sham electron eigenenergy spectrum
$\varepsilon_{i}^{\mbox{\scriptsize ASIC}}(n)$ of the alkane-
naphthalenediimide molecule used as our final trap design (Fig. 2), with the
net charge $Q(n)=-e(n-n_{0})=-e$, as a function of the applied voltage $V$.
(Here $i$ is the spin-orbital index; see Sec. III below.) Point colors in Fig.
4a crudely represent the spatial localization of the orbitals: blue
corresponds to their localization at the trapping (acceptor) group, while red
marks the localization at the alkane chain’s part close to the source
electrode. Figure 4b shows the probability density of the working orbital
$\psi_{W}^{\mbox{\scriptsize ASIC}}=\psi_{n_{0}+1}^{\mbox{\scriptsize
ASIC}}(n_{0}+1)$ of the molecular trap, integrated over the directions
perpendicular to the molecule’s axis, with blue lines corresponding to the
probability density at the most negative applied voltage. At the equilibrating
voltage $V_{e}\approx 2.2$ V, the working orbital is well localized at the
acceptor group, and is isolated from the source electrode by a $\sim 4.5$-eV-
high energy barrier. However, as Fig. 4b shows, the probability density of the
orbital decays into the alkane group rather slowly, with the exponent
coefficient $\beta\approx 0.4a_{\mbox{\scriptsize B}}^{-1}$, corresponding (in
the parabolic approximation of the dispersion relation) to the effective
electron mass $m_{ef}\approx 0.1m_{0}$ 333Experiments (for a recent summary,
see, e.g., Table 1 in mcdermott09 ) give for the exponent coefficient $\beta$
a wide range $(0.26-0.53)a_{\mbox{\scriptsize B}}^{-1}$ corresponding to the
effective mass range $(0.05-0.2)m_{0}$ (assuming a rectangular, 4.5-eV-high
energy barrier). It has been suggested mcdermott09 that such a large
variation is due to a complex dispersion law inside the alkane bandgap, making
$\beta$ a strong function of the tunneling electron energy.. As a result, a
long ($\sim 5$ nm) alkane chain is needed to ensure an acceptable retention
time of the trap. (The 2-nm free-space separation between the other side of
the molecule and the control/drain electrode, shown in Fig. 2, is quite
sufficient for preventing electron escape in that direction.)
Figure 4: ASIC density-functional-theory (DFT) results (corrected for level
“freezing” at high positive and negative voltages — see Sec. III for details)
for the singly-negatively charged alkane-naphthalenediimide trap molecule. (a)
Kohn-Sham energy spectrum as a function of the applied voltage $V$, with
colors representing the spatial localization of the corresponding orbitals —
see the legend bar on the right. The vertical lines mark voltage values
$V_{\leftarrow}$, $V_{e}$, and $V_{\rightarrow}$ corresponding to the left,
middle, and right panels of Fig. 3b. The dashed lines labeled
$E_{\mbox{\scriptsize F}}(s)$ and $E_{\mbox{\scriptsize F}}(c)$ show the Fermi
levels of the source and control/drain electrodes whose workfunction was
assumed to equal 5 eV. (b) The “working” orbital’s probability density,
integrated over the directions perpendicular to molecule’s axis, for a series
of applied voltages $V$ — see the legend bar on the right of the panel.
At a sufficiently high forward/reverse bias voltage, the working orbital
energy $\varepsilon_{W}$ crosses into the conduction/valence band of the
alkane chain, so that the orbital partly hybridizes with the states localized
near the source electrode interface, described by the rise of $|\psi_{W}|^{2}$
at larger values of $z$ — see Fig. 4b. This rise enables fast electron
tunneling to/from the source electrode, i.e. a fast switching of the device to
the counterpart charge state, in a manner similar to that of the conventional
single-electron trap, as shown schematically on the left and right panels of
Fig. 3b. Thus the long molecular chain, with a sufficiently large HOMO-LUMO
gap, may indeed play the roles of both the tunnel junction and intermediate
islands of the “orthodox” single-electron trap.
For the design of the second component of the switch, the molecular single-
electron transistor, the most important challenge is to satisfy the ON and OFF
state current requirements. In particular, the ON current should not be too
large to keep the power dissipation in the circuit at a manageable level, but
simultaneously not too small, so that the device output does not vanish in the
noise of the sense amplifier (for memory applications StrukovLikharevMemory05
; strukov07 ) or the CMOS invertor (in hybrid logic circuits
StrukovLikharevLogic05 ). Also, the ON/OFF current ratio should be
sufficiently high to suppress current “sneak paths” in large crossbar arrays
franzon05 ; strukov07 . In addition, the transistor molecule should be
geometrically and chemically compatible with the trap molecule, enabling their
chemical assembly as a unimolecular device, with their single-electron island
groups sufficiently close to provide substantial electrostatic coupling.
(Without it, the charge of the trap would not provide a substantial modulation
of the transistor current.) At the same time, the molecules must not be too
close, in order to prevent a parasitic discharge of the trap via electron
cotunneling through the transistor into one of the electrodes. The chemical
compatibility strongly favors the use of similar chains as the transistor’s
tunnel junctions.
We have analyzed several alkane-chain based transistor devices with
naphthalenediimide, perylenediimide and benzobisoxazole acceptor groups as
transistor islands. However, in all these cases the long alkane chains, needed
to match the lengths of the transistor and trap molecules, make ON currents
too low. Finally, we have decided to use an unusually long ($\sim 4.3$-nm)
benzene-benzobisoxazole krausea88 island group — see Fig. 2 and Fig. 5b.
Figure 5a shows the Kohn-Sham electron eigenenergy spectrum
$\varepsilon_{i}^{\mbox{\scriptsize ASIC}}(l_{0}+1)$ of this molecule as a
function of voltage $V$ (where $l_{0}$ is the number of protons in the
transistor molecule). Blue/red colored points correspond to the orbitals
localized at the left/right alkane chain, while green color points denote the
orbitals extended over the whole acceptor group. This extension is clearly
visible in Fig. 5b, which shows the probability density of the working orbital
$\varepsilon_{W^{\prime}}^{\mbox{\scriptsize
ASIC}}=\varepsilon_{l_{0}+1}^{\mbox{\scriptsize ASIC}}(l_{0}+1)$ of the
transistor molecule. During transistor operation, the tunneling electron may
populate any of several group-localized orbitals, resembling the operation of
the usual (metallic) single-electron transistor. As a result of such island
extension, alkane chains of the transistor could be substantially shortened,
to $\sim 1.5$-nm-long C11H25, enabling low but still acceptable ON currents of
the order of 0.1 pA, even if a small (0.25-nm) vacuum gap between the alkane
chain and the source electrode is included into calculations to give a
phenomenological description of the experimentally observed current reduction
due to unknown interfacial chemistry simonian07 .
Figure 5: ASIC results for the single-negatively charged benzene-
benzobisoxazole transistor molecule. (a) Kohn-Sham energy spectrum as a
function of the applied voltage $V$, with colors representing the spatial
localization (within the junction) of the corresponding orbitals — see the
legend bar on the right. The dashed lines labelled $E_{\mbox{\scriptsize
F}}(s)$ and $E_{\mbox{\scriptsize F}}(c)$ show the Fermi levels of the source
and control/drain electrodes whose workfunction was assumed to equal 5 eV. (b)
Probability density of the working orbital, integrated over the directions
perpendicular to the molecular axis, for a series of applied voltages — see
the legend bar on the right.
## III Theoretical model and approximations
Each molecule used in our device has a discrete set of possible excited
states, and hence the electron transport is not limited to a single channel.
In order to take into account all of these channels, we have used the “quasi-
single-particle approximation” whose simplest version had been first
formulated by Averin and Korotkov for semiconductor quantum dots
AverinKorotkov90 ; averin91 and which was recently generalized simonian07 to
be more applicable to molecular structures. In this approximation, the energy
of an arbitrary state $k=\\{n,i\\}$ of the molecule equals
$E_{k}=E_{gr}(n)+\sum_{i>n}\varepsilon_{i}(n)p_{i}-\sum_{i\leq
n}\varepsilon_{i}(n)(1-p_{i}),$ (3)
where coefficients $\varepsilon_{i}(n)$ have the physical meaning of single-
particle excitation energies of an $n$-electron ion, and numbers $p_{i}$
(equal to either 0 or 1) are the single-particle energy level occupancies. The
condition of elastic tunneling, leading to a transition between states $k$ and
$k^{\prime}$, is given by the natural generalization of Eqs. (1) and (2):
$\varepsilon_{k\rightarrow k^{\prime}}=W-e\gamma V,$ (4)
where the single-electron recharging/excitation energy is now defined as
$\varepsilon_{k\rightarrow k^{\prime}}\equiv E_{k}-E_{k^{\prime}}.$ (5)
Because of the large size and complexity of the molecules used in our design,
the only practical way to calculate their electronic structure is to use a
software package (such as SIESTA soler02 ) 444Initially, we made an attempt to
use NRLMOL pederson00 which had been successfully employed in our previous
study of single-electron tunneling through smaller molecules simonian07 .
However, we have found the performance of SIESTA (with the “standard” double-
Zeta polarized basis set) for our current problem to be substantially higher,
though the results obtained from NRLMOL may be slightly more accurate., based
on the density-functional-theory (DFT) jones89 , which may provide a
reasonably accurate ground state energy $E_{gr}^{\mbox{\scriptsize DFT}}(n)$
and a single-particle spectrum $\varepsilon_{i}^{\mbox{\scriptsize DFT}}(n)$
at a fraction of the computational cost of more correct ab-initio methods.
Unfortunately, for such strongly correlated electronic systems as molecules
considered in this paper, results obtained using standard DFT software
packages 555this is valid not only for the DFT packages based on the local
spin density approximation (LSDA), such as the standard version of SIESTA.
Another popular DFT functional, the generalized gradient approximation (GGA)
PerdewBurkeErnzerhof96 , does not provide much improvement on these results
have significant self-interaction errors pedrew81 .
We believe the source of such errors is that the approximate treatment of the
exchange-correlation term in the Kohn-Sham Hamiltonian does not completely
cancel the self-interaction energy present in the “Hartree term” of the
Hamiltonian 666In contrast, in the Hartree-Fock theory the exchange energy is
exact (of course, in the usual sense of the first approximation of the
perturbation theory), and the self-interaction errors are absent pedrew81 ..
Indeed, the standard DFT approach leads to errors, in the key energies (1) and
(5), of the order of the single-electron charging energy $e^{2}/2C$, where $C$
is the effective capacitance of the island group — see Appendix A for details.
This error may be rather substantial; for example in the naphthalenediimide-
based trap molecule (Fig. 2), it is approximately equal to 1.8 eV. For this
reason, the electron affinity $E_{gr}(n_{0}+1)-E_{gr}(n_{0})$, calculated
using the LSDA DFT for the singly-negatively charged ion of the molecular
trap, is significantly (by $\sim 3.2$ eV) larger than the experimental value
of similar molecules bhozale08 ; singh06 . The LSDA energies may be readily
corrected to yield a much better agreement with experiments (see Table 1 in
Appendix A), however, it is not quite clear how such a theory may be used for
a self-consistent calculation of the corresponding working orbital
$\psi_{W}(\mathbf{r})$.
We have found that a significant improvement may be achieved by using the
recently proposed Atomic Self-Interaction Corrected DFT scheme (dubbed ASIC
pemmaraju07 ) implemented in a custom version of the SIESTA software package.
For the molecules that we have considered here, this approach gives the Kohn-
Sham energy $\varepsilon_{n_{0}+1}(n_{0}+1)$ very close to the experimental
electron affinity. However, we have found that using even this advanced
approach for our task faces two challenges.
First, the algorithm gives (at least for our molecular trap states with
$n=n_{0}+1$ and $n=n_{0}+2$ electrons) substantial deviations from the
relation $\varepsilon_{W}=\varepsilon_{n+1}(n+1)$ for $n=n_{0}$ (which has to
be satisfied in any exact theory janak78 ; pemmaraju07 ), with the ground
energy difference (1) close to the LSDA DFT results. This means that Eq. (5)
cannot be directly used with the ASIC results; instead, for the electron
transfer energy between adjacent ions $n$ and $n-1$ we have used the following
expression:
$\varepsilon_{k\rightarrow
k^{\prime}}=\varepsilon_{i^{\prime}}^{\mbox{\scriptsize ASIC}}(n).$ (6)
This relation implies that the differences
$\varepsilon_{i^{\prime}}^{\mbox{\scriptsize
ASIC}}(n)-\varepsilon_{n}^{\mbox{\scriptsize ASIC}}(n)$ describe all possible
single-particle excitations within the acceptor group, if the index
$i^{\prime}$ is restricted to orbitals localized on the group. (Other
orbitals, localized on the alkane chain are irrelevant for our calculations
since they do not contribute to the elastic tunneling between the molecular
group and the electrode.)
In order to appreciate the second problem, look at Fig. 6 which shows the
voltage-dependent Kohn-Sham spectra of the singly-negatively charged molecular
trap, calculated using the ASIC SIESTA package for $T>0$ K. Notice that above
voltage $V_{t}\approx 13$V, and below voltage $V_{t}^{\prime}\approx-7$V, the
eigenenergy spectrum is virtually “frozen”. (The LSDA SIESTA gives similar
results.). As explained in Appendix B using a simple but reasonable model
(similar to that used in Appendix A), at $V>V_{t}$ such “freezing” originates
from the spurious self-interaction of an electron whose wavefunction cloud is
gradually shifted from the top occupied orbital of the valence band of the
chain, with energy $\varepsilon_{v}$, into the initially empty group-localized
orbital with energy $\varepsilon_{W+1}$. (A similar freeze at voltages
$V<V_{t}^{\prime}$, is due to the spurious gradual transfer of the electron
wavefunction cloud from the working orbital, localized at the acceptor group,
with energy $\varepsilon_{W}$, to the lowest orbital of the conduction band of
the chain, with energy $\varepsilon_{c}$.) It is somewhat surprising that this
spurious effect (which should not be present in any consistent quantum-
mechanical approach — see Appendix B) is so strongly expressed in the ASIC
version of the SIESTA code, which was purposely designed to get rid of the
self-interaction in the first place. Being no SIESTA experts, we may only
speculate that the nature of this artifact is related to the smoothing of the
derivative discontinuity present in the ASIC method as the electron number
passes through an integer value, which is mentioned in pemmaraju07 — see also
Fig. 7 in that paper.
Figure 6: The Kohn-Sham spectra of the singly-negatively charged molecular
trap, calculated with the ASIC SIESTA at $T=10$ K. At voltages below
$V_{t}^{\prime}\approx-7$ V, the spectrum is virtually frozen due to a
spurious gradual shift of the highest-energy electron from the “working”
orbital (with energy $\varepsilon_{W}$, shown with a solid blue line)
localized on the acceptor group, to the lowest orbital (with energy
$\varepsilon_{c}$, shown with a solid red line) of the conduction band of the
alkane chain. As a result, the calculated spectrum is virtually voltage-
insensitive (“frozen”). In the voltage range $V_{t}^{\prime}<V<V_{t}\approx
13$ V, ASIC SIESTA gives apparently correct solutions, with the working
orbital $\varepsilon_{W}$ fully occupied, and the next group-localized orbital
(with energy $\varepsilon_{W+1}$, the dashed blue line) completely unoccupied.
However, at $V>V_{t}$ the package describes a similar spurious gradual shift
of the highest-energy electron from the highest level $\varepsilon_{v}$ of the
valence band of the chain to orbital $\varepsilon_{W+1}$, resulting in a
similar spectrum “freeze”. The spectrum evolution, calculated after the
(approximate) correction of this spurious “freezing” effect, is shown in Fig.
4a above.
Fortunately, there is a way to correct this error very substantially by
following the iterative process of self-consistent energy minimization within
ASIC SIESTA. Indeed, for a fixed temperature $T>0$ K (when the program
automatically populates molecular orbitals in accordance with the single-
particle Fermi-Dirac statistics) and voltages $V>V_{t}\approx 13$ V and
$V<V_{t}^{\prime}\approx-7$ V, its iterative process converges to a wrong
solution with the energy levels frozen at their $V_{t}$ and $V_{t}^{\prime}$
values, as is discussed above — see Fig. 6. However, if the temperature in
that program is fixed at $T=0$ K, its iterative process ends up in quasi-
periodic oscillations between different solutions — most of them with frozen
levels (just like in Fig. 6), but some of them with the group localized
energies like the working orbital energies $\varepsilon_{W}$,
$\varepsilon_{W+1}$ and the valence/conduction band edge energies
$\varepsilon_{v}$, $\varepsilon_{c}$ close to their expected (unfrozen)
values. (Those values were obtained by a linear extrapolation of their voltage
behavior calculated at $V_{t}^{\prime}<V<V_{t}$.) Since such a solution is
repeated almost exactly at each iterative cycle (see the vertical boxes in
Fig. 7), we believe that it is close to the correct solution expected from the
self-consistent quantum-mechanical theory — see Appendix B. These approximate
solutions were used in our calculations both above $V_{t}$ and below
$V_{t}^{\prime}$; they are illustrated in Fig. 4a, where we have substituted
the incorrect “frozen” solutions for $T>0$ K with solutions for $T=0$ K, with
$\varepsilon_{W}\approx\varepsilon_{W}^{\mbox{\scriptsize fit}}$,
$\varepsilon_{W+1}\approx\varepsilon_{W+1}^{\mbox{\scriptsize fit}}$ and
$\varepsilon_{c}\approx\varepsilon_{c}^{\mbox{\scriptsize fit}}$ at
$V<V_{t}^{\prime}$ or
$\varepsilon_{v}\approx\varepsilon_{v}^{\mbox{\scriptsize fit}}$ at $V>V_{t}$.
Let us emphasize that the approximate nature of these solutions may have
affected our calculations (we believe, rather insignificantly), only at
$V>V_{t}\approx 13$ V and $V<V_{t}^{\prime}\approx-7$ V, i.e. only the device
recharging time results, but not the most important retention time
calculations for smaller voltages — see Fig. 9 below.
Figure 7: The Kohn-Sham energy spectrum of our trap molecule, as calculated by
successive iterations within ASIC SIESTA for $T=0$ K and $V=14.9$ V, i.e.
above the threshold voltage $V_{t}\approx 13$ V. Vertical boxes mark the
apparently correct solutions with energies of the working orbital
($\varepsilon_{W}^{\mbox{\scriptsize ASIC}}$), the next group-localized
orbital ($\varepsilon_{W+1}^{\mbox{\scriptsize ASIC}}$), and the highest
orbital of the valence band of the alkane chain
($\varepsilon_{v}^{\mbox{\scriptsize ASIC}}$) all close to their respective
values $\varepsilon_{W}^{\mbox{\scriptsize fit}}$,
$\varepsilon_{W+1}^{\mbox{\scriptsize fit}}$ and
$\varepsilon_{v}^{\mbox{\scriptsize fit}}$ obtained by a linear extrapolation
of their voltage dependence calculated at $V_{t}^{\prime}<V<V_{t}$. Just like
in Figs. 4a, 5a and 6, point colors represent the spatial localization of the
corresponding orbitals. Lines are only guides for the eye.
With the electron orbitals and eigenenergies calculated, we have described
dynamics of both the trap and the transistor, just as in our first work
simonian07 , by a set of master equations for state probabilities averin91 ,
which are valid because of the incoherent character of single-electron
tunneling to/from the continuum of electronic states in metallic electrodes
likharev99 . Moreover, for the inelastic relaxation rates
$\Gamma_{\mbox{\scriptsize inel}}$ and the rates $\Gamma_{\leftarrow}$ and
$\Gamma_{\rightarrow}$ of the elastic tunneling between the molecular group
and the metallic electrodes (see arrows in Fig. 3), the following strong
inequality,
$\Gamma_{\mbox{\scriptsize inel}}\gg\Gamma_{\leftarrow},\Gamma_{\rightarrow}$
(7)
is well fulfilled. (Indeed, the rates $\Gamma_{\mbox{\scriptsize inel}}$ are
crudely of the order of $10^{12}$ 1/s in both molecules and metals. On the
other hand, our results, described in Sec. IV below, yield transistor currents
$I\sim 10^{-13}$ A, meaning that $\Gamma_{\leftarrow}$ and
$\Gamma_{\rightarrow}$ are of the order of $I/e\sim 10^{6}$ 1/s in the
transistor; the rates are even much lower than that in the trap — see Fig.
9b.) Relation (7) allows us to account only for the tunneling events starting
from thermal equilibrium, and ensures that the rates
$\Gamma_{\mbox{\scriptsize inel}}$ drop out of the calculations.
In comparison with simonian07 , one more new element of this work is the
electrostatic coupling between the trap and the transistor which features
similar but much more frequent single-charge transitions. This rate hierarchy
allows the trap to be described by averaging rates $\Gamma$ of tunneling
events in it over a time interval much longer than the average time period
between tunneling events in the transistor, but still much shorter than
$1/\Gamma$. These average rates may be calculated as
$\left<\Gamma_{\rightarrow}\right>=\sum_{l}\sigma_{n_{0}}(l)w_{l,+}(n_{0}),$
(8)
for electron tunneling from the source into the trap molecule, and
$\left<\Gamma_{\leftarrow}\right>=\sum_{l}\sigma_{n_{0}+1}(l)w_{l,-}(n_{0}+1),$
(9)
for the reciprocal event. Here $\sigma_{n}(l)$ are the conditional
probabilities of certain charge states $l$ of the transistor island provided
that the trap is in the $n$-electron charge state (with $n$ equal to either
$n_{0}$ or $n_{0}+1$), while $w_{l,\pm}(n)$ are the total rates of single
electron tunneling between the trap in its initial charge state $n$ and the
source electrode. These rates have been calculated using Eq. (11) in
simonian07 , with an extra index $l$ added to account for the transistor’s
state. The conditional probabilities $\sigma_{n}(l)$ satisfy the usual
normalization condition:
$\sum_{l}\sigma_{n}(l)=1,$ (10)
and (together with the dc current $I$ flowing through the transistor) have
been calculated as in simonian07 , by combining the master equations of
single-electronics AverinKorotkov90 ; averin91 with ab-initio calculations of
molecular orbitals and spectra, and the Bardeen formula bardeen61 for
tunneling rates.
Figure 8: A schematic view of charge densities participating in Eq. (11).
The electrostatic interaction between the two molecules is taken into account
by an iterative incorporation of the numerically calculated Coulomb potential
created by both molecules (as well as by the series of their charge images in
the metallic electrodes of the system, which we have assumed to be plane,
infinite surfaces — see Fig. 8) into the Kohn-Sham potentials. From the
elementary electrostatics, this potential may be expressed as
$\displaystyle\begin{split}\phi_{s}(\mathbf{r})=&\int\frac{\rho_{c}(\mathbf{r_{0}})}{\left|\mathbf{r}-\mathbf{r_{0}}\right|}d^{3}r_{0}\\\
&+\sum_{j\neq
0}(-1)^{j}\int\frac{\rho_{c}(\mathbf{r_{j}})+\rho_{s}(\mathbf{r_{j}})}{\left|\mathbf{r}-\mathbf{r_{j}}\right|}d^{3}r_{j}\\\
&-V\frac{z-d/2}{d},\end{split}$
$\displaystyle\mathbf{r}_{j}\equiv\mathbf{r}_{0}+\mathbf{n}_{z}\times\left\\{\begin{split}jd,&\mbox{
for $j$ even,}\\\ (j+1)d-2z,&\mbox{ for $j$ odd.}\end{split}\right.$ (11)
where $\rho_{s}(\mathbf{r}_{0})$, $\rho_{c}(\mathbf{r}_{0})$ are the total
charge distributions of the molecule under analysis and the complementary
molecule, and $\rho_{s}(\mathbf{r}_{j})$, $\rho_{c}(\mathbf{r}_{j})$ are the
corresponding charge images in the source $(j>0)$ and the drain $(j<0)$
electrodes — see Fig. 8. The first term in Eq. (11) is the potential created
by the complement molecule, the second term describes the potential of the
infinite set of charge images of both molecules in the source and
control/drain electrodes, and the third term is the potential created by the
applied source-drain voltage. At the 0-th iteration, the first two terms are
taken equal to zero. Fortunately, the iterations give rapidly converging
results, so that there was actually no need to go beyond the second iteration
— see Fig. 9.
Figure 9: Differences between the energies of the working orbital of the
molecular trap, calculated with ASIC SIESTA at the $k$-th and $(k-1)$-th
iterations of the Coulomb interaction potential, given by Eq. (11), as
functions of the applied voltage.
Another important change introduced into our calculations is that the charge
transfer rates (see Fig. 4c in simonian07 ) are calculated in a simpler way.
Namely, one of the key conditions of validity of the Bardeen formula for the
tunneling matrix elements
$T_{[s,c],i}=\frac{\hbar^{2}}{2m}\int_{S}\left(\psi_{s,c}^{\ast}\frac{\partial\psi_{i}}{\partial
z}-\psi_{i}^{\ast}\frac{\partial\psi_{s,c}}{\partial z}\right)dS$ (12)
(where $\psi_{i}$ is the molecular orbital, $\psi_{s,c}$ are the wavefunctions
of electrons located inside the source or control/drain electrodes, and $S$ is
an arbitrary surface separating the single-electron island from the
corresponding electrode) is that the result given by Eq. (12) is independent
of the position of the surface $S$. Due to electrostatic screening of the
electric field by the electrode, the Kohn-Sham potential becomes very close to
the vacuum potential at just a few Bohr radii $a_{\mbox{\scriptsize B}}$ away
from the molecule’s last atom. Therefore, if the surface $S$ is selected
inside the vacuum gap between the molecule’s end and the electrode surface,
the effect of the molecule on wavefunctions $\psi_{s,c}$ is negligible with
good accuracy (corresponding to a fraction of one order of magnitude in the
resulting current). Hence, these wavefunctions may be calculated analytically
to describe the usual exponential 1D decay into vacuum, instead of a numerical
solution of a Schrödinger equation, as it had been done in simonian07 .
On top of the ab-initio calculation scheme, we have performed the following
check of the component molecule spacing. As was mentioned in Sec. II, the
molecules have to be placed sufficiently far from each other to prevent a
parasitic discharge of the trap via the elastic cotunneling through the
transistor island, into one of the electrodes. This effect may be estimated
using the following formula stoof96 ; averin00 :
$\Gamma_{\rightarrow}^{\mbox{\scriptsize
cot}}=\frac{\Delta^{2}w_{\rightarrow}}{\hbar^{2}w_{\rightarrow}^{2}+2\Delta^{2}+4\varepsilon^{2}},$
(13)
where $\Delta$ is the matrix element of electron tunneling between the trap
and the transistor islands (that exponentially depends on the distance $d_{p}$
between the molecules, see Fig. 2), $\varepsilon$ is the difference between
eigenenergies of these states, and $w_{\rightarrow}$ is the rate of tunneling
between the transistor island and its electrodes. We have found that in order
for the cotunneling rate $\Gamma_{\rightarrow}^{\mbox{\scriptsize cot}}$ to be
below the retention rate of the trap $\Gamma_{r}$, the distance $d_{p}$ should
not be lower than $\sim 1.5$ nm. Such relatively large separation justifies
separate DFT calculations of the electronic structures of the trap and the
transistor (with the molecular geometry of each component device initially
relaxed, using LSDA SIESTA 777The geometry relaxation was done for isolated
neutral molecules only, at no applied bias voltage and without the account of
the possible image charge effects; in the relaxed geometry all force
components on the atoms are smaller than $0.05$ eV/$\AA$. To justify this
procedure, we have verified that trap charging and discharging rates,
calculated using the trap geometry relaxed in the presence of a high (8 V)
bias, do not significantly differ from the rates (shown in Fig. 10b)
calculated for its relaxation at $V=0$.), related only by their electrostatic
interaction described by Eq. (11).
## IV Simulation results for a single resistive switch
Figure 10 shows our main results for the resistive switch shown in Fig. 2, for
temperature $T=300$ K. They include the dc $I-V$ curves of the transistor,
plotted in Fig. 10a for both charge states of the trap, and the rates of
transitions between the neutral and single-negatively charged states of the
trap, with and without the account of the transistor effect on the trap
molecule (Fig. 10b). The plots show that the resulting $I-V$ curves fit our
initial specifications rather well, with a broad voltage window (from $\sim
2.0$ to $\sim 2.5$ V) for the trap state readout, a large ON/OFF current ratio
within that window (inset in Fig. 10a), and the ON current $I\sim 0.2$ pA.
Figure 10: (a) Calculated dc $I-V$ curves of the transistor for two possible
charge states of the trap molecule. The inset shows the ON/OFF current ratio
of the transistor on a semi-log scale, within the most important voltage
interval. (b) The trap switching rates, calculated with (solid lines) and
without (dashed lines) taking into account the transistor’s back action, as
functions of the applied voltage. Red dashed lines on panel (b) show the trap
switching rates calculated without the level “freezing” correction (Sec. III
and Appendix B) and without taking into the account the transistors’ back
action.
Figure 10b shows that the trap features a high retention time,
$\tau_{r}>10^{8}$s, for both charge states, within a broad voltage range,
$-2$V $<V<$ $+5$V. (It is somewhat surprising how little is the trap retention
affected by the electrostatic “shot noise” generated by fast, quasi-periodic
charging and discharging of the transistor island, which is taken into account
by our theory.) The range includes point $V=0$, so that the device may be
considered a nonvolatile memory cell.
At the same time, the device may be switched between its states relatively
quickly by applied voltages outside of this window. The price being paid for
using alkane chains with their large HOMO-LUMO gap is that the voltages
necessary for fast switching are large — they must align the valence or
conduction band of the alkane chain with the group-localized working orbital —
see Fig. 4a.
## V SAMs of resistive switches
Probably the largest problem of molecular electronics tour03 ; cuniberti05 is
the low reproducibility of interfaces between molecules and metallic
electrodes. However, recent results akkerman06 indicate that this challenge
may be met at least for self-assembled monolayers (SAMs) encapsulated using
special organic counter-electrodes. This is why we have explored properties of
SAMs consisting of square arrays of $N\times N$ resistive switches described
above — see Fig. 11. In order to increase the tolerance of the resulting SAM
devices to self-assembly defects and charged impurities, it is beneficial to
place the component molecular assemblies (Fig. 2) as close to each other as
possible, say at distances comparable to that ($\sim 1.5$ nm) between the trap
and transistor. In this case, the Coulomb interactions between the component
molecules are very substantial, and properties of the system have to be
calculated taking these interactions into account.
Figure 11: Schematic view of a $5\times 5$-switch SAM sandwiched between two
electrodes.
A system of $N\times N$ resistive switches has $2N\times 2N$ single-electron
islands and hence at least $2^{2N\times 2N}$ possible charge states, which
would require solving that many master equations for their exact description.
Even for relatively small $N$, this approach is impracticable, and virtually
the only way to explore the properties of the system is to perform its Monte
Carlo simulations likharev89 ; likharev99 . In this method a random number
generator is used twice for each state change: first, to calculate the random
time of some state change (which obeys Poissonian statistics), and second, to
calculate the charge transition type (if several transitions are possible
simultaneously). The procedure requires a prior calculation of rates of
transitions between all pairs of charge configurations which differ by one
single-electron tunneling event.
Figure 12: Effect of a single charge of a trap molecule on the electron
affinity of another molecule, located at distance $r$ without and with the
account of the electric field screening by the common metallic electrodes.
As was discussed above, the peculiarity of our particular system is that it
features two very different time scales: the first one (for our devices,
$\tau_{t}\approx e/I_{ON}\sim 10^{-4}-10^{-6}$ s) characterizes fast charge
tunneling through single-electron transistors, and the second one corresponds
to the lifetimes of trap states ($\tau_{r}=1/\Gamma\sim 10^{8}-10^{-2}$ s). In
order to gather reasonable statistics of the switching rates, our data
accumulation time, for each parameter set, corresponded to the physical times
of up to 10 s, i.e., included up to a million transition tunneling events in
the system’s transistors.
Figure 13: Trap tunnel rates as functions of the applied voltage for two
quasi-similar nearest-neighbor charge configurations shown in the insets.
As a check of the validity of the procedure, the Monte Carlo algorithm was
first applied to a single resistive switch, and it indeed gave virtually the
same result as the master equation solution. We then used the approach for a
direct simulation of SAM fragments with two and more coupled resistive
switches. As the fragment is increased beyond a $2\times 2$ switch array, even
the Monte Carlo method runs into computer limitations, because of the
exponentially growing number of the possible charge configurations. The
calculations may be very significantly sped up by using the approximation in
which each molecule’s state affects the potential of only its nearest
neighbors. This approximation has turned out to be very reasonable (Fig. 12)
and may be justified by the fact that metallic electrodes of the system
substantially screen the Coulomb potential of the charges of distant
molecules: the distance between the acceptor group centers and the electrodes,
$d/2\approx 4$ nm, is of the same order as the 3-nm distance between the
molecule and its next-next neighbors. In this nearest-neighbor approximation,
each molecule (a trap or a transistor) is still affected by 8 other molecules.
To limit the number of the charge configurations even further, we have treated
all “essentially similar” of them (having charge pairs at equal distances,
irrespective of their angular position) as identical — see Fig. 13.
Figure 14: Monte-Carlo simulated dc $I-V$ curves of a 25-switch SAM. The top
inset shows the fraction $\beta$ of single-negatively charged traps, averaged
over 40 sweeps of applied voltage between -8V and 13V. The bottom inset shows
the ON/OFF current ratio averaged over the voltage sweeps, and its maximum
sweep-to-sweep spread.
Figure 14 shows the results of calculations, based on this approach, for a
$5\times 5$-switch SAM, of the total area close to $10\times 10$ nm2. The
switching and state readout properties are very comparable with those of a
single switch (Fig. 10), despite a significant mutual repulsion between single
electrons charging neighboring traps. In order to better understand why this
repulsion does not have adverse effects on the operation of the SAM as a
whole, we have calculated the correlation coefficients of charging of two
molecules in the SAM as a function of the distance between them. At voltages
above the transistor Coulomb blockade, transistor molecules switch their
charge state fast and the correlation coefficient $K(r)$ between two
transistor molecules may be calculated directly from their time evolution
records at a constant $V$. On the other hand, trap molecules have quasi-
stationary charge states, so that the correlation between two trap molecules
has to be calculated from a set of snapshots of their charge states (at some
voltage of interest) taken at repeated, slow sweeping of the applied voltage
throughout the whole voltage range.
Figure 15: The average correlation between two traps (green) and two
transistors (blue) as a function of distance between them in a $5\times 5$
device SAM.
Figure 15 shows the resulting average correlation between molecules (and its
fluctuations) as a function of the distance between them in the $5\times
5$-switch SAM. The charge states of neighboring traps are significantly
anticorrelated, while the next-next neighbor charge states are positively
correlated. This means that the switching is due to a nearly-simultaneous
entry of electrons into roughly every other trap 888there is virtually no
correlation between the transistor molecules, just as with their
autocorrelation in time korotkov94 , because at least two transition channels
are open at any time.. This explains why in the top inset in Fig. 14 the
average fraction of charged traps is close to 1/2. Thus the only adverse
effect of the Coulomb interaction between individual resistive switches is the
approximately two-fold reduction of the average ON current per device. Figure
16 presents a summary characterization of the SAM operation as a function of
its size (and hence its area).
Figure 16: Summary of Monte Carlo simulations of SAMs of various area: (a) the
average ON currents at voltages providing certain ON/OFF ratios; (b) the
average fraction of negatively charged traps at the equilibrating voltage
$V_{e}$.
The fact that even the fractional charging of traps in SAMs is sufficient for
a very good modulation of their net current suggests that these devices should
have a high tolerance to defects and stray electric charges likharev99 . In
order to verify this, we have carried out a preliminary evaluation of the
defect tolerance by artificially fixing charge states of certain, randomly
selected component molecules. The results, shown in Fig. 17, are rather
encouraging, implying that the switches may provide the ON/OFF current ratios
above 100 at defective switch fractions up to $\sim 10$%, and at a comparable
concentration of random offset charges.
Figure 17: Defect tolerance of the $5\times 5$ SAM switch: ON current as a
function of a number of molecules held artificially in a fixed, random charge
state, at random locations, at the applied voltage values necessary to ensure
a certain level of the ON/OFF current ratio. Error bars show the r.m.s. spread
of results.
## VI Conclusion
Despite the problems with the description of single-electron charging in the
density-functional theory, described in detail in Appendices A and B, we have
managed to combine its advanced (ASIC) version to analyze the possibility of
using single-electron tunneling effects in molecular assemblies for the
implementation of bistable memristive devices (“resistive switches”). Our
results indicate that chemically-plausible molecules and self-assembled
monolayers of such molecules may indeed operate, at room temperature, as
nonvolatile resistive switches which would combine multi-year retention times
with sub-second switching times, and have ON/OFF current ratios in excess of
$10^{3}$. Moreover, we have obtained strong evidence that operation of the SAM
version of the device is tolerant to a rather high concentration of defects
and randomly located charged impurities. The ON current of a single device
($\sim 0.1$ pA at $V\approx 2$ V) corresponds to a very reasonable density
($\sim 4$ W/cm2) of the power dissipated in an open SAM switch, potentially
enabling 3D integration of hybrid CMOS/nano circuits ChengStrukov12 . (Note
that the average power density in a crossbar is at least 4 times lower because
of the necessary crosspoint device spacing (Fig. 1d); besides that, in all
applications we are aware of, at least 50% of the switches (and frequently
much more) are closed, decreasing the power even further.)
However, even our best design (Fig. 2) still requires additional work. First,
proper spatial positions of the functional molecules have to be enforced by
some additional molecular support groups which have not been taken into
account in our analysis yet. If the spacer groups fixing the relative spatial
arrangement of the islands can be constructed from saturated molecular units
similar to the alkane chains used to separate the islands from the electrodes,
then the calculations presented here should be applicable to complete devices,
but this expectation still has to be verified.
Second, we feel that there is room for improvement in the choice of molecular
chains used as tunnel barriers and intermediate islands. For example, the low
calculated effective mass, $m_{ef}\approx 0.1m_{0}$, of electrons tunneling
along alkane chains makes it necessary to use rather long chains, despite
their large HOMO-LUMO gaps (which, in turn, require large switching voltages —
see Figs. 4, 10). The use of a molecular chain with a higher $m_{ef}$ and a
narrower gap would decrease switching voltages (and hence energy dissipation
at switching), and also reduce the total device length, resulting in shorter
switching times (at the same charge retention).
Third, the defect tolerance of SAM-based switches should be evaluated in more
detail, for charged impurities located not only on the molecular acceptor
groups, but also between them — say, inside the (still unspecified) support
groups.
Finally, an experimental verification of our predictions looks imperative for
the further progress of work towards practicable molecular resistive switches.
## Acknowledgment
This work was supported by the Air Force Office of Scientific Research. The
supercomputer resources used in this work were provided by DOD’s HPCMP.
Valuable comments by P. Allen, D. Averin and M. Fernandez-Serra are gratefully
acknowledged. We would also like to thank C. Pammaraju and S. Sanvito for
their generous help with the ASIC SIESTA software package.
## Appendix A Single-electron charging correction
Let us consider a simple but reasonable model of a well-conducting (say,
metallic) island, of a size well above the Thomas-Fermi screening length, in
which the single-electron addition energies are simply
$\Delta E(i)=K_{i}-e\phi_{i},$ (14)
where $K_{i}$ is $i$-th electron’s kinetic energy (which, as well as the
island capacitance $C$, is assumed to be independent of other electron state
occupancies, but is an arbitrary function of $i$), and the second term
describes the potential energy of that electron in the net electrostatic
potential of all other charges,
$\phi_{i}=\phi_{0}-(i-1)\frac{e}{C},$ (15)
where $\phi_{0}$ is the background potential of the nuclei, and the second
term is due to the previously added electrons. In this model the total ground-
state energy of an $n$-electron ion (besides the electron-independent
contributions) is
$\displaystyle E_{gr}(n)$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\Delta
E(i)$ (16) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}K_{i}-en\phi_{0}+\frac{e^{2}}{2C}n(n-1),$
so that the energy difference created by the last charging is
$\displaystyle\Delta E(n)$ $\displaystyle=$ $\displaystyle
E_{gr}(n)-E_{gr}(n-1)$ (17) $\displaystyle=$ $\displaystyle
K_{n}-e\phi_{0}+\frac{e^{2}}{C}(n-1).$
On the other hand, in a hypothetical naïve DFT theory, without the partial
self-interaction corrections present in its LSDA, GGA and ASIC versions, the
single-particle (Kohn-Sham) energies of ion $n$ of this model are written as
$\varepsilon_{i}^{\mbox{\scriptsize DFT}}(n)=K_{i}-e\phi_{n},\mbox{
}\phi_{n}=\phi_{0}-\frac{e}{C}n.$ (18)
For the calculation of the full ground-state energy of ion $n$, such generic
DFT sums up these energies from $i=1$ to $i=n$, adding the “double-counting
correction” term jones89 , in the Gaussian units equal to
$E_{\mbox{\scriptsize corr}}=-\frac{1}{2}\int d^{3}r\int
d^{3}r^{\prime}\frac{\rho(\textbf{r})\rho(\textbf{r}^{\prime})}{\left|\textbf{r}-\textbf{r}^{\prime}\right|},$
(19)
where $\rho(\textbf{r})$ is the total electron charge density at point r. For
our simple model, this correction is just $–e^{2}n^{2}/2C$, so that
$\displaystyle E_{gr}^{\mbox{\scriptsize DFT}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\varepsilon_{i}^{\mbox{\scriptsize
DFT}}(n)-\frac{e^{2}n^{2}}{2C}$ (20) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}K_{i}-en\phi_{0}+\frac{e^{2}n^{2}}{2C},$
and
$\displaystyle\Delta E^{\mbox{\scriptsize DFT}}(n)$ $\displaystyle\equiv$
$\displaystyle E_{gr}^{\mbox{\scriptsize DFT}}(n)-E_{gr}^{\mbox{\scriptsize
DFT}}(n-1)$ (21) $\displaystyle=$ $\displaystyle
K_{n}-e\phi_{0}+\frac{e^{2}}{C}\left(n-\frac{1}{2}\right).$
Comparing this result with Eq. (17), we obtain the following relation:
$\Delta E(n)=\Delta E^{\mbox{\scriptsize DFT}}(n)-\frac{e^{2}}{2C}.$ (22)
Thus in the naïve DFT theory, the single-electron addition energy differs from
the correct expression (17) by $e^{2}/2C$. Moreover, it does not satisfy the
fundamental Eq. (1). Indeed, for $i=n$, Eq. (22) gives the following result,
$\varepsilon_{n}^{\mbox{\scriptsize DFT}}(n)=K_{n}-e\phi_{0}+\frac{e^{2}}{C}n$
(23)
which, according to Eqs. (17) and (21) may be rewritten either as
$\Delta E(n)=\varepsilon_{n}^{\mbox{\scriptsize DFT}}(n)-\frac{e^{2}}{C},$
(24)
or as
$\Delta E^{\mbox{\scriptsize DFT}}(n)=\varepsilon_{n}^{\mbox{\scriptsize
DFT}}(n)-\frac{e^{2}}{2C}.$ (25)
This error is natural, because such DFT version ignores the fundamental
physical fact that an electron does not interact with itself, even if it is
quantum-mechanically spread over a finite volume. This difference can become
quite substantial in small objects such as molecular groups. For example,
Table 1 shows the results using LSDA SIESTA calculations for two different
ions of our trap molecule (Fig. 2), with $n=n_{0}+1$ and $n=n_{0}+2$, where
$n_{0}=330$ is the total number of protons in the molecule. The results show
that the inconsistency described by Eq. (25) is indeed very substantial and is
independent (as it should be) of the applied voltage $V$ in the range keeping
the working orbital’s energy inside the HOMO-LUMO gap of the alkane chain. The
two last columns of the tables show the values of $e^{2}/2C$, calculated in
two different ways: from the relation following from Eq. (18):
$\frac{e^{2}}{2C}=\frac{\varepsilon_{n}^{\mbox{\scriptsize
DFT}}(n)-\varepsilon_{n}^{\mbox{\scriptsize DFT}}(n-1)}{2},$ (26)
and from the direct electrostatic expression
$\frac{e^{2}}{2C}=\frac{1}{2}\int\phi_{n}(\textbf{r})\left|\psi_{n}^{\mbox{\scriptsize
DFT}}(\textbf{r})\right|^{2}d^{3}r,$ (27)
where $\phi_{n}(\textbf{r})$ is the part of the electrostatic potential,
created by the electron of the $n$-th orbital of the $n$-th ion. The values
are very close to each other and correspond to capacitance $C\approx 4.5\times
10^{-20}$ F which a perfectly conducting sphere of diameter $d\approx 0.8$ nm
would have. The last number is in a very reasonable correspondence with the
size of the acceptor group of the molecule — see Fig. 2.
Table 1: Columns 2 and 3: values of the single-electron transfer energy
$\Delta E(n)$ for the trap molecule ions with $n=n_{0}+1$ and $n=n_{0}+2$
electrons, calculated in LSDA SIESTA and then self-interaction corrected as
discussed in Appendix A, as functions of the applied voltage (Column 1).
Columns 3 and 4 list the values of parameter $e^{2}/C$ , calculated as
discussed in Appendix A.
Voltage $V$ (V) | $\Delta E(n)$ from Eq. (22) (eV) | $\Delta E(n)$ from Eq. (24) (eV) | $e^{2}/2C$ from Eq. (26) (eV) | $e^{2}/2C$ from Eq. (27) (eV)
---|---|---|---|---
$n=n_{0}+1$
-2.36 | -3.08 | -3.01 | 1.84 | 1.79
-1.18 | -3.38 | -3.37 | 1.84 | 1.79
0.00 | -3.73${}^{\mbox{\scriptsize(a)}}$ | -3.73${}^{\mbox{\scriptsize(a)}}$ | 1.84 | 1.79
1.18 | -4.07 | -4.10 | 1.84 | 1.79
2.36 | -4.42 | -4.46 | 1.84 | 1.79
3.53 | -4.77 | -4.82 | 1.84 | 1.79
$n=n_{0}+2$
7.07 | -1.91 | -2.02 | 1.82 | 1.79
8.24 | -2.29 | -2.39 | 1.82 | 1.79
9.42 | -2.61 | -2.75 | 1.82 | 1.79
10.60 | -2.97 | -3.11 | 1.82 | 1.79
11.78 | -3.32 | -3.47 | 1.82 | 1.79
${}^{\mbox{\scriptsize(a)}}$ The numbers to be compared with experimental
values of electron affinity: -3.31 eV Ref. bhozale08 and -3.57 eV Ref.
singh06 .
The second and third columns of the table present the genuine electron
addition energies $\Delta E(n)$ calculated from, respectively, Eq. (22) and
(24), using the average of the above values of $e^{2}/2C$. Not only do these
values coincide very well; they are in a remarkable agreement with
experimentally measured electron affinities singh06 ; bhozale08 of molecules
similar to our molecular trap.
We believe that these results show that, first, LSDA SIESTA provides very
small compensation of the self-interaction effects in the key energy $\Delta
E(n)$ and, second, that (at least for the lowest negative ions of our trap
molecules), an effective compensation may be provided using any of the simple
relations (22) and (24).
## Appendix B Level freezing in DFT
For the analysis of the fictitious “level freezing” predicted by a naïve DFT
at $V>V_{t}$ (see Fig. 6), let us consider the following simple model: a
molecule consisting of a small acceptor group with just one essential energy
level, and a spatially separated chain with a quasi-continuous valence band.
Figure 18 shows the energy spectrum of the system at $V<V_{t}$. (As before,
the occupied levels are shown in black, while the unoccupied ones are shown in
green.)
Figure 18: The schematic energy spectrum of our model at a voltage $V$ below
voltage $V_{t}$ that aligns the group localized level $\varepsilon$ with the
valence band edge $\varepsilon_{v}$.
The edge $\varepsilon_{v}$ of the band is separated from the first unoccupied
level in the group by energy $-e(V-V_{t})$, where $V$ is the fraction of the
voltage drop between the centers of the group and the tail of a molecule, and
$V_{t}$ is its value which aligns the level with $\varepsilon_{v}$. Now let
$V$ be close to $V_{t}$, so that the occupancy $p$ of the discrete level is
noticeable. If the effect of group charging on the exchange-correlation energy
is negligible, a generic DFT theory (e.g., LSDA) would describe the system
energy as
$\displaystyle E$ $\displaystyle=$ $\displaystyle E_{0}-e(V-V_{t})p$ (28)
$\displaystyle+$ $\displaystyle\frac{1}{2}\int d^{3}r\int
d^{3}r^{\prime}\frac{\rho(\textbf{r})\rho(\textbf{r}^{\prime})-\rho_{0}(\textbf{r})\rho_{0}(\textbf{r}^{\prime})}{\left|\textbf{r}-\textbf{r}^{\prime}\right|},$
where index 0 marks the variable values at $p=0$.
Now let us simplify Eq. (28) by assuming that due to a small size of the
acceptor group, the Coulomb interaction of electrons localized on it is much
larger than that on the chain, so that the latter may be neglected. (For the
trap molecule shown in Fig. 2, this assumption is true within $\sim$5%.) Then
Eq. (28) is reduced to
$\displaystyle E$ $\displaystyle\approx$ $\displaystyle
E_{0}-e(V-V_{t})p-\frac{e}{2}\int_{\mbox{\scriptsize
group}}\phi(\textbf{r})\left|\psi(\textbf{r})\right|^{2}d^{3}r,$
$\displaystyle p$ $\displaystyle=$ $\displaystyle\int_{\mbox{\scriptsize
group}}\left|\psi(\textbf{r})\right|^{2}d^{3}r,$ (29)
where $\phi(\textbf{r})$ is the electrostatic potential created by the part of
the electronic wavefunction that resides on the group. In the simple
capacitive model of the group (used in particular in Appendix A),
$\phi(\textbf{r})=-ep/C$, where $C$ is the effective capacitance of the group,
so that
$E\approx E_{0}-e(V-V_{t})p+\frac{e^{2}p^{2}}{2C}.$ (30)
On the other hand, in accordance with Eq. (20), the total energy in the DFT
may also be presented in the form
$E=\sum_{i}p_{i}\varepsilon_{i}-\frac{1}{2}\int d^{3}r\int
d^{3}r^{\prime}\frac{\rho(\textbf{r})\rho(\textbf{r}^{\prime})}{\left|\textbf{r}-\textbf{r}^{\prime}\right|},$
(31)
where $\varepsilon_{i}$ are all occupied (or partially occupied) single
particle energies, so that in our simple model
$E\approx E_{0}+(\varepsilon-\varepsilon_{v})p-\frac{e^{2}p^{2}}{2C}.$ (32)
Comparing Eqs. (30) and (32), we arrive at the following expression:
$\varepsilon-\varepsilon_{v}\approx-e(V-V_{t})+\frac{e^{2}p}{C}.$ (33)
In most DFT packages, level occupancies $p_{i}$ are calculated from the
single-particle Fermi distribution,
$p_{i}=\frac{1}{\mbox{exp}\left\\{(\varepsilon_{i}-\mu)/k_{\mbox{\scriptsize
B}}T\right\\}+1};$ (34)
for our simple model, index $i$ may be dropped, and (due to the valence band
multiplicity) $\mu\approx\varepsilon_{v}$.
As is evident from the sketch of Eqs. (33) and (34), in Fig. 19, if the
thermal fluctuation scale $k_{\mbox{\scriptsize B}}T$ is much lower than the
charging energy scale $e^{2}/C$, then almost within the whole range
$V_{t}<V<V_{t}+e/C$, the approximate solution of the system of these equations
is
$p\approx\frac{C}{e}(V-V_{t}),\mbox{ }\varepsilon\approx\varepsilon_{v}.$ (35)
Figure 19: A sketch of Eqs. (33) and (34).
Panel (a) in Fig. 20 shows (schematically) the resulting dependence of the
energy spectrum of our model system on the applied voltage $V$, with level
freezing in the range $V_{t}<V<V_{t}+e/C$. The dashed black-green line
indicates the region with a partial occupancy $0<p<1$ of the group-localized
orbital. In panel (b) in (Fig. 20) we show the evolution which should follow
from the correct quantum-mechanical theory, in which electrons do not self-
interact and as a result there is the usual anticrossing of energy levels
$\varepsilon$ and $\varepsilon_{v}$ at $V=V_{t}$. (For clarity, Fig. 20
strongly exaggerates the anticrossing width, which is less than $10^{-3}$ eV
for our trap 999Direct SIESTA calculation has shown that the anticrossing
energy splitting is less than the calculation error (of the order of $10^{-3}$
eV). An indirect calculation using Eq. (12), with $\psi_{W}$ and $\psi_{v}$
substituted instead of $\psi_{i}$ and $\psi_{s,c}$, suggests that this overlap
is as small as $\sim 10^{-8}$ eV..)
Figure 20: (a) A sketch of the evolution of the energy spectrum from Fig. 18
as a function of the applied voltage $V$, illustrating the self-interaction
errors giving rise to a spurious level freezing in the $V_{t}<V<V_{t}+e/C$
voltage range. The dashed black-green line indicates the region with a partial
occupancy $0<p<1$ of the group-localized orbital. (b) A sketch of the
evolution of the same energy spectrum in a correct quantum-mechanical theory,
in which electrons do not self-interact.
The actual spectrum of our molecular trap is somewhat more complex than that
of the simple model above — see Figs. 4a and 6. First, not only the valence
energy band of the alkane chain, but also its conduction band is important for
electron transfer in our voltage range. Second, the molecular group has not
one, but a series of discrete energy levels, with the most important of them
corresponding to the working orbital (energy $\varepsilon_{W}$), and one more
group-localized orbital with energy
$\varepsilon_{W+1}\approx\varepsilon_{W}+0.7$ eV.
Figure 21: (a) A sketch of the evolution of the molecular energy spectrum of
our trap molecule as a function of the applied voltage $V$, illustrating the
self-interaction errors giving rise to a spurious level freezing in voltage
ranges $V_{t}^{\prime}-e/C<V<V_{t}^{\prime}$ and $V_{t}<V<V_{t}+e/C$. The
dashed black-green line indicates the region with a partial occupancy $0<p<1$
of the group-localized orbitals with energies $\varepsilon_{W}$ or
$\varepsilon_{W+1}$. (b) A sketch of the evolution of the molecular energy
spectrum but in a correct quantum-mechanical theory, in which electrons do not
self-interact.
Nevertheless, the behavior of the spectrum, predicted by uncorrected versions
of DFT (Fig. 6) may still be well understood using our model. Just as was
discussed above, for voltages $V$ above the threshold value $V_{t}$ (which now
corresponds to the alignment of $\varepsilon_{v}$ with $\varepsilon_{W+1}$
rather than $\varepsilon_{W}$), it describes a gradual transfer of an electron
between the top level of the valence band and the second group-localized
orbital, with its occupation number $p_{W+1}$ gradually growing in accordance
with Eq. (35) — see panel (a) in Fig. 21. Similarly, at voltages $V$ below
$V_{t}^{\prime}$ (which corresponds to the alignment of the working orbital’s
energy $\varepsilon_{W}$ with the lowest level $\varepsilon_{c}$ of the
chain’s conduction band), there is a similar spurious gradual transfer of an
electron between the corresponding orbitals. In both voltage ranges, a
spurious internal electrostatic potential is created; as is described by Eq.
(35), it closely compensates the changes of the applied external potential,
thus “freezing” all orbital energies of the system at their levels reached at
thresholds $V_{t}^{\prime}$ and $V_{t}$ — see panel (a) in Fig. 21. Figure 22
shows that results of both the LSDA and ASIC DFT calculations at $V>V_{t}$
agree well with Eq. (35), with a value $C=4.5\times 10^{-20}$ F calculated as
discussed in Appendix A, indicating that the electron self-interaction effects
remain almost uncompensated in these software packages, at least for complex
molecules such as our trap.
Figure 22: The DFT-calculated occupancy $p_{W+1}$ of the $(W+1)$’st orbital of
the acceptor group of our trap molecule at voltages above the threshold
voltage $V_{t}$ of the alignment of its energy $\varepsilon_{W+1}$ with alkane
chain’s valence band edge $\varepsilon_{v}$. Black lines show results of two
versions of DFT theory, for two ion states: the singly-negatively charged ion
and the neutral molecule, while the red line shows the result given by Eq.
(35) with $C=4.5\times 10^{-20}$ F.
Again, in the correct quantum-mechanical theory, there should be a simple (and
in our molecules, extremely narrow) anticrossing between the effective single-
particle levels of the acceptor group and the alkane chain — see panel (b) in
Fig. 21. As described in Sec. III of the main text, we have succeeded to
describe this behavior rather closely, using the internal iteration dynamics
of ASIC SIESTA with $T=0$ K.
## References
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* (3) D. Strukov and H. Kohlstedt, MRS Bulletin 37, 108 (2012)
* (4) S. Fölling, Ö. Türel, and K. K. Likharev, in _Proc. of the Int. Joint Conference on Neural Networks_ (Int. Neural Network Society, Mount Royal, NY, 2001) pp. 216–221
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|
arxiv-papers
| 2012-10-18T20:23:34 |
2024-09-04T02:49:36.787875
|
{
"license": "Public Domain",
"authors": "Nikita Simonian, Andreas Mayr and Konstantin K. Likharev",
"submitter": "Nikita Simonian",
"url": "https://arxiv.org/abs/1210.5253"
}
|
1210.5259
|
# K-essence scalar field as dynamical dark energy
L. A. García lagarciape@unal.edu.co J. M. Tejeiro jmtejeiros@unal.edu.co L.
Castañeda lcastanedac@unal.edu.co Observatorio Astronómico Nacional,
Universidad Nacional de Colombia.
###### Abstract
We study an early dark energy (EDE) model as a K-essence scalar field in the
framework of FLRW universe using an effective parametrization of the state
equation as a function of the redshift $z$ with the tracker condition during
radiation domination, but also demanding an accelerated expansion of the
universe at late times emulating cosmological constant. We found all the
dynamical variables of the EDE system. We use the luminosity distances of the
SNIA to get the best estimations for the free parameters of the model and
also, we constrain the model using primordial abundances of light nuclei in
BBN theory. We summarize the necessary conditions to achieve BBN predictions
and the accelerated expansion of the universe at late times.
###### pacs:
26.35.+c, 95.36.+d.
## I Introduction
Recent observations of the luminosity distances of the SNIa perlmutter reveal
that the expansion of the universe is accelerated and there is an unknown
matter–energy contribution as about 70% of the critical density, which is
smooth and has negative pressure. In order to explain this phenomenon, it has
been proposed many plausible solutions: the cosmological constant $\Lambda$
which is related to the vacuum energy of the quantum fields carroll ,
Quintessence fields (with the state equation
$\omega=\frac{p_{Q}}{\rho_{Q}}$=constant), Kessence, Taquionic fields,
frustrated topological defects, extra–dimensions, massive (or massless)
fermionic fields, galileons, effective parametrizations of the state equation,
primordial magnetic fields, holographic models, etc. All these proposals have
been used to model this contribution predicted by the Friedmann equations in
the framework of the General Relativity. Also, there are other possible
options as Modified Gravity, where the accelerated expansion effect is
geometric and not as a matter–energy form.
Nowadays, the current paradigm is the $\Lambda$CDM model, which is so far the
best fitting to the present observations, even though the conceptual problems
that persist with the nature of $\Lambda$.
On the other hand, primordial abundances of the light nuclei that were formed
during Big Bang Nucleosynthesis (BBN) are well observed and quantified with
astrophysical methods, specially the mass fraction of the ${}^{4}He$.
Actually, there is an extended theory of BBN, introduced by Alpher, Bethe and
Gamov alpher ; gamov , and it has been developed many numerical codes that
resolve the Boltzmann equation for each isotope including the cosmological
background (based on Kawano code). There is a good agreement between the
predicted abundances and the observations, however, the main problem remains
with ${}^{7}Li$. Despite the efforts that have been made, including
corrections on the cross sections of this element, the discrepancies are not
negligible.
With the aim to enhance the primordial abundances calculated from BBN and
also, give a plausible explanation of the accelerated expansion of the
universe, we have proposed a model of dark energy which has a non–null
contribution at early times to increase the Hubble radius during radiation
domination era and influence the Boltzmann equations that determine the
evolution of the light abundances. All the conditions that allow us to
describe the early dark energy are achieved with a K-essence scalar field,
which is characterized by its state equation that overcomes the attractors
defined by a dynamical system.
In the first section, we expose the main conditions imposed to the scalar
field and we resolve the dynamical system that appears from the cosmological
assumptions. At the second part, we propose the effective parametrization of
the state equation, determinate the best estimations of the free parameters of
the model using the luminosity distances of the SNIa from the Union dataset
and derive analytical expressions for the dynamical variables of the K-essence
system. In the IV section, we test our model with some standard proves and
summarize the main differences between this model and $\Lambda$CDM.
Taking into account that there is a non-negligible energy density of the
K-essence field during radiation era, we compute the BBN abundances, including
the scalar field degrees of freedom in Hubble parameter, following the
derivation and discussion made by Bernstein bernstein2 and using the
available codes for Nucleosynthesis. Finally, we find the values of
$\Omega_{B}$ and $\eta_{B}$ predicted by our model as a result of the BBN
calculations.
## II K-essence scalar field and the dynamical system
The K-essence scalar fields appeared at the late 90s with the K-inflation
model proposed by Armendariz-Picon armendariz1 ; armendariz2 . However, the
idea was extended to describe a dynamical dark energy contribution, taking
into account that this field can track during radiation domination epoch and
also, it could avoid the fine–tuning of the initial values of the field and
its velocity. These features are well known for the system, so the challenge
is resolve the evolution equations of the field in the FLRW spatially flat
universe and without cosmological constant.
The lagrangian of the K-essence scalar field is given by:
$p(X,\phi)=K(\phi)L(X),$ (1)
where $X=-\frac{1}{2}\nabla^{\alpha}\phi\nabla_{\alpha}\phi$ is the kinetic
energy of the field and $v$ its velocity $v=\frac{d\phi}{dt}=\sqrt{-2X}>0$.
Rewriting (1) in terms of $v$, the lagrangian has the following form :
$p(v,\phi)=K(\phi)Q(v).$ (2)
The most general action that describes the K-essence field in a cosmological
plasm is given by:
$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{2\kappa^{2}}+p(\phi,X)\right)+S_{B},$
(3)
with $\kappa^{2}=8\pi G$ and $S_{B}$ is the action of the background matter.
The signature that is used is $\\{-1,+1,+1,+1\\}$ The equations of motion of
the field come from the variation of the lagragian with respect to the field
$\phi$:
$\left(\frac{1}{v}\frac{\partial p}{\partial v}+v\frac{\partial}{\partial
v}\left(\frac{1}{v}\frac{\partial p}{\partial
v}\right)\right)\ddot{\phi}+\frac{\partial p}{\partial
v}(3H)+v\frac{\partial^{2}p}{\partial\phi\partial v}-\frac{\partial
p}{\partial\phi}=0.$ (4)
From (3), it is obtained the energy–momentum tensor:
$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-p(\phi,v)g_{\mu\nu}.$ (5)
Since the scalar field can be described as a perfect fluid, the energy density
and the pressure are defined by:
$\rho_{\phi}=K(\phi)\left(v\frac{\partial Q}{\partial v}-Q\right),$ (6)
$p_{\phi}=K(\phi)Q(v).$ (7)
In addition, the adiabatic velocity of sound for the K-essence field is given
by:
$C_{s}^{2}=\dfrac{Q^{\prime}}{vQ^{\prime\prime}}.$ (8)
where $\prime$ denotes derivative with respect to the velocity of the field
$v$. $C_{s}^{2}$ gives relevant information of the stability of the
perturbations associated with the K-essence field.
In order to resolve the (4) and find an explicit form of (6) and (7), there
have been suggested many alternatives: fixing a specific function of $\phi(t)$
or $v(t)$ chiba , making redefinition of the field to face a modified
lagrangian $\bar{Q}(v)$ copeland , considering pure kinetic K-essence model
stiff ; linder ; nojiri ; yang ; scherrer2 or imposing Slow roll conditions
on $K(\phi)$.
However, we want to resolve the complete dynamical system defined by the
Friedmann and the continuity equations for non-interactive fluids in the
cosmological background: a matter (or radiation component) and the K-essence
scalar field.
$H=\frac{\dot{a}}{a}=\kappa\sqrt{\rho_{\phi}+\rho_{m}},\>\>\>\>\>\>$ (9)
$\frac{\ddot{a}}{a}=-\frac{\kappa^{2}}{6}((1+3\omega_{\phi})\rho_{\phi}+(1+3\omega_{m})\rho_{m}),\>\>\>\>\>\>$
(10)
$\dot{\rho_{m}}=-3H(1+\omega_{m})\rho_{m},\>\>\>\>\>\>\>\>\>\>\>\>\dot{\rho_{\phi}}=-3H(1+\omega_{\phi})\rho_{\phi},\>\>\>\>\>\>$
(11)
Reexpressing the equiations (9), (10), (11) in terms of the velocity and the
dimensionless variable $F=\frac{\rho_{m}}{\rho_{\phi}+\rho_{m}}$ (matter
energy density fraction):
$\frac{dv}{d\phi}=-C_{s}^{2}\left[\frac{(lnK)_{,\phi}v}{1+\omega_{\phi}}+3\kappa\left(\frac{K\tilde{\rho_{\phi}}}{1-F}\right)^{1/2}\right],$
(12)
$\frac{dF}{d\phi}=-\frac{3\kappa}{v}F\sqrt{1-F}\sqrt{K\tilde{\rho_{\phi}}}\left(\omega_{m}-\omega_{\phi}\right).$
(13)
In addition, it must be fulfilled the following conditions:
$\frac{d\tilde{\rho_{\phi}}}{dv}=\frac{(1+\omega_{\phi})}{vC_{s}^{2}}\tilde{\rho_{\phi}},$
(14)
$\frac{d\omega_{\phi}}{dv}=\frac{1+\omega_{\phi}}{v}\left(1-\frac{\omega_{\phi}}{C_{s}^{2}}\right).$
(15)
It is assumed an asymptotic behaviour for the function $K(\phi)$ kang :
$K(\phi)=\frac{1+K_{0}(\phi)}{\phi^{2}},\>\>\>\>\>\>\>\>\>\lim_{\phi\to\infty}K_{0}(\phi)=0.$
(16)
To reach an attractor during radiation domination epoch, the dynamical system
$\\{v(\phi),R(\phi)\\}$ must fulfill the ansatz:
$v(\phi)=v_{0}-A(\phi),\>\>\>\>\>\>\>\>\>F(\phi)=F_{0}-B(\phi).$ (17)
where $A(\phi),B(\phi)\rightarrow 0$ monotonically for $\phi\to\infty$ (or
equivalently, $v=v_{rad}$ and $F\sim 0$). The ansatz implies the following
physical conditions on the system:
$\omega_{m}=\omega_{\phi}(v_{0}),\>\>\>\>\>\>\>\>\tilde{\rho_{\phi}}(\phi)\neq
0,\>\>\>\>\>\>\>\>C_{s}^{2}(v_{0})>\omega_{m}$ (18)
$\omega_{\phi}^{\prime}(v_{0})=\left(1-\frac{\omega_{m}}{C_{s}^{2}(v_{0})}\right)\frac{1+\omega_{m}}{v_{0}}\neq
0.$ (19)
There is also a De-Sitter attractor given by the condition $R\sim 0$, which
guarantees the existence of a accelerated expansion at late times:
$v(\phi)=v_{s}-A(\phi),\>\>\>\>\>\>\>\>\>R(\phi)=B(\phi).$ (20)
The last ansatz entails a condition on the state equation on the vicinity of
the De-Sitter attractor:
$\dfrac{{1-\omega_{\phi}(v_{s})}}{{1+\omega_{\phi}(v_{s})}}>0,\>\>\>\>\>\>\>\>|\omega_{\phi}(v_{s})|<1.$
(21)
## III Effective parametrization of $\omega_{\phi}$
In order to get a general solution for the dynamical system defined by (14),
(15), (18), (19) and (21), it has been proposed an effective parametrization
of the state equation from $z<10^{15}$ given by:
$\omega_{\phi}(z)=\frac{4/3}{\left(\frac{1+z_{d}}{1+z}\right)^{m}+1}-1,$ (22)
where $m$ is factor that modules the transitions between the attractors,
$z_{d}$ is a redshift in matter domination epoch defined by
$z_{d}=\dfrac{z_{eq}+z_{*}}{2}$ and $z_{*}$, the redshift where the De-Sitter
domination -accelerated expansion- begins.
The parametrization (22) respects all the conditions previously mentioned,
hence it is possible to resolve the functions related to the K-essence
lagrangian.
The energy density of the field $\rho_{\phi}$ (6) results:
$\int_{\rho}^{\rho_{0}}\frac{d\rho^{\prime}}{\rho^{\prime}}=-3\int_{a}^{1}\frac{(1+\omega_{\phi}(a^{\prime}))}{a^{\prime}}da^{\prime},$
(23)
integrating (23), it is obtained:
$\rho=\rho_{0}\cdot(1+z)^{4}\left[\frac{\left(\left(\frac{1+z_{d}}{1+z}\right)^{m}+1\right)}{\left(\left(1+z_{d}\right)^{m}+1\right)}\right]^{4/m}=\rho_{0}\cdot
f(z).$ (24)
with
$\displaystyle f(a)$
$\displaystyle=exp\left[-3\int_{a}^{1}\frac{(1+\omega_{\phi}(a^{\prime}))}{a^{\prime}}da^{\prime}\right]$
(25)
$\displaystyle=a^{-4}\left[\frac{\left(\left(\frac{a}{a_{d}}\right)^{m}+1\right)}{\left(\left(\frac{1}{a_{d}}\right)^{m}+1\right)}\right]^{4/m},$
(26)
Meanwhile, the fraction of the dark energy density
$\Omega_{\phi}=1-F=\frac{\rho_{\phi}}{\rho_{cr}}$:
$\Omega_{\phi}=\frac{\Omega_{\phi 0}\cdot f(a)}{\Omega_{\phi 0}\cdot
f(a)+\Omega_{m0}\cdot a^{-3}},$ (27)
The formal solution of (22) will be obtained with the best estimation of the
free parameters of the model $\\{\Omega_{\phi_{0}},m,z_{*}\\}$. For this
reason, we use the luminosity distances of the SNIa from the survey Supernova
Cosmology Project with $z>0.8$ to minimize the function $\chi^{2}$:
$\chi^{2}=\sum_{i=1}^{N\sim 59}\frac{[\mu_{i}-\mu(z_{i})]^{2}}{\sigma^{2}}$
(28)
with the distance modulus given by the expression
$\mu=m-M=5(log_{10}d_{L}(z)-1)$ and the luminosity distance of our model:
$d_{L}(z)=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{B(z)}.$ (29)
$B(z)=(\Omega_{\phi_{0}}f(z^{\prime};m,z_{*})+(1-\Omega_{\phi_{0}})(1+z^{\prime})^{3})^{1/2}$
(28) must be resolved together with the constraints:
$\sum_{i=1}^{N\sim
59}\left(\frac{\mu_{i}-\mu(z_{i})}{\sigma^{2}}\right)\left(\frac{\partial\mu(z_{i};\Omega_{\phi_{0}},m,z_{*})}{\partial
x}\right)=0,$ (30)
where $x=\Omega_{\phi_{0}},m,z_{*}$. In addition, we have imposed other 2
conditions for the parameters: the deceleration parameter has to be zero at
$z_{*}$, therefore $q(z_{*})=0$, but also $f(z\sim z_{BBN}\approx 10^{9})$ has
to overcome the maximum value at the Primordial Nucleosynthesis to contribute
with some relativistic degrees of freedom and enhance the predicted primordial
abundances.
Resolving (28) simultaneously as (30), it is found the following values for
the free parameters:
| K-essence (22)
---|---
$\Omega_{\phi_{0}}$ | 0.69
$m$ | 1.0
$z_{*}$ | 1.48
$\omega_{0}$ | \- 0.99
In the figure 1 is plotted the evolution of the state equation $\omega$ as
function of $a$. The field emulates radiation during this epoch and then it
evolves to the next attractor: De-Sitter. Here is clear the tracker behaviour
imposed in the dynamical system. In addition, it is shown that the field is
relaxing to the asymptotic $\Lambda$CDM model for late times (during De-Sitter
attractor.)
Figure 1: State equation of K-essence field as a function of $a$.
Figure 2 shows the behaviour of the the function $f(z)$ that characterizes the
dark energy density evolution in the model. During radiation domination epoch,
the field scales as radiation $\rho\propto(1+z)^{4}$ until $z_{eq}$. After
that, $\rho$ has a complex behaviour which guarantees the second attractor
will be reached. At this point, the K-essence scalar field evolves in the De-
Sitter attractor and its state equation goes asymptotically to $-1$ (as
cosmological constant).
Figure 2: Function $f(z)$: dark energy density evolution.
The figure 3 displays the luminosity distance for the model compared with the
predicted by $\Lambda$CDM. The shift between the curves shows that the
luminosity distance is upper than the associated with the $\Lambda$CDM model,
because the matter density today predicted by our model is higher than the
second one (compared with WMAP-7 $\\{\Omega_{\phi
0},\Omega_{m0}\\}=\\{0.734\pm 0.029,0.266\pm 0.029\\}$
111lambda.gsfc.nasa.gov/product/map/dr4/parameters.cfm.
However, for low redshifts the luminosity distance grows linearly independent
on the model at this regime.
Figure 3: Luminosity distance as a function of $z$.
On the other hand, figure 4 displays the evolution of the dark energy density
fraction for the model with the estimations of the free parameters (III):
Figure 4: Evolution of the dark energy density fraction as a function of $z$.
Up to now, we are interested to resolve the dynamical variables of the
K-essence according to the effective parametrization we have proposed. The
first one of these quantities is the adiabatic velocity of the sound
$C_{s}^{2}$, which has a behaviour defined by equation (31).
$C_{s}^{2}=\omega_{\phi}+\frac{m}{4}\left(\frac{1+z_{d}}{1+z}\right)^{m}(\omega_{\phi}+1).$
(31)
Using (22), we complete the evolution of $C_{s}^{2}$. The behaviour of the
adiabatic velocity is plotted in the figure 5.
Figure 5: Adiabatic sound velocity evolution within the model.
$C_{s}^{2}$ fulfills the condition (18) in the radiation domination epoch
attractor. Whatsoever, there is not a straightforward physical interpretation
of the adiabatic velocity during De-Sitter attractor, because this value
implies a complex Young module of the plasma perturbations.
It might be that the field is unstable when $C_{s}^{2}<0$, however, it is
necessary to compute the evolution of the field perturbations and the time of
the stability condition to conclude something in this respect.
On the other hand, the velocity of the field (the time evolution of the field)
has a solution in terms on the scale factor given by:
$v=v_{0}\cdot(1+z)\left(\frac{\left(\frac{1+z_{d}}{1+z}\right)^{m}+1}{\left(1+z_{d}\right)^{m}+1}\right)^{4/m-1}.$
(32)
where $v_{0}$ is the field velocity today.
Finally, the non-canonical term of the action results:
$Q=Q_{0}\cdot(1+z)^{4}\left(\frac{\left(\frac{1+z_{d}}{1+z}\right)^{m}+1}{\left(1+z_{d}\right)^{m}+1}\right)^{4/m+1}\left(\frac{\left(1+z_{d}\right)^{m}-1/3}{\left(\frac{1+z_{d}}{1+z}\right)^{m}-1/3}\right).$
(33)
That is the predicted behaviour for the non-canonical term of the lagrangian
in terms on $z$. More interesting is that $Q(v)$ evolves as radiation in the
first attractor and then acquires a more complicated dependence during the
transition between the attractors, ensuring the continuity of the function.
## IV Standard cosmological proves
### Age of the universe with this model
The age of the universe according to our model is given by the expression:
$t_{0}=\frac{1}{H_{0}}\int_{0}^{1}\frac{a^{-1}da}{(0.31a^{-3}+0.69f(a))^{1/2}},\\\
t_{0}=1.2987\times 10^{10}\text{years}.$
This is just an approximation of the age of the universe; however, this is an
excellent result taking into account that the parametrization was figured out
for $z<10^{15}$.
On the other hand, the existence of this scalar field implies that the
universe evolves faster than in the standard model, because for a more
negative $\omega$, more accelerated is the expansion and older is the universe
for a given $H_{0}$.
### Matter inhomogeneities evolution
When the Meszaros equation is resolved within the model with the K-essence
scalar field, one solution for the inhomogeneous modes grow with the Hubble
factor de Hubble. Those are the modes which maintain their amplitude after
cross the horizon.
There are many effects on the density of perturbations: it is suppressed the
linear growth $\omega(z)\leq-1/2$, with respect $\Lambda$CDM model, where the
growth is proportional to the scale factor $a$; this suppression raises with
higher $\omega$ at matter domination epoch and entails an earlier beginning to
the dark energy domination epoch (accelerated expansion).
Otherwise, considering the CMB anisotropies spectrum, it is possible to verify
that for values of $m>1.0$, there is formation of the first two acoustics
peaks, then the model has a strong influence during radiation and matter
domination epochs, which not correspond with the observations. On the other
hand, for $m$ values lower than $0.5$, the spectrum goes faster to
$\Lambda$CMB, and they are rejected because do not scale as radiation in that
domination but tend $\omega_{\phi}\sim-1$ for $a<<a_{*}$.
The acoustic peaks correspond to the modes which at the CMB decoupling epoch
were in the maximum compression (odds peaks) or rarefaction (even) and their
position is really sensitive with the state equation of the dark energy.
Actually, the first peak depends on $\Omega_{de}$ monotonically and taking
into account that our model predicts a lower value for this parameter compared
with the $\Lambda$CDM, hence the first peak undergoes a shifting to lower
multipole moments. However, there is a degeneracy that must be broken down in
this phenomenon: the existence of this kind of model implies a rise in the
number of baryons $\Omega_{B}$, that also entails a shifting to lower
multipole moments of the peaks. Furthermore, the larger $\Omega_{B}$ is, the
higher would be the first peak.
Therefore, if $\Omega_{m}$ increases its value within the model, the relative
distance between the peaks decreases, because the mass associated with the
baryons makes that the oscillations occur faster.
### Statefinder parameters
In order to distinguish between our model and $\Lambda$CDM, rs has proposed a
test using Statefinder parameters, defined by:
$\displaystyle r$
$\displaystyle=1+\frac{9}{2}\Omega_{\phi}\omega_{\phi}(1+\omega_{\phi})-\frac{3}{2}\Omega_{\phi}\frac{\dot{\omega_{\phi}}}{H},$
(34) $\displaystyle s$
$\displaystyle=1+\omega_{\phi}-\frac{1}{3}\frac{\dot{\omega_{\phi}}}{H\omega_{\phi}},$
(35)
When the parameters are computed, we concluded that the K-essence scalar field
model is slightly off $\Lambda$CDM, because the prediction of the values
$\Omega_{m0}$, $\Omega_{\phi 0}$ and $z_{*}$ are upper than the obtained with
cosmological constant. However, as we have argued in this section, the model
is relaxed and tends to $\Lambda$CDM for redshifts during De-Sitter domination
epoch.
### CMB shift parameter $R$
This parameter measures the shifting of the acoustic peaks from from BAO and
it is defined as the comoving distance between the last scattering surface and
today:
$R=(\Omega_{m}H_{0}^{2})^{1/2}\int_{0}^{1089}\frac{dz}{H(z)},$ (36)
with $H(z)$ depends strongly on the model.
The measured value for this parameter is $R=1.719\pm 0.019$ panotopoulos ,
meanwhile the numerical calculations made with our model give $R_{cal}=1.75$;
the convergence to this value is insured for $z<200$.
## V Constrains with BBN abundances
As it has been proposed in the introduction of this paper, the main goal with
the effective parametrization of the state equation for the K-essence scalar
field is to avoid the need to fine tuning the initial conditions, but also
reproduce the BBN assess enhancing the predicted abundances. The devise is
really simple, we include the field energy density as relativistic degrees of
freedom during radiation epoch, then compute the Hubble factor and
consequently the Boltzmann equations for the light nuclei.
Firstly, we set out a fiducial baryon to photon ratio $\eta=6.2\times 10^{-9}$
222lambda.gsfc.nasa.gov/product/map/dr4/parameters.cfm and calculate the
capture temperature that deuterium began to be formed via nuclear reactions
given by:
$T_{c,\gamma}=\epsilon_{D}/26=0.088MeV.$ (37)
For this temperature the deuterium is not longer photodissociate bernstein2 ,
where $\epsilon_{D}$ is the deuterium binding energy. Temperatures lower than
$T_{c;\gamma}$ the whole chains related with primordial nucleosynthesis could
be run.
Following the discussion of Bernstein , we assess analytically the value of
the mass fraction $Y_{{}^{4}He}$ and then, using FastBBN kawano ; kellogg ,
Public BBN and BBNreactions bbnreactions , we compute numerically the light
abundances.
In order to introduce the effective degrees of freedom of the K-essence scalar
field, we impose the condition mentioned in section III to restrict the value
of the m-parameter: we want to achieve the largest contribution of the energy
density, been subdominant with respect to the radiation energy density. The
condition can be quantified as:
$\rho_{\phi}|_{rad}=b\cdot\rho_{rad}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>0\leq b<1.$
(38)
We introduce this energy density in the Hubble parameter during radiation era
and executed the time capture, the neutron’s fraction at this time and
finally, the mass fraction, obtaining the following results:
| $b=0$ | Model with a contribution of $b=0.2$
---|---|---
$H(b)$ ($s^{-1}$) | 1.13 | 1.2379
$t_{c}$ ($s$) | 182 | 169.8
$X_{n}(t_{c})$ | 0.123 | 0.125
$Y_{4}$ | 0.247 | 0.249
The values in the V are compared with the calculated values within the
standard model (where $b=0$, i.e. null contribution of the K-essence scalar
field).
The abundances for the other nuclei are not precisely computed to be reported
using this analytical method, because they depend on the coupled Boltzmann
chains.
However, it is notable that the predicted values for the ${}^{4}He$ is in
agreed with the observational boundaries, therefore our model is an excellent
candidate for dynamical dark energy model in FRWL.
Using FastBBN kawano ; kellogg , Public BBN and BBNreactions bbnreactions , we
compute numerically the light abundances including the relativistic degrees of
freedom of the field as $b$:
Códe | $b$ | $D/H\times 10^{-5}$ | ${}^{3}He/H\times 10^{-5}$ | $Y_{p}$ | ${}^{7}Li/H\times 10^{-10}$
---|---|---|---|---|---
Fast BBN | $0.0$ | $2.335$ | $1.546$ | $0.241$ | $1.268$
Fast BBN | $5.0\times 10^{-3}$ | $2.345$ | $1.548$ | $0.241$ | $1.261$
Fast BBN | $5.0\times 10^{-2}$ | $2.436$ | $1.570$ | $0.245$ | $1.263$
Fast BBN | $0.1$ | $2.537$ | $1.594$ | $0.249$ | $1.261$
Fast BBN | $0.2$ | $2.741$ | $1.639$ | $0.256$ | $1.269$
BBN reactions | $0.0$ | $4.043$ | $2.363$ | $0.243$ | $1.543$
Public Big bang | $0.0$ | $1.542$ | $3.000$ | $0.242$ | $4.884$
Public Big bang | $5.0\times 10^{-3}$ | $1.547$ | $3.002$ | $0.242$ | $4.907$
Public Big bang | $5.0\times 10^{-2}$ | $1.5845$ | $3.015$ | $0.246$ | $5.111$
Public Big bang | $0.1$ | $1.629$ | $3.030$ | $0.250$ | $5.345$
Public Big bang | $0.2$ | $1.717$ | $3.059$ | $0.257$ | $5.825$
our EDE model | $0.2$ | – | – | $0.249$ | –
wmap1 | $0.0$ | $2.75\pm 0.24$ | $0.93\pm 0.055$ | $0.2484\pm 0.0004$ | $3.82\pm 0.66$
wmap2 | $0.0$ | $2.60\pm 0.18$ | $1.04\pm 0.04$ | $0.2479\pm 0.0004$ | $4.15\pm 0.47$
Moreover, it is possible to assess the baryon to photon ratio from the CMB
temperature anisotropies spectrum, taking into account the relation:
$\eta_{B}=\frac{n_{B}}{n_{\gamma}}=5.5\times
10^{-10}\left(\frac{\Omega_{B}h^{2}}{0.022}\right)$ (39)
In the table 4 are shown the values of the parameters related with BBN
($\Omega_{B}$ and $\eta_{B}$) in terms of $b$, where $b=0.0$ is the associated
value to $\Lambda$CDM and $b=0.2$, the maximum contribution of the field
during radiation domination epoch:
$\rho_{\phi}$ | $\Omega_{B}$ | $\Omega_{B}h^{2}$ | $\eta_{B}\times 10^{-10}$
---|---|---|---
$b=0.0$ | 0.044 | 0.02218 | 6.20
$b=0.2$ | 0.053 | 0.02692 | 6.73
wmap1 | $0.0449\pm 0.0028$ | $0.02258\pm 0.00057$ | $6.190\pm 0.145$
According to the results shown in the table V, we plot the abundances for the
light nuclei including ${}^{4}He$, for different contribution of the field,
implemented as effective relativistic degrees of freedom in the code Public
BBN.
Figure 6: $b=0.0$ (solid line), $b=5.0\times 10^{-3}$ (dashed line),
$b=5.0\times 10^{-2}$ (dotted line), $b=0.1$ (width line) and $b=0.2$ (dashed
dotted line).
## VI Conclusions
In this work, we have made a general description of the dark energy component
as a K-essence scalar field that evolves during radiation as a tracker
($\omega_{\phi}|_{rad}=\frac{1}{3}$) and at late times achieves the De-Sitter
attractor $\omega_{\phi}\rightarrow-1$, emulating cosmological constant
$\Lambda$. All the description was made in the hot cosmological plasm
including matter-radiation components and assuming a spacially flat universe.
With these assumptions for the field, we have proposed an effective
parametrization for the state equation $\omega_{\phi}$ (22), that its free
parameters estimations were obtained by minimizing the function $\chi^{2}$
with the distances modulus of the Type Ia Supernovae.
We have rewritten the velocity of the field and the non-canonical kinetic term
of the K-essence lagrangian in terms of $z$, obtaining the completed behaviour
of the field during the thermal history and this let us say that K-essence
system includes different classes of Quintessence, when it is considered
specific cases of the non-canonical term. Another advantage with this
formulation is that avoids the Fine-tuning of the initials conditions of the
field and its velocity.
As it was expected, during radiation and matter domination epochs the field
has upper predictions for some observables $\Omega_{m0}$, $\Omega_{\phi 0}$
and $z_{*}$ (but respecting the observational bounds for them) comparing with
$\Lambda$CDM model. However, at late times $z\sim 0$ tends asymptotically the
standard model, after it has evolved from the radiation to the second
attractor. All these results are in agreement with the conditions that have
been imposed to resolve the dynamical system kang .
On the other hand, when it is included the non-null contribution of the field
during radiation domination era, the Hubble factor is affected, but also the
time capture and therefore, the primordial light nuclei abundances, because
there were more neutrons out of the equilibrium to form 4He by two body-
reactions. Actually, the whole reactions occurred faster, such that the
production of the nuclei are more effective and drives in an upper mass
fraction.
The predicted value for the 4He abundance prediction according to our model is
inside the observational bounds, then, our model is an excellent candidate to
a dynamic dark energy with a subdominant contribution during radiation and
matter epoch.
Finally, it is remarkable that the results of this paper can be compared with
other kind of models, because the field degrees of freedom can be treated as
effective degrees of some other component (for instance, a Quintessence scalar
field or even, sterile neutrinos lua ). In fact, we have include the field
contribution in the numerical codes in these way, and it let us to compare
degenerations of different kind of models.
###### Acknowledgements.
This work is supported and developed by the Observatorio Astronómico Nacional
under the auspices of Universidad Nacional de Colombia.
We also want to thank Daniel Molano, Carlos Cedeño and all the professors who
give us an advise to enhance this paper in the international events that this
work was presented.
## References
* (1) S. M. Carroll, Living Rev. Relativity 3 (2001).
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* (5) L. García, J. Tejeiro, L. Castañeda. Proceedings of the International School of Physics Enrico Fermi 178, 309,(2011).
* (6) A. Rendall. arXiv:gr-qc/0511.158,(2005).
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* (8) C. Armendariz-Picon, V. Mukhanov, T. Damour. Phys. Lett.B 458, 209),(1999).
* (9) L. Kawano. Fermilab preprint FERMILAB-PUB-92/04-A,(1992).
* (10) R. J. Scherrer. Phys. Rev. Lett. 93 011301,(2004).
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* (13) R. Saitou, S. Nojiri. arXiv:hep-th/1104.0558,(2011).
* (14) X. Gao, R. Yang. arXiv:gr-qc/1003.2786v1,(2010).
* (15) R. Yang, X. Gao. arXiv:gr-qc/1006.4986,(2010).
* (16) J. Kang, V. Vanchurin, S. Winitzki. Phys. Rev. D 76, 083511,(2007).
* (17) C. Kellogg. PRadiation Lab preprint OAP-714.
* (18) J. MacDonald, D. J. Mullan. Physics Rev. D 80, 043507,(2009).
* (19) A. Riess, A. Filippenko, W. Li, B. Schmidt. Astrophys.J. 536, 62,(2000).
* (20) R. H. Cyburt, B. D. Fields, K. A. Olive. Physics Letters B 567, 227-234,(2003).
* (21) A. Coc, E. Vangioni-Flam, P. Descouvemont, A. Adahchour, C. Angulo. The Astrophysical Journal, 600:544 552,(2001).
* (22) E. Komatsu, et. al. ApJS, 192, 18,(2010).
* (23) E. J. Copeland, M. Sami, S. Tsujikawa. arXiv:hep-th/0603057 ,(2006).
* (24) G. Steigman. Proceedings IAU Symposium No. 265, (2009).
* (25) J. Bernstein, L. S. Brown, G. Feinberg. Reviews of Modern physics, Vol. 61, No. 1,(1989).
* (26) G. Steigman, J.P. Kneller. arXiv:astro-ph/0210500,(2002).
* (27) G. Panotopoulos. arXiv:gr-qc/1107.4475v1,(2011).
* (28) T. Chiba. arXiv:astro-ph/0206298,(2002).
|
arxiv-papers
| 2012-10-18T20:41:39 |
2024-09-04T02:49:36.802166
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Garc\\'ia, J. Tejeiro, L. Casta\\~neda",
"submitter": "Leonardo Castaneda",
"url": "https://arxiv.org/abs/1210.5259"
}
|
1210.5274
|
# Results of the First IPTA Closed Mock Data Challenge
J. Ellis Department of Physics, University of Wisconsin-Milwaukee, Milwaukee,
WI 53201, USA X. Siemens Department of Physics, University of Wisconsin-
Milwaukee, Milwaukee, WI 53201, USA S. Chamberlin Department of Physics,
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
###### Abstract
The 2012 International Pulsar Timing Array (IPTA) Mock Data Challenge (MDC) is
designed to test current Gravitational Wave (GW) detection algorithms. Here we
will briefly outline two detection algorithms for a stochastic background of
gravitational waves, namely, a first-order likelihood method and an optimal
statistic method and present our results from the closed MDC data sets.
## I Generating the Residuals
The first step in any data analysis pipeline used in this paper is to generate
the residuals and make sure that all of the fits converged. Here we use tempo2
with the general2 plugin to generate the residuals. For our analysis we will
also need the design matrix for each pulsar which is obtained using the
designmatrix plugin developed by R. van Haasteren and modified by J. Ellis.
From inspection of the residuals, it is immediately obvious that all three
datasets contain strong red noise in many of the residuals. We can also see
“by-eye” that the residuals of dataset 1 have similar errorbars and that the
errorbars of datasets 2 and 3 seem to vary quite a bit from pulsar to pulsar.
## II Noise Estimation
The next step in our analysis pipelines is to estimate the red and white noise
levels in each set of residuals. A detailed description of the process is in
preparation (Ellis et al., 2012). Here we will simply review the method and
present the results from the Mock Data Challenge. We model the noise in the
data as a sum of three processes: a red component with a power law power
spectrum, a systematic white noise component that multiplies the residual
error bars (EFAC), and an extra white noise component independent of the error
bars (EQUAD). We work in the time domain, so these noise processes can be
completely described by their covariance matrices. Here we model the _pre-fit_
covariance matrix as
$\Sigma_{y}=\langle\mathbf{y}\mathbf{y}^{T}\rangle=C_{y}^{\rm red}+C_{y}^{\rm
EFAC}+C_{y}^{\rm EQUAD}.$ (1)
Details of the various components and transformations into a _post-fit_ basis
will be described elsewhere (Ellis et al., 2012). For this discussion it is
only important to note that we maximize the likelihood function
$\mathcal{L}(\vec{\theta}|\mathbf{r})=\frac{1}{\sqrt{\det(2\pi\Sigma_{r})}}\exp(-\frac{1}{2}\mathbf{r}^{T}\Sigma_{r}^{-1}\mathbf{r})$
(2)
over the parameters
$\vec{\theta}=\\{\mathcal{A},\gamma,\mathcal{F},\mathcal{Q}\\}$, where
$\mathcal{A}$ and $\gamma$ are the amplitude and spectral index of the red
noise power spectrum and $\mathcal{F}$ and $\mathcal{Q}$ are the EFAC and
EQUAD parameters, respectively. In the above expression, $\Sigma_{r}$ and
$\mathbf{r}$ are the _post-fit_ covariance matrix and residuals, respectively.
We have analyzed each pulsar from each dataset and compiled the results in
Figures 1–3. For this four parameter search, typical runtimes are on the order
of 30 minutes per pulsar. When run in parallel using MPI, these runtimes
decrease to a few minutes per pulsar.
Figure 1: Results of our noise estimation algorithm on individual dataset 1
pulsars. The left plot is the amplitude of a red noise signal converted to
units of GWB strain vs. the power spectral index. The plot on the right is the
EFAC parameter plotted against the EQUAD parameter in microseconds. Figure 2:
Results of our noise estimation algorithm on individual dataset 2 pulsars. The
left plot is the amplitude of a red noise signal converted to units of GWB
strain vs. the power spectral index. Note that the red points show upper
limits on the amplitude and EQUAD parameters respectively. This is done
because these points are consistent with 0 at the 1-sigma level. Here, 32 of
the 36 pulsars show evidence for red noise in the individual case indicating
the presence of a strong red noise signal in nearly all pulsars. The plot on
the right is the EFAC parameter plotted against the EQUAD parameter in
microseconds. Figure 3: Results of our noise estimation algorithm on
individual dataset 3 pulsars. The left plot is the amplitude of a red noise
signal converted to units of GWB strain vs. the power spectral index. Note
that the red points show upper limits on the amplitude and EQUAD parameters
respectively. This is done because these points are consistent with 0 at the
1-sigma level. Here, only 15 of the 36 pulsars show evidence for red noise in
the individual case indicating that many pulsars are white noise dominated.
The plot on the right is the EFAC parameter plotted against the EQUAD
parameter in microseconds.
In these figures we plot the red noise amplitude in familiar GWB strain units
vs. the power spectral index. For example, in these units a GWB from
supermassive black hole binaries SMBHBs with a characteristic spectral index
of -2/3 will have a power spectral index of 13/3. We also plot the EFAC
parameter vs. the EQUAD parameter. In general if the white noise were
completely described by the errorbars then the EFAC parameter is unity and the
EQUAD parameter is 0. However, since these datasets have equal errorbars on
all TOAs, EFAC and EQUAD are degenerate and may not result in the above
situation. However, from these results we can conclude that there is no
evidence for any white noise other than radiometer noise contained in the
error bars. Red points indicate that the amplitude or EQUAD is consistent with
0 at the one sigma level and we simply plot upper limits on those parameters
(red triangles). Here we notice that MDC 1 has strong red noise in all the
residuals clustered around an Amplitude of $A\sim 1\times 10^{-14}$ and a
spectral index of $\gamma\sim 4.0$. MDC 2 is centered around a higher
amplitude and similar spectral index but with a larger spread. In contrast,
many of the pulsars in MDC 3 show little evidence for red noise, with quite a
large spread in both spectral index and amplitude.
## III Analysis Methods
### III.1 Optimal Statistic
The optimal statistic (OS) used for this work is based on that published in
Anholm et al. (2009) and will be further discussed in a future work. Here we
will briefly review the algorithm. In general, we wish to construct a
statistic that gives some measure of the significance of the expected
(Hellings-Downs curve) correlations among pulsars giving larger weight to
those residuals with lower noise levels. It is also useful to construct this
statistic, such that, on average its value is the amplitude of the GWB for
some given spectral index. It is shown in Anholm et al. (2009) that this
statistic is both an optimal filter and maximizes the likelihood function in
the low-signal limit. The optimal statistic can be written as
$\hat{\Omega}=\frac{\sum_{I,J}r_{I}^{T}P_{I}^{-1}S_{IJ}P_{J}^{-1}r_{J}}{\sum_{I,J}{\rm
tr}\left[P_{I}^{-1}S_{IJ}P_{J}^{-1}S_{JI}\right]}$ (3)
with an SNR of
$\hat{\rho}=\frac{\sum_{I,J}r_{I}^{T}P_{I}^{-1}S_{IJ}P_{J}^{-1}r_{J}}{\sqrt{\sum_{I,J}{\rm
tr}\left[P_{I}^{-1}S_{IJ}P_{J}^{-1}S_{JI}\right]}},$ (4)
where $\sum_{I,J}$ denotes the sum over pulsar pairs, $r_{I}$, $P_{I}$, and
$S_{IJ}$ are the residuals for pulsar $I$, the auto-covariance matrix of
pulsar $I$ and the amplitude free cross covariance matrix of pulsar pair $IJ$.
We use the amplitude free cross covariance matrix to ensure that
$\langle\hat{\Omega}\rangle=\Omega_{\rm true}$ under the assumption that
$\langle r_{I}r_{J}^{T}\rangle=\Omega_{\rm true}S_{IJ}$. The SNR used here
gives a measure of how strong the _correlated_ signal is compared to an
_uncorrelated_ signal of the same strength.
For this analysis we use two versions of the optimal statistic: the zero-order
and the iterative optimal statistic. The zero-order optimal statistic consists
of estimating the maximum likelihood noise parameters, as discussed above, and
forming the auto-covariance matrices as input into the code. In general, this
method will give a biased result for the estimate of the GWB amplitude
$\Omega$ because we are ignoring the presence of a common GWB signal in all
pulsars. As we can see from Figures 1–3, there is quite a large spread on the
red noise parameters and this large spread adds to the bias in the optimal
statistic. However, if we iterate this procedure we can obtain a much better
estimate of $\Omega$ and a higher SNR. The iterative procedure is as follows:
1. 1.
Run the zero-order likelihood with the auto-covariance matrices produced from
the noise estimation procedure to obtain an estimate of the GWB amplitude,
$\Omega_{0}$ (here we assume a known spectral index in the construction of the
cross-covariance matrices).
2. 2.
Now fix the spectral index for both the auto and cross covariance matrices and
use the estimate $\Omega_{0}$ from the previous step to produce new auto-
covariance matrices.
3. 3.
Run the optimal statistic code again, now with the new auto-covariance
matrices and produce a new estimate of the amplitude $\Omega_{i}$
4. 4.
Repeat these steps until the new estimate of the amplitude changes from the
previous estimate by some amount $\varepsilon$111For this analysis,
$\varepsilon$ was chosen to be 0.005..
It is important to note that for the iterative method to be valid we must have
a priori knowledge about any red noise intrinsic to the pulsars. However, for
the MDC datasets, no pulsars show evidence of intrinsic red noise, so it is a
good approximation to assume that the only red noise source is the GWB.
Results for both the zero-order and iterative optimal statistics are shown in
Table LABEL:tab:results and will be discussed in the final section.
### III.2 First-Order Likelihood
The first-order likelihood combines elements of van Haasteren et al. (2009)
and Anholm et al. (2009) and will be published in a future paper. For a pulsar
timing array with $M$ pulsars we define the probability distribution function
of the presumed Gaussian noise as multivariate Gaussian222Here we will assume
that all noise processes are stochastic and that there are no deterministic
signals present in the data
$p(\mathbf{r})=\frac{1}{\sqrt{\det
2\pi\boldsymbol{\Sigma}}}\exp\left(-\frac{1}{2}\mathbf{r}^{T}\boldsymbol{\Sigma}^{-1}\mathbf{r}\right),$
(5)
where
$\mathbf{r}=\begin{bmatrix}{r}_{1}\\\ {r}_{2}\\\ \vdots\\\
{r}_{M}\end{bmatrix}$ (6)
is a vector of the residual time-series, $r_{\alpha}(t)$, for all pulsars,
$\boldsymbol{\Sigma}=\begin{bmatrix}P_{1}&S_{12}&\ldots&S_{1M}\\\
S_{21}&P_{2}&\ldots&S_{2M}\\\ \vdots&\vdots&\ddots&\vdots\\\
S_{M1}&S_{M2}&\ldots&P_{M}\end{bmatrix}$ (7)
is a multivariate block covariance matrix, and
$\displaystyle P_{{\alpha}}$ $\displaystyle=\langle
r_{\alpha}r_{\alpha}^{T}\rangle$ (8) $\displaystyle S_{\alpha\beta}$
$\displaystyle=\langle
r_{\alpha}r_{\beta}^{T}\rangle\big{|}_{\alpha\neq\beta},$ (9)
are the auto-covariance and cross-covariance matrices, respectively, for each
set of residuals.
Here we are interested in measuring the spectral index, $\gamma_{\rm gw}$, and
amplitude, $\Omega$, of the stochastic background. These parameters will be
common among all pulsars, however, as mentioned above, each pulsar will have
intrinsic noise parameters as well: an amplitude $A_{\alpha}$ and spectral
index $\gamma_{\alpha}$ for a power law red noise process, and EFAC and EQUAD
parameters, $\mathcal{F}_{\alpha}$ and $\mathcal{Q}_{\alpha}$, for white noise
processes. Therefore, we write our auto-covariance as a sum of a common GWB
term and a pulsar dependent term
$P_{\alpha}=R_{\alpha}+S_{a\alpha},$ (10)
where $R_{\alpha}$ is the intrinsic noise auto-covariance matrix and
$S_{a\alpha}$ is the common GWB auto-covariance matrix. It is convenient to
work in a block matrix notation where
$\mathbf{\Sigma}=\mathbf{P}+\mathbf{S}_{c},$ (11)
where $\mathbf{P}$ is a block diagonal matrix with diagonals $P_{\alpha}$ and
$\mathbf{S}_{c}$ is block matrix with off diagonals $S_{\alpha\beta}$ and zero
matrices on the diagonal.
In general, we write the log likelihood function as
$\ln\,\mathcal{L}=-\frac{1}{2}\left[\operatorname{Tr}\,\ln\mathbf{\Sigma}+\mathbf{r}^{T}\mathbf{\Sigma}^{-1}\mathbf{r}\right],$
(12)
where we have used the fact that $\ln\det(A)=\operatorname{Tr}\ln(A)$.
Figure 4: (from left to right) Results from our first-order likelihood method
run on the “best 12” pulsars for closed mock datasets 1, 2 and 3. Here the
solid, dashed and dash-dot lines are the 68%, 95%, and 99% contours,
respectively. The points with errorbars are the estimates of the amplitude
obtained from the iterative optimal statistic with 1-sigma uncertainties.
In practice the matrix $\mathbf{\Sigma}$ is quite large and therefore,
computationally prohibitive to invert. Since many multi-frequency residual
datasets now have on the order of $10^{3}$ points, for many modern PTAs the
matrix $\mathbf{\Sigma}$ will be of order $10^{4}\times 10^{4}$. To avoid
inverting the full covariance matrix we can expand the inverse out in a
Neumann series to obtain
$\mathbf{\Sigma}^{-1}=\mathbf{P}^{-1}-\mathbf{P}^{-1}\mathbf{S}_{c}\mathbf{P}^{-1}+\mathcal{O}(\epsilon^{2}).$
(13)
where $\epsilon$ is an order parameter. Physically, $\epsilon$ can represent
the amplitude of the GWB, $\Omega$, since it is a _small_ constant multiplier
of the elements of $\mathbf{S}_{c}$ Since the diagonal elements of
$\mathbf{\Sigma}$ will always be less than the diagonal terms, we can keep
only up to first order in $\epsilon$. The determinant can also be approximated
(to first order in $\epsilon$) as
$\ln\det\mathbf{\Sigma}=\operatorname{Tr}\ln\mathbf{P}+\mathcal{O}(\epsilon^{2}).$
(14)
With these approximations, it is now possible to write the approximate log-
likelihood
$\begin{split}\ln\mathcal{L}&=-\frac{1}{2}\left[\operatorname{Tr}\ln\mathbf{P}+\mathbf{r}^{T}\mathbf{P}^{-1}\mathbf{r}-\mathbf{r}^{T}\mathbf{P}^{-1}\mathbf{S}_{c}\mathbf{P}^{-1}\mathbf{r}\right]\\\
&=-\frac{1}{2}\sum_{I}\left[\operatorname{Tr}\ln
P_{I}+r_{I}^{T}P_{I}^{-1}r_{I}\right]+\Omega\sum_{I,J}r_{I}^{T}P_{I}^{-1}S_{IJ}P_{J}^{-1}r_{J},\end{split}$
(15)
where again, we have used the notation that $\sum_{I,J}$ denotes a sum of all
_unique_ pulsar pairs and $S_{IJ}$ is the _amplitude-free_ cross covariance
matrix. The results of evaluating the first order likelihood for MDC’s 1, 2,
and 3 are shown in Figure 4 and will be discussed in the next section.
## IV Executive Summary
Here we will discuss our results in detail and use this information to make a
decision as to what GWB signals are present in the data. As mentioned above,
first we ran the noise estimation analysis on all pulsars of all datasets. For
the purposes here we do not directly use the red noise information obtained
from this search in subsequent analyses, however, we do record the white noise
levels for all pulsars and find that the errorbars give a good estimate of the
total white noise level. We will use these white noise values throughout our
GWB analyses as the white noise components are not correlated with red noise
components, thus, the measurements of the white noise are independent of red
noise model assuming the data is well described by a two component noise
model.
We have shown above that the first-order likelihood is designed for signals
that are noise dominated. However, this is not the case for most residuals in
the three datasets. For both MDC 2 and MDC 3 we have split the datasets up
into thirds sorted by the white noise level. Using the open datasets and the
noise estimation results as a gauge, we have found that the best 12 pulsars
(sorted by the white noise level) is sufficiently noise dominated (across the
12 pulsar span) that the first-order likelihood performs well with no
discernible biases. We have not used all 36 pulsars because we see biases in
our results in the open datasets that we do not currently understand when
using all 36 pulsars, and we have not chosen the worst 12 because they are
noise dominated and show little evidence of a GWB. However, for MDC 1 all of
the pulsars are in the large signal limit thus we simply choose a random group
of 12 pulsars just to be consistent with the searches carried out for MDC’s 2
and 3. We also use 12 pulsars in an attempt to keep the number of search
parameters to a minimum (26 parameters in the case of 12 pulsars) and we have
found that our analysis method using MultiNest is slowly convergent in large
dimensional parameter spaces and in fact becomes computationally prohibitive
in the 36 pulsar case. Other samplers are being explored but were not tested
in time for this challenge.
Table 1: “best 12” pulsars for each dataset. MDC1 | MDC2 | MDC3
---|---|---
J0030+0451 | J1939+2134 | J1939+2134
J1738+0333 | J0437-4715 | J0437-4715
J1741+1351 | J1713+0747 | J1713+0747
J1744-1134 | J1909-3744 | J1909-3744
J1751-2857 | J1744-1134 | J1744-1134
J1853+1303 | J1910+1256 | J1910+1256
J1857+0943 | J1853+1303 | J1853+1303
J1732-5049 | J1955+2908 | J1955+2908
J1909-3744 | J1741+1351 | J1741+1351
J1918-0642 | J1640+2224 | J1640+2224
J1939+2134 | J1600-3053 | J1600-3053
J1955+2908 | J1738+0333 | J1738+0333
With these datasets now chosen we have run our first-order likelihood assuming
three different models:
1. 1.
Stochastic GWB with amplitud$A$, spectral index $\gamma_{\rm gw}$ with
Hellings-Downs correlation coefficients plus intrinsic red noise with a power
law red noise with amplitude $A_{i}$ and spectral index $\gamma_{i}$ and white
noise given by the noise estimation estimates. We use flat priors in the GW
amplitude and spectral index parameters and the noise amplitude and spectral
index with $A\in(0,5\times 10^{-13})$, $\gamma_{\rm gw}\in[1,7]$,
$A_{i}\in(0,5\times 10^{-13})$333Here we use a red noise amplitude that is
analogous to the traditional GW strain amplitude, and $\gamma_{i}\in[1,7]$.
This results in a 2+$2M$ dimensional search, where $M$ is the number of
pulsars used (two parameters for the GWB and 2$M$ for the intrinsic noise
parameters).
2. 2.
Null hypothesis where all red noise processes are intrinsic. Again we assume
an intrinsic amplitude $A_{i}$ and spectral index $\gamma_{i}$ with noise
estimated white noise parameters and no GWB signal. This results in a $2M$
parameter search.
3. 3.
Common red noise signal with amplitude $\mathcal{B}_{\rm common}$ and spectral
index $\gamma_{\rm common}$ but no spatial correlations and intrinsic red
noise parameters as described above. Again, this results in a $2+2M$ parameter
search.
For each model we evaluate the Bayesian evidence and then compute the Bayes
factors $\mathcal{B}_{0}$ and $\mathcal{B}_{\rm common}$, which compare the
GWB hypothesis to the null hypothesis and uncorrelated common red noise
hypothesis, respectively. All of these searches were run on the Nemo Computing
Cluster at UWM using the MultiNest (Feroz et al., 2009) algorithm. Typical
runtimes using 20 CPUs for each dataset (12 pulsars) are around 2 hours.
Table 2: Table of results from many analysis methods. Here we have estimates of the GWB amplitude $A$ and the corresponding SNR using the zero-order optimal statistic and iterative optimal statistic methods described above. The uncertainty on $A$ represent the 1-$\sigma$ confidence levels. We have also listed the results of our first-order likelihood method, also described above. We quote the maximum of the marginalized posterior distribution for both the GWB amplitude, $A$, and the spectral index $\gamma$ as well as the natural logarithm of the two Bayes factors described in the text. The uncertainties on $A$ and $\gamma$ represent the 68% credible regions. | Zero Order OSa | Iterative OS | | First-Order Likelihoodb
---|---|---|---|---
| $A[\times 10^{-14}]$ | SNR | $A[\times 10^{-14}]$ | SNR | | $A[\times 10^{-14}]$ | $\gamma$ | $\ln\,\mathcal{B}_{0}$ | $\ln\,\mathcal{B}_{\rm common}$
MDC 1 | $0.45$ | 4.3 | $0.92\pm 0.2$ | 9.2 | | $1.2^{+0.19}_{-0.16}$ | $4.26^{+0.22}_{-0.27}$ | 33.5 | 16.8
MDC 2 | $1.8$ | 2.0 | $4.3\pm 1.4$ | 9.5 | | $6.87^{+0.40}_{-0.65}$ | $4.16^{+0.18}_{-0.15}$ | 153.4 | 10.5
MDC 3 | $0.15$ | 1.2 | $0.63\pm 0.38$ | 5.5 | | $0.88^{+0.20}_{-0.22}$ | $3.58^{+0.48}_{-0.44}$ | 31.6 | 20.6
a An estimate of the amplitude is required to fully calculate the uncertainty, thus we do not include it here. |
b Note that these results are only for the “best 12” pulsars. |
The results of the first order likelihood run on the “best 12” pulsars is
shown in Table LABEL:tab:results and the list of pulsars used in each dataset
is shown in table LABEL:tab:psrs. Here we quote the maximum of the
marginalized posterior distribution for both the amplitude (written in
characteristic strain units) and spectral index as well as the 68% credible
regions and the logarithm of the two Bayes factors, $\mathcal{B}_{0}$ and
$\mathcal{B}_{\rm common}$. Firstly, we note that all three datasets (using
the “best 12” pulsars) have large Bayes factors in favor of a GWB model over a
model with just _intrinsic_ red noise444Remember, a Bayes factor $>10$ is
strong evidence against the competing hypothesis.. We also note that all 3
datasets also strongly favor a GWB model over an uncorrelated common red noise
signal. As we will note shortly, we can use the optimal statistic in
combination with the first-order likelihood to make a model decision in
situations where the evidence for a particular model is weak. We also see that
the most likely spectral index is consistent with $13/3$ (SMBHB spectrum) for
all three datasets. We also note that there was no evidence for red noise in
MDC1 or MDC2, however, we did find evidence for intrinsic red noise in closed
dataset 3. Although there is evidence for red noise, we cannot accurately
characterize it in many cases. For PSRs J1939+2134, J0437-4715 and J1955+2908,
the 1-sigma contours do are not consistent with red noise of 0 amplitude,
however for all others we are unable to characterize the red noise. Contour
plots of the red noise parameters for MDC3 are shown in Figure 5.
Figure 5: Noise contours for the “best 12” pulsars of MDC3. The $x$-axis is
the amplitude measured in GW strain units normalized by $10^{-14}$ and the
$y$-axis is the power spectral index. The solid, dashed, and dot-dashed lines
represent the 1, 2, and 3 sigma confidence contours, respectively.
We have also run both the zero-order and iterative optimal statistics on the
_full_ datasets. Typical run times are on the order of 5 minutes on a modern
workstation. As mentioned above, for the zero order OS we use the noise
estimation results shown in Figures 1–3 to construct the auto-covariance
matrices. The amplitude (written in characteristic strain units) and
corresponding SNR are shown in the second and third columns of Table
LABEL:tab:results. We see that there is a significant detection for MDC 1 and
marginal detections for MDC’s 2 and 3 using this method. Intuitively, this
makes sense because there is more white noise in pulsars from MDC’s 2 and 3
and thus a larger spread in the estimated amplitudes and spectral indices of
the GWB (based on the first order likelihood results we assume that any
measured red noise is due to the GWB) and thus a more biased result from the
optimal statistic. When we now use the iterative method we make confident
detections for all three datasets and recover amplitudes that are consistent
with the first-order likelihood at the 95% level or better.
## References
* Anholm et al. (2009) Anholm, M., Ballmer, S., Creighton, J. D. E., Price, L. R., & Siemens, X. 2009, Phys. Rev. D, 79, 084030
* Ellis et al. (2012) Ellis, J. A., et al. 2012, in preparation
* Feroz et al. (2009) Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601
* van Haasteren et al. (2009) van Haasteren, R., Levin, Y., McDonald, P., & Lu, T. 2009, MNRAS, 395, 1005
|
arxiv-papers
| 2012-10-18T22:47:48 |
2024-09-04T02:49:36.811037
|
{
"license": "Public Domain",
"authors": "Justin Ellis, Xavier Siemens, Sydney Chamberlin",
"submitter": "Justin Ellis",
"url": "https://arxiv.org/abs/1210.5274"
}
|
1210.5279
|
IC/HEP/12-03 March 18, 2013
The effect of S-wave interference on the $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ angular observables
Thomas Blakea, Ulrik Egedeb, Alex Shiresb111Corresponding author.
_a CERN, Geneva, Switzerland_
_b Imperial College London, London SW7 2AZ, United Kingdom_
The rare decay $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ is a flavour
changing neutral current decay with a high sensitivity to physics beyond the
Standard Model. Nearly all theoretical predictions and all experimental
measurements so far have assumed a $K^{*0}$ P-wave that decays into the
$K^{+}\pi^{-}$ final state. In this paper the addition of an S-wave within the
$K^{+}\pi^{-}$ system of $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ and the
subsequent impact of this on the angular distribution of the final state
particles is explored. The inclusion of the S-wave causes a distinction
between the values of the angular observables obtained from counting
experiments and those obtained from fits to the angular distribution. The
effect of a non-zero S-wave on an angular analysis of $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ is assessed as a function of dataset size and the
relative size of the S-wave amplitude. An S-wave contribution, equivalent to
what is measured in $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ at BaBar, leads to a significant bias on the angular observables
for datasets of above 200 signal decays. Any future experimental analysis of
the $K^{+}\pi^{-}\ell^{+}\ell^{-}$ final state will have to take the S-wave
contribution into account.
Keywords : B-Physics, Rare Decays
## 1 Introduction
The description of flavour physics in the Standard Model (SM) has so far
accurately matched the observations in the data from the $B$ factories, the
Tevatron and the LHC very well. However, there are several fundamental
questions which do not have an explanation within the SM such as the mass
hierarchy of the quarks and why there are three generations. To avoid creating
large flavour changing neutral currents, any physics beyond the SM that
contains new degrees of freedom that couple to the flavour sector is required
to be at an energy scale of multiple $\mathrm{\,Te\kern-1.00006ptV}$ or to
have small couplings between the generations, i.e. couplings that closely
mimic those of the SM. The measurement of the inclusive $b\\!\rightarrow
s\gamma$ width [1] is one of the strongest constraints on new physics from the
flavour sector; for the exclusive decays, $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ is of major importance.
The analysis of $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ is based on the
evaluating the angular distribution of the daughter particles [2]. How to
extract the maximal amount of information from the decay while keeping
uncertainties from QCD minimal has recently attracted much interest [3, 4, 5,
6, 7, 8]. The results from the experimental analyses of $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ [9, 10, 11, 12] have focused on the forward backward
asymmetry of the dimuon system ($A_{\mathrm{FB}}$) and the fraction of
longitudinal polarisation of the $K^{*0}$ ($F_{\mathrm{L}}$) as a function of
the dimuon invariant mass.
With the acquisition of large data sets of $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ decays, scrutiny is required of assumptions that have
been made in current experiments. Nearly all theoretical papers to date use
the narrow width assumption for the $K^{+}\pi^{-}$ system meaning that the
natural width of the $K^{*0}(892)$ is ignored. This means there is no
interference with other $K^{+}\pi^{-}$ resonances. Existing
$B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ analyses consider
$B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ signal with $K^{+}\pi^{-}$
candidates in a narrow mass window around the $K^{*0}(892)$. However, in this
region there is evidence of a broad S-wave below the $K^{*0}(892)$ and higher
mass states which decay strongly to $K^{+}\pi^{-}$, such as the S-wave
$K^{*0}_{\mathrm{0}}(1430)$ and the D-wave $K^{*0}_{\mathrm{2}}(1430)$ [13].
The best understanding of the low mass S-wave contribution comes from the
analysis of $K^{+}\pi^{-}$ scattering at the LASS experiment [14].
The interference of an S-wave in a predominantly P-wave system has previously
been used to disambiguate otherwise equivalent solutions for the value of the
$C\\!P$-violating phase in $B^{0}$ [15] and $B^{0}_{s}$ [16] oscillations. In
the determination of $\varphi_{s}$ in the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decay it was also shown that it is required to take the S-wave contribution
into account [17] and this has subsequently been done for the experimental
measurements [18, 19, 20]. The interference of a $K^{+}\pi^{-}$ S-wave in the
angular analysis of $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ has previously
been considered in Refs. [21, 22]. In both references, the authors show that
the presence of the S-wave can introduce significant biases to angular
observables in the decay. We extend these studies to explore the consequences
of the S-wave contribution for the present and future experimental analyses.
Further, we explore the interplay between statistical and systematical
uncertainties for different analysis approaches.
In this paper, we detail how a generic $K^{+}\pi^{-}$ S-wave contribution to
$B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ can be included in the angular
analysis. Firstly, we develop the formalism set out in [23] to explicitly
include a spin-0 S-wave and a spin-1 P-wave state in the $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$ angular distribution. Here $K^{*0}$ is used for
any neutral kaon state which decays to $K^{+}\pi^{-}$. The impact of an S-wave
contribution on the determination of the theoretical observables is evaluated
in two ways: in the first we look for the minimum sample size in which an
S-wave contribution (such as measured in [15]) significantly biases the
angular observables; secondly we determine, for a given sample size, the
minimum S-wave contribution needed to bias the angular observables. We then
demonstrate how the S-wave contribution can be correctly taken into account
and evaluate the effect of this on the statistical precision that can be
obtained on the angular observables with a given number of signal events.
## 2 The $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ angular distribution
The differential angular distribution for $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ is expressed as a function of the five kinematic
variables ($\cos{\theta_{l}}$, $\cos{\theta_{K}}$, $\phi$, $p^{2}$ and
$q^{2}$). The angle $\theta_{K}$ is defined as the angle between the $K^{+}$
and the $B^{0}$ momentum vector in the rest frame of the $K^{*0}$. The angle
$\theta_{l}$ is similarly defined between the $\ell^{+}$ in the rest frame of
the dilepton pair and the momentum vector of the $B^{0}$. The angle $\phi$ is
defined as the signed angle between the planes, in the rest frame of the
$B^{0}$, formed by the dilepton pair and the $K^{+}\pi^{-}$ pair
respectively.222This is the same sign convention for $\cos{\theta_{l}}$ and
$\cos{\theta_{K}}$ as used by the BaBar, Belle, CDF and LHCb experiments [9,
10, 11, 12] and the same $\phi$ convention as used in LHCb [24]. The mass
squared of the $K^{+}\pi^{-}$ system is denoted $p^{2}$ and the mass squared
of the dilepton pair $q^{2}$. The angular distribution is given as a function
of $\cos{\theta_{l}}$, $\cos{\theta_{K}}$ and $\phi$ as
$\displaystyle\frac{\text{d}^{5}\Gamma}{\text{d}q^{2}\text{d}p^{2}\mathrm{dcos}{\theta_{K}}\mathrm{dcos}{\theta_{l}}\mathrm{d}\phi}=$
$\displaystyle\frac{3}{8}\bigg{(}I_{1}^{c}+2I_{1}^{s}+(I_{2}^{c}+2I_{2}^{s})\cos
2\theta_{l}+2I_{3}\sin^{2}{\theta_{l}}\cos 2\phi$
$\displaystyle+2\sqrt{2}I_{4}\sin
2\theta_{l}\cos\phi+2\sqrt{2}I_{5}\sin{\theta_{l}}\cos\phi+2I_{6}\cos{\theta_{l}}$
(2.1) $\displaystyle+2\sqrt{2}I_{7}\sin{\theta_{l}}\sin\phi+2\sqrt{2}I_{8}\sin
2\theta_{l}\sin\phi+2\sqrt{2}I_{9}\sin^{2}{\theta_{l}}\sin 2\phi\bigg{)}$
Ignoring scalar and tensor contributions, the complete set of angular terms
are
$\displaystyle I_{1}^{c}$
$\displaystyle=|\mathcal{A}_{0L}|^{2}+|\mathcal{A}_{0R}|^{2}+8\frac{m_{l}^{2}}{q^{2}}\Re\left(\mathcal{A}_{L0}\mathcal{A}_{R0}^{*}\right)+4\frac{m_{l}^{2}}{q^{2}}|\mathcal{A}_{t}|^{2}\frac{}{},$
$\displaystyle I_{1}^{s}$
$\displaystyle=\frac{3}{4}\left(|\mathcal{A}_{L||}|^{2}+|\mathcal{A}_{L\bot}|^{2}+(L\rightarrow
R)\right)\left(1-\frac{4m_{l}^{2}}{q^{2}}\right)+\frac{4m_{l}^{2}}{q^{2}}\Re\left(\mathcal{A}_{L\bot}\mathcal{A}_{R\bot}+\mathcal{A}_{L||}\mathcal{A}_{R||}\right)\frac{}{},$
$\displaystyle I_{2}^{c}$
$\displaystyle=-\beta_{l}^{2}\left(|\mathcal{A}_{L0}|^{2}+|\mathcal{A}_{R0}|^{2}\right)\frac{}{},$
$\displaystyle I_{2}^{s}$
$\displaystyle=\frac{1}{4}\beta_{l}^{2}\left(|\mathcal{A}_{L||}|^{2}+|\mathcal{A}_{L\bot}|^{2}+|\mathcal{A}_{R||}|^{2}+|\mathcal{A}_{R\bot}|^{2}\right)\frac{}{},$
$\displaystyle I_{3}$
$\displaystyle=\frac{1}{2}\beta_{l}^{2}\left(|\mathcal{A}_{L\bot}|^{2}-|\mathcal{A}_{L||}|^{2}+|\mathcal{A}_{R\bot}|^{2}-|\mathcal{A}_{R||}|^{2}\right)\frac{}{},$
$\displaystyle I_{4}$
$\displaystyle=\frac{1}{\sqrt{2}}\beta_{l}^{2}\left(\Re(\mathcal{A}_{L0}\mathcal{A}_{L||}^{*})+(L\rightarrow
R)\right)\frac{}{},$ (2.2) $\displaystyle I_{5}$
$\displaystyle=\sqrt{2}\beta_{l}\left(\Re(\mathcal{A}_{L0}\mathcal{A}_{L\bot}^{*})-(L\rightarrow
R)\right)\frac{}{},$ $\displaystyle I_{6}$
$\displaystyle=2\beta_{l}\left(\Re(\mathcal{A}_{L||}\mathcal{A}_{L\bot}^{*})-(L\rightarrow
R)\right)\frac{}{},$ $\displaystyle I_{7}$
$\displaystyle=\sqrt{2}\beta_{l}\left(\Im(\mathcal{A}_{L0}\mathcal{A}_{L||}^{*})-(L\rightarrow
R)\right)\frac{}{},$ $\displaystyle I_{8}$
$\displaystyle=\frac{1}{\sqrt{2}}\beta_{l}^{2}\left(\Im(\mathcal{A}_{L0}\mathcal{A}_{L\bot}^{*})+(L\rightarrow
R)\right)\frac{}{},$ $\displaystyle I_{9}$
$\displaystyle=\beta_{l}^{2}\left(\Im(\mathcal{A}_{L||}\mathcal{A}_{L\bot}^{*})+(L\rightarrow
R)\right)\frac{}{},$
where $\mathcal{A}_{H(0,||,\bot,t)}$ are the $K^{*0}$ helicity amplitudes and
$\beta_{l}^{2}=1-4m_{l}^{2}/q^{2}$ [2]. In this paper the lepton mass is
assumed to be insignificant, such that the angular terms with
$m^{2}_{l}/q^{2}$ dependence can be neglected and $\beta_{l}=1$ such that
$I_{1}$ and $I_{2}$ can be related by $I_{2}^{c}=-I_{1}^{c}$ and
$I_{2}^{s}=\frac{1}{3}I_{1}^{s}$.
For a $K^{+}\pi^{-}$ state which is a combination of different spin states,
the amplitudes for a given handedness ($H=L,R$) can be expressed as a sum over
the resonances ($J$)
$\displaystyle\mathcal{A}_{H,0/t}(p^{2},q^{2})$ $\displaystyle=\sum_{J\geq
0}\sqrt{N_{J}}\ M_{J,H,0}(q^{2})\ P_{J}(p^{2})\ Y_{J}^{0}(\theta_{K},0),$
(2.3) $\displaystyle\mathcal{A}_{H,||/\bot}(p^{2},q^{2})$
$\displaystyle=\sum_{J\geq 1}\sqrt{N_{J}}\ M_{J,H,||/\bot}(q^{2})\
P_{J}(p^{2})\ Y_{J}^{-1}(\theta_{K},0),$
where $Y_{J}^{m}(\theta_{K},0)$ are the spherical harmonics, $M$ is the matrix
element and $P_{J}(p^{2})$ is the propagator of the spin state which
encompasses the $p^{2}$ dependence. A detailed description of the spin-
dependent matrix elements and normalisation factors can be found in Ref. [23].
## 3 Angular distribution of $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$ for a combined S- and P-wave
For $K^{+}\pi^{-}$ masses below $1200\mathrm{\,Me\kern-1.00006ptV}$,333Natural
units are assumed throughout this paper the contribution to the amplitudes
from the D-wave $K^{*0}(1430)$ is so small that it can be ignored [14] and
only the $J=0,1$ terms in the sums of Eq. 2.3 will be considered. The S-wave
contribution to these amplitudes only enters in $\mathcal{A}_{0}$ giving
$\displaystyle\mathcal{A}_{H,0}$
$\displaystyle=\frac{1}{\sqrt{4\pi}}A_{0,H,0}+\sqrt{\frac{3}{4\pi}}A_{1,H,0}\cos{\theta_{K}},$
$\displaystyle\mathcal{A}_{H,||}$
$\displaystyle=\sqrt{\frac{3}{4\pi}}A_{1,H,||}\cos{\theta_{K}},$ (3.1)
$\displaystyle\mathcal{A}_{H,\bot}$
$\displaystyle=\sqrt{\frac{3}{8\pi}}A_{1,H,\bot}\sin{\theta_{K}},$
where the spherical harmonics have been expanded out, leaving the propagator
and the matrix element as part of the spin-dependent amplitudes
$\begin{split}A_{0,H,0}&\propto\ M_{0,H,0}(q^{2})\ P_{0}(p^{2}),\\\
A_{1,H,0}&\propto\ M_{1,H,0}(q^{2})\ P_{1}(p^{2}),\\\ A_{1,H,\bot}&\propto\
M_{1,H,\bot}(q^{2})\ P_{1}(p^{2}),\\\ A_{1,H,||}&\propto\ M_{1,H,||}(q^{2})\
P_{1}(p^{2}),\end{split}$ (3.2)
where the first index denotes the spin and the normalisation from the three-
body phase space factor is omitted. The propagator for the P-wave is described
by a relativistic Breit-Wigner distribution with the amplitude given by
$\displaystyle
P_{1}(p^{2})=\frac{m_{K^{*0}_{\mathrm{1}}}\Gamma_{K^{*0}_{\mathrm{1}}}(p^{2})}{m_{K^{*0}_{\mathrm{1}}}^{2}-p^{2}+i\
m_{K^{*0}_{\mathrm{1}}}\Gamma_{K^{*0}_{\mathrm{1}}}(p^{2})}$ (3.3)
where $m_{K^{*0}_{\mathrm{1}}}$ is the resonant mass and
$\displaystyle\Gamma_{K^{*0}_{\mathrm{1}}}(p^{2})=\Gamma_{K^{*0}_{\mathrm{1}}}^{0}\left(\frac{t}{t_{0}}\right)^{3}\left(\frac{m_{K^{*0}_{\mathrm{1}}}}{p}\right)\frac{B\left(tR_{P}\right)}{B\left(t_{0}R_{P}\right)}$
(3.4)
the running width. Here $t$ is the $K^{+}$ momentum in the rest frame of the
$K^{+}\pi^{-}$ system and $t_{0}$ is t evaluated at the $K^{+}\pi^{-}$ pole
mass. $B$ is the Blatt-Weisskopf damping factor [25] with a radius $R_{P}$.
The amplitude can be defined in terms of a phase ($\delta$) through the
substitution
$\displaystyle\cot\delta=\frac{m_{K^{*0}_{\mathrm{1}}}^{2}-p^{2}}{\Gamma_{m_{K^{*0}_{\mathrm{1}}}}(p^{2})m_{K^{*0}_{\mathrm{1}}}}$
(3.5)
to give the polar form of the relativistic Breit-Wigner propagator
$\displaystyle P_{1}(p^{2})=\frac{1}{\cot\delta-i}$ (3.6)
The LASS parametrisation of the S-wave [14] can be used to describe a generic
$K^{+}\pi^{-}$ S-wave. In this parametrisation, the S-wave propagator is
defined as
$\displaystyle
P_{0}(p^{2})=\frac{p}{t}\left(\frac{1}{\cot\delta_{B}-i}+e^{2i\delta_{B}}(\frac{1}{\cot\delta_{R}-i})\right)$
(3.7)
where the first term is an empirical term from inelastic scattering and the
second term is the resonant contribution with a phase factor to retain
unitarity. The first phase factor is defined as
$\displaystyle\cot\delta_{B}=\frac{1}{ta}+\frac{1}{2}rt,$ (3.8)
where $r$ and $a$ are free parameters and $t$ is defined previously, while the
second phase factor describes the $K^{*0}_{\mathrm{0}}(1430)$ through
$\displaystyle\cot\delta_{R}=\frac{m_{\mathrm{S}}^{2}-p^{2}}{\Gamma_{\mathrm{S}}(p^{2})m_{\mathrm{S}}}.$
(3.9)
Here, $m_{\mathrm{S}}$ is the S-wave pole mass and $\Gamma_{\mathrm{S}}$ is
the running width using the pole mass of the $K^{*0}_{\mathrm{0}}(1430)$. The
overall strong phase shift between the results from the LASS scattering
experiment and measured values for $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $K^{+}\pi^{-}$ has been found
to be consistent with $\pi$ [15]. The parameters for the $p^{2}$ spectrum used
in this paper are given in Table 1.
Table 1: Parameters of the $K^{+}\pi^{-}$ resonances used to generate toy data sets. The $K^{*}$ masses and widths are taken from Ref. [13] and the $K^{*0}_{\mathrm{1}}$ Blatt-Weisskopf radius and the LASS parameters are taken from Ref. [26] State. | mass | $\Gamma$ | R | $r$ | $a$ | $\delta_{\mathrm{S}}$
---|---|---|---|---|---|---
| ($\mathrm{\,Me\kern-1.00006ptV}$) | $(\mathrm{\,Me\kern-1.00006ptV})$ | ($\mathrm{\,Ge\kern-1.00006ptV})^{-1}$ | $(\mathrm{\,Ge\kern-1.00006ptV})^{-1}$ | $(\mathrm{\,Ge\kern-1.00006ptV})^{-1}$ |
$K^{*0}_{\mathrm{1}}$ | $894.94\pm 0.22$ | $48.7\pm 0.8$ | $3.0$ | | |
$K^{*0}_{\mathrm{0}}$ | $1425\pm 50$ | $270\pm 80$ | $1.0$ | $1.94$ | $1.73$ | $\pi$
The angular terms modified by the inclusion of the S-wave are
$I_{1,2,4,5,7,8}$ and the complete set of angular terms expressed in terms of
the spin-dependent amplitudes is
$\displaystyle I_{1}^{c}$
$\displaystyle=\frac{1}{4\pi}|A_{0L0}|^{2}+\frac{3}{4\pi}|A_{1L0}|^{2}\cos^{2}{\theta_{K}}+2\frac{\sqrt{3}}{4\pi}|A_{0L0}||A_{1L0}|\cos\delta_{0,0}^{L}\cos{\theta_{K}}+(L\rightarrow
R)\frac{}{}$ $\displaystyle I_{1}^{s}$
$\displaystyle=\frac{3}{4}\frac{3}{8\pi}\left(|A_{1L||}|^{2}+|A_{1L\bot}|^{2}+(L\rightarrow
R)\right)\frac{}{}\sin^{2}{\theta_{K}}$ $\displaystyle I_{2}^{c}$
$\displaystyle=-I_{1}^{c},\qquad\,I_{2}^{s}=\frac{1}{3}I_{1}^{s}$
$\displaystyle I_{3}$
$\displaystyle=\frac{1}{2}\frac{3}{8\pi}\left(|A_{1L\bot}|^{2}-|A_{1L||}|^{2}+(L\rightarrow
R)\right)\sin^{2}{\theta_{K}}\frac{}{}$ $\displaystyle I_{4}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\frac{1}{4\pi}\sqrt{\frac{3}{2}}\Re(A_{0L0}A_{1L||}^{*})\cos\delta_{0,||}^{L}\sin{\theta_{K}}\right.\frac{}{}$
$\displaystyle+\left.\frac{3}{4\pi}\sqrt{\frac{1}{2}}\Re(A_{1L0}A_{1L||}^{*})\sin{\theta_{K}}\cos{\theta_{K}}+(L\rightarrow
R)\right]\frac{}{}$ $\displaystyle I_{5}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\frac{1}{4\pi}\sqrt{\frac{3}{2}}\Re(A_{0L0}A_{1L\bot}^{*})\cos\delta_{0,\bot}^{L}\sin{\theta_{K}}\right.\frac{}{}$
$\displaystyle+\left.\frac{3}{4\pi}\sqrt{\frac{1}{2}}\Re(A_{1L0}A_{1L\bot}^{*})\sin{\theta_{K}}\cos{\theta_{K}}-(L\rightarrow
R)\right]\frac{}{}$ (3.10) $\displaystyle I_{6}$
$\displaystyle=2\frac{3}{8\pi}\left(\Re(A_{1L||}A_{1L\bot}^{*})-(L\rightarrow
R)\right)\sin^{2}{\theta_{K}}\frac{}{}$ $\displaystyle I_{7}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\frac{1}{4\pi}\sqrt{\frac{3}{2}}\Im(A_{0L0}A_{1L||}^{*})\cos\delta_{0,||}^{L}\sin{\theta_{K}}\right.\frac{}{}$
$\displaystyle+\left.\frac{3}{4\pi}\sqrt{\frac{1}{2}}\Im(A_{1L0}A_{1L||}^{*})\sin{\theta_{K}}\cos{\theta_{K}}-(L\rightarrow
R)\right]\frac{}{}$ $\displaystyle I_{8}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\frac{1}{4\pi}\sqrt{\frac{3}{2}}\Im(A_{0L0}A_{1L\bot}^{*})\cos\delta_{0,\bot}^{L}\sin{\theta_{K}}\right.\frac{}{}$
$\displaystyle+\left.\frac{3}{4\pi}\sqrt{\frac{1}{2}}\Im(A_{1L0}A_{1L\bot}^{*})\sin{\theta_{K}}\cos{\theta_{K}}+(L\rightarrow
R)\right]\frac{}{}$ $\displaystyle I_{9}$
$\displaystyle=\frac{3}{8\pi}\left(\Im(A_{1L||}A_{1L\bot}^{*})+(L\rightarrow
R)\right)\sin^{2}{\theta_{K}}\frac{}{}$
The interference term of $I_{1}$ shows how this parametrisation encompasses
the strong phase difference between the S and P-wave state. The left handed
part of the interference term for $I_{1}$ can be written as
$\displaystyle 2|A_{0L0}||A_{1L0}|\cos\delta_{0,0}^{L}\propto
2\,|M_{0,L,0}||P_{0}(p^{2})||M_{1,L,0}||P_{1}(p^{2})|\cos(\delta_{0,0}^{L})$
(3.11)
where
$\displaystyle\delta_{0,0}^{L}=\delta_{M_{0L0}}+\delta_{P_{0}}-\delta_{M_{1L0}}-\delta_{P_{1}}.$
(3.12)
where $\delta_{M_{JL0}}$ is the phase of the longitudinal matrix element and
$\delta_{P_{J}}$ is the phase of the propagator. The phases in the
interference terms for $I_{4,5,7,8}$ can be similarly defined. For real matrix
elements, i.e. nearly true in the Standard Model, the phases are equal for
both handed interference terms $\delta^{L}=\delta^{R}$. The phase difference
between the S-wave and the P-wave propagators can be expressed as a single
strong phase, $\delta_{\mathrm{S}}$.
The $p^{2}$ spectrum for the $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$ angular distribution can be calculated by
summing over the S- and P-waves and integrating out the $\cos{\theta_{l}}$,
$\cos{\theta_{K}}$ and $\phi$ dependence. This is illustrated in Fig. 1
Figure 1: An illustration of the $p^{2}$ spectrum for the P-wave (dashed) and
the S-wave (dash-dotted). The total distribution from both states is the solid
line. The values were calculated at $q^{2}=6\mathrm{\,Ge\kern-1.00006ptV}^{2}$
by integrating out the angular distribution of $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$ using equal matrix elements for each state. The
S-wave fraction here is 16% between $800<p<1000\mathrm{\,Me\kern-1.00006ptV}$
where the matrix elements from Refs [4, 6] at a $q^{2}$ value of
$6\mathrm{\,Ge\kern-1.00006ptV}^{2}$ are used. Here the S-wave amplitude is
assumed to be equivalent to the longitudinal P-wave amplitude. The S-wave
fraction in the $800<p<1000\mathrm{\,Me\kern-1.00006ptV}$ window around around
the P-wave is calculated to be 16% when using this approximation. As will be
seen later there are no interference terms left in the angular distribution
after the integral over $\cos{\theta_{K}}$.
## 4 The effect on $B^{0}\\!\rightarrow K^{+}\pi^{-}\ell^{+}\ell^{-}$
observables
So far the forward-backward asymmetry ($A_{\mathrm{FB}}$), the fraction of the
$K^{*0}$ longitudinal polarisation ($F_{\mathrm{L}}$) and two combinations of
the transverse amplitudes ($A_{\mathrm{T}}^{2}$ and $A_{\mathrm{Im}}$) have
been measured. As such, these are the observables that will be concentrated on
here.
$A_{\mathrm{FB}}$ is defined in terms of the amplitudes as
$\displaystyle A_{\mathrm{FB}}(q^{2})$
$\displaystyle=\frac{3}{2}\frac{\Re(A_{1L||}A_{1L\bot}^{*})-\Re(A_{1R||}A_{1R\bot}^{*})}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}}$
(4.1)
for a pure P-wave state where the generic combination of amplitudes
$A_{Ji}A_{Ji}^{*}$ is defined as
$\displaystyle A_{Ji}A_{Ji}^{*}$
$\displaystyle=A_{JiL}A_{JiL}^{*}+A_{JiR}A_{JiR}^{*}.$ (4.2)
where $i\in\\{0,||,\bot,t\\}$ and $J=0,1$. The factorisation of the amplitudes
into matrix elements and the propagators removes the $p^{2}$ dependence from
the theoretical observables. In a similar way, $F_{\mathrm{L}}$,
$A_{\mathrm{T}}^{2}$ and $A_{\mathrm{Im}}$ are defined as
$\begin{split}F_{\mathrm{L}}(q^{2})&=\frac{|A_{10}|^{2}}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}},\\\
A_{\mathrm{T}}^{2}(q^{2})&=\frac{|A_{1\bot}|^{2}-|A_{1||}|^{2}}{|A_{1\bot}|^{2}+|A_{1||}|^{2}}=(1-F_{\mathrm{L}})\frac{|A_{1\bot}|^{2}-|A_{1||}|^{2}}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}},\\\
A_{\mathrm{Im}}(q^{2})&=\frac{\Im(A_{1L||}A_{1L\bot}^{*})+\Im(A_{1R||}A_{1R\bot}^{*})}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}}.\end{split}$
(4.3)
These theoretical observables are normalised to the sum of the spin-1
amplitudes. In terms of the angular distribution, $A_{\mathrm{FB}}$ can also
be expressed as the difference between the number of ‘forward-going’ $\mu^{+}$
and the number of ‘backward-going’ $\mu^{+}$ in the rest frame of the $B^{0}$,
$\displaystyle\left[\int_{0}^{1}-\int_{-1}^{0}\right]\text{d}\cos{\theta_{l}}\frac{\text{d}\Gamma}{\text{d}q^{2}\text{d}\cos{\theta_{l}}}/\frac{\text{d}\Gamma}{\text{d}q^{2}}$
(4.4)
which explains the name of the observable. In Ref [24], this expression was
used to determine the zero-crossing point of $A_{\mathrm{FB}}$. The inclusion
of the S-wave in the complete angular distribution means that
$A_{\mathrm{FB}}$ can no longer be determined by experimentally counting the
number of events with forward-going and backward-going leptons, as Eqs. 4.1
and 4.4 are no longer equivalent. However, as the S-wave has no forward-
backward asymmetry, no bias occurs in the determination of the zero-crossing
point by ignoring the S-wave. The total normalisation for the angular
distribution changes to the sum of S- and P-wave amplitudes,
$\displaystyle\Gamma^{{}^{\prime\prime}}$
$\displaystyle\equiv\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}p^{2}\mathrm{d}q^{2}}=|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}+|A_{00}|^{2}.$
(4.5)
such that there is a factor of
$\displaystyle\mathcal{F}_{\mathrm{P}}(p^{2},q^{2})$
$\displaystyle=\left(\frac{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}+|A_{00}|^{2}}\right)$
(4.6)
between the pure P-wave and the admixture of the S and the P-wave. This is the
fraction of the yield coming from the P-wave at a given value of $p^{2}$ and
$q^{2}$. Similarly, the S-wave fraction is defined as
$\displaystyle\mathcal{F}_{\mathrm{S}}(p^{2},q^{2})$
$\displaystyle=\left(\frac{|A_{00}|^{2}}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}+|A_{00}|^{2}}\right)$
(4.7)
and the interference between the S-wave and the P-wave as
$\displaystyle\mathcal{A}_{\mathrm{S}}(p^{2},q^{2})$
$\displaystyle=\frac{\sqrt{3}}{2}\left(\frac{|A_{0L0}||A_{1L0}|\cos\delta_{L}+(L\rightarrow
R)}{|A_{10}|^{2}+|A_{1||}|^{2}+|A_{1\bot}|^{2}+|A_{00}|^{2}}\right)$ (4.8)
Substituting the above observables into the angular terms gives
$\displaystyle\frac{I_{1}^{c}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=\frac{1}{4\pi}\mathcal{F}_{\mathrm{S}}+\frac{3}{4\pi}\mathcal{F}_{\mathrm{P}}F_{\mathrm{L}}\cos^{2}{\theta_{K}}+\frac{3}{4\pi}\mathcal{A}_{\mathrm{S}}\cos{\theta_{K}},$
$\displaystyle\frac{I_{1}^{s}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=\frac{3}{4}\frac{3}{8\pi}\mathcal{F}_{\mathrm{P}}\left(1-F_{\mathrm{L}}\right)\left(1-\cos^{2}{\theta_{K}}\right),$
$\displaystyle\frac{I_{2}^{c}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=-\left(\frac{1}{4\pi}\right.\mathcal{F}_{\mathrm{S}}+\frac{3}{4\pi}\mathcal{F}_{\mathrm{P}}\left(1-F_{\mathrm{L}}\right)\cos^{2}{\theta_{K}}+\left.\frac{3}{4\pi}\mathcal{A}_{\mathrm{S}}\cos{\theta_{K}}\cos{\theta_{K}}\right),$
$\displaystyle\frac{I_{2}^{s}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=\frac{1}{4}\frac{3}{8\pi}\mathcal{F}_{\mathrm{P}}\left(1-F_{\mathrm{L}}\right)\left(1-\cos^{2}{\theta_{K}}\right),$
(4.9) $\displaystyle\frac{I_{3}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=\frac{1}{2}\frac{3}{8\pi}\mathcal{F}_{\mathrm{P}}A_{\mathrm{T}}^{2}\left(1-\cos^{2}{\theta_{K}}\right),$
$\displaystyle\frac{I_{6}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=2\frac{3}{8\pi}\frac{4}{3}\mathcal{F}_{\mathrm{P}}A_{\mathrm{FB}}\left(1-\cos^{2}{\theta_{K}}\right),$
$\displaystyle\frac{I_{9}}{\Gamma^{{}^{\prime\prime}}}$
$\displaystyle=\frac{3}{8\pi}\mathcal{F}_{\mathrm{P}}A_{\mathrm{Im}}\left(1-\cos^{2}{\theta_{K}}\right).$
For the purpose of this paper, a simplification of the angular distribution
can be achieved by folding the distribution in $\phi$ such that
$\phi^{{}^{\prime}}=\phi-\pi$ for $\phi<0$ [27]. The $I_{4,5,7,8}$ angular
terms which are dependent on $\cos\phi$ or $\sin\phi$ are cancelled leaving
$I_{1,2,3,6,9}$ in the angular distribution:
$\displaystyle\frac{\text{d}^{5}\Gamma}{\text{d}q^{2}\text{d}p^{2}\mathrm{dcos}{\theta_{K}}\mathrm{dcos}{\theta_{l}}\text{d}\phi^{{}^{\prime}}}=$
$\displaystyle\frac{3}{8}\left(I_{1}^{c}+2I_{1}^{s}+(I_{2}^{c}+2I_{2}^{s})\cos
2\theta_{l}+2I_{3}\sin^{2}{\theta_{l}}\cos 2\phi^{{}^{\prime}}\right.$
$\displaystyle\left.+2I_{6}\cos{\theta_{l}}+2\sqrt{2}I_{9}\sin^{2}{\theta_{l}}\sin
2\phi^{{}^{\prime}}\frac{}{}\right).$ (4.10)
Combining Equation 4 with 4 gives the differential decay distribution,
$\begin{split}\frac{1}{\Gamma^{{}^{\prime\prime}}}\frac{\text{d}^{5}\Gamma}{\text{d}q^{2}\text{d}p^{2}\mathrm{dcos}{\theta_{K}}\mathrm{dcos}{\theta_{l}}\text{d}\phi^{{}^{\prime}}}=\frac{9}{16\pi}\Bigg{(}&\left(\frac{2}{3}\mathcal{F}_{\mathrm{S}}+\frac{4}{3}\mathcal{A}_{\mathrm{S}}\cos{\theta_{K}}\right)(1-\cos^{2}{\theta_{l}})\\\
&\ \ \ \
+\mathcal{F}_{\mathrm{P}}\bigg{[}2F_{\mathrm{L}}\cos^{2}{\theta_{K}}(1-\cos^{2}{\theta_{l}})\\\
&\ \ \ \ \ \ \ \
+\frac{1}{2}(1-F_{\mathrm{L}})(1-\cos^{2}{\theta_{K}})(1+\cos^{2}{\theta_{l}})\\\
&\ \ \ \ \ \ \ \
+\frac{1}{2}(1-F_{\mathrm{L}})A_{\mathrm{T}}^{2}(1-\cos^{2}{\theta_{K}})(1-\cos^{2}{\theta_{l}})\cos
2\phi^{{}^{\prime}}\\\ &\ \ \ \ \ \ \ \
+\frac{4}{3}A_{\mathrm{FB}}(1-\cos^{2}{\theta_{K}})\cos{\theta_{l}}\\\ &\ \ \
\ \ \ \ \ +A_{\mathrm{Im}}(1-\cos^{2}{\theta_{K}})(1-\cos^{2}{\theta_{l}})\sin
2\phi^{{}^{\prime}}\bigg{]}\Bigg{)}.\end{split}$ (4.11)
The angular distribution as a function of $\cos{\theta_{l}}$ and
$\cos{\theta_{K}}$ is given by integrating over $\phi$ in Eq. 4.11
$\begin{split}\frac{1}{\Gamma^{{}^{\prime\prime}}}\frac{\text{d}^{4}\Gamma}{\text{d}q^{2}\text{d}p^{2}\text{d}\cos{\theta_{K}}\text{d}\cos{\theta_{l}}}=\frac{9}{16}&\Bigg{(}\left(\frac{2}{3}\mathcal{F}_{\mathrm{S}}+\frac{4}{3}\mathcal{A}_{\mathrm{S}}\cos{\theta_{K}}\right)(1-\cos^{2}{\theta_{l}})\\\
&+\mathcal{F}_{\mathrm{P}}\bigg{[}2F_{\mathrm{L}}\cos^{2}{\theta_{K}}(1-\cos^{2}{\theta_{l}})\\\
&\ \ \ \ \
+\frac{1}{2}(1-F_{\mathrm{L}})(1-\cos^{2}{\theta_{K}})(1+\cos^{2}{\theta_{l}})\\\
&\ \ \ \ \
+\frac{4}{3}A_{\mathrm{FB}}(1-\cos^{2}{\theta_{K}})\cos{\theta_{l}}\bigg{]}\Bigg{)}\end{split}$
(4.12)
and further integration from Equation 4.11 yields the angular distribution for
each of the angles,
$\begin{split}\frac{1}{\Gamma^{{}^{\prime\prime}}}\frac{\text{d}^{3}\Gamma}{\text{d}q^{2}\text{d}p^{2}\mathrm{dcos}{\theta_{l}}}&=\frac{3}{4}\mathcal{F}_{\mathrm{S}}(1-\cos^{2}{\theta_{l}})+\mathcal{F}_{\mathrm{P}}\bigg{[}\frac{3}{4}F_{\mathrm{L}}(1-\cos^{2}{\theta_{l}})\\\
&+\frac{3}{8}(1-F_{\mathrm{L}})(1+\cos^{2}{\theta_{l}})+A_{\mathrm{FB}}\cos{\theta_{l}}\bigg{]},\\\
\frac{1}{\Gamma^{{}^{\prime\prime}}}\frac{\text{d}^{3}\Gamma}{\text{d}q^{2}\text{d}p^{2}\mathrm{dcos}{\theta_{K}}}&=\frac{1}{2}\mathcal{F}_{\mathrm{S}}+\mathcal{A}_{\mathrm{S}}\cos{\theta_{K}}\\\
&+\mathcal{F}_{\mathrm{P}}\bigg{[}\frac{3}{2}F_{\mathrm{L}}\cos^{2}{\theta_{K}}+\frac{3}{4}(1-F_{\mathrm{L}})(1-\cos^{2}{\theta_{K}})\bigg{]},\\\
\frac{1}{\Gamma^{{}^{\prime\prime}}}\frac{\text{d}^{3}\Gamma}{\text{d}q^{2}\text{d}p^{2}\text{d}\phi^{{}^{\prime}}}&=\frac{1}{\pi}\Bigg{(}1+\frac{3}{4}\mathcal{F}_{\mathrm{S}}+\mathcal{F}_{\mathrm{P}}\bigg{[}F_{\mathrm{L}}+\frac{1}{2}(1-F_{\mathrm{L}})A_{\mathrm{T}}^{2}\cos
2\phi^{{}^{\prime}}+A_{\mathrm{Im}}\sin
2\phi^{{}^{\prime}}\bigg{]}\Bigg{)}.\end{split}$ (4.13)
The angular distribution can be integrated over $p^{2}$ using the weighted
integral
$\displaystyle O(q^{2})$
$\displaystyle=\frac{\int\mathcal{O}(p^{2},q^{2})\frac{\text{d}^{2}\Gamma}{\text{d}p^{2}\text{d}q^{2}}\text{d}p^{2}}{\int\frac{\text{d}^{2}\Gamma}{\text{d}p^{2}\text{d}q^{2}}\text{d}p^{2}}$
(4.14)
for the value of the observables integrated over a given region in $p^{2}$.
This leads to the integrated observables $F_{\mathrm{P}}$, $F_{\mathrm{S}}$
and $A_{\mathrm{S}}$ which are solely dependant on $q^{2}$. By definition, the
fraction of the S-wave and the P-wave sum to one,
$F_{\mathrm{S}}+F_{\mathrm{P}}=1$. The complete angular distribution without
any $p^{2}$ dependence is given by
$\displaystyle\frac{1}{\Gamma^{{}^{\prime}}}\frac{\text{d}^{5}\Gamma}{\text{d}q^{2}\mathrm{dcos}{\theta_{K}}\mathrm{dcos}{\theta_{l}}\text{d}\phi^{{}^{\prime}}}=\frac{9}{16\pi}$
$\displaystyle\Bigg{(}\left(\frac{2}{3}F_{\mathrm{S}}+\frac{4}{3}A_{\mathrm{S}}\cos{\theta_{K}}\right)(1-\cos^{2}{\theta_{l}})$
$\displaystyle+(1-F_{\mathrm{S}})\bigg{[}2F_{\mathrm{L}}\cos^{2}{\theta_{K}}(1-\cos^{2}{\theta_{l}})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{2}(1-F_{\mathrm{L}})(1-\cos^{2}{\theta_{K}})(1+\cos^{2}{\theta_{l}})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{2}(1-F_{\mathrm{L}})A_{\mathrm{T}}^{2}(1-\cos^{2}{\theta_{K}})(1-\cos^{2}{\theta_{l}})\cos
2\phi^{{}^{\prime}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{4}{3}A_{\mathrm{FB}}(1-\cos^{2}{\theta_{K}})\cos{\theta_{l}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+A_{\mathrm{Im}}(1-\cos^{2}{\theta_{K}})(1-\cos^{2}{\theta_{l}})\sin
2\phi^{{}^{\prime}}\bigg{]}\Bigg{)}.$ (4.15)
where the normalisation of the angular distribution is given by
$\displaystyle\Gamma^{{}^{\prime}}=\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}$
(4.16)
The ‘dilution’ effect of the S-wave can clearly be seen from the factor of
(1-$F_{\mathrm{S}}$) that appears in front of the observables in Eq. 4.
The effect of an S-wave on the angular distribution as a function of
$\cos{\theta_{K}}$, $\cos{\theta_{l}}$ and $\phi^{{}^{\prime}}$ as illustrated
in Figure 2.
Figure 2: One-dimensional projections of (a) $\cos{\theta_{l}}$, (b)
$\cos{\theta_{K}}$, (c) $\phi^{{}^{\prime}}$ for the angular distribution of
$B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ with (blue-dashed) and without
(red-solid) an S-wave component of 7%. The dilution effect of the S-wave on
the asymmetry in $\cos{\theta_{l}}$ and the asymmetric effect in
$\cos{\theta_{K}}$ can be clearly seen.
Here it is possible to see that the asymmetry in $\cos{\theta_{l}}$, given by
$A_{\mathrm{FB}}$, has decreased and that there is an asymmetry in
$\cos{\theta_{K}}$ introduced by the interference term.
## 5 Effect of an S-wave on the angular analysis
In an angular analysis of $B^{0}\\!\rightarrow K^{+}\pi^{-}\ell^{+}\ell^{-}$,
the S-wave can be considered to be a systematic effect that could bias the
results of the angular observables. The implications of this systematic effect
are tested by generating toy Monte Carlo experiments and fitting the angular
distribution to them. The results of the fit to the observables are evaluated
for multiple toy datasets.
The effect of the S-wave is evaluated for two different cases. Firstly, the
effect of S-wave interference is examined as a function of the size of the
dataset used. The aim of this is to give an idea of the current situation and
the possible implications on future measurements of $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$. Datasets of sizes between 50 and 1000 events
are tested. For comparison, the latest results from LHCb [24] have between 20
and 200 signal events in the 6 different $q^{2}$ bins considered. Secondly,
the effect of different levels of S-wave contribution is examined. At present,
the only information about the S-wave fraction is the measurement of
$F_{\mathrm{S}}$ of approximately 7% in the decay
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-}$
from [15] for the range $800<p<1000\mathrm{\,Me\kern-1.00006ptV}$. As the
value may be different in $B^{0}\\!\rightarrow K^{+}\pi^{-}\ell^{+}\ell^{-}$,
we consider values of $F_{\mathrm{S}}$ in this region ranging from 1% to 40%.
The fraction of the S-wave, $F_{\mathrm{S}}$, is expected to have some $q^{2}$
dependence because of the $q^{2}$ dependence of the transverse P-wave
amplitudes.
The parameters used to generate the toy datasets are summarised in Tables 1
and 2. The values of the angular observables used to generate toy Monte Carlo
simulations are taken from the LHCb angular analysis of $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ in the $1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}$ bin
[24]. Within errors, these measurements are compatible with the Standard Model
prediction for $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ and the central
value of the measurement is used. The nominal magnitude and phase difference
of the S-wave contribution are taken from the angular analysis of $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $K^{+}\pi^{-}$
[15].
Table 2: Parameters used to generate toy datasets. $A_{\mathrm{FB}}$,
$F_{\mathrm{L}}$, $A_{\mathrm{T}}^{2}$ and $A_{\mathrm{Im}}$ are taken from
Ref. [24] in the $1<q^{2}<6~{}(\mathrm{\,Ge\kern-1.00006ptV}^{2})$ bin. The
$F_{\mathrm{S}}$ value is taken from Ref. [15]
. Obs. $A_{\mathrm{FB}}$ $F_{\mathrm{L}}$ $A_{\mathrm{T}}^{2}$
$A_{\mathrm{Im}}$ $F_{\mathrm{S}}$ Value $-0.18$ $0.66$ $0.294$ $0.07$ $0.07$
The toy datasets are generated as a function of the $\cos{\theta_{l}}$,
$\cos{\theta_{K}}$, $\phi$ and $p^{2}$ using the angular distribution given in
Eq. 4.11. For each set of input parameters 1000 toy datasets were generated.
For each of these toy datasets, an unbinned log likelihood fit is performed
that returns the best fit value of the observables and an estimate of their
error. The expected experimental resolution is obtained by plotting the best
fit values of an observable for the ensemble of toy simulations as illustrated
for $A_{\mathrm{FB}}$ in Fig. 3 (left) The pull value for an observable ($O$)
is defined as
$\displaystyle
p_{O}^{i}=\frac{O_{\text{fit}}^{i}-O_{\text{gen}}^{i}}{\sigma_{O}^{i}}$ (5.1)
where $\sigma_{O}^{i}$ is the estimated error on the fit to the observable
$O^{i}$. This distribution is seen in Fig. 3 (right). The mean and the width
are extracted from a Gaussian fit. For a well performing fit without bias, the
pull distribution should have zero mean and unit width. A negative pull value
implies that the result is underestimated and a positive pull value implies
overestimation of the true observable.
Figure 3: Distribution of (left) the $A_{\mathrm{FB}}$ results and (right)
pull values for fits to 1000 toy simulations each containing 1000 events. The
S-wave is ignored in these fits. The resolution obtained is $(0.026\pm
0.001)$. Since the S-wave is ignored there is a non-zero pull mean at
$(0.26\pm 0.02)\sigma$ . The width of the pull distribution is consistent with
unity at $(1.01\pm 0.01)\sigma$.
### 5.1 The impact of ignoring the S-wave in an angular analysis of
$B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$
Firstly, the effect of an S-wave was tested as a function of dataset size in
order to find a minimum dataset at which the bias from the S-wave in the
angular observables becomes significant. Datasets were generated for sample
sizes ranging from 50 and 1000 events and analysed assuming a pure P-wave
state. The results are shown in Fig. 4.
Figure 4: Resolution (left) and pull mean (right) of 1000 toy datasets
analysed as a pure P-wave state as a function of dataset size. It can be seen
that the bias on the observable increases dramatically as the sample size
increases. This is because the statistical error decreases increasing the
sensitivity to the S-wave contribution. The bias of $A_{\mathrm{FB}}$ is
positive because $A_{\mathrm{FB}}$ in negative in the $q^{2}$ bin chosen.
From Eq. 4.12, it can be seen that $A_{\mathrm{T}}^{2}$ has a factor of
(1-$F_{\mathrm{L}}$) in front of it. The large value of $F_{\mathrm{L}}$ used
in generated the datasets is in turn causing $A_{\mathrm{T}}^{2}$ to have a
much worse resolution than $A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and
$A_{\mathrm{Im}}$. There is significant bias (non-zero mean) of the pull
distribution for all observables when the S-wave is ignored for datasets of
more than 200 events. This corresponds to a change of 0.2$\sigma$ in
$F_{\mathrm{L}}$ for a dataset of 200 events. The behaviour can be understood
in terms of the $(1-F_{\mathrm{S}})$ factor in Eq 4.12. It gives an offset to
the fitted value of the observables which are proportional to the value of
$F_{\mathrm{S}}$.
Secondly, the angular fit was performed on toy datasets with an increasing
S-wave contribution. Datasets of 500 events were generated with a varying
S-wave contribution in the narrow $p^{2}$ mass window of
($800<p<1000\mathrm{\,Me\kern-1.00006ptV}$) from no S-wave up to a
$F_{\mathrm{S}}$ value of $0.6$. The resolution, the mean and width of the
pull distribution for each of the four observables ($A_{\mathrm{FB}}$,
$F_{\mathrm{L}}$, $A_{\mathrm{T}}^{2}$, $A_{\mathrm{Im}}$) were calculated and
the results are shown in Fig. 5.
Figure 5: Mean of the pull distribution of 1000 toy datasets analysed as a
pure P-wave state as a function of S-wave contribution. The bias can be seen
to increase with the size of the S-wave contribution in a linear fashion.
Significant bias is seen in the angular observable for an S-wave magnitude of
greater than 5%. The linear increase in the bias is another consequence of the
(1-$F_{\mathrm{S}}$) factor.
### 5.2 Measuring the S-wave in $B^{0}\\!\rightarrow
K^{+}\pi^{-}\ell^{+}\ell^{-}$
Obtaining unbiased values for the angular observables beyond the limits shown
requires a measurement of the S-wave contribution rather than ignoring it.
With the formalism developed in Sect.4, three options are explored for
measuring this. The first option is to ignore the $p^{2}$ dependence and
simply fit for $p^{2}$-averaged values of $F_{\mathrm{S}}$ and
$A_{\mathrm{S}}$. The second option is to fit the $p^{2}$ line-shape
simultaneously with the angular distribution. This can be done in a small $p$
window between $800$ and $1000\mathrm{\,Me\kern-1.00006ptV}$ or in the region
from the lower kinematic threshold to $1200\mathrm{\,Me\kern-1.00006ptV}$. In
all cases the datasets used to perform the studies are identical to those used
in Sect. 5.1. The difference is in how the fit is performed. In each case, the
dataset and the S-wave sizes refer to the number of events in the smaller
$p^{2}$ window.
The angular distribution without $p^{2}$ dependence is given in Eq. 4. for
each set of samples, we look at the resolution, the mean and the width of the
pull distribution of the angular observables.
The change in the resolution obtained on the angular observables for the three
methods of including the S-wave in the angular distribution is demonstrated by
plotting the ratio with respect to the resolution obtained when a single
P-wave state is assumed.
The resolutions and the mean of the pull distributions for the three different
fit methods (ignoring the $p^{2}$ dependence, fitting a narrow $p^{2}$ window
and fitting a wide $p^{2}$ window) relative to the resolution and mean
obtained using the assumption of a pure P-wave state. The ratio between the
fit methods including the S-wave in angular distribution and assuming a P-wave
state as a function of dataset size are shown in Fig 6.
(a) $A_{\mathrm{FB}}$ (b) $F_{\mathrm{L}}$ (c) $A_{\mathrm{T}}^{2}$ (d)
$A_{\mathrm{Im}}$
Figure 6: Resolutions for three different methods to incorporate the S-wave
relative to the resolution obtained when the S-wave is ignored. It can be seen
that the best resolution is obtained when using the largest $p^{2}$ window.
The original resolution is recovered to within 10%.
The pull mean for all four fit methods is shown in Fig 7.
(a) $A_{\mathrm{FB}}$ (b) $F_{\mathrm{L}}$ (c) $A_{\mathrm{T}}^{2}$ (d)
$A_{\mathrm{Im}}$
Figure 7: Pull mean for the three different methods to incorporate the S-wave
and when the S-wave is ignored. There is a slight bias when the S-wave is
included for datasets of less than 200 events but this bias is removed from
all the observables when the S-wave is included in the fit for datasets of
over 500 events.
For all observables, it can be seen that the resolution degrades when the
S-wave is included and the $p^{2}$ dependence is ignored. The resolution
degrades by a smaller amount when the $p^{2}$ dependence is included in a
small bin and the original resolution is recovered to within 10% when using
the large $p^{2}$ range. There are two effects contributing to the improvement
of the resolution. There are more P-wave events in the larger range and the
wider mass window allows for the S-wave to be constrained by using the
information from above and below the P-wave resonance. This results in the
best resolution when the S-wave is included in the angular distribution.
For all the observables, the pull mean approaches zero for datasets of greater
than 300 events implying that the bias present in all the observables when a
pure P-wave state is removed when an S-wave is included in the angular
distribution. This means that the inclusion of the S-wave component will be
mandatory for all future experimental analyses.
Another approach to reduce the bias from the S-wave is to ignore it in fits
but to only include data from a narrower window in $p$ arounnd the
$K^{*0}(892)$ resonance. By reducing the window from
200$\mathrm{\,Me\kern-1.00006ptV}$ to 100$\mathrm{\,Me\kern-1.00006ptV}$, the
P-wave component is reduced by 20% while the S-wave component is roughly
halved. Conducting the same tests as described above shows, as expected, a 10%
increase in the statistical error of the observables while the bias for a
given dataset is reduced by a factor two. Given what has been shown in this
paper, the experimental datasets will in the future be so large that the best
approach is to fit the S-wave rather than half the bias and accepting an
increased statistical uncertainty.
Until now the lineshape of the S-wave has been parameterised according to the
LASS model (Eqs. 3.7-3.9). We asses the model dependence of this assumption by
using the alternative isobar model [28] for generating the S-wave component
while keeping the same fit model. This only has an effect on the fits where a
fit is performed to the $p^{2}$ dependence. The results of this show that the
systematic uncertainly due to the model dependence is much smaller than the
statistical error for all observables for all sample sizes we studied.
## 6 Conclusion
In summary, the inclusion of a resonant $K^{+}\pi^{-}$ S-wave in the angular
analysis of $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ has been formalised
and the complete angular distribution for both an S- and P-wave state
described. We find that the inclusion of an S-wave state has an overall
dilution effect on the theoretical observables. The impact of an S-wave on an
angular analysis is evaluated using toy Monte Carlo datasets. We find that the
S-wave contribution can only be ignored for datasets of less than 200 events.
The bias on the angular observables incurred by assuming a pure P-wave
$K^{+}\pi^{-}$ state can be removed by including the S-wave in the angular
distribution. The degradation in resolution on the angular observables from
fitting a more complicated angular distribution can be minimised by performing
the fit in a wide region around the $K^{*0}(892)$ resonance. The systematic
uncertainty introduced by the model dependence of the S-wave lineshape is
minimal and can be ignored.
## Acknowledgements
T.B. acknowledges support from CERN. U.E and A.S acknowledge support from the
Science and Technologies Facilities Council under grant numbers ST/K001280/1
and ST/F007027/1.
## References
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* [4] U. Egede et al., New observables in the decay mode $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$, JHEP 11 (2008) 032, arXiv:0807.2589
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* [27] J. Lefrancois and M. H. Schune, Measuring the photon polarization in $b\\!\rightarrow s\gamma$ using the $B^{0}\rightarrow K^{*0}e^{+}e^{-}$ decay channel, LHCb-PUB-2009-008
* [28] E791 collaboration, E. M. Aitala et al., Dalitz plot analysis of the decay ${D}^{+}\rightarrow{}{K}^{-}{\pi{}}^{+}{\pi{}}^{+}$ and indication of a low-mass scalar $k\pi{}$ resonance, Phys. Rev. Lett. 89 (2002) 121801
|
arxiv-papers
| 2012-10-18T23:25:57 |
2024-09-04T02:49:36.819999
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Blake, Ulrik Egede, Alex Shires",
"submitter": "Ulrik Egede",
"url": "https://arxiv.org/abs/1210.5279"
}
|
1210.5312
|
# On the dimension of splines spaces over T-meshes with smoothing cofactor-
conformality method
X. Li
Department of Mathematics, USTC,
Hefei, Anhui Province 230026, P. R. China.
lixustc@ustc.edu.cn
###### Abstract
The present paper provides a general formula for the dimension of spline space
over T-meshes using smoothing cofactor-conformality method. And we introduce a
new notion, Diagonalizable T-mesh, over which the dimension formula is only
associated with the topological information of the T-mesh. A necessary and
sufficient condition for characterization a diagonalizable T-mesh is also
provided. Using this technique, we find that the dimension is possible
instable under the condition of [1] and we also provide a new correction
theorem.
Keywords: Dimension, T-splines, Spline space, T-mesh, smoothing cofactor-
conformality method
## 1 Introduction
NURBS (Non-Uniform Rational B-Spline) is the standard for generating and
representing free-form curves and surfaces, which is a basic tool for using in
CAD [2] and is also a desirable tool for iso-geometric analysis [3]. A well-
known and significant disadvantage of NURBS is that it is based on a tensor
product structure with a global knot insertion operation. It is desirable to
generalize NURBS to spline space which can hander hanging nodes.
Spline spaces over T-meshes
$\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})$ were first introduced in
[4], which is a bi-degree $(d_{1},d_{2})$ piecewise polynomial spline space
over T-mesh $\mathcal{T}$ with smoothness order $\alpha$ and $\beta$ in two
directions. Spline space over T-meshes have been applied in fitting [5],
stitching [6], simplification [7], isogeometric analysis [8], [9], solving
elliptic equations [10] and spline space over triangulations with hanging
nodes [11]. Spline over T-mesh has several advantages. For example, it has a
simple local refinement which will never introduce additional refinement
according to the definition. And it is a polynomial in each face which has
simple and efficient integration for numerical analysis.
NURBS with local refinement is discovered before spline over T-mesh, called
”T-spline”, which are defined on a T-mesh by certain collections of B-splines
functions defined on the mesh [12], [13]. T-splines have been proved to be a
powerful free-form geometric shape technology that solves most of the
limitations inherent in NURBS. T-splines have several advantages over NURBS
such as local refinement [13] and watertightness [14], and they are forward
and backward compatible with NURBS. These capabilities make T-splines
attractive both for CAD and also desirable for iso-geometric analysis [15].
However, it is very difficult to unravel the mystery of T-spline spaces which
have important implications in establishing approximation, stability, and
error estimates [16]. And till now, only a sub-class of T-spline space,
analysis-suitable T-splines space [17], [18], [19] has been discovered using
the technology from spline space over T-meshes.
Thus, it is very important to understand the spline space of piecewise
polynomials of a given smoothness on a T-mesh, which foundation but non-
trivial step is to calculate the dimension of the space. Till now, many
different methods have been applied to tackle these issues, such as B-net [4],
minimal determining set [20], smoothing cofactor-conformality method [1] and
homological technique [21]. B-net and minimal determining set methods are
suitable for spline space with reduced regularity, i.e., the degrees for the
polynomial is larger enough than the smoothness order. Smooth cofactor-
conformality method [22], [23] is a power tool for calculating the dimension
of spline space in multi-variate splines theory. It transfers the smoothness
conditions into algebraic forms and calculate the dimension using linear
algebraic tools. Homological technique is another power tool for calculating
the dimension using the similar idea except regarding the smoothness
conditions as the kernel of a certain linear maps. The present paper is
focusing on smoothing cofactor-conformality method.
All the existing dimension results are forcing the T-mesh to be special
nesting structure, such as hierarchical, regular T-subdivision and no cycles
[20]. With such structure, one can calculate the dimension level by level. Our
approach is based on a different point of view. As smoothing cofactor-
conformality method can convert the smoothness conditions into algebraic
forms, so we directly focus on the algebraic forms and study the condition,
under which the dimension of the linear system doesn’t associated with knot
values. And then we use the condition to find a new notion, diagonalizable
T-meshes, is the corresponding T-meshes. We believe that this new notion is
the key condition to compute the dimension using smoothing cofactor-
conformality method. The main contribution of the present paper includes,
* •
We provide a general formula for the dimension of spline space over any
regular T-meshes without holes using smoothing cofactor-conformality method.
* •
We provide a new notion, diagonalizable T-mesh, over which the dimension
formula is only associated with the topological information of the T-mesh. We
also provide a necessary and sufficient condition for characterization
diagonalizable T-meshes;
* •
We discover that the dimension is possible instable under the condition of [1]
and we also provide a new correction theorem.
The remaining paper is structured as follows. Pertinent background on spline
space over T-mesh is reviewed in Section 2. In Section 3, we review how to use
smoothing cofactor-conformality method to analysis the dimension of spline
space over T-meshes. In section 4, we provide a condition under which we can
calculate the dimension of spline space over T-mesh without considering the
knot values. In section 5, we first show that the dimension result in [1] is
not correct and we provide a new correction based on this new technology. The
last section is conclusion and future work.
## 2 T-mesh and spline over T-mesh
In this section, we briefly review the notion of spline spaces over T-meshes
and the related dimension results of the corresponding spline space.
### 2.1 T-mesh
A T-mesh $\mathcal{T}$ is a collection of axis-aligned rectangles $F_{i}$ such
that the interior of the domain $\Omega$ is $\cup F_{i}$, and the distinct
rectangles $F_{i}$ and $F_{j}$ can only intersect at points on their edges.
The rectangles $F_{i}$ are also called the _face_ or _cell_ of the T-mesh. The
vertices of the rectangles are called the _nodes_ or _vertices_ for a T-mesh.
The line segment connecting two adjacent vertices on a grid line is called an
_edge_ of the T-mesh. T-meshes include tensor-product meshes as a special
case. However, in contrast to tensor-product meshes, T-meshes are allowed to
have _T-junctions_ , or _T-nodes_ , which are vertices of one rectangle that
lies in the interior of an edge of another rectangle. The domain $\Omega$ need
not be rectangular, which may have holes, concave corners. For example the
T-mesh in Figure 1, the grey region is a hole and vertices $V_{38}$ and
$V_{88}$ are both concave corners. In the present paper, we require the
T-meshes to be regular and without holes. Here regular means that the set of
all rectangles for a T-mesh containing a vertex has a connected interior [24].
Figure 1: A T-mesh.
The vertices, edges can be divided into two parts. If a vertex is on the
boundary grid line of the T-mesh, then is called a _boundary vertex_.
Otherwise, it is called an _interior vertex_. If both vertices of an edge are
boundary vertices, then it is called a _boundary edge_ ; otherwise it is
called an _interior edge_. An _l-edge_ is a line segment which consists of
several interior edges. It is the longest possible line segment, which
interior edges are connected and two end points being T-junctions or boundary
vertices. l-edges have three different classes. If the two end vertices of a
l-edge are interior vertices, then the l-edge is called _interior l-edge_. If
two end vertices of a l-edge are both boundary vertices, then the l-edge is
called a _cross-cut_. Otherwise, if one end vertex is boundary vertex and the
other is interior vertex, then the l-edge is called a _ray_. A _mono-vertex_
is the intersection vertex of an interior l-edge and a cross-cut or a ray and
a _free-vertex_ is the intersection between cross-cuts and rays. For example,
in Figure 1, vertices $V_{53}$, $V_{44}$ and $V_{46}$ are interior vertices,
and $V_{15}$, $V_{16}$ and $V_{95}$ are boundary vertices. The l-edge
$V_{15}V_{95}$ is a cross-cut, while $V_{16}V_{46}$ is a ray, and
$V_{44}V_{84}$ and $V_{57}V_{58}$ are interior l-edges.
For later use we introduce some notations for a T-mesh as shown in Table 1
Table 1: Notations for a T-mesh $F$ | number of faces in $\mathcal{T}$
---|---
$E^{h}$ | number of horizonal interior edges in $\mathcal{T}$
$E^{v}$ | number of vertical interior edges in $\mathcal{T}$
$V$ | number of interior vertices in $\mathcal{T}$
$C^{h}$ | number of horizonal cross-cuts in $\mathcal{T}$
$C^{v}$ | number of vertical cross-cuts in $\mathcal{T}$
$T^{h}$ | number of horizonal interior l-edges in $\mathcal{T}$
$T^{v}$ | number of vertical interior l-edges in $\mathcal{T}$
$n_{e}$ | number of interior l-edges in $\mathcal{T}$ ($T^{h}+T^{v}$)
$V^{+}$ | number of free-vertices in $\mathcal{T}$
$N^{h}$ | minimal integer larger or equal to $\frac{d_{1}+1}{d_{1}-\alpha}$
$N^{v}$ | minimal integer larger or equal to $\frac{d_{2}+1}{d_{2}-\beta}$
$n_{c}$ | $(d_{1}-\alpha)(d_{2}-\beta)(V-V^{+})$
$n_{r}$ | $(d_{1}+1)(d_{2}-\beta)T^{h}+(d_{2}+1)(d_{1}-\alpha)T^{v}$
### 2.2 Spline space over T-mesh
Given a T-mesh $\mathcal{T}\in\mathbb{R}^{2}$, let $\mathcal{F}$ denote all
the cells in $\mathcal{T}$ and $\Omega$ the region occupied by all the cells
in $\mathcal{T}$. The bi-degree $(d_{1},d_{2})$ polynomial spline space over
T-mesh $\mathcal{T}$ with smoothness order $\alpha$ and $\beta$ is defined as
$\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T}):=\Big{\\{}f(x,y)\in
C^{\alpha,\beta}(\Omega)\Big{|}f|_{\phi}\in\mathrm{P}_{d_{1}d_{2}},\forall\phi\in\mathcal{F}\Big{\\}},$
where $\mathrm{P}_{d_{1}d_{2}}$ is the space of all the polynomials with bi-
degree $(d_{1},d_{2})$, and $C^{\alpha,\beta}(\Omega)$ is the space consisting
of all the bivariate functions which are continuous in $\Omega$ with order
$\alpha$ along $x$ direction and with order $\beta$ along $y$ direction. It is
obvious that $\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})$ is a linear
space, which is called the spline space over the given T-mesh $\mathcal{T}$.
Until now, several articles have been studied to analysis the dimension of the
spline space over some special families of T-meshes.
* •
Reduced regularity:
In 2006, [4] studied the dimension of the spline space under the constrains
that the order of the smoothness is less than half of the degree of the spline
functions. According to Theorem 4.2 in [4], it follows that if $d_{1}\geq
2\alpha+1$ and $d_{2}\geq 2\beta+1$,
$\displaystyle\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})=$
$\displaystyle F(d_{1}+1)(d_{2}+1)-E_{h}(d_{1}+1)(\beta+1)-$ $\displaystyle
E_{v}(d_{2}+1)(\alpha+1)+V(\alpha+1)(\beta+1),$
where $F$, $E_{h}$, $E_{v}$, and $V$ are defined in Table 1. [24] also proved
this result using minimal determining set method. And later, [25] analysis a
special T-spline with reducing regularity using the dimension in [4].
* •
Enough mono-vertices
In 2006, [1] calculated the dimension of spline space over a T-mesh if each
interior l-edges have enough mono-vertices. In the T-mesh, if the interior of
each horizontal interior l-edge has at least $N^{h}-2$ mono-vertices and the
interior of each vertical interior edge segment has at least $N^{v}-2$ mono-
vertices, then the dimension of spline space over the T-mesh is,
$\displaystyle\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})=$
$\displaystyle(d_{1}+1)(d_{2}+1)+(C_{h}-T_{h})(d_{1}+1)(d_{2}-\beta)+$
$\displaystyle(C_{v}-T_{v})(d_{2}+1)(d_{1}-\alpha)+V(d_{1}-\alpha)(d_{2}-\beta),$
here $C_{h}$, $C_{v}$, $T_{h}$, $T_{v}$ and $V$ are defined in Table 1.
However, we will show that the condition in this paper is not right.
* •
Instability:
In 2011, [26, 27] discovered that the dimension of the associated spline space
is instability over some particular T-meshes, i.e, the dimension is not only
associated with the topological information of the T-mesh but also associated
with the geometric information of the T-mesh.
* •
Analysis-suitable T-splines:
In 2011, [17, 18] provided a mildly restricted subset of T-splines, which
optimized to meet the needs of both design and analysis. And [19] compute the
dimension of the spline space $\mathcal{S}(d,d,d-1,d-1,\mathcal{T})$ if the
T-mesh $\mathcal{T}$ is an extended T-mesh of an analysis-suitable T-mesh.
* •
Regular T-subdivision:
[21] studied the dimension for spline space
$\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})$ when the T-mesh is a
regular T-subdivision by exploiting homological techniques, which is a special
case of the present paper.
* •
Special hierarchical T-mesh:
[28] provided the dimension for spline space
$\mathcal{S}(d,d,d-1,d-1,\mathcal{T})$ over a special hierarchical T-mesh
using homological algebra technique.
## 3 Smoothing cofactor-conformality method
In this section, we will review one of the main methods, smooth cofactor-
conformality method introduced in [22] and [23], for computing the dimension
of spline space over T-meshes.
In the theory of multi-variate splines, in order to calculate the dimension of
a spline space, one first needs to transfer the smoothness conditions into
algebraic forms.
Referring to Figure 2, for any interior vertex $v_{i,j}=(x_{i},y_{j})$,
suppose the four surrounding bi-degree $(d_{1},d_{2})$ polynomial patches are
$p_{i,j}^{k}(x,y),k=0,1,\dots,3$ respectively (if the vertex $v_{i,j}$ is a
T-junction, then some of the polynomial patches are identical). For example
for the left T-junction in Figure 2 b, patches $p_{i,j}^{1}(x,y)$,
$p_{i,j}^{2}(x,y)$ are identical.
Figure 2: smoothing cofactors around a vertex.
As $p_{i,j}^{0}(x,y)$ and $p_{i,j}^{1}(x,y)$ are $C^{\alpha}$ continuity, so
there exists a bi-degree $(d_{1}-\alpha-1,d_{2})$ polynomial
$\lambda_{i,j}^{2}(y)$ such that
$p_{i,j}^{1}(x,y)-p_{i,j}^{0}(x,y)=\lambda_{i,j}^{2}(x,y)(x-x_{i})^{\alpha+1},$
(1)
Here $\lambda_{i,j}^{2}(x,y)$ is called edge cofactor for the common edge of
patches $p_{i,j}^{0}(x,y)$ and $p_{i,j}^{1}(x,y)$. If two patches are
identical, then the edge cofactor is zero.
Similarly, there also exist bi-degree $(d_{1}-\alpha-1,d_{2})$ polynomial
$\lambda_{i,j}^{1}(x,y)$, bi-degree $(d_{1},d_{2}-\beta-1)$ polynomials
$\mu_{i,j}^{1}(x,y)$ and $\mu_{i,j}^{2}(x,y)$, such that
$\displaystyle p_{i,j}^{2}(x,y)-p_{i,j}^{1}(x,y)$
$\displaystyle=\mu_{i,j}^{1}(x,y)(y-y_{j})^{\beta+1},$ (2) $\displaystyle
p_{i,j}^{3}(x,y)-p_{i,j}^{2}(x,y)$
$\displaystyle=-\lambda_{i,j}^{1}(x,y)(x-x_{i})^{\alpha+1},$ (3)
$\displaystyle p_{i,j}^{0}(x,y)-p_{i,j}^{3}(x,y)$
$\displaystyle=-\mu_{i,j}^{2}(x,y)(y-y_{j})^{\beta+1}.$ (4)
Sum with all these equations, we have
$(\lambda_{i,j}^{1}(x,y)-\lambda_{i,j}^{2}(x,y))(x-x_{i})^{\alpha+1}=(\mu_{i,j}^{1}(x,y)-\mu_{i,j}^{2}(x,y))(y-y_{j})^{\beta+1}.$
(5)
Since $(x-x_{i})^{\alpha+1}$ and $(y-y_{j})^{\beta+1}$ are prime to each
other, so there exist bi-degree $(d_{1}-\alpha-1,d_{2}-\beta-1)$ polynomial
$d_{i,j}(x,y)$, such that,
$\lambda_{i,j}^{1}(x,y)-\lambda_{i,j}^{2}(x,y)=d_{i,j}(x,y)(y-y_{j})^{\beta+1},\quad\mu_{i,j}^{1}(x,y)-\mu_{i,j}^{2}(x,y)=d_{i,j}(x,y)(x-x_{i})^{\alpha+1}.$
Let
$d_{i,j}(x,y)=\sum_{p=0}^{d_{1}-\alpha-1}\sum_{q=0}^{d_{2}-\beta-1}d_{i,j}^{p,q}(x-x_{i})^{p}(y-y_{j})^{q},$
We call these $d_{i,j}^{p,q}$ vertex cofactor. Denote $\hat{\textbf{d}}_{i,j}$
to be a vector contains all coefficients $d_{i,j}^{p,q}$, i.e.,
$\hat{\textbf{d}}_{i,j}=(d_{i,j}^{0,0},d_{i,j}^{0,1},\dots,d_{i,j}^{d_{1}-\alpha-1,d_{2}-\beta-2},d_{i,j}^{d_{1}-\alpha-1,d_{2}-\beta-1}).$
For boundary vertices, there is a little different. Since in this case, we
only have parts of the four equations such as (1),(2),(3),(4). So we don’t
need to assign the bi-degree $(d_{1}-\alpha-1,d_{2}-\beta-1)$ polynomial
$d_{i,j}(x,y)$ for the boundary vertex. Instead, we will assign the edge
cofactor for the corresponding edge. For example, in Figure 3a, we need two
edge cofactors $\lambda_{i,j}^{1}$ and $\mu_{i,j}^{1}$ and for the figure b,
we need only one edge cofactors $\mu_{i,j}^{1}$.
Figure 3: smoothing cofactors around a boundary vertex. The grey regions are
outside of the T-mesh.
The vertex cofactors for the interior vertices are not totally free, there are
other constrains for the continuity condition along each l-edge in the T-mesh.
(a) a.
Figure 4: smoothing cofactors along a horizonal edge segment.
We first consider a horizontal _interior l-edge_ referring to Figure 4 with
$k+1$ vertices $\nu_{i_{t},j},t=0,1,\dots,k$. According to equation (3), we
have
$\displaystyle\mu_{i_{0},j}^{1}(x,y)-0$
$\displaystyle=d_{i_{0},j}(x,y)(x-x_{i_{0}})^{\alpha+1},$ $\displaystyle\dots$
$\displaystyle\mu_{i_{t},j}^{1}(x,y)-\mu_{i_{t},j}^{2}(x,y)$
$\displaystyle=d_{i_{t},j}(x,y)(x-x_{i_{t}})^{\alpha+1},$ (6)
$\displaystyle\dots$ $\displaystyle 0-\mu_{i_{k},j}^{2}(x,y)$
$\displaystyle=d_{i_{k},j}(x,y)(x-x_{i_{k}})^{\alpha+1}.$
and
$\mu_{i_{t-1},j}^{2}(x,y)=\mu_{i_{t},j}^{1}(x,y).$
Sum all these equations, we have the following equation,
$\sum_{t=0}^{k}d_{i_{t},j}(x,y)(x-x_{i_{t}})^{\alpha+1}=0.$ (7)
Similarly, the constrains for a vertical interior l-edge is
$\sum_{t=0}^{l}d_{i,j_{t}}(x,y)(y-y_{j_{t}})^{\beta+1}=0.$ (8)
The above equations are called edge conformality conditions.
###### Lemma 3.1.
Assume each $x_{i_{t}}$ and $y_{j_{t}}$ are distinct, then the dimensions of
solution space of equation (7) and (8) are
$(d_{2}-\beta)(l(d_{1}-\alpha)-d_{1}-1)_{+}$ and
$(d_{1}-\alpha)(k(d_{2}-\beta)-d_{2}-1)_{+}$ respectively.
###### Proof.
See [1] for more details. ∎
###### Remark 3.2.
If the number of the vertices in a horizontal l-edge is less than $N^{h}$, or
the number of the vertices in a vertical l-edge is less than $N^{v}$, then the
l-edge will not contribute the dimension of the spline space, i.e., we can
delete the l-edge without altering the spline space. Thus, we called such
l-edge vanished l-edge. In the following, we assume that all l-edges are not
vanished l-edges.
Now we consider a horizontal ray with $r+1$ vertices
$\nu_{i_{t},j},t=0,1,\dots,k$. Without loss generalization, we assume
$\nu_{i_{0},j}$ is a boundary vertex. According to equation (3), we have
$\displaystyle\mu_{i_{1},j}^{1}(x,y)-\mu_{i_{1},j}^{2}(x,y)$
$\displaystyle=d_{i_{1},j}(x,y)(x-x_{i_{1}})^{\alpha+1},,$
$\displaystyle\dots$
$\displaystyle\mu_{i_{t},j}^{1}(x,y)-\mu_{i_{t},j}^{2}(x,y)$
$\displaystyle=d_{i_{t},j}(x,y)(x-x_{i_{t}})^{\alpha+1},$ (9)
$\displaystyle\dots$ $\displaystyle 0-\mu_{i_{k},j}^{2}(x,y)$
$\displaystyle=d_{i_{k},j}(x,y)(x-x_{i_{k}})^{\alpha+1}.$
and
$\mu_{i_{t-1},j}^{2}(x,y)=\mu_{i_{t},j}^{1}(x,y).$
Sum all these equations, we have the following equation,
$\sum_{t=1}^{k}d_{i_{t},j}(x,y)(x-x_{i_{t}})^{\alpha+1}=\mu_{i_{1},j}^{2}(x,y).$
(10)
The main different of the constraint from (7) and (8) is that these is one
edge cofactor $\mu_{i_{1},j}^{2}(x,y)$ in the constraints. For any
$d_{i_{t},j}(x,y)$ assigned for the interior vertices, we can assign
$\mu_{i_{1},j}^{2}(x,y)$ as
$\sum_{t=1}^{k}d_{i_{t},j}(x,y)(x-x_{i_{t}})^{\alpha+1}$ to satisfy the
constraint.
For a horizontal cross-cut in the T-mesh, since it has two boundary vertices,
we can conclude that it has $(d_{1}+1)(d_{2}-\beta)$ degrees of freedoms.
Similarly, a vertical cross-cut has $(d_{2}+1)(d_{1}-\alpha)$ degrees of
freedoms.
The linear systems as (7) or (8), associated with each interior l-edges can be
assembled into a global system as $\mathcal{M}{\bf x}=0$. Here $\mathcal{M}$
is a $n_{r}\times n_{c}$ matrix, which is called conformality conditions
matrix. And ${\bf x}$ is a column vector whose elements are all the vertex
cofactors for the interior vertices in the T-mesh.
And according to the above anlaysis, we can get the dimension for spline space
over any general T-mesh without holes according to smoothing cofactor-
conformality method which is stated in the following theorem.
###### Theorem 3.3.
Given a T-mesh $\mathcal{T}$ which has no vanished l-edges and holes, let
matrix $\mathcal{M}$ be the conformality conditions matrix, then the dimension
of spline space over the T-mesh is,
$\displaystyle\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})=$
$\displaystyle(d_{1}+1)(d_{2}+1)+C^{h}(d_{1}+1)(d_{2}-\beta)+$ $\displaystyle
C^{v}(d_{2}+1)(d_{1}-\alpha)+V^{+}(d_{1}-\alpha)(d_{2}-\beta)+\dim\mathcal{M},$
## 4 Diagonalizable
Theorem 3.3 indicates that the main difficult to compute the dimension of
spline space over T-mesh is to calculate the rank of conformality condition
matrix $\mathcal{M}$. It is obvious that the structure of the matrix is
associated with the order of edge conformality conditions and it is also
associated with the order of vertices cofactors. Most of existing method study
the dimension of the matrix by forcing the T-mesh to be nest structure, such
as T-subdivision, hierarchical. However, nesting structure should be fine for
application, but it is hiding some essential properties for T-mesh. What we
wish to answer is that under what condition can we compute the dimension
regardless the knot intervals for a given T-mesh and why such condition is
important.
###### Definition 4.1.
Given a T-mesh $\mathcal{T}$, suppose we order the all the interior l-edges as
$e_{i_{j}},j=1,2,\dots,n_{e}$, then we can compute the new-vertex-vector
$\nu^{i}$. Here $\nu^{i}_{1}$ is the number of vertices on l-edge $e_{i_{1}}$
and $\nu^{i}_{j}$ is the number of vertices on l-edge $e_{i_{j}}$ but not on
l-edges $e_{i_{k}},k<j$.
###### Definition 4.2.
A T-mesh is called Diagonalizable if there exists an order of l-edges
$e_{i_{j}},j=1,2,\dots,n_{e}$ such that the _new-vertex-vector_ $\nu^{i}$
satisfies that $\nu_{j}^{i}\geq N^{h}$ if $e_{i_{j}}$ is horizonal and
$\nu_{j}^{i}\geq N^{v}$ if $e_{i_{j}}$ is vertical.
###### Lemma 4.3.
If a T-mesh is diagonalizable, then the matrix $\mathcal{M}$ has full column
rank regardless the knot intervals.
###### Proof.
Since we assume that the T-mesh has no vanished l-edges, the matrix
$\mathcal{M}$ has more columns than rows, i.e., $n_{c}\leq n_{r}$. After
arranging the order of edge conformality conditions and the order of vertex
cofactors, an appropriate partition of the linear system of constraints is
$\left[\begin{array}[]{c|c}\mathcal{M}_{1}&\mathcal{M}_{2}\end{array}\right]\left[\begin{array}[]{c}\textbf{x}_{1}\vspace{2pt}\\\
\hline\cr\vspace{-10pt}\hfil\\\ \textbf{x}_{2}\end{array}\right]=\mathbf{0}$
(11)
where $\mathcal{M}_{1}$ is a $n_{r}\times n_{r}$ matrix and $\mathcal{M}_{2}$
is a $n_{r}\times(n_{c}-n_{r})$ matrix, $\textbf{x}_{1}^{T}$ is a vector of
the first $n_{r}$ vertex cofactors, and $\textbf{x}_{2}$ is a vector of the
remaining vertex cofactors.
Since the T-mesh is diagonalizable, so there exist the order for the interior
l-edges satisfy the condition stated in definition 4.2. Without loss of
generalization, we assume the order for the l-edges are
$e_{i},i=1,2,\ldots,n_{e}$. Then we arrange the order of the edge conformality
conditions corresponding to l-edge from $e_{n_{e}}$ to $e_{1}$. The order of
the vertex cofactors can be placed in the following fashion. For l-edge
$e_{n_{e}}$, if it is a horizonal l-edge, then there exist $N^{h}$ vertices
which are not appeared in the other l-edges. Each vertex corresponds
$(d_{1}-\alpha)\times(d_{2}-\beta)$ cofactors, so these $N^{h}$ vertices
correspond $(d_{1}-\alpha)\times(d_{2}-\beta)N^{h}\geq(d_{1}+1)(d_{2}-\beta)$
cofactors. Select any $(d_{1}+1)(d_{2}-\beta)$ cofactors and put in the
beginning of $\textbf{x}_{1}^{T}$. If the l-edge is a vertical l-edge, we will
select $(d_{2}+1)(d_{1}-\alpha)$ cofactors and put them in the beginning of
$\textbf{x}_{1}^{T}$. These process can be applied for the remaining l-edges.
And then the matrix $\mathcal{M}_{1}$ is in upper block triangular form and
according to Lemma 3.1 each diagonal block
$(d_{2}-\beta)(d_{1}+1)\times(d_{2}-\beta)(d_{1}+1)$ or
$(d_{1}-\alpha)(d_{2}+1)\times(d_{1}-\alpha)(d_{2}+1)$ matrix is full rank,
thus matrix $\mathcal{M}_{1}$ is obviously of full rank. ∎
###### Theorem 4.4.
Suppose T-mesh $\mathcal{T}$ is diagonalizable and has no vanished l-edges and
holes, then the dimension of spline space over the T-mesh is,
$\displaystyle\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})=$
$\displaystyle(d_{1}+1)(d_{2}+1)+(C^{h}-T^{h})(d_{1}+1)(d_{2}-\beta)+$
$\displaystyle(C^{v}-T^{v})(d_{2}+1)(d_{1}-\alpha)+V(d_{1}-\alpha)(d_{2}-\beta).$
###### Proof.
As T-mesh $\mathcal{T}$ is diagonalizable, so
$\dim\mathcal{M}=n_{c}-n_{r}=(d_{1}-\alpha)(d_{2}-\beta)(V-V^{+})-(d_{1}+1)(d_{2}-\beta)E^{h}-(d_{2}+1)(d_{1}-\alpha)E^{v}.$
According to theorem 3.3, we have the dimension
$\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})$ is
$\displaystyle dim=$
$\displaystyle(d_{1}+1)(d_{2}+1)+C^{h}(d_{1}+1)(d_{2}-\beta)+C^{v}(d_{2}+1)(d_{1}-\alpha)+V^{+}(d_{1}-\alpha)(d_{2}-\beta)+$
$\displaystyle(d_{1}-\alpha)(d_{2}-\beta)(V-V^{+})-(d_{1}+1)(d_{2}-\beta)E^{h}-(d_{2}+1)(d_{1}-\alpha)E^{v}$
$\displaystyle=$
$\displaystyle(d_{1}+1)(d_{2}+1)+(C^{h}-T^{h})(d_{1}+1)(d_{2}-\beta)+(C^{v}-T^{v})(d_{2}+1)(d_{1}-\alpha)+$
$\displaystyle V(d_{1}-\alpha)(d_{2}-\beta).$
∎
Now we provide several examples for dimension of spline space over the
following T-meshes.
(a) a. (b) b.
Figure 5: Two example T-meshes for spline space
$\mathcal{S}(3,3,2,2,\mathcal{T})$.
###### Example 4.1.
The first examples are spline space $\mathcal{S}(3,3,2,2,\mathcal{T})$ over
the two T-meshes in Figure 5.
The first T-mesh has some concave corners and one interior l-edge which is a
vanished l-edge since it has only two vertices. And the T-mesh has two cross-
cut. So according to theorem 3.3, the dimension of the spline space
$\mathcal{S}(3,3,2,2,\mathcal{T})$ over the first T-mesh is $16+2\times 4=24$.
The second T-mesh is much more complex. It has four interior l-edges
$e_{i},i=1,\dots,4$, four cross-cuts and five rays. If we arrange l-edges as
$e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, then the new-vertex-vector corresponds
this order is $(5,5,4,3)$. But we can arrange the l-edges as order $e_{1}$,
$e_{4}$, $e_{3}$, $e_{2}$, then the new-vertex-vector corresponds this order
is $(5,4,4,4)$, which means the T-mesh is diagonalizable. In the T-mesh, we
have $4$ cross-cuts, $2$ horizonal and $2$ vertical interior l-edges and $31$
interior vertices, so according to theorem 4.4, the dimension of the spline
space $\mathcal{S}(3,3,2,2,\mathcal{T})$ over the first T-mesh is
$16+4(4-4)+31=47$. We should mention that non of existing method can calculate
the dimension for this example since the T-mesh is not T-subdivision or
hierarchical.
Figure 6: Non-diagonalizable vs. diagonalizable for spline space
$\mathcal{S}(3,3,1,1,\mathcal{T})$.
###### Example 4.2.
The second example is associated with spline space
$\mathcal{S}(3,3,1,1,\mathcal{T})$ over the two T-meshes in Figure 6.
The first T-mesh has four interior l-edges. And we can see that it is not
diagonalizable since the new-vertex-vector could be $(3,2,2,1)$ or $(3,3,1,1)$
for different orders. Thus, we cannot complete the dimension using the current
method. Actually, according to our knowledge, no existing method can compute
the dimension of such spline space.
The second T-mesh also has four interior l-edges, but according to our
knowledge, no existing method can compute the dimension of such spline space.
But if we arrange the order of the l-edges to be $e_{1}$, $e_{2}$, $e_{3}$ and
$e_{4}$, we can see that the new-vertex-vector is $(3,2,2,2)$, i.e., the
T-mesh is diagonalizable. Since the T-mesh has 8 cross-cut, $4$ interior
l-edges, and $27$ interior vertices. So the dimension of the spline space
$\mathcal{S}(3,3,1,1,\mathcal{T})$ over the second T-mesh is $16+4\times
2\times(8-4)+27\times 4=156$.
Figure 7: The dimension of bi-cubic spline space over the T-mesh is instable,
which is generated from the left T-mesh by moving two red l-edges.
###### Example 4.3.
The third example is associated with spline space
$\mathcal{S}(3,3,2,2,\mathcal{T})$ over the two T-meshes in Figure 7. Both
T-meshes have four interior l-edges and we can check that the T-meshes are not
diagonalizble. For the first T-mesh, the dimension is $65$ and the dimension
for the second T-mesh is instable, i.e., the dimension is associated with the
value of the knots.
We can see that the second T-mesh can be constructed by moving two red l-edges
of the first T-mesh. Although the two T-meshes have different structure, but
if we look at the structure of the conformality conditions matrix, we can see
that the structure of two matrixes are identical, except the two red l-edges
using the knots of $s_{2}$ to $s_{6}$ instead of $s_{1}$ to $s_{5}$. So
diagonalizble is a concept for the structure of the conformality conditions
matrix not T-mesh itself.
This example also tells us that if a T-mesh is not diagonalizable, then we
have to consider the knot values in order to analysis the dimension of the
spline space using smoothing cofactor-conformality method.
###### Remark 4.5.
A similar result has been abstained in [21], which provides the dimension for
a special T-mesh, called regular T-subdivision. The main difference between
these two method is that the diagonalizable T-meshes don’t need to be nested
structure. Regular T-subdivision is a special case of diagonalizable T-mesh.
### 4.1 Characterization
In this section, we will provide a necessary and sufficient condition for
characterization a diagonalizable T-mesh.
###### Lemma 4.6.
A a necessary and sufficient condition for a T-mesh to be diagonalizable is
for any interior l-edges set $\mathbb{S}$, there at least exists one horizonal
l-edge such that the number of vertices on this l-edge but not on the other
l-edge in $\mathbb{S}$ is at least $N^{h}$, or there at least exists one
vertical l-edge such that the number of vertices on this l-edge but not on the
other l-edge in $\mathbb{S}$ is at least $N^{v}$.
###### Proof.
First, we prove the condition is necessary using reduction to absurdity. IF
the T-mesh is diagonalizable, but there exists a set of l-edges
$\\{e_{i_{1}},e_{i_{2}},\dots,d_{i_{s}}\\}$ such that any horizonal l-edges in
the set at most have $N^{h}$ vertices which are not on the other l-edges in
the set and any vertical l-edges at most have $N^{v}$ vertices which are not
on the other l-edges in the set. And since the T-mesh is diagonalizable, so
without loss of generalization, we assume when the order of the interior
l-edges is $e_{1},e_{2},\dots,e_{n_{e}}$, in which order the l-edges satisfy
the condition of diagonalizable. Let $k$ to the maximal index for all
$i_{j},j=1,\dots,s$. Now, we consider l-edge $e_{k}$, since it has at most
$N^{h}-1$ or $N^{v}-1$ vertices which are not on the other l-edges in the set,
so it also has at most $N^{h}-1$ or $N^{v}-1$ vertices are not on the l-edges
for $e_{1},\dots,e_{k-1}$ since
$\mathbb{S}\subseteq\\{e_{1},\dots,e_{k-1},e_{k}\\}$, which violates the
assumption of diagonalizable. Thus, the condition is necessary.
Now we prove the condition is sufficient. For the set of l-edges
$e_{i},i=1,\dots,n_{e}$, according to the assumption, there exist one l-edge
which has enough vertices on the l-edge but not on the others. Without loss of
generalization, we assume it is $e_{1}$. Suppose we have ordered the l-edges
as $e_{1},e_{2},\dots,e_{j}$ satisfy the diagonalizable condition, then for
set $\\{e_{j+1},\dots,e_{n_{e}}\\}$, according to the assumption, there exist
one l-edge which has enough vertices on the l-edge but not on the others.
Without loss of generalization, we assume it is $e_{j+1}$. With this process,
we can order the l-edges such that it satisfy the diagonalizable condition,
which completes the proof. ∎
## 5 Correction for [1]
In this section, we will show that the dimension result in [1] is not right.
Precisely, the dimension under the condition of [1] is possible instable. And
we also provide a new modified theorem using the technology builded in the
last section.
### 5.1 A instable example under the condition of [1]
Figure 8: Counterexample for paper [1].
Consider the spline space $\mathcal{S}(3,3,2,2,\mathcal{T})$ over the T-mesh
illustrated in Figure 8. There are four interior l-edges in the T-mesh and
each one have two mocro-vertices in the interior, which satisfy the condition
in [1]. However, we will analysis the dimension in the following and show that
the dimension of the spline space over the T-mesh is instable.
Actually, we arrange the interior l-edges as the order of $e_{1}$, $e_{2}$,
$e_{3}$ and $e_{4}$. And we arrange the order of the vertices as $v_{2,2}$,
$v_{3,2}$, $v_{4,2}$, $v_{8,2}$, $v_{7,2}$, $v_{7,3}$, $v_{7,4}$, $v_{7,8}$,
$v_{7,7}$, $v_{6,7}$, $v_{5,7}$, $v_{1,7}$, $v_{2,7}$, $v_{2,6}$, $v_{2,5}$,
$v_{2,1}$ and $v_{2,2}$, then we can get the sparse matrix $\mathcal{M}$ which
has the following form,
$\mathcal{M}=\left(\begin{array}[]{cccccccccccccccc}1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0\\\
s_{2}&s_{3}&s_{4}&s_{8}&s_{7}&0&0&0&0&0&0&0&0&0&0&0\\\
s_{2}^{2}&s_{3}^{2}&s_{4}^{2}&s_{8}^{2}&s_{7}^{2}&0&0&0&0&0&0&0&0&0&0&0\\\
s_{2}^{3}&s_{3}^{3}&s_{4}^{3}&s_{8}^{3}&s_{7}^{3}&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&1&0&0&0&0&0&0&0\\\
0&0&0&0&t_{2}&t_{3}&t_{4}&t_{8}&t_{7}&0&0&0&0&0&0&0\\\
0&0&0&0&t_{2}&t_{3}^{2}&t_{4}^{2}&t_{8}^{2}&t_{7}^{2}&0&0&0&0&0&0&0\\\
0&0&0&0&t_{2}&t_{3}^{3}&t_{4}^{3}&t_{8}^{3}&t_{7}^{3}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&1&1&1&1&1&0&0&0\\\
0&0&0&0&0&0&0&0&s_{7}&s_{6}&s_{5}&s_{1}&s_{2}&0&0&0\\\
0&0&0&0&0&0&0&0&s_{7}^{2}&s_{6}^{2}&s_{5}^{2}&s_{1}^{2}&s_{2}^{2}&0&0&0\\\
0&0&0&0&0&0&0&0&s_{7}^{3}&s_{6}^{3}&s_{5}^{3}&s_{1}^{3}&s_{2}^{3}&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1\\\
t_{2}&0&0&0&0&0&0&0&0&0&0&0&t_{7}&t_{6}&t_{5}&t_{1}\\\
t_{2}^{2}&0&0&0&0&0&0&0&0&0&0&0&t_{7}^{2}&t_{6}^{2}&t_{5}^{2}&t_{1}^{2}\\\
t_{2}^{3}&0&0&0&0&0&0&0&0&0&0&0&t_{7}^{3}&t_{6}^{3}&t_{5}^{3}&t_{1}^{3}\\\
\end{array}\right)$
If $s_{i}=i$ and $t_{j}=j$, we can verify that the rank of the matrix is $15$,
thus the dimension of the spline space is $49$. If we perturb one of the knots
a little bit, such as $s_{3}=3.0+\epsilon$, where $\epsilon$ is an arbitrary
small value, then the dimension is $48$. In other words, the dimension is
instable.
### 5.2 Correction theorem
In this section, we will provide a new theorem to correct the theorem in [1]
using the method in the last section.
###### Theorem 5.1.
Given a regular T-mesh $\mathcal{T}$ without holes, if each horizontal l-edges
has at least $N^{h}-1$ mono-vertices and each vertical l-edge has at least
$N^{v}-1$ mono-vertices except two end vertices, then the dimension of the
spline space defined on $\mathcal{T}$ is
$\displaystyle\dim\mathcal{S}(d_{1},d_{2},\alpha,\beta,\mathcal{T})=$
$\displaystyle(d_{1}+1)(d_{2}+1)+(C^{h}-T^{h})(d_{1}+1)(d_{2}-\beta)+$
$\displaystyle(C^{v}-T^{v})(d_{2}+1)(d_{1}-\alpha)+V(d_{1}-\alpha)(d_{2}-\beta).$
Figure 9: The new correction theorem for [1].
###### Proof.
We will prove that under the condition, the T-mesh is diagonalizable using
reduction to absurdity. Actually, if the T-mesh is not diagonalizable. Then
for any set of l-edges, it is bounded. Suppose $e_{i}$ is the most bottom
horizonal l-edges in the set (if there are more than one l-edge in the set, we
will pick the leftmost one), see Figure 9 as an illustration. According to the
assumption, $e_{i}$ has at least $N^{h}-1$ mono-vertices (black rectangle
vertices in the figure), if one of the two end vertices is not on the other
l-edges of in the set, then $e_{i}$ at least has $N^{h}$ vertices which are
not on the other l-edges in the set. According to Lemma 4.6, the T-mesh is
diagonalizable which violates the assumption. Thus, there exists a vertical
l-edge, $e_{j}$, which contains one of the end vertices of $e_{i}$. Without
loss of generalization, we assume the right end vertex is on the l-edge
$e_{j}$ in the set. Now, we consider l-edge $e_{j}$, the bottom end vertex of
the l-edge cannot lie on the other l-edges in the set because it is on the
bottom of $e_{i}$ which is the bottommost l-edges in the set. So $e_{j}$ at
least has $N^{v}$ vertices which are not on the other l-edges in the set.
According to Lemma 3.1, the T-mesh is diagonalizable which violates the
assumption. Using Theorem 4.4, we prove the theorem. ∎
## 6 Conclusion and Future work
In the present paper, we introduce a class of T-meshes, diagonalizable
T-meshes, over which the dimension of spline spaces is stable. We also provide
a necessary and sufficient condition to characterize this class of T-meshes.
The dimension result in the present paper can cover all the existing dimension
results as special cases.
The paper leaves several open problems for further research. As we have
provided the dimension of the spline space, so there are many problems which
need to be solved, such as construction of a set of basis functions with good
properties, geometric operations and properties of the splines over T-meshes,
etc. We will explore these problems in detail in future papers. It is also an
important and interesting question to find out other general with fix
dimension T-meshes.
## Acknowledgements
The authors are supported by the NSF of China (No.11031007, No.60903148),
Chinese Universities Scientific Fund, SRF for ROCS SE and Chinese Academy of
Science (Startup Scientific Research Foundation).
## References
* [1] C. J. Li, R. H. Wang, F. Zhang, Improvement on the Dimensions of Spline Spaces on T-Mesh, Journal of Information & Computational Science 3 (2) (2006) 235–244.
* [2] G. Farin, NURBS Curves and Surfaces: from Projective Geometry to Practical Use, Fourht Edition, A. K. Peters, Ltd., Natick, MA, 2002.
* [3] J. A. Cottrell, T. J. R. Hughes, Y. Bazilevs, Isogeometric analysis: Toward Integration of CAD and FEA, Wiley, Chichester, 2009.
* [4] J. Deng, F. Chen, Y. Feng, Dimensions of spline spaces over t-meshes, Journal of Computational and Applied Mathematics 194 (2006) 267–283.
* [5] J. Deng, F. Chen, X. Li, C. Hu, W. Tong, Z. Yang, Y. Feng, Polynomial splines over hierarchical t-meshes, Graphical Models 74 (2008) 76–86.
* [6] X. Li, J. Deng, F. Chen, Surface modeling with polynomial splines over hierarchical t-meshes, The Visual Computer 23 (2007) 1027–1033.
* [7] X. Li, J. Deng, F. Chen, Polynomial splines over general t-meshes, The Visual Computer 26 (2010) 277–286.
* [8] N. Nguyen-Thanh, H. Nguyen-Xuan, S. P. A. Bordas, T. Rabczuk, Isogeometric analysis using polynomial splines over hierarchical t-meshes for two-dimensional elastic solids, Computer Methods in Applied Mechanics and Engineering 200 (2011) 1892 C1908.
* [9] P. Wang, J. Xu, J. Deng, F. Chen, Adaptiveisogeometricanalysis using rationalpht-splines, Computer-Aided Design 43 (2011) 1438–1448.
* [10] L. Tian, F. Chen, Q. Du, Adaptive finite element methods for elliptic equations over hierarchical t-meshes, J. Comput. Appl. Math. 236 (2011) 878–891.
* [11] L. L. Schumaker, L. Wang, Splines on triangulations with hanging vertices, Constructive Approximationdoi:10.1007/s00365-012-9167-x.
* [12] T. W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCSs, ACM Transactions on Graphics 22 (3) (2003) 477–484.
* [13] T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng, T. Lyche, T-spline simplification and local refinement, ACM Transactions on Graphics 23 (3) (2004) 276–283.
* [14] T. W. Sederberg, G. T. Finnigan, X. Li, H. Lin, H. Ipson, Watertight trimmed NURBS, ACM Transactions on Graphics 27 (3) (2008) Article no. 79.
* [15] Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. A. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott, T. W. Sederberg, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering 199 (5-8) (2010) 229 – 263.
* [16] Y. Bazilevs, L. Beirao de Veiga, J. Cottrell, T. Hughes, G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for $h$-refined meshes, Mathematical Models and Methods in Applied Sciences 16 (2006) 1031–1090.
* [17] X. Li, J. Zheng, T. W. Sederberg, T. J. R. Hughes, M. A. Scott, On the linear independence of T-splines blending functions, Computer Aided Geometric Design, 29 (2012) 63–76.
* [18] M. A. Scott, X. Li, T. W. Sederberg, T. J. R. Hughes, Local refinement of analysis-suitable T-splines, Computer Methods in Applied Mechanics and Engineering 213-216 (2012) 206–222.
* [19] X. Li, M. A. Scott, Analysis-suitable t-splines: Characterization, refinablility and approximation, submitted Mathematical Models and Methods in Applied Sciences for publish.
* [20] L. L. Schumaker, L. Wang, Spline spaces on tr-meshes with hanging vertices, Numerische Mathematik 118 (2011) 531–548.
* [21] B. Mourrain, On the dimension of spline spaces on planar t-subdivisions, Arxiv preprint arXiv:1011.1752.
* [22] R.-H. Wang, Multivariate Spline Functions and Their Applications, Science Press/ Kluwer Academic Publishers, 2001.
* [23] L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, in: Multivariate Approximation Theory, In: Schempp, W., Zeller, K. (Eds.), Birkhauser Verlag, Basel, 1979, pp. 396–412.
* [24] L. L. Schumaker, L. Wang, Approximation power of polynomial splines on t-meshes, Computer Aided Geometric Designdoi:http://dx.doi.org/10.1016/j.cagd.2012.04.003.
* [25] A. Buffa, D. Cho, M. Kumar, Characterization of t-splines with reduced continuity order on t-meshes, Comput. Methods Appl. Mech. Engrg. 201-204 (2012) 112–126.
* [26] X. Li, F. Chen, On the instability in the dimension of spline space over particular t-meshes, Computer Aided Geometric Design 28 (2011) 420–426.
* [27] D. Berdinskya, M. jae Oha, T. wan Kima, B. Mourrain, On the problem of instability in the dimension of a spline space over a t-mesh, Computers Graphics 36(2) (2012) 507–513.
* [28] M. Wu, J. Deng, F. Chen, The dimension of spline spaces with highest order smoothness over hierarchical t-meshes, Arxiv preprint arXiv:1112.1144.
|
arxiv-papers
| 2012-10-19T04:37:42 |
2024-09-04T02:49:36.831114
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1210.5312"
}
|
1210.5329
|
# The optimal division between sample and background measurement time for
photon counting experiments
Brian Richard Pauw brian@stack.nl – Samuel Tardif samuel.tardif@gmail.com –
(August 27, 2024)
## I Outline
_Note by the author: It was found after this derivation, that the work
presented here had been derived differently but with similar conclusions by
Steinhart and Plestil Steinhart and Plestil (1993). We nevertheless believe
that this simplified derivation may be of use to some readers._
Usually, equal time is given to measuring the background and the sample, or
even a longer background measurement is taken as it has so few counts. While
this seems the right thing to do, the relative error after background
subtraction improves when more time is spent counting the measurement with the
highest amount of scattering. As the available measurement time is always
limited, a good division must be found between measuring the background and
sample, so that the uncertainty of the background-subtracted intensity is as
low as possible.
Herein outlined is the method to determine how best to divide measurement time
between a sample and the background, in order to minimize the relative
uncertainty. Also given is the relative reduction in uncertainty to be gained
from the considered division. It is particularly useful in the case of
scanning diffractometers, including the likes of Bonse-Hart cameras, where the
measurement time division for each point can be optimized depending on the
signal-to-noise ratio.
The optimum setting for machines with photon-counting two-dimensional
detectors has to be further evaluated, but the intention is to include that in
this note.
## II The calculation
We assume that the number of background intensity photons $I_{b}$, measured
for a time $t_{b}$ is subtracted from the sample measurement photon count
$I_{s}$ which was measured for a time $t_{s}$, to result in the background
subtracted count rate $C_{bs}$:
$C_{bs}=\frac{I_{s}}{t_{s}}-\frac{I_{b}}{t_{b}}$ (1)
Defining the sample uncertainty as $\Delta I_{s}=\sqrt{I_{s}}$ and the
background uncertainty similarly as $\Delta I_{b}=\sqrt{I_{b}}$, the
uncertainty $\Delta C_{bs}$ would then be:
$\Delta C_{bs}=\sqrt{\left(\frac{\Delta
I_{s}}{t_{s}}\right)^{2}+\left(\frac{\Delta
I_{b}}{t_{b}}\right)^{2}}=\sqrt{\frac{I_{s}}{t_{s}^{2}}+\frac{I_{b}}{t_{b}^{2}}}$
(2)
Defining the number of counted photons $I$ to be a multiplication of the count
rate $C$ and the measurement time, we get $I_{b}=C_{b}t_{b}$ and
$I_{s}=C_{s}t_{s}$. Further defining the signal-to-noise ratio
$g=\frac{C_{s}}{C_{b}}$, the total time $t_{t}=t_{b}+t_{s}$ and the fraction
of time spent measuring the sample $f=\frac{t_{s}}{t_{t}}$, we can define our
relative uncertainty in terms of signal-to-noise ratio and time fraction:
$\frac{\Delta
C_{bs}}{C_{bs}}=\sqrt{\frac{\frac{C_{s}}{t_{s}}+\frac{C_{b}}{t_{b}}}{(C_{s}-C_{b})^{2}}}=\sqrt{\frac{1}{C_{b}t_{t}}}\sqrt{\frac{\frac{g}{f}+\frac{1}{(1-f)}}{(g-1)^{2}}}$
(3)
We then can try to find the optimum by locating the value for $f$ where the
derivative of equation LABEL:eq:sErel2 is zero:
$\frac{\partial\frac{\Delta C_{bs}}{C_{bs}}}{\partial
f}=\frac{\partial}{\partial
f}\sqrt{\frac{1}{C_{b}t_{t}}}\sqrt{\frac{\frac{g}{f}+\frac{1}{(1-f)}}{(g-1)^{2}}}=0$
(4) $0=\frac{\partial}{\partial
f}\sqrt{\frac{\frac{g}{f}+\frac{1}{(1-f)}}{(g-1)^{2}}}$ (5)
which, given $0<f<1$, is true for
$f=\frac{g-\sqrt{g}}{g-1}$ (6)
We can calculate the relative reduction in uncertainty compared to the 50/50
case (i.e. equal time spent on background and sample measurements) as:
$\frac{\frac{\Delta C_{bs}}{C_{bs}}\mid_{\mathrm{50/50}}-\frac{\Delta
C_{bs}}{C_{bs}}\mid_{\mathrm{optimal}}}{\frac{\Delta
C_{bs}}{C_{bs}}\mid_{\mathrm{50/50}}}=\frac{\sqrt{\frac{2g+2}{(g-1)^{2}}}-\sqrt{\frac{\frac{g^{2}-g}{g-\sqrt{g}}+\frac{g-1}{\sqrt{g}-1}}{(g-1)^{2}}}}{\sqrt{\frac{2g+2}{(g-1)^{2}}}}$
(7)
Figure 1: The optimal fraction of time $f$ spent measuring the sample as
opposed to measuring the background, as a function of the signal-to-noise
ratio $g$. Figure 2: The reduction in error that can be obtained by dividing
the time optimally between sample and background measurement, as a function of
the signal-to-noise ratio $g$.
## III Conclusions
Figures 1 and 2 show the optimal division of time between sample and
background, and the reduction in uncertainty obtained through this
optimization, respectively. These figures clarify that the reduction in
uncertainty may be worth the trouble of a quick determination of the signal-
to-noise ratio, especially in areas where this ratio strongly deviates from
unity.
A quick scan of sample and background may be used to automatically select the
most optimal use of measurement time, in particular for scanning (small- and
wide-angle) diffractometers (including oddities such as Bonse-Hart cameras)
where the measurement time _per point_ can be freely tuned.
## References
* Steinhart and Plestil (1993) M. Steinhart and J. Plestil, Journal of Applied Crystallography 26, 591 (1993).
|
arxiv-papers
| 2012-10-19T07:46:17 |
2024-09-04T02:49:36.841226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Brian Richard Pauw, Samuel Tardif",
"submitter": "Brian Pauw",
"url": "https://arxiv.org/abs/1210.5329"
}
|
1210.5502
|
OpenCFU, a New Free and Open-Source Software to Count Cell Colonies and Other
Circular Objects
Quentin Geissmann1,2,∗
1 Department of Animal and Plant Sciences, University of Sheffield, Sheffield,
United Kingdom, 2 Institute for Biology, Free University of Berlin, Berlin,
Germany.
$\ast$ E-mail: Corresponding q.geissmann@fu-berlin.de
## Abstract
Counting circular objects such as cell colonies is an important source of
information for biologists. Although this task is often time-consuming and
subjective, it is still predominantly performed manually. The aim of the
present work is to provide a new tool to enumerate circular objects from
digital pictures and video streams. Here, I demonstrate that the created
program, OpenCFU, is very robust, accurate and fast. In addition, it provides
control over the processing parameters and is implemented in an intuitive and
modern interface. OpenCFU is a cross-platform and open-source software freely
available at http://opencfu.sourceforge.net.
## Introduction
Counting objects has always formed an important element of data collection in
many fields of biology. It is therefore very common for biologists to
enumerate objects such as pollen[1], eggs[2], seeds[3], nuclei[4], cells[5] or
organisms[6]. Given that such tasks are time-consuming and, to some extent,
subjective, it is surprising that automation is still infrequent. Effectively,
enumerating objects is a two-part process: image capture and image analysis.
Nowadays, technologies such as digital cameras and webcams provide an
increasingly high image quality and are increasingly inexpensive.
Simultaneously, many optimised image processing algorithms and open-source
libraries can be used on laptops and desktop computers.
In different fields of microbiology, immunology and cellular biology, counting
colonies of cells growing on agar plates is routine. Automating such counting
procedures is not simple since colonies must first be isolated from the
background and then, if they overlap, be separated. In addition, such methods
must be capable of rejecting common artefacts such as imperfections in the
agar, dust and edges of Petri dishes. However, since cell colonies are
topologically fairly simple objects, solutions to enumerate them from pictures
have long since been considered [7, 8]. Commercial tools have been
developed[9], but remain expensive. Furthermore, the fact that the programs
they provide are proprietary (_i.e._ not open-source) makes them very
restrictive: it is rarely possible to know the precise nature of the analysis
being performed, nor to modify or share these programs. In contrast, some
authors have recently published open-source methods to count colonies from
digital images [10, 11, 12, 13, 14, 15, 16]. These use a variety of techniques
such as generalised Hough transform[11], template matching algorithm [5],
advanced illumination correction and particle filter [15], watershed algorithm
on a thresholded image [14, 10] and, more recently, a two-pass thresholding
procedure in combination with statistical models [16]. They were either
implemented as standalone programs capable of analysing image files [12, 11,
5, 13], extensions to existing image processing programs [15, 14, 10] or
integrated (hardware-software) counting systems [16].
All these methods claim to give results very comparable to human counts, to
improve objectivity and save time. However, none of them have been widely
adopted. In addition to the fact that some biologists are not aware of the
existence of such tools, their apparent failure to be adopted could be
explained by shortcomings in performance, such as an inability to split merged
colonies or long processing times. It is also very important for a method to
be robust and versatile: it must be able to perform well without changing
parameters when optimal conditions vary slightly, and the presence of
artefacts should only have minor consequences. Another reason for non-adoption
of automatic methods could be their lack of user-friendliness: even a perfect
method may fail to be adopted if the software that implements it does not
provide a modern user interface. Such an interface should enable the user to
easily submit a list of images for analysis and to manually alter the values
of the processing parameters. Additionally, the user should be able to select
a region of interest and visually check the results. Since many users do not
have deep knowledge of the underlying image processing, it is also important
that the parameters are intuitive and few. A final reason could be the lack of
public availability and maintenance. It is not rare that authors publish work
about a program without mentioning a download link, releasing the software on
a public repository or enclosing it as a supplementary material[11, 5]. In
addition, a program will most likely need to be updated to correct unpredicted
behaviour or avoid reliance on obsolete dependencies.
The aim of the work presented herein is to provide an alternative open-source
tool that features very robust, accurate and fast image processing as well as
a modern and functional user interface. Thanks to the optimised OpenCV
library[17], a rapid implementation of image processing functions has been
possible. This renders the analysis of very large and numerous picture-sets
easy and offers integration of capture devices (such as webcams). In order to
assess the relative efficacy and usefulness of the new software, a comparison
to two other available tools was undertaken.
The NIST’s Integrated Colony Enumerator (NICE)[13] software implements a
combination of threshold and extended minima in order to improve robustness.
The program comes with a functional user interface relying on few intuitive
parameters and provides accurate results. However, it is relatively slow and
lacks postprocessing filters. More recently, Cai _et al._ published[10] a
short ImageJ macro which they described as very accurate and only depending on
two parameters. The method relies on adaptive thresholding, watershed
algorithm and subsequent particle filtering. It does not provide a functional
user interface or indicate a way to open a list of files and display the
outlines of the detected colonies, nor does it allow postprocessing of the
detected colonies. Since ImageJ macro language is flexible, it is possible,
but requires some familiarity with the language.
My results show that OpenCFU, the software created, is faster, more accurate
and more robust to the presence of usual artefacts than NICE and Cai _et al._
’s macro. As well as efficiently counting bacterial colonies, the program can
also be used to enumerate other circular objects such as seeds or pollen.
## Algorithm and Implementation
### Algorithm
Methods relying on direct thresholding of a grey-scale image followed by
morphological segmentation are likely to detect high contrast artefacts such
as parts of the edges of Petri dishes and bubbles. For a given value of
threshold, some portions of these artefacts could, by chance, be
morphologically similar to colonies and will result in detection of false
positives. The algorithm proposed here aims to increase robustness by
virtually testing all possible values of threshold and keeping only regions
that were recurrently morphologically valid. A detailed flowchart is provided
in figure 1 and portions of images at different processing stages are shown in
figure 2. Briefly, the colour image is split into three channels from which
the background is independently estimated using a local median filter. In
order to enhance discrimination of foreground objects, the positive Laplacian
of Gaussian of each channel’s foreground mask is subtracted from itself and
three preprocessed channels are normalised and merged to form a new grey-scale
image.
The first pass of the processing involves thresholding the resulting grey-
scale image by multiple values. For each value, the algorithm will search for
connected components in the binary image. Each component is assessed by a
particle filter that takes into account relationships between variables such
as area, perimeter, convexity, aspect ratio and hollowness in order to
determine whether or not a region is likely to be valid (_i.e_ made from one
or more circular objects). Over the range of threshold values, every time a
valid region is found, all its pixels are incremented in a “score-map”. The
score-map can be understood as a representation of how recurrently (over the
iterations of threshold) pixels are part of a circular region. Finally, a
user-defined (or automatic) threshold is applied to the score-map. During the
second pass of the processing, a similar particle filter is applied but this
time it classifies the connected components as “invalid”, “individual object”
or “multiple objects”. “Individual objects” are accepted and “invalid” ones
are rejected immediately, whilst “multiple objects” are morphologically
segmented using a variant of the watershed algorithm on their distance-
map[18]. Finally, all segmented objects are reassessed by the particle filter.
Optionally, a normal distribution is fitted to the relative colour intensities
of objects that were not split (since they are less often falsely positive).
This distribution then serves to compute the likelihood of each object to be
valid. Finally, a simple likelihood user-controlled threshold is applied to
exclude marginal objects.
### Implementation and User Interface
For performance reasons, OpenCFU was programmed in C++. The image processing
was implemented using OpenCV framework[17] which offers highly optimised image
processing functions. The time-consuming loops were optimised further for
multi-core architecture using OpenMP library[19]. The graphical user interface
was designed using GTKmm. These three libraries are open-source, cross
platform and regularly maintained.
OpenCFU was designed in order to accelerate the calibration phase by having a
fast processing time and by immediately displaying results after parameters
have been changed. In addition, when a parameter is changed, OpenCFU
dynamically restarts the analysis from the first step involving this variable,
as opposed to systematically re-analysing the image from the beginning. For
instance, if the value of a postprocessing filter is altered, the whole
processing will not be re-run. The software also integrates an interface to
video devices and other features such as optional automatic dish detection
based on Hough circle transform.
The program can deliver two different types of result: a summary or a detailed
output. In the summary, each row of data contains the name of the analysed
image, the number of colonies detected in this image and, if a mask was used,
the surface of the mask. In the detailed output, each row of data corresponds
to a different colony. Each colony is characterised by the name of the image
it comes from, the surface of the mask used for this image, the position (X,Y)
of its centre, its corrected median values of colour intensity, its area, its
perimeter and the number of colonies that were in the same cluster as this
colony. This latter output is helpful for users needing to perform advanced
analysis.
## Results
### Speed
Since algorithms are likely to iteratively process each of the foreground
objects, their speed could differ according to their size and number.
Therefore, in order to assess the method speed, two types of images were
generated from the same template: “scaled-up” images featuring a constant
number of increasingly big bacterial colonies, or “tiled” images with an
increasing number of constant-sized colonies (fig. 3A).
The processing time for OpenCFU, NICE[13] and an ImageJ macro by Cai and
colleagues [10] (IJM), were compared while analysing the same images (fig.
3B). Under the tested conditions, the three algorithms performed in linear
time ($O(n)$). OpenCFU was the fastest, followed by IJM and NICE. For
instance, for a typical picture of 1.6$\times{}$1.6kpx the tools would take
approximatively 0.69, 1.22 and 3.0 seconds, respectively.
### Accuracy
In order to assess the accuracy of OpenCFU, 19 plates containing between 10
and 1000 _Staphylococcus aureus_ colonies were prepared, independently
enumerated by seven trained humans and then photographed with a high-
definition camera. The pictures were analysed by OpenCFU, NICE and IJM. The
results obtained by humans and automatic methods were then compared. The
deviations of the results of each agent from the medians of human counts, the
reference, were calculated.
NICE and, to a lesser extent, IJM tended to overestimate the number of
colonies when few are present whilst underestimating the high-density plates
(fig. 4A). The significance of the slopes of the linear regressions was
assessed by a t-test: $a=-34.56\%$ ($P$-$value=4.52\cdot 10^{-5}$) and
$a=-10.15\%$ ($P$-$value=1.98\cdot 10^{-4}$), respectively. Analysis by
OpenCFU did not result in a significant bias: $a=-2.58\%$ ($P$-$value=0.169$).
The average of absolute deviations from the reference was used as a measure of
error and compared between agents (fig. 4B). Images of the same plates were
also taken with a low-cost webcam and analysed in order to estimate the impact
of poor quality images on the accuracy of the three methods. The median error
was $0.93\%$ for human agents. In order to assess the inaccuracy of the
automatic methods, their errors were compared to the human errors by
performing a Wilcoxon test: NICE (median error $=9.93\%$) and IJM (median
error $=6.64\%$) had a significantly higher inaccuracy than humans
($P$-$value=4.65\cdot 10^{-10}$ and $1.37\cdot 10^{-5}$, respectively). In
comparison, OpenCFU (median error $=1.93\%$) was not less accurate
($P$-$value=0.44$). As expected, using poor quality pictures increased the
error for OpenCFU (median error $=2.78\%$; $P$-$value=0.0495$), NICE (median
error $=13.0\%$; $P$-$value=1.92\cdot 10^{-10}$) and IJM (median error
$=11.1\%$; $P$-$value=2.91\cdot 10^{-6}$);
### Robustness
In order to assess how robust the three methods were, pictures of plates
featuring typical artefacts were analysed. Figure 5A shows qualitative results
of this approach. Both NICE and IJM seemed likely to falsely count bubbles,
edges, cracks or dust, whilst OpenCFU appeared unaffected.
To quantify the robustness of the three methods to the presence of edge, the
images used for the accuracy test (fig. 4) were translated by 25 pixels to the
top-left corner to simulate a slight (1.7mm) mispositioning during acquisition
(fig. 5B), and were re-analysed with the same parameters. The bias induced was
assessed by measuring the difference between the results before and after
perturbation. A paired t-test was performed between the original results of
the three methods and their respective results after perturbations. In these
conditions, IJM and NICE overestimated their own result by $26.0$
($sd=19.7,P$-$value=9.01\cdot 10^{-6}$) and $7.37$
($sd=7.38,P$-$value=1.9\cdot 10^{-4}$) colonies, respectively. OpenCFU was not
affected: $0.159$ ($sd=1.64,P$-$value=0.34$) colonies.
In order to quantify the extent to which the presence of bubbles in the agar
matrix would impact on the precision of the methods, 18 plates containing
exclusively bubbles (between 0 and 20) were analysed. The relationships
between number of bubbles and number of detected colonies is represented in
figure 5C. Linear regressions were performed and the significance of the
slopes was assessed by a t-test. The number of objects detected by IJM and
NICE were positively related to the number of bubbles, $R^{2}=0.66$
($a=1.18,P$-$value=2.56\cdot 10^{-5}$) and $R^{2}=0.65$
($a=0.94,P$-$value=3.35\cdot 10^{-5}$) , respectively. This was not the case
for OpenCFU, $R^{2}=0.00$ ($a=0.03,P$-$value=0.90)$ . In order to
qualitatively assess the ability of OpenCFU to process images of diverse
nature, representative pictures of circular biological objects were analysed.
Graphical outputs are shown in figure 6. These results suggest that the
algorithm is versatile enough to enumerate circular objects from very
different images.
## Discussion
In the present study, a new algorithm based on recursive research of circular
regions over values of threshold of a grey-scale image has been presented,
implemented and compared to two alternative methods[13, 10]. Comparison with
additional software tools was considered, but, for different reasons, could
not be formally performed. Among them, Clono-counter [12] can certainly be
useful in some situations; however, it seems inappropriate for analysing
numerous or large images. The program does not allow the user to analyse
successively several files, and, for instance, using the same mask for
different images is not possible. In addition, large images need to be down-
scaled before analysis; as an objective comparison between programs requires
all methods to process the same images, it was not possible to use Clono-
counter. Both CHiTA[11] and Arraycount[5] have probably been very useful to
some researchers in the past, but they were both unavailable for download and
could not be compared to the present method. The method based on Cell profiler
[15] could, in my case, not be adapted through minor alterations in the
proposed pipeline. The ImageJ plug-in developed by Sieuwerts _et al._ [14]
involved several human interventions (converting to grey-scale and
thresholding). In addition, a general, rather than adaptive, threshold on the
grey-scale image was, in my case, not satisfying since background intensity
varied between and within plates. For this reason, this plug-in was not
included either. Finally, the results presented by S. Brugger and co-workers
[16] are promising, but they proposed a colony counter relying on a specific
acquisition platform rather than a standalone program capable of processing a
range of pictures from different devices.
Despite the intensive nature of the image processing, the implementation of
the described algorithm has been shown to perform even faster than methods
that are supposedly less intensive [13, 10] (fig. 3). This is most likely due
to the use of the optimised OpenCV library[17] in combination with custom C++
functions. For a standard picture, the three tested methods performed in less
than three seconds. In comparison, much slower methods, such as the one
presented by Vokes and Carpenter[15], would take approximately two minutes for
the same image (data not shown). Since, on modern hardware, the processing
time will rarely be much longer than the total acquisition time, OpenCFU’s
faster performance is unlikely to provide a decisive direct gain of time over
NICE[13] and the ImageJ macro published by Cai and colleagues (IJM)[10].
However, authors have acknowledged that calibration is the most time-consuming
step of semi-automatic methods [15, 10]. Since, during calibration, a human
will essentially try multiple values of parameters on different sample images,
a fast processing speed coupled with an immediate display of the result will
certainly provide an advantage. In addition, OpenCFU will save calibration
time by only reprocessing an image from the first step involving the parameter
that has been changed rather than from the beginning.
OpenCFU was shown to be very accurate and, with high-definition pictures, did
not generate more errors than the average human error (fig. 4). In contrast,
NICE and IJM were shown to overestimate plates with few (less than 50)
colonies and underestimate plates with large numbers of colonies (fig. 4A).
They also had a significantly higher inaccuracy than humans (fig. 4B). NICE
and IJM had to be used with regions of interest drawn inside the dish. This
practice excludes colonies from the analysis and could result in a consistent
underestimation. But such bias could also be a consequence of an inefficiency
to detect or segregate small colonies. The overestimation of low-load plates
was however probably the result of false positives arising from the presence
of artefacts which are pragmatically hard to avoid. Using NICE with Otsu’s
method for thresholding gave overall good results (fig. 4A). However, it
assumes foreground objects exist and would be inappropriate if only few (or
no) colonies are present. One solution could be to manually count plates that
have few colonies and use an image processing-based method for high-load
plates. This will surely result in logistic complications and errors. Another
solution is to individually verify each processed image and reprocess some
with a different threshold. This is not easy with NICE because the user has to
specifically click on image names one-by-one and then click each time to query
a display of the colonies. Finally, if an image is judged to be misprocessed,
there is no obvious manner to reprocess it exclusively. IJM does not provide a
way to load a list of images and overlay results on top of the original images
for visual verification. Achieving this is possible but involves some
knowledge of ImageJ macro language.
Authors have emphasised the need to have a low-cost platform to count colonies
and have, for instance, used desktop-scanners[18, 9, 13, 10] that are
generally rather slow. In this study, a low-cost webcam used in combination
with OpenCFU provided very satisfying results (fig. 4B). OpenCFU integrates
video devices such as webcams, USB microscopes or firewire cameras in order to
facilitate this approach.
For most biologists, a moderate average deviation (lower than 20%) will often
be negligible compared to noise generated by other experimental factors.
Rather than trying to provide a perfect similarity to human counts, methods
based on digital image processing should focus on robustness and safety. In
this study, OpenCFU has been shown to be very robust to the presence of
artefacts in comparison to alternative methods (fig. 5A). A simulation of a
small, but likely, physical perturbation of the settings — mispositioning the
dish during acquisition — was proven to lead to a large overestimation of the
number of colonies by alternative methods (fig. 5B). Even if a method is very
accurate in perfect conditions, a biologist may be reluctant to use it if
moving the acquisition set-up by less than two millimetres could add more than
twenty colonies to the result. In order to limit the impact of such an
eventuality, the experimenter will need to systematically check and sometimes
reprocess images. Vokes and Carpenter [15] solved this problem by using a
template mask and computing its best alignment to the actual image. As OpenCFU
is, by design, very robust to the presence of edges, the use of a mask is
rarely needed. In fact, the regions of interest defined in this study
completely included the outside boundaries of the dishes. In addition, the
software provides automatic detection of a Petri dish in the images which
corrects for shifts of the dish. This feature also avoids the subjectivity of
manually drawing a mask and could indirectly save acquisition time since the
user can afford to take less care in positioning the dishes.
Unlike alternatives, OpenCFU was not affected by common artefacts such as
bubbles in the agar matrix (fig. 5C). _Staphylococcus aureus_ colonies, used
in this study, are opaque and therefore produced well contrasted areas. As
NICE and IJM do not perform a stringent morphological analysis of detected
objects, their sensitivity to artefacts can be suspected to be even higher
when analysing less contrasted objects. Although it is possible to verify that
no artefacts are present and to redraw the mask if necessary, this step would
be an additional subjective and time-consuming human intervention. Another
advantage resulting from the robustness of the algorithm is the ability of
OpenCFU to process a diverse range of images. Preliminary results (fig. 6)
indicate that the algorithm is capable of enumerating different types of
colonies, as well as objects such as round seeds and pollen from very
different pictures.
In this study, humans took 39.3 (sd=6.1) minutes in order to count the 19
plates. Theoretically, the three methods could achieve the same result in less
than five minutes (including acquisition time). However, the lack of
robustness of NICE and IJM could require systematic verification and
rectification of the results after processing. The user interface of NICE does
not make this task easy and IJM does not natively provide this function. Such
flaws could negate the time-gain they would offer in perfect conditions.
OpenCFU provides postprocessing filters to eliminate detected objects
according to their intensity and colour similarity. This makes the method even
more robust since it can be used to exclude circular artefacts such as
contaminant bacteria. Most frequently, biologists will write in a small area
of the dish or in the edge; these filters can also be used to remove all
writing, as it often has a very specific colour. OpenCFU can also produce
detailed output containing information about each detected colony. This is
useful for users who, for instance, wish to calculate the respective number of
distinct populations of cells.
In this study, a new algorithm was used to count bacterial colonies and
implemented in a modern and functional interface. In the tested conditions,
OpenCFU has been shown to be faster, more accurate, and more robust to common
perturbations than the two tested alternatives. OpenCFU will help to save time
and reduce subjectivity of colony counting. Since many other biological
objects (for instance, seeds, pollen, cells, nuclei and eggs) are circular and
well defined from the background, they can also be counted without any
modification of the software.
## Availability and Future Directions
OpenCFU is an open-source program distributed under the conditions of the GNU
General Public Licence version 3. It is available on Sourceforge
(http://opencfu.sourceforge.net). A user manual and video tutorial are also
provided. In addition, an increasingly large list of image samples and
corresponding results will be maintained in order to help users calibrate the
method. The program will be improved and updated as users request features and
bugs are pointed out. Planned improvements of the program include support for
multiple regions of interest, user-supervised exclusion of outlier colonies
and the availability of a command line version.
## Materials and Methods
### Image Processing
Details of the processing pipeline and its implementation can be freely
viewed, modified and redistributed from the source code. The file
“robustCircularRegions.cpp” implements the new algorithm. The particle filter
and custom watershed function are in “metaContourFilter.cpp”. All the
processing functions used are either directly provided by OpenCV or written in
C++ (using OpenCV framework) in the same file.
The particle filter first rejects objects that do not fulfil all of the
following conditions:
$Perimeter>2\times{}minRad\times{}\pi$
$Area>\pi\times{}{minRad}^{2}$
$Height>minRad\times{}2$
Where,
$minRad$ is the user-defined minimal radius.
Then, it decides if a region is a single object (colony) or multiple clustered
objects. To be a single object, it must fulfil at least one of the following
criteria:
$Solidity>0.95$
$Area<1.6\times{}\pi\times{}{minRad}^{2}$
$Perimeter<1.6\times{}minRad\times{}2$
Where,
$Solidity=\frac{Area_{shape}}{Area_{ConvexHull}}$
Otherwise, it is decided to be multiple objects.
If it is a single object, it must fulfil all of the following criteria:
$Aspect$ $Ratio<1.7$
$Area<\pi\times{}{maxRad}^{2}$
$\frac{Perimeter^{2}}{Area\times{}4\pi}<1.6$
Where,
$maxRad$ is the user-defined (or calculated from the image dimensions) maximal
radius for an object.
If it is a cluster of objects, it must fulfil all of the following criteria:
$Aspect$ $Ratio<5$
$Width>1.4\times{}2\times{}minRad$
$Area<10\times{}\pi\times{}{maxRad}^{2}$
$\frac{Perimeter^{2}}{Area\times{}4\pi}<4.0$
To segregate clustered objects — during the second pass of the processing — a
distance transform is computed on a 5$\times{}$5 pixel neighbourhood with a=1,
b=1.4, c=2.1969. The peaks of the distance transform serve as markers for a
custom watershed function. Briefly, the watershed-like function works as
follows:
All markers correspond to the local maxima in the distance-map.
As long as marked regions can grow:
If a marked pixel has a higher or equal value than a neighbour and the
neighbour is not marked, the neighbour becomes marked with the same label.
In addition, marked regions are not allowed to expand their area over a limit
value $V$:
$V=1.6\times{}\pi\times{}{r}^{2}$
And the distance between the original marker of a region and any pixel of this
region must be lower than $D$:
$D=1.6\times{}{r}$
Where,
$r$ is the value of the corresponding peak in the distance-map (_i.e._ the
presumptive radius).
### Plates Preparation and Manual Counting
LB broth supplemented with 1.5% agar was poured into 19 standard 90mm plates.
An overnight culture of _Staphylococcus aureus_ was diluted and 100$\mu{}$L
were plated. The bacterial solution was spread using ten 2mm glass beads. The
plated bacteria were grown overnight at 37∘. Seven trained individuals were
given the 19 plates in a random order. The experiment was blinded so that no
subject could know the results of any other before counting. The total time
they took (excluding copying data to an electronic file) was recorded. Plates
with bubbles (fig. 5C) were obtained by injecting between 0.5 and 1$\mu{}$L of
air in the agar matrix. Sometimes, bubbles were surrounded by one or two
smaller adjacent bubbles. Under this scenario, only the largest was counted.
### Semi-Automatic Methods
OpenCFU version 3.3 was used with a “threshold” value of 12 for high-
definition images and 7 for webcam images. The mask was drawn automatically
for all plates by choosing the option “Auto-Petri” with a margin of -25px. The
software NICE and the necessary proprietary MATLAB Complier Runtime were both
downloaded from http://www.nist.gov/pml/div682/grp01/nice.cfm. An elliptic
region of interest was manually drawn and applied for all plates in order to
exclude the edge of the Petri dishes and the Otsu thresholding method was used
with the “high resolution” setting for high-definition pictures and “medium
resolution” for the webcam images. The ImageJ macro was adapted from Cai’s
publication[10] with minor modifications. The threshold ($z1$) was 205 and the
minimal size ($z2$) was 15.
### Hardware
The high quality pictures were taken using QIMAGING Micropublisher 3.3RTV
device with a TAMRON 1:1.4 25mm ø30.5 lens. The pictures generated were
1536$\times{}$1538px well-contrasted images. The webcam used as a capture
device for the real-time enumeration was a Sweex Blackberry Black WC250
(1600$\times{}$1200px, 30fps). A white trans-illuminator was used to optimise
contrast in both cases.
The processing time of OpenCFU, NICE and IJM were assessed on the same
machine: a “System76 Gazelle Professional” (CPU = intel i7-2630QM, 2.00GHz).
OpenCFU and IJM were used under GNU/Linux operating sytem (Linux Mint 13,
kernel 3.2). OpenCFU was compiled with g++ 4.6.2 and OpenCV 2.4.2. IJM was
used with ImageJ 1.46 (Java 1.6.0_24). NICE was used under
Windows7-professional (64bit).
### Statistical analysis
In order to assess the effect of the number of colonies on the deviation from
the reference (fig. 4A), a linear model between deviation and
$log_{10}(Reference)$, was fitted. A t-test was performed on the slope of the
regression line.
The deviation in the count of each plate was given by:
$Deviation_{p}=100\times{}\frac{(Agent_{p}-Reference_{p})}{Reference_{p}}$
Where,
$p$ is the plate
and $Reference$ the median of human counts for $p$
The absolute deviations from the reference (fig. 4B) were compared between
each agent and the pooled human group by performing a Wilcoxon test. In order
to assess the significance of the greater number of detected colonies after
translation of images (fig. 5B), a one-tailed paired t-test comparing the
numbers before and after perturbation was performed. In order to quantify the
effect of the number of bubbles on the number of detected colonies (fig. 5C),
a linear model was fitted and a t-test was performed on the slope of the
regression line. Statistical analysis was performed using R software[20] .
## Acknowledgments
I am very thankful to Jens Rolff for his support and to Clayton Costa for
proving the pollen picture of figure 6.
## References
* 1. Costa CM, Yang S (2009) Counting pollen grains using readily available, free image processing and analysis software. Annals of Botany 104: 1005–1010.
* 2. Mello CAB, dos Santos WP, Rodrigues MAB, Candeias ALB, Gusmão CMG (2008) Image segmentation of ovitraps for automatic counting of _Aedes Aegypti_ eggs. Conference proceedings: Annual International Conference of the IEEE Engineering in Medicine and Biology Society IEEE Engineering in Medicine and Biology Society Conference 2008: 3103–3106.
* 3. Severini AD, Borrás L, Cirilo AG (2011) Counting maize kernels through digital image analysis. Crop Science 51: 2796.
* 4. Forero MG, Pennack JA, Hidalgo A (2010) DeadEasy neurons: automatic counting of HB9 neuronal nuclei in _Drosophila_. Cytometry Part A: the journal of the International Society for Analytical Cytology 77: 371–378.
* 5. Kachouie NN, Kang L, Khademhosseini A (2009) Arraycount, an algorithm for automatic cell counting in microwell arrays. BioTechniques 47: x–xvi.
* 6. Yati A, Dey S (2011) FlyCounter: a simple software for counting large populations of small clumped objects in the laboratory. BioTechniques 51: 347–348.
* 7. Mansberg HP (1957) Automatic particle and bacterial colony counter. Science 126: 823 –827.
* 8. Mukherjee DP, Pal A, Sarma S, Majumder D (1995) Bacterial colony counting using distance transform. International Journal of Bio-Medical Computing 38: 131–140.
* 9. Putman M, Burton R, Nahm MH (2005) Simplified method to automatically count bacterial colony forming unit. Journal of Immunological Methods 302: 99–102.
* 10. Cai Z, Chattopadhyay N, Liu WJ, Chan C, Pignol JP, et al. (2011) Optimized digital counting colonies of clonogenic assays using ImageJ software and customized macros: comparison with manual counting. International Journal of Radiation Biology 87: 1135–1146.
* 11. Bewes JM, Suchowerska N, McKenzie DR (2008) Automated cell colony counting and analysis using the circular hough image transform algorithm (CHiTA). Physics in Medicine and Biology 53: 5991–6008.
* 12. Niyazi M, Niyazi I, Belka C (2007) Counting colonies of clonogenic assays by using densitometric software. Radiation Oncology 2: 4.
* 13. Clarke ML, Burton RL, Hill AN, Litorja M, Nahm MH, et al. (2010) Low‐cost, high‐throughput, automated counting of bacterial colonies. Cytometry Part A 77A: 790–797.
* 14. Sieuwerts S, De Bok FA, Mols E, De Vos WM, Van Hylckama Vlieg JE (2008) A simple and fast method for determining colony forming units. Letters in Applied Microbiology 47: 275–278.
* 15. Vokes MS, Carpenter AE (2008) Using CellProfiler for automatic identification and measurement of biological objects in images. Current protocols in molecular biology edited by Frederick M Ausubel et al Chapter 14: Unit 14.17.
* 16. Brugger SD, Baumberger C, Jost M, Jenni W, Brugger U, et al. (2012) Automated counting of bacterial colony forming units on agar plates. PLoS ONE 7: e33695.
* 17. Bradski G (2000) The OpenCV Library. Dr Dobb’s Journal of Software Tools .
* 18. Marotz J, Lübbert C, Eisenbeiß W (2001) Effective object recognition for automated counting of colonies in petri dishes (automated colony counting). Computer Methods and Programs in Biomedicine 66: 183–198.
* 19. OpenMP Architecture Review Board (2011). OpenMP application program interface version 3.1. URL http://www.openmp.org/mp-documents/OpenMP3.1.pdf.
* 20. R Core Team (2012) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org. ISBN 3-900051-07-0.
## Figure Legends
Figure 1: Flowchart representing the processing steps. The image is
preprocessed (1) in order to correct for gradual changes in background
intensity and increase the contrast. The first pass of the processing (2)
generates a score-map by iteratively annotating valid regions. The second pass
(3) involves finding connected components in the thresholded score-map and
segmenting them using a distance transform/watershed approach. Optional
postprocessing filters (4) can be performed by OpenCFU or, using the raw data,
by the user.
Figure 2: Illustration of the processing steps performed on three sample
images. Each channel of the original image (A) is preprocessed individually
and merged to a grey-scale image (B). A score-map is generated by recursive
thresholding an annotation of circular regions (C). This excludes regions that
were morphologically unlikely to be colonies (_i.e._ arrows 1 and 2). The
score-map is then thresholded by a user-defined or automatically calculated
value (D). The objects identified as merged colonies (on the basis of their
morphological features) are segmented using a watershed variant on their
distance transform (E). Arrow 3 shows objects that have been successfully
segmented. Finally, the morphologically valid objects can be assessed further
using intensity and colour filters. Arrow 4 shows a minority contaminant
bacteria that was excluded using postprocessing filter and represented by
crossed-out red ellipses. Arrow 5 shows valid colonies represented by yellow
and blue ellipses. For the purpose of explanation, only representative areas
(200$\times{}$200 pixels) of three processed images are shown here.
Figure 3: Processing time of OpenCFU, NICE[13] and an ImageJ macro[10] for
images of different size. An original arbitrary square image was either tiled
to itself or scaled-up (A) in order to obtain a range of square images
featuring an increasing number of colonies or increasingly large colonies,
respectively. The processing time of the three methods for these images was
estimated in both cases (B). On the tested range of resolutions, OpenCFU was
faster than both NICE and the ImageJ macro (IJM). The segments joining points
do not represent data, but only aid readability.
Figure 4: Comparison of accuracy between OpenCFU, NICE[13] and an ImageJ
macro[10]. The medians of seven humans counts were used as a reference to
measure deviation. The effect of the number of colonies on the deviation from
the reference was assessed (A). For NICE and the ImageJ macro (IJM), the slope
was significantly negative. The dotted line represents the reference. The
absolute deviation from the reference was used as a measure of error (B).
Error for the best human, the worst human and the three methods were compared
to the pooled human group. With high-definition images (HD), NICE and IJM had
a higher error than the pooled human group (Pool) while OpenCFU (OFU) did not.
Using low-definition pictures (LD) from a low-cost webcam increased the error
for the three methods.
Figure 5: Comparison of robustness to common perturbations between OpenCFU,
NICE[13] and an ImageJ macro[10]. A qualitative assessment of robustness was
undertaken by analysing pictures containing artefacts (A). Representative
portions of 1.7cm by 1.7cm (200$\times{}$200 pixels) illustrate the results of
the presence of bubbles (1), cracks in the agar (2), dust (3) and edge of dish
(4) in the region of interest. Objects detected by OpenCFU, NICE and the
ImageJ macro (IJM) are represented by ellipses, crosses and arbitrary colours,
respectively. NICE and IJM but not OpenCFU seemed to consistently detect
artefacts as colonies.
A quantitative analysis of robustness to plate mispositioning was conducted
(B). OpenCFU, NICE and IJM were used to count the number of colonies in the
pictures of 19 plates. Then, all the images were translated by 1.7mm (25px)
and analysed with the same region of interest as the original. This procedure
induced a significant bias for NICE, $7.37$ ($sd=7.38,P$-$value=1.9\cdot
10^{-4}$) colonies and IJM $26.0$ ($sd=19.7,P$-$value=9.01\cdot 10^{-6}$)
colonies, but not for OpenCFU $0.159$ ($sd=1.64,P$-$value=0.34$) colonies
(one-sided paired t-test).
The impact of the presence of bubbles in the agar was measured by analysing
pictures of 18 plates containing exclusively bubbles (C). A linear regression
between the number of bubbles and the number of detected objects was
performed. NICE and IJM counts were both positively related to the number of
bubbles, $R^{2}=0.65$ ($a=0.94,P$-$value=3.35\cdot 10^{-5}$) and $R^{2}=0.66$
($a=1.18,P$-$value=2.56\cdot 10^{-5}$) , respectively. OpenCFU was not
affected: $R^{2}=0.00$ ($a=0.03,P$-$value=0.90)$ .
Figure 6: Versatility of OpenCFU. A qualitative assessment of the versatility
of OpenCFU was undertaken by analysing pictures of different circular
biological objects: a clear (A) and a poor quality (B) picture of
_Staphylococcus aureus_ colonies, a low-contrasted picture of _Escherichia
coli_ (C), a noisy picture of mustard seeds (D), a noisy picture of soy-bean
seeds (E), and a micrography of _Carduus sp._ pollen (F). For the purpose of
explanation, only representative areas (200$\times{}$200 pixels) of six
processed images are shown here. Original portions of images are on the left
and correspond to the graphical results obtained using OpenCFU on the right.
|
arxiv-papers
| 2012-10-18T14:05:17 |
2024-09-04T02:49:36.851202
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quentin Geissmann",
"submitter": "Quentin Geissmann",
"url": "https://arxiv.org/abs/1210.5502"
}
|
1210.5557
|
# Applications of fixed point theorems in the theory of invariant subspaces
Rafa Espínola and Miguel Lacruz Rafa Espínola - espinola@us.es Miguel Lacruz-
lacruz@us.es Departamento de Análisis Matemático, Facultad de Matemáticas,
Universidad de Sevilla, Avenida Reina Mercedes, 41012 Seville (SPAIN)
###### Abstract
We survey several applications of fixed point theorems in the theory of
invariant subspaces. The general idea is that a fixed point theorem applied to
a suitable map yields the existence of invariant subspaces for an operator on
a Banach space.
###### Key words and phrases:
Invariant subspace; Fixed point
###### 1991 Mathematics Subject Classification:
47A15, 47H10
## 1\. Introduction
One of the most recalcitrant unsolved problems in operator theory is the
invariant subspace problem. The question has an easy formulation. Does every
operator on an infinite dimensional, separable complex Hilbert space have a
non trivial invariant subspace?
Despite the simplicity of its statement, this is a very difficult problem and
it has generated a very large amount of literature. We refer the reader to the
expository paper of Yadav [1] for a detailed account of results related to the
invariant subspace problem.
In this survey we discuss some applications of fixed point theorems in the
theory of invariant subspaces. The general idea is that a fixed point theorem
applied to a suitable map yields the existence of invariant subspaces for an
operator on a Banach space.
In Section 2 we consider the striking theorem of Lomonosov [2] about the
existence of invariant subspaces for algebras containing compact operators.
The proof of this theorem is based on the Schauder fixed point theorem.
In Section 3 we present a recent result of Lomonosov, Radjavi, and Troitsky
[3] about the existence of invariant subspaces for localizing algebras. The
proof of this result is based on the Ky Fan fixed point theorem for
multivalued maps. The idea of using fixed point theorems for multivalued maps
in the search for invariant subspaces was first introduced by Androulakis [4].
In Section 4 we consider an extension of Burnside’s theorem to infinite
dimensional Banach spaces. This result is originally due to Lomonosov [5]. We
present a proof of it in a special case that was obtained independently by
Scott Brown [6] and that once again is based on the Schauder fixed point
theorem.
In Section 5 we address the existence of invariant subspaces for operators on
Krein space of indefinite product, and we present a result of Albeverio,
Makarov, and Motovilov [7] whose proof uses the Banach fixed point theorem.
The rest of this section contains some notation, a precise statement of the
invariant subspace problem, and a few historical remarks.
Let $E$ be an infinite dimensional, complex Banach space and let
${\mathcal{B}}(E)$ denote the algebra of all bounded linear operators on $E.$
A subspace of $E$ is by definition a closed linear manifold in $E.$
A subspace $M\subseteq E$ is said to be invariant under an operator
$T\in{\mathcal{B}}(E)$ provided that $TM\subseteq M,$ and a subspace
$M\subseteq E$ is said to be invariant under a subalgebra
${\mathcal{R}}\subseteq{\mathcal{B}}(H)$ provided that $M$ is invariant under
every $R\in{\mathcal{R}}.$ A subalgebra
${\mathcal{R}}\subseteq{\mathcal{B}}(H)$ is said to be transitive provided
that the only subspaces invariant under ${\mathcal{R}}$ are the trivial ones,
$M=\\{0\\}$ and $M=E.$ This is equivalent to say that the subspace
$\\{Rx:R\in{\mathcal{R}}\\}$ is dense in $E$ for each $x\in
E\backslash\\{0\\}.$
The commutant of a set of operators ${\mathcal{S}}\subseteq{\mathcal{B}}(E)$
is the subalgebra ${\mathcal{S}}^{\prime}$ of all operators
$R\in{\mathcal{B}}(E)$ such that $SR=RS$ for all $S\in{\mathcal{S}}.$ A
subspace $M\subseteq E$ is said to be hyperinvariant under an operator
$T\in{\mathcal{B}}(E)$ provided that $M$ is invariant under
$\\{T\\}^{\prime}.$
The invariant subspace problem is the question of whether every operator in
${\mathcal{B}}(E)$ has a non trivial invariant subspace. This is one of the
most important open problems in operator theory.
The origin of this question goes back to 1935, when von Neumann proved the
unpublished result that any compact operator on a Hilbert space has a non
trivial invariant subspace. Aronszajn and Smith [8] extended this result in
1954 to general Banach spaces. Bernstein and Robinson [9] used non standard
analysis to prove in 1966 that every polynomially compact operator on a
Hilbert space has a non trivial invariant subspace. Halmos [10] obtained a
proof of the same result using classical methods.
Lomonosov [2] proved in 1973 that any non scalar operator on a Banach space
that commutes with a non zero compact operator has a non trivial
hyperinvariant subspace. The result of Lomonosov came into the scene like a
lightning bolt in a clear sky, generalizing all the previously known results,
and introducing the use of the Schauder fixed point theorem as a new technique
to produce invariant subspaces.
Enflo [11] constructed in 1976 the first example of an operator on a Banach
space without non trivial invariant subspaces. The example circulated in
preprint form and it did not appear published until 1987, when it was
recognized as correct work [12]. In the meantime, Beauzamy [13] simplified the
technique, and further examples were given by Read [14, 15].
Very recently, Argyros and Haydon [16] constructed an example of an infinite
dimensional, separable Banach space such that every continuous operator is the
sum of a compact operator and a scalar operator, so that every operator on it
has a non trivial invariant subspace.
However, after so many decades, the question about the existence of invariant
subspaces for operators on Hilbert space is still an open problem.
## 2\. Invariant subspaces for algebras containing compact operators
We start with a fixed point theorem that is the key for the main result in
this section. The use of this result is one of the main ideas in the technique
of Lomonosov. We shall denote by $\overline{{\rm conv}}(S)$ the closed convex
hull of a subset $S\subseteq E.$
###### Proposition 2.1.
[17, Proposition 1] Let $E$ be a Banach space, let $C\subseteq E$ be a closed
convex set, and let $\Phi:C\to E$ be a continuous mapping such that $\Phi(C)$
is a relatively compact subset of $C.$ Then there is a point $x_{0}\in C$ such
that $\Phi(x_{0})=x_{0}.$
###### Proof.
Let $Q$ denote the closure of $\Phi(C).$ It follows from a theorem of Mazur
that $\overline{{\rm conv}}(Q)$ is a compact, convex subset of $E,$ and since
$C$ is closed and convex, we have $\overline{{\rm conv}}(Q)\subseteq C.$ Since
$\Phi(C)\subseteq Q,$ we have $\Phi(\overline{{\rm conv}}(Q))\subseteq
Q\subseteq\overline{{\rm conv}}(Q),$ and now the result follows from the
Schauder fixed point theorem. ∎
###### Theorem 2.2.
[17, Theorem 2] Let ${\mathcal{R}}\subseteq{\mathcal{B}}(E)$ be a transitive
algebra and let $K\in{\mathcal{B}}(E)$ be a non zero compact operator. Then
there is a operator $R\in{\mathcal{R}}$ and there is a vector $x_{1}\in E$
such that $RKx_{1}=x_{1}.$
###### Proof.
We may assume without loss of generality that $\|K\|=1.$ Choose an $x_{0}\in
E$ such that $\|Kx_{0}\|>1,$ so that $\|x_{0}\|>1.$ Consider the closed ball
$B=\\{x\in E:\|x-x_{0}\|\leq 1\\}.$ Then, for each $R\in{\mathcal{R}},$
consider the open set $G_{R}=\\{y\in E:\|Ry-x_{0}\|<1\\}.$ Since
${\mathcal{R}}$ is a transitive algebra, we have
$\bigcup_{R\in{\mathcal{R}}}G_{R}=E\backslash\\{0\\}.$
Since $K$ is a compact operator, $\overline{KB}$ is a compact subset of $E,$
and since $\|K\|=1$ and $\|Kx_{0}\|>1,$ we have $0\notin\overline{KB}.$ Thus,
the family $\\{G_{R}:R\in{\mathcal{R}}\\}$ is an open cover of
$\overline{KB}.$ Hence, there exist finitely many operators
$R_{1},\ldots,R_{n}\in{\mathcal{R}}$ such that
$\overline{KB}\subseteq\bigcup_{i=1}^{n}G_{R_{i}}.$
Next, for each $y\in\overline{KB}$ and $i=1,\ldots,n$ we define
$\alpha_{i}(y)=\max\\{0,1-\|R_{i}y-x_{0}\|\\}.$ Then $0\leq\alpha_{i}(y)\leq
1,$ and for each $y\in\overline{KB}$ there is an $i=1,\ldots,n$ such that
$y\in G_{R_{i}},$ so that $\alpha_{i}(y)>0.$ Thus
$\displaystyle{\sum_{i=1}^{n}\alpha_{i}(y)>0}$ for each $y\in\overline{KB},$
and we may define
$\beta_{i}(y)=\frac{\alpha_{i}(y)}{\displaystyle{\sum_{j=1}^{n}\alpha_{j}(y)}}$
for $i=1,\ldots,n$ and $y\in\overline{KB}.$ Now, each $\beta_{i}$ is a
continuous function from $\overline{KB}$ into ${\mathbb{R}}.$ Hence, we may
define a continuous mapping $\Phi:B\to E$ by the expression
$\Phi(x)=\sum_{i=1}^{n}\beta_{i}(Kx)R_{i}Kx.$
We claim that $\Phi(B)\subseteq B.$ Indeed, for each $x\in B$ we have
$\displaystyle{\sum_{i=1}^{n}\beta_{i}(Kx)=1,}$ so that
$\|\Phi(x)-x_{0}\|=\left\|\sum_{i=1}^{n}\beta_{i}(Kx)(R_{i}Kx-
x_{0})\right\|\leq\sum_{i=1}^{n}\beta_{i}(Kx)\|R_{i}Kx-x_{0}\|.$
If $\|R_{i}Kx-x_{0}\|>1$ then $\alpha_{i}(Kx)=0$ and therefore
$\beta_{i}(Kx)=0.$ Hence
$\|\Phi(x)-x_{0}\|\leq\sum_{i=1}^{n}\beta_{i}(Kx)=1,$
and this completes the proof of our claim. Finally, each operator $R_{i}K$ is
compact, so that each $R_{i}KB$ is relatively compact, and it follows from an
earlier mentioned theorem of Mazur that $\displaystyle{Q=\overline{{\rm
conv}}\bigcup_{i=1}^{n}R_{i}KB}$ is compact. Since $\Phi(B)\subseteq Q,$ the
set $\Phi(B)$ is a relatively compact subset of $B.$ Now we apply Proposition
2.1 to find a vector $x_{1}\in B$ such that $\Phi(x_{1})=x_{1}.$ Since
$0\notin B,$ we have $x_{1}\neq 0.$ Then we consider the operator defined by
$Rx=\sum_{i=1}^{n}\beta_{i}(Kx_{1})R_{i}x,$
and we conclude that $R\in{\mathcal{R}}$ and $RKx_{1}=x_{1},$ as we wanted. ∎
###### Corollary 2.3.
[2], [17, Theorem 3] Every non scalar operator that commutes with a non zero
compact operator has a non trivial, hyperinvariant subspace.
###### Proof.
Let $T\in{\mathcal{B}}(E)$ be a non scalar operator and suppose that $T$
commutes with a nonzero compact operator $K.$ We must show that the commutant
$\\{T\\}^{\prime}$ is non transitive. Suppose, on the contrary, that
$\\{T\\}^{\prime}$ is transitive. We can apply Theorem 2.2 to find an operator
$R\in\\{T\\}^{\prime}$ such that $\lambda=1$ is an eigenvalue of the compact
operator $RK$ with associated finite dimensional eigenspace $F=\ker(RK-I).$
Since $T$ commutes with $RK,$ we observe that $T$ maps $F$ into itself and
therefore $T$ must have an eigenvalue. Since $T$ is non scalar, the
corresponding eigenspace $M$ cannot be the whole $E$, and it is invariant
under $\\{T\\}^{\prime}.$ The contradiction has arrived. ∎
## 3\. Invariant subspaces for localizing algebras
In this section we use the following fixed point theorem of Ky Fan [18].
Recall that if $\Omega$ is a topological space and
$\Phi:\Omega\to{\mathcal{P}}(\Omega)$ is a point to set map from $\Omega$ to
the power set of $\Omega,$ then $\Phi$ is said to be upper semicontinuous if
for every $x_{0}\in\Omega$ and every open set $U\subseteq\Omega$ such that
$\Phi(x_{0})\subseteq U$ there is a neighborhood $V$ of $x_{0}$ such that
$\Phi(x)\subseteq U$ for every $x\in V.$ In terms of convergence of nets, this
definition is equivalent to say that for every $x\in\Omega,$ for every net
$(x_{\alpha})$ with $x_{\alpha}\to x,$ and for every
$y_{\alpha}\in\Phi(x_{\alpha})$ such that the net $(y_{\alpha})$ converges to
some $y\in\Omega,$ we have $y\in\Phi(x).$
###### Theorem 3.1 (Ky Fan fixed point theorem [18]).
Let $C$ be a compact convex subset of a locally convex space and let
$\Phi\colon C\to{\mathcal{P}}(C)$ be an upper semicontinuous mapping such that
$\Phi(x)$ is a non empty, closed convex set for every $x\in C.$ Then there is
an $x_{0}\in C$ such that $x_{0}\in\Phi(x_{0}).$
A subalgebra ${\mathcal{R}}\subseteq{\mathcal{B}}(H)$ is said to be strongly
compact if its unit ball is precompact in the strong operator topology. An
important example of a strongly compact algebra is the commutant of a compact
operator with dense range. We shall denote by ${\rm ball}(\mathcal{R})$ the
unit ball of $\mathcal{R}.$
This notion was introduced by Lomonosov [19] as a means to prove the existence
of invariant subspaces for essentially normal operators on Hilbert spaces.
Recall that an operator $T$ on a Hilbert space is said to be essentially
normal if $T^{\ast}T-TT^{\ast}$ is a compact operator. Lomonosov showed that
if an essentially normal operator $T$ has the property that both its commutant
$\\{T\\}^{\prime}$ and the commutant of its adjoint $\\{T^{\ast}\\}^{\prime}$
fail to be strongly compact, then $T$ has a nontrivial invariant subspace.
Thus, in order to solve the invariant subspace problem for essentially normal
operators, it suffices to consider only operators with a strongly compact
commutant.
Lomonosov, Radjavi and Troitsky [3] obtained a result about the existence of
invariant subspaces for an operator with a strongly compact commutant under
the additional assumption that the commutant of the adjoint is a localizing
algebra.
A subalgebra ${\mathcal{R}}\subseteq{\mathcal{B}}(E)$ is said to be localizing
provided that there is a closed ball $B\subseteq E$ such that $0\notin B$ and
such that for every sequence $(x_{n})$ in $B$ there is a subsequence
$(x_{n_{j}})$ and a sequence of operators $(R_{j})$ in ${\mathcal{R}}$ such
that $\|R_{j}\|\leq 1$ and $(R_{j}x_{n_{j}})$ converges in norm to some non
zero vector. An important example of a localizing algebra is any algebra
containing a non zero compact operator.
###### Proposition 3.2.
[3, Proof of Theorem 2.3] Let ${\mathcal{R}}\subseteq{\mathcal{B}}(E)$ be a
transitive localizing algebra, let $B\subseteq E$ be a closed ball as above,
and let $T\in{\mathcal{R}}^{\prime}$ be a nonzero operator. Then there exists
an $r>0$ such that for every $x\in B$ we have $r\,{\rm
ball}({\mathcal{R}})(Tx)\cap B\neq\emptyset.$
###### Proof.
First, $T$ is one to one, because ${\mathcal{R}}$ is transitive and $\ker T$
is invariant under ${\mathcal{R}}.$ If this is not so, then for every $n\geq
1$ there is a vector $x_{n}\in B$ such that $\|R\|\geq n$, whenever
$R\in{\mathcal{R}}$ and $RTx_{n}\in B$. Since ${\mathcal{R}}$ is localizing,
there is a subsequence $(x_{n_{j}})$ and a sequence $(R_{j})$ in
${\mathcal{R}}$ such that $\|R_{j}\|\leq 1$ and $(R_{j}x_{n_{j}})$ converges
in norm to some nonzero vector $x\in X$. We have $TR_{j}=R_{j}T$ for all
$j\geq 1$, so that $(R_{j}Tx_{n_{j}})$ converges to $Tx$ in norm. Now $Tx\neq
0$ because $T$ is injective and $x\neq 0$. Since ${\mathcal{R}}$ is
transitive, there is an operator $R\in{\mathcal{R}}$ such that $RTx\in{\rm
int}\,B$. It follows that there is a $j_{0}\geq 1$ such that
$RR_{j}Tx_{n_{j}}\in{\rm int}\,B$ for every $j\geq j_{0}$. Since
$RR_{j}\in{\mathcal{R}}$, the choice of the sequence $(x_{n})$ implies that
$\|RR_{j}\|\geq n_{j}$ for every $j\geq j_{0}$, and this is a contradiction
because $\|RR_{j}\|\leq\|R\|$ for every $j\geq 1$. ∎
If $E$ is a Banach space then $E^{\ast}$ denotes its dual space. If
${\mathcal{R}}\subseteq{\mathcal{B}}(E)$ is a subalgebra, then
${\mathcal{R}}^{\ast}$ denotes the subalgebra of ${\mathcal{B}}(E^{\ast})$ of
the adjoints of the elements of ${\mathcal{R}},$ that is
${\mathcal{R}}^{\ast}=\\{R^{\ast}:R\in{\mathcal{R}}\\}.$
###### Theorem 3.3.
[3, Theorem 2.3] Let $E$ be a complex Banach space, let
${\mathcal{R}}\subseteq{\mathcal{B}}(E)$ be a strongly compact subalgebra such
that ${\mathcal{R}}^{\ast}$ is a transitive localizing algebra and it is
closed in the weak-$\ast$ operator topology. If $T\in{\mathcal{R}}^{\prime}$
is a non zero operator then there is an operator $R\in{\mathcal{R}}$ and there
is a non zero vector $x^{\ast}\in E^{\ast}$ such that
$R^{\ast}T^{\ast}x^{\ast}=x^{\ast}.$ Moreover, the operator $T^{\ast}$ has a
non trivial invariant subspace.
###### Proof.
We shall apply Proposition 3.2 to the algebra ${\mathcal{R}}^{\ast}.$ Let
$B^{\ast}\subseteq E^{\ast}$ be a closed ball as in the definition of a
localizing algebra, let $r>0$ be a positive number as in Proposition 3.2, and
define a multivalued map $\Phi:B^{\ast}\to{\mathcal{P}}(B^{\ast})$ by the
expression
$\Phi(x^{\ast})=r\,{\rm ball}({\mathcal{R}}^{\ast})(T^{\ast}x^{\ast})\cap
B^{\ast}.$
Then, $\Phi(x^{\ast})$ is a non empty, convex subset of $B^{\ast}.$ Also,
$\Phi(x^{\ast})$ is weak-$\ast$ closed because ${\rm
ball}({\mathcal{R}}^{\ast})(T^{\ast}x^{\ast})$ is weak-$\ast$ compact as the
image of ${\rm ball}({\mathcal{R}}^{\ast})$ under the map $R^{\ast}\to
R^{\ast}T^{\ast}x^{\ast},$ which is continuous from ${\mathcal{B}}(E^{\ast})$
with the weak-$\ast$ operator topology into $E^{\ast}$ with the weak-$\ast$
topology, and ${\rm ball}({\mathcal{R}}^{\ast})$ is compact in the weak-$\ast$
operator topology.
We claim that $\Phi$ is upper semicontinuous for the weak-$\ast$ topology.
Indeed, let $x^{\ast},y^{\ast}\in B^{\ast},$ and let $(x^{\ast}_{\alpha})$ and
$(y^{\ast}_{\alpha})$ be two nets in $B^{\ast}$ with $x^{\ast}_{\alpha}\to
x^{\ast},$ $y^{\ast}_{\alpha}\to y^{\ast}$ in the weak-$\ast$ topology, and
such that $y^{\ast}_{\alpha}\in\Phi(x^{\ast}_{\alpha}).$ We must show that
$y^{\ast}\in\Phi(x^{\ast}).$ Since
$y^{\ast}_{\alpha}\in\Phi(x^{\ast}_{\alpha}),$ there is an
$R^{\ast}_{\alpha}\in\,{\rm ball}({\mathcal{R}}^{\ast})$ such that
$y^{\ast}_{\alpha}=rR^{\ast}_{\alpha}T^{\ast}x^{\ast}_{\alpha}.$ Since ${\rm
ball}({\mathcal{R}})$ is precompact in the strong operator topology, there
exists a subnet $(R_{\alpha_{\beta}})$ that converges in the strong operator
topology to some $R\in{\mathcal{B}}(E).$ Thus, $R^{\ast}_{\alpha_{\beta}}\to
R^{\ast}$ in the weak-$\ast$ operator topology. Notice that ${\rm
ball}({\mathcal{R}}^{\ast})$ is compact in this topology because ${\rm
ball}({\mathcal{B}}(E^{\ast}))$ is compact in this topology and
${\mathcal{R}}^{\ast}$ is closed in this topology. It follows that
$R^{\ast}\in{\rm ball}({\mathcal{R}}^{\ast}).$ Let $x\in E$ and notice that
$\|TR_{\alpha_{\beta}}x-TRx\|\to 0.$ Then
$\displaystyle\langle x,y^{\ast}_{\alpha_{\beta}}\rangle$ $\displaystyle=$
$\displaystyle\langle
x,rR^{\ast}_{\alpha_{\beta}}T^{\ast}x^{\ast}_{\alpha_{\beta}}\rangle$
$\displaystyle=$ $\displaystyle r\langle
TR_{\alpha_{\beta}}x,x^{\ast}_{\alpha_{\beta}}\rangle$ $\displaystyle=$
$\displaystyle r\langle
TR_{\alpha_{\beta}}x-TRx,x^{\ast}_{\alpha_{\beta}}\rangle+r\langle
TRx,x^{\ast}_{\alpha_{\beta}}\rangle.$
We have $\langle TR_{\alpha_{\beta}}x-TRx,x^{\ast}_{\alpha_{\beta}}\rangle\to
0$ and $\langle TRx,x^{\ast}_{\alpha_{\beta}}\rangle\to\langle
TRx,x^{\ast}\rangle=\langle x,R^{\ast}T^{\ast}x^{\ast}\rangle,$ so that
$\langle x,y^{\ast}_{\alpha_{\beta}}\rangle\to\langle
x,rR^{\ast}T^{\ast}x^{\ast}\rangle.$
Since $x\in E$ is arbitrary, $y^{\ast}_{\alpha_{\beta}}\to
rR^{\ast}T^{\ast}x^{\ast}$ in the weak-$\ast$ topology, and it follows that
$y^{\ast}=rR^{\ast}T^{\ast}x^{\ast}.$ This shows that
$y^{\ast}\in\Phi(x^{\ast}),$ and the proof of our claim is complete.
Since the map $\Phi$ is upper semicontinuous and $B^{\ast}$ is compact in the
weak-$\ast$ topology, it follows from the Ky Fan fixed point theorem that
there is a vector $x^{\ast}\in B^{\ast}$ such that
$x^{\ast}\in\Phi(x^{\ast}),$ that is, there is an operator $R\in{\rm
ball}({\mathcal{R}})$ such that $x^{\ast}=rR^{\ast}T^{\ast}x^{\ast}.$ ∎
###### Corollary 3.4.
[3, Corollary 2.4] Let $T\in{\mathcal{B}}(E)$ be an operator such that
$\\{T\\}^{\prime}$ is a strongly compact algebra and $\\{T^{\ast}\\}^{\prime}$
is a localizing algebra. Then $T^{\ast}$ has a nontrivial invariant subspace.
###### Proof.
If $T^{\ast}$ has a hyperinvariant subspace then there is nothing to prove,
and otherwise $\\{T^{\ast}\\}^{\prime}$ is a transitive algebra, so that
Theorem 3.3 applies. ∎
Notice that the assumptions of Corollary 3.4 are met whenever $T$ is a compact
operator with dense range.
## 4\. An infinite dimensional version of Burnside’s theorem
Burnside’s classical theorem is the assertion that for a finite dimensional
linear space $F,$ the only transitive subalgebra of ${\mathcal{B}}(F)$ is the
whole algebra ${\mathcal{B}}(F).$ Lomonosov [5] obtained a generalization of
Burnside’s theorem to infinite dimensional Banach spaces. Scott Brown [6]
proved the same result independently for the special case of a Hilbert space
and a commutative algebra. Lindström and Schlüchtermann [20] provided a
relatively short proof of the Lomonosov result in full generality. In this
section we present a proof of the Scott Brown result that is based on Schauder
fixed point theorem.
Let $H$ be a complex, infinite dimensional, separable Hilbert space. Let
$T\in{\mathcal{B}}(H)$ and let $\|T\|_{e}$ denote the essential norm of $T,$
that is, the distance from $T$ to the space of compact operators.
###### Theorem 4.1.
[6, Theorem 1.1] Let ${\mathcal{R}}$ be a commutative subalgebra of
${\mathcal{B}}(H)$. Then there exist nonzero vectors $x,y\in H$ such that for
any $R\in{\mathcal{R}}$ we have $|\langle Rx,y\rangle|\leq\|R\|_{e}.$
###### Proof.
Consider the set ${\mathcal{E}}=\\{R\in{\mathcal{R}}:\|R\|_{e}\leq 1/16\\}.$
We claim that there is some $x\in H\backslash\\{0\\}$ such that the set
${\mathcal{E}}x$ is not dense in $H.$ The result then follows easily because
in that case there is some $y\in H\backslash\\{0\\}$ such that $|\langle
Rx,y\rangle|\leq 1$ for all $R\in{\mathcal{E}}.$ Now, for the proof of our
claim, we proceed by contradiction. Suppose that the set ${\mathcal{E}}x$ is
dense in $H$ for every $x\in H\backslash\\{0\\}.$ Choose $x_{0}\in H$ with
$\|x_{0}\|=2$ and consider the closed ball $B=\\{x\in H:\|x-x_{0}\|\leq 1\\}.$
Then, for every vector $x\in B$ there is an operator $R_{x}\in{\mathcal{E}}$
such that $\|R_{x}x-x_{0}\|<1/2.$ Next, there is a bounded operator $T_{x}$
and a compact operator $K_{x}$ such that $R_{x}=T_{x}+K_{x}$ and
$\|T_{x}\|\leq 1/8.$ Since $K_{x}$ is a compact operator, it is weak to norm
continuous on bounded sets, so that there exists an open neighborhood of $x$
in the weak topology, say $V_{x}\subseteq H,$ such that
$\|K_{x}y-K_{x}x\|<1/4$ for all $y\in V_{x}\ \cap B.$ Then consider the set
$U_{x}=V_{x}\cap B$ and notice that $U_{x}$ is an open neighborhood of $x$ in
the weak topology relative to $B.$ Moreover, for $y\in U_{x}$ we have
$\|R_{x}y-R_{x}x\|\leq\|T_{x}y-T_{x}x\|+\|K_{x}y-K_{x}x\|<2\cdot\frac{1}{8}+\frac{1}{4}=\frac{1}{2},$
and therefore $\|R_{x}y-x_{0}\|<1.$ Hence, $R_{x}U_{x}\subseteq B.$ Since $B$
is compact in the weak topology, there exist finitely many vectors
$x_{1},\ldots x_{n}\in B$ such that
$B\subseteq\bigcup_{j=1}^{n}U_{x_{j}}.$
Choose some weakly continuous functions $f_{1},\ldots,f_{n}$ on $B$ such that
${\rm supp}(f_{j})\subseteq U_{j},$ $0\leq f_{j}(x)\leq 1,$ and
$\sum_{j=1}^{n}f_{j}(x)=1\qquad\text{for all }x\in B.$
Define a weakly continuous mapping $\Phi:B\to B$ by the expression
$\Phi(x)=\sum_{j=1}^{n}f_{j}(x)R_{x_{j}}x\qquad\text{for all }x\in B,$
and apply the Schauder fixed point theorem to find a vector $y_{0}\in B$ such
that $\Phi(y_{0})=y_{0}.$ Finally, consider the operator $R\in{\mathcal{R}}$
defined by the expression
$R=\sum_{j=1}^{n}f_{j}(y_{0})R_{x_{j}}.$
Hence, $Ry_{0}=y_{0}.$ Notice that $R\neq I$ because $\|R\|_{e}\leq 1/8.$
Then, the eigenspace $M=\\{x\in H:Rx=x\\}$ is a closed nontrivial invariant
subspace for the algebra ${\mathcal{R}}.$ Thus, any vector $x\in M$ has the
property that the set ${\mathcal{E}}x$ is not dense in $H.$ The contradiction
has arrived. ∎
## 5\. Invariant subspaces for operators on Krein space
Let $H_{1},H_{2}$ be two Hilbert spaces and consider the orthogonal direct sum
$H=H_{1}\oplus H_{2}.$ Let $P_{1},P_{2}$ denote the orthogonal projections
from $H$ onto $H_{1},H_{2},$ respectively. Consider the operator
$J:=P_{1}-P_{2}.$ The Krein space is the space $H$ provided with the
indefinite product
$[x,y]:=\langle Jx,y\rangle,\qquad x,y\in H.$
Notice that $J$ is a selfadjoint involution, that is, $J^{\ast}=J$ and
$J^{2}=I.$ The operator $J$ is sometimes called the fundamental symmetry of
the Krein space.
A vector $x\in H$ is said to be non negative provided that $[x,x]\geq 0,$ and
a subspace $M\subseteq H$ is said to be non negative provided that $[x,x]\geq
0$ for all $x\in M.$
Every operator $T\in{\mathcal{B}}(H)$ has a matrix representation
$T=\left[\begin{array}[]{cc}T_{11}&T_{12}\\\ T_{21}&T_{22}\end{array}\right]$
with respect to the decomposition $H=H_{1}\oplus H_{2}.$
There is a natural, one to one and onto correspondence between the maximal non
negative invariant subspaces $M$ of an operator $T\in{\mathcal{B}}(H)$ and the
contractive solutions $X\in{\mathcal{B}}(H_{1},H_{2})$ of the so called
operator Riccati equation
$XT_{12}X+XT_{11}-T_{22}X-T_{21}=0.$
The correspondence $X\leftrightarrow M$ is given by $M=\\{x_{1}\oplus
Xx_{1}:x_{1}\in H_{1}\\},$ where $\|X\|\leq 1.$ The operator $T$ is usually
called the hamiltonian operator of the operator Ricatti equation.
An operator $T\in{\mathcal{B}}(H)$ is said to be $J$-selfadjoint provided that
$[Tx,y]=[x,Ty]$ for every $x,y\in H.$ This is equivalent to say that
$JT=T^{\ast}J,$ or in other words, $T_{11}^{\ast}=T_{11},$
$T_{22}^{\ast}=T_{22},$ and $T_{12}^{\ast}=-T_{21}.$
A classical theorem of Krein is the assertion that, if the hamiltonian
operator $T$ is $J$-selfadjoint and the corner operator $T_{12}$ is compact,
then there exists a maximal non negative invariant subspace for $T.$
Albeverio, Makarov, and Motovilov [7] addressed the question of the existence
and uniqueness of contractive solutions to the operator Riccati equation under
the condition that the diagonal entries in the hamiltonian operator have
disjoint spectra, that is, $\sigma(T_{11})\cap\sigma(T_{22})=\emptyset.$ They
proved the following
###### Theorem 5.1.
[7, Theorem 3.6 and Lemma 3.11] There is some universal constant $c>0$ such
that whenever the corner operator $T_{12}$ satisfies the condition
$\|T_{12}\|<c\cdot{\rm dist}[\sigma(T_{11}),\sigma(T_{22})],$
there is a unique solution $X$ to the operator Riccati equation with
$\|X\|\leq 1.$
An earlier result in this direction was given by Motovilov [21, Corollary 1]
with the stronger assumption that the corner operator $T_{12}$ is Hilbert-
Schmidt. Adamjan, Langer, and Tretter [22] extended the technique to the case
that the hamiltonian operator is not $J$-selfadjoint. Kostrykin, Makarov, and
Motovilov [23] adopted the assumption that $\sigma(T_{11})$ lies in a gap of
$\sigma(T_{22})$ and they showed that the best constant, in that context, is
$c=\sqrt{2}.$
We present a proof of Theorem 5.2 that is based on Banach fixed point theorem.
This method can be found in the paper of Albeverio, Motovilov and Shkalikov
[24, Theorem 4.1]. A basic tool is the bounded linear operator $R$ defined for
$X\in{\mathcal{B}}(H_{1},H_{2})$ by the expression
$R(X):=T_{22}X-XT_{11}.$
It follows from the Rosenblum theorem that the map $R$ is invertible. The main
result is the following
###### Theorem 5.2.
[24, Theorem 4.1] If the operators $T_{11},T_{22}$ have disjoint spectra and
the corner operator $T_{12}$ satisfies the estimate
$\|T_{12}\|<\frac{1}{2\|R^{-1}\|}$
then there is a unique solution $X$ to the operator Riccati equation with
$\|X\|\leq 1.$
The following upper bound on the norm of the inverse $R^{-1}$ can be found in
the work of Albeverio, Makarov, and Motovilov [7, Theorem 2.7]. See also the
paper by Bhatia and Rosenthal [4, p.15] for this interesting result and other
related issues.
###### Theorem 5.3.
[7, Theorem 2.7] If the operators $T_{11},T_{22}$ have disjoint spectra then
$\|R^{-1}\|\leq\frac{\pi}{2}\cdot\frac{1}{{\rm
dist}[\sigma(T_{11}),\sigma(T_{22})]}$
Notice that Theorem 5.1 becomes a corollary of Theorem 5.2 and Theorem 5.3
with the constant $c=1/\pi.$
###### Proof of Theorem 5.2..
Consider the quadratic map $Q$ defined for $X\in{\mathcal{B}}(H_{1},H_{2})$ by
the expression
$Q(X):=XT_{12}X-T_{21}.$
It is clear that the operator Riccati equation can be expressed as
$Q(X)-R(X)=0,$
or equivalently, $X=R^{-1}(Q(X)).$ Thus, the solutions of the operator Riccati
equation are the fixed points of the map $S:=R^{-1}\circ Q.$ Now, let us check
that the map $S$ takes the unit ball of ${\mathcal{B}}(H_{1},H_{2})$ into
itself. Indeed, if $\|X\|\leq 1$ then
$\displaystyle\|S(X)\|$ $\displaystyle=$
$\displaystyle\|R^{-1}(Q(X))\|\leq\|R^{-1}\|\cdot\|Q(X)\|$ $\displaystyle\leq$
$\displaystyle\|R^{-1}\|\cdot(\|T_{12}\|\cdot\|X\|^{2}+\|T_{21}\|)$
$\displaystyle\leq$
$\displaystyle\|R^{-1}\|\cdot(\|T_{12}\|+\|T_{21}\|)=2\|R^{-1}\|\cdot\|T_{12}\|<1.$
Also, the map $S$ is contractive, for if $\|X\|,\|Y\|\leq 1$ then
$\displaystyle\|Q(X)-Q(Y)\|$ $\displaystyle=$
$\displaystyle\|XT_{12}X-YT_{12}Y\|$ $\displaystyle\leq$
$\displaystyle\|XT_{12}X-XT_{12}Y\|+\|XT_{12}Y-YT_{12}Y\|$ $\displaystyle\leq$
$\displaystyle(\|X\|+\|Y\|)\cdot\|T_{12}\|\cdot\|X-Y\|\leq
2\|T_{12}\|\cdot\|X-Y\|,$
and from this inequality it follows that
$\displaystyle\|S(X)-S(Y)\|$ $\displaystyle=$
$\displaystyle\|R^{-1}(Q(X)-Q(Y))\|$ $\displaystyle\leq$
$\displaystyle\|R^{-1}\|\cdot\|Q(X)-Q(Y)\|\leq
2\|R^{-1}\|\cdot\|T_{12}\|\cdot\|X-Y\|,$
so that the map $S$ satisfies a Lipschitz condition with a Lipschitz constant
$2\|R^{-1}\|\cdot\|T_{12}\|<1.$ The result now follows at once as a
consequence of the Banach fixed point theorem. ∎
## Competing interests
The authors declare that they have no competing interests.
## Author’s contributions
Both authors contributed equally and significantly in writing this article.
Both authors read and approved the final manuscript.
## Acknowledgements
This research was partially supported by Junta de Andalucía under projects
FQM-127 and FQM-3737, and by Ministerio de Educación, Cultura y Deporte under
projects MTM2012-34847C02-01 and MTM2009-08934.
## References
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|
arxiv-papers
| 2012-10-19T22:38:22 |
2024-09-04T02:49:36.862325
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rafa Esp\\'inola and Miguel Lacruz",
"submitter": "Miguel Lacruz",
"url": "https://arxiv.org/abs/1210.5557"
}
|
1210.5652
|
# Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function,
Fractal Strings, and a Finite Reflection Formula
Stephen Crowley
###### Abstract.
The harmonic sawtooth map $w\left(x\right)$ of the unit interval onto itself
is defined. It is shown that its fixed points
$\left\\{x:w\left(x\right)=x\right\\}$ are enumerated by the $n$-th
derivatives of a Meijer-G function and Lerch transcendent, serving as
exponential and ordinary generating functions respectively, and involving the
golden ratio in their parameters. The appropriately scaled Mellin transform of
$w\left(x\right)$ is an analytic continuation of the Riemann zeta function
$\zeta(s)$ valid $\forall-\operatorname{Re}(s)\not\in\mathbbm{N}$. The series
expansion of the inverse scaling function associated to the Mellin transform
of $w\left(x\right)$ has coefficients enumerating the Large Schröder Numbers
$S_{n}$, defined as the number of perfect matchings in a triangular grid of
$n$ squares and expressible as a hypergeometric function. A finite-sum
approximation to $\zeta\left(s\right)$ denoted by $\zeta_{w}\left(N;s\right)$
is examined and an associated function $\chi\left(N;s\right)$ is found which
solves the reflection formula
$\zeta_{w}\left(N;1-s\right)=\chi\left(N;s\right)\zeta_{w}\left(N;s\right)$.
The function $\chi\left(N;s\right)$ is singular at $s=0$ and the residue at
this point changes sign from negative to positive between the values of
$N=176$ and $N=177$. Some rather elegant graphs of the reflection functions
$\chi\left(N;s\right)$ are also provided. The Mellin and Laplace transforms of
the individual component functions of the infinite sums and their roots are
compared. The Gauss map $h\left(x\right)$ is recalled so that its fixed points
and Mellin transform can be contrasted to those of $w\left(x\right)$. The
geometric counting function
$N_{\mathcal{L}_{w}}(x)=\left\lfloor\frac{\sqrt{2x+1}}{2}-\frac{1}{2}\right\rfloor$
of the fractal string $\mathcal{L}_{w}$ associated to the lengths of the
harmonic sawtooth map components
$\left\\{w_{n}\left(x\right)\right\\}_{n=1}^{\infty}$ happens to coincide with
the counting function for the number of Pythagorean triangles of the form
$\\{(a,b,b+1):(b+1)\leqslant x\\}$. The volume of the inner tubular
neighborhood of the boundary of the map $\partial\mathcal{L}_{w}$ with radius
$\varepsilon$ is shown to have the particuarly simple closed-form
$V_{\mathcal{L}_{w}}(\varepsilon)=\frac{4\varepsilon
v(\varepsilon)^{2}-4\varepsilon v(\varepsilon)+1}{2v(\varepsilon)}$ where
$v\left(\varepsilon\right)=\left\lfloor\frac{\varepsilon+\sqrt{\varepsilon^{2}+\varepsilon}}{2\varepsilon}\right\rfloor$.
Also, the Minkowski content of $\mathcal{L}_{w}$ is shown to be
$\mathcal{M}_{\mathcal{L}_{w}}=2$ and the Minkowski dimension to be
$D_{\mathcal{L}_{w}}=\frac{1}{2}$ and thus not invertible. The geometric zeta
function, which is the Mellin transform of the geometric counting function
$N_{\mathcal{L}_{w}}(x)$, is calculated and shown to have a rather unusual
closed form involving a finite sum of Riemann zeta functions and binomioal
coefficients. Some definitions from the theory of fractal strings and
membranes are also recalled.
Email: stephen.crowley@mavs.uta.edu
###### Contents
1. 1 Unit Interval Mappings
1. 1.1 The $n$-th Harmonic Sawtooth Function $w_{n}(x)$
1. 1.1.1 Infinite Sum Decomposition
2. 1.1.2 The Fixed Points of $w(x)$ as an Iterated Function System
2. 1.2 Integrals Transforms of w(x)
1. 1.2.1 $\operatorname{Dirichlet}\operatorname{Polynomial}\operatorname{Series}$ and the Mellin Transform of $w(x)$
2. 1.2.2 The Reflection Formula for $\zeta_{w}\left(N;s\right)$
3. 1.2.3 The Laplace Transforms $L[w_{n}(x);x\rightarrow s]$
4. 1.2.4 The Roots $\rho^{M}_{w_{n}}\left(m\right)$ of $M[w_{n}(x);x\rightarrow s]$
5. 1.2.5 Quotients and Differences of $\rho^{M}_{w_{n}}\left(m\right)$
6. 1.2.6 The Laplace Transform $L\left[(s-1)M[w_{n}(x);x\rightarrow s\right];s\rightarrow t]$
3. 1.3 The Gauss Map $h(x)$
1. 1.3.1 $\operatorname{Continued}\operatorname{Fractions}$
2. 1.3.2 The Mellin Transform of $h(x)$
4. 1.4 The Harmonic Sawtooth Map w(x) as an Ordinary Fractal String
1. 1.4.1 Definition and Length
2. 1.4.2 Geometry and Volume of the Inner Tubular Neighborhood
3. 1.4.3 The Geometric Zeta Function $\zeta_{\mathcal{L}_{w}}(s)$
2. 2 Fractal Strings and Dynamical Zeta Functions
1. 2.1 Fractal Strings
1. 2.1.1 The Minkowski Dimension $D_{\mathcal{L}}$ and Content $\mathcal{M}_{\mathcal{L}}$
2. 2.1.2 The Geometric Zeta Function $\zeta_{\mathcal{L}}(s)$
3. 2.1.3 Complex Dimensions, Screens and Windows
4. 2.1.4 Frequencies of Fractal Strings and Spectral Zeta Functions
5. 2.1.5 Generalized Fractal Strings and Dirichlet Integrals
2. 2.2 Fractal Membranes and Spectral Partitions
1. 2.2.1 Complex Dimensions of Dynamical Zeta Functions
2. 2.2.2 Dynamical Zeta Functions of Fractal Membranes
3. 3 Special Functions, Definitions, and Conventions
1. 3.1 Special Functions
1. 3.1.1 The Interval Indicator (Characteristic) Function $\chi(x,I)$
2. 3.1.2 “Harmonic” Intervals
3. 3.1.3 The Laplace Transform $L_{a}^{b}[f(x);x\rightarrow s]$
4. 3.1.4 The Mellin Transform $M_{a}^{b}[f(x);x\rightarrow s]$
5. 3.1.5 The Lambert W Function $W(k,x)$
6. 3.1.6 The Lerch Transcendent $\Phi(z,a,v)$
2. 3.2 Applications of w(x)
1. 3.2.1 Expansion of $\gamma$
3. 3.3 Conventions and Symbols
## 1\. Unit Interval Mappings
### 1.1. The $n$-th Harmonic Sawtooth Function $w_{n}(x)$
#### 1.1.1. Infinite Sum Decomposition
Let the $n$-th component$w_{n}(x)\in[0,1]\forall x\in[0,1]$ of the harmonic
sawtooth function $w(x$) [9, 2.3] be defined as
(1)
$\begin{array}[]{ll}w_{n}(x)&=n(xn+x-1)\chi\left(x,I^{H}_{n}\right)\end{array}$
where
(2)
$\chi\left(x,I^{H}_{n}\right)=\left\\{\begin{array}[]{ll}1&\frac{1}{n+1}<x\leqslant\frac{1}{n}\\\
0&\operatorname{otherwise}\end{array}\right.$
is the characteristic function of the $n$-th harmonic interval(132). By
setting $n=\left\lfloor x^{-1}\right\rfloor$ as in (134) we get the unit
interval mapping
(3) $\begin{array}[]{ll}w(x)&=w_{\left\lfloor x^{-1}\right\rfloor}(x)\\\
&=\left\lfloor x^{-1}\right\rfloor(x\left\lfloor
x^{-1}\right\rfloor+x-1)\chi\left(x,I^{H}_{\left\lfloor
x^{-1}\right\rfloor}\right)\\\ &=\sum_{n=1}^{\infty}w_{n}(x)\\\
&=\sum_{n=1}^{\infty}n(xn+x-1)\chi\left(x,I^{H}_{n}\right)\\\ &=\left\lfloor
x^{-1}\right\rfloor\left(x\left\lfloor x^{-1}\right\rfloor+x-1\right)\\\
&=x\left\lfloor x^{-1}\right\rfloor^{2}+x\left\lfloor
x^{-1}\right\rfloor-\left\lfloor x^{-1}\right\rfloor\end{array}$
Figure 1. The Harmonic Sawtooth map
As can be seen in Figure 1, $w(x)$ is discontinuous at a countably infinite
set of points of Lebesgue measure zero
(4) $\begin{array}[]{ll}\text{$\mathbbm{H}$}&=\left\\{y:\lim_{x\rightarrow
y^{-}}w(x)\neq\lim_{x\rightarrow y^{+}}w(x)\right\\}\\\
&=\left\\{0,\frac{1}{n}:n\in\mathbbm{Z}\right\\}\end{array}$
The left and right limits at the discontinuous points are
(5) $\begin{array}[]{lll}\lim_{x\rightarrow\in\mathbbm{H}^{-}}w(x)&=1&\\\
\lim_{x\rightarrow\in\mathbbm{H}^{+}}w(x)&=0&\end{array}$
#### 1.1.2. The Fixed Points of $w(x)$ as an Iterated Function System
The iterates of the map
(6)
$[w(x),w(w(x)),w(w(w(x))),w(w(w(w(x)))),\ldots]=\left[\overset{}{w^{1}}(x),\overset{}{w^{2}}(x),\overset{}{w^{3}}(x),\overset{}{w^{4}}(x),\ldots\right]$
have the form
(7) $\begin{array}[]{ll}w^{r}(x)&=a_{r}-(x-c)b_{r}\end{array}$
where $\\{a_{r},b_{r}\in\mathbbm{Z}:r\in\mathbbm{N}\\}$ is a pair of integer
sequences and $c\in\mathbbm{R}$ is some constant. The sequence of quotients
$\frac{a_{r}}{b_{r}}$ converges rather quickly to the fixed value
(8)
$\begin{array}[]{l}\lim_{r\rightarrow\infty}\frac{a_{r}}{b_{r}}=x-c\end{array}\forall
x\in[0,1]$
The explicit equation(sometimes called Schröder’s equation) for the fixed
points of $w(x)$, is
(9) $\begin{array}[]{ll}\operatorname{Fix}_{w}^{n}&=\\{x:w_{n}(x)=x\\}\\\
&=\\{x:n(xn+x-1)\chi(x,I^{H}_{n})=x\\}\\\ &=\\{x:n(xn+x-1)=x\\}\\\
&=\lim_{x\rightarrow
0}\frac{\frac{\mathrm{d}}{\mathrm{d}x^{n}}\operatorname{Fix}_{w}(x)}{n!}\\\
&=\lim_{x\rightarrow
0}\frac{x^{1-n}}{n!}G_{4,4}^{1,4}\left(-x|\begin{array}[]{llll}-1&-1&-\phi&\phi-1\\\
0&n-1&2-\phi&-1-\phi\end{array}\right)\\\ &=\frac{n}{n^{2}+n-1}\end{array}$
where $G$ is the Meijer-G function and $\operatorname{Fix}_{w}(x)$ is the
generation function
(10)
$\begin{array}[]{ll}\operatorname{Fix}_{w}(x)&=\begin{array}[]{l}\sum_{n=1}^{\infty}\end{array}\operatorname{Fix}^{n}_{w}x^{n}\\\
&=\sum_{n=1}^{\infty}\frac{nx^{n}}{n^{2}+n-1}\\\
&=\frac{1}{10}\left((5-\sqrt{5})\Phi\left(x,1,1-\phi\right)+(5-\sqrt{5})\Phi\left(x,1,\phi\right)\right)\end{array}$
where $\Phi(z,a,v)$ is the Lerch Transcendent (159), and $\phi$ is the Golden
Ratio, which is the ratio of two numbers having the property that the ratio of
the sum to the larger equals the ratio of the larger to the smaller. [7,
Ch.XX][35, I.7][16, p.50][28] The number $\phi$ can be called the “most
irrational number” because its continued fraction expansion, given by
iterations of the Gauss map (78), converges more slowly than any other number.
The constant $\phi$ satisfies the simple identities
(11) $\phi=\frac{1}{\phi-1}=\frac{\phi}{\phi+1}$ (12) $\phi^{2}-\phi-1=0$
An interesting fact is that the density of a motif in a certain noncommutative
space described in [23, 5.1] must necessarily be an element of the group
$\mathbbm{Z}+\phi\mathbbm{Z}$.
### 1.2. Integrals Transforms of w(x)
#### 1.2.1.
$\operatorname{Dirichlet}\operatorname{Polynomial}\operatorname{Series}$ and
the Mellin Transform of $w(x)$
The Mellin transform of the harmonic saw map $w(x)$, multiplied by
(13) $\tau(s)=s\frac{s+1}{s-1}$
is an analytic continuation of the Riemann zeta function
$\zeta(s)\forall-\operatorname{Re}(s)\not\in\mathbbm{N}$. This form of the
zeta function, denoted by $\zeta_{w}(s)$, is the infinite sum of the Mellin
transformations of the component functions.
(14) $\begin{array}[]{ll}\left.M[w_{n}(x);x\rightarrow
s\right]&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}w_{n}(x)x^{s-1}\mathrm{d}x\\\
&=\int_{n}^{n+1}w_{n}(x^{-1})x^{-s-1}\mathrm{d}x\\\
&=\int_{0}^{1}n(xn+x-1)\chi(x,I^{H}_{n})x^{s-1}\mathrm{d}x\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)x^{s-1}\mathrm{d}x\\\
&=-\frac{e^{\ln(n)(1-s)}-ne^{s\ln\left(\frac{1}{n+1}\right)}-se^{s\ln\left(\frac{1}{n}\right)}}{s^{2}+s}\\\
&=\frac{n^{s}n+s(n+1)^{s}-(n+1)^{s}n}{n^{s}s(n+1)^{s}+n^{s}s^{2}(n+1)^{s}}\\\
&=-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\end{array}$
There is a conjugate pair of inverse branches of $\tau(s)$ found by solving
(15) $\begin{array}[]{ll}\tau_{\pm}^{-1}(t)&=\\{s:\tau(s)=t\\}\\\
&=\left\\{s:s\frac{s+1}{s-1}=t\right\\}\\\
&=\frac{t}{2}-\frac{1}{2}\pm\frac{\sqrt{1-6t+t^{2}}}{2}\end{array}$
where $\tau_{+}^{-1}(t)$ and $\tau_{-}^{-1}(t)$ denote the positive and
negative solutions respectively. The coefficients in the series expansions are
integers enumerating the large Schröder numbers $S_{n}$ which count the number
of perfect matchings in a triangular grid of $n$ squares, named after Ernst
Schröder(1841-1902).[17, A006318][4, p.340][10]
(16)
$\begin{array}[]{ll}S_{n}&={{}_{2}F_{1}}\left(\begin{array}[]{ll}n+1&2-n\\\
2&\end{array}|-1\right)2\end{array}$
where ${}_{p}F_{q}$ is a hypergeometric function. We have
(17) $\begin{array}[]{ll}\lim_{t\rightarrow
0}\frac{\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\text{$\tau_{+}^{-1}(t)$}}{n!}&=\left\\{\begin{array}[]{ll}0&n=0\\\
-1&n=1\\\ -S_{n}&n\geqslant 2\end{array}\right.\\\ \lim_{t\rightarrow
0}\frac{\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\text{$\tau_{-}^{-1}(t)$}}{n!}&=\left\\{\begin{array}[]{ll}-1&n=0\\\
2&n=1\\\ S_{n}&n\geqslant 2\end{array}\right.\end{array}$ (18)
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
n&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\\ \hline\cr
S_{n}&1&1&2&6&22&90&394&1806&8558&41586&206098&1037718&5293446&27297738&142078746\\\
\hline\cr\end{array}$
The residue at the singular point $s=1$ of $\tau(s)$ is
(19)
$\begin{array}[]{ll}\underset{}{\underset{s=1}{\operatorname{Res}}\left(\tau(s)M[w_{n}(x);x\rightarrow
s]\right)}&=\underset{}{\underset{s=1}{\operatorname{Res}}\left(\frac{s^{2}+s}{s-1}\left(-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\right)\right)}\\\
&=\underset{}{\underset{s=1}{\operatorname{Res}}\left(\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}\right)}\\\
&=2\int_{\frac{1}{n+1}}^{\frac{1}{n}}w_{n}(x)\mathrm{d}x\\\
&=2\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)\mathrm{d}x\\\
&=\frac{1}{n^{2}+n}\end{array}$
The infinite sum of the Mellin transforms multiplied by $\tau(s)$ analytically
continues $\zeta(s)\text{$\forall(-\operatorname{Re}(s))\not\in\mathbbm{N}$}$
(20) $\begin{array}[]{lll}\zeta_{w}(s)&=\zeta(s)&\\\
&=\tau(s)M[w(x);x\rightarrow s]&\\\
&=\tau(s)\int_{0}^{1}w(x)x^{s-1}\mathrm{d}x&\\\
&=s\frac{s+1}{s-1}\int_{0}^{1}\left\lfloor
x^{-1}\right\rfloor\left(x\left\lfloor
x^{-1}\right\rfloor+x-1\right)x^{s-1}\mathrm{d}x&\\\
&=s\frac{s+1}{s-1}\sum_{n=1}^{\infty}M[w_{n}(x);x\rightarrow s]&\\\
&=s\frac{s+1}{s-1}\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)x^{s-1}\mathrm{d}x&\text{
}\\\
&=\sum_{n=1}^{\infty}s\frac{s+1}{s-1}\left(-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s\left(s+1\right)}\right)&\\\
&=\sum_{n=1}^{\infty}\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}&\\\
&=\frac{1}{s-1}\sum_{n=1}^{\infty}n(n+1)^{-s}-n^{1-s}+sn^{-s}&\end{array}$
Unlike the Mellin transform of the Gauss map (78), which must be multiplied by
the factor $s$ then subtracted from $\frac{s}{s-1}$ before it equals
$\zeta(s$), the “harmonic sawtooth continuation” $\zeta_{w}(s)$ of the zeta
function $\zeta(s$) has the fortuitous property that it equals $\zeta(s$)
after multiplying $M[w(x);x\rightarrow s]$ by $\tau(s$). This property of
$w(x$) allows us to put $\tau(s$) inside the sum to get an expression, denoted
by $\zeta_{w}(N;s)$, involving the difference of two Dirichlet polynomials,
one of which is scaled by $s$. The substition $\infty\rightarrow N$ is made in
the infinite sum appearing the expression for $\zeta_{w}(s$) to get
(21)
$\begin{array}[]{ll}\zeta_{w}(N;s)&=\tau(s)\sum_{n=1}^{N}M[w_{n}(x);x\rightarrow
s]\\\ &=\frac{1}{s-1}\sum_{n=1}^{N}n(n+1)^{-s}-n^{1-s}+sn^{-s}\\\
&=\frac{1}{s-1}\left(s+(N+1)^{1-s}-1+s\sum_{n=2}^{N}n^{-s}-\sum_{n=2}^{N+1}n^{-s}\right)\\\
&=\frac{N}{\left(s-1\right)\left(N+1\right)^{s}}-\frac{\cos\left(\pi
s\right)\Psi\left(s-1,N+1\right)}{\Gamma\left(s\right)}+\zeta\left(s\right)\forall
s\in\mathbbm{N}^{\ast}\end{array}$
with equality in the limit except at the negative integers
(22)
$\begin{array}[]{ll}\lim_{N\rightarrow\infty}\zeta_{w}(N;s)&=\zeta(s)\forall-s\not\in\mathbbm{N}^{\ast}\end{array}$
The functions $\zeta_{w}\left(N;s\right)$ have real roots at $s=0$ and $s=-1$.
That is
(23) $\lim_{s\rightarrow
0}\zeta_{w}\left(N;s\right)=\lim_{s\rightarrow-1}\zeta_{w}\left(N;s\right)=0$
The residue of $\zeta_{w}(N;s)$ at $s=1$ is given by
(24)
$\begin{array}[]{ll}\underset{}{\underset{s=1}{\operatorname{Res}}(\zeta_{w}(N;s))}&=\underset{}{\underset{s=1}{\operatorname{Res}}\left(\tau(s)\sum_{n=1}^{N}M[w_{n}(x);x\rightarrow
s]\right)}\\\
&=\sum_{n=1}^{N}\underset{s=1}{\operatorname{Res}}\left(\underset{}{\underset{}{}\tau(s)}M[w_{n}(x);x\rightarrow
s]\right)\\\ &=\sum_{n=1}^{N}\frac{1}{n^{2}+n}\\\ &=\frac{N}{N+1}\end{array}$
Thus, as required
(25)
$\begin{array}[]{ll}\lim_{N\rightarrow\infty}\underset{}{\underset{s=1}{\operatorname{Res}}(\zeta_{w}(N;s))}&=\lim_{N\rightarrow\infty}\sum_{n=1}^{N}\frac{1}{n^{2}+n}\\\
&=\lim_{N\rightarrow\infty}\frac{N}{N+1}\\\ &=1\end{array}$
The function $\tau(s)$ has zeros at $-1$ and $0$ and a simple pole at $s=1$
with residue
(26)
$\begin{array}[]{ll}\underset{s=1}{\operatorname{Res}}\left(\tau(s)\right)&=\underset{}{\underset{s=1}{\operatorname{Res}}\left(s\frac{s+1}{s-1}\right)}=2\end{array}$
The Mellin transform of $\tau(s)$ has an interesting Laurent series,
convergent on the unit disc, given by
(27) $\begin{array}[]{lll}M[\tau(s);s\rightarrow
t]&=\int_{0}^{\infty}\tau(s)s^{t-1}\mathrm{d}s&\\\
&=\int_{0}^{\infty}s\frac{s+1}{s-1}s^{t-1}\mathrm{d}s&\\\
&=\sum_{n=1}^{\infty}4\zeta(2n-2)t^{2n-3}&\text{$\forall|t|<1$}\\\
&=-4i\pi\frac{\left(\frac{1}{2}e^{i\pi t}+\frac{1}{2}e^{-i\pi
t}\right)}{e^{i\pi t}-e^{-i\pi t}}&\\\ &=-2\pi\frac{\cos(\pi t)}{\sin(\pi
t)}&\end{array}$
The transformations $M[w_{n}(x);x\rightarrow s]$ have removable singularities
at $-1$ and $0$ where the limits are given by
(28)
$\begin{array}[]{ll}\underset{s\rightarrow-1}{\lim}\underset{}{M[w_{n}(x);x\rightarrow
s]}&=n^{2}\ln\left(\frac{n+1}{n}\right)+n\ln\left(\frac{n+1}{n}\right)-n\\\
\underset{s\rightarrow 0}{\lim}\underset{}{M[w_{n}(x);x\rightarrow
s]}&=1+n\ln\left(\frac{n+1}{n}\right)\end{array}$ (29)
$\begin{array}[]{ll}\lim_{s\rightarrow 0}e^{-M[w_{n}(x);x\rightarrow
s]}&=(n+1)^{n}n^{-n}e^{-1}\\\
\begin{array}[]{l}\lim_{s\rightarrow-1}e^{-M[w_{n}(x);x\rightarrow
s]}\end{array}&=n^{(n^{2}+n)}(n+1)^{(-n^{2}-n)}e^{n}\end{array}$
So, (14) can be rewritten as
(30) $\begin{array}[]{ll}M\left[w_{n}(x);x\rightarrow
s\right]&=\left\\{\begin{array}[]{ll}n^{2}\ln\left(\frac{n+1}{n}\right)+n\ln\left(\frac{n+1}{n}\right)-n&s=-1\\\
1+n\ln\left(\frac{n+1}{n}\right)&s=0\\\
\begin{array}[]{l}-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\end{array}&\operatorname{otherwise}\end{array}\right.\end{array}$
Furthermore, we have the limits
(31)
$\begin{array}[]{ll}\lim_{n\rightarrow\infty}\underset{s\rightarrow-1}{\lim}M\left[w_{n}(x);x\rightarrow
s\right]&=\lim_{n\rightarrow\infty}\lim_{s\rightarrow-1}-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\\\
&=\lim_{n\rightarrow\infty}n^{2}\ln\left(\frac{n+1}{n}\right)+n\ln\left(\frac{n+1}{n}\right)-n\\\
&=\frac{1}{2}\end{array}$
and
(32) $\begin{array}[]{ll}\lim_{n\rightarrow\infty}\underset{s\rightarrow
0}{\lim}M\left[w_{n}(x);x\rightarrow
s\right]&=\lim_{n\rightarrow\infty}\lim_{s\rightarrow
0}-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\\\
&=\lim_{n\rightarrow\infty}1+n\ln\left(\frac{n+1}{n}\right)\\\ &=0\end{array}$
The sum of limits over $n$ at $s=-1$ is Euler’s constant. [13][14, 1.1]
(33)
$\begin{array}[]{ll}\sum_{n=1}^{\infty}\frac{M\left[w_{n}(x);x\rightarrow-1\right]}{n}&=\sum_{n=1}^{\infty}\frac{1+n\ln\left(\frac{n+1}{n}\right)}{n}\\\
&=\lim_{s\mathop{\rightarrow}\limits 1}\zeta(s)-\frac{1}{s-1}\\\
&=\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\\\ &=\gamma\\\
&\cong 0.577215664901533\ldots\end{array}$
#### 1.2.2. The Reflection Formula for $\zeta_{w}\left(N;s\right)$
There is a reflection equation for the finite-sum approximation
$\zeta_{w}(N;s)$ which is similiar to the well-known formula
$\zeta\left(1-s\right)=\chi\left(s\right)\zeta\left(s\right)$ with
$\chi\left(s\right)=2\left(2\pi\right)^{-s}\cos\left(\frac{\pi
s}{2}\right)\Gamma\left(s\right)$. The solution to
(34)
$\zeta_{w}\left(N;1-s\right)=\chi\left(N;s\right)\zeta_{w}\left(N;s\right)$
is given by the expression
(35)
$\begin{array}[]{ll}\chi\left(N;s\right)&=\frac{\zeta_{w}\left(N;1-s\right)}{\zeta_{w}\left(N;s\right)}\\\
&=\frac{\sum_{n=1}^{N}-\frac{-n^{s}+\left(n+1\right)^{s-1}n+n^{s-1}-n^{s-1}s}{s}}{\sum_{n=1}^{N}\frac{-n^{1-s}+\left(n+1\right)^{-s}n+n^{-s}s}{s-1}}\\\
&=-\frac{\left(s-1\right)\sum_{n=1}^{N}-n^{s}+\left(n+1\right)^{s-1}n+n^{s-1}-n^{s-1}s}{s\sum_{n=1}^{N}-n^{1-s}+\left(n+1\right)^{-s}n+n^{-s}s}\end{array}$
which satisfies
(36) $\chi\left(N;1-s\right)=\chi\left(N;s\right)^{-1}$
The functions $\chi\left(N;s\right)$, indexed by $N$, have singularities at
$s=0$. Let
(37)
$\begin{array}[]{ll}a\left(N\right)&=\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\\\
b\left(N\right)&=\sum_{n=1}^{N}\frac{\ln\left(n\right)n^{2}-\ln\left(n+1\right)n^{2}-\ln\left(n\right)}{n\left(n+1\right)}\\\
c\left(N\right)&=\frac{1}{2}\sum_{n=1}^{N}n\left(\ln\left(n+1\right)^{2}-\ln\left(n\right)^{2}\right)\end{array}$
then the residue at the singular point $s=0$ is given by the expression
(38)
$\begin{array}[]{ll}\underset{s=0}{\operatorname{Res}}(\chi(N;s))&=-\underset{s=1}{\operatorname{Res}}(\chi(N;s)^{-1})\\\
&=\frac{1+\gamma+\Psi\left(n+2\right)-\frac{2}{N+1}+b\left(N\right)-\frac{N\left(\ln\left(\Gamma\left(N+1\right)\right)-c\left(N\right)\right)}{\left(N-a\left(N\right)\right)\left(N+1\right)}}{a\left(N\right)-N}\\\
&=\frac{1+\gamma+\Psi\left(n+2\right)-\frac{2}{N+1}+\sum_{n=1}^{N}\frac{\ln\left(n\right)n^{2}-\ln\left(n+1\right)n^{2}-\ln\left(n\right)}{n\left(n+1\right)}-\frac{N\left(\ln\left(\Gamma\left(N+1\right)\right)-\frac{1}{2}\sum_{n=1}^{N}n\left(\ln\left(n+1\right)^{2}-\ln\left(n\right)^{2}\right)\right)}{\left(N-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)\left(N+1\right)}}{\left(\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)-N}\end{array}$
which has the limit
(39)
$\lim_{N\rightarrow\infty}\underset{s=0}{\operatorname{Res}}(\chi(N;s))=1$
We also have the residue of the reciprocal at $s=2$
(40)
$\begin{array}[]{ll}\underset{s=2}{\operatorname{Res}}(\chi(N;s)^{-1})&=\frac{\frac{2N}{\left(N+1\right)^{2}}-2\Psi\left(1,N+1\right)+2\zeta\left(2\right)}{\frac{\left(N+1\right)^{2}}{2}-\frac{N}{2}-\frac{1}{2}-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)+\ln\left(n+1\right)n-\ln\left(n\right)-n\ln\left(n\right)\right)}\end{array}$
which vanishes as $N$ tends to infinity
(41)
$\lim_{N\rightarrow\infty}\underset{s=2}{\operatorname{Res}}(\chi(N;s)^{-1})=0$
As can be seen in the figures below, the residue at $s=0$ changes sign from
negative to positive between the values of $N=176$ and $N=177$.
Figure 2. $\left\\{\underset{s=0}{\operatorname{Res}}(\chi(N;s)):N=1\ldots
250\right\\}$
Figure 3.
$\left\\{\underset{s=0}{\operatorname{Res}}(\chi(N;s))^{-1}:N=1\ldots
250\right\\}$
For any positive integer N, we have the limits
(42) $\begin{array}[]{ll}\lim_{s\rightarrow 0}\chi\left(N;s\right)&=\infty\\\
\lim_{s\rightarrow
0}\frac{\mathrm{d}^{n}}{\mathrm{d}s^{n}}\chi\left(N;s\right)&=\infty\\\
\lim_{s\rightarrow\frac{1}{2}}\chi\left(N;s\right)&=1\\\ \lim_{s\rightarrow
1}\chi\left(N;s\right)&=0\\\ \lim_{s\rightarrow 2}\chi\left(N;s\right)&=0\\\
\lim_{s\rightarrow
1}\frac{\mathrm{d}}{\mathrm{d}s}\chi\left(N;s\right)&=0\end{array}$
The line $\operatorname{Re}\left(s\right)=\frac{1}{2}$ has a constant modulus
(43) $\left|\chi\left(N;\frac{1}{2}+is\right)\right|=1$
There is also the complex conjugate symmetry
(44) $\chi\left(N;x+iy\right)=\overline{\chi\left(N;x-iy\right)}$
If $s=n\in\mathbbm{N}^{\ast}$ is a positive integer then
$\chi\left(N;n\right)$ can be written as
(45)
$\begin{array}[]{cc}\chi\left(N;n\right)&=\frac{\zeta_{w}\left(N;1-n\right)}{\zeta_{w}\left(N;n\right)}\\\
&=\frac{\sum_{m=1}^{N}-\sum_{k=1}^{n-2}\frac{m^{k}}{n}\binom{n-1}{k-1}}{\frac{N}{\left(n-1\right)\left(N+1\right)^{n}}-\frac{\cos\left(\pi
n\right)\Psi\left(n-1,N+1\right)}{\Gamma\left(n\right)}+\zeta\left(n\right)}\\\
&=\frac{-\sum_{m=1}^{N}\frac{1}{n}\left(\left(n-1\right)m^{n-1}+m^{n}-\left(m+1\right)^{n-1}m\right)}{\frac{N}{\left(n-1\right)\left(N+1\right)^{n}}-\frac{\cos\left(\pi
n\right)\Psi\left(n-1,N+1\right)}{\Gamma\left(n\right)}+\zeta\left(n\right)}\end{array}$
The Bernoulli numbers[1] make an appearance since
(46)
$\begin{array}[]{ll}\chi\left(N;2n\right)\zeta_{w}(N;2n)&=B_{2n}\left(N+1\right)^{2}\frac{\left(2n+1\right)}{2}+\ldots\end{array}$
The denominator of $\chi\left(N;n\right)$ has the limits
(47)
$\begin{array}[]{cl}\lim_{N\rightarrow\infty}\zeta_{w}\left(N;n\right)&=\zeta\left(n\right)\\\
\lim_{n\rightarrow\infty}\zeta_{w}\left(N;n\right)&=1\end{array}$
Another interesting formula gives the limit at $s=1$ of the quotient of
successive functions
(48)
$\begin{array}[]{cl}\lim_{s=1}\frac{\chi\left(N+1;s\right)}{\chi\left(N;s\right)}&=\frac{\left(N+2\right)N\left(N+1-a\left(N+1\right)\right)}{\left(N+1\right)^{2}\left(N-a\left(N\right)\right)}\\\
&=\frac{\left(N+2\right)N\left(N+1-\sum_{n=1}^{N+1}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)}{\left(N+1\right)^{2}\left(N-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)}\end{array}$
Figure 4. $\left\\{\chi\left(N;s\right):s=1\ldots 2,N=1\ldots 25\right\\}$
Figure 5. $\left\\{\chi\left(N;s\right):s=\frac{1}{2}\ldots 2,N=1\ldots
100\right\\}$
Let
(49)
$\nu\left(s\right)=\chi\left(\infty;s\right)=\frac{\zeta\left(1-s\right)}{\zeta\left(s\right)}$
Then the residue at the even negative integers is
(50)
$\underset{s=-n}{\operatorname{Res}}(\nu(s))=\left\\{\begin{array}[]{ll}\frac{\zeta\left(1-n\right)}{\frac{\mathrm{d}}{\mathrm{d}s}\zeta\left(s\right)|_{s=-n}}&n\operatorname{even}\\\
0&n\operatorname{odd}\end{array}\right.$
#### 1.2.3. The Laplace Transforms $L[w_{n}(x);x\rightarrow s]$
The Laplace transform $L[w_{n}(x);x\rightarrow s]$ and its roots are
calculated to shed light on the behaviour roots of the Mellin transforms
$M[w_{n}(x);x\rightarrow s]$ but it is unclear whether this is accomplished.
The Laplace transform (135) of the $n$-th component function is given by
(51) $\begin{array}[]{ll}L[w_{n}(x);x\rightarrow
s]&=\int_{0}^{1}w_{n}(x)e^{-xs}\mathrm{d}x\\\
&=\int_{0}^{1}n(xn+x-1)\chi(x,I^{H}_{n})e^{-xs}\mathrm{d}x\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)e^{-xs}\mathrm{d}x\\\
&=\frac{n(n+1)e^{-\frac{s}{n+1}}-(n^{2}+n+s)e^{-\frac{s}{n}}}{s^{2}}\end{array}$
There is a removable singularity at $s=0$ which has the limit
(52) $\begin{array}[]{ll}L[w_{n}(x);x\rightarrow 0]&=\lim_{s\rightarrow
0}L[w_{n}(x);x\rightarrow s]\\\ &=\lim_{s\rightarrow
0}\frac{n(n+1)e^{-\frac{s}{n+1}}-(n^{2}+n+s)e^{-\frac{s}{n}}}{s^{2}}\\\
&=\frac{1}{2}\underset{s=1}{\operatorname{Res}}\left(\tau(s)M[w_{n}(x);x\rightarrow
s]\right)\\\ &=\frac{1}{2n\left(n+1\right)}\end{array}$
Additionally,
(53) $\begin{array}[]{ll}\sum_{n=1}^{\infty}L[w_{n}(x);x\rightarrow
0]&=\sum_{n=1}^{\infty}\lim_{s\rightarrow
0}\frac{n(n+1)e^{-\frac{s}{n+1}}-(n^{2}+n+s)e^{-\frac{s}{n}}}{s^{2}}\\\
&=\sum_{n=1}^{\infty}\frac{1}{2n\left(n+1\right)}\\\
&=\lim_{n\rightarrow\infty}\underset{s\rightarrow-1}{\lim}M[w_{n}(x);x\rightarrow
s]\\\
&=\lim_{n\rightarrow\infty}\lim_{s\rightarrow-1}-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s^{2}+s}\\\
&=\frac{1}{2}\end{array}$
The roots of $L[w_{n}(x);x\rightarrow s]$ are enumerated by
(54)
$\begin{array}[]{ll}\rho^{L}_{w_{n}}(m)&\text{$=\left\\{s:\text{$L[w_{n}(x);x\rightarrow
s]$}=0\right\\}$}=-n(n+1)W(m,-e^{-1})-n(n+1)\end{array}$
where $W(m,x)$ is the Lambert W function (146) and
$m\in\mathbbm{Z},\text{$n\in\mathbbm{N}$}$. It can be verified that
(55)
$\begin{array}[]{ll}L[w_{n}(z);x\rightarrow\rho^{L}_{w_{n}}(m)]&=\frac{e^{(n+1)(1+W(m,-e^{-1}))}W(m,-e^{-1})+e^{n(1+W(m,-e^{-1}))}}{n(1+W(m,-e^{-1}))^{2}(n+1)}=0\end{array}$
where $-e^{-1}$ is expressed as the continued fraction via its quotient
sequence
(56)
$-e^{-1}=[-1,1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,1,1,18,1,1,20,...]$
The roots $\rho^{L}_{w_{n}}(m)$ satisfy a functional reflection equation with
respect to $m$
(57)
$\begin{array}[]{ll}\rho^{L}_{w_{n}}(m)&=\overline{\rho^{L}_{w_{n}}(-m-1)}\\\
-n(n+1)W(m,-e^{-1})-n(n+1)&=-n(n+1)\overline{W(-m-1,-e^{-1})}-n(n+1)\end{array}$
where $\bar{x}=\operatorname{Re}(x)-\operatorname{Im}(x)$ denotes complex
conjugation. The quotients of the roots of consecutive transforms is
(58)
$\begin{array}[]{ll}\frac{\rho^{L}_{w_{n}}(m)}{\rho^{L}_{w_{n-1}}(m)}&=\frac{n+1}{n-1}\end{array}$
Thus
(59)
$\begin{array}[]{ll}\lim_{n\rightarrow\infty}\frac{\rho^{L}_{w_{n}}(m)}{\rho^{L}_{w_{n-1}}(m)}&=\lim_{n\rightarrow\infty}\frac{n+1}{n-1}=1\end{array}$
Figure 6. The Roots $\rho^{L}_{w_{n}}(m)$ of $\left\\{L[w_{n}(x);x\rightarrow
s]\right.\left.:n\in 1\ldots 9\right\\}$
#### 1.2.4. The Roots $\rho^{M}_{w_{n}}\left(m\right)$ of
$M[w_{n}(x);x\rightarrow s]$
Define
(60) $\begin{array}[]{ll}M_{\tau
w}(s,n)&=\text{$\tau(s)\text{$M[w_{n}(x);x\rightarrow s]$}$}\\\
&=\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}\end{array}$
as in (21) and its infinte number of inverse branches (which are currently
lacking closed-form expression if such a thing is possible except when $n=1$),
where the branches are indexed by $m$
(61) $\begin{array}[]{ll}M^{-1}_{\tau
w}(z,n,m)&=\\{s:\tau(s)\text{$M[w_{n}(x);x\rightarrow s]$}=z\\}\\\
&=\left\\{s:\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}=z\right\\}\end{array}$
then we see that the first function where $n=1$, $M^{-1}_{\tau w}(z,1,m)$, has
the closed-form
(62) $\begin{array}[]{ll}\text{$M^{-1}_{\tau
w}(z,1,m)$}&=\frac{W\left(m,\frac{\ln(\sqrt{2})}{z-1}\right)+\ln(2)}{\ln(2)}\end{array}$
where $W$ is the Lambert W function (146). It is verified that
(63) $\begin{array}[]{ll}\text{$M_{\tau w}(M^{-1}_{\tau
w}(z,1,m),1,m)$}&=-\frac{1-2^{-\frac{W\left(m,\frac{\ln(\sqrt{2})}{z-1}\right)+\ln(2)}{\ln(2)}}-\frac{W\left(m,\frac{\ln(\sqrt{2})}{z-1}\right)+\ln(2)}{\ln(2)}}{\frac{W\left(m,\frac{\ln(\sqrt{2})}{z-1}\right)+\ln(2)}{\ln(2)}-1}=z\end{array}$
Furthermore, let $\rho^{M}_{w_{n}}\left(m\right)$ denote the $m$-th root of
$M[w_{n}(x);x\rightarrow s]$
(64)
$\begin{array}[]{ll}\rho_{w_{n}}^{M}(m)&=\\{s:\text{$M[w_{n}(x);x\rightarrow
s]$}=0\\}\\\ &=M^{-1}_{w}(0,n,m)\end{array}$
which satisfies
(65)
$\begin{array}[]{l}\operatorname{Im}\left(\rho^{M}_{w_{n}}\left(m\right)\right)>\operatorname{Im}\left(\rho^{M}_{w_{n-1}}\left(m\right)\right)\end{array}$
(66)
$\begin{array}[]{ll}\lim_{m\rightarrow\pm\infty}\operatorname{Re}(\rho^{M}_{w_{n}}\left(m\right))&=0\end{array}$
Thus,
(67)
$\begin{array}[]{ll}\lim_{m\rightarrow\pm\infty}\arg\left(\rho^{M}_{w_{n}}\left(m\right)\right)&=\frac{\pi}{2}\end{array}$
#### 1.2.5. Quotients and Differences of $\rho^{M}_{w_{n}}\left(m\right)$
Let
(68)
$\begin{array}[]{ll}\Delta\rho^{M}_{w_{n}}(m)&=\rho^{M}_{w_{n}}(m+1)-\rho^{M}_{w_{n}}\left(m\right)\end{array}$
be the forward difference of consecutive roots of $M[w_{n}(x);x\rightarrow
s]$. The limiting difference between consecutive roots is the countably
infinite set of solutions to the equation
$n^{\frac{s}{2}}+(n+1)^{\frac{s}{2}}=0$ given by
(69)
$\begin{array}[]{ll}\Delta\rho^{M}_{w_{n}}(\pm\infty)&=\lim_{m\rightarrow\pm\infty}\overset{}{\Delta}\rho^{M}_{w_{n}}(m)\\\
&=\left\\{s:n^{\frac{s}{2}}+(n+1)^{\frac{s}{2}}=0\right\\}\\\
&=\lim_{m\rightarrow\pm\infty}\left(\rho^{M}_{w_{n}}(m+1)-\rho^{M}_{w_{n}}(m)\right)\\\
&=\lim_{m\rightarrow\pm\infty}\left(\frac{\rho^{M}_{w_{n}}(m)}{m}\right)\\\
&=\frac{2\pi i}{M[\chi\left(x,I^{H}_{n}\right);x\rightarrow 0]}\\\
&=\frac{2\pi i}{\lim_{s\rightarrow
0}\left(\frac{n^{-s}-(n+1)^{-s}}{s}\right)}\\\ &=\frac{2\pi
i}{\ln(n+1)-\ln(n)}\end{array}\text{}$
Let $\mathcal{Q}^{\infty}_{\rho^{M}_{w_{n}}}$ denote the limit
(70)
$\begin{array}[]{ll}\mathcal{Q}^{\infty}_{\rho^{M}_{w_{n}}}&=\lim_{m\rightarrow\pm\infty}\frac{\rho^{M}_{w_{n}}(m)}{\rho^{M}_{w_{n-1}}(m)}\\\
&=\frac{\Delta\rho^{M}_{w_{n}}(\pm\infty)}{\Delta\rho^{M}_{w_{n-1}}(\pm\infty)}\\\
&=\frac{\overset{}{M}\left[\chi\left(x,\left(\frac{1}{n+1},\frac{1}{n}\right)\right);x\rightarrow
0\right]}{\overset{}{M}\left[\chi\left(x,\left(\frac{1}{n},\frac{1}{n-1}\right)\right);x\rightarrow
0\right]}\\\ &=\frac{\ln(n)-\ln(n-1)}{\ln(n+1)-\ln(n)}\end{array}$
then we also have the limit of the limits
$\mathcal{Q}^{\infty}_{\rho^{M}_{w_{n}}}$ as $n\rightarrow\infty$ given by
(71)
$\begin{array}[]{ll}\lim_{n\rightarrow\pm\infty}\text{$\mathcal{Q}^{\infty}_{\rho^{M}_{w_{n}}}$
}&=\lim_{n\rightarrow\pm\infty}\left(\frac{\Delta\rho^{M}_{w_{n}}(\pm\infty)}{\Delta\rho^{M}_{w_{n-1}}(\pm\infty)}\right)\\\
&=\lim_{n\rightarrow\pm\infty}\left(\frac{\frac{\ln(n)-\ln(n-1)}{\ln(n+1)-\ln(n)}}{\frac{\ln(n-1)-\ln(n-2)}{\ln(n)-\ln(n-2)}}\right)\\\
&=\lim_{n\rightarrow\pm\infty}\left(\frac{\ln(n)-\ln(n-1)}{\ln(n+1)-\ln(n)}\right)\\\
&=1\end{array}$
Figure 7. $\\{\rho^{M}_{w_{n}}(m):m=1\ldots 5\\}$
The limiting quotients
$\frac{e^{-\rho^{M}_{w_{n}}(m+1)}}{e^{-\rho^{M}_{w_{n}}(m)}}=e^{\rho^{M}_{w_{n}}(m)-\rho^{M}_{w_{n}}(m+1)}$
as $m\rightarrow\infty$ are given by
(72)
$\begin{array}[]{ll}\begin{array}[]{l}\lim_{m\rightarrow\infty}e^{\rho^{M}_{w_{n}}(m)-\rho^{M}_{w_{n}}(m+1)}\end{array}&=1-2\sin\left(\frac{\pi}{\ln(n+1)-\ln(n)}\right)^{2}-2i\cos\left(\frac{\pi}{\ln(n+1)-\ln(n)}\right)\sin\left(\frac{\pi}{\ln(n+1)-\ln(n)}\right)\\\
&=e^{-\frac{2\pi i}{\ln(n+1)-\ln(n)}}\end{array}$
where we have
(73)
$\begin{array}[]{lll}|\lim_{m\rightarrow\infty}e^{\rho^{M}_{w_{n}}(m)-\rho^{M}_{w_{n}}(m+1)}|&=\lim_{m\rightarrow\infty}|e^{\rho^{M}_{w_{n}}(m)-\rho^{M}_{w_{n}}(m+1)}|&\\\
&=\sqrt{e^{-\frac{2\pi i}{\ln(n+1)-\ln(n)}}}&\\\ &=1&\end{array}$
and
(74)
$\begin{array}[]{l}\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}e^{\rho^{M}_{w_{n}}(m)-\rho^{M}_{w_{n}}(m+1)}=-1\end{array}$
#### 1.2.6. The Laplace Transform $L\left[(s-1)M[w_{n}(x);x\rightarrow
s\right];s\rightarrow t]$
The Laplace transform of $(s-1)M[w_{n}(x);x\rightarrow s]$ defined by
(75) $\begin{array}[]{lll}\left.L[(s-1)M[w_{n}(x);x\rightarrow
s\right];s\rightarrow t]&&=L[n(n+1)^{-s}+sn^{-s}-n^{1-s};s\rightarrow t]\\\
&&=\int_{0}^{\infty}n(n+1)^{-s}+sn^{-s}-n^{1-s}e^{-st}\mathrm{d}s\\\
&&=\frac{t+\ln\left(\frac{n^{n}}{(n+1)^{n}}\right)t+\ln(n+1)+n\ln(n)^{2}-n\ln(n)\ln(n+1)}{(\ln(n)+t)^{2}(\ln(n+1)+t)}\end{array}$
has poles at $-\ln(n)$ and $-\ln(n+1)$ with residues
(76)
$\begin{array}[]{ll}\underset{}{\underset{t=-\ln(n)}{\operatorname{Res}}\left(\underset{}{L}[(s-1)M[w_{n}(x);x\rightarrow
s];s\rightarrow t]\right)}&=-n\\\
\underset{}{\underset{t=-\ln(n+1)}{\operatorname{Res}}\left(\underset{}{L}[(s-1)M[w_{n}(x);x\rightarrow
s];s\rightarrow t]\right)}&=n\end{array}$
### 1.3. The Gauss Map $h(x)$
#### 1.3.1. $\operatorname{Continued}\operatorname{Fractions}$
The Gauss map $h(x)$, also known as the Gauss function or Gauss
transformation, maps unit intervals onto unit intervals and by iteration gives
the continued fraction expansion of a real number [38, A.1.7][35, I.1][12, X]
The $n$-th component function $h_{n}(x)$ of the map $h(x)$ is given by
(77)
$\begin{array}[]{ll}h_{n}(x)&=\frac{1-xn}{x}\chi\left(x,I^{H}_{n}\right)\end{array}$
The infinite sum of the component functions is the Gauss map
(78) $\begin{array}[]{ll}h(x)&=\sum_{n=1}^{\infty}h_{n}(x)\\\
&=\sum_{n=1}^{\infty}\frac{1-xn}{x}\chi\left(x,I^{H}_{n}\right)\\\
&=x^{-1}-\left\lfloor x^{-1}\right\rfloor\\\ &=\\{x^{-1}\\}\end{array}$
Figure 8. The Gauss Map
The fixed points of $h(x)$ are the (positive) solutions to the equation
$h_{n}(x)=x$ given by
(79) $\begin{array}[]{ll}\operatorname{Fix}_{h}^{n}&=\\{x:h_{n}(x)=x\\}\\\
&=\left\\{x:\frac{1-xn}{x}\chi\left(x,I^{H}_{n}\right)=x\right\\}\\\
&=\left\\{x:\frac{1-xn}{x}=x\right\\}\\\
&=\frac{\sqrt{n^{2}+4}}{2}-\frac{n}{2}\end{array}$
#### 1.3.2. The Mellin Transform of $h(x)$
The Mellin transform (138) of the Gauss map $h(x$) over the unit interval,
scaled by $s$ then subtracted from $\frac{s}{s-1}$, is an analytic
continuation of $\zeta(s$), denoted by $\zeta_{h}(s$), valid for
all$-\operatorname{Re}(s))\not\in\mathbbm{N}$. The transfer operator and
thermodynamic aspects of the Gauss map are discussed in [42][41][40][39][36].
The Mellin transform of the $n$-th component function $w_{n}(x$) is given by
(80) $\begin{array}[]{ll}M[h_{n}(x);x\rightarrow
s]&=\int_{0}^{1}h_{n}(x)x^{s-1}\mathrm{d}x\\\
&=\int_{0}^{1}\frac{1-xn}{x}\chi\left(x,I^{H}_{n}\right)x^{s-1}\mathrm{d}x\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\left(x^{-1}-\left\lfloor
x^{-1}\right\rfloor\right)x^{s-1}\mathrm{d}x\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\frac{1-xn}{x}x^{s-1}\mathrm{d}x\\\
&=-\frac{n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s}}{s^{2}-s}\end{array}$
which provides an analytic continuation
$\zeta_{h}(s)=\zeta(s)\forall(-\operatorname{Re}(s))\not\in\mathbbm{N}$
(81) $\begin{array}[]{ll}\zeta_{h}(s)&=\frac{s}{s-1}-sM[h(x);x\rightarrow
s]\\\ &=\frac{s}{s-1}-s\int_{0}^{1}h(x)x^{s-1}\mathrm{d}x\\\
&=\frac{s}{s-1}-s\int_{0}^{1}\left(x^{-1}-\left\lfloor
x^{-1}\right\rfloor\right)x^{s-1}\mathrm{d}x\\\
&=\frac{s}{s-1}-s\sum_{n=1}^{\infty}M[h_{n}(x);x\rightarrow s]\\\
&=\frac{s}{s-1}-\frac{1}{s-1}\sum_{n=1}^{\infty}-(n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s})\\\
&=\frac{s}{s-1}-\frac{1}{s-1}\sum_{n=1}^{\infty}n^{1-s}-n(n+1)^{-s}-s(n+1)^{-s}\end{array}$
### 1.4. The Harmonic Sawtooth Map w(x) as an Ordinary Fractal String
#### 1.4.1. Definition and Length
Let
(82)
$\begin{array}[]{ll}I^{H}_{n}&=\left(\frac{1}{n+1},\frac{1}{n}\right)\end{array}$
be the $n$-th harmonic interval, then
$\\{w(x)\in\mathcal{L}_{w}:x\in\Omega\\}$ is the piecewise monotone mapping of
the unit interval onto itself. The fractal string $\mathcal{L}_{w}$ associated
with $w(x)$ is the set of connected component functions $w_{n}(x)\subset w(x)$
where each $w_{n}(x)$ maps$I_{n}^{H}$ onto $(0,1)$ and vanishes when $x\not\in
I_{n}^{H}$. Thus, the disjoint union of the connected components of
$\mathcal{L}_{w}$ is the infinite sum $w(x)=\sum_{n=1}^{\infty}w_{n}(x)$ where
only 1 of the $w_{n}(x)$ is nonzero for each $x$, thus $w(x)$ maps entire unit
interval onto itself uniqely except except for the points of discontinuity on
the boundary
$\partial\mathcal{L}_{w}=\\{0,\frac{1}{n}:n\in\mathbbm{N}^{\ast}\\}$ where a
choice is to be made between 0 and 1 depending on the direction in which the
limit is approached. Let
(83) $\begin{array}[]{ll}w_{n}(x)&=n(xn+x-1)\chi(x,I^{H}_{n})\end{array}$
where $\chi(x,I^{H}_{n})$ is the $n$-th harmonic interval indicator (127)
(84)
$\begin{array}[]{ll}\chi(x,I_{n}^{H})&=\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\end{array}$
The substitution $n\rightarrow\left\lfloor\frac{1}{x}\right\rfloor$ can be
made in (132) where it is seen that
(85) $\begin{array}[]{ll}\text{$\chi\left(x,I^{H}_{\left\lfloor
x^{-1}\right\rfloor}\right)$}&=\theta\left(\frac{x\text{$\left\lfloor
x^{-1}\right\rfloor$}+x-1}{\text{$\left\lfloor
x^{-1}\right\rfloor$}+1}\right)-\theta\left(\frac{x\left\lfloor
x^{-1}\right\rfloor\text{}-1}{\left\lfloor
x^{-1}\right\rfloor}\right)=1\end{array}$
and so making the same substitution in (83) gives
(86) $\begin{array}[]{ll}w(x)&=\sum_{n=1}^{\infty}w_{n}(x)\\\
&=\sum_{n=1}^{\infty}n(xn+x-1)\chi(x,I^{H}_{n})\\\ &=\left\lfloor
x^{-1}\right\rfloor\text{}\left(x\left\lfloor
x^{-1}\right\rfloor\text{}+x-1\right)\\\ &=x\left\lfloor
x^{-1}\right]^{2}+x\left\lfloor x^{-1}\right\rfloor\text{}-\left\lfloor
x^{-1}\right\rfloor\text{}\end{array}$
Figure 9. The Harmonic Sawtooth Map
The intervals $I_{n}^{w}$ will be defined such that $\ell
w_{n}=\left|I_{n}^{w}\right|=\left|w_{n}(x)\right|$. Let
(87)
$\begin{array}[]{ll}\mathfrak{h}_{n}&=\int_{I^{H}_{n}}x(n+1)n\mathrm{d}x\\\
&=\frac{\left(\frac{1}{n+1}+\frac{1}{n}\right)}{2}\\\
&=\frac{2n+1}{2n(n+1)}\end{array}$
be the midpoint of $I^{H}_{n}$ then
(88)
$\begin{array}[]{ll}I_{n}^{w}&=\left(\mathfrak{h}_{n}-\frac{\left|w_{n}(x)\right|}{2},\mathfrak{h}_{n}+\frac{\left|w_{n}(x)\right|}{2}\right)\\\
&=\left(\frac{4n+1}{4n(n+1)},\frac{4n+3}{4n(n+1)}\right)\end{array}$
so that
(89) $\begin{array}[]{ll}\ell w_{n}&=\left|w_{n}(x)\right|\\\
&=\int_{0}^{1}n(xn+x-1)\chi(x,I^{H}_{n})\mathrm{d}x\\\
&=\int^{\frac{1}{n}}_{\frac{1}{n+1}}w(x)\mathrm{d}x\\\
&=\left|I_{n}^{w}\right|\\\ &=\int_{0}^{1}\chi(x,I_{n}^{w})\mathrm{d}x\\\
&=\frac{4n+3}{4n(n+1)}-\frac{4n+1}{4n(n+1)}\\\ &=\frac{1}{2n(n+1)}\end{array}$
Figure 10. Reciprocal lengths $\ell w_{n}^{-1}$
The total length of $\mathcal{L}_{w}$ is
(90) $\begin{array}[]{ll}|\mathcal{L}_{w}|&=\int_{0}^{1}w(x)\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}\ell w_{n}\\\ &=\sum_{n=1}^{\infty}\frac{1}{2n(n+1)}\\\
&=\frac{1}{2}\end{array}$
Figure 11. $\chi(x,I_{n}^{w})$ green and $w_{n}(x)$ blue for $n=1\ldots 3$ and
$x=\frac{1}{4}..1$
#### 1.4.2. Geometry and Volume of the Inner Tubular Neighborhood
The geometric counting function (109) of $\mathcal{L}_{w}$ is
(91) $\begin{array}[]{ll}N_{\mathcal{L}_{w}}(x)&=\\#\\{n\geqslant 1:\ell
w_{n}^{-1}\leqslant x\\}\\\ &=\\#\\{n\geqslant 1:2(n+1)n\leqslant x\\}\\\
&=\left\lfloor\frac{\sqrt{2x+1}}{2}-\frac{1}{2}\right\rfloor\end{array}$
which is used to calculate the limiting constant (111) $C_{w}$ appearing in
the equation for the Minkowski content
(92)
$\begin{array}[]{ll}C_{w}&=\lim_{x\rightarrow\infty}\frac{N_{\mathcal{L}_{w}}(x)}{x^{D_{\mathcal{L}_{w}}}}\\\
&=\lim_{x\rightarrow\infty}\frac{\frac{\sqrt{2x+1}}{2}-\frac{1}{2}}{\sqrt{x}}\\\
&=\frac{\sqrt{2}}{2}\end{array}$
The function $N_{\mathcal{L}_{w}}(x)$ happens to coincide with [17, A095861],
which is the number of primitive Pythagorean triangles of the form
$\\{(a,b,b+1):(b+1)\leqslant n\\}$. [6, 171-176][37, 10.1][18, 11.2-11.5] Let
(93) $\begin{array}[]{lll}v(\varepsilon)&=\min(j:\ell
w_{j}<2\varepsilon)&=\left\lfloor\frac{\varepsilon+\sqrt{\varepsilon^{2}+\varepsilon}}{2\varepsilon}\right\rfloor\end{array}$
which is the floor of the solution to the inverse length equation
(94)
$\begin{array}[]{ll}\frac{\varepsilon+\sqrt{\varepsilon^{2}+\varepsilon}}{2\varepsilon}=\left\\{n:\ell
w_{n-1}=2\varepsilon\right\\}=\left\\{n:\frac{1}{2n(n-1)}=2\varepsilon\right\\}&\\\
\frac{\ell
w_{\frac{\varepsilon+\sqrt{\varepsilon^{2}+\varepsilon}}{2\varepsilon}-1}}{2}=\varepsilon&\end{array}$
Then the volume of the inner tubular neighborhood of $\partial\mathcal{L}_{w}$
with radius $\varepsilon$ (108) is
(95) $\begin{array}[]{ll}V_{\mathcal{L}_{w}}(\varepsilon)&=2\varepsilon
N_{\mathcal{L}_{w}}\left(\frac{1}{2\varepsilon}\right)+\sum_{j}^{\ell
w_{j}<2\varepsilon}\ell w_{j}\\\ &=2\varepsilon
N_{\mathcal{L}_{w}}\left(\frac{1}{2\varepsilon}\right)+\sum_{\text{\scriptsize{$\begin{array}[]{l}n=v(\varepsilon)\end{array}$}}}^{\infty}\frac{1}{2(n+1)n}\\\
&=2\varepsilon
N_{\mathcal{L}_{w}}\left(\frac{1}{2\varepsilon}\right)+\frac{1}{2v(\varepsilon)}\\\
&=2\varepsilon\left\lfloor\frac{\sqrt{\frac{1}{\varepsilon}+1}}{2}-\frac{1}{2}\right\rfloor+\frac{1}{2v(\varepsilon)}\\\
&=\frac{4\varepsilon v(\varepsilon)^{2}-4\varepsilon
v(\varepsilon)+1}{2v(\varepsilon)}\end{array}$
since
(96)
$\begin{array}[]{ll}\sum_{n=m}^{\infty}\frac{1}{2n(n+1)}&=\frac{1}{2m}\end{array}$
and by defintion we have
(97) $\begin{array}[]{ll}\lim_{\varepsilon\rightarrow
0^{+}}V_{\mathcal{L}_{w}}(\varepsilon)&=0\\\
\lim_{\varepsilon\rightarrow\infty}V_{\mathcal{L}_{w}}(\varepsilon)&=|\mathcal{L}_{w}|=\frac{1}{2}\end{array}$
Thus, using (92) and (95), the Minkowski content (110) of $\mathcal{L}_{w}$ is
(98) $\begin{array}[]{lll}\mathcal{M}_{\mathcal{L}_{w}}&=\lim_{e\rightarrow
0^{+}}\frac{V_{\mathcal{L}w}(\varepsilon)}{\varepsilon^{1-D_{\mathcal{L}_{w}}}}&\\\
&=\lim_{e\rightarrow
0^{+}}\frac{1}{\sqrt{\varepsilon}}\left(2\varepsilon\left\lfloor\frac{\sqrt{\frac{1}{\varepsilon}+1}}{2}-\frac{1}{2}\right\rfloor+\frac{1}{2}\left\lfloor\frac{\varepsilon+\sqrt{\varepsilon^{2}+\varepsilon}}{2\varepsilon}\right\rfloor^{-1}\right)&\\\
&=\frac{C_{w}2^{1-D_{\mathcal{L}_{w}}}}{1-D_{\mathcal{L}_{w}}}&\\\
&=\frac{\frac{\sqrt{2}}{2}2^{1-\frac{1}{2}}}{1-\frac{1}{2}}&\\\
&=2&\end{array}$
Figure 12. Geometric Counting Function $N_{\mathcal{L}_{w}}(x)$ of $w(x)$
Figure 13. $\\{v(\varepsilon)=\min(n:\ell
w_{n}<2\varepsilon):\varepsilon=\frac{1}{1000}\ldots\frac{1}{4}\\}$
Figure 14. Volume of the inner tubular neighborhood of
$\partial\mathcal{L}_{w}$ with radius $\varepsilon$
$\left\\{V_{\mathcal{L}_{w}}(\varepsilon):\varepsilon=0\ldots\frac{1}{8}\right\\}$
Figure 15.
$\left\\{\frac{V\mathcal{L}_{w}(\varepsilon)}{\sqrt{\varepsilon}}:\varepsilon=0\ldots\frac{1}{8}\right\\}$
and $\frac{V\mathcal{L}_{w}(8^{-1})}{\sqrt{8^{-1}}}=\sqrt{2}$
#### 1.4.3. The Geometric Zeta Function $\zeta_{\mathcal{L}_{w}}(s)$
The geometric zeta function (112) of $w(x$) is the Dirichlet series of the
lengths $\ell w_{n}$ (89) and also an integral over the geometric length
counting function (91) $N_{\mathcal{L}_{w}}(x)$
(99) $\begin{array}[]{ll}\zeta_{\mathcal{L}_{w}}(s)&=\sum_{n=1}^{\infty}\ell
w_{j}^{s}\\\ &=\sum_{n=1}^{\infty}\left(\frac{1}{2n(n+1)}\right)^{s}\\\
&=\sum_{n=1}^{\infty}2^{-s}(n+1)^{-s}n^{-s}\\\
&=s\int_{0}^{\infty}N_{\mathcal{L}_{w}}(x)x^{-s-1}\mathrm{d}x\\\
&=s\int_{0}^{\infty}\left\lfloor\frac{\sqrt{2x+1}}{2}-\frac{1}{2}\right\rfloor
x^{-s-1}\mathrm{d}x\end{array}$
The residue (115) of $\zeta_{\mathcal{L}_{w}}(s$) at $D_{\mathcal{L}_{w}}$ is
(100)
$\begin{array}[]{ll}\underset{s=D_{\mathcal{L}}}{\operatorname{Res}(\zeta_{\mathcal{L}}(s))}&=\lim_{s\rightarrow
D_{\mathcal{L}_{w}}^{+}}(s-D_{\mathcal{L}})\zeta_{\mathcal{L}}(s)\\\
&=\lim_{s\rightarrow\frac{1}{2}^{+}}\left(s-\frac{1}{2}\right)\zeta_{\mathcal{L}}(s)\\\
&=\lim_{s\rightarrow\frac{1}{2}^{+}}\left(s-\frac{1}{2}\right)\sum_{n=1}^{\infty}2^{-s}(n+1)^{-s}n^{-s}\\\
&=\lim_{s\rightarrow\frac{1}{2}^{+}}\left(s-\frac{1}{2}\right)s\int_{0}^{\infty}\left\lfloor\frac{\sqrt{2x+1}}{2}-\frac{1}{2}\right\rfloor
x^{-s-1}\mathrm{d}x\\\ &=0\end{array}$
The values of $\zeta_{\mathcal{L}_{w}}(n)$ at positive integer values
$n\in\mathbbm{N}^{\ast}$ are given explicitly by a rather unwieldy sum of
binomial coefficients and the Riemann zeta function $\zeta(n)$ at even integer
values. First, define
(101)
$\begin{array}[]{ll}a_{n}&=\frac{\left(n-1\right)\left(1-\left(-1\right)^{n+1}\right)}{2}\\\
b_{n}&=\frac{\left(-1\right)^{n+1}\left(n-1\right)}{2}+n-\frac{7}{4}+\frac{(-1)^{n}}{4}\\\
c_{n}&=(-1)^{n}(n-1)\\\ d_{n}&=\frac{(-1)^{n}}{2}\end{array}$
then
(102)
$\begin{array}[]{ll}\zeta_{\mathcal{L}_{w}}(n)&=\frac{(-1)^{n}\binom{2n-1}{n-1}}{2^{n}}+\sum_{m=a_{n}}^{b_{n}}\frac{2(-1)^{n}\binom{2m+c_{n}-d_{n}+\frac{1}{2}}{n-1}\zeta\left(d_{n}+2n-\frac{3}{2}-2m-c_{n}\right)}{2^{n}}\end{array}$
The terms of $\zeta_{\mathcal{L}_{w}}(n)$ from $n=1$ to $10$ are shown below
in Table 1.
$\left(\begin{array}[]{llllll}+\frac{1}{2}&&&&&\\\
-\frac{3}{4}&+\frac{1}{2}\hskip 2.5pt\zeta\left(2\right)&&&&\\\
+\frac{5}{3}&-\frac{3}{4}\hskip 2.5pt\zeta\left(2\right)&&&&\\\
-\frac{35}{16}&+\frac{5}{4}\hskip 2.5pt\zeta\left(2\right)&+\frac{1}{8}\hskip
2.5pt\zeta\left(4\right)&&&\\\ +\frac{63}{16}&-\frac{35}{16}\hskip
2.5pt\zeta\left(2\right)&-\frac{5}{16}\hskip 2.5pt\zeta\left(4\right)&&&\\\
-\frac{231}{32}&+\frac{63}{16}\hskip
2.5pt\zeta\left(2\right)&+\frac{21}{32}\hskip
2.5pt\zeta\left(4\right)&+\frac{1}{32}\zeta\left(6\right)&&\\\
+\frac{429}{32}&-\frac{231}{32}\hskip
2.5pt\zeta\left(2\right)&-\frac{21}{16}\hskip
2.5pt\zeta\left(4\right)&-\frac{7}{64}\hskip 2.5pt\zeta\left(6\right)&&\\\
-\frac{6435}{256}&+\frac{429}{32}\hskip
2.5pt\zeta\left(2\right)&+\frac{165}{64}\hskip
2.5pt\zeta\left(4\right)&+\frac{9}{32}\hskip
2.5pt\zeta\left(6\right)&+\frac{1}{128}\hskip 2.5pt\zeta\left(8\right)&\\\
+\frac{12155}{256}&-\frac{6435}{256}\hskip
2.5pt\zeta\left(2\right)&-\frac{1287}{256}\hskip
2.5pt\zeta\left(4\right)&-\frac{165}{256}\hskip
2.5pt\zeta\left(6\right)&-\frac{9}{256}\hskip 2.5pt\zeta\left(8\right)&\\\
-\frac{46189}{512}&+\frac{12155}{256}\hskip
2.5pt\zeta\left(2\right)&+\frac{5005}{512}\hskip
2.5pt\zeta\left(4\right)&+\frac{715}{512}\hskip
2.5pt\zeta\left(6\right)&+\frac{55}{512}\hskip
2.5pt\zeta\left(8\right)+&\frac{1}{512\mathfrak{}}\hskip
2.5pt\zeta\mathbb{}\left(\mathfrak{}10\mathsf{}\right)\end{array}\right)$
Table 1.
$\\{\text{$\zeta_{\mathcal{L}_{w}}(n)=\Sigma\operatorname{row}_{n}$:n=1..10\\}}$
## 2\. Fractal Strings and Dynamical Zeta Functions
### 2.1. Fractal Strings
A a fractal string $\mathcal{L}$ is defined as a nonempty bounded open subset
of the real line $\mathcal{L}\subseteq\mathbbm{R}$ consisting of a countable
disjoint union of open intervals $I_{j}$
(103) $\mathcal{L}=\bigcup_{j=1}^{\infty}I_{j}$
The length of the $j$-th interval $I_{j}$ is denoted by
(104) $\ell_{j}=\left|I_{j}\right|$
where $\left.\left.\right|\cdot\right|$ is the $1$-dimensional Lebesgue
measure. The lengths $\ell_{j}$ must form a nonnegative monotically decreasing
sequence and the total length must be finite, that is
(105)
$\begin{array}[]{l}|\mathcal{L}|_{1}=\sum_{j=1}^{\infty}\ell_{j}<\infty\\\
\ell_{1}\geqslant\ell_{2}\geqslant\ldots\geqslant\text{$\ell_{j}\geqslant\ell_{j+1}\geqslant\cdots\geqslant
0$}\end{array}$
The case when $\ell_{j}=0$ for any $j$ will be excluded here since $\ell_{j}$
is a finite sequence. The fractal string is defined completely by its sequence
of lengths so it can be denoted
(106)
$\begin{array}[]{ll}\mathcal{L}&=\\{\ell_{j}\\}_{j=1}^{\infty}\end{array}$
The boundary of $\mathcal{L}$ in $\mathbbm{R}$, denoted by
$\partial\mathcal{L}\subset\Omega$, is a totally disconnected bounded perfect
subset which can be represented as a string of finite length, and generally
any compact subset of $\mathbbm{R}$ also has this property. The boundary
$\partial\mathcal{L}$ is said to be perfect since it is closed and each of its
points is a limit point. Since the Cantor-Bendixon lemma states that there
exists a perfect set $P\subset\partial\mathcal{L}$ such that
$\partial\mathcal{L}-P$ is a most countable, we can define $\mathcal{L}$ as
the complenent of $\partial\mathcal{L}$ in its closed convex hull. The
connected components of the bounded open set
$\mathcal{L}\backslash\partial\mathcal{L}$ are the intervals $I_{j}$. [25,
1.2][32, 2.2 Ex17][23, 3.1][30][21][20][15][22][11][19][29]
#### 2.1.1. The Minkowski Dimension $D_{\mathcal{L}}$ and Content
$\mathcal{M}_{\mathcal{L}}$
The Minkowski dimension $D_{\mathcal{L}}\in[0,1]$, also known as the box
dimension, is maximum value of $V(\varepsilon)$
(107) $\begin{array}[]{ll}D_{\mathcal{L}}&=\inf\\{\alpha\geqslant
0:V(\varepsilon)=O(\varepsilon^{1-\alpha})\operatorname{as}\varepsilon\rightarrow
0^{+}\\}=\zeta_{\mathcal{L}}(1)=\sum_{n=1}^{\infty}\ell_{j}\end{array}$
where $V(\varepsilon)$ is the volume of the inner tubular neighborhoods of
$\partial\mathcal{L}$ with radius $\varepsilon$
(108)
$\begin{array}[]{ll}V_{\mathcal{L}}(\varepsilon)&\left.=|x\in\mathcal{L}:d(x,\partial\mathcal{L})<\varepsilon\right|\\\
&=\sum_{j}^{\ell_{j}\geqslant
2\varepsilon}2\varepsilon+\sum_{j}^{\ell_{j}<2\varepsilon}\ell_{j}\\\
&=2\varepsilon
N_{\mathcal{L}}\left(\frac{1}{2\varepsilon}\right)+\sum_{j}^{\ell_{j}<2\varepsilon}\ell_{j}\end{array}$
and $N_{\mathcal{L}}(x)$ is the geometric counting function which is the
number of components with their reciprocal length being less than or equal to
$x$.
(109) $\begin{array}[]{ll}N_{\mathcal{L}}(x)&=\\#\\{j\geqslant
1:\ell_{j}^{-1}\leqslant x\\}\\\ &=\sum^{\ell_{j}^{-1}\leqslant
x}_{\text{\scriptsize{$\begin{array}[]{l}j\geqslant
1\end{array}$}}}1\end{array}$
The Minkowski content of $\mathcal{L}$ is then defined as
(110) $\begin{array}[]{ll}\mathcal{M}_{\mathcal{L}}&=\lim_{e\rightarrow
0^{+}}\frac{V_{\mathcal{L}}(\varepsilon)}{\varepsilon^{1-D_{\mathcal{L}}}}\\\
&=\frac{C_{\mathcal{L}}2^{1-D_{\mathcal{L}}}}{1-D_{\mathcal{L}}}\\\
&=\frac{\operatorname{Res}(\zeta_{\mathcal{L}}(s);D_{\mathcal{L}})2^{1-D_{\mathcal{L}}}}{D_{\mathcal{L}}(1-D_{\mathcal{L}})}\end{array}$
where $C_{\mathcal{L}}$ is the constant
(111)
$\begin{array}[]{ll}C_{\mathcal{L}}=&\lim_{x\rightarrow\infty}\frac{N_{\mathcal{L}}(x)}{x^{D_{\mathcal{L}}}}\end{array}$
If $\mathcal{M}_{\mathcal{L}}\in(0,\infty)$ exists then $\mathcal{L}$ is said
to be Minkowski measurable which necessarily means that the geometry of
$\mathcal{L}$ does not oscillate and vice versa. [27, 1] [3][24][26, 6.2]
#### 2.1.2. The Geometric Zeta Function $\zeta_{\mathcal{L}}(s)$
The geometric Zeta function $\zeta_{\mathcal{L}}(s)$ of $\mathcal{L}$ is the
Dirichlet series
(112)
$\begin{array}[]{ll}\zeta_{\mathcal{L}}(s)&=\sum_{n=1}^{\infty}\ell_{j}^{s}\\\
&=s\int_{0}^{\infty}N_{\mathcal{L}_{w}}(x)x^{-s-1}\mathrm{d}x\end{array}$
which is holomorphic for $\operatorname{Re}(s)>D_{\mathcal{L}}$. If
$\mathcal{L}$ is Minkowski measurable then $0<D_{\mathcal{L}}<1$ is the simple
unique pole of $\zeta_{\mathcal{L}}(s)$ on the vertical line
$\operatorname{Re}(s)=D_{\mathcal{L}}$. Assuming $\zeta_{\mathcal{L}}(s)$ has
a meromorphic extension to a neighboorhood of $D_{\mathcal{L}}$ then
$\zeta_{\mathcal{L}}(s)$ has a simple pole at
$\zeta_{\mathcal{L}}(D_{\mathcal{L}})$ if
(113)
$N_{\mathcal{L}}(s)=O(s^{D_{\mathcal{L}}})\operatorname{as}s\rightarrow\infty$
or if the volume of the tubular neighborhoods satisfies
(114)
$V_{\mathcal{L}}(\varepsilon)=O(\varepsilon^{1-D_{\mathcal{L}}})\operatorname{as}\varepsilon\rightarrow
0^{+}$
It can be possible that the residue of $\zeta_{\mathcal{L}}(s)$ at
$s=D_{\mathcal{L}}$ is positive and finite
(115) $0<\lim_{s\rightarrow
D_{\mathcal{L}}}(s-D_{\mathcal{L}})\zeta_{\mathcal{L}}(s)<\infty$
even if $N_{\mathcal{L}}(s)$ is not of order $s^{D_{\mathcal{L}}}$ as
$s\rightarrow\infty$ and $V_{\mathcal{L}}(\varepsilon)$ is not of order
$\varepsilon^{1-D_{\mathcal{L}}}$, however this does not contradict the
Minkowski measurability of $\mathcal{L}$.
#### 2.1.3. Complex Dimensions, Screens and Windows
The set of visible complex dimensions of $\mathcal{L}$, denoted by
$\mathcal{D}_{\mathcal{L}}(W)$, is a discrete subset of $\mathbbm{C}$
consisting of the poles of $\\{\zeta_{\mathcal{L}}(s):s\in W\\}$.
(116) $\begin{array}[]{ll}\mathcal{D}_{\mathcal{L}}(W)&=\\{w\in
W:\zeta_{\mathcal{L}}(w)\operatorname{is}a\operatorname{pole}\\}\end{array}$
When $W$ is the entire complex plane then the set
$\mathcal{D}_{\mathcal{L}}(\mathbbm{C})=\mathcal{D}_{\mathcal{L}}$ is simply
called the set of complex dimensions of $\mathcal{L}$. The presence of
oscillations in $V(\varepsilon)$ implies the presence of imaginary complex
dimensions with $\operatorname{Re}(\cdot)=D_{\mathcal{L}}$ and vice versa.
More generally, the complex dimensions of a fractal string $\mathcal{L}$
describe its geometric and spectral oscillations.
#### 2.1.4. Frequencies of Fractal Strings and Spectral Zeta Functions
The eigenvalues $\lambda_{n}$ of the Dirichlet Laplacian $\Delta
u(x)=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}u(x)$ on a bounded open set
$\Omega\subset\mathbbm{R}$ correspond to the normalized frequencies
$f_{n}=\frac{\sqrt{\lambda_{n}}}{\pi}$ of a fractal string. The frequencies of
the unit interval are the natural numbers $n\in\mathbbm{N}^{\ast}$ and the
frequencies of an interval of length $\ell$ are $n\ell^{-1}$. The frequencies
of $\mathcal{L}$ are the numbers
(117) $\begin{array}[]{l}f_{k,j}=k\ell_{j}^{-1}\forall\text{
$k,j\in\mathbbm{N}^{\ast}$}\end{array}$
The spectral counting function $N_{v\mathcal{L}}(x)$ counts the frequencies of
$\mathcal{L}$ with multiplicity
(118)
$\begin{array}[]{ll}N_{v\mathcal{L}}(x)&=\sum_{j=1}^{\infty}N_{\mathcal{L}}\left(\frac{x}{j}\right)\\\
&=\sum_{j=1}^{\infty}\left\lfloor x\ell_{j}\right\rfloor\end{array}$
The spectral zeta function $\zeta_{\upsilon\mathcal{L}}(s)$ of $\mathcal{L}$
is connected to the Riemann zeta function (LABEL:zeta) by
(119)
$\begin{array}[]{ll}\zeta_{\upsilon\mathcal{L}}(s)&=\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}k^{-s}\ell^{s}_{j}\\\
&=\zeta(s)\sum_{n=1}^{\infty}\ell_{j}^{s}\\\
&=\zeta(s)\zeta_{\mathcal{L}}(s)\end{array}$
[27, 1.1][25, 1.2.1]
#### 2.1.5. Generalized Fractal Strings and Dirichlet Integrals
A generalized fractal string is a positive or complex or local measure
$\eta(x)$ on $(0,\infty)$ such that
(120) $\int_{0}^{x_{0}}\eta(x)\mathrm{d}x=0$
for some $x_{0}>0$. A local positive measure is a standard positive Borel
measure $\eta(J)$ on $(0,\infty)$ where $J$ is the set of all bounded
subintervals of $(0,\infty)$ in which case $\eta(x)=|\eta(x)|$. More
generally, a meausre $\eta(x)$ is a local complex measure if $\eta(A)$ is
well-defined for any subset $A\subset[a,b]$ where $[a,b]\subset[0,\infty]$ is
a bounded subset of the positive half-line $(0,\infty)$ and the restriction of
$\eta$ to the Borel subsets of $[a,b]$ is a complex measure on $[a,b]$ in the
traditional sense. The geometric counting function of $\eta(x)$ is defined as
(121)
$\begin{array}[]{ll}N_{\eta}(x)&=\int_{0}^{x}\eta(x)\mathrm{d}x\end{array}$
The dimension $D_{\eta}$ is the abscissa of conergence of the Dirichlet
integral
(122)
$\zeta_{|\eta|}(\sigma)=\int_{0}^{\infty}x^{-\sigma}|\eta(x)|\mathrm{d}x$
In other terms, it is the smallest real positive $\sigma$ such that the
improper Riemann-Lebesgue converges to a finite value. The geometric zeta
function is defined as the Mellin transform
(123)
$\begin{array}[]{ll}\zeta_{\eta}(s)&=\end{array}\int_{0}^{\infty}x^{-s}\eta(x)\mathrm{d}x$
where $\operatorname{Re}(s)>D_{\eta}$.
### 2.2. Fractal Membranes and Spectral Partitions
#### 2.2.1. Complex Dimensions of Dynamical Zeta Functions
The fractal membrane $\mathcal{T}_{L}$ associated with $\mathcal{L}$ is the
adelic product
(124) $\mathcal{T}_{L}=\coprod_{j=1}^{\infty}\mathcal{T}_{j}$
where each $\mathcal{T}_{j}$ is an interval $I_{j}$ of length
$\log(\ell_{j}^{-1})^{-1}$. To each $\mathcal{T}_{j}$ is associated a Hilbert
space $\mathcal{H}_{j}=L^{2}(I_{j})$ of square integrable functions on
$I_{j}$. The spectral partition function $Z_{\mathcal{L}}(s)$ of $\mathcal{L}$
is a Euler product expansion which has no zeros or poles in
$\operatorname{Re}(s)>D_{M}(\mathcal{L})$.
(125)
$\begin{array}[]{ll}Z_{\mathcal{L}}(s)&=\prod_{j=1}^{\infty}\frac{1}{1-\ell_{j}^{s}}\\\
&=\prod_{j=1}^{\infty}Z_{\mathcal{L}_{j}}(s)\end{array}$
where $D_{M}(\mathcal{L})$ is the Minkowski dimension of $\mathcal{L}$ and
$Z_{\mathcal{L}_{j}}(s)=\frac{1}{1-\ell_{j}^{s}}$ is the $j$-th Euler factor,
the partition function of the $j$-th component of the fractal membrane. [23,
3.2.2]
#### 2.2.2. Dynamical Zeta Functions of Fractal Membranes
The dynamical zeta function of a fractal membrane $\mathcal{L}$ is the
negative of the logarithmic derivative of the Zeta function associated with
$\mathcal{L}$.
(126)
$\begin{array}[]{ll}Z_{\mathcal{L}}(s)&=-\frac{\mathrm{d}}{\mathrm{d}s}\ln(\zeta_{\mathcal{L}}(s))\\\
&=-\frac{\frac{\mathrm{d}}{\mathrm{d}s}\zeta_{\mathcal{L}}(s)}{\zeta_{\mathcal{L}}(s)}\end{array}$
## 3\. Special Functions, Definitions, and Conventions
### 3.1. Special Functions
#### 3.1.1. The Interval Indicator (Characteristic) Function $\chi(x,I)$
The (left-open, right-closed) interval indicator function is $\chi(x,I)$ where
$I=(a,b]$
(127) $\begin{array}[]{ll}\chi(x,I)&=\left\\{\begin{array}[]{ll}1&x\in I\\\
0&x\not\in I\end{array}\right.\\\ &=\left\\{\begin{array}[]{ll}1&a<x\leqslant
b\\\ 0&\operatorname{otherwise}\end{array}\right.\\\
&=\theta(x-a)-\theta(x-a)\theta(x-b)\end{array}\text{}$
and $\theta$ is the Heaviside unit step function, the derivative of which is
the Dirac delta function $\delta$
(128) $\begin{array}[]{ll}\text{$\int\delta(x)\mathrm{d}x$}&=\theta(x)\\\
\theta(x)&=\left\\{\begin{array}[]{ll}0&x<0\\\ 1&x\geqslant
0\end{array}\right.\end{array}$
The discontinous point of $\theta(x)$ has the limiting values
(129) $\begin{array}[]{ll}\lim_{x\rightarrow 0^{-}}\theta(x)&=0\\\
\lim_{x\rightarrow 0^{+}}\theta(x)&=1\end{array}$
thus the values of $\chi(x,(a,b))$ on the boundary can be chosen according to
which side the limit is regarded as being approached from.
(130) $\begin{array}[]{ll}\lim_{x\rightarrow
a^{-}}\text{$\chi(x,(a,b])$}&=0\\\ \lim_{x\rightarrow
a^{+}}\text{$\chi(x,(a,b])$}&=1-\theta(a-b)\\\ \lim_{x\rightarrow
b^{-}}\text{$\chi(x,(a,b])$}&=\theta(b-a)\\\ \lim_{x\rightarrow
b^{+}}\text{$\chi(x,(a,b])$}&=0\end{array}$
#### 3.1.2. “Harmonic” Intervals
Let the $n$-th harmonic (left-open, right-closed) interval be defined as
(131)
$\begin{array}[]{ll}I^{H}_{n}&=\left(\frac{1}{n+1},\frac{1}{n}\right]\end{array}$
then its characteristic function is
(132)
$\begin{array}[]{lll}\chi(x,I_{n}^{H})&&=\theta\left(x-\frac{1}{n+1}\right)-\theta\left(x-\frac{1}{n}\right)\\\
&&=\theta\left(x-\frac{n}{n+1}\right)-\theta\left(x-n\right)\\\
&&=\left\\{\begin{array}[]{ll}1&\frac{1}{n+1}<x\leqslant\frac{1}{n}\\\
0&\operatorname{otherwise}\end{array}\right.\end{array}$
As can be seen
(133)
$\begin{array}[]{llll}\bigcup_{n=1}^{\infty}I_{n}^{H}&=&\bigcup_{n=1}^{\infty}\left(\frac{1}{n+1},\frac{1}{n}\right]&=(0,1]\\\
\sum_{n=1}^{\infty}\chi(x,I_{n}^{H})&=&\sum_{n=1}^{\infty}\chi\left(x,\left(\frac{1}{n+1},\frac{1}{n}\right]\right)&=\chi(x,(0,1])\end{array}$
The substitution $n\rightarrow\left\lfloor\frac{1}{x}\right\rfloor$ can be
made in (132) where it is seen that
(134) $\begin{array}[]{lll}\text{$\chi\left(x,I^{H}_{\left\lfloor
x^{-1}\right\rfloor}\right)$}&=\theta\left(\frac{x\text{$\left\lfloor
x^{-1}\right\rfloor$}+x-1}{\text{$\left\lfloor
x^{-1}\right\rfloor$}\text{}+1}\right)-\theta\left(\frac{x\text{$\left\lfloor
x^{-1}\right\rfloor$}-1}{\left\lfloor x^{-1}\right\rfloor}\right)=1&\forall
x\in[-1,+1]\end{array}$
#### 3.1.3. The Laplace Transform $L_{a}^{b}[f(x);x\rightarrow s]$
The Laplace transform [2, 1.5] is defined as
(135) $\begin{array}[]{ll}\text{}\text{$L_{a}^{b}[f(x);x\rightarrow
s]$}&=\int_{a}^{b}f(x)e^{-xs}\mathrm{d}x\end{array}$
where the unilateral Laplace transform is over the interval ($a,b)=(0,\infty$)
and the bilateral transform is over ($a,b)=(-\infty,\infty$). When ($a,b$) is
not specified, it is assumed to range over the support of $f(x$) if the
support is an interval. If the support of $f(x$) is not an interval then
($a,b$) must be specified. Applying $L$ to the interval indicator function
(127) gives
(136) $\begin{array}[]{lll}\text{$L_{a}^{b}[\chi(x,(a,b));x\rightarrow
s]$}&=\int_{a}^{b}\chi(x,(a,b))e^{-xs}\mathrm{d}x&\\\
&=\int_{a}^{b}(\theta(x-a)-\theta(x-b)\theta(x-a))e^{-xs}\mathrm{d}x&\\\
&=\frac{e^{bs}-e^{\operatorname{as}}}{se^{bs}e^{as}}&\\\
&=-\frac{(e^{as}-e^{bs})e^{-s(b+a)}}{s}&\\\ &&\end{array}$
The limit at the singular point $s=0$ is
(137) $\begin{array}[]{ll}\lim_{s\rightarrow
0}\text{$L_{a}^{b}[\chi(x,(a,b));x\rightarrow s]$}&=\lim_{s\rightarrow
0}-\frac{(e^{as}-e^{bs})e^{-s(b+a)}}{s}\\\ &=b-a\end{array}$
#### 3.1.4. The Mellin Transform $M_{a}^{b}[f(x);x\rightarrow s]$
The Mellin transform [31, 3.2][33, II.10.8][5, 3.6] is defined as
(138) $\begin{array}[]{ll}\text{$M_{a}^{b}[f(x);x\rightarrow
s]$}&=\int_{a}^{b}f(x)x^{s-1}\mathrm{d}x\end{array}$
where the standard Mellin transform is over the interval ($a,b)=(0,\infty$).
Again, as with the notation for Laplace transform, the integral is over the
support of $f(x$) if the support is an interval and ($a,b$) is not specified,
otherwise ($a,b$) must be specified. Applying $M$ to the interval indicator
function (127) gives
(139) $\begin{array}[]{lll}&M_{a}^{b}[\chi(x,(a,b));x\rightarrow
s]&=\int_{a}^{b}\chi(x,(a,b))x^{s-1}\mathrm{d}x\\\
&&=\int_{a}^{b}(\theta(x-a)-\theta(x-b)\theta(x-a))x^{s-1}\mathrm{d}x\\\
&&=\frac{b^{s}-a^{s}}{s}\end{array}$
The limit at the singular point $s=0$ is
(140) $\begin{array}[]{ll}M_{a}^{b}\left[\chi(x,(a,b));x\rightarrow
0\right]&=M\left[\chi(x,(a,b));x\rightarrow 0\right]\\\ &=\lim_{s\rightarrow
0}M_{a}^{b}[\chi(x,(a,b));x\rightarrow s]\\\ &=\lim_{s\rightarrow
0}\frac{b^{s}-a^{s}}{s}\\\ &=\ln(b)-\ln(a)\end{array}$
The Mellin transform has several identities [31, 3.1.2], including but not
limited to
(141) $\begin{array}[]{ll}M[f(\alpha x);x\rightarrow
s]&=\alpha^{-s}M[f(x);x\rightarrow s]\\\ M[x^{\alpha}f(x);x\rightarrow
s]&=M[f(x);x\rightarrow s+\alpha]\\\ M[f(x^{\alpha});x\rightarrow
s]&=\frac{1}{a}M\left[f(x);x\rightarrow\frac{s}{\alpha}\right]\\\
M[f(x^{-\alpha});x\rightarrow
s]&=\frac{1}{a}M\left[f(x);x\rightarrow-\frac{s}{\alpha}\right]\\\
M[x^{\alpha}f(x^{\mu});x\rightarrow
s]&=\frac{1}{\mu}M\left[f(x);x\rightarrow\frac{s+\alpha}{\mu}\right]\\\
M[x^{\alpha}f(x^{-\mu});x\rightarrow
s]&=\frac{1}{\mu}M\left[f(x);x\rightarrow-\frac{s+\alpha}{\mu}\right]\\\
M[\ln(x)^{n}f(x);x\rightarrow
s]&=\frac{\mathrm{d}^{n}}{\mathrm{d}s^{n}}M\left[f(x);x\rightarrow
s\right]\end{array}$
where $\alpha>0,\mu>0$, and $n\in\mathbbm{N}$. The Mellin transform of the
harmonic interval indicator function (132) is
(142) $\begin{array}[]{lll}M\left[\chi\left(x,I^{H}_{n}\right);x\rightarrow
0\right]&&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\chi\left(x,\left(\frac{1}{n+1},\frac{1}{n}\right)\right)x^{s-1}\mathrm{d}x\\\
&&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\left(\theta\left(t-\frac{1}{n+1}\right)-\theta\left(t-\frac{1}{n}\right)\right)x^{s-1}\mathrm{d}x\\\
&&=\frac{n^{-s}-(n+1)^{-s}}{s}\end{array}$
which has the limit
(143) $\begin{array}[]{ll}M\left[\chi\left(x,I^{H}_{n}\right);x\rightarrow
0\right]&=\lim_{s\rightarrow
0}M\left[\chi\left(x,I^{H}_{n}\right);x\rightarrow s\right]\\\
&=\lim_{s\rightarrow 0}\frac{n^{-s}-(n+1)^{-s}}{s}\\\
&=\ln(n+1)-\ln(n)\end{array}$
The Mellin and bilaterial Laplace transforms are related by the change of
variables $x\rightarrow-\ln(y)$ resulting in the identity [31, 3.1.1]
(144) $\begin{array}[]{lll}M_{0}^{\infty}[f(-\ln(x));x\rightarrow
s]&=L_{-\infty}^{+\infty}[f(y);y\rightarrow s]&\\\
\int_{0}^{\infty}f(-\ln(x))x^{s-1}\mathrm{d}x&=\int_{-\infty}^{-\infty}f(y)e^{-ys}\mathrm{d}y&\end{array}$
#### 3.1.5. The Lambert W Function $W(k,x)$
The Lambert W function [8][34] is the inverse of $xe^{x}$ given by
(145) $\begin{array}[]{ll}W(z)&=\\{x:xe^{x}=z\\}\\\ &=W(0,z)\\\
&=1+(\ln(z)-1)\exp\left(\frac{i}{2\pi}\int_{0}^{\infty}\frac{1}{x+1}\ln\left(\frac{x-i\pi-\ln(x)+\ln(z)}{x+i\pi-\ln(x)+\ln(z)}\right)\mathrm{d}x\right)\\\
&=\sum_{k=1}^{\infty}\frac{(-k)^{k-1}z^{k}}{k!}\end{array}$
where $W(a,z)\forall a\in\mathbbm{Z},z\not\in\\{0,-e^{-1}\\}$ is
(146) $\begin{array}[]{ll}W(a,z)&=1+(2i\pi
a+\ln(z)-1)\exp\left(\frac{i}{2\pi}\int_{0}^{\infty}\frac{1}{x+1}\ln\left(\frac{x+\left(2\hskip
1.75pta-1\right)i\pi-\ln\left(x\right)+\ln\left(z\right)}{x+\left(2\hskip
1.75pta+1\right)i\pi-\ln\left(x\right)+\ln\left(z\right)}\right)\mathrm{d}x\right)\end{array}$
A generaliztion of (145) is solved by
(147)
$\begin{array}[]{ll}\\{x:xb^{x}=z\\}&=\frac{W(\ln(b)z)}{\ln(b)}\end{array}$
The W function satisifes several identities
(148) $\begin{array}[]{lll}W(z)e^{W(z)}&=z&\\\ W(z\ln(z))&=\ln(z)&\forall
z<1\\\ |W(z)|&=W(|z|)&\\\ e^{nW(z)}&=z^{n}W(z)^{-n}&\\\
\ln(W(n,z))&=\ln(z)-W(n,z)+2i\pi n&\\\
W\left(-\frac{\ln(z)}{z}\right)&=-\ln(z)&\forall z\in[0,e]\\\
\frac{W(-\ln(z))}{-\ln(z)}&=z^{z^{z^{z^{.^{.^{.}}}}}}&\end{array}$
where $n\in\mathbbm{Z}$. Some special values are
(149) $\begin{array}[]{lll}W\left(-1,-e^{-1}\right)&=-1&\\\ W(-e^{-1})&=-1&\\\
W(e)&=1&\\\ W(0)&=0&\\\ W(\infty)&=\infty&\\\ W(-\infty)&=\infty+i\pi&\\\
W\left(-\frac{\pi}{2}\right)&=\frac{i\pi}{2}&\\\
W\left(-\ln(\sqrt{2})\right)&=-\ln(2)&\\\
W\left(-1,-\ln(\sqrt{2})\right)&=-2\ln(2)&\end{array}$
We also have the limit
(150) $\begin{array}[]{ll}\lim_{a\rightarrow\pm\infty}\frac{W(a,x)}{a}&=2\pi
i\end{array}$
and differential
(151)
$\begin{array}[]{lll}\frac{\mathrm{d}}{\mathrm{d}z}W(a,f(z))&&=\frac{W(a,f(z))\frac{\mathrm{d}}{\mathrm{d}z}f(z)}{f(z)(1+W(a,f(z)))}\end{array}$
as well as the obvious integral
(152)
$\begin{array}[]{ll}\int_{0}^{1}W\left(-\frac{\ln(x)}{x}\right)\mathrm{d}x&=\int_{0}^{1}-\ln(x)\mathrm{d}x=1\end{array}$
Let us define, for the sake of brevity, the function
(153) $\begin{array}[]{ll}W_{\ln}(z)&=W\left(-1,-\frac{\ln(z)}{z}\right)\\\
&=1+\left(\ln\left(-\frac{\ln(z)}{z}\right)-1-2\pi
i\right)\exp\left(\frac{i}{2\pi}\int_{0}^{\infty}\frac{1}{x+1}\ln\left(\frac{x-3i\pi-\ln\left(x\right)+\ln\left(-\frac{\ln(z)}{z}\right)}{x-i\pi-\ln\left(x\right)+\ln\left(-\frac{\ln(z)}{z}\right)}\right)\mathrm{d}x\right)\end{array}$
Figure 16. $W\left(-\frac{\ln(x)}{x}\right)=-\ln(x)$ and
$W_{\ln}(x)=W\left(-1,-\frac{\ln(x)}{x}\right)$
Then we have the limits
(154) $\begin{array}[]{ll}\lim_{x\rightarrow-\infty}W_{\ln}(x)&=0\\\
\lim_{x\rightarrow+\infty}W_{\ln}(x)&=-\infty\end{array}$
and
(155)
$\operatorname{Im}\left(W_{\ln}(x)\right)=\left\\{\begin{array}[]{ll}-\pi&-\infty<x<0\\\
\ldots&0\leqslant x\leqslant 1\\\ 0&1<x<\infty\end{array}\right.$ (156)
$\begin{array}[]{lll}W_{\ln}(x)&=-\ln(x)&\forall x\not\in[0,e]\end{array}$
The root of $\operatorname{Re}\left(W_{\ln}(x)\right)$ is given by
(157)
$\begin{array}[]{ll}\left\\{x:\operatorname{Re}\left(W_{\ln}(x))=0\right\\}\right.&=\\{-x:\left(x^{2}\right)^{\frac{1}{x}}=e^{3\pi}\\}\\\
&=\frac{2}{3}\pi W\left(\frac{3}{2}\pi\right)\\\ &\cong
0.27441063190284810044\ldots\end{array}$
where the imaginary part of the value at the root of the real part of
$W_{\ln}(z)$ is
(158) $\begin{array}[]{ll}W_{\ln}\left(\frac{2}{3}\pi
W\left(\frac{3}{2}\pi\right)\right)&=W\left(-1,-\frac{\ln\left(\frac{2}{3}\pi
W\left(\frac{3}{2}\pi\right)\right)}{\frac{2}{3}\pi
W\left(\frac{3}{2}\pi\right)}\right)\\\ &=W\left(-1,\frac{3\pi}{2}\right)\\\
&=\frac{3\pi i}{2}\\\ &\cong i4.712388980384689857\ldots\end{array}$
#### 3.1.6. The Lerch Transcendent $\Phi(z,a,v)$
The Lerch Transcendent [14, 1.11] is defined by
(159)
$\begin{array}[]{lll}\Phi(z,a,v)&=\sum_{n=0}^{\infty}\frac{z^{n}}{(v+n)^{a}}&\forall\\{|z|<1\\}\operatorname{or}\\{|z|=1\operatorname{and}\operatorname{Re}(a)>1\\}\end{array}$
The Riemann zeta function is the special case
(160)
$\begin{array}[]{ll}\zeta(s)&=\Phi(1,s,1)=\sum_{n=0}^{\infty}\frac{1}{(1+n)^{s}}\end{array}$
### 3.2. Applications of w(x)
#### 3.2.1. Expansion of $\gamma$
Consider Euler’s constant $\gamma=0.577215664901533\ldots$ (33)
(161) $\begin{array}[]{ll}w^{n}(\gamma)&=a_{n}-b_{n}\gamma\end{array}$
whereupon iteration we see that
(162) $\begin{array}[]{ll}\left(\begin{array}[]{c}n\\\ -a_{n}\\\
-b_{n}\end{array}\right)&\text{$=\left(\begin{array}[]{cccccccccccc}0&1&2&3&4&5&6&7&8&9&10&\ldots\\\
0&1&48&290&581&1163&2327&13964&7492468716&14984937433&1078915495184&\ldots\\\
1&2&84&504&1008&2016&4032&24192&12980362752&25960725504&1869172236288&\ldots\end{array}\right)$}\end{array}$
### 3.3. Conventions and Symbols
Many of these symbols are from [23, p491].
(163) $\begin{array}[]{ll}i&\sqrt{-1}\\\
\mathbbm{R}&\\{x:-\infty<x<\infty\\}\\\
\bar{\mathbbm{R}}&\\{x:-\infty\leqslant x\leqslant\infty\\}\\\
\mathbbm{R}^{+}&\\{x:0\leqslant x<\infty\\}\\\ \mathbbm{R}^{d}&\\{x_{1}\ldots
x_{d}:-\infty<x_{i}<\infty\\}\\\ \mathbbm{C}&\\{x+iy:x,y\in\mathbbm{R}\\}\\\
\mathbbm{Z}&\\{\ldots,-2,-1,0,1,2,\ldots\\}\\\
\mathbbm{N}&\\{0,1,2,3,\ldots.\\}\\\ \mathbbm{N}^{\ast}&\\{1,2,3,\ldots.\\}\\\
\text{$\mathbbm{H}$}&\left\\{0,\frac{1}{n}:n\in\mathbbm{Z}\right\\}\\\
f(x)=O(g(x))&\frac{f(x)}{g(x)}<\infty\\\
f(x)=o(g(x))&\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0\\\ f(x)\asymp
g(x)&\left\\{a\leqslant\frac{f(x)}{g(x)}\leqslant b:\\{a,b\\}>0\right\\}\\\
\\#A&\operatorname{numbers}\operatorname{of}\operatorname{elements}\operatorname{in}\operatorname{the}\operatorname{finite}\operatorname{set}A\\\
|A|_{d}&d\text{-dimensional}\operatorname{Lebesgue}\operatorname{measure}(\operatorname{volume})\operatorname{of}A\subseteq\mathbbm{R}^{d}\\\
\text{$d(x,A)$}&\\{\min(|x-y|):y\in
A\\}\operatorname{Euclidean}\operatorname{distance}\operatorname{between}x\operatorname{and}\operatorname{the}\operatorname{nearest}\operatorname{point}\operatorname{of}A\\\
\exp(x)&\operatorname{exponential}e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\\\
\underset{}{\underset{x=y}{\operatorname{Res}}(f(x))}&\operatorname{complex}\text{residue
of $f(x)$ at $x=y$}\\\ \left\lfloor
x\right\rfloor&\operatorname{floor},\operatorname{the}\operatorname{greatest}\operatorname{integer}\leqslant
x\\\ \\{x\\}&x-\left\lfloor
x\right\rfloor,\operatorname{the}\operatorname{fractional}\operatorname{part}\operatorname{of}x\\\
\text{$\bar{x}$}&\operatorname{complex}\operatorname{conjugate},\operatorname{Re}(x)-\operatorname{Im}(x)\\\
\operatorname{Fix}_{f}^{n}&\text{$n$-th}\operatorname{fixed}\operatorname{point}\operatorname{of}\operatorname{the}\operatorname{map}f(x),\text{$n$-th}\operatorname{solution}\operatorname{to}f(x)=x\\\
p_{k}&\text{$k$-th}\operatorname{prime}\operatorname{number}\\\
\ln_{b}(a)&\frac{\ln(a)}{\ln(b)}\end{array}$
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|
arxiv-papers
| 2012-10-20T19:44:08 |
2024-09-04T02:49:36.882535
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephen Crowley",
"submitter": "Stephen Crowley",
"url": "https://arxiv.org/abs/1210.5652"
}
|
1210.5698
|
# Noise spectrum of a quantum dot-resonator lasing circuit
Jinshuang Jin Institut für Theoretische Festkörperphysik, Karlsruhe Institute
of Technology, 76128 Karlsruhe, Germany Department of Physics, Hangzhou
Normal University, Hangzhou 310036, China Michael Marthaler Institut für
Theoretische Festkörperphysik, Karlsruhe Institute of Technology, 76128
Karlsruhe, Germany Pei-Qing Jin Institut für Theoretische Festkörperphysik,
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Institute of
Logistics Engineering, Shanghai Maritime University, Shanghai 201306, China
Dmitry Golubev Institute für Nanotechnologie, Karlsruhe Institute of
Technology, 76021 Karlsruhe, Germany Gerd Schön Institut für Theoretische
Festkörperphysik, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Institute für Nanotechnologie, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany DFG Center for Functional Nanostructures, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany
###### Abstract
Single-electron tunneling processes through a double quantum dot can induce a
lasing state in an electromagnetic resonator which is coupled coherently to
the dot system. Here we study the noise properties of the transport current in
the lasing regime, i.e., both the zero-frequency shot noise as well as the
noise spectrum. The former shows a remarkable super-Poissonian behavior when
the system approaches the lasing transition, but sub-Poissonian behavior deep
in the lasing state. The noise spectrum contains information about the
coherent dynamics of the coupled dot-resonator system. It shows pronounced
structures at frequencies matching that of the resonator due to the excitation
of photons. For strong interdot Coulomb interaction we observe asymmetries in
the auto-correlation noise spectra of the left and right junctions, which we
trace back to asymmetries in the incoherent tunneling channels.
## I Introduction
A variety of fundamental quantum effects and phenomena characteristic for
cavity quantum electrodynamics (QED) have been demonstrated in superconducting
circuit QED. Wal04162 ; Chi04159 ; Bla04062320 ; Ili03097906 The equivalent
of single-atom lasing has been observed, with frequencies in the few GHz
range, when a single Josephson charge qubit is strongly coupled to a
superconducting transmission line resonator. Ast07588 ; Hau08037003 This
progress stimulated the study of a different circuit QED setup where the
superconducting qubit is replaced by a semiconductor double quantum dot with
discrete charge states. Incoherent single-electron tunneling through the
double dot sandwiched between two electrodes can lead to a population
inversion in the dot levels and, as a consequence, induce a lasing state in
the resonator. Chi04042302 ; Jin11035322 The potential advantages of quantum
dots are their high tunability, both of the couplings and energy levels.
Fuj98932 ; Pet052180 ; Now071430 In addition, larger frequencies are
accessible since the restriction to frequencies below the superconducting gap
is no longer needed. Experimental progress has been made recently towards
coupling semiconductor quantum dots to a GHz-frequency high quality
transmission line resonator. Fre12046807 ; Fre11262105 ; Del11256804
The double quantum dot – resonator circuit lasing setup differs from the more
familiar interband-transition semiconductor laser, Ben994756 ; Str11607 where
the cavity mode is coupled to the lowest quantum dot interband transition, and
which is driven by carrier injection in a p-n-junction or via optical pumping.
Since the circuit, considered here, is driven by single-electron tunneling,
the lasing state correlates with electron transport properties. This fact
allows probing the former via a current measurement. Jin11035322 Further
information about the system is contained in the current fluctuations. Due to
the charge discreteness the noise is shot noise. which has been studied
extensively. Her927061 ; Din9715521 ; Bla001 ; Naz03 ; Kie07206602 ;
Gus06076605 ; Fuj061634 For the double dot – resonator lasing circuit it is
therefore important to compare the electron shot noise with the fluctuations
of the photons in the resonator.
Although more difficult, experimental progress has also been made towards
measuring the finite-frequency noise spectrum of electron transport. Ubb121
It contains information about the full dynamics of the system, including the
relevant time scales that characterize the transport processes. In the present
work, we therefore investigate the frequency-dependent noise spectrum of the
transport current through the system in and near the lasing regime. It shows
pronounced characteristic signals at frequencies close to the eigen-Rabi
frequency of the coupled system or matching that of the resonator.
The present paper is organized as follows. In Sec. II, we introduce the model
of a quantum dot-resonator lasing circuit and the methods. We extend the work
of Ref. Jin11035322, , where strong interdot Coulomb interaction was assumed,
to arbitrary strength interaction. Van031 The method used for the calculation
of the noise spectrum is based on a master equation combined with the quantum
regression theorem. In Sec. III, the stationary properties of the resonator,
the average current, and the zero-frequency noise are studied. The finite-
frequency noise spectrum is evaluated in Sec. IV in the low- and high-
frequency regimes, both for strong and weak interdot Coulomb interaction. We
find characteristic symmetric and asymmetric features in the frequency-
dependent noise spectrum. We conclude with a discussion in Sec. V.
## II Methodology
### II.1 Model
We consider the electron transport setup schematically shown in Fig. 1, where
electrons tunnel through a semiconducting double quantum dot coupled to a
high-$Q$ electromagnetic resonator such as a superconducting transmission
line. The Hamiltonian includes the interacting dot-resonator system,
$H_{S}=H_{\rm d}+H_{\rm r}+H_{\rm I}$, which is responsible for the coherent
dynamics. The double dot is described by
$\displaystyle H_{\rm d}$
$\displaystyle=\sum_{j}\varepsilon_{j}d^{\dagger}_{j}d_{j}+Ud^{\dagger}_{l}d_{l}d^{\dagger}_{r}d_{r}+\frac{t_{c}}{2}\big{(}d^{\dagger}_{l}d_{r}+d^{\dagger}_{r}d_{l}\big{)},$
(1)
with $d_{j}^{\dagger}$ being the electron creation operators for the two
levels in the dots $j$ ($j=l,r$) with energies $\varepsilon_{j}$, separated by
$\varepsilon=\varepsilon_{l}-\varepsilon_{r}$, which are coupled coherently
with strength $t_{c}$. Both $\varepsilon_{j}$ and $t_{c}$ can be tuned by gate
voltages. Oos98873 ; Har10195310 ; Hay03226804 ; Pet052180 ; Now071430 The
interdot Coulomb interaction is denoted by $U$. The transmission line can be
modeled as a harmonic oscillator, $H_{\rm r}=\omega_{r}a^{\dagger}a$, with
frequency $\omega_{\rm r}$ and $a^{\dagger}$ denoting the creation operator of
photons in the resonator. The dipole moment induces an interaction between the
resonator and the double dot, $H_{\rm I}$, which will be specified below.
We further account for electron tunneling between the dots and electrodes,
$H_{\rm
t}=\sum_{k}(V_{Lk}c^{\dagger}_{Lk}d_{l}+V_{Rk}c^{\dagger}_{Rk}d_{r}+\mbox{H.c.})$,
with tunneling amplitudes $V_{\alpha k}$ (with $\alpha=L,R$). The electrodes
with $H_{\rm b}=\sum_{\alpha k}\varepsilon_{\alpha k}c^{\dagger}_{\alpha
k}c_{\alpha k}$ act as baths. Here $c_{\alpha k}^{\dagger}$ is the electron
creation operator for an electron state in the electrode $\alpha$. Below, the
tunneling between the electrodes and the dots is assumed to be an incoherent
process.
The double dot can be biased such that at most one electron occupies each dot.
The two charge states $|L\rangle$ and $|R\rangle$ serve as basis of a charge
qubit. Li01012302 ; Gur9715215 In the present work, we consider two limits,
(i) strong $U$ and (ii) weak $U$, respectively. In case (i) transport through
the double dots involves only one extra third state, namely the empty-dot
$|0\rangle$, while in case (ii) two extra states, $|0\rangle$ and the double
occupation state $|2\rangle\equiv|LR\rangle$, are involved in the transport.
In both limits the dipole interaction between the resonator and the double dot
is, $H_{\rm I}=\hbar g_{0}(a^{\dagger}+a)\tau_{z}$, with Pauli matrices acting
in the space of the two charge states, $\tau_{z}=|L\rangle\langle
L|-|R\rangle\langle R|$.
Figure 1: (Color online) A double quantum dot–resonator lasing circuit. The
dot is placed at a maximum of the electric field of the transmission line in
order to maximize the dipole interaction. The population inversion in the dot
levels, leading to the lasing state, is created by incoherent electron
tunneling through the dots, driven by the bias voltage, which is assumed to be
high, $eV=\mu_{L}-\mu_{R}\gg\omega_{r}$.
In the eigenbasis of the double dot and within rotating wave approximation,
the Hamiltonian of the coupled dot-resonator system, for strong interdot
Coulomb interaction, can be reduced to
$\displaystyle H_{S}=\frac{\hbar\omega_{0}}{2}\sigma_{z}+\hbar\omega_{\rm
r}a^{\dagger}a+\hbar g(a^{\dagger}\sigma_{-}+a\sigma_{+}),$ (2)
while for weak interdot interaction an extra term $U|2\rangle\langle 2|$ is to
be included. In the restricted space of states we have $d_{l}=|0\rangle\langle
L|+|R\rangle\langle 2|$ and $d_{r}=|0\rangle\langle R|-|L\rangle\langle 2|$,
and the Pauli matrix operates in the eigenbasis, i.e.,
$\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$ with
$\displaystyle|e\rangle$ $\displaystyle=$
$\displaystyle\cos\left(\theta/2\right)|L\rangle+\sin\left(\theta/2\right)|R\rangle,$
$\displaystyle|g\rangle$ $\displaystyle=$
$\displaystyle\sin\left(\theta/2\right)|L\rangle-\cos\left(\theta/2\right)|R\rangle.$
(3)
Here, we fix the zero energy level by $\varepsilon_{l}+\varepsilon_{r}=0$. The
angle $\theta=\arctan(t_{c}/\varepsilon)$ characterizes the mixture of the
pure charge states, the coupling strength is $g=g_{0}\sin\theta$, and
$\omega_{0}=\sqrt{\varepsilon^{2}+t^{2}_{c}}/\hbar$ denotes the level spacing
of the two eigenstates. It can be tuned via gate voltages, which allows
control of the detuning $\Delta=\omega_{0}-\omega_{\rm r}$ from the resonator
frequency.
### II.2 Master equation
The dynamics of the coupled dot-resonator system, which is assumed to be
weakly coupled to the electron reservoirs with smooth spectral density, can be
described by a master equation for the reduced density matrix $\rho$ in the
Born-Markov approximation. Gar04 ; Car02 Throughout this paper we consider
low temperatures, $T=0$, with vanishing thermal photon number and excitation
rates. Consequently, the master equation is
$\displaystyle\dot{\rho}$
$\displaystyle=-\frac{i}{\hbar}\left[H_{S},\rho\right]+\mathcal{L}_{\rm
L}\,\rho+\mathcal{L}_{\rm R}\,\rho+\mathcal{L}_{\rm
r}\,\rho\equiv\mathcal{L}_{\rm tot}\,\rho,$ (4a) where the dissipative
dynamics is described by Lindblad operators of the form ${\cal
L}_{i}\rho=\frac{\Gamma_{i}}{2}\left(2L_{i}\rho
L_{i}^{{\dagger}}-L_{i}^{{\dagger}}L_{i}\rho-\rho
L_{i}^{{\dagger}}L_{i}\right).$ (4b)
The first two terms $\mathcal{L}_{\rm L/R}$ account for the incoherent
sequential tunneling between the electrodes and the dots with
$\Gamma_{\alpha}(\omega)=2\pi\sum_{k}|V_{\alpha
k}|^{2}\delta(\omega-\varepsilon_{\alpha k})\equiv\Gamma_{\alpha}$. For the
assumed high voltage and low temperature, i.e., in the absence of reverse
tunneling processes, we have $L_{\rm L}=d^{\dagger}_{l}$ and $L_{\rm R}=d_{r}$
with tunneling rates $\Gamma_{L}$ and $\Gamma_{R}$, respectively. For the
oscillator we take the standard decay term $L_{\rm r}=a$ with rate
$\Gamma_{\rm r}=\kappa$. Here, we ignore other dissipative effects, such as
relaxation and dephasing of the two charge states, which were studied in Ref.
Jin11035322, , since such effects only weakly affect the main points we wish
to study.
From the definition $I_{\alpha}(t)\equiv-e\frac{d\langle
n_{\alpha}(t)\rangle}{dt}$ with $n_{\alpha}=\sum_{k}c^{\dagger}_{\alpha
k}c_{\alpha k}$, it is straightforward to obtain the transport current from
the electrodes to the dots, Lam08214302 ; Li05205304 $I_{\alpha}(t)={\rm
Tr}\big{[}\hat{I}_{\rm\alpha}\rho(t)\big{]}$, with current operators given by
$\displaystyle\hat{I}_{\rm L}\rho(t)$
$\displaystyle=\frac{e}{\hbar}\Gamma_{\rm L}d^{\dagger}_{l}\rho(t)d_{l},$ (5a)
$\displaystyle\hat{I}_{R}\rho(t)$ $\displaystyle=-\frac{e}{\hbar}\Gamma_{\rm
R}d_{r}\rho(t)d^{\dagger}_{r},$ (5b)
In the stationary limit, $t\rightarrow\infty$, the average current satisfies
$I=\frac{1}{2}(I_{L}-I_{R})=I_{L}=-I_{R}$, consistent with charge
conservation.
### II.3 Current noise spectrum
We consider the symmetrized current noise spectrum
$\displaystyle S(\omega)$ $\displaystyle={\cal
F}\langle\\{\delta\hat{I}(t),\delta\hat{I}(0)\\}\rangle$
$\displaystyle\equiv\int^{\infty}_{-\infty}dt\,e^{i\omega
t}\langle\\{\delta\hat{I}(t),\delta\hat{I}(0)\\}\rangle$
$\displaystyle=2\,{\rm
Re}\big{\\{}\widetilde{G}_{I}(\omega)+\widetilde{G}_{I}(-\omega)\big{\\}},$
(6)
where $\delta\hat{I}(t)=\hat{I}(t)-I$ and
$\widetilde{G}_{I}(\pm\omega)=\int^{\infty}_{0}dt\,e^{\pm i\omega t}G_{I}(t)$
with $G_{I}(t)=\langle\delta\hat{I}(t)\delta\hat{I}(0)\rangle$. In Born-Markov
approximation, the current noise spectrum can be calculated via the widely
used MacDonald’s formula Mac62 or the quantum regression theorem. Gar04
Since we already know the current operators, as expressed in Eq. (5), it is
more convenient to calculate the current correlation function via the quantum
regression theorem,
$G_{I}(t)={\rm Tr}\big{[}\hat{I}e^{{\cal L}_{\rm tot}t}\hat{I}\rho^{\rm
st}\big{]}-I^{2},$ (7)
where $\rho^{\rm st}$ denotes the steady-state density matrix.
According to the Ramo–Shockley theorem, the measured quantity in most
experiments Bla001 is the total circuit current $I(t)=aI_{L}(t)-bI_{R}(t)$,
with coefficients, $a+b=1$, depending on the symmetry of the transport setup
(e.g., the junction capacitances). The circuit noise spectrum is thus composed
of three components:
$S(\omega)=a^{2}S_{L}(\omega)+b^{2}S_{R}(\omega)-2abS_{LR}(\omega)$, Bla001 ;
Eng04136602 where $S_{\rm\alpha}(\omega)={\cal
F}\langle\\{\delta\hat{I}_{\alpha}(t),\delta\hat{I}_{\alpha}(0)\\}\rangle$ are
the auto-correlation noise spectra of the current from lead-$\alpha$, and
$S_{\rm LR}(\omega)=({\cal
F}\langle\\{\delta\hat{I}_{L}(t),\delta\hat{I}_{R}(0)\\}\rangle+{\cal
F}\langle\\{\delta\hat{I}_{L}(t),\delta\hat{I}_{R}(0)\\}\rangle$)/2 is the
current cross-correlation noise spectrum between different leads.
Alternatively, in view of the charge conservation, i.e., $I_{\rm L}=I_{\rm
R}+dQ/dt$, where $Q$ is the charge on the central dots, the circuit noise
spectrum can be expressed as Moz02161313 ; Agu04206601 ; Luo07085325
$S(\omega)=aS_{\rm L}(\omega)+bS_{\rm R}(\omega)-ab\,S_{\rm C}(\omega)$ with
$S_{\rm C}(\omega)\equiv{\cal
F}\langle\\{\delta\dot{Q}(t),\delta\dot{Q}(0)\\}\rangle=2S_{\rm
LR}(\omega)+S_{\rm L}(\omega)+S_{\rm R}(\omega)$. Jin11053704 Thus, from the
behavior of the auto-correlation and cross-correlation noise spectra, which
will be studied in the following, we can fully understand the circuit noise
spectrum even including the charge fluctuation spectrum in the central dots,
$S_{C}(\omega)$. At zero frequency, we have
$S(0)=S_{L}(0)=S_{R}(0)=-S_{LR}(0)$ and $S_{\rm C}(0)=0$ due to current
conservation in the steady-state.
Figure 2: (Color online) Incoherent electron tunneling induces transitions
between different states in the dot. The upper panels, (a) and (b), and the
lower ones, (c) and (d), describe the incoherent transitions in the dot-basis
and eigen-basis (including the interaction with the resonator), respectively.
Panels (a) and (c) corresponde to asymmetric transition channels for strong
interdot Coulomb interaction, and (b) and (d) to symmetric transition channels
for weak interdot Coulomb interaction. Furthermore, we have
$\Gamma^{+}_{\alpha}=\Gamma_{\alpha}\cos^{2}(\theta/2)$ and
$\Gamma^{-}_{\alpha}=\Gamma_{\alpha}\sin^{2}(\theta/2)$ with $\alpha={\rm
L,R}$.
## III Stationary properties
Let us first recall the parameter regime for which, according to Ref.
Jin11035322, , lasing can be induced for the present setup. Fre12046807 ;
Fre11262105 ; Del11256804 We consider the level spacing in the dots
comparable to the resonator frequency $\omega_{r}$ in the range of few GHZ,
and a high quality resonator with $Q$ factor assumed to be $5\times 10^{4}$,
corresponding to a decay rate $\kappa=2\times 10^{-5}\omega_{r}$. The coupling
of dot and resonator, chosen as $g_{0}=10^{-3}\omega_{r}$, is strong enough
compared to the photon decay rate in the resonator, and we assume weak
incoherent tunneling with $\Gamma_{\rm L}=\Gamma_{\rm
R}=\Gamma=10^{-3}\omega_{r}$ to be a few MHz throughout of the paper, unless
otherwise stated.
Figure 3: (Color online) (a) Average photon number $\langle n\rangle$, (b)
Fano factor $F_{n}=(\langle n^{2}\rangle-\langle n\rangle^{2})/\langle
n\rangle^{2}$ of photons in the resonator, (c) average current $I$, and (d)
Fano factor of the current $F_{I}=S(0)/2I$, as function of detuning for weak
Coulomb interaction (red solid line) and strong interaction (black dashed
line). Throughout this paper we choose the tunneling rate
$\Gamma=10^{-3}\omega_{r}$ and interdot coupling strength
$t_{c}=0.3\omega_{r}$.
A crucial prerequisite for lasing is a pumping mechanism, Mu925944 ; Ast07588
involving a third state, which creates a population inversion in the two-level
system. In Ref. Jin11035322, , the empty state $|0\rangle$ in the double dot
was considered as the single third state under the assumption of strong
charging energy, $\varepsilon_{j}+U>\mu_{L}>\varepsilon_{j}>\mu_{R}$. This
limit, which we call case (i), is sketched in Fig. 2 (a). On the other hand,
the interdot Coulomb interaction may also be weaker compared to the level
spacing of the charge states. In the tunneling regime we have
$\mu_{L}>\varepsilon_{j},\varepsilon_{j}+U>\mu_{R}$. This limit, called case
(ii), where two extra states are involved in the incoherent tunneling, is
illustrated in Fig. 2 (b). The question arises, which case is better for
lasing.
Let us first consider the key factor for lasing, i.e., the population
inversion defined by $\tau_{0}=\left(\rho^{\rm st}_{e}-\rho^{\rm
st}_{g}\right)/\left(\rho^{\rm st}_{e}+\rho^{\rm st}_{g}\right)$, with
$\rho^{\rm st}_{i}=\sum_{n}\langle i,n|\rho^{\rm st}|i,n\rangle$ being the
stationary population of the state of the dots ($i=e,g$). Explicitly, we find
$\displaystyle\tau_{0}$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{(\Gamma^{2}_{\rm
R}/\omega^{2}_{0}+4)\cos\theta}{\Gamma^{2}_{\rm R}/\omega^{2}_{0}+3+\cos
2\theta};&\qquad\quad\text{ for case (i)}\\\ \\\
\frac{(\Gamma^{2}_{0}/\omega^{2}_{0}+4)\cos\theta}{\Gamma^{2}_{0}/\omega^{2}_{0}+3+\cos
2\theta};&\qquad\quad\text{ for case (ii)}\end{array}\right.,$ (11)
with $\Gamma_{0}=\Gamma_{\rm L}+\Gamma_{\rm R}$. The population inversion does
not depend on $\Gamma_{\rm L}$ for case (i). But it depends on both tunneling
rates for case (ii), suggesting that in this case the population inversion is
driven by transitions from $|R\rangle$ to both extra states $|0\rangle$ and
$|2\rangle$. See Fig. 2 (a) and (b) for case (i) and (ii), respectively.
Although in general, an additional incoherent tunneling channel reduces the
population inversion slightly, for the parameters studied in the present work,
i.e., $\Gamma\ll\omega_{r}$, it approaches the same value for both cases (i)
and (ii), $\tau_{0}\approx 4\cos\theta/(3+\cos 2\theta)$, which reaches a
maximum, $\tau_{0}\rightarrow 1$, for $\theta\rightarrow\pi/2$. To balance the
effective dot-resonator coupling $g=g_{0}\sin\theta$ and the population
inversion $\tau_{0}$, following the consideration in Ref. Jin11035322, , we
set the interdot coupling strength $t_{c}=0.3\omega_{r}$ throughout this work.
The properties of the resonator can be characterized by the average number of
photons $\langle n\rangle$ and the Fano factor $F_{n}\equiv\left(\langle
n^{2}\rangle-\langle n\rangle^{2}\right)/\langle n\rangle^{2}$ describing
their fluctuations. Str11607 When reducing the detuning between dot and
resonator from large values to zero, we observe that the system undergoes a
transition from the nonlasing regime, where $\langle n\rangle<1$ and
$F_{n}=\langle n\rangle+1$, to a lasing state with a sharp increase in the
photon number. Before we reach the lasing state the photon number distribution
has a thermal shape, which explains the value of the Fano-Factor. At the
transition to the lasing regime the amplitude fluctuations lead to a peak in
the Fano factor, as shown in Fig. 3 (b). In the lasing state the photon number
is saturated, and the Fano factor drops to $F_{n}<1$, indicating a squeezed
photon number distribution in the resonator. Interestingly, the average photon
number in the lasing regime, as well as the corresponding peak in the Fano
factor at the lasing transition are larger for weak interdot interaction, case
(ii), than for strong one, case (i). Approximately, we obtain the average
photon number Mar09 for case (ii)
$\langle
n\rangle\simeq\frac{\Gamma\cos\theta}{2\kappa}-\frac{\Gamma^{2}+4\Delta^{2}}{8g^{2}}.$
(12)
Compared to case (i), where Jin11035322 $\langle
n\rangle_{i}\simeq\frac{\Gamma\cos\theta}{3\kappa}-\frac{\Gamma^{2}+4\Delta^{2}}{96g^{2}}(7+\cos\theta)$,
we find an increase to $\langle n\rangle_{ii}\approx\langle
n\rangle_{i}+\frac{\Gamma}{6\kappa}$, showing that case (ii) with four levels
is more suited for lasing. The difference is due to the existence of one more
incoherent tunneling channel, driven as illustrated in Fig. 2 (b) and (d).
Since photons in the resonator are excited by the incoherent tunneling between
the dot and the electrodes, the lasing state closely correlates with the
transport current. The current can be expressed approximately (for
$\kappa\ll\Gamma$ and small $\theta$) for case (i) as Jin11035322
$I(\Delta)\simeq
e\Gamma\sum_{n=0}P(n)\frac{(n+1)}{3(n+1)+(\Gamma^{2}+4\Delta^{2})/4g^{2}}$
(13)
with
$P(n)\simeq(\Gamma/\kappa)P(0)\prod^{n}_{l=1}[3l+(\Gamma^{2}+4\Delta^{2})/4g^{2}]^{-1}$
being the probability of $n$ photons in the resonator (in Ref. Jin11035322, a
factor $\frac{1}{2}$ was missing). As shown in Fig. 3 (c), the transport
current as function of the detuning follows closely the behavior of the
average photon number. Similarly, the corresponding transport current for case
(ii) is obtained as,
$I(\Delta)\simeq
e\Gamma\sum_{n=0}P(n)\frac{(n+1)}{2(n+1)+(\Gamma^{2}+4\Delta^{2})/4g^{2}},$
(14)
where
$P(n)\simeq(\Gamma/\kappa)P(0)\prod^{n}_{l=1}[2l+(\Gamma^{2}+4\Delta^{2})/4g^{2}]^{-1}$.
Both the average photon number of the resonator as well as the current are
larger for case (ii) than for case (i).
As had been pointed out in Ref. Har08024513, , for a superconducting single-
electron transistor (SSET) coupled to a resonator, the noise spectra of the
fluctuations of the photons are correlated with the zero-frequency shot noise
of the current. This fact is illustrated for the Fano factor $F_{I}=S(0)/2I$
in Fig. 3 (d). For strong detuning in the nonlasing regime, where the dots
effectively do not interact with the resonator, the shot noise shows a
Poissonian distribution, i.e., $F_{I}\simeq 1$. Near the lasing transition the
shot noise is enhanced strongly with a super-Poissonian distribution. Compared
to the Fano factor of the photons, the signal in the shot noise is stronger
with a narrower transition window and sharper peak. In the lasing state, where
the photons are saturated and the transport current reaches the maximum value,
we find sub-Poissonian current noise, $F_{I}\simeq 0.5$, while the photon Fano
factor $F_{n}$ describes a squeezed state of the radiation field in this
nonclassical regime, differing from a conventional coherent state with
Poissonian distribution. The cross-correlation noise (not displayed in the
figures) shows a similar behavior with the opposite sign due to the relation
of $S_{L}(0)=S_{R}(0)=-S_{LR}(0)$.
Figure 4: (Color online) The noise spectra in the low-frequency regime for
strong interdot Coulomb interaction. (a) Auto-correlation
$S_{L}(\omega)=S_{R}(\omega)$. (b) Cross-correlation $S_{LR}(\omega)$.
Different colors of the plotted noise spectra refer to different values of the
detuning, as denoted in the inset of (a) by color circles. The other
parameters are the same as in Fig. 3.
## IV Noise spectrum
Since in the nonlasing regime the noise spectrum displays only trivial
features, we focus in the following on the finite-frequency noise spectra in
the lasing regime and at the lasing transition, as shown in the inset of Fig.
4(a). For tunneling dissipative operators ${\cal L}_{L}$ and ${\cal L}_{R}$ as
defined after Eq. (4b) it has been demonstrated Ema07161404 that all
correlation functions can be expanded in terms of the eigenvalues
$\lambda_{k}$ of ${\cal L}_{\rm tot}$ and the coefficients
$c_{k}=[\hat{V}^{-1}\hat{I}_{\alpha}\hat{V}]_{kk}$. Here $\hat{V}$ is built
from the eigenvectors of ${\cal L}_{\rm tot}$, and $\hat{I}_{\alpha}$ is the
current operator described in Eq. (5). E.g., we have
$\displaystyle\frac{S_{\alpha}(\omega)}{2I}$
$\displaystyle=1-2\sum_{k}\frac{{\rm Re}(c_{k}){\rm Re}(\lambda_{k})+{\rm
Im}(c_{k})[\omega+{\rm Im}(\lambda_{k})]}{[\omega+{\rm
Im}(\lambda_{k})]^{2}+[{\rm Re}(\lambda_{k})]^{2}},$ (15)
where the imaginary part ${\rm Im}(\lambda_{k})$ and real part ${\rm
Re}(\lambda_{k})$ represent the coherent and dissipative dynamics,
respectively. The coherent dynamics follows from the Jaynes-Cummings
Hamiltonian, Eq. (2), with eigenstates Bla04062320 ; Har92
$\displaystyle|+,n\rangle$
$\displaystyle=\cos\theta_{n}|e,n\rangle+\sin\theta_{n}|g,n+1\rangle,$ (16)
$\displaystyle|-,n\rangle$
$\displaystyle=\sin\theta_{n}|e,n\rangle-\cos\theta_{n}|g,n+1\rangle,$ (17)
and eigenergies
$E_{\pm,n}=(n+1)\omega_{r}\pm\frac{1}{2}\sqrt{4g^{2}(n+1)+\Delta^{2}}.$ (18)
with
$\theta_{n}=\frac{1}{2}\tan^{-1}\left(\frac{2g\sqrt{n+1}}{\Delta}\right)$. The
typical signal in the noise spectrum is dominated by these eigenenergies,
while the linewidth of the signal follows from the jump operators in Eq. (4b).
Figure 5: (Color online) The low-frequency auto-correlation and cross-
correlation noise spectrum for different values of the detuning in the lasing
regime. Panel (a) and (b) are for strong interdot Coulomb interaction, and (c)
and (d) for weak interdot Coulomb interaction. The other parameters are the
same as in Fig. 3.
### IV.1 Low-frequency regime
Let us first consider the low frequency regime around $\omega\sim 0$ displayed
in Fig. 4. We find a zero-frequency peak and dip in the auto- and cross-
correlation noise spectra, respectively. Both decrease and finally disappear
when one approaches the lasing state. The height of the zero-frequency peak as
function of a detuning is shown in Fig. 3 (d). Since in the absence of a
resonator we have $S_{\alpha}(\omega\approx 0)/2I=-S_{LR}(\omega\approx
0)/2I\simeq 1$, the peak/dip feature at zero-frequency in the noise spectra
must be the effect of the resonator.
The noise spectra in Fig. 4 have a Lorentzian shape with linewidth
$\gamma_{0}\sim\kappa$, determined by the emission spectrum of the photons.
Car99 In the regime around zero-frequency, corresponding to the long-time
limit, the noise spectra are determined by the single minimum eigenvalue
$\lambda_{\rm min}$ with real part dominated by the weakest decay rate, i.e.,
$\kappa$. For weak interdot Coulomb interaction, where we have to account for
one more incoherent tunneling channel (see Fig. 2 (b) and (d)) the low-
frequency noise spectra display a similar behavior as in Fig. 4 (d), except
for the enhancement of the zero-frequency peak as shown in Fig. 3. It is worth
noting that in this low-frequency regime, the relation $S_{L}(\omega\sim
0)=S_{R}(\omega\sim 0)=-S_{LR}(\omega\sim 0)$ is still satisfied. However, as
we will show below, the cross-correlation noise changes sign beyond the low-
frequency regime.
At higher frequencies but still within the range $|\omega|<\omega_{r}$, the
spectra are no longer Lorentzian due to the contributions from several
$\lambda_{k}$ in Eq. (15). We find characteristic features showing a step and
peak in the auto- and cross-correlation noise spectra, respectively, as shown
in Fig. 5. The position of the step/peak is nearly independent of the
detuning, while the magnitude is sensitive to it. With increasing dot-
resonator interaction, both the step and peak are shifted as shown in Fig. 6.
These characteristics are a consequence of the coherent dynamics of the
coupled dot-resonator system. The step/peak occurs at $\omega=\delta E$, where
$\delta E=|E_{+,\langle n\rangle}-E_{-,\langle n\rangle}|=\sqrt{4g^{2}(\langle
n\rangle+1)+\Delta^{2}}\approx 2g\big{(}\langle n\rangle+1\big{)}$ is the Rabi
frequency corresponing to the photon number $\langle n\rangle$. As expected
this coherent signal of the step/peak becomes weak and even disappears with
increasing incoherent tunneling rate $\Gamma$ (not shown in the figure).
Interestingly, as shown in Fig. 6 (b), we find that with increasing dot-
resonator interaction, the coherent signal for weak interdot Coulomb
interaction is not only shifted, but the step also turns into a dip. This is
consistent with the coherent signal of the Rabi frequency in the double dot in
the absence of the resonator showing a dip and peak in the auto- and cross-
correlation noise spectra, respectively. Luo07085325 ; Jin1105 It arises from
the recovered symmetrical transition tunneling channels [Fig. 2 (b) and (d)].
In the low-frequency regime, $\omega<|\omega_{r}|$, the auto-correlation noise
spectra of left and right lead satisfy $S_{L}(\omega)=S_{R}(\omega)$. This is
no longer true in the high-frequency regime $\omega\gtrsim|\omega_{r}|$ for
strong interdot Coulomb interaction, as will be shown in the following
subsection.
Figure 6: (Color online) The low-frequency noise spectra for different dot-
resonator coupling strength in the lasing state at $\Delta=0$, (a) for strong
interdot Coulomb interaction, and (b) for weak interdot Coulomb interaction.
The other parameters are the same as in Fig. 3.
### IV.2 Regime close to the resonator frequency
Before addressing the noise spectrum in the high-frequency regime, let us
briefly discuss its properties in the absence of the resonator. It has been
demonstrated Agu04206601 ; Luo07085325 that the signal of the intrinsic Rabi
frequency $\omega_{0}$ of the double dots can be extracted from the noise
spectra. For instance, the auto-correlation noise spectrum shows a dip-peak
structure and a dip at $\omega=\omega_{0}$ for strong and weak interdot
Coulomb interaction, respectively. Agu04206601 ; Luo07085325 Considering the
present parameter regime, where lasing is induced for
$\omega_{0}\approx\omega_{r}$ with very weak incoherent tunneling,
$\Gamma=10^{-3}\omega_{0}$, we find in the strong Coulomb interaction case
nearly Poissonian noise in the full-frequency regime, with a small correction
due to a weak coherent Rabi signal, i.e., $S_{\alpha}(\omega_{0})/2I\sim 1\pm
5\times 10^{-5}$. The correction can be neglected compared to the signal
induced by the coupled resonator as shown in Fig. 7.
Figure 7: (Color online) The finite-frequency noise spectra for strong
Coulomb interaction in the lasing regime. Panel (a) and (b) for strong
interdot Coulomb interaction, and (c) and (b) for weak interdot Coulomb
interaction. The other parameters are the same as in Fig. 3.
The signals in the high-frequency noise spectrum arise because of transitions
with the energy $E_{\pm,n}-E_{\pm,n-1}\approx\omega_{r}$. They depend on the
detuning in the same way as the spectrum of the oscillator. And10053802
Namely for positive detuning we find a signal at frequencies somewhat higher
than $\omega_{r}$ and for negative detuning at frequencies below $\omega_{r}$.
In contrast to the low-frequency case, for high frequencies the spectra of the
current in the left and right junction, $S_{L}(\omega)$ and $S_{R}(\omega)$,
do not have to be identical due to the overall symmetry of the circuit broken
by the resonator. This feature has been demonstrated by the previous studies
in Refs. Arm04165315, ; Har10104514, for investigating the spectral
properties of a resonator coupled to a SET and a SSET, respectively, in non-
lasing regime. For the present studied setup in lasing regime, in this case we
find significant differences between the cases (i) and (ii) of strong and weak
Coulomb interaction, as illustrated in the left and right columns of Fig.7,
respectively. For strong Coulomb interaction the correlators are
$\displaystyle\langle I_{L}(t)I_{L}(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}\langle n|\langle 0|\rho_{I_{L}}(t)|0\rangle|n\rangle$
$\displaystyle\langle I_{R}(t)I_{R}(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}\langle n|\langle
R|\rho_{I_{R}}(t)|R\rangle|n\rangle\,,$ (19)
while for weak Coulomb interaction we have
$\displaystyle\langle I_{L}(t)I_{L}(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}\langle n|\left[\langle
0|\rho_{I_{L}}(t)|0\rangle+\langle R|\rho_{I_{L}}(t)|R\rangle\right]|n\rangle$
$\displaystyle\langle I_{R}(t)I_{R}(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}\langle n|\left[\langle
R|\rho_{I_{R}}(t)|R\rangle+\langle
2|\rho_{I_{R}}(t)|2\rangle\right]|n\rangle\,.$
Here we introduced the density matrix $\rho_{I_{i}}(t)$ which satisfies the
master equation (4) with the initial condition
$\rho_{I_{i}}(0)=\hat{I}_{i}\rho^{\rm st}$ ($i=L,R$).
For strong Coulomb interaction only $S_{R}(\omega)$ couples directly to the
state $|R\rangle$, which in turn couples resonantly to the oscillator. As a
result we observe the signal at $\omega\approx\omega_{r}$ only in
$S_{R}(\omega)$, while $S_{L}(\omega)\approx 1$ is unaffected by the
oscillator. In contrast, in case (ii), where we allow the state $|2\rangle$ to
participate, we again find a symmetry between the currents through the right
and left junction and $S_{L}(\omega)=S_{R}(\omega)$, as well as the anti-
symmetry between the auto- and cross-correlation noise spectrums, i.e.,
roughly $S_{\alpha}(\omega)/2I\approx 1+\Delta S(\omega)$ and
$S_{LR}(\omega)/2I\approx-\Delta S(\omega)$ with the signal $\Delta S(\omega)$
changing sign leading to a peak and dip as function of frequency. Furthermore,
in contrast to the low-frequency regime, the noise spectrum at high frequency
shows a Fano-resonance profile. It displays the same mechanism as observed by
Rodrigues Rod09067202 that the detector (here the double quantum dot) feels
the force in two ways, namely the original one from the voltage-driven
tunneling and the one from the resonator. It arises from a destructive
inference between the two transition paths between $|g\rangle$ and
$|e\rangle$, i.e, a direct tunneling channel through the leads and a
transition assisted by the absorption and emission at the detection frequency.
Still, we like to mention that the present Fano-resonance profile occurs in
the current noise spectra differs from the result presented in Ref.
Rod09067202, where the resonator coupled to SET showed the Fano-resonance in
the SET charge noise spectra.
## V summary and discussion
We have evaluated the frequency-dependent noise spectrum of the transport
current through a coupled dot-resonator system in the lasing regime, in a
situation when incoherent tunneling induces a population inversion. We
considered both strong and weak interdot Coulomb interactions, in the latter
case taking into account the doubly occupied state as well. Both situations
lead to a similar behavior of the zero-frequency shot noise but to different
features in the finite-frequency noise spectrum.
When the system approaches the lasing regime the zero-frequency shot noise is
enhanced strongly showing a remarkable super-Poissonian distribution. When the
resonator is in the lasing state, the shot noise displays sub-Poissonian
characteristics. The current follows here the behaviour of the photon
distribution, which is also super-Poissonian as one approaches the lasing
regime and becomes sub-Poissonian near resonance.
We found that the average photon number and the corresponding Fano factor, as
well as the average current and its Fano factor in the lasing regime is larger
for weak interdot Coulomb interaction than for strong interaction. The zero-
frequency shot noise could be detected with current experimental technologies.
E.g., a quantum point contact coupled to the dots has been demonstrated to
detect in real-time single electron tunneling through the double dot.
Gus06076605 ; Fuj061634
Considering the finite-frequency noise spectra we found pronounced
characteristic structures in the low- and high-frequency regimes reflecting
the coherent dynamics of the coupled dot-resonator system. At low but finite
frequencies the coherent dynamics of the oscillator leads to a peak at the
eigen Rabi frequency of the coupled system. At frequencies close to that of
the resonator, due to the excitations of the photons in the resonator, we
found for strong interdot Coulomb interaction a strongly asymmetric signal in
the auto-correlation noise spectra of the left and right junction. Symmetry is
restored for weak interdot Coulomb interaction. The difference arises from the
asymmetrical and symmetrical incoherent tunneling channels induced by strong
and weak interdot Coulomb interactions, respectively.
###### Acknowledgements.
J. S. Jin acknowledges support from the National Natural Science Foundation of
China NSFC (10904029 and 11274085) as well as the support by the Ministry of
Science, Research and the Arts of the State of Baden-Württemberg.
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|
arxiv-papers
| 2012-10-21T06:42:19 |
2024-09-04T02:49:36.905459
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinshuang Jin, Michael Marthaler, Pei-Qing Jin, Dmitry Golubev, and\n Gerd Sch\\\"on",
"submitter": "Jinshuang Jin",
"url": "https://arxiv.org/abs/1210.5698"
}
|
1210.5713
|
# Klein-Gordon equations for energy-momentum of relativistic particle in
rapidity space
Robert M. Yamaleev
Joint Institute for Nuclear Research, LIT,Dubna, Russia
Universidad Nacional Autonoma de México, México.
Email:yamaleev@jinr.ru
###### Abstract
The notion of four- rapidity is defined as a four-vector with one time-like
and three space-like coordinates. It is proved, the energy and momentum
defined in the space of four-rapidity obey Klein-Gordon equations constrained
by the classical trajectory of a relativistic particle. It is shown, for small
values of a proper mass influence of the constraint is weakened and the
classical motion gains features of a wave motion.
## 1 Introduction
The goal of this paper is to prove that energy and momentum defined in the
space of four- rapidity obey the differential equation similar the Klein-
Gordon equation. This method is based on a key-formula connecting the fraction
with an exponential function the argument of which is proportional the
difference between numerator and denominator. The energy and momentum can be
defined either as functions of the hyperbolic angle, or as functions of the
circular angle. In a covariant formulation we arrive to concept of rapidity
expressed as a four-vector the time-like part of which is presented by the
hyperbolic angle. The circular angle is extended into three quantities
functionally depending of the hyperbolic angle. It is shown, the energy and
momentum defined in a such way satisfy the Klein-Gordon equations written in
four-dimensional space with Minkowskii signature.
## 2 Key-formulae linking an exponential function with ratio of two
quantities
2.1 Parametrization of relativistic evolution with respect to hyperbolic
angle.
The dynamical variables of the relativistic particle, the energy $p_{0}$, the
momentum $p$ and the proper mass satisfy a mass-shell equation [1]
$p_{0}^{2}=p^{2}+m^{2},$ $None$
where the speed of light $c$ taken in unit $c=1$.
Our construction is based on the Key-formula which establishes some natural
interrelation between the ratio of a pair of quantities and an exponential
function.
Consider general form of the complex number given by unit ${\mathbf{e}}$
obeying the quadratic equation [2]:
${\mathbf{e}}^{2}-2p_{0}{\mathbf{e}}+p^{2}=0,$ $None$
with distinct positive real roots $x_{1},x_{2}$, so that,
$2p_{0}=x_{1}+x_{2},~{}~{}p^{2}=x_{1}x_{2}.$ $None$
The coefficients $p_{0},p^{2}$ are real numbers and $p_{0}^{2}>p^{2}$. The
solutions of equation (2.1) are defined by
$x_{1}=p_{0}+m,~{}~{}x_{2}=p_{0}-m,~{}~{}m=+\sqrt{p_{0}^{2}-p^{2}}.$ $None$
The normal form of the matrix obeying equation (2.1) is given by
$E=\left(\begin{array}[]{cc}0&-p^{2}\\\ 1&2p_{0}\end{array}\right).$ $None$
Consider an evolution generated by matrix $E$. Write the Euler formula
$\exp(E\phi)=E~{}g_{1}(\phi)+I~{}g_{0}(\phi),$ $None$
$I$-is a unit matrix. Diagonal form of this matrix equation consists of two
equations
$\exp(x_{2}\phi)=x_{2}~{}g_{1}(\phi)+g_{0}(\phi),~{}~{}\exp(x_{1}\phi)=x_{1}~{}g_{1}(\phi)+g_{0}(\phi).$
$None$
Notice, the modified cosine-sine functions $g_{0}(\phi),~{}g_{1}(\phi)$ depend
also of coefficients $p_{0},p^{2}$. Form the following ratio
$\exp((x_{2}-x_{1})\phi)=\frac{x_{2}~{}g_{1}(\phi)+g_{0}(\phi)}{x_{1}~{}g_{1}(\phi)+g_{0}(\phi)}=\frac{x_{2}+D}{x_{1}+D},$
$None$
where
$D=\frac{g_{0}(\phi)}{g_{1}(\phi)}.$ $None$
Let $\phi=\phi_{0}$ be the point where $g_{0}(\phi_{0})=0$. Then,
$\exp((x_{2}-x_{1})\phi_{0})=\frac{x_{2}}{x_{1}}.$ $None$
From (2.3) it follows
$m=\frac{1}{2}(x_{2}-x_{1}),~{}~{}p_{0}=\frac{1}{2}(x_{1}+x_{2}),~{}p^{2}=x_{1}x_{2}.$
$None$
Let $\phi=\phi_{0}$ be an initial point. Then according with (2.9) we conclude
that the roots and the coefficients of equation (2.1) $x_{1},x_{2}$ and
$p_{0},p$ are functions of $\phi_{0}$, however the difference $2m=x_{2}-x_{1}$
does not depend of $\phi_{0}$:
$x_{2}(\phi_{0})=\exp(m\phi_{0})\frac{m}{\sinh(m\phi_{0})},~{}x_{1}(\phi_{0})=\exp(-m\phi_{0})\frac{m}{\sinh(m\phi_{0})},$
$None$
Use these formulae in (2.9). Then,
$\exp((x_{2}-x_{1})\phi_{0})=\frac{p_{0}+m}{p_{0}-m}.$ $None$
Consequently, we have the following dependence $p_{0},p$ of $\phi_{0}$:
$p_{0}(\phi_{0})=m\coth(m\phi_{0}),~{}~{}p(\phi_{0})=\frac{m}{\sinh(m\phi_{0})}.$
$None$
In Refs.[3, 4, 5] formula (2.9) has been denominated as Key-formula.
2.1 Parametrization of evolution with respect to periodic angle.
Now, consider general complex algebra with generator ${\mathbf{e}}$ obeying
the quadratic equation
${\mathbf{e}}^{2}-2p{\mathbf{e}}+p_{0}^{2}=0.$ $None$
which differs from (2.1) by transposition of the coefficients $p_{0}$ and $p$.
Since $p_{0}^{2}>p^{2}$, two solutions of equation (2.14) are given by complex
conjugated numbers:
$y_{2}=p+im,~{}~{}y_{1}=p-im,~{}~{}m=+\sqrt{p_{0}^{2}-p^{2}}.$ $None$
Exponential function at solutions of this equation is defined by expansions
$\exp(y_{2}\theta)=y_{2}~{}f_{1}(\theta)+f_{0}(\theta),~{}~{}\exp(y_{1}\theta)=y_{1}~{}f_{1}(\theta)+f_{0}(\theta),$
$None$
where functions $f_{0}(\theta),f_{1}(\theta)$ depend of coefficients
$p,p_{0}^{2}$. Form the following ratio
$\exp(i2m\theta)=\frac{y_{2}~{}f_{1}(\theta)+f_{0}(\theta)}{y_{1}~{}f_{1}(\theta)+f_{0}(\theta)}=\frac{y_{2}+F}{y_{1}+F}$
$=\frac{p+im+F}{p-im+F},$ $None$
where
$F=\frac{f_{0}}{f_{1}}.$ $None$
Let $\theta=\theta_{0}$ be the initial point where $f_{0}(m\theta_{0})=0$.
Then, formula (2.17) is reduced into the following relationship
$\exp(i2m\theta_{0})=\frac{p(\theta_{0})+im}{p(\theta_{0})-im}.$ $None$
From this formula it follows
$p(\theta_{0})=m\cot(m\theta_{0}),~{}~{}p_{0}(\theta_{0})=m\frac{1}{\sin(m\theta_{0})}.$
$None$
The roots $y_{1},y_{2}$ also are functions of $\theta_{0}$,
$y_{1}=\exp(-im\theta_{0})\frac{m}{sin(m\theta_{0})},~{}y_{2}=\exp(im\theta_{0})\frac{m}{sin(m\theta_{0})}.$
$None$
Thus, we obtained two representations for the energy- momentum. The first one
is done via hyperbolic trigonometric functions,
$p_{0}(\phi)=m\coth(m\phi),~{}~{}~{}p(\phi)=m\frac{1}{\sinh(m\phi)},$ $None$
and the other one is defined by ordinary periodic trigonometric functions
$p_{0}(\theta)=m\frac{1}{\sin(m\theta)},~{}~{}~{}p(\chi)=m\cot(m\theta).$
$None$
In the both representations the arguments of the trigonometric functions are
proportional to mass $m$.
Since formulae (2.22) and (2.23) are related to same physical quantities, we
come to the next relationships between hyperbolic and periodic trigonometric
functions
$\tanh(m\phi)=\sin(m\theta),~{}\mbox{or},~{}\sinh(m\phi)=\tan(m\theta).$
$None$
Notice, when $m=0$, $\phi=\theta$.
The relationships between $\phi$ and $\theta$ can be presented also as follows
$\exp(m\phi)=\frac{1+\sin(m\theta)}{1-\sin(m\theta)}=\frac{1+\tan\frac{m\theta}{2}}{1-\tan\frac{m\theta}{2}},$
$None$
$\exp(im\theta)=\frac{1+i\sinh(m\phi)}{1-i\sinh(m\phi)}=\frac{1+i\tanh\frac{m\phi}{2}}{1-i\tanh\frac{m\phi}{2}}.$
$None$
Also, it is important to notice that the differential relationship between
variables $\theta$ and $\phi$ coincides with the definition of the velocity:
$\frac{d\theta}{d\phi}=\frac{dr}{dt}=\frac{p}{p_{0}}.$ $None$
These formulae express a general interrelation between periodic and hyperbolic
trigonometry. Let $\triangle ABC$ be a right-angled triangle with right angle
at $C$. Denote the sides by $a=BC,b=AC$, the hypotenuse $AB$ by $c$. If we
make a geometrical motion by moving the point $A$ along line $AC$, then this
motion changes the sides $c,b$, but remains invariant the side $a$. The angle
$A$ can be used in quality of parameter this evolution. In accordance with the
Key-formula (2.19) we write
$\frac{b+ia}{b-ia}=\exp(2ia\theta),~{}~{}~{}b=a\cot(a\theta),~{}~{}c=a\frac{1}{\sin(a\theta)}.$
$None$
It is easily seen that $a\theta=A$. On the other hand, In accordance with Key-
formula (2.12) we have
$\frac{c+a}{c-a}=\exp(2a\phi),~{}~{}c=a\coth(a\phi),~{}~{}b=\frac{a}{\sinh(a\phi)}.$
$None$
## 3 Pythagoras theorem and two dimensional Fermi-like oscillator
For the sake of convenience in this section let us use for derivatives short
notations
$\frac{d}{d\phi}=d,~{}~{}~{}\frac{d}{d\theta}=\partial.$
Then differentiating formula (2.22) and (2.23) we come to the following system
of differential equations
$dp_{0}=-p^{2},~{}~{}~{}dp=-pp_{0},~{}~{}~{}\partial
p_{0}=-pp_{0},~{}~{}~{}\partial p=-p^{2}_{0}.$ $None$
The operators $d$ and $\partial$ do not commute. Introduce two dimensional
vector of a state by
$\Phi(p_{0},p)=\left(\begin{array}[]{c}p_{0}\\\ p\end{array}\right).$ $None$
Calculate actions of the operators $d^{2}-\partial^{2}$ and
$d\partial-\partial d$ on this vector:
$(d^{2}-\partial^{2})\Phi(p_{0},p)=m^{2}~{}\Phi(p_{0},p),$ $None$ $(\partial
d-d\partial)\Phi(p_{0},p)=m^{2}~{}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\Phi(p_{0},p).$ $None$
Introduce operators
$a^{-}=d-\partial,~{}~{}a^{+}=d+\partial,$
with following commutation and anti-commutation rules
$\frac{1}{2}(a^{-}a^{+}+a^{+}a^{-})=m^{2}~{}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),$ $None$
$a^{-}a^{+}-a^{+}a^{-}=2m^{2}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ $None$
It is seen, we deal with some kind of two dimensional Fermi-like oscillator
with Hamilton operator
$H=\frac{1}{2}(a^{-}a^{+}-a^{+}a^{-}),$ $None$
and with anti-commutation relation given by
$(a^{-}a^{+}+a^{+}a^{-})=2m^{2},$ $None$
acting on the state $\Phi_{0}(p_{0},p)$. The Hamilton operator possesses with
two eigenvalues
$H\Phi_{n}=E_{n}\Phi_{n},~{}n=1,2,$ $None$
where
$E_{1}=+m^{2},~{}~{}E_{2}=-m^{2},$
so that,
$a^{+}\Phi_{1}=\Phi_{2},~{}~{}a^{-}\Phi_{2}=\Phi_{1},$
and, due to anti-commutation relations (3.8),
$a^{\pm}H+Ha^{\pm}=0.$
## 4 Klein-Gordon equations for energy-momentum of classical relativistic
particle in the space of rapidity
It is seen that equation (3.3) is nothing else than two dimensional Klein-
Gordon equation. Comparing this equation with two dimensional Klein-Gordon
equation written in terms of space-time coordinates we come to conclusion that
the parameter $\phi$\- is a time-like parameter, whereas the parameter
$\theta$ is an analogue of a space coordinate. In order to pass to the Klein
-Gordon equation in four-dimensional Minkowski space with signature $(+---)$
we shall extend the parameter $\theta$ till to three dimensional vector. In
this way we arrive to covariant formulation of evolution equations.
The momentum is a spatial part of the four-vector energy-momentum with
components $p_{k},k=1,2,3$. Now, instead of $\phi$ we will use the letter
$\rho_{0}$, and $\theta$ has to be replaced by spatial part of four-vector of
rapidity containing components $\rho_{1},\rho_{2},\rho_{3}$.
In these variables the evolution equations have to be written in a Lorentz-
covariant form. The evolution equations we shall extend as follows. The single
variable $p$ is replaced by the components of three-vector of momentum,
$p_{k},k=1,2,3$. The square $p^{2}$ means $p^{2}=-p^{k}p_{k}$. In this way we
arrive to the following set of equations
$(a)~{}~{}~{}d^{0}~{}p_{0}=~{}-p^{k}p_{k},~{}~{}~{}(b)~{}~{}d^{0}~{}p_{k}=p_{k}p^{0},~{}~{}k=1,2,3.$
$None$
$(a)~{}~{}~{}\partial_{k}~{}p_{0}=p_{k}p_{0},~{}~{}~{}(b)~{}~{}~{}~{}\partial^{k}~{}p_{k}=-p^{2}_{0}.$
$None$
Hereafter we use the following notations for derivatives
$\partial^{k}=\frac{\partial}{\partial\rho_{k}},~{}~{}~{}d^{0}=\frac{d}{d\rho_{0}},$
and adopt, so-called, a summation convention, according to which any repeated
index in one term, once up, once down, implies summation over all its values.
Remember, however, that there exist some functional dependence between
$\rho_{0}$ and $\rho_{k},k=1,2,3$, so that the spatial variables are functions
of the time-like parameter, i.e., $\rho_{k}=\rho_{k}(\rho_{0}),k=1,2,3.$ This
means, the full derivative with respect to $\rho_{0}$ is
$d^{0}p_{0}=-p^{k}p_{k}=~{}\frac{d\rho_{k}}{d\rho_{0}}\frac{\partial}{\partial\rho_{k}}p_{0}.$
$None$
On making use of equations (4.1)-(4.2), we get
$d^{0}p_{0}=p^{2}=-p^{k}p_{k}=~{}p_{k}\frac{d\rho_{k}}{d\rho_{0}}~{}p_{0}.$
$None$
In order to provide this equality we have to take
$~{}p_{k}\frac{d\rho_{k}}{d\rho_{0}}=\frac{p^{2}}{p_{0}}.$ $None$
Our purpose is to complete the evolution equations (4.1)-(4.2) with an
equation containing the following derivative
$~{}~{}\frac{\partial}{\partial\rho_{n}}p_{k}.$
For that reason let us re-write equation (4.1b) as follows
$~{}\frac{d}{d\rho_{0}}~{}p_{k}=~{}\frac{d\rho_{n}}{d\rho_{0}}\frac{\partial}{\partial\rho_{n}}~{}p_{k}=p_{k}p_{0}.$
In order to provide this equality we have to suppose that
$~{}\frac{\partial}{\partial\rho_{n}}p_{k}=p_{k}p^{n}\frac{p_{0}^{2}}{p^{2}}.$
$None$
One may easily check that formula (4.6) is in accordance with (4.1) and (4.2).
Equations with second order derivatives.
Firstly, calculate the second order derivatives of $p_{0}$ and $p$ with
respect to time-like variable $\rho_{0}$. We have,
$\frac{d}{d\rho^{0}}\frac{d}{d\rho_{0}}p_{0}=-2p^{k}p_{k}p_{0}=2p^{2}p_{0}.$
$None$
Secondly, calculate action of the Laplace operator on $p_{0}$. Define the
Laplace operator by
$\Delta=\frac{\partial}{\partial\rho_{k}}\frac{\partial}{\partial\rho^{k}}.$
By taking into account (4.2a) we obtain
$\partial^{k}\partial_{k}~{}p_{0}=-p_{0}^{2}p_{0}+p^{k}p_{k}p_{0}=-p_{0}^{3}-p^{2}p_{0}.$
$None$
Joining this equation with (4.7) we come to Klein-Gordon equation for $p_{0}$:
$d^{0}d_{0}~{}p_{0}+\partial^{k}\partial_{k}~{}p_{0}=-m^{2}~{}p_{0}.$ $None$
Now calculate action of operator $\Delta$ on $p_{k}$ by using formulae (4.6)
and (4.2b).
$\partial^{n}\partial_{n}~{}p_{k}=\partial^{n}~{}(~{}p_{k}p_{n}\frac{p_{0}^{2}}{p^{2}}~{})$
$=(\partial^{n}~{}(~{}p_{k}p_{n}))~{}\frac{p_{0}^{2}}{p^{2}}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=\frac{p_{0}^{2}}{p^{2}}~{}(p^{n}p_{n})~{}\frac{p_{0}^{2}}{p^{2}}+p_{k}(\partial^{n}p_{n})~{}\frac{p_{0}^{2}}{p^{2}}~{}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}~{}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}~{}~{}-2p_{k}p^{n}\frac{p_{n}p_{0}^{2}m^{2}}{p^{4}}$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}+~{}~{}2p_{k}\frac{p_{0}^{2}m^{2}}{p^{2}}$
$=(~{}~{}-2\frac{p_{0}^{4}}{p^{2}}p_{k}+~{}2p_{k}\frac{p_{0}^{4}}{p^{2}}~{}~{})=-2p_{k}p_{0}^{2}.$
Joining this result with
$\frac{d}{d\rho^{0}}\frac{d}{d\rho_{0}}~{}p_{k}=p_{k}p_{0}^{2}-p_{k}p^{n}p_{n}=p_{k}p_{0}^{2}+p_{k}p^{2},$
we come to analogue of Klein-Gordon equation for $p_{k}$:
$\Delta~{}p_{k}+d^{0}d_{0}~{}p_{k}~{}=-m^{2}~{}p_{k}.$
Comparison with Klein-Gordon equation used in relativistic quantum mechanics.
From formula (4.5)
$~{}p_{k}\frac{d\rho^{k}}{d\rho_{0}}=\frac{p^{2}}{p_{0}},$
we may conclude that
$\frac{d\rho^{k}}{d\rho_{0}}=v^{k}+M^{kl}p_{l},$
where
$v^{k}=\frac{dx^{k}}{dx^{0}}$
is the velocity with respect to coordinate time, $M^{kl}=-M^{lk}$ is an
arbitrary anti-symmetric tensor.
In the relativistic quantum mechanics the Klein-Gordon equation is obtained
simply by using some conventional receipt according to which components of
four-momentum in the mass-shell equation are replaced by corresponding
differential operators as follows [6]
$p_{k}=-i\hbar\frac{\partial}{\partial
x^{k}},~{}p_{0}=i\hbar\frac{\partial}{\partial x^{0}}.$
So, we come to the following correspondence
$\hbar\rho^{\mu}\Rightarrow
x^{\mu},~{}~{}\frac{\partial}{\partial\rho^{\mu}}=\hbar\frac{\partial}{\partial
x^{\mu}}.$
## 5 Concluding remarks
1\. We considered two ways of description an evolution constrained by
Pythagoras formula. The first one is given by the hyperbolic angle, and the
second one, by the periodic angle. The both angles are proportional to the
fixed side of the right angled triangle. In the case of relativistic
mechanics, the hypotenuse is the energy, the fixed side is the mass and the
moving side is the momentum.
2\. The derivative of the periodic angle with respect to the hyperbolic angle
is equal to a ratio of the moving side to the hypotenuse, in the case of
relativistic mechanics, this ratio is the velocity
$\frac{d\theta}{d\phi}=\frac{p}{p_{0}}=\frac{v}{c}.$
3\. This relationship prompts us to conclude that the hyperbolic angle $\phi$
is the time-like parameter, whereas the periodic angle $\theta$ is the space-
like parameter.
4\. The evolution equations admit an extension to the case of three (or more)
dimensions, however, in this case we could not find an explicit expression for
the momenta.
5\. Near the point where the fixed side of the triangle ( mass) becomes
infinitesimal and according to the Klein-Gordon equation it is conjectured
that this motion will display features of the wave motion.
## References
* [1] A.O. Barut, Electrodynamics and classical theory of fields and particles (Dover Publ., New York, 1980).
* [2] R.M. Yamaleev, J. Math. Anal. Appl. 340, 1046 (2008),doi:10.1016/j.jmaa.2007.09.018.
* [3] R.M. Yamaleev, Phys. Atom. Nucl., 74, 1775 (2011).
* [4] R.M. Yamaleev J. Mod. Phys., 2, 849 (2011).
* [5] R.M. Yamaleev, hep-th/0905.0234.
* [6] L.H. Ryder, Quantum Field Theory (Cambridge Univ. Press., Cambridge, 1985).
|
arxiv-papers
| 2012-10-21T10:25:02 |
2024-09-04T02:49:36.915868
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert M. Yamaleev",
"submitter": "Robert Yamaleev Masgutovich",
"url": "https://arxiv.org/abs/1210.5713"
}
|
1210.5817
|
# Amplitude and phase control of gain without inversion in a four-level atomic
system using loop-transition
Jinhua Zou,111Author for correspondence. jhzou@yangtzeu.edu.cn Dahai Xu and
Huafeng Zhang Department of Physics, Yangtze University, Jingzhou 434023,
People’s Republic of China
###### Abstract
Amplitude and phase control of gain without inversion is investigated in a
four level loop-structure atomic system. Two features are presented. One is
that gain without inversion can be obtained through the amplitude control of
the applied fields. The other is that gain without inversion show a phase-
dependence on the relative phase between the fields applied on the two two-
photon transitions. Gain and phase-dependence originate from interference
induced by such a loop-transition structure.
###### pacs:
PACS numbers: 42.50.Gy, 42.50.Lc, 03.67.Mn, 03.65.Ud
Gain without inversion corresponds to the gain at the emission peak. For the
case of emission, there is no population inversion between the upper atomic
state and the lower atomic state. If this involved transition is a lasing
transition, then laser without inversion can be obtained $\left[1-6\right]$.
That is to say, gain without inversion may lead to laser without inversion in
such systems. As laser without inversion is an important coherence phenomena,
it has been paid considerable interest $[7-12]$. For a traditional laser
action, population inversion between the laser transition is needed. Later it
was suggested that lasing without inversion can be realized through
interference between different channels $\left[7-9\right]$. In these schemes,
the lasing transition is adjusted by the interference terms induced by the
driving fields or the initial coherence. Through the variation of the
interference, lasing without inversion can be achieved. The origin of laser
without inversion is the gain at the emission peak we explained before. Gain
without inversion has been realized in many system, such as atomic systems
$\left[1-6\right]$ and semiconductor nanostructures $[13]$. Among these
schemes, most of the schemes put the effort to obtain the condition of laser
without inversion $[7-12]$ and seldom schemes are trying to deal with the
phase control of the laser without inversion $[6]$. As the amplitude and phase
control is a strong way to adjust the response of atomic-field system, it may
also can be used to control gain without inversion in certain interacting
atom-field systems.
It is acceptable that in a loop structure, the relative phase of the applied
fields does contribute a lot to the response of the probe field. It means that
in such cases, coherent population trapping $[14]$, the absorption spectra
$\left[15,16\right]$, the steady state population $\left[17\right]$, or the
spontaneous emission spectra $\left[18-19\right]$ will have a phase-
dependence, which can not be found in non-loop transition structures. There
are two ways to form the loop-structure. One is to drive the dipole-allowed
transitions in simple atomic system $[20-21]$. The second one is to use
microwave fields to drive the dipole-prohibited transition together with the
dipole-allowed transitions $[6,17,22]$. Among the three methods, the first one
is the best one if just simple atomic system is considered. Here we choose the
first way to use four level atom with double middle states, one ground state
and one excited state to form a loop-transition.
In this paper we are going to investigate the amplitude and phase control of
gain without inversion in a four-level loop-structure system. The motivation
lies in the fact that in a loop-structure atomic system, relative phase plays
an essential role in the response. The aim of this paper is to present the
phase control action of the gain without inversion. The considered loop-
structure contains two two-photon transitions, which will be shown later. The
main results are as follows: (i) Gain without inversion can be obtained by
varing the amplitude of applied fields. (ii) Gain without inversion displays a
phase dependence on the relative phase between the fields applied on the two
two-photon transitions. As the phase changes from 0 to $\pi$, gain and
absorption zones exchange. Phase-dependent gain without inversion in such a
system originates from interference induced by two two-photon loop-transition
structure. Gain and phase-dependence are attributed to coherence and
interference induced by such a loop-transition.
The considered atomic system with a loop-transition is shown in Fig. 1. Four
coherent driving fields are applied on dipole-allowed transitions
respectively. The transitions can be divided into two kinds of two-photon
transitions: $|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$ and
$|1\rangle\leftrightarrow|2^{{}^{\prime}}\rangle\leftrightarrow|3\rangle$.
There are two possible coupling schemes according to the probing transition,
which are shown in Fig. 1(a) and Fig. 1(b). We first concentrate on the low-
level probing case shown in Fig. 1(a). Coherent fields with frequencies
$\omega_{c}$ and $\omega_{p}$ are applied on the transition
$|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$, and coherent
fields with frequencies $\omega_{1}$ and $\omega_{2}$ are applied on the
transition
$|1\rangle\leftrightarrow|2^{{}^{\prime}}\rangle\leftrightarrow|3\rangle$,
respectively. We call this case low-level probing as the probe field is
applied on the transition including the lower level. The Hamiltonian of the
whole atom-field system can be written as
$H=H_{0}+H_{I}$ (1)
$H_{0}=\Delta_{p}\sigma_{22}+(\Delta_{1}+\Delta_{2})\sigma_{33}+\Delta_{2}\sigma_{2^{{}^{\prime}}2^{{}^{\prime}}}$
(2)
$H_{I}=-\Omega_{p}\sigma_{21}-\Omega_{c}\sigma_{32}-\Omega_{1}\sigma_{32^{{}^{\prime}}}-\Omega_{2}\sigma_{2^{{}^{\prime}}1}+h.c.$
(3)
where h.c. symbols the hermit conjugate of the front terms. The detunings
$\Delta_{j}(j=p,c,1,2)$ are defined as the frequency differences between the
applied fields and the corresponding transitions. For specific,
$\Delta_{p}=\omega_{21}-\omega_{p}$, $\Delta_{c}=\omega_{32}-\omega_{c}$,
$\Delta_{1}=\omega_{32^{{}^{\prime}}}-\omega_{1}$, and
$\Delta_{c}=\omega_{2^{{}^{\prime}}1}-\omega_{2}$. In order to fulfill the
rotating transformation the four detunings should satisfy
$\Delta_{p}+\Delta_{c}=\Delta_{1}+\Delta_{2}$, i.e., the two two-photon
transitions have the same sum detuning. When Operators $\sigma_{ij}$ are
population operators when $i=j$ ($j=1,2,2^{{}^{\prime}},3$), and $\sigma_{ij}$
are flip operators when $i\neq j$ ($i$, $j=1,2,2^{{}^{\prime}},3$). And
$\Omega_{j}$ ($j=p,c,1,2$) are Rabi frequencies of the coherent driving fields
and generally they are assumed to be complex values. The dynamic behavior of
the system can be described by the master equation of the density matrix
$\rho$ as $\left[23\right]$
$\dot{\rho}=-i[H,\rho]+L\rho$ (4)
where the term $L\rho$ describes the contribution of the atomic decay terms,
and it has the form
$\displaystyle L\rho$ $\displaystyle=$
$\displaystyle\sum\frac{\gamma_{jk}}{2}(2\sigma_{jk}\rho\sigma_{kj}-\sigma_{kj}\sigma_{jk}\rho-\rho\sigma_{kj}\sigma_{jk}),\text{\quad}$
$\displaystyle jk$ $\displaystyle=$ $\displaystyle
2^{{}^{\prime}}3,12^{{}^{\prime}},23,12$ (5)
Figure 1: (a) Low-level probing scheme for two two-photon transitions atom-
field system. (b) Up-level probing scheme for two two-photon transitions atom-
field system.
The elements of the density matrix can be obtained directly from the master
equation. It is known that for loop-transition, the relative of the fields
plays important role in the result. The complex Rabi frequencies are defined
as
$\Omega_{c}=\Omega_{c}^{0}e^{i\phi_{c}},\Omega_{p}=\Omega_{p}^{0}e^{i\phi_{p}},\Omega_{1}=\Omega_{1}^{0}e^{i\phi_{1}},\Omega_{2}=\Omega_{2}^{0}e^{i\phi_{2}}$,
where $\Omega_{j}^{0}$($j=p,c,1,2$) are strength of the Rabi frequencies.. In
order to see the relative phase, we make the transformation as
$\rho_{32}=\sigma_{32}e^{i\phi_{c}},\rho_{32^{{}^{\prime}}}=\sigma_{32^{{}^{\prime}}}e^{i\phi_{1}},\rho_{21}=\sigma_{21}e^{i\phi_{p}},\rho_{2^{{}^{\prime}}1}=\sigma_{2^{{}^{\prime}}1}e^{i\phi_{2}},\rho_{31}=\sigma_{31}e^{i(\phi_{1}+\phi_{2})},\rho_{22^{{}^{\prime}}}=\sigma_{22^{{}^{\prime}}}e^{i(\phi_{1}-\phi_{c})}$
and $\rho_{jj}=\sigma_{jj}$ $(j=1,2,2^{{}^{\prime}},3)$. After the
transformation the motion of the elements are as following
$\displaystyle\dot{\sigma}_{33}$ $\displaystyle=$
$\displaystyle-(\gamma_{1}+\gamma_{c})\sigma_{33}-i\Omega_{c}^{0}\sigma_{32}-i\Omega_{1}^{0}\rho_{32^{{}^{\prime}}}+i\Omega_{c}^{0}\sigma_{23}$
$\displaystyle+i\Omega_{1}^{0}\sigma_{2^{{}^{\prime}}3}$
$\displaystyle\dot{\sigma}_{22}$ $\displaystyle=$
$\displaystyle\gamma_{c}\sigma_{33}-\gamma_{p}\sigma_{22}+i\Omega_{c}^{0}\sigma_{32}-i\Omega_{p}^{0}\sigma_{21}-i\Omega_{c}^{0}\sigma_{23}$
$\displaystyle+i\Omega_{p}^{0}\sigma_{12}$
$\displaystyle\dot{\sigma}_{2^{{}^{\prime}}2^{{}^{\prime}}}$ $\displaystyle=$
$\displaystyle\gamma_{1}\sigma_{33}-\gamma_{2}\sigma_{2^{{}^{\prime}}2^{{}^{\prime}}}+i\Omega_{1}^{0}\sigma_{32^{{}^{\prime}}}-i\Omega_{2}^{0}\sigma_{2^{{}^{\prime}}1}-i\Omega_{1}^{0}\sigma_{2^{\prime}3}$
$\displaystyle+i\Omega_{2}^{0}\sigma_{12^{{}^{\prime}}}$
$\displaystyle\dot{\sigma}_{32}$ $\displaystyle=$
$\displaystyle-\Gamma_{32}\sigma_{32}-i\Omega_{c}^{0}(\sigma_{33}-\sigma_{22})-i\Omega_{p}^{0}e^{-i\phi}\sigma_{31}$
$\displaystyle+i\Omega_{1}^{0}\sigma_{2^{{}^{\prime}}2}$
$\displaystyle\dot{\sigma}_{32^{{}^{\prime}}}$ $\displaystyle=$
$\displaystyle-\Gamma_{32^{{}^{\prime}}}\sigma_{32^{{}^{\prime}}}-i\Omega_{1}^{0}(\sigma_{33}-\sigma_{2^{{}^{\prime}}2^{{}^{\prime}}})-i\Omega_{2}^{0}\sigma_{31}+i\Omega_{c}^{0}\sigma_{22^{{}^{\prime}}}$
$\displaystyle+i\Omega_{p}^{0}\sigma_{12}$ $\displaystyle\dot{\sigma}_{31}$
$\displaystyle=$
$\displaystyle-\Gamma_{31}\sigma_{31}-i\Omega_{p}^{0}e^{-i\phi}\sigma_{32}-i\Omega_{2}^{0}\sigma_{32^{{}^{\prime}}}+i\Omega_{c}^{0}e^{-i\phi}\sigma_{21}$
$\displaystyle+i\Omega_{1}^{0}\sigma_{2^{{}^{\prime}}1}$
$\displaystyle\dot{\sigma}_{22^{{}^{\prime}}}$ $\displaystyle=$
$\displaystyle-\Gamma_{22^{{}^{\prime}}}\sigma_{22^{{}^{\prime}}}+i\Omega_{c}^{0}\sigma_{32^{{}^{\prime}}}-i\Omega_{2}^{0}e^{-i\phi}\sigma_{21}-i\Omega_{1}^{0}\sigma_{23}$
$\displaystyle+i\Omega_{p}^{0}e^{-i\phi}\sigma_{12^{{}^{\prime}}}$
$\displaystyle\dot{\sigma}_{21}$ $\displaystyle=$
$\displaystyle-\Gamma_{21}\sigma_{21}-i\Omega_{p}^{0}(\sigma_{22}-\sigma_{11})+i\Omega_{c}^{0}e^{i\phi}\sigma_{31}$
$\displaystyle-i\Omega_{2}^{0}e^{i\phi}\sigma_{22^{{}^{\prime}}}$
$\displaystyle\dot{\sigma}_{2^{{}^{\prime}}1}$ $\displaystyle=$
$\displaystyle-\Gamma_{2^{{}^{\prime}}1}\sigma_{2^{\prime}1}-i\Omega_{2}^{0}(\sigma_{2^{{}^{\prime}}2^{{}^{\prime}}}-\sigma_{11})+i\Omega_{1}^{0}\sigma_{31}$
$\displaystyle-i\Omega_{p}^{0}e^{-i\phi}\sigma_{2^{{}^{\prime}}2}$
where we have used the closed relation of the population
$1=\rho_{11}+\rho_{22}+\rho_{2^{{}^{\prime}}2^{{}^{\prime}}}+\rho_{33}$, and
$\rho_{jk}=\rho_{kj}^{*},k\neq j$ to obtain other elements with unlisted
equations. We also defined
$\gamma_{2^{{}^{\prime}}3}=\gamma_{1},\gamma_{23}=\gamma_{c},\gamma_{12^{{}^{\prime}}}=\gamma_{2},\gamma_{12}=\gamma_{p}$,
and the related effective decay rates are
$\Gamma_{32}=\frac{1}{2}(\gamma_{1}+\gamma_{c}+\gamma_{p})+i\Delta_{c}$,
$\Gamma_{32^{{}^{\prime}}}=\frac{1}{2}(\gamma_{1}+\gamma_{c}+\gamma_{2})+i\Delta_{1}$,
$\Gamma_{31}=\frac{1}{2}(\gamma_{1}+\gamma_{c})+i(\Delta_{p}+\Delta_{c})$,
$\Gamma_{22^{{}^{\prime}}}=\frac{1}{2}(\gamma_{2}+\gamma_{p})+i(\Delta_{p}-\Delta_{2})$
and
$\Gamma_{21}=\frac{1}{2}\gamma_{p}+i\Delta_{p},\Gamma_{2^{{}^{\prime}}1}=\frac{1}{2}\gamma_{2}+i\Delta_{2}$.
$\phi=\phi_{1}+\phi_{2}-\phi_{c}-\phi_{p}$ is the relative phase of the
applied fields. From the definition above the phase $\phi$ can also be
understood as the relative phase between the two two-photon transitions
$|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$ and
$|1\rangle\leftrightarrow|2^{{}^{\prime}}\rangle\leftrightarrow|3\rangle$. As
will shown below the relative phase plays a crucial role in the gain spectra.
Figure 2: (a) Variation of Im($\sigma_{21}$) with probe detuning $\Delta_{p}$
for $\phi=0$ (solid line) and $\phi=\pi$ (dotted line) for $\Omega_{c}^{0}$
$=10$, $\Omega_{p}^{0}$ $=0.05$, $\Delta_{c}=\Delta_{1}=0$,
$\Omega_{1}^{0}=10$ and $\Omega_{2}^{0}=1$. (c) Population difference
$\sigma_{22}-\sigma_{11}$ vesus probe detuning $\Delta_{p}$ corresponding to
gain in (a).
Steady state solution of the master equation can be obtained by setting
$\dot{\sigma}_{ij}=0$. Absorption behavior of the weak probe field
$\Omega_{p}$ is described by Im($\sigma_{21}$). When Im($\sigma_{21}$)$<0$, it
symbols gain behavior. In our calculation we set $\gamma_{j}=\gamma=1$
($j=p,c,1,2$), and scale all the $\Omega_{j}^{0}$ and detunings $\Delta_{j}$
($j=p,c,1,2$) in units of $\gamma$. We choose the probe field to be weak and
real. The main results are shown in Fig. 2.
Phase-dependent gain and corresponding population difference
$\sigma_{22}-\sigma_{11}$ vesus probe detuning $\Delta_{p}$ is shown in Fig.
2. Im($\sigma_{21}$) vesus probe detuning $\Delta_{p}$ for $\phi=0$ (solid
line) and $\phi=\pi$ (dotted line) is plotted in Fig. 2(a). Other parameters
are chosen as $\Omega_{c}^{0}$ $=10$, $\Omega_{p}^{0}=0.05$,
$\Delta_{c}=\Delta_{1}=0$, $\Omega_{1}^{0}=10$ and $\Omega_{2}^{0}=1$.
Corresponding population difference $\sigma_{22}-\sigma_{11}$ vesus probe
detuning $\Delta_{p}$ in (a) is shown in Fig. 2(b). Seen from Fig. 2(a), it is
clear that when the relative phase $\phi$ changes from $0$ to $\pi$, the probe
field experiences different gain shapes. Gain and absorption zones exchange.
The most remarkable change lies in the fact that the gain zones has increased
from a single zone near resonant point $\Delta_{p}=0$ to two separate
frequency zones localling around $\Delta_{p}=\pm 15.9$. This means that the
number of gain zones are controlled by the relative phase of the applied
fields. And seen fro Fig. 2(b), population difference
$\sigma_{22}-\sigma_{11}<0$ always holds, which suggests that population
inversion is impossible during the interaction. So the gain behavior does not
come from population inversion between levels $|2\rangle$ and $|1\rangle$.
Figure 3: (a) Variation of Im($\sigma_{21}$) with probe detuning $\Delta_{p}$
for $\phi=0$ (solid line) and $\phi=\pi$ (dotted line) for $\Omega_{c}^{0}$
$=10$, $\Omega_{p}^{0}$ $=0.05$, $\Delta_{c}=\Delta_{1}=0$,
$\Omega_{1}^{0}=0.1$ and $\Omega_{2}^{0}=0.1$. The inner graph in Fig. 3(a) is
the absorption spectrum vesus $\Delta_{p}$ for three-level cascade driving
without the transitions coupled by the fields $\Omega_{1}$ and $\Omega_{2}$.
(b) Probe gain versus $\Omega_{2}^{0}$ for $\phi=0,\Delta_{p}=0$ (solid line)
and $\phi=\pi$, $\Delta_{p}=15.9$ (dotted line). The other parameters are the
same as those in Fig. 2(a). Figure 4: (a) Variation of Im($\sigma_{32}$) with
probe detuning $\Delta_{p}$ for $\phi=0$ (solid line) and $\phi=\pi$ (dotted
line) for $\Omega_{c}^{0}$ $=10$, $\Omega_{p}^{0}$ $=0.05$,
$\Delta_{c}=\Delta_{1}=0$, $\Omega_{1}^{0}=10$ and $\Omega_{2}^{0}=1$ (b)
Population difference $\sigma_{33}-\sigma_{22}$ vesus probe detuning
$\Delta_{p}$ corresponding to gain in (a).
Fig. 3(a) shows the case with small driving of one two-photon transition,
i.e., $\Omega_{1}^{0}$ $=\Omega_{2}^{0}$ $=0.1$ for $\phi=0$ (solid lines) and
$\phi=\pi$ (dotted lines) , and the other parameters are the same as those in
Fig. 2(a). Two feathers are presented. One is that the probe absorption
domains with two remarkable absorption peaks. The other is that when the phase
changes from $0$ to $\pi$, the probe gain around the resonant point goes into
the probe absorption while the dominated absorption keeps unchanged. In order
to see the effects of the two two-photon transitions, we plot the probe
absorption for the case of one two-photon transition
$|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$ with the same
parameters. The inner graph in Fig. 3(a) is the absorption spectrum vesus
$\Delta_{p}$ for three-level cascade driving without the transitions coupled
by the fields $\Omega_{1}$ and $\Omega_{2}$. The results show that
electromagnetically induced transparency can be obtained and no phase
dependence and no gain are found for single two-photon transition. Thus it is
the two two-photon transitions is the origin of the probe gain and the phase
dependence. In Fig. 3(b) the probe absorption Im($\sigma_{21}$) vesus
$\Omega_{2}^{0}$, the amplitude of $\Omega_{2}$ is also plotted for
$\phi=0,\Delta_{p}=0$ (solid line) and $\phi=\pi$, $\Delta_{p}=15.9$ (dotted
line). The other parameters are the same as those in Fig. 2(a). It is easy to
see that the parameters we choose in Fig. 2(a) are optimal for gain behavior.
Figure 5: (a) Variation of Im($\sigma_{32}$) with probe detuning $\Delta_{p}$
for $\phi=0$ (solid line) and $\phi=\pi$ (dotted line) for $\Omega_{c}^{0}$
$=10$, $\Omega_{p}^{0}$ $=0.05$, $\Delta_{c}=\Delta_{1}=0$,
$\Omega_{1}^{0}=0.1$ and $\Omega_{2}^{0}=0.1$. The inner graph in Fig. 5(a) is
the absorption spectrum vesus $\Delta_{p}$ for three-level cascade driving
without the transitions coupled by the fields $\Omega_{1}$ and $\Omega_{2}$.
(b) Probe gain versus $\Omega_{2}^{0}$ for $\phi=0,\Delta_{p}=20.2$ (solid
line) and $\phi=\pi$, $\Delta_{p}=0$ (dotted line). The other parameters are
the same as those in Fig. 4(a).
When we exchange the probe field $\Omega_{p}$ and the driving field
$\Omega_{c}$ with coupling scheme shown in Fig. $1$(b), gain without inversion
can also be obtained. We call this case up-level probing. The results are
presented in Fig. 4 and Fig. 5 with similar parameters as those in Fig. 2 and
Fig. 3. It is clear that gain without inversion also exhibits in such a system
and the phase dependence of the gain on relative phase $\phi$ is presented as
well. There are two differences between the low level probing and the up level
probing case. One is that the number of gain zone is different with the same
phase. For specific, when $\phi=0$, the low-level coupling has single gain
without inversion zone at $\Delta_{p}=0$ while the up-level coupling has two
gain without inversion zones at $\Delta_{p}=\pm 20.2$; when $\phi=\pi$, the
low-level coupling has two gain without inversion zones at $\Delta_{p}=\pm
15.9$ while the up-level coupling has only one gain without inversion zones at
$\Delta_{p}=0$. The other feature is that no gain is presented when the two-
photon transition
$|1\rangle\leftrightarrow|2^{{}^{\prime}}\rangle\leftrightarrow|3\rangle$ is
small when the phase changes from $0$ to $\pi$ and the phase just makes the
shift of the absorption a little. In a word the absorption spectra act like
the probe absorption for the case of one two-photon transition
$|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$ with the same
parameters except the phase influence. In Fig. 5(b) the dependence of probe
gain on $\Omega_{2}^{0}$ is also presented. From this, one can see that the
optimal parameters are used in Fig. 4(a).
To under stand the above results, we can simply use the steady state
expression of two terms as following
$\displaystyle\mathop{\rm Im}(\sigma_{21})$ $\displaystyle=$
$\displaystyle\frac{1}{D}\mathop{\rm
Im}[(\frac{\gamma_{p}}{2}-i\Delta_{p})(i\Omega_{c}^{0}\sigma_{31}-i\Omega_{2}^{0}\sigma_{22^{{}^{\prime}}})e^{i\phi}]$
(6)
$\displaystyle-\frac{\frac{1}{2}\Omega_{p}^{0}\gamma_{p}(\sigma_{22}-\sigma_{11})}{\frac{1}{4}\gamma_{p}^{2}+\Delta_{p}^{2}}$
$\displaystyle\mathop{\rm Im}(\sigma_{32})$ $\displaystyle=$
$\displaystyle\frac{1}{D}\mathop{\rm
Im}[(\frac{\gamma_{c}}{2}-i\Delta_{c})(-i\Omega_{p}^{0}e^{-i\phi}\sigma_{31}+i\Omega_{1}^{0}\sigma_{2^{{}^{\prime}}2})]$
(7)
$\displaystyle-\frac{\frac{1}{2}\Omega_{c}^{0}\gamma_{c}(\sigma_{33}-\sigma_{22})}{\frac{1}{4}\gamma_{c}^{2}+\Delta_{c}^{2}}$
where $D=\frac{1}{4}\gamma_{p}^{2}+\Delta_{p}^{2}$ . From eq. (6), it is easy
to see that when no population inversion happens, i.e.,
$(\sigma_{22}-\sigma_{11})<0$, the first term is always positive. So when gain
occurs (Im$(\sigma_{21})<0$), the second term contributes. Similar results
also hold for up-level probing case just by make the replace of $\sigma_{21}$
by $\sigma_{32}$ and exchange the field $\Omega_{c}$ and $\Omega_{p}$. Due to
pure absorption and no phase-dependent behavior of three-level cascade driving
$|1\rangle\leftrightarrow|2\rangle\leftrightarrow|3\rangle$ by $\Omega_{c}$
and $\Omega_{p}$, we can conclude that it is the two-photon transition
$|1\rangle\leftrightarrow|2^{{}^{\prime}}\rangle\leftrightarrow|3\rangle$
driving by $\Omega_{1}$ and $\Omega_{2}$ that induces the gain without
inversion and the two two-photon transitions
$|1\rangle\leftrightarrow|2(2^{{}^{\prime}})\rangle\leftrightarrow|3\rangle$
loop structure induces the phase dependence of the gain behavior.
In conclusion, gain without inversion in a four level loop-transition atomic
system has been investigated. The main results are two features: (i) Gain
without inversion is exhibited by varing the amplitude of coupled fields. (ii)
Gain without inversion displays a phase dependence on the relative phase
between the fields applied on the two two-photon transitions. As the phase
changes from 0 to $\pi$, gain and absorption zones exchange. Population
inversionless holds for all the case. Gain without inversion and phase-
dependence are attributed to interference induced by such a loop-transition
structure.
Acknowledgments This work is supported by the Scientific Research Plan of the
Provincial Education Department in Hubei (Grant No. Q20101304) and NSFC under
Grant No. 11147153.
## References
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|
arxiv-papers
| 2012-10-22T07:13:50 |
2024-09-04T02:49:36.925045
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinhua Zou, Dahai Xu and Huafeng Zhang",
"submitter": "Jin-hua Zou",
"url": "https://arxiv.org/abs/1210.5817"
}
|
1210.5882
|
# vanishing of cohomology over complete intersection rings
Arash Sadeghi School of Mathematics, Institute for Research in Fundamental
Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran. sadeghiarash61@gmail.com
(Date: March 18, 2013.)
###### Abstract.
Let $R$ be a complete intersection ring and let $M$ and $N$ be $R$–modules. It
is shown that the vanishing of $\operatorname{Ext}^{i}_{R}(M,N)$ for a certain
number of consecutive values of $i$ starting at $n$ forces the complete
intersection dimension of $M$ to be at most $n-1$. We also estimate the
complete intersection dimension of $M^{*}$, the dual of $M$, in terms of
vanishing of the cohomology modules, $\operatorname{Ext}^{i}_{R}(M,N)$.
###### Key words and phrases:
Complete intersection dimension, Complexity, Vanishing of cohomology,
Grothendieck group.
###### 2000 Mathematics Subject Classification:
13D07, 13H10
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Vanishing of Ext for rigid modules
4. 4 Vanishing of Ext over complete intersection rings
## 1\. Introduction
In this paper, we study the relationship between the vanishing of
$\operatorname{Ext}^{i}_{R}(M,N)$ for various consecutive values of $i$, and
the complete intersection dimensions of $M$ and $M^{*}$, the dual of $M$. The
vanishing of homology was first studied by Auslander [3]. For two finitely
generated modules $M$ and $N$ over an unramified regular local ring $R$, he
proved that if $\operatorname{Tor}_{i}^{R}(M,N)=0$ for some $i>0$, then
$\operatorname{Tor}_{n}^{R}(M,N)=0$ for all $i\geq n$. In [17], Lichtenbaum
settled the ramified case. It is easy to see that a similar statement is not
true in general, with Tor replaced by Ext. In [15], Jothilingam studied the
vanishing of cohomology by using the rigidity Theorem of Auslander. For two
nonzero modules $M$ and $N$ over a regular local ring $R$, he proved that if
$M$ satisfies $(S_{n})$ for some $n\geq 0$ and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$, then $\operatorname{Ext}^{j}_{R}(M,N)=0$
for all $j\geq i$. In [16], Jothilingam and Duraivel studied the relationship
between the vanishing of $\operatorname{Ext}^{i}_{R}(M,N)$ and the freeness of
$M^{*}$. For two nonzero modules $M$ and $N$ over a regular local ring $R$,
they proved that if $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free. In this
paper we are going to generalize these results.
An $R$–module $M$ is said to be c-rigid if for all $R$–modules $N$,
$\operatorname{Tor}_{i+1}^{R}(M,N)=\operatorname{Tor}_{i+2}^{R}(M,N)=\cdots=\operatorname{Tor}_{i+c}^{R}(M,N)=0$
for some $i\geq 0$ implies that $\operatorname{Tor}_{n}^{R}(M,N)=0$ for all
$n>i$. If $c=1$ then we simply say that $M$ is rigid.
The aim of this paper is to study the following question.
###### Question 1.1.
Let $R$ be a Gorenstein local ring and let $M$ and $N$ be $R$–modules such
that $N$ has reducible complexity. Assume that $n\geq 0$, $c>0$ are integers
and that $N$ is $c$-rigid. If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$,
$1\leq i\leq\max\\{c,\operatorname{depth}_{R}(N)-n\\}$, then what can we say
about the Gorenstein dimensions of $M$ and $M^{*}$?
In section 2, we collect necessary notations, definitions and some known
results which will be used in this paper.
In section 3, we study the Question 1.1 for rigid modules. Over a Gorenstein
local ring $R$, given nonzero $R$-modules $M$ and $N$ such that $N$ has
reducible complexity, we show that if $N$ is rigid and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$ and some $n\geq 2$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$, which is a
generalization of [16, Theorem 1]. In particular, if $M$ satisfies $(S_{n})$,
then $\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$ (see Theorem 3.2). As a
consequence, for two nonzero modules $M$ and $N$ over a complete intersection
ring $R$, it is shown that if $N$ is rigid and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer
$i\geq\operatorname{depth}_{R}(N)$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<i$ (see
Theorem 3.5).
In section 4, we generalize [15, Corollary 1] for modules over a complete
intersection ring. For two modules $M$ and $N$ over a complete intersection
ring $R$ with codimension $c$, it is shown that if $M$ satisfies $(S_{t})$ for
some $t\geq 0$, $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $n\leq i\leq
n+c$ and some $n>0$ and $\operatorname{depth}_{R}(N)\leq n+c+t$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ (see
Corollary 4.3).
## 2\. Preliminaries
Throughout the paper, $(R,\mathfrak{m})$ is a commutative Noetherian local
ring and all modules are finite (i.e. finitely generated) $R$–modules. The
codimension of $R$ is defined to be the non-negative integer
$\operatorname{embdim}(R)-\dim(R)$ where $\operatorname{embdim}(R)$, the
embedding dimension of $R$, is the minimal number of generators of
$\mathfrak{m}$. Recall that $R$ is said to be a complete intersection if the
$\mathfrak{m}$-adic completion $\widehat{R}$ of $R$ has the form $Q/(f)$,
where $f$ is a regular sequence of $Q$ and $Q$ is a regular local ring. A
complete intersection of codimension one is called a hypersurface. A local
ring $R$ is said to be an admissible complete intersection if the
$\mathfrak{m}$-adic completion $\widehat{R}$ of $R$ has the form $Q/(f)$,
where $f$ is a regular sequence of $Q$ and $Q$ is a power series ring over a
field or a discrete valuation ring. Let
$\cdots\rightarrow F_{n+1}\rightarrow F_{n}\rightarrow
F_{n-1}\rightarrow\cdots\rightarrow F_{0}\rightarrow M\rightarrow 0$
be the minimal free resolution of $M$. Recall that the $n^{\text{th}}$ syzygy
of an $R$–module $M$ is the cokernel of the $F_{n+1}\rightarrow F_{n}$ and
denoted by $\Omega^{n}M$, and it is unique up to isomorphism. The
$n^{\text{th}}$ Betti number, denoted $\beta_{n}^{R}(M)$, is the rank of the
free $R$–module $F_{n}$. The complexity of $M$ is defined as follows.
$\operatorname{cx}_{R}(M)=\inf\\{i\in\mathbb{N}\cup
0\mid\exists\gamma\in\mathbb{R}\text{ such that }\beta_{n}^{R}(M)\leq\gamma
n^{i-1}\text{ for }n\gg 0\\}.$
Note that $\operatorname{cx}_{R}(M)=\operatorname{cx}_{R}(\Omega^{i}M)$ for
every $i\geq 0$. It follows from the definition that
$\operatorname{cx}_{R}(M)=0$ if and only if $\operatorname{pd}_{R}(M)<\infty$.
If $R$ is a complete intersection, then the complexity of $M$ is less than or
equal to the codimension of $R$ (see [12]). The complete intersection
dimension was introduced by Avramov, Gasharov and Peeva [6]. A module of
finite complete intersection dimension behaves homologically like a module
over a complete intersection. Recall that a quasi-deformation of $R$ is a
diagram $R\rightarrow A\twoheadleftarrow Q$ of local homomorphisms, in which
$R\rightarrow A$ is faithfully flat, and $A\twoheadleftarrow Q$ is surjective
with kernel generated by a regular sequence. The module $M$ has finite
complete intersection dimension if there exists such a quasi-deformation for
which $\operatorname{pd}_{Q}(M\otimes_{R}A)$ is finite. The complete
intersection dimension of $M$, denoted $\operatorname{\textnormal{CI-
dim}}_{R}(M)$, is defined as follows.
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\inf\\{\operatorname{pd}_{Q}(M\otimes_{R}A)-\operatorname{pd}_{Q}(A)\mid
R\rightarrow A\twoheadleftarrow Q\text{ is a quasi-deformation }\\}.$
The complete intersection dimension of $M$ is bounded above by the projective
dimension, $\operatorname{pd}_{R}(M)$, of $M$ and if
$\operatorname{pd}_{R}(M)<\infty$, then the equality holds (see [6, Theorem
1.4]). Every module of finite complete intersection dimension has finite
complexity (see [6, Theorem 5.3]).
The concept of modules with reducible complexity was introduced by Bergh [7].
Let $M$ and $N$ be $R$–modules and consider a homogeneous element $\eta$ in
the graded $R$–module
$\operatorname{Ext}^{*}_{R}(M,N)=\bigoplus^{\infty}_{i=0}\operatorname{Ext}^{i}_{R}(M,N)$.
Choose a map $f_{\eta}:\Omega^{|\eta|}_{R}(M)\rightarrow N$ representing
$\eta$, and denote by $K_{\eta}$ the pushout of this map and the inclusion
$\Omega^{|\eta|}_{R}(M)\hookrightarrow F_{|\eta|-1}$. Therefore we obtain a
commutative diagram
$\setcounter{MaxMatrixCols}{14}\begin{CD}&&&&&&&&\\\ \ \
&&&&0@>{}>{}>\Omega^{|\eta|}M@>{}>{}>F_{|\eta|-1}@>{}>{}>\Omega^{|\eta|-1}M@>{}>{}>0&\\\
&&&&&&@V{}V{f_{\eta}}V@V{}V{}V@V{}V{{\parallel}}V\\\ \ \
&&&&0@>{}>{}>N@>{}>{}>K_{\eta}@>{}>{}>\Omega^{|\eta|-1}M@>{}>{}>0.&\\\
\end{CD}$
with exact rows. Note that the module $K_{\eta}$ is independent, up to
isomorphism, of the map $f_{\eta}$ chosen to represent ${\eta}$.
###### Definition 2.1.
The full subcategory of $R$-modules consisting of the modules having reducible
complexity is defined inductively as follows:
* (i)
Every $R$-module of finite projective dimension has reducible complexity.
* (ii)
An $R$-module $M$ of finite positive complexity has reducible complexity if
there exists a homogeneous element $\eta\in\operatorname{Ext}^{*}_{R}(M,M)$,
of positive degree, such that
$\operatorname{cx}_{R}(K_{\eta})<\operatorname{cx}_{R}(M)$,
$\operatorname{depth}_{R}(M)=\operatorname{depth}_{R}(K_{\eta})$ and
$K_{\eta}$ has reducible complexity.
By [7, Proposition 2.2(i)], every module of finite complete intersection
dimension has reducible complexity. In particular, every module over a local
complete intersection ring has reducible complexity. On the other hand, there
are modules having reducible complexity but whose complete intersection
dimension is infinite (see for example, [9, Corollarry 4.7]).
The notion of the Gorenstein(or G-) dimension was introduced by Auslander [2],
and developed by Auslander and Bridger in [4].
###### Definition 2.2.
An $R$–module $M$ is said to be of $G$-dimension zero whenever
* (i)
_the biduality map $M\rightarrow M^{**}$ is an isomorphism._
* (ii)
_$\operatorname{Ext}^{i}_{R}(M,R)=0$ for all $i>0$._
* (iii)
_$\operatorname{Ext}^{i}_{R}(M^{*},R)=0$ for all $i>0$._
The Gorenstein dimension of $M$, denoted
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)$, is defined to be the infimum
of all nonnegative integers $n$, such that there exists an exact sequence
$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow M\rightarrow
0$
in which all the $G_{i}$ have $G$-dimension zero. By [4, Theorem 4.13], if $M$
has finite Gorenstein dimension, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=\operatorname{depth}R-\operatorname{depth}_{R}(M)$.
By [6, Theorem 1.4], $\operatorname{\textnormal{G-dim}}-dim_{R}(M)$ is bounded
above by the complete intersection dimension, $\operatorname{\textnormal{CI-
dim}}_{R}(M)$, of $M$ and if $\operatorname{\textnormal{CI-
dim}}_{R}(M)<\infty$, then the equality holds.
Let $R$ be a local ring and let $M$ and $N$ be finite nonzero $R$-modules. We
say the pair $(M,N)$ satisfies the depth formula provided:
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N).$
The depth formula was first studied by Auslander [3] for finite modules of
finite projective dimension. In [13], Huneke and Wiegand proved that the depth
formula holds for $M$ and $N$ over complete intersection rings $R$ provided
$\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>0$. In [9], Bergh and Jorgensen
generalize this result for modules with reducible complexity over a local
Gorenstein ring. More precisely, they proved the following result:
###### Theorem 2.3.
[9, Corollary 3.4] Let $R$ be a Gorenstein local ring and let $M$ and $N$ be
nonzero $R$–modules. If $M$ has reducible complexity and
$\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>0$, then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.
We denote by $G(R)$ the Grothendieck group of finite modules over $R$, that
is, the quotient of the free abelian group of all isomorphism classes of
finite $R$–modules by the subgroup generated by the relations coming from
short exact sequences of finite $R$-modules. We also denote by
$\overline{G}(R)=G(R)/[R]$, the reduced Grothendieck group. For an abelian
group $G$, we set $G_{\mathbb{Q}}=G\otimes_{\mathbb{Z}}\mathbb{Q}$.
Let $P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow M\rightarrow 0$ be a finite
projective presentation of $M$. The transpose of $M$, $\operatorname{Tr}M$, is
defined to be $\operatorname{coker}f^{*}$, where
$(-)^{*}:=\operatorname{Hom}_{R}(-,R)$, which satisfies in the exact sequence
(2.1) $0\rightarrow M^{*}\rightarrow P_{0}^{*}\rightarrow
P_{1}^{*}\rightarrow\operatorname{Tr}M\rightarrow 0$
and is unique up to projective equivalence. Thus the minimal projective
presentations of $M$ represent isomorphic transposes of $M$. Two modules $M$
and $N$ are called _stably isomorphic_ and write $M\approx N$ if $M\oplus
P\cong N\oplus Q$ for some projective modules $P$ and $Q$. Note that
$M^{*}\approx\Omega^{2}\operatorname{Tr}M$ by the exact sequence (2.1).
The composed functors $\mathcal{T}_{k}:=\operatorname{Tr}\Omega^{k-1}$ for
$k>0$ introduced by Auslander and Bridger in [4]. If
$\operatorname{Ext}^{i}_{R}(M,R)=0$ for some $i>0$, then it is easy to see
that $\mathcal{T}_{i}M\approx\Omega\mathcal{T}_{i+1}M$.
We frequently use the following Theorem of Auslander and Bridger.
###### Theorem 2.4.
[4, Theorem 2.8] Let $M$ be an $R$–module and $n\geq 0$ an integer. Then there
are exact sequences of functors:
(2.4.1)
$0\rightarrow\operatorname{Ext}^{1}_{R}(\mathcal{T}_{n+1}M,-)\rightarrow\operatorname{Tor}_{n}^{R}(M,-)\rightarrow\operatorname{Hom}_{R}(\operatorname{Ext}^{n}_{R}(M,R),-)\rightarrow\operatorname{Ext}^{2}_{R}(\mathcal{T}_{n+1}M,-),$
(2.4.2)
$\operatorname{Tor}_{2}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow(\operatorname{Ext}^{n}_{R}(M,R)\otimes_{R}-)\rightarrow\operatorname{Ext}^{n}_{R}(M,-)\rightarrow\operatorname{Tor}_{1}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow
0.$
For an integer $n\geq 0$, we say $M$ satisfies $(S_{n})$ if
$\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\\{n,\dim(R_{\mathfrak{p}})\\}$
for all $\mathfrak{p}\in\operatorname{Spec}(R)$. If $R$ is Gorenstein, then
$M$ satisfies $(S_{n})$ if and only if
$\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)=0$ for all $1\leq i\leq n$
(see [4, Theorem 4.25]). In particular, $M$ satisfies $(S_{2})$ if and only if
it is reflexive, i.e., the natural map $M\rightarrow M^{**}$ is bijective,
where $M^{*}=\operatorname{Hom}_{R}(M,R)$ (see [11, Theorem 3.6]).
The following results will be used throughout the paper.
###### Theorem 2.5.
Let $R$ be a local complete intersection ring and let $M$ and $N$ be
$R$–modules. Then $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\gg 0$ if and
only if $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$. Moreover, if $R$
is a hypersurface, then either $\operatorname{pd}_{R}(M)<\infty$ or
$\operatorname{pd}_{R}(N)<\infty$.
###### Proof.
See [5, Theorem 6.1] and [5, Proposition 5.12]. ∎
###### Theorem 2.6.
Let $R$ be a local ring and let $M$ and $N$ be nonzero $R$–modules. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$ and
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)<\infty$, then the following
statements hold true.
1. (i)
If $\operatorname{\textnormal{CI-dim}}_{R}(M)<\infty$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}$.
2. (ii)
If $\operatorname{\textnormal{CI-dim}}_{R}(N)<\infty$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq
0\\}$.
###### Proof.
See [1, Theorem 4.2] and [21, Theorem 4.4]. ∎
###### Theorem 2.7.
Let $R$ be a local ring, and $M$, $N$ two $R$–modules. If
$\operatorname{\textnormal{CI-dim}}_{R}(M)=0$, then
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$ if and only if
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$.
###### Proof.
First note that $\operatorname{\textnormal{CI-dim}}_{R}(\operatorname{Tr}M)=0$
by [21, Lemma 3.3] and $M\approx\operatorname{Tr}\operatorname{Tr}M$. Now the
assertion is clear by [21, Proposition 3.4]. ∎
## 3\. Vanishing of Ext for rigid modules
We start this section by estimate the Gorenstein dimension of the transpose of
$M$ in terms of vanishing of the cohomology modules,
$\operatorname{Ext}^{i}_{R}(M,N)$.
###### Lemma 3.1.
Let $R$ be a Gorenstein ring and let $M$ and $N$ be nonzero $R$–modules.
Assume that $n\geq 0$ is an integer and that the following conditions hold.
1. (1)
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$.
2. (2)
$N$ is rigid.
3. (3)
$N$ has reducible complexity.
Then $\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ and
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$.
###### Proof.
If $\operatorname{Tr}M=0$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ and we have
nothing to prove so let $\operatorname{Tr}M\neq 0$. As
$\operatorname{Ext}^{1}_{R}(M,N)=0$,
$\operatorname{Tor}_{1}^{R}(\mathcal{T}_{2}M,N)=0$ by the exact sequence
(2.4.2). Since $N$ is rigid, we have
$\operatorname{Tor}_{i}^{R}(\mathcal{T}_{2}M,N)=0$ for all $i>0$. It follows
from the exact sequence (2.4.2) again that
$\operatorname{Ext}^{1}_{R}(M,R)\otimes_{R}N=0$ and since $N$ is nonzero,
$\operatorname{Ext}^{1}_{R}(M,R)=0$. Now it is easy to see that
$\mathcal{T}_{1}M\approx\Omega\mathcal{T}_{2}M$ and so
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$. Therefore
we have the following equality.
(3.1.1)
$\operatorname{depth}_{R}(\operatorname{Tr}M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(\operatorname{Tr}M)+\operatorname{depth}_{R}(N),$
by Theorem 2.3. Set $t=\operatorname{depth}_{R}(N)-n$. We argue by induction
on $t$. If $t\leq 1$, then $\operatorname{depth}_{R}(N)\leq n+1$. If
$\operatorname{depth}_{R}(N)=0$, then it is clear that
$\operatorname{depth}_{R}(\operatorname{Tr}M)=\operatorname{depth}R$ by
(3.1.1) and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ by
Auslander-Bridger formula. Now let $0<\operatorname{depth}_{R}(N)\leq n+1$. As
$M\approx\operatorname{Tr}\operatorname{Tr}M$, we obtain the following exact
sequence
$0\rightarrow\operatorname{Ext}^{1}_{R}(M,N)\rightarrow\operatorname{Tr}M\otimes_{R}N\rightarrow\operatorname{Hom}_{R}((\operatorname{Tr}M)^{*},N)\rightarrow\operatorname{Ext}^{2}_{R}(M,N),$
from the exact sequence (2.4.1). As $\operatorname{Ext}^{1}_{R}(M,N)=0$, we
get the following exact sequence.
(3.1.2)
$0\rightarrow\operatorname{Tr}M\otimes_{R}N\rightarrow\operatorname{Hom}_{R}((\operatorname{Tr}M)^{*},N)\rightarrow\operatorname{Ext}^{2}_{R}(M,N).$
Therefore
$\operatorname{Ass}_{R}(\operatorname{Tr}M\otimes_{R}N)\subseteq\operatorname{Ass}_{R}(\operatorname{Hom}_{R}((\operatorname{Tr}M)^{*},N))\subseteq\operatorname{Ass}_{R}(N)$,
by the exact sequence (3.1.2). Hence
$\operatorname{depth}_{R}(\operatorname{Tr}M\otimes_{R}N)>0$. Now by (3.1.1),
it is easy to see that
$\operatorname{depth}_{R}(\operatorname{Tr}M)\geq\operatorname{depth}R-n$ and
so $\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$.
Now suppose that $t>1$ and consider the following exact sequence
(3.1.3) $0\rightarrow\Omega M\rightarrow F\rightarrow M\rightarrow 0,$
where $F$ is a free $R$–module. From the exact sequence (3.1.3), we obtain the
following exact sequence
$0\rightarrow M^{*}\rightarrow F^{*}\rightarrow(\Omega
M)^{*}\rightarrow\mathbb{D}(M)\rightarrow\mathbb{D}(F)\rightarrow\mathbb{D}(\Omega
M)\rightarrow 0.$
Where $\mathbb{D}(X)\approx\operatorname{Tr}X$ for all $R$–modules $X$ by [4,
Lemma 3.9]. As $\operatorname{Ext}^{1}_{R}(M,R)=0$, we get the following exact
sequence
(3.1.4)
$0\rightarrow\mathbb{D}(M)\rightarrow\mathbb{D}(F)\rightarrow\mathbb{D}(\Omega
M)\rightarrow 0.$
Note that $\mathbb{D}(F)$ is free. As $\operatorname{Ext}^{i}_{R}(\Omega
M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)=0$ for all $1\leq
i\leq\operatorname{depth}_{R}(N)-n-1$, we have
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}\Omega M)\leq n+1$
by induction hypothesis. Therefore,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ by the
exact sequence (3.1.4). ∎
###### Theorem 3.2.
Let $R$ be a Gorenstein ring and let $M$ and $N$ be nonzero $R$–modules such
that $N$ has reducible complexity. Assume that $N$ is rigid and that $n\geq 0$
is an integer. Then the following statements hold true.
1. (i)
If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$ and $M$ satisfies $(S_{n})$,
then $\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$.
2. (ii)
If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$.
###### Proof.
(i). First note that
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)\neq
0\\}$ by [4, Theorem 4.13]. As $M$ satisfies $(S_{n})$,
$\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)=0$ for all $1\leq i\leq n$
by [4, Theorem 4.25]. On the other hand,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ by Lemma
3.1. Therefore
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$ by [4, Lemmm 4.9].
(ii). Note that $M^{*}\approx\Omega^{2}\operatorname{Tr}M$. By Lemma 3.1,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$. ∎
The following is a generalization of [16, Theorem 1].
###### Corollary 3.3.
Let $R$ be a complete intersection and let $M$ and $N$ be nonzero $R$–modules.
Assume that $N$ is a rigid module of maximal complexity. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free.
###### Proof.
By Lemma 3.1, $\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all
$i>0$. As $M^{*}\approx\Omega^{2}\operatorname{Tr}M$,
$\operatorname{Tor}_{i}^{R}(M^{*},N)=0$ for all $i>0$ and so
$\operatorname{cx}_{R}(M^{*})+\operatorname{cx}_{R}(N)\leq\operatorname{codim}R$
by [5, Theorem II]. Since $N$ has maximal complexity, it follows that
$\operatorname{cx}_{R}(M^{*})=0$. Therefore,
$\operatorname{pd}_{R}(M^{*})=\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})=0$
by Theorem 3.2(ii). ∎
It is well-known that over a regular local ring every finite module is rigid.
In the following we collect some other examples of rigid modules.
###### Example 3.4.
1. (i)
A class of rigid modules was discovered by Peskine and Szpiro [19]. They
proved that if $R$ is local, and the minimal free resolution of $M$ over $R$
is of the form
$0\rightarrow R^{m}\rightarrow R^{k+m}\rightarrow R^{k}\rightarrow 0,$
for some $m>0$ and $k>0$, then $M$ is rigid. In [22], Tchernev discovered a
new class of rigid modules. He showed that if $R$ is local, and the minimal
free resolution of $M$ over $R$ is of the form
$0\rightarrow R^{k}\rightarrow R^{m+1}\rightarrow R^{m}\rightarrow 0,$
for some $m>0$ and $k>0$, then $M$ is rigid ([22, Theorem 3.6]).
2. (ii)
Let $R$ be an admissible hypersurface with isolated singularity and let $N$ be
an $R$–module. If $[N]=0$ in $\overline{G}(R)_{\mathbb{Q}}$, then $N$ is rigid
[10, Corollary 4.2].
3. (iii)
Let $(R,\mathfrak{m})$ be a local hypersurface ring such that
$\widehat{R}=S/(f)$ where $(S,\mathfrak{n})$ is a complete unramified regular
local ring and $f$ is a regular element of $S$ contained in $n^{2}$. Let $M$
be an $R$-module of finite projective dimension. Then $M$ is rigid [17,
Theorem 3].
In the following, we generalize [15, Corollary 1].
###### Theorem 3.5.
Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero
$R$–modules. Assume the following conditions hold.
1. (i)
$N$ is rigid.
2. (ii)
$M$ satisfies $(S_{n})$ for some $n\geq 0$.
3. (iii)
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$.
Then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{j\mid\operatorname{Ext}^{j}_{R}(M,N)\neq 0\\}<i$.
###### Proof.
Set $L=\Omega^{i-1}M$. Note that $L$ satisfies $(S_{n+i-1})$ and
$\operatorname{Ext}^{1}_{R}(L,N)=0$. Now by Theorem 3.2(i),
$\operatorname{\textnormal{CI-
dim}}_{R}(L)=\operatorname{\textnormal{G-dim}}-dim_{R}(L)=0$. By Lemma 3.1,
$\operatorname{Tor}_{j}^{R}(\operatorname{Tr}L,N)=0$ for all $j>0$ and so
$\operatorname{Ext}^{j}_{R}(L,N)=0$ for all $j>0$ by Theorem 2.7. Therefore
$\operatorname{Ext}^{j}_{R}(M,N)=0$ for all $j\geq i$ and so
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{j\mid\operatorname{Ext}^{j}_{R}(M,N)\neq 0\\}<i$ by
Theorem 2.6. ∎
The following is a generalization of [15, Corollary 2]
###### Theorem 3.6.
Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero
$R$–modules. Suppose that $N$ is rigid and that $M$ satisfies $(S_{n})$ for
some $n\geq 0$. If
$\operatorname{depth}_{R}(N)-n\leq\operatorname{\textnormal{CI-dim}}_{R}(M)$,
then for all $i>0$ in the range $\operatorname{depth}_{R}(N)-n\leq
i\leq\operatorname{\textnormal{CI-dim}}_{R}(M)$, we have
$\operatorname{Ext}^{i}_{R}(M,N)\neq 0$.
###### Proof.
If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for some
$\operatorname{depth}_{R}(N)-n\leq i\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then
$\operatorname{Ext}^{1}_{R}(\Omega^{i-1}M,N)\cong\operatorname{Ext}^{i}_{R}(M,N)=0$.
Note that $\Omega^{i-1}M$ satisfies $(S_{n+i-1})$. Now by Theorem 3.2(i), we
have $\operatorname{\textnormal{CI-
dim}}_{R}(\Omega^{i-1}M)=\operatorname{\textnormal{G-dim}}-dim_{R}(\Omega^{i-1}M)=0$.
Therefore $\operatorname{\textnormal{CI-dim}}_{R}(M)<i$ by [6, Lemma 1.9],
which is a contradiction. ∎
Let $R$ be a hypersurface and let $M$ and $N$ be $R$–modules such that
$\operatorname{length}_{R}(N)<\infty$. It is well-known that if
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some $i>\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then $\operatorname{Ext}^{n}_{R}(M,N)=0$ for all
$n>\operatorname{\textnormal{CI-dim}}_{R}(M)$ (see for example [8, Corollary
3.5]). In special cases, we can remove the condition that
$i>\operatorname{\textnormal{CI-dim}}_{R}(M)$.
###### Corollary 3.7.
Let $(R,\mathfrak{m})$ be a local hypersurface ring such that
$\widehat{R}=S/(f)$ where $(S,\mathfrak{n})$ is a complete unramified regular
local ring and $f$ is a regular element of $S$ contained in
${\mathfrak{n}}^{2}$. Let $M$ and $N$ be nonzero $R$-modules such that
$\operatorname{length}_{R}(N)<\infty$. If $\operatorname{Ext}^{n}_{R}(M,N)=0$
for some $n\geq 1$, then the following statements hold true.
1. (i)
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.
2. (ii)
either $\operatorname{pd}_{R}(M)<\infty$ or $\operatorname{pd}_{R}(N)<\infty$.
###### Proof.
First note that $N$ is rigid by [13, Theorem 2.4]. It follows from Theorem 3.5
that $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$. As
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$, either
$\operatorname{pd}_{R}(M)<\infty$ or $\operatorname{pd}_{R}(N)<\infty$ by
Theorem 2.5. ∎
As an application of Theorem 3.2, we have the following result.
###### Corollary 3.8.
Let $R$ be an admissible hypersurface and let $M$ and $N$ be nonzero
$R$–modules such that $\operatorname{cx}_{R}(N)=1$. Assume that the minimal
free resolution of $N$ is eventually periodic of period one and that $M$
satisfies $(S_{n})$ for some $n\geq 0$. Then the following statements hold
true.
1. (i)
If $\operatorname{depth}_{R}(N)-n\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then for all $i>0$ in the range
$\operatorname{depth}_{R}(N)-n\leq i\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, we have $\operatorname{Ext}^{i}_{R}(M,N)\neq 0$.
2. (ii)
If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$, then $\operatorname{pd}_{R}(M)<i$.
3. (iii)
If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free.
###### Proof.
Note that $N$ is rigid by [10, Corollary 5.6]. Now the first assertion is
clear by Theorem 3.6.
(ii). By Theorem 3.5, $\operatorname{Ext}^{j}_{R}(M,N)=0$ for all $j\geq i$.
Therefore, $\operatorname{pd}_{R}(M)<\infty$ by Theorem 2.5 and so
$\operatorname{pd}_{R}(M)<i$.
(iii). Note that $N$ has maximal complexity. Therefore, the assertion is clear
by Corollary 3.3. ∎
Let $R$ be an admissible hypersurface with isolated singularity of dimension
$d>1$. By [10, Theorem 3.4], every $R$–module of dimension less than or equal
one is rigid. As an immediate consequence of Theorem 3.5, we have the
following result.
###### Corollary 3.9.
Let $R$ be an admissible hypersurface with isolated singularity of dimension
$d>1$ and let $M$ and $N$ be nonzero $R$–modules such that $\dim_{R}(N)\leq
1$. If $\operatorname{Ext}^{n}_{R}(M,N)=0$ for some $n>0$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.
Moreover, either $\operatorname{pd}_{R}(M)<\infty$ or
$\operatorname{pd}_{R}(N)<\infty$.
In the dimension 2 case, we have the following result.
###### Proposition 3.10.
Let $R$ be an admissible hypersurface of dimension $2$. Assume further that
$R$ is normal. Let $M$ and $N$ be nonzero $R$–modules such that
$\operatorname{depth}_{R}(N)\leq\operatorname{depth}_{R}(M)+1$. If
$\operatorname{Ext}^{1}_{R}(M,N)=0$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=0$ and $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$.
Moreover, either $M$ is free or $N$ has finite projective dimension.
###### Proof.
First note that $N$ is rigid by [10, Corollary 3.6]. If
$\operatorname{depth}_{R}(N)\leq 1$, then the assertion is clear by Theorem
3.5. Now let $N$ be maximal Cohen-Macaulay. Then
$\operatorname{depth}_{R}(M)>0$ and so
(3.10.1) $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,R)\neq
0\\}=2-\operatorname{depth}_{R}(M)\leq 1.$
By Theorem 2.4, $\operatorname{Tor}_{1}^{R}(\mathcal{T}_{2}M,N)=0$. As $N$ is
rigid, $\operatorname{Tor}_{i}^{R}(\mathcal{T}_{2}M,N)=0$ for all $i>0$. It
follows from Theorem 2.4 again that $\operatorname{Ext}^{1}_{R}(M,R)=0$ and so
$M$ is maximal Cohen-Macaulay by (3.10.1). Now it is easy to see that
$\operatorname{Tr}M\approx\Omega\mathcal{T}_{2}M$ and so
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$. Therefore,
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$ by Theorem 2.7 and so either
$M$ is free or $N$ has finite projective dimension by Theorem 2.5. ∎
## 4\. Vanishing of Ext over complete intersection rings
Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be $R$–modules. In [18], Murthy proved that if
$\operatorname{Tor}_{n}^{R}(M,N)=\operatorname{Tor}_{n+1}^{R}(M,N)=\cdots=\operatorname{Tor}_{n+c}^{R}(M,N)=0$
for some $n>0$, then $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\geq n$. It
is easy to see that a similar statement is not true in general, with Tor
replaced by Ext. In the following, we prove a similar result for Ext with an
extra hypothesis. The following result is a generalization of [14, Corollary].
###### Theorem 4.1.
Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be nonzero $R$–modules. Assume $n$ is a positive integer. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$, for all $i$, $n\leq i\leq n+c$ and
$\operatorname{depth}_{R}(N)\leq n+c$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.
###### Proof.
Without lose of generality we may assume that $R$ is complete. We have
$R=Q/(x)$ with $Q$ a complete regular local ring and $x$ an $Q$-sequence of
length $c$ contained in the square of the maximal ideal of $Q$. We argue by
induction on $c$. If $c=0$, then $R$ is a regular local ring and so
$\operatorname{pd}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq
0\\}<n$ by [15, Corollary 1]. For $c>0$, set $S=Q/(x_{1},\ldots,x_{c-1})$.
Therefore $R\cong S/(x_{c})$. Note that
$\operatorname{depth}_{R}(N)=\operatorname{depth}_{S}(N)$.
The change of rings spectral sequence (see [20, Theorem 11.66])
$\operatorname{Ext}^{p}_{R}(M,\operatorname{Ext}^{q}_{S}(R,N))\underset{p}{\Rightarrow}\operatorname{Ext}^{p+q}_{S}(M,N)$
degenerates into a long exact sequence
$\cdots\rightarrow\operatorname{Ext}^{i}_{R}(M,N)\rightarrow\operatorname{Ext}^{i}_{S}(M,N)\rightarrow\operatorname{Ext}^{i-1}_{R}(M,N)\rightarrow\operatorname{Ext}^{i+1}_{R}(M,N)\rightarrow\cdots.$
It follows that $\operatorname{Ext}^{i}_{S}(M,N)=0$ for all $i$, $n+1\leq
i\leq n+c$, and so by induction hypothesis we conclude that
$\operatorname{\textnormal{CI-
dim}}_{S}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{S}(M,N)\neq 0\\}<n+1$.
Therefore,
$\operatorname{Ext}^{i-1}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)$ for
all $i>n$. As $c>0$, it is clear that $\operatorname{Ext}^{i}_{R}(M,N)=0$ for
all $i\geq n$ and so $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ by
Theorem 2.6. ∎
In special cases, one can improve the Theorem 4.1 slightly. The following is a
generalization of corollary 3.7.
###### Proposition 4.2.
Let $(R,\mathfrak{m})$ be a local ring such that $\widehat{R}=S/(f)$ where
$(S,\mathfrak{n})$ is a complete unramified regular local ring and
$f=f_{1},f_{2},\ldots,f_{c}$ is a regular sequence of $S$ contained in
${\mathfrak{n}}^{2}$. Assume that $n\geq 0$ is an integer and that $M$ and $N$
are nonzero finite $R$-modules such that
$\operatorname{length}_{R}(N)<\infty$. If $\operatorname{Ext}^{i}_{R}(M,N)=0$
for all $i$, $n+1\leq i\leq n+c$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}\leq n$.
###### Proof.
Without lose of generality we may assume that $R$ is complete and $R=S/(f)$
where $(S,\mathfrak{n})$ is a complete unramified regular local ring and
$f=f_{1},f_{2},\ldots,f_{c}$ is a regular sequence of $S$ contained in
${\mathfrak{n}}^{2}$. We argue by induction on $c$. If $c=1$, then the
assertion holds by Corollary 3.7. For $c>1$, set $Q=S/(f_{1},\ldots,f_{c-1})$.
Therefore, $R\cong Q/(f_{c})$. Note that
$\operatorname{length}_{Q}(N)<\infty$. The change of rings spectral sequence
$\operatorname{Ext}^{p}_{R}(M,\operatorname{Ext}^{q}_{Q}(R,N))\underset{p}{\Rightarrow}\operatorname{Ext}^{p+q}_{Q}(M,N)$
degenerates into a long exact sequence
$\cdots\rightarrow\operatorname{Ext}^{i}_{R}(M,N)\rightarrow\operatorname{Ext}^{i}_{Q}(M,N)\rightarrow\operatorname{Ext}^{i-1}_{R}(M,N)\rightarrow\operatorname{Ext}^{i+1}_{R}(M,N)\rightarrow\cdots.$
It follows that $\operatorname{Ext}^{i}_{Q}(M,N)=0$ for all $i$, $n+2\leq
i\leq n+c$, and so by induction hypothesis we conclude that
$\operatorname{\textnormal{CI-dim}}_{Q}(M)\leq n+1$ and
$\operatorname{Ext}^{i}_{Q}(M,N)=0$ for all $i>n+1$. Therefore,
$\operatorname{Ext}^{i-1}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)$ for
all $i>n+1$. As $c>1$, it is clear that $\operatorname{Ext}^{i}_{R}(M,N)=0$
for all $i>n$ and so $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}\leq n$ by
Theorem 2.6. ∎
As an application of Theorem 4.1, we can generalize [15, Corollary 1] as
follows.
###### Corollary 4.3.
Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be nonzero $R$–modules. Assume that $n>0$ and $t\geq 0$ are integers
and that the following conditions hold.
1. (i)
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $n\leq i\leq n+c$.
2. (ii)
$M$ satisfies $(S_{t})$.
3. (iii)
$\operatorname{depth}_{R}(N)\leq n+c+t$.
Then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.
###### Proof.
We argue by induction on $t$. If $t=0$, then the assertion is clear by Theorem
4.1. Now suppose that $t>0$ and consider the universal pushforward of $M$,
(4.3.1) $0\rightarrow M\rightarrow F\rightarrow M_{1}\rightarrow 0,$
where $F$ is free. It is easy to see that $M_{1}$ satisfies $(S_{t-1})$. From
the exact sequence (4.3.1), it is clear that
(4.3.2)
$\operatorname{Ext}^{i}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M_{1},N)\text{
for all }i>0.$
Therefore, $\operatorname{Ext}^{i}_{R}(M_{1},N)=0$ for all $i$, $n+1\leq i\leq
n+c+1$. By induction hypothesis, we conclude that
$\operatorname{Ext}^{i}_{R}(M_{1},N)=0$ for all $i>n$. By (4.3.2),
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\geq n$ and so
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ by
Theorem 2.6. ∎
Acknowledgements. I would like to thank Olgur Celikbas and my thesis adviser
Mohammad Taghi Dibaei for valuable suggestions and comments on this paper.
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|
arxiv-papers
| 2012-10-22T13:02:15 |
2024-09-04T02:49:36.938405
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arash Sadeghi",
"submitter": "Arash Sadeghi",
"url": "https://arxiv.org/abs/1210.5882"
}
|
1210.6065
|
# Performance of Polarization-based Stereoscopy Screens
Xiaozhu Zhang Kristian Hantke Max Planck Institute for Dynamics and Self-
Organization (MPIDS), 37077 Goettingen, Germany Cornelius Fischer Georg-
August-Universität Göttingen, 37077 Göttingen, Germany Matthias Schröter
matthias.schroeter@ds.mpg.de Max Planck Institute for Dynamics and Self-
Organization (MPIDS), 37077 Goettingen, Germany
###### Abstract
The screen is a key part of stereoscopic display systems using polarization to
separate the different channels for each eye. The system crosstalk,
characterizing the imperfection of the screen in terms of preserving the
polarization of the incoming signal, and the scattering rate, characterizing
the ability of the screen to deliver the incoming light to the viewers,
determine the image quality of the system. Both values will depend on the
viewing angle. In this work we measure the performance of three silver screens
and three rear-projection screens. Additionally, we measure the surface
texture of the screens using white-light interferometry. While part of our
optical results can be explained by the surface roughness, more work is needed
to understand the optical properties of the screens from a microscopic model.
stereoscopic projection and polarization and screen and system crosstalk and
ghosting and surface texture
## I Introduction
Displaying 3D content is not only an important issue in the entertainment
industry, it is also of increasing importance in science where new numeric and
experimental methods have created a wealth of three-dimensional datasets. Many
stereoscopic display system are based on polarization filtering: the visual
information for each eye is oppositely polarized, projected to and scattered
or transmitted by the screen, and finally filtered by the viewer’s glasses
which consist of two polarizers admitting only the correctly polarized light
to each eye iizuka:06 ; janssen:08 ; kim:10 . There are two options for
polarization filtering: linear and circular polarized light. While linear
polarizers are simpler to manufacture, circular polarization has the advantage
that head tilting will not impair the quality of the image
An ideal screen would completely preserve the polarization of the incoming
light. However, in practice there is always some amount of “ghosting”
resulting from the change of polarization at the screen. A measure for
ghosting is the system crosstalk $C$. It is defined as the ratio between the
intensity of light that leaks from the unintended channel to the intended one
and the intensity of the intended channel woods:11 . According to measurements
of Huang et al. huang:03 , the maximal acceptable system crosstalk for a
typical viewer to still experience a stereo sensation is 0.1. (Lower values
down to $10^{-4}$ can still be detected by careful visual inspection). While
it is also known on a theoretical basis that the viewing angle will influence
the amount of system crosstalk richards:10 , to our knowledge no measurements
of the angle-dependent system crosstalk of different screen types have been
published up to now. Neither has the question been studied how the inclination
angle (between the incoming light from the projector and the surface normal of
the screen) influences the system crosstalk.
A second measure for the quality of a screen is the brightness of the image,
which depends on the amount and angular distribution of the reflectance (for
silver screens) or transparency (for rear-projection screens) of the screen.
For silver screens this is typically quoted as the screen gain, the intensity
measured at normal incidence normalized by the intensity of a Lambertian
source brennesholtz:08 . Here we measure the angle dependent scattering rate
$S$ for both silver screens and rear-projection screens. $S$ is defined as the
ratio of the intensity received by a viewer in a certain angle to the
intensity of the incoming light, normalized by the solid angle.
In this paper we present measurements of the angular dependence of system
crosstalk and scattering rate for three samples of silver screens (labeled SS1
to SS3) and three rear-projection screens (RP1 to RP3). Additionally, we
determine the surface texture of the samples using white-light interferometry;
this information provides some qualitative insight into our optical results.
## II Experimental Setup
| |
---|---|---
(a) | (b) | (c)
Figure 1: Sketch of the experimental setup using circular polarized light:
panels a) and b) are used for silver screens and a) and c) for rear-projection
screens. The generation and the detection of linearly polarized light is
achieved by removing the devices colored in red.
Figure 1 shows the experimental setup used for measuring the angular
dependence of the system crosstalk and the scattering rate. Diode pumped solid
state lasers (DPGL-2050 from Photop and Verdi V5 SF from Coherent) with a
wavelength of 532nm were used as light sources for the experiments. Passing a
beam expander, the diameter of the laser beam was increased to 3.4mm (FWHM),
whereas the typical size of structural inhomogeneities on the screen surface
is at most a few hundred micrometers as shown below. This ensured that the
measured data for different spots on the screen are reproducible within $\pm
5$%.
The laser light was linearly polarized by passing a polarizer or circularly
polarized by passing an additional Babinet-Soleil compensator (from B. Halle).
The screen sample is irradiated by the laser at normal incidence and the
scattered laser light of the silver and rear projection screens is detected by
an detection unit in reflection (1b) and in transmission (1c), respectively.
The detection unit consists of a power meter (PM100D with sensor S130C from
Thorlabs), an analyzer and in case of the circular polarization an additional
quarter-wave plate (both from B. Halle). A long and narrow tube was placed in
front of the power meter, ensuring that only the photons are detected that
scatter from the irradiated spot on the screen along the viewing axis of the
sensor in a solid angle $\Omega$ of 2$\cdot$10-4 sr. The detection unit was
placed on a rotatable rail with the rotational axis being fixed in such a way
that the normal viewing axis of the sensor intercepts always with the
illuminated area on the screen during rotation. The viewing angle $\theta$ can
be varied from -20° to 80°. For silver screens the range of $\pm$6° is
inaccessible in order to not block the incoming beam.
In both cases of linear and circular polarization the incoming laser intensity
$I_{in}$ was measured just in front of the sample. Furthermore, the intensity
of the scattered light $I_{out}$ was measured for the intended channel with
the polarization being the same direction as the incoming one ($I_{out}^{sp}$,
analyzer and polarizer parallel) and for the unintended channel with the
polarization being the opposite direction ($I_{out}^{op}$, analyzer and
polarizer perpendicular) for different viewing angles $\theta$. From this data
one can compute the crosstalk $C(\theta)$:
$C(\theta)=\frac{I_{out}^{op}(\theta)}{I_{out}^{sp}(\theta)}$ (1)
and the scattering rate $S(\theta)$:
$S(\theta)=\frac{I_{out}^{sp}(\theta)+I_{out}^{op}(\theta)}{I_{in}\;\Omega}$
(2)
The precision of the measurement for the crosstalk depends strongly on the
purity of the initial laser polarization, whereas the scattering rate is not
affected within our measurement precision. Analyzing the crosstalk without any
screen sample (i.e. putting the laser directly in front of the analyzer
system) we found the lower resolution limit in the linear case to be less than
$3\times 10^{-3}$. In the circular case the degree of polarization results in
a lower resolution limit of $7.5\times 10^{-3}$.
The screen samples were obtained from the company Screenlab (Elmshorn,
Germany), their specifications and brand names are listed in table 1.
Table 1: Sample labels, brand names, manufacturer information on gain and transmission, and surface properties measured by white light interferometry: the root mean square roughness $R_{q}$ and the ratio between the surface area and the projected area $F$. Sample | brand name | gain | transmission | $R_{q}[\mu m]$ | $F$
---|---|---|---|---|---
RP1 | BS XRP3 | | 41.8 | 3.8 | 1.03
RP2 | WS XRP3 | | 88.8 | 3 | 1.13
RP3 | BS RP2 | | 41.2 | 4.2 | 1.2
SS1 | SH120 | 2.4 | | 5 | 2.3
SS2 | SF120 | 2.4 | | 8 | 2.6
SS3 | WA160 | 1.3 | | 22 | 2.2
Figure 2: Angular dependence of the crosstalk caused by the screen using a)
circular polarized light and b) linear polarization. The grey dash line
corresponds to the threshold for still acceptable stereo fusion according to
Huang et al. huang:03 .
The surface topography of the screens was measured using a ZeMapper whitelight
vertical scanning interferometer (Zemetrics, Tucson, USA): the focal plane of
an interference pattern is vertically scanned through the sample topography,
then a height map is calculated from the collected amplitude maps of the
interference patterns. The vertical resolution of the instrument is better
than 1 nm; the maximum field of view applied in this study is 1.4 mm. For more
information on the instrument see darbha:10 . Prior to the measurement the
rear projection screens where sputter coated with a 40 nm gold layer to
increase surface reflectivity.
## III Results
Figure 2 displays the system crosstalk of the six screen samples, both with
circular and linear polarized light. Based on the criterion found by Huang et
al. huang:03 , all screens allow stereo vision for viewing angles $\theta$
smaller than 40° (circular) or 48° (linear). In practice this range will be
smaller due to the additional crosstalk originating from the glasses and the
inclination angle of the incoming light; the latter effect will be described
below.
In general, the screens seem to fall into two categories; they are either
optimized for a large range of acceptable crosstalk or a minimized crosstalk
at small $\theta$. In both categories the silver screens are outperformed by
the rear projection screens: RP3 has a smaller $C$ at small $\theta$ than SS1
while RP1 has a broader range of acceptable viewing angles than SS3.
Regarding the polarization mode, linear polarization has for each screen a
clear advantage over circular. $C_{circ}/C_{lin}$ measured at $\theta$ = 10°
varies between 1.1 (RP1) and 4 (RP3) as shown in figure 4. Please observe,
that our measurements of the crosstalk of RP3 at small angles might be limited
by our experimental resolution.
Figure 3: Ratio of crosstalk measured with circular and linear polarization of
the incoming light.
Figure 4: Dependence of the crosstalk on the inclination angle $\phi$.
Measured on screen SS1 using linear polarized light.
Under real world conditions it is quite likely that the incoming light itself
will have an inclination angle $\phi$ to the surface normal of the screen. To
quantify the additional crosstalk created this way, we modified the
experimental setup by adding a periscope in front of the polarizer. Figure 4
shows the crosstalk for sample SS1 in the case of linear polarization for
inclination angles between 0°and 15°. While there is a clear increase of
crosstalk with $\phi$, the range of acceptable viewing angles $\theta$ is
reduced by only 5°.
Figure 5: Angular dependence of the scattering rate measured with circular
polarization. Scattering rate values for linear polarization agree within 5.4
percent.
The scattering rates $S$ of the screen samples with circular polarization are
shown in figure 5. Deviations of $S$ measured with linear or the circular
polarization are within our errorbars. For high luminosities at small viewing
angles SS1 and RP2 are the best choice, in terms of best homogeneity SS3 comes
closest to a Lambertian source.
Figure 6: Perspective images of the surface texture of the screen samples.
Please note the different horizontal and vertical scales for sample SS3.
Images contain between 0.7% (RP1) and 30% (SS3) interpolated pixels.
From a theoretical side the performance of a screen will depend both on its
material and its surface texture jin:10 . While we do not have information on
the electromagnetic properties of the screen material, the surface texture can
be measured with white light interferometry. Perspective images of the
surfaces of SS1, SS3, RP1, RP3 are shown in figure 6. The RMS (root mean
square) roughness $R_{q}$ and the ratio between the surface area and the
projected area $F$ of all six screen samples is listed in table 1.
A comparison of the angular dependence of $S$ with these values hints at
$R_{q}$ as a predictor for the deviation from a Lambertian source. This is
particularly shown by SS3 which has by far the highest value of $R_{q}$ and
the smallest $\theta$ dependence of $S$. Regarding the system crosstalk a
similar correlation between $R_{q}$ and the slope of $C$ at large angles might
exist. On the other side we do not find a clear correlation between the
optical properties and $F$.
## IV Conclusion
All screens allow effective stereo projection for viewing angles up to 40°. At
larger angles the crosstalk of rear projection screens is considerably smaller
than that of silver screens. Also for each screen the crosstalk was larger
with circular polarization than with linear. However, when planing a display
system additional factors have to be taken into account like the available
space behind the screen or the sensitivity of the system against the viewers
tilting their heads. Consequentially, no optimal solution for all possible
scenarios exist. While the roughness of the screens influences their large
viewing angle behavior, clearly more research is needed for a quantitative
understanding.
Acknowledgements: We would like to thank Günter Daszinnies from the company
Screenlabs for providing the test samples.
## References
* (1) Brennesholtz, M.S., Stupp, E.H.: Projection Displays. Wiley (2008)
* (2) Darbha, G., Schäfer, T., Heberling, F., Lüttge, A., Fischer, C.: Retention of latex colloids on calcite as a function of surface roughness and topography. Langmuir 26, 4743–4752 (2010)
* (3) Huang, K.C., Yuan, J.C., Tsai, C.H., Hsueh, W.J., Wang, N.Y.: A study of how crosstalk affects stereopsis in stereoscopic displays. In: Proceedings of SPIE-IS&T, vol. 5006, pp. 247–253 (2003)
* (4) Iizuka, K.: Welcome to the wonderful world of 3D: introduction, principles and history. Optics and Photonics News 17, 42–51 (2006)
* (5) Janssen, J.K.: 3D 2.0, Neuer Anlauf für Stereoskopie im Kino. c’t 16, 72–75 (2008)
* (6) Jin, L., Kasahara, M., Gelloz, B., Takizawa, K.: Polarization properties of scattered light from macrorough surfaces. Optics Letters 35, 595–597 (2010)
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|
arxiv-papers
| 2012-10-22T20:46:41 |
2024-09-04T02:49:36.959145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaozhu Zhang, Kristian Hantke, Cornelius Fischer, Matthias Schr\\\"oter",
"submitter": "Kristian Hantke Ph. D.",
"url": "https://arxiv.org/abs/1210.6065"
}
|
1210.6289
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-314 LHCb-PAPER-2012-029 October $23$, $2012$
A study of the $Z$ production cross-section in $pp$ collisions at
$\sqrt{s}=7~{}\mathrm{Te\kern-2.07413ptV}$ using tau final states
The LHCb collaboration†††Authors are listed on the following pages.
A measurement of the inclusive $Z\to\tau\tau$ cross-section in $pp$ collisions
at ${\sqrt{s}=7~{}\mathrm{Te\kern-1.00006ptV}}$ is presented based on a
dataset of $1.0~{}\mathrm{fb}^{-1}$ collected by the LHCb detector. Candidates
for $Z\to\tau\tau$ decays are identified through reconstructed final states
with two muons, a muon and an electron, a muon and a hadron, or an electron
and a hadron. The production cross-section for $Z$ bosons, with invariant mass
between $60$ and $120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$, which decay to
$\tau$ leptons with transverse momenta greater than
$20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ and pseudorapidities between $2.0$ and
$4.5$, is measured to be $\sigma_{pp\to Z\to\tau\tau}=71.4\pm 3.5\pm 2.8\pm
2.5~{}\mathrm{pb}$; the first uncertainty is statistical, the second is
systematic, and the third is due to the uncertainty on the integrated
luminosity. The ratio of the cross-sections for $Z\to\tau\tau$ to $Z\to\mu\mu$
is determined to be $0.93\pm 0.09$, where the uncertainty is the combination
of statistical, systematic, and luminosity uncertainties of the two
measurements.
Submitted to JHEP.
LHCb collaboration
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Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
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Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
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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, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
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F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
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Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N.
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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
25National Center for Nuclear Research (NCBJ), 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
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The measurement of the production cross-section for $Z$ bosons‡‡‡Here, the $Z$
is used to indicate production from $Z$ bosons, photons, and their
interference. in proton-proton ($pp$) collisions constitutes an important
verification of Standard Model predictions. Since lepton universality in $Z$
decays has been tested to better than $1\%$ at LEP [1], any deviation observed
at the LHC would be evidence for additional physics effects producing final
state leptons. In particular, $\tau$-lepton pairs can be important signatures
for supersymmetry, extra gauge bosons, or extra dimensions [2, 3, 4]. The LHCb
experiment has previously measured the cross-section for $Z\to\mu\mu$ [5] with
both leptons having transverse momentum ($p_{\mathrm{T}}$) above
$20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ and an invariant mass between $60$ and
$120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$. Here a complementary measurement
in the decay mode $Z\to\tau\tau$ is presented. This measurement extends the
$Z\to\tau\tau$ cross-section measurements from the central pseudorapidity
range covered by ATLAS $\left(\left|\eta\right|<2.4\right)$ [6] and CMS
$\left(\left|\eta\right|<2.3\right)$ [7] into the forward region covered by
the LHCb experiment $(2<\eta<4.5)$.
## 2 Detector and datasets
The LHCb detector [8] is a single-arm forward spectrometer 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
($\mathrm{VELO}$) surrounding the $pp$ interaction region, a large-area
silicon-strip detector ($\mathrm{TT}$) located upstream of a dipole magnet
with a bending power of about $4~{}\mathrm{Tm}$, and three stations of
silicon-strip detectors ($\mathrm{IT}$) and straw drift tubes ($\mathrm{OT}$)
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 resolution
of $20~{}\mu\mathrm{m}$ for tracks with high $p_{\mathrm{T}}$.
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 ($\mathrm{SPD}$) and pre-shower detectors
($\mathrm{PRS}$), an electromagnetic calorimeter ($\mathrm{ECAL}$) and a
hadronic calorimeter ($\mathrm{HCAL}$). Muons are identified by a 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 that applies a full
event reconstruction. The hardware stage imposes a global event requirement
($\mathrm{GEC}$) on the hit multiplicities of most sub-detectors used in the
pattern recognition algorithms to avoid overloading of the software trigger by
high occupancy events.
This analysis uses data, corresponding to an integrated luminosity of $1028\pm
36~{}\mathrm{pb}^{-1}$, taken at a centre-of-mass energy of
$7~{}\mathrm{Te\kern-1.00006ptV}$. The absolute luminosity scale was measured
periodically throughout the data taking period using Van der Meer scans [9]
where the beam profile is determined by moving the beams transversely across
one another. A beam-gas imaging method was also used where the beam profile is
determined through reconstructing beam-gas interaction vertices near the beam
crossing point [10]. Both methods provide similar results and the integrated
luminosity is determined from the average of the two, with an estimated
systematic uncertainty of $3.5\%$ [11]. The primary systematic uncertainty of
$2.7\%$ is due to the beam current measurement, shared between the two
methods.
Simulated data samples are used to develop the event selection, determine
efficiencies, and estimate systematic uncertainties. Each sample was generated
using an LHCb configuration [12] of Pythia $6.4$ [13] with the CTEQ$6$L$1$
leading-order PDF set [14] and passed through a Geant4 [15, *geantb] based
simulation of the LHCb detector [17]. Trigger emulation and full event
reconstruction were performed using the LHCb reconstruction software [18].
Additional samples, without detector simulation or event reconstruction, are
used to study the signal acceptance and were generated using Pythia $8.1.55$
[19], Herwig++ $2.5.1$ [20], and Herwig++ with the Powheg method [21].
## 3 Event selection
The signatures for $Z\to\tau\tau$ decays considered in this analysis are two
oppositely-charged tracks, consistent with an electron, muon, or hadron
hypothesis, having large impact parameters with respect to the primary vertex
of the event.
Tracks are reconstructed in the $\mathrm{VELO}$ and extrapolated to the
$\mathrm{IT}$/$\mathrm{OT}$ sub-detectors; any $\mathrm{TT}$ sub-detector hits
consistent with the track are added and a full track fit is performed. Only
tracks with fit probabilities greater than $0.001$ are considered.
Tracks are extrapolated to the calorimeters and matched with calorimeter
clusters. Electron candidates are required to have a $\mathrm{PRS}$ energy
greater than $0.05~{}\mathrm{Ge\kern-1.00006ptV}$, a ratio of $\mathrm{ECAL}$
energy to candidate momentum, $E/pc$, greater than $0.1$, and a ratio of
$\mathrm{HCAL}$ energy to candidate momentum less than $0.05$. Any electron
candidate momentum is corrected using bremsstrahlung photon recovery [22].
Since the $\mathrm{ECAL}$ is designed to register particles from $b$-hadron
decays, calorimeter cells with transverse energy above
$10~{}\mathrm{Ge\kern-1.00006ptV}$ saturate the electronics, and lead to
degradation in the electron energy resolution.
Hadron candidates are identified by requiring the ratio of $\mathrm{HCAL}$
energy to track momentum to be greater than $0.05$. Due to the limited
$\mathrm{HCAL}$ acceptance, the candidate track is required to have a
pseudorapidity of $2.25\leq\eta\leq 3.75$.
Muon candidates are identified by extrapolating tracks to the muon system
downstream of the calorimeters and matching them with compatible hits. Muon
candidates are required to have a hit in each of the four stations and
consequently will have traversed over $20$ hadronic interaction lengths of
material.
The data have been collected using two triggers: a trigger which selects muon
candidates with a $p_{\mathrm{T}}$ greater than
$10~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$; and a trigger which selects electron
candidates with $p_{\mathrm{T}}$ greater than
$15~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$.
The analysis is divided into five streams, labelled $\tau_{\mu}\tau_{\mu}$,
$\tau_{\mu}\tau_{e}$, $\tau_{e}\tau_{\mu}$, $\tau_{\mu}\tau_{h}$, and
$\tau_{e}\tau_{h}$, defined such that the streams are exclusive. The first
$\tau$ lepton decay product candidate is required to have
$p_{\mathrm{T}}>20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ and the second is
required to have $p_{\mathrm{T}}>5~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$. The
following additional kinematic and particle identification requirements are
specific to each analysis stream:
* •
$\tau_{\mu}\tau_{\mu}$ requires two oppositely-charged muons where at least
one triggered the event. The muon with the larger $p_{\mathrm{T}}$ is
considered as the first $\tau$ lepton decay product candidate.
* •
$\tau_{\mu}\tau_{e}$ requires a muon that triggered the event and an
oppositely-charged electron.
* •
$\tau_{e}\tau_{\mu}$ requires an electron and an oppositely-charged muon with
$p_{\mathrm{T}}<20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$. Either lepton can
trigger the event.
* •
$\tau_{\mu}\tau_{h}$ requires a muon that triggered the event and an
oppositely-charged hadron.
* •
$\tau_{e}\tau_{h}$ requires an electron that triggered the event and an
oppositely-charged hadron.
In $pp$ collisions the cross-section for hadronic QCD processes is very large.
These events can pass the above requirements either due to semileptonic $c$\-
or $b$-hadron decays or through the misidentification of hadrons as leptons.
Signal decays, coming from an on-shell $Z$, tend to have back-to-back isolated
tracks in the transverse plane with a higher invariant mass than tracks in QCD
events. The absolute difference in azimuthal angle of the two $\tau$ lepton
decay product candidates, ${|\Delta\Phi|}$, is required to be greater than
$2.7~{}\mathrm{radians}$ and their invariant mass is required to be greater
than $20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$.
Tracks in QCD events also tend to be associated with jet activity, in contrast
to signal events where they are isolated. An isolation variable,
${I_{p_{\mathrm{T}}}}$, is defined as the transverse component of the
vectorial sum of all track momenta that satisfy
$\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}<0.5$, where $\Delta\phi$ and
$\Delta\eta$ are the differences in $\phi$ and $\eta$ between the $\tau$
lepton decay product candidate and the track. The track of the $\tau$ lepton
decay product candidate is excluded from the sum. Both $\tau$ lepton decay
product candidates are required to have
${I_{p_{\mathrm{T}}}}<2~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ for the
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{e}$, and $\tau_{e}\tau_{\mu}$
analysis streams and
${I_{p_{\mathrm{T}}}}<1~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ for
$\tau_{\mu}\tau_{h}$ and $\tau_{e}\tau_{h}$ due to the larger $\mathrm{QCD}$
backgrounds.
The lifetime of the $\tau$ lepton is used to separate signal from prompt
backgrounds. The signed impact parameter for a track is defined as the
magnitude of the track vector of closest approach to the primary vertex signed
by the $z$-component of the cross product between this vector and the track
momentum. The impact parameter significance, ${\mathrm{IPS}}$, is then defined
as the absolute sum of the signed impact parameters of the two $\tau$ lepton
decay product candidates, divided by their combined uncertainty. The
${\mathrm{IPS}}$ is required to be greater than $9$ for the
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ analysis
streams while no ${\mathrm{IPS}}$ requirement is placed on the
$\tau_{\mu}\tau_{e}$ or $\tau_{e}\tau_{\mu}$ streams.
In the $\tau_{\mu}\tau_{\mu}$ analysis stream an additional background
component arises from $Z\to\mu\mu$ events. This produces two muons with
similar $p_{\mathrm{T}}$, most of which also have an invariant mass close to
the $Z$ mass. In contrast, signal events tend to have unbalanced
$p_{\mathrm{T}}$ and a lower invariant mass due to unreconstructed energy from
neutrinos and neutral hadrons. The $p_{\mathrm{T}}$ asymmetry,
${A_{p_{\mathrm{T}}}}$, is defined as the absolute difference between the
$p_{\mathrm{T}}$ of the two candidates divided by their sum. For the
$\tau_{\mu}\tau_{\mu}$ analysis stream the ${A_{p_{\mathrm{T}}}}$ is required
to be greater than $0.3$ and the di-muon invariant mass must lie outside the
range $80<M_{\mu\mu}<100~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$.
## 4 Background estimation
The invariant mass distributions for the selected $Z\to\tau\tau$ candidates,
the simulated signal, and the estimated backgrounds for the five analysis
streams are shown in Fig. 1, where no candidates are observed with a mass
above $120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$. Five types of background
have been considered: generic QCD; electroweak, where a high $p_{\mathrm{T}}$
lepton is produced by a $W$ or $Z$ boson and the second candidate $\tau$
lepton decay product is misidentified from the underlying event; ${t\bar{t}}$,
where two hard leptons are produced from top decays; ${WW}$, where each $W$
decays to a lepton; and $Z\to\ell\ell$ for the $\tau_{\mu}\tau_{\mu}$,
$\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ streams, where for the
$\tau_{\mu}\tau_{h}$ stream a single muon is misidentified as a hadron and for
the $\tau_{e}\tau_{h}$ stream an electron is misidentified. The ${t\bar{t}}$
and ${WW}$ backgrounds are estimated using simulation and found to be small
for all final states.
(a)
(b)
(c)
(d)
(e)
Figure 1: Invariant mass distributions for the LABEL:sub@fig:mumu
$\tau_{\mu}\tau_{\mu}$, LABEL:sub@fig:mue $\tau_{\mu}\tau_{e}$,
LABEL:sub@fig:emu $\tau_{e}\tau_{\mu}$, LABEL:sub@fig:muh
$\tau_{\mu}\tau_{h}$, and LABEL:sub@fig:eh $\tau_{e}\tau_{h}$ candidates with
the excluded mass range indicated for $\tau_{\mu}\tau_{\mu}$. The
$Z\to\tau\tau$ simulation (solid red) is normalised to the number of signal
events. The $\mathrm{QCD}$ (horizontal green), electroweak (vertical blue),
and $Z$ (solid cyan) backgrounds are estimated from data. The ${t\bar{t}}$
(vertical orange) and ${WW}$ (horizontal magenta) backgrounds are estimated
from simulation and generally not visible.
The QCD and electroweak backgrounds are estimated from data. A signal-depleted
control sample is created by applying all selection criteria but requiring
that the $\tau$ lepton decay product candidates have the same-sign
($\mathrm{SS}$) charge. The QCD and electroweak background events in this
sample, $N_{\mathrm{QCD}}^{\mathrm{SS}}$ and $N_{\mathrm{EWK}}^{\mathrm{SS}}$,
are obtained by fitting template shapes to the distribution of the difference
between the $p_{\mathrm{T}}$ of the first and second $\tau$ lepton decay
product candidates. The template shape for the electroweak background is taken
from simulation. To determine the shape of the QCD contribution, the isolation
requirement is reversed such that
${I_{p_{\mathrm{T}}}}>10~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$. The number of
candidates for each background category in the signal sample is then
calculated as
$N_{\mathrm{QCD}}=f_{\mathrm{QCD}}N^{\mathrm{SS}}_{\mathrm{QCD}}$ and
$N_{\mathrm{EWK}}=f_{\mathrm{EWK}}N^{\mathrm{SS}}_{\mathrm{EWK}}$, where
$f_{\mathrm{QCD}}$ and $f_{\mathrm{EWK}}$ are the ratio of opposite-sign to
same-sign events for QCD and electroweak events respectively. Both
$f_{\mathrm{QCD}}$ and $f_{\mathrm{EWK}}$ are determined as the ratio of
opposite-sign to same-sign events satisfying the template requirements. The
uncertainties on the $\mathrm{QCD}$ and electroweak backgrounds are estimated
by combining the statistical uncertainty on the fraction with the
uncertainties from the fit used to determine $N_{\mathrm{QCD}}^{\mathrm{SS}}$
and $N_{\mathrm{EWK}}^{\mathrm{SS}}$.
The number of $Z\to\mu\mu$ background events for the $\tau_{\mu}\tau_{\mu}$
stream is obtained by applying all selection criteria except for the
$80<M_{\mu\mu}<100~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$ requirement. This
produces a sample with a clear peak around the $Z$ mass as shown in Fig.
LABEL:sub@fig:mumu. A template for $Z\to\mu\mu$ events is obtained from data
by applying the event selection, but requiring prompt events with
${\mathrm{IPS}}<1$. The template is normalised to the number of events within
the $\tau_{\mu}\tau_{\mu}$ sample with ${\mathrm{IPS}}>9$ and within the
invariant mass range
$80<M_{\mu\mu}<100~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$. The $Z\to\mu\mu$
background is the number of events in the normalised template outside this
mass range. The uncertainty on this background is estimated from the
statistical uncertainty on the normalisation factor.
The $Z\to\mu\mu$ process also contributes a small background to the
$\tau_{\mu}\tau_{h}$ stream when one of the muons is misidentified as a
hadron. This is evaluated by applying the $\tau_{\mu}\tau_{h}$ selection but
requiring a second identified muon rather than a hadron, and scaling this by
the probability for a muon to be misidentified as a hadron. The latter is
found from a sample of $Z\to\mu\mu$ events that have been selected by
requiring a single well defined muon and a second isolated track, which give
an invariant mass between $80$ and
$100~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$; $(0.06\pm 0.01)\%$ of these
tracks pass the hadron identification requirement.
Similarly, a small $Z\to ee$ background can contribute to the
$\tau_{e}\tau_{h}$ stream when one of the electrons is misidentified as a
hadron. This is evaluated by applying the $\tau_{e}\tau_{h}$ selection but
requiring a second identified electron rather than a hadron, and scaling this
by the probability for an electron to be misidentified as a hadron. The
electron mis-identification is found from simulated $Z\to ee$ events to be
$(0.63\pm 0.02)\%$.
## 5 Cross-section measurement
The $pp\to Z\to\tau\tau$ cross-section is calculated within the kinematic
region $60<M_{\tau\tau}<120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$,
$2.0\leq\eta^{\tau}\leq 4.5$, and
$p_{\mathrm{T}}^{\tau}>20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ using
$\sigma_{pp\to
Z\to\tau\tau}=\frac{\sum_{i=1}^{N}1/\varepsilon_{\mathrm{rec}}^{i}-\sum_{j}N^{j}_{\mathrm{bkg}}\langle
1/\varepsilon_{\mathrm{rec}}\rangle^{j}}{\mathscr{L}\cdot\mathcal{A}\cdot\mathcal{B}\cdot{\varepsilon_{\mathrm{sel}}}}$
(1)
where $N$ is the number of observed candidates and $N_{\mathrm{bkg}}^{j}$ is
the estimated background from source $j$. The integrated luminosity is given
by $\mathscr{L}$, $\mathcal{A}$ is an acceptance and final state radiation
correction factor, $\mathcal{B}$ is the branching fraction for the
$\tau$-lepton pair to decay to the final state, and
${\varepsilon_{\mathrm{sel}}}$ is the selection efficiency. A summary of these
values for each final state is given in Table 1. The reconstruction
efficiency, ${\varepsilon_{\mathrm{rec}}}$, is calculated using simulation or
data for each event, assuming that it is signal, and depends on the momentum
and pseudorapidity of the $\tau$ lepton decay product candidates. $\langle
1/\varepsilon_{\mathrm{rec}}\rangle^{j}$ indicates the average value of
$1/{\varepsilon_{\mathrm{rec}}}$ for background source $j$.
Table 1: Acceptance factors, branching fractions, selection efficiencies, numbers of background and observed events for each $Z\to\tau\tau$ analysis stream. Stream | $\mathcal{A}$ | $\mathcal{B}$ $[\%]$ | ${\varepsilon_{\mathrm{sel}}}$ | $N_{\mathrm{bkg}}$ | $N$
---|---|---|---|---|---
$\tau_{\mu}\tau_{\mu}$ | $0.405\pm 0.006$ | $3.031\pm 0.014$ | $0.138\pm 0.006$ | $41.6\pm 8.5$ | $124$
$\tau_{\mu}\tau_{e}$ | $0.248\pm 0.004$ | $6.208\pm 0.020$ | $0.517\pm 0.012$ | $129.7\pm 4.9$ | $421$
$\tau_{e}\tau_{\mu}$ | $0.152\pm 0.002$ | $6.208\pm 0.020$ | $0.344\pm 0.016$ | $56.6\pm 3.3$ | $155$
$\tau_{\mu}\tau_{h}$ | $0.182\pm 0.002$ | $16.933\pm 0.056$ | $0.135\pm 0.004$ | $53.3\pm 0.8$ | $189$
$\tau_{e}\tau_{h}$ | $0.180\pm 0.002$ | $17.341\pm 0.057$ | $0.082\pm 0.004$ | $36.6\pm 0.9$ | $101$
The integrated luminosity of the datasets for the $\tau_{\mu}\tau_{\mu}$,
$\tau_{\mu}\tau_{e}$, and $\tau_{\mu}\tau_{h}$ samples is $1028\pm
36~{}\mathrm{pb}^{-1}$, while the $\tau_{e}\tau_{\mu}$ and $\tau_{e}\tau_{h}$
final state datasets have an integrated luminosity of $955\pm
33~{}\mathrm{pb}^{-1}$.
### 5.1 Acceptances and branching fractions
The acceptance factor, $\mathcal{A}$, is used to correct the kinematics of
each analysis stream to the kinematic region
$60<M_{\tau\tau}<120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$,
$2.0\leq\eta^{\tau}\leq 4.5$, and
$p_{\mathrm{T}}^{\tau}>20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$. This region
corresponds to the detector fiducial acceptance and allows a comparison with
the LHCb $Z\to\mu\mu$ measurement [5]. The acceptance factor is taken from
simulation and is defined as the number of $Z\to\tau\tau$ events where the
generated $\tau$ lepton decay products fulfil the kinematic requirements
described in Sect. 3, divided by the number of $Z\to\tau\tau$ events where the
generated $\tau$ leptons lie within the kinematic region defined above.
For each final state the acceptance factors are calculated at leading-order
using fully modelled hadronic decay currents and spin correlated $\tau$ lepton
decays with final state radiation in Pythia $8$ and Herwig++, and at next-to-
leading-order using the Powheg method implemented in Herwig++. For Pythia $8$
the CTEQ$5$L leading-order PDF set [23] was used, while for Herwig++ the
MSTW$08$ PDF set [24] was used. The mean of the maximum and minimum values
from the three generators is taken as the acceptance factor and is given in
Table 1. The uncertainty is taken as half the difference between the maximum
and minimum values.
The branching fractions are calculated using the world averages [25] and are
given in Table 1. The $\tau$ lepton to single charged-hadron branching
fraction is the sum of all $\tau$ lepton decays containing a single charged
hadron. The final states presented in this analysis account for $44\%$ of all
expected $Z\to\tau\tau$ decays.
### 5.2 Selection efficiency
The event selection efficiency, ${\varepsilon_{\mathrm{sel}}}$, is the product
of the efficiencies described below. Each efficiency is determined from either
data, or simulation which has been calibrated using data. The resulting
${\varepsilon_{\mathrm{sel}}}$ for each stream is given in Table 1.
The kinematic efficiency, ${\varepsilon_{\mathrm{kin}}}$, is obtained from
simulation and is the number of events fulfilling the kinematic requirements
of Sect. 3 at both the simulated and reconstructed level divided by the number
of events passing the requirements at the simulated level. The efficiency is
consistent with unity for the $\tau_{\mu}\tau_{\mu}$ and $\tau_{\mu}\tau_{h}$
analysis streams. For streams involving electrons,
${\varepsilon_{\mathrm{kin}}}$ is significantly lower due to the saturation of
the $\mathrm{ECAL}$. This results in electrons being reconstructed with lower
momenta than their true momenta due to incomplete bremsstrahlung recovery. In
the $\tau_{\mu}\tau_{e}$, $\tau_{e}\tau_{\mu}$, and $\tau_{e}\tau_{h}$
streams, ${\varepsilon_{\mathrm{kin}}}$ is $(99.3\pm 1.0)\%$, $(66.8\pm
1.9)\%$, and $(67.0\pm 1.3)\%$ respectively. The uncertainties come from the
statistical uncertainty of the $Z\to\tau\tau$ simulation and the calibration
of the electron momentum scale which has been obtained by comparing the
$p_{\mathrm{T}}$ spectrum of $Z\to ee$ events in data and simulation [26].
The efficiency of the isolation requirement,
${\varepsilon_{{I_{p_{\mathrm{T}}}}}}$, for each analysis stream is taken from
$Z\to\tau\tau$ simulation, and calibrated to data by multiplying
${\varepsilon_{{I_{p_{\mathrm{T}}}}}}$ by the ratio of the efficiency obtained
in $Z\to\mu\mu$ data to $Z\to\tau_{\mu}\tau_{\mu}$ simulation. The systematic
uncertainty on ${\varepsilon_{{I_{p_{\mathrm{T}}}}}}$ is estimated as the
difference between the efficiencies obtained from $Z\to\mu\mu$ simulation and
$Z\to\tau_{\mu}\tau_{\mu}$ simulation.
The efficiency of the impact parameter significance requirement,
${\varepsilon_{{\mathrm{IPS}}}}$, is evaluated from $Z\to\tau\tau$ simulation.
A comparison of the ${\mathrm{IPS}}$ distributions in $Z\to\mu\mu$ events from
data and simulation show that the impact parameter resolution is
underestimated by $(12\pm 1)\%$ in simulation, and so the simulated
$Z\to\tau\tau$ events are corrected by this factor. The systematic uncertainty
on ${\varepsilon_{{\mathrm{IPS}}}}$ is determined by re-calculating the
efficiency in $Z\to\tau\tau$ simulation with the scale factor varied by its
uncertainty.
The efficiency of the azimuthal angle separation requirement,
${\varepsilon_{{|\Delta\Phi|}}}$, and $p_{\mathrm{T}}$ asymmetry efficiency
requirement, ${\varepsilon_{{A_{p_{\mathrm{T}}}}}}$, are evaluated from
simulation. The systematic uncertainty on each is taken as the difference in
the evaluation of these efficiencies in $Z\to\mu\mu$ data and simulation,
combined in quadrature with the statistical uncertainty from the
$Z\to\tau\tau$ simulation.
### 5.3 Reconstruction efficiency
The reconstruction efficiency, ${\varepsilon_{\mathrm{rec}}}$, is the product
of the $\mathrm{GEC}$, trigger, and tracking and identification efficiencies
for both $\tau$ lepton decay product candidates. The tracking efficiency is
the probability for reconstructing the track and the identification efficiency
is the probability for the track to be identified by the relevant sub-
detectors. All efficiencies determined from data have been checked against
simulation and found to agree within the percent level.
The $\mathrm{GEC}$ efficiency, ${\varepsilon_{\mathrm{GEC}}}$, is a correction
for the loss due to the rejection by the hardware trigger of events with an
$\mathrm{SPD}$ multiplicity of greater than $600~{}\mathrm{hits}$. For muon
triggered events, the efficiency has been evaluated to be $(95.5\pm 0.1)\%$
from $Z\to\mu\mu$ data events using a hardware di-muon trigger with a relaxed
$\mathrm{SPD}$ requirement of $900~{}\mathrm{hits}$. For electron trigger
events, the efficiency is estimated to be $(95.1\pm 0.1)\%$ by comparing the
hit multiplicities in $Z\to\mu\mu$ and $Z\to ee$ events.
The muon and electron trigger efficiencies, ${\varepsilon_{\mathrm{trg}}}$,
are evaluated in bins of momentum using a tag-and-probe method on
$Z\to\ell\ell$ data events, which have been selected requiring two
reconstructed and identified muon or electron candidates with an invariant
mass consistent with that of the $Z$. In the events the triggered lepton is
taken as the tag lepton, and the other as the probe lepton. The event
topologies for $Z\to\ell\ell$ and $Z\to\tau\tau$ events are nearly identical
except for the momenta of the final state particles and so the trigger
efficiency is calculated only as a function of the probe momentum below
$500~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$. The trigger efficiency is the
fraction of events where the probe has also triggered, and varies as a
function of probe momentum between $75\%$ and $80\%$ for the muon trigger and
between $62\%$ and $75\%$ for the electron trigger. The trigger efficiency
uncertainty for each bin in momenta is taken as the statistical uncertainty.
The tracking efficiency, ${\varepsilon_{\mathrm{trk}}}$, is also evaluated for
muons using a tag-and-probe method on the $Z\to\mu\mu$ data. The tag must
satisfy all the muon reconstruction and identification requirements. The probe
is reconstructed from a track segment in the muon chambers that has been
associated to a hit in the $\mathrm{TT}$ sub-detector, which is not required
in the track reconstruction. Events with a tag and probe mass consistent with
the on-shell $Z$ mass are used. The tracking efficiency is evaluated as the
number of events with a reconstructed probe track over the total number of
events. For lower $p_{\mathrm{T}}$ tracks, masses consistent with the $J/\psi$
are used.
The $J/\psi\to\mu\mu$ topology differs from the $Z\to\tau\tau$ topology in
both pseudorapidity and momentum, and so the $J/\psi$ muon tracking
efficiencies are evaluated in bins of both variables. The muon tracking
efficiency is found to vary between $85\%$ and $93\%$. The uncertainty on the
tracking efficiency is given by the statistical precision and the knowledge of
the purity of the sample of $J/\psi\to\mu\mu$ candidates. The purity is
estimated by fitting the di-muon invariant mass distribution of the
$J/\psi\to\mu\mu$ candidates with a Crystal Ball function [27] to describe the
signal shape and a linear background. An alternative estimate is obtained by
fitting only the linear background on either side of the di-muon resonance.
The difference in the efficiency evaluated using the two purity methods is
taken as the systematic uncertainty.
All particles pass through approximately $20\%$ of a hadronic interaction
length of material prior to the final tracking station. Early showering of
hadrons reduces the hadron tracking efficiency compared to the muon tracking
efficiency. An additional correction factor to the muon tracking efficiency of
$(84.3\pm 1.5)\%$ for hadrons is applied which has been estimated using the
full detector simulation, where the uncertainty on this correction corresponds
to an uncertainty of $10\%$ in the material budget [28].
The electron tracking efficiency uses a tag-and-probe method on $Z\to ee$ data
events. The tag must satisfy all the electron reconstruction and
identification requirements and the probe is selected as the highest energy
$\mathrm{ECAL}$ cluster in the event not associated with the tag. The purity
of the sample is found, from simulation, to depend on the $p_{\mathrm{T}}$ of
the tag. The dependence of the purity is fitted with signal and background
templates obtained from same-sign and opposite-sign events from data. No
momentum information is available for the probe, so the tag-and-probe
technique only provides an overall tracking efficiency for the electrons,
which is measured to be $(83\pm 3)\%$. The momentum dependence is taken from
$Z\to ee$ and $Z\to\tau\tau$ simulation. The electron tracking efficiency
uncertainty is taken from the fit uncertainty added in quadrature to the
statistical uncertainty.
The identification efficiency, ${\varepsilon_{\mathrm{id}}}$, is measured for
muons with the tag-and-probe method for the $Z\to\mu\mu$ data, using a
reconstructed track as the probe lepton and evaluated as a function of the
probe momentum. For low momenta the efficiency is evaluated using a
$J/\psi\to\mu\mu$ sample as a function of both probe pseudorapidity and
momentum. The muon identification efficiency is found to vary between $93\%$
and $99\%$ in pseudorapidity and momentum. The muon
${\varepsilon_{\mathrm{id}}}$ uncertainty is evaluated with the same method
used for the muon ${\varepsilon_{\mathrm{trk}}}$ uncertainty.
The electron identification efficiency is measured as a function of probe
momentum using the tag-and-probe method on $Z\to ee$ data and simulation
events. The isolation requirement introduces a bias of $1\%-4\%$ in data and
reconstructed simulation and so simulation without the isolation criteria is
used instead. The electron identification efficiency is found to vary between
$85\%$ and $96\%$, with an uncertainty in each bin estimated as the difference
in the biased efficiencies from data and simulation.
The hadron identification efficiency is determined using events triggered on a
single $\mathrm{VELO}$ track. The highest $p_{\mathrm{T}}$ track in each
minimum bias event is assumed to be a hadron, as verified by simulation. The
hadron identification efficiency is taken as the fraction of tracks fulfilling
the hadron identification requirements. Although the minimum bias topology
differs significantly from the $Z\to\tau\tau$ topology, an efficiency
dependence is observed only in pseudorapidity and so the efficiency is
evaluated as a function of pseudorapidity and found to vary between $92\%$ and
$95\%$. The uncertainty for each bin of pseudorapidity is estimated as the
statistical uncertainty of the bin. A summary of the systematic uncertainties
is given in Table 2.
Table 2: Systematic uncertainties expressed as a percentage of the cross-section for each $Z\to\tau\tau$ analysis stream. Contributions from acceptance $\mathcal{A}$, branching fractions $\mathcal{B}$, number of background events $N_{\mathrm{bkg}}$, reconstruction efficiencies ${\varepsilon_{\mathrm{rec}}}$, and selection efficiencies ${\varepsilon_{\mathrm{sel}}}$ are listed. The superscripts on ${\varepsilon_{\mathrm{trk}}}^{(i)}$ and ${\varepsilon_{\mathrm{id}}}^{(i)}$ indicate the first or second $\tau$ lepton decay product candidate. The percentage uncertainties on the cross-section for $N_{\mathrm{bkg}}$ are quoted for each individual background, as well as the total background. The efficiency uncertainties are split in a similar fashion. Stream | $\Delta\sigma_{pp\to Z\to\tau\tau}~{}[\%]$
---|---
$\tau_{\mu}\tau_{\mu}$ | $\tau_{\mu}\tau_{e}$ | $\tau_{e}\tau_{\mu}$ | $\tau_{\mu}\tau_{h}$ | $\tau_{e}\tau_{h}$
$\mathcal{A}$ | $1.48$ | $1.61$ | $1.32$ | $1.10$ | $1.11$
$\mathcal{B}$ | $0.46$ | $0.32$ | $0.32$ | $0.32$ | $0.33$
$N_{\mathrm{bkg}}$ | $N_{\mathrm{QCD}}$ | $4.33$ | $0.80$ | $3.08$ | $0.40$ | $0.92$
$N_{\mathrm{EWK}}$ | $4.22$ | $1.54$ | $1.52$ | $0.40$ | $0.72$
$N_{t\bar{t}}$ | $0.02$ | $0.08$ | $0.12$ | $0.00$ | $0.58$
$N_{WW}$ | $0.02$ | $0.14$ | $0.13$ | $0.09$ | $0.08$
$N_{Z}$ | $8.00$ | $-$ | $-$ | $0.22$ | $0.23$
Total $N_{\mathrm{bkg}}$ | $10.03$ | $1.75$ | $3.44$ | $0.61$ | $1.32$
${\varepsilon_{\mathrm{rec}}}$ | ${\varepsilon_{\mathrm{GEC}}}$ | $0.10$ | $0.10$ | $0.10$ | $0.10$ | $0.10$
${\varepsilon_{\mathrm{trg}}}$ | $0.88$ | $0.71$ | $2.29$ | $0.72$ | $4.30$
${\varepsilon_{\mathrm{trk}}}^{(1)}$ | $0.71$ | $0.74$ | $3.67$ | $0.79$ | $3.67$
${\varepsilon_{\mathrm{trk}}}^{(2)}$ | $0.34$ | $3.67$ | $0.61$ | $1.76$ | $1.68$
${\varepsilon_{\mathrm{id}}}^{(1)}$ | $0.38$ | $0.28$ | $1.72$ | $0.29$ | $1.73$
${\varepsilon_{\mathrm{id}}}^{(2)}$ | $0.78$ | $0.18$ | $0.56$ | $0.03$ | $0.09$
Total ${\varepsilon_{\mathrm{rec}}}$ | $1.47$ | $4.21$ | $4.73$ | $2.08$ | $6.15$
${\varepsilon_{\mathrm{sel}}}$ | ${\varepsilon_{\mathrm{kin}}}$ | $-$ | $1.04$ | $2.89$ | $-$ | $1.91$
${\varepsilon_{{I_{p_{\mathrm{T}}}}}}$ | $1.79$ | $1.91$ | $3.19$ | $1.65$ | $2.75$
${\varepsilon_{{|\Delta\Phi|}}}$ | $1.08$ | $1.03$ | $1.86$ | $0.60$ | $0.97$
${\varepsilon_{{\mathrm{IPS}}}}$ | $2.70$ | $-$ | $-$ | $1.92$ | $2.85$
${\varepsilon_{{A_{p_{\mathrm{T}}}}}}$ | $2.03$ | $-$ | $-$ | $-$ | $-$
Total ${\varepsilon_{\mathrm{sel}}}$ | $3.97$ | $2.41$ | $4.69$ | $2.60$ | $4.50$
Total systematic | $11.13$ | $5.41$ | $7.56$ | $3.88$ | $7.88$
## 6 Results
The cross-sections for each analysis stream are determined using Eq. 1, the
values given in Table 1, and the systematic uncertainties presented in Table
2. The results are
$\displaystyle\sigma_{pp\to Z\to\tau\tau}~{}(\tau_{\mu}\tau_{\mu}){}$
$\displaystyle=77.4\pm 10.4\pm 8.6\pm 2.7~{}\mathrm{pb}$
$\displaystyle\sigma_{pp\to Z\to\tau\tau}~{}(\tau_{\mu}\tau_{e}){}$
$\displaystyle=75.2\pm\phantom{1}5.4\pm 4.1\pm 2.6~{}\mathrm{pb}$
$\displaystyle\sigma_{pp\to Z\to\tau\tau}~{}(\tau_{e}\tau_{\mu}){}$
$\displaystyle=64.2\pm\phantom{1}8.2\pm 4.9\pm 2.2~{}\mathrm{pb}$
$\displaystyle\sigma_{pp\to Z\to\tau\tau}~{}(\tau_{\mu}\tau_{h}){}$
$\displaystyle=68.3\pm\phantom{1}7.0\pm 2.6\pm 2.4~{}\mathrm{pb}$
$\displaystyle\sigma_{pp\to Z\to\tau\tau}~{}(\tau_{e}\tau_{h}){}$
$\displaystyle=77.9\pm 12.2\pm 6.1\pm 2.7~{}\mathrm{pb}$
where the first uncertainty is statistical, the second uncertainty is
systematic, and the third is due to the uncertainty on the integrated
luminosity.
A global fit is performed using a best linear unbiased estimator [29]
including correlations between the final states, and a combined result of
$\sigma_{pp\to Z\to\tau\tau}=71.4\pm 3.5\pm 2.8\pm 2.5~{}\mathrm{pb}$
is obtained, with a $\chi^{2}$ per degree of freedom of $0.43$. The
statistical uncertainties are assumed to be uncorrelated as each analysis
stream contains mutually exclusive datasets. The luminosity and any shared
selection or reconstruction efficiencies are assumed to be fully correlated.
A graphical summary of the individual final state measurements, the combined
measurement, the $Z\to\mu\mu$ measurement of Ref. [5], and a theory prediction
is shown in Fig. 2. The theory calculation uses Dynnlo [30] with the MSTW$08$
next-to-next-leading-order (NNLO) PDF set [24], and is found to be
$74.3^{+1.9}_{-2.1}~{}\mathrm{pb}$.
The ratio of the combined cross-section to the LHCb $Z\to\mu\mu$ cross-section
measurement [5] is found to be
$\frac{\sigma_{pp\to Z\to\tau\tau}}{\sigma_{pp\to Z\to\mu\mu}}=0.93\pm 0.09$
where the uncertainty is the combined statistical, systematic, and luminosity
uncertainties from both measured cross-sections, which are assumed to be
uncorrelated.
Figure 2: Measured cross-sections for the $Z$ decaying to the final states
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{e}$, $\tau_{e}\tau_{\mu}$,
$\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ (open points) compared with
theory (yellow band) and the combined $Z\to\tau\tau$ and LHCb $Z\to\mu\mu$
measurements (closed points) where $p_{\mathrm{T}}$ and $\eta$ are the
transverse momentum and pseudorapidity of the leptons, and $M$ is the di-
lepton invariant mass. The inner error bars represent statistical uncertainty
while the outer error bars represent combined statistical, systematic, and
luminosity uncertainties. The central theory value is given by the light
yellow line while the associated uncertainty by the orange band.
## 7 Conclusions
Measurements of inclusive $Z\to\tau\tau$ production in $pp$ collisions at
$\sqrt{s}=7~{}\mathrm{Te\kern-1.00006ptV}$ in final states containing two
muons, a muon and an electron, a muon and a hadron, and an electron and a
hadron have been performed using a dataset corresponding to an integrated
luminosity of $1028\pm 36~{}\mathrm{pb}^{-1}.$ The cross-sections for the
individual states have been measured in the forward region of
$2.0\leq\eta^{\tau}\leq 4.5$ with
$p_{\mathrm{T}}^{\tau}>20~{}\mathrm{Ge\kern-1.00006ptV\\!/}c$ and
$60<M_{\tau\tau}<120~{}\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$, and a combined
result calculated. The results have been compared to Standard Model NNLO
theory predictions and with the LHCb $Z\to\mu\mu$ cross-section measurement.
The individual measurements, the combined result, the $Z\to\mu\mu$ cross-
section, and the theory prediction are all in good agreement. The ratio of the
$Z\to\mu\mu$ cross-section to the $Z\to\tau\tau$ cross-section is found to be
consistent with lepton universality.
## 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2012-10-23T16:58:21 |
2024-09-04T02:49:36.987375
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B.\n Adeva, M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F.\n Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves\n Jr, S. Amato, Y. Amhis, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, 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, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, 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, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, 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, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n 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, I.R. Kenyon, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, Y.M. Kim, O. Kochebina, V. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, 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, Y. Li, L. Li Gioi, 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, H. Luo, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n 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, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D.A. Milanes, M.-N. Minard, J.\n Molina 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, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, 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, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, 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, M. Smith, K. Sobczak,\n F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes,\n P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, 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, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, 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, S. Wright, S. Wu, K. Wyllie,\n Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M.\n Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Philip Ilten",
"url": "https://arxiv.org/abs/1210.6289"
}
|
1210.6369
|
# Long-lived heavy quarks : a review
Mathieu Buchkremer mathieu.buchkremer@uclouvain.be Centre for Cosmology,
Particle Physics and Phenomenology (CP3),
Université catholique de Louvain,
Chemin du Cyclotron, 2, B-1348, Louvain-la-Neuve, Belgium. Alexander Schmidt
Alexander.Schmidt@cern.ch Institut für Experimentalphysik
Universität Hamburg
Luruper Chaussee 149
22761 Hamburg, Germany Centre for Cosmology, Particle Physics and
Phenomenology (CP3)
Université catholique de Louvain
Chemin du Cyclotron 2
B-1348, Louvain-la-Neuve, Belgium
email: mathieu.buchkremer@uclouvain.be Institut für Experimentalphysik
Universität Hamburg
Luruper Chaussee 149
22761 Hamburg, Germany
email: Alexander.Schmidt@cern.ch
###### Abstract
We review the theoretical and experimental situation for long-lived heavy
quarks, or bound states thereof, arising in simple extensions of the Standard
Model. If these particles propagate large distances before their decay, they
give rise to specific signatures requiring dedicated analysis methods. In
particular, vector-like quarks with negligible couplings to the three known
families could have eluded the past experimental searches. While most analyses
assume prompt decays at the production vertex, novel heavy quarks might lead
to signatures involving displaced vertices, new hadronic bound states, or
decays happening outside of the detector acceptance. We perform
reinterpretations of existing searches for short- and long-lived particles,
and give suggestions on how to extend their reach to long-lived heavy quarks.
††preprint: a††preprint: a
## I Introduction
Over the past decades, we have discovered that Nature consists of a given
number of elementary particles, deeply connected with the known fundamental
forces driving our universe. The recent observation of a new particle
resembling the long-sought Higgs boson at the Large Hadron Collider (LHC) now
provides us with strong evidence for the validity of the Standard Model (SM)
Aad _et al._ (2012a); Chatrchyan _et al._ (2012a). Yet, while it is
generally acknowledged that the latter comprises three generations of chiral
quarks and leptons, various fundamental problems do not find their answers
within this framework. For instance, long-standing issues such as the origin
of the fermion mass hierarchy or the nature of the Cabibbo-Kobayashi-Maskawa
(CKM) mixing matrix hint at the possible need for new physics. While the SM is
expected to loose its predictive power as the experimental searches now reach
the TeV scale, models suggesting new heavy fermions might offer solutions to
these problems in the near future.
Although a fourth sequential family of quarks is now disfavoured by the recent
results of the searches for the Higgs boson Eberhardt _et al._ (2012), other
models predicting new heavy particles beyond the top quark are still
consistent with the current experimental measurements. In particular, the
possibility for vector-like quarks, i.e., quarks having their left- and right-
handed components transforming identically under the electroweak gauge group,
are a common feature in many scenarios going beyond the SM, e.g. extra
dimensional models, Little Higgs models, grand unified theories, and so forth
Frampton _et al._ (2000). Non-chiral quarks also have the peculiarity to
decouple in the heavy mass limit, leading to SM-like signals. Should the
mixings of these with the light SM fermions be suppressed, the Higgs
production rates would not be easily distinguishable from the SM expectations
Dawson and Furlan (2012). These new quarks could be sufficiently long-lived to
behave like effectively stable particles, evading the current searches as they
propagate over sizeable distances.
From the collider searches it is clear that if new states with masses less
than a hundred GeV had existed, they would have been observed. On the other
hand, the experimental reach for detecting novel heavy stable particles above
the TeV scale depends on how readily identifiable such states are at the LHC.
While particles with nanoseconds lifetimes could fly away from the primary
interaction point before they decay to ordinary particles and lead to
displaced vertices, they can also hadronise, allowing for a possibly rich
spectrum of new exotic and heavy bound states. Such stable massive particles
are anticipated in many new physics models, either due to the presence of a
new conserved quantum number (e.g., $R$ parity in supersymmetric models), or
because the decays are suppressed by kinematics or small couplings (see
Fairbairn _et al._ (2007) and references therein for an exhaustive review).
Although the majority of the past studies focused on stable states in
supersymmetry, the possibility for new stable quarks received few attention at
the LHC. While the CMS and ATLAS experiments are now setting strong limits on
long-lived gluino, stop and stau pair production (see, for instance,
Chatrchyan _et al._ (2012b) and Aad _et al._ (2011)), scenarios involving
quasi-stable heavy quarks have been barely investigated at the LHC.
In the following, we review the phenomenology of new, long-lived heavy quarks.
Assuming a general parametrisation, Section II first examines their production
and decay modes at the LHC, considering both the chiral and vector-like
scenarios. The possible signatures for displaced vertices and stable massive
hadrons are then covered within the context of negligible mixings with the SM
fermions. The experimental aspects of the collider searches for new long-lived
quarks are presented in Section III. We review the situation for prompt decay
searches, displaced vertices and Heavy Stable Charged Particles (hereafter,
HSCPs) and highlight some of their limitations. Reinterpretations of the
existing searches are then presented for short- and long-lived particles.
Considering the possibility to improve the sensitivity of the existing
analyses to long-lived quarks beyond the TeV scale, some alternative search
topologies are finally described. Our conclusions are given in Section IV.
## II Long-lived quarks signatures
### II.1 Production
As we review the possibility for long-lived quarks in general, two different
scenarios will be distinguished in this work. New chiral fermions, for which
the left- and right-handed chiralities have different charges under the
electroweak gauge group, gain their masses from the electroweak symmetry
breaking mechanism. If heavy, a sequential fourth generation quark doublet
$(t^{\prime},b^{\prime})_{L}$ can therefore induce large corrections to loop
observables. Such a scenario is now severely constrained from the electroweak
precision data and the Higgs searches results. Given that chiral fermions
couple to the Higgs boson with a strength proportional to their Yukawa
couplings, they do not decouple from its production, as the corresponding rate
should increase by a factor of about 9 due to the additional fermion loops
occurring in gluon fusion. The unobserved enhancement thus now strongly
disfavours a fourth generation Standard Model Eberhardt _et al._ (2012).
Non-chiral quarks, on the other hand, still provide a viable extension to the
SM and certainly require a dedicated discussion. Considering the introduction
of new vector-like heavy partners mixing with the SM quarks, additional
parameters are allowed from $SU(2)_{L}\times U(1)_{Y}$ invariant Yukawa
interactions and Dirac mass terms in the SM Lagrangian. The light quark
couplings to the Higgs and electroweak bosons are consequently modified, along
with the introduction of new couplings between the heavy and the SM quarks.
Still, it is important to emphasise that such new states cannot have arbitrary
quantum numbers, since they can only mix with the SM quarks through a limited
number of gauge-invariant couplings. Classifying them into $SU(2)_{L}$
multiplets, their Yukawa terms only allow for seven distinct possibilities,
_i.e._ , the two singlets, the three doublets and the two triplets displayed
in Table 1.
$Q_{q}$ | $T_{\frac{2}{3}}$ | $B_{-\frac{1}{3}}$ | $\begin{pmatrix}X_{\frac{5}{3}}\\\ T_{\frac{2}{3}}\end{pmatrix}$ | $\begin{pmatrix}T_{\frac{2}{3}}\\\ B_{-\frac{1}{3}}\end{pmatrix}$ | $\begin{pmatrix}B_{-\frac{1}{3}}\\\ Y_{-\frac{4}{3}}\end{pmatrix}$ | $\begin{pmatrix}X_{\frac{5}{3}}\\\ T_{\frac{2}{3}}\\\ B_{-\frac{1}{3}}\end{pmatrix}$ | $\begin{pmatrix}T_{\frac{2}{3}}\\\ B_{-\frac{1}{3}}\\\ Y_{-\frac{4}{3}}\end{pmatrix}$
---|---|---|---|---|---|---|---
$T_{3}$ | $0$ | $0$ | $1/2$ | $1/2$ | $1/2$ | $1$ | $1$
$Y$ | $2/3$ | $-1/3$ | $7/6$ | $1/6$ | $-5/6$ | $2/3$ | $-1/3$
Table 1: Vector-like multiplets allowed to mix with the SM quarks through
Yukawa couplings. The electric charge is the sum of the third component of
isospin $T_{3}$ and of the hypercharge $Y$. Adapted from del Aguila _et al._
(2000).
The production of such new heavy quarks, either chiral or vector-like, is
usually assumed to proceed dominantly at the LHC through gluon fusion,
$gg\rightarrow Q\bar{Q}$. Nevertheless, depending on the model at hand,
electroweak single production can also provide an alternative mechanism as it
is not affected by the large phase-space suppression from which pair
production suffers. In particular, new heavy quarks can be produced singly in
flavour-changing processes via the electroweak interaction through
$q_{i}\overset{\textbf{(---)}}{q_{j}}\rightarrow V^{\ast}\rightarrow q_{k}Q$,
where $V=W,Z$. A comparison between the dominant production modes can be found
in Campbell _et al._ (2009); Buchkremer _et al._ (2012) for fourth
generation, and in Atre _et al._ (2011) for vector-like quarks. Assuming
order unity couplings, benchmark cross-sections have been evaluated for
various mass scenarios and reproduced in Table 2.
channel | $\sqrt{s}=7\ $TeV | $\sqrt{s}=14\ $TeV
---|---|---
| $m_{Q}=900\ $GeV/$c^{2}$ | $m_{Q}=1800\ $GeV/$c^{2}$
$pp\rightarrow W^{\ast}\rightarrow jX$ | 1.4 | 0.36
$pp\rightarrow W^{\ast}\rightarrow j\bar{X}$ | 0.037 | 0.0092
$pp\rightarrow W^{\ast}\rightarrow jT$ | 0.61 | 0.16
$pp\rightarrow W^{\ast}\rightarrow j\bar{T}$ | 0.052 | 0.013
$pp\rightarrow Z^{\ast}\rightarrow jT$ | 0.43 | 0.11
$pp\rightarrow Z^{\ast}\rightarrow j\bar{T}$ | 0.025 | 0.0064
$pp\rightarrow W^{\ast}\rightarrow jB$ | 0.69 | 0.18
$pp\rightarrow W^{\ast}\rightarrow j\bar{B}$ | 0.089 | 0.022
$pp\rightarrow Z^{\ast}\rightarrow jB$ | 0.18 | 0.047
$pp\rightarrow Z^{\ast}\rightarrow j\bar{B}$ | 0.034 | 0.0088
$pp\rightarrow W^{\ast}\rightarrow jY$ | 0.29 | 0.074
$pp\rightarrow W^{\ast}\rightarrow j\bar{Y}$ | 0.12 | 0.031
Table 2: Electroweak single production cross-sections (in pb) as a function of
the mass $m_{Q}$ of the heavy quark, assuming order unity couplings as
computed in Atre _et al._ (2011). In contrast, strong pair production
$pp\rightarrow Q\bar{Q}$ yields $3.61$ ($0.63$) fb for $m_{Q}=900$ ($1800$)
GeV/$c^{2}$ at $\sqrt{s}=7$ $(14)$ TeV.
As described in Atre _et al._ (2011), the leading contributions to vector-
like $T$ and $B$ single production arise from $ud\rightarrow
W^{\ast}\rightarrow jT$, $du\rightarrow W^{\ast}\rightarrow jB$,
$qd\rightarrow Z^{\ast}\rightarrow jB$ and $qu\rightarrow Z^{\ast}\rightarrow
jT$, where $q$ denotes a valence quark parton and $j$ a generic light quark
jet. Unlike QCD pair production, the above processes however scale with the
$Q-q$ quark coupling squared, and can be strongly suppressed if the associated
mixings are negligibly small. On the other hand, Figs. $2-4$ in Atre _et al._
(2011) indicate that $\sigma(pp\rightarrow Q\bar{Q})$ falls off faster than
the single production cross-sections as soon as $m_{Q}$ reaches the TeV scale,
while the electroweak production channels displayed in Table 2 are enhanced by
a factor $O(m_{Q}^{2}/m_{V}^{2})$, originating from the longitudinal
polarisation of the gauge boson $V$. Furthermore, the relative rates of the
$jQ$ and $j\bar{Q}$ channels strongly depend on the valence quark density in
the initial state. Given the difference in the PDFs of valence and sea quarks
in the initial states, new heavy quarks are produced singly at a much higher
rate than antiquarks for increasing $m_{Q}$. While the $u$ quark contributes
more than the $d$ quark, the neutral current contributions are weaker than the
charged current ones. Finally, we underline the absence of the tree-level
processes $pp\rightarrow Z^{\ast}\rightarrow jQ$ for vector-like quarks
involving exotic charges, given that they only interact with other states via
tree-level charged currents Atre _et al._ (2011). Although most of the above
statements are model-dependent, we conclude that the large running energy of
the LHC makes single electroweak production a promising discovery channel when
considering new heavy quarks searches with $m_{Q}\gtrsim 1$ TeV/$c^{2}$.
### II.2 Decay
In the search of new fermions, the associated decay modes require careful
attention. The exact partial width for a new sequential heavy quark $Q$
decaying on-shell to a light quark $q$ through a charged current can be
written as
$\Gamma(Q\rightarrow qW)=\frac{G_{F}m_{Q}^{3}}{8\pi\sqrt{2}}\text{
}|\kappa_{Qq}|^{2}\text{ }f_{2}(\frac{m_{q}}{m_{Q}},\frac{m_{W}}{m_{Q}}),\\\ $
(1)
$f_{2}(\alpha,\beta)=[(1-\alpha^{2})^{2}+\beta^{2}(1+\alpha^{2})-2\beta^{4}]\text{
}\sqrt{[1-(\alpha+\beta)^{2}][1-(\alpha-\beta)^{2}]},$ (2)
where $\kappa_{Qq}$ denotes the generic $Q-q$ quark coupling, equal to the CKM
mixing matrix element $V_{Qq}$ when considering a new sequential family of
quarks. Interestingly, the width (1) holds for both chiral and vector-like
singlet quarks, given that all heavy-to-light charged current quark decays
occur to be pure $V-A$ processes with identical rates Frampton and Hung
(1998). If $m_{Q}\gg m_{q}$, the above partial width can be shown to reach the
asymptotic form
$\Gamma(Q\rightarrow qW)\simeq 170.5\text{
MeV}\times|\kappa_{Qq}|^{2}\times\frac{m_{Q}^{3}}{m_{W}^{3}}.$ (3)
Considering a new heavy quark $Q$ decaying exclusively via the above 2-body
decay mode, we evaluate in Table 3 the corresponding decay lengths, with
$c\tau\simeq 2\times 10^{-10}$ GeV$/\Gamma($GeV) $\mu$m and $|\kappa_{Qq}|=1$.
With increasing mass, the probability for a new heavy quark $Q$ to decay
weakly thus becomes more and more important as its lifetime decreases.
However, small couplings to the lighter SM quarks strongly affect this
statement as the corresponding widths are suppressed by $|\kappa_{Qq}|^{2}$.
As we will see in Section II.4, this could lead to a very different
phenomenology as the quarks $Q$, if very heavy, would form bound states and
possibly decay hadronically.
$m_{Q}$ (GeV/$c^{2}$) | $\Gamma$ (GeV) | $c\tau$ $\times 10^{12}$ ($\mu$m)
---|---|---
300 | 8.73 (2.39) | 22.91 (83.85)
400 | 20.90 (11.14) | 9.57 (17.96)
500 | 40.94 (27.86) | 4.89 (7.18)
600 | 70.81 (54.52) | 2.82 (3.67)
700 | 112.50 (93.06) | 1.78 (2.15)
800 | 167.97 (147.43) | 1.19 (1.38)
900 | 239.18 (213.57) | 83.62 (9.36) $\times 10^{-1}$
1000 | 328.12 (299.46) | 6.10 (6.68) $\times 10^{-1}$
1100 | 436.74 (405.05) | 4.58 (4.94) $\times 10^{-1}$
1200 | 567.02 (532.31) | 3.53 (3.76) $\times 10^{-1}$
1300 | 720.93 (683.21) | 2.77 (2.92) $\times 10^{-1}$
1400 | 900.43 (859.72) | 2.22 (2.33) $\times 10^{-1}$
1500 | 1107.5 (1063.79) | 1.81 (1.88) $\times 10^{-1}$
1600 | 1344.1 (1297.4) | 1.49 (1.54) $\times 10^{-1}$
1700 | 1612.2 (1562.5) | 1.24 (1.28) $\times 10^{-1}$
1800 | 1913.8 (1861.1) | 1.05 (1.07) $\times 10^{-1}$
Table 3: Partial widths and decay lengths for a new heavy quark $Q$ decaying
to light quarks via on-shell charged currents $Q\rightarrow qW$, with
$m_{q}=0$ (173.2) GeV/$c^{2}$ and $|\kappa_{Qq}|=1$, as computed from Eq. (1).
Several conclusions can already be drawn at this stage. As far as a heavy
fourth quark family $(t^{\prime},b^{\prime})_{L}$ is concerned, we highlight
that the ratios of the charged current decay rates to the lighter families
only depend on the off-diagonal mixing elements. Assuming that the extended
$4\times 4$ CKM matrix is unitary, the CKM mixing elements $|V_{Qq}|$ imply
non-unity branching ratios for fourth generation quarks in general.
The possibility for tree-level Flavour Changing Neutral Currents lead to
similar conclusions for models involving new vector-like quarks interacting
with the SM families. Depending on their Yukawa couplings, they can undergo
tree-level transitions to lighter quarks and $W$, $Z$ and Higgs bosons.
Considering vector-like quarks $Q=T,B$ with $m_{Q}\gg m_{q}$, the neutral
currents $Q\rightarrow qZ$ and $Q\rightarrow qH$ can occur with a rate of the
same magnitude as given by (3), if they are allowed to mix with the light SM
quarks $q$. If no assumption is made on the hierarchy of the $Q-q$ coupling
strengths, no exclusive decay mode to the first, second or third family should
be preferred.
For non-chiral quarks decaying to $Wq$, $Zq$ and $Hq$, long lifetimes can be
expected in the $|\kappa_{Qq}|\simeq 0$ limit. Additionally, it is interesting
to notice that $\Gamma(Q\rightarrow qZ)/\Gamma(Q\rightarrow qH)\simeq 1$ holds
in the large mass limit $m_{Q}\gg m_{q}$ for most scenarios, while the
charged-to-neutral current ratios $\Gamma(Q\rightarrow qW)/\Gamma(Q\rightarrow
q(Z,H))$ approach either $2$ or $0$ depending on the vector-like
representation Cacciapaglia _et al._ (2010).
If more than one new heavy quark is considered, and if they are allowed to
decay into each other, the partial width (3) becomes inaccurate due to the
possible competition with the heavy-to-heavy transitions $Q_{1}\rightarrow
Q_{2}V^{(\ast)}$, where $V=W,Z,H$, depending on the model at hand. Indeed, if
the decay rates of the daughter particles are significant with respect to the
mass of the parent, the decay of an unstable particle is allowed through
threshold effects, even if kinematically forbidden. For both chiral and
vector-like quarks, real and virtual $V$ emission near threshold proceed with
the rate
$\Gamma(Q_{1}\rightarrow
Q_{2}V^{(\ast)})=\frac{G_{F}^{2}m_{Q_{1}}^{5}}{192\pi^{3}}\text{
}|\kappa_{Q_{1}Q_{2}}|^{2}\text{
}f_{3}\Big{(}\frac{m_{Q_{1}}^{2}}{m_{V}^{2}},\frac{m_{Q_{2}}^{2}}{m_{V}^{2}},\frac{\Gamma_{V}^{2}}{m_{V}^{2}}\Big{)},$
(4) $f_{3}(\alpha,\beta,\gamma)=2\int_{0}^{(1-\sqrt{\beta})^{2}}dx\text{
}\frac{[(1-\beta)^{2}+x\text{
}(1+\beta)-2x^{2}]\sqrt{\lambda(1,x,\beta)}}{[(1-\alpha x)^{2}+\gamma^{2}]},$
(5)
where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ac$ Frampton _et al._
(2000). While the $Q\rightarrow qV$ rates essentially depend on the $Q-q$
quark couplings, the heavy quark mass splittings $|m_{Q_{1}}-m_{Q_{2}}|$ drive
the strength of the heavy-to-heavy transitions Chao _et al._ (2011);
Buchkremer _et al._ (2012). If the quarks $Q_{1}$ and $Q_{2}$ belong to the
same weak multiplet with a large mass splitting, the $Q_{1}\rightarrow Q_{2}V$
decay could be very rapid and dominate the other decay modes if there is no
$|\kappa_{Q_{1}Q_{2}}|$ suppression. On the other hand, the lightest partner
can only decay into a SM family quark, and is thus plausibly long-lived for
small mixings. Should this be the case, we will see in Sections II.4 and II.5
that the latter could hadronise, leading to a different decay phenomenology.
This occurs specifically for $SU(2)_{L}$ vector-like singlets, which free
quark decays can only proceed via non-zero mixing into the lighter
generations.
As far as the doublet and triplet representations are concerned, new vector-
like partners should be very close in mass if one assumes that isospin
conservation is respected. Although it may be broken by higher order
corrections, such a degeneracy has been shown in Sher (1995) to require a mass
difference between $70$ and $110$ MeV for non-chiral doublets, allowing for
nanoseconds lifetimes. If their couplings to the lighter quarks are
negligible, the partial widths for $X_{5/3}$, $T_{2/3}$, $B_{-1/3}$ and
$Y_{-4/3}$ quarks are thus pausibly small.
Summarising these results, the range of the observable couplings to which
long-lived particles searches are sensitive at the LHC can be evaluated. From
the approximation (3), the partial decay length for a vector-like quark reads
$\lambda\simeq 2\text{ cm
}\times\Big{(}\frac{10^{-8}}{|\kappa_{Qq}|}\Big{)}^{2}\Big{(}\frac{500\text{
GeV/$c^{2}$}}{m_{Q}}\Big{)}^{3}.$ (6)
Assuming $m_{Q}\lesssim 1$ TeV/$c^{2}$ and an experimental resolution of at
least 25 $\mu$m, the heavy quark decays are prompt only if their associated
couplings are above the $10^{-7}$ level. If $|\kappa_{Qq}|\lesssim 10^{-7}$,
the corresponding decay lengths could be observed over a few tens of
centimeters. In the limit of such small mixings with the SM fermions, it is
thus natural to motivate searches for displaced vertices.
### II.3 Classification of signatures
We now detail the possible situations giving rise to the observable signatures
of long-lived quarks at the LHC.
The classification given in Table 4 summarises the long-lived quark scenarios
corresponding to the small mixing scenarios treated in this work. We emphasise
that if mixing with all SM quark families is permitted, direct searches should
aim at being sensitive to all light quarks $q$ in the observed final states.
Given that their interactions are allowed through arbitrary Yukawa couplings,
the searches for long-lived vector-like quarks should be as inclusive as
possible in order to cover the full spectrum of possibilities at the LHC. Such
a program is important to set new constraints on the $Q\rightarrow
qV^{(\ast)}$ decay modes, independently of any assumptions on the $Q-q$ mixing
sector. The relative importance of the $Q\rightarrow(u,c,d,s)V$ channels is
most relevant compared to the $Q\rightarrow(t,b)V$ charged currents, as no
exclusive decay mode should be preferred for $V=W,Z,H$.
| $(i)$ | $(ii)$ | $(iii)$
---|---|---|---
| Short-lived | Intermediate | Long-lived
$|\kappa_{Qq}|$ | $\gtrsim 10^{-7}$ | $\left.\begin{array}[]{c}\lesssim 10^{-7}\\\ \gtrsim 10^{-9}\end{array}\right.$ | $\lesssim 10^{-9}$
$\Gamma$ (GeV) | $\gtrsim 10^{-12}$ | $\left.\begin{array}[]{c}\lesssim 10^{-12}\\\ \gtrsim 10^{-16}\end{array}\right.$ | $\lesssim 10^{-16}$
$\tau$ (s) | $\lesssim 10^{-13}$ | $\left.\begin{array}[]{c}\gtrsim 10^{-13}\\\ \lesssim 10^{-9}\end{array}\right.$ | $\gtrsim 10^{-9}$
Table 4: Possible decay signatures for new long-lived quarks ($m_{Q}=1$
TeV/$c^{2}$), as discussed in the text. Figure 1: Lifetime $\tau$ in seconds
($\gamma\beta\simeq 1$) for a sequential heavy quark $Q$ decaying to the
lighter families by emitting a real $W$ boson, for $m_{Q}=300\ $GeV/$c^{2}$
(in blue) and $m_{Q}=1800$ GeV/$c^{2}$ (in red). The three regions of interest
refer to the conventions given in Table 4.
As illustrated in Figure 1, three cases for new long-lived quarks can be
distinguished depending on their mixing parameters with the SM fermions :
* •
Firstly, if all couplings with the known SM fermions are larger than the
$10^{-7}$ level, all heavy-to-light decays $Q\rightarrow qV$ are prompt, over
distances smaller than 25 $\mu$m. This defines the short-lived scenario _(i)_
, with no experimental difference compared to the direct searches.
* •
The second scenario _(ii)_ arises for intermediate decay lengths ranging
between a few microns and centimetric distances. If there is at least one
light quark $q$ such that $10^{-7}\lesssim|\kappa_{Qq}|\lesssim 10^{-9}$,
displaced events could be observed, yet with a partial width suppressed by
$|\kappa_{Qq}|^{2}$. Consequently, if all the couplings but one are below the
$10^{-7}$ level, a single exclusive channel would lead to a prompt decay
signature corresponding to a short-lived heavy quark. A possible exception,
though, is the case for new heavy multiplets with sizeable mass splittings, as
allowed in extra generation models and possible extensions Buchkremer _et
al._ (2012); Dighe _et al._ (2012); Bar-Shalom _et al._ (2011); Asilar _et
al._ (2012). If $m_{Q_{1}}\gtrsim m_{Q_{2}}+m_{V}$, the heaviest quark $Q_{1}$
can be short-lived and decay semi-weakly to $Q_{2}V^{(\ast)}$, while the
lightest partner is likely stable if all its decay modes suffer severe
suppression. If $m_{Q_{1}}\simeq m_{Q_{2}}$, on the other hand, all heavy-to-
heavy transitions are suppressed and both quarks could be long-lived on
detector scales.
* •
For particles with decay lengths larger than the detector dimensions lies the
long-lived region _(iii)_ , the details of which will be discussed in Section
II.4. If all heavy quarks couplings with the SM fermions are below the
$10^{-9}$ level, the stable case becomes a relevant scenario, possibly in
conjunction with displaced events.
We emphasise that the scenarios for which prompt decays and displaced vertices
could be observed only call for one of the allowed decay modes to fulfill the
associated conditions. The stable scenario, however, requires all decay
lengths and partial widths to lie in the ranges given in Table 4. As we will
see in Section II.4, if all $Q-q$ quark couplings to the SM families are
smaller than $10^{-2}$, such new heavy fermions could hadronise. As a result,
the annihilation decays and hadronic transitions between the formed bound
states would dominate, while the single quark decay events might be
suppressed.
### II.4 Signatures for heavy stable particles : heavy quarkonia
Previously, we have discussed how displaced events could occur in the single
quark decays transitions in the case of small $Q-q$ couplings. In this
section, we describe the possibility for new heavy quarks to bind into hadrons
if their lifetime is large enough, providing us with an interesting
alternative for new signal searches at the LHC.
While new coloured particles with masses heavier than the top quark are
usually assumed to decay as free particles, their possible formation into
baryons and mesons is an important issue to consider in the case of small
couplings with the SM fermions.
Should the partial widths $Q\rightarrow q(W,Z,H)$ be suppressed, the quark $Q$
might be stable and hadronise. Assuming that the binding force in a fourth
generation bound state is of Coulombic nature, the seminal condition
$m_{Q}<125\text{ GeV}(100\text{ GeV})\times|V_{Qq}|^{-2/3}$ (7)
was derived in Bigi _et al._ (1986) for $Q\bar{Q}$ ($Q\bar{q}$) formation.
According to Eq. (7), new heavy quarks form hadronic states if their mixing
with known quarks is sufficiently small. Mixings roughly smaller than 0.2 are
typically required for $300$ GeV/$c^{2}$ new quarks to form bound states,
which remains perfectly consistent with the current bounds from the
electroweak precision observables. As shown on Figure 2 for a new up-like
fourth generation quark $t^{\prime}$, the electroweak precision fits typically
restrict the mixing matrix element $|V_{t^{\prime}b}|\simeq|V_{tb^{\prime}}|$
to be smaller than $10^{-1}$ for $m_{t^{\prime}}>700$ GeV/$c^{2}$. If lighter,
the $t^{\prime}$ total width can be smaller than $10$ MeV Buchkremer _et al._
(2012), thus allowing new chiral quarks to form bound states with nanosecond
lifetimes.
Figure 2: Reproduction of the $\chi^{2}$ fit of the oblique parameters $S$ and
$T$ as given from Fig. 2 in Buchkremer _et al._ (2012), considering a chiral
fourth generation quark $t^{\prime}$. The upper limits at 68% (darker blue)
and 95% (lighter blue) CL on the $t^{\prime}$ mass are depicted as a function
of the CKM mixing $|V_{t^{\prime}b}|$, freely varied over [300,800]
GeV/$c^{2}$ and [0,0.3], respectively. The solid (dashed) curve denotes the
upper bound $m_{t^{\prime}}(|V_{t^{\prime}b}|)$ under which
$t^{\prime}\bar{t}^{\prime}$ and $t^{\prime}\bar{q}$ ($\bar{t^{\prime}}q$)
bound states can form, as defined by relation (7).
As far as non-chiral fermions are concerned, the electroweak precision
constraints provide upper bounds on the mixing parameters, but are not as
restrictive given that their effects decouple in the limit of large masses
Lavoura and Silva (1993). Although vector-like quarks are usually considered
to couple dominantly to the heaviest third quark family (see Atre _et al._
(2011); Agashe _et al._ (2006) for some exceptions), they are allowed, in
principle, to mix with all up- (down-) type quarks Cacciapaglia _et al._
(2010).
In any case, $|\kappa_{Qq}|\lesssim 10^{-2}$ is used as a rule of thumb in
most of the allowed vector-like representations Cacciapaglia _et al._ (2010).
Furthermore, corrections to the CKM matrix elements are required to be small
for $SU(2)_{L}$ singlets Barger _et al._ (1995), doublets and triplets
Yoshikawa (1996). In this framework, the hadronic production of new bound
states involving sequential or non-chiral quarks is a conceivable possibility
at the LHC. Assuming that (7) is satisfied due to small $Q-q$ couplings, their
lifetime can be longer than the orbital period of the corresponding $Q\bar{Q}$
($Q\bar{q}$) bound state, allowing to build up new quarkonium resonances,
open-flavour mesons and baryons Bigi _et al._ (1986).
If their formation is governed by the same strong interactions that are
responsible for the existence of the ordinary hadronic spectroscopy, the
associated mass spectrum can be determined from the known properties of low-
energy hadron physics Mackeprang and Milstead (2010). Considering a simplified
model of QCD-like hadrons, new heavy quarks would form $Q\bar{Q}$ bound states
(hereafter defined as $\eta_{Q}$) from induced, strongly attractive
potentials. Conjointly, quarkonium formation can arise from Higgs boson
exchange, as the corrections from Yukawa-type forces cannot be neglected for
large quark masses Ishiwata and Wise (2011); Flambaum and Kuchiev (2011). In
Hung and Xiong (2011), Hung detailed the numerical resolution of a general
non-relativistic Higgs exchange potential, with interesting consequences on
the associated scalar spectrum. More recently, Enkhbat et al. gave a
preliminary discussion of the fourth generation Yukawa bound states
phenomenology at the LHC, considering a ultra-heavy degenerate new quark
family Enkhbat _et al._ (2011). Although binding energies of Yukawa origin
certainly provide a more consistent framework for a dedicated analysis, we
restrict the present discussion to the context of Coulomb-like potential
models, and assume that the short-distance behaviour of the quark-antiquark
potential dominates (we refer the reader to Barger _et al._ (1987) and Kuhn
and Zerwas (1988) for previous and more exhaustive phenomenological reviews).
In this context, the effects of the quarkonium wave function must be taken
into account as we consider the corresponding production and decay modes. In
perturbative QCD, the dominant contribution to $\eta_{Q}$ production cross-
section in $pp$ collisions reads Barger _et al._ (1987)
$\displaystyle\sigma(pp$ $\displaystyle\rightarrow$ $\displaystyle
gg\rightarrow\eta_{Q})=\frac{\pi^{2}\tau}{8m_{\eta}^{3}}\Gamma(\eta_{Q}\rightarrow
gg)\text{ }\text{ }[\tau\frac{dL}{d\tau}]_{gg},$ (8)
$\displaystyle\Gamma(\eta_{Q}$ $\displaystyle\rightarrow$ $\displaystyle
gg)=\frac{8\alpha_{s}^{2}(m_{\eta}^{2})}{3m_{\eta}^{2}}|R_{S}(0)|^{2},$ (9)
where $m_{\eta}=2m_{Q}$, and $[\tau\frac{dL}{d\tau}]_{gg}$ denotes the gluon
luminosity with $\tau=m_{\eta}^{2}/s$. The full NLO partonic cross-sections
for the $gg$, $qg$ and $q\bar{q}$ initiated reactions are provided in Kuhn and
Mirkes (1993a), and have been updated in Arik _et al._ (2002). Interestingly,
they are all proportional to the square of the $Q\bar{Q}$ radial wave function
$R_{S}$ at the origin, which might drive the heavy quarkonium production
cross-section $\sigma(gg\rightarrow\eta_{Q})\ $ to be substantially smaller
than $\sigma(gg\rightarrow Q\bar{Q})$. Indeed, the quarkonium production rate
typically amounts to a few percent of the pair production cross-section Kuhn
and Mirkes (1993a); Arik _et al._ (2002); Hagiwara _et al._ (2008).
Depending whether the decay rate of the constituent particles, $\Gamma_{Q}$,
is larger or smaller than the bound state annihilation rate, two different
cases can be considered as the bound states can either undergo single quark
decays or annihilate hadronically. As detailed in Kats and Schwartz (2010),
the strength of the annihilation decay signal is enhanced when the intrinsic
width of the heavy quark $Q$ is decreased. Should its rate be of the same
order of magnitude as the binding energy of the $\eta_{Q}$ state, the
quarkonium can be broad and display little evidence of any resonance behaviour
over the continuum. On the other hand, if its 2-body decays are suppressed due
to kinematical suppression or small couplings as discussed in Section II.2,
$Q$ can only decay through off-shell intermediate states. In such a case, the
signature would correspond to an annihilation signal, namely, a hadronic final
state markedly distinguishable from pair production and decay.
We now give a brief and qualitative description of some of the expected
$\eta_{Q}$ hadronic signatures at the LHC, considering either a chiral or
vector-like heavy quark $Q$. While similar results hold for all $S$\- and
$P$-wave quarkonium states, we limit our analysis to the case of a stable,
neutral $J^{PC}=0^{-+}$ pseudoscalar state $\eta_{Q}$. In particular, a
${}^{1}S_{0}$ quarkonium state formed from down-type fourth generation
$b^{\prime}$ quarks is known to provide a possible candidate if the condition
(7) is satisfied Arik _et al._ (2002). The latter is mainly produced through
the gluon fusion process (9) with a production cross-section two orders of
magnitude larger than the $J^{PC}=1^{--}$ vector state Arik _et al._ (2002).
If $b^{\prime}$ is the lightest fourth generation quark, a $\eta_{b^{\prime}}$
bound state can decay either through $q-b^{\prime}$ family mixing with
$q=u,c,t$, or to boson pairs. Whether the single quark decays would compete
with the $\eta_{b^{\prime}}$ hadronic modes depends on the heavy quark masses
and mixings. If $m_{b^{\prime}}>1$ TeV/$c^{2}$, a fourth-generation down-type
quark dominantly decays into fermion-antifermion pairs if
$|V_{qb^{\prime}}|\gtrsim 10^{-2}$. For $m_{b^{\prime}}<1$ TeV/$c^{2}$, the
$\eta_{b^{\prime}}$ total width lies below $1$ GeV/$c^{2}$, and the quarkonium
state decays proceed dominantly via the annihilation diagrams depicted in
Figure 3 Arik _et al._ (2002). Decays to gauge boson pairs, fermion-
antifermion pairs, gauge-Higgs and Higgs boson pairs can occur through
$\gamma,Z$ and $H$ exchange in the $s-$ channel, or proceed via quark mixing
in the $t-$ and $u-$ channels, if allowed. Decays to Higgs boson pairs
($\eta_{Q}\rightarrow HH$) are forbidden by CP conservation in the case of
$J^{PC}=0^{-+}$ pseudoscalars. While the processes involving charged currents
are suppressed in the case of small quark couplings, the
$\eta_{b^{\prime}}\rightarrow WW$ mode is allowed at loop-level via the
exchange of the $SU(2)_{L}$ partner $t^{\prime}$, even if
$|V_{tb^{\prime}}|\simeq 0$. Decays to $WW$ pairs are not allowed for
$\eta_{T,B}$ bound states involving singlet vector-like quarks.
Figure 3: First order diagrams for the $\eta_{Q}\rightarrow p_{1}p_{2}$
decays, where $p_{1}$ and $p_{2}$ can be a gauge boson pair, a fermion-
antifermion pair, a gauge-Higgs boson pair, or a pair of Higgs bosons. The
strong decay mode $\eta_{Q}\rightarrow gg$ is also allowed but not shown
above. The (a), (b), and (c) diagrams denote decays via $\gamma$, $Z$ or $H$
exchange in the $s-$channel, while (d) and (e) correspond to the $p_{1}p_{2}$
decays involving quark exchange in the $t-$ and $u-$ channels. Figure from
Barger _et al._ (1987).
An exhaustive study of the production and subsequent decays of the $S-$ and
$P-$ wave quarkonium states of a heavy chiral quark is available in Barger
_et al._ (1987) and Kuhn and Mirkes (1993b), where all the allowed production
mechanisms and decay patterns have been thoroughly investigated. As an
illustration, we compare in Figure 4 the branching ratios of the allowed decay
modes for quarkonium states formed by a chiral fourth generation quark
$b^{\prime}$ and a vector-like isosinglet $B$ quark Kuhn and Mirkes (1993b).
The quarkonium annihilation decay rates depend markedly on the mass scale.
Figure 4: Branching ratios as given in Kuhn and Mirkes (1993b) for $\eta_{Q}$
quarkonium, considering a chiral $b^{\prime}$ quark (top) and a down-type
singlet vector-like $B$ quark (bottom), with $m_{H}=125$ GeV/$c^{2}$.
$|V_{tb^{\prime}}|\simeq 1$ is assumed in the fourth generation case.
In the chiral case, the main decay modes are $ZH$, $gg$, $WW$, $ZZ$, $Z\gamma$
and $\gamma\gamma$ respectively. The fermionic decays
$\eta_{b^{\prime}}\rightarrow q\bar{q}$ are also allowed if
$|V_{qb^{\prime}}|\neq 0$. Substantial decay rates to $q\bar{q}$ fermion pairs
are allowed in case of large $Q-q$ couplings. For pseudoscalar
$\eta_{b^{\prime}}$ masses lighter than $450$ GeV/$c^{2}$, the dominant
annihilation mode proceeds via the strong interaction, _i.e._ ,
$\eta_{b^{\prime}}\rightarrow gg$. For larger masses, the width into two
gluons is decreasing, whereas the annihilation mode into $ZH$ pairs takes
over. This follows from the fact that the decay rates into Higgs and
longitudinal gauge bosons are enhanced by the large Yukawa coupling of the
heavy $b^{\prime}$ quark, so that the $ZH$ branching ratio becomes sizeable
for increasing masses. The latter channel then leads to a salient signature
for heavy fourth generation bound states.
While fourth generation pseudoscalar quarkonia decays allow for striking
signals, $T\bar{T}$ and $B\bar{B}$ bound states of vector-like quark singlets
can only decay to $gg$, $\gamma\gamma$, $\gamma Z$ and $ZZ$ pairs, as the
$\eta_{Q}\rightarrow ZH$ mode is absent Kuhn and Mirkes (1993b). This follows
from the fact that the $ZH$ mode proceeds through the axial part of the
neutral current coupling, which in turn is proportional to the third component
of the weak isospin. Only $\gamma\gamma$, $Z\gamma$ and $ZZ$ would then signal
the visible hadronic modes with branching ratios of about $10^{-4}$.
$T\bar{T}$ and $B\bar{B}$ bound states of $SU(2)_{L}$ vector-like singlet
quarks are thus hardly observable at the LHC as the gluon mode dominates. On
the other hand, $T$ and $B$ vector-like doublet (triplet) partners with non-
vanishing weak isospin can have large branching ratios to $ZH$ via $\gamma$,
$Z$ or $H$ exchange in the $s-$channel. The $\eta_{X}$ and $\eta_{Y}$
quarkonia provide a noticeable exception, given that the exotic quarks
$X_{5/3}$ and $Y_{-4/3}$ do not couple to neutral currents at tree-level.
Since they only mix with the other states via charged currents, the
annihilation modes including $\gamma$, $Z$ and $H$ exchange are thus forbidden
for $\eta_{X}$ and $\eta_{Y}$, which then mainly decay to $gg$ and $WW$
bosons. Although it is suppressed below the percent level, the distinctive
diphoton decay mode also provides a possible golden channel for discovery. If
new stable, coloured particles are produced at the LHC, the resulting final
states could indeed allow for resonant signatures including photons and
leptons, from which their quantum numbers and masses can be identified
independently of their decay modes. As the decay rate for such bound states
might be small, model-independent searches for new resonances could be
competitive with the direct searches Kats and Schwartz (2010).
### II.5 Signatures for heavy stable particles : open-flavour mesons
While $Q\bar{Q}$ quarkonium resonances might be challenging to observe, the
possibility for “open-flavour”hadrons $Q\bar{q}$ ($\bar{Q}q$) and $Qqq$
($\bar{Q}\bar{q}\bar{q}$), with $q$ being a light SM quark, also cover a broad
spectrum of new heavy mesons and baryons to search for. Should they be
produced at the LHC, such states can form by capturing a light quark partner
from the vacuum. As they pass through the detector, they can transform into
various slowly-moving heavy states. Assuming that they hadronise with $u$, $d$
and $s$ quarks after being produced, Table 5 lists the corresponding allowed
stable mesons and baryons.
The properties of new hypothetical fourth generation quarkonium and open-
flavour mesons have been examined previously in Ikhdair and Sever (2006)
considering large-$N$ expansion techniques. In Bashiry _et al._ (2012), their
masses and decay constants have been estimated from the experimental
measurements of the ordinary $b\bar{b}$ ($c\bar{c}$) hadron masses, using the
QCD sum rules. Interestingly, both approaches indicate that the corresponding
spectra share numerous features with most of the scenarios predicting stable
bound states of heavy quarks beyond the Standard Model, depending on their
masses, charges and angular momenta.
Interestingly, the interactions of such open-flavour mesons with the material
occur to be similar to that of R-hadrons, _i.e._ , stable hadrons composed of
a supersymmetric particle and at least one SM quark (see Kraan (2004);
Mackeprang and Rizzi (2007) and references therein for related works). The 1/2
difference in spin with respect to stop hadrons has little impact on the
search strategies, which depend mostly on the interactions between the lighter
quarks and the matter of the detector. This is in substance the description
given by the spectator model Bjorken (1978); Suzuki (1977), for which the
allowed transitions of heavy and light quarks within a given bound state are
known to be flavour and spin independent Isgur and Wise (1989, 1991). In this
context, the heavy quark $Q$ can be considered as a source of kinetic energy
which sole function is to give mass and momentum to the underlying hadron.
Such new heavy bound states then behave as rather passive objects, consisting
of a non-interacting heavy component, accompanied by lighter constituents ($u$
and $d$ quarks, mostly). While the partons are being scattered within the
detector, their cross-sections vary with their inverse mass in perturbative
QCD Fairbairn _et al._ (2007); Mackeprang and Milstead (2010). A heavy
$Q\bar{q}$ ($Q\bar{q}$) bound state is then expected to suffer small energy
losses when interacting with the material, given that only its light quark
half is responsible for the hadronic interactions with the detector.
We list in Table 6 the various allowed $2\rightarrow 2$ and $2\rightarrow 3$
transitions involving meson-to-meson and meson-to-baryon conversions. In
general, these interactions lead to simultaneous pion emission as the initial
mesons convert into slowly-moving baryons.
Charge | Mesons | Baryons
---|---|---
$Q=0$ | $\mathbf{T\bar{u}},\mathbf{\bar{T}u},\mathbf{B\bar{d}},\mathbf{\bar{B}d},B\bar{s},\bar{B}s$ | $Tdd,Tds,Tss,\mathbf{Bud},\bar{B}\bar{u}\bar{d},Bus,\bar{B}\bar{u}\bar{s},Yuu,\bar{Y}\bar{u}\bar{u}$
$Q=1$ | $\mathbf{X\bar{u}},\mathbf{T\bar{d}},T\bar{s},\mathbf{\bar{B}u},\mathbf{\bar{Y}d},\bar{Y}s$ | $Xdd,Xds,Xss,\mathbf{Tud},Tus,Buu,\bar{B}\bar{d}\bar{d},\bar{B}\bar{d}\bar{s},\bar{B}\bar{s}\bar{s},\bar{Y}\bar{u}\bar{d},\bar{Y}\bar{u}\bar{s}$
$Q=2$ | $\mathbf{X\bar{d}},X\bar{s},\mathbf{\bar{Y}u}$ | $\mathbf{Xud},Xus,Tuu,\bar{Y}\bar{d}\bar{d},\bar{Y}\bar{d}\bar{s},\bar{Y}\bar{s}\bar{s}$
$Q=3$ | - | $Xuu$
Table 5: $Q\bar{q}$ ($\bar{Q}q$) mesons and $Qqq$ ($\bar{Q}\bar{q}\bar{q}$)
(anti-)baryons involving $Q=X_{5/3}$, $T_{2/3}$, $B_{-1/3}$ and $Y_{-4/3}$
vector-like quarks. The states in bold font are hadrons which yields are
expected to be substantial at the LHC, as predicted in Mackeprang and Milstead
(2010) for penetration depths between 0 and 3 meters. For simplicity, only the
neutral and positively charged hadrons are displayed.
Meson-to-Meson $2\rightarrow 2$ processes:
---
Charge exchange:
$Q\bar{u}+p\longleftrightarrow Q\bar{d}+n$
Elastic scattering:
$Q\bar{q}+p/n\rightarrow Q\bar{q}+p/n$
Meson-to-Baryon $2\rightarrow 2$ processes: Baryon exchange:
$Q\bar{u}+n\rightarrow Qud+\pi^{-}$ $Q\bar{u}+n\rightarrow Qdd+\pi^{0}$
$Q\bar{u}+p\rightarrow Quu+\pi^{-}$ $Q\bar{u}+p\rightarrow Qud+\pi^{0}$
$Q\bar{d}+n\rightarrow Qud+\pi^{0}$ $Q\bar{d}+n\rightarrow Qdd+\pi^{+}$
$Q\bar{d}+p\rightarrow Quu+\pi^{0}$ $Q\bar{d}+p\rightarrow Qud+\pi^{+}$
Meson-to-Meson $2\rightarrow 3$ processes: Inelastic scattering:
$Q\bar{u}+n\rightarrow Q\bar{d}+n+\pi^{-}$ $Q\bar{u}+n\rightarrow
Q\bar{u}+n+\pi^{0}$ $Q\bar{u}+n\rightarrow Q\bar{u}+p+\pi^{-}$
$Q\bar{u}+p\rightarrow Q\bar{d}+p+\pi^{-}$ $Q\bar{u}+p\rightarrow
Q\bar{u}+p+\pi^{0}$ $Q\bar{u}+p\rightarrow Q\bar{u}+n+\pi^{+}$
$Q\bar{u}+p\rightarrow Q\bar{d}+n+\pi^{0}$ $Q\bar{d}+n\rightarrow
Q\bar{u}+n+\pi^{+}$ $Q\bar{d}+n\rightarrow Q\bar{d}+n+\pi^{0}$
$Q\bar{d}+n\rightarrow Q\bar{u}+p+\pi^{0}$ $Q\bar{d}+n\rightarrow
Q\bar{d}+p+\pi^{-}$ $Q\bar{d}+p\rightarrow Q\bar{u}+p+\pi^{+}$
$Q\bar{d}+p\rightarrow Q\bar{d}+p+\pi^{0}$ $Q\bar{d}+p\rightarrow
Q\bar{d}+n+\pi^{+}$ Meson-to-Baryon $2\rightarrow 3$ processes: Baryon
exchange: $Q\bar{u}+n\rightarrow Qud+\pi^{-}+\pi^{0}$ $Q\bar{u}+n\rightarrow
Qdd+\pi^{-}+\pi^{+}$ $Q\bar{u}+n\rightarrow Qdd+\pi^{0}+\pi^{0}$
$Q\bar{u}+p\rightarrow Quu+\pi^{-}+\pi^{0}$ $Q\bar{u}+p\rightarrow
Qud+\pi^{-}+\pi^{+}$ $Q\bar{u}+p\rightarrow Qud+\pi^{0}+\pi^{0}$
$Q\bar{u}+p\rightarrow Qdd+\pi^{+}+\pi^{0}$ $Q\bar{d}+n\rightarrow
Qud+\pi^{-}+\pi^{+}$ $Q\bar{d}+n\rightarrow Qud+\pi^{0}+\pi^{0}$
$Q\bar{d}+n\rightarrow Qdd+\pi^{0}+\pi^{+}$ $Q\bar{d}+p\rightarrow
Quu+\pi^{-}+\pi^{+}$ $Q\bar{d}+p\rightarrow Quu+\pi^{0}+\pi^{0}$
$Q\bar{d}+p\rightarrow Qud+\pi^{0}+\pi^{+}$
Table 6: Meson-to-Meson and Meson-to-Baryon processes for $Q\bar{u}$ and
$Q\bar{d}$ mesonic states, where $Q=X_{5/3},T_{2/3},B_{-1/3},Y_{-4/3}$ and
$q=u,d$. Similar transitions can be obtained for the charge conjugated
processes involving $\bar{Q}q$. Assuming that the cross-sections for all the
transitions are of the same order of magnitude, Meson-to-Baryon processes to
$Qud$ states are more likely than those involving $Quu$ and $Qdd$ in the final
state.
While they interact with the nuclear matter, most of the new states suffer
multiple scattering and convert into baryons, allowing for $Quu$, $Qud$ and
$Qdd$ states, as new heavy mesons are kinematically favoured to increase their
baryon number by emitting one or more pions Mackeprang and Milstead (2010).
Due to the lack of these in the material, the reverse reaction is known to be
less favoured.
Interestingly, new mesons which convert at the beginning of the scattering
chain could generate tracks with “dashed-lines”, signalling possible baryonic
or electric charge exchange. Indeed, $Q\bar{q}$ and $\bar{Q}q$ bound states
traversing a medium composed of light quarks likely flip their electric
charge, frequently interchanging their parton constituents with those of the
material nuclei. Eventually, they could leave the detector as heavy stable
neutral particles and hence not be observable in muon detectors. The modeling
of the nuclear interactions of new heavy hadrons traveling through matter, a
survey of which is given in Mackeprang and Milstead (2010); de Boer _et al._
(2008), actually favours scenarios with significant charge suppression if
heavy open-flavour mesons have a sizeable probability to transform into
neutral particles while traversing the detector. This poses a serious
challenge for the experimental searches, since the most conservative scenarios
assume a complete charge suppression where 100% of the produced hadrons become
neutral before reaching the muon detectors. Taking a specific example, we
consider pair-produced $T_{2/3}$ quarks hadronising into the heavy states
$T\bar{q}$ and $\bar{T}q$ immediately after production. For convenience, we
assume that a majority ($\gtrsim 90\%$) of the new heavy quarks initially
hadronise into mesons, while a smaller amount ($\lesssim 10\%$) form baryons
Fairbairn _et al._ (2007). As they interact farther in the detector, most of
the $T\bar{q}$ mesons eventually exit the detector as $T\bar{u}$ and
$T\bar{d}$ mesonic states, or as $Tud$ baryons. Baryon-to-meson conversions
are allowed as well, as these states can also annihilate back into $T\bar{u}$
and $T\bar{d}$ mesons, yet with a smaller rate. $\bar{T}q$ mesons of vector-
like antiquarks, on the other hand, unlikely give rise to antibaryons, but can
still flip their charge through the exchange of $u$ and $d$ quarks with the
material. Possibly large flavour fractions for $\bar{T}u$ and $\bar{T}d$ can
thus be expected in the detector, with a negligible amount of
$\bar{T}\bar{q}\bar{q}$ states. If such antibaryons were to form, they would
quickly annihilate by baryon-to-meson interactions, producing pions Kang _et
al._ (2008).
After travelling through the detectors a few nuclear lengths away from the
production vertex, the novel hadrons transform in roughly 100% to the
positively charged baryon $Tud$, whereas one half of the $\bar{T}-$ states
thus transform into $\bar{T}u$ mesons, and the other half in $\bar{T}d$. The
corresponding hadronic flavour decompositions, evaluated as a function of the
penetration depth in the detector, can be read from fig. 5 of Mackeprang and
Milstead (2010) for R-hadrons involving stable scalar top and antitop quarks.
We notice that similar results are obtained when replacing stop $\tilde{t}$
squarks by stable quarks $T$ of equal mass. The $Tud$, $T\bar{u}$ ($\bar{T}u$)
and $T\bar{d}$ ($\bar{T}d$) states are then expected to retain the largest
flavour composition fractions when considering penetration depths larger than
$3$ meters, and are thus the most likely observable states, which we
emphasised in bold font in Table 5.
Except for $Yuu$ and $\bar{Y}\bar{u}\bar{u}$, it is interesting to notice that
none of the states listed in Table 5 is neutral if involving vector-like
quarks with exotic charges. The corresponding bound states might give rise to
possibly large fractions of slowly moving charged particles $Xud$ and $Yud$
with $Q=+2e$ and $-e$ respectively, accompanied by a small amount of
$X\bar{u}$, $X\bar{d}$, $Y\bar{u}$ and $Y\bar{d}$ mesons (and charge
conjugates). These particles are all electrically charged as they go across
the detector, emitting charged and neutral pions in the process together with
losing small amounts of energy. Given the small expected interaction cross-
section from the Meson-to-Baryon transitions, the processes shown in Table VI
leave small energy deposits in the calorimeters, of the order of $O(1)$ GeV
Fairbairn _et al._ (2007). On the other hand, such stable hadrons would lead
to observable tracks due to the ionisation energy losses, allowing for
signatures similar to slow-moving muons with high transverse momentum.
Searches for stable charged particles, if adapted to such a case, thus provide
a promising strategy to rule out the possibility for novel exotic quarks with
large lifetimes. Given these very specific signals, the aforementioned
signatures can certainly be discriminated at the LHC.
## III Experimental aspects
In this section, we review the past and current experimental searches for new
long-lived quarks. Limitations of these searches are then discussed along with
possible extensions to enhance sensitivity to heavy quarks. Reinterpretations
of direct searches for heavy quarks as well as general searches for long lived
particles are given in the context of long lived heavy quarks.
### III.1 Previous searches at Tevatron
The searches undertaken at the Tevatron resulted in already stringent mass
bounds on long-lived heavy quarks, yet not without assumptions. Looking for
long-lived parents of the $Z$ boson in displaced vertices from $p\bar{p}$
collisions at $\sqrt{s}=1.8$ TeV, CDF set limits on the cross-section of a
fourth generation charge $-1/3e$ quark as a function of its lifetime with an
integrated luminosity of $90$ pb-1 Abe _et al._ (1998). Isolated electron-
positron pairs originating from $Z$ decays were searched for in the exclusive
channel $b^{\prime}\rightarrow bZ$ with $m_{b^{\prime}}>m_{Z}$. Finding no
evidence for new long-lived particles, CDF excluded $m_{b^{\prime}}<148$
GeV/$c^{2}$ for $c\tau=$ 1 cm at 95% CL. This limit drops to
$m_{Z}+m_{b}\simeq 96$ GeV/$c^{2}$ if $c\tau>$ 22 cm or $c\tau<$ 0.009 cm.
Previously, the D0 collaboration already ruled out the range
$m_{Z}/2<m_{b^{\prime}}<m_{Z}+m_{b}$ from $b^{\prime}\rightarrow b\gamma$
searches in Abachi _et al._ (1997) for all proper lifetimes, while a
$b^{\prime}$ quark with a mass lower than $m_{Z}/2$ was previously dismissed
by the LEP direct searches Decamp _et al._ (1990). Within a 90 pb-1 data
sample of $p\bar{p}$ collisions recorded during 1994-95, CDF performed a
search for low velocity massive charged stable particles leaving large amounts
of energy in the calorimeters Acosta _et al._ (2003).
Assuming a muonlike penetration and searching for an anomalously high
ionisation energy loss signature, the data was found to agree with background
expectations, and upper limits of the order of 1 pb were derived on the
production cross-section. Sensitive to long-lived fourth generation quarks
scenarios, the lower bounds $m_{b^{\prime}}>190$ GeV/$c^{2}$ and
$m_{t^{\prime}}>220$ GeV/$c^{2}$ have been obtained for $q=-1/3e$ and $q=2/3e$
stable quarks respectively, with no observed excess over background. More
recently, D0 studied a 1.1 fb-1 data sample and looked for $Z\rightarrow
e^{-}e^{+}$ decays assumed to follow from a long-lived $b^{\prime}$ quark
parent with BR($b^{\prime}\rightarrow bZ)=100\%$ Abazov _et al._ (2008).
Assuming that the electromagnetic showers were originating from the same
vertex, the analysis found no hint away from the $p\bar{p}$ interaction point.
D0 excluded $m_{b^{\prime}}<190$ GeV/$c^{2}$ at the 95% confidence level for
decay lengths between 3.2 mm and 7 m. With the same integrated luminosity, CDF
excluded $m_{b^{\prime}}<268$ GeV/$c^{2}$ at 95% C.L., considering a long-
lived $b^{\prime}$ quark decaying exclusively into a $Z$ boson and a $b$ jet
Aaltonen _et al._ (2007). However, it has been emphasised in Hung and Sher
(2008) that the assumption BR($b^{\prime}\rightarrow bZ)=100\%$ is inaccurate
for $m_{b^{\prime}}>255\ $GeV/$c^{2}$, given that the $b^{\prime}$ decay mode
$b^{\prime}\rightarrow tW$ should take over if $m_{b^{\prime}}>m_{t}+m_{W}$.
Furthermore, the processes $b^{\prime}\rightarrow(u,c)W$ were hinted to
proceed with non-negligible rates, leading to an even lower branching ratio.
Indeed, even if the $b^{\prime}\rightarrow tW$ decay dominates, the
aforementioned mass limits depend sensitively on the CKM mixing elements
between the fourth and the first three generations. If non-unity couplings
between the fourth and the lighter quarks are allowed, the CDF lower bound on
$m_{b^{\prime}}$ can be significantly affected.
Additionally, it is known that the above CDF limits do not apply for long-
lived heavy quarks decaying between roughly 1 cm and 3 m within the detector
Hung and Sher (2008). Should their couplings to SM quarks lie in the
corresponding range $4.5\times 10^{-9}-7.8\times 10^{-8}$, there exists no
bounds on the $t^{\prime}$ and $b^{\prime}$ masses in this uncovered region.
For decay lengths larger than the detector dimensions, the lower bounds drop
to $m_{t^{\prime}}>220$ GeV/$c^{2}$ and $m_{b^{\prime}}>190$ GeV/$c^{2}$, as
obtained from the stable quark searches Acosta _et al._ (2003); Abazov _et
al._ (2008).
### III.2 LHC
#### III.2.1 Limitations of direct searches
While dedicated results for long-lived bound states of vector-like heavy
quarks are still missing at the LHC, various searches for heavy quarks of zero
lifetime have already been performed by the ATLAS and CMS experiments. In this
section, we discuss how the results of searches for prompt production could be
reinterpreted for lifetimes larger than $10^{-10}$ s.
The main aspects of reinterpreting these results are the branching ratios to
the investigated final states, which may be different in alternative models,
and of course the lifetime of the heavy particle.
Considering here a specific example, the current best limit on $b^{\prime}$
production, published by the CMS collaboration, excludes production cross-
sections of $\sigma>0.1$ pb at 95% C.L. with an integrated luminosity of 4.9
fb-1 at $\sqrt{s}=7$ TeV Chatrchyan _et al._ (2012c), assuming a branching
ratio ${\rm BR}(b^{\prime}\to tW)=100\%$. As we have seen in Section II.2, the
assumption of exclusive branching ratios is not generally valid for new heavy
quarks. Nevertheless, the fraction to which a hypothetical signature occurs
can be used to recalculate the cross-section limit in the most naïve
approximation
$\sigma_{\rm true}=\sigma_{\rm excluded}\times\frac{{\rm BR}_{\rm
assumed}}{{\rm BR}_{\rm true}},$ (10)
where $\sigma_{\rm excluded}$ is the excluded cross-section under the
assumption of ${\rm BR}_{\rm assumed}$, and ${\rm BR}_{\rm true}$ is the
branching ratio in the given model.
While this statement is a trivial estimation, the reinterpretation for longer
lifetimes requires more thought. In particular, the implicit assumption of
heavy quarks decaying promptly at the production vertex is valid only for very
short lifetimes. If these particles form bound states and propagate certain
distances before their decay, they can potentially escape the direct searches.
In such a scenario, the published limits could be considerably weaker,
depending on the particle’s lifetimes, simply because they would fly too far
to be detected at the primary vertex.
For intermediate lifetimes with displaced decay vertices within the detector
volume (cfr. the region (_ii_) discussed in Section II.3), a recalculation of
the published limits can be attempted. Assuming an exponential decay function,
a fraction of the decays will always happen in the vicinity of the beam line,
so that the prompt searches will pick them up. Based on this fraction, one can
recalculate the limits as a function of the heavy quark lifetime. However, in
order to do this correctly, one requires the exact selection efficiency as a
function of the displacement from the beam line. Unfortunately, such
information has not been made available by the experiments. Still, it can be
estimated from an educated guess under specific assumptions. For instance, the
analysis in Chatrchyan _et al._ (2012c) applies $b-$jet tagging, which
requires high-quality tracks to originate within the inner detector volume.
The CMS innermost part, the pixel detector consists of three barrel layers
with the innermost layer at a distance of $4.4$ cm from the beam line. This
represents the technical limitation to this analysis, so that we can assume
that the selection efficiency vanishes at a transverse displacement of around
$4$ cm.
Even in analyses without application of $b-$jet tagging, stringent quality
cuts are usually required to select reconstructed detector objects originating
from the primary vertex. One of the motivations for these requirements is the
rejection of pileup. Jets from displaced decays which do not point to the
production vertex are mostly removed by the pileup cleaning procedures. We
therefore make the assumption that events are kept in the direct searches if
the decay happens at less than $4$ cm from the primary interaction vertex, and
lost otherwise. The fraction of the lost events can be determined from the
distance these particles travel before their decay. This distance $d$ depends
on the mass, the momentum and the proper lifetime $\tau$. It is defined in the
laboratory frame as $d=\gamma\cdot\beta\cdot c\cdot\tau$. The fraction is
obtained by convoluting $d$ with an exponential decay function and the
momentum spectrum which determines $\gamma$ and $\beta$.
We take the momentum spectrum of pair-produced heavy quarks as predicted by
the MadGraph Alwall _et al._ (2011) Monte Carlo generator. The assumed center
of mass energy is $\sqrt{s}=8$ TeV. The momentum is needed for a precise
calculation of the decay distance on an event by event basis because the
velocity may be significantly smaller than $c$. The $p_{T}$ distribution for
various quark masses is shown in Figure 5.
Figure 5: Transverse momentum distribution of pair-produced heavy quarks as
predicted by MadGraph.
The distribution of the velocity in units of $c$ is shown in Figure 6.
Figure 6: Velocity distribution of pair-produced heavy quarks in units of
$\beta=v/c$ as predicted by MadGraph.
Our simulation assumes that the production mechanism and kinematics are
independent of the lifetime of the particle. Possible limitations of that
assumption are discussed in Section III.2.4. Calculating the distribution of
decay vertices in the detector frame, we obtain the fraction of decays outside
of the geometrical acceptance of $4$ cm as a function of the proper lifetime
$\tau$. Table 7 summarises our results for four benchmark points in the
parameter space. The results are displayed as a function of the proper
lifetime in Figure 7.
Lifetime | $10^{-10}$ s | $10^{-9}$ s | $10^{-8}$ s
---|---|---|---
$m=300$ GeV/$c^{2}$ | 32% | 86.5% | 98.5%
$m=500$ GeV/$c^{2}$ | 26% | 84.1% | 98.3%
$m=700$ GeV/$c^{2}$ | 21% | 82.3% | 97.9%
$m=1000$ GeV/$c^{2}$ | 16% | 79.4% | 97.5%
Table 7: Fraction of rejected decays of heavy quarks outside of the
geometrical acceptance of $4$ cm around the primary interaction vertex for
three different lifetimes and four different heavy quark masses. Figure 7:
Fraction of rejected decays of heavy quarks outside of the geometrical
acceptance of $4$ cm around the primary interaction vertex as a function of
the proper lifetime $\tau$.
We conclude that direct searches are only valid for lifetimes which are
considerably shorter than about $10^{-10}$ s, otherwise the particles would
propagate too far and be rejected by the selection criteria. We also stress
that this result is relatively independent of the particle masses within our
approximations.
The published limits on the heavy quark production cross-sections can be
simply reinterpreted in the region $10^{-8}$ s $<\tau<10^{-10}$ s by
$\sigma_{\rm true}=\sigma_{\rm excluded}\times(1-f)$ (11)
where $f$ is the fraction of lost events given in Table 7 and Figure 7. The
recalculation of the mass limits can then be done by comparing the
recalculated cross-sections with predicted cross-sections as a function of the
mass.
#### III.2.2 Displaced vertices
The immediate question that arises from the previous results is whether
lifetimes larger than $10^{-10}$ s are covered by dedicated searches for long-
lived particles, either through displaced vertices or signatures of stable
particles propagating through the full detector.
Displaced topologies have been searched for at LHC and published, for instance
in Chatrchyan _et al._ (2012d) and Aad _et al._ (2012b). These searches are
indirectly applicable to heavy quarks and will be discussed in this section.
The CMS analysis Chatrchyan _et al._ (2012d) is looking for massive long-
lived spinless neutral $\chi$ bosons, produced in decays of Higgs bosons
($H\to\chi\chi$). The $\chi$ bosons are decaying to di-leptons $\chi\to
l^{+}l^{-}$. This analysis is feasible thanks to the CMS track reconstruction
algorithm which is able to identify very displaced tracks not originating from
the primary interaction vertex. By fitting tracks of two oppositely charged
leptons to a common vertex, transverse displacements up to $50$ cm from the
beam line can be reconstructed. This analysis is very sensitive and it is able
to put limits on production cross-sections of the order of
$\sigma\times\mathrm{BR<10^{-3}}$ pb at 95% C.L., depending on the assumed
masses of the $\chi$ bosons and their mother particles. The decays of heavy
quarks such as $Q\rightarrow tW$, for instance, can lead to displaced di-
lepton vertices as well, because both $W$ bosons can decay leptonically.
In this analysis, the momentum of the vertex is required to be parallel to the
vector pointing from the primary vertex to the $\chi$ boson decay vertex in
the transverse plane. This “collinearity” cut is 0.2 (0.8) radians for muon
(electron) final states. A direct interpretation of the published limits in
context of long-lived heavy quarks is therefore not straightforward. Due to
the presence of the neutrinos and of the $b$-quark jet in $Q\rightarrow tW$
decays, the momentum of the di-lepton vertex will not be parallel to the
vector pointing from the primary to the secondary vertex. However, we can
attempt to estimate the efficiency of this collinearity cut based on simulated
events. We use again our MadGraph simulation from Section III.2.1 and
calculate the angle between the momentum of a heavy quark $Q$ and the di-muon
momentum in leptonic $Q\rightarrow tW$ decays. It is found that the
distribution of this collinearity angle is fairly independent of the mass of
the heavy quark. The collinearity cut has an efficiency of 68% for electrons
and 25% for muons in leptonic $Q\rightarrow tW$ decays which represents a
moderate decrease. A reinterpretation of the published results, taking these
efficiencies into account, will be considered in the following.
A central component that would be necessary to derive trustworthy limits on
heavy quark production from displaced vertex searches is the Monte Carlo
simulation of the signal process. The interactions of these heavy particles
with the detector material and the efficiency of their reconstruction and
selection need to be estimated in order to facilitate the calculation of
limits. Difficulties may arise from missing implementations of certain exotic
models in the simulation software. Existing simulations of heavy stable
particles such as those of R-hadrons may provide good approximations though
Mackeprang and Milstead (2010).
We see in Chatrchyan _et al._ (2012d) that different assumptions for the
masses $m_{H}$ and $m_{\chi}$ lead to different exclusion limits due to the
kinematical properties of the final state. A direct matching of the mass of a
heavy quark to $m_{H}$ and $m_{\chi}$ is not straightforward. The assumption
that the results from Chatrchyan _et al._ (2012d) are applicable for heavy
quarks can therefore only be understood as an approximation. In the following
we assume that the limits for higher $m_{H}$ and $m_{\chi}$ values are more
applicable to heavy quarks, so we only consider those mass points with
$m_{H}\geq 400$ GeV/$c^{2}$ and $m_{\chi}\geq 50$ GeV/$c^{2}$. Fortunately,
the resulting exclusion limits vary only in a small window between $5\cdot
10^{-4}$ pb and $2\cdot 10^{-3}$ pb in the dimuon channel for $1$ cm $<\rm
c\tau<10$ cm when scanning the various mass points for $m_{H}$ and $m_{\chi}$.
Following the strategy of a worst case scenario we use the worst limit of
$2\cdot 10^{-3}$ pb which weakens to $8\cdot 10^{-3}$ pb taking the effect of
the collinearity cut into account. Now the $tW$ final state has only a 1%
branching ratio into dimuons which results in an excluded cross-section of
$\sigma\cdot BR(Q\to tW)>0.8$ pb for $1$ cm $<\rm c\tau<10$ cm. In the context
of QCD pair production of heavy quarks, the value of 0.8 pb corresponds to a
mass of $m_{Q}=435$ GeV/$c^{2}$ at $\sqrt{s}=7$ TeV Cacciari _et al._ (2012).
This can therefore be assumed to be the exclusion limit under the assumption
of a branching fraction $BR(Q\to tW)=100\%$. Comparing this to the searches at
CDF and D0 quoted in Section III.1 we see a clear improvement of the limits,
at least in the range $1$ cm $<\rm c\tau<10$ cm. While the reinterpretation is
difficult due to the discussed reasons, much stronger bounds would certainly
be possible by including the heavy quark signatures into the displaced vertex
analysis.
A displaced vertex analysis has also been performed by the ATLAS collaboration
Aad _et al._ (2013), following a slightly different strategy. Displaced
vertices are reconstructed in an inclusive way, using all displaced tracks in
the event as potential seeds. In contrast to CMS, where only opposite-sign di-
muon vertices have been used, the ATLAS approach requires at least five tracks
at the vertex and an invariant mass of the vertex of at least 10 GeV/$c^{2}$.
To ensure a good fit quality, only vertices within the fiducial barrel pixel
detector volume are considered, which means that transverse displacements up
to $18$ cm are considered. In addition, one muon with $p_{T}>50$ GeV/$c$ is
required to be associated with the signal vertex. This analysis has been
carried out in the context of an $R$-parity violating supersymmetric scenario,
deriving limits on $\sigma\cdot\mathrm{BR}^{2}$, where $\mathrm{BR}^{2}$ is
the square of the branching ratio for produced squark pairs to decay via long-
lived neutralinos to muons and quarks. The requirement that the triggering
muon candidate is associated with the displaced vertex ensures that the
selection efficiency for each neutralino is independent of the rest of the
event. This facilitates a reinterpretation for scenarios with different
numbers of long-lived neutralinos in the event.
Also in this analysis the limits depend on masses and kinematics of the
hypothetical long-lived particles and their mother particles. In the most
pessimistic scenario the excluded cross-sections are approximately
$\sigma\cdot\mathrm{BR}^{2}>10^{-2}$ pb for lifetimes of 3 mm $<{\rm
c}\tau<10^{2}$ mm using an integrated luminosity of 4.4 fb-1 at $\sqrt{s}=7$
TeV. For even larger lifetimes of $10^{2}$ mm $<{\rm c}\tau<10^{3}$ mm the
limits are better than $\sigma\cdot\mathrm{BR}^{2}>10^{-1}$ pb.
The interpretation of these results for decays such as the $Q\rightarrow tW$
or even $Q\rightarrow bW$ is possible because these signatures would certainly
give rise to decay vertices with very high invariant mass and high track
multiplicity. The $tW$ final states has at least one muon in 19% of the cases
which means that the excluded values are $\sigma\cdot BR(Q\to tW)>0.05$ pb for
3 mm $<{\rm c}\tau<10^{2}$ mm. This is about an order of magnitude better than
our reinterpretation of the CMS result. The value of 0.05 pb corresponds to
$m_{Q}=650$ GeV/$c^{2}$ Berger and Cao (2010).
#### III.2.3 Heavy Stable Charged Particles
For very long-lived scenarios (cfr. the region (_iii_) discussed in Section
II.3), the new heavy states do not decay, but travel through the full
detector. In the following we review recent results by CMS Chatrchyan _et
al._ (2012e) along with an assessment of their relevance for HSCPs.
The main identification criteria for HSCPs are high momentum, high ionization
and long time of flight (TOF). The energy loss $dE/dx$ along the track is
calculated from the charge deposits in the silicon tracker, while the TOF is
obtained from the arrival time in the muon system. These quantities are
uncorrelated for SM particles. This noncorrelation is used to estimate the
background yields in signal-depleted regions.
Models of charge suppression due to interaction of HSCPs with the detector
material are considered as well. If charged particles become neutral while
propagating through the detector they do not reach the muon chambers. The TOF
criterion cannot be used in this case. Therefore, the results have also been
estimated without the TOF requirement at the cost of weaker exclusion limits.
The obtained limits for a given mass can be very different depending on the
assumed particle type. This is due to the predicted kinematics of the particle
and its expected detector signal. For instance, the limit on pair production
of a scalar top quark with a mass between $300$ and $800$ GeV/$c^{2}$ is
$\sigma<$ 3 fb for an integrated luminosity of $5$ fb-1 at $\sqrt{s}=7$ TeV.
Limits for long-lived heavy quarks have not been considered explicitly,
unfortunately. However, as an approximation, we make the assumption that a
long-lived bound state of a stop would behave similar or equal to a bound
state of a heavy up-type quark Mackeprang and Milstead (2010). This concerns
both the interaction with the detector material and the production mechanism
which is assumed to proceed via the strong interaction. It is then possible to
use the stop exclusion limits directly for stable heavy quarks. Limitations in
that assumption are discussed in Section III.2.4.
The predicted production cross-sections of heavy quarks are quite large for
low masses (Section II.1). Even with a large systematic safety margin, these
cross-sections can be considered to be excluded by the HSCP searches. The
heavy quark pair production cross section for $m_{Q}=800$ GeV/$c^{2}$ (the
highest mass value considered in Chatrchyan _et al._ (2012e)), computed at
NLO, is $\sigma=9.7$ fb at $\sqrt{s}=7$ TeV Cacciari _et al._ (2012), which
can therefore be assumed to be excluded for sufficiently long lifetimes. Heavy
quark masses of the order of $m_{Q}\approx 1$ TeV/$c^{2}$ are not yet excluded
though, neither in direct searches for prompt decays nor in long-lived
searches.
To make a quantitative statement about the excluded lifetimes by the HSCP
searches, we repeat our simulation from Section III.2.1. To be detected by
these analyses, the particles have to be produced at the primary vertex and
they have to traverse the full tracking devices, and the muon chambers for the
combined tracker+TOF analysis (or the tracker for the tracker-only analysis).
The CMS tracker has a length of about $6$ m and a diameter of $2.8$ m
Chatrchyan _et al._ (2008). We can calculate the fraction of decays inside
the tracker volume and assume that these decays will not be reconstructed in
the HSCP analyses because of missing hits in the outermost tracker layers.
Tables 8 and Figure 8 summarise our estimations for a few benchmark points and
as a function of the proper lifetime. The transition between the two extrema
of being fully efficient and losing all decays happens within two orders of
magnitudes of the lifetime between $10^{-9}$ s and $10^{-7}$ s.
Lifetime | $2\cdot 10^{-9}$ s | $10^{-8}$ s | $10^{-7}$ s
---|---|---|---
$m=300$ GeV/$c^{2}$ | 90% | 47% | 7%
$m=500$ GeV/$c^{2}$ | 94% | 53% | 8%
$m=700$ GeV/$c^{2}$ | 96% | 58% | 10%
$m=1000$ GeV/$c^{2}$ | 98% | 64% | 11%
Table 8: Fraction of rejected decays of heavy quarks within the geometrical
acceptance of the CMS tracking detector. Decays within the tracker are
rejected because the particle’s trajectory does not reach the outermost
tracker layers. Figure 8: Fraction of rejected decays of heavy quarks within
the volume of the CMS tracking detector as a function of the proper lifetime
$\tau$.
From these results we conclude that heavy quark masses reaching $m_{Q}=800$
GeV/$c^{2}$ can be excluded at 95% C.L. for lifetimes longer than $10^{-7}$ s.
As the selection efficiency of the HSCP searches is about 50% for a lifetime
of $\tau=10^{-8}$ s, the published limit by the HSCP searches weakens by a
factor of two, which is still sufficient to exclude the 800 GeV/$c^{2}$. The
efficiency drops quickly for even shorter lifetimes.
To conclude this section, we would like to mention one additional very
interesting experimental strategy. As discussed in detail in Graham _et al._
(2012) heavy metastable particles may be stopped in the detectors and give
out-of-time decays. This kind of signature has been investigated by the CMS
collaboration Chatrchyan _et al._ (2012b) using gaps of no collisions between
the proton bunches. Particles of extremely long lifetimes (between 10 $\mu$s
and 1000 s) may produce jets which can be triggered by the calorimeters during
periods of no proton collisions. Using a data set in which CMS recorded an
integrated luminosity of 4 fb-1 and a search interval corresponding to 246
hours of trigger live time, 12 events have been observed with a background
prediction of $8.6\pm 2.4$ events. This result has been interpreted in the
context of long-lived gluinos (R-hadrons) and scalar top quarks. The best
limit is obtained for lifetimes larger than $10^{-6}$ s where the upper limit
on the production of stop quarks has been found to be
$\sigma(pp\to\tilde{t}\tilde{t})\cdot BR(\tilde{t}\to t\tilde{\chi}^{0})>0.7$
pb. When interpreting this for pair production of heavy quarks, the cross-
section of 0.7 pb corresponds to $m_{Q}=445$ GeV/$c^{2}$ Cacciari _et al._
(2012). These results are clearly less stringent than those of the stable HSCP
searches discussed above, however they are able to probe signatures which can
potentially escape the HSCP searches in case the particle propagates with an
extremely low velocity or does not reach the outermost tracker layers before
it stops.
#### III.2.4 Discussion of the results
In this section we summarise our findings from Sections III.2.1 to III.2.3,
followed by a discussion of possible shortcomings and alternative search
strategies.
The following list gives an overview of the different types of searches for
heavy or long-lived particles along with the results of our reinterpretations:
* •
Prompt decays of heavy quarks: by far, the largest number of published
searches for heavy quarks assume their prompt production and decay at the
proton-proton interaction vertex. We estimated that this assumption is valid
for lifetimes up to $\tau<10^{-10}$ s (c$\tau<3$ cm). The selection efficiency
decreases for longer lifetimes and drops to 10% - 20% for $\tau>10^{-9}$ s
(c$\tau>30$ cm).
* •
Displaced vertices: our reinterpretation of the ATLAS results Aad _et al._
(2013) for $tW$ final states rules out cross-sections of $\sigma\cdot BR(Q\to
tW)>0.05$ pb for lifetimes of $\rm 10^{-11}$ s $<\tau<3\cdot 10^{-10}$ s (3 mm
$<{\rm c}\tau<10$ cm), corresponding to an excluded heavy quark mass of
$m_{Q}<650$ GeV/$c^{2}$.
* •
Heavy Stable Charged Particles: these searches require the heavy state to
propagate through the full tracking detectors to be detected. Our
reinterpretation of the CMS search Chatrchyan _et al._ (2012e), assuming
similar behaviour as for scalar long-lived top quarks, excludes cross-sections
of $\sigma>3$ fb for very long lifetimes $\tau>10^{-7}$ s (c$\tau>30$ m). The
mass limit $m_{Q}<800$ GeV/$c^{2}$ is still valid for lifetimes $\tau>10^{-8}$
s (c$\tau>3$ m).
This is obviously still a patchwork of different sensitivities for different
lifetimes and different models and methods. We also see that a small
geometrical window of inefficiency remains. This is the region beyond
displacements of more than about 50 cm but within the volume of the tracking
devices. In this region, both the searches for displaced vertices and the
searches for HSCP do not reconstruct any events. The exact size of this
geometrical region depends on the experiment. We estimate the fraction of
decays of heavy quarks within this geometrical region based on our simulation
from the previous sections. The result is shown in Figure 9. We see that in
the worst case scenario only 50% of the decays are lost for a lifetime of
$\tau=3\cdot 10^{-9}$ s. This represents a moderate decrease of the selection
efficiency for a narrow window, so that we can expect a full coverage of the
lifetime spectrum in the future.
Figure 9: Fraction of decays of heavy quarks between the acceptance of the
displaced vertex analysis and HSCP analyses.
Yet, some of the assumptions made in this section may have certain
shortcomings. First of all, we considered a general framework in which the new
heavy quarks are pair-produced, with kinematics independent of the lifetime of
the particle. Our motivation was that the current searches for long-lived
R-hadrons formed from stable stops generally assume that strong pair
production dominates. In supersymmetric models, however, the situation changes
drastically when the squark masses become closer to the TeV scale. As pointed
out in Johansen _et al._ (2010), the squark pair production from light quarks
can dominate the gluon fusion mechanism for large masses due to the decreasing
probability for the gluons to carry large fractions of the proton momentum.
For scenarios where either the squark mass or the mass of the gluino
propagator reaches $1$ TeV, Higgs and squark weak production start to
dominate. The production channels then become more and more model-dependent
for increasing masses, arguing for dedicated MC simulations above such mass
values.
Secondly, we assumed that the heavy quark pair production cross-sections were
of the same magnitude as the rate of particles hadronising into bound states,
while we stressed in Section II.4 that this is not the case in general. In
particular, the $\eta_{Q}$ quarkonium production rate is expected to be
smaller than the $Q\bar{Q}$ pair production cross-section, due to the possibly
large suppression arising from the bound state wave function, which also leads
to a different decay phenomenology. Precise calculations would be needed to
refine our analysis and the above discussion might need to be adjusted in case
of strongly bound states.
As a possible suggestion to extend the reach of the above analyses, another
search topology may be the case in which a heavy quark decays within the
tracker volume. The signature to be searched for may then be one or more
displaced jets, which trigger the event, and originate from the heavy quark
decay. In addition, a short track with a few hits and large d$E$/d$x$ pointing
to the decay vertex could be required. This signature may have relatively low
backgrounds as the jets should have a large transverse momentum. The tracks
within these jets are not required to arise from the primary interaction
vertex which makes track reconstruction difficult. To circumvent this, one
could perform a regional search for short tracklets pointing towards the
energy deposit in the calorimeter.
## IV Summary and conclusions
The current searches for heavy quarks at the LHC mostly ignore the option of
long lifetimes. Although a new chiral family of fermions is now strongly
disfavoured from the recent Higgs search results, such a scenario remains of
prime importance when considering vector-like quarks beyond the Standard
Model. As non-chiral fermions decouple in the limit of vanishing mixing with
the lighter generations, they can be long-lived and form bound states.
Dedicated searches for novel heavy $Q\bar{Q}$ quarkonia, $Qqq$ baryons and
$Q\bar{q}$ ($\bar{Q}q$) open-flavour mesons might provide promising strategies
for future investigations.
As we have detailed, our reinterpretation of the HSCP searches indicates that
quark masses of $800$ GeV/$c^{2}$ can be excluded at $95\%$ CL for lifetimes
longer than $10^{-8}$ s. This result however assumes that the interactions of
the heavy quarks with the detector material are similar to that of stop
particles. As discussed in this review, there are potential restrictions to
such an assumption, and the forthcoming searches should be interpreted in a
model-specific context. Additionally, it is usually considered that the bound
states production cross-sections are comparable to that of pair-produced heavy
quarks. This is not necessarily the case for the reasons we presented.
Finally, searches for prompt decays are still valid for long lifetimes to a
certain extent, but the assumptions about branching ratios and the potential
loss of efficiency need to be considered. We showed that these analyses start
to become insensitive for lifetimes longer than $10^{-10}$ s, corresponding to
$Q-q$ quark couplings below the $10^{-9}$ level. For lifetimes beyond
$10^{-10}$ s, searches for displaced vertices in the tracker volume have been
shown to play an important role.
Acknowledgements The work of M.B. is supported by the National Fund for
Scientific Research (F.R.S.-FNRS, Belgium) under a FRIA grant. A.S. is
supported by the German Research Foundation (DFG) under an Emmy Noether grant.
The authors would like to thank Daniel Stolarski and Yevgeny Kats for their
comments on the early version of this manuscript. We acknowledge the
organisers of the ”Focus Workshop on Heavy Quarks at LHC” in Taipei (NTU) for
the discussions that initiated this work.
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|
arxiv-papers
| 2012-10-23T20:25:48 |
2024-09-04T02:49:37.001692
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mathieu Buchkremer, Alexander Schmidt",
"submitter": "Mathieu Buchkremer Mr.",
"url": "https://arxiv.org/abs/1210.6369"
}
|
1210.6467
|
# Desynchronizing Networks Using Phase Resetting
Jon Borresen J.Borresen@mmu.ac.uk David Broomhead School of Computing,
Mathematics and Digital Technology, Manchester Metropolitan University,
Manchester UK M1 5GD School of Mathematics, The University of Manchester,
Manchester UK M60 1QD
###### Abstract
Understanding complex systems which exhibit desynchronization as an emergent
property should have important implications, particularly in treating
neurological disorders and designing efficient communication networks. Here
were demonstrate how, using a system similar to the pulse coupling used to
model firefly interactions, phase desynchronization can be achieved in pulse
coupled oscillator systems, for a variety of network architectures, with
symmetric and non symmetric internal oscillator frequencies and with both
instantaneous and time delayed coupling.
###### keywords:
Desynchronization , Phase resetting , Pulse Coupling, , Oscillator Dynamics
###### MSC:
34C15 ,
###### MSC:
36D06
††journal: Physics Letters A
## 1 Introduction
Synchronization, both full and partial, has been observed and studied in a
wide variety of systems and is known to be an essential feature of the
collective dynamics of interacting systems [1, 2, 3, 4, 5]. Desynchronization
is likewise of considerable interest. This is the case particularly in
neuroscience where it has implications for the treatment of Parkinson’s
disease [6, 7] and other neurological disorders such as epilepsy [8, 9, 10].
Consequently, research has focussed primarily on breaking synchronization in
neural systems via a variety of methods [11, 12, 13, 14, 15, 16] or
investigating phase resetting, cluster splitting and the stability of cluster
dynamics in general [17, 18, 19, 20, 21, 22]. Desynchronization is also a
useful concept in designing asynchronous communication systems [23, 24, 25,
26].
Arguably, desynchronization is a ubiquitous naturally occurring emergent
property. Here we demonstrate a general mechanism, similar to that found in
the pulse coupling model of synchronization in firefly interactions, which
generates desynchronization in a variety of coupled oscillator network
architectures, having either uniform or non-uniform distributions of
frequencies and either instantaneous or time-delayed coupling. We use an
adaptation of the Mirollo-Strogatz model [27], which is itself an adaptation
of the Peskin [28] model for synchronization in heart pacemaker cells. (See
also [29, 30, 31, 32, 33] for investigations of coupled systems with time
delays.)
## 2 The Basic Model
Consider a network of $N$ coupled oscillators as an undirected graph, $(V,E)$,
where the vertices, $V$, represent individual oscillators and the edges, $E$,
the network connections between them. The state of each vertex $v_{i}\in V$,
where $i\in(1,\ldots,N)$ is described by a phase $x_{i}\in\mathcal{S}^{1}$. In
this basic model, it will be assumed that the internal oscillator frequencies
are identical and that the coupling is instantaneous. This model will later be
extended to cover variable frequencies and networks with time delays. In a
manner similar to [27] coupling is through phase resetting. Whenever the phase
of an individual oscillator passes through $x_{i}=1$ it is reset to $x_{i}=0$
($0\sim 1$ in $\mathcal{S}^{1}$) and each oscillator connected directly to
this oscillator has its phase reset according to a smooth function
$f:\mathcal{S}^{1}\rightarrow\mathcal{S}^{1}$ where $Df(x)>0$ for all
$x\in\mathcal{S}^{1}$. For desynchronization to occur, $f$ will be a nonlinear
function having the asynchronous state as a stable (under iteration) fixed
point with all other fixed points unstable. For instance
$f(x)=\frac{1}{4}\left(\ln(1+(e^{2}-1)x)-\ln(1+(e^{-2}-1)x)\right),$ (1)
(inspired by the rise function of the Mirollo-Strogatz model) has an unstable
fixed point at $0$ (the synchronous state) and a stable fixed point at $0.5$
(see figure 1).
Figure 1: Interaction function $f$ (Equation 1) has a stable fixed point at
$x=0.5$ and an unstable fixed point at $x=0$.
For a network consisting of oscillators with identical frequencies and
instantaneous coupling, steady-state behaviour will be determined by the
network topology and the form of $f$. In the simplest network, consisting of
two coupled oscillators with phases ($x_{1},x_{2}$), the dynamics is a flow on
a $2$-torus (seen as the unit square with the opposing edges identified).
Since the oscillators advance at the same rate, the flow is parallel to the
diagonal $x_{1}=x_{2}$. The identification of the opposite edges means that
trajectories leaving the right and top edges of the square reappear at the
left and bottom edges respectively (equivalent to resetting $x_{i}=1$ to
$x_{i}=0$). Function evaluations $f(x)$ are performed at this time. The form
of $f$ yields a repelling closed orbit $x_{1}=x_{2}$ (as the gradient
$f^{\prime}(1)>1$ ) and an attracting closed orbit $x_{1}=x_{2}-0.5$, since
$0<f^{\prime}(0.5)<1$.
Considering the dynamics in the rotating $x_{1}$ frame, the $2-$dimensional
system may be reduced to a $1-$dimensional map describing the phase
difference, $d=x_{1}-x_{2}$. We observe that the phase rotation is an isometry
which swaps the upper and right edges of the unit square, where the function
evaluations occur. The phase difference after each rotation of $x_{1}$ is
given by a single discontinuous function, $F$, which is the composition of the
phase rotation and $f$ (see Figure 2).
$F(d)=\left\\{\begin{array}[]{ll}f(d+1)&-1<d<0\\\ 0&d=0\\\ -f(1-d)&0<d\leq
1\end{array}\right.$ (2)
Figure 2: Discrete map function for two oscillator model describing phase
difference $d=x_{1}-x_{2}$ in the rotating $x_{1}$ frame. This represents the
second return map for the flow and function evaluations for each phase
rotation of $x_{1}$.
$F(d)$ has a globally attracting period-$2$ orbit $d=\pm 0.5$ (the
desynchronized state $x_{1}=x_{2}-0.5$). The other (isolated) fixed point
($d=0$) is unstable. For the two oscillator model, therefore, the asymptotic
dynamics are a globally stable limit cycle at $x_{1}=x_{2}-0.5$; apart from
initial synchronization, all initial conditions result in phase
desynchronization whereby the difference between the phases of the oscillators
is maximal.
For a system of $N$ globally coupled oscillators—with phases
$\mathrm{x}=(x_{1},\ldots,x_{N})$—the dynamics can be represented as flow in
an $N$-torus, seen as an $N$-cube with opposing faces identified. The flow is
parallel to the $x_{1}=\ldots=x_{N}$ diagonal and function evaluations occur
on the exiting faces. As with the two oscillator model, we determine the
steady state behavior by considering a discrete map equivalent of the dynamics
constructed from the phase differences in the rotating $x_{1}$ frame i.e.
$d_{i}=x_{1}-x_{i+1}$. The transformation from $\mathrm{x}$ to $\mathrm{d}$ is
accomplished by a $(N-1)\times N$-matrix $M=(\mathrm{e}:-\mathbf{1}_{N-1})$
where $\mathrm{e}$ is a column vector with all components equal to unity and
$\mathbf{1}_{N-1}$ is the $(N-1)\times(N-1)$ unit matrix. Between transitions
each component of $\mathrm{d}$ is conserved because the flow is parallel to
the diagonal.
If the components of $\mathrm{x}$ are ordered so that $x_{1}>\ldots>x_{N}$,
the first transition occurs when $x_{1}=1$ and the effect of the functional
mapping is $\mathrm{d}\mapsto G(\mathrm{d})$ where
$G_{k}(\mathrm{d})=-f(1-d_{k})$. With all-to-all coupling (since
$x_{i}>x_{i+1}$ implies $f(x_{i})>f(x_{i+1})$), following the transition the
cyclic permutation
$P_{\ast}(x_{1},x_{2},\ldots,x_{N})=(x_{2},x_{3},\ldots,x_{1})$ restores the
assumed ordering of the components of $\mathrm{x}$. Given a general
permutation $P$, we would like to find a corresponding transformation $T$ on
$\mathrm{d}$, where $TM=MP$. Since $MP$ is a matrix consisting of the permuted
columns of $M$, we write $MP=(\mathrm{v}:M^{\prime})$ where $\mathrm{v}$ is
the column coming first in the permutation, and $M^{\prime}$ is the rest of
the matrix. Now $TM=T(\mathrm{e}:-\mathbf{1}_{N-1})=(T\mathrm{e}:-T)$ which
implies that $T=-M^{\prime}$. For this to be a solution, it must be true that
$\mathrm{v}=-M^{\prime}\mathrm{e}$. This follows from the fact that the row
sums of $M$ are all zero. The transformations $\\{T\\}$ inherit the group
structure of the set of permutation matrices $\\{P\\}$. In particular, if
$\\{P\\}=<P_{\ast}>$, the cyclic group of order $N$ generated by $P_{\ast}$,
then the group of transformations, $\\{T\\}$, is isomorphic, consisting of the
powers of $T_{\ast}$ where $T_{\ast}M=MP_{\ast}$. Given any initial condition
which is consistent with the initial ordering of the phases we can evolve
$\mathrm{d}$ in time
$\mathrm{d}(t+k)=T^{1-k}_{\ast}\circ G\circ(T_{\ast}\circ
G)^{k-1}(\mathrm{d}(t)).$
When $k=N$, the order of the cyclic group, this reduces to a simple iteration
$\mathrm{d}(t+N)=(T_{\ast}\circ G)^{N}(\mathrm{d}(t)).$
Figure 3: The projection of the 3-cube along its principle diagonal
$x_{1}=x_{2}=x_{3}$. The plane is parameterised by
$\mathrm{d}=(x_{1}-x_{2},x_{1}-x_{3})$ . Points on different faces of the cube
with the same vector $\mathrm{d}$ are identified by the projection. For
example, the shaded region A, when interpreted as being in the bottom face of
the cube is the set $\\{x_{1}>x_{2}>x_{3}=0\\}$; interpreted as the top face
it is the set $\\{1=x_{1}>x_{2}>x_{3}\\}$.
The three oscillator all-to-all network is illustrated in Figure 3 which shows
a projection of the 3-cube along its principle diagonal $x_{1}=x_{2}=x_{3}$.
In this projection the flow reduces to a field of fixed points. Points on
different faces of the cube which correspond to the same vector $\mathrm{d}$
are identified by the projection. The shaded region A, when interpreted as
being in the bottom face of the cube is the set $x_{1}>x_{2}>x_{3}=0$. The
flow identifies this with the top face $1=x_{1}>x_{2}>x_{3}$ interpretation of
A. Resetting $x_{1}$ to zero and applying $f$ to the other two phases maps
this region onto B (interpreted as the back face). The same argument now shows
that B maps onto C and thus back to A. Region A is therefore invariant under
the action of $(T_{\ast}\circ G)^{3}$. The closure of A contains synchronized
or partially synchronized states in its boundary. However, the choice of $f$
ensures that these are all unstable. Numerical work suggests that given this
form of $f$, the interior of A contains a unique attracting fixed point. That
is, we find a unique attracting orbit which visits the regions A, B and C
cyclically. The unshaded regions contain a second attracting orbit which is
generated by assuming a different initial ordering of the phases,
$x_{2}>x_{1}>x_{3}$ for example. In the general $N$ clock setting, assuming
the unique attractor suggested by the numerics, the group theoretical
underpinning of this system allows us to count the number of distinct orbits
corresponding to desynchronized states. The order of the symmetric group
consisting of all permutations of $N$ objects is $N!$ and, by Lagrange’s
theorem, the number of permutations which are not mapped to each other when
acted upon by $<P_{\ast}>$ is $(N-1)!$. For each of these initial permutations
there is an attracting desynchronized orbit and a repelling synchronized orbit
(excluding symmetry).
## 3 Systems with Non-identical Frequencies
The heart of the previous analysis is that the behaviour of the $N$-oscillator
system is based on a cyclic permutation of the initial ordering of the phases.
We might suppose that this property is robust to a sufficiently small
variation in the frequencies of the oscillators. Consider a system of two
oscillators with frequencies $\omega_{1}$ and $\omega_{2}$. We can demonstrate
graphically that if $\rho=\frac{\omega_{1}}{\omega_{2}}$ is less than some
critical value $\rho_{c}$ then limit cycles as described above will occur.
Figure 4 demonstrates that if a stable limit cycle exists outside a region
determined by the ratio of the oscillators’ frequencies then the dynamics will
pass through alternate faces of the torus in succession. As the change in flow
does not affect the existence of the globally stable limit cycle (as the
interaction function $f$ has not changed), it is only required that the limit
cycle lie within this region.
Figure 4: When the flow is not parallel to the diagonal $x_{1}=x_{2}$, the
oscillators will alternately pass through $x_{i}=1$, if the limit cycle is
outside the hashed region. In this case, desynchronization will occur. The
size of the hashed region is determined by the interaction function $f(x)$ and
the angle of flow.
It is straightforward to derive the orbit of the limit cycle to be
$x_{i}=f(\rho(1-f(\rho(1-x_{i}))))$ for all $x_{i}$. For $f$ given in equation
1 the limit cycle is within the ‘non-overtaking’ region (see figure 4) for
$\rho\leq 1.11$. This value increases as the gradient of equation 1 becomes
more pronounced.
Using a similar argument, it can be shown that for any network it is
sufficient that the ratio of any two connected oscillators’ frequencies be
less than $\rho_{c}$ for desynchronization to occur. Again, the value of
$\rho_{c}$ is determined by the interaction function $f$. For networks which
are not globally connected, we can, via an argument identical to that used
above, demonstrate that any two connected oscillators will desynchronize.
However some consideration must be given to the reducibility of such networks
via symmetry arguments (see [34]). For non-globally connected networks, phase
coupling of the form proposed will result in local asymptotic
desynchronization across the network i.e. each oscillator will desynchronize
with each of those to which it is connected.
## 4 Time Delays
The final modeling assumption that will be relaxed concerns time delays across
the network. When considering desynchronization using the previous coupling
function (equation 1) the time delay can reverse the stability of the fixed
point at $x=0$. Consider two oscillators separated by a time delay of $\tau$.
If the difference in their phases $x_{1}-x_{2}<\tau$, when the first
oscillator sends a pulse on crossing $x=1$, the second oscillator passes $x=1$
before receiving the pulse. In this case the second oscillator is perturbed
closer to the first and with each cycle the oscillators move closer together.
This difficulty can be overcome by redesigning the interaction function $f$.
From the argument described above synchronization will only occur (using a
continuous interaction function) if the oscillators’ frequencies differ by
less than the propagation delay between them. It is required, therefore, that
should this occur, the perturbed oscillator (the one receiving the data)
should not be perturbed closer to the transmitting oscillator’s time. We can
derive a new interaction function which has this property and still retains a
stable fixed point at $d=0.5$ using a shifted cubic curve. Interpolating
through the fixed point at $d=0.5$, the curve can be expressed as follows (see
also Figure 5) if $\tau$ is the maximum propagation delay, and $\beta\in(0,1)$
the gradient at the fixed point:
$f(x)=a(\tau,\beta)x^{3}+b(\tau,\beta)x^{2}+c(\tau,\beta)x-\tau\,\mathrm{mod}\,1.$
(3)
This function is a continuous mapping of the circle $f:S^{1}\rightarrow S^{1}$
but appears discontinuous in the interval $x\in[0,1)$.
Figure 5: Discontinuous interaction functions with a stable fixed point at
$x=0.5$ and no fixed point at $x=0,1$. Left figure $\tau=0.1$ and $\beta=0.5$
and right $\tau=0.05$ and $\beta=0.25$.
There are important factors within the transient dynamics, which may have some
impact on the functioning of such time delayed systems. For instance, the
existence of transient chaos cannot, at this stage, be excluded nor other
dynamical effects present in systems of interacting oscillators and
particularly the presence of unstable attractors within the dynamics needs to
be considered [35]. However, the above arguments can be applied to suggest
that the only asymptotically stable dynamics of such systems would be the
desynchronized state, where each oscillator pulses in turn.
Figure 6: Time series for simulations of $5$ globally coupled oscillators
(top) and the associated time series for the order parameter $P$ (bottom) with
near synchronous initial conditions: (a) identical oscillator frequencies
period $=1$, no time delays; (b) normally distributed frequencies (mean period
$=1$ standard deviation $=0.05$); (c) identical oscillator frequencies,
uniform time delay $\tau=0.01$; (d) normally distributed frequencies (mean
period $=1$ standard deviation $=0.05$) uniform time delay $\tau=0.01$.
## 5 Simulations
The model has been simulated for a variety of networks, both homogeneous and
inhomogeneous, using identical internal frequencies, distributed internal
frequencies and for networks with small, uniform time delays. In all cases
local asymptotic desynchronization was observed, in accordance with the above
(see Figure 6). For time delayed systems the duration of transient behaviour
grows considerably as the number of oscillators is increased and it is
possible to observe clustering if a suitable choice of the interaction
function is not made, however, the long term dynamics appear in all cases to
converge to the desynchronized limit cycle previously described.
Figure 6 also shows the time series for an order parameter $P$ which gives a
measure of the total coherence in the network [36, 37].
$P=\frac{1}{N}|\sum_{k=1}^{N}e^{2\pi ix_{k}}|,$ (4)
where $P=1$ corresponds to synchronous oscillation and for any other state
$0\leq P<1$. As can be observed, with initial conditions near the ordered
synchronous state, the oscillator dynamics rapidly desynchronize and the order
of each oscillator as it is distributed around the phase remains unchanged.
## 6 Summary
The analysis as presented here demonstrates how, via pulse coupling, a network
of connected oscillators may be forced to achieve phase desynchronization as a
collective dynamic. The model is intended to demonstrate ‘proof of principle’
of the design of an emergent property. We conjecture that the method of pulse
coupling applied here would be equally applicable to weakly coupled
oscillators exhibiting synchronization. The applications of such a concept may
be far reaching, particularly when applied to digital communication systems,
the design of neural based computers and in the treatment of Parkinson’s
disease and epilepsy. Acknowledgements The authors would like to thank J.
Shapiro, M. Sorea, S. Furber and L.O.Gowrie for their advice, discussions and
encouragement. This work was sponsored by EPSRC grant GR/T11258/01.
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|
arxiv-papers
| 2012-10-24T09:19:02 |
2024-09-04T02:49:37.017828
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Borresen and D. Broomhead",
"submitter": "Jon Borresen",
"url": "https://arxiv.org/abs/1210.6467"
}
|
1210.6659
|
# Quantum Monte Carlo Approaches to Nuclear and Atomic Physics
J. Carlson 1 Stefano Gandolfi1 and Alexandros Gezerlis2,3 1Theoretical
Division1Theoretical Division Los Alamos National Laboratory
Los Alamos Los Alamos National Laboratory
Los Alamos NM 87545 NM 87545 USA
2ExtreMe Matter Institute EMMI USA
2ExtreMe Matter Institute EMMI GSI Helmholtzzentrum für Schwerionenforschung
GmbH GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291 Darmstadt 64291
Darmstadt Germany
3Institut für Kernphysik Germany
3Institut für Kernphysik Technische Universität Darmstadt Technische
Universität Darmstadt 64289 Darmstadt 64289 Darmstadt Germany Germany
(today)
###### Abstract
Quantum Monte Carlo methods have proven to be valuable in the study of
strongly correlated quantum systems, particularly nuclear physics and cold
atomic gases. Historically, such ab initio simulations have been used to study
properties of light nuclei, including spectra and form factors, low-energy
scattering, and high-momentum properties including inclusive scattering and
one- and two-body momentum distributions. More recently they have been used to
study the properties of homogeneous and inhomogeneous neutron matter and cold
atomic gases. There are close analogies between these seemingly diverse
systems, including the equation of state, superfluid pairing, and linear
response to external probes. In this paper, we compare and contrast results
found in nuclear and cold atom physics. We show updated lattice results for
the energy of the homogeneous unitary Fermi gas and comparisons with neutron
matter, as well as for the dependence of the cold atom energy on the mass
ratio between paired particles, which yields insights on the structure of the
ground state. We also provide new lattice and continuum results for the
harmonically trapped unitary gas, again comparing neutron matter and cold
atoms.
## 1 Introduction
Quantum Monte Carlo methods[(1), (2)] have been proven very valuable in
studying a host of strongly correlated quantum systems, including solid and
liquid Helium,[(3), (4), (5), (6)] electronic systems[(7), (8)], light
nuclei[(9), (10), (11)] and more recently neutron matter[(12), (13), (16),
(14), (15)] and cold atomic Fermi gases.[(17), (18), (19), (20)] Many of these
studies are formulated in the continuum to be able to describe the short-range
repulsion between, for example, helium atoms, nucleons, or electrons. For low-
density neutron matter and cold Fermi gases, where the dominant interaction is
a relatively short-range attraction, lattice methods have also proven
valuable.[(21), (22), (23), (24)] In this paper we describe applications of
Quantum Monte Carlo methods to the equation of state, superfluid pairing gap,
and related properties of neutron matter and cold atomic gases.
We concentrate on the zero temperature properties of cold atoms and neutron
matter. Cold Fermi atoms have a deceptively simple, essentially zero-range
interaction, $H=\sum_{i}-\hbar/2m\nabla^{2}_{i}+\sum_{i,j}V_{0}\delta({\bf
r}_{i}-{\bf r}_{j})$, the strength of which ($V_{0}$) can be tuned to produce
a very rich set of physics described by a relatively small set of universal
parameters. As the strength of the (attractive) interaction $V_{0}$ increases,
one goes from a weak BCS pairing regime to the BEC regime of strongly bound
pairs. Many simulations and experiments[(20)] are performed near and at the
unitary limit, which is where the two-body system produces a nearly zero-
energy bound state.
The $s$-wave interaction between two neutrons is also very attractive, nearly
producing a bound state. For very dilute neutron matter, the neutron matter
and cold atom equations-of-state should be very similar as a function of the
product of the fermi momentum $k_{F}\equiv(3\pi^{2}\rho)^{1/3}$ times the
magnitude $a$ of the scattering length. The $s$-wave interaction between
neutrons also has a significant effective range which causes the equation of
state of neutron matter and cold atoms to diverge at relatively modest
densities. The effective range of the interaction also affects the pairing
gap, etc. It may be possible to use narrow Feshbach resonances to more
directly mimic neutron matter and study the dependence of the equation of
state on the effective range.[(25)] There are, of course, additional $p-$wave
interactions between neutrons, though these are relatively modest at low
densities.
## 2 Monte Carlo Methods
Monte Carlo methods have proven quite effective in dealing with strongly
correlated quantum systems. They are most efficient when they incorporate as
much knowledge of the physical system to be studied as possible. The zero-
temperature Diffusion Monte Carlo (DMC) and Auxiliary Field Monte Carlo (AFMC)
methods we employ use Monte Carlo to propagate a trial wave function to the
true ground state of quantum systems:
$|\Psi_{0}\rangle\ =\ \exp[-H\tau]\ |\Psi_{T}\rangle\ =\ \prod_{i=1}^{N}\
\exp[-H(\tau/N)]\ |\Psi_{T}\rangle,$ (1)
where the imaginary time propagation is split into small imaginary time steps
$\delta\tau=\tau/N$. For small $\delta\tau$ the propagator can be evaluated
accurately in terms of the two-body propagator:
$\displaystyle\langle{\bf R}^{\prime}\ |\ \exp[-H\delta\tau]\ |\ {\bf
R}\rangle$ $\displaystyle=$ $\displaystyle\langle{\bf R}^{\prime}\
|\prod_{i}\exp[-h_{i}^{0}\delta\tau]\ \ {\cal S}\prod_{i<j}\frac{g_{ij}({\bf
r}^{\prime}_{ij},{\bf r}_{ij})}{g_{ij}^{0}({\bf r}^{\prime}_{ij},{\bf
r}_{ij})}\ |{\bf R}\rangle$ $\displaystyle g_{ij}({\bf r}^{\prime}_{ij},{\bf
r}_{ij})\ $ $\displaystyle=$ $\displaystyle\langle{\bf r}^{\prime}|\
\exp[-h_{ij}\delta\tau]\ |{\bf r}\rangle,$ (2)
where $g_{ij}$ and $g_{ij}^{0}$ are the interacting and free two-particle
propagators respectively, determined by the eigenstates of the interacting and
free two-particle Hamiltonian: $h_{ij}=h_{ij}^{0}+v(r_{ij})$ and
$h_{ij}^{0}=-\frac{\hbar^{2}}{m}\nabla_{ij}^{2}$. The factor
$\exp[-h_{i}^{0}\delta\tau]$ is the free single particle propagator, a simple
gaussian, and ${\bf R}$ is a 3$N$-dimensional vector containing the
coordinate-space positions of all the particles.
Diffusion Monte Carlo and Auxiliary Field Monte Carlo operate in different
spaces: DMC performs the simulations in coordinate space, Monte Carlo methods
are used to sample the spatial integrals. AFMC calculations are carried out on
a lattice, and can be viewed as the evolution of single-particle orbitals in
imaginary time. In these lattice calculations the single-particle orbitals can
be transformed between coordinate and momentum space through fast fourier
transforms. In AFMC the Monte Carlo is incorporated by evaluating the
potential matrix elements in terms of fluctuating auxiliary fields.[(23)]
In either case, it is possible to use a very accurate short-time two-body
propagator. In the continuum DMC simulations and in the limit of zero-range
interactions the propagator at unitarity takes the simple form[(26)]:
$g_{ij}({\bf r}^{\prime}_{ij},{\bf r}_{ij})\ =\ g_{ij}^{0}({\bf
r}^{\prime}_{ij},{\bf
r}_{ij})+\frac{\sqrt{m\pi/(\hbar^{2}\delta\tau)}}{4\pi^{2}r_{ij}r^{\prime}_{ij}}\exp[-m/(\hbar^{2}\delta\tau)({r_{ij}}^{2}+{r^{\prime}_{ij}}^{2})/4],$
(3)
where the correction to the free particle propagator arises from the pair
propagating to the same point and then diffusion plus multiple scattering
terms. In actual simulations typically an analytic potential is used and a
short-time approximation used for the propagator:
$g_{ij}({\bf r}^{\prime}_{ij},{\bf r}_{ij})\
=\exp[-V(r^{\prime}_{ij})\delta\tau/2]\ \ g_{ij}^{0}({\bf
r}^{\prime}_{ij},{\bf r}_{ij})\exp[-V(r_{ij})\delta\tau/2].$ (4)
This expression is accurate to order $(\delta\tau)^{2}$.
For lattice calculations, the simplest propagator used is for an on-site
attractive interaction and either a Hubbard-like hopping Hamiltonian or a
$k^{2}/(2m)$ kinetic term in an expression analogous to Eq. 4:
$g_{ij}({\bf r}^{\prime}_{ij},{\bf r}_{ij})\ =\exp[-T\delta\tau/2]\
\exp[-V\delta\tau]\ \exp[-T\delta\tau/2].$ (5)
The potential is evaluated by Monte Carlo sampling of auxiliary fields and the
kinetic energy exactly through the use of fast fourier transforms.[(21)] This
simplified interaction yields a finite range of the order of the lattice
spacing. One can remove this residual effective range by altering either the
kinetic term $T$, adding higher-order momentum terms in the kinetic energy, or
by introducing additional auxiliary fields to give the correct low-energy two-
body spectrum.[(27), (28)] Lattice methods cannot be fully galilean invariant
as there is a lattice cutoff at the high-momentum scale, and simple
implementations of the improved actions can have similar effects to a finite
effective range for pairs with non-zero momentum.[(29)]. These corrections
typically vanish in the limit of large lattices.
The AFMC lattice simulations have the great advantage that they do not suffer
from a sign problem for purely attractive two-body interactions. The up and
down spin evolution can be factorized in AFMC and for unpolarized systems the
spin up and spin down determinants are real and equal and the product is
positive. Therefore, the Monte Carlo results should be exact within
statistical errors if we can perform simulations that are sufficiently dilute,
for sufficiently large number of particles, and at low-enough temperatures
(large $\tau$).
The DMC continuum simulations, in contrast, suffer from a sign problem as the
spin up and spin down determinants are independent and the product can be
positive or negative. The overlap of the propagated wave function with the
trial function has a statistical error that grows with respect to the
propagation time and the number of particles. The fixed-node algorithm,
though, provides an accurate upper bound to the energy by requiring the paths
in the evolution not to cross planes where the trial wave function is zero.
This can be proved to provide an upper bound to the true ground state energy,
and hence the trial wave function can be optimized as the one that produces
the lowest energy. The advantage of the DMC simulations is that the fixed-node
algorithm is equally applicable to polarized or unpolarized systems, to
systems with unequal masses, etc., where the AFMC method also suffers from a
sign problem. Similar constrained-path algorithms exist for AFMC
methods,[(30), (31)] but do not typically provide upper bounds.
In both DMC and AFMC simulations it is very important to use good trial
functions in strongly correlated ground state calculations. In DMC the fixed-
node results are quite accurate if a good trial function of BCS type is
employed, as we shall discuss later. In AFMC the results should be correct for
any trial function that has a finite overlap with the true ground state.
However the statistical errors are dramatically reduced if one uses a BCS
trial function for the trial state, as described in Ref. 23. It is also
extremely valuable to use a branching random walk algorithm to limit
statistical errors in both the DMC simulations, as traditionally done, and in
the AFMC simulations.zhang1995constrained ; zhang1997constrained In zero-
temperature calculations we are trying to reach the eigenstate of the transfer
matrix $\exp[-H\tau]$, the branching random walk algorithm is a Markov chain
algorithm, that is, only the most recent history of the path is required for
performing the next step. Consequently one can iterate to very low
temperatures/large imaginary times compared to other algorithms.
For quantities other than the energy, we often evaluate matrix elements of the
form:
$\langle
O(\tau)\rangle=\frac{\langle\Psi_{T}|O\exp[-H\tau]|\Psi_{i}\rangle}{\langle\Psi_{T}|\exp[-H\tau]|\Psi_{i}\rangle},$
(6)
where $\Psi_{i}$ is an initial state used to start the simulation and
$\Psi_{T}$ is a trial state incorporating as much knowledge as possible of the
ground state. It is often possible to take $\Psi_{i}=\Psi_{T}$, though this is
not required. In the limit of large imaginary time $\tau$, the $O(\tau)$ can
be used to determine the ground-state properties of the system. The energy is
the simplest: in that case the Hamiltonian commutes with the propagator
$\exp[-H\tau]$ and one can calculate the ground state expectation value. For
other properties one must insert the propagation symmetrically between the
initial and final state or create a new Hamiltonian $H^{\prime}=H+\epsilon O$,
in either case evaluating the expectation value of the operator:
$\langle
O(\tau,\tau^{\prime})\rangle=\frac{\langle\Psi_{T}|\exp[-H\tau]O\exp[-H\tau^{\prime}]|\Psi_{i}\rangle}{\langle\Psi_{T}|\exp[-H(\tau+\tau^{\prime})]|\Psi_{i}\rangle},$
(7)
which in the limit of large $\tau$ and $\tau^{\prime}$ gives the true ground
state expectation value.
## 3 Equation of State
The equations of state for cold atoms and for neutron matter have been
extensively studied theoretically, and the cold atom system has been
extensively studied experimentally as well. The cold atom system is very
simple: for a zero-range interaction the equation of state is a function of
only the product of the Fermi momentum and the scattering length ($k_{F}a$).
More specifically, the energy can be written as a function of $k_{F}a$ times
the energy of a non-interacting Fermi Gas
($E_{FG}=(3/5)(\hbar^{2}/2m)k_{F}^{2}$) at the same density. Here the Fermi
momentum is defined through the density of the corresponding non-interacting
gas: $k_{F}=(3\pi^{2}\rho)^{1/3}$. At unitarity (infinite scattering length)
the ratio of the energies of interacting and noninteracting Fermi gases
$E/E_{FG}$ is typically called the Bertsch parameter $\xi$.bertsch1998
### 3.1 Unitarity
Figure 1: AFMC lattice calculations of the unitary Fermi Gas $\xi$ parameter,
updated from Ref. 23. Symbols are for different kinetic terms as a function of
particle number and lattice size. The lattice spacing is denoted as $\alpha$.
Simulations have been performed with $L^{3}$ lattices, for different values of
lattice length $L$ in each direction; open symbols are for even L=16,20,24;
closed are for odd L (see text). All extrapolations are consistent with
$\xi=0.372(5)$.
A history of results for the Bertsch parameter is given in Ref. 28. The first
DMC calculation used up to 40 particles and a modified Poeschl-Teller
potential with $k_{F}r_{e}\approx 0.3$, where $r_{e}$ is the effective range
of the interaction, and yielded a fixed-node energy of
$\xi=0.44(1)$.carlson2003superfluid Subsequent DMC calculations used improved
trial functions, larger particle numbers, and better extrapolations to
$k_{F}r_{e}\rightarrow 0$ to yield $\xi=0.40(1)$.carlson2008superfluid The
best present DMC result is from the calculations of Ref. 34, while an updated
extrapolation to $r_{e}\rightarrow 0$ gives $\xi=0.390(1)$ gandolfi2011bec
for an upper bound. This calculation also carefully compared results at finite
particle number to a superfluid Local Density Approximation (LDA) to
extrapolate to large N. It was found that calculations for $N=38$ or larger
are very close to the thermodynamic limit.
There is also a substantial history of lattice simulations, both for the
ground-state,lee2006 ; lee2006ground ; lee2007superfluidity ; lee2008ground ;
abe2009from and at finite temperature.bulgac2006 ; bulgac2008quantum The
earliest ground-state calculations estimated $\xi=0.25(3)$, for systems up to
22 particles on lattices up to $6^{3}$. The recent calculations of Ref. 23 use
branching random walks and a BCS trial function and importance sampling for
systems of 66 particles on lattices up to $27^{3}$ and obtain $\xi=0.372(5)$
for several different actions. Updated results for these calculations are
shown in Figure 1.
In the figure, the upper curves use a $k^{2}$ dispersion relation tuned to
unitarity. This $k^{2}$ dispersion has a finite positive effective range of
$0.337\ \alpha$, where $\alpha$ is the lattice spacing. The middle set of
curves adopt a $k^{2}+k^{4}$ dispersion that is tuned to zero effective range,
and the lower curves use a Hubbard dispersion relation, which has a negative
effective range of $-0.306\ \alpha$. The $k^{2}+k^{4}$ results show a set of
simulations with even L as open symbols, while simulations at odd L are shown
as filled symbols. The two sets of results are slightly displaced; similar
displacements have been found with limited statistics for the other
dispersions. All extrapolate to the same value of $\xi$ within statistical
errors; we return to the dependence on effective range below. A new lattice
calculation in Ref. 28 reports a higher value of $\xi$, above the upper bound
found in DMC calculations.
There have also been a large number of experimental determinations of $\xi$:
the original measurementsbartenstein2004collective ; kinast2005heat ;
partridge2006pairing have found qualitative agreement with the DMC
calculations listed above. More precise recent experiments have found
$\xi=0.39(2)$luo2009thermodynamic and $\xi=0.41(1)$navon2010equation with a
smaller value of $\xi=0.375(5)$ found most recently.ku2012revealing This
experimental value is quite precise and overlaps our lattice results.
### 3.2 Equation of State: Cold Atoms and Neutron Matter
Of course the full equation of state ($E/E_{FG}$) as a function of $k_{F}a$ is
required to compare with neutron matter, which has a fixed, large effective
range and must be studied by varying the density. The most recent DMC results
for the full equation of state are presented in Fig. 2, and compared to the
lattice results and the most recent experimental result. These results are
quite smooth as a function of $k_{F}a$ and extrapolate correctly in both the
BCS and BEC regimes.
Figure 2: Equation of state of cold atoms versus $1/(k_{F}a)$. Blue circles
are DMC calculations, the red square and green diamond are lattice and
experimental values at unitarity $1/(k_{F}a)$ = 0. The insert shows the
corrections from finite effective range near unitarity (see text).
Because the cold-atom interaction is short-ranged, the derivative of the
energy with respect to $k_{F}a$ is given completely by short-range physics, as
originally written down by Tan in a series of papers.tan2008generalized ;
tan2008degenerate ; tan2008energetics The derivative of the energy per
particle with respect to $k_{F}a$ is given, using the Hellman-Feynman theorem,
by:
$\frac{dE}{da^{-1}}\ =\ \frac{N}{2}\int d^{3}{\bf
r}g_{\uparrow\downarrow}(r)\frac{dV(r)}{da^{-1}}$ (8)
The pair distribution $g_{\uparrow\downarrow}(r)\rightarrow 0$ goes like
$A^{2}/r^{2}$ at unitarity for small $r$, with
$g_{\uparrow\downarrow}(r)\rightarrow 1/2$ at large r. The change in energy
with respect to $a^{-1}$ is
$\frac{dE}{da^{-1}}=\ -\ \frac{\hbar^{2}2\pi\rho A^{2}}{m}\rightarrow
C=8\pi^{2}\rho^{2}A^{2},$ (9)
where $C$ is Tan’s contact parameter. Near unitarity the EOS is conventionally
parametrized as
$\frac{E}{E_{FG}}\ =\ \xi-\frac{\zeta}{k_{F}a}+...,$ (10)
with $\zeta=(5\pi/2)C/k_{F}^{4}$. We return to the contact parameter in the
discussion of short-range physics below.
Figure 3: Comparison of the equation of state of cold atoms and neutron matter
at low density. Neutron matter calculations are from Ref. 14. Differences at
low density are primarily due to the effective range of the neutron-neutron
interaction. The solid line is a fit to the cold atom results, the dashed line
includes an estimate of effective range effects (see text).
In Fig. 3 these cold atom results are compared to the QMC for neutron
matter,gezerlis2010low and to the analytic expression available at small
$k_{F}a$. At low densities, the neutron matter and cold atom results agree,
they also agree with a simple extrapolation of the analytic results near
$k_{F}a=0$. At higher densities, the cold atom and neutron matter equations of
state start to diverge somewhat as the effective range becomes important. The
dependence of the equation of state on effective range can be made explicit,
as we discuss below. This dependence gives a quantitative picture of the
difference between neutron matter and cold atoms that could perhaps be tested
in cold atom experiments with narrow resonances. We will return to the finite-
range corrections below.
### 3.3 Equation of State: Unequal Masses
Cold atom experiments can also be performed with species of different mass,
providing important information about the structure of the ground state of the
unitary Fermi Gas. For species of different mass $m_{\uparrow}$ and
$m_{\downarrow}$, if we normalize the $\xi$ parameter by the reduced mass
$E_{FG}=\frac{\hbar^{2}k_{F}^{2}}{4\mu}$, BCS theory would give a value of
$\xi$ independent of the mass ratio $r=m_{\uparrow}/m_{\downarrow}$. The
difference in Hamiltonians for equal $(r=1,m_{\uparrow}=m_{\downarrow}=m)$
masses and unequal masses is
$\displaystyle\Delta H$ $\displaystyle=$ $\displaystyle\
\sum_{i=1}^{N_{\uparrow}}-\
\frac{\hbar^{2}\nabla_{i}^{2}}{2m_{\uparrow}}+\sum_{i=1}^{N_{\downarrow}}-\
\frac{\hbar^{2}\nabla_{i}^{2}}{2m_{\downarrow}}-\sum_{i=1}^{N_{\uparrow}+N_{\downarrow}}-\
\frac{\hbar^{2}\nabla_{i}^{2}}{2m}$ (11) $\displaystyle=$
$\displaystyle\sum_{i=1}^{\\#pairs}-\frac{\nabla_{i}^{2}}{4m}\frac{(r-1)^{2}}{(r+1)^{2}},$
where in the last line the particles have arbitrarily been divided into $N/2$
spin up - spin down pairs.
Figure 4: Dependence of the unitary Fermi Gas equation of state on mass ratio
for fixed reduced mass
$\mu=m_{\uparrow}m_{\downarrow}/(m_{\uparrow}+m_{\downarrow})$, plotted versus
$(r-1)^{2}/(r+1)^{2}$, where $r$ is the mass ratio.
Figure 4 shows the DMC calculations of $\xi$ for different mass ratios.
Initial calculations for different mass ratios were reported in Ref. 49. From
Eq. 11 we can see that the energy change can be evaluated in perturbation
theory near r=1.
$\Delta(E/N)\ =\ \langle 0|\Delta H|\ 0\rangle=(1/2)\langle
P_{ij}^{2}/(4m)\rangle|_{r=1}\frac{(r-1)^{2}}{(r+1)^{2}},$ (12)
where the 1/2 comes from the number of pairs (N/2), $P_{ij}$ is the total
momentum of a pair, and the expectation value is to be taken in the ground
state of the equal mass unitary gas. Note that these calculations were
performed for small but finite value of the effective range, yielding a
slightly larger value of $\xi$ than at zero effective range. This rather
asymmetric way of writing the energy difference is valuable because it tells
us something about the character of the state. For a BCS-like state with all
pairs at $P=0$ the energy difference is zero in first-order perturbation
theory. Of course the free Fermi Gas can also be written in this manner. The
difference is finite for the case when the ground state wave function does not
have a spin down particle at $-{\bf p}$ for every spin up particle at momentum
${\bf p}$.
Fig. 4 shows the DMC calculations as points with error bars, and a quadratic
fit to the data. The linear coefficient in this fit is very small, consistent
with zero within statistical errors. Thus to a very good approximation the
ground state of the unitary gas can be written as a state of pairs with zero
momentum. To confirm this result it would be important to have experimental
measurements of the energy for several different mass ratios.
### 3.4 Equation of State: effective range
As is apparent from Fig. 1, the equation of state for finite effective range
$r_{e}$ varies linearly with $k_{F}r_{e}$ at small effective range. In Ref.
23, the equation of state at unitarity for different effective ranges was
found to be:
$\xi(k_{F}r_{e})=\xi(0)+Sk_{F}r_{e},$ (13)
where $\xi(0)$ characterizes the ground-state energy at zero effective range,
and $S$ is the slope parameter giving the linear dependence on $k_{F}r_{e}$.
The slope parameter was extracted from both DMC and AFMC
calculations,carlson2011auxiliary and found to be $S=0.12(3)$. The results
for different effective ranges are shown in Figure 5. More recent DMC results
for a variety of interactions have recently appeared,forbes2012effective they
find $S=0.127(4)$ using a variety of interactions. These calculations further
demonstrate that $S$ is a universal parameter, as originally conjectured in
the original version of Ref. 29. In this manuscript, the authors also claim
that lattice results will in general have a dependence on the total momentum
$P$ of a pair. For the unitary gas, however, the expectation value of $\langle
P^{2}\rangle$ is approximately zero, as shown above. Therefore the lattice and
continuum results are both in agreement with Eq. 13.
Figure 5: Dependence of the unitary Fermi Gas equation of state on Fermi
momentum vs. effective range $(k_{F}r_{e})$. Shaded bands are fits to the
lattice results, and dashed lines give DMC results.
In DMC calculations the slope parameter $S$ is not too sensitive to $k_{F}a$
near unitarity. Fig. 2 shows, in the inset, the slope parameter $S$ evaluated
from DMC calculations near unitarity. It is positive and approximately $0.1$
near unitarity, but changes significantly in the BCS and BEC regimes. The
difference between the cold atom EOS and neutron matter at sufficiently small
densities should be approximately $\xi_{neutrons}-\xi_{atoms}\approx
Sk_{F}r_{e}$, or approximately 0.05 at $-k_{F}a=5$ since the neutron-neutron
effective range is expected to be approximately 2.7 fm. Fig. 3 shows a fit to
the cold atom results at zero effective range as a solid line. The dashed line
adds an effective range correction with $S\ =\ 0.1$. This should be the
dominant correction at $\ k_{F}\leq 0.25$ fm-1 , near $k_{F}\approx 0.5$ fm-1
one would have $k_{F}r_{e}\approx 1$ and higher order corrections in $s-$ and
$p-$wave interactions could be important.
## 4 Pairing Gap
Both low-density neutron matter and cold atoms are strongly paired Fermi
systems, they exhibit some of the largest pairing gaps of any systems known
when measured in terms of the Fermi energy. We define the pairing gap at T=0
as the difference between the energy of an odd particle system and the average
of the two nearby even particle systems in periodic boundary conditions:
$\Delta\ =\ E(N+1)-(E(N)+E(N+2))/2,$ (14)
with the universal parameter $\delta$ defined as the pairing gap divided by
the Fermi energy $E_{F}=\hbar^{2}k_{F}^{2}/2m$. For simulations of a large
enough number of particles this should correspond to the traditional
definition of the pairing gap.
The fact that the pairing gap is so large, a sizable fraction of the Fermi
energy, makes it possible to use QMC methods to accurately calculate the gap
by separately calculating the energies of the even and odd particle systems.
In addition, the fact that the energy per particle shows no significant shell
effects for reasonably small systems $(N>30)$ makes it much easier to approach
the continuum limit. Though there is an upper bound principle for the even and
odd systems, there is no specific bound on the pairing gap.
The original calculations of the pairing gap in cold atoms at unitarity found
$\Delta/E_{FG}\approx 0.9$ or $\delta=0.55(5)$.carlson2003superfluid
Subsequent improvements to the wave functioncarlson2005asymmetric found a
slightly reduced value for the gap, $\delta=0.50(5)$. These results can be
compared to an extraction of the pairing gap from the measured density
distributions in partially spin-polarized trapped cold
atomscarlson2008superfluid and measurements of the RF response in such
systemsschirotzek2008determination , who find $\delta=0.45(5)$ and
$\delta=0.44(3)$, respectively.
The pairing gap in neutron matter has historically been the subject of a great
deal of interest and theoretical activity.Gandolfi:2008 ; Gandolfi:2009 QMC
calculations of the pairing gap were performed in 13 and 14. These
calculations used the $s$-wave and $s$\- & $p$-wave components of the AV18
interaction, respectively. A summary of the results are shown in Fig.
6.gezerlis2008strongly
Figure 6: Pairing gap in cold atoms and neutron matter. BCS mean-field results
are shown as solid lines, DMC results are shown as symbols. The pairing gaps
are divided with the relevant one-body quantity, namely the Fermi energy
$E_{F}$ (analogously to the ground-state energy in Fig. 11 being divided with
$E_{FG}=3E_{F}/5$.
In the figure, BCS results are given by solid lines. In the weak-coupling
limit, the pairing gap is expected to be reduced from the BCS value by
$(1/4e)^{1/3}\approx 0.45$ from the Gorkov polarization correction.gorkov1961
It is difficult for QMC calculations to calculate the pairing gap at coupling
weaker than $k_{F}a=-1$ because of the delicate cancellations. At this
coupling, though, we find a suppression in the gap roughly compatible with the
Gorkov suppression. At stronger coupling the suppression diminishes smoothly,
and the gap reaches a value of $0.50(05)$ at unitarity. In the BEC regime the
pairing gap approaches half the binding energy of the pair as reproduced by
the BCS equation.
The calculated gaps in neutron matter are considerably smaller than in cold
atoms, but still reach a maximum of nearly $0.3E_{F}$ at $k_{F}a=-5$. The
effective range in the neutron-neutron interaction reduces the gap
significantly, as shown in the comparison of BCS results and in the Monte
Carlo calculations. These pairing gaps are considerably larger than found in
many diagrammatic approaches,gezerlis2010low but in agreement with the
lattice results of Ref. 24.
## 5 Short-range physics
One can also investigate the short-range (high-momentum) physics in cold atom
experiments. Because of the simple short-range interaction, this short range
physics is directly related to the equation of state discussed above. The pair
distribution function and off-diagonal density matrix at short distances and
the momentum distribution at high momenta are all governed by the contact
parameter.
The pair distribution function is shown in Fig. 7; the contact governs the
huge spike near $r=0$. In the inset the pair distribution function is
multiplied by $(k_{F}r)^{2}$ to show the behavior near $r=0$, the dip at very
short distances is due to the finite range interaction used in the
simulations. The behavior in the region of the vertical dashed line and
slightly beyond is governed by universal physics. In the figure the red points
are variational Monte Carlo (VMC) results obtained from the trial wave
function $|\Psi_{T}\rangle$, the green are the mixed estimates of the form
obtained from Eq. 6 and blue are the full extrapolated DMC results, as is
apparent in the figure the VMC calculation is correctly capturing the basic
physics.
Figure 7: Pair distribution function $g_{\uparrow\downarrow}(r)$ for cold
atoms at unitarity. Inset shows the behavior at short distances scaled by
$r^{2}$, the magnitude of the contact determines the value of this quantity
(see text).
The momentum distribution scaled by $k^{4}$ is plotted in Fig. 8. The momentum
distribution at large $k$ is proportional to the contact. The horizontal line
in the figure is the value that would be expected from calculations of the
equation of state: $\zeta=0.901(2)$. Extractions of the contact from all these
observables are consistent with this value, though of course some are noisier
than others. It will be interesting to see what information on the short-range
physics of neutron matter can be obtained from theory and experiments with
narrow resonances with a significant effective range.
Figure 8: Momentum distribution scaled by $k^{4}$ for cold atoms at unitarity.
Initial experiments on the spin and density response of cold atoms have also
been performed.hoinka2012dynamic These and future results will be very
interesting as they can tell us about the propagation of particles and spin in
the unitary gas. The response functions can be written as:
$\displaystyle S_{\rho}(q,\omega)\ $ $\displaystyle=$ $\displaystyle\sum_{f}\
\langle 0|\sum_{i}\exp[-i{\bf q}\cdot{\bf r}_{i}]|f\ \rangle\langle
f|\sum_{j}\exp[i{\bf q}\cdot{{\bf r}_{j}}]|0\rangle\
\delta(\omega-(E_{f}-E_{0}))$ $\displaystyle S_{\sigma}(q,\omega)\ $
$\displaystyle=$ $\displaystyle\sum_{f}\ \langle 0|\sum_{i}\exp[-i{\bf
q}\cdot{\bf r}_{i}]\ {\mbox{\boldmath$\sigma$}}_{i}\ |f\ \rangle\cdot\langle
f|\sum_{j}\exp[i{\bf q}\cdot{{\bf r}_{j}}]\ {\mbox{\boldmath$\sigma$}}_{j}\
|0\rangle\ \delta(\omega-(E_{f}-E_{0}))$
These response functions have been calculated at high momenta in terms of the
operator production and related high-momentum expansions.son2010short ;
goldberger2012structure ; hofmann2011current ; hu2012universal ;
nishida2012probing The experiments show a two-peak structure in the density
response, one at $\omega=q^{2}/(2m)$ associated with the breaking of a pair
and one at $\omega=q^{2}/(4m)$ associated with the propagation of a pair. The
spin response requires breaking of a pair. The initial experiments are at
rather high momentum transfer, many times the Fermi momentum, and hence probe
the short-range physics. It will be interesting to see how these response
functions evolve at lower momenta. One can also compare calculations and
experiments on the sum rules associated with the contact parameter by
integrating the response over $\omega$.
## 6 Inhomogeneous Matter
Finally, we turn to the properties of inhomogeneous matter. This is the regime
with perhaps the closest connection between nuclear physics and cold atom
physics. In the inner crust of a neutron star the neutrons form a gas between
the neutron-rich nuclei that exist in a lattice structure. This neutron matter
is very low density and is inhomogeneous, and for many of the transport
properties we would like to understand the behavior of this gas. The
properties of inhomogeneous neutron matter, particularly the gradient terms,
are very difficult to determine from the binding energies of atomic nuclei.
The isovector gradient term is one of the least constrained parameters in
nuclear density functionals, and ab-initio calculations can provide valuable
guidance.
### 6.1 Inhomogeneous Matter: Bulk Properties
We have calculated the properties of finite systems of neutrons bound in
harmonic and Woods-Saxon wells.gandolfi2010cold Original calculations of
these dropspudliner1996neutron were limited to N=8 neutrons because of the
spin-dependence of the nuclear interaction. These more recent calculations use
Auxiliary Field Diffusion Monte Carlosarsa2003neutron ; gandolfi2007quantum ;
gandolfi2009quantum methods and Green’s function Monte Carlo methods; the
former have been used to treat up to N=50 neutrons. The AV18 NN interaction
plus the UIX three-nucleon interaction have been used for these calculations.
As these wells produce fairly modest densities for 50 particles or less, the
three-neutron interaction is not very important.
Figure 9: Scaled energies for neutrons bound in a harmonic
well.gandolfi2010cold The upper and lower straight dashed lines are the
Thomas-Fermi mean-field results for free fermions and for a scaled EOS
$E/N=\xi E_{FG}$ with $\xi=0.5$. The open symbols are calculations with
previous generation density functionals, and the filled symbols are GFMC and
AFDMC calculations. The jagged line shows the results for the $SLy4$ density
functional. The upper panel is for a harmonic trap frequency of 10 MeV and the
lower for 5 MeV.
The energies in Fig. 9 have been scaled by $1/(\hbar\omega N^{4/3})$, the
expected behavior in the Thomas-Fermi approximation for an EOS of the form
$E/E_{FG}=\xi$. The upper horizontal dashed lines are for $\xi=1$ (free
fermions), and the lower for $\xi=0.5$. Neutron matter over a considerable
range of densities is roughly consistent with $\xi=0.5$, though it is less
attractive at low and high densities. The traps have harmonic frequencies of
$\omega=10$ MeV (upper panel) and $5$ MeV ( lower panel). Results for several
typical older-generation density functionals are shown as open circles, and
calculations using SLY4 as solid lines.
The Skyrme interactions typically give significantly lower energies than the
microscopic calculations, particularly for the 10 MeV well. The greater
difference for the 10 MeV well suggests that isovector gradient terms in the
density functional should be more repulsive. The curves marked “SLY4-adj” in
the figure are obtained by adjusting the isovector gradient, the isovector
pairing, and the isovector spin-orbit terms in the interaction. At the closed
shells ($N=8,20,40$) only the change in the isovector gradient term is
important. A reasonable fit can be obtained to the closed-shell energies in
the 5 and 10 MeV harmonic wells and the Woods-Saxon wells with a single
adjustment to the isovector gradient term. This adjustment also better
reproduces the rms radii and mass distributions of the ab-initio
calculations.gandolfi2010cold
It is interesting to compare the neutron drop results to those obtained for
cold atoms. In Fig. 10 we plot the scaled energies of cold atom systems
obtained in DMC and AFMC calculations as well as the results of previous
calculations.chang2007unitary ; blume2007universal ; endres2011 Two previous
DMC calculations used fairly simple trial wave functions,chang2007unitary ;
blume2007universal the first using an orbital basis for the trial function
and the second using a simple $1/r$ BCS pairing function as the trial wave
function. Both resulted in energies far above what would be expected in local-
density (Thomas Fermi) theory with $\xi\approx 0.4$. New lattice
resultsendres2011 yield somewhat lower energies, but significant shell
structure. They are also above the energies expected from the measured and
calculated values of $\xi$ in the continuum, indicating either unnaturally
large gradient corrections or other effects.
Our new DMC and AFMC calculations produce energies considerably lower than
previous results. The DMC energies are very smooth as a function of N as
compared to neutron drop results, indicating a lack of shell closures for the
unitary Fermi gas with zero effective range. These DMC calculations use a more
sophisticated trial function incorporating both a single-particle shell model
basis and substantial short-range pairing into the trial wave function. The
resulting energy is somewhat higher than the expectations from the bulk for N
up to 50 particles.
It has been shownwerner2006unitary that the cold atoms trapped in harmonic
wells have a breathing mode associated with the scale invariance of the
Hamiltonian of exactly $2\ \hbar\omega$, independent of particle number N,
giving further evidence that cold atoms at unitarity have a very weak shell
structure, if any. Initial AFMC lattice calculations give energies for 30
fermions much closer to the expected bulk limit, indicating a smooth and rapid
transition from few-particle systems to the bulk. Further DMC and AFMC
calculations are being pursued.
Figure 10: Comparison of different calculations of the harmonically confined
unitary Fermi Gas.chang2007unitary ; blume2007universal ; endres2011 ;
gandolfi2012prep ; forbes2012effective The vertical axis is the square of the
energy divided by Thomas-Fermi energy $E_{TF}=\omega(3N)^{4/3}/4$, the
horizontal axis is the number of particles. For a very large system
$(E/E_{TF})^{2}$ should approach $\xi$.
In Fig. 11 we compare cold atom results to neutron drops in the same traps as
a function of particle number $N$. We plot the square of the energies because,
in the local density approximation, the square of the energy of the confined
system is proportional to the energy of uniform matter determined by $\xi$.
The bulk limit as obtained from the lattice calculations shown in Fig. 1 is
shown as an arrow at the lower right of the figure. The energies of neutron
drops are considerably higher than those of cold atoms, this is at least
partially the result of the effective range in neutron matter. The gradient
terms are likely also important, however. These can be more precisely
constrained by performing a local density calculation using a realistic
equation of state for neutron matter.
Figure 11: Comparison of harmonically confined neutron drops and cold atoms.
The neutron drops have significant shell closures at N= 8, 20, 40, etc.
because of the finite effective range and further corrections.
### 6.2 Inhomogeneous Matter: pairing
We have also performed calculations of the pairing gap in neutron drops, shown
in Fig. 12. In atomic nuclei pairing is, at least predominantly, a bulk
effect, in that the coherence length appears to be comparable to the size of
the nucleus. This is to be expected in a regime where the effective range is
comparable to the interparticle spacing, in such cases it should be possible
to construct a mean-field theory that produces a qualitatively correct
picture.
Figure 12: Calculated pairing gaps in harmonically confined neutron drops, at
harmonic frequencies $\omega=5$ MeV (left) and $10$ MeV (right). The large
values at N $=8$ are associated with the shell closure. Gaps are calculated
from the mass difference formula with a phase factor of $-1^{N+1}$ to take
into the account the fact that the unpaired particles have a higher energy
associated with the pairing gap.
For cold atoms the conclusions will be quite different; for large enough
systems the unpaired atom will necessarily sit outside the center of the drop.
The gap is simply too high in the high-density central region for the unpaired
particle to penetrate. It will be very instructive to compare theories and
experiments as a function of scattering length, effective range, and particle
number.
## 7 Conclusions
Cold atom experiments and theory provide many valuable insights into our
understanding of strongly correlated fermions, and in particular have a close
relationship with low-density neutron matter. The equations of state for low-
density neutron matter and cold atoms are by now well understood and very
similar as a function of $k_{F}a$, and the difference is understood in terms
of a correction proportional to the fermi momentum times the effective range.
Pairing gaps in cold atoms demonstrate a smooth transition from the BEC to BCS
regime, and indicate that a sizable pairing gap is to be expected. Further
experimental and theoretical studies as a function of $k_{F}r_{e}$ at and near
unitarity would be very valuable in providing explicit confirmation and a more
precise understanding.
A better understanding of the linear response of cold Fermi atoms could lead
to new insights into the dynamic response of neutron matter. It would be
particularly valuable to map out both the density and spin response of cold
atoms as a function of momentum transfer. Analogies to topics such as neutrino
propagation in dense matter are clear, though there is not a direct
correspondence as there is for the equation of state and pairing gap.
Inhomogeneous matter is also quite intriguing, including small systems of
trapped fermions, fermions in optical lattices, and the transition from three
to two-dimensions. Inhomogeneous cold atom systems also have close analogies
in nuclear physics, including the physics of nuclei and the neutron star
crust. Studies of narrow resonances with finite effective range could help us
understand the evolution of pairing from a local to a bulk phenomenon. The
rapidly expanding scope of cold atom experiments and Quantum Monte Carlo will
undoubtedly reveal intriguing new physics and close correlations with nuclear
physics.
## Acknowledgements
The authors would like to thank Yusuke Nishida and Steven Pieper for valuable
conversations and insights. The authors are grateful to the LANL Institutional
computing program, the NERSC computing facility at Berkeley National
Laboratory, and the Oak Ridge Leadership Computing Facility located in the Oak
Ridge National Laboratory, which is supported by the Office of Science of the
Department of Energy under Contract DE-AC05-00OR22725. The work of J. Carlson
and Stefano Gandolfi is supported by the Nuclear Physics program at the DOE
Office of Science, and Stefano Gandolfi is also supported by the LANL LDRD
program. The work of Alexandros Gezerlis is supported by the Helmholtz
Alliance Program of the Helmholtz Association, contract HA216/EMMI “Extremes
of Density and Temperature: Cosmic Matter in the Laboratory”.
## Note
This is a pre-copy-editing, author-produced PDF of an article accepted for
publication in Progress of Theoretical and Experimental Physics following peer
review. The definitive publisher-authenticated version Prog. Theor. Exp. Phys.
2012, 01A209 is available online here.
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|
arxiv-papers
| 2012-10-24T20:00:10 |
2024-09-04T02:49:37.033952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Carlson, Stefano Gandolfi, Alexandros Gezerlis",
"submitter": "Alexandros Gezerlis",
"url": "https://arxiv.org/abs/1210.6659"
}
|
1210.6669
|
aainstitutetext: Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Förhringer Ring 6, 80805 München, Germany. bbinstitutetext: State Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Science, Beijing 100190, People’s Republic of China
# Magnetic field induced lattice ground states from holography
Yan-Yan Bu a Johanna Erdmenger a Jonathan P. Shock a Migael Strydom
(yybu,jke,jonshock,mstrydom)@mppmu.mpg.de
###### Abstract
We study the holographic field theory dual of a probe $SU(2)$ Yang-Mills field
in a background $(4+1)$-dimensional asymptotically Anti-de Sitter space. We
find a new ground state when a magnetic component of the gauge field is larger
than a critical value. The ground state forms a triangular Abrikosov lattice
in the spatial directions perpendicular to the magnetic field. The lattice is
composed of superconducting vortices induced by the condensation of a charged
vector operator. We perform this calculation both at finite temperature and at
zero temperature with a hard wall cutoff dual to a confining gauge theory. The
study of this state may be of relevance to both holographic condensed matter
models as well as to heavy ion physics. The results shown here provide support
for the proposal that such a ground state may be found in the QCD vacuum when
a large magnetic field is present.
###### Keywords:
Gauge-gravity correspondence, Holography and condensed matter physics(AdS/CMT)
††preprint: MPP-2012-144
## 1 Introduction
The study of black hole instabilities is an important research topic that has
led to very interesting results. In particular, within gauge/gravity duality,
the study of Anti-de Sitter black hole solutions and their stability
properties is important for understanding thermal states on the gauge theory
side. Last year in Ammon:2011je , some of the authors of the present paper
studied an $SU(2)$ Einstein-Yang-Mills model at finite temperature in
asymptotically AdS space. They found that when a magnetic component of the
gauge field reaches a critical value in units of the temperature, the system
becomes unstable. Though the critical value of the magnetic field for onset of
the instability was calculated, the new ground state of the system was not
known. In the current work we calculate a ground state solution, using a
perturbative analysis similar to the one performed by Abrikosov in
Abrikosov:1956sx for type II superconductors. In agreement with the work of
Abrikosov we find that the ground state is a triangular lattice. There have
been many attempts recently to model lattices holographically with the goal of
providing more realistic models for condensed matter systems Horowitz:2012ky ;
Flauger:2010tv , and this novel procedure for generating a lattice dynamically
adds to these developments. Moreover, our holographic model provides support
for recent QCD studies of $\rho$ meson condensation from a strong magnetic
field Chernodub:2010qx ; Chernodub:2010zw ; Chernodub:2011gs . The effect
described here is similar to the Nielsen-Olsen solution for gluon condensation
Nielsen:1978rm and to magnetically catalysed W boson condensation
Ambjorn:1988tm ; Ambjorn:1989bd ; Ambjorn:1989sz .
Although this is the first holographic calculation to explicitly uncover an
Abrikosov lattice in 3+1 dimensions, it is not the first to examine spatially
inhomogeneous phases of strongly coupled field theories. In Horowitz:2012ky ,
the authors studied the holographic construction of an Einstein-Maxwell-scalar
theory at finite temperature and density. They looked at the gauge theory
optical conductivity, which is the conductivity in the direction of an applied
electric field. They broke the translational invariance explicitly by imposing
scalar field boundary conditions in the form of a lattice modulated in one of
the Minkowski spatial directions. A fully backreacted solution was found which
thus induces a spatially inhomogeneous black hole solution. This leads to an
extremely rich behaviour of the frequency dependent optical conductivity. At
low frequencies there appears a Drude peak. A Drude peak is a broadening of
the zero frequency delta peak in the conductivity. In real materials this is
due to impurities and finite temperature effects. The Drude peak is not
present when translational invariance is unbroken. The solution also exhibits
a power law behaviour at frequencies intermediate with respect to the
temperature, and a constant value in the high frequency regime. The power law
behaviour is the same as that found experimentally in cuprates, while the
constant value at high frequencies is expected from conformal invariance. The
setup in this context was a (2+1)-dimensional model where the lattice was
periodic in only one of the spatial dimensions. A more realistic lattice
structure would be highly desirable.
While the lattice in the approach of Horowitz:2012ky was implemented in the
boundary conditions, there are a number of other mechanisms known that lead
dynamically to ground states without translational symmetry. One approach was
pioneered in Domokos:2007kt by studying a Yang-Mills-Chern-Simons theory in
the gauge/gravity context. It was shown that the Chern-Simons term can induce
an instability which leads to a ground state with both translational and
rotational symmetry breaking. Such work was continued in Nakamura:2009tf ;
Chuang:2010ku ; Bergman:2011rf ; Bayona:2011ab ; Takeuchi:2011uk where the
spatially homogeneous phase was found to be unstable in a variety of
gravitational contexts in the presence of Chern-Simons couplings. The
perturbative analysis of quasinormal modes that become tachyonic at finite
momentum gives a relatively simple computational tool for finding
instabilities to ground states without translational symmetry. These solutions
are found to induce a helical current Ooguri:2010kt ; Donos:2012gg ;
Donos:2012wi . Interestingly, a Chern-Simons term is not always enough to
induce such an instability. It was shown by Ammon:2011hz that this type of
instability does not exist in the D3/D7 system.
Translational invariance can also be broken with a magnetic field or with
magnetic monopoles. The former was first studied by Gauntlett et al. in
Donos:2011qt and Donos:2011bh where the instability of the magnetically
charged black hole in a top-down framework was studied in detail. In the
latter work, an infinite family of solutions coming from $D=11$ supergravity
was shown to exhibit a magnetically catalysed instability. Such work is
important as it proves that these instabilities can also come from real string
theory constructions. The subject of magnetic monopoles in $(3+1)$-dimensional
AdS space was studied in Bolognesi:2010nb and Sutcliffe:2011sr . These
magnetic monopoles are solutions to the scalar field in a Yang-Mills-Higgs
theory with gauge group $SU(2)$. In a certain limit where the monopole
magnetic charge becomes large and a “monopole wall” is formed, it was shown in
Bolognesi:2010nb that there is a W boson instability. In Sutcliffe:2011sr a
hexagonal lattice ground state of these monopole walls was found numerically.
In Allahbakhshi:2011nh the holographic dual of a self-gravitating Julia-Zee
Dyon was constructed, and it was shown to contain a vortex condensate.
There are some holographic models exhibiting a superconducting phase
transition that results in a vortex lattice ground state. The first we mention
involves an $s$-wave superconductor. In Maeda:2009vf a type II superconductor
was modelled using a $(3+1)$-dimensional gravitational setup. A type II
superconductor is one for which the external applied magnetic field has two
critical values. When the magnitude of the magnetic field increases beyond the
lower of the two critical values, the field starts to penetrate the
superconducting condensate. Some of the condensate remains until the magnitude
of the field is increased beyond the upper critical value, at which point
superconductivity is completely destroyed. Just before the upper critical
value is reached from below, the ground state of the system is a triangular
Abrikosov lattice Abrikosov:1956sx . The authors of Maeda:2009vf constructed
a holographic superconductor modelling the behaviour of a type II
superconductor near the upper critical value of the magnetic field and found
the Abrikosov lattice ground state explicitly.111 The model of Maeda:2009vf
does not display the transition at the lower critical magnetic field value
because the gauge field is not dynamical. See Domenech:2010nf for adding
dynamical gauge fields to holographic superconductors. The transition at the
upper critical value is present however because there the condensate is small
so the backreaction is negligible. In Murray:2011gr it was shown how to
construct a similar vortex lattice solution in a model describing a $p$-wave
superconductor. There the authors used a holographic model with an $SU(2)$
gauge field similar to the one described in the current paper. Both of theses
examples are different from our model, however, because here we find a
superconducting Abrikosov lattice ground state that is induced by an $SU(2)$
magnetic field, rather than being destroyed by it. Moreover, in contrast to
these models, we do not need a finite density. Our model is a cousin of
holographic p-wave superconductors where the condensation is induced by a
finite isospin density, holographically realised by a non-trivial temporal
component of the $SU(2)$ gauge field (see Gubser:2008wv and Ammon:2008fc ;
Ammon:2009fe as well as the recent Chunlen:2012zy ). Here, in contrast, a
spatial component of this gauge field has a non-trivial profile. Whereas in
Ammon:2008fc ; Ammon:2009fe , a Meissner effect is shown to occur by which a
magnetic field reduces the transition temperature, here it is again the
magnetic field which induces condensation at zero density.
In addition to being interesting in the broader context of holographic
lattices, the model we discuss serves as supporting evidence for a phenomenon
first described by Chernodub et al. in Chernodub:2010qx ; Chernodub:2010zw .
There it was proposed that the QCD$\times$QED vacuum may itself be susceptible
to a superconducting transition when a magnetic field of the order of the QCD
scale is present. Such extreme conditions are rare but they may be present for
a few femtoseconds during highly off-centre heavy ion collisions. The
discovery of this phase came about through the study of an effective field
theory description (the DSGS model proposed by Djukanovic, Schindler, Gegelia
and Scherer in Djukanovic:2005ag ) of $\rho$ mesons interacting with a
magnetic field. A destabilisation of the vacuum was shown that would clearly
lead to the condensation of charged and neutral $\rho$ mesons. This breaks the
$U(1)$ gauge symmetry and leads to a superconductor with the quark-antiquark
pairs in the mesons acting as Cooper pairs. The instability was also found
using an extended Nambu–Jona-Lasinio model with $SU(3)$ colour and $SU(2)$
flavour in Chernodub:2011mc . Lattice gauge theory studies were then performed
looking at QCD in strong magnetic fields and these indicate the same
instability. Moreover, using the DSGS model and guided by the Ginzburg-Landau
model of type II superconductors, a solution was found in which the $\rho$
meson condensate forms an Abrikosov lattice made up of superconducting
vortices Chernodub:2011gs . This may be relevant experimentally. Evidence has
mounted at both RHIC and the ALICE experiment at CERN that strong magnetic
fields may contribute to the physics of the strongly coupled quark gluon
plasma as charges are quickly accelerated during the interaction period
Skokov:2009qp ; Bzdak:2011yy . The importance of these effects remains a
contested topic because the time scales involved are small. However, given
that strong magnetic fields may be present, it is interesting to ask if traces
of this $\rho$ meson condensate could be detected.
In the current paper we find a possible ground state of the system in
Ammon:2011je . As mentioned above, this system was shown to be unstable under
the imposition of a large $SU(2)$ magnetic field.222It was shown in
Callebaut:2011ab that the same sort of instability occurs in the Sakai-
Sugimoto model, but there the ground state has also not been found. We show
that it has very similar properties to the ground state of a type II
superconductor near the upper critical magnetic field as well as to the ground
state in the model of Chernodub et al. In other words, the ground state is a
triangular Abrikosov lattice. We take here a very simple model of a strongly
coupled finite temperature quantum field theory in $(3+1)$-dimensions with a
global $SU(2)$ symmetry. The dual gravity theory is an $SU(2)$ Einstein-Yang-
Mills theory in $(4+1)$-dimensions with a magnetic component of the $SU(2)$
switched on. We work entirely within the probe approximation, which means that
the Yang-Mills term is small compared to the Einstein-Hilbert term in the
action. We also fix the gauge in such a way that the gauge theory condensate
is transformed under a $U(1)$ subgroup of the global $SU(2)$ symmetry. There
appears to be a certain universality to the triangular lattice ground state.
Here we show that it forms in both the AdS Schwarzschild background (dual to a
finite temperature field theory) as well as the hard wall cutoff model (dual
to a confining field theory). It would be interesting to uncover exactly how
universal these results are.
The two holographic models that we study have several important differences
from QCD. In the finite temperature model there is no confinement or chiral
symmetry breaking and so there are no goldstone bosons (pions) present which
are the normal decay modes of the $\rho$ meson in QCD. The hard wall model has
its conformal symmetry broken only by an IR boundary condition which sets a
confinement scale. However, the phenomenology of these two models appears to
be close enough to that of QCD to compare qualitatively with the models of
Chernodub et al.
In section 2 we provide the details of the holographic setup. There we also
explain the strategy behind the perturbative expansion of the $SU(2)$ gauge
field near the critical magnetic field. Since we follow the philosophy of
Abrikosov’s calculation of the ground state in type II superconductors, which
was done in the Ginzburg-Landau model, in section 3 we give a brief outline of
this approach and then follow it to solve perturbatively up to third order. In
section 4 we discuss the numerical results and analyse the free energy of the
different lattice solutions, showing that the triangular lattice has the
lowest free energy of all the Abrikosov solutions studied. It is important to
note that we are not able to show conclusively that we have found _the_ ground
state but we are able to find a state with lower free energy than the
translationally invariant state and that has lowest energy within a large
class of lattice solutions. In section 5 we conclude and give an outline of
important future work.
## 2 Holographic setup
### 2.1 The finite temperature and hard wall backgrounds
The system we study is an Einstein-Yang-Mills theory on the (Poincaré patch
of) an asymptotically AdS5 geometry with an $SU(2)$ gauge field. The action is
$S=\int d^{5}x\sqrt{-g}~{}\left\\{\frac{1}{16\pi
G_{N}}\left(R+\frac{12}{L^{2}}\right)-\frac{1}{4\hat{g}^{2}}\mathrm{tr}\left(F_{\mu\nu}F^{\mu\nu}\right)\right\\}~{},$
(1)
where $\hat{g}$ is the Yang-Mills coupling, $G_{N}$ is the 5D gravitational
constant and $L$ is the AdS5 radius. $R$ and $F$ are the Ricci scalar and
Yang-Mills field strength respectively.
We consider the probe approximation, where the Yang-Mills term is small
compared to the Einstein-Hilbert term, so that the backreaction of the gauge
fields on the geometry can be neglected. We thus choose a fixed 5-dimensional
background metric, given by
$ds^{2}=\frac{L^{2}}{u^{2}}\left(-f(u)dt^{2}+dx^{2}+dy^{2}+dz^{2}+\frac{du^{2}}{f(u)}\right)~{},$
(2)
where the asymptotically AdS region is at $u\rightarrow 0$. We study two
different models. The first is a finite temperature model where the background
is AdS Schwarzschild, first proposed in Witten:1998qj . In this case,
$f(u)=1-\frac{u^{4}}{u_{H}^{4}}$, where $u_{H}$ is the location of the planar
black hole horizon. The Hawking temperature of the black hole is $T=1/\pi
u_{H}$. The second model is the hard wall cutoff model, proposed in
Erlich:2005qh ; DaRold:2005zs , where $f(u)=1$ and the geometry terminates at
a radial distance $u_{C}$. This model corresponds to a zero temperature theory
($u_{H}=\infty$), but it still has a scale $u_{C}$ which corresponds to a
confinement scale in the gauge theory. The intrinsic scales in these theories
allow us to form a dimensionless magnitude for the magnetic field. This will
be the parameter that we tune in order to find the instability of the
spatially invariant ground state. Without loss of generality we can choose
units where $u_{H}=1$ in the finite temperature theory and $u_{C}=1$ in the
confining theory. Factors of $u_{H}$ and $u_{C}$ can then be restored through
dimensional analysis. In the following the exact form of the metric is not
important until we come to solving the numerical equations in the radial
direction of AdS.
### 2.2 The Yang-Mills action
The relevant part of the action simplifies to
$S=-\frac{1}{4\hat{g}^{2}}\int
d^{5}x\sqrt{-g}~{}\mathrm{tr}\left(F_{\mu\nu}F^{\mu\nu}\right)~{},$ (3)
with the equations of motion
$\nabla^{\mu}F^{a}_{\mu\nu}+\epsilon^{abc}\mathcal{A}^{b\mu}F^{c}_{\mu\nu}=0~{}.$
(4)
The $SU(2)$ gauge field is
$\mathcal{A}=\mathcal{A}_{\mu}^{a}\tau^{a}dx^{\mu}$, for $a=1\dots 3$. We use
the convention where the Lie algebra basis is given by
$\tau^{a}=\frac{\sigma^{a}}{2i}$, with $\sigma^{a}$ the Pauli matrices, and
the structure constants $f^{abc}$ are defined by
$[\tau^{a},\tau^{b}]=\epsilon^{abc}\tau^{c}$ so that $f^{abc}=\epsilon^{abc}$.
With these definitions, the components of the field-strength tensor
$F=d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}$ become
$F^{a}_{\mu\nu}=\partial_{\mu}\mathcal{A}^{a}_{\nu}-\partial_{\nu}\mathcal{A}^{a}_{\mu}+\epsilon^{abc}\mathcal{A}^{b}_{\mu}\mathcal{A}^{c}_{\nu}~{}.$
(5)
It will be important to understand how gauge transformations affect the
system. Under a gauge transformation $e^{i\Lambda(x^{\mu})}$, $\mathcal{A}$
transforms as
$\mathcal{A}_{\mu}\rightarrow\mathcal{A}_{\mu}+\delta\mathcal{A}_{\mu}=e^{i\Lambda}\mathcal{A}_{\mu}e^{-i\Lambda}-i\partial_{\mu}e^{i\Lambda}e^{-i\Lambda}~{}.$
(6)
When $\Lambda(x^{\mu})$ is an infinitesimal transformation, this becomes
$\delta\mathcal{A}^{a}_{\mu}=\mathcal{D}_{\mu}\Lambda^{a}=\partial_{\mu}\Lambda^{a}+\epsilon^{abc}\mathcal{A}^{b}_{\mu}\Lambda^{c}~{}.$
(7)
The gauge transformations give us the freedom to fix the gauge
$\mathcal{A}^{a}_{u}=0$. We work in this gauge from now on.
In this paper we look at the effect of a strong (flavour-)magnetic field given
by $F^{3}_{xy}=B$, with all other components of $F^{a}_{\mu\nu}$ vanishing. As
we will see, when $B$ becomes large333Since we have chosen the units where
$u_{H}=1$ or $u_{C}=1$, $B$ is a dimensionless quantity. Restoring the units,
the statement is that $Bu_{H}^{2}=B/(\pi T)^{2}$ or $Bu_{C}^{2}\sim
B/\Lambda_{QCD}^{2}$ is large, or that $B$ is large compared to the radial
scale of the background., other components of $F$ become non-zero dynamically.
To get a consistent set of equations we therefore consider a gauge field
$\mathcal{A}$ of the form
$\mathcal{A}=\sum_{a=1,2,3,\mu=x,y}\mathcal{A}^{a}_{\mu}(x,y,u)\tau^{a}dx^{\mu}~{}.$
(8)
It turns out that we can turn off the $t$ and $z$ dependence of the gauge
field and still have consistent equations. This simplifies the equations.
Turning off the $t$ dependence guarantees a static solution. Turning off the
$z$ dependence, where the $z$ direction is parallel to the magnetic field,
yields a lattice in the $x,y$-plane.
The action 3 has an $SU(2)$ gauge freedom. Choosing the solution
$F^{3}_{xy}=B$, with all other components vanishing, breaks this symmetry.
Only $U(1)$ transformations of the form $\Lambda=\Lambda^{3}\tau^{3}$ leave it
invariant. For $B$ large enough, all the components in 8 become nonzero due to
the dynamics. We thus claim to have a superconductor, because the $U(1)$
symmetry is broken dynamically. Note however that it is technically a
superfluid because the $U(1)$ gauge symmetry in the bulk theory gets mapped to
a global symmetry in the field theory. Taking the linear combinations
$\mathcal{E}^{\pm}_{\mu}=\mathcal{A}^{1}_{\mu}\pm i\mathcal{A}^{2}_{\mu}$
gives fields that transform in the fundamental of the remaining gauge
symmetry. It can be checked from 7 that $\mathcal{E}^{\pm}_{\mu}\rightarrow\mp
i\Lambda^{3}\mathcal{E}^{\pm}_{\mu}$ whenever $\Lambda=\Lambda^{3}\tau^{3}$.
Later on we work only with the fields $\mathcal{E}^{+}$, which we rename to
$\mathcal{E}$.
### 2.3 Perturbative expansion of the gauge fields
Substituting the ansatz 8 into equation 4 yields nine coupled partial
differential equations in the variables $x$, $y$ and $u$. Of these nine
equations of motion, six are dynamical equations for each field
$\mathcal{A}^{1,2,3}_{x,y}$, and three equations are constraints. The
constraint equations arise from the equations of motion for the components
$\mathcal{A}^{1,2,3}_{u}$, which were chosen to be zero using gauge symmetry.
In solving the PDE’s, we follow the strategy of Abrikosov:1956sx ;
AbrikosovBook , which works as follows. When the magnetic field $B$ is smaller
than some critical value $B_{c}$, the field configuration
$\mathcal{A}^{3}_{y}=xB$, $\mathcal{A}^{3}_{x}=0$ and
$\mathcal{A}^{1,2}_{x,y}=0$ solves the equations of motion. This is the normal
phase of the superconductor. As shown in Ammon:2011je , the system enters a
new phase when the magnetic field is increased beyond some critical value
$B_{c}$. In this phase, the superconducting phase, the ground state has a non-
trivial profile for all fields in the ansatz equation 8. We look for this
configuration at some value of $B$ infinitesimally above $B_{c}$, where the
condensate is still small. This lets us do a perturbative expansion in a small
parameter $\varepsilon\sim\frac{B-B_{c}}{B_{c}}$. For notational convenience
we leave this parameter $\varepsilon$ explicit when studying the expansion.
However, it will be absorbed into the definition of the perturbative
corrections to the fields when we come to minimising the energy. We thus write
an ansatz for the expansion in the form
$\displaystyle\mathcal{A}^{3}_{y}$ $\displaystyle=xB_{c}+\varepsilon
A^{3}_{y}+\varepsilon^{2}a^{3}_{y}+\dots,$ (9)
$\displaystyle\mathcal{A}^{a}_{\mu}$ $\displaystyle=\varepsilon
A^{a}_{\mu}+\varepsilon^{2}a^{a}_{\mu}+\dots~{}~{}~{}~{}~{}\mathrm{for~{}}(a,\mu)\neq(3,y)~{},$
(10)
and solve the equations order by order in $\varepsilon$, as detailed in
section 3.
### 2.4 Gauge field boundary conditions
The holographic dictionary relates field theory operators to gravity theory
fields through the relation
$\displaystyle e^{-W_{\mathrm{CFT}}[\mathcal{A}^{(0)}]}=\langle
e^{\int_{\partial
AdS}\mathcal{A}^{(0)}_{\mu}J^{\mu}}\rangle=e^{-S_{\mathrm{on-shell}}}~{}.$
(11)
The minus sign on the right-hand side is because we are in Euclidean space for
simplicity. Here $\mathcal{A}^{(0)}$ is the value of the gauge field
$\mathcal{A}$ at the AdS boundary. It acts as a source in the boundary field
theory. In our setup, the only source we want in the field theory comes from
the component $\mathcal{A}^{3}_{y}=xB$, producing the magnetic field. For the
other components in 8, there should be no explicit source because we want to
model spontaneous symmetry breaking. The spontaneous symmetry breaking results
in a vev444We also need to take holographic renormalisation into account to
yield a finite on-shell action.,
$\displaystyle\langle J^{\mu}\rangle=\left.\frac{\delta
W_{\mathrm{CFT}}}{\delta\mathcal{A}^{(0)}_{\mu}}\right|_{\mathcal{A}^{(0)}_{\mu}=0}=\left.\frac{\delta
S_{\mathrm{on-
shell}}}{\delta\mathcal{A}^{(0)}_{\mu}}\right|_{\mathcal{A}^{(0)}_{\mu}=0}=\left.-\int
d^{4}x\frac{\partial\mathcal{L}}{\partial\left(\partial_{u}\mathcal{A}_{\mu}\right)}\right|_{u=0}$
(12)
The second equality is a generalisation to the radial coordinate of one of the
steps in deriving the Hamilton-Jacobi equation. It relates the variation of
final value of a generalised coordinate with respect to the on-shell action
and the conjugate momentum at the final time.
It is interesting to note that the on-shell action can be written as
$\displaystyle S_{\mathrm{on-shell}}=$
$\displaystyle-\frac{1}{2\hat{g}^{2}}\int_{\partial
AdS}d^{d}x\sqrt{-\gamma}n_{\mu}A^{a}_{\nu}F^{a\mu\nu}+\frac{1}{4\hat{g}^{2}}\int_{AdS}d^{d+1}x\sqrt{-g}\epsilon^{abc}A^{a}_{\mu}A^{b}_{\nu}F^{c\mu\nu}~{},$
(13)
where we integrated by parts and substituted in the equations of motion. The
second term on the right-hand side, the bulk term, is not present in non-
interacting theories. In our case, however, it is present and nonzero even
after using ansatz 8. This bulk term should seemingly influence the
calculation of the condensate when varying with respect to the boundary value.
It turns out that, due to the formula at the right of equality 12, it makes no
contribution.
Equations 11 and 12 imply that in an expansion of the gauge fields near the
AdS boundary, the leading term is the source and the subleading term is
proportional to the vev. The field $\mathcal{A}_{x,y}^{3}$ has a boundary
expansion given by
$\left.\mathcal{A}_{x,y}^{3}\right|_{u\rightarrow
0}=s_{x,y}^{(3)}+v_{x,y}^{(3)}u^{2}+\dots~{},$ (14)
where $s_{x,y}^{(3)}$ is the value of the source, which in this case is the
externally applied magnetic field potential. $v_{x,y}^{(3)}$ is proportional
to the vev corresponding to the magnetisation. We set the boundary conditions
so that the applied magnetic field is not corrected by the higher order
perturbations in $\varepsilon$, whereas the magnetisation will obtain a non-
zero value.
Similarly, the fields $A_{x,y}^{1,2}$ have a boundary expansion given by
$\left.{\mathcal{A}_{x,y}^{1,2}}\right|_{u\rightarrow
0}=s_{x,y}^{(1,2)}+v_{x,y}^{(1,2)}u^{2}+\dots~{},$ (15)
where $s^{(1,2)}_{x,y}$ corresponds to the source of the operator that will
condense to break the $U(1)$ symmetry. We adjust the boundary conditions in
such a way that this vanishes. This means that the symmetry breaking is
spontaneous. $v_{x,y}^{(1,2)}$ is proportional to the vacuum expectation value
of this operator, which we read off to find the resulting supercurrent in the
superconducting phase.
Boundary conditions are also imposed on the fields in the IR. In the case of
the black hole background, we impose regularity at the horizon and in the case
of the hard wall model we impose Neumann boundary conditions.
### 2.5 The gauge theory ground state energy
In finding the ground state, it is important to be able to calculate the
energy of the field theory solution from the action. We would like to compare
the solutions in the normal phase to those in the superconducting phase. The
energy $\mathcal{F}$ of the gauge theory solution is found by using the
holographic dictionary. In the case of the finite temperature solution, we are
in the canonical ensemble and we calculate the free energy, which is
$\mathcal{F}/T=-\ln\mathcal{Z}=-S_{cl}$ with our conventions. Here
$S_{cl}=-\frac{1}{4\hat{g}^{2}}\int d^{5}x\sqrt{-g}F^{a}_{\mu\nu}F^{a\mu\nu}$
is the classical action. In the hard wall case, we are simply calculating the
energy of the field configuration, which is defined in terms of the classical
action in the same way. Since we are only interested in whether the energy of
a particular superconducting solution is lower than that of the normal phase
solution, we can simply calculate the difference
$\Delta\mathcal{F}=\mathcal{F}_{s}-\mathcal{F}_{n}$ and thus do not need to
implement holographic renormalisation. Here $\mathcal{F}_{s}$ is the energy of
the superconducting phase, while $\mathcal{F}_{n}$ is the normal phase energy
with $\mathcal{A}^{3}_{y}=xB$ and all other components zero. We also need to
take care of the fact that $S_{cl}$ diverges when we perform the integral over
the Minkowski directions. This is easy to fix by considering the energy
density555We divide the free energy by $T$ in the finite temperature model to
get a dimensionless $\Omega$. This means that in both models, our total
dimensionless energy is simply -$S_{cl}$. $\Omega$, which is obtained by
integrating $S_{cl}$ only over the world volume of one lattice cell and
dividing by its volume. Having explained how to calculate the energy of a
field configuration, in the next section we turn to the problem of solving the
equations of motion to find the ground state.
## 3 Solving the equations
### 3.1 The comparison with Ginzburg-Landau theory
Before turning to the equations of motion, it helps to first look at the
Ginzburg-Landau equations for an analogy. In some suitable units defined in
Abrikosov:1956sx ; AbrikosovBook , they are
$\displaystyle\left(-i\nabla-\vec{A}\right)^{2}\psi-\psi+|\psi|^{2}\psi=0~{},$
(16)
$\displaystyle~{}\nabla\times\nabla\times\vec{A}=-i\left(\bar{\psi}\nabla\psi-\psi\nabla\bar{\psi}\right)-|\psi|^{2}\vec{A}~{}.$
(17)
Only the structure of these equations is important, so we have ignored
constant factors. Here $\psi$ is the wave function of Cooper pairs, and
$\vec{A}$ is the electromagnetic vector potential. The nine equations of
motion in our system can be split into two groups that roughly correspond to
the two equations above.
The first of the two groups, hereafter called the condensate equations,
contains the six equations for the fields $\mathcal{A}^{1,2}_{x,y,u}$. The
superconducting condensate of the dual field theory, which is like $\psi$
above, is found by differentiating the on-shell action with respect to the
boundary values of $\mathcal{A}^{1,2}_{x,y}$, as in equation 12. Of the six
equations in this group, the dynamical equations are for
$\mathcal{A}^{1,2}_{x,y}$ and the constraint666Recall that we have set
$\mathcal{A}^{a}_{u}=0$. However, its equations of motion still impose
constraints on the other fields. equations are for $\mathcal{A}^{1,2}_{u}$.
So this first group is analogous to equation 16. The analogy can be made more
clear. As mentioned above, we can make the field definitions
$\mathcal{E}_{x,y}=\mathcal{A}^{1}_{x,y}+i\mathcal{A}^{2}_{x,y}$. Doing so
allows us to combine the six real equations into three complex equations, two
dynamical and one constraint. The constraint equation relates
$\mathcal{E}_{x}$ and $\mathcal{E}_{y}$ such that there is only one complex
degree of freedom left, which is analogous to the state $\psi$. All this is
hard to see at the non-perturbative level, but it illustrates the strategy we
follow for solving the equations at each order: we use the constraint equation
to reduce the two dynamical equations into one, and then solve it.
The second group of equations, which we call the magnetic field equations, is
for the fields $\mathcal{A}^{3}_{x,y,u}$, corresponding to $\vec{A}$ in
equation 17 above. There are three such equations, one of which is a
constraint. At each order we will be able to use the constraint to separate
the equations into one for $\mathcal{A}^{3}_{x}$ and one for
$\mathcal{A}^{3}_{y}$.
### 3.2 The gauge field perturbative expansion in more detail
Having defined the ansatz for our gauge potential in equation 9 we can learn
more about the perturbative expansion by studying the non-linear structure of
the equations of motion. The equation for $\mathcal{A}^{3}_{u}$ is
$\displaystyle-\mathcal{A}^{2}_{x}\partial_{u}\mathcal{A}^{1}_{x}-\mathcal{A}^{2}_{y}\partial_{u}\mathcal{A}^{1}_{y}+\mathcal{A}^{1}_{x}\partial_{u}\mathcal{A}^{2}_{x}+\mathcal{A}^{1}_{y}\partial_{u}\mathcal{A}^{2}_{y}+\partial_{y}\partial_{u}\mathcal{A}^{3}_{y}+\partial_{x}\partial_{u}\mathcal{A}^{3}_{x}=0~{}.$
(18)
We see that the magnetic field components appear in the linear terms, while
the condensate components appear in quadratic terms. This suggests that a
contribution to the condensate components that is first order in the
perturbative expansion influences a second order contribution in the magnetic
field components. More generally, an odd order contribution to the condensate
components influences an even order contribution to the magnetic field
components.
This structure is common throughout all the equations of motion. It turns out
that terms in the perturbative expansion of the magnetic field components that
have an odd order vanish. The even order terms in the condensate components
can then also be set to zero. We can thus constrain the expansion ansatz of
equation 9 to
$\displaystyle\mathcal{E}_{x,y}$ $\displaystyle=\varepsilon
E_{x,y}+\varepsilon^{3}e_{x,y}+\mathcal{O}(\varepsilon^{5})~{},$
$\displaystyle\mathcal{A}^{3}_{y}$
$\displaystyle=xB_{c}+\varepsilon^{2}a^{3}_{y}+\mathcal{O}(\varepsilon^{4})~{},$
(19) $\displaystyle\mathcal{A}^{3}_{x}$
$\displaystyle=\varepsilon^{2}a^{3}_{x}+\mathcal{O}(\varepsilon^{4})~{}.$
Here the calligraphic letters denote the non-perturbative fields. $E_{x,y}$
and $e_{x,y}$ are first and third order contributions to the condensate
components, respectively, while $a^{3}_{x,y}$ are second order corrections to
$\mathcal{A}^{3}_{x,y}$.
Because of this convenient expansion of the fields, the condensate components
and the magnetic components decouple at each order. That means that at each
order, we only need to work with fields we have already solved at previous
orders. Our strategy is thus to solve for the fields in the following
sequence:
$\textstyle{{\mathcal{E}_{x,y}}=~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{\varepsilon
E_{x,y}}$$\textstyle{+}$$\textstyle{\varepsilon^{3}e_{x,y}}$$\textstyle{+~{}~{}\mathcal{O}(\varepsilon^{5})~{},}$$\textstyle{{\begin{array}[]{c}\mathcal{A}^{3}_{y}\\\
{}\hfil\\\ \mathcal{A}^{3}_{x}\end{array}}{\begin{array}[]{c}=\\\ {}\hfil\\\
=\end{array}}{\begin{array}[]{c}xB_{c}\\\ {}\hfil\\\
{}\hfil\end{array}}}$$\textstyle{\begin{array}[]{c}+\\\ {}\hfil\\\
{}\hfil\end{array}}$$\textstyle{\begin{array}[]{c}\varepsilon^{2}a^{3}_{y}\\\
{}\hfil\\\ \varepsilon^{2}a^{3}_{x}\end{array}}$
$\textstyle{{\begin{array}[]{c}+~{}~{}\mathcal{O}(\varepsilon^{4})~{},\\\
{}\hfil\\\ +~{}~{}\mathcal{O}(\varepsilon^{4})~{}.\end{array}}}$
In the next section we start with the linear order solution, which will shed
more light on the procedure that must be implemented at higher orders.
### 3.3 Solving the equations to linear order
Using the expansion 19 and keeping terms to linear order, we find that there
are six remaining equations given (in complex form) by
$\displaystyle 0$
$\displaystyle=-iB_{c}x\partial_{u}E_{y}-\partial_{y}\partial_{u}E_{y}-\partial_{x}\partial_{u}E_{x}~{},$
(20) $\displaystyle 0$
$\displaystyle=B_{c}^{2}x^{2}E_{x}-iB_{c}E_{y}+\left(\frac{f}{u}-f^{\prime}\right)\partial_{u}E_{x}-f\partial_{u}^{2}E_{x}-2iB_{c}x\partial_{y}E_{x}$
$\displaystyle~{}-\partial_{y}^{2}E_{x}+iB_{c}x\partial_{x}E_{y}+\partial_{x}\partial_{y}E_{y}~{},$
(21) $\displaystyle 0$
$\displaystyle=2iB_{c}E_{x}+\left(\frac{f}{u}-f^{\prime}\right)\partial_{u}E_{y}-f\partial_{u}^{2}E_{y}+iB_{c}x\partial_{x}E_{x}+\partial_{x}\partial_{y}E_{x}-\partial_{x}^{2}E_{y}~{}.$
(22)
Here, as above, $f(u)=1-u^{4}$ for the AdS Schwarzschild model and $f(u)=1$
for the hard wall model.
We can solve these equations by following Abrikosov AbrikosovBook . The
solution is given by
$\displaystyle E_{y}$ $\displaystyle=-iE_{x}~{},$ (23) $\displaystyle E_{x}$
$\displaystyle=\sum_{n=-\infty}^{\infty}C_{n}e^{-inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}U(u)~{}.$
(24)
$U(u)$ is determined by solving
$U^{\prime\prime}+\left(\frac{f^{\prime}(u)}{f(u)}-\frac{1}{u}\right)U^{\prime}+\frac{B_{c}}{f(u)}U=0~{},$
(25)
subject to the constraints $U(0)=0$ and $U^{\prime}(1)=0$. For the AdS
Schwarzschild model ($f(u)=1-u^{4}$), the latter constraint comes from
imposing regularity at the horizon. It is possible to calculate $B_{c}$ by
numerically finding the value at which $U(u)$ satisfies these constraints.
There is an infinite tower of solutions to $B_{c}$, but we are only interested
in the lowest one, which is where the phase transition occurs. For further
details on solving this equation in the AdS Schwarzschild model, see
Ammon:2011je . For the hard wall model, $f(u)=1$ so the equation simplifies to
the extent that it can be solved analytically, the solution being a Bessel
function. Qualitatively the solutions for $U(u)$ in both models look very
similar and we are only interested in their numerical form. For the AdS
Schwarzschild model, we get $B_{c}\approx 5.1$, while we get $B_{c}\approx
5.8$777This is the zero of the Bessel function of the first kind
$J_{0}(\sqrt{B})$. for the hard wall model.
It should be noted that the solution 24 for $E_{x}$ agrees precisely (except
for the factor of $U(u)$) with the linear order solution for the order
parameter close to the upper critical magnetic field $H_{c2}$ in the theory of
type II superconductors, as seen in AbrikosovBook . It is also the result
found by Chernodub et al. in Chernodub:2011gs . Depending on the values of the
parameters $C_{n}$ and $k$ (to be determined by the higher order equations in
the perturbative expansion), $E_{x}$ corresponds to different inhomogeneous
functions in the $x,y$-plane. We are particularly interested in finding those
with lattice symmetries that represent evenly spaced vortices running in the
$z$ direction in the gauge theory.
### 3.4 The Abrikosov lattice solution
Before going beyond linear order, we discuss the possible solutions we can
expect. The number of coefficients specifying a configuration can make the
problem of finding the lowest energy solution unmanageable without making use
of some symmetries. We can argue that, since nothing in the setup is
explicitly breaking translational invariance in the $x,y$-directions, the
solution should be a highly symmetric lattice. A nice review of how lattices
can be formed from the Abrikosov solution (24) is given in RosensteinLi2010 .
There the authors explain that in order for $|E_{x}|$ to be a lattice
solution, the coefficients $C_{n}$ must have the same magnitude $|C_{n}|$ and
moreover be periodic in some integer $P$, that is, $C_{n}=C_{n+P}$.
In Abrikosov:1956sx , Abrikosov first studied the simplest solution, a square
lattice. In this case, $P=1$, implying that $C_{n}=C$ for all $n$, and
$k=\sqrt{2\pi B_{c}}$. Later Kleiner et al. in Kleiner:1964 generalised the
analysis by looking at $P=2$, with $C_{1}=\pm iC_{0}=\pm iC$. This choice of
coefficients specifies a general rhombic lattice, with the shape of the
rhombus controlled by varying $k$. In particular, a square lattice can be
obtained by choosing $k=\sqrt{\pi B_{c}}$. This square lattice is the same as
Abrikosov’s solution with $P=1$, but it is rotated by $\pi/4$ and translated.
A triangular lattice is obtained by choosing $k=3^{\frac{1}{4}}\sqrt{\pi
B_{c}}$.
To show how this works, we first substitute $P=2$ and $C_{1}=iC_{0}=iC$ into
the solution for $E_{x}$, which simplifies to
$\displaystyle E_{x}$
$\displaystyle=C\sum_{n=-\infty}^{\infty}e^{i\frac{\pi}{2}n^{2}-inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}U(u)~{}.$
(26)
It is then easy to see the symmetries
$|E_{x}(x+[m+\frac{1}{2}q]L_{x},y+[n+\frac{1}{2}q]L_{y})|=|E_{x}(x,y)|$ for
integers $m$, $n$ and $q$. $L_{x}$ and $L_{y}$ are the lengths of the lattice
cell in the $x$ and $y$ directions, and are given by $L_{x}=2k/B_{c}$ and
$L_{y}=2\pi/k$. See figure 1.
Figure 1: A lattice cell, illustrating the meanings of $L_{x}$ and $L_{y}$ for
a fixed area cell.
We follow the approach of Kleiner et al, which is to compute the energy
density of the lattice for a range of values of the ratio
$L_{x}/L_{y}=k^{2}/\pi B_{c}$. This essentially means that we vary $k$. The
energy is computed numerically from the analytic expressions we obtain at each
order in the following sections. What we find agrees with their result that
the triangular lattice has the lowest energy of the $P\leq 2$ solutions. When
doing this, magnetic flux conservation is an important constraint. The total
applied magnetic field per unit area is constant, and each lattice cell
corresponds to a vortex with a single quantum of magnetic flux. This means
that when comparing the energy of different lattices, we should make sure that
they have the same magnetic flux per unit area, which in turn means that their
lattice cells have the same area. Fortunately with this ansatz that is always
the case since the area $L_{x}L_{y}=4\pi/B_{c}$ is independent of $k$.
In the following sections we calculate analytic expressions for the higher
order corrections to the gauge field. We keep $P$ and the coefficients $C_{n}$
general, except for imposing the periodicity condition $C_{n}=C_{n+P}$.
### 3.5 Higher order contributions to the energy
In order to find the ground state solution we must calculate the energy of the
superconducting solutions and compare them to the normal phase solution. We
can study the form of the energy as defined in section 2.5 to see how far we
must go in the perturbative expansion of the gauge fields. The energy has
terms that are quadratic and quartic in the gauge potential. The quartic term
ensures that the energy is bounded below, because it has a positive
coefficient. The quartic terms have lowest perturbative contributions of order
$\varepsilon^{4}$. One might expect contributions of order $\varepsilon^{3}$
coming from the zeroth order magnetic field contribution multiplied by three
first order corrections. However, from equation 19 it can be shown that such
terms do not arise. Thus we should expect to expand to third order in
$\mathcal{A}_{x,y}^{1,2}$ and fourth order in $\mathcal{A}_{x,y}^{3}$ .
However, it turns out that going to fourth order is not necessary because
inserting ansatz 19 into the action of equation 3, we find that the only
fourth order terms from $\mathcal{A}_{x,y}^{3}$ that appear at the fourth
order of the action are proportional to
$\sim\partial_{y}a^{(4)3}_{x}-\partial_{x}a^{(4)3}_{y}$. Here $a^{(4)3}_{x}$
and $a^{(4)3}_{y}$ are the fourth order corrections to $\mathcal{A}^{3}_{x}$
and $\mathcal{A}^{3}_{y}$, respectively. This term respects the lattice
symmetries, thus on performing the integration over the lattice cell to get
the free energy density, it vanishes by Stokes’ theorem.
We saw above that the parameters $k$ and $C_{n}$ in the solution 24 are not
fixed by the equations of motion to linear order. This is due to the fact that
to linear order, the different vortices do not interact. We can therefore not
expect to fix any of the coefficients $C_{n}$ or the spacing parameter $k$ at
this order. In fact, trying to calculate $\Delta\Omega$ to this order, which
has no quartic terms in $\mathcal{A}$, one finds that the free energy density
is not bounded below; increasing the overall magnitude of the condensate
always decreases $\Delta\Omega$. To see which configuration, that is, which
set of values for $C_{n}$ and $k$, is energetically favourable, we clearly
have to go beyond linear order.
### 3.6 Solving the equations to higher orders
In this section we solve the equations of motion up to third order in the
perturbation parameter.
The second order corrections to the gauge fields contribute to the potentials
$\mathcal{A}^{3}_{x}$ and $\mathcal{A}^{3}_{y}$, that is, $a^{3}_{x}$ and
$a^{3}_{y}$ in 19. These fields source the external magnetic field and the
magnetisation. We impose that these corrections must vanish at the AdS
boundary, so that the dual field theory has a constant applied magnetic field.
We find however that they do not vanish throughout the bulk. In particular
they develop non-vanishing subleading terms in the boundary expansion,
representing a magnetisation in the field theory.
In appendix A we explain how the equations for the Fourier modes of the fields
$a^{3}_{x}$ and $a^{3}_{y}$ can be decoupled. This yields the following
equations
$\displaystyle
u\partial_{u}\left(\frac{f}{u}\partial_{u}\hat{a}^{3}_{x,y}(m,n,u)\right)-\left(k^{2}n^{2}+\frac{4B_{c}^{2}m^{2}\pi^{2}}{k^{2}P^{2}}\right)\hat{a}^{3}_{x,y}(m,n,u)$
$\displaystyle+T_{x,y}e^{-\frac{k^{2}n^{2}}{4B_{c}}+\frac{inm\pi}{P}-\frac{B_{c}m^{2}\pi^{2}}{k^{2}P^{2}}}\left(\sum_{l=0}^{P-1}e^{\frac{2ilm\pi}{P}}\bar{C}_{l}C_{l+n}\right)U^{2}=0~{},$
(27)
where
$T_{x}=-i\frac{\sqrt{B_{c}\pi}}{P}n~{},~{}~{}~{}T_{y}=2i\frac{\pi^{3/2}B_{c}^{3/2}}{k^{2}P^{2}}m~{},$
(28)
and
$a^{3}_{x,y}(x,y,u)=\sum_{m}\sum_{n}e^{-i\frac{2\pi
mB_{c}}{Pk}x-inky}~{}\hat{a}^{3}_{x,y}(m,n,u)~{}.$ (29)
As before, $P$ defines the periodicity in the $C_{n}$. The parameters $m$ and
$n$ correspond to the Fourier space levels of these fields. In order to
calculate the solution $a_{x,y}^{3}(x,y,u)$ we will in theory need to solve
these equations for all values of $m$ and $n$. However, it will turn out to be
sufficient to only study the first few Fourier modes. The numerical procedure
for solving these will be explained in section 3.7
At third order we are studying the perturbative corrections to the condensate.
Here we calculate the corrections $e_{x}$ and $e_{y}$. It is reasonable to
assume that the answer is of the form
$\displaystyle\varepsilon E_{x}+\varepsilon^{3}e_{x}$
$\displaystyle=\varepsilon\sum_{n=-\infty}^{\infty}\left(C_{n}U(u)+\varepsilon^{2}c_{x,n}(u)\right)e^{-inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}~{},$
(30) $\displaystyle\varepsilon E_{y}+\varepsilon^{3}e_{y}$
$\displaystyle=\varepsilon\sum_{n=-\infty}^{\infty}\left(-iC_{n}U(u)+\varepsilon^{2}c_{y,n}(u)\right)e^{-inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}~{},$
(31)
where we have made use of equation 23 to relate the first order terms
$C_{n}U(u)$ in $\mathcal{E}_{x}$ and $\mathcal{E}_{y}$. $c_{(x,y),n}(u)$ is
the first perturbative correction to the condensate where the $u$ dependence
is a function of $n$ in contrast to the first order term.
We can write $e_{x}$ and $e_{y}$ in Fourier space, then use the three
condensate equations discussed in section 3 to calculate these corrections.
The one constraint equation can be used to decouple the other two equations.
We then have one equation for $c_{x,n}(u)$ and one for $c_{y,n}(u)$. Further
details are provided in appendix B.
### 3.7 Numerical solutions
Having separated the equations into ordinary differential equations in $u$ by
the method outlined in the appendices, we can now solve them numerically. Both
the second and third order equations take the same general form, given by
$\displaystyle
u\partial_{u}\left(\frac{f}{u}\partial_{u}\phi\right)+G(m,n)\phi+H(m,n,u)=0~{}.$
(32)
This equation can be solved numerically by picking some parameters for $C_{n}$
and $k$ that give a particular lattice and then using a shooting method to
integrate from $u=1$ (the horizon/hard wall cutoff) to $u=0$ (the AdS
boundary). It is an inhomogeneous second order differential equation, so there
are two integration constants. The first is fixed by imposing regularity at
the horizon or Neumann boundary conditions at the hard wall cutoff. This fixes
the value of $\partial_{u}\phi(1)$. The second constant is obtained by
demanding that $\phi(0)=0$, so that the fields vanish at the AdS boundary.
This vanishing corresponds to both the magnetic field strength corrections and
the source for the condensate being set to zero. We fulfil this boundary
condition by adjusting $\phi(1)$. Unlike in the case of the first order
equations, the equations here are not homogeneous and thus the source sets a
scale with which the value $\phi(1)$ can be compared. Changing $\phi(1)$ in
this case thus acts as more than just a scaling for the solution and so is
used as the tuning parameter to satisfy the UV constraint.
For all of the equations, we can implement this procedure for arbitrary
integers $m$ and $n$, corresponding to the different Fourier modes of the
gauge fields. This will then give a Fourier coefficient
$\hat{a}^{3}_{x,y}(m,n,u)$ that can be used to determine $a^{3}_{x,y}(x,y,u)$.
Fortunately we do not have to do the calculation for many different values of
$m$ and $n$, because as the values get large, the source term gets suppressed
exponentially. This can be seen in equation 27 for the second order terms and
is true also for the third order equation. For a vanishing source, the
equations for $\hat{a}^{3}_{x}$ or $\hat{a}^{3}_{y}$ have only the trivial
solution. This means that $\hat{a}^{3}_{x,y}(m,n,u)$ is negligibly small for
large $m$ or $n$, and we can therefore truncate the Fourier series for
$a^{3}_{x,y}$ beyond $m,n\approx 3$.
## 4 Results
### 4.1 Finding the minimum energy state
As explained above, we wish to find the values of the parameters $k$, $P$ and
$C_{n}=C_{n+P}$ that give the minimum energy state. These parameters define
the shape of the lattice. Our analysis is only valid for $B$ slightly above
$B_{c}$, where $B_{c}$ was determined in section 3.3. The first step is thus
to pick a value for $B$ in this vicinity. We then choose a set of lattice
parameters that give us the lattice solution we wish to consider. As mentioned
in RosensteinLi2010 , for lattice solutions all the $C_{n}$ must have the same
magnitude $C$. We can therefore fix $C_{n}$ up to the normalisation $C$, along
with a value of $k$, according to the discussion in section 3.4. We then
substitute these values into the energy density that was defined in section
2.5. It takes the form $\Delta\Omega=a_{1}\varepsilon
C+a_{2}\varepsilon^{2}C^{2}+\dots~{}$. At this point we see that we can
redefine $C$ by absorbing a factor of $\varepsilon$, which we call
$C_{\varepsilon}$. $C_{\varepsilon}$ is the only parameter left unfixed up to
this point in the analysis. Here the $a_{i}$ are values that are calculated
numerically from substituting the solutions to the equations of motion into
the expression for the energy derived in appendix C. $\Delta\Omega$ forms a
Mexican hat potential, which is easy to minimise numerically. An illustration
of this procedure is shown in figure 2.
Figure 2: The change in energy density in units of temperature as a function
of $C_{\varepsilon}$, the overall condensate scale. The leftmost curve
corresponds to $B=B_{c}$, which is never negative for nonzero condensate.
Curves for $B<B_{c}$ are similar. Increasing $B$ beyond $B_{c}$ yields the
curves to the right, and we see the formation of a clear minimum of the energy
that is lower than the energy of the normal phase. The dashed line traces out
the minimum of each of these curves, which corresponds to the energetically
preferred size of the condensate as a function of $B$. This plot was generated
in the AdS Schwarzschild model for $P=2$ and $k=3^{\frac{1}{4}}\sqrt{\pi B}$,
corresponding to a triangular lattice. $B$ takes the values $B\approx
B_{c},1.04B_{c},1.07B_{c},1.1B_{c}$ from left to right. Changing $P$ and $k$
to correspond to different lattices or using the hard wall model yields
qualitatively similar results.
The plot in figure 3 shows the energy-minimising value of $C_{\varepsilon}$ as
a function of magnetic field near the phase transition at $B_{c}$. It shows
that $C_{\varepsilon}\sim(B-B_{c})^{\frac{1}{2}}$, so the condensate888Note
that only the combination $\varepsilon C$ is physically relevant, not $C$ or
$\varepsilon$ independently. has a critical exponent of $1/2$. A fit to the
numerical data for the triangular lattice gives that
$C_{\varepsilon}=0.58(B-B_{c})^{\frac{1}{2}}$ in the AdS Schwarzschild model
and $C_{\varepsilon}=0.53(B-B_{c})^{\frac{1}{2}}$ in the hard wall model.
Figure 3: $C_{\varepsilon}\sim$ the overall condensate size for the AdS
Schwarzschild solution in units of the temperature, as a function of the
external magnetic field $B$. For $B<B_{c}$, the condensate is zero, and for
$B$ slightly above $B_{c}$, we see a $(B-B_{c})^{\frac{1}{2}}$ scaling
behaviour. This plot was generated for $P=2$ and $k=3^{\frac{1}{4}}\sqrt{\pi
B}$, corresponding to a triangular lattice. The plot for different lattices in
both the AdS Schwarzschild and hard wall models is the same, up to a scaling
of the $B$ and $C_{\varepsilon}$ axes. For the triangular lattice, the AdS
Schwarzschild model has scaling behaviour
$C_{\varepsilon}=0.58(B-5.1)^{\frac{1}{2}}$ and the hard wall model has
$C_{\varepsilon}=0.53(B-5.8)^{\frac{1}{2}}$.
Having minimised with respect to $C_{\varepsilon}$ for a given value of $B$
and a given lattice configuration, we can plot the difference in the energy
between the normal and superconducting states. Figure 4 shows $\Delta\Omega$,
the difference between the energy density in the superconducting and normal
phases, as a function of external magnetic field for two different lattices.
The first lattice is square, and the second is triangular. Both are described
in section 3.4.
Figure 4: The change in energy density (compared to the normal phase) for the
triangular and square lattices as the external applied magnetic field is
varied. The phase transition happens at $B_{c}\approx 5.1$, which is where the
coordinate axes are centred.
$\Delta\Omega_{\text{square}}-\Delta\Omega_{\text{triangle}}$ is so small that
the two plots are almost on top of each other. This is for the AdS
Schwarzschild model, but the plots for the hard wall model are identical
except for the scale on the axes. In the hard wall model, $B_{c}\approx 5.8$.
The curves in figure 4 are the result of calculations in the AdS Schwarzschild
model, but we get the same results up to a rescaling of the axes for the hard
wall model. In the AdS Schwarzschild model, the critical magnetic field
$B_{c}\approx 5.1$, while in the hard wall model $B_{c}\approx 5.8$. Each
curve shows that the free energy density is proportional to
$\left(B-B_{c}\right)^{2}$. This shows that the phase transition is second
order, as expected if one looks at the analogous case in Ginzburg-Landau
theory. There one can show (TinkhamBook ) that the free energy is proportional
to $\left(T-T_{c}\right)^{2}$, where $T_{c}$ is the phase transition critical
temperature.
### 4.2 An analysis of $P=2$ solutions
We now specialise to the case where the periodicity of the $C_{n}$ is $P=2$.
This describes a general rhombic lattice solution which includes both the
triangular and square lattices. The $P=1$ square lattice can be found within
the $P=2$ solutions up to translation and rotation. We here perform the
analysis done in Kleiner:1964 as described at the end of section 3.4.
The energy difference as a function of $R$ is plotted in figure 5. By looking
at the form of equation 26, it is possible to see that the triangular lattice
occurs for $R=L_{x}/L_{y}=\sqrt{3}$ and $R=1/\sqrt{3}$. In general, $R$ and
$1/R$ give the same lattice but with the $x$ and $y$ directions flipped. This
is why figure 5 displays the symmetry $\Delta\Omega(R)=\Delta\Omega(1/R)$. The
triangular lattice corresponds to a global minimum of the energy as a function
of $R$, as seen from the figure. There is a local maximum for the square
lattice, which is when $R=1$. As $R\rightarrow\infty$ (or $R\rightarrow 0$),
the free energy increases. Intuitively one can understand this by making use
of the properties of Abrikosov vortices that we understand from type II
superconductors. These vortices repel. Since $R\rightarrow\infty$ and
$R\rightarrow 0$ correspond to elongating the rhombic lattice cell (while
keeping the area constant) neighbouring vortices are squeezed together, and
since they repel, this is energetically unfavourable.
Figure 5: The change in free energy density as a function of $R=L_{x}/L_{y}$,
the ratio of side lengths of a constant area lattice cell. This plot is for
the AdS Schwarzschild model, but the plot for the hard wall model is identical
up to a rescaling of the axes. When $R=1$, the lattice is square and the free
energy achieves a local maximum. When $R=\sqrt{3}$ and $1/{\sqrt{3}}$, the
lattice is triangular and the free energy is at a global minimum. Note that
the plot has the symmetry $\Delta\Omega(R)=\Delta\Omega(1/R)$, which simply
corresponds to swapping the $x,y$-axes.
We can calculate the condensate in the minimum energy state using equation 12.
The result, to linear order in $\varepsilon$, is
$\langle J^{+}_{x}\rangle\equiv\frac{\delta S_{\mathrm{on-shell}}}{\delta
E^{(0)}_{x}}=\frac{L}{2\hat{g}^{2}}U_{\mathrm{sub}}C_{\varepsilon}\sum_{n=-\infty}^{\infty}e^{-i\frac{\pi}{2}n^{2}+inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}$
(33)
The AdS radius can be related to field theory quantities through the relation
$L^{4}=2\lambda{\alpha^{\prime}}^{2}$, where $\lambda$ is the ’t Hooft
coupling and $\alpha^{\prime}$ the string tension. The factor
$U_{\mathrm{sub}}$ is equal to the subleading term in the boundary expansion
of $U(u)$. Using equation 25 it is possible to show that
$U_{\mathrm{sub}}=B_{c}\int_{0}^{u_{H}}\frac{U(u)}{u}du~{},$ (34)
so it can be determined numerically. In figure 6 we present the contour plot
of
$3^{\frac{1}{4}}\sqrt{8}\frac{\hat{g}^{4}}{L^{2}U_{\mathrm{sub}}^{2}C_{\varepsilon}^{2}}\left|\langle
J^{+}_{x}\rangle\right|^{2}$, the modulus squared of the condensate in the
$x,y$-plane for the minimum energy solution corresponding to the triangular
lattice. The factors are chosen so that the maximum value is 1. Substituting
in the numerical values, we find that the maximum value the condensate takes
is $|\langle
J^{+}_{x}\rangle|=1.0\frac{L}{\hat{g}^{2}}\left(B-B_{c}\right)^{\frac{1}{2}}$
for the AdS Schwarzschild model, where $B_{c}\approx 5.1$, and $|\langle
J^{+}_{x}\rangle|=1.3\frac{L}{\hat{g}^{2}}\left(B-B_{c}\right)^{\frac{1}{2}}$
for the hard wall model, where $B_{c}\approx 5.8$.
Figure 6: A contour plot of
$3^{\frac{1}{4}}\sqrt{8}\frac{\hat{g}^{4}}{L^{2}U_{\mathrm{sub}}^{2}C_{\varepsilon}^{2}}\left|\langle
J^{+}_{x}\rangle\right|^{2}$, the modulus squared of the field theory
condensate dual to $E_{x}$ in the ground state triangular lattice. At the
center of the dark vortices, the condensate vanishes.
We could also plot the magnetisation of the ground state, which is found from
the normalisable term in the boundary value expansion of
$\partial_{x}a^{3}_{y}-\partial_{y}a^{3}_{x}$. However, it takes the same form
as the condensate and the numerics indicate that it differs only up to a
scale.
## 5 Conclusion
In this work we have found a likely ground state for the black hole Yang-Mills
instability analysed in Ammon:2011je . The solution, being of a lattice form,
clearly has much potential for analysis in condensed matter models, where the
breaking of translational invariance has already been shown to be very
important in getting realistic phenomenology. As we have explained, it also
has possible implications for heavy ion collider physics. There are a number
of interesting areas where we could apply similar techniques and perform
further calculations to elucidate the phenomenology of the ground state found
here.
It is certainly important to understand exactly how universal this result is.
We have seen that the difference between the normal phase and superconducting
phase energy density of this solution is, up to a scale, independent of the
background geometry in the two models that we have studied here. It would be
very useful to understand precisely where this universality stems from and
find how much we can deform the gravity solutions until the Abrikosov lattice
is no longer the ground state.
In the present work we have analysed lattices with $P=1$ and $P=2$,
corresponding to square and rhombic forms, respectively. Going to $P=3$
requires a large increase in computational power. While this would be an
interesting further calculation, the analogous cases of type II
superconductivity and the model of Chernodub et al point to the triangular
lattice being the true ground state. We thus expect higher $P$ lattices to be
energetically disfavoured.
Having found this solution, there are some extensions that can be made. It
will now be possible to study time dependent fluctuations about the ground
state. In order to do this we would have to introduce a second perturbative
parameter in addition to the parameter $\varepsilon$ used in the current work.
This would be analogous to the parameter $\alpha^{\prime}$ in a D-brane
construction. This would allow us to study the transport properties of the
lattice ground state, by looking at current-current correlation functions. If
we wish to study the effect of the lattice on the shear viscosity to entropy
density ratio, we would have to introduce gravitational back reaction in our
model. Clearly this will be a much more involved calculation, with many non-
linear couplings, but if we want to study the theory in a more realistic
scenario, where the stress energy tensor also has a lattice structure, such a
calculation would clearly be important.
It is expected that if the QCD vacuum is unstable to $\rho$ meson condensation
in extremely off-centre heavy ion collisions, then the timescale of the
instability would not be enough to form a well-defined lattice. Abrikosov
vortices may form, but the magnetic field would likely drop below the critical
value before they had time to arrange themselves into a lattice. It would be
very interesting to perform a real-time calculation in order study the
formation of the vortices and their movements as the magnetic field increased
and decreased through the lifetime of a single off-centre collision.
## Acknowledgements
Y.Y.B. was supported by MPG-CAS Doctoral Promotion Programme. The work of J.S.
was supported by the European Union through a Marie Curie Fellowship. The work
of J.E. and M.S. was supported in part by the DFG cluster of excellence
‘Origin and Structure of the Universe’ (www.universe-cluster.de). We thank
Maxim Chernodub for useful discussions.
## Appendix A Deriving the equations for $a^{3}_{x,y}$
We substitute the ansatz 19 into the full equations of motion and neglect
terms beyond quadratic order in $\varepsilon$. To get rid of all appearances
of $E_{y}$, we use the relation that $E_{y}=-iE_{x}$ from 23. Then we find
that there are only three equations in which the fluctuations $a^{3}_{x,y}$
appear. We focus on those three equations.
The simplest equation of the three is the constraint equation, which came from
the equation of motion for $A^{3}_{u}$. To quadratic order, this equation is
simply
$\partial_{u}\partial_{x}a^{3}_{x}+\partial_{u}\partial_{y}a^{3}_{y}=0~{}.$
(35)
The first thing to do is integrate by $u$. This gives an integration constant,
but by the fact that both $a^{3}_{x}$ and $a^{3}_{y}$ must vanish at $u=0$,
this integration constant vanishes. So the even simpler constraint is
$\partial_{x}a^{3}_{x}+\partial_{y}a^{3}_{y}=0~{}.$ (36)
This is all we need to decouple the other two equations in $a^{3}_{x}$ and
$a^{3}_{y}$. These equations now become
$\displaystyle 0=$
$\displaystyle\frac{3}{2}E_{x}\partial_{y}\bar{E}_{x}+\frac{3}{2}\bar{E}_{x}\partial_{y}E_{x}-\frac{1}{2}iE_{x}\partial_{x}\bar{E}_{x}+\frac{1}{2}i\bar{E}_{x}\partial_{x}E_{x}$
$\displaystyle+u\partial_{u}\left(\frac{f}{u}\partial_{u}a^{3}_{x}\right)+\partial^{2}_{y}a^{3}_{x}+\partial^{2}_{x}a^{3}_{x}~{},$
(37) $\displaystyle 0=$ $\displaystyle-
B_{c}x\bar{E}_{x}E_{x}-\frac{1}{2}iE_{x}\partial_{y}\bar{E}_{x}+\frac{1}{2}i\bar{E}_{x}\partial_{y}E_{x}$
$\displaystyle-\frac{3}{2}E_{x}\partial_{x}\bar{E}_{x}-\frac{3}{2}\bar{E}_{x}\partial_{x}E_{x}+u\partial_{u}\left(\frac{f}{u}\partial_{u}a^{3}_{y}\right)+\partial^{2}_{y}a^{3}_{y}+\partial^{2}_{x}a^{3}_{y}~{},$
(38)
which are partial differential equations with sources that come from the
linear order solutions. These two equations only differ by their source terms,
so we will focus on $a^{3}_{x}$. $a^{3}_{y}$ should be similar. Using the
expression 24, we can see that the source term is periodic with $y\sim
y+\frac{2\pi}{k}$. $a^{3}_{x}$ must have the same periodicity, so we can write
it as a Fourier series,
$\displaystyle
a^{3}_{x}(x,y,u)=\sum_{n=-\infty}^{\infty}e^{-inky}\tilde{a}^{3}_{x}(x,n,u)~{}.$
(39)
The equation becomes
$\displaystyle\sum_{m}-ie^{-\frac{1}{2}B_{c}\left(-\frac{km}{B_{c}}+x\right)^{2}-\frac{1}{2}B_{c}\left(-\frac{k(n+m)}{B_{c}}+x\right)^{2}}kn\bar{C}_{m}C_{n+m}U^{2}$
$\displaystyle-k^{2}n^{2}\tilde{a}^{3}_{x}+u\partial_{u}\left(\frac{f}{u}\partial_{u}\tilde{a}^{3}_{x}\right)+\partial^{2}_{x}\tilde{a}^{3}_{x}$
$\displaystyle=0~{}.$ (40)
We notice that the source term in this equation is periodic in the
$x$-direction; $x\sim x+\frac{Pk}{B_{c}}$. This lets us expand
$\tilde{a}^{3}_{x}$ as a Fourier series in $x$ as well:
$\displaystyle\tilde{a}^{3}_{x}=\sum_{m}e^{-i\frac{2\pi
mB_{c}}{Pk}x}\hat{a}^{3}_{x}(m,n,u)~{}.$ (41)
Writing the source term as a series lets us then obtain the equation 27 for
the coefficients $\hat{a}^{3}_{x}(m,n,u)$.
Calling the source term $S(x)$, the naïve way of finding its Fourier
coefficients is to use the formula
$\displaystyle\tilde{S}_{n}=\frac{B_{c}}{Pk}\int_{0}^{\frac{Pk}{B_{c}}}e^{i\frac{2\pi
nB_{c}}{Pk}x}S(x)~{}.$ (42)
However, the source terms contains Gaussians, and those are much easier to
integrate when the domain of integration is the entire real line. So we do the
following trick. Doing a continuous Fourier transform on a periodic function
gives a sum of $\delta$-functions,
$\displaystyle\int dx~{}e^{ipx}S(x)$ $\displaystyle=\int
dx~{}e^{ipx}\sum_{m}e^{-i\frac{2\pi mB_{c}}{Pk}x}\tilde{S}_{n}$
$\displaystyle=2\pi\sum_{m}\tilde{S}_{n}\delta\left(p-\frac{2\pi
mB_{c}}{Pk}\right)~{}.$ (43)
The coefficients in front of the $\delta$-functions are what we are looking
for. We get
$\displaystyle\int dx~{}e^{ipx}S(x)$
$\displaystyle=-\sqrt{\frac{\pi}{B_{c}}}\sum_{m,n}ie^{-\frac{k^{2}n^{2}}{4B_{c}}+\frac{ikmp}{B_{c}}+\frac{iknp}{2B_{c}}-\frac{p^{2}}{4B_{c}}}kn\bar{C}_{m}C_{m+n}U^{2}~{}.$
(44)
Using
$\displaystyle\sum_{m=-\infty}^{\infty}f(m)=\sum_{m=-\infty}^{\infty}\sum_{l=0}^{P-1}f(Pm+l)$
(45)
and then using the symmetry $C_{i+P}=C_{i}$, the only $m$-dependence remaining
in the sum comes from $e^{\frac{ikPmp}{B_{c}}}$. Making use of the identity
$\sum_{m=-\infty}^{\infty}e^{imq}=2\pi\sum_{m=-\infty}^{\infty}\delta(q-2\pi
m)$ (46)
and $\delta(\alpha x)=\frac{\delta(x)}{|\alpha|}$ gives us the sum over
$\delta$-functions from 43. Then we can simply read off the coefficients
$\tilde{S}_{n}$. This gives us the equation 27.
## Appendix B Deriving the equations for $c_{x,n}$, $c_{y,n}$
The third order equations of motion are
$\displaystyle 0$
$\displaystyle=ia^{3}_{x}\partial_{u}E_{x}+a^{3}_{y}\partial_{u}E_{x}-iE_{x}\partial_{u}a^{3}_{x}-E_{x}\partial_{u}a^{3}_{y}+iB_{c}x\partial_{u}e_{y}+\partial_{y}\partial_{u}e_{y}+\partial_{x}\partial_{u}e_{x}~{},$
(47) $\displaystyle 0$
$\displaystyle=-iB_{c}xa^{3}_{x}E_{x}-2B_{c}xa^{3}_{y}E_{x}-\bar{E}_{x}E_{x}^{2}-a^{3}_{x}\partial_{y}E_{x}+2ia^{3}_{y}\partial_{y}E_{x}-a^{3}_{y}\partial_{x}E_{x}$
$\displaystyle~{}-2E_{x}\partial_{y}a^{3}_{x}+iE_{x}\partial_{y}a^{3}_{y}+E_{x}\partial_{x}a^{3}_{y}+iB_{c}e_{y}-iB_{c}x\partial_{x}e_{y}-\partial_{x}\partial_{y}e_{y}$
$\displaystyle~{}-B_{c}^{2}x^{2}e_{x}+2iB_{c}x\partial_{y}e_{x}+\partial^{2}_{y}e_{x}+u\partial_{u}\left(\frac{f}{u}\partial_{u}e_{x}\right)$
(48) $\displaystyle 0$
$\displaystyle=B_{c}xa^{3}_{x}E_{x}+i\bar{E}_{x}E_{x}^{2}-ia^{3}_{x}\partial_{y}E_{x}+2a^{3}_{x}\partial_{x}E_{x}-ia^{3}_{y}\partial_{x}E_{x}$
$\displaystyle~{}+iE_{x}\partial_{y}a^{3}_{x}+E_{x}\partial_{x}a^{3}_{x}-2iE_{x}\partial_{x}a^{3}_{y}-2iB_{c}e_{x}-iB_{c}x\partial_{x}e_{x}-\partial_{x}\partial_{y}e_{x}+\partial^{2}_{x}e_{y}+u\partial_{u}\left(\frac{f}{u}\partial_{u}e_{y}\right)~{}.$
(49)
The first of these is the constraint equation. We use it to relate $e_{x}$ and
$e_{y}$. In order to do this, we first simplify it by noticing that, since
$\displaystyle e_{y}$
$\displaystyle=\sum_{n=-\infty}^{\infty}c_{y,n}(u)e^{-inky-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}~{},$
(50)
we have that
$iB_{c}x\partial_{u}e_{y}+\partial_{y}\partial_{u}e_{y}=-i\partial_{x}\partial_{u}e_{y}$.
We can then integrate the entire equation by $u$, imposing vanishing boundary
conditions at the AdS boundary. The constraint equation then simplifies to
$\displaystyle 0$
$\displaystyle=-2i\frac{E_{x}}{U}J_{x}-2\frac{E_{x}}{U}J_{y}+ia^{3}_{x}E_{x}+a^{3}_{y}E_{x}+\partial_{x}e_{x}-i\partial_{x}e_{y}~{},$
(51)
where
$\displaystyle
J_{x,y}(x,y,u)=\int_{0}^{u}U(\tilde{u})\partial_{\tilde{u}}a^{3}_{x,y}(x,y,\tilde{u})d\tilde{u}~{}.$
(52)
This allows us to eliminate $e_{x}$ in equation 49 (after differentiating it
by $x$). We write each function as a Fourier series in $y$ and find an
equation for the coefficients $c_{y,n}$. At this point the equation still has
an $x$ dependence, which can be eliminated by multiplying the equation by
$(nk-B_{c}x)$ to make it an even function in $x$ and then integrating
$\int_{-\infty}^{\infty}dx$. In doing so we use the solution for $E_{x}$ and
the form for $e_{y}$ given by 50, as well as the Fourier series representation
of the other functions. Once this is done, we are left with an equation for
$e_{y}$ in the form 32.
The resulting equation for $c_{y,n}$ is
$\displaystyle 0$
$\displaystyle=\sum_{q,r=-\infty}^{\infty}\left\\{e^{-\frac{2\pi
q\left(ik^{2}P(n-r)+B_{c}\pi
q\right)}{k^{2}P^{2}}}\left[\frac{C_{n-r}\left(-2\left(k^{2}Pr+2iB_{c}\pi
q\right)\hat{J}_{x,qr}\right)}{kP}\right.\right.$
$\displaystyle\left.+\frac{C_{n-r}\left(\left(2ik^{2}Pr-4B_{c}\pi
q\right)\hat{J}_{y,qr}+\left(2iB_{c}\pi
q\hat{a}^{3}_{x}+\left(-ik^{2}Pr+4B_{c}\pi
q\right)\hat{a}^{3}_{y}\right)U\right)}{kP}\right]$
$\displaystyle\left.-\frac{ie^{-\frac{k^{2}\left(3r^{2}-3rq+q^{2}\right)}{3B_{c}}}\left(3B_{c}+2k^{2}q(-2r+q)\right)\bar{C}_{n+q}C_{n+r}C_{n-r+q}U^{3}}{3\sqrt{3}B_{c}}\right\\}$
$\displaystyle-
B_{c}c_{y,n}+u\partial_{u}\left(\frac{f}{u}\partial_{u}c_{y,n}\right)~{},$
(53)
where
$\displaystyle\hat{J}_{i,qr}(u)$
$\displaystyle=\int_{0}^{u}U(\tilde{u})\partial_{\tilde{u}}\hat{a}^{3}_{i}(q,r,\tilde{u})d\tilde{u}~{},$
(54)
for $i=x,y$.
A similar procedure gives the constraint equation in terms of the
coefficients,
$\displaystyle 0$ $\displaystyle=c_{x,n}(u)-ic_{y,n}(u)$
$\displaystyle~{}~{}~{}+\frac{1}{PkB_{c}}\sum_{q,r=-\infty}^{\infty}\Biggl{\\{}e^{-\frac{2\pi
q\left(ik^{2}P(n-r)+B_{c}\pi q\right)}{k^{2}P^{2}}}\left(-ik^{2}Pr+2B_{c}\pi
q\right)C_{n-r}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(2\hat{J}_{x,qr}-2i\hat{J}_{y,qr}-(\hat{a}^{3}_{x,qr}-i\hat{a}^{3}_{y,qr})U(u)\right)\Biggr{\\}}~{}.$
(55)
Once the coefficients $c_{y,n}$ are found, we use this to calculate $c_{x,n}$.
## Appendix C Calculating the energy
The difference between the energy of the superconducting phase and that of the
normal phase is
$\displaystyle\Delta\mathcal{F}$ $\displaystyle=\frac{1}{4\hat{g}^{2}}\int
d^{5}x\sqrt{-g}\left(\left.F^{a}_{\mu\nu}F^{a\mu\nu}\right|_{superconducting}-\left.F^{a}_{\mu\nu}F^{a\mu\nu}\right|_{normal}\right)~{}.$
(56)
Note that for the AdS Schwarzschild model we implicitly divided by the
temperature to make the energy dimensionless. We calculate the energy density
by averaging over the domain $0\leq y<\frac{2\pi}{k}$, $0\leq
x<\frac{Pk}{B_{c}}$, $0\leq u\leq 1$ and $t,z\in\mathbb{R}$. Since the
integrand is independent of $t$ and $z$, the averaging amounts to simply
dropping the integration over those variables. In the following expression we
use
$\displaystyle\mathcal{E}_{x,y}=\mathcal{A}^{1}_{x,y}+i\mathcal{A}^{2}_{x,y}=\sum_{n}\mathcal{C}_{(x,y),n}(u)e^{-ikny-\frac{1}{2}B_{c}\left(x-\frac{nk}{B_{c}}\right)^{2}}~{},$
(57)
we write $\mathcal{A}^{3}_{x}=a^{3}_{x}$ and
$\mathcal{A}^{3}_{y}=xB+a^{3}_{y}$, and call the averaged energy
$\Delta\Omega$. The result is
$\displaystyle 4\hat{g}^{2}\Delta\Omega$ $\displaystyle=\int
du\left\\{\Omega_{1}(u)+\sum_{m,n=-\infty}^{\infty}\left[\Omega_{2}(m,n,u)+\Omega_{3}(m,n,u)+\Omega_{4}(m,n,u)\right]\right.$
$\displaystyle~{}\left.\sum_{m,n,p,q=-\infty}^{\infty}\Omega_{5}(m,n,q,r,u)\right\\}~{},$
(58)
where
$\displaystyle\Omega_{1}=$ $\displaystyle\frac{\sqrt{\pi
B}}{kPu}\sum_{l=0}^{P-1}\frac{B}{2}\left(\sum_{j=x,y}\left(f\partial_{u}\bar{\mathcal{C}}_{j,l}\partial_{u}\mathcal{C}_{j,l}+\bar{\mathcal{C}}_{j,l}\mathcal{C}_{j,l}\right)+3(i\bar{\mathcal{C}}_{y,l}\mathcal{C}_{x,l}-i\bar{\mathcal{C}}_{x,l}\mathcal{C}_{y,l})\right)~{},$
(59) $\displaystyle\Omega_{2}=$
$\displaystyle\frac{1}{u}\left\|kn\hat{a}^{3}_{x}(m,n,u)-\frac{2Bm\pi}{kP}\hat{a}^{3}_{y}(m,n,u)\right\|^{2}+\frac{f}{u}\sum_{j=x,y}\left\|\partial_{u}\hat{a}^{3}_{j}(m,n,u)\right\|^{2}~{},$
(60) $\displaystyle\Omega_{3}=$ $\displaystyle\frac{\sqrt{\pi
B}}{2k^{2}P^{2}u}\sum_{l=0}^{P-1}e^{-\frac{k^{2}m^{2}}{4B}-\frac{i(2l+m)n\pi}{P}-\frac{Bn^{2}\pi^{2}}{k^{2}P^{2}}}\Bigl{(}\left(3k^{2}mP+2iBn\pi\right)\hat{a}^{3}_{x}(n,m,u)\bar{\mathcal{C}}_{x,l+m}\mathcal{C}_{y,l}$
$\displaystyle+\hat{a}^{3}_{x}(n,-m,u)\bar{\mathcal{C}}_{y,l}\left(\left(3k^{2}mP+2iBn\pi\right)\mathcal{C}_{x,l+m}-2ik^{2}mP\mathcal{C}_{y,l+m}\right)$
$\displaystyle+\hat{a}^{3}_{y}(n,-m,u)\mathcal{C}_{x,l+m}\left(-4iBn\pi\bar{\mathcal{C}}_{x,l}+\left(ik^{2}mP+6Bn\pi\right)\bar{\mathcal{C}}_{y,l}\right)$
$\displaystyle+\hat{a}^{3}_{y}(n,m,u)\bar{\mathcal{C}}_{x,l+m}\mathcal{C}_{y,l}\left(-ik^{2}mP-6Bn\pi\right)\Bigr{)}~{},$
(61) $\displaystyle\Omega_{4}=$ $\displaystyle-\frac{1}{4kPu}\sqrt{\frac{\pi
B}{2}}e^{-\frac{k^{2}\left(m^{2}+n^{2}\right)}{2B}}\times$
$\displaystyle\sum_{l=0}^{P-1}\left(\bar{\mathcal{C}}_{y,l+m}\bar{\mathcal{C}}_{y,l+n}\mathcal{C}_{x,l}\mathcal{C}_{x,l+m+n}-2\bar{\mathcal{C}}_{x,l+m}\bar{\mathcal{C}}_{y,l+n}\mathcal{C}_{x,l+m+n}\mathcal{C}_{y,l}+\bar{\mathcal{C}}_{x,l}\bar{\mathcal{C}}_{x,l+m+n}\mathcal{C}_{y,l+m}\mathcal{C}_{y,l+n}\right)~{},$
(62) $\displaystyle\Omega_{5}=$ $\displaystyle\frac{\sqrt{\pi
B}}{Pku}\sum_{l=0}^{P-1}e^{-\frac{k^{2}m^{2}}{4B}-\frac{i(2l+m)n\pi}{P}-\frac{Bn^{2}\pi^{2}}{k^{2}P^{2}}}\times$
$\displaystyle\left(\hat{a}^{3}_{y}(n-q,-(m+r),u)\hat{a}^{3}_{y}(q,r,u)\bar{\mathcal{C}}_{x,l}\mathcal{C}_{x,l+m}-\hat{a}^{3}_{x}(n-q,r,u)\hat{a}^{3}_{y}(q,m-r,u)\bar{\mathcal{C}}_{x,l+m}\mathcal{C}_{y,l}\right.$
$\displaystyle\left.-\hat{a}^{3}_{x}(n-q,r,u)\hat{a}^{3}_{y}(q,-(m+r),u)\bar{\mathcal{C}}_{y,l}\mathcal{C}_{x,l+m}+\hat{a}^{3}_{x}(n-q,-(m+r),u)\hat{a}^{3}_{x}(q,r,u)\bar{\mathcal{C}}_{y,l}\mathcal{C}_{y,l+m}\right)~{}.$
(63)
In these expressions, $\mathcal{C}_{x,n}$ and $\mathcal{C}_{y,n}$ are
functions of $u$. Their complex conjugates are given by
$\bar{\mathcal{C}}_{x,n}$ and $\bar{\mathcal{C}}_{y,n}$, respectively. All the
infinite sums in the energy C can be terminated at a small finite value
because of exponential suppression in the $\Omega_{1\dots 5}$ terms.
In deriving this, it helps to make use of the formulae
$\displaystyle\int_{0}^{L}dx$
$\displaystyle\sum_{m=-\infty}^{\infty}e^{-\frac{B_{c}}{2}\left(x-mL\right)^{2}}=\int_{-\infty}^{\infty}dxe^{-\frac{B_{c}}{2}x^{2}}~{},$
(64) $\displaystyle\int_{0}^{L}dx$
$\displaystyle\sum_{m,n=-\infty}^{\infty}e^{-\frac{B_{c}}{2}\left(x-\frac{mL}{P}\right)^{2}-\frac{B_{c}}{2}\left(x-\frac{nL}{P}\right)^{2}}h(x,m,n)$
(65)
$\displaystyle=\int_{-\infty}^{\infty}dx\sum_{l=0}^{P-1}\sum_{m=-\infty}^{\infty}e^{-\frac{B_{c}}{2}\left(x-\frac{mL}{P}\right)^{2}-\frac{B_{c}}{2}\left(x-\frac{lL}{P}\right)^{2}}h(x,m,l)~{},$
(66)
where the latter is valid whenever $h(x,m,n)=h(x+L,m+P,n+P)$.
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|
arxiv-papers
| 2012-10-24T20:07:31 |
2024-09-04T02:49:37.046334
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan-Yan Bu, Johanna Erdmenger, Jonathan P. Shock, Migael Strydom",
"submitter": "Migael Strydom",
"url": "https://arxiv.org/abs/1210.6669"
}
|
1210.6676
|
# Radio Detection of the Fermi LAT Blind Search Millisecond Pulsar
J1311$-$3430
P. S. Ray11affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375-5352, USA 22affiliation: email: Paul.Ray@nrl.navy.mil ,
S. M. Ransom33affiliation: National Radio Astronomy Observatory (NRAO),
Charlottesville, VA 22903, USA , C. C. Cheung44affiliation: National Research
Council Research Associate, National Academy of Sciences, Washington, DC
20001, resident at Naval Research Laboratory, Washington, DC 20375, USA , M.
Giroletti55affiliation: INAF Istituto di Radioastronomia, 40129 Bologna, Italy
, I. Cognard66affiliation: Laboratoire de Physique et Chimie de
l’Environnement, LPCE UMR 6115 CNRS, F-45071 Orléans Cedex 02, and Station de
radioastronomie de Nançay, Observatoire de Paris, CNRS/INSU, F-18330 Nançay,
France , F. Camilo77affiliation: Columbia Astrophysics Laboratory, Columbia
University, New York, NY 10027, USA 1818affiliation: Arecibo Observatory, HC3
Box 53995, Arecibo, PR 00612, USA , B. Bhattacharyya88affiliation: Inter-
University Centre for Astronomy and Astrophysics, Pune 411 007, India , J.
Roy99affiliation: National Centre for Radio Astrophysics, Tata Institute of
Fundamental Research, Pune 411 007, India , R. W. Romani1010affiliation: W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA , E. C.
Ferrara1111affiliation: NASA Goddard Space Flight Center, Greenbelt, MD 20771,
USA , L. Guillemot1212affiliation: Max-Planck-Institut für Radioastronomie,
Auf dem Hügel 69, 53121 Bonn, Germany , S. Johnston1313affiliation: CSIRO
Astronomy and Space Science, Australia Telescope National Facility, Epping NSW
1710, Australia , M. Keith1313affiliation: CSIRO Astronomy and Space Science,
Australia Telescope National Facility, Epping NSW 1710, Australia , M.
Kerr1010affiliation: W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA , M. Kramer1414affiliation: Jodrell Bank Centre for Astrophysics, School
of Physics and Astronomy, The University of Manchester, M13 9PL, UK
1212affiliation: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany , H. J. Pletsch1515affiliation: Albert-Einstein-Institut,
Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany
1616affiliation: Leibniz Universität Hannover, D-30167 Hannover, Germany , P.
M. Saz Parkinson1717affiliation: Santa Cruz Institute for Particle Physics,
Department of Physics and Department of Astronomy and Astrophysics, University
of California at Santa Cruz, Santa Cruz, CA 95064, USA , K. S.
Wood11affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375-5352, USA
###### Abstract
We report the detection of radio emission from PSR J1311$-$3430, the first
millisecond pulsar discovered in a blind search of Fermi Large Area Telescope
(LAT) gamma-ray data. We detected radio pulsations at 2 GHz, visible for
$<$10% of $\sim$4.5 hours of observations using the Green Bank Telescope
(GBT). Observations at 5 GHz with the GBT and at several lower frequencies
with Parkes, Nançay, and the Giant Metrewave Radio Telescope resulted in non-
detections. We also report the faint detection of a steep spectrum continuum
radio source (0.1 mJy at 5 GHz) in interferometric imaging observations with
the Jansky Very Large Array. These detections demonstrate that PSR
J1311$-$3430 is not radio quiet, and provides additional evidence that radio-
quiet millisecond pulsars are rare. The radio dispersion measure of 37.8 pc
cm-3 provides a distance estimate of 1.4 kpc for the system, yielding a gamma-
ray efficiency of 30%, typical of LAT-detected MSPs. We see apparent excess
delay in the radio pulses as the pulsar appears from eclipse and we speculate
on possible mechanisms for the non-detections of the pulse at other orbital
phases and observing frequencies.
###### Subject headings:
pulsars: individual (PSR J1311$-$3430)
## 1\. Introduction
The Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope has been
surveying the gamma-ray sky in the 20 MeV to 300 GeV band since August 2008.
The LAT survey has vastly increased the population of known gamma-ray emitting
sources including pulsars, blazars, supernova remnants, pulsar wind nebulae,
and more. However, in the second LAT source catalog (Nolan et al., 2012,
hereafter 2FGL) 575 (31%) of the 1873 sources were not associated with any
known counterpart. The properties of the unassociated source population have
been studied (Ackermann et al., 2012) and they roughly fall into two
categories of ‘pulsar-like’ (non-variable with spectra that exhibit
significant curvature) and ‘blazar-like’ (highly variable, spectra without
strong cutoffs).
Beyond studying their gamma-ray emission properties, the most promising way to
make progress on understanding and identifying the unassociated sources is
through multi-wavelength observations. As a striking example, deep searches
for radio pulsars in the pulsar-like population have led to the discovery of
43 new millisecond pulsars (MSPs) and 4 young and middle-aged pulsars (Ray et
al., 2012). Of course, since all of these pulsars were discovered in radio
searches, these discoveries do not probe the possible radio-quiet MSP
population, which must be discovered in other wavelengths.
The first strong candidate radio-quiet MSP was found through X-ray and optical
studies of the bright unassociated source 2FGL J2339.6$-$0532 (Romani & Shaw,
2011; Kong et al., 2012). The orbital modulation of the optical emission
suggests the system comprises a “black widow”-type MSP (King et al., 2005) in
an 4.6-hr orbit around a low-mass companion. Very recently, radio pulsations
from this source were reported (Ray et al., 2012), confirming that it is
indeed an MSP whose wind is evaporating its companion.
More recently, Romani (2012) identified another such system associated with
2FGL J1311.7$-$3429 (hereafter J1311; see also Kataoka et al. (2012)). In this
case, the optical modulation revealed an orbital period of 1.56 hours, the
shortest of any known MSP, with a companion that is very hydrogen poor. Using
the optical source position and information about the orbital parameters from
the optical measurements, Pletsch et al. (2012) discovered the first gamma-ray
MSP found in a blind search, PSR J1311$-$3430, with period $P=2.56$ ms. This
discovery confirmed the black widow pulsar nature of the system and yielded
important parameters such as the spin-down luminosity ($\dot{E}=4.9\times
10^{34}$ erg s-1) of the pulsar and the precise orbital parameters, including
the projected semi-major axis of the orbit (10.6 lt-ms). As the pulsar was
discovered in the gamma-ray band, it was a candidate for being the first
radio-quiet MSP. Analysis of archival 820 MHz and 350 MHz radio observations
with the Robert C. Byrd Green Bank Telescope (GBT) resulted in no detections
(Pletsch et al., 2012). However, as Romani (2012) pointed out, the low
frequency radio emission could be scattered or absorbed and thus be
undetectable from Earth, even if the pulsar radio beam was directed at Earth.
Because scattering and absorption of the radio emission are strong functions
of frequency, this motivates further radio observations at higher frequencies.
In this Letter, we report the detection of radio emission from PSR
J1311$-$3430, both in continuum imaging observations with the Karl G. Jansky
Very Large Array (VLA) and in a radio pulsation search with the GBT at 2 GHz.
This confirms that PSR J1311$-$3430 is indeed a radio MSP with the beam
sweeping across Earth, completing the chain from unassociated gamma-ray source
to optical identification to gamma-ray pulsation detection to radio detection.
## 2\. Observations and Results
### 2.1. VLA Imaging Observations
We observed the J1311 field on 2011 January 14 with the VLA (Perley et al.,
2011) while in the C-array (program S2270). Observations were made at 1.4 and
5 GHz using two 128 MHz wide intermediate frequencies centered at 1.327 +
1.455 GHz and 4.895 + 5.023 GHz, respectively. In the 1-hr observing run, we
obtained two 9-min target scans per frequency bracketed with scans of a phase
calibrator (J1316$-$3338), while the primary flux calibrator was 3C 286.
An analysis of the 5 GHz data revealed an unresolved (ratio of the peak to
integrated flux density $\sim 1$; Condon (1997)) $0.10\pm 0.04$ mJy point
source at a position (J2000.0), R.A. = 13h11m45s.78 ($\pm 1.0\arcsec$) and
Decl. = $-34^{\circ}30^{\prime}31.2^{\prime\prime}$ ($\pm 1.7\arcsec$),
consistent with the pulsar (i.e. the pulsar is within the VLA synthesized
beam). In the 1.4 GHz image, there is a $0.99\pm 0.44$ mJy source (peak =
$0.33\pm 0.11$ mJy/beam) that shows evidence of extension, with a centroid
that is slightly offset ($\sim 13^{\prime\prime}$) from the 5 GHz position
(Figure 1). Imaging the two scans of J1311 at each frequency separately (see
Figure 2 for the corresponding orbital phases) did not reveal any
statistically significant flux changes.
Figure 1.— VLA radio contours at 5 GHz (white; 80 and 113 $\mu$Jy/beam) and
1.4 GHz (green; 250 and 354 $\mu$Jy/beam) overlaid onto a Chandra $0.5-7$ keV
image of the region showing the X-ray point source, CXOU J131145.7$-$343030
from Cheung et al. (2012), found coincident with the pulsar. The synthesized
beams are $16.2\arcsec\times 3.5\arcsec$ (position angle, $PA=-26.1\deg$) and
$49.1\arcsec\times 14.7\arcsec$ ($PA=-22.9\deg$), respectively. Figure 2.—
Figure showing the orbit of the pulsar with the phase range of the 2 GHz radio
pulsation detection marked in gray. The VLA observation intervals centered at
phases 0.51 and 0.87 (5 GHz; cyan) and 0.64 and 1.00 (1.4 GHz; red) are
marked. Orbital phase is measured as a fraction of the orbit from the
ascending node (when the pulsar crosses the plane of the sky heading away);
orbital phases of 0.0, 0.25, 0.5, and 0.75 are noted. Superior conjunction
(e.g. mid-eclipse time) is also shown.
### 2.2. Radio Pulsar Observations
After the discovery of the gamma-ray pulsations by Pletsch et al. (2012), we
performed a series of radio observations to search for radio pulsations from
the pulsar. Given the non-detections at 820 and 350 MHz (Ransom et al., 2011;
Pletsch et al., 2012), and the 1.4 and 5 GHz VLA detections (§2.1), the
priority was on higher frequency observations to minimize the effects of
scattering, dispersion, and absorption. For completeness, we include the
earlier GBT observations in our analysis. The observations performed are
listed in Table 1. All observations are made with 2 summed polarizations. In
all cases, we compute the pulsed flux density limits using the modified
radiometer equation as described by Ray et al. (2011). We assume a pulse duty
cycle of 0.1 and a signal to noise threshold for detection of 5$\sigma$.
#### 2.2.1 Green Bank Telescope
We observed the J1311 field with the GBT (project GBT/12A-487) with the GUPPI
backend (DuPlain et al., 2008) on 2012 July 28 for 1.4 hours at 2 GHz and 1.3
hours at 5 GHz with a bandwidth of 800 MHz and 40.96 $\mu$s sampling. At 2 GHz
we used 2048 frequency channels, while at 5 GHz we used 1024 channels. We
obtained a second GBT observation on 2012 August 19 spanning 3 hours at 2 GHz,
with the same observing setup. At the start of each observation, we observed
the pulsar B1257+12, which was detected each time.
We analyzed the data using PRESTO (Ransom, 2001) to first excise strong radio
frequency interference (RFI) signals and then fold the data using an ephemeris
based on the parameters of the gamma-ray pulsation discovery. The RFI
environment during these observations was rather benign, requiring blanking of
some narrow band signals amounting to $<7$% of the data. The only free
parameter in the search was dispersion measure (DM). In our first 2 GHz
observation, we detected radio pulsations during an $\sim 1100$-s interval
beginning at MJD (UTC) 56136.908 with a DM of $37.84\pm 0.26$ pc cm-3 (see
Figure 3). The significance of the detection over the 1100 seconds where the
pulse is visible, after correcting for the 6400 trials in the DM search, is
10.3 $\sigma$. This significance is computed from the profile using the
$\chi^{2}$ test for excess variance. It differs from the value shown in Figure
3 because it is based on a subset of the data that exclude the first 600
seconds where no signal is apparent. Based on the detection significance and
the sensitivity of our observations, we estimate a mean flux density of $60\pm
30$ $\mu$Jy. No pulsations were seen in the 5 GHz observation or the second 2
GHz observation.
As seen in Figure 3, the signal significance peaks at a frequency slightly
offset ($\Delta\nu\sim 5\times 10^{-3}$ Hz) from that predicted by the gamma-
ray ephemeris. The orbital phase at the start of the interval in Figure 3 is
0.24, when the pulsar is expected to be at mid-eclipse, so this is apparently
caused by excess delay in the radio pulse as the pulsar is emerging from
eclipse, as is seen in many eclipsing MSPs. To test this, we folded only the
data starting after 600 seconds and the detection significance peaks at the
predicted period and period derivative. In addition, we generated 15 times of
arrival (TOAs) across the observation, with 114 seconds of integration per
TOA. We discarded the first three with no apparent pulsed detection. Looking
at only the last 10 TOAs, the measured period is consistent with that
predicted from the gamma-ray ephemeris. The two earlier TOAs appear to show a
changing delay that is about 0.6 ms (1/4 of a pulse period) just as the pulse
becomes visible.
Figure 3.— Radio detection of PSR J1311$-$3430 using the GBT at 2 GHz. This
shows a span of 1711 seconds folded using the ephemeris from Pletsch et al.
(2012). The pulsar is very weak during the first 600 seconds of this interval
and appears to show some excess delay as it comes out of eclipse. When the
first 600 seconds are excluded from the analysis, the detection significance
is above 10 $\sigma$. The fact that the detection is strongest in a frequency
band of $\sim 150$ MHz and weak in the rest of the band is consistent with
interstellar scintillation.
#### 2.2.2 Parkes
We made a total of 7 observations at 1.4 GHz using the Parkes Telescope with
the Analog Filter Bank (AFB) backend recording 256 MHz of bandwidth (0.5 MHz
channels sampled every 125 $\mu$s) and analyzed them in the same manner as
described above. No pulsations were detected in any of the observations.
#### 2.2.3 Nançay
We performed 5 observations with the Nançay radio telescope (NRT) around 1.5
GHz ranging from 700 to 3000 s. We used a 512-MHz bandwidth divided into 1024
channels with 64-$\mu$s sampling. Dedispersion at a DM of 37.84 pc cm-3 and
folding using the known parameters did not produce any detectable pulsations.
The offsets from the gamma-ray pulsation position are negligible when compared
to the beam size of the telescope.
#### 2.2.4 Giant Metrewave Radio Telescope
We analyzed two archival 607-MHz (32 MHz bandwidth) observations made using
the Giant Metrewave Radio Telescope (GMRT) incoherent array in early 2011 as
part of a pulsar search of LAT unassociated sources. Folding these data with
the timing model from the LAT discovery resulted in no detections. Later with
the precise pulsar position, we made sensitive coherent array observations at
322 MHz, also with 32 MHz bandwidth (Roy et al., 2010). Again, no pulsations
were detected.
Table 1Radio Pulsar Observations Obs Code | Date | $t_{\mathrm{obs}}$ | Orb Phase | $T_{\mathrm{sys}}$aaSystem temperature including receiver temperature ($T_{\mathrm{rec}}$) and sky temperature ($T_{\mathrm{sky}}$) from scaling the 408 MHz Haslam map to the observing frequency and adding 2.7 K for the cold sky background. | $S_{\mathrm{min}}$bbFlux density limits are computed for a standard integration time of 1000 s at the observing frequency (corresponding to the duration where pulses were seen in the one detection), with the equivalent limit at 1400 MHz in parentheses, assuming a spectral index of $-1.8$. The bold value is the detected flux density in this observation, scaled to 1.4 GHz.
---|---|---|---|---|---
| | (s) | | (K) | ($\mu$Jy)
GBT-820 | 2011 Jan 15 | 2577 | 0.092 – 0.550 | 38 | 53 (22)
GBT-820 | 2011 Dec 10 | 1723 | 0.346 – 0.652 | 38 | 53 (22)
GBT-820 | 2012 Aug 24 | 900 | 0.295 – 0.455 | 38 | 53 (22)
GBT-820 | 2012 Aug 25 | 900 | 0.977 – 0.137 | 38 | 53 (22)
GBT-350 | 2012 Feb 24 | 1287 | 0.339 – 0.568 | 104 | 203 (22)
GBT-S | 2012 Jul 28 | 4939 | 0.859 – 0.737 | 26 | 114 $\pm$ 60
GBT-S | 2012 Aug 19 | 11697 | 2 orbits | 26 | 20 (36)
GBT-S | 2012 Aug 25 | 900 | 0.346 – 0.506 | 26 | 20 (36)
GBT-C | 2012 Jul 28 | 4552 | 0.931 – 0.740 | 22 | 16 (115)
GMRT-607 | 2011 Feb 15 | 3386 | 0.108 – 0.710 | 108 | 445 (117)
GMRT-607 | 2011 Jun 23 | 7246 | 0.349 – 0.637 | 108 | 445 (117)
GMRT-322 | 2012 Aug 15 | 3607 | 0.844 – 0.485 | 137 | 184 (18)
Parkes-AFB | 2012 Mar 22 | 3600 | 0.606 – 0.246 | 29 | 115 (114)
Parkes-AFB | 2012 Jul 11 | 3600 | 0.337 – 0.977 | 29 | 115 (114)
Parkes-AFB | 2012 Jul 12 | 3600 | 0.025 – 0.665 | 29 | 115 (114)
Parkes-AFB | 2012 Jul 16 | 3600 | 0.092 – 0.732 | 29 | 115 (114)
Parkes-AFB | 2012 Aug 17 | 1635 | 0.403 – 0.694 | 29 | 115 (114)
Parkes-AFB | 2012 Aug 17 | 5357 | 0.766 – 0.718 | 29 | 115 (114)
Parkes-AFB | 2012 Aug 24 | 4500 | 0.130 – 0.930 | 29 | 115 (114)
NRT-L | 2012 Jul 16 | 2512 | 0.182–0.628 | 39 | 48 (54)
NRT-L | 2012 Jul 21 | 2997 | 0.563–0.096 | 39 | 48 (54)
NRT-L | 2012 Aug 23 | 736 | 0.368–0.499 | 39 | 48 (54)
NRT-L | 2012 Sep 1 | 2996 | 0.788–0.320 | 39 | 48 (54)
NRT-L | 2012 Sep 24 | 1257 | 0.094–0.317 | 39 | 48 (54)
### 2.3. Radio/Gamma-Ray Profiles
Figure 4.— Phase-aligned radio and gamma-ray profiles. The gamma-ray profiles
are weighted counts with energies $>100$ MeV and the radio profile is from the
GBT detection of pulsations at 2 GHz. The dotted lines are the same profile
phase shifted forward and back by 0.1 in phase, comparable to the DM
uncertainty.
We computed phase-aligned radio and gamma-ray profiles, shown in Figure 4. The
gamma-ray profiles are computed from the same weighted LAT events as in
Pletsch et al. (2012) spanning 2008 August 4 to 2012 July 10. The phase was
computed according to the pulsar ephemeris of Pletsch et al. (2012), with the
fiducial phase shifted to correctly align with the radio profile, using a
dispersion measure of 37.84 pc cm-3. The radio profile at 2 GHz was
constructed from the GBT observation at 2 GHz using the 1100 seconds of data
where the pulsar signal is apparent.
Both the radio and gamma-ray profiles show a double peak structure and are
roughly aligned. In comparing the radio to gamma-ray profile alignment, it is
important to consider the uncertainty in absolute phasing due to the
uncertainty in the DM measurement. The uncertainty in the DM is 0.26 pc cm-3.
A change in the DM of this magnitude corresponds to a shift in the relative
phasing of the radio and gamma-ray profiles of 0.1 in phase. So, within the
uncertainties, either the first radio peak could be aligned with the first
gamma-ray peak, or the second peaks could be aligned. However, the peak
separation in the radio profile is $0.28\pm 0.02$ and the gamma-ray peaks are
separated by $0.35\pm 0.01$, so it not possible for both peaks to be precisely
aligned. Finally, the radio profile also may be broadened by scattering but
without a higher quality profile, or detections at other frequencies, this is
hard to quantify.
## 3\. Discussion
This detection demonstrates that although this pulsar would have been
extraordinarily difficult to detect in a radio search, it is indeed visible
from Earth in the radio band, at least when the conditions in the system
permit. The measurement of the pulsar dispersion measure yields a distance
estimate from the NE2001 galactic free electron density model (Cordes & Lazio,
2002) of 1.4 kpc, typical of the LAT-detected MSP sample. Using this distance
and the measured proper motion of 8 mas yr-1 we compute that the contribution
to the observed period derivative from the Shklovskii effect (Shklovskii,
1970) is only 2.7%. Using the DM distance to compute the gamma-ray luminosity
gives an efficiency $\eta=L_{\gamma}/\dot{E}=30$%, assuming that the beaming
correction $f_{\Omega}$ is 1.
To gauge PSR J1311$-$3430’s significance for the gamma-ray pulsar population,
we follow Romani (2012) and restrict our attention to the brightest $\sim 250$
Fermi sources, for which extensive counterpart studies (including repeated
radio pulse searches) have been made. These include 14 gamma-ray MSPs, all of
which have now been detected in radio plus the newly-discovered PSR
J2339$-$0533\. There are four remaining unassociated objects in this sample –
their gamma-ray properties (a significant high-energy spectral cut-off and low
spectral variability) mark them as likely pulsars. However some are at lower
Galactic latitudes and so, unlike J1311$-$3430 and J2339$-$0533, may be
powered by young pulsars. Even if we count all these sources as MSPs
undetectable in the radio, the fraction is only $4/18=0.22$. This is
dramatically lower than the $27/42=0.64$ fraction of radio-quiet young pulsars
in this sample. We conclude that, for MSPs, radio beams cover nearly as much
sky as the gamma-ray beams, and largely overlap. This agrees with previous
results showing that the beaming fraction of MSPs is near unity and likely
from a fan beam (Lorimer, 2008, and references therein). Conversely the lack
of a large number of black-widow-like LAT sources remaining to be detected
suggests that there is not a large phase space for binary gamma-ray pulsars
that cannot (at least occasionally) be detected in the radio; by the time the
wind from the companion is strong enough to completely bury the radio signal,
the pulsar accelerator itself may be quenched.
The small fraction of time that the radio pulsations are detectable is
notable. There are several possible explanations that are difficult to
differentiate between with only one relatively low signal-to-noise detection.
One reason could be simply interstellar scintillation. High frequency
observations of nearby MSPs are particularly susceptible to scintillation. For
some of the MSPs found in LAT-directed surveys at Parkes, the observed flux
density varies by a factor of 20 from observation to observation and some of
the pulsars are detectable in only about one third of the observations (Camilo
et al. 2012, in preparation). Alternatively, the pulsations could be eclipsed
by material local to the system itself. There are several possible mechanisms
for such eclipses (see Thompson et al., 1994, for a review) and which
mechanism is operative may be different for different systems. In this case,
the continuum detections of a radio point source suggest that the pulsations
may be scattered out of existence (e.g. scattering by plasma turbulence)
rather than the radio waves being actually absorbed. Variations in the
scattering medium may be responsible for the non-detections at other times and
frequencies. The presence of a red, flaring optical component, likely from the
pulsar flux reprocessed in the companion wind (Romani et al., 2012) gives
additional evidence for a wind of variable density and covering fraction. The
timescale and magnitude of dispersion measure variations can be a useful
diagnostic of these processes. We see some evidence for DM growing to just
over 38.0 pc cm-3 in the later part of our observation but the significance is
too limited to draw any firm conclusions. Also, it is difficult to constrain
the presence of a scattering tail in the pulse profile for the same reasons.
Considering the VLA imaging observations, if a significant portion of the flux
at 1.4 GHz is from the point source, it would imply a spectral index of
$-1.8$, typical of known pulsars (Maron et al., 2000), although there is an
uncertain contribution from, e.g., a bow shock or pulsar wind nebula. Deeper
imaging in a higher resolution array configuration is required to characterize
the continuum source and its variability properties.
Within the uncertainties in the absolute phasing, the radio and gamma-ray
pulse profiles are approximately aligned, although their peak separations are
slightly different, with the radio peaks being closer together than the gamma-
ray ones. This could indicate that both are formed in the same geometric
region, but at different altitudes.
This detection illustrates that additional observations, particularly at high
frequency, are needed to search for radio pulsations from the other strong
radio-quiet MSP candidates. Of course, these searches would be much more
sensitive with a known pulsation from a gamma-ray detection, as was the case
with J1311. Exhaustive searches of the LAT data as well as repeated radio
observations are clearly warranted in these cases.
The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. The Parkes Observatory is part of the Australia Telescope National
Facility, which is funded by the Commonwealth of Australia for operation as a
National Facility managed by CSIRO. The GMRT is run by the National Centre for
Radio Astrophysics of the Tata Institute of Fundamental Research. We thank the
staff of the GMRT for help with the observations. The Nançay Radio Observatory
is operated by the Paris Observatory, associated with the French Centre
National de la Recherche Scientifique (CNRS). The Fermi LAT Collaboration
acknowledges support from a number of agencies and institutes for both
development and the operation of the LAT as well as scientific data analysis.
These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in
France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A.
Wallenberg Foundation, the Swedish Research Council and the National Space
Board in Sweden. Additional support from INAF in Italy and CNES in France for
science analysis during the operations phase is also gratefully acknowledged.
This work was partially supported by the Fermi Guest Observer Program,
administered by NASA. C.C.C.’s work was completed while under contract with
NRL and supported by NASA DPR S-15633-Y. M.G. acknowledges financial
contribution from the agreement ASI-INAF I/009/10/0. We thank John Sarkissian
for help with observations at Parkes. Facilities: GBT (GUPPI), VLA, GMRT,
Nançay, Parkes
## References
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|
arxiv-papers
| 2012-10-24T20:33:29 |
2024-09-04T02:49:37.060357
|
{
"license": "Public Domain",
"authors": "P. S. Ray and S. M. Ransom, C. C. Cheung, M. Giroletti, I. Cognard, F.\n Camilo, B. Bhattacharyya, J. Roy, R. W. Romani, E. C. Ferrara, L. Guillemot,\n S. Johnston, M. Keith, M. Kerr, M. Kramer, H. J. Pletsch, P. M. Saz Parkinson\n and K. S. Wood",
"submitter": "Paul S. Ray",
"url": "https://arxiv.org/abs/1210.6676"
}
|
1210.6693
|
# Theory of the magnetic and metal-insulator transitions in RNiO3 bulk and
layered structures.
Bayo Lau and Andrew J. Millis Department of Physics, Columbia University, 538
West 120th Street, New York, NY, USA 10027
###### Abstract
A slave rotor-Hartree Fock formalism is presented for studying the properties
of the p-d model describing perovskite transition metal oxides, and a flexible
and efficient numerical formalism is developed for its solution. The
methodology is shown to yield, within an unified formulation, the significant
aspects of the rare earth nickelate phase diagram, including the paramagnetic
metal state observed for the LaNiO3 and the correct ground-state magnetic
order of insulating compounds. It is then used to elucidate ground state
changes occurring as morphology is varied from bulk to strained and un-
strained thin-film form. For ultrathin films, epitaxial strain and charge-
transfer to the apical out-of-plane oxygen sites are shown to have significant
impact on the phase diagram.
###### pacs:
71.30.+h,73.21.-b,75.25.-j,75.25.Dk
Understanding the unusual electronic behavior of transition-metal oxides has
been a long-standing question in condensed matter physics,mit and interest
has intensified following the demonstration Ohtomo02 that the materials could
be used as components of atomically precise oxide heterostructures. Mannhart08
; Hwang12 ; Chakhalian12 The theoretical challenge posed by the materials is
to treat simultaneously the strong local correlations in the transition metal
d-orbitals and their substantial hybridization with oxygen $p$ orbitals. In
some systems, the $p$ orbitals can be integrated out and the physics
represented in terms of a (possibly multi band) Hubbard model representing the
$d$ orbitals only, for which many theoretical methods are availablemit .
However, in many cases the charge transfer between $p$ and $d$ orbitals is
large enough that the $p$ states cannot be neglected. This “negative charge
transfer energy” regimezsa has been less extensively studied. While much
useful information has been provided by density functional theory (DFT)
Jones89 and extensions such as DFT+UAnisimov93 ; dft0 ; dft1 ; Han11 ; Han12
; Blanca-romero11 ; Anisimov99 , hybrid functionals Gou11 ; Puggioni12 , and
DFT+DMFT Kotliar06 ; Park12 , these methods are computationally intensive, so
that the large supercells required for long-period ordered phases are
difficult to study. Furthermore, the variety of experimental bulk and
superlattice configurations and of many-body phenomena emphasizes the need for
a model-system treatment that encapsulates the essential physics so the
importance of different contributions can be disentangled.
Figure 1: (Top to bottom) Bulk, 2-, and 1-layer phase diagram on the
$u$-$\delta d_{pd}$ plane. Slight charge ordering accompanies the spin
ordering for $\delta d_{pd}\neq$0\. $\Uparrow$/$\uparrow$/0 denote large
moment/small moment/non-magnetic sites of the long-period magnetic ordering
patterns detailed in the text. In the lower two panels the light dotted line
indicates the phase boundary obtained without the energy shift on the apical
oxygen.
The rare earth nickelatesnio3_rev , $ReNiO_{3}$, are an important case in
point. According to standard valence counting arguments, Ni is in the 3+
oxidation state with a $d^{7}$ valence configuration. However, photoemission
experiments mizokawa reported that the dominant GS configuration is
$d^{8}\underline{L}$, in agreement with unrestricted Hartree-Fock
(UHF)mizokawa ; hf1 and DFT+UBlanca-romero11 ; Han12 ; Anisimov99 , and
DFT+DMFTPark12 approaches. Additionally, O 1s x-ray absorption experiments
have found significant hole concentration on the oxygen $p$ orbitalsmizokawa ;
xas0 ; xas1 . This large degree of charge transfer places the material near
the negative charge-transfer regimezsa ; hf1 . As $Re$ is varied across the
lanathanum row of the periodic table, the ground state (GS) of bulk materials
changes from paramagnetic metal (PM-M) to correlated insulator mit ; nio3_rev
; Torrance92 . The correlated insulator phases exhibit a rock-salt-pattern
lattice distortion garcia1 ; alonso0 ; alonso1 ; medarde0 ; Medarde08 and a
nontrivial long-period magnetic ordering alonso0 ; alonso1 ; garcia0 ;
scagnoli0 ; scagnoli1 ; bodenthin . In ultra-thin films a metal-insulator
transition which is apparently unaccompanied by rock-salt-pattern lattice
distortion occurs as film thickness and strain are varied liu ; liu1 ; keimer
; chakhalian ; son0 ; son1 ; moon . Understanding how these apparently
different transitions can occur within a single formulation is an important
open theoretical challengedft0 ; dft1 ; mazin ; Gou11 ; Han11 ; Han12 ;
Puggioni12 ; Blanca-romero11 ; chakhalian ; Anisimov99 ; hf1 ; balents_prl ;
balents_prb ; Park12 .
In this paper, we develop a mean-field (MF) approach based on a combination of
slave-rotor (SR)rotor0 ; rotor1 ; rotor2 ; rotorj0 ; rotorj1 and Hartree-Fock
(HF) methods, along with an efficient and flexible numerical formalism for
solving the MF equations. The method is powerful enough to permit the
examination of the large supercells needed to investigate long-period ordering
patterns in the context of realistic crystal structures. The results reconcile
the bulk and film phase diagrams and allows us to identify the key role played
by the oxygen degrees of freedom and the lattice distortions.
Figure 1 summarizes our key new results. It presents phase diagrams in the
space of Ni charging energy $U$ and amplitude $\delta d_{pd}$ of rocksalt-type
lattice distortion (defined more precisely below). The upper panel shows that,
in the absence of oxygen breathing distortion, bulk materials are PM-M at any
value of $U$, consistent with experimentnio3_rev . As the lattice is
distorted, transitions occur, first to a metallic magnetic phase and then to a
magnetic insulator. The nontrivial ordering wavevector found in
experimentalonso0 ; alonso1 ; garcia0 ; scagnoli0 ; scagnoli1 ; bodenthin is
correctly obtained as a 3D 16-formula-unit magnetic pattern of the type
$\Uparrow$0$\Downarrow$0, denoting the z-projection of magnetic moment
detailed below. The 4 $\Uparrow$ and 4 $\Downarrow$ Ni sites are surrounded by
6 spin-0 Ni sites, which in our interpretation form singlets with its 6 oxygen
neighbors. Each of the 8 spin-0 Ni sites are connected to 3 $\Uparrow$ and 3
$\Downarrow$ Ni sites via oxygen bonds. States with the
$\uparrow\uparrow\downarrow\downarrow$ pattern are unstable against PM or
$\Uparrow 0\Downarrow 0$ solutions, and indeed are not observed in
experimentsscagnoli0 . In agreement with symmetry-based argumentsbalents_prl ,
slight charge ordering always accompanies the spin ordering. For reasonable
parameter values the change in electronic structure across the $Re$ series is
properly accounted for, with one exception: the magnetic metallic phase found
in theory is not observed in experiment. This is discussed in more detail
below. The lower panels show the evolution of the phase diagram with film
thickness $M$ and (bottom panel) with applied strain, revealing that in ultra-
thin films an insulating phase can occur even in the absence of rocksalt-type
lattice distortions.
Our theoretical approach begins from a $p$-$d$ lattice model of the form
$H=\sum_{k}H^{pd}_{k}+\sum_{l}\left(U_{l}+J_{l}\right)$, with $k$ and $l$ sum
over crystal momenta and Ni sites, respectively. The parameters of this model
may be obtained from, e.g. maximially localized Wannier fits to DFT
resultsMarzari97 , but the precise form is not important for this paper (see
also Ref. Wang11, for another example of the insensitivity of results to the
precise p-d model band parameters). We adopt a simplified scenario in which we
retain the Ni $e_{g}$ and O $2p\sigma$ orbitals, with nearest-neighbor $p$-$d$
and $p$-$p$ hopping. The difference between the bare Ni-d and O-2p energies
$\varepsilon_{d}$-$\varepsilon_{p}$ is important, as discussed below.
The structure of the ReNiO3 materials is derived from the ideal ABO3 cubic
perovskite structure which is a lattice of corner-sharing oxygen octahedra,
each containing a Ni site at the center. The crystal structure of the actual
materials is distorted from this structure by rotations of the octahedra which
are not important for our purposes and, in the insulating cases, by a two-
sublattice distortion in which adjacent Ni’s have significant different Ni-O
bond lengthsgarcia1 ; alonso0 ; alonso1 ; Medarde08 . To incorporate the bond
disproportionation in the Hamiltonian under the cubic approximation, we scale
the hopping according to the Harrison ruleharrison ,
$t_{pd}=t_{pd}^{0}\left(1+\delta d_{pd}/d_{pd}^{0}\right)^{-4}$ and
$t_{pp}=t_{pp}^{0}\left(1+\delta d_{pp}/d_{pp}^{0}\right)^{-3}$, with
$d_{pd}^{0}=1.95\AA$ and $d_{pp}^{0}=\sqrt{2}d_{pd}^{0}$. An additional effect
may occur in layered structures. Liu et al. liu showed that the presence of
Al at the interface would deplete holes on the out-of-layer oxygen sites
linking Al and Ni, raising the charge-transfer energy from those apical sites
by $\sim$1eV. We model the $M$-layer 2D structures using supercells with $M$
NiO3 units in the z-direction, terminated on both ends with apical oxygen
sites whose $e_{p}$ are shifted by -1eV. For reference we also performed some
calculations without such shift.
The interaction terms $U_{l}$ and $J_{l}$ contain the on-site repulsion and
the Hund’s interactions in the rotationally-invariant Slater-Kanamori formmit
.
$U_{l}=U\sum_{(m\sigma)>(m^{\prime}\sigma^{\prime})}n_{m\sigma}n_{m^{\prime}\sigma^{\prime}}$
(1) $\displaystyle J_{l}=$ $\displaystyle-$
$\displaystyle\frac{4}{3}j\sum_{\sigma}n_{a\sigma}n_{b\sigma}+\frac{5}{3}j\sum_{m}n_{m\uparrow}n_{m\downarrow}$
(2) $\displaystyle-$ $\displaystyle
j\sum_{\sigma}\left(\frac{1}{3}n_{a\sigma}n_{b-\sigma}+d_{a\sigma}^{\dagger}d_{b-\sigma}^{\dagger}d_{b\sigma}d_{a-\sigma}\right)$
$\displaystyle+$ $\displaystyle
j\left(d_{a\uparrow}^{\dagger}d_{a\downarrow}^{\dagger}d_{b\downarrow}d_{b\uparrow}+h.c.\right)$
In this expression, $J_{l}$ differentiates only states with the same occupancy
but inequivalent configurations.
To treat the on-site Coulomb interactions, we adopt the slave rotor (SR)
approachrotor0 ; rotor1 ; rotor2 ; rotorj0 ; rotorj1 . The method was
originally applied to the Hubbard model and then to other $d$-only models with
Hund’s-like interactionsrotorj0 ; rotorj1 under the $j<<U$ approximation. We
apply it here to the p-d model, noting that for RNiO3 j$\sim$1eV is much
smaller than either the d-d repulsion and the electron bandwidth. For each
$e_{g}$ site, the approach introduces an auxiliary SR field,
$\theta_{l}\in[0,2\pi)$, and decomposes electron operators as
$d^{\dagger}_{l\alpha\sigma}\rightarrow
f^{\dagger}_{l\alpha\sigma}e^{i\theta_{l}}$. The consistency of SR state and
d-occupancy is enforced by the constrain
$\widehat{L}_{l}=\frac{\partial}{i\partial\theta_{l}}=\sum_{\alpha\sigma}\left(f^{\dagger}_{\alpha\sigma}f_{\alpha\sigma}-\frac{1}{2}\right)$
(3)
The spectrum of $U_{l}^{(\theta)}\propto\widehat{L}_{l}^{2}$. We treat the
large $U_{l}$ with $\theta$ and the small j-scale $J_{l}$ with $f^{\dagger}$
using the weak coupling Hartree-Fock approximation. That is, we solve the
$p$-$d$ model with the single-Slater-determinant ansatz
$|MF\rangle=|p,f\rangle|\theta\rangle$. We follow previous SR applications
using Lagrange multipliers, $h_{l}$. Along with constraint Eq. 3 and up to a
constant, the system of equations reads
$\displaystyle H_{p,f}=$
$\displaystyle\sum_{l}J^{(f)}_{l}+\sum_{l\alpha\sigma}(\epsilon_{d}-\epsilon_{p}+\frac{3}{2}U-h_{l})f^{\dagger}_{l\alpha\sigma}f_{l\alpha\sigma}$
(4) $\displaystyle+$ $\displaystyle\sum_{l\epsilon\alpha\sigma}\langle
e^{-i\theta_{l}}\rangle
V_{\epsilon\alpha}p^{\dagger}_{l+\epsilon\sigma}f_{l\alpha\sigma}+h.c.$
$\displaystyle+$
$\displaystyle\sum_{l\epsilon\delta\sigma}t_{pp,\epsilon,\delta}p^{\dagger}_{l+\epsilon\sigma}p_{l+\epsilon+\delta,\sigma}+h.c.$
$\displaystyle H_{\theta}=$
$\displaystyle\sum_{l}\frac{U}{2}\widehat{L}_{l}^{2}+h_{l}\widehat{L}_{l}$ (5)
$\displaystyle+$
$\displaystyle\sum_{l}e^{i\theta_{l}}\sum_{\alpha\epsilon\sigma}V_{\epsilon\alpha}\langle
p^{\dagger}_{l+\epsilon\sigma}f_{l\sigma}\rangle+h.c.$
where $V^{0}_{\epsilon\alpha}$ is given by the product of $t^{0}_{pd}$ and the
Slater-Koster orbital symmetry factors in Table I of Ref. Slater54, .
We solve the MF equations without any restriction other than the single-
Slater-determinant assumption. A 3D Bravais lattice of Ni16O48 supercells is
required for unrestricted modeling of the $(1/2,0,1/2)$ pattern with respect
to the orthorhombic unit cell. To systematically capture the effects of
dimensionality, we model the $M$-layer structures using a 2D Bravais lattice
of Ni4MO12M+4 supercells, which is connected to the 3D Ni16O48 supercell for
$M\rightarrow\infty$. The system is solved by the standard T=0 iterative
procedure for up to 65536 supercells. Without much optimization, the worst
case initial condition with bi-partite charge-, orbital-, and magnetic-order
would converge into a PM-M or $(1/2,0,1/2)$-ordered insulator within 30 cpu
hours on an Opteron-2350 cluster.
We perform bulk calculations for (u,$e_{d}$) such that $\langle
n_{e_{g}}\rangle\sim 2$ as found in UHF with parameters fitted to
photoemission spectrumhf1 , DFTBlanca-romero11 ; Han12 ; Anisimov99 , and
DMFT+DFTPark12 . For the reference set of $U=5.3$, $j=1$,
$t_{p_{\sigma},d_{x^{2}-y^{2}}}=1.5$, $t_{pp}=0.5$,
$\Delta_{pd}=e_{d}-e_{p}+U=0.3$, the undistorted 3D lattice has $\langle
n_{e_{g}}\rangle=1.95$, slightly less than those of the above calculations. To
isolate dimensionality effects, we use the bulk’s ($u$,$e_{d}$) pairs, in
addition to the aforementioned apical $e_{p}$ shift, for layered calculations.
Figure 2: (Left) Stable solutions’ MF energy. (Right) rotor renormalization
and insulating gap for the bulk system at $U=5.3$.
The left panel of Fig. 2 shows representative results for the energies of
different locally stable phases computed at interaction (U,j)=(5.3,1) as a
function of $(\delta d_{pd})^{2}$. We see that in agreement with experiment,
the undistorted lattice is a PM-M. As oxygen breathing disproportionation
$\delta d_{pd}$ is increased a transition to a period-4 metallic magnetic
state occurs. At yet larger distortion a metal-insulator transition occurs.
The ferromagnetic states found in UHF mizokawa ; hf1 , DFT+U Blanca-romero11 ;
Han12 ; Anisimov99 and DFT+DMFT Park12 become stable only at larger $\delta
d_{pd}$ and gain less energy than does the experimentally observed ordering
pattern. Also, $E$$\sim$$-\delta d_{pd}^{2}$ implies an equilibrium distortion
ultimately determined by the aharmonic lattice restoration force beyond the
scope of this study.
The dashed and dotted lines in the right panel of Fig. 2 are the SR
renormalizations of the p-d hopping (Eq. 4) on the two sublattices of the
distorted structure. We see that the Coulomb renormalizations are not large,
and are only weakly dependent on sublattice. This renormalization has a
moderate effect on physical properties, e.g. Fermi velocity renormalization at
(U,j)=(5.3,1) is $v_{5.3,1}/v_{0,0}\sim$0.65. The relatively modest SR
renormalizations confirms that near the “negative charge transfer” region of
the phase diagram zsa the on-site Coulomb correlations do not drive the phase
transition in an orthodox way, in particular the physics is far from the
$\langle e^{i\theta}\rangle\rightarrow 0$ Brinkman-Rice insulator. Even then,
the renormalized kinetic terms, combined with lattice distortion and the
magnetic order opens a gap in the density of states. We stress that SR
renormalization $\langle e^{i\theta}\rangle$ due to $U$ is required for
insulation within physical parameter range.
Parameters scaled for different Re bulk materialsMedarde08 ; garcia1 ; alonso0
; alonso1 are marked on the phase diagram in top panel of Fig. 1. It captures
all experimentally observed phases over the Re series, with the exception of
Re=Pr, which is predicted to be a metal instead of an insulator, albeit with
the correct charge and magnetic order. This can be understood by noting that
our approach underestimates the insulating gap, which is 160meV at
$\frac{\delta d_{pd}}{d_{pd}^{0}}$=2.5% compared to $\sim$200meV in a 2-site
DFT+DMFT studyPark12 , and that Re=Pr is also extrapolated to be a metal from
those Re=Lu results.
In agreement with experiments, the $\Uparrow$0$\Downarrow$0 insulator (top
panel of Fig. 1) has a (1/2,0,1/2) magnetic structure with respect to the
orthorhombic unit cell, a rock-salt charge ordering pattern, but without
orbital ordering. Let us identify the Ni sites with longer/shorter Ni-O bond
length as Ni${}_{\mbox{long}}$ and Ni${}_{\mbox{short}}$ and consider the
$\frac{\delta d_{pd}}{d_{pd}^{0}}$=2.5% solution. The occupancy at those sites
are $n_{\mbox{long}}=2.03$ and $n_{\mbox{short}}$=1.90 . This
disproportionation can be understood by measuring the Ni-O hybridization order
parameter
$\widehat{O}_{l}=\sum_{\epsilon\alpha\sigma}V^{0}_{l\epsilon\alpha}e^{-i\theta_{l}}p^{\dagger}_{l+\epsilon\sigma}f_{l\alpha\sigma}+h.c.$
We found that $\frac{\langle O_{\mbox{short}}\rangle}{\langle
O_{\mbox{long}}\rangle}=1.4$, which is greater than expected from the
corresponding $\delta t_{pd}/t_{pd}^{0}\sim$10% hopping modulation. This shows
that Ni${}_{\mbox{short}}$ is strongly hybridized with the oxygen bands while
Ni${}_{\mbox{long}}$ retains its d8 characteristic. The magnetic moments are
$m_{\mbox{long}}$=1.1, which agrees with experimentalonso0 ; alonso1 , and
$m_{\mbox{short}}$=0. For charge-transfer insulators, the effective physics of
strong p-d hybridization is a strong antiferromagnetic
$\overline{S}_{d}\cdot\overline{S}_{p}$ spin correlation. In this context,
having $m_{\mbox{short}}=0$ in the single-Slater-determinant approximation is
evident of a d8-high-spin state forming a singlet with its oxygen neighbors.
The formation of $p$-$d$ singlets has also been reported in a two-site
DFT+DMFT study which predicted FM GSPark12 .
Figure 3: Results for $u=7$. (Top) $d$-occupancy for the 1-layer structure.
(Bottom) $m_{long}$ magnetic moment for bulk, middle layer of 3-layer, 2- and
1-layer structures.
We now discuss the effects of dimensional confinement. The bulk and $M\geq 3$
layers structures have similar phase diagrams, with increasing ordered and
insulating region for decreasing $M$ and increasing $U$. In particular, the
lower panel of Fig. 3 shows that PM-M exists for $M\geq 3$. The phase diagram
of 2-, and 1-layer ultra-thin films are shown in the lower panels of Fig 1. We
see that, for $M\leq 2$ layers, the ground state is always magnetically
ordered, even in the absence of the bond disproportionation. Further, the
distortion amplitude required for the transition to the insulating phase
decreases with increasing $u$ and decreasing $M$. At (u,j)=(5.3,1), the metal-
insulator boundary $(\frac{\delta d_{pd}}{d_{pd}^{0}})_{MIT}$ is reduced from
1.9% to 0.96% and 0.44% for 2- and 1-layer, respectively. A comparison between
the solid and dotted lines in Fig. 1 shows that charge transfer to the out-of-
plane orbitalsliu greatly enlarge the insulating regions.
To study the effects of epitaxial strain, we focus on 1-layer structure for
concreteness. We include the strain effects by scaling hopping integrals
according to the Harrison ruleharrison for the geometry specified in Ref.
keimer, , which introduced compressive (tensile) strain with LaSrAlO4 (SrTiO3)
such that the lattice parameters are a=b=3.769 (3.853)Å and c = 3.853 (3.790)Å
. Breathing-mode distortion is then introduced as discussed above. In
agreement with experiments on ultra-thin layerskeimer ; liu1 ; moon , Fig. 1
shows that compressive (tensile) epitaxial strain enlarges (shrinks) the
insulating region. Under realistic values of $U$ and compressive stain,
ordered insulator is possible for $\delta d_{pd}$=0, agreeing with experiments
which found breathing-mode distortion only under tensile strainchakhalian .
Note that further shift in apical oxygen $e_{p}$ also decreases $\delta
d_{pd}$ required for insulation.
Figure 3 demonstrates an orbital polarization of $\sim$5%. We also found that
the polarization decreases (increases) with compressive (tensile) strain, in
agreement with a DFT+U studyHan12 ; Han11 . For example, LaSrAlO4 (SrTiO3)
substrate changes the polarization to $\sim$4 (6)%. While DFT+U predicted FM
order, our $M$=1,2 GS have an AFM in-plane order of
$\Uparrow\uparrow\Downarrow\downarrow$, which differs slightly from the bulk’s
$\Uparrow$0$\Downarrow$0 pattern by having small $m_{short}$$\neq$0 which
decreases with increasing distortion. The 2-layer structure has an additional
magnetic transition between FM and AFM ordering in the z-direction.
Lastly, we compare our results with other theoretical works. In our
formulation, the bulk PM-M develops into $\Uparrow$0$\Downarrow$0-I for
$\frac{\delta d_{pd}}{d_{pd}^{0}}>$1.9% compared to UHF’s $\frac{\delta
d_{pd}}{d_{pd}^{0}}>$7.5% transition from ferromagnetic to MCO state with
similar patternhf1 . The pattern is also similar to the $j/U>1$ S-SDW state in
the $d$-only modelbalents_prb , but we have $j/U<1/5$ and a bond and charge
disproportionation with $d^{8+\delta}\underline{L}d^{8-\delta}\underline{L}$
involving strong $pd$ hybridization. Unlike the $d$-only model, our results
are sensitive to dimensional confinement. Even though different DFT
implementations have been employed to capture the bulk LaNiO3 PM-MGou11 and
LaNiO3/LaAlO3 ordered insulating layerPuggioni12 , the demonstration of all
bulk and layered phases by a single formulation had been elusive to the best
of our knowledge.
Our formulation bridges the gap between HF-like approaches and the expensive
cluster DMFT. The results connect all ReNiO3 delicate phases in bulk and layer
form and also provide detailed insights about them. This demonstrates a viable
pathway of treating systems with different superstructures as well as
compounds such as Fe and Co oxides with strong $p$-$d$ hybridization and large
number of partially filled strongly correlated orbitals. For example, Sr2FeO4
has also been shown to exhibit strong hybridization and non-trivial magnetic
orderRozenberg98 .
We thank G. A. Sawatzky, H. Chen, H. T. Dang, R. Fernandez, S. Park, and D.
Zgid for helpful discussions. This effort is supported by US National Science
Foundation under grant NSF-DMR-1006282.
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|
arxiv-papers
| 2012-10-24T22:09:01 |
2024-09-04T02:49:37.069784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bayo Lau and Andrew J. Millis",
"submitter": "Bayo Lau",
"url": "https://arxiv.org/abs/1210.6693"
}
|
1210.6725
|
Rational curves and special metrics on twistor spaces
Misha Verbitsky111Partially supported by RFBR grants 12-01-00944- ,
10-01-93113-NCNIL-a, and AG Laboratory NRI-HSE, RF government grant, ag.
11.G34.31.0023.
verbit@verbit.ru
To Professor H. Blaine Lawson at his 70th birthday
Abstract
A Hermitian metric $\omega$ on a complex manifold is called SKT or pluriclosed
if $dd^{c}\omega=0$. Let $M$ be a twistor space of a compact, anti-selfdual
Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that
in this case $M$ is Kähler, hence isomorphic to ${\mathbb{C}}P^{3}$ or a flag
space. This result is obtained from rational connectedness of the twistor
space, due to F. Campana. As an aside, we prove that the moduli space of
rational curves on the twistor space of a K3 surface is Stein.
###### Contents
1. 1 Introduction
1. 1.1 Special Hermitian metrics on complex manifolds
2. 1.2 Strongly Gauduchon and symplectic Hermitian metrics
3. 1.3 Rational curves and pluriclosed metrics
2. 2 Twistor spaces for 4-dimensional Riemannian manifolds and the space of rational curves
1. 2.1 Twistor spaces for 4-dimensional Riemannian manifolds: definition and basic results
2. 2.2 Rational curves on $\operatorname{Tw}(M)$
3. 2.3 Rational curves and plurinegative metrics
4. 2.4 An aside: rational lines on the twistor space of a K3 surface
5. 2.5 Quasilines and Moishezon manifolds
3. 3 Pluriclosed and Hermitian symplectic metrics on twistor spaces
## 1 Introduction
### 1.1 Special Hermitian metrics on complex manifolds
The world of non-Kähler complex geometry is infinitely bigger than that
inhabited by Kähler manifolds. For instance, as shown by Taubes [T] (see also
[PP], any finitely-generated group can be realized as a fundamental group of a
compact complex manifold. On contrary, the Kähler condition puts big
restrictions on the fundamental group.
However, there are not many constructions which lead to explicit non-Kähler
complex manifolds. There are many homogeneous and locally homogeneous
manifolds (such as complex nilmanifolds), which are known to be non-Kähler.
The locally conformally Kähler manifolds are non-Kähler by a theorem of
Vaisman ([Va]). Kodaira class VII surfaces (forming a vast and still not
completely understood class of complex surfaces) are never Kähler. Finally,
the twistor spaces, as shown by Hitchin, are never Kähler, except two
examples: ${\mathbb{C}}P^{3}$, being a twistor space of $S^{4}$, and the flag
space, being a twistor space of ${\mathbb{C}}P^{2}$ ([Hi]).
There are many ways to weaken the Kähler condition $d\omega=0$. Given a
Hermitian form $\omega$ on a complex $n$-manifold, one may consider an
equation $d(\omega^{k})=0$. For $1<k<n-1$, this equation is equivalent to
$d\omega=0$, but the equation $d(\omega^{n-1})$ is quite non-trivial. Such
metrics are called balanced. All twistor spaces are balanced ([M]); also, all
Moishezon manifolds are balanced ([AB2]).
Another way to weaken the Kähler condition is to consider the equation
$dd^{c}(\omega^{k})=0$, where $d^{c}=-IdI$; this equation is non-trivial for
all $0<k<n$. When $k=1$, a metric satisfying $dd^{c}\omega=0$ is called
pluriclosed, or strong Kähler torsion (SKT) metric; such metrics are quite
important in physics and in generalized complex geometry.
A Hermitian metric satisfying $dd^{c}(\omega^{n-1})=0$ is called Gauduchon. As
shown by P. Gauduchon ([Ga]), every Hermitian metric is conformally equivalent
to a Gauduchon metric, which is unique in its conformal class up to a constant
multiplier.
Since a twistor space has complex dimension 3, and is balanced, the only non-
trivial metric condition (among those mentioned above) for the twistor space
is $dd^{c}(\omega)=0$.
The main result of this paper is the following theorem, similar to Hitchin’s
theorem on non-Kählerianity of twistor spaces.
Theorem 1.1: Let $M$ be a twistor space of a compact 4-dimensional anti-
selfdual Riemannian manifold. Assume that $M$ admits a pluriclosed Hermitian
form $\omega$: $dd^{c}(\omega)=0.$ Then $M$ is Kaehler.
Proof: 3.
### 1.2 Strongly Gauduchon and symplectic Hermitian metrics
The Gauduchon, pluriclosed and all the rest of the $dd^{c}(\omega^{k})=0$
Hermitian metrics have an interesting variation of a cohomological nature.
Definition 1.2: Let $(M,I)$ be a complex manifold, and $\omega$ a Hermitian
form. We say that $\omega^{k}$ is strongly pluriclosed if any of the following
equivalent conditions are satisfied.
(i)
$d(\omega^{k})$ is $dd^{c}$-exact.
(ii)
$\omega^{k}$ is a $(k,k)$-part of a closed $2k$-form.
Notice that either of these conditions easily implies $dd^{c}(\omega^{k})=0$,
but these conditions are significantly stronger.
For $k=1$ and $n-1$ this condition is especially interesting. When a
pluriclosed Hermitian form $\omega$ is (1,1)-part of a closed (and hence
symplectic) form $\tilde{\omega}$, $\omega$ is called taming or Hermitian
symplectic, and when $(\omega)^{n-1}$ is $(n-1,n-1)$-part of a closed form,
$\omega$ is called strongly Gauduchon ([Po]).
In the paper [ST] Streets and Tian have constructed a parabolic flow for
Hermitian symplectic metric, analoguous to the Kähler-Ricci flow. They asked
whether there exists a compact complex Hermitian symplectic manifold not
admitting a Kähler structure. This question was considered in [EFG] and [EFV]
for complex nilmanifolds. In [EFV] it was shown that complex nilmanifolds
cannot admit Hermitian symplectic metrics. However, the pluriclosed metrics
exist on many complex nilmanifolds.
The present paper grew as an attempt to answer the Streets-Tian’s question for
twistor spaces. However, it was found that the twistor spaces are not only
never Hermitian symplectic, they never admit a pluriclosed metric unless they
are Kähler.
### 1.3 Rational curves and pluriclosed metrics
The results of the present paper are based on the study of the moduli of
rational curves. Unlike many complex non-algebraic manifolds, the twistor
spaces are very rich in curves: there exists a smooth rational curve passing
through any finite subset of a twistor space (2.2).
For an almost complex structure $I$ equipped with a taming symplectic form,
all components of the space of complex curves are compact, by Gromov’s
compactness theorem ([Gr, AL]). I will show that the same is true for
pluriclosed metrics, if $I$ is integrable (2.3). This is used to prove that a
twistor space admitting a pluriclosed metric is actually Moishezon (2.5).
However, Moishezon varieties satisfy the $dd^{c}$-lemma. This is used to show
that any pluriclosed metric is in fact Hermitian symplectic.
Finally, by using the Peternell’s theorem from [Pe], we prove that no
Moishezon manifold can be Hermitian symplectic (3).
## 2 Twistor spaces for 4-dimensional Riemannian manifolds and the space of
rational curves
### 2.1 Twistor spaces for 4-dimensional Riemannian manifolds: definition and
basic results
Definition 2.1: Let $M$ be a Riemannian 4-manifold. Consider the action of the
Hodge $*$-operator: $*:\;\Lambda^{2}M{\>\longrightarrow\>}\Lambda^{2}M$. Since
$*^{2}=1$, the eigenvalues are $\pm 1$, and one has a decomposition
$\Lambda^{2}M=\Lambda^{+}M\oplus\Lambda^{-}M$ onto selfdual ($*\eta=\eta$) and
anti-selfdual ($*\eta=-\eta$) forms.
Remark 2.2: If one changes the orientation of $M$, leaving metric the same,
$\Lambda^{+}M$ and $\Lambda^{-}M$ are exchanged. Therefore,
$\dim\Lambda^{2}M=6$ implies $\dim\Lambda^{\pm}(M)=3$.
Remark 2.3: Using the isomorphism $\Lambda^{2}M=\mathfrak{so}(TM)$, we
interpret $\eta\in\Lambda^{2}_{m}M$ as an endomorphisms of $T_{m}M$. Then the
unit vectors $\eta\in\Lambda^{+}_{m}M$ correspond to oriented, orthogonal
complex structures on $T_{m}M$.
Definition 2.4: Let $\operatorname{Tw}(M):=S\Lambda^{+}M$ be the set of unit
vectors in $\Lambda^{+}M$. At each point $(m,s)\in\operatorname{Tw}(M)$,
consider the decomposition $T_{m,s}\operatorname{Tw}(M)=T_{m}M\oplus
T_{s}S\Lambda^{+}_{m}M$, induced by the Levi-Civita connection. Let $I_{s}$ be
the complex structure on $T_{m}M$ induced by $s$, $I_{S\Lambda^{+}_{m}M}$ the
complex structure on $S\Lambda^{+}_{m}M=S^{2}$ induced by the metrics and
orientation, and ${\cal
I}:\;T_{m,s}\operatorname{Tw}(M){\>\longrightarrow\>}T_{m,s}\operatorname{Tw}(M)$
be equal to ${\cal I}_{s}\oplus I_{S\Lambda^{+}_{m}M}$. An almost complex
manifold $(\operatorname{Tw}(M),{\cal I})$ is called the twistor space of $M$.
The following results about twistor spaces are well known (see e.g. [Bes]).
Theorem 2.5: The almost complex structure on $(\operatorname{Tw}(M),{\cal I})$
is a conformal invariant of $M$. Moreover, one can reconstruct the conformal
structure on $M$ from the almost complex structure on $\operatorname{Tw}(M)$
and its anticomplex involution $(m,s){\>\longrightarrow\>}(m,-s)$.
Theorem 2.6: $(\operatorname{Tw}(M),{\cal I})$ is a complex manifold if and
only if $W^{+}=0$, where $W^{+}$ (“self-dual conformal curvature”) is an
autodual component of the curvature tensor. Such manifolds are called
conformally half-flat or ASD (anti-selfdual).
### 2.2 Rational curves on $\operatorname{Tw}(M)$
Definition 2.7: An ample rational curve on a complex manifold $M$ is a smooth
curve $S\cong{\mathbb{C}}P^{1}\subset M$ such that
$NS=\bigoplus_{k=1}^{n-1}{\cal O}(i_{k})$, with $i_{k}>0$. It is called a
quasi-line if all $i_{k}=1$.
Claim 2.8: Let $M$ be a compact complex manifold containing a an ample
rational line. Then any $N$ points $z_{1},...,z_{N}$ can be connected by an
ample rational curve.
Proof: This fact is well known in algebraic geometry (see [Ko]). However, its
proof is valid for all complex manifolds.
Claim 2.9: Let $M$ be a Riemannian 4-manifold,
$\operatorname{Tw}(M)\stackrel{{\scriptstyle\sigma}}{{{\>\longrightarrow\>}}}M$
its twistor space, $m\in M$ a point, and
$S_{m}:=\sigma^{-1}(m)=S\Lambda^{+}_{m}(M)$ the corresponding $S^{2}$ in
$\operatorname{Tw}(M)$. Then $S_{m}$ is a quasi-line.
Proof: Since the claim is essentially infinitesimal, it suffices to check it
when $M$ is flat. Then $\operatorname{Tw}(M)=\operatorname{Tot}({\cal
O}(1)^{\oplus 2})\cong{\mathbb{C}}P^{3}\backslash{\mathbb{C}}P^{1}$, and
$S_{m}$ is a section of ${\cal O}(1)^{\oplus 2}$.
Corollary 2.10: Any $N$ points $z_{1},...,z_{N}$ on a twistor space can be
connected by an smooth, ample rational curve
### 2.3 Rational curves and plurinegative metrics
For other applications of Gromov’s compactness theorem on manifolds with
pluriclosed metrics, please see [I].
Definition 2.11: Let $S$ be a complex curve on a Hermitian manifold
$(M,I,g,\omega)$. Define the Riemannian volume as
$\operatorname{Vol}(S):=\int_{S}\omega$.
Definition 2.12: A Hermitian form $\omega$ is called plurinegative
(pluripositive) if the (2,2)-form $dd^{c}\omega$ is negative (positive).
Remark 2.13: The notion of “positive $(k,k)$-form” comes in two flavours:
weakly positive and strongly positive ([D]). When $k=1$ or
$k=\dim_{\mathbb{C}}M-1$, these two notions coincide. Since in the present
paper we are interesned mostly in 3-dimensional complex manifolds, this
distinction becomes irrelevant. For the sake of a definition, we shall
consider, in the present paper, “positive” as a synonym to “strongly
positive”.
Of course, pluriclosed Hermitian metrics are both pluripositive and
plurinegative.
As shown in [KV, (8.12)], a standard Hermitian form on a twistor space of a
hyperkähler manifold is pluripositive.
Claim 2.14: Let $X$ be a component of the moduli of complex curves on a given
complex manifold, $\tilde{X}$ the set of pairs $\\{S\in X,z\in S\subset M\\}$,
(“the universal family”), and $\pi_{M}:\;\tilde{X}{\>\longrightarrow\>}M$,
$\pi_{X}:\;\tilde{X}{\>\longrightarrow\>}X$ the forgetful maps. Then the
volume function $\operatorname{Vol}:\;X{\>\longrightarrow\>}{\mathbb{R}}^{>0}$
can be expressed as $\operatorname{Vol}=(\pi_{X})_{*}\pi_{M}^{*}\omega$.
Remark 2.15: Since pullback and pushforward of differential forms commute with
$d$, $d^{c}$, this gives
$dd^{c}\operatorname{Vol}=(\pi_{X})_{*}\pi_{M}^{*}(dd^{c}\omega)$. Therefore,
$-\operatorname{Vol}$ is plurisubharmonic on $X$ whenever $\omega$ is
plurinegative.
Theorem 2.16: (Gromov) Let $M$ be a compact Hermitian almost complex
manifold, ${\mathfrak{X}}$ the space of all complex curves on $M$, and
${\mathfrak{X}}\stackrel{{\scriptstyle\operatorname{Vol}}}{{{\>\longrightarrow\>}}}{\mathbb{R}}^{>0}$
the volume function. Then $\operatorname{Vol}$ is proper (that is, preimage of
a compact set is compact).
Proof: [Gr], [AL].
Corollary 2.17: Let $M$ be a complex manifold, equipped with a plurinegative
Hermitian form $\omega$, and $X$ a component of the moduli of complex curves.
Then the function
$\operatorname{Vol}:\;X{\>\longrightarrow\>}{\mathbb{R}}^{>0}$ is constant,
and $X$ is compact.
Proof: Since $\operatorname{Vol}\geqslant 0$, the set
$\operatorname{Vol}^{-1}(]-\infty,C])$ is compact for all $C\in{\mathbb{R}}$,
hence $-\operatorname{Vol}$ has a maximum somewhere in $X$. However, a
plurisubharmonic function which has a maximum is necessarily constant by E.
Hopf’s strong maximum principle. Therefore, $\operatorname{Vol}$ is constant:
$\operatorname{Vol}=A$. Now, compactness of $X=\operatorname{Vol}^{-1}(A)$
follows from Gromov’s theorem.
### 2.4 An aside: rational lines on the twistor space of a K3 surface
For a complex manifold $Z$ equipped with a pluripositive Hermitian form, the
same argument implies that any component of the moduli of curves on $Z$ is
pseudoconvex. In particular, this is true on twistor spaces of hyperkähler
manifolds ([KV], [DDM]). For a twistor space of K3, a stronger result can be
achieved.
Theorem 2.18: Let $M$ be a K3 surface equipped with a hyperkähler metric, and
$\operatorname{Tw}(M)$ its twistor space. Denote by $X$ a connected component
of the moduli of rational curves on $\operatorname{Tw}(M)$. Then $X$ is Stein.
Proof: The proof is based on the following useful theorem of Fornæss-
Narasimhan.
Definition 2.19: Let $X$ be a complex variety (possibly singular), and
$\varphi:\;X{\>\longrightarrow\>}[-\infty,\infty[$ an upper semicontinuous
function. We say that $\varphi$ is plurisubharmonic (in the weak sense) if for
any holomorphic map $D\stackrel{{\scriptstyle f}}{{{\>\longrightarrow\>}}}X$
from a disc in ${\mathbb{C}}$, the composition
$f\circ\varphi:\;D{\>\longrightarrow\>}{\mathbb{R}}$ is plurisubharmonic (or
identically $-\infty$). This function is called strongly plurisubharmonic if
any petrurbation of $\varphi$ which is small in $C^{2}$-topology remains
plurisubharmonic.111To define precisely what it means “small in
$C^{2}$-topology”, we embed an open subset $U\subset X$ to ${\mathbb{C}}^{n}$.
Suppose that there exists $\varepsilon>0$ such that for any function $f$ on
${\mathbb{C}}^{n}$ with $|f|_{C^{2}}<\varepsilon$, the sum of
$\varphi+f{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is plurisubharmonic.
Then $\varphi$ is called strongly plurisubharmonic in $U$. If $X$ admits a
covering by such $U$, then $\varphi$ is called strongly plurisubharmonic in
$X$.
Theorem 2.20: Let $X$ be a complex variety admitting an exhaustion function
which is strictly plurisubharmonic. Then $X$ is Stein.
Proof: [FN, Theorem 6.1].
Now we can prove 2.4. By Gromov’s compactness (2.3),
$\operatorname{Vol}:\;X{\>\longrightarrow\>}{\mathbb{R}}$ is exhaustion, and
by [KV] it is plurisubharmonic. It remains to show that this function is
strictly plurisubharmonic.
Let $\omega$ be the standard Hermitian form on
$\operatorname{Tw}(M)=M\times{\mathbb{C}}P^{1}$. Denote by
$\omega_{{\mathbb{C}}P^{1}}$ the Fubini-Study 2-form on ${\mathbb{C}}P^{1}$,
and let $\pi^{*}\omega_{{\mathbb{C}}P^{1}}$ be its lifting to
$\operatorname{Tw}(M)$, where
$\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ is the
twistor projection. Then
$dd^{c}\omega=\omega\wedge\pi^{*}\omega_{{\mathbb{C}}P^{1}}$ ([KV, (8.12)]).
Let $v\in Z_{x}X$ be a vector from the Zariski tangent cone of $X$. The strict
plurisubharmonicity of $\operatorname{Vol}$ would follow if the second
derivative
$\sqrt{-1}\>\operatorname{Lie}_{v}\operatorname{Lie}_{\overline{v}}\operatorname{Vol}$
is positive for all $v\neq 0$.
Now, let $x=[S]\in X$ be a point represented by a curve $S$. Then $Z_{x}X$ is
a subspace of $H^{0}(NS)$, where $NS$ is the normal sheaf of $S$. A priori,
$S$ can have several irreducible components, some of them sitting in the
fibers of the twistor projection
$\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$, others
transversal to these fibers. However, all components sitting in the fibers of
$\pi$ are fixed, because the rational curves on K3 are fixed. Therefore, $v$
in non-trivial along the fibers of $\pi$. Now,
$\sqrt{-1}\>\operatorname{Lie}_{v}\operatorname{Lie}_{\overline{v}}\operatorname{Vol}=dd^{c}\operatorname{Vol}(v,\overline{v})=\int_{S}(dd^{c}\omega)(v,\overline{v})\geqslant\int_{S}\pi^{*}\omega_{{\mathbb{C}}P^{1}}\cdot\omega(v,\overline{v}).$
The last integral is positive, because $v$ vanishes on those components of $S$
which belong to the fibers of $\pi$, hence $v\neq 0$ on a component $S_{1}$
which is transversal to $\pi$. Then,
$\int_{S}\pi^{*}\omega_{{\mathbb{C}}P^{1}}\cdot\omega(v,\overline{v})\geqslant\int_{S_{1}}\pi^{*}\omega_{{\mathbb{C}}P^{1}}\cdot\omega(v,\overline{v}),$
but this integral is positive, because $\pi^{*}\omega_{{\mathbb{C}}P^{1}}$ is
positive on each transversal component of $S$. This proves that
$\sqrt{-1}\>\operatorname{Lie}_{v}\operatorname{Lie}_{\overline{v}}\operatorname{Vol}>0$,
implying strict plurisubharmonicity of $\operatorname{Vol}$
Remark 2.21: The variety $X$, which is shown to be Stein in 2.4, could be
singular (a complex variety is Stein if it admits a closed holomorphic
embedding to ${\mathbb{C}}^{n}$). However, there is not a single known example
of a singular point in any component of the space ${\cal S}(M)$ of rational
curves on $\operatorname{Tw}(M)$, when $M$ is a K3. It is not hard to see that
${\cal S}(M)$ is smooth when $M$ is a compact torus. It is not entirely
impossible that it is also smooth for a K3.
### 2.5 Quasilines and Moishezon manifolds
Let $M$ be a compact complex manifold, and $S\subset M$ an ample rational
curve. Assume that the space of deformations of $S$ in $M$ is compact. From
[C1, Theorem 3] it follows that $M$ is Moishezon ([C2, Remark 3.2 and Theorem
4.5]). For the convenience of the reader, I will give an independent proof of
this result here.
Recall that a quasiline is a smooth rational curve $S\subset M$ such that its
normal bundle is isomorphic to ${\cal O}(1)^{n}$. A neighbourhood of a
quasiline shares many properties with a neighbourhood of a line in
${\mathbb{C}}P^{n}$. Heuristically, this can be stated as follows.
An imprecise statement. Let $S\subset M$ be a quasi-line. Then, for an
appropriate tubular neighbourhood $U\subset M$ of $S$, “for every two points
$x,y\in U$ close to $S$ and far from each other, there is a unique deformation
of $S$ containing $X$ and $Y$.”
More precisely:
Claim 2.22: Let $S\subset M$ be a quasi-line. Then, for any sufficiently
small tubular neighbourhood $U\subset M$ of $S$, there exists a smaller
tubular neighbourhood $W\subset U$, satisfying the following condition. Let
$\Delta_{S}$ be the image of the diagonal embedding
$\Delta_{S}:\;S{\>\longrightarrow\>}W\times W$. Then there exists an open
neighbourhood $V$ of $\Delta_{S}$, properly contained in $W\times W$, such
that for any pair $(x,y)\in W\times W\backslash V$, there exists a unique
deformation $S^{\prime}\subset U$ of $S$ containing $x$ and $y$.
Claim 2.23: A small deformation $S^{\prime}\subset U$ of $S$ passing through
$z\in S$ is uniquely determined by a 1-jet of $S^{\prime}$ at $z$.
Both of these claims follow from a general results of deformation theory: the
first cohomology of a normal bundle $NS$ vanish, hence there are no
obstructions to a deformation, and the deformation space is locally modeled on
a space of sections of $NS$. However, since $NS={\cal O}(1)^{n}$, any section
is uniquely determined by its values in two different points, or by its 1-jet
at any given point.
Further on, we shall need the following simple lemma.
Lemma 2.24: Let $X{\>\longrightarrow\>}Y$ be a domimant map of complex
varieties, which is finite at a general point. Assume that $X$ is Moishezon.
Then $Y$ is also Moishezon.
Proof: Replacing $X$ by its ramified covering, we may assume that outside of
its singularities, the map $X{\>\longrightarrow\>}Y$ is a Galois covering,
with the Galois group $G$. Then $X/G{\>\longrightarrow\>}Y$ is bimeromorphic.
Replacing $X$ by its resolution, we can also assume that $X$ is projective.
Then $X/G$ is also projective, by Noether’s theorem on invariant rings.
Now we can prove the main result of this subsection (see also [C1, Theorem
3]).
Theorem 2.25: Let $M$ be a complex manifold, $S\subset M$ a quasi-line, and
$W$ its deformation space. Assume that $W$ is compact. Then $M$ is Moishezon.
Proof. Step 1: Let $z\in M$ a point, containing a quasi-line $S\in W$, $W_{z}$
the set of all curves $S_{1}\in W$ containing $z$, and $\tilde{W}_{z}$ – the
set of all pairs $\\{x\in S_{1},S_{1}\in W_{z}\\}$. From 2.5, it follows that
the map $\tilde{W}_{z}{\>\longrightarrow\>}M$, $(S,x){\>\longrightarrow\>}x$
is surjective and finite at generic point.
Step 2: By 2.5, it would suffice to prove that $\tilde{W}_{z}$ is Moishezon.
Step 3: After an appropriate bimeromorphic transform, we may assume that
$\tilde{W}_{z}{\>\longrightarrow\>}W_{z}$ is a smooth, proper map with
rational, 1-dimensional fibers. Then $\tilde{W}_{z}$ is Moishezon
$\Leftrightarrow$ $W_{z}$ is Moishezon. Indeed, the space of cycles in a
Moishezon manifold is Moishezon.
Step 4: By 2.5, the map from $W_{z}$ to ${\mathbb{P}}T_{z}M$ mapping a quasi-
line to its 1-jet is also generically finite. Therefore, $W_{z}$ is Moishezon.
Corollary 2.26: Let $M$ be a twistor space admitting a pluriclosed (or
plurinegative) Hermitian metric. Then $M$ is Moishezon.
## 3 Pluriclosed and Hermitian symplectic metrics on twistor spaces
Recall the following classical theorem of Harvey and Lawson ([HL]).
Theorem 3.1: Let $M$ be a compact, complex $n$-manifold. Then the following
conditions are equivalent.
(i)
$M$ does not admit a Kähler metric.
(ii)
$M$ has a non-zero, positive $(n-1,n-1)$-current $\Theta$ which is an
$(n-1,n-1)$-part of a closed current.
The same argument, applied to pluriclosed or taming metrics, brings the
following (see also [AB1]).
Theorem 3.2: Let $M$ be a compact, complex $n$-manifold. Then
(a)
$M$ admits no Hermitian symplectic metrics $\Leftrightarrow$ $M$ admits a
positive, exact, non-zero $(n-1,n-1)$-current.
(b)
$M$ admits no pluriclosed metrics $\Leftrightarrow$ $M$ admits a positive,
non-zero, $dd^{c}$-exact $(n-1,n-1)$-current.
Proof of (a): Let $A\subset\Lambda^{2}(M)$ be a cone of real 2-forms $\eta$
such that the $(1,1)$-part $\eta^{1,1}$ is strictly positive. Then $A\cap\ker
d=0$ is equivalent, by Hahn-Banach theorem, to existence of a $(2n-2)$-current
vanishing on $\ker d$ (hence, exact) and positive on $A$, hence of type
$(n-1,n-1)$ and positive.
Proof of (b): Let $A$ be the same as above. Then $A\cap\ker dd^{c}=0$ is
equivalent, by Hahn-Banach, to existence of a $(2n-2)$-current $\Theta$
positive on $A$ (hence, of type $(n-1,n-1)$ and positive) and vanishing on
$\ker dd^{c}$. A positive $dd^{c}$-exact $(n-1,n-1)$-current clearly vanishes
on $\ker dd^{c}$. It remains to show, conversely, that $\Theta$ is
$dd^{c}$-exact whenever $\Theta$ vanishes on $\ker dd^{c}$.
Since $\ker dd^{c}$ contains the space $\ker d$, the current $\Theta$ is
exact. Let
$H^{n-1,n-1}_{BC}(M,{\mathbb{R}}):=\frac{\ker
d{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{n-1,n-1}(M)}}\right.}}{dd^{c}(\Lambda^{n-2,n-2}(M))}$
be the Bott-Chern cohomology group, and
$H^{1,1}_{AE}(M,{\mathbb{R}}):=\frac{\ker
dd^{c}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{1,1}(M)}}\right.}}{\operatorname{im}d\cap\Lambda^{1,1}(M)}$
be the Aepli cohomology ([Ae], [Sch]). The exterior multiplication induces a
pairing between these two groups, and it is not hard to see that they are
dual. Since $\Theta$ vanishes on $\ker dd^{c}$, the pairing with its Bott-
Chern cohomology class
$\langle[\Theta],\cdot\rangle:\;H^{1,1}_{AE}(M,{\mathbb{R}}){\>\longrightarrow\>}{\mathbb{R}}$
vanishes. Therefore, the class of $\Theta$ in
$H^{n-1,n-1}_{BC}(M,{\mathbb{R}})$ is equal zero, hence
$\eta\in\operatorname{im}dd^{c}$. 3 is proved.
This leads to the following useful proposition.
Proposition 3.3: Any twistor space $M$ which admits a pluriclosed metric also
admits a Hermitian symplectic structure.
Proof: By 2.5, $M$ is Moishezon. Then, [DGMS] implies that $M$ satisfies
$dd^{c}$-lemma. Therefore, any exact (2,2)-current is $dd^{c}$-exact. Applying
3, we obtain that $M$ is Hermitian symplectic.
Corollary 3.4: Let $M$ be a twistor space admitting a pluriclosed (or
Hermitian symplectic) metric. Then $M$ is Kähler.
Proof: Th. Peternell ([Pe, Corollary 2.3]) has shown that any compact non-
Kähler Moishezon $n$-manifold admits an exact, positive $(n-1,n-1)$-current.
Therefore, it is never Hermitian symplectic (3). By 3, $M$ cannot be
pluriclosed.
Acknowledgements: Many thanks to Gang Tian for asking the questions which lead
to the writing of this paper, and to Frederic Campana and Thomas Peternell for
insightful comments and the reference. Part of this paper was written during
my visit to the Simons Center for Geometry and Physics; I wish to thank SCGP
for hospitality and stimulating atmosphere. I am grateful to Nigel Hitchin for
the kind interest and the inspiration.
## References
* [Ae] A. Aeppli, Some exact sequences in cohomology theory for Kähler manifolds, Pacific J. Math., 12, 1962.
* [AB1] L. Alessandrini and G. Bassanelli, Positive $\partial\overline{\partial}$-closed currents and non-Kähler geometry, J. Geom. Anal. 2 (1992), no. 4, 291-316.
* [AB2] L. Alessandrini and G. Bassanelli, Metric properties of manifolds bimeromorphic to compact Kähler spaces, J. Diff. Geom. 37(1993), 95-121.
* [AL] M. Audin and J. Lafontaine ed., Holomorphic Curves in Symplectic Geometry, Progress in Math. 117, Birkhäuser, 1994.
* [Bes] Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)
* [C1] Campana, F., Algébricité et compacité dans les espaces de cycles, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 3, A195eA197.
* [C2] Campana, F., On twistor spaces of the class C, J. Differential Geom. 33 (1991), no. 2, 541-549.
* [D] Demailly, Jean-Pierre, Analytic methods in algebraic geometry, Lecture Notes, École d’éé de Mathematiques de Grenoble ”Géométrie des variétés projectives complexes : programme du modèle minimal” (June-July 2007)
* [DDM] Guillaume Deschamps, Noël Le Du, Christophe Mourougane, Hessian of the metric form on twistor spaces, arXiv:1202.0183
* [DGMS] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274.
* [FN] Fornæss, John Erik; Narasimhan, Raghavan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), no. 1, 47-72.
* [EFG] Nicola Enrietti, Anna Fino, Gueo Grantcharov, Tamed Symplectic forms and Generalized Geometry, arXiv:1112.2592
* [EFV] N. Enrietti, A. Fino and L. Vezzoni, Hermitian Symplectic structures and SKT metrics, J. Symplectic Geom. 10, n. 2 (2012), 203-223.
* [Ga] P. Gauduchon, La 1-forme de torsion d’une variete hermitienne compacte, Math. Ann., 267 (1984), 495–518.
* [Gr] Gromov, M. Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307-347.
* [HL] R. Harvey and H. B. Lawson, ”An intrinsic characterisation of Kähler manifolds,” Invent. Math 74 (1983) 169-198.
* [Hi] N. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981) 133-150.
* [I] S. Ivashkovich, Complex Plateau problem in non Kähler manifolds, arXiv:math/9804008.
* [KV] Kaledin, D., Verbitsky, M., Non-Hermitian Yang-Mills connections, Selecta Math. (N.S.) 4 (1998), no. 2, 279–320.
* [Ko] Kollár, J., Rational curves on algebraic varieties, Springer, 1996, viii+320 pp..
* [M] M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), 261-295.
* [PP] D. Panov, A. Petrunin Telescopic actions, arXiv:1104.4814.
* [Pe] Peternell, Th., Algebraicity criteria for compact complex manifolds. Math. Ann. 275 (1986), no. 4, 653-672.
* [Po] Dan Popovici, Stability of Strongly Gauduchon Manifolds under Modifications, arXiv:1009.5408.
* [Sch] Michel Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528, 28 pages.
* [ST] Jeffrey Streets, Gang Tian Symplectic curvature flow, arXiv:1012.2104.
* [T] Taubes, C. The existence of anti-self-dual conformal structures, J. Differential Geom. 36 (1992), no. 1, 163-253.
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Misha Verbitsky
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
7 Vavilova Str. Moscow, Russia
|
arxiv-papers
| 2012-10-25T02:22:31 |
2024-09-04T02:49:37.078962
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1210.6725"
}
|
1210.6736
|
# Noether current of the surface term of Einstein-Hilbert action, Virasoro
algebra and entropy
Bibhas Ranjan Majhi
IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
E-mail bibhas@iucaa.ernet.in
###### Abstract:
A derivation of Noether current from the surface term of Einstein-Hilbert
action is given. We show that the corresponding charge, calculated on the
horizon, is related to the Bekenstein-Hawking entropy. Also using the charge,
the same entropy is found based on the Virasoro algebra and Cardy formula
approach. In this approach, the relevant diffeomorphisms are found by imposing
a very simple physical argument: diffeomorphisms keep the horizon structure
invariant. This complements similar earlier results [27](arXiv:1204.1422)
obtained from York-Gibbons-Hawking surface term. Finally we discuss the
technical simplicities and improvements over the earlier attempts and also
various important physical implications.
Classical Theories of Gravity, Virasoro Algebra, Cardy formula, Entropy
## 1 Introduction
The thermodynamic properties of horizon arises from the combination of the
general theory of relativity and the quantum field theory. This was first
observed in the case of black holes [1, 2]. Now it is evident that it is much
more general and a local Rindler observer can attribute temperature and
entropy to the null surfaces in the context of the emergent paradigm of
gravity [3, 4]. Such a generality might provide us a deeper insight towards
the quantum nature of the spacetime. So far several attempts have been made to
know the microscopic origin of the entropy, but every method has its own
merits and demerits. Among others, Carlip made an attempt [5, 6] in the
context of Virasoro algebra to illuminate this aspect which is basically the
generalisation of the method by Brown and Hanneaux [7]. In brief, in this
method one first defines a bracket among the Noether charges and calculate it
for certain diffeomorphisms, chosen by some physical considerations. It turns
out that the algebra is identical to the Virasoro algebra. The central charge
and the zero mode eigenvalue of the Fourier modes of the charge are then
automatically identified which after substituting in the Cardy formula [8, 9]
one finds the Bekenstein-Hawking entropy 111For a complete list of works which
lead to further development of this method, see [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25].. In all the previous attempts, the
Noether current was taken related to the Einstein-Hilbert (EH) action and the
analysis was on-shell, i.e. equation of motion has been used explicitly. Later
an off-shell analysis and a generalization to Lanczos-Lovelock gravity have
been presented in [26]. Earlier [27], based on the Virasoro algebra approach,
we showed that the entropy can also be obtained from the Noether current
corresponding to the York-Gibbons-Hawking surface term. But it is not clear if
the same can be achived from the surface term of the Einstein-Hilbert action,
since they are not exactly identical. So it is necessary to investigate this
issue in the light of Virasoro algebra context, particulary because both the
surface terms lead to the same entropy on the horizon. This will complement
our earlier work [27].
In this paper we will use the Noether current associated to the surface term
of EH action. Before going into the motivations for taking the surface term
only, let us first highlight some peculiar facts of EH action which are
essential for the present purpose. $\bullet$ It is an unavoidable fact that to
obtain the equation of motion in the Lagrangian formalism one has to impose
some extra prescription, like adding extra boundary term (in this case York-
Gibbons-Hawking term). This is because the action contains second order
derivative of metric tensor $g_{ab}$. But unfortunately the choice of the
surface term is not unique. This is quite different from other well known
field theories. $\bullet$ The EH action can be separated into two terms: one
contains the squires of the Christoffel connections ( i. e. it is in
$\Gamma\Gamma-\Gamma\Gamma$ structure) and the other one contains the total
derivation of $\Gamma$ ($\partial\Gamma-\partial\Gamma$ structure). We will
call them as $L_{quad}$ and $L_{sur}$, respectively. Interestingly, Einstein’s
equation of motion can be obtained solely from $L_{quad}$ by using the usual
variation principle where no additional prescription is not required [28].
$\bullet$ The most important one is that these two terms are related by an
algebraic relation, usually known as holographic relation [29, 30].
Interestingly, all the above features are happened to be common even for the
Lanczos-Lovelock theory [31]. For a recent review in this direction, see [32].
Although an extensive study on the Noether current of gravity has been done
starting from Wald [33], discussion on the current derived from $L_{sur}$ is
still lacking. To motivate why one should be interested, let us summarise
below the already observed facts. $\bullet$ It is expected that the entropy is
associated to the degrees of freedom around or on the relevant null surface
rather than the bulk geometry of spacetime. $\bullet$ This surface term
calculated on the Rindler horizon gives exactly the Bekenstein-Hawking entropy
[28]. $\bullet$ Extremization of the surface term with respect to the
diffeomorphism parameter whose norm is a constant, leads to the Einstein’s
equation [29]. $\bullet$ Another interesting fact is that in a small region
around an event, EH action reduces to a pure surface term when evaluated in
the Riemann normal coordinates. All these indicate that either the bulk and
the surface terms are duplicating all the information or the actual dynamics
is stored in surface term rather than in bulk term. To illuminate more on this
issue, one needs to study every aspect of the surface term.
In this paper, we shall discuss the Noether realization of the surface term of
the EH action, particularly we shall examine if the Noether current represents
the Virasoro algebra for a certain class of diffeomorphisms. This is necessary
to have a deeper understanding of the role of the surface term in the gravity.
Also it will give a further insight towards the earlier claim: the actual
information of the gravity is stored in the surface. To do this explicitly, we
shall consider the form of the metric close to the null surface in the local
Rindler frame around some event. This is given by the Rindler metric. The
reasons for choosing such metric are as follows. According to equivalence
principle, gravity can be mimicked by an accelerated observer and an uniformly
accelerated frame will have Rindler metric. Apart from that, it is a relevant
frame for an observer sitting very near to the black hole horizon. Hence any
thermodynamic feature of the null surface can be attributed by this metric and
it provides a general description which was originally obtained only for the
black hole horizon. Moreover, all the quantities will be observer dependent.
In this paper we shall proceed as follows. First a detailed derivation of the
Noether current for a diffeomorphism $x^{a}\rightarrow x^{a}+\xi^{a}$,
corresponding to $L_{sur}$, will be given. This is important because it has
not been done earlier and therefore the properties of the current have not
been explored. Here we will show that the corresponding charge $Q[\xi]$,
calculated on the null surface for $\xi^{a}$ to be Killing, yields exactly one
quarter of the horizon area after multiplying it by $2\pi/\kappa$ where
$\kappa$ is the acceleration of the observer or the surface gravity in the
case of a black hole. Next, a definition of the bracket among the charges will
be given. This will be done by taking variation of the charge $Q[\xi_{1}]$ for
another transformation $x^{a}\rightarrow x^{a}+\xi_{2}^{a}$. Finally, we need
to calculate all these quantities for a particular diffeomorphism. To identify
the relevant diffeomorphisms from which the algebra has to be constructed,
following our earlier work [27], we use the criterion that the diffeomorphism
should leave the near horizon form of the metric invariant in some non-
singular coordinate system. This will lead to a set of diffeomorphism vectors
for which the Fourier components of the bracket among the charges will be
exactly similar to Virasoro algebra. It is then very easy to identify the zero
mode eigenvalue and the central extension. Substitution of all these values in
the Cardy formula [8, 9] will yield exactly the Bekenstein-Hawking entropy [1,
2]. A similar analysis was done in [25] based on the Noether current
corresponding to $L_{bulk}$ [34, 35, 36]. In this calculation, to obtain the
correct value of the entropy, a particular boundary condition (Dirichlet or
Neumann) was used. But the physical significance of it is not well understood.
Before going into the main calculation, let us summarize the main features of
the present analysis. $\bullet$ First is the technical aspect. To obtain the
correct entropy, in most of the earlier works, one had to either shift the
zero mode eigenvalue [6] or choose a parameter contained in the Fourier modes
of $\xi^{a}$ as the surface gravity $\kappa$ [25] or both [26]. Here we shall
show that none of ad hoc prescriptions will be required. $\bullet$ The
important one is the simplicity of the criterion (near horizon structure of
the metric remains invariant in some non-singular coordinate system) to find
the relevant diffeoprphisms for which we obtain the Virasoro algebra. This was
first introduced by us [27] in this context. The significance of this choice
is that the full set of diffeomorphism symmetry of the theory is now reduced
to a subset which respects the existence of horizon in a given coordinate
system. Hence it may happen that some of the original gauge degrees of freedom
(which could have been eliminated by certain diffeomorphisms which are now
disallowed) now being effectively upgraded to physical degrees of freedom as
far as a particular class of observers are concerned. So all the thermodynamic
quantities, attributed to the horizon, become observer dependent. $\bullet$ In
our present analysis we will not need any use of boundary condition like
Dirichlet or Neumann to obtain the exact form of the entropy. $\bullet$ Since
our analysis will be completely based on $L_{sur}$ where no information about
$L_{bulk}$ is needed, it will definitely illuminate the emergent paradigm of
gravity, particularly the holographic aspects in the action. We will discuss
later more on different aspects and significance of our results.
The organization of the paper as follows. In section 2, the derivation of the
Noether current for the $L_{sur}$ will be presented explicitly. Next we shall
give the definition of the bracket among the charges and the relevant
diffeomorphims based on the invariance of horizon structure criterion. Section
4 will be devoted to show that the Fourier mode of the bracket is exactly like
the Virasoro algebra which by the Cardy formula will lead to Bekenstein-
Hawking entropy. Finally, we shall conclude.
## 2 Derivation of Noether current from the surface term of Einstein-Hilbert
action
In this section, a detailed derivation of the Noether current and the
potential corresponding to the surface term of EH action will be presented.
Then we shall calculate the charge on the Rindler horizon.
The Lagrangian corresponding to the surface term is given by [28],
$\displaystyle L_{sur}=\partial_{a}(\sqrt{-g}S^{a})~{},$ (1)
where
$\displaystyle S^{a}=2Q^{ad}_{ck}g^{bk}\Gamma^{c}_{bd};\,\,\,\
Q^{ad}_{ck}=\frac{1}{2}(\delta^{a}_{c}\delta^{d}_{k}-\delta^{a}_{k}\delta^{d}_{c})~{}.$
(2)
Here the normalization $1/16\pi G$ is omitted and it will be inserted where
necessary. Now our task is to find the variations of both sides of (1) for a
diffeomorphism $x^{\prime a}=x^{a}+\xi^{a}$ and then equate them. The
variation we shall consider here as the Lie variation which is defined, in
general, as
$\displaystyle\delta A=A(x^{\prime})-A^{\prime}(x^{\prime})~{},$ (3)
where $A(x^{\prime})=A(x+\xi)=A(x)+\xi^{a}\partial_{a}A(x)$, $A(x)$ and
$A^{\prime}(x^{\prime})$ are the evaluated in two different coordinate systems
$x$ and $x^{\prime}$, respectively. In the following, for the notational
simplicity, we shall denote $A(x)$ as $A$.
The variation of the right hand side of (1) is given by,
$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\partial_{a}[\delta(\sqrt{-g}S^{a})]=\partial_{a}\Big{[}S^{a}\delta(\sqrt{-g})+\sqrt{-g}\delta
S^{a}\Big{]}$ (4) $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}\frac{S^{a}}{2}\sqrt{-g}g^{bc}\delta
g_{bc}+\sqrt{-g}\delta S^{a}\Big{]}~{}.$
Since $g_{ab}$ is a tensor, for the Lie variation, $\delta g_{ab}$ is
expressed by the Lie derivative and is given by
$\displaystyle\delta g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}~{}.$ (5)
Therefore,
$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}S^{a}\partial_{b}(\sqrt{-g}\xi^{b})+\sqrt{-g}\delta
S^{a}\Big{]}~{}.$ (6)
On the other hand, since $S^{a}$ is not a tensor, the variation of it can not
be expressed by simple Lie derivative. To find $\delta S^{a}$ we shall used
the general definition (3). Let us first calculate $S^{\prime a}(x^{\prime})$.
Under the change $x^{\prime a}=x^{a}+\xi^{a}$ we have,
$\displaystyle\frac{\partial x^{\prime a}}{\partial
x^{b}}=\delta^{a}_{b}+\partial_{b}\xi^{a}~{};$ $\displaystyle\frac{\partial
x^{b}}{\partial x^{\prime a}}=\delta^{b}_{a}-\partial_{a}\xi^{b}~{}.$ (7)
Here we considered infinitesimal change and so the terms from
$\partial\xi\partial\xi$ have been ignored. This will be followed in the later
analysis. Hence,
$\displaystyle\Gamma^{\prime
a}_{bc}(x^{\prime})=\Gamma^{a}_{bc}-\Gamma^{a}_{bd}\partial_{c}\xi^{d}-\Gamma^{a}_{cd}\partial_{b}\xi^{d}+\Gamma^{d}_{bc}\partial_{d}\xi^{a}-\partial_{b}\partial_{c}\xi^{a}~{},$
$\displaystyle g^{\prime
bk}(x^{\prime})=g^{bk}+g^{bf}\partial_{f}\xi^{k}+g^{kf}\partial_{f}\xi^{b}~{},$
$\displaystyle Q^{\prime ad}_{ck}(x^{\prime})=Q^{ad}_{ck}~{}.$ (8)
Substitution of these in $S^{\prime a}(x^{\prime})=2Q^{\prime
ad}_{ck}(x^{\prime})g^{\prime bk}(x^{\prime})\Gamma^{\prime
c}_{bd}(x^{\prime})$ lead to,
$\displaystyle S^{\prime
a}(x^{\prime})=S^{a}+S^{b}\partial_{b}\xi^{a}-g^{bd}\partial_{b}\partial_{d}\xi^{a}+g^{ab}\partial_{b}\partial_{c}\xi^{c}~{}.$
(9)
Other one is given by
$\displaystyle
S^{a}(x^{\prime})=S^{a}(x^{b}+\xi^{b})=S^{a}+\xi^{b}\partial_{b}S^{a}~{}.$
(10)
Therefore, according to (3), the Lie variation of $S^{a}$ due to the
diffeomorphism is
$\displaystyle\delta S^{a}=S^{a}(x^{\prime})-S^{\prime
a}(x^{\prime})=\xi^{b}\partial_{b}S^{a}-S^{b}\partial_{b}\xi^{a}+M^{a}~{},$
(11)
where
$\displaystyle
M^{a}=g^{bd}\partial_{b}\partial_{d}\xi^{a}-g^{ab}\partial_{b}\partial_{c}\xi^{c}~{}.$
(12)
Substituting this in (6) we obtain the variation of right hand side of (1) as,
$\displaystyle\delta
L_{sur}=\partial_{a}\Big{[}\partial_{b}(\sqrt{-g}S^{a}\xi^{b})-\sqrt{-g}S^{b}\partial_{b}\xi^{a}+\sqrt{-g}M^{a}\Big{]}~{}.$
(13)
Next we find the variation of left hand side of (1); i.e. $L_{sur}$. For this
we will start from the following relation:
$\displaystyle L_{sur}=\sqrt{-g}(L_{g}-L_{quad})~{},$ (14)
where
$\displaystyle L_{g}=R;\,\,\,\
L_{quad}=2Q^{bcd}_{a}\Gamma^{a}_{dk}\Gamma^{k}_{bc}~{},$ (15)
with
$Q^{bcd}_{a}=\frac{1}{2}\Big{(}\delta^{c}_{a}g^{bd}-\delta^{d}_{a}g^{bc}\Big{)}$.
Since $L_{g}$ is a scalar, by the definition of Lie derivative $\delta
L_{g}=\xi^{a}\partial_{a}L_{g}$. Therefore using (5) we find
$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\delta(\sqrt{-g}L_{g})-\delta(\sqrt{-g}L_{quad})$ (16)
$\displaystyle=$
$\displaystyle\partial_{a}(\sqrt{-g}\xi^{a}L_{g})-\partial_{a}(\sqrt{-g}\xi^{a})L_{quad}-\sqrt{-g}\delta
L_{quad}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}\sqrt{-g}\xi^{a}(L_{g}-L_{quad})\Big{]}+\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\sqrt{-g}\delta
L_{quad}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{(}\xi^{a}L_{sur}\Big{)}+\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\sqrt{-g}\delta
L_{quad}~{}.$
To find $\delta L_{quad}$, we will proceed as earlier. Under the change
$x^{\prime a}=x^{a}+\xi^{a}$, $L^{\prime}_{quad}(x^{\prime})$ is calculated as
$\displaystyle L^{\prime}_{quad}(x^{\prime})$ $\displaystyle=$ $\displaystyle
2Q^{\prime bcd}_{a}(x^{\prime})\Gamma^{\prime
a}_{dk}(x^{\prime})\Gamma^{\prime k}_{bc}(x^{\prime})$ (17) $\displaystyle=$
$\displaystyle
L_{quad}+g^{bc}\Gamma^{k}_{bc}\partial_{d}\partial_{k}\xi^{d}+g^{bc}\Gamma^{d}_{dk}\partial_{b}\partial_{c}\xi^{k}-g^{bd}\Gamma^{c}_{dk}\partial_{b}\partial_{c}\xi^{k}~{},$
where (8) has been used. This can be expressed in terms of $M^{a}$ in the
following way. Second term on the right hand side can be expressed in the
following form
$\displaystyle\sqrt{-g}g^{bc}\Gamma^{k}_{bc}\partial_{d}\partial_{k}\xi^{d}$
$\displaystyle=$
$\displaystyle\Big{[}\sqrt{-g}g^{bc}g^{ak}\partial_{b}g_{ac}-g^{ak}\partial_{a}(\sqrt{-g})\Big{]}\partial_{d}\partial_{k}\xi^{d}$
(18) $\displaystyle=$
$\displaystyle-\partial_{a}(\sqrt{-g}g^{ak})\partial_{d}\partial_{k}\xi^{d}~{},$
where in the above we used
$g^{bc}g^{ak}\partial_{b}g_{ac}=-\partial_{a}g^{ak}$. Third term of (17)
reduces to
$\displaystyle\sqrt{-g}g^{bc}\Gamma^{d}_{dk}\partial_{b}\partial_{c}\xi^{k}=\partial_{k}(\sqrt{-g})g^{bc}\partial_{b}\partial_{c}\xi^{k}~{}.$
(19)
Similarly, the last term can be expressed as
$\displaystyle 2\sqrt{-g}g^{bd}\Gamma^{c}_{dk}\partial_{b}\partial_{c}\xi^{k}$
$\displaystyle=$
$\displaystyle\sqrt{-g}g^{bd}g^{ca}\partial_{k}g_{ad}\partial_{b}\partial_{c}\xi^{k}$
(20) $\displaystyle=$
$\displaystyle-\sqrt{-g}\partial_{k}(g^{bc})\partial_{b}\partial_{c}\xi^{k}~{},$
where in the last line $g^{bd}g^{ca}\partial_{k}g_{ad}=-\partial_{k}g^{bc}$
has been used. Substituting all these in (17) we obtain
$\displaystyle L^{\prime}_{quad}(x^{\prime})$ $\displaystyle=$ $\displaystyle
L_{quad}-\frac{1}{\sqrt{-g}}\Big{[}\partial_{a}(\sqrt{-g}g^{ak})\partial_{d}\partial_{k}\xi^{d}+\partial_{k}(\sqrt{-g}g^{bc})\partial_{b}\partial_{c}\xi^{k}\Big{]}$
(21) $\displaystyle=$ $\displaystyle
L_{quad}+\frac{1}{\sqrt{-g}}\partial_{a}(\sqrt{-g}M^{a})$
On the other hand,
$\displaystyle
L_{quad}(x^{\prime})=L_{quad}(x^{a}+\xi^{a})=L_{quad}+\xi^{a}\partial_{a}L_{quad}~{}.$
(22)
Hence
$\displaystyle\sqrt{-g}\delta
L_{quad}=\sqrt{-g}L_{quad}(x^{\prime})-\sqrt{-g}L^{\prime}_{quad}(x^{\prime})=\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\partial_{a}(\sqrt{-g}M^{a})~{}.$
(23)
Substituting this in (16) we obtain
$\displaystyle\delta
L_{sur}=\partial_{a}\Big{(}\xi^{a}L_{sur}+\sqrt{-g}M^{a}\Big{)}~{}.$ (24)
Now equating (13) and (24) we obtain $\partial_{a}J^{a}[\xi]=0$, where the
conserved Noether current $J^{a}[\xi]$ is given by
$\displaystyle
J^{a}[\xi]=-\partial_{b}(\sqrt{-g}S^{a}\xi^{b})+\sqrt{-g}S^{b}\partial_{b}\xi^{a}+\xi^{a}L_{sur}~{}.$
(25)
Finally, using $L_{sur}=\partial_{a}(\sqrt{-g}S^{a})$ in the above, we can
express the current as the divergence of a anti-symmetric two index quantity:
$\displaystyle
J^{a}[\xi]=\partial_{b}\Big{[}\sqrt{-g}(\xi^{a}S^{b}-\xi^{b}S^{a})\Big{]}=\partial_{b}\Big{[}\sqrt{-g}J^{ab}[\xi]\Big{]}~{}.$
(26)
It is evident that the anti-symmetric object $J^{ab}[\xi]$ is not a tensor and
it is usually called the Noether potential. Therefore, inserting the proper
normalization, the charge is given by
$\displaystyle Q[\xi]=\frac{1}{32\pi
G}\int_{\cal{H}}d\Sigma_{ab}\sqrt{h}J^{ab}[\xi]~{},$ (27)
where $d\Sigma_{ab}=-d^{2}x(N_{a}M_{b}-N_{b}M_{a})$ is the surface element of
the $2$-dimensional surface $\cal{H}$ and $h$ is the determinant of the
corresponding metric. Since our present discussion will be near the horizon,
we choose the unit normals $N_{a}$ and $M_{a}$ as spacelike and timelike
respectively.
Now we shall calculate the charge (27) explicitly on the horizon. This will be
done by considering the form of the metric near the horizon,
$\displaystyle ds^{2}=-2\kappa xdt^{2}+\frac{1}{2\kappa
x}dx^{2}+dx^{2}_{\perp}~{},$ (28)
where $x_{\perp}$ represents the transverse coordinates. The metric has a
timelike Killing vector $\chi^{a}=(1,0,0,0)$ and the Killing horizon is given
by $\chi^{2}=0$; i.e. $x=0$. The non-zero Christoffer connections are
$\displaystyle\Gamma^{t}_{tx}=\frac{1}{2x};\,\,\,\
\Gamma^{x}_{tt}=2\kappa^{2}x;\,\,\,\ \Gamma^{x}_{xx}=-\frac{1}{2x}~{}.$ (29)
For the metric (28) we find
$\displaystyle N^{a}=(0,\sqrt{2\kappa x},0,0);\,\,\
M^{a}=(\frac{1}{\sqrt{2\kappa x}},0,0,0)~{},$ (30)
and hence $d\Sigma_{tx}=-d^{2}x$. Also, (2) yields
$\displaystyle S^{t}=0;\,\,\ S^{x}=-2\kappa~{}.$ (31)
Therefore,
$\displaystyle J^{tx}=(\xi^{t}S^{x}-\xi^{x}S^{t})=-2\kappa\xi^{t}~{}.$ (32)
Now if $\xi^{a}$ is a Killing vector, then $\xi^{t}=\chi^{t}=1$ and so
calculating the charge (27) explicitly we find
$\displaystyle Q[\xi=\chi]=\frac{\kappa A_{\perp}}{8\pi G}~{},$ (33)
where $A_{\perp}=\int_{\cal{H}}d^{2}x$ is the horizon cross-section area.
Multiplying it by the periodicity of time coordinate $2\pi/\kappa$ we obtain
exactly the entropy: one quarter of horizon area. Moreover, the above can be
expressed as $Q[\xi=\chi]=TS$, where $T=\kappa/2\pi$ is the temperature of the
horizon and $S=A_{\perp/4G}$ is the entropy. Therefore one can call it as the
Noether energy. Such interpretaion was done earlier in [37, 38].
So far we found that the Noether charge corresponding to the surface term of
EH action alone led to the entropy of the Rindler horizon. This was shown
earlier for the charge coming from the total EH action [33]. Therefore, the
present analysis reveled that it may be possible that the information is
actually encoded in the surface term rather than the bulk term. Then the
natural question arises: What are the degrees of freedom responsible for this
entropy? So far it is not known. In the next couple of sections we shall give
an idea on the nature of the possible degrees of freedom in the context of
Virasoro algebra and Cardy formula.
## 3 Bracket among the charges and the diffeomorphism generators
In the previous section, we have given the expression for the charge (see Eq.
(27)) for an arbitrary diffeomorphism. Here we shall define the bracket among
the charges. The relevant diffeomorphisms will be chosen by imposing a minimum
condition on the spacetime metric. The charge and the bracket will be then
expressed in terms of these generators.
We shall find the bracket following our earlier works [26, 27]. For this let
us first calculate the following:
$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\delta_{\xi_{1}}(\sqrt{-g})J^{ab}[\xi_{2}]+\sqrt{-g}\delta_{\xi_{1}}(J^{ab}[\xi_{2}])$
(34) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sqrt{-g}g_{mn}\delta_{\xi_{1}}g^{mn}J^{ab}[\xi_{2}]$
$\displaystyle+$
$\displaystyle\sqrt{-g}\Big{[}(\delta_{\xi_{1}}\xi_{2}^{a})S^{b}+\xi_{2}^{a}(\delta_{\xi_{1}}S^{b})-(a\leftrightarrow
b)\Big{]}~{}.$
Using
$\displaystyle\delta_{\xi}g^{ab}$ $\displaystyle=$
$\displaystyle\pounds_{\xi}g^{ab}=-\nabla^{a}\xi^{b}-\nabla^{b}\xi^{a};$
$\displaystyle\delta_{\xi}\Gamma^{a}_{bc}$ $\displaystyle=$
$\displaystyle\nabla_{b}\nabla_{c}\xi^{a}+R^{a}_{cmb}\xi^{m}~{},$ (35)
and in addition the expression for $S^{a}$, given by (2), we obtain,
$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\sqrt{-g}\Big{[}\nabla_{m}\xi_{1}^{m}J^{ab}[\xi_{2}]+\\{(\xi_{1}^{m}\nabla_{m}\xi_{2}^{a}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{a})S^{b}$
(36) $\displaystyle+$
$\displaystyle\xi_{2}^{a}\Big{(}-2\Gamma^{b}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{b}+2R^{b}_{m}\xi_{1}^{m}$
$\displaystyle-$
$\displaystyle\Gamma^{n}_{nm}(\nabla^{b}\xi_{1}^{m}+\nabla^{m}\xi_{1}^{b})-\nabla_{m}\nabla^{b}\xi_{1}^{m}\Big{)}-(a\leftrightarrow
b)\\}\Big{]}~{}.$
For the present metric (28), $g=-1$, $R^{a}_{b}=0$ and hence
$\sqrt{-g}\Gamma^{n}_{nm}=\partial_{m}(\sqrt{-}g)=0$. Therefore
$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\Big{[}\nabla_{m}\xi_{1}^{m}J^{ab}[\xi_{2}]+\frac{1}{16\pi
G}\\{(\xi_{1}^{m}\nabla_{m}\xi_{2}^{a}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{a})S^{b}$
$\displaystyle+$
$\displaystyle\xi_{2}^{a}\Big{(}-2\Gamma^{b}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{b}-\nabla_{m}\nabla^{b}\xi_{1}^{m}\Big{)}-(a\leftrightarrow
b)\\}\Big{]}$ $\displaystyle\equiv K^{ab}_{12}~{}.$ (37)
Finally we define a bracket as:
$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:=\frac{1}{2}\int_{\cal{H}}d\Sigma_{ab}\sqrt{h}\Big{[}K^{ab}_{12}-(1\leftrightarrow
2)\Big{]}~{},$ (38)
which for the present metric (28) reduces to
$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:=-\int_{\cal{H}}d^{2}x\Big{[}K^{tx}_{12}-(1\leftrightarrow
2)\Big{]}~{}.$ (39)
To calculate the above bracket we need to know about the generators $\xi^{a}$.
We shall determine them by using the condition that the horizon structure
remains invariant in some nonsingular coordinate system. For that let us first
express the metric (28) in Gaussian null (or Bondi like) coordinates,
$\displaystyle du=dt-\frac{dx}{2\kappa x};\,\,\ dX$ $\displaystyle=$
$\displaystyle dx~{}.$ (40)
In these coordinates the metric reduces to the following form:
$\displaystyle ds^{2}=-2\kappa Xdu^{2}-2dudX+dx^{2}_{\perp}~{}.$ (41)
Now impose the condition that the metric coefficients $g_{XX}$ and $g_{uX}$ do
not change under the diffeomorphism, i.e.
$\displaystyle\pounds_{\tilde{\xi}}g_{XX}=0;\,\,\,\
\pounds_{\tilde{\xi}}g_{uX}=0~{},$ (42)
where $\pounds_{\tilde{\xi}}$ is the Lie derivative along the vector
$\tilde{\xi}$. These lead to,
$\displaystyle\pounds_{\tilde{\xi}}g_{XX}=-2\partial_{X}\tilde{\xi}^{u}=0;$
$\displaystyle\pounds_{\tilde{\xi}}g_{uX}=-\partial_{u}\tilde{\xi}^{u}-2\kappa
X\partial_{X}\tilde{\xi}^{u}-\partial_{X}\tilde{\xi}^{X}=0~{}.$ (43)
The solutions are:
$\displaystyle\tilde{\xi}^{u}=F(u,x_{\perp});$
$\displaystyle\tilde{\xi}^{X}=-X\partial_{u}F(u,x_{\perp})~{}.$ (44)
The condition $\pounds_{\tilde{\xi}}g_{uu}=0$ automatically satisfied near the
horizon, because use of the above solutions lead to
$\pounds_{\tilde{\xi}}g_{uu}={\cal{O}}(X)$. These conditions were appeared
earlier in [39] in the context of late time symmetry near the black hole
horizon. Finally expressing (44) in the old coordinates ($t,x$) we find
$\displaystyle\xi^{t}=T-\frac{1}{2\kappa}\partial_{t}T;\,\,\,\
\xi^{x}=-x\partial_{t}T~{},$ (45)
where $T(t,x,x_{\perp})=F(u,x_{\perp})$.
Next we calculate $K^{tx}_{12}$ from (37) for our present case. Since
$S^{t}=0$, we find
$\displaystyle K^{tx}_{12}$ $\displaystyle=$
$\displaystyle\nabla_{m}\xi_{1}^{m}\xi_{2}^{t}S^{x}+(\xi_{1}^{m}\nabla_{m}\xi_{2}^{t}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{t})S^{x}$
(46) $\displaystyle+$
$\displaystyle\xi_{2}^{t}\Big{(}-2\Gamma^{x}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{x}-\nabla_{m}\nabla^{x}\xi_{1}^{m}\Big{)}$
$\displaystyle-$
$\displaystyle\xi_{2}^{x}\Big{(}-2\Gamma^{t}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{t}-\nabla_{m}\nabla^{t}\xi_{1}^{m}\Big{)}~{}.$
Now since the integration (39) will ultimately be evaluated on the horizon, we
shall find the value of each term of the above very near the horizon.
Therefore, using (29), (31) and the form of the generators (45) we obtain the
values of each term of the above expression near the horizon $x=0$ as,
$\displaystyle\nabla_{m}\xi_{1}^{m}\xi_{2}^{t}S^{x}=T_{2}\partial_{t}^{2}T_{1}-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{1}^{m}\nabla_{m}\xi_{2}^{t}S^{x}=-\kappa
T_{1}\partial_{t}T_{2}+\kappa
T_{2}\partial_{t}T_{1}+T_{1}\partial^{2}_{t}T_{2}-\frac{1}{2\kappa}\partial_{t}T_{1}\partial^{2}_{t}T_{2}$
$\displaystyle-\xi_{2}^{m}\nabla_{m}\xi_{1}^{t}S^{x}=\kappa
T_{2}\partial_{t}T_{1}-\kappa
T_{1}\partial_{t}T_{2}-T_{2}\partial^{2}_{t}T_{1}+\frac{1}{2\kappa}\partial_{t}T_{2}\partial^{2}_{t}T_{1}$
$\displaystyle-2\xi_{2}^{t}\Gamma^{x}_{mn}\nabla^{m}\xi_{1}^{n}=-T_{2}\partial^{2}_{t}T_{1}+\frac{1}{2\kappa}\partial^{2}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{2}^{t}\nabla_{m}\nabla^{m}\xi_{1}^{x}=\frac{1}{2\kappa}T_{2}\partial^{3}_{t}T_{1}-\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}-2\kappa
T_{2}\partial_{t}T_{1}+\partial_{t}T_{1}\partial_{t}T_{2}+T_{2}\partial^{2}_{t}T_{1}-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle-\xi_{2}^{t}\nabla_{m}\nabla^{x}\xi_{1}^{m}=0$ $\displaystyle
2\xi_{2}^{x}\Gamma^{t}_{mn}\nabla^{m}\xi_{1}^{n}=-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}-2\kappa
x\partial_{x}T_{1}\partial_{t}T_{2}$
$\displaystyle-\xi_{2}^{x}\nabla_{m}\nabla^{m}\xi_{1}^{t}=\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{2}^{x}\nabla_{m}\nabla^{t}\xi_{1}^{m}=-\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}$
(47)
So near the horizon (46) reduces to
$\displaystyle K^{tx}_{12}$ $\displaystyle=$ $\displaystyle-2\kappa
T_{1}\partial_{t}T_{2}+T_{1}\partial_{t}^{2}T_{2}-\frac{1}{2\kappa}\Big{(}\partial_{t}T_{1}\partial^{2}_{t}T_{2}+\partial^{2}_{t}T_{1}\partial_{t}T_{2}\Big{)}+\frac{1}{2\kappa}T_{2}\partial^{3}_{t}T_{1}+\partial_{t}T_{1}\partial_{t}T_{2}$
(48) $\displaystyle-\frac{1}{4\kappa^{2}}\partial^{3}T_{1}\partial_{t}T_{2}$
Substituting this in (39) and inserting the normalization factor, we obtain
the expression for the bracket
$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{1}\partial_{t}T_{2}-T_{2}\partial_{t}T_{1}\Big{)}-\Big{(}T_{1}\partial^{2}_{t}T_{2}-T_{2}\partial^{2}_{t}T_{1}\Big{)}$
(49) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{1}\partial^{3}_{t}T_{2}-T_{2}\partial^{3}_{t}T_{1}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{1}\partial_{t}T_{2}-\partial^{3}_{t}T_{2}\partial_{t}T_{1}\Big{)}\Big{]}~{}.$
Similarly, (27) yields,
$\displaystyle Q[\xi]=\frac{1}{8\pi G}\int d^{2}x\Big{(}\kappa
T-\frac{1}{2}\partial_{t}T\Big{)}~{}.$ (50)
A couple of comments are in order. It must be noted that in finding the
expression for the bracket (49), no use of boundary conditions (Dirichlet or
Neumann) has been used. Earlier this was used for the case of $L_{bulk}$ to
throw away the non-covariant terms in the bracket without giving any physical
meaning [25]. Also, we did not use the condition
$\delta_{\xi_{1}}\xi^{a}_{2}=0$ (see Eq. (34)) which was adopted in earlier
works. For instance, see [6, 25]. This is logically correct since
$\delta_{\xi_{1}}\xi^{a}_{2}=0$ contradicts the algebra among the Fourier
modes of the diffeomorphisms (see Eq. (52), in next section).
## 4 Virasoro algebra and entropy
In this section, the Fourier modes of the bracket and the charge will be found
out. We shall show that for a particular ansatz for the the Fourier modes of
the generators will lead to the Virasoro algebra. Finally using the Cardy
formula, the entropy will be calculated.
Consider the Fourier decompositions of $T_{1}$ and $T_{2}$:
$\displaystyle T_{1}=\displaystyle\sum_{m}A_{m}T_{m};\,\,\
T_{2}=\displaystyle\sum_{n}B_{n}T_{n}~{},$ (51)
where $A^{*}_{m}=A_{-m}$; $B^{*}_{n}=B_{-n}$. The Fourier modes $T_{m}$ will
be chosen such that the Fourier modes of the diffeomorphisms (44) obey one
sub-algebra isomorphic to Diff. $S^{1}$:
$\displaystyle i\\{\xi_{m},\xi_{n}\\}^{a}=(m-n)\xi_{m+n}^{a}~{},$ (52)
where $\\{,\\}$ is the Lie bracket. Now with the use of (51), let us first
find the Fourier modes of the bracket (49) and the charge (50). Substitution
of (51) in (49) yields,
$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{m,n}\frac{C_{m,n}}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{m}\partial_{t}T_{n}-T_{n}\partial_{t}T_{m}\Big{)}-\Big{(}T_{m}\partial^{2}_{t}T_{n}-T_{n}\partial^{2}_{t}T_{m}\Big{)}$
(53) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{m}\partial^{3}_{t}T_{n}-T_{n}\partial^{3}_{t}T_{m}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{m}\partial_{t}T_{n}-\partial^{3}_{t}T_{n}\partial_{t}T_{m}\Big{)}\Big{]}~{},$
where $C_{m,n}=A_{m}B_{n}$ and so $C^{*}_{m,n}=C_{-m,-n}$. Next defining the
Fourier modes of $[Q[\xi_{1}],Q[\xi_{2}]]$ as
$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]=\displaystyle\sum_{m,n}C_{m,n}[Q_{m},Q_{n}]~{}.$
(54)
we find
$\displaystyle[Q_{m},Q_{n}]:$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{m}\partial_{t}T_{n}-T_{n}\partial_{t}T_{m}\Big{)}-\Big{(}T_{m}\partial^{2}_{t}T_{n}-T_{n}\partial^{2}_{t}T_{m}\Big{)}$
(55) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{m}\partial^{3}_{t}T_{n}-T_{n}\partial^{3}_{t}T_{m}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{m}\partial_{t}T_{n}-\partial^{3}_{t}T_{n}\partial_{t}T_{m}\Big{)}\Big{]}~{}.$
Similarly from (50), the Fourier modes of the charge are given by
$\displaystyle Q_{m}=\frac{1}{8\pi G}\int d^{2}x\Big{(}\kappa
T_{m}-\frac{1}{2}\partial_{t}T_{m}\Big{)}~{},$ (56)
where $Q[\xi]=\displaystyle\sum_{m}A_{m}Q_{m}$. It must be noted that the
present expression (56) is exactly identical to that obtained in [27] for the
York-Gobbons-Hawking surface term whereas the other expression (55) is
different by some terms. This may be because these two surface terms are not
exactly same. But we shall show that final result for the bracket is identical
to the earlier analysis.
To calculate the above expressions (55) and (56) explicitly we need to have
$T_{m}$’s. Following the earlier arguments, we choose
$\displaystyle T_{m}=\frac{1}{\alpha}e^{im\Big{(}\alpha
t+g(x)+p.x_{\perp}\Big{)}}$ (57)
such that they satisfy the algebra (52). Here $\alpha$ is a constant, $p$ is
an integer and $g(x)=G(X=x)=-\alpha\int\frac{dx}{2\kappa x}$. This is a
standard choice in these computations and has been used several times in the
literature [5, 6, 25]. It must be noted that the transverse directions are
non-compact due to our Rindler approximations and so we will assume that
$T_{m}$ is periodic in the transverse coordinates with the periodicities
$L_{y}$ and $L_{z}$ on $y$ and $z$, respectively. Now substituting (57) in
(55) and (56) and then integrating over the cross-sectional area
$A_{\perp}=L_{y}L_{z}$ we obtain
$\displaystyle Q_{m}=\frac{A_{\perp}}{8\pi
G}\frac{\kappa}{\alpha}\delta_{m,0};$ (58) $\displaystyle
i[Q_{m},Q_{n}]:=\frac{A_{\perp}}{8\pi
G}\frac{\kappa}{\alpha}(m-n)\delta_{m+n,0}+n^{3}\frac{A_{\perp}}{16\pi
G}\frac{\alpha}{\kappa}\delta_{m+n,0}~{}.$ (59)
Using (58), (59) can be re-expressed as
$\displaystyle i[Q_{m},Q_{n}]:=(m-n)Q_{m+n}+n^{3}\frac{A_{\perp}}{16\pi
G}\frac{\alpha}{\kappa}\delta_{m+n,0}~{}.$ (60)
This is exactly identical to Virasoro algebra with the central charge $C$ is
identified as
$\displaystyle\frac{C}{12}=\frac{A_{\perp}}{16\pi G}\frac{\alpha}{\kappa}~{}.$
(61)
The zero mode eigenvalue is evaluated from (58) for $m=0$:
$\displaystyle Q_{0}=\frac{A_{\perp}}{8\pi G}\frac{\kappa}{\alpha}~{}.$ (62)
Finally using the Cardy formula [8, 9], we obtain the entropy as
$\displaystyle S=2\pi\sqrt{\frac{CQ_{0}}{6}}=\frac{A_{\perp}}{4G}~{},$ (63)
which is exactly the Bekenstein-Hawking entropy.
## 5 Conclusions
It has already been observed that several interesting features and
informations can be obtained from the surface term without incorporating the
bulk term of the gravity action. In this paper we studied the surface term of
the Einstein-Hilbert (EH) action in the context of Noether current. So far we
know, this has not been attempted before. First the current was derived for an
arbitrary diffeomorphism by using Noether prescription. Then we showed that
the charge evaluated on the horizon for a Killing vector led to the
Bekenstein-Hawking entropy after multiplied it by $2\pi/\kappa$. But till now,
it is not known about the degrees of freedom responsible for the entropy. Here
we addressed the issue and try to shed some light. This has been discussed in
the context of Virasoro algebra and Cardy formula.
In this paper, we defined the bracket among the charges. It was done, in the
sprite of our earlier works [26, 27], by taking the variation of the Noether
potential $J^{ab}[\xi_{1}]$ for a different diffeomorphism $x^{a}\rightarrow
x^{a}+\xi^{a}_{2}$, then an anti-symmetric combination between the indices $1$
and $2$ and integrating over the horizon surface. To achieve the final form,
we did not use Einstein equation of motion or any ambiguous prescription, like
vanishing of the variation of diffeomorphism parameter $\xi^{a}$, certain
boundary conditions (e.g. Dirichlet or Neumann), etc. For explicit evaluation
of our bracket, the spacetime was considered as the Rindler metric. The
relevant diffeomorphisms were identified by using a very simple, physically
motivated condition: the diffeomorphisms keep the horizon structure of the
metric invariant in some non-singular coordinate system. It turned out that
the Fourier modes of the bracket is similar to the Virasoro algebra.
Identifying the central charge and the zero mode eigenvalue and then using
these in Cardy formula we obtained exactly the Bekenstein-Hawking entropy.
Let us now discuss in details what we have achieved in this paper. We first
tabulate couple of technical points. $\bullet$ To obtain the exact expression
for entropy we did not need any hand waving prescriptions, like shifting of
the value of the zero mode eigenvalue or the specific choice of the value of
the parameter $\alpha$ appeared in Fourier modes of $T$ or both. $\bullet$ The
relevant diffeomorphisms for invariance of the horizon structure can be
obtained various ways. Here our idea was to impose minimum constraints so that
the bracket led to Virasoro algebra. It is also possible to have other choices
of constraints to find the vectors $\xi^{a}$. For instance, the whole metric
is invariant and the diffeomorphisms come out to be the Killing vectors which
in general do not exist for a general spacetime. Finally, we discuss several
conceptual aspects. The analysis presents a nice connection between the
horizon entropy and the degrees of freedom which are responsible for it. In
the usual cases, one always find that the concepts of degrees of freedom and
entropy are absolute. These does not have any observer dependent description.
But in the case of gravity, as we know, the notion of temperature and entropy
is observer dependent and hence one can expect that the degrees of freedom may
not be absolute. Here we showed that a certain class of observers which can
see the horizon and keep the horizon structure invariant, always attribute
entropy. This signifies the fact that among all the diffeomorphisms, some of
them upgraded to real degrees of freedom which were originally gauge degrees
of freedom and they have observer dependent notion. Also, everything what we
achieved here, was done from surface term. This again illustrates the
holographic nature of the gravity actions - either the bulk and the surface
terms may duplicate the same information or the surface term alone contains
all the information about the theory of gravity. Moreover, the methodology is
general enough to discuss other theories of gravity [40].
Acknowledgement
I thank T. Padmanabhan for several useful discussions.
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|
arxiv-papers
| 2012-10-25T04:10:29 |
2024-09-04T02:49:37.088443
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1210.6736"
}
|
1210.6750
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-315 LHCb-PAPER-2012-032 March 23, 2013
Measurement of the $B^{0}$–$\kern 3.73305pt\overline{\kern-3.73305ptB}{}^{0}$
oscillation frequency $\Delta m_{d}$ with the decays $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ and $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$
The LHCb collaboration†††Authors are listed on the following pages.
The $B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ oscillation
frequency $\Delta m_{d}$ is measured by the LHCb experiment using a dataset
corresponding to an integrated luminosity of $1.0\,\mathrm{fb^{-1}}$ of
proton-proton collisions at $\sqrt{s}=7\,\mathrm{TeV}$, and is found to be
$\Delta m_{d}=0.5156\pm 0.0051\,(\mathrm{stat.})\pm
0.0033\,(\mathrm{syst.}){\rm\,ps^{-1}}$. The measurement is based on results
from analyses of the decays $B^{0}\\!\rightarrow D^{-}\pi^{+}$
($D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$) and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$K^{*0}\rightarrow K^{+}\pi^{-}$) and their charge conjugated modes.
Published in Physics Letters B 719 (2013), pp. 318-325
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, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
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, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, 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,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, 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. Contu52,15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
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Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
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Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, 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.
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A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O.
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N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
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Santos35, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
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Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
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S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
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R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K.
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Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, 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
25National Center for Nuclear Research (NCBJ), 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
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The frequency $\Delta m_{d}$ of oscillations between $B^{0}$ mesons and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons also describes the mass
difference $\Delta m_{d}$ between the physical eigenstates in the
$B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ system, and has
been measured at LEP [1], the Tevatron [2, 3], and the $B$ factories [4, 5].
The current world average is $\Delta m_{d}=0.507\pm 0.004\,\mathrm{ps^{-1}}$
[6], whilst the best single measurement prior to this Letter is by the Belle
experiment, $\Delta m_{d}=0.511\pm 0.005\,(\mathrm{stat.})\pm
0.006\,(\mathrm{syst.})\,\mathrm{ps^{-1}}$ [5]. In this document the
convention $\hbar=c=1$ is used for all units.
With increasing accuracy of the measurement of $\Delta m_{s}$, the counterpart
of $\Delta m_{d}$ in the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system [7], a more precise
knowledge of $\Delta m_{d}$ becomes important, as the ratio $\Delta
m_{d}/\Delta m_{s}$ together with input from lattice QCD calculations [8, 9]
constrains the apex of the CKM unitarity triangle [10, 11]. Therefore, the
measurement of $\Delta m_{d}$ provides an important test of the Standard
Model[12, 13]. Furthermore, $\Delta m_{d}$ is an input parameter in the
determination of $\sin 2\beta$ at LHCb [14].
This Letter presents a measurement of $\Delta m_{d}$, using a dataset
corresponding to $1.0\,\mathrm{fb^{-1}}$ of $pp$ collisions at
$\sqrt{s}=7\,\mathrm{TeV}$, using the decay channels $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ ($D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$) and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$K^{*0}\rightarrow K^{+}\pi^{-}$) and their charge conjugated modes.
For a measurement of $\Delta m_{d}$, the flavour of the $B^{0}$ meson at
production and decay must be known. The flavour at decay is determined in both
decay channels from the charge of the final state kaon; contributions from
suppressed $B^{0}\rightarrow D^{+}\pi^{-}$ amplitudes are negligible. The
determination of the flavour at production is achieved by the flavour tagging
algorithms which are described in more detail in Sect. 4.
The $B^{0}$ meson is defined as unmixed (mixed) if the production flavour is
equal (not equal) to the flavour at decay. With this knowledge, the
oscillation frequency $\Delta m_{d}$ of the $B^{0}$ meson can be determined
using the time dependent mixing asymmetry
$\displaystyle\mathcal{A}_{\mathrm{mix}}^{\mathrm{signal}}(t)=\frac{N_{\text{unmixed}}(t)-N_{\text{mixed}}(t)}{N_{\text{unmixed}}(t)+N_{\text{mixed}}(t)}=\cos(\Delta
m_{d}t),$ (1)
where $t$ is the $B^{0}$ decay time and $N_{\text{(un)mixed}}$ is the number
of (un)mixed events.
## 2 Experimental setup and datasets
The LHCb detector [15] 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 $4$ 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$$\mathrm{\,Ge\kern-1.00006ptV}$ to $0.6$ % at
100$\mathrm{\,Ge\kern-1.00006ptV}$, and an impact parameter (IP) resolution of
$20$$\,\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 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.
Events including $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays are required to
have tracks with high transverse momentum $p_{\rm T}$ to pass the hardware
trigger. The software trigger requires a two-, three- or four-track secondary
vertex with a large sum of the $p_{\rm T}$ of the tracks, significant
displacement from the associated primary vertex (PV), and at least one track
with $\mbox{$p_{\rm T}$}>1.7\mathrm{\,Ge\kern-1.00006ptV}$ and a large impact
parameter with respect to that PV, and a good track fit. A multivariate
algorithm is used for the identification of the secondary vertices [16].
Events in the decay $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ are first required to pass a hardware trigger which selects a
single muon with $\mbox{$p_{\rm T}$}>1.48\mathrm{\,Ge\kern-1.00006ptV}$. In
the subsequent software trigger [16], at least one of the final state
particles is required to have $\mbox{$p_{\rm
T}$}>0.8\mathrm{\,Ge\kern-1.00006ptV}$ and a large IP with respect to all PVs
in the event. Finally, the tracks of two or more of the final state particles
are required to form a vertex which is significantly displaced from the PVs in
the event.
For the simulation studies, $pp$ collisions are generated using Pythia 6.4
[17] with a specific LHCb configuration [18]. Decays of hadronic particles are
described by EvtGen [19] in which final state radiation is generated using
Photos [20]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [21, 22] as described in
Ref. [23].
## 3 Selection
The decay time $t$ of a $B^{0}$ candidate is evaluated from the measured
momenta and from a vertex fit that constrains the $B^{0}$ candidate to
originate from the associated PV [24], and using $t=\ell\cdot m(B^{0})/p$,
with the flight distance $\ell$. The associated PV is the primary vertex that
is closest to the decaying $B^{0}$ meson. No mass constraints on the
intermediate resonances are applied. For the calculation of the invariant mass
$m$, no mass constraints are used in the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
channel, while the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass is
constrained to the world average [6] in the analysis of the decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$.
All kaons, pions and muons are required to have large $p_{\rm T}$ and well
reconstructed tracks and vertices. In addition to this, particle
identification is used to distinguish between pion, kaon and proton tracks.
The $B^{0}\\!\rightarrow D^{-}\pi^{+}$ selection requires that the $D^{-}$
reconstructed mass be in a range of $\pm 100\,\mathrm{MeV}$ around the world
average [6]. Furthermore, the $D^{-}$ decay vertex is required to be
downstream of the PV associated to the $B^{0}$ candidate.
The sum of the $D^{-}$ and $\pi^{+}$ $p_{\rm T}$ must be larger than
$5\,\mathrm{GeV}$. The $B^{0}$ candidate invariant mass must be in the
interval $5000\,\mathrm{}\leq
m(K^{+}\pi^{-}\pi^{-}\pi^{+})<5700\,\mathrm{MeV}$. Additionally, the cosine of
the pointing angle between the $B^{0}$ momentum vector and the line segment
between PV and secondary vertex is required to be larger than $0.999$.
Candidates are classified by a boosted decision tree (BDT) [25, 26] with the
AdaBoost algorithm[27]. The BDT is trained with $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}$ candidates with no particle ID criteria applied to the
daughter pions and kaons. The cut on the BDT classifier is optimised in order
to maximise the significance of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ signal.
Several input variables are used: the IP significance, the flight distance
perpendicular to the beam axis, the vertex quality of the $B^{0}$ and the
$D^{-}$ candidate, the angle between the $B^{0}$ momentum and the line segment
between PV and $B^{0}$ decay vertex, the angle between the $D^{-}$ momentum
and the line segment between PV and the $D^{-}$ decay vertex, the angle
between the $D^{-}$ momentum and the line segment between the $B^{0}$ decay
vertex and $D^{-}$ decay vertex, the IP and $p_{\rm T}$ of the $\pi^{+}$
track, and the angle between the $\pi^{+}$ momentum and the line segment
between PV and $B^{0}$ decay vertex. Only $B^{0}$ candidates with a decay time
$t>0.3\,\mathrm{ps}$ are accepted.
To suppress potential background from misidentified kaons in
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$ decays, all $D^{-}$ candidates are
removed if they have a daughter pion candidate that might pass a loose kaon
selection and are within a $\pm 25\,\mathrm{MeV}$ mass window (the $D^{-}$
mass resolution is smaller than 10 MeV) around the $D^{-}_{s}$ mass when that
pion is reconstructed under the kaon mass hypothesis.
Remaining background comes from $B^{0}\rightarrow D^{-}\rho^{+}$ and
$B^{0}\rightarrow D^{*-}\pi^{+}$ decays. In both cases the final state is
similar to the signal, except for an additional neutral pion that is not
reconstructed. This leads to two additional peaking components with invariant
masses lower than those of the signal candidates. Therefore, for the
measurement of $\Delta m_{d}$ only candidates with an invariant mass in the
range $5200\leq m<5450\,\mathrm{MeV}$ are used.
The $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
selection requires that the $K^{*0}$ candidate has a $\mbox{$p_{\rm
T}$}>2\,\mathrm{GeV}$ and $826\,\mathrm{}\leq
m(K^{+}\pi^{-})<966\,\mathrm{MeV}$.
The unconstrained $\mu^{+}\mu^{-}$ invariant mass must be within $\pm
80\,\mathrm{MeV}$ of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
[6]. $B^{0}$ candidates are required to have a large IP with respect to other
PVs in the event and the $B^{0}$ decay vertex must be significantly separated
from the PV. Additionally, $B^{0}$ candidates are required to have a
reconstructed decay time $t>0.3\,\mathrm{ps}$ and an invariant mass in the
range $5230\,\mathrm{}\leq m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\pi^{-})<5330\,\mathrm{MeV}$. To suppress potential background from
misidentified $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays, all candidates are removed for which the $K^{+}\pi^{-}$
mass is within a $\pm 10\,\mathrm{MeV}$ window around the nominal $\phi(1020)$
mass when computed under the kaon mass hypothesis for the pion. The resulting
mass distributions for the two decay channels are shown in Fig. 1.
Figure 1: Distribution of the $B^{0}$ candidate mass (black points). (Left)
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ candidates with the invariant mass pdf as
described in Sect. 6 and two additional components for the physics background
taken from MC simulated events.The blue dashed line shows the fit projection
of the signal, the dotted orange line corresponds to the combinatorial
background, the filled areas represent the physics background, and the black
solid line corresponds to the fit projection. (Right)
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
candidates, with the results of the fits described in Sect. 6 superimposed.
The blue dashed line shows the fit projection of the signal, the dotted orange
line corresponds to the combinatorial background with long lifetime and the
dash dotted red line shows the combinatorial background with short lifetime.
The black solid line corresponds to the fit projection.
## 4 Flavour tagging
This analysis makes use of a combination of opposite side taggers and the same
side pion tagger to determine the flavour of the $B^{0}$ meson at production.
The opposite side taggers, which use decay products of the $b$ quark not
belonging to the signal decay, are described in detail in Ref.[28].
The same side pion tagger uses the charge of a pion that originates from the
fragmentation process of the $B^{0}$ meson or from decays of charged excited
$B$ mesons. Pion tagging candidates are required to fulfil criteria on $p_{\rm
T}$ and particle identification, as well as their IP significance and the
difference between the $B^{0}$ candidate mass and the combined mass of the
$B^{0}$ candidate and the pion [29].
Depending on the tagging decision, a mixing state $q$ is assigned to each
candidate, to distinguish the unmixed ($q=+1$) from the mixed ($q=-1$).
Untagged events ($q=0$) are not used in this analysis. The tag and its
predicted wrong tag probability $\eta_{c}$ are evaluated for each event using
a neural network calibrated and optimized on
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ events.
To take into account a possible difference in the overall tagging performance
between the calibration channels and the decay channels used in this analysis,
the corrected wrong tag probability $\omega$ assigned to each event is
parametrised as a linear function of $\eta_{c}$ (the method is described and
tested in Ref. [28])
$\displaystyle\omega(\eta_{c}|p_{0},p_{1})=p_{0}+p_{1}(\eta_{c}-\langle\eta_{c}\rangle),$
(2)
where $p_{0}$ and $p_{1}$ are free parameters in the fit for $\Delta m_{d}$
described in Sect. 6. In this way, uncertainties due to the overall
calibration of the tagging performance are absorbed in the statistical
uncertainty on $\Delta m_{d}$ returned by the fit.
## 5 Decay time resolution and acceptance
The decay time resolution of the detector is around $0.05\,\mathrm{ps}$ [30].
This is small compared to the $B^{0}$ oscillation period of about
$12\,\mathrm{ps}$ and does not have significant impact on the measurement of
$\Delta m_{d}$. The resolution is accounted for by convolving a Gaussian
function $G(t;\sigma_{t})$, using a fixed width
$\sigma_{t}=0.05\,\mathrm{ps}$, with the signal probability density function
(PDF) from Eq. (6). Possible systematic uncertainties introduced by the
resolution are discussed in Sect. 7.
Trigger, reconstruction and selection criteria introduce efficiency effects
that depend on the decay time. While these effects cancel in the asymmetry of
Eq. (1) for signal events, they can be important for event samples that
include background. As will be shown in Sect. 6, the only relevant background
in the $B^{0}$ signal region is combinatorial in nature. For this background
the asymmetry $N_{q=1}^{\mathrm{bkg}}(t)-N_{q=-1}^{\mathrm{bkg}}(t)$ is
expected to cancel to first order as $q$ has no physical meaning. Therefore,
$\displaystyle\mathcal{A}_{\mathrm{mix}}(t)$
$\displaystyle\propto\frac{(N_{q=1}^{\mathrm{sig}}(t)+N_{q=1}^{\mathrm{bkg}}(t))-(N_{q=-1}^{\mathrm{sig}}(t)+N_{q=-1}^{\mathrm{bkg}}(t))}{(N_{q=1}^{\mathrm{sig}}(t)+N_{q=1}^{\mathrm{bkg}}(t))+(N_{q=-1}^{\mathrm{sig}}(t)+N_{q=-1}^{\mathrm{bkg}}(t))}$
(3) $\displaystyle\propto\frac{S(t)}{S(t)+B(t)}\cos(\Delta m_{d}t),$
where $N_{q=\pm 1}^{\mathrm{sig,bkg}}(t)$ denotes the number of unmixed or
mixed signal (sig) and background (bkg) events. $S(t)$ and $B(t)$ denote the
number of signal and background events as a function of the decay time. Thus,
the shapes of $S(t)$ and $B(t)$ have to be known to account for the time
dependent amplitude of the asymmetry function.
In the analysis of decays
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, the
decay time acceptance is determined from data, using a control sample of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
that is collected without applying any of the decay time biasing selection
criteria. The decay time acceptance is evaluated in bins of $t$ and is
implemented in the fit described in Sect. 6.
In the decay $B^{0}\\!\rightarrow D^{-}\pi^{+}$ there is no control dataset
that can be used to measure the decay time acceptance. From an analysis of
simulated events, it is determined that the decay time acceptance can be
described by the empirical function
$\displaystyle\epsilon_{\mathrm{acc}}(t|a_{1},a_{2})=\arctan(a_{1}\exp(a_{2}t)),$
(4)
where the parameters $a_{1}$ and $a_{2}$ are both free in the maximum
likelihood fit for $\Delta m_{d}$ described in Sect. 6.
## 6 Measurement of $\Delta m_{d}$
The value of $\Delta m_{d}$ is measured using a multi-dimensional extended
maximum likelihood fit. The $B^{0}\\!\rightarrow D^{-}\pi^{+}$ data are
described by a two component PDF in which one component describes the signal
and the other describes the combinatorial background. The signal component
consists of the sum of a Gaussian function and a Crystal Ball function [31]
with a common mean for the mass distribution, multiplied by a function
$\mathcal{P}_{\mathrm{sig}}^{t}$ to describe the decay time distribution,
$\displaystyle\mathcal{P}_{\mathrm{sig}}^{t}(t,q;\tau,\Delta
m_{d},\omega,\sigma_{t},a_{1},a_{2})\propto$
$\displaystyle\left[\Theta(t-0.3\,\mathrm{ps})\cdot\mathrm{e}^{-\frac{t}{\tau}}\left(1+q(1-2\omega(\eta_{c}|p_{0},p_{1}))\cos\left(\Delta
m_{d}t\right)\right)\otimes G(t;\sigma_{t})\right]$
$\displaystyle\cdot\epsilon_{\mathrm{acc}}(t|a_{1},a_{2}).$ (5)
Here, $\Theta(t)$ is the step function, while the $B^{0}$ lifetime $\tau$ is a
free fit parameter and the average decay time resolution $\sigma_{t}$ is
fixed. Other fit parameters are $a_{1}$ and $a_{2}$ from the decay time
acceptance function $\epsilon_{\mathrm{acc}}(t|a_{1},a_{2})$ described in
Sect. 5, as well as the parameters $p_{0}$ and $p_{1}$ from the tagging
calibration function $\omega(\eta_{c}|p_{0},p_{1})$ described in Sect. 4. Any
$B^{0}$/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production
asymmetry cancels in the mixing asymmetry function, and is neglected in this
analysis.
The combinatorial background component consists of an exponential PDF
describing the mass distribution and the decay time PDF
$\displaystyle\mathcal{P}_{\mathrm{bkg}}^{t}(t,q;\tau_{\text{bkg}},\omega_{\text{bkg}},\sigma_{t})\propto$
$\displaystyle\left[\Theta(t-0.3\,\mathrm{ps})\cdot\mathrm{e}^{-\frac{t}{\tau_{\text{bkg}}}}\left(1+q(1-2\omega_{\text{bkg}})\right)\otimes
G(t;\sigma_{t})\right].$ (6)
The PDF is similar to the signal decay time PDF with $\Delta m_{d}$ fixed to
zero. The parameter $\omega_{\text{bkg}}$ allows the PDF to reflect a possible
asymmetry in the number of events tagged with $q=\pm 1$ in the background. The
effective lifetime $\tau_{\text{bkg}}$ of the long-lived background component
is allowed to vary independently in the fit.
Possible backgrounds from misidentified or partially reconstructed decays are
studied using mass templates determined from simulation. These are found to be
negligible in the mass window $5200\,\mathrm{}\leq
m(K^{+}\pi^{-}\pi^{-}\pi^{+})<5450\,\mathrm{MeV}$ that is used in the fit
(c.f. Fig. 1).
In the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
analysis, the signal mass distribution is modelled by a double Gaussian
function with a common mean and the decay time PDF is the same as described in
Eq. (6), except for the decay time acceptance
$\epsilon_{\mathrm{acc}}(t|a_{1},a_{2})$ that is replaced by the acceptance
histogram described in Sect. 5 and has no free parameters. The mass
distribution of the combinatorial background in
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
is also described by an exponential function. However, the decay time
distribution includes a second component of shorter lifetime to account for
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates passing the
selection. The long-lived component is described by the same function as the
combinatorial background in $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays as in
Eq. (6), whereas the short-lived component is described by a simple
exponential function. No other significant source of background is found.
The resulting values for $\Delta m_{d}$ are $0.5178\pm
0.0061\,\mathrm{ps^{-1}}$ and $0.5096\pm 0.0114\,\mathrm{ps^{-1}}$ in the
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decay
modes respectively. The fit yields $87\,724\pm 321$ signal decays for
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $39\,148\pm 316$ signal decays for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$. The
fit projections onto the decay time distributions are displayed in Fig. 2 and
the resulting asymmetries are shown in Fig. 3.
No result for the $B^{0}$ lifetime is quoted, since it is affected by possible
biases due to acceptance corrections. These acceptance effects do not
influence the measurement of $\Delta m_{d}$.
Figure 2: Distribution of the decay time (black points) for (left)
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and (right)
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
candidates. The blue dashed line shows the fit projection of the signal, the
dotted orange line corresponds to the combinatorial background with long
lifetime and the dash dotted red line shows the combinatorial background with
short lifetime (only in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ mode).
The black solid line corresponds to the projection of the combined PDF.
Figure 3: Raw mixing asymmetry $\mathcal{A}_{\mathrm{mix}}$ (black points) for
(left) $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and (right)
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
candidates. The solid black line is the projection of the mixing asymmetry of
the combined PDF.
## 7 Systematic uncertainties
As explained in Sect. 5, systematic effects due to decay time resolution are
expected to be small. This is tested using samples of simulated events that
are generated with decay time distributions given by the result of the fit to
data and convolved with the average measured decay time resolution of
$0.05\,\mathrm{ps}$. The event samples are then fitted with the PDF described
in Sect. 6, with the decay time resolution parameter fixed either to zero or
to $\sigma_{t}=0.10\,\mathrm{ps}$. The maximum observed bias on $\Delta m_{d}$
of $0.0002\,\mathrm{ps^{-1}}$ is assigned as systematic uncertainty.
Systematic effects due to decay time acceptance are estimated in a similar
study, generating samples of simulated events according to the nominal decay
time acceptance functions described in Sect. 5. These samples are then fitted
with the PDF described in Sect. 6, but neglecting the decay time acceptance
function in the fit. The average observed shift of $0.0004\,\mathrm{ps^{-1}}$
($0.0001\,\mathrm{ps^{-1}}$) in $B^{0}\\!\rightarrow D^{-}\pi^{+}$
($B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$)
decays is taken as systematic uncertainty. The influence of event-by-event
variation of the decay time resolution is found to be negligible.
In order to estimate systematic effects due to the parametrisation of the
decay time PDFs for signal and background, an alternative parametrisation is
derived with a data-driven method, using sWeights[32] from a fit to the mass
distribution. The sWeighted decay time distributions for the signal and
background components are then described by Gaussian kernel PDFs, which
replace the exponential terms of the decay time PDF. This leads to a
description of the data which is independent of a model for the decay time and
its acceptance, that can be used to fit for $\Delta m_{d}$. The resulting
shifts of $0.0037\,\mathrm{ps^{-1}}$ ($0.0022\,\mathrm{ps^{-1}}$) in the decay
$B^{0}\\!\rightarrow D^{-}\pi^{+}$
($B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$) are
taken as the systematic uncertainty due to the fit model.
Uncertainties in the geometric description of the detector lead to
uncertainties in the measurement of flight distances and the momenta of final
state particles. From alignment measurements on the vertex detector, the
relative uncertainty on the length scale is known to be smaller than
$0.1\,\mathrm{\%}$. This uncertainty translates directly into a relative
systematic uncertainty on $\Delta m_{d}$, yielding an absolute uncertainty of
$0.0005\,\mathrm{ps^{-1}}$.
From measurements of biases in the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass in several run periods,
the relative uncertainty on the uncalibrated momentum scale is measured to be
smaller than $0.15\,\mathrm{\%}$. This uncertainty, however, cancels to a
large extent in the calculation of the $B^{0}$ decay time, as it affects both
the reconstructed $B^{0}$ momentum and its reconstructed mass, which is
dominated by the measured momenta of the final state particles. The remaining
systematic uncertainty on the decay time is found to be an order of magnitude
smaller than that due to the length scale and is neglected.
A summary of the systematic uncertainties can be found in Table 1. The
systematic uncertainty on the combined $\Delta m_{d}$ result is calculated
using a weighted average of the combined uncorrelated uncertainties in both
channels. The uncertainty on the length scale is fully correlated across the
channels and therefore added after the combination.
Table 1: Systematic uncertainties on $\Delta m_{d}$ in $\text{ps}^{-1}$ | $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $B^{0}\\!\rightarrow D^{-}\pi^{+}$
---|---|---
Acceptance | $0.0001$ | $0.0004$
Decay time resolution | $0.0002$ | $0.0002$
Fit model | $0.0022$ | $0.0037$
Total uncorrelated | $0.0022$ | $0.0037$
Length scale | $0.0005$ | $0.0005$
Total including correlated | $0.0023$ | $0.0037$
## 8 Conclusion
The $B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ oscillation
frequency $\Delta m_{d}$ has been measured using samples of
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
collected in $1.0\,\mathrm{\mbox{\,fb}^{-1}}$ of $pp$ collisions at
$\sqrt{s}=7\,\mathrm{TeV}$ and is found to be
$\displaystyle\Delta m_{d}(B^{0}\\!\rightarrow D^{-}\pi^{+})$
$\displaystyle=0.5178\pm 0.0061\,(\mathrm{stat.})\pm
0.0037\,(\mathrm{syst.})\,\mathrm{ps^{-1}}\ \mathrm{and}$ $\displaystyle\Delta
m_{d}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0})$
$\displaystyle=0.5096\pm 0.0114\,(\mathrm{stat.})\pm
0.0022\,(\mathrm{syst.})\,\mathrm{ps^{-1}}.$
The combined value for $\Delta m_{d}$ is calculated as the weighted average of
the individual results taking correlated systematic uncertainties into account
$\displaystyle\Delta m_{d}=0.5156\pm 0.0051\,(\mathrm{stat.})\pm
0.0033\,(\mathrm{syst.})\,\mathrm{ps^{-1}}.$
It is currently the most precise measurement of this parameter. The relative
uncertainty on $\Delta m_{d}$ is $1.2\,\mathrm{\%}$, where it is around
$0.6\,\mathrm{\%}$ for $\Delta m_{s}$ [7]. Thus, the uncertainty on the ratio
${\Delta m_{d}}/{\Delta m_{s}}$ is dominated by $\Delta m_{d}$. As the
systematic uncertainties in the $\Delta m_{d}$ and $\Delta m_{s}$ measurements
are small, the error on the ratio can be further improved with more data.
## 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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arxiv-papers
| 2012-10-25T07:12:15 |
2024-09-04T02:49:37.099184
|
{
"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, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, 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, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, 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, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, 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, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n 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, I.R. Kenyon, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, Y.M. Kim, O. Kochebina, V. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, 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, Y. Li, L. Li Gioi, 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, H. Luo, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n 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, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D.A. Milanes, M.-N. Minard, J.\n Molina 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, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, 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, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, 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, M. Smith, K. Sobczak,\n F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes,\n P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, 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, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, 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, S. Wright, S. Wu, K. Wyllie,\n Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M.\n Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Tobias Brambach",
"url": "https://arxiv.org/abs/1210.6750"
}
|
1210.6850
|
# A New Correlation between GRB X-Ray Flares and the Prompt Emission
E. Sonbas11affiliationmark: 22affiliationmark: 1University of Adiyaman,
Department of Physics, 02040 Adiyaman, Turkey 2NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA G. A. MacLachlan33affiliationmark: , A.
Shenoy33affiliationmark: , K.S. Dhuga33affiliationmark: and W. C.
Parke33affiliationmark: 3Department of Physics, The George Washington
University, Washington, DC 20052, USA edasonbas@yahoo.com
###### Abstract
From a sample of GRBs detected by the $Fermi$ and $Swift$ missions, we have
extracted the minimum variability time scales for temporal structures in the
light curves associated with the prompt emission and X-ray flares. A
comparison of this variability time scale with pulse parameters such as rise
times, determined via pulse-fitting procedures, and spectral lags, extracted
via the cross-correlation function (CCF), indicate a tight correlation between
these temporal features for both the X-ray flares and the prompt emission.
These correlations suggests a common origin for the production of X-ray flares
and the prompt emission in GRBs.
###### Subject headings:
X-ray flares, Gamma-ray bursts
## 1\. Introduction
The occurrence of X-ray flares (XRFs), associated with a large percentage of
the GRBs detected by $Swift$ is now well established (Burrows et al. 2005;
Romano et al. 2006; Falcone et al. 2006; Chincarini et al. 2007). Interest now
concentrates on how this flaring is related to the physics of the prompt
emission, early afterglow, the transition between these phases via internal
(and possible external) shock activities of GRBs, and the variability of the
central engine itself (Rees & Meszaros 1994; Kobayashi et al. 1997; Panaitescu
et al. 1999; Zhang et al. 2006; Maxham & Zhang 2009; Yu & Dai 2009).
As has been noted in a number of studies (Burrows et al. 2005; Nousek et al.
2006; O’Brien et al. 2006; Willingale et al. 2007), the X-ray light curves
observed by $Swift$/XRT follow a similar pattern; essentially comprised of a
prompt exponential decay followed by a steep power-law decay over a certain
time scale. For most GRBs, the steep decay is followed by a shallow plateau
that gradually gives way to another decreasing phase during which the X-ray
flux decays according to a different power-law over a time scale that is
significantly longer compared to the prompt emission and the early afterglow.
XRFs are known to occur predominantly during the steeply declining phase of
the X-ray light curve but flares during the plateau portion of the light curve
are not uncommon. Empirically, the behavior of the composite light curve is
consistent with the presence of two emission processes that overlap in time
(Willingale et al. 2007): a short-duration episode in addition to an episode
longer in duration but lower in luminosity. However, the underlying mechanisms
that produce the flaring activity are not fully understood. Typical questions
that arise include (a) are XRFs related to the late activity of the central
engine, and (b) is the same (internal) shock mechanism responsible for both
the prompt emission and the flaring activity? These questions have been
tackled in different ways but focus primarily on linking the observed temporal
and spectral properties of prompt emission in long bursts to similar
properties seen in bursts exhibiting XRFs: Examples of these properties and/or
relations include extending the lag-luminosity relation to X-ray flares,
comparing pulse-profiles of temporal structures in the prompt emission and
X-ray flares, and studies of evolution of spectral lag and the comparison of
spectral hardness of XRFs with that of the underlying afterglow.
The lag-luminosity relation for XRFs has been investigated by Margutti et al.
2010 and was found to be consistent with the existing relation for the prompt
emission (Ukwatta et al. 2012; Norris et al. 2007) suggesting that XRFs may
share common origins with prompt emission. A very similar study (Sultana et
al. 2012) makes a connection between the prompt emission data and the late
afterglow X-ray data and suggests that the lag-luminosity relation is valid
over a time scale well beyond the early steep-declining phase of the X-ray
light curve. Maxham & Zhang 2009 present a summary of the salient properties
of XRFs and also show, using an internal shell collision model, that the main
time histories of XRFs can be explained by the late activity of the central
engine. Another study that hints at a connection between the prompt emission
and the X-ray afterglow is that of Kocevski et al. 2007 in which the authors
examined the evolution of pulse widths of the flares and found that the
correlation between the widths of the pulses and time is consistent with the
effects of internal shocks at ever increasing collision radii. Other
techniques that seem to hold promise include the study of Epeak evolution
(Sonbas et al. 2012), the investigation of the relations predicted by various
curvature models (Liang et al. 2006, Shenoy et al. 2012), and the time
variability of bursts.
In a recent wavelet analysis (MacLachlan et al. 2013) of the gamma-ray prompt
emission from a sizable sample of long and short GRBs detected by the
$Fermi$/GBM satellite, it was shown that a variability (related to the minimum
time scale that separates white noise from red noise) of a few milliseconds is
quite common. Moreover, it was demonstrated that there is a direct link
between the shortest pulse structures as determined by the minimum time scale
and pulse-fit parameters such as rise times. This type of analysis is quite
easily extended to a larger sample of (long-duration) $Swift$ bursts, where
the time variability and pulse-fit parameters for both the prompt emission and
XRFs can be extracted and compared. In this paper, we report on the results of
such an analysis.
## 2\. Data and Methodology
The prompt emission light curves for a sample of GBM bursts were taken
directly from the work of MacLachlan et al. 2013, who used a technique based
on wavelets to determine the minimum time scale (MTS) at which scaling
processes dominate over random noise processes. The authors associate this
time scale with a transition from red-noise processes to parts of the power
spectrum dominated by white-noise or random noise components. Accordingly, the
authors note that this time scale is the shortest resolvable variability time
for physical processes intrinsic to the GRB. We have used the same technique
to extract the time scaling characteristics of the XRFs for a small sample of
$Swift$ bursts (see Table 1). For the extraction of X-ray light curves, we
used the method developed by Evans et al. 2009 in WT (windows timing) mode.
Using the software tools available directly from their website
($\texttt{h}ttp://www.swift.ac.uk/userobjects/$), we extracted X-ray-flare
light curves with different time bins. By constructing log-scale diagrams
(plot of log(variance) of the signal vs. inverse frequency in octaves) for the
sample, we have determined the minimum time scale above which scaling
processes dominate over random intrinsic noise processes. A typical example of
a bright XRF light curve and the associated log-scale diagram is shown in
Figure 1. Note that the white-noise region (plateau region of the log-scale
diagram) intersects the red-noise region (scaling region) at around octave 3.5
which corresponds to approximately 6 seconds for the light curve in question.
As noted by MacLachlan et al. 2013, the time scale for the transition from the
scaling region to the plateau region provides a measure of the smallest time
variation for physical processes intrinsic to the GRB. We associate this time
scale with the variability of the burst.
Table 1Minimum variability times for the XRFs in the sample GRB Name | $\tau$ [sec] | $\delta\tau{{}^{-}}$ [sec] | $\delta\tau{{}^{+}}$ [sec]
---|---|---|---
GRB 050502B | 10.35 | 1.89 | 2.98
GRB 050713A | 1.43 | 0.53 | 2.11
GRB 050730 | 10.93 | 1.81 | 2.70
GRB 050822 | 5.83 | 2.13 | 7.99
GRB 051117A | 13.14 | 2.91 | 5.23
GRB 060111A | 6.25 | 1.17 | 1.87
GRB 060124 | 3.02 | 1.17 | 5.33
GRB 060204B | 2.64 | 0.69 | 1.43
GRB 060210 | 3.78 | 0.97 | 2.02
GRB 060312 | 1.21 | 0.46 | 1.97
GRB 060418 | 2.13 | 0.78 | 2.94
GRB 060526 | 2.87 | 0.93 | 2.69
GRB 060607A | 3.35 | 0.78 | 1.44
GRB 060714 | 2.20 | 0.75 | 2.36
GRB 060904A | 2.92 | 1.03 | 3.57
GRB 060904B | 3.02 | 1.21 | 6.20
GRB 060929 | 9.59 | 1.67 | 2.56
GRB 070520B | 5.94 | 1.09 | 1.73
GRB 070704 | 6.89 | 1.41 | 2.40
Figure 1.— Logscale diagram (and light curve; see inset) for the bright X-ray
flare in GRB070520B: Log(Variance) of signal as a function of octave (inverse
frequency). Plateau region is white noise and the sloped region is red noise.
Using a particular functional form for pulse shapes, Margutti et al. 2010,
have extracted a set of key pulse-fit parameters such as rise times, decay
times, widths, and times since trigger for a set of bright XRFs detected by
$Swift$/XRT. Their prime interest lay in the testing of (and extending) the
validity of the lag-luminosity relation for XRFs. Our immediate interest in
this study, however, focuses on their results for the various pulse-fit
parameters such as rise times and pulse widths, because we can use these
directly to compare with the variability time scales that we have extracted
for the prompt emission and the X-ray flares. We note that the pulse rise
times are invariably shorter than the pulse widths or decay times. To augment
our sample we have also used the pulse-fit parameters from the work by
Kocevski et al. 2007. The appropriate pulse-fit parameters for the prompt
emission data were taken from the catalog produced by Bhat et al. 2012.
## 3\. Results and Discussion
Following the work of MacLachlan et al. 2012, we have used a technique based
on wavelets to extract a minimum time scale for a sample of GRBs detected by
the $Fermi$ and $Swift$ missions. Shown in Figure 2 is a plot of the pulse
rise-times versus the minimum time scale for the GRBs in our sample. We have
plotted the data as observer-frame quantities because the redshift-dependent
time dilation factor is the same for both variables: Black data points
indicate the prompt emission data (with the pulse-fit parameters from Bhat et
al. 2012); the blue and green points depict the XRF data with pulse-fit
parameters taken from Kocevski et al. 2007 and Margutti et al. 2010)
respectively. Also shown in the figure is a line depicting the equality of
time scales. The best-fit line (not shown) leads to a slope of 1.26 $\pm$
0.05. The data show a strong correlation (Spearman correlation of 0.96 $\pm$
0.02 and a Kendall correlation of 0.79 $\pm$ 0.02) between pulse rise times
and minimum time scales all the way from prompt emission to X-ray flares, i.e.
more than three decades of variability time. This result extends the work of
MacLachlan et al (2012), who examined prompt emission only, to the temporal
domain covered by XRFs and reinforces their main conclusion that the two
techniques, wavelets and pulse-fitting, can be used independently to extract a
minimum time scale for physical processes of interest as long as close
attention is paid to time binning and the proper identification of distinct
pulses. In order to pursue the apparent connection between the temporal
properties of prompt emission and the XRFs, we explore below the possible link
between another temporal property, that of spectral lags, and the MTS.
Figure 2.— Rise time versus the minimum variability time scale in the observer
frame for a sample of GRBs: Black points (prompt emission); green and blue
points (XRF data). The solid line indicates the equality of the respective
temporal scales.
For the prompt emission data, we extracted spectral lags for various observer-
frame energy bands using the CCF method described in detail by Ukwatta et al.
(2012). Some of these results have been presented by Sonbas et al. 2012. Using
the flare peak times reported by Margutti et al. (2010), we have also
extracted the spectral lags for the XRFs between the energy bands 0.3-1 keV
and the 3-10 keV respectively. A plot of the spectral lags versus the minimum
time scale for the GRBs in our sample is shown in Figure 3. Black and magenta
data points depict the prompt emission for long and short bursts; the blue
points represent the XRF data. The red line indicates the best-fit (a slope of
1.44 $\pm$ 0.07) through the combined data set. The results clearly indicate a
strong positive correlation (a Spearman correlation of 0.96 $\pm$ 0.05 and a
Kendall correlation of 0.86 $\pm$ 0.05) between the two temporal features,
spectral lag and the MTS. Also shown in Figure 3 (see insert) is a plot of the
pulse-rise times as function of the spectral lags. As expected, a positive
correlation is observed but the scatter appears to be relatively large at the
small time scales possibly indicating the difficulty in the identification and
fitting of pulses at these scales. In addition, we note, as did MacLachlan et
al 2012, that the uncertainties in the pulse rise times, quoted by Bhat et al
2012, are in many cases significantly smaller than the time binning of the
lightcurves. We follow MacLachlan et al 2012 and adjust the uncertainties in
the rise times by folding in quadrature the bin widths to the uncertainties
given by Bhat et al 2012. With this minor adjustment, we argue that the
observed correlations, taken as a whole, are suggestive of more than a trivial
connection between the prompt emission and the XRFs.
Figure 3.— Observer frame spectral lags and minimum variability time scales
are plotted for prompt and flare emission: Black points (prompt emission for
long bursts); magenta point prompt emission for short burst); and blue points
(XRF data). (insert) Observer frame rise times as function of spectral lags
for prompt and flare emission: Black points (prompt emission for long bursts);
and blue points (XRF data). In both cases, the solid line indicates the best-
fit to the data.
It is relatively straightforward to interpret the correlation between pulse
parameters and the MTS in terms of the internal shock model in which the basic
units of emission are assumed to be pulses that are produced via the collision
of relativistic shells emitted by the central engine. Quilligan et al. 2002 in
their study of the brightest BATSE bursts identified and fitted distinct
pulses and showed a strong positive correlation between the number of pulses
and the duration of the burst. More recent studies (Bhat & Guiriec 2011;
Hakkila & Cumbee 2009; Hakkila & Preece 2011) have provided further evidence
for the pulse paradigm view of the prompt emission in GRBs. Maxham & Zhang
2009, use the internal shell collision model to probe the spectral and
temporal connection between the prompt emission and the XRFs. By assuming the
Band function for the spectrum, an empirical temporal profile for the flares,
and arbitrary central engine activity, they are able to explain the major
temporal features of the XRFs, in particular, they note that the XRF time
history reflects the time history of the central engine, which reactivates
multiple times after the main prompt emission phase. Other authors (Narayan &
Kumar 2009) invoke relativistic outflow mechanisms to suggest that local
turbulence amplified through Lorentz boosting leads to causally disconnected
regions that in turn act as independent centers for the observed prompt
emission. In more recent developments (Zhang & Yan 2011, Vetere et al. 2006,
Gao et al. 2012, MacLachlan et al. 2013) there is a suggestion that the
variability may be composed of two distinct time scales; a rapidly varying
component (order of milliseconds) embedded on a slower component (order of
seconds) with the implication that these two components probe distinctly
different aspects of GRB production and propagation. Similarly, simulation
studies (Morsony et al. 2010) of the propagation of a jet through stellar
material indicate that the temporal variability at different time scales is
possibly related to the central engine and the propagation of the jet itself,
and is measurable from the prompt emission. In the model reported by Zhang and
Yan (2011), the authors invoke a magnetically dominated relativistic outflow
to suggest that it is the slow component of the variability that is linked to
the activity of the central engine and that the more rapidly varying component
is associated with magnetic turbulence. While we are not in a position to
distinguish between the aforementioned models, which incidentally are
typically used to describe the variability only in the prompt emission, it is
intriguing nonetheless that the observed correlation particularly that between
the spectral lag and the MTS connects both the prompt emission and the flaring
activity.
Kocevski et al. (2007) suggest that the rise time of the X-ray flare pulse is
related to the shell thickness as two shells collide after the second (faster)
shell catches up with the (slower) first shell. The observed rise-time is
estimated by: $\Delta t_{r}\approx\frac{\delta R}{2c\Gamma^{2}_{m}}$, where
$\delta R$ is the thickness, $\Gamma_{m}$ is the relative Lorentz factor of
the merged shells. Higher-latitude emission (for viewing angles less than the
opening angle, $\theta$, of a conical jet) will be detected as broader pulses
than lower-latitude ones. Following Zhang et al 2006, one can determine the
decay time scale as the difference in light-travel time between photons
emitted along the line of sight and the photons emitted at an angle along a
shell of a given radius, R.
$\Delta t_{decay}\approx(R/c)(\theta^{2}/2)$ (1)
For simplicity, we have omitted the redshift-dependent dilation factor. If we
can assume the decay time scale is the spectral lag due to curvature, then the
above arguments suggest a correlation between the lag and some measure of the
variability which we associate with the MTS. While our interpretation is
obviously speculative, the existence of the strong correlation, which we
contend to be of astrophysical significance, warrants detailed theoretical
investigation.
As far as the extraction of the time variability directly from data is
concerned, the wavelet method of MacLachlan et al. 2013 does not assume any
temporal profile nor does it rely on identifying distinct pulses but instead
uses the multi-resolution capacity of the wavelet technique to resolve the
smallest significant temporal scale present in the light curves (of prompt
emission and XRFs). These authors showed that the shortest pulse structures
and the MTS track each other very closely for the prompt emission. In this
work we have demonstrated that the spectral lag too tracks the MTS. Moreover,
we have extended the work of MacLachlan et al 2012 to include both the prompt
emission and the XRFs. This result, depicted in Figure 3 (supported by the
data in Figure 2), provides new and compelling evidence that, as far as these
temporal measures are concerned, the XRFs appear to be simple ’temporal
extensions’ of the pulse structures observed in the prompt emission.
## 4\. Conclusions
For a sample of long-duration GRBs detected by the $Fermi$/GBM and $Swift$
missions, we have extracted the minimum variability time scales and spectral
lags for both prompt emission and XRF light curves. In addition, we have
utilized the pulse-fit parameters presented by Margutti et al. 2010 and
Kocevski et al. 2007 from their respective studies of XRFs. We compare the
minimum variability time scale, extracted through a technique based on
wavelets, both with the pulse rise times extracted through a fitting
procedure, and spectral lags extracted via the CCF method. With these combined
results, we have studied the relationship between key parameters that describe
the temporal properties of a sample of prompt-emission and XRF light curves.
Our main results are summarized as follows; (1) The prompt emission and the
XRFs exhibit a significant positive correlation between pulse rise times and
the minimum time scale, with time scales ranging from several milliseconds to
a few seconds respectively, (2) The short-time variabilities in the prompt
emission scale over time into the short-time variabilities in XRFs and (3) The
spectral lag for both the prompt emission and the XRFs shows a strong positive
correlation with the minimum variability time scale. Taken together these
results are highly suggestive of a direct link between the mechanisms that
lead to the production of XRFs and prompt emission in GRBs.
## 5\. Acknowledgements
This work made use of data supplied by the UK $Swift$ Science Data Centre at
the University of Leicester. The work of ES was partially supported through
the $Swift$ Mission (PI: N. Gehrels) and is gratefully acknowledged.
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|
arxiv-papers
| 2012-10-25T14:19:01 |
2024-09-04T02:49:37.115925
|
{
"license": "Public Domain",
"authors": "E. Sonbas, G. A. MacLachlan, A. Shenoy, K.S. Dhuga and W. C. Parke",
"submitter": "Eda Sonbas",
"url": "https://arxiv.org/abs/1210.6850"
}
|
1210.7186
|
Recovering physical properties from narrow-band photometry
W. Schoenell1,2, R. Cid Fernandes2, N. Benítez1 and N. Vale Asari3,4
1 Instituto de Astrofísica de Andalucía (CSIC)
2 Departamento de Física - CFM - Universidade Federal de Santa Catarina
3 Institute of Astronomy, University of Cambridge
4 CAPES Foundation, Ministry of Education of Brazil
## Abstract
Our aim in this work is to answer, using simulated narrow-band photometry
data, the following general question: What can we learn about galaxies from
these new generation cosmological surveys? For instance, can we estimate
stellar age and metallicity distributions? Can we separate star-forming
galaxies from AGN? Can we measure emission lines, nebular abundances and
extinction? With what precision?
To accomplish this, we selected a sample of about 300k galaxies with good S/N
from the SDSS and divided them in two groups: 200k objects and a template
library of 100k. We corrected the spectra to $z=0$ and converted them to
filter fluxes. Using a statistical approach, we calculated a Probability
Distribution Function (PDF) for each property of each object and the library.
Since we have the properties of all the data from the starlight-SDSS database,
we could compare them with the results obtained from summaries of the PDF
(mean, median, etc).
Our results shows that we retrieve the weighted average of the log of the
galaxy age with a good error margin ($\sigma\approx 0.1-0.2$ dex), and
similarly for the physical properties such as mass-to-light ratio, mean
stellar metallicity, etc. Furthermore, our main result is that we can derive
emission line intensities and ratios with similar precision. This makes this
method unique in comparison to the other methods on the market to analyze
photometry data and shows that, from the point of view of galaxy studies,
future photometric surveys will be much more useful than anticipated.
## 1 Introduction
This paper is essencially motivated due to the J-PAS project http://j-pas.org,
which in a near future will produce 8000 sq degrees of imaging in 56
filterbands of 135Å up to magnitude $i_{AB}\approx 24$. The survey, besides
producing a very high-quality photometric redshift catalog, will provide
highly informative data important to other astronomical communities as galaxy
evolution.
Here we describe a bayesian method to “boost the spectral resolution” of these
kind of photometric data. The idea, which will be described in more detail on
section 3, is basically that if a high-resolution degrated spectra (or a set
of them) is similar to an observed j-spectra111We define a j-spectrum by the
set of photometric measurements on J-PAS filter system. then its physical
properties (and even emission lines) would be similar.
To test the efficiency of this method, we used the starlight-SDSS [2]
http://starlight.ufsc.br/ database as a sandbox. We downloaded the 926246
galaxies where there are measured physical properties and emission lines.
## 2 From SDSS to JPAS
Given a SDSS observed spectra, one can convert its energy distribution
$O_{\lambda}$ to an arbitrary observed photometric filter $l$ defined by its
transmission curve $T_{l,\lambda}$ using the simple conversion
$J_{l}=\frac{\int O_{\lambda}T_{l,\lambda}d\lambda}{\int
T_{l,\lambda}d\lambda}$ and the error on the filter $l$, $\sigma^{2}(J_{l})$,
with the relation
$\sigma^{2}(J_{l})=\Lambda_{l}\langle\sigma(n_{\lambda})^{2}t_{l,\lambda}^{2}\rangle\Delta\lambda$
where $\Lambda_{l}\equiv N_{\lambda,l}\Delta\lambda$ is the effective filter
size and $n_{\lambda}$ is the spectral error in each point.
The filtersystem curves to JPAS considering an airmass of 1.3 and the expected
CCD plus Telescope efficiencies are shown on figure 1. There will be 56
filters, but due to the spectral coverage of SDSS we removed the first one in
the blue and the four last filters giving us an filtersystem of 51 filters
plotted in solid lines. For comparsion, we plotted right below our
filtersystem the SDSS filterset u, g, r, i and z sensitivities through the
same air mass.
Figure 1: JPAS 56 filtersystem sensitivity curves. The 51 solid lines are the
filters used on this work assuming mirror and CCD characteristics and an
airmass of 1.3. We also plotted in solid fill the SDSS u, g, r, i and z
filters in the same conditions.
The main idea of this work is to obtain high precision properties from low
resolution data by taking a shortcut from what we measure from high resolution
spectra. We consider that if an pair object-template j-spectra are similar,
then their measured properties will be similar as well. On fig 2 we show two
examples of matching two objects with their five best matches in respect to
our template library library222In this paper all the galaxies which belongs to
the comparsion sample we call library template galaxies.. In green we plotted
the SDSS object and in blue its four best matches accordingly to their
$\chi^{2}$ calculated over their j-spectrum (plotted as red dots). On the
right panel we have an example of a star-forming galaxy and on the left an
early-type.
This idea is similar to the adopted by [5] and [6] and others, but with an
important difference: Here we do not compare an observed spectrum with a set
of models but we compare it to a set of another observed spectra. This not
only permits us to measure the standard physical properties such as mean age
and stellar masses that can be measured by ohter methodos, but it also allows
us to measure indirectly emission lines on data which we evidently do not have
enough resolution. This is the most relevant advantage of this method.
Figure 2: Two examples of j-spectrum and its 5 best matches based on
$\chi^{2}$. At the top, in green, the two observed spectra examples: an early-
type on left panel and a star-forming galaxy on the the right. In blue, from
top to bottom, the first five best matches. Their j-spectrum is represented by
the connected red dots. Yellow points are flagged as bad pixels and were not
considered. Stars represent the points were j-spectrum is not available.
## 3 Method
To derive the properties $p$ listed down on table 1, we calculated for 100k
objects and two samples of comparsion their likelihood functions ${\cal
L}_{i,j}=e^{-f_{L}\frac{1}{2}\chi^{2}_{i,j}}$. Where
$\chi^{2}_{i,j}=\sum_{l}\frac{1}{N_{\rm
good}}(O_{l,i}-a_{i,j}B_{l,j})^{2}w_{l,i,j}^{2}$ with a scaling factor
$a_{i,j}=\frac{\sum_{l}O_{l,i}B_{l,j}w_{l,i,j}^{2}}{\sum_{l}B_{l,j}^{2}w_{l,i,j}^{2}}$
which is determined by interacting the calculation of $a$ and $\chi^{2}$ until
a convergence critery of $\Delta a_{i,j}<10^{-5}$ is accomplished. In our
simulations, this takes no more than four interactions. The $f_{L}$ term
adjusts the width of the PDF. It can be adjusted to minimize errors, but we
will not treat it here.
The weighting used to the spectra was defined by
$w^{2}_{l,i,j}=\left(\frac{\langle
O^{2}_{l,i}\rangle}{O^{2}_{l,i}}\right)\left(\frac{1}{\sigma^{2}(O_{l,i})+a^{2}_{i,j}\sigma^{2}(B_{l,i})}\right)$.
This was selected to have an unbiased weight in the amplitude of the spectra.
In other words, we would like to give the same importance to the parts with
high fluxes (e.g. emission lines) than that to that regions that have lower
ones (e.g. continuum).
With the likelihood for each object and base, we calculated as the output of
our method a PDF estimator. In our case, to estimate the output, we used the
likelihood-weighted average $\overline{p_{i}}=\frac{\sum_{j}p_{j}{\cal
L}_{i,j}}{\sum_{j}{\cal L}_{i,j}}=p_{out,i}$
Figure 3: Examples of normalized PDF distributions to age, extinction,
H$\alpha$ and [N ii]/H$\alpha$. The left and right boxes corresponds to the
left and right observed galaxaies shown on fig. 2. In blue dashed line we show
the distribution of our base of templates, in magenta the likelihood
distribution (or the posterior) and the horizontal lines represents in solid
black the 16th and 84th percentiles, in red dashed the average and in solid
green the value measured by starlight directly on the spectrum.
## 4 Sample selection
Our main sample was retrieved from the starlight-SDSS database based on very
wide criteria, trying to get all kinds of objects. Firstly, we separated from
the database the galaxies which were in the SDSS main galaxy sample [7], then
we selected those who do not have any bad pixel in intervals of 31Å centered
on the emission lines H$\alpha$, H$\beta$, [N ii]$\lambda$6584, [O
ii]$\lambda$3727 and [O iii]$\lambda$5007 to assure that when we do not
measure an emission line it is because it is too weak to measure and not
because of any observational error. A last selection in redshift ($0.01\leq
z\leq 0.11$) was applied in order to have spectral coverage on the wavelength
interval where JPAS will observe. Those criteria reduced the total number of
galaxies on our sample from 926246 to 299253 galaxies. All these galaxies were
corrected to the rest-frame. As a quick-look test, we will not treat the
redshift as a variable here and all results will be shown to $z=0$.
We then converted all the observed spectra to J-spectra. In case of problems
on the SDSS spectra (e.g. bad pixels), we changed the observed flux to the
best fit from the starlight in the cases where the filters have less bad
pixels than 50% of the filter width. Otherwise, the filter is flagged to be
neglected.
Then, we divided the sample in two sub-samples: one with the galaxies with
$S/N>20$ (113821 galaxies) which we call mother library and other one with
galaxies with $S/N<20$ (185432 galaxies) which we call object sample. All
comparsions made in this paper will be in respect of a set of objects and
their PDFs calculated over a given library which is a set of galaxies from the
mother library. From the mother library, we selected two samples of galaxies
based on two independent diagrams.
The first one, which we call CMD, is based on the physical analog to the
color-magnitude diagram to galaxies, the $\log\ M_{\star}$ – $\langle\log\
t\rangle$ diagram. We chopped this diagram in boxes of 0.1 dex, and on each
box we got 10% of the objects distributed along the $A_{V}$ axis. This gives
us 11952 galaxies on this library.
The second library, which we call WHAN, is based on the WHAN diagnostic
diagram introduced by [3] and divide the galaxies basically between star-
forming, active nuclei and passive or retired galaxies. We did the same cut on
this diagram as we did on CMD library, but here we changed the $A_{V}$ to the
emission line ratio $\log[\rm{N}\,\textsc{ii}]/{\rm H}alpha$. This gives us
27537 galaxies on this library.
## 5 Results
To estimate our method’s precision, we compared the properties derived by our
method (output) on the low-resolution j-spectra with the values derived using
starlight on the high-resolution SDSS spectra (input). So, to do this
comparsion, we evaluate, for each property $p$ and object $i$, the $\Delta
p_{i}=p_{i,\rm output}-p_{i,\rm input}$.
This experiment shows the potential of the proposed method and accomplish with
our initial objective which is to test the precision of galaxy properties with
it. From table 1, we can resume the precision of our method: It can measure
physical properties like age and extinction with typical precision of $0.2$
dex (or mag, in the case of $A_{V}$) and emission lines with $0.3$ dex and our
library selection does not affected the final result, probably because we have
a number of templates which are in both libraries and/or beacause we have
oversampled libraries with $N_{\rm galaxies}$ of about 10k galaxies.
Once more, our main result is that we can measure emission lines without
having sufficient spectral resolution to do it directly on our data.
Table 1: Results Property | $\overline{\Delta p}$ | $\sigma(\Delta p)$ (CMD) | $\overline{\Delta p}$ | $\sigma(\Delta p)$ (WHAN)
---|---|---|---|---
$A_{V}$ | 0.023 | 0.106 | 0.023 | 0.101
$\langle\log t_{\star}\rangle_{L}$ | -0.018 | 0.199 | -0.013 | 0.192
$\langle\log Z_{\star}\rangle_{L}$ | -0.021 | 0.144 | -0.020 | 0.141
$\log M/L_{r}$ | -0.045 | 0.114 | -0.040 | 0.110
$\log W_{[O_{II}]}$ | 0.051 | 0.223 | 0.040 | 0.218
$\log W_{H\beta}$ | 0.024 | 0.145 | 0.030 | 0.143
$\log W_{[O_{III}]}$ | 0.046 | 0.245 | 0.029 | 0.232
$\log W_{H\alpha}$ | 0.010 | 0.160 | 0.022 | 0.157
$\log W_{[\rm{N}II]}$ | -0.028 | 0.159 | -0.010 | 0.156
$\log[N_{II}]/H_{\alpha}$ | -0.045 | 0.141 | -0.044 | 0.146
$\log[O_{III}]/H_{\beta}$ | 0.026 | 0.250 | -0.000 | 0.238
$\log H_{\alpha}/H_{\beta}$ | -0.011 | 0.107 | -0.010 | 0.114
$\log S_{II}/H_{\alpha}$ | -0.006 | 0.172 | -0.016 | 0.174
$\log[O_{II}]/H_{\beta}$ | 0.036 | 0.202 | 0.016 | 0.198
$\log[O_{III}]/[N_{II}]$ | 0.075 | 0.265 | 0.044 | 0.252
## Acknowledgments
We thank financial support from CNPq, FAPESP, Instituto de Astronomía de
Andalucía and the CNPq’s Instituto Nacional de Ciência e Tecnologia -
Astrofísica. NVA has been supported by CAPES (proc. no. 6382-10-0). WS has
been supported by Spanish Ministerio de Economía y Competitividad (grant
AYA2010-22111-C03-01) and Brazilian INCT-A. The SEAGal Team wishes to thank
all researchers involved in the Sloan Digital Sky Survey for their dedication
to a project which has made the present work possible.
## References
* [1] Benítez, N., 2000, ApJ, 536, 571
* [2] Cid Fernandes R. et al, 2005, MNRAS 358, 363
* [3] Cid Fernandes R. et al, 2010, MNRAS, 403, 1036
* [4] Driver, S. P, 2011, arXiv:1112.6244
* [5] Gallazzi, A. et al, 2005, MNRAS 362, 41
* [6] Kauffmann, G. et al, D., 2003, MNRAS, 341, 33
* [7] York, D. G et al, 2000, AJ, 120, 1579
|
arxiv-papers
| 2012-10-26T16:47:04 |
2024-09-04T02:49:37.138731
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "William Schoenell, Roberto Cid Fernandes, Narciso Ben\\'itez and\n Natalia Vale Asari",
"submitter": "William Schoenell",
"url": "https://arxiv.org/abs/1210.7186"
}
|
1210.7191
|
1]Met Office Hadley Centre, FitzRoy Road, Exeter, EX1 3PB, UK 2]Cooperative
Institute for Climate and Satellites, North Carolina State University and
NOAA’s National Climatic Data Center, Patton Avenue, Asheville, NC, 28801, USA
3]NOAA’s National Climatic Data Center, Patton Avenue, Asheville, NC, 28801,
USA 4]National Center for Atmospheric Research (NCAR), P.O. Box 3000, Boulder,
CO 80307, USA *]formerly at: Met Office Hadley Centre, FitzRoy Road, Exeter,
EX1 3PB, UK
R. J. H. Dunn (robert.dunn@metoffice.gov.uk)
21 May 2012
25 October 2012 8 5
# HadISD: a quality-controlled global synoptic report database for selected
variables at long-term stations from 1973–2011
R. J. H. Dunn K. M. Willett P. W. Thorne E. V. Woolley I. Durre A. Dai
D. E. Parker R. S. Vose [ [ [ [ [
(19 April 2012; 24 September 2012; 26 September 2012; 2012)
###### Abstract
This paper describes the creation of HadISD: an automatically quality-
controlled synoptic resolution dataset of temperature, dewpoint temperature,
sea-level pressure, wind speed, wind direction and cloud cover from global
weather stations for 1973–2011. The full dataset consists of over 6000
stations, with 3427 long-term stations deemed to have sufficient sampling and
quality for climate applications requiring sub-daily resolution. As with other
surface datasets, coverage is heavily skewed towards Northern Hemisphere mid-
latitudes.
The dataset is constructed from a large pre-existing ASCII flatfile data bank
that represents over a decade of substantial effort at data retrieval,
reformatting and provision. These raw data have had varying levels of quality
control applied to them by individual data providers. The work proceeded in
several steps: merging stations with multiple reporting identifiers;
reformatting to netCDF; quality control; and then filtering to form a final
dataset. Particular attention has been paid to maintaining true extreme values
where possible within an automated, objective process. Detailed validation has
been performed on a subset of global stations and also on UK data using known
extreme events to help finalise the QC tests. Further validation was performed
on a selection of extreme events world-wide (Hurricane Katrina in 2005, the
cold snap in Alaska in 1989 and heat waves in SE Australia in 2009). Some very
initial analyses are performed to illustrate some of the types of problems to
which the final data could be applied. Although the filtering has removed the
poorest station records, no attempt has been made to homogenise the data thus
far, due to the complexity of retaining the true distribution of high-
resolution data when applying adjustments. Hence non-climatic, time-varying
errors may still exist in many of the individual station records and care is
needed in inferring long-term trends from these data.
This dataset will allow the study of high frequency variations of temperature,
pressure and humidity on a global basis over the last four decades. Both
individual extremes and the overall population of extreme events could be
investigated in detail to allow for comparison with past and projected
climate. A version-control system has been constructed for this dataset to
allow for the clear documentation of any updates and corrections in the
future.
The Integrated Surface Database (ISD) held at NOAA’s National Climatic Data
Center is an archive of synoptic reports from a large number of global surface
stations (Smith et al., 2011; Ame, 2004; see
http://www.ncdc.noaa.gov/oa/climate/isd/index.php). It is a rich source of
data useful for the study of climate variations, individual meteorological
events and historical climate impacts. For example, these data have been
applied to quantify precipitation frequency (Dai, 2001a) and its diurnal cycle
(Dai, 2001b), diurnal variations in surface winds and divergence field (Dai
and Deser, 1999), and recent changes in surface humidity (Dai, 2006; Willett
et al., 2008), cloudiness (Dai et al., 2006) and wind speed (Peterson et al.,
2011).
The collation of ISD, merging and reformatting to a single format from over
100 constituent sources and three major databanks represented a substantial
and ground-breaking effort undertaken over more than a decade at NOAA NCDC.
The database is updated in near real-time. A number of automated quality
control (QC) tests are applied to the data that largely consider internal
station series consistency and are geographically invariant in their
application (i.e. threshold values are the same for all stations regardless of
the local climatology). These procedures are briefly outlined in Ame (2004)
and (Smith et al., 2011). The tests concentrate on the most widely used
variables and consist of a mix of logical consistency checks and outlier type
checks. Values are flagged rather than deleted. Automated checks are essential
as it is impractical to manually check thousands of individual station records
that could each consist of several tens of thousands of individual
observations. It should be noted that the raw data in many cases have been
previously quality controlled manually by the data providers, so the raw data
are not necessarily completely raw for all stations.
The ISD database is non-trivial for the non-expert to access and use, as each
station consists of a series of annual ASCII flatfiles (with each year being a
separate directory) with each observation representing a row in a format akin
to the synoptic reporting codes that is not immediately intuitive or amenable
to easy machine reading (http://www1.ncdc.noaa.gov/pub/data/ish/ish-format-
document.pdf). NCDC, however, provides access to the ISD database using a GIS
interface. This does give the ability for users to select parameters and
stations and output the results to a text file. Also, a subset of the ISD
variables (air temperature, dewpoint temperature, sea level pressure, wind
direction, wind speed, total cloud cover, one-hour accumulated liquid
precipitation, six-hour accumulated liquid precipitation) is available as ISD-
Lite in fixed-width format ASCII files. However, there has been no selection
on data or station quality. In this paper we outline the steps undertaken to
provide a new quality-controlled version, called HadISD, which is based on the
raw ISD records, in netCDF format for selected variables for a subset of the
stations with long records. This new dataset will allow the easy study of the
behaviour of short-timescale climate phenomena in recent decades, with the
subsequent comparison to past climates and future climate projections.
One of the primary uses of a sub-daily resolution database will be the
characterisation of extreme events for specific locations, and so it is
imperative that multiple, independent efforts be undertaken to assess the
fundamental quality of individual observations. We also therefore undertake a
new and comprehensive quality control of the ISD, based upon the raw holdings,
which should be seen as complementary to that which already exists. In the
same way that multiple independent homogenisation efforts have informed our
understanding of true long-term trends in variables such as tropospheric
temperatures (Thorne et al., 2011), numerous independent QC efforts will be
required to fully understand changes in extremes. Arguably, in this context
structural uncertainty (Thorne et al., 2005) in quality control choices will
be as important as that in any homogenisation processes that were to be
applied in ensuring an adequate portrayal of our true degree of uncertainty in
extremes behaviour. Poorly applied quality control processes could certainly
have a more detrimental effect than poor homogenisation processes. Too
aggressive and the real tails are removed; too liberal and data artefacts
remain to be misinterpreted by the unwary. As we are unable to know for
certain whether a given value is truly valid, it is impossible to
unambiguously determine the prevalence of type-I and type-II errors for any
candidate QC algorithm. In this work, type-I errors occur when a good value is
flagged, and type-II errors are when a bad value is not flagged.
Quality control is therefore an increasingly important aspect of climate
dataset construction as the focus moves towards regional- and local-scale
impacts and mitigation in support of climate services (Doherty et al., 2008).
The data required to support these applications need to be at a much finer
temporal and spatial resolution than is typically the case for most climate
datasets, free of gross errors and homogenised in such a way as to retain the
high as well as low temporal frequency characteristics of the record.
Homogenisation at the individual observation level is a separate and arguably
substantially more complex challenge. Here we describe solely the data
preparation and QC. The methodology is loosely based upon that developed in
Durre et al. (2010) for daily data from the Global Historical Climatology
Network. Further discussion of the data QC problem, previous efforts and
references can be found therein. These historical issues are not covered in
any detail here.
Section 1 describes how stations that report under varying identifiers were
combined, an issue that was found to be globally insidious and particularly
prevalent in certain regions. Section 2 outlines selection of an initial set
of stations for subsequent QC. Section 3 outlines the intra- and inter-station
QC procedures developed and summarises their impact. We validate the final
quality-controlled dataset in Sect. 4. Section LABEL:sec:finalselection
briefly summarises the final selection of stations, and Sect.
LABEL:sec:nomenclature describes our version numbering system. Section
LABEL:sec:uses outlines some very simple analyses of the data to illustrate
their likely utility, whilst Sect. LABEL:sec:summary concludes.
The final data are available through http://www.metoffice.gov.uk/hadobs/hadisd
along with the large volume of process metadata that cannot reasonably be
appended to this paper. The database covers 1973 to end-2011, because
availability drops off substantially prior to 1973 (Willett et al., 2008). In
future periodic updates are planned to keep the dataset up-to-date.
Figure 1: Top: locations of assigned composite stations from the ISD database
before any station selection and filtering. Only 943 of these 1504 stations
were passed into the QC process. Bottom: locations of 83 duplicated stations
identified by the inter-station duplicate check (Sect. 3.1.1, test 1.)
Figure 2: Station distributions for different minimum reporting frequencies
for a 1976–2005 climatology period. For presentational purposes we show the
number of stations within $1.5^{\circ}\times 1.5^{\circ}$ grid boxes. Hourly
(top panel); 3-hourly (middle panel) and 12-hourly (bottom panel).
## 1 Compositing stations
The ISD database archives according to the station identifier (ID) appended to
the report transmission, resulting in around 28 000 individual station IDs.
Despite efforts by the ISD dataset creators, this causes issues for stations
that have changed their reporting ID frequently or that have reported
simultaneously under multiple IDs to different ISD source databanks (i.e.
using a WMO identifier over the GTS and a national identifier to a local
repository). Many such station records exist in multiple independent station
files within the ISD database despite in reality being a single station
record. In some regions, e.g. Canada and parts of Eastern Europe, WMO station
ID changes have been ubiquitous, so compositing is essential for record
completeness.
Table 1: Hierarchical criteria for deciding whether given pairs of stations in the ISD master listing were potentially the same station and therefore needed assessing further. The final value arising for a given pair of stations is the sum of the values for all hierarchical criteria met, e.g. a station pair that agrees within the elevation and latitude/longitude bounds but for no other criteria will have a value of 7. | Hierarchical
---|---
Criteria | criteria value
Reported elevation within 50 m | 1
Latitude within 0.05∘ | 2
Longitude within 0.05∘ | 4
Same country | 8
WMO identifier agrees and not missing, same country | 16
USAF identifier agrees in first 5 numbers and not missing | 32
Station name agrees and country either the same or missing | 64
METAR (Civil aviation) station call sign agrees | 128
|
Station location and ID information were read from the ISD station inventory,
and the potential for station matches assessed by pairwise comparisons using a
hierarchical scoring system (Table 1). The inventory is used instead of within
data file location information as the latter had been found to be
substantially more questionable (Neal Lott, personal communication, 2008).
Scores are high for those elements which, if identical, would give high
confidence that the stations are the same. For example it is highly
implausible that a METAR call sign will have been recycled between
geographically distinct stations. Station pairs that exceeded a total score of
14 are selected for further analysis (see Table 1). Therefore, a candidate
pair for consideration must at an absolute minimum be close in distance and
elevation and from the same country, or have the same ID or name. Several
stations appeared in more than one unique pairing of potential composites.
These cases were combined to form consolidated sets of potential matches. Some
of these sets comprise as many as five apparently unique station IDs in the
ISD database.
Figure 3: Flow diagram of the testing procedure. Final output is available on
www.metoffice.gov.uk/hadobs/hadisd. Other outputs (yellow trapezoids) are
available on request.
Figure 4: Frequent value check (test 4) for station 037930-99999, Anvil
Green, Kent, UK (51.22∘ N, 1.000∘ E, 140 m), showing temperature. Top:
Histogram with logarithmic y-axis for entire station record showing the bins
which have been identified as being likely frequent values. Bottom: red points
show values removed by this test and blue points by other tests for the years
1977, 1980 and 1983. The panel below each year indicates which station the
observations come from in the case of a composite (not relevant here but is
relevant in other station plots so included in all).
For each potential station match set, in addition to the hierarchical scoring
system value (Table 1), were considered graphically the following quantities:
00:00 UTC temperature anomalies from the ISD-lite database
(http://www.ncdc.noaa.gov/oa/climate/isd/index.php) using anomalies relative
to the mean of the entire set of candidate station records; the ISD-lite data
count by month; and the daily distribution of observing times. This required
in-depth manual input taking roughly a calendar month to complete resulting in
1504 likely composite sets assigned as matches (comprising 3353 unique station
IDs, Fig. 1). Of these just over half are very obviously the same station. For
example, data ceased from one identifier simultaneously with data commencing
from the other where the data are clearly not substantially inhomogeneous
across the break; or the different identifiers report at different synoptic
hours, but all other details are the same. Other cases were less clear, in
most cases because data overlap implied potentially distinct stations or
discontinuities yielding larger uncertainties in assignment. Assigned sets
were merged giving initial preference to longer record segments but allowing
infilling of missing elements where records overlap from the shorter segment
records to maximise record completeness. This matching of stations was carried
out on an earlier extraction of the ISD dataset spanning 1973 to 2007. The
final dataset is based on an extraction from the ISD of data spanning 1973 to
end-2011, and the station assignments have been carried over with no
reanalysis.
There may well be assigned composites that should be separate stations,
especially in densely sampled regions of the globe. If the merge were being
done for the raw ISD archive that constitutes the baseline synoptic dataset
held in the designated WMO World Data Centre, then far more meticulous
analysis would be required. For this value added product a few false station
merges can be tolerated and later amended/removed if detected. The station IDs
that were combined to form a single record are noted in the metadata of the
final output file where appropriate. A list of the identifiers of the 943
stations in the final dataset, which are assigned composites as well as their
component station IDs, can be found on the HadISD website.
Figure 5: Schematic for the diurnal cycle check. (a) An example time series
for a given day. There are observations in more than 3 quartiles of the day
and the diurnal range is more than 5 ∘C so the test will run. (b) A sine curve
is fitted to the day observations. In this schematic case, the best fit that
occurs has a 9-h shift. The cost function used to calculate the best fit is
indicated by the dotted vertical lines. (c) The cost function distribution for
each of the possible 24 offsets of the sine curve for this day. The terciles
of the distribution are shown by horizontal black dotted lines. Where the cost
function values enter the second tercile determines the uncertainty (vertical
blue lines). The larger of the two differences (in this case 9 to 15 = 6 h) is
chosen as the uncertainty. So if the climatological value is between 3 and 15
h, then this day does not have an anomalous diurnal cycle phase. Figure 6:
Distributional gap check (test 6) example for composite station 714740-99999,
Clinton, BC, Canada (51.15∘ N, 121.50∘ W, 1057 m), showing temperature for the
years 1974, 1975 and 1984. Red points show values removed by this test and
blue points by other tests (none for the years shown). The panel below each
year shows whether the data in the composited station come from the named
station (blue) or a matched station (green). There is no change in source
station within 1975, and so the compositing has not caused the clear offset
observed therein, but the source station has changed for 1984 compared to the
other two years.
## 2 Selection and retrieval of an initial set of stations
The ISD consists of a large number of stations, some of which have reported
only rarely. Of the 30 000 stations, about 2/3 have observations for 30 yr or
fewer and several thousand have small total file sizes, corresponding to few
observations. However, almost 2000 stations have long records extending 60 or
more years between 1901 and end-2011. Most of these have large total file
sizes indicating quasi-continuous records, rather than only a few observations
per year. To simplify selection, only stations that may plausibly have records
suitable for climate applications were considered, using two key requirements:
length of record and reporting frequency. The latter is important for
characterisation of extremes, as too infrequent observing will greatly reduce
the potential to capture both truly extreme events and the diurnal cycle
characteristics. A degree of pre-screening was therefore deemed necessary
prior to application of QC tests to winnow out those records that would be
grossly inappropriate for climate studies.
To maximise spatial coverage, network distributions for four climatology
periods (1976–2005, 1981–2000, 1986–2005 and 1991–2000) and four different
average time steps between consecutive reports (hourly, 3-hourly, 6-hourly,
12-hourly) were compared. For a station to qualify for a climatology period,
at least half of the years within the climatology period must have a
corresponding data file regardless of its size. No attempt was made at this
very initial screening stage to ensure these are well distributed within the
climatological period. To assign the reporting frequency, (up to) the first
250 observations of each annual file were used to work out the average
interval between consecutive observations. With hourly frequency, stipulation
coverage collapses to essentially NW Europe and North America (Fig. 2). Three-
hourly frequency yields a much more globally complete distribution. There is
little additional coverage or station density derived by further coarsening to
6- (not shown) or 12-hourly except in parts of Australia, South America and
the Pacific. Sensitivity to choice of climatology period is much smaller (not
shown), so a 1976–2005 climatology period and a 3-hourly reporting frequency
were chosen as a minimum requirement. This selection resulted in 6187 stations
selected for further analysis.
Table 2: Variables extracted from the ISD database and converted to netCDF
for subsequent potential analysis. The second column indicates whether the
value is an instantaneous measure or a time-averaged quantity. The third
column shows the subset that we quality controlled and the fourth column the
set included within the final files which includes some non-quality controlled
variables.
| Instantaneous (I) | Subse- | Output
---|---|---|---
| or past period (P) | quent | in final
Variable | measurement | QC | dataset
Temperature | I | Y | Y
Dewpoint | I | Y | Y
SLP | I | Y | Y
Total cloud cover | I | Y | Y
High cloud cover | I | Y | Y
Medium cloud cover | I | Y | Y
Low cloud cover | I | Y | Y
Cloud base | I | N | Y
Wind speed | I | Y | Y
Wind direction | I | Y | Y
Present significant weather | I | N | N
Past significant weather #1 | P | N | Y
Past significant weather #2 | P | N | N
Precipitation report #1 | P | N | Y
Precipitation report #2 | P | N | N
Precipitation report #3 | P | N | N
Precipitation report #4 | P | N | N
Extreme temperature report #1 | P | N | N
Extreme temperature report #2 | P | N | N
Sunshine duration | P | N | N
| | |
ISD raw data files are (potentially) very large ASCII flat files – one per
station per year. The stations data were converted to hourly resolution netCDF
files for a subset of the variables including both WMO-designated mandatory
and optional reporting parameters. Details of all variables retrieved and
those considered further in the current quality control suite are given in
Table 2. There are some stations which for part of the analysed period report
at sub-hourly frequencies. As both temperature and dewpoint temperature are
required to be measured simultaneously for any study on humidity to be
reliably carried out, reports that have both temperature and dewpoint
temperature observations are favoured (under the assumption that the readings
were taken at close proximity in space and time) over those reports that have
one or the other (but not both), even if the reports with both observations
are further from the full hour. In cases where observations only have
temperature or dewpoint temperature (and never both), then those with
temperature are favoured, even if these are further from the full hour (00
min). All variables in a single HadISD hourly time step always derive from a
single ISD time step, with no blending between the various within-hour
reports. However the HadISD times are always converted to the nearest whole
hour. To minimise data storage the time axis is collapsed in the netCDF files
so that only time steps with observations are retained.
Figure 7: Distributional gap check (test 6) example when comparing all of a
given calendar month in the dataset for composite station 476960-43323,
Yokosuka, Japan (35.58∘ N, 139.667∘ E, 530 m), for (top) temperature and
(middle) dewpoint temperature for the years 1973, 1974 and 1985. Red points
show values removed by this test and blue points by other tests (in this case,
mainly the diurnal cycle check). The problem for this station affects both
variables, but the tests are applied separately. There is no change in source
station in any of the years, and so compositing has not caused the bad data
quality of this station. Figure 8: Distribution of the observations from all
Januaries in the station record for composite station 476960-43323, Yokosuka,
Japan (35.58∘ N, 139.667∘ E, 530 m). The population highlighted in red is
removed by the distributional gap check (test 6), as shown in Fig. 7. Note
logarithmic y-axis.
## 3 Quality control steps and analysis
An individual hourly station record with full temporal sampling from 1973 to
2011 could contain in excess of 340 000 observations and there are $>6$ 000
candidate stations. Hence, a fully automated quality-control procedure was
essential. A similar approach to that of GHCND (Durre et al., 2010) was taken.
Intra-station tests were initially trained against a single (UK) case-study
station series with bad data deliberately introduced to ensure that the tests,
at least to first order, behaved as expected. Both intra- and inter-station
tests were then further designed, developed and validated based upon expert
judgment and analysis using a set of 76 stations from across the globe (listed
on the HadISD website). This set included both stations with proportionally
large data removals in early versions of the tests and GCOS (Global Climate
Observing System) Surface Network stations known to be highly equipped and
well staffed so that major problems are unlikely. The test software suite took
a number of iterations to obtain a satisfactorily small expert judgement false
positive rate (type I error rate) and, on subjective assessment, a clean
dataset for these stations. In addition, geographical maps of detection rates
were viewed for each test and in total to ensure that rejection rates did not
appear to have a real physical basis for any given test or variable. Deeper
validation on UK stations (IDs beginning 03) was carried out using the well-
documented 2003 heat wave and storms of 1987 and 1990. This resulted in a
further round of refining, resulting in the tests as presented below.
Wherever distributional assumptions were made, an indicator that is robust to
outliers was required. Pervasive data issues can lead to an unduly large
standard deviation ($\sigma$) being calculated which results in the tests
being too conservative. So, the inter-quartile range (IQR) or the median
absolute deviation (MAD) was used instead; these sample solely the (presumably
reasonable) core portion of the distribution. The IQR samples 50 per cent of
the population, whereas $\pm 1$$\sigma$ encapsulates 68 per cent of the
population for a truly normal distribution. One IQR is 1.35$\sigma$, and one
MAD is 0.67$\sigma$ if the underlying data are truly normally distributed.
Figure 9: Repeated streaks/unusual streak frequency check (test 8) example
for composite station 724797-23176 (Milford, UT, USA; 38.44∘ N, 112.038∘ W,
1534 m), for dewpoint temperature in 1982, illustrating frequent short
streaks. Red points show values removed by this test and blue points by other
tests. The panel below each year shows whether the data in the composited
station come from the named station (blue) or a matched station (orange).
There is no change in source station in 1982, and so the compositing has not
caused the streaks observed in 1982, but a different station is used in 1998
compared to the other two years. Figure 10: Climatological outlier check
(test 9) for 040180-16201 (Keflavik, Iceland, 63.97∘ N, 22.6∘ W, 50 m) for
temperature showing the distribution for May. Note logarithmic y-axis. The
threshold values are shown by the vertical lines. The right-hand side shows
the flagged values which occur further from the centre of the distribution
than the gap and the threshold value. The left-hand side shows observations
which have been tentatively flagged, as they are only further from the centre
of the distribution than the threshold value. It is therefore not clear if the
large tail is real or an artefact.
The Durre et al. (2010) method applies tests in a deliberate order, removing
bad data progressively. Here, a slightly different approach is taken including
a multi-level flagging system. All bad data have associated flags identifying
the tests that they failed. Some tests result in instantaneous data removal
(latitude-longitude and station duplicate checks), whereas most just flag the
data. Flagged, but retained, data are not used for any further derivations of
test thresholds. However, all retained data undergo each test such that an
individual observation may receive multiple flags. Furthermore, some of the
tests outlined in the next section set tentative flags. These values can be
reinstated using comparisons with neighbouring stations in a later test, which
reduces the chances of removing true local or regional extremes. The tests are
conducted in a specified order such that large chunks of bad data are removed
from the test threshold derivations first and so the tests become
progressively more sensitive. After an initial latitude-longitude check (which
removed one station) and a duplicate station check, intra-station tests are
applied to the station in isolation, followed by inter-station neighbour
comparisons. A subset of the intra-station tests is then re-run, followed by
the inter-station checks again and then a final clean-up (Fig. 3).
### 3.1 QC tests
#### 3.1.1 Test 1: inter-station duplicate check
It is possible that two unique station identifiers actually contain identical
data. This may be simple data management error or an artefact of dummy station
files intended for temporary data storage. To detect these, each station’s
temperature time series is compared iteratively with that of every other
station. To account for reporting time ($t$) issues, the series are offset by
1 h steps between $t-11$ and $t+11$ h. Series with $>1000$ coincident non-
missing data points, of which over 25 per cent are flagged as exact
duplicates, are listed for further consideration. This computer-intensive
check resulted in 280 stations being put forward for manual scrutiny.
Figure 11: Spike check (test 10) schematic, showing the requirements on the
first differences inside and outside of a multi-point spike. The inset shows
the spike of three observations clearly above the rest of the time series. The
first difference value leading into the spike has to be greater than the
threshold value, $t$, and the first difference value coming out of the spike
has to be of the opposite direction and at least half the threshold value
($t/2$). The differences outside and inside the spike (as pointed to by the
red arrows) have to be less than half the threshold value.
All duplicate pairs and groups were then manually assessed using the match
statistics, reporting frequencies, separation distance and time series of the
stations involved. If a station pair had exact matches on $\geq 70$ per cent
of potential occasions, then the shortest station of the pair was removed.
This results in a further loss of stations. As this test is searching for
duplicates after the merging of composite stations (Sect. 2), any stations
found by this test did not previously meet the requirements for stations to be
merged, but still have significant periods where the observations are
duplicated. Therefore the removal of data is the safest course of action.
Stations that appeared in the potential duplicates list twice or more were
also removed. A further subjective decision was taken to remove any stations
having a very patchy or obscure time series, for example with very high
variance. This set of checks removed a total of 83 stations (Fig. 1), leaving
6103 to go forward into the rest of the QC procedure.
#### 3.1.2 Test 2: duplicate months check
Given day-to-day weather, an exact match of synoptic data for a month with any
other month in that station is highly unlikely. This test checks for exact
replicas of whole months of temperature data where at least 20 observations
are present. Each month is pattern-matched for data presence with all other
months, and any months with exact duplicates for each matched value are
flagged. As it cannot be known a priori which month is correct, both are
flagged. Although the test was successful at detecting deliberately engineered
duplication in a case study station, no occurrences of such errors were found
within the real data. The test was retained for completeness and also because
such an error may occur in future updates of HadISD.
Figure 12: Spike check (test 10) for composite station 718936-99999 (49.95∘
N, 125.267∘ W, 106 m, Campbell River, BC, Canada), for dewpoint temperature
showing the removal of a ghost station. Red points show values removed by this
test and blue points by other tests. The panel below each year shows whether
the data in the composited station come from the named station (blue) or a
matched station (red). In 1988 and 2006 a single station is used for the data,
but in 1996 there is clearly a blend between two stations (718936-99999 and
712050-99999). In this case the compositing has caused the ghosting; however,
both these stations are labelled in the ISD history file as Campbell River,
with identical latitudes and longitudes. An earlier period of merger between
these two stations did not lead to any ghosting effects.
Figure 13: Unusual variance check (test 13) for (top) 912180-99999 (13.57∘ N,
144.917∘ E, 162 m, Andersen Air Force Base, Guam) for dewpoint temperature and
(bottom) 133530-99999 (43.82∘ N, 18.33∘ E, 511 m, Sarajevo, Bosnia-
Herzegovina) for temperature. Red points show values removed by this test and
blue points by other tests (none for the years and variables shown).
#### 3.1.3 Test 3: odd cluster check
A number of time series exhibit isolated clusters of data. An instrument that
reports sporadically is of questionable scientific value. Furthermore, with
little or no surrounding data it is much more difficult to determine whether
individual observations are valid. Hence, any short clusters of up to 6 h
within a 24 h period separated by 48 h or longer from all other data are
flagged. This applies to temperature, dewpoint temperature and sea-level
pressure elements individually. These flags can be undone if the neighbouring
stations have concurrent, unflagged observations whose range encompasses the
observations in question (see Sect. 3.1.14).
#### 3.1.4 Test 4: frequent value check
The problem of frequent values found in Durre et al. (2010) also extends to
synoptic data. Some stations contain far more observations of a given value
than would be reasonably expected. This could be the use of zero to signify
missing data, or the occurrence of some other local data-issue identifier111A
“local data-issue identifier” is where a physically valid but locally
implausible value is used to mark a problem with a particular data point. On
subsequent ingestion into the ISD, this value has been interpreted as a real
measurement rather than a flag. that has been mistakenly ingested into the
database as a true value. This test identifies suspect values using the entire
record and then scans for each value on a year-by-year basis to flag only if
they are a problem within that year.
This test is also run seasonally (JF + D, MAM, JJA, SON), using a similar
approach as above. Each set of three months is scanned over the entire record
to identify problem values (e.g. all MAMs over the entire record), but flags
applied on an annual basis using just the three months on their own (e.g. each
MAM individually, scanning for values highlighted in the previous step). As
indicated by JF + D, the January and February are combined with the following
December (from the same calendar year) to create a season, rather than working
with the December from the previous calendar year. Performing a seasonal
version, although having fewer observations to work with, is more powerful
because the seasonal shift in the distribution of the temperatures and
dewpoints can reveal previously hidden frequent values.
For the filtered (where previously flagged observations are not included)
temperature, dewpoint and sea-level pressure data, histograms are created with
0.5 or 1.0 ∘C or hPa increments (depending on the reporting accuracy of the
measurement) and each histogram bin compared to the three on either side. If
this bin contains more than half of the total population of the seven bins
combined and also more than 30 observations over the station record (20 for
the seasonal scan), then the histogram bin interval is highlighted for further
investigation (Fig. 4). The minimum number limit was imposed to avoid removing
true tails of the distribution.
Figure 14: Nearest neighbour data check (test 14) for 912180-99999 (13.57∘ N,
144.917∘ E, 162 m, Andersen Air Force Base, Guam) for sea-level pressure. Red
points show values removed by this test (none for the years shown) and blue
points by other tests. The spikes for the hurricanes in 1976 and 1977 are kept
in the dataset. February 1976 is removed by the variance check – this February
has higher variance than expected when compared to all other Februaries for
this station.
After this identification stage, the unfiltered distribution is studied on a
yearly basis. If the highlighted bins are prominent (contain $>50$ per cent of
the observations of all seven bins and more than 20 observations in the year,
or $90$ per cent of the observations of all seven bins and more than 10
observations in the year) in any year, then they are flagged (the bin sizes
are reduced to 15 and 10 respectively for the seasonal scan). This two-stage
process was designed to avoid removing too many valid observations (type II
errors). However, even with this method, by flagging all values within a bin
it is likely that some real data are flagged if the values are sufficiently
close to the mean of the overall data distribution. Also, frequent values that
are pervasive for only a few years out of a longer record and are close to the
distribution peak may not be identified with this method (type I errors).
However, alternative solutions were found to be too computationally
inefficient. Station 037930-99999 (Anvil Green, Kent, UK) shows severe
problems from frequent values in the temperature data for 1980 (Fig. 4).
Temperature and dewpoint flags are synergistically applied, i.e. temperature
flags are applied to both temperature and dewpoint data, and vice versa.
#### 3.1.5 Test 5: diurnal cycle check
All ISD data are archived as UTC; conversion has generally taken place from
local time at some point during recording, reporting and archiving the data.
Errors could introduce large biases into the data for some applications that
consider changes in the diurnal characteristics. The test is only applied to
stations at latitudes below 60∘ N/S as above these latitudes the diurnal cycle
in temperature can be weak or absent, and obvious robust geographical patterns
across political borders were apparent in the test failure rates when it was
applied in these regions.
This test is run on temperature only as this variable has the most robust
diurnal cycle, but it flags data for all variables. Firstly, a diurnal cycle
is calculated for each day with at least four observations spread across at
least three quartiles of the day (see Fig. 5). This is done by fitting a sine
curve with amplitude equal to half the spread of reported temperatures on that
day. The phase of the sine curve is determined to the nearest hour by
minimising a cost function, namely the mean squared deviations of the
observations from the curve (see Fig. 5). The climatologically expected phase
for a given calendar month is that with which the largest number of individual
days phases agrees. If a day’s temperature range is less than 5 ∘C, no attempt
is made to determine the diurnal cycle for that day.
Figure 15: Passage of low pressure core over the British Isles during the
night of 15–16 October 1987. Green points (highlighted by circles) are
stations where the observation for that hour has been removed. There are two,
at 05:00 and 06:00 UTC, on 16 October 1987 in the north-east of England. These
flagged observations are investigated in Fig. 16. Figure 16: Sea level
pressure data from station 032450-99999 (Newcastle Weather Centre, 54.967∘ N,
$-$1.617∘ W, 47 m) during mid-October 1987. The two observations that have
triggered the spike check are clearly visible and are distinct from the rest
of the data. Given their values (994.6 and 993.1 hPa), the two flagged
observations are clearly separate from their adjacent ones (966.4 and 963.3
hPa). It is possible that a keying error in the SYNOP report led to 946 and
931 being reported, rather than 646 and 631. However, we make no attempt in
this dataset to rescue flagged values.
It is then assessed whether a given day’s fitted phase matches the expected
phase within an uncertainty estimate. This uncertainty estimate is the larger
of the number of hours by which the day’s phase must be advanced or retarded
for the cost function to cross into the middle tercile of its distribution
over all 24 possible phase-hours for that day. The uncertainty is assigned as
symmetric (see Fig. 5). Any periods $>30$ days where the diurnal cycle
deviates from the expected phase by more than this uncertainty, without three
consecutive good or missing days or six consecutive days consisting of a mix
of only good or missing values, are deemed dubious and the entire period of
data (including all non-temperature elements) is flagged.
Small deviations, such as daylight saving time (DST) reporting hour changes,
are not detected by this test. This type of problem has been found for a
number of Australian stations where during DST the local time of observing
remains constant, resulting in changes in the common GMT reporting hours
across the year222Such an error has been noted and reported back to the ISD
team at NCDC.. Such changes in reporting frequency and also the hours on which
the reports are taken are noted in the metadata of the netCDF file.
Figure 17: Passage of low pressure core of Hurricane Katrina during its
landfall in 2005. Every second hour is shown. Green points are observations
which have been removed, in this case by the neighbour outlier check (see test
14).
#### 3.1.6 Test 6: distributional gap check
Portions of a time series may be erroneous, perhaps originating from station
ID issues, recording or reporting errors, or instrument malfunction. To
capture these, monthly medians $M_{ij}$ are created from the filtered data for
calendar month $i$ in year $j$. All monthly medians are converted to anomalies
$A_{ij}\equiv M_{ij}-M_{i}$ from the calendar monthly median $M_{i}$ and
standardised by the calendar month inter-quartile range IQRi (inflated to 4 ∘C
or hPa for those months with very small IQRi) to account for any seasonal
cycle in variance. The station’s series of standardised anomalies
$S_{ij}A_{ij}/\textrm{IQR}_{i}$ is then ranked, and the median, $\acute{S}$,
obtained.
Figure 18: Left: Alaskan daily mean temperature in 1989 (green curve) shown
against the climatological daily average temperature (black line) and the 5th
and 95th percentile region, red curves and yellow shading. The cold spell in
late January is clearly visible. Right: similar plots, but showing the sub-
daily resolution data for a two month period starting in January 1989\. The
climatology, 5th and 95th percentile lines have been smoothed using an
11-point binomial filter in all four plots. Top: McGrath (702310-99999, 62.95∘
N, 155.60∘ W, 103 m), bottom: Fairbanks (702610-26411, 64.82∘ N, 147.86∘ W,
138 m).
Firstly, all observations in any month and year with $S_{ij}$ outside the
range $\pm 5$ (in units of the IQRi) from $\acute{S}$ are flagged, to remove
gross outliers. Then, proceeding outwards from $\acute{S}$, pairs of $S_{ij}$
above and below ($S_{iu}$, $S_{iv}$) it are compared in a step-wise fashion.
Flagging is triggered if one anomaly $S_{iu}$ is at least twice the other
$S_{iv}$ and both are at least 1.5IQRi from $\acute{S}$. All observations are
flagged for the months for which $S_{ij}$ exceeds $S_{iu}$ and has the same
sign. This flags one entire tail of the distribution. This test should
identify stations that have a gap in the data distribution, which is
unrealistic. Later checks should find any issues existing in the remaining
tail. Station 714740-99999 (Clinton, BC, Canada, an assigned composite) shows
an example of the effectiveness of this test at highlighting a significantly
outlying period in temperature between 1975 and 1976 (Fig. 6).
Figure 19: Left: daily mean temperature in southern Australia in 2009 (green
curve) with climatological average (black line) and 5th and 95th percentiles
(red lines and yellow shading). The exceptionally high temperatures in late
January/early February and mid-November can clearly be seen. Right: similar
plots showing the full sub-daily resolution data for a two month period
starting in January 2009. The climatology, 5th and 95th percentile lines have
been smoothed using an 11-point binomial filter in all four plots. Top:
Adelaide (946725-99999, 34.93∘ S, 138.53∘ E, 4 m), bottom: Melbourne
(948660-99999, 37.67∘ S, 144.85∘ E, 119 m).
An extension of this test compares all the observations for a given calendar
month over all years to look for outliers or secondary populations. A
histogram is created from all observations within a calendar month. To
characterise the width of the distribution for this month, a Gaussian curve is
fitted. The positions where this expected distribution crosses the $y=0.1$
line are noted333When the Gaussian crosses the $y=0.1$ line, assuming a
Gaussian distribution for the data, the expectation is that there would be
less than 1/10th of an observation in the entire data series for values beyond
this point for this data distribution. Hence we would not expect to see any
observations in the data further from the mean if the distribution was
perfectly Gaussian. Therefore, any observations that are significantly further
from the mean and are separated from the rest of the observations may be
suspect. In Fig. 7 this crossing occurs at around 2.5IQR. Rounding up and
adding one results in a threshold of 4IQR. There is a gap of greater than 2
bin widths prior to the beginning of the second population at 4IQR, and so the
secondary population is flagged., and rounded outwards to the next integer-
plus-one to create a threshold value. From the centre outwards, the histogram
is scanned for gaps, i.e. bins which have a value of zero. When a gap is
found, and it is large enough (at least twice the bin width), then any bins
beyond the end of the gap, which are also beyond the threshold value, are
flagged.
Figure 20: Rejection rates by variable for each station. Top panel:
temperature, Middle panel: dewpoint temperature, and Lower Panel: sea-level
pressure. Different rejection rates are shown by different colours, and the
key in each panel provides the total number of stations in each band.
Although a Gaussian fit may not be optimal or appropriate, it will account for
the spread of the majority of observations for each station, and the
contiguous portion of the distribution will be retained. For Station
476960-43323 (Yokosuka, Japan, an assigned composite) this part of the test
flags a number of observations. In fact, during the winter all temperature
measurements below 0 ∘C appear to be measured in Fahrenheit (see Fig.
7)444Such an error has been noted and reported back to NCDC.. In months that
have a mixture of above and below 0 ∘C data (possibly Celsius and Fahrenheit
data), the monthly median may not show a large anomaly, so this extension is
needed to capture the bad data. Figure 7 shows that the two clusters of red
points in January and October 1973 are captured by this portion of the test.
By comparing the observations for a given calendar month over all years, the
difference between the two populations is clear (see bottom panel in Fig. 8).
If there are two, approximately equally sized distributions in the station
record, then this test will not be able to choose between them.
Figure 21: The results of the final filtering to select climate quality
stations. Top: the selected stations that pass the filtering, with red for
composite stations (556/3427). Bottom: the rejected stations. Of these,
1234/2676 fail to meet the daily, monthly, annual or interannual requirements
(D/M/A/IA); 689/2676 begin after 1980 or end before 2000; 626/2676 have a gap
exceeding two years after the daily, monthly and annual completeness criteria
have been applied; and 127 fail because one of the three main variables has a
high proportion of flags.
To prevent the low pressure extremes associated with tropical cyclones being
excessively flagged, any low SLP observation identified by this second part of
the test is only tentatively flagged. Simultaneous wind speed observations, if
present, are used to identify any storms present, in which case low SLP
anomalies are likely to be true. If the simultaneous wind speed observations
exceed the median wind speed for that calendar month by 4.5 MADs, then
storminess is assumed and the SLP flags are unset. If there are no wind data
present, the neighbouring stations can be used to unset these tentative flags
in test 14. The tentative flags are only used for SLP observations in this
test.
#### 3.1.7 Test 7: known records check
Absolute limits are assigned based on recognised and documented world and
regional records (Table 3). All hourly observations outside these limits are
flagged. If temperature observations exceed a record, the dewpoints are
synergistically flagged. Recent analyses of the record Libyan temperature have
resulted in a change to the global and African temperature record (Fadli et
al., 2012). Any observations that would be flagged using the new value but not
by the old are likely to have been flagged by another test. This only affects
African observations, and those not assigned to the WMO regions outlined in
Table 3. The value used by this test will be updated in a future release of
HadISD.
Figure 22: The median diurnal temperature ranges recorded by each station
(using the selected 3427 stations) for each of the four three-month seasons.
Top-left for December-January-February, top-right for March-April-May, bottom-
left for June-July-August and bottom-right for September-October-November.
Figure 23: The temperature for each station on the 23 June 2003 at 00:00, 06:00, 12:00 and 18:00 UT using all 6103 stations. Table 3: Extreme limits for observed variables gained from http://wmo.asu.edu (the official WMO climate extremes repository) and the GHCND tests. Dewpoint minima are estimates based upon the record temperature minimum for each region. First element in each cell is the minimum and the second the maximum legal value. Regions follow WMO regional definitions and are given at: http://weather.noaa.gov/tg/site.shtml. Global values are used for any station where the assigned WMO identifier is missing or does not fall within the region categorization. Wind speed and sea-level pressure records are not currently documented regionally so global values are used throughout. We note that the value for the African and global maximum temperature has changed (Fadli et al., 2012). This will be updated in a future version of HadISD. Region | Temperature | | Dewpoint Temperature | | Wind speed | | Sea-level pressure
---|---|---|---|---|---|---|---
| (∘C) | | (∘C) | | (m s-1) | | (hPa)
| max | min | | max | min | | max | min | | max | min
Global | $-$89.2 | 57.8 | | $-$100.0 | 57.8 | | 0.0 | 113.3 | | 870 | 1083.3
Africa | $-$23.0 | 57.8 | | $-$50.0 | 57.8 | | – | – | | – | –
Asia | $-$67.8 | 53.9 | | $-$100.0 | 53.9 | | – | – | | – | –
S. America | $-$32.8 | 48.9 | | $-$60.0 | 48.9 | | – | – | | – | –
N. America | $-$63.0 | 56.7 | | $-$100.0 | 56.7 | | – | – | | – | –
Pacific | $-$23.0 | 50.7 | | $-$50.0 | 50.7 | | – | – | | – | –
Europe | $-$58.1 | 48.0 | | $-$100.0 | 48.0 | | – | – | | – | –
Antarctica | $-$89.2 | 15.0 | | $-$100.0 | 15.0 | | – | – | | – | –
| | | | | | | | | | |
#### 3.1.8 Test 8: repeated streaks/unusual spell frequency
This test searches for consecutive observation replication, same hour
observation replication over, a number of days (either using a threshold of a
certain number of observations, or for sparser records, a number of days
during which all the observations have the same value) and also whole day
replication for a streak of days. All three tests are conditional upon the
typical reporting precision as coarser precision reporting (e.g. temperatures
only to the nearest whole degree) will increase the chances of a streak
arising by chance (Table 4). For wind speed, all values below 0.5 ms-1 (or 1
ms-1 for coarse recording resolution) are also discounted in the streak search
given that this variable is not normally distributed and there could be long
streaks of calm conditions.
Table 4: Streak check criteria and their assigned sensitivity to typical within-station reporting resolution for each variable. | Reporting | Straight repeat | Hour repeat | Day repeat
---|---|---|---|---
Variable | resolution | streak criteria | streak criteria | streak criteria
| 1 ∘C | 40 values of 14 days | 25 days | 10 days
Temperature | 0.5 ∘C | 30 values or 10 days | 20 days | 7 days
| 0.1 ∘C | 24 values or 7 days | 15 days | 5 days
| 1 ∘C | 80 values of 14 days | 25 days | 10 days
Dewpoint | 0.5 ∘C | 60 values or 10 days | 20 days | 7 days
| 0.1 ∘C | 48 values or 7 days | 15 days | 5 days
| 1 hPa | 120 values of 28 days | 25 days | 10 days
SLP | 0.5 hPa | 100 values or 21 days | 20 days | 7 days
| 0.1 hPa | 72 values or 14 days | 15 days | 5 days
| 1 ms-1 | 40 values of 14 days | 25 days | 10 days
Wind speed | 0.5 ms-1 | 30 values or 10 days | 20 days | 7 days
| 0.1 ms-1 | 24 values or 7 days | 15 days | 5 days
| | | |
During development of the test a number of station time series were found to
exhibit an alarming frequency of streaks shorter than the assigned critical
lengths in some years. An extra criterion was added to flag all streaks in a
given year when consecutive value streaks of $>10$ elements occur with
extraordinary frequency ($>5$ times the median annual frequency). Station
724797-23176 (Milford, UT, USA, an assigned composite) exhibits a propensity
for streaks during 1981 and 1982 in the dewpoint temperature (Fig. 9), which
is not seen in any other years or nearby stations.
#### 3.1.9 Test 9: climatological outlier check
Individual gross outliers from the general station distribution are a common
error in observational data caused by random recording, reporting, formatting
or instrumental errors (Fiebrich and Crawford, 2009). This test uses
individual observation deviations derived from the monthly mean climatology
calculated for each hour of the day. These climatologies are calculated using
observations that have been winsorised555Winsorising is the process by which
all values beyond a threshold value from the mean are set to that threshold
value (5 and 95 per cent in this instance). The number of data values in the
population therefore remains the same, unlike trimming, where the data further
from the mean are removed from the population (Afifi and Azen, 1979). to
remove the initial effects of outliers. The raw, un-winsorised observations
are anomalised using these climatologies and standardised by the IQR for that
month and hour. Values are subsequently low-pass filtered to remove any
climate change signal that would cause overzealous removal at the ends of the
time series. In an analogous way to the distributional gap check, a Gaussian
is fitted to the histogram of these anomalies for each month, and a threshold
value, rounded outwards, is set where this crosses the $y=0.1$ line. The
distribution beyond this threshold value is scanned for a gap (equal to the
bin width or more), and all values beyond any gap are flagged. Observations
that fall between the critical threshold value and the gap or the critical
threshold value and the end of the distribution are tentatively flagged, as
they fall outside of the expected distribution (assuming it is Gaussian; see
Fig. 10). These may be later reinstated on comparison with good data from
neighbouring stations (see Sect. 3.1.14). A caveat to protect low-variance
stations is added whereby the IQR cannot be less than 1.5 ∘C. When applied to
sea-level pressure, this test frequently flags storm signals, which are likely
to be of high interest to many users, and so this test is not applied to the
pressure data.
As for the distributional gap check, the Gaussian may not be the best fit or
even appropriate for the distribution, but by fitting to the observed
distribution, the spread of the majority of the observations for the station
is accounted for, and searching for a gap means that the contiguous portion of
distribution is retained.
#### 3.1.10 Test 10: spike check
Unlike the operational ISD product, which uses a fixed value for all stations
(Lott et al., 2001), this test uses the filtered station time series to decide
what constitutes a “spike”, given the statistics of the series. This should
avoid over zealous flagging of data in high variance locations but at a
potential cost for stations where false data spikes are truly pervasive. A
first difference series is created from the filtered data for each time step
(hourly, 2-hourly, 3-hourly) where data exist within the past three hours.
These differences for each month over all years are then ranked and the IQR
calculated. Critical values of 6 times the rounded-up IQR are calculated for
one-, two- and three-hourly differences on a monthly basis to account for
large seasonal cycles in some regions. There is a caveat that no critical
value is smaller than 1 ∘C or hPa (conceivable in some regions but below the
typically expected reported resolution). Also hourly critical values are
compared with two hourly critical values to ensure that hourly values are not
less than 66 per cent of two hourly values. Spikes of up to three sequential
observations in the unfiltered data are defined by satisfying the following
criteria. The first difference change into the spike has to exceed the
threshold and then have a change out of the spike of the opposite sign and at
least half the critical amplitude. The first differences just outside of the
spike have to be under the critical values, and those within a multi-
observation spike have to be under half the critical value (see Fig. 11
highlighting the various thresholds). These checks ensure that noisy high
variance stations are not overly flagged by this test. Observations at the
beginning or end of a contiguous set are also checked for spikes by comparing
against the median of the subsequent or previous 10 observations. Spike check
is particularly efficient at flagging an apparently duplicate period of record
for station 718936-99999 (Campbell River, Canada, an assigned composite
station), together with the climatological check (Fig. 12).
#### 3.1.11 Test 11: temperature and dewpoint temperature cross-check
Following (Willett et al., 2008), this test is specific to humidity-related
errors and searches for three different scenarios:
1. 1.
Supersaturation (dewpoint temperature $>$ temperature), although physically
plausible especially in very cold and humid climates (Makkonen and Laakso,
2005), is highly unlikely in most regions. Furthermore, standard
meteorological instruments are unreliable at measuring this accurately.
2. 2.
Wet-bulb reservoir drying (due to evaporation or freezing) is very common in
all climates, especially in automated stations. It is evidenced by extended
periods of temperature equal to dewpoint temperature (dewpoint depression of 0
∘C).
3. 3.
Cutoffs of dewpoint temperatures at temperature extremes. Systematic flagging
of dewpoint temperatures when the simultaneous temperature exceeds a threshold
(specific to individual National Meteorological Services’ recording methods)
has been a common practice historically with radiosondes (Elliott, 1995;
McCarthy et al., 2009). This has also been found in surface stations both for
hot and cold extremes (Willett et al., 2008).
For supersaturation, only the dewpoint temperature is flagged if the dewpoint
temperature exceeds the temperature. The temperature data may still be
desirable for some users. However, if this occurs for 20 per cent or more of
the data within a month, then the whole month is flagged. In fact, no values
are flagged by this test and a later, independent check run at NCDC showed
that there were no episodes of supersaturation in the raw ISD (Neal Lott,
personal communication). However it is retained for completeness. For wet-bulb
reservoir drying, all continuous streaks of absolute dewpoint depression
$<0.25$ ∘C are noted. The leeway of $\pm 0.25$ ∘C allows for small systematic
differences between the thermometers. If a streak is $>24$ h with $\geq$ four
observations present, then all the observations of dewpoint temperature are
flagged unless there are simultaneous precipitation or fog observations for
more than one-third of the continuous streak. We use a cloud base measurement
of $<1000$ feet to indicate fog as well as the present weather information.
This attempts to avoid over zealous flagging in fog- or rain-prone regions
(which would dry-bias the observations if many fog or rain events were
removed). However, it is not perfect as not all stations include these
variables. For cutoffs, all observations within a month are binned into 10 ∘C
temperature bins from $-$90 ∘C to 70 ∘C (a range that extends wider than
recognised historically recorded global extremes). For any month where at
least 50 per cent of temperatures within a bin do not have a simultaneous
dewpoint temperature, all temperature and dewpoint data within the bin are
flagged. Reporting frequencies of temperature and dewpoint are identified for
the month, and removals are not applied where frequencies differ significantly
between the variables. The cutoffs part of this test can flag good dewpoint
data even if only a small portion of the month has problems, or if there are
gaps in the dewpoint series that are not present in the temperature
observations.
#### 3.1.12 Test 12: cloud coverage logical checks
Synoptic cloud data are a priori a very difficult parameter to test for
quality and homogeneity. Traditionally, cloud base height and coverage of each
layer (low, mid, and high) in oktas were estimated by eye. Now cloud is
observed in many countries primarily using a ceilometer which takes a single
180∘ scan across the sky with a very narrow off-scan field-of-view. Depending
on cloud type and cloud orientation, this could easily under- or over-estimate
actual sky coverage. Worse, most ceilometers can only observe low or at best
mid-level clouds. Here, a conservative approach has been taken where simple
cross checking on cloud layer totals is used to infer basic data quality. This
should flag the most glaring issues but does not guarantee a high quality
database.
Six tests are applied to the data. If coverage at any level is given as 9 or
10, which officially mean sky obscured and partial obstruction respectively,
that individual value is flagged666All ISD values greater than 10, which
signify scattered, broken and full cloud for 11, 12 and 13 respectively, have
been converted to 2, 4 and 8 oktas respectively during netCDF conversion prior
to QC.. If total cloud cover is less than the sum of low, middle and high
level cloud cover, then all are flagged. If low cloud is given as 8 oktas
(full coverage) but middle or high level clouds have a value, then, as it is
not immediately apparent which observations are at fault, the low, middle
and/or high cloud cover values are flagged. If middle layer cloud is given as
8 oktas (full coverage) but high level clouds have a value, then, similarly,
both the middle and high cloud cover value are flagged. If the cloud base
height is given as 22 000, this means that the cloud base is unobservable (sky
is clear). This value is then set to $-$10 for computational reasons. Finally,
cloud coverage can only be from 0 to 8 oktas. Any value of total, low, middle
layer or high cloud that is outside these bounds is flagged.
#### 3.1.13 Test 13: unusual variance check
The variance check flags whole months of temperature, dewpoint temperature and
sea-level pressure where the within month variance of the normalised anomalies
(as described for climatological check) is sufficiently greater than the
median variance over the full station series for that month based on
winsorised data (Afifi and Azen, 1979). The variance is taken as the MAD of
the normalised anomalies in each individual month with $\geq 120$
observations. Where there is sufficient representation of that calendar month
within the time series (10 months each with $\geq 120$ observations), a median
variance and IQR of the variances are calculated. Months that differ by more
than 8 IQR (temperatures and dewpoints) or 6 IQR (sea-level pressures) from
the station month median are flagged. This threshold is increased to 10 or 8
IQR respectively if there is a reduction in reporting frequency or resolution
for the month relative to the majority of the time series.
Sea-level pressure is accorded special treatment to reduce the removal of
storm signals (extreme low pressure). The first difference series is taken.
Any month where the largest consecutive negative or positive streak in the
difference series exceeds 10 data points is not considered for removal as this
identifies a spike in the data that is progressive rather than transient.
Where possible, the wind speed data are also included, and the median found
for a given month over all years of data. The presence of a storm is
determined from the wind speed data in combination with the sea-level pressure
profile. When the wind speed climbs above 4.5 MADs from the median wind speed
value for that month and if this peak is coincident with a minimum of the sea-
level pressure ($\pm 24$ h), which is also more than 4.5 MADs from the median
pressure for that month, then storminess is assumed. If these criteria are
satisfied, then no flag is set. This test for storminess includes an
additional test for unusually low SLP values, as initially this QC test only
identifies periods of high variance. Figure 13, for station 912180-99999
(Andersen Air Force Base, Guam), illustrates how this check is flagging
obviously dubious dewpoints that previous tests had failed to identify.
#### 3.1.14 Test 14: nearest neighbour data checks
Recording, reporting or instrument error is unlikely to be replicated across
networks. Such an error may not be detectable from the intra-station
distribution, which is inherently quite noisy. However, it may stand out
against simultaneous neighbour observations if the correlation decay distance
(Briffa and Jones, 1993) is large compared to the actual distance between
stations and therefore the noise in the difference series is comparatively
low. This is usually true for temperature, dewpoint and pressure. However the
check is less powerful for localised features such as convective precipitation
or storms.
Table 5: Summary of tests applied to the data.
Test | Applies to | Test failure | Notes
---|---|---|---
(Number) | T | Td | SLP | ws | wd | clouds | criteria |
Intra-station | | | | | | | |
Duplicate months check (2) | X | X | X | X | X | X | Complete match to least temporally complete month’s record for T |
Odd cluster check (3) | X | X | X | X | X | | $\leq 6$ values in 24 h separated from any other data by $>48$ h | Wind direction removed using wind speed characteristics
Frequent values check (4) | X | X | X | | | | Initially $>50$ % of all data in current 0.5 ∘C or hPa bin out of this and $\pm 3$ bins for all data to highlight, with $\geq 30$ in the bin. Then on yearly basis using highlighted bins with $>50$ % of data and $\geq 20$ observations in this and $\pm 3$ bins OR $>90$ % data and $\geq 10$ observations in this and $\pm 3$ bins. For seasons, the bin size thresholds are reduced to 20, 15 and 10 respectively. | Histogram approach for computational expediency. T and Td synergistically removed, if T is bad, then Td is removed and vice versa.
Diurnal cycle check (5) | X | X | X | X | X | X | 30 days without 3 consecutive good fit/missing or 6 days mix of these to T diurnal cycle. |
Distributional gap check (6) | X | X | X | | | | Monthly median anomaly $>5$IQR from median. Monthly median anomaly at least twice the distance from the median as the other tail and $>1.5$ IQR. Data outside of the Gaussian distribution for each calendar month over all years, separated from the main population. | All months in tail with apparent gap in the distribution are removed beyond the assigned gap for the variable in question. Using the distribution for all calendar months, tentative flags set if further from mean than threshold value. To keep storms, low SLP observations are only tentatively
Known record check (7) | X | X | X | X | | | See Table 3 | Td flagged if T flagged.
Repeated streaks/unusual spell frequency check (8) | X | X | X | X | | | See Table 4 |
Climatological outliers check (9) | X | X | | | | | Distribution of normalised (by IQR) anomalies investigated for outliers using same method as for distributional gap test. | To keep low variance stations, minimum IQR is 1.5 ∘C
Spike check (10) | X | X | X | | | | Spikes of up to 3 consecutive points allowed. Critical value of 6IQR (minimum 1 ∘C or hPa) of first difference at start of spike, at least half as large and in opposite direction at end. | First differences outside and inside a spike have to be under the critical and half the critical values respectively
T and Td cross-check: Supersaturation (11) | | X | | | | | Td $>$ T | Both variables removed, all data removed for a month if $>20$ % of data fails
T and Td cross-check: Wet bulb drying (11) | | X | | | | | T $=$ Td $>24$ h and $>4$ observations unless rain / fog (low cloud base) reported for $>1/3$ of string | 0.25 ∘C leeway allowed.
T and Td cross-check: Wet bulb cutoffs (11) | | X | | | | | $>20$ % of T has no Td within a 10 ∘C T bin | Takes into account that Td at many stations reported less frequently than T.
Cloud coverage logical checks (12) | | | | | | X | Simple logical criteria (see Sect. 3.1.12) |
Unusual variance check (13) | X | X | X | | | | Winsorised normalised (by IQR) anomalies exceeding 6 IQR after filtering | 8 IQR if there is a change in reporting frequency or resolution. For SLP first difference series used to find spikes (storms). Wind speed also used to identify storms
| | | | | | | |
Table 5: Continued.
Test | Applies to | Test failure | Notes
---|---|---|---
(Number) | T | Td | SLP | ws | wd | clouds | criteria |
Intra-station | | | | | | | |
Inter-station duplicate check (1) | X | | | | | | $>1000$ valid points and $>25$ % exact match over $t-11$ to $t+11$ window, followed by manual assessment of identified series | Stations identified as duplicates removed in entirety.
Nearest neighbour data check (14) | X | X | X | | | | $>2/3$ of station comparisons suggest the value is anomalous within the difference series at the 5 IQR level. | At least three and up to ten neighbours within 300 km and 500 m, with preference given to filling directional quadrants over distance in neighbour selection. Pressure has additional caveat to ensure against removal of severe storms.
Station clean-up (15) | X | X | X | X | X | | $<20$ values per month or $>40$ % of values in a given month flagged for the variable | Results in removal of whole month for that variable
| | | | | | | |
For each station, up to ten nearest neighbours (within 500 m elevation and 300
km distance) are identified. Where possible, all four quadrants (northeast,
southeast, southwest and northwest) surrounding the station must be
represented by at least two neighbours to prevent geographical biases arising
in areas of substantial gradients such as frontal regions. Where there are
less than three valid neighbours, the nearest neighbour check is not applied.
In such cases the station ID is noted, and these stations can be found on the
HadISD website. The station may be of questionable value in any subsequent
homogenisation procedure that uses neighbour comparisons. A difference series
is created for each candidate station minus neighbour pair. Any observation
associated with a difference exceeding 5IQR of the whole difference series is
flagged as potentially dubious. For each time step, if the ratio of dubious
candidate-neighbour differences flagged to candidate-neighbour differences
present exceeds 0.67 (2 in 3 comparisons yield a dubious value), and there are
three or more neighbours present, then the candidate observation differs
substantially from most of its neighbours and is flagged. Observations where
there are fewer than three neighbours that have valid data are noted in the
flag array.
For sea-level pressure in the tropics, this check would remove some negative
spikes which are real storms as the low pressure core can be narrow. So, any
candidate-neighbour pair with a distance greater than 100 km between is
assessed. If 2/3 or more of the difference series flags (over the entire
record) are negative (indicating that this site is liable to be affected by
tropical storms), then only the positive differences are counted towards the
potential neighbour outlier removals when all neighbours are combined. This
succeeds in retaining many storm signals in the record. However, very large
negative spikes in sea-level pressure (tropical storms) at coastal stations
may still be prone to removal especially just after landfall in relatively
station dense regions (see Sect. 4.1). Here, station distances may not be
large enough to switch off the negative difference flags but distant enough to
experience large differences as the storm passes. Isolated island stations are
not as susceptible to this effect, as only the station in question will be
within the low-pressure core and the switch off of negative difference flags
will be activated. Station 912180-99999 (Anderson, Guam) in the western
Tropical Pacific has many storm signals in the sea-level pressure (Fig. 14).
It is important that these extremes are not removed.
Flags from the spike, gap (tentative low SLP flags only; see Sect. 3.1.6),
climatological (tentative flags only; see Sect. 3.1.9), odd cluster and
dewpoint depression tests (test numbers 3, 6, 9, 10 & 11) can be unset by the
nearest neighbour data check. For the first four tests this occurs if there
are three or more neighbouring stations that have simultaneous observations
that have not been flagged. If the difference between the observation for the
station in question and the median of the simultaneous neighbouring
observations is less than the threshold value of 4.5 MADs777As calculated from
the neighbours observations, approximately 3$\sigma$., then the flag is
removed. These criteria are to ensure that only observations that are likely
to be good can have their flags removed.
In cases where there are few neighbouring stations with unflagged
observations, their distribution can be very narrow. This narrow distribution,
when combined with poor instrumental reporting accuracy, can lead to an
artificially small MAD, and so to the erroneous retention of flags. Therefore,
the MAD is restricted to a minimum of 0.5 times the worst reporting accuracy
of all the stations involved with this test. So, for example, for a station
where one neighbour has 1 ∘C reporting, the threshold value is 2.25 ∘C = 0.5
$\times$ 1 ∘C $\times$ 4.5.
Wet-bulb reservoir drying flags can also be unset if more than two-thirds of
the neighbours also have that flag set. Reservoir drying should be an isolated
event, and so simultaneous flagging across stations suggests an actual high
humidity event. The tentative climatological flags are also unset if there are
insufficient neighbours. As these flags are only tentative, without sufficient
neighbours there can be no definitive indication that the observations are
bad, and so they need to be retained.
#### 3.1.15 Test 15: station clean-up
A final test is applied to remove data for any month where there are $<20$
observations remaining or $>40$ per cent of observations removed by the QC.
This check is not applied to cloud data as errors in cloud data are most
likely due to isolated manual errors.
### 3.2 Test order
The order of the tests has been chosen both for computational convenience
(intra-station checks taking place before inter-station checks) and also so
that the most glaring errors are removed early on such that distributional
checks (which are based on observations that have been filtered according the
flags set thus far) are not biased. Inter-station duplicate check (test 1) is
run only once, followed by the latitude and longitude check. Tests 2 to 13 are
run through in sequence followed by test 14, the neighbour check. At this
point the flags are applied creating a masked, preliminary, quality-controlled
dataset, and the flagged values copied to a separate store in case any user
wishes to retrieve them at a later date. In the main data stream these flagged
observations are marked with a flagged data indicator, different from the
missing data indicator.
Then the spike (test 10) and odd-cluster (test 3) tests are re-run on this
masked data. New spikes may be found using the masked data to set the
threshold values, and odd clusters may have been left after the removal of bad
data. Test 14 is re-run to assess any further changes and reinstate any
tentative flags from the rerun of tests 3 and 10 where appropriate. Then the
clean-up of bad months, test 15, is run and the flags applied as above
creating a final quality-controlled dataset. A simple flow diagram is shown in
Fig. 3 indicating the order in which the tests are applied. Table 3.1.14
summarises which tests are applied to which data, what critical values were
applied, and any other relevant notes. Although the final quality-controlled
suite includes wind speed, direction and cloud data, the tests concentrate
upon SLP, temperature and dewpoint temperature and it is these data that
therefore are likely to have the highest quality; so users of the remaining
variables should take great care. The typical reporting resolution and
frequency are also extracted and stored in the output netCDF file header
fields.
Table 6: Summary of removal of data from individual stations by the different
tests for the 6103 stations considered in detailed analysis. The final column
denotes any geographical prevalence. A version of this table in percent is
presented in Table LABEL:table:9.
Test | Variable | Stations with detection rate band (% of total original observations) | Notes on geographical prevalence
---|---|---|---
(Number) | | 0 | 0–0.1 | 0.1–0.2 | 0.2–0.5 | 0.5–1.0 | 1.0–2.0 | 2.0–5.0 | 5.0 | of extreme removals
Duplicate months check (2) | All | 6103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Odd cluster check (3) | T | 2041 | 2789 | 484 | 413 | 213 | 126 | 34 | 3 | Ethiopia, Cameroon, Uganda, Ukraine, Baltic states, Pacific coast of Colombia, Indonesian Guinea
| Td | 1855 | 2946 | 485 | 439 | 214 | 128 | 35 | 1 | As for temperature
| SLP | 1586 | 3149 | 567 | 487 | 203 | 94 | 17 | 0 | Cameroon, Ukraine, Bulgaria, Baltic states, Indonesian Guinea
| ws | 1959 | 2851 | 480 | 435 | 218 | 125 | 32 | 3 | As for temperature
Frequent values check (4) | T | 5980 | 93 | 14 | 7 | 4 | 3 | 2 | 0 | Largely random. Generally more prevalent in tropics, particularly Kenya
| Td | 5941 | 91 | 19 | 17 | 14 | 8 | 8 | 5 | Largely random, particularly bad in Sahel region and Philippines.
| SLP | 5998 | 27 | 8 | 8 | 9 | 4 | 31 | 18 | Almost exclusively Mexican stations. Also a few UK stations.
Diurnal cycle check (5) | All | 5765 | 1 | 16 | 179 | 70 | 35 | 24 | 12 | Mainly NE N. America, central Canada and central Russia regions
Distributional gap check (6) | T | 2570 | 3253 | 42 | 81 | 77 | 33 | 38 | 9 | Mainly mid- to high-latitudes, more in N. America and central Asia
| Td | 1155 | 4204 | 298 | 245 | 114 | 45 | 36 | 6 | Mainly mid- to high-latitudes, more in N. America and central Asia
| SLP | 2736 | 3096 | 73 | 90 | 55 | 28 | 18 | 7 | Scattered
Known records check (7) | T | 5313 | 785 | 1 | 4 | 0 | 0 | 0 | 0 | S. America, central Europe
| Td | 6090 | 12 | 0 | 1 | 0 | 0 | 0 | 0 |
| SLP | 4872 | 1228 | 2 | 1 | 0 | 0 | 0 | 0 | Worldwide apart from N. America, Australia, E China, Scandinavia
| ws | 6103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| | | | | | | | | |
Table 6: Continued.
Test | Variable | Stations with detection rate band (% of total original observations) | Notes on geographical prevalence
---|---|---|---
(Number) | | 0 | 0–0.1 | 0.1–0.2 | 0.2–0.5 | 0.5–1.0 | 1.0–2.0 | 2.0–5.0 | $<5.0$ | of extreme removals
Repeated streaks/unusual spell frequency check (8) | T | 4785 | 259 | 183 | 284 | 218 | 212 | 146 | 16 | Particularly Germany, Japan, UK, Finland and NE. America and Pacific Canada/Alaska
| Td | 4343 | 220 | 190 | 349 | 344 | 353 | 272 | 32 | Similar to T, but more prevalent, additional cluster in Caribbean.
| SLP | 5996 | 26 | 13 | 15 | 8 | 7 | 22 | 16 | Almost exclusively Mexican stations
| ws | 5414 | 243 | 147 | 135 | 74 | 60 | 23 | 7 | Central & northern South America, Eastern Africa, SE Europe, S Asia, Mongolia
Climatological outliers check (9) | T | 1295 | 4382 | 217 | 159 | 36 | 13 | 1 | 0 | Fairly uniform, but higher in tropics
| Td | 1064 | 4538 | 238 | 192 | 49 | 19 | 3 | 0 | As for temperature
Spike check (10) | T | 95 | 3650 | 1270 | 992 | 92 | 2 | 2 | 0 | Fairly uniform, higher in Asia, especially eastern China
| Td | 38 | 3567 | 1486 | 940 | 66 | 4 | 2 | 0 | As for temperature
| SLP | 760 | 3437 | 1068 | 802 | 33 | 3 | 0 | 0 | Fairly uniform, few flags in southern Africa and western China
T and Td cross-check: Supersaturation (11) | T, Td | 6103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
T and Td cross-check: Wet bulb drying (11) | Td | 3982 | 1721 | 194 | 140 | 37 | 22 | 6 | 1 | Almost exclusively NH extra-tropical, concentrations, Russian high Arctic, Scandinavia, Romania.
T and Td cross-check: Wet bulb cutoffs (11) | Td | 5055 | 114 | 211 | 319 | 175 | 128 | 69 | 32 | Mainly high latitude/elevation stations, particularly Scandinavia, Alaska, Mongolia, Algeria, USA.
Cloud coverage logical check (12) | Cloud variables | 1 | 682 | 471 | 979 | 1124 | 1357 | 1173 | 315 | Worst in Central/Eastern Europe, Russian and Chinese coastal sites, USA, Mexico, eastern central Africa.
Unusual variance check (13) | T | 5658 | 13 | 80 | 298 | 50 | 4 | 0 | 0 | Most prevalent in parts of Europe, US Gulf and west coasts
| Td | 5605 | 13 | 78 | 334 | 60 | 10 | 3 | 0 | Largely Europe, SE Asia and Caribbean/Gulf of Mexico
| SLP | 5263 | 33 | 118 | 494 | 150 | 26 | 10 | 9 | Almost exclusively tropics, particularly prevalent in sub-Saharan Africa, Ukraine, also eastern China
Nearest neighbour data check (14) | T | 1549 | 4369 | 94 | 35 | 27 | 19 | 12 | 1 | Fairly uniform, worst in Ukraine, UK, Alaska
| Td | 1456 | 4368 | 159 | 58 | 36 | 16 | 10 | 0 | As for temperature
| SLP | 1823 | 3995 | 203 | 58 | 13 | 4 | 6 | 1 | Fairly uniform, worst in Ukraine, UK, eastern Arctic Russia
Station clean-up (15) | T | 3865 | 1546 | 239 | 219 | 138 | 60 | 27 | 9 | High latitude N. America, Vietnam, eastern Europe, Siberia
| Td | 3756 | 1526 | 240 | 277 | 159 | 85 | 46 | 14 | Very similar to that for temperatures
| SLP | 3244 | 2242 | 212 | 224 | 107 | 40 | 29 | 5 | Many in Central America, Vietnam, Baltic states.
| ws | 3900 | 1584 | 214 | 226 | 108 | 40 | 24 | 7 | Western tropical coasts of Central America, central & eastern Africa, Myanmar, Indonesia
| | | | | | | | | |
### 3.3 Fine-tuning
In order to fine-tune the tests and their critical and threshold values, the
entire suite was first tested on the 167 stations in the British Isles. To
ensure that the tests were still capturing known and well-documented extremes,
three such events were studied in detail: the European heat wave in August
2003 and the storms of October 1987 and January 1990. During the course of
these analyses, it was noted that the tests (in their then current version)
were not performing as expected and were removing true extreme values as
documented in official Met Office records and literature for those events.
This led to further fine-tuning and additions resulting in the tests as
presented above. All analyses and diagrams are from the quality control
procedure after the updates from this fine-tuning.
As an example Fig. 15 shows the passage of the low pressure core of the 1987
storm. The low pressure minimum is clearly not excluded by the tests as they
now stand, whereas previously a large number of valid observations around the
low pressure minimum were flagged. The two removed observations come from a
single station and were flagged by the spike test (they are clear anomalies
above the remaining SLP observations; see Fig. 16).
Any pervasive issues with the data or individual stations will be reported to
the ISD team at NCDC to allow for the improvement of the data for all users.
We encourage users of HadISD who discover suspect data in the product to
contact the authors to allow the station to be investigated and any
improvements to the raw data or the QC suite to be applied.
NCDC provide a list of known issues with the ISD database
(http://www1.ncdc.noaa.gov/pub/data/ish/isd-problems.pdf). Of the 27 problems
known at the time of writing (31 July 2012), most are for stations, variables
or time periods which are not included in the above study. Of the four that
relate to data issues that could be captured by the present analysis, all the
bad data were successfully identified and removed (numbers 6, 7, 8 and 25,
stations 718790, 722053, 722051 and 722010). Number 22 has been solved during
the compositing process (our station 725765-24061 contains both 725765-99999
and 726720-99999). However, number 24 (station 725020-14734) cannot be
detected by the QC suite as this error relates to the reporting accuracy of
the instrument.
## 4 Validation and analysis of quality control results
To determine how well the dataset captures extremes, a number of known extreme
climate events from around the globe were studied to determine the success of
the QC procedure in retaining extreme values while removing bad data. This
also allows the limitations of the QC procedure to be assessed. It also
ensures that the fine-tuning outlined in Sect. 4.2 did not lead to at least
gross over-tuning being based upon the climatic characteristics of a single
relatively small region of the globe.
### 4.1 Hurricane Katrina, September 2005
Katrina formed over the Bahamas on 23 August 2005 and crossed southern Florida
as a moderate Category 1 hurricane, causing some deaths and flooding. It
rapidly strengthened in the Gulf of Mexico, reaching Category 5 within a few
hours. The storm weakened before making its second landfall as a Category 3
storm in southeast Louisiana. It was one of the strongest storms to hit the
USA, with sustained winds of 127 mph at landfall, equivalent to a Category 3
storm on the Saffir-Simpson scale (Graumann et al., 2006). After causing over
$100 billion of damage and 1800 deaths in Mississippi and Louisiana, the core
moved northwards before being absorbed into a front around the Great Lakes.
Figure 17 shows the passage of the low pressure core of Katrina over the
southern part of the USA on 29 and 30 August 2005. This passage can clearly be
tracked across the country. There are a number of observations which have been
removed by the QC, highlighted in the figure. These observations have been
removed by the neighbour check. This identifies the issue raised in Sect.
3.1.14 (test 14), where even stations close by can experience very different
simultaneous sea-level pressures with the passing of very strong storms.
However the passage of this pressure system can still be characterised from
this dataset.
### 4.2 Alaskan cold spell, February 1989
The last two weeks of January 1989 were extremely cold throughout Alaska
except the Alaska Panhandle and Aleutian Islands. A number of new minimum
temperature records were set (e.g. $-$60.0 ∘C at Tanana and $-$59.4 ∘C at
McGrath; Tanaka and MILKOVICH, 1990). Records were also set for the number of
days below a certain temperature threshold (e.g. 6 days of less than $-$40.0
∘C at Fairbanks; Tanaka and MILKOVICH, 1990).
The period of low temperatures was caused by a large static high-pressure
system which remained over the state for two weeks before moving southwards,
breaking records in the lower 48 states as it went (Tanaka and MILKOVICH,
1990). The period immediately following this cold snap, in early February, was
then much warmer than average (by 18 ∘C for the monthly mean in Barrow).
The daily average temperatures for 1989 show this period of exceptionally low
temperatures clearly for McGrath and Fairbanks (Fig. 18). The traces include
the short period of warming during the middle of the cold snap which was
reported in Fairbanks. The rapid warming and subsequent high temperatures are
also detected at both stations. Figure 18 also shows the synoptic resolution
data for January and February 1989. These show the full extent of the cold
snap. The minimum temperature in HadISD for this period in McGrath was $-$58.9
∘C (only 0.5 ∘C warmer than the new record) and $-$46.1 ∘C at Fairbanks. As
HadISD is a sub-daily resolution dataset, then the true minimum values are
likely to have been missed, but the dataset still captures the very cold
temperatures of this event. Some observations over the two week period were
flagged, from a mixture of the gap, climatological, spike and odd cluster
checks, and some were removed by the month clean-up. However, they do not
prevent the detailed analysis of the event.
### 4.3 Australian heat waves, January & November 2009
South-eastern Australia experienced two heat waves during 2009. The first,
starting in late January, lasted approximately two weeks. The highest
temperature recorded was 48.8 ∘C in Hopetoun, Victoria, a new state record,
and Melbourne reached 46.4 ∘C, also a record for the city. The duration of the
heat wave is shown by the record set in Mildura, Victoria, which had 12 days
where the temperature rose to over 40 ∘C.
The second heat wave struck in mid-November, and although not as extreme as
the previous, still broke records for November temperatures. Only a few
stations recorded maxima over 40 ∘C but many reached over 35 ∘C.
In Fig. 19 we show the average daily temperature calculated from the HadISD
data for Adelaide and Melbourne and also the full synoptic resolution data for
January and February 2009. Although these plots are complicated by the diurnal
cycle variation, the very warm temperatures in this period stand out as
exceptional. The maximum temperatures recorded in the HadISD in Adelaide are
44.0 ∘C and 46.1 ∘C in Melbourne. The maximum temperature for Melbourne in the
HadISD is only 0.3 ∘C lower than the true maximum temperature. However, some
observations over each of the two week periods were flagged, from a mixture of
the gap, climatological, spike and odd cluster checks, but they do not prevent
the detailed analysis of the event.
### 4.4 Global overview of the quality control procedure
The overall observation flagging rates as a percentage of total number of
observations are given in Fig. 20 for temperature, dewpoint temperature and
sea-level pressure. Disaggregated results for each test and variable are
summarised in Table 6. For all variables the majority of stations have $<1$
per cent of the total number of observations flagged. Flagging patterns are
spatially distinct for many of the individual tests and often follow
geopolitical rather than physically plausible patterns (Table 6, final
column), lending credence to a non-physical origin. For example, Mexican
stations are almost ubiquitously poor for sea-level pressure measurements. For
the three plotted variables, rejection rates are also broadly inversely
proportional to natural climate variability (Fig. 20). This is unsurprising
because it will always be easier to find an error of a given absolute
magnitude in a time series of intrinsically lower variability. From these
analyses we contend that the QC procedure is adequate and unlikely to be over-
aggressive.
In a number of cases, stations that had apparently high flagging rates for
certain tests were also composite stations (see figures for the tests). In
order to check whether the compositing has caused more problems than it
solved, 20 composite stations were selected at random to see if there were any
obvious discontinuities across their entire record using the raw, non-quality-
controlled data. No such problems were found in these 20 stations. Secondly,
we compared the flagging prevalence (as in Table LABEL:table:9) for each of
the different tests focussing on the three main variables. For most tests the
difference in flagging percentages between composite and non-composite
stations is small. The most common change is that there are fewer composite
stations with 0 per cent of data flagged and more stations with 0–0.1 per cent
of data flagged than non-composites. We do not believe these differences
substantiate any concern. However, there are some cases of note. In the case
of the dewpoint cut-off test, there is a large tail out to higher failure
fractions, with a correspondingly much smaller 0 per cent flagging rate in the
case of composite stations. There is a reduction in the prevalence of stations
which have high flagging rates in the isolated odd cluster test in the
composite stations versus the non-composite stations. The number of flagging
due to streaks of all types is elevated in the composite stations.
Despite no pervasive large differences being found in apparent data quality
between composited stations and non-composited stations, there are likely to
be some isolated cases where the compositing has caused a degrading of the
data quality. Should any issues become apparent to the user, feedback to the
authors is strongly encouraged so that amendments can be made where possible.
Edited by: H. Goosse
## References
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* Durre et al. (2010) I. Durre, M.J. Menne, B.E. Gleason, T.G. Houston, and R.S. Vose. Comprehensive automated quality assurance of daily surface observations. _Journal of Applied Meteorology and Climatology_ , 49(8):1615–1633, 2010.
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|
arxiv-papers
| 2012-10-26T16:57:09 |
2024-09-04T02:49:37.147500
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert J. H. Dunn (1), Kate M. Willett (1), Peter W. Thorne (2,3),\n Emma V. Woolley, Imke Durre (3), Aiguo Dai (4), David E. Parker (1), Russ E.\n Vose (3) ((1) Met Office Hadley Centre, Exeter, UK, (2) CICS-NC, Asheville,\n NC, (3) NOAA NCDC, Asheville, NC, (4) NCAR, Boulder, CO)",
"submitter": "Robert Dunn",
"url": "https://arxiv.org/abs/1210.7191"
}
|
1210.7259
|
###### Abstract
This work addresses the study of the oscillator algebra, defined by four
parameters $p$, $q$, $\alpha$, and $\nu$. The time-independent Schrödinger
equation for the induced deformed harmonic oscillator is solved; explicit
analytic expressions of the energy spectrum are given. Deformed states are
built and discussed with respect to the criteria of coherent state
construction. Various commutators involving annihilation and creation
operators and their combinatorics are computed and analyzed. Finally, the
correlation functions of matrix elements of main normal and antinormal forms,
pertinent for quantum optics analysis, are computed.
New Deformation of quantum oscillator algebra: Representation and some
applications
S. Arjika†,1, D. Ousmane Samary‡,1,2,
E. Baloïtcha∗,1 and M. N. Hounkonnou∗∗,1
$1$-International Chair in Mathematical Physics and Applications
(ICMPA-UNESCO Chair), University of Abomey-Calavi,
072B.P.50, Cotonou, Rep. of Benin
$2$-CNRS - Université Lyon 1, Institut Camille Jordan
(Bat. Jean Braconnier, bd du 11 novembre 1918),
F-69622 Villeurbanne Cedex, France
E-mails: †rjksama2008@gmail.com, ‡ousmanesamarydine@yahoo.fr,
∗ezinvi.baloitcha@cipma.uac.bj, ∗∗norbert.hounkonnou@cipma.uac.bj
[Ref-preprint] CIPMA-MPA/031/2012
## 1 Introduction
The deformation of the harmonic oscillator algebra whose applications in
physics are presently rather technical but nonetheless very promising,
possesses an important and useful representation theory in connection to that
of their classical limit algebra. From the other side, there are some hopes
that, in physical studies of nonlinear phenomena, the deformed oscillator can
play the same role as the usual boson oscillator in usual nonrelativistic
quantum mechanics. This could explain why various quantum deformations of
boson oscillator commutation relations have attracted a great attention during
the last few years (see [1]\- [17] and references therein). This might be also
due to the fact that there exist correspondences between quantum groups,
quantum algebras, statistical mechanics, quantum field theory, conformal field
theory, quantum and nonlinear optics and noncommutative geometry, etc.
Furthermore, such a connection is extended to coherent states (cs) deducible
from the study of quantum groups and, therefore, from the deformation of
Heisenberg algebra. As a pertinent application in quantum optics, cs can be
used to compute matrix elements, ${}_{\nu}\langle
z|a^{{\dagger}m}a^{n}|z\rangle_{\nu},$ ${}_{\nu}\langle
z|a^{n}a^{{\dagger}m}|z\rangle_{\nu},$ corresponding to normal and antinormal
forms, respectively [17], (also called symmetric form), by using the normal
product technique.
Note that the most spread, in the literature, multiparameter deformation of
the harmonic oscillator is the so-called $(p,q)-$deformed oscillator. For more
details, see [5]. To cite an example, let us mention the $(p,q)-$deformed
oscillator algebra defined by the following Chakrabarti and Jagannathan’s
commutation relations [7]:
$aa^{\dagger}-qa^{\dagger}a=p^{-N},\quad
aa^{\dagger}-p^{-1}a^{\dagger}a=q^{N},$
$[N,a]=-a,\quad[N,a^{\dagger}]=a^{\dagger},$ where $p,q\in\mathbb{R};$
$a,a^{\dagger}$ and $N=a^{\dagger}a$ are the annihilation, creation and number
operators, respectively. This deformation has been generalized in various ways
in the literature [1] [6] and [8].
This paper extend the result on [8] by introducing two new positive parameters
functions $\phi_{1}$ and $\phi_{2}$. The spectrum and states of this new class
of quantum oscillator algebra are given. We also analyzed the coherent states
and correlation functions useful for the study of quantum optics properties.
The work is organized as follows. In section 2, we give the preliminary and
definitions permitted to define properly the mean result of our work. Section
3 is devoted to compute related deformed states. Relevant correlation
functions as well as new identities are also built in section 4. We give in
Section 5 a discussion of a particular case of algebra (4). We end with the
concluding and remarks in section 6.
## 2 New class of deformation quantum oscillator algebra
We start from the following mains definitions of the $(p,q)-$deformed
oscillator algebra and its generalization.
$\bullet$ The $(p,q)-$oscillator algebra is generated by three elements
$a,\;a^{\dagger}$ and $N$ obeying the relation [7]
$\displaystyle
aa^{\dagger}-p^{-1}a^{\dagger}a=q^{N},\;\;aa^{\dagger}-qa^{\dagger}a=p^{-N},\;\;[N,a]=-a,\;\;[N,a^{\dagger}]=a^{\dagger}.$
(1)
$\bullet$ The $(q,p;\alpha,\beta,l)-$deformed canonical commutation relations
defined in [6] is written as follows
$\displaystyle aa^{\dagger}-q^{l}a^{\dagger}a=p^{-\alpha
N-\beta},\;aa^{\dagger}-p^{-l}a^{\dagger}a=q^{\alpha N+\beta}$ (2)
$\displaystyle[N,a]=-\frac{l}{\alpha}a,\;[N,a^{\dagger}]=\frac{l}{\alpha}a^{\dagger}.$
(3)
It is worth noticing that further generalization involving a new parameters
$\phi_{1}$ and $\phi_{2}$ generate a richer algebra, with novel interesting
properties, as we will see in the sequel. In this case, we arrive at the
following definition.
Definition1: (The deformed algebra) The deformed algebra generated by the
operators $\\{1,a,a^{\dagger},N\\}$ is defined in this paper by the relations
$\displaystyle aa^{\dagger}-q^{\nu}a^{\dagger}a=\phi_{1}(p,q)p^{-\alpha
N},\quad aa^{\dagger}-p^{-\nu}a^{\dagger}a=\phi_{2}(p,q)q^{\alpha N},$ (4)
$\displaystyle[N,a^{\dagger}]=\frac{\nu}{\alpha}a^{\dagger},\;\;\;[N,a]=-\frac{\nu}{\alpha}a.$
(5)
where $\phi_{1}(p,q)$ and $\phi_{2}(p,q)$ are two non singular and real valued
positive functions of deformation parameters $p$ and $q$.
Remark1: It’s important to notify immediately that in the limit when
$\nu,\alpha\to 1$ and $\phi_{1}(p,q)=\phi_{2}(p,q)=1$, one recovers the
algebra studied by Chakrabarti et al [7]. In the same manner if
$\phi_{1}(p,q)=p^{-\beta},\phi_{2}(p,q)=q^{\beta}$, we recover the well known
algebra investigated by Burban [6]. So the introduction of such regular
functions $\phi_{1}$ and $\phi_{2}$ generalized the previous oscilator
algebras existing in the literature.
From (4), one can easily deduce that
$\displaystyle aa^{\dagger}=\frac{\phi_{1}(p,q)p^{-\alpha
N-\nu}-\phi_{2}(p,q)q^{\alpha
N+\nu}}{p^{-\nu}-q^{\nu}},\,a^{\dagger}a=\frac{\phi_{1}(p,q)p^{-\alpha
N}-\phi_{2}(p,q)q^{\alpha N}}{p^{-\nu}-q^{\nu}}.$ (6)
Let $\mathcal{F}$ be a Fock space spanned by the orthogonal basis
$\\{|n\rangle,n=0,1,2\ldots\\}$. We defined the vacuum state $|0\rangle$ as
the eigen-vector of $N$ such that $N|0\rangle=\chi_{0}|0\rangle.$ Due to the
commutativity of $aa^{\dagger}$ and $a^{\dagger}a$ with $N$, we may assume
that $aa^{\dagger}|0\rangle=\mu_{0}|0\rangle,\quad
a^{\dagger}a|0\rangle=\lambda_{0}|0\rangle$, $\chi_{0},\,\mu_{0}$ and
$\lambda_{0}$ are three reals numbers. We will also impose the normalization
condition $\langle 0|0\rangle=1.$
Proposition1 The states $|n\rangle$ are built as follows: if $pq<1$ and
$\phi_{2}(p,q)<\phi_{1}(p,q)$
$\displaystyle|n\rangle=\frac{p^{n\nu(n-1)/4}}{\sqrt{\tau_{p^{-1},q}^{n}(\phi_{1})(\phi_{1}^{-1}(p,q)\phi_{2}(p,q)(pq)^{\alpha\chi_{0}+\nu};(pq)^{\nu})_{n}}}a^{{\dagger}n}|0\rangle,\quad
n\geq 0,$ (7)
and if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$,
$\displaystyle|n\rangle=\frac{q^{-n\nu(n-1)/4}}{\sqrt{\tau_{q,p^{-1}}^{n}(\phi_{2})(\phi_{2}^{-1}(p,q)\phi_{1}(p,q)(pq)^{-\alpha\chi_{0}-\nu};(pq)^{-\nu})_{n}}}a^{{\dagger}n}|0\rangle,\quad
n\geq 0.$ (8)
The states (7) and (8) satisfy the orthogonality and completeness conditions
$\displaystyle\langle
m|n\rangle=\delta_{mn},\quad\sum_{n=0}^{\infty}|n\rangle\langle
n|=\mathbf{1}.$ (9)
In order to understand and study properly this new oscillator algebra we
calculate the actions of the deformed operators $a$, $a^{\dagger}$ and $N$ on
$\mathcal{F}$ and get
* •
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$
$\displaystyle
a|n\rangle=\tau_{p^{-1},q}^{1/2}(\phi_{1}(p,q))p^{-(n-1)\nu/2}\sqrt{1-\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha\chi_{0}+n\nu}}\,|n-1\rangle.$
(10)
* •
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$
$\displaystyle
a|n\rangle=\tau_{q,p^{-1}}^{1/2}(\phi_{2}(p,q))q^{(n-1)\nu/2}\sqrt{1-\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha\chi_{0}-n\nu}}\,|n-1\rangle.$
(11)
* •
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$
$\displaystyle
a^{\dagger}|n\rangle=\tau_{p^{-1},q}^{1/2}(\phi_{1}(p,q))p^{-n\nu/2}\sqrt{1-\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha\chi_{0}+(n+1)\nu}}\,|n+1\rangle.$
(12)
* •
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$
$\displaystyle
a^{\dagger}|n\rangle=\tau_{q,p^{-1}}^{1/2}(\phi_{2}(p,q))q^{n\nu/2}\sqrt{1-\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha\chi_{0}-(n+1)\nu}}\,|n+1\rangle.$
(13)
Let
$\displaystyle\tau_{p,q}(t)=\frac{tp^{\alpha\chi_{0}+\nu}}{p^{\nu}-q^{\nu}},$
it gets even
$\displaystyle N|n\rangle$ $\displaystyle=$
$\displaystyle(\chi_{0}+n)|n\rangle.$ (14)
From (7)-(13), one can deduce that
$\displaystyle aa^{\dagger}|n\rangle$ $\displaystyle=$
$\displaystyle\frac{\phi_{1}(p,q)p^{-\alpha\chi_{0}-(n+1)\nu}-\phi_{2}(p,q)q^{\alpha\chi_{0}+(n+1)\nu}}{p^{-\nu}-q^{\nu}}|n\rangle,$
(15) $\displaystyle a^{\dagger}a|n\rangle$ $\displaystyle=$
$\displaystyle\frac{\phi_{1}(p,q)p^{-\alpha\chi_{0}-n\nu}-\phi_{2}(p,q)q^{\alpha\chi_{0}+n\nu}}{p^{-\nu}-q^{\nu}}|n\rangle.$
(16)
Finally we may conclude that, the states $|n\rangle$ solve the time-
independent Schrödringer equation of the deformed oscillator Hamiltonian
$H=a^{\dagger}a+aa^{\dagger},$ i.e.
$H|n\rangle=E_{n,\alpha}^{\nu}(p,q)|n\rangle$, with the corresponding
eigenvalue $E_{n,\alpha}^{\nu}(p,q)$ given by
$\displaystyle E_{n,\alpha}^{\nu}(p,q)$ $\displaystyle=$
$\displaystyle\tau_{p^{-1},q}(\phi_{1}(p,q))p^{-(n-1)\nu}\Bigg{\\{}1+p^{-\nu}-\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha\chi_{0}+n\nu}(1+q^{\nu})\Bigg{\\}},$
$\displaystyle\mbox{ with }\,pq<1\mbox{ and }\phi_{2}(p,q)<\phi_{1}(p,q)$
$\displaystyle E_{n,\alpha}^{\nu}(p,q)$ $\displaystyle=$
$\displaystyle\tau_{q,p^{-1}}(\phi_{2}(p,q))q^{(n-1)\nu}\Bigg{\\{}1+q^{\nu}-\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{\alpha\chi_{0}+n\nu}(1+p^{-\nu})\Bigg{\\}},$
(19) $\displaystyle\mbox{ with }\,pq>1\mbox{ and
}\phi_{1}(p,q)<\phi_{2}(p,q).$
It follows from (2) and (19) that the spectrum of the deformed Hamiltonian $H$
is symmetric under the change $q\to p^{-1},\phi_{1}(p,q)\to\phi_{2}(p,q).$
Proposition2 For $A$ and $B\in L(\mathcal{L})$, where $L(\mathcal{L})$ is the
set of linears operators acting on Fock space $\mathcal{L}$ we consider the
$q^{\nu}$ and $p^{-\nu}$ commutators defined by:
$[A,B]_{q^{\nu}}=AB-q^{\nu}BA$ and $[A,B]_{p^{-\nu}}=AB-p^{-\nu}BA,$ the
following brackets hold
$\displaystyle[a,a^{{\dagger}m+1}]_{q^{\nu}}=a^{{\dagger}m}\Bigg{(}\frac{\phi_{1}(p,q)p^{-\alpha
N}(p^{-(m+1)\nu}-q^{\nu})-\phi_{2}(p,q)q^{\alpha
N}(q^{(m+1)\nu}-q^{\nu})}{p^{-\nu}-q^{\nu}}\Bigg{)},$ (20)
$\displaystyle[a,a^{{\dagger}m+1}]_{p^{-\nu}}=a^{{\dagger}m}\Bigg{(}\frac{\phi_{1}(p,q)p^{-\alpha
N}(p^{-(m+1)\nu}-p^{-\nu})-\phi_{2}(p,q)q^{\alpha
N}(q^{(m+1)\nu}-p^{-\nu})}{p^{-\nu}-q^{\nu}}\Bigg{)}.$ (21)
Proof The proof can be performed by using equation (6). $\blacksquare$
For $n,m\in\mathbb{N}\diagdown\\{0\\}$, the expressions of operators
$a^{n}a^{{\dagger}m}$ and $a^{{\dagger}m}a^{n}$ are given in different cases
by the following relations
$\bullet\,\,\,\,\,\,$For $n<m$
$\displaystyle
a^{n}a^{{\dagger}m}=p^{-\nu\binom{n}{2}}\mathcal{T}_{p^{-1},q}^{n}(\phi_{1}(p,q))\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N+\nu};(pq)^{\nu}\Big{)}_{n}a^{{\dagger}m-n}$ (22)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{n}a^{{\dagger}m}=q^{\nu\binom{n}{2}}\mathcal{T}_{q,p^{-1}}^{n}(\phi_{2}(p,q))\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N-\nu};(pq)^{-\nu}\Big{)}_{n}a^{{\dagger}m-n},$ (23)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$.
$\displaystyle
a^{{\dagger}m}a^{n}=p^{-\nu\binom{n}{2}}\Big{[}{{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{-m}\mathcal{T}_{p^{-1},q}(\phi_{1}(p,q))\Big{]}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N+\nu};(pq)^{\nu}\Big{)}_{n}a^{{\dagger}m-n},$ (24)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{{\dagger}m}a^{n}=q^{\nu\binom{n}{2}}{{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{-m}\mathcal{T}_{q,p^{-1}}^{n}(\phi_{2}(p,q))\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N-\nu};(pq)^{-\nu}\Big{)}_{n}a^{{\dagger}m-n},$ (25)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$.
$\bullet\,\,\,\,\,\,$If $n>m$,
$\displaystyle
a^{n}a^{{\dagger}m}=p^{\nu\binom{m}{2}}{{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{n}\Bigg{(}\Big{[}{{}_{p}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{p^{-1},q}^{m}((\phi_{1}(p,q))\Big{]}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N};(pq)^{-\nu}\Big{)}_{m}\Bigg{)}a^{n-m},$ (26)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{n}a^{{\dagger}m}=q^{-\nu\binom{m}{2}}{{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{n}\Bigg{(}\Big{[}{{}_{q}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{q,p^{-1}}^{m}((\phi_{2}(p,q))\Big{]}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N};(pq)^{\nu}\Big{)}_{m}\Bigg{)}a^{n-m},$ (27)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$.
$\displaystyle a^{{\dagger}m}a^{n}$ $\displaystyle=$ $\displaystyle
p^{\nu\binom{m}{2}}\Big{[}{{}_{p}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{p^{-1},q}^{m}(\phi_{1}(p,q))\Big{]}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N};(pq)^{-\nu}\Big{)}_{m}a^{n-m},$ (28)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{{\dagger}m}a^{n}=q^{-\nu\binom{m}{2}}\Big{[}{{}_{q}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{q,p^{-1}}^{m}(\phi_{2}(p,q))\Big{]}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N};(pq)^{\nu}\Big{)}_{m}a^{n-m},$ (29)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$.
$\bullet\,\,\,\,\,\,$ If $n=m$,
$\displaystyle
a^{n}a^{{\dagger}n}=p^{-\nu\binom{n}{2}}\mathcal{T}_{p^{-1},q}^{n}(\phi_{1}(p,q))\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N+\nu};(pq)^{\nu}\Big{)}_{n},$ (30)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{n}a^{{\dagger}n}=q^{\nu\binom{n}{2}}\mathcal{T}_{q,p^{-1}}^{n}(\phi_{2}(p,q))\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N-\nu};(pq)^{-\nu}\Big{)}_{n},$ (31)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{{\dagger}n}a^{n}=p^{\nu\binom{n}{2}}\Big{[}{{}_{p}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{p^{-1},q}(\phi_{1}(p,q))\Big{]}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N};(pq)^{-\nu}\Big{)}_{n},$ (32)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$.
$\displaystyle
a^{{\dagger}n}a^{n}=q^{-\nu\binom{n}{2}}\Big{[}{{}_{q}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{q,p^{-1}}(\phi_{1}(p,q))\Big{]}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{-\alpha
N};(pq)^{\nu}\Big{)}_{n},$ (33)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q)$. In the above expressions,
${}_{(p,q)}\mathcal{S}_{a^{\dagger}}$ is the translation operator defined as
follows
$\,\,{}_{(p,q)}\mathcal{S}_{a^{\dagger}}={{}_{p}}\mathcal{S}_{a^{\dagger}}\,{{}_{q}}\mathcal{S}_{a^{\dagger}},\quad_{p}\mathcal{S}_{a^{\dagger}}^{n}p^{\alpha
N}=p^{\alpha N+n\nu},\quad{{}_{q}}\mathcal{S}_{a^{\dagger}}^{n}q^{\alpha
N}=q^{\alpha N+n\nu},\quad\forall\,n\in\mathbb{N}.$ (34)
The operator $\mathcal{T}_{p,q}(t)$ acts on the vacuum state $|0\rangle$ as
follows
$\displaystyle\mathcal{T}_{p,q}(t)|0\rangle=\frac{tp^{\alpha\chi_{0}+\nu}}{p^{\nu}-q^{\nu}}|0\rangle,$
(35)
where the product $(a;q)_{l}=(1-a)(1-aq)\ldots(1-aq^{l-1})$, $l=1,2,3\ldots$.
This results reveal to be useful for deducing the commutators between $a^{n}$
and $a^{{\dagger}m}$
$\bullet\,\,\,$ For $n<m$,
$\displaystyle[a^{n},a^{{\dagger}m}]_{q^{\nu}}$ $\displaystyle=$
$\displaystyle
p^{-\nu\binom{n}{2}}(1-q^{\nu}{{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{m})\mathcal{T}_{p^{-1},q}^{n}(\phi_{1}(p,q))$
$\displaystyle\times$
$\displaystyle\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N+\nu};(pq)^{\nu}\Big{)}_{n}a^{{\dagger}m-n},$
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$ and
$\displaystyle[a^{n},a^{{\dagger}m}]_{q^{\nu}}$ $\displaystyle=$
$\displaystyle
q^{\nu\binom{n}{2}}(1-q^{\nu}\mathcal{S}_{a^{\dagger}}^{-m})\mathcal{T}_{q,p^{-1}}^{n}(\phi_{2}(p,q))$
$\displaystyle\times$
$\displaystyle\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N-\nu};(pq)^{-\nu}\Big{)}_{n}a^{{\dagger}m-n},$
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q).$
$\bullet\,\,\,$ For $n>m$,
$\displaystyle[a^{n},a^{{\dagger}m}]_{q^{\nu}}$ $\displaystyle=$
$\displaystyle
p^{\nu\binom{m}{2}}({{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{n}-q^{\nu}){{}_{p}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{p^{-1},q}^{m}(\phi_{1}(p,q))$
(36) $\displaystyle\times$
$\displaystyle\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\alpha
N};(pq)^{-\nu}\Big{)}_{m}a^{n-m},$ (37)
if $pq<1$ and $\phi_{2}(p,q)<\phi_{1}(p,q)$ and
$\displaystyle[a^{n},a^{{\dagger}m}]_{q^{\nu}}$ $\displaystyle=$
$\displaystyle
q^{-\nu\binom{m}{2}}({{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}^{n}-q^{\nu})\Big{[}{{}_{q}}\mathcal{S}_{a^{\dagger}}^{-1}\mathcal{T}_{q,p^{-1}}^{m}(\phi_{2}(p,q))\Big{]}$
(38) $\displaystyle\times$
$\displaystyle\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\alpha
N};(pq)^{\nu}\Big{)}_{m}a^{n-m},$ (39)
if $pq>1$ and $\phi_{1}(p,q)<\phi_{2}(p,q).$ Noting that by similar
computation we can derive the $p^{-\nu}$ commutators between $a^{n}$ and
$a^{{\dagger}m}$.
## 3 Coherent states
In this part we investigate some class of deformed states. These states are
the eigenstates of the annihilation operator so called coherent states.
Proposition3 The CS associated with the algebra (4) and (5) with ($\chi_{0}=0$
and $\nu=\alpha$ ) are given in different case by
1. 1.
if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q)$
$\displaystyle|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1/2}(|z|^{2})\sum_{n=0}^{\infty}\frac{p^{n\nu(n-1)/4}z^{n}}{\sqrt{\tau^{n}(\phi_{1}^{-1}(p,q)\phi_{2}(p,q)(pq)^{\nu};(pq)^{\nu})_{n}}}|n\rangle,\,z\in\mathcal{D}_{\nu},$
(40)
where
$\displaystyle\mathcal{N}_{\nu}(x)=\sum_{n=0}^{\infty}\frac{p^{\nu\binom{n}{2}}}{(\phi_{1}^{-1}(p,q)\phi_{2}(p,q)(pq)^{\nu};(pq)^{\nu})_{n}}\Big{(}\frac{1-(pq)^{\nu}}{\phi_{1}(p,q)}x\Big{)}^{n},$
(41)
2. 2.
if $pq>1$, $\phi_{1}(p,q)<\phi_{2}(p,q)$
$\displaystyle|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1/2}(|z|^{2})\sum_{n=0}^{\infty}\frac{q^{-n\nu(n-1)/4}z^{n}}{\sqrt{\tau^{n}(\phi_{2}^{-1}(p,q)\phi_{1}(p,q)(pq)^{-\nu};(pq)^{-\nu})_{n}}}|n\rangle,\,z\in\mathcal{D}_{\nu},$
(42)
where
$\displaystyle\mathcal{N}_{\nu}(x)=\sum_{n=0}^{\infty}\frac{q^{-\nu\binom{n}{2}}}{(\phi_{2}^{-1}(p,q)\phi_{1}(p,q)(pq)^{-\nu};(pq)^{-\nu})_{n}}\Big{(}\frac{1-(pq)^{-\nu}}{\phi_{2}(p,q)}x\Big{)}^{n},$
(43)
In the above expressions the convergence domain $\mathcal{D}_{\nu}$ of the
serie $\mathcal{N}_{\nu}(x)$ is given by
$\displaystyle\mathcal{D}_{\nu}=\\{z\in\mathbb{C}:|z|^{2}<R_{\nu}\\},\,\mbox{
with }R_{\nu}=\infty,\;\nu>0.$ (44)
$R_{\nu}$ is the radius associated with the same series.
In the limit, when $\phi_{1}(p,q)=1=\phi_{1}(p,q)$, $p,\,q\rightarrow 1$, the
series $\mathcal{N}_{\nu}(x)$ is reduced to the usual exponential function
$e^{x}$. Also, when $\phi_{1}(p,q)=1=\phi_{1}(p,q)$, $p\rightarrow
1,\nu\rightarrow 1$ the series $\mathcal{N}_{\nu}(x)$ is reduced to the
$q-$exponential function $e_{q}(x)$. The most important of property of
$|z\rangle_{\nu}$ is that if $|z^{\prime}\rangle_{\nu}$ is another CS, then
${}_{\nu}\langle
z^{\prime}|z\rangle_{\nu}=\frac{\mathcal{N}_{\nu}(z\bar{z}^{\prime})}{\sqrt{\mathcal{N}_{\nu}(|z|^{2})\mathcal{N}_{\nu}(|z^{\prime}|^{2})}},$
(45)
which means that such states are not orthogonal.
Proposition The CS defined in (40) and (42) are normalized, are continuous in
$z$. and solve the unity i.e,
$\int_{\mathcal{D}_{\nu}}\frac{d^{2}}{\pi}\mathcal{W}_{\nu}(|z|^{2})|z\rangle_{\nu}\,{{}_{\nu}}\langle
z|={\bf 1}.$ The resolution of unity assumes the existence of a positive
weight a function $\mathcal{W}_{\nu}(|z|^{2})$ such that
$\mathcal{W}_{\nu}(x)=\mathcal{N}_{\nu}(x)\tilde{\mathcal{W}}_{\nu}(x),\;x=|z|^{2}.$
Proof: From (45), when $z^{\prime}\to z,$ (40) and (42) are normalized. For
the continuity we can see that
$|||z^{\prime}\rangle_{\nu}-|z\rangle_{\nu}||^{2}=2(1-\mathcal{R}e\,_{\nu}\langle
z^{\prime}|z\rangle_{\nu}),$ so
$|||z^{\prime}\rangle_{\nu}-|z\rangle_{\nu}||^{2}\rightarrow 0\mbox{ as
}|z-z^{\prime}|\rightarrow 0,$
since ${}_{\nu}\langle z^{\prime}|z\rangle_{\nu}\rightarrow 0$ as
$|z^{\prime}-z|\rightarrow 0.$ The resolution of the unity can be seem by
using the relation
$\displaystyle\int_{\mathcal{D}_{\nu}}\frac{d^{2}}{\pi}\mathcal{W}_{\nu}(|z|^{2})|z\rangle_{\nu}\,{{}_{\nu}}\langle
z|$ $\displaystyle=$
$\displaystyle\sum_{n,m=0}^{\infty}\frac{p^{n\nu(n-1)/4+m\nu(m-1)/4}\tau^{-\frac{n+m}{2}}}{\sqrt{(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{n}(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{m}}}$
$\displaystyle\times$
$\displaystyle\int_{\mathcal{D}_{\nu}}\bar{z}^{m}z^{n}\frac{d^{2}\mathcal{W}_{\nu}(|z|^{2})}{\pi\mathcal{N}_{\nu}(|z|^{2})}.$
where $\tilde{\mathcal{W}}_{\nu}(x)$ has to be determined from the equations
$\displaystyle\int_{\mathcal{D}_{\nu}}x^{n}\tilde{\mathcal{W}}_{\nu}(x)dx=p^{-n\nu(n-1)/2}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu}\Big{)}_{n}\Big{(}\frac{\phi_{1}(p,q)}{1-(pq)^{\nu}}\Big{)}^{n},$
(46)
if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q)$ and
$\displaystyle\int_{\mathcal{D}_{\nu}}x^{n}\tilde{\mathcal{W}}_{\nu}(x)dx=q^{n\nu(n-1)/2}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu}\Big{)}_{n}\Big{(}\frac{\phi_{2}(p,q)}{1-(pq)^{-\nu}}\Big{)}^{n},$
(47)
if $pq>1,\;\phi_{1}(p,q)<\phi_{2}(p,q)$. If $n$ is extended to
$s-1,\;s\in\mathbb{C},$ then the problem can be formulated to the classical
Stieltjes power moment problem when $0<pq<1$ or Hausdorff power moment problem
when $pq>1.\;\blacksquare$
## 4 Matrix elements
Let us now compute correlation functions with matrix elements of normal and
antinormal forms pertaining to quantum optics.
$\bullet$ For $n<m$, if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q),$ the normal form
is defined by
${}_{\nu}\langle z|a^{\dagger
m}a^{n}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{p^{r\nu(r-1)/4+s\nu(s-1)/4}\tau^{-\frac{r+s}{2}}}{\sqrt{(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{r}(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{s}}}\mathcal{F}_{mn}^{rs},$
(48)
where the matrix elements $\mathcal{F}_{mn}^{rs}$ are given by
$\displaystyle\mathcal{F}_{mn}^{rs}=\langle r|a^{{\dagger}m}a^{n}|s\rangle$
$\displaystyle=C_{s}C_{m-n+s}^{-1}p^{\nu\binom{n}{2}}\Big{(}\frac{\phi_{1}(p,q)p^{(1-s)\nu}}{1-(pq)^{\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{s\nu};(pq)^{-\nu}\Big{)}_{n}\delta_{r,m-n+s},$
(49)
and
$\displaystyle
C_{n}^{2}=p^{\nu\binom{n}{2}}\Big{(}\frac{1-(pq)^{\nu}}{\phi_{1}(p,q)}\Big{)}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu}\Big{)}_{n}^{-1},$
(50)
if $pq>1,\;\phi_{1}(p,q)<\phi_{2}(p,q)$
${}_{\nu}\langle z|a^{\dagger
m}a^{n}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{q^{-r\nu(r-1)/4-s\nu(s-1)/4}\tilde{\tau}^{-\frac{r+s}{2}}\quad\mathbf{F}_{mn}^{rs}}{\sqrt{(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{r}(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{s}}},$
where the matrix elements $\mathbf{F}_{mn}^{rs}$ are given by
$\displaystyle\mathbf{F}_{mn}^{rs}=\langle r|a^{{\dagger}m}a^{n}|s\rangle$
$\displaystyle=\tilde{C}_{s}\tilde{C}_{m-n+s}^{-1}q^{-\nu\binom{n}{2}}\Big{(}\frac{\phi_{2}(p,q)q^{(s-1)\nu}}{1-(pq)^{-\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-s\nu};(pq)^{\nu}\Big{)}_{n}\delta_{r,m-n+s},$
(51)
$\displaystyle\tilde{C}_{n}^{2}=q^{-\nu\binom{n}{2}}\Big{(}\frac{1-(pq)^{-\nu}}{\phi_{2}(p,q)}\Big{)}^{n}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu}\Big{)}_{n}^{-1}.$
(52)
The antinormal form if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q),$ is given by
${}_{\nu}\langle
z|a^{n}a^{{\dagger}m}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{p^{r\nu(r-1)/4+s\nu(s-1)/4}\tau^{-\frac{r+s}{2}}}{\sqrt{(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{r}(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{s}}}\mathcal{G}_{mn}^{rs}$
(53)
where the matrix elements $\mathcal{G}_{mn}^{rs}$ are given by
$\displaystyle\mathcal{G}_{mn}^{rs}=\langle r|a^{n}a^{{\dagger}m}|s\rangle$
$\displaystyle=C_{s}C_{m-n+s}^{-1}p^{\nu\binom{n}{2}}\Big{(}\frac{\phi_{1}(p,q)p^{(1-s-m)\nu}}{1-(pq)^{\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{(s+m)\nu};(pq)^{-\nu}\Big{)}_{n}\delta_{r,m-n+s}.$
(54)
if $pq>1,\;\phi_{1}(p,q)<\phi_{2}(p,q)$
${}_{\nu}\langle
z|a^{n}a^{{\dagger}m}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{q^{-r\nu(r-1)/4-s\nu(s-1)/4}\tilde{\tau}^{-\frac{r+s}{2}}\;\;\mathbf{G}_{mn}^{rs}}{\sqrt{(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{r}(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{s}}}$
where the matrix elements $\mathbf{G}_{mn}^{rs}$ are given by
$\displaystyle\mathbf{G}_{mn}^{rs}=\langle r|a^{n}a^{{\dagger}m}|s\rangle$
$\displaystyle=\tilde{C}_{s}\tilde{C}_{m-n+s}^{-1}q^{-\nu\binom{n}{2}}\Big{(}\frac{\phi_{2}(p,q)q^{(s+m-1)\nu}}{1-(pq)^{-\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-(s+m)\nu};(pq)^{\nu}\Big{)}_{n}\delta_{r,m-n+s}.$
(55)
$\bullet$ For $n>m,$ if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q),$ the normal form
is defined by
${}_{\nu}\langle
z|a^{{\dagger}m}a^{n}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{p^{r\nu(r-1)/4+s\nu(s-1)/4}\tau^{-\frac{r+s}{2}}}{\sqrt{(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{r}(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{s}}}\mathcal{\tilde{G}}_{mn}^{rs},$
(56)
where the matrix elements $\mathcal{\tilde{G}}_{mn}^{rs}$ are given by
$\displaystyle\mathcal{\tilde{G}}_{mn}^{rs}=\langle
r|a^{{\dagger}m}a^{n}|s\rangle$
$\displaystyle=C_{r}C_{r+n-m}^{-1}p^{-\nu\binom{m}{2}}\Big{(}\frac{\phi_{1}(p,q)p^{(n-s)\nu}}{1-(pq)^{\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{(s-m+1)\nu};(pq)^{\nu}\Big{)}_{m}\delta_{s,r+n-m},$
(57)
if $pq>1,\;\phi_{1}(p,q)<\phi_{2}(p,q),$
${}_{\nu}\langle
z|a^{{\dagger}m}a^{n}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{q^{-r\nu(r-1)/4-s\nu(s-1)/4}\tilde{\tau}^{-\frac{r+s}{2}}\;\mathbf{\tilde{G}}_{mn}^{rs}}{\sqrt{(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{r}(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{s}}},$
where the matrix elements $\mathbf{\tilde{G}}_{mn}^{rs}$ are given by
$\displaystyle\mathbf{\tilde{G}}_{mn}^{rs}=\langle
r|a^{{\dagger}m}a^{n}|s\rangle$
$\displaystyle=\tilde{C}_{r}\tilde{C}_{r+n-m}^{-1}q^{\nu\binom{m}{2}}\Big{(}\frac{\phi_{2}(p,q)q^{(s-n)\nu}}{1-(pq)^{-\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{(n-s-1)\nu};(pq)^{-\nu}\Big{)}_{m}\delta_{s,r+n-m},$
(58)
The antinormal form if $pq<1,\;\phi_{2}(p,q)<\phi_{1}(p,q),$ is expressed by
the formula:
${}_{\nu}\langle
z|a^{n}a^{{\dagger}m}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{p^{r\nu(r-1)/4+s\nu(s-1)/4}\tau^{-\frac{r+s}{2}}}{\sqrt{(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{r}(\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{\nu};(pq)^{\nu})_{s}}}\mathcal{\tilde{F}}_{mn}^{rs},$
(59)
where the matrix elements $\mathcal{\tilde{F}}_{mn}^{rs}$ are given by
$\displaystyle\mathcal{\tilde{F}}_{mn}^{rs}=\langle
r|a^{n}a^{{\dagger}m}|s\rangle$
$\displaystyle=C_{s}C_{r+n-m}^{-1}p^{-\nu\binom{m}{2}}\Big{(}\frac{\phi_{1}(p,q)p^{-s\nu}}{1-(pq)^{\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{2}(p,q)}{\phi_{1}(p,q)}(pq)^{(s+1)\nu};(pq)^{\nu}\Big{)}_{m}\delta_{s,r+n-m}.$
(60)
if $pq>1,\;\phi_{1}(p,q)<\phi_{2}(p,q)$
${}_{\nu}\langle
z|a^{n}a^{{\dagger}m}|z\rangle_{\nu}=\mathcal{N}_{\nu}^{-1}(|z|^{2})\sum_{r,s=0}^{\infty}\frac{q^{-r\nu(r-1)/4-s\nu(s-1)/4}\tilde{\tau}^{-\frac{r+s}{2}}\;\mathbf{\tilde{G}}_{mn}^{rs}}{\sqrt{(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{r}(\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-\nu};(pq)^{-\nu})_{s}}},$
where the matrix elements $\mathbf{\tilde{F}}_{mn}^{rs}$ are given by
$\displaystyle\mathbf{\tilde{F}}_{mn}^{rs}=\langle
r|a^{n}a^{{\dagger}m}|s\rangle$
$\displaystyle=\tilde{C}_{s}\tilde{C}_{r+n-m}^{-1}q^{\nu\binom{m}{2}}\Big{(}\frac{\phi_{2}(p,q)q^{s\nu}}{1-(pq)^{-\nu}}\Big{)}^{n}\Big{(}\frac{\phi_{1}(p,q)}{\phi_{2}(p,q)}(pq)^{-(s+1)\nu};(pq)^{-\nu}\Big{)}_{m}\delta_{s,r+n-m}.$
(61)
## 5 Discussions
This part addressed the study of the previous deformation in the particular
case that we present. Let us reexpressed the relations (4) as
$\displaystyle aa^{{\dagger}n}-q^{n\nu}a^{{\dagger}n}a$ $\displaystyle=$
$\displaystyle[n\nu]_{p,q,\nu}^{1,1}a^{{\dagger}n-1}p^{-\alpha N}$ (62)
$\displaystyle aa^{{\dagger}n}-p^{-n\nu}a^{{\dagger}n}a$ $\displaystyle=$
$\displaystyle[n\nu]_{p,q,\nu}^{1,1}a^{{\dagger}n-1}q^{\alpha
N},\;\;\forall\,n\,\in\mathbb{N}\diagdown\\{0\\},$ (63)
where the quantity $[n]_{p,q,\nu}^{\phi_{1},\phi_{2}}$ is the deformed number
given by
$\displaystyle[n\nu]_{p,q,\nu}^{\phi_{1},\phi_{2}}=\frac{\phi_{1}(p,q)p^{-n\nu}-\phi_{2}(p,q)q^{n\nu}}{p^{-\nu}-q^{\nu}}.$
(64)
Then it appears important to emphasize that: By setting $C_{1}$ as the Casimir
operator and defined the regular operators functions
$\displaystyle\hat{\phi}_{1}(p,q,C_{1})=(1+2\gamma
C_{1})p^{-\beta},\,\,\,\,\hat{\phi}_{2}(p,q,C_{1})=(1+2\gamma C_{1})q^{\beta}$
(65)
such that $\hat{\phi}_{1}|n\rangle=(1+2\gamma\omega^{2})p^{-\beta}|n\rangle$
and $\hat{\phi}_{2}|n\rangle=(1+2\gamma\omega^{2})q^{\beta}|n\rangle$, where
$w^{2}$ is the eigenvalue of the operator $C_{1}$. The commutation relations
(4) take the form
$\displaystyle
aa^{\dagger}-q^{\nu}a^{\dagger}a=\hat{\phi}_{1}(p,q,C_{1})p^{-\alpha
N},\,\,aa^{\dagger}-p^{-\nu}a^{\dagger}a=\hat{\phi}_{2}(p,q,C_{1})q^{\alpha
N}.$ (66)
The eigen-equation associated with the Hamiltonian operator
$H_{\alpha,\beta}^{\nu,\gamma}(p,q)=a^{\dagger}a+aa^{\dagger},$ i.e.
$H_{\alpha,\beta}^{\nu,\gamma}(p,q)|n\rangle=E_{n,\alpha,\beta}^{\nu,\gamma}(p,q)|n\rangle$,
is such that the corresponding eigenvalue
$E_{n,\alpha,\beta}^{\nu,\gamma}(p,q)$ is written as
$\displaystyle E_{n,\alpha,\beta}^{\nu,\gamma}(p,q)$ $\displaystyle=$
$\displaystyle(1+2\gamma
w^{2})\\{q^{\alpha\chi_{0}+n\nu+\beta}+(1+p^{-\nu})[n\nu+\alpha\chi_{0}+\beta]_{(p,q;\nu)}\\}$
(67)
or, equivalently,
$\displaystyle E_{n,\alpha,\beta}^{\nu,\gamma}(p,q)$ $\displaystyle=$
$\displaystyle(1+2\gamma
w^{2})\\{p^{-\alpha\chi_{0}-n\nu-\beta}+(1+q^{\nu})[n\nu+\alpha\chi_{0}+\beta]_{(p,q;\nu)}\\},$
(68)
where
$\displaystyle[n;\gamma C_{1}]_{(p,q;\nu)}!=\Big{(}1+2\gamma
C_{1}\Big{)}^{n}\frac{((p^{-\nu},q^{\nu});(p^{-\nu},q^{\nu}))_{n}}{(p^{-\nu}-q^{\nu})^{n}}.$
(69)
Indeed, for $n,m\in\mathbb{N}\diagdown\\{0\\}$, we derive the next
commutators:
$\bullet\,\,\,$ For $n<m,\,p<1$ and $q>1$,
$\displaystyle[a^{n},(a^{\dagger})^{m}]_{q^{\nu}}$ $\displaystyle=$
$\displaystyle\frac{(1+2\gamma
C_{1})^{n}}{(p^{-\nu}-q^{\nu})^{n}}\Bigg{[}((p^{-\alpha N-\beta-\nu},q^{\alpha
N+\beta+\nu});(p^{-\nu},q^{\nu}))_{n}$ $\displaystyle-$ $\displaystyle
q^{\nu}((p^{-\alpha N-\beta+(m-1)\nu},q^{\alpha
N+\beta-(m-1)\nu});(p^{-\nu},q^{\nu}))_{n}\Bigg{]}(a^{\dagger})^{m-n}.$
There follows:
$\displaystyle\sum\limits_{n=0}^{m-1}\frac{[a^{n},a^{\dagger
m}]_{q^{\nu}}}{[n;\gamma
C_{1}]_{(p,q;\nu)}!}={{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}\Big{(}1-q^{\nu}_{(p,q)}\mathcal{S}_{a^{\dagger}}^{-m}\Big{)}\sum\limits_{n=0}^{m-1}\frac{((p^{-\alpha
N-\beta},q^{\alpha
N+\beta});(p^{-\nu},q^{\nu}))_{n}}{((p^{-\nu},q^{\nu});(p^{-\nu},q^{\nu}))_{n}}{a^{\dagger}}^{m-n}$
(70)
$\displaystyle={{}_{(p,q)}}\mathcal{S}_{a^{\dagger}}\Big{(}1-q^{\nu}_{(p,q)}\mathcal{S}_{a^{\dagger}}^{-m}\Big{)}\mathcal{L}_{m-1}[(p^{-\alpha
N-\beta},q^{\alpha
N+\beta});(p^{-\nu},q^{\nu});{a^{\dagger}}^{-1}]{a^{\dagger}}^{m},$ (71)
where $\mathcal{L}_{m}$ is a deformed hypergeometric function given by
$\displaystyle\mathcal{L}_{m}[(\lambda,\sigma);(p,q);z]:=\sum\limits_{n=0}^{m}\frac{((\lambda,\sigma);(p,q))_{n}}{((p,q);(p,q))_{n}}z^{n},$
(72)
By using the $(p,q)$-binomial theorem given by
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{((a,b);(p,q))_{n}}{((p,q);(p,q))_{n}}z^{n}=\frac{((p,bz);(p,q))_{\infty}}{((p,az);(p,q))_{\infty}},$
(73)
and for $m\rightarrow\infty,$ the relation (70) is reduced to the
$\displaystyle\mathcal{L}_{\infty}$ $\displaystyle=$
$\displaystyle\frac{\Big{(}(p^{-\nu},q^{\alpha
N+\beta}a^{\dagger-1});(p^{-\nu},q^{\nu})\Big{)}_{\infty}}{((p^{-\nu},p^{-\alpha
N-\beta}a^{\dagger-1});(p^{-\nu},q^{\nu}))_{\infty}},\quad||a^{\dagger-1}||<1,$
(74)
$\bullet\,\,\,$ For $n>m$, by using the identity
$\displaystyle[m;\gamma C_{1}]_{(p,q;\nu)}!=(-1)^{m}(1+2\gamma
C_{1})^{m}(p^{-1}q)^{m\nu+\nu
m(m-1)/2}\frac{((p^{\nu},q^{-\nu});(p^{\nu},q^{-\nu}))_{m}}{(p^{-\nu}-q^{\nu})^{m}},$
(76)
we infer
$\displaystyle\sum\limits_{m=0}^{n-1}\frac{[a^{n},a^{\dagger
m}]_{q^{\nu}}((p^{\nu},0);(p^{\nu},q^{-\nu}))_{m}}{[m;\gamma
C_{1}]_{(p,q;\nu)}!}=(_{(p,q)}T_{a}^{n}-q^{\nu})\mathcal{L}_{n-1}[(p^{-\alpha
N-\beta},q^{\alpha N+\beta});(p^{\nu},q^{-\nu});a^{-1}]a^{n},$ (77)
where
$\displaystyle\mathcal{L}_{n-1}[(p^{-\alpha N-\beta},q^{\alpha
N+\beta});(p^{\nu},q^{-\nu});a^{-1}]=\sum\limits_{m=0}^{n-1}\frac{((p^{-\alpha
N-\beta},q^{\alpha
N+\beta});(p^{\nu},q^{-\nu}))_{m}}{((0,q^{-\nu}),(p^{\nu},q^{-\nu});(p^{\nu},q^{-\nu}))_{m}}a^{-m}.$
(79)
When $n\rightarrow\infty$, (79) becomes
$\displaystyle\mathcal{L}_{\infty}$ $\displaystyle=$
$\displaystyle{{}_{2}}\phi_{1}\left(\left.\begin{array}[]{c}(p^{-\alpha
N-\beta},q^{\alpha N+\beta}),0\\\
(0,q^{-\nu})\end{array}\right|(p^{\nu},q^{-\nu});a^{-1}\right)||a^{-1}||<1.$
(82)
$\bullet\,\,\,$ For $n=m,$ we obtain
$\sum\limits_{n=0}^{\infty}{{((1,0),(p^{-\nu},q^{\nu}))_{n}a^{n}(a^{\dagger})^{2n}a^{n}(p^{-\nu}-q^{\nu})^{n}}\over{[n;\gamma
C_{1}]!}}\;=$
$\displaystyle{{}_{2}}\phi_{1}\Bigg{(}\left.\begin{array}[]{cc}(p^{-\alpha
N-\beta-\nu},q^{\alpha N+\beta+\nu}),(p^{\alpha N+\beta},q^{-\alpha
N-\beta})\\\ (0,1)\end{array}\right|(p^{-\nu},q^{\nu});(1+2\gamma
C_{1})(qp^{-1})^{\alpha N+\beta}\Bigg{)},$ (86)
## 6 Conclusion
In this paper, associated relevant properties have been investigated for the
induced deformed harmonic oscillator; the energy spectrum has been explicitly
computed. Deformed coherent states have been built and discussed with respect
to the criteria of coherent state construction. Various commutators involving
annihilation and creation operators and their combinatorics have been computed
and analyzed. Finally, the correlation functions of matrix elements of main
normal and antinormal forms, pertinent for quantum optics analysis, have been
computed.
## Acknowledgments
This work is partially supported by the Abdus Salam International Centre for
Theoretical Physics (ICTP, Trieste, Italy) through the Office of External
Activities (OEA) - Prj-15. The ICMPA is also in partnership with the Daniel
Iagolnitzer Foundation (DIF), France. The authors thank Plyushchay M. S. for
the fruitful discussions. D. Ousmane Samary thanks the Centre international de
mathématiques pures et appliquées (CIMPA) for financial supports.
## References
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* [3] Burban I. M. Ukr. J. Phys. 9 2012.
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* [5] Burban I. M. and Klimyk A. U. Integral Transforms and Special Functions. 2 1994 15–36.
* [6] Burban I. M. arxiv: 1110.1225 2011.
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* [8] El Kinani E. H., Ouhadan A. and Bermi L. El Int. Journal of Algebra. 2 2008 751-756.
* [9] Hounkonnou M. N. and Ngompe Nkouankam E. B. J. Phys. A: Math. Theor. 40 2007 883543.
* [10] Plyushchay M. S. Nucl. Phys. B4911997619-634.
* [11] Akhiezer N. I. Olivier and Boyd London 1965
* [12] Klauder R. J. J. Math. Phys. 4 1963 1055–1058.
* [13] Koekoek R. and Swarttouw R. F. Report 98-17, TU Delft, 1998.
* [14] Odzijewicz A. Commun. Math. Phys. 192 1998 183–215.
* [15] Quesne C. J. Phys. A: Math. Gen. 35 2002 9213–26.
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|
arxiv-papers
| 2012-10-26T21:56:38 |
2024-09-04T02:49:37.170502
|
{
"license": "Public Domain",
"authors": "Sama Arjika, Dine Ousmane Samary, Ezinvi Baloitcha and Mahouton\n Norbert Hounkonnou",
"submitter": "Dine Ousmane Samary",
"url": "https://arxiv.org/abs/1210.7259"
}
|
1210.7335
|
# Professional diversity and the productivity of cities
Luís M. A. Bettencourt,1 Horacio Samaniego,2,3111To whom correspondence should
be addressed; E-mail: horacio@ecoinformatica.cl HyeJin Youn,1
1 Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe NM 87501, USA,
2 Center for Non Linear Studies, Theoretical Division MS B284,
Los Alamos National Laboratory, NM 87545, USA,
3Facultad de Ciencias Forestales & Recursos Naturales,
Universidad Austral de Chile, Valdivia, Chile
###### Abstract
The relationships between diversity, productivity and scale determine much of
the structure and robustness of complex biological and social systems[1, 2].
While arguments for the link between specialization and productivity are
common[3, 5, 6, 7, 4], diversity has often been invoked as a hedging
strategy[8, 9], allowing systems to evolve in response to environmental
change[8, 9, 10]. Despite their general appeal, these arguments have not
typically produced quantitative predictions for optimal levels of functional
diversity consistent with observations. One important reason why these
relationships have resisted formalization is the idiosyncratic nature of
diversity measures, which depend on given classification schemes[11, 12].
Here, we address these issues by analyzing the statistics of professions in
cities and show how their probability distribution takes a universal scale-
invariant form, common to all cities, obtained in the limit of infinite
resolution of given taxonomies. We propose a model that generates the form and
parameters of this distribution via the introduction of new occupations at a
rate leading to individual specialization subject to the preservation of
access to overall function via their ego social networks. This perspective
unifies ideas about the importance of network structure in ecology and of
innovation as a recombinatory process with economic concepts of productivity
gains obtained through the division and coordination of labor, stimulated by
scale.
A fundamental theme across many complex systems[1, 2] \- from ecosystems[13]
to human behavior[14] and socio-economic organization[15, 16] \- deals with
understanding the mechanisms by which diversity arises and is sustained. In
contemporary human societies socioeconomic diversity is associated primarily
with cities[17] accounting for their role in producing new ideas and
stimulating development[17, 10, 4]. However, counter arguments have also been
made noting that specialized cities are sometimes more productive[5, 6, 7, 4].
Familiar examples are contemporary Silicon Valley or manufacturing cities in
their heyday. Nevertheless, these questions remain far from settled, in part
because of the difficulties inherent to measuring diversity in any complex
system[11, 12].
Measures of diversity typically account for the presence, and sometimes the
relative proportion[18, 19, 20], of different functional types, for example
different professions or business types in cities or nations, or different
species in an ecosystem[21]. Such measures, are inevitably linked to
particular classification schemes or taxonomies. To appreciate this point
consider the question: How many different professions are there in a large
city, like New York? In general, there is no objective answer to this question
as it depends on how finely one differentiates similar functions. Here, we
show that under specific conditions a limit of infinite resolution can be
obtained in a way similar to the treatment of physical quantities close to
phase transitions[22] and that, in this limit, scheme-independent measures of
diversity can emerge.
The simplest measure of diversity, $D(N)$, counts the number of distinct
professions present in a city. Fig. 1A shows $D$, for US metropolitan areas
vs. their total employment, $N_{e}$. Because $N_{e}$ is, on average,
proportional to population[23], $N$, we use the two measures of scale
interchangeably. $D$ increases with $N_{e}$ initially and then saturates for
large cities and is well fit by
$D(N_{e})=d_{0}\frac{\left(\frac{N_{e}}{N_{0}}\right)^{\gamma}}{1+\left(\frac{N_{e}}{N_{0}}\right)^{\gamma}}.$
(1)
Eq. 1 holds over time and for different levels of resolution, $r$, in the
occupations hierarchical classification scheme (see SI). The parameters in Eq.
1 are, in general, functions of $r$. The scale $d_{0}(r)$ is the effective
size of the classification scheme at resolution $r$, $N_{0}(r)$ is a
characteristic size of a city at which saturation starts. $\gamma$,
empirically independent of $r$ (see SI), is a scaling exponent giving the
proportionality between the population growth rate and that of new occupations
in the city, in the absence of saturation.
Coarsening the hierarchical classification leads to similar saturation at each
of the scheme’s size $d_{0}(r_{6}),d_{0}(r_{5}),d_{0}(r_{4})$, etc (Fig. 1B).
This behavior is the hallmark of a finite resolution artefact, a phenomenon
well understood in terms of finite size scaling at phase transitions[22]. The
explicit dependence of $d_{0}$ on $r$ means that given classification schemes
are too coarse to capture the professional diversity of large US cities[24],
beyond $N_{0}\sim 10^{5}$. Nevertheless, we can use the variation of the
statistics of occupations with $r$ to derive classification scheme independent
results. We reconcile all curves for $D(N)$ at different $r$ and extract their
limit as $r\rightarrow\infty$. We define a dimensionless function $h(N$ $/$
$N_{0},\gamma)$ such that
$D(N)=d_{0}~{}h\left(\frac{N}{N_{0}}\right)~{}\left(\frac{N}{N_{0}}\right)^{\gamma}\rightarrow\quad\begin{cases}D_{0}~{}N^{\gamma},\qquad
N<<N_{0},\\\ d_{0}(r),\qquad\quad N>>N_{0},\end{cases}$
where $D_{0}$ is a constant. Comparison with Eq. 1 tells us that in the limit
$\frac{N}{N_{0}}\rightarrow 0$, $h\rightarrow 1$,
$D_{0}\rightarrow\frac{d_{0}}{N_{0}^{\gamma}}$, and in the limit
$\frac{N}{N_{0}}\rightarrow\infty$,
$h\rightarrow\left(\frac{N_{0}}{N}\right)^{\gamma}$. A universal scaling
regime exists if and only if the quantity
$D_{0}=\frac{d_{0}(r)}{N_{0}(r)^{\gamma}}$ becomes a constant, independent of
$r$, as $r\rightarrow\infty$ (Fig. 1B). Fig. 1C shows $d_{0}$ vs.
$N_{0}^{\gamma}$ across $r$ and over time. The relationship is well described
by a straight-line with slope of $D_{0}=0.05$ across all years. These results
suggest the existence of a resolution independent, scale-invariant limit for
$D(N)$ and show that the occupational diversity of cities is in fact open-
ended: the number of distinct occupations in US cities increases by $\sim
85\%$ with each doubling of its labor force, meaning that larger cities are at
once more diverse in absolute terms and more specialized per capita. These
insights can be proven as simple theorem (see SI).
Beyond analyzing the presence or absence of professions, which gives only a
crude measure of urban diversity, we can characterize their frequency
distribution. The analysis of the frequency of different types in complex
systems, from word frequency to city size, is naturally described in terms of
their (Zipfian) rank-frequency distribution. To derive this distribution we
identify $D(N)$ with the maximum rank at each value of $N$, which has
probability $p(D)=\frac{1}{N}$. Inverting this relation and generalizing it to
all ranks, $i$, leads to the occupational frequency, $f(i)$:
$\displaystyle
f(i)=\frac{N_{e}}{N_{0}}\left(\frac{d_{0}-i}{i}\right)^{1/\gamma}.$ (2)
This is also scheme independent in the large resolution limit and can be used
to derive the probability density, $p(i)$, as
$\displaystyle
p(i)=\frac{f(i)}{\sum_{i=1}^{D}f(i)}=\frac{1-\gamma}{\gamma}\frac{i^{-1/\gamma}}{1-D(N)^{-\frac{1-\gamma}{\gamma}}};$
(3)
which is also independent of $r$. The occupational probability has a residual
dependence on $N$ through $D(N)$ because the rarest professions cannot have
less than one person. This is the only source of city size dependence of
traditional measures of diversity such as the Herfindahl-Hirschman index or
the Shannon entropy (see SI)[18, 19, 21], which are functionals of $p(i)$.
Given Eq. 3, both measures express increases in diversity (the Herfindahl-
Hirschman index decreases, the Shannon entropy increases) towards a finite
limit at infinite $N$. For large cities the approach to this limit is
controlled by a term $\sim N^{\delta}$, with $\delta=1-\gamma$ (see SI for
derivation).
Fig. 2 shows that the distribution of occupations for different cities is
universal: When adjusted for scale, $N_{e}$, all frequency curves collapse
onto a single line. This shows that there is an expected nested sequence of
occupations, predicted by city size, as expected by the hierarchy principle of
central place theory[25, 20] and in analogy to products vs. level of economic
development at the national level[15, 16].
A simple model that predicts the form of the occupational diversity
distribution, Eq. 3, is a version of the Yule-Simon mechanism of preferential
attachment[27, 26]: as the city grows by one more job, $\Delta N_{e}=1$, it
creates a new occupation with probability $\alpha=\frac{dD}{dN_{e}}=\gamma
D_{0}N_{e}^{\gamma-1}$, or it takes up an existing profession, proportionally
to its frequency, with probability $1-\alpha$. For large $N_{e}$ this predicts
an exponent[27, 26] in the occupational distribution of
$\gamma=\frac{N_{e}}{1-\alpha(N_{e})}\frac{\alpha(N_{e})}{D(N_{e})}$, note
that even for small $N_{e}$, $\alpha<0.04<<1$.
Given the results so far, we may expect economic productivity to be inversely
proportional to professional diversity. Consider that indicators of economic
productivity (wages, GDP) scale, on average, superlinearly[23] with $N$,
$W(N,t)=W_{0}(t)N(t)^{\beta}$, with $W_{0}(t)$ and $\beta\simeq 1+\delta>1$
independent of $N$ (see SI). An average wage per capita is, then,
$w(N)=W_{0}N^{\delta}$, where $\delta\sim 1/6\simeq 1-\gamma$. This result has
been derived from a general theoretical framework that defines cities as co-
located social networks, subject to infrastructural efficiency
constraints[28], with $w=Gk(N)$, where $G$ is a constant in $N$, involving a
balance between people and infrastructural properties, and
$k(N)=k_{0}N^{\delta}$ is the average social connectivity (degree) per person,
which has been observed in urban telecommunication networks[29]. Similarly,
diversity per capita, $d(N)=D(N)/N=D_{0}N^{\gamma-1}=D_{0}N^{-\delta}$. Hence,
we conclude that $w\sim 1/d$. This relation is an expression of the abundant
evidence in economics for specialization (a decrease in $d(N)$) as the source
of increases in productivity[3, 7]. However, no city has become rich by
reducing its occupational diversity to a single activity: What then is the
optimal level of diversity that maximizes the economic productivity of a city?
To answer this question we observe that the process of specialization, by
which an individual sheds tasks to others, requires that such functions remain
tightly integrated so that overall functionality is preserved. This implements
a form of comparative advantage at the individual level, where increases in
productivity at each node, gained through specialization, remain integrated
with other necessary functions via social network links. We may therefore
require that the number of functions directly accessible to each person is
preserved as the city grows. The number of functions that each individual
reaches directly through its social network is $N_{f}=d.k$, which we require
to stay constant, $A$, in $N$, $d.k=A$. We now reconcile the expectation that
$w\sim 1/d$ with the claim that, like other urban socioeconomic outputs, $w$
be proportional to social connectivity[28], $w\sim k$. We write $w=g(kd)/d$,
with $g$ an analytic function, independent of $N$. We now maximize wages
subject to the conservation of functionality across social links by a Lagrange
multiplier procedure (see SI) to show that $d=A/k$ and that
$w(N)=g(A)/d(N)=g(A)/A~{}k(N)$. Then,
$D(N)=A/k(N)N=A/k_{0}N^{1-\delta}=D_{0}N^{\gamma}$, which predicts the form of
the scaling of occupational diversity with city size. As shown above, this
relation, taken across all $N$, also predicts the rank-size distribution of
urban occupations and associated measures of diversity.
In summary, we showed that the patterns of occupational diversity and economic
productivity observed in US metropolitan areas can be derived from an
integrated view of cities as socioeconomic networks that promote a systematic
division and coordination of labor without loss of overall functionalities
available to individuals. Similar quantitative patterns characterize the
technological complexity of simpler human societies[30] and may be a property
of networked systems that can experience open-ended increases in their
productivity with scale. The reversibility of these processes, e.g. the
existence of hysteresis in the externalization and reabsorption of functions
by individuals and networks, may also underlie the resilience of many complex
systems[13, 12, 9] under unexpected functional change or population loss.
## Acknowledgements
We thank Doug Erwin, Ricardo Hausmann, Cesar Hidalgo, José Lobo and Geoffrey
West for discussions. This research is partially supported by the Rockefeller
Foundation, the James S. McDonnell Foundation (grant no. 220020195), the
National Science Foundation (grant no. 103522), the John Templeton Foundation
(grant no. 15705), the U.S. Department of Energy through the LANL/LDRD Program
(contract no. DE-AC52-06NA25396), and by a gift from the Bryan J. and June B.
Zwan Foundation.
## References
* [1] May, R. M. _Stability and Complexity in Model Ecosystems._ (Princeton University Press, 2001).
* [2] Bonner, J. T. _The Evolution of Complexity by Means of Natural Selection_ (Princeton University Press, 1988).
* [3] Smith, A. _An Inquiry into the Nature and Causes of The Wealth of Nations_ (Mobi Classics, 2010).
* [4] Duranton, G. & Puga, D. Diversity and Specialisation in Cities: Why, Where and When Does it Matter? _Urban Stud._ 37, 533–555 (2000).
* [5] Henderson, J. V. The Sizes and Types of Cities. _Am. Econ. Rev._ 64, 640–656 (1974).
* [6] Glaeser, E. L., Kallal, H. D., Scheinkman, J. A. & Shleifer, A. Growth in Cities. _J. Polit. Econ._ 100, 1126–1152 (1992).
* [7] Henderson, J. V., Kuncoro, A. & Turner, M. Industrial Development in Cities. _J. Pol. Econ._ 103, 1067–1090 (1995).
* [8] Haldane, A. G. & May R. M. Systemic Risk in Banking Ecosystems, _Nature_ 469 351–355 (2011).
* [9] Scheffer, M. et al. Anticipating critical transitions. Science 338, 344–348 (2012).
* [10] Quigley, J. M. Urban diversity and economic growth. _J. Econ. Perspect._ 12, 127–138 (1998).
* [11] May, R. M. How Many Species Are There on Earth? Science 241, 1441–1449 (1988).
* [12] Marshall, C. R. Marine Biodiversity Dynamics over Deep Time. _Science_ 329, 1156–1157 (2010).
* [13] Tilman, D. _et al._ Diversity and Productivity in a Long-Term Grassland Experiment. _Science_ 294, 843–845 (2001).
* [14] Eagle, N., Macy, M. & Claxton, R. Network diversity and economic development. _Science_ 328, 1029–1031 (2010).
* [15] Hidalgo, C. A., Klinger, B., Barabási, A.-L. & Hausmann, R. The Product Space Conditions the Development of Nations. _Science_ 317, 482–487 (2007).
* [16] Hidalgo, C. A. & Hausmann, R. The Building Blocks of Economic Complexity. _Proc. Natl. Acad. Sci. U.S.A._ 106, 10570–10575 (2009).
* [17] Jacobs, J. _The Death and Life of Great American Cities_ (Random House, 1961).
* [18] Hirschman, A. O. _National Power and the Structure of Foreign Trade_ (University of California Press, 1945).
* [19] Cover, T. M. & Thomas, J. A. Elements of Information Theory (Wiley & Sons, 2006).
* [20] Mori, T., Nishikimi, K. & Smith, T. E. The Number-Average Size Rule: A New Empirical Relationship Between Industrial Location and City Size. _J. Reg. Sci._ 48 165-211 (2008).
* [21] Magurran, A E. Measuring Biological Diversity (Wiley-Blackwell, 2003).
* [22] Stanley, H. E. _Introduction to Phase Transitions and Critical Phenomena_ (Oxford University Press, 1971).
* [23] Bettencourt, L. M. A., Lobo, J., Helbing, D., Kühnert, C. & West, G. B. Growth, Innovation, Scaling, and the Pace of Life in Cities. _Proc. Natl. Acad. Sci. U.S.A._ 104, 7301–7306 (2007).
* [24] Mameli, F., Faggian, A. & McCann, P. Employment Growth in Italian Local Labour Systems: Issues of Model Specification and Sectoral Aggregation. _Spatial Economic Analysis_ 3, 343–360 (2008).
* [25] Christaller, W. _Central Places in Southern Germany_ (Prentice Hall, Englewood Cliffs, NJ, 1966).
* [26] D. H. Zanette & Montemurro, M. A. Dynamics of Text Generation with Realistic Zipf Distribution. _J. Quant. Ling._ 12, 29–40 (2005).
* [27] Simon, H. A. Biometrika Trust On a Class of Skew Distribution Functions. _Biometrika_ 42, 425–440 (1955).
* [28] Bettencourt, L. M. A. The Origins of Scaling in Cities. (2012). Available online at http://www.santafe.edu/research/working-papers/
* [29] Schläpfer, M. et al. The scaling of human interactions with city size. _Arxiv preprint_ 1210.5215 (2012).
* [30] Kline, M. A. & Boyd, R. Population Size Predicts Technological Complexity in Oceania. _Proc. R. Soc. B._ 277, 2559–2564 (2010).
### Fig. 1.
The number of distinct occupations in US Metropolitan Statistical Areas vs.
total employment. (A) The relationship between the number of professions
present for each city (orange dots) and city size is well described by
$D(N_{e})=d_{0}\frac{(N_{e}/N_{0})^{\gamma}}{1+(N_{e}/N_{0})^{\gamma}}$, with
$d_{0}=686$, $\gamma=0.84$, $N_{0}=1.48\times 10^{5}$ (blue line). (B)
$D(N_{e})$ at different labels of resolution of the occupational
classification scheme, $r_{i}$, with $i=6$ the finest and $i=3$ the coarsest.
(C) $d_{0}$ is proportional to $N_{0}^{\gamma}$ across levels of
classification scheme resolution and time, suggesting the there is a
$r$-independent limit to the form of the occupational diversity of cities and
that $D$ is open-ended. In this limit, $D(N_{e})=D_{0}N_{e}^{\gamma}$ and
larger cities are always more diverse as a whole, but more specialized per
capita.
### Fig. 2.
The distribution of occupations in US metropolitan areas is universal. (A)
Frequency distribution for several cities with different population sizes only
differ in their amplitude, which is set by city size and the extent to which
they probe rare occupations. The horizontal grey line shows the minimum number
of professions (thirty) reported. (B) The rank-probability distributions for
different cities collapse on each other when adjusted for city size (total
employment). The yellow line shows the fit of the universal form to
$f(i)/N_{e}=\frac{1}{N_{0}}\left(\frac{d_{0}-(i+i_{0})}{(i+i_{0})}\right)^{1/\gamma}$,
where we introduces a scale $i_{0}\simeq 3$ at small ranks. The black line is
the form of $f(i)/N_{e}$ in the absence of saturation.
|
arxiv-papers
| 2012-10-27T15:24:59 |
2024-09-04T02:49:37.180783
|
{
"license": "Public Domain",
"authors": "Lu\\'is M. A. Bettencourt and Horacio Samaniego and HyeJin Youn",
"submitter": "Horacio Samaniego",
"url": "https://arxiv.org/abs/1210.7335"
}
|
1210.7340
|
# On $L^{p}$ Estimates in Homogenization
of Elliptic Equations of Maxwell’s Type
Zhongwei Shen Liang Song
###### Abstract
For a family of second-order elliptic systems of Maxwell’s type with rapidly
oscillating periodic coefficients in a $C^{1,\alpha}$ domain $\Omega$, we
establish uniform estimates of solutions $u_{\varepsilon}$ and $\nabla\times
u_{\varepsilon}$ in $L^{p}(\Omega)$ for $1<p\leq\infty$. The proof relies on
the uniform $W^{1,p}$ and Lipschitz estimates for solutions of scalar elliptic
equations with periodic coefficients.
Keywords: Homogenization; Maxwell’s Equations; $L^{p}$ Estimates.
## 1 Introduction
Let $\Omega$ be a bounded $C^{1,\alpha}$ domain in $\mathbb{R}^{3}$ for some
$\alpha>0$ and $n$ denote the outward unit normal to $\partial\Omega$. Let
$A(y)=(a_{ij}(y))$ and $B(y)=(b_{ij}(y))$ be two $3\times 3$ matrices with
real entries satisfying the ellipticity conditions:
$\mu|\xi|^{2}\leq
a_{ij}(y)\xi_{i}\xi_{j}\leq\frac{1}{\mu}|\xi|^{2},\quad\mu|\xi|^{2}\leq
b_{ij}(y)\xi_{i}\xi_{j}\leq\frac{1}{\mu}|\xi|^{2}$ (1.1)
for any $\xi,y\in\mathbb{R}^{3}$ and some $\mu>0$. Consider the second-order
elliptic system of Maxwell’s type:
$\nabla\times\big{(}A(x/\varepsilon)\nabla\times
u_{\varepsilon}\big{)}+B(x/\varepsilon)u_{\varepsilon}=F+\nabla\times
G\quad\text{ in }\Omega,$ (1.2)
where $u_{\varepsilon}$ is a vector field in $\Omega$ and $\varepsilon>0$ a
small parameter. Given $F,G\in L^{2}(\Omega;\mathbb{R}^{3})$, it follows
readily from the Lax-Milgram Theorem that the elliptic system (1.2) has a
unique (weak) solution in
$V^{2}_{0}(\Omega)=\big{\\{}u\in L^{2}(\Omega;\mathbb{R}^{3}):\ \nabla\times
u\in L^{2}(\Omega;\mathbb{R}^{3})\text{ and }n\times u=0\text{ on
}\partial\Omega\big{\\}}.$ (1.3)
Moreover, the solution satisfies the estimate
$\|u_{\varepsilon}\|_{L^{2}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\left\\{\|F\|_{L^{2}(\Omega)}+\|G\|_{L^{2}(\Omega)}\right\\},$ (1.4)
where $C$ depends only on $\mu$ and $\Omega$. Suppose, in addition, that the
matrices $A(y)$ and $B(y)$ are periodic with respect to $\mathbb{Z}^{3}$:
$A(y+z)=A(y)\quad\text{ and }\quad B(y+z)=B(y)\quad\text{ for any
}y\in\mathbb{R}^{3},z\in\mathbb{Z}^{3}.$ (1.5)
It follows from the theory of homogenization that $u_{\varepsilon}\to u_{0}$
weakly in $V^{2}_{0}(\Omega)$ as $\varepsilon\to 0$, and $u_{0}$ is the unique
solution in $V^{2}_{0}(\Omega)$ of the homogenized system:
$\nabla\times\big{(}A_{0}\nabla\times u_{0}\big{)}+B_{0}u_{0}=F+\nabla\times
G\quad\text{ in }\Omega,\\\ $ (1.6)
where $A_{0}$ and $B_{0}$ are constant (effective) matrices given by
$A_{0}=\big{(}\mathcal{H}(A^{-1})\big{)}^{-1}\quad\text{ and }\quad
B_{0}=\mathcal{H}(B).$
We refer the reader to [3, pp.81-91] for the definition of $A_{0}$ and $B_{0}$
as well as the homogenization theory for (1.2).
In this paper we consider the boundary value problem for the elliptic system
(1.2):
$\left\\{\begin{aligned} \nabla\times\big{(}A(x/\varepsilon)\nabla\times
u_{\varepsilon}\big{)}+B(x/\varepsilon)u_{\varepsilon}&=F+\nabla\times
G&\quad&\text{ in }\Omega,\\\ n\times u_{\varepsilon}&=f&\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$ (1.7)
We shall be interested in the estimates of $u_{\varepsilon}$ and $\nabla\times
u_{\varepsilon}$, which are uniform in $\varepsilon>0$, in $L^{p}(\Omega)$ for
$1<p\leq\infty$, under the ellipticity and periodicity conditions on $A$ and
$B$.
For $1<p<\infty$, let
$V^{p}(\Omega)=\big{\\{}u\in L^{p}(\Omega;\mathbb{R}^{3}):\,\nabla\times u\in
L^{p}(\Omega;\mathbb{R}^{3})\big{\\}}.$ (1.8)
If $f\in L^{p}(\partial\Omega;\mathbb{R}^{3})$ and $n\cdot f=0$ on
$\partial\Omega$, we will use Div$(f)$ to denote the surface divergence of $f$
on $\partial\Omega$, defined by
$<\text{Div}(f),\psi>_{W^{-1,p}(\partial\Omega)\times
W^{1,p^{\prime}}(\partial\Omega)}=-\int_{\partial\Omega}<f,\nabla_{\tan}\psi>\,d\sigma,$
(1.9)
where $\psi\in C^{\infty}(\mathbb{R}^{d})$ and
$\nabla_{\tan}\psi=\nabla\psi-<\nabla\psi,n>n$ denotes the tangential gradient
of $\psi$ on $\partial\Omega$. The following are the main results of the
paper.
###### Theorem 1.1.
Let $1<p<\infty$ and $\Omega$ be a bounded, simply connected, $C^{1,\alpha}$
domain in $\mathbb{R}^{3}$ with connected boundary. Suppose that $A$ and $B$
satisfy conditions (1.1) and (1.5) and that $A,B$ are Hölder continuous. Let
$F,G\in L^{p}(\Omega;\mathbb{R}^{3})$ and $f\in
L^{p}(\partial\Omega;\mathbb{R}^{3})$ with $n\cdot f=0$ on $\partial\Omega$
and Div$(f)\in W^{-\frac{1}{p},p}(\partial\Omega)$. Then the boundary value
problem (1.7) has a unique solution in $V^{p}(\Omega)$. Moreover, the solution
$u_{\varepsilon}$ satisfies
$\displaystyle\|u_{\varepsilon}\|_{L^{p}(\Omega)}$
$\displaystyle+\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}$ (1.10)
$\displaystyle\leq
C_{p}\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|f\|_{L^{p}(\partial\Omega)}+\|\text{\rm
Div}(f)\|_{W^{-\frac{1}{p},p}(\partial\Omega)}\right\\},$
where the constant $C_{p}$ is independent of $\varepsilon>0$.
###### Theorem 1.2.
Let $\Omega$ be a bounded, simply connected, $C^{1,\alpha}$ domain in
$\mathbb{R}^{3}$ with connected boundary. Suppose that $A$ and $B$ satisfy
conditions (1.1) and (1.5) and that $A$, $B$ are Hölder continuous. Also
assume that $A$ is symmetric. Let $F,G\in C^{\gamma}(\Omega;\mathbb{R}^{3})$
and $f\in C^{\gamma}(\partial\Omega;\mathbb{R}^{3})$ with $n\cdot f=0$ on
$\partial\Omega$ and Div$(f)\in C^{\gamma}(\partial\Omega)$ for some
$\gamma>0$. Let $u_{\varepsilon}$ be the unique solution of (1.7) in
$V^{2}(\Omega)$. Then $u_{\varepsilon},\nabla\times u_{\varepsilon}\in
L^{\infty}(\Omega;\mathbb{R}^{3})$, and
$\displaystyle\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}$
$\displaystyle+\|\nabla\times u_{\varepsilon}\|_{L^{\infty}(\Omega)}$ (1.11)
$\displaystyle\leq
C_{\gamma}\left\\{\|F\|_{C^{\gamma}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|f\|_{C^{\gamma}(\partial\Omega)}+\|\text{\rm
Div}(f)\|_{C^{\gamma}(\partial\Omega)}\right\\},$
where the constant $C_{\gamma}$ is independent of $\varepsilon>0$.
Besides the interest in their own rights, uniform regularity estimates are an
important tool in the study of convergence problems for solutions
$u_{\varepsilon}$, eigenfunctions, and eigenvalues in the theory of
homogenization. For the elliptic systems
$-\text{\rm div}\big{(}A(x/\varepsilon)\nabla
u_{\varepsilon}\big{)}=F\quad\text{ in }\Omega,$ (1.12)
where $A(y)=\big{(}a_{ij}^{\alpha\beta}(y)\big{)}$ with $1\leq i,j\leq d$ and
$1\leq\alpha,\beta\leq m$ is uniform elliptic, periodic, and Hölder
continuous, uniform $W^{1,p}$ estimates, Hölder estimates, and Lipschitz
estimates were established in [1] [2] for solutions in $C^{1,\alpha}$ domains
with the Dirichlet boundary condition. Analogous results for solutions in
$C^{1,\alpha}$ domains with the Neumann boundary conditions were recently
obtained in [8]. We mention that for suitable solutions of
$\text{div}\big{(}A(x/\varepsilon)\nabla u_{\varepsilon}\big{)}=0$ in a
Lipschitz domain $\Omega$, under the additional symmetry condition
$a_{ij}^{\alpha\beta}(y)=a_{ji}^{\beta\alpha}(y)$, the following uniform
$L^{2}$ Rellich estimates:
$\left\|\frac{\partial
u_{\varepsilon}}{\partial\nu_{\varepsilon}}\right\|_{L^{2}(\partial\Omega)}\approx\|\nabla_{\tan}u_{\varepsilon}\|_{L^{2}(\partial\Omega)}$
(1.13)
were proved in [9] [10], where $\partial
u_{\varepsilon}/\partial\nu_{\varepsilon}$ and $\nabla_{\tan}u_{\varepsilon}$
denote the conormal derivative and tangential gradient of $u_{\varepsilon}$ on
$\partial\Omega$, respectively. The proof for the Lipschitz estimates in [8]
relies on the $L^{2}$ Relllich estimates in [10]. As a result, the Lipschitz
estimates in [8] for solutions with the Neumann boundary conditions, which are
used in the proof of Theorem 1.2, were established under the additional
symmetry condition.
To prove Theorems 1.1 and 1.2, our basic idea is to reduce the study of (1.2)
to that of a scalar uniform elliptic equation of divergence form. This uses
the well-known fact that on a simply connected domain $\Omega$ in
$\mathbb{R}^{3}$, $u\in L^{2}(\Omega;\mathbb{R}^{3})$ and $\nabla\times u=0$
in $\Omega$ imply that $u=\nabla P$ in $\Omega$ for some scalar function $P\in
H^{1}(\Omega)$. It also relies on the fact that on a bounded $C^{1}$ domain
$\Omega$ with connected boundary, $u\in L^{p}(\Omega;\mathbb{R}^{3})$ and
div$(u)=0$ in $\Omega$ imply that $u=\nabla\times v$ in $\Omega$ for some
$v\in W^{1,p}(\Omega;\mathbb{R}^{3})$. The approach allows us to reduce the
estimates (1.10) and (1.11) to the $W^{1,p}$ and Lipschitz estimates for
solutions of the scalar elliptic equation
$-\text{\rm div}\big{(}A(x/\varepsilon)(\nabla
w_{\varepsilon}+g)\big{)}=F\quad\text{ in }\Omega.$ (1.14)
We point out that both the Dirichlet condition and the Neumann condition for
the elliptic equation (1.14) are needed to handle the system (1.2).
The rest of the paper is organized as follows. In Section 2 we collect some
basic facts related to the divergence and curl operators, which will be needed
in Section 4. In Section 3 we establish the $W^{1,p}$ and Lipschitz estimates
for (1.14) in a bounded $C^{1,\alpha}$ domain. While the $W^{1,p}$ estimates
for (1.14) follow readily from those for (1.12) with $m=1$ in [1, 2] and [8],
the desired Lipschitz estimates require some additional argument, involving
the Green and Neumann functions for (1.12). The proof of Theorem 1.1 is given
in Section 4, and the proof of Theorem 1.2 in Section 5. Finally, we point out
that under the additional assumption that $A$ is Lipschitz continuous, $B$ is
a constant matrix, and $\Omega$ is $C^{1,1}$, it follows from the estimate
(1.10) that
$\|u_{\varepsilon}\|_{W^{1,p}(\Omega)}\leq
C_{p}\left\\{\|F\|_{L^{p}(\Omega)}+\|\text{\rm
div}(F)\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|f\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\}$
(1.15)
for $1<p<\infty$ (see Remark 2.4).
## 2 Some preliminaries
The materials in this section are more or less known.
###### Theorem 2.1.
Let $\Omega$ be a bounded, simply-connected, Lipschitz domain in
$\mathbb{R}^{3}$. Suppose that $u\in L^{p}(\Omega;\mathbb{R}^{3})$ for some
$1<p<\infty$ and $\nabla\times u=0$ in $\Omega$. Then $u=\nabla P$ in $\Omega$
for some $P\in W^{1,p}(\Omega)$.
###### Proof.
The case $p>2$ follows directly from the case $p=2$, which is well known. The
case $p<2$ may be proved in the same manner as in the case $p=2$ (see e.g. [7,
pp.31-32]). ∎
We will use $W^{t,p}(\partial\Omega)$ to denote the Sobolev-Besov space of
order $t$ and exponent $p$ on $\partial\Omega$ for $-1<t<1$ and $1<p<\infty$.
Note that the dual of $W^{t,p}(\partial\Omega)$ is given by
$W^{-t,q}(\partial\Omega)$, where $q=p^{\prime}=\frac{p}{p-1}$.
###### Theorem 2.2.
Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{3}$ with connected
boundary. Let $g\in L^{p}(\Omega;\mathbb{R}^{3})$ for some $1<p<\infty$.
Suppose that $\text{\rm div}(g)=0$ in $\Omega$. Then there exists $h\in
W^{1,p}(\Omega;\mathbb{R}^{3})$ such that $\nabla\times h=g$ in $\Omega$.
Moreover, $\text{\rm div}(h)=0$ in $\Omega$ and $\|h\|_{W^{1,p}(\Omega)}\leq
C_{p}\,\|g\|_{L^{p}(\Omega)}$, where $C_{p}$ depends only on $p$ and $\Omega$.
###### Proof.
The result is well known for smooth domains. The proof for the case of $C^{1}$
domains is similar. We provide a proof, which follows the lines in [7] and
[4], for the sake of completeness.
We first note that if $u\in L^{p}(\mathbb{R}^{3};\mathbb{R}^{3})$ with
supp$(u)\subset B=B(0,R)$ and $\text{div}(u)=0$ in $\mathbb{R}^{3}$, then
there exists $v\in W^{1,p}(B;\mathbb{R}^{3})$ such that $\nabla\times v=u$ in
$B$, $\text{div}(v)=0$ in $B$, and $\|v\|_{W^{1,p}(B)}\leq
C_{p}\,\|u\|_{L^{p}(\mathbb{R}^{3})}$. To see this, we let $v=\nabla\times w$,
where
$w(x)=\int_{\mathbb{R}^{3}}\Gamma(x-y)u(y)\,dy,$
and $\Gamma(x)=(4\pi|x|)^{-1}$ is the fundamental solution for $-\Delta$ in
$\mathbb{R}^{3}$, with pole at the origin. It follows from $\text{div}(u)=0$
in $\mathbb{R}^{3}$ that $\text{div}(w)=0$ in $\mathbb{R}^{3}$. Hence,
$\nabla\times v=\nabla\times(\nabla\times w)=-\Delta
w+\nabla(\text{div}(w))=-\Delta w=u.$
Clearly, div$(v)=0$ in $\mathbb{R}^{3}$. Also, by the Calderón-Zygmund
estimate and fractional intergal estimate,
$\|v\|_{W^{1,p}(B)}\leq C\,\|\nabla w\|_{W^{1,p}(B)}\leq
C_{p}\,\|u\|_{L^{p}(\mathbb{R}^{3})},$
where $C_{p}$ may depend on $R$.
We now consider the case where $\Omega$ is a bounded $C^{1}$ domain with
connected boundary. Choose a ball $B=B(0,R)$ such that
$\overline{\Omega}\subset B(0,R/4)$. Since $\partial\Omega$ is connected,
$B\setminus\overline{\Omega}$ is a bounded (connected) $C^{1}$ domain. Also,
$g\in L^{p}(\Omega;\mathbb{R}^{3})$ and div$(g)=0$ in $\Omega$ imply that
$n\cdot g\in W^{-\frac{1}{p},p}(\partial\Omega)$ and
$<n\cdot g,1>_{W^{-\frac{1}{p},p}(\partial\Omega)\times
W^{\frac{1}{p},p^{\prime}}(\partial\Omega)}=0.$
It follows from [5] that there exists $f\in
W^{1,p}(B\setminus\overline{\Omega})$ such that $\Delta f=0$ in
$B\setminus\overline{\Omega}$, $\frac{\partial f}{\partial n}=n\cdot g$ on
$\partial\Omega$, and $\frac{\partial f}{\partial n}=0$ on $\partial B$.
Moreover,
$\|\nabla f\|_{L^{p}(B\setminus\overline{\Omega})}\leq C_{p}\,\|n\cdot
g\|_{W^{-\frac{1}{p},p}(\partial\Omega)}\leq C_{p}\,\|g\|_{L^{p}(\Omega)}.$
(2.1)
Define
$\widetilde{g}=\left\\{\begin{aligned} &g&\quad&\text{ in }\Omega,\\\ &\nabla
f&\quad&\text{ in }B\setminus\overline{\Omega},\\\ &0&\quad&\text{ in
}\mathbb{R}^{3}\setminus B.\end{aligned}\right.$
Note that $\widetilde{g}\in L^{p}(\mathbb{R}^{3};\mathbb{R}^{3})$ and for any
$\psi\in C_{0}^{\infty}(\mathbb{R}^{3})$,
$\displaystyle\int_{\mathbb{R}^{3}}\widetilde{g}\cdot\nabla\psi\,dx$
$\displaystyle=\int_{\Omega}g\cdot\nabla\psi\,dx+\int_{B\setminus\overline{\Omega}}\nabla
f\cdot\nabla\psi\,dx$ $\displaystyle=<n\cdot
g,\psi>_{W^{-\frac{1}{p},p}(\partial\Omega)\times
W^{\frac{1}{p},p^{\prime}}(\partial\Omega)}+\int_{B\setminus\overline{\Omega}}\nabla
f\cdot\nabla\psi\,dx$ $\displaystyle=0.$
Thus, div$(\widetilde{g})=0$ in $\mathbb{R}^{3}$. It follows from the first
part of the proof that $\widetilde{g}=\nabla\times h$ in $B$ for some $h\in
W^{1,p}(B;\mathbb{R}^{3})$ with div$(h)=0$ in $B$. Furthermore,
$\|h\|_{W^{1,p}(\Omega)}\leq\|h\|_{W^{1,p}(B)}\leq
C_{p}\,\|\widetilde{g}\|_{L^{p}(B)}\leq C_{p}\,\|g\|_{L^{p}(\Omega)},$
where we have used (2.1) for the last inequality. This competes the proof. ∎
###### Theorem 2.3.
Let $1<p<\infty$ and $\Omega$ be a bounded, simply-connected, $C^{1,1}$ domain
in $\mathbb{R}^{3}$ with connected boundary. Let $A=A(x)$ be a $3\times 3$
matrix in $\mathbb{R}^{3}$ satisfying the ellipticity condition (3.1). Also
assume that $A$ is Lipschitz continuous. Then, for any $u\in
L^{p}(\Omega;\mathbb{R}^{3})$ such that the right hand side of (2.2) is
finite,
$\|\nabla u\|_{L^{p}(\Omega)}\leq C_{p}\bigg{\\{}\|\nabla\times
u\|_{L^{p}(\Omega)}+\|\text{\rm div}(Au)\|_{L^{p}(\Omega)}+\|n\times
u\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\bigg{\\}},$ (2.2)
where $C_{p}$ depends only on $p$, $\Omega$, and $A$.
###### Proof.
Let $u$ be a function in $L^{p}(\Omega;\mathbb{R}^{3})$ such that the right
hand side of (2.2) is finite. Let $g=\nabla\times u$ in $\Omega$. Then $g\in
L^{p}(\Omega;\mathbb{R}^{3})$ and div$(g)=0$ in $\Omega$. In view of Theorem
2.2, there exists $h\in W^{1,p}(\Omega;\mathbb{R}^{3})$ such that
$g=\nabla\times h$ in $\Omega$, div$(h)=0$ in $\Omega$, and
$\|h\|_{W^{1,p}(\Omega)}\leq C_{p}\,\|\nabla\times u\|_{L^{p}(\Omega)}.$ (2.3)
Let $w=u-h$ in $\Omega$. Note that $w\in L^{p}(\Omega;\mathbb{R}^{3})$ and
$\nabla\times w=0$ in $\Omega$. It then follows from Theorem 2.1 that there
exists $P\in W^{1,p}(\Omega)$ such that $w=\nabla P$ in $\Omega$.
We now observe that
$\text{\rm div}(A\nabla P)=\text{\rm div}(Au)-\text{\rm div}(Ah)\in
L^{p}(\Omega)$ (2.4)
and
$n\times\nabla P=n\times u-n\times h\in W^{1-\frac{1}{p},p}(\partial\Omega),$
(2.5)
where we have used the fact that $\Omega$ is $C^{1,1}$ and
$\|n\times h\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\leq
C\,\|h\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\leq
C\,\|h\|_{W^{1,p}(\Omega)}\leq C\,\|\nabla\times u\|_{L^{p}(\Omega)}.$ (2.6)
Finally, we note that if $\int_{\partial\Omega}P=0$,
$\|P\|_{W^{2-\frac{1}{p},p}(\partial\Omega)}\leq
C\,\|\nabla_{\tan}P\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\leq
C\,\|n\times\nabla P\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}.$
It follows from the $W^{2,p}$ estimates for elliptic equations in $C^{1,1}$
domains (see e.g. [6]) that
$\displaystyle\|u\|_{W^{1,p}(\Omega)}$
$\displaystyle\leq\|h\|_{W^{1,p}(\Omega)}+\|\nabla P\|_{W^{1,p}(\Omega)}$
(2.7) $\displaystyle\leq C\left\\{\|\nabla\times
u\|_{L^{p}(\Omega)}+\|\text{\rm div}(Au)\|_{L^{p}(\Omega)}+\|n\times
u\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\}.$
This completes the proof. ∎
###### Remark 2.4.
Assume that $\Omega$ is a bounded, simply-connected, $C^{1,1}$ domain in
$\mathbb{R}^{3}$ with connected boundary and that $B$ is a positive-definite
constant matrix. Let $u_{\varepsilon}$ be a solution of (1.7). It follows from
(2.2) that
$\displaystyle\|\nabla u_{\varepsilon}\|_{W^{1,p}(\Omega)}$ $\displaystyle\leq
C_{p}\left\\{\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}+\|\text{\rm
div}(Bu_{\varepsilon})\|_{L^{p}(\Omega)}+\|n\times
u_{\varepsilon}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\}$
$\displaystyle=C_{p}\left\\{\|\nabla\times
u_{\varepsilon}\|_{L^{p}(\Omega)}+\|\text{\rm
div}(F)\|_{L^{p}(\Omega)}+\|n\times
u_{\varepsilon}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\}.$
This, together with (1.10), gives (1.15).
## 3 Uniform estimates for scalar elliptic equations with periodic
coefficients
In this section we establish the $W^{1,p}$ and Lipschitz estimates for
solutions of the elliptic equation (1.14). These estimates will be used in the
proof of Theorems 1.1 and 1.2.
Let $A=A(y)=(a_{ij}(y))$ be a $d\times d$ matrix in $\mathbb{R}^{d}$, $d\geq
2$. We say $A\in\Lambda(\mu,\lambda,\tau)$ for some $\mu>0$, $\tau\in(0,1]$,
and $\lambda\geq 0$, if $A$ satisfies the ellipticity condition,
$\mu|\xi|^{2}\leq a_{ij}(y)\xi_{i}\xi_{j}\leq\frac{1}{\mu}|\xi|^{2}\quad\text{
for any }y\in\mathbb{R}^{d}\text{ and }\xi\in\mathbb{R}^{d},$ (3.1)
the periodicity condition,
$A(y+z)=A(y)\quad\text{ for any }y\in\mathbb{R}^{d}\text{ and
}z\in\mathbb{Z}^{d},$ (3.2)
and the smoothness condition,
$|A(x)-A(y)|\leq\lambda|x-y|^{\tau}\text{ for any }x,y\in\mathbb{R}^{d}.$
(3.3)
We start out with the $W^{1,p}$ estimate for solutions of the Dirichlet
problem.
###### Theorem 3.1.
Let $1<p<\infty$ and $\Omega$ be a bounded $C^{1,\alpha}$ domain in
$\mathbb{R}^{d}$, $d\geq 2$. Suppose that $A\in\Lambda(\mu,\lambda,\tau)$. Let
$w_{\varepsilon}\in W^{1,p}(\Omega)$ be the solution of the Dirichlet problem:
$\left\\{\begin{aligned} \text{\rm div}\left\\{A(x/\varepsilon)(\nabla
w_{\varepsilon}+g)\right\\}&=\text{\rm div}(F)&\quad&\text{ in }\Omega,\\\
w_{\varepsilon}&=f&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (3.4)
where $g\in L^{p}(\Omega;\mathbb{R}^{d})$, $F\in
L^{p}(\Omega;\mathbb{R}^{d})$, and $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$.
Then,
$\|w_{\varepsilon}\|_{W^{1,p}(\Omega)}\leq
C_{p}\left\\{\|g\|_{L^{p}(\Omega)}+\|F\|_{L^{p}(\Omega)}+\|f\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\},$
(3.5)
where $C_{p}$ depends only on $p$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
###### Proof.
Rewrite the elliptic equation in (3.4) as
$\text{div}\left\\{A(x/\varepsilon)\nabla
w_{\varepsilon}\right\\}=\text{div}\big{(}F-A(x/\varepsilon)g\big{)}.$ (3.6)
The estimate (3.5) is a simple consequence of [2, Theorem C]. ∎
The next theorem establishes the Lipschitz estimate for solutions of the
Dirichlet problem.
###### Theorem 3.2.
Suppose that $A$ and $\Omega$ satisfy the same assumptions as in Theorem 3.1.
Let $g\in C^{\gamma}(\Omega;\mathbb{R}^{d})$, $F\in
C^{\gamma}(\Omega;\mathbb{R}^{d})$, and $f\in C^{1,\gamma}(\partial\Omega)$
for some $0<\gamma<\alpha$. Let $w_{\varepsilon}\in H^{1}(\Omega)$ be the
solution of the Dirichlet problem (3.4). Then $\nabla w_{\varepsilon}\in
L^{\infty}(\Omega)$ and
$\|\nabla w_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C_{\gamma}\left\\{\|g\|_{C^{\gamma}(\Omega)}+\|F\|_{C^{\gamma}(\Omega)}+\|f\|_{C^{1,\gamma}(\partial\Omega)}\right\\},$
(3.7)
where $C_{\gamma}$ depends only on $\gamma$, $\mu$, $\lambda$, $\tau$, and
$\Omega$.
###### Proof.
We begin by choosing $h\in C^{1,\gamma}(\overline{\Omega})$ so that $h=f$ on
$\partial\Omega$ and $\|h\|_{C^{1,\gamma}(\Omega)}\leq
C\,\|f\|_{C^{1,\gamma}(\partial\Omega)}$. By considering $w_{\varepsilon}-h$,
we may assume that $f=0$. Next, in view of (3.6), we may write
$\displaystyle w_{\varepsilon}(x)$
$\displaystyle=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}G_{\varepsilon}(x,y)\big{\\}}a_{ij}(y/\varepsilon)g_{j}(y)\,dy+\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}G_{\varepsilon}(x,y)\big{\\}}F_{i}(y)\,dy$ (3.8)
$\displaystyle=w_{\varepsilon}^{(1)}(x)+w_{\varepsilon}^{(2)}(x),$
where $F=(F_{1},\dots,F_{d})$ and $G_{\varepsilon}(x,y)$ denotes the Green
function for the operator $-\text{div}(A(x/\varepsilon)\nabla)$ in $\Omega$,
with pole at $y$. It follows from [1] that for any $x,y\in\Omega$,
$\displaystyle|G_{\varepsilon}(x,y)|$ $\displaystyle\leq C|x-y|^{2-d},$ (3.9)
$\displaystyle|\nabla_{x}G_{\varepsilon}(x,y)|+|\nabla_{y}G_{\varepsilon}(x,y)|$
$\displaystyle\leq C|x-y|^{1-d},$
$\displaystyle|\nabla_{x}\nabla_{y}G_{\varepsilon}(x,y)|$ $\displaystyle\leq
C|x-y|^{-d},$
where $C$ depends only on $\mu$, $\lambda$, $\tau$, and $\Omega$. We note that
if $d=2$, the first inequality in (3.9) should be replaced by
$|G_{\varepsilon}(x,y)|\leq C(1+\log|x-y|)$. Using (3.9), we see that for any
$x\in\Omega$,
$\displaystyle|\nabla w_{\varepsilon}^{(2)}(x)|$
$\displaystyle=\left|\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}G_{\varepsilon}(x,y)\big{\\}}\big{\\{}F_{i}(y)-F_{i}(x)\big{\\}}\,dy\right|$
(3.10) $\displaystyle\leq
C\,\|F\|_{C^{\gamma}(\Omega)}\int_{\Omega}\frac{dy}{|x-y|^{d-\gamma}}$
$\displaystyle\leq C\,\|F\|_{C^{\gamma}(\Omega)}.$
Finally, to estimate $\nabla w_{\varepsilon}^{(1)}$, we let
$\Phi_{\varepsilon}(x)$ be the Dirichlet corrector for the operator
$-\text{div}(A(x/\varepsilon)\nabla)$ in $\Omega$; i.e.,
$\Phi_{\varepsilon}=(\Phi_{\varepsilon,1}(x),\dots,\Phi_{\varepsilon,d}(x))$
is the function in $H^{1}(\Omega;\mathbb{R}^{d})$ satisfying
$\left\\{\begin{aligned}
\text{div}\big{(}A(x/\varepsilon)\nabla\Phi_{\varepsilon,k}\big{)}&=0&\quad&\text{
in }\Omega,\\\ \Phi_{\varepsilon,k}&=x_{k}&\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$ (3.11)
Since $\Phi_{\varepsilon,k}-x_{k}=0$ on $\partial\Omega$ and
$-\text{div}\big{\\{}A(x/\varepsilon)\nabla\big{(}\Phi_{\varepsilon,k}-x_{k}\big{)}\big{\\}}=\frac{\partial}{\partial
x_{i}}\big{\\{}a_{ik}(x/\varepsilon)\big{\\}}\quad\text{ in }\Omega,$
we see that
$\Phi_{\varepsilon,k}(x)-x_{k}=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}G_{\varepsilon}(x,y)\big{\\}}a_{ik}(y/\varepsilon)\,dy$ (3.12)
for $1\leq k\leq d$. It follows that
$\displaystyle|\nabla w_{\varepsilon}^{(1)}(x)|$
$\displaystyle=\left|\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}G_{\varepsilon}(x,y)\big{\\}}\big{\\{}a_{ij}(y/\varepsilon)g_{j}(y)-a_{ij}(x/\varepsilon)g_{j}(x)\big{\\}}\,dy\right|$
$\displaystyle\leq\int_{\Omega}|\nabla_{x}\nabla_{y}G_{\varepsilon}(x,y)|\,|A(y/\varepsilon)|\,|g(y)-g(x)|\,dy$
$\displaystyle\qquad+\left|g_{j}(x)\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}G_{\varepsilon}(x,y)\big{\\}}\big{\\{}a_{ij}(y/\varepsilon)-a_{ij}(x/\varepsilon)\big{\\}}\,dy\right|$
$\displaystyle\leq
C\,\|g\|_{C^{\gamma}(\Omega)}+|g_{j}(x)|\,|\nabla\left\\{\Phi_{\varepsilon,j}(x)-x_{j}\right\\}|,$
where we have used (3.12) and the estimate
$|\nabla_{x}\nabla_{y}G_{\varepsilon}(x,y)|\leq C|x-y|^{-d}$ for the last
inequality. This, together with the Lipschitz estimate
$\|\nabla\Phi_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq C$, established in [1],
yields $\|\nabla w_{\varepsilon}^{(1)}\|_{L^{\infty}(\Omega)}\leq
C\,\|g\|_{C^{\gamma}(\Omega)}$. The proof is complete. ∎
We now turn to the $W^{1,p}$ estimate for solutions of the Neumann problem.
###### Theorem 3.3.
Let $1<p<\infty$ and $\Omega$ be a bounded $C^{1,\alpha}$ domain in
$\mathbb{R}^{d}$, $d\geq 2$. Suppose that $A\in\Lambda(\mu,\lambda,\tau)$. Let
$w_{\varepsilon}\in W^{1,p}(\Omega)$ be a solution of the Neumann problem:
$\left\\{\begin{aligned} \text{\rm div}\left\\{A(x/\varepsilon)(\nabla
w_{\varepsilon}+g)\right\\}&=0&\quad&\text{ in }\Omega,\\\ n\cdot
A(x/\varepsilon)(\nabla w_{\varepsilon}+g)&=f&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$ (3.13)
where $g\in L^{p}(\Omega;\mathbb{R}^{d})$, $f\in
W^{-\frac{1}{p},p}(\partial\Omega)$ and $<f,1>=0$. Then
$\|\nabla w_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C_{p}\left\\{\|g\|_{L^{p}(\Omega)}+\|f\|_{W^{-\frac{1}{p},p}(\partial\Omega)}\right\\},$
(3.14)
where $C_{p}$ depends only on $p$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
###### Proof.
This is a direct consequence of Theorem 1.1 in [8]. ∎
The next theorem gives the Lipschitz estimate for solutions of the Neumann
problem (3.13). Note that in addition to the ellipticity and periodicity
conditions, we also assume that $A^{*}=A$; i.e., $a_{ij}(y)=a_{ji}(y)$.
###### Theorem 3.4.
Let $\Omega$ be a bounded $C^{1,\alpha}$ domain in $\mathbb{R}^{d}$, $d\geq
2$. Suppose that $A\in\Lambda(\mu,\lambda,\tau)$ and $A^{*}=A$. Let $g\in
C^{\gamma}(\Omega;\mathbb{R}^{d})$, and $f\in C^{\gamma}(\partial\Omega)$ with
mean value zero, for some $0<\gamma<\alpha$. Let $w_{\varepsilon}\in
H^{1}(\Omega)$ be a solution of the Neumann problem (3.13). Then $\nabla
u_{\varepsilon}\in L^{\infty}(\Omega)$, and
$\|\nabla u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C_{\gamma}\left\\{\|g\|_{C^{\gamma}(\Omega)}+\|f\|_{C^{\gamma}(\partial\Omega)}\right\\},$
(3.15)
where $C_{\gamma}$ depends only on $\gamma$, $\mu$, $\lambda$, $\tau$, and
$\Omega$.
###### Proof.
Let $v_{\varepsilon}\in H^{1}(\Omega)$ be a solution of the Neumann problem:
$\text{div}\big{(}A(x/\varepsilon)\nabla v_{\varepsilon}\big{)}=0$ in $\Omega$
and $n\cdot A(x/\varepsilon)\nabla v_{\varepsilon}=f$ on $\partial\Omega$. It
follows from [8, Theorem 1.2] that
$\|\nabla v_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C\,\|f\|_{C^{\gamma}(\partial\Omega)},$
where $C$ depends only on $\gamma$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
Thus, by considering $w_{\varepsilon}-v_{\varepsilon}$, we may assume that
$f=0$.
Let $g=(g_{1},\dots,g_{d})\in C^{\gamma}(\Omega;\mathbb{R}^{d})$ and
$w_{\varepsilon}$ be a solution of (3.13) with $f=0$. Then
$w_{\varepsilon}(x)=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}N_{\varepsilon}(x,y)\big{\\}}a_{ij}(y/\varepsilon)g_{j}(y)\,dy+E$
for some constant $E$, where $N_{\varepsilon}(x,y)$ denotes the Neumann
function for the elliptic operator $-\text{div}(A(x/\varepsilon)\nabla)$ in
$\Omega$, with pole at $y$. Under the assumption that
$A\in\Lambda(\mu,\lambda,\tau)$ and $A^{*}=A$, it was proved in [8] that for
$d\geq 3$,
$\displaystyle|N_{\varepsilon}(x,y)$ $\displaystyle\leq C|x-y|^{2-d},$ (3.16)
$\displaystyle|\nabla_{x}N_{\varepsilon}(x,y)|+|\nabla_{y}N_{\varepsilon}(x,y)|$
$\displaystyle\leq C|x-y|^{1-d},$
$\displaystyle|\nabla_{x}\nabla_{y}N_{\varepsilon}(x,y)|$ $\displaystyle\leq
C|x-y|^{-d},$
where $C$ depends only on $\mu$, $\lambda$, $\tau$, and $\Omega$. If $d=2$,
one obtains $|N_{\varepsilon}(x,y)|\leq C_{\eta}|x-y|^{-\eta}$,
$|\nabla_{x}N_{\varepsilon}(x,y)|+|\nabla_{y}N_{\varepsilon}(x,y)|\leq
C_{\eta}|x-y|^{-1-\eta}$, and $|\nabla_{x}\nabla_{y}N_{\varepsilon}(x,y)|\leq
C_{\eta}|x-y|^{-2-\eta}$ for any $\eta>0$ (this is not sharp, but enough for
the proof of this theorem). It follows that for any $x\in\Omega$,
$\displaystyle\nabla w_{\varepsilon}(x)$
$\displaystyle=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}N_{\varepsilon}(x,y)\big{\\}}\big{[}a_{ij}(y/\varepsilon)g_{j}(y)-a_{ij}(x/\varepsilon)g_{j}(x)\big{]}\,dy$
(3.17) $\displaystyle\qquad-
a_{ij}(x/\varepsilon)g_{j}(x)\int_{\partial\Omega}n_{i}(y)\nabla_{x}N_{\varepsilon}(x,y)\,d\sigma(y)$
$\displaystyle=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}N_{\varepsilon}(x,y)\big{\\}}\big{[}g_{j}(y)-g_{j}(x)\big{]}a_{ij}(y/\varepsilon)\,dy$
$\displaystyle\qquad-g_{j}(x)\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}N_{\varepsilon}(x,y)\big{\\}}\big{[}a_{ij}(y/\varepsilon)-a_{ij}(x/\varepsilon)\big{]}\,dy$
$\displaystyle\qquad-
a_{ij}(x/\varepsilon)g_{j}(x)\int_{\partial\Omega}n_{i}(y)\nabla_{x}N_{\varepsilon}(x,y)\,d\sigma(y).$
Note that if $g_{j}(x)=-\delta_{jk}$, then $w_{\varepsilon}(x)=x_{k}$ is a
solution of (3.13) with $f=0$. In view of (3.17), this implies that
$\displaystyle\nabla(x_{k})$
$\displaystyle=\delta_{jk}\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}N_{\varepsilon}(x,y)\big{\\}}\big{[}a_{ij}(y/\varepsilon)-a_{ij}(x/\varepsilon)\big{]}\,dy$
(3.18)
$\displaystyle\qquad+a_{ij}(x/\varepsilon)\delta_{jk}\int_{\partial\Omega}n_{i}(y)\nabla_{x}N_{\varepsilon}(x,y)\,d\sigma(y).$
By combining (3.17) and (3.18) we obtain
$\nabla
w_{\varepsilon}(x)+g_{j}(x)\nabla(x_{j})=-\int_{\Omega}\frac{\partial}{\partial
y_{i}}\big{\\{}\nabla_{x}N_{\varepsilon}(x,y)\big{\\}}\big{[}g_{j}(y)-g_{j}(x)\big{]}a_{ij}(y/\varepsilon)\,dy.$
As a result, for any $x\in\Omega$,
$\displaystyle|\nabla w_{\varepsilon}(x)|$ $\displaystyle\leq
C\|g\|_{L^{\infty}(\Omega)}+C\,\|g\|_{C^{\gamma}(\Omega)}\int_{\Omega}\frac{dy}{|x-y|^{d-\gamma}}$
$\displaystyle\leq C\|g\|_{C^{\gamma}(\Omega)},$
where we have used the estimate
$|\nabla_{x}\nabla_{y}N_{\varepsilon}(x,y)|\leq C|x-y|^{-d}$ (the case $d=2$
may be handled in a similar manner). This finishes the proof. ∎
## 4 $L^{p}$ estimates
The goal of this section is to prove Theorem 1.1. Throughout this section we
will assume that $\Omega$ is a bounded, simply connected, $C^{1,\alpha}$
domain in $\mathbb{R}^{3}$ with connected boundary, and that
$A,B\in\Lambda(\mu,\lambda,\tau)$.
###### Lemma 4.1.
Let $2\leq q<3$ and $2\leq p\leq p_{0}$, where
$\frac{1}{p_{0}}=\frac{1}{q}-\frac{1}{3}$. Given $F\in
L^{q}(\Omega;\mathbb{R}^{3})$ and $G\in L^{p}(\Omega;\mathbb{R}^{3})$, let
$u_{\varepsilon}$ be the unique solution in $V_{0}^{2}(\Omega)$ of (1.7) with
$f=0$. Suppose that $u_{\varepsilon}\in L^{q}(\Omega;\mathbb{R}^{3})$. Then
$\nabla\times u_{\varepsilon}\in L^{p}(\Omega;\mathbb{R}^{3})$, and
$\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$
(4.1)
where $C$ depends only on $q$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
###### Proof.
It follows from the elliptic system in (1.2) that
$\text{div}\big{(}B(x/\varepsilon)u_{\varepsilon}-F)=0\quad\text{ in }\Omega.$
Since $B(x/\varepsilon)u_{\varepsilon}-F\in L^{q}(\Omega;\mathbb{R}^{3})$, by
Theorem 2.2, there exists $h_{\varepsilon}\in W^{1,q}(\Omega;\mathbb{R}^{3})$
such that
$\nabla\times h_{\varepsilon}=B(x/\varepsilon)u_{\varepsilon}-F\quad\text{ in
}\Omega,$ (4.2)
and
$\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}\leq
C_{q}\,\|B(x/\varepsilon)u_{\varepsilon}-F\|_{L^{q}(\Omega)}.$ (4.3)
Thus,
$\nabla\times\big{\\{}A(x/\varepsilon)\nabla\times
u_{\varepsilon}+h_{\varepsilon}-G\big{\\}}=0\quad\text{ in }\Omega.$
Since $\Omega$ is simply connected, there exists $P_{\varepsilon}\in
W^{1,2}(\Omega)$ such that
$A(x/\varepsilon)\nabla\times u_{\varepsilon}+h_{\varepsilon}-G=\nabla
P_{\varepsilon}\quad\text{ in }\Omega.$ (4.4)
It follows that
$A^{-1}(x/\varepsilon)\nabla P_{\varepsilon}=\nabla\times
u_{\varepsilon}+A^{-1}(x/\varepsilon)\big{(}h_{\varepsilon}-G\big{)}\quad\text{
in }\Omega.$ (4.5)
Thus, $P_{\varepsilon}\in W^{1,2}(\Omega)$ is the solution of
$\left\\{\begin{aligned} \text{\rm
div}\big{\\{}A^{-1}(x/\varepsilon)\big{(}\nabla
P_{\varepsilon}-h_{\varepsilon}+G\big{)}\big{\\}}&=0&\quad&\text{ in
}\Omega,\\\ n\cdot A^{-1}(x/\varepsilon)\big{(}\nabla
P_{\varepsilon}-h_{\varepsilon}+G\big{)}&=0&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$ (4.6)
where we have used the fact that
$n\cdot(\nabla\times u_{\varepsilon})=-\text{Div}(n\times
u_{\varepsilon})=0\quad\text{ on }\partial\Omega.$
In view of Theorem 3.3, we obtain
$\|\nabla P_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\left\\{\|h_{\varepsilon}\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\},$
where $C$ depends only on $p$, $\mu$, $\lambda$, $\tau$, and $\Omega$. This,
together with (4.5) and the estimate (4.3), gives
$\displaystyle\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C\left\\{\|h_{\varepsilon}\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\}$
(4.7) $\displaystyle\leq
C\left\\{\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$
where we also used the Sobolev imbedding for the second inequality. ∎
###### Remark 4.2.
Let $F\in L^{q}(\Omega;\mathbb{R}^{3})$, $G\in
C^{\gamma}(\Omega;\mathbb{R}^{3})$, and $f\in
C^{\gamma}(\partial\Omega;\mathbb{R}^{3})$ with $n\cdot f=0$ on
$\partial\Omega$ and Div$(f)\in C^{\gamma}(\partial\Omega)$, where
$3<q<\infty$ and $\gamma=1-\frac{3}{q}<\alpha$. Let $u_{\varepsilon}$ be the
solution in $V^{2}(\Omega)$ of (1.7). Suppose that $A$ is symmetric. Then
$\|\nabla\times u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|\text{\rm
Div}(f)\|_{C^{\gamma}(\partial\Omega)}+\|u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$
(4.8)
where $C$ depends only $q$, $\mu$, $\lambda$, $\tau$, and $\Omega$. To see
this, we let $h_{\varepsilon}\in W^{1,q}(\Omega;\mathbb{R}^{3})$ and
$P_{\varepsilon}\in W^{1,2}(\Omega)$ be the same functions as in the proof of
Lemma 4.1. It follows from (4.4), (4.6) and Theorem 3.4 that
$\displaystyle\|\nabla\times u_{\varepsilon}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq C\left\\{\|\nabla
P_{\varepsilon}\|_{L^{\infty}(\Omega)}+\|h_{\varepsilon}\|_{L^{\infty}(\Omega)}+\|G\|_{L^{\infty}(\Omega)}\right\\}$
$\displaystyle\leq
C\left\\{\|h_{\varepsilon}\|_{C^{\gamma}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|\text{\rm
Div}(f)\|_{C^{\gamma}(\partial\Omega)}\right\\}$ $\displaystyle\leq
C\left\\{\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|\text{\rm
Div}(f)\|_{C^{\gamma}(\partial\Omega)}\right\\}$ $\displaystyle\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|\text{\rm
Div}(f)\|_{C^{\gamma}(\partial\Omega)}+\|u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$
where we have used the Sobolev imbedding for the third inequality and (4.3)
for the last.
Next we reverse the roles of $u_{\varepsilon}$ and $\nabla\times
u_{\varepsilon}$ in the estimate (4.1).
###### Lemma 4.3.
Let $2\leq q<3$ and $2\leq p\leq p_{0}$, where
$\frac{1}{p_{0}}=\frac{1}{q}-\frac{1}{3}$. Given $F\in
L^{p}(\Omega;\mathbb{R}^{3})$ and $G\in L^{q}(\Omega;\mathbb{R}^{3})$, let
$u_{\varepsilon}$ be the unique solution in $V_{0}^{2}(\Omega)$ of (1.7) with
$f=0$. Suppose that $\nabla\times u_{\varepsilon}\in
L^{q}(\Omega;\mathbb{R}^{3})$. Then $u_{\varepsilon}\in
L^{p}(\Omega;\mathbb{R}^{3})$ and
$\|u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$ (4.9)
where $C$ depends only on $q$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
###### Proof.
Let
$v_{\varepsilon}=A(x/\varepsilon)\nabla\times u_{\varepsilon}-G\quad\text{ in
}\Omega.$ (4.10)
Then $\nabla\times
u_{\varepsilon}=A^{-1}(x/\varepsilon)\big{(}v_{\varepsilon}+G\big{)}$ in
$\Omega$. It follows that
$\text{\rm
div}\big{(}A^{-1}(x/\varepsilon)(v_{\varepsilon}+G)\big{)}=0\quad\text{ in
}\Omega.$
By Theorem 2.2 there exists $h_{\varepsilon}\in
W^{1,q}(\Omega;\mathbb{R}^{3})$ such that
$A^{-1}(x/\varepsilon)(v_{\varepsilon}+G)=\nabla\times
h_{\varepsilon}\quad\text{ in }\Omega$ (4.11)
and
$\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}\leq
C\left\\{\|v_{\varepsilon}\|_{L^{q}(\Omega)}+\|G\|_{L^{q}(\Omega)}\right\\}\leq
C\left\\{\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}+\|G\|_{L^{q}(\Omega)}\right\\}.$ (4.12)
Note that by the elliptic system (1.2), $\nabla\times
v_{\varepsilon}=-B(x/\varepsilon)u_{\varepsilon}+F\text{ in }\Omega$. Thus,
$u_{\varepsilon}=B^{-1}(x/\varepsilon)F-B^{-1}(x/\varepsilon)\nabla\times
v_{\varepsilon}\quad\text{ in }\Omega.$ (4.13)
Since $\Omega$ is simply connected and $\nabla\times
h_{\varepsilon}=\nabla\times u_{\varepsilon}$ in $\Omega$, there exists
$Q_{\varepsilon}\in W^{1,2}(\Omega)$ such that $\nabla
Q_{\varepsilon}=u_{\varepsilon}-h_{\varepsilon}$ in $\Omega$. Thus,
$B(x/\varepsilon)\big{(}\nabla
Q_{\varepsilon}+h_{\varepsilon}\big{)}=B(x/\varepsilon)u_{\varepsilon}=F-\nabla\times
v_{\varepsilon}\quad\text{ in }\Omega.$
It follows that
$\text{div}\big{\\{}B(x/\varepsilon)\big{(}\nabla
Q_{\varepsilon}+h_{\varepsilon}\big{)}\big{\\}}=\text{div}\big{(}F\big{)}\quad\text{
in }\Omega.$ (4.14)
In view of Theorem 3.1 we obtain
$\|\nabla Q_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\left\\{\|h_{\varepsilon}\|_{L^{p}(\Omega)}+\|F\|_{L^{p}(\Omega)}+\|Q_{\varepsilon}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\right\\},$
(4.15)
where $C$ depends only on $p$, $\mu$, $\lambda$, $\tau$, and $\Omega$.
Finally, we note that
$n\times\nabla Q_{\varepsilon}=n\times u_{\varepsilon}-n\times
h_{\varepsilon}=-n\times h_{\varepsilon}\quad\text{ on }\partial\Omega.$
(4.16)
By subtracting a constant we may assume that
$\int_{\partial\Omega}Q_{\varepsilon}\,d\sigma=0$. Since
$|\nabla_{\tan}Q_{\varepsilon}|=|n\times\nabla Q_{\varepsilon}|$ on
$\partial\Omega$, we see that
$\displaystyle\|Q_{\varepsilon}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}$
$\displaystyle\leq C\,\|n\times\nabla
Q_{\varepsilon}\|_{L^{q_{1}}(\partial\Omega)}=C\,\|h_{\varepsilon}\|_{L^{q_{1}}(\partial\Omega)}$
$\displaystyle\leq
C\,\|h_{\varepsilon}\|_{W^{1-\frac{1}{q},q}(\partial\Omega)}\leq
C\,\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}$ $\displaystyle\leq
C\left\\{\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}+\|G\|_{L^{q}(\Omega)}\right\\},$
where $q_{1}=2p/3$ and we have used the Sobolev imbedding on $\partial\Omega$
as well as the estimate (4.12). This, together with (4.15) and (4.12), gives
$\displaystyle\|u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq\|\nabla
Q_{\varepsilon}\|_{L^{p}(\Omega)}+\|h_{\varepsilon}\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$
and completes the proof. ∎
###### Remark 4.4.
Let $F\in C^{\gamma}(\Omega;\mathbb{R}^{3})$, $G\in
L^{q}(\Omega;\mathbb{R}^{3})$, and $f\in
C^{\gamma}(\partial\Omega;\mathbb{R}^{3})$ with $n\cdot f=0$ on
$\partial\Omega$, where $3<q<\infty$ and $\gamma=1-\frac{3}{q}<\alpha$. Let
$u_{\varepsilon}$ be the solution in $V^{2}(\Omega)$ of (1.7). Then
$\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C\left\\{\|F\|_{C^{\gamma}(\Omega)}+\|G\|_{L^{q}(\Omega)}+\|f\|_{C^{\gamma}(\partial\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\},$ (4.17)
where $C$ depends only on $q$, $\mu$, $\lambda$, $\tau$, and $\Omega$. To see
this, we let $h_{\varepsilon}\in W^{1,q}(\Omega;\mathbb{R}^{3})$ and
$Q_{\varepsilon}\in W^{1,2}(\Omega)$ be the same functions as in the proof of
Lemma 4.3. It follows from Theorem 3.2 that
$\displaystyle\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq\|\nabla
Q_{\varepsilon}\|_{L^{\infty}(\Omega)}+\|h_{\varepsilon}\|_{L^{\infty}(\Omega)}$
(4.18) $\displaystyle\leq
C\left\\{\|F\|_{C^{\gamma}(\Omega)}+\|h_{\varepsilon}\|_{C^{\gamma}(\Omega)}+\|\nabla_{\tan}Q_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}\right\\}.$
$\displaystyle\leq
C\left\\{\|F\|_{C^{\gamma}(\Omega)}+\|h\|_{W^{1,q}(\Omega)}+\|\nabla_{\tan}Q_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}\right\\}$
$\displaystyle\leq
C\left\\{\|F\|_{C^{\gamma}(\Omega)}+\|G\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}+\|\nabla_{\tan}Q_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}\right\\},$
where we have used the Sobolev embedding for the third inequality and (4.12)
for the fourth. This, together with the estimate
$\displaystyle\|\nabla_{\tan}Q_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}$
$\displaystyle\leq\|n\times\nabla
Q_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}$ $\displaystyle\leq\|n\times
u_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}+\|n\times
h_{\varepsilon}\|_{C^{\gamma}(\partial\Omega)}$
$\displaystyle\leq\|f\|_{C^{\gamma}(\partial\Omega)}+C\,\|h_{\varepsilon}\|_{W^{1,q}(\Omega)}$
$\displaystyle\leq\|f\|_{C^{\gamma}(\partial\Omega)}+C\left\\{\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}+\|G\|_{L^{q}(\Omega)}\right\\},$
gives (4.17).
We are now ready to prove Theorem 1.1.
###### Proof of Theorem 1.1.
Given $f\in L^{p}(\partial\Omega;\mathbb{R}^{3})$ with $n\cdot f=0$ on
$\partial\Omega$ and Div$(f)\in W^{-\frac{1}{p},p}(\partial\Omega)$, it
follows from [11, Theorem 11.6] that there exists $u\in V^{p}(\Omega)$ such
that $n\times u=f$ on $\partial\Omega$ and
$\|u\|_{L^{p}(\Omega)}+\|\nabla\times u\|_{L^{p}(\Omega)}\leq
C_{p}\left\\{\|f\|_{L^{p}(\partial\Omega)}+\|\text{Div}(f)\|_{W^{-\frac{1}{p},p}(\partial\Omega)}\right\\}.$
Consequently, by considering $u_{\varepsilon}-u$ in $\Omega$, we may assume
that $f=0$.
We first consider the case $p>2$. The uniqueness follows from the uniqueness
in the case $p=2$. Let $F\in L^{p}(\Omega;\mathbb{R}^{3})$, $G\in
L^{p}(\Omega;\mathbb{R}^{3})$, and $u_{\varepsilon}$ be the unique solution of
(1.2) in $V_{0}^{2}(\Omega)$. To establish the $L^{p}$ estimate (1.10), we
further assume that $2<p\leq 6$. Since
$\|u_{\varepsilon}\|_{L^{2}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\left\\{\|F\|_{L^{2}(\Omega)}+\|G\|_{L^{2}(\Omega)}\right\\}$, it follows
from (4.1) that
$\displaystyle\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{2}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\}.$
Similarly, by the estimate (4.9),
$\displaystyle\|u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{2}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$ $\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\}.$
Suppose now that $p>6$. Let $\frac{1}{q}=\frac{1}{p}+\frac{1}{3}$. Then
$2<q<3$ and we have proved that
$\|u_{\varepsilon}\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{L^{q}(\Omega)}\right\\}.$
As before, we may use estimates (4.1) and (4.9) to obtain
$\displaystyle\|\nabla\times u_{\varepsilon}\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{q}(\Omega)}+\|G\|_{L^{p}(\Omega)}+\|u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\}$
$\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\},$
and
$\displaystyle\|u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\right\\}$ $\displaystyle\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\}.$
Finally, we handle the case $1<p<2$ by a duality argument. Let $F,G\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{3})$ and $u_{\varepsilon}$ be the solution
in $V_{0}^{2}(\Omega)$ of (1.2). Let $F_{1}$, $G_{1}\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{3})$ and $v_{\varepsilon}$ be the solution
in $V_{0}^{2}(\Omega)$ of
$\left\\{\begin{aligned} \nabla\times\big{(}A^{*}(x/\varepsilon)\nabla\times
v_{\varepsilon}\big{)}+B^{*}(x/\varepsilon)v_{\varepsilon}&=F_{1}+\nabla\times
G_{1}&\quad&\text{ in }\Omega,\\\ n\times v_{\varepsilon}&=0&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$
where $A^{*}$ and $B^{*}$ are the adjoints of $A$ and $B$, respectively. Since
$A^{*}$ and $B^{*}$ satisfy the same conditions as $A$ and $B$, we see that
$\|v_{\varepsilon}\|_{L^{p^{\prime}}(\Omega)}+\|\nabla\times
v_{\varepsilon}\|_{L^{p^{\prime}}(\Omega)}\leq
C\left\\{\|F_{1}\|_{L^{p^{\prime}}(\Omega)}+\|G_{1}\|_{L^{p^{\prime}}(\Omega)}\right\\}.$
(4.19)
Note that
$\displaystyle\int_{\Omega}F_{1}\cdot
u_{\varepsilon}\,dx+\int_{\Omega}G_{1}\cdot\nabla\times u_{\varepsilon}\,dx$
(4.20) $\displaystyle=\int_{\Omega}A(x/\varepsilon)\nabla\times
u_{\varepsilon}\cdot\nabla\times
v_{\varepsilon}\,dx+\int_{\Omega}B(x/\varepsilon)u_{\varepsilon}\cdot
v_{\varepsilon}\,dx$ $\displaystyle=\int_{\Omega}F\cdot
v_{\varepsilon}\,dx+\int_{\Omega}G\cdot\nabla\times v_{\varepsilon}\,dx.$
This, together with (4.19), yields
$\|u_{\varepsilon}\|_{L^{p}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\left\\{\|F\|_{L^{p}(\Omega)}+\|G\|_{L^{p}(\Omega)}\right\\},$ (4.21)
by duality. With the estimate (4.21) at our disposal, the existence of
solutions in $V^{p}(\Omega)$ as well as the estimate (1.10) for arbitrary data
$F,G\in L^{p}(\Omega;\mathbb{R}^{3})$ follows readily by a density argument.
Observe that the duality relation (4.20) holds as long as $u_{\varepsilon}\in
V^{p}(\Omega)$ and $v_{\varepsilon}\in V^{p^{\prime}}(\Omega)$ are solutions
of (1.2) and its adjoint system, respectively. The uniqueness for $p<2$ also
follows from (4.20) and (4.19) by duality. This completes the proof of Theorem
1.1. ∎
## 5 Proof of Theorem 1.2
Choose $q>3$ such that $\gamma=1-\frac{3}{q}$. It follows from estimates (4.8)
and (4.17) that
$\displaystyle\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}$
$\displaystyle+\|\nabla\times u_{\varepsilon}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq
C\big{\\{}\|F\|_{C^{\gamma}(\Omega)}+\|G\|_{C^{\gamma}(\Omega)}+\|f\|_{C^{\gamma}(\partial\Omega)}+\|\text{Div}(f)\|_{C^{\gamma}(\partial\Omega)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\|u_{\varepsilon}\|_{L^{q}(\Omega)}+\|\nabla\times
u_{\varepsilon}\|_{L^{q}(\Omega)}\big{\\}}.$
This, together with the $L^{q}$ estimate of $u_{\varepsilon}$ and
$\nabla\times u_{\varepsilon}$ in Theorem 1.1, gives (1.11).
## References
* [1] M. Avellaneda and F. Lin, _Compactness methods in the theory of homogenization_ , Comm. Pure Appl. Math. 40 (1987), 803–847.
* [2] , _${L}^{p}$ bounds on singular integrals in homogenization_, Comm. Pure Appl. Math. 44 (1991), 897–910.
* [3] A. Bensoussan, J.-L. Lions, and G.C. Papanicolaou, _Asymptotic Analysis for Periodic Structures_ , AMS Chelsea Publishing, Providence, Rhode Island, 2011.
* [4] M. Costabel, _A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains_ , Math. Methods Appl. Sci. 12 (1990), 365–368.
* [5] E. Fabes, O. Mendez, and M. Mitrea, _Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains_ , J. Funct. Anal. 159 (1998), 323–368.
* [6] D. Gilbarg and N.S. Trudinger, _Elliptic Partial Differential Equations of Second Order_ , Springer-Verlag, Berlin, 1983.
* [7] V. Girault and P. Ravirart, _Finite Element Methods for Navier-Stokes Equations_ , Springer-Verlag, Berlin, 1985.
* [8] C. Kenig, F. Lin, and Z. Shen, _Homogenization of elliptic systems with Neumann boundary conditions_ , Preprint, arXiv:1010.6114 (2010).
* [9] C. Kenig and Z. Shen, _Homogenization of elliptic boundary value problems in Lipschitz domains_ , Math. Ann. 350 (2011), 867–917.
* [10] , _Layer potential methods for elliptic homogenization problems_ , Comm. Pure Appl. Math. 64 (2011), 1–44.
* [11] D. Mitrea, M. Mitrea, and J. Pipher, _Vector potential theory on nonsmooth domains in $\mathbb{R}^{3}$ and applications to electromagnetic scattering_, J. Fourier Anal. Appl. 3 (1997), 131–192.
Zhongwei Shen, Department of Mathematics, University of Kentucky, Lexington,
Kentucky 40506, USA.
E-mail: zshen2@uky.edu
Liang Song, Department of Mathematics, Sun Yat-sen University, Guangzhou,
510275, P. R. China.
E-mail: songl@mail.sysu.edu.cn
|
arxiv-papers
| 2012-10-27T15:52:13 |
2024-09-04T02:49:37.188186
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhongwei Shen and Liang Song",
"submitter": "Liang Song",
"url": "https://arxiv.org/abs/1210.7340"
}
|
1210.7357
|
# A Finite Reflection Formula For A Polynomial Approximation To The Riemann
Zeta Function
Stephen Crowley
###### Abstract.
The Riemann zeta function can be written as the Mellin transform of the unit
interval map $w\left(x\right)=\left\lfloor
x^{-1}\right\rfloor\left(x\left\lfloor x^{-1}\right\rfloor+x-1\right)$
multiplied by $s\frac{s+1}{s-1}$. A finite-sum approximation to
$\zeta\left(s\right)$ denoted by $\zeta_{w}\left(N;s\right)$ which has real
roots at $s=-1$ and $s=0$ is examined and an associated function
$\chi\left(N;s\right)$ is found which solves the reflection formula
$\zeta_{w}\left(N;1-s\right)=\chi\left(N;s\right)\zeta_{w}\left(N;s\right)$. A
closed-form expression for the integral of $\zeta_{w}\left(N;s\right)$ over
the interval $s=-1\ldots 0$ is given. The function $\chi\left(N;s\right)$ is
singular at $s=0$ and the residue at this point changes sign from negative to
positive between the values of $N=176$ and $N=177$. Some rather elegant graphs
of $\zeta_{w}\left(N;s\right)$ and the reflection functions
$\chi\left(N;s\right)$ are also provided. The values
$\zeta_{w}\left(N;1-n\right)$ for integer values of $n$ are found to be
related to the Bernoulli numbers.
Email: stephen.crowley@mavs.uta.edu
###### Contents
1. 1 The Riemann Zeta Function as the Mellin Transform of a Unit Interval Map
1. 1.1 The Truncated Zeta Function
1. 1.1.1 Integrating Over the Critical Strip
2. 1.1.2 The Reflection Formula
## 1\. The Riemann Zeta Function as the Mellin Transform of a Unit Interval
Map
The Riemann zeta function can be written as the Mellin transform of the unit
interval map $w\left(x\right)=\left\lfloor
x^{-1}\right\rfloor\left(x\left\lfloor x^{-1}\right\rfloor+x-1\right)$
multiplied by $s\frac{s+1}{s-1}$. [3][2]
Figure 1. The Harmonic Sawtooth map
(1)
$\begin{array}[]{lll}\zeta_{w}(s)&=\zeta(s)\forall-s\not\in\mathbbm{N}^{\ast}&\\\
&=s\frac{s+1}{s-1}\int_{0}^{1}\left\lfloor
x^{-1}\right\rfloor\left(x\left\lfloor
x^{-1}\right\rfloor+x-1\right)x^{s-1}\mathrm{d}x&\\\
&=s\frac{s+1}{s-1}\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)x^{s-1}\mathrm{d}x&\text{
}\\\
&=\sum_{n=1}^{\infty}s\frac{s+1}{s-1}\left(-\frac{n^{1-s}-n(n+1)^{-s}-sn^{-s}}{s\left(s+1\right)}\right)&\\\
&=\sum_{n=1}^{\infty}\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}&\\\
&=\frac{1}{s-1}\sum_{n=1}^{\infty}n(n+1)^{-s}-n^{1-s}+sn^{-s}&\end{array}$
### 1.1. The Truncated Zeta Function
The substition $\infty\rightarrow N$ is made in the infinite sum appearing the
expression for $\zeta_{w}(s$) to get a finite polynomial approximation
(2)
$\begin{array}[]{ll}\zeta_{w}(N;s)&=\frac{1}{s-1}\sum_{n=1}^{N}n(n+1)^{-s}-n^{1-s}+sn^{-s}\\\
&=\frac{1}{s-1}\left(s+(N+1)^{1-s}-1+s\sum_{n=2}^{N}n^{-s}-\sum_{n=2}^{N+1}n^{-s}\right)\\\
&=\frac{N}{\left(s-1\right)\left(N+1\right)^{s}}-\frac{\cos\left(\pi
s\right)\Psi\left(s-1,N+1\right)}{\Gamma\left(s\right)}+\zeta\left(s\right)\forall
s\in\mathbbm{N}^{\ast}\end{array}$
with equality in the limit except at the negative integers
(3)
$\begin{array}[]{ll}\lim_{N\rightarrow\infty}\zeta_{w}(N;s)&=\zeta(s)\forall-s\not\in\mathbbm{N}^{\ast}\end{array}$
and where
$\Psi\left(x,n\right)=\frac{\mathrm{d}}{\mathrm{d}x^{n}}\Psi\left(x\right)$ is
the polygamma function and
$\Psi\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}\ln\left(\Gamma\left(x\right)\right)$
is the digamma function. The functions $\zeta_{w}\left(N;s\right)$ have real
zeros at $s=-1$ and $s=0$, that is
(4) $\lim_{s\rightarrow-1}\zeta_{w}\left(N;s\right)=\lim_{s\rightarrow
0}\zeta_{w}\left(N;s\right)=0$
One possible idea is that the functions $\zeta_{w}\left(N;s\right)$ can be
orthonormalized over the interval $s=-1\ldots 0$ via the Gram-Schmidt
process[4] and that the result might possibly shed some light on the zeroes of
$\zeta\left(s\right)$. Let the logarithmic integral be defined
(5)
$\begin{array}[]{ll}\operatorname{Li}\left(x\right)&=\end{array}\int_{0}^{\ln\left(x\right)}\frac{e^{y}-1}{y}\mathrm{d}y+\ln\left(\ln\left(x\right)\right)+\gamma$
where $\gamma=0.57721\ldots$ is Euler’s constant, then the normalization
factors are given by the integral
(6)
$\begin{array}[]{ll}\int_{-1}^{0}\zeta_{w}\left(N;s\right)\mathrm{d}s&=\int_{-1}^{0}\sum_{n=1}^{N}\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}\mathrm{d}s\\\
&=1+\frac{N}{N+1}\left(\operatorname{Li}\left(N+1\right)-\operatorname{Li}\left(\left(N+1\right)^{2}\right)\right)+\sum_{n=1}^{N-1}\frac{n}{\ln\left(n+1\right)}\end{array}$
Figure 2. $\left\\{\zeta_{w}\left(N;s\right):s=-1\ldots 0,N=1\ldots
25\right\\}$
The following table lists the values of $\zeta_{w}\left(N;1-n\right)$ for
$n=2\ldots 12$.
$\left[\begin{array}[]{c}0\\\ -\frac{1}{6}\hskip 2.5ptN-\frac{1}{6}\hskip
2.5ptN^{2}\\\ -\frac{1}{4}N-\frac{1}{2}N^{2}-\frac{1}{4}\hskip 2.5ptN^{3}\\\
-\frac{7}{30}N-\frac{4}{5}N^{2}-\frac{13}{15}\hskip
2.5ptN^{3}-\frac{3}{10}N^{4}\\\ -\frac{1}{6}\hskip 2.5ptN-\frac{11}{12}\hskip
2.5ptN^{2}-\frac{5}{3}N^{3}-\frac{5}{4}N^{4}-\frac{1}{3}N^{5}\\\
-\frac{5}{42}\hskip 2.5ptN-\frac{6}{7}N^{2}-\frac{97}{42}\hskip
2.5ptN^{3}-\frac{20}{7}\hskip 2.5ptN^{4}-\frac{23}{14}\hskip
2.5ptN^{5}-\frac{5}{14}\hskip 2.5ptN^{6}\\\ -\frac{1}{8}N-\frac{19}{24}\hskip
2.5ptN^{2}-\frac{21}{8}\hskip 2.5ptN^{3}-\frac{14}{3}\hskip
2.5ptN^{4}-\frac{35}{8}\hskip 2.5ptN^{5}-\frac{49}{24}\hskip
2.5ptN^{6}-\frac{3}{8}N^{7}\\\ -\frac{13}{90}\hskip 2.5ptN-\frac{8}{9}\hskip
2.5ptN^{2}-\frac{26}{9}\hskip 2.5ptN^{3}-\frac{56}{9}\hskip
2.5ptN^{4}-\frac{371}{45}\hskip 2.5ptN^{5}-\frac{56}{9}\hskip
2.5ptN^{6}-\frac{22}{9}\hskip 2.5ptN^{7}-\frac{7}{18}\hskip 2.5ptN^{8}\\\
-\frac{1}{10}N-\frac{21}{20}\hskip 2.5ptN^{2}-\frac{18}{5}\hskip
2.5ptN^{3}-\frac{79}{10}\hskip 2.5ptN^{4}-\frac{63}{5}\hskip
2.5ptN^{5}-\frac{133}{10}\hskip 2.5ptN^{6}-\frac{42}{5}\hskip
2.5ptN^{7}-\frac{57}{20}\hskip 2.5ptN^{8}-\frac{2}{5}N^{9}\\\
-\frac{1}{66}\hskip 2.5ptN-\frac{10}{11}\hskip 2.5ptN^{2}-\frac{101}{22}\hskip
2.5ptN^{3}-\frac{120}{11}\hskip 2.5ptN^{4}-\frac{199}{11}\hskip
2.5ptN^{5}-\frac{252}{11}\hskip 2.5ptN^{6}-\frac{221}{11}\hskip
2.5ptN^{7}-\frac{120}{11}\hskip 2.5ptN^{8}-\frac{215}{66}\hskip
2.5ptN^{9}-\frac{9}{22}\hskip 2.5ptN^{10}\\\
-\frac{1}{12}N-\frac{1}{2}N^{2}-\frac{55}{12}\hskip
2.5ptN^{3}-\frac{121}{8}N^{4}-\frac{55}{2}N^{5}-\frac{110}{3}\hskip
2.5ptN^{6}-\frac{77}{2}\hskip 2.5ptN^{7}-\frac{231}{8}\hskip
2.5ptN^{8}-\frac{55}{4}\hskip 2.5ptN^{9}-\frac{11}{3}N^{10}-\frac{5}{12}\hskip
2.5ptN^{11}\end{array}\right]$
#### 1.1.1. Integrating Over the Critical Strip
There is a formula similiar to (6) which gives the integral of
$\zeta_{w}\left(N;s\right)$ over the critical strip
$0\leqslant\operatorname{Re}\left(s\right)\leqslant 1$.
(7)
$\begin{array}[]{ll}\int_{0}^{1}\zeta_{w}\left(N;c+is\right)\mathrm{d}c&=1+\frac{N}{N+1}\left(\operatorname{Ei}_{1}\left(is\ln\left(N+1\right)-\ln\left(N+1\right)\right)-\operatorname{Ei}_{1}\left(is\ln\left(N+1\right)\right)\right)+\sum_{n=1}^{N-1}\frac{n\left(n+1\right)^{-is}}{\left(n+1\right)\ln\left(n+1\right)}\end{array}$
where $\operatorname{Ei}_{1}\left(t\right)$ is the exponential integral
defined by
(8)
$\operatorname{Ei}_{1}\left(t\right)=t\int_{0}^{1}\int_{0}^{1}e^{-txy}\mathrm{d}y\mathrm{d}x-\gamma-\ln\left(t\right)$
The contribution from the $\operatorname{Ei}$ term vanishes as
$s\rightarrow\infty$, that is
(9)
$\lim_{s\rightarrow\infty}\frac{N}{N+1}\left(\operatorname{Ei}_{1}\left(is\ln\left(N+1\right)-\ln\left(N+1\right)\right)-\operatorname{Ei}_{1}\left(is\ln\left(N+1\right)\right)\right)=0$
#### 1.1.2. The Reflection Formula
There is a reflection equation for the finite-sum approximation
$\zeta_{w}(N;s)$ which is similiar to the well-known formula
$\zeta\left(1-s\right)=\chi\left(s\right)\zeta\left(s\right)$ with
$\chi\left(s\right)=2\left(2\pi\right)^{-s}\cos\left(\frac{\pi
s}{2}\right)\Gamma\left(s\right)$. The solution to
(10)
$\zeta_{w}\left(N;1-s\right)=\chi\left(N;s\right)\zeta_{w}\left(N;s\right)$
is given by the expression
(11)
$\begin{array}[]{ll}\chi\left(N;s\right)&=\frac{\zeta_{w}\left(N;1-s\right)}{\zeta_{w}\left(N;s\right)}\\\
&=\frac{\sum_{n=1}^{N}-\frac{-n^{s}+\left(n+1\right)^{s-1}n+n^{s-1}-n^{s-1}s}{s}}{\sum_{n=1}^{N}\frac{-n^{1-s}+\left(n+1\right)^{-s}n+n^{-s}s}{s-1}}\\\
&=-\frac{\left(s-1\right)\sum_{n=1}^{N}-n^{s}+\left(n+1\right)^{s-1}n+n^{s-1}-n^{s-1}s}{s\sum_{n=1}^{N}-n^{1-s}+\left(n+1\right)^{-s}n+n^{-s}s}\end{array}$
which satisfies
(12) $\chi\left(N;1-s\right)=\chi\left(N;s\right)^{-1}$
The functions $\chi\left(N;s\right)$, indexed by $N$, have singularities at
$s=0$. Let
(13)
$\begin{array}[]{ll}a\left(N\right)&=\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\\\
b\left(N\right)&=\sum_{n=1}^{N}\frac{\ln\left(n\right)n^{2}-\ln\left(n+1\right)n^{2}-\ln\left(n\right)}{n\left(n+1\right)}\\\
c\left(N\right)&=\frac{1}{2}\sum_{n=1}^{N}n\left(\ln\left(n+1\right)^{2}-\ln\left(n\right)^{2}\right)\end{array}$
then the residue at the singular point $s=0$ is given by the expression
(14)
$\begin{array}[]{ll}\underset{s=0}{\operatorname{Res}}(\chi(N;s))&=-\underset{s=1}{\operatorname{Res}}(\chi(N;s)^{-1})\\\
&=\frac{1+\gamma+\Psi\left(n+2\right)-\frac{2}{N+1}+b\left(N\right)-\frac{N\left(\ln\left(\Gamma\left(N+1\right)\right)-c\left(N\right)\right)}{\left(N-a\left(N\right)\right)\left(N+1\right)}}{a\left(N\right)-N}\\\
&=\frac{1+\gamma+\Psi\left(n+2\right)-\frac{2}{N+1}+\sum_{n=1}^{N}\frac{\ln\left(n\right)n^{2}-\ln\left(n+1\right)n^{2}-\ln\left(n\right)}{n\left(n+1\right)}-\frac{N\left(\ln\left(\Gamma\left(N+1\right)\right)-\frac{1}{2}\sum_{n=1}^{N}n\left(\ln\left(n+1\right)^{2}-\ln\left(n\right)^{2}\right)\right)}{\left(N-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)\left(N+1\right)}}{\left(\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)-N}\end{array}$
which has the limit
(15)
$\lim_{N\rightarrow\infty}\underset{s=0}{\operatorname{Res}}(\chi(N;s))=1$
We also have the residue of the reciprocal at $s=2$
(16)
$\begin{array}[]{ll}\underset{s=2}{\operatorname{Res}}(\chi(N;s)^{-1})&=\frac{\frac{2N}{\left(N+1\right)^{2}}-2\Psi\left(1,N+1\right)+2\zeta\left(2\right)}{\frac{\left(N+1\right)^{2}}{2}-\frac{N}{2}-\frac{1}{2}-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)+\ln\left(n+1\right)n-\ln\left(n\right)-n\ln\left(n\right)\right)}\end{array}$
which vanishes as $N$ tends to infinity
(17)
$\lim_{N\rightarrow\infty}\underset{s=2}{\operatorname{Res}}(\chi(N;s)^{-1})=0$
As can be seen in the figures below, the residue at $s=0$ changes sign from
negative to positive between the values of $N=176$ and $N=177$.
Figure 3. $\left\\{\underset{s=0}{\operatorname{Res}}(\chi(N;s)):N=1\ldots
250\right\\}$
Figure 4.
$\left\\{\underset{s=0}{\operatorname{Res}}(\chi(N;s))^{-1}:N=1\ldots
250\right\\}$
For any positive integer N, we have the limits
(18) $\begin{array}[]{ll}\lim_{s\rightarrow 0}\chi\left(N;s\right)&=\infty\\\
\lim_{s\rightarrow
0}\frac{\mathrm{d}^{n}}{\mathrm{d}s^{n}}\chi\left(N;s\right)&=\infty\\\
\lim_{s\rightarrow\frac{1}{2}}\chi\left(N;s\right)&=1\\\ \lim_{s\rightarrow
1}\chi\left(N;s\right)&=0\\\ \lim_{s\rightarrow 2}\chi\left(N;s\right)&=0\\\
\lim_{s\rightarrow
1}\frac{\mathrm{d}}{\mathrm{d}s}\chi\left(N;s\right)&=0\end{array}$
The line $\operatorname{Re}\left(s\right)=\frac{1}{2}$ has a constant modulus
(19) $\left|\chi\left(N;\frac{1}{2}+is\right)\right|=1$
There is also the complex conjugate symmetry
(20) $\chi\left(N;x+iy\right)=\overline{\chi\left(N;x-iy\right)}$
If $s=n\in\mathbbm{N}^{\ast}$ is a positive integer then
$\chi\left(N;n\right)$ can be written as
(21)
$\begin{array}[]{cc}\chi\left(N;n\right)&=\frac{\zeta_{w}\left(N;1-n\right)}{\zeta_{w}\left(N;n\right)}\\\
&=\frac{\sum_{m=1}^{N}-\sum_{k=1}^{n-2}\frac{m^{k}}{n}\binom{n-1}{k-1}}{\frac{N}{\left(n-1\right)\left(N+1\right)^{n}}-\frac{\cos\left(\pi
n\right)\Psi\left(n-1,N+1\right)}{\Gamma\left(n\right)}+\zeta\left(n\right)}\\\
&=\frac{-\sum_{m=1}^{N}\frac{1}{n}\left(\left(n-1\right)m^{n-1}+m^{n}-\left(m+1\right)^{n-1}m\right)}{\frac{N}{\left(n-1\right)\left(N+1\right)^{n}}-\frac{\cos\left(\pi
n\right)\Psi\left(n-1,N+1\right)}{\Gamma\left(n\right)}+\zeta\left(n\right)}\end{array}$
where $\binom{n-1}{k-1}$ is of course a binomial. The Bernoulli numbers[1]
make an appearance since
(22)
$\begin{array}[]{ll}\chi\left(N;2n\right)\zeta_{w}(N;2n)&=B_{2n}\left(N+1\right)^{2}\frac{\left(2n+1\right)}{2}+\ldots\end{array}$
The denominator of $\chi\left(N;n\right)$ has the limits
(23)
$\begin{array}[]{cl}\lim_{N\rightarrow\infty}\zeta_{w}\left(N;n\right)&=\zeta\left(n\right)\\\
\lim_{n\rightarrow\infty}\zeta_{w}\left(N;n\right)&=1\end{array}$
Another interesting formula gives the limit at $s=1$ of the quotient of
successive functions
(24)
$\begin{array}[]{cl}\lim_{s=1}\frac{\chi\left(N+1;s\right)}{\chi\left(N;s\right)}&=\frac{\left(N+2\right)N\left(N+1-a\left(N+1\right)\right)}{\left(N+1\right)^{2}\left(N-a\left(N\right)\right)}\\\
&=\frac{\left(N+2\right)N\left(N+1-\sum_{n=1}^{N+1}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)}{\left(N+1\right)^{2}\left(N-\sum_{n=1}^{N}n\left(\ln\left(n+1\right)-\ln\left(n\right)\right)\right)}\end{array}$
Figure 5. $\left\\{\chi\left(N;s\right):s=1\ldots 2,N=1\ldots 25\right\\}$
Figure 6. $\left\\{\chi\left(N;s\right):s=\frac{1}{2}\ldots 2,N=1\ldots
100\right\\}$
Let
(25)
$\nu\left(s\right)=\chi\left(\infty;s\right)=\frac{\zeta\left(1-s\right)}{\zeta\left(s\right)}$
Then the residue at the even negative integers is
(26)
$\underset{s=-n}{\operatorname{Res}}(\nu(s))=\left\\{\begin{array}[]{ll}\frac{\zeta\left(1-n\right)}{\frac{\mathrm{d}}{\mathrm{d}s}\zeta\left(s\right)|_{s=-n}}&n\operatorname{even}\\\
0&n\operatorname{odd}\end{array}\right.$
## References
* [1] G Arfken. Mathematical Methods for Physicists, 3rd ed., chapter 5.9, Bernoulli Numbers, Euler-Maclaurin Formula., pages 327–338. Academic Press, 1985.
* [2] Stephen Crowley. Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, Fractal Strings, and a Finite Reflection Formula http://arxiv.org/abs/1210.5652, October 2012.
* [3] Stephen Crowley. Two new zeta constants: Fractal string, continued fraction, and hypergeometric aspects of the riemann zeta function. http://arxiv.org/abs/1207.1126, July 2012\.
* [4] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins, 3 edition, 1996.
|
arxiv-papers
| 2012-10-27T18:42:58 |
2024-09-04T02:49:37.197641
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephen Crowley",
"submitter": "Stephen Crowley",
"url": "https://arxiv.org/abs/1210.7357"
}
|
1210.7363
|
# Convex solutions to the power-of-mean curvature flow††thanks: ©2012 by the
author.
Shibing Chen Department of Mathematics, University of Toronto, Toronto,
Ontario Canada M5S 2E4 sbchen@math.toronto.edu.
###### Abstract
We prove some estimates for convex ancient solutions (the existence time for
the solution starts from $-\infty$) to the power-of-mean curvature flow, when
the power is strictly greater than $\frac{1}{2}$. As an application, we prove
that in two dimension, the blow-down of the entire convex translating
solution, namely $u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x),$ locally
uniformly converges to $\frac{1}{1+\alpha}|x|^{1+\alpha}$ as
$h\rightarrow\infty$. Another application is that for generalized curve
shortening flow (convex curve evolving in its normal direction with speed
equal to a power of its curvature), if a convex compact ancient solution
sweeps $\textbf{R}^{2}$, it has to be a shrinking circle. Otherwise the
solution is defined in a strip region.
## 1 Introduction
Recently, classifying ancient convex solution to mean curvature flow has
attracted much interest, due to its importance in studying the singularities
of mean curvature flow. Some important progress was made by Wang [12], and
Daskalopoulos, Hamilton and Sesum [5]. In [12] Wang proved that an entire
convex translating solution to mean curvature flow must be rotationally
symmetric which was a conjecture formulated explicitly by White in [11]. Wang
also constructed some entire convex translating solution with level set
neither spherical nor cylindrical in dimension greater or equal to 3. In the
same paper, Wang also proved that if a convex ancient solution to the curve
shortening flow sweeps the whole space $\textbf{R}^{2}$, it must be a
shrinking circle, otherwise the convex ancient solution must be defined in a
strip region and he indeed constructed such solutions by some compactness
argument. Daskalopoulos, Hamilton and Sesum [5] showed that besides the
shrinking circle, the so called Angenent oval (a convex ancient solution of
the curve shortening flow discovered by Angenent that decomposes into two
translating solutions of the flow) is the only other embedded convex compact
ancient solution of the curve shortening flow. That means the corresponding
curve shortening solution defined in a strip region constructed by Wang is
exactly the “Angenent oval”.
The power-of-mean curvature flow, in which a hypersurface evolves in its
normal direction with speed equal to a power $\alpha$ of its mean curvature
$H$, was studied by Andrews [1], [2], [3], Schulze [8], Chou and Zhu [4] and
Sheng and Wu [10] . Schulze [8] called it $H^{\alpha}$-flow. In the following,
we will also call the one dimensional power-of-curvature flow the generalized
curve shortening flow. Similar to the mean curvature flow, when one blows up
the flow near the type II singularity appropriately, a convex translating
solution will arise, see [10] for details. It will be very interesting if one
could classify the ancient convex solutions. In this paper, we will use the
method developed by Wang [12] to study the geometric asymptotic behavior of
ancient convex solutions to $H^{\alpha}$-flow. The general equation for
$H^{\alpha}$-flow is $\frac{\partial F}{\partial t}=-H^{\alpha}\vec{v}$, where
$F:M\times[0,T)\rightarrow\textbf{R}^{n+1}$ is a time-dependent embedding of
the evolving hypersurface, $\vec{v}$ is the unit normal vector to the
hypersurface $F(M,t)$ in $\textbf{R}^{n+1}$ and $H$ is its mean curvature. If
the evolving hypersurface can be represented as a graph of a function $u(x,t)$
over some domain in $\textbf{R}^{n}$, then we can project the evolution
equation to the $(n+1)$th coordinate direction of $\textbf{R}^{n+1}$ and the
equation becomes
$u_{t}=\sqrt{1+|Du|^{2}}\left(\text{div}(\frac{Du}{\sqrt{1+|Du|^{2}}})\right)^{\alpha}.$
Then a translating solution to the $H^{\alpha}$-flow will satisfy the equation
$\sqrt{1+|Du|^{2}}(\text{div}(\frac{Du}{\sqrt{1+|Du|^{2}}}))^{\alpha}=1,$
which is equivalent to the following equation (3) when $\sigma=1$,
$\displaystyle L_{\sigma}(u)$ $\displaystyle=$
$\displaystyle(\sqrt{\sigma+|Du|^{2}})^{\frac{1}{\alpha}}\text{div}(\frac{Du}{\sqrt{\sigma+|Du|^{2}}})$
(1) $\displaystyle=$
$\displaystyle(\sigma+|Du|^{2})^{\frac{1}{2\alpha}-\frac{1}{2}}\displaystyle{\sum_{i,j=1}^{n}}(\delta_{ij}-\frac{u_{i}u_{j}}{\sigma+|Du|^{2}})u_{ij}$
(2) $\displaystyle=$ $\displaystyle 1,$ (3)
where $\sigma\in[0,1]$, $\alpha\in(\frac{1}{2},\infty]$ is a constant, $n=2$
is the dimension of $\textbf{R}^{2}$. If $u$ is a convex solution of (3), then
$u+t$, as a function of $(x,t)\in\textbf{R}^{2}\times\textbf{R}$, is a
translating solution to the flow
$\displaystyle
u_{t}=\sqrt{\sigma+|Du|^{2}}(\text{div}(\frac{Du}{\sqrt{\sigma+|Du|^{2}}}))^{\alpha}.$
(4)
When $\sigma=1$, equation (4) is the non-parametric power-of-mean curvature
flow. When $\sigma=0$, the level set $\\{u=-t\\}$, where $-\infty<t<-\inf u$,
evolves by the power-of-mean curvature.
In the following we will assume $\sigma\in[0,1]$,
$\alpha\in(\frac{1}{2},\infty]$ and the dimension $n=2$, although some of the
estimates do hold in high dimension. The main results of this paper are the
following theorems.
###### Theorem 1.
Suppose $u$ be an entire convex solution of (3). Let
$u_{h}(x)=h^{-1}u(h^{\frac{1}{1+\alpha}}x).$ Then $u_{h}$ locally uniformly
converges to $\frac{1}{1+\alpha}|x|^{1+\alpha}$, as $h\rightarrow\infty.$
###### Theorem 2.
Let $u_{\sigma}$ be an entire convex solution of (3). Then
$u_{0}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}$ up to a a translation of the
coordinate system. When $\sigma\in(0,1]$, if
$|D^{2}u(x)|=O(|x|^{3\alpha-1-2\alpha\beta})\ \text{as}\
|x|\rightarrow\infty,$ for some constant $\beta$ satisfying
$\frac{1}{2\alpha}<\beta<\min\\{1,\frac{1+\alpha}{2\alpha}\\},$ then
$u_{\sigma}$ is rotationally symmetric after a suitable translation of the
coordinate system.
###### Corollary 1.
A convex compact ancient solution to the generalized curve shortening flow
which sweeps $\textbf{R}^{2}$ must be a shrinking circle.
_Remark 1_. The condition $\alpha>\frac{1}{2}$ is necessary for our results.
One can consider the translating solution $v(x)$ to (3) with $\sigma=1$ in one
dimension. In fact when $\alpha\leq\frac{1}{2}$, the translating solution
$v(x)$ is a convex function defined on the entire real line ([4] page 28).
Then one can construct a function $u(x,y)=v(x)-y$ defined on the entire plane,
and $u$ will satisfy (3) with $\sigma=0$ and it is obviously not rotationally
symmetric. We can also let $u(x,y)=v(x)$, which is an entire solution to (3)
with $\sigma=1$ and it is not rotationally symmetric. When the dimension $n$
is higher than two, similar examples can be given: we can take an entire
rotationally symmetric solution $v(x)$ to (1) with dimension $n\geq 2$ and
$\sigma=1$, and then again let $u(x,y)=v(x)-y$, here $y$ is the $(n+1)$th
coordinate for $R^{n+1}$. It is easy to see that $u$ will satisfy (3) with $n$
replaced by $n+1$ and $\sigma=0$, and the level set of $u$ is neither a sphere
nor a cylinder.
Before embarking on the argument, we would like to point out that this
elementary construction can be used to give a slight simplification of Wang’s
proof for Theorem 2.1 in [12]( corresponding to our Corollary 2 for
$\alpha=1$). Let $v_{\sigma}$ be an entire convex solution to (3) in dimension
$n$ with $\sigma\in(0,1]$. Then $u(x,y)=v_{\sigma}(x)-\sqrt{\sigma}y$ will be
an entire convex solution to (3) in dimension $n+1$ with $\sigma=0$. Hence if
one has proved the estimate in Corollary 2 for $\sigma=0$ in all dimensions,
the estimates for $\sigma\in(0,1]$ follows immediately from the above
construction.
The remainder of the paper is divided into three sections. The first contains
the proof of Theorem 1 and the first part of Theorem 2. The second section
establish Corollary 1 and the last section completes the proof of Theorem 2.
## 2 Proof of Theorem 1
For a given constant $h>0$, we denote
$\displaystyle\Gamma_{h}$ $\displaystyle=\\{x\in R^{n}:u(x)=h\\},$
$\displaystyle\Omega_{h}$ $\displaystyle=\\{x\in R^{n}:u(x)<h\\},$
so that $\Gamma_{h}$ is the boundary of $\Omega_{h}.$ Denote $\kappa$ as the
curvature of the level curve $\Gamma_{h}$. We have
$\displaystyle L_{\sigma}(u)$ $\displaystyle=$
$\displaystyle(\sigma+u_{\gamma}^{2})^{\frac{1}{2\alpha}-\frac{1}{2}}(\kappa
u_{\gamma}+\frac{\sigma u_{\gamma\gamma}}{\sigma+u_{\gamma}^{2}})$ (5)
$\displaystyle\geq$ $\displaystyle\kappa
u_{\gamma}^{\frac{1}{\alpha}}=L_{0}(u),$ (6)
where $\gamma$ is the unit outward normal to $\Omega_{h}$, and
$u_{\gamma\gamma}=\gamma_{i}\gamma_{j}u_{ij}$.
###### Lemma 1.
Suppose $u$ is a complete convex solution of (3). Suppose $u(0)=0$ and
$\inf\\{|x|:x\in\Gamma_{1}\\}$ is achieved at
$x_{0}=(0,-\delta)\in\Gamma_{1}$, for some $\delta>0$ very small. Let $D_{1}$
be the projection of $\Gamma_{1}$ on the axis $\\{x_{2}=0\\}$. Then the
interval $(-R,R)$ is contained in $D_{1}$ with
$\displaystyle R\geq C_{1}(-\log\delta-C_{2})^{\frac{\alpha}{\alpha+1}},$ (7)
where $C_{1},C_{2}>0$ are independent of $\delta$.
_Proof._ We will prove the lemma when $\frac{1}{2}<\alpha\leq 1$ and indicate
the small change needed for the case $\alpha>1$. Suppose locally around
$x_{0}$, $\Gamma_{1}$ is given by $z_{2}=f(z_{1})$. Then $f$ is a convex
function satisfying $f(0)=-\delta$ and $f^{\prime}(0)=0$. Take $a>0$ be a
constant such that $f^{\prime}(b)=1$. To prove (7) it is enough to prove
$\displaystyle a\geq C_{1}(-\log\delta-C_{2})^{\frac{\alpha}{\alpha+1}}.$ (8)
For any $z=(z_{1},z_{2})\in\Gamma_{1}$, where $z_{1}\in[0,a]$, let
$\xi=\frac{z}{|z|}$, from [12] we have
$\displaystyle u_{\gamma}(z)\geq\frac{\sqrt{1+f^{\prime
2}}}{z_{1}f^{\prime}-z_{2}},$ (9)
. Since $L_{0}u\leq 1$, we have
$\displaystyle\frac{f^{\prime\prime}}{(1+f^{\prime
2})^{\frac{3}{2}}}\frac{(1+f^{\prime
2})^{\frac{1}{2\alpha}}}{(z_{1}f^{\prime}-z_{2})^{\frac{1}{\alpha}}}\leq\kappa
u_{\gamma}^{\frac{1}{\alpha}}\leq 1.$ (10)
Hence
$\displaystyle f^{\prime\prime}(z_{1})$ $\displaystyle\leq$
$\displaystyle(1+f^{\prime
2})^{\frac{3}{2}-\frac{1}{2\alpha}}(z_{1}f^{\prime}-z_{2})^{\frac{1}{\alpha}}$
(11) $\displaystyle\leq$ $\displaystyle
10z_{1}^{\frac{1}{\alpha}}f^{\prime}+10\delta$ (12)
where $z_{2}=f(z_{1})$ and $f^{\prime}(z_{1})\leq 1$ for $z_{1}\in(0,b)$. The
inequality from (11) to (12) is trivial when $z_{2}\geq 0.$ When $z_{2}\leq
0$, since $|z_{2}|\leq\delta$, we have either $z_{1}f^{\prime}\leq\delta$ or
$z_{1}f^{\prime}>\delta$, for the former
$(z_{1}f^{\prime}-z_{2})^{\frac{1}{\alpha}}\leq(2\delta)^{\frac{1}{\alpha}}\leq
4\delta$, for the latter
$(z_{1}f^{\prime}-z_{2})^{\frac{1}{\alpha}}\leq(2z_{1}f^{\prime})^{\frac{1}{\alpha}}\leq
4z_{1}^{\frac{1}{\alpha}}f^{\prime}$, since $f^{\prime}(z_{1})\leq 1$. We
consider the equation
$\displaystyle\rho^{\prime\prime}(t)=10t^{\frac{1}{\alpha}}\rho^{\prime}+10\delta$
(13)
with initial conditions $\rho(0)=-\delta$ and $\rho^{\prime}(0)=0$. Then for
$t\in(0,b)$ we have
$\displaystyle\rho^{\prime}(t)=10\delta
e^{\frac{10\alpha}{\alpha+1}t^{\frac{\alpha+1}{\alpha}}}\int_{0}^{t}e^{-\frac{10\alpha}{\alpha+1}s^{\frac{\alpha+1}{\alpha}}}ds.$
(14)
Since
$\int_{0}^{\infty}e^{-\frac{10\alpha}{\alpha+1}s^{\frac{\alpha+1}{\alpha}}}ds$
is bounded above by some constant $C$, we have
$\displaystyle 1=\rho^{\prime}(a)$ $\displaystyle=$ $\displaystyle 10\delta
e^{\frac{10\alpha}{\alpha+1}a^{\frac{\alpha+1}{\alpha}}}\int_{0}^{a}e^{-\frac{10\alpha}{\alpha+1}s^{\frac{\alpha+1}{\alpha}}}ds.$
$\displaystyle\leq$ $\displaystyle C_{1}\delta
e^{\frac{10\alpha}{\alpha+1}a^{\frac{\alpha+1}{\alpha}}},$
from where (8) follows. When $\alpha>1$, we need only to introduce a number
$c$ such that $f^{\prime}(c)=\frac{1}{2}$ and then we can find the lower bound
of $a-c$ in a similar way.
_Remark 2_. It follows from Lemma 1 that when $\delta$ is sufficiently small,
by convexity and in view of Figure 1, we see that $\Omega_{1}$ contains the
shadowed region. Then it is easy to check that $\Omega_{1}$ contains an
ellipse
$\displaystyle
E=\\{(x_{1},x_{2})|\frac{x_{1}^{2}}{(\frac{R}{6})^{2}}+\frac{(x_{2}-\frac{7\delta^{*}-5\delta}{12})^{2}}{(\frac{\delta^{*}+\delta}{4})^{2}}=1\\},$
(15)
where $\delta^{*}$ is a positive constant such that $u(0,\delta^{*})=1$ and
$R$ is defined in the Lemma 1.
Figure 1: $\Gamma_{1}$ contains the shadow part.
_Remark 3._ One can also establish similar lemma in higher dimensions, which
says $D_{1}$ (convex set with dimension greater than 1) contains a ball
centered at the origin with radius $R\geq
C_{n}(-\log\delta-C)^{\frac{\alpha}{\alpha+1}}$. For the details of how to
reduce the situation to lower dimensional case we refer the reader to the
proof of Lemma 2.6 in [12].
###### Lemma 2.
Suppose $u$ is a complete convex solution of (3). Suppose $u(0)=0$, $\delta$
and $\delta^{*}$ are defined as in Lemma 1 and Remark 2. Then if $\delta$ and
$\delta^{*}$ are small enough, $u$ is defined in a strip region.
The proof of Lemma 2 is based on a careful study of the shape of the level
curve of $u$, we will give an important corollary first.
###### Corollary 2.
Suppose $u$ is an entire convex solution of (3) in $\textbf{R}^{2}$, then
$\displaystyle u(x)\leq C(1+|x|^{1+\alpha}),$ (16)
for some constant $C$ depending only on the upper bound of $u(0)$ and
$|Du(0)|$.
_Proof._ By subtracting a constant we may assume $u(0)=0$. It is enough to
prove that $\text{dist}(0,\Gamma_{h})\geq Ch^{\frac{1}{1+\alpha}}$ for all
large $h$. By the rescaling $u_{h}(x)=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x)$
we need only to prove $\text{dist}(0,\Gamma_{1,u_{h}})\geq C$. Notice that
$|Du_{h}(0)|=\frac{1}{h^{\frac{\alpha}{1+\alpha}}}|Du(0)|\rightarrow
0,\text{as}\ h\rightarrow\infty.$ Hence by convexity $\inf_{B_{r}(0)}u_{h}$
goes to 0 uniformly for fixed radius $r$. Note also that $u_{h}$ satisfies
equation (3) with $\sigma\rightarrow 0\ \text{as}\ \ h\rightarrow\infty$.
If the estimate
$\text{dist}(0,\Gamma_{1,u_{h}})\geq C,\ \text{for all large}\ h$
fails, we can find a sequence $h_{k}\rightarrow\infty$ such that
$\delta_{k}=\inf\\{|x|:x\in\Gamma_{1,u_{h_{k}}}\\}\rightarrow 0$. Now, we take
$\delta_{k}^{*}$ as in Remark 2 with respect to $u_{h_{k}}$. $\delta_{k}^{*}$
has a positive lower bound $\delta^{*}$, otherwise by Lemma 2 $u_{h_{k}}$ can
not be an entire solution for large $k$.
If $\delta_{k}^{*}\leq 1000$ for all large $k$, since the ellipse $E_{k}$
defined for $u_{h_{k}}$ as in Remark 2 is contained in $\Omega_{1,u_{h_{k}}}$
and the distance between the center $O_{k}$ of $E_{k}$ and the origin is
bounded above by 1000, by the previous discussion we know $u_{h_{k}}(O_{k})$
is bounded bellow by $-1$ when $k$ is large. Let $E_{k}(t)$ be the solution to
the generalized curve shortening flow starting from time $t=-1$ , with initial
data $E_{k}(-1)=E_{k}$. (1) When $\sigma=0$, $\partial\Omega_{-t,u_{h_{k}}}$
evolves under the generalized curve shortening flow, we have the inclusion
$E_{k}(t)\subset\partial\Omega_{-t,u_{h_{k}}}$ for all $t>-1$. Hence
$\inf_{B_{1000}(0)}u_{h_{k}}$ is smaller than 1 minus the time needed for
$E_{k}$ to shrink to $O_{k}$. However, by the size of $E_{k}$, the time needed
for it to shrink to a point goes to infinity as $k$ goes to infinity, which is
contradictory to the discussion at the beginning of the proof that $u_{h_{k}}$
converges to 0 uniformly in the ball $B_{1000}(0)$ as $h_{k}$ goes to
infinity. (2)When $\sigma\in(0,1]$, we can take $v_{k}$ as the solution of
$L_{\sigma_{k}}v=1$ in $E_{k}$ with $v=1$ on $\partial E_{k}$, where
$\sigma_{k}=h_{k}^{-\frac{2\alpha}{1+\alpha}}$. Passing to a subsequence and
adjusting the size of $E_{k}$ if necessary, we can assume $E_{k}$ converge to
some ellipse $E$ with the length of its long axis very large, the length of
its short axis bigger than some fixed positive number and the distance from
its center to the origin is less than 1000. Then $v_{k}$ converges to a
solution of the generalized curve shortening flow, and a contradiction can be
made as for the case $\sigma=0$
Otherwise, by the definition of $b$ in the proof of Lemma 1 and the convexity
of $\Omega_{1,u_{h}}$ we can find a disc $B_{k}$ with center $O=(0,50)$ and
radius 20 inside $\Omega_{1,u_{h_{k}}}$, obviously it will take time more than
2 for $B_{k}$ to shrink to $O$. We can take $B_{k}(t)$ as a solution to the
generalized curve shortening flow starting from time $t=-1$ with
$B_{k}(-1)=B_{k}$, then a similar contradiction will be made as before.
_Remark 4_. The estimate in Corollary 1 is also true for higher dimensions,
one can prove it by reducing the problem to two dimensional case similar to
the corresponding part in [12].
_Proof of Lemma 2._ By rotating coordinates we assume the axial directions of
$E$ in Remark 2 are the same with those of the coordinate system. Denote
$M_{u}$ as the graph of $u$, and as in [12] we divide it into two parts,
$M^{+}$ and $M^{-}$, where $M^{+}=\\{(x,u(x))\in\textbf{R}^{3}:u_{x_{2}}\geq
0\\}$ and $M^{-}=\\{(x,u(x))\in\textbf{R}^{3}:u_{x_{2}}\leq 0\\}$. Then
$M^{\pm}$ are the graphs of functions $f^{\pm}$, namely the graph of
$x_{2}=f^{\pm}(x_{1},x_{3}),(x_{1},x_{2})\in D$ and $D$ is the projection of
$M_{u}$ on the $x_{1}x_{3}$ plane. The function $f^{+}$ is concave and $f^{-}$
is convex, and we have $x_{3}=u(x_{1},f^{\pm}(x_{1},x_{3}))$. Let
$\displaystyle f=f^{+}-f^{-}.$ (17)
Now it is easy to see that $f$ is positive and concave in $D$. Also note that
$f$ is vanishing on $\partial{D}$. For any $h>0$ we also denote
$f_{h}(x_{1})=f(x_{1},h),$ $f^{\pm}_{h}(x_{1})=f(x_{1},h)$, and
$D_{h}=\\{x_{1}\in\textbf{R}^{1}:(x_{1},h)\in D\\}$. Then $f_{h}$ is a
positive, concave function in $D_{h}$, vanishing on $\partial{D_{h}}$, and
$D_{h}=(-\underline{a}_{h},\overline{a}_{h})$ is an interval containing the
origin. We denote $b_{h}=f_{h}(0)$. We will consider the case $\sigma=0$
first.
_Claim 1:_ suppose $h$ large, $f_{1}(0)=\delta^{*}+\delta$ small, $b_{h}\leq
4$ and $\underline{a}_{h},\overline{a}_{h}\geq b_{h}$. Then
$\overline{a}_{h}\geq\frac{1}{1000}\frac{h}{b_{h}^{\alpha}}$ for $\alpha\leq
1$ and
$\overline{a}_{h}\geq\frac{1}{1000}\frac{h^{\frac{1}{2\alpha-1}}}{b_{h}^{\frac{1}{2\alpha-1}}}$
for $\alpha>1$.
_Proof._ Without loss of generality, we assume
$\overline{a}_{h}\leq\underline{a}_{h}$. Denote
$U_{h}=\Omega_{h}\cap\\{x_{1}>0\\}$. Similar to that in [12], we have that the
arc-length of the image of $\Gamma_{s}\cap\\{x_{1}>0\\}$ under Gauss map is
bigger then $\frac{\pi}{6}.$ Notice that $\Omega_{1}$ contains $E$, which was
defined in Remark 2. When $\delta$ and $\delta^{*}$ are very small, $E$ is
very thin and long. The centre of $E$ is very close to the origin, in fact for
our purpose we can just pretend $E$ is centered at the origin. By convexity of
$\Omega_{h}$ and in view of Figure 2, we see that
$\Gamma_{s}\cap\\{x_{1}>0\\}$ is trapped between two lines $\ell_{1}$ and
$\ell_{2}$, and the slopes of $\ell_{1}$ and $\ell_{2}$ are very close to 0
when $E$ is very long and thin. Then it is clear that the largest distance
from the points on $\Gamma_{s}\cap\\{x_{1}>0\\}$ to the origin can not be
bigger than $10\overline{a}_{h}$. By convexity of $u$, we have
$u_{\gamma}(x)\geq\frac{h}{20\overline{a}_{h}},$ for
$x\in\Gamma_{s}\cap\\{x_{1}>0\\}$. Since $\Gamma_{s}\cap\\{x_{1}>0\\}$ evolves
under the generalized curve shortening flow, when $\alpha\leq 1$ we have the
following estimate
$\displaystyle\frac{d}{ds}(|U_{s}|)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}\cap\\{x_{1}>0\\}}\kappa^{\alpha}d\xi$ (18)
$\displaystyle=$
$\displaystyle\int_{\Gamma_{s}\cap\\{x_{1}>0\\}}u_{\gamma}^{\frac{1}{\alpha}-1}\kappa
d\xi$ (19) $\displaystyle\geq$
$\displaystyle\frac{1}{50}(\frac{h}{\overline{a}_{h}})^{\frac{1}{\alpha}-1}\frac{\pi}{6},$
(20)
from (18) to (19) we used the equation $\kappa
u_{\gamma}^{\frac{1}{\alpha}}=1$. The claim follows by the simple fact
$\frac{3}{2}b_{h}\overline{a}_{h}\geq|U_{h}|\geq\frac{1}{50}(\frac{h}{\overline{a}_{h}})^{\frac{1}{\alpha}-1}\frac{\pi}{6}\frac{h}{2}$.
Figure 2: $\Gamma_{s}\cap\\{x_{1}>0\\}$ is trapped between two lines
When $\alpha>1$, denote $l_{s}$ as the arc length of
$\Gamma_{s}\cap\\{x_{1}>0\\}$, by the above discussion, it is not hard to see
that $l_{s}\approx C\overline{a}_{h}$. Then by a simple application of
Jensen’s inequality, we have
$\displaystyle\frac{d}{ds}(|U_{s}|)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}\cap\\{x_{1}>0\\}}\kappa^{\alpha}d\xi$
$\displaystyle=$
$\displaystyle\l_{s}\int_{\Gamma_{s}\cap\\{x_{1}>0\\}}\kappa^{\alpha}\frac{1}{l_{s}}d\xi$
$\displaystyle\geq$ $\displaystyle
l_{s}(\int_{\Gamma_{s}\cap\\{x_{1}>0\\}}\frac{\kappa}{l_{s}}d\xi)^{\alpha}\geq
Cl_{s}^{1-\alpha}\geq C\overline{a}_{h}^{1-\alpha},$
then by the simple fact that $\frac{3}{2}b_{h}\overline{a}_{h}\geq|U_{h}|$ we
can finish the proof in the same way as the previous case.
_Claim 2:_ Denote $f_{k}=f_{h_{k}}$. Then
$\displaystyle f_{k}(0)\leq f_{k-1}(0)+C_{0}2^{\frac{-k}{C}}\ \text{for all}\
k\ \text{large},$ (21)
where $C_{0}$ is a fixed constant, and $C$ depends only on $\alpha$.
Lemma 2 follows from Claim 1 and Claim 2 in the following way. Let the convex
set $P$ be the projection of the graph of $g$ on the plane $\\{x_{3}=0\\}$, by
Claim 2 and the fact that $P$ contains $x_{1}$-axis (it follows from Claim 1),
$P$ must equal to $I\times\textbf{R}$ for some interval
$I\subset[0,\displaystyle{\lim_{k\rightarrow\infty}}g_{k}(0)].$ Then, by (17)
$\mathcal{M}_{u}$ is also contained in a strip region as stated in Lemma 2.
The proof of Claim 2 can be done by following the lines in [12] closely, but
one should be very careful for choosing the proper constants in the proof
which is very different from the case in [12].
_Proof of Theorem 1 and the first part of Theorem 2._ First we prove that one
can find a subsequence of $u_{h},$ where
$u_{h}(x)=h^{-1}u(h^{\frac{1}{1+\alpha}}x),$ which converges to
$\frac{1}{1+\alpha}|x|^{1+\alpha}.$
By subtracting a constant we may suppose $u(0)=0.$ Suppose $x_{n+1}=b\cdot x$
is the tangent plane of $u$ at $0.$ By Corollary 2 and the convexity of $u$ we
have
$b\cdot x\leq u(x)\leq C(1+|x|^{1+\alpha}).$
Hence,
$h^{-\frac{\alpha}{1+\alpha}}b\cdot x\leq u_{h}(x)\leq
C(\frac{1}{h}+|x|^{1+\alpha}).$
It is easy to see that $Du_{h}$ is locally uniformly bounded. Hence $u_{h}$
sub-converges to a convex function $u_{0}$ which satisfies $u_{0}(0)=0,$ and
$0\leq u_{0}(x)\leq C|x|^{1+\alpha}.$
Then it is easy to check that $u_{0}$ is an entire convex viscosity solution
to equation (3) with $\sigma=0,$ and the comparison principle holds on any
bounded domain. By using comparison principle it is easy to prove
$\\{x|u_{0}(x)=0\\}=\\{0\\}.$
Now since $\\{x|u_{0}(x)=0\\}=\\{0\\},$ $\Gamma_{1,u_{0}}=\\{x|u_{0}(x)=1\\}$
is a bounded convex curve, and the level set $\\{x|u_{0}(x)=-t\\}$, with time
$t\in(-\infty,0)$, evolves under the generalized curve shortening flow, from
[1], [2] we have the following asymptotic behavior of the convex solution
$u_{0}$ of $L_{0}u=1$
$\displaystyle u_{0}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}+\varphi(x),\
\text{where}\ \varphi(x)=o(x^{1+\alpha}),\ \text{for}\ x\neq 0\ \text{near the
origin}.$ (22)
In fact, if the initial level curve is in a sufficiently small neighborhood of
circle, by Lemma 4 in the beginning of the fourth section, we have that
$|\varphi(x)|\leq C|x|^{1+\alpha+\eta}$ for some small positive $\eta,$ where
$C$ is a constant depending only on the initial closeness to the circle.
Hence, given any $\epsilon>0$, for small enough $h>0$, we have
$B_{(1-\epsilon)r}(0)\subset\Omega_{h,u_{0}}\subset B_{(1+\epsilon)r}(0),$
where $r=\left((1+\alpha)h\right)^{\frac{1}{1+\alpha}}.$ So there is a
sequence $h_{m}\rightarrow\infty$ such that
$B_{(1-\frac{1}{m})r_{m,i}}(0)\subset\Omega_{h_{m},u}\subset
B_{(1+\frac{1}{m})r_{m,i}}(0),$
where
$r_{m,i}=\left((1+\alpha)ih_{m}\right)^{\frac{1}{1+\alpha}},i=1,\cdots,m.$
Then $u_{h_{m}}$ sub-converges to $\frac{1}{1+\alpha}|x|^{1+\alpha}.$
Since $u_{0}$ is an entire convex solution to $L_{0}u=1,$ from the above
argument, we can find a sequence $h_{m},$ such that
$u_{0h_{m}}(x)=\frac{1}{h_{m}}u_{0}(h_{m}^{\frac{1}{1+\alpha}}x)$ locally
uniformly converges to $\frac{1}{1+\alpha}|x|^{1+\alpha}.$ Hence, the sublevel
set $\Omega_{\frac{1}{1+\alpha},u_{0h_{m}}}$ satisfies
$B_{1-\epsilon_{m}}(0)\subset\Omega_{\frac{1}{1+\alpha},u_{0h_{m}}}\subset
B_{1+\epsilon_{m}}(0),$
where $\epsilon_{m}\rightarrow 0$ as $m\rightarrow\infty.$ By the discussion
below (54), we have
$u_{0h_{m}}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}+\varphi(x),$
where $|\varphi(x)|\leq C|x|^{1+\alpha+\eta}$ for some fixed small positive
$\eta,$ and the constant $C$ is independent of $m.$ Replacing $x$ by
$h_{m}^{-\frac{1}{1+\alpha}}x$ in the above asymptotic formula, we have
$u_{0}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}+h_{m}\varphi(h_{m}^{-\frac{1}{1+\alpha}}x),$
where for a given $x,$ $h_{m}\varphi(h_{m}^{-\frac{1}{1+\alpha}}x)\rightarrow
0.$ Hence $u_{0}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}.$ So we have proved
Theorem 1 and the first part of Theorem 2.
## 3 Proof of Corollary 1
We will follow the lines in the section 4 of [12]. It will be accomplished by
the following lemma which is also true for higher dimensions, but we will only
state it for $\textbf{R}^{2}$.
###### Lemma 3.
Suppose $\Omega$ is a smooth, convex, bounded domain in $\textbf{R}^{2}$. Let
$u$ be a solution of (3) with $\sigma=0$, satisfying $u=0$ on
$\partial\Omega$. Then $-\log(-u)$ is a convex function.
_Proof._ Observe $\varphi:=-\log(-u)$ satisfies
$|D\varphi|^{\frac{1}{\alpha}-1}\displaystyle{\sum_{i,j=1}^{2}}(\delta_{ij}-\frac{\varphi_{i}\varphi_{j}}{|D\varphi|^{2}})\varphi_{ij}=e^{\frac{1}{\alpha}\varphi}.$
Since $\varphi(x)\rightarrow+\infty$ as $x\rightarrow\partial\Omega$, the
result in [7](Theorem 3.13) implies $\varphi$ is convex. One may notice that
two of the conditions required in [8] are the strict convexity of domain and
the $C^{2}$ smoothness of solution. The first one can be resolved by using
strictly convex domains to approximate the convex domain. For the smoothness
condition, one may worry about the minimum point where the gradient vanishes
and the equation is singular. Moreover, in view of the solution
$u=\frac{1}{1+\alpha}|x|^{1+\alpha}$, we see when $\alpha<1$ it is not $C^{2}$
at the origin. However, by examining the proof in [8], one can see that the
argument is made away from the minimum point, which means it can still be
applied to our situation.
With the above lemma and the Lemma 4.4 in [12], we know that any convex
compact ancient solution to the generalized curve shortening flow can be
represented as a convex solution $u$ to equation (3) with $\sigma=0$, and if
the solution to the flow sweeps the whole space, the corresponding $u$ will be
an entire solution. Thus Theorem 2 implies Corollary 1 immediately.
_Remark 6._ We can also use the method in the section 4 of [12] to construct a
non-rotationally symmetric convex compact ancient solution for generalized
curve shortening flow with power $\alpha\in(\frac{1}{2},1)$, and in fact the
solution will be defined in a strip region. All we need to do is replace Lemma
4.2, 4.3 and 4.4 in [12] for mean curvature flow by the corresponding lemmas
for the generalized curve shortening flow.
## 4 Proof of the second part of Theorem 2
First of all, we would like to point out that instead of using Gage and
Hamilton’s exponential convergence of the curve shortening flow in [6] we need
to use the corresponding exponential convergence for the generalized curve
shortening flow and we will state it as a lemma which is corresponding to
lemma 3.2 in [12].
###### Lemma 4.
Suppose $\\{C_{t}\\}$ be a convex solution to the generalized curve shortening
flow with initial curve $\\{C_{0}\\}$ uniformly convex. Suppose
$\\{C_{t}\\}\subset N_{\delta_{0}}S^{1}$ for some unit circle $S^{1}$ ,
$\\{C_{t}\\}$ shrinks to the origin at $t=\frac{1}{1+\alpha}$. Denote
$\widetilde{C_{t}}=(1-(1+\alpha)t)^{-\frac{1}{1+\alpha}}C_{t}$ as a
normalization of $C_{t}$. Then
$\widetilde{C_{t}}\subset N_{\delta_{t}}S^{1},$
with
$\delta_{t}\leq C\delta_{0}(\frac{1}{1+\alpha}-t)^{\iota}$
for some small positive constant $\iota$.
The proof of the above lemma is similar to the proof of lemma 3.2 in [12].
Using the condition that the initial curve is uniformly convex and the
estimates in section II of [1], we can apply Schauder’s estimates safely for
$\alpha>\frac{1}{2}$ as in [12], which says that for
$t\in(\frac{1}{4\alpha+4},\frac{1}{2\alpha+2})$,
$\|\widetilde{\ell_{t}}-S^{1}\|_{C^{k}}\leq C\delta_{0}.$
Although the constant $C$ will depend on the lower and upper bound of the
curvature of the initial curve, it is not a problem for our purpose, since
when we blow down the solution for $\sigma=0$, the norm of the gradient
$Du_{h}$ on the curve $\\{u_{h}(x)=1\\}$ approaches to 1. By the equation
$\kappa u_{\gamma}^{\frac{1}{\alpha}}=1$ we see that the curvature $\kappa$ is
also very close to 1 on that curve. However, the estimates in section II of
[1] also shows that when $\alpha\leq 1$ the uniformly convex condition (though
the convexity is still needed) is not needed, and the constant $C$ in the
above lemma is independent of the bound on the curvature of the initial curve.
For the exponential decay rate of the derivatives of curvature, one can
imitate the proof in Hamilton and Gage [6], and our corresponding estimate
will be $|\kappa^{\prime}(\tau)|\leq C\delta_{0}e^{-\iota\tau}$ for some small
positive number $\iota$, where
$\tau=-\frac{1}{1+\alpha}\log(\frac{1}{1+\alpha}-t)$. This estimate
immediately implies our lemma.
An alternative way to see that is by writing down the normalized evolution
equation for the generalized curve shortening flow by using support function
$s(\theta,\tau)$ as following
$s_{\tau}=-(s_{\theta\theta})^{-\alpha}+s,$
here we still take the origin as the limiting point of the original
generalized curve shortening flow. Then the linearized equation of the flow
about the circle solution is
$s_{\tau}=\alpha(s_{\theta\theta}+s)+s.$
The rate of convergence is governed by the eigenvalues of the right hand side.
The constant eigenfunction corresponds to scaling, which is factored out,
while the $\sin\theta$ and $\cos\theta$ correspond to translations, which are
also factored out. The next is $\cos(2\theta)$, which gives eigenvalue
$1-3\alpha$. So when $\alpha>\frac{1}{3},$ we have exponential convergence of
the normalized solution to the limiting circle with exponent $1-3\alpha.$ The
author learned this from professor Ben Andrews.
In the following we will consider the case when $\sigma=1$ . Without loss of
generality we can assume $u(0)=\inf u$. Let
$u_{h}(x)=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x)$. Then $u_{h}$ satisfies the
equation $L_{\sigma}u_{h}=1$ with $\sigma=h^{-\frac{2\alpha}{1+\alpha}}.$ By
Theorem 1, $\Gamma_{\frac{1}{1+\alpha},u_{h}}$ converges to the unit circle as
$h\rightarrow\infty$.
###### Lemma 5.
$u(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}+O(|x|^{1+\alpha-2\alpha\beta})$ (23)
where $C$ is a fixed constant and the constant $\beta$ is chosen such that
$\frac{1}{2\alpha}<\beta<\min\\{1,\frac{1+\alpha}{2\alpha}\\}$.
For small $\delta_{0}>0$, taking $h$ large enough so that
$\displaystyle\Gamma_{\frac{1}{1+\alpha},u_{h}}\subset N_{\delta_{0}}(S^{1})$
(24)
for unit circle $S^{1}$ with center $p_{0}$. Note that when $h$ is large,
$\delta_{0}$ is very close to 0. Then we will prove the following claim,
_Claim 3_. For small fixed $\tau$,
$\displaystyle\Gamma_{\tau,u_{h}}\subset((1+\alpha)\tau)^{\frac{1}{1+\alpha}}N_{\delta_{\tau}}((1+\frac{a_{0}}{\tau})^{\frac{1}{1+\alpha}}S^{1})$
(25)
with
$\displaystyle\delta_{\tau}\leq
C_{1}(\tau)\sigma^{\beta}+C_{2}\delta_{0}\tau^{\eta},$ (26)
where the constants $C_{1}$ and $C_{2}$ are independent of $\delta_{0}$ and
$h$, and $C_{2}$ is also independent of $\tau$, $\eta$ is a small positive
constant. $u_{0}$ is the solution of $L_{0}(u)=1$ in
$\Omega_{\frac{1}{1+\alpha},u_{h}}$ satisfying
$u_{0}=u_{h}=\frac{1}{1+\alpha}$ on
$\partial\Omega_{\frac{1}{1+\alpha},u_{h}}$, $a_{0}=|\inf u_{0}|$ and the
center of $(1+\frac{a_{0}}{\tau})^{\frac{1}{1+\alpha}}S^{1}$ is the minimum
point of $u_{0}$ times a factor $((1+\alpha)\tau)^{-\frac{1}{1+\alpha}}$.
_Proof of Claim 3._ It is equivalent to prove
$\displaystyle\text{dist}\left((1+\alpha)^{\frac{1}{1+\alpha}}(\tau+a_{0})^{\frac{1}{1+\alpha}}S^{1},\Gamma_{\tau,u}\right)\leq
C_{1}(\tau)\sigma^{\beta}+C_{2}\delta_{0}\tau^{\frac{1}{1+\alpha}+\eta},$ (27)
where $\eta$ is some small positive constant, $C_{2}$ is independent of
$\tau$. by Theorem 1 we know $u_{h}$ converges to
$\frac{1}{1+\alpha}|x|^{1+\alpha}$ uniformly on any compact subset of
$\textbf{R}^{2}$, then by the convexity of $u_{h}$, we have that when
$x\in\\{x\in\Omega_{\frac{1}{1+\alpha},u_{h}}:\tau_{0}\leq
u_{h}<\frac{1}{1+\alpha}\\},$
$|Du_{h}|$ is bounded above and below by some constants depending on
$\tau_{0}$ for large $h$, by the growth condition for $D^{2}u$ in Theorem 2 we
have $\sigma(u_{h})_{\gamma\gamma}\leq C\sigma^{\beta},$ where $C$ is a
constant depending on $\tau_{0}$. Therefore we have
$\kappa(u_{h})_{\gamma}^{\frac{1}{\alpha}}\approx 1-C\sigma^{\beta}\
\text{on}\ \\{x\in\Omega_{\frac{1}{1+\alpha},u_{h}}:\tau\leq
u_{h}<\frac{1}{1+\alpha}\\},$ where $C$ depends on $\tau_{0}$. Denote
$\widetilde{u}_{0}=(1-C\sigma^{\beta})^{\alpha}(u_{0}-\frac{1}{1+\alpha})+\frac{1}{1+\alpha},$
then
$L_{0}(\widetilde{u}_{0})=1-C\sigma^{\beta}\ \text{in}\
\Omega_{\frac{1}{1+\alpha},u_{h}}$
with $\widetilde{u}_{0}=u_{h}=\frac{1}{1+\alpha}$ on
$\partial\Omega_{\frac{1}{1+\alpha},u_{h}}$. Now by comparison principle we
have
$\Omega_{\tau,u_{0}}\subset\Omega_{\tau,u_{h}}\subset\Omega_{\tau,\widetilde{u}_{0}}$,
and by the asymptotic behavior of $u_{0}$ we have
$\Gamma_{\tau,u_{0}}\subset N_{\zeta}((\tau+a_{0})^{\frac{1}{1+\alpha}}S^{1})\
\text{and}\ \Gamma_{\tau,\widetilde{u}_{0}}\subset
N_{\zeta}((\tau+a_{0}-C\sigma^{\beta})^{\frac{1}{1+\alpha}}S^{1}),$
where $\zeta={C\delta_{0}(\tau+a_{0})^{\eta}}$. Denote
$\ell_{1}=(\tau+a_{0})^{\frac{1}{1+\alpha}}S^{1}$,
$\ell_{2}=(\tau+a_{0}-C\sigma^{\beta})^{\frac{1}{1+\alpha}}S^{1}$, both of
them are centered at $p_{1}$, which is the minimum point of $u_{0}$. Hence
$\text{dist}((\tau+a_{0})^{\frac{1}{1+\alpha}}S^{1},\Gamma_{\tau,u_{h}})\leq\text{dist}(\ell_{1},\ell_{2})+C\delta_{0}(\tau+a_{0})^{\frac{1}{1+\alpha}+\eta}$,
where $\text{dist}(\ell_{1},\ell_{2})$ can be bounded by
$C_{1}(\tau)\sigma^{\beta}$, hence (26) follows from the above discussion. Now
we will use an iteration argument to prove the following Claim 4, which will
enable us to simplify (25) and (26).
_Claim 4_ :
$a_{0}\leq\begin{cases}C\sigma|\log(\sigma)|&\text{if}\ \alpha\leq 1\\\
C\sigma^{\frac{1+\alpha}{2\alpha}}&\text{if}\ \alpha>1\end{cases}$ (28)
_Proof of Claim 4._ We fix a large constant $A$ such that
$\\{u_{\frac{A}{\tau}}=\frac{1}{1+\alpha}\\}$ is very close to a unit circle.
Let $u_{0,\tau^{k}}$ solve $L_{0}u=1$ with boundary condition $u=\tau^{k}$ on
$\\{u_{h}=\tau^{k}\\}$. Denote $a_{k}=|\inf u_{0,\tau^{k}}|.$ From the proof
of Claim 3 we see that
$\\{u_{0}<\tau\\}\supset\\{u_{0,\tau}<\tau\\}\supset\\{\widetilde{u}_{0}<\tau\\},$
by comparison principle, we have $\inf u_{0}<\inf
u_{0,\tau}<\inf\widetilde{u}_{0}.$ So by the construction of
$\widetilde{u}_{0}$ and a simple computation, we have
$a_{0}-a_{1}\leq\inf\widetilde{u}_{0}-\inf u_{0}\leq C\sigma.$ When
$\tau^{k}\geq\frac{A}{h}$, we can iterate this argument for $u_{0,\tau^{k}}$
and $u_{0,\tau^{k+1}}$ by rescaling them to
$\frac{1}{1+\alpha}\tau^{-k}u_{0,\tau^{k}}\left((1+\alpha)^{\frac{1}{1+\alpha}}\tau^{\frac{k}{1+\alpha}}x\right)$
and
$\frac{1}{1+\alpha}\tau^{-k}u_{0,\tau^{k+1}}\left((1+\alpha)^{\frac{1}{1+\alpha}}\tau^{\frac{k}{1+\alpha}}x\right)$
respectively, after rescaling back, we have $a_{k}-a_{k+1}\leq C\sigma$. Note
that the choice of $A$ and the condition $\tau^{k}\geq\frac{A}{h}$ ensure the
uniform gradient bound needed in the above argument. Let $k_{0}$ be an integer
satisfying $\tau^{k_{0}}\geq\frac{A}{h}\geq\tau^{k_{0}+1}$, after $k_{0}$
steps we stop the iteration, and notice that
$\\{u_{h}=\frac{A}{h}\\}=\frac{1}{h^{\frac{1}{1+\alpha}}}\\{u=A\\}$ is
contained in a circle with radius $Ch^{-\frac{1}{1+\alpha}}$ for some constant
$C$, so it takes at most time $Ch^{-1}=C\sigma^{\frac{1+\alpha}{2\alpha}}$ for
$\\{u_{h}=\frac{A}{h}\\}$ shrink into a point. Claim 4 follows from the above
discussion.
By omitting the lower order term we can rewrite (25) and (26) as
$\displaystyle\Gamma_{\tau,u_{h}}\subset((1+\alpha)\tau)^{\frac{1}{1+\alpha}}N_{\delta_{\tau}}(S^{1})$
with
$\delta_{\tau}\leq C_{1}(\tau)\sigma^{\beta}+C_{2}\delta_{0}\tau^{\eta}.$ (29)
If we take $\tau$ small such that $C_{2}\tau^{\eta}\leq\frac{1}{4}$, (29)
becomes
$\delta_{\tau}\leq C_{1}(\tau)\sigma^{\beta}+\frac{1}{4}\delta_{0}.$ (30)
Now we can carry out an iteration argument similar as that in [12]. We start
at the level $\frac{1}{1+\alpha}\tau^{-k_{0}}$ for $k_{0}$ very large. Denote
$\Omega_{k}=\tau^{\frac{k}{1+\alpha}}\Omega_{\frac{1}{1+\alpha}\tau^{-k},u}$
and $\Gamma_{k}=\partial\Omega_{k}$. Define $\delta_{k}$ similarly to that in
[12]. By (30) we have
$\delta_{k-1}\leq
C_{1}(\tau)\tau^{(k-1)\frac{2\alpha\beta}{1+\alpha}}+\frac{1}{4}\delta_{k}$
(31)
for $k=k_{0},k_{0}+1,\cdots$. Then we have
$\displaystyle\Gamma_{j}\subset N_{\delta_{j}}(S^{1})$ (32)
with
$\delta_{j}\leq C\tau^{j\frac{2\alpha\beta}{1+\alpha}}$ (33)
It follows that
$\displaystyle\Gamma_{\frac{1}{1+\alpha}\tau^{-j},u}\subset
N_{\widetilde{\delta}_{j}}(\tau^{\frac{-j}{1+\alpha}}S^{1})$ (34)
with
$\widetilde{\delta}_{j}\leq C\tau^{\frac{2\alpha\beta-1}{1+\alpha}j}$ (35)
where $\tau^{\frac{-j}{1+\alpha}}S^{1}$ is centered at
$z_{j}=\tau^{\frac{-j}{1+\alpha}}y_{j}$. From Lemma 3 and (30) it is not hard
to see that we have
$|z_{j}-z_{j-1}|\leq C\tau^{\frac{2\alpha\beta-1}{1+\alpha}j}$ (36)
Denote $z_{0}=\lim_{j\rightarrow\infty}z_{j}$. Then
$|z_{j}-z_{0}|\leq C\tau^{\frac{2\alpha\beta-1}{1+\alpha}j},$ (37)
which means in (34) we can assume the circle is centered at $z_{0}$ by
changing the constant $C$ a little bit. In fact when we choose different
$\tau$, the corresponding $z_{0}$ will not change, so we can assume $z_{0}=0$.
Hence for $h=\frac{1}{1+\alpha}\tau^{-j}$,
$\Gamma_{h,u}\subset
N_{\delta}\left((1+\alpha)^{\frac{1}{1+\alpha}}h^{\frac{1}{1+\alpha}}S^{1}\right),$
where
$\delta\leq Ch^{\frac{1-2\alpha\beta}{1+\alpha}}$ (38)
and $S^{1}$ is centered at the origin. By using different $\tau$, it is easy
to see that estimate holds for any large $h$. Lemma 5 follows from the above
estimates.
Now we can finish the proof of Theorem 2 in the following way.
_Proof of the second part of Theorem 2._ Denote $u^{*}$ as the Legendre
transform of $u$. Then $u^{*}$ satisfies the following equation
$\displaystyle
G(x,D^{2}u^{*})=\frac{\det{D^{2}u^{*}}}{(\delta_{ij}-\frac{x_{i}x_{j}}{1+|x|^{2}})F^{ij}(u^{*})}=(1+|x|^{2})^{\frac{1}{2\alpha}-\frac{1}{2}},$
(39)
where $F^{ij}(u^{*})=\frac{\partial\det{r}}{\partial r_{ij}}$, at
$r=D^{2}(u^{*}).$ We have
$\displaystyle
u^{*}(x)=C(\alpha)|x|^{1+\alpha}+O(|x|^{\frac{1+\alpha-2\alpha\beta}{\alpha}}),$
(40)
where $C(\alpha)$ is a constant depending only on $\alpha$. In fact, for big
$h$, by Lemma 5 we have
$u_{h}(x)=\frac{1}{1+\alpha}|x|^{1+\alpha}+O(|h|^{\frac{-2\alpha\beta}{1+\alpha}})$
in $B_{1}(0).$ Denote $u_{h}^{*}$ as the Legendre transforms of $u_{h}$. Then
$u_{h}^{*}(x)=C(\alpha)|x|^{1+\frac{1}{\alpha}}+O(|h|^{\frac{-2\alpha\beta}{1+\alpha}}),$
where $C(\alpha)$ is a constant depending only on $\alpha$ and in fact it is
comes from the Legendre transform of the function
$\frac{1}{1+\alpha}|x|^{1+\alpha}.$ Note that
$u_{h}^{*}(x)=h^{-1}u^{*}(h^{\frac{\alpha}{1+\alpha}}x)$, we obtain (40).
Let $u_{0}$ be the unique radial solution of (3) with $\sigma=1$, and let
$u_{0}^{*}$ be the Legendre transform of $u_{0}$. Similar to (40) we have
$\displaystyle
u_{0}^{*}(x)=C(\alpha)|x|^{1+\alpha}+O(|x|^{\frac{1+\alpha-2\alpha\beta}{\alpha}}).$
(41)
Since both $u^{*}$ and $u_{0}^{*}$ satisfy equation (39), $v=u^{*}-u_{0}^{*}$
satisfies the following elliptic equation
$\displaystyle{\sum_{i,j=1}^{n}}a_{ij}(x)v_{ij}=0\ \text{in}\ \textbf{R}^{2},$
where
$a_{ij}=\int_{0}^{1}G^{ij}(x,D^{2}u_{0}^{*}+t(D^{2}u^{*}-D^{2}u_{0}^{*})dt,$
here $G^{ij}=\frac{\partial G(x,r)}{\partial r_{ij}}$ for any symmetric matrix
$r$. Note that by the choice of $\beta$,
$\frac{1+\alpha-2\alpha\beta}{\alpha}<1$, so by (40) and (41)
$v=O(|x|^{\frac{1+\alpha-2\alpha\beta}{\alpha}})=o(|x|)$, as
$|x|\rightarrow\infty$. Using the Liouville Theorem by Bernstein [9] (p.245)
we conclude that $v$ is a constant.
## References
* [1] B. Andrews. Evolving convex curves. _Calc. Var. Partial Differential Equations_ 7 (1998), no 3, 315-371.
* [2] B. Andrews. Classification of limiting shapes for isotropic curve flows. _J. Amer. Math. Soc._ 16 (2003), no. 2, 443-459.
* [3] B. Andrews. Non-convergence and instability in the asymptotic behavior of curves evolving by curvature. _Comm. Anal. & Geom._,10 (2002), no. 2, 409-449.
* [4] K.S.Chou, X.P.Zhu. The Curve Shortening Problem. _CRC Press.Boca Raton_ (2001).
* [5] P. Daskalopoulos, R. Hamilton, N. Sesum. Classification of compact ancient solutions to the curve shortening flow. _J. Differential Geometry_ 84 (2010), 455-464.
* [6] M. Gage and R.S. Hamilton. The heat equation shrinking convex plane curves. _J. Diff. Geom._ 23 (1986), 6996.
* [7] B. Kawohl. Rearrangements and convexity of level sets in PDE, _Lecture Notes in Math._ , 1150. Springer, Berlin (1985).
* [8] F.Schulze. Evolution of convex hypersurfaces by powers of the mean curvature. _Math. Z._ , 251, (2005), 721-733.
* [9] L. Simon. The minimal surface equation. Geometry V, Encyclopaedia Math. Sci., 90 (1997), 239-272.
* [10] W. M. Sheng and C. Wu. On asymptotic behavior for singularities of powers of mean curvature flow. _Chin. Ann. Math._ Ser. B 30 (2009), 51-66.
* [11] B. White. The size of the singular set in mean curvature flow of mean-convex sets. _J. Amer. Math. Soc._ 13 (2000), 665-695.
* [12] X.J.Wang. Convex solutions to the mean curvature flow. _Ann. of Math._ (3) 173 (2011), no. 1, 1185-1239.
|
arxiv-papers
| 2012-10-27T19:26:13 |
2024-09-04T02:49:37.205181
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shibing Chen",
"submitter": "Shibing Chen",
"url": "https://arxiv.org/abs/1210.7363"
}
|
1210.7572
|
# Exploration of Network Scaling: Variations on Optimal Channel Networks
Lily Briggs Mukkai Krishnamoorthy
###### Abstract
Metabolic allometry, a common pattern in nature, is a close-to-3/4-power
scaling law between metabolic rate and body mass in organisms, across and
within species. An analogous relationship between metabolic rate and water
volume in river networks has also been observed. Optimal Channel Networks
(OCNs), at local optima, accurately model many scaling properties of river
systems, including metabolic allometry. OCNs are embedded in two-dimensional
space; this work extends the model to three dimensions. In this paper we
compare characteristics of 3d OCNs with 2d OCNs and with organic metabolic
networks, studying the scaling behaviors of area, length, volume, and energy.
In addition, we take a preliminary look at comparing Steiner trees with OCNs.
We find that the three-dimensional OCN has predictable characteristics
analogous to those of the two-dimensional version, as well as scaling
properties similar to metabolic networks in biological organisms.
## 1 Introduction
Fractal Networks have been widely used to study well known network behaviors.
Some of the properties commonly used in the study of fractal networks are
scaling factors (or laws); scaling laws can describe how lengths are related
to area and areas are related to volumes. In this paper, we further study
these allometric scaling laws. We use simulation and elementary mathematical
bounding techniques to study such laws.
A common pattern in nature, known as metabolic allometry, is a 3/4-power
scaling law between metabolic rate ($M$) and body mass ($B$) in organisms,
across and within species: $B\propto M^{3/4}$. An analogous relationship
between water mass ($C$) and contributing area ($A$) in river networks has
also been observed: $A\propto C^{2/3}$.
The relationship in biological organisms was first observed by Max Kleiber in
1932 and has garnered a lot of attention in the last decade and a half from
biologists, mathematicians, and physicists alike [13]. The areas this
relationship finds application in include models of climate change effects in
rivers [19] and oceans [15], analyses of river stability [17], and in studies
of information networks [18]. Many theories concerning the origins and nature
of this phenomenon have been proposed, and there has been much debate about
how ubiquitous the scaling law really is.
In 1997, West, Brown and Enquist (WBE hereafter) introduced a theory
purporting to explain metabolic scaling [26]. Their theory is based on the
idea of a fractal space-filling hierarchical network of resource distribution,
with certain assumptions based on simplified biology. The geometry of the
network (e.g. branching ratios, vessel radius relationships) are important to
the development of their result. The authors claimed the model is applicable
even to organisms without a physical blood vessel network since they can be
treated as having a virtual distribution network [27].
Subsequent studies questioned the WBE results, by showing that data does not
appear to strictly follow a 3/4-power scaling law after all; the theoretical
approach taken by WBE has also been criticized [1, 28]. Some authors claim
that an exponent of 2/3 fits the data better [9], while others assert that no
universal scaling law exists at all [14]. Savage et al. pointed out that the
WBE model predicts a curvilinear relationship between $logB$ and $logM$, and
only predicts a constant power-law relationship in the infinite limit of body
size; however, even with finite size adjustments and variations in the
structural properties of the model, they found the results were still
inconsistent with data [25]. Some other modifications to the WBE model do
yield results somewhat more consistent with data, such as taking into account
body temperature [5] or fluid velocity [4]. Kolokotrones et al. use regression
on large set of organism data to show a quadratic, rather than linear,
relationship between $logB$ and $logM$ [14]. Also, their results reported a
lot of variation. They build off the modifications to WBE in [25] and find
that a model network with a proportional transition between area-increasing
and area-preserving branching yields a fit to the data almost as good as their
empirically-derived quadratic model.
Other theories and models have been put forth, some based on network
properties like the WBE model, and others on other biological properties.
Dodds [10] proposed a model based on virtual networks of metabolite
transportation rather than physical networks.
Banavar et al. proposed a general model for any efficient transportation
network, predicting $B\propto M^{D/(D+1)}$ (at most) where $D$ is the
dimension of the network (i.e. $D=3$ for organism metabolism, and $D=2$ for
river networks) [3]. Thus, they predict that if the network were as efficient
as possible, organism metabolism should scale as $B\propto M^{3/4}$, and
deviations of the exponent below $3/4$ might be explained by some inefficiency
in the metabolic distribution network. Later the authors added to this model a
supply and demand mechanism that accounts for the deviations [2].
Isaac et al. propose a less strict understanding of the “3/4-power law”:
though the 3/4 exponent might not be a consistent, universal constant, it is
still an observable trend; if there isn’t a single unifying principle
explaining why it appears, there still may be multiple reasons for it worth
studying [12].
In the context of river networks, the analogous properties to metabolic rate
and body mass are contributing area (A) and water mass (C). It is well-
observed that the properties obey a power-law relationship with an exponent of
slightly more than 2/3: $A\propto C^{2/3}$. This is consistent with the
general resource transportation model of Banavar et al. [3] for systems in two
dimensions, and also with the virtual network model of Dodds [10].
Rinaldo et al. devised a model of river networks called Optimal Channel
Networks (OCNs) [21]. This model uses a grid network to represent the area of
a river basin, and constructs a spanning network on the grid that minimizes a
functional representing the energy required to drain the area of land through
river channels. The energy functional is derived both from theoretical
physical characteristics and observed characteristics of river basins [23].
Statistical properties of Optimal Channel Networks (including metabolic
allometry) correspond excellently with observed properties of real river
networks [16, 22].
This paper explores a three-dimensional version of OCNs, using simulations to
compare characteristics of 3d OCNs with 2d OCNs and with organic metabolic
networks and study the scaling behavior of area, length, volume, and energy.
We do not attempt to explain 3/4-power scaling in nature, but rather explore
some interesting and potentially useful connections, somewhat in the spirit of
Isaac et al.’s proposed understanding of the scaling law [12].
We further study an alternative model using Steiner trees for OCNs and we
compare the scaling behaviors between these two models on small network sizes.
Section 2 describes definitions and methodologies used in our paper. In
Section 3, we derive lower bound results for various scaling laws. In section
4, we describe our simulation results and show Energy, Length and Volume
Scaling with respect to area and compare it with the lower bound results
described in Section 3. In Section 5, we describe an alternative model based
on Steiner Trees and derive analytical results for small network sizes.
Section 6 gives conclusions and suggestions for future work.
## 2 Definitions and Methods
Consider an area of land, and embed a grid network in it such that every node
in the network is associated with a unit area of land. Construct a spanning
tree on the grid and direct it so that all nodes have a path in the tree to a
single root (call it the outlet). Define a partial order on the nodes such
that $x\leq y$ iff $x$ is on the path from $y$ to the root. If $x<y$, we will
say $y$ is upstream of $x$. The terms area, volume, upstream length, link
length, and energy are defined here in this context.
Note that each node in the spanning tree, except for the root, has exactly one
link directed out of it. Thus we can name each link according to its
corresponding node; that is, when we speak of link $x$ we are referring to the
link out of node $x$. All the properties defined here for nodes will be
extended to their corresponding links. For example, the area of link $x$ is
defined to be the area of node $x$.
The area of node $x$, $A_{x}$, is the total number of nodes whose paths to the
outlet include $x$; that is, the number of nodes $y$ such that $x\leq y$. This
is equivalent to the sum of the areas of each node directly linked to $x$ plus
one for $x$ itself:
$A_{x}\>=\>\sum_{y}A_{y}\>+\>1$ (1)
where y ranges over the nodes directly linked to x [21].
The volume of node $x$, $C_{x}$, is the sum of all the areas of all nodes
upstream of $x$ [3]:
$C_{x}\>=\>\sum_{y|x<y}A_{y}$ (2)
Equation 2 can be transformed into a recursive definition for volume:
$C_{x}\>=\>\sum_{y}(C_{y}+A_{y})$ (3)
where y ranges over the nodes directly linked to x.
Upstream length is another quantity of interest; it is defined as the number
of links in the path from a node $x$ to a source node, where at each step
upstream the path taken goes to the node with the largest area [7].
Statistically, it is equivalent to define it as the distance (counted in
number of links) to the farthest away upstream source [20].
Link length is a weight assigned to each link, generally representing the
geometric length of the link in the embedding. The length of link $i$ is
denoted by $l_{i}$.
Energy is a function of the link lengths and areas of all the links. The
energy of a given tree configuration $s$ is given by:
$H_{\gamma}(s)=\sum_{i\in links(s)}{A_{i}^{\gamma}l_{i}}$ (4)
with $0\leq\gamma\leq 1$. A tree network that achieves the minimum
$H_{\gamma}(s)$ over all possible tree configurations on a given grid is an
optimal channel network [21].
Figure 1 shows a grid and a possible tree configuration $s_{1}$. $s_{1}$
minimizes $H_{1/2}(s)$ for all configurations on this grid. Area, volume and
upstream length values are also depicted for the nodes in $s_{1}$.
(a) Underlying grid
(b) Areas of nodes
in $s_{1}$
(c) Volumes of nodes
in $s_{1}$
(d) Upstream lengths
of links in $s_{1}$
Figure 1: The outlets are in the top left corners. Tree $s_{1}$ minimizes
$H_{1/2}(s)$ for the grid in 1(a). $H_{1/2}(s_{1})=24.3024964149$.
Rinaldo et al. [21] derived a gamma value of 1/2 from estimations of physical
properties of rivers, and hypothesized that natural rivers would minimize
$H_{1/2}(s)$. A remarkable result of [21] is that while the global optima of
OCNs do not match well statistically with real river networks, the local
optima are an excellent match. The idea that nature tends towards a local
optimum makes sense, so this suggests that OCNs are a good model for what
actually happens in a river [23].
When $\gamma=1$ and all links are of unit length, the energy is equivalent to
the volume. When $\gamma=0$, the energy is equivalent to the total length of
all links, so minimizing the energy is equivalent to finding a minimum
spanning tree (which, when all links are of unit length, would be _any_
spanning tree).
In most studies of OCNs, the network of potential links is a 2-dimensional
lattice with additional links between diagonally-adjacent pairs of nodes (Fig.
2(a)). For the sake of simplicity, all the links, diagonal and orthogonal, are
here considered to have unit length, because it has been shown ([23]) that the
properties with which this paper is concerned are independent of whether the
diagonal links are given realistic length or not.
(a) 7x7 network of potential links
(b) 2x2x2 3d network of potential links
Figure 2:
In this study we also analyzed a three dimensional version of optimal channel
networks, built on a 3-dimensional grid (Fig. 2(b)).
To make an OCN, we first generated a random spanning tree on a given grid. We
used Prim’s algorithm for minimum spanning trees for this. Essentially, the
algorithm builds the tree from a root node, maintaining a set of frontier
edges from the grid that are incident on exactly one node in the tree. Our
algorithm randomly chooses one edge at a time from this set, adds it to the
tree, and updates the set. Rinaldo et al. also began with random spanning
trees, though they did not specify their method for generating them [21].
The random tree was then optimized using a version of Lin’s algorithm for TSP,
as in [21]. That algorithm is as follows:
Input: Undirected graph $G$; random spanning tree $T$, directed so that there
exists a path in $T$ from each node to a single root
Output: Locally optimal spanning tree on $G$
while _Convergence has not been reached_ do
Step 1: Choose a random node $i$ in $T$
Step 2: Choose a random neighbor of $i$ in $G$ distinct from the node $i$
links to in $T$
Step 3: Construct $T^{*}$ from $T$ by redirecting $i$ to link to this new node
if _$T^{*}$ contains a loop_ then discard $T^{*}$ ;
else if _$H_{\gamma}(T^{*}) <H_{\gamma}(T)$_ then discard $T$ and let $T^{*}$
be the new $T$ ;
else discard $T^{*}$;
The condition for convergence was that the number of improvements made be only
1% of the total number of iterations (or 2% for larger networks). See Figure 3
for an illustration of the process.
(a) Initial tree configuration $T$. The outlet is in the top left corner.
$H_{\gamma}(T)=9.65028$.
(b) Step 1: Node $i$ has been chosen; the unused links to its neighbours in
$G$ are highlighted.
(c) Step 3: A new link for $i$ in the tree has been chosen, creating $T^{*}$.
$H_{\gamma}(T^{*})=9.8637>H_{\gamma}(T)$, so this tree will be discarded.
(d) the next iteration, Step 3: A different new link for $i$ in the tree has
been chosen, creating $T^{*}$. $H_{\gamma}(T^{*})=9.44949<H_{\gamma}(T)$, so
this tree will become the new $T$.
Figure 3: Optimization algorithm
(a)
(b)
Figure 4: A random tree spanning a 60x60 grid (a) and the resulting locally
optimal OCN (b). The outlets are in the top left corners. Figure 5: An OCN on
a 3x3x3 grid; the red links are in the OCN.
Figure 4(a) shows a random tree on a 60x60 grid and Figure 4(b) shows the
result of using the above algorithm to optimize the tree. In order to make the
structure of the network clearer, links are drawn with a thickness
proportional to the log of their area. Figure 5 depicts a three-dimensional
OCN on a 3x3x3 grid.
A range of grid sizes were analyzed. Two-dimensional $n$x$n$ grids with $n=$
$10$, $20$, $30$, $40$, $50$, $60$, $70$, and $80$, and three-dimensional
$n$x$n$x$n$ grids with $n=8$, $10$, $12$, $14$, $16$, $18$, and $20$ were
used. For the smaller grids, data was averaged from at least $4$ realizations
of each size, while for the larger grids at least $3$ realizations were used
for each. The quantities measured were distributions of area and length in
whole basins, and average length, volume, and energy per area of subbasins and
whole basins.
## 3 Analytical Proofs
### 3.1 Energy bounds
In [8], Colaiori et al. proved lower bounds for the energy of OCNs on an
orthogonal grid (one with no diagonal links). Here, we look at lower bounds
for the energy of OCNs on an eight-neighbor grid. The main result for this
section (Theorem 3) is that for an $n\times n$ 8-way grid,
$\frac{3}{2}n^{2}-\frac{7}{2}n+1$ is a lower bound on $H_{0.5}(s)$.
###### Lemma 1.
Let (P) be the optimization problem
$\begin{array}[]{rl}\min&\displaystyle\sum_{i=1}^{n}A_{i}^{\gamma}\\\
\mbox{s.t.}&\displaystyle\sum_{i=1}^{n}A_{i}=n+m\\\ &A_{i}\geq 1\>\>\forall
i\\\ &A_{i}\in\mathbb{Z}\>\>\forall i\end{array}$
where $n$ is the number of entries in the vector $A$, $m$ is some natural
number, and $0\leq\gamma\leq 1$.
$A_{1}=A_{2}=...=A_{n-1}=1,A_{n}=m+1$ is an optimal solution to $(P)$.
###### Proof.
Suppose A is an optimal solution for $(P)$ and $\exists j$ s. t.
$1<A_{j}<m+1$.
$\exists k\>\>s.t.\>\>1<A_{k}<m+1,j\neq k$.
We can construct $A^{*}$ such that $A^{*}_{i}=A_{i}\>\>\forall i\neq j,k$;
$A^{*}_{j}=1$; $A^{*}_{k}=A_{j}+A_{k}-1$.
Then
$\sum_{i=1}^{n}{A^{*}_{i}}^{\gamma}=(\sum_{i=1}^{n}A_{i}^{\gamma})-A_{j}^{\gamma}-A_{k}^{\gamma}+1+(A_{j}+A_{k}-1)^{\gamma}$.
Since $f(x)=x^{\gamma}$ when $0\leq\gamma\leq 1$ is concave, and
$A_{j},A_{k}>1$, $(A_{j}+A_{k}-1)^{\gamma}\leq
A_{j}^{\gamma}+A_{k}^{\gamma}-1$. So
$\sum_{i=1}^{n}{A^{*}_{i}}^{\gamma}\leq\sum_{i=1}^{n}A_{i}^{\gamma}$. Thus
$A^{*}$ must be an optimal solution for $(P)$. By an iteration of this
argument, there must be an optimal solution $A^{+}$ with all entries except
for one equal to $1$. Without loss of generality, the entries can be
rearranged so that $A^{+}_{n}$ is the largest (only non-unit) entry. ∎
###### Theorem 2.
For an $n\times n$ eight-neighbor grid $G$, the optimal $H_{1}(s)$ is
$\frac{4n^{3}-3n^{2}-n}{6}$.
###### Proof.
Figure 6: The five “stripes” of a $5\times 5$ 8-way grid with the outlet in
the bottom left corner.
Consider the subsets of vertices $C_{k}$ where vertex $i\in C_{k}$ iff the
distance from $i$ to the outlet equals $k-1$ (Figure 6). In an $n\times n$
8-way grid there are n such stripes. Assuming the outlet is located in a
corner, $|C_{k}|=2k-1$. Clearly,
$H_{\gamma}(s)=\sum_{i\in links(s)}A_{i}^{\gamma}=\sum_{k=2}^{n}\sum_{i\in
C_{k}}A_{i}^{\gamma}$ (5)
Recall that the area of node $i$ is the number of nodes upstream of $i$, plus
$1$ for $i$ itself. So for a given $k$, $\sum_{i\in C_{k}}A_{i}^{\gamma}$ will
have to include the number of nodes in $C_{k}$ and the total number of nodes
upstream of all $i\in C_{k}$, which must include all the nodes
$j\in\bigcup_{l>k}C_{l}$, since their paths to the outlet must pass through
stripe $C_{k}$ at some point. $\bigcup_{l>k}C_{l}$ is simply the set of all
the nodes in the grid minus those within the $k$x$k$ section enclosed by
$C_{k}$, which is $n^{2}-k^{2}$ nodes. Thus the total amount of area being
passed through $C_{k}$ is at least
$|C_{k}|+n^{2}-k^{2}$ (6)
Then we have
$H_{1}(s)\>=\>\sum_{k=2}^{n}\sum_{i\in
C_{k}}A_{i}\>\geq\>\sum_{k=2}^{n}(2k-1+n^{2}-k^{2})$
The rightmost sum simplifies to $\frac{1}{6}(4n^{3}-3n^{2}-n)$; therefore this
is a lower bound on $H_{1}(s)$.
Consider the network $t$ where every link from node $i$ to node $j$ is such
that if $i\in C_{k}$ then $j\in C_{k-1}$. Here, quantity 6 is exactly the area
passing through stripe $C_{k}$, so $H_{1}(t)=\frac{1}{6}(4n^{3}-3n^{2}-n)$.
Hence the lower bound is achievable.
∎
###### Theorem 3.
For an $n\times n$ eight-neighbor grid, $\frac{3}{2}n^{2}-\frac{7}{2}n+1$ is a
lower bound on $H_{0.5}(s)$.
###### Proof.
Following the same reasoning as above, we have that
$H_{\gamma}(s)=\sum_{k=2}^{n}\sum_{i\in C_{k}}A_{i}^{\gamma}$ (7)
and the total amount of area being passed through $C_{k}$ is at least
$|C_{k}|+n^{2}-k^{2}$.
By Lemma 1, the optimal way to distribute this area over the vertices in
$C_{k}$ is to send all the upstream area, $n^{2}-k^{2}$, through one vertex in
$C_{k}$, leaving each of the other vertices in $C_{k}$ with area 1. (Note that
this is not usually feasible, but it is a lower bound.)
Thus for a given $k$,
$\sum_{i\in C_{k}}A_{i}^{\gamma}\geq|C_{k}|-1+(n^{2}-k^{2}+1)^{\gamma}$
Since $f(x)=x^{\gamma}$ is concave,
$\sum_{k=2}^{n}(n^{2}-k^{2}+1)^{\gamma}\>\geq\>\sum_{k=2}^{n}[n^{2\gamma}-(k^{2}-1)^{\gamma}]\>\geq\>n^{2\gamma+1}-n^{2\gamma}-\sum_{k=2}^{n}(k^{2})^{\gamma}$
So
$\sum_{k=2}^{n}\sum_{i\in
C_{k}}A_{i}^{\gamma}\>\geq\>\sum_{k=2}^{n}[2k-2+(n^{2}-k^{2}+1)^{\gamma}]\>\geq\>n^{2}-3n+n^{2\gamma+1}-n^{2\gamma}-\sum_{k=2}^{n}k^{2\gamma}$
For $\gamma=0.5$, this evaluates to
$2n^{2}-4n-(\frac{n^{2}-n}{2}-1)\>=\>\frac{3}{2}n^{2}-\frac{7}{2}n+1$. ∎
### 3.2 Volume Scaling
In [16], Maritan et al. make an analytical prediction for the scaling of
volume with area in rivers. Their analysis depends on the relationship
$\langle L_{x}\rangle\propto A_{x}^{h}$, where $\langle L_{x}\rangle$ is the
mean distance from vertices upstream of $x$ to $x$, and $h$ is Hack’s
exponent, from Hack’s law (a power-law relationship between basin length and
area) [11]. This is combined with the equation $V_{x}=A_{x}\langle
L_{x}\rangle$ to arrive at the relationship $V_{x}\propto A_{x}^{1+h}$. This
relationship is verified by their data on real river networks as well as OCNs,
where (for both of which) $h$ is typically close to $0.57$ [16].
If $h$ were to equal exactly $0.5$, this would imply isometric scaling of
length and area in river basins, as $A_{x}$ would scale directly with
$l_{x}^{2}$, and shape would be preserved. This would line up with the
prediction in [3] that $V\propto A_{x}^{{D+1}/D}$ in the most efficient
networks for $D=2$. However, since river basins tend to elongate with growth,
getting proportionally narrower as they get larger, $h$ is usually slightly
greater than $0.5$, and the lower bound predicted in [3] is rarely reached.
Shifting focus to three dimensions, we can look for evidence of a
corresponding Hack’s law for three dimensional OCNs (that is, evidence that
area scales with length to some constant power). If there is a relatively
constant scaling exponent $h$, the analysis in [16] can be extended to three
dimensions, since the steps taken there do not otherwise depend on dimension.
Thus, we might expect to see $V_{x}\propto A_{x}^{1+h}$ for some $h$;
according to the analysis in [3], $V_{x}$ should scale at least as
$A_{x}^{4/3}$, so we might expect $1/3$ to be a lower bound for $h$, with $h$
slightly greater than $1/3$ if three dimensional OCNs elongate in a similar
way to two dimensional ones. These predictions are borne out in the simulation
results that follow.
## 4 Experimental Results
### 4.1 Energy Scaling
Figure 7: Log-log plot of area vs. energy for 2-dimensional OCNs, showing both
analytical lower bound (lower, blue) and observed values (upper, red).
Fig. 7 shows how empirically observed minimum possible energy scales with area
in two-dimensional OCNs. The estimated line of best fit is shown, as well as
the analytical lower bound derived in Section 3. The observed values of energy
follow a power law with a small exponent, while the lower bound is essentially
linear with respect to area.
### 4.2 Length Scaling
Scaling exponents $h$ where $l\propto A^{h}$ in two and three dimensions is
shown in Table 1. The observed values of the exponents for two dimensions are
within the bounds found in other studies [11, 16]. For three dimensions, the
exponent is also fairly consistent, though less so than in two dimensions. It
also deviates slightly more from the isometric value of 1/3 than the two
dimensional version does from 1/2. The fact that it is higher than 1/3 fulfils
the hypothesis that three-dimensional basins elongate like two-dimensional
ones.
basins analyzed | $h$
---|---
2d whole basins | 0.6066895125
2d all subbasins | 0.5808520176
3d whole basins | 0.3645961454
3d all subbasins | 0.4230582779
Table 1: Length scaling exponents
### 4.3 Volume Scaling
Fig. 8 shows the scaling of volume with area in two-dimensional OCNs, for
whole networks and for all subbasins. Note that the exponents are very
similar, indicating that the scaling behavior is the same within basins as
across different sizes of basins. The exponents are both close to 1.57, as in
[16].
(a)
(b)
Figure 8: Volume plotted with area for whole nxn basins (8(a)) and all
subbasins (8(b)) of 2-dimensional OCNs.
(a) Whole nxn basins
(b) all subbasins
(c) subbasins of 10x10x10 OCNs
(d) all subbasins with Area $\leq$ 1000
Figure 9: Volume plotted with area for 3-dimensional OCNs. basins analyzed | $\alpha$
---|---
whole networks | 1.386
all subbasins | 1.424
all subbasins of 20x20x20 network | 1.437
subbasins with area $>$ 1000 | 1.3776
subbasins of 20x20x20 network with area $>$ 1000 | 1.342
all subbasins with area $\leq$ 1000 | 1.44456
subbasins of 20x20x20 network with area $\leq$ 100 | 1.457
subbasins of 10x10x10 network | 1.4506
Table 2: Volume scaling exponents for three-dimensional OCNs
The three-dimensional results for volume scaling are displayed in Fig. 9 and
summarized in Table 2. The differences in exponents between the different data
sets show higher scaling behavior in the sets of smaller basins; the
difference between the exponent for whole basins (9(a)) and all subbasins
(9(b)) is due to the greater concentration of smaller basins in the set of all
subbasins, skewing the slope. Note that the exponents in 9(c) and 9(d) are
similar, showing that scaling behavior in the subbasins is the same for the
same distributions of area independent of the size of the encompassing whole
basin. In Section 3 we predicted that $\alpha$ would equal $1+h$, using the
analysis in [16], and this relationship is clear in Tables 1 and 2.
The exponent $\alpha$ ranges from $\approx$1.34 to $\approx$1.46, with a
higher exponent for collections of smaller basins. These results are not
inconsistent with the data for metabolic allometry in biological organisms,
where a power-law fit to data from a wide range of sizes approximates that
metabolic rate scales with mass to the power of 0.70 ($1/\alpha$) [25]. Not
only is this value within the range found in the OCNs, but in organisms as
well it is found that sets of smaller organisms yield a higher exponent than
sets of larger ones [14, 24].
## 5 Alternative Models
### 5.1 Steiner Tree Model
In addition to studying OCNs, we took a preliminary look at comparing OCNs
with Steiner trees.
Steiner trees are minimum-weight trees that connect a given subset of nodes in
a graph. The specified nodes are called terminals. Steiner trees may include
non-terminal nodes; these are called Steiner points (or Steiner nodes).
In [6], Steiner trees built on a grid are considered. The underlying graph is
all points in a Euclidean plane, completely connected, and the terminals are
the points of a grid. We began comparing Steiner trees with OCNs by looking at
the relative optimality of OCNs minimizing $H_{0.5}(s)$ and normal Steiner
trees on grids of the same sizes.
In order to evaluate the energy of a Steiner tree, the tree needs to be
directed; this can be done by choosing the outlet to be an arbitrary corner
node and directing all the edges towards the outlet.
For the purposes of this study area was defined as the number of terminal
nodes in the subtree rooted at a given node. This way, the terminal nodes have
exactly one unit of area associated with each of them, just as the grid nodes
do in the OCN.
The OCNs here have realistic link lengths, where diagonal links are length
$\sqrt{2}$, to make them comparable to the Steiner trees.
OCNs $s$ of size 2x2, 3x3, 4x4, and 5x5 were constructed and $H_{\gamma}(s)$
for each were compared with $H_{\gamma}(t)$ for Steiner trees $t$ of the same
size. The Steiner trees used were found in [6]. Figure 10 depicts the pairs of
trees that were compared. On the left of each pair is the Steiner tree, with
open circles for the Steiner nodes, and on the right of each is the
corresponding OCN.
(a) 2x2
(b) 3x3
(c) 4x4
(d) 5x5
Figure 10: Steiner trees and OCNs for different nxn grids.
Since Steiner trees minimize total link length, it was clear that they would
always have lowest energy for $\gamma=0$. For larger values of gamma, however,
where area becomes a bigger factor than link length, we hypothesized that the
OCNs would have a lower energy value than the Steiner trees. Finding whether
this was true, and where the crossover point would be, was the goal of the
comparisons.
### 5.2 Analytic Results
(a) 2x2
(b) 3x3
(c) 4x4
(d) 5x5
Figure 11: Plots of $H_{\gamma}(s)$ (blue) and $H_{\gamma}(t)$ (red), where
$s$ is an OCN and $t$ a Steiner tree of the given size. In each plot, the
horizontal axis is $\gamma$ and the vertical axis is energy.
Figure 11 shows plots of the energy for corresponding Steiner trees and OCNs,
$0\leq\gamma\leq 1$. As expected, the Steiner trees have lower energy than the
OCNs for smaller $\gamma$, while for $\gamma$ past a certain crossover point
the OCNs are better. Table 3 lists the crossover values of $\gamma$ for each
size.
size | crossover $\gamma$
---|---
2x2 | 0.55433153188825812933
3x3 | 0.63036154802337965422
4x4 | 0.01907039679465522977
5x5 | 0.32264708810531986705
Table 3: Crossover values of $\gamma$
### 5.3 Interpretation
From the sizes studied, it is not possible to detect a trend in crossover
points.
The next step would be to combine the two models, constructing Steiner-type
trees that minimize $H_{\gamma}(s)$ for varying $\gamma$ and comparing them
with real river networks and with normal OCNs.
## 6 Conclusion
We have looked at three dimensional OCNs as they compare with metabolite
distribution networks in organisms. In the organism context, volume is
interpreted as volume of blood (which is directly proportional to the mass or
volume of an organism) and area is proportional to the number of capillaries
[25]. In this context, the grid used to create OCNs no longer represents fixed
space (as it does in the two-dimensional/river context) and the amount of
physical area directly associated with each node may not be the same over
different body masses. This would only have a linear effect on the
relationship between area and mass, however, so this has no effect on the
scaling exponent.
Though the three-dimensional networks do not actually look like the metabolic
networks in organisms, we have shown that they have similar area/volume
scaling behavior. This adds credence to the idea that the observed metabolic
scaling is a result of characteristics of the distribution network.
Another way to study the possible similarities between OCNs and metabolic
networks would be to derive an energy functional for three dimensional OCNs
from metabolic networks themselves, rather than using the one derived from
rivers. This could also include consideration of heterogeneity. Heterogeneity
in the river context is addressed in [23] and [8]; possibly this work could be
extended to three dimensions.
## 7 Acknowledgments
The first author was supported by a Graduate Fellowship from Rensselaer
Polytechnic Institute during the completion of this work.
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* [24] V. M. Savage, J. F. Gillooly, W. H. Woodruff, G. B. West, A. P. Allen, B. J. Enquist, and J. H. Brown. The predominance of quarter-power scaling in biology. Functional Ecology, 18(2):257–282, 2004.
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|
arxiv-papers
| 2012-10-29T06:03:08 |
2024-09-04T02:49:37.223093
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lily Briggs, Mukkai Krishnamoorthy",
"submitter": "Lily Briggs",
"url": "https://arxiv.org/abs/1210.7572"
}
|
1210.7656
|
# Efficient rounding for the noncommutative
Grothendieck inequality
Assaf Naor Courant Institute, New York University, 251 Mercer Street, New York
NY 10012, USA. Supported by NSF grant CCF-0832795, BSF grant 2010021, the
Packard Foundation and the Simons Foundation. Part of this work was completed
while A. N. was visiting Université de Paris Est Marne-la-Vallée. Oded Regev
Courant Institute, New York University. Supported by a European Research
Council (ERC) Starting Grant. Part of the work done while the author was with
the CNRS, DI, ENS, Paris. Thomas Vidick Computer Science and Artificial
Intelligence Laboratory, Massachusetts Institute of Technology. Supported by
the National Science Foundation under Grant No. 0844626.
###### Abstract
The classical Grothendieck inequality has applications to the design of
approximation algorithms for NP-hard optimization problems. We show that an
algorithmic interpretation may also be given for a _noncommutative_
generalization of the Grothendieck inequality due to Pisier and Haagerup. Our
main result, an efficient rounding procedure for this inequality, leads to a
constant-factor polynomial time approximation algorithm for an optimization
problem which generalizes the Cut Norm problem of Frieze and Kannan, and is
shown here to have additional applications to robust principle component
analysis and the orthogonal Procrustes problem.
###### Contents
1. 1 Introduction
1. 1.1 Applications of Theorem 1
1. 1.1.1 The Grothendieck problem
2. 1.1.2 Robust PCA
3. 1.1.3 The orthogonal Procrustes problem
4. 1.1.4 A Frieze-Kannan decomposition for $4$-tensors
5. 1.1.5 Quantum XOR games
2. 1.2 The noncommutative Grothendieck inequality
1. 1.2.1 The complex case
2. 1.2.2 The rounding algorithm
3. 1.2.3 An intuitive description of the rounding procedure in the commutative case
2. 2 Proof of Theorem 4
1. 2.1 Analysis of the rounding procedure
2. 2.2 Derandomized rounding
3. 2.3 The rounding procedure in the case of (18)
3. 3 The real and Hermitian cases
1. 3.1 Two-dimensional rounding
2. 3.2 Rounding in the Hermitian case
3. 3.3 Proof of Theorem 10
4. 4 Direct rounding in the real case
5. 5 Some applications
1. 5.1 Constant-factor algorithm for robust PCA problems
2. 5.2 A constant-factor algorithm for the orthogonal Procrustes problem
3. 5.3 An algorithmic noncommutative dense regularity lemma
## 1 Introduction
In what follows, the standard scalar product on $\mathbb{C}^{n}$ is denoted
$\langle\cdot,\cdot\rangle$, i.e, $\langle
x,y\rangle=\sum_{i=1}^{n}x_{i}\overline{y_{i}}$ for all
$x,y\in\mathbb{C}^{n}$. We always think of $\mathbb{R}^{n}$ as canonically
embedded in $\mathbb{C}^{n}$; in particular the restriction of
$\langle\cdot,\cdot\rangle$ to $\mathbb{R}^{n}$ is the standard scalar product
on $\mathbb{R}^{n}$. Given a set $S$, the space $M_{n}(S)$ stands for all the
matrices $M=(M_{ij})_{i,j=1}^{n}$ with $M_{ij}\in S$ for all
$i,j\in\\{1,\ldots,n\\}$. Thus, $M_{n}(M_{n}(\mathbb{R}))$ is naturally
identified with the $n^{4}$-dimensional space of all $4$-tensors
$M=(M_{ijkl})_{i,j,k,l=1}^{n}$ with $M_{ijkl}\in\mathbb{R}$ for all
$i,j,k,l\in\\{1,\ldots,n\\}$. The set of all $n\times n$ orthogonal matrices
is denoted $\mathcal{O}_{n}\subseteq M_{n}(\mathbb{R})$, and the set of all
$n\times n$ unitary matrices is denoted $\mathcal{U}_{n}\subseteq
M_{n}(\mathbb{C})$.
Given $M=(M_{ijkl})\in M_{n}(M_{n}(\mathbb{R}))$ denote
$\mathrm{Opt}_{\mathbb{R}}(M)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{U,V\in\mathcal{O}_{n}}\sum_{i,j,k,l=1}^{n}M_{ijkl}U_{ij}V_{kl}.$
and similarly, for $M=(M_{ijkl})\in M_{n}(M_{n}(\mathbb{C}))$ denote
$\mathrm{Opt}_{\mathbb{C}}(M)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{U,V\in\mathcal{U}_{n}}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}U_{ij}\overline{V_{kl}}\Big{|}.$
###### Theorem 1.
There exists a polynomial time algorithm that takes as input $M\in
M_{n}(M_{n}(\mathbb{R}))$ and outputs $U,V\in\mathcal{O}_{n}$ such that
$\mathrm{Opt}_{\mathbb{R}}(M)\leqslant
O(1)\sum_{i,j,k,l=1}^{n}M_{ijkl}U_{ij}V_{kl}.$
Respectively, there exists a polynomial time algorithm that takes as input
$M\in M_{n}(M_{n}(\mathbb{C}))$ and outputs $U,V\in\mathcal{U}_{n}$ such that
$\mathrm{Opt}_{\mathbb{C}}(M)\leqslant
O(1)\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}U_{ij}\overline{V_{kl}}\Big{|}.$
We will explain the ideas that go into the proof of Theorem 1 later, and it
suffices to say at this juncture that our algorithm is based on a rounding
procedure for semidefinite programs that is markedly different from rounding
algorithms that have been previously used in the optimization literature, and
as such it indicates the availability of techniques that have thus far
remained untapped for the purpose of algorithm design. Prior to explaining the
proof of Theorem 1 we list below some of its applications as an indication of
its usefulness.
###### Remark 2.
The implied constants in the $O(1)$ terms of Theorem 1 can be taken to be any
number greater than $2\sqrt{2}$ in the real case, and any number greater than
$2$ in the complex case. There is no reason to believe that the factor
$2\sqrt{2}$ in the real case is optimal, but the factor $2$ in the complex
case is sharp in a certain natural sense that will become clear later. The
main content of Theorem 1 is the availability of a constant factor algorithm
rather than the value of the constant itself. In particular, the novelty of
the applications to combinatorial optimization that are described below is the
mere existence of a constant-factor approximation algorithm.
### 1.1 Applications of Theorem 1
We now describe some examples demonstrating the usefulness of Theorem 1. The
first example does not lead to a new result, and is meant to put Theorem 1 in
context. All the other examples lead to new algorithmic results. Many of the
applications below follow from a more versatile reformulation of Theorem 1
that is presented in Section 5 (see Proposition 20).
#### 1.1.1 The Grothendieck problem
The Grothendieck optimization problem takes as input a matrix $A\in
M_{n}(\mathbb{R})$ and aims to efficiently compute (or estimate) the quantity
$\max_{\varepsilon,\delta\in\\{-1,1\\}^{n}}\sum_{i,j=1}^{n}A_{ij}\varepsilon_{i}\delta_{j}.$
(1)
This problem falls into the framework of Theorem 1 by considering the
$4$-tensor $M\in M_{n}(M_{n}(\mathbb{R}))$ given by $M_{iijj}=A_{ij}$ and
$M_{ijkl}=0$ if either $i\neq j$ or $k\neq l$. Indeed,
$\mathrm{Opt}_{\mathbb{R}}(M)=\max_{U,V\in\mathcal{O}_{n}}\sum_{i,j=1}^{n}A_{ij}U_{ii}V_{jj}=\max_{x,y\in[-1,1]^{n}}\sum_{i,j=1}^{n}A_{ij}x_{i}y_{j}=\max_{\varepsilon,\delta\in\\{-1,1\\}^{n}}\sum_{i,j=1}^{n}A_{ij}\varepsilon_{i}\delta_{j}.$
A constant-factor polynomial time approximation algorithm for the Grothendieck
problem was designed in [AN04], where it was also shown that it is NP-hard to
approximate this problem within a factor less that $1+\varepsilon_{0}$ for
some $\varepsilon_{0}\in(0,1)$. A simple transformation [AN04] relates the
Grothendieck problem to the Frieze-Kannan Cut Norm problem [FK99] (this
transformation can be made to have no loss in the approximation guarantee
[KN12, Sec. 2.1]), and as such the constant-factor approximation algorithm for
the Grothendieck problem has found a variety of applications in combinatorial
optimization; see the survey [KN12] for much more on this topic. In another
direction, based on important work of Tsirelson [Tsi87], the Grothendieck
problem has found applications to quantum information theory [CHTW04]. Since
the problem of computing $\mathrm{Opt}_{\mathbb{R}}(\cdot)$ contains the
Grothendieck problem as a special case, Theorem 1 encompasses all of these
applications, albeit with the approximation factor being a larger constant.
#### 1.1.2 Robust PCA
The input to the classical principal component analysis (PCA) problem is
$K,n\in\mathbb{N}$ a set of points $a_{1},\ldots,a_{N}\in\mathbb{R}^{n}$. The
goal is to find a $K$-dimensional subspace maximizing the sum of the squared
$\ell_{2}$ norms of the projections of the $a_{i}$ on the subspace.
Equivalently, the problem is to find the maximizing vectors in
$\max_{\begin{subarray}{c}y_{1},\ldots,y_{K}\in\mathbb{R}^{n}\\\ \langle
y_{i},y_{j}\rangle=\delta_{ij}\end{subarray}}\,\sum_{i=1}^{N}\sum_{j=1}^{K}\,|\langle
a_{i},y_{j}\rangle|^{2},$ (2)
where here, and in what follows, $\delta_{ij}$ is the Kronecker delta. This
question has a closed-form solution in terms of the singular values of the
$N\times n$ matrix whose $i$-th row contains the coefficients of the point
$a_{i}$.
The fact that the quantity appearing in (2) is the maximum of the sum of the
_squared_ norms of the projected points makes it somewhat non-robust to
outliers, in the sense that a single long vector can have a large effect on
the maximum. Several more robust versions of PCA were suggested in the
literature. One variant, known as “R1-PCA,” is due to Ding, Zhou, He, and Zha
[DZHZ06], and aims to maximize the sum of the Euclidean norms of the projected
points, namely,
$\max_{\begin{subarray}{c}y_{1},\ldots,y_{K}\in\mathbb{R}^{n}\\\ \langle
y_{i},y_{j}\rangle=\delta_{ij}\end{subarray}}\,\sum_{i=1}^{N}\Big{(}\sum_{j=1}^{K}\,|\langle
a_{i},y_{j}\rangle|^{2}\Big{)}^{1/2}.$ (3)
We are not aware of any prior efficient algorithm for this problem that
achieves a guaranteed approximation factor. Another robust variant of PCA,
known as “L1-PCA”, was suggested by Kwak [Kwa08], and further studied by McCoy
and Tropp [MT12] (see Section 2.7 in [MT12] in particular). Here the goal is
to maximize the sum of the $\ell_{1}$ norms of the projected points, namely,
$\max_{\begin{subarray}{c}y_{1},\ldots,y_{K}\in\mathbb{R}^{n}\\\ \langle
y_{i},y_{j}\rangle=\delta_{ij}\end{subarray}}\,\sum_{i=1}^{N}\sum_{j=1}^{K}\,|\langle
a_{i},y_{j}\rangle|.$ (4)
In [MT12] a constant factor approximation algorithm for the above problem is
obtained for $K=1$ based on [AN04], and for general $K$ an approximation
algorithm with an approximation guarantee of $O(\log n)$ is obtained based on
prior work by So [So11].
In Section 5.1 we show that both of the above robust versions of PCA can be
cast as special cases of Theorem 1, thus yielding constant-factor
approximation algorithms for both problems and all $K\in\\{1,\ldots,n\\}$.
#### 1.1.3 The orthogonal Procrustes problem
Let $n,d\geqslant 1$ and $K\geqslant 2$ be integers. Suppose that
$S_{1},\ldots,S_{K}\subseteq\mathbb{R}^{d}$ are $n$-point subsets of
$\mathbb{R}^{d}$. The goal of the _generalized orthogonal Procrustes problem_
is to rotate each of the $S_{k}$ separately so as to best align them.
Formally, write $S_{k}=\\{x^{k}_{1},x_{2}^{k},\ldots,x_{n}^{k}\\}$. The goal
is to find $K$ orthogonal matrices $U_{1},\ldots,U_{K}\in\mathcal{O}_{d}$ that
maximize the quantity
$\sum_{i=1}^{n}\Big{\|}\sum_{k=1}^{K}U_{k}x^{k}_{i}\Big{\|}_{2}^{2}.$ (5)
If one focuses on a single summand appearing in (5), say
$\sum_{k=1}^{K}U_{k}x^{k}_{1}$, then it is clear that in order to maximize its
length one would want to rotate each of the $x_{1}^{k}$ so that they would all
point in the same direction, i.e., they would all be positive multiples of the
same vector. The above problem aims to achieve the best possible such
alignment (in aggregate) for multiple summands of this type. We note that by
expanding the squares one sees that $U_{1},\ldots,U_{K}\in\mathcal{O}_{d}$
maximize the quantity appearing in (5) if and only if they minimize the
quantity
$\sum_{i=1}^{n}\sum_{k,l=1}^{K}\|U_{k}x^{k}_{i}-U_{l}x^{l}_{i}\|_{2}^{2}$.
The term “generalized” was used above because the _orthogonal Procrustes
problem_ refers to the case $K=2$, which has a closed-form solution. The name
“Procustes” is a (macabre) reference to Greek mythology (see
http://en.wikipedia.org/wiki/Procrustes).
The generalized orthogonal Procrustes problem has been extensively studied
since the 1970s, initially in the psychometric literature (see, e.g., [CC71,
Gow75, TB77]), and more recent applications of it are to areas such as image
and shape analysis, market research and biometric identification; see the
books [GD04, DM98], the lecture notes [SG02], and [MP07] for much more
information on this topic.
The generalized orthogonal Procrustes problem is known to be intractable, and
it has been investigated algorithmically in, e.g., [TB77, Bou82, SB88]. A
rigorous analysis of a polynomial-time approximation algorithm for this
problem appears in the work of Nemirovski [Nem07], where the generalized
orthogonal Procrustes problem is treated as an important special case of a
more general family of problems called “quadratic optimization under
orthogonality constraints”, for which he obtains a $O(\sqrt[3]{n+d}+\log K)$
approximation algorithm. This was subsequently improved by So [So11] to
$O(\log(n+d+K))$. In Section 5.2 we use Theorem 1 to improve the approximation
guarantee for the generalized orthogonal Procrustes problem as defined above
to a constant approximation factor. See also Section 5.2 for a more complete
discussion of variants of this problem considered in [Nem07, So11] and how
they compare to our work.
#### 1.1.4 A Frieze-Kannan decomposition for $4$-tensors
In [FK99] Frieze and Kannan designed an algorithm which decomposes every
(appropriately defined) “dense” matrix into a sum of a few “cut matrices” plus
an error matrix that has small cut-norm. We refer to [FK99] and also Section
2.1.2 in the survey [KN12] for a precise formulation of this statement, as
well as its extension, due to [AN04], to an algorithm that allows sub-constant
errors. In Section 5.3 we apply Theorem 1 to prove the following result, which
can be viewed as a noncommutative variant of the Frieze-Kannan decomposition.
For the purpose of the statement below it is convenient to identify the space
$M_{n}(M_{n}(\mathbb{C}))$ of all $4$-tensors with $M_{n}(\mathbb{C})\otimes
M_{n}(\mathbb{C})$. Also, for $M\in M_{n}(\mathbb{C})\otimes
M_{n}(\mathbb{C})$ we denote from now on its Frobenius (Hilbert-Schmidt) norm
by
$\|M\|_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sqrt{\sum_{i,j,k,l=1}^{n}|M_{ijkl}|^{2}}.$
###### Theorem 3.
There exists a universal constant $c\in(0,\infty)$ with the following
property. Suppose that $M\in M_{n}(\mathbb{C})\otimes M_{n}(\mathbb{C})$ and
$0<\varepsilon\leqslant 1/2$, and let
$T\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\lceil\frac{cn^{2}\|M\|_{2}^{2}}{\varepsilon^{2}\mathrm{Opt}_{\mathbb{C}}(M)^{2}}\right\rceil.$
(6)
One can compute in time ${\mathrm{poly}}(n,1/\varepsilon)$ a decomposition
$M\,=\,\sum_{t=1}^{T}\,\alpha_{t}(A_{t}\otimes B_{t})+E,$ (7)
such that $A_{t},B_{t}\in\mathcal{U}_{n}$, the coefficients
$\alpha_{t}\in\mathbb{C}$ satisfy $|\alpha_{t}|=O(\|M\|_{2}/n)$, and
$\mathrm{Opt}_{\mathbb{C}}(E)\leqslant\varepsilon\mathrm{Opt}_{\mathbb{C}}(M)$.
Moreover, if $M\in M_{n}(\mathbb{R})\otimes M_{n}(\mathbb{R})$ then one can
replace $\mathrm{Opt}_{\mathbb{C}}(M)$ in (6) by
$\mathrm{Opt}_{\mathbb{R}}(M)$, take the coefficients $\alpha_{t}$ to be real,
$A_{t},B_{t}\in\mathcal{O}_{n}$ and $E$ such that
$\mathrm{Opt}_{\mathbb{R}}(E)\leqslant\varepsilon\mathrm{Opt}_{\mathbb{R}}(M)$.
Theorem 3 contains as a special case its commutative counterpart, as studied
in [FK99, AN04]. Here we are given $A\in M_{n}(\mathbb{R})$ with
$|a_{ij}|\leqslant 1$ for all $i,j\in\\{1,\ldots,n\\}$, and we aim for an
error $\varepsilon n^{2}$. Define $M_{iijj}=a_{ij}$ and $M_{ijkl}=0$ if $i\neq
j$ or $k\neq l$. Then $\|M\|_{2}\leqslant n$. An application of Theorem 3 (in
the real case) with $\varepsilon$ replaced by $\varepsilon
n^{2}/\mathrm{Opt}_{\mathbb{R}}(M)$ yields a decomposition
$A=\sum_{t=1}^{T}\alpha_{t}(a_{t}b_{t}^{*})+E$ with $a_{t},b_{t}\in[-1,1]^{n}$
and $E\in M_{n}(\mathbb{R})$ satisfying
$\sup_{\varepsilon,\delta\in\\{-1,1\\}}\sum_{ij=1}^{n}E_{ij}\varepsilon_{i}\delta_{j}\leqslant\varepsilon
n^{2}$. Moreover, the number of terms is $T=O(1/\varepsilon^{2})$.
Theorem 3 is proved in Section 5.3 via an iterative application of Theorem 1,
following the “energy decrement” strategy as formulated by Lovász and Szegedy
[LS07] in the context of general weak regularity lemmas. Other than being a
structural statement of interest in its own right, we show in Section 5.3 that
Theorem 3 can be used to enhance the constant factor approximation of Theorem
1 to a PTAS for computing $\mathrm{Opt}_{\mathbb{C}}(M)$ when
$\mathrm{Opt}_{\mathbb{C}}(M)=\Omega(n\|M\|_{2})$. Specifically, if
$\mathrm{Opt}_{\mathbb{C}}(M)\geqslant\kappa n\|M\|_{2}$ then one can compute
a $(1+\varepsilon)$-factor approximation to $\mathrm{Opt}_{\mathbb{C}}(M)$ in
time $2^{{\mathrm{poly}}(1/(\kappa\varepsilon))}{\mathrm{poly}}(n)$. This is
reminiscent of the Frieze-Kannan algorithmic framework [FK99] for dense graph
and matrix problems.
#### 1.1.5 Quantum XOR games
As we already noted, the Grothendieck problem (recall Section 1.1.1) also has
consequences in quantum information theory [CHTW04], and more specifically to
bounding the power of entanglement in so-called “XOR games”, which are two-
player one-round games in which the players each answer with a bit and the
referee bases her decision on the XOR of the two bits. As will be explained in
detail in Section 1.2 below, the literature on the Grothendieck problem relies
on a classical inequality of Grothendieck [Gro53], while our work relies on a
more recent yet by now classical noncommutative Grothendieck inequality of
Pisier [Pis78] (and its sharp form due to Haagerup [Haa85]). Even more
recently, the Grothendieck inequality has been generalized to another setting,
that of _completely bounded_ linear maps defined on operator spaces [PS02,
HM08]. While we do not discuss the operator space Grothendieck inequality
here, we remark that in [RV12a] the operator space Grothendieck inequality is
proved by reducing it to the Pisier-Haagerup noncommutative Grothendieck
inequality. Without going into details, we note that this reduction is also
algorithmic. Combined with our results, it leads to an algorithmic proof of
the operator space Grothendieck inequality, together with an accompanying
rounding procedure.
In the preprint [RV12b] written by the last two named authors, the
noncommutative and operator space Grothendieck inequalities are shown to have
consequences in a setting that generalizes that of classical XOR games, called
“quantum XOR games”: in such games, the questions to the players may be
quantum states (and the answers are still a single classical bit). The results
in [RV12b] derive an efficient factor-$2$ approximation algorithm for the
maximum success probability of players in such a game, in three settings:
players sharing an arbitrary quantum state, players sharing a maximally
entangled state, and players not sharing any entanglement. Theorem 1 implies
that in all three cases a good strategy for the players, achieving a success
that is a factor $2$ from optimal, may be found in polynomial time. These
matters are taken up in [RV12b] and will not be discussed further here.
### 1.2 The noncommutative Grothendieck inequality
The natural semidefinite relaxation of (1) is
$\sup_{d\in\mathbb{N}}\sup_{x,y\in(S^{d-1})^{n}}\sum_{i,j=1}^{n}A_{ij}\langle
x_{i},y_{j}\rangle,$ (8)
where $S^{d-1}$ is the unit sphere of $\mathbb{R}^{d}$. Since, being a
semidefinite program (SDP), the quantity appearing in (8) can be computed in
polynomial time with arbitrarily good precision (see [GLS93]), the fact that
the Grothendieck optimization problem admits a constant-factor polynomial time
approximation algorithm follows from the following inequality, which is a
classical inequality of Grothendieck of major importance to several
mathematical disciplines (see Pisier’s survey [Pis12] and the references
therein for much more on this topic; the formulation of the inequality as
below is due to Lindenstrauss and Pełczyński [LP68]).
$\sup_{d\in\mathbb{N}}\sup_{x,y\in(S^{d-1})^{n}}\sum_{i,j=1}^{n}A_{ij}\langle
x_{i},y_{j}\rangle\leqslant
K_{G}\sup_{\varepsilon,\delta\in\\{-1,1\\}^{n}}\sum_{i,j=1}^{n}A_{ij}\varepsilon_{i}\delta_{j}.$
(9)
Here $K_{G}\in(0,\infty)$, which is understood to be the infimum over those
constants for which (9) holds true for all $n\in\mathbb{N}$ and all $A\in
M_{n}(\mathbb{R})$, is a universal constant known as the (real) Grothendieck
constant. Its exact value remains unknown, the best available bounds [Ree91,
BMMN11] being $1.676<K_{G}<1.783$. In order to actually find an assignment
$\varepsilon,\delta$ to (1) that is within a constant factor of the optimum
one needs to argue that a proof of (9) can be turned into an efficient
rounding algorithm; this is done in [AN04].
If one wishes to mimic the above algorithmic success of the Grothendieck
inequality in the context of efficient computation of
$\mathrm{Opt}_{\mathbb{R}}(\cdot)$, the following natural strategy presents
itself: one should replace real entries of matrices by vectors in $\ell_{2}$,
i.e., consider elements of $M_{n}(\ell_{2})$, and replace the orthogonality
constraints underlying the inclusion $U\in\mathcal{O}_{n}$, namely,
$\forall\,i,j\in\\{1,\ldots,n\\},\quad\sum_{k=1}^{n}U_{ik}U_{jk}=\sum_{k=1}^{n}U_{ki}U_{kj}=\delta_{ij},$
by the corresponding constraints using scalar product. Specifically, given an
$n\times n$ vector-valued matrix $X\in M_{n}(\ell_{2})$ define two real
matrices $XX^{*},X^{*}X\in M_{n}(\mathbb{R})$ by
$\forall\,i,j\in\\{1,\ldots,n\\},\quad(XX^{*})_{ij}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{k=1}^{n}\langle
X_{ik},X_{jk}\rangle,\quad\mathrm{and}\quad(X^{*}X)_{ij}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{k=1}^{n}\langle
X_{ki},X_{kj}\rangle,$ (10)
and let the set of $d$-dimensional vector-valued orthogonal matrices be given
by
$\mathcal{O}_{n}(\mathbb{R}^{d})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{X\in
M_{n}(\mathbb{R}^{d}):\ XX^{*}=X^{*}X=I\right\\}.$ (11)
One then considers the following quantity associated to every $M\in
M_{n}(M_{n}(\mathbb{R}))$,
$\mathrm{SDP}_{\mathbb{R}}(M)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{d\in\mathbb{N}}\sup_{X,Y\in\mathcal{O}_{n}(\mathbb{R}^{d})}\sum_{i,j,k,l=1}^{n}M_{ijkl}\left\langle
X_{ij},Y_{kl}\right\rangle.$ (12)
Since the constraints that underlie the inclusion
$X,Y\in\mathcal{O}_{n}(\mathbb{R}^{d})$ are linear equalities in the pairwise
scalar products of the entries of $X$ and $Y$, the quantity
$\mathrm{SDP}_{\mathbb{R}}(M)$ is a semidefinite program and can therefore be
computed in polynomial time with arbitrarily good precision. One would
therefore aim to prove the following noncommutative variant of the
Grothendieck inequality (9),
$\forall\,n\in\mathbb{N},\ \forall\,M\in
M_{n}(M_{n}(\mathbb{R})),\quad\mathrm{SDP}_{\mathbb{R}}(M)\leqslant
O(1)\cdot\mathrm{Opt}_{\mathbb{R}}(M).$ (13)
The term “noncommutative” refers here to the fact that
$\mathrm{Opt}_{\mathbb{R}}(M)$ is an optimization problem over the
noncommutative group $\mathcal{O}_{n}$, while the classical Grothendieck
inequality addresses an optimization problem over the commutative group
$\\{-1,1\\}^{n}$. In the same vein, noncommutativity is manifested by the fact
that the classical Grothendieck inequality corresponds to the special case of
“diagonal” $4$-tensors $M\in M_{n}(M_{n}(\mathbb{R}))$, i.e., those that
satisfy $M_{ijkl}=0$ whenever $i\neq j$ or $k\neq l$.
Grothendieck conjectured [Gro53] the validity of (13) in 1953, a conjecture
that remained open until its 1978 affirmative solution by Pisier [Pis78]. A
simpler, yet still highly nontrivial proof of the noncommutative Grothendieck
inequality (13) was obtained by Kaijser [Kai83]. In Section 4 we design a
rounding algorithm corresponding to (13) based on Kaijser’s approach. This
settles the case of real $4$-tensors of Theorem 1, albeit with worse
approximation guarantee than the one claimed in Remark 2. The algorithm
modeled on Kaijser’s proof is interesting in its own right, and seems to be
versatile and applicable to other problems, such as possible non-bipartite
extensions of the noncommutative Grothendieck inequality in the spirit of
[AMMN06]; we shall not pursue this direction here.
A better approximation guarantee, and arguably an even more striking rounding
algorithm, arises from the work of Haagerup [Haa85] on the complex version of
(13). In Section 3 we show how the real case of Theorem 1 follows formally
from our results on its complex counterpart, so from now on we focus our
attention on the complex case.
#### 1.2.1 The complex case
In what follows we let $S^{d-1}_{\mathbb{C}}$ denote the unit sphere of
$\mathbb{C}^{d}$ (thus $S^{0}_{\mathbb{C}}$ can be identified with the unit
circle $S^{1}\subseteq\mathbb{R}^{2}$). The classical complex Grothendieck
inequality [Gro53, LP68] asserts that there exists $K\in(0,\infty)$ such that
$\forall\,n\in\mathbb{N},\ \forall\,A\in
M_{n}(\mathbb{C}),\quad\sup_{x,y\in(S^{2n-1}_{\mathbb{C}})^{n}}\Big{|}\sum_{i,j=1}^{n}A_{ij}\langle
x_{i},y_{j}\rangle\Big{|}\leqslant
O(1)\sup_{\alpha,\beta\in(S_{\mathbb{C}}^{0})^{n}}\Big{|}\sum_{i,j=1}^{n}A_{ij}\alpha_{i}\overline{\beta_{j}}\Big{|}.$
(14)
Let $K_{G}^{\mathbb{C}}$ denote the infimum over those $K\in(0,\infty)$ for
which (14) holds true. The exact value of $K_{G}^{\mathbb{C}}$ remains
unknown, the best available bounds being $1.338<K_{G}^{\mathbb{C}}<1.4049$
(the left inequality is due to unpublished work of Davie, and the right one is
due to Haagerup [Haa87]).
For $M\in M_{n}(M_{n}(\mathbb{C}))$ we define
$\mathrm{SDP}_{\mathbb{C}}(M)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{d\in\mathbb{N}}\sup_{X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\left\langle
X_{ij},Y_{kl}\right\rangle\Big{|},$ (15)
where analogously to (11) we set
$\mathcal{U}_{n}(\mathbb{C}^{d})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{X\in
M_{n}(\mathbb{C}^{d}):\ XX^{*}=X^{*}X=I\right\\}.$
Here for $X\in M_{n}(\mathbb{C}^{d})$ the complex matrices $XX^{*},X^{*}X\in
M_{n}(\mathbb{C})$ are defined exactly as in (10), with the scalar product
being the complex scalar product. Haagerup proved [Haa85] that
$\forall\,n\in\mathbb{N},\ \forall\,M\in
M_{n}(M_{n}(\mathbb{C})),\quad\mathrm{SDP}_{\mathbb{C}}(M)\leqslant
2\cdot\mathrm{Opt}_{\mathbb{C}}(M).$ (16)
Our main algorithm is an efficient rounding scheme corresponding to inequality
(16). The constant $2$ in (16) is sharp, as shown in [HI95] (see also [Pis12,
Sec. 12]).
We note that the noncommutative Grothendieck inequality, as it usually appears
in the literature, involves a slightly more relaxed semidefinite program. In
order to describe it, we first remark that instead of maximizing over
$X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$ in (15) we could equivalently maximize
over $X,Y\in M_{n}(\mathbb{C}^{d})$ satisfying
$XX^{*},X^{*}X,YY^{*},Y^{*}Y\leqslant I$, which is the same as the requirement
$\|XX^{*}\|,\|X^{*}X\|,\|YY^{*}\|,\|Y^{*}Y\|\leqslant 1$, where here and in
what follows $\|\cdot\|$ denotes the operator norm of matrices. This fact is
made formal in Lemma 6 below. By relaxing the constraints to
$\|XX^{*}\|+\|X^{*}X\|\leqslant 2$ and $\|YY^{*}\|+\|Y^{*}Y\|\leqslant 2$, we
obtain the following quantity, which can be shown to still be a semidefinite
program.
$\|M\|_{nc}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{d\in\mathbb{N}}\sup_{\begin{subarray}{c}X,Y\in
M_{n}(\mathbb{C}^{d})\\\ \|XX^{*}\|+\|X^{*}X\|\leqslant 2\\\
\|YY^{*}\|+\|Y^{*}Y\|\leqslant
2\end{subarray}}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\left\langle
X_{ij},Y_{kl}\right\rangle\Big{|}.$ (17)
Clearly $\|M\|_{nc}\geqslant\mathrm{SDP}_{\mathbb{C}}(M)$ for all $M\in
M_{n}(M_{n}(\mathbb{C}))$. Haagerup proved [Haa85] that the following stronger
inequality holds true for all $n\in\mathbb{N}$ and $M\in
M_{n}(M_{n}(\mathbb{C}))$.
$\|M\|_{nc}\leqslant 2\cdot\mathrm{Opt}_{\mathbb{C}}(M).$ (18)
As our main focus is algorithmic, in the following discussion we will
establish a rounding algorithm for the tightest relaxation (16). In Section
2.3 we show that the same rounding procedure can be used to obtain an
algorithmic analogue of (18) as well.
#### 1.2.2 The rounding algorithm
Our main algorithm is an efficient rounding scheme corresponding to (16). In
order to describe it, we first introduce the following notation. Let
$\varphi:\mathbb{R}\to\mathbb{R}^{+}$ be given by
$\varphi(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{2}\text{sech}\Big{(}\frac{\pi}{2}t\Big{)}\,=\,\frac{1}{e^{\pi
t/2}+e^{-\pi t/2}}.$ (19)
One computes that $\int_{\mathbb{R}}\varphi(t)\text{d}t=1$, so $\varphi$ is a
density of a probability measure $\mu$ on $\mathbb{R}$, known as the
hyperbolic secant distribution. By [JKB95, Sec. 23.11] we have
$\forall\,a\in(0,\infty),\quad\int_{\mathbb{R}}a^{it}\varphi(t)\text{d}t=\frac{2a}{1+a^{2}}.$
(20)
It is possible to efficiently sample from $\mu$ using standard techniques;
see, e.g., [Dev86, Ch. IX.7].
In what follows, given $X\in M_{n}(\mathbb{C}^{d})$ and $z\in\mathbb{C}^{d}$
we denote by $\langle X,z\rangle\in M_{n}(\mathbb{C})$ the matrix whose
entries are $\langle X,z\rangle_{jk}=\langle X_{jk},z\rangle$.
Rounding procedure
1. 1.
Let $X,Y\in M_{n}(\mathbb{C}^{d})$ be given as input. Choose
$z\in\\{1,-1,i,-i\\}^{d}$ uniformly at random, and sample $t\in\mathbb{R}$
according to the hyperbolic secant distribution $\mu$.
2. 2.
Set $X_{z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{\sqrt{2}}\langle
X,z\rangle\in M_{n}(\mathbb{C})$ and
$Y_{z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{\sqrt{2}}\left\langle
Y,z\right\rangle\in M_{n}(\mathbb{C})$.
3. 3.
Output the pair of matrices
$(A,B)=(A(z,t),B(z,t))\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(U_{z}|X_{z}|^{it},V_{z}|Y_{z}|^{-it})\in\mathcal{U}_{n}\times\mathcal{U}_{n}$
where $X_{z}=U_{z}|X_{z}|$ and $Y_{z}=V_{z}|Y_{z}|$ are the polar
decompositions of $X_{{z}}$ and $Y_{{z}}$, respectively.
Figure 1: The rounding algorithm takes as input a pair of vector-valued
matrices $X,Y\in M_{n}(\mathbb{C}^{d})$. It outputs two matrices
$A,B\in\mathcal{U}_{n}(\mathbb{C})$.
###### Theorem 4.
Fix $n,d\in\mathbb{N}$ and $\varepsilon\in(0,1)$. Suppose that $M\in
M_{n}(M_{n}(\mathbb{C}))$ and that $X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$ are
such that
$\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\langle
X_{ij},Y_{kl}\rangle\Big{|}\,\geqslant\,(1-\varepsilon)\mathrm{SDP}_{\mathbb{C}}(M),$
(21)
where $\mathrm{SDP}_{\mathbb{C}}(M)$ is given in (15). Then the rounding
procedure described in Figure 1 outputs a pair of matrices
$A,B\in\mathcal{U}_{n}$ such that
$\mathbb{E}\Big{[}\,\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\,A_{ij}\overline{B_{kl}}\Big{|}\,\Big{]}\,\geqslant\,\Big{(}\frac{1}{2}-\varepsilon\Big{)}\mathrm{SDP}_{\mathbb{C}}(M).$
(22)
Moreover, rounding can be performed in time polynomial in $n$ and
$\log(1/\varepsilon)$, and can be derandomized in time
${\mathrm{poly}}(n,1/\varepsilon)$.
While the rounding procedure of Figure 1 and the proof of Theorem 4 (contained
in Section 2 below) appear to be different from Haagerup’s original proof of
(18) in [Haa85], we derived them using Haagerup’s ideas. One source of
difference arises from changes that we introduced in order to work with the
quantity $\mathrm{SDP}_{\mathbb{C}}(M)$, while Haagerup’s argument treats the
quantity $\|M\|_{nc}$. A second source of difference is that Haagerup’s proof
of (18) is rather indirect and nonconstructive, while it is crucial to the
algorithmic applications that were already mentioned in Section 1.1 for us to
formulate a polynomial-time rounding procedure. Specifically, Haagerup
establishes the _dual_ formulation of (18), through a repeated use of duality,
and he uses a bootstrapping argument that relies on nonconstructive tools from
complex analysis. The third step in Figure 1 originates from Haagerup’s
complex-analytic considerations. Readers who are accustomed to semidefinite
rounding techniques will immediately notice that this step is unusual; we give
intuition for it in Section 1.2.3 below, focusing for simplicity on applying
the rounding procedure to vectors rather than matrices (i.e., the more
familiar setting of the classical Grothendieck inequality).
#### 1.2.3 An intuitive description of the rounding procedure in the
commutative case
Consider the effect of the rounding procedure in the commutative case, i.e.,
when $X,Y\in M_{n}(\mathbb{C}^{d})$ are diagonal matrices. Let the diagonals
of $X,Y$ be $x,y\in(\mathbb{C}^{d})^{n}$, respectively. The first step
consists in performing a random projection: for $j\in\\{1,\ldots,n\\}$ let
$\alpha_{j}=\left\langle x_{j},z\right\rangle/\sqrt{2}\in\mathbb{C}$ and
$\beta_{j}=\left\langle y_{j},z\right\rangle/\sqrt{2}\in\mathbb{C}$, where $z$
is chosen uniformly at random from $\\{1,-1,i,-i\\}^{n}$ (alternatively, with
minor modifications to the proof one may choose i.i.d. $z_{j}$ uniformly from
the unit circle, as was done by Haagerup [Haa85], or use standard complex
Gaussians). This step results in sequences of complex numbers whose pairwise
products $\alpha_{k}\overline{\beta_{j}}$, in expectation, exactly reproduce
the pairwise scalar products $\langle x_{k},y_{j}\rangle$. However, in general
the resulting complex numbers $\alpha_{k}$ and $\beta_{j}$ may have modulus
larger than $1$. Extending the “sign” rounding performed in, say, the Goemans-
Williamson algorithm for MAXCUT [GW95] to the complex domain, one could then
round each $\alpha_{k}$ and $\beta_{j}$ independently by simply replacing them
by their respective complex phase.
The procedure that we consider differs from this standard practice by taking
into account potential information contained in the modulus of the random
complex numbers $\alpha_{k},\beta_{j}$. Writing in polar coordinates
$\alpha_{k}=r_{k}e^{i\theta_{k}}$ and $\beta_{j}=s_{j}e^{i\phi_{j}}$ we sample
a real $t$ according to a specific distribution (the hyperbolic secant
distribution $\mu$), and round each $\alpha_{k}$ and each $\beta_{j}$ to
$a_{k}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{i(\theta_{k}+t\log
r_{k})}\in S_{\mathbb{C}}^{0},\quad\mathrm{and}\quad
b_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{i(\phi_{j}-t\log s_{j})}\in
S_{\mathbb{C}}^{0},$
respectively. Observe that this step performs a _correlated_ rounding: the
parameter $t$ is the same for all $j,k\in\\{1,\ldots,n\\}$.
The proof presented in [Haa85] uses the maximum modulus principle to show the
_existence_ of a real $t$ for which $a_{k},b_{j}$ as defined above provide a
good assignment. Intuition for the existence of such a good $t$ can be given
as follows. Varying $t$ along the real line corresponds to rotating the phases
of the complex numbers $\alpha_{j},\beta_{k}$ at a speed proportional to the
logarithm of their modulus: elements with very small modulus vary very fast,
those with modulus $1$ are left unchanged, and elements with relatively large
modulus are again varied at (logarithmically) increasing speeds. This means
that the rounding procedure takes into account the fact that an element with
modulus away from $1$ is a “miss”: that particular element’s phase is probably
irrelevant, and should be changed. However, elements with modulus close to $1$
are “good”: their phase can be kept essentially unchanged.
We identify a specific distribution $\mu$ such that a random $t$ distributed
according to $\mu$ is good, in expectation. This results in a variation on the
usual “sign” rounding technique: instead of directly keeping the phases
obtained in the initial step of random projection, they are synchronously
rotated for a random time $t$, at speeds depending on the associated moduli,
resulting in a provably good pair of sequences $a_{k},b_{j}$ of complex
numbers with modulus $1$.
##### Roadmap:
In Section 2 we prove Theorem 4 both as stated in Section 1.2.2 and in a form
based on (18). The real case as well as a closely related Hermitian case are
treated next, first in Section 3 as a corollary of Theorem 4, and then using
an alternative direct rounding procedure in Section 4. Section 5 presents the
applications that were outlined in Section 1.1.
## 2 Proof of Theorem 4
In this section we prove Theorem 4. The rounding procedure described in Figure
1 is analyzed in Section 2.1, while the derandomized version is presented in
Section 2.2. The efficiency of the procedure is clear; we refer to Section 2.2
for a discussion on how to discretize the choice of $t$. In Section 2.3 we
show how the analysis can be modified to the case of $\|M\|_{nc}$ and (18).
In what follows, it will be convenient to use the following notation. Given
$M\in M_{n}(M_{n}(\mathbb{C}))$ and $X,Y\in M_{n}(\mathbb{C}^{d})$, define
$M(X,Y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i,j,k,l=1}^{n}M_{ijkl}\left\langle
X_{ij},Y_{kl}\right\rangle\in\mathbb{C}.$ (23)
Thus $M(\cdot,\cdot)$ is a sesquilinear form on $M_{n}(\mathbb{C}^{d})\times
M_{n}(\mathbb{C}^{d})$, i.e., $M(\alpha X,\beta
Y)=\alpha\overline{\beta}M(X,Y)$ for all $X,Y\in M_{n}(\mathbb{C}^{d})$ and
$\alpha,\beta\in\mathbb{C}$. Observe that if $A,B\in M_{n}(\mathbb{C})$ then
$M(A,B)=\sum_{i,j,k,l=1}^{n}M_{ijkl}A_{ij}\overline{B_{kl}}=\sum_{i,j,k,l=1}^{n}M_{ijkl}\left(A\otimes\overline{B}\right)_{(ij),(kl)}.$
(24)
### 2.1 Analysis of the rounding procedure
###### Proof of (22).
Let $X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$ be vector-valued matrices
satisfying (21). Let $z\in\\{1,-1,i,-i\\}^{d}$ be chosen uniformly at random,
and
$X_{{z}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{\sqrt{2}}\left\langle
X,z\right\rangle\qquad\text{and}\qquad
Y_{{z}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{\sqrt{2}}\left\langle
Y,z\right\rangle$
be random variables taking values in $M_{n}(\mathbb{C})$ defined as in the
second step of the rounding procedure (see Figure 1). Then,
$\mathbb{E}_{z}\big{[}\,M(X_{{z}},Y_{{z}})\,\big{]}=\frac{1}{2}\mathbb{E}_{z}\Big{[}\,\sum_{r,s=1}^{d}\overline{z_{r}}{z_{s}}\sum_{i,j,k,l=1}^{n}M_{ijkl}(X_{ij})_{r}\overline{(Y_{kl})_{s}}\,\Big{]}=\frac{1}{2}M(X,Y),$
(25)
where we used the fact that $\mathbb{E}[\overline{z_{r}}z_{s}]=\delta_{rs}$
for every $r,s\in\\{1,\ldots,d\\}$.
Observe that (20) implies that
$\forall\,a\in(0,\infty),\quad\mathbb{E}_{t}[a^{it}]=2a-\mathbb{E}_{t}[a^{2+it}].$
Applying this identity to the nonzero singular values of
$X_{{z}}\otimes\overline{Y_{{z}}}$, we deduce the matrix equality
$\displaystyle\mathbb{E}_{t}\left(A\otimes\overline{B}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}\left[\,\left(U_{z}|X_{{z}}|^{it}\right)\otimes\left(\overline{V_{z}}|Y_{{z}}|^{it}\right)\,\right]$
(26) $\displaystyle=$ $\displaystyle
2X_{{z}}\otimes\overline{Y_{{z}}}-\mathbb{E}_{t}\left[\,\left(U_{z}|X_{{z}}|^{2+it}\right)\otimes\left(\overline{V_{z}}|Y_{{z}}|^{2+it}\right)\,\right]$
$\displaystyle=$ $\displaystyle
2X_{{z}}\otimes\overline{Y_{{z}}}-\mathbb{E}_{t}\left[\,\left(U_{z}|X_{{z}}|^{2+it}\right)\otimes\left(\overline{V_{z}|Y_{{z}}|^{2-it}}\right)\,\right],$
where $U_{z},V_{z}\in\mathcal{U}_{n}$ are such that $X_{{z}}=U_{z}|X_{{z}}|$
and $Y_{{z}}=V_{z}|Y_{{z}}|$ are the polar decompositions of $X_{{z}}$ and
$Y_{{z}}$, respectively (and therefore the polar decomposition of
$\overline{Y_{z}}$ is $\overline{Y_{z}}=\overline{V_{z}}|Y_{z}|$), and we
recall that the output of our rounding scheme as described in Figure 1 is
$A=U_{z}|X_{{z}}|^{it}$ and $B=V_{z}|Y_{{z}}|^{-it}$.
It follows from (23), (24), (25) and (26) that
$\mathbb{E}_{z,t}\big{[}\,M(A,B)\,\big{]}=M(X,Y)-\mathbb{E}_{z,t}\big{[}\,M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}\,\big{]}.$
(27)
Our goal from now on is to bound the second, “error” term on the right-hand
side of (27). Specifically, the rest of the proof is devoted to showing that
for any fixed $t\in\mathbb{R}$ we have
$\left|\mathbb{E}_{z}\big{[}\,M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}\,\big{]}\right|\leqslant\frac{1}{2}\mathrm{SDP}_{\mathbb{C}}(M).$
(28)
Once established, the estimate (28) completes the proof of the desired
expectation bound (22) since
$\mathbb{E}_{z,t}\big{[}\,|M(A,B)|\,\big{]}\stackrel{{\scriptstyle\eqref{eq:an-2}\wedge\eqref{eq:half
sdp
statement}}}{{\geqslant}}M(X,Y)-\frac{1}{2}\mathrm{SDP}_{\mathbb{C}}(M)\stackrel{{\scriptstyle\eqref{eq:maximizer
assumption}}}{{\geqslant}}\left(\frac{1}{2}-\varepsilon\right)\mathrm{SDP}_{\mathbb{C}}(M).$
So, for the rest of the proof, fix some $t\in\mathbb{R}$. As a first step
towards (28) we state the following claim.
###### Claim 5.
Let $W\in M_{n}(\mathbb{C}^{d})$ be a vector-valued matrix, and for every
$r\in\\{1,\ldots,d\\}$ define $W_{r}\in M_{n}(\mathbb{C})$ by
$(W_{r})_{ij}=(W_{ij})_{r}$. Let $z\in\\{1,-1,i,-i\\}^{d}$ be chosen uniformly
at random. Writing $W_{z}=\langle W,z\rangle\in M_{n}(\mathbb{C})$, we have
$\displaystyle\mathbb{E}_{z}\left[(W_{z}W_{z}^{*})^{2}\right]$
$\displaystyle=(WW^{*})^{2}+\sum_{r=1}^{d}W_{r}(W^{*}W-W_{r}^{*}W_{r})W_{r}^{*},$
(29) $\displaystyle\mathbb{E}_{z}\left[(W_{z}^{*}W_{z})^{2}\right]$
$\displaystyle=(W^{*}W)^{2}+\sum_{r=1}^{d}W_{r}^{*}(WW^{*}-W_{r}W_{r}^{*})W_{r}.$
(30)
###### Proof.
By definition $W_{z}=\sum_{r=1}^{d}\overline{z_{r}}{W_{r}}$, and recalling
(10) we have $WW^{*}=\sum_{r=1}^{d}W_{r}W_{r}^{*}$ and
$W^{*}W=\sum_{r=1}^{d}W_{r}^{*}W_{r}$. Consequently,
$\displaystyle\mathbb{E}_{z}\left[(W_{z}W_{z}^{*})^{2}\right]$
$\displaystyle=\mathbb{E}_{z}\Big{[}\,\sum_{p,q,r,s=1}^{d}\overline{z_{p}}{z_{q}}\overline{z_{r}}{z_{s}}{W_{p}}W_{q}^{*}{W_{r}}W_{s}^{*}\,\Big{]}$
$\displaystyle=\sum_{p=1}^{d}W_{p}W_{p}^{*}W_{p}W_{p}^{*}+\sum_{\begin{subarray}{c}p,q\in\\{1,\ldots,d\\}\\\
p\neq
q\end{subarray}}\left(W_{p}W_{p}^{*}W_{q}W_{q}^{*}+W_{p}W_{q}^{*}W_{q}W_{p}^{*}\right)$
$\displaystyle=\sum_{p,q=1}^{d}W_{p}W_{p}^{*}W_{q}W_{q}^{*}+\sum_{p,q=1}^{d}W_{p}W_{q}^{*}W_{q}W_{p}^{*}-\sum_{p=1}^{d}W_{p}W_{p}^{*}W_{p}W_{p}^{*}$
$\displaystyle=\Big{(}\sum_{p=1}^{d}W_{p}W_{p}^{*}\Big{)}^{2}+\sum_{p=1}^{d}W_{p}\Big{(}\sum_{q=1}^{d}W_{q}^{*}W_{q}\Big{)}W_{p}^{*}-\sum_{p=1}^{d}W_{p}W_{p}^{*}W_{p}W_{p}^{*},$
proving (29). A similar calculation yields (30). ∎
Now, for every $t\in\mathbb{R}$ define two vector-valued matrices
$F(t),G(t)\in M_{n}\left(\mathbb{C}^{\\{1,-1,i,-i\\}^{d}}\right)$
by setting for every $j,k\in\\{1,\ldots,n\\}$ and $z\in\\{1,-1,i,-i\\}^{d}$,
$(F(t)_{jk})_{z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{2^{d}}\left(U_{z}|X_{{z}}|^{2+it}\right)_{jk}\qquad\text{and}\qquad(G(t)_{jk})_{z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{2^{d}}\left(V_{z}|Y_{{z}}|^{2-it}\right)_{jk}.$
(31)
Thus,
$M(F(t),G(t))=\frac{1}{4^{d}}\sum_{z\in\\{1,-1,i,-i\\}^{d}}M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}=\mathbb{E}_{z}\big{[}\,M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}\,\big{]}.$
(32)
Moreover, recalling that $X_{z}=U_{z}|X_{z}|$ is the polar decomposition of
$X_{z}$, we have
$F(t)F(t)^{*}=\frac{1}{4^{d}}\sum_{z\in\\{1,-1,i,-i\\}^{d}}U_{z}|X_{z}|^{4}U_{z}^{*}=\mathbb{E}_{z}\left[U_{z}|X_{z}|^{4}U_{z}^{*}\right]=\mathbb{E}_{z}\left[\left(X_{z}X_{z}^{*}\right)^{2}\right].$
(33)
Similarly
$F(t)^{*}F(t)=\mathbb{E}_{z}\left[\left(X_{z}^{*}X_{z}\right)^{2}\right]$, so
that an application of Claim 5 with $W=\frac{1}{\sqrt{2}}X$ yields, using
$XX^{*}=X^{*}X=I$ since $X\in\mathcal{U}_{n}(\mathbb{C}^{d})$,
$F(t)F(t)^{*}+\frac{1}{4}\sum_{r=1}^{d}X_{r}X_{r}^{*}X_{r}X_{r}^{*}=F(t)^{*}F(t)+\frac{1}{4}\sum_{r=1}^{d}X_{r}^{*}X_{r}X_{r}^{*}X_{r}=\frac{1}{2}I.$
(34)
Analogously,
$G(t)G(t)^{*}+\frac{1}{4}\sum_{r=1}^{d}Y_{r}Y_{r}^{*}Y_{r}Y_{r}^{*}=G(t)^{*}G(t)+\frac{1}{4}\sum_{r=1}^{d}Y_{r}^{*}Y_{r}Y_{r}^{*}Y_{r}=\frac{1}{2}I.$
(35)
The two equations above imply that $F(t),G(t)$ satisfy the norm bounds
$\max\big{\\{}\|F(t)F(t)^{*}\|,\,\|F(t)^{*}F(t)\|,\,\|G(t)G(t)^{*}\|,\,\|G(t)^{*}G(t)\|\big{\\}}\leqslant\frac{1}{2}.$
(36)
As shown in Lemma 6 below, (36) implies that there exists a pair of vector-
valued matrices
$R(t),S(t)\in\mathcal{U}_{n}(\mathbb{C}^{d+2n^{2}})$
such that
$M(R(t),S(t))=M(\sqrt{2}F(t),\sqrt{2}G(t)).$ (37)
(This fact can also be derived directly using (34) and (35).) Recalling the
definition of $\mathrm{SDP}_{\mathbb{C}}(M)$ in (15), it follows that for
every $t\in\mathbb{R}$,
$\left|\mathbb{E}_{z}\big{[}\,M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}\,\big{]}\right|\stackrel{{\scriptstyle\eqref{eq:MFG}}}{{=}}|M(F(t),G(t))|\stackrel{{\scriptstyle\eqref{eq:MRS}}}{{=}}\frac{1}{2}|M(R(t),S(t))|\leqslant\frac{1}{2}\mathrm{SDP}_{\mathbb{C}}(M),$
(38)
completing the proof of (28). ∎
###### Lemma 6.
Let $X,Y\in M_{n}(\mathbb{C}^{d})$ be such that
$\max(\|X^{*}X\|,\|XX^{*}\|,\|Y^{*}Y\|,\|YY^{*}\|)\leqslant 1$. Then there
exist $R,S\in\mathcal{U}_{n}(\mathbb{C}^{d+2n^{2}})$ such that for every $M\in
M_{n}(M_{n}(\mathbb{C}))$ we have $M(R,S)=M(X,Y)$. Moreover, $R$ and $S$ can
be computed from $X$ and $Y$ in time ${\mathrm{poly}}(n,d)$.
###### Proof.
Let $A=I-XX^{*}$ and $B=I-X^{*}X$, and note that $A,B\geqslant 0$ and
$\mbox{\rm Tr}(A)=\mbox{\rm Tr}(B)$. Write the spectral decompositions of $A$
and $B$ as $A=\sum_{i=1}^{n}\lambda_{i}(u_{i}u_{i}^{*})$ and
$B=\sum_{j=1}^{n}\mu_{j}(v_{j}v_{j}^{*})$ respectively. Set
$\sigma=\sum_{i=1}^{n}\lambda_{i}=\sum_{j=1}^{n}\mu_{j}$, and define
$R\stackrel{{\scriptstyle\mathrm{def}}}{{=}}X\oplus\Big{(}\bigoplus_{i,j=1}^{n}\sqrt{\frac{\lambda_{i}\mu_{j}}{\sigma}}\,(u_{i}v_{j}^{*})\Big{)}\oplus\Big{(}0_{M_{n}(\mathbb{C}^{n^{2}})}\Big{)}\in
M_{n}(\mathbb{C}^{d}\oplus\mathbb{C}^{n^{2}}\oplus\mathbb{C}^{n^{2}}).$
With this definition we have $RR^{*}=XX^{*}+A=I$ and $R^{*}R=X^{*}X+B=I$, so
$R\in\mathcal{U}_{n}(\mathbb{C}^{d+2n^{2}})$. Let
$S\in\mathcal{U}_{n}(\mathbb{C}^{d+2n^{2}})$ be defined analogously from $Y$,
with the last two blocks of $n^{2}$ coordinates permuted. One checks that
$M(R,S)=M(X,Y)$, as required.
Finally, $A,B$, their spectral decomposition, and the resulting $R,S$ can all
be computed in time ${\mathrm{poly}}(n,d)$ from $X,Y$. ∎
### 2.2 Derandomized rounding
Note that we can always assume that $X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$,
where $d\leqslant 2n^{2}$. We start by slightly changing the projection step.
Define $X_{{z}}^{\prime}$ to be the projection
$X_{{z}}=\frac{1}{\sqrt{2}}\langle X,z\rangle$, after we replace all singular
values of $X_{{z}}$ that are smaller than $\varepsilon$ with $\varepsilon$.
Then, writing
$2X_{{z}}^{\prime}\otimes\overline{Y_{{z}}}^{\prime}=2X_{{z}}\otimes\overline{Y_{{z}}}+2(X_{{z}}^{\prime}-X_{{z}})\otimes\overline{Y_{{z}}}+2X_{{z}}^{\prime}\otimes(\overline{Y_{{z}}^{\prime}}-\overline{Y_{{z}}}),$
we see that in the analogue of (27) the first term is at least
$M(X,Y)-4\varepsilon d\mathrm{SDP}_{\mathbb{C}}(M)$. Here we use that
$\|X_{{z}}\|,\|Y_{{z}}\|\leqslant d$ which follows by the triangle inequality
and $X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$. For the second term in (27), the
previous analysis remains unchanged, provided we prove an analogue of (36).
Using (33) and the analogous equations for $F(t)^{*}F(t),G(t)G(t)^{*}$ and
$G(t)^{*}G(t)$, it will suffice to bound four expressions such as
$\big{\|}\mathbb{E}_{z}\big{[}\,(X_{{z}}^{\prime}(X_{{z}}^{\prime})^{*})^{2}\,\big{]}\big{\|}$.
One checks that the modification to the rounding we did can only increase this
by $\varepsilon^{4}$ (even for each ${z}$), hence following the previous
analysis we get the bound in (38) with $\mathrm{SDP}_{\mathbb{C}}(M)/2$
replaced by $(1+\varepsilon^{4})\mathrm{SDP}_{\mathbb{C}}(M)/2$.
Next, we observe that the coordinates of ${z}$ need not be independent, and it
suffices if they are chosen from a four-wise independent distribution. As a
result, there are only ${\mathrm{poly}}(n)$ possible values of ${z}$ and they
can be enumerated efficiently. Therefore, we can assume that we have a value
${z}\in\\{1,-1,i,-i\\}^{d}$ for which
$\displaystyle\big{|}\mathbb{E}_{t}\big{[}\,M\big{(}U_{z}|X_{{z}}^{\prime}|^{it},V_{z}|Y_{{z}}^{\prime}|^{-it}\big{)}\,\big{]}\big{|}\,\geqslant\,\Big{(}\frac{1}{2}-\varepsilon\Big{)}\mathrm{SDP}_{\mathbb{C}}(M).$
(39)
Notice that for some universal constant $c>0$, with probability at least
$1-\varepsilon$, a sample from the hyperbolic secant distribution is at most
$c\log(1/\varepsilon)$ in absolute value. Therefore, denoting the restriction
of the hyperbolic secant distribution to the interval
$[-c\log(1/\varepsilon),c\log(1/\varepsilon)]$ by $\mu^{\prime}$, and using
the fact that the expression inside the expectation in (39) is never larger
than $\mathrm{SDP}_{\mathbb{C}}(M)$, for $t^{\prime}$ distributed according to
$\mu^{\prime}$ we have
$\big{|}\mathbb{E}_{t^{\prime}}\big{[}\,M\big{(}U_{z}|X_{{z}}^{\prime}|^{it^{\prime}},V_{z}|Y_{{z}}^{\prime}|^{-it^{\prime}}\big{)}\,\big{]}\big{|}\,\geqslant\,\Big{(}\frac{1}{2}-2\varepsilon\Big{)}\mathrm{SDP}_{\mathbb{C}}(M).$
Moreover, for any $t,t^{\prime}\in\mathbb{R}$,
$\displaystyle\big{|}M\big{(}U_{z}|X_{{z}}^{\prime}|^{it},V_{z}|Y_{{z}}^{\prime}|^{-it}\big{)}-M\big{(}U_{z}|X_{{z}}^{\prime}|^{it^{\prime}},V_{z}|Y_{{z}}^{\prime}|^{-it^{\prime}}\big{)}\big{|}$
$\displaystyle\qquad=\big{|}M\big{(}U_{z}(|X_{{z}}^{\prime}|^{it}-|X_{{z}}^{\prime}|^{it^{\prime}}),V_{z}|Y_{{z}}^{\prime}|^{-it}\big{)}+M\big{(}U_{z}|X_{{z}}^{\prime}|^{it^{\prime}},V_{z}(|Y_{{z}}^{\prime}|^{-it}-|Y_{{z}}^{\prime}|^{-it^{\prime}})\big{)}\big{|}$
$\displaystyle\qquad\leqslant\big{|}M\big{(}U_{z}(|X_{{z}}^{\prime}|^{it}-|X_{{z}}^{\prime}|^{it^{\prime}}),V_{z}|Y_{{z}}^{\prime}|^{-it}\big{)}\big{|}+\big{|}M\big{(}U_{z}|X_{{z}}^{\prime}|^{it^{\prime}},V_{z}(|Y_{{z}}^{\prime}|^{-it}-|Y_{{z}}^{\prime}|^{-it^{\prime}})\big{)}\big{|}.$
(40)
The first absolute value in (40) is at most
$\displaystyle\mathrm{SDP}_{\mathbb{C}}(M)\cdot\|U_{z}(|X_{{z}}^{\prime}|^{it}-|X_{{z}}^{\prime}|^{it^{\prime}})\|\cdot\|V_{z}|Y_{{z}}^{\prime}|^{-it}\|$
$\displaystyle=\mathrm{SDP}_{\mathbb{C}}(M)\cdot\||X_{{z}}^{\prime}|^{it}-|X_{{z}}^{\prime}|^{it^{\prime}}\|$
$\displaystyle\leqslant\mathrm{SDP}_{\mathbb{C}}(M)\cdot\log(\max\\{\|X_{{z}}^{\prime}\|,\|(X_{{z}}^{\prime})^{-1}\|\\})\cdot|t-t^{\prime}|.$
We have $\|X_{{z}}^{\prime}\|\leqslant d$ as explained above, and
$\|(X_{{z}}^{\prime})^{-1}\|\leqslant 1/\varepsilon$ by the way our modified
rounding procedure was defined. Similar bounds hold for $Y_{{z}}^{\prime}$. It
therefore suffices to pick $t$ from a grid of size
$O(\log(1/\varepsilon)\max(1/\varepsilon^{2},d/\varepsilon))$.
### 2.3 The rounding procedure in the case of (18)
Theorem 4 addressed the performance of the rounding procedure described in
Figure 1 with respect to inequality (16). Here we prove that this rounding
procedure has the same performance with respect to the noncommutative
Grothendieck inequality (18) as well. This is the content of the following
theorem.
###### Theorem 7.
Fix $n,d\in\mathbb{N}$ and $\varepsilon\in(0,1)$. Suppose that $M\in
M_{n}(M_{n}(\mathbb{C}))$ and that $X,Y\in M_{n}(\mathbb{C}^{d})$ satisfy
$\max\left\\{\|XX^{*}\|+\|X^{*}X\|,\|YY^{*}\|+\|Y^{*}Y\|\right\\}\leqslant 2,$
(41)
and
$\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\langle
X_{ij},Y_{kl}\rangle\Big{|}\,\geqslant\,(1-\varepsilon)\|M\|_{nc},$ (42)
where $\|M\|_{nc}$ was defined in (18). Then the rounding procedure described
in Figure 1 outputs a pair of matrices $A,B\in\mathcal{U}_{n}$ such that
$\mathbb{E}\Big{[}\,\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}\,A_{ij}\overline{B_{kl}}\Big{|}\,\Big{]}\,\geqslant\,\Big{(}\frac{1}{2}-\varepsilon\Big{)}\|M\|_{nc}.$
(43)
###### Proof.
We shall explain how to modify the argument presented in Section 2.1, relying
on the notation that was introduced there. All we need to do is to replace
(38) by the assertion
$\forall\,t\in\mathbb{R},\qquad|M(F(t),G(t))|\leqslant\frac{1}{2}\|M\|_{nc}.$
(44)
To this end we use the following corollary of Claim 5, which is a slight
variant of [Haa85, Lem. 4.1].
###### Corollary 8.
Let $W\in M_{n}(\mathbb{C}^{d})$ be a vector-valued matrix and
$z\in\\{1,-1,i,-i\\}^{d}$ chosen uniformly at random. Writing $W_{z}=\langle
W,z\rangle\in M_{n}(\mathbb{C})$, we have
$\left\|\mathbb{E}_{z}\left[(W_{z}W_{z}^{*})^{2}\right]\right\|+\left\|\mathbb{E}_{z}\left[(W_{z}^{*}W_{z})^{2}\right]\right\|\leqslant\big{(}\left\|WW^{*}\right\|+\left\|W^{*}W\right\|\big{)}^{2}.$
(45)
###### Proof.
Starting from (29) and noting that
$\Big{\|}\sum_{r=1}^{d}W_{r}(W^{*}W-W_{r}^{*}W_{r})W_{r}^{*}\Big{\|}\leqslant\|W^{*}W\|\cdot\Big{\|}\sum_{r=1}^{d}W_{r}W_{r}^{*}\Big{\|}=\left\|W^{*}W\right\|\cdot\left\|WW^{*}\right\|,$
we obtain the inequality
$\left\|\mathbb{E}_{z}\left[(W_{z}W_{z}^{*})^{2}\right]\right\|\leqslant\left\|WW^{*}\right\|^{2}+\left\|W^{*}W\right\|\cdot\left\|WW^{*}\right\|.$
(46)
Similarly, from (30) we get
$\left\|\mathbb{E}_{z}\left[(W_{z}^{*}W_{z})^{2}\right]\right\|\leqslant\left\|W^{*}W\right\|^{2}+\left\|WW^{*}\right\|\cdot\left\|W^{*}W\right\|.$
(47)
By summing (46) and (47) one deduces (45). ∎
Combining (33) with (45) for $W=X/\sqrt{2}$, we have
$\|F(t)F(t)^{*}\|+\|F(t)^{*}F(t)\|\leqslant\frac{1}{4}(\|XX^{*}\|+\|X^{*}X\|)^{2}\stackrel{{\scriptstyle\eqref{eq:max
constraint}}}{{\leqslant}}1.$ (48)
Analogously,
$\|G(t)G(t)^{*}\|+\|G(t)^{*}G(t)\|\leqslant\frac{1}{4}(\|YY^{*}\|+\|Y^{*}Y\|)^{2}\leqslant
1.$ (49)
Recalling the definition of $\|M\|_{nc}$, it follows from (48) and (49) that
for every $t\in\mathbb{R}$,
$\left|\mathbb{E}_{z}\big{[}\,M\big{(}U_{z}|X_{{z}}|^{2+it},V_{z}|Y_{{z}}|^{2-it}\big{)}\,\big{]}\right|\stackrel{{\scriptstyle\eqref{eq:MFG}}}{{=}}|M(F(t),G(t))|\leqslant\frac{1}{2}\|M\|_{nc}.$
(50)
Hence,
$\mathbb{E}_{z,t}\big{[}\,|M(A,B)|\,\big{]}\stackrel{{\scriptstyle\eqref{eq:an-2}\wedge\eqref{eq:half
nc}}}{{\geqslant}}M(X,Y)-\frac{1}{2}\|M\|_{nc}\stackrel{{\scriptstyle\eqref{eq:maximizer
assumption-nc}}}{{\geqslant}}\left(\frac{1}{2}-\varepsilon\right)\|M\|_{nc},$
completing the proof of the desired expectation bound (43). ∎
###### Remark 9.
The following example, due to Haagerup [Haa85], shows that the factor $2$
approximation guarantee obtained in Theorem 7 is optimal: the best constant in
(18) equals $2$. Let $M\in M_{n}(M_{n}(\mathbb{C}))$ be given by
$M_{1jk1}=\delta_{jk}$, and $M_{ijkl}=0$ if $(i,l)\neq(1,1)$. A direct
computation shows that $\mathrm{Opt}_{\mathbb{C}}(M)=1$. Define $X,Y\in
M_{n}(\mathbb{C}^{n})$ by $X_{1j}=Y_{j1}=\sqrt{2/(n+1)}\
e_{j}\in\mathbb{C}^{n}$ for $j\in\\{1,\ldots,n\\}$ and all other entries of
$X$ and $Y$ vanish (here $e_{1},\ldots,e_{n}$ is the standard basis of
$\mathbb{C}^{n}$). Using these two vector-valued matrices one shows that
$\|M\|_{nc}\geqslant 2n/(n+1)$. Recall that in the Introduction we mentioned
that it was shown in [HI95] that the best constant in the weaker inequality
(16) is also $2$, but the example exhibiting this stronger fact is more
involved.
## 3 The real and Hermitian cases
The $n\times n$ Hermitian matrices are denoted $H_{n}$. A $4$-tensor $M\in
M_{n}(M_{n}(\mathbb{C}))\cong M_{n}(\mathbb{C})\otimes M_{n}(\mathbb{C})$ is
said to be Hermitian if $M_{ijkl}=\overline{M_{jilk}}$ for all
$i,j,k,l\in\\{1,\ldots,n\\}$. Investigating the noncommutative Grothendieck
inequality in the setting of Hermitian $M$ is most natural in applications to
quantum information, while problems in real optimization as described in the
Introduction lead to real $M\in M_{n}(M_{n}(\mathbb{R}))$. These special cases
are treated in this section.
Consider the following Hermitian analogue of the quantity
$\mathrm{Opt}_{\mathbb{C}}(M)$.
$\mathrm{Opt}^{*}_{\mathbb{C}}(M)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{\begin{subarray}{c}A,B\in
H_{n}\\\ \|A\|,\|B\|\leqslant
1\end{subarray}}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}A_{ij}\overline{B_{kl}}\Big{|}.$
Note that the convex hull of $\mathcal{U}_{n}$ consists of all the matrices
$A\in M_{n}(\mathbb{C})$ with $\|A\|\leqslant 1$, so by convexity for every
$M\in M_{n}(M_{n}(\mathbb{C}))$ we have
$\mathrm{Opt}_{\mathbb{C}}(M)=\sup_{\begin{subarray}{c}A,B\in
M_{n}(\mathbb{C})\\\ \|A\|,\|B\|\leqslant
1\end{subarray}}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}A_{ij}\overline{B_{kl}}\Big{|}.$
(51)
This explains why $\mathrm{Opt}^{*}_{\mathbb{C}}(M)$ should indeed be viewed
as a Hermitian analogue of $\mathrm{Opt}_{\mathbb{C}}(M)$. The real analogue
of (51) is that, due to the fact that the convex hull of $\mathcal{O}_{n}$
consists of all the matrices $A\in M_{n}(\mathbb{R})$ with $\|A\|\leqslant 1$,
for every $M\in M_{n}(M_{n}(\mathbb{R}))$ we have
$\mathrm{Opt}_{\mathbb{R}}(M)=\sup_{\begin{subarray}{c}A,B\in
M_{n}(\mathbb{R})\\\ \|A\|,\|B\|\leqslant
1\end{subarray}}\Big{|}\sum_{i,j,k,l=1}^{n}M_{ijkl}A_{ij}B_{kl}\Big{|}.$ (52)
The following theorem establishes an algorithmic equivalence between the
problems of approximating either of these two quantities.
###### Theorem 10.
For every $K\in[1,\infty)$ the following two assertions are equivalent.
1. 1.
There exists a polynomial time algorithm $\mathrm{Alg}^{*}$ that takes as
input a Hermitian $M\in M_{n}(M_{n}(\mathbb{C}))$ and outputs $A,B\in H_{n}$
with $\max\\{\|A\|,\|B\|\\}\leqslant 1$ and
$\mathrm{Opt}_{\mathbb{C}}^{*}(M)\leqslant K|M(A,B)|$.
2. 2.
There exists a polynomial time algorithm $\mathrm{Alg}$ that takes as input
$M\in M_{n}(M_{n}(\mathbb{R}))$ and outputs $U,V\in\mathcal{O}_{n}$ such that
$\mathrm{Opt}_{\mathbb{R}}(M)\leqslant KM(U,V)$.
In Section 3.2 we show that for every $K>2\sqrt{2}$ there exists an algorithm
$\mathrm{Alg}^{*}$ as in part $1)$ of Theorem 10. Consequently, we obtain the
algorithm for computing $\mathrm{Opt}_{\mathbb{R}}(M)$ whose existence was
claimed in Theorem 1. The implication $1)\implies 2)$ of Theorem 10 is the
only part of Theorem 10 that will be used in this article; the reverse
direction $2)\implies 1)$ is included here for completeness. Both directions
of Theorem 10 are proved in Section 3.3.
### 3.1 Two-dimensional rounding
In this section we give an algorithmic version of Krivine’s proof [Kri79] that
the $2$-dimensional real Grothendieck constant satisfies
$K_{G}(2)\leqslant\sqrt{2}$. The following theorem is implicit in the proof of
[Kri79, Thm. 1].
###### Theorem 11 (Krivine).
Let $g:\mathbb{R}\to\mathbb{R}$ be defined by $g(x)=\mathrm{sign}(\cos(x))$,
and let $f:[0,\pi/2)\to\mathbb{R}$ be given by
$f(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\begin{cases}1&\text{ if\
$0\leqslant t\leqslant\frac{\pi}{4}$,}\\\
\frac{6}{\pi}\Big{(}\frac{\pi}{2}-t\Big{)}-\frac{1}{2}\Big{(}\frac{4}{\pi}\Big{)}^{3}\Big{(}\frac{\pi}{2}-t\Big{)}^{3}&\text{
if\ $\frac{\pi}{4}\leqslant t<\frac{\pi}{2}$.}\end{cases}$
Extend $f$ to a function defined on all of $\mathbb{R}$ by requiring that it
is even and $f(x+\pi)=-f(x)$ for all $x\in\mathbb{R}$. There exists a sequence
$\\{b_{2\ell+1}\\}_{\ell=0}^{\infty}\in\mathbb{R}^{\mathbb{N}}$ such that for
every $L\in\mathbb{N}$ the numbers $\\{b_{0},\ldots,b_{2L+1}\\}$ can be
computed in ${\mathrm{poly}}(L)$ time,
$\sum_{\ell=L+1}^{\infty}|b_{2\ell+1}|\leqslant C/L$ for some universal
constant $C$, $\sum_{\ell=0}^{\infty}|b_{2\ell+1}|=1$, and
$\forall
x,y\in\mathbb{R},\qquad\cos(x-y)\,=\,\sqrt{2}\sum_{\ell=0}^{\infty}b_{2\ell+1}\frac{1}{2\pi}\int_{-\pi}^{\pi}f\big{(}(2\ell+1)x-t\big{)}g\big{(}t-(2\ell+1)y\big{)}{\rm
d}t.$
An explicit formula for the sequence $\\{b_{2\ell+1}\\}_{\ell=0}^{\infty}$ can
be extracted as follows from the proof of [Kri79, Thm. 1]. For any
$\ell\geqslant 0$, define $a_{2\ell}=0$,
$a_{2\ell+1}\,=\,(-1)^{\ell}\cos\Big{(}\frac{(2\ell+1)\pi}{4}\Big{)}\frac{16}{\pi^{2}(2\ell+1)^{4}}\Big{(}\frac{1}{2\ell+1}-(-1)^{\ell}\frac{\pi}{4}\Big{)},$
$b_{1}=\sqrt{2}(\pi/4)^{3}/(3a_{1})$, and for $\ell>0$,
$b_{2\ell+1}\,=\,-\frac{1}{a_{1}}\sum_{\begin{subarray}{c}d|(2\ell+1)\\\ d\neq
1\end{subarray}}a_{d}b_{\frac{2\ell+1}{d}}.$
Then $|a_{2\ell+1}|=O(1/\ell^{4})$, from which one deduces the crude bound
$|b_{2\ell+1}|=O(1/\ell^{2})$.
Two-dimensional rounding procedure
1. 1.
Let $\varepsilon>0$ and, for $j,k\in\\{1,\ldots,n\\}$ let
$x_{j},y_{k}\in\mathbb{C}$ with $|x_{j}|=|y_{k}|=1$, be given as input. Let
$f,g,C$ and $\\{b_{2\ell+1}\\}_{\ell=0}^{\infty}$ be as in Theorem 11.
2. 2.
For every $j,k$ let $\theta_{j}\in[0,2\pi)$ (resp. $\phi_{k}\in[0,2\pi)$) be
the angle that $x_{j}$ (resp. $y_{k}$) makes with the $x$-axis.
3. 3.
Select $t\in[-\pi,\pi]$ uniformly at random. Let $L=\lceil
C/\varepsilon\rceil$ and $p=1-\sum_{\ell=L+1}^{\infty}|b_{2\ell+1}|$. Select
$\ell\in\\{-1,0,\ldots,L\\}$ with probability $\Pr(-1)=1-p$ and
$\Pr(\ell)=|b_{2\ell+1}|$ for $\ell\in\\{0,\ldots,L\\}$.
4. 4.
For every $j,k$, if $\ell\geqslant 0$ then set
$\lambda_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{sign}(b_{2\ell+1})f((2\ell+1)\theta_{j}-t)$
and $\mu_{k}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}g(t-(2\ell+1)\phi_{k})$.
Otherwise, set $\lambda_{j}=0$, $\mu_{k}=0$.
5. 5.
Return $(\lambda_{j})_{j\in\\{1,\ldots,n\\}}$ and
$(\mu_{k})_{k\in\\{1,\ldots,n\\}}$.
Figure 2: The two-dimensional rounding algorithm takes as input real
$2$-dimensional unit vectors. It returns real numbers of absolute value at
most $1$.
Figure 2 describes a two-dimensional rounding scheme derived from Theorem 11.
The following claim states its correctness in a way that will be useful for us
later.
###### Claim 12.
Let $\varepsilon>0$ and for every $j,k\in\\{1,\ldots,n\\}$ let
$x_{j},y_{k}\in\mathbb{C}$ satisfy $|x_{j}|=|y_{k}|=1$. Then the rounding
procedure described in Figure 2 runs in time
${\mathrm{poly}}(n,1/\varepsilon)$ and returns
$\lambda_{j},\mu_{k}\in\mathbb{R}$ with $|\lambda_{j}|,|\mu_{k}|\leqslant 1$
for every $j,k\in\\{1,\ldots,n\\}$, and
$\mathbb{E}\big{[}\,\lambda_{j}\,\mu_{k}\,\big{]}\,=\,\frac{1}{\sqrt{2}}\,\Re\left(x_{j}\overline{y_{k}}\right)+\varepsilon\langle
x_{j}^{\prime},y_{k}^{\prime}\rangle,$ (53)
where $x_{j}^{\prime},y_{k}^{\prime}\in L_{2}(\mathbb{R})$ are such that
$\|x_{j}^{\prime}\|_{2},\|y_{k}^{\prime}\|_{2}\leqslant 1$.
###### Proof.
Fix $j,k\in\\{1,\ldots,n\\}$ and let $\theta_{j},\phi_{k}$ and
$\lambda_{j},\mu_{k}$ be as defined in Steps 2 and 4 of the rounding
procedure, respectively. Applying Theorem 11,
$\displaystyle\mathbb{E}\big{[}\,\lambda_{j}\,\mu_{k}\,\big{]}$
$\displaystyle=\sum_{\ell=0}^{L}b_{2\ell+1}\frac{1}{2\pi}\int_{-\pi}^{\pi}f\big{(}(2\ell+1)\theta_{j}-t\big{)}g\big{(}t-(2\ell+1)\phi_{k}\big{)}{\rm
d}t=\frac{1}{\sqrt{2}}\cos(\theta_{j}-\phi_{k})+\eta_{jk},$
where
$\displaystyle\eta_{jk}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}-\sum_{\ell=L+1}^{\infty}b_{2\ell+1}\frac{1}{2\pi}\int_{-\pi}^{\pi}f\big{(}(2\ell+1)\theta_{j}-t\big{)}g\big{(}t-(2\ell+1)\phi_{k}\big{)}{\rm
d}t.$
By definition,
$\cos(\theta_{j}-\phi_{k})=\Re\left(x_{j}\overline{y_{k}}\right)$. Using
$1-\varepsilon\leqslant p\leqslant 1$, which follows from the bound stated in
Theorem 11, $\eta_{jk}$ equals $1-p\leqslant\varepsilon$ times a weighted
average of the product of certain values taken by $f$ and $g$, the former only
depending on $\theta_{j}$ and the latter on $\phi_{k}$. Equivalently, this
weighted average can be written as the inner product of two vectors
$x_{j}^{\prime}$ and $y_{k}^{\prime}$ of norm at most $1$. Finally, all steps
of the rounding procedure can be performed in time polynomial in $n$ and
$1/\varepsilon$. ∎
### 3.2 Rounding in the Hermitian case
Let $M\in M_{n}(M_{n}(\mathbb{C}))$ be Hermitian, and
$X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$. For every $r\in\\{1,\ldots,d\\}$
define as usual $X_{r},Y_{r}\in M_{n}(\mathbb{C})$ by
$(X_{r})_{jk}=(X_{jk})_{r}$ and $(Y_{r})_{jk}=(Y_{jk})_{r}$. Define
$X^{\prime},Y^{\prime}\in M_{n}(\mathbb{C}^{2d})$ by
$X^{\prime}_{jk}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{p=1}^{d}\left(\left(\frac{X_{p}+X_{p}^{*}}{2}\right)_{jk}e_{2p-1}+i\left(\frac{X_{p}-X_{p}^{*}}{2}\right)_{jk}e_{2p}\right)\in\mathbb{C}^{2d},$
and
$Y^{\prime}_{jk}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{p=1}^{d}\left(\left(\frac{Y_{p}+Y_{p}^{*}}{2}\right)_{jk}e_{2p-1}+i\left(\frac{Y_{p}-Y_{p}^{*}}{2}\right)_{jk}e_{2p}\right)\in\mathbb{C}^{2d}.$
Then
$(X^{\prime})(X^{\prime})^{*}=(X^{\prime})^{*}(X^{\prime})=(XX^{*}+X^{*}X)/2=I$,
so $X^{\prime}\in\mathcal{U}_{n}(\mathbb{C}^{2d})$ and similarly
$Y^{\prime}\in\mathcal{U}_{n}(\mathbb{C}^{2d})$. Moreover, since $M$ is
Hermitian, $|M(X,Y)|=|M(X^{\prime},Y^{\prime})|$. This shows that for the
purpose of proving the noncommutative Grothendieck inequality for Hermitian
$M$ we may assume without loss of generality that the “component matrices” of
$X,Y$ are Hermitian themselves. Nevertheless, even in this case the rounding
algorithm described in Figure 1 returns unitary matrices $A,B$ that are not
necessarily Hermitian. The following simple argument shows how Krivine’s two-
dimensional rounding scheme can be applied on the eigenvalues of $A,B$ to
obtain Hermitian matrices of norm $1$, at the loss of a factor $\sqrt{2}$ in
the approximation. A similar argument, albeit not explicitly algorithmic, also
appears in [RV12b, Claim 4.7].
Hermitian rounding procedure
1. 1.
Let $X,Y\in M_{n}(\mathbb{C}^{d})$ and $\varepsilon>0$ be given as input.
2. 2.
Let $A,B\in M_{n}(\mathbb{C})$ be the unitary matrices returned by the complex
rounding procedure described in Figure 1. If necessary, multiply $A$ by a
complex phase to ensure that $M(A,B)$ is real. Write the spectral
decompositions of $A,B$ as
$A=\sum_{j=1}^{n}e^{i\theta_{j}}u_{j}u_{j}^{*}\qquad\mathrm{and}\qquad
B=\sum_{k=1}^{n}e^{i\phi_{k}}v_{k}v_{k}^{*},$
where $\theta_{j},\phi_{k}\in\mathbb{R}$ and $u_{j},v_{k}\in\mathbb{C}^{n}$.
3. 3.
Apply the two-dimensional rounding algorithm from Figure 2 to the vectors
$x_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{i\theta_{j}}$ and
$y_{k}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{i\phi_{k}}$. Let
$\lambda_{j},\mu_{k}$ be the results.
4. 4.
Output
$A^{\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{n}\lambda_{j}u_{j}u_{j}^{*}\qquad\mathrm{and}\qquad
B^{\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{k=1}^{n}\mu_{k}v_{k}v_{k}^{*}.$
Figure 3: The Hermitian rounding algorithm takes as input a pair of vector-
valued matrices $X,Y\in M_{n}(\mathbb{C}^{d})$. It outputs two Hermitian
matrices $A^{\prime},B^{\prime}\in M_{n}(\mathbb{C})$ of norm at most $1$.
###### Theorem 13.
Let $n$ be an integer, $M\in M_{n}(M_{n}(\mathbb{C}))$ Hermitian,
$\varepsilon\in(0,1)$ and $X,Y\in\mathcal{U}_{n}(\mathbb{C}^{d})$ such that
$\big{|}\,M\big{(}X,\,Y\big{)}\,\big{|}\,\geqslant\,(1-\varepsilon)\mathrm{SDP}_{\mathbb{C}}(M).$
Then the rounding procedure described in Figure 3 runs in time polynomial in
$n$ and $1/\varepsilon$ and outputs a pair of Hermitian matrices
$A^{\prime},B^{\prime}\in M_{n}(\mathbb{C})$ with norm at most $1$ such that
$\mathbb{E}\Big{[}\,\big{|}\,M\big{(}A^{\prime},\,B^{\prime}\big{)}\,\big{|}\,\Big{]}\,\geqslant\,\Big{(}\frac{1}{2\sqrt{2}}-\Big{(}1+\frac{1}{\sqrt{2}}\Big{)}\varepsilon\Big{)}\mathrm{SDP}_{\mathbb{C}}(M).$
###### Proof.
Let $A,B\in M_{n}(\mathbb{C})$ be as defined as in Step 2 of Figure 3, and
assume as in Figure 3 that $M(A,B)$ is real. By Theorem 4 we have
$\mathbb{E}[|M(A,B)|]\geqslant\left(\frac{1}{2}-\varepsilon\right)\mathrm{SDP}_{\mathbb{C}}(M)$.
Hence to conclude it will suffice to show that for any fixed pair of matrices
$A,B\in M_{n}(\mathbb{C})$,
$\mathbb{E}\Big{[}\,\big{|}M(A^{\prime},B^{\prime})\big{|}\,\Big{]}\,\geqslant\,\frac{1}{\sqrt{2}}\big{|}M(A,B)\big{|}-\varepsilon\,\mathrm{SDP}_{\mathbb{C}}(M),$
(54)
where $A^{\prime},B^{\prime}$ are as returned by the rounding procedure and
the expectation is over the random choices made in the two-dimensional
rounding step, i.e., Step 3 of Figure 3. Applying Claim 12,
$\displaystyle\mathbb{E}\big{[}\,\big{|}M(A^{\prime},B^{\prime})\big{|}\,]$
$\displaystyle\geqslant$
$\displaystyle\Big{|}\sum_{j,k=1}^{n}M(u_{j}u_{j}^{*},v_{k}v_{k}^{*})\mathbb{E}\big{[}\lambda_{j}\mu_{k}\,\big{]}\Big{|}$
(55) $\displaystyle\stackrel{{\scriptstyle\eqref{eq:lambda mu
identity}}}{{\geqslant}}$
$\displaystyle\Big{|}\frac{1}{\sqrt{2}}\sum_{j,k=1}^{n}M(u_{j}u_{j}^{*},v_{k}v_{k}^{*})\Re\big{(}e^{i(\theta_{j}-\phi_{k})}\big{)}\,\big{]}\Big{|}-\varepsilon\,\Big{|}\sum_{j,k=1}^{n}\langle
x_{j}^{\prime},y_{k}^{\prime}\rangle\,M(u_{j}u_{j}^{*},v_{k}v_{k}^{*})\Big{|}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{|}M(A,B)\big{|}-\varepsilon\big{|}M(W,Z)\big{|},$
where in the last inequality, for the first term we used that
$M(u_{j}u_{j}^{*},v_{k}v_{k}^{*})$ is real since $M$ is Hermitian, and for the
second term we defined the vector-valued matrices
$W\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{n}\,x_{j}^{\prime}\,u_{j}u_{j}^{*}\in
M_{n}(L_{2}(\mathbb{C}))\qquad\text{and}\qquad
Z\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{k=1}^{n}\,y_{k}^{\prime}\,v_{k}v_{k}^{*}\in
M_{n}(L_{2}(\mathbb{C})).$
One checks that
$WW^{*}=\sum_{j=1}^{n}\|x_{j}^{\prime}\|_{2}^{2}u_{j}u_{j}^{*}$. Since
$\|x_{j}^{\prime}\|_{2}\leqslant 1$ for all $j\in\\{1,\ldots,n\\}$, it follows
that $\|WW^{*}\|\leqslant 1$. Similarly
$\max\\{\|W^{*}W\|,\|ZZ^{*}\|,\|Z^{*}Z\|\\}\leqslant 1$. Applying Lemma 6 we
obtain $R,S\in\mathcal{U}_{n}(\ell_{2}(\mathbb{C}))$ such that
$M(W,Z)=M(R,S)$, hence $|M(W,Z)|\leqslant\mathrm{SDP}_{\mathbb{C}}(M)$. Eq.
(55) then implies the desired estimate (54). ∎
### 3.3 Proof of Theorem 10
In this section we prove Theorem 10. We first record for future use the
following simple lemma, which is an algorithmic version of (52).
###### Lemma 14.
There exists a polynomial time algorithm that takes as input a $4$-tensor
$M\in M_{n}(M_{n}(\mathbb{R}))$ and two matrices $A,B\in M_{n}(\mathbb{R})$
with $\max\\{\|A\|,\|B\|\\}\leqslant 1$ and outputs two orthogonal matrices
$U,V\in\mathcal{O}_{n}$ such that
$\sum_{i,j,k,l=1}^{n}M_{ijkl}A_{ij}B_{kl}\leqslant\sum_{i,j,k,l=1}^{n}M_{ijkl}U_{ij}V_{kl}.$
###### Proof.
Write the singular value decompositions of $A,B$ as
$A=\sum_{i=1}^{n}\sigma_{i}e_{i}f_{i}^{*}$ and
$B=\sum_{i=1}^{n}\tau_{i}g_{i}h_{i}^{*}$, where each of the sequences
$(e_{i})_{i=1}^{n}$, $(f_{i})_{i=1}^{n}$,
$(g_{i})_{i=1}^{n},(h_{i})_{i=1}^{n}\subseteq\mathbb{R}^{n}$ is orthonormal,
and $\sigma,\tau\in[0,1]^{n}$ (since $\max\\{\|A\|,\|B\|\\}\leqslant 1$). Now,
$M(A,B)$ is given by
$\sum_{i,j=1}^{n}\sigma_{i}\tau_{j}M(e_{i}f_{i}^{*},g_{j}h_{j}^{*}).$ (56)
Fixing all $\sigma_{i},\tau_{j}$ but, say, $\sigma_{1}$, (56) is a linear
function of $\sigma_{1}$, and thus we can shift $\sigma_{1}$ to either $-1$ or
$1$, without decreasing (56). Proceeding in this way with the other variables
$\sigma_{2},\ldots,\sigma_{n},\tau_{1},\ldots,\tau_{n}$, each one in its turn,
we obtain $\varepsilon,\delta\in\\{-1,1\\}^{n}$ such that if we define
$U,V\in\mathcal{O}_{n}$ by $U=\sum_{i=1}^{n}\varepsilon_{i}(e_{i}f_{i}^{*})$
and $V=\sum_{i=1}^{n}\delta_{i}(g_{i}h_{i}^{*})$ then $M(A,B)\leqslant
M(U,V)$, as required. ∎
###### Proof of Theorem 10.
We first prove the implication $1)\implies 2)$. For any $A\in
M_{2n}(\mathbb{C})$ define $A_{1},A_{2},A_{3},A_{4}\in M_{n}(\mathbb{C})$
through the block decomposition $A=\left(\begin{smallmatrix}A_{1}&A_{2}\\\
A_{3}&A_{4}\end{smallmatrix}\right)$. Given $M\in M_{n}(M_{n}(\mathbb{R}))$
let $M^{\prime}\in M_{2n}(M_{2n}(\mathbb{R}))$ be such that
$M^{\prime}(A,B)=M(\Re A_{2},\Re B_{2})$ for every $A,B\in H_{2n}$. Formally,
for every $i,j,k,l\in\\{1,\ldots,2n\\}$ we have
$M^{\prime}_{ijkl}=\frac{1}{4}\left\\{\begin{array}[]{ll}M_{i,j-n,k,l-n}&\mathrm{if}\
(i,k)\in\\{1,\ldots,n\\}^{2}\ \mathrm{and}\
(j,l)\in\\{n+1,\ldots,2n\\}^{2},\\\ M_{j,i-n,l,k-n}&\mathrm{if}\
(j,l)\in\\{1,\ldots,n\\}^{2}\ \mathrm{and}\
(i,k)\in\\{n+1,\ldots,2n\\}^{2},\\\ M_{i,j-n,l,k-n}&\mathrm{if}\
(i,l)\in\\{1,\ldots,n\\}^{2}\ \mathrm{and}\
(j,k)\in\\{n+1,\ldots,2n\\}^{2},\\\ M_{j,i-n,k,l-n}&\mathrm{if}\
(j,k)\in\\{1,\ldots,n\\}^{2}\ \mathrm{and}\
(i,l)\in\\{n+1,\ldots,2n\\}^{2},\\\ 0&\mathrm{otherwise.}\end{array}\right.$
Then $M^{\prime}$ is Hermitian. Apply the algorithm $\mathrm{Alg}^{*}$ (whose
existence is the premise of part $1)$ of Theorem 10) to $M^{\prime}$ to get
$A,B\in H_{2n}$ such that $\mathrm{Opt}_{\mathbb{C}}^{*}(M)\leqslant
K|M(A,B)|$. Since $A,B\in H_{2n}$ we have $A_{3}=A_{2}^{*}$ and
$B_{3}=B_{2}^{*}$. Because $\max\\{\|A\|,\|B\|\\}\leqslant 1$ also
$\max\\{\|\Re A\|,\|\Re B\|\\}\leqslant 1$. By Lemma 14 we can therefore
efficiently find $U,V\in\mathcal{O}_{n}$ such that
$KM(U,V)\geqslant K|M(\Re A,\Re
B)|=K|M^{\prime}(A,B)|\geqslant\mathrm{Opt}_{\mathbb{C}}^{*}(M^{\prime})\\\
=\sup_{\begin{subarray}{c}C,D\in H_{2n}\\\ \|C\|,\|D\|\leqslant
1\end{subarray}}\left|M^{\prime}(C,D)\right|\geqslant\sup_{S,T\in\mathcal{O}_{n}}\left|M^{\prime}\left(\left(\begin{smallmatrix}0&S\\\
S^{*}&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&T\\\
T^{*}&0\end{smallmatrix}\right)\right)\right|=\sup_{S,T\in
O_{n}}M(S,T)=\mathrm{Opt}_{\mathbb{R}}(M).$
To prove the reverse implication $2)\implies 1)$, define
$\psi:M_{2n}(\mathbb{R})\to H_{n}$ by
$\psi\left(\left(\begin{smallmatrix}A_{1}&A_{2}\\\
A_{3}&A_{4}\end{smallmatrix}\right)\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{4}\left(A_{1}+A_{1}^{*}+A_{4}+A_{4}^{*}\right)+\frac{i}{4}\left(A_{2}-A_{2}^{*}+A_{3}^{*}-A_{3}\right)\in
H_{n}.$ (57)
Suppose that $M\in M_{n}(M_{n}(\mathbb{C}))$ is Hermitian and define
$M^{\prime\prime}\in M_{2n}(M_{2n}(\mathbb{R}))$ by requiring that
$M^{\prime\prime}(A,B)=M\left(\psi(A),\psi(B)\right)$ for every $A,B\in
M_{2n}(\mathbb{R})$. Note that $M(\psi(A),\psi(B))\in\mathbb{R}$ since
$M,\psi(A),\psi(B)$ are all Hermitian. Apply the algorithm $\mathrm{Alg}$ of
part $2)$ to $M^{\prime\prime}$, obtaining two orthogonal matrices
$U,V\in\mathcal{O}_{2n}$ satisfying
$\mathrm{Opt}_{\mathbb{R}}(M^{\prime\prime})\leqslant
KM^{\prime\prime}(U,V)=KM(\psi(U),\psi(V))$. By Lemma 15 below we have
$\max\\{\|\psi(U)\|,\|\psi(V)\|\\}\leqslant 1$. Moreover, using Lemma 16 below
we have
$\mathrm{Opt}_{\mathbb{R}}(M^{\prime\prime})=\sup_{\begin{subarray}{c}A,B\in
M_{2n}(\mathbb{R})\\\ \|A\|,\|B\|\leqslant
1\end{subarray}}\big{|}M(\psi(A),\psi(B))\big{|}\geqslant\sup_{\begin{subarray}{c}X,Y\in
H_{n}\\\ \|X\|,\|Y\|\leqslant
1\end{subarray}}\left|M\left(\psi\left(\left(\begin{smallmatrix}\Re X&\Im X\\\
-\Im X&\Re
X\end{smallmatrix}\right)\right),\psi\left(\left(\begin{smallmatrix}\Re Y&\Im
Y\\\ -\Im Y&\Im Y\end{smallmatrix}\right)\right)\right)\right|.$ (58)
Observe that for every $X\in H_{n}$ we have
$\psi\left(\left(\begin{smallmatrix}\Re X&\Im X\\\ -\Im X&\Re
X\end{smallmatrix}\right)\right)=X$. Consequently the rightmost term in (58)
equals $\mathrm{Opt}_{\mathbb{C}}^{*}(M)$. Therefore
$\mathrm{Opt}_{\mathbb{C}}^{*}(M)\leqslant K|M(\psi(U),\psi(V))|$, so that the
algorithm that outputs $\psi(U),\psi(V)$ has the desired approximation factor.
∎
###### Lemma 15.
Let $\psi:M_{2n}(\mathbb{R})\to H_{n}$ be given as in (57). Then
$\|\psi(Y)\|\leqslant\|Y\|$ for all $Y\in M_{2n}(\mathbb{R})$.
###### Proof.
For $Y\in M_{2n}(\mathbb{R})$ write $Z=(Y+Y^{*})/2$. Setting
$Z=\left(\begin{smallmatrix}Z_{1}&Z_{2}\\\
Z_{3}&Z_{4}\end{smallmatrix}\right)$, where $Z_{1},Z_{2},Z_{3},Z_{4}\in
M_{n}(\mathbb{R})$, we then have $\psi(Y)=(Z_{1}+Z_{4})/2+i(Z_{2}-Z_{3})/2$.
Take $z\in\mathbb{C}^{n}$ and write $z=x+iy$, where $x,y\in\mathbb{R}^{n}$.
Then
$\displaystyle\Re\big{(}\langle\psi(Y)z,z\rangle\big{)}$
$\displaystyle=\frac{\langle Z_{1}x,x\rangle+\langle Z_{4}y,y\rangle-\langle
Z_{2}y,x\rangle-\langle Z_{3}x,y\rangle}{2}+\frac{\langle
Z_{1}y,y\rangle+\langle Z_{4}x,x\rangle+\langle Z_{2}x,y\rangle+\langle
Z_{3}y,x\rangle}{2}$ $\displaystyle=\frac{1}{2}\left\langle
Z\begin{pmatrix}x\\\ -y\end{pmatrix},\begin{pmatrix}x\\\
-y\end{pmatrix}\right\rangle+\frac{1}{2}\left\langle Z\begin{pmatrix}y\\\
x\end{pmatrix},\begin{pmatrix}y\\\ x\end{pmatrix}\right\rangle.$
Since $\psi(Y)$ is Hermitian, it follows that
$\|\psi(Y)\|\leqslant\|Z\|\leqslant\|Y\|$. ∎
###### Lemma 16.
For every $X\in H_{n}$ we have $\left\|\left(\begin{smallmatrix}\Re X&\Im X\\\
-\Im X&\Re X\end{smallmatrix}\right)\right\|=\|X\|$.
###### Proof.
Write $Z=\left(\begin{smallmatrix}\Re X&\Im X\\\ -\Im X&\Re
X\end{smallmatrix}\right)\in M_{2n}(\mathbb{R})$. Since $X$ is Hermitian, $Z$
is symmetric. For every $a,b\in\mathbb{R}^{n}$,
$\displaystyle\left\langle Z\begin{pmatrix}a\\\
b\end{pmatrix},\begin{pmatrix}a\\\ b\end{pmatrix}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle(\Re
X)a,a\right\rangle+\left\langle(\Im X)b,a\right\rangle-\left\langle(\Im
X)a,b\right\rangle+\left\langle(\Re X)b,b\right\rangle$ $\displaystyle=$
$\displaystyle\Re\left(\langle X(a-ib),a-ib\rangle\right).$
Since $Z$ is symmetric and $X$ is Hermitian, it follows that $\|Z\|=\|X\|$. ∎
## 4 Direct rounding in the real case
We now describe and analyze a different rounding procedure for the real case
of the noncommutative Grothendieck inequality. The argument is based on the
proof of the noncommutative Grothendieck inequality due to Kaijser [Kai83],
which itself has a structure similar to the proof of the classical
Grothendieck inequality due to Krivine [Kri74] (see also [Jam85], the “Notes
and Remarks” section of Chapter 1 of [DJT95], and the survey article [JL01]),
and uses ideas from [Pis78] to extend that proof to the non-commutative
setting.
Real rounding procedure
1. 1.
Let $X,Y\in M_{n}(\mathbb{R}^{d})$ be given as input, and let
$\varepsilon\in\\{-1,1\\}^{d}$ be chosen uniformly at random.
2. 2.
Set
$Y_{{\varepsilon}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\langle\varepsilon,Y\rangle$.
Write the singular value decomposition of $Y_{{\varepsilon}}$ as
$Y_{{\varepsilon}}=\sum_{i=1}^{n}\,t_{i}(\varepsilon)\,u_{i}(\varepsilon)v_{i}(\varepsilon)^{*}$,
where $t_{i}(\varepsilon)\in[0,\infty)$ and
$u_{i}(\varepsilon),v_{i}(\varepsilon)\in\mathbb{R}^{n}$. Define
$(Y_{{\varepsilon}})_{\tau}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{n}\min\\{t_{i}(\varepsilon),\tau\\}u_{i}(\varepsilon)v_{i}(\varepsilon)^{*},$
where $\tau\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sqrt{3}/2$.
3. 3.
Let $X(\varepsilon)\in M_{n}(\mathbb{R})$ of norm at most $1$ be such that
$M(X(\varepsilon),(Y_{{\varepsilon}})_{\tau})=\max_{\begin{subarray}{c}X\in
M_{n}(\mathbb{R})\\\ \|X\|\leqslant
1\end{subarray}}|M(X,(Y_{{\varepsilon}})_{\tau})|.$
4. 4.
Output the pair $A=X(\varepsilon)$ and
$B=\frac{1}{\tau}(Y_{{\varepsilon}})_{\tau}$.
Figure 4: The real rounding algorithm takes as input $X,Y\in
M_{n}(\mathbb{R}^{d})$. It outputs two real matrices $A,B\in
M_{n}(\mathbb{R})$ of norm at most $1$.
###### Theorem 17.
Given $n\in\mathbb{N}$, $M\in M_{n}(M_{n}(\mathbb{R}))$ and $\eta\in(0,1/2)$,
suppose that $X,Y\in\mathcal{O}_{n}(\mathbb{R}^{d})$ are such that
$\big{|}\,M\big{(}X,\,Y\big{)}\,\big{|}\,\geqslant\,(1-\eta)\mathrm{SDP}_{\mathbb{R}}(M),$
(59)
where $\mathrm{SDP}_{\mathbb{R}}(M)$ is defined in (12). Then the rounding
procedure described in Figure 4 runs in time polynomial in $n$ and outputs a
pair of real matrices $A,B\in M_{n}(\mathbb{R})$ with norm at most $1$ such
that
$\mathbb{E}\Big{[}\,\big{|}\,M\big{(}A,\,B\big{)}\,\big{|}\,\Big{]}\,\geqslant\,\frac{(1-2\eta)^{2}}{3\sqrt{3}}\mathrm{SDP}_{\mathbb{R}}(M).$
(60)
Note that by Lemma 14 we can also efficiently convert the matrices $A,B$ to
orthogonal matrices $U,V\in\mathcal{O}_{n}$ without changing the approximation
guarantee.
The proof of Theorem 17 relies on two claims that are used to bound the error
that results from the truncation step (step 2) in the rounding procedure in
Figure 4. The first claim plays the same role as Claim 5 did in the complex
case.
###### Claim 18.
Fix $X\in M_{n}(\mathbb{R}^{d})$ and let $\varepsilon\in\\{-1,1\\}^{d}$ be
chosen uniformly at random. Set
$X_{{\varepsilon}}=\langle\varepsilon,X\rangle$. Then
$\displaystyle\mathbb{E}_{\varepsilon}\big{[}(X_{{\varepsilon}}X_{{\varepsilon}}^{*})^{2}\big{]}$
$\displaystyle\leqslant(XX^{*})^{2}+2\|X^{*}X\|XX^{*},$
$\displaystyle\mathbb{E}_{\varepsilon}\big{[}(X_{{\varepsilon}}^{*}X_{{\varepsilon}})^{2}\big{]}$
$\displaystyle\leqslant(X^{*}X)^{2}+2\|XX^{*}\|X^{*}X.$
###### Proof.
Define the symmetric real vector-valued matrix $Z$ by
$Z\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\begin{pmatrix}0&X\\\
X^{*}&0\end{pmatrix}.$
Following the proof of Lemma 1.1 in [Pis78] (which establishes a bound
analogous to the one proved in Claim 8 for the case of i.i.d. $\\{\pm 1\\}$
random variables) and defining as usual $Z_{r}\in M_{n}(\mathbb{R})$ by
$(Z_{r})_{ij}=(Z_{ij})_{r}$ for every $r\in\\{1,\ldots,d\\}$, we have
$\displaystyle\mathbb{E}_{\varepsilon}\big{[}\langle\varepsilon,Z\rangle^{4}\big{]}$
$\displaystyle=\sum_{r=1}^{d}Z_{r}^{4}+\sum_{\begin{subarray}{c}r,s\in\\{1,\ldots,d\\}\\\
r\neq
s\end{subarray}}\,\big{(}Z_{r}^{2}Z_{s}^{2}+Z_{r}Z_{s}Z_{r}Z_{s}+Z_{r}Z_{s}^{2}Z_{r}\big{)}$
$\displaystyle=\Big{(}\sum_{r=1}^{d}Z_{r}^{2}\Big{)}^{2}+\sum_{\begin{subarray}{c}r,s\in\\{1,\ldots,d\\}\\\
r<s\end{subarray}}\,\big{(}Z_{r}Z_{s}+Z_{s}Z_{r}\big{)}^{2}.$ (61)
Using the inequality $(A+B)(A+B)^{*}\leqslant 2(AA^{*}+BB^{*})$, which holds
for all $A,B\in M_{n}(\mathbb{R})$, we can bound the second sum in (4) as
follows.
$\displaystyle\sum_{\begin{subarray}{c}r,s\in\\{1,\ldots,d\\}\\\
r<s\end{subarray}}\,\big{(}Z_{r}Z_{s}+Z_{s}Z_{r}\big{)}^{2}$
$\displaystyle\leqslant
2\,\Big{(}\sum_{r=1}^{d}\,Z_{r}\Big{(}\sum_{\begin{subarray}{c}s\in\\{1,\ldots,d\\}\\\
s>r\end{subarray}}Z_{s}^{2}\Big{)}Z_{r}+\sum_{s=1}^{d}\,Z_{s}\Big{(}\sum_{\begin{subarray}{c}r\in\\{1,\ldots,d\\}\\\
r<s\end{subarray}}Z_{r}^{2}\Big{)}Z_{s}\Big{)}$
$\displaystyle=2\sum_{r=1}^{d}\,Z_{r}\Big{(}\sum_{\begin{subarray}{c}s\in\\{1,\ldots,d\\}\\\
s\neq r\end{subarray}}Z_{s}^{2}\Big{)}Z_{r}$ $\displaystyle\leqslant
2\sum_{r=1}^{d}Z_{r}\Big{(}\sum_{s=1}^{d}Z_{s}^{2}\Big{)}Z_{r}.$
Replacing $Z_{r}$ by its definition and using $ABA^{*}\leqslant\|B\|AA^{*}$,
which holds true for every positive semidefinite $B\in M_{n}(\mathbb{R})$, we
arrive at the following matrix inequality:
$\displaystyle\mathbb{E}_{\varepsilon}\big{[}\langle\varepsilon,Z\rangle^{4}\big{]}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\mathbb{E}_{\varepsilon}\big{[}(X_{{\varepsilon}}X_{{\varepsilon}}^{*})^{2}\big{]}&0\\\
0&\mathbb{E}_{\varepsilon}\big{[}(X_{{\varepsilon}}^{*}X_{{\varepsilon}})^{2}\big{]}\end{pmatrix}$
$\displaystyle\leqslant$ $\displaystyle\begin{pmatrix}(XX^{*})^{2}&0\\\
0&(X^{*}X)^{2}\end{pmatrix}+2\,\begin{pmatrix}\|X^{*}X\|\,XX^{*}&0\\\
0&\|XX^{*}\|\,X^{*}X\end{pmatrix}.$
The inequality above implies a separate matrix inequality for both diagonal
blocks, proving the claim. ∎
The second claim appears as Lemma 2.3 in [Kai83] (in the complex case). We
include a short proof for the sake of completeness.
###### Claim 19.
Let $Y\in M_{n}(\mathbb{R})$. For any $\tau>0$, there exists a decomposition
$Y=Y_{\tau}+Y^{\tau}$ such that $\|Y_{\tau}\|_{\infty}\leqslant\tau$ and
$Y^{\tau}(Y^{\tau})^{*}\leqslant\frac{1}{(4\tau)^{2}}(YY^{*})^{2}\qquad\text{and}\qquad(Y^{\tau})^{*}Y^{\tau}\leqslant\frac{1}{(4\tau)^{2}}(Y^{*}Y)^{2}.$
###### Proof.
Define $Y_{\tau}$ by “truncating” the singular values of $Y$ at $\tau$, as is
done in step 2 of the rounding procedure described in Figure 4, so that
$\|Y_{\tau}\|\leqslant\tau$. Define $Y^{\tau}=Y-Y_{\tau}$. By definition, $Y$,
$Y_{\tau}$ and $Y^{\tau}$ all have the same singular vectors, and the non-zero
singular values of $Y^{\tau}$ are of the form $\mu-\tau$, where
$\mu\geqslant\tau$ is a singular value of $Y$. Using the bound
$|\mu-\tau|\leqslant\mu^{2}/(4\tau)$ valid for any $\mu\geqslant\tau$, any
nonzero eigenvalue $\lambda=(\mu-\tau)^{2}$ of $Y^{\tau}(Y^{\tau})^{*}$ (resp.
$(Y^{\tau})^{*}Y^{\tau}$) satisfies $\lambda\leqslant\mu^{4}/(4\tau)^{2}$,
which proves the claim. ∎
###### Proof of Theorem 17.
We shall continue using the notation introduced in Figure 4, Claim 18 and
Claim 19. Let $X,Y\in\mathcal{O}_{n}(\mathbb{R}^{d})$ satisfying (59). For
every $\varepsilon\in\\{-1,1\\}^{d}$ and any $\tau>0$ let
$Y_{{\varepsilon}}=\langle\varepsilon,Y\rangle=(Y_{{\varepsilon}})_{\tau}+Y_{{\varepsilon}}^{\tau}$
be the decomposition promised by Claim 19. Combining the bound from Claim 18
with the one from Claim 19, we see that
$\big{\|}\mathbb{E}_{\varepsilon}\big{[}Y_{{\varepsilon}}^{\tau}(Y_{{\varepsilon}}^{\tau})^{*}\big{]}\big{\|}\leqslant\frac{1}{(4\tau)^{2}}\big{(}\|YY^{*}\|^{2}+2\|Y^{*}Y\|\|YY^{*}\|\big{)}=\frac{3}{(4\tau)^{2}},$
(62)
where the final step of (62) follows from
$Y\in\mathcal{O}_{n}(\mathbb{R}^{d})$, and the same bound holds on
$\|\mathbb{E}_{\varepsilon}\big{[}(Y_{{\varepsilon}}^{\tau})^{*}Y_{{\varepsilon}}^{\tau}\big{]}\|$.
We have
$\displaystyle\big{|}M\big{(}X,\,Y\big{)}\big{|}$
$\displaystyle=\big{|}\mathbb{E}_{\varepsilon}\big{[}M(X_{{\varepsilon}},\,Y_{{\varepsilon}})\big{]}\big{|}\leqslant\big{|}\mathbb{E}_{\varepsilon}\big{[}M(X_{{\varepsilon}},\,(Y_{{\varepsilon}})_{\tau})\big{]}\big{|}+\big{|}\mathbb{E}_{\varepsilon}\big{[}M(X_{{\varepsilon}},\,Y_{{\varepsilon}}^{\tau})\big{]}\big{|}$
$\displaystyle\leqslant\big{|}\mathbb{E}_{\varepsilon}\big{[}M(X_{{\varepsilon}},\,(Y_{{\varepsilon}})_{\tau})\big{]}\big{|}+\frac{\sqrt{3}}{4\tau}\mathrm{SDP}_{\mathbb{R}}(M),$
(63)
where the last inequality in (63) follows from the definition of
$\mathrm{SDP}_{\mathbb{R}}(M)$ and (62).
To bound the first term in (63), let
$X_{{\varepsilon}}=\sum_{i=1}^{n}s_{i}(\varepsilon)w_{i}(\varepsilon)z_{i}(\varepsilon)^{*}$,
$(Y_{{\varepsilon}})_{\tau}=\sum_{i=1}^{n}t_{i}^{\prime}(\varepsilon)u_{i}(\varepsilon)v_{i}(\varepsilon)^{*}$,
where $s_{i}(\varepsilon),t_{i}^{\prime}(\varepsilon)\geqslant 0$ and
$u_{i}(\varepsilon),v_{i}(\varepsilon),w_{i}(\varepsilon),z_{i}(\varepsilon)\in\mathbb{R}^{n}$,
be the singular value decompositions of
$X_{{\varepsilon}},(Y_{{\varepsilon}})_{\tau}$, respectively. Set
$M_{i,j}(\varepsilon)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}M(w_{i}(\varepsilon)z_{i}(\varepsilon)^{*},u_{j}(\varepsilon)v_{j}(\varepsilon)^{*})$.
With these definitions,
$\displaystyle\big{|}\mathbb{E}_{\varepsilon}\left[M(X_{{\varepsilon}},\,(Y_{{\varepsilon}})_{\tau})\right]\big{|}$
$\displaystyle=\Big{|}\mathbb{E}_{\varepsilon}\Big{[}\sum_{i,j=1}^{n}\,M_{i,j}(\varepsilon)s_{i}(\varepsilon)t_{j}^{\prime}(\varepsilon)\Big{]}\Big{|}\leqslant\mathbb{E}_{\varepsilon}\Big{[}\sum_{i=1}^{n}\,s_{i}(\varepsilon)\,\Big{|}\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)\Big{|}\Big{]}$
$\displaystyle\leqslant\Big{(}\mathbb{E}_{\varepsilon}\Big{[}\sum_{i=1}^{n}\,s_{i}(\varepsilon)^{2}\Big{|}\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)\Big{|}\Big{]}\Big{)}^{1/2}\Big{(}\mathbb{E}_{\varepsilon}\Big{[}\sum_{i=1}^{n}\Big{|}\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)\Big{|}\Big{]}\Big{)}^{1/2}.$
(64)
Note that the rightmost term in (64) is at most
$\Big{(}\tau\mathbb{E}_{\varepsilon}\Big{[}M(X(\varepsilon),B(\varepsilon))\Big{]}\Big{)}^{1/2}$,
where $B(\varepsilon)=\frac{1}{\tau}(Y_{{\varepsilon}})_{\tau}$ and
$X(\varepsilon)$ is as defined in the rounding procedure in Figure 4. To bound
the leftmost term in (64), define
$W_{\varepsilon}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{n}(r_{i}(\varepsilon)s_{i}(\varepsilon)^{2})w_{i}(\varepsilon)z_{i}(\varepsilon)^{*}$,
where $r_{i}(\varepsilon)$ is the sign of
$\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)$, so that
$\sum_{i=1}^{n}\,s_{i}(\varepsilon)^{2}|\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)|=M(W_{\varepsilon},(Y_{{\varepsilon}})_{\tau}).$
Moreover, by definition
$W_{\varepsilon}W_{\varepsilon}^{*}=(X_{{\varepsilon}}X_{{\varepsilon}}^{*})^{2}$,
so using Claim 18 we have
$\displaystyle\big{\|}\mathbb{E}_{\varepsilon}\big{[}W_{\varepsilon}^{*}W_{\varepsilon}\big{]}\big{\|}$
$\displaystyle=\big{\|}\mathbb{E}_{\varepsilon}\left[(X_{{\varepsilon}}^{*}X_{{\varepsilon}})^{2}\right]\big{\|}\leqslant\|X^{*}X\|^{2}+2\|XX^{*}\|\|X^{*}X\|=3,$
and the same bound holds for
$\|\mathbb{E}_{\varepsilon}\big{[}W_{\varepsilon}W_{\varepsilon}^{*}\big{]}\|$.
By the definition of $(Y_{\varepsilon})_{\tau}$ we also have
$\max\Big{\\{}\big{\|}\mathbb{E}_{\varepsilon}\left[(Y_{{\varepsilon}})_{\tau}^{*}(Y_{{\varepsilon}})_{\tau}\right]\big{\|},\,\big{\|}\mathbb{E}_{\varepsilon}\left[(Y_{{\varepsilon}})_{\tau}(Y_{{\varepsilon}})_{\tau}^{*}\right]\big{\|}\Big{\\}}\leqslant\tau^{2}.$
Hence
$\mathbb{E}_{\varepsilon}\Big{[}\sum_{i=1}^{n}\,s_{i}(\varepsilon)^{2}\Big{|}\sum_{j=1}^{n}M_{i,j}(\varepsilon)t_{j}^{\prime}(\varepsilon)\Big{|}\Big{]}\,=\,\sqrt{3}\tau\,\mathbb{E}_{\varepsilon}\Big{[}M\Big{(}\frac{W_{\varepsilon}}{\sqrt{3}},\,\frac{(Y_{{\varepsilon}})_{\tau}}{\tau}\Big{)}\Big{]}\,\leqslant\,\sqrt{3}\tau\,\mathrm{SDP}_{\mathbb{R}}(M),$
where we used the definition of $\mathrm{SDP}_{\mathbb{R}}(M)$. Finally,
combining (63) and (64) with the bounds shown above we obtain
$(1-\eta)\mathrm{SDP}_{\mathbb{R}}(M)\stackrel{{\scriptstyle\eqref{eq:eta
asusmption}}}{{\leqslant}}\big{|}M\big{(}X,\,Y\big{)}\big{|}\,\leqslant\,\sqrt[4]{3}\tau\sqrt{\mathrm{SDP}_{\mathbb{R}}(M)\cdot\mathbb{E}_{\varepsilon}\left[M(X(\varepsilon),B(\varepsilon))\right]}+\frac{\sqrt{3}}{4\tau}\mathrm{SDP}_{\mathbb{R}}(M).$
(65)
Setting $\tau=\sqrt{3}/2$ in (65) and simplifying leads to the desired bound
(60).
All steps in the rounding procedure can be performed efficiently. The
calculation of $X(\varepsilon)$ in the third step of Figure 4 can be expressed
as a semidefinite program and solved in time polynomial in $n$. Alternatively,
one may directly compute $X(\varepsilon)$ by writing the polar decomposition
$\mbox{\rm Tr}_{2}(M^{*}(I\otimes B))=QP\in M_{n}(\mathbb{R}),$
where the partial trace is taken with respect to the second tensor, $Q$ is an
orthogonal matrix and $P$ is positive semidefinite, and taking
$X(\varepsilon)=Q^{*}$. ∎
## 5 Some applications
Before presenting the details of our applications of Theorem 1, we observe
that the problem of computing $\mathrm{Opt}_{\mathbb{R}}(M)$ is a rather
versatile optimization problem, perhaps more so than what one might initially
guess from its definition. The main observation is that by considering
matrices $M$ which only act non-trivially on certain diagonal blocks of the
two variables $U,V$ that appear in the definition of
$\mathrm{Opt}_{\mathbb{R}}(M)$, these variables can each be thought of as a
sequence of multiple matrix variables, possibly of different shapes but all
with operator norm at most $1$. This allows for some flexibility in adapting
the noncommutative Grothendieck optimization problem to concrete settings, and
we explain the transformation in detail next.
For every $n,m\geqslant 1$, let $M_{m,n}(\mathbb{R})$ be the vector space of
real $m\times n$ matrices. Given integers $k,\ell\geqslant 1$ and sequences of
integers $(m_{i}),(n_{i})\in\mathbb{N}^{k}$,
$(p_{j}),(q_{j})\in\mathbb{N}^{\ell}$, we define
$\textrm{Bil}_{\mathbb{R}}(k,\ell;(m_{i}),(n_{i}),(p_{j}),(q_{j}))$, or simply
$\textrm{Bil}_{\mathbb{R}}(k,\ell)$ when the remaining sequences are clear
from context, as the set of all
$f:\Big{(}\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})\Big{)}\times\Big{(}\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})\Big{)}\,\to\,\mathbb{R}$
that are linear in both arguments. Concretely,
$f\in\textrm{Bil}_{\mathbb{R}}(k,\ell)$ if and only if there exists real
coefficients $\alpha_{irs,juv}$ such that for every
$(A_{i})\in\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})$ and
$(B_{j})\in\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})$,
$f\big{(}(A_{i})_{i\in\\{1,\ldots,k\\}},(B_{j})_{j\in\\{1,\ldots,\ell\\}}\big{)}=\sum_{i=1}^{k}\sum_{j=1}^{\ell}\sum_{r=1}^{m_{i}}\sum_{s=1}^{n_{i}}\sum_{u=1}^{p_{j}}\sum_{v=1}^{q_{j}}\,\alpha_{irs,juv}\,(A_{i})_{rs}(B_{j})_{uv}.$
(66)
For integers $m,n\geqslant 1$, let $\mathcal{O}_{m,n}\subset
M_{m,n}(\mathbb{R})$ denote the set of all $m\times n$ real matrices $U$ such
that $UU^{*}=I$ if $m\leqslant n$ and $U^{*}U=I$ if $m\geqslant n$. If $m=n$
then $\mathcal{O}_{n,n}=\mathcal{O}_{n}$ is the set of orthogonal matrices;
$\mathcal{O}_{n,1}$ is the set of all $n$-dimensional unit vectors;
$\mathcal{O}_{1,1}$ is simply the set $\\{-1,1\\}$. Given
$f\in\textrm{Bil}_{\mathbb{R}}(k,\ell)$, consider the quantity
$\mathrm{Opt}_{\mathbb{R}}(f)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{\begin{subarray}{c}(U_{i})\in\bigoplus_{i=1}^{k}\mathcal{O}_{m_{i},n_{i}}\\\
(V_{j})\in\bigoplus_{j=1}^{\ell}\mathcal{O}_{p_{j},q_{j}}\end{subarray}}f\big{(}(U_{i}),(V_{j})\big{)}.$
Note that this definition coincides with the definition of
$\mathrm{Opt}_{\mathbb{R}}(f)$ given in the introduction whenever
$f\in\textrm{Bil}_{\mathbb{R}}(1,1;n,n,n,n)$. The proof of the following
proposition shows that the new optimization problem still belongs the
framework of the noncommutative Grothendieck problem.
###### Proposition 20.
There exists a polynomial time algorithm that takes as input
$k,\ell\in\mathbb{N}$, $(m_{i}),(n_{i})\in\mathbb{N}^{k}$,
$(p_{j}),(q_{j})\in\mathbb{N}^{\ell}$ and
$f\in\textrm{Bil}_{\mathbb{R}}(k,\ell;(m_{i}),(n_{i}),(p_{j}),(q_{j}))$ and
outputs $(U_{i})\in\bigoplus_{i=1}^{k}\mathcal{O}_{m_{i},n_{i}}$ and
$(V_{j})\in\bigoplus_{j=1}^{\ell}\mathcal{O}_{p_{j},q_{j}}$ such that
$\mathrm{Opt}_{\mathbb{R}}(f)\,\leqslant\,O(1)\cdot
f\big{(}(U_{i}),(V_{j})\big{)}.$
Moreover, the implied constant in the $O(1)$ term can be taken to be any
number larger than $2\sqrt{2}$.
###### Proof.
Let $k,\ell\in\mathbb{N}$, $(m_{i}),(n_{i})\in\mathbb{N}^{k}$,
$(p_{j}),(q_{j})\in\mathbb{N}^{\ell}$ be given, and define
$m\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{k}m_{i},\qquad
n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{k}n_{i},\qquad
p\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{\ell}p_{j},\qquad
q\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{\ell}q_{j},$
and $t\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\\{m,n,p,q\\}$. We first
describe how $\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})$ (respectively
$\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})$) can be identified with a
subset of $M_{t}(\mathbb{R})$ consisting of block diagonal matrices. For any
$i\in\\{1,\ldots,k\\}$ and $r\in\\{1,\ldots,m_{i}\\}$,
$s\in\\{1,\ldots,n_{i}\\}$, let $F_{r,s}^{i}\in M_{t}(\mathbb{R})$ be the
matrix that has all entries equal to $0$ except the entry in position
$(r+\sum_{j<i}m_{j},s+\sum_{j<i}n_{j})$, which equals $1$. Similarly, for any
$j\in\\{1,\ldots,\ell\\}$ and $u\in\\{1,\ldots,p_{j}\\}$,
$v\in\\{1,\ldots,q_{j}\\}$ we let $G_{u,v}^{j}\in M_{t}(\mathbb{R})$ be the
matrix that has all entries equal to $0$ except the entry in position
$(u+\sum_{i<j}p_{i},v+\sum_{i<j}q_{i})$, which equals $1$. Define linear maps
$\Phi:\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})\to M_{t}(\mathbb{R})$ and
$\Psi:\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})\to M_{t}(\mathbb{R})$
by
$\Phi\big{(}(A_{i})\big{)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{k}\sum_{r=1}^{m_{i}}\sum_{s=1}^{n_{i}}\,(A_{i})_{r,s}\,F^{i}_{r,s}\qquad\text{and}\qquad\Psi\big{(}(B_{j})\big{)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{\ell}\sum_{u=1}^{p_{j}}\sum_{v=1}^{q_{j}}\,(B_{j})_{u,v}\,G^{j}_{u,v}.$
From the definition, one verifies that
$\|\Phi((A_{i}))\|=\max_{i\in\\{1,\ldots,k\\}}\|A_{i}\|\qquad\text{and}\qquad\|\Psi((B_{j}))\|=\max_{j\in\\{1,\ldots,\ell\\}}\|B_{j}\|.$
(67)
Let $f\in\textrm{Bil}_{\mathbb{R}}(k,\ell)$ and $(\alpha_{irs,juv})$ real
coefficients as in (66). Define $M\in M_{t}(M_{t}(\mathbb{R}))$ by
$M(F_{r,s}^{i},G_{u,v}^{j})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\alpha_{irs,juv},$
and $M(A,B)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}0$ if $A$ or $B$ are not
in the linear span of the $F_{r,s}^{i}$ or $G_{u,v}^{j}$ respectively. (Recall
that the notation $M(A,B)$ was introduced at the beginning of Section 2.) With
this definition, for any
$(A_{i})\in\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})$ and
$(B_{j})\in\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})$ we have
$M\big{(}\Phi((A_{i})),\Psi((B_{j}))\big{)}\,=\,f\big{(}(A_{i}),(B_{j})\big{)}.$
We claim that $\mathrm{Opt}_{\mathbb{R}}(f)=\mathrm{Opt}_{\mathbb{R}}(M)$,
where $\mathrm{Opt}_{\mathbb{R}}(M)$ is as defined in the Introduction.
Indeed, by (67) for any
$(U_{i})\in\bigoplus_{i=1}^{k}\mathcal{O}_{m_{i},n_{i}}$ and
$(V_{j})\in\bigoplus_{j=1}^{\ell}\mathcal{O}_{p_{j},q_{j}}$ we have
$\|\Phi((U_{i}))\|,\|\Psi((V_{j}))\|\leqslant 1$, hence
$\mathrm{Opt}_{\mathbb{R}}(f)\leqslant\mathrm{Opt}_{\mathbb{R}}(M)$.
Conversely, let $U,V\in\mathcal{O}_{t}$ be arbitrary, and
$U^{\prime},V^{\prime}$ their orthogonal projections on $\mathrm{Im}(\Phi)$,
$\mathrm{Im}(\Psi)$ respectively. Then $M(U^{\prime},V^{\prime})=M(U,V)$.
Moreover, if
$(U^{\prime}_{i})\in\bigoplus_{i=1}^{k}M_{m_{i},n_{i}}(\mathbb{R})\qquad\mathrm{and}\qquad(V^{\prime}_{j})\in\bigoplus_{j=1}^{\ell}M_{p_{j},q_{j}}(\mathbb{R})$
are such that $\Phi((U^{\prime}_{i}))=U^{\prime}$ and
$\Psi((V^{\prime}_{j}))=V^{\prime}$ then by (67)
$\max_{i\in\\{1,\ldots,k\\}}\|U^{\prime}_{i}\|=\|U^{\prime}\|\leqslant\|U\|=1$,
and similarly $\max_{j\in\\{1,\ldots,\ell\\}}\|V^{\prime}_{j}\|\leqslant 1$.
As in the proof of Lemma 14, we may then argue that there exists
$U_{i}\in\mathcal{O}_{m_{i},n_{i}}$, $V_{j}\in\mathcal{O}_{p_{j},q_{j}}$ such
that
$f((U_{i}),(V_{j}))\geqslant
f((U^{\prime}_{i}),(V^{\prime}_{j}))=M(U^{\prime},V^{\prime})=M(U,V),$
proving the reverse inequality
$\mathrm{Opt}_{\mathbb{R}}(f)\geqslant\mathrm{Opt}_{\mathbb{R}}(M)$.
To conclude the proof of Proposition 20 it remains to note that the algorithm
of Theorem 1, when applied to $M$, produces in polynomial time
$U,V\in\mathcal{O}_{t}$ such that $\mathrm{Opt}_{\mathbb{R}}(M)\leqslant
O(1)\cdot M(U,V)$. Arguing as in Lemma 14, $(U_{i})$ and $(V_{j})$ can be
computed from $U,V$ in polynomial time and constitute the output of the
algorithm. ∎
### 5.1 Constant-factor algorithm for robust PCA problems
We start with the R1-PCA problem, as described in (3). Let
$a_{1},\ldots,a_{N}\in\mathbb{R}^{n}$ be given, and define
$f\in\textrm{Bil}_{\mathbb{R}}(1,N;(K),(n),(1,\ldots,1),(K,\ldots,K))$ by
$f\big{(}Y,(Z_{1},\ldots,Z_{N})\big{)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{N}\sum_{k=1}^{K}\,(Z_{i})_{1k}\,\Big{(}\sum_{j=1}^{n}\,Y_{kj}(a_{i})_{j}\Big{)}.$
The condition $Z_{i}\in\mathcal{O}_{1,K}$ is equivalent to $Z_{i}$ being a
unit vector, while $Y\in\mathcal{O}_{K,n}$ is equivalent to the $K$ rows of
$Y$ being orthonormal. Using that the $\ell_{2}$ norm of a vector
$u\in\mathbb{R}^{K}$ is equal to $\max_{v\in S^{K-1}}\langle u,v\rangle$, the
quantity appearing in (3) is equal to
$\sup_{\begin{subarray}{c}Y\in\mathcal{O}_{K,n}\\\
Z_{1},\ldots,Z_{N}\in\mathcal{O}_{1,K}\end{subarray}}\,f\big{(}Y,(Z_{1},\ldots,Z_{N})\big{)},$
which by definition is equal to $\mathrm{Opt}_{\mathbb{R}}(f)$. An
approximation algorithm for the R1-PCA problem then follows immediately from
Proposition 20. The algorithm for L1-PCA follows similarly. Letting
$g\in\textrm{Bil}_{\mathbb{R}}(1,NK;(K),(n),(1,\ldots,1),(1,\ldots,1))$ be
defined as
$g\big{(}Y,(Z_{1,1},\ldots,Z_{N,K})\big{)}\,=\,\sum_{i=1}^{N}\sum_{k=1}^{K}\,Z_{ik}\,\Big{(}\sum_{j=1}^{n}\,Y_{kj}(a_{i})_{j}\Big{)},$
and using that $Z_{ik}\in\mathcal{O}_{1,1}$ is equivalent to
$Z_{ik}\in\\{-1,1\\}$, the quantity appearing in (4) is equal to
$\sup_{\begin{subarray}{c}Y\in\mathcal{O}_{K,n}\\\
Z_{1,1},\ldots,Z_{N,K}\in\mathcal{O}_{1,1}\end{subarray}}\,g\big{(}Y,(Z_{1,1},\ldots,Z_{N,K})\big{)}\,=\,\mathrm{Opt}_{\mathbb{R}}(g),$
which again fits into the framework of Proposition 20.
### 5.2 A constant-factor algorithm for the orthogonal Procrustes problem
The generalized orthogonal Procustes problem was described in Section (1.1.3).
Continuing with the notation introduced there, let $A_{1},\ldots,A_{K}$ be
$d\times n$ real matrices such that the $i$-th row of $A_{k}$ is the vector
$x^{k}_{i}\in\mathbb{R}^{d}$. Our goal is then to efficiently approximate
$P(A_{1},\ldots
A_{K})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max_{U_{1},\ldots
U_{K}\in\mathcal{O}_{d}}\Big{\|}\sum_{k=1}^{K}U_{k}A_{k}\Big{\|}_{2}^{2}=\max_{U_{1},\ldots
U_{K}\in\mathcal{O}_{d}}\sum_{k,l=1}^{K}\langle U_{k}A_{k},U_{l}A_{l}\rangle,$
(68)
where we set $\langle
A,B\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{d}\sum_{j=1}^{n}A_{ij}B_{ij}$
for every two $d\times n$ real matrices $A,B$. We first observe that (68) is
equal to
$\max_{U_{1},\ldots U_{K}\in\mathcal{O}_{d}}\max_{V_{1},\ldots
V_{K}\in\mathcal{O}_{d}}\Big{\langle}\sum_{k=1}^{K}U_{k}A_{k},\sum_{l=1}^{K}V_{l}A_{l}\Big{\rangle}.$
(69)
It is clear that (69) is at least as large as (68). For the other direction,
using Cauchy-Schwarz,
$\Big{\langle}\sum_{k=1}^{K}U_{k}A_{k},\sum_{l=1}^{K}V_{l}A_{l}\Big{\rangle}\leqslant\Big{\|}\sum_{k=1}^{K}U_{k}A_{k}\Big{\|}_{2}\cdot\Big{\|}\sum_{l=1}^{K}V_{l}A_{l}\Big{\|}_{2},$
so either $U_{1},\ldots,U_{K}$ or $V_{1},\ldots,V_{K}$ achieve a value in (68)
that is at least as large as (69). The desired algorithm now follows by noting
that (69) falls into the framework of Proposition 20: it suffices to define
$f\in\textrm{Bil}_{\mathbb{R}}(K,K;(d,\ldots,d),(d,\ldots,d),(d,\ldots,d),(d,\ldots,d))$
by
$f\big{(}(U_{1},\ldots,U_{K}),(V_{1},\ldots,V_{K})\big{)}\,=\,\Big{\langle}\sum_{k=1}^{K}U_{k}A_{k},\sum_{l=1}^{K}V_{l}A_{l}\Big{\rangle}.$
Finally, from an assignment to (69) we can efficiently extract an assignment
to (68) achieving at least as high a value by choosing the one among the
$(U_{k})$ or the $(V_{l})$ that leads to a higher value.
##### Comparison with previous work.
To compare the approximation guarantee that we obtained with the literature,
we first note that $P(A_{1},\ldots,A_{K})$ is attained at
$U_{1},\ldots,U_{K}\in\mathcal{O}_{d}$ if and only if $U_{1},\ldots,U_{K}$ are
maximizers of the following quantity over all
$V_{1},\ldots,V_{K}\in\mathcal{O}_{d}$.
$\sum_{\begin{subarray}{c}k,l\in\\{1,\ldots,K\\}\\\ k\neq
l\end{subarray}}\langle V_{k}A_{k},V_{l}A_{l}\rangle$ (70)
Indeed, the diagonal term $\langle
V_{k}A_{k},V_{k}A_{k}\rangle=\|V_{k}A_{k}\|_{2}^{2}$ equals
$\|A_{k}\|_{2}^{2}$, since $V_{k}$ is orthogonal. Consequently, the quantity
appearing in (70) differs from the quantity defining $P(A_{1},\ldots,A_{K})$
by an additive term that does not depend on $V_{1},\ldots,V_{K}$. In the same
vein, as already mentioned in Section (1.1.3), $P(A_{1},\ldots,A_{K})$ is
attained at $U_{1},\ldots,U_{K}\in\mathcal{O}_{d}$ if and only if
$U_{1},\ldots,U_{K}$ are minimizers of the following quantity over all
$V_{1},\ldots,V_{K}\in\mathcal{O}_{d}$.
$\sum_{k,l=1}^{K}\|V_{k}A_{k}-V_{l}A_{l}\|_{2}^{2}.$ (71)
While the optimization problems appearing in (68), (70) and (71) have the same
exact solutions, this is no longer the case when it comes to approximation
algorithms. To the best of our knowledge the polynomial time approximability
of the minimization of the quantity appearing in (71) was not investigated in
the literature. Nemirovski [Nem07] and So [So11] studied the polynomial time
approximability of the maximization of the quantity appearing in (70): the
best known algorithm for this problem [So11] has approximation factor
$O(\log(n+d+K))$. This immediately translates to the same approximation factor
for computing $P(A_{1},\ldots,A_{K})$, which was the previously best known
algorithm for this problem. Our constant-factor approximation algorithm for
$P(A_{1},\ldots,A_{K})$ yields an approximation for the maximization of the
quantity appearing in (70) that has a better approximation factor than that of
[So11] unless the additive difference $\sum_{k=1}^{K}\|A_{k}\|_{2}^{2}$ is too
large. Precisely, our algorithm becomes applicable in this context as long as
this term is at most a factor $(1-1/C)$ smaller than $P(A_{1},\ldots,A_{K})$.
This will be the case for typical applications in which one may think of each
$A_{k}$ as obtained from a common $A$ by applying a small perturbation
followed by an arbitrary rotation: in that case it is reasonable to expect the
optimal solution to satisfy $\langle
U_{k}A_{k},U_{l}A_{l}\rangle\approx\|A\|_{2}$ for every
$l,k\in\\{1,\ldots,K\\}$; see, e.g., [Nem07, Sec. 4.3].
### 5.3 An algorithmic noncommutative dense regularity lemma
Our goal here is to prove Theorem 3, but before doing so we note that it leads
to a PTAS for computing $\mathrm{Opt}_{\mathbb{C}}(M)$ whenever
$\mathrm{Opt}_{\mathbb{C}}(M)\geqslant\kappa n\|M\|_{2}$ for some constant
$\kappa>0$ (we shall use below the notation introduced in the statement of
Theorem 3).
The idea for this is exactly as in Section 3 of [FK99]. The main point is that
given such an $M$ and $\varepsilon\in(0,1)$ the decomposition in Theorem 3
only involves $T=O(1/(\kappa\varepsilon)^{2})$ terms, which can be computed in
polynomial time. Given such a decomposition, we will exhaustively search over
a suitably discretized set of values $a_{t},b_{t}$ for $\mbox{\rm Tr}(A_{t}X)$
and $\mbox{\rm Tr}(B_{t}Y)$ respectively. For each such choice of values,
verifying whether it is achievable using an $X$ and $Y$ of operator norm at
most $1$ can be cast as a semidefinite program. The final approximation to
$\mathrm{Opt}_{\mathbb{C}}(M)$ is given by the maximum value of
$|\sum_{t=1}^{T}\alpha_{t}a_{t}b_{t}|$, among those sequences
$(a_{t}),(b_{t})$ that were determined to be feasible.
In slightly more detail, first note that for any $X,Y$ of operator norm at
most $1$ the values of $\mbox{\rm Tr}(A_{t}X)$ and $\mbox{\rm Tr}(B_{t}Y)$ lie
in the complex disc with radius $n$. Given our assumption on
$\mathrm{Opt}_{\mathbb{C}}(M)$, the bound on $\alpha_{t}$ stated in Theorem 3
implies $|\alpha_{t}|=O(\mathrm{Opt}_{\mathbb{C}}(M)/(\kappa n^{2}))$. Hence
an approximation of each $\mbox{\rm Tr}(A_{t}X)$ and $\mbox{\rm Tr}(B_{t}Y)$
up to additive error $\varepsilon\kappa n/T$ will translate to an additive
approximation error to $M(X,Y)$ of
$O(\varepsilon\mathrm{Opt}_{\mathbb{C}}(M))$. As a result, to obtain a
multiplicative $(1\pm\varepsilon)$ approximation to
$\mathrm{Opt}_{\mathbb{C}}(M)$ it will suffice to exhaustively enumerate among
$(O((n\cdot T/(\varepsilon\kappa n))^{2}))^{2T}$ possible values for the
sequences $(a_{t}),(b_{t})$.
Finally, to decide whether a given sequence of values can be achieved, it
suffices to decide two independent feasibility problems of the following form:
given $n\times n$ complex matrices $(A_{t})_{t=1}^{T}$ of norm at most $1$ and
$(a_{t})_{t=1}^{T}\in\mathbb{C}^{T}$, does there exist $X\in
M_{n}(\mathbb{C})$ of norm at most $1$ such that
$\max_{t\in\\{1,\ldots,T\\}}\max\\{|\mathrm{Re}(\mbox{\rm
Tr}(A_{t}X)-a_{t})|,|\mathrm{Im}(\mbox{\rm
Tr}(A_{t}X)-a_{t})|\\}\leqslant\frac{\varepsilon\kappa n}{T}\,?$
This problem can be cast as a semidefinite program, and feasibility can be
decided in time that is polynomial in $n$ and $T$.
###### Proof of Theorem 3.
The argument is iterative. Assume that
$\\{A_{t},B_{t}\\}_{t=1}^{\tau-1}\subseteq\mathcal{U}_{n}$ have already been
constructed. Write $M_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}M$ and
$M_{\tau}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}M-\sum_{t=1}^{\tau-1}\alpha_{t}(A_{t}\otimes
B_{t}).$
If
$\mathrm{Opt}_{\mathbb{C}}(M_{\tau})\leqslant\varepsilon\mathrm{Opt}_{\mathbb{C}}(M)$
then we may stop the construction. Otherwise, by Theorem 1 and multiplying by
an appropriate complex phase if necessary, we can find
$A_{\tau},B_{\tau}\in\mathcal{U}_{n}$ such that
$M_{\tau}(A_{\tau},B_{\tau})\,\geqslant\,\Big{(}\frac{1}{2}-\frac{\varepsilon}{2}\Big{)}\mathrm{Opt}_{\mathbb{C}}(M_{\tau}),$
(72)
with the left-hand side of (72) real. Set
$\alpha_{\tau}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{M_{\tau}(A_{\tau},B_{\tau})}{\|A_{\tau}\|_{2}^{2}\cdot\|B_{\tau}\|_{2}^{2}},$
and define
$M_{\tau+1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}M_{\tau}-\alpha_{\tau}(A_{\tau}\otimes
B_{\tau})$. By expanding the square we have
$\displaystyle\big{\|}M_{\tau+1}\big{\|}_{2}^{2}$
$\displaystyle=\big{\|}M_{\tau}\big{\|}_{2}^{2}-\frac{M_{\tau}(A_{\tau},B_{\tau})^{2}}{\|A_{\tau}\|_{2}^{2}\|B_{\tau}\|_{2}^{2}}$
$\displaystyle\leqslant\big{\|}M_{\tau}\big{\|}_{2}^{2}-\frac{(1-\varepsilon)^{2}}{4n^{2}}\mathrm{Opt}_{\mathbb{C}}(M_{\tau})^{2}$
$\displaystyle\leqslant\big{\|}M_{\tau}\big{\|}_{2}^{2}-\frac{\varepsilon^{2}(1-\varepsilon)^{2}}{4n^{2}}\mathrm{Opt}_{\mathbb{C}}(M)^{2}.$
(73)
It follows from (5.3) that as long as this process continues,
$\|M_{\tau}\|_{2}^{2}$ decreases by an additive term of at least
$\varepsilon^{2}(1-\varepsilon)^{2}\mathrm{Opt}_{\mathbb{C}}(M)^{2}/(4n^{2})$
at each step. This process must therefore terminate after at most $T\leqslant
4n^{2}\|M\|_{2}^{2}/(\varepsilon(1-\varepsilon)\mathrm{Opt}_{\mathbb{C}}(M))^{2}$
steps.
The “moreover” part of Theorem 3 follows immediately by using the part of
Theorem 1 pertaining to $\mathrm{Opt}_{\mathbb{R}}(M)$ (note that the specific
approximation factor does not matter here). ∎
Acknowledgements. We thank Daniel Dadush and Raghu Meka for useful
discussions.
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|
arxiv-papers
| 2012-10-29T13:29:52 |
2024-09-04T02:49:37.240457
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Assaf Naor, Oded Regev, Thomas Vidick",
"submitter": "Oded Regev",
"url": "https://arxiv.org/abs/1210.7656"
}
|
1210.7671
|
# Sup-norm estimates for parabolic systems with dynamic boundary conditions
###### Abstract.
We consider parabolic systems with nonlinear dynamic boundary conditions, for
which we give a rigorous derivation. Then, we give them several physical
interpretations which includes an interpretation for the porous-medium
equation, and for certain reaction-diffusion systems that occur in
mathematical biology and ecology. We devise several strategies which imply
(uniform) $L^{p}$ and $L^{\infty}$ estimates on the solutions for the initial
value problems considered.
###### Key words and phrases:
parabolic systems, dynamic boundary conditions, uniform estimates, porous
medium equation, ecology
###### 1991 Mathematics Subject Classification:
Primary: 35K59, 35B45, 35Q92 ; Secondary: 92B05
Ciprian G. Gal
Department of Mathematics, Florida International University
Miami, FL, 33199, USA
## 1\. Introduction
In this article, we consider the following system of quasilinear parabolic
equations
(1.1)
$\partial_{t}u_{i}-\Delta\left(A_{i}\left(u_{i}\right)\right)+f_{i}\left(x,t,\overrightarrow{u}\right)=0\text{,
in }\Omega\times\left(0,\infty\right),$
for $i=1,...,m$, where $\overrightarrow{u}=\left(u_{1},...,u_{m}\right),$
$\Omega$ is a bounded domain in $\mathbb{R}^{N},$ $N\geq 1$, with sufficiently
smooth boundary $\Gamma:=\partial\Omega$ (which is at least of class
$\mathcal{C}^{2}$), for some given functions $A_{i}$ and $f_{i}.$ Denote by
$\mathbb{N}_{m}=\left\\{1,...,m\right\\}$ and consider two mutually disjoint
(possibly empty) subsets $I_{m},J_{m}\subseteq\mathbb{N}_{m}$ such that
$I_{m}\cup J_{m}=\mathbb{N}_{m}$. Equation (1.1) is subject to the following
set of boundary conditions
(1.2)
$\partial_{\mathbf{n}}u_{i}+h_{i}\left(x,t,\overrightarrow{u}\right)=0,\text{
on }\Gamma\times\left(0,\infty\right),\text{ \thinspace}i\in I_{m}$
and
(1.3)
$\delta_{i}\partial_{t}u_{i}+\partial_{\mathbf{n}}\left(A_{i}\left(u_{i}\right)\right)+g_{i}\left(x,t,\overrightarrow{u}\right)=0,\text{
on }\Gamma\times\left(0,\infty\right),\text{ \thinspace}i\in J_{m},$
for some given functions $g_{i}$ and $h_{i}$. Here $\delta_{i}>0$ for $i\in
J_{m}$, and we may assume, without loss of generality, that $\delta_{i}=0,$
for $i\in I_{m}$. The boundary conditions in (1.2)-(1.3) may be also mixed,
that is, the boundary $\Gamma$ may consists of two disjoint open subsets
$\Gamma_{1}$ and $\Gamma_{2}$ on which the boundary conditions may be either
of Dirichlet type or of the form (1.2) and (1.3), respectively. Finally, the
model (1.1)-(1.3) could be also generalized be letting the reaction terms
depend on advection, by allowing the diffusion rates depend also on $x$ and
$t$, or in other various ways. As usual, we equip the system (1.1)-(1.3) with
the initial conditions
(1.4) $u_{i\mid t=0}=u_{i0}\text{ in }\Omega,\text{ }u_{i\mid
t=0}=v_{i0}\text{ on }\Gamma,\text{ }i\in\mathbb{N}_{m},$
where in general, we may have $u_{i0\mid\Gamma}\neq v_{i0},$
$i\in\mathbb{N}_{m}$ (i.e., if $u_{i0}$ is well-defined in the trace sense).
We aim to give some results which allow to deduce $L^{\infty}$-estimates for
solutions of (1.1)-(1.4) assuming that some sort of energy estimate is apriori
known in $L^{p}$-norm for some finite $p$. The main tool will be an iterative
argument following a well-known Alikakos-Moser technique combined with a
suitable form of Gronwall’s inequality, which can then be used to prove that
the $L^{p}$-$L^{\infty}$ smoothing property holds for any solutions of the
_non-degenerate_ parabolic system (1.1)-(1.4) (e.g., at least when
$a_{i}\left(u_{i}\right):=A_{i}^{{}^{\prime}}\left(u_{i}\right)$ satisfies
(1.5) below). In order to deal with the full degenerate case (1.1)-(1.4) (at
least in the case when $a_{i}\left(u_{i}\right)=\left|u_{i}\right|^{p_{i}},$
$p_{i}>0$), we employ DeGiorgi’s truncation method to prove the
$L^{p}$-$L^{\infty}$ smoothing property. The precise statements of these
results can be found in Section 3, see Theorems 3.1 and 3.2. A rigorous
derivation and physical interpretation of the system (1.1)-(1.4) shall be
given below in Section 2.
Why is it important to establish apriori (possibly, uniform in time)
$L^{\infty}$-estimates from some given $L^{p}$-estimate? To better give an
idea of our larger scope let us take a look at some history for problems of
the form (1.1)-(1.4). Problems such as (1.1)-(1.4) have already been
investigated in a number of papers [16, 17, 8, 31] assuming that diffusion
rates $a_{i}\left(u_{i}\right):=A_{i}^{{}^{\prime}}\left(u_{i}\right)$ satisfy
(1.5) $a_{i}\left(u_{i}\right)\geq d_{i}>0,\text{ for all
}u_{i}\in\mathbb{R}\text{ and }i\in\mathbb{N}_{m}.$
For instance, Constantin and Escher [16, 17] show that unique (classical)
maximal solutions exist in some Bessel potential spaces under suitable
assumptions on the nonlinearities $f_{i},$ $g_{i}$ and $h_{i}$. Such results
also enable the authors to investigate other qualitative properties concerning
global existence and blow-up phenomena (see, also [8]). These results are also
improved by Meyries [31], still in the non-degenerate case (1.5), by assuming
more general boundary conditions (by also incorporating surface diffusion in
(1.3)), and by requiring that the functions
$f_{i}\left(\overrightarrow{u}\right),$
$h_{i}\left(\overrightarrow{u}\right),$ $g_{i}\left(\overrightarrow{u}\right)$
are dissipative in a certain sense. However, none of these contributions deal
with the degenerate case for equation (1.1), that is, when
$a_{i}\left(u_{i}\right)$ is allowed to have a polynomial degeneracy at zero
for some (if not all) $i\in\mathbb{N}_{m}$; for instance, one can take
(1.6) $a_{i}\left(u_{i}\right)=\left|u_{i}\right|^{p_{i}},\text{ }p_{i}>0.$
Moreover, it is well-known in the scalar case $m=1$, that when at least one of
the source terms, the bulk nonlinear term $f_{1}$ or the boundary term $g_{1}$
is present in (1.1)-(1.2), conditions can be derived on their growth rates
which imply either the global existence of solutions or blow-up in finite time
[21]. Namely in the non-degenerate case, for $\lambda,\mu\in\left\\{0,\pm
1\right\\}$ with $\max\left\\{\lambda,\mu\right\\}=1$,
$f_{1}\left(s\right):=-\lambda\left|s\right|^{r_{1}-1}s$ and
$g_{1}\left(s\right):=-\mu\left|s\right|^{r_{2}-1}s$, solutions of
(1.7) $\partial_{t}u-\nu\Delta
u+f_{1}\left(u\right)=h_{1}\left(x\right),\text{ in }\Omega\times(0,+\infty),$
subject to the dynamic condition
(1.8) $\partial_{t}u+\nu
b\partial_{\mathbf{n}}u+g_{1}\left(u\right)=h_{2}\left(x\right),\text{ on
}\Gamma\times\left(0,\infty\right),$
are globally well-defined, for every given (sufficiently smooth) initial data
(1.4), if $r_{1}r_{2}>1$ and $\lambda r_{1}+\mu r_{2}>0$. Furthermore, [21]
shows that if we further restrict the growths of $r_{1}$,$r_{2}$ so that
$r_{1}<\left(N+2\right)/\left(N-2\right)$ and $r_{2}\leq N/\left(N-2\right)$,
then the global solutions are also bounded. These restrictions can be
eventually removed and more general conditions on $f_{1},g_{1}$ can be deduced
(see, e.g., [23]). On the other hand, if $\lambda=0$, $\mu=1,$ then some
solutions blowup in finite time with blowup occurring in the $L^{\infty}$-norm
at a rate $\left(t-T_{\ast}\right)^{-\left(r_{2}-1\right)},$ for some
additional conditions on $u_{0}$ and $r_{2}$. In the same way, when $\mu=0$
and $\lambda=1$, then some solutions blowup in finite time with a blowup rate
which depends on $r_{1}$ and $u_{0}$ (see [3]). In the case when both
$\mu\in\mathbb{R}$ and $\lambda>0$ are nonzero, blowup may still occur for
superlinear growth of $f_{1}$ and any growth of $g$ (see [23]). The occurrence
of blow up phenomena is closely related to the blowup problem for the ordinary
differential equation
$u_{t}+H\left(u\right)=0,$
where either $H=f_{1}$ or $H=g_{1}$. More precisely, it is easy to see that
solutions of the ODE are spatially homogeneous solutions of either equation
(1.7) or (1.8), and so if these solutions blowup in finite time so do the
solutions of (1.7)-(1.8). It is worth mentioning that in [8] a criterion for
the global existence of a (classical) maximal solution (on some interval
$[0,t_{+})$) to (1.7)-(1.8) is established using a variation of parameter
formula. In particular, it is shown that if $t_{+}<\infty$ then necessarily we
must have
$\underset{t\rightarrow
t_{+}}{\lim\sup}\left\|u\left(t\right)\right\|_{L^{\infty}\left(\Omega\right)}=\infty.$
Therefore, it appears that in order to deduce global existence of classical
solutions to systems of the form (1.1)-(1.4), (1.5), it is generally required
that we should deduce bounds on the solutions in $L^{\infty}$-norm (see [31]
also).
Finally, the $L^{p}$-$L^{\infty}$ smoothing property also becomes an essential
tool in attractor theory where it can be used to establish the existence of an
absorbing set in $L^{\infty}$-norm if this property can be deduced easily in
$L^{p}$-norm for some finite $p$ (in many applications in physics and
mechanics, $p$ is equal to either $1$ or $2$). Recall that a subset
$\mathcal{B}\subset\mathcal{H}$, where $\mathcal{H}$ is a topological space
endowed with a given metric, is called _absorbing_ if the orbits corresponding
to bounded sets $\mathcal{V}$ of initial data enter into $\mathcal{B}$ after a
certain time (which may depend on the set $\mathcal{V}$) and will stay there
forever. Moreover, we note that in order to study the long term behavior of
the parabolic system (1.1)-(1.2), if the absorbing property holds in
$L^{\infty}$-norm, the growth rate of the nonlinearities $f_{i}$, $g_{i}$ and
$h_{i}$ with respect to $u_{i}$ becomes nonessential for further
investigations of attractors. Indeed, the absorbing property can be also
established in higher-order $W^{s,p}$-norms with relative ease provided that
it is known in $L^{\infty}$-norm. For the application of this property to
attractor theory for parabolic equations of the form (1.7), (1.8), see [22,
23], where explicit dimension estimates for the global attractor for
(1.7)-(1.8) are obtained.
The main goal of this paper is to deduce sufficiently general conditions on
the diffusions and sources in (1.1)-(1.3), which would prevent blowup of any
solution in the $L^{\infty}$-norm, and show that the parabolic system under
consideration is dissipative in a suitable sense. We outline the plan of the
paper, as follows. In Section 2, we give the full derivation of systems of the
form (1.1)-(1.3), and give physical interpretations to the dynamic boundary
condition (1.2) for the porous-medium equation, and some models in ecology. In
Section 3, after we introduce some notations and preliminary facts, we give
the statements of our main results and some further applications. Finally, in
Section 4 we provide the full proofs of these results.
## 2\. Derivation and interpretation
Let $\Gamma\subset\mathbb{R}^{N-1}$ consists of two disjoint open subsets
$\Gamma_{1}$ and $\Gamma_{2}$, each
$\overline{\Gamma}_{i}\backslash\Gamma_{i}$ is a $S$-null subset of $\Gamma$
and $\Gamma=\overline{\Gamma}_{1}\cup\overline{\Gamma}_{2}$ with
$\Gamma_{1}\subseteq\Gamma$. We shall only give the derivation in the case of
the scalar equation
(2.1) $\partial_{t}u-\text{div}\left(a\left(u\right)\nabla
u\right)+f\left(u\right)=h_{1}\left(x\right),$
equipped with (nonlinear) dynamic boundary conditions
(2.2) $\partial_{t}u+a\left(u\right)\nabla
u\cdot\mathbf{n}+g\left(u\right)=h_{2}\left(x\right),$
on $\Gamma_{1},$ and Dirichlet boundary conditions
(2.3) $u_{\mid\Gamma_{2}}=0.$
We can easily extend our arguments to systems as well (see below). Equations
(2.1)-(2.3) are also subject to the initial conditions
(2.4) $u_{\mid t=0}=u_{0}\text{ in }\Omega,\quad u_{\mid t=0}=v_{0}\text{ on
}\Gamma.$
Standard derivations of the porous medium equation always use the principle
“amount of fluid in equals amount of fluid out” over a region $\Omega$,
occupied by either a liquid or gas, and is based on the fact that this fluid
diffuses from locations of higher to those of lower pressure. In the
traditional approach, the porous medium equation is assumed to hold in the
region $\Omega$ and then the boundary conditions are appended later. There are
three standard boundary conditions that specify the density on the boundary of
$\Omega$; they are Dirichlet and Neumann-Robin type of boundary conditions
(see, e.g., [38]). Dynamic boundary conditions for porous medium equations
seem to have appeared before in different contexts [6, 19, 36, 35]. For
instance, [6] deals with the modelling of the rain water infiltration through
the soil above an aquifer in regimes where there is runoff at the ground
surface. In general, all rain water infiltrates into the soil, but if the
rainfall event is particularly intense, the maximum draining capacity of the
soil is exceeded. In this case, dynamic boundary conditions (see (2.8) below)
are needed to describe the saturation of layers near the ground surface (cf.
also [19, 36]). Porous-medium like systems (2.1)-(2.3) can also be found as
part of (larger) coupled systems of partial differential equations (such as,
(1.1)-(1.4)) describing the vertical movement of water and salt in a domain
splitted in two parts: a water reservoir and a saturated porous medium below
it, in which a continuous extraction of fresh water takes place (for instance,
by the roots of mangroves) [24]. Such problems are formulated in terms of
equations for the salt concentration and the water flow in the porous medium,
with a dynamic boundary condition which connects both subdomains. Finally,
dynamic boundary conditions similar to (2.2) also appear in certain classes of
parabolic equations with boundary hysteresis (see, e.g., [35, Section 4] and
the references therein). For some applications of dynamic boundary conditions
for physiologically structured populations with diffusion we refer the reader,
for instance, to [20].
For all the phenomena of the kind discussed here, the method of introducing
dynamic boundary conditions seems ad hoc. It would be more natural if such
boundary conditions could be derived in the context of energy balance and
constitutive laws. Moreover, the usual derivation of the porous medium
equation with standard boundary conditions does not show how to model, for
instance, a water source, which is located on the boundary of $\Omega$. To
this end, we shall rethink the usual derivation of the porous medium equation
(2.1), by making essential connections between the differential equation (2.1)
and the boundary conditions (2.2)-(2.3), and, thus, try to convince the reader
that our new perspective is more natural than the traditional way. Let
$p\left(x,t\right)$ denote the pressure of fluid at $x\in\Omega$ and time
$t>0$. Consider the mass of fluid in an element of volume $V$ given by
$\int_{V}\alpha\left(x\right)u\left(x,t\right)dx,$
where $\alpha\left(x\right)>0$ defines the porosity of medium at the point
$x\in\Omega$. Similarly, we define
$\int_{\Gamma}\beta\left(x\right)v\left(x,t\right)dS$
as the mass of fluid across the surface $\Gamma$, where $\beta\left(x\right)$
is such that
$\Gamma_{3}:=\left\\{x\in\Gamma_{1}:\beta\left(x\right)>0\right\\}$
is a set of positive measure and $\Gamma_{3}\subseteq\Gamma_{1}$. In what
follows, we shall take $\Gamma_{2}=\varnothing$ for the sake of exposition, so
that $\Gamma_{1}\equiv\Gamma$. The flux $\mathbb{J}\left(x,t\right)$, at which
the fluid moves across a surface element $S$ with normal $\mathbf{n}$, is
given by
$\int_{S}\mathbb{J}\left(x,t\right)\cdot\mathbf{n}dS.$
Suppose now there is a source on the boundary $\Gamma$ to be represented by a
function
$\Psi=\Psi(t,x,u,\nabla u).$
The amount of fluid leaving the region is still given by
$\int_{\Gamma}\mathbb{J}\left(x,t\right)\cdot\mathbf{n}dS$, but the amount of
fluid leaving into the region must also take into account the action of the
source $\Psi$ on $\Gamma$. It is worth pointing out that, in practice, when
rainfall only partially infiltrates the soil, the water will accumulate on the
ground surface $\Gamma_{1}$ as the surface layer becomes saturated; hence,
necessarily, $\Psi\neq 0$ on $\Gamma$. We use the measure space
$\left(\overline{\Omega},d\mu\right)$ which we redefine as
$\left(\Omega,dx\right)\oplus\left(\Gamma,dS\right)$. Then the conservation of
fluid in $\overline{\Omega}$ takes the form
(2.5) $\displaystyle\partial_{t}\left(\int_{\Omega}\alpha\left(x\right)\rho
dx+\int_{\Gamma}\beta\left(x\right)\eta
dS\right)+\int_{\Gamma}\mathbb{J}\cdot\mathbf{n}dS$
$\displaystyle=\int_{\Omega}\Xi dx+\int_{\Gamma}\Psi dS,$
where $\Xi=\Xi\left(x,t,u\right)$ denotes any volume source density function.
Notice that equation (2.5) must also account for a term like
$\int_{\Gamma}\beta\left(x\right)vdS,$
due to the presence of the source density $\Psi$ at $\Gamma$. Assuming that
$\mathbb{J}$ is sufficiently smooth and applying the divergence theorem in
(2.5), we deduce
(2.6)
$\displaystyle\partial_{t}\left(\int_{\Omega}\alpha\left(x\right)udx+\int_{\Gamma}\beta\left(x\right)udS\right)+\int_{\Omega}div\left(\mathbb{J}\right)dx$
$\displaystyle=\int_{\Omega}\Xi dx+\int_{\Gamma}\Psi dS.$
Assuming that the density functions $u,v$ are also differentiable with respect
to $t>0$ and since (2.6) holds for any subdomain $\Omega_{0}\subseteq\Omega$,
the usual argument yields the following differential equation
(2.7)
$\partial_{t}\left(\alpha\left(x\right)u\left(x,t\right)\right)=-\text{div}\left(\mathbb{J}\left(x,t\right)\right)+\Xi\left(x,t,u\left(x,t\right)\right)\text{,
}x\in\Omega,\text{ }t>0.$
Then, from (2.6) the boundary condition becomes
$\int_{\Gamma}\left[\partial_{t}\left(\beta\left(x\right)u\left(x,t\right)\right)-\Psi\left(x,t,u\left(x,t\right),\nabla
u\left(x,t\right)\right)\right]dS=0,\text{ }t>0,$
which clearly holds if
(2.8)
$\partial_{t}\left(\beta\left(x\right)u\left(x,t\right)\right)-\Psi\left(x,t,u\left(x,t\right),\nabla
u\left(x,t\right)\right)=0\text{, for }x\in\Gamma,\text{ }t>0.$
Darcy’s law states that the flux $\mathbb{J}$ depends on the pressure gradient
so it takes the form
(2.9)
$\mathbb{J}\left(x,t\right)=-\frac{\mathcal{K}\left(x\right)}{\nu}u\left(x,t\right)\nabla
p\left(x,t\right),\text{ }x\in\Omega,\text{ }t>0,$
where $\nu>0$ is the viscosity of the fluid and $\mathcal{K}\left(x\right)$
defines the permeability of the porous medium. Finally, if one also makes the
assumption that the pressure $p$ is described by an equation of state
involving the density, $p=b\left(u\right)$, then substituting the appropriate
quantities in (2.7), we obtain the following porous medium equation
(2.10)
$\partial_{t}\left(\alpha\left(x\right)u\left(x,t\right)\right)=\frac{1}{\nu}\text{div}\left(\mathcal{K}\left(x\right)a\left(u\left(x,t\right)\right)\nabla
u\left(x,t\right)\right)+\Xi\left(x,t,u\left(x,t\right)\right),\text{ in
}\Omega,\text{ }t>0,$
where we have set $a\left(t\right)\equiv tb\left(t\right)$. The function $b$
that relates the density to pressure is, in general, monotone and, in fact,
strictly increasing in many applications of the type of fluid being considered
in the literature.
Finally, let us now focus on the boundary condition (2.8). We now show that a
quite large class of boundary conditions for equation (2.1) can be written in
this way for various choices of $\Psi$. We emphasize that in this formulation
the boundary conditions arise naturally in the formulation of the problem.
Suppose first $\Gamma_{3}\equiv\Gamma_{1}$ (i.e., $\beta\left(x\right)>0$ a.e.
in $\Gamma_{1}$) and $\beta\in C^{1}\left(\Gamma_{1}\right).$ Choosing
$\Psi\equiv 0$ (i.e., no source is located at $\Gamma_{1}$), so that
$\partial_{t}u\equiv 0$ on $\Gamma_{1}=\Gamma,$ therefore
(2.11) $u\left(x,t\right)=u_{0}\left(x\right),$
for $x\in\Gamma_{1}$ and $t\geq 0,$ where $u_{0}$ is the initial condition
associated with equation (2.1). Thus, we obtain a Dirichlet boundary condition
for $u$. In order to derive an inhomogeneous Neumann boundary condition, we
suppose that $\Psi$ only depends on $t$. Then, if $u$ is sufficiently regular,
we have
$\partial_{t}u(x,t)=\left(1/\beta\left(x\right)\right)\Psi\left(t\right)$ on
$\Gamma_{1},$ for any $t>0$. Hence, if $\Gamma_{1}$ is smooth enough as well,
we have $\partial_{t}\left(\nabla
u\right)=\nabla\left(\partial_{t}u\right)=\mathbf{\gamma}\left(x\right)\Psi\left(t\right)$
on $\overline{\Omega}\times(0,+\infty),$ for some
$\mathbf{\gamma}\left(x\right)\in\mathbb{R}^{N}$. This entails that $\nabla
u\left(x,t\right)=\mathbf{H}\left(x,t\right)$ holds for
$(x,t)\in\Gamma_{1}\times(0,+\infty)$ and some smooth function
$\mathbf{H}\in\mathbb{R}^{N}.$ Therefore, we have
(2.12) $\nabla
u\left(x,t\right)\cdot\mathbf{n}=\mathbf{H}\left(x,t\right)\cdot\mathbf{n},\quad(x,t)\in\Gamma_{1}\times(0,+\infty).$
To obtain a Robin boundary condition, we set $\Psi\left(x,t,u,\nabla
u\right)=\beta\left(x\right)e^{Cr}q\left(t\right),$ for some $C\in\mathbb{R}$,
where $r$ is defined as the parameter describing the line $\ell$ which passes
through $x$ and contains ${\mathbf{n}}$ such that $r>0$ at all points on
$\ell\cap\Omega$ which are close to $x.$ We thus have
$\partial_{t}u\left(x,t\right)=e^{Cr}q\left(t\right)$ which implies
$\nabla\left(\partial_{t}u\left(x,t\right)\right)\cdot\mathbf{n}\left(x,t\right)=\partial_{t}\left(\nabla
u\left(x,t\right)\cdot\mathbf{n}\right)\left(x,t\right)=Ce^{Cr}q\left(t\right),$
for $(x,t)\in\Gamma_{1}\times(0,+\infty)$. Therefore, we infer
$\partial_{t}\left(\nabla
u\left(x,t\right)\cdot\mathbf{n}\right)\left(x,t\right)-C\partial_{t}u\left(x,t\right)=0,\qquad\text{
on }\Gamma_{1}\times(0,+\infty),$
so that
(2.13) $\nabla
u\left(x,t\right)\cdot\mathbf{n}-Cu(x,t)=j\left(x\right),\quad(x,t)\in\Gamma_{1}\times(0,+\infty).$
We have thus recovered the most common boundary conditions. On the other hand,
in order to model a water source placed on the boundary $\Gamma_{1},$ we may
assume that $\Psi$ only depends nonlinearly on the flux $\nabla
u\cdot\mathbf{n}$ across the boundary as well as on a nonlinear source
$g\left(u\right),$ which may represent effects of reaction or absorption at
$\Gamma_{1}$. That is, we let
$\Psi\left(x,u,\nabla u\right)=-a\left(u\right)\nabla
u\cdot\mathbf{n}-g\left(u\right)-h_{2}\left(x\right).$
Then, the resulting boundary condition for equation (2.1) becomes
(2.14)
$\beta\left(x\right)\partial_{t}u\left(x,t\right)+a\left(u\left(x,t\right)\right)\nabla
u\left(x,t\right)\cdot\mathbf{n}+g\left(u\left(x,t\right)\right)=h_{2}\left(x\right),$
for $\left(x,t\right)\in\Gamma_{1}\times(0,+\infty)$. This is a reasonably
general condition which contains the usual (homogeneous) ones along with the
so-called dynamic boundary condition when $\Gamma_{3}=\Gamma_{1}\equiv\Gamma$.
When ponding or surface runoff occurs at the surface $\Gamma_{1}$, it also
includes the dynamic boundary condition contained in Filo-Luckhaus [19].
Finally, we notice that $\beta\left(x\right)$ may be such that
$\Gamma_{3}\neq\Gamma_{1}$, so that we can also cover the case where the
boundary conditions (2.14) are dynamic only on a part of the boundary of
$\Gamma_{1}$. If the source $\Psi$ also depends on $v$ in the tangential
coordinates of the boundary, i.e., $\Psi=\Psi\left(x,u,\nabla
u,\nabla_{\Gamma_{1}}u\right),$ then we can also model local diffusion in
(2.14) by incorporating the elliptic Laplace-Beltrami operator
$\Delta_{\Gamma_{1}}$ (or other nonlinear differential operators) on the
manifold $\Gamma_{1}\subset\mathbb{R}^{N-1}$.
We will now give a physical interpretation of the effect of a water source on
the patch $\Gamma_{1}$, at least in some cases. A similar approach was also
used in the derivation of heat and wave equations (see, e.g., [25]). We will
mainly focus on the following boundary condition
(2.15) $\partial_{t}u+a\left(u\right)\nabla u\cdot\mathbf{n}=0,\text{ on
}\Gamma_{1}\times\left(0,\infty\right).$
We work in an infinitesimal region on the boundary. Choose a point
$x\in\Gamma_{1}$ and let $B_{\kappa}\left(x\right)$ be a ball of radius
$\kappa>0$ about $x$. Since $\Gamma_{1}$ is regular, we can choose a
coordinate system for $B_{\kappa}\left(x\right)\cap\overline{\Omega}$ so that
the boundary of $B_{\kappa}\left(x\right)\cap\overline{\Omega}$ in the
transformed coordinate system is flat, $x$ is mapped to
$\overline{x}=\left(x_{1},x_{2},...,x_{N-1},0\right)$, that is, the boundary
$\Gamma$, at least locally near $x$ lies on the hyperplane $x_{N}=0$. Then the
outward unit normal $\mathbf{n}$ to $\Gamma$ at $x$ is the unit vector in the
direction of $e_{N}$ which we will denote by $r$. Then, locally near $x,$
(2.15) becomes
(2.16) $\partial_{t}u+\partial_{r}\left(A\left(u\right)\right)=0,\text{
}\left(r,t\right)\in\left(0,r_{0}\right)\times\left(0,t_{0}\right),$
for some sufficiently small positive constants $r_{0}$, $t_{0}$. Observe that
(2.16) resembles nothing more than a scalar _conservation law_ , where we have
set $A\left(u\right)=\int_{0}^{u}a\left(t\right)dt$. Equation (2.16), subject
to initial condition $u\left(r,0\right)=v_{0}\left(r\right),$
$r\in\left(0,r_{0}\right),$ possesses interesting types of fundamental
solutions such as, travelling waves that describe the movement of a mass of
fluid in the direction of the unit normal $\mathbf{n}\in\mathbb{R}^{N}$, and
source-type solutions starting from a finite mass concentrated at a single
point of space, say,
$v_{0}\left(r\right)=\mathcal{C}\delta_{0}\left(r\right),$ $\mathcal{C}>0$.
In the latter case, explicit self-similar solutions of the form
$u(r,t)=\Theta\left(r/t\right)$ are well-known to exist, as a consequence of
the invariance of (2.16) under the scaling
$\left(r,t\right)\longmapsto\left(\lambda r,\lambda t\right),$
$\lambda\in\mathbb{R}$. More precisely, from (2.16) $\Theta$ clearly has to
satisfy
$\zeta\Theta^{{}^{\prime}}\left(\zeta\right)-\left(A\left(\Theta\left(\zeta\right)\right)\right)^{{}^{\prime}}=0$
and this yields, formally, to
$u\left(r,t\right)=\Theta\left(r/t\right)=(A^{{}^{\prime}})^{-1}\left(r/t\right),$
as long as, $(A^{{}^{\prime}})^{-1}$ is well defined at least in a
sufficiently small real interval. In the former case, one can search for
particular solutions of (2.16) in the form
$u\left(r,t\right)=\eta\left(r-ct\right),$ where $c\in\mathbb{R}$ is the speed
of the travelling wave and $\eta$ has to be determined. Substituting the
expression for $u\left(r,t\right)$ in (2.16), we deduce that $c$ is an
eigenvalue (with $\eta^{{}^{\prime}}$ as eigenvector) for
$\left(-c+a\left(\eta\left(r-ct\right)\right)\right)\eta^{{}^{\prime}}\left(r-ct\right)=0.$
Under appropriate structural conditions on $a\left(\cdot\right)$ (see, e.g.,
Section 3), this eigenvalue problem is strictly hyperbolic (cf., e.g., [18,
Chapter 11]) with speed $c=c\left(\eta\right)>0,$ hence a wave-like solution
$\eta=\eta\left(r-ct\right)$ to (2.16) can be found. This is a unidirectional
wave which travels _into_ the region $\Omega$. We can now map back to our
original coordinate system to find that
$u\left(r,t\right)=\eta\left(x-ct\mathbf{n}\right)$ is a solution to (2.15).
In plain physical terms, the mass of fluid is carried by the wave $\eta$ into
an infinitesimal layer near the boundary $\Gamma$. This wave will cease to
exist after some small time since once inside $\Omega$, the primary process is
governed by nonlinear diffusion in the porous medium equation (2.10).
It is easy to extend our derivation to systems of the form (1.1)-(1.4) and to
give them phyiscal interpretations. For instance, these systems also occur in
the pharmaceutical industry by mathematical modells for the development of
blood coagulation treatments with specific coagulation factors [14, 28, 32].
The systems (1.1)-(1.4) are also motivated by diffusion processes on metric
graphs and ramified spaces, which yield interface problems for quantum graphs
with coupled dynamic boundary conditions at the nodes (see, e.g., [33] and
references therein). On the other hand, the reaction-diffusion equations (1.1)
(for $A_{i}\left(u_{i}\right)=d_{i}u_{i},$ $i\in\mathbb{N}_{m}$) arise as
models for the densities $u_{i},$ $i\in\mathbb{N}_{m}$ of substances or
organisms that disperse through space by Brownian motion, random walks,
hydronamic turbulance or similar mechanisms. These equations are widely used
as models to account for spatial effects in ecological enviroments [9]. For
equations (1.1), the Dirichlet boundary condition (2.11) specifies the density
$u_{i}$ of species at the boundary $\Gamma$, with an interpretation that
anything that reaches the boundary $\Gamma$ of $\Omega$ leaves and does not
return. If $u_{0}\equiv 0,$ then (2.11) may be interpreted as if the species
suffers extiction if say the patch $\Gamma_{1}$ where the individuals live is
toxic. The (homogeneous, $\mathbf{H}\equiv\mathbf{0}$) Neumann boundary
condition (2.12) says that nothing can cross the boundary of $\Omega.$ Another
relation in ecological models is the Robin boundary condition (2.13) with
$j\left(x\right)\equiv 0$ and $C=C_{i}\in\mathbb{R}^{\ast},$
$i\in\mathbb{N}_{m}$ which can be interpreted as saying that when organisms
reach the boundary some leave it but some do _not_ depending on the sign of
$C_{i}$. Finally, the other _not_ so common condition is the Wentzell-type
(dynamic) boundary condition (2.14) which states that change in the density of
individuals at $\Gamma_{1}$ is a function of their flux in the normal
direction across $\Gamma_{1}$ and some other function of density if no
dispersive effects along $\Gamma_{1}$ are taken into account. Following our
reasonning above, this type of boundary condition (1.3) can be interpreted as
saying that some individuals may choose to live on the patch $\Gamma_{1}$ but
some may _not_ and can choose to return to the region $\Omega$, where spatial
diffusion coupled with reaction in the bulk $\Omega$ is the main mechanism for
population movement and interaction. Suppose, for instance, that certain
critical resources for a specific population $u_{i},$ $i\in\mathbb{N}_{m},$
are available only on $\Gamma_{1}$. Then $u_{i}$ must obey the rule
(2.17) $\partial_{t}u_{i}+d_{i}\nabla
u_{i}\cdot\mathbf{n}+h\left(x\right)u_{i}=0,\text{ on
}\Gamma_{1}\times\left(0,\infty\right),$
which says that the density $u_{i}$ diffuses (in an infinitesimal layer near
$\Gamma_{1}$) toward the patch $\Gamma_{1}$ in the direction of normal flux.
Again the main mechanism for this behavior here is the influence of external
forces on $\Gamma$ on a particular population $u_{i}$. Of course, in this
context the function $h\left(x\right)$ plays the role of a resource density
function on $\Gamma_{1}$, and it can generally depend also on time. In fact,
it is not hard to imagine a typical scenario where predatory individuals are
preferentially concentrated around valued resources on $\Gamma_{1}$ where the
likelihood of prey is greatest. Hence, in the more general case of (2.14) the
state densities $u_{i}$ may be also allowed to carry mass on $\Gamma_{1}$ in
contrast to the usual Robin condition for which the mass is always zero. This
general description (1.3) along the patch $\Gamma_{1}$ can have substantial
consequences on the dynamics of various ecological enviroments modelled by
reaction-diffusion systems. We give a short reasonning for this behavior as
follows. In the case of a scalar (non-degenerate) diffusion equation ($m=1,$
$a_{1}\left(\cdot\right)\equiv d_{1}$), we have shown in [22, 23] (say, in
dimension $N\geq 3$) that problem (1.7)-(1.8) posseses a finite dimensional
global attractor $\mathcal{A}_{\text{dyn}}$ whose dimension is essentially of
_different order_ than the dimension of the global attractor
$\mathcal{A}_{\text{D-N-R}}$ for the same parabolic problem (1.7) with a
Dirichlet/Neumann-Robin boundary condition (2.11)-(2.13) (cf., also [23]). In
particular, the correct asymptotics for the Hausdorff and, respectively, the
fractal dimensions of $\mathcal{A}_{\text{dyn}}$ are
(2.18) $\dim_{H}\mathcal{A}_{\text{dyn}}\sim
C\left(f_{1},g_{1}\right)\frac{\left|\Gamma\right|}{\left(\nu
b\right)^{N-1}}\text{, }\dim_{F}\mathcal{A}_{\text{dyn}}\sim
C\left(f_{1},g_{1}\right)\frac{\left|\Gamma\right|}{\left(\nu
b\right)^{N-1}},$
as long as $\nu\rightarrow 0^{+}$. Here, $C=C\left(f_{1},g_{1}\right)$ is a
positive constant that is independent of the size of $\Omega,$ but depends
only on $f_{1}$ and $g_{1}$, and $\left|\Gamma\right|$ denotes the natural
Lebesgue surface measure of $\Gamma\subset\mathbb{R}^{2}$. Note that the
asymptotics for the dimension of $\mathcal{A}_{\text{D-N-R}}$ is actually
$C\left(f_{1}\right)\left|\Omega\right|/\left(\nu b\right)^{N/2}$, as
$\nu\rightarrow 0^{+}$ (see, e.g., [5]) suggesting that the dynamics on
$\mathcal{A}_{\text{dyn}}$ is _qualitatively_ different than that on
$\mathcal{A}_{\text{D-N-R}}$ even though both systems are gradient like [23]
(i.e., both problems possess a global Lyapunov function). The asymptotic
estimates in (2.18) are essentially determined by the instability indices of a
properly chosen family of (hyperbolic) equilibria $u_{\ast}$ (see, e.g., [5],
[22]). One achieves a lower bound like (2.18) by computing the dimension of
the unstable eigenspace $E^{u}$ of the linearization of (1.7)-(1.8) around a
constant equilibrium $u_{\ast}$. In this case, the linearized system possesses
at least $n\sim C\left(f_{1},g_{1}\right)\left|\Gamma\right|/\left(\nu
b\right)^{N-1}$ (as $\nu\rightarrow 0^{+}$) unstable solutions. This points
out once again to the destabilizing nature of the dynamic boundary condition
(2.17) even when the dynamics in the bulk $\Omega$ is essentially strictly
_linear_ (see, Appendix). We emphasize that this kind of behavior _cannot_
hold for the Dirichlet/Neumann-Robin boundary condition (2.11)-(2.13).
## 3\. Main results
The natural phase-space for problems of the form (1.1)-(1.4) is
$\mathbb{X}^{s_{1},s_{2}}:=L^{s_{1}}(\Omega)\oplus
L^{s_{2}}(\Gamma)=\\{U=\binom{u_{1}}{u_{2}}:\;u_{1}\in
L^{s_{1}}(\Omega),\;u_{2}\in L^{s_{2}}(\Gamma)\\},$
$s_{1},s_{2}\in\left[1,+\infty\right],$ endowed with norm
(3.1)
$\left\|U\right\|_{\mathbb{X}^{s_{1},s_{2}}}=\left(\int_{\Omega}\left|u_{1}\left(x\right)\right|^{s_{1}}dx\right)^{1/s_{1}}+\left(\int_{\Gamma}\left|u_{2}(x)\right|^{s_{2}}dS_{x}\right)^{1/s_{2}},$
if $s_{1},s_{2}\in[1,\infty),$ and
$\displaystyle\|U\|_{\mathbb{X}^{\infty}}$
$\displaystyle:=\max\\{\|u_{1}\|_{L^{\infty}(\Omega)},\|u_{2}\|_{L^{\infty}(\Gamma)}\\}$
$\displaystyle\simeq\|u_{1}\|_{L^{\infty}(\Omega)}+\|u_{2}\|_{L^{\infty}(\Gamma)}.$
We agree to denote by $\mathbb{X}^{s}$ the space $\mathbb{X}^{s,s}.$ Moreover,
we have
(3.2) $\mathbb{X}^{s}=L^{s}\left(\overline{\Omega},d\mu\right),\text{
}s\in\left[1,+\infty\right],$
where the measure $d\mu=dx_{\mid\Omega}\oplus dS_{x}{}_{\mid\Gamma}$ on
$\overline{\Omega}$ is defined for any measurable set
$A\subset\overline{\Omega}$ by
(3.3) $\mu(A)=|A\cap\Omega|+S(A\cap\Gamma).$
Identifying each function $\theta\in C\left(\overline{\Omega}\right)$ with the
vector $\Theta=\binom{\theta_{\mid\Omega}}{\theta_{\mid\Gamma}}$, we have that
$C(\overline{\Omega})$ is a dense subspace of $\mathbb{X}^{s}$ for every
$s\in[1,\infty)$ and a closed subspace of $\mathbb{X}^{\infty}$. In general,
any vector $\theta\in\mathbb{X}^{s}$ will be of the form
$\binom{\theta_{1}}{\theta_{2}}$ with $\theta_{1}\in
L^{s}\left(\Omega,dx\right)$ and $\theta_{2}\in L^{s}\left(\Gamma,dS\right),$
and there need not be any connection between $\theta_{1}$ and $\theta_{2}$.
Next, we set
$\mathcal{X}^{s_{1},s_{2}}:=\prod\nolimits_{i\in
I_{m}}L^{s_{1}}\left(\Omega\right)\times\prod\nolimits_{i\in
J_{m}}\mathbb{X}^{s_{1},s_{2}},\text{ for any
}s_{1},s_{2}\in\left[1,+\infty\right],$
where, for any given set $X,$
$\prod\nolimits_{i\in I}X:=\underset{\left|I\right|\text{
times}}{\underbrace{X\times...\times X}}.$
The norm in the space $\mathcal{X}^{s_{1},s_{2}},$ for any
$s_{1},s_{2}\in[1,+\infty)$ is
(3.4)
$\left\|\overrightarrow{u}\right\|_{\mathcal{X}^{s_{1},s_{2}}}:=\sum\nolimits_{i\in
I_{m}}\left\|u_{i}\right\|_{L^{s_{1}}\left(\Omega\right)}+\sum\nolimits_{i\in
J_{m}}\left(\left\|u_{i}\right\|_{L^{s_{1}}\left(\Omega\right)}+\delta_{i}\left\|u_{i}\right\|_{L^{s_{2}}\left(\Gamma\right)}\right),$
where $\overrightarrow{u}=\left(u_{1},...,u_{m}\right)$, while the norm in the
space $\mathcal{X}^{\infty}:=\mathcal{X}^{\infty,\infty}$ is naturally given
by
$\left\|\overrightarrow{u}\right\|_{\mathcal{X}^{\infty}}:=\max\\{\max_{i\in\mathbb{N}_{m}}\left\|u_{i}\right\|_{L^{\infty}\left(\Omega\right)},\max_{i\in
J_{m}}\left\|u_{i}\right\|_{L^{\infty}\left(\Gamma\right)}\\}.$
If $s_{1}=s_{2}=s,$ we will simply write $\mathcal{X}^{s}$ instead of
$\mathcal{X}^{s_{1},s_{2}}$. Finally, without further abuse of notation, we
will also refer to $\mathcal{X}^{\overrightarrow{r}},$
$\overrightarrow{r}=\left(r_{1},...,r_{m}\right)$, as the following Banach
space
$\mathcal{X}^{\overrightarrow{r}}:=\prod\nolimits_{i\in
I_{m}}L^{r_{i}}\left(\Omega\right)\times\prod\nolimits_{i\in
J_{m}}\mathbb{X}^{r_{i},r_{i}},$
endowed with the natural norm in (3.4).
Let us now state our main hypotheses on the source terms $f_{i},g_{i},$
$h_{i}$ and nonlinear diffusions $a_{i}$, for each $i\in\mathbb{N}_{m}.$
Conditions on $a_{i}:$ The Carathéodory functions $a_{i}$ (with values in
$\mathbb{R}$) satisfy the condition: $\exists\alpha_{i}>0,$ $\forall
s_{i}\in\mathbb{R}$ such that
(3.5) $a_{i}\left(s_{i}\right)\geq\alpha_{i}\left|s_{i}\right|^{p_{i}},\text{
}i\in\mathbb{N}_{m},$
for some nonnegative $p_{i}.$
Conditions on $f_{i}$, $g_{i},$ $h_{i}:$ The Carathéodory functions $f_{i}$,
$g_{i},$ $h_{i}$ (with values in $\mathbb{R}$) satisfy the conditions:
$\exists C_{f_{i}},C_{g_{i}},C_{h_{i}}>0,$ for almost all $\left(x,t\right),$
$\forall s_{i}\in\mathbb{R}$, such that
(3.6)
$\left\\{\begin{array}[]{l}\sum\nolimits_{i\in\mathbb{N}_{m}}f_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\geq-\sum\nolimits_{i\in\mathbb{N}_{m}}C_{f_{i}}\left|s_{i}\right|^{2}-\widetilde{C}_{f},\\\
\sum\nolimits_{i\in
J_{m}}g_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\geq-\sum\nolimits_{i\in
J_{m}}C_{g_{i}}\left|s_{i}\right|^{2}-\widetilde{C}_{g},\\\
\sum\nolimits_{i\in
I_{m}}h_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\geq-\sum\nolimits_{i\in
I_{m}}C_{h_{i}}\left|s_{i}\right|^{2}-\widetilde{C}_{h},\end{array}\right.$
for some nonnegative $\widetilde{C}_{f},$ $\widetilde{C}_{g}$ and
$\widetilde{C}_{h}$.
The question of global existence and the $L^{p}$-$L^{\infty}$ smoothing
property for solutions of (1.1)-(1.4) can be stated for the function
$\overrightarrow{u}=\left(u_{1},...,u_{m}\right)$, as follows. We say that the
parabolic system (1.1)-(1.4) satisfies Property P$\left(r_{1},r_{2}\right)$,
for some finite $r_{1},r_{2}\geq 1$, if, for all $i\in\mathbb{N}_{m}$, any of
the following conditions holds:
(i) There exists a positive function $Q,$ independent of initial data, and a
positive constant $\eta$, such that
(3.7)
$\sup_{t\geq\eta>0}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{r_{1},r_{2}}}\leq
Q\left(\eta\right).$
(ii) There exists a positive constant $\mathcal{C}$, independent of initial
data, such that
(3.8)
$\underset{t\rightarrow\infty}{\lim\sup}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{r_{1},r_{2}}}\leq\mathcal{C}.$
(iii) If $\overrightarrow{u}_{0}=\left(u_{10},...,u_{m0}\right)\in$
$\mathcal{X}^{\infty},$ $i\in\mathbb{N}_{m}$, there exists a positive function
$Q$, independent of initial data, such that
(3.9) $\sup_{t\geq
0}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{r_{1},r_{2}}}\leq
Q\left(\left\|\overrightarrow{u}_{0}\right\|_{\mathcal{X}^{\infty}}\right).$
The first result concerning the $L^{\infty}$-estimate for solutions of
(1.1)-(1.4) in the non-degenerate case, shows that if either one of the
properties for P$\left(s_{1},s_{2}\right)$ above holds apriori for some finite
$s_{1},s_{2}\geq 1$, then it also holds for $s_{1}=s_{2}=\infty$.
###### Theorem 3.1.
Let the assumptions (3.5), (3.6) be satisfied such that $p_{i}=0$, for all
$i\in\mathbb{N}_{m}$ (i.e., (1.5) holds). Suppose that the system (1.1)-(1.4)
satisfies the property P$\left(1,1\right)$-(i) (respectively, (ii) or (iii)),
then it also satisfies P$\left(\infty,\infty\right)$-(i) (respectively, (ii)
or (iii)).
###### Remark 3.1.
Note that we can also consider the more general case in which
$a_{i}\left(u_{i}\right)$ is replaced by
$a_{i}\left(x,t,\overrightarrow{u}\right),$ $i\in\mathbb{N}_{m}$, that is, the
equations (1.1) are strongly coupled in their diffusions. In this case, we
must replace assumption (3.5) by
(3.10)
$a_{i}\left(x,t,\overrightarrow{s}\right)\geq\alpha_{i}\left|\overrightarrow{s}\right|^{p_{i}},\text{
}\forall\overrightarrow{s}\in\mathbb{R}^{m},$
for almost all $\left(x,t\right)$, and notice that all the computations
performed in the proof of Theorem 3.1 hold automatically since
$\left|\overrightarrow{s}\right|^{p_{i}}\geq\left|s_{i}\right|^{p_{i}}$ for
any $\overrightarrow{s}=\left(s_{1},...,s_{m}\right)\in\mathbb{R}^{m}$ (of
course, in that case $p_{i}=0$ by assumption, for all $i\in\mathbb{N}_{m}$).
Theorem 3.1 can be also extended to systems with $p$-Laplacian diffusions,
i.e.,
$a_{i}=a_{i}\left(u_{i},\left|\nabla
u_{i}\right|^{\varrho_{i}-2}\right)\geq\alpha_{i}\left|\nabla
u_{i}\right|^{\varrho_{i}-2},$
for some $\varrho_{i}\geq 2$, $i\in\mathbb{N}_{m}$, by following, for
instance, [23].
The second result is concerned with the full degenerate case (1.1)-(1.4) when
the assumptions of Theorem 3.1 do not hold (in particular, if it happens that
$p_{i}>0$ for some $i\in\mathbb{N}_{m}$). The proof is based on a truncation
technique which was originally developed by DeGiorgi to study the regularity
of solutions to elliptic equations, and then extensively used by many authors
to study weak solutions to degenerate parabolic systems, subject to the usual
static boundary conditions (see, e.g., [10, 27]). Here, we extend DeGiorgi’s
method to problems of the form (1.1)-(1.4). In order to avoid additional
technicalities due to the different conditions that one can assign on the
boundary $\Gamma$ for each $u_{i},$ $i\in\mathbb{N}_{m}$, we shall focus our
attention to the case $J_{m}=\mathbb{N}_{m}$ only (i.e., we will assume that
$I_{m}=\varnothing$). In this case, we require that the following growth
assumptions hold:
(3.11)
$\left\\{\begin{array}[]{l}\left|f_{i}\left(x,t,s_{1},...,s_{m}\right)\right|\leq
C_{f}\left(\sum\nolimits_{i\in\mathbb{N}_{m}}\left|s_{i}\right|^{\theta_{i}}+1\right),\\\
\left|g_{i}\left(x,t,s_{1},...,s_{m}\right)\right|\leq
C_{g}\left(\sum\nolimits_{i\in\mathbb{N}_{m}}\left|s_{i}\right|^{\beta_{i}}+1\right),\end{array}\right.$
for some $\theta_{i},\beta_{i}>0$ and some positive constants $C_{f},C_{g}.$
###### Theorem 3.2.
Let (3.11) hold, and assume that $\exists\alpha_{i},\sigma_{i}>0$ such that
(3.12) $\alpha_{i}\left|\overrightarrow{s}\right|^{p_{i}}\leq
a_{i}\left(x,t,\overrightarrow{s}\right)\leq\sigma_{i}\left|\overrightarrow{s}\right|^{p_{i}},\text{
}i\in\mathbb{N}_{m},$
for any $\overrightarrow{s}=\left(s_{1},...,s_{m}\right)\in\mathbb{R}^{m}$.
Let
(3.13)
$\delta:=\max_{i\in\mathbb{N}_{m}}\left\\{2,\theta_{i}+1,\frac{p_{i}}{2}+1\right\\},\text{
}\gamma:=\max_{i\in\mathbb{N}_{m}}\left\\{2,\beta_{i}+1,\frac{p_{i}}{2}+1\right\\}.$
Suppose that the system (1.1)-(1.4) satisfies the property
P$\left(\delta,\gamma\right)$-(i) (respectively, (ii) or (iii)), then it also
satisfies P$\left(\infty,\infty\right)$-(i) (respectively, (ii) or (iii)). In
particular, for every $i\in\mathbb{N}_{m}$ and $T,\tau>0$ such that
$T-2\tau>0,$ the following estimate holds:
(3.14)
$\displaystyle\sup_{\left(x,t\right)\in\left[T-\tau,T\right]\times\overline{\Omega}}\left|u_{i}\left(x,t\right)\right|$
$\displaystyle\leq
Q\left(1+\left\|\overrightarrow{u}\right\|_{L^{\delta}\left(\left[T-2\tau,T\right]\times\Omega\right)}+\left\|\overrightarrow{u}\right\|_{L^{\gamma}\left(\left[T-2\tau,T\right]\times\Gamma\right)}\right),$
for some positive function $Q$ which is independent of $\overrightarrow{u},$
time and the initial data. The function $Q$ can be computed explicitly in
terms of the physical paramaters of the problem.
We will now show how to deduce the property P$\left(s_{1},s_{2}\right),$ for
some finite $s_{1},s_{2}\geq 1,$ for the problem (1.1)-(1.2) subject to a
dynamic boundary condition of the form (1.4). We shall first consider a
special case. Let $\Gamma\subset\mathbb{R}^{N-1}$ consists of two disjoint
open subsets $\Gamma_{1}$ and $\Gamma_{2}$, each
$\overline{\Gamma}_{i}\backslash\Gamma_{i}$ is a $S$-null subset of $\Gamma$
and $\Gamma=\overline{\Gamma}_{1}\cup\overline{\Gamma}_{2}$, such that
$u_{i},$ $i\in\mathbb{N}_{m}$ satisfy (1.3) on
$\Gamma_{1}\times\left(0,\infty\right),$ and
(3.15) $u_{i}=0\text{, on }\Gamma_{2}\times\left(0,\infty\right),\text{ for
}i\in\mathbb{N}_{m}.$
We assume that $\Gamma_{2}$ is a set of positive surface measure, and that the
nonlinearities $f_{i},g_{i}$ satisfy the following special form of (3.6), that
is,
(3.16)
$\left\\{\begin{array}[]{l}\sum\nolimits_{i\in\mathbb{N}_{m}}f_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\left|s_{i}\right|^{m_{i}-2}\geq-\sum\nolimits_{i\in\mathbb{N}_{m}}C_{f_{i}}\left|s_{i}\right|^{m_{i}}-\widetilde{C}_{f},\\\
\sum\nolimits_{i\in\mathbb{N}_{m}}g_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\left|s_{i}\right|^{m_{i}-2}\geq-\sum\nolimits_{i\in\mathbb{N}_{m}}C_{g_{i}}\left|s_{i}\right|^{m_{i}}-\widetilde{C}_{g},\end{array}\right.$
for some $m_{i}\geq 1,$ and some positive constants $C_{f_{i}},C_{g_{i}}$ and
$\widetilde{C}_{f},\widetilde{C}_{g}\geq 0.$ Note that (3.6) is equivalent to
(3.16) for $m_{i}=2.$
The second main result gives a $\mathcal{X}^{\overrightarrow{r}}$-dissipative
estimate for solutions of (1.1), (1.3), (1.4), (3.15).
###### Theorem 3.3.
Suppose $\Gamma_{2}$ is a set of positive surface measure. Let the assumptions
(3.5), (3.16) be satisfied, and let $p_{i}>0$ for all $i\in\mathbb{N}_{m}$.
Then, the system (1.1), (1.3), (1.4), (3.15) satisfies property
P$\left(\overrightarrow{r}\right)$-(i) for
$\overrightarrow{r}=\left(r_{1},...,r_{m}\right)$ with $r_{i}=m_{i},$
$i\in\mathbb{N}_{m}.$ Moreover, if
$\overrightarrow{u}_{0}\in\mathcal{X}^{\overrightarrow{r}},$ then there also
exists a positive function $Q$, independent of initial data and time, such
that
$\sup_{t\geq
0}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{\overrightarrow{r}}}\leq
Q\left(\left\|\overrightarrow{u}_{0}\right\|_{\mathcal{X}^{\overrightarrow{r}}}\right).$
###### Remark 3.2.
Theorem 3.3 _only_ holds if $\Gamma\neq\Gamma_{1}$, i.e., when the boundary
$\Gamma_{2}$ has positive measure (cf. also Remark 4.2 below). We require
different arguments for the case when $\Gamma_{2}\equiv\varnothing$.
We shall now derive another dissipative estimate for solutions of (1.1)-(1.4)
in $\mathcal{X}^{\overrightarrow{r}}$-norm which also covers Dirichlet
boundary conditions (for $i\in I_{m}$) and applies to the case when
$\Gamma_{2}\equiv\varnothing$, without enforcing any further sign restrictions
on all the $p_{i}$’s (compare with the assumptions in Theorem 3.3). Analogous
to (3.16), we shall assume that the functions $f_{i},g_{i}$ satisfy
(3.17)
$\left\\{\begin{array}[]{l}\sum\nolimits_{i\in\mathbb{N}_{m}}f_{i}\left(x,t,s_{1},...,s_{m},\zeta_{i}\right)s_{i}\left|s_{i}\right|^{m_{i}-2}\geq-\sum\nolimits_{i\in\mathbb{N}_{m}}C_{f_{i}}\left|s_{i}\right|^{m_{i}+p_{i}}-\widetilde{C}_{f},\\\
\sum\nolimits_{i\in\mathbb{N}_{m}}g_{i}\left(x,t,s_{1},...,s_{m}\right)s_{i}\left|s_{i}\right|^{m_{i}-2}\geq-\sum\nolimits_{i\in\mathbb{N}_{m}}C_{g_{i}}\left|s_{i}\right|^{m_{i}+p_{i}}-\widetilde{C}_{g},\end{array}\right.$
for some $m_{i}>1,$ and some _real_ constants $C_{f_{i}},C_{g_{i}}$ and
$\widetilde{C}_{f},\widetilde{C}_{g}\geq 0.$ Moreover, consider the (self-
adjoint) eigenvalue problem for so-called Wentzell Laplacians $\Delta_{W,i}$
(see [22, Appendix]), as follows:
(3.18) $-a_{i}\Delta\varphi_{i}-C_{f_{i}}\varphi_{i}=\Lambda_{i}\varphi\text{
in }\Omega,$
where
$a_{i}:=\alpha_{i}\left(m_{i}-1\right)\left(\frac{2}{m_{i}+p_{i}}\right)^{2},$
with a boundary condition that depends on the eigenvalue $\Lambda_{i}$
explicitly,
(3.19)
$a_{i}\partial_{\mathbf{n}}\varphi_{i}-C_{g_{i}}\varphi_{i}=\Lambda_{i}\varphi_{i}\text{
on }\Gamma_{1},$
such that
(3.20) $\varphi_{i}=0\text{ on }\Gamma_{2}.$
Here $\Gamma_{2}$ is assumed to be a set of nonnegative surface measure (the
case $\Gamma_{2}=\varnothing$ may be also allowed). Our second condition on
the nonlinearities $f_{i},$ $g_{i}$ is concerned with some sign assumptions on
$C_{f_{i}}$ and $C_{g_{i}}$. In particular, we assume that
(3.21) $\Lambda_{1}:=\inf_{i\in\mathbb{N}_{m}}\Lambda_{1,i}>0.$
Observe that, if $C_{f_{i}}<0,$ $C_{g_{i}}<0,$ for all $i\in\mathbb{N}_{m},$
then the first eigenvalue $\Lambda_{1,i}$ of (3.18)-(3.20) is always positive,
for any $i\in\mathbb{N}_{m}.$ Otherwise, we note that even when at least one
of $C_{f_{i}}$ or $C_{g_{i}}$ is positive, it may still happen that (3.21)
holds. In this sense, our system (1.1)-(1.4) becomes dissipative even when at
least one of the terms $f_{i},g_{i}$ has the wrong sign at infinity, but the
bad sign is compensated by the other term (see, also [31] for similar
assumptions).
###### Theorem 3.4.
Suppose that $\Gamma_{2}$ is a set of nonnegative surface measure. Let (3.5),
(3.17), (3.21) hold, and assume that $p_{i}>0$ for all $i\in\mathbb{N}_{m}$.
Then, the system (1.1), (1.4), (3.15) satisfies property
P$\left(\overrightarrow{r}\right)$-(i), for
$\overrightarrow{r}=\left(r_{1},...,r_{m}\right)$ with $r_{i}=m_{i},$
$i\in\mathbb{N}_{m}.$ On the other hand, if $p_{i}=0$ for some
$i\in\mathbb{N}_{m}$, and if
$\overrightarrow{u}_{0}\in\mathcal{X}^{\overrightarrow{r}},$ then this system
satisfies P$\left(\overrightarrow{r}\right)$-(ii) instead.
###### Remark 3.3.
All the above results can be extended to models which also incorporate
advection effects in the domain $\Omega$ and on the boundary $\Gamma$ (i.e.,
the reaction terms $f_{i},g_{i}$ and $h_{i}$ may also depend on $\nabla u_{i}$
and $\nabla_{\Gamma}u_{i}$, respectively). Indeed, by making similar
assumptions to (3.6)-(3.17), any integral over $\left|\nabla u_{i}\right|$ may
be absorbed by diffusion in the bulk $\Omega$ with an appropiate application
of suitable Young and Sobolev inequalities (cf. Section 4). We will return to
these questions elsewhere.
## 4\. Proof of main results
### 4.1. Proof of Theorem 3.1
In order to justify our computations, we need to construct an approximation
scheme which relies on the existence of classical (smooth) solutions to the
non-degenerate analogue of (1.1)-(1.4) (if at least one $p_{i}\neq 0$). One of
the advantages of this construction is that now every (possibly, very weak)
solution can be approximated by regular ones and the justification of our
estimates for such solutions is immediate. To this end, for each $\epsilon>0$,
let us consider the following non-degenerate parabolic system:
(4.1)
$\partial_{t}u_{i}-\text{div}\left(a_{i}^{\epsilon}\left(u_{i}\right)\nabla
u_{i}\right)+f_{i}\left(x,t,\overrightarrow{u}\right)=0\text{, in
}\Omega\times\left(0,\infty\right),$
for $i=1,...,m,$ where
$a_{i}^{\epsilon}\left(s_{i}\right):=a_{i}\left(s_{i}+\epsilon\right)\geq
O\left(\epsilon\right)>0,$ subject to the following set of boundary conditions
(4.2)
$\partial_{\mathbf{n}}u_{i}+h_{i}\left(x,t,\overrightarrow{u}\right)=0,\text{
on }\Gamma\times\left(0,\infty\right),\text{ \thinspace}i\in I_{m}$
and
(4.3)
$\delta_{i}\partial_{t}u_{i}+a_{\epsilon}\left(u_{i}\right)\partial_{\mathbf{n}}\left(u_{i}\right)+g_{i}\left(x,t,\overrightarrow{u}\right)=0,\text{
on }\Gamma\times\left(0,\infty\right),\text{ \thinspace}i\in J_{m}.$
We equip the system (1.1)-(1.3) with the initial conditions
(4.4) $u_{i\mid t=0}^{\epsilon}=u_{i0}^{\epsilon}\text{ in }\Omega,\text{
}u_{i\mid t=0}^{\epsilon}=u_{i0}^{\epsilon}\text{ on }\Gamma,$
for $u_{i}^{\epsilon}\left(0\right):=u_{i0}^{\epsilon}\in
C^{\infty}\left(\overline{\Omega}\right),$ $i\in\mathbb{N}_{m}$, such that
$u_{i}^{\epsilon}\left(0\right)\rightarrow u_{i0}\text{ in
}L^{s}\left(\Omega\right)\text{,
}u_{i}^{\epsilon}\left(0\right)_{\mid\Gamma}\rightarrow v_{i0}\text{ in
}L^{s}\left(\Gamma\right),$
for some given $s\geq 1.$ Then, the approximate problem (4.1)-(4.4) admits a
unique (smooth) classical solution with
$\overrightarrow{u}^{\epsilon}=\left(u_{1}^{\epsilon},...,u_{m}^{\epsilon}\right)\in
C^{1}\left(\left[0,t_{\ast}\right];\left(C^{\infty}\left(\overline{\Omega}\right)\right)^{m}\right),$
for some $t_{\ast}>0$ and each $\epsilon>0$ (see [16, 17, 8, 31]). Being
pedants, we cannot apply the main results of [16, 31] (cf. also [17]) directly
to equations (4.1)-(4.4) since the functions $a_{i}^{\epsilon},$ $f_{i},$
$g_{i}$ and $h_{i}$ are not smooth enough. Moreover, the solutions constructed
this way may only exists locally in time for some interval
$\left[0,t_{\ast}\right]$. However, by approximating the functions
$a_{i}^{\epsilon},$ $f_{i},$ $g_{i}$, $h_{i}$ by smooth ones, say, in
$C^{\infty}\left(\mathbb{R},\mathbb{R}\right)$, we may apply Remark 4.1 below
for the solutions of the approximate equations (4.1)-(4.4), and deduce the
existence of a globally well-defined solution on $\left[0,T\right]$, for any
$T>0$. Indeed, an apriori global bound in Hölder-norm for
$\overrightarrow{u}^{\epsilon}$ guarantees the global existence of classical
solutions (see, e.g., [31]). Nevertheless, even when these bounds are not
available apriori, we may still choose to work with locally-defined in
$\left[0,t_{\ast}\right],$ smooth solutions $\overrightarrow{u}^{\epsilon}$,
that are globally defined on $\mathbb{R}_{+}$ in the lower-order
$L^{p}$-norms; this turns out to be sufficient for our purposes. Indeed, if
the solution $\overrightarrow{u}^{\epsilon}$ is globally-defined on
$\left[0,T\right]$ in $\mathcal{X}^{r}$-norm, then it will also be global in
$\mathcal{X}^{\infty}$-norm by (iii). As we shall see in this section, a
global bound in $\mathcal{X}^{r}$-norm can be established for the same
assumptions (3.5)-(3.17) on the nonlinearities.
We begin with the proof of Theorem 3.1, by following similar arguments to [11,
13] for the system (1.1) with static boundary conditions. From now on, $c$
will denote a positive constant that is independent of $t,$ $\epsilon$, $n,$
$\overrightarrow{u}$ and initial data, which only depends on the other
structural parameters of the problem. Such a constant may vary even from line
to line. Moreover, we shall denote by $Q_{\tau}\left(n\right)$ a monotone
nondecreasing function in $n$ of order $\tau,$ for some nonnegative constant
$\tau,$ independent of $n.$ More precisely, $Q_{\tau}\left(n\right)\sim
cn^{\tau}$ as $n\rightarrow+\infty.$ We begin by showing that the
$\mathcal{X}^{n}$-norm of $\overrightarrow{u}=\overrightarrow{u}^{\epsilon}$
satisfies a local recursive relation which can be used to perform an iterative
argument. We divide the proof of Theorem 3.1 into several steps.
Step 1 (The basic energy estimate in $\mathcal{X}^{n+1}$). We multiply (4.1)
by $\left|u_{i}\right|^{n-1}u_{i},$ $n\geq 1,$ and integrate over $\Omega$,
for each $i\in\mathbb{N}_{m}$. We obtain
(4.5)
$\displaystyle\frac{1}{\left(n+1\right)}\frac{d}{dt}\left\|u_{i}\right\|_{L^{n+1}\left(\Omega\right)}^{n+1}+\left\langle
f_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Omega\right)}$
$\displaystyle+n\int_{\Omega}a_{i}^{\epsilon}\left(u_{i}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{n-1}dx$
$\displaystyle=\int_{\Gamma}a_{i}^{\epsilon}\left(u_{i}\right)\partial_{\mathbf{n}}u_{i}\left|u_{i}\right|^{n-1}u_{i}dS.$
Similarly, we multiply (4.2) and (4.3) by $\left|u_{i}\right|^{n-1}u_{i}$ and
integrate each relation over $\Gamma$. We have
(4.6)
$\displaystyle\frac{\delta_{i}}{\left(n+1\right)}\frac{d}{dt}\left\|u_{i}\right\|_{L^{n+1}\left(\Gamma\right)}^{n+1}+\int_{\Gamma}a_{i}^{\epsilon}\left(u_{i}\right)\partial_{\mathbf{n}}u_{i}\left|u_{i}\right|^{n-1}u_{i}dS$
$\displaystyle+\left\langle
g_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Gamma\right)}$
$\displaystyle=0,$
for each $i\in J_{m}$, and
(4.7)
$\int_{\Gamma}a_{i}^{\epsilon}\left(u_{i}\right)\partial_{\mathbf{n}}u_{i}\left|u_{i}\right|^{n-1}u_{i}dS+\left\langle
a_{i}^{\epsilon}\left(u_{i}\right)h_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Gamma\right)}=0,\text{
}i\in I_{m}.$
In the case when (1.2) is replaced by a Dirichlet boundary condition for
$u_{i},$ $i\in I_{m},$ equation (4.7) still holds since $h_{i}\equiv 0$ in
that case.
Let us first observe that, in light of assumption (3.5), we have
$a_{i}^{\epsilon}\left(s_{i}\right)\geq\alpha_{i},$ $\forall
s_{i}\in\mathbb{R}$, $\epsilon>0$ (recall that $p_{i}=0$), which immediately
implies
(4.8) $\int_{\Omega}a_{i}^{\epsilon}\left(u_{i}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{n-1}dx\geq\alpha_{i}\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{n-1}dx,\text{ }i\in\mathbb{N}_{m}.$
Moreover, on account of of the assumptions (3.6) for $f_{i}$, $g_{i}$ and a
basic application of Hölder and Young inequalities, we deduce
(4.9) $\sum\nolimits_{i\in\mathbb{N}_{m}}\left\langle
f_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Omega\right)}\geq-c\sum\nolimits_{i\in\mathbb{N}_{m}}\left\|u_{i}\right\|_{L^{n+1}\left(\Omega\right)}^{n+1}-c,$
and
(4.10) $\sum\nolimits_{i\in J_{m}}\left\langle
g_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Gamma\right)}\geq-c\sum\nolimits_{i\in
J_{m}}\delta_{i}\left\|u_{i}\right\|_{L^{n+1}\left(\Gamma\right)}^{n+1}-c.$
In order to estimate the source terms involving $h_{i}$ on the boundary (4.7),
we need the following lemma which allows us to control surface integrals in
terms of volume integrals (see, e.g., [23]).
###### Lemma 4.1.
Let $n\geq 1$, $p\geq 0,$ $s>-1.$ Then for every $\varepsilon>0,$ there holds
$\int_{\Gamma}\left|u\right|^{s+n}dS\leq\varepsilon\left(s+n\right)\int_{\Omega}\left|\nabla
u\right|^{2}\left|u\right|^{p+n-1}dx+\frac{C}{\varepsilon}\left(s+n\right)\left(\left\|u\right\|_{L^{s+n}\left(\Omega\right)}^{s+n}+1\right),$
for some positive constant $C=C\left(p,s\right)$ independent of $u,$
$\varepsilon$ and $n.$
Applying Lemma 4.1 to $u=u_{i},$ with $s=1$ and $p=0,$ $i\in I_{m}$, we have
(4.11) $\displaystyle\int_{\Gamma}\left|u_{i}\right|^{1+n}dS$
$\displaystyle\leq\varepsilon\left(n+1\right)\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\left|u\right|^{n-1}dx$
$\displaystyle+\frac{C}{\varepsilon}\left(n+1\right)\left(\left\|u_{i}\right\|_{L^{n+1}\left(\Omega\right)}^{n+1}+1\right).$
Thus, in light of the third assumption of (3.6), using (3.5), we find
(4.12) $\displaystyle\sum\nolimits_{i\in I_{m}}\left\langle
a_{i}^{\epsilon}\left(u_{i}\right)h_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{n-1}u_{i}\right\rangle_{L^{2}\left(\Gamma\right)}$
$\displaystyle\geq-\sum\nolimits_{i\in
I_{m}}\left(\alpha_{i}C_{h_{i}}\left\|u_{i}\right\|_{L^{n+1}\left(\Gamma\right)}^{n+1}+\widetilde{C}_{h_{i}}\left\|u_{i}\right\|_{L^{n-1}\left(\Gamma\right)}^{n-1}\right)$
$\displaystyle\geq-c\sum\nolimits_{i\in
I_{m}}\alpha_{i}\left\|u_{i}\right\|_{L^{n+1}\left(\Gamma\right)}^{n+1}-c$
which can be bounded below by
(4.13) $\displaystyle-c\varepsilon\sum\nolimits_{i\in
I_{m}}\alpha_{i}\left(n+1\right)\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{n-1}dx$
$\displaystyle-\frac{c}{\varepsilon}\sum\nolimits_{i\in
I_{m}}\left(n+1\right)(\left\|u_{i}\right\|_{L^{n+1}\left(\Omega\right)}^{n+1}+1),$
exploiting the estimate (4.11). Choose now $\varepsilon=\varepsilon_{n}>0$ in
(4.13) such that
(4.14)
$\varepsilon_{n}=\max_{i\in\mathbb{N}_{m}}\frac{cn\alpha_{i}}{2\left(n+1\right)},\text{
}\forall n\geq 1,$
and note that $\varepsilon_{n}\leq c$, uniformly as $n\rightarrow\infty$.
Summing the equations (4.5) (respectively, (4.6) and (4.7)) over the sets
$i\in\mathbb{N}_{m}$ (respectively, $i\in J_{m}$ and $i\in I_{m}$), then
adding the relations that we obtain, on account of (4.9)-(4.10), (4.12)-(4.13)
we deduce
(4.15)
$\displaystyle\frac{d}{dt}\left\|\overrightarrow{u}\right\|_{\mathcal{X}^{n+1}}^{n+1}+cn\left(n+1\right)\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{n-1}dx$ $\displaystyle\leq
Q_{2}\left(n\right)\left(\left\|\overrightarrow{u}\right\|_{\mathcal{X}^{n+1}}^{n+1}+1\right),$
for any $n\geq 1$. Here, the function $Q_{2}\left(n\right)\sim n^{2}$ as
$n\rightarrow\infty$.
Step 2 (The local relation). Set $n_{k}=2^{k}-1,$ $k\geq 0$, and define
(4.16)
$\mathcal{Y}_{k}\left(t\right):=\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{n_{k}+1}}^{n_{k}+1},$
for all $k\geq 0$. Then, using the basic identity for $u=u_{i},$
(4.17) $\int_{\Omega}\left|\nabla
u\right|^{2}\left|u\right|^{n-1}dx=\left(\frac{2}{n+1}\right)^{2}\int_{\Omega}\left|\nabla\left|u\right|^{\frac{n+1}{2}}\right|^{2}dx,$
from (4.15) it holds
(4.18)
$\frac{d}{dt}\mathcal{Y}_{k}\left(t\right)+\sum\nolimits_{i\in\mathbb{N}_{m}}\gamma_{n_{k}}\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{n_{k}+1}{2}}\right|^{2}dx\leq
Q_{2}\left(n_{k}\right)\left(\mathcal{Y}_{k}\left(t\right)+1\right),$
for all $k\geq 0$, where $0<\gamma_{0}\leq\gamma_{n_{k}}\sim c$, as
$n_{k}\rightarrow\infty$. Let $t,\mu$ be two positive constants such that
$t-\mu/n_{k}>0$. Their precise values will be chosen later. We claim that
(4.19) $\mathcal{Y}_{k}\left(t\right)\leq
M_{k}\left(t,\mu\right):=c_{0}\left(\mu\right)\left(n^{k}\right)^{\sigma}(\sup_{s\geq
t-\mu/n_{k}}\mathcal{Y}_{k-1}\left(s\right)+1)^{\theta_{k}},\text{ }\forall
k\geq 1,$
where $c_{0},$ $\sigma$ are positive constants independent of $k,$ and
$\theta_{k}\geq 1$ is a bounded sequence for all $k.$ The constant
$c_{0}\left(\mu\right)$ is bounded if $\mu$ is bounded away from zero.
We will now prove (4.19) when $2<N.$ The case $N\leq 2$ requires only minor
adjustments. We will follow an argument similar to the proof of [22, Theorem
2.3] (cf. also [23]). For each $k\geq 0$, we define
$r_{i,k}:=\frac{N\left(n_{k}+1\right)-\left(N-2\right)\left(1+n_{k}\right)}{N\left(n_{k}+1\right)-\left(N-2\right)\left(1+n_{k-1}\right)},\text{
}s_{i,k}:=1-r_{i,k}.$
We aim to estimate the term on the right-hand side of (4.15) in terms of the
$\mathcal{X}^{1+n_{k-1}}$-norm of $\overrightarrow{u}.$ First, Hölder and
Sobolev inequalities (with the equivalent norm of Sobolev spaces in
$W^{1,2}\left(\Omega\right)\subset L^{p_{s}}\left(\Omega\right)$,
$p_{s}=2N/\left(N-2\right)$) yield
(4.20) $\displaystyle\int_{\Omega}\left|u_{i}\right|^{1+n_{k}}dx$
$\displaystyle\leq\left(\int_{\Omega}\left|u_{i}\right|^{\frac{\left(n_{k}+1\right)N}{N-2}}dx\right)^{s_{i,k}}\left(\int_{\Omega}\left|u_{i}\right|^{1+n_{k-1}}dx\right)^{r_{i,k}}$
$\displaystyle\leq
c\left(\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{\left(n_{k}+1\right)}{2}}\right|^{2}dx+\int_{\Omega}\left|u_{i}\right|^{1+n_{k}}dx\right)^{\overline{s}_{i,k}}$
$\displaystyle\times\left(\int_{\Omega}\left|u_{i}\right|^{1+n_{k-1}}dx\right)^{r_{i,k}},$
with $\overline{s}_{i,k}:=s_{i,k}N/\left(N-2\right)\in\left(0,1\right)$.
Applying Young’s inequality on the right-hand side of (4.20), we get
(4.21)
$\int_{\Omega}\left|u_{i}\right|^{1+n_{k}}dx\leq\frac{\gamma_{n_{k}}}{4}\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{n_{k}+1}{2}}\right|^{2}dx+Q_{\tau_{1}}\left(n_{k}\right)\left(\int_{\Omega}\left|u_{i}\right|^{1+n_{k-1}}dx\right)^{z_{i,k}},$
for some positive constant $\tau_{1}$ independent of $n_{k},$ and where
$z_{i,k}:=r_{i,k}/\left(1-\overline{s}_{i,k}\right)\geq 1$
is bounded for all $k$. Note that we can choose $\tau_{2}$ to be some fixed
positive number since $Q_{\tau_{2}}$ also depends on $\gamma_{n_{k}}\sim c$.
To treat the boundary terms on the right-hand side of (4.15), we define for
$k\geq 0,$
$y_{i,k}:=\frac{\left(N-1\right)\left(n_{k}+1\right)-\left(N-2\right)\left(1+n_{k}\right)}{\left(N-1\right)\left(n_{k}+1\right)-\left(N-2\right)\left(1+n_{k-1}\right)},\text{
}x_{i,k}:=1-y_{i,k}.$
On account of Hölder and Sobolev inequalities (e.g.,
$W^{1,2}\left(\Omega\right)\subset L^{q_{s}}\left(\Gamma\right),$
$q_{s}=2\left(N-1\right)/\left(N-2\right)$), we obtain
(4.22) $\displaystyle\int_{\Gamma}\left|u_{i}\right|^{1+n_{k}}dS$
$\displaystyle\leq
c\left(\int_{\Gamma}\left|u_{i}\right|^{\frac{\left(N-1\right)\left(n_{k}+1\right)}{N-2}}dS\right)^{x_{i,k}}\left(\int_{\Gamma}\left|u_{i}\right|^{1+n_{k-1}}dS\right)^{y_{i,k}}$
$\displaystyle\leq
c\left(\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{\left(n_{k}+1\right)}{2}}\right|^{2}dx+\int_{\Omega}\left|u_{i}\right|^{1+n_{k}}dx\right)^{\overline{x}_{i,k}}$
$\displaystyle\times\left(\int_{\Gamma}\left|u_{i}\right|^{1+n_{k-1}}dS\right)^{y_{i,k}},$
with $\overline{x}_{i,k}:=x_{i,k}\left(N-1\right)/\left(N-2\right)$. Since
$\overline{x}_{i,k}\in\left(0,1\right)$, we can apply Young’s inequality on
the right-hand side of (4.22), use the estimate for the
$L^{1+n_{k}}\left(\Omega\right)$-norm of $u_{i}$ from (4.20) in order to
deduce the following estimate:
(4.23)
$\int_{\Gamma}\left|u_{i}\right|^{1+n_{k}}dS\leq\frac{\gamma_{n_{k}}}{4}\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{n_{k}+1}{2}}\right|^{2}dx+Q_{\tau_{2}}\left(n_{k}\right)\left(\int_{\Omega}\left|u_{i}\right|^{1+n_{k-1}}dx\right)^{l_{i,k}},$
for some positive constant $\tau_{2}$ depending on $\tau_{1},$ but which is
independent of $n_{k}$, and where
$l_{i,k}:=\frac{y_{i,k}}{\left(1-\overline{x}_{i,k}\right)}\geq 1$
is bounded for all $k\geq 0$. Inserting estimates (4.21)-(4.23) on the right-
hand side of (4.18), we obtain the following inequality:
(4.24)
$\partial_{t}\mathcal{Y}_{k}\left(t\right)+\sum\nolimits_{i\in\mathbb{N}_{m}}\gamma_{n_{k}}\int_{\Omega}\left|\nabla\left|u_{i}\right|^{\frac{n_{k}+1}{2}}\right|^{2}dx\leq
c\left(n_{k}\right)^{\sigma_{1}}\left(\mathcal{Y}_{k-1}+1\right)^{\theta_{k}},$
where $c,$ $\sigma_{1}$ are positive constants independent of $k,$ and
$\theta_{k}:=\max(\max_{i}\left\\{z_{i,k}\right\\},\max_{i}\left\\{l_{i,k}\right\\})\geq
1$
is a bounded sequence for all $k.$
We are now ready to prove (4.19) using (4.24). To this end, let
$\zeta\left(s\right)$ be a positive function
$\zeta:\mathbb{R}_{+}\rightarrow\left[0,1\right]$ such that
$\zeta\left(s\right)=0$ for $s\in\left[0,t-\mu/n_{k}\right],$
$\zeta\left(s\right)=1$ if $s\in\left[t,+\infty\right)$ and
$\left|d\zeta/ds\right|\leq n_{k}/\mu$, if $s\in\left(t-\mu/n_{k},t\right)$.
We define
$Z_{k}\left(s\right)=\zeta\left(s\right)\mathcal{Y}_{k}\left(s\right)$ and
notice that
$\frac{d}{ds}Z_{k}\left(s\right)\leq\frac{n_{k}}{\mu}\mathcal{Y}_{k}\left(s\right)+\zeta\left(s\right)\frac{d}{ds}\mathcal{Y}_{k}\left(s\right).$
Combining this estimate with (4.18), (4.21), (4.23) and noticing that
$Z_{k}\leq\mathcal{Y}_{k}$, we deduce the following estimate for $Z_{k}$:
(4.25)
$\frac{d}{ds}Z_{k}\left(s\right)+C\left(\mu\right)n_{k}Z_{k}\left(s\right)\leq
M_{k}\left(t,\mu\right),\text{ for all }s\in\left[t-\mu/n_{k},+\infty\right),$
for some positive constant $C$ independent of $k$. Integrating (4.25) with
respect to $s$ from $t-\mu/n_{k}$ to $t$ and taking into account the fact that
$Z_{k}\left(t-\mu/n_{k}\right)=0,$ we obtain that
$\mathcal{Y}_{k}\left(t\right)=Z_{k}\left(t\right)\leq
M_{k}\left(t,\mu\right)\left(1-e^{-C\mu}\right),$
which proves the claim (4.19).
Step 3 (The iterative argument). Let now $\tau^{{}^{\prime}}>\tau>0$ be given;
define $\mu=2(\tau^{{}^{\prime}}-\tau),$ $t_{0}=\tau^{{}^{\prime}}$ and
$t_{k}=t_{k-1}-\mu/n_{k},$ $k\geq 1$. Using (4.19), we have
(4.26) $\sup_{t\geq t_{k-1}}\mathcal{Y}_{k}\left(t\right)\leq
c_{0}\left(n_{k}\right)^{\sigma}(\sup_{s\geq
t_{k}}\mathcal{Y}_{k-1}\left(s\right)+1)^{\theta_{k}},\text{ }k\geq 1.$
Here $c_{0}=c_{0}\left(\mu\right)$ depends only on $\mu$. Now let us define
(4.27) $\overline{C}:=\sup_{s\geq
t_{1}=\tau}\left(\mathcal{Y}_{0}\left(s\right)+1\right)=\sup_{s\geq
t_{1}=\tau}\left(\left\|\overrightarrow{u}\left(s\right)\right\|_{\mathcal{X}^{1}}+1\right).$
Thus, we can iterate in (4.26) with respect to $k\geq 1$ and obtain that
(4.28) $\displaystyle\sup_{t\geq t_{k-1}}\mathcal{Y}_{k}\left(t\right)$
$\displaystyle\leq\left(c_{0}n_{k}^{\sigma}\right)\left(c_{0}n_{k-1}^{\sigma}\right)^{\theta_{k}}\left(c_{0}n_{k-2}^{\sigma}\right)^{\theta_{k}\theta_{k-1}}\cdot...\cdot\left(c_{0}n_{0}^{\sigma}\right)^{\theta_{k}\theta_{k-1}...\theta_{0}}(\overline{C})^{\xi_{k}}$
$\displaystyle\leq c_{0}^{A_{k}}2^{\sigma
B_{k}}\left(\overline{C}\right)^{\xi_{k}},$
where $\xi_{k}:=\theta_{k}\theta_{k-1}...\theta_{0},$ and
(4.29)
$A_{k}:=1+\theta_{k}+\theta_{k}\theta_{k-1}+...+\theta_{k}\theta_{k-1}...\theta_{0},$
(4.30)
$B_{k}:=k+\theta_{k}\left(k-1\right)+\theta_{k}\theta_{k-1}\left(k-2\right)+...+\theta_{k}\theta_{k-1}...\theta_{0}.$
We can easily show that
(4.31)
$A_{k}\leq\left(c_{1}+n_{k}\right)\sum_{j=1}^{\infty}\frac{1}{c_{1}+n_{j}}\text{
and
}B_{k}\leq\left(c_{2}+n_{k}\right)\sum_{j=1}^{\infty}\frac{j}{c_{2}+n_{j}},$
for some positive constants $c_{1},c_{2}$ independent of $k,$ $\mu$.
Therefore, since
(4.32) $\sup_{t\geq t_{0}}\mathcal{Y}_{k}\left(t\right)\leq\sup_{t\geq
t_{k-1}}\mathcal{Y}_{k}\left(t\right)\leq c_{0}^{A_{k}}2^{\sigma
B_{k}}\left(\overline{C}\right)^{\xi_{k}}$
and the series in (4.31) are convergent, we can take the $1+n_{k}$-root on
both sides of (4.32) and let $k\rightarrow+\infty$. We deduce
$\sup_{t\geq
t_{0}=\tau^{{}^{\prime}}}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{\infty}}\leq\lim_{k\rightarrow+\infty}\sup_{t\geq
t_{0}}\left(\mathcal{Y}_{k}\left(t\right)\right)^{1/\left(1+n_{k}\right)},$
which, on account of (4.32), yields
(4.33) $\sup_{t\geq
t_{0}=\tau^{{}^{\prime}}}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{\infty}}\leq
C\left(\mu\right)\left(\overline{C}\right)^{1/c_{3}},$
for some positive constant $c_{3}$ independent of $t,$ $k$,
$\overrightarrow{u},$ $\epsilon,$ $\mu,$ and initial data. Note that
$\overline{C}$ depends on $\tau$ (see (4.27)).
Step 4 (The final argument). Let us first assume that Property
P$\left(1\right)$-(i) holds. Then we already know that the
$\mathcal{X}^{1}$-norm of $\overrightarrow{u}\left(t\right)$ is bounded
independently of the initial data, for each $t\geq\tau$. Therefore, from
(4.33) we also obtain the claim for the $\mathcal{X}^{\infty}$-norm of
$\overrightarrow{u}\left(t\right)$, i.e., Property P$\left(\infty\right)$-(i)
holds as well. If, on the other hand, Property P$\left(1\right)$-(ii) holds,
we can choose $\tau^{{}^{\prime}}=\tau+2\mu$ with $\mu=1$ so that
$\overline{C}$ and $C\left(\mu\right)$ are bounded uniformly with respect to
initial data as $\tau\rightarrow\infty$. Hence, Property
P$\left(\infty\right)$-(ii) is also satisfied by letting
$\tau\rightarrow\infty$ in (4.33). In order to show the final property (iii),
taking advantage of the fact that the initial data
$\overrightarrow{u}_{0}\in\mathcal{X}^{\infty}$, it suffices to note that in
place of the inequality (4.19), we may use instead the inequality
$\mathcal{Y}_{k}\left(t\right)\leq
Q(\left\|\overrightarrow{u}_{0}\right\|_{\mathcal{X}^{\infty}},\sup_{t>0}M_{k}\left(t,\mu\right)),$
which is an immediate consequence of (4.24). Arguing analogously as in [31,
Lemma 5.5.3], we obtain the claim. The proof of Theorem 4.33 is now complete.
###### Remark 4.1.
It was proven in [31, Section 5] that maximal $L^{p}$-regularity for
(4.1)-(4.4) can be used to reduce the question of global existence of the
solutions $\overrightarrow{u}^{\epsilon}$ in a space of maximal regularity, to
the boundedness of $\overrightarrow{u}^{\epsilon}$ in a Hölder norm
$C^{0,\beta}\left(\overline{\Omega}\right)$, $\beta>0$. It should be possible
to prove, under the natural assumptions of Theorem 3.1, that every classical
solution of problem (4.1)-(4.4) is globally Hölder continuous on
$\overline{\Omega}$. Establishing global Hölder continuity for solutions to
systems with dynamic boundary conditions requires a more detailed analysis,
involving careful local estimates of the solution near the boundary. Of
course, as in the case of Dirichlet/Robin boundary conditions for (4.1) (see,
e.g., [12]), these Hölder bounds should apriori depend on the
$\,L^{\infty}$-norm of the solution. Thus, our analysis constitutes only the
first step in proving boundedness in Hölder norm
$C^{0,\beta}\left(\overline{\Omega}\right)$. This question remains open for
now.
### 4.2. Proof of Theorem 3.2
We shall divide the proof into several steps. As in Section 4.1, we can
justify our computations by exploiting the approximation scheme (4.1)-(4.4).
As before, $c$ will denote a positive constant that is independent of $t,$
$\epsilon$, $n,$ $\overrightarrow{u}$ and initial data, which only depends on
the other structural parameters of the problem. Such a constant may vary even
from line to line. Without loss of generality, we may assume that
$\delta_{i}=1$, for all $i\in J_{m}=\mathbb{N}_{m}.$
Let $T,\tau$ and $L$ be positive numbers such that $T-2\tau>0$ and $L\geq 1.$
We set $t_{0}=T-2\tau$ and define the sequences
$t_{n}=t_{n-1}+\frac{\tau}{2^{n}},\text{
}k_{n}=L\left(2-\frac{1}{2^{n}}\right),\text{ for all }n\geq 1.$
Consider the (smooth) cut-off functions $\eta_{n}\in
C^{1}\left(\mathbb{R},\left[0,1\right]\right)$ with the property that
$\eta_{n}\left(t\right)=\left\\{\begin{array}[]{ll}1,&t\geq t_{n,}\\\
0,&t<t_{n-1}.\end{array}\right.$
Next, denote $Q_{n}:=I_{n}\times\overline{\Omega}$, where
$I_{n}:=\left[t_{n-1},T\right]$, and the sets
$A_{i,n}^{\Omega}:=\left\\{\left(x,t\right)\in
I_{n}\times\Omega:u_{i}\left(x,t\right)>k_{n}\right\\},\text{
}A_{i,n}^{\Gamma}:=\left\\{\left(x,t\right)\in
I_{n}\times\Gamma:u_{i}\left(x,t\right)>k_{n}\right\\}.$
Let $\overline{A}_{i,n}=A_{i,n}^{\Omega}\cup A_{i,n}^{\Gamma}$, and note that
$\overline{A}_{i,n}=\left\\{\left(x,t\right)\in
Q_{n}:u_{i}\left(x,t\right)>k_{n}\right\\}.$
Finally, we denote by $\left|A_{i,n}^{\Omega}\right|$ the ($N+1$-dimensional)
Lebesgue measure of the set $A_{i,n}^{\Omega}$, and by
$\left|A_{i,n}^{\Gamma}\right|$, the ($N$-dimensional) Lebesgue measure of
$A_{i,n}^{\Gamma}$, respectively. We note that, according to (3.2)-(3.3), we
have
$\left|Q_{n}\right|=\left|I_{n}\right|\mu\left(\overline{\Omega}\right)=\left|I_{n}\right|\left(\left|\Omega\right|+\left|\Gamma\right|\right),$
and we can do so similarly for the set $\overline{A}_{i,n}$.
Step 1. (The energy inequality). We define the truncated functions
$u_{i,n}\left(x,t\right):=\max\left\\{u_{i}\left(x,t\right)-k_{n},0\right\\}=\left(u_{i}-k_{n}\right)_{+}.$
We begin by multiplying equation (4.1) by $u_{i,n}\eta_{n}^{2}\left(t\right)$
and integrating the resulting identity over $I_{n}\times\Omega$. Then, we
multiply (4.3) by $u_{i,n}\eta_{n}^{2}\left(t\right)$ and integrate over
$I_{n}\times\Gamma$. Adding as usual (cf., e.g., Section 4.1), then exploiting
the growth assumptions (3.11) on $f_{i}$ and $g_{i}$, the fact that
$\left|\eta_{n}^{{}^{\prime}}\left(t\right)\right|\leq 2^{n}/\tau$, we obtain
after standard transformations
(4.34) $\displaystyle\max_{t\in
I_{n}}\left(\int_{\Omega}u_{i,n}^{2}\left(x,t\right)dx+\int_{\Gamma}u_{i,n}^{2}\left(x,t\right)dS\right)+\iint\nolimits_{I_{n}\times\Omega}a_{i}\left(u_{i}\right)\left|\nabla\left(u_{i,n}\eta_{n}\right)\left(x,t\right)\right|^{2}dxdt$
$\displaystyle\leq
c\iint\nolimits_{A_{i,n}^{\Omega}}\left(\frac{2^{n}}{\tau}u_{i,n}^{2}\left(x,t\right)\eta_{n}+\sum\nolimits_{j\in\mathbb{N}_{m}}\left|u_{j}\left(x,t\right)\right|^{\theta}u_{i,n}\left(x,t\right)\eta_{n}^{2}\left(t\right)+u_{i,n}\left(x,t\right)\eta_{n}^{2}\right)dxdt$
$\displaystyle+c\iint\nolimits_{A_{i,n}^{\Gamma}}\left(\frac{2^{n}}{\tau}u_{i,n}^{2}\left(x,t\right)\eta_{n}+\sum\nolimits_{j\in\mathbb{N}_{m}}\left|u_{j}\left(x,t\right)\right|^{\beta}u_{i,n}\left(x,t\right)\eta_{n}^{2}+u_{i,n}\left(x,t\right)\eta_{n}^{2}\right)dSdt,$
where we have set $\theta:=\max_{i\in\mathbb{N}_{m}}\theta_{i}$ and
$\beta:=\max_{i\in\mathbb{N}_{m}}\beta_{i}$. Now, we wish to estimate the
terms on the right-hand side of (4.34). To this end, set
$\mathcal{A}_{n}^{\Omega}:=\cup_{k\geq 1}A_{k,n}^{\Omega}\text{,
}\mathcal{A}_{n}^{\Gamma}:=\cup_{k\geq 1}A_{k,n}^{\Gamma}$
and note that on $\mathcal{A}_{n}^{\Omega}\backslash A_{i,n}^{\Omega}$ and
$\mathcal{A}_{n}^{\Gamma}\backslash A_{i,n}^{\Gamma}$, respectively, we have
$u_{i}\left(x,t\right)\leq k_{n}\leq 2L$ and
$u_{i}\left(x,t\right)_{\mid\Gamma}\leq k_{n}\leq 2L$, respectively.
Therefore,
(4.35) $\iint\nolimits_{\mathcal{A}_{n}^{\Omega}\backprime
A_{i,n}^{\Omega}}\left|u_{i}\right|^{\theta+1}dxdt\leq
cL^{\theta+1}\iint\nolimits_{\mathcal{A}_{n}^{\Omega}\backprime
A_{i,n}^{\Omega}}\left(1\right)dxdt\leq
cL^{\theta+1}\sum\nolimits_{j\in\mathbb{N}_{m}}\left|A_{j,n}^{\Omega}\right|,$
and, analogously, for the trace of $u_{i}$ we have
(4.36) $\iint\nolimits_{\mathcal{A}_{n}^{\Gamma}\backprime
A_{i,n}^{\Gamma}}\left|u_{i}\right|^{\beta+1}dSdt\leq
cL^{\beta+1}\iint\nolimits_{\mathcal{A}_{n}^{\Gamma}\backprime
A_{i,n}^{\Gamma}}\left(1\right)dSdt\leq
cL^{\beta+1}\sum\nolimits_{j\in\mathbb{N}_{m}}\left|A_{j,n}^{\Gamma}\right|,$
Since $k_{n}\sim L$, as $n\rightarrow\infty$, it is easy to see that the
following inequalities hold:
$\left\\{\begin{array}[]{l}L^{\alpha+1}\left|A_{j,n}^{\Omega}\right|\leq
ck_{n}^{\alpha+1}\left|A_{j,n}^{\Omega}\right|\leq
c\iint\nolimits_{A_{j,n}^{\Omega}}\left|u_{j}\right|^{\alpha+1}dxdt,\\\
L^{\beta+1}\left|A_{j,n}^{\Gamma}\right|\leq
ck_{n}^{\beta+1}\left|A_{j,n}^{\Gamma}\right|\leq
c\iint\nolimits_{A_{j,n}^{\Gamma}}\left|u_{j}\right|^{\beta+1}dSdt.\end{array}\right.$
From these estimates, we thus find that
(4.37)
$\displaystyle\iint\nolimits_{I_{n}\times\Omega}\left|u_{j}\right|^{\theta}u_{i,n}dxdt$
$\displaystyle\leq\iint\nolimits_{\mathcal{A}_{n}^{\Omega}}\left(\left|u_{j}\right|^{\theta+1}+\left|u_{i}\right|^{\theta+1}\right)dxdt$
$\displaystyle\leq
c\sum\nolimits_{j\in\mathbb{N}_{m}}\iint\nolimits_{A_{j,n}^{\Omega}}\left|u_{j}\right|^{\theta+1}dxdt$
and, similarly,
(4.38)
$\displaystyle\iint\nolimits_{I_{n}\times\Gamma}\left|u_{j}\right|^{\beta}u_{i,n}dSdt$
$\displaystyle\leq\iint\nolimits_{\mathcal{A}_{n}^{\Gamma}}\left(\left|u_{j}\right|^{\beta+1}+\left|u_{i}\right|^{\beta+1}\right)dxdt$
$\displaystyle\leq
c\sum\nolimits_{j\in\mathbb{N}_{m}}\iint\nolimits_{A_{j,n}^{\Gamma}}\left|u_{j}\right|^{\beta+1}dSdt.$
Hence, using the above inequalities (4.37)-(4.38) on the right-hand side of
(4.34), and summing the resulting relation over $i\in\mathbb{N}_{m}$, we
deduce
(4.39) $\displaystyle\max_{t\in
I_{n}}\left(\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Omega}u_{i,n}^{2}\left(x,t\right)dx+\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Gamma}u_{i,n}^{2}\left(x,t\right)dS\right)$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{I_{n}\times\Omega}\alpha_{i}\left|u_{i}\right|^{p_{i}}\left|\nabla\left(u_{i,n}\eta_{n}\right)\right|^{2}dxdt$
$\displaystyle\leq\frac{2^{n}c}{\tau}\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{A_{i,n}^{\Omega}}\left|u_{i}\left(x,t\right)\right|^{\delta}dxdt+\frac{2^{n}c}{\tau}\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{A_{i,n}^{\Gamma}}\left|u_{i}\left(x,t\right)\right|^{\gamma}dSdt,$
where $\delta$ and $\gamma$ are defined as in (3.13). Here we have also used
assumption (3.5).
Step 2. (Additional estimates). From the definition of $k_{n}$, we see that
$1-k_{n}/k_{n+1}\geq 2^{-\left(n+2\right)}$, which yields
(4.40)
$\displaystyle\iint\nolimits_{A_{i,n+1}^{\Omega}}\left|u_{i}\right|^{\delta}dxdt$
$\displaystyle\leq
2^{\left(n+2\right)\delta}\iint\nolimits_{A_{i,n+1}^{\Omega}}\left|u_{i}\right|^{\delta}\left(1-\frac{k_{n}}{k_{n+1}}\right)dxdt$
$\displaystyle\leq
2^{n\delta}c\iint\nolimits_{A_{i,n+1}^{\Omega}}\left(u_{i}-k_{n}\right)_{+}^{\delta}dxdt.$
Moreover, the same argument gives
(4.41) $\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left|u_{i}\right|^{\gamma}dSdt\leq
2^{n\gamma}c\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left(u_{i}-k_{n}\right)_{+}^{\gamma}dSdt.$
On the other hand, since on $A_{i,n+1}^{\Omega}\cup A_{i,n+1}^{\Gamma},$ we
have $\left(u_{i}-k_{n}\right)_{+}\geq k_{n+1}-k_{n}$, there holds
(4.42)
$\displaystyle\iint\nolimits_{A_{i,n+1}^{\Omega}}\left(u_{i}-k_{n}\right)_{+}^{\delta}dxdt$
$\displaystyle\geq\left(k_{n+1}-k_{n}\right)^{\delta}\left|A_{i,n+1}^{\Omega}\right|\geq
c\frac{L^{\delta}}{2^{n\delta}}\left|A_{i,n+1}^{\Omega}\right|,\text{ }$
$\displaystyle\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left(u_{i}-k_{n}\right)_{+}^{\gamma}dSdt$
$\displaystyle\geq\left(k_{n+1}-k_{n}\right)^{\gamma}\left|A_{i,n+1}^{\Gamma}\right|\geq
c\frac{L^{\gamma}}{2^{n\gamma}}\left|A_{i,n+1}^{\Gamma}\right|.$
Because of these two inequalities (4.42), for any positive number $\lambda$
such that, if $\lambda<\delta$ and $\lambda<\gamma$, on account of Holder’s
inequality, it also holds
(4.43)
$\displaystyle\iint\nolimits_{A_{i,n+1}^{\Omega}}\left(u_{i}-k_{n+1}\right)_{+}^{\lambda}dxdt$
$\displaystyle\leq\left(\iint\nolimits_{A_{i,n+1}^{\Omega}}\left(u_{i}-k_{n+1}\right)_{+}^{\delta}dxdt\right)^{\lambda/\delta}\left|A_{i,n+1}^{\Omega}\right|^{1-\lambda/\delta}$
$\displaystyle\leq\frac{c2^{n\left(\delta-\lambda\right)}}{L^{\delta-\lambda}}\iint\nolimits_{A_{i,n+1}^{\Omega}}\left(u_{i}-k_{n}\right)_{+}^{\delta}dxdt,$
and
(4.44)
$\displaystyle\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left(u_{i}-k_{n+1}\right)_{+}^{\lambda}dSdt$
$\displaystyle\leq\left(\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left(u_{i}-k_{n+1}\right)_{+}^{\gamma}dxdt\right)^{\lambda/\gamma}\left|A_{i,n+1}^{\Gamma}\right|^{1-\lambda/\gamma}$
$\displaystyle\leq\frac{c2^{n\left(\gamma-\lambda\right)}}{L^{\gamma-\lambda}}\iint\nolimits_{A_{i,n+1}^{\Gamma}}\left(u_{i}-k_{n}\right)_{+}^{\gamma}dSdt.$
Next, we will collect some useful inequalities which follow from the following
well-known embeddings: $H^{1}\left(\Omega\right)\subset L^{p_{s}},$
$p_{s}:=2N/\left(N-2\right)$ and $H^{1}\left(\Omega\right)\subset
L^{q_{s}}\left(\Gamma\right),$ $q_{s}:=2\left(N-1\right)/\left(N-2\right)$. We
give the argument for $N>2$, the case $N\leq 2$ can be treated analogously.
Suppressing the dependance on the subscript $n$ for the moment, from Hölder’s
inequality and these Sobolev embeddings, for every $v^{M_{i}}\in
W^{1,2}\left(I\times\Omega\right)$ we have
(4.45) $\displaystyle\iint\nolimits_{{}^{I\times\Omega}}v^{s}dxdt$
$\displaystyle\leq\int\nolimits_{I}\left(\int\nolimits_{\Omega}v^{M_{i}p_{s}}dx\right)^{\frac{N-2}{N}}\left(\int\nolimits_{\Omega}v^{2}dx\right)^{\frac{2}{N}}dt$
$\displaystyle\leq\left(\iint\nolimits_{{}^{I\times\Omega}}\left(\left|\nabla
v^{M_{i}}\right|^{2}+\left|v\right|^{M_{i}}\right)dxdt\right)\times\left(\max_{t\in
I}\int\nolimits_{\Omega}v^{2}dx\right)^{\frac{2}{N}},$
where, for each $M_{i}>0$, we have set
$s=2M_{i}+\frac{4}{N}.$
Similarly, for each $M_{i}>0$ and $l=2M_{i}+2/\left(N-1\right),$ we have
(4.46) $\displaystyle\iint\nolimits_{{}^{I\times\Gamma}}v^{s}dS$
$\displaystyle\leq\int\nolimits_{I}\left(\int\nolimits_{\Gamma}v^{M_{i}q_{s}}dS\right)^{\frac{N-2}{N-1}}\left(\int\nolimits_{\Gamma}v^{2}dS\right)^{\frac{1}{N-1}}dt$
$\displaystyle\leq\left(\iint\nolimits_{I\times\Omega}\left(\left|\nabla
v^{M_{i}}\right|^{2}+\left|v\right|^{M_{i}}\right)dxdt\right)\times\left(\max_{t\in
I}\int\nolimits_{\Gamma}v^{2}dS\right)^{\frac{1}{N-1}}.$
Exploiting (4.45)-(4.46) with $M_{i}=p_{i}/2+1,$ $I=I_{n+1}$,
$v=\left(u_{i}-k_{n+1}\right)_{+}$, we get
(4.47)
$\displaystyle\iint\nolimits_{I_{n+1}\times\Omega}\left(u_{i}-k_{n+1}\right)_{+}^{s}dxdt$
$\displaystyle\leq\left(\iint\nolimits_{{}^{I_{n+1}\times\Omega}}\left(\left|u_{i}\right|^{p_{i}}\left|\nabla
u_{i,n+1}\right|^{2}+u_{i,n+1}^{M_{i}}\right)dxdt\right)\times\left(\max_{t\in
I_{n+1}}\int\nolimits_{\Omega}u_{i,n+1}^{2}dx\right)^{\frac{2}{N}}$
and
(4.48)
$\displaystyle\iint\nolimits_{{}^{I_{n+1}\times\Gamma}}\left(u_{i}-k_{n+1}\right)_{+}^{l}dSdt$
$\displaystyle\leq\left(\iint\nolimits_{{}^{I_{n+1}\times\Omega}}\left(\left|u_{i}\right|^{p_{i}}\left|\nabla
u_{i,n+1}\right|^{2}+u_{i,n+1}^{M_{i}}\right)dxdt\right)\times\left(\max_{t\in
I_{n+1}}\int\nolimits_{\Gamma}u_{i,n+1}^{2}dx\right)^{\frac{1}{N-1}}.$
Finally, from (4.39) we see that estimates (4.40)-(4.41) yield the following
inequality
(4.49) $\displaystyle\max_{t\in
I_{n}}\left(\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Omega}u_{i,n}^{2}\left(x,t\right)dx+\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Gamma}u_{i,n}^{2}\left(x,t\right)dS\right)$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{I_{n}\times\Omega}\alpha_{i}\left|u_{i}\right|^{p_{i}}\left|\nabla\left(u_{i,n}\eta_{n}\right)\right|^{2}dxdt$
$\displaystyle\leq\frac{2^{n\left(\delta+1\right)}c}{\tau}\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{A_{i,n}^{\Omega}}\left(u_{i}-k_{n-1}\right)_{+}^{\delta}dxdt$
$\displaystyle+\frac{2^{n\left(\gamma+1\right)}c}{\tau}\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{A_{i,n}^{\Gamma}}\left(u_{i}-k_{n-1}\right)_{+}^{\gamma}dSdt.$
Step 3. (The iterative argument). We continue our main argument by first
recalling the following result (see, e.g., [10, Lemma 4.1]).
###### Lemma 4.2.
Let $\left\\{\mathcal{Y}_{n}\right\\}$ be a sequence of positive numbers such
that it satisfies
(4.50) $\mathcal{Y}_{n+1}\leq Cb^{n}\mathcal{Y}_{n}^{1+\kappa},$
for some constants $C,b,\kappa>0$. If $\mathcal{Y}_{0}\leq
C^{-1/\kappa}b^{-1/\kappa^{2}}$, then $\mathcal{Y}_{n}\rightarrow 0$ as
$n\rightarrow\infty.$
Define
$\mathcal{Y}_{i,n}:=\frac{1}{\left|Q_{n}\right|}\left(\iint\nolimits_{I_{n}\times\Omega}\left(u_{i}-k_{n}\right)_{+}^{\delta}dxdt+\iint\nolimits_{I_{n}\times\Gamma}\left(u_{i}-k_{n}\right)_{+}^{\gamma}dSdt\right),$
where we recall that $Q_{n}=I_{n}\times\overline{\Omega}$ and
$\left|Q_{n}\right|=\left|I_{n}\times\Omega\right|+\left|I_{n}\times\Gamma\right|.$
Set $\mathcal{Y}_{n}=\sum\nolimits_{i=1}^{m}\mathcal{Y}_{i,n}$. The goal now
is to show that the sequence $\left\\{\mathcal{Y}_{n}\right\\}$ satisfies a
recursive relation of the form (4.50). First, using the definition of
$\mathcal{Y}_{n},$ we can rewrite (4.49) as the following inequality:
(4.51) $\displaystyle\max_{t\in
I_{n+1}}\left(\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Omega}u_{i,n+1}^{2}\left(x,t\right)dx+\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Gamma}u_{i,n+1}^{2}\left(x,t\right)dS\right)$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\iint\nolimits_{I_{n+1}\times\Omega}\alpha_{i}\left|u_{i}\right|^{p_{i}}\left|\nabla\left(u_{i,n+1}\eta_{n+1}\right)\right|^{2}dxdt$
$\displaystyle\leq
c2^{\left(n+1\right)\left(\rho+1\right)}\tau^{-1}\left|Q_{n+1}\right|\mathcal{Y}_{n},$
for $\rho:=\max\left(\gamma,\delta\right)>1.$ Secondly, applying (4.43) to the
bulk integral over $u_{i,n+1}^{M_{i}}$ (where $M_{i}:=p_{i}/2+1$), which
occurs in the integrals in (4.47)-(4.48), and then using (4.51) to estimate
the second terms in those products, we obtain
(4.52) $\displaystyle\iint\nolimits_{{}^{I_{n+1}\times\Omega}}$
$\displaystyle\left(u_{i}-k_{n+1}\right)_{+}^{s}dxdt$ $\displaystyle\leq
c\left(\frac{2^{n\rho}}{\tau}+\frac{2^{n\left(\delta-M_{i}\right)}}{L^{\delta-
M_{i}}}\right)\left|Q_{n+1}\right|\mathcal{Y}_{n}\left(\frac{c2^{n\rho}}{\tau}\left|Q_{n+1}\right|\mathcal{Y}_{n}\right)^{\frac{2}{N}},$
and
(4.53)
$\displaystyle\iint\nolimits_{{}^{I_{n+1}\times\Gamma}}\left(u_{i}-k_{n+1}\right)_{+}^{l}dSdt$
$\displaystyle\leq c\left(\frac{2^{n\rho}}{\tau}+\frac{2^{n\left(\gamma-
M_{i}\right)}}{L^{\gamma-
M_{i}}}\right)\left|Q_{n+1}\right|\mathcal{Y}_{n}\left(\frac{c2^{n\rho}}{\tau}\left|Q_{n+1}\right|\mathcal{Y}_{n}\right)^{\frac{1}{N-1}}.$
Hölder’s inequality applied to $\mathcal{Y}_{i,n+1}$ yields
(4.54) $\displaystyle\mathcal{Y}_{i,n+1}$
$\displaystyle\leq\frac{1}{\left|Q_{n+1}\right|}\left(\iint\nolimits_{{}^{I_{n+1}\times\Omega}}\left(u_{i}-k_{n+1}\right)_{+}^{s}dxdt\right)^{\delta/s}\left|A_{i,n+1}^{\Omega}\right|^{1-\delta/s}$
$\displaystyle+\frac{1}{\left|Q_{n+1}\right|}\left(\iint\nolimits_{{}^{I_{n+1}\times\Gamma}}\left(u_{i}-k_{n+1}\right)_{+}^{l}dSdt\right)^{\gamma/l}\left|A_{i,n+1}^{\Gamma}\right|^{1-\gamma/l}.$
Inserting the estimates for $\left|A_{i,n+1}^{\Omega}\right|$ and
$\left|A_{i,n+1}^{\Gamma}\right|$, respectively, from (4.42), on the right-
hand side of (4.54), we deduce
$\displaystyle\mathcal{Y}_{i,n+1}$ $\displaystyle\leq
c\left|Q_{n+1}\right|^{\frac{2\delta}{Ns}}\mathcal{Y}_{n}^{1+\frac{2\delta}{Ns}}\left(\frac{2^{n\rho}}{\tau}+\frac{2^{n\left(\delta-
M_{i}\right)}}{L^{\delta-M_{i}}}\right)^{\delta/s}$
$\displaystyle\times\left(\frac{c2^{n\rho}}{\tau}\right)^{\frac{2\delta}{Ns}}\left(\frac{2^{n\delta}}{L^{\delta}}\right)^{1-\delta/s}$
$\displaystyle+c\left|Q_{n+1}\right|^{\frac{\gamma}{\left(N-1\right)l}}\mathcal{Y}_{n}^{1+\frac{\gamma}{\left(N-1\right)l}}\left(\frac{2^{n\rho}}{\tau}+\frac{2^{n\left(\gamma-
M_{i}\right)}}{L^{\gamma-M_{i}}}\right)^{\gamma/l}$
$\displaystyle\times\left(\frac{c2^{n\rho}}{\tau}\right)^{\frac{\gamma}{\left(N-1\right)l}}\left(\frac{2^{n\gamma}}{L^{\gamma}}\right)^{1-\gamma/l}.$
Henceforth, by setting
$\kappa:=\kappa\left(\delta,\gamma\right)=\left\\{\begin{array}[]{ll}\max\left(\frac{2\delta}{Ns},\frac{\gamma}{\left(N-1\right)l}\right),&\text{if
}\mathcal{Y}_{n}\geq 1,\\\
\min\left(\frac{2\delta}{Ns},\frac{\gamma}{\left(N-1\right)l}\right),&\text{if
}\mathcal{Y}_{n}<1,\end{array}\right.$
the above inequality yields the recursive relation
$\mathcal{Y}_{n+1}\leq Cb^{n}\mathcal{Y}_{n}^{1+\kappa},\text{ with
}\kappa>0,$
where $C\sim L^{-\sigma}$ depends on $\tau^{-1}$ and $b\sim 2^{\zeta}$, for
some positive constants $\sigma,\zeta$ depending on $\delta,\gamma,s$, $l.$
Therefore, if we choose $L\geq 1$ sufficiently large so there holds
$\mathcal{Y}_{0}\leq C^{-1/\kappa}b^{-1/\kappa^{2}}\lessapprox
L^{\sigma/\kappa}b^{-1/\kappa^{2}},$
then by Lemma 4.2, it follows that $\mathcal{Y}_{n}\rightarrow 0$ as
$n\rightarrow\infty$. This implies that
$\sup_{\left(x,t\right)\in\left[T-\tau,T\right]\times\overline{\Omega}}u_{i}\left(x,t\right)\leq\lim_{n\rightarrow\infty}k_{n}\leq
2L.$
In order to estimate $u_{i}\left(x,t\right)$ from below it suffices to apply
the result just obtained to the functions
$\widetilde{u}_{i}\left(x,t\right)=-u_{i}\left(x,t\right)$, which satisfies a
system of the same type as for $u_{i}\left(x,t\right),$ with nonlinearities
$\widetilde{a}_{i}\left(x,t,\widetilde{u}_{i}\right)=-a_{i}\left(x,t,-u_{i}\right),$
$\widetilde{f}_{i}\left(x,t,\widetilde{u}_{i}\right)=-f_{i}\left(x,t,-u_{i}\right)$
and
$\widetilde{g}_{i}\left(x,t,\widetilde{u}_{i}\right)=-g_{i}\left(x,t,-u_{i}\right)$,
respectively. These functions are subject to the same conditions (3.12),
(3.11). This yields the desired estimate (3.14). Finally, we may conclude that
if $T$ is sufficiently large, we can take $\tau=1$ in (3.14), which also
immediately gives the first conclusion in the theorem. The proof is finished.
### 4.3. Proof of Theorems 3.3, 3.4
In order to justify our computations for problem (1.1), (1.3), (1.4), (3.15),
it is not clear how to use the scheme introduced at the beginning of the
section due to the nature of the boundary domain (indeed,
$\overline{\Gamma}_{1}\cap\overline{\Gamma}_{2}\neq\varnothing$ in that case,
so we _cannot_ exploit maximal regularity theory to construct smooth solutions
unless $\Gamma_{2}\equiv\varnothing$). However, the proof can be based on the
application of a Galerkin approximation scheme which is not standard due to
the nature of the boundary condition (1.3). We refer the reader to, e.g., [7],
for further details, where a system of reaction-diffusion equations for the
phase-field equations with dynamic boundary conditions were considered (cf.
also [26] in a degenerate case).
We begin the proof of Theorem 3.3 with a result for the eigenvalue problem for
so-called Wentzell Laplacian $\Delta_{W}$ (see [22, Appendix]). More
precisely, let us consider the equation
(4.55) $-\Delta\varphi=\Lambda\varphi\text{ in }\Omega,$
with a boundary condition that depends on the eigenvalue $\Lambda$ explicitly,
(4.56) $\partial_{\mathbf{n}}\varphi=\Lambda\varphi\text{ on }\Gamma_{1},$
such that
(4.57) $\partial_{\mathbf{n}}\varphi+\varphi=0\text{ on }\Gamma_{2}.$
(recall that $\Gamma_{2}$ is assumed to be a set of positive measure and that
(4.57) holds where a Dirichlet boundary condition for $u=u_{i}$ is satisfied
on $\Gamma_{2}\times\mathbb{R}_{+}$). Such a function $\varphi$ will be called
an eigenfunction associated with $\Lambda$ and the set of all eigenvalues
$\Lambda$ of (4.55)-(4.57) will be denoted by $\Lambda_{j},$ $j\in\mathbb{N}$.
Let $\varphi_{1}\in C^{2}\left(\overline{\Omega}\right)$ and $\Lambda_{1}$,
denote the principal eigenfunction and eigenvalue of (4.55)-(4.57),
respectively. We have the following.
###### Proposition 4.1.
For the spectral problem (4.55)-(4.57), $\Lambda_{1}>0$ is simple and
$\varphi_{1}>0$ in $\overline{\Omega}$.
###### Proof.
Using the standard characterization for the eigenvalues $\Lambda_{j}$ of
$\Delta_{W}$ (see, e.g., [22]), we obtain that the following min-max principle
holds:
(4.58) $\Lambda_{j}=\min_{\begin{subarray}{c}Y_{j}\subset
H^{1}\left(\Omega\right),\\\ \dim Y_{j}=j\end{subarray}}\max_{0\neq\varphi\in
Y_{j}}R_{W}\left(\varphi,\varphi\right),\text{ }j\in\mathbb{N}\text{,}$
where the Rayleigh quotient $R_{W}$, for the (boundary perturbed) Wentzell
operators $\Delta_{W}$, is given by
(4.59)
$R_{W}\left(\varphi,\varphi\right):=\frac{\left\|\nabla\varphi\right\|_{2}^{2}+\left\langle\varphi,\varphi\right\rangle_{L^{2}\left(\Gamma_{2}\right)}}{\left\|\varphi\right\|_{\mathbb{X}^{2}}^{2}},\text{
}0\neq\varphi\in H^{1}\left(\Omega\right).$
Exploiting a well-known Friedrichs-Poincare’s inequality, we have
$R_{W}\left(\varphi,\varphi\right)\geq
c_{W}\left\|\varphi\right\|_{\mathbb{X}^{2}}^{2},$ for some positive constant
$c_{W},$ which implies that $\Lambda_{j}>0$, for any $j\in\mathbb{N}$. By the
maximum principle, $\varphi_{1}$ is positive in $\overline{\Omega}$ since
$\Gamma_{2}$ has positive surface measure. The fact that $\Lambda_{1}$ is
simple, follows again from the maximum principle (see, e.g., [4]).
We are now ready to give the proof of Theorem 3.3. Without loss of generality,
we can take $\delta_{i}=1,$ for all $i\in J_{m}$. We multiply (1.1) by
$\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1},$ and integrate
over $\Omega$, for each $i\in\mathbb{N}_{m}$. We obtain
$\displaystyle\frac{1}{m_{i}}\frac{d}{dt}\int_{\Omega}\left|u_{i}\right|^{m_{i}}\varphi_{1}dx+\left\langle
f_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}\right\rangle_{L^{2}\left(\Omega\right)}$
$\displaystyle-\int_{\Omega}div\left(a_{i}\left(u_{i}\right)\nabla
u_{i}\right)\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}dx$
$\displaystyle=$ $\displaystyle 0.$
Similarly, we multiply (1.3) by
$\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}$ and integrate
the relation over $\Gamma$ (recall that (3.15) holds over $\Gamma_{2}$). We
have
(4.61)
$\displaystyle\frac{1}{m_{i}}\frac{d}{dt}\int_{\Gamma_{1}}\left|u_{i}\right|^{m_{i}}\varphi_{1}dS+\int_{\Gamma}a_{i}\left(u_{i}\right)\partial_{\mathbf{n}}u_{i}\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}dS$
$\displaystyle+\left\langle
g_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}\right\rangle_{L^{2}\left(\Gamma_{1}\right)}$
$\displaystyle=0,$
for each $i\in\mathbb{N}_{m}$. Consider the following real-valued function
(4.62)
$\mathcal{E}\left(\overrightarrow{u}\left(x,t\right)\right)=\sum\nolimits_{i\in\mathbb{N}_{m}}\frac{1}{m_{i}}\left|u_{i}\left(x,t\right)\right|^{m_{i}}.$
Integrating by parts in (4.3), then using (4.61), on account of the following
computation
$\displaystyle\int_{\Omega}div\left(a_{i}\left(u_{i}\right)\nabla
u_{i}\right)\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}dx$
$\displaystyle=-\left(m_{i}-1\right)\int_{\Omega}a_{i}\left(u_{i}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{m_{i}-2}\varphi_{1}dx$
$\displaystyle-\int_{\Omega}a_{i}\left(u_{i}\right)\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\nabla
u_{i}\cdot\nabla\varphi_{1}dx$
$\displaystyle+\int_{\Gamma}a_{i}\left(u_{i}\right)\partial_{\mathbf{n}}u_{i}\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\varphi_{1}dS,$
we deduce the following inequality
$\displaystyle\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu+\sum\nolimits_{i\in\mathbb{N}_{m}}\left(m_{i}-1\right)\int_{\Omega}a_{i}\left(u_{i}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{m_{i}-2}\varphi_{1}dx$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\Omega}a_{i}\left(u_{i}\right)\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\nabla
u_{i}\cdot\nabla\varphi_{1}dx$ $\displaystyle\leq$ $\displaystyle
c\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu+c.$
Here we have employed (3.16) to estimate the terms involving $f_{i},g_{i}$ in
(4.3)-(4.61). Let us now estimate the third integral term on the left-hand
side of (4.3). Exploiting the assumption (3.5) on $a_{i}$,
$i\in\mathbb{N}_{m}$, we deduce for $u=u_{i}$, $p=p_{i},$ $M=m_{i}$ that
(4.64)
$\displaystyle\int_{\Omega}a\left(u\right)\left|u\right|^{p-1}sgn\left(u\right)\nabla
u\cdot\nabla\varphi_{1}dx$
$\displaystyle\geq\int_{\Omega}\left|u\right|^{p+M-1}sgn\left(u\right)\nabla
u\cdot\nabla\varphi_{1}dx$
$\displaystyle=\int_{\Omega}\nabla\overline{a}\left(u\right)\cdot\nabla\varphi_{1}dx,$
where we have set
$\overline{a}\left(u\right):=\int_{0}^{\left|u\right|}a\left(s\right)\left|s\right|^{p-1}ds\geq
c\left|u\right|^{M+p}.$
Integrating by parts in (4.64) once more and noting that
$\overline{a}\left(0\right)=0$, we obtain
(4.65)
$\displaystyle\int_{\Omega}\nabla\overline{a}\left(u\right)\cdot\nabla\varphi_{1}dx$
$\displaystyle=\int_{\Gamma}\overline{a}\left(u\right)\partial_{\mathbf{n}}\varphi_{1}dS-\int_{\Omega}\overline{a}\left(u\right)\Delta\varphi_{1}dx$
$\displaystyle=\int_{\Gamma_{1}}\overline{a}\left(u\right)\partial_{\mathbf{n}}\varphi_{1}dS+\int_{\Gamma_{2}}\overline{a}\left(u\right)\partial_{\mathbf{n}}\varphi_{1}dS$
$\displaystyle-\int_{\Omega}\overline{a}\left(u\right)\Delta\varphi_{1}dx$
$\displaystyle=\int_{\Gamma_{1}}\overline{a}\left(u\right)\Lambda_{1}\varphi_{1}dS+\int_{\Omega}\overline{a}\left(u\right)\Lambda_{1}\varphi_{1}dx$
$\displaystyle\geq\Lambda_{1}\int_{\overline{\Omega}}\left|u\right|^{M+p}\varphi_{1}d\mu,$
since $\left(\Lambda_{1},\varphi_{1}\right)$ satisfies (4.55)-(4.57).
Inserting the above estimates in (4.3), we get the following inequality
(4.66)
$\displaystyle\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu+\Lambda_{1}\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\overline{\Omega}}\left|u_{i}\right|^{m_{i}+p_{i}}\varphi_{1}d\mu$
$\displaystyle\leq
c\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu+c.$
Since all $p_{i}$’s are positive, we can absorb the integral term on the
right-hand side of (4.66), using the Young’s inequality as follows:
$\sum\nolimits_{i\in\mathbb{N}_{m}}\left|u_{i}\right|^{m_{i}}\leq\varepsilon\sum\nolimits_{i\in\mathbb{N}_{m}}\left|u_{i}\right|^{m_{i}+p_{i}}+C_{\varepsilon},$
for a sufficiently small $\varepsilon\in\left(0,\Lambda_{1}\right)$ and some
positive constant $C_{\varepsilon},$ independent of $u_{i},t.$ Moreover,
setting $\nu=\min_{i\in\mathbb{N}_{m}}\left(m_{i}/p_{i}\right)+1>1$, we
immediatelly have from the above inequality, that
(4.67)
$\int_{\overline{\Omega}}\left(\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\right)^{\nu}\varphi_{1}d\mu\leq
c\sum\nolimits_{i\in\mathbb{N}_{m}}\int_{\overline{\Omega}}\left|u_{i}\right|^{m_{i}+p_{i}}\varphi_{1}d\mu+c,$
for some positive constant $c$ independent of $\overrightarrow{u}$, $t$ and
initial data. Using (4.67), we see that (4.66) yields the following estimate
(4.68)
$\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu+c\int_{\overline{\Omega}}\left(\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\right)^{\nu}\varphi_{1}d\mu\leq
c,$
by an appropriate choice of $\varepsilon\leq\Lambda_{1}/2.$ By normalizing the
eigenfunction $\varphi_{1}$ in (4.68) such that
$\left\|\varphi_{1}\right\|_{L^{1}\left(\overline{\Omega},d\mu\right)}=1$, on
account of Jensen’s inequality, it follows that
$\left(\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu\right)^{\nu}\leq\int_{\overline{\Omega}}\left(\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\right)^{\nu}\varphi_{1}d\mu,$
which gives the following estimate:
(4.69)
$\partial_{t}Y\left(t\right)+c_{1}\left(Y\left(t\right)\right)^{\nu}\leq
c_{2},$
for some positive constants $c_{1},c_{2}$, where we have set
$Y\left(t\right):=\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)\varphi_{1}d\mu.$
We can now use the Gronwall’s inequality (see, e.g., [37, Chapter III, Lemma
5.1]), applied to (4.69) to deduce that
(4.70)
$Y\left(t\right)\leq\left(\frac{c_{2}}{c_{1}}\right)^{\nu}+\left(c_{1}\left(\nu-1\right)t\right)^{-\frac{1}{\nu-1}},\text{
}\forall t>0,$
which yields the desired claim. The proof of the theorem is complete.
###### Remark 4.2.
In the case when $\overrightarrow{u}_{0}\in\mathcal{X}^{\overrightarrow{r}}$,
$Y\left(0\right)=\lim_{t\rightarrow 0^{+}}Y\left(t\right)$ is finite, so a
similar argument to [37, Chapter III, Lemma 5.1] gives
$Y\left(t\right)\leq\max\left\\{Y\left(0\right),\left(\frac{c_{2}}{c_{1}}\right)^{\nu}\right\\},\text{
}\forall t\geq 0.$
Thus, the second assertion in Theorem 3.3 also follows. We also note that the
above argument relies entirely on the fact that the boundary $\Gamma_{2}$ has
positive measure and this gives $\Lambda_{1}>0$. The proof fails to work if
for instance, $\Gamma\equiv\Gamma_{1}$ (i.e., when $\Gamma_{2}=\varnothing$).
We shall require different arguments for this case (see below). Finally, if at
least one $p_{i}=0$, for some $i\in\mathbb{N}_{m}$, the above argument can
still be used to derive the following bound
$\sup_{t\geq
0}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{\overrightarrow{r}}}\leq
Q\left(\left\|\overrightarrow{u}_{0}\right\|_{\mathcal{X}^{\overrightarrow{r}}}e^{ct}\right).$
Indeed, this follows from a standard application of Gronwall’s inequality to
(4.66).
We now continue with the proof of Theorem 3.4. As in the proof of Theorem 3.3,
we multiply (1.1) by $\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right),$ and
integrate over $\Omega$, for each $i\in\mathbb{N}_{m}$. Then, we multiply both
equations (1.3) and (1.2) by
$\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)$ and integrate the
relations that we obtain over $\Gamma$. Analogously to (4.3)-(4.61) and
arguing in a standard way as in (4.7), we deduce the following identity:
(4.71)
$\displaystyle\frac{d}{dt}\sum\nolimits_{i\in\mathbb{N}_{m}}\frac{1}{m_{i}}\left(\int_{\Omega}\left|u_{i}\right|^{m_{i}}dx+\int_{\Gamma}\left|u_{i}\right|^{m_{i}}dS\right)$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\left\langle
f_{i}\left(x,t,\overrightarrow{u}\right),\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\right\rangle_{L^{2}\left(\Omega\right)}$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\left\langle
g_{i}\left(x,t,\overrightarrow{y}\right),\left|u_{i}\right|^{m_{i}-1}sgn\left(u_{i}\right)\right\rangle_{L^{2}\left(\Gamma\right)}$
$\displaystyle=-\sum\nolimits_{i\in\mathbb{N}_{m}}\left(m_{i}-1\right)\int_{\Omega}a_{i}\left(x,t,\overrightarrow{u}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{m_{i}-2}dx.$
By assumption (3.5), we can estimate the term on the right-hand side of (4.71)
as follows:
(4.72)
$\displaystyle\int_{\Omega}a_{i}\left(t,\overrightarrow{u}\right)\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{m_{i}-2}dx$
$\displaystyle\geq\alpha_{i}\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\left|u_{i}\right|^{m_{i}+p_{i}-2}dx$
$\displaystyle=\alpha_{i}\left(\frac{2}{m_{i}+p_{i}}\right)^{2}\int_{\Omega}\left|\nabla\left|u\right|^{\frac{m_{i}+p_{i}}{2}}\right|^{2}dx.$
To estimate the nonlinear terms on the left-hand side of (4.71), we may
exploit (3.17). On account of (4.72), we have
(4.73)
$\displaystyle\frac{d}{dt}\sum\nolimits_{i\in\mathbb{N}_{m}}\frac{1}{m_{i}}\left(\int_{\Omega}\left|u_{i}\right|^{m_{i}}dx+\int_{\Gamma}\left|u_{i}\right|^{m_{i}}dS\right)$
$\displaystyle+\sum\nolimits_{i\in\mathbb{N}_{m}}\alpha_{i}\left(\frac{2}{m_{i}+p_{i}}\right)^{2}\left(m_{i}-1\right)\int_{\Omega}\left|\nabla\left|u\right|^{\frac{m_{i}+p_{i}}{2}}\right|^{2}dx$
$\displaystyle-\sum\nolimits_{i\in\mathbb{N}_{m}}\left(C_{f_{i}}\int_{\Omega}\left|u_{i}\right|^{m_{i}+p_{i}}dx+C_{g_{i}}\int_{\Gamma}\left|u_{i}\right|^{m_{i}+p_{i}}dS\right)$
$\displaystyle\leq c.$
By assumption (3.21), it follows that, for all $i\in\mathbb{N}_{m}$, it holds
(4.74) $\displaystyle
a_{i}\left\|\nabla\varphi_{i}\right\|_{L^{2}\left(\Omega\right)}^{2}-C_{f_{i}}\left\|\varphi_{i}\right\|_{L^{2}\left(\Omega\right)}^{2}-C_{g_{i}}\left\|\varphi_{i}\right\|_{L^{2}\left(\Gamma\right)}^{2}$
$\displaystyle\geq\Lambda_{1,i}\left(\left\|\varphi_{i}\right\|_{L^{2}\left(\Omega\right)}^{2}+\left\|\varphi_{i}\right\|_{L^{2}\left(\Gamma\right)}^{2}\right),$
for all $\varphi_{i}\in H^{1}\left(\Omega\right)$. Thus, by choosing
$\varphi_{i}=\left|u_{i}\right|^{\left(m_{i}+p_{i}\right)/2}$ in (4.74), and
recalling (4.62), from (4.73), we obtain the following inequality:
(4.75)
$\displaystyle\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu+\Lambda_{1}\sum\nolimits_{i\in\mathbb{N}_{m}}\left(\int_{\Omega}\left|u_{i}\right|^{m_{i}+p_{i}}dx+C_{g_{i}}\int_{\Gamma}\left|u_{i}\right|^{m_{i}+p_{i}}dS\right)$
$\displaystyle\leq c.$
Let us assume that $p_{i}>0$, for all $i\in\mathbb{N}_{m}$. Arguing now as in
the proof of Theorem 3.3 (see (4.66)-(4.69)), it is not hard to see that we
arrive at the following inequality
(4.76)
$\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu+c\left(\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu\right)^{v}\leq
c,$
where $v=\min_{i\in\mathbb{N}_{m}}\left(m_{i}/p_{i}\right)+1>1.$ Thus, we can
use the Gronwall’s inequality as before (see (4.70)) to derive the estimate
(4.77)
$\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu\leq
c\left(1+t^{-\frac{1}{\nu-1}}\right),\text{ }\forall t>0.$
Hence, the first claim of the theorem follows from (4.77). On the other hand,
if at least one $p_{i}=0$ for some $i\in\mathbb{N}_{m},$ we obtain the
following analogue of (4.76):
(4.78)
$\partial_{t}\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu+c\int_{\overline{\Omega}}\mathcal{E}\left(\overrightarrow{u}\left(t\right)\right)d\mu\leq
c,$
which yields the second claim of the theorem once more on account of
Gronwall’s inequality. In particular, there exists a positive function $Q$,
independent of initial data and time, such that
(4.79) $\sup_{t\geq
0}\left\|\overrightarrow{u}\left(t\right)\right\|_{\mathcal{X}^{\overrightarrow{r}}}\leq
Q\left(\left\|\overrightarrow{u}_{0}\right\|_{\mathcal{X}^{\overrightarrow{r}}}\right)e^{-c_{0}t}+C_{0},$
for some positive constants $c_{0},C_{0}$ independent of initial data and
time. The proof of Theorem 3.4 is now complete.
## 5\. Appendix
We will consider a more general problem than (1.7)-(1.8) by taking
$f_{1}\left(s\right)=f\left(s\right)-\lambda s$,
$g_{1}\left(s\right)=g\left(s\right)-\gamma s,$ provided that
$\lambda,\gamma>0$ are sufficiently large, and
$f\left(s\right)\geq-c_{f}\text{, }g\left(s\right)\geq-c_{g}\text{, }\forall
s\in\mathbb{R}\text{,}$
for some positive constants $c_{f},c_{g}.$ If $f\left(s\right)\sim
C_{f}\left|s\right|^{p}s$ and $g\left(s\right)\sim C_{g}\left|s\right|^{q}s$,
as $\left|s\right|\rightarrow\infty$, for $p,q>1$ and some positive constants
$C_{f},C_{g}$, it is well-known [23] that problem
(5.1) $\partial_{t}u-\nu\Delta u+f\left(u\right)-\lambda u=0,\text{ in
}\Omega\times(0,+\infty),$
subject to the dynamic condition
(5.2) $\partial_{t}u+\nu\partial_{\mathbf{n}}u+g\left(u\right)-\gamma
u=0,\text{ on }\Gamma\times\left(0,\infty\right),$
and initial condition
(5.3) $u_{\mid t=0}=u_{0}\text{ in }\overline{\Omega},$
possesses a finite dimensional global attractor $\mathcal{A}_{\text{dyn}}$
which is bounded in $H^{2}\left(\Omega\right)\cap\mathbb{X}^{\infty}$.
Let $u_{\ast}$ be a constant (hyperbolic) equilbrium for the system
(5.1)-(5.2) (see [22, Section 3]). We linearize (5.1)-(5.2) around $u_{\ast}.$
We obtain
(5.4) $\partial_{t}u=\nu\Delta
u-\left(f^{{}^{\prime}}\left(u_{\ast}\right)-\lambda\right)u,\text{ in
}\Omega\times(0,+\infty),$
subject to the dynamic condition
(5.5)
$\partial_{t}u=-\nu\partial_{\mathbf{n}}u-\left(g^{{}^{\prime}}\left(u_{\ast}\right)-\gamma\right)u,\text{
on }\Gamma\times\left(0,\infty\right).$
We aim to better understand the nature of the (invariant) unstable eigenspace
$E^{u}$ which corresponds to the following (matrix) operator
$\mathbf{L}\left(u_{\ast}\right)W=\binom{\nu\Delta
w-f^{{}^{\prime}}\left(u_{\ast}\right)w+\lambda
w}{-\nu\partial_{\mathbf{n}}w-g^{{}^{\prime}}\left(u_{\ast}\right)w+\gamma
w},\text{ }W=\left(\begin{array}[]{c}w\\\ w_{\mid\Gamma}\end{array}\right),$
with
$\sigma\left(\mathbf{L}\left(u_{\ast}\right)\right)\subset\left\\{\zeta:\zeta>0\right\\}$.
We note that
$\left(\mathbf{L}\left(u_{\ast}\right),\text{dom}\left(\mathbf{L}\left(u_{\ast}\right)\right)\right)$
is self-adjoint on $\mathbb{X}^{2}$ with spectrum contained in
$\left(-\infty,C_{\lambda,\gamma}\right],$ for some $C_{\lambda,\gamma}>0$
which depends only on $f,g,\lambda$ and $\gamma$ (see, e.g., [22] and
references therein). Next, let
$\left\\{\varphi_{j}\left(x\right)\right\\}_{j\in\mathbb{N}_{0}}$ be an
orthonormal basis in $\mathbb{X}^{2}$ consisting of eigenfunctions of the
(positive) Wentzell Laplacian $\Delta_{W}$ operator (see [22, Theorem 5.1])
(5.6) $\Delta_{W}\varphi_{j}=\Lambda_{j}\varphi_{j},\text{
}j\in\mathbb{N}_{0},\text{
}\varphi_{j}\in\text{dom}\left(\Delta_{W}\right)\cap
C\left(\overline{\Omega}\right)$
such that
$0=\Lambda_{0}<\Lambda_{1}\leq\Lambda_{2}\leq...\leq\Lambda_{,j}\leq\Lambda_{j+1}\leq....\rightarrow+\infty.$
We shall seek for eigenvectors
$W_{j}=\binom{w_{j}}{w_{j\mid\Gamma}}\in\mathbb{X}^{2}$, of the form
$w_{j}\left(x\right)=\varphi_{j}\left(x\right)p_{j},$ $p_{j}\in\mathbb{R}$,
satisfying equation
(5.7) $\mathbf{L}\left(u_{\ast}\right)W_{j}=\zeta_{j}W_{j},\text{
}W_{j}\in\text{dom}\left(\mathbf{L}\left(u_{\ast}\right)\right):=\text{dom}\left(\Delta_{W}\right).$
Note that for $W_{j}\in dom\left(\mathbf{L}\left(u_{\ast}\right)\right)\subset
H^{1}\left(\Omega\right)\times L^{2}\left(\Gamma\right),$ the trace of $w_{j}$
makes sense as an element of $H^{1/2}\left(\Gamma\right)$. Substituting such
$w_{j}$ into (5.7), taking into account (5.6) and the fact that
$\mathbf{L}\left(u_{\ast}\right)W_{j}=-\nu\Delta_{W}W_{j}+\Pi_{\lambda,\gamma}W_{j},\text{
}\Pi_{\lambda,\gamma}W_{j}:=\binom{(-f^{{}^{\prime}}\left(u_{\ast}\right)+\lambda)w_{j}}{(-g^{{}^{\prime}}\left(u_{\ast}\right)+\gamma)w_{j\mid\Gamma}},$
we obtain the equation
(5.8)
$\left(-\nu\Lambda_{j}I+\Pi_{\lambda,\gamma}\right)p_{j}=\zeta_{j}p_{j},\text{
}\Pi_{\lambda,\gamma}=\left(\begin{array}[]{cc}-f^{{}^{\prime}}\left(u_{\ast}\right)+\lambda&0\\\
0&-g^{{}^{\prime}}\left(u_{\ast}\right)+\gamma\end{array}\right).$
A nonzero $p_{j}$ exists if $\zeta=\zeta_{j}$ is a root of the equation
(5.9) $\det\left(-\nu\Lambda_{j}I+\Pi_{\lambda,\gamma}-\zeta I\right)=0,\text{
}\zeta>0.$
When $\nu=0,$ this equation has at least one root $\zeta>0$ provided that at
least one of $\lambda$ and $\gamma$ is sufficiently large, i.e., either
$\lambda>f^{{}^{\prime}}\left(u_{\ast}\right)$ or
$\gamma>g^{{}^{\prime}}\left(u_{\ast}\right)$ (in fact the roots are
$\zeta=\lambda-f^{{}^{\prime}}\left(u_{\ast}\right)$ and
$\zeta=\gamma-g^{{}^{\prime}}\left(u_{\ast}\right)$, respectively). Therefore,
there exists $\delta>0$ such that when $\nu\Lambda_{j}<\delta,$ the equation
(5.9) has a root $\zeta_{j}\left(\mathbf{L}\right)=\zeta_{j}\left(\nu\right)$
with $\zeta_{j}>0$. Therefore, to any such root $\zeta_{j}$, we can assign a
nontrivial $p_{j},$ which is a solution of (5.8), and thus an eigenvector
$W_{j}$. Let us now compute how many $j$’s satisfy the inequality
$\nu\Lambda_{j}<\delta$. When $N\geq 3$, the asymptotic behavior of
$\Lambda_{j}$ is
(5.10) $\Lambda_{j}\sim C_{S}\left(\Gamma\right)j^{1/\left(N-1\right)}\text{
as }j\rightarrow\infty$
(see [22, Theorem 5.4]). The inequality $\nu\Lambda_{j}<\delta$ certainly
holds when
(5.11) $1\leq j\leq
C_{\lambda,\gamma}\delta^{n-1}\left(C_{S}\left(\Gamma\right)\nu\right)^{1-N}=C_{\lambda,\gamma}^{{}^{\prime}}\left|\Gamma\right|\left(\frac{1}{\nu}\right)^{N-1},\text{
for }N\geq 3,$
where the positive constants
$C_{\lambda,\gamma},C_{\lambda,\gamma}^{{}^{\prime}}$ depend only on
$\lambda,\gamma$ and $N$.
###### Remark 5.1.
Note that the number of unstable mode solutions to (5.4)-(5.5) obeys the same
relation (5.11) even when $f\equiv 0$ and $\lambda=0$ in (5.1) (i.e., the
dynamics of $u$ inside the bulk $\Omega$ is strictly linear). Finally, we note
that both
$C_{\lambda,\gamma},C_{\lambda,\gamma}^{{}^{\prime}}\rightarrow+\infty$ if
either $\gamma\rightarrow+\infty$ or $\lambda\rightarrow+\infty$ (cf. also
[22, Section 3]). In this case the instability index of $u_{\ast}$ is
$N_{+}\left(u_{\ast}\right)\sim
C_{\lambda,\gamma}^{{}^{\prime}}\left|\Gamma\right|\left(\frac{1}{\nu}\right)^{N-1},\text{
}N\geq 3.$
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|
arxiv-papers
| 2012-10-29T14:09:14 |
2024-09-04T02:49:37.264563
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ciprian G. Gal",
"submitter": "Ciprian Gal",
"url": "https://arxiv.org/abs/1210.7671"
}
|
1210.7734
|
.
# A formula representing phase deformed magnetic Berezin transforms as
functions of the magnetic Laplacian on $\mathbb{C}^{n}$
Nour eddine Askour ∗ Department of Mathematics, Sultan My Slimane University,
Faculty of Sciences and Technics (M’Ghila), Beni Mellal, Morocco.
askour@fstbm.ac.ma
###### Abstract.
we introduce a new class of a phase deformed magnetic berezin transforms and
we give two formulae representing these transforms as a functions of the
magnetic laplacian on $\mathbb{C}^{n}.$ As a consequence, we give an
inequality of diamagnetic type.
###### Key words and phrases:
Berezin transform, magnetic Laplacian, spectral function, Laguerre polynomials
###### 2010 Mathematics Subject Classification:
47G10;47B35,46N50;47N50.
## 1\. Introduction
The Berezin transform was introduced by Berezin [6] and [7] for certain
classical domains in $\mathbb{C}$. This transform links the Berezin symbols
and the symbols for Toeplitz operators. The formula represented the Berezin
transform as a function of the Laplace-Beltrami operator plays an important
role in the Berezin quantization theory. Classically this transform is defined
as follows. consider a domain $\Omega$ $\subset\mathbb{C}^{n}$ and a Borel
measure $d\mu$ on $\Omega$. Let $\mathfrak{H}$ be a closed subspace of
$L^{2}\left(\Omega,d\mu\right)$ consisting of continuous function and assume
that $\mathfrak{H}$ has a reproducing kernel $K\left(.,.\right)$. The Berezin
symbol $\widehat{A}$ of a bounded operator $A$ on $\mathfrak{H}$ is the
function defined on $\Omega$ by
$\hat{A}\left(z\right)=\frac{\left\langle
AK\left(.,z\right),K\left(.,z\right)\right\rangle}{K\left(z,z\right)},\qquad
z\in\Omega.$ (1.1)
For each $\varphi$ such that $\varphi\mathfrak{H}\in
L^{2}\left(\Omega,d\mu\right)-$ for instance for any $\varphi\in
L^{\infty}(\Omega)$, the Toeplitz operator $T_{\varphi}$ with symbol $\varphi$
is the operator on $\mathfrak{H}$ given by $T_{\varphi}[f]=P(f\varphi)$;
$f\in\mathfrak{H}$, where P is the orthogonal projector on $\mathfrak{H}$. By
definition the Berezin transform B is the integral transform defined by:
$B[\varphi](z):=\widehat{T_{\varphi}}(z)=\int_{\Omega}\frac{|K(z,\omega)|^{2}}{K(z,z)}\varphi(\omega)d\mu(\omega).$
(1.2)
Now, taking into a count that the Berezin transform can be defined provided
that there is a given closed subspace, which possesses a reproducing kernel,
we are here concerned with the eigenspaces
$A_{m}^{2}=\\{\varphi\in
L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z)),\tilde{\triangle}\varphi=E_{m}\varphi\\}$
(1.3)
of the second order differential operator
$\tilde{\triangle}=-\sum_{j=1}^{n}\frac{\partial^{2}}{\partial
z_{j}\partial\overline{z}_{j}}+\sum_{j=1}^{n}\overline{z}_{j}\frac{\partial}{\partial\overline{z}_{j}}$
(1.4)
corresponding to the eigenvalues $E_{m}=m$, m=1,2,…in (1,1), $d\nu$ denotes
the Lebesgue measure on $\mathbb{C}^{n}$. These eigenspaces called generalized
Bargmann spaces are reproducing kernel Hilbert space with reproducing kernels
given by
$K_{m}(z,w)=\frac{1}{\pi^{n}}e^{<z,w>}L_{m}^{n-1}{(|z-w|^{2})},z,w\in\mathbb{C}^{n}$
(1.5)
(see [5]) It is known, that the space $A^{2}_{m}$, corresponding to $m=0$
coincides with the Bargmann-Fock space $\mathfrak{F(\mathbb{C})}$, of
halomorphic function that are $e^{-|z|^{2}}d\nu$-integrable, while for $m\neq
0$, The space $A_{m}^{2}$ which can be viewed as as a Kernel spaces of the
Hypoelliptic differential operator $\widetilde{\Delta}-m$, consists of non
holomorphic functions. The Berezin transform $B_{m}$ is given by the following
convolution operator,
$B_{m}[\varphi](z)=\frac{m!}{\pi^{n}(n)_{m}}e^{-|w|^{2}}(L_{m}^{n-1}(|w|^{2}))^{2}\ast\varphi(z),\varphi\in
L^{2}\left(\mathbb{C}^{n},d\nu\right)$ (1.6)
This Berezin transform can be expressed in terms of the Euclidean Laplacian as
$B_{m}=\frac{e^{\frac{{\Delta}_{\mathbb{C}^{n}}}{4}}}{(n)_{m}}\sum_{k=0}^{m}\frac{(n-1)_{k}(m-k)!}{k!}(\frac{\Delta_{\mathbb{C}^{n}}}{4})^{k}L_{m-k}^{k}(\frac{\Delta_{\mathbb{C}^{n}}}{4})L_{m-k}^{n-1+k}(\frac{\Delta_{\mathbb{C}^{n}}}{4})$
(1.7)
[2] where $\triangle_{\mathbb{C}^{n}}$ the Euclidean Laplacian on
$\mathbb{C}^{n}$ and $(\alpha)_{j})$ denotes the Pochammer symbol and
$L^{(\alpha)}_{j}$ the well know Laguerre polynomials.
In this paper, we will be concerned with the following phase-deformed magnetic
Berezin transform,
$\mathfrak{B}_{m}[\varphi](z):=\int_{\mathbb{C}^{n}}e^{<z,w>}\frac{|K_{m}(z,w)|^{2}}{K_{m}(z,z)K_{m}(w,w)}\varphi(w)e^{-|w|^{2}}d\nu(w).$
(1.8)
acting on the Hilbert space $L^{2}(\mathbb{C}^{n},e^{-|w|^{2}}d\nu(w))$.
Explicitly,
$\mathfrak{B}_{m}[\varphi](z):=(\frac{m!}{(n)_{m}})^{2}\int_{\mathbb{C}^{n}}e^{(\left\langle
z,w\right\rangle-|z-w|^{2})}(L_{m}^{n-1}(|z-w|^{2})^{2}\varphi(w)e^{-|w|^{2}}d\nu(w).$
(1.9)
Here our aim is to express the ”phase deformed magnetic Berezin transform” as
a function of the magnetic Laplacian $\widetilde{\Delta}$ defined by (1.4).
The method used is based on the concrete $L^{2}$-spectral theory of the
operator $\widetilde{\Delta}$ [3], together with the functional calculus for
unbounded self-adjoint operator on complex Hilbert Space [8],[1], Precisely we
establish the following results:
$\mathfrak{B}_{m}=\pi^{n}(m!)^{2}2^{2m-n}\sum_{j=m+1}^{2m}\frac{2^{-j}}{j!(2m-j)!\Gamma(j-m+1)^{2}}\times$
(1.10)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n,j-m+1,j-m+1;1)\times$
$\frac{\Gamma(j+n+\widetilde{\Delta})}{\Gamma(n+\widetilde{\Delta})}e^{-\log(2)\widetilde{\Delta}}.$
$\mathfrak{B}_{m}=\frac{2^{-n}\pi^{n}}{\Gamma(n)}(\frac{m!}{(n)_{m}})^{2}\sum_{l=0}^{2m}2^{-l}\sigma^{n,m}_{l}\frac{\Gamma(n+l)}{l!}F(\widetilde{\Delta},n+l,n;\frac{1}{2}).$
(1.11)
where the coefficients $\sigma^{n,m}_{j}$ are given by:
$\sigma^{n,m}_{l}=(-1)^{l}\sum_{i=0}^{l}\binom{l}{i}\binom{n+m-1}{m-l+i}\binom{n+m-1}{m-i}.$
(1.12)
This paper is organized as follows. In section 2, we recall briefly some
spectral properties of the operator $\mathfrak{B}_{m},$ and as a consequence
we give its spectral functions. In section 3, the first part will be devoted
for the proof of the boundedness of the magnetic phase deformed Berezin
transform. In the second part, we give the proof of the main results(1.10) and
(1.11).
## 2\. $L^{2}$-CONCRETE SPECTRAL ANALYSIS OF THE MAGNETIC LAPLACIAN
$\widetilde{\Delta}$
In this section, we recall some spectral properties on the $L^{2}-$ concrete
spectral analysis of the magnetic Laplacian in the space of the $L^{2}-$
function on $\mathbb{C}^{n},$ with respect to the Gaussian measure [4], [5],
and [3].
Let us fix some notations. For $p,q\in\mathbb{Z}_{+}$ and let $H(p,q)$ denotes
the space of restriction to the sphere
$\mathcal{S}^{2n-1}=\\{\omega\in\mathbb{C}^{n},|\omega|=1\\}$ of the Euclidean
harmonic polynomials on $\mathbb{C}^{n}$ which are homogenous of degree $p$ in
$z$ and degree $q$ in $\overline{z}.$ The dimension $d(n,p,q)$ of $H(p,q)$ is
as follows. For $n$=2,3,…, we have
$d(n,p,q)=\frac{(p+q-1)(p+n-2)(q+n-2)}{p!q!(n-1)!(n-2)!},$ (2.1)
and for $n$=1, we make the convention $pq=0$; then $d(1,p,q)=1$; See [10].
For a complex number $\tau$, we denote by $A_{\tau}^{2}(\mathbb{C}^{n})$ the
space of eigenfunctions $f$ of $\widetilde{\Delta}$. That is
$A_{\tau}^{2}(\mathbb{C}^{n})=\\{\psi\in
L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z)),\widetilde{\Delta}\psi=\tau\psi\\}.$
(2.2)
The concrete description of this eigenspaces is given by the following.
proposition[5] Let $\tau\in\mathbb{C}.$ Then, we have:
i) For $\tau\neq 0,1,2,3,...$, the space $A_{\tau}^{2}(\mathbb{C}^{n})$ is
trivial,
ii)if $\tau=m\in\mathbb{Z_{+}}$ then the complex-valued function $f$ belongs
to $A_{m}^{2}(\mathbb{C}^{n})$ if and only if it can be expanded in the form
$f(z)=\sum_{(p,q)\in\Xi}a_{p,q}F(-m+q,n+p+q;r^{2})r^{p+q}h_{p,q}(\omega),z=r\omega,|\omega|=1,$
(2.3)
where $F(\alpha,\gamma,x)$ is the confluent hypergeometric function,
$\Xi=\\{(p,q)\in\mathbb{Z}\times\mathbb{Z},p\geq 0,0\leq q\leq m\\},$
and
$h_{p,q}=\\{h_{p,q}^{j}\\}_{1\leq j\leq d(n;p;q)}$ an orthonormal basis of
$H(p,q)$ with $a_{p,q}\in\mathbb{C}^{d(n,p,q)}$ are such that,
$\sum_{(p,q)\in\Xi}\gamma(n,m;p,q)|a_{p;q}|^{2}<+\infty,$ (2.4)
where
$\gamma(n,m;p,q)=\frac{(m-q)!(p+q+n-1)!}{2}\frac{\Gamma(n+p+q)}{\Gamma(m+p+q)},$
(2.5)
In [3], we have proved the essentially self-adjointnees of the the operator
$\widetilde{\Delta}$ with the space $C_{0}^{\infty}(\mathbb{C}^{n})$ of
$\mathbb{C}-$ valued $C^{\infty}-$ function with a compact support on
$\mathbb{C}^{n},$ as its natural regular domain $D(\widetilde{\Delta}),$ in
$L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z)).$ Also, the corresponding spectral
family [20] of $\widetilde{\Delta}$ has been given by the following
proposition.
###### Proposition 2.1.
The spectral family of the operator $\widetilde{\Delta}$ is given the
following integral operator on $L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z)):$
$E_{\lambda}[f](z)=\pi^{-n}\int_{\mathbb{C}^{n}}e^{<z,w>}L^{n}_{[\lambda]}(|z-w|^{2})f(w)e^{-|w|^{2}}d\nu(z)$
(2.6)
if $\lambda\geq 0$ and $E_{\lambda}=0,$ if $\lambda<0$.
($[\lambda]$=the greatest integer not exceeding $\lambda.$)
For our goal, that of expressing the phase-deformed magnetic Berezin transform
in termes of magnetic Laplacian $\widetilde{\Delta},$ we will need to give its
spectral function [9]. Precisely, we have the following proposition.
###### Proposition 2.2.
The spectral function associated the self-adjoint operator
$\widetilde{\Delta}$ is given by:
$e(\lambda,z,w)=(\pi)^{-n}e^{<z,w>}\sum_{m\in\mathbb{Z}_{+}}L_{m}^{n-1}(|z-w|^{2})\delta(\lambda-m),$
where $\delta(\lambda-m)$ is the Dirac delta generalized function at the point
$m.$
proof. By a simple derivation of the equation (2.6) in the distributional
sense, with the use of the jumps formula (see formula (6.16) in [19], p.101),
is not hard to see that the kernel (spectral function)
$e(\lambda,z,w)\in\mathcal{D}^{\prime}(\mathbb{R},\mathcal{D}^{\prime}(\mathbb{C}^{n}\times\mathbb{C}^{n}))$
of spectral density
$\frac{dE_{\lambda}}{d\lambda}\in\mathcal{D^{\prime}}(\mathbb{R},L(D(\widetilde{\Delta}),L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z)))$
is given by:
$e(\lambda,z,w)=(\pi)^{-n}e^{<z,w>}\sum_{k\in\mathbb{Z}_{+}}(L_{k}^{n}(|z-w|^{2})-L_{k-1}^{n}(|z-w|^{2}))\delta(\lambda-m)$
(2.7)
Now, using the following formula ([13],p241)
$L_{j}^{(\alpha)}(x)=L_{j}^{(\alpha+1)}(x)-L_{j-1}^{(\alpha+1)}(x)$ (2.8)
for $j=k$ and $\alpha=n-1$, the equation (2.7) becomes:
$e(\lambda,z,w)=(\pi)^{-n}e^{<z,w>}\sum_{m\in\mathbb{Z}_{+}}L_{m}^{n-1}(|z-w|^{2})\delta(\lambda-m).$
(2.9)
This ends the proof.
###### Remark 2.1.
For a suitable function $g:\mathbb{R}\rightarrow\mathbb{C},$ the operator
$g(\widetilde{\Delta})$ acts on $L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))$ by
the following formula:
$g(\widetilde{\Delta})[\varphi](z)=\int_{\mathbb{C}^{n}}\Psi_{g}(z,w)\varphi(w)e^{-|w|^{2}}d\nu(w),$
(2.10)
where $\Psi_{g}(z,w)$, is the Schwartz kernel given by:
$\Psi_{g}(z,w)=(\pi)^{-n}e^{\nu<z,w>}\sum_{m\in\mathbb{Z}_{+}}L_{m}^{n-1}(|z-w|^{2})g(m).$
(2.11)
Where the right hand side of the equations (2.10) and (2.11) are understood in
the distributional sense.
Remark 2.2. Let
$\widetilde{H}=\frac{-1}{4}\sum_{j=1}^{n}((\frac{\partial}{\partial
x_{j}}+iy_{j})^{2}+(\frac{\partial}{\partial y_{j}}-ix_{j})^{2}),$
be the Schrödinger operator with uniform field on $\mathbb{R}^{2n},$ we can
transform $\widetilde{H}$ to obtain the Magnetic Laplacian
$\widetilde{\Delta}$. Precisely, we have:
$Q\circ(\widetilde{H}-\frac{n}{2})\circ Q^{-1}=\widetilde{\Delta},$ (2.12)
Where
$Qf=exp(\frac{1}{2}\sum_{j=1}^{n}(x_{j}^{2}+y_{j}^{2}))f,$ (2.13)
$f\in L^{2}(\mathbb{R}^{2n},dx_{1}...dx_{n}dy_{1}...dy_{n}).$
## 3\. phase deformed magnetic Berezin transform and the magnetic Laplacian
In a first part of this section, we show that the phase-deformed magnetic
Berezin transform is a bounded operator. The second part will be reserved to
establish the two formulas expressing the phase-deformed magnetic Berezin
transform as a function of the magnetic Laplacian operator.
As given in the introduction, we recall that the phase-deformed magnetic
Berezin transform acting on Hilbert space
$L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z)),$ is given by:
$\mathfrak{B}_{m}[\varphi](z):=(\frac{m!}{(n)_{m}})^{2}\int_{\mathbb{C}^{n}}e^{(\left\langle
z,w\right\rangle-|z-w|^{2})}(L_{m}^{n-1}(|z-w|^{2})^{2}\varphi(w)e^{-|w|^{2}}d\nu(w).$
(3.1)
We have the following proposition.
###### Proposition 3.1.
The phase-deformed magnetic Berezin transform $\mathfrak{B}_{m},$ is a bounded
operator. Precisely, we have the following estimate:
$\|\mathfrak{B}_{m}[\varphi]\|_{L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))}\leq\frac{m!}{(n)_{m}}\|\varphi\|_{L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))}.$
(3.2)
Proof. From (3.1),for $\varphi\in L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z)),$ we
have:
$|\mathfrak{B}_{m}[\varphi](z)|\leq(\frac{m!}{(n)_{m}})^{2}\int_{\mathbb{C}^{n}}e^{Re<z,w>-|z-w|^{2}}(L_{m}^{n-1}(|z-w|^{2})^{2}|\varphi(w)|e^{-|w|^{2}}d\nu_{\nu}(w).$
(3.3)
By using the well known inequality,
$Re(<z,w>)\leq\frac{1}{2}(|z|^{2}+|w|^{2}),$
the inequality (3.3) becomes:
$e^{\frac{-1}{2}|z|^{2}}|\mathfrak{B}_{m}[\varphi](z)|\leq(\frac{m!}{(n)_{m}})^{2}\int_{\mathbb{C}^{n}}e^{-|z-w|^{2}}(L_{m}^{n-1}(|z-w|^{2})^{2}e^{-\frac{1}{2}|w|^{2}}|\varphi(w)|d\nu(w).$
(3.4)
The inequality (3.4) can be rewritten as:
$|e^{\frac{-1}{2}|z|^{2}}\mathfrak{B}_{m}[\varphi](z)|\leq\frac{\pi^{n}m!}{(n)_{m}}B_{m}[e^{-\frac{|.|}{2}}|\varphi|(.)](z).$
(3.5)
where, $B_{m}$ is the Berezin transform defined in (1.6). From the inequality
(3.5), we obtain:
$\|\mathfrak{B}_{m}[\varphi]\|_{L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))}\leq\frac{\pi^{n}m!}{(n)_{m}}\|B_{m}[e^{-\frac{|.|}{2}}|\varphi|(.)]\|_{L^{2}(\mathbb{C},d\nu(z))}$
(3.6)
Then, by using the following inequality see [2], p.4,
$\|B_{m}[\psi]\|_{L^{2}(\mathbb{C},d\nu(z))}\leq\pi^{-n}\|\psi\|_{L^{2}(\mathbb{C},d\nu(z))},\psi\in
L^{2}(\mathbb{C},d\nu(z)),$ (3.7)
for $\psi(.)=e^{-\frac{|.|}{2}}|\varphi|(.)],$ we get:
$\|\mathfrak{B}_{m}[\varphi]\|_{L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))}\leq\frac{m!}{(n)_{m}}\|\varphi\|_{L^{2}(\mathbb{C}^{n},e^{-|z|^{2}}d\nu(z))}.$
(3.8)
This ends the proof.
Now, we shall express the deformed Berezin transform as a function of the
magnetic Laplacian on $\mathbb{C}^{n}.$
###### Theorem 3.1.
Let $m\in\mathbb{Z}_{+}$, then the phase deformed magnetic Berezin transform
$\mathfrak{B}_{m}$ can be expressed in terms of the magnetic Laplacian
$\widetilde{\Delta}$ as
$\mathfrak{B}_{m}=\frac{2^{2m-n}(m!)^{3}}{(n)_{m}}\sum_{j=m+1}^{2m}\frac{2^{-j}}{j!(2m-j)!\Gamma(j-m+1)^{2}}\times$
(3.9)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n,j-m+1,j-m+1;1)\times$
$\frac{\Gamma(j+n+\widetilde{\Delta})}{\Gamma(n+\widetilde{\Delta})}e^{-\log(2)\widetilde{\Delta}}.$
Proof.Let $g=g_{m;n}:\mathbb{R}\rightarrow\mathbb{C},$ be a Borel function
such that
$\mathfrak{B}_{m}=g(\widetilde{\Delta}).$ (3.10)
Recalling the expression of $\mathfrak{B_{m}}$ given in (1.9),
$\mathfrak{B}_{m}[\varphi](z)=(\frac{m!}{(n)_{m}})^{2}\int_{\mathbb{C}^{n}}e^{(\left\langle
z,w\right\rangle-|z-w|^{2})}(L_{m}^{n-1}(|z-w|^{2})^{2}\varphi(w)e^{-|w|^{2}}d\nu(w).$
By using (2.11), we are lead to consider the following equality:
$(\frac{m!}{(n)_{m}})^{2}e^{<z,w>-|z-w|^{2}}(L^{n-1}_{m}(|z-w|^{2})^{2}=\pi^{-n}e^{<z,w>}\sum_{k\in\mathbb{Z}_{+}}L_{k}^{n-1}(|z-w|^{2})g(k).$
(3.11)
The equation (3.11) can be rewritten as:
$e^{-x}(L^{n-1}_{m}(x))^{2}=\sum_{k\in\mathbb{Z}_{+}}h(k)L^{n-1}_{k}(x),$
(3.12)
where we have set $x=|z-w|^{2}$ and
$h(k)=\pi^{-n}(\frac{(n)_{m}}{m!})^{2}g(k).$ (3.13)
By using the orthogonality relation of the Laguerre polynomials
$\\{L^{n-1}_{k}\\}_{k\in\mathbb{Z}_{+}}$ in the Hilbert space
$L^{2}(\mathbb{R},x^{n-1}e^{-x}dx),$ [[14], p.37,56] the Fourier coefficient
$h(k),$ is given by:
$h(k)=\|L_{k}^{n-1}\|_{L^{-2}(\mathbb{R},x^{n-1}e^{-x}dx)}^{-2}\int_{0}^{+\infty}x^{n-1}e^{-2x}(L_{m}^{n-1}(x))^{2}L_{k}^{n-1}(x)dx.$
(3.14)
Making use of the formula [14], p.56 :
$\|L^{\alpha}_{j}\|^{2}=\frac{\Gamma(j+\alpha+1)}{j!},$ (3.15)
for $\alpha=n-1$ and $j=k,$ to obtain:
$h(k)=\frac{k!}{\Gamma(n+k)}\int_{0}^{+\infty}x^{n-1}e^{-2x}(L_{m}^{n-1}(x))^{2}L_{k}^{n-1}(x)dx.$
(3.16)
Now, we make use the following linearization of the product of Laguerre
polynomials ([[17], P 7361]):
$L_{p}^{(\alpha}(x)L_{q}^{(\alpha}(x)=\sum_{j=|p-q|}^{p+q}l_{p,q,j}L^{(\alpha)}_{j}(x),$
(3.17)
where the coefficients are given in terms of$\quad{}_{3}F_{2}$ hypergeometric
function [[15], p.159] as
$l_{p,q,j}=\frac{2^{p+q-j}p!q!}{(p+q-j)!\Gamma(j-p+1)\Gamma(j-q+1)}\times$
(3.18)
$\quad{}_{3}F_{2}(\frac{j-p-q}{2},\frac{j-p-q+1}{2},j+\alpha+1,j-p+1,j-q+1;1)$
for the particular case $\alpha=n-1$ and $p=q=m.$ We obtain
$(L^{n-1}_{m}(x))^{2}=\sum_{j=m-1}^{2m}\frac{(m!)^{2}2^{2m-j}}{(2m-j)!(\Gamma(j-m+1)^{2}}\times$
(3.19)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n,j-m+1,j-m+1;1)L^{n-1}_{m}(x).$
By inserting (3.19) in (3.16), we obtain:
$h(k)=\frac{k!}{\Gamma(n+k)}\sum_{j=m-1}^{2m}\frac{(m!)^{2}2^{2m-j}}{(2m-j)!(\Gamma(j-m+1)^{2}}\times$
(3.20)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n,j-m+1,j-m+1;1)\times$
$\int_{0}^{+\infty}x^{n-1}e^{-2x}L^{n-1}_{j}(x)L^{n-1}_{k}(x)dx.$
Next, making use of the identity [[11],p. 809]:
$\int_{0}^{+\infty}e^{-bx}x^{\alpha}L^{(\alpha)}_{j}(\lambda
x)L^{(\alpha)}_{j}(\mu x)dx=\frac{\Gamma(j+k+\alpha+1}{j!k!}\times$ (3.21)
$\frac{(b-\lambda)^{j}(n-\mu)^{k}}{b^{j+k+\alpha+1}}\quad_{2}F_{1}(-j,-k;-j-k-\alpha;\frac{b(b-\lambda-\mu)}{(b-\lambda)(b-\mu)})$
$Re(\alpha)>-1,Re(b)>0,$ for $\alpha=n-1,$ $\lambda=\mu=1,b=2,$ the integral
in (3.20) takes the form
$\int_{0}^{+\infty}x^{n-1}e^{-2x}L^{n-1}_{j}(x)L^{n-1}_{k}(x)dx=\frac{\Gamma(j+k+n)}{j!k!2^{j+k+n}}.$
(3.22)
and (3.16) becomes
$h(k)=\sum_{j=m-1}^{2m}\frac{(m!)^{2}2^{2(m-j)-k-n}}{j!(2m-j)!(\Gamma(j-m+1))^{2}}\times$
(3.23)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n;j-m+1,j-m+1;1)\frac{\Gamma(j+k+n)}{\Gamma(k+n)}.$
Now, if we extend the function $h$ to the set of real numbers by:
$h(\lambda)=2^{-\lambda}\sum_{j=m-1}^{2m}\frac{(m!)^{2}2^{2(m-j)-n}}{j!(2m-j)!(\Gamma(j-m+1))^{2}}\times$
(3.24)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n;j-m+1,j-m+1;1)\frac{\Gamma(j+\lambda+n)}{\Gamma(\lambda+n)}.$
For $\lambda\geq 0$ and $h(\lambda)=0$ if $\lambda<0.$
This function is well defined and satisfies the equation (3.12).
Then, as function $g$ satisfying (3.13), we can set:
$g(\lambda)=\frac{2^{2m-n}(m!)^{3}}{(n)_{m}}\sum_{j=m-1}^{2m}\frac{2^{-j}}{j!(2m-j)!(\Gamma(j-m+1))^{2}}\times$
(3.25)
$\quad{}_{3}F_{2}(\frac{j-2m}{2},\frac{j-2m+1}{2},j+n;j-m+1,j-m+1;1)\frac{\Gamma(j+\lambda+n)}{\Gamma(\lambda+n)}e^{-\log(2)\lambda};$
For $\lambda\geq 0$ and $g(\lambda)=0$ if $\lambda<0.$
Now, is not hard to see that the coefficient $g(k)$ satisfies the following
estimate:
$|g(k)|\leq\tau_{n,m}P_{n,m}(k)e^{-log(2)k}.$ (3.26)
where, $\tau_{n,m}$ is a positive constant and $P_{n,m}(k)$ is a polynomial of
degree at most equal to $2m+n.$
On the other hand, according to the equation (7.6.11) in [[18],p173], we have
for a fixed $\alpha>-1$ and $\omega$ a positive constant, the following
asymptotic behavior,
$L_{k}^{\alpha}(x)=O(k^{a}),a=max(\frac{1}{2}\alpha-\frac{1}{4},\alpha),k\rightarrow+\infty,$
(3.27)
where $x$ is a fixed number in $[0,\omega].$
Then, by the estimate (3.26) and the asymptotic behavior (3.27) considered for
$\alpha=n-1,$ $x=|z-w|^{2}$ and $\omega$ be a fixed number such that $0\leq
x\leq\omega,$ one can easily show that we have the following inequality:
$|g(k)|L^{n-1}_{k}(|z-w|^{2})\leq\frac{Cn,m}{k^{2}},$ (3.28)
where $Cn,m$ is a positive constant and $k$ sufficiently large.
This last inequality ensures that the right hand side of the equation (3.11)
and the kernel $\Psi_{g}(z,w)$ given by (2.11) are well defined.
To close the proof of the theorem, it remains to justify that the operator
$g(\widetilde{\Delta})$ given by (2.10) is well defined as densely operator.
For this, let us consider a $C^{\infty}-$ function $\varphi$ with compact
support in $\mathbb{C}^{n}.$
Recall that the right hand side of the equation (2.10) is understood in the
distributional sense. So, the action of the operator
$g(\widetilde{\Delta})$ on the function $\varphi$ is given by:
$g(\widetilde{\Delta})[\varphi](z)=\sum_{k\in\mathbb{Z}_{+}}g(k)\int_{\mathbb{C}^{n}}e^{<z,w>}L^{n-1}_{k}(|z-w|^{2})\varphi(w)e^{-|w|^{2}}d\nu(w).$
(3.29)
We have,
$|\int_{\mathbb{C}^{n}}e^{<z,w>}L^{n-1}_{k}(|z-w|^{2})\varphi(w)e^{-|w|^{2}}d\nu(w)|\leq\|\varphi\|_{\infty}\int_{\mathbb{C}^{n}}e^{2Re<z,w>}|L^{n-1}_{k}(|z-w|^{2})|e^{-|w|^{2}}d\nu(w)$
(3.30)
$\leq\|\varphi\|_{\infty}e^{|z|^{2}}\int_{\mathbb{C}^{n}}e^{-|z-w|^{2}}|L^{n-1}_{k}(|z-w|^{2})|d\nu(w)$
(3.31)
where $\|\varphi\|_{\infty}=\sup_{w\in supp(\varphi)}|\varphi(w)|$ and
$supp(\varphi)$ means the compact support of the function $\varphi.$
By using the change of variable $\xi=z-w,$ in the involved integral in the
right hand side of the inequality (3.31), we obtain:
$\int_{\mathbb{C}^{n}}e^{-|z-w|^{2}}|L^{n-1}_{k}(|z-w|^{2})|d\nu(w)=\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}|L^{n-1}_{k}(|\xi|^{2})|d\nu(\xi)$
(3.32)
By applying the cauchy-Schwartz inequality to this last integral, we get:
$\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}|L^{n-1}_{k}(|\xi|^{2})|d\nu(\xi)\leq(\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}(L^{n-1}_{k}(|\xi|^{2}))^{2}d\nu(\xi))^{\frac{1}{2}}(\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}d\nu(\xi))^{\frac{1}{2}}.$
(3.33)
Now, let us compute the two involved integrals in the right hand side of the
inequality (3.33). For this, we use the polar coordinates $z=\rho\omega,$
$\rho>0$ and $\omega\in S^{2n-1}.$ Then, the first integral takes the
following form
$\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}(L^{n-1}_{k}(|\xi|^{2})=\Omega_{2n}\int_{0}^{+\infty}e^{-\rho^{2}}(L^{(n-1)}_{k}(\rho^{2})\rho^{2n-1}d\rho$
(3.34)
where $\Omega_{2n}=\int_{S^{2n-1}}d\sigma(\omega)=\frac{2\pi^{n}}{\Gamma(n)},$
is the area surface of the unit sphere in $\mathbb{C}^{n}.$
Next, by using the change of variable $s=\varrho^{2},$ the last integral in
(3.34) becomes
$\int_{0}^{+\infty}e^{-\rho^{2}}(L^{(n-1)}_{k}(\rho^{2})\rho^{2n-1}d\rho=\frac{1}{2}\int_{0}^{+\infty}(L^{n-1}_{k}(s))^{2}s^{n-1}e^{-s}ds.$
(3.35)
$=\frac{1}{2}\|L_{k}^{n-1}\|_{L^{2}(\mathbb{R},s^{n-1}e^{-s}ds)}^{2}.$ (3.36)
Finally, by using the formula (3.15), for $j=k$ and $\alpha=n-1,$ the integral
given in (3.35) becomes:
$\int_{0}^{+\infty}e^{-\rho^{2}}(L^{(n-1)}_{k}(\rho^{2})\rho^{2n-1}d\rho=\frac{1}{2}\frac{\Gamma(k+n)}{k!}.$
(3.37)
For the second integral involved in the right hand side of the inequality
(3.33), we recognizing the well known Gaussian integral:
$\int_{\mathbb{C}^{n}}e^{-|\xi|^{2}}d\nu(\xi)=\pi^{n}.$ (3.38)
Now, taking into account of the previous expressions from (3.30), we get the
following inequality:
$|\int_{\mathbb{C}^{n}}e^{<z,w>}L^{n-1}_{k}(|z-w|^{2})\varphi(w)e^{-|w|^{2}}d\nu(w)|\leq\kappa(\varphi,n,z)\frac{\Gamma(n+k)}{k!},$
(3.39)
where
$\kappa(\varphi,n,z)=\frac{2\pi^{n}}{\Gamma(n)}\|\varphi\|_{\infty}e^{|z|^{2}}.$
Since the right hand side of (3.39) has a polynomial growth with respect to
$k$ as variable, then by using the inequality (3.26) is not hard to show that
we have the following estimate
$|g(k)\int_{\mathbb{C}^{n}}e^{<z,w>}L^{n-1}_{k}(|z-w|^{2})\varphi(w)e^{-|w|^{2}}d\nu(w)|\leq\vartheta(z,\varphi,n)\frac{1}{k^{2}},$
(3.40)
where $\vartheta(z,\varphi,n)$ is a positive constant and $k$ sufficiently
large. This last inequality ensures that the right hand side of the equation
(2.10) is well defined, that is the operator $g(\widetilde{\Delta})$ is
densely defined on $L^{2}(\mathbb{C},e^{-|z|^{2}}d\nu(z))$ with
$C_{0}^{\infty}(\mathbb{C}^{n})$ as its natural regular domain and satisfies
the equation (3.10) . This ends the proof.
###### Remark 3.1.
The equation (3.12) is a particular case of the following identity [12],
p.263,
$e^{-ax}L^{(\alpha)}_{p}(x)e^{-ax}L^{(\alpha)}_{q}(x)e^{-ax}=\sum_{k=0}^{+\infty}c_{p,q,k}e^{-ax}L_{k}^{(\alpha)}(x),$
for, $p=q=m,$ $a=\frac{1}{2},$ and $\alpha=n-1$. This identity has been
considered by Szego (1933) and Askey, and also by Gasper in connection with
K.O. Frederic and H. Lewy problem.
Another way to write the phase-deformed Berezin transform $\mathfrak{B}_{m},$
as a function of the magnetic Laplacian $\widetilde{\Delta}$ is as follows.
###### Theorem 3.2.
Let $m\in\mathbb{Z}_{+}$, then the phase deformed magnetic Berezin transform
$\mathfrak{B}_{m}$ can be expressed in terms of the magnetic Laplacian
$\widetilde{\Delta}$ as
$\mathfrak{B}_{m}=\frac{2^{-n}\pi^{n}}{\Gamma(n)}(\frac{m!}{(n)_{m}})^{2}\sum_{l=0}^{2m}2^{-l}\sigma^{n,m}_{l}\frac{\Gamma(n+l)}{l!}F(-\widetilde{\Delta},n+l,n;\frac{1}{2}),$
(3.41)
with $F(a,b,c;x)$ is the Gauss hypergeometric function and the coefficients
$\sigma^{n,m}_{j}$ are given by:
$\sigma^{n,m}_{l}=(-1)^{l}\sum_{i=0}^{l}\binom{l}{i}\binom{n+m-1}{m-l+i}\binom{n+m-1}{m-i}.$
(3.42)
Proof. we return back to (3.16) and we make use of the Feldheim formula [16],
which expresses the product of Laguerre polynomial as a sum of monomial terms
$L^{(\alpha)}_{p}(x)L^{(\beta)}_{q}(x)=(-1)^{p+q}\sum_{l=0}^{p+q}B_{l}(p,q,\alpha,\beta)\frac{x^{l}}{l!},$
(3.43)
where the coefficient are given by:
$B_{l}(p,q,\alpha,\beta)=(-1)^{p+q+l}\sum_{i=0}^{l}\binom{l}{i}\binom{\beta-q}{q-l+i}\binom{\alpha+p}{p-i},$
(3.44)
for the particular case $\alpha=\beta=n-1,$ $p=q=m.$ We obtain:
$(L^{(n-1)}_{m}(x))^{2}=\sum_{l=0}^{2m}\sigma^{n,m}_{l}\frac{x^{l}}{l!}.$
(3.45)
where,
$\sigma^{n,m}_{l}=(-1)^{l}\sum_{i=0}^{l}\binom{l}{i}\binom{n+m-1}{m-l+i}\binom{n+m-1}{m-i}.$
(3.46)
Therefore, equation (3.16) takes the form
$h(k)=\frac{k!}{\Gamma(n+k)}\sum_{l=0}^{l=2m}\frac{\sigma^{n,m}_{l}}{l!}\int_{0}^{+\infty}x^{n+l-1}e^{-2x}L^{n-1}_{k}(x)dx.$
(3.47)
Next, making use of the identity ([[11],p.809]):
$\int_{0}^{+\infty}e^{-sx}x^{\beta}L^{(\alpha)}_{j}(x)dx=\frac{\Gamma(\beta+1)\Gamma(\alpha+j+1)}{j!\Gamma(\alpha+1)}s^{-(\beta+1)}F(-j,\beta+1;\alpha+1;\frac{1}{s}),$
(3.48)
$Re(\beta)>-1,Re(s)>0$ for $s=2,\beta=n+l-1,\alpha=n-1$ and $j=k,$ the
integral in (3.48)takes the form:
$\int_{0}^{+\infty}x^{n+l-1}e^{-2x}L^{n-1}_{k}(x)dx=\frac{\Gamma(n+l)\Gamma(n+k)}{k!\Gamma(n)}2^{-(n+l)}F(-k,n+l,n;\frac{1}{2})$
(3.49)
and (3.47) becomes
$h(k)=\frac{2^{-n}}{\Gamma(n)}\sum_{l=0}^{2m}2^{-l}\sigma^{n,m}_{l}\frac{\Gamma(n+l)}{l!}F(-k,n+l,n;\frac{1}{2}).$
(3.50)
Also as before, this function can be extended to the whole real line by
setting:
$h(\lambda)=\frac{2^{-n}}{\Gamma(n)}\sum_{l=0}^{2m}2^{-l}\sigma^{n,m}_{l}\frac{\Gamma(n+l)}{l!}F(-\lambda,n+l,n;\frac{1}{2}).$
(3.51)
if $\lambda>0$ and $h(\lambda)=0,$ for $\lambda<0.$ This function is well
defined and satisfies the equation (3.12).
Now, according to the equation (3.13), we can set:
$g(\lambda)=\frac{2^{-n}\pi^{n}}{\Gamma(n)}(\frac{m!}{(n)_{m}})^{2}\sum_{l=0}^{2m}2^{-l}\sigma^{n,m}_{l}\frac{\Gamma(n+l)}{l!}F(-\lambda,n+l,n;\frac{1}{2}).$
(3.52)
for $\lambda>0$ and $g(\lambda)=0,$ for $\lambda<0.$
According to (3.13) and (3.16) it is natural that the function $g$ defined by
(3.25) and that defined by (3.52) coincides on the set $\mathbb{Z}_{+}.$ Then,
it follows that the rest of the proof is exactly the same as that of theorem
(3.1) from the inequality (3.26). This ends the proof.
###### Remark 3.2.
Return buck to inequality (3.5), and replacing the phase-deformed magnetic
berezin transform $\mathfrak{B}_{m}$ by its expression in terms of the
magnetic laplacian $\widetilde{\Delta}$ and the magnetic Berezin transform
$B_{m}$ by its expressions in terms the Euclidean Laplacian $\Delta_{C^{n}}$
then, we obtain the following inequality of Diamagnetic type:
$|e^{\frac{-1}{2}|z|^{2}}g_{m}(\widetilde{\Delta})[\varphi](z)|\leq\frac{\pi^{n}m!}{(n)_{m}}f_{m}(\Delta_{C^{n}})[e^{-\frac{|.|}{2}}|\varphi|(.)](z).$
(3.53)
where $g_{m}$ is one of the functions defined by (3.25) or (3.52) and $f_{m}$
is the function defined by (1.7).
## References
* [1] N. Akhiezer, I. Glazman, and I. Glazman. Theory of linear operators in Hilbert space. Dover books on advanced mathematics. Dover Publications, 1993.
* [2] N. Askour, A. Intissar, and Z. Mouayn. A formula representing magnetic Berezin transforms as functions of the Laplacian on $\mathbb{C}^{n}$. Integral Transforms Spec. Funct., 22(11):841–849, 2011.
* [3] N. Askour and Z. Mouayn. Spectral decomposition and resolvent kernel for a magnetic Laplacian in $\mathbb{C}^{n}$. J. Math. Phys., 41(10):6937–6943, 2000.
* [4] N. e. Askour, A. Intissar, and Z. Mouayn. Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants. C. R. Acad. Sci. Paris Sér. I Math., 325(7):707–712, 1997.
* [5] N. E. Askour, A. Intissar, and Z. Mouayn. Explicit formulas for reproducing kernels of generalized Bargmann spaces of ${\bf C}^{n}$. J. Math. Phys., 41(5):3057–3067, 2000.
* [6] F. A. Berezin. Quantization. Izv. Akad. Nauk SSSR Ser. Mat., 38:1116–1175, 1974.
* [7] F. A. Berezin. Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR Ser. Mat., 39(2):363–402, 472, 1975.
* [8] M. Birman and M. Solomi͡a͡k. Spectral theory of self-adjoint operators in Hilbert space. Mathematics and its applications: Soviet series. D. Reidel Pub. Co., 1987\.
* [9] R. Estrada and S. A. Fulling. Distributional asymptotic expansions of spectral functions and of the associated Green kernels. Electron. J. Differential Equations, pages No. 07, 37 pp. (electronic), 1999.
* [10] G. B. Folland. Spherical harmonic expansion of the Poisson-Szegő kernel for the ball. Proc. Amer. Math. Soc., 47:401–408, 1975.
* [11] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
* [12] M. Ismail. Classical And Quantum Orthogonal Polynomials In One Variable. Number vol. 13 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 2005.
* [13] W. Magnus, F. Oberhettinger, and R. P. Soni. Formulas and Theorems for the Special Functions of Mathematical Physics. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Volume 52. Springer-Verlag, 1966.
* [14] A. Nikiforov, V. Ouvarov, and V. Kotliar. Fonctions spéciales de la physique mathématique. Traduit du russe. Mir, 1983.
* [15] F. Olver. Asymptotics and special functions. AKP classics. A.K. Peters, 1974.
* [16] B. S. Popov and H. M. Srivastava. Linearization of a product of two polynomials of different orthogonal systems. Facta Univ., Ser. Math. Inf., 18:1–8, 2003.
* [17] J. Sánchez-Ruiz, P. L. Artés, A. Martínez-Finkelshtein, and J. S. Dehesa. General linearization formulae for products of continuous hypergeometric-type polynomials. Journal of Physics A: Mathematical and General, 32(42):7345, 1999\.
* [18] G. Szegö. Orthogonal Polynomials. Colloquium Publications - American Mathematical Society. American Mathematical Society, 1939.
* [19] V. Vladimirov. Equations of mathematical physics. Translated from Russian. Mir, 1984.
* [20] K. Yosida. Functional analysis. Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123. Springer-Verlag New York Inc., New York, 1968.
|
arxiv-papers
| 2012-10-29T17:15:51 |
2024-09-04T02:49:37.285418
|
{
"license": "Public Domain",
"authors": "N. Askour",
"submitter": "Nour Eddine Askour",
"url": "https://arxiv.org/abs/1210.7734"
}
|
1210.7772
|
# Homotopy idempotent functors on classifying spaces
Natàlia Castellana and Ramón Flores
###### Abstract
Fix a prime $p$. Since their definition in the context of Localization Theory,
the homotopy functors $P_{B{\mathbb{Z}}/p}$ and $CW_{B{\mathbb{Z}}/p}$ have
shown to be powerful tools to understand and describe the mod $p$ structure of
a space. In this paper, we study the effect of these functors on a wide class
of spaces which includes classifying spaces of compact Lie groups and their
homotopical analogues. Moreover, we investigate their relationship in this
context with other relevant functors in the analysis of the mod $p$ homotopy,
such as Bousfield-Kan completion and Bousfield homological localization.
††N. Castellana: Departamento de Matemáticas, Universidad Autónoma de
Barcelona, 08193 Bellaterra, Spain; e-mail: natalia@mat.uab.es
R. Flores: Departamento de Estadística, Universidad Carlos III, 28029
Colmenarejo, Spain; e-mail: rflores@est-econ.uc3m.es (corresponding author)††
_Mathematics Subject Classification (2010):_ Primary 55P20; Secondary 55P80
## 1 Introduction
Let $A$ and $X$ be two connected topological spaces. The study of the
homotopical properties of $X$ that are visible through the mapping space ${\rm
map\,}(A,X)$ is called the $A$-homotopy theory of $X$ and was proposed by E.
Dror-Farjoun in [Far96]. In this context, it is particularly important to
describe the behaviour of the nullification $P_{\Sigma^{i}A}$ and the
cellularization ${CW}_{\Sigma^{i}A}$ (see definitions in Section 2), which are
functors that play in $A$-homotopy theory the same role as the connected
covers and Postnikov pieces play in classical ($S^{0}$-) homotopy theory.
Let $p$ be a prime. If $X$ is a space and we are interested in describing the
$p$-primary part of $X$ through its $A$-homotopy theory for some space $A$,
there are some choices of $A$ that become apparent. Probably the easiest one
are the Moore spaces $M({\mathbb{Z}}/p^{n},1)$ and their suspensions; this
task was undertaken in the nineties by Rodriguez-Scherer in the case of
cellularization [RS01] and Bousfield [Bou97], who did not only described
${P}_{M({\mathbb{Z}}/p^{m},n)}X$ for a wide number of spaces, but remarked the
close relationship between these functors and the $v_{n}$-periodic homotopy
theory.
In this paper we deal with the case $A=B{\mathbb{Z}}/p$. After Miller’s
solution of Sullivan conjecture [Mil84], and subsequent work of Lannes, Dwyer-
Zabrodsky and others, a number of new powerful tools were available to
researchers attempting to study the mapping space ${\rm
map\,}(B{\mathbb{Z}}/p,X)$, and overwhelming success was reached, particularly
for nilpotent spaces $X$. In the framework we are interested in, we should
emphasize the work of Neisendorfer [Nei95], where the author proves that the
functor ${P}_{B{\mathbb{Z}}/p}$ can often recover the $p$-primary homotopy of
$X$ from that of its $n$-connected cover, or [CCS07-2], about the
$B{\mathbb{Z}}/p$-homotopy of $H$-spaces.
The first motivation for our work came from two different sources: the study
undertaken by W. Dwyer in [Dwy96] concerning $B{\mathbb{Z}}/p$-nullification
of classifying spaces of compact Lie groups whose group of components is a
$p$-group, and its relationship with the homology decompositions of $BG$; and
Question 11 in Farjoun’s book ([Far96, page 175]), where he asked about the
cellularity of the $p$-completion of $BG$. This seemed a natural extension of
the problems considered by the second author concerning
$B{\mathbb{Z}}/p$-homotopy of finite groups (see [Flo07], [FS07], and [FF]),
so it was natural to investigate this structure with similar methods.
We show that under certain hypothesis, we are able to characterize the effect
of the nullification functor $P_{B{\mathbb{Z}}/p}$ by means of a fibration.
Theorem 4.1. Let $X$ be a connected space with finite fundamental group and
such that $(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle))^{\wedge}_{p}\simeq*$. Then
there is a fibration
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$
where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$, and $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ denotes
Bousfield homological localization of $X_{p}$ with respect to
$H^{*}(-;{\mathbb{Z}}[\frac{1}{p}])$.
In particular, one can compute the homotopy groups of $P_{B{\mathbb{Z}}/p}(X)$
in terms of those of $X$ if $X$ is good enough. This result is quite general,
and in fact describes in a single statement a phenomenon which was previously
known for finite groups, $p$-compact groups and some compact Lie groups, but
not for $p$-local compact or Kač-Moody groups (Corollary 4.13); so, it can be
read then as a common property of a big family of homotopy meaningful spaces.
Moreover, finite loop spaces also satisfy this property (Corollary 4.12).
Another source of examples is the theory of infinite loop spaces. McGibbon
[McG] shows that infinite loop spaces satisfy the hypothesis of Theorem 4.1
(see Corollary 4.14).
We also obtain some interesting consequences of these results, including a
detailed analysis of the relationship between the
$B{\mathbb{Z}}/p$-nullification and ${\mathbb{Z}}[1/p]$-localization of these
spaces -which is very much in the spirit of [Dwy96, Section 6]\- and the
commutation of nullity functors on them, a situation that was discussed in
[RS00] in a general framework.
The second part of the paper deals with the effect of the cellularization
functor $CW_{B{\mathbb{Z}}/p}$ on classifying spaces of compact Lie groups. We
show a Serre-type dichotomy theorem.
Theorem 6.9. Let $G$ be a compact connected Lie group. If there exists a non
$p$-cohomologically central element of order $p$, then the
$B{\mathbb{Z}}/p$-cellullarization of $BG$ has infinitely many nonzero
homotopy groups. Otherwise, it has the homotopy type of a $K(V,1)$, where $V$
is a finite elementary abelian $p$-group.
This statement is in fact a consequence of a more general statement which
extends Proposition 2.3 in [FS07].
Theorem 6.1. Let $X$ be a connected nilpotent $\Sigma^{n}B{\mathbb{Z}}/p$-null
space for some $n\geq 0$. Then the $B{\mathbb{Z}}/p$-cellullarization of $X$
has the homotopy type of a Postnikov piece with homotopy groups are
concentrated in degrees $1$ to $n$ , or else it has infinitely many nonzero
homotopy groups. Moreover, if $X$ is $1$-connected of finite type, then the
fundamental group $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finite elementary
abelian $p$-group.
This result opens the way to describe with precision (up to $p$-completion)
the $B{\mathbb{Z}}/p$-cellularization of $BG$ for an ample class of Lie groups
which includes p-toral groups and their discrete approximations, the 3-sphere,
extensions of elementary abelian groups by groups of order prime to $p$ -which
generalize [FS07, Corollary 3.3]-, or $BSO(3)$. In particular, we find
examples of both cases of the dichotomy statement. It is interesting to remark
here that we use intensively the fact that $CW_{A}$ preserves nilpotent spaces
(Lemma 2.5), a fact that was conjectured in [Far96], but which to our
knowledge has not been so far exploited in the literature.
Notation: Let $R$ be a commutative ring, $R_{\infty}(X)$ denotes Bousfield-Kan
$p$-completion of a space $X$ ([BK72]). When $R={\mathbb{Z}}/p$ for a prime
$p$, $R_{\infty}(X)$ will be used instead of $X^{\wedge}_{p}$. Moreover,
$L_{R}(X)$ denotes the $HR$-localization of Bousfield ([Bou75]). All spaces
are assumed to have the homotopy type of a $CW$-complex.
## 2 The cellularization and nullification functors
Let $A$ be a connected space. In this section we will define the functors
$CW_{A}$ and $P_{A}$, which are the main tools we use to describe the
$p$-primary structure of the spaces of interest in our work. Only some
particular features of these functors, that will be crucial in our further
developments, will be described while their relationship with Bousfield-Kan
completion will be studied in the next section. A thorough account to these
constructions can be found in [Far96].
###### Definition 2.1.
Let $A$ and $X$ be spaces. Then $X$ is called $A$-_null_ if the inclusion of
constant maps $X\hookrightarrow{\rm map\,}(A,X)$ is a weak equivalence.
This is equivalent to the condition that ${\rm map\,}_{*}(A,X)$ is weakly
contractible when $X$ is connected. Dror-Farjoun defines a coaugmented and
idempotent functor ${P}_{A}:\mathbf{Spaces}\rightarrow\mathbf{Spaces}$ where
${P}_{A}X$ is $A$-null for every $X$, and such that the coaugmentation
$X\rightarrow{P}_{A}X$ induces a weak equivalence ${\rm
map\,}({P}_{A}X,Y)\rightarrow{\rm map\,}(X,Y)$ for every $A$-null space $Y$.
The corresponding definitions in the pointed context are completely analogous.
Note that in the language of homotopy localization, $P_{A}$ is the
localization with regard to the constant map $A\rightarrow{*}$, and the
notation comes from Postnikov sections, which are in fact
$S^{n}$-nullifications. Moreover, a space $X$ such that $P_{A}X\simeq*$ is
called $A$-_acyclic_.
Now we consider the cellular construction, which is somewhat dual of the
previous construction, although not completely (see Theorem 2.3 below).
###### Definition 2.2.
Given pointed spaces $A$ and $X$, $X$ is said $A$-_cellular_ if it can be
built from $A$ by means of pointed homotopy colimits, possibly iterated.
Moreover, a map $X\rightarrow Y$ is said to be an $A$-_equivalence_ if it
induces a weak equivalence ${\rm map\,}_{*}(A,X)\rightarrow{\rm
map\,}_{*}(A,Y)$.
The $A$-cellularization (or $A$-cellular approximation) is a canonical way of
turning every space into an $A$-cellular space from the point of view of
$A$-equivalences, which generalizes the classic process of cellular
approximation. There exists an augmented endofunctor ${CW}_{A}$ of the
category of pointed spaces, such that for every space $X$ the augmentation
${CW}_{A}X\rightarrow X$ is an $A$-equivalence, and in initial among all maps
$Y\rightarrow X$ which induce $A$-equivalence. Unlike ${P}_{A}$, this functor
only makes sense in the pointed context ([Cha96, 7.4]), and can be
characterized in several ways [Far96, 2.E.8].
The remaining of the section is devoted to describe some properties of these
functors that we will frequently use later. We begin with a theorem of W.
Chachólski that can be considered the most powerful tool to compute
cellularization of spaces in an explicit way. The proof can be found in
[Cha96, 20.3].
###### Theorem 2.3.
Let $A$ and $X$ be pointed spaces, and let $C$ be the homotopy cofibre of the
evaluation $\bigvee_{[A,X]_{*}}A\rightarrow X$, where the wedge is taken over
all the homotopy classes of maps $A\rightarrow X$. Then $CW_{A}X$ has the
homotopy type of the fibre of the map $X\rightarrow P_{\Sigma A}C$.
Next we will describe two preservation properties, that will be used
extensively as we will frequently focus our interest in simply connected
spaces and, more generally, nilpotent spaces.
###### Lemma 2.4.
[Bou94, 2.9] If $X$ is $1$-connected then $P_{A}(X)$ is also $1$-connected.
In particular, note that, according to a famous result of Neisendorfer [Nei95,
Thm 0.1], there is no analogous result for higher degrees of connectivity.
The second preservation property concerns to cellularization and it answers
question $7$ stated by Dror-Farjoun in his book [Far96, p.175]. It is
remarkable that the analogous problem in the category of groups was solved in
[FGS07].
###### Lemma 2.5.
If $X$ is a nilpotent space then $CW_{A}(X)$ is also nilpotent.
###### Proof.
Apply [BK72, V.5.2] to the fibration $CW_{A}(X)\rightarrow X\rightarrow
P_{\Sigma A}C$ in Theorem 2.3. ∎
From the definitions, one can check that if $X$ is $A$-null then
$CW_{A}(X)\simeq*$ since $*\hookrightarrow X$ is an $A$-equivalence. In
general, the $A$-cellularization functor also preserves $\Sigma^{n}A$-nullity
for $n\geq 1$.
###### Lemma 2.6.
Let $X$ be a space which is $\Sigma^{n}A$-null for some $n\geq 1$ then
$CW_{A}(X)$ is also $\Sigma^{n}A$-null.
###### Proof.
Again from Theorem 2.3 we have a fibre sequence $CW_{A}(X)\rightarrow
X\rightarrow P_{\Sigma A}(X)$. Since the base space is $\Sigma A$-null, it is
also $\Sigma^{n}A$-null for any $n\geq 1$. The result follows since the
nullification functor $P_{\Sigma A}$ preserves then the fibration [Far96,
3.D.3]. ∎
If we specialize now to $A=B{\mathbb{Z}}/p$, which is the case of interest in
this paper, and we turn our attention to Eilenberg-MacLane spaces, it is
interesting to observe that given an arbitrary group $G$, the
$B{\mathbb{Z}}/p$-nullity properties of $K(G,n)$ for small values of $n$ imply
the $B{\mathbb{Z}}/p$-nullity for $every$ value of $n$, as well as some group-
theoretic features of $G$.
###### Lemma 2.7.
Let $G$ be an abelian discrete group. $K(G,2)$ is $B{\mathbb{Z}}/p$-null if
and only if $p$ is invertible in $G$ and $K(G,n)$ is $B{\mathbb{Z}}/p$-null
for all $n\geq 1$.
###### Proof.
We only need to show that if $K(G,2)$ is $B{\mathbb{Z}}/p$-null then $p$ is
invertible in $G$ and $K(G,n)$ is $B{\mathbb{Z}}/p$-null for all $n\geq 1$.
Since $K(G,1)\simeq\Omega K(G,2)$ is $B{\mathbb{Z}}/p$-null,
$Hom({\mathbb{Z}}/p,G)=[B{\mathbb{Z}}/p,BG]_{*}=0$. Therefore $G$ has no
elements of order $p$. Then, multiplication by $p$ gives rise to a short exact
sequence $0\rightarrow G\stackrel{{\scriptstyle p}}{{\rightarrow}}G\rightarrow
G/pG\rightarrow 0$. Now consider the induced fibration $K(G,1)\rightarrow
K(G/pG,1)\rightarrow K(G,2)$. Since both $K(G,1)$ and $K(G,2)$ are
$B{\mathbb{Z}}/p$-null, by [Far96, 3.D.3], we see that $B(G/pG)$ is also
$B{\mathbb{Z}}/p$-null. Therefore $G/pG$ has no elements of order $p$, so it
must be trivial. That is $G\stackrel{{\scriptstyle p}}{{\rightarrow}}G$ is an
isomorphism and $p$ is invertible in $G$.
A standard transfer argument (see e.g. [Bro82, Prop III.10.1]) shows that
$\tilde{H}^{*}(B{\mathbb{Z}}/p;G)$ is trivial. In particular, ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,K(G,n))$ is weakly contractible for all $n\geq 1$.
∎
We finish this preliminary section by describing a context in which we can
obtain information about the homology and homotopy groups of the
cellularization.
###### Lemma 2.8.
If $R$ is a ring of coefficients and $\tilde{H}^{*}(A;R)=0$, then
$\tilde{H}^{*}(CW_{A}(X);R)=0$. If $X$ is nilpotent and $R\subset{\mathbb{Q}}$
then $\pi_{i}(CW_{A}(X))\otimes R=0$ for $i>0$.
###### Proof.
Under the hypothesis of the theorem, $K(R,n)$ is $A$-local for $n>0$, then the
space ${\rm map\,}_{*}(CW_{A}(X),K(R,n))$ is weakly contractible. By Lemma
2.5, we can apply [BK72, V.3.1]. ∎
## 3 $B{\mathbb{Z}}/p$-homotopy and $p$-completion
We devote this section to the description of the behaviour of the functors
$CW_{A}$ and $P_{A}$ with respect to the $p$-completion functor of Bousfield
and Kan. In particular, if $\eta\colon X\rightarrow X^{\wedge}_{p}$ is the
$p$-completion, we want to characterize when the maps $CW_{A}(\eta)$ and
$P_{A}(\eta)$ are mod $p$ equivalences. This will be fundamental in our
approach to the $B{\mathbb{Z}}/p$-nullification and
$B{\mathbb{Z}}/p$-cellularization of classifying spaces, which will be
undertaken in the last two sections and is the main goal of our note.
A first approximation to these kind of questions appears in the work of Miller
in the solution of the Sullivan Conjecture, which implies immediately a
statement about $B{\mathbb{Z}}/p$-nullity.
###### Theorem 3.1.
[Mil84, Thm 1.5] Let $W$ be a connected space with
$\tilde{H}^{*}(W;{\mathbb{Z}}[\frac{1}{p}])=0$ and let $X$ be a nilpotent
space. Then $\eta\colon X\rightarrow X^{\wedge}_{p}$ is a $W$-equivalence.
###### Corollary 3.2.
If $X$ is a nilpotent space, the $p$-completion $\eta\colon X\rightarrow
X^{\wedge}_{p}$ is a $\emph{B}{\mathbb{Z}}/p$-equivalence.
Observe that if $X$ is $1$-connected, we can $p$-complete our target space, if
necessary, before computing $CW_{B{\mathbb{Z}}/p}X$. This statement, and the
fact that the $B{\mathbb{Z}}/p$-cellularization is constructed using copies of
$B{\mathbb{Z}}/p$ as pieces, may lead to think that $CW_{B{\mathbb{Z}}/p}X$ is
always a $p$-complete space. Next lemma shows that this is true in certain
cases but, as we will see in Example 3.4, not always.
###### Lemma 3.3.
If $X$ is a nilpotent space, then $CW_{B{\mathbb{Z}}/p}(X)$ is $p$-complete if
and only if
$\tilde{H}_{*}(CW_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p};{\mathbb{Q}})=0$.
###### Proof.
Since $B{\mathbb{Z}}/p$ is both $\mathbb{Q}$-acyclic and
$\mathbb{F}_{q}$-acyclic for $q\neq p$, $CW_{B{\mathbb{Z}}/p}(X)$ is so
([Far96, D.2.5] or Lemma 2.8), and then the rationalization and
$q$-completions of $CW_{B{\mathbb{Z}}/p}(X)$ are trivial. By Lemma 2.5,
$CW_{B{\mathbb{Z}}/p}(X)$ is also a nilpotent space, so it admits a Sullivan
arithmetic square decomposition. The result follows. ∎
###### Example 3.4.
Consider the space $X=K({\mathbb{Z}}/p^{\infty},2)$. $X$ is
$B{\mathbb{Z}}/p$-cellular by [CCS07-2, Lemma 3.3], but it is not $p$-complete
since $X^{\wedge}_{p}\simeq K({\mathbb{Z}}^{\wedge}_{p},3)$ and
$\tilde{H}^{*}(X^{\wedge}_{p};{\mathbb{Q}})\neq 0$. In fact, the
$p$-completion $\eta\colon K({\mathbb{Z}}/p^{\infty},2)\rightarrow
K({\mathbb{Z}}^{\wedge}_{p},3)$ induces a $B{\mathbb{Z}}/p$-cellular
equivalence, then $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq X$. On the other
hand, taking for example $p=2$ and $X=B\Sigma_{3}$, the classifying space of
the symmetric group in three letters, it is not nilpotent, and the
cellularization is not complete. See [FS07, Example 2.6] for details.
We proceed now to a systematic study of the induced map
$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(X)\rightarrow
CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$. We want to show under which conditions
it becomes a mod $p$ equivalence. The first step is a reduction concerning the
fundamental group, for which we need the following definition.
###### Definition 3.5.
We say that an element $x\in\pi_{1}(X)$ lifts to $X$ if there exists a
homotopy lift
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\emph{B}(\langle
x\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\langle
x\rangle}}$$\textstyle{\emph{B}(\pi_{1}(X)).}$
###### Proposition 3.6.
Let X be a connected space. There is a fibration
$CW_{B{\mathbb{Z}}/p}(X)\rightarrow X\rightarrow Z$
with $\pi_{1}(Z)\cong\pi_{1}(X)/S$, where $S$ is the normal subgroup generated
by the elements of order $p$ which lift to $X$.
###### Proof.
The fibration in the proposition is the one constructed by Chachólski (see
Theorem 2.3) where $Z=P_{\Sigma B{\mathbb{Z}}/p}(C_{X})$. The subgroup $S$ is
constructed in [CCS07-2, Prop. 2.1] in a way that $E\rightarrow X$ is a
$B{\mathbb{Z}}/p$-cellular equivalence, where $E$ is the homotopy pullback
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Bi}$$\textstyle{B(\pi_{1}(X)).}$
By construction $\pi_{1}(E)\cong S$ is generated by elements of order $p$
which lift to $E$. Then the Chachólski’s cofibre $C_{E}$ (see Theorem 2.3) is
$1$-connected and $P_{\Sigma B{\mathbb{Z}}/p}(C_{E})$ is too by Lemma 2.4.
Since $E\rightarrow X$ is a $B{\mathbb{Z}}/p$-equivalence, from the following
diagram of fibrations
$\textstyle{CW_{B{\mathbb{Z}}/p}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{CW_{B{\mathbb{Z}}/p}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C).}$
where $C$ is Chachólski’s cofibre for $X$, we see that the fundamental group
of $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is $\pi_{1}(X)/S$. ∎
###### Corollary 3.7.
Let $X$ be a connected space such that $\pi_{1}(X)$ is generated by elements
of order $p$ which lift to $X$. There is a bijection
$[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]\cong[B{\mathbb{Z}}/p,X]$ between
unpointed homotopy classes of maps.
###### Proof.
Since $CW_{B{\mathbb{Z}}/p}(X)\rightarrow X$ is a $B{\mathbb{Z}}/p$-homotopy
equivalence, there is a bijection
$[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]_{*}\cong[B{\mathbb{Z}}/p,X]_{*}$
between pointed homotopy classes of maps. The following diagram
$\textstyle{[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,X]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,X]}$
shows that the quotient map is also a bijection since the induced morphism on
fundamental groups $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))\rightarrow\pi_{1}(X)$ is
an epimorphism by Proposition 3.6. ∎
We can get information about the fundamental group of the cellularization
since being $B{\mathbb{Z}}/p$-cellular imposes some restrictions on the
fundamental group of the space.
###### Lemma 3.8.
If $X$ is a $B{\mathbb{Z}}/p$-cellular space, its fundamental group is
generated by elements of order $p$ which lift to $X$. Moreover, if $X$ is a
finite type $1$-connected space,then $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a
finitely generated abelian generated by elements of order $p$ which lift to
$CW_{B{\mathbb{Z}}/p}(X)$.
###### Proof.
Let $S$ be the normal subgroup of $\pi_{1}(X)$ generated by elements of order
$p$ which lift to $X$. Consider the pullback diagram
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Bi}$$\textstyle{B\pi_{1}(X).}$
By [CCS07-2, Prop 2.1], the map $E\rightarrow X$ is a
$B{\mathbb{Z}}/p$-cellular equivalence. Since $X$ is
$B{\mathbb{Z}}/p$-cellular, there exists a map $f\colon E\rightarrow
CW_{B{\mathbb{Z}}/p}(E)$ such that $i\circ f\simeq id$ where $i\colon
CW_{B{\mathbb{Z}}/p}(E)\rightarrow E$. In fact, this implies that $p\colon
E\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C_{E})$ is nullhomotopic, $p\simeq
p\circ i\circ f\simeq*\circ f\simeq*$, therefore
$CW_{B{\mathbb{Z}}/p}(E)\simeq E\times P_{B{\mathbb{Z}}/p}(\Omega C_{E})$. But
this implies that $E$ is $B{\mathbb{Z}}/p$-cellular since
$CW_{B{\mathbb{Z}}/p}(E)$ is $B{\mathbb{Z}}/p$-acyclic, and then $E\simeq X$.
In particular, $\pi_{1}(X)=S$.
To prove the second statement, it remains to prove that
$\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finitely generated abelian group.
Since $X$ is $1$-connected, then the Chachólski’s cofibre $C_{X}$ is
$1$-connected and $P_{\Sigma B{\mathbb{Z}}/p}(C_{X})$ is too by Lemma 2.4.
Then, we see that $\pi_{2}(P_{\Sigma B{\mathbb{Z}}/p}(C_{X}))\cong
H_{2}(P_{\Sigma B{\mathbb{Z}}/p}(C_{X});{\mathbb{Z}})$ is a quotient of
$H_{2}(C_{X};{\mathbb{Z}})$, which in turn is a quotient of the finitely
generated group $H_{2}(X;{\mathbb{Z}})$. ∎
Next we need a technical lemma which describes the somewhat intrincate
relationship between completion and nullification and that is a key result to
understand under which conditions $P_{A}(\eta)\colon P_{A}(X)\rightarrow
P_{A}(X^{\wedge}_{p})$ is a mod $p$ equivalence (Corollary 3.11).
###### Lemma 3.9.
Let $A$ be a connected space, and let $X$ such that $P_{A}(X^{\wedge}_{p})$
and $P_{A}(X)$ are $p$-good spaces. Assume that $P_{A}(X)^{\wedge}_{p}$ and
$P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ are $A$-null spaces. Then the
$p$-completion map $\eta_{X}\colon X\rightarrow X^{\wedge}_{p}$ induces a mod
$p$ equivalence $P_{A}(\eta)\colon P_{A}(X)\rightarrow P_{A}(X^{\wedge}_{p})$.
###### Proof.
Let $\epsilon\colon P_{A}(X^{\wedge}_{p})\rightarrow(P_{A}(X))^{\wedge}_{p}$
be the unique map up to homotopy such that the right square of the following
diagram commutes:
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{X}}$$\scriptstyle{\iota_{X}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\iota_{X^{\wedge}_{p}}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{X})^{\wedge}_{p}}$$\textstyle{P_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P_{A}(X)^{\wedge}_{p}.}$
Note that $\epsilon$ exists because $P_{A}(X)^{\wedge}_{p}$ is $A$-null by
hypothesis. The left square commutes by naturality, so
$(\iota_{X})^{\wedge}_{p}\circ\eta_{X}\simeq\epsilon\circ
P_{A}(\eta_{X})\circ\iota_{X}$. But also,
$(\iota_{X})^{\wedge}_{p}\circ\eta_{X}\simeq\eta_{P_{A}(X)}\circ\iota_{X}$ by
naturality of the completion. Because of the universal property of the
nullification functor, $\epsilon\circ P_{A}(\eta_{X})\simeq\eta_{P_{A}(X)}$.
Since $P_{A}(X)$ is $p$-good, $\eta_{P_{A}(X)}^{*}$ is an isomorphism in mod
$p$ cohomology. In particular, $\epsilon^{*}$ is a monomorphism and
$P_{A}({\eta}_{X})^{*}$ is an epimorphism.
Now consider the following commutative diagram:
$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\iota_{X^{\wedge}_{p}}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{X})^{\wedge}_{p}}$$\scriptstyle{(\iota_{X})^{\wedge}_{p}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P_{A}(X)^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})^{\wedge}_{p}}$$\textstyle{P_{A}(X^{\wedge}_{p})^{\wedge}_{p}.}$
That is
$(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}\circ(\eta_{X})^{\wedge}_{p}\simeq
P_{A}(\eta)^{\wedge}_{p}\circ\epsilon\circ\iota_{X^{\wedge}_{p}}$. But we also
have
$(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}\circ(\eta_{X})^{\wedge}_{p}\simeq(\eta_{P_{A}(X^{\wedge}_{p})})\circ\iota_{X^{\wedge}_{p}}$.
By hypothesis $P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ is $A$-null, then the
universal property of the nullification functor implies that
$P_{A}({\eta}_{X})^{\wedge}_{p}\circ\epsilon\simeq\eta_{P_{A}(X^{\wedge}_{p})}$.
Since $P_{A}(X^{\wedge}_{p})$ is $p$-good,
$(\eta_{P_{A}(X^{\wedge}_{p})})^{*}$ is an isomorphism and hence
$(P_{A}(\eta_{X})^{\wedge}_{p})^{*}$ is a monomorphism. Therefore
$P_{A}(\eta_{X})^{*}$ is so, and we are done. ∎
###### Remark 3.10.
If $X$ has finite fundamental group, then both $P_{A}(X^{\wedge}_{p})$ and
$P_{A}(X)$ are $p$-good spaces since they also have finite fundamental groups.
###### Corollary 3.11.
If $X$ is a $1$-connected space and $A$ is such that
$\tilde{H}_{*}(A;{\mathbb{Z}}[\frac{1}{p}])=0$ then $P_{A}(\eta)\colon
P_{A}(X)\rightarrow P_{A}(X^{\wedge}_{p})$ is a mod $p$ equivalence.
###### Proof.
If $X$ is $1$-connected then $X^{\wedge}_{p}$ is also $1$-connected and both
spaces are $p$-good. Moreover the $B{\mathbb{Z}}/p$-nullification of a
$1$-connected space is also $1$-connected. Miller’s theorem [Mil84, Thm 1.5]
implies that the spaces $P_{A}(X)^{\wedge}_{p}$ and
$P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ are $A$-null. The hypothesis of Lemma 3.9
are then satisfied. ∎
We can also describe a general situation in which the nullification of a mod
$p$ equivalence is so.
###### Corollary 3.12.
Let $A$ be a space such that $\tilde{H}_{*}(A;{\mathbb{Z}}[\frac{1}{p}])=0$.
If $f\colon X\rightarrow Y$ is a mod $p$ equivalence between $1$-connected
spaces then $P_{A}(f)\colon P_{A}(X)\rightarrow P_{A}(Y)$ is a mod $p$
equivalence.
###### Proof.
If $f$ is a mod $p$ equivalence, then $f^{\wedge}_{p}$ is an equivalence. Then
the following diagram commutes
$\textstyle{P_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})}$$\scriptstyle{P_{A}(f)}$$\textstyle{P_{A}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{Y})}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(f^{\wedge}_{p})}$$\textstyle{P_{A}(Y^{\wedge}_{p})}$
By Corollary 3.11, the two vertical arrows are mod $p$ equivalences and the
bottom horizontal map is an equivalence. Then $P_{A}(f)$ is a mod $p$
equivalence. ∎
###### Remark 3.13.
Note that Corollaries 3.11 and 3.12 hold when $A=B{\mathbb{Z}}/p$. In fact, in
Corollary 3.12, one can relax the assumptions on $1$-connectivity by checking
that both spaces $X$ and $Y$ satisfy the assumptions of Lemma 3.9.
Now we follow the parallelism giving a condition for the analogous equivalence
between cellularizations to hold. According to Proposition 3.6, the hypothesis
of lifting elements in the fundamental group is not a real restriction.
###### Proposition 3.14.
Let $X$ be a space whose fundamental group $\pi_{1}(X)$ is finite and
generated by elements of order $p$ which lift to $X$. Assume that there is a
bijection $[B{\mathbb{Z}}/p,X]=[B{\mathbb{Z}}/p,X^{\wedge}_{p}]$, then the map
induced by the $p$-completion
$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(X)\rightarrow
CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$
is a mod $p$ equivalence.
###### Proof.
Since $\pi_{1}(X)$ is finite, $X$ is $p$-good [BK72, VII.5.1]. There is an
epimorphism $\pi_{1}(X)\rightarrow\pi_{1}(X^{\wedge}_{p})$ and, by assumption,
$[B{\mathbb{Z}}/p,X]\cong[B{\mathbb{Z}}/p,X^{\wedge}_{p}]$. In order to
compute the cellularization, we first analyze Chachólski’s cofibres
$\textstyle{\vee
B{\mathbb{Z}}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{id}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\vee
B{\mathbb{Z}}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D.}$
Since $\pi_{1}(X)$ is finite and generated by elements of order $p$ which lift
to $X$, the maps $h_{1}$ and $h_{2}$ induce epimorphisms on the fundamental
group and then $C$ and $D$ are 1-connected spaces. Moreover $g$ is a mod $p$
equivalence. Now, the cellularization fits in the following diagram of
fibrations:
$\textstyle{CW_{B{\mathbb{Z}}/p}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{CW_{B{\mathbb{Z}}/p}(\eta)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{\Sigma
B{\mathbb{Z}}/p}(g)}$$\textstyle{CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(D),}$
where $P_{\Sigma B{\mathbb{Z}}/p}(C)$ and $P_{\Sigma B{\mathbb{Z}}/p}(D)$ are
$1$-connected. All the spaces in the previous diagram are $p$-good. Therefore,
to show that $CW_{B{\mathbb{Z}}/p}(\eta)$ is a mod $p$ equivalence, it is
enough to prove that $P_{\Sigma B{\mathbb{Z}}/p}(g)$ is so. This follows from
the previous Corollary 3.12 since $g$ is a mod $p$-equivalence. ∎
###### Remark 3.15.
We note in Example 3.4 that $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$ does not
need to be $p$-complete and a condition for this to be true was stated in
Lemma 3.3. If $X$ satisfies the hypothesis of Proposition 3.14, we see from
the proof that $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$ is $p$-complete if
$P_{\Sigma B{\mathbb{Z}}/p}(D)$ is so. This last space is $1$-connected and,
using an arithmetic Sullivan square argument, we see that this is the case if
$X^{\wedge}_{p}\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(D)^{\wedge}_{p}$ is a
rational equivalence. Examples of this situation are provided by classifying
spaces of finite groups, since $(BG^{\wedge}_{p})_{\mathbb{Q}}\simeq*$ and
$P_{\Sigma B{\mathbb{Z}}/p}(D)^{\wedge}_{p}\simeq P_{\Sigma
B{\mathbb{Z}}/p}(C)^{\wedge}_{p}$ is homotopic to the $p$-completion of the
classifying space of a finite group by [FS07, Proposition 5.5] and [FF,
Theorem 4.3].
###### Remark 3.16.
The hypothesis of Proposition 3.14 are satisfied if $\pi_{1}(X)$ is a finite
$p$-group generated by elements of order $p$ which lift to $X$ (see [DZ87,
Proof of 3.1]).
###### Remark 3.17.
Let $P$ be a $p$-toral group. Then $P$ is an extension of a finite $p$-group
$\pi$ by a torus $(S^{1})^{n}$. Assume that $\pi$ is generated by elements of
order $p$ which lift to $BP$. The arguments of [DZ87, proof of 3.1] applied to
the fibration $B(S^{1})^{n}\rightarrow BP\rightarrow B\pi$ show that
$[B{\mathbb{Z}}/p,BP]=[B{\mathbb{Z}}/p,BP^{\wedge}_{p}]$. Then
$BP^{\wedge}_{p}$ is a $p$-compact toral group, and
$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(BP)\rightarrow
CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is a mod $p$ equivalence.
By [DW94, Proposition 6.9], there exists a discrete $p$-toral group
$P_{\infty}$, that is, an extension of a finite $p$-group $\pi$ by a finite
sum of Prüfer groups $({\mathbb{Z}}/p^{\infty})^{n}$, such that
$BP_{\infty}\rightarrow BP^{\wedge}_{p}$ is a mod $p$ equivalence. Moreover,
$[B{\mathbb{Z}}/p,BP_{\infty}]\cong[B{\mathbb{Z}}/p,(BP_{\infty})^{\wedge}_{p}]$
by [DW94, Remark 6.12], so we should study, up to $p$-completion, the
$B{\mathbb{Z}}/p$-cellullarization of discrete $p$-toral groups. See Example
6.16.
## 4 $B{\mathbb{Z}}/p$-nullification of classifying spaces
In this section, we are concerned with $B{\mathbb{Z}}/p$-nullification. The
original motivating example for our study were classifying spaces of compact
Lie groups, for which Dwyer computed in [Dwy96] the value of
$P_{B{\mathbb{Z}}/p}BG$ for the case in which $\pi_{0}(G)$ is a (finite)
$p$-group. For this sake, he used an induction principle based on the
centralizer decomposition of $BG$, a method that also solve the problem when
we take a $p$-compact group $X$ instead of $G$. However, the hypothesis over
the fundamental group is essential and cannot be removed from his proof, so we
need to follow a completely different path to solve the general case. In fact,
our new strategy was useful to describe $P_{B{\mathbb{Z}}/p}X$ for a bigger
family of spaces, which in particular need not to be classifying spaces.
Recall that, if $S$ is a set of primes, the $S$-radical subgroup $T_{S}(G)$ of
a finite group $G$ is the smallest normal subgroup of $G$ which contains all
the $S$-torsion. This is the last ingredient we need to state the main result
of this section.
###### Theorem 4.1.
Let $X$ be a connected space with finite fundamental group and such that
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\simeq*$. Then there is a fibration
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$
where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$, and $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ denotes the
homological localization of $X_{p}$ in the ring ${\mathbb{Z}}[\frac{1}{p}]$.
Theorem 4.1 will be a consequence of the following result.
###### Theorem 4.2.
Let $X$ be a connected space with finite fundamental group generated by
$p$-torsion elements which lift to $X$ and such that
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\simeq*$. Then there is an equivalence
$P_{B{\mathbb{Z}}/p}(X)\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X),$ where
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X)$.
Now in order to prove Theorem 4.2 we follow the strategy of the second author
in [Flo07] when dealing with classifying spaces of finite groups, although now
there is rational information that is absent in the finite case. Before,
however, we will be deal with some issues concerning to the fundamental group
of $X$ which will be crucial in the proof.
###### Lemma 4.3.
Let $G$ be a finite group and $S$ a set of primes that divide the order of
$G$. If $G=\emph{T}_{S}G$, then $G$ is $S^{-1}$-perfect. In particular, if $X$
is a space with finite fundamental group such that
$\pi_{1}X=T_{S}(\pi_{1}(X))$, then $L_{\mathbb{Z}[S^{-1}]}(X)$ is simply-
connected.
###### Proof.
The first statement follows from the fact that, since $G$ is generated by
$S$-torsion, $G_{ab}$ is an abelian finite $S$-torsion subgroup, and then
$\mathbb{Z}[S^{-1}]\otimes G^{ab}=0$.
For the second statement, observe that as $G$ is $S^{-1}$-perfect, then $X$ is
a $\mathbb{Z}[S^{-1}]$-good space, and the ${\mathbb{Z}}[S^{-1}]$-completion
of $X$ is $1$-connected, by [BK72, VII.3.2]. But for a connected
${\mathbb{Z}}[\frac{1}{p}]$-good space $X$, the
${\mathbb{Z}}[\frac{1}{p}]$-completion is an
$H_{*}(-;{\mathbb{Z}}[\frac{1}{p}])$-localization (see [BK72, page 205]). ∎
In particular, if $X$ is a connected space such that its fundamental group is
finite and equal to its $\mathbb{Z}/p$-radical, then $L_{\mathbb{Z}[1/p]}X$ is
a simply-connected space.
###### Lemma 4.4.
Let $X$ be a connected space and $p$ a prime. Then the coaugmentation
$X\rightarrow L_{{\mathbb{Z}}[1/p]}X$ is an $\mathbb{F}_{q}$-equivalence and a
${\mathbb{Q}}$-equivalence where $q$ is a prime such that $(q,p)=1$. If
$L_{{\mathbb{Z}}[1/p]}X$ is $1$-connected then $L_{{\mathbb{Z}}[1/p]}X$ is
${\mathbb{F}}_{p}$-acyclic.
###### Proof.
By universal coefficient theorem (e.g. see [Spa66, 5.2.15]), the
coaugmentation $X\rightarrow L_{{\mathbb{Z}}[1/p]}X$ is a $G$-equivalence for
any ${\mathbb{Z}}[\frac{1}{p}]$-module $G$. The last statement follows form
[Dwy96, Lemma 6.2]. ∎
###### Lemma 4.5.
Let $Z$ be a $B{\mathbb{Z}}/p$-null space and $X$ be a connected space such
that $\pi_{1}(X)$ is a finite group generated by $p$-torsion elements which
lift to $X$. Then for any $f\colon X\rightarrow Z$, the composite
$X\rightarrow Z\rightarrow B\pi_{1}(Z)$ is nullhomotopic.
###### Proof.
Let $f\colon X\rightarrow Z$ be any map. We must check that $\pi_{1}(f)$ is
the trivial morphism. It is enough to show that the map between unpointed
homotopy classes $[S^{1},X]\rightarrow[S^{1},Z]$ is trivial.
Let $x\in\pi_{1}(X)$ be a generator, $\langle
x\rangle\cong{\mathbb{Z}}/p^{n}\subseteq\pi_{1}(X)$, we need to show that the
composite $B{\mathbb{Z}}/p^{n}\rightarrow B\pi_{1}(X)\rightarrow B\pi_{1}(Z)$
is nullhomotopic for any generator $x$.
By hypothesis, there is a lift
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B{\mathbb{Z}}/p^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(Z).}$
But since $Z$ is $B{\mathbb{Z}}/p$-null and
$P_{B{\mathbb{Z}}/p}(B{\mathbb{Z}}/p^{n})\simeq*$, it follows that the top
composite $B{\mathbb{Z}}/p^{n}\rightarrow X\rightarrow Z$ is nullhomotopic,
and therefore $\pi_{1}(f)(x)=0$. ∎
The hypothesis in Theorem 4.1 concerning the pointed mapping space from the
universal cover of $X$ is also satisfied by connected covers of $X$.
###### Lemma 4.6.
Let $X$ be a connected space.
1. 1.
Assume that $X$ is $1$-connected. Then
$P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$ iff ${\rm map\,}_{*}(X,Z)\simeq*$
for any connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$.
2. 2.
Assume that $X$ has a finite fundamental group. Let $Y$ be a connected cover
of $X$. If $Z$ is a connected $B{\mathbb{Z}}/p$-null and $p$-complete space,
then the equivalence ${\rm map\,}_{*}(Y,Z)\simeq*$ implies ${\rm
map\,}_{*}(X,Z)\simeq*$.
###### Proof.
1. 1.
Note that if $Z$ is a connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$,
there are weak homotopy equivalences
${\rm map\,}_{*}(P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p},Z)\simeq{\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X),Z)\simeq{\rm map\,}(X,Z)$
If $P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$, then ${\rm
map\,}(X,Z)\simeq*$ for any connected $B{\mathbb{Z}}/p$-null $p$-complete
space $Z$. On the other hand, assume that ${\rm map\,}_{*}(X,Z)\simeq*$ for
any connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$. Since
$P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}$ is a $p$-complete $B{\mathbb{Z}}/p$-null
space by Corollary 3.2, ${\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p},P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p})\simeq*$,
therefore $P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$.
2. 2.
Consider a fibration $Y\rightarrow X\rightarrow BG$ where $G$ is a finite
group. Zabrodsky’s Lemma (see [Mil84, 9.5]) tells us that there is an
equivalence of pointed mapping spaces ${\rm map\,}_{*}(X,Z)\simeq{\rm
map\,}_{*}(BG,Z)$ since ${\rm map\,}_{*}(Y,Z)\simeq*$. Finally, this mapping
space is contractible since we have weak homotopy equivalences ${\rm
map\,}_{*}(BG,Z)\simeq{\rm map\,}_{*}(BG^{\wedge}_{p},Z)$ and
$P_{B{\mathbb{Z}}/p}((BG)^{\wedge}_{p})\simeq*$ by [Flo07, 3.14].
∎
Now we are ready to undertake the proof of Theorem 4.2.
###### Proof of Theorem 4.2.
By hypothesis, $\pi_{1}(X)$ has no quotients whose order is prime to $p$,
which amounts to say that $\pi_{1}(X)$ is equal to its $\mathbb{Z}/p$-radical
$T_{p}(\pi_{1}(X))$.
First of all, notice that $L_{\mathbb{Z}[1/p]}X$ is $B{\mathbb{Z}}/p$-null by
Lemma 4.3 and [Dwy96, Lemma 6.2]. In order to show that
$P_{B{\mathbb{Z}}/p}(X)\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X)$ is a weak
equivalence, since $L_{\mathbb{Z}[1/p]}X$ is $B{\mathbb{Z}}/p$-null, we must
show that for every $B\mathbb{Z}/p$-null space $Y$ the natural coaugmentation
$X\longrightarrow L_{\mathbb{Z}[1/p]}X$ gives a weak equivalence ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}X,Y)\simeq{\rm map\,}_{*}(X,Y)$.
So let $Y$ be a $B{\mathbb{Z}}/p$-null space. Assume first that $Y$ is simply-
connected. By Miller’s Theorem 3.1, $Y^{\wedge}_{p}$ is also
$B{\mathbb{Z}}/p$-null. According to Bousfield-Kan fracture lemmas ([BK72,
V.6]), we must prove that, for every prime $q$, there is a weak homotopy
equivalence ${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y_{q}^{\wedge})\simeq{\rm
map\,}_{*}(X,Y_{q}^{\wedge})$, and ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y_{\mathbb{Q}})\simeq{\rm
map\,}_{*}(X,Y_{\mathbb{Q}})$. By Lemmas 4.4 and 4.6, this is a consequence of
the assumption of the theorem, so we finish the situation in which $Y$ is
simply connected.
Now let $Y$ be a $B\mathbb{Z}/p$-null space and $\tilde{Y}$ its universal
cover. The coaugmentation $X\longrightarrow L_{\mathbb{Z}[1/p]}X$ induces a
diagram of fibrations over the component of the constant map
where ${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)_{\\{c\\}}$ and ${\rm
map\,}_{*}(X,Y)_{\\{c\\}}$ are those components such that $\rho$ induce the
constant map when composing with $Y\rightarrow B\pi_{1}(Y)$.
The top horizontal map is an equivalence because of the previous argument
since $\tilde{Y}$ is a simply connected $B{\mathbb{Z}}/p$-null space. For any
connected space $A$ and a discrete group $H$, ${\rm map\,}_{*}(A,BH)$ is a
homotopically discrete space, and then ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),B\pi_{1}(Y))_{c}$ and ${\rm
map\,}_{*}(X,B\pi_{1}(Y))_{c}$ are contractible. Thus, the bottom horizontal
arrow in the diagram is also a weak equivalence.
To finish the proof we need to show that there are weak homotopy equivalences
${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)_{\\{c\\}}\simeq{\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)$ and ${\rm
map\,}_{*}(X,Y)_{\\{c\\}}\simeq{\rm map\,}_{*}(X,Y)$. The first equivalence
follows now from the fact that $L_{\mathbb{Z}[1/p]}(X)$ is simply connected by
Lemma 4.3, while the second follows from Lemma 4.5. ∎
###### Proof of Theorem 4.1.
Theorem 4.2 applied to the universal cover of $X$ implies that the map in
[Dwy96, 1.6], $P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(X\langle 1\rangle)$, is an equivalence.
Let $X_{p}$ be the covering space of $X$ with fundamental group
$T_{p}(\pi_{1}(X))$. There is a fibration $X_{p}\rightarrow X\rightarrow
B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$. Since the base space of this fibration
$B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$ is $B{\mathbb{Z}}/p$-null, the nullification
functor preserves the fibration by [Far96, 3.D.3] and there is another
fibration
$P_{B{\mathbb{Z}}/p}(X_{p})\rightarrow P_{B{\mathbb{Z}}/p}(X)\rightarrow
B(\pi_{1}(X)/T_{p}(\pi_{1}(X))).$
To prove the theorem we shall show that the natural map
$P_{B{\mathbb{Z}}/p}(X_{p})\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$,
which exists because $B{\mathbb{Z}}/p$ is $H{\mathbb{Z}}[\frac{1}{p}]$-acyclic
and then $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is $B{\mathbb{Z}}/p$-null, is
a homotopy equivalence. Note also that $(X_{p})\langle 1\rangle\simeq X\langle
1\rangle$. Therefore $X_{p}$ also satisfies the hypothesis of the theorem.
From now on we assume that $\pi_{1}(X)$ has no quotients whose order is prime
to $p$, which amounts to say that $\pi_{1}(X)$ is equal to its
$\mathbb{Z}/p$-radical $T_{p}(\pi_{1}(X))$.
Consider the fibration $X\langle 1\rangle\rightarrow X\rightarrow B\pi_{1}(X)$
and its fibrewise nullfication (see [Far96, 1.F]) which gives a diagram of
fibrations
$\textstyle{X\langle
1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\xi}}$$\textstyle{B\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p}(X\langle
1\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(X).}$
where $\bar{\xi}$ is an equivalence after $B{\mathbb{Z}}/p$-nullification.
Then, by [Dwy96, 1.6], $\bar{\xi}$ is a
${\mathbb{Z}}[\frac{1}{p}]$-equivalence. Note that it is enough to show that
the map $P_{B{\mathbb{Z}}/p}(\bar{X})\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{X})$ is an equivalence since there is a
chain
$P_{B{\mathbb{Z}}/p}(X)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}P_{B{\mathbb{Z}}/p}(\bar{X})\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{X})\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}L_{{\mathbb{Z}}[\frac{1}{p}]}({X}).$
Moreover, $\pi_{1}(\bar{X})\cong\pi_{1}(X)$ because the fibre
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)$ is $1$-connected, then the universal
cover of $\bar{X}$ is $P_{B{\mathbb{Z}}/p}(X)$ and
$\pi_{1}(\bar{X})=T_{p}(\pi_{1}(\bar{X}))$. For each generator
$x\in\pi_{1}(\bar{X})$, the obstructions to lift the map $B(\langle
x\rangle)\simeq B{\mathbb{Z}}/p^{n}\rightarrow B\pi_{1}(\bar{X})$ to $\bar{X}$
lie in the twisted cohomology groups
$H^{i+1}(B{\mathbb{Z}}/p^{n};\pi_{i}(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)))$
for $i\geq 1$, and these groups are trivial since the homotopy groups
$\pi_{i}(P_{B{\mathbb{Z}}/p}(X\langle
1\rangle))\cong\pi_{i}(L_{{\mathbb{Z}}[\frac{1}{p}]}(X\langle 1\rangle))$ are
${\mathbb{Z}}[\frac{1}{p}]$-modules. That is, $\bar{X}$ is a connected space
with finite fundamental group generated by $p$-torsion whose generators lift
to $\bar{X}$.
In order to apply Theorem 4.2, it remains to check that ${\rm
map\,}_{*}(\bar{X}\langle 1\rangle,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. Recall that $\bar{X}\langle
1\rangle\simeq P_{B{\mathbb{Z}}/p}({X}\langle 1\rangle)$. Then ${\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle),Z)\simeq{\rm
map\,}_{*}({X}\langle 1\rangle,Z)\simeq*$ where the last equivalence follows
by hypothesis. ∎
###### Remark 4.7.
The proof of Theorem 4.1 also holds if we replace the analysis at one prime
$p$ for a set of primes $S$ and imposing that the hypothesis on pointed
mapping spaces are satisfied for any prime $p$ in the set $S$. In that case we
have to replace $L_{{\mathbb{Z}}[\frac{1}{p}]}$ by $L_{{\mathbb{Z}}[S^{-1}]}$,
and $P_{B{\mathbb{Z}}/p}$ by $P_{W}$ where $W=\vee_{p\in S}B{\mathbb{Z}}/p$.
### 4.1 Examples
We want to explore the implications of these results on classifying spaces of
Lie groups, which was the original motivation for our work. For this sake we
need the following Lemma, which was proved by Dwyer [Dwy96, Theorem 1.2] using
an induction. We include here a shorter proof, based on the homology
decomposition of $BG$ via $p$-toral subgroups. The key point here is that this
decomposition is indexed over an mod $p$ _acyclic_ category, and this opens
the way for computing $P_{B{\mathbb{Z}}/p}$ for a more general class of
$p$-good spaces (see Corollary 4.11).
###### Lemma 4.8.
Let $Z$ be a connected $B{\mathbb{Z}}/p$-null and $p$-complete space. Let
$F\colon\mathcal{C}\rightarrow Top$ be a functor such that for each object
$c\in\mathcal{C}$, $F(c)$ is connected and
$P_{B{\mathbb{Z}}/p}(F(c)^{\wedge}_{p})$ is mod $p$ acyclic. If
$|\mathcal{C}|^{\wedge}_{p}\simeq*$, then ${\rm
map\,}_{*}(\operatornamewithlimits{hocolim}_{\mathcal{C}}F(c),Z)\simeq*$.
###### Proof.
The statement follows from a sequence of equivalences:
${\rm
map\,}(\operatornamewithlimits{hocolim}_{\mathcal{C}}F(c),Z)\simeq\operatornamewithlimits{holim}_{\mathcal{C}}{\rm
map\,}(F(c),Z)\simeq\operatornamewithlimits{holim}_{\mathcal{C}}{\rm
map\,}(P_{B{\mathbb{Z}}/p}(F(c)^{\wedge}_{p}),Z).$
Under the hypothesis of the lemma, this last mapping space is homotopy
equivalent to ${\rm map\,}(|\mathcal{C}|,Z)\simeq Z$ if
$|\mathcal{C}|^{\wedge}_{p}\simeq*$. ∎
###### Corollary 4.9.
Let $p$ be a prime. If $G$ is a compact Lie group and $X$ is a connected
$p$-complete $B{\mathbb{Z}}/p$-null space, then ${\rm
map\,}_{*}(BG^{\wedge}_{p},X)$ is weakly contractible.
###### Proof.
The proof is divided into two steps. In the first one we assume that $G$ is a
$p$-toral group, and then we use the existence of mod $p$ homology
decompositions of $BG$ with respect to certain families of $p$-toral subgroups
of $G$, see [JMO90], to undertake the general case.
Consider first when $G=T=(S^{1})^{n}$. In this case, $BT^{\wedge}_{p}\simeq
K({\mathbb{Z}}^{\wedge}_{p},2)^{n}\simeq(B({\mathbb{Z}}/{p^{\infty}})^{n})^{\wedge}_{p}$.
As $X$ is a $p$-complete space, we have the weak homotopy equivalence ${\rm
map\,}_{*}(BT^{\wedge}_{p},X)\simeq{\rm
map\,}_{*}(B({\mathbb{Z}}/p^{\infty})^{n},X)$. This mapping space is
contractible because ${\mathbb{Z}}/p^{\infty}\cong\lim{\mathbb{Z}}/p^{r}$
defined by inclusions ${\mathbb{Z}}/p^{n}\subset{\mathbb{Z}}/p^{n+1}$, and
since ${\mathbb{Z}}/p^{r}$ is a $p$-group, $B{\mathbb{Z}}/p^{r}$ is
$B{\mathbb{Z}}/p$-acyclic and ${\rm map\,}_{*}(B{\mathbb{Z}}/p^{r},X)\simeq*$,
and we can apply Lemma 4.8. Now, if $G=P$ is a $p$-toral group given by a
group extension $T^{n}\hookrightarrow P\mbox{$\to$$\to$}\pi$, Dwyer and
Wilkerson show that $BG$ admits a $p$-discrete approximation [DW94, Prop 6.9].
There is a sequence of finite $p$-groups $P_{0}\subset P_{1}\subset\ldots$
such that $BP\simeq\operatornamewithlimits{hocolim}BP_{n}$. Again, by Lemma
4.8, we obtain that ${\rm map\,}_{*}(BP,X)\simeq*$.
Let us go now through the general case. Our goal will be to prove that the
inclusion of constant maps induces an equivalence $X\simeq{\rm
map\,}(BG^{\wedge}_{p},X)$. By work of Jackowski-McClure-Oliver ([JMO90, Thm
4]), the space $BG$ is mod $p$ equivalent to
$\operatornamewithlimits{hocolim}_{\mathcal{O}_{p}G}F$, where
$\mathcal{O}_{p}G$ is the orbit category of stubborn $p$-toral subgroups of
$G$ and $F$ is a functor whose values have the homotopy type of classifying
spaces of stubborn $p$-toral subgroups of $G$. Since the statement holds for
$p$-toral groups, by Lemma 4.8 it is enough to observe that $\mathcal{O}_{p}G$
is $\mathbb{F}_{p}$-acyclic, see [JMO90, Prop 6.1], and we are done. ∎
Now we are ready to prove the desired result, which was previously known for
finite groups ([Flo07, Theorem 3.5]).
###### Theorem 4.10.
Let $G$ be a compact Lie group and $\pi$ its group of components. Let $G_{p}$
be the subgroup of $G$ whose group of components is $T_{p}(\pi)$. Then the
$B\mathbb{Z}/p$-nullification of $BG$ fits in the following covering
fibration:
$L_{\mathbb{Z}[1/p]}BG_{p}\longrightarrow P_{B\mathbb{Z}/p}BG\longrightarrow
B(\pi/\emph{T}_{p}\pi)$
###### Proof.
We have to check that the assumptions on Theorem 4.1 are satisfied when
$X=BG$. Since the universal cover of $BG$ is $BG_{0}$ is again the classifying
space of a compact Lie group, by Corollary 4.9 the hypothesis of Theorem 4.2
are satisfied. ∎
The proof of Corollary 4.9 applies to other type of spaces which admits mod
$p$ homology decompositions, for example $p$-compact groups (see [CLN07]). The
theory of $p$-local compact groups introduced by Broto, Levi and Oliver in
[BLO07] includes both the theory of $p$-compact groups [DW94] and $p$-local
finite groups [BLO03]. Roughly speaking, a $p$-local compact is a triple
$(S,\mathcal{F},\mathcal{L})$ where $S$ is a discrete $p$-toral group and
$\mathcal{F}$ and $\mathcal{L}$ are categories which model conjugacy relations
among subgroups of $S$. The classifying space of a $p$-local compact group is
$|\mathcal{L}|^{\wedge}_{p}$, and one of the main features of $p$-local
compact groups is that this space admits mod $p$-homology decompositions in
terms of classifying spaces of $p$-compact toral subgroups over mod $p$
acyclic orbit categories (see [BLO07, Proposition 4.6] and [BLO07, Corollary
5.6]).
###### Proposition 4.11.
Let $p$ be a prime, and $(S,\mathcal{F},\mathcal{L})$ a $p$-local compact
group. Then there is an equivalence
$L_{\mathbb{Z}[1/p]}(|\mathcal{L}|^{\wedge}_{p})\simeq{P}_{\emph{B}\mathbb{Z}/p}(|\mathcal{L}|^{\wedge}_{p})$.
###### Proof.
First of all, $\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$ is a finite $p$-group by
[BLO07, Proposition 4.4], therefore
$T_{p}(\pi_{1}(|\mathcal{L}|^{\wedge}_{p}))=\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$.
Then, we only need to check the hypothesis in Theorem 4.2. That is, ${\rm
map\,}_{*}(X\langle 1\rangle,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$.
The same argument used in the proof of Corollary 4.9 using mod $p$ homology
decompositions can be applied and it shows that ${\rm
map\,}_{*}(|\mathcal{L}|^{\wedge}_{p},Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. But it is not known in general
if the universal cover $|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ is the
classifying space of a $p$-local compact group. Instead, we will check that
the proof of Corollary 4.9 applies by showing that
$|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ admits a description, up to
$p$-completion, as a homotopy colimit of $B{\mathbb{Z}}/p$-acyclic spaces over
a mod $p$-acyclic category.
Let $P\leq S$ be an object in $\mathcal{O}(\mathcal{F}_{0})$ (see [BLO07,
Proposition 4.6]) and let $E_{P\leq S}$ be the pullback of
$|\mathcal{L}|^{\wedge}_{p}\langle
1\rangle\rightarrow|\mathcal{L}|^{\wedge}_{p}$ along $\tilde{B}P\rightarrow
BS$. Then, by naturality there is a map
$\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}E_{P\leq
S}\rightarrow|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ which fits in a
diagram of fibrations by Puppe’s theorem (e.g. [Far96, Appendix]),
$\textstyle{|\mathcal{L}|^{\wedge}_{p}\langle
1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|\mathcal{L}|^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})}$$\textstyle{\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}E_{P\leq
S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}\tilde{B}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{id}$
Since the middle vertical arrow is a mod $p$-equivalence, it follows that the
left vertical arrow is also a mod $p$-equivalence. Moreover,
$(\tilde{B}P)^{\wedge}_{p}$ is the classifying space of a $p$-compact toral
group, so it follows from the fibration $E_{P\leq
S}\rightarrow\tilde{B}P\rightarrow B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$ that
$(E_{P\leq S})^{\wedge}_{p}$ is also the classifying space of a $p$-compact
toral group, and therefore $B{\mathbb{Z}}/p$-acyclic. Then the proof of
Corollary 4.9 applies. ∎
A finite loop space is a triple $(X,BX,e)$ where $e\colon X\rightarrow\Omega
BX$ is a weak equivalence and $H^{*}(X;{\mathbb{Z}})$ is finite. Note that if
$X$ is a finite loop space, then $BX\langle 1\rangle^{\wedge}_{p}$ is the
classifying space of connected $p$-compact group since $H^{*}(\Omega BX\langle
1\rangle;{\mathbb{F}}_{p})$ is finite dimensional (see [DW94]).
###### Corollary 4.12.
Let $X$ be a finite loop space. Then there is a fibration
$L_{\mathbb{Z}[1/p]}BX_{p}\longrightarrow P_{B\mathbb{Z}/p}BX\longrightarrow
B(\pi/\emph{T}_{p}\pi)$
where $\pi=\pi_{0}(X)$ and $BX_{p}$ is the covering of $X$ whose fundamental
group is $T_{p}(\pi_{0}(X))$.
###### Proof.
Since $BX\langle 1\rangle^{\wedge}_{p}$ is the classifying space of a
$p$-compact group, Proposition 4.11 shows that $P_{B{\mathbb{Z}}/p}(BX\langle
1\rangle)^{\wedge}_{p}\simeq*$. Therefore we can apply Theorem 4.1. ∎
We finish with a somewhat two different examples.
###### Corollary 4.13.
Let $p$ be a prime. Let $K$ be a Kač-Moody group with a finite group of
components. Then there is a fibration
$L_{\mathbb{Z}[1/p]}BK_{p}\longrightarrow
P_{\emph{B}\mathbb{Z}/p}BK\longrightarrow\emph{B}(\pi_{1}(BK)/{T}_{p}(\pi_{1}(BK))$
where $K_{p}$ is the subgroup of $K$ whose group of components is
$T_{p}(\pi_{1}(BK)$
###### Proof.
First of all, the universal cover of $BK$ is $BK_{0}$ where $K_{0}$ is the
connected component of the unit in $K$. It is shown in Nitu Kitchloo thesis
(see [BK02]) that $BK$ is homotopy equivalent to a colimit over a contractible
category of classifying spaces of compact Lie groups. Then Corollary 4.9 and
its proof apply to show that ${\rm map\,}_{*}(BK,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. ∎
A different kind of example arises from the theory of infinite loop spaces and
it is a consequence of Theorem $2$ in [McG].
###### Corollary 4.14.
Let $X$ be a connected infinite loop space with finite fundamental group. Then
there is a fibration
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$
where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$.
## 5 Relation of nullification functors with other idempotent functors
In this section we compare the effect of nullification $P_{B{\mathbb{Z}}/p}$
on spaces which satisfy the hypothesis of Theorem 4.1 with the effect of some
completions or localizations on it. We analyse both functors that are supposed
to kill the $p$-torsion, like $L_{{\mathbb{Z}}[1/p]}$ or
${\mathbb{Z}}[1/p]_{\infty}$, and functors that usually preserve it, as
$L_{{\mathbb{Z}}[1/q]}$ and $p$-completion do.
###### Lemma 5.1.
Let $X$ be a connected space with finite fundamental group, $p$ and $q$
different primes. Then $(X^{\wedge}_{p})^{\wedge}_{q}$ is contractible.
###### Proof.
If $X$ is $1$-connected the case of $\mathbb{F}_{q}$-completion is described
in [BK72, VI.5.1]. If $X$ is not simply-connected, consider the fibration
$X^{\wedge}_{p}\langle 1\rangle\rightarrow X^{\wedge}_{p}\rightarrow
B\pi_{1}(X^{\wedge}_{p})$ and its fibrewise $q$-completion,
$(X^{\wedge}_{p}\langle 1\rangle)^{\wedge}_{q}\rightarrow Y\rightarrow
B\pi_{1}(X^{\wedge}_{p}).$
Since the fibre is a $1$-connected $p$-complete space completed at $q$ it is
contractible, then $Y\simeq B\pi_{1}(X^{\wedge}_{p})$. But then
$(X^{\wedge}_{p})^{\wedge}_{q}\simeq Y^{\wedge}_{q}\simeq
B\pi_{1}(X^{\wedge}_{p})^{\wedge}_{q}$ which is contractible since
$B\pi_{1}(X^{\wedge}_{p})$ is the classifying space of a finite $p$-group. ∎
We start by showing some direct direct consequences of Theorem 4.1.
###### Proposition 5.2.
Let $X$ be a space which satisfies the hypothesis of Theorem 4.1. Then
$\pi_{1}(P_{B{\mathbb{Z}}/p}(X))=\pi_{1}(X)/T_{p}(\pi_{1}(X))$ and
$(P_{B{\mathbb{Z}}/p}(X))^{\wedge}_{p}\simeq{*}$. Moreover
$P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(X^{\wedge}_{p})\simeq(X^{\wedge}_{p})_{\mathbb{Q}}$
is $1$-connected.
###### Proof.
Since $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is $1$-connected by Lemma 4.3, it
is clear from the fibration in Theorem 4.1 that
$\pi_{1}(P_{B{\mathbb{Z}}/p}(X))=\pi_{1}(X)/T_{p}(\pi_{1}(X))$. The space
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is mod $p$ acyclic by [Dwy96, Lemma
6.2] and the order of $\pi_{1}(X)/T_{p}(\pi_{1}(X))$ is prime to $p$, it
follows that $(P_{B{\mathbb{Z}}/p}(X))^{\wedge}_{p}$ is weakly contractible.
The second statement follows from applying Theorem 4.1 to $X^{\wedge}_{p}$.
Observe that if $X$ satisfies the hypothesis of the theorem, then
$X^{\wedge}_{p}$ also does. Moreover, $\pi_{1}(X^{\wedge}_{p})$ is a finite
$p$-group, then $P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(X^{\wedge}_{p})$. It remains to prove that they
are equivalent to $(X^{\wedge}_{p})_{\mathbb{Q}}$. Since they are
$1$-connected we can apply Sullivan’s arithmetic square.
We have proved that $(P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p}))^{\wedge}_{p}$ is
weakly contractible. Moreover, if $q\neq p$ then
$(P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p}))^{\wedge}_{q}\simeq(X^{\wedge}_{p})^{\wedge}_{q}$
which is weakly contractible by Lemma 5.1. Then
$P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})_{\mathbb{Q}}\simeq(X^{\wedge}_{p})_{\mathbb{Q}}$.
∎
We start by showing that $B{\mathbb{Z}}/p$-nullification and $p$-completion
behave like opposite functors in this context.
###### Remark 5.3.
If we complete in one prime $q$ and $B{\mathbb{Z}}/p$-nullify with regard to a
different prime $p$, then $X^{\wedge}_{q}$ is $B{\mathbb{Z}}/p$-null and the
coaugmentation $X\rightarrow P_{B{\mathbb{Z}}/p}X$ is an equivalence after
$q$-completion if $X$ satisfies the hypothesis of Lemma 3.9.
###### Remark 5.4.
Note that in general a connected space $X$ could be ${\mathbb{Z}}[1/p]$-bad if
$X$ is not $1$-connected, and then it is not possible in general to replace
completion by localization in the previous results. If we know in advance that
$X$ is ${\mathbb{Z}}[1/p]$-good (this happens, for example, if its fundamental
group is ${\mathbb{Z}}[1/p]$-perfect) then we can do the replacement, and
moreover ${\mathbb{Z}}[1/p]_{\infty}X\simeq L_{{\mathbb{Z}}[1/p]}X$. See for
example ([Far96, 1.E]) for more information about the relation between
$R$-localization and $R$-completion.
###### Proposition 5.5.
Let $X$ be a space which satisfies the hypothesis of Theorem 4.1 and such that
$\pi_{1}(X)\cong T_{q}(\pi_{1}(X))$. Then there are homotopy equivalences
$P_{B\mathbb{Z}/p}L_{\mathbb{Z}[1/q]}X\simeq L_{\mathbb{Z}[1/p,1/q]}X\simeq
L_{\mathbb{Z}[1/p]}P_{B\mathbb{Z}/q}X.$
###### Proof.
Since $\pi_{1}(X)\cong T_{q}(\pi_{1}(X))$,
$L_{{\mathbb{Z}}[\frac{1}{q}]}(X)\simeq P_{B{\mathbb{Z}}/q}(X)$ is
$1$-connected. Then, by Theorem 4.1, we have
$P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))\simeq
P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X))\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X))\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(P_{B{\mathbb{Z}}/q}(X))$. ∎
We finish by establishing the commutativity of the functors
$P_{B{\mathbb{Z}}/p}$ and $P_{B{\mathbb{Z}}/q}$. The problem of commutation of
localization functors was extensively studied in [RS00].
###### Proposition 5.6.
Let $X$ be a connected space, $p$ and $q$ two different primes. Assume that
$X$ satisfies the hypothesis of Theorem 4.1 for both primes $p$ and $q$. Then
there are homotopy equivalences
$P_{B\mathbb{Z}/p}P_{B\mathbb{Z}/q}X\simeq P_{B\mathbb{Z}/p\vee
B\mathbb{Z}/q}X\simeq P_{B\mathbb{Z}/q}P_{B\mathbb{Z}/p}X.$
###### Proof.
It is enough to show the first equivalence since the other one will follow by
symmetry.
Consider the set of primes $S=\\{p,q\\}$. By pulling back the universal
fibration, there is a fibration $X_{S}\rightarrow X\rightarrow
B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))$, where $\pi_{1}(X_{S})=T_{S}(\pi_{1}(X))$.
Since the order of $\pi_{1}(X)/T_{S}(\pi_{1}(X))$ is prime to both $p$ and
$q$, the space $B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))$ is both
$B{\mathbb{Z}}/p$-null and $B{\mathbb{Z}}/q$-null (in particular it is also
$B{\mathbb{Z}}/p\vee B{\mathbb{Z}}/q$-null) the composite of functors
$P_{B{\mathbb{Z}}/p}\circ P_{B{\mathbb{Z}}/q}$ and $P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}$ preserve the fibration [Far96, 3.D.3], and there is a
diagram of fibrations
$\textstyle{P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X_{S}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}(X_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi_{1}(X)/(T_{S}(\pi_{1}(X))).}$
where the first two vertical maps exist because if a space $Y$ is
$B{\mathbb{Z}}/p\vee B{\mathbb{Z}}/q$-null, and then it is also
$B{\mathbb{Z}}/p$-null and $B{\mathbb{Z}}/q$-null. Then we can assume that the
group $\pi_{1}(X)=T_{S}(\pi_{1}(X))$, which we simply denote by $\pi$ in the
sequel, is generated by $p$ and $q$ torsion.
By [RS00, Prop 1.1], we need to show that
$P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$ is $B{\mathbb{Z}}/q$-null and
conversely $P_{B{\mathbb{Z}}/q}(P_{B{\mathbb{Z}}/p}(X))$ is
$B{\mathbb{Z}}/p$-null. In our situation, by symmetry, it is enough to check
one of the two conditions.
Let us see first that $P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$ is
$1$-connected. We can apply the fibrewise $B{\mathbb{Z}}/p$-nullification to
the fibration in Theorem 4.1,
$\textstyle{L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/q}(X_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi/T_{q}(\pi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi/T_{q}(\pi)),}$
where $L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})$ is $1$-connected and
$P_{B{\mathbb{Z}}/p}(\bar{P})\simeq
P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X_{q}))$;ˇ then $\bar{P}$ has
fundamental group $\pi/T_{q}(\pi)$ which is generated by $p$ torsion. If
$\bar{P}$ satisfies the hypothesis of Theorem 4.1 for the prime $p$ then
$P_{B{\mathbb{Z}}/p}(\bar{P})\simeq L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{P})$ is
$1$-connected. We need to check that for any connected space $Z$ which is
$p$-complete and $B{\mathbb{Z}}/p$-null, ${\rm map\,}_{*}(\bar{P}\langle
1\rangle,Z)$ is weakly contractible. Note that $\bar{P}\langle 1\rangle\simeq
P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}))$, and then
$\textrm{map}_{*}(P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})),Z)\simeq\textrm{map}_{*}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}),Z)\simeq\textrm{map}_{*}(P_{B{\mathbb{Z}}/q}(X_{q}),Z)\simeq$
$\simeq\textrm{map}_{*}(X_{q},Z),$
where the last equivalence follows because
$(P_{B{\mathbb{Z}}/q}(X_{q}))^{\wedge}_{p}\simeq(X_{q})^{\wedge}_{p}$ and $Z$
is $p$-complete. Finally Lemma 4.6 tells us that this last mapping space is
weakly contractible.
We denote by $Y$ the space $P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$, and
we finally should check that it is $B{\mathbb{Z}}/q$-null. Since it is a
$1$-connected space, we can use Sullivan’s arithmetic square and check that
the mapping spaces ${\rm map\,}_{*}(B{\mathbb{Z}}/q,Y_{\mathbb{Q}})$ and ${\rm
map\,}_{*}(B{\mathbb{Z}}/q,Y^{\wedge}_{r})$ are weakly contractible for any
prime $r$.
If $r\neq q$, ${\rm map\,}_{*}(B{\mathbb{Z}}/q,Y^{\wedge}_{r})\simeq*$ because
$(B{\mathbb{Z}}/q)^{\wedge}_{r}\simeq*$. Also, since
$(B{\mathbb{Z}}/q)_{\mathbb{Q}}\simeq*$, ${\rm
map\,}_{*}(B{\mathbb{Z}}/q,Y_{\mathbb{Q}})\simeq*$. We are left to the case
$r=q$, and
$Y^{\wedge}_{q}=(P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X)))^{\wedge}_{q}\simeq(P_{B{\mathbb{Z}}/q}(X))^{\wedge}_{q}\simeq*$
by Proposition 5.2. So we are done. ∎
For example, given a compact Lie group, $BG$ satisfies the hypothesis of
Theorem 4.1 for any prime $p$.
###### Remark 5.7.
The same proof remains valid if we apply in succession over $X$ a finite
number of $B{\mathbb{Z}}/p$-nullification functors for different primes
assuming $X$ satisifes the hypothesis of Theorem 4.1 for each prime. On the
other hand, it is likely that that the nullification of $X$ with regard to the
wedge of the classifying spaces of _all_ primes is homotopy equivalent to the
rational localization of $X$. See [Flo07, Section 3.2] for details.
## 6 $B{\mathbb{Z}}/p$-cellularization of classifying spaces
In this section we will give a Serre-type general dichotomy theorem (Theorem
6.1), which is very much in the spirit of [FS07]. Then, we will use this
statement to describe several examples concerning the
$B{\mathbb{Z}}/p$-cellularization of some families of classifying spaces of
remarkable groups, such as $p$-toral groups, finite groups with a $p$-subgroup
of $p^{\prime}$-index, $BS^{3}$ or $BSO(3)$ (at the prime $2$). Our
considerations are also based in the results of the previous sections relating
cellularization, nullification and completion.
### 6.1 The dichotomy theorem
The main result of this section is:
###### Theorem 6.1.
Let $X$ be a connected nilpotent $\Sigma^{n}B{\mathbb{Z}}/p$-null space for
some $n\geq 0$. Then the $B{\mathbb{Z}}/p$-cellullarization of $X$ has the
homotopy type of a Postnikov piece with homotopy groups are concentrated in
degrees $1$ to $n$ , or else it has infinitely many nonzero homotopy groups.
Moreover, if $X$ is $1$-connected of finite type, then the fundamental group
$\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finite elementary abelian $p$-group.
The rest of the subsection is devoted to the proof of Theorem 6.1. Even if the
statement is similar to the one in [FS07, Proposition 2.3], the authors deal
with the situation in which the space is torsion, and this is not the case for
$BG$ where $G$ is a compact connected Lie group. The strategy used in [FS07]
for classifying spaces of finite groups can be summarized as follows.
###### Proposition 6.2.
Let $X$ be a torsion Postnikov piece whose fundamental group is generated by
elements of order $p$ which lift to $X$. Assume there exists a prime $q\neq p$
such that $X^{\wedge}_{q}$ is torsion and it has infinitely many non-trivial
homotopy groups. Then $CW_{B{\mathbb{Z}}/p}(X)$ also has infinitely many non-
trivial homotopy groups.
###### Proof.
Consider the fibration $CW_{B{\mathbb{Z}}/p}(X)\rightarrow X\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ from Theorem 2.3. Note that $P_{\Sigma
B{\mathbb{Z}}/p}(C)$ is $1$-connected since $C$ is so. To prove the statement,
we will show that $P_{\Sigma B{\mathbb{Z}}/p}(C)$ has infinitely many non-
trivial homotopy groups. We apply Sullivan’s arithmetic square to $P_{\Sigma
B{\mathbb{Z}}/p}(C)$ to obtain a pullback diagram
$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\prod_{r\neq
p}X^{\wedge}_{r})\times(P_{\Sigma
B{\mathbb{Z}}/p}(C))^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$*$\textstyle{(\prod_{r\neq
q,p}X^{\wedge}_{r})_{\mathbb{Q}}\times((P_{\Sigma
B{\mathbb{Z}}/p}(C))^{\wedge}_{p})_{\mathbb{Q}}.}$
which allow us to construct a map $s\colon X^{\wedge}_{q}\rightarrow P_{\Sigma
B{\mathbb{Z}}/p}(C)$ such that $\eta\circ s\simeq id$. That is $s$ is a
section of the $q$-completion. Then for $n\geq 2$ we have that
$\pi_{n}(X^{\wedge}_{q})$ is a direct summand of $\pi_{n}(P_{\Sigma
B{\mathbb{Z}}/p}(C))$. ∎
For example, by Levi’s work in [Lev95], the previous theorem applies when $X$
is the classifying space of a finite group.
Now we need to state some general results concerning to the cellularization of
$\Sigma B{\mathbb{Z}}/p$-null spaces, that deal with the consequences of
imposing that $CW_{B{\mathbb{Z}}/p}(X)$ is a Postnikov piece for a certain
space $X$. Note that this is the “forbidden” case in Theorem 6.1.
###### Lemma 6.3.
Let $P[n]$ be a connected $\Sigma B{\mathbb{Z}}/p$-null Postnikov piece with
$n\geq 3$ then $p$ is invertible in $\pi_{n}(P[n])$.
###### Proof.
Note that if $P[n]$ is $\Sigma B{\mathbb{Z}}/p$-null then $\Omega P[n]$ is
$B{\mathbb{Z}}/p$-null, and also $\Omega^{n-1}P[n]$ is so. Since the connected
component of the constant in $\Omega^{n-1}P[n]$ is an Eilenberg-MacLane space
$K(\pi_{n}(P[n]),1)$, we see that $K(\pi_{n}(P[n]),1)$ is also
$B{\mathbb{Z}}/p$-null.
Similarly, the connected component $E$ of $\Omega^{n-2}P[n]$ is
$B{\mathbb{Z}}/p$-null. There is a fibration $K(\pi_{n}(P[n]),2)\rightarrow
E\rightarrow K(\pi_{n-1}(P[n]),1)$. Since the pointed mapping spaces ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,K(\pi_{n-1}(P[n]),1))_{c}$ and ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,E)$ are weakly contractible, we obtain from the
previous fibration that ${\rm map\,}(B{\mathbb{Z}}/p,K(\pi_{n}(P[n]),2))$ is
also weakly contractible. That is $K(\pi_{n}(P[n]),2)$ is
$B{\mathbb{Z}}/p$-null. The conclusion now follows from Lemma 2.7. ∎
###### Proposition 6.4.
Let $X$ be a connected nilpotent $\Sigma B{\mathbb{Z}}/p$-null space such that
${\mathbb{Z}}[\frac{1}{p}]_{\infty}(X)\simeq*$. Then $X$ has the homotopy type
of a $K(G,1)$ or it has infinitely many nonzero homotopy groups.
###### Proof.
Assume that $X\simeq P[n]$ is a finite Postnikov piece. First we show that
$n\leq 2$. Since ${\mathbb{Z}}[\frac{1}{p}]_{\infty}(X)$ is weakly
contractible (see Lemma 2.8), then
$\pi_{i}(P[n])\otimes{\mathbb{Z}}[\frac{1}{p}]=0$ for all $i>0$ ([BK72,
V.4.1]). But if $n\geq 3$, $p$ is invertible in $\pi_{n}(P[n])$ by the
previous Proposition 6.3, then $\pi_{n}(P[n])=0$. We can apply this argument
as long as $n\geq 3$.
From now on we assume $n=2$. Next we prove that $\pi_{2}(P[2])$ has no
$p$-torsion. There is a fibration $K(\pi_{2}(P[2]),2)\rightarrow
P[2]\rightarrow K(\pi_{1}(P[2]),1)$, which induces a covering
$K(\pi_{1}(P[2]),0)\rightarrow K(\pi_{2}(P[2]),2)\rightarrow P[2]$. If
$\pi_{2}(P[2])$ had $p$-torsion, then a non-trivial homomorphism
${\mathbb{Z}}/p\rightarrow\pi_{2}(P[2])$ would induce a nontrivial map
$f\colon\Sigma B{\mathbb{Z}}/p\rightarrow K(\pi_{2}(P[2]),2)$ which is
nullhomotopic when composed with $K(\pi_{2}(P[2]),2)\rightarrow P[2]$ since
$P[2]$ is $\Sigma B{\mathbb{Z}}/p$-null (see Lemma 2.6). Then we obtain a
contradiction since $f$ must be nullhomotopic. So $\pi_{2}(P[2])$ has no
$p$-torsion, and this is again a contradiction since
$\pi_{2}(P[2])\otimes{\mathbb{Z}}[\frac{1}{p}]=0$. ∎
We now state our dichotomy theorem for nilpotent $\Sigma B{\mathbb{Z}}/p$-null
spaces.
###### Theorem 6.5.
Let $X$ be a connected nilpotent $\Sigma B{\mathbb{Z}}/p$-null space. Then the
$B{\mathbb{Z}}/p$-cellulla-rization of $X$ has the homotopy type of a $K(G,1)$
or it has infinitely many nontrivial homotopy groups. Moreover, if $X$ is
$1$-connected of finite type, then $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a
finite elementary abelian $p$-group.
###### Proof.
By Lemma 2.5, $CW_{B{\mathbb{Z}}/p}(X)$ is also nilpotent. Moreover, by Lemma
2.8 we have an equivalence
${\mathbb{Z}}[\frac{1}{p}]_{\infty}(CW_{B{\mathbb{Z}}/p}(X))\simeq*$ , and
then we can apply Proposition 6.4 to $CW_{B{\mathbb{Z}}/p}(X)$. ∎
###### Proof of Theorem 6.1.
Assume that $n\geq 2$, we can assume that $X$ is $B{\mathbb{Z}}/p$-cellular
and $\Sigma^{n}B{\mathbb{Z}}/p$-null. By [Bou94, Theorem 7.2], there is a
principal fibration
$K(P,n)\rightarrow X\rightarrow P_{\Sigma^{n-1}}(X)$
where $P$ is a $p$-torsion grup. Since $K(P,n)$ is $B{\mathbb{Z}}/p$-cellular
([CCS07-2, Lemma 3.3]) and $X$ too by assumption, then $P_{\Sigma^{n-1}}(X)$
is also $B{\mathbb{Z}}/p$-cellular by [Cha96, Theorem 4.7]. By induction we
reduce to the case in which $n=1$, which is proved in Theorem 6.5. ∎
The next question we need to refer concerning Theorem 6.1 is when the
cellularization of a classifying space is again a classifying space, not
necessarily of a discrete group. This is important to understand the first
part of the previous dichotomy.
###### Proposition 6.6.
Let $X$ be a space. If $CW_{B{\mathbb{Z}}/p}(X)\simeq BH$ for some compact Lie
group $H$, then it must be a finite $p$-group generated by order $p$ elements.
###### Proof.
Since the pointed homotopy colimit of acyclics is acyclic for any cohomology
theory ([Far96, 2.D.2.5]), it is clear that
$\tilde{H}^{*}(BH;{\mathbb{Q}})=\tilde{H}^{*}(CW_{B{\mathbb{Z}}/p}X;\mathbb{Q})=0$.
On the other hand, it is well-known that the rational cohomology of $BH$ are
the invariants of the rational cohomology of the classifying space of the
maximal torus $T$ under the action of the Weyl group $W$. In fact
$\tilde{H}^{*}(BH;{\mathbb{Q}})=0$ if and only if $H$ is a finite group.
Finally, the functor $CW$ is idempotent, so $BH$ must be
$B{\mathbb{Z}}/p$-cellular. Thus, we can apply [Flo07, Prop 4.14 ] to finish
the proof. ∎
###### Remark 6.7.
The arguments in Proposition 6.6 also work if
$CW_{B{\mathbb{Z}}/p}(X)\simeq(BH)^{\wedge}_{p}$ where $H$ is a compact Lie
group. It is clear then that
$\tilde{H}^{*}(BH;{\mathbb{Z}}^{\wedge}_{p})\otimes{\mathbb{Q}}=0$. But again
this is only possible if $H$ is discrete. If $H$ is in particular finite,
conditions are known (see [FS07, Corollary 3.3]) under which
$(BH)^{\wedge}_{p}$ is $B{\mathbb{Z}}/p$-cellular. See Example 6.14 below.
###### Remark 6.8.
When $X$ is an $H$-space satisfying the hypothesis of Theorem 6.1, Castellana,
Crespo and Scherer proved in [CCS07-2] that the
$B{\mathbb{Z}}/p$-cellularization of $X$ is always a Postnikov piece. Examples
of such spaces are given by $H$-spaces whose mod $p$ cohomology is finitely
generated as an algebra over the Steenrod algebra (see [CCS07-1]).
### 6.2 Examples
In this subsection we concentrate in the description of the
$B{\mathbb{Z}}/p$-cellularization of classifying spaces of compact Lie groups,
generalizing to the continuous case work of the second author in the finite
case ([Flo07] and [FS07]). In the study of the homotopy type of classifying
spaces of Lie groups, a very useful strategy is to isolate the information at
every prime.
Theorem 6.1 implies automatically the following dichotomy theorem for
classifying spaces of compact Lie groups. We say that an element $g\in G$ is
$p$-cohomologically central if the map induced by the inclusion
$BC_{G}(x)\rightarrow BG$ is a mod $p$ homology isomorphism. Mislin in [Mis92]
shows that there is a natural bijection between the set of conjugacy classes
of $p$-cohomologically central elements of order $p$ in $G$ with
$pZ(G/O_{p^{\prime}}(G))$ where $O_{p^{\prime}}(G)$ is the largest normal
$p^{\prime}$-subgroup of $G$ and $pZ(G)$ are the elements of order $p$ in
$Z(G)$.
###### Theorem 6.9.
Let $G$ be a compact connected Lie group. If there exists a non
$p$-cohomologically central element of order $p$, then the
$B{\mathbb{Z}}/p$-cellullarization of $BG$ has infinitely many nonzero
homotopy groups. Otherwise, it has the homotopy type of a $K(V,1)$, where $V$
is a finite elementary abelian $p$-group.
###### Proof.
Since $G$ is assumed to be connected, $BG$ is simply connected. Moreover it is
of finite type, and $\Sigma B{\mathbb{Z}}/p$-null because of Miller’s solution
of the Sullivan conjecture because $\Omega BG\simeq G$ is a finite complex.
Now, we apply Theorem 6.1. If $CW_{BZ/p}(BG)$ is an Eilenberg-MacLane space
$K(V,1)$, then ${\rm map\,}_{*}(B{\mathbb{Z}}/p,BG)$ is homotopically
discrete. Since $BG$ is simply connected,
$[B{\mathbb{Z}}/p,BG]_{*}\cong[B{\mathbb{Z}}/p,BG]\cong{\rm
Rep\,}({\mathbb{Z}}/p,G)$. If ${\rm map\,}_{*}(B{\mathbb{Z}}/p,BG)$ is
homotopically discrete, then for each $\rho\in{\rm Rep\,}({\mathbb{Z}}/p,G)$,
the evaluation ${\rm map\,}(B{\mathbb{Z}}/p,BG)_{B\rho}\rightarrow BG$ induces
an ${\mathbb{F}}_{p}$-homology equivalence
$BC_{G}(\rho({\mathbb{Z}}/p))\rightarrow BG$ by [DZ87]. And this only happens
if all the elements of order $p$ are $p$-cohomologically central. ∎
In the continuous case, there are paradigmatic examples of $BG$ whose
cellularization is again a classifying space.
###### Example 6.10.
If $X=BS^{1}=K({\mathbb{Z}},2)$, it is clear comparing pointed mapping spaces
that $CW_{B{\mathbb{Z}}/p}BS^{1}=B{\mathbb{Z}}/p$ since ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,BS^{1})$ is homotopically discrete with components
${\rm Hom\,}({\mathbb{Z}}/p,S^{1})$. Let us now consider $BS^{3}$ the
classifying space of the $3$-sphere. Lemma 3.2 reduces the computation of
$CW_{B{\mathbb{Z}}/p}(BS^{3})$ to that of
$CW_{B{\mathbb{Z}}/p}((BS^{3})^{\wedge}_{p})$. The mapping space from
$B{\mathbb{Z}}/p$ into $(BS^{3})^{\wedge}_{p}$ has been well studied. If
$p=2$, the inclusion of the centre $B{\mathbb{Z}}/2\rightarrow BS^{3}$ induces
a homotopy equivalence ${\rm
map\,}(B{\mathbb{Z}}/2,B{\mathbb{Z}}/2)\rightarrow{\rm
map\,}(B{\mathbb{Z}}/2,(BS^{3})^{\wedge}_{2})$ since ${\rm
map\,}(B{\mathbb{Z}}/2,(BS^{3})^{\wedge}_{2})_{f}\simeq(BC_{S^{3}}(f))^{\wedge}_{2}$
(see [DMW87]), and therefore $CW_{B{\mathbb{Z}}/2}(BS^{3})\simeq
B{\mathbb{Z}}/2$. If $p$ is odd, then $(BS^{3})^{\wedge}_{p}\simeq
BN(T)^{\wedge}_{p}$, and this case will be studied in Example 6.15.
Sometimes, if we are unable to describe $CW_{B{\mathbb{Z}}/p}BG$, we can at
least identify it with another classifying space at a prime.
###### Example 6.11.
Let $BO(2)$ be the classifying space of the orthogonal group $O(2)$. There is
a mod $2$ equivalence $BD_{2^{\infty}}\rightarrow BO(2)$ where
$D_{2^{\infty}}=colim_{n}D_{2^{n}}$. Moreover $BD_{2^{\infty}}$ is
$B{\mathbb{Z}}/2$-cellular by [Flo07, Example 5.1]. Since
$\pi_{1}(BO(2))={\mathbb{Z}}/2$ is generated by an element of order $2$ which
lifts to $BO(2)$, we are in the situation of Remark 3.17. This will be used in
particular in Proposition 6.17.
We devote the remaining of the section to study some families of Lie groups
which show different and interesting features in this context. We begin with
extensions of elementary abelian $p$-groups by a finite group of order prime
to $p$, which provide an example in which Proposition 3.14 does not hold.
Compare with [FS07]. We start with a situation which deals with fibrations.
###### Proposition 6.12.
Let $F\rightarrow E\rightarrow B$ be a fibration of $p$-good connected spaces
such that $F$ is $B{\mathbb{Z}}/p$-cellular, $B$ is $B{\mathbb{Z}}/p$-null
and, $B^{\wedge}_{p}$ is $\Sigma B{\mathbb{Z}}/p$-null. Assume that
$[B{\mathbb{Z}}/p,E]\rightarrow[B{\mathbb{Z}}/p,E^{\wedge}_{p}]$ is exhaustive
and $\pi_{1}(F)\rightarrow\pi_{1}(E^{\wedge}_{p})$ is an epimorphism. Then
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}$ is the homotopy fiber of
$E^{\wedge}_{p}\rightarrow B^{\wedge}_{p}$.
###### Proof.
First of all, note that since $B$ is $B{\mathbb{Z}}/p$-null, then
$F\rightarrow E$ is a $B{\mathbb{Z}}/p$-equivalence, and thus $F\simeq
CW_{B{\mathbb{Z}}/p}(E)$. To compute the cellularization of $E^{\wedge}_{p}$
we proceed by applying Chachólski’s strategy (Theorem 2.3, see also [Cha96,
Section 7] for the slightly general formulation we use here). Consider the
following diagram of horizontal cofibrations,
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{E^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$
where $D$ is $1$-connected since $i$ is an epimorphism on fundamental groups.
Since $E$ is $p$-good, $g$ induces an homotopy equivalence
$C^{\wedge}_{p}\simeq D^{\wedge}_{p}$, and therefore $C^{\wedge}_{p}$ is also
$1$-connected.
Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the fibration
$F\rightarrow E\rightarrow B$ and the composite map $E\rightarrow C\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ implies that there is a map $B\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ which fits in a diagram of fibrations:
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C.}$
where the first vertical map is a homotopy equivalence,
$CW_{B{\mathbb{Z}}/p}(E)\simeq F$. The long exact sequence for homotopy groups
shows that the last vertical arrow is also a homotopy equivalence. Now
consider the diagram of fibre sequences
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C\simeq
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}D.}$
The spaces $P_{\Sigma B{\mathbb{Z}}/p}(C)\simeq B$ and $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ are $p$-good spaces (note that $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected by Lemma 2.4) and Miller’s
theorem apply to show that $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ is $\Sigma
B{\mathbb{Z}}/p$-null. Also $P_{\Sigma B{\mathbb{Z}}/p}(C)^{\wedge}_{p}\simeq
B^{\wedge}_{p}$ is $\Sigma B{\mathbb{Z}}/p$-null by hypothesis. Applying Lemma
3.9 and the proof of Corollary 3.12, we obtain that the composite $P_{\Sigma
B{\mathbb{Z}}/p}(C)\simeq B\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(D)$ is a mod
$p$ equivalence, and therefore we conclude that the $p$-completion
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}$ is the homotopy fiber of
$E^{\wedge}_{p}\rightarrow B^{\wedge}_{p}$ is a mod $p$ equivalence by
$p$-completion of the fibration
$CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})\rightarrow E^{\wedge}_{p}\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(D)$. ∎
###### Corollary 6.13.
Let $F\rightarrow E\rightarrow B$ be a fibration of connected spaces such that
$F$ is $B{\mathbb{Z}}/p$-cellular, $B$ is $B{\mathbb{Z}}/p$-null and
$B^{\wedge}_{p}\simeq*$. Assume that
$[B{\mathbb{Z}}/p,E]\rightarrow[B{\mathbb{Z}}/p,E^{\wedge}_{p}]$ is
exhaustive, $\pi_{1}(F)\rightarrow\pi_{1}(E^{\wedge}_{p})$ is an epimorphism
and $\pi_{i}(E)$ are finite groups for all $i\geq 1$. Then $E^{\wedge}_{p}$ is
$B{\mathbb{Z}}/p$-cellular.
###### Proof.
By Proposition 6.12, we know that
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}\simeq E^{\wedge}_{p}$. We
will prove that $CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is $p$-complete. Since
$\pi_{i}(E)$ are finite groups for all $i\geq 1$, $\pi_{i}(E^{\wedge}_{p})$
are all finite $p$-groups and $E^{\wedge}_{p}$ is nilpotent ([BK72, VII.4.3]).
Therefore $CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is nilpotent by Lemma 2.5. A
Sullivan’s arithmetic square argument shows that
$CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is $p$-complete since
$((CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p})_{\mathbb{Q}}\simeq(E^{\wedge}_{p})_{\mathbb{Q}}\simeq*$.
∎
###### Example 6.14.
Let $G$ be a finite group which is an extension $H\rightarrow G\rightarrow W$
where $BH$ is $B{\mathbb{Z}}/p$-cellular and $(|W|,p)=1$. Then
$CW_{B{\mathbb{Z}}/p}(BG)\simeq BH$ and $BG^{\wedge}_{p}$ is
$B{\mathbb{Z}}/p$-cellular by the previous result. Note that $G$ does not need
to be generated by elements of order $p$; compare with [FS07, Section 4].
Other examples are provided by nilpotent Postnikov pieces whose fundamental
group is of order prime to $p$ and the $1$-connected cover is $p$-torsion.
###### Example 6.15.
Let $N$ be an extension of a finite group of order prime to $p$ with a torus,
that is, we have a fibration $BT\rightarrow BN\rightarrow BW$ where
$T\cong(S^{1})^{n}$ and $(|W|,p)=1$. From this fibration we see that
$CW_{B{\mathbb{Z}}/p}(BN)\simeq CW_{B{\mathbb{Z}}/p}(BT)\simeq BV$ where
$V\cong({\mathbb{Z}}/p)^{n}$, as $BW$ is $B{\mathbb{Z}}/p$-null and
$BT\rightarrow BN$ is a $B{\mathbb{Z}}/p$-equivalence.
Next we compute the cellularization of $(BN)^{\wedge}_{p}$. First, by [BK02,
Prop. 7.5], there is a bijection
$[B{\mathbb{Z}}/p,BN]\cong[B{\mathbb{Z}}/p,BN^{\wedge}_{p}]$. Consider the
following diagram of horizontal cofibrations,
$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{BN^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D.}$
where $D$ is $1$-connected since $BN^{\wedge}_{p}$ is also $1$-connected.
Therefore $P_{\Sigma B{\mathbb{Z}}/p}(D)$ is also $1$-connected by Lemma 2.4.
Since $\pi_{1}(C)$ is finite, $C$ is $p$-good. Moreover, $g$ is a mod $p$
equivalence, therefore $C^{\wedge}_{p}$ is $1$-connected. Now consider the
following diagram of fibrations:
$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(BN^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{BN^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}D.}$
We will show that $P_{\Sigma B{\mathbb{Z}}/p}(g)\colon P_{\Sigma
B{\mathbb{Z}}/p}C\rightarrow P_{\Sigma B{\mathbb{Z}}/p}D$ is a mod $p$
equivalence. Since $g$ is a mod $p$ equivalence, and also $P_{\Sigma
B{\mathbb{Z}}/p}(\eta_{D})\colon P_{\Sigma B{\mathbb{Z}}/p}(D)\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(D^{\wedge}_{p})$ by Corollary 3.11, we only need to
prove that $P_{\Sigma B{\mathbb{Z}}/p}(\eta_{C})\colon P_{\Sigma
B{\mathbb{Z}}/p}(C)\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is
also a mod $p$ equivalence by checking that $C$ satisfies the hypothesis of
Lemma 3.9.
Consider the following diagram of fibrations
$\textstyle{T/V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(T/V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(T/V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{BW.}$
where $f$ exists by Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the
fibration $BV\rightarrow BN\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C)$ and the
map $BN\rightarrow BW$. It implies that there is a map $P_{\Sigma
B{\mathbb{Z}}/p}(C)\rightarrow BG$ which fits in a diagram of fibrations. The
bottom fibration shows that $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is homotopy
equivalent to the classifying space of a compact Lie group whose fundamental
group is $G$.
Now we check that $C$ satisfies the hypothesis of Lemma 3.9. First, $C$ and
$P_{\Sigma B{\mathbb{Z}}/p}(C)$ are $p$-good since they have finite
fundamental groups ([BK72, VII.5.1]), $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected and therefore it is also
$p$-good. It remains to check that $P_{\Sigma
B{\mathbb{Z}}/p}(C)^{\wedge}_{p}$ and $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ are $\Sigma
B{\mathbb{Z}}/p$-null spaces. Since $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is
homotopy equivalent to the classifying space of a compact Lie group, its
$p$-completion is $\Sigma B{\mathbb{Z}}/p$-null (see e.g. [BK02, Prop 7.5]).
Finally $P_{\Sigma B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ is also
$\Sigma B{\mathbb{Z}}/p$-null since $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected by Theorem 3.1.
Summarizing, $(CW_{B{\mathbb{Z}}/p}(BN^{\wedge}_{p}))^{\wedge}_{p}$ is the
homotopy fiber of $BN^{\wedge}_{p}\rightarrow BK^{\wedge}_{p}$ where $K$ is an
extension of $W$ by $T/V$ and $V$ is the maximal elementary abelian
$p$-subgroup in the torus $T$.
Our next example concerns $p$-toral groups. Recall that a $p$-toral group is
an extension of a torus by a finite $p$-group. A $p$-compact toral group is an
extension of a $p$-compact torus by a finite $p$-group, and a discrete
$p$-toral group is a group $P$ with normal subgroup $T$ such that $T$ is
isomorphic to a finite product of copies of ${\mathbb{Z}}/p^{\infty}$ and
$P/T$ is a finite $p$-group.
Since $CW_{B{\mathbb{Z}}/p}(BT^{\wedge}_{p})\simeq CW_{B{\mathbb{Z}}/p}(BT)$
by Lemma 3.2 and $CW_{B{\mathbb{Z}}/p}(BT)\simeq BV$ where $V$ is the subgroup
of elements of order $p$, the following is also true for $p$-compact toral
groups.
###### Example 6.16.
Let $P$ be a $p$-toral group with group of components $\pi$. First of all, by
[CCS07-2, Proposition 2.1], we can assume that $\pi$ is a finite $p$-group
generated by elements of order $p$ which lift to $BP$. By Proposition 3.14 and
Remark 3.17, there is a mod $p$ equivalence
$CW_{B{\mathbb{Z}}/p}(BP)\rightarrow CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$.
Dwyer and Wilkerson show in [DW94] that there exists a discrete $p$-toral
group $P_{\infty}$ such that $BP_{\infty}\rightarrow BP$ is a mod $p$
equivalence. We are reduced then to study the cellularization of discrete
$p$-toral groups.
Following [FS07, Section 4], we consider $\Omega_{1}(P_{\infty})$, the
subgroup generated by the elements of order $p$. Since a subgroup of a
$p$-toral discrete group is also a $p$-toral discrete group and the map
$B\Omega_{1}(P_{\infty})\rightarrow BP_{\infty}$ is a
$B{\mathbb{Z}}/p$-cellular equivalence (note that ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,BP_{\infty})\simeq
Hom({\mathbb{Z}}/p,P_{\infty})$), we can assume that $P_{\infty}$ is generated
by elements of order $p$. For any $p$-discrete toral group there is an
increasing sequence $P_{0}\leq P_{1}\leq\cdots$ such that $P_{\infty}=\cup
P_{n}$. Take a countable set of generators of order $p$ for $P_{\infty}$,
$\\{g_{i}|i=1,\ldots,n\\}$; then the subgroups $Q_{n}=\langle
g_{1},\ldots,g_{n}\rangle$ satisfy that $P_{\infty}=\cup Q_{n}$ and each
$Q_{n}$ is a finite $p$-group generated by elements of order $p$, so by
[Flo07, Prop 4.14, Prop 4.8], $BQ_{n}$ is $B{\mathbb{Z}}/p$-cellular and
therefore $BP_{\infty}$ is so.
Finally the space $B\Omega_{1}(P_{\infty})$ is $B{\mathbb{Z}}/p$-cellular, so
it remains to check that $B\Omega_{1}(P_{\infty})\rightarrow
CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is a mod $p$ equivalence. Let
$C_{BP_{\infty}}$ and $C_{BP^{\wedge}_{p}}$ be the corresponding Chach-ólski’s
cofibres. Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the fibration
$B\Omega_{1}(P_{\infty})\rightarrow BP_{\infty}\rightarrow P_{\Sigma
B{\mathbb{Z}}/p}(C_{BP_{\infty}})$ and the map $BP_{\infty}\rightarrow
B(P_{\infty}/\Omega_{1}(P_{\infty}))$ shows that there is a homotopy
equivalence $P_{\Sigma B{\mathbb{Z}}/p}(C_{BP_{\infty}})\simeq
B(P_{\infty}/\Omega_{1}(P_{\infty}))$. In particular, $C_{BP\infty}$ satisfies
the hypothesis of Lemma 3.9. Moreover, $C_{BP^{\wedge}_{p}}$ is $1$-connected.
The map $g\colon C_{BP\infty}\rightarrow C_{BP^{\wedge}_{p}}$ is a mod $p$
equivalence and, by Remark 3.13 and Corollary 3.12, $P_{\Sigma
B{\mathbb{Z}}/p}(g)$ is a mod $p$ equivalence. Finally, Proposition 3.14
combined with the previous results, show that
$B\Omega_{1}(P_{\infty})\rightarrow CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is
a mod $p$ equivalence.
In particular, from Example 6.11 we obtain that there are mod $2$ equivalences
$BD_{2^{\infty}}\rightarrow CW_{B{\mathbb{Z}}/2}(BO(2))\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$, and hence a chain of homotopy
equivalences $CW_{B{\mathbb{Z}}/2}(BO(2))^{\wedge}_{2}\simeq
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})^{\wedge}_{2}\simeq
BO(2)^{\wedge}_{2}$.
We finish the section with a last example in which we can observe a completely
different pattern, and where the cellularization is obtained by combining in
an adequate way some nice push-out decompositions.
###### Proposition 6.17.
The $\emph{B}{\mathbb{Z}}/2$-cellularization of ${B}SO(3)$ fits in a fibration
$({CW}_{\emph{B}{\mathbb{Z}}/2}{B}SO(3))^{\wedge}_{2}\rightarrow{B}SO(3)^{\wedge}_{2}\rightarrow({B}SO(3)^{\wedge}_{2})_{\mathbb{Q}}.$
###### Proof.
Since $SO(3)$ is connected, by Lemma 3.2 the $p$-completion induces a homotopy
equivalence
${CW}_{B{\mathbb{Z}}/p}BSO(3)\simeq{CW}_{B{\mathbb{Z}}/p}(BSO(3)^{\wedge}_{p})$.
According to [DMW87, Cor 4.2], $BSO(3)$ is equivalent at the prime 2 to the
pushout $X$ of the following diagram:
$\textstyle{BD_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\textstyle{BO(2)^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(B\Sigma_{4})^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X,}$
where $f_{1}$ is induced by inclusion of the 2-Sylow subgroup, and $f_{2}$ is
given by the map of extensions
$\textstyle{{\mathbb{Z}}/4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{{\mathbb{Z}}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{SO(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{O(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}/2.}$
Our strategy will be to cellularize the previous diagram, and compare the
respective pushouts. Now, recall that $BD_{8}$ is $B{\mathbb{Z}}/2$-cellular
([Flo07, 4.14]) and moreover the 2-completion of $B\Sigma_{4}$ is so ([FS07,
Thm 4.4]). On the other hand, $D_{2^{\infty}}$ is a $2$-discrete approximation
of $O(2)$ -i.e.$BD_{2^{\infty}}\rightarrow BO(2)$ is a mod $2$ equivalence-,
so the previous Example 6.16 implies $BD_{2^{\infty}}\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2))$ is a mod $2$ equivalence. Moreover, by
Proposition 3.14 and Remark 3.17, there is also a mod $2$ equivalence
$CW_{B{\mathbb{Z}}/2}(BO(2))\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$. So, we can consider another pushout
diagram by applying the functor $CW_{B{\mathbb{Z}}/2}$ to the previous one,
$\textstyle{BD_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(B\Sigma_{4})^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$
There exists a map $g\colon Y\rightarrow X$ induced by the augmentation map
from the cellularization which is a mod $2$ equivalence since
$CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})\rightarrow BO(2)^{\wedge}_{2}$ is
so.
Now we attempt to compute the $B{\mathbb{Z}}/2$-cellularization of
$X^{\wedge}_{2}$ by using the cofibre of the map $k\colon Y\rightarrow
X^{\wedge}_{2}$. In order to do this, a result of Chachólski [Cha96, Thm 20.3]
together with [FS07, Thm 1.1] tells us that we need to check that
$[B{\mathbb{Z}}/2,Y]\rightarrow[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]\cong[B{\mathbb{Z}}/2,X^{\wedge}_{2}]$
is exhaustive and $Y$ is $B{\mathbb{Z}}/2$-cellular. $Y$ is
$B{\mathbb{Z}}/2$-cellular since it is a pushout of $B{\mathbb{Z}}/2$-cellular
spaces. It remains to check that
$[B{\mathbb{Z}}/2,Y]\rightarrow[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]$ is
exhaustive.
Let $\mathcal{P}$ be the category $1\leftarrow 0\rightarrow 2$ describing a
pushout diagram, and let $F\colon\mathcal{P}\rightarrow Top$ be the functor
describing the pushout for $Y$, that is, $F(1)=(B\Sigma_{4})^{\wedge}_{2}$,
$F(0)=BD_{8}$ and $F(2)=CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$ with the
corresponding morphisms. There is a commutative diagram of sets
$\textstyle{\operatornamewithlimits{\hbox{$\varinjlim$}}[B{\mathbb{Z}}/2,F]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{F})_{*}}$$\textstyle{[B{\mathbb{Z}}/2,Y]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{*}}$$\textstyle{\operatornamewithlimits{\hbox{$\varinjlim$}}[B{\mathbb{Z}}/2,F^{\wedge}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]}$
where the vertical maps are induced by $2$-completion of the target. Since the
spaces ${\rm map\,}(B{\mathbb{Z}}/2,F^{\wedge}_{2})$ are $2$-complete (see
[BK02, Proposition 7.5]), by [BLO03, Lemma 4.2] the bottom horizontal map is a
bijection. To prove that $\eta_{*}$ is exhaustive, it is enough to show that
$(\eta_{F})_{*}$ is so. But then, looking at the diagram, it reduces to check
that
$[B{\mathbb{Z}}/2,CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})]\rightarrow[B{\mathbb{Z}}/2,CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})^{\wedge}_{2}]\cong[B{\mathbb{Z}}/2,BO(2)^{\wedge}_{2}]$
is exhaustive (see Example 6.16) and this follows from Corollary 3.7.
Let $C$ be the cofibre of $k$. We know that $C$ is mod $2$ acyclic and
$1$-connected. Now if $q$ is an odd prime, since $Y$ is
$B{\mathbb{Z}}/2$-cellular, $Y$ is mod $q$ acyclic and
$C^{\wedge}_{q}\simeq(BSO(3)^{\wedge}_{2})^{\wedge}_{q}$ is contractible.
Finally $C_{\mathbb{Q}}\simeq(BSO(3)^{\wedge}_{2})_{\mathbb{Q}}$. Then, by a
Sullivan arithmetic square argument,
$C\simeq(BSO(3)^{\wedge}_{2})_{\mathbb{Q}}$ which is, in turn,
$B{\mathbb{Z}}/2$-null. In particular $C$ is $\Sigma B{\mathbb{Z}}/2$-null.
Therefore, the fibration of the theorem follows from Chachólski’s fibration
describing the cellularization (Theorem 2.3). ∎
###### Remark 6.18.
Note that if $p$ is an odd prime, then $BSO(3)^{\wedge}_{p}\simeq
BN(T)^{\wedge}_{p}$, where $N(T)$ is the normalizer of the maximal torus, and
we analyzed this case in Example 6.15.
It seems natural to ask if the problem of computing $CW_{B{\mathbb{Z}}/p}(BG)$
for any compact Lie group $G$ is accessible at this point. A strategy was
developed for finite groups in [FF], based in the description of the strongly
closed subgroups of $G$, which are classified. Recent research has remarked
the role of the strongly closed subgroups of discrete $p$-toral groups in the
homotopy theory of compact Lie groups and, more generally, $p$-local compact
groups [Gon10], but to our knowledge there is no available classification of
these objects. On the other hand, the nontrivial rational homotopy of $BG$
seems an important obstacle to generalize the arithmetic square arguments of
the strategy. We plan to undertake these issues in subsequent work, and, in
particular, an intriguing question which arises in a natural way from the last
example:
Question: For which class of classifying spaces of compact Lie groups (or
spaces in general) is the $B{\mathbb{Z}}/p$-cellularization equivalent to the
homotopy fibre of the rationalization, up to $p$-completion?
Acknowledgements. We would like to thank Carles Broto and Jérôme Scherer for
interesting conversations on this subject.
## References
* [1]
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* [BK02] Broto, C., Kitchloo, N.: Classifying spaces of Kač-Moody groups. Math. Z. 240, 621–649 (2002)
* [BLO03] Broto, C., Levi, R., Oliver, R.: The homotopy theory of fusion systems. J. Amer. Math. Soc. 16, 779–856 (2003)
* [BLO07] Broto, C., Levi, R., Oliver, R.: Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups. Geom. Topol. 11, 315–427 (2007)
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* [Bou97] Bousfield, A. K.: Homotopical localizations of spaces. Amer. J. Math. 119, 1321–1354 (1997)
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* [CLN07] Castellana, N., Levi, R., Notbohm, D.: Homology decompositions for $p$-compact groups. Adv. Math. 216, 491–432 (2007).
* [Cha96] Chachólski, W.: On the functors $CW_{A}$ and $P_{A}$. Duke Math. J. 84, 599–631 (1996)
* [DMW87] Dwyer, W. G., Miller, H. R., Wilkerson, C. W.: The homotopic uniqueness of $BS^{3}$. In: Algebraic topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, Berlin, 90–105 (1987)
* [DW94] Dwyer, W. G., Wilkerson, C. W.: Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. of Math. (2) 139, 395–442 (1994)
* [Dwy96] Dwyer, W. G.: The centralizer decomposition of $BG$. In: Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994), Progr. Math. 136, Birkhäuser, Basel, 167–184 (1996)
* [DZ87] Dwyer, W., Zabrodsky, A.: Maps between classifying spaces. In: Algebraic topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, Berlin, 106–119 (1987)
* [Far96] Farjoun, E.: Cellular spaces, null spaces and homotopy localization. Lecture Notes in Math. 1622, Springer, Berlin (1996)
* [FF] Flores, R., Foote, R. M.: The cellular structure of the classifying spaces of finite groups. To appear in Israel J. Math.
* [FGS07] Farjoun, E., Göbel, R., Segev, Yoav.: Cellular covers of groups. J. Pure Appl. Algebra 208, 61–76 (2007).
* [Flo07] Flores, R.: Nullification and cellularization of classifying spaces of finite groups. Trans. Amer. Math. Soc. 359, 1791–1816 (2007)
* [FS07] Flores, R., Scherer, J.: Cellularization of classifying spaces and fusion properties of finite groups. J. Lond. Math. Soc. (2) 76, 41–56 (2007)
* [Gon10] González, A.: The structure of $p$-local compact groups. PhD thesis (2010)
* [JMO90] Jackowski, S., McClure, J. E., Oliver, R.: Self-maps of classifying spaces of compact simple Lie groups. Bull. Amer. Math. Soc. (N.S.) 22, 65–72 (1990)
* [Lev95] Levi, R.: On finite groups and homotopy theory. Mem. Amer. Math. Soc., 118 (1995)
* [McG] McGibbon, C. A.: Infinite loop spaces and Neisendorfer localization. Proc. Amer. Math. Soc. (1), 125, 309–313 (1995)
* [Mil84] Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120, 39–87 (1984)
* [Mis92] Mislin, G.: Cohomologically central elements and fusion in groups. Algebraic topology (San Feliu de Guíxols, 1990), Lecture Notes in Math. 1509, 294–300, Springer-Berlin, 1992.
* [Nei95] Neisendorfer, J. A.: Localization and connected covers of finite complexes. In: The Čech centennial (Boston, MA, 1993), Contemp. Math. 181, Amer. Math. Soc., Providence, RI, 385–390 (1995)
* [RS00] Rodríguez, J. L., Scevenels, D.: Iterating series of localization functors. In: Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Contemp. Math. 265, Amer. Math. Soc., Providence, RI, 211–221 (2000)
* [RS01] Rodríguez, J. L., Scherer, J.: Cellular approximations using Moore spaces. In: Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math. 196, Amer. Math. Soc., Basel, 357–374 (2001)
* [Spa66] Spanier, E. H.: Algebraic topology. Springer-Verlag, New York (1966)
|
arxiv-papers
| 2012-10-29T18:49:58 |
2024-09-04T02:49:37.297418
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nat\\`alia Castellana and Ram\\'on Flores",
"submitter": "Ram\\'on Flores",
"url": "https://arxiv.org/abs/1210.7772"
}
|
1210.7909
|
0em plus .3em minus .15em
0em plus .3em minus .15em
0em plus .2em minus .1em
0em plus .2em minus .1em
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arxiv-papers
| 2012-10-30T06:11:56 |
2024-09-04T02:49:37.315162
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ricardo Andrade",
"submitter": "Ricardo Andrade",
"url": "https://arxiv.org/abs/1210.7909"
}
|
1210.7925
|
Irregular conformal block and its matrix model
Chaiho Rim***email: rimpine@sogang.ac.kr
Department of Physics and Center for Quantum Spacetime (CQUeST)
Sogang University, Seoul 121-742, Korea
###### Abstract
Irregular conformal block is a new tool to study Argyres-Douglas theory, whose
irregular vector is represented as a simultaneous eigenstate of a set of
positive Virasoro generators. One way to find the irregular conformal block is
to use the partition function of the $\beta$-ensemble of hermitian matrix
model. So far the method is limited to the case of irregular singularity of
even degree. In this letter, we present a new matrix model for the case of odd
degree and calculate its partition function. The model is different from the
previous one in that its potential has additional factor of square root of
matrix.
###### Contents
1. 1 Introduction
2. 2 Partition function and loop equation
3. 3 Solution to the lowest order in $\hbar$
4. 4 Discussion and comments
5. A Explicit form of $f(z)$
## 1 Introduction
Irregular conformal block (ICB) is obtained from the conformal block by taking
the colliding limit which puts some of primary fields at the same point
maintaining its limit finite [1]. ICB generalizes the AGT relation [2] and is
considered to reproduce [1, 3] the Argyres-Douglas (AD) theory [4].
ICB is first obtained as the eigenstate of Virasoro generator $L_{1}$ or
$(L_{1},L_{2})$ in [5]. The eigenstate is not annihilated by $L_{k}$ with
$k>0$ but is an eigenstate. This coherent state is called irregular vector.
This idea is generalized so that the irregular vector is the simultaneous
eigenstate of $L_{k}$’s ($0<n\leq k\leq 2n$) and is constructed in terms of
the free field representation in the presence of background charge in [1]. In
fact, this ICB is described by the Seiberg-Witten curve $x^{2}=\phi_{2}$ where
$\phi_{2}$ is a quadratic differential with a pole of degree $2n+2$, which is
associated to a punctured Riemann sphere and is classified as $D_{2n}$ of AD
theory [7].
On the other hand, it is noted in [6] that ICB is conveniently obtained from
the hermitian matrix model whose potential consists of inverse powers of
matrix and logarithmic one. Its partition function is identified with the
inner product of an irregular vector at the origin and a regular one at
infinity. Because this matrix model can be obtained as the appropriate limit
of the Penner-type matrix model [8, 9, 10], this matrix model approach shares
the same idea as that of the colliding limit of the regular conformal block.
The matrix model is, however, so far limited to the case of even degree. The
presence of the pole of degree $2n+1$ requires the irregular state annihilated
by $L_{2n}$ but simultaneous eigenstates of $L_{k}$ with $0<n\leq k\leq 2n-1$.
This ICB may be constructed in terms of the twisted free field modes which
introduce a new square root branch-cut at the origin as shown in [1].
In this letter, we present a new matrix model with square root potential and
calculate the corresponding partition function. In section 2, we define the
matrix model and obtain the loop equation. The square root potential not only
introduces the branch-cut at the origin but also changes the details of the
loop equation. In section 3, we provide the solution to the simplest non-
trivial case namely, the case $n=2$ (or $D_{3}$) explicitly. It is
straightforward to generalize to the case with $n>2$. Section 4 is for
discussion and a detailed derivation of the loop equation is provided in
appendix A.
## 2 Partition function and loop equation
We consider the partition function of a hermitian matrix
$Z_{M}=\int\left[\prod_{I=1}^{N}dz_{I}\right]\Delta(\lambda_{I})^{2\beta}e^{\frac{\sqrt{\beta}}{g}\sum_{I}V(\lambda_{I})}$
(2.1)
where $\Delta(\lambda_{I})$ is the Vandermonde determinant. $\lambda_{I}$ is
the eigenvalue of the matrix and $N$ is the size of the matrix. The potential
has the form
$\displaystyle V(z)=\alpha\log z-\sum_{s\in S}\frac{c_{s}}{sz^{s}}$ (2.2)
where $S=\\{1/2,3/2,\cdots,n-1/2\\}$ is a finite set of $n$ half-odd integers.
It is convenient to denote the potential as $V(z)=V_{e}(z)+V_{o}(z)$ where
$V_{e}(z)=\alpha\log z$ and $V_{o}(z)=-\sum_{s\in S}\frac{c_{s}}{sz^{s}}$ so
that $V_{o}(z)$ has a cut $\Gamma$ along the negative real axis. We assume
that there is an appropriate domain of the parameter space
$\\{\alpha,c_{S}\\}$ so that the partition function is well defined when the
integration range lies along the positive real axis.
The saddle point equation holds on the positive real axis
$2w(\lambda_{I})+V^{\prime}(\lambda_{I})=0\,,~{}~{}~{}~{}w(\lambda_{I})=g\sqrt{\beta}\sum_{J\neq
I}\left(\frac{1}{\lambda_{I}-\lambda_{J}}\right)\,.$ (2.3)
A nice way to find the functional relation is to introduce the resolvent [11],
$W(z)=g\sqrt{\beta}\left\langle\sum_{J}\frac{1}{z-\lambda_{J}}\right\rangle$.
Here $\langle\cdots\rangle$ stands for the expectation value with the given
potential $V(z)$. The saddle point equation (2.3) shows that the resolvent is
discontinuous along a certain integration range $\Lambda$ of the partition
function
$W(\xi+i0)+W(\xi-i0)+V^{\prime}(\xi)=0~{}~{}~{}{\rm when}~{}\xi\in\Lambda\,.$
(2.4)
This discontinuity is encoded in terms of $G(z)\equiv~{}2W(z)+V^{\prime}(z)$
so that $G(\xi-i\epsilon)+G(\xi+i\epsilon)=0$ is automatically satisfied on
the square-root branch cut. Furthermore, $G(z)$ is not continuous on the
branch cut $\Gamma$ and consists of the monodromy even and odd parts.
The resolvent satisfies the loop equation of the form
$f(z)=4W(z)^{2}+4V^{\prime}(z)W(z)+2\hbar QW^{\prime}(z)-\hbar^{2}W(z,z)\,.$
(2.5)
where we switch the notation from $g$ and $\beta$ to $\sqrt{\beta}=-ib$,
$\hbar=-2ig$ and$Q=b+1/b$.
$W(z_{1},z_{2})=\beta\left\langle\sum_{I_{1},I_{2}}\frac{1}{z_{1}-\lambda_{I_{1}}}\frac{1}{z_{2}-\lambda_{I_{2}}}\right\rangle_{\\!\\!\rm
conn}$ is a two-point (connected) resolvent and $f(z)$ is the quantum
correction defined as $f(z)=2\hbar
b\sum_{I=1}^{N}\left\langle\frac{V^{\prime}(z)-V^{\prime}(\lambda_{I})}{z-\lambda_{I}}\right\rangle$.
As shown in appendix A, its explicit form is
$\displaystyle f(z)$
$\displaystyle=\sum_{k=0}^{n-1}\frac{d_{k}}{z^{2+k}}+4\frac{V_{o}^{\prime}(z)}{\sqrt{z}}~{}\tau(z)$
(2.6)
where $d_{k}=v_{k}(-\hbar^{2}\log Z_{M})$ and $v_{k}=\sum_{s\in
S}sc_{s+k}\frac{\partial}{\partial c_{s}}$. Here we use the definition
$c_{s}=0$ if $s$ does not belong to the set $S$. $\tau(z)=\frac{\hbar
b}{2}\sum_{I}\left\langle\frac{1}{\sqrt{z}+\sqrt{\lambda_{I}}}\right\rangle$
is a new analytic function, continuous on $\Lambda$ but discontinuous on
$\Gamma$.
## 3 Solution to the lowest order in $\hbar$
The coefficient $d_{k}$ in (2.6) is the key element to find the partition
function. Once $d_{k}$’s are known, one can find the partition function using
the differential equation with respect to the parameters of the potential. Our
strategy is to find $d_{k}$’s from the loop equation order by oder in $\hbar$.
To make the expansion feasible, we assume that $V^{\prime}(z)$ is the order of
$\hbar$ so that the expansion is equivalent to the large $N$ expansion
($\hbar\propto 1/N$). This is achieved if $\alpha$ and $c_{S}$ are
proportional to $\hbar$.
To the lowest order in $\hbar$ (planar limit), $G(z)$ satisfies the loop
equation (2.5)
$G(z)^{2}-2V_{o}^{\prime}(z)H(z)=\varphi_{2}(z)$ (3.1)
where $H(z)=2{\tau(z)}/{\sqrt{z}}+V_{e}^{\prime}(z)$ and
$\varphi_{2}(z)=\sum_{k=0}^{n-1}\frac{d_{k}}{z^{2+k}}+V_{e}^{\prime}(z)^{2}+V_{o}^{\prime}(z)^{2}$
is the expectation value of the energy-momentum tensor which has the pole of
odd degree $2n+1$. At large $z$, the dominant term of (3.1) is the order of
$1/z^{2}$ and its coefficient provides an identity
$(\hbar bN+\alpha)^{2}=d_{0}+\alpha^{2}\,.$ (3.2)
For the simplest case, $n=1$,
$d_{0}=\frac{1}{2}c_{1/2}\frac{\partial}{\partial c_{1/2}}(-\hbar^{2}\log
Z_{M})$ determines the partition function,
$Z_{M}=\zeta~{}(c_{1/2})^{-2\left((bN)^{2}+2bN\alpha/\hbar\right)}$. $\zeta$
is a $c_{1/2}$-independent constant and can be absorbed to the definition of
the partition function. Furthermore, one has $G(z)=\frac{\hbar
bN+\alpha}{z}+\frac{c_{1}/2}{z^{3/2}}$ and $H(z)=\frac{\hbar bN+\alpha}{z}$
which shows no no branch cut $\Lambda$. In fact, $\Lambda$ reduces to the
origin $z=0$ as seen from the $n=2$ case below (if $c_{3/2}$ is turned off).
When $n\geq 2$, the solution is more involved. It is natural to assume there
are $n-1$ cuts $\Lambda_{k}$’s each of which lies between $a_{k}$ and $b_{k}$
($0<a_{k}<b_{k}$) on the positive real axis. In addition, according to the
monodromy at $z=0$, one may put $G(z)=G_{e}(z)+G_{o}(z)$;
$G_{e}(z)=\frac{g_{e}(z)}{z^{n}}{\prod_{k}^{n-1}\sqrt{(z-a_{k})(z-b_{k})}}\,,~{}~{}~{}G_{o}(z)=\frac{g_{o}(z)}{z^{n+1/2}}\prod_{k}^{n-1}\sqrt{(z-a_{k})(z-b_{b})}\,.$
(3.3)
$g_{e}(z)$ and $g_{o}(z)$ are holomorphic functions whose large $z$ behavior
is $g_{e}(z)=O(z^{0})$ and $g_{o}(z)=O(z^{0})$. The loop equation (3.1) is
split into two parts
$\displaystyle
G_{e}(z)^{2}+G_{o}(z)^{2}=\varphi_{2}(z)\,,~{}~{}G_{e}(z)G_{o}(z)=V_{o}^{\prime}(z)H(z)\,.$
(3.4)
The way of splitting is consistent with the fact that $H(z)$ has no cut along
$\Lambda$.
For $n=2$, the non-trivial simplest case, the partition function satisfies the
differential equations $d_{0}=v_{0}(-\hbar^{2}\log Z_{M})$ and
$d_{1}=v_{1}(-\hbar^{2}\log Z_{M})$. It is convenient to re-parametrize as
$t=c_{3/2}/c_{1/2}^{3}$ and $x=c_{3/2}$ and regard the partition function as
the function of $t$ and $x$. The merit of parametrization is that any function
of $t$ is the homogeneous solution of $v_{0}$ since $v_{0}(t)=0$. Employing
the fact that $v_{0}(x)=3x/2$, $v_{1}(x)=0$ and
$v_{1}(t)=-(3/2)t^{2}c_{1/2}^{2}$, one has the differential equations
$d_{0}=-\frac{3\hbar^{2}}{2}x\frac{\partial}{\partial x}(\log
Z_{M})\,,~{}~{}~{}~{}\delta_{1}\equiv
d_{1}/c_{1/2}^{2}=\frac{3\hbar^{2}}{2}t^{2}\frac{\partial}{\partial t}(\log
Z_{M})\,.$ (3.5)
With $d_{0}$ in (3.2), we put $\log Z_{M}$ as
$\log Z_{M}=\frac{2}{3\hbar^{2}}\Big{(}-d_{0}\log x+Y(t)\Big{)}$ (3.6)
and $Y(t)$ satisfies the differential equation,
$t^{2}{dY(t)}/{dt}=\delta_{1}$. This implies that $\delta_{1}$ is the function
of $t$ only and vanishes as $t\to 0$.
It is noted that $\varphi_{2}(z)$ in (3.4) is cast into the form
$\varphi_{2}(z)={P_{3}(z)}/{z^{5}}$ where
$\displaystyle
P_{3}(z)=(d_{0}+\alpha^{2})z^{3}+(d_{1}+c_{1/2}^{2})z^{2}+2c_{3/2}c_{1/2}z+c_{3/2}^{2}\,.$
(3.7)
Rescaling $z=uc_{1/2}^{2}$ and ${P_{3}(z)}=c_{1/2}^{6}Q_{3}(u)$, one puts
$Q_{3}(u)$ in terms of $t$ and $x$,
$\displaystyle Q_{3}(u)$
$\displaystyle=(d_{0}+\alpha^{2})u^{3}+(\delta_{1}+1)u^{2}+2tu+t^{2}$
$\displaystyle\equiv(d_{0}+\alpha^{2})(u-A)(u-B)(u+\gamma^{2})\,.$ (3.8)
$Q(u)=0$ has two positive roots $A$ and $B$ and one negative root
$-\gamma^{2}$. Two positive roots are related with branch points
$a=Ac_{1/2}^{2}$ and $b=Bc_{1/2}^{2}\,.$ The negative root is related with
holomorphic function $g_{e}(z)$ and $g_{o}(z)$ through the loop equation (3.4)
$g_{e}(z)^{2}u+{g_{o}(z)^{2}}/{c_{1/2}^{2}}=(d_{0}+\alpha^{2})(u+\gamma^{2})\,.$
(3.9)
Noting that $g_{e}(z)^{2}$ and $g_{o}(z)^{2}$ are even degree of $u$, we
conclude that they are constant and $g_{e}(z)=\sqrt{d_{0}+\alpha^{2}}$ and
$g_{o}(z)=c_{1/2}{\gamma}\sqrt{d_{0}+\alpha^{2}}$.
The relation between three roots are obtained from (3.8)
$\displaystyle
t^{2}=(d_{0}+\alpha^{2})\gamma^{2}AB\,,~{}~{}~{}2t=(d_{0}+\alpha^{2})(AB-(A+B)\gamma^{2})$
$\displaystyle\delta_{1}+1=(d_{0}+\alpha^{2})(\gamma^{2}-(A+B))\,.$ (3.10)
In addition, $H(z)$ in (3.4) is the order of $1/z$ whose coefficient provides
an additional identity; $\hbar bN+\alpha=\gamma(d_{0}+\alpha^{2})$ or
$\gamma(\hbar bN+\alpha)=1$. This together with (3.10) solves the algebraic
equations
$\displaystyle AB$
$\displaystyle=t^{2}\,,~{}~{}A+B=-2t+t^{2}(d_{0}+\alpha^{2})$
$\displaystyle\delta_{1}$
$\displaystyle=2(d_{0}+\alpha^{2})t-(d_{0}+\alpha^{2})^{2}t^{2}\,.$ (3.11)
Therefore, $Y(t)$ in (3.6) is determined to the lowest order in $\hbar$ to
give
$\displaystyle\log(Z_{M}/\zeta)=\frac{2}{3\hbar^{2}}\Big{(}-d_{0}\log
x+2(d_{0}+\alpha^{2})\log t-(d_{0}+\alpha^{2})^{2}t\Big{)}$ $\displaystyle
Z_{M}/\zeta=(c_{3/2})^{2((bN+\hat{\alpha})^{2}+\hat{\alpha}^{2})/3}\left(c_{1/2}\right)^{-4(bN+\hat{\alpha})^{2}}\exp\left(-\frac{2c_{3/2}(bN+\hat{\alpha}^{2})^{4}\hbar^{2}}{3c_{1/2}^{3}}\right)$
(3.12)
where $\hat{\alpha}=\alpha/\hbar$.
## 4 Discussion and comments
We presented how to find the partition function of matrix model which
corresponds to the irregular conformal block in the presence of the irregular
singularity of odd degree. Explicit solution is obtained for $n=1$ and $n=2$
cases.
It is obvious to generalize to $n>2$. One may have
$\varphi_{2}(z)=P_{2n-1}(z)/z^{2n+1}$ in (3.4) where $P_{2n-1}(z)$ is a
polynomial of degree $(2n-1)$. The $2(n-1)$ positive zeros of $P_{2n-1}(z)$
will provide the $2(n-1)$ branch points. The remaining factor linear in $z$
will fix the holomorphic functions $g_{e}(z)$ and $g_{o}(z)$. $d_{0}$ is fixed
as in (3.2) and other $d_{k}$’s are determined from the filling fractions,
$\oint_{\Lambda_{k}}\frac{dz}{\pi i}~{}W(z)=\hbar bN_{k}$ with $\sum N_{k}=N$.
It is noted that the contour surrounding all $\Lambda_{k}$’s is trivial.
Instead, one additional constraint comes from the $1/z$ behavior of $H(z)$ in
(3.4).
One may also consider more complicated potential which may contain polynomials
together with inverse powers and a logarithmic one. Its partition function can
be regarded as the inner product of the irregular states. Details of this
consideration and the calculation of higher orders in $\hbar$ will appear
elsewhere.
Finally, it is noted that the matrix model we presented in this letter reduces
to the $O(n)$ matrix model on the random surface with $n=-2$ when $\beta=1$
[12]. It will be interesting to understand the results in terms of $O(n)$
matrix model.
### Acknowledgments
The author thanks Ivan Kostov and Takahiro Nishinaka for useful discussion.
This work is supported in part by the National Research Foundation (NRF) of
Korea funded by the Korea government (MEST) 2005-0049409.
## Appendix A Explicit form of $f(z)$
We provide the explicit form of $f(z)$ presented in (2.6). Using the potential
(2.2) one has
$\displaystyle f(z)$
$\displaystyle=4g\sqrt{\beta}\sum_{I=1}^{N}\left\langle\frac{V^{\prime}(z)-V^{\prime}(\lambda_{I})}{z-\lambda_{I}}\right\rangle$
$\displaystyle=4g\sqrt{\beta}\sum_{I=1}^{N}\left\langle\frac{\alpha(1/z-1/\lambda_{I})+\sum_{s\in
S}{c_{s}}(1/z^{s+1}-1/\lambda_{I}^{s+1})}{z-\lambda_{I}}\right\rangle$ (A.1)
where $S=\\{1/2,3/2,\cdots,n-1/2\\}$. Let us eliminate the $\alpha$ term by
using the identity
$0=\left\langle\sum_{I=1}^{N}V^{\prime}(\lambda_{I})\right\rangle=\sum_{I=1}^{N}\left\langle\frac{\alpha}{\lambda_{I}}+\sum_{s}\frac{c_{s}}{\lambda_{I}^{s+1}}\right\rangle$
(A.2)
which stands for the invariance of the shift of the integration. One may
wonder if the identity may fail since in our case, the integration is along
the positive real line. However, this is not the case thanks to the
Vandermonde determinant because the infinitesimal shift $\epsilon$ around the
origin contributes to the order of $\epsilon^{N(N-1)/2}$ which does not
contribute as $N\gg 1$.
$\displaystyle f(z)$
$\displaystyle=-4g\sqrt{\beta}\sum_{I,s}\left\langle~{}\frac{c_{s}(z^{s}-\lambda_{I}^{s})}{z^{s+1}\lambda_{I}^{s}(z-\lambda_{I})}\right\rangle$
$\displaystyle=-\frac{4g\sqrt{\beta}}{z^{3/2}}\sum_{I,s}\left\langle\frac{c_{s}}{\lambda_{I}^{s}}\left(\sum_{m=even\geq
0}^{2s-1}+\sum_{m=odd\geq
1}^{2s-2}\right)\frac{(\lambda_{I}/z)^{m/2}}{(\sqrt{z}+\sqrt{\lambda_{I}})}\right\rangle\,.$
(A.3)
Inside the sum in even $m$, one may add and subtract $1/\sqrt{z}$ to
$1/(\sqrt{z}+\sqrt{\lambda_{I}})$ to rewrite
$\displaystyle f(z)$
$\displaystyle=-\frac{4g\sqrt{\beta}}{z^{2}}\sum_{I,s}\left\langle\sum_{m=even}\frac{c_{s}}{\lambda_{I}^{s-m/2}z^{m/2}}\right\rangle$
$\displaystyle~{}~{}~{}+\frac{4g\sqrt{\beta}}{z^{3/2}}\sum_{I,s}\left\langle\frac{c_{s}}{\lambda_{I}^{s}(\sqrt{z}+\sqrt{\lambda_{I}})}\left(\sum_{m=even}\frac{\lambda_{I}^{(m+1)/2}}{z^{(m+1)/2}}-\sum_{m=odd}\frac{\lambda_{I}^{m/2}}{z^{m/2}}\right)\right\rangle$
$\displaystyle=-\frac{4g\sqrt{\beta}}{z^{2}}\sum_{I,s}\left\langle\sum_{k=integer\geq
0}^{s-1/2}\frac{1}{\lambda_{I}^{s-k}z^{k}}\right\rangle+\frac{4g\sqrt{\beta}}{z^{3/2}}\sum_{I,s}\left\langle\frac{c_{s}}{z^{s}(\sqrt{z}+\sqrt{\lambda_{I}})}\right\rangle$
(A.4)
Re-arranging the order of the summation over $s$ and $k$, one has
$\displaystyle f(z)$ $\displaystyle=4g^{2}\sum_{k=0}^{n-1}\frac{v_{k}(\log
Z_{M})}{z^{2+k}}+4g\sqrt{\beta}\left\langle\sum_{I}\frac{1}{\sqrt{z}+\sqrt{\lambda_{I}}}\right\rangle~{}\frac{V_{o}^{\prime}(z)}{\sqrt{z}}$
(A.5)
where $v_{k}=\sum_{s}sc_{s+k}\frac{\partial}{\partial c_{s}}$ with the
definition $c_{s}=0$ if $s$ does not belong to $S$.
## References
* [1] D. Gaiotto and J. Teschner, “Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I”, arXiv:1203.1052 [hep-th].
* [2] L. Alday, D. Gaiotto, Y.Tachikawa , “Liouville Correlation Functions from Four-dimensional Gauge Theories”, Lett. Math. Phys.91:167-197 (2010).
* [3] G. Bonelli, K. Maruyoshi and A. Tanzini, “Wild Quiver Gauge Theories,” JHEP 1202 (2012) 031.
* [4] Philip C. Argyres and Michael R. Douglas, “New Phenomena in SU(3) Supersymmetric Gauge Theory”, Nucl. Phys. B448 (1995) 93.
* [5] D. Gaiotto, “Asymptotically free N=2 theories and irregular conformal blocks”, arXiv:0908.0307 [hep-th].
* [6] T. Nishinaka and C. Rim, “Matrix models for irregular conformal blocks and Argyres-Douglas theories”, JHEP 10 (2012) 138.
* [7] S. Cecotti and C. Vafa, “Classification of complete N=2 supersymmetric theories in 4 dimensions,” arXiv:1103.5832 [hep-th].
* [8] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems,” arXiv:0909.2453 [hep-th].
* [9] T. Eguchi and K. Maruyoshi, “Penner Type Matrix Model and Seiberg-Witten Theory,” JHEP 1002 (2010) 022.
* [10] T. Nishinaka and C. Rim, “$\beta$-Deformed Matrix Model and Nekrasov Partition Function,” JHEP 1202 (2012) 114.
* [11] See e. g. P. Ginsparg and G. Moore, ” Lectures on 2D gravity and 2D string theory”, arXiv:hep-th/9304011 and F. Di Francesco, P. Ginsparg and Z. Zinn-Justin, arXiv:hep-th/9306153.
* [12] B. Eynard and J. Zinn-Justin, ”The O(n) model on a random surface: critical points and large-order behaviour”, Nucl. Phys. B386 (1992) 558.
|
arxiv-papers
| 2012-10-30T08:24:07 |
2024-09-04T02:49:37.321539
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chaiho Rim",
"submitter": "Chaiho Rim Prof",
"url": "https://arxiv.org/abs/1210.7925"
}
|
1210.7975
|
# Warm inflection
Rafael Cerezo cerezo@ugr.es Departamento de Física Teórica y del Cosmos and
CAFPE
Universidad de Granada, Granada-18071, Spain João G. Rosa joao.rosa@ua.pt
Departamento de Física da Universidade de Aveiro and I3N
Campus de Santiago, 3810-183 Aveiro, Portugal
###### Abstract
While ubiquitous in supersymmetric and string theory models, inflationary
scenarios near an inflection point in the scalar potential generically require
a severe fine-tuning of a priori unrelated supersymmetry breaking effects. We
show that this can be significantly alleviated by the inclusion of dissipative
effects that damp the inflaton’s motion and produce a nearly-thermal radiation
bath. We focus on the case where the slow-rolling inflaton directly excites
heavy virtual modes that then decay into light degrees of freedom, although
our main qualitative results should apply in other regimes. Furthermore, our
analysis shows that the minimum amount of dissipation required to keep the
temperature of the radiation bath above the Hubble rate during inflation is
largely independent of the degree of flatness of the potential, although it
depends on the field value at the inflection point. We then discuss the
relevance of this result to warm inflation model building.
## I Introduction
Inflation is an extremely successful paradigm in addressing the shortcomings
of standard Big Bang cosmology, providing an elegant solution to the horizon
and flatness problems by considering a period of accelerated expansion in the
early universe inflation . The simplest and most commonly considered scenario
is that of a scalar field, the inflaton $\phi$, that slowly rolls down its
potential, acting as an effective cosmological constant for a finite period.
This typically ends with the field rolling towards the minimum of its
potential and, while oscillating about it, reheating the universe through
perturbative and non-perturbative particle production. This scenario also
provides a seed for the observed temperature anisotropies in the Cosmic
Microwave Background and the Large Scale Structure of the universe, as the
accelerated expansion amplifies small quantum fluctuations of the scalar
field.
The slow-roll inflationary paradigm requires the scalar potential to have a
very flat region, where its motion is overdamped and the nearly constant
potential is the dominant source of energy density in the early universe.
While a plethora of phenomenological potentials with this property have been
constructed in the literature, the main challenge has been to embed the
inflationary dynamics within a more fundamental theory that reduces to the
Standard Model at low energies. This is important not only in establishing a
connection between inflation and low-energy particle phenomenology but also
due to the sensitivity of the inflationary dynamics to ultraviolet effects
close to the Planck scale.
This has motivated a search for inflaton candidates in supersymmetric (SUSY)
theories, in particular in the context of supergravity/string theory (see e.g.
Baumann:2009ni ), which provides the best-known candidate for a fundamental
theory of quantum gravity. These scenarios have the appealing feature of
naturally including several additional scalar fields, in particular the
superpartners of the Standard Model fermions and also the Higgs fields, as
well as a plethora of extra-dimensional moduli. Moreover, these models
generically exhibit a multitude of directions in field space along which the
scalar potential is completely flat in the supersymmetric limit and which are
uplifted by different SUSY breaking effects. For example, even the simplest
supersymmetric extension of the Standard Model, the MSSM, includes nearly 300
flat directions corresponding to gauge invariant combinations of the matter
and Higgs superfields Gherghetta:1995dv .
Flat directions can be lifted by a variety of effects, including soft terms
from SUSY breaking in a hidden/sequestered sector, renormalizable and non-
renormalizable terms in the superpotential, as well as non-perturbative
effects (e.g. gaugino condensation). In the context of string theory, these
effects are generically related to the geometry and topology of the
compactified extra-dimensions, which typically involves different fluxes
and/or D-brane configurations. All these different effects may a priori yield
both attractive and repulsive contributions to the scalar potential, which may
conspire to produce an inflection point or even a saddle point in the
potential.
The resulting flatness thus provides a very attractive setup for inflationary
dynamics, and several successful models have been constructed in the
literature. In the context of the MSSM flat directions, inflection points may
for example result from the interplay between repulsive soft trilinear
$A$-terms and (non-)renormalizable terms in the superpotential, providing not
only inflationary models consistent with observational data but also
interesting connections to low-energy phenomenology, such as neutrino masses,
natural dark matter candidates and the recent Higgs mass from LHC
Allahverdi:2006iq ; Allahverdi:2006cx ; Bueno Sanchez:2006xk ;
Allahverdi:2006we ; Allahverdi:2007vy ; Allahverdi:2007wt ; Lalak:2007rsa ;
Allahverdi:2010zp ; Enqvist:2010vd ; Allahverdi:2011su ; Chatterjee:2011qr ;
Boehm:2012rh . Several different flat directions in the MSSM field space have
been analyzed so far, including simple extensions such as additional singlet
fields leading to hybrid inflation models Hotchkiss:2011am , as well as taking
into account supergravity corrections Enqvist:2007tf ; Mazumdar:2011ih and
possible embeddings within the string theory landscape Allahverdi:2007wh . In
the context of string theory, several new possibilities arise, as for example
the case of warped D-brane inflation Kachru:2003sx ; warped_brane_potentials ,
where the D-brane potential receives a broad array of contributions such as
Coulomb-like interactions in brane-antibrane pairs, couplings to the four-
dimensional scalar curvature and several different moduli stabilization
effects in the bulk of the compactification. A recent statistical analysis of
these contributions has shown that successful models typically occur near an
inflection point in the potential Agarwal:2011wm . Such features may also be
found in closed-string moduli dynamics, for example in the context of
racetrack models BlancoPillado:2004ns and the so-called accidental inflation
scenarios Linde:2007jn ; BlancoPillado:2012cb .
Inflection point inflation thus appears in a broad range of different setups,
being quite successful in terms of consistency with observational data, as
well as providing a natural embedding within ultraviolet completions of the
Standard Model and desirable links to low-energy phenomenology. However, these
models are far from generic and typically require a fine-tuning of the
different contributions to the scalar potential, making inflation rather
special within the vast landscape of different possibilities.
In this work, we revisit inflationary dynamics near an inflection point in the
potential taking into account the effects of dissipation in the inflaton’s
motion. Dissipation is a natural outcome of the interactions between the
inflaton field and other degrees of freedom, which are in fact required to
ensure a graceful exit into a radiation-dominated era, and is intrinsically
associated with particle production. If such dissipative effects are
sufficiently strong, particle production may actually balance the dilution
effect of the accelerated expansion, resulting in an inflationary state that
is far from the supercooled vacuum that is conventionally considered. In
particular, if the resulting particles have sufficiently strong interactions
between them, they can possibly reach a nearly-thermal state at a temperature
$T>H$, where $H$ is the inflationary Hubble rate, thus potentially changing
the dynamics of inflation. Although it encompasses a much more general class
of models where fluctuation-dissipation effects are significant during
inflation, the resulting paradigm is thus known as warm inflation, having been
originally proposed in wi ; Berera:1996nv following on earlier work in
earlydissp .
Most scenarios considered in the literature so far in the context of quantum
field theory Berera:1998gx ; Yokoyama:1998ju ; Berera:1999ws ; BR1 ;
Berera:2002sp ; Hall:2004zr ; br05 ; Moss:2006gt ; Berera:2008ar ; Moss:2008yb
; Graham:2008vu ; BasteroGil:2009ec ; BasteroGil:2010pb ; BasteroGil:2012cm ,
as well as phenomenological models ph consider the case where a nearly-
thermal bath of radiation is produced concurrently with accelerated expansion
from the adiabatic motion of the inflaton field, which may be analyzed within
the framework of linear response theory. In this regime, the non-local effects
of dissipation yield, to leading order, an additional friction term in the
inflaton’s equation of motion that helps overdamping its trajectory, thus
allowing for longer periods of slow-roll inflation and alleviating the need
for a very flat potential (see e.g. BasteroGil:2009ec ), which is particularly
important in the context of supergravity and string theory warm_string ;
warm_brane , where one typically finds a severe ‘eta-problem’. In this sense,
we expect the inclusion of dissipative effects to minimize the fine-tuning of
different terms in the scalar potential required for a sufficiently long
period of inflation near an inflection point. On the other hand, most studies
of warm inflation so far have focused on relatively steep potentials,
typically suffering from an eta-problem, where dissipative effects may play a
more prominent role, so that this will allow us to explore a new regime of
warm inflationary dynamics with a very flat potential.
Warm inflation has several other attractive features, such as providing an
alternative to the graceful exit problem, since radiation, although
necessarily sub-leading during inflation, may become the dominant component of
the energy density as soon as the conditions for slow-roll evolution break
down. The spectrum of primordial perturbations is also typically modified in
this context, as thermal fluctuations of the inflaton field overcome vacuum
fluctuations for $T>H$ wi ; Berera:1999ws ; Taylor:2000ze ; Hall:2003zp ;
Moss:2008yb . This generically suppresses the amount of primordial gravity
waves produced during inflation and induces a significant non-gaussian
component in the spectrum, which is generically expected to be within the
reach of the Planck mission Gupta:2002kn ; Chen:2007gd ; Moss:2007cv ;
Moss:2011qc . Furthermore, the inclusion of dissipative effects and the finite
temperature during inflation may address other outstanding problems in modern
cosmology, such as the generation of a baryon asymmetry BasteroGil:2011cx and
the overproduction of gravitinos in supersymmetric models Taylor:2000jw ;
Sanchez:2010vj ; Bartrum:2012tg .
This work is organized as follows. In the next section, we review the
conventional dynamics of inflation near an inflection point in the potential,
taking as a working example that we will follow throughout our discussion a
renormalizable flat direction in a $U(1)_{\mathrm{B-L}}$ extension of the
MSSM. In section III we give a basic review of the generic features and
conditions for a successful model of warm inflation, applying these to a
potential with an inflection point in section IV. We then perform numerical
simulations of the evolution of the coupled inflaton-radiation system as a
function of the parameters characterizing the potential and the dissipation
coefficient, presenting our results in section V. Finally, in section VI we
summarize our main conclusions and discuss their impact on inflationary model
building, as well as prospects for future research in this topic.
## II Cold inflation near an inflection point
As discussed earlier, scalar potentials exhibiting an inflection point may
arise in a variety of models in supersymmetric theories and
supergravity/string theory models. For concreteness, we will consider
throughout most of our discussion a simple example introduced in
Allahverdi:2006iq ; Allahverdi:2007vy and also considered in Hotchkiss:2011am
, consisting of a low-scale extension of the MSSM with an additional
$U(1)_{\mathrm{B-L}}$ symmetry and right-handed neutrino superfields. In
particular, we focus on the scalar potential induced for the $NH_{u}L$ flat
direction, parametrized by a scalar field $\phi$ that plays the role of the
inflaton and which, without loss of generality, we take to be real. This flat
direction is lifted by a renormalizable term in the superpotential and by
soft-SUSY breaking terms, yielding:
$V(\phi)=\frac{1}{2}m_{\phi}^{2}\phi^{2}+\frac{h^{2}}{12}\phi^{4}-\frac{Ah}{6\sqrt{3}}\phi^{3}~{}.$
(1)
For $A\simeq 4m_{\phi}$, this potential exhibits an approximate saddle point
for a field value $\phi_{0}\simeq\sqrt{3}m_{\phi}/h$, such that
$V^{\prime}(\phi_{0})\simeq V^{\prime\prime}(\phi_{0})\simeq 0$, which is thus
suitable for inflation. We may then define Hotchkiss:2011am :
$A=4m_{\phi}\sqrt{1-{\beta^{2}\over 4}}$ (2)
and expand the potential about the generic point of inflection, yielding for
$\beta\ll 1$, to lowest order:
$V(\phi)\simeq
V_{0}\left(1+3\beta^{2}\left({\phi-\phi_{0}\over\phi_{0}}\right)+4\left({\phi-\phi_{0}\over\phi_{0}}\right)^{3}\right)~{},$
(3)
where $V_{0}=V(\phi_{0})$. This clearly shows that, for $\beta=0$, $\phi_{0}$
is a saddle point in the potential, with $\beta$ determining the deviations
from this case, i.e. the fine-tuning of the parameters in the potential
required for a sufficiently flat inflationary potential. Note that for real
values of $\beta$, the potential exhibits an inflection point at $\phi_{0}$,
whereas for imaginary values of $\beta$ it develops a local minimum at
$\phi>\phi_{0}$, as illustrated in Fig. 1. This latter option could be suited
for inflation with the field trapped in the false vacuum and then tunneling
into the true minimum, as in the old inflationary picture. However, this does
not lead to a graceful exit into a radiation-dominated era, so we will not
consider this case in the remainder of our discussion.
Figure 1: Normalized scalar potential for different values of the fine-tuning
parameter $\beta$.
The inflationary dynamics, in the absence of dissipation, is determined by the
slow-roll parameters, which are in this case given by:
$\displaystyle\epsilon_{\phi}$ $\displaystyle=$ $\displaystyle{1\over
2}m_{P}^{2}\left({V^{\prime}(\phi)\over V(\phi)}\right)^{2}\simeq{1\over
2}\left({m_{P}\over\phi_{0}}\right)^{2}\left(3\beta^{2}+12\Delta_{\phi}^{2}\right)^{2}~{},$
$\displaystyle\eta_{\phi}$ $\displaystyle=$ $\displaystyle
m_{P}^{2}{V^{\prime\prime}(\phi)\over V(\phi)}\simeq
24\left({m_{P}\over\phi_{0}}\right)^{2}\Delta_{\phi}~{},$ (4)
where $m_{P}=2.4\times 10^{18}$ GeV is the reduced Planck mass,
$\Delta_{\phi}=(\phi-\phi_{0})/\phi_{0}$ and we have taken $V(\phi)\simeq
V_{0}$, which holds for $\Delta_{\phi},\beta\ll 1$. From these quantities we
may determine the amplitude and tilt of the resulting spectrum of density
perturbations, given by:
$\displaystyle\mathcal{P}_{R}$ $\displaystyle=$ $\displaystyle{1\over
24\pi^{2}}{V_{0}/m_{P}^{4}\over\epsilon_{\phi_{*}}}~{},$ $\displaystyle n_{s}$
$\displaystyle=$ $\displaystyle 1+2\eta_{\phi_{*}}-6\epsilon_{\phi_{*}}\simeq
1+48\left({m_{P}\over\phi_{0}}\right)^{2}\Delta_{\phi_{*}}~{},$ (5)
where $\phi_{*}$ denotes the value of the field when the relevant CMB scales
left the horizon about 40-60 e-folds before the end of inflation, and we have
used that $|\eta_{\phi_{*}}|\gg\epsilon_{\phi_{*}}$ for
$\Delta_{\phi},\beta\ll 1$. These two conditions can be used to determine the
constant term in the potential $V_{0}$ and $\phi_{*}$, leaving $\phi_{0}$ and
$\beta$ as the only undetermined parameters.
The dynamics of inflation is governed by the slow-roll equation:
$3H\dot{\phi}\simeq-V^{\prime}(\phi)~{},$ (6)
with $H^{2}\simeq V(\phi)/3m_{P}^{2}$. Inflation ends in this case when the
slow-roll condition $|\eta_{\phi}|<1$ is violated, such that
$\Delta_{\phi_{e}}\simeq-(\phi_{0}/m_{P})^{2}/24$. This allows us to compute
the total number of e-folds of inflation from horizon-crossing, which is then
given by:
$N_{e}=\int_{t_{*}}^{t_{e}}Hdt\simeq-\int_{\phi_{*}}^{\phi_{e}}{3H^{2}\over
V^{\prime}(\phi)}d\phi\simeq{\arctan(1/2\xi)+\arctan\left((n_{s}-1)/4\xi\right)\over\xi}~{},$
(7)
where $\xi=6\beta(m_{P}/\phi_{0})^{2}$. Thus, in the limit $\beta\rightarrow
0$ for any given $\phi_{0}$, this yields a maximum of $N_{e}\simeq 119$
e-folds of inflation for $n_{s}=0.967$ Komatsu:2010fb , which exceeds the
observationally required range. We can invert this to determine the value of
$\beta$ required for 40-60 e-folds of inflation, yielding:
$\beta\simeq(3.1-5.2)\times 10^{-3}\left({\phi_{0}\over m_{P}}\right)^{2}~{},$
(8)
with smaller values of $\beta$ yielding longer periods of inflation, since the
resulting potential is flatter. This illustrates the generic fine-tuning
problem of inflection point models, and in this particular case the soft
inflaton mass and the trilinear term in Eqs. (1) and (2) have to compensate
each other to at least one part in $10^{6}$ for a successful model with sub-
planckian inflaton values, as can be seen by inserting Eq. (8) in Eq. (2).
## III Warm inflation dynamics
In warm inflation, the interactions of the inflaton field with other degrees
of freedom can, as discussed above, significantly affect its dynamics and the
standard inflationary observables, such as the amplitude and the tilt of the
power spectrum, its non-gaussianity and the spectrum of primordial
gravitational waves. The time non-local component of these interactions leads
to a transfer of energy from the inflaton field to other degrees of freedom,
yielding a dissipative process which is in general described by non-
equilibrium dynamics. However, in the slow-roll regime, the motion of inflaton
may be sufficiently slow compared to the overall relaxation time of the system
to allow for an adiabatic description of the evolution, giving to leading
order an additional friction term in the scalar equation of motion:
$\ddot{\phi}+(3H+\Upsilon)\dot{\phi}+V^{\prime}(\phi)=0.$ (9)
The dissipative coefficient can be computed microscopically given a particular
particle physics realization of inflation Berera:2008ar and may in general
depend on the value of the inflaton field and the properties of the multi-
particle state produced by dissipation. Successful realizations of warm
inflation have been constructed in a regime where the fields coupled to the
inflaton acquire large masses during inflation, proportional to the inflaton
value, and then decay into light degrees of freedom, which is generically
known as the two-stage mechanism Berera:2002sp . For sufficiently strong
interactions the produced particles may then thermalize in less than a Hubble
time, producing a quasi-thermal bath of radiation at a temperature $T$
BasteroGil:2012cm . If the intermediate fields are sufficiently heavy compared
to the temperature of the radiation, thermal corrections to the inflaton mass
are Boltzmann-suppressed, thus preserving the tree-level flatness of the
inflaton potential, in particular in the case of supersymmetric flat
directions. Such large masses are naturally obtained for large values of the
inflaton field, hence avoiding the problems of early realizations of warm
inflation in a high-temperature regime Berera:1998gx ; Yokoyama:1998ju .
Supersymmetry also provides a natural framework for keeping quantum and
thermal corrections to the scalar potential under control Hall:2004zr , even
though it is broken by finite temperature and energy density effects during
inflation. A generic superpotential realizing the two-stage interactions is
given by BasteroGil:2009ec ; Moss:2008yb :
$W=W(\Phi)+g\Phi X^{2}+hXY^{2}~{},$ (10)
where the scalar component of $\Phi$ is the inflaton field, its expectation
value giving large masses to the bosonic and fermionic components of the
intermediate superfield(s) $X$. They decay in turn into the components of the
superfield $Y$, which remain light and form the radiation fluid. For $T\ll
m_{X}$ and a broad range of couplings and field multiplicities, the dominant
contribution to the dissipative coefficient corresponds to virtual excitations
of the intermediate scalar fields decaying into the bosonic $Y$ components and
has the form Moss:2008yb ; BasteroGil:2010pb ; BasteroGil:2012cm :
$\Upsilon\approx C_{\phi}\frac{T^{3}}{\phi^{2}}~{},$ (11)
where $C_{\phi}$ is a constant that depends on the coupling $h$ and the field
multiplicities in the $X$ and $Y$ sectors and which, for the purposes of our
discussion, we will take as a free parameter of the model. Note that
renormalizable superpotentials of the form in Eq. (10) are ubiquitous in
supersymmetric models, such as for example the NMSSM, where the additional
singlet could play the role of the inflaton and dissipate its energy into
(s)quarks and (s)leptons through the Higgs portal, e.g. $W=g\Phi
H_{u}H_{d}+hQH_{u}U+\ldots$ Notice, however, that a much larger number of
fields is required with the dissipation coefficient in Eq. (11) to achieve a
sufficient number of e-folds of inflation than in the MSSM. More generically,
one expects fields coupled to supersymmetric flat directions to acquire large
masses during inflation (see e.g. Allahverdi:2007zz ), while fields that do
not couple directly to flat directions should remain light. Such a
superpotential also arises in D-brane constructions, where dissipative effects
have been shown to play an important role in overcoming the associated eta-
problem warm_brane .
The thermalized radiation fluid has an energy density
$\rho_{r}\simeq\frac{\pi^{2}}{30}g_{*}T^{4}~{},$ (12)
where $g_{*}$ is the effective number of light degrees of freedom, and is
sourced by the dissipative motion of the inflaton field, yielding
$\dot{\rho}_{r}+3H(\rho_{r}+p_{r})=\Upsilon\dot{\phi}^{2},$ (13)
where $p_{r}$ is the pressure associated with the radiation fluid. In warm
inflation, this fluid is not redshifted away during inflation, due to the
additional dissipative source term wi ; wi2 . The radiation energy density
needs, however, to be subdominant to achieve a period of accelerated
expansion, i.e. $\rho_{r}\ll\rho_{\phi}$, where
$\rho_{\phi}=\dot{\phi}^{2}/2+V(\phi)$. However, the associated temperature
may be larger than the expansion rate, $T>H$, which allows one to neglect the
effects of expansion in computing the dissipation coefficient and modifies the
evolution of inflaton fluctuations. Otherwise, when $T<H$, dissipative effects
can be neglected and the standard cold inflation scenario is recovered.
Slow-roll inflation, whether cold or warm, requires an overdamped evolution of
the inflaton field. In warm inflation this can be achieved due to the friction
term $\Upsilon$ in addition to Hubble damping. Once the field $\phi$ is in the
slow-roll regime, the evolution of the radiation fluid is also generically
damped, and the equations of motion reduce to
$\displaystyle 3H(1+Q)\dot{\phi}$ $\displaystyle\simeq-V^{\prime}(\phi)\,,$
(14) $\displaystyle 4\rho_{R}$ $\displaystyle\simeq 3Q\dot{\phi}^{2}\,,$ (15)
where we have introduced the dissipative ratio $Q=\Upsilon/(3H)$, which,
depending on the particular model, may increase or decrease during inflation.
The slow-roll conditions are now given by $\epsilon_{\phi},\eta_{\phi}\ll
1+Q$, alleviating the need for a very flat potential, and additionally one
requires that Moss:2008yb :
$\displaystyle\beta_{\Upsilon}$ $\displaystyle=$ $\displaystyle
m_{P}^{2}\left(\frac{\Upsilon_{\phi}V_{\phi}}{\Upsilon V}\right)\ll 1+Q\,,$
(16) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{TV_{T\phi}}{V_{\phi}}<1\,,$ (17)
where $\beta_{\Upsilon}$ measures the variation of the dissipation coefficient
with respect to the inflaton field and the last condition ensures that thermal
corrections to the potential are small, which is the case in the regime where
Eq. (11) is valid.
## IV Warm inflation near an inflection point
As previously discussed, the additional friction term in Eq. (9) alleviates
the flatness of the potential required in order to achieve a sufficient amount
of inflation. In the context of inflection point inflation, we have seen that
the $\beta$ parameter determines the shape of the potential in the vicinity of
the inflection point, measuring the fine-tuning of the underlying parameters.
Therefore, we expect that a warm realization of these models can naturally
reduce the amount of fine-tuning required.
We will use the Eq. (1) as a working example of a potential with an inflection
point to analyze the generic dynamics of warm inflation in this context,
although this does not correspond to a concrete realization of warm inflation
in the MSSM. Writting Eq. (1) in the form of Eq. (3), the dynamics is
described by six independent parameters, in particular the value of the field
at the inflection point $\phi_{0}$, the corresponding height of the potential
$V_{0}$, the fine-tuning parameter $\beta$, the value of the field at horizon-
crossing $\phi_{*}$, the dissipative constant $C_{\phi}$ and the effective
number of light degrees of freedom $g_{*}$. We can use the WMAP 7-year results
giving a power spectrum with an amplitude $\mathcal{P}_{\mathcal{R}}=(2.43\pm
0.11)\times 10^{-9}$ and a spectral index $n_{s}=0.967\pm 0.014$
Komatsu:2010fb to determine $V_{0}$ and $\phi_{*}$. As mentioned above, for
$T>H$ the dominant contribution to the spectrum of primordial perturbations
are the thermal fluctuations in the coupled inflaton-radiation system, and the
resulting amplitude of the power spectrum in the slow-roll regime is given by
wi ; Berera:1999ws ; Hall:2003zp :
$\mathcal{P}^{1/2}_{\mathcal{R}}\simeq\left(\frac{H}{2\pi}\right)\left(\frac{3H^{2}}{V^{\prime}(\phi)}\right)(1+Q)^{5/4}\left(\frac{T}{H}\right)^{1/2}~{},$
(18)
where all quantities are implicitly evaluated at horizon-crossing. In order to
solve this equation, it is useful to write it in a more convenient way. Using
the slow-roll equations (14) and (15), one obtains
$Q_{*}(1+Q_{*})^{13/2}\simeq\mathcal{P_{R}}\left(\frac{C_{\phi}}{3}\right)\left(\frac{\pi
C_{\phi}}{2C_{R}}\right)^{2}(2\epsilon_{\phi_{*}})^{3}\left(\frac{m_{P}}{\phi_{*}}\right)^{6}~{},$
(19)
where $C_{R}=g_{*}\pi^{2}/30$. Eq. (19) and the expression for the spectral
index BasteroGil:2009ec
$(1+Q_{*})(1+7Q_{*})(n_{s}-1)+(2+9Q_{*})\epsilon_{\phi_{*}}+3Q_{*}\eta_{\phi_{*}}+(1+9Q_{*})\beta_{\Upsilon_{*}}\simeq
0$ (20)
form a coupled system of equations for $Q_{*}$ and $\phi_{*}$ that needs to be
solved numerically for given values of $\phi_{0}$, $\beta$, $C_{\phi}$ and
$g_{*}$. Once the system is solved, we can obtain the value of $V_{0}$ using
Eq. (18):
$V_{0}\simeq\left(\frac{C_{R}}{C_{\phi}}\right)\frac{144\pi^{2}\mathcal{P_{R}}\phi^{2}_{*}m^{2}_{P}}{\sqrt{1+Q_{*}}\left(1+3\beta^{2}\left({\phi_{*}-\phi_{0}\over\phi_{0}}\right)+4\left({\phi_{*}-\phi_{0}\over\phi_{0}}\right)^{3}\right)}~{}.$
(21)
The system has in general three possible solutions satisfying the
observational constraints, and we have consistently chosen the one that
maximizes the difference $\phi_{*}-\phi_{0}$, since as we discuss below this
minimizes the amount of dissipation required for a sufficiently long period of
inflation. To simplify the numerical procedure, one can use the approximate
solutions in the strong and weak dissipative regimes, $Q_{*}\ll 1$ and
$Q_{*}\gg 1$, respectively, where the equations decouple, to find the initial
guess required to calculate numerically the full solution to the coupled
system of equations. In the intermediate regime, $Q_{*}\sim 1$, it is
sufficient to use an initial guess in this range.
## V Numerical results
Having determined $V_{0}$ and $\phi_{*}$ from the observational constraints,
we may now study the evolution of the coupled inflaton-radiation system as a
function of the remaining parameters, $C_{\phi}$, $\beta$, $\phi_{0}$ and
$g_{*}$. For concreteness, we first fix the number of relativistic degrees of
freedom $g_{*}=100$, corresponding to the order of magnitude of the number of
MSSM scalar fields, although we study the effect of varying this parameter at
the end of this section. Our main goal is to determine which is the lowest
value of $C_{\phi}$ required for a sufficiently long period of inflation as a
function of the fine-tuning parameter $\beta$ and for different values of
$\phi_{0}$. The number of e-folds of warm inflation can be computed by
including the effects of dissipation in Eq. (7):
$N_{e}\simeq-\int_{\phi_{*}}^{\phi_{e}}{\frac{3H^{2}(1+Q)}{V^{\prime}(\phi)}d\phi}.$
(22)
However, due to the $T$\- and $\phi$-dependent dissipative ratio $Q$, this
integral cannot be solved analytically as in the cold inflation case. Besides,
the value of the field at the end of inflation cannot be calculated a priori.
Hence, the equations of motion for both the inflaton and the radiation fluid
have to be integrated numerically. In most areas of the parameter space, the
inflaton field is always in the slow-roll regime and therefore we may
integrate Eq. (14). However, in some regions of the parameter space the
radiation fluid is not slow-rolling, in which case we integrate the full
equation (13). The consistency of our analysis is determined by three main
conditions:
* •
$\epsilon_{H}=-\frac{\dot{H}}{H}<1$, which is required for accelerated
expansion and generalizes the slow-roll conditions;
* •
$m_{X}\gg T$, for which the form of the dissipative coefficient in Eq. (11) is
valid.
* •
$T>H$, which allows one to neglect the effects of expansion in computing the
dissipation coefficient and generically defines the regime where inflation is
warm.
These conditions need to hold for $40-60$ e-folds of inflation in order to
solve the horizon and flatness problems, and in Fig. 2 we show the regions in
the plane $C_{\phi}-\beta$ where this is obtained for different values of the
inflection point $\phi_{0}$.
Figure 2: Values of $C_{\phi}$ and $\beta$ required to obtain
$N_{e}\in[40,60]$ for $g_{*}=100$ and different values of $\phi_{0}/m_{P}$.
As one can easily see in this figure, lower values of $\phi_{0}$ require more
dissipation in order to obtain the same number of e-folds, which is related to
the associated increase in the slow-roll parameters in Eq. (II), due to a
steeper shape of the potential. In addition, we find two distinct regions of
parameter space in Fig. 2, corresponding to small and large values of the
fine-tuning parameter $\beta$. The separation between these regions depends on
the value of $\phi_{0}$, with the small-$\beta$ region moving to lower values
of $\beta$ for smaller $\phi_{0}$.
In the small-$\beta$ region, the potential is extremely flat and intuitively
one would expect less friction to be required for a given period of
accelerated expansion. However, Fig. 2 clearly shows that the required value
of $C_{\phi}$ becomes constant for low values of $\beta$, which suggests
taking a closer look at the physical mechanism behind dissipation. Since it is
the motion of the inflaton field that produces light particles in a quasi-
thermal bath, the amount of radiation produced depends on how fast the
inflaton is rolling, as can be explicitly seen in Eq. (13). If the potential
is too flat, the inflaton will roll too slowly, which suppresses the amount of
radiation produced and consequently decreases the temperature of the thermal
bath. In fact, it is the condition $T>H$ that determines the end of warm
inflation in this region of parameter space, as one can see in Fig. 3, where
we plot the evolution of the relevant quantities in this regime. This also
explains why the initial condition farther away from the inflection point
yields the lowest value of $C_{\phi}$, since an initially steeper potential
can more easily produce a radiation bath with $T>H$.
(a) $(\phi-\phi_{0})/\phi_{0}$
(b) $\epsilon_{H}$ and $\rho_{r}/\rho_{\phi}$
(c) $T/H$
Figure 3: Evolution with the number of e-folds of $(\phi-\phi_{0})/\phi_{0}$,
$\epsilon_{H}$, $\rho_{r}/\rho_{\phi}$ and $T/H$ for $\phi/m_{P}=10^{-2}$,
$g_{*}=100$ and $\beta=10^{-7}$ when inflation lasts for $40$ e-folds.
In Fig. 3(a), one can see that the inflaton starts above the inflection point
and ends close to the latter, with the temperature dropping below the Hubble
rate after 40 e-folds of inflation. Notice, however, that inflation does not
necessarily end at this point, since $\epsilon_{H}<1$ and decreasing, but our
analysis is no longer consistent at this stage since de Sitter effects may
modify the dissipation coefficient. It may, in fact, be possible for an
additional period of cold inflation to follow, thus decreasing the amount of
dissipation required to achieve the desired number of e-folds, although a
detailed analysis of this possibility is beyond the scope of this work.
Finally, in Fig. 3 we see that $\epsilon_{H}$ follows closely the evolution of
the radiation energy density, which in this case is becoming more and more
sub-leading compared to the inflaton field, thus requiring an additional
reheating stage.
In the large-$\beta$ region the potential is steeper, therefore the production
of radiation is enhanced and $T>H$ is no longer the dominant constraint. In
fact, in this regime radiation tends to be overproduced and dominate the
energy density, thus allowing for a graceful exit from inflation, as shown in
Fig. 4 where we plot the evolution with the number of e-folds of the relevant
quantities in the large-$\beta$ region.
(a) $(\phi-\phi_{0})/\phi_{0}$
(b) $\epsilon_{H}$ and $\rho_{r}/\rho_{\phi}$
(c) $T/H$
Figure 4: Evolution with the number of e-folds of $(\phi-\phi_{0})/\phi_{0}$,
$\epsilon_{H}$, $\rho_{r}/\rho_{\phi}$ and $T/H$ for $\phi/m_{P}=10^{-2}$,
$g_{*}=100$ and $\beta=10^{-2}$ when inflation lasts for $40$ e-folds.
In Fig. 4(a), one can see that the inflaton field starts away from the
inflection point, remains close to it for a few e-folds but that, due to the
slope of the potential, inflation ends beyond the point of inflection, in
contrast with the small-$\beta$ behavior. Notice that $\rho_{r}/\rho_{\phi}$
decreases sharply when the field slows down close to the inflection point, in
agreement with the discussion above, but then increases as the field moves to
lower values and eventually ends inflation with a smooth exit into a
radiation-dominated era. In Fig. 4(c) it is also clear that $T>H$ for the
whole duration of inflation.
Although we have not plotted the condition $m_{X}\gg T$ in Figs. 3 and 4, we
have checked that it is satisfied in all the parameter space shown, for
couplings $g\sim 1$. On the other hand, we may consider more general
potentials, associated with different SUSY breaking effects, yielding a
different value for the numerical coefficient of the slow-roll parameter
$\eta_{\phi}$ in Eq. (II). We then find that, for lower values of this
coefficient, the condition $m_{X}\gg T$ is more stringent than $T>H$. However,
the amount of dissipation required does not change significantly even for an
order of magnitude change in this coefficient, so we do not explore this
possibility in more detail.
Finally, we analyze the effect of the number of relativistic degrees of
freedom on the amount of dissipation required for successful inflation. In
Fig. 5 we show the $C_{\phi}-\beta$ region where 40-60 e-folds of inflation
are obtained with different values of $g_{*}$.
Figure 5: Values of $C_{\phi}$ and $\beta$ required to obtain
$N_{e}\in[40,60]$ for $\phi_{0}/m_{P}=1$ and $g_{*}=1,10^{4}$.
In Fig. 5, it can be observed that the required value of $C_{\phi}$ decreases
for smaller $g_{*}$. In order to understand this behavior, we compute the
explicit dependence of the dissipation coefficient on $g_{*}$ by substituting
Eq. (12) into Eq. (11):
$\Upsilon=\frac{30^{3/4}C_{\phi}}{\pi^{3/2}g_{*}^{3/4}}\frac{\rho^{3/4}_{r}}{\phi^{2}}.$
(23)
Hence, the relevant quantity is an effective dissipation constant:
$\tilde{C}_{\phi}=\frac{C_{\phi}}{g^{3/4}_{*}}$ (24)
that remains constant in Fig. 5 for the different values of $g_{*}$, which is
also the case for smaller (sub-planckian) values of the inflection point.
## VI Summary and future prospects
In this work we have analyzed the dynamics of inflation close to an inflection
point in the scalar potential taking into account dissipative effects
resulting from the coupling of the inflaton field to other degrees of freedom.
We have focused on supersymmetric models, where the plethora of available flat
directions may be lifted by competing SUSY-breaking effects, producing
inflection and even saddle points in the potential, although at the expense of
fine-tuning a priori unrelated parameters. Moreover, supersymmetry provides a
natural framework for warm inflation, helping to protect the flatness of the
potential against both thermal and radiative corrections, in particular in the
regime where the fields coupled to the inflaton acquire large masses and their
effects are Boltzmann-suppressed, nevertheless allowing for the dissipative
production of virtual excitations that may decay into light fields in a quasi-
thermal bath. For concreteness, we have focused on the $NH_{u}L$ flat
direction in a low scale extension of the MSSM, although the resulting scalar
potential has a sufficiently generic form and our main results should apply to
other realizations of inflection point inflation.
Our numerical simulations of the dissipative dynamics of inflation in this
model have lead us to two main conclusions. Firstly, if dissipative effects
are sufficiently strong, a sufficiently long period of inflation may occur
independently of the fine-tuning of the parameters in the potential, which was
expected since the additional friction alleviates the need for a very flat
potential. Secondly, and more surprisingly, the required amount of dissipation
does not decrease arbitrarily for flatter potentials, given that if the scalar
potential is too flat and the inflaton evolves too slowly, it becomes more
difficult to sustain a radiation bath with a temperature above the Hubble
rate, which is required for consistency of our analysis. This results in a
field-dependent critical value of the fine-tuning parameter $\beta$ below
which the required dissipation parameter $C_{\phi}$ becomes constant. Above
this value, the potential is sufficiently steep to ensure that $T>H$
throughout inflation, with steeper potentials requiring larger values of the
dissipation parameter.
The value of $C_{\phi}$ depends on the coupling between the intermediate
fields and the light degrees of freedom, as well as on the multiplicities of
both heavy and light fields. The minimum value of $C_{\phi}\gtrsim 10^{6}$
obtained for $g_{*}=100$ is of the same order as that obtained for other forms
of the inflaton potential, such as monomial or hybrid models BasteroGil:2009ec
, which implies large couplings and field multiplicities, so one may ask
whether there is any gain from the model building perspective in trading a
large fine-tuning in the parameters of the potential for large couplings and a
large number of fields. On one hand, fine-tuning makes inflation less generic,
since it isolates a small region of the available parameter space, whereas
inflation should provide an explanation for the otherwise finely-tuned
conditions in the early universe. On the other hand, a large number of degrees
of freedom during inflation points towards more complicated beyond the
Standard Model scenarios, e.g. with fields in large representations, which may
be realized in generic GUT constructions or D-brane models warm_brane . As
discussed earlier in this work, strong dissipative effects may have other
interesting consequences in the dynamics of the early universe and, moreover,
we have seen that the minimum value of $C_{\phi}$ may be substantially reduced
in models with a smaller number of relativistic degrees of freedom, where the
temperature of the radiation bath is consequently larger, pointing towards
constructions with several fields coupled directly to the inflaton but with
few distinct decay channels.
One should point out that we have focused on a particular form of the
dissipation coefficient, corresponding to the decay of virtual excitations of
the heavy fields coupled to the rolling inflaton. In BasteroGil:2012cm , it
was pointed out that this is not the only possibility, as the excitation of
real modes becomes the dominant contribution to dissipative effects for
$m_{X}/T\lesssim 10$, yielding a much larger dissipation coefficient than
virtual excitations for $\mathcal{O}(1)$ couplings and field multiplicities,
nevertheless suppressing quantum and thermal corrections to the scalar
potential. While the dynamics of warm inflation in this regime remains
unexplored, it suggests a much more promising avenue from the model building
point of view, and we expect our main qualitative results to apply in this
case as well, since they depend more on the presence of significant
dissipative effects than on the specific form of the dissipation coefficient.
Whereas we intend to explore this regime in more detail in the future, our
results show that dissipative effects are an interesting and well-motivated
alternative to fine-tuning in inflationary models and we hope to motivate
further investigation of this topic in other inflationary scenarios.
###### Acknowledgements.
We would like to thank Mar Bastero-Gil, Arjun Berera and Anupam Mazumdar for
their useful comments and suggestions. R.C. is partially supported by MICINN
(FIS2010-17395) and “Junta de Andalucía” (FQM101) and would like to
acknowledge the hospitality of the Particle Theory Group at the University of
Edinburgh during the completion of this work. The work of J.G.R. was partially
supported by the STFC Particle Theory Group grant at the University of
Edinburgh (United Kingdom), the FCT grant PTDC/FIS/116625/2010 (Portugal) and
the Marie Curie action NRHEP 295189- FP7-PEOPLE-2011-IRSES.
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* (57) E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arXiv:1001.4538 [astro-ph.CO]].
* (58) R. Allahverdi and A. Mazumdar, Phys. Rev. D 76, 103526 (2007) [hep-ph/0603244].
|
arxiv-papers
| 2012-10-30T12:11:30 |
2024-09-04T02:49:37.330444
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rafael Cerezo, Jo\\~ao G. Rosa",
"submitter": "Rafael Cerezo",
"url": "https://arxiv.org/abs/1210.7975"
}
|
1210.7991
|
# A NEW APPROACH TO LINAC RESONANCES AND EQUIPARTITION ?
I. Hofmann GSI-HI-Jena 64291 Darmstadt Germany
###### Abstract
In this note we refer to a recent paper “Equipartition, Reality or Swindle?”
[1] presented by Lagniel at HB2012, Beijing, which claims to challenge the
currently used approach to describe space charge resonances and emittance
exchange with the help of linac-specific stability charts. On the one hand we
agree with the general observation that enforcing equipartition (EP) would be
an unnecessary constraint; however, we find that the heuristic single-particle
arguments and examples presented by Lagniel are speculative and cannot be
reconciled with results from self-consistent computer simulation. Thus, we see
no justification for Lagniel’s suggestions, which include a modified EP
definition (treating $x$ and $y$ as correlated). Instead, we suggest to
maintain the current approach and to continue using the “conventional” EP
definition. With our findings we also respond in some detail to Lagniel’s
“Topics of Discussion”.
## 1 Introduction
One of the widely accepted criteria in high-current linac design is to use
linac-specific stability charts to identify parameter regions, where emittance
exchange between the longitudinal and transverse degrees of freedom might
occur. It is common understanding that this exchange is caused by space charge
resonances. Plotting the rms tune ratios and tune depressions from simulation
codes on these stability charts is a widely used approach in new linac
projects in order to deal with the problem of undesirable emittance exchange.
These charts include the physics of phase space flow on a perturbational
level, and under the effect of coupling between two degrees of freedom due to
space charge “pseudo-multipoles” [2]. The resulting resonance stop-bands
proliferate not only the location of possible resonances lines (as ring
resonance diagrams normally do), they also represent their space charge
dependent driving terms. Obviously, particle-in-cell (PIC) simulation is
necessary to examine the validity of the charts, which was carried out
successfully under a great variety of conditions (for a recent discussion
including relevant references see Ref. [3]).
Lagniel’s paper is based on three arguments mainly:
(1) The law of EP in a rigorous sense holds only for ergodic systems
(undeniable - see our comments in the before last section).
(2) If used at all, the “conventional” EP condition employing an rms energy
ratio $T$ as shown in Eq. 1
$T\equiv\frac{\epsilon_{z}k_{z}}{\epsilon_{x}k_{x}}=1,$ (1)
(all quantities understood as rms quantities) “is wrong” and should have a
factor 2 in the denominator, with the argument that the sum of transverse
energies should be in balance with the longitudinal energy (see Eq. (5) in
Ref. [1]).
(3) The conventional EP-condition does not prevent resonant emittance
exchange.
Below we examine first assertion (3) in the next section, followed by a
critical discussion of (2), and concluding with comments on Lagniel’s “Topics
of discussion” as well as some final remarks.
## 2 PIC examination of Lagniel’s examples
In the following we undertake a careful examination of the two examples of
equipartitioned and non-equipartitioned beams by using the TRACEWIN particle-
in-cell simulation of realistic bunched beams in a periodic FODO lattice, with
no acceleration and RF gaps to keep the bunches longitudinally. We employ
100.000 simulation particles and an input distribution following the TRACEWIN
standard option of randomly generated particles in the 4d transverse hyper
space as well as randomly in the longitudinal phase plane ellipse.
The parameters of the exactly equipartitioned example ($T=1$ by the
conventional definition of Eq. 1) described in Eqs. (6)-(9) of Ref. [1] are
identically chosen as $\epsilon_{z}/\epsilon_{x}=1.18$, $k_{z}/k_{x}=0.85$ ,
$k_{x}/k_{0x}=0.75$ and $k_{0z}=73^{o}$ (per cell). The beam is matched - as
much as possible with TRACEWIN - and then propagated over 100 cells (45 m).
The result is shown in Fig. 1 including the tune footprint from TRACEWIN on
the stability chart for $\epsilon_{z}/\epsilon_{x}=1.18$. Note that the
vertical line at $k_{z}/k_{x}=1$ (the “main” or “fourth order” resonance
$2k_{z}-2k_{x}=0$) in the stability charts corresponds to the diagonal in
Lagniel’s Figure1.
Figure 1: Evolution of rms emittances for example on the EP condition, with
stability chart for $\epsilon_{z}/\epsilon_{x}=1.18$ (dotted line is EP
condition $T=1$).
It is noted that there is a 2-3% initial transverse emittance growth, whereas
the longitudinal emittance is only oscillating around its initial value
(1.18). Running this case over twice the distance we find that $\epsilon_{z}$
still remains constant within $\pm 0.1\%$ and $\epsilon_{x},\epsilon_{y}$ grow
slowly, but $<1\%$. Hence, we find absence of emittance exchange - consistent
with the “safe” distance of the tune footprint from the stop-band in the
stability chart. Note here that the width of the stop-band of the main
resonance (near $k_{z}/k_{x}=1$) is shrinking to zero for
$\epsilon_{z}/\epsilon_{x}\rightarrow 1$ (see Ref. [3]).
Thus, contrary to Ref. [1], we find that the proximity of the EP working point
to the main resonance $k_{z}/k_{x}=1$ is not adversary to the stability of the
rms emittances. In fact, Lagniel is drawing his conclusions from a schematic
picture of a tune footprint in his Figure 1, which he finds indicative of an
overlap with the resonance $k_{z}/k_{x}=1$. Firstly, the square box tune
footprint is un-physical - particles not seeing any space charge in one
direction (as the ones on the two sides adjacent to the right upper corner,
which is given by the zero-current tunes) don’t exist. Consistent tune
footprints in a $k_{z}-k_{x}$ plane are actually necktie-shaped rather than
square-boxed, which reduces the overlap. Secondly, it is necessary to consider
the response of the bunch as a whole, i.e. the property of the ensemble versus
that of single particles. Individual particles may have growing amplitudes in
one direction, which can be compensated by other particles with shrinking
amplitudes, and no net effect.
Not surprisingly there is a fast ($<20$ cells) and pronounced emittance
exchange of about 10% , if we lower the transverse focusing such that
$k_{z}/k_{x}=1.02$ and the working point sits exactly on the stop-band, while
the beam is initially weakly non-equipartitioned with $T=1.2$ (Fig. 2).
Figure 2: Same as Fig. 1, but $k_{z}/k_{x}=1.02$.
As far as the second example, following Eq. (10) of Ref. [1] (with $k_{0x}$
further lowered to 50o), we agree, in principle, that no emittance exchange
should be expected. The distance to the fourth order resonance line
$k_{z}/k_{x}=1$ is large enough, equally to the third order line
$k_{z}/k_{x}=2$ (i.e. $k_{z}-2k_{x}=0$). It should be noted, however, that the
driving term for this third order mode requires a sextupolar component in the
space charge potential. Such a third order term is absent in the matched
initial beam, where only even powers in the space charge potential exist.
Therefore we cannot see how it enters into Lagniel’s frame of discussion, if
the driving term is absent. In PIC simulation however, equally in our
stability chart, this driving term evolves self-consistently from a resonant
unstable behaviour of the third order mode building up from initial noise. An
example for the effect of this third order mode on the extended stability
chart including $k_{z}/k_{x}=2$ is given below in Fig. 4.
## 3 Do we need a new EP-formula?
In the discussion preceding his Eq. (5), Lagniel argues that the conventional
EP condition Eq. 1 “is wrong” and should have a factor 2 in the denominator to
account for “a total correlation between the two radial degrees of freedom”.
We cannot follow this argument, because dynamically speaking each degree of
freedom is independent - no matter what its initial tune and emittance values
are. But let us use simulation to help decide between the conventional EP
condition and Lagniel’s proposition. To this end let us call Lagniel’s
modified energy ratio $T^{*}$ ($=T/2$ and the condition $T^{*}=1$ the modified
equipartition condition EP∗.
Let us start with Case 1: fulfilling the “conventional” EP condition ($T=1$,
but $T^{*}=1/2$), with $\epsilon_{z}/\epsilon_{x}=1$, $k_{z}/k_{x}=1$ (also
$k_{0z}/k_{0x}=1$) and $k_{0z}=73^{o}$ per cell as well as a transverse tune
depression of $k_{x}/k_{0x}=0.72$. The resulting rms emittances as well as the
footprint of tunes on the stability chart are shown in Fig. 3.
Figure 3: Case 1: Evolution of rms emittances and stability chart for
$\epsilon_{z}/\epsilon_{x}=1$ (dotted line is EP condition $T=1$).
Besides the usual fluctuations there is no real emittance transfer -
consistent with the stability chart. Note that there is an indication that the
two transverse emittances actually behave as independent and undergo small
deviations varying in time - in spite of “identical” starting conditions.
Now we switch to Case 2: a weaker transverse focusing (but same emittance
ratio), such that $k_{z}/k_{x}=2$ and EP∗ is fulfilled ($T^{*}=1$, whereas
$T=2$ ). Results are shown in Fig. 4.
Figure 4: Case 2: Weaker transverse focusing and stability chart for
$\epsilon_{z}/\epsilon_{x}=1$.
There is a real emittance transfer from the longitudinal direction into
transverse - obviously induced by the third order resonance discussed in the
end of Section 2. For the (conventional) energy ratio we find $T=2\to 1.3$.
Note that the “independent” behaviour of transverse emittances is even more
pronounced than in Fig. 3, hence Lagniel’s argument of “total correlation”
between $x$ and $y$ is not supported.
We have also examined a Case 3: with $\epsilon_{z}/\epsilon_{x}=2$ and
$k_{z}/k_{x}=1$ we find the expected main resonance, which leads to a fast
emittance transfer during the first 10 cells already, with a final evolution
$T=2\rightarrow 1.2$. Below we summarize the impact of these three simulations
on the energy ratios $T$ as well as $T^{*}$:
$\begin{array}[]{cccc}&Case1&Case2&Case3\\\ T&1\rightarrow 1&2\rightarrow
1.3&2\rightarrow 1.2\\\ T^{*}&0.5\rightarrow 0.5&1\rightarrow
0.65&1\rightarrow 0.6\\\ \end{array}$
In view of all this we find it straightforward to continue with the
conventional definition of EP as $T=1$ (using $T$ defined in Eq. 1), which
considers all degrees of freedom as independent. This is supported by the
“splitting” of transverse emittances; furthermore by the fact that we have not
found (by simulation, and avoiding extreme tune depression) a single case,
where $T=1$ is subject to emittance transfer - in contrast with the assertions
in Ref. [1]. The initial $T^{*}=1$ as in Case 2, instead, is unstable.
## 4 Lagniel’s Topics of Discussion
Based on the above findings we attempt to respond to the discussion opened in
the last section of Lagniel’s paper by referring to his six points and
starting with the original quotations from Ref.[1].
1- ”The linac beams are out of the EQP theorem validity limit, to apply the
EQP rule designing a linac is a mistake.” It is undeniable that “true”
equipartition can be applied to ergodic systems only. As we have a general
difficulty to describe and measure distributions in 6D phase space, the
concept of projections into 2D planes and of rms values in 2D was developed -
successfully so far. In the same spirit it has become common practice to
employ an rms energy ratio ($T$ in Eq.1 as a reduced, but well-defined
quantity) and call the special case $T=1$ equipartitioned. Whether or not
$T=1$ is a practically helpful requirement is a different question.
2- ”The application of the EQP rule does not prevent emittance exchanges
induced by coupling resonances.” We find this statement is a speculative
interpretation of fictitious beam footprints and not supported by our PIC
simulation, also not by the stability charts. We have simulated exactly the
same case as in Lagniel’s example and find that definitely no rms emittance
exchange occurs (similarly for a variety of other initial emittance ratios,
still equipartitioned).
3- ”Safe tunes with beam footprints out of the coupling resonances can be
found when the EQP rule is not respected.” \- a well-established recognition
in the linac community.
4- ”The constraint imposed by the EQP rule on a linac design can lead to a non
optimized beam dynamics and higher construction and operation costs.” \- out
of question.
5- ”The question of energy exchange / emittance transfer must be analyzed as
done in circular machines (tune diagram, evaluation of the resonance
excitation strength).” Authors should feel free to introduce different kinds
of tune diagrams as long as they prove they are viable. We have suggested
linac stability charts as they include tune depression (intensity) and tune
ratios. Circular machine diagrams with $k_{z}$, $k_{x}$ (or $k_{y}$) separate
may be fine, but would require a third dimension to include intensity.
Actually, as we need to worry only about difference resonances of the kind
$nk_{z}-mk_{x}=0$, the tune ratios suffice. The resonance driving terms are
already part of the stop-band widths of the stability charts and need not to
be evaluated separately.
6- ”The modern physics tools developed to characterize the level of disorder
(chaos) present in nonlinear Hamiltonian systems could be applied to
characterize and optimize our beams.” It should certainly be welcomed to
continue using all the great tools developed in nonlinear dynamics.
Finally, we would like to comment also on Lagniel’s question at the end of his
before last section: ”Why the belief in EQP did not pollute the synchrotron
world?” Synchrotrons indeed have many resonances to worry about. However, as
early as 1968, Montague already warned about the effect of horizontal-vertical
emittance exchange by a space charge pseudo-octupole resonance on the main
diagonal of the CERN Proton Synchrotron ($2Q_{x}-2Q_{y}=0$) [4]. Owed to its
possible importance for high-current operation at CERN, the subject was
carefully studied experimentally only much later - with excellent agreement
with theory [5].
## 5 Final Remarks
We have shown that Lagniel’s assertions on EP and on linac resonances are not
supported by PIC simulations, therefore a new approach to this topic on the
ground of the presented arguments cannot be seen. Independent of this it is
known since many years that there is no necessity to enforce EP, as most of
the parameter space is filled by non-equipartitioned regions, where no
emittance coupling is found - as shown by the linac stability charts.
It should be emphasized here that the notion of EP or non-EP in our context is
based on rms quantities (emittances, tunes). Such an approach obviously cannot
say anything about the question - also raised by Lagniel - of energy
equipartition on surfaces in a multi-dimensional phase space. It would
certainly be welcomed by everybody if future analysis would go beyond rms
measures, for example including halo distributions and the question of
coupling in the tail distributions, and thus open a new dimension of this
problem to the scientific discussion. At the time being, however, linac
designers may continue to work with their validated tools and feel free to be
on EP, or not to be on EP - as long as they have a convincing reason for it.
Acknowledgment: The author is indebted to G. Franchetti for valuable
discussions.
## References
* [1] J.-M. Lagniel, HB2012 conference, Beijing, paper TUO3A03 (2012)
* [2] I. Hofmann, Phys. Rev. E 57, 4713 (1998).
* [3] I. Hofmann, HB2012 conference, Beijing, paper TUO3A01 (2012)
* [4] B.W. Montague, CERN-Report No. 68-38, CERN (1968)
* [5] E. Metral et al., Proc. of EPAC 2004, p. 1894 (2004)
|
arxiv-papers
| 2012-10-30T13:06:44 |
2024-09-04T02:49:37.342026
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ingo Hofmann",
"submitter": "Ingo Hofmann",
"url": "https://arxiv.org/abs/1210.7991"
}
|
1210.8193
|
# Decision dynamics in complex networks subject to mass media and social
contact transmission mechanisms
Carlos Rodríguez Lucatero, Luis Alarcón, Roberto Bernal Jaquez∗, Alexander
Schaum
###### Abstract
The dynamics of decisions in complex networks is studied within a Markov
process framework using numerical simulations combined with mathematical
insight into the process mechanisms. A mathematical discrete-time model is
derived based on a set of basic assumptions on the convincing mechanisms
associated to two opinions. The model is analyzed with respect to multiplicity
of critical points, illustrating in this way the main behavior to be expected
in the network. Particular interest is focussed on the effect of social
network and exogenous mass media-based influences on the decision behavior. A
set of numerical simulation results is provided illustrating how these
mechanisms impact the final decision results. The analysis reveals (i) the
presence of fixed-point multiplicity (with a maximum of four different fixed
points), multistability, and sensitivity with respect to process parameters,
and (ii) that mass media have a strong impact on the decision behavior.
## 1 Introduction
Recently, the problem of modeling, analysis and simulation of rumor and
decision dynamics in complex networks has obtained increasing attention,
including studies on rumor spread in social and small-world networks [1], and
decision dynamics in scale-free networks [2]. While the problem of rumor
spreading [1] considers the time evolution of rumor spreaders and stiflers,
the decision dynamics [2] is concerned with the time evolution of different
decisions in terms of the associated competitive interplay, similar to the one
observed in biological studies on different predators competing for the same
prey, or chemical species competing for the same reactant. The most notable
difference with respect to these classic fields of competition dynamics
consists in the fact that in social system dynamics the topology of the
underlying contact network is substantial. For example, social phenomena
involve different personalities with different number of contacts each, so
that the dynamics take place with an intrinsic distributed character and, in
particular, emergence of nearly homogeneous groups (so called clusters) is
possible. These differences make it necessary to analyze social behavior from
the view point of dynamic networks. It is particularly noteworthy that it has
been observed that the underlying network topology, unless changing in time,
always fulfills the same characteristics in terms of node degree distribution
(see e.g. [3, 4, 5]). For the purpose of understanding the mechanisms
underlying dynamic phenomena over dynamic networks, mathematical models can be
developed and analyzed formally as well as with numerical simulation studies.
With particular emphasis on the dynamics of decision competition in scale-free
networks, in [2] the regularity of spreading of information and public
opinions towards two competing products in complex networks was analyzed. They
proposed a simple linear model and simulated the associated decision
trajectories over time. It was highlighted that, in contrast to most models
studied so far via modified SIS and SIR models [6, 7, 8, 9, 10, 11, 12, 13,
14], in a real life setting there are frequently various information being
spread simultaneously (diverse virus, multiple opinions, rumors, etc.), and,
in contrast to the single opinion spread dynamics, these may be mutually
strenghtened or even annihilated. The competition-based dynamics of two
opinions which freely flow in a complex network was studied, showing some
interesting and important facts concerning the behavior of decision
competition. Nevertheless, studying the underlying mechanisms of mutual
influence between nodes in the network is expected to lead to a nonlinear
dynamics, just as in the SIS and SIR case mentioned above. Accordingly, it may
be supposed that the inherent competition mechanisms lead to classical
nonlinear phenomena such as multiple attractors and parameter sensitivity.
In the present study a mathematical model is introduced and analyzed for the
dynamics of two competing opinions (with neutral intermediate state) which
explicitly accounts for the kind of nonlinear interactions inherent to the
dynamics of competing opinions in complex social networks. A generalized model
for the interaction mechanisms is employed which has been recently proposed in
[13] and provides intermediate cases between the classical contact and
reactive process, and the influence of exogenous mass media opinion
propagation into the network is considered explicitly. The mathematical model
is analyzed with respect to attractor multiplicity, delimiting in this way
some system inherent behavior possibilities. In particular the study focusses
on the effect of opinion adaptation through (i) social contacts, and (ii)
exogenous mass media influence. In order to analyze these effects, a series of
numerical simulations is provided for the two extreme cases of contact and
reactive process [13], illustrating typical behavior in presence of:
* •
static contacts
* •
dynamic contacts
* •
exogenous mass media propagation of one opinion
* •
exogenous mass media propagation of both opinions.
The presented results illustrate (i) typical nonlinear behavior such as
attractor multiplicity, and sensitivity with respect to process parameters,
and (ii) the strong impact mass media have on the decision behavior. The
presented results establish an extension of those reported in [2].
From a methodological point of view, we are putting together the quantitative
and qualitative potential of mathematical modeling and simulation, with some
basic concepts of dynamical systems theory on attractor multiplicity and
stability on the basis of numerical simulation studies.
The paper is organized as follows. In Section 2, the nonlinear opinion
dynamics model for dynamic networks is derived based on a set of basic
assumptions, and the fixed points are established for the corresponding static
and unperturbed version. In Section 3, numerical simulations are presented
which illustrate the main behavior observed in decision dynamics in social
networks. In Section 4, the main contributions of the study are summarized.
## 2 The decision dynamics model
In this section a mathematical model for the prevalence of two opinions,
$\mathcal{A}$ and $\mathcal{B}$, with a neutral intermediate state
$\mathcal{N}$ is derived based on the Markov process assumption [15] that the
state at any time instant $t\geq 1$ ($t\in\mathbb{N}$) depends exclusively on
the state at the immediately preceding time instant $t-1$. The result is a
discrete time $3N$-dimensional nonlinear model, where $N$ is the number of
nodes in the network.
### 2.1 Basic assumptions and nomenclature
In order to derive the model we introduce the following basic assumptions:
* (A1)
Two opinions $\mathcal{A}$ and $\mathcal{B}$ are being propagated in a social
network with $N$ nodes and Power Law degree ($k$) distribution [16]
$\displaystyle P[k]\sim k^{-\gamma}$ (1)
(see Figure 1). The corresponding network topology is reflected in the square
adjacency matrix $A(t)$ with time-varying entries $A_{ij}(t)\in\\{0,1\\}$.
Figure 1: Power law degree distributed network with $N=100$ nodes, and
$P[k]\sim k^{-2.63}$.
* (A2)
The total number of nodes $N$ is constant.
* (A3)
Any node $i$ can be in one of three states: $\mathcal{A}$ if it has the
opinion $\mathcal{A}$, $\mathcal{B}$ if it has the opinion $\mathcal{B}$, and
$\mathcal{N}$ if it is neutral with respect to the opinions $\mathcal{A}$ and
$\mathcal{B}$.
* (A4)
In order to change from opinion $\mathcal{A}$ to $\mathcal{B}$ (or vice
versa), the node has to pass through the neutral state $\mathcal{N}$, i.e.
there is no direct connection between the states $\mathcal{A}$ and
$\mathcal{B}$. The associated state-diagram is depicted in Figure 2.
* (A5)
The probability for a node $i$ to be in the state $\mathcal{A}$,
$\mathcal{B}$, or $\mathcal{N}$, is denoted by $a_{i}$, $b_{i}$ and $n_{i}$,
respectively. It holds that $a_{i},b_{i},n_{i}\in[0,1]$ for any
$i=1,\ldots,N$.
* (A6)
If the node $i$ has the opinion $\mathcal{A}$ (or $\mathcal{B}$), it will
convince of opinion $\mathcal{A}$ (or $\mathcal{B}$) the nodes connected with
him with probability $\kappa_{A}$ (or $\kappa_{B}$).
* (A7)
A neutral node $i$ (i.e. a node which is in the state $\mathcal{N}$), does not
convince any neighbor node.
* (A8)
The impact any neighbor node $j\neq i$ has on node $i$ is independent of $j$.
* (A9)
The influences of any two nodes $j\neq k\neq i$ on node $i$ are mutually
independent. The total effect of all neighbors of node $i$ is given by the
mean influence.
* (A10)
The opinion $\mathcal{A}$ (or $\mathcal{B}$) is interchanged between two nodes
$i$ and $j$ $\lambda_{A}$ (or $\lambda_{B}$) times in each time step [13].
* (A11)
The mass media influence can be modeled in terms of an exogenous perturbation
with period of appearance $T$, taking into account the underlying automata
transition possibilities (see assumption A4 and Figure 2).
Figure 2: State transition diagram.
A direct consequence of assumptions A2 and A4 is that
$\displaystyle a_{i}(t)+b_{i}(t)+n_{i}(t)=1.$ (2)
Assumption A8 is motivated by the fact that any node $j$ has the same impact
on $i$ (assumption A7) and the final state of node $i$ after contacting his
neighbors will be a weighted sum of all particular contacts. Assumption A9 is
associated to the type of interchange mechanism [13]. Two extreme cases have
been reported in the literature: the contact process (with
$\lambda_{k}=1,\,k=A,B$), and the reactive process (with
$\lambda_{k}\to\infty,\,k=A,B$). In [13] a generalized model for the contact
rate was proposed covering all intermediate scenarios. In the present study
the model proposed in [13] for the contact rate is adapted by defining the
transmission rates associated to opinion $A$ and $B$ as
$\displaystyle
r_{ij}^{A}(t)=1-\left(1-\dfrac{1}{N_{i}(t)}\right)^{\lambda_{A}},\quad
r_{ij}^{B}(t)=1-\left(1-\dfrac{1}{N_{i}(t)}\right)^{\lambda_{B}}$ (3)
where $N_{i}(t)$ is the total number of neighbours of node $i$ at time $t$,
i,e.
$\displaystyle N_{i}(t)=\sum_{j\neq i}A_{i,j}(t).$ (4)
For the reactive process ($\lambda_{k}\to\infty,\,k=A,B$)
$r_{ij}^{k}=1,\,k=A,B$. The transition probabilities associated to the process
are denoted as follows:
* •
If the node $i$ is in the state $\mathcal{A}$, $\mathcal{B}$ or $\mathcal{N}$,
the probability for node $i$ of remaining in $\mathcal{A}$, $\mathcal{B}$, or
$\mathcal{N}$ is denoted by $\alpha_{i}$, $\beta_{i}$, or $\nu_{i}$,
respectively.
* •
If the node $i$ is in the neutral state $\mathcal{N}$, the probability of
being convinced of opinion $\mathcal{A}$ (or $\mathcal{B}$) is denoted by
$\mu_{i}$ (or $\sigma_{i}$).
### 2.2 Derivation of the Markov process model
Having as point of departure the preceding assumptions, the transition
probabilities $\alpha_{i},\beta_{i},\nu_{i},\mu_{i}$ and $\sigma_{i}$ are
determined.
As representative case, suppose that a node $i$ is of opinion $\mathcal{A}$
and has contact with a node $j\neq i$. The following events are possible:
* •
The node $j$ is in the state $\mathcal{A}$ (or $\mathcal{N}$) with probability
$a_{j}$ (or $n_{j}$), and consequently will not convince node $i$ of changing
its opinion.
* •
The node $j$ is in the state $\mathcal{B}$ with probability $b_{j}$, and
consequently is trying to convince node $i$ with probability of success given
by $r_{ij}^{B}\kappa_{B}$, and will fail to convince $i$ of opinion
$\mathcal{B}$ with probability $1-r_{ij}^{B}\kappa_{B}$, with $r_{ij}^{B}$
given in (3).
On the basis of these considerations, the probability for node $i$ of not
being convinced by node $j$ of changing from state $\mathcal{A}$ to
$\mathcal{N}$ (the only alternative to remaining in $\mathcal{A}$), is given
by the sum
$\displaystyle\alpha_{ij}=a_{j}+n_{j}+(1-r_{ij}^{B}\kappa_{B})b_{j}=1-r_{ij}^{B}\kappa_{B}b_{j},$
(5)
where the last equality is a consequence of property (2).
The same reasoning applies to the probability $\beta_{ij}$ of node $i$ being
in state $\mathcal{A}$ of not being convinced to change from the state
$\mathcal{B}$ to the state $\mathcal{N}$ by its neighbor $j\neq i$ [having in
mind property (2)]:
$\displaystyle\beta_{ij}=b_{j}+n_{j}+(1-r_{ij}^{A}\kappa_{A})a_{j}=1-r_{ij}^{A}\kappa_{A}a_{j}.$
(6)
In the case that node $i$ is in the state $\mathcal{N}$ and in contact with a
node $j\neq i$, the following events imply that $i$ remains in $\mathcal{N}$:
* •
The node $j$ is of opinion $\mathcal{N}$ with probability $n_{j}$, and does
not convince $i$ of changing its opinion.
* •
The node $j$ is of opinion $\mathcal{A}$ (or $\mathcal{B}$) with probability
$a_{j}$ (or $b_{j}$, and does convince $i$ of opinion $\mathcal{A}$ (or
$\mathcal{B}$) with probability $r_{ij}^{A}\kappa_{A}$ (or
$r_{ij}^{B}\kappa_{B}$), and fail with probability $1-r_{ij}^{A}\kappa_{A}$
(or $1-r_{ij}^{B}\kappa_{B}$).
Accordingly, having in mind property (2), the probability for node $i$ of not
being convinced by node $j\neq i$ is given by
$\displaystyle\nu_{ij}=n_{j}+(1-r_{ij}^{A}\kappa_{A})a_{j}+(1-r_{ij}^{B}\kappa_{B})b_{j}=1-r_{ij}^{A}\kappa_{A}a_{j}-r_{ij}^{B}\kappa_{B}b_{j},$
(7)
and the probabilities of node $j\neq i$ convincing node $i$ of opinion $A$ or
opinion $B$ are given respectively by
$\displaystyle\mu_{ij}=r_{ij}^{A}\kappa_{A}a_{j},\quad\sigma_{ij}=r_{ij}^{B}\kappa_{B}b_{j}.$
(8)
Next, to take into account the combined effect of all neighbors of node $i$ on
the transition probabilities, recall assumption A8. The overall transition
probabilities for node $i$ at time $t$ are given by:
$\displaystyle\begin{array}[c]{l}\alpha_{i}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}\alpha_{ij}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}[1-r_{ij}^{B}\kappa_{B}b_{j}(t)],\\\
\beta_{i}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}\beta_{ij}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}[1-r_{ij}^{A}(t)\kappa_{A}a_{j}(t)],\\\
\nu_{i}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}\nu_{ij}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}[1-r_{ij}^{A}(t)\kappa_{A}a_{j}(t)-r_{ij}^{B}(t)\kappa_{B}b_{j}(t)],\\\
\mu_{i}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}\mu_{ij}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}r_{ij}^{A}(t)\kappa_{A}a_{j}(t),\\\
\sigma_{i}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}\sigma_{ij}(t)=\dfrac{1}{N_{i}(t)}\sum_{j\neq
i}r_{ij}^{B}(t)\kappa_{B}b_{j}(t).\end{array}$ (14)
The associated discrete time model for the dynamics of decisions between
$\mathcal{A}$ and $\mathcal{B}$ in the network is given by
$\displaystyle\begin{array}[c]{l}a_{i}(t+1)=\alpha_{i}(t)a_{i}(t)+\mu_{i}(t)n_{i}(t),\quad\quad
a_{i}(0)=a_{i0}\\\
n_{i}(t+1)=\nu_{i}(t)n_{i}(t)+[1-\alpha_{i}(t)]a_{i}(t)+[1-\beta_{i}(t)]b_{i}(t),\quad
n_{i}(0)=n_{i0}\\\
b_{i}(t+1)=\beta_{i}(t)b_{i}(t)+\sigma_{i}(t)n_{i}(t),\quad\quad
b_{i}(0)=b_{i0}\\\ 0=1-[a_{i}(t)+b_{i}(t)+n_{i}(t)].\end{array}$ (19)
Taking into account the identities [see (14)]
$\displaystyle\mu_{i}(t)=1-\beta_{i}(t),\quad\sigma_{i}(t)=1-\alpha_{i}(t)$
(20)
model (19) is equivalent to
$\displaystyle\begin{array}[c]{l}a_{i}(t+1)=\alpha_{i}(t)a_{i}(t)+[1-\beta_{i}(t)][1-a_{i}(t)-b_{i}(t)],\quad\quad
a_{i}(0)=a_{i0}\\\
b_{i}(t+1)=\beta_{i}(t)b_{i}(t)+[1-\alpha_{i}(t)][1-a_{i}(t)-b_{i}(t)],\quad\quad
b_{i}(0)=b_{i0}\\\ n_{i}(t)=1-a_{i}(t)-b_{i}(t)\end{array}$ (24)
with $\alpha_{i}(t)$ and $\beta_{i}(t)$ defined in (14). In vector notation
the preceding dynamics are written as
$\displaystyle x_{i}(t+1)=f[x_{i}(t)],\quad x_{i}(0)=x_{i0},\quad
x_{i}=[a_{i},b_{i},n_{i}]\in T,\quad i=1,\ldots,N,$ (25)
where $T$ is the two-dimensional triangle set (see Figure 3)
$\displaystyle
T=\\{z=[z_{1},z_{2},z_{3}]^{\prime}\in[0,1]^{3}\subset\mathbb{R}^{3}\,|\,z_{1}+z_{2},+z_{3}=1\\}.$
(26)
Introduce the mean decision probabilities
$\displaystyle\rho_{A}=\dfrac{1}{N}\sum_{i=1}^{\mathcal{N}}a_{i},\quad\rho_{B}=\dfrac{1}{N}\sum_{i=1}^{\mathcal{N}}b_{i},\quad\rho_{N}=\dfrac{1}{N}\sum_{i=1}^{\mathcal{N}}n_{i}=1-\rho_{A}-\rho_{B},$
(27)
and the associated mean probability
$\rho(t)=[\rho_{a}(t),\rho_{b}(t),\rho_{n}(t)]^{\prime}$, it can be easily
seen that
$\displaystyle\rho(t)\in T,\quad\forall\,t\geq 0.$ (28)
In order to analyze the influence of mass media on the decision behavior,
consider the time-varying index subset
$\displaystyle I_{k}(t)\subset\\{1,\ldots,N\\},\quad\\#I_{k}=[\eta_{k}N],\quad
k=\mathcal{A},\mathcal{B},$ (29)
affecting $\eta\%$ of the total population (here $\\#I$ indicates the
cardinality of the set $I$), and introduce the associated characteristic
function
$\displaystyle\chi_{i}^{k}(t)=\left\\{\begin{array}[]{cc}1,&i\in I_{k}(t)\\\
0,&i\notin I_{k}(t)\end{array}\right.,\quad k=A,B.$ (32)
In terms of the index subset $I_{k},\,k=\mathcal{A},\mathcal{B}$ the mass
media influence can be modeled as follows:
$\displaystyle\begin{array}[c]{l}a_{i}(t+1)=\alpha_{i}(t)a_{i}(t)+[1-\beta_{i}(t)][1-a_{i}(t)-b_{i}(t)]+\\\
\hskip
142.26378pt+\chi_{i}^{A}(t)\upsilon^{A}(t)n_{i}(t)-\chi_{i}^{B}(t)\upsilon^{B}(t)a_{i}(t),\quad\quad
a_{i}(0)=a_{i0}\\\
b_{i}(t+1)=\beta_{i}(t)b_{i}(t)+[1-\alpha_{i}(t)][1-a_{i}(t)-b_{i}(t)]+\\\
\hskip
142.26378pt+\chi_{i}^{B}(t)\upsilon^{B}(t)n_{i}(t)-\chi_{i}^{A}(t)\upsilon^{A}(t)b_{i}(t),\quad\quad
b_{i}(0)=b_{i0}\\\ \end{array}$ (37)
where
$\displaystyle\upsilon^{k}(t)=\left\\{\begin{array}[]{cc}0,&t\neq
sT_{k},\,s\in\mathbb{N}\\\
1,&t=sT_{k},\,s\in\mathbb{N}\end{array}\right.,\quad k=A,B$ (40)
is a function which is nonzero only in discrete time instants
$sT_{k},s\in\mathbb{N},k=A,B$ with period $T_{k}$.
It should be noted that, in comparison to the study presented in [2], there
are some substantial differences to the present work. First, the model built
in [2] is linear while the model (37) is intrinsically nonlinear. Second, the
model (37) allows to take into account different modalities of transmission
process between the two extreme cases of contact process ($\lambda=1$) and
reactive process ($\lambda\to\infty$) as proposed in [13]. Third, in (37) the
influence of exogenous opinion propagation into the network is explicitly
considered.
### 2.3 Basic characterization of the dynamic behavior
Assume for the moment that the connections between individuals do not vary
over time (i.e., the adjacency matrix $A$ is constant, and hence $N_{i}$ is
constant for any node $i$). Based on this assumption the fixed points
associated to the dynamics (19) can de determined in order to establish limit
case conditions for the time evolution of the opinions $\mathcal{A}$ and
$\mathcal{B}$ in the network.
The fixed point condition read
$\displaystyle a_{i}(t+1)=a_{i}(t)=a_{i},\quad b_{i}(t+1)=b_{i}(t)=b_{i},\quad
n_{i}(t+1)=n_{i}(t)=n_{i}$ (41)
where $a_{i},b_{i},n_{i}$ are the fixed point values of the stochastic
variables $a_{i}(t),b_{i}(t)$ and $n_{i}$, respectively. It follows that
$\displaystyle\begin{array}[c]{l}0=(\alpha_{i}-1)a_{i}+(1-\beta_{i})n_{i},\\\
0=(\beta_{i}-1)b_{i}+(1-\alpha_{i})n_{i},\\\
0=-(1-\alpha_{i})a_{i}-(1-\beta_{i})b_{i}+(2-\alpha_{i}-\beta_{i})n_{i}.\end{array}$
(45)
Given that for $a_{i}=0$ (or $b_{i}=0$) for all $i$ it holds that
$\beta_{i}=1$ (or $\alpha_{i}=1$), it follows that the three vertexes
$\displaystyle x=\left[\begin{array}[]{c}1\\\ 0\\\ 0\end{array}\right],\quad
x=\left[\begin{array}[]{c}0\\\ 1\\\ 0\end{array}\right],\quad
x=\left[\begin{array}[]{c}0\\\ 0\\\ 1\end{array}\right]$ (55)
of the triangle set $T$ (26) (Figure 3) are particular fixed points.
Furthermore, solving the equation set (45) in general with respect to
$a_{i},b_{i}$ and $n_{i}$ yields the following set of implicit solutions
$\displaystyle a_{i}=\dfrac{1}{\xi_{i}^{2}+\xi_{i}+1},\quad
b_{i}=\dfrac{\xi_{i}^{2}}{\xi_{i}^{2}+\xi_{i}+1},\quad
n_{i}=1-a_{i}-b_{i}=\dfrac{\xi_{i}}{\xi_{i}^{2}+\xi_{i}+1},\quad
i=1,\ldots,N,$ (56)
parameterized by the scalar
$\displaystyle\xi_{i}=\dfrac{1-\alpha_{i}}{1-\beta_{i}}\in\mathbb{R},\quad\alpha_{i}=\dfrac{1}{N_{i}}\sum_{j\neq
i}(1-\kappa_{b}b_{j}),\,\beta_{i}=\dfrac{1}{N_{i}}\sum_{j\neq
i}(1-\kappa_{a}a_{j}).$ (57)
Introducing the solution in vector notation
$\displaystyle s_{i}=[a_{i},b_{i},n_{i}]^{\prime}\in T,$ (58)
the following limit cases can be directly derived:
$\displaystyle\kappa_{A}\to 0,\,\Rightarrow\,\xi_{i}\to\infty\,\text{ and
}\,s_{i}\to\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right],\quad\kappa_{B}\to 0\,\Rightarrow\,\xi_{i}\to 0\,\text{
and }\,s_{i}\to\left[\begin{array}[]{c}0\\\ 1\\\ 0\end{array}\right],$ (65)
corresponding to the two bottom vertexes of the triangle set $T$ (26) (Figure
3). In terms of the opinion density $\rho$ (28) there are four associated
fixed points
$\displaystyle\rho\in\left\\{\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right],\left[\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right],\dfrac{1}{N}\left[\begin{array}[]{c}\sum_{i=1}^{N}a_{i}\\\
\sum_{i=1}^{N}b_{i}\\\ \sum_{i=1}^{N}n_{i}\end{array}\right]\right\\},$ (78)
where $a_{i},b_{i}$ and $n_{i}$ are given in (56). Naturally, the limit cases
(65) can be expressed in terms of the mean probability solution $s_{\rho}$ as
$\displaystyle\kappa_{A}\to
0,\,\Rightarrow\,s_{\rho}\to\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right],\quad\kappa_{B}\to
0\,\Rightarrow\,s_{\rho}\to\left[\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right],\quad s_{\rho}=\dfrac{1}{N}\sum_{j\neq i}s_{i}.$ (85)
Figure 3: Sketch of the triangle set $T$ (26) where the decision probabilities
of each node $i$ and the mean probabilities $\rho$ evolve, and the curve $S$
parameterized by the scalar $\xi$ (57).
Summarizing, for any parameter combination
$(\kappa_{A},\kappa_{B},\lambda_{A},\lambda_{B})>0$ there are four fixed
points, and in the limit $\kappa_{A}=0$ (or $\kappa_{B}=0$) there are three
fixed points. This fact illustrates the multiplicity of fixed points and
sensitivity with respect to the dynamic’s parameters, and reveals that the
nonlinearity introduced by the interchange mechanisms is too strong to be
neglected. This fact establishes a main difference with respect to the recent
study [2], where a linear model was used for approximating the decision
dynamics. A formal analysis of the stability properties of the four fixed
points nevertheless goes beyond the scope of the present study, and here we
circumscribe ourselves to numerical simulation studies in order to find the
main behavior expected in decision dynamics in social networks, and study the
impact of mass media exogenous perturbations.
## 3 Simulation results
In this section, the dynamic behavior of decisions in social networks is
analyzed for four different scenarios:
* •
Static network, i.e. static adjacency matrix with power law (scale free)
distribution.
* •
Dynamic network, i.e. time-varying adjacency matrix with power law
distribution over any time interval.
* •
Dynamic network with exogenous mass media influence of one opinion on 18 $\%$
of the population with period $T_{A}=8$ time units.
* •
Dynamic network with exogenous mass media influence of both opinion on 10 $\%$
of the population with period $T_{A}=8$ time units and $T_{B}=16$ time units.
The simulations were taken out for a total population of $N=10.000$ nodes, and
a power law node degree $k$ distribution (1) with [16]
$\displaystyle\gamma=2.16.$ (86)
### 3.1 Static network without exogenous perturbations
In order to illustrate the main (nonlinear) behavior of the decision dynamics
process according to the fixed point multiplicity discussed in Section 2.3,
the two limit cases with $\lambda_{k}=1,k=A,B$ (contact process) and
$\lambda_{k}\to\infty,k=A,B$ (reactive process) are illustrated for three
different parameter scenarios. For the purpose at hand the projection of the
trajectories in the triangle set $T$ (26) onto the $(\rho_{a},\rho_{b}$)-plane
is presented. Note that accordingly it is possible that trajectories seem to
intersect in the projection.
For the purpose at hand consider the following three parameter sets:
$\displaystyle\kappa_{A}=0.1,\kappa_{B}=0.9\,(\kappa_{A}<\kappa_{B}),\quad\kappa_{A}=\kappa_{B}=0.5\,(\kappa_{A}=\kappa_{B}),\quad\kappa_{A}=0.9,\kappa_{B}=0.1\,(\kappa_{A}>\kappa_{B})$
(87)
and that $\mathcal{A}$ and $\mathcal{B}$ are propagated via contact processes
(i.e., $\lambda_{A}=\lambda_{B}=1$). The corresponding simulation results are
presented in Figure 4.
Figure 4: Projection onto the $(\rho_{a},\rho_{b})$-plane of the trajectories
of the mean decision probability $\rho$ (28) for the three different parameter
sets given in (87) and a contact transmission process.
It can be seen in Figure 4 that:
* (i)
for the first parameter set $(\kappa_{A}<\kappa_{B}$) all the trajectories
move towards an attractor in the upper left corner of the
($\rho_{A},\rho_{B}$)-plane.
* (ii)
for the second parameter set $(\kappa_{A}=\kappa_{B}$) there are two
attractors (one in the upper left and one in the lower right corner) towards
which the decision trajectory $\rho$ may converge, in dependence on the
initial value.
* (iii)
for the third parameter set $(\kappa_{A}>\kappa_{B}$) all the trajectories
move towards an attractor in the lower right corner of the
($\rho_{A},\rho_{B}$)-plane.
According to these results, there is a strong sensitivity, in the kind of
structural instability [17], of the decision behavior in the network on the
convincing parameter pair ($\kappa_{A},\kappa_{B}$), manifesting itself in
particular through the fact that, in the passage from the first to third
parameter set in (87), the attractor at the lower right corner initially is
unique, than coexists with a second attractor in the upper left corner, and
finally dissapears leaving the attractor in the upper left corner as the
unique one. Hence, for some parameter combinations the final decision state
will depend strongly on the initial condition.
It should be noticed that close to the fixed-point, the dynamics becomes very
slow in correspondence to the fact that almost the whole network is of one
single opinion, so that the existing contacts only confirm the present
decision state. This behavior is predicted by the model (24), given that close
to the fixed point the functions $\alpha_{i},\beta_{i}\approx 1$ for any node
$i$, and hence the system attains a slowly varying (i.e., quasi steady-state)
behavior according to
$\displaystyle a_{i}(t+1)\approx a_{i}(t),\quad b_{i}(t+1)\approx
b_{i}(t),\quad n_{i}(t+1)\approx n_{i}(t).$ (88)
Our conjecture is that this fact is also related to the generation of decision
clusters within the network.
In Figure 5 are presented the simulation results for the three parameter
combinations defined in (87), but for reactive processes with
$\lambda_{A}=\lambda_{B}\to\infty$, or equivalently $r_{ij}^{A}=r_{ij}^{B}=1$.
It can be observed that the global over-all behavior is quite similar to the
one observed for the contact process. Comparing the behavior over time, the
only substantial difference consists in that the convergence speed is
considerably faster for the reactive process, in accordance to the fact that
the transmission rate is much higher.
Figure 5: Proyection onto the $(\rho_{a},\rho_{b})$-plane of the trajectories
of the mean decision probability $\rho$ (28) for the three different parameter
sets given in (87) and a reactive transmission process.
### 3.2 Dynamic network without exogenous perturbations
Next, consider the case that the network connections vary with time in such a
way that the overall degree distribution (1) is maintained. In this case the
system dynamics (24) are no longer autonomous, and thus no fixed points exist
for the system (unless in the mean, i.e. in terms of $\rho_{k},k=A,B,N$,
asymptotical stability-like behavior may be observed).
The associated decision time evolution is compared in Figure 6 to the static
network case for the parameter value and initial condition set
$\displaystyle\kappa_{A}=0.1,\quad\kappa_{B}=0.9,\quad\rho_{a}(0)=\rho_{b}(0)=0.5,\quad\rho_{n}(0)=0$
(89)
and a reactive process (i.e., $r_{ij}^{k}=1,k=A,B$). It can be observed that
the over-all behavior is quite similar, with the important difference that for
the dynamic network case the trajectories are smoother and converge in about
$9$ time steps, while in the static network set-up still the fixed-point is
not reached. These observations are in line with the results presented in
[18], and are probably a result of the fact that in the dynamic network set-
up, i.e. with contacts changing at each time instant, the formation of
decision clusters is less probable than in the static network set-up. This
conjecture should be carefully studied in future research.
Figure 6: Time evolution of the mean decision trajectories associated to
decision $\mathcal{A}$ (thin grey line) and $\mathcal{B}$ (thick black line)
for the parameter value and initial condition set (89) and a reactive
transmission process in a scale-free dynamic network.
### 3.3 Dynamic network with mass media influence of one single opinion
To analyze the impact of mass media on the decision behavior of a dynamic
network, the case of exogenous propagation of opinion $\mathcal{A}$ is studied
based on the model (37) with
$\displaystyle T_{A}=8\text{ time units },T_{B}=\infty.$ (90)
In Figure 7 are presented the simulation results for the parameter values and
initial conditions
$\displaystyle\kappa_{A}=0.3,\,\kappa_{B}=0.6,\quad\rho_{A}(0)=0.8,\quad\rho_{B}(0)=0.2,\quad\rho_{N}(0)=0.$
(91)
If there were no external forces the trajectory would converge towards the
upper left corner corresponding to decision $\mathcal{B}$, as can be seen in
the corresponding time-evolution shown in the left sub-figure of Figure 7, and
in the projection onto the $(\rho_{a},\rho_{b}$)-plane shown in the right sub-
figure by the thick blue line. For the case that $\mathcal{A}$ is propagated
through an exogenous force with arbitrary $18\%$ of the population being
impacted each $T_{A}=8$ time units the corresponding time evolution is shown
in the central sub-figure of Figure 7, and the projection onto the
($\rho_{a},\rho_{b}$)-plane is shown in the right sub-figure by the black
line. It can be seen that at each period there is a force pushing $18\%$ of
the population towards decision $\mathcal{A}$. Accordingly, the trajectory
does not reach the attractor (corresponding to the decision $\mathcal{B}$),
but is maintained in an oscillatory regime about some intermediate decision
distribution. This illustrates the impact of exogenous forces on the dynamic
behavior in the network and the influence on the overall (mean) decision
vector $\rho$.
Figure 7: Left: Time evolution of the unperturbed decision dynamics on a
dynamic network for the parameter values and initial conditions given in (91)
and a reactive transmission process. Center: Time-evolution of the associated
decision dynamics with exogenous influence on $18\%$ of the population each
$T_{A}=8$ time units. Right: Projection onto the $(\rho_{a},\rho_{b})$-plane
of the trajectories of the mean decision probability $\rho$ (28) for the
unperturbed dynamics (thick blue line), and the one with exogenous
perturbation (black line).
### 3.4 Dynamic network with mass media influence of both opinions
In order to illustrate the competition of two external forces, consider the
case that both decisions are propagated through external forces into the
network (e.g. by exogenous mass media) but with different periods
$\displaystyle T_{A}=8\text{ time units},\quad T_{B}=16\text{ time units}.$
(92)
It is considered that both mechanisms affect $10\%$ of the total population to
their favor. The parameter values and initial conditions were set to
$\displaystyle\kappa_{A}=\kappa_{B}=0.5,\quad\rho_{1}(0)=[0.25,0.15,0.6]^{\prime},\quad\rho_{2}(0)=[0.15,0.25,0.6]^{\prime}$
(93)
corresponding to two representative cases as shown in Figure 8.
Figure 8: Projection onto the $(\rho_{a},\rho_{b})$-plane of two
representative trajectories of the mean decision probability $\rho$ (28) for
the parameter values and initial conditions given in (93) and a reactive
transmission process. The unperturbed trajectories are represented by the
thick blue lines, and the ones with exogenous perturbation by the black lines.
As illustrated by the thick blue lines, in the unperturbed case, the
trajectory starting at $\rho_{1}(0)$ (93) converges towards the attractor at
the upper left corner, and the trajectory starting at $\rho_{2}(0)$ (93)
converges towards the attractor at the lower rightt corner. In presence of the
exogenous perturbations with periods given in (92) and affecting both a total
number of $10\%$ of the population to their favor, the trajectories are given
by the black lines. It can be seen that the trajectory starting at
$\rho_{1}(0)$ does no longer converge towards the attractor at the upper left
corner, but enters into an oscillatory regime about some intermediate mean
decision value $\rho$, and the trajectory starting at $\rho_{2}(0)$ almost
converges towards the lower right corner but enters into an oscillatory regime
close to the corresponding attractor for the unperturbed trajectory.
These results illustrate the differences that may be caused by exogenous
propagation mechanisms on the decision dynamics in complex networks. In
particular it reveals that the final behavior still represents nonlinear
character implying that the final decision state will strongly depend on the
initial condition.
## 4 Concluding remarks
The decision dynamics in a scale-free network has been analyzed with respect
to inherent nonlinear convincing mechanisms, impact of network variations over
time, and mass media favoritism of one single or both opinions. A mathematical
model was derived on the basis of some assumptions establishing a quantitative
means to analyze the main mechanisms present in decision dynamics. Based on a
preliminary analysis of the system’s fixed points, it was shown that for a
static network there are between three and four fixed points, as a consequence
of the system nonlinearity. Numerical simulation studies were presented
illustrating the predicted behavior for static and dynamics scale-free
networks, which is in correspondence to the one expected in real networks. The
impact that mass media have on the decision dynamics was illustrated in
different characteristic situations.
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* [3] A.-L. Barabási and R. Albert. Emergence of scaling in random graphs. Science, 286:509–512, 1999.
* [4] M. Faloutsos, P. Faloutsos, and C. Faloutsos. On power-law relationships of the internet topology. In In Proceedings Sigcomm 1999, 1999.
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|
arxiv-papers
| 2012-10-30T22:51:38 |
2024-09-04T02:49:37.366444
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Rodr\\'iguez Lucatero, Luis Alarc\\'on, Roberto Bernal Jaquez,\n Alexander Schaum",
"submitter": "Roberto Bernal-Jaquez",
"url": "https://arxiv.org/abs/1210.8193"
}
|
1210.8214
|
# Well-posedness of a 3 D Parabolic-hyperbolic Keller-Segel System in the
Sobolev Space Framework
Chao Deng, Tong Li
###### Abstract
We study the global strong solutions to a 3-dimensional parabolic-hyperbolic
Keller-Segel model with initial data close to a stable equilibrium with
perturbations belonging to $L^{2}(\mathbb{R}^{3})\times
H^{1}(\mathbb{R}^{3})$. We obtain global well-posedness and decay property.
Furthermore, if the mean value of initial cell density is smaller than a
suitabale constant, then the chemical concentration decays exponentially to
zero as $t$ goes to infinity. Proofs of the main results are based on an
application of Fourier analysis method to uniform estimates for a linearized
parabolic-hyperbolic system and also based on the smoothing effect of the cell
density as well as the damping effect of the chemical concentration.
Keywords: Keller-Seger system; Sobolev space framework; parabolic-hyperbolic
system; well-posedness; Fourier analysis method.
Mathematics Subject Classification: 35Q92, 35B40, 35G55, 92C17
## 1 Introduction
In this paper, we study the following normalized 3-dimensional chemotaxis
model
$\displaystyle\left\\{\begin{aligned} &\partial_{t}u=\Delta
u+\nabla\cdot(u\nabla\ln v),\ \ \ \ \ \ \ \ \ \\\ &\partial_{t}v=uv-\mu
v\end{aligned}\right.$ (1.1)
for $t>0$ and $x\in\mathbb{R}^{3}$, where $u(x,t)$, $v(x,t)$ denote the cell
density and the chemical concentration, respectively. System (1.1) was
proposed by Othmer and Stevens [29] to describe the chemotactic movement of
particles where the chemicals are non-diffusible and can modify the local
environment for succeeding passages. For example, myxobacteria produce slime
over which their cohorts can move more readily and ants can follow trails left
by predecessors [10]. One direct application of (1.1) is to model haptotaxis
where cells move towards an increasing concentration of immobilized signals
such as surface or matrix-bound adhesive molecules.
With no loss of generality, by setting $w=\mu{t}+\ln{v}$ in (1.1), we get
$\displaystyle\left\\{\begin{aligned} &\partial_{t}u=\Delta
u+\nabla\cdot(u\nabla{w}),\\\ &\partial_{t}{w}=u,\\\
&(u,w)|_{t=0}=(u_{0},w_{0})\end{aligned}\right.\hskip 33.57404pt$ (1.2)
for $t>0$ and $x\in\mathbb{R}^{3}$. System (1.2) was studied in [34] in one-
dimensional case and was extended to multidimensional cases in [23, 24]. It
was studied in [29] and a comprehensive qualitative and numerical analysis was
provided there. We refer readers to Refs. [5, 7, 8, 10, 13, 14, 15, 16, 21,
22, 23, 25, 26, 27, 28, 30, 31, 34, 35, 36, 37] for more discussions in this
direction. Recently, in [23], the local and global existence of the classical
solution to (1.2) was studied when initial data
$(u_{0}-\bar{u},\nabla{w}_{0})\in
H^{\frac{5}{2}+}(\mathbb{R}^{3})\times{H^{\frac{5}{2}+}}(\mathbb{R}^{3})$ with
$\bar{u}$ being the mean value of $u_{0}$ ($s=\frac{5}{2}+$ stands for
$s>\\!\frac{5}{2}$ and similar conventions are applied throughout this paper).
Later on, Hao [12] studied global existence and uniqueness of global mild
solution for initial data close to some constant state in critical Besov space
with minimal regularity where the proof is in the Chemin-Lerner space
framework which was introduced by Chemin and Lerner [5] and aferwards
developed in a series of works (see e.g. [9]).
Noticing that the Cauchy problem of system (1.2) is invariant under the
following scaling transformations
$\Big{(}u(t,x),\;(\nabla{w})(t,x)\Big{)}\rightarrow\Big{(}\lambda^{2}u(\lambda^{2}t,\lambda
x),\;\lambda({\nabla}w)(\lambda^{2}t,\lambda{x})\Big{)}$
and
$\Big{(}u_{0}(x),\;({\nabla}w_{0})(x)\Big{)}\rightarrow\Big{(}\lambda^{2}u_{0}(\lambda
x),\;\lambda({\nabla}w_{0})(\lambda{x})\Big{)}.$
The idea of using a functional setting invariant by the scaling is now
classical and was originated from many works (see e.g. [3]). It is clear that
the critical Sobolev space for $(u_{0},\nabla{w}_{0})$ is $\dot{H}^{\hskip
0.28436pt-\hskip
0.28436pt\frac{1}{2}}(\mathbb{R}^{3})\times\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3})$
and, correspondingly,
$\dot{H}^{s-2}(\mathbb{R}^{3})\times\dot{H}^{s-1}(\mathbb{R}^{3})$
($s>\frac{3}{2}$) is the subcritical Sobolev space.
As for the critical case, it seems to be difficult to prove global existence
of mild solution to system (1.2) with
$(u_{0},\nabla{w}_{0})\in\dot{H}^{-\frac{1}{2}}(\mathbb{R}^{3})\times\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3})$
due to the invalidity of
$\dot{H}^{\frac{3}{2}}(\mathbb{R}^{3})\\!\hookrightarrow\\!L^{\infty}(\mathbb{R}^{3})$.
Thus a suitably smaller initial data space$--$the hybrid Besov space
$\dot{B}^{-\frac{1}{2}}_{2,1}(\mathbb{R}^{3})\times(\dot{B}^{\frac{1}{2}}_{2,1}(\mathbb{R}^{3})\cap\dot{B}^{\frac{3}{2}}_{2,1}(\mathbb{R}^{3}))$
and
$\dot{B}^{\frac{3}{2}}_{2,1}(\mathbb{R}^{3})\\!\hookrightarrow\\!L^{\infty}(\mathbb{R}^{3})$
were used in [12].
As for the subcritical case, we observe that $L^{2}(\mathbb{R}^{3})$ function
$(1+|x|^{2})^{\hskip-0.28436pt-\hskip-0.28436pt1}$ neither belongs to
$\dot{H}^{\hskip-0.28436pt-\hskip-0.28436pt\frac{1}{2}}(\mathbb{R}^{3})$ nor
to
$\dot{B}^{\hskip-0.56917pt-\hskip-0.28436pt\frac{1}{2}}_{2,1}(\mathbb{R}^{3})$.
Hence the case of $(u_{0},\nabla{w}_{0})\in\\!L^{2}(\mathbb{R}^{3})\times
H^{1}(\mathbb{R}^{3})$ can not be treated directly by applying results of the
critical case, cf. [12]. We believe that the Chemin-Lerner space framework can
be modified slightly to handle the subcritical cases. However, we do not
proceed to this way but consider well-posedness of mild solution in the
$L^{2}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ framework and Fourier
multiplier theory provides us with another option. Recalling the well known
weak solution theory for heat equation, we observe that searching a solution
$u$ in $C([0,\infty);L^{2}(\mathbb{R}^{3}))\cap
L^{2}(0,\infty;\dot{H}^{1}(\mathbb{R}^{3}))$ is also important to understand
(1.2). Meanwhile, assuming $\nabla{w}_{0}\in H^{1}(\mathbb{R}^{3})$ is
convenient to study the decay property of $w$.
Following similar energy arguments as in [23], one can decrease the indices of
the solution space $H^{s}(\mathbb{R}^{3})$ from $s=\frac{5}{2}+$ to $s=2$ or
even to $s=\frac{3}{2}+$, where $s=\frac{3}{2}$ seems to be unreachable for
energy arguments. Indeed, by multiplying (1.2) by some proper terms of $u$ and
$w$, then integrating by parts, we get
$\displaystyle\frac{d}{dt}\Big{(}\|u\|_{L^{2}}^{2}\\!+\\!\|\nabla
w\|_{L^{2}}^{2}\\!+\\!\|\Lambda^{{\frac{3}{2}+}}u\|_{L^{2}}^{2}\\!+\\!\|\Lambda^{{\frac{3}{2}+}}\nabla{w}\|_{L^{2}}^{2}\Big{)}\\!\leq\\!-2\|{\nabla}u\|_{L^{2}}^{2}+\|\nabla{(u^{2})}\cdot\nabla{w}\|_{L^{1}}$
$\displaystyle\;\;+\\!2\|\nabla{w}\cdot\nabla{u}\|_{L^{1}}-\\!2\|\Lambda^{{\frac{5}{2}+}}u\|_{L^{2}}^{2}+\|\nabla\Lambda^{{\frac{3}{2}+}}\\!p\cdot\\!\Lambda^{\frac{3}{2}+}(pq)\|_{L^{1}}\\!+\\!\|\nabla\Lambda^{\frac{3}{2}+}\\!{u}\cdot\\!\nabla\Lambda^{\frac{3}{2}+}w\|_{L^{1}}.$
Applying Hölder’s inequality and
$H^{\frac{3}{2}+}(\mathbb{R}^{3})\\!\hookrightarrow\\!L^{\\!\infty}(\mathbb{R}^{3})$
to the above inequality and following similar arguments of the proof of
Theorem 1.1 in [23], we get
$\displaystyle\frac{d}{dt}(\|u\|_{H^{\frac{3}{2}+}}^{2}+\|\nabla
w\|_{H^{\frac{3}{2}+}}^{2}+1)\lesssim-\|\nabla{u}\|_{H^{\frac{3}{2}+}}^{2}+(\|u\|_{H^{\frac{3}{2}+}}^{2}+\|\nabla{w}\|_{H^{\frac{3}{2}+}}^{2}+1)^{2}.$
Then a simple Gronwall argument yields local well-posedness of (1.2) for
$(u_{0},\nabla w_{0})\in H^{\frac{3}{2}+}(\mathbb{R}^{3})\times
H^{\frac{3}{2}+}(\mathbb{R}^{3})$. The methods used in the present article
serve as a supplement to the energy method.
Meanwhile, in system (1.2), one needs to consider two major terms $\Delta{u}$
and $u\Delta{w}$. It suffices to assume that all the second derivatives of $u$
and $w$ exist almost everywhere, although maybe certain higher derivatives
will not exist. Consequently, we also expect to establish well-posedness of
such solution to system (1.2) with initial data $(u_{0},\nabla{w}_{0})\in
H^{2}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$. Precisely, we will show
that the Cauchy problem of system (1.1) has a unique solution
$(u-\bar{u},\nabla((\mu-\bar{u})t+\ln v))$ in
$C([0,\infty);H^{2}(\mathbb{R}^{3}))\times
C([0,\infty);H^{1}(\mathbb{R}^{3}))$ provided that the initial data
$(u_{0},\nabla\ln v_{0})$ is close to some constant equilibrium state
$(\bar{u},0)$ and the difference $(u_{0}-\bar{u},\nabla\ln{v}_{0})$ belongs to
$\in{H}^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$ ($\bar{u}$ is
defined in (1.3)).
In the 4-dimensional case, scaling invariant discussion suggests that the
initial data space $L^{2}(\mathbb{R}^{4})\times H^{1}(\mathbb{R}^{4})$ is
critical. It is an interesting question whether the 4D model (1.2) has a
solution even locally in time with $(u_{0},\nabla w_{0})\in
L^{2}(\mathbb{R}^{4})\times{H}^{1}(\mathbb{R}^{4})$.
By modifying the definition of the mean value of $u$ in bounded domains, we
define
$\displaystyle\bar{u}=\lim_{R\rightarrow\infty}\frac{1}{|B_{R}|}\int_{B_{R}}u_{0}(x)dx,\hskip
28.45274pt$ (1.3)
where $B_{R}\subset\mathbb{R}^{3}$ is a ball centered at the origin with
radius $R$ and $u_{0}$ is the initial cell density. Applying $p=u-\bar{u}$,
$h=(\mu-\bar{u})t+\ln{v}$ and $\bar{u}=1$ to (1.1), we get
$\displaystyle\left\\{\begin{aligned} &\partial_{t}p=\Delta p+\Delta
h+\nabla\cdot(p\nabla h),\\\ &\partial_{t}h=p.\end{aligned}\right.\hskip
23.04666pt$
It is easy to check that for any positive constant $c$, if $(p,h)$ is a
solution to the above system, then $(p,h+\ln c)$ is also a solution. Or
equivalently, if $(u,v)$ is a solution to system (1.1), then $(u,cv)$ is also
a solution to system (1.1). It is natural to think $\nabla{h}$ as a new
unknown function whence $\nabla{h}$ is uniquely determined. Setting
$\Lambda=\sqrt{-\Delta}$, $q=-\Lambda h$ and
$G=\Lambda^{-1}\nabla\cdot(p\nabla\Lambda^{-1}q)$, we obtain the following
model
$\displaystyle\left\\{\begin{aligned} &\partial_{t}p=\Delta p+\Lambda
q-\Lambda G,\ \ \ \\\ &\partial_{t}q=-\Lambda p\ \ \end{aligned}\right.\hskip
45.23978pt$ (1.4)
for $t>0$ and $x\in\mathbb{R}^{3}$.
In this paper, we study system (1.4) with initial data $(p_{0}(x),q_{0}(x))\in
H^{k}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$ ($k=0,2$). More precisely,
we prove the global well-posedness of system (1.4) with small initial data
satisfying $(p_{0},q_{0})\in
H^{k}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$ ($k=0,2$, see Theorems 1.1
and 1.2). The main tools are Fourier transformation theory and the smoothing
properties of parabolic-hyperbolic coupled systems (see inequalities
(3.17)–(3.19) below for details). Particularly, from (3.12) and (3.17) as well
as definition of $m_{1}(t,\xi)$ for $|\xi|>2$ in ($M$), we observe that if
$|\xi|>4$, then
$\displaystyle
m_{1}(t,\xi)=\frac{e^{-\frac{t(1+\Xi)|\xi|^{2}}{2}}}{\frac{2\Xi}{\Xi+1}}-\frac{e^{-\frac{2t}{1+\Xi}}}{\frac{\Xi(\Xi+1)|\xi|^{2}}{2}}\quad\text{
with }\;\Xi=\sqrt{1-\frac{4}{|\xi|^{2}}}\in(\frac{\sqrt{3}}{2},1).$ (1.5)
Considering the smoothing effects, we need to study
$\partial_{t}^{k}\partial^{\alpha}m_{1}(t,D)$ with symbol
$\displaystyle\partial_{t}^{k}\xi^{\alpha}m_{1}(t,\xi)=-\frac{(1\\!+\\!\Xi)^{k+1}}{{(-2)^{k+1}\Xi}}|\xi|^{2k}\xi^{\alpha}{e^{-\frac{t(1+\Xi)|\xi|^{2}}{2}}}+\frac{(-2)^{k+1}}{{\Xi(\Xi+1)^{k+1}}}{\xi^{\alpha}}{|\xi|^{-2}}{e^{-\frac{2t}{1+\Xi}}},$
(1.6)
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\xi_{2}^{\alpha_{2}}\xi_{3}^{\alpha_{3}}$,
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}$, $k\in\mathbb{N}$
and $|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}\leq 2$. Indeed, for any
$p_{0}\in L^{2}\\!,$ $t>0$ and $|\xi|>4$, from (1.5) and (1.6) we have the
following smoothing property
$\displaystyle\|\partial_{t}^{k}\partial^{\alpha}m_{1}(t,D)p_{0}\|_{L^{2}}\leq\Big{(}C_{1}(k)\,t^{-\frac{|\alpha|}{2}-k}+C_{2}(k)\,e^{-t}\Big{)}\|p_{0}\|_{L^{2}}.$
(1.7)
However, following similar arguments as in (1.5)–(1.7), if $t>0$, $|\xi|>4$
and $q_{0}\in H^{1}$, then from (3.12), ($M$) and (3.17) we can only get
$\displaystyle\|\partial_{t}^{k}m_{2}(t,D)q_{0}\|_{H^{1}}\leq
C_{3}(k)\|q_{0}\|_{H^{1}},$ (1.8)
where no smoothing effect exists for spatial variable. Considering the low
frequency piece, smoothing properties of $m_{1}(t,D)$ and
$m_{2}(t,D)(-\Delta)^{-\frac{1}{2}}$ are similar to $e^{t\Delta}$, hence it is
omitted. In some cases, this special coupled system with such smoothing
effects is also called weak dissipative structure, see for instance [19]. For
various aspects of the smoothing properties, we refer the readers to see, for
instance [6, 9] and the references therein. The proof here is based on a
combination of the Fourier transform and estimates of the eigenvalues of the
corresponding characteristic matrix (see (3.7)–(3.19) below for details). The
different decay properties of the eigenvalues of the characteristic matrix
enable us to take advantages of the smoothing property of the high frequency
piece 111Definitions of the low, medium and high frequency pieces of a
function are given by (1.11). of $p$, i.e., $p\in
L^{1}(0,\infty;\dot{H}^{{7}/{4}}_{\psi})$ instead of that of $q$ since the
high frequency piece of $q$ does not have spatial smoothing effect (see (1.8)
above). The introduced $L^{1}(0,\infty;\dot{H}^{{7}/{4}}_{\psi})$ space is the
new point of this article. The main difficulty is to estimate
$\|p\nabla{q}\|_{L^{1}(0,\infty;L^{2})}$, which forces us to use frequency
decomposition or partition of unit and smoothing effect of the high frequency
piece of $p$ (see Lemma 3.2 below). Once
$\|p\nabla{q}\|_{L^{1}(0,\infty;L^{2})}$ being estimated, the desired result
follows from a standard fixed point argument. As for the decay property of $v$
in system (1.1), we apply the limiting case of the Sobolev inequality in $BMO$
(cf. for instance, [18]) to $v=c\,e^{(\bar{u}-\mu)t}e^{-\Lambda^{-1}q}$ hence
obtain lower and upper bounds for its $L^{\infty}$ norm which are stated in
(1.18)–(1.19).
Before stating the main results, we define the partition of unit. Let us
briefly explain how it may be built in $\mathbb{R}^{3}$. Let
$\mathcal{S}(\mathbb{R}^{3})$ be the Schwarz class and $(\eta,\varphi,\psi)$
be three smooth radially symmetric functions valued in $[0,1]$ such that
$\displaystyle{\rm
supp}\;\psi\subset\\{\xi\in\mathbb{R}^{3};\;|\xi|>2^{4}\\},$
$\displaystyle{\rm
supp}\;\varphi\subset\\{\xi\in\mathbb{R}^{3};\;1<|\xi|<2^{5}\\},$ (1.9)
$\displaystyle{\rm supp}\;\eta\subset\\{\xi\in\mathbb{R}^{3};\;|\xi|<2\\},$
$\displaystyle\eta(\xi)+\varphi(\xi)+\psi(\xi)=1,\ \forall\
\xi\\!\in\\!\mathbb{R}^{3}.$ (1.10)
For $f\in\mathcal{S}^{\prime}(\mathbb{R}^{3})$, we define the low, medium and
high frequency operators as follows:222
$f^{l}=\eta(D)f=\mathcal{F}^{-1}(\eta(\xi)\widehat{f}(\xi))$ and similar
conventions are applied throughout this paper.
$\displaystyle f^{l}=\eta(D){f},\hskip 11.38092ptf^{m}=\varphi(D){f},\hskip
11.38092ptf^{h}=\psi(D){f},\hskip 11.38092pt\eta(D)\psi(D){f}\equiv 0$ (1.11)
with $\eta(\xi),\varphi(\xi)$ and $\psi(\xi)$ being symbols of
$\eta(D),\varphi(D)$ and $\psi(D)$, respectively.
Throughout this paper, $\mathcal{F}f$ and $\widehat{f}$ stand for Fourier
transform of $f$ with respect to space variable and $\mathcal{F}^{-1}$ stands
for the corresponding inverse Fourier transform. For any $s\geq 0$ and any
function $f$, we shall define the fractional Riesz potential $\Lambda^{s}$ and
Bessel potential $\langle\Lambda\rangle^{s}:=(1-\Delta)^{\frac{s}{2}}$ via
$\displaystyle\widehat{\Lambda^{s}f}(\xi)=|\xi|^{s}\widehat{f}(\xi)\ \text{
and }\
\widehat{\langle\Lambda\rangle^{s}f}(\xi)=\langle\xi\rangle^{s}\widehat{f}(\xi)=({1+|\xi|^{2}})^{\frac{s}{2}}\widehat{f}(\xi),$
(1.12)
respectively. $\|\cdot\|_{L^{2}}$, $\|\cdot\|_{L^{\infty}}$,
$\|\cdot\|_{H^{s}}$ and $\|\cdot\|_{\dot{H}^{s}}$ denote the norms of the
usual Lebesgue measurable function spaces $L^{2}$, $L^{\infty}$, the usual
Bessel potential space
$\displaystyle H^{s}:=\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\
\|\langle\Lambda\rangle^{s}f\|_{L^{2}}<\infty\\}$ (1.13)
and Riesz potential space
$\displaystyle\dot{H^{s}}:=\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\
\|\Lambda^{s}f\|_{L^{2}}<\infty\\},$ (1.14)
respectively. Moreover, from (1.13) and (1.14), we observe that for any $s>0$,
there holds $H^{s}=\dot{H}^{s}\cap L^{2}$. For simplicity, for any
$s\in\mathbb{R}$, we define
$\displaystyle\dot{H}^{s}_{\psi}=\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\
\|f\|_{\dot{H}^{s}_{\psi}}=\|\Lambda^{s}\psi(D)f\|_{L^{2}}=\|\Lambda^{s}f^{h}\|_{L^{2}}<\infty\\},$
(1.15)
where $\dot{H}^{s}_{\psi}$ itself is not a Banach space since from (1.11) one
can prove that for any $g\in\mathcal{S}(\mathbb{R}^{3})$ satisfying ${\rm
supp}\,\widehat{g}\subset\\{\xi\in\mathbb{R}^{3};\;|\xi|<2^{4}\\}$ and
$f\in\dot{H}^{s}_{\psi}$, there holds
$\|f\|_{\dot{H}^{s}_{\psi}}=\|f+g\|_{\dot{H}^{s}_{\psi}}$. Hence we need to
introduce another Banach space $Z$ to get an intersection space
$Z\cap\dot{H}^{s}_{\psi}$ which forms a Banach space.
The function space $C([0,\infty);X)$ is equipped with norm
$\|f\|_{L^{\infty}_{t}X}$, where $X$ stands for some Banach space. For any two
quantities $A$ and $B$, we shall use the notation $A\lesssim B$ when $A\leq
CB$ for some positive constant $C$. The dependence of $C$ on various
parameters is usually clear from the context. $A\sim B$ if and only if
$A\lesssim B$ and $B\lesssim A$. For any $1\leq{\rho},r\leq\infty$, we denote
$L^{\rho}(0,\infty)$ and $L^{\rho}(0,\infty;L^{r})$ by $L^{\rho}_{t}$ and
$L^{\rho}_{t}L^{r}$, respectively.
We state the main results as follows.
###### Theorem 1.1.
For any initial data $(p_{0},{q}_{0})\in
L^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$, there exist positive
constants $C$ and $\varepsilon_{0}$ such that if
$\|(p_{0},{q}_{0})\|_{L^{2}\times H^{1}}\leq\varepsilon_{0}$, then system
(1.4) has a unique global solution $\,\,(p,q)\in{C}([0,\infty);L^{2})\times
C([0,\infty);H^{1})\,\,$ satisfying
$\displaystyle\|(p,\,q)\|_{L^{\infty}_{t}L^{2}\times{L^{\infty}_{t}H^{1}}}+\|(\nabla{p},\nabla{q})\|_{L^{2}_{t}L^{2}\times{L^{2}_{t}L^{2}}}+\|p\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\leq{C}\,\varepsilon_{0}.$
###### Theorem 1.2.
For any initial data $(p_{0},{q}_{0})\in
H^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})\\!,$ there exist positive
constants $C$ and $\varepsilon_{0}$ such that if
$\|(p_{0},{q}_{0})\|_{H^{2}\times H^{1}}\leq\varepsilon_{0}$, then system
(1.4) has a unique global solution
$\,\,(p,q)\in{C}([0,\infty);H^{2})\times{C}([0,\infty);H^{1})\,\,$ satisfying
$\displaystyle\|(p,q)\|_{L^{\\!\infty}_{t}H^{2}\times{L^{\\!\infty}_{t}H}^{1}}+\sup_{t>0}\,(1\\!+t)^{\frac{1}{2}}\|(\nabla{p},\nabla{q})\|_{L^{2}\times
L^{2}}+\sup_{t>0}\,(1\\!+t)^{\frac{7}{8}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}\leq{C}\varepsilon_{0}.$
Theorem 1.2 proves well-posedness of (1.4) with data
$(p_{0},q_{0})\\!\in\\!{H}^{2}(\mathbb{R}^{3})\\!\times\\!{H}^{\hskip-0.42677pt1}(\mathbb{R}^{3}).$
Notice that $L^{\infty}$ is also natural setting for cell density $u$ and
chemical concentration $v$. Based on the transformation of $(u,v)$ and
$(p,q)$, in order to study the $L^{\infty}$ norm decay of $(u,v)$, we add
${\displaystyle{\sup_{t>0}}}\,(1\\!+t)^{\frac{1}{2}}\|(\nabla{p},\nabla{q})\|_{L^{2}\times{L}^{2}}$
and
${\displaystyle{\sup_{t>0}}}\,(1\\!+t)^{\frac{7}{8}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}$.
Recall that if $(u,v)$ solves the system (1.1), then for any positive constant
$c$, $(u,cv)$ also solves system (1.1). Hence from the unique solution $(p,q)$
of (1.4), we have a sequence of solutions $(u,cv)$ such that
$v=c\,e^{(\bar{u}-\mu)t}e^{-\Lambda^{-1}q}$. Keeping this in mind and from
embedding theorems $\dot{H}^{\frac{5}{4}}\hookrightarrow{L}^{12}$,
$\dot{H}^{\frac{1}{2}}\hookrightarrow L^{3}\hookrightarrow{BMO}^{-1}$ as well
as Lemma 2.5 below, we get $v=c\,e^{(\bar{u}-\mu)t}e^{-\Lambda^{-1}q}$ and 333
We refer the readers to [32] to see definition of $BMO$ space and [17] to see
definition of $BMO^{-1}$ as well as embedding theorem $L^{n}\hookrightarrow
BMO^{-1}$.
$\displaystyle\|\Lambda^{-1}q\|_{L^{\infty}}$
$\displaystyle\leq{C}(1+\|\Lambda^{-1}q\|_{BMO}(1+\max\\{0,\ln{\|\Lambda^{-1}q\|_{W^{\frac{3}{4},12}}}\\})\,)$
$\displaystyle\leq{C}(1+\|q\|_{BMO^{-1}}(1+\max\\{0,\ln(\|\Lambda^{-1}q\|_{L^{12}}+\|\Lambda^{-\frac{1}{4}}q\|_{L^{12}})\\})\,)$
$\displaystyle\leq{C}(1+\|q\|_{\dot{H}^{\frac{1}{2}}}(1+\max\\{0,\ln(\|q\|_{\dot{H}^{\frac{1}{4}}}+\|q\|_{\dot{H}^{1}})\\})\,)$
$\displaystyle\leq{C}(1+\|\nabla{q}\|_{L^{2}}^{\frac{1}{2}}\|q\|_{L^{2}}^{\frac{1}{2}}(1+\max\\{0,\ln\|q\|_{{H}^{1}}\\})\,).$
(1.16)
In Theorem 1.2, we chose $\varepsilon_{0}$ such that $C\varepsilon_{0}\leq 1$.
Then from (1.16), we obtain that
$\displaystyle\|\Lambda^{-1}q\|_{L^{\infty}}\leq{C}(1+\|\nabla{q}\|_{L^{2}}^{\frac{1}{2}}\|q\|_{L^{2}}^{\frac{1}{2}})\leq{C}(1+\|\nabla{q}\|_{L^{2}}^{\frac{1}{2}}).$
(1.17)
Making use of definition of Banach valued series $e^{f}$, we observe that
$e^{f}$ is well defined if $f\in L^{\infty}$. Applying (1.17) to
$v=c\,e^{(\bar{u}-\mu)t}e^{-\Lambda^{-1}q}$, we get
$\displaystyle\frac{1}{c}\|v\|_{L^{\infty}}$
$\displaystyle=e^{(\bar{u}-\mu)t}\|e^{-\Lambda^{-1}q}\|_{L^{\infty}}\leq{e}^{(\bar{u}-\mu)t}\,e^{\,\|\Lambda^{-1}q\|_{L^{\infty}}}\leq{e}^{(\bar{u}-\mu)t}\,e^{\,C(1+\|\nabla{q}\|_{L^{2}}^{\frac{1}{2}})},$
(1.18) $\displaystyle\frac{1}{c}\|v\|_{L^{\infty}}$
$\displaystyle=e^{(\bar{u}-\mu)t}\|e^{-\Lambda^{-1}q}\|_{L^{\infty}}\geq{e}^{(\bar{u}-\mu)t}e^{-\|\Lambda^{-1}q\|_{L^{\infty}}}\\!\geq\\!{e}^{(\bar{u}-\mu)t}{e}^{-C(1+\|\nabla{q}\|_{L^{2}}^{\frac{1}{2}})}.$
(1.19)
From Theorem 1.2 and (1.18)–(1.19), we have the following result.
###### Corollary 1.3.
If initial data $(u_{0}-\bar{u},{v}_{0})$ satisfying
$(u_{0}-\bar{u},\nabla\ln{v_{0}})\in{H}^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$
and if there exists constant $\varepsilon_{0}>0$ such that
$\|(u_{0}-\bar{u},\nabla\ln v_{0})\|_{H^{2}\times{H}^{1}}\leq\varepsilon_{0}$,
then system (1.1) has a global solution $(u,v)$ satisfying
$\|(u-\bar{u},\nabla{\ln
v})\|_{L^{\\!\infty}_{t}H^{2}\times{L^{\\!\infty}_{t}H}^{1}}\\!\lesssim\varepsilon_{0}$,
$(u-\bar{u},\nabla\ln{v})\in{C}([0,\infty)\,;H^{2})\times{C}([0,\infty)\,;H^{1})$
and
$\sup_{t>0}\,(1\\!+t)^{\frac{1}{2}}\|(\nabla{u},\Delta{\ln}v)\|_{L^{2}\times
L^{2}}+\sup_{t>0}\,(1\\!+t)^{\frac{7}{8}}\|\Lambda^{\frac{7}{4}}u\|_{L^{2}}\lesssim\varepsilon_{0}$
and moreover
$\displaystyle\|u-\bar{u}\|_{L^{\infty}}\lesssim(1+t)^{-\frac{1}{2}}\text{ as
}t\rightarrow\infty\,;\quad\|v\|_{L^{\infty}}\sim{e}^{(\bar{u}-\mu)t}\text{ as
}t\rightarrow\infty.$ (1.20)
Plan of the paper: In Sect. 2 we introduce several preliminaries lemmas,
while in Sect. 3 we prove Theorems 1.1 and 1.2 and Corollary 1.3.
## 2 Preliminary lemmas
In this section, we list several known lemmas and prove some key lemma which
will be used in proving the well-posedness of the parabolic-hyperbolic
chemotaxis. The first lemma given below is concerned with functions whose
Fourier transforms are supported in low, medium and high frequency areas in
the frequency space. We note that the first two results are the well-known
Bernstein’s inequalities (cf. [20] Proposition 3.2 on page 24, or [1] Lemma
2.1 on page 52) and the last one is a direct applications of the Sobolev
embedding theorem.
###### Lemma 2.1.
If $(s,a,b)\in[0,\infty)\times[1,\infty]^{2}$, $a\leq{b}$ and
$f(x)\in{L}^{a}$, then for any two positive constants $c_{1}$ and $c_{2}$
there exists positive constant $c$ such that
$\displaystyle{\rm
supp}\;\widehat{f}\\!\subset\\{\xi\in\mathbb{R}^{3};\;|\xi|\leq
c_{2}\\},\hskip 40.40285pt\|\Lambda^{{s}}{f}\|_{L^{b}}\leq
c\;\\!c_{2}^{s+{n}(\frac{1}{a}-\frac{1}{b})}\,\|f\|_{L^{a}},$ (2.1)
$\displaystyle{\rm
supp}\;\widehat{f}\\!\subset\\{\xi\in\mathbb{R}^{3};\;c_{1}\\!<\\!|\xi|\\!<\\!c_{2}\\},\quad\quad\frac{c}{\kappa^{c_{1}^{\,s}}}\|{f}\|_{L^{a}}\leq\|\Lambda^{s}f\|_{L^{a}}\leq{c}\kappa^{c_{2}^{\,s}}\|f\|_{L^{a}},$
(2.2) $\displaystyle{\rm
supp}\;\widehat{f}\\!\subset\\{\xi\in\mathbb{R}^{3};\;|\xi|\geq
c_{1}\\},\hskip
40.68723pt\|{f}\|_{L^{a}}\leq\|\langle\Lambda\rangle^{s}f\|_{L^{a}}=\|f\|_{W^{s,p}},$
(2.3)
where $\kappa=\ln\frac{c_{2}}{c_{1}}$ and $W^{s,p}$ is the fractional Sobolev
space.
###### Proof.
The first two results are direct consequences of Proposition 3.2 of [20] and
Lemma 2.1 of [1] by using Littlewood-Paley decomposition, while the third
inequality is also a direct consequence of Sobolev embedding theorem. Hence we
finish the proof. ∎
Applying Lemma 2.1 with $2=a\leq{b}\leq\infty$ and $s\geq 0$ to $\eta(D)f$ and
$\psi(D)f$, we get
$\|\eta(D)f\|_{L^{b}}\lesssim\|f\|_{L^{2}}\;\text{ and
}\;\|\psi(D)f\|_{L^{2}}\lesssim\|\Lambda^{s}f\|_{L^{2}}.$
From (2.1)–(2.2), we have the following lemma concerning the $L^{2}$ Fourier
multiplier.
###### Lemma 2.2.
If $\;r\in[1,\infty]$, $v\in{L}^{2}$,
$m(t,\xi)\in\\!{L}^{r}_{t}L^{\\!\infty}_{\xi}$ and
$m(t,D)v=\\!\mathcal{F}^{-1}m(t,\xi)\widehat{v}(\xi)$, then we get
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}L^{2}}\leq\|m\|_{{L}^{r}_{t}L^{\\!\infty}_{\xi}}\|v\|_{L^{2}};$
(2.4)
if $\;r\\!\in\\![2,\infty]$, $v\\!\in\\!{L}^{2}$,
$m(t,\xi)|\xi|^{s}\\!\in\\!{L^{\\!\infty}_{\xi}{L}^{r}_{t}}$ and
$m(t,D)v\\!=\\!\mathcal{F}^{-1}m(t,\xi)\widehat{v}(\xi)$, then we get
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}\dot{H}^{s}}\leq\sup_{\xi\in\mathbb{R}^{3}}\Big{(}|\xi|^{s}\|m(\cdot,\xi)\|_{L^{r}_{t}}\Big{)}\|v\|_{L^{2}}.$
(2.5)
###### Proof.
The proof of (2.4) follows from the classical Fourier multiplier theory and
for readers convenience, we give the proof as follows
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}L^{2}}$
$\displaystyle=\|m(t,\cdot)\widehat{v}(\cdot)\|_{L^{r}_{t}L^{2}_{\xi}}\leq\|\|m(t,\cdot)\|_{L^{\infty}_{\xi}}\|_{L^{r}_{t}}\|\widehat{v}\|_{L^{2}_{\xi}}$
$\displaystyle\leq\|m\|_{L^{r}_{t}L^{\infty}_{\xi}}\|v\|_{L^{2}}.$
In order to prove (2.5), we need to use Plancherel equality, Minkowski’s
inequality, Hölder’s inequality and Plancherel equality again, i.e.,
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}\dot{H}^{s}}$
$\displaystyle=\|m(t,\cdot)|\cdot\\!|^{s}\widehat{v}(\cdot)\|_{L^{r}_{t}L^{2}_{\xi}}\lesssim\|m(t,\cdot)|\cdot\\!|^{s}\widehat{v}(\cdot)\|_{L^{2}_{\xi}{L^{r}_{t}}}$
$\displaystyle\leq\sup_{\xi\in\mathbb{R}^{3}}\Big{(}\|m(\cdot,\xi)\|_{L^{r}_{t}}|\xi|^{s}\Big{)}\|\widehat{v}\|_{L^{2}_{\xi}}.$
Hence we finish the proof. ∎
The skill we used in proving Lemma 2.2 will be used repeatedly in the
following subsections. In this paper, the multipliers satisfying the
assumptions of Lemma 2.2 are $e^{-ct|\xi|^{2}}$ and
$e^{-ct}\frac{1}{1+|\xi|^{2}}$ as well as $e^{-ct}$. The next lemma is devoted
to estimate the bilinear term which is known as the maximal
$L^{r}_{t}L^{\rho}$ regularity result for heat kernel (cf. [20], Chapter 7).
The operator $A$ defined by
$u(t,x)\mapsto Au(t,x)=\int_{0}^{t}e^{(t-\tau)\Delta}\Delta{u}(\tau,x)d\tau$
is bounded from $L^{r}_{t}L^{\rho}$ to $L^{r}_{t}L^{\rho}$ with
$1<r,\rho<\infty$. In this paper, we also need to establish a similar result
whose proof is even simpler in Sobolev spaces and hence we list it as the
following lemma.
###### Lemma 2.3.
If $|m(t,\xi)|\leq c_{1}(\frac{e^{-ct}}{1+|\xi|^{2}}+e^{-ct|\xi|^{2}})$ and
$|\mu(t,\xi)|\leq c_{1}e^{-ct}$ with positive constants $c$ and $c_{1}$,
$2\\!\leq\\!\rho\\!\leq\\!\infty$, $1\leq\\!{r}\leq\\!\rho_{1}\leq\\!\infty$
and $(u,v)\\!\in L^{2}_{t}L^{2}\\!\times\\!{L}^{r}_{t}L^{2}$, then we get
$\displaystyle\|\int_{0}^{t}m(t-\tau,D)u(\tau)d\tau\|_{L^{\rho}_{t}\dot{H}^{1+\frac{2}{\rho}}}\lesssim\|u\|_{L^{2}_{t}L^{2}},$
(2.6)
$\displaystyle\|\int_{0}^{t}\mu(t-\tau,D)u(\tau)d\tau\|_{L^{\rho_{1}}_{t}L^{2}}\lesssim\|u\|_{L^{r}_{t}L^{2}}.$
(2.7)
###### Proof.
By applying Plancherel equality, Lemma 2.2 with
$\|m(t,\cdot)\|_{L^{\infty}_{\xi}}\leq c_{1}e^{-ct}$ and integrability of
$e^{-ct}$, we see that the proof of (2.7) is quite straightforward. Hence it
suffices to prove (2.6). Noticing that $1+\frac{2}{\rho}\in[0,2]$, then by
making use of definition of Fourier transformation and Fubini theorem, we have
$\displaystyle\|\int_{0}^{t}m(t-\tau,D)$ $\displaystyle
u(\tau)d\tau\|_{L^{\rho}_{t}\dot{H}^{1+\frac{2}{\rho}}}=\|\int_{0}^{t}m(t-\tau,\xi)|\xi|^{1+\frac{2}{\rho}}\widehat{u}(\tau)d\tau\|_{L^{\rho}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|\int_{0}^{t}m(t-\tau,\xi)|\xi|^{1+\frac{2}{\rho}}\widehat{u}(\tau)d\tau\|_{L^{2}_{\xi}{L}^{\rho}_{t}}$
$\displaystyle\lesssim\|\int_{0}^{t}(e^{-c(t-\tau)|\xi|^{2}}|\xi|^{1+\frac{2}{\rho}}+e^{-c(t-\tau)})|\widehat{u}(\tau)|d\tau\|_{L^{2}_{\xi}{L}^{\rho}_{t}}$
$\displaystyle\lesssim\|\|\widehat{u}\|_{L^{2}_{t}}\|_{L^{2}_{\xi}}\sim\|\widehat{u}\|_{L^{2}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|u\|_{L^{2}_{t}L^{2}},$
where in the second, fourth and fifth inequalities we have used Minkowski,
Young’s inequality, Fubini theorem and Plancherel equality. Hence we finish
the proof ∎
The next lemma is about the Picard contraction argument (see e.g. [3]). We
will use this lemma to prove the main results concerning well-posedness of
system (1.4) with $(p_{0},q_{0})$ being chosen as $(x_{10},x_{20})$ and
initial data space $X_{10}\times{X}_{20}$ being
$L^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$ or
$H^{2}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$.
###### Lemma 2.4.
Let $({X_{10}}\times{X_{20}},\ \|\cdot\|_{{X_{10}}}+\|\cdot\|_{{X_{20}}})$ and
$({X_{1}}\times{X_{2}},\ \|\cdot\|_{{X_{1}}}+\|\cdot\|_{{X_{2}}})$ be abstract
Banach product spaces, $L_{1}:X_{10}\times{X_{20}}\rightarrow X_{1}$,
$L_{2}:X_{10}\times{X_{20}}\rightarrow{X_{2}}$,
$B_{1}:{X_{1}}\times{X_{2}}\rightarrow{X_{1}}$ and
$B_{2}:{X_{1}}\times{X_{2}}\rightarrow{X_{2}}$ are two linear and two bilinear
operators such that for any $(x_{10},x_{20})\in X_{10}\times{X}_{20}$,
$(x_{1},x_{2})\in{X_{1}}\times{X_{2}}$, $c>0$ and $i=1,2$, if
$\displaystyle\|L_{i}(x_{10},x_{20})\|_{X_{i}}\leq{c}(\|x_{10}\|_{{X_{10}}}+\|x_{20}\|_{{X_{20}}})\
\text{and}\
\|B_{i}(x_{1},x_{2})\|_{X_{i}}\leq{c}\|x_{1}\|_{{X}_{1}}\|x_{2}\|_{{X}_{2}},$
then for any $(x_{10},x_{20})\\!\in\\!{X_{10}}\\!\times\\!{X_{20}}$ with
$\|(x_{10},{x_{20}})\|_{{X_{10}}\\!\times\\!{X_{20}}}\\!<\\!\frac{1}{4c^{2}}$,
the following system
$(x_{1},x_{2})=(L_{1}(x_{10},x_{20}),L_{2}(x_{10},x_{20}))+(B_{1}(x_{1},x_{2}),\
B_{2}(x_{1},x_{2}))$
has a solution $(x_{1},x_{2})$ in ${X_{1}}\times{X_{2}}$. In particular, the
solution is such that
$\displaystyle\|(x_{1},{x_{2}})\|_{{X_{1}}\times{X_{2}}}\leq{4c\|(x_{10},{x_{20}})\|_{{X_{10}}\times{X_{20}}}}$
and it is the only one such that
$\|(x_{1},{x_{2}})\|_{{X_{1}}\times{X_{2}}}<\frac{1}{c}.$
The last lemma is the limiting case of the Sobolev inequality in $BMO$, see
[18].
###### Lemma 2.5.
For $n=\\!3$ and $s>\\!\frac{1}{4}$, there exists a constant $C$ depending on
$s$ so that
$\displaystyle\|f\|_{L^{\infty}}\leq{C}(1+\|f\|_{BMO}(1+\max\\{0,\ln{\|f\|_{W^{s,\,12}}}\\}))\quad\text{for
all}\;f\in W^{s,12}.$
## 3 Cauchy problem of parabolic-hyperbolic system (1.4)
In this section, we mainly use Fourier transformation framework to study the
well-posedness of (1.4) with initial data in Sobolev the space.
### 3.1 Linearization of (1.4) and the corresponding integral equations
In this subsection, we first study the linearized system of (1.4) around
$(p_{0},q_{0})$
$\displaystyle\frac{d}{dt}\left(\\!\begin{array}[]{c}p\\\ q\\\
\end{array}\\!\right)=\left(\\!\begin{array}[]{cc}-\Lambda^{2}&\Lambda\\\
-\Lambda&0\\\ \end{array}\\!\right)\left(\\!\begin{array}[]{c}p\\\ q\\\
\end{array}\\!\right).\hskip 36.98866pt$ (3.7)
Taking Fourier transform of (3.7) with respect to the space variable yields
$\displaystyle\frac{d}{dt}\left(\\!\begin{array}[]{c}\widehat{p}\\\
\widehat{q}\\\
\end{array}\\!\right)=L(\xi)\left(\\!\begin{array}[]{c}\widehat{p}\\\
\widehat{q}\\\ \end{array}\\!\right)\quad\text{with
}L(\xi)=\left(\\!\begin{array}[]{cc}-|\xi|^{2}&|\xi|\\\ -|\xi|&0\\\
\end{array}\\!\right).$
The characteristic polynomial of $L(\xi)$ is $X^{2}+|\xi|^{2}X+|\xi|^{2}$.
According to the size of $|\xi|$, we have the following three subcases:
$\bullet$ If $|\xi|>2$, then the characteristic polynomial possesses two
distinct real roots: $\lambda_{+}=\frac{|\xi|^{2}}{2}(-1+\Xi)$ and
$\lambda_{-}=\frac{|\xi|^{2}}{2}(-1-\Xi)$ with
$\Xi:=\sqrt{1-\frac{4}{|\xi|^{2}}}$. Since $\lambda_{1}\neq\lambda_{2}$, the
matrix $L(\xi)$ is diagonalizable. After computing the associated eigenspaces,
we find that
$\displaystyle\widehat{p}$
$\displaystyle=\big{(}\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}+\frac{e^{t\lambda_{-}}-e^{t\lambda_{+}}}{2\Xi}\big{)}\widehat{p}_{0}+\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{\Xi}\frac{\widehat{q}_{0}}{|\xi|},$
(3.8) $\displaystyle\widehat{q}$
$\displaystyle=\frac{e^{t\lambda_{-}}-e^{t\lambda_{+}}}{\Xi}\frac{\widehat{p}_{0}}{|\xi|}+\big{(}\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}+\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{2\Xi}\big{)}\widehat{q}_{0},$
(3.9)
where, for simplicity, we denote $\frac{e^{t\lambda_{+}}+e^{t\lambda_{-}}}{2}$
and $\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{2\Xi}$ by $\Omega_{1,t}(\xi)$
and $\Omega_{2,t}(\xi)$, respectively. Moreover, if there is no confusion, we
will denote $\Omega_{1,t}(\xi)$ and $\Omega_{2,t}(\xi)$ by $\Omega_{1,t}$ and
$\Omega_{2,t}$, respectively.
$\bullet$ If $|\xi|<2$, then the characteristic polynomial has two distinct
complex roots: $\lambda_{+}=-\frac{|\xi|^{2}}{2}-i\frac{\Theta|\xi|^{2}}{2}$
and $\lambda_{-}=-\frac{|\xi|^{2}}{2}+i\frac{\Theta|\xi|^{2}}{2}$ with
$\Theta:=\sqrt{-1+\frac{4}{|\xi|^{2}}}$. Noticing that
$\lambda_{1}\neq\lambda_{2}$, hence the matrix $L(\xi)$ is also
diagonalizable. After computing the associated eigenspaces, we get
$\displaystyle\widehat{p}$
$\displaystyle=\big{(}\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}+\frac{e^{t\lambda_{-}}-e^{t\lambda_{+}}}{-2i\Theta}\big{)}\widehat{p}_{0}+\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{-i\Theta}\frac{\widehat{q}_{0}}{|\xi|},$
(3.10) $\displaystyle\widehat{q}$
$\displaystyle=\frac{e^{t\lambda_{-}}-e^{t\lambda_{+}}}{-i\Theta}\frac{\widehat{p}_{0}}{|\xi|}+\big{(}\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}+\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{-2i\Theta}\big{)}\widehat{q}_{0},$
(3.11)
where, for simplicity, we denote $\frac{e^{t\lambda_{+}}+e^{t\lambda_{-}}}{2}$
and $\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{-2i\Theta}$ by $\Omega_{3,t}$
and $\Omega_{4,t}$, respectively.
$\bullet$ If $|\xi|=2$, $L(\xi)$ is not diagonalizable. However, this case can
be defined via $\lim_{|\xi|\rightarrow 2^{+}}$ and $\lim_{|\xi|\rightarrow
2^{-}}$ since the two limits not only exist, but also coincide.
Analysis of multipliers in (3.8)–(3.11) is rewriten into the following five
subcases:
1. $\bullet$ If $|\xi|>4$, then we obtain that $\frac{\sqrt{3}}{2}\\!<\Xi\\!<1$, $\lambda_{+}\\!=-\frac{2}{1+\Xi}$, $\lambda_{-}=-\frac{(1+\Xi)|\xi|^{2}}{2}$, $\Omega_{2,t}=\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{2\Xi}=\frac{e^{-\frac{2t}{1+\Xi}}(1-e^{-t|\xi|^{2}\Xi})}{2\Xi}$ and $\Omega_{1,t}-\Omega_{2,t}=\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}+\frac{e^{t\lambda_{-}}-e^{t\lambda_{+}}}{2\Xi}=\frac{e^{-\frac{t(1+\Xi)|\xi|^{2}}{2}}}{\frac{2\Xi}{\Xi+1}}-\frac{e^{-\frac{2t}{1+\Xi}}}{\frac{\Xi(\Xi+1)|\xi|^{2}}{2}}$ which yields that
$\displaystyle|\Omega_{1,t}-\Omega_{2,t}|\leq
2e^{-\frac{t|\xi|^{2}}{2}}+3e^{-t}\frac{1}{1+|\xi|^{2}}\quad\text{and}\quad|\Omega_{2,t}|\leq
e^{-t}.$ (3.12)
2. $\bullet$ If $2<|\xi|\leq 4$, then we have $0<\Xi\leq\frac{\sqrt{3}}{2}$, $\lambda_{+}\\!=-\frac{2}{1+\Xi}$, $\lambda_{-}\\!=-\frac{(1+\Xi)|\xi|^{2}}{2}$ and $\Omega_{1,t}=\frac{e^{t\lambda_{-}}+e^{t\lambda_{+}}}{2}=\frac{e^{-\frac{t(1+\Xi)|\xi|^{2}}{2}}+{e^{-\frac{2t}{1+\Xi}}}}{2}$. Applying ${1-e^{-|x|}}\leq{|x|}$ to $\Omega_{2,t}$ and noticing that $4\\!<\\!|\xi|^{2}\\!\leq\\!16$, there holds
$\displaystyle|\Omega_{1,t}|\leq e^{-t}\quad\text{and}\quad|\Omega_{2,t}|\leq
16e^{-\frac{t}{2}}.$ (3.13)
3. $\bullet$ If $1\leq|\xi|<2$, then we obtain that $0<\Theta|\xi|\leq\sqrt{3}$, $\lambda_{\pm}\>=-\frac{|\xi|^{2}}{2}\mp i\frac{\Theta|\xi|^{2}}{2}$, $\Omega_{3,t}=\frac{e^{t\lambda_{+}}+e^{t\lambda_{-}}}{2}=e^{-\frac{t|\xi|^{2}}{2}}{\cos{\frac{\Theta|\xi|^{2}t}{2}}}$ and $\frac{\Omega_{4,t}}{|\xi|}=\frac{e^{t\lambda_{+}}-e^{t\lambda_{-}}}{-2i\Theta|\xi|}=\frac{1}{2}e^{-\frac{t|\xi|^{2}}{2}}\frac{\sin{\frac{\Theta|\xi|^{2}t}{2}}}{\frac{\Theta|\xi|}{2}}=\frac{1}{2}e^{-\frac{t|\xi|^{2}}{2}}\frac{\sin{\frac{\Theta|\xi|^{2}t}{2}}}{\frac{\Theta|\xi|^{2}t}{2}}t|\xi|$. Applying $|\sin{x}|\leq|x|$, $|\cos x|\leq 1$ to $\Omega_{3,t}$ and $\Omega_{4,t}$, we get
$\displaystyle\frac{|\Omega_{4,t}|}{|\xi|}\leq 4e^{-\frac{t}{4}},\ \
|\Omega_{4,t}|\leq 8e^{-\frac{t}{4}}\ \ \text{and}\ \ |\Omega_{3,t}|\leq
e^{-\frac{t}{2}}.$ (3.14)
4. $\bullet$ If $|\xi|<1$, then we can prove that $\sqrt{3}<\Theta|\xi|<2$, $\lambda_{\pm}\>=\\!-\frac{|\xi|^{2}}{2}\mp i\frac{\Theta|\xi|^{2}}{2}$,
$\displaystyle\frac{|\Omega_{4,t}|}{|\xi|}\leq e^{-\frac{t|\xi|^{2}}{2}},\ \
|\Omega_{4,t}|\leq 4e^{-\frac{t|\xi|^{2}}{4}}\ \text{ and }\
|\Omega_{3,t}|\leq 2e^{-\frac{t|\xi|^{2}}{2}}.$ (3.15)
5. $\bullet$ If $|\xi|\\!\rightarrow\\!2$, then we obtain that $\lim_{|\xi|\rightarrow 2^{+}}\Xi=\lim_{|\xi|\rightarrow 2^{-}}\Theta=0$, $\lim_{|\xi|\rightarrow 2}\lambda_{+}=\lim_{|\xi|\rightarrow 2}\lambda_{-}=\\!-2$ and
$\displaystyle\lim_{|\xi|\rightarrow 2^{+}}\Omega_{1,t}=\lim_{|\xi|\rightarrow
2^{-}}\Omega_{3,t}=e^{-2t},\ \lim_{|\xi|\rightarrow
2^{+}}\Omega_{2,t}=\lim_{|\xi|\rightarrow 2^{-}}\Omega_{4,t}=2te^{-2t}.$
(3.16)
For simplicity, we define the following two multipliers:
$m_{1}(t,\xi)=\\!\left\\{\begin{aligned} &\Omega_{1,t}-\Omega_{2,t}&\text{if
}\hskip 2.84544pt&|\xi|>2,\\\ &e^{-2t}\\!-\\!2te^{\\!-2t}&\text{if }\hskip
2.84544pt&|\xi|=2,\ \\\ &\Omega_{3,t}-\Omega_{4,t}&\text{if }\hskip
2.84544pt&|\xi|<2,\end{aligned}\right.$
$m_{2}(t,\xi)=\\!\left\\{\begin{aligned} &{\Omega_{2,t}}&\text{if }\hskip
2.84544pt&|\xi|>2,\\\ &2te^{-2t}&\text{if }\hskip
2.84544pt&|\xi|=2,\quad\quad(M)\\\ &{\Omega_{4,t}}&\text{if }\hskip
2.84544pt&|\xi|<2.\end{aligned}\right.$ Applying (3.12)–(3.16) to
$m_{1}(t,\xi)$ and $m_{2}(t,\xi)$, we observe that $m_{1}(t,\xi)$ and
$m_{2}(t,\xi)$ are not only radial but also continuous with respect to
frequency variable $\xi$. Moreover, there exist constants $c$ and $c_{1}$ such
that if $|\xi|>2^{4}$, then we get
$\displaystyle|m_{1}(t,\xi)|\leq
c_{1}(e^{-ct|\xi|^{2}}+e^{-ct}\frac{1}{1\\!+\\!|\xi|^{2}})\ \text{ and }\
|m_{2}(t,\xi)|\leq c_{1}e^{-ct};$ (3.17)
if $1<|\xi|<2^{5}$, then we get
$\displaystyle|m_{1}(t,\xi)|+|m_{2}(t,\xi)|\leq c_{1}e^{-ct};$ (3.18)
else if $|\xi|<2$, then we get
$\displaystyle|m_{1}(t,\xi)|+|m_{2}(t,\xi)|+\frac{|m_{2}(t,\xi)|}{|\xi|}\leq
c_{1}e^{-ct|\xi|^{2}}.$ (3.19)
Next we study system (1.4) with data $(p_{0},q_{0})$ and write it into
equivalent integral equations. Taking Fourier transform of (1.4) with respect
to the space variable, applying the well-known Duhamel principle to
(3.8)–(3.11) and then applying the inverse Fourier transform, we get
$\displaystyle{p}=$
$\displaystyle\;m_{1}(t,D){p}_{0}+{2m_{2}(t,D)}\Lambda^{-1}{q}_{0}\\!-\\!\\!\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)\Lambda{G}(\tau)d\tau,$
(3.20) $\displaystyle{q}=$
$\displaystyle\\!-\\!{2m_{2}(t,D)}{\Lambda^{\\!\\!-1}}{{p}_{0}}+(m_{1}(t,D)+2m_{2}(t,D)){q}_{0}\\!-\\!2\\!\\!\int_{0}^{t}\\!\\!m_{2}(t\\!-\\!\tau,D){G}(\tau)d\tau,$
(3.21)
where $m_{1}(t,D)$ and $m_{2}(t,D)$ are symbols of $m_{1}(t,\xi)$ and
$m_{2}(t,\xi)$, respectively. From (3.20) and (3.21), for any
$(p_{0},q_{0})\in L^{2}\times H^{1}$, we define a map $\mathfrak{F}$ such that
$\displaystyle\mathfrak{F}({p,q})$
$\displaystyle=(\mathfrak{F}_{1}(p,q),\mathfrak{F}_{2}(p,q))=(\text{``r.h.s."
of (3.14)},\ \text{``r.h.s." of (3.15)}),$ (3.22)
where “r.h.s.” stands for “right hand side”.
The proof of Theorem 1.2 is similar but simpler than that of Theorem 1.1, thus
we prove Theorem 1.1 first.
### 3.2 Proof of Theorem 1.1
In this subsection, we first prove several a priori estimates including the
crucial bilinear estimates. We define the corresponding resolution spaces as
follows
$\displaystyle X\times
Y=\Big{\\{}(p,q)\in\\!C([0,\infty);L^{2})\\!\times\\!{C}([0,\infty);H^{1})\text{
and}\ \|p\|_{X}+\|q\|_{Y}\\!<\\!\infty\Big{\\}},$ (3.23)
where
$\|p\|_{X}:=\|p\|_{L^{\infty}_{t}L^{2}}+\|p\|_{L^{2}_{t}\dot{H}^{1}}+\|p\|_{{L}^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\;\text{
and }\;\|q\|_{Y}:=\|q\|_{L^{\infty}_{t}{H}^{1}}+\|q\|_{L^{2}_{t}\dot{H}^{1}}$.
In what follows, we prove several key estimates.
###### Proposition 3.1.
Let $(p,q)$ be a solution to system (1.4) with $(p_{0},q_{0})\in
L^{2}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ and $\mathfrak{F}$ and
$\mathfrak{F_{1}}$ be defined as in (3.22). Then there hold
$\displaystyle\|\mathfrak{F}({p,q})\|_{L^{\\!\infty}_{t}L^{2}\times{L}^{\\!\infty}_{t}H^{1}}\lesssim\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}\\!+\|G\|_{L^{2}_{t}L^{2}}+\|G\|_{{L}^{1}_{t}\dot{H}^{1}},$
(3.24)
$\displaystyle\|\mathfrak{F}({p,q})\|_{L^{2}_{t}\dot{H}^{1}\times{L^{2}_{t}\dot{H}^{1}}}\hskip
2.27626pt\lesssim\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}+\|G\|_{L^{2}_{t}L^{2}}+\|G\|_{L^{1}_{t}\dot{H}^{1}},$
(3.25)
$\displaystyle\|\mathfrak{F}_{1}({p,q})\|_{{L}^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\lesssim\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}+\|G\|_{{L}^{1}_{t}\dot{H}^{1}}.$
(3.26)
###### Proof.
In order to prove (3.24)–(3.26), from (3.21)–(3.23) we observe that we have to
establish several estimates whose proof will be divided into three parts.
Part I. Estimate of
$\|\mathfrak{F}({p,q})\|_{L^{\infty}_{t}L^{2}\times{L^{\infty}_{t}H^{1}}}$.
First, we derive the estimate for $\mathfrak{F_{1}}({p,q})$ defined in (3.20)
(3.22).
Noticing that any $L^{\infty}_{\xi}$ function $m(\xi)$ is an $H^{s}$ (or
$\dot{H}^{s}$) Fourier multiplier which means that for any $H^{s}$ (or
$\dot{H}^{s}$) function $f(x)$ (or $g$), there hold
$\displaystyle\left\\{\begin{aligned}
&\|m(D){f}\|_{H^{s}}=\|\langle\cdot\rangle^{s}m(\cdot)\widehat{f}(\cdot)\|_{L^{2}_{\xi}}\lesssim\|m\|_{L^{\infty}_{\xi}}\|f\|_{H^{s}},\\\
&\|m(D){g}\|_{\dot{H}^{s}}=\||\cdot\\!|^{s}m(\cdot)\widehat{g}(\cdot)\|_{L^{2}_{\xi}}\lesssim\|m\|_{L^{\infty}_{\xi}}\|g\|_{\dot{H}^{s}},\end{aligned}\right.$
(3.27)
where
$\|\langle\cdot\rangle^{s}\widehat{f}(\cdot)\|_{L^{2}_{\xi}}=\|f\|_{H^{s}}$
and
$\||\\!\cdot\\!|^{s}\widehat{f}(\cdot)\|_{L^{2}_{\xi}}=\|f\|_{\dot{H}^{s}}$.
For $m_{1}(t,D)$ and $m_{2}(t,D)\Lambda^{-1}$, from (3.17)–(3.19) and a simple
calculation, we have that $m_{1}(t,\xi),\frac{2m_{2}(t,\xi)}{|\xi|}\in
L^{\infty}_{t}L^{\infty}_{\xi}$. Hence by applying (2.4) with $r=\infty$ and
$s=0$ to $m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}$, we get
$\displaystyle\|m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}+\|q_{0}\|_{L^{2}}.$ (3.28)
As for $m_{1}(t,D)+2m_{2}(t,D)$ and
$m_{2}(t,D)\Lambda^{-1}\langle\Lambda\rangle$, from (3.17)–(3.19) and (3.28),
we observe that
$m_{1}(t,\xi)+2m_{2}(t,\xi),\frac{\langle\xi\rangle{m}_{2}(t,\xi)}{|\xi|}\in
L^{\\!\infty}_{t}L^{\\!\infty}_{\xi}$. Hence applying (2.4) with $r=\infty$
and $s=1$ to $m_{2}(t,D)\Lambda^{-1}p_{0}+(m_{1}(t,D)+2m_{2}(t,D))q_{0}$, we
get
$\displaystyle\|-2m_{2}(t,D)\Lambda^{-1}p_{0}+(m_{1}(t,D)+2m_{2}(t,D))q_{0}\|_{L^{\infty}_{t}H^{1}}$
$\displaystyle\lesssim\|2m_{2}(t,D)\Lambda^{-1}\langle\Lambda\rangle
p_{0}\|_{L^{\infty}_{t}L^{2}}+\|(m_{1}(t,D)+2m_{2}(t,D)){q}_{0}\|_{L^{\infty}_{t}H^{1}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}+\|q_{0}\|_{H^{1}}.$ (3.29)
Then we deal with the third term of (3.20). Applying (3.17)–(3.19) and (2.7)
with $r=2$, $\rho=\infty$, $s=1$ and $m(t,\xi)=m_{1}(t,\xi)$ to $G$, we get
$\displaystyle\|\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)\Lambda
G(\tau)d\tau\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle=\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)G(\tau)d\tau\|_{L^{\infty}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|G\|_{L^{2}_{t}L^{2}}.$ (3.30)
It remains to derive the estimate for $\mathfrak{F_{2}}({p,q})$ defined in
(3.21) (3.22). By partition of unit, we have $G=G^{l}+G^{m}+G^{h}$. Then from
Lemma 2.1, we get
$\displaystyle\|\int_{0}^{t}\\!m_{2}(t-\tau,D)G(\tau)d\tau\|_{L^{\infty}_{t}H^{1}}=\|\int_{0}^{t}\\!m_{2}(t-\tau,D)\langle\Lambda\rangle
G(\tau)d\tau\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\leq\|\int_{0}^{t}(m_{2}(t-\tau,D)\Lambda^{-1})\Lambda\langle\Lambda\rangle(G^{l}+G^{m})d\tau+m_{2}(t-\tau,D)\langle\Lambda\rangle{G}^{h}d\tau\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\leq\|\\!\int_{0}^{t}\\!\\!m_{2}(t-\tau,D)\Lambda^{-1}\langle\Lambda\rangle({G}^{l}\\!+\\!{G}^{m})d\tau\|_{L^{\infty}_{t}\dot{H}^{1}}+\|\\!\int_{0}^{t}\\!\\!m_{2}(t-\tau,D){G}^{h}d\tau\|_{L^{\infty}_{t}H^{1}}$
$\displaystyle:=I_{11}+I_{12}.$
Applying (3.17)–(3.19), (2.7) and Bernstein inequalities to $I_{11}$ and
$I_{12}$, we get
$\displaystyle I_{11}$
$\displaystyle=\|\int_{0}^{t}m_{2}(t-\tau,D)\Lambda^{-1}\langle\Lambda\rangle({G}^{l}+{G}^{m})d\tau\|_{L^{\infty}_{t}\dot{H}^{1}}$
$\displaystyle\leq\|\langle\Lambda\rangle({G}^{l}+{G}^{m})\|_{L^{2}_{t}L^{2}}$
$\displaystyle\lesssim\|\langle\cdot\rangle\eta(\cdot)+\langle\cdot\rangle\varphi(\cdot)\|_{L^{\infty}_{\xi}}\|G\|_{L^{2}_{t}L^{2}}$
$\displaystyle\lesssim\|G\|_{L^{2}_{t}L^{2}}$ (3.31)
and
$\displaystyle I_{12}$
$\displaystyle=\|\int_{0}^{t}m_{2}(t-\tau,D)\langle\Lambda\rangle{G}^{h}d\tau\|_{L^{\infty}_{t}L^{2}}\hskip
55.48277pt$
$\displaystyle\leq\|\Lambda^{-1}\langle\Lambda\rangle\psi(D){G}\|_{L^{1}_{t}\dot{H}^{1}}$
$\displaystyle\leq\|\||\cdot|^{-1}\langle\cdot\rangle\psi(\cdot)\|_{L^{\infty}_{\xi}}\|G\|_{\dot{H}^{1}}\|_{L^{1}_{t}}$
$\displaystyle\lesssim\|G\|_{L^{1}_{t}\dot{H}^{1}}$ (3.32)
where in (3.2) we have used the fact that
$0\leq\frac{\langle\xi\rangle\psi(\xi)}{|\xi|}\leq 2$.
Part II. Estimate of
$\|\mathfrak{F}({p,q})\|_{L^{2}_{t}\dot{H}^{1}\times{L^{2}_{t}\dot{H}^{1}}}$.
We first derive the estimate for $\mathfrak{F_{1}}({p,q})$ defined in (3.20)
(3.22).
For $m_{1}(t,D)$ and $m_{2}(t,D)\Lambda^{-1}$, from (3.17)–(3.19) and a simple
calculation, we observe that $|\xi|m_{1}(t,\xi)+{2m_{2}(t,\xi)}\in
L^{\infty}_{\xi}{L}^{2}_{t}$. Hence by applying (2.5) with $r=2$ and $s=1$ to
$m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}$, we get
$\displaystyle\|m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}+\|q_{0}\|_{L^{2}}.$ (3.33)
As for $m_{1}(t,D)+2m_{2}(t,D)$ and $m_{2}(t,D)\Lambda^{-1}$, from
(3.17)–(3.19) and (3.28), we observe that $|\xi|m_{1}(t,\xi)\in
L^{\infty}_{t}{L}^{2}_{\xi}$ and $m_{2}(t,\xi)\in
L^{\infty}_{\xi}{L^{2}_{t}}$. Hence applying (2.5) with $r=2$ and $s=1$ to
$m_{2}(t,D)\Lambda^{-1}p_{0}+(m_{1}(t,D)+2m_{2}(t,D))q_{0}$, we get
$\displaystyle\|-2m_{2}(t,D)\Lambda^{-1}p_{0}+m_{1}(t,D)q_{0}+2m_{2}(t,D)q_{0}\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}+\|q_{0}\|_{L^{2}}+\|\Lambda{q}_{0}\|_{L^{2}}$
$\displaystyle\lesssim\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}.$ (3.34)
We first deal with the third term on the r.h.s. of (3.20). Applying
(3.17)–(3.19) and (2.7) with $r=2$, $\rho=2$, $s=2$ and
$m(t,\xi)=m_{1}(t,\xi)$ to the term of $G$, we get
$\displaystyle\|\int_{0}^{t}m_{1}(t-\tau,D)\Lambda
G(\tau)d\tau\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle=\|\int_{0}^{t}\\!m_{1}(t-\tau,D)G(\tau)d\tau\|_{L^{2}_{t}\dot{H}^{2}}$
$\displaystyle\lesssim\|G\|_{L^{2}_{t}L^{2}}.$ (3.35)
It remains to derive the estimate for $\mathfrak{F_{2}}({p,q})$ defined in
(3.21) (3.22). Using similar ways in proving (3.2) and (3.2), we get
$\displaystyle\|\int_{0}^{t}\\!m_{2}(t\\!-\\!\tau,D)G(\tau)d\tau\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle=\|\int_{0}^{t}\\!m_{2}(t\\!-\\!\tau,D)\Lambda{G}(\tau)d\tau\|_{L^{2}_{t}L^{2}}$
$\displaystyle\lesssim\|G\|_{L^{2}_{t}L^{2}}+\|G^{h}\|_{L^{1}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|G\|_{L^{2}_{t}L^{2}}+\|G\|_{L^{1}_{t}\dot{H}^{1}}.$
(3.36)
Part III. Estimate of
$\|\mathfrak{F}_{1}({p,q})\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}$.
From maximal regularity results, (3.20) and (3.22), we observe that
$\displaystyle\|\mathfrak{F}_{1}(p,q)\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\leq$
$\displaystyle\
\|m_{1}(t,D)p_{0}\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}+2\|m_{2}(t,D)\Lambda^{-1}q_{0}\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}$
$\displaystyle+\|\int_{0}^{t}m_{1}(t-\tau)\Lambda{G}(\tau)d\tau\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}$
$\displaystyle:=$ $\displaystyle\ I_{21}+I_{22}+I_{23}.$ (3.37)
As for $I_{21}$, applying (3.17)–(3.18) and Lemma 2.1 to $I_{21}$ with
$|\xi|\geq 2^{4}$, we claim that
$\displaystyle\hskip 39.83368pt\,I_{21}$
$\displaystyle=\|m_{1}(t,D)p_{0}\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}$
$\displaystyle\lesssim\|e^{-ct|\xi|^{2}}|\xi|^{\frac{7}{4}}\psi(\xi)\widehat{p}_{0}\|_{L^{1}_{t}L^{2}_{\xi}}+\|e^{-ct}\psi(\xi)|\xi|^{\frac{7}{4}}\langle\xi\rangle^{-2}\|_{L^{1}_{t}L^{\infty}_{\xi}}\|\widehat{p}_{0}\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}.$ (3.38)
In order to show (3.2), it suffices to estimate
$\|e^{-ct|\xi|^{2}}|\xi|^{\frac{7}{4}}\psi(\xi)\widehat{p}_{0}\|_{L^{1}_{t}L^{2}_{\xi}}$
as follows
$\displaystyle\|e^{-ct|\xi|^{2}}|\xi|^{\frac{7}{4}}\psi($
$\displaystyle\xi)\widehat{p}_{0}\|_{L^{1}_{t}L^{2}_{\xi}}=\int_{0}^{1}\\!(\int_{\mathbb{R}^{3}}e^{-2ct|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)|\widehat{p}_{0}|^{2}d\xi)^{\frac{1}{2}}dt$
$\displaystyle+\int_{1}^{\infty}(\int_{|\xi|>2^{4}}e^{-2ct|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)|\widehat{p}_{0}|^{2}d\xi)^{\frac{1}{2}}dt$
$\displaystyle:=I_{211}+I_{212}.$
Applying Plancherel equality to $I_{211}$, we have
$\displaystyle{\sup_{\xi\in\mathbb{R}^{3}}}e^{-2ct|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)\lesssim
t^{-\frac{7}{4}}$ and
$\displaystyle I_{211}$
$\displaystyle=\int_{0}^{1}(\int_{\mathbb{R}^{3}}e^{-2ct|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)|\widehat{p}_{0}|^{2}d\xi)^{\frac{1}{2}}dt\quad\quad\;\;$
$\displaystyle\lesssim\int_{0}^{1}t^{-\frac{7}{8}}dt\|\widehat{p}_{0}\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}.$
For $|\xi|>2^{4}$ and $t>1$, we get
$e^{-2ct|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)\leq{e}^{-ct}e^{-c|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)\lesssim{e}^{-ct}$
and
$\displaystyle I_{212}$
$\displaystyle\lesssim\int_{1}^{\infty}(\int_{|\xi|>2^{4}}e^{-ct}e^{-c|\xi|^{2}}|\xi|^{\frac{7}{2}}\psi(\xi)|\widehat{p}_{0}|^{2}d\xi)^{\frac{1}{2}}dt$
$\displaystyle\lesssim\int_{1}^{\infty}{e}^{-ct}dt\|\widehat{p}_{0}\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|p_{0}\|_{L^{2}}.$
Similarly, applying (3.17)–(3.18) and Lemma 2.1 to $I_{22}$, we get
$\displaystyle\hskip 36.98866ptI_{22}$
$\displaystyle=\|m_{2}(t,D)\Lambda^{-1}q_{0}\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\lesssim\|e^{-ct}\psi(\xi)\|_{L^{1}_{t}L^{\infty}_{\xi}}\|q_{0}\|_{\dot{H}^{\frac{3}{4}}}$
$\displaystyle\lesssim\|q_{0}\|_{H^{1}}.$ (3.39)
It remains to estimate $I_{23}$. Noticing that the multipliers below can be
estimated as follows:
$m_{1}(t-\tau,\xi)|\xi|^{\frac{7}{4}}\lesssim(t-\tau)^{-\frac{7}{8}}$ and
$m_{1}(t-\tau,\xi)|\xi|^{\frac{11}{4}}\lesssim(t-\tau)^{-\frac{11}{8}}$,
respectively. Hence we get
$\displaystyle\hskip 39.83368ptI_{23}$
$\displaystyle=\|\int_{0}^{t}m_{1}(t-\tau)\Lambda{G}(\tau)d\tau\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}$
$\displaystyle=\|\int_{0}^{t}m_{1}(t-\tau)\Lambda^{\frac{7}{4}}\Lambda\psi(D){G}(\tau)d\tau\|_{L^{1}_{t}L^{2}}$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\int_{0}^{t}\\!\min\Big{\\{}(t-\tau)^{-\frac{7}{8}}\|\Lambda\psi(D){G}(\tau)\|_{L^{2}},\;(t-\tau)^{-\frac{11}{4}}\|\psi(D)G\|_{L^{2}}\Big{\\}}d\tau{dt}$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\int_{0}^{t}\\!\min\Big{\\{}(t-\tau)^{-\frac{7}{8}},\;(t-\tau)^{-\frac{11}{4}}\Big{\\}}\|{G}(\tau)\|_{\dot{H}^{1}}d\tau{dt}$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\int_{\tau}^{\infty}\\!\min\Big{\\{}(t-\tau)^{-\frac{7}{8}},\;(t-\tau)^{-\frac{11}{4}}\Big{\\}}dt\|{G}(\tau)\|_{\dot{H}^{1}}d\tau$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\min\Big{\\{}t^{-\frac{7}{8}},\;t^{-\frac{11}{4}}\Big{\\}}dt\|{G}(\tau)\|_{\dot{H}^{1}}d\tau$
$\displaystyle\lesssim\|G\|_{L^{1}_{t}\dot{H}^{1}}$ (3.40)
where in the fourth inequality we have applied (2.3) to $\psi(D)G$ with $s=1$
and $a=2$.
Combining the above arguments, we finish the proof. ∎
Recalling that $G=\Lambda^{-1}\nabla\cdot(p\nabla\Lambda^{-1}q)$ and Riesz
transforms are bounded in $L^{2}$, thus we need to estimate
$\|\nabla\cdot(p\nabla\Lambda^{-1}q)\|_{L^{1}_{t}L^{2}}=\|G\|_{L^{1}_{t}\dot{H}^{1}}$.
The following key lemma is devoted to estimating
$\|\nabla{p}\cdot\nabla\Lambda^{-1}{q}\|_{L^{1}_{t}L^{2}}$ and
$\|p\Lambda{q}\|_{L^{1}_{t}L^{2}}$, where
$\displaystyle\nabla\cdot(p\nabla\Lambda^{\\!-1}{q})=\nabla{p}\cdot\nabla\Lambda^{\\!-1}{q}-p\,\Lambda{q}.$
(3.41)
###### Lemma 3.2.
Let $X\times Y$ be defined in (3.23). If $u\in X$ and $v\in Y$, then we get
$\displaystyle\|u\nabla{v}\|_{L^{1}_{t}L^{2}}+\|{\nabla}uv\|_{L^{1}_{t}L^{2}}\lesssim\|u\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{2}_{t}\dot{H}^{1}}+\|u\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\|v\|_{L^{\infty}_{t}{H}^{1}},$
(3.42)
$\displaystyle\|uv\|_{L^{2}_{t}L^{2}}\lesssim\|u\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{\infty}_{t}H^{1}}.$
(3.43)
###### Proof.
At first, we prove (3.42). Recall that
$u{\nabla}v=(u^{l}+u^{m}){\nabla}v+u^{h}{\nabla}v$. By making use of Hölder’s
inequality, we have
$\displaystyle\|u{\nabla}v\|_{L^{1}_{t}L^{2}}$
$\displaystyle\leq\|(u^{l}+u^{m}){\nabla}v\|_{L^{1}_{t}L^{2}}+\|u^{h}{\nabla}v\|_{L^{1}_{t}L^{2}}$
$\displaystyle\lesssim\|u^{l}+u^{m}\|_{L^{2}_{t}L^{\infty}}\|{\nabla}v\|_{L^{2}_{t}L^{2}}+\|u^{h}\|_{L^{1}_{t}L^{\infty}}\|{\nabla}v\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle:\\!$ $\displaystyle=I_{31}+I_{32}$
where from (2.1) with $|\xi|<2^{5}$ and Sobolev embedding theorem, there holds
$\displaystyle I_{31}$
$\displaystyle=\|u^{l}+u^{m}\|_{L^{2}_{t}L^{\infty}}\|{\nabla}v\|_{L^{2}_{t}L^{2}}$
$\displaystyle\lesssim\|u^{l}+u^{m}\|_{L^{2}_{t}L^{6}}\|v\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|u^{l}+u^{m}\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|u\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{2}_{t}\dot{H}^{1}}$
(3.44)
where in the fourth inequality, we used the fact that $\eta(\xi)+\varphi(\xi)$
is an $L^{2}$-multiplier; From (2.3) with $|\xi|>2^{4}$ and Sobolev embedding
theorem $H^{\frac{7}{4}}\hookrightarrow L^{\infty}$, we get
$\displaystyle I_{32}$
$\displaystyle=\|u^{h}\|_{L^{1}_{t}L^{\infty}}\|{\nabla}v\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\lesssim\|u^{h}\|_{L^{1}_{t}{H}^{\frac{7}{4}}}\|v\|_{L^{\infty}_{t}H^{1}}\quad\quad\quad$
$\displaystyle\lesssim\|u\|_{L^{1}_{t}\dot{H}^{\frac{7}{4}}_{\psi}}\|v\|_{L^{\infty}_{t}H^{1}}.$
(3.45)
Estimate of ${\|\nabla}uv\|_{L^{1}_{t}L^{2}}$ is rather simple. By making use
of Hölder’s inequality, we get
$\displaystyle\|{\nabla}uv\|_{L^{1}_{t}L^{2}}$
$\displaystyle\lesssim\|\nabla{u}\|_{L^{2}_{t}L^{3}}\|v\|_{L^{2}_{t}L^{6}}\lesssim\|{u}\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{2}_{t}\dot{H}^{1}}.$
(3.46)
This proves (3.42).
It remains to prove (3.43). By making use of Hölder’s inequality, we get
$\displaystyle\|uv\|_{L^{2}_{t}L^{2}}$
$\displaystyle\lesssim\|u\|_{L^{2}_{t}L^{6}}\|v\|_{L^{\infty}_{t}L^{3}}\lesssim\|u\|_{L^{2}_{t}\dot{H}^{1}}\|v\|_{L^{\infty}_{t}H^{1}}.$
(3.47)
Finally, combining (3.2)–(3.47), we prove all the desired results. ∎
Applying (3.42) and (3.43) to
$\nabla{p}\cdot\nabla\Lambda^{-1}q-p\,\Lambda{q}$ and
$\Lambda^{-1}\nabla\cdot(p\nabla\Lambda^{-1}q)$, respectively, combining
Proposition 3.1, Lemma 3.2 and (3.23), we have the following a-priori
estimates.
###### Corollary 3.3.
Let $(p,q)$ be a solution to system (1.4) with
$(p_{0},q_{0})\in{L}^{2}(\mathbb{R}^{3})\times{H}^{1}(\mathbb{R}^{3})$ and
$\mathfrak{F}$ be defined as in (3.22). Then there holds
$\displaystyle\|\mathfrak{F}({p,q})\|_{X\times
Y}\lesssim\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}+\|(p,q)\|_{X\times{Y}}^{2}.$
Proof of Theorem 1.1. Applying Lemma 2.4, Corollary 3.3 and following a
standard fixed point argument, we prove Theorem 1.1 provided that
$\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}$ is small.
### 3.3 Proof of Theorem 1.2
In this subsection, we first prove the a priori estimates including the
crucial bilinear estimates as follows.
###### Proposition 3.4.
Let $(p,q)$ be a solution to system (1.4) with $(p_{0},q_{0})\in
H^{(}\mathbb{R}^{3})2\times H^{1}(\mathbb{R}^{3})$ and $\mathfrak{F}$ be
defined as in (3.22). Then there hold
$\displaystyle\|\mathfrak{F}({p,q})\|_{L^{\infty}_{t}H^{2}\times{L}^{\infty}_{t}H^{1}}\lesssim\|(p_{0},q_{0})\|_{H^{2}\times{H}^{1}}+\|p\|_{L^{\infty}_{t}H^{2}}\|q\|_{L^{\infty}_{t}H^{1}}.$
(3.48)
###### Proof.
We first derive the estimate for $\mathfrak{F_{1}}({p,q})$ defined in (3.20)
(3.22).
Applying $m_{1}(t,\xi),\frac{m_{2}(t,\xi)}{|\xi|}\in
L^{\infty}_{t}L^{\infty}_{\xi}$, $\frac{\langle\xi\rangle\psi(\xi)}{|\xi|}\in
L^{\infty}_{\xi}$ and (2.4) with $r=\infty$ and $s=0$ to
$m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}$, we get
$\displaystyle\|m_{1}(t,D)p_{0}+2m_{2}(t,D)\Lambda^{-1}q_{0}\|_{H^{2}}$
$\displaystyle\lesssim\|m_{1}(t,\xi)\langle\xi\rangle^{2}\widehat{p_{0}}\|_{L^{2}_{\xi}}+\|\frac{m_{2}(t,\xi)\langle\xi\rangle^{2}}{|\xi|}\widehat{q_{0}}\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|p_{0}\|_{H^{2}}+\|q_{0}\|_{H^{1}}.$ (3.49)
Similarly, noticing that
$m_{1}(t,\xi)+2m_{2}(t,\xi),\frac{\langle\xi\rangle{m}_{2}(t,\xi)}{|\xi|}\in
L^{\\!\infty}_{t}L^{\\!\infty}_{\xi}$, applying (2.4) with $r=\infty$ and
$s=1$ to $m_{2}(t,D)\Lambda^{-1}p_{0}+(m_{1}(t,D)+2m_{2}(t,D))q_{0}$, we get
$\displaystyle\|2m_{2}(t,D)\Lambda^{-1}p_{0}-(m_{1}(t,D)+2m_{2}(t,D))q_{0}\|_{L^{\infty}_{t}H^{1}}\lesssim\|p_{0}\|_{L^{2}}+\|q_{0}\|_{H^{1}}.$
(3.50)
Now we deal with the third term on the r.h.s. of (3.20). Applying
(3.17)–(3.19) and (2.7) with $r=2$, $\rho=\infty$, $s=1$ and
$m(t,\xi)=m_{1}(t,\xi)$ to $G$, we get
$\displaystyle\|\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)G(\tau)d\tau\|_{L^{\infty}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\|G\|_{L^{\infty}_{t}\dot{H}^{-1}}\lesssim\|G\|_{L^{\infty}_{t}L^{\frac{3}{2}}}$
$\displaystyle\lesssim\|p\|_{L^{\infty}_{t}H^{2}}\|q\|_{L^{\infty}_{t}H^{1}}$
(3.51)
and
$\displaystyle\|\int_{0}^{t}\\!$ $\displaystyle
m_{1}(t\\!-\\!\tau,D)\Lambda{G}(\tau)d\tau\|_{L^{\infty}_{t}\dot{H}^{2}}=\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)G(\tau)d\tau\|_{L^{\infty}_{t}\dot{H}^{3}}$
$\displaystyle\lesssim\|G\|_{L^{\infty}_{t}\dot{H}^{1}}\lesssim\|\nabla{p}\|_{L^{\infty}_{t}L^{6}}\|q\|_{L^{\infty}_{t}L^{3}}+\|p\|_{L^{\infty}_{t}L^{\infty}}\|\nabla{q}\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\lesssim\|p\|_{L^{\infty}_{t}H^{2}}\|q\|_{L^{\infty}_{t}H^{1}}.$
(3.52)
It remains to derive the estimate for $\mathfrak{F_{2}}({p,q})$ defined in
(3.21) (3.22). Using similar ways in proving (3.2) and (3.2), we get
$\displaystyle\|\int_{0}^{t}m_{2}(t-\tau,D)\Lambda^{-1}\langle\Lambda\rangle({G}^{l}+{G}^{m})d\tau\|_{L^{\infty}_{t}\dot{H}^{1}}\lesssim\|G\|_{L^{\infty}_{t}\dot{H}^{-1}}\lesssim\|G\|_{L^{\infty}_{t}L^{\frac{3}{2}}}$
(3.53)
and
$\displaystyle\|\int_{0}^{t}m_{2}(t-\tau,D)\langle\Lambda\rangle{G}^{h}d\tau\|_{L^{\infty}_{t}L^{2}}\lesssim\|G\|_{L^{\infty}_{t}H^{1}}$
(3.54)
where we have used the damping property of $G^{h}$, i.e.,
$\psi(\xi)m_{2}(t,\xi)\lesssim e^{-ct}$.
Combining the above arguments, we finish the proof. ∎
The following proposition is used to prove decay estimates of solutions to
(1.1).
###### Proposition 3.5.
Let $(p,q)$ be a solution to system (1.4) with $(p_{0},q_{0})\in
H^{2}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ and $\mathfrak{F}$ be
defined as in (3.22). Then there hold
$\displaystyle(1+t)^{\frac{1}{2}}\|\nabla\mathfrak{F}({p,q})\|_{L^{2}}+(1+t)^{\frac{7}{8}}\|\Lambda^{\frac{7}{4}}\mathfrak{F}_{1}(p,q)\|_{L^{2}}$
$\displaystyle\lesssim\|(p_{0},q_{0})\|_{H^{2}\times{H}^{1}}+\sup_{t>0}\Big{(}(1+t)^{\frac{1}{2}}\|(\nabla
p,\nabla q)\|_{L^{2}\times
L^{2}}\Big{)}^{2}+\sup_{t>0}\Big{(}(1+t)^{\frac{7}{8}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}\Big{)}^{2}.$
###### Proof.
Noticing that $m_{1}(t,\xi)|\xi|+{m_{2}(t,\xi)}\lesssim
e^{-ct|\xi|^{2}}|\xi|+e^{-ct}$, we have
$\displaystyle\|m_{1}(t,D)\Lambda{p}_{0}+2m_{2}(t,D)q_{0}\|_{L^{2}}$
$\displaystyle\lesssim\|m_{1}(t,\xi)|\xi|\widehat{p_{0}}\|_{L^{2}_{\xi}}+\|m_{2}(t,\xi)\widehat{q_{0}}\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim(1+t)^{-\frac{1}{2}}(\|p_{0}\|_{H^{1}}+\|q_{0}\|_{H^{1}})$
and
$\|m_{1}(t,D)\Lambda^{\frac{7}{4}}{p}_{0}+2m_{2}(t,D)\Lambda^{\frac{3}{4}}q_{0}\|_{L^{2}}\lesssim(1+t)^{-\frac{7}{8}}(\|p_{0}\|_{H^{2}}+\|q_{0}\|_{H^{1}}).$
Similarly,
$\displaystyle\|2m_{2}(t,D)p_{0}-(m_{1}(t,D)\Lambda+2m_{2}(t,D)\Lambda)q_{0}\|_{L^{2}}\lesssim(1+t)^{-\frac{1}{2}}(\|p_{0}\|_{H^{1}}+\|q_{0}\|_{H^{1}}).$
As for the third term on the r.h.s. of (3.20), by using
$m_{1}(t,\xi)|\xi|\lesssim e^{-ct|\xi|^{2}}|\xi|+e^{-ct}$, chain rule,
Plancherel equality and Sobolev embedding, we get
$\displaystyle\|\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)\Delta{G}(\tau)d\tau\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}(1+\tau)^{-\frac{5}{4}}d\tau\sup_{\tau>0}(1+\tau)^{\frac{5}{4}}(\|{\nabla}p\|_{L^{3}}\|q\|_{L^{6}}+\|\nabla{q}\|_{L^{2}}\|p\|_{L^{\infty}})$
$\displaystyle\lesssim(1+t)^{-\frac{1}{2}}\sup_{\tau>0}\Big{(}(1+\tau)^{\frac{3}{4}}\|\Lambda^{\frac{3}{2}}p\|_{L^{2}}+(1+\tau)^{\frac{7}{4}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}\Big{)}(1+\tau)^{\frac{1}{2}}\|\nabla
q\|_{L^{2}}.$
Similarly, we have
$\displaystyle\|\int_{0}^{t}\\!m_{1}(t\\!-\\!\tau,D)\Lambda^{\frac{7}{4}}\nabla{G}(\tau)d\tau\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}(t-\tau)^{-\frac{7}{8}}(1+\tau)^{-\frac{5}{4}}d\tau\sup_{\tau>0}(1+\tau)^{\frac{5}{4}}(\|{\nabla}p\|_{L^{3}}\|q\|_{L^{6}}+\|\nabla{q}\|_{L^{2}}\|p\|_{L^{\infty}})$
$\displaystyle\lesssim(1+t)^{-\frac{7}{8}}\sup_{\tau>0}\Big{(}(1+\tau)^{\frac{3}{4}}\|\Lambda^{\frac{3}{2}}p\|_{L^{2}}+(1+\tau)^{\frac{7}{4}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}\Big{)}(1+\tau)^{\frac{1}{2}}\|\nabla
q\|_{L^{2}},$
and
$\displaystyle\|\int_{0}^{t}m_{2}(t-\tau,D)\nabla{G}d\tau\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}(1+\tau)^{-\frac{5}{4}}d\tau\sup_{\tau>0}(1+\tau)^{\frac{5}{4}}(\|{\nabla}p\|_{L^{3}}\|q\|_{L^{6}}+\|\nabla{q}\|_{L^{2}}\|p\|_{L^{\infty}})$
$\displaystyle\lesssim(1+t)^{-\frac{1}{2}}\sup_{\tau>0}\Big{(}(1+\tau)^{\frac{3}{4}}\|\Lambda^{\frac{3}{2}}p\|_{L^{2}}+(1+\tau)^{\frac{7}{4}}\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}\Big{)}(1+\tau)^{\frac{1}{2}}\|\nabla
q\|_{L^{2}}.$
Combining the above arguments and
$\|\Lambda^{\frac{3}{2}}p\|_{L^{2}}\lesssim\|\Lambda^{\frac{7}{4}}p\|_{L^{2}}^{\frac{2}{3}}\|\nabla
p\|_{L^{2}}^{\frac{1}{3}}$, we finish the proof. ∎
Proof of Theorem 1.2. Applying Lemma 2.4, Propositions 3.4 and 3.5, following
standard fixed point argument, we prove Theorem 1.2 provides that
$\|(p_{0},q_{0})\|_{L^{2}\times{H}^{1}}$ is small.
### 3.4 Proof of Corollary 1.3
Proof of Corollary 1.3. Applying Lemma 2.4 and Corollary 3.3 to system (1.1),
we prove the existence results of Corollary 1.3. As for the decay property of
$v$, we use (1.18)–(1.19). We omit the details.
Acknowledgmens: Chao Deng is supported by PAPD of Jiangsu Higher Education
Institutions, by JSNU under Grant No. 9212112101, and by the NSFC under Grant
No. 11171357 & 11271166; he would like to express his gratitude to Professor
Congming Li’s invitation to the Colorado University at Boulder where part of
this work was done. Tong Li would like to thank Congming Li for his
friendship.
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| Chao Deng
---|---
| Math Department, Jiangsu Normal University,
| Xuzhou, Jiangsu 221116, China
| Email: deng315@yahoo.com.cn
|
| Tong Li
| Math Department, University of Iowa,
| Iowa city, IA 52242
| Email: tong-li@uiowa.edu
|
arxiv-papers
| 2012-10-31T02:02:39 |
2024-09-04T02:49:37.376719
|
{
"license": "Public Domain",
"authors": "Chao Deng and Tong Li",
"submitter": "Chao Deng",
"url": "https://arxiv.org/abs/1210.8214"
}
|
1210.8285
|
# On Poincaré series of unicritical polynomials at the critical point
Juan Rivera-Letelier† Juan Rivera-Letelier, Facultad de Matemáticas,
Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860,
Santiago, Chile riveraletelier@mat.puc.cl and Weixiao Shen‡ Weixiao Shen,
Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
matsw@nus.edu.sg
###### Abstract.
In this paper, we show that for a unicritical polynomial having _a priori_
bounds, the unique conformal measure of minimal exponent has no atom at the
critical point. For a conformal measure of higher exponent, we give a
necessary and sufficient condition for the critical point to be an atom, in
terms of the growth rate of the derivatives at the critical value.
† Partially supported by FONDECYT grant 1100922, Chile.
‡ Partially supported by National University of Singapore grant
C-146-000-032-001
## 1\. Introduction
Let $f:\mathbb{C}\to\mathbb{C}$ be a polynomial of degree at least $2$, and
let $J(f)$ denote its Julia set. An important tool to study the fractal
dimensions of the Julia set is the Patterson-Sullivan conformal measures.
Given $t>0$, a _conformal measure of exponent $t$ for $f$_ is a Borel
probability measure $\mu$ supported on $J(f)$, such that for each Borel set
$E$ on which $f$ is injective we have
$\mu(f(E))=\int_{E}|Df|^{t}\ d\mu.$
Sullivan showed that for every polynomial there is $t>0$ and a conformal
measure of $f$ of exponent $t$, see [Sul83]. Moreover, Denker, Przytycki, and
Urbański showed that the minimal exponent for which such a measure exists
coincides with the _hyperbolic dimension of $f$_, defined by
$\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f):=\\{\operatorname{HD}(X):X\emph{
hyperbolic set of~{}$f$}\\},$
where $\operatorname{HD}(X)$ denotes the Hausdorff dimension of $X$, see
[DU91, Prz93], and also [McM00, PU10, Urb03]. In the uniformly hyperbolic
case, there is a unique conformal measure of minimal exponent, and this
measure coincides with the normalized Hausdorff measure of dimension equal to
$\operatorname{HD}(J(f))$. Under certain non-uniformly hyperbolicity
assumptions, it has also been proved that there is a unique conformal measure
of minimal exponent, and that this measure is supported on the “conical Julia
set”, which roughly speaking is the expanding part of the Julia set, see for
example [Prz99, GS09, PRL07, RLS10, Urb03] and references therein. In these
cases, the conformal measure of minimal exponent is used to prove that the
hyperbolic dimension coincides with the Hausdorff and the box dimensions of
the Julia set. However, there are conformal measures of minimal exponent that
are not supported on the conical Julia set, see [Urb03].
In this paper, we shall study atoms of conformal measures of polynomials
having precisely $1$ critical point; we call such a polynomial _unicritical_.
Note that if $f$ is a unicritical polynomial, then its degree $d$ is at least
$2$, and there is $c$ in $\mathbb{C}$ such that $f$ is affine conjugate to the
polynomial $z^{d}+c$. A unicritical polynomial written in this form is
_normalized_. We shall make the following assumption.
###### Definition 1.
Let $f$ be a unicritical polynomial whose critical point is non-periodic and
recurrent. Assume for simplicity that $f$ is normalized, so its critical point
is $0$. Then we say $f$ has _a priori bounds_ , if there exists $\tau>0$ such
that for each $\varepsilon>0$ there exists a topological disk $V$ containing
$0$, satisfying $\operatorname{diam}(V)<\varepsilon$, and such that the
following holds: For each integer $n\geq 1$ such that $f^{n}(0)$ is in $V$,
the connected component $U$ of $f^{-n}(V)$ that contains $0$ satisfies
$\overline{U}\subset V$, and there is annulus $A$ contained in
$V\setminus\overline{U}$, enclosing $U$, and whose modulus is at least $\tau$.
Examples of unicritical polynomials having _a priori_ bounds include at most
finitely renormalizable polynomials without neutral cycles [Hub93, KL09,
KvS09], all real polynomials [LvS98, LvS00], and some infinitely
renormalizable complex polynomials [KL08, Lyu97]. Recall that for an integer
$s\geq 1$, a unicritical polynomial $f$ of degree $d\geq 2$ is _renormalizable
of period $s$_, if there are Jordan disks $U$ and $V$ such that $\overline{U}$
is contained in $V$, and such that the following properties hold:
* •
$f^{s}:U\to V$ is $d$-to-$1$;
* •
$U$ contains the critical point of $f$, and for each $j$ in
$\\{1,\ldots,s-1\\}$ the set $f^{j}(U)$ does not contain it;
* •
The set $\\{z\in U:f^{sn}(z)\in U,n=0,1,\ldots\\}$ is a connected proper
subset of $J(f)$.
Moreover, $f$ is _infinitely renormalizable_ , if there are infinitely many
integers $s$ such that $f$ is renormalizable of period $s$.
In the statement of the following theorem we use the fact that for a
unicritical polynomial $f$ having _a priori_ bounds there is a unique
conformal measure of exponent
$t=\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$ for $f$, see
[Pra98, Theorem $1$].
###### Theorem A.
Let $f$ be a unicritical polynomial having _a priori_ bounds. Then the
conformal measure $\mu$ of minimal exponent of $f$ does not have an atom at
the critical point of $f$.
We remark that in this result, it is essential that we consider the conformal
measure of minimal exponent, as opposed to a conformal measure of higher
exponent. In fact, every unicritical map $f$ that is not uniformly hyperbolic
has, for each $t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$, a
conformal measure of exponent $t$, see [GS09] for the case $f$ satisfies the
Collet-Eckmann condition, and [PRLS03, PRLS04] for the case $f$ does not. In
many cases, even if $f$ has _a priori_ bounds, for each
$t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$ there is
conformal measure of exponent $t$ that is supported on the backward orbit of
the critical point.
The existence of an atom at a point $w$ for a conformal measure of $f$ of
exponent $t$, is closely related to convergence of the following series:
$\mathcal{P}(w,t):=\sum_{n=0}^{\infty}\sum_{z\in f^{-n}(w)}|Df^{n}(z)|^{-t};$
it is the _Poincaré series at $w$ of exponent $t$_. In fact:
* •
If a conformal measure of exponent $t$ has an atom at $w$, then
$\mathcal{P}(w,t)<\infty$;
* •
A conformal measure of exponent $t$ has an atom at the critical point $c_{0}$
if and only if $\mathcal{P}(c_{0},t)<\infty$.
###### Theorem B.
Let $f$ be a unicritical polynomial having _a priori_ bounds. Assume $f$ is
normalized so its critical point is $0$. Then for each
$t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$, the series
$\mathcal{P}(0,t)$ is finite, if and only if
$\sum_{n=0}^{\infty}|Df^{n}(f(0))|^{-t/d}$ is finite.
### 1.1. Strategy and organization
We derive Theorem A from Theorem B arguing by contradiction: If the conformal
measure of minimal exponent had an atom at the critical point, by Theorem B
the derivatives at the critical value would grow to infinity. Together with
the _a priori_ bounds hypothesis, this implies that the map is “backward
contracting” (Theorem 1), so the results of [RLS10] apply to this map; in
particular, the Poincaré series at the critical point diverges. This
contradicts the existence of an atom at the critical point. In §2 we fix some
notation and terminology and recall the definition of backward contracting
maps. We deduce Theorem A from Theorem B in §3, after proving Theorem 1.
To prove Theorem B, we show that in either case the map is backward
contracting (part $1$ of Theorem 1). This allows us to use the
characterization for backward contracting maps of the summability condition
given in [LS11]. We recall this result in §3, as part $2$ of Theorem 1. To
prove the direct implication in Theorem B, we divide the integral of the
backward contraction function, characterizing the summability condition, into
intervals bounded by consecutive close return scales. Then we show that each
of these integrals is bounded by one of the terms in the Poincaré series up to
a multiplicative constant (Lemma 5). This is done in §4. The proof of the
reverse implication in Theorem B occupies §5 and is more involved. We use a
discretized sequence of scales to code each iterated preimage of the critical
point. To do this, we consider the largest scale whose pull-back is conformal,
and consider the “critical hits” when pulling back the previous scale, as a
code. One of the crucial estimates is the contribution in the Poincaré series
of those iterated preimages of the critical point for which a certain ball can
be pulled back conformally (Lemma 10). This estimate is done using one of the
main results of [RLS10]: For a backward contracting map the diameter of pull-
backs decreases at a super-polynomial rate.
## 2\. Preliminaries
We endow $\mathbb{C}$ with its norm distance, and for a bounded subset $W$ of
$\mathbb{C}$ we denote by $\operatorname{diam}(W)$ the diameter of $W$.
Moreover, for $\delta>0$ and for a point $z$ of $\mathbb{C}$, we denote by
$B(z,\delta)$ the ball of $\mathbb{C}$ centered at $z$ and of radius $\delta$.
A _topological disk_ is an open subset of $\mathbb{C}$ homeomorphic to the
unit disk, and that is not equal to $\mathbb{C}$. We endow such a set with its
hyperbolic metric. If $V$ and $V^{\prime}$ are topological disks such that
$\overline{V^{\prime}}\subset V$, we denote by
$\operatorname{mod}(V;V^{\prime})$ the supremum of the moduli of all annuli
contained in $V\setminus V^{\prime}$ that enclose $V^{\prime}$. See for
example [Mil06, CG93] for background.
Throughout the rest of this paper we fix an integer $d\geq 2$, a parameter $c$
in $\mathbb{C}$, and put $f(z):=z^{d}+c$. Moreover, we assume $0$ is non-
periodic and recurrent by $f$; this implies that $0$ is in the Julia set of
$J(f)$. Given a subset $V$ of $\mathbb{C}$, and an integer $n\geq 0$, each
connected component of $f^{-n}(V)$ is called a _a pull-back of $V$ by
$f^{n}$_. When $n\geq 1$, a pull-back of $V$ is _critical_ if it contains $0$.
### 2.1. Backward contraction
In this section we recall the definition of “backward contraction” from
[RL07], and compare it with a variant from [LS11].
For each $\delta>0$, put $\widetilde{B}(\delta):=B(0,\delta^{1/d})$, and
$r(\delta):=\sup\left\\{r>0:\begin{array}[]{ll}\text{ for every pull-
back~{}$U$ of~{}$\widetilde{B}(\delta r)$,}\\\ \text{
$\operatorname{dist}(W,c)\leq\delta$
implies~{}$\operatorname{diam}(W)<\delta$}\end{array}\right\\}.$
Given $r_{0}>1$, the map $f$ is _backward contracting with constant $r_{0}$_,
if for every sufficiently small $\delta$ we have $r(\delta)>r_{0}$. Moreover,
$f$ is _backward contracting_ if $r(\delta)\to\infty$ as $\delta\to 0$.
For each $\delta>0$, put
$R(\delta):=\inf\left\\{\frac{\delta}{\operatorname{diam}(U)}:U\text{ pull-
back of $\widetilde{B}(\delta)$ containing~{}$c$}\right\\}.$
Clearly, for every $\delta>0$ and $\delta^{\prime}>\delta$ we have
$R(\delta)\geq(\delta/\delta^{\prime})R(\delta^{\prime})$. Moreover, for every
$\delta>0$ and every $r$ in $(0,r(\delta))$, we have $R(\delta r)\geq r$. So,
if $f$ is backward contracting with some constant $r_{0}>1$, then for every
sufficiently small $\delta$ we have $R(\delta)\geq r_{0}$. In particular, if
$f$ is backward contracting, then $R(\delta)\to\infty$ as $\delta\to 0$.
###### Lemma 1.
There is a constant $\rho_{0}>1$ independent of $f$ and $d$, such that for
every $\delta>0$ satisfying $R(\delta)\geq\rho_{0}^{d}$, and every
$\delta^{\prime}$ in $[\delta/R(\delta),\delta)$, we have
$r(\delta^{\prime})\geq\rho_{0}^{-d}\delta/\delta^{\prime}$. In particular, if
there is $R_{0}\geq\rho_{0}^{d}$ such that for every sufficiently small
$\delta>0$ we have $R(\delta)>R_{0}$, then $f$ is backward contracting with
constant $\rho_{0}^{-d}R_{0}$. Moreover, $f$ is backward contracting if and
only if $R(\delta)\to\infty$ as $\delta\to\infty$.
###### Proof.
Let $\rho_{0}>1$ be sufficiently large so that for every pair of topological
disks $U$ and $V$ such that
$\overline{U}\subset V\text{ and }\operatorname{mod}(V;U)\geq\log\rho_{0},$
we have $\operatorname{diam}(U)<\operatorname{dist}(U,\partial V)$.
Fix $\delta>0$ such that $R(\delta)\geq\rho_{0}^{d}$, and $\delta^{\prime}$ in
$[\delta/R(\delta),\delta)$. To prove the lemma it suffices to show that for
every integer $n\geq 0$, and every pull-back $W$ of
$\widetilde{B}(\rho_{0}^{-d}\delta)$ by $f^{n}$ satisfying
$\operatorname{dist}(W,c)\leq\delta^{\prime}$, we have
$\operatorname{diam}(W)<\delta^{\prime}$. We proceed by induction in $n$. When
$n=0$, we have $W=\widetilde{B}(\rho_{0}^{-d}\delta)$. If $c$ is in
$\widetilde{B}(\delta)$, then by the definition of $R(\delta)$ we have
$\operatorname{diam}(W)<\operatorname{diam}(\widetilde{B}(\delta))\leq\delta/R(\delta)\leq\delta^{\prime}.$
Otherwise, by the definition of $\rho_{0}$ we have
$\operatorname{diam}(W)<\operatorname{dist}(W,\partial\widetilde{B}(\delta))\leq\operatorname{dist}(W,c)\leq\delta^{\prime}.$
This proves the desired assertion when $n=0$. Let $n\geq 1$ be an integer such
that the desired assertion holds with $n$ replaced by each element of
$\\{0,\ldots,n-1\\}$. Let $W$ be a pull-back of
$\widetilde{B}(\rho_{0}^{-d}\delta)$ by $f^{n}$ satisfying
$\operatorname{dist}(W,c)\leq\delta^{\prime}$. If the pull-back $\widehat{W}$
of $\widetilde{B}(\delta)$ containing $W$ contains $c$, then
$\operatorname{mod}(\widehat{W};W)=\operatorname{mod}(\widetilde{B}(\delta);\widetilde{B}(\rho_{0}^{-d}\delta))=\log\rho_{0}$
so by the definition of $R(\delta)$ we have
$\operatorname{diam}(W)<\operatorname{diam}(\widehat{W})\leq\delta/R(\delta)\leq\delta^{\prime}.$
So we suppose $\widehat{W}$ does not contain $c$. If $f^{n}$ is conformal on
$\widehat{W}$, then by the definition of $\rho_{0}$ we have
$\operatorname{diam}(W)<\operatorname{dist}(W,\partial\widehat{W})\leq\operatorname{dist}(W,c)\leq\delta^{\prime}.$
It remains to consider the case where $f^{n}$ is not conformal on
$\widehat{W}$, so there is $m$ in $\\{0,\ldots,n-1\\}$ such that
$f^{m}(\widehat{W})$ contains $0$. Then $f^{m+1}(\widehat{W})$ contains $c$,
and by definition of $R(\delta)$ we have
$\operatorname{diam}(f^{m+1}(\widehat{W}))\leq\delta/R(\delta)\leq\rho_{0}^{-d}\delta.$
This implies that $f^{m+1}(\widehat{W})$ is contained in
$B(c,\rho_{0}^{-d}\delta)$, and therefore that $f^{m}(\widehat{W})$ is
contained in $\widetilde{B}(\rho_{0}^{-d}\delta)$. Using the induction
hypothesis, we conclude that $\operatorname{diam}(W)<\delta^{\prime}$. This
completes the proof of the induction step and of the lemma. ∎
### 2.2. Nice sets and children
The purpose of this section is to prove a general lemma about backward
contracting maps that is used in §5.1 to prove Theorem B.
A topological disk $V$ containing $0$ is a _nice set for $f$_ if for every
integer $n\geq 1$ the set $f^{n}(\partial V)$ is disjoint from $V$.
For an integer $m\geq 1$, and a connected open set $V$, a pull-back $W$ of $V$
by $f^{m}$ is a _child of $V$_, if $f^{m}$ maps $W$ onto $V$ as a $d$-to-$1$
map and $W\ni 0$.
The following lemma is a more precise version of [RLS10, Lemma $3.15$], with
the same proof.
###### Lemma 2.
Suppose $f$ is backward contracting. Then for every $s>0$ there is
$\delta_{0}>0$, such that for every $\delta$ in $(0,\delta_{0}]$ there is a
nice set $V$ for $f$ satisfying $\widetilde{B}(\delta/2)\subset
V\subset\widetilde{B}(\delta)$, and such that
$\sum_{Y:\text{ child of
}V}\operatorname{diam}(f(Y))^{s}\leq(1-2^{-s})^{-1}\left(R(\delta)^{-1}\delta\right)^{s}.$
###### Proof.
Let $\lambda>0$ be sufficiently large so that for every pair of topological
discs $Y$ and $Y^{\prime}$ satisfying
$\overline{Y^{\prime}}\subset Y\text{ and
}\operatorname{mod}(Y;Y^{\prime})\geq\lambda/d,$
we have $\operatorname{diam}(Y^{\prime})\leq\operatorname{diam}(Y)/2$, see for
example [McM94, Theorem $2.1$]. In view of [RLS10, Lemma $3.13$], for every
sufficiently small $\delta>0$ there is a $\lambda$-nice set $V$ for $f$ such
that $\widetilde{B}(\delta/2)\subset V\subset\widetilde{B}(\delta)$. We prove
the desired assertion hold for this choice of $V$.
For each integer $k\geq 1$, let $Y_{k}$ be the $k$-th smallest child of $V$
and let $s_{k}$ be the integer such that $f^{s_{k}}(Y_{k})=V$. By the backward
contracting property, we have $\operatorname{diam}(f(Y_{1}))\leq
R(\delta)^{-1}\delta$. Note that for each $k\geq 1$ the set
$f^{s_{k}}(Y_{k+1})$ is contained in a return domain of $V$, so
$\operatorname{mod}(V;f^{s_{k}}(Y_{k+1}))\geq\lambda$. By the definition of
child, the map $f^{s_{k}}:Y_{k}\to V$ is $d$-to-$1$, thus
$\operatorname{mod}(Y_{k};Y_{k+1})\geq\lambda/d$ and therefore
$\operatorname{diam}(Y_{k+1})\leq\operatorname{diam}(Y_{k})/2$. The conclusion
of the lemma follows. ∎
## 3\. _A priori_ bounds and backward contraction
In this section we derive Theorem A from Theorem B. To do so, we first
establish a sufficient criterion for a unicritical map having _a priori_
bounds to be backward contracting (Theorem 1 below).
In the following theorem we summarize and complement results in [LS10, LS11],
when restricted to unicritical maps. We state it in a stronger form than what
is needed for this section.
###### Theorem 1.
For each $\tau>0$ there is a constant $\eta>1$, such that if $f$ has _a
priori_ bounds with constant $\tau$, then the following properties hold.
1. 1.
Let $R_{0}>1$ be such that for every sufficiently large $n$ we have either
$|Df^{n}(c)|\geq\eta R_{0},\text{ or }\min_{\zeta\in
f^{-n}(0)}|Df^{n}(\zeta)|\geq(\eta R_{0})^{1/d}.$
Then for every sufficiently small $\delta>0$ we have $R(\delta)\geq R_{0}$.
2. 2.
For each $t>0$, the sum $\sum_{n=0}^{\infty}|Df^{n}(c)|^{-t}$ is finite if and
only if $\int_{0+}^{\infty}R(\delta)^{-t}\ \frac{d\delta}{\delta}$ is finite.
When $|Df^{n}(c)|$ is eventually bounded from below by $\eta R_{0}$, part $1$
is given by (the proof of) [LS10, Theorem A]. Part $2$ is a direct consequence
of part $1$ and [LS11, Theorems $1.3$ and $1.4$], together with Lemma 1 and
[RL07, Corollary $8.3$].
To prove this theorem, we shall use the following variation of the Koebe
distortion theorem.
###### Lemma 3.
For each $\tau_{0}>0$ there is a constant $C_{@}>1$, such that for every pair
of topological disks $V$ and $V^{\prime}$ containing $0$, and satisfying
$\overline{V^{\prime}}\subset V\text{ and
}\operatorname{mod}(V;V^{\prime})\geq\tau_{0},$
the following property holds. Let $s\geq 1$ be a integer such that $f^{s}(0)$
is in $V^{\prime}$, and such that $f^{s-1}$ maps a neighborhood $W$ of $c$
conformally onto $V$. Moreover, let $W^{\prime}$ be the pull-back of
$V^{\prime}$ by $f^{s-1}$ contained in $W$, and let $\zeta$ in $f^{-s}(0)$ be
such that $f(\zeta)$ is in $W^{\prime}$. Then we have
$\operatorname{diam}(W^{\prime})\leq
C_{@}\max\left\\{|Df^{s}(c)|,|Df^{s}(\zeta)|^{d}\right\\}^{-1}\operatorname{diam}(f(V^{\prime})).$
###### Proof.
By Koebe distortion theorem there is a constant $K_{1}>1$ that only depends on
$\tau_{0}$, such that the distortion of $f^{s-1}$ on $W^{\prime}$ is bounded
by $K_{1}$. We thus have,
$\operatorname{diam}(W^{\prime})\leq
K_{1}|Df^{s-1}(c)|^{-1}\operatorname{diam}(V^{\prime})\\\
=K_{1}|Df^{s}(c)|^{-1}|Df(f^{s-1}(c))|\operatorname{diam}(V^{\prime}).$
Since $|Df(f^{s-1}(c))|=d|f^{s-1}(c)|^{d-1}\leq
d\operatorname{diam}(V^{\prime})^{d-1}$, we obtain
(1) $\operatorname{diam}(W^{\prime})\leq
dK_{1}|Df^{s}(c)|^{-1}\operatorname{diam}(V^{\prime})^{d}\leq
d2^{d}K_{1}|Df^{s}(c)|^{-1}\operatorname{diam}(f(V^{\prime})).$
On the other hand, we have $|Df^{s-1}(f(\zeta))|\leq K_{1}|Df^{s-1}(c)|$.
Using the formula of $f$, we obtain
(2) $|Df^{s}(\zeta)|\leq K_{1}(|\zeta|/|f^{s-1}(c)|)^{d-1}|Df^{s}(c)|.$
Using Koebe distortion theorem and the formula of $f$ again, we obtain
$|f^{s-1}(c)|\geq
K_{2}^{-1}|Df^{s-1}(f(\zeta))|\cdot|f(\zeta)-c|=(dK_{2})^{-1}|Df^{s}(\zeta)|\cdot|\zeta|,$
where $K_{2}>1$ is a constant depending only on $\tau_{0}$. Together with (2),
we obtain $|Df^{s}(\zeta)|^{d}\leq d^{d-1}K^{d}|Df^{s}(c)|$, where
$K=\max\\{K_{1},K_{2}\\}$. Combined with (1), this implies the desired
inequality with $C_{@}=(2d)^{d}K^{d+1}$. ∎
For each topological disk $V$ containing $0$, put
$M_{+}(V):=\inf\left\\{|Df^{n}(c)|:n\geq 1,f^{n}(0)\in V\right\\},$
$M_{-}(V):=\inf\left\\{|Df^{n}(\zeta)|^{d}:n\geq 1,\zeta\in
V,f^{n}(\zeta)=0\right\\},$
and
$M(V):=\max\\{M_{+}(V),M_{-}(V)\\}.$
###### Lemma 4.
For every constant $\tau_{0}>0$ there is a constant $C_{!}>0$ such that the
following property holds. Let $V$ and $V^{\prime}$ be topological disks
containing $0$, such that $\overline{V^{\prime}}\subset V$,
$\operatorname{mod}(V;V^{\prime})\geq\tau_{0}$, and such that every critical
pull-back of $V$ is contained in $V^{\prime}$. Then for every critical pull-
back $U^{\prime}$ of $V^{\prime}$, we have
$\operatorname{diam}(f(U^{\prime}))\leq
C_{!}M(V)^{-1}\operatorname{diam}(f(V^{\prime})).$
###### Proof.
Let $C_{@}$ be the constant given by Lemma 3. Let $n\geq 1$ be an integer such
that $f^{n}(0)$ is in $V^{\prime}$, and let $U$ (resp. $U^{\prime}$) be the
pull-back of $V$ (resp. $V^{\prime}$) by $f^{n}$ containing $0$. It suffices
to consider the case that $U$ is not contained in any critical pull-back of
$V^{\prime}$. In this case we claim that $f^{n}:U\to V$ is $d$-to-$1$.
Otherwise, there would exist $m$ in $\\{1,\ldots,n-1\\}$ such that $f^{m}(U)$
contains $0$. This implies that $f^{m}(U)$ is contained in the pull-back of
$V$ by $f^{n-m}$ containing $0$, which is contained in $V^{\prime}$. Thus
$f^{m}(U)$ is contained in $V^{\prime}$, and therefore $U$ is contained in the
pull-back of $V^{\prime}$ by $f^{m}$ containing $0$. We thus obtain a
contradiction that shows that $f^{n}:U\to V$ is $d$-to-$1$. Then the lemma
follows from Lemma 3 with $C_{!}=C_{@}$. ∎
Given a topological disk $V$, a point $x$ of $V$, and $\Delta>0$, denote by
$B_{V}(x,\Delta)$ the ball for the hyperbolic metric of $V$ centered at $x$
and of radius $\Delta$. By Koebe distortion theorem, for every $\Delta>0$
there is $\xi>1$ such that for every topological disk $V$, and every $x$ in
$V$, we have
$\sup_{z\in\partial B_{V}(x,\Delta)}|z-x|\leq\xi\inf_{z\in\partial
B_{V}(x,\Delta)}|z-x|.$
###### Proof of Theorem 1.
As mentioned above, part $2$ is a direct consequence of part $1$, and the
combination of [LS11, Theorems $1.3$ and $1.4$], Lemma 1, and [RL07, Corollary
$8.3$].
To prove part $1$, note that for $\delta>0$ the number
$\operatorname{mod}(\widetilde{B}(\delta);\widetilde{B}(\delta/2))$ is
independent of $\delta$; denote it by $\tau_{1}$. On the other hand, let
$\tau$ be the constant given by the _a priori_ bounds hypothesis. Note that
there is a constant $\Delta>0$ such that if $U$ is a topological disk
satisfying $\overline{U}\subset V$ and $\operatorname{mod}(V;U)\geq\tau$, then
the diameter of $U$ with respect to the hyperbolic metric of $V$ is bounded by
$\Delta$. Moreover, the number $\operatorname{mod}(V;B_{V}(0,\Delta))$ is
bounded from below by a constant $\tau_{0}>0$ that is independent of $V$. Let
$C_{!}$ be the constant given by Lemma 4 with $\tau_{0}$ replaced by
$\min\\{\tau_{0},\tau_{1}\\}$. Let $\xi>1$ be the constant defined above for
this choice of $\Delta$, and put $\eta:=4C_{!}\xi^{d}$.
Let $\varepsilon>0$ be sufficiently small so that for every topological disk
$V$ containing $0$ and satisfying $\operatorname{diam}(V)<\varepsilon$, we
have $M(V)\geq\eta R_{0}$. By the _a priori_ bounds hypothesis, there is such
a topological disk so that in addition for every critical pull-back $U$ of $V$
we have $\overline{U}\subset V$ and $\operatorname{mod}(V;U)\geq\tau$. By our
choice of $\Delta$, this implies that $U$ is contained in
$V^{\prime}:=B_{V}(0,\Delta)$. So the hypotheses of Lemma 4 are satisfied for
these choices of $V$, and $V^{\prime}$. It follows that for every critical
pull-back $U^{\prime}$ of $V^{\prime}$, we have
$\operatorname{diam}(f(U^{\prime}))\leq
C_{!}M(V)^{-1}\operatorname{diam}(f(V^{\prime}))\leq
4^{-1}\xi^{-d}R_{0}^{-1}\operatorname{diam}(f(V^{\prime})).$
Thus, if we put $\delta_{0}:=\inf_{z\in\partial V^{\prime}}|z|^{d}$, then
$\operatorname{diam}(f(V^{\prime}))\leq 2\xi^{d}\delta_{0}$, and for every
critical pull-back $U^{\prime\prime}$ of $\widetilde{B}(\delta_{0})$ we have
$\operatorname{diam}(f(U^{\prime\prime}))\leq(2R_{0})^{-1}\delta_{0}.$
This proves $R(\delta_{0})\geq 2R_{0}$.
To complete the proof part $1$, for each integer $n\geq 1$ put
$\delta_{n}:=2^{-n}\delta_{0}$. We prove by induction that for each $n$ we
have $R(\delta_{n})\geq 2R_{0}$. The case $n=0$ is shown above. Let $n\geq 1$
be an integer such that $R(\delta_{n-1})\geq 2R_{0}>2$. Then every critical
pull-back $\widetilde{B}(\delta_{n-1})$ is contained in
$\widetilde{B}(\delta_{n})$. So by Lemma 4, for every critical pull-back
$U^{\prime}$ of $\widetilde{B}(\delta_{n})$ we have
$\operatorname{diam}(f(U^{\prime}))\leq
2C_{!}M(\widetilde{B}(\delta_{n-1}))^{-1}\delta_{n}\leq(2R_{0})^{-1}\delta_{n}.$
This proves $R(\delta_{n})\geq 2R_{0}$ and completes the proof of the
induction step. Thus for every $n\geq 0$ we have $R(\delta_{n})\geq 2R_{0}$,
and therefore for every $\delta$ in $(0,\delta_{0})$ we have $R(\delta)\geq
R_{0}$. This completes the proof of part $1$ and of the theorem. ∎
###### Proof of Theorem A assuming Theorem B.
Let $\mu$ denote the conformal measure of minimal exponent
$h=\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$. Assume by
contraction that $\mu$ has an atom at $0$. Then $\mathcal{P}(0,h)<\infty$,
hence for each $t>h$ we have $\mathcal{P}(0,t)<\infty$. By Theorem B, it
follows that $\sum_{n=0}^{\infty}|Df^{n}(c)|^{-t/d}<\infty$, hence
$|Df^{n}(c)|\to\infty$ as $n\to\infty$. By Lemma 1 and part $1$ of Theorem 1,
the map $f$ is backward contracting. But then, [RLS10, Proposition $7.3$]
implies $\mathcal{P}(0,h)=\infty$. We thus obtain a contradiction that
completes the proof of the theorem. ∎
## 4\. Convergence of Poincaré series implies forward summability
The purpose of this section is to prove the following proposition, giving one
of the implications in Theorem B.
###### Proposition 1.
Suppose $f$ satisfies the _a priori_ bounds condition. Then for each $t>0$
such that $\mathcal{P}(0,t)$ is finite, the sum
$\sum_{n=1}^{\infty}|Df^{n}(c)|^{-t/d}$ is also finite.
For each $\delta>0$, let $n(\delta)$ be the minimal integer $n\geq 1$ such
that $f^{n}(0)$ is in $\widetilde{B}(\delta)$, let $U_{\delta}$ denote the
pull-back of $\widetilde{B}(\delta)$ by $f^{n(\delta)}$ that contains $0$, and
let $\zeta(\delta)$ be a point of $f^{-n(\delta)}(0)$ in $U_{\delta}$.
Clearly, $n(\delta)$ is non-increasing with $\delta$, left continuous, and we
have $n(\delta)\to\infty$ as $\delta\to 0$. In view of Theorem 1, Proposition
1 is a direct consequence of the following lemma.
###### Lemma 5.
There is $\delta_{0}>0$ such that for each $t>0$ there is $C_{{\dagger}}>0$
such that the following property holds: For every $\delta_{1}$ and
$\delta_{2}$ in $(0,\delta_{0})$ satisfying $\delta_{1}<\delta_{2}$, and such
that $n(\cdot)$ is constant on $(\delta_{1},\delta_{2}]$, we have
$\int_{\delta_{1}}^{\delta_{2}}R(\delta)^{-t/d}\ \frac{d\delta}{\delta}\leq
C_{{\dagger}}|Df^{n(\delta_{2})}(\zeta(\delta_{2}))|^{-t}.$
The proof of this lemma is after the following one.
###### Lemma 6.
Assume that $f$ is backward contracting, and for each $\delta>0$ let
$W_{\delta}$ be the pull-back of $\widetilde{B}(2\delta)$ by $f^{n(\delta)-1}$
containing $c$. Then for every sufficiently small $\delta$ we have
$R(\delta)\geq\delta/\operatorname{diam}(W_{\delta})$.
###### Proof.
Let $\delta_{0}>0$ be sufficiently small so that for every $\delta$ in
$(0,\delta_{0})$ we have $r(\delta)>2$. It suffices to show that for each
$\delta$ in $(0,\delta_{0})$, and every integer $m\geq 0$ such that $f^{m}(c)$
is in $\widetilde{B}(\delta)$, the pull-back $W$ of $\widetilde{B}(\delta)$ by
$f^{m}$ containing $c$ is contained in $W_{\delta}$. Clearly $m\geq
n(\delta)-1$, and when $m=n(\delta)-1$ we have $W\subset W_{\delta}$. If
$m\geq n(\delta)$, then our hypothesis $r(\delta)>2$ implies that
$f^{n(\delta)-1}(W)$ is contained in $\widetilde{B}(2\delta)$. This shows that
$W$ is contained in $W_{\delta}$, as wanted. ∎
###### Proof of Lemma 5.
By Koebe Distortion Theorem there is a constant $K>1$ such that for every
$\delta>0$, every integer $n\geq 1$, and every pull-back $W$ of
$\widetilde{B}(2\delta)$ by $f^{n}$ for which the corresponding pull-back of
$\widetilde{B}(4\delta)$ is conformal, the distortion of $f^{n}$ on $W$ is
bounded by $K$. Let $\delta_{0}>0$ be such that the conclusion of Lemma 6
holds for every $\delta$ in $(0,\delta_{0})$. Reducing $\delta_{0}$ if
necessary, assume that for each $\delta$ in $(0,4\delta_{0})$ we have
$R(4\delta)\geq 4$.
To prove the lemma, let $\delta_{1}$ and $\delta_{2}$ in $(0,\delta_{0})$ be
such that $n(\cdot)$ is constant on $(\delta_{1},\delta_{2}]$. Put
$n:=n(\delta_{2})\text{ and }\zeta:=\zeta(\delta_{2}),$
and note that $|f^{n-1}(c)|=|f^{n}(0)|\leq\delta_{1}^{1/d}$. Let $\delta$ in
$(\delta_{1},\delta_{2}]$, and let $W_{\delta}$ (resp. $\widehat{W}_{\delta}$)
be the pull-back of $\widetilde{B}(2\delta)$ (resp. $\widetilde{B}(4\delta)$)
by $f^{n-1}$ containing $c$. We claim that $f^{n-1}$ is conformal on
$\widehat{W}_{\delta}$. Otherwise, we would have $n\geq 2$, and there would
exist $m$ in $\\{1,\ldots,n-1\\}$ such that $f^{m}(\widehat{W}_{\delta})$
contains $c$. However, this implies that
$\operatorname{diam}(f^{m}(\widehat{W}_{\delta}))\leq
4\delta/R(4\delta)\leq\delta,$
and therefore that $f^{m-1}(\widehat{W}_{\delta})$, and hence $f^{m-1}(0)$, is
contained in $\widetilde{B}(\delta)$; we thus obtain a contradiction with the
definition of $n=n(\delta)$ that proves that $f^{n-1}$ is conformal on
$\widehat{W}_{\delta}$.
Using Koebe distortion theorem and the formula of $f$, we have
(3) $\operatorname{diam}(W_{\delta})\leq
K(2\delta)^{1/d}|Df^{n-1}(f(\zeta))|^{-1}=dK(2\delta)^{1/d}|Df^{n}(\zeta)|^{-1}|\zeta|^{d-1},$
and
$|\zeta|^{d}=|f(\zeta)-c|\leq K|Df^{n-1}(f(\zeta))|^{-1}|f^{n-1}(c)|\\\ \leq
K\delta_{1}^{1/d}|Df^{n-1}(f(\zeta))|^{-1}=dK\delta_{1}^{1/d}|Df^{n}(\zeta)|^{-1}|\zeta|^{d-1},$
and therefore $|\zeta|\leq dK\delta_{1}^{1/d}|Df^{n}(\zeta)|^{-1}$. Hence,
letting $C_{{\ddagger}}:=2^{1/d}(dK)^{d}$, by (3) we have
$\operatorname{diam}(W_{\delta})\leq
C_{{\ddagger}}\delta_{1}(\delta/\delta_{1})^{1/d}|Df^{n}(\zeta)|^{-d}.$
Together with Lemma 6, this implies
$R(\delta)\geq
C_{{\ddagger}}^{-1}(\delta/\delta_{1})^{1-1/d}|Df^{n}(\zeta)|^{d}.$
The desired assertion follows with
$C_{{\dagger}}=C_{{\ddagger}}^{t/d}\int_{1}^{\infty}\eta^{-1-(t/d)(1-1/d)}\
d\eta$. ∎
## 5\. Forward summability implies backward summability
In this section we complete the proof of Theorem B. After some estimates in
§5.1, the proof of Theorem B is in §5.2.
### 5.1. Thickened grandchildren
Throughout this section, assume $f$ is backward contracting, so that
$R(\delta)\to\infty$ as $\delta\to 0$, see §2.1. Moreover, fix $t>0$, put
$\tau:=2^{-d}$, and let $\delta_{0}>0$ be given by Lemma 2 with $s=t/d$. For
every integer $q\geq 0$, let $V_{q}$ be given by Lemma 2 with
$\delta=\tau^{q}\delta_{0}$, so that
$\widetilde{B}(\tau^{q}\delta_{0}/2)\subset
V_{q}\subset\widetilde{B}(\tau^{q}\delta_{0}).$
Note that for every $q\geq 0$ the set $\overline{V_{q+1}}$ is contained in
$V_{q}$ and $\operatorname{mod}(V_{q};V_{q+1})$ is bounded from below
independently of $q$. Reducing $\delta_{0}$ if necessary, assume that for
every $\delta$ in $(0,\delta_{0}]$ we have $R(\delta)>2\tau^{-1}$.
For each integer $q\geq 0$, let $\mathcal{N}(q)$ be the set of all integers
$n\geq 1$ such that $f^{n}(0)$ is in $V_{q}$, and such that the pull-back
$W_{n}(q)$ of $V_{q}$ by $f^{n}$ containing $0$ is a child of $V_{q}$. By our
choice of $\delta_{0}$, the set $W_{n}(q)$ is contained in
$\widetilde{B}(\tau^{q+1}\delta_{0}/2)$, and hence in $V_{q+1}$; denote by
$p_{n}(q)$ the largest integer $p\geq q$, such that $W_{n}(q)$ is contained in
$V_{p+1}$. Then we have
$\operatorname{diam}(f(W_{n}(q)))\geq\tau^{p_{n}(q)+2}\delta_{0}/2.$
Combined with Lemma 2, this implies
(4) $\sum_{n\in\mathcal{N}(q)}\tau^{t\cdot
p_{n}(q)/d}\leq(1-2^{-t/d})^{-1}\left(2R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}.$
Given an integer $q\geq 0$, let $\mathcal{S}(q)$ be the set of all finite
sequences of integers $(n_{1},\ldots,n_{k})$, such that there are integers
$p_{0}:=q,p_{1},\ldots,p_{k},$
satisfying the following property: For each $i$ in $\\{1,\ldots,k\\}$ the
integer $n_{i}$ is in $\mathcal{N}(p_{i-1})$, and $p_{i}=p_{n_{i}}(p_{i-1})$.
Note that in this case for each $i$ in $\\{1,\ldots,k\\}$ we have $p_{i}\geq
p_{i-1}+1$. In the situation above, we put
$p_{(n_{1},\ldots,n_{k})}(q):=p_{k}$.
Let $q\geq 0$ be an integer. For each $k\geq 1$ and
$\textbf{n}=(n_{1},\ldots,n_{k})$ in $\mathcal{S}(q)$, put
$|\textbf{n}|:=n_{1}+\cdots+n_{k}\text{ and }k(\textbf{n}):=k.$
Moreover, let $Z_{\textbf{n}}(q)$ be the set of all points $z$ of
$f^{-|\textbf{n}|}(0)$ such that $f^{|\textbf{n}|}$ maps a neighborhood of $z$
conformally onto $V_{q+1}$, and such that in the case $k\geq 2$, for each $i$
in $\\{2,\ldots,k\\}$ the point $f^{n_{k}+\cdots+n_{i+1}}(z)$ is in the pull-
back of $V_{p_{(n_{1},\ldots,n_{i-1})}}$ by $f^{n_{i}}$ containing $0$. Note
that $\\#Z_{\textbf{n}}(q)\leq d^{k(\textbf{n})}$, and that
$Z_{\textbf{n}}(q)$ is contained in $V_{p_{\textbf{n}}(q)+1}$.
The purpose of this section is to prove the following.
###### Proposition 2.
Suppose $f$ is backward contracting, and put
$C_{\\#}:=2^{1+2t/d}d(1-2^{-t/d})^{-1}\tau^{-2t/d}.$
Then, for all $q\geq 0$ sufficiently large we have
$\sum_{\textbf{n}\in\mathcal{S}(q)}\sum_{z\in
Z_{\textbf{n}}(q)}\left|Df^{|\textbf{n}|}(z)\right|^{-t}\leq
C_{\\#}R(\tau^{q}\delta_{0})^{-t/d}.$
The proof of this proposition is given after the following lemma.
###### Lemma 7.
Suppose $f$ is backward contracting. Then, for every integer $q\geq 0$ and
every n in $\mathcal{S}(q)$ we have
$\sum_{z\in Z_{\textbf{n}}(q)}\left|Df^{|\textbf{n}|}(z)\right|^{-t}\leq
2^{t/d}d^{k(\textbf{n})}\left(\tau^{p_{\textbf{n}}(q)-q}\right)^{t/d}.$
###### Proof.
By definition, for each $z$ in $Z_{\textbf{n}}(q)$ the map $f^{|\textbf{n}|}$
maps a neighborhood $U$ of $z$ conformally onto $V_{q+1}$. Noting that $U$ is
contained in $V_{p_{\textbf{n}}(q)+1}$, and hence in
$\widetilde{B}(\tau^{p_{\textbf{n}}(q)+1}\delta_{0})$, by Schwarz’ lemma we
have
$\left|Df^{|\textbf{n}|}(z)\right|\geq\frac{(\tau^{q+1}\delta_{0}/2)^{1/d}}{(\tau^{p_{\textbf{n}}(q)+1}\delta_{0})^{1/d}}\geq\frac{1}{2^{1/d}}\tau^{-(p_{\textbf{n}}(q)-q)/d}.$
Since $\\#Z_{\textbf{n}}(q)\leq d^{k(\textbf{n})}$, the desired conclusion
follows. ∎
In view of Lemma 7, Proposition 2 is a direct consequence of the following
lemma.
###### Lemma 8.
Suppose $f$ is backward contracting. Then for every sufficiently large $q\geq
0$ we have
$\sum_{\textbf{n}\in\mathcal{S}(q)}d^{k(\textbf{n})}\tau^{t\cdot
p_{\textbf{n}}(q)/q}\leq
2^{1+t/d}d(1-2^{-t/d})^{-1}\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}.$
###### Proof.
Put $C_{*}:=d(1-2^{-t/d})^{-1}2^{t/d}$ and let $q_{0}\geq 0$ be sufficiently
large so that for every $q\geq q_{0}$ we have
(5) $R(\tau^{q}\delta_{0})>\tau^{-2}\left(2C_{*}\right)^{d/t}.$
For each pair of integers $q\geq 0$ and $m\geq 1$, put
$\Xi_{t}(q,m):=\sum_{\begin{subarray}{c}\textbf{n}\in\mathcal{S}(q)\\\
|\textbf{n}|\leq m\end{subarray}}d^{k(\textbf{n})}\tau^{t\cdot
p_{\textbf{n}}(q)/q},$
and $\Xi_{t}(q,0):=0$. To prove the lemma, it is enough to show that for every
pair of integers $q\geq q_{0}$ and $m\geq 0$ we have
(6) $\Xi_{t}(q,m)\leq
2C_{*}\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}.$
We proceed by induction in $m$. By definition, for every $q$ we have
$\Xi_{t}(q,0)=0$, so inequality (6) holds trivially when $m=0$. Let $m\geq 1$
be an integer and suppose that for every $q\geq q_{0}$ the inequality (6)
holds with $m$ replaced by $m-1$. Let $q\geq q_{0}$ be given and note that for
every $k\geq 2$, and every $\textbf{n}=(n_{1},n_{2},\ldots,n_{k})$ in
$\mathcal{S}(q)$, we have
$(n_{2},\ldots,n_{k})\in\mathcal{S}(p_{n_{1}}(q))\text{ and
}p_{(n_{2},\ldots,n_{k})}(p_{n_{1}}(q))=p_{(n_{1},\ldots,n_{k})}(q).$
Conversely, for every integer $n$ in $\mathcal{N}(q)$, and every
$k^{\prime}\geq 1$ and $(n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime})$ in
$\mathcal{S}(p_{n}(q))$, we have
$(n,n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime})\in\mathcal{S}(q)\text{ and
}p_{(n,n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime})}(q)=p_{(n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime})}(p_{n}(q)).$
Thus,
$\begin{split}\Xi_{t}(q,m)&=\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\
n\leq m\end{subarray}}d\left(\tau^{t\cdot
p_{n}(q)/d}+\Xi_{t}\left(p_{n}(q),m-n\right)\right)\\\
&\leq\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\ n\leq
m\end{subarray}}d\tau^{t\cdot
p_{n}(q)/d}+d\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\ n\leq
m-1\end{subarray}}\Xi_{t}\left(p_{n}(q),m-1\right).\end{split}$
Together with (4), and the induction hypothesis, this implies
$\Xi_{t}(q,m)\leq
C_{*}\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}+d\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\
n\leq m-1\end{subarray}}\Xi_{t}(p_{n}(q),m-1)\\\ \leq
C_{*}\left[\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}+2d\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\
n\leq
m-1\end{subarray}}\left(R(\tau^{p_{n}(q)}\delta_{0})^{-1}\tau^{p_{n}(q)-2}\right)^{t/d}\right]$
Using (5), and then (4) again, we obtain
$\begin{split}\Xi_{t}(q,m)&\leq
C_{*}\left[\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}+(1-2^{-t/d})2^{-t/d}\sum_{\begin{subarray}{c}n\in\mathcal{N}(q)\\\
n\leq m\end{subarray}}\tau^{t\cdot p_{n}(q)/d}\right]\\\ &\leq
2C_{*}\left(R(\tau^{q}\delta_{0})^{-1}\tau^{q-2}\right)^{t/d}.\end{split}$
This completes the proof of the induction step and of the lemma. ∎
### 5.2. Proof of Theorem B
The proof of Theorem B is at the end of this section, after a couple of
lemmas.
Assume $f$ is backward contracting, fix
$t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(J(f))$, and consider
the notation introduced in §5.1 for this choice of $t$. Put
$\mathcal{O}^{-}(0):=\bigcup_{m=1}^{\infty}f^{-m}(0)$, and for each $z$ in
this set denote by $m(z)\geq 1$ the unique integer $m\geq 1$ such that
$f^{m}(z)=0$. Note that for every $z$ in $\mathcal{O}^{-}(0)$ there is an
integer $q\geq 0$ such that the pull-back of $V_{q}$ by $f^{m(z)}$ containing
$z$ is conformal. Denote by $q(z)$ the least integer $q\geq 0$ with this
property.
###### Lemma 9.
There is a constant $C_{\&}>0$ such that for every $z$ in $\mathcal{O}^{-}(0)$
satisfying $q(z)\geq 1$, and every $q$ in $\\{0,\ldots,q(z)-1\\}$, there
exists n in $\mathcal{S}(q)$ such that the following hold:
* •
$m(z)\geq|\textbf{n}|$, and $\zeta(z):=f^{m(z)-|\textbf{n}|}(z)$ is in
$Z_{\textbf{n}}(q)$, and hence in $V_{p_{\textbf{n}}(q)+1}$;
* •
$f^{m(z)-|\textbf{n}|}$ maps a neighborhood $U(z)$ of $z$ conformally onto
$V_{p_{\textbf{n}}(q)}$;
* •
Denoting by $\zeta^{\prime}(z)$ the unique point in $U(z)$ such that
$f^{m(z)-|\textbf{n}|}(\zeta^{\prime}(z))=0$, we have
$|Df^{m(z)}(z)|\geq
C_{\&}\left|Df^{|\textbf{n}|}(\zeta(z))\right|\left|Df^{m(z)-|\textbf{n}|}(\zeta^{\prime}(z))\right|.$
###### Proof.
The third assertion follows from the first and the second, together with Koebe
distortion theorem. To prove the first and second assertions, we proceed by
induction in $m(z)$. Let $z$ be a point in $\mathcal{O}^{-}(0)$ such that
$q(z)\geq 1$ and $m(z)=1$, and let $q$ be in $\\{0,\ldots,q(z)-1\\}$. Then $1$
is in $\mathcal{N}(q)$, and the desired assertions are easily seen to be
satisfied with $\textbf{n}=(1)$. Let $m\geq 2$ be an integer and suppose the
desired assertions are satisfied for every $z$ in $\mathcal{O}^{-}(0)$ such
that $q(z)\geq 1$ and $m(z)\leq m-1$, and for every $q$ in
$\\{0,\ldots,q(z)-1\\}$. Let $z$ be a point in $\mathcal{O}^{-}(0)$ such that
$q(z)\geq 1$ and $m(z)=m$, and let $q$ be in $\\{0,\ldots,q(z)-1\\}$. Note
that $f(z)$ is in $\mathcal{O}^{-}(0)$ and $m(f(z))=m(z)-1$. If $q(f(z))\leq
q$, then the pull-back of $V_{q}$ by $f^{m(z)-1}$ containing $f(z)$ is
conformal. This implies that $m(z)$ is in $\mathcal{N}(q)$, and then the
desired assertions are verified with $\textbf{n}=(m(z))$. If $q(f(z))\geq
q+1$, then we can apply the induction hypothesis with $z$ replaced by $f(z)$;
let $\textbf{n}^{\prime}=(n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime})$ be
the corresponding element of $\mathcal{S}(q)$. If the pull-back of $U(f(z))$
by $f$ containing $z$ is conformal, then the desired assertions are verified
with $\textbf{n}=\textbf{n}^{\prime}$. Otherwise,
$n^{\prime}:=m(z)-|\textbf{n}^{\prime}|$ is in
$\mathcal{N}(p_{\textbf{n}^{\prime}}(q))$, and then the desired assertions are
verified with
$\textbf{n}=(n_{1}^{\prime},\ldots,n_{k^{\prime}}^{\prime},n^{\prime})$. ∎
###### Lemma 10.
Assume that $f$ is backward contracting. Then for each $q\geq 0$ and
$t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(J(f))$, we have
$\sum_{\begin{subarray}{c}z\in\mathcal{O}^{-}(0)\\\ q(z)\leq
q\end{subarray}}|Df^{m(z)}(z)|^{-t}<\infty.$
###### Proof.
Let $\mu$ denote a conformal measure of $f$ of exponent
$h:=\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$. For each $z$ in
$\mathcal{O}^{-}(0)$, let $B(z)$ denote the pull-back of $V_{q+1}$ by
$f^{m(z)}$ that contains $z$. By Koebe distortion theorem, there is a constant
$C>1$ such that for every $z$ in $\mathcal{O}^{-}(0)$ satisfying $q(z)\leq q$,
the distortion of $f^{m(z)}$ on $B(z)$ is bounded by $C$. Then for every such
$z$ we have $|Df^{m(z)}(z)|\geq C^{-1}\operatorname{diam}(B(z))^{t}$, and by
conformality of $\mu$, we also have
$\mu(B(z))\geq C^{-h}|Df^{m(z)}(z)|^{-h}\mu(V_{q+1}).$
Since for a fixed integer $m\geq 1$ the sets $(B(z))_{z\in f^{-m}(0)}$ are
pairwise disjoint, we have
$\begin{split}\sum_{\begin{subarray}{c}z\in f^{-m}(0)\\\ q(z)\leq
q\end{subarray}}|Df^{m}(z)|^{-t}&\leq
C^{t}\mu(V_{q+1})^{-1}\sum_{\begin{subarray}{c}z\in f^{-m}(0)\\\ q(z)\leq
q\end{subarray}}\operatorname{diam}(B(z))^{t-h}\mu(B(z))\\\
&\leq\mu(V_{q+1})^{-1}\max_{z\in
f^{-m}(0)}\operatorname{diam}(B(z))^{t-h}.\end{split}$
By [RLS10, Theorem A], for every $\beta>0$ the sequence $\left(\max_{z\in
f^{-m}(0)}\operatorname{diam}(B(z))\right)_{m=1}^{\infty}$ decreases faster
than the sequence $(m^{-\beta})_{m=1}^{\infty}$. The proposition follows
summing over $m\geq 1$. ∎
###### Proof of Theorem B.
One of the implications of the theorem is given by Proposition 1. To prove the
reverse implication, fix
$t>\operatorname{\operatorname{HD}_{\operatorname{hyp}}}(f)$ such that
$\sum_{n=1}^{\infty}|Df^{n}(c)|^{-t/d}<+\infty.$
By part $2$ of Theorem 1 we have
$\sum_{q=0}^{\infty}R(\tau^{q}\delta_{0})^{-t/d}<+\infty.$
Thus, if we denote by $C_{\\#}$ and $C_{\&}$ the constants given by
Proposition 2 and Lemma 9, respectively, then there is $Q\geq 1$ so that
$C_{\\#}C_{\&}^{-t}\sum_{q=Q+1}^{\infty}R(\tau^{q-1}\delta_{0})^{-t/d}\leq\frac{1}{2}.$
Taking $Q$ larger if necessary, assume that for each $z$ in $f^{-1}(0)$ we
have $Q\geq q(z)$. For every pair of integers $q\geq 0$ and $m\geq 1$, put
$\mathcal{P}_{t}(q,m):=\sum_{\begin{subarray}{c}z\in\mathcal{O}^{-}(0)\\\
q(z)=q,m(z)\leq m\end{subarray}}|Df^{m(z)}(z)|^{-t},$
and note that by Lemma 10 we have
$C^{\prime}:=\sum_{q=0}^{Q}\sup\\{\mathcal{P}_{t}(q,m):m\geq 1\\}<\infty.$
To prove that $\mathcal{P}(0,t)$ is finite, it is enough to prove that for
every integer $m$ we have
(7) $\sum_{q=0}^{+\infty}\mathcal{P}_{t}(q,m)\leq 2C^{\prime}.$
We proceed by induction. By our choice of $Q$, for each $z$ in $f^{-1}(0)$ we
have $q(z)\leq Q$. So when $m=1$ inequality (7) follows from our definition of
$C^{\prime}$. Let $m\geq 2$ be an integer and suppose (7) holds with $m$
replaced by $m-1$. Using the notation of Lemma 9, for each $q\geq Q+1$ we have
$\mathcal{P}_{t}(q,m)\\\ \leq
C_{\&}^{-t}\left(\sum_{\textbf{n}\in\mathcal{S}(q-1)}\sum_{\zeta\in
Z_{\textbf{n}}(q-1)}|Df^{m(\zeta)}(\zeta)|^{-t}\right)\cdot\left(\sum_{\begin{subarray}{c}\zeta^{\prime}\in\mathcal{O}^{-}(0)\\\
m(\zeta^{\prime})\leq
m-1\end{subarray}}|Df^{m(\zeta^{\prime})}(\zeta^{\prime})|^{-t}\right).$
So by Proposition 2 we have
$\mathcal{P}_{t}(q,m)\leq
C_{\\#}C_{\&}^{-t}R(\tau^{q-1}\delta_{0})^{-t/d}\sum_{p=0}^{\infty}\mathcal{P}_{t}(p,m-1).$
Summing over $q\geq Q+1$, we obtain by the induction hypothesis
$\sum_{q=Q+1}^{\infty}\mathcal{P}_{t}(q,m)\leq\frac{1}{2}\sum_{p=0}^{\infty}\mathcal{P}_{t}(p,m-1)\leq
C^{\prime}.$
Using the definition of $C^{\prime}$, this implies (7). This completes the
proof of the induction step and of the theorem. ∎
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|
arxiv-papers
| 2012-10-31T10:20:23 |
2024-09-04T02:49:37.395618
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan Rivera-Letelier, Weixiao Shen",
"submitter": "Juan Rivera-Letelier",
"url": "https://arxiv.org/abs/1210.8285"
}
|
1210.8299
|
# Quantum-criticality-induced strong Kerr nonlinearities in optomechanical
systems
Xin-You Lü1 Wei-Min Zhang1,4 Sahel Ashhab1,2 Ying Wu3 Franco Nori1,2 1Advanced
Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan
2Physics Department, The University of Michigan, Ann Arbor, Michigan
48109-1040, USA
3Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, People’s Republic of China
4 Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
###### Abstract
We investigate a hybrid electro-optomechanical system that allows us to obtain
controllable strong Kerr nonlinearities in the weak-coupling regime. We show
that when the controllable electromechanical subsystem is close to its quantum
critical point, strong photon-photon interactions can be generated by
adjusting the intensity (or frequency) of the microwave driving field.
Nonlinear optical phenomena, such as the appearance of the photon blockade and
the generation of nonclassical states (e.g., Schrödinger cat states), are
predicted in the weak-coupling regime, which is feasible for most current
optomechanical experiments.
###### pacs:
42.50.Wk; 07.10.Cm; 42.65.-k
Introduction.— Strong optical nonlinearity, as one of the central issues of
quantum optics, give rise to many strictly quantum effects, such as photon
blockade 1 ; 1.5 , optical solitons 3.1 , quantum phase transitions 4 ; 4.1 ,
quantum squeezing 4.2 and optical switching with single photon 5 . These
nonlinear optical effects have been demonstrated in cavity QED systems, where
the quantum coherence in the atom 1 ; 1.5 (or artificial atom 6 ; 6.1 ; 6.2 ;
6.3 ) generates strong effective photon-nonlinearities.
Recently, cavity optomechanics has become a rapidly developing research field
exploring nonlinear couplings via radiation pressure between the
electromagnetic and mechanical systems 7 ; 7.1 ; 7.2 . It has been shown
theoretically that strong optical nonlinear effects (and relevant
applications, such as generating nonclassical state, photon blockade, multiple
sidebands, photon-phonon transistors, and optomechanical photon measurement)
can be realized in single-mode 7.3 ; 8 ; 9 ; 9.1 ; 10 ; 10.1 ; 10.2 or two-
mode optomechanical systems (OMSs) 11 ; 12 in the single-photon strong-
coupling regime, where the optomechanical coupling at the single-photon level
$g_{a}$ exceeds the cavity decay rate $\kappa_{a}$ ($g_{a}>\kappa_{a}$).
However, in most experiments to date 12.2 ; 12.3 ; 12.4 , $g_{a}$ is much
smaller than $\kappa_{a}$ ($g_{a}/\kappa_{a}\sim 10^{-3}$). Only a few new-
type optomechanical setups, using ultracold atoms in optical resonators
($g_{a}/\kappa_{a}\sim 10^{-1}$) 13 or optomechanical crystals
($g_{a}/\kappa_{a}\sim 10^{-2}$) 14 , can approach the single-photon strong-
coupling regime. On the other hand, a strong optical driving field may enhance
the optomechanical coupling by a factor $\sqrt{n}$, where $n$ is the mean
photon number in the cavity 15 ; 16 ; 17 . But such enhancement comes at the
cost of losing the nonlinear character of the photon-photon interaction.
Figure 1: (Color online) Schematic sketch of the hybrid electro-optomechanical
system, where a mechanical oscillator couples to both an optical cavity and a
microwave LC resonator.
Given the above, it is highly desirable to find a new method for obtaining
strong Kerr nonlinearities in OMSs in the weak-coupling regime
($g_{a}\ll\kappa_{a}$). In this Letter, we investigate the Kerr nonlinear
effects of the optical field in a hybrid electro-optomechanical system
containing a mechanical oscillator coupled to both an optical cavity and a
microwave LC resonator (see Fig. 1) 22 ; 24 ; 24.5 . We find that the
eletromechanical subsystem (the mechanical oscillator and microwave resonator)
displays quantum criticality. One can drive the electromechanical subsystem
close to the quantum critical regime by applying a strong microwave field to
the microwave resonator. The quantum criticality can induce a strong Kerr
nonlinearity in the optical cavity, even if the optomechanical systems (the
optical cavity and mechanical oscillator) is in the weak-coupling regime. This
strong Kerr nonlinearity can be demonstrated by the existences of photon
blockade and nonclassical states (e.g., Schrödinger cat states) of the cavity
field when the electronechanical subsystem approaches the quantum critical
point. Furthermore, the strong Kerr nonlinearity can also be controlled easily
by tuning the intensity (or frequency) of the microwave driving field. This
provides a promising route for experimentally observing strong Kerr
nonlinearities in OMSs in the weak-coupling regime.
Quantum criticality of the electromechanical subsystem.— In the hybrid
electro-optomechanical system of Fig. 1, the mechanical oscillator is
parametrically coupled to both the optical cavity and the microwave resonator.
The microwave resonator is driven by a strong field with amplitude
$\varepsilon_{c}$ and frequency $\omega_{ci}$, where $\varepsilon_{c}$ is
related to the input microwave power $P$ and microwave decay rate $\kappa_{c}$
by $|\varepsilon_{c}|=\sqrt{2P\kappa_{c}/\hbar\omega_{ci}}$. In the frame
rotating with frequency $\omega_{ci}$, the Hamiltonian for the hybrid systems
is written as 24.6
$\displaystyle\hat{H}/\hbar$
$\displaystyle=\delta_{c}\hat{c}^{\dagger}\hat{c}+\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+g_{a}\hat{a}^{\dagger}\hat{a}\left(\hat{b}^{\dagger}+\hat{b}\right)$
$\displaystyle\;\;\;\;+g_{c}\hat{c}^{\dagger}\hat{c}\left(\hat{b}^{\dagger}+\hat{b}\right)+\varepsilon_{c}\left(\hat{c}^{\dagger}+\hat{c}\right),$
(1)
where the detuning $\delta_{c}=\omega_{c}-\omega_{ci}$ and the microwave
frequency $\omega_{c}=1/\sqrt{LC}$, $g_{a}$ ($g_{c}$) denotes the
optomechanical (electromechanical) coupling strength at the single-photon
level, and $\hat{a}$ ($\hat{b}$ or $\hat{c}$) is the annihilation operator of
the optical cavity (the mechanical oscillator or the microwave resonator).
Taking a strong microwave driving field and following the standard
linearization procedure (shifting $\hat{c}$ and $\hat{b}$ with their stable-
state mean value $\alpha$ and $\beta$) 25 ; 26 ; 27 , the Hamiltonian can be
transformed into
$\displaystyle\hat{H}_{\rm opt}/\hbar$
$\displaystyle=\Delta_{c}\hat{c}^{{\dagger}}\hat{c}+\tilde{\omega}_{a}\hat{a}^{{\dagger}}\hat{a}+\omega_{b}\hat{b}^{{\dagger}}\hat{b}$
$\displaystyle\;\;\;\;+g_{a}\hat{a}^{{\dagger}}\hat{a}\left(\hat{b}^{\dagger}+\hat{b}\right)-G\left(\hat{c}^{{\dagger}}+\hat{c}\right)\left(\hat{b}^{{\dagger}}+\hat{b}\right),$
(2)
where
$G=g_{c}\sqrt{\frac{\varepsilon^{2}_{c}}{\kappa^{2}_{c}+\Delta^{2}_{c}}}$ is
the linearized electromechanical coupling strength;
$\Delta_{c}=\delta_{c}-\frac{2g^{2}_{c}\varepsilon^{2}_{c}}{\omega_{b}(\kappa_{c}^{2}+\Delta^{2}_{c})}$
and
$\tilde{\omega}_{a}=\omega_{a}-\frac{2g^{2}_{c}\varepsilon^{2}_{c}}{\omega_{b}(\kappa_{c}^{2}+\Delta^{2}_{c})}$
are, respectively, the effective microwave detuning and optical frequency
including the radiation-pressure-induced optical resonance shift. Notice that
$G$ and $\Delta_{c}$ can be easily controlled by tuning the power and
frequency of the microwave driving field 30 .
Figure 2: (Color online) (a,b) Quantum criticality of the electromechanical
subsystem, characterized by the normal-mode frequency
$\omega_{\pm}/\omega_{b}$. (c,d) Strong Kerr-nonlinearity given by the photon-
photon interaction strength $\eta$ in the optical cavity, as a function of the
adjustable parameters $G$ and $\Delta_{c}$ controlled by the microwave driving
field. The red circles and shaded area in (c,d) correspond, respectively, to
the regimes $\eta=\kappa_{a}$ and $\eta>\kappa_{a}$. The black dot-dashed
vertical lines indicate the quantum critical points $G_{cp}$ and
$\Delta_{cp}$. Other system parameters are taken as: $\omega_{b}$/2$\pi$=10
MHz, $g_{a}/\omega_{b}$=$g_{c}/\omega_{b}$=$10^{-3}$,
$\kappa_{a}/\omega_{b}=0.1$, $\kappa_{c}/\omega_{b}=0.127$,
$\Delta_{c}/\omega_{b}=1.251$ (a,c), and $G/\omega_{b}=0.5595$ (b,d).
To show quantum criticality in the electromechanical subsystem through the
control of the microwave deriving field, we first diagonalize the
electromechanical subsystem by a Bogoliubov transformation $\hat{R}=M\hat{B}$.
Here, the canonical operators are
$\hat{R}^{T}=(\hat{c},\hat{c}^{{\dagger}},\hat{b},\hat{b}^{{\dagger}})$ and
$\hat{B}^{T}=(\hat{B}_{-},\hat{B}_{-}^{{\dagger}},\hat{B}_{+},\hat{B}_{+}^{{\dagger}})$,
and $M$ is the transformation matrix (the explicit form of $M$ is shown in the
supplemental material 30 ). Then, the Hamiltonian $\hat{H}_{\rm opt}$ becomes
$\displaystyle\hat{H}_{\rm opt}/\hbar$
$\displaystyle=\omega_{-}\hat{B}_{-}^{{\dagger}}\newline
\hat{B}_{-}+\omega_{+}\hat{B}_{+}^{{\dagger}}\newline
\hat{B}_{+}+\tilde{\omega}_{a}\hat{a}^{{\dagger}}\hat{a}$ $\displaystyle-
g_{-}\hat{a}^{\dagger}\hat{a}\left(\hat{B}_{-}^{\dagger}\newline
+\hat{B}_{-}\right)+g_{+}\hat{a}^{\dagger}\hat{a}\left(\hat{B}_{+}^{{\dagger}}\newline
+\hat{B}_{+}\right),$ (3)
where $\omega_{\pm}$ are the normal mode frequencies of the electromechanical
subsystem, and $g_{\pm}$ are the effective coupling strengths between the
optical photon and the normal modes,
$\displaystyle\omega_{\pm}^{2}=\frac{1}{2}\left(\Delta_{c}^{2}+\omega_{b}^{2}\pm\sqrt{\left(\omega_{b}^{2}-\Delta_{c}^{2}\right)^{2}+16G^{2}\Delta_{c}\omega_{b}}\right)$
(4a) $\displaystyle g_{\pm}=g_{a}\sqrt{\frac{\omega_{b}(1\pm{\rm
cos2\theta})}{2\omega_{\pm}}},\;\;\;\;{\rm
tan}2\theta=\frac{4G_{c}\sqrt{\Delta_{c}\omega_{b}}}{\Delta^{2}_{c}-\omega^{2}_{b}}.$
(4b)
Equation (4) shows that $\omega^{2}_{-}$ becomes zero (negative) when
$G=G_{cp}=\sqrt{\Delta_{c}\omega_{b}}/2$ ($G>G_{cp}$), as shown in Fig. 2 (c).
This corresponds to a quantum criticality 31 , namely, the normal mode
$\omega_{-}$ will change from a standard harmonic oscillator ($G<G_{cp}$) to a
free particle, and further becomes dynamically unstable ($G>G_{cp}$) as $G$
crosses its critical value $G_{cp}$, as shown in Fig. 2 (a). Physically, when
$G$ approaches (or exceeds) $G_{cp}$, the effective potential of the normal
mode $\omega_{-}$ becomes increasingly flat (or inverted). Since $G$ can be
easily varied by tuning the power and frequency of the microwave driving
field, this quantum criticality can be easily realized in experiments.
Quantum-criticality-induced strong Kerr nonlinearities.— We find that when the
electromechanical subsystem approaches its quantum critical region, the
optical cavity shows a strong Kerr nonlinearity. To show this quantum-
criticality-induced strong Kerr nonlinearity, the Hamiltonian $\hat{H}_{\rm
opt}$ should be further diagonalized in a displaced-oscillator representation,
$\hat{H}_{\rm opt}\rightarrow\hat{V}^{{\dagger}}\hat{H}_{\rm opt}\hat{V}$,
where $\hat{V}=e^{{\hat{\mathcal{P}}\hat{a}^{\dagger}\hat{a}}}$ and
$\hat{\mathcal{P}}=\zeta_{-}\hat{\mathcal{P}}_{-}-\zeta_{+}\hat{\mathcal{P}}_{+}$
with $\hat{\mathcal{P}}_{j}=\hat{B}^{\dagger}_{j}-\hat{B}_{j}$ $(j=\pm)$,
$\zeta_{\pm}=g_{\pm}/\omega_{\pm}$. The result is
$\displaystyle\frac{\hat{H}_{\rm opt}}{\hbar}$
$\displaystyle=\tilde{\omega}_{a}\hat{a}^{{\dagger}}\hat{a}-\eta\
\hat{a}^{{\dagger}}\hat{a}\hat{a}^{{\dagger}}\hat{a}+\omega_{-}\hat{B}_{-}^{{\dagger}}\newline
\hat{B}_{-}+\omega_{+}\hat{B}_{+}^{{\dagger}}\newline \hat{B}_{+}.$ (5)
Here the photon-photon interaction strength
$\displaystyle\eta=\frac{g^{2}_{a}}{\omega_{b}-4G^{2}/\Delta_{c}}.$ (6)
It can be seen in Figs. 2 (c,d) that even in the weak-coupling regime
$g_{m}\ll\kappa_{m}$ ($m=a,c$), a large photon-photon interaction $\eta$
($\eta>\kappa_{a}$) can still be obtained when $G$ (or $\Delta_{c}$) is in the
quantum critical regime. In particular, Fig. 2 shows that when the coupling
strength $G$ (or the detuning $\Delta_{c}$) is close to its quantum critical
point, a very small normal mode frequency $\omega_{-}$ is obtained, which
induces a large photon-photon interaction with $\eta\propto 1/\omega_{-}$. The
interesting ranges of $G$ and $\Delta_{c}$ are respectively on the order of
$0.1$ kHz and $1$ kHz for the quantum critical region $\eta>\kappa_{a}$
(shaded area in Fig. 2), and this parameter precision is experimentally
realizable 32 .
More importantly, the obtained large photon-photon interaction directly
characterizes a strong optical Kerr nonlinearity. This is because, as we will
show later, the quantum criticality also significantly suppresses the sideband
phonon transitions in the optomechanical subsystem. Thus, the quantum-
criticality-induced strong self-Kerr nonlinearity is very different from
previous investigations in the usual OMSs, where the strong self-Kerr
nonlinearity is reachable only in the single-photon strong-coupling
($g_{a}>\kappa_{a}$) and the resolved sideband ($\kappa_{a}\ll\omega_{b}$)
regimes 8 ; 11 ; 12 . To demonstrate the strong Kerr nonlinearity in the
present system, we should calculate the steady-state second-order correlation
function of the optical field $g^{(2)}(0)$, and show explicitly the photon
blockade effect [$g^{(2)}(0)\rightarrow 0$] in the weak-coupling regime, as
can be experimentally detected by a Hanbury-Brown-Twiss Interferometer 1.5 .
We will also calculate the dynamical evolution of the cavity field and show
the periodic generation of noclassical states, which are experimentally
detectable via quantum state tomography. Notice that the photon blockade 8 ;
11 and nonclassical states 7.3 , as evidences of strong Kerr nonlinearities,
were obtained in the OMSs only in the single-photon strong-coupling regime and
the resolved sideband regime.
Figure 3: (Color online) Equal-time second-order correlation function
$g^{(2)}(0)$ versus: (a) coupling strength $G$, and (b) detuning $\Delta_{c}$,
for the decay rates $\kappa_{-}/2\pi=(500,250,50)$ kHz (corresponding to
$\kappa_{c}/2\pi=(1270,620,110)$ kHz and $\kappa_{b}/2\pi=1$ kHz). The system
parameters are the same as in Fig. 2 except for $\Delta_{a}=\eta$,
$\kappa_{+}/2\pi=500$ kHz.
Photon blockade.— We now drive the optical cavity with a weak laser field of
frequency $\omega_{ai}$ and amplitude $\varepsilon_{a}$. The Hamiltonian of
the system becomes
$\displaystyle\hat{H}_{\rm opt}^{\prime}/\hbar$
$\displaystyle=\Delta_{a}\hat{a}^{{\dagger}}\hat{a}-\eta\hat{a}^{{\dagger}}\hat{a}\hat{a}^{{\dagger}}\hat{a}+\varepsilon_{a}\left(\hat{a}^{\dagger}e^{-\hat{\mathcal{P}}}+e^{\hat{\mathcal{P}}}\hat{a}\right)$
$\displaystyle+\omega_{-}\hat{B}_{-}^{{\dagger}}\newline
\hat{B}_{-}+\omega_{+}\hat{B}_{+}^{{\dagger}}\newline \hat{B}_{+},$ (7)
where all the similarity transformations used before have been taken into
account, and $\Delta_{a}=\tilde{\omega}_{a}-\omega_{ai}$. Also, we may include
the damping effect arising from the coupling of the optical field to the
electromagnetic vacuum modes of the environment. Then, the dissipative
dynamics of cavity mode $\hat{a}$ can be described by the quantum Langevin
equation,
$\displaystyle\frac{\partial}{\partial
t}\hat{a}(t)=\frac{i}{\hbar}\left[\hat{H}_{\rm
opt}^{\prime},\hat{a}(t)\right]-\kappa_{a}\hat{a}(t)-\sqrt{2\kappa_{a}}e^{-\hat{\mathcal{P}}}\hat{f}_{\rm
in}(t).$ (8)
Here $\kappa_{a}$ is the decay rate of cavity mode $\hat{a}$ and $\hat{f}_{\rm
in}$ is a vacuum noise operator satisfying $\langle\hat{f}_{\rm
in}\hat{f}_{\rm in}^{\dagger}\rangle=\delta(t-t^{{}^{\prime}})$,
$\langle\hat{f}^{\dagger}_{\rm in}\hat{f}_{\rm in}\rangle=0$.
For a weak optical driving field, the quantum Langevin equations can be solved
by truncating them to the lowest relevant order in $\varepsilon_{a}$ 8 . The
resulting two-photon correlation is given by $g^{(2)}(0)={\rm
lim}_{t\rightarrow\infty}\langle\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rangle(t)/\langle\hat{a}^{\dagger}\hat{a}\rangle^{2}(t)$
with
$\displaystyle{\rm lim}_{t\rightarrow\infty}\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle(t)=\frac{2\varepsilon^{4}_{a}}{\kappa_{a}}{\rm
Re}\\!\int^{\infty}_{0}\\!\\!\\!\\!\\!\\!d\tau_{1}\int^{\infty}_{0}\\!\\!\\!\\!\\!\\!d\tau_{2}\int^{\infty}_{0}\\!\\!\\!\\!\\!\\!d\tau_{3}$
$\displaystyle\times
e^{2(-i\tilde{\Delta}_{a}+i\eta-\kappa_{a})\tau_{1}}e^{(i\tilde{\Delta}_{a}-\kappa_{a})\tau_{2}}e^{(-i\tilde{\Delta}_{a}-\kappa_{a})\tau_{3}}e^{-\Phi_{4}},$
(9a) $\displaystyle{\rm
lim}_{t\rightarrow\infty}\langle\hat{a}^{\dagger}\hat{a}\rangle(t)=\frac{\varepsilon^{2}_{a}}{\kappa_{a}}{\rm
Re}\int^{\infty}_{0}\\!\\!\\!\\!\\!d\tau\,e^{-(i\tilde{\Delta}_{a}+\kappa_{a})\tau}e^{-\Phi_{2}},$
(9b)
where $e^{-\Phi_{4}}=\langle
e^{\hat{\mathcal{P}}(\tau_{1}-\tau_{2})}e^{\hat{\mathcal{P}}(\tau_{1})}e^{-\hat{\mathcal{P}}(0)}e^{-\hat{\mathcal{P}}(-\tau_{3})}\rangle$,
$e^{-\Phi_{2}}=\langle
e^{\hat{\mathcal{P}}(\tau)}e^{-\hat{\mathcal{P}}(0)}\rangle$, and
$\tilde{\Delta}_{a}=\Delta_{a}-\eta$. Note that
$\hat{\mathcal{P}}=\zeta_{-}\hat{\mathcal{P}}_{-}-\zeta_{+}\hat{\mathcal{P}}_{+}$
is a complex operator including the microwave field $\hat{c}$ and the
mechanical mode $\hat{b}$. The dynamics of $\hat{\mathcal{P}}_{j}(t)$
$(j=\pm)$ is given by
$\hat{\mathcal{P}}_{j}(t)=e^{-\frac{i}{\hbar}(\hat{H}_{\rm
opt}^{\prime}-i\kappa_{j}/2)t}\hat{\mathcal{P}}_{j}(0)e^{\frac{i}{\hbar}(\hat{H}_{\rm
opt}^{\prime}+i\kappa_{j}/2)t}$, where the $\kappa_{j}$ are the effective
decay rates of the electromechanical normal modes (we have also assumed a
white vacuum noise on the microwave cavity and a thermal white noise bath
coupling to the mechanical oscillator, so that the effective decay rates
$\kappa_{j}$ are proportional to the original decay rates of the microwave
resonator and the mechanical oscillator 30 ).
Assuming the microwave (mechanical) mode is initially in the coherent state
$|\alpha\rangle$ ($|\beta\rangle$), and the optical field in the vacuum state,
then the two-point correlation function $e^{-\Phi_{2}}$ and the four-point
correlation function $e^{-\Phi_{4}}$ can be calculated 30 . We numerically
integrate Eqs. (9) and show the dependence of $g^{(2)}(0)$ on both $G$ and
$\Delta_{c}$ for different decay rates $\kappa_{-}$ in Fig. 3, while the
effective decay rate $\kappa_{+}$ is a fixed value due to its negligible
effect on $g^{(2)}(0)$. Fig. 3 shows that the photon blockade
[$g^{(2)}(0)\rightarrow 0$] occurs when the tunable parameter $G$ (or
$\Delta_{c}$) approaches its quantum critical value even if the optomechanical
coupling $g_{a}$ is very weak.
Furthermore, we find that the photon antibunching effect [$g^{(2)}(0)<1$]
disappears when $\kappa_{-}\ll\kappa_{a}$ (see the insets in Fig. 3). The
physical meaning of this result can be explained as follows. In the hybrid
OMS, a relatively large decay rate $\kappa_{-}$ ($\kappa_{-}\sim\kappa_{a}$)
with respect to the effective mechanical mode $\omega_{-}$ occurs when the
electromechanical subsystem approaches the quantum critical point. This decay
will significantly suppress the steady-state sideband transition in the
electromechanical subsystem. Meanwhile, the very small $\omega_{-}$ near the
quantum critical point effectively enhances the photon-photon interaction to
$\eta>\kappa_{a}$ because $\eta\propto 1/\omega_{-}$. Thus, the photon
blockade can still be obtained in our system even if $\omega_{-}<\kappa_{a}$.
However, for the usual OMSs, when the frequency of the mechanical oscillator
is smaller than the decay rate of the cavity mode (out of the resolved
sideband regime), the photon blockade will disappear due to the strong phonon
sideband transition 8 ; 11 .
Figure 4: (Color online) Parameter regimes (a) for obtaining the two- (b),
three- (c) and four-component (d) Schrödinger cat state. The quadratures
variables are $x=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$,
$y=-i(\hat{a}-\hat{a}^{\dagger})/\sqrt{2}$. The system parameters are the same
as in Fig. 2 except for $\Upsilon=2$.
Nonclassical states.— As demonstrated in previous studies 7.3 , strong Kerr
nonlinearities generally lead to the periodic in time generation of a broad
variety of nonclassical states, (i.e., cat states) of the cavity field. With
the help of the Hamiltonian (5), we can obtain the time evolution operator in
the interaction picture,
$\displaystyle\hat{U}(t)=\hat{V}e^{-i(\hat{H}_{\rm
opt}-\tilde{\omega}_{a}\hat{a}^{\dagger}\hat{a})t}\hat{V}^{\dagger}$
$\displaystyle\approx
e^{i\eta\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}t}\left\\{e^{\zeta_{-}\hat{a}^{\dagger}\hat{a}\left[\hat{B}^{\dagger}_{-}(1-e^{-i\omega_{-}t})-\hat{B}_{-}(1-e^{i\omega_{-}t})\right]}\right\\},$
(10)
where the term corresponding to $\zeta_{+}$ has been omitted due to its
negligible effect on the evolution of the cavity mode $\hat{a}$
($\zeta_{+}/\omega_{b}\sim 10^{-4}$) near the quantum critical point. If the
cavity field $\hat{a}$ is initially in a coherent state $|\Upsilon\rangle$,
the cavity field at time $t_{n}$, $t_{n}=2n\pi/\omega_{-}$ $(n=1,2...)$ will
evolve into the state
$\displaystyle|\Psi_{a}(t_{n})\rangle=e^{-|\Upsilon|^{2}/2}\sum^{\infty}_{m=0}\frac{\Upsilon^{m}}{\sqrt{m!}}e^{i\frac{2n\pi\eta}{\omega_{-}}m^{2}}|m\rangle_{a}.$
(11)
The state $|\Psi_{a}(t_{n})\rangle$ is a multi-component cat state, depending
on the value of $\eta/\omega_{-}$. Fig. 4 shows the different multi-component
cat states for different values of $\eta/\omega_{-}$ near the quantum critical
point. Figs. 4 (b,c,d) present the specific realization of two-, three- and
four-component cat states, respectively. Here we should point out that the
system’s damping (given by $\kappa_{a},\kappa_{c},\kappa_{b}$) has been
ignored. In principle, this approximation is valid when the cut-off time
$t_{n}\ll 1/\kappa_{a},1/\kappa_{c},1/\kappa_{b}$. The above result indicates
that the quantum-criticality-induced strong Kerr nonlinearities in this hybrid
OMS can generate nonclassical states by cutting off the optomechanical
interaction at the appropriate time, which can be detected via Wigner
tomography.
Conclusion.–In summary, we have identified a mechanism for obtaining strong
Kerr nonlinear effects in a hybrid OMS in the weak-coupling regime. The
photon-photon interaction is controllable through the adjustable parameters
$G$ and $\Delta_{c}$, and the sideband phonon transitions can be suppressed.
The photon blockade and nonclassical states are demonstrated near the quantum
critical point. This may provide a new avenue for experimentally realizing
strong optical nonlinearities in the weak-coupling regime and largely enrich
the parameter scope for implementing quantum information processing and
quantum metrology with cavity OMSs.
Acknowledgements— XYL thanks Jing Liu, Jie-Qiao Liao, Guang-Ri Jing, Jing
Zhang and Wei Cui for valuable discussions. FN was partially supported by the
ARO, JSPS-RFBR contract No. 12-02-92100, Grant-in-Aid for Scientific Research
(S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS via its FIRST program.
YW is was partially supported by the National Science Foundation (NSF) of
China (Grants No. 10975054), the National Fundamental Research Program of
China (Grant No. 2012CB922103). WMZ is supported by the National Science
Council of ROC under Contract No. NSC-99-2112-M-006-008-MY3, and the National
Center for Theoretical Science of Taiwan. XYL was supported by Japanese
Society for the Promotion of Science (JSPS) Foreign Postdoctoral Fellowship
No. P12204 and the NSF of China under grant number 11005057.
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|
arxiv-papers
| 2012-10-31T11:28:48 |
2024-09-04T02:49:37.406275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin-You L\\\"u, Wei-Min Zhang, Sahel Ashhab, Ying Wu, and Franco Nori",
"submitter": "Xinyou Lu Dr.",
"url": "https://arxiv.org/abs/1210.8299"
}
|
1210.8385
|
# First Experiments with PowerPlay
Rupesh Kumar Srivastava, Bas R. Steunebrink and Jürgen Schmidhuber
The Swiss AI Lab IDSIA, Galleria 2, 6928 Manno-Lugano
University of Lugano & SUPSI, Switzerland
(2012)
###### Abstract
Like a scientist or a playing child, PowerPlay [24] not only learns new skills
to solve given problems, but also invents new interesting problems by itself.
By design, it continually comes up with the fastest to find, initially novel,
but eventually solvable tasks. It also continually simplifies or compresses or
speeds up solutions to previous tasks. Here we describe first experiments with
PowerPlay. A self-delimiting recurrent neural network SLIM RNN [25] is used as
a general computational problem solving architecture. Its connection weights
can encode arbitrary, self-delimiting, halting or non-halting programs
affecting both environment (through effectors) and internal states encoding
abstractions of event sequences. Our PowerPlay-driven SLIM RNN learns to
become an increasingly general solver of self-invented problems, continually
adding new problem solving procedures to its growing skill repertoire.
Extending a recent conference paper [28], we identify interesting, emerging,
developmental stages of our open-ended system. We also show how it
automatically self-modularizes, frequently re-using code for previously
invented skills, always trying to invent novel tasks that can be quickly
validated because they do not require too many weight changes affecting too
many previous tasks.
## 1 Introduction
To automatically construct an increasingly general problem solver, the recent
PowerPlay framework [24] incrementally and efficiently searches the space of
possible pairs of (1) new task descriptions (from the set of all computable
task descriptions), and (2) modifications of the current problem solver. The
search continues until the first pair is discovered for which (i) the current
solver cannot solve the new task, and (ii) the modified solver provably solves
all previously learned tasks plus the new one. Here a new task may actually be
to simplify, compress, or speed up previous solutions, which in turn may
invoke or partially re-use solutions to other tasks. The above process of
discovering and solving a novel task can be repeated forever in open-ended
fashion.
As a concrete implementation of the solver, we use a special neural network
(NN) [2] architecture called the Self-Delimiting NN or SLIM NN [25]. Given a
SLIM NN that can already solve a finite known set of previously learned tasks,
an asymptotically optimal program search algorithm [9, 26, 20, 21] can be used
to find a new pair that provably has properties (i) and (ii). Once such a pair
is found, the cycle repeats itself. This results in a continually growing set
of tasks solvable by an increasingly more powerful solver. The resulting
repertoire of self-invented problem-solving procedures or skills can be
exploited at any time to solve externally posed tasks.
The SLIM NN has modifiable components, namely, its connection weights. By
keeping track of which tasks depend on each connection, PowerPlay can reduce
the time required for testing previously solved tasks with certain newly
modified connection weights, because only tasks that depend on the changed
connections need to be retested. If the solution of the most recently invented
task does not require changes of many weights, and if the changed connections
do not affect many previous tasks, then validation may be very efficient.
Since PowerPlay’s efficient search process has a built-in bias towards tasks
whose validity check requires little computational effort, there is an
implicit incentive to generate weight modifications that do not impact too
many previous tasks. This leads to a natural decomposition of the space of
tasks and their solutions into more or less independent regions. Thus, divide
and conquer strategies are natural by-products of PowerPlay.
Note that active learning methods [5] such as AdaBoost [6] have a totally
different set-up and purpose: there the user provides a set of samples to be
learned, then each new classifier in a series of classifiers focuses on
samples badly classified by previous classifiers. In open-ended PowerPlay,
however, all computational tasks (not necessarily classification tasks) can be
self-invented; there is no need for a pre-defined global set of tasks that
each new solver tries to solve better, instead the task set continually grows
based on which task is easy to invent and validate, given what is already
known.
Unlike our first implementations of curious / creative / playful agents from
the 1990s [17, 29, 18] (_cf_. [1, 4, 13, 11]), PowerPlay provably (by design)
does not have any problems with online learning—it cannot forget previously
learned skills, automatically segmenting its life into a sequence of clearly
identified tasks with explicitly recorded solutions. Unlike the task search of
theoretically optimal creative agents [22, 23], PowerPlay’s task search is
greedy, yet practically feasible. Here we present first experiments, extending
recent work [28].
## 2 Notation & Algorithmic Framework for PowerPlay (Variant II)
We use the notation of the original paper [24], and briefly review the basics
relevant here. $B^{*}$ denotes the set of finite bit strings over the binary
alphabet $B=\\{0,1\\}$, $\mathbb{N}$ the natural numbers, $\mathbb{R}$ the
real numbers. The computational architecture of PowerPlay’s problem solver may
be a deterministic universal computer, or a more limited device such as a
feedforward NN. All problem solvers can be uniquely encoded [7] or implemented
on universal computers such as universal Turing Machines (TM) [31]. Therefore,
without loss of generality, we can assume a fixed universal reference computer
whose inputs and outputs are elements of $B^{*}$. User-defined subsets ${\cal
S},{\cal T}\subset B^{*}$ define the sets of possible problem solvers and task
descriptions. For example, $\cal T$ may be the infinite set of all computable
tasks, or a small subset thereof. ${\cal P}\subset B^{*}$ defines a set of
possible programs which may be used to generate or modify members of $\cal S$
or $\cal T$. If our solver is a feedforward NN, then $\cal S$ could be a
highly restricted subset of programs encoding the NN’s possible topologies and
weights, $\cal T$ could be encodings of input-output pairs for a supervised
learning task, and $\cal P$ could be an algorithm that modifies the weights of
the network.
The problem solver’s initial program is called $s_{0}$. A particular sequence
of unique task descriptions $T_{1},T_{2},\ldots$ (where each $T_{i}\in{\cal
T}$) is chosen or “invented” by a search method (see below) such that the
solutions of $T_{1},\ldots,T_{i}$ can be computed by $s_{i}$, the $i$-th
instance of the program, but $T_{i}$ cannot be solved by $s_{i-1}$. Each
$T_{i}$ consists of a unique problem identifier that can be read by $s_{i}$
through some built-in mechanism (e.g., input neurons of an NN as in Sec. 3 and
4), and a unique description of a deterministic procedure for deciding whether
the problem has been solved. For example, a simple task may require the solver
to answer a particular input pattern with a particular output pattern. Or it
may require the solver to steer a robot towards a goal through a sequence of
actions. Denote $T_{\leq i}=\\{T_{1},\ldots,T_{i}\\}$;
$T_{<i}=\\{T_{1},\ldots,T_{i-1}\\}$. A valid task $T_{i}$ ($i>1$) may require
solving at least one previously solved task $T_{k}$ ($k<i$) more efficiently,
by using less resources such as storage space, computation time, energy, etc.
quantified by the function $Cost(s,T)$. The algorithmic framework (Alg. 1)
incrementally trains the problem solver by finding $p\in{\cal P}$ that
increase the set of solvable tasks. For more details, the reader is encouraged
to refer to the original report [24].
Algorithm 1 PowerPlay Framework (Variant II)
Initialize $s_{0}$ in some way
for $i:=1,2,\ldots$ do
Declare new global variables $T_{i}\in{\cal T}$, $s_{i}\in{\cal S}$,
$p_{i}\in{\cal P}$, $c_{i},c^{*}_{i}\in\mathbb{R}$ (all unassigned)
repeat
Let a search algorithm (e.g., Section 3) set $p_{i}$, a new candidate program.
Give $p_{i}$ limited time to do:
* Task Invention: Unless the user specifies $T_{i}$, let $p_{i}$ set $T_{i}$.
* Solver Modification: Let $p_{i}$ set $s_{i}$ by computing a modification of $s_{i-1}$.
* Correctness Demonstration: Let $p_{i}$ compute $c_{i}:=Cost(s_{i},T_{\leq i})$ and $c^{*}_{i}:=Cost(s_{i-1},T_{\leq i})$
until $c^{*}_{i}-c_{i}>\epsilon$ (minimal savings of costs such as
time/space/etc on all tasks so far)
Freeze/store forever $p_{i},T_{i},s_{i},c_{i},c^{*}_{i}$
end for
## 3 Experiment 1: Self-Invented Pattern Recognition Tasks
We start with pattern classification tasks. In this setup, $s$ encodes an
arbitrary set of weights for a fixed-topology multi-layer perceptron (MLP).
The MLP maps two-dimensional, real-valued input vectors from the unit square
to binary labels; i.e., $s$: $[0,1)\times[0,1)\rightarrow{0,1}$. The output
label is 0 or 1 depending on whether or not the real-valued activation of the
MLP’s single output neuron exceeds 0.5. Binary programs $p\in{\cal P}$ of
length $\mathit{length}(p)$ compute tasks and modify $s$ as follows. If
$p^{1}$ (the first bit of $p$) is 0, this will specify that the current task
is to simplify $s$ by weight decay, under the assumption that smaller weights
are simpler. Such programs implement compression tasks. But if $p^{1}$ is 1,
then the target label of the current task candidate $T$ will be given by the
next bit $p^{2}$, and $T$’s two-dimensional input vector will be uniquely
encoded by the remainder of $p$’s bit string, $p^{3}p^{4}\ldots p^{n}$, as
follows. The string $p^{3}p^{4}\ldots p^{n}$ is taken as the binary
representation of an integer $N$. Then a 2D Gaussian pseudo-random number
generator is used to generate numbers $(x_{1},y_{1}),(x_{2},y_{2}),\ldots$,
where $x$ and $y$ are used as 2D coordinates in the unit square. Now the task
is to label the coordinates $(x_{N},y_{N})$ as $p_{2}$.
The random number generator is re-seeded by the same seed every time a new
task search begins, thus ensuring a deterministic search order. Since we only
have two labels in this experiment, we do not need $p^{2}$ as we can choose
the target label to be different from the label currently assigned by the MLP
to the encoded input. To run $p$ for $t$ steps (on a training set of $i$
patterns so far) means to execute $\lfloor t/2i\rfloor$ epochs of gradient
descent on the training set and check whether the patterns are correctly
classified. Here one step always refers to the processing of a single pattern
(either a forward or backward pass), regardless of the task.
Assume now that PowerPlay has already learned a version of $s$ called
$s_{i-1}$ able to classify $i-1$ previously invented training patterns
($i>1$). Then the next task is defined by a simple enumerative search in the
style of universal search [10, 26, 21], which combines task simplification and
systematic run-time growth (see Alg. 2).
Algorithm 2 PowerPlay implementation for experiment 1
Initialize $s_{0}$ in some way
for $i:=1,2,\ldots$ do
for $m:=1,2,\ldots$ do
for all candidate programs $p$ s.t. $\mathit{length}(p)\leq m$ do
Run $p$ for at most $2^{m-\mathit{length}(p)}$ steps
if $p$ creates $s_{i}$ from $s_{i-1}$ correctly classifying all $i$ training
patterns so far and ($s_{i}$ is substantially simpler than $s_{i-1}$or $s_{i}$
can classify a newly found pattern misclassified by $s_{i-1}$) then
Set $p_{i}$ := $p$ (store the candidate)
exit $m$ loop;
end if
end for
end for
end for
(a) After 1 task (b) After 16 tasks (c) After 17 tasks
(d) After 25 tasks (e) After 33 tasks (f) After 43 tasks
Figure 1: Experiment 1. Right after initialization, before the first
compressions, the decision boundary may be arbitrary and possibly non-linear.
The drive to compress and simplify, however, first encourages linear
separability (top row). As more associations are invented, it becomes harder
and harder to learn new ones that break the previous solver’s generalization
ability, while maintaining a linear boundary. Eventually this causes the
decision boundary to become non-linear (bottom row). The decision boundary
becomes increasingly non-linear, as more and more associations are invented
and learned.
Since the compression task code is the single bit ‘0’, roughly half of the
total search time is spent on simplification, the rest is spent on the
invention of new training patterns that break the MLP’s current generalization
ability.
To monitor the evolution of the solver’s generalization map, after each
successful search for a new task, the labels of grid points are plotted in a
rather dense grid on the unit square (Fig. 1), to see how the MLP maps
$[0,1)\times[0,1)$ to ${0,1}$. As expected, the experiments show that in the
beginning PowerPlay prefers to invent and learn simple linear functions.
However, there is a phase transition to more complex non-linear functions
after a few tasks, indicating a new developmental stage [14, 19, 12]. This is
a natural by-product of the search for simple tasks—they are easier to invent
and verify than more complex non-linear tasks. As learning proceeds, we
observe that the decision boundary becomes increasingly non-linear, because
the system has to come up with tasks which the solver cannot solve yet, but
the solver becomes increasingly more powerful, so the system has to invent
increasingly harder tasks. On the other hand, the search time for solutions to
harder and harder tasks need not grow over time, since new solutions do not
have to be learnt from scratch, but may re-use previous solutions encoded as
parts of the previous solver.
## 4 Experiment 2: Self-Invented Tasks Involving Motor Control and Internal
Abstractions
### 4.1 Self-Delimiting (SLIM) Programs Run on A Recurrent Neural Network
(RNN)
Here we describe experiments with a PowerPlay-based RNN that continually
invents novel sequences of actions affecting an external environment, over
time becoming a more and more general solver of self-invented problems.
RNNs are general computers that allow for both sequential and parallel
computations. Given enough neurons and an appropriate weight matrix, an RNN
can compute any function computable by a standard PC [16]. We use a particular
RNN named SLIM RNN [25] to define $\cal S$ for our experiment. Here we briefly
review its basics.
The $k$-th computational unit or neuron of our SLIM RNN is denoted $u^{k}$
($0<k\leq n(u)\in\mathbb{N}$). $w^{lk}$ is the real-valued weight on the
directed connection $c^{lk}$ from $u^{l}$ to $u^{k}$. At discrete time step
$t=1,2,\ldots,t_{end}$ of a finite interaction sequence with the environment,
$u^{k}(t)$ denotes the real-valued activation of $u^{k}$. There are designated
neurons serving as _online inputs_ , which read real-valued observations from
the environment, and _outputs_ whose activations encode actions in the
environment, e.g., the movement commands for a robot. We initialize all
$u^{k}(1)$ = 0 and compute $u^{k}(t+1)=f^{k}(\sum_{l}w^{lk}u^{l}(t))$ where
$f$ may be of the form $f^{k}(x)=1/(1+e^{-x})$, or $f^{k}(x)=x$, or
$f^{k}(x)=1$ if $x\geq 0$ and 0 otherwise. To program the SLIM RNN means to
set the weight matrix $\langle w^{lk}\rangle$.
A special feature of the SLIM RNN is that it has a single _halt_ neuron with a
fixed _halt-threshold_. If at any time $t$ its activation exceeds the _halt-
threshold_ , the network’s computation stops. Thus, any network topology in
which there exists a path from the online or task inputs to the halt neuron
can run self-delimiting programs [10, 3, 26, 21] studied in the theory of
Kolmogorov complexity and algorithmic probability [27, 8]. Inspired by a
previous architecture [15], neurons other than the inputs and outputs in our
RNN are arranged in winner-take-all subsets (WITAS) of $n_{\mathit{witas}}$
neurons each ($n_{\mathit{witas}}=4$ was used for this experiment). At each
time step $t$, $u^{k}(t)$ is set to 1 if $u^{k}$ is a winning neuron in some
WITAS (the one with the highest activation), and to 0 otherwise. This feature
gives the SLIM RNN the potential to modularize itself, since neurons can act
as _gates_ to various self-determined regions of the network. By regulating
the information flow, the network may use only a fraction of the weights
$\langle w^{lk}\rangle$ for a given task.
(a) $t$ = 1 (b) $t$ = 2
(c) $t$ = 3 (d) $t$ = 4
Figure 2: SLIM RNN activation scheme. At various time steps, active/winning
neurons and their outgoing connections are highlighted. At each step, at most
one neuron per WITAS can become active and propagate activations through its
outgoing connections.
Apart from the online input, output and halt neurons, a fixed number $n_{ti}$
of neurons are set to be _task inputs_. These inputs remain constant for
$1\leq t<t_{end}$ and serve as self-generated task specifications. Finally,
there is a subset of $n_{s}$ _internal state_ neurons whose activations are
considered as the final outcome when the program halts. Thus a _non-
compression_ task is: Given a particular task input, interact with the
environment (read online inputs, produce outputs) until the network halts and
produces a particular internal state—the abstract goal—which is read from the
internal state neurons. Since the SLIM RNN is a general computer, it can
represent essentially arbitrary computable tasks in this way. Fig. 2
illustrates the network’s activation spreading for a particular task. A more
detailed discussion of SLIM RNNs and their efficient implementation can be
found in the original report [25].
The SLIM RNN is trained on the fovea environment described in Sec. 4.2 using
the PowerPlay framework according to Algorithm 3 below. The difference to
Algorithm 2 lies in task set-specific details such as the encoding of task
inputs and the definition of ‘inventing and learning’ a task. The bit string
$p$ now encodes a set of $n_{ti}$ real numbers between 0 and 1 which denote
the constant task inputs for this program. Given a new set of task inputs, the
new task is considered learned if the network halts and reaches a particular
internal state (potentially after interacting with the environment), and
remains able to properly reproduce the saved internal states for all
previously learned tasks. This is implemented by first checking if the network
can halt and produce an internal state on the newly generated task inputs.
Only if the network cannot halt within a chosen fraction of the time budget
dictated by $\mathit{length}(p)$, the length of program $p$, the remaining
budget is used for trying to learn the task using a simple mutation rule, by
modifying a few weights of the network. When $p$ is the single bit ‘0’, the
task is interpreted as a _compression_ task. Here compression either means a
reduction of the sum of squared weights without increasing the total number of
connection usages by all previously learned tasks, or a reduction of the total
number of connection usages on all previously learned tasks without increasing
the sum of squared weights.
Algorithm 3 PowerPlay implementation for experiment 2
Initialize $s_{0}$ in some way
for $i:=1,2,\ldots$ do
for $m:=1,2,\ldots$ do
for all candidate programs $p$ s.t. $\mathit{length}(p)\leq m$ do
Set $\mathit{time\\_budget}:=2^{m-\mathit{length}(p)}$
if $p$ encodes a compression task then
Set $s_{\mathit{temp}}:=s_{i-1}$
while $\mathit{time\\_budget}>0$ do
Create $s_{i}$ from $s_{\mathit{temp}}$ through random perturbation of a few
connection weights
if compression is successful and $\mathit{time\\_budget}\geq 0$ then
Set $s_{\mathit{temp}}:=s_{i}$
end if
end while
else
while $\mathit{time\\_budget}>0$ do
Create $s_{i}$ from $s_{i-1}$ through random perturbation of a few connection
weights
From $p$ generate task $k$
if $s_{i-1}$ does not solve $k$ and $s_{i}$ solves $k$ and $s_{i}$ solves all
previous tasks in the repertoire and $\mathit{time\\_budget}\geq 0$ then
Add the pair ($k$, internal state) to the repertoire
exit $m$ loop
end if
end while
end if
end for
end for
end for
Since our Powerplay variant methodically increases search time, half of which
is used for compression, it automatically encourages the network to invent
novel tasks that do not require many changes of weights used by many previous
tasks.
Our SLIM RNN implementation efficiently resets activations computed by the
numerous unsuccessful tested candidate programs. We keep track of used
connections and active (winner) neurons at each time step, to reset
activations such that tracking/undoing effects of programs essentially does
not cost more than their execution.
### 4.2 RNN-Controlled Fovea Environment
(a)
(b)
Figure 3: (a) Fovea design. Pixel intensities over each square are averaged to
produce a real valued input. The smallest squares in the center are of size
$3\times 3$. (b) The RNN controls the fovea movement over a static image, in
our experiments this photo of the city of Lugano.
The environment for this experiment consists of a static image which is
observed sequentially by the RNN through a fovea, whose movement it can
control at each time step. The size of the fovea is $81\times 81$ pixels; it
produces 25 real valued online inputs (normalized to $[0,1]$) by averaging the
pixel intensities over regions of varying sizes such that it has higher
resolution at the center and lower resolution in the periphery (Fig. 3). The
fovea is controlled using 8 real-valued outputs of the network, and a
parameter _win-threshold_. Out of the first four outputs, the one with the
highest value greater than _win-threshold_ is interpreted as a movement
command: up, down, left, or right. If none of the first four outputs exceeds
the threshold, the fovea does not move. Similarily, the next four outputs are
interpreted as the fovea step size on the image (3, 9, 27 or 81 pixels in case
of exceeding the threshold, 1 pixel otherwise).
### 4.3 Results
Figure 4: For the first five self-invented non-compression tasks, we plot the
number of connection usages per task. In this run, solutions to 340 self-
generated tasks were learned. 67 of them were non-compression tasks (marked by
small black lines at the top); the rest resulted in successful compressions of
the SLIM RNN’s weight matrix. Over time, previously learned skills tend to
require less and less computational resources, i.e., the SLIM RNN-based solver
learns to speed up its solutions to previous self-invented tasks. Although
some plot lines occasionally go up, this is compensated for by a decrease of
connection usages for dozens of other tasks (not shown here to prevent
clutter).
The network’s internal states can be viewed as abstract summaries of its
trajectories through the fovea environment and its parallel “internal
thoughts.” The system invents more and more novel skills, each breaking the
generalization ability of its previous SLIM NN weight matrix, without
forgetting previously learned skills. Within 8 hours on a standard PC, a SLIM
RNN consisting of 20 WITAS, with 4 neurons in each WITAS, invented 67 novel
action sequences guiding the fovea before halting. These varied in length,
consuming up to 27 steps. Over time the SLIM NN not only invented new skills
to solve novel tasks, but also learned to speed up solutions to previously
learned tasks, as shown in Fig. 4. For clarity, all figures presented here
depict aspects of this same run, though results were consistent over many
different runs.
Figure 5: For only six selected tasks (to prevent clutter), we plot the number
of interactions with the environment, over a run where 67 novel non-
compression tasks were learned, besides numerous additional compression tasks
ignored here. Here an interaction is a SLIM NN computation step that produces
at least one non-zero output neuron activation. The total number of
interactions cannot exceed the number of steps until the halt neuron is
activated.
The SLIM NN also learns to reduce the interactions with the environment. Fig.
5 shows the number of interactions required to solve certain previously
learned fovea control tasks. Here an “interaction” is a SLIM NN computation
step that produces at least one non-zero output neuron activation. General
trend over different tasks and runs: the interactions decrease over time. That
is, the SLIM NN essentially learns to build internal representations of its
interaction with the environment, due to PowerPlay’s continual built-in
pressure to speed up and simplify and generalize.
Figure 6: Connection usage ratios for all SLIM RNN connections after learning
227 out of the 340 total self-invented tasks, 50 of them non-compression tasks
forming the so-called task repertoire, the rest compression tasks. The _usage
ratio_ on the $y$-axis is the number of repertoire tasks using the connection,
divided by the number of repertoire tasks. This ratio is 1 for the first 110
connections, which are frequently used outgoing connections from task and
online inputs. The network learns to better utilize its own architecture by
using different connections for different tasks, thus reducing the number of
connections with high usage ratio. Such modularization can help to speed up
task search in later stages.
The SLIM NN often uses partially overlapping subsets of connection weights for
generating different self-invented trajectories. Fig. 6 shows that not all
connections are used for all tasks, and that the connections used to solve
individual tasks can become progressively more separated. In general, the
variation in degree of separation depends on network parameters and
environment.
Figure 7: For each self-invented non-compression task, we plot the number of
modified SLIM NN weights needed to learn it without forgetting solutions to
old tasks. During task search, the number of connections to modify is chosen
randomly. Once the growing repertoire has reached a significant size, however,
_successfully_ learned additional tasks tend to require few weight changes
affecting few previous tasks (especially tasks with computationally expensive
solutions). This is due to PowerPlay’s bias towards tasks that are fast to
find and validate on the entire repertoire. See text for details.
As expected, PowerPlay-based SLIM NNs prefer to modify only few connections
per novel task. Randomly choosing one to fifteen weight modifications per
task, on average only $2.9$ weights were changed to invent a new skill—see
Fig. 7. Why? Because PowerPlay is always going for the novel task that is
fastest to find and validate, and fewer weight changes tend to affect fewer
previously learned tasks; that is, less time is needed to re-validate
performance on previous tasks. In this way PowerPlay avoids a naively expected
slowdown linear in the number of tasks. Although the number of skills that
must not be forgotten grows all the time, the search time for new skills does
not at all have to grow in proportion to the number of previously solved
tasks.
As a consequence of its bias towards fast-to-validate solutions, the
PowerPlay-based SLIM NN automatically self-modularizes. The SLIM RNN tested
above had 1120 connections. Typically, 600 of them were used to solve a
particular task, but on average less than three of them were changed. This
means that for each newly invented task, the system re-uses a lot of
previously acquired knowledge without modification. The truly novel aspects of
the task and its solution often can be encoded within just a handful of bits.
This type of self-modularization is more general than what can be found in
traditional (non-inventive) modular reinforcement learning (RL) systems whose
action sequences are chunked into macros to be re-used by higher-level macros,
like in the options framework [30], or in hierarchical RL [32]. Since the SLIM
RNN is a general computer, and its weights are its program, subsets of the
weights can be viewed as sub-programs, and new sub-programs can be formed from
old ones in essentially arbitrary computable ways, like in general incremental
program search [21].
## 5 Discussion and Outlook
PowerPlay for SLIM RNN represents a greedy implementation of central aspects
of the Formal Theory of Fun and Creativity [22, 23]. The setup permits
practically feasible, curious/creative agents that learn hierarchically and
modularly, using general computational problem solving architectures. Each new
task invention either breaks the solver’s present generalization ability, or
compresses the solver, or speeds it up.
We can know precisely what is learned by PowerPlaying SLIM NN. The self-
invented tasks are clearly defined by inputs and abstract internal outcomes /
results. Human interpretation of the NN’s weight changes, however, may be
difficult, a bit like with a baby that generates new internal representations
and skills or skill fragments during play. What is their “meaning” in the eyes
of the parents, to whom the baby’s internal state is a black box? For example,
in case of the fovea tasks the learner invents certain input-dependent
movements as well as abstractions of trajectories in the environment (limited
by its vocabulary of internal states). The RNN weights at any stage encode the
agent’s present (possibly limited) understanding of the environment and what
can be done in it.
PowerPlay has no problems with noisy inputs from the environment. However, a
noisy version of an old, previously solved task must be considered as a new
task, because in general we do not know what is noise and what is not. But
over time PowerPlay can automatically learn to generalize away the “noise,”
eventually finding a compact solver that solves all “noisy” instances seen so
far.
Our first experiments focused on developmental stages of purely creative
systems, and did not involve any externally posed tasks yet. Future work will
test the hypothesis that systems that have been running PowerPlay for a while
will be faster at solving many user-provided tasks than systems without such
purely explorative components. This hypothesis is inspired by babies who
creatively seem to invent and learn many skills autonomously, which then helps
them to learn additional teacher-defined external tasks. We intend to identify
conditions under which such knowledge transfer can be expected.
## Appendix A Appendix: Implementation details
The SLIM RNN used for Experiment 2 (fovea control) is constructed as follows:
Let the number of input, output and state neurons in the network be _n_input_
, _n_output_ and _n_state_ , respectively. Let _nb_comp_ = number of
computation blocks each with _block_size_ neurons. Thus there are _nb_comp_
$\times$_block_size_ computation neurons in the network.
The network is wired as follows. Each task input neuron is connected to
_nb_comp_ computation neurons at random. Each online input neuron is connected
to _nb_comp_ /10 neurons at random. Each internal state neuron receives
connections from _nb_comp_ /2 random computation neurons. The halt neuron
recieves connections from _nb_comp_ /2 random computation neurons. _nb_comp_
$\times$_n_output_ random computation neurons are connected to random output
neurons. Each neuron in each computation block is randomly connected to
_block_size_ other computation neurons.
We used _nb_comp = 20_ , _block_size = 4_ , and _n_state = 3_ with _n_input =
25_ and _n_output = 8 _missing__ for the fovea control task. The _halt-
threshold_ was set to 3, and the WITAS and fovea control _win-threshold_ s
were set to $0.00001$. All connection weights were initialized to random
values in $[-1,1]$. The cost of using a connection (consuming part of the
_time_budget_) was set to $0.1$ for all connections. The mutation rule was as
follows. For non-compression tasks, the network is first run using the new
task inputs to check if the task can already be solved by generalization. If
not, we randomly generate an integer number $m$ between 1 and 1/50th of all
connections used during the unsuccessful run, and randomly change $m$ weights
by adding to them a uniformly random number in $[-0.5,0.5]$. For compression
tasks, we randomly generate a number $m$ between 1 and 1/50th of all
connections used for any of the tasks in the current repertoire, and randomly
modify $m$ of those connections.
The time budget fraction available to check whether a candidate task is not
yet solvable by $s_{i-1}$ was chosen randomly between 0 and _time_budget_ /2.
For compression tasks, the sum of squared weights had to decrease by at least
a factor of 1/1000 to be acceptable.
## Acknowledgments
PowerPlay [24] and self-delimiting recurrent neural networks (SLIM RNN) [25]
were developed by J. Schmidhuber and implemented by R.K. Srivastava and B.R.
Steunebrink. We thank M. Stollenga and N.E. Toklu for their help with the
implementations. This research was funded by the following EU projects: IM-
CLeVeR (FP7-ICT-IP-231722) and WAY (FP7-ICT-288551).
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|
arxiv-papers
| 2012-10-31T16:41:37 |
2024-09-04T02:49:37.416346
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rupesh Kumar Srivastava, Bas R. Steunebrink and J\\\"urgen Schmidhuber",
"submitter": "Rupesh Kumar Srivastava",
"url": "https://arxiv.org/abs/1210.8385"
}
|
1210.8405
|
# Lead monoxide $\alpha$-PbO: electronic properties and point defect
formation.
J. Berashevich ‡ Thunder Bay Regional Research Institute, 290 Munro St.,
Thunder Bay, ON, P7B 5E1, Canada O. Semeniuk Thunder Bay Regional Research
Institute, 290 Munro St., Thunder Bay, ON, P7B 5E1, Canada Department of
Physics, Lakehead University, 955 Oliver Road, Thunder Bay, ON, P7B 5E1 O.
Rubel Thunder Bay Regional Research Institute, 290 Munro St., Thunder Bay,
ON, P7B 5E1, Canada Department of Physics, Lakehead University, 955 Oliver
Road, Thunder Bay, ON, P7B 5E1 J.A. Rowlands Thunder Bay Regional Research
Institute, 290 Munro St., Thunder Bay, ON, P7B 5E1, Canada A. Reznik Thunder
Bay Regional Research Institute, 290 Munro St., Thunder Bay, ON, P7B 5E1,
Canada Department of Physics, Lakehead University, 955 Oliver Road, Thunder
Bay, ON, P7B 5E1
###### Abstract
The electronic properties of polycrystalline lead oxide consisting of a
network of single-crystalline $\alpha$-PbO platelets and the formation of the
native point defects in $\alpha$-PbO crystal lattice are studied using first
principles calculations. The $\alpha$-PbO lattice consists of coupled layers
interaction between which is too low to produce high efficiency interlayer
charge transfer. In practice, the polycrystalline nature of $\alpha$-PbO
causes the formation of lattice defects in such a high concentration that
defect-related conductivity becomes the dominant factor in the interlayer
charge transition. We found that the formation energy for the O vacancies is
low, such vacancies are occupied by two electrons in the zero charge state and
tend to initiate the ionization interactions with the Pb vacancies. The
vacancies introduce localized states in the band gap which can affect charge
transport. The O vacancy forms a defect state at 1.03 eV above the valence
band which can act as a deep trap for electrons, while the Pb vacancy forms a
shallow trap for holes located just 0.1 eV above the valence band. Charge de-
trapping from O vacancies can be accounted for the experimentally found dark
current decay in ITO/PbO/Au structures.
## I Introduction
Polycrystalline lead oxide (PbO) is one of the most promising photoconductive
materials for use as a x-ray-to-charge transducer in direct conversion x-ray
detectors simon . Since the direct conversion detection scheme offers a number
of advantages over the indirect conversion, Sasha , photoconductive materials
for x-ray imaging have recently attracted much interest. The four most
important criteria when potential x-ray photoconductors are considered
include: (1) high conversion gain; (2) high x-ray absorption efficiency; (3)
compatibility with large area detector technology and (4) good photoconductive
properties. PbO satisfies the first three criteria. However, thick PbO does
not show adequate transport properties and this results in poor temporal
characteristics (signal propagation/delay time) simon ; hughes . This is the
primary issue in the development of PbO in x-ray medical imaging detectors.
For use in x-ray detectors, photoconductive layers are deposited directly over
an imaging matrix. When PbO layer is deposited in the evaporation process, it
condenses in very thin platelets a few micrometers in size which have a
porosity of around 50$\%$ simon . On the mesoscopic scale a single platelet is
a collection of the stacked PbO layers lec (Fig. 1). It is expected that a
layered structure will result in anisotropy in the transport properties and
inter-platelets and/or interlayer charge transfer will limit the overall
charge mobility.
Figure 1: The crystal structure of the tetragonal $\alpha$-PbO of the space
group 129P4/nmm.
Another common reason for the low charge drift mobility in polycrystalline
compounds that must be considered in PbO is charge carrier trapping. Traps
capture the photogenerated carriers or act as the scattering or recombination
centers kabir . Generally, the effect of the carrier trapping on the transport
properties depends on the trap concentration. The process of carrier de-
trapping from shallow traps is fast in comparison to the deep traps which tend
to hold the carriers longer, thereby significantly impairing the temporal
characteristics of the material. For x-ray application, the worst situation is
when the carriers de-trapping becomes longer than the collection time of the
X-ray signal and as such the de-trapped carriers contribute to the appearance
of image artifacts known as ’ghosting’ rowland . This limits the application
of the PbO-based detectors in real-time imaging procedures.
Thermally deposited PbO has an oxygen deficiency chienes ; bigelow ; scanlon .
Although the effect of the O vacancies on transport in PbO is not clearly
understood, it was found that thermal annealing in pure oxygen increases the
electrical conductivity that can be attributed to the reduction of the oxygen
vacancy concentration chienes . Unfortunately, high-temperature annealing is
not practical for PbO layers deposited over an imaging matrix. Therefore,
comprehensive studies of PbO structure and defect formation in thermally
evaporated PbO layers are needed to improve the temporal characteristics of
the PbO compound. A growth process must be developed to reduce the trap
concentration and ultimately to improve the performance of PbO for real time
x-ray detector applications. We started our investigations by modelling
$\alpha$-PbO crystal lattice, revealing the crystal structure parameters and
verifying them with available theoretical and experimental data. Due to
limitation of the software used, the dispersive interactions have been
neglected in our studies. The effect of this assumption on results is
discussed in Sec. II. Further, the vacancies have been induced into the
lattice of $\alpha$-PbO and defect formation energies, energetic location of
the defects within the band gap and potential of their transition to the
charged states have been investigated. In order to show the consistency of our
results with the experimental data, we completed our work with measurements of
the dark current on the ITO/PbO/Au samples with an analysis of the trap
participation in the charge transport process.
## II Calculation technique and experiment
### II.1 Calculation technique
In our study we applied the density functional theory (DFT) available in the
Wien2k package wien which utilizes the full-potential augmented plane-wave
method. All calculations of the electronic structure were performed with the
Perdew-Burke-Ernzerhof parametrization GGA of the generalized gradient
approximation (GGA) to DFT.
Calculations of the formation energy of defects (the cohesive properties) with
DFT methods require attention to many parameters and calculations with high
precision have to be implemented. Applying the appropriate energy cut off to
separate the core and valence states we treated $5p$, $5d$, $6s$ and $6p$
electrons of Pb atoms and $2s$ and $2p$ electrons of O atoms as valence
electrons. The inclusion of the $5d$ states of Pb atom in the valence states
(not in the core states) is required to introduce a proper description of
bonding and orbital hybridization of Pb atoms with O atoms. Indeed, the
orbital energy of the $5d$ electrons in Pb atoms and $2p$ electrons in O atom
are located in the same energy range terp . There is one more parameter
needing special attention when the cohesive properties are studied, the so-
called $RK_{max}$ parameter, which is the product of the atomic sphere radius
and the plane-wave cutoff in $k$-space. In our calculation it was assigned as
8. The Brillouin zone of a primitive cell for most of the calculations was set
to a 11$\times$11$\times$8 Monkhorst-Pack mesh. When the supercell procedure
was used, the size of the Monkhorst-Pack mesh was adjusted to the size of the
supercell, i.e. the mesh was appropriately reduced as the supercell was
enlarged. The calculations of the formation energy of the native point defects
have been done with the sufficiently large supercell of 108-atom size
(3$\times$3$\times$3 array of the primitive unit cells) and
4$\times$4$\times$3 Monkhorst-Pack mesh.
The optimization procedure for the $\alpha$-PbO lattice was performed based on
minimization of the forces forces and it provided us with the following
lattice parameters: lattice constants $a_{0}=4.06$ Å (within the layer) and
$c_{0}=5.51$ Å (inter-layer distance) with the ratio $c_{0}/a_{0}=1.357$ and
the Pb-O bond length of 2.35 Å. The achieved magnitude for $a_{0}$ is
identical to the value obtained with GGA in Ref. venk and is in excellent
agreement with the experimental value of 3.96 Åobtained in Ref. lec . The
calculated length of Pb-O bond is in agreement with other theoretical studies
walsh and correlates well with an experimental value of 2.32 Å lec . The
second lattice parameter $c_{0}$ agrees very well with GGA calculations
performed in Ref.walsh ; Oleg but is larger than the experimentally
determined value of 5.07 Å venk .
The mismatch in lattice parameter $c_{0}$ occurs due to limitations of GGA
functional which does not include into account the dispersive interactions. In
attempt to compensate for this limitation, we have used the experimental value
of the lattice parameter $c_{0}$=5.07 Å venk that induces a reduction in
layer separation. As a result, an increase in the interlayer interaction
strength has contributed into a raise of the layers binding energy from 0.013
eV/atom to 0.016 eV/atom being surprisingly small (calculated as the
difference in the total energy between the systems of a single layer and two
layers of $\alpha$-PbO). However, it has caused significant suppression in the
band gap size. If for the optimized lattice constant $c_{0}=5.51$ Å the
indirect gap $\bm{\Gamma}-\mathbf{M}^{*}$ is 1.8 eV mine which is in good
agreement with experimentally found optical gap of 1.9$\pm$0.1 eV thang , with
implementation of the lattice parameter $c_{0}$=5.07 Å venk , the gap shrinks
by 0.22 eV. The important distinction of the $\alpha$-PbO crystal structure is
that the layers are held together by the weak orbital overlap of the $6s^{2}$
lone pairs terp while reduction of $c_{0}$ leads to an overestimation of the
interlayer overlap of these orbitals by GGA. Since the indirect gap is defined
by strength of the interlayer interactions, it decreases with $c_{0}$
suppression mine . Similar was observed for SnO which has the same lattice
type as $\alpha$-PbO (129P4/nmm space group) for which inclusion of the
dispersive interactions while correcting the lattice parameters caused
unreasonable reduction in the band gap sizes $E_{G}$ allen . However, the
formation energies of vacancies did not show strong dependence on the lattice
parameter $c_{0}$. The vacancy states are localized entirely within the single
layer, such as their formation energy is affected strongly only by the lattice
constant $a_{0}$ and length of the Pb-O bond, but contribution of the
interlayer interactions defined by $c_{0}$ is negligibly small. For the
purposes of this work, the large discrepancy in the band gap size makes it
difficult to define correctly an appearance of defects inside the band gap.
Therefore, in order to achieve the meaningful results on both, location of the
defect states inside the band gap and their formation energy, we found more
justified to use the optimized lattice constant $c_{0}=5.51$ Å (see Sec. III).
The value of the interlayer interactions in order of 0.013 eV/atom (0.016
eV/atom for $c_{0}$=5.07 Å) is lower than thermal energy at room temperature
$kT\simeq 0.026$ eV and much lower than in most other solid state materials.
For example, in graphite which consists of stacked graphene layers, the
interlayer interactions were found to be 0.052 eV/atom graphite . The
extremely low magnitude of interactions in $\alpha$-PbO is similar to the
inter-molecular interactions in $\pi$-conjugate organic systems DNA . For
$\pi$-conjugate organic systems the low interlayer interaction is the primary
reason for the extremely low carrier mobility. Since in PbO the interlayer
interaction is half of $kT$ at room temperature, we anticipate that the
interlayer electron transport in $\alpha$-PbO by all potential transport
mechanisms would be insufficient, meaning that electron mobility in this
direction is extremely low. This statement is supported by the almost
dispersionless valence bands observed in the band diagram of $\alpha$-PbO terp
.
### II.2 Experimental details
Thick (40 $\mu$m) PbO layers were deposited on ITO-covered Corning glass
substrates by thermal evaporation of the PbO powder (purity 99.9999 $\%$) in a
vacuum of 0.2 Pa under additional molecular oxygen flow. The growth rate was 1
$\mu$m/min. The substrate was kept at 120∘ C to suppress the growth of
$\beta$-PbO simon (grown PbO layers may contain a trace of other lead-
compounds) and to achieve good adhesion to the substrate. Subsequently, gold
contacts were deposited through the contact mask by a spattering technique.
The resulting ITO/PbO/Au structures were biased to different electric fields
(ITO was negatively biased) and time dependence of the dark current density
was measured automatically every second for 50 minutes.
## III Results and discussion
### III.1 Native point defects
As it was mentioned above, in practice thermally grown polycrystalline
$\alpha$-PbO contains large amount of the O vacancies chienes ; bigelow ;
scanlon . Therefore, in this work we consider the formation of the native
point defects (i.e. O or Pb vacancies) assuming that no impurities responsible
for formation of other type of defects are present. This assumption is quite
reasonable when PbO is grown in vacuum employing only molecular oxygen flow.
We applied a supercell approach in which the larger is the supercell size, the
smaller is the interaction between the repeated units (neighbouring supercells
each containing a defect). A truly isolated defect should show the dispersion-
less flat band on the band diagram which was indeed received with a supercell
of 108-atom size (3$\times$3$\times$3 array).
Figure 2: Colour on-line. Density of states calculated for defect free
$\alpha$-PbO (dashed line) and the same system with defects (solid lines): DOS
for the O vacancy is shown by red solid line while for the Pb vacancy it is
marked by a black line. The energetic location of the vacancy states inside
the band gap relatively the top of the valence band is
$E_{D}$($V^{\operatorname{O}}$)=1.03 eV and
$E_{D}$($V^{\operatorname{Pb}}$)=0.01 eV for the O vacancy and, the Pb
vacancy, respectively.
We examined the formation of vacancy defect by removing one of the
corresponding atoms from the $\alpha$-PbO lattice and then optimized its
geometry with respect to the internal degrees of freedom. As a single defect
is induced, the lattice rearrangement around the defect site occurs. Within
the $\alpha$-PbO lattice, the layer of O atoms are tightly sandwiched between
Pb atoms (see Fig. 1), i.e. the Pb atoms are located on the side of the layers
holding the skeleton of each layer. Therefore, removal of the Pb atom from the
lattice induces a significant lattice rearrangement, while in the case the O
atom of small atomic radius is removed, the distortion of the lattice is
minimal. Thus, removal of Pb atom ($V^{\operatorname{Pb}}$) initiates the
enlargement of the distance between O atoms which were bonded to Pb site, and
each O atom moves apart from the defect site by 0.22 Å due to a repulsion felt
by the O ions. In contrast, the lattice modification induced by the O vacancy
is almost unnoticeable: Pb atoms move only by 0.07 Å toward the vacancy site.
We show in Fig. 2 an alteration in the density of states (DOS) as defects are
induced into the $\alpha$-PbO lattice. Insertion of either O or Pb vacancy
creates the defect level inside the band gap and the electronic density for
the defect state is strongly localized in both cases. The electron density
distribution outside the band gap is not significantly affected by the
presence of defects since only O:2p4 and Pb:6p2 electrons participate in
formation of the Pb-O bond (these states are located close to the band gap
edges). Our results indicate that O vacancy ($V^{\operatorname{O}}$) forms the
defect level near the midgap. The energetic location of this defect level is
$E_{D}$($V^{\operatorname{O}}$)=1.03 eV above the top of the valence band
$E_{V}$ as shown in Fig. 2. The O vacancy in its uncharged state is already
filled with two electrons. Because O atom forms 4 bonds with nearest Pb atoms
in the $\alpha$-PbO lattice, a removal of a single O atom leaves 0.5 unbounded
electrons on each Pb atom that overall results in occupation of this defect
level by two electrons. The Pb vacancy ($V^{\operatorname{Pb}}$) induces the
defect energetically located close to the top of the valence band with
$E_{D}$($V^{\operatorname{Pb}}$)=0.1 eV and this vacancy is not filled with
electrons.
### III.2 The formation energy of the vacancies
The formation energy of a vacancy is an important parameter as it determines
how likely a vacancy will be generated in the compound under given growth
conditions. The formation energy is mainly defined by several parameters:
($i$) type of the crystal structure as energy required to remove an atom from
the crystal structure depends on the strength of the electronic interactions
within the lattice, ($ii$) the final state of the removed species, ($iii$) the
characteristics of the environment, i.e. growth conditions.
The formation energy of a defect $D$ in charge state $q$ can be defined as
walle :
$\Delta
E^{f}(D)=E_{tot}(D^{q})-E_{tot}(S)+\sum_{i}n_{i}\mu_{i}+q(E_{F}+E_{V}+\Delta
V)$ (1)
where $E_{tot}(D^{q})$ and $E_{tot}(S)$ are the total energy of the system
containing the single defect and defect-free system, respectively. $n_{i}$
indicates a number of $i$-atoms removed while $\mu_{i}$ is the chemical
potential of those atoms. ($E_{F}+E_{V}$) is the position of the Fermi level
relative to the valence band maximum ($E_{V}$). $q$ defines the charge of the
state (+2/+1/0/-1/-2). For the charged point defects, the position of the
valence band $E_{V}$ has to be corrected with $\Delta V$ calculated through
alignment of the reference potential in defective supercell with that in bulk
$\alpha$-PbO (for details see Ref. walle ). The formation energy of the
defects has been also corrected with so-called band gap error $\Delta E_{G}$
togo which is defined as a difference between the direct band gap
$\bm{\Gamma}$-$\bm{\Gamma}^{*}$=1.94 eV and the experimental optical band gap
of 1.9 eV thang . In this particular case, the contribution of $\Delta
E_{G}$=-0.04 eV into the formation energy is minor because of good agreement
of the theoretical value and experimental data (a value of the corrected band
gap of suprecell togo is 1.89 eV). The formation energy has been increased by
this band gap correction $m\Delta E_{G}$, where $m$ is the number of the
electron at the defect site.
The chemical potentials are defined as following
$\mu_{i}=E_{tot}(i)+\mu_{i}^{*}$, where $\mu_{i}^{*}=\mu_{i}^{0}+kT\cdot
ln(p/p^{0})$ is the part related to real growth conditions: the partial
pressure $p$ and temperature $T$ ($\mu_{i}^{0}$ is an alteration to the
chemical potential induced by change of temperature from 0 to T under the
standard pressure $p^{0}$). However, not willing to speculate on the Pb and O
partial pressures used during deposition, we consider the extreme cases, i.e.
the Pb-rich or O-rich growth conditions
($\mu_{\operatorname{(Pb)}}$=$\mu_{\operatorname{(Pb)[bulk]}}$ and
$\mu_{\operatorname{(O)[O_{2}]}}$, respectively) as it was suggested in Refs.
walle ; togo ; allen ; zheng . The chemical potentials for the extreme cases
can be evaluated through the standard enthalpy of formation
$\Delta_{f}H^{0}(\operatorname{PbO})$ as walle :
$E_{tot}(\operatorname{PbO})=\mu_{\operatorname{(Pb)[bulk]}}+\mu_{\operatorname{(O)[O_{2}]}}+\Delta_{f}H^{0}(\operatorname{PbO})$
(2)
where $E_{tot}(\operatorname{PbO})$ is the total energy of the product;
$\mu_{\operatorname{(Pb)[bulk]}}$ and $\mu_{\operatorname{(O)[O_{2}]}}$ are
the chemical potentials of bulk Pb and O2 molecule, respectively. Therefore,
$\Delta_{f}H^{0}(\operatorname{PbO})$ is an important parameter in definition
of the chemical potentials. In the calculation of $\Delta_{f}H^{0}$ for oxides
the main discrepancy between the theoretical and experimental data is known to
come from the binding energy of the O2 molecule ($\Delta_{f}H^{0}$(O2)) used
in definition of the chemical potential
$\mu_{\operatorname{(O)}[O_{2}]}=\frac{1}{2}(2E_{tot}(\operatorname{O})+\Delta_{f}H^{0}$(O2)+$\mu_{(\operatorname{O}_{2})}^{*})$
wang ; zheng ; hammer . To disregard this error in our calculations we used
the experimental value of $\Delta_{f}H^{0}$(O2)=-5.23 eV exp (best
theoretical estimation is $\Delta_{f}H^{0}$(O2)=-6.01 eV wang ). With that
assumption we obtained $\Delta_{f}H^{0}$(PbO)=-2.92 eV per Pb-O pair which is
in appropriate agreement with the experimental value of
$\Delta_{f}H^{0}$(crystal $\alpha$-PbO)=-2.29 eV JANAF .
It is known that regardless the deposition techniques used, the PbO layers are
not stoichiometric and has deficit of oxygen chienes ; bigelow ; scanlon .
Hence, we consider the Pb-rich/O-poor conditions for which the O-poor limit
has been assigned to
$\mu_{\operatorname{(O)}}^{*}=\Delta_{f}H^{0}(\operatorname{PbO})$ togo
($\mu_{\operatorname{(O)}}^{*}$=-2.92 eV), while Pb-rich limit has been found
from relation
$\Delta_{f}H^{0}(\operatorname{PbO})=\mu_{\operatorname{(Pb)}}^{*}+\mu_{\operatorname{(O)}}^{*}$
($\mu_{\operatorname{Pb}}^{*}$=0 eV). The formation energy of the defects
$\Delta E^{f}(D)$ for $V^{\operatorname{O}}$ and $V^{\operatorname{Pb}}$ for
the different charge states calculated with help of Eq. 1 are presented in
Fig. 3.
Figure 3: The formation energy of the defects $\Delta E^{f}(D)$ for
$V^{\operatorname{O}}$ and $V^{\operatorname{Pb}}$ for Pb-rich/O-poor limit.
The charge states for which added electron or hole remains localized on the
vacancy site are shown (1+/2+ states for the $V^{\operatorname{O}}$ vacancy
and 2$-$/1$-$ states for the $V^{\operatorname{Pb}}$ vacancy).
The O vacancy in its neutral state is occupied by two electrons. If the O
vacancy drops electron (1+ charged states), its formation energy is reduced.
The Pb vacancy in its uncharged state is empty $V^{\operatorname{Pb(0)}}$ and
its formation energy is comparably high, but is lowered if vacancy accepts
electrons (1$-$/2$-$ charged states). Therefore, both the O and Pb vacancies
($V^{\operatorname{O}}$ and $V^{\operatorname{Pb}}$) intend to appear in the
opposite charged states. The neutral O vacancy would prefer to give away one
electron to reduce its formation energy. To conserve the electroneutrality of
material there are only the Pb vacancies that can accept electrons under the
equilibrium conditions. This makes the Pb vacancy a compensation center for
the O vacancy. The considered mechanisms of the charge exchange between
vacancies are presented Fig. 4 (a) and (b). In case of thermodynamic
equilibrium, when formation energies for both types of vacancies are equal
and, therefore, their concentrations are equal as well, vacancies would become
doubly ionized ($V^{\operatorname{Pb(2-)}}$ and $V^{\operatorname{O(2+)}}$)
utilizing the mechanism of the electron exchange as presented in Fig. 4 (a).
Figure 4: Schemes showing the mechanisms of ionization of the neutral
vacancies through the electron exchange ($V^{\operatorname{Pb}}$ accepts two
or single electron occupying the $V^{\operatorname{O}}$ vacancy).
However, for the Pb-rich/O-poor limit considered in present work the formation
energies of the vacancies are different from the thermodynamic equilibrium
(see Fig. 3). As seen in Fig. 3, the formation energy of the O vacancy is much
lower than the Pb vacancy. This presumes much higher concentration of the O
vacancies so that they will be only partially compensated by the Pb vacancies.
Based on achieved magnitude of the formation energies we expect the Pb
vacancies to appear preferably in their (2$-$) states while the O vacancies to
be formed in the different charged states, (0)/(1+). In this case, ionization
of the Pb vacancy to the $V^{\operatorname{Pb(2-)}}$ state can occur with
participation of two O vacancies such as each $V^{\operatorname{O(0)}}$
donates single electron becoming ionized only to the
$V^{\operatorname{O(1+)}}$ state as presented in Fig. 4 (b). The O vacancies
in the $V^{\operatorname{O(1+)}}$ state are still occupied by one electron,
and the larger the difference in O/Pb vacancy concentration, the larger amount
of the non-compensated O vacancies. This behaviour helps in understanding of
the experimentally observed $n$-conductivity of PbO chienes ; bigelow ;
scanlon . Both neutral and singly charged O vacancies
($V^{\operatorname{O(0)}}$ and $V^{\operatorname{O(1+)}}$) act as $n$-type
donor. Moreover, in this case, a pinning of the Fermi level position slightly
above the midgap (0.95 eV below the conduction band) observed experimentally
broek can be assigned to $n$-type doping induced by the O vacancies.
Previously, the pinning was associated with the surface states at the
crystallites boundaries but nature of those states was unknown broek . A
remarkable agreement between the Fermi level position predicted in Ref.broek
and position of the O vacancy states found here (see Fig. 2) suggests that the
Fermi level is stabilized by the presence of the O vacancies.
Therefore, we anticipate that the O vacancies would affect the transport and
photogeneration in lead oxide more significantly than the Pb vacancies.
Indeed, shallow traps for holes created by the ionized Pb vacancies might
slightly reduce the hole mobility which is already low due to the extremely
heavy holes mine , but much deeper O vacancies when they are ionized (see Fig.
2) would not only slow down the electron propagation in the conduction band
through trapping, but can additionally act as the recombination centers.
Therefore, because a contribution of the deep traps in the charge transport is
known to impair significantly the current decay kabir1 ; street , the temporal
behaviour of the dark conductivity can be used to confirm a presence of the O
vacancies.
### III.3 Dark current kinetics
The dark current kinetics is a sensitive measure of the electronic properties
of a material and is used here to describe the effect of point defects on the
conductivity in PbO layers. The results of time dependence of the dark current
density for selected biases are shown in Fig. 5. As it is seen from Fig. 5,
after bias voltages are applied dark current decays slowly reaching a steady
state value after about 250 minutes. The steady-state current density depends
on electric field and increases by a factor of 2 when bias is increased from 3
to 7 V$\mu$m. Similar behaviour of the dark current was observed by Mahmood
and Kabir in amorphous selenium (a-Se) multilayer $n-i-p$ structures kabir1
and by Street in hydrogenated amorphous silicon (a-Si:H) $p-i-n$ structures
street ; street3 . Mahmood and Kabir explain dark current decay in a-Se
multilayer structure by carrier trapping within comparatively thick (few
$\mu$m rev ) $n$\- and $p$\- layers which induces screening of the electric
field at the metal/$n$\- or $p$-layer interfaces. The subsequent
redistribution of the electric field suppresses carrier injection from metal
contacts and reduces the dark current which is mainly controlled by the
injection.
Figure 5: Colour on-line. The time dependence of the dark current density for
the different bias applied. $I_{st}$ is the steady state current which is
reached at 1.5$\times 10^{4}$ sec.
An alternative model is suggested by Street who explains similar dark current
kinetics by depletion of charge from the bulk of $i$-layer assuming that dark
current is controlled by thermal generation from defects states in the gap. As
traps are depleting, the quasi Fermi level moves toward the midgap and the
thermal generation current decreases street . As PbO samples studied here are
uniform, we have to assume that the electric field is also uniform across the
layer (neglecting thin pre-contact areas). Therefore, the model developed by
Street is more applicable in our case. Neglecting charge carrier injection
under the applied low biases, we can assume that the thermal generation
current arises from the excitation from the O vacancies occupied by electrons
to the conduction band. Hence the dark current decay can be described by the
time dependent quasi Fermi level position as is was suggested in Ref. street .
Without speculating on the degree of compensation in our layers we assume that
quasi Fermi level is initially located at 1.03 eV from the valence band, i.e.
the activation energy is 0.77 eV. After an electric field is applied, the
occupancy of the O vacancies changes as electrons are emitted to the
conduction band and the quasi Fermi level shifts toward the midgap. Once traps
are fully depleted quasi Fermi level approaches the equilibrium Fermi level
and the thermal generated current saturates at electric field dependent
steady-state value.
Although at that point we provide just qualitative analysis of dark current
kinetics in PbO layers, it allows us to link defects (namely, the O vacancies)
with transport properties in this material. It has to be mentioned that
steady-state dark current is extremely low (and much lower than in a-Se kabir1
) that is very encouraging for the application of PbO layers in direct
conversion medical x-ray imaging detectors detectors .
## IV Conclusion
First-principles density-functional calculations were used to calculate the
electronic properties of polycrystalline $\alpha$-PbO and the formation of
native point defects (namely, O and Pb vacancies) in this material. It was
found that the O vacancies induce very deep donor level close to the midgap at
1.03 eV above the valence band. In contrast, the Pb vacancies create shallow
defect level at just 0.1 eV above the valence band which acts as acceptor.
Under applied bias the ionized O vacancies in PbO would act as the deep traps
and recombination centers for the electrons in the conduction band, while the
Pb vacancies are the shallow traps for holes in the valence band.
The formation energies of the defects in their neutral charge states are
comparatively small: 0.85 eV for the O vacancy and 6.64 eV for the Pb vacancy
($E_{F}$ is assigned to the midgap and the Pb-rich/O-poor growth conditions
are considered) and are further reduced if a vacancy appears in its
energetically favourable charged state. For example, for the doubly ionized
(2-) Pb vacancy the formation energy is reduced to 4.99 eV. The electron
exchange between vacancies initiates ionization of the vacancies, but for Pb-
rich/O-poor growth conditions when concentration of the vacancies is not
balanced, most of the O vacancies remains unionized, i.e. in their (0)
uncharged state or (1+) charged state in which the O vacancies are occupied
with electrons.
The presence of defects and differential trapping of electrons and holes were
predicted by others to explain space charge limited photoconductivity in PbO
hughes and to model x-ray sensitivity, modulation transfer function (MTF) and
detective quantum efficiency (DQE) of the PbO x-ray detector kabir . The
results presented here agree well with these studies and provide insight into
the nature of defects in PbO clarifying their electronic and charge states and
explaining why vacancies exist in high concentration in thermally evaporated
PbO layers. Moreover, our own experimental results on time dependence of the
dark current density suggest that this is the field dependent occupancy of O
vacancies that governs the dark current kinetics. Thus, the O vacancies are
occupied with electrons and because these centers are located close to the
midgap of PbO, a process of detrapping of the vacancies is slow thus impairing
the temporal characteristics of compound.
Since O vacancies play more essential role in the transport properties of PbO
layers, material science solutions must be found to improve PbO layers
deposition techniques in order to suppress their appearance. Methods to
consider include thermal evaporation with optional low energy O ion
bombardment or passivation of vacancies by post-growth annealing in oxygen
atmosphere.
## Acknowledgement
Authors are thankful to Dr. Matthias Simon (X-ray Imaging Systems, Philips
Research) for numerous stimulating discussions and Giovanni DeCrescenzo for
technical support in conducting the dark-current measurements. Financial
support of Ontario Research Fund- Research Excellence program is highly
acknowledged.
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|
arxiv-papers
| 2012-10-30T18:08:37 |
2024-09-04T02:49:37.426760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Berashevich, O. Semeniuk, O. Rubel, J. A. Rowlands, A. Reznik",
"submitter": "Julia Berashevich",
"url": "https://arxiv.org/abs/1210.8405"
}
|
1211.0043
|
# Hereditarily Normal Wijsman Hyperspaces
Are Metrizable∗
Jiling Cao School of Computing and Mathematical Sciences, Auckland University
of Technology, Private Bag 92006, Auckland 1142, New Zealand
jiling.cao@aut.ac.nz and Heikki J. K. Junnila Department of Mathematics and
Statistics, The University of Helsinki, P. O. Box 68, FI-00014, Helsinki,
Finland heikki.junnila@helsinki.fi Dedicated to Professor Mitrofan Choban and
Professor Stoyan Nedev
for their 70 birthday.
###### Abstract.
In this paper, we study normality and metrizability of Wijsman hyperspaces. We
show that every hereditarily normal Wijsman hyperspace is metrizable. This
provides a partially answer to a problem of Di Maio and Meccariello in 1998.
2010 _Mathematics Subject Classification._ Primary 54E35; Secondary 54B20,
54D15.
_Keywords_. Embedding, hereditarily normal, hyperspace, metrizable, normal,
Wijsman topology.
∗This paper was initially and partially written when the first author was in a
Research and Study Leave from July to December 2009, and visited the second
author in August 2009. The paper was eventually completed when the two authors
met and discussed at the International Conference on Topology and the Related
Fields, held at Nanjing, China, 22-25 September 2012. The two authors would
like to thank the School of Computing and Mathematical Sciences at the
Auckland University of Technology, and the Department of Mathematics and
Statistics at the University of Helsinki for their financial supports.
## 1\. Introduction
Throughout this paper, $2^{X}$ denotes the family of all nonempty closed
subsets of a given topological space $X$. For a metric space $(X,d)$, let
$d(x,A)=\inf\\{d(x,a):a\in A\\}$ denote the distance between a point $x\in X$
and a nonempty subset $A$ of $(X,d)$, and $S_{d}(A,\varepsilon)=\\{x\in
X:d(x,A)<\varepsilon\\}$. A net $\\{A_{\alpha}:\alpha\in D\\}$ in $2^{X}$ is
said to be _Wijsman convergent to_ some $A$ in $2^{X}$ if $d(x,A_{\alpha})\to
d(x,A)$ for every $x\in X$. The Wijsman topology on $2^{X}$ induced by $d$,
denoted by $\tau_{w(d)}$, is the weakest topology such that for every $x\in
X$, the distance functional $d(x,\cdot)$ is continuous. To see the structure
of this topology, for any $E\subseteq X$, let $E^{-}=\\{A\in 2^{X}:A\cap
E\neq\emptyset\\}$. It can be seen easily that the Wijsman topology on $2^{X}$
induced by $d$ has the family
$\left\\{U^{-}:U\mbox{ is open in }X\right\\}\cup\left\\{\\{A\in
2^{X}:d(x,A)>\varepsilon\\}:x\in X,\varepsilon>0\right\\}$
as a subbase, refer to [2]. Moreover, for a finite subset $E\subseteq X$ and
$\varepsilon>0$, let
${\mathcal{N}}_{A,E,\varepsilon}=\left\\{B\in
2^{X}:|d(x,A)-d(x,B)|<\varepsilon\mbox{ for all }x\in E\right\\}.$
Then for any $A\in 2^{X}$, the collection
$\\{{\mathcal{N}}_{A,E,\varepsilon}:E\subseteq X\mbox{ is finite and
}\varepsilon>0\\}$
forms a neighborhood base of $A$ in $\tau_{w(d)}$. This type of convergence
was first introduced by Wijsman in [20] for sequences of closed convex sets in
Euclidean space $\mathbb{R}^{n}$, when he studied optimum properties of the
sequential probability ratio test. It was [17] where Wijsman convergence was
considered in the general framework of a metric space, and the metrizability
of the Wijsman topology of a separable metric space was established. Since
then, there has been a considerable effort to explore various topological
properties of Wijsman hyperspaces. For example, Beer [1] and Costantini [8]
studied Polishness of Wijsman hyperspaces, Cao and Tomita [6] as well as
Zsilinszky [21] investigated Baireness of Wijsman hyperspaces, Cao and Junnila
[4] studied Amsterdam properties of Wijsman spaces. However, Wijsman
hyperspaces are far to be completely understood, and still there are many
problems concerning fundamental properties of these objects unsolved. This
motivates the authors to continue their study of Wijsman hyperspaces in the
present paper.
Note that all Wijsman topologies are Tychonoff, since they are weak
topologies. In a more recent paper, Cao, Junnila and Moors [5] showed that
Wijsman hyperspaces are universal Tychonoff spaces in the sense that every
Tychonoff space is embeddable as a closed subspace in the Wijsman hyperspace
of a complete metric space which is locally $\mathbb{R}$. Thus, one of the
fundamental problems is to determine when a Wijsman hyperspace is normal. The
problem was first mentioned by Di Maio and Meccariello in [9], where it was
asked whether the normality of a Wijsman hyperspace is equivalent to its
metrizability. A partial solution to this problem, which asserts that the
answer is “yes” when the base space of a Wijsman hyperspace is a normed linear
space, was recently observed by Holá and Novotný in [13]. The main purpose of
this paper is to give another partial answer to this problem. By using
techniques similar to those of Keesling in [15], we are able to establish that
a Wijsman hyperspace is hereditarily normal if and only if it is metrizable.
The rest of this paper is organized as follows. In Section 2, an overview on
the normality and metrizability of basic types of hyperspaces is provided. The
main result and its proof are given in Section 3. Our terminology and notation
are standard. For undefined terms, refer to [2], [3] or [10].
## 2\. Normality and metrizability of hyperspaces
It has been an interesting and challenging problem in general topology to
characterize normality of the hyperspace of a topological space. In 1955,
Ivanova [14] showed that $2^{\mathbb{N}}$ with the Vietoris topology is not
normal, where $\mathbb{N}$ is equipped with the discrete topology. Continuing
in this direction, Keesling [15] proved that under the CH (Continuum
Hypothesis), for a Tychonoff space $X$, $(2^{X},\tau_{V})$ is normal if and
only if $(2^{X},\tau_{V})$ is compact (and thus if and only if $X$ is
compact), where $\tau_{V}$ denotes the Vietoris topology on $2^{X}$. In
addition, he also showed in [16] that for a regular $T_{1}$ space $X$, a
number of covering properties of $(2^{X},\tau_{V})$ (including Lindelöfness,
paracompactness, metacompactness, and meta-Lindelöfness) are equivalent to
compactness of $(2^{X},\tau_{V})$. Finally, Veličko [19] further showed that
Keesling’s result on normality also holds without the CH. This completely
solved the normality problem of Vietoris hyperspaces. The normality problem of
Fell hyperspaces was settled by Holá, Levi and Pelant in [12], where they
showed that $(2^{X},\tau_{F})$ is normal if and only if $(2^{X},\tau_{F})$ is
Lindelöf, if and only if $X$ is locally compact and Lindelöf, here $\tau_{F}$
denotes the Fell topology on $2^{X}$. Since in general the Wijsman topology
induced by a metric is coarser than the Vietoris topology but finer than the
Fell topology induced by the same metric, the following natural question
arises.
###### Problem 2.1.
Let $(X,d)$ be a metric space. When is the Wijsman hyperspace
$\left(2^{X},\tau_{w(d)}\right)$ a normal space?
Let us temporarily put the normality problem of Wijsman hyperspaces aside.
Instead, let us recall when a hyperspace is metrizable. A classical result
claims that for a $T_{1}$ space $X$, $(2^{X},\tau_{V})$ is metrizable if and
only if $X$ is compact and metrizable, refer to [10, p.298]. A corresponding
result for the Fell topology states that, for a Hausforff space $X$,
$(2^{X},\tau_{F})$ is metrizable if and only if $X$ is locally compact and
second countable (and thus Lindelöf), refer to [11]. For Wijsman hyperspaces,
we have the following classical result.
###### Theorem 2.2 ([17]).
Let $(X,d)$ be a metric space. Then $\left(2^{X},\tau_{w(d)}\right)$ is
metrizable if and only if $(X,d)$ is separable.
Indeed, if $\\{x_{n}:n\in\mathbb{N}\\}$ is any dense subset of $X$, then it
can be checked that
$\varrho_{d}(A,B)=\sum_{n=1}^{\infty}\frac{|d(x_{n},A)-d(x_{n},B)|\wedge
1}{2^{n}}$
defines a metric on $2^{X}$ that is compatible with $\tau_{w(d)}$. As a
consequence of this result, $\left(2^{X},\tau_{w(d)}\right)$ is Lindelöf if
and only if it is metrizable. Note that for any metric space $(X,d)$, we have
that $\left(2^{X},\tau_{w(d)}\right)$ is countably compact if and only if
$\left(2^{X},\tau_{w(d)}\right)$ is compact. Theorem 3.5 in [17] also claims
that $\left(2^{X},\tau_{w(d)}\right)$ is metrizable if and only if $\\{X\\}$
is a $G_{\delta}$-point of $\left(2^{X},\tau_{w(d)}\right)$. As a consequence,
the metrizability of $\left(2^{X},\tau_{w(d)}\right)$ is equivalent to a large
number of generalized metric properties. For example,
$\left(2^{X},\tau_{w(d)}\right)$ is metrizable if and only if it has a
$G_{\delta}$-diagonal or it is semi-stratifiable.
In the light of the work of Keesling in [16] and Veličko in [19] on the
normality of Vietoris hyperspaces as well as the work of Holá et al. on the
normality of Fell hyperspaces, one may wonder whether the paracompactness and
the metrizability of Wijsman hyperspaces are equivalent. These facts motivated
Di Maio and Meccariello to pose the following natural problem in 1998, which
also brings the normality and the metrizability of Wijsman hyperspaces
together.
###### Problem 2.3 ([9]).
It is known that if $(X,d)$ is a separable metric space, then
$\left(2^{X},\tau_{w(d)}\right)$ is metrizable and so paracompact and normal.
Is the opposite true? Is $\left(2^{X},\tau_{w(d)}\right)$ normal if, and only
if, $\left(2^{X},\tau_{w(d)}\right)$ is metrizable?
Note that neither Problem 2.1 nor Problem 2.3 is completely solved. An
affirmative solution to Problem 2.3 would also solve Problem 2.1. The
following partial answer to Problem 2.3 was recently established by Holá and
Novotný in [13].
###### Theorem 2.4 ([13]).
Let $(X,\|\cdot\|)$ be a normed linear space, and let $d$ be the metric on $X$
induced by the norm $\|\cdot\|$. Then $\left(2^{X},\tau_{w(d)}\right)$ is
normal if and only if it is metrizable.
Given a topological space $X$, define ${\rm nlc}(X)$ by
${\rm nlc}(X)=\\{x\in X:\ x\mbox{ has no compact neighbourhood in }X\\}.$
The following result was established by Chaber and Pol in [7].
###### Theorem 2.5 ([7]).
If $(X,d)$ is a metric space such that ${\rm nlc}(X)$ is non-separable, then
$\left(2^{X},\tau_{w(d)}\right)$ contains a closed copy of
${\mathbb{N}}^{\omega_{1}}$.
Since ${\mathbb{N}}^{\omega_{1}}$ is non-normal, as a corollary of Theorem
2.5, if $(X,d)$ is a metric space such that ${\rm nlc}(X)$ is non-separable,
then $\left(2^{X},\tau_{w(d)}\right)$ is non-normal. In particular, if
$(X,\|\cdot\|)$ is a non-separable normed linear space and $d$ is the metric
on $X$ induced by the norm $\|\cdot\|$, then $\left(2^{X},\tau_{w(d)}\right)$
is non-normal. Therefore, Theorem 2.4 can be viewed as a corollary of Theorem
2.5.
## 3\. The main result
In this section, we shall prove the following main result of this paper, which
can be treated as a partial answer to Problem 2.1 and Problem 2.3.
###### Theorem 3.1.
Let $(X,d)$ be a metric space. The following are equivalent.
* (i)
$\left(2^{X},\tau_{w(d)}\right)$ is metrizable.
* (ii)
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ is paracompact.
* (iii)
$\left(2^{X},\tau_{w(d)}\right)$ is hereditarily normal.
To prove the above theorem, we use the embedding techniques, similar to those
used by Keesling in [15]. In what follows, the ordinals $\omega_{1}$ and
$\omega_{1}+1$ are viewed as topological spaces equipped with the order
topology.
###### Proposition 3.2.
Let $(X,d)$ be a non-separable metric space. Then for any $n\geq 1$, the
Wijsman hyperspace $\left(2^{X},\tau_{w(d)}\right)$ contains a copy of
$(\omega_{1}+1)^{n}$.
###### Proof.
Let $Y_{n}=(\omega_{1}+1)^{n}$. Since $(X,d)$ is non-separable, there exist
$\varepsilon>0$ and a set $D\subseteq X$ with $|D|=\aleph_{1}$ which is
$\varepsilon$-discrete, that is, $d(x,y)\geq\varepsilon$ for all distinct
$x,y\in D$. Let $n\geq 1$. We express $D$ as the disjoint union
$D=\bigcup_{i=0}^{n}D_{i}$ such that $D_{0}=\\{d\\}$ and $|D_{i}|=\aleph_{1}$
for all $1\leq i\leq n$. Next, for $1\leq i\leq n$, we enumerate $D_{i}$ as
$D_{i}=\\{x_{\alpha}^{i}:\alpha<\omega_{1}\\}$, and for each
$\alpha\leq\omega_{1}$, we put $L_{\alpha}^{i}=\\{x_{\lambda}^{i}\in
D_{i}:\lambda<\alpha\\}$. Obviously, each $L_{\alpha}^{i}$ is closed in $X$.
Define a mapping $\varphi:Y_{n}\to\left(2^{X},\tau_{w(d)}\right)$ by the
formula $\varphi(\alpha_{i})=D_{0}\cup\bigcup_{i=1}^{n}L_{\alpha_{i}}^{i}$. It
is clear that $\varphi$ is one-to-one.
To see that $\varphi$ is continuous, suppose $\varphi(\alpha_{i})\cap
V\neq\emptyset$ for some open set $V\subseteq X$. If $D_{0}\cap
V\neq\emptyset$, there is nothing to verify. So, we assume that
$L_{\alpha_{i}}^{i}\cap V\neq\emptyset$ for some $1\leq i\leq n$. Hence, there
is some non-limit ordinal $\lambda_{i}<\alpha_{i}$ such that
$L_{\lambda_{i}}^{i}\cap V\neq\emptyset$. It follows that for any neighborhood
$N(\alpha_{j})$ of $\alpha_{j}$ with $j\neq i$, we have
$\varphi\left(\prod_{j<i}N(\alpha_{j})\times(\lambda_{i},\alpha_{i}]\times\prod_{j>i}N(\alpha_{j})\right)\subseteq
V^{-}.$
On the other hand, if $d(x,\varphi(\alpha_{i}))>r$ for some $x\in X$ and
$r>0$, then for any $\lambda_{i}\leq\alpha_{i}$ we have
$d(x,\varphi(\lambda_{i}))>r$. Thus, we have verified that $\varphi$ is
continuous at any point $(\alpha_{i})\in Y_{n}$.
Since $Y_{n}$ is compact, the continuous one-to-one mapping $\varphi$ is an
embedding. ∎
###### Corollary 3.3.
Let $(X,d)$ be a metric space. Then for the space
$\left(2^{X},\tau_{w(d)}\right)$, metrizability is equivalent to each one of
the following properties: Fréchetness, sequentiality, countable tightness.
###### Proof.
Proposition 3.2 shows, in particular, that if $(X,d)$ is non-separable, then
$\left(2^{X},\tau_{w(d)}\right)$ contains a copy of $\omega_{1}+1$. Since
$\omega_{1}+1$ does not have countable tightness, the conclusion follows. ∎
###### Question 3.4.
Let $(X,d)$ be a non-separable metric space. Can
$\left(2^{X},\tau_{w(d)}\right)$ contain a copy of $(\omega_{1}+1)^{\omega}$
or $(\omega_{1}+1)^{\omega_{1}}$?
For the proof of our next proposition, we need an auxiliary result.
###### Lemma 3.5.
Let $(X,d)$ be a metric space. If ${\mathcal{F}}$ is a directed family in
$2^{X}$ such that $H=\bigcup{\mathcal{F}}$ is closed, then $H$ is a limit
point of the net $(\mathcal{F},\subseteq)$ in
$\left(2^{X},\tau_{w(d)}\right)$.
###### Proof.
The proof of this lemma is straightforward, and thus is omitted. ∎
In his proof of the equivalence of normality and compactness for Vietoris
hyperspaces (under the CH), Keesling [15] established that, for a noncompact
Tychonoff space $X$, $(2^{X},\tau_{V})$ contains a closed copy of the space
$\omega_{1}\times(\omega_{1}+1)$. We are not able to obtain a similar
embedding in the Wijsman hyperspace of a non-separable metric space $(X,d)$.
However, we have the following result.
###### Proposition 3.6.
Let $(X,d)$ be a non-separable metric space. Then the subspace
$2^{X}\setminus\\{X\\}$ of $\left(2^{X},\tau_{w(d)}\right)$ contains a closed
copy of the space $\omega_{1}\times(\omega_{1}+1)$.
###### Proof.
Since $(X,d)$ is non-separable, there exist $\varepsilon>0$ and an
$\varepsilon$-discrete proper subset
$D=\\{x_{\alpha,\beta}:\alpha<\omega_{1}\mbox{ and }\beta\leq\omega_{1}\\}$ of
$X$, with $x_{\alpha,\beta}\neq x_{\alpha^{\prime},\beta^{\prime}}$ for
$(\alpha,\beta)\neq(\alpha^{\prime},\beta^{\prime})$. For every
$\alpha<\omega_{1}$, let
$D_{\alpha}=\\{x_{\alpha,\beta}:\beta\leq\omega_{1}\\}$ and
$G_{\alpha}=S_{d}(D_{\alpha},\frac{\varepsilon}{4})$. For every
$\alpha<\omega_{1}$ and each $\beta\leq\omega_{1}$, let
$F_{\alpha}=X\setminus\left(\bigcup_{\gamma\geq\alpha}G_{\gamma}\right)$
and
$S_{\beta}=\\{x_{\gamma,\delta}:\gamma<\omega_{1}\mbox{ and }\delta<\beta\\}.$
Note that the families ${\mathcal{F}}=\\{F_{\alpha}:\alpha<\omega_{1}\\}$ and
$\mathcal{S}=\\{S_{\beta}:\beta\leq\omega_{1}\\}$ are “continuously
increasing”, in the sense that
$F_{\alpha}=\bigcup\\{F_{\gamma+1}:\gamma<\alpha\\}$ and
$S_{\beta}=\bigcup\\{S_{\delta+1}:\delta<\beta\\}$ for all
$0<\alpha<\omega_{1}$ and $0<\beta\leq\omega_{1}$. We show that the subspace
${\mathcal{H}}=\\{F_{\alpha}\cup S_{\beta}:\alpha<\omega_{1}\mbox{ and
}\beta\leq\omega_{1}\\}$
of $(2^{X},\tau_{w(d)})$ is homeomorphic to the product space
$\omega_{1}\times(\omega_{1}+1)$. Define a mapping
$\varphi:\omega_{1}\times(\omega_{1}+1)\to{\mathcal{H}}$ by the formula
$\varphi(\alpha,\beta)=F_{\alpha}\cup S_{\beta}$, and note that $\varphi$ is
one-to-one and onto.
To show that $\varphi$ is continuous, let
$A\subseteq\omega_{1}\times(\omega_{1}+1)$, and let
$(\alpha,\beta)\in\overline{A}$. We show that
$\varphi(\alpha,\beta)\in\overline{\varphi(A)}$. Let
$A^{\prime}=\\{(\gamma,\delta)\in A:\gamma\leq\alpha\ {\rm and}\
\delta\leq\beta\\}$, and note that $(\alpha,\beta)\in\overline{A^{\prime}}$.
Note that, for all $(\gamma,\delta),(\gamma^{\prime},\delta^{\prime})\in
A^{\prime}$, there exists $(\mu,\nu)\in A^{\prime}$ such that
$\mu\geq\max(\gamma,\gamma^{\prime})$ and
$\nu\geq\max(\delta,\delta^{\prime})$. As a consequence, the family
$\\{F_{\gamma}\cup S_{\delta}:(\gamma,\delta)\in A^{\prime}\\}$ is directed.
Since $(\alpha,\beta)\in\overline{A^{\prime}}$, we have for all
$\alpha^{\prime}<\alpha$ and $\beta^{\prime}<\beta$ that there exists
$(\gamma,\delta)\in A^{\prime}$ such that $\gamma\geq\alpha^{\prime}$ and
$\delta\geq\beta^{\prime}$. As $\mathcal{F}$ and $\mathcal{S}$ are
continuously increasing, it follows that
$\bigcup\\{F_{\gamma}\cup S_{\delta}:(\gamma,\delta)\in
A^{\prime}\\}=F_{\alpha}\cup S_{\beta}.$
From the foregoing it follows by Lemma 3.5 that the net $(\\{F_{\gamma}\cup
S_{\delta}:(\gamma,\delta)\in A^{\prime}\\},\subseteq)$ converges to
$F_{\alpha}\cup S_{\beta}$ in $\tau_{w(d)}$. As a consequence,
$\varphi(\alpha,\beta)\in\overline{\varphi(A^{\prime})}\subseteq\overline{\varphi(A)}$.
We have shown that $\varphi$ is continuous.
Next, we show that $\varphi$ is open. Let $W$ be an open subset of
$\omega_{1}\times(\omega_{1}+1)$. To show that $\varphi(W)$ is open in
$\mathcal{H}$, let $(\alpha,\beta)\in W$. Denote by $J$ the element
$\varphi(\alpha,\beta)=F_{\alpha}\cup S_{\beta}$ of the set $\varphi(W)$.
There exist $\gamma<\alpha$ and $\delta<\beta$ such that
$(\gamma,\alpha]\times(\delta,\beta]\subseteq W$. Let
$E=\\{x_{\alpha,\beta},x_{\alpha,\delta},x_{\gamma,\beta}\\}$ and
${\mathcal{N}}_{J,E,\varepsilon/2}=\left\\{H\in\mathcal{H}:|d(x,H)-d(x,J)|<\frac{\varepsilon}{2}\mbox{
for every }x\in E\right\\}.$
Note that ${\mathcal{N}}_{J,E,\varepsilon/2}$ is a neighborhood of $J$ in
$\mathcal{H}$. We show that
${\mathcal{N}}_{J,E,\varepsilon/2}\subseteq\varphi(W)$. Let
$H\in{\mathcal{N}}_{J,E,\varepsilon/2}$, and let $\mu<\omega_{1}$ and
$\nu\leq\omega_{1}$ be such that $H=F_{\mu}\cup S_{\nu}$. To show that
$H\in\varphi(W)$, we need to show that the inequalities $\gamma<\mu\leq\alpha$
and $\delta<\nu\leq\beta$ hold. For the element $x_{\alpha,\beta}$ of $E$, we
have $x_{\alpha,\beta}\in G_{\alpha}\subseteq X\setminus F_{\alpha}$ and
$x_{\alpha,\beta}\not\in S_{\beta}$. It follows that $x_{\alpha,\beta}\not\in
J$ and hence that $d(x_{\alpha,\beta},J)\geq\varepsilon$. As a consequence,
$d(x_{\alpha,\beta},H)\geq
d(x_{\alpha,\beta},J)-|d(x_{\alpha,\beta},H)-d(x_{\alpha,\beta},J)|\geq\varepsilon-\frac{\varepsilon}{2}>0.$
By the foregoing, we have that $x_{\alpha,\beta}\not\in H$, and this means
that $x_{\alpha,\beta}\not\in F_{\mu}$ and $x_{\alpha,\beta}\not\in S_{\nu}$.
It follows that we have $\mu\leq\alpha$ and $\nu\leq\beta$. For the element
$x_{\alpha,\delta}$ of $E$, we have $x_{\alpha,\delta}\in S_{\beta}\subseteq
J$, and hence $d(x_{\alpha,\delta},J)=0$. It follows that
$d(x_{\alpha,\delta},H)=|d(x_{\alpha,\delta},H)-d(x_{\alpha,\delta},J)|<\frac{\varepsilon}{2}.$
Since $H\subseteq D$ and $x_{\alpha,\delta}\in D$, it follows further, by
$\varepsilon$-discreteness of $D$, that $x_{\alpha,\delta}\in H$. Since
$H=F_{\mu}\cup S_{\nu}$, we have either $x_{\alpha,\delta}\in F_{\mu}$ or
$x_{\alpha,\delta}\in S_{\nu}$. In the first case, since $x_{\alpha,\delta}\in
D_{\alpha}\subseteq G_{\alpha}$, we would have that $\alpha<\mu$; however, we
showed above that $\mu\leq\alpha$. Hence we must have that
$x_{\alpha,\delta}\in S_{\nu}$. It follows that $\delta<\nu$. We have shown
that $\delta<\nu\leq\beta$. Similarly, for the element $x_{\gamma,\beta}$ of
$E$, we have that $x_{\gamma,\beta}\in F_{\alpha}\subseteq J$, and hence that
$d(x_{\gamma,\beta},J)=0$. It follows that
$d(x_{\gamma,\beta},H)<\frac{\varepsilon}{2}$, and further, that
$x_{\gamma,\beta}\in H$. As a consequence, we have either $x_{\gamma,\beta}\in
F_{\mu}$ or $x_{\gamma,\beta}\in S_{\nu}$. In the second case we would have
that $\beta<\nu$, but this does not hold, since we showed above that
$\nu\leq\beta$. Hence we must have $x_{\gamma,\beta}\in F_{\mu}$, and it
follows from this that $\gamma<\mu$. We have shown that
$\gamma<\mu\leq\alpha$. This completes the proof of openness of $\varphi$.
We have shown that the subspace $\mathcal{H}$ of
$\left(2^{X},\tau_{w(d)}\right)$ is homeomorphic to the space
$\omega_{1}\times(\omega_{1}+1)$. Note that $\bigcup\mathcal{H}=D$. As a
consequence, $X\not\in\mathcal{H}$. To complete the proof, we show that
$\mathcal{H}$ is closed in the subspace $2^{X}\setminus\\{X\\}$ of
$\left(2^{X},\tau_{w(d)}\right)$. Let
$K\in\overline{\mathcal{H}}\setminus\mathcal{H}$. To show that $K=X$, assume
on the contrary that $X\setminus K\neq\emptyset$. Let $y\in X\setminus K$.
There exists $\alpha_{0}<\omega_{1}$ such that
$y\not\in\bigcup_{\gamma>\alpha_{0}}G_{\gamma}$. Note that $y\in F_{\alpha}$
for each $\alpha>\alpha_{0}$. The subset $\mathcal{H}_{0}=\\{F_{\alpha}\cup
S_{\beta}:\alpha\leq\alpha_{0}\mbox{ and }\beta\leq\omega_{1}\\}$ of
$\mathcal{H}$ is compact, because it is homeomorphic to
$[0,\alpha_{0}]\times[0,\omega_{1}]$. Since
$K\in\overline{\mathcal{H}}\setminus\mathcal{H}$, it follows that
$K\in\overline{\mathcal{H}\setminus\mathcal{H}_{0}}$. Let $r=d(y,K)$ and
consider the neighborhood
${\mathcal{M}}_{K,\\{y\\},r}=\\{H\in 2^{X}:|d(y,H)-d(y,K)|<r\\}$
of $K$ in $\left(2^{X},\tau_{w(d)}\right)$. It follows from the foregoing,
that there exist $\alpha>\alpha_{0}$ and $\beta\leq\omega_{1}$ such that
$F_{\alpha}\cup S_{\beta}\in{\mathcal{M}}_{K,\\{y\\},r}$. However, now we have
that $y\in F_{\alpha}$ and hence $d(y,F_{\alpha}\cup S_{\beta})=0$. Since
$F_{\alpha}\cup S_{\beta}\in{\mathcal{M}}_{K,\\{y\\},r}$, we have
$d(y,K)<r\,$; this, however, contradicts with the definition of $r$. We have
shown that $\overline{\mathcal{H}}\setminus\mathcal{H}\subseteq\\{X\\}$ and
hence that $\mathcal{H}$ is closed in $2^{X}\setminus\\{X\\}$. ∎
###### Corollary 3.7.
Let $(X,d)$ be a metric space. The following are equivalent.
* (i)
$\left(2^{X},\tau_{w(d)}\right)$ is metrizable.
* (ii)
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ is metacompact.
* (iii)
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ is meta-Lindelöf.
* (iv)
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ is orthocompact.
###### Proof.
We only need to show that (iv) $\Rightarrow$ (i). Assume that $(X,d)$ is not
separable. By Proposition 3.6,
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ contains a closed copy of
$\omega_{1}\times(\omega_{1}+1)$. As
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ is orthocompact, then
$\omega_{1}\times(\omega_{1}+1)$ is orthocompact, which contradicts with a
result of Scott in [18]. ∎
We now use Proposition 3.6 to prove Theorem 3.1.
###### Proof of Theorem 3.1.
We only need to prove that (iii) implies (i). Assume that (iii) holds, but
$\left(2^{X},\tau_{w(d)}\right)$ is not metrizable. By Theorem 2.2, $(X,d)$ is
not separable. By Proposition 3.6,
$\left(2^{X}\setminus\\{X\\},\tau_{w(d)}\right)$ contains a closed copy of
$\omega_{1}\times(\omega_{1}+1)$. Since $\omega_{1}\times(\omega_{1}+1)$ is
not normal, (iii) does not hold. This is a contradiction. ∎
We conclude this paper with the following open question.
###### Question 3.8.
Let $(X,d)$ be a metric space. If $\left(2^{X},\tau_{w(d)}\right)$ is non-
normal, does $\left(2^{X},\tau_{w(d)}\right)$ contain a closed copy of
$\omega_{1}\times(\omega_{1}+1)$?
Note that there exists a metric space $(X,d)$ such that
$\left(2^{X},\tau_{w(d)}\right)$ is non-normal, but
$\left(2^{X},\tau_{w(d)}\right)$ contains no closed copy of
${\mathbb{N}}^{\omega_{1}}$. Indeed, take any set $X$ with $|X|=\omega_{1}$
and equip $X$ with the 0-1 metric $d$. By Remark 3.1 of [7],
$\left(2^{X},\tau_{w(d)}\right)$ is homeomorphic to
$\\{0,1\\}^{\omega_{1}}\setminus\\{\bf 0\\}$, which is locally compact. Thus,
$\left(2^{X},\tau_{w(d)}\right)$ contains no closed copy of
${\mathbb{N}}^{\omega_{1}}$.
## References
* [1] G. Beer, _A Polish topology for the closed subsets of a Polish space_ , Proc. Amer. Math. Soc. 113 (1991), 1123–1133.
* [2] G. Beer, _Topologies on closed and closed convex sets_ , Kluwer, Dordrecht, 1993.
* [3] D. K. Burke, _Covering properties_ , in Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 347 422.
* [4] J. Cao and H. J. K. Junnila, _Amsterdam properties of Wijsman hyperspaces_ , Proc. Amer. Math. Soc. 138 (2010), 769–776.
* [5] J. Cao, H. J. K. Junnila and W. B. Moors, _Wijsman hyperspaces: subspaces and embeddings_ , Topology Appl. 159 (2012), 1620–1624.
* [6] J. Cao and A. H. Tomita, _The Wijsman hyperspace of a metric hereditarily Baire space is Baire_ , Topology Appl. 157 (2010), 145–151.
* [7] J. Chaber and R. Pol, _Note on the Wijsman hyperspaces of completely metrizable spaces_ , Boll. U. M. I. 85-B (2002), 827–832.
* [8] C. Costantini, _Every Wijsman topology relative to a Polish space is Polish_ , Proc. Amer. Math. Soc. 123 (1995), 2569–2574.
* [9] G. Di Maio and E. Meccariello, _Wjisman topology_ , Recent Progress in Function spaces, pp.55–91, Quad. Mat. 3, Dept. Math., Seconda Univ. Napoli, Caserta, 1998.
* [10] R. Engelking, General topology, Revised and completed edition, Heldermann Verlag, Berlin, 1989\.
* [11] J. Flachsmeyer, _Versschiedene topologisierungen im raum der abgeschlossenen teilmengen_ , Math. Nachr. 26 (1964), 321–337.
* [12] L. Holá, S. Levi and J. Pelant, _Normality and paracompactness of the Fell topology_ , Proc. Amer. Math. Soc. 127 (1999), 2193–2197.
* [13] L. Holá and B. Novotný, _On normality of the Wijsman topology_ , Annali di Matematica, forthcoimg. DOI 10.1007/s10231-011-0227-9
* [14] V. M. Ivanova, _On the theory of the spaces of subsets_ , Dokl. Akad. Nauk SSSR 101 (1955), 601–603.
* [15] J. Keesling, _On the equivalence of normality and compactness in hyperspaces_ , Pacific J. Math. 33 (1970), 657–667.
* [16] J. Keesling, _Normality and properties related to compactness in hyperspaces_ , Proc. Amer. Math. Soc. 24 (1970), 760–766.
* [17] A. Lechicki and S. Levi, _Wijsman convergence in the hyperspace of a metric space_ , Boll. Un. Mat. Ital. (7) 1-B (1987), 439–452.
* [18] B. M. Scott, _Toward a product theory for orthocompactness_ , Studies in topology, pp. 517–537, Academic Press, New York, 1975.
* [19] N. V. Velichko, _On spaces of closed subsets_ , Siberian Math. J. 16 (1975), 484–486.
* [20] R. Wijsman, _Convergence of sequences of convex sets, cones and functions II_ , Trans. Amer. Math. Soc. 123 (1966), 32–45.
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|
arxiv-papers
| 2012-10-31T21:49:48 |
2024-09-04T02:49:37.441347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiling Cao and Heikki J. K. Junnila",
"submitter": "Jiling Cao",
"url": "https://arxiv.org/abs/1211.0043"
}
|
1211.0047
|
# Aggregate Preferred Correspondence and
the Existence of a Maximin REE
Anuj Bhowmik School of Computing and Mathematical Sciences, Auckland
University of Technology, Private Bag 92006, Auckland 1142, New Zealand
anuj.bhowmik@aut.ac.nz , Jiling Cao School of Computing and Mathematical
Sciences, Auckland University of Technology, Private Bag 92006, Auckland 1142,
New Zealand jiling.cao@aut.ac.nz and Nicholas C. Yannelis Department of
Economics, Henry B. Tippie College of Business, The University of Iowa, IA
52242-1994, USA, and Economics - School of Social Sciences, The University of
Manchester, Manchester M13 9PL, UK nicholasyannelis@gmail.com
###### Abstract.
In this paper, a general model of a pure exchange differential information
economy is studied. In this economic model, the space of states of nature is a
complete probability measure space, the space of agents is a measure space
with a finite measure, and the commodity space is the Euclidean space. Under
appropriate and standard assumptions on agents’ characteristics, results on
continuity and measurability of the aggregate preferred correspondence in the
sense of Aumann in [4] are established. These results together with other
techniques are then employed to prove the existence of a maximin rational
expectations equilibrium (maximin REE) of the economic model.
_JEL Classification Numbers:_ D51, D82.
_Keywords_. Aggregate correspondence; Budget correspondence; Differential
information; Hausdorff continuous; Lower measurable; Maximin rational
expectations allocation; Walrasian equilibrium.
## 1\. Introduction
When traders come to a market with different information about the items to be
traded, the resulting market prices may reveal to some traders information
originally available only to others. The possibility for such inferences rests
upon traders having expectations of how equilibrium prices are related to
initial information. This endogenous relationship was considered by Radner in
his seminal paper [14], where he introduced the concept of a rational
expectations equilibrium by imposing on agents the Bayesian (subjective
expected utility) decision doctrine. Under the Bayesian decision making,
agents maximize their subjective expected utilities conditioned on their own
private information and also on the information that the equilibrium prices
generate. The resulting equilibrium allocations are measurable with respect to
the private information of each individual and also with respect to the
information the equilibrium prices generate and clear the market for every
state of nature. In both papers [1] and [14], conditions on the existence of a
Bayesian rational expectations equilibrium (REE) were studied and some generic
existence results were proved. However, Kreps [12] provided an example that
shows that the Bayesian REE may not exist universally. In addition, a Bayesian
REE may fail to be fully Pareto optimal and incentive compatible and may not
be implementable as a perfect Bayesian equilibrium of an extensive form game,
refer to Glycopantis et al. [9] for more details.
In a recent paper [7], de Castro et al. introduced a new notion of REE by a
careful examination of Krep’s example of the nonexistence of the Bayesian REE.
In this formulation, the Bayesian decision making adopted in the papers of
Allen [1] and Radner [14] was abandoned and replaced by the maximin expected
utility (MEU) (see Gilboa and Schmeidler [8]). In this new setup, agents
maximize their MEU conditioned on their own private information and also on
the information the equilibrium prices have generated. Contrary to the
Bayesian REE, the resulting maximin REE may not be measurable with respect to
the private information of each individual or the information that the
equilibrium prices generate.
Although Bayesian REE and maximin REE coincide in some special cases (e.g.,
fully revealing Bayesian REE and maximin REE), these two concepts are in
general not equivalent. Nonetheless, the introduction of the MEU into the
general equilibrium modeling enables de Castro et al. to prove that the
maximin REE exists universally under the standard continuity and concavity
assumptions on the utility functions of agents. Furthermore, they showed that
the maximin REE is incentive compatible and efficient. Note that in the
economic model considered in [7], it is assumed that there are finitely many
states of nature and finitely many agents, and the commodity space is finite-
dimensional. Thus, one of open questions is whether their existence theorem
can be extended to more general cases. The main motivation of this paper is to
tackle this question. In this paper, a general model of a pure exchange
differential information economy is studied. In this economic model, the space
of states of nature is a complete probability measure space, and the space of
agents is a measure space with a finite measure. Under appropriate and
standard assumptions on agents’ characteristics, the existence of a maximin
rational expectations equilibrium (maximin REE) of this general economic model
is established.
This paper is organized as follows. In Section 2, the economic model, some
notation and assumptions are introduced and explained. Section 3, some results
on continuity and measurability of agents’ aggregate preferred correspondence
in the sense of Aumann in [4] are established. These results are key
techniques employed to prove the existence of a maximin rational expectations
equilibrium of the economic model in Section 4. In Section 5, results and
techniques appeared in this paper are compared with relevant results in the
literature. Finally, some mathematical preliminaries and key facts used in
this paper are presented in the appendix appeared at the end of the paper.
## 2\. Differential information economies
In this paper, a model of a pure exchange economy $\mathscr{E}$ with
differential information is considered. The space of states of nature is a
complete probability measure space $(\Omega,\mathscr{F},\nu)$. The space of
agents is a measure space $(T,\Sigma,\mu)$ with a finite measure $\mu$. The
commodity space is the $\ell$-dimensional Euclidean space
${\mathbb{R}}^{\ell}$, and the positive cone ${\mathbb{R}}^{\ell}_{+}$ is the
_consumption set_ for each agent $t\in T$ in every state of nature
$\omega\in\Omega$. Each agent $t\in T$ is associated with her/his
_characteristics_ $({\mathscr{F}}_{t},U(t,\cdot,\cdot),a(t,\cdot),q_{t})$,
where ${\mathscr{F}}_{t}$ is the $\sigma$-algebra generated by a partition
$\Pi_{t}$ of $\Omega$ representing the _private information_ of $t$;
$U(t,\cdot,\cdot):\Omega\times{\mathbb{R}}^{\ell}_{+}\to\mathbb{R}$ is a
_random utility function_ of $t$;
$a(t,\cdot):\Omega\rightarrow\mathbb{R}^{\ell}_{+}$ is the _random initial
endowment_ of $t$ and $q_{t}$ is a probability measure on $\Omega$ giving the
_prior_ of $t$. The economy extends over two time periods $\tau=1,2$. At the
ex ante stage ($\tau=0$), only the above description of the economy is a
common knowledge. At the stage $\tau=1$, agent $t$ only knows that the
realized state of nature belongs to the event
${\mathscr{F}}_{t}(\omega^{\ast})$, where $\omega^{\ast}$ is the true state of
nature at $\tau=2$. With this information (or with the information acquired
through prices), agents trade. At the ex post stage ($\tau=2$), agents execute
the trades according to the contract agreed at period $\tau=1$, and
consumption takes place. The coordinate-wise order on ${\mathbb{R}}^{\ell}$ is
denoted by $\leq$ and the symbol $x\gg 0$ means that $x$ is an interior point
of ${\mathbb{R}}^{\ell}_{+}$, and $x>0$ means that $x\geq 0$ but $x\neq 0$.
Let $L_{1}\left(\mu,{\mathbb{R}}^{\ell}_{+}\right)$ be the set of Lebesgue
integrable functions from $T$ to ${\mathbb{R}}^{\ell}_{+}$. An _allocation_ in
$\mathscr{E}$ is a function
$f:T\times\Omega\rightarrow{\mathbb{R}}^{\ell}_{+}$ such that
$f(\cdot,\omega)\in L_{1}\left(\mu,{\mathbb{R}}^{\ell}_{+}\right)$ for all
$\omega\in\Omega$. An allocation $f$ in $\mathscr{E}$ is _feasible_ if
$\int_{T}f(\cdot,\omega)d\mu=\int_{T}a(\cdot,\omega)d\mu$
for all $\omega\in\Omega$.
The following standard assumptions on agents’ characteristics shall be used:
(${\bf A}_{1}$) The initial endowment function
$a:T\times\Omega\rightarrow\mathbb{R}^{\ell}_{+}$ is jointly measurable such
that $\int_{T}a(\cdot,\omega)d\mu\gg 0$ for each $\omega\in\Omega$.
(${\bf A}_{2}$) $U(\cdot,\cdot,x)$ is jointly measurable for all
$x\in\mathbb{R}^{\ell}_{+}$ and $U(t,\omega,\cdot)$ is continuous for all
$(t,\omega)\in T\times\Omega$.
(${\bf A}_{3}$) For each $(t,\omega)\in T\times\Omega$, $U(t,\omega,\cdot)$ is
monotone in the sense that if $x,y\in\mathbb{R}^{\ell}_{+}$ with $y>0$, then
$U(t,\omega,x+y)>U(t,\omega,x)$.
(${\bf A}_{4}$) For each $(t,\omega)\in T\times\Omega$, $U(t,\omega,\cdot)$ is
concave.
Let
$\Delta=\left\\{p\in\mathbb{R}^{\ell}_{+}:p\gg 0\mbox{ and
}\sum_{h=1}^{\ell}p^{h}=1\right\\}.$
Then, each element $p\in\Delta$ is viewed as a (normalized) _price system_ in
the deterministic case. The _budget correspondence_
$B:T\times\Omega\times\Delta\rightrightarrows\mathbb{R}^{\ell}_{+}$ is defined
by
$B(t,\omega,p)=\left\\{x\in\mathbb{R}^{\ell}_{+}:\langle p,x\rangle\leq\langle
p,a(t,\omega)\rangle\right\\}$
for all $(t,\omega,p)\in T\times\Omega\times\Delta$. Obviously, $B$ is non-
empty and closed-valued. For each $\omega\in\Omega$, by Theorem 2 in [10,
p.151], there are $p(\omega)\in\Delta$ and an allocation $f(\cdot,\omega)$
such that $(f(\cdot,\omega),p(\omega))$ is a Walrasian equilibrium of the
deterministic economy $\mathscr{E}(\omega)$, given by
$\mathscr{E}(\omega)=((T,\Sigma,\mu);\mathbb{R}^{\ell}_{+};(U(t,\omega,\cdot),a(t,\omega)):t\in
T).$
Define a function $\delta:\Delta\to\mathbb{R}_{+}$ by
$\delta(p)=\min\left\\{p^{h}:1\leq h\leq\ell\right\\},$
where $p=(p^{1},...,p^{\ell})\in\Delta$. For any $(t,\omega,p)\in
T\times\Omega\times\Delta$, let
$\gamma(t,\omega,p)=\frac{1}{\delta(p)}\sum_{h=1}^{\ell}a^{h}(t,\omega)$
and
$b(t,\omega,p)=(\gamma(t,\omega,p),...,\gamma(t,\omega,p)).$
Define $X:T\times\Omega\times\Delta\rightrightarrows\mathbb{R}^{\ell}_{+}$ by
$X(t,\omega,p)=\\{x\in\mathbb{R}^{\ell}_{+}:x\leq b(t,\omega,p)\\}$
for all $(t,\omega,p)\in T\times\Omega\times\Delta$. Note that $X$ is non-
empty, compact- and convex-valued such that $B(t,\omega,p)\subseteq
X(t,\omega,p)$ for all $(t,\omega,p)\in T\times\Omega\times\Delta$. It can be
readily verified that for every $(t,\omega)\in T\times\Omega$, the
correspondence
$X(t,\omega,\cdot):\Delta\rightrightarrows{\mathbb{R}}^{\ell}_{+}$ is
Hausdorff continuous. Define two correspondences $C,\
C^{X}:T\times\Omega\times\Delta\rightrightarrows\mathbb{R}^{\ell}_{+}$ by
$C(t,\omega,p)=\left\\{y\in\mathbb{R}^{\ell}_{+}:U(t,\omega,y)\geq
U(t,\omega,x)\mbox{ for all }x\in B(t,\omega,p)\right\\}$
and
$C^{X}(t,\omega,p)=C(t,\omega,p)\cap X(t,\omega,p).$
Obviously,
$B(t,\omega,p)\cap C(t,\omega,p)=B(t,\omega,p)\cap C^{X}(t,\omega,p)$
holds for all $(t,\omega,p)\in T\times\Omega\times\Delta$. Note that under
(${\bf A}_{2}$), $U(t,\omega,\cdot)$ is continuous on the non-empty compact
set $B(t,\omega,p)$. Thus, one has
$B(t,\omega,p)\cap C(t,\omega,p)\neq\emptyset$
for all $(t,\omega,p)\in T\times\Omega\times\Delta$.
###### Proposition 2.1.
Let $(t,\omega,p)\in T\times\Omega\times\Delta$. Under (${\bf A}_{3}$),
$\langle p,x\rangle\geq\langle p,a(t,\omega)\rangle$ for every point $x\in
C^{X}(t,\omega,p)$.
###### Proof.
Assume that $\langle p,x_{0}\rangle<\langle p,a(t,\omega)\rangle$ for some
point $x_{0}\in C^{X}(t,\omega,p)$. Then, one can choose some
$y\in\mathbb{R}^{\ell}_{+}$ such that $y>0$ and $\langle
p,x_{0}+y\rangle<\langle p,a(t,\omega)\rangle$. Thus, $x_{0}+y\in
B(t,\omega,p)$. Since $x_{0}\in C^{X}(t,\omega,p)$, one has
$U(t,\omega,x_{0})>U(t,\omega,x_{0}+y)$. However, (${\bf A}_{3}$) implies
$U(t,\omega,x_{0}+y)>U(t,\omega,x_{0})$. This is a contradiction, which
completes the proof. ∎
Following Aumann in [4], $C^{X}(t,\omega,p)$ is called the _preferred set_ of
agent $t$ at the price $p$ and state of nature $\omega$, and
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ is called the _aggregate preferred set_ at
the price $p$ and state of nature $\omega$. Moreover, we shall call
$\int_{T}C^{X}(\cdot,\cdot,\cdot)d\mu:\Omega\times\Delta\rightrightarrows\mathbb{R}^{\ell}_{+}$
the _aggregate preferred correspondence_. In the next section, we shall
discuss some descriptive properties of this object. These properties will be
used to derive the existence of a maximin rational expectations equilibrium of
our economic model.
## 3\. Properties of the aggregate preferred correspondence
In this section, some results on continuity and measurability of the aggregate
preferred correspondence are presented.
###### Proposition 3.1.
Under _( A1)_, for every $(\omega,p)\in\Omega\times\Delta$,
$B(\cdot,\omega,p):T\rightrightarrows\mathbb{R}^{\ell}_{+}$ and
$X(\cdot,\omega,p):T\rightrightarrows\mathbb{R}^{\ell}_{+}$ are lower
measurable.
###### Proof.
Here, only the proof of lower measurability of $B(\cdot,\omega,p)$ is
provided. The other case can be done analogously. Fix
$(\omega,p)\in\Omega\times\Delta$. Define a function
$h:T\times\mathbb{R}^{\ell}_{+}\to\mathbb{R}$ by letting
$h(t,x)=\langle p,x\rangle-\langle p,a(t,\omega)\rangle$
for all $(t,x)\in T\times\mathbb{R}^{\ell}_{+}$. Then, $h(\cdot,x)$ is
measurable for all $x\in\mathbb{R}^{\ell}_{+}$. Note that
$B(t,\omega,p)=h(t,\cdot)^{-1}((-\infty,0]).$
Let $V\subseteq\mathbb{R}^{\ell}_{+}$ be a non-empty open subset, and put
$V\cap\mathbb{Q}_{+}^{\ell}=\\{x_{k}:k\geq 1\\}.$
It is worth to point out that if $x\in B(t,\omega,p)\cap V$, then $x_{k}\in
B(t,\omega,p)$ for some $k\geq 1$. Since $h(\cdot,x_{k})$ is measurable, one
has
$\left\\{t\in T:h(t,x_{k})\in(-\infty,0]\right\\}\in\Sigma$
for all $k\geq 1$. Thus,
$\displaystyle B(\cdot,\omega,p)^{-1}(V)$ $\displaystyle=$
$\displaystyle\bigcup_{k\geq 1}\left\\{t\in T:x_{k}\in B(t,\omega,p)\right\\}$
$\displaystyle=$ $\displaystyle\bigcup_{k\geq 1}\left\\{t\in
T:h(t,x_{k})\in(-\infty,0]\right\\}$
belongs to $\Sigma$. It follows that $B(\cdot,\omega,p)$ is lower measurable.
∎
###### Proposition 3.2.
Under _( A1)-(A3)_, $\int_{T}C^{X}(\cdot,\cdot,\cdot)d\mu$ is non-empty
compact-valued.
###### Proof.
Fix $(\omega,p)\in\Omega\times\Delta$. By (A2), $C^{X}(t,\omega,p)$ is non-
empty closed for all $t\in T$. By the lower measurability of
$B(\cdot,\omega,p)$, there exists a sequence $\\{f_{n}:n\geq 1\\}$ of
measurable functions from $T$ to ${\mathbb{R}}^{\ell}_{+}$ such that
$B(t,\omega,p)=\overline{\\{f_{n}(t):n\geq 1\\}}$
for all $t\in T$. For each $n\geq 1$, define a correspondence
$C_{n}:T\rightrightarrows\mathbb{R}^{\ell}_{+}$ by letting
$C_{n}(t)=\left\\{x\in\mathbb{R}^{\ell}_{+}:U(t,\omega,x)\geq
U(t,\omega,f_{n}(t))\right\\}$
for all $t\in T$. Obviously, one has
$C(t,\omega,p)\subseteq\bigcap_{n\geq 1}C_{n}(t)$
for all $t\in T$. If $x\in{\mathbb{R}}^{\ell}_{+}\setminus C(t,\omega,p)$ for
some $t\in T$, there exists a point $y\in B(t,\omega,p)$ such that
$U(t,\omega,y)>U(t,\omega,x)$. By (A2), there exists an $n_{0}\geq 1$ such
that $U(t,\omega,f_{n_{0}}(t))>U(t,\omega,x)$. This implies $x\notin
C_{n_{0}}(t)$. Thus, it is verified that
$C(t,\omega,p)=\bigcap_{n\geq 1}C_{n}(t)$
for all $t\in T$. Fix $n\geq 1$, and define a function
$h:T\times\mathbb{R}^{\ell}_{+}\to\mathbb{R}$ by
$h(t,x)=U(t,\omega,f_{n}(t))-U(t,\omega,x).$
Clearly, $h$ is Carathéodory. Similar to Proposition 3.1, one can show that
$C_{n}$ is lower measurable. Since $X(\cdot,\omega,p)$ is compact-valued and
$C^{X}(\cdot,\omega,p)=\bigcap_{n\geq 1}C_{n}(\cdot)\cap X(\cdot,\omega,p),$
then $C^{X}(\cdot,\omega,p)$ is lower measurable. By the Kuratowski-Ryll-
Nardzewski measurable selection theorem in [13], $C^{X}(\cdot,\omega,p)$ has a
measurable selection which is also integrable, as $b(\cdot,\omega,p)$ is so.
Since $C^{X}(\cdot,\omega,p)$ is closed-valued and integrably bounded,
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ is compact-valued. ∎
Next, we establish Hausdorff continuity of the aggregate preferred
correspondence with respect to the variable $p\in\Delta$.
###### Theorem 3.3.
Assume _( A1)-(A3)_. For each $\omega\in\Omega$,
$\int_{T}C^{X}(\cdot,\omega,\cdot)d\mu:\Delta\rightrightarrows{\mathbb{R}}^{\ell}_{+}$
is Hausdorff continuous
###### Proof.
Fix $\omega\in\Omega$. Let $\\{p_{n}:n\geq 1\\}\subseteq\Delta$ converge to
$p\in\Delta$. Choose $\varepsilon>0$ and $N\geq 1$ such that
$\varepsilon<\delta(p)$ and $\varepsilon<\delta(p_{n})$ for all $n\geq N$. Let
$c=\min\left\\{\delta(p_{n}),\varepsilon:n=1,2,\cdots,N-1\right\\},$
$d(t,\omega)=\frac{1}{c}\sum_{h=1}^{\ell}a^{h}(t,\omega),$
and
$\xi(t,\omega)=(d(t,\omega),...,d(t,\omega)).$
Define $M(\omega)$ by
$M(\omega)=\left\\{x\in\mathbb{R}^{\ell}_{+}:x\leq\int_{T}\xi(\cdot,\omega)d\mu\right\\}.$
Since $X(\cdot,\omega,p_{n})$ and $X(\cdot,\omega,p)$ are upper bounded by
$\xi(\cdot,\omega)$, then $\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu$ and
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ are contained in the compact subset
$M(\omega)$ of $\mathbb{R}^{\ell}_{+}$ . Thus, one only needs to show that
$\left\\{\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu:n\geq 1\right\\}$ converges to
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ in the Hausdorff metric topology on
${\mathscr{K}}_{0}(M(\omega))$, which is equivalent to
${\rm Li}\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu={\rm
Ls}\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu=\int_{T}C^{X}(\cdot,\omega,p)d\mu.$
The verification of the above equation can be split into two steps. First, one
verifies
${\rm
Ls}\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu\subseteq\int_{T}C^{X}(\cdot,\omega,p)d\mu.$
To do this, it is enough to verify that for any $t\in T$,
${\rm Ls}C^{X}(t,\omega,p_{n})\subseteq C^{X}(t,\omega,p).$
Pick $t\in T$ and $x\in{\rm Ls}C^{X}(t,\omega,p_{n})$. Then, there exist
positive integers $n_{1}<n_{2}<n_{3}<\cdots$ and for each $k$ a point
$x_{k}\in C^{X}(t,\omega,p_{n_{k}})$ such that $\\{x_{k}:k\geq 1\\}$ converges
to $x$. It is obvious that $x\in X(t,\omega,p)$. If $x\notin
C^{X}(t,\omega,p)$, by the continuity of $U(t,\omega,\cdot)$, one can choose
some $y\in\mathbb{R}^{\ell}_{+}$ such that $\langle p,y\rangle<\langle
p,a(t,\omega)\rangle$ and $U(t,\omega,y)>U(t,\omega,x)$. By the Hausdorff
continuity of $X(t,\omega,\cdot)$, $\\{X(t,\omega,p_{n_{k}}):k\geq 1\\}$
converges to $X(t,\omega,p)$ in the Hausdorff metric topology. Since $y\in
X(t,\omega,p)$, there exists a sequence $\\{y_{k}:k\geq 1\\}$ such that
$y_{k}\in X(t,\omega,p_{n_{k}})$ for all $k\geq 1$ and $\\{y_{k}:k\geq 1\\}$
converges to $y$. It follows that $U(t,\omega,y_{k})>U(t,\omega,x_{k})$ and
$\langle p_{n_{k}},y_{k}\rangle<\langle p_{n_{k}},a(t,\omega)\rangle$ for all
sufficiently large $k$, which is a contradiction with $x_{k}\in
C^{X}\left(t,\omega,p_{n_{k}}\right)$ for all $k\geq 1$. Therefore, one must
have $x\in C^{X}(t,\omega,p)$. Secondly, one needs to verify
$\int_{T}C^{X}(\cdot,\omega,p)d\mu\subseteq{\rm
Li}\int_{T}C^{X}(\cdot,\omega,p_{n})d\mu.$
It is enough to verify that for all $t\in T$,
$C^{X}(t,\omega,p)\subseteq{\rm Li}C^{X}(t,\omega,p_{n}).$
Fix $t\in T$ and pick $d\in C^{X}(t,\omega,p)$. If $d=b(t,\omega,p)$, then
$b(t,\omega,p_{n})\in C^{X}(t,\omega,p_{n})$ and the sequence
$\\{b(t,\omega,p_{n}):n\geq 1\\}$ converges to $d$. Assume $d<b(t,\omega,p)$.
Select $\delta>0$ such that
$d+(0,\cdots,\delta,\cdots,0)\leq b(t,\omega,p).$
Further, choose a sequence $\\{\delta_{i}:i\geq 1\\}$ in $(0,\delta]$
converging to $0$. For each $i\geq 1$, let
$d^{i}=d+(0,\cdots,\delta_{i},\cdots,0),$
and choose a sequence $\\{d_{n}^{i}:n\geq 1\\}$ such that for each $n$,
$d_{n}^{i}\in X(t,\omega,p_{n})$ and $\\{d_{n}^{i}:n\geq 1\\}$ converges to
$d^{i}$. It is claimed that for each $i\geq 1$, $d_{n}^{i}\in
C^{X}(t,\omega,p_{n})$ for sufficiently large $n$. Otherwise, there must exist
an $i_{0}$ and a subsequence $\\{d_{n_{k}}^{i_{0}}:k\geq 1\\}$ of
$\\{d_{n}^{i_{0}}:n\geq 1\\}$ such that $d_{n_{k}}^{i_{0}}\notin
C^{X}(t,\omega,p_{n_{k}})$. Let $b_{k}\in B(t,\omega,p_{n_{k}})$ and
$U(t,\omega,b_{k})>U(t,\omega,d_{n_{k}}^{i_{0}})$ for all $k\geq 1$. Then
$\\{b_{k}:k\geq 1\\}$ has a subsequence converging to some $b\in
B(t,\omega,p)$. Applying (A2) and (A3), one can obtain
$U(t,\omega,b)\geq U(t,\omega,d^{i_{0}})>U(t,\omega,d),$
which contradicts with the fact $d\in C^{X}(t,\omega,p)$. To complete the
proof, note that the previous claim implies that for each $i$, $\\{{\rm
dist}(d^{i},C^{X}(t,\omega,p_{n})):n\geq 1\\}$ converges to $0$. Since
$\\{d^{i}:i\geq 1\\}$ converges to $d$, one concludes that $\\{{\rm
dist}(d,C^{X}(t,\omega,p_{n})):n\geq 1\\}$ converges to $0$. This means that
$d\in{\rm Li}C^{X}(t,\omega,p_{n})$. ∎
The next result is crucial for the existence theorem in Section 4. In its
proof, the following characterization of lower measurability of
correspondences in [2] is used: A correspondence
$F:(\Omega,{\mathscr{F}},\nu)\rightrightarrows\mathbb{R}^{\ell}_{+}$ is lower
measurable if and only if for all $y\in\mathbb{R}^{\ell}_{+}$, ${\rm
dist}(y,F(\cdot)):(\Omega,{\mathscr{F}},\nu)\to\mathbb{R}_{+}$ is a measurable
function.
###### Theorem 3.4.
Assume _( A1)-(A3)_. For each $p\in\Delta$,
$\int_{T}C^{X}(\cdot,\cdot,p)d\mu:\Omega\rightrightarrows\mathbb{R}^{\ell}_{+}$
is lower measurable.
###### Proof.
Fix $p\in\Delta$. Since $a$ and $U$ are $\Sigma\otimes\mathscr{F}$-measurable
and
$\Sigma\otimes\mathscr{F}\otimes\mathfrak{B}(\mathbb{R}^{\ell}_{+})$-measurable
respectively, by the argument of a result in [16, 131], there exist two
sequences $\\{a_{n}:n\geq 1\\}$ and $\\{\psi_{n}:n\geq 1\\}$ of simple
$\Sigma\otimes\mathscr{F}$-measurable and simple
$\Sigma\otimes\mathscr{F}\otimes\mathfrak{B}(\mathbb{R}^{\ell}_{+})$-measurable
functions respectively such that $\\{a_{n}:n\geq 1\\}$ uniformly converges to
$a$ on $T\times\Omega$ and $\\{\psi_{n}:n\geq 1\\}$ uniformly converges to $U$
on $T\times\Omega\times\mathbb{R}^{\ell}_{+}$. For each $n\geq 1$, write
$a_{n}$ and $\psi_{n}$ as
$a_{n}=\sum_{i\geq 1}e_{i}\chi_{T_{i}^{n}\times\Omega_{i}^{n}}$
and
$\psi_{n}=\sum_{i\geq 1}v_{i}\chi_{T_{i}^{n}\times\Omega_{i}^{n}\times
B_{i}^{n}},$
where $e_{i}\in\mathbb{R}^{\ell}_{+}$, $v_{i}\in\mathbb{R}$, and
$\\{T_{i}^{n}\times\Omega_{i}^{n}\times B_{i}^{n}:i\geq 1\\}$ is a partition
of $T\times\Omega\times\mathbb{R}^{\ell}_{+}$ for all $n\geq 1$. Choose $N\geq
1$ such that $\|a_{n}-a\|_{\infty}<1$ for all $n\geq N$. By the measurability
of $a_{n}(\cdot,\omega)$, $a_{n}(\cdot,\omega)\in
L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)$ for all $\omega\in\Omega$ and all
$n\geq 1$ (replacing $a_{n}$ for all $1\leq n<N$ by some constant functions,
if necessary). Let
$\gamma_{n}(t,\omega)=\frac{1}{\delta(p)}\sum_{h=1}^{\ell}a_{n}^{h}(t,\omega)$
and
$b_{n}(t,\omega)=(\gamma_{n}(t,\omega),\cdots,\gamma_{n}(t,\omega)).$
Define $X_{n},B_{n},C_{n}:T\times\Omega\rightrightarrows\mathbb{R}^{\ell}_{+}$
such that
$X_{n}(t,\omega)=\left\\{x\in\mathbb{R}^{\ell}_{+}:x\leq
b_{n}(t,\omega)\right\\},$
$B_{n}(t,\omega)=\left\\{x\in\mathbb{R}^{\ell}_{+}:\langle
p,x\rangle\leq\langle p,a_{n}(t,\omega)\rangle\right\\}$
and
$C_{n}(t,\omega)=\left\\{y\in\mathbb{R}^{\ell}_{+}:\psi_{n}(t,\omega,y)\geq\psi_{n}(t,\omega,x)\mbox{
for all }x\in B_{n}(t,\omega)\right\\}.$
In addition, define
$C_{n}^{X}:T\times\Omega\rightrightarrows\mathbb{R}^{\ell}_{+}$ such that for
all $(t,\omega)\in T\times\Omega$,
$C_{n}^{X}(t,\omega)=\left(C_{n}(t,\omega)\cup\\{b_{n}(t,\omega)\\}\right)\cap
X_{n}(t,\omega).$
For every $n\geq 1$, define the correspondence
$H_{n}:(\Omega,\mathscr{F},\nu)\rightrightarrows
L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)$ by letting
$H_{n}(\omega)={\mathscr{S}}_{C_{n}^{X}(\cdot,\omega)}$. Obviously,
$H_{n}(\omega)\neq\emptyset$ for all $\omega\in\Omega$.
Claim 1. For each $n\geq 1$, $H_{n}$ is lower measurable. For convenience, let
$\Theta:L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)\times\Omega\to\mathbb{R}_{+}$
be the function such that
$\Theta(g,\omega)={\rm dist}(g,H_{n}(\omega))$
for all $g\in L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)$ and
$\omega\in\Omega$. To verify the claim, one needs to verify that for all $g\in
L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)$, $\Theta(g,\cdot)$ is measurable.
Since
$\Theta(\cdot,\omega):L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)\to\mathbb{R}_{+}$
is norm-continuous, it suffices to show that
$\Theta(g,\cdot):\Omega\to\mathbb{R}_{+}$ is measurable for every simple
function $g=\sum_{j=1}^{r}x_{j}\chi_{T_{j}}$, where
$x_{j}\in\mathbb{R}^{\ell}_{+}$. To this end, consider the function
$\Gamma:T\times\Omega\to\mathbb{R}_{+}$ such that
$\Gamma(t,\omega)={\rm dist}\left(g(t),C_{n}^{X}(t,\omega)\right)$
for all $(t,\omega)\in T\times\Omega$. Since $\Gamma$ is constant on each
$(T^{n}_{i}\cap T_{j})\times\Omega_{i}^{n}$, it is jointly measurable. Note
that
$\Gamma(t,\omega)\leq\|g(t)-b_{n}(t,\omega)\|$
for all $(t,\omega)\in T\times\Omega$. This implies for all $\omega\in\Omega$,
$\Gamma(\cdot,\omega)$ is integrable. Thus, $\Theta(g,\cdot)$ is measurable
and the claim is verified if one can show for all $\omega\in\Omega$,
$\int_{T}\Gamma(\cdot,\omega)d\mu=\Theta(g,\omega).$
Assume that
$\int_{T}\Gamma(\cdot,\omega_{0})d\mu<\Theta(g,\omega_{0})$
for some $\omega_{0}\in\Omega$. Pick $\varepsilon>0$ such that
$\int_{T}\Gamma(\cdot,\omega_{0})d\mu+\varepsilon\mu(T)<\Theta(g,\omega_{0}).$
Further, pick $t\in T_{i}^{n}\cap T_{j}$ and $y_{(i,j)}\in
C_{n}^{X}(t,\omega_{0})$ so that
$\|x_{j}-y_{(i,j)}\|<\Gamma(t,\omega_{0})+\varepsilon.$
Then the function $\zeta:T\to\mathbb{R}^{\ell}_{+}$, defined by
$\zeta(t)=y_{(i,j)}$ if $t\in T_{i}^{n}\cap T_{j}$, belongs to
$H_{n}(\omega_{0})$ and
$\|g-\zeta\|_{1}<\int_{T}\Gamma(\cdot,\omega_{0})d\mu+\varepsilon\mu(T),$
which is a contradiction.
Claim 2. The correspondence
$\int_{T}C_{n}^{X}(\cdot,\cdot)d\mu:(\Omega,\mathscr{F},\nu)\rightrightarrows\mathbb{R}^{\ell}_{+}$
is lower measurable. Consider the function
$\xi:L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)\to\mathbb{R}^{\ell}_{+}$
defined by $\xi(f)=\int_{T}fd\mu$ for all $f\in
L_{1}\left(\mu,\mathbb{R}^{\ell}_{+}\right)$. Let $V$ be an open subset of
$\mathbb{R}^{\ell}_{+}$. Note that
$\xi\circ H_{n}(\omega)=\int_{T}C_{n}^{X}(\cdot,\omega)d\mu$
for all $\omega\in\Omega$, and
$(\xi\circ
H_{n})^{-1}(V)=\\{\omega\in\Omega:H_{n}(\omega)\cap\xi^{-1}(V)\neq\emptyset\\}.$
Since $\xi$ is norm-continuous, by Claim 1, $(\xi\circ
H_{n})^{-1}(V)\in\mathscr{F}$. This verifies the claim.
Claim 3. For each $\omega\in\Omega$,
${\rm Li}\int_{T}C_{n}^{X}(\cdot,\omega)d\mu={\rm
Ls}\int_{T}C_{n}^{X}(\cdot,\omega)d\mu=\int_{T}C^{X}(\cdot,\omega,p)d\mu.$
To see this, for each $\omega\in\Omega$, put
$\alpha(\cdot,\omega)=\sup\left\\{b_{1}(\cdot,\omega),\cdots,b_{N-1}(\cdot,\omega),b(\cdot,\omega,p)+\left(\frac{\ell}{\delta(p)},\cdots,\frac{\ell}{\delta(p)}\right)\right\\}.$
Then, $C^{X}(\cdot,\omega,p)$ and all $C_{n}^{X}(\cdot,\omega)$ are integrably
bounded by $\alpha(\cdot,\omega)$. Now, it suffices to verify that for all
$t\in T$,
${\rm Ls}C_{n}^{X}(t,\omega)\subseteq C^{X}(t,\omega,p),$
and
$C^{X}(t,\omega,p)\subseteq{\rm Li}C_{n}^{X}(t,\omega).$
First, let $x\in{\rm Ls}C_{n}^{X}(t,\omega)$. If $x=b(t,\omega,p)$, then
$\\{b_{n}(t,\omega):n\geq 1\\}$ converges to $x$ and $b_{n}(t,\omega)\in
C_{n}^{X}(\cdot,\omega)$ for all $n\geq 1$. Otherwise, there exist positive
integers $n_{1}<n_{2}<n_{3}<\cdots$ and for each $k$ a point $x_{k}\in
C_{n_{k}}^{X}(t,\omega)$ such that $\\{x_{k}:k\geq 1\\}$ converges to $x$.
Obviously, $x_{k}\neq b_{n_{k}}(t,\omega)$ for all sufficiently large $k$, and
$x\in X(t,\omega,p)$. If $x\notin C^{X}(t,\omega,p)$, there exists some $y\in
B(t,\omega,p)$ such that $U(t,\omega,y)>U(t,\omega,x)$. By the continuity of
$U(t,\omega,\cdot)$, $y$ can be chosen so that $\langle p,y\rangle<\langle
p,a(t,\omega)\rangle$. Since $\\{X_{n_{k}}(t,\omega):k\geq 1\\}$ converges to
$X(t,\omega,p)$ in the Hausdorff metric topology, there exists a sequence
$\\{y_{k}:k\geq 1\\}$ such that $y_{k}\in X_{n_{k}}(t,\omega)$ for all $k\geq
1$ and $\\{y_{k}:k\geq 1\\}$ converges to $y$. By the inequality
$\displaystyle|U(t,\omega,x)-\psi_{n_{k}}(t,\omega,x_{k})|$ $\displaystyle<$
$\displaystyle|U(t,\omega,x)-U(t,\omega,x_{k})|$ $\displaystyle+$
$\displaystyle|U(t,\omega,x_{k})-\psi_{n_{k}}(t,\omega,x_{k})|,$
the continuity of $U(t,\omega,\cdot)$ and the uniform convergence of
$\psi_{n_{k}}(t,\omega,\cdot)$ to $U(t,\omega,\cdot)$, one concludes that
$\psi_{n_{k}}(t,\omega,y_{k})>\psi_{n_{k}}(t,\omega,x_{k})$ and $\langle
p,y_{k}\rangle<\langle p,a_{n_{k}}(t,\omega)\rangle$ for sufficiently large
$k$, which contradicts with the fact that $x_{k}\in C_{n_{k}}^{X}(t,\omega)$
for all $k\geq 1$. Hence, $x\in C^{X}(t,\omega,p)$. Now, let $d\in
C^{X}(t,\omega,p)$. If $d=b(t,\omega,p)$, there is nothing to verify. Thus,
$d\in{\rm Li}C_{n}^{X}(t,\omega)$. Assume $d<b(t,\omega,p)$. Similar to that
in the proof of Theorem 3.3, one can show that $d\in{\rm
Li}C_{n}^{X}(t,\omega)$.
To complete the proof, for each $\omega\in\Omega$, put
$M(\omega)=\left\\{x\in\mathbb{R}^{\ell}_{+}:x\leq\int_{T}\alpha(\cdot,\omega)d\mu\right\\}.$
Clearly, $\int_{T}C_{n}^{X}(\cdot,\omega)d\mu$ and
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ are contained in the compact set
$M(\omega)$. It follows from Claim 3 that
$\left\\{\int_{T}C_{n}^{X}(\cdot,\omega)d\mu:n\geq 1\right\\}$ converges to
$\int_{T}C^{X}(\cdot,\omega,p)d\mu$ in $M(\omega)$ in the Hausdorff metric
topology. It is well known that a nonempty compact-valued correspondence is
lower measurable if and only if it is measurable when viewed as a single-
valued function whose range space is the space of nonempty compact sets
endowed with the Hausdorff metric topology. By Claim 2,
$\int_{T}C^{X}(\cdot,\cdot,p)d\mu$ is lower measurable. ∎
###### Corollary 3.5.
Assume _( A1)-(A3)_. Then
$\int_{T}C^{X}(\cdot,\cdot,\cdot)d\mu:\Omega\times\Delta\rightarrow{\mathscr{K}}_{0}(\mathbb{R}^{\ell}_{+})$
is a jointly measurable function, where
${\mathscr{K}}_{0}(\mathbb{R}^{\ell}_{+})$ is endowed with the Hausdorff
metric topology.
###### Proof.
By Theorem 3.3, for every $\omega\in\Omega$,
$\int_{T}C^{X}(\cdot,\omega,\cdot)d\mu:\Omega\times\Delta\to{\mathscr{K}}_{0}\left(\mathbb{R}^{\ell}_{+}\right)$
is continuous. Furthermore, by Theorem 3.4, for every $p\in\Delta$, the
correspondence
$\int_{T}C^{X}(\cdot,\cdot,p)d\mu:\Omega\rightrightarrows\mathbb{R}^{\ell}_{+}$
is lower measurable. Hence, for every $p\in\Delta$,
$\int_{T}C^{X}(\cdot,\cdot,p)d\mu:\Omega\rightarrow{\mathscr{K}}_{0}(\mathbb{R}^{\ell}_{+})$
is measurable. This means that
$\int_{T}C^{X}(\cdot,\cdot,\cdot)d\mu:\Omega\times\Delta\rightarrow{\mathscr{K}}_{0}(\mathbb{R}^{\ell}_{+})$
is Carathéodory, and therefore is jointly measurable. ∎
## 4\. The existence of a maximin REE
A _price system_ of $\mathscr{E}$ is a measurable function
$\pi:(\Omega,\mathscr{F},\nu)\to\Delta$. Let $\sigma(\pi)$ be the smallest
sub-algebra of $\mathscr{F}$ such that $\pi$ is measurable and let
${\mathscr{G}}_{t}={\mathscr{F}}_{t}\vee\sigma(\pi)$. For each
$\omega\in\Omega$, let ${\mathscr{G}}_{t}(\omega)$ denote the smallest element
of ${\mathscr{G}}_{t}$ containing $\omega$. Given $t\in T$, $\omega\in\Omega$
and a price system $\pi$, let $B^{REE}(t,\omega,\pi)$ be defined by
$B^{REE}(t,\omega,\pi)=\left\\{x\in(\mathbb{R}^{\ell}_{+})^{\Omega}:x(\omega^{\prime})\in
B(t,\omega^{\prime},\pi(\omega^{\prime}))\mbox{ for all
}\omega^{\prime}\in\mathscr{G}_{t}(\omega)\right\\}.$
The _maximin utility_ of each agent $t\in T$ with respect to $\mathscr{G}_{t}$
at an allocation $f:T\times\Omega\to\mathbb{R}^{\ell}_{+}$ in state
$\omega\in\Omega$, denoted by $\b{\it U}^{REE}(t,\omega,f(t,\cdot))$, is
defined by
$\b{\it
U}^{REE}(t,\omega,f(t,\cdot))=\inf_{\omega^{\prime}\in\mathscr{G}_{t}(\omega)}U(t,\omega^{\prime},f(t,\omega^{\prime})).$
###### Definition 4.1.
Given a feasible allocation $f$ and a price system $\pi$, the pair $(f,\pi)$
is called a _maximin rational expectations equilibrium_ of $\mathscr{E}$ if
$f(t,\omega)\in B(t,\omega,\pi(\omega))$ for almost all $(t,\omega)\in
T\times\Omega$, and $f(t,\cdot)$ maximizes $\b{\it U}^{REE}(t,\omega,\cdot)$
on $B^{REE}(t,\omega,\pi)$.
An allocation $f$ is called a _maximin rational expectations allocation_ if
there exists a price system $\pi$ such that $(f,\pi)$ is a maximin rational
expectations equilibrium. The set of maximin rational expectations allocations
is denoted by $MREE(\mathscr{E})$. Recently, de Castro et al. [7] showed that
$MREE(\mathscr{E})\neq\emptyset$ when $\Omega$ and $T$ are finite. Our next
theorem extends their result to a more general case.
###### Theorem 4.2.
Under _( A1)-(A4)_, $MREE(\mathscr{E})\neq\emptyset$.
###### Proof.
Consider the correspondence
$Z:\Omega\times\Delta\rightrightarrows\mathbb{R}^{\ell}$ defined by
$Z(\omega,p)=\int_{T}C^{X}(\cdot,\omega,p)d\mu-\int_{T}a(\cdot,\omega)d\mu.$
By Proposition 3.2, $Z$ is non-empty compact-valued. In addition, by Corollary
3.5 and (A1),
$Z:\Omega\times\Delta\rightarrow\mathbb{\mathscr{K}}_{0}(\mathbb{R}^{\ell})$
is jointly measurable. Define another correspondence
$F:\Omega\rightrightarrows\Delta$ such that
$F(\omega)=\\{p\in\Delta:Z(\omega,p)\cap\\{0\\}\neq\emptyset\\}.$
By Theorem 2 in [10, p.151], $F$ is non-empty valued. Since
${\rm Gr}_{F}=Z^{-1}(\\{0\\}),{\rm
Gr}_{F}\in\mathscr{F}\otimes\mathfrak{B}(\Delta),$
by the Aumann-Saint-Beuve measurable selection theorem, there exists a
measurable function $\hat{\pi}:\Omega\to\Delta$ such that
$\hat{\pi}(\omega)\in F(\omega)$ for all $\omega\in\Omega$. By the definition
of $Z$, there exists an allocation $f$ such that $f(t,\omega)\in
C^{X}(t,\omega,\hat{\pi}(\omega))$ and
$\int_{T}f(\cdot,\omega)d\mu=\int_{T}a(\cdot,\omega)d\mu$
for almost all $t\in T$ and all $\omega\in\Omega$. By Proposition 2.1, one has
$\langle\hat{\pi}(\omega),f(t,\omega)\rangle\geq\langle\hat{\pi}(\omega),a(t,\omega)\rangle$
for almost all $t\in T$ and all $\omega\in\Omega$. Then, the previous equation
implies
$\langle\hat{\pi}(\omega),f(t,\omega)\rangle=\langle\hat{\pi}(\omega),a(t,\omega)\rangle$
for almost all $t\in T$ and all $\omega\in\Omega$. Thus, $f(t,\omega)\in
B(t,\omega,\hat{\pi}(\omega))$ for almost all $t\in T$ and all
$\omega\in\Omega$. For every $\omega\in\Omega$, define $T_{\omega}\subseteq T$
by
$T_{\omega}=\\{t\in T:f(t,\omega)\in B(t,\omega,\hat{\pi}(\omega))\cap
C(t,\omega,\hat{\pi}(\omega))\\}.$
Then, $\mu(T_{\omega})=\mu(T)$ for all $\omega\in\Omega$. Next, for every
$\omega\in\Omega$ and every $t\in T\setminus T_{\omega}$, as
$B(t,\omega,\hat{\pi}(\omega))\cap
C(t,\omega,\hat{\pi}(\omega))\neq\emptyset$, one can pick a point
$h(t,\omega)\in B(t,\omega,\hat{\pi}(\omega))\cap
C(t,\omega,\hat{\pi}(\omega)),$
and then define a function $\hat{f}:T\times\Omega\to\mathbb{R}^{\ell}_{+}$
such that
$\hat{f}(t,\omega)=\left\\{\begin{array}[]{ll}f(t,\omega),&\mbox{if $t\in
T_{\omega}$;}\\\\[5.0pt] h(t,\omega),&\mbox{if $t\in T\setminus
T_{\omega}$.}\end{array}\right.$
It is obvious that $\hat{f}(t,\omega)\in B(t,\omega,\hat{\pi}(\omega))\cap
C(t,\omega,\hat{\pi}(\omega))$ for all $(t,\omega)\in T\times\Omega$. Assume
that there are an agent $t_{0}\in T$, a state of nature
$\omega_{t_{0}}\in\Omega$ and an element $y(t_{0},\cdot)\in
B^{REE}(t_{0},\omega_{t_{0}},\hat{\pi})$ such that
$\b{\it U}^{REE}(t_{0},\omega_{t_{0}},y(t_{0},\cdot))>\b{\it
U}^{REE}(t_{0},\omega_{t_{0}},\hat{f}(t_{0},\cdot)).$
Then, one obtains
$U(t_{0},\omega_{t_{0}}^{\prime},y(t_{0},\omega_{t_{0}}^{\prime}))>U(t_{0},\omega_{t_{0}}^{\prime},\hat{f}(t_{0},\omega_{t_{0}}^{\prime}))$
for some $\omega_{t_{0}}^{\prime}\in\mathscr{G}_{t_{0}}(\omega_{t_{0}})$,
which contradicts with $\hat{f}(t_{0},\omega_{t_{0}}^{\prime})\in
C(t_{0},\omega_{t_{0}}^{\prime},\hat{\pi}(\omega_{t_{0}}^{\prime}))$. This
verifies that $(\hat{f},\hat{\pi})$ is a maximin rational expectations
equilibrium of $\mathscr{E}$. ∎
## 5\. Conclusion
The first application of maximin expected utility functions to the general
equilibrium theory with differential information appeared in Correia da Silva
and Hervés Beloso [6], where an existence theorem for a Walrasian equilibrium
in an economy was established. However, their MEU formulation is in the ex-
ante sense, and REE notion was not considered.
In our paper, an existence theorem on a maximin rational expectations
equilibrium (maximin REE) for an exchange differential information economy is
proved. Comparing with the existence result on maximin REE in [7], our theorem
applies to a more general economic model with an arbitrary finite measure
space of agents and an arbitrary complete probability measure space as the
space of states of nature, while the later applies only to an economic model
which has finitely many agents and finitely many states of nature. Assumptions
in our paper are similar to those in [7], except the joint measurability and
continuity of utility functions, and the joint measurability of the initial
endowment function. The proof techniques in this paper are quite different
from those in [7]. Since there are only finitely many agents and states of
nature in the model considered in [7], neither measurability nor continuity of
utility functions and the initial endowment function plays any role in the
proof of the existence of a maximin REE. Instead, the existence of a
competitive equilibrium for complete information economies is applied. In
contrast, both measurability and continuity of utility functions and the
initial endowment function play key roles in this paper. To establish the
existence theorem, Aumann’s techniques in [4] are adopted, measurability and
continuity of the aggregate preferred correspondence are investigated.
However, for special cases, the techniques can be simplified. For instance, if
there are finitely many states of nature, one can still apply the approach
employed in [7] and obtains an existence theorem. On the other hand, if there
are finitely many agents, then one can show that the demand of each agent is
${\mathscr{F}}\otimes{\mathfrak{B}}(\Delta)$-measurable and so is the
aggregate demand. Then, an approach similar to that in the proof of Theorem
4.2 can be applied to establish the existence theorem.
Appendix
Let $G$ be a non-empty set and $\mathbb{R}^{\ell}$ be the $\ell$-dimensional
Euclidean space. On $\mathbb{R}^{\ell}$, two different but equivalent standard
norms $\|\cdot\|_{\infty}$ and $\|\cdot\|_{1}$ are used in this paper, where
for each pint $x=(x_{1},x_{2},\cdots,x_{\ell})\in\mathbb{R}^{\ell}$,
$\|x\|_{\infty}=\max\\{|x_{i}|:1\leq i\leq\ell\\},$
and
$\|x\|_{1}=\sum_{1\leq i\leq\ell}|x_{i}|.$
A _correspondence_ $F:G\rightrightarrows\mathbb{R}^{\ell}$ from $G$ to
$\mathbb{R}^{\ell}$ assigns to each $x\in G$ a subset $F(x)$ of
$\mathbb{R}^{\ell}$. Meanwhile, $F$ can also be viewed as a function $F:G\to
2^{\mathbb{R}^{\ell}}$, where $2^{\mathbb{R}^{\ell}}$ denotes the power set of
$\mathbb{R}^{\ell}$. Further, $F$ is called non-empty valued (resp. closed-
valued, compact-valued, convex-valued) if $F(x)$ is a non-empty (resp. closed,
compact, convex) subset of $\mathbb{R}^{\ell}$ for all $x\in G$. The _graph_
of $F$, denoted by ${\rm Gr}_{F}$, is defined by
${\rm Gr}_{F}=\\{(x,y)\in G\times\mathbb{R}^{\ell}:y\in F(x)\ {\rm and}\ x\in
G\\}.$
For each point $x\in{\mathbb{R}}^{\ell}$ and a subset $A\in
2^{\mathbb{R}^{\ell}}\setminus\\{\emptyset\\}$, define
${\rm dist}(x,A)=\inf\\{d(x,y):y\in A\\},$
where $d$ is the Euclidean metric on $\mathbb{R}^{\ell}$. Let
${\mathscr{K}}_{0}\left(\mathbb{R}^{\ell}\right)$ be the family of non-empty
compact subsets of $\mathbb{R}^{\ell}$. Recall that the _Hausdorff metric_ $H$
on ${\mathscr{K}}_{0}\left(\mathbb{R}^{\ell}\right)$ is defined such that for
any two $A,B\in{\mathscr{K}}_{0}\left(\mathbb{R}^{\ell}\right)$,
$H(A,B)=\max\left\\{\sup_{a\in A}\ {\rm dist}(a,B),\ \sup_{b\in B}\ {\rm
dist}(b,A)\right\\}.$
For equivalent definitions of $H$, refer to [2]. The topology
${\mathscr{T}}_{H}$ on ${\mathscr{K}}_{0}(\mathbb{R}^{\ell})$, generated by
$H$, is called the _Hausdorff metric topology_. For a closed subset $M$ of
$\mathbb{R}^{\ell}$, ${\mathscr{K}}_{0}(M)$ and the Hausdorff metric $H$ on
${\mathscr{K}}_{0}(M)$ can be defined similarly. When $G$ is a topological
space, a non-empty compact-valued correspondence
$F:G\rightrightarrows\mathbb{R}^{\ell}$ is called _Hausdorff continuous_ if
$F:G\to\left({\mathscr{K}}_{0}\left(\mathbb{R}^{\ell}\right),{\mathscr{T}}_{H}\right)$
is continuous. This statement also holds when $\mathbb{R}^{\ell}$ is replaced
by a closed subset $M$ of $\mathbb{R}^{\ell}$.
Let $\\{A_{n}:n\geq 1\\}$ be a sequence of non-empty subsets of
$\mathbb{R}^{\ell}$. A point $x\in\mathbb{R}^{\ell}$ is called a _limit point_
of $\\{A_{n}:n\geq 1\\}$ if there exist $N\geq 1$ and points $x_{n}\in A_{n}$
for each $n\geq N$ such that $\\{x_{n}:n\geq N\\}$ converges to $x$. The set
of all limit points of $\\{A_{n}:n\geq 1\\}$ is denoted by ${\rm Li}A_{n}$.
Similarly, a point $x\in\mathbb{R}^{\ell}$ is called a _cluster point_ of
$\\{A_{n}:n\geq 1\\}$ if there exist positive integers $n_{1}<n_{2}<\cdots$
and for each $k$ a point $x_{k}\in A_{n_{k}}$ such that $\\{x_{k}:k\geq 1\\}$
converges to $x$. The set of all cluster points of $\\{A_{n}:n\geq 1\\}$ is
denoted by ${\rm Ls}A_{n}$. It is clear that ${\rm Li}A_{n}\subseteq{\rm
Ls}A_{n}$. If ${\rm Ls}A_{n}\subseteq{\rm Li}A_{n}$,
${\rm Li}A_{n}={\rm Ls}A_{n}=A$
is called the _limit_ of the sequence $\\{A_{n}:n\geq 1\\}$. If $A$ and all
$A_{n}^{\prime}$s are closed and contained in a compact subset
$M\subseteq\mathbb{R}^{\ell}$, then it is well known that
${\rm Li}A_{n}={\rm Ls}A_{n}=A$
if and only if $\\{A_{n}:n\geq 1\\}$ converges to $A$ in Hausdorff metric
topology on ${\mathscr{K}}_{0}(M)$, refer to [2].
Let $(T,\Sigma,\mu)$ be a measure space and $\\{F_{n}:n\geq
1\\},F:(T,\Sigma,\mu)\rightrightarrows\mathbb{R}^{\ell}_{+}$ be
correspondences. Recall that
$F:(T,\Sigma,\mu)\rightrightarrows\mathbb{R}^{\ell}$ is said to be _lower
measurable_ if
$F^{-1}(V)=\\{t\in T:F(t)\cap V\neq\emptyset\\}\in\Sigma$
for every open subset $V$ of $\mathbb{R}^{\ell}$. It is well known that a non-
empty closed-valued correspondence
$F:(T,\Sigma,\mu)\rightrightarrows\mathbb{R}^{\ell}$ is lower measurable if
and only if there exists a sequence of measurable selections $\\{f_{n}:n\geq
1\\}$ of $F$ such that for all $t\in T$,
$F(t)=\overline{\\{f_{n}(t):n\geq 1\\}}.$
If all $F_{n}^{\prime}s$ are non-empty closed-valued and lower measurable and
at least one of $F_{n}^{\prime}s$ is compact-valued, $\bigcap_{n\geq 1}F_{n}$
is lower measurable, refer to [11]. If all $F_{n}^{\prime}s$ are integrably
bounded by the same function, then
${\rm Ls}\int_{T}F_{n}d\mu\subseteq\int_{T}{\rm Ls}F_{n}d\mu,$
and
$\int_{T}{\rm Li}F_{n}d\mu\subseteq{\rm Li}\int_{T}F_{n}d\mu.$
If $(S,\mathscr{S},\nu)$ is another measure space and $f:T\times
S\to\mathbb{R}^{\ell}$ is a jointly measurable function, then it is well known
that $\int_{T}:f(\cdot,\cdot)d\mu:S\to\mathbb{R}^{\ell}$ is a measurable
function. Let $M\subseteq\mathbb{R}^{\ell}$ be endowed with the relative
Euclidean topology, and $(Y,\varrho)$ be a metric space. A function $f:T\times
M\to(Y,\varrho)$ is called _Carathéodory_ if $f(\cdot,x)$ is measurable for
all $x\in M$, and $f(t,\cdot)$ is continuous for all $t\in T$. It is known
that any Carathéodory function is jointly measurable with respect to the Borel
structure on $M$. A _selection_ of $F$ is a single-valued function
$f:(T,\Sigma,\mu)\to\mathbb{R}^{\ell}$ such that $f(t)\in F(t)$ for almost all
$t\in T$. If a selection $f$ of $F$ is measurable (resp. integrable), then it
is called a _measurable_ (resp. an _integrable_) _selection_. Let
${\mathscr{S}}_{F}$ denote the set of all integrable selections of $F$. The
_integration_ of $F$ over $T$ in the sense of Aumann in [3] is a subset of
$\mathbb{R}^{\ell}$, defined as
$\int_{T}Fd\mu=\left\\{\int_{T}fd\mu:f\in{\mathscr{S}}_{F}\right\\}.$
If $F$ is non-empty closed-valued and integrably bounded, then $\int_{T}Fd\mu$
is compact, refer to [10]. The following two theorems on measurable selections
have been employed in this paper.
Kuratowski-Ryll-Nardzewski Measurable Selection Theorem ([11, 13]). _If
$F:(T,\Sigma,\mu)\rightrightarrows\mathbb{R}^{\ell}$ is a closed-valued and
lower measurable correspondence, then it has a measurable selection._
Aumann-Saint-Beuve Measurable Selection Theorem ([5, 15]). _Let
$(T,\Sigma,\mu)$ be a complete finite measure space,
$B\subseteq\mathbb{R}^{\ell}$ be a Borel subset. If $F:T\rightrightarrows B$
has a measurable graph, there exists a measurable function $f:T\to B$ such
that $f(t)\in F(t)$ for all $t\in T$”._
Of course, the above two theorems were stated in more general forms in the
literature. But here, they are adapted and presented in particular and simpler
forms to fit in this paper.
Acknowledgement. The authors are very grateful to He Wei for his valuable
comments and suggestions on the early draft of the paper.
## References
* [1] Allen, B.: Generic existence of completely revealing equilibria with uncertainty, when prices convey information, Econometrica 49, 1173–1199 (1981)
* [2] Aubin, J.P., Frankowska, H.: Set-valued analysiss. Springer (2008).
* [3] Aumann, Robert J.: Integrals of set-valued functions, J. Math. Anal. Appl. 12, 1- 12 (1965).
* [4] Aumann, R. J.: Existence of competitive equilibria in markets with a continuum of traders, Econometrica 34, 1–17 (1966)
* [5] Aumann, R. J.: Measurable utility and the measurable choice theorem, La Décision Colloque Internationaux du C.N.R.S., Paris 171, 15–26 (1969)
* [6] Correia-da-Silva, J., Hervés-Beloso, C.: Prudent expectations equilibrium in economies with uncertain delivery, Econ. Theory 39, 67–92 (2009)
* [7] de Castro, L. I., Pesce, M., Yannelis, N. C.: A new perspective to rational expectations: maximin rational expectation equilibrium, under preparation
* [8] Gilboa, I., Schmeidler, D.: Maxmin expected utility with a non-unique prior, J. Math. Econ. 18, 141–153 (1989)
* [9] Glycopantis, D., Muir, A., Yannelis, N.C.: Non-implementation of rational expectations as a perfect Bayesian equilibrium, Econ. Theory 26, 765–791 (2005)
* [10] Hildenbrand, W.: Core and equilibria in large economies, Priceton University Press, 1974
* [11] Himmelberg, C. J.: Measurable relations, Fund. Math. 87, 53 -72 (1975)
* [12] Kreps, D. M.: A note on ‘fulfilled expectations’ equilibrium, J. Econ. Theory 14(1), 32–43 (1977)
* [13] Kuratowski, K., Ryll-Nardzewski, C.: A general theorem on selectors, Bull. Acad. Polon. Sci. 13, 397–403 (1965)
* [14] Radner, R.: Rational expectation equilibrium: generic existence and information revealed by prices, Econometrica 47, 655–678 (1979)
* [15] Saint-Beuve, M.-F.: On the extension of von Neumann-Aumann’s theorem, J. Funct. Anal. 17, 112–129 (1974)
* [16] Yosida, K.: Functional analysis, Sixth edition. Springer (1980).
|
arxiv-papers
| 2012-10-31T22:11:54 |
2024-09-04T02:49:37.448858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anuj Bhowmik, Jiling Cao and Nicholas C. Yannelis",
"submitter": "Jiling Cao",
"url": "https://arxiv.org/abs/1211.0047"
}
|
1211.0055
|
###### Abstract
Hyperspectral image is a substitution of more than a hundred images, called
bands, of the same region. They are taken at juxtaposed frequencies. The
reference image of the region is called Ground Truth map (GT). the problematic
is how to find the good bands to classify the pixels of regions; because the
bands can be not only redundant, but a source of confusion, and decreasing so
the accuracy of classification. Some methods use Mutual Information (MI) and
threshold, to select relevant bands without treatement of redundancy; others
consider the neighbors having sensibly the same MI with the GT as redundant
and so discarded. This is the most inconvenient of this method, because this
avoids the advantage of hyperspectral images: some precious information can be
discarded. In this paper well make difference between useful and useless
redundancy. A band contains useful redundancy if it contributes to decreasing
error probability. According to this scheme, we introduce new algorithm using
also mutual information, but it retains only the bands minimizing the error
probability of classification. To control redundancy, we introduce a
complementary threshold. So the good band candidate must contribute to
decrease the last error probability augmented by the threshold. This process
is a wrapper strategy; it gets high performance of classification accuracy but
it is expensive than filter strategy.
Applied Mathematical Sciences, Vol. 6, 2012, no. 102, 5073 - 5084
Dimensionality Reduction and Classification Feature
Using Mutual Information Applied to Hyperspectral
Images: A Wrapper Strategy Algorithm Based on
Minimizing the Error Probability Using
the Inequality of Fano
ELkebir Sarhrouni*, Ahmed Hammouch** and Driss Aboutajdine*
*LRIT, Faculty of Sciences, Mohamed V - Agdal University, Morocco
**LRGE, ENSET, Mohamed V - Souissi University, Morocco
sarhrouni436@yahoo.fr
Keywords: Hyperspectral images, classification, feature selection, mutual
information, error probability, redundancy
## 1 Introduction
In the feature classification domain, the choice of data affects widely the
results. For the Hyperspectral image, the bands dont all contain the
information; some bands are irrelevant like those affected by various
atmospheric effects, see Figure.4, and decrease the classification accuracy.
And there exist redundant bands to complicate the learning system and product
incorrect prediction [14]. Even the bands contain enough information about the
scene they may can’t predict the classes correctly if the dimension of space
images, see Figure.3, is so large that needs many cases to detect the
relationship between the bands and the scene (Hughes phenomenon) [10]. We can
reduce the dimensionality of hyperspectral images by selecting only the
relevant bands (feature selection or subset selection methodology), or
extracting, from the original bands, new bands containing the maximal
information about the classes, using any functions, logical or numerical
(feature extraction methodology) [11][9]. Here we focus on the feature
selection using mutual information. Hyperspectral images have three advantages
regarding the multispectral images [6],
Assertion: when we reduce hyperspectral images dimensionality, any method used
must save the precision and high discrimination of substances given by
hyperspectral image.
Figure 1: Precision an dicrimination added by hyperspectral images
In this paper we use the Hyperspectral image AVIRIS 92AV3C (Airborne Visible
Infrared Imaging Spectrometer). [2]. It contains 220 images taken on the
region ”Indiana Pine” at ”north-western Indiana”, USA [1]. The 220 called
bands are taken between 0.4m and 2.5m. Each band has 145 lines and 145
columns. The ground truth map is also provided, but only 10366 pixels are
labeled fro 1 to 16. Each label indicates one from 16 classes. The zeros
indicate pixels how are not classified yet, Figure.2.
The hyperspectral image AVIRIS 92AV3C contains numbers between 955 and 9406\.
Each pixel of the ground truth map has a set of 220 numbers (measures) along
the hyperspectral image. This numbers (measures) represent the reflectance of
the pixel in each band. So the pixel is shown as vector off 220 components.
Figure .3.
Figure 2: The Ground Truth map of AVIRIS 92AV3C and the 16 classes
The hyperspectral image AVIRIS 92AV3C contains numbers between 955 and 9406.
Each pixel of the ground truth map has a set of 220 numbers (measures) along
the hyperspectral image. This numbers (measures) represent the reflectance of
the pixel in each band. So the pixel is shown as vector off 220 components.
Figure.3 shows the vector pixel’s notion [7]. So reducing dimensionality means
selecting only the dimensions caring a lot of information regarding the
classes.
Figure 3: The notion of pixel vector
We can also note that not all classes are carrier of information. In Figure.
4, for example, we can show the effects of atmospheric affects on bands: 155,
220 and other bands. This hyperspectral image presents the problematic of
dimensionality reduction.
## 2 Mutual Information based feature selection
### 2.1 Definition of mutual information
This is a measure of exchanged information between tow ensembles of random
variables A and B :
$I(A,B)=\sum\;log_{2}\;p(A,B)\;\frac{p(A,B)}{p(A).p(B)}$
Considering the ground truth map, and bands as ensembles of random variables,
we calculate their interdependence.
Geo [3] uses also the average of bands 170 to 210, to product an estimated
ground truth map, and use it instead of the real truth map. Their curves are
similar. This is shown at Figure 4.
Figure 4: Mutual information of AVIRIS with the Ground Truth map (solid line)
and with the ground apporoximated by averaging bands 170 to 210 (dashed line)
.
### 2.2 The mesure of error probability
Fano [14] has demonstrated that as soon as mutual information of already
selected feature has high value, the error probability of classification is
decreasing, according to the formula bellow:
$\;\frac{H(C/X)-1}{Log_{2}(N_{c})}\leq\;P_{e}\leq\frac{H(C/X)}{Log_{2}}\;$
with :
$\;\frac{H(C/X)-1}{Log_{2}(N_{c})}=\frac{H(C)-I(C;X)-1}{Log_{2}(N_{c})}\;$
and :
$P_{e}\leq\frac{H(C)-I(C;X)}{Log_{2}}=\frac{H(C/X)}{Log_{2}}\;$
The expression of conditional entropy H(C/X) is calculated between the ground
truth map (i.e. the classes C) and the subset of bands candidates X. Nc is the
number of classes. So when the features X have a higher value of mutual
information with the ground truth map, (is more near to the ground truth map),
the error probability will be lower. But it’s difficult to compute this
conjoint mutual information I(C;X), regarding the high dimensionality
[14].This propriety makes Mutual Information a good criterion to measure
resemblance between too bands, like it’s exploited in section II. Furthermore,
we will interest at case of one feature candidate X.
Corollary: for one feature X, as X approaches the ground truth map, the
interval Pe is very small.
## 3 The principe of proposed algorithm based on inequality of Fano
Our idea is based on this observation: the band that has higher value of
Mutual Information with the ground truth map can be a good approximation of
it. So we note that the subset of selected bands are the good ones, if thy can
generate an estimated reference map, sensibly equal the ground truth map. It’s
clearly that’s an Incremental Wrapper-based Subset Selection (IWSS)
approach[16] [13].
Our process of band selection will be as following: we order the bands
according to value of its mutual information with the ground truth map. Then
we initialize the selected bands ensemble with the band that has highest value
of MI. At a point of process, we build an approximated reference map Cest with
the already selected bands, and we put it instead of $X$ for computing the
error probability (Pe); the latest band added (at those already selected) must
make Pe decreasing, if else it will be discarded from the ensemble retained.
Then we introduce a complementary threshold Th to control redundancy. So the
band to be selected must make error probability less than ( Pe \- Th) , where
Pe is calculated before adding it. The algorithm following shows more detail
of the process:
Let SS be the ensemble of bands already selected and S the band candidate to
be selected. Build${}_{estimated}C()$ is a procedure to construct the
estimated reference map. Pe is initialized with a value P${}_{e}^{*}$ . X the
number of bands to be selected, $SS$ is empty and $R={1..220}$.
Algorithm 1 Proposed for Dimentionality Reduction and Redundancy control
while $|SS|<X$ do
Select band indexs $S$=argmaxs MI(s)
$SS\leftarrow\textit{SS}\cup\textit{S}$ and
$R\leftarrow\textit{R}\setminus\textit{S}$
Cest= Build${}_{estimated}C(SS)$
$Pe=\frac{H(C/C_{est})}{Log_{2}}\
-\frac{H(C/C_{est})-1}{Log_{2}(N_{c})};\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
if $Pe\leq Pe^{*}-Th$ then
$Pe\leftarrow Pe^{*}$
else
$SS\leftarrow\textit{SS}\setminus\textit{S}$
end if
end while
## 4 Results and analysis
We apply this algorithm on the hyperspectral image AVIRIS 92AV3C [1], 50% of
the labeled pixels are randomly chosen and used in training; and the other 50%
are used for testing classifcation [3]. The classifer used is the SVM [5] [12]
[4].
The procedure to construct the estimated reference map Cest is the same SVM
classifier used for classification. So Cest is the output of classification.
### 4.1 Results
Table. I shows the results obtained for several thresholds. We can see the
effectiveness selection bands of our algorithm, and the important effect of
avoiding redundancy.
Table 1: Results illustrate elimination of Redundancy using algorithm based on inequality of Fano, for thresholds ($Th$) ${\mathrm{B}ands}$ | | | | $Th$ | |
---|---|---|---|---|---|---
${\mathrm{r}etained}$ | 0.00 | 0.001 | 0.008 | 0.015 | 0.020 | 0.030
10 | 55.43 | 55.43 | 55.58 | 53.09 | 60.06 | 71.62
18 | 59.09 | 59.09 | 64.41 | 73.70 | 82.62 | 90.00
20 | 63.08 | 63.08 | 68.50 | 76.15 | 84.36 | -
25 | 66.02 | 66.12 | 74.62 | 84.41 | 89.06 | -
27 | 69.47 | 69.47 | 76.00 | 86.73 | 91.70 | -
30 | 73.54 | 73.54 | 79.04 | 88.68 | - | -
35 | 76.06 | 76.06 | 81.38 | 92.36 | - | -
40 | 78.96 | 79.41 | 86.48 | - | - | -
45 | 80.58 | 80.60 | 89.09 | - | - | -
50 | 81.63 | 81.20 | 91.14 | - | - | -
53 | 82.27 | 81.22 | 92.67 | - | - | -
60 | 86.13 | 86.23 | - | - | - | -
70 | 86.97 | 87.55 | - | - | - | -
80 | 89.11 | 89.42 | - | - | - | -
90 | 90.55 | 90.92 | - | - | - | -
100 | 92.50 | 93.18 | - | - | - | -
102 | 92.62 | 93.34 | - | - | - | -
114 | 93.34 | - | - | - | - | -
Figure.5 shows more detail of the accuracy curves, versus number of bands
retained, for several thresholds. This covers all behaviors of the algorithm.
Figure 5: Accuracy of classification using the algorithm based on inequality
of Fano, using numerous thresholds.
### 4.2 Analysis
Table.I and Figure.5 allow us to comment four cases:
First: For the highest threshold values (0.1, 0.05, 0.03 and 0.02) we obtain a
hard selection: a few more informative bands are selected; the accuracy of
classification is 90% with less than 20 bands selected.
Second: For the medium threshold values (0.015, 0.012, 0.010, 0.008, 0.006),
some redundancy is allowed, in order to made increasing the classification
accuracy.
Tired: For the small threshold values (0.001 and 0), the redundancy allowed
becomes useless, we have the same accuracy with more bands.
Finally: for the negative thresholds, for example -0.01, we allow all bands to
be selected, and we have no action of the algorithm. This corresponds at
selection bay ordering bands on mutual information . The performance is low.
We can not here that [15] uses two axioms to characterize feature selection.
Sufficiency axiom: the subset selected feature must be able to reproduce the
training simples without losing information. The necessity axiom ”simplest
among different alternatives is preferred for prediction”. In the algorithm
proposed, reducing error probability between the truth map and the estimated
minimize the information loosed for the samples training and also the
predicate ones.
We note also that we can use the number of features selected like condition to
stop the search; so we can obtain an hybrid approach filter-wrapper[16].
Partial conclusion:
The algorithm proposed is a very good method to reduce dimensionality of
hyperspectral images.
We illustrate in Figure .6, the Ground Truth map originally displayed, like at
Figure .1, and the scene classified with our method, for threshold as 0.03, so
18 bands selected.
Figure 6: Original Grand Truth map(in the left) and the map produced bay our
algorithm according to the threshold 0.03 i.e 18 bands (in the right).
Acuracy=90%.
Table II indicates the classification accuracy of each class, for several
thresholds.
Table 2: Accuracy of classification(%) of each class for numerous thresholds ($Th$) ${\mathrm{C}lass}$ | $Total$ | | | | $Th$ | |
---|---|---|---|---|---|---|---
| ${\mathrm{p}ixels}$ | 0.00 | 0.001 | 0.008 | 0.015 | 0.020 | 0.030
1 : | 54 | 86.96 | 82.61 | 86.96 | 83.96 | 78.26 | 86.96
2 : | 1434 | 91.07 | 89.40 | 89.54 | 89.12 | 88.01 | 83.96
3 : | 834 | 89.93 | 90.89 | 89.69 | 86.09 | 83.69 | 81.53
4 : | 234 | 96.32 | 83.76 | 86.32 | 87.18 | 87.18 | 86.32
5 : | 597 | 95.93 | 95.53 | 94.34 | 95.93 | 95.93 | 95.53
6 : | 747 | 98.60 | 98.60 | 98.32 | 98.60 | 98.32 | 98.32
7 : | 26 | 84.62 | 84.62 | 84.62 | 84.62 | 84.62 | 84.62
8 : | 489 | 98.37 | 98.37 | 98.78 | 97.96 | 98.78 | 98.78
9 : | 20 | 100 | 100 | 100 | 100 | 100 | 100
10: | 968 | 92.15 | 92.98 | 91.32 | 91.74 | 90.91 | 89.05
11: | 2468 | 93.84 | 94.17 | 92.54 | 92.71 | 91.90 | 91.25
12: | 614 | 91.21 | 93.49 | 92.83 | 92.18 | 88.93 | 87.30
13: | 212 | 98.06 | 98.06 | 98.06 | 98.06 | 98.06 | 98.06
14: | 1294 | 97.53 | 97.86 | 97.22 | 97.84 | 97.99 | 97.53
15: | 390 | 79.52 | 77.71 | 75.90 | 74.10 | 78.92 | 64.46
16: | 95 | 93.48 | 93.48 | 93.48 | 93.48 | 93.48 | 93.48
Comments:
First :we can not the effectiveness of this algorithm for particularly the
classes with a few number of pixels, for example class number 9.
Second: we can not that 18 bands (i.e. threshold 0.03) are sufficient to
detect materials contained in the region. It’s also shown in Figure .6
Tired: one of important comment is that most of class accuracy change lately
when the threshold changes between 0.03 and 0.015
## 5 Conclusion
In this paper we presented the necessity to reduce the number of bands, in
classification of Hyperspectral images. Then we have introduce the mutual
information based scheme. We carried out their effectiveness to select bands
able to classify the pixels of ground truth. We introduce an algorithm also
based on mutual information and using a measure of error probability
(inequality of Fano). To choice a band, it must contributes to reduce error
probability. A complementary threshold is added to avoid redundancy. So each
band retained has to contribute to reduce error probability by a step equal to
threshold even if it caries a redundant information. We can tell that we
conserve the useful redundancy by adjusting the complementary threshold. The
process introduced is able to select the good bands to classification for also
the classes that have a few number of pixels. This algorithm is a feature
selection methodology. But it’s a wrapper approach, because we use the
classifier to make the estimated reference map. This is expenssive than Filter
strategy, but it can be used for application that need more precision. This
scheme is very interesting to investigate and improve, considering its
performance.
## References
* [1] D. Landgrebe, “On information extraction principles for hyperspectral data: A white paper,” Purdue University, West Lafayette, IN, Technical Report, School of Electrical and Computer Engineering, 1997. Téléchargeable ici : http://dynamo.ecn.purdue.edu/ landgreb/whitepaper.pdf.
* [2] ftp://ftp.ecn.purdue.edu/biehl/MultiSpec/
* [3] Baofeng Guo, Steve R. Gunn, R. I. Damper Senior Member, ”Band Selection for Hyperspectral Image Classification Using Mutual Information” , IEEE and J. D. B. Nelson. IEEE GEOSCIENCE AND REMOTE SINSING LETTERS, Vol .3, NO .4, OCTOBER 2006.
* [4] Baofeng Guo, Steve R. Gunn, R. I. Damper, Senior Member, IEEE, and James D. B. Nelson.”Customizing Kernel Functions for SVM-Based Hyperspectral Image Classification”, IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 4, APRIL 2008.
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* [6] Nathalie GORRETTA-MONTEIRO , Proposition d’une approche de’segmentation d’images hyperspectrales. PhD thesis. Universite Montpellier II. Février 2009.
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* [9] Nojun Kwak and C. Kim,”Dimensionality Reduction Based on ICA Regression Problem”. ARTIFICIAL NEURAL NETWORKS-ICANN 2006. Lecture Notes in Computer Science, 2006, Isbn 978-3-540-38625-4, Volume 1431/2006.
* [10] Huges, G. Information Thaory,”On the mean accuracy of statistical pattern recognizers”. IEEE Transactionon Jan 1968, Volume 14, Issue:1, p:55-63, ISSN 0018-9448 DOI: 10.1109/TIT.1968.1054102.
* [11] YANG, Yiming, and Jan O. PEDERSEN, 1997.A comparative study of feature selection in text categorization. In: ICML. 97: Proceedings of the Fourteenth International Conference on Machine Learning. San Francisco, CA, USA:Morgan Kaufmann Publishers Inc., pp. 412.420.
* [12] Chih-Wei Hsu; Chih-Jen Lin,”A comparison of methods for multiclass support vector machines” ;Dept. of Comput. Sci. Inf. Eng., Nat Taiwan Univ. Taipei Mar 2002, Volume: 13 I:2;pages: 415 - 425 ISSN: 1045-9227, IAN: 7224559, DOI: 10.1109/72.991427
* [13] Bermejo, P.; Gamez, J.A.; Puerta, J.M.”Incremental Wrapper-based subset Selection with replacement: An advantageous alternative to sequential forward selection” ;Comput. Syst. Dept., Univ. de Castilla-La Mancha, Albacete; Computational Intelligence and Data Mining, 2009. CIDM ’09. IEEE Symposium on March 30 2009-April 2 2009, pages: 367 - 374, ISBN: 978-1-4244-2765-9, IAN: 10647089, DOI: 10.1109/CIDM.2009.4938673
* [14] Lei Yu, Huan Liu,”Efficient Feature Selection via Analysis of Relevance and Redundancy”, Department of Computer Science and Engineering; Arizona State University, Tempe, AZ 85287-8809, USA, Journal of Machine Learning Research 5 (2004) 1205-1224.
* [15] Hui Wang, David Bell, and Fionn Murtagh, ”Feature subset selection based on relevance” , Vistas in Astronomy, Volume 41, Issue 3, 1997, Pages 387-396.
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Received: March, 2012
|
arxiv-papers
| 2012-10-31T23:30:59 |
2024-09-04T02:49:37.458190
|
{
"license": "Public Domain",
"authors": "Elkebir Sarhrouni, Ahmed Hammouch and Driss Aboutajdine",
"submitter": "ELkebir Sarhrouni",
"url": "https://arxiv.org/abs/1211.0055"
}
|
1211.0093
|
# Investigation of target in C-ADS and IAEA ADS benchmark 111Supported by
National Natural Science Foundation of China (11045003 & 10975150)
Guojun Hu 222huguojun@mail.ustc.edu.cn Hongli Wu Tian Jing
Xiangqi Wang333Corresponding author. wangxaqi@ustc.edu.cn Jingyu Tang
National Synchrotron Radiation Laboratory,USTC School of Nuclear Science and
Technology, USTC Institute of High Energy Physics Chinese Academy of Sciences
Department of Modern Physics, USTC
###### Abstract
The spatial and energy distribution of spallation neutrons have an effect on
the performance of Accelerator Driven Subcritical systems. In this work, the
spatial, energy distribution of spallation neutrons and the effect of these
factors on proton efficiency was studied. When the radius of spallation region
increases, backward neutrons were found to have a rather big ratio and have a
positive effect on proton efficiency. By making better use of these neutrons,
we may increase the radius of target to satisfy some other requirements in the
design of subcritical core.
###### keywords:
C-ADS, Spallation target, Backward neutrons
††journal: arXiv.org
## 1 Introduction
China ADS (C-ADS) consists of a high-intensity proton accelerator in CW
(continuous wave) mode with proton energy of 0.6 GeV in phase three and 1.5
GeV in phase four, a subcritical core, and a spallation target of liquid Lead-
Bismuth Eutectic (LBE or Pb-Bi), which is favored in projects for its low
melting point, high boiling point and low neutron absorption cross section
[1].
The target should have such a size that it incepts the main part of the high-
energy cascade and then we get a optimized neutron multiplicity[4]. Neutron
multiplicity is defined to be the number of neutrons produced per beam
particle. Increasing the size of target will affect the spatial and energy
distribution of spallation neutrons and thus affecting the performance of the
total system. So in this work, we discussed the effect of the spatial and
energy distribution of spallation neutrons.
Neutron multiplicity, spatial and energy distribution were simulated with
FLUKA [5, 6] in this work. In these simulations, Intra-nuclear cascade, pre-
equilibrium, evaporation and fission models were activated. Proton efficiency
of the system was partly simulated with RMC(Reactor Monte Carlo code)[7, 8].
IAEA ADS benchmark [9] was an ADS model and was used in these simulations,
shown in Fig. 2.
## 2 Validation of FLUKA
Double-differential distribution of spallation neutrons were measured in the
Laboratoire National Sturne experiment [10]. In order to observe the
availability of FLUKA in simulating spallation reaction, this experiment was
simulated with FLUKA.
Table 1: Parameters of Saturne experiment[10] Beam | Target | Detectors
---|---|---
Projectile | Proton | Material | Lead | $R$ | 800 cm
$E_{\rm{p}}$ | 1.6 GeV | $R_{\rm{s}}$ | 1.5 cm | $dS$ | 1963.5 $\rm{cm}^{2}$
Direction | (0,0,1) | $N$ | $3.30\times 10^{28}\rm{m^{-3}}$ | $d\Omega$ | $3.068\times 10^{-3}\rm{sr}$
$R_{\rm{b}}$ | 1.4 cm | $L_{\rm{s}}$ | 2.0 cm | |
Fig. 1: Double-differential neutron production cross-section. Data of
different polar angle had been scaled with different ratio marked in the
figure. $1\sigma<5\%$
Parameters of target, beam and detectors in this simulation are listed in
Table 1. Detectors are in a sphere surface with radius of $R=800\ \rm{cm}$.
Detectors are set in polar angle of
$\theta=0^{\circ},10^{\circ},55^{\circ}$,$85^{\circ},130^{\circ},160^{\circ}$.
Double-differential cross-section of neutron production is obtained with
Eq.(1):
$\frac{{\rm d}^{2}\sigma}{{\rm d}\Omega{\rm d}E}=\frac{{\rm
d}N_{\rm{ns}}}{N_{\rm{primary}}\frac{{\rm d}S}{R^{2}}{\rm
d}E}\frac{1}{NL_{\rm{s}}}=\frac{{\rm d}N_{\rm{ns}}}{N_{\rm{primary}}{\rm
d}S{\rm d}E}R^{2}\frac{1}{NL_{\rm{s}}}$ (1)
Where, ${\rm d}S$ , ${\rm d}\Omega={\rm d}S/R^{2}$ , $N$ and $L_{\rm{s}}$ is
area of detector, solid angle, atomic density and thickness of target in Table
1. ${\rm d}N_{\rm{ns}}/({N_{\rm{primary}}{\rm d}S{\rm d}E})$ is scored with
USRBDX card.
Comparison between this simulation and the experiment is shown in Fig. 1. Data
of the experiment was obtained from EXFOR/CRISRS. The result of FLUKA agrees
well with the experiment. There is difference in high energy due to the defect
of physical models and the statistic error of simulation. FlUKA is suitable to
simulate the spallation reaction.
## 3 Description of ADS benchmark and proton efficiency
### 3.1 Description of IAEA ADS benchmark
IAEA ADS benchmark was chosen to simulate the effect of spallation neutrons.
Geometry and parameters of IAEA ADS benchmark is show in Fig. 2 and Table 2.
In regions ”core1” and ”core2”, the ratio between 233-U and 232-Th is 1:9 and
effective multiplication factor is: $k_{\rm{eff}}=0.96455\pm 0.00062$.
Fig. 2: RZ view of IAEA ADS benchmark. The different regions are:(1)Target;
(2)Buffer; (3)Core1; (4)Core2; (5)Core3; (6)Reflector; (7)Beam pipe;
(8),(10)Moderator Table 2: Parameters of ADS benchmark[9] The element of IAEA
ADS benchmark at BOL (20 ∘C) $10^{24}\rm{cm}^{-3}$
---
Nuclides | Core1 | Core2 | Core3 | Reflector | Moderator, Target, Buffer
232-Th | | | 7.45 | |
232-Th,233-U | 6.350 | 7.450 | | |
O | 12.70 | 14.90 | 14.90 | |
Fe | 8.100 | 8.870 | 8.870 | 6.630 |
Cr | 1.120 | 1.060 | 1.060 | 0.800 |
Mn | 0.046 | 0.051 | 0.051 | 0.038 |
W | 0.046 | 0.051 | 0.051 | 0.038 |
Pb | 17.70 | 15.60 | 15.60 | 0.024 | 30.50
The source of IAEA ADS benchmark
Projectile | Proton
Kinetic Energy:$E_{\rm{p}}$ | 1.0 GeV
Beam radius:$R_{\rm{b}}$ | 10.0 cm
Liquid Lead is the material of moderator and target in ADS benchmark.
Spallation region is bounded by the subcritical core and beam pipe. We divided
the spallation region into ”target” and ”buffer” as used in ”Energy
Amplifier”[2]. $R_{\rm{s}}$ and $L_{\rm{s}}$ is the size of spallation region,
see Fig. 2. In simulations of FLUKA, except for region ”target”, all other
regions were filled with vacuum.
### 3.2 Neutron source efficiency and proton efficiency
Neutron flux $\phi_{\rm{s}}$ in the subcritical core is the solution to the
inhomogeneous steady-state neutron transport equation Eq.(2):
$A\phi_{\rm{s}}=F\phi_{\rm{s}}+S_{\rm{n}}$ (2)
Where $F$ is fission operator, $A$ is net neutron loss operator and
$S_{\rm{n}}$ is external neutron source.
The subcritical multiplication factor $k_{\rm{s}}$ [11] is defined as Eq.(3):
$k_{\rm{s}}=\frac{<F\phi_{\rm{s}}>}{<F\phi_{\rm{s}}>+<S_{\rm{n}}>}$ (3)
$k_{\rm{s}}$ describes the number of fission neutrons produced per lost
neutron in the subcritical system.
Neutron source efficiency, usually denoted as $\varphi^{*}$ [11], represents
the efficiency of the external source neutrons and can be expressed as Eq.
(4):
$\varphi^{*}=(\frac{1}{k_{\rm{eff}}-1})\frac{<F\phi_{\rm{s}}>}{S_{\rm{n}}}$
(4)
$<F\phi_{\rm{s}}>$ is total number of neutrons produced by fission.
$<S_{\rm{n}}>$ is total number of external source neutrons. For a given values
of $k_{\rm{eff}}$ and $<S_{\rm{n}}>$, the larger $\varphi^{*}$, the bigger the
fission power. Definition of neutron source efficiency needs the definition of
source neutrons $<S_{\rm{n}}>$. In this work, $<S_{\rm{n}}>$ means neutrons
escaping the spallation region.
Relationship between $k_{\rm{keff}}$ and $k_{\rm{s}}$ [11] is:
$(1-\frac{1}{k_{\rm{keff}}})=\varphi^{*}(1-\frac{1}{k_{\rm{s}}})$ (5)
Proton efficiency[12], denoted as $\psi^{*}$, is defined to be:
$\psi^{*}=\varphi^{*}\frac{<S_{\rm{n}}>}{<S_{\rm{p}}>}=(\frac{1}{k_{\rm{eff}}-1})\frac{<F\phi_{\rm{s}}>}{<S_{\rm{p}}>}$
(6)
$S_{\rm{p}}$ is total number of source protons.
To study proton efficiency of the system, simulations had been performed in
two steps:
* 1.
protons impact on the target, neutron multiplicity and neutrons escaping the
target are recorded with FLUKA.
* 2.
the source neutrons recorded in the first step are then processed as an
external neutron source and transported in the system.
In order to reduce the time of simulation, the height and diameter of the
total system was cut to be 360 cm.
## 4 Spatial and energy distribution of spallation neutrons
### 4.1 Description of spatial and energy distribution of spallation neutrons
Spatial and energy distribution of spallation neutrons have an effect on
proton efficiency. 232-Th,238-U, Am can be used as fuels of ADS. An external
neutron hardly ever induce fissions of these fuels when its kinetic energy is
below 1 MeV[12]. The energy spectrum of neutrons in subcritical core is
different from that of spallation neutrons; the former is determined by
geometry and composition of the system, while the later is mostly determined
by the size of spallation region. The spatial distribution of spallation
neutrons determines the distribution of neutron flux in the subcritical core
and thus determining the proton efficiency of the system.
Fig. 3: RZ view of target model used in FLUKA. The different regions are:
(1)Target; (2)Subcritical core; (3)Beam pipe; (4)Moderator; (5)Void. In
simulations of FLUKA, region Moderator and Subcritical core is filled with
vacuum. Spallation neutrons include : Backward(1$\rightarrow$3,4);
Forward(1$\rightarrow$5); Radial(1$\rightarrow$2); Survival(4$\rightarrow$2);
Leak(3,4$\rightarrow$5)
In this work, spatial and energy distribution of spallation neutrons was
simulated with FLUKA. The target model used in FLUKA is shown in Fig. 3.
Neutrons are divided into forward, radial, backward neutrons according to the
surface the neutrons cross when they escape the target. Radial neutrons enter
the subcritical core directly. Forward neutrons hardly ever enter the
subcritical core. Some backward neutrons make it to the subcritical core
through the moderator between the beam pipe and the subcritical core (region
”10” in Fig. 2), called survival neutrons; other backward neutrons enter the
reflector or leak out from the beam pipe, called leak neutrons. A clear
explanation of these neutrons is shown in Fig. 3. Except for region ”target”,
all other regions are filled with vacuum.
As for spatial distribution, what we concern here was the neutron current
surface density $I_{\rm{r}}(Z)$ in the side surface of target. $Z$ is the
distance between a point in the side surface and the top surface of target. We
also cared about the number of backward, forward and radial neutrons. As for
energy distribution, the energy spectrum of radial neutrons were mostly
discussed.
### 4.2 Spatial and energy distribution of radial neutrons in C-ADS
Radial neutrons are the main part of spallation neutrons and their spatial
distribution is closely related to the neutron flux in the subcritical core.
So we studied this factor quantitatively with FlUKA. Remember that we mainly
care about neutron current surface density here.
In this simulation, proton source is a circular uniform beam with radius of
$R_{\rm{b}}=15\ \rm{cm}$ in C-ADS HEBT (High Energy Beam Transport ) [13].
Simulation had been performed with parameters listed in Table 3. To obtain the
neutron current surface density, subroutine mgdraw.f was modified and
activated to record neutrons escaping the ”target” region in Fig. 3. Then we
constructed the spatial and energy distribution of neutrons with the
information recorded. The result of this simulation is shown in Fig. 4.
Average kinetic energy of spallation neutrons in different position was used
in calculating energy distribution: $\overline{E_{\rm{r}}(Z)}$.
Table 3: Parameters of target and beam of C-ADS [13] Target characteristic | Beam
---|---
Material | 50%Pb,50%Bi | Projectile | Proton
$R_{\rm{s}}$ | 25.0 cm | $E_{\rm{p}}$ | 1.5 GeV
$L_{\rm{s}}$ | 100.0 cm | Source | HEBT
As can be seen in Fig. 4, as $Z$ increase: $\overline{E_{\rm{r}}(Z)}$ reaches
a minimum and then saturates; $I_{\rm{r}}(Z)$ reaches a maximum and then
decreases to zero. Based on this result, if the position of maximum
$I_{\rm{r}}(Z)$ locates at the axial center of subcritical core(that is $z=0\
\rm{cm}$), neutron flux $\phi_{\rm{s}}(z)$ in the subcritical core will be
symmetric (with axis at $z=0\ \rm{cm}$). Generally, an symmetric distribution
of $\phi_{\rm{s}}(z)$ is a key factor in designing the system. In this way, we
may optimize the position(relative to the subcritical core) of spallation
target, discussed in Sec. 4.4.1.
Fig. 4: Neutron current surface density $I_{\rm{r}}(Z)$ and average kinetic
energy$\overline{E_{\rm{r}}(Z)}$ as a function of $Z$. $I_{\rm{r}}(Z)$ is
normalized to per source proton.
In this case, $\overline{E_{\rm{r}}(Z)}$ changes slightly,
$(E_{\rm(rmax)}-E_{\rm{rmin}})/E_{\rm(raverage)}\approx 0.141$. Energy
distribution of radial spallation neutrons has a influence on the neutron flux
of the subcritical core, but it is a less important factor than the spatial
distribution of radial neutrons. Detailed effect of the energy distribution
needs further research.
### 4.3 Backward neutrons and radial neutrons as a function of $R_{\rm{s}}$
The size of subcritical core changes with the radius of spallation region. It
was proved that proton efficiency decrease when radius of spallation region
(called target in [12]) increases due to the soft and leakage of neutrons
[12]. Changing the size of the subcritical core will bring to change of
several factors that influence the proton efficiency. In this work, as is
shown in Fig. 5, a ”void” region is inserted between the spallation region and
the subcritical core to study the effect of backward neutrons. Changing the
thickness of this ”void” region, we could simulate different $R_{\rm{s}}$
without changing other parameters, such as the size of the subcritical core.
This ”void” region does not exist in practical design of ADS, but it is useful
to study the effect of backward neutrons.
Fig. 5: RZ view of modified ADS benchmark. The different regions
are:(1)Target; (2)Buffer; (3)Core1; (4)Core2; (5)Core3; (6)Reflector; (7)Beam
pipe; (8)Moderator; (9)Void.
To study the effect of backward and radial neutrons on proton efficiency with
ADS model, parameters of IAEA ADS benchmark were used, as listed in Table 2;
$L_{\rm{s}}=100\ \rm{cm}$. We use the target model in Fig. 3 to study the
number of backward neutrons and the soft of radial neutrons when $R_{\rm{s}}$
changes. Four different thickness of ”Void” region were chosen to get
$R_{\rm{s}}=12,18,24,30\ \rm{cm}$. The result is listed in Table 4. Remember
that in all simulations with FLUKA, region subcritical core is filled with
vacuum.
Table 4: Characteristic of spallation neutrons, $1\sigma<1.0\%$ Number of
spallation neutrons in different direction
---
$R_{\rm{s}}/\rm{cm}$ | 12 | 18 | 24 | 30
Radial | 20.665 | 21.838 | 21.219 | 20.199
Forward | 0.007 | 0.034 | 0.110 | 0.263
Backward: | 3.729 | 6.368 | 8.702 | 10.767
Survival | 3.530 | 5.697 | 7.270 | 8.281
Leak | 0.199 | 0.671 | 1.432 | 2.486
Ratio of high-energy Radial neutrons
$>100\ \rm{MeV}\%$ | 1.70 | 0.99 | 0.62 | 0.41
$>20\ \rm{MeV}\%$ | 1.26 | 0.86 | 0.58 | 0.39
$>1\ \rm{MeV}\%$ | 59.62 | 48.3 | 38.97 | 31.04
In Table 4, as $R_{\rm{s}}$, we find that: (1) the number of radial neutrons
changes slightly and the spectrum of radial neutrons becomes softer (ratio of
neutrons with kinetic energy higher than 1 MeV decrease from 59.62% to
31.04%); (2) the number of backward neutrons increases greatly (from 3.729 to
10.767); (3) the number of survival neutrons increases considerably (from
3.530 to 8.821). Explanation to these results may be: most neutrons,
especially high energy neutrons are produced near the top surface of target;
as $R_{\rm{s}}$ increases, it is more likely that these neutrons make it to
the top surface of target and then leak out in backward direction. In
conclusion, when the radius of spallation region increases, number of survival
neutrons becomes considerably big, which will contribute a positive effect to
proton efficiency; radial neutrons become softer, which will contribute a
negative effect to proton efficiency. Comparison of these two effects are
discussed below.
In this simulation, material of region ”moderator” in Fig. 3 was vacuum, not
Lead; replacing the vacuum with Lead would bring a slight decrease to the
number of survival neutrons due to the scattering and absorption of neutrons
in the ”moderator”. Another simulation had been performed; for
$R_{\rm{s}}=12,18,24,30\ \rm{cm}$, survival neutrons decrease to 3.409, 5.098,
6.162, 6.838 per source proton. This difference does not severely affect the
fact that a rather big number of backward neutrons will enter the subcritical
core when the radius of spallation region is big.
### 4.4 Effect of spatial and energy distribution of spallation neutrons
#### 4.4.1 Optimizing target position with spatial distribution of radial
neutrons
The spatial distribution of radial neutrons may be used to optimize the axial
position of target, as discussed in Sec. 4.2 . In this work, target position
is determined by the top surface of target $Z_{\rm{top}}$. Optimization of
$Z_{\rm{top}}$ had been performed with the following considerations:
* 1.
$L_{\rm{}s}$ will be larger than the range of proton in Lead, about 100 cm for
1.0-1.5 GeV protons. In this way, $I_{\rm{r}}(Z)$ and neutron multiplicity
will not change when $Z_{\rm{top}}$ changes.
* 2.
The bottom of target, $Z_{\rm{bottom}}$, remains at 75 cm; $Z_{\rm{top}}$
changes from -60 cm to -15 cm and $L_{\rm{}s}$ varies from 90 cm to 135 cm.
* 3.
All other parameters do not change, such as $R_{\rm{s}}=32.5\ \rm{cm}$ and
composition of the subcritical core. $k_{\rm{eff}}$ has a little change and
will be considered in the calculation of proton efficiency.
* 4.
$L_{\rm{s}}$ reaches its saturation, so $<S_{\rm{n}}>/{<S_{\rm{p}}>}$ does not
change when $Z_{\rm{top}}$ changes. $\varphi^{*}$ is equal to $\psi^{*}$.
The position of maximum $I_{\rm{r}}(Z)$ varies with the radius of spallation
region and the energy of protons. For ADS benchmark, $R_{\rm{s}}=32.5\
\rm{cm}$, $L_{\rm{}s}=100\ \rm{cm}$, $E_{\rm{p}}=1.0\ GeV$, the maximum
reaches at $Z\approx 20\ \rm{cm}$. So we expected a maximum proton efficiency
when $Z_{\rm(top)}=-20\ \rm{cm}$. Parameters of this simulation are listed in
Table 3. $\varphi^{*}$ and $\phi_{\rm{s}}(z)$ was obtained as $Z_{\rm{top}}$
changed. The result is shown in Fig. 6 and Fig. 7.
Fig. 6: Neutron source efficiency $\varphi^{*}$, subcritical multiplication
factor $k_{\rm{s}}$ and effective multiplication factor $k_{\rm{eff}}$ as a
function of $Z_{\rm{top}}$
Fig. 7: Neutron flux $\phi_{\rm{s}}(z)$ as a function of $Z_{\rm{top}}$
In Fig. 6, we find that both $k_{\rm{s}}$ and $\varphi^{*}$ reaches a maximum
value at $Z_{\rm{top}}=-25\ \rm{cm}$. This result is a bit smaller than the
position, -20 cm, we expected. This difference may be explained by the fact
that the distribution of $I_{\rm{r}}(Z)$ is not perfectly symmetric, there is
a ”tail” at high $Z$. In Fig. 7, as $Z_{\rm{top}}$ departs from $-25\
\rm{cm}$, the symmetry axis of $\phi_{\rm{s}}(z)$ departs from $z=0\ \rm{cm}$.
This deviation will increase the leakage of neutrons from the subcritical core
to the reflector and thus decreasing the proton efficiency.
Survival neutrons also influence $\varphi^{*}$. The number of survival
neutrons will decrease when $Z_{\rm{top}}$ becomes more negative and this will
bring a decrease to $\varphi^{*}$. The effect of $I_{\rm{r}}(Z)$ and survival
neutrons are coupled. But when $Z_{\rm{top}}$ becomes more positive, the
effect of survival neutrons becomes smaller. In conclusion, both
$\phi_{\rm{s}}(z)$ and $\varphi^{*}$ support the assumption about target
position. The effect of survival neutrons makes the assumption less persuasive
and will be discussed below.
#### 4.4.2 Effect of radial and backward neutrons on proton efficiency
Change of $R_{\rm{s}}$ will bring a change to the characteristic of radial and
backward neutrons as discussed in Sec. 4.3. These two factors are coupled.
Based on simulations in Sec. 4.3, proton efficiency $\psi^{*}$ of the modified
ADS benchmark was simulated for different $R_{\rm{s}}$. Parameters of these
simulations are listed in Table 2, the result is listed in Table 5. Changing
the thickness of void region brought to a slight change of $k_{\rm{eff}}$.
This change was considered in the calculation of $\psi^{*}$.
Table 5: Proton efficiency varies with $R_{\rm{s}}$, $1\sigma<1.0\%$ Proton
efficiency $R_{\rm{s}}$
---
$R_{\rm{s}}/\rm{cm}$ | 12 | 18 | 24 | 30
| 0.94828 | 0.94886 | 0.94972 | 0.95148
$k_{\rm{eff}}\pm std$ | 0.00026 | 0.00026 | 0.00025 | 0.00026
$\psi^{*}$ | 26.803 | 27.499 | 27.882 | 28.002
We find that: as $R_{\rm{s}}$ increases, $\psi^{*}$ increases slightly. This
result seems paradox to the conclusion that the bigger of target radius, the
smaller of proton efficiency in [12]. In fact,as is seen in Table 4, it is the
trade-off between the soft of radial neutrons and the increase of survival
neutrons. Source efficiency of backward neutrons in inducing fissions is lower
than these radial neutrons, but the rather big number of survival neutrons
neutralizes the negative effect of radial neutrons. This result does indicate
that the effect of backward neutrons is pronounced. When radius of spallation
region increase, number of neutrons produced per beam particle does increase,
but these extra neutrons are not effectively used. Based on this simulation,
it is difficult to draw the conclusion that the radius of spallation region
should be larger; but if we need to increase radius of spallation region to
satisfy other requirements in designing the system, such as decreasing the
damage of high-energy spallation neutrons to structural material, we may try
to make better use of these backward neutrons and avoid severely affecting the
proton efficiency. At present, we may replace the moderator between the beam
pipe and the subcritical core with fuels and increase the radius of spallation
region; or we may design an annular target (ratio between area of top surface
and side surface is smaller than that of a cylinder target) to reduce the
number of backward neutrons.
## 5 Conclusion
In this work, spallation region and subcritical core in ADS are combined to
study effect of the spatial, energy distribution of spallation neutrons. The
characteristic of radial and backward neutrons were mainly discussed.
As for radial neutrons, neutron current surface density $I_{\rm{r}}(Z)$ was
simulated with FLUKA and used to optimize the position of target. An
assumption about this optimization was made and tested. As for the design of
C-ADS, the considerations made in Sec. 4.4.1 may be used to optimize position
of target. As for backward neutrons, survival neutrons were mainly discussed
and found to have a considerable ratio when radius of spallation region
increases. These survival neutrons have a positive effect on the proton
efficiency. If we make better use of these survival neutrons, we may increase
the radius of spallation region to satisfy other requirements in the design of
subcritical core without affecting proton efficiency severely.
## Acknowledgements
Numerical Simulation received the support and help of Supercomputing Center of
USTC. The authors acknowledge the members in the PDSP group of NSRL for their
help and discussion.
## References
* [1] G.S.Bauer, Target Design and Technology for Research Spallation Neutron Sources, Workshop on Technology and Application of Accelerator Driven Systems (ADS),17-28 October 2005
* [2] C.Rubbia, J.A, S.Buono,Conceptual design of a fast neutron operated high power energy amplifier, CERN/AT/95-44(ET), September 29, 1995.
* [3] H. Nifenecker, O. M plan, S. David :Accelerator Driven Subcritical Reactors ,Institute of Physics Publishing,London (2003)
* [4] Antonin Krasa : Spallation Reaction Physics, Chapter 1 Spallation reaction
* [5] A. Ferrari, J. Ranft, and P.R. Sala, ”FLUKA: a multipar-ticle transport code”, CERN-2005-10 (2005), INFN/TC 05/11, SLAC-R-773
* [6] G. Battistoni, S. Muraro, P.R. Sala, F. Cerutti, A. Ferrari,S. Roesler, A. Fasso‘, J. Ranft,”The FLUKA code: Description and benchmarking”,Proceedings of the Hadronic Shower Simulation Workshop 2006,Fermilab 6-8 September 2006, M. Albrow, R. Raja eds.,AIP Conference Proceeding 896, 31-49, (2007)
* [7] Ding SHE, Qi XU, Kan WANG et al., ”RMC1.0 - Development of Monte Carlo Code for Reactor Analysis”, Proceedings of the 18th International Conference on Nuclear Engineering (ICONE18), Xi an, China: May 17-21 (2010).
* [8] Wang, K., Li, Z.G., She D., et al., Progress on RMC - A Monte Carlo Neutron Transport Code for Reactor Analysis”, M&C 2011, Rio de Janeiro, RJ, Brazil, May 8-12 (2011).
* [9] Slessarev I.,Tchistiakov A. IAEA ADS-BENCHMARK Results and Analysis[A]. TCM-Meeting, Madrid, 17-19 September 1997
* [10] X. Ledoux, F. Borne et al.,Spallation Neutron Production by 0.8, 1.2, and 1.6 GeV Protons on Pb Targets, Physical review letters [0031-9007]
* [11] Hesham Shahbunder, Cheol HoPyeon, et al. subcritical multiplication factor and source efficiency in accelerator-driven system, Annals of Nuclear Energy Volume 37, Issue 9, September 2010, Pages 1214 C1222
* [12] Seltborg et al., 2003 P. Seltborg, J. Wallenius, K. Tucek, W. Gudowski,Definition and application of proton source efficiency in accelerator-driven system, Nuclear Science and Engineering, 145 (3) (2003), pp. 390 C399
* [13] Wang Xiang-qi, Luo Huan-li, et al. Distribution transformation by rotating dipole magnetic field and beam optics design for downstream of hurling magnet in C-ADS LTBT, China Physics C(2012)
|
arxiv-papers
| 2012-11-01T04:54:25 |
2024-09-04T02:49:37.465066
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guojun Hu, Hongli Wu, Tian Jing, Xiangqi Wang, Jingyu Tang",
"submitter": "Guojun Hu",
"url": "https://arxiv.org/abs/1211.0093"
}
|
1211.0308
|
${\theta}(\hat{x},\hat{p})-$deformation of the harmonic oscillator in a
$2D-$phase space
M. N. Hounkonnou†,1 D. Ousmane Samary‡,1,2,
E. Baloïtcha∗,1 and , S. Arjika∗∗,1
$1$-International Chair in Mathematical Physics and Applications
(ICMPA-UNESCO Chair), University of Abomey-Calavi,
072B.P.50, Cotonou, Rep. of Benin
$2$-CNRS - Université Lyon 1, Institut Camille Jordan
(Bat. Jean Braconnier, bd du 11 novembre 1918),
F-69622 Villeurbanne Cedex, France
E-mails: †norbert.hounkonnou@cipma.uac.bj , ‡ousmanesamarydine@yahoo.fr,
∗ezinvi.baloitcha@cipma.uac.bj, ∗∗rjksama2008@gmail.com
[Ref-preprint] CIPMA-MPA/014/2012
This work addresses a ${\theta}(\hat{x},\hat{p})-$deformation of the harmonic
oscillator in a $2D-$phase space. Specifically, it concerns a quantum
mechanics of the harmonic oscillator based on a noncanonical commutation
relation depending on the phase space coordinates. A reformulation of this
deformation is considered in terms of a $q-$deformation allowing to easily
deduce the energy spectrum of the induced deformed harmonic oscillator. Then,
it is proved that the deformed position and momentum operators admit a one-
parameter family of self-adjoint extensions. These operators engender new
families of deformed Hermite polynomials generalizing usual $q-$ Hermite
polynomials. Relevant matrix elements are computed. Finally, a $su(2)-$algebra
representation of the considered deformation is investigated and discussed.
Pacs numbers:
Key words: Harmonic oscillator, energy spectrum, q-deformation, Hermite
polynomials, matrix
elements, $su(2)-$algebra
## 1 Introduction
Consider a $2D$ phase space $\mathcal{P}\subset\mathbb{R}^{2}.$ Coordinates of
position and momentum are denoted by $x$ and $p.$ Corresponding Hilbert space
quantum operators $\hat{x}$ and $\hat{p}$ satisfy the following commutation
relation
$\displaystyle[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\theta(\hat{x},\hat{p})$
(1)
where $\theta$ is the deformation parameter encoding the noncommutativity of
the phase space: $\theta(x,p)=1+\alpha x^{2}+\beta
p^{2},\,\,\alpha,\beta\in\mathbb{R}_{+}$ with the uncertainty relation:
$\displaystyle\Delta\hat{x}\Delta\hat{p}\geq\frac{1}{2}\Big{(}1+\alpha(\Delta\hat{x})^{2}+\beta(\Delta\hat{p})^{2}\Big{)}$
(2)
where the parameters $\alpha$ and $\beta$ are real positive numbers.
The motivations for this study derive from a series of works devoted to the
relation (1). Indeed, already in [11] Kempf et al investigated (1) for the
particular case $\alpha=0$ with
$\displaystyle\Delta\hat{x}\Delta\hat{p}\geq\frac{1}{2}(1+\beta(\Delta\hat{p})^{2})$
(3)
and led to the conclusion that the energy levels of a given system can deviate
significantly from the usual quantum mechanical case once energy scales become
comparable to the scale $\sqrt{\beta}.$ Although the onset of this scale is an
empirical question, it is presumably set by quantum gravitational effects. In
another work [8], Kempf, for the same model with $\alpha=0$, led to the
conclusion that the anomalies observed with fields over unsharp coordinates
might be testable if the onset of strong gravity effects is not too far above
the currently experimentally accessible scale about $10^{-18}m$, rather than
at the Planck scale of $10^{-35}m.$ More recently, in [10], it was shown that
similar relation can be applied to discrete models of matter or space-time,
including loop quantum cosmology. For more motivations, see [6], [1] and [5],
but also [7], [13], [12] and [14] and references therein.
In this work, we investigate how such a deformation may affect main
properties, e.g. energy spectrum, of a simple physical system like a harmonic
oscillator.
The paper is organized as follows. In section 2, we introduce a reformulation
of the $\theta(\hat{x},\hat{p})-$deformation in terms of a $q-$deformation
allowing to easily deduce the energy spectrum of the induced deformed harmonic
oscillator. Then it is proved that the deformed position and momentum
operators admit a one-parameter family of self-adjoint extensions. These
operators engender new families of deformed Hermite polynomials generalizing
usual $q-$ Hermite polynomials. Section 3 is devoted to the matrix element
computation. Finally, in section 4, we provide a $su(2)-$algebra
representation of the considered deformation. Section 5 deals with concluding
remarks.
## 2 $q$-like realization
It is worth noticing that such a $\theta(x,p)-$deformation (1) admits an
interesting $q-$like realization via the following re-parameterization of
deformed creation and annihilation operators:
$\displaystyle\hat{b}^{\dagger}=\frac{1}{2}(m_{\alpha}\hat{x}-im_{\beta}\hat{p}),\quad\hat{b}=\frac{1}{2}(m_{\alpha}\hat{x}+im_{\beta}\hat{p})$
(4)
satisfying the peculiar q-Heisenberg commutation relation:
$\displaystyle\hat{b}\hat{b}^{\dagger}-q\hat{b}^{\dagger}\hat{b}=1$ (5)
where the parameter $q$ is written in the form
$\displaystyle q=\frac{1+\sqrt{\alpha\beta}}{1-\sqrt{\alpha\beta}}\geq 1$ (6)
and the quantities $m_{\alpha}$ and $m_{\beta}$ are given by
$\displaystyle
m_{\alpha}=\sqrt{2\alpha\Big{(}\frac{1}{\sqrt{\alpha\beta}}-1\Big{)}},\quad
m_{\beta}=\sqrt{2\beta\Big{(}\frac{1}{\sqrt{\alpha\beta}}-1\Big{)}}.$ (7)
With this consideration, the spectrum of the induced harmonic oscillator
Hamiltonian $\hat{H}=\hat{b}\hat{b}^{\dagger}+\hat{b}^{\dagger}\hat{b}$ is
given by
$\displaystyle\mathcal{E}_{n}=\frac{1}{2}([n]_{q}+[n+1]_{q})$ (8)
where the $q-$number $[n]_{q}$ is defined by $[n]_{q}=\frac{1-q^{n}}{1-q}$.
Let $\mathcal{F}$ be a $q-$ deformed Fock space and
$\\{|n,q\rangle\;|\;n\in\mathbb{N}\bigcup\\{0\\}\\}$ be its orthonormal basis.
The actions of $\hat{b}$, $\hat{b}^{\dagger}$ on $\mathcal{F}$ are given by
$\displaystyle\hat{b}|n,q\rangle=\sqrt{[n]_{q}}|n-1,q\rangle,\;\mbox{ and
}\;\;\hat{b}^{\dagger}|n,q\rangle=\sqrt{[n+1]_{q}}|n+1,q\rangle,\;$ (9)
where $|0,q\rangle$ is a normalized vacuum:
$\displaystyle\hat{b}|0,q\rangle=0,\qquad\langle q,0|0,q\rangle=1.$ (10)
The Hamiltonian operator $\hat{H}$ acts on the states $|n,q\rangle$ to give:
$\hat{H}|n,q\rangle=\mathcal{E}_{n}|n,q\rangle.$
###### Theorem 2.1
The position operator $\hat{x}$ and momentum operator $\hat{p},$ defined on
the Fock space $\mathcal{F},$ are not essentially self-adjoint, but have a
one-parameter family of self-adjoint extensions.
Proof: Consider the following matrix elements of the position operator
$\hat{x}$ and momentum operator $\hat{p}$ :
$\displaystyle<m,q|\hat{x}|n,q>$ $\displaystyle:=$
$\displaystyle<m,q|\frac{1}{m_{\alpha}}(\hat{b}^{\dagger}+\hat{b})|n,q>$ (11)
$\displaystyle=$
$\displaystyle\frac{1}{m_{\alpha}}\sqrt{[n+1]_{q}}\delta_{m,n+1}+\frac{1}{m_{\alpha}}\sqrt{[n]_{q}}\delta_{m,n-1}$
(12) $\displaystyle<m,q|\hat{p}|n,q>$ $\displaystyle:=$
$\displaystyle<m,q|\frac{i}{m_{\beta}}(\hat{b}^{\dagger}-\hat{b})|n,q>$ (13)
$\displaystyle=$
$\displaystyle\frac{i}{m_{\beta}}\sqrt{[n+1]_{q}}\delta_{m,n+1}-\frac{i}{m_{\beta}}\sqrt{[n]_{q}}\delta_{m,n-1}.$
(14)
Setting $x_{n,\alpha}=\frac{1}{m_{\alpha}}\sqrt{[n]_{q}}$ and
$x_{n,\beta}=\frac{1}{m_{\beta}}\sqrt{[n]_{q}}$, then the position operator
$\hat{x}$ and momentum operator $\hat{p}$ can be represented by the two
following symmetric Jacobi matrices, respectively:
$\displaystyle
M_{\hat{x},q,\alpha}=\left(\begin{array}[]{cccccc}0&x_{1,\alpha}&0&0&0&\cdots\\\
x_{1,\alpha}&0&x_{2,\alpha}&0&0&\cdots\\\
0&x_{2,\alpha}&0&x_{3,\alpha}&0&\cdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\end{array}\right)$ (19)
and
$\displaystyle
M_{\hat{p},q,\beta}=\left(\begin{array}[]{cccccc}0&-ix_{1,\beta}&0&0&0&\cdots\\\
ix_{1,\beta}&0&-ix_{2,\beta}&0&0&\cdots\\\
0&ix_{2,\beta}&0&-ix_{3,\beta}&0&\cdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\end{array}\right).$ (24)
The quantity
$|x_{n,\alpha}|=\frac{1}{m_{\alpha}}\Big{|}\frac{1-q^{n}}{1-q}\Big{|}^{1/2}$
is not bounded since $q>1$ by definition, and
$\lim\limits_{n\rightarrow\infty}|x_{n,\alpha}|=\infty.$ Considering the
series $y_{\alpha}=\sum\limits_{n=0}^{\infty}1/x_{n,\alpha},$ we get
$\displaystyle\overline{\lim_{n\to\infty}}\left(\frac{1/x_{n+1}}{1/x_{n}}\right)=q^{-1/2}<1$
and, hence, the series $y_{\alpha}$ converges. Besides, as the quantity
$q^{-1}+q>2,$
$\displaystyle
0<x_{n+1,\alpha}x_{n-1,\alpha}=\frac{1}{m_{\alpha}^{2}(1-q)}\Big{[}1-q^{n}(q^{-1}+q)+q^{2n}\Big{]}^{1/2}<x_{n,\alpha}^{2}$
(25)
Hence, the Jacobi matrices in (19) and (24) are not self-adjoint (Theorem
1.5., Chapter VII in Ref. [2]). The deficiency indices of these operators are
equal to $(1,1)$. One concludes that the position operator $\hat{x}$ and the
momentum operator $\hat{p}$ are no longer essentially self-adjoint but have
each a one-parameter family of self-adjoint extensions instead. This means
that their deficiency subspaces are one-dimensional. $\square$
Besides, in this case, the deficiency subspaces $N_{z}$, $Imz\neq 0,$ are
defined by the generalized vectors
$|z\rangle=\sum\limits_{0}^{\infty}P_{n}(z)||n,q\rangle$ such that [2], [3]:
$\displaystyle\sqrt{[n]_{q}}P_{n-1}(z)+\sqrt{[n+1]_{q}}P_{n+1}(z)=zP_{n}(z)$
(26)
with the initial conditions $P_{-1}(z)=0,\;\;P_{0}(z)=1$.
* •
In the position representation, the states $|x,q>$ such that
$\displaystyle\hat{x}|x,q>=x|x,q>,\mbox{
and}\,\,|x,q>=\sum_{n=0}^{\infty}P_{n,q}(x)|n,q>,$ (27)
transforms the relation (26) into
$\displaystyle m_{\alpha}xP_{n,q}(x)$ $\displaystyle=$
$\displaystyle\sqrt{[n+1]_{q}}P_{n+1,q}(x)+\sqrt{[n]_{q}}P_{n-1,q}(x)$ (28)
$\displaystyle n$ $\displaystyle=$ $\displaystyle
0,1,\ldots;\;P_{-1,q}(x)=0,\;P_{0,q}(x)=1$ (29)
giving
$\displaystyle
2\gamma(x,q)P_{n,q}\Big{(}\frac{2\gamma(x,q)}{(1-q)^{1/2}m_{\alpha}}\Big{)}$
$\displaystyle=$
$\displaystyle(1-q^{n+1})^{\frac{1}{2}}P_{n+1,q}\Big{(}\frac{2\gamma(x)}{(1-q)^{1/2}m_{\alpha}}\Big{)}$
(30) $\displaystyle+$
$\displaystyle(1-q^{n})^{\frac{1}{2}}P_{n-1,q}\Big{(}\frac{2\gamma(x,q)}{(1-q)^{1/2}m_{\alpha}}\Big{)}$
(31)
where $2\gamma(x,q)=(1-q)^{1/2}m_{\alpha}x.$
Setting
$\widetilde{P}_{n,q}(\gamma(x,q))=P_{n,q}\Big{(}\frac{2\gamma(x,q)}{(1-q)^{1/2}m_{\alpha}}\Big{)}$,
the equation (30) can be re-expressed as
$\displaystyle 2\gamma(x,q)\widetilde{P}_{n,q}(\gamma(x,q))$ $\displaystyle=$
$\displaystyle(1-q^{n+1})^{\frac{1}{2}}\widetilde{P}_{n+1,q}(\gamma(x,q))$
(32) $\displaystyle+$
$\displaystyle(1-q^{n})^{\frac{1}{2}}\widetilde{P}_{n-1,q}(\gamma(x,q)).$ (33)
Finally, putting $(q;q)_{n}^{1/2}\widetilde{P}_{n,q}(\gamma(x,q))=H_{n,q}(x),$
the formula (32) recalls the recurrence relation satisfied by $q-$Hermite
polynomials:
$\displaystyle 2xH_{n,q}(x)=H_{n+1,q}(x)+(1-q^{n})H_{n-1,q}(x)$ (34)
where $(q;q)_{n}=(1-q)(1-q^{2})\cdots(1-q^{n}).$
* •
In the momentum representation, the state $|p,q>$ such that
$\displaystyle\hat{p}|p,q>=p|p,q>,\mbox{
and}\,\,|p,q>=\sum_{n=0}^{\infty}Q_{n,q}(p)|n,q>$ (35)
leads to the following reccurence relation between functions $Q_{n}(x,q):$
$\displaystyle im_{\beta}pQ_{n,q}(p)$ $\displaystyle=$
$\displaystyle\sqrt{[n+1]_{q}}Q_{n+1,q}(p)-\sqrt{[n]_{q}}Q_{n-1,q}(p)$ (36)
$\displaystyle n$ $\displaystyle=$ $\displaystyle
0,1,\ldots;\;Q_{-1,q}(p)=0,\;Q_{0,q}(p)=1.$ (37)
This equation can be also re-expressed as
$\displaystyle\tilde{\gamma}(\tilde{p},q)Q_{n,q}(p)=(1-q^{n+1})^{\frac{1}{2}}Q_{n+1,q}(p)-(1-q^{n})^{\frac{1}{2}}Q_{n-1,q}(p)$
(38)
where $\tilde{\gamma}(\tilde{p},q)=(1-q)^{1/2}m_{\beta}\tilde{p},$
$\tilde{p}=ip.$
Setting
$\widetilde{Q}_{n,q}(\tilde{\gamma}(\tilde{p},q))=Q_{n,q}\Big{(}\frac{2\tilde{\gamma}(\tilde{p},q)}{(1-q)^{1/2}m_{\alpha}}\Big{)}$,
then the equation (38) yields
$\displaystyle
2\tilde{\gamma}(\tilde{p},q)\widetilde{Q}_{n,q}(\tilde{\gamma}(\tilde{p},q))$
$\displaystyle=$
$\displaystyle(1-q^{n+1})^{\frac{1}{2}}\widetilde{Q}_{n+1,q}(\tilde{\gamma}(\tilde{p},q))$
(40)
$\displaystyle-(1-q^{n})^{\frac{1}{2}}\widetilde{Q}_{n-1,q}(\tilde{\gamma}(\tilde{p},q)).$
Letting
$(q;q)_{n}^{1/2}\widetilde{Q}_{n,q}(\tilde{\gamma}(\tilde{p},q))=H_{n,q}(ip),$
we arrive at the recurrence relation satisfied by the complex $q-$polynomials
$H_{n,q}(ip)$ given by
$\displaystyle 2ipH_{n,q}(ip)=H_{n+1,q}(ip)-(1-q^{n})H_{n-1,q}(ip).$ (41)
###### Remark 2.1
The following is worthy of attention:
1. (i)
In the $x-$space where the momentum operator is defined by the relation
$\displaystyle\hat{p}:=-i\theta(\hat{x},\hat{p})\partial_{x},$ (42)
any function $\Psi_{q}(x)$ in $x-$representation can be expressed in terms of
its analog $\Psi_{q}(p)$ in $p-$representation by the relation
$\displaystyle\Psi_{q}(x)=\int_{-\infty}^{\infty}dp\,\exp\Big{(}\frac{ip}{\alpha\sigma(p)}\arctan\frac{x}{\sigma(p)}\Big{)}\Psi_{q}(p),$
(43)
where $\sigma(p)=\sqrt{p^{2}+\frac{1}{\alpha}}$. Defining the Hilbert space
inner product as
$\displaystyle<f,g>=\int\,\frac{dx}{\theta(\hat{x},\hat{p})}\bar{f}_{q}(x)g_{q}(x)$
(44)
one can readily prove that $\hat{p}$ reverts the property of a Hermitian
operator.
2. (ii)
Analogously, in the $p-$space
$\displaystyle\hat{x}:=i\theta(\hat{x},\hat{p})\partial_{p}$ (45)
and
$\displaystyle\Psi_{q}(p)=\int_{-\infty}^{\infty}dx\,\exp\Big{(}\frac{-ix}{\alpha\sigma(x)}\arctan\frac{p}{\sigma(x)}\Big{)}\Psi_{q}(x).$
(46)
The appropriate inner product, in the momentum space, rendering the operator
$\hat{x}$ Hermitian is defined as
$\displaystyle<f,g>=\int\,\frac{dp}{\theta(\hat{x},\hat{p})}\bar{f}_{q}(p)g_{q}(p)$
(47)
with the condition
$\lim\limits_{x\rightarrow-\infty}\Psi_{q}(x)=\lim\limits_{x\rightarrow\infty}\Psi_{q}(x)=0.$
## 3 Matrix elements
From the natural actions of $q-$deformed position operator $\hat{x}$ and
momentum operator $\hat{p}$ on the basis vectors $|n,q>\in\mathcal{F}:$
$\displaystyle\hat{x}|n,q>=\frac{1}{m_{\alpha}}(\hat{b}+\hat{b}^{\dagger})|n,q>,\quad\hat{p}|n,q>=\frac{i}{m_{\beta}}(\hat{b}^{\dagger}-\hat{b})|n,q>$
(48)
we immediately deduce the matrix elements
$\displaystyle<m,q|\hat{b}^{{\dagger}l}\hat{b}^{r}|n,q>=\sqrt{\frac{\Gamma_{q}(n+1)\Gamma_{q}(n-r+l+1)}{\Gamma_{q}(n-r+1)\Gamma_{q}(n-r+1)}}\delta_{m,n-r+l}$
(49)
$\displaystyle<m,q|\hat{b}^{r}\hat{b}^{{\dagger}l}|n,q>=\sqrt{\frac{\Gamma_{q}(n+l+1)\Gamma_{q}(n+l+1)}{\Gamma_{q}(n+1)\Gamma_{q}(n-r+l+1)}}\delta_{m,n-r+l}$
(50)
where $\Gamma_{q}(.)$ is the $q$-Gamma function. Denoting by $:\,:$ the normal
ordering, then the expectation value of normal ordering product of
$\hat{x}^{l}\hat{p}^{r}$ can be computed by the following relation:
$\displaystyle<m,q|:\hat{x}^{l}\hat{p}^{r}:|n,q>=\frac{i^{r}}{m_{\alpha}^{l}m_{\beta}^{r}}\sum_{s=0}^{l}\sum_{t=0}^{r}C_{l}^{s}C_{r}^{t}<m,q|\hat{b}^{{\dagger}l-s+t}\hat{b}^{s+r-t}|n,q>$
(51)
which can be given explicitly by using relation (49). Then it becomes a matter
of computation to determine the basis operators in terms of $\hat{b}$ and
$\hat{b}^{\dagger}$ as follows:
$\displaystyle|m,q><n,q|=:\frac{\hat{b}^{{\dagger}m}}{\sqrt{[m]_{q}!}}e^{-\hat{b}^{\dagger}\hat{b}}\frac{\hat{b}^{n}}{\sqrt{[n]_{q}!}}:.$
(53)
## 4 $su(2)-$algebra representation
Turning back to the standard expression of the harmonic oscillator Hamiltonian
operator, i.e. $\hat{H}=\hat{a}^{\dagger}\hat{a},$ such that
$\displaystyle\hat{a}=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p}),\,\,\,\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}(\hat{x}-i\hat{p}),$
(54)
we get explicitly
$\displaystyle\hat{H}=\frac{1}{2}\Big{[}(1+\alpha)\hat{x}^{2}+(1+\beta)\hat{p}^{2}+1\Big{]}$
(55)
giving the simpler form $\hat{H}=\frac{1}{2}$ when $\alpha=-1$ and $\beta=-1.$
From (42) and (45), the Hamiltonian $H$ can be considered as non-local and we
can define
$\displaystyle\hat{H}^{\mbox{loc}}:=\hat{H}(\theta,\partial_{x}\theta,\partial_{x}^{2}\theta\cdots,x,\partial_{x},\partial_{x}^{2},\cdots,\alpha,\beta)$
(56)
with
$\displaystyle\theta,\partial_{x}\theta,\cdots=f(\theta,\partial_{x}\theta,\partial_{x}^{2}\theta\cdots,x,\partial_{x},\partial_{x}^{2},\cdots,\alpha,\beta).$
(57)
adding some nonlinearity to the Hamiltonian operator nonlocality.
Assume the parameters $\alpha$ and $\beta$ satisfy the condition:
$|\alpha|<<1,|\beta|<<1$ and put $\tilde{\alpha}=\alpha$ and
$\tilde{\beta}=-\beta.$ Then $\hat{x},\,\hat{p},\,\theta$ can be viewed as the
elements of $su(2)-$algebra, i.e.
$\displaystyle[\hat{x},\hat{p}]=i\theta,\,\,\,[\hat{p},\theta]=i\alpha\\{\hat{x},\theta\\}=i\tilde{\alpha}\hat{x},\,\,\,[\theta,\hat{x}]=-i\beta\\{\hat{p},\theta\\}=i\tilde{\beta}\hat{p}.$
(58)
Let $\hat{\overrightarrow{J}}:=(\hat{x},\hat{p},\theta)$ be the angular
momentum such that there exist states $|j,m>\in\mathcal{F}$ satisfying the
condition $<j,m|j,m^{\prime}>=\delta_{mm^{\prime}}$. Define the operators
$\hat{J}_{+}$ and $\hat{J}_{-}$ by
$\displaystyle\hat{J}_{+}:=\frac{1}{\beta}\hat{x}+\frac{i}{\alpha}\hat{p},\,\,\,\,\hat{J}_{-}:=\frac{1}{\beta}\hat{x}-\frac{i}{\alpha}\hat{p}.$
(59)
###### Proposition 4.1
There exists an arbitrary number $\nu$ such that
$\displaystyle\hat{J}_{-}|j,m>=C_{-}(m,j)|j,m-\nu>,\,\,\,\,\hat{J}_{+}|j,m>=C_{+}(m,j)|j,m+\nu>,$
(60) $\displaystyle\theta|j,m>=f(m,j)|j,m>$ (61)
where $C_{-}(m,j),$ $C_{+}(m,j)$ and $f(m,j)$ are three constants depending on
$j$ and $m$.
The parameters $j$ and $m$ depend on $\alpha$ and $\beta.$ The unitary
representation of $su(2)-$ algebra, $\\{|j,m>,\,\,j\in\mathbb{N},\,\,-j\leq
m\leq j\\},$ is infinite dimensional. The operators
$\\{\hat{x},\hat{p},\theta\\}$ act on the Fock space
$\mathcal{H}=\\{|j,m>/m\in\mathbb{N}\cup\\{0\\}\\}$ following (60). Note that
$\theta$ and
$\hat{\overrightarrow{J}^{2}}=(1+2\alpha)\hat{x}^{2}+(1+2\beta)\hat{p}^{2}+1$
commute. Therefore, $\hat{\overrightarrow{J}^{2}}$ and $\hat{H}$ commute too.
Besides, the following commutation relations are in order:
$\displaystyle[\theta,\hat{J}_{+}]=\hat{x}+i\hat{p},\,\,\,\,[\theta,\hat{J}_{-},]=-(\hat{x}-i\hat{p}).$
(62)
In the interesting particular case where $\alpha=\beta$, the relations (62)
are reduced to
$\displaystyle[\theta,\hat{J}_{+}]=\alpha\hat{J}_{+},\,\,\,\,[\theta,\hat{J}_{-},]=-\alpha\hat{J}_{-},\,\,\,\,[\hat{J}_{+},\hat{J}_{-}]=2\alpha^{-2}\theta.$
(63)
Taking $f(m,j)=m$ yields the condition
$\displaystyle\theta\hat{J}_{+}|j,m>=(m+\alpha)\hat{J}_{+}|j,m>,\,\,\,\theta\hat{J}_{-}|j,m>=(m-\alpha)\hat{J}_{+}|j,m>.$
(64)
Besides, we have
$\displaystyle\hat{J}_{+}|j,m>=C_{+}|j,m+\alpha>,\,\,\,\hat{J}_{-}|j,m>=C_{-}|j,m-\alpha>$
(65)
where
$\displaystyle C_{+}=\sqrt{(j-m)(j+m+\alpha)},\quad
C_{-}=\sqrt{(j+m)(j-m+\alpha)}.$ (66)
The eigenfunctions of the Hamiltonian $\hat{H}$ in the position and momentum
representations are given, respectively, by
$\displaystyle\Psi_{j,m}(x)=<x|j,m>,\,\,\,\,\Psi_{j,m}(p)=<p|j,m>$ (67)
solution of the equation
$\displaystyle\hat{H}\Psi_{j,m}(x)=\frac{\alpha^{2}}{2}\sqrt{(j+m)(j-m)(j+m+\alpha)(j-m+\alpha)}\Psi_{j,m}(x)$
(68)
easily obtainable by solving the eigenvalue problem for the Casimir operator
$\hat{J}_{+}\hat{J}_{-}$. Furthermore, we get
$\displaystyle\hat{J}^{2}\Psi_{j,m}(x)=2\alpha^{2}\sqrt{(j+m)(j-m)(j+m+2\alpha)(j-m+2\alpha)}\Psi_{j,m}(x).$
(69)
## 5 Concluding remarks
In work, we have introduced a reformulation of the
${\theta}(\hat{x},\hat{p})-$deformation in terms of a $q-$deformation allowing
to easily deduce the energy spectrum of the induced deformed harmonic
oscillator. Then we have proved that the deformed position and momentum
operators admit a one-parameter family of self-adjoint extensions. These
operators have engendered new families of deformed Hermite polynomials
generalizing usual $q-$ Hermite polynomials. We have also computed relevant
matrix elements. Finally, a $su(2)-$algebra representation of the considered
deformation has been investigated and discussed.
### Acknowledgment
This work is partially supported by the ICTP through the OEA-ICMPA-Prj-15. The
ICMPA is in partnership with the Daniel Iagolnitzer Foundation (DIF), France.
MNH expresses his gratefulness to Professor A. Odzijewicz and all his staff
for their hospitality and the good organization of the Workshops in Geometric
Methods in Physics.
## References
* [1] D. Amati, M. Ciafaloni, G. Veneziano.: (1989), Can spacetime be probed below the string size Phys. lett. B 216 41.
* [2] Ju. M. Berezanskií, Expansions in Eigenfunctions of Selfadjoint Operators, (Amer. Math. Soc., Providence, Rhode Island, 1968).
* [3] Burban I. M.: Generalized $q-$deformed oscillators, $q-$Hermite polynomials, generalized coherent states.
* [4] Doplicher, S. Fredenhagen, K. and E. Roberts, J.: (1995), The Quantum Structure of Spacetime at the Planck Scale and Quantum Fields, Comm. Math. Phys. 172, pp. 187-220.
* [5] L. J. Garay.: (1995) Models of neutrino masses and mixings, Int. J. Mod. Phys. A10, 145
* [6] D. J. Gross, P. F. Mende.: (1988)The Minimal Length in String Theory, Nucl. Phys. B303, 407.
* [7] Hirshfeld, A.C. and Henselder, P.: (2002), Deformation quantization in the teaching of quantum mechanics, American Journal of Physics, 70 (5) pp. 537–547, May.
* [8] Kempf, A.: (2000), A Generalized Shannon Sampling Theorem, Fields at the Planck Scale as Bandlimited Signals, Phys. Rev. Lett. 85, pp. 2873 [e-print hep-th/9905114].
* [9] Kempf, A.: (1998), On the Structure of Space-Time at the Planck Scale , [e-print hep-th/9810215].
* [10] Kempf, A.: (2011), Generalized uncertainty principles and localization in discrete space, [e-print hep-th/1112.0994].
* [11] Kempf, A. Mangano, G. and Mann, R.B.: (1995), Hilbert Space Representation of the Minimal Length Uncertainty Relation, J. Phys. D. 52, pp. 1108.
* [12] Lay Nam Chang, Dlordje Minic, Naotoshi Okamura and Tatsu Takeuchi : Exact solutions of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations, [arXiv:hep-th/0111181].
* [13] Quesne C. and Tkachuk V. M.: Lorentz-covariant deformed algebra with minimal length, [arXiv:hep-th/0612093].
* [14] Sketesko M. M and Tkachuk V. M.: Perturbative hydrogen-atom spectrum in deformed space with minimal length, [arXiv:hep-th/0603042].
|
arxiv-papers
| 2012-11-01T21:05:50 |
2024-09-04T02:49:37.475978
|
{
"license": "Public Domain",
"authors": "M. N. Hounkonnou, D. Ousmane Samary, E. Baloitcha and S. Arjika",
"submitter": "Dine Ousmane Samary",
"url": "https://arxiv.org/abs/1211.0308"
}
|
1211.0324
|
# PHENIX Highlights
Takao Sakaguchi (for the PHENIX111A list of members of the PHENIX
Collaboration and acknowledgements can be found at the end of this
issue.Collaboration) Brookhaven National Laboratory, Physics Department,
Upton, NY 11973, USA
###### Abstract
PHENIX reports on electromagnetic and hadronic observables in large data sets
of p+p, d+Au and Au+Au collisions at various cms energies. Initial state
effects in cold nuclear matter are quantified by centrality dependent
$\pi^{0}$, $\eta$, reconstructed jets and $\psi\prime$ measurements. Using the
first run of the new EBIS ion source at RHIC, we report first results for
particle flow ($v_{1}$ and $v_{2}$) and quarkonium production in U+U and Cu+Au
collisions. Hot matter created in Au+Au is characterized using event-plane
dependent HBT and dielectrons. Parton-medium interactions are investigated
using high $p_{T}$ single hadrons, $\gamma$-hadron correlations and heavy
flavor decay electrons identified with the newly installed VTX detector.
††journal: Nuclear Physics A
## 1 Introduction
RHIC experiments have demonstrated that quark-gluon plasma (QGP) is created in
Au+Au collisions. Subsequent experimental goals focus on characterizing the
QGP, and exploring the phase transition and a possible critical point by
varying the colliding nuclei and collision energy. We present the latest
results from the PHENIX experiment at RHIC. The results are categorized
according to time within the collision: initial state effects, hot matter
dynamics, and parton-medium interactions.
## 2 The baseline for QGP properties
Initial state effects on the primordial hard scatterings which produce hard
probes of the hot dense medium reflect parton distribution functions in
nucleons and nuclei, as well as possible energy loss or bound state
dissociation in cold matter. p+p collisions provide the baseline, while d+Au
collisions reflect additional cold nuclear matter (CNM) effects arising from
the presence of a nucleus in the collision. Direct photon spectra in d+Au
collisions show little or no modification compared to expectations from p+p
collisions, though an isospin effect is consistent with the data [1, 2]. On
the other hand, hadronic probes display centrality-dependent CNM effects.
In Fig. 1, the nuclear modification factors ($R_{dA}$) for $\pi^{0}$, $\eta$
and fully-reconstructed jets in d+Au collisions at $\sqrt{s_{NN}}$=200 GeV are
shown [3].
Figure 1: Nuclear modification factors ($R_{dA}$) for $\pi^{0}$, $\eta$ and
fully-reconstructed jets in d+Au collisions.
These three probes result from different analysis techniques, but show
remarkable consistency. While a slight suppression is found in the most
central collisions, there is significant enhancement at high $p_{T}$ in the
peripheral collisions; such a feature is not predicted by available models. We
note that centrality is determined in the data by the particle multiplicity in
the beam-beam counter (BBC) at 3.1$<|\eta|<$3.9, while the models use impact
parameters. Therefore, the centrality classes between data and models may
differ somewhat, particularly for events with high $p_{T}$ hadrons or jets
($p_{T}>$10 GeV).
Quarkonia offer key probes of initial state effects. The high statistics 2008
data enable measurement of $\psi\prime$ production as a function of collision
centrality in d+Au at $\sqrt{s_{NN}}$=200 GeV. The dielectron mass spectrum in
the $J/\psi$ and $\psi\prime$ region is shown in Fig. 3 [3].
Figure 2: $\psi\prime$ peak seen in dielectron mass spectra in minimum bias
d+Au collisions.
Figure 3: $R_{dA}$ of $\psi\prime$ and $J/\psi$ in d+Au collisions as a
function of centrality. $\psi\prime$ is more suppressed than $J/\psi$.
The ratio of $\psi\prime/(J/\psi)$ is $\sim 0.8$ %, smaller than in p+p (2.1
%). The centrality dependence of the ratio is shown in Fig. 3. The yield of
$J/\psi$ is somewhat suppressed in d+Au collisions compared to p+p, but
$\psi\prime$ is more heavily suppressed. The relative suppression at FNAL
fixed target energy can be reasonably well explained by considering the time
the precursor $c\overline{c}$ state spends in the nucleus. However, this
explanation completely fails at RHIC energy.
## 3 New collision systems for QGP characterization
In 2012, first data were taken with the EBIS ion source, providing new ion
species at RHIC.
### 3.1 U+U collisions
Uranium is a highly deformed nucleus. Therefore, selecting collisions where
two uranium nuclei are aligning with their long axes head-on (tip-tip) should
access higher energy density at RHIC energy. PHENIX collected data in U+U
collisions at $\sqrt{s_{NN}}$=193 GeV. First results show that the Bjorken
energy density for 0-2 % centrality (tip-tip enriched events) is $\sim$30 %
higher than the most central Au+Au collisions at 200GeV [4]. The elliptic flow
of identified charged pions and protons in U+U collisions are compared to that
in Au+Au at 0-10 % centrality in Fig. 5 [5].
Figure 4: Elliptic flow of charged pions and protons in (a) Au+Au and (b) U+U
collisions at 0-10 % centrality.
Figure 5: Elliptic flow of charged pions and protons in (a) 0-2 % and (b) 6-10
% centrality in U+U collisions.
For pions in U+U, $v_{2}$ is flatter with $p_{T}$ for $p_{T}<$1 GeV,
reminiscent of observations in Pb+Pb at LHC [6]. This can be interpreted as a
consequence of very strong radial flow for higher energy density matter. The
finer centrality classes shown in Fig. 5 show that this flattening is most
pronounced in 0-2 %, where tip-tip events represent a larger fraction of the
total, and is nearly nonexistent for 6-10 % centrality. It is intriguing to
see this phenomenon at RHIC, as the energy density reached at LHC is $\sim$3
times higher than at RHIC.
### 3.2 Cu+Au collisions
Figure 6: $v_{1}$ of charged pions in Cu+Au collisions.
Figure 7: $v_{2}$ of charged pions and protons in Cu+Au collisions.
In Cu+Au collisions, one can select events in which the Cu is fully contained
within the Au nucleus. Furthermore, the pressure gradient developed should be
asymmetric, resulting in an intrinsic triangularity in the collective motion.
This triangularity generates a $v_{3}$ that is geometric, rather than
fluctuation driven. PHENIX has already analyzed $\sim$20 % of the 2012 Cu+Au
data. The shower-max detector in the zero-degree calorimeters is used to
determine the event plane (reaction plane $\Psi_{1}$), allowing a measurement
of $v_{1}$. We defined the Au-spectator going side as positive $\Psi_{1}$.
Fig. 7 and 7 show $v_{1}$ and $v_{2}$ for identified pions and protons. A
sizable positive $v_{1}$ is observed; the sign and magnitude of $v_{1}$ differ
from predictions of the the AMPT model (HIJING+parton cascade) [7], although
AMPT describes symmetric collision systems very well.
Utilizing the muon arms at 1.2$<|\eta|<$2.2, PHENIX measured $J/\psi$ spectra
in Cu+Au collisions via the di-muon decay channel. This allowed determination
of $J/\psi$ $R_{AA}$ in Cu+Au, and comparison to the the $J/\psi$ suppression
observed in Cu+Cu and Au+Au collisions. The results and comparison are shown
in Fig. 8 [8].
Figure 8: $R_{AA}$ of $J/\psi$ in Cu+Au, Cu+Cu and Au+Au collisions.
The suppression of the $J/\psi$ yield as a function of $N_{part}$ on the Au-
going side fits well with the $N_{part}$ dependence in Au+Au and Cu+Cu
collisions. However, on the Cu-going side, the suppression is significantly
stronger. The difference of the nuclear PDFs in Cu and Au nuclei partly
explains this trend, but additional final state effects may need to be
considered in order to understand the magnitude on the Cu-going side.
## 4 Characterizing hot dense matter by new probes
Two boson correlations as pioneered by Hanbury-Brown and Twiss (colloquially
known as HBT correlations in our field) provide information on the space-time
evolution of the particle emission source. PHENIX has measured the angular
dependence of HBT radii with respect to the (second order) event plane
($R^{2}_{s}(\Delta\phi)=R^{2}_{s,0}+2\times\Sigma
R^{2}_{s,n}cos(n\Delta\phi)$). This analysis shows that the eccentricity,
defined as $\epsilon_{HBT}=2R^{2}_{s,2}/R^{2}_{s,0}$, obtained from the Kaon
HBT correlations is consistent with the initial eccentricities. This implies
early freezeout of kaons from the expanding hadronic phase late in the
collision.
PHENIX studied the HBT radii with respect to the 3rd order event plane for the
first time. Fig. 9 shows $R_{s}$ and $R_{o}$ for charged pions as a function
of the emission angle with respect to the 2nd and 3rd order event plane in
0-10 % in Au+Au collisions [9].
Figure 9: Angular dependence of HBT radii ($R_{s}$ and $R_{o}$) with respect
to 2nd ($\Psi_{2}$) and 3rd ($\Psi_{3}$) order event plane in 0-10 % Au+Au
collsion at $\sqrt{s_{NN}}$=200 GeV.
The average radii are 10 and 5 for $\Psi_{2}$ and $\Psi_{3}$, respectively.
While $R_{s}$ shows only a weak dependence on orientation with respect to the
event plane, $R_{o}$ indicates a clear temporal variation as a function of
angle for both the $\Psi_{2}$ and $\Psi_{3}$ event planes. Moreover, the
magnitude of the oscillation for $\Psi_{3}$ is almost the same as $\Psi_{2}$.
Since $R_{o}$ includes the emission duration in addition to the spatial
homogeneity length in the transverse direction, this measurement reflects the
space-time evolution of ellipticity and triangularity of the collision system.
Turning to thermal radiation, we report on a unique and penetrating probe of
thermodynamic quantities. Measuring thermal radiation gives access, for
example, to the temperature of the hot dense medium produced in the collision.
Using the Hadron Blind Detector (HBD) installed in 2010, PHENIX performed a
new measurement of thermal di-electron radiation. In all bins from peripheral
to semi-central, the new result is consistent with that from the 2004 data.
The most central bin is still being analyzed. For further details, see [10].
## 5 Detailed study of energy loss of partons
### 5.1 $\gamma$-h correlation
Figure 10: Ratio of per-trigger away-side hadron yield in 200 GeV Au+Au and
p+p $\gamma$-hadron measurements as a function of $\xi$. The legend identifies
the three different integrating angular ranges.
$\gamma$-hadron correlations are often considered a ”golden channel” to
evaluate the energy loss of partons in the medium. Since direct photons
reflect the momentum of the original scattered partons, they tag the initial
momentum of the scattered parton that traverses the medium. Consequently one
can deduce the energy loss via $\delta p_{T}=p_{T}(\gamma)-p_{T}(hadrons)$.
PHENIX has measured the hadron yield opposing direct photons as a function of
$\xi$($\equiv-ln(z_{T})$) for the 2007 and 2010 Au+Au data sets together. The
results are shown in Fig. 10 [11]. As the angular range in which the away-side
hadrons are integrated is increased from $|\delta\phi|>5\pi/6$ to
$|\delta\phi|>\pi/2$, the per-trigger away-side yield increases compared to
p+p collisions. The increase is in the larger $\xi$ or small $z_{T}$ region,
implying that the softer away-side jet fragments are distributed over a wider
angle than fragments carrying a large fraction of the jet energy. This is
expected from medium-enhanced splitting, and is qualitatively consistent with
the angular broadening observed in hadron-hadron correlations at RHIC and the
jet shape broadening observed at LHC.
### 5.2 High $p_{T}$ $\pi^{0}$ to determine fractional momentum shift
Using the large 2007 Au+Au data set at $\sqrt{s_{NN}}$ = 200 GeV and a new
algorithm to correct for shower merging, PHENIX extended the $p_{T}$ range of
$\pi^{0}$ spectra [12]. Fig. 11 compares $\pi^{0}$ $R_{AA}$ in 200 GeV Au+Au
collisions from RHIC and charged hadron $R_{AA}$ measured in 2.76 TeV Pb+Pb at
the LHC (ALICE experiment) [13]. For both centralities and over the entire
$p_{T}$ range shown, the two data sets are rather similar.
Figure 11: Comparison of $R_{AA}$ of $\pi^{0}$ from PHENIX at RHIC and charged
hadrons from ALICE experiments at LHC. Two data sets from ALICE experiments
correspond to two different p+p references.
However, this does not necessarily mean that the energy loss of the partons is
the same, as the spectral shapes differ. PHENIX determines the average
fractional momentum shift ($S_{loss}$) of high $p_{T}$ $\pi^{0}$ to estimate
the average fractional energy loss of the initial parton. $S_{loss}$ is
defined as $\delta p_{T}/p_{T}$, where $\delta p_{T}$ is the difference of the
momentum in p+p collisions ($p_{\rm T,pp}$) and Au+Au ($p_{\rm T,AuAu}$);
$p_{T}$ in the denominator is $p_{\rm T,pp}$. The calculation method is
schematically depicted in Fig. 13.
Figure 12: Demonstration of calculating the fractional momentum loss
($S_{loss}$).
Figure 13: $S_{loss}$ for PHENIX $\pi^{0}$ and ALICE charged hadrons.
First, the $\pi^{0}$ cross-section in p+p ($f(p_{\rm T})$) is scaled by
$T_{AA}$ corresponding to the centrality selection of the Au+Au data
($g(p_{\rm T})$). Next, the scaled p+p cross-section ($T_{\rm AA}f(p_{\rm
T})$) is fit with a power-law function ($h(p_{\rm T})$). Third, a scaled p+p
point closest in yield ($p_{\rm T,pp}^{\prime}$) to the Au+Au point of
interest ($p_{\rm T,AuAu}$) is shifted using the fit function as
$\displaystyle T_{\rm AA}f(p_{\rm T,pp})=h(p_{\rm T,pp})/h(p_{\rm
T,pp}^{\prime})\times T_{\rm AA}f(p_{\rm T,pp}^{\prime}),$ (1)
where $p_{\rm T,pp}$ is chosen to satisfy the relation
$\displaystyle T_{\rm AA}f(p_{\rm T,pp})=g(p_{\rm T,AuAu}).$ (2)
The $\delta p_{T}$ is calculated as $p_{\rm T,pp}-p_{\rm T,AuAu}$. For
obtaining $S_{loss}$, the $\delta p_{\rm T}$ is divided by the $p_{\rm T,pp}$.
We show the comparison of $S_{loss}$ for PHENIX $\pi^{0}$’s in Au+Au and ALICE
charged hadrons in Pb+Pb in Fig. 13. We took the corresponding Pb+Pb and p+p
data for black points in Fig. 11 for calculating the $S_{loss}$ for ALICE
charged hadrons. The $S_{loss}$ for the ALICE charged hadrons is found to be
$\sim$25 % higher than that for the PHENIX $\pi^{0}$’s. PHENIX has also
measured $S_{loss}$ in Au+Au at 39 and 62.4 GeV cms energy. The change in
$S_{loss}$ reaches approximately a factor of 4 from 200 GeV to 39 GeV [14,
15].
## 6 Heavy flavor electrons
The suppression of non-photonic electron yields was found to be nearly as
large as that of $\pi^{0}$’s. The question has been whether the suppression
arises from the suppression of charm and/or bottom quarks. Theoretical
predictions about this differ greatly. Consequently, PHENIX constructed a
micro vertex detector to separate electrons from charm and bottom decays. This
detector, the VTX, consists of 2 layers each of strip-pixel and pixel silicon
detectors. The measurement of the distance of closest approach (DCA) of
electrons to the collision vertex allows separation according to the parent
hadrons. Fig. 15 shows a DCA distribution measured in the VTX detector in p+p
collisions.
Figure 14: DCA distribution in p+p collisions for 2$<p^{e}_{T}<$2.5 GeV/$c$,
and its deconvolution to contributions from parent hadrons.
Figure 15: Ratio of electron yields from bottom quarks to bottom and charm
quarks in p+p collisions.
The distribution was deconvoluted to determine the relative contributions of
the heavy flavor parent hadrons. Fig. 15 shows the ratio of electron yields
from b-quarks, and b + c quarks for p+p collisions [8]. The points with large
error bars are from $K\pi$ correlation measurements [16]. Our new result is
consistent with both the $K\pi$ correlation analysis and with FONLL
calculations. We also analyzed electrons in Au+Au collisions to obtain the
$R_{AA}$ for electrons from charm and bottom quarks [8]. The analysis assumed
that the parent hadrons (e.g. D, B) follow the $p_{T}$ distributions given by
PYTHIA. The results demonstrate that the Au+Au data are inconsistent with
these input assumptions unless there is also a large suppression of electrons
from bottom across the measured $p^{e}_{T}$ range. However, this large
suppression implies a change in the parent hadron $p_{T}$ distributions, which
results in changes in the electron DCA distributions. We are actively working
on evaluation of the uncertainties caused by this fact.
## 7 Summary
PHENIX has reported new findings in collisions at various energies and species
available at RHIC. There is a significant enhancement of high $p_{T}$ hadrons
and jets in the peripheral d+Au collisions. The yield of $\psi\prime$ is more
heavily suppressed than that of $J/\psi$ in d+Au collisions. A strong radial
flow is seen in tip-tip enriched 0-2 % U+U collisions. A sizable positive
$v_{1}$ is observed in Cu+Au collisions. In the same system, the $J/\psi$
yield measured in 1.2$<|\eta|<$2.2 as a function of $N_{part}$ shows larger
suppression in the Cu-going side than the Au-going side. The space-time
evolution of the ellipticity and triangularity of the Au+Au collision system
is observed in angular dependent HBT measurement. PHENIX has seen an angular
broadening of the away-side soft particles in $\gamma$-h measurement, similar
to h-h correlations and jet shape measurements from RHIC and LHC. The high
$p_{T}$ hadron spectra showed that the fractional momentum shift under the
presence of hot dense matter is $\sim$25 % higher at LHC compared to the RHIC
top energy. Identification of electrons from bottom and charm quarks has been
successful at RHIC using the VTX detector. The ratio of ($b\rightarrow
e$)/($b,c\rightarrow e$) for p+p collisions is consistent with the previous
measurement and well described by a FONLL calculation.
## References
* [1] A. Adare et al. [PHENIX Collaboration], arXiv:1205.5533 [hep-ex].
* [2] A. Adare, et al. [PHENIX Collaboration], arXiv:1208.1234 [nucl-ex].
* [3] M. Wysocki [PHENIX Collaboration], these proceedings.
* [4] J. Mitchell [PHENIX Collaboration], these proceedings.
* [5] S. Huang [PHENIX Collaboration], these proceedings.
* [6] K. Safarik, [ALICE Collaboration], these proceedings.
* [7] Z. -W. Lin, C. M. Ko, B. -A. Li, B. Zhang and S. Pal, Phys. Rev. C 72, 064901 (2005).
* [8] M. Rosati [PHENIX Collaboration], these proceedings.
* [9] T. Niida [PHENIX Collaboration], these proceedings.
* [10] I. Tserruya [PHENIX Collaboration], these proceedings.
* [11] M. McCumber [PHENIX Collaboration], these proceedings.
* [12] A. Adare, et al. [PHENIX Collaboration], arXiv:1208.2254 [nucl-ex].
* [13] K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 696, 30 (2011).
* [14] A. Adare, et al. [PHENIX Collaboration], arXiv:1204.1526 [nucl-ex].
* [15] E. O’Brien [PHENIX Collaboration], these proceedings.
* [16] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 103, 082002 (2009)
|
arxiv-papers
| 2012-11-01T22:46:05 |
2024-09-04T02:49:37.482558
|
{
"license": "Public Domain",
"authors": "Takao Sakaguchi (for the PHENIX Collaboration)",
"submitter": "Takao Sakaguchi",
"url": "https://arxiv.org/abs/1211.0324"
}
|
1211.0392
|
arxiv-papers
| 2012-11-02T08:27:32 |
2024-09-04T02:49:37.490014
|
{
"license": "Public Domain",
"authors": "S. Taj, B. Manaut, M. El Idrissi, Y. Attaourti and L. Oufni",
"submitter": "Bouzid Manaut",
"url": "https://arxiv.org/abs/1211.0392"
}
|
|
1211.0409
|
# __
Spin effects in laser-assisted semirelativistic excitation of atomic hydrogen
by electronic impact
S. Taj1, B. Manaut1,2, M. El Idrissi1, Y. Attaourti2 and L. Oufni3
1 Université Sultan Moulay Slimane, FPBM, LIRST, BP : 523, 23000, Béni Mellal,
Morocco.
2 Université Cadi Ayyad, FSSM, LPHEA, BP : 2390, 40000, Marrakech, Morocco
3 Université Sultan Moulay Slimane, FSTBM, LPMM, BP : 523, 23000, Béni Mellal,
Morocco. b.manaut@usms.ma
###### Abstract
New insights into our understanding of the semirelativistic excitation of
atomic hydrogen by electronic impact have been made possible by combining the
use of polarized electron beams and intense laser field. The paper reviews
relativistic theoretical treatment in laser-assisted electron scattering with
particular emphasis upon spin effects. Different spin configurations for
inelastic electron-atom collisions is also discussed. The role of laser field
in such collision is of major importance and reveals new information on the
dynamics of the collision process. The examined modern theoretical
investigations of such relativistic laser-assisted collisions have shown that
the need for experimental data is of a paramount importance in order to asses
the accuracy of our calculations.
PACS number(s): 34.80.Dp, 12.20.Ds
## 1 Introduction
The spin is not only an indispensable ingredient in atomic physics but also
responsible for many phenomena observed in solid-state physics. In addition to
the uses of polarized electrons in studies of atomic physics, there have been
numerous studies of polarized electron scattering and polarized electron
emission from ferromagnetic solids over the past decade. In 1975, the purpose
of the Spin-Polarized Electron Source was to describe how this effect, which
had been discovered in spin-polarized photoemission experiments by Pierce et
al at the ETH-Zurich [1] , could be used to provide a compact spin-polarized
electron gun. Later on, an experimental work has been done to produce electron
beams in which the spin has a preferential orientation. They are called
polarized electron beams [2] in analogy to polarized light, in which the field
vectors have a preferred orientation. Extensive theoretical works have been
performed by introducing relativistic and spin effects in the collision
between incident particles and atoms [3-6].There are many reasons for the
interest in polarized electrons. One important reason is that in physical
experiments one endeavors to define as exactly as possible the initial and/or
final states of the systems being considered.
Since the 1960s when lasers became a worldwide-used laboratory equipment and
also large polarization effects in low-energy electron scattering were
ascertained, experimental and theoretical studies of laser-matter interaction
have witnessed continuous progress. By virtue of the increasing progress in
the availability of more powerful and tunable lasers , such processes are
nowadays being observed in laboratories [7-10]. Most experimental and
theoretical studies of laser-assisted electron-atom collisions were restricted
to the nonrelativistic regime and low-frequency fields, where it has been
already recognized that, as a general consequence of the infrared divergence
of QED, large numbers of photons can be exchanged between the field and the
projectile-target system. An extension of the first-Born nonrelativistic
treatment [11] to the relativistic domain was formally derived for unpolarized
electrons [12]. There have been also theoretical investigations of
relativistic scattering in multimode fields [13].
In the present paper we have to extended our previous results [14] to the case
of laser assisted inelastic excitation $1s$ to $2s$ of the atomic hydrogen by
polarized electrons. Therefore, we have begun with the most basic results of
our work using atomic units (a.u) in which one has ($\hbar=m_{e}=e=1$), where
$m_{e}$ is the electron mass at rest. We have used the metric tensor
$g^{\mu\nu}=diag(1,-1,-1,-1)$ and the Lorentz scalar product defined by
$(a.b)=a^{\mu}b_{\mu}$. The organization of this paper is as follows : the
presentation of the necessary formalism of this work in section 2, the result
and discussion in section 3 and at last a brief conclusion in section 4.
## 2 Theory
The transition matrix element corresponding to the laser assisted inelastic
excitation of atomic hydrogen by electronic impact from the initial state $i$
to the final state $f$ is given by
$\displaystyle S_{fi}=-i\int
dt\;\langle\psi_{q_{f}}(\mathbf{r}_{1})\phi_{f}(\mathbf{r}_{2})|V_{d}|\psi_{q_{i}}(\mathbf{r}_{1})\phi_{i}(\mathbf{r}_{2})\rangle$
(1)
where $V_{d}=1/r_{12}-Z/r_{1}$ is the interaction potential, $\mathbf{r}_{1}$
are the coordinates of the incident and scattered electron, $\mathbf{r}_{2}$
the atomic electron coordinates, $r_{12}=|\mathbf{r}_{1}-\mathbf{r}_{2}|$ and
$r_{1}=|\mathbf{r}_{1}|$. Before we present the most interesting results of
our investigation regarding laser-assisted inelastic excitation of atomic
hydrogen by electronic impact, we sketch the principal steps of our
theoretical treatment. The solutions of the Dirac equation for an electron
with four-momentum $p^{\mu}$ inside an electromagnetic plane wave are well
known [15]. They read for the case of circular polarization of the field
propagating along the $Oz$ direction
$\displaystyle\psi_{q}(x)=\left(1+\frac{k\\!\\!\\!/A\\!\\!\\!/}{2c(kq)}\right)\frac{u(p,s)}{\sqrt{2VQ_{0}}}\exp\left[-i(qx)-i\int_{0}^{kx}\frac{(Ap)}{c(kq)}d\phi\right].$
(2)
where $u$ represents a free electron bispinor satisfying the Dirac equation
without field and which is normalized by
$\overline{u}u=u^{\dagger}\gamma^{0}u=2c^{2}$. Here the Feynman slash notation
is used, and $V$ is the normalization volume. The physical significance of
$q^{\mu}=(Q/c,\mathbf{q})$ is the averaged four-momentum (dressed momentum) of
the particle inside the laser field having a vector potential
$A^{\mu}=(0,a_{1}\cos(kx),a_{2}\sin(kx),0)$ with wave four-vector $k^{\mu}$ :
$q^{\mu}=p^{\mu}-k^{\mu}[A^{2}/2(kp)c^{2}]$.
In inelastic scattering, it is not only the state of the electron that is
changed, but also the state of the atom. Before starting the calculations, we
clarified the different configurations appearing with the orientations of the
electron’s spin polarizations. We have many possible scattering scenarios
1 | $e^{(\uparrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\uparrow)}(2s)$
---|---
2 | $e^{(\downarrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\uparrow)}(2s)$
3 | $e^{(\uparrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\uparrow)}(2s)$
4 | $e^{(\uparrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\downarrow)}(2s)$
6 | $e^{(\downarrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\downarrow)}(2s)$
7 | $e^{(\downarrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\downarrow)}(2s)$
⋮ | ⋮
Here, the up and down arrows indicate the direction of the electron’s and
atom’s spin polarization relative to some fixed axis. During the interaction,
the products of states with spin non flip are the same
$\varphi_{2s}^{{\dagger}(\uparrow})(\mathbf{r}_{2})\varphi_{1s}^{(\uparrow)}(\mathbf{r}_{2})=\varphi_{2s}^{{\dagger}(\downarrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\downarrow)}(\mathbf{r}_{2})$
and the product of states with spin flip gives zero
$\displaystyle\varphi_{2s}^{{\dagger}(\uparrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\downarrow)}(\mathbf{r}_{2})$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{llll}2-r_{2},&0,&\frac{-i(4-r_{2})}{4r_{2}c}z,&\frac{(4-r_{2})}{4r_{2}c}(-y-ix)\\\
\end{array}\right)\left(\begin{array}[]{c}0\\\ 1\\\ \frac{i}{2cr_{2}}(x-iy)\\\
-\frac{i}{2cr_{2}}z\end{array}\right)\frac{1}{4\sqrt{2}\pi}e^{-2r_{2}}$
$\displaystyle=$
$\displaystyle\varphi_{2s}^{{\dagger}(\downarrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\uparrow)}(\mathbf{r}_{2})$
$\displaystyle=$ $\displaystyle 0$ (9)
with $\varphi_{2s}(\mathbf{r}_{2})$ and $\varphi_{1s}(\mathbf{r}_{2})$ are the
wave functions of atomic hydrogen corresponding to $2s$ and $1s$ states
respectively. In this case, the probability that the bound electron changes
the orientation of its spin in the transition from the initial state $1s$ to
the final state $2s$ is zero. Thus, the number of realistic configurations
reduces many more. We review first the basics needed for the description of
spin polarization. Free electrons with four-momentum $p$ and spin $s$ are
described by the free spinors $u(p,s)$, the vector $s^{\mu}$ is defined by
$s^{\mu}=\frac{1}{c}\left(\left|\mathbf{p}\right|,\frac{E}{c}\widehat{\mathbf{p}}\right),$
(10)
(with $\widehat{\mathbf{p}}=\mathbf{p}/|\mathbf{p}|$ ) is a Lorentz vector in
a frame in which the particle moves with momentum $\mathbf{p}$. One easily
checks the normalization and the orthogonality conditions respectively
$s.s=s_{\mu}.s^{\mu}=-1\quad;\quad p.s=p_{\mu}.s^{\mu}=0.$ (11)
In practical calculations of quantum electrodynamic (QED) processes, we become
acquainted with a technique of calculation which allows the simple treatment
of complicated expressions especially the calculation of traces of products of
many $\gamma$ matrices. It is based on a projection procedure. The appropriate
operators which achieve this are called spin projection operators. In the
relativistic case, it is given by
$\widehat{\Sigma}(s)=\frac{1}{2}(1+\gamma_{5}s\\!\\!\\!/),$ (12)
with
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$.
This operator has the following properties
$\widehat{\Sigma}(s)u(p,\pm s)=\pm u(p,s).$ (13)
One can also apply this formalism to helicity states where the spin points in
the direction of the momentum $\mathbf{p}$
$s_{\lambda}^{\prime}=\lambda\frac{\mathbf{p}}{\left|\mathbf{p}\right|}=\lambda\widehat{\mathbf{p}}\,\lambda=\pm
1.$ (14)
We can then define a four spin vector as
$s_{\lambda}^{\mu}=\frac{\lambda}{c}\left(\left|\mathbf{p}\right|,\frac{E}{c}\widehat{\mathbf{p}}\right).$
(15)
The starting point of our calculation is the laser-assisted DCS for atomic
hydrogen by an electron with well defined momentum $p_{i}$ and well defined
spin $s_{i}$. If the final spin $s_{f}$ is also measured, the polarized DCS
then reads as
$\displaystyle\frac{d\sigma}{d\Omega_{f}}(\lambda_{i},\lambda_{f})=\sum_{n=-\infty}^{+\infty}\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i},\lambda_{f}),$
(16)
with
$\displaystyle\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i},\lambda_{f})=\left.\frac{|\mathbf{q}_{f}|}{|\mathbf{q}_{i}|}\frac{1}{(4\pi
c^{2})^{2}}\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}\left|H_{inel}(\Delta_{s})\right|^{2}\right|_{Q_{f}=Q_{i}+n\omega+E_{1s1/2}-E_{2s1/2}}.$
(17)
The quantity $H_{inel}(\Delta_{s})$ which represents the integral part is
given by :
$\displaystyle H_{inel}(\Delta)=-\frac{4\pi}{\sqrt{2}}(I_{1}+I_{2}+I_{3})$
(18)
with $I_{1}$, $I_{2}$ and $I_{3}$ are as follow :
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{4}{27c^{2}}\int_{0}^{+\infty}dr_{1}\;r_{1}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{4}{27c^{2}}\frac{1}{((3/2)^{2}+\mathbf{\Delta}^{2})}$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{6}{27}(\frac{1}{c^{2}}-4)\int_{0}^{+\infty}dr_{1}\;r_{1}^{2}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{2}{27}(\frac{1}{c^{2}}-4)\frac{3}{((3/2)^{2}+\mathbf{\Delta}^{2})^{2}}$
(19) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\frac{4}{9}(1+\frac{1}{8c^{2}})\int_{0}^{+\infty}dr_{1}\;r_{1}^{3}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{8}{9}(1+\frac{1}{8c^{2}})\frac{\mathbf{\Delta}^{2}-27/4}{((3/2)^{2}+\mathbf{\Delta}^{2})^{3}}.$
Using REDUCE [17], the spinorial part obtained after tedious calculations
reads as
$\displaystyle\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}$
$\displaystyle=$
$\displaystyle\textbf{Tr}\\{\Gamma_{n}\frac{(1+\lambda_{i}\gamma_{5}s\\!\\!\\!/_{i})}{2}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}\frac{(1+\lambda_{f}\gamma_{5}s\\!\\!\\!/_{f})}{2}(cp\\!\\!\\!/_{f}+c^{2})\\},$
(20) $\displaystyle=$
$\displaystyle\big{\\{}J_{n}^{2}(z)\mathcal{A}+\big{(}J^{2}_{n+1}(z)+J^{2}_{n-1}(z)\big{)}\mathcal{B}+\big{(}J_{n+1}(z)J_{n-1}(z)\big{)}\mathcal{C}$
$\displaystyle+J_{n}(z)\big{(}J_{n-1}(z)+J_{n+1}(z)\big{)}\mathcal{D}\big{\\}}.$
with $\overline{\Gamma}_{n}=\gamma^{0}\Gamma_{n}^{{\dagger}}\gamma^{0}$ and
$\Gamma_{n}$ is explicitly detailed in our previous work [18].
Before presenting our analytical results, we would like to emphasize that the
REDUCE code we have written for obtaining the four coefficients $\mathcal{A}$,
$\mathcal{B}$, $\mathcal{C}$ and $\mathcal{D}$ gave very long analytical
expressions which were difficult to incorporate in the corresponding latex
manuscript. Thus, we prefer to give below, for example, just the coefficient
$\mathcal{A}$ multiplying the Bessel function $J_{n}^{2}(z)$.
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{(2(k.p_{f})^{2}(k.p_{i})^{2}c^{8})}\Big{[}2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{8}\cos(\theta_{if})-2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}$
$\displaystyle\times
c^{6}\cos(\theta_{if})E_{i}^{2}+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{10}-2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{i}}|^{2}c^{6}\cos(\theta_{if})E_{f}^{2}$
$\displaystyle+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}c^{8}\cos(\theta_{if})E_{f}E_{i}+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}c^{4}\cos(\theta_{if})E_{f}^{2}E_{i}^{2}+2(k.p_{f})^{2}$
$\displaystyle\times(k.p_{i})^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{10}\cos(\theta_{if})+2(k.p_{f})^{2}(k.p_{i})^{2}c^{12}+2(k.p_{f})^{2}(k.p_{i})^{2}c^{8}E_{f}E_{i}+2(k.p_{f})^{2}(k.p_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})E_{f}\omega-2(k.p_{f})^{2}(k.p_{i})\lambda_{f}\lambda_{i}|a|c^{2}\cos(\theta_{if})E_{f}E_{i}^{2}\omega-2(k.p_{f})^{2}(k.p_{i})|a|c^{6}$
$\displaystyle\times
E_{i}\omega+2(k.p_{f})(k.p_{i})^{2}\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}c^{4}\cos(\theta_{if})E_{i}\omega-2(k.p_{f})(k.p_{i})^{2}\lambda_{f}\lambda_{i}|a|c^{2}\cos(\theta_{if})E_{f}^{2}E_{i}\omega$
$\displaystyle-2(k.p_{f})(k.p_{i})^{2}|a|c^{6}E_{f}\omega-2(k.p_{f})(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}||\mathbf{p_{i}}|^{2}c^{6}\cos(\theta_{if})\omega+2(k.p_{f})(k.p_{i})$
$\displaystyle\times(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|c^{4}\cos(\theta_{if})E_{i}^{2}\omega-2(k.p_{f})(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|c^{8}\omega-2(k.p_{f})(k.p_{i})(k.s_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|c^{6}\cos(\theta_{if})\omega-2(k.p_{f})(k.p_{i})(k.s_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|c^{8}\omega+2(k.p_{f})(k.p_{i})(k.s_{i})\lambda_{f}\lambda_{i}$
$\displaystyle\times|a||\mathbf{p_{i}}|c^{4}\cos(\theta_{if})E_{f}^{2}\omega-(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{2}\omega
a^{2}+k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}a^{2}\cos(\theta_{if})E_{f}E_{i}\omega^{2}$
$\displaystyle+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})\omega^{2}-2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}c^{2}\cos(\theta_{if})E_{i}^{2}\omega^{2}$
$\displaystyle+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\omega^{2}-2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|^{2}c^{2}\cos(\theta_{if})E_{f}^{2}\omega^{2}-2(k.p_{f})(k.p_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a|c^{4}\cos(\theta_{if})E_{f}E_{i}\omega^{2}+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a|\cos(\theta_{if})E_{f}^{2}E_{i}^{2}\omega^{2}+(k.p_{f})(k.p_{i})a^{2}c^{4}\omega^{2}$
$\displaystyle-2(k.p_{f})(k.p_{i})|a||\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\cos(\theta_{if})\omega^{2}-2(k.p_{f})(k.p_{i})|a|c^{8}\omega^{2}+2(k.p_{f})(k.p_{i})|a|c^{4}E_{f}E_{i}\omega^{2}$
$\displaystyle+(k.p_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}|c^{2}E_{i}\omega^{2}-(k.p_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{i}}|c^{2}\cos(\theta_{if})E_{f}\omega^{2}-(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}$
$\displaystyle\times
a^{2}|\mathbf{p_{f}}|c^{2}\cos(\theta_{if})E_{i}\omega^{2}+(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{i}}|c^{2}E_{f}\omega^{2}+(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{4}\cos(\theta_{if})$
$\displaystyle\times\omega^{2}+(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}c^{6}\omega^{2}-(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}c^{2}E_{f}E_{i}\omega^{2}\Big{]}$
At this stage, note that if $\lambda_{i}\lambda_{f}=1$ during the scattering
process, which physically means that there is no helicity flip occurring but
if $\lambda_{i}\lambda_{f}=-1$, this means that a helicity flip occurred. In
the absence of the laser field ($|a|=0)$, this coefficient reduces to
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{4}}\Big{[}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})-\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}c^{2}\cos(\theta_{if})E_{i}^{2}+\lambda_{f}\lambda_{i}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}-\lambda_{f}\lambda_{i}|\mathbf{p_{i}}|^{2}c^{2}\cos(\theta_{if})E_{f}^{2}$
$\displaystyle+\lambda_{f}\lambda_{i}c^{4}\cos(\theta_{if})E_{f}E_{i}+\lambda_{f}\lambda_{i}\cos(\theta_{if})E_{f}^{2}E_{i}^{2}+|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\cos(\theta_{if})+c^{8}+c^{4}E_{f}E_{i}\Big{]}$
We note that without laser field, equations (16) and (17) reduce to the well-
known polarized (first-Born) differential cross section of inelastic
excitation of atomic hydrogen by electronic impact.
Figure 1: The different TDCSs (Unpolarized DCS, Spin polarized DCS with
($\lambda_{i}=\lambda_{f}=0$)) scaled in $10^{-9}$ as a function of the angle
$\theta_{f}$ . The relativistic parameter is $\gamma=1.0053$, the electrical
field strength is $\mathcal{E}=0.05\;a.u$. The geometric parameters are
$\theta_{i}=45^{\circ}$, $\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and the
number of photons exchanged are $n=\pm 100$.
## 3 Results and discussions
The spin-polarized differential cross sections $d\sigma^{(n)}/d\Omega_{f}$ are
computed for the laser-assisted semirelativistic excitation of atomic hydrogen
by electronic impact, where $n$ denotes the number of photons absorbed or
emitted. With a view to qualitative comparison with our previous work [14],
kinematics and the geometry parameters are chosen in accordance with those
used in [14]. The direction of the laser field is chosen parallel to $Oz$
axis. The corresponding spin unpolarized differential cross section
$d\overline{\sigma}^{(n)}/d\Omega_{f}$ results are also presented for
comparison.
Figure 2: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-9}$ as a function of the angle $\theta_{f}$ . The
relativistic parameter is $\gamma=1.0053$, the electrical field strength is
$\mathcal{E}=0.05\;a.u$. The geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and the number of photons
exchanged are $n=\pm 100$.
In view of equation (20) if the mathematical condition
($\lambda_{i}=\lambda_{f}=0$) is used, the spinorial part takes the following
form
$\displaystyle\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}=\frac{1}{4}\textbf{Tr}\\{\Gamma_{n}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}(cp\\!\\!\\!/_{f}+c^{2})\\}$
(21)
In comparison with the unpolarized spinorial part
$\displaystyle\frac{1}{2}\sum_{s_{i}s_{f}}\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}=\textbf{Tr}\\{\Gamma_{n}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}(cp\\!\\!\\!/_{f}+c^{2})\\}$
(22)
it may be noted from equations (21) and (22) that the spin unpolarized DCS is
equal to two times to the spin polarized DCS but only under the condition
($\lambda_{i}=\lambda_{f}=0$)
$\displaystyle\frac{d\overline{\sigma}^{(n)}}{d\Omega_{f}}=2\times\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i}=0,\lambda_{f}=0),$
(23)
this equation represents our first consistency check. The numerical results
are displayed in figure 1, where we have plotted the two DCSs (2 times spin
polarized DCS with ($\lambda_{i}=\lambda_{f}=0$) and spin unpolarized DCS ) by
varying the angle $\theta_{f}$ of the scattered electron. As it is seen in
this figure, we have indistinguishable curves. The most important result
(second consistency check) is the sum of polarized DCS (spin flip) and
polarized DCS (spin non flip) always gives the spin unpolarized DCS as this is
shown in figure 2.
Figure 3: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-13}$ as a function of the relativistic parameter
$\gamma$, the geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and $\theta_{f}=45^{\circ}$. The
electrical field strength is $\mathcal{E}=0.5\;a.u$ and the number of photons
exchanged are $n=\pm 50$.
Figure 3 shows the different spin polarized and unpolarized DCSs versus the
relativistic parameter $\gamma$. It is apparent from this figure that the
kinetic energy of the incident electron has an important effect in the spin
orientation. For much of the high energy range, the spin polarized DCS (spin
non flip) is approximately equal to the spin unpolarized DCS and the spin
polarized DCS (spin flip) is equal to zero. The physical meaning of this
result is that : in high energy, the probability that the incident electron
changes its spin is zero.
Figure 4: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-22}$ as a function of the relativistic parameter
$\gamma$, the geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and $\theta_{f}=45^{\circ}$. The
electrical field strength is $\mathcal{E}=0.5\;a.u$ and the number of photons
exchanged are $n=\pm 50$.
In order to clarify the situation in which we have seemingly overlapping
curves for the three approaches in figure 3, we give in figure 4 the three
approaches (spin polarized (spin flip and spin non flip) and the spin
polarized DCS). As it is noticed, the spin polarized DCS (spin non flip) and
the spin unpolarized DCS overlap but the spin polarized DCS (spin flip)
converges to zero.
Figure 5: The behavior of the degree of polarization $P$ as a function of the
angle $\theta_{f}$ varying from $-180^{\circ}$ to $180^{\circ}$ and the
relativistic parameter $\gamma$ scaled in $10^{-2}$ in absence of the laser
field.
Our main interest is the polarization degree of the electron after the event.
This quantity is defined as
$\displaystyle P=\frac{\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin non
flip})-\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin
flip})}{\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin non
flip})+\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin flip})},$ (24)
figure 5 shows the polarization degree, which is related by equation (24) to
the spin polarized differential cross section ratio. A tree dimensional plot
of this quantity is given versus the angle $\theta_{f}$ and the relativistic
parameter $\gamma$. The first observation that can be made concerns the shape
of the polarization degree that is strongly changed with the relativistic
parameter $\gamma$. An interesting behavior emerges with increasing $\gamma$,
particularly for $\theta_{f}=0^{\circ}$, where the polarization degree
performs a plateau-like behavior. This emphasizes the fact that the spin
polarized DCS is very sensitive to the variation of the relativistic parameter
$\gamma$ and this fact has to remain true for the case in the presence of the
laser field. This is consistent with the results shown in figure 3 and 4.
## 4 Conclusion
We have studied the laser assisted inelastic excitation of atomic hydrogen by
polarized electrons. We unraveled the influence of the orientation of the
spin-polarization of the incoming and scattered electrons relative to the
orientation of the spin-polarization of the bound electron. These spin effects
depend strongly on the energies of the incoming electron. It is observed that
in the transition from $1s$ state to $2s$ state, the electron’s probability
for changing its spin is zero.
## References
* [1] D.T. Pierce, F. Meier, and P. Z rcher, Appl. Phys. Lett. 26, 670 (1975).
* [2] J. Kessler, Polarized Electrons, 2nd ed. (Springer, Berlin, 1985).
* [3] V Zeman, R P McEachran and A D Stauffe, J. Phys. B: At. Mol. Opt. Phys. 30 3475, (1997).
* [4] Rajesh Srivastava, Sachin Saxena,Radiation Physics and Chemistry 75, 2136 2150, (2006)
* [5] Kapil K. Sharmaa, Neerjab, and R. P. Vatsa, J. At. Mol. Sci. 3, 197-208, (2012).
* [6] Alvaro Muñoz-castro and Ramiro Arratia-Perez, Phys. Chem. Chem. Phys. 14, 1408-1411, (2012).
* [7] S. Luan, R. Hippler , and H. O. Lutz , J. Phys. B. , 24 , 3241 ( 1991 ) .
* [8] B. WallBank , J. K. Holmes , S. Maclsaac and A. Weingartshofer J. Phys. B. , 25 , 1265 , ( 1992 ).
* [9] B. WallBank , J. K. Holmes , J. Phys. B., 27 , 1221 ( 1994 ).
* [10] B. WallBank , J. K. Holmes , J. Phys. B., 27 , 5405 ( 1994 ).
* [11] Bunkin, F.V. and Fedorov, M.V., 1965, Zh. Eksp. Teor. Fiz., 49, 1215 [1966, Sov. Phys. JETP, 22, 844].
* [12] Denisov, M.M. and Fedorov, M.V., 1967, Zh. Eksp. Teor. Fiz., 53, 1340 [1968, Sov. Phys. JETP, 26 779].
* [13] Roshchupkin, S.P., 1996, Zh. Eksp. Teor. Fiz., 109, 337 [1996, JETP, 82, 177]; Zhou, F. and Rosenberg, L., 1992, Phys. Rev. A, 45, 7818.
* [14] S. Taj, B. Manaut, M. El Idrissi and L. Oufni, Chin. Journal of Phys. 49, 1164-1177, (2011)
* [15] D. M. Volkov, Z. Phys 94, 250 (1935).
* [16] S. Taj, B. Manaut and L. Oufni, Acta Physica Polonica A, 119, 769-773 (2011)
* [17] A. G. Grozin, Using REDUCE in High Energy Physics (Cambridge University, Cambridge, England,1997).
* [18] Y. Attaourti and B. Manaut, Phys. Rev. A 68, 067401 (2003).
|
arxiv-papers
| 2012-11-02T10:26:30 |
2024-09-04T02:49:37.494854
|
{
"license": "Public Domain",
"authors": "S. Taj, B. Manaut, M. El Idrissi, Y. Attaourti and L. Oufni",
"submitter": "Bouzid Manaut",
"url": "https://arxiv.org/abs/1211.0409"
}
|
1211.0483
|
# __
Semirelativistic $1s-2s$ excitation of atomic hydrogen by electron impact
S. Taj1, B. Manaut1 and L. Oufni2
1 Université Sultan Moulay Slimane, FPBM, LIRST, BP : 523, 23000, Béni Mellal,
Morocco.
2 Université Sultan Moulay Slimane, FSTBM, LPMM, BP : 523, 23000, Béni Mellal,
Morocco. b.manaut@usms.ma
###### Abstract
In the framework of the first Born approximation, we present a
semirelativistic theoretical study of the inelastic excitation
($1s_{1/2}\longrightarrow 2s_{1/2}$) of hydrogen atom by electronic impact.
The incident and scattered electrons are described by a free Dirac spinor and
the hydrogen atom target is described by the Darwin wave function.
Relativistic and spin effects are examined in the relativistic regime. A
detailed study has been devoted to the nonrelativistic regime as well as the
moderate relativistic regime. Some aspects of this dependence as well as the
dynamic behavior of the DCS in the relativistic regime have been addressed.
PACS number(s): 34.80.Dp, 12.20.Ds
## 1 Introduction
The theoretical study of relativistic electron-atom collisions is fundamental
to our understanding of many aspects in plasma physics and astrophysics. The
development of electron-atom collision studies has also been strongly
motivated by the need of data for testing and developing suitable theories of
the scattering and collision process, and providing a tool for obtaining
detailed information on the structure of the target atoms and molecules. Many
authors have studied this process using numerical tools. Thus, Kisielius et
al. [1] employed, the R-matrix method with nonrelativistic and relativistic
approximations for the hydrogen like $He^{+}$, $Fe^{25+}$ and $U^{91+}$ ions,
where the case of transitions $1s\longrightarrow 2s$ and $1s\longrightarrow
2p$ as well as those between fine structure $n=2$ levels was considered.
Andersen et al. [2] have applied the semirelativistic Breit Pauli R-matrix to
calculate the electron-impact excitation of the ${}^{2}S_{1/2}$
$\longrightarrow$ ${}^{2}P^{o}_{1/2,3/2}$ resonance transitions in heavy
alkali atoms. Payne et al. [3] have studied the electron-impact excitation of
the $5s\longrightarrow 5p$ resonance transition in rubidium by using a semi-
relativistic Breit Pauli R-matrix with pseudo-states (close-coupling)
approach. Attaourti et al. [4] have investigated the exact analytical
relativistic excitation $1S_{1/2}\longrightarrow 1S_{1/2}$ of atomic hydrogen,
by electron impact in the presence of a laser field. They have found that a
simple formal analogy links the analytical expressions of the unpolarized
differential cross section without laser and the unpolarized differential
cross section in the presence of a laser field.
The aim of this contribution is to add some new physical insights and to show
that the non-relativistic formalism becomes enable to describe particles with
hight kinetic energies. Before we present the results of our investigation, we
first begin by sketching the main steps of our treatment. For pedagogical
purposes, we begin by the most basic results of our work using atomic units
(a.u) in which one has ($\hbar=m_{e}=e=1$), where $m_{e}$ is the electron mass
at rest, and which will be used throughout this work. We will also work with
the metric tensor $g^{\mu\nu}=diag(1,-1,-1,-1)$ and the Lorentz scalar product
which is defined by $(a.b)=a^{\mu}b_{\mu}$. The layout of this paper is as
follows. We present the necessary formalism of this work in section [2,3 and
4], the result and discussion in section 5 and we end by a brief conclusion in
section 6.
## 2 Theory of the inelastic collision $1s_{1/2}\longrightarrow 2s_{1/2}$
In this section, we calculate the exact analytical expression of the
semirelativistic unpolarized DCS for the relativistic excitation of atomic
hydrogen by electron impact. The transition matrix element for the direct
channel (exchange effects are neglected) is given by
$\displaystyle S_{fi}$ $\displaystyle=$ $\displaystyle-i\int
dt\langle\psi_{p_{f}}(x_{1})\phi_{f}(x_{2})\mid
V_{d}\mid\psi_{p_{i}}(x_{1})\phi_{i}(x_{2})\rangle$ (1) $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\int
d\mathbf{r}_{1}\overline{\psi}_{p_{f}}(t,\mathbf{r}_{1})\gamma^{0}\psi_{p_{i}}(t,\mathbf{r}_{1})\langle\phi_{f}(x_{2})\mid
V_{d}\mid\phi_{i}(x_{2})\rangle$
where
$V_{d}=\frac{1}{r_{12}}-\frac{Z}{r_{1}}$ (2)
is the direct interaction potential, $\mathbf{r}_{1}$ are the coordinates of
the incident and scattered electron, $\mathbf{r}_{2}$ the atomic electron
coordinates, $r_{12}=$ $\mid\mathbf{r}_{1}-\mathbf{r}_{2}\mid$ and
$r_{1}=\mid\mathbf{r}_{1}\mid$. The function
$\psi_{p_{i}}(x_{1})=\psi_{p}(t,\mathbf{r}_{1})=u(p,s)\exp(-ip.x)/\sqrt{2EV}$
is the electron wave function, described by a free Dirac spinor normalized to
the volume $V$, and $\phi_{i,f}(x_{2})=\phi_{i,f}(t,\mathbf{r}_{2})$ are the
semirelativistic wave functions of the hydrogen atom where the index $i$ and
$f$ stand for the initial and final states respectively. The semirelativistic
wave function of the atomic hydrogen is the Darwin wave function for bound
states [5], which is given by :
$\phi_{i}(t,\mathbf{r}_{2})=\exp(-i\mathcal{E}_{b}(1s_{1/2})t)\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})$
(3)
where $\mathcal{E}_{b}(1s_{1/2})$ is the binding energy of the ground state of
atomic hydrogen and $\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})$ is given by :
$\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})=(\mathsf{1}_{4}-\frac{i}{2c}\mathbf{\alpha.\nabla}_{(2)})u^{(\pm)}\varphi_{0}(\mathbf{r}_{2})$
(4)
it represents a quasi relativistic bound state wave function, accurate to
first order in $Z/c$ in the relativistic corrections (and normalized to the
same order), with $\varphi_{0}$ being the non-relativistic bound state
hydrogenic function. The spinors $u^{(\pm)}$ are such that
$u^{(+)}=(1,0,0,0)^{T}$ and $u^{(-)}=(0,1,0,0)^{T}$ and represent the basic
four-component spinors for a particle at rest with spin-up and spin-down,
respectively. The matrix differential operator $\alpha.\Delta$ is given by :
$\displaystyle\alpha.\Delta=\begin{pmatrix}0&0&\partial_{z}&\partial_{x}-i\partial_{y}\\\
0&0&\partial_{x}+i\partial_{y}&-\partial_{z}\\\
\partial_{z}&\partial_{x}-i\partial_{y}&0&0\\\
\partial_{x}+i\partial_{y}&-\partial_{z}&0&0\end{pmatrix}$ (5)
For the spin up, we have :
$\displaystyle\varphi_{1s}^{(+)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{1}}\left(\begin{array}[]{c}1\\\ 0\\\
\frac{i}{2cr_{2}}z\\\
\frac{i}{2cr_{2}}(x+iy)\end{array}\right)\frac{1}{\sqrt{\pi}}e^{-r_{2}}$ (10)
and for the spin down, we have :
$\displaystyle\varphi_{1s}^{(-)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{1}}\left(\begin{array}[]{c}0\\\ 1\\\
\frac{i}{2cr_{2}}(x-iy)\\\
-\frac{i}{2cr_{2}}z\end{array}\right)\frac{1}{\sqrt{\pi}}e^{-r_{2}}$ (15)
where
$N_{D_{1}}=2c/\sqrt{4c^{2}+1}$ (16)
is a normalization constant lower but very close to 1. Let us mention that the
function $\phi_{f}(t,\mathbf{r}_{2})$ in Eq. (1) is the Darwin wave function
for bound states [6], which is also accurate to the order $Z/c$ in the
relativistic corrections. This is expressed as
$\phi_{f}(t,\mathbf{r}_{2})=\exp(-i\mathcal{E}_{b}(2s_{1/2})t)\varphi_{2s}^{(\pm)}(\mathbf{r}_{2})$
with $\mathcal{E}_{b}(2s_{1/2})$ as the binding energy of the $2s_{1/2}$ state
of atomic hydrogen.
$\displaystyle\varphi_{2s}^{(+)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{2}}\left(\begin{array}[]{c}2-r_{2}\\\ 0\\\
\frac{i(4-r_{2})}{4r_{2}c}z\\\
\frac{(4-r_{2})}{4rc}(-y+ix)\end{array}\right)\frac{1}{4\sqrt{2\pi}}e^{-r_{2}}$
(21)
for the spin up and
$\displaystyle\varphi_{2s}^{(-)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{2}}\left(\begin{array}[]{c}0\\\ 2-r_{2}\\\
\frac{4-r_{2}}{4cr_{2}}(y+ix)\\\
i\frac{(r_{2}-4)}{4cr_{2}}z\end{array}\right)\frac{1}{4\sqrt{2\pi}}e^{-r_{2}}$
(26)
for the spin down. The transition matrix element in Eq. (1) becomes :
$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\int
d\mathbf{r}_{1}d\mathbf{r}_{2}\overline{\psi}_{p_{f}}(t,\mathbf{r}_{1})\gamma^{0}\psi_{p_{i}}(t,\mathbf{r}_{1})\phi^{{\dagger}}_{f}(t,r_{2})\phi_{i}(t,r_{2})V_{d}$
(27)
and it is straightforward to get, for the transition amplitude,
$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle-i\frac{\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})}{2V\sqrt{E_{f}E_{i}}}2\pi
H_{inel}(\Delta)\delta\big{(}E_{f}+\mathcal{E}(2s_{1/2})-E_{i}-\mathcal{E}(1s_{1/2})\big{)}$
where $\Delta=|p_{i}-p_{f}|$ and $\gamma^{0}$ is the Dirac matrix. Using the
standard technique of the QED, we find for the unpolarized DCS
$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}$ $\displaystyle=$
$\displaystyle\frac{|\mathbf{p}_{f}|}{|\mathbf{p}_{i}|}\frac{1}{(4\pi
c^{2})^{2}}\left(\frac{1}{2}\sum_{s_{i}s_{f}}|\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})|^{2}\right)\left|H_{inel}(\Delta)\right|^{2}$
(29)
## 3 Calculation of the integral part
The function $H_{inel}(\Delta)$ is found if one performs the various integrals
:
$\displaystyle
H_{inel}(\Delta)=\int_{0}^{+\infty}d\mathbf{r}_{1}e^{i\mathbf{\Delta}\mathbf{r}_{1}}I(\mathbf{r}_{1})$
(30)
### 3.1 Integral over $\mathbf{r}_{2}$
The quantity $I(\mathbf{r}_{1})$ is easily evaluated in the following way. We
first write the explicit form of $I(\mathbf{r}_{1})$ :
$\displaystyle
I(\mathbf{r}_{1})=\int_{0}^{+\infty}d\mathbf{r}_{2}\phi^{{\dagger}}_{2s}(\mathbf{r}_{2})\left[\frac{1}{r_{12}}-\frac{Z}{r_{1}}\right]\phi_{1s}(\mathbf{r}_{2})$
(31)
Next, we develop the quantity $r^{-1}_{12}$ in spherical harmonics as
$\displaystyle\frac{1}{\mathbf{r}_{12}}=4\pi\sum_{lm}\frac{Y_{lm}(\widehat{r}_{1})Y_{lm}^{*}(\widehat{r}_{2})}{2l+1}\frac{(\mathbf{r}_{<})^{l}}{(\mathbf{r}_{>})^{l+1}}$
(32)
where $r_{>}$ is the greater of $r_{1}$ and $r_{2}$, and $r_{<}$ the lesser of
them. The angular coordinates of the vectors $\mathbf{r}_{1}$ and
$\mathbf{r}_{2}$ are such that : $\widehat{r}_{1}=(\theta_{1},\varphi_{1})$
and $\widehat{r}_{2}=(\theta_{2},\varphi_{2})$. We use the well known integral
[7]
$\displaystyle\int_{x}^{+\infty}du\;u^{m}e^{-\alpha
u}=\frac{m!}{\alpha^{m+1}}e^{-\alpha
x}\sum_{\mu=0}^{m}\frac{\alpha^{\mu}x^{\mu}}{\mu!}\qquad\qquad Re(\alpha)>0$
(33)
then, after some analytic calculations, we get for $I(\mathbf{r}_{1})$ :
$\displaystyle
I(\mathbf{r}_{1})=\frac{6}{27}(\frac{1}{c^{2}}-4)+\frac{4}{27c^{2}}\frac{1}{\mathbf{r}_{1}}-\frac{4}{9}(1+\frac{1}{8c^{2}})\mathbf{r}_{1}$
(34)
### 3.2 Integral over $\mathbf{r}_{1}$
The integration over $\mathbf{r}_{1}$ gives rise to the following formula :
$\displaystyle
H_{inel}(\Delta)=\int_{0}^{+\infty}d\mathbf{r}_{1}e^{i\mathbf{\Delta}\mathbf{r}_{1}}I(\mathbf{r}_{1})=-\frac{4\pi}{\sqrt{2}}(I_{1}+I_{2}+I_{3})$
(35)
the angular integrals are performed by expanding the plane wave
$e^{i\mathbf{\Delta}\mathbf{r}_{1}}$ in spherical harmonics as :
$\displaystyle e^{i\mathbf{\Delta}\mathbf{r}_{1}}=\sum_{lm}4\pi
i^{l}j_{l}(\mathbf{\Delta}\mathbf{r}_{1})Y_{lm}(\widehat{\mathbf{\Delta}})Y_{lm}^{*}(\widehat{\mathbf{r}}_{1})$
(36)
with $\mathbf{\Delta}=\mathbf{p}_{i}-\mathbf{p}_{f}$ is the relativistic
momentum transfer and $\widehat{\mathbf{\Delta}}$ is the angular coordinates
of the vector $\mathbf{\Delta}$. Then, after some analytic computations, we
get for $I_{1}$, $I_{2}$ and $I_{3}$ the following result :
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{4}{27c^{2}}\int_{0}^{+\infty}dr_{1}\;r_{1}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{4}{27c^{2}}\frac{1}{((3/2)^{2}+\mathbf{\Delta}^{2})}$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{6}{27}(\frac{1}{c^{2}}-4)\int_{0}^{+\infty}dr_{1}\;r_{1}^{2}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{2}{27}(\frac{1}{c^{2}}-4)\frac{3}{((3/2)^{2}+\mathbf{\Delta}^{2})^{2}}$
(37) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\frac{4}{9}(1+\frac{1}{8c^{2}})\int_{0}^{+\infty}dr_{1}\;r_{1}^{3}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{8}{9}(1+\frac{1}{8c^{2}})\frac{\mathbf{\Delta}^{2}-27/4}{((3/2)^{2}+\mathbf{\Delta}^{2})^{3}}$
It is clear that the situation is different than in elastic collision [4],
since we have no singularity in the case ($\mathbf{\Delta}\to 0$)
Figure 1: Behavior of the probability density for radial Darwin wave function
compared with that of the Dirac wave function for small distances and for
increasing values of the atomic charge number.
## 4 Calculation of the spinorial part
The calculation is now reduced to the computation of traces of $\gamma$
matrices. This is routinely done using Reduce [8]. We consider the unpolarized
DCS. Therefore, the various polarization states have the same probability and
the actual calculated spinorial part is given by summing over the final
polarization $s_{f}$ and averaging aver the initial polariztion $s_{i}$.
Therfore, the spinorial part is given by :
$\displaystyle\frac{1}{2}\sum_{s_{i}s_{f}}|\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})|^{2}$
$\displaystyle=$
$\displaystyle\text{Tr}\left\\{\gamma^{0}(p\\!\\!\\!/_{i}c+c^{2})\gamma^{0}(p\\!\\!\\!/_{f}c+c^{2})\right\\}$
(38) $\displaystyle=$ $\displaystyle
2c^{2}[\frac{2E_{f}E_{i}}{c^{2}}-(p_{i}.p_{f})+c^{2}]$
We must, of course, recover the result in the nonrelativistic limit
($\gamma\longrightarrow 1$), situation of which the differential cross section
can simply given by :
$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}=\frac{|\mathbf{K}_{f}|}{|\mathbf{K}_{i}|}\frac{128}{\left(|\mathbf{\Delta_{nr}}|^{2}+\frac{9}{4}\right)^{6}}$
(39)
with $|\mathbf{\Delta_{nr}}|=|\mathbf{K}_{i}-\mathbf{K}_{f}|$ is the
nonrelativistic momentum transfer and the momentum vectors ($\mathbf{K}_{i}$,
$\mathbf{K}_{f}$) are related by the following formula :
$\displaystyle\mathbf{K}_{f}=(|\mathbf{K}_{i}|^{2}-3/4)^{1/2}$ (40)
Figure 2: The long-dashed line represents the semi-relativistic DCS, the
solid line represents the corresponding non-relativistic DCS for a
relativistic parameter ($\gamma=1.5$) as functions of the scattering angle
$\theta$.
Figure 3: The solid line represents the semi-relativistic DCS, the long-
dashed line represents the corresponding non-relativistic DCS for various
values of the relativistic parameter ($\gamma=1.5$, $\gamma=2$ and
$\gamma=2.5$) as functions of the scattering angle $\theta$.
## 5 Results and discussions
In presenting our results it is convenient to consider separately those
corresponding to non-relativistic regime (the relativistic parameter
$\gamma\simeq 1$) and those related to relativistic one (the relativistic
parameter $\gamma\simeq 2$). Before beginning the discussion of the obtained
results, it is worthwhile to recall the meaning of some abbreviation that will
appear throughout this section. The NRDCS stands for the nonrelativistic
differential cross section, where nonrelativistic plane wave are used to
describe the incident and scattered electrons. The SRDCS stands for the
semirelativistic differential cross section.
We begin our numerical work, by the study of the dependence of the probability
density for radial Darwin and Dirac wave functions, on the atomic charge
number $Z$.
Figure 4: The solid line represents the semi-relativistic DCS, the long-
dashed line represents the corresponding non-relativistic DCS for a
relativistic parameter $\gamma=1.00053$ as functions of the scattering angle
$\theta$.
Figure 5: The variation of the SRDCSs with respect to $\theta$, for various
kinetic energies.
As long as the condition $Z\alpha\ll 1$ is verified, the use of Darwin wave
function do not have any influence at all on the results at least in the first
order of perturbation theory. So, the semi-relativistic treatment when $Z$
increases may generate large errors but not in the case of this work. In this
paper, we can not have numerical instabilities since there are none. For the
sake of illustration, we give below the behavior of the probability density
for radial Darwin wave functions as well as that of the exact relativistic
Dirac wave functions for different values of $Z$. As you may see, even if it
is not noticeable on the figure 1, there are growing discrepancies for $Z=10$
and these become more pronounced when $Z=20$. The QED formulation shows that
there are relativistic and spin effects at the relativistic domain and the non
relativistic formulation is no longer valid.
Figure 6: The variation of the differential $1s-2s$ cross section of
$e^{-}-H$ scattering at $200\;eV$. The dots are the observed values of J. F.
Wiliams (1981) ; the solid line represents the semi-relativistic approximation
and the long-dashed line corresponds to the non-relativistic DCS.
In the relativistic regime, the semirelativistic differential cross section
results obtained for the $1s\longrightarrow 2s$ transition in atomic hydrogen
by electron impact, are displayed in figures 2 and 3. In this regime, there
are no theoretical models and experimental data for comparison as in
nonrelativistic regime. In such a situation, it appears from figures 3 and 3
that in the limit of high electron kinetic energy, the effects of the
additional spin terms and the relativity begin to be noticeable and that the
non-relativistic formalism is no longer applicable. Also a pick in the
vicinity of $\theta_{f}=0^{\circ}$ is clearly observed.
The investigation in the nonrelativistic regime were conducted with $\gamma$
as a relativistic parameter and $\theta$ as a scattering angle. In atomic
units, the kinetic energy is related to $\gamma$ by the following relation :
$E_{k}=c^{2}(\gamma-1)$. Figure 4 shows the dependence of DCS, obtained in two
models (SRDCS, NRDCS), on scattering angle $\theta$. In this regime, it
appears clearly that there is no difference between these models. Figure 5
shows the variation of the SRDCS with $\theta$ for various energies. It also
shows approximatively in the interval [-5, 5], the SRDCS increases with
$\gamma$, but decreases elsewhere. Figure 6 presents the observed and
calculated angular dependence of $1s-2s$ differential cross section of
$e^{-}-H$ scattering at incident energie $200\;eV$. Results obtained in two
approaches semirelativistic and non-relativistic approximations are
indistinguishable and in good agreement with the experimental data provided by
J. F. Williams [9].
## 6 Conclusion
In this paper we have presented the results of a semirelativistic excitation
of atomic hydrogen by electronic impact. We have used the simple
semirelativistic Darwin wave function that allows to obtain analytical results
in an exact and closed form within the framework of the first Born
approximation. This model gives good results if the condition $Z/c\ll 1$ is
fulfilled. We have compared our results with previous nonrelativistic results
and have found that the agreement between the different theoretical approaches
is good in the nonrelativistic regime. We have also showed that the non-
relativistic treatment is no longer reliable for energies higher. We hope that
we will be able to compare our theoretical results with forthcoming
experimental data in the relativistic regime.
ACKNOWLEDGMENT
I would like to thank Professor N. BOURIMA for his help in maintaining the
text linguistically acceptable.
## References
* [1] R. Kisielius, K.A. Berrington and P.H. Norrington J. Phys. B, 28, 2459-2471, (1995).
* [2] N. Andersen and K. Bartschat, J. Phys. B, 35 4507-4525 (2002).
* [3] Daniel Payne, Benjamin Krueger and Klaus Bartschat J. Phys. B, 38 3349-3357 (2005) .
* [4] Y. Attaourti, B. Manaut and A. Makhoute, Phys. Rev. A 69, 063407 (2004).
* [5] J. Eichler and W.E. Meyerhof, Relativistic Atomic Collisions, Academic Press, (1995).
* [6] F.W. Jr Byron and C.J. Joachain, Phys. Rep. 179, 211, (1989).
* [7] Gradstein, L S., Rizik, I.M. : Tables of Integrals, Sutures, Sets and Their Products. Moscow: Nauka. (1971).
* [8] A. G. Grozin, Using REDUCE in High Energy Physics (Cambridge University, Cambridge, England, 1997).
* [9] J. F. Williams, J. Phys. B 14, 1197 (1981).
|
arxiv-papers
| 2012-11-02T16:17:38 |
2024-09-04T02:49:37.502645
|
{
"license": "Public Domain",
"authors": "S. Taj, B. Manaut and L. Oufni",
"submitter": "Bouzid Manaut",
"url": "https://arxiv.org/abs/1211.0483"
}
|
1211.0550
|
# __
Laser-Assisted Semi Relativistic Excitation of Atomic Hydrogen by Electronic
Impact.
S. Taj1, B. Manaut1, M. El Idrissi1 and L. Oufni 2
1 Faculté Polydisciplinaire, Université Sultan Moulay Slimane,
Laboratoire Interdisciplinaire de Recherche en Sciences et Techniques (LIRST),
Boite Postale 523, 23000 Béni Mellal, Morocco.
2 Université Sultan Moulay Slimane, Faculté des Sciences et Techniques,
Département de Physique, LPMM-ERM, Boite Postale 523, 23000 Béni Mellal,
Morocco. manaut@fstbm.ac.ma
###### Abstract
The excitation of H ($1s-2s$) by electron impact in the presence and in the
absence of the laser field is studied in the framework of the first Born
approximation. The angular variation of the laser-assisted differential cross
section (DCS) for atomic hydrogen by electronic impact is presented at various
kinetic energies for the incident electron. The use of Darwin wave function as
a semirelativistic state to represent the atomic hydrogen gives interesting
results when the condition $z/c\ll 1$ is fulfilled. A comparison with the non
relativistic theory and experimental data gives good agreement. It was
observed that beyond ($2700$ $eV$) which represents the limit between the two
approaches, the non relativistic theory does not yield close agreement with
our theory and that, over certain ranges of energy, it can be in error by
several orders of magnitude. The sum rule given by Bunkin and Fedorov and by
Kroll and Watson [23] has been verified in both nonrelativistic and
relativistic regimes.
PACS number(s): 34.80.Dp, 12.20.Ds
## 1 Introduction
Relativistic laser-atom physics has become recently as a new research area and
constitute a new field of systematic experimental and theoretical study. In
particular, the laser-asssited electron-atom scattering becomes a rapidly
growing subject. These processes are however of fundamental interest and
paramount importance, for instance in the laser heating of plasmas and high-
power gaz lasers. One of its most remarkable features is the possibility of
exciting the target via the absorption of one or more photons. In recent years
considerable attention has been given to these topics. Most of the works are
theoretical and there is very little experimental work at present. Several
experiments have been performed, in which the exchange of one or more photons
between the electron-target and the laser field has been observed in laser-
assisted elastic [1] and inelastic scattering [2-5]. In particular, the
excitation processes have been largely investigated in the literature by
several authors [6-9], mainly in the perturbative weak-field limit. The first
theoretical studies on the inelastic scattering were inspired from the
pioneering works [10-12], in which the interaction between the free electron
and the field can be treated exactly by using the exact Volkov waves [13].
Another investigation have been done for the exact analytical relativistic
excitation $1S_{1/2}\longrightarrow 1S_{1/2}$ of atomic hydrogen, by electron
impact in the presence of a laser field, see [14]. For a summary of
relativistic laser-atom collisions, see [15]. In the absence of the laser
field, many authors have studied this process using numerical tools. Thus,
Kisielius et al. [16] employed, the R-matrix method with nonrelativistic and
relativistic approximations for the hydrogen like $He^{+}$, $Fe^{25+}$ and
$U^{91+}$ ions, where the case of transitions $1s\longrightarrow 2s$ and
$1s\longrightarrow 2p$ as well as those between fine structure $n=2$ levels
was considered. Andersen et al. [17] have applied the semirelativistic Breit
Pauli R-matrix to calculate the electron-impact excitation of the
${}^{2}S_{1/2}$ $\longrightarrow$ ${}^{2}P^{o}_{1/2,3/2}$ resonance
transitions in heavy alkali atoms. Payne et al. [18] have studied the
electron-impact excitation of the $5s\longrightarrow 5p$ resonance transition
in rubidium by using a semi-relativistic Breit Pauli R-matrix with pseudo-
states (close-coupling) approach. S. Taj et al. [19] have presented a
theoretical semirelativistic model and have found that the semirelativistic
Coulomb Born approximation (SRCBA) in a closed and exact form for the
description of the ionization of atomic hydrogen by electron impact in the
first Born approximation is valid for all geometries. The aim of this
contribution is to add some new physical insights and to give the analytical
formula for the excitation differential cross section for hydrogen from the 1s
ground state to the 2s excited state in the absence and in the presence of the
laser field. Before we present the results of our investigation, we first
begin by sketching the main steps of our treatment. For pedagogical purposes,
we begin by the most basic results of our work using atomic units (a.u) in
which one has ($\hbar=m_{e}=e=1$), where $m_{e}$ is the electron mass at rest,
and which will be used throughout this work. We will also work with the metric
tensor $g^{\mu\nu}=diag(1,-1,-1,-1)$ and the Lorentz scalar product which is
defined by $(a.b)=a^{\mu}b_{\mu}$. The layout of this paper is as follows :
the presentation of the necessary formalism of this work in section [2 and 3],
the result and discussion in section 4 and at last a brief conclusion in
section 5.
## 2 Theory of the inelastic collision in the absence of de laser field
In this section, we calculate the exact analytical expression of the
semirelativistic unpolarized DCS for the relativistic excitation of atomic
hydrogen by electron impact. The transition matrix element for the direct
channel (exchange effects are neglected) is given by
$\displaystyle S_{fi}$ $\displaystyle=$ $\displaystyle-i\int
dt\langle\psi_{p_{f}}(x_{1})\phi_{f}(x_{2})\mid
V_{d}\mid\psi_{p_{i}}(x_{1})\phi_{i}(x_{2})\rangle$ (1) $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\int
d\mathbf{r}_{1}\overline{\psi}_{p_{f}}(t,\mathbf{r}_{1})\gamma^{0}\psi_{p_{i}}(t,\mathbf{r}_{1})\langle\phi_{f}(x_{2})\mid
V_{d}\mid\phi_{i}(x_{2})\rangle$
where
$V_{d}=\frac{1}{r_{12}}-\frac{Z}{r_{1}}$ (2)
is the direct interaction potential, $\mathbf{r}_{1}$ are the coordinates of
the incident and scattered electron, $\mathbf{r}_{2}$ the atomic electron
coordinates, $r_{12}=$ $\mid\mathbf{r}_{1}-\mathbf{r}_{2}\mid$ and
$r_{1}=\mid\mathbf{r}_{1}\mid$. The function
$\psi_{p_{i}}(x_{1})=\psi_{p}(t,\mathbf{r}_{1})=u(p,s)\exp(-ip.x)/\sqrt{2EV}$
is the electron wave function, described by a free Dirac spinor normalized to
the volume $V$, and $\phi_{i,f}(x_{2})=\phi_{i,f}(t,\mathbf{r}_{2})$ are the
semirelativistic wave functions of the hydrogen atom where the index $i$ and
$f$ stand for the initial and final states respectively. The semirelativistic
wave function of the atomic hydrogen is the Darwin wave function for bound
states [17], which is given by :
$\phi_{i}(t,\mathbf{r}_{2})=\exp(-i\mathcal{E}_{b}(1s_{1/2})t)\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})$
(3)
where $\mathcal{E}_{b}(1s_{1/2})$ is the binding energy of the ground state of
atomic hydrogen and $\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})$ is given by :
$\varphi_{1s}^{(\pm)}(\mathbf{r}_{2})=(\mathsf{1}_{4}-\frac{i}{2c}\mathbf{\alpha.\nabla}_{(2)})u^{(\pm)}\varphi_{0}(\mathbf{r}_{2})$
(4)
it represents a quasi relativistic bound state wave function, accurate to
first order in $Z/c$ in the relativistic corrections (and normalized to the
same order), with $\varphi_{0}$ being the non-relativistic bound state
hydrogenic function. The spinors $u^{(\pm)}$ are such that
$u^{(+)}=(1,0,0,0)^{T}$ and $u^{(-)}=(0,1,0,0)^{T}$ and represent the basic
four-component spinors for a particle at rest with spin-up and spin-down,
respectively. The matrix differential operator $\alpha.\nabla$ is given by :
$\displaystyle\alpha.\nabla=\left(\begin{array}[]{cccc}0&0&\partial_{z}&\partial_{x}-i\partial_{y}\\\
0&0&\partial_{x}+i\partial_{y}&-\partial_{z}\\\
\partial_{z}&\partial_{x}-i\partial_{y}&0&0\\\
\partial_{x}+i\partial_{y}&-\partial_{z}&0&0\end{array}\right)$ (9)
For the spin up, we have :
$\displaystyle\varphi_{1s}^{(+)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{1}}\left(\begin{array}[]{c}1\\\ 0\\\
\frac{i}{2cr_{2}}z\\\
\frac{i}{2cr_{2}}(x+iy)\end{array}\right)\frac{1}{\sqrt{\pi}}e^{-r_{2}}$ (14)
and for the spin down, we have :
$\displaystyle\varphi_{1s}^{(-)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{1}}\left(\begin{array}[]{c}0\\\ 1\\\
\frac{i}{2cr_{2}}(x-iy)\\\
-\frac{i}{2cr_{2}}z\end{array}\right)\frac{1}{\sqrt{\pi}}e^{-r_{2}}$ (19)
where
$N_{D_{1}}=2c/\sqrt{4c^{2}+1}$ (20)
is a normalization constant lower but very close to 1. Let us mention that the
function $\phi_{f}(t,\mathbf{r}_{2})$ in Eq. (1) is the Darwin wave function
for bound states [18], which is also accurate to the order $Z/c$ in the
relativistic corrections. This is expressed as
$\phi_{f}(t,\mathbf{r}_{2})=\exp(-i\mathcal{E}_{b}(2s_{1/2})t)\varphi_{2s}^{(\pm)}(\mathbf{r}_{2})$
with $\mathcal{E}_{b}(2s_{1/2})$ as the binding energy of the $2s_{1/2}$ state
of atomic hydrogen.
$\displaystyle\varphi_{2s}^{(+)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{2}}\left(\begin{array}[]{c}2-r_{2}\\\ 0\\\
\frac{i(4-r_{2})}{4r_{2}c}z\\\
\frac{(4-r_{2})}{4rc}(-y+ix)\end{array}\right)\frac{1}{4\sqrt{2\pi}}e^{-r_{2}}$
(25)
for the spin up and
$\displaystyle\varphi_{2s}^{(-)}(\mathbf{r}_{2})$ $\displaystyle=$
$\displaystyle N_{D_{2}}\left(\begin{array}[]{c}0\\\ 2-r_{2}\\\
\frac{4-r_{2}}{4cr_{2}}(y+ix)\\\
i\frac{(r_{2}-4)}{4cr_{2}}z\end{array}\right)\frac{1}{4\sqrt{2\pi}}e^{-r_{2}}$
(30)
for the spin down. The transition matrix element in Eq. (1) becomes :
$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\int
d\mathbf{r}_{1}d\mathbf{r}_{2}\overline{\psi}_{p_{f}}(t,\mathbf{r}_{1})\gamma^{0}\psi_{p_{i}}(t,\mathbf{r}_{1})\phi^{{\dagger}}_{f}(t,r_{2})\phi_{i}(t,r_{2})V_{d}$
(31)
and it is straightforward to get, for the transition amplitude,
$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle-i\frac{\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})}{2V\sqrt{E_{f}E_{i}}}2\pi
H_{inel}(\Delta)\delta\big{(}E_{f}+\mathcal{E}(2s_{1/2})-E_{i}-\mathcal{E}(1s_{1/2})\big{)}$
where $\Delta=|p_{i}-p_{f}|$ and $\gamma^{0}$ is the Dirac matrix. Using the
standard technique of the QED, we find for the unpolarized DCS
$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}$ $\displaystyle=$
$\displaystyle\frac{|\mathbf{p}_{f}|}{|\mathbf{p}_{i}|}\frac{1}{(4\pi
c^{2})^{2}}\left(\frac{1}{2}\sum_{s_{i}s_{f}}|\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})|^{2}\right)\left|H_{inel}(\Delta)\right|^{2}$
(33)
## 3 Calculation of the integral part
The function $H_{inel}(\Delta)$ is found if one performs the various integrals
:
$\displaystyle
H_{inel}(\Delta)=\int_{0}^{+\infty}d\mathbf{r}_{1}e^{i\mathbf{\Delta}\mathbf{r}_{1}}I(\mathbf{r}_{1})$
(34)
The quantity $I(\mathbf{r}_{1})$ is easily evaluated in the following way. We
first write the explicit form of $I(\mathbf{r}_{1})$ :
$\displaystyle
I(\mathbf{r}_{1})=\int_{0}^{+\infty}d\mathbf{r}_{2}\phi^{{\dagger}}_{2s}(\mathbf{r}_{2})\left[\frac{1}{r_{12}}-\frac{Z}{r_{1}}\right]\phi_{1s}(\mathbf{r}_{2})$
(35)
Next, we develop the quantity $r^{-1}_{12}$ in spherical harmonics as
$\displaystyle\frac{1}{\mathbf{r}_{12}}=4\pi\sum_{lm}\frac{Y_{lm}(\widehat{r}_{1})Y_{lm}^{*}(\widehat{r}_{2})}{2l+1}\frac{(\mathbf{r}_{<})^{l}}{(\mathbf{r}_{>})^{l+1}}$
(36)
where $r_{>}$ is the greater of $r_{1}$ and $r_{2}$, and $r_{<}$ the lesser of
them. The angular coordinates of the vectors $\mathbf{r}_{1}$ and
$\mathbf{r}_{2}$ are such that : $\widehat{r}_{1}=(\theta_{1},\varphi_{1})$
and $\widehat{r}_{2}=(\theta_{2},\varphi_{2})$. We use the well known integral
[19]
$\displaystyle\int_{x}^{+\infty}du\;u^{m}e^{-\alpha
u}=\frac{m!}{\alpha^{m+1}}e^{-\alpha
x}\sum_{\mu=0}^{m}\frac{\alpha^{\mu}x^{\mu}}{\mu!}\qquad\qquad Re(\alpha)>0$
(37)
then, after some analytic calculations, we get for $I(\mathbf{r}_{1})$ :
$\displaystyle
I(\mathbf{r}_{1})=\frac{6}{27}(\frac{1}{c^{2}}-4)+\frac{4}{27c^{2}}\frac{1}{\mathbf{r}_{1}}-\frac{4}{9}(1+\frac{1}{8c^{2}})\mathbf{r}_{1}$
(38)
The integration over $\mathbf{r}_{1}$ gives rise to the following formula :
$\displaystyle
H_{inel}(\Delta)=\int_{0}^{+\infty}d\mathbf{r}_{1}e^{i\mathbf{\Delta}\mathbf{r}_{1}}I(\mathbf{r}_{1})=-\frac{4\pi}{\sqrt{2}}(I_{1}+I_{2}+I_{3})$
(39)
the angular integrals are performed by expanding the plane wave
$e^{i\mathbf{\Delta}\mathbf{r}_{1}}$ in spherical harmonics as :
$\displaystyle e^{i\mathbf{\Delta}\mathbf{r}_{1}}=\sum_{lm}4\pi
i^{l}j_{l}(\mathbf{\Delta}\mathbf{r}_{1})Y_{lm}(\widehat{\mathbf{\Delta}})Y_{lm}^{*}(\widehat{\mathbf{r}}_{1})$
(40)
with $\mathbf{\Delta}=\mathbf{p}_{i}-\mathbf{p}_{f}$ is the relativistic
momentum transfer and $\widehat{\mathbf{\Delta}}$ is the angular coordinates
of the vector $\mathbf{\Delta}$. Then, after some analytic computations, we
get for $I_{1}$, $I_{2}$ and $I_{3}$ the following result :
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{4}{27c^{2}}\int_{0}^{+\infty}dr_{1}\;r_{1}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{4}{27c^{2}}\frac{1}{((3/2)^{2}+\mathbf{\Delta}^{2})}$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{6}{27}(\frac{1}{c^{2}}-4)\int_{0}^{+\infty}dr_{1}\;r_{1}^{2}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{2}{27}(\frac{1}{c^{2}}-4)\frac{3}{((3/2)^{2}+\mathbf{\Delta}^{2})^{2}}$
(41) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\frac{4}{9}(1+\frac{1}{8c^{2}})\int_{0}^{+\infty}dr_{1}\;r_{1}^{3}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{8}{9}(1+\frac{1}{8c^{2}})\frac{\mathbf{\Delta}^{2}-27/4}{((3/2)^{2}+\mathbf{\Delta}^{2})^{3}}$
It is clear that the situation is different than in elastic collision [14],
since we have no singularity in the case ($\mathbf{\Delta}\to 0$)
The calculation the spinorial part is reduced to the computation of traces of
$\gamma$ matrices. This is routinely done using Reduce [20]. We consider the
unpolarized DCS. Therefore, the various polarization states have the same
probability and the actual calculated spinorial part is given by summing over
the final polarization $s_{f}$ and averaging aver the initial polariztion
$s_{i}$. Therfore, the spinorial part is given by :
$\displaystyle\frac{1}{2}\sum_{s_{i}s_{f}}|\overline{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})|^{2}$
$\displaystyle=$
$\displaystyle\textbf{Tr}\left\\{\gamma^{0}(p\\!\\!\\!/_{i}c+c^{2})\gamma^{0}(p\\!\\!\\!/_{f}c+c^{2})\right\\}$
(42) $\displaystyle=$ $\displaystyle
2c^{2}[\frac{2E_{f}E_{i}}{c^{2}}-(p_{i}.p_{f})+c^{2}]$
We must, of course, recover the result in the nonrelativistic limit
($\gamma\longrightarrow 1$), situation of which the differential cross section
can simply given by :
$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}=\frac{|\mathbf{K}_{f}|}{|\mathbf{K}_{i}|}\frac{128}{\left(|\mathbf{\Delta_{nr}}|^{2}+\frac{9}{4}\right)^{6}}$
(43)
with $|\mathbf{\Delta_{nr}}|=|\mathbf{K}_{i}-\mathbf{K}_{f}|$ is the
nonrelativistic momentum transfer and the momentum vectors ($\mathbf{K}_{i}$,
$\mathbf{K}_{f}$) are related by the following formula :
$\displaystyle\mathbf{K}_{f}=(|\mathbf{K}_{i}|^{2}-3/4)^{1/2}$ (44)
## 4 Theory of the inelastic collision in the presence of de laser field
The second-order Dirac equation for the elctron in the presence of an external
electromagnetic field is given by :
$\displaystyle\left[\left(p-\frac{1}{c}A\right)^{2}-c^{2}-\frac{i}{2c}F_{\mu\nu}\sigma^{\mu\nu}\right]\psi(x)=0$
(45)
where $\sigma^{\mu\nu}=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$ is the tensors
related to Dirac matrices $\gamma^{\mu}$ and
$F^{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor. $A^{\mu}$ is the four-vector potential. The
plane wave solution of the second-order equation is known as the Volkov state
[13].
$\displaystyle\psi(x)=\left(1+\frac{k\\!\\!\\!/A\\!\\!\\!/}{2c(kp)}\right)\frac{u(p,s)}{\sqrt{2VQ_{0}}}\exp\left[-i(qx)-i\int_{0}^{kx}\frac{(Ap)}{c(kp)}d\phi\right]$
(46)
We turn now to the calculation of the laser-assisted transition amplitude. The
instantaneous interaction potential is given by
$\displaystyle V_{d}=\frac{1}{\mathbf{r}_{12}}-\frac{Z}{\mathbf{r}_{1}}$ (47)
where $\mathbf{r}_{12}=|\mathbf{r}_{1}-\mathbf{r}_{2}|$; $\mathbf{r}_{1}$ are
the electron coordinates and $\mathbf{r}_{2}$ are the atomic coordinates. The
transition matrix element corresponding to the process of laser assisted
electron-atomic hydrogen from the initial state $i$ to the final state $f$ is
given by
$\displaystyle S_{fi}=-i\int
dt\;\langle\overline{\psi}_{q_{f}}(\mathbf{r}_{1})\phi_{f}(\mathbf{r}_{2})|V_{d}|\psi_{q_{i}}(\mathbf{r}_{1})\phi_{i}(\mathbf{r}_{2})\rangle$
(48)
Proceeding along the lines of standard calculations in QED, one has for the
DCS
$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}=\sum_{s=-\infty}^{+\infty}\frac{d\overline{\sigma}^{(s)}}{d\Omega_{f}}$
(49)
with
$\displaystyle\frac{d\overline{\sigma}^{(s)}}{d\Omega_{f}}=\left.\frac{|\mathbf{q}_{f}|}{|\mathbf{q}_{i}|}\frac{1}{(4\pi
c^{2})^{2}}\left(\frac{1}{2}\sum_{s_{i}s_{f}}|M_{fi}^{(s)}|^{2}\right)\left|H_{inel}(\Delta_{s})\right|^{2}\right|_{Q_{f}=Q_{i}+s\omega+E_{1s1/2}-E_{2s1/2}}.$
(50)
The novelty in the various stages of the calculations is contained in the
spinorial part $\frac{1}{2}\sum_{s_{i}}\sum_{s_{f}}|M_{fi}^{n}|^{2}$ that
contains all the information about the spin and the laser-interaction effects.
This quantity can be obtained using REDUCE [20] and are explicitly given in
our previous work [21]. After some analytical calculations, the integral part
$H_{inel}(\Delta_{s})$ reduces to
$\displaystyle
H_{inel}(\Delta_{s})=-\frac{4\pi}{\sqrt{2}}[I_{1}(s)+I_{2}(s)+I_{3}(s)]$ (51)
with
$\displaystyle I_{1}(s)$ $\displaystyle=$
$\displaystyle\frac{4}{27c^{2}}\frac{1}{((3/2)^{2}+\mathbf{\Delta_{s}}^{2})}$
$\displaystyle I_{2}(s)$ $\displaystyle=$
$\displaystyle\frac{2}{27}(\frac{1}{c^{2}}-4)\frac{3}{((3/2)^{2}+\mathbf{\Delta_{s}}^{2})^{2}}$
(52) $\displaystyle I_{3}(s)$ $\displaystyle=$
$\displaystyle\frac{8}{9}(1+\frac{1}{8c^{2}})\frac{\mathbf{\Delta_{s}}^{2}-27/4}{((3/2)^{2}+\mathbf{\Delta_{s}}^{2})^{3}}$
with $\mathbf{\Delta}_{s}=\mathbf{q}_{f}-\mathbf{q}_{i}-s\mathbf{k}$ is the
momentum transfer in the presence of the laser field.
## 5 Results and discussions
### 5.1 In the absence of the laser field
Figure 1: Behavior of the probability density for radial Darwin wave function
compared with that of the Dirac wave function for small distances and for
increasing values of the atomic charge number.
In presenting our results it is convenient to consider separately those
corresponding to non-relativistic regime (the relativistic parameter
$\gamma\simeq 1$) and those related to relativistic one (the relativistic
parameter $\gamma\simeq 2$). Before beginning the discussion of the obtained
results, it is worthwhile to recall the meaning of some abbreviation that will
appear throughout this section. The NRDCS stands for the nonrelativistic
differential cross section, where nonrelativistic plane wave are used to
describe the incident and scattered electrons. The SRDCS stands for the
semirelativistic differential cross section.
We begin our numerical work with the study of the dependence of the
probability density for radial Darwin and Dirac wave functions on the atomic
charge number $Z$.
Figure 2: The long-dashed line represents the semi-relativistic DCS, the
solid line represents the corresponding non-relativistic DCS for a
relativistic parameter ($\gamma=1.5$) as functions of the scattering angle
$\theta$.
Figure 3: The solid line represents the semi-relativistic DCS, the long-
dashed line represents the corresponding non-relativistic DCS for various
values of the relativistic parameter ($\gamma=1.5$, $\gamma=2$ and
$\gamma=2.5$) as functions of the scattering angle $\theta$.
Figure 4: The solid line represents the semi-relativistic DCS, the long-
dashed line represents the corresponding non-relativistic DCS for a
relativistic parameter $\gamma=1.00053$ as functions of the scattering angle
$\theta$.
Figure 5: The variation of the SRDCSs with respect to $\theta$, for various
kinetic energies.
So long as the condition $Z\alpha\ll 1$ is verified, the use of Darwin wave
function do not have any influence at all on the results at least in the first
order of perturbation theory. So, the semirelativistic treatment when $Z$
increases may generate large errors but not in the case of this work. In this
paper, we can not have numerical instabilities since there are none. For the
sake of illustration, we give in figure 1 the behavior of the probability
density for radial Darwin wave functions as well as that of the exact
relativistic Dirac wave functions for different values of $Z$. As you may see,
even if it is not noticeable in figure 1, there are growing discrepancies for
$Z=10$ and these become more pronounced when $Z=20$. The QED formulation shows
that there are relativistic and spin effects at the relativistic domain and
the non relativistic formulation is no longer valid.
Figure 6: The variation of the differential $1s-2s$ cross section of
$e^{-}-H$ scattering at $200\;eV$. The dots are the observed values of J. F.
Wiliams (1981) ; the solid line represents the semi-relativistic approximation
and the long-dashed line corresponds to the non-relativistic DCS.
In the relativistic regime, the semirelativistic differential cross section
results obtained for the $1s\longrightarrow 2s$ transition in atomic hydrogen
by electron impact, are displayed in figures 2 and 3. In this regime, there
are no theoretical models and experimental data for comparison as in
nonrelativistic regime. In such a situation, it appears from figures 3 and 3
that in the limit of high electron kinetic energy, the effects of the
additional spin terms and the relativity begin to be noticeable and that the
non-relativistic formalism is no longer applicable. Also a peak in the
vicinity of $\theta_{f}=0^{\circ}$ is clearly observed.
The investigation in the nonrelativistic regime was carried out with $\gamma$
as a relativistic parameter and $\theta$ as a scattering angle. In atomic
units, the kinetic energy is related to $\gamma$ by the following relation :
$E_{k}=c^{2}(\gamma-1)$. Figure 4 shows the dependence of DCS, obtained in two
models (SRDCS, NRDCS), on scattering angle $\theta$. In this regime, it
appears clearly that there is no difference between these models. Figure 5
shows the variation of the SRDCS with $\theta$ for various energies. It also
shows approximatively in the interval [-5, 5] that the SRDCS increases with
$\gamma$, but decreases elsewhere. Figure 6 presents the observed and
calculated angular dependence of $1s-2s$ differential cross section of
$e^{-}-H$ scattering at incident energie $200\;eV$. Results obtained in the
approaches (semirelativistic and nonrelativistic approximations) are
indistinguishable and in good agreement with the experimental data provided by
J. F. Williams [22].
### 5.2 In the presence of the laser field
For the description of the scattering geometry, we work in a coordinate system
in which $k||\widehat{e}_{z}$. This means that the direction of the laser
propagation is along $Oz$ axis. We begin by defining our scattering geometry.
The undressed angular coordinates of the incoming electron are
$\theta_{i}=90^{\circ}$, $\phi_{i}=45^{\circ}$ and for the scattered electron,
we have $0^{\circ}\leq\theta_{f}\leq 180^{\circ}$ and $\phi_{f}=45^{\circ}$.
When we take the zero electric field strength ($\mathcal{E}=0\;a.u$), one
finds overlapping curves for the tree approaches [SRDCS (without laser), SRDCS
(with laser) and the Non relativistic DCS]. It represents our first
consistency check of our calculations and it is shown in figure 7.
Figure 7: The various DCSs as a function of $\theta_{f}$ for an electrical
field strength of $\mathcal{E}=0\;a.u$, a relativistic parameter
$\gamma=1.0053$
Figure 9 shows the envelope of the laser assisted SRDCS as a function of the
net number of photons exchanged. The cutoffs are $s\simeq-50$ photons for the
negative part of the envelope and $s\simeq+50$ photons for the positive part.
In figure 9, for the geometry $\theta_{i}=90^{\circ}$, $\phi_{i}=45^{\circ}$
and $\phi_{f}=45^{\circ}$, we have made simulations concerning the laser-
assisted SRDCS for a set of net number photons exchanged. These sets ($\pm
10$, $\pm 20$, $\pm 30$, $\pm 40$, $\pm 45$ and $\pm 50$) show that at $\pm
50$ the SRDCS (with laser) is almost close to the SRDCS (without laser).
Figure 8: Envelope of the SRDCS in the non relativistic regime
($\gamma=1.0053$ and $\mathcal{E}=0.05\;a.u$) for a number of net photons
exchanged $\pm 100$.
Figure 9: The variation of the SRDCSs with and without laser for various
numbers of photons exchanged ($\pm 10$, $\pm 20$, $\pm 20$, $\pm 40$, $\pm
45$) and $\pm 50$).
But for instance at $\pm 10$, we have several orders of magnitude as a result
of the difference between the two approaches. We return to the first case $\pm
50$, the convergence reached here is called the sum-rule that was shown by
Bunkin and Fedorov as well as by Kroll and Watson [23]. The figures (9 and 9)
correctly introduce correlation in the net number of photons exchanged that
reaches the well-known sum-rule. As you can see from the figure 9, the SRDCS
full off abruptly beyond the interval [$-50$, $+50$] and figure 9 shows
clearly that beyond $\pm 50$, the sum-rule is clearly checked.
Figure 10: Envelope of the SRDCS in the relativistic regime ($\gamma=2.0$ and
$\mathcal{E}=0.05\;a.u$) for a number of net photons exchanged $\pm 30000$.
Figure 10 show clearly that for the relativistic regime ($\gamma=2.0$ and
$\mathcal{E}=0.05\;a.u$) and for the geometry ($\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and $\theta_{f}=45^{\circ}$), the
value of the cutoffs have been changed to $s=-13986$ for the negative part of
the envelope and $s=+13912$ for the positive part. In this regime, when the
number of photons exchanged increases, the convergence of the two SRDCSs (with
and witout laser) will be reached at approximatively $\pm 13945$, but as our
computational capacity is limited, it is not possible, actually, to give the
figure which illustrates such a situation.
## 6 Conclusion
This paper display the results of a semirelativistic excitation of atomic
hydrogen by electronic impact in the presence and absence of the laser field.
The simple semirelativistic Darwin wave function that allows to obtain
analytical results in an exact and closed form within the framework of the
first Born approximation is used. Our results have been compared with previous
nonrelativistic results, revealing that the agreement between the different
theoretical approaches is good in the nonrelativistic regime. The
nonrelativistic treatment has been shown to be no longer reliable for high
energies. It has been also found that a simple formal analogy links the
analytical expressions of the unpolarized differential cross section in the
absence of the laser field and the laser-assisted unpolarized differential
cross section.
## References
* [1] A. Weingartshofer, J. K. Holmes, J. Sabbagh and S. I. Chu, J. Phys. B 16, 1805 (1983). See also, B. Wallbank, J. K. Holmes, J. Phys. B 27, 1221 (1994).
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* [5] B. Wallbank, J. K. Holmes and A. Weingartshofer, Phys. Rev. A 40, 5461, (1989); J.Phys. B 23, 2997 (1990).
* [6] N. K. Rahman and F. H. M. Faisal, J. Phys. B 11, 2003 (1978).
* [7] S. Jetzke, F. H. M. Faisal, R. Hippler and O. H. Lutz, Z. Phys. A 315, 271, (1984).
* [8] S. Jetzke, J. Broad and A. Maquet, J. Phys. B 20, 2887 (1987).
* [9] R. S. Pundir and K. C. Mathur, Z. Phys. D 1, 385 (1986).
* [10] F. W. Byron Jr, P. Francken and C. J. Joachain, J. Phys. B 20 5487 (1987).
* [11] F. W. Byron Jr and C. J. Joachain, Phys. Rev. A 35 1590 (1987).
* [12] P. Francken, Y. Attaourti and C. J. Joachain, Phys. Rev. A 38, 1785 (1988).
* [13] D. M. Volkov, Z. Phys 94, 250 (1935).
* [14] Y. Attaourti, B. Manaut and A. Makhoute, Phys. Rev. A 69, 063407 (2004).
* [15] Alfred Maquet, Richard Ta eb and Val rie V niard, Springer Series in Optical Sciences, Volume 134, 477-496, (2008).
* [16] R. Kisielius, K.A. Berrington and P.H. Norrington J. Phys. B, 28, 2459-2471, (1995).
* [17] J. Eichler and W.E. Meyerhof, Relativistic Atomic Collisions, Academic Press, (1995).
* [18] F.W. Jr Byron and C.J. Joachain, Phys. Rep. 179, 211, (1989).
* [19] Gradstein, L S., Rizik, I.M. : Tables of Integrals, Sutures, Sets and Their Products. Moscow: Nauka. (1971).
* [20] A. G. Grozin, Using REDUCE in High Energy Physics (Cambridge University, Cambridge, England, 1997).
* [21] Y. Attaourti and B. Manaut Phys. Rev. A, 68, 067401 (2003)
* [22] J. F. Williams, J. Phys. B 14, 1197 (1981).
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|
arxiv-papers
| 2012-11-02T20:03:33 |
2024-09-04T02:49:37.510124
|
{
"license": "Public Domain",
"authors": "S. Taj, B. Manaut, M. El Idrissi and L. Oufni",
"submitter": "Bouzid Manaut",
"url": "https://arxiv.org/abs/1211.0550"
}
|
1211.0566
|
# SLKMC-II study of self-diffusion of small Ni clusters on Ni (111) surface
Syed Islamuddin Shah islamuddin@knights.ucf.edu Giridhar Nandipati
giridhar.nandipati@ucf.edu Abdelkader Kara abdelkader.kara@ucf.edu Talat S.
Rahman talat.rahman@ucf.edu Department of Physics, University of Central
Florida, Orlando, FL 32816
###### Abstract
We studied self-diffusion of small 2D Ni islands (consisting of up to $10$
atoms) on Ni (111) surface using a self-learning kinetic Monte Carlo (SLKMC-
II) method with an improved pattern-recognition scheme that allows inclusion
of both fcc and hcp sites in the simulations. In an SLKMC simulation, a
database holds information about the local neighborhood of an atom and
associated processes that is accumulated on-the-fly as the simulation
proceeds. In this study, these diffusion processes were identified using the
drag method, and their activation barriers calculated using a semi-empirical
interaction potential based on the embedded-atom method. Although a variety of
concerted, multi-atom and single-atom processes were automatically revealed in
our simulations, we found that these small islands diffuse primarily via
concerted diffusion processes. We report diffusion coefficients for each
island size at various tepmratures, the effective energy barrier for islands
of each size and the processes most responsible for diffusion of islands of
various sizes, including concerted and multi-atom processes that are not
accessible under SLKMC-I or in short time-scale MD simulations.
###### pacs:
68.35.Fx, 68.43.Jk,81.15.Aa,68.37.-d
## I Introduction
Surface diffusion is of interest not only because it is so different from
diffusion in bulk solids Antczak and Ehrlich (2010) but because diffusion of
adatoms on metal surfaces, individually or as a group via multi-atom or
concerted diffusion processes plays an essential role in a wide variety of
such surface phenomena as heterogenous catalysis, epitaxial crystal growth,
surface reconstruction, phase transitions, segregation, and sintering.Zangwill
(1988) A precise knowledge of diffusion mechanisms is essential for
understanding and control of these phenomena.Kaxiras (1996) Adatoms can
diffuse on a substrate in a variety of ways, and competition between various
types of diffusion processes (due to the differences in their rates)
determines the shapes of the islands formed and (on macroscopic times scales),
the morphological evolution of thin films. Hence a great deal of effort has
been devoted to investigation of self-diffusion of adatom islands on metal
surfaces, initially using field ion microscopy (FIM),Bassett (1976); TSong and
Casanova (1980); Wang and Ehrlich (1990, 1992); Wang _et al._ (1998); Kellogg
(1993) and more recently scanning tunneling microscopy (STM).Wen _et al._
(1994); Pai _et al._ (1997); Giesen and Ibach (2003); Giesen _et al._
(1998); Fern _et al._ (2000); der Vegt _et al._ (1995); Busse _et al._
(2003); Muller _et al._ (2005); Liu and Adams (1992) Because of inherent
differences in the microscopic processes responsible for island diffusion on
different metal surfaces, this is still an on-going research problem. Both
experimental and theoretical studies for various systems have succeeded in
finding the activation barriers and prefactors for a single-adatom diffusion
processes Tung and Graham (1980); Fu and Tsong (2000); Flahive and Graham
(1980); Liu _et al._ (1991); Rice _et al._ (1992); Stoltze (1994); Li and
DePristo (1996); Mortensen _et al._ (1996); Chang _et al._ (2000); Haug and
Jenkins (2000); Kurpick (2001); Bulou and Massobrio (2005); Kim _et al._
(2006). Ref Antczak and Ehrlich, 2010 provides a good survey of those efforts.
To the best of our knowledge, however, there has so far been no systematic
experimental or theoretical effort to identify the diffusion mechanisms
responsible for diffusion of small 2D Ni islands on Ni(111) and to calculate
their activation barriers. In this article we report our results of doing so
for such islands, ranging in size from 1 to 10 atoms.
Arrangement of atoms in the substrate of an fcc(111) surface results in two
types of three-fold hollow sites for an adatom: the regular fcc site (with no
atom beneath it in the second layer), and an hcp site (with an atom beneath it
in the second layer). Occupancy of adatoms at fcc sites maintains the crystal
stacking order (ABC stacking) of fcc structure, while occupancy of hcp sites
leads to a stacking fault. Depending on its relative occupation energy, which
is material dependent, an adatom can occupy one or the other of these sites.
Which site is preferred on the fcc(111) surface affects the way diffusion and
hence growth proceeds. It is therefore important to understand whether the
diffusion proceeds via movement of atoms from fcc-to-fcc or hcp-to-hcp or fcc-
to-hcp hcp-to-fcc sites. It has been observed experimetally that for smaller
clusters mixed occupancy Repp _et al._ (2003) of fcc & hcp sites is possible.
A host of studies has been devoted to problems of self-diffusion and diffusion
mechanisms on metal fcc(111) surfaces, almost exclusively, however, with
either a preconceived set of processes or merely approximate activation
barriers. It is nevertheless crucial to discover the full range of processes
at work and to accurately establish the activation barrier of each. It is also
well known that the fcc(111) surface, being atomically flat, has the least
corrugated potential energy surface of any fcc surface, resulting in low
diffusion barriers even for clusters to diffuse as a whole. Consequently,
studies of diffusion processes on fcc(111) surfaces is a challenging problem
for both experiment and simulation even to this day. For a monomer and smaller
islands like dimer, trimer and up to certain extent, tetramer, all possible
diffusion processes may be guessed. But as islands further increase in size,
it becomes more difficult to enumerate all possible diffusion processes a
priori. An alternative is to resort to molecular dynamics (MD) simulation. But
because diffusion processes are rare events, an MD simulation cannot capture
every microscopic process possible, as most of the computational time is spent
in simulating atomic vibration of atoms. Instead, to do a systematic study of
small Ni island diffusion on Ni (111) surface we resorted to an on-lattice
self-learning kinetic Monte Carlo (SLKMC-II) method, which enables us to study
longer time-scales than are feasible with MD yet to find all the relevant
atomic processes and their activation barriers on-the-fly, as KMC methods
limited to a priori set of processes cannot do. Moreover, whereas previous
studies have used an on-lattice SLKMC method, Trushin _et al._ (2005); Karim
_et al._ (2006); Nandipati _et al._ (2009, 2011) in which adatoms were
restricted to fcc occupancy, in the present study both fcc and hcp occupancies
are allowed, and are detectible by our recently developed improved pattern-
recognition schemeShah _et al._ (2012).
The remainder of the paper is organized as follows. In Section II we discuss
the details of our SLKMC-II simulations, with particular attention to the way
we find diffusion processes and calculate their activation barriers. In
Section III we present details of concerted, important multi-atom and single-
atom diffusion processes responsible for the diffusion of Ni islands as a
function of island size. In Section IV we present a quantitative analysis of
diffusion coefficients at various temperatures and of effective energy
barriers as a function of island size. In Section V we present our
conclusions.
## II Simulation Details
To study Ni island diffusion on fcc Ni(111) surface, we carried out SLKMC
simulations using the pattern-recognition scheme we developed recently Shah
_et al._ (2012) that includes both fcc and hcp sites in the identification of
an atom’s neighborhood. Various types of diffusion processes are possible, and
their activation barrier depends on the atom’s local neighborhood. Whenever a
new neighborhood around an atom is identified, a saddle-point search is
carried out to find all the possible atomic processes and calculate their
activation barriers – provided that it has at least one similar empty site in
the second ringShah _et al._ (2012), since when an atom occupies an fcc (or
alternatively an hcp) site, the nearest neighbor (NN) hcp (or,
correspondingly, fcc) sites cannot be occupied. In our simulations we used a
system size of 16x16x5 with the bottom 2 layers fixed, and carried out saddle-
point searches using the drag method. In this method a central or active atom
is dragged in small steps towards a probable final position. If the central
atom is on an fcc (hcp) site, then it is dragged towards a NN vacant fcc (hcp)
site in the second ring. Since atoms are allowed to occupy either hcp or fcc
sites, an atom being dragged from an fcc (hcp) site to a neighboring similar
site is allowed to relax to an intermediate hcp (fcc) site in between the two
fcc (hcp) sites. In other words, processes are possible in which atoms in an
island may occupy fcc, hcp or both fcc & hcp sites simultaneously. In the drag
method, the atom being dragged is always constrained in the direction of the
reaction coordinate but allowed to relax along its other degrees of freedom
(those perpendicular to the reaction coordinate), while all the other atoms in
the system are allowed to relax in all degrees of freedom. Once the transition
state is found, the entire system is completely relaxed to find the final
state of the process. The activation barrier of the process is the difference
between the energies of the transition and initial states. We have verified
the activation barriers of some of the key processes found by the drag method
using the (more accurate but computationaly expensive) nudged-elastic band
(NEB) methodJónsson _et al._ (1998), and found no significant difference. For
inter-atomic interactions, we used an interaction potential based on the
embedded-atom method (EAM) as developed by Foiles et al.Foiles _et al._
(1986). In all our SLKMC simulations we used the same pre-exponential factor
of $10^{12}$s-1, which has been demonstrated to be a good assumption for such
systems as the one under examination here.Yildirim _et al._ (2007, 2005)
For the small islands under study here (1-10 atoms), we found that when an
atom is dragged rest of the atoms in the island usually follow. For very small
islands (1-4 atoms), $\it all$ of the processes identified by the drag method
were concerted-diffusion processes. As island size increases we found single-
atom and multi-atom processes as well. For islands of size 5-6, even single-
atom detachment processes are identified and stored in the database (even
though they are not allowed in our simulations). To account for all types of
processes associated with both compact and non-compact shapes – especially
concerted processes and multi-atom processes – we used $10$ rings to identify
the neighborhood around an active atom in our SLKMC simulations. Using $10$
rings corresponds to including fifth nearest-neighbor interactions. To make
sure we identified all the single-atom processes, we also carried out saddle-
point searches with all of the atoms fixed except the atom being dragged.
Although there is no infallible method for discovering all possible processes,
we did exhaust the search for possible processes identifiable using the drag
method.
In order to save computational time, we first carried out SLKMC simulations at
700K for each island size, and used the database thus generated to carry out
our simulations for the same size at lower tempratures (300, 400, 500 and 600K
). The rationale for this approach is that an island goes through many more
shapes at higher tempratures: when a simulation is carried out at a lower
temprature starting out with a database generated at a higher temprature, it
only rarely finds an unknown configuration. It is not possible, however, to
economize on computational time by using, for the smaller islands under study
here, a database generated for (say) the larger among them, because the types
of processes possible (along with their respective barriers) are dependent on
an island’s particular size.
## III Results
As mentioned above, all of the processes for a given island are identified and
their activation barriers calculated, and stored in a database on-the-fly. We
discuss in this section, however, only key processes of the various general
types (concerted, multi-atom and single-atom).
Fig. 1(a) is a sketch of the fcc(111) surface with its adsorption sites marked
as fcc and hcp. Determining whether an adatom is on an fcc or on an hcp site
on this surface requires knowledge of at least $2$ substrate layers below the
adatom layer. In all our figures we show only the adatom layer and the layer
below (the top substrate layer) with the convention that the center of an
upward-pointing triangle (along the y-axis) formed by the (top layer)
substrate atoms is an fcc site, while the center of a downward-pointing
triangle pointing triangle is an hcp site. An island on an fcc (111) surface
can be on fcc sites or on hcp sites or a combination of both sites (some atoms
of the island sitting on fcc sites and the rest on hcp sites). Depending on
the type of material either the fcc or the hcp site will be energetically
favorable. As we shall see for each island size under study here, the fcc site
for Ni(111) is always at least slightly more favorable than the hcp site.
Figure 1: (a) fcc and hcp sites on an fcc(111) surface, with corresponding
directions for concerted diffusion processes; (b) A-type and B-type step edges
(here, for an all-hcp island) for the same surface.
A compact adatom island on an fcc(111) surface can move in the three
directions shown in Fig.1(a). Note that the numbering scheme for the
directions open to an atom on an fcc site is inverse to that for those open to
an atom on an hcp site (see Fig. 1(a)). We follow the enumeration convention
for directions distinguished in Fig. 1(a) throughout the article in tabulating
activation barriers for concerted processes for islands of various sizes and
shapes. Concerted processes involve all atoms moving together from all-fcc
sites to all-hcp sites or vice-versa. In a concerted diffusion process a
cluster can either translate in one of the three directions shown in Fig. 1
(concerted translation) or rotate around an axis (around the center of mass),
either clockwise or anti-clockwise (concerted rotation). Since concerted
rotational processes do not produce any displacement in the center of mass of
an island, they do not contribute to island diffusion. Depending on the size
of the island and its shape, activation barriers for the processes in these
three directions can be different.
Activation barriers for single-atom processes, however, depend on the type of
step-edge along which atom diffuses. Fig. 1(b) shows, using the example of a
6-atom hcp island, how an A-type step-edge $\\--$ a (100) micro-step differs
from a B-type step-edge $\\--$ a (111) micro-step. We discuss important
single-atom diffusion processes systematically and in detail in Sub-section
III.11.
As island size increases not only does the frequency of single-atom processes
increase but the frequency of multi-atom processes does so as well. All multi-
atom mechanisms involve shearing. A special case is reptation mechanismChirita
_et al._ (2000, 1999), a two-step shearing process that moves the cluster from
all-fcc to all-hcp sites or the reverse: first, part of the island moves from
fcc to hcp sites; then the rest of the island moves from fcc to hcp. Hence at
the intermediate stage, the island has mixed fcc-hcp occupancy. In case of Ni-
island diffusion, reptation processes occur only when the shape of the island
becomes non-compact. We will discuss reptation in detail when we take up
islands of size 8-10.
### III.1 Monomer
As mentioned earlier, much work has been done to determine activation barrier
for Ni monomer diffusion on Ni (111) surfaceAntczak and Ehrlich (2010). A
monomer on fcc (111) surface can adsorb either on an fcc or an hcp site. We
find that adsorption of an adatom on an fcc site is slightly favored over than
on an hcp site by $0.002$ eV – in good agreement with the value reported in
Ref Liu _et al._ , 1991. Diffusion of a monomer occurs through hopping
between fcc sites via an intermediate hcp site. We find the activation barrier
for a monomer’s hopping from an fcc site to a neighboring hcp site to be
$0.059$ eV while that for the reverse process is $0.057$ eV. The effective
energy barrier for monomer is found to be 0.057 eV, which is consistent with
the result reported by Liu et al.Liu and Adams (1992) of $0.056$ eV.
### III.2 Dimer
On any fcc(111) surface a dimer (of the same species) can have three possible
arrangements: both atoms on fcc sites (an FF-dimer, Fig. 2(b)), both on hcp
sites (an HH-dimer, Fig. 2(a)) or one atom on an fcc and the other on an hcp
site (an FH-dimer, Fig. 2(c)). We find that the FF-dimer is energetically more
favorable than the HH-dimer by $0.005$eV and the FH-dimer the least favorable
by $0.011$ eV. We find that both FF and HH dimers diffuse via concerted as
well as single-atom processes, whereas the FH-dimer diffuses via single-atom
processes only.
Figure 2: Possible configurations for a dimer, with activation barriers for concerted diffusion processes. (a) FF dimer (both atoms on fcc sites); (b) HH dimer (both atoms on hcp sites); (c) FH dimer (one atom on an fcc and the other on an hcp site); (d) FF dimer in concerted clockwise rotation and (e) HH dimer in concerted counter-clockwise rotation. Table 1: Activation barriers (in eV) of concerted processes for dimer diffusion. Direction | fcc | hcp
---|---|---
1 | 0.071 | 0.066
2 | 0.148 | 0.143
3 | 0.148 | 0.143
In concerted diffusion processes, both atoms in a FF (HH) dimer move from fcc
(hcp) to the nearest hcp (fcc) sites as shown in Fig. 2 (a)&(b) and Fig. 2
(d)&(e), thereby converting an FF (HH) dimer into an HH (FF) dimer. In the
case of an FF (HH) dimer, the activation barrier for concerted translational
(Fig. 2 (a)&(b)) is $0.148$ eV ($0.143$ eV) while that for concerted rotation
(Fig. 2 (d)&(e)) is $0.038$ eV. In concerted dimer rotation, the activation
barriers for both clockwise and anti-clockwise directions are the same, as
they are symmetric to each other. Activation barriers for translational
concerted diffusion processes in all three directions (see Fig. 1) both for FF
and HH dimers are reported in Table. 1. Our results for concerted processes
are $0.028$ eV higher than the corresponding activation barriers for a dimer
reported in Ref. Liu and Adams, 1992. (This difference $\\--$ as with those in
what follows $\\--$ may be due to the different inter-atomic potential
employed in their study and ours.)
Single-atom processes transform both FF and HH dimers into an FH-dimer. In
this case one of the fcc atoms in an FF-dimer or an hcp atom in an HH-dimer
moves to a nearest-neighbor hcp or an fcc site respectively, as shown in Fig.
2 (a)&(b) with the double-headed arrow. The activation barriers are $0.034$ eV
and $0.035$ eV for hcp and fcc dimer, respectively. In the case of an FH-
dimer, two types of single-atom diffusion processes are possible, as shown in
Fig. 2: an fcc atom moves to the nearest hcp site in the direction of the open
arrowhead, forming an HH-dimer, or an hcp atom moves in the direction of the
solid arrowhead to the nearest fcc site, forming an FF-dimer. The activation
barriers for these processes are $0.028$ eV and $0.024$ eV, respectively.
### III.3 Trimer
Depending on where a third atom is attached to the dimers shown in Fig. 2(a &
b), there are four possible arrangements of atoms in a compact trimer: two
types of fcc timers – one centered around an hcp site (F3H), the other
centered around a top site (F3T) (see Figs. 3(a) & (d)) $\\--$ and two types
of hcp trimers $\\--$ one centered around an fcc site (H3F), the other
centered around a top site (H3T) (Figs. 3(c) & (b)). Although all four trimers
have the same shape, their local environment is different, so that their
adsorption energies are distinct, as are the activation barriers for their
possible diffusion processes. F3T timer is the most energetically favorable:
F3H, H3T and H3F are less energetically favorable by $0.006$, $0.007$ and
$0.0013$ eV, respectively. It should also be noted that although trimers can
take on non-compact shapes, the configurations depicted in Fig. 3 are the most
frequently observed in our trimer simulations.
Figure 3: Possible arrangements of atoms in a trimer, with possible concerted
diffusion processes and their activation barriers. (a)$\\--$(d) Concerted
translation: (a) F3H-all atoms on fcc sites centered around an hcp site; (b)
H3T-all atoms on hcp sites centered around a top site; (c) H3F-all atoms on
hcp sites centered around an fcc site; (d) F3T-all atoms on fcc sites centered
around a top site. (e) & (f) Concerted rotation: F3T and H3T respectively.
In the case of F3T and H3T trimers, two types of concerted processes were
observed, a non-diffusive concerted rotation (clockwise and anti-clockwise)
(Fig 3 (e) & (f)) and a diffusive concerted translation (in all three
directions) (Fig 3 (d) & (b)). Concerted rotation processes transform an H3T
timer into a F3T trimer and vice versa. The activation barrier for the
concerted rotation processes for F3T trimer is $0.150$ eV while that for those
of the H3T trimer is $0.144$ eV. Translation transforms an F3T timer into a
H3F timer and vice versa. The activation barrier for the concerted
translations possible for these two trimers are $0.200$ eV and $0.193$ eV for
F3T and H3T, respectively. The activation barrier of $0.200$ eV for
translational motion of F3T trimer is in agreement with the value reported in
Ref Liu and Adams, 1992. For F3H and H3F trimers, only concerted translation
processes are possible; their activation barriers are $0.194$ eV and $0.186$
eV (the value reported for the same process in Ref Liu and Adams, 1992 is
$0.187$ eV), respectively. Fig. 3 (a-b) & (c-d) reveal that these concerted
diffusion processes transform an F3H into an H3T timer and an H3F to an F3T
trimer. Since the shape of these trimers is symmetric (see Fig 3 (a) & (c)),
the activation barriers for their diffusion in all $3$ possible directions are
the same.
Table 2: Activation barriers (in eV) for single-atom diffusion processes for an H3T compact trimer in the directions shown in Fig. 4. Type | 1 | 2 | 3 | 4
---|---|---|---|---
F3T | 0.439 | 0.858 | 0.858 | 0.439
F3H | 0.432 | 0.875 | 0.875 | 0.432
H3T | 0.436 | 0.856 | 0.856 | 0.436
H3F | 0.429 | 0.872 | 0.872 | 0.429
As for single-atom processes in the case of a trimer: an atom can move in $4$
different directions as shown in Fig. 4, resulting $2$ different types of
single-atom processes: directions $1$ & $4$ correspond to edge-diffusion
processes which open up the trimer; directions $2$ & $3$ correspond to
detachment processes (excluded from the present study, which is confined to
diffusion of single whole islands, in which an island’s integrity [and hence
its size] is maintained). We note that these processes move atoms from fcc
(hcp) to nearest fcc (hcp) site. Activation barriers for processes in these
$4$ directions for different types of trimers are given in Table. 2. Because
these activation barriers are so high relative to those for concerted
processes, single-atom processes were rarely observed in our simulations of
trimer diffusion.
Figure 4: Single-atom processes possible for an H3T trimer. Activation
barriers for the processes in these 4 directions for the 4 posible trimer
configurations are given in Table. 2
We note that as island size increases, possible types of single-atom processes
increases as well (though with the decamer, basically all possible types have
appeared). Accordingly, it is convenient to defer detailed discussion of
single-atom processes until later (Sub-section. III.11)
### III.4 Tetramer
Adding another atom to any of the trimers shown in Fig. 3 (a - d) results in
the formation of a compact tetramer, diamond-shaped $\\--$ with a long
diagonal (along the line joining farthest atoms) and a short one
(perpendicular to the long one), as shown in Fig. 5. Once again the fcc island
(Fig. 5(a)) is energetically more favorable than the hcp one $\\--$ in this
case, by $0.009$ eV. Three types of translational concerted diffusion
processes are possible for each of the fcc and hcp tetramers, that is, one
along each of the three directions specified in Fig.1. An example of a
cncerted fcc-to-hcp process (along direction 1) for a tetramer is shown in
Fig. 5(a); its activation barrier is 0.213 eV. The reverse process (hcp to
fcc) is shown in Fig. 5(b); its activation barrier is 0.204 eV (the value
reported in Ref. 18 is 0.210 eV). Because the process in direction 3 is
symmetric to that in direction 1, the energy barriers for these processes are
identical, as are those for the reverse processes. The energy barrier along
direction 2 is 0.313 eV from fcc to hcp and 0.304 eV from hcp to fcc. These
values are systematically displayed in Table. 3.
Although the multi-atom processes shown in Fig. 5(c & d) have activation
barriers lower than those of single-atom processes, they could not be found
using the drag method. These processes thus had to be added manually to the
database and their activation barriers obtained using the nudged elastic band
method (NEB). In these multi-atom processes, two atoms move togehter in the
same direction, the result is a shearing mechanism as shown in Fig. 5(c) &
(d). For this shearing process, from fcc to hcp, the activation barrier is
0.285 eV; that for the reverse process from hcp to fcc is 0.276 eV. The drag
method also finds single-atom processes, but because in tetramers (as in
islands of size 3 $\\--$ 7) these have higher activation barriers than those
of concerted processes, they were not observed during the simulations.
Figure 5: Diffusion processes possible for a tetramer: (a) & (b) concerted diffusion along the short diagonal; (c) & (d) shearing processes. Table 3: Activation barriers (eV) for the concerted tetramer translation processes shown in Fig. 5(a)&(b). Direction | fcc | hcp
---|---|---
1 | 0.213 | 0.204
2 | 0.313 | 0.304
3 | 0.213 | 0.204
### III.5 Pentamer
Figure 6: Examples of concerted diffusion processes for a compact pentamer,
along with their activation barriers.
The compact shapes of a pentamer can be obtained by attaching an atom to a
diamond-shaped tetramer. Although the geometries of compact pentamer clusters
thus obtained are the same, the island’s diffusion is crucially affected by
where this additional atom is placed: attachment of an atom to an A-type step-
edge of an fcc tetramer results in the long A-type step-edge pentamer shown in
Fig. 6(c); attachment of an atom to a B-type step-edge of the same tetramer
results in the long B-type step-edge pentamer shown in Fig. 6(a)); the
corresponding results of attaching an atom to an hcp tetramer are shown in
Figs. 6(b) and (d), respectively. The most energetically favorable of these is
the fcc pentamer with a long A-type step-edge (Fig. 6(c)); less favorlable by
$0.005$ eV are the fcc pentamer with a long B-type step-edge (Fig. 6(a)), by
$0.011$ eV the hcp pentamer with a long A-type step-edge, and by $0.017$ eV
the hcp pentamer with a long B-type step-edge. That is: as usual, fcc islands
are more stable than hcp ones. And, within each of those types, pentamers with
a long A-type step-edge are more stable than those with a long B-type step-
edge.
Figure 7: The single-atom processes that convert the long A-type step-edge pentamer to the long B-type step-edge pentamer. Table 4: Activation barriers (eV) without parantheses are for concerted-translation processes of pentamers with a long A-type step-edge, as shown in Figs. 6 (c)&(b); barriers in parentheses are for such processes for pentamers with a long B-type step-edge, as shown in Figs. 6 (a)&(d). Directions | fcc | hcp
---|---|---
| A (B) | A (B)
1 | 0.348 (0.301) | 0.342 (0.284)
2 | 0.348 (0.353) | 0.342 (0.337)
3 | 0.295 (0.353) | 0.289 (0.337)
In our simulations we found that compact pentamers diffuse mostly via
concerted diffusion processes, which displace the island as a whole from fcc-
to-hcp or vice-versa. Fig. 6 shows concerted diffusion processes along
direction 1 for the long B-type step-edge pentamer and along direction 3 for
the long A-type step-edge pentamer. Table 4 displays activation barriers for
concerted processes in all 3 directions for both types of pentamer. Fig. 7
(a),(b) & (c) shows the single-atom processes that transform an fcc pentamer
from long A-type (= short B-type) to a short A-type (= long B-type) cluster,
with the activation barrier for each.
### III.6 Hexamer
Depending on whether a sixth atom is attached to a long A-type or to a long
B-type step-edge pentamer, there are $3$ possible compact shapes for a
hexamer: (1) when an atom is added in such a way as to extend the shorter edge
of either a long A-type or a long B-type step-edge pentamer, the result is one
of the parallelogramic hexamers shown in Fig. 8; (2) when an atom is attached
to the long edge of either type of pentamer, the result is the one of the
irregular hexamers shown in Fig. 9; ($3$) when an atom is added to the shorter
edge of either type of pentamer, the result is one of the triangular hexamers
with all step edges of either the A-type (Figs. 10(b) &(c)) or of the B-type
(Figs. 10(a) &(d)).
Figure 8: Parallelogramic hexamers obtained by extending (by one atom) the shorter edge of either the long A-type or long B-type pentamers: (a) fcc cluster; (b) hcp cluster. The activation barriers indicated are for concerted diffusion in direction $1$. Table 5: Relative stabilities of the hexamers most frequently observed in our simulations. P = Parallelogram; I = Irregular; T = Triangular. Type | Shape | Description | Energy (eV) | Reference
---|---|---|---|---
fcc | P | equal A & B steps | 0 | Fig. 8(a)
fcc | I | edge atom on A step | 0.011 | Fig. 9(b)
fcc | I | edge atom on B step | 0.011 | not shown
hcp | P | equal A & B steps | 0.014 | Fig. 8(b)
fcc | T | all A steps | 0.019 | Fig. 10(c)
hcp | I | edge atom on B step | 0.024 | Fig. 9(b)
hcp | I | edge atom on A step | 0.024 | not shown
fcc | T | all B steps | 0.029 | Fig. 10(a)
hcp | T | all A steps | 0.032 | Fig. 10(b)
hcp | T | all B steps | 0.042 | Fig. 10(d)
Table. 5 shows the order of relative stabilties of the hexamers most
frequently obseved in our simulations. It reveals that, for $\it a$ $\it
given$ $\it shape$, hexamers on fcc sites are more energetically favored than
those on hcp sites, and that among hexamers on fcc sites (as for those on hcp
sites), clusters in which A steps are longer than B steps are more stable.
Figs. 8-10 also show concerted diffusion processes (in direction 1) for these
hexamers, together with the activation barriers for each. Tables. 6 & 7 give
activation barriers for the hexamers shown in Figs. 8 & 9, respectively. Since
triangular hexamers (Fig. 10) are symmetric, their activation barriers for
concerted diffusion are same in all three directions.
Figure 9: Irregular hexamers obtained by attaching an atom to the long edge of a pentamer: (a) fcc cluster; (b) hcp cluster. The activation barriers indicated are for concerted diffusion in direction $1$. Figure 10: Triangular hexamers obtained by adding an atom to the short edge of a pentamer: (a) fcc hexamer with B-type step edges; (b) hcp hexamer with A-type step edges; (c) fcc hexamer with A-type step edges; (d) hcp hexamer with B-type step edges. The activation barriers here are for concerted diffusion in direction $1$. Table 6: Activation barriers (eV) of the concerted translation processes in all three directions for the hexamer shown in Fig.8 Directions | fcc | hcp
---|---|---
1 | 0.374 | 0.360
2 | 0.466 | 0.451
3 | 0.254 | 0.240
Table 7: Activation barriers (eV) of concerted translations processes in all $3$ directions for the hexamers shown in Fig.9 Directions | fcc | hcp
---|---|---
1 | 0.393 | 0.380
2 | 0.397 | 0.383
3 | 0.391 | 0.378
Fig. 11 shows the most frequently observed multi-atom processes for a hexamer
– shearing processes in which a dimer moves along the A-type step-edge of the
cluster from sites of one type to the nearest-neighbor sites of the same type.
Figs. 11 (a) & (b) show this kind of diffusion process for an hcp cluster and
Figs. 11 (c) & (d) for an fcc cluster. Although this dimer shearing process
does not much displace the center of mass of a hexamer; it does have a
striking consequence: it converts a parallelogramic hexamer (Fig. 8) into an
irregular hexamer, (Fig. 9) and vice-versa.
Figure 11: Dimer shearing processes in case of a hexamer along with their
activation barriers: (a) & (b) all-hcp hexamers; (c) & (d) all-fcc clusters.
### III.7 Heptamer
Figure 12: Concerted diffusion processes and their activation barriers in
direction $1$ for a heptamer.
On an fcc(111) surface, an heptamer has a compact closed-shell structure with
each edge atom having at least three nearest-neighbor bonds, as shown in Fig.
12. Our SLKMC simulations (keep in mind here the range of temperatures to
which they were confined) found that heptamer diffuses exclusively via
concerted diffusion processes, which displace the cluster from fcc-to-hcp and
vice versa; the barriers for which are shown in Fig. 12. That these processes
will predominate can also be concluded from the fact that the effective energy
barrier for heptamer diffusion (cf. Table. 23) is close to the average of the
activation barriers shown in Fig. 12.
Since the compact heptamer has a symmetric shape, activation barriers in all
three directions are the same as those shown in Fig. 12. Again, the fcc island
is more energetically favorable than its hcp counterpart – in this case by
$0.015$ eV.
### III.8 Octamer
Figure 13: Possible orientations for a compact fcc octamer: (a) with long
A-type step edges; (b) with long B-type step edges.
Compact octamers have two distinct orientations, one with two long A-type
step-edges, the other with two long B-type step-edges, as shown in Fig. 13 (a)
& (b) for an fcc octamer. Octamers with long A-type step-edges (Fig. 13 (a))
can be obtained by attaching an atom to any B-type step-edge of a compact
heptamer, while a compact octamer with long B-type step-edges results from
attaching one to any A-type step-edge. Again: the fcc islands are more
energetically favorable than the hcp ones, and within each type, islands with
long A-type step edges are more stable than those with long B-type step-edges.
A compact octamer diffuses via concerted diffusion processes, as shown in Fig
14. The activation barrier of a concerted diffusion process depends on whether
the octamer has long A- or long B-type step-edges. As Fig. 14 shows, a
concerted diffusion process converts a long A-type step-edge fcc octamer into
a long B-type step-edge hcp cluster, and vice-versa. Table. 8 reports the
activation barriers for concerted diffusion processes in all $3$ directions
for both orientations for an fcc as well as an hcp (see Fig. 13) octamer.
Concerted diffusion processes in directions $2$ and $3$ are the most
frequently observed processes in octamer diffusion.
Figure 14: Examples of concerted diffusion processes in direction 2 for an octamer: (a) compact fcc octamer with long A-type step-edges; (b) compact hcp octamer with long B-type step-edges. Table 8: Activation barriers (in eV) of the twelve concerted diffusion processes for compact octamers. Directions | fcc | hcp
---|---|---
| A (B) | A (B)
1 | 0.589 (0.585) | 0.567 (0.571)
2 | 0.491 (0.484) | 0.468 (0.469)
3 | 0.491 (0.484) | 0.468 (0.469)
Although an octamer diffuses primarily via concerted processes we found in our
simulations that both multi-atom and single-atom processes are also relatively
common. As mentioned before, we defer comprehensive discussion of single-atom
processes to section III.11. Here we discuss multi-atom processes particular
to octamers.
Figure 15: Example of shearing diffusion processes within an octamer along
with their activation barriers.
Multi-atom processes involving shearing and reptation are shown in Figs. 15 &
16, respectively. In shearing processes, $\it part$ of the island (more than
one atom) moves from fcc to the nearest fcc sites, if all the island is
initially on fcc sites – or from hcp to hcp sites, if all of it initially sits
on hcp sites. Figs. 15 (a) & (b) show trimer shearing processes within an
octamer, along with their activation barriers, while Figs. 15 (c) & (d) show
dimer shearing processes, with their activation barriers.
Reptation is a 2-step diffusion process. In the case of an fcc island, the
entire island diffuses from fcc to nearest-neighbor hcp sites in two steps. In
the first, part of the island moves from fcc to nearest-neighbor hcp sites,
leaving part of the island on fcc sites and part on hcp sites. In the next,
the remainder of the island initially on fcc sites moves to hcp sites. Figs.
16 (a)-(d) shows various steps (subprocesses) of a reptation process, with
their activation barriers.
Figure 16: Successive sub-processes (or steps) involved in an octamer
reptation diffusion mechanism.
### III.9 Nonamer
Figure 17: Concerted diffusion processes and their activation barriers for
nonamers. Figure 18: Dimer shearing processes and their activation barriers
for compact nonamers.
For a nonamer, we observed all types of diffusion processes – single-atom,
multi-atom and concerted. The most frequently observed were two types of
single-atom mechanisms: edge-diffusion processes along an A- or a B-type step-
edge and corner rounding (Fig. 21). The nonamer is the smallest island for
which concerted processes are not the most frequently picked (the concerted
processes for the most frequently observed nonamer configurations [compact or
nearly so] are shown in Fig. 17). Even so, concerted processes contribute the
most to $\it island$ diffusion: that is, the displacement they produce in the
nonamer’s center of mass is far greater than that produced by single-atom
processes, despite the far greater frequency of the latter. This is reflected
in the fact that the effective energy barrier for nonamer (cf. Table 12) is
much closer to the average activation barrier for concerted processes (cf.
Table. 9) than for that of single-atom processes (cf. Table 9). The fact that
the effective activation barrier is slightly higher than the average energy
barrier for concerted processes is due mainly to the contribution of kink
processes, which do contribute somewhat to island diffusion.
Table 9: Activation barriers (eV) of concerted translations processes in all
$3$ directions for the nonamers shown in Figs.17(a–d).
Directions | fcc | hcp
---|---|---
| (a) | (b)
1 | 0.520 | 0.499
2 | 0.626 | 0.605
3 | 0.605 | 0.583
fcc | hcp
---|---
(c) | (d)
0.486 | 0.465
0.486 | 0.465
0.693 | 0.672
The most frequently observed multi-atom processes are the four forms of dimer
shearing along an A-type step-edge shown in Fig. 18(c)-(f), as have been
discussed above for island of sizes 6 & 8\. The activation barriers for these
dimer shearing processes are lower than those for single-atom diffusion
processes along an edge and also for some corner rounding processes. Reptation
processes also show up, but only when the nonamer is non-compact (we do not
illustrate these here) Shah _et al._ (2012).
### III.10 Decamer
Even in the case of a decamer, we have observed single-atom, multi-atom and
concerted diffusion processes. Single-atom diffusion processes are the most
frequently observed. The most frequently observed compact shape of decamer
during our simulations is that shape shown in Fig. 19, which has the same
number of A- and B-type step edges. As usual, an fcc cluster is energetically
more favorable than an hcp cluster.
Table 10: Activation barriers (eV) of concerted diffusion processes in all $3$ directions of decamer as shown in Fig.19 (a) & (b). Directions | fcc | hcp
---|---|---
1 | 0.661 | 0.638
2 | 0.700 | 0.677
3 | 0.700 | 0.677
For the shape shown in Fig. 19, the most frequently observed concerted
diffusion processes are those shown in the same figure, along with their
activation barriers reported in Table 10. It can be seen from the Table 23,
that the effective energy for decamer diffusion is close to that of the
average energy barrier of these concerted processes. That is why decamer
diffusion is dominated by concerted processes.
Figure 19: Frequent concerted diffusion processes and their activation
barriers for compact decamers. Figure 20: Diffusion steps (sub-processes) in
decamer reptation. Note that the decamers in (b) and (c) are identical, though
the arrows indicate different processes.
As with the nonamer, a decamer also undergoes multi-atom processes (shearing
and reptation). Of these, the most frequently observed is dimer shearing along
an A-type step-edge, similar to what has been discussed for clusters of size
6, 8 and 9. Fig. 20 shows the sub-processes in a reptation process, along with
the activation barrier of each.
### III.11 Single-atom Processes
In this section we provide detail about single-atom processes: edge-diffusion,
corner rounding, kink attachment, and kink detachment, as shown in Fig. 21 for
an hcp island. Their corresponding activation barriers and those for their fcc
analogues are given in Table 11. In each single-atom process, an atom on an
fcc site moves to a nearest-neighbor fcc site, while an atom on hcp site moves
to a nearest-neighbor hcp site. The activation barriers for single-atom
processes depend not only on whether the atom is part of an fcc sland or an
hcp island, but also on whether the diffusing atom is on an A-type or a B-type
step-edge.
In classifying single-atom processes in Table 11 we have used the notation
X${}_{n_{i}}$U $\rightarrow$ Y${}_{n_{f}}$V, where where X or Y = A (for an
A-type step-edge) or B (for a B-type step-edge) or K (for kink) or C (for
corner) or M (for monomer); ni = the number of nearest-neighbors of the
diffusing atom before the process; nf = the number of that atom’s nearest
neighbors after the process. U or V = A or B (for corner or kink processes) or
null (for all other other process types).
For example, process 1, B2 $\rightarrow$ B2, is a single-atom B-step edge
process in which the diffusing atom has 2 nearest-neighbors before and after
the process. Process 3, C1B $\rightarrow$ B2, is a corner rounding process
towards a B-step, the diffusing atom starting on the corner of a B-step with
one nearest-neighbor and ending up on the B-step with two nearest-neighbors.
In process 10, C2A $\rightarrow$ C1B, the diffusing atom begins on the corner
of an A-step having two nearest-neighbors and ends up on the corner of a
B-step with only one nearest-neighbor.
Figure 21: Single-atom processes for an hcp island (Though analogous processes occur for an fcc island, we do not illustrate them here). The index numbers designate the processes described in Table 11, which gives the activation barriers for each. Table 11: Activation barriers (eV) of single-atom processes for both fcc and hcp islands. The index numbers refer to the types of processes illustrated in Fig. 21. See text for explaination of the notation used to classify the process types. Index no. | Process type | fcc | hcp
---|---|---|---
1 | B2 $\rightarrow$ B2 | 0.454 | 0.448
2 | B2 $\rightarrow$ M | 0.821 | 0.815
3 | C1B $\rightarrow$ B2 | 0.177 | 0.173
4 | C1B $\rightarrow$ M | 0.458 | 0.455
5 | C1B $\rightarrow$ A2 | 0.040 | 0.038
6 | C1B $\rightarrow$ C1B | 0.540 | 0.585
7 | C1B $\rightarrow$ M | 0.811 | 0.809
8 | A2 $\rightarrow$ M | 0.795 | 0.794
9 | A2 $\rightarrow$ A2 | 0.326 | 0.307
10 | C2A $\rightarrow$ C1B | 0.399 | 0.397
11 | C2A $\rightarrow$ M | 0.787 | 0.785
12 | B2 $\rightarrow$ K3B | 0.415 | 0.298
13 | K3A $\rightarrow$ A2 | 0.601 | 0.701
14 | A2 $\rightarrow$ K3A | 0.302 | 0.389
15 | K3B $\rightarrow$ B2 | 0.729 | 0.597
16 | K3B $\rightarrow$ C1B | 0.731 | 0.787
17 | K3B $\rightarrow$ M | 1.138 | 1.150
18 | K3B $\rightarrow$ C1B | 0.820 | 0.759
## IV Diffusion coefficients and effective energy barriers
We start our SLKMC simulations with an empty database. Every time a new
configuration (or neighborhood) is turned up, SLKMC-II finds on the fly all
possible processes using the drag method, calculates their activation barriers
and stores them in the database as the simulation proceeds. Calculation of
energetics occurs at each KMC step during initial stages of the simulation
when the database is empty or nearly so, and ever less frequently later on.
Recall that the types of processes and their activation barriers are dependent
on island size, each one of which requires a separate database that cannot be
derived from that for islands of other sizes.
Figure 22: Arrhenious plots for 1$\\--$10 atom islands.
We carried out 107 KMC steps for each island size at temperatures 300K, 400K,
500K, 600K and 700K. We calculated the diffusion coefficient of an island of a
given size using Einstein Equation:Einstein (1905) $D=\lim_{t\to\infty}\langle
R_{CM}(t)-R_{CM}(0)]^{2}\rangle/2dt$, where $R_{CM}(t)$ is the position of the
center of mass of the island at time $t$, and $d$ is the dimensionality of the
system, which in our case is $2$. Diffusion coefficients thus obtained for
island sizes $1-10$ at various temperatures are summarized in Table 12. At
$300$ K, diffusion coefficients range from $1.63\times 10^{11}$ Å2/s for a
monomer to $8.66\times 10^{01}$ Å2/s for a decamer. Effective energy barriers
for islands are extracted from their respective Arrhenius plots (Fig. 22) and
also summarized in Table 12. Fig. 23 plots effective energy barrier as a
function of island size. It can be seen that the effective energy barrier
increases almost linearly with island size.
Figure 23: Effective energy barriers of 1$\\--$10 atom islands as a function of island size. Table 12: Diffusion coefficients ((Å2/s) at various temperatures and effective energy barriers for Ni islands. Island Size | 300K | 400K | 500K | 600K | 700K | $E_{eff}$ (eV)
---|---|---|---|---|---|---
1 | $1.63\times 10^{11}$ | $2.85\times 10^{11}$ | $3.99\times 10^{11}$ | $5.00\times 10^{11}$ | $5.87\times 10^{11}$ | 0.058
2 | $8.14\times 10^{09}$ | $2.88\times 10^{10}$ | $6.54\times 10^{10}$ | $1.13\times 10^{11}$ | $1.64\times 10^{11}$ | 0.136
3 | $1.09\times 10^{09}$ | $7.42\times 10^{09}$ | $2.30\times 10^{10}$ | $5.00\times 10^{10}$ | $8.23\times 10^{10}$ | 0.196
4 | $7.64\times 10^{08}$ | $5.83\times 10^{09}$ | $2.02\times 10^{10}$ | $4.91\times 10^{10}$ | $9.93\times 10^{10}$ | 0.217
5 | $2.90\times 10^{06}$ | $8.42\times 10^{07}$ | $6.59\times 10^{08}$ | $2.55\times 10^{09}$ | $7.36\times 10^{09}$ | 0.353
6 | $1.08\times 10^{06}$ | $7.39\times 10^{07}$ | $5.66\times 10^{08}$ | $2.69\times 10^{09}$ | $8.12\times 10^{09}$ | 0.400
7 | $6.24\times 10^{04}$ | $6.40\times 10^{06}$ | $1.01\times 10^{08}$ | $6.37\times 10^{08}$ | $2.37\times 10^{09}$ | 0.477
8 | $1.38\times 10^{04}$ | $1.50\times 10^{06}$ | $2.31\times 10^{07}$ | $1.57\times 10^{08}$ | $5.91\times 10^{08}$ | 0.482
9 | $5.48\times 10^{02}$ | $8.77\times 10^{04}$ | $1.89\times 10^{06}$ | $1.35\times 10^{07}$ | $6.23\times 10^{07}$ | 0.525
10 | $8.66\times 10^{01}$ | $2.25\times 10^{04}$ | $7.20\times 10^{05}$ | $7.97\times 10^{06}$ | $4.95\times 10^{07}$ | 0.627
## V Further Discussion And Conclusions
To summarize: we have performed a systematic study of the diffusion of small
Ni islands (1-10 atoms) on Ni(111), using a self-learning KMC method with a
newly-developed pattern recognition scheme (SLKMC-II) in which the system is
allowed to evolve through mechanisms of its choice on the basis of a self-
generated database of single-atom, multiple-atom and concerted diffusion
processes (each with its particular activation barrier) involving fcc-fcc,
fcc-hcp and hcp-hcp jumps. We find that concerted diffusion processes
contribute the most to the displacement of the center of mass (i.e., to island
diffusion), while single-atom processes contribute the least. As for multi-
atom processes (reptation or shearing): while these produce more displacement
than the latter, they are hardly ever selected. Though the energy barriers for
reptation processes are small compared to those for concerted diffusion
processes, reptation occurs only when an island is transformed into a non-
compact shape, as happens only rarely in the temprature range to which our
study is confined. In contrast, though shearing occurs with close-to-compact
shapes (which appear more frequently than do non-compact shapes, but with
islands of certain sizes), the barriers for these processes are higher than
those for reptation. Finally, although for all island sizes, island diffusion
is primarily dominated by concerted diffusion processes, the frequency of
occurrence of both single-atom and multi-atom processes does increase with
increase in island size, owing to increase in the activation barrier for
concerted diffusion processes with island size.
Figure 24: Distribution of single-atom, multi-atom, concerted and total
processes for 1-10 atom islands accumulated in the database during SLKMC
simulations. Inset shows the log-linear plot for up to the 6-atom island.
Fig. 24 shows the number of each type of diffusion process (single-, multi-
atom and concerted) collected during our SLKMC-II simulations for each island
size (1-10), together with the total number of processes of all types for each
island size. (For the sake of clarity, the insert shows a log-linear plot of
these quantities for island sizes 1-6). It can be seen that the number of
processes accumulated increases with island size, and significantly so beyond
the tetramer. It can also be seen from Fig. 24 that the overall increase in
number of processes with island size is constituted predominantly by
significant increases in single-atom processes and (to a lesser degree) multi-
atom processes. Meanwhile the number of concerted processes accumulated in the
database increases at a much slower pace with island size.
This significant increase in single-atom processes is mainly due to the use of
$10$ rings to identify the neighborhood of an atom. Use of $10$ rings
corresponds to inclusion of $5$ nearest-neighbor fcc-fcc or hcp-hcp
interactions. Elsewhere we show that $6$ rings (which corresponds to $3$
nearest-neighbor interactions) offer a range of interaction sufficient for
accurately calculating the activation barriers for single-atom processes.Shah
_et al._ (shed) But it is essential to include the long-range interaction (and
hence 10 rings) if one aims to accurately take into account multi-atom and
concerted processes, the latter of which predominate in small-island
diffusion. This significant increase in the number of processes with island
size also justifies resorting to an automatic way of finding all the possible
processes during simulations instead of using a fixed (and thus necessarily
preconceived) list of events.
As mentioned earlier, with increasing island size not only does the number of
accumulated single-atom processes and multi-atom processes increase but also
their frequency of occurrence. Still island diffusion is primarily due to
concerted diffusion processes, since it is these that produce largest
displacement of center of mass. This can be easily observed by comparing, for
each island size, the effective diffusion barriers given in Table 12 with the
activation barriers in the tables given in Sect. III for concerted diffusion
processes: effective diffusion barriers more or less closely follow activation
barriers for concerted diffusion processes – except for the 9-atom island, in
which the contribution of single-atom processes to the island’s diffusion is
significantly larger than for other island sizes.
In conclusion, Ni small-island diffusion on Ni (111) is primarily due to to
concerted diffusion processes, even though the frequency of their occurrence
decreases with increase in island size.
###### Acknowledgements.
We would like to acknowledge computational resources provided by University of
Central Florida. We also thank Lyman Baker for critical reading of the
manuscript and Oleg Trushin for his suggestions during initial stages of this
work.
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|
arxiv-papers
| 2012-11-02T21:27:15 |
2024-09-04T02:49:37.516753
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Syed Islamuddin Shah, Giridhar Nandipati, Abdelkader Kara, and Talat\n S. Rahman",
"submitter": "Syed Islamuddin Shah",
"url": "https://arxiv.org/abs/1211.0566"
}
|
1211.0709
|
# Shaping Operations to Attack Robust Terror Networks
Devon Callahan, Paulo Shakarian
Network Science Center and
Dept. of Electrical Engineering and Computer Science
United States Military Academy
West Point, NY 10996
Email: devon.callahan[at]usma.edu,
paulo[at]shakarian.net Jeffrey Nielsen, Anthony N. Johnson
Network Science Center and
Dept. of Mathematical Science
United States Military Academy
West Point, NY 10996
Email: jeffrey.nielsen[at]usma.edu,
anthony.johnson[at]usma.edu
###### Abstract
Security organizations often attempt to disrupt terror or insurgent networks
by targeting “high value targets” (HVT’s). However, there have been numerous
examples that illustrate how such networks are able to quickly re-generate
leadership after such an operation. Here, we introduce the notion of a shaping
operation in which the terrorist network is first targeted for the purpose of
reducing its leadership re-generation ability before targeting HVT’s. We look
to conduct shaping by maximizing the network-wide degree centrality through
node removal. We formally define this problem and prove solving it is NP-
Complete. We introduce a mixed integer-linear program that solves this problem
exactly as well as a greedy heuristic for more practical use. We implement the
greedy heuristic and found in examining five real-world terrorist networks
that removing only $12\%$ of nodes can increase the network-wide centrality
between $17\%$ and $45\%$. We also show our algorithm can scale to large
social networks of $1,133$ nodes and $5,541$ edges on commodity hardware.
## I Introduction
Terrorist and insurgent networks are known for their ability to regenerate
leadership after targeted attacks. For example, the infamous Al Qaeda in Iraq
terrorist leader Abu Musab al-Zarqawi was killed on June 8th, 2006
111http://www.nytimes.com/2006/06/08/world/middleeast/08cnd-iraq.html?_r=1
only to be replaced with Abu Ayyub al-Masri about a week later.
222http://articles.cnn.com/2006-06-15/world/iraq.main_1_al-zarqawi-al-qaeda-
leader-zawahiri?_s=PM:WORLD Here, we introduce the notion of a shaping
operation in which the terrorist network is first targeted for the purpose of
reducing its leadership re-generation ability. Such shaping operations would
then be followed by normal attacks against high value targets – however the
network would be less likely to recover due to the initial shaping operations.
In this paper, we look to shape such networks by increasing network-wide
centrality, first introduced in [1]. Intuitively, this measure provides
insight into the criticality of high-degree nodes. Hence, a network with a low
network-wide centrality is a more decentralized organization and likely to
regenerate leadership. In the shaping operations introduced in this paper, we
seek to target nodes that will maximize this measure - making follow-on
attacks against leadership more effective. Previous work has primarily dealt
with the problem of leadership regeneration by focusing on individuals likely
to emerge as new leaders [2]. However, targeting or obtaining information
about certain individuals may not always be possible. Hence, in this paper, we
target nodes that affect the reduce the network’s ability regenerate
leadership as a whole.
The main contributions of this paper is the introduction of a formal problem
we call FRAGILITY (Section II) which seeks to find a set of nodes whose
removal would maximize the network-wide centrality. We also included in the
problem a “no strike list” - nodes in the network that cannot be targeted for
various reasons. This is because real-world targeting of terrorist or
insurgent networks often includes restrictions against certain individuals. We
also prove that this problem is NP-complete (and the associated optimization
problem is NP-hard) which means that an efficient algorithm to solve it
optimally is currently unknown. We then provide two algorithms for solving
this problem (Section III). Our first algorithm is an integer program that
ensures an exact solution and, though intractable by our complexity result,
may be amenable to an integer program solver. Then we introduce a greedy
heuristic that we show experimentally (in Section IV) to provide good results
in practice (as we demonstrate on six different real-world terrorist networks)
and scales to networks of $1,133$ nodes and $5,541$ edges. In examining five
real-world terrorist networks, we found that successful targetting operations
against only $12\%$ (or less) of nodes can increase the network-wide
centrality between $17\%$ and $45\%$. Additionally, we discuss related work
further in Section V.
We would like to note that the targeting of individuals in a terrorist or
insurgent network does not necessarily mean to that they should be killed. In
fact, for “shaping operations” as the ones described in this paper, the
killing of certain individuals in the network may be counter-productive. This
is due to the fact that the capture of individuals who are likely emergent
leaders may provide further intelligence on the organization in question.
## II Technical Preliminaries and Computational Complexity
We assume that an undirected social network is represented by the graph
$G=(V,E)$. Additionally, we assume a “no strike” set, $S\subseteq V$.
Intuitively, these are nodes in a terrorist/insurgent network that cannot be
targeted. This set is a key part of our framework, as real-world targeting of
terrorist and/or insurgents in a terrorist/insurgent network is often
accompanied by real-world constraints. For example, consider the following:
* •
We may know an individual’s relationships in the terrorist/insurgent network,
but may not have enough information (i.e. where he or she may reside, enough
evidence, etc.) to actually target him or her.
* •
The potential target may be politically sensitive.
* •
The potential target may have fled the country or area of operations but still
maintains his or her role in the terrorist/insurgent network through
electronic communication.
* •
The potential “target” may actually be a source of intelligence and/or part of
an ongoing counter-intelligence operation (i.e. as described in [3]).
Throughout this paper we will also use the following notation. The symbols
$N_{G},M_{G}$ will denote the sizes of $V,E$ respectively. For each $i\in V$,
we will use $d_{i}$ to denote the degree of that node (the number of
individuals he/she is connected to) and $\eta_{i}$ to denote the set of
neighbors and we extend this notation for subsets of $V$ (for
$V^{\prime}\subseteq V,\eta(V^{\prime})=\bigcup_{i\in V^{\prime}}\eta_{i}$).
We will use the notation $\kappa_{i}$ to denote all edges in $E$ that are
adjacent to node $i$ and the notation $d^{*}_{G}$ to denote the maximum degree
of the network. Given some subset $V^{\prime}\subseteq V$, we will use the
notation $G(V^{\prime})$ to denote the subgraph of $G$ induced by
$V^{\prime}$. We describe an example network in Example II.1.
###### Example II.1
Consider network $G_{sam}$ in Figure 1. Nodes a and b may be leaders of a
strategic cell that provides guidance to attack cells (nodes c-f and g-j).
Note that no members in the attack cells are linked to each other. Also note
that if node a is the leader, and targeted, he could easily be replaced by b.
Figure 1: Sample network ($G_{sam}$) for Example II.1.
### A Network-Wide Degree Centrality
We now introduce the notion of network-wide degree centrality as per [1]. The
key intuition of this paper is to use this centrality as a measure of the
network’s ability to re-generate leadership.
###### Definition II.1 (Network-Wide Degree Centrality [1])
The degree centrality of a network $G$, denoted $C_{G}$ is defined as:
$\displaystyle C_{G}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}d^{*}_{G}-d_{i}}{(N_{G}-1)(N_{G}-2)}$ (1)
We note that there are other types of network-wide centrality (i.e. network-
wide betweenness, closeness, etc.). We leave the consideration of these
alternate definitions of network-wide centrality to future work. Freeman [1]
shows that for a star network, the quantity $\sum_{i}d^{*}_{G}-d_{i}$ equals
$(N_{G}-1)(N_{G}-2)$ \- and this is the maximum possible value for this
quantity. Hence, the value for $C_{G}$ can be at most $1$. As this equation is
clearly always positive, network-wide degree centrality is a scalar in
$[0,1]$. Turning back to Example II.1, we can compute $C_{G_{sam}}=0.38$ \-
which seems to indicate that in this particular terrorist/insurgent network
that, after leadership is targeted, there is a cadre of second-tier
individuals who can eventually take control of the organization. Throughout
this paper, we find it useful to manipulate Equation 1 as follows.
$\displaystyle C_{G}$ $\displaystyle=$
$\displaystyle\frac{N_{G}d^{*}_{G}-2M_{G}}{(N_{G}-1)(N_{G}-2)}$ (2)
We notice that the centrality of a network really depends on three things:
number of nodes, number of edges, and the highest degree of any node in the
network. We leverage this re-arranged equation in many of our proofs. Further,
we will use the function $fragile_{G}:V\rightarrow\Re$ to denote the level of
network-wide of the graph after some set of nodes is removed. Hence,
$fragile_{G}(V^{\prime})=C_{G(V-V^{\prime})}$. We note that this function has
some interesting characteristics. For example, for some subset
$V^{\prime}\subset V$ and element $i\in V-V^{\prime}$, it is possible that
$fragile_{G}(V^{\prime})>fragile(V^{\prime}\cup\\{i\\})$ or
$fragile_{G}(V^{\prime})<fragile(V^{\prime}\cup\\{i\\})$, hence $fragile_{G}$
is not necessarily monotonic or anti-monotonic in this sense. Further, given
some additional element $j\in V-V^{\prime}$, it is possible that
$fragile_{G}(V^{\prime}\cup\\{j\\})-fragile_{G}(V^{\prime})>fragile_{G}(V^{\prime}\cup\\{i,j\\})-fragile_{G}(V^{\prime}\cup\\{j\\})$
or
$fragile_{G}(V^{\prime}\cup\\{j\\})-fragile_{G}(V^{\prime})<fragile_{G}(V^{\prime}\cup\\{i,j\\})-fragile_{G}(V^{\prime}\cup\\{j\\})$.
Hence, $fragile_{G}$ is not necessarily sub- or super- modular either.
Consider Example II.2.
###### Example II.2
Consider the network $G_{sam}$ in Figure 1. Here,
$fragile_{G_{sam}}(\emptyset)=0.33$,
$fragile_{G_{sam}}(\\{a\\})=fragile_{G_{sam}}(\\{b\\})=0.57$,
$fragile_{G_{sam}}(\\{c\\})=0.30$, and $fragile_{G_{sam}}(\\{a,b\\})=0.0$. The
fact that $fragile_{G_{sam}}(\\{c\\}<fragile_{G_{sam}}(\emptyset)$ and
$fragile_{G_{sam}}(\\{a\\}>fragile_{G_{sam}}(\emptyset)$ illustrate that
$fragile_{G_{sam}}$ is not necessarily monotonic or anti-monotonic. Now let us
consider the incremental increase of adding an additional element. Adding $a$
to $\emptyset$ causes $fragile_{G_{sam}}$ to increase by $0.24$ while adding
$a$ to $\\{b\\}\supset\emptyset$ causes $fragile_{G_{sam}}$ to decrease by
$0.57$ \- implying sub-modularity. However, adding $c$ to $\emptyset$ causes
$fragile_{G_{sam}}$ to decrease by $0.03$ while adding $c$ to set
$\\{a,b\\}\supset\emptyset$ causes $fragile_{G_{sam}}$ to increase by $0.1$
(as $fragile_{G_{sam}}(\\{a,b,c\\}=0.1$) - implying super-modularity. Hence,
$fragile_{G_{sam}}$ is not necessarily sub- or super- modular.
### B Problems and Complexity Results
We now have all the pieces to introduce our problems of interest. We include
decision and optimization versions.
$FRAGILITY(k,x,G,S)$:
INPUT: Natural number $k$, real number $x$, network $G=(V,E)$, and no-strike
set $S$
OUTPUT: “Yes” if there exists set $V^{\prime}\subseteq V-S$ s.t.
$|V^{\prime}|\leq k$ and $fragile_{G}(V^{\prime})>x$ – “no” otherwise.
$FRAGILITY\\_OPT(k,G,S)$:
INPUT: Natural number $k$, network $G=(V,E)$, and no-strike set $S$
OUTPUT: Set $V^{\prime}\subseteq V-S$ s.t. $|V^{\prime}|\leq k$ s.t.
$\not\exists V^{\prime\prime}\subseteq V-S$ s.t. $|V^{\prime\prime}|\leq k$
and $fragile_{G}(V^{\prime\prime})>fragile_{G}(V^{\prime})$.
As our problems seek to find sets of nodes, rather than individual ones, it
raises the question of “how difficult are these problems.” We prove that
$FRAGILITY$ is NP-Complete - meaning an efficient algorithm to solve it
optimally is currently unknown. Following directly from this result is the NP-
hardness of $FRAGILITY\\_OPT$. Below we state and prove this result.
###### Theorem 1 (Complexity of $FRAGILITY$)
$FRAGILITY$ is NP-Complete.
#### Proof.
Membership in NP is trivial, consider a set $V^{\prime}$ of size $k$, –
clearly we can calculate $fragile_{G}(V^{\prime})$ in polynomial time.
Next we consider the vertex-cover ($VC$) problem and show that it can be
embedded into an instance of $FRAGILITY$. In the $VC$ problem, the input
consists of undirected graph $G^{*}=(V^{*},E^{*})$ and natural number $k$. The
output is “yes” iff there is a set $V^{**}\subseteq V^{*}$ of size at most $k$
s.t. for all $(i,j)\in E^{*}$, either $i$ or $j$ (or both) are in $V^{*}$.
This problem is well-known to be NP-hard. First we create a new network
$G=(V,E)$ which consists of graph $G^{*}$ but with $N_{G^{*}}+2$ additional
nodes which form a star that is disconnected from the rest of the network. All
of the new nodes are put in the no-strike set $S$ (part of the input of
$FRAGILITY$). Clearly, the center of this star is always the most central node
in the graph, no matter what is removed from set $V-S$. This allows us to
treat $d^{*}_{G}$ as a constant equal to $N_{G}+1$. Also note that with this
construction, for both problems, if a solution exists of less than size $k$,
there also exists a solution of exactly size $k$. Further, we note that for
any subset of $V$ whose removal does not affect the overall maximal degree of
the network (which is any node outside the set $S$ \- hence in some
corresponding subset of $V^{*}$ in the graph of the dominating set problem),
when some set $V^{\prime}$ (of size $k$) is removed from $V$, the network-wide
degree centrality for the resulting graph can be expressed as follows:
$fragile_{G}(V^{\prime})=\frac{(N_{G}-k)(N_{G}+1)-2(M_{G}-|\bigcup_{i\in
V^{\prime}}\kappa_{i}|)}{(N_{G}-k-1)(N_{G}-k-2)}$.
The proof of correctness of the embedding rests on proving that a “yes” answer
is returned for the vertex cover problem iff
$FRAGILITY(k,\frac{(N_{G}-k)(N_{G}+1)-2N_{G^{*}}-2}{(N_{G}-k-1)(N_{G}-k-2)},G,S)=\textit{``yes''}$.
First, suppose by way of contradiction (BWOC) there is a “yes” answer to the
$VC$ problem and a “no” answer to the corresponding $FRAGILITY$ problem. Let
$V^{**}$ be the set of nodes that cause a “yes” answer to $VC$. If we remove
the corresponding nodes from $G$, there are $N_{G^{*}}+1$ edges left in that
network. Hence, as this is a set of size $k$ (thus, meeting the cardinality
requirement of $FRAGILITY$ then
$fragile_{G}(V^{**})=\frac{(N_{G}-k)(N_{G}+1)-2N_{G^{*}}-4}{(N_{G}-k-1)(N_{G}-k-2)}$
which would cause a “yes” answer for $FRAGILITY$ – hence a contradiction.
Going the other direction, suppose BWOC there is a “yes” answer to the
$FRAGILITY$ problem and a “no” answer to the corresponding $VC$ problem. Let
$V^{\prime}$ be the nodes in the solution to $FRAGILITY$. Clearly, this set is
of size $k$ and by how we set up the no-strike list ($S$), there are
corresponding nodes in $G^{**}$. As these nodes cause a “yes” answer to
$FRAGILITY$, they result in the removal of $M_{G^{*}}$ number of edges in $G$.
By the construction, none of these edges are adjacent to nodes in $S$. Hence,
there are corresponding edges in $G^{*}$. As this is also the number of edges
in $G^{*}$, then this set is also a vertex cover - hence a contradiction.
Hence, as we have shown membership in NP and that this problem is at least as
hard as the dominating set problem (resulting in NP hardness), the statement
of the theorem follows.
###### Corollary 1 (Hardness of $FRAGILITY\\_OPT$)
$FRAGILITY\\_OPT$ is NP-hard
#### Proof.
Follows directly from Theorem 1.
## III Algorithms
Now with the problems and their complexity identified, we proceed to develop
algorithms to solve them. First, we develop an integer program that, if solved
exactly, will produce an optimal solution. We note that solving a general
integer program is also NP-hard. Hence, an exact solution will likely take
exponential time. However, good approximation techniques such as branch-and-
bound exist and mature tools such as QSopt and CPLEX can readily take and
approximate solutions to integer programs. We follow our integer program
formulation with a greedy heuristic. Though we cannot guarantee that the
greedy heuristic provides an optimal solution, it often provides a natural
approach to approximating many NP-hard optimization problems.
### A Integer Program
Our first algorithm is presented in the form of an integer program. The idea
is that certain variables in the integer program correspond with the nodes in
the original network that can be set to either $0$ or $1$. An objective
function, which mirrors the $fragile$ function is then maximized. When this
function is maximized, all nodes associated with a $1$ variable are picked as
the solution.
###### Definition III.1 ($FRAGILITY\\_IP$)
For each $i\in V$, create variables $X_{i},Z_{i}$. For each undirected edge
$ij\in E$, create three variables: $Y_{ij},Q_{ij},Q_{ji}$. Note that the edge
is considered in only “one direction” for the $Y$ variables and both
directions for the $Q$ variables. We define the $FRAGILITY\\_IP$ integer
program as follows:
$\displaystyle\max$
$\displaystyle\frac{(N_{G}-\sum_{i}X_{i})\sum_{ij}Q_{ij}-2\sum_{ij}Y_{ij}}{(N_{G}-1-\sum_{i}X_{i})(N_{G}-2-\sum_{i}X_{i})}$
Subject to:
$\displaystyle\sum_{i}X_{i}\leq k$ (3) $\displaystyle\sum_{i}Z_{i}=1$ (4)
$\displaystyle\forall ij\in E$ $\displaystyle Y_{ij}\leq 1-X_{i}$ (5)
$\displaystyle\forall ij\in E$ $\displaystyle Y_{ij}\leq 1-X_{j}$ (6)
$\displaystyle\forall ij\in E$ $\displaystyle Q_{ij}\leq Y_{ij}$ (7)
$\displaystyle\forall ij\in E$ $\displaystyle Q_{ij}\leq Y_{ji}$ (8)
$\displaystyle\forall ij\in E$ $\displaystyle Q_{ij}\leq Z_{i}$ (9)
$\displaystyle\forall i\in V$ $\displaystyle Z_{i}\in\\{0,1\\}$ (10)
$\displaystyle\forall i\in S$ $\displaystyle X_{i}=0$ (11)
$\displaystyle\forall i\in V-S$ $\displaystyle X_{i}\in\\{0,1\\}$ (12)
Next we prove how many variables and constraints $FRAGILITY\\_IP$ requires as
well as prove that it provides a correct solution to $FRAGILITY\\_OPT$.
###### Proposition III.1
$FRAGILITY\\_IP$ has $2N_{G}+3M_{G}$ variables and $2+2N_{G}+5M_{G}$
constraints.
###### Proposition III.2
(1.) Given the vector $X$ returned by $FRAGILITY\\_IP$, the set
$\bigcup_{X_{i}=1}i$ is a solution to $FRAGILITY\\_OPT$.
(2.)Given a solution $V^{\prime}$ to $FRAGILITY\\_OPT$, $\forall i\in S,X=1$
and $\forall i\notin S,X=0$ will maximize $FRAGILITY\\_IP$.
#### Proof.
(1.) Suppose, BWOC, $\bigcup_{X_{i}=1}i$ is not an optimal solution to
$FRAGILITY\\_OPT$. Then there is some $V^{\prime}\neq\bigcup_{X_{i}=1}i$ that
is. Suppose $\forall i\in S,X=1$ and $\forall i\notin S,X=0$. Clearly, by the
definition of a solution to $FRAGILITY\\_OPT$, constraints 3,11 and 12 are all
met. Constraints 5 and 6 set variables associated with edges adjacent to nodes
not in $V^{\prime}$ to $1$. Hence, the quantity $\sum_{ij}Y_{ij}$ is equal to
the number of edges in the network. The $Y$ edge variables (both of them for
each edge) are also set in a similar manner. Constraints 4,10 ensures that
only one set of such edge variables are set to $1$. Hence, the quantity
$\sum_{i}X_{i})\sum_{ij}Q_{ij}$ is the degree of one node in the network. As
this quantity is present in the objective function and non-negative, it
corresponds to the $d^{*}_{G}$. As we note that $\sum_{i}X_{i}$ is equal to
the number of nodes in $G$ when $V^{\prime}$ is removed, we see that this
function is $fragile_{G}$. As this quantity is maximized, we have a
contradiction.
(2.) Suppose, BWOC, $\forall i\in S,X=1$ and $\forall i\notin S,X=0$ is not an
optimal solution to $FRAGILITY\\_IP$. Using the same line of reasoning as
above, we see that the objective function of $FRAGILITY\\_IP$ is the same as
$fragile_{G}$, which also gives us a contradiction.
Note that this integer program does not have a linear objective function.
However, this can be accommodated for by instead solving $k$ different integer
programs and taking the solution from whichever one returns the greatest value
for the objective function (that is greater than the initial network-wide
degree centrality, of course). In this case, each integer program is
identified with a natural number $i\in\\{1,\ldots,k\\}$ and the $i$th integer
program has the following objective function:
$\displaystyle\max$
$\displaystyle\frac{(N_{G}-i)\sum_{ij}Q_{ij}-2\sum_{ij}Y_{ij}}{(N_{G}-1-i)(N_{G}-2-i)}$
(13)
As well as constraint 3 as follows:
$\displaystyle\sum_{i}X_{i}\leq i$ (14)
Notice that now the quantities $(N_{G}-i)$ and $(N_{G}-1-i)(N_{G}-2-i)$ can be
treated as constants, making the objective function linear. However, for
networks with a heterogeneous degree distribution where $N_{G}>>k$, it is
likely that only the integer program for the case where $i=k$ is needed as
removing any node with edges that is unconnected to a maximal degree node will
result in an increase in network-wide degree centrality.
Again, we stress that $FRAGILITY\\_IP$ provides an exact solution. As integer-
programming is also NP-hard, solving these constraints is likely intractable
unless $P=NP$. However, techniques such as branch-and-bound and mature solvers
such as QSopt and CPLEX can provide good approximate solutions to such
constraints. Even if the integer program must be linear, we can use the
techniques described above to solve $k$ smaller integer programs or obtaining
an approximation by treating the terms involving the total number of nodes in
the resulting graph (in the objective function) as constants. Additionally, a
relaxation of the above constraints where $Z_{i}$ and $X_{i}$ variables lie in
the interval $[0,1]$ is solvable in polynomial time and would provide a lower-
bound on the solution to the problem (although this would likely be a loose
bound in many cases).
### B A Greedy Heuristic
The integer program introduced in the last section can be leveraged by an
integer-program solver for an approximate solution to $FRAGILITY\\_OPT$.
However, it likely will not scale well to extremely large networks. Therefore,
we introduce a greedy heuristic to find an approximate solution. The ideas is
to iteratively pick the node in the network that provides the greatest
increase in $fragile$ \- and does not cause a decrease.
Algorithm 1 GREEDY_FRAGILE
0: Network $G=(V,E)$, no-strike set $S\subseteq V$, cardinality constraint $k$
0: Subset $V^{\prime}$
1: $V^{\prime}=\emptyset$
2: $flag=TRUE$
3: while $|V^{\prime}|\leq k$ and $flag$ do
4: $curBest=null$, $curBestScore=0$, $haveValidScore=FALSE$
5: for $i\in V-(V^{\prime}\cup S)$ do
6: $curScore=fragile_{G}(V^{\prime}\cup\\{i\\})-fragile_{G}(V^{\prime})$
7: if $curScore\geq curBestScore$ then
8: $curBest=i$
9: $curBestScore=curScore$
10: $haveValidScore=TRUE$
11: end if
12: end for
13: if $haveValidScore=FALSE$ then
14: $flag=FALSE$
15: else
16: $V^{\prime}=V^{\prime}\cup\\{curBest\\}$
17: end if
18: end while
19: return $V^{\prime}$.
The following two propositions describe characteristics of the output and run-
time of $GREEDY\\_FRAGILE$, respectively.
###### Proposition III.3
If $GREEDY\\_FRAGILE$ returns a non-empty solution ($V^{\prime}$), then
$|V^{\prime}|\leq k$ and $fragile_{G}(V^{\prime})>fragile_{G}(\emptyset)$.
#### Proof.
As the algorithm terminates its main loop once the cardinality of the solution
reaches $k$ and as in each iteration, the variable $curBestScore$ is
initialized as zero, the statement follows.
###### Proposition III.4
$GREEDY\\_FRAGILE$ runs in $O(kN_{G}^{2})$ time.
#### Proof.
We note that $fragile$ is computed in $O(N_{G})$ time as it must update the
node with the maximum degree. As the outer loop of the algorithm iterates at
most $k$ times and the inner loop iterates $N_{G}$ times, the statement
follows.
Though our guarantees on $GREEDY\\_FRAGILE$ are limited, we show that it
performs well experimentally in the next section.
###### Example III.1
Following from Examples II.1-II.2 using the terrorist/insurgent network
$G_{sam}$ from Figure 1, suppose a user wants to identify $3$ nodes that will
cause the network to become “as fragile as possible” and is able to target any
node. Hence, he would like to solve $FRAGILE\\_OPT(3,G_{sam},\emptyset)$ and
decides to do so using $GREEDY\\_FRAGILE$. Initially,
$fragile_{G_{sam}}(\emptyset)=0.33$. In the first iteration, it selects and
removes node a, increasing the fragility ($fragile_{G_{sam}}(\\{a\\})=0.57$).
In the next iteration, it selects node j, giving us
$fragile_{G_{sam}}(\\{a,j\\})=0.57$. Finally, in the third iteration, it picks
node c. This results in $fragile_{G_{sam}}(\\{a,j,c\\})=0.6$. The algorithm
then terminates.
## IV Implementation and Experiments
All experiments were run on a computer equipped with an Intel Core 2 Duo CPU
T9550 processor operating at $2.66$ GHz (only one core was used). The machine
was running Microsoft Windows 7 (32 bit) and equipped with $4.0$ GB of
physical memory. We implemented the $\textsf{GREEDY}\\_\textsf{FRAGILE}$
algorithm using Python 2.6 in under $30$ lines of code that leveraged the
NetworkX library available from http://networkx.lanl.gov/.
We compared the results of the $\textsf{GREEDY}\\_\textsf{FRAGILE}$ to three
other more traditional approaches to targeting that rely on centrality
measures from the literature. Specifically, we look at the top closeness and
betweenness nodes in the network. Given node $i$, its closeness is the inverse
of the average shortest path length from node $i$ to all other nodes in the
graph. Betweenness, on the other hand, is defined as the number of shortest
paths between node pairs that pass through $i$. Formal definitions of both of
these measures can be found in [4].
### A Datasets
We studied the effects of our algorithm on five different datasets. The
network Tanzania [5] is a social network of the individuals involved with the
Al Qaeda bombing of the U.S. embassy in Dar es Salaam in 1998. It was
collected from newspaper accounts by subject matter experts in the field. The
remainder networks, GenTerrorNw1-GenTerrorNw4 are terrorist networks generated
from real-world classified datasets[6, 7]. The Tanzania and the
GenTerrorNw1-GenTerrorNw4 datasets used in our analysis were multi-modal
networks, meaning they contain multiple node classes such as Agents,
Resources, Locations, etc. The presence of the different node classes generate
multiple or meta networks, which, in their original state, do not provide the
single-mode Agent by Agent network needed to test our algorithms. Johnson and
McCulloh [8] demonstrated a mathematical technique to convert meta networks
into single-mode networks without losing critical information. Using this
methodology, we were able to derive distant relationships between nodes as a
series of basic matrix algebra operations on all five networks. The result is
an agent based social network of potential terrorist. Characteristics of the
transformed networks of agent node class only can be found in Table 1.
Table 1: Network Datasets Name | Nodes | Edges | Density | Avg. Degree
---|---|---|---|---
Tanzania | $17$ | $29$ | $0.213$ | $3.412$
GenTerrorNw1 | $57$ | $162$ | $0.102$ | $5.684$
GenTerrorNw2 | $102$ | $388$ | $0.0753$ | $7.608$
GenTerrorNw3 | $105$ | $590$ | $0.108$ | $11.238$
GenTerrorNw4 | $135$ | $556$ | $0.0615$ | $8.237$
URV E-Mail | $1,133$ | $5,541$ | $0.00864$ | $9.781$
CA-NetSci | $1,463$ | $2,743$ | $0.00256$ | $3.750$
### B Increasing the Fragility of Networks
In our experiments, we showed that our algorithm was able to significantly
increase the network-wide degree centrality by removing nodes - hence
increasing the $fragile$ function with respect to a given network. In each of
the five real-world terrorist networks that we examined, removal of only
$12\%$ of nodes can increase the network-wide centrality between $17\%$ and
$45\%$ (see Figures 3-7). In Figure 2 we show a visualization of how the
Tanzania network becomes more “star-like” with subsequent removal of nodes by
the greedy algorithm.
Figure 2: Visualization of the Tanzania network after nodes removed by
$\textsf{GREEDY}\\_\textsf{FRAGILE}$. Panel A shows the original network.
Panel B shows the network after $3$ nodes are removed, panel C shows the
network after $5$ nodes are removed, and panel D shows the network after $9$
nodes are removed. Notice that the network becomes more “star-like” after
subsequent node removals. In our experiment, after
$\textsf{GREEDY}\\_\textsf{FRAGILE}$ removed $11$ of the nodes in the network,
it took the topology of a star.
For comparison, we also looked at the removal of high degree, closeness, and
betweenness nodes. Removal of high-degree, closeness, or betweenness nodes
tended to increase the network-wide centrality. In other words, traditional
efforts of targeting leadership without first conducting shaping operations
may actually increase the organization’s ability to regenerate leadership - as
such targeting operations effectively cause an organization to de-centralize.
We display these results graphically in Figures 3-7. Notice that
$\textsf{GREEDY}\\_\textsf{FRAGILE}$ consistently causes an increase in the
network-wide degree centrality. An analysis of variance (ANOVA) reveals that
there is a significant difference in the performance among our algorithm and
the centrality measures with respect to increase or decrease in network-wide
degree centrality ($p$-value less than $2.2\cdot 10^{-16}$, calculated with R
version 2.13). Additionally, pairwise analysis conducted using Tukey’s Honest
Significant Difference (HSD) test indicates that the results of our algorithm
differ significantly from any of the three centrality measures with a
probability approaching $1.0$ ($95\%$ confidence, calculated with R version
2.13). Typically, the ratio of percent increase in fragility to the percent of
removed nodes is typically $2:1$ or greater.
Figure 3: Percent of nodes removed vs. percent increase in fragility for the
Tanzania network using $\textsf{GREEDY}\\_\textsf{FRAGILE}$, top degree, top
closeness, and top betweenness. The scale of the x-axis is positioned at
$0\%$. Figure 4: Percent of nodes removed vs. percent increase in fragility
for the GenTerrorNet1 network using $\textsf{GREEDY}\\_\textsf{FRAGILE}$, top
degree, top closeness, and top betweenness. The scale of the x-axis is
positioned at $0\%$. Figure 5: Percent of nodes removed vs. percent increase
in fragility for the GenTerrorNw2 network using
$\textsf{GREEDY}\\_\textsf{FRAGILE}$, top degree, top closeness, and top
betweenness. The scale of the x-axis is positioned at $0\%$. Figure 6: Percent
of nodes removed vs. percent increase in fragility for the GenTerrorNw3
network using $\textsf{GREEDY}\\_\textsf{FRAGILE}$, top degree, top closeness,
and top betweenness. The scale of the x-axis is positioned at $0\%$. Figure 7:
Percent of nodes removed vs. percent increase in fragility for the
GenTerrorNw4 network using $\textsf{GREEDY}\\_\textsf{FRAGILE}$, top degree,
top closeness, and top betweenness. The scale of the x-axis is positioned at
$0\%$.
### C Runtime
We also evaluated the run-time of the $\textsf{GREEDY}\\_\textsf{FRAGILE}$
algorithm. With the largest terror network considered (GenTerrorNw4), we
achieved short runtime (under $7$ seconds) on standard commodity hardware (see
Figure 8). Hence, in terms of runtime, our algorithm is practical for use by a
real-world analyst. As predicted in our time complexity result, we found that
the runtime of $\textsf{GREEDY}\\_\textsf{FRAGILE}$ increases with the number
of nodes removed. We note that the implementations of top degree, closeness,
and betweenness calculate those measures for the entire network at once -
hence increasing the number of nodes to remove does not affect their runtime.
Figure 8: Number of nodes removed vs. runtime for the GenTerrorNw4 network
using $\textsf{GREEDY}\\_\textsf{FRAGILE}$, top degree, top closeness, and top
betweenness.
### D Experiments on Large Data-Sets
To study the scalability of $\textsf{GREEDY}\\_\textsf{FRAGILE}$, we also
employed it on two large social networks. Note that these datasets are not
terrorist or insurgent networks. However, the larger size of these datasets is
meant to illustrate how well our approach scales. For these experiments, we
used an e-mail network from University Rovira i Virgili (URV E-Mail) [9] and a
Network Science collaboration network (CA-NetSci) from [10] (see Table 1). In
Figure 9 we show the percentage of nodes removed vs. the percent increase in
fragility. We note that $2:1$ ratio of percent increase in fragility to the
percent of removed nodes appears to be maintained even in these large
datasets. In Figure 10 we show the runtime for
$\textsf{GREEDY}\\_\textsf{FRAGILE}$ on the two large networks. We note that
the behavior of runtime vs. number of nodes removed resembles that of the
GenTerrorNw4 network from the previous section. Also of interest is that the
algorithm was able to handle networks of over a thousand nodes in about $20$
minutes on commodity hardware.
Figure 9: Percent of nodes removed vs. percent increase in fragility for the
URV E-Mail and CA-NetSci networks using $\textsf{GREEDY}\\_\textsf{FRAGILE}$.
Figure 10: Number of nodes removed vs. runtime for the URV E-Mail and CA-
NetSci networks using $\textsf{GREEDY}\\_\textsf{FRAGILE}$.
## V Related Work
Various aspects of the resiliency of terrorist networks have been previously
explored in the literature. For instance, [11] studies the ability such
network to facilitate communication while maintaining secrecy while [12]
studies how such networks are resilient to cascades. However, to our
knowledge, the network-wide degree centrality in such networks - and how to
increase this property - has not been previously studied.
There has been much work dealing with the removal of nodes from a network to
maximize fragmentation [13, 14, 15] where the nodes removed are mean to either
increase fragmentation of the network or reduce the size of the largest
connected component. While this work has many applications, it is important to
note that there are special considerations of terrorist and insurgent networks
that we must account for in a targeting strategy. For instance, if conducting
a counter-intelligence operation while targeting, as in the case of [3], it
may be desirable to preserve some amount of connectivity in the network.
Additionally, fragmentation of a network may result in the splintering of an
organization into smaller, but more radical and deadly organizations. This
happens because in some cases, it may be desirable to keep certain terrorist
or insurgent leaders in place to restrain certain, more radical elements of
their organization. Such splinter was observed for the insurgent organization
Jaysh al-Mahdi in Iraq [16]. Further, these techniques do not specifically
address the issue of emerging leaders. Hence, if they were to be used for
counter-terrorism or counter-insurgency, they would likely still benefit from
a shaping operation to reduce organization’s ability to regenerate leadership.
There has been some previous work on identifying emerging leaders in terrorist
networks. Although such an approach could be useful in identifying certain
leaders, it does not account the organizations ability as a whole to
regenerate leadership. In [2], the topic of cognitive demand is studied. The
cognitive load of an individual deals with their ability to handle multiple
demands on their time and work on complex tasks. Typically, this can be
obtained by studying networks where the nodes may represent more than
individual people - but tasks, events, and responsibilities. However, it may
often be the case that this type of information is often limited or non-
existent in many situations. Additionally, as discussed throughout this paper,
the targeting of individual nodes may often not be possible for various
reasons. Hence, our framework, that focuses on the network’s ability to
regenerate leadership as opposed to finding individual emerging leaders may be
more useful as we can restrict the available nodes in our search using the “no
strike list.” By removing these nodes from targeting consideration - but by
still considering their structural role - our framework allows a security
force to reduce the regenerative ability of a terror network by “working
around” individuals that may not be targeted.
In more recent work [17] looks at the problem of removing leadership nodes
from a terrorist or criminal network in a manner that accounts for new links
created in the aftermath of an operation. Additionally, [18] look at
identifying leaders in covert terrorist network who attempt to minimize their
communication due to the clandestine nature of their operations. They do this
by introducing a new centrality measure called “covertness centrality.” Both
of these approaches are complementary to ours as they focus on the leadership
of the terrorist or insurgent group - as this approach focuses on the networks
ability to re-generate leadership. A more complete integration of this
approach leadership targeting method such as these (i.e. using a network-wide
version of covertness centrality) is an obvious direction for future work.
## VI Conclusions
In this paper we described how to target nodes in a terrorist or insurgent
network as part of a shaping operation designed to reduce the organization’s
ability to regenerate leadership. Our key intuition was to increase the
network-wide degree centrality which would likely have the effect of
eliminating emerging leaders as maximizing this quantity would intuitively
increase the organization’s reliance on a single leader. In this paper, we
found that though identifying a set of nodes to maximize this network-wide
degree centrality is NP-hard, our greedy approach proved to be a viable
heuristic for this problem, increasing this quantity between $17\%-45\%$ in
our experiments. Future work could include an examination of other types of
network-wide centrality – for instance network-wide closeness centrality –
instead of network-wide degree centrality. Another aspect that we are
considering in ongoing research is determining the effectiveness of the
shaping strategy when we have observed only part of the terrorist or insurgent
organization – as is often the case as such networks are created from
intelligence data.
## Acknowledgements
We would like to thank Jon Bentley and Charles Weko for their feedback on an
earlier version of this paper.
Some of the authors are supported under by the Army Research Office (project
2GDATXR042). The opinions in this paper are those of the authors and do not
necessarily reflect the opinions of the funders, the U.S. Military Academy, or
the U.S. Army.
## References
* [1] L. Freeman, “Centrality in social networks conceptual clarification,” _Social networks_ , vol. 1, no. 3, pp. 215–239, 1979.
* [2] K. Carley, “Estimating Vulnerabilities in Large Covert Networks,” Carnegie Mellon University, Tech. Report, 2004.
* [3] O. Deforest and D. Chanoff, _Slow Burn: The Rise and Bitter Fall of American Intelligence in Vietnam._ Simon and Schuster, 1990.
* [4] S. Wasserman and K. Faust, _Social Network Analysis: Methods and Applications_ , 1st ed., ser. Structural analysis in the social sciences. Cambridge University Press, 1994, no. 8.
* [5] I.-C. Moon, “Destabilization of adversarial organizations with strategic interventions,” Ph.D. dissertation, Carnegie Mellon University, Pittsburgh, PA, USA, Jun. 2008.
* [6] K. M. Carley, “FICTA data,” _Center for Computational Analysis of Social and Organizational Systems_ , 2009.
* [7] _Dynamic Network Analysis (DNA) and ORA_. San Francisco, CA: 2nd International Conference on Cross-Cultural Decision Making: Focus 2012, Jul. 2012.
* [8] A. N. Johnson and I. A. McCulloh, _Advanced Network Analysis and Targeting (ANAT)_ , 1st ed., Joint Training Counter IED Operational Integration Center, Washington D.C., Jan. 2009.
* [9] A. Arenas, “Network data sets,” 2012. [Online]. Available: http://deim.urv.cat/ãarenas/data/welcome.htm
* [10] M. Newman, “Network data,” 2011. [Online]. Available: http://www-personal.umich.edu/~mejn/netdata/
* [11] R. Lindelauf, P. Borm, and H. Hamers, “The influence of secrecy on the communication structure of covert networks,” _Social Networks_ , vol. 31, no. 2, pp. 126 – 137, 2009.
* [12] A. Gutfraind, “Optimizing topological cascade resilience based on the structure of terrorist networks,” _PLoS ONE_ , vol. 5, no. 11, p. e13448, 11 2010.
* [13] R. Albert, H. Jeong, and A. Barabási, “Error and attack tolerance of complex networks,” _Nature_ , vol. 406, pp. 378–382, 2000.
* [14] S. Borgatti, “Identifying sets of key players in a social network,” _Computational and Mathematical Organization Theory_ , vol. 12, no. 1, 2006.
* [15] A. Arulselvan, C. Commander, L. Elefteriadou, and P. M. Pardalos, “Detecting critical nodes in sparse graphs,” _Computers and Operations Research_ , vol. 36, 2009.
* [16] M. Cochrane, “The Fragmentation of the Sadrist Movement,” The Institute for the Study of War, Iraq Report 12, Jan. 2009.
* [17] R. Petersen, C. Rhodes, and U. Wiil, “Node removal in criminal networks,” in _Intelligence and Security Informatics Conference (EISIC), 2011 European_ , Sep. 2011, pp. 360 –365.
* [18] M. Ovelgonne, C. Kang, A. Sawant, and V. Subrahmanian, “Covertness centrality in networks,” in _Proc. 2012 Intl. Symposium on Foundations of Open Source Intelligence and Security Informatics_ , Aug. 2012.
|
arxiv-papers
| 2012-11-04T18:12:37 |
2024-09-04T02:49:37.532561
|
{
"license": "Public Domain",
"authors": "Devon Callahan, Paulo Shakarian, Jeffrey Nielsen, Anthony N. Johnson",
"submitter": "Paulo Shakarian",
"url": "https://arxiv.org/abs/1211.0709"
}
|
1211.0757
|
# Efficient Point-to-Subspace Query in $\ell^{1}$: Theory and Applications in
Computer Vision††thanks: This work is published in the $12^{th}$ European
Conference on Computer Vision (ECCV 2012) [19]. A full version with technical
details is available online http://arxiv.org/abs/1208.0432 [20].
Ju Sun, Yuqian Zhang, and John Wright
Department of Electrical Engineering, Columbia University, NY, USA
{jusun, yuqianzhang, johnwright}@ee.columbia.edu
###### Abstract
Motivated by vision tasks such as robust face and object recognition, we
consider the following general problem: _given a collection of low-dimensional
linear subspaces in a high-dimensional ambient (image) space and a query point
(image), efficiently determine the nearest subspace to the query in $\ell^{1}$
distance_. We show in theory that Cauchy random embedding of the objects into
significantly-lower-dimensional spaces helps preserve the identity of the
nearest subspace with constant probability. This offers the possibility of
efficiently selecting several candidates for accurate search. We sketch
preliminary experiments on robust face and digit recognition to corroborate
our theory.
## 1 Introduction
Big data often come with prominent dimensionality and volume. Statistical
learning and inference may be hard at such scales, due not only to conceivable
computational burdens, but to stability issues. Nevertheless, parsimony that
almost invariably dominates data-generating process dictates that structures
associated with big data may be significant. For example, all images of a
Lambertian convex object of fixed pose undergoing (remote) illumination
changes lie approximately on a very low ($\approx 9$)-dimensional linear
subspace, in the image space of dimensionality equal to the number of pixels
per image (normally $\sim 10000$) [4]. Examples that have low-dimensional
manifold structures abound (see, e.g., [17, 9]). By assuming reasonable
structures for data, we can then turn the inference problem as an instance of
nearest structure search, of which _nearest subspace search_ may serve as a
basic building block:
###### Problem 1 (Nearest Subspace Search)
Given $n$ linear subspaces $\mathcal{S}_{1},\dots,\mathcal{S}_{n}$ of
dimension $r$ in $\mathds{R}^{D}$ and a query point
$\mathbf{q}\in\mathds{R}^{D}$, determine the nearest $\mathcal{S}_{i}$ to
$\mathbf{q}$.
Exploiting structures in big data has greatly helped in providing attractively
simple formulations for learning and inference, and the remaining tasks are to
make concrete the measure of “nearness” and to design efficient algorithm to
solve the search problem.
### Measure of Nearness.
Typically, one adopts a metric $d(\cdot,\cdot)$ on $\mathds{R}^{D}$, and then
sets
$d(\mathbf{q},\mathcal{S}_{i})=\min_{\mathbf{v}\in\mathcal{S}_{i}}d(\mathbf{q},\mathbf{v}).$
Certainly the appropriate choice of metric $d$ depends on our prior knowledge.
For example, if the observation $\mathbf{q}$ is known to be perturbed by
i.i.d. Gaussian noise from its originating subspace, minimizing the $\ell^{2}$
norm $d(\mathbf{q},\mathbf{v})=\|\mathbf{q}-\mathbf{v}\|_{2}$ yields a maximum
likelihood estimator. However, in practice other norms may be more
appropriate: particularly in situations where the data may have sparse but
significant errors, the $\ell^{1}$ norm is a more robust alternative [7, 23].
For images, such errors are due to factors such as occlusions, shadows,
specularities. We focus on the choice of $\ell^{1}$ norm here and our main
problem is
###### Problem 2 (Main, Nearest Subspace Search in $\ell^{1}$)
Given $n$ linear subspaces $\mathcal{S}_{1},\dots,\mathcal{S}_{n}$ of
dimension $r$ in $\mathds{R}^{D}$ and a query point
$\mathbf{q}\in\mathds{R}^{D}$, determine the nearest $\mathcal{S}_{i}$ to
$\mathbf{q}$ in $\ell^{1}$ distance.
### Efficiency of Algorithms.
We would like solve Problem 2 using computational resources that depend as
gracefully as possible on the ambient dimension $D$ and the number of models
$n$, both of which could be very large for big data. The straightforward
solution proceeds by solving a sequence of $n$ $\ell^{1}$ regression problems
$d_{\ell^{1}}\left(\mathbf{q},\mathcal{S}_{i}\right)=\min_{\mathbf{v}\in\mathcal{S}_{i}}\|\mathbf{q}-\mathbf{v}\|_{1}.$
(1)
The total cost is $O(n\cdot T_{\ell^{1}}(D,r))$, where $T_{\ell^{1}}(D,r)$ is
the time required to solve (1). The best known complexity guarantees for
solving (1), based on scalable first-order methods [11, 5, 25, 24], are
superlinear in $D$, though linear runtimes may be achievable when the residual
$\mathbf{q}-\mathbf{v}_{\star}$ is very sparse [10] or the problem is
otherwise well-structured [1]. So even in the best case, the straightforward
solution has complexity $\Omega(nD)$. When both terms are large, this
dependence is prohibitive: Although Problem 2 is simple to state and easy to
solve in polynomial time, achieving real-time performance or scaling massive
databases appears to require a more careful study.
### Dealing with $D$ by Cauchy Random Embedding.
We present a very simple, practical approach to Problem 2 with much improved
dependency on $D$111The reason that we do not deal with $n$ concurrently is
discussed in Sec 3. . Rather than working directly in the high-dimensional
space $\mathds{R}^{D}$, we randomly embed the query $\mathbf{q}$ and subspaces
$\mathcal{S}_{i}$ into $\mathds{R}^{d}$, with $d\ll D$. The random embedding
is given by a $d\times D$ matrix $\mathbf{P}$ whose entries are i.i.d.
standard Cauchy’s. That is to say, instead of solving (1), we solve
$d_{\ell^{1}}\left(\mathbf{P}\mathbf{q},\mathbf{P}\mathcal{S}_{i}\right)=\min_{\mathbf{v}\in\mathcal{S}_{i}}\|\mathbf{P}\mathbf{q}-\mathbf{P}\mathbf{v}\|_{1}.$
(2)
We prove that if the embedded dimension $d$ is sufficiently large – say
$d=\mathrm{poly}(r\log n)$, then with constant probability the model
$\mathcal{S}_{i}$ obtained from (2) is the same as the one obtained from the
original optimization (1). The required dimension $d$ does not depend in any
way on the ambient dimension $D$, and is often significantly smaller: e.g.,
$d=25$ vs. $D=32,000$ for one typical example of face recognition. The
resulting (small) $\ell^{1}$ regression problems are amenable to customized
interior point solvers (e.g., [16]). The price paid for this improved
complexity is a small increase in the probability of failure to locate the
nearest subspace. Our theory quantifies how large $d$ needs to be to render
this probability of error under control. Repeated trials with independent
projections $\mathbf{P}$ can then be used to make the probability of failure
as small as desired.
## 2 Cauchy Random Embedding and Theoretical Analysis
Our entire algorithm relies on the standard Cauchy distribution with p.d.f.
$p_{\mathcal{C}}(x)=1/\left[\pi\left(1+x^{2}\right)\right],$ (3)
which is $1$-stable [21] and heavy-tailed (shown in Figure 1). The core of our
algorithm is summarized as follows (right).
Figure 1: Standard Cauchy Input: $n$ subspaces
$\mathcal{S}_{1},\cdots,\mathcal{S}_{n}$ of dimension $r$ and query
$\mathbf{q}$ Output: Identity of the closest subspace $\mathcal{S}_{\star}$ to
$\mathbf{q}$ in $\ell^{1}$ distance Preprocessing: Generate
$\mathbf{P}\in\mathds{R}^{d\times D}$ with i.i.d. Cauchy RV’s ($d\ll D$) and
Compute the projections $\mathbf{P}\mathcal{S}_{1}$, $\cdots$,
$\mathbf{P}\mathcal{S}_{n}$ Test: Compute the projection
$\mathbf{P}\mathbf{q}$, and compute its $\ell^{1}$ distance to each of
$\mathbf{P}\mathcal{S}_{i}$
Our main theoretical result states that if $d$ is chosen appropriately, with
at least constant probability, the subspace $\mathcal{S}_{i_{\star}}$ selected
will be the original closest subspace $\mathcal{S}_{\star}$:
###### Theorem 3
Suppose we are given $n$ linear subspaces
$\left\\{\mathcal{S}_{1},\cdots,\mathcal{S}_{n}\right\\}$ of dimension $r$ in
$\mathds{R}^{D}$ and any query point $\mathbf{q}$, and that the $\ell^{1}$
distances of $\mathbf{q}$ to each of
$\left\\{\mathcal{S}_{1},\cdots,\mathcal{S}_{n}\right\\}$ are
$\xi_{1^{\prime}}\leq\cdots\leq\xi_{n^{\prime}}$ when arranged in ascending
order, with $\xi_{2^{\prime}}/\xi_{1^{\prime}}\geq\eta>1$. For any fixed
$\alpha<1-1/\eta$, there exists $d\sim O\left[\left(r\log
n\right)^{1/\alpha}\right]$ (assuming $n>r$), if
$\mathbf{P}\in\mathds{R}^{d\times D}$ is iid Cauchy, we have
$\mathop{\arg\min}_{i\in[n]}d_{\ell^{1}}\left(\mathbf{P}\mathbf{q},\mathbf{P}\mathcal{S}_{i}\right)=\mathop{\arg\min}_{i\in[n]}d_{\ell^{1}}\left(\mathbf{q},\mathcal{S}_{i}\right)$
(4)
with (nonzero) constant probability.
At least two things are interesting about Theorem 3: 1) $d$ depends on the
relative gap $\eta$, ratio of distances to the closest and to the second
closest subspaces. Notice that $\eta\in[1,\infty)$, and that the exponent
$1/\alpha$ becomes large as $\eta$ approaches one. This suggests that our
dimensionality reduction will be most effective when the relative gap is
nonnegligible; 2) $d$ depends on the number of models $n$ only through its
logarithm. This rather weak dependence is a strong point, and, interestingly,
mirrors the Johnson-Lindenstrauss lemma for dimensionality reduction in
$\ell^{2}$, even though JL-syle embeddings are impossible for $\ell^{1}$.
Additional practical implications of Theorem 3 are in order: 1) First, Theorem
3 only guarantees success with constant probability. This probability is
easily amplified by taking a _small number_ of independent trials. Each of
these trials generates one or more candidate subspaces $\mathbf{S}_{i}$. We
can then perform $\ell^{1}$ regression in $\mathds{R}^{D}$ to determine which
of these candidates is actually nearest to the query; 2) Since the gap $\eta$
is one important factor controlling the resource demanded, if we have reason
to believe that $\eta$ will be especially small, we may instead set $d$
according to the gap between $\xi_{1^{\prime}}$ and $\xi_{k^{\prime}}$, for
some $k^{\prime}>2$. With this choice, Theorem 3 implies that with constant
probability the desired subspace is amongst the $k^{\prime}-1$ nearest (saved
for further examination) to the query. If $k^{\prime}\ll n$, this is still a
significant saving over the naive approach.
### Idea of the Analysis.
We present the full technical details in the report [20], while highlight the
intuition behind analysis now. Figure 2 shows a histogram of the random
variable
$\psi=d_{\ell^{1}}\left(\mathbf{P}\mathbf{q},\mathbf{P}\mathcal{S}\right)$
($d_{\ell^{1}}\left(\mathbf{q},\mathbf{S}\right)$ is normalized), over
randomly generated Cauchy matrices $\mathbf{P}$, for two different
configurations of query $\mathbf{q}$ and subspace $\mathcal{S}$. Two
properties are especially noteworthy. First, the upper tail of the
distribution can be quite heavy: with non-negligible probability, $\psi$ may
significantly exceed its median. In constrat, the lower tail is much better
behaved: with very high probability, $\psi$ is not significantly smaller than
its median.
Figure 2: An illustration of how random Cauchy embedding changes the query-to-
subspace $\ell^{1}$ distance in statistics.
This inhomogeneous behavior (in particular, the heavy upper tail) precludes
very tight distance-preserving embeddings using the Cauchy. However, our goal
is not to find an (near-isometric) embedding of the data, per se, but rather
to find the nearest subspace, to the query. In fact, it suffices to show that
with nontrivial constant probability
* •
$\mathbf{P}$ does not increase the distance from $\mathbf{q}$ to
$\mathcal{S}_{\star}$ too much; and,
* •
$\mathbf{P}$ does not shrink the distance from $\mathbf{q}$ to any of the
other subspaces $\mathcal{S}_{i}$ too much.
The observed inhomogeneous behavior is much less of an obstacle to
establishing the desired results.
## 3 Related Work
Problem 2 is an example of a subspace search problem. In $\ell^{2}$, for $r=0$
and $r=1$, efficient algorithms with _sublinear_ query complexity in $n$ exist
for the approximate versions [8, 2]. For $r>1$, recent attempts [3, 13]
offered promising numerical examples, but not sublinear complexity guarantees.
Results in theoretical computer science suggest that these limitations may be
intrinsic to the problem [22].
These attempts exploit special properties of the $\ell^{2}$ version of Problem
2, and do not apply to its $\ell^{1}$ variant. However, the $\ell^{1}$ variant
retains the aforementioned difficulties, suggesting that an algorithm for
$\ell^{1}$ near subspace search with sublinear dependence on $n$ is unlikely
as well.222Although it could be possible if we are willing to accept time and
space complexity exponential in $r$ or $D$, ala [15]. This motivates us to
focus on ameliorating the dependence on $D$. Our approach is very simple and
very natural: Cauchy projections are chosen because the Cauchy is the unique
$1$-stable distribution, a property which has been widely exploited in
previous algorithmic work [8, 14, 18].
However, on a technical level, it is not obvious that Cauchy embedding should
succeed for this problem. The Cauchy is a heavy tailed distribution, and
because of this it does not yield embeddings that very tightly preserve
distances between points, as in the Johnson-Lindenstrauss lemma. In fact, for
$\ell^{1}$, there exist lower bounds showing that certain point sets in
$\ell^{1}$ cannot be embedded in significantly lower-dimensional spaces
without incurring non-negligble distortion [6]. For a single subspace,
embedding results exist – most notably due to Soehler and Woodruff [18], but
the distortion incurred is so large as to render them inapplicable to Problem
2.
## 4 Experimental Verification
Again we highlight part of our experiments here and more details can be found
in the report [20]. We take the The Extended Yale B face dataset [12] ($n=38$,
$D=168*192\sim 30000$) and treat the facial images of one person as lying on a
_9-dimensional_ linear subspace (as argued in [4] and practiced in [23]). For
each subject, we take half of the images for training ($1205$ in total) and
the others for testing ($1209$ in total). To better illustrate the behavior of
our algorithm, we strategically divided the test set into two subsets:
moderately illuminated ($909$, Subset M) and extremely illuminated ($300$,
Subset E).
Figure. 3 presents typical evolution of recognition rate on Subset M as the
projection dimension ($d$) grows _with only one repetition of the projection_.
The high-dimensional NS (HDS) in $\ell^{1}$ achieves perfect ($100\%$)
recognition, and the recognition rate (also probability of success as in
Theorem 3) stays stable above $95\%$ with $d\geq 25$. Suppose the distance gap
is significant such that $1/\alpha\to 1$, our theorem predicts $d=r\log
n=9*\log 38\approx 33$.
Figure 3: Recognition rate versus projection dimension ($d$) _with one
repetition_ on Subset M face images of EYB. Figure 4: Samples of
moderately/extremely illuminated face images and their $\ell^{1}$ distances to
other subject subspaces.
For extremely illuminated face images, the $\ell^{1}$ distance gap between the
first and second nearest subspaces is much less significant (one example shown
in Figure 4). Our theory suggests $d$ should be increased to compensate for
the weak gap (because the exponent $1/\alpha$ becomes significant). Our
experimental results in Table 1 confirm this prediction.
Table 1: Recognition Rate on Subset E of EYB with varying $d$ and $N_{back}$ ($\\#$ candidates for further test). | HDS | $d=25$ | $d=50$ | $d=70$
---|---|---|---|---
$r=15,N_{back}=5$ | 94.7% | 79.3% | 87.7% | 92.3%
$r=15,N_{back}=10$ | 94.7% | 87.3% | 92.0% | 94.0%
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|
arxiv-papers
| 2012-11-05T04:15:25 |
2024-09-04T02:49:37.541601
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ju Sun, Yuqian Zhang, John Wright",
"submitter": "Ju Sun",
"url": "https://arxiv.org/abs/1211.0757"
}
|
1211.0842
|
# Depth of some square free monomial ideals
Dorin Popescu and Andrei Zarojanu Dorin Popescu, ”Simion Stoilow” Institute
of Mathematics of Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest
014700, Romania dorin.popescu@imar.ro Andrei Zarojanu, Faculty of Mathematics
and Computer Sciences, University of Bucharest, Str. Academiei 14, Bucharest,
Romania andrei_zarojanu@yahoo.com
###### Abstract.
Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra
over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when
$I$ contains precisely one variable, the other generators having degrees $\geq
2$, if the Stanley depth of $I/J$ is $\leq 2$ then the usual depth of $I/J$ is
$\leq 2$ too, that is the Stanley Conjecture holds in these cases.
Key words : Monomial Ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.
## Introduction
Let $K$ be a field, $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$
variables over $K$ and $I\supsetneq J$ two square free monomial ideals of $S$.
We assume that $I$, $J$ are generated by square free monomials of degrees
$\geq d$, resp. $\geq d+1$ for some $d\in{\bf N}$. Then
$\operatorname{depth}_{S}I/J\geq d$ (see [4, Proposition 3.1], [12, Lemma
1.1]). Upper bounds of $\operatorname{depth}_{S}I/J$ are given by numerical
conditions in [11], [12, Theorem 2.2], [13, Theorem 1.3] and [15, Theorem
2.4]. An important tool in the proofs is the Koszul homology, except in the
last quoted paper, where the results are stronger, but the proofs are
extremely short relying completely on some results concerning the Hilbert
depth, which proves there to be a very strong tool (see [2], [17] and [6]).
These results are inspired by the so called the Stanley Conjecture, which we
explain below.
Let $P_{I\setminus J}$ be the poset of all square free monomials of
$I\setminus J$ (a finite set) with the order given by the divisibility. Let
${\mathcal{P}}$ be a partition of $P_{I\setminus J}$ in intervals
$[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$, let us say $P_{I\setminus
J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and the so called Stanley depth of $I/J$ given by
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I\setminus J}$
(see [4], [16]). The Stanley depth is not easy to handle, see [4], [14], [7],
[5] for some of its properties.
Stanley’s Conjecture says that
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. Thus the
Stanley depth of $I/J$ is a natural combinatorial upper bound of
$\operatorname{depth}_{S}I/J$ and the above results give numerical conditions
to imply upper bounds of $\operatorname{sdepth}_{S}I/J$. When $J=0$ the
Stanley Conjecture holds either when $n\leq 5$ by [9], or when $I$ is an
intersection of four monomial prime ideals by [8], [10], or when $I$ is an
intersection of three primary ideals by [18], or when $I$ is an almost
complete intersection by [3].
Let $r$ be the number of the square free monomials of degree $d$ of $I$ and
$B$ (resp. $C$) be the set of the square free monomials of degrees $d+1$
(resp. $d+2$) of $I\setminus J$. Set $s=|B|$, $q=|C|$. If either $s>r+q$, or
$r>q$, or $s<2r$ then $\operatorname{sdepth}_{S}I/J\leq d+1$ and if the
Stanley Conjecture holds then any of these numerical conditions would imply
$\operatorname{depth}_{S}I/J\leq d+1$. In particular this was proved directly
in [13] and [15].
Now suppose that $I$ is generated by one variable and some square free
monomials of degrees $\geq 2$. It is the purpose of our paper to show that
almost always if $\operatorname{sdepth}_{S}I/J\leq 2$ then
$\operatorname{depth}_{S}I/J\leq 2$ (see our Theorem 1.10). It is known
already that $\operatorname{sdepth}_{S}I/J\leq 1$ implies
$\operatorname{depth}_{S}I/J\leq 1$ (see [12, Theorem 4.3]) and so our Theorem
1.10 could be seen as a new step (small but difficult) in the study of
Stanley’s Conjecture.
## 1\. Stanley depth of some square free monomial ideals
Let $I\supsetneq J$ be two square free monomial ideals of $S$. We assume that
$I$, $J$ are generated by square free monomials of degrees $\geq d$, resp.
$\geq d+1$ for some $d\in{\bf N}$. As above $B$ (resp. $C$) denotes the set of
the square free monomials of degrees $d+1$ (resp. $d+2$) of $I\setminus J$.
###### Lemma 1.1.
Suppose that $d=1$, $I=(x_{1},\ldots,x_{r})$ for some $1\leq r<n$ and
$J\subset I$ be a square free monomial ideal generated in degree $\geq 2$. Let
$B$ be the set of all square free monomials of degrees $2$ from $I\setminus
J$. Suppose that $\operatorname{depth}_{S}I/(J+((x_{j})\cap B))=1$ for some
$r<j\leq n$. Then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
Since $I/(J+((x_{j})\cap B))$ has a square free, multigraded free resolution
we see that only the components of square free degrees of
$\operatorname{Tor}_{n-1}^{S}(K,I/(J+(x_{j})\cap B)))\cong
H_{n-1}(x;I/(J+(x_{j})\cap B))$
are nonzero. Thus we may find
$z=\sum_{i=1}^{r}y_{i}x_{i}e_{[n]\setminus\\{\i\\}}\in
K_{n-1}(x;I/(J+(x_{j})\cap B)$, $y_{i}\in K$ inducing a nonzero element in
$H_{n-1}(x;I/(J+(x_{j})\cap B)$. Here we denoted $e_{\tau}=\wedge_{j\in\tau}\
e_{j}$ for a subset $\tau\subset[n]$. Then we see that
$z^{\prime}=\sum_{i=1}^{r}y_{i}x_{i}e_{[n]\setminus\\{i,j\\}}\in
K_{n-2}(x;I/J)$
induces a nonzero element in $H_{n-2}(x;I/J)$. Thus
$\operatorname{depth}_{S}I/J\leq 2$ (see [1, Theorem 1.6.17]).
###### Example 1.2.
Let $n=4$, $r=2$, $d=1$, $I=(x_{1},x_{2})$, $J=(x_{1}x_{2})$,
$B=\\{x_{1}x_{3},x_{1}x_{4},x_{2}x_{3},x_{2}x_{4}\\}$. Then
$F=I/(J+(x_{1})\cap B)\cong(x_{1},x_{2})/((x_{1})\cap(x_{2},x_{3},x_{4}))$ has
sdepth and depth $=1$, but $\operatorname{depth}_{S}I/J=3$. Thus the statement
of the above lemma can be false if $j<r$. More precisely,
$\operatorname{depth}_{S}F=1$ because $z=x_{1}e_{234}$ induces a nonzero
element in $H_{3}(x;F)$ but $e_{1}$ is not present in $e_{234}$.
###### Proposition 1.3.
Suppose that $I\subset S$ is generated by $\\{x_{1},\ldots,x_{r}\\}$ for some
$1\leq r\leq n$ and some square free monomials of degrees $\geq 2$, and
$x_{i}x_{t}x_{k}\in J$ for all $i\in[r]$ and $r<t<k\leq n$. Then
$\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
First suppose that $I=(x_{1},...,x_{r})$. If there exists $j>r$ such that
$\operatorname{depth}_{S}I/(J+(x_{j})\cap B)=1$ then we may apply the above
lemma. Thus we may suppose that $\operatorname{depth}_{S}I/(J+(x_{j})\cap
B)\geq 2$ for all $j>r$. Assume that $\operatorname{depth}_{S}I/J>2$. By
decreasing induction on $r<t\leq n$ we show that
$\operatorname{depth}_{S}I/(J+(x_{t},\ldots,x_{n}))\cap B)\geq 2$. We assume
that $t<n$ and $\operatorname{depth}_{S}I/(J+(x_{t+1},\ldots,x_{n}))\cap
B)\geq 2$, $\operatorname{depth}_{S}I/(J+(x_{t},\ldots,x_{n}))\cap B)=1$. Set
$L=(J+(x_{t})\cap B)\cap(J+(x_{t+1},\ldots,x_{n})\cap B)$. In the following
exact sequence
$0\rightarrow I/L\rightarrow I/(J+(x_{t})\cap B)\oplus
I/(J+(x_{t+1},\ldots,x_{n})\cap B)\rightarrow I/(J+(x_{t},\ldots,x_{n})\cap
B)\rightarrow 0$
the last term has the depth $1$ and the middle the depth $\geq 2$. By the
Depth Lemma we get $\operatorname{depth}_{S}I/L=2$.
Remains to show that
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}I/L$. Note that there
exist no $c\in C$ multiple of $x_{t}x_{j}$ for some $r<t<j\leq n$ by our
hypothesis. Thus $L=J$. Then it follows $\operatorname{depth}_{S}I/J=2$ which
contradicts our assumption. The induction ends for $t=r+1$ and we get
$\operatorname{depth}_{S}I/(J+(x_{r+1},\ldots,x_{n})\cap B)=2$; but this is
not possible (see for example [12, Lemma 1.8]).
Now suppose that $I=U+V$, where $U=(x_{1},...,x_{r})$ and $V$ is generated by
some square free monomials of degrees $\geq 2$. In the following exact
sequence
$0\rightarrow U/(U\cap J)\rightarrow I/J\rightarrow I/(U+J)\rightarrow 0$
the first term has depth $\leq 2$ from above and the last term is isomorphic
with $V/(V\cap(U+J))$ and has depth $\geq 2$ by [12, Lemma 1.1]. So by the
Depth Lemma it follows that $\operatorname{depth}I/J\leq 2$.
###### Example 1.4.
Let $n=4$, $I=(x_{1},x_{2},x_{3})$, $J=(x_{1}x_{3})$. Clearly,
$B_{1}=\emptyset$,
$B=\\{x_{1}x_{2},x_{1}x_{4},x_{2}x_{3},x_{2}x_{4},x_{3}x_{4}\\}$ and
$C=\\{x_{1}x_{2}x_{4},x_{2}x_{3}x_{4}\\}$. We have $s=5$, $r=3$, $q=2$ and so
$s=r+q$. Note that each $c\in C$ is a multiple of a monomial of the form
$x_{i}x_{j}$ for some $1\leq i<j\leq 3$ and so
$\operatorname{depth}_{S}I/J\leq 2$ by the above proposition. On the other
hand, it is easy to see that $z=x_{1}e_{2}\wedge e_{3}-x_{2}e_{1}\wedge
e_{3}+x_{3}e_{1}\wedge e_{2}$ induces a nonzero element in $H_{2}(x;I/J)$ and
so again $\operatorname{depth}_{S}I/J\leq 2$.
###### Lemma 1.5.
If a monomial $u$ of degree $k$ from $I\setminus J$ has all multiples of
degrees $k+1$ in $J$ then $\operatorname{depth}I/J\leq k$.
###### Proof.
Renumbering the variables $x$ we may suppose that $u=x_{1}\cdots x_{k}$. Then
we see that $u(x_{k+1},...,x_{n})=0$ so
$\operatorname{Ann}_{S}u=(x_{k+1},...,x_{n})\in\operatorname{Ass}_{S}I/J.$
Thus $\operatorname{depth}I/J\leq k$.
###### Lemma 1.6.
Suppose that $J\subset I$ are square free monomial ideals generated in degree
$\geq d+1$, respectively $\geq d$ and let $V$ be an ideal generated by $e$
square free monomials of degrees $\geq d+2$, which are not in $I$. Then
$\operatorname{sdepth}_{S}(I+V)/J\leq d+1$ (resp.
$\operatorname{depth}_{S}(I+V)/J\leq d+1$) implies that
$\operatorname{sdepth}_{S}I/J\leq d+1$ (resp. $\operatorname{depth}_{S}I/J\leq
d+1$). For the depth the converse is also true.
###### Proof.
By induction on $e$, we may consider only the case $e=1$, that is $V=\\{v\\}$.
In the following exact sequence
$0\rightarrow I/J\rightarrow(I+V)/J\rightarrow(I+V)/(I+J)\rightarrow 0$
the last term is isomorphic with $(v)/((v)\cap(I+J))$ and has depth and sdepth
$\geq d+2$. Then the first term has sdepth $\leq d+1$ by [14, Lemma 2.2] and
depth $\leq d+1$ by the Depth Lemma.
###### Lemma 1.7.
Suppose that $I\subset S$ is generated by $x_{1},\ldots,x_{r}$ and a nonempty
set $E$ of square free monomials of degrees $2$ in the variables
$x_{r+1},\ldots,x_{n}$, and $\operatorname{sdepth}_{S}I/J=2$. Let
$x_{1}x_{t}\in B$ for some $t$, $r<t\leq n$,
$I^{\prime}=(x_{2},\ldots,x_{r})+(B\setminus\\{x_{1}x_{t}\\})$,
$J^{\prime}=J\cap I^{\prime}$ and $\mathcal{P}$ a partition of
$I^{\prime}/J^{\prime}$ with sdepth $3$. Assume that any square free monomial
$u\in S$ of degree $2$, which is not in $I$, satisfies $x_{1}u\in J$. Then
1. (1)
For any $a\in(B\setminus(x_{2},\ldots,x_{r},x_{1}x_{t}))\cap(x_{t})$ with
$x_{1}a\not\in J$ the interval $[a,x_{1}a]$ is in $\mathcal{P}$.
2. (2)
If $c=x_{t}x_{i}x_{j}\not\in J$, $r<i<j\leq n$, $i,j\not=t$ and
$x_{1}x_{t}x_{i},x_{1}x_{t}x_{j}\not\in J$ then $b=c/x_{t}\in B$ and if
moreover $x_{1}b\not\in J$ then $c$ is not present in an interval $[a,c]$,
$a\in B$ of $\mathcal{P}$.
###### Proof.
Let $a=x_{t}x_{\nu}$ be a monomial of
$B\setminus(x_{2},\ldots,x_{r},x_{1}x_{t}))$ with satisfies $x_{1}a\not\in J$.
Suppose that the interval $[a,x_{1}a]$ is not in $\mathcal{P}$. Then there
exists in $\mathcal{P}$ an interval $[a,c]$ with $c\in C$. Thus $x_{1}x_{\nu}$
is in $B$ and so in $\mathcal{P}$ there exists an interval
$[x_{1}x_{\nu},c^{\prime}]$, $c^{\prime}\in C$,. We replace the interval
$[x_{1}x_{\nu},c^{\prime}]$ by $[x_{1},x_{1}a]$ to get a partition of $I/J$
with sdepth $\geq 3$. However, such partition of $I/J$ is not possible because
$\operatorname{sdepth}_{S}I/J=2$. Thus the interval $[a,x_{1}a]$ is in
$\mathcal{P}$.
Now, let $c$ be as in $(2)$. We will show that $b=c/x_{t}\in B$. Indeed, if
$b\not\in B$ then $b\not\in(x_{1},\ldots,x_{r})$ because otherwise $b\in J$,
which is false. Thus $c$ can enter only in an interval $[a,c]$ for let us say
$a=x_{t}x_{i}$. But this interval is not in $\mathcal{P}$ because $a$ belongs
to the interval $[a,x_{1}a]$. Contradiction! Thus $c$ does not appear in the
intervals of $\mathcal{P}$. Replacing $[a,x_{1}a]$ with $[a,c]$ in
$\mathcal{P}$ we get another partition of $I^{\prime}/J^{\prime}$ with sdepth
$3$, where the interval $[a,x_{1}a]$ is not present, contradicting $(1)$.
Moreover suppose that $x_{1}b\not\in J$. By $(1)$, $c$ can appear only in the
interval $[b,c]$ because we have already the intervals
$[x_{t}x_{i},x_{1}x_{t}x_{i}]$, $[x_{t}x_{j},x_{1}x_{t}x_{j}]$ in
$\mathcal{P}$. Then we cannot have an interval $[b,x_{1}b]$ in $\mathcal{P}$
and so $x_{1}b$ could appear in the interval, let us say
$[x_{1}x_{i},x_{1}b]$. Certainly, it is possible that $x_{1}b$ will not appear
at all in an interval of $\mathcal{P}$, but we may modify $\mathcal{P}$ to get
this. Replace in $\mathcal{P}$ the intervals $[x_{1}x_{i},x_{1}b]$, $[b,c]$,
$[x_{t}x_{i},x_{1}x_{t}x_{i}]$ by the intervals $[b,x_{1}b]$,
$[x_{t}x_{i},c]$, $[x_{1}x_{i},x_{1}x_{t}x_{i}]$ and we get another partition
of $I^{\prime}/J^{\prime}$ with sdepth $3$ but without the interval
$[x_{t}x_{i},x_{1}x_{t}x_{i}]$, contradicting again (1).
###### Lemma 1.8.
Suppose that $I\subset S$ is generated by $x_{1}$ and a nonempty set $E$ of
square free monomials of degrees $2$ in $x_{2},\ldots,x_{n}$ and
$\operatorname{sdepth}_{S}I/J=2$. Assume that $x_{1}a\not\in J$ for all $a\in
E$ and any square free monomial $u\in S$ of degree $2$, which is not in $I$,
satisfies $x_{1}u\in J$. Then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
Let $1<t\leq n$ be such that $x_{1}x_{t}\in B$. We may assume that
$a_{1},\ldots,a_{k}$, are all monomials of
$(E\cap(x_{t}))\setminus\\{x_{1}x_{t}\\}$. Set
$I_{t}=(B\setminus\\{x_{1}x_{t}\\})$ and $J_{t}=J\cap I_{t}$. In the exact
sequence
$0\rightarrow I_{t}/J_{t}\rightarrow I/J\rightarrow I/J+I_{t}\rightarrow 0$
the last term has depth $\geq 2$ because it is isomorphic with
$(x_{1})/(x_{1})\cap(J+I_{t})$ and $x_{1}x_{t}\not\in J+I_{t}$. If
$\operatorname{sdepth}_{S}I_{t}/J_{t}\leq 2$ then we get
$\operatorname{depth}_{S}I_{t}/J_{t}\leq 2$ by [12, Theorem 4.3]. Applying the
Depth Lemma we get $\operatorname{depth}_{S}I/J\leq 2$.
Thus we may assume that $\operatorname{sdepth}_{S}I_{t}/J_{t}\geq 3$ for all
$1<t\leq n$ such that $x_{1}x_{t}\in B$. Let ${\mathcal{P}}={\mathcal{P}_{t}}$
be a partition of $I_{t}/J_{t}$ with sdepth $=3$. By the above lemma the
intervals $[a_{j},x_{1}a_{j}]$, $1\leq j\leq k$ are in $\mathcal{P}$.
Suppose that $c=x_{i}x_{j}x_{t}\in C$, $i,j,t>1$ and $x_{j}x_{t},x_{i}x_{t}\in
E$. Then $a=x_{i}x_{j}\in E$ by the above lemma. By our hypothesis we have
$x_{1}a,x_{1}x_{j}x_{t},x_{1}x_{i}x_{t}\in C$. Thus $c$ cannot appear in an
interval of $\mathcal{P}$ using again the above lemma.
For $b=x_{1}x_{i}\in B$, $\mathcal{P}$ must contain some intervals of the form
$[x_{1}x_{i},x_{1}a^{\prime}_{i}]$ for some $a^{\prime}_{i}\in E$. Certainly
$a^{\prime}_{i}\not\in(x_{t})$ because we saw that all $a_{j}$, $1\leq j\leq
k$ enter already in the intervals $[a_{j},x_{1}a_{j}]$. Then these
$a^{\prime}_{i}$ enter in some intervals $[a^{\prime}_{i},c^{\prime}_{i}]$
with $c^{\prime}_{i}\in(C\setminus(x_{1}))$. If $c^{\prime}_{i}\in(a_{j})$ for
some $a_{j}$, $1\leq j\leq k$ then the third divisor of $c^{\prime}_{i}$ of
degrees $2$ is in $B$ too, and as above $c^{\prime}_{i}$ cannot appear in an
interval of ${\mathcal{P}}$. Contradiction! Thus
$c^{\prime}_{i}\in(C\setminus(x_{1},a_{1},\ldots,a_{k}))$.
Let $I^{\prime}=(x_{1}x_{t},a_{1},\ldots,a_{k})$, $J^{\prime}=J\cap
I^{\prime}$. We have seen that $c^{\prime}_{i}\not\in I^{\prime}$. In the
exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/J+I^{\prime}\rightarrow 0$
we show that the last term has sdepth $\geq 3$. Let
$a^{\prime}_{i}=x_{i}x_{\nu_{i}}\in B$ for some $1<\nu_{i}\leq n$. We may
suppose that $t>2$, $x_{1}x_{2}\in B$ and we see that the intervals
$[x_{1},x_{1}a^{\prime}_{2}]$, $[x_{1}x_{i},x_{1}a^{\prime}_{i}]$, $i>2$,
$i\not=\nu_{2}$, $[a^{\prime}_{i},c^{\prime}_{i}]$ induce with the help of
$\mathcal{P}$ a partition of $I/J+I^{\prime}$ with sdepth $3$. Indeed, the
only possible problem is that in $\mathcal{P}$ could appear some intervals of
type $[a,ax_{t}]$ for some $a\in(E\setminus(x_{t}))$, $c=ax_{t}$ being the
least common multiple of two $(a_{j})$. But this is not possible as we saw
above. By [14, Lemma 2.2] we get
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq 2$ and so
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq 2$ by [12, Theorem 4.3].
Applying the Depth Lemma we get as $\operatorname{depth}_{S}I/J\leq 2$.
###### Proposition 1.9.
Suppose that $I\subset S$ is generated by $x_{1}$ and a nonempty set $E$ of
square free monomials of degrees $2$ in $x_{2},\ldots,x_{n}$ and
$\operatorname{sdepth}_{S}I/J=2$. Let $E^{\prime}=\\{a\in E:x_{1}a\in C\\}$
and $E^{\prime\prime}=E\setminus E^{\prime}$. Assume that any square free
monomial $u\in S$ of degree $2$, which is not in $I$, satisfies $x_{1}u\in J$
and one of the following conditions hold:
1. (1)
$|E^{\prime\prime}|\leq|C\setminus(x_{1},E^{\prime})|$
2. (2)
$|E^{\prime\prime}|>|C\setminus(x_{1},E^{\prime})|$ and $|B|\not=|C|+1$.
Then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
If $E^{\prime\prime}=\emptyset$ then we apply the above lemma. Apply induction
on $|E^{\prime\prime}|$. If $E^{\prime}=\emptyset$ then
$C\cap(x_{1})=\emptyset$ and the conclusion follows from Lemma 1.5. Let
$E^{\prime\prime}=\\{a_{1},\ldots,a_{k}\\}$, $k>0$. We claim that we may
reduce our problem to the case when
$(C\setminus(x_{1}))\subset(E^{\prime\prime})$. Indeed, otherwise let
$c\in(C\setminus(x_{1},E^{\prime\prime}))$. Then there exists $b\in
E^{\prime}$ such that $c\in(b)$. Choose $t$, $1<t\leq n$ such that $x_{t}|b$.
Then $x_{1}x_{t}$ divides $x_{1}b\in C$ and so it is in $B$. Set
$I^{\prime}=(B\setminus\\{x_{1}x_{t}\\})$, $J^{\prime}=J\cap I^{\prime}$. In
the following exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(I^{\prime}+J)\rightarrow 0$
the last term is isomorphic with $(x_{1})/(x_{1})\cap(I^{\prime}+J)$ and has
depth $\geq 2$ because $x_{1}x_{t}\not\in(I^{\prime}+J)$. If
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq 2$ then by [12, Theorem
4.3] we get $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq 2$ and using
the Depth Lemma it follows $\operatorname{depth}_{S}I/J\leq 2$.
Thus we may suppose that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\geq
3$ and let ${\mathcal{P}}={\mathcal{P}}_{t}$ be a partition of
$I^{\prime}/J^{\prime}$ with sdepth $3$. By Lemma 1.7 (see also the above
lemma), $\mathcal{P}$ may contain some disjoint intervals
$[x_{1}x_{i},x_{1}b^{\prime}_{i}]$, $[b^{\prime}_{i},c^{\prime}_{i}]$, for
some $b^{\prime}_{i}\in E^{\prime}$, $c^{\prime}_{i}\in C\setminus(x_{1})$,
$i\not=1,t$ with $x_{1}x_{i}\in B$, $[b^{\prime},x_{1}b^{\prime}]$ for
$b^{\prime}\in E^{\prime}\setminus\\{\\{b^{\prime}_{i}\\}\\}$ and
$[a_{j},c_{j}]$, $j\in[k]$, $c_{j}\in C$. As in the proof of the above lemma
we have $b^{\prime}_{i}\not\in(x_{t})$. Thus the above $b$ is not one of
$b^{\prime}_{i}$ and enters in $\mathcal{P}$ in the interval $[b,x_{1}b]$.
Note that $c$ is not among $\\{\\{c_{j}\\}\\}$ because is not in
$(E^{\prime\prime})$. On the other hand, if $c=c^{\prime}_{i}$ then should be
divisible by $b$ and $b^{\prime}_{i}$, both being from $E^{\prime}$. Then by
Lemma 1.7 applied for a $t^{\prime}$ given by the only one common variable
$x_{t^{\prime}}$ of $b$, $b^{\prime}_{i}$, the third divisor
$u=c/x_{t^{\prime}}$ of degree $2$ of $c$ is in $E$, and $x_{1}u\in J$ because
$c$ can enter in an interval $[u,c]$ of a partition
${\mathcal{P}}_{t^{\prime}}$. Thus $u\in E^{\prime\prime}$ and so
$c\in(E^{\prime\prime})$, which is false. Then we may replace the interval
$[b,x_{1}b]$ by $[b,c]$, which is again false because all intervals
$[b^{\prime},x_{1}b^{\prime}]$, $b^{\prime}\in(E^{\prime})\cap(x_{t})$ should
be present in $\mathcal{P}$ by Lemma 1.7. This proves our claim. Also note
that $|C\setminus(x_{1})|\geq|B\cap(x_{1})|-1+k$.
Then we may assume that $(C\setminus(x_{1}))\subset(E^{\prime\prime})$. We may
suppose that $c_{i}\in(E^{\prime})$ if and only if $p<i\leq k$ for some $0\leq
p\leq k$. Moreover, we will arrange to have as many as possible $c_{j}$
outside $(E^{\prime})$. If $c^{\prime}\in(C\setminus(x_{1}))$ is a multiple of
let say $a_{p+1}$, but $c^{\prime}\not\in(E^{\prime})$. We may replace in the
above intervals $c_{p+1}$ by $c^{\prime}$, the effect being the increasing of
$p$. Thus after such procedure we may suppose that either $p=k$, or there
exist no $c$ in
$(C\setminus(x_{1},c_{1},\ldots,c_{p}))\cap(a_{p+1},\ldots,a_{k})$ which is
not in $(E^{\prime})$.
If $p=k$ then set $I^{\prime\prime}=(x_{1},E^{\prime})$,
$J^{\prime\prime}=I^{\prime\prime}\cap J$ and see that in the exact sequence
$0\rightarrow I^{\prime\prime}/J^{\prime\prime}\rightarrow I/J\rightarrow
I/(I^{\prime\prime}+J)\rightarrow 0$
the last term is isomorphic with
$(E^{\prime\prime})/(E^{\prime\prime})\cap(I^{\prime\prime}+J)$ and has sdepth
$3$ because the intervals $[a_{j},c_{j}]$, $j\in[k]$ gives a partition with
sdepth $3$. Then
$\operatorname{sdepth}_{S}I^{\prime\prime}/J^{\prime\prime}\leq 2$ by [14,
Proposition 2.2] and we get
$\operatorname{depth}_{S}I^{\prime\prime}/J^{\prime\prime}\leq 2$ by Lemma
1.8. Using the Depth Lemma it follows $\operatorname{depth}_{S}I/J\leq 2$.
Next suppose that $p<k$. Then
$(C\setminus(x_{1},c_{1},\ldots,c_{p}))\cap(a_{p+1},\ldots,a_{k})\subset(E^{\prime})$.
We may choose $c_{1},\ldots,c_{p}$ from the beginning (it is possible to make
such changes in $\mathcal{P}$) such that
$e=|\\{i:c_{i}\not\in(a_{p+1},\ldots,a_{k})\\}|$ is maxim possible and
renumbering $a_{j}$, $j\leq p$ we may suppose that
$c_{i}\not\in(a_{p+1},\ldots,a_{k})$ if and only if $i\in[e]$ for some $0\leq
e\leq p$.
Suppose that there exists $c\in C\setminus(x_{1},c_{1},\ldots,c_{p})$ such
that $c\not\in E^{\prime}$. Then $c$ is not in $(a_{p+1},\ldots,a_{k})$ and
necessary $c\in(a_{1},\ldots,a_{p})$. Assume that $c\in(a_{i})$ for some
$i\in[p]$. If $i>e$ then $c_{i}\in(a_{p+1},\ldots,a_{k})$, let us say
$c_{i}\in(a_{j})$ for some $j>p$ and we may change $c_{j}$ by $c_{i}$ and
replace $c_{i}$ by $c$ increasing $p$ because $c_{i}\not\in E^{\prime}$. This
is not possible since $p$ was maxim given. Thus $i\leq e$ and so $e>0$. If
$c_{i}\in(a_{e+1},\ldots,a_{p})$, let us say $c_{i}\in(a_{p})$ then we may
replace $c_{p}$ by $c_{i}$ and $c_{i}$ by $c$ increasing $e$ which is also not
possible. Thus $c_{i}\not\in(a_{e+1},\ldots,a_{p})$.
Then set $I_{e}=(x_{1},B\setminus\\{a_{1},\ldots,a_{e}\\})$, $J_{e}=I_{e}\cap
J$. In the exact sequence
$0\rightarrow I_{e}/J_{e}\rightarrow I/J\rightarrow I/(I_{e}+J)\rightarrow 0$
the last term has sdepth $3$ because we may write there the intervals
$[a_{i},c_{i}],i\in[e]$ since $c_{i}\not\in I_{e}$. By [14, Proposition 2.2]
it follows that $\operatorname{sdepth}_{S}I_{e}/J_{e}\leq 2$ and so
$\operatorname{depth}_{S}I_{e}/J_{e}\leq 2$ by induction hypothesis on
$|E^{\prime\prime}|$. Using the Depth Lemma it follows
$\operatorname{depth}_{S}I/J\leq 2$.
Now suppose that there exist no such $c$, that is
$C\setminus(x_{1},E^{\prime})=\\{c_{1},\ldots,c_{p}\\}$. Thus
$p=|C\setminus(x_{1},E^{\prime})|$ and so we end the case when the condition
(1) holds. Now suppose that the condition (2) holds, in particular $k>p$ and
$s=|B|\not=1+q$ for $q=|C|$. If $s>1+q$ then we end with [13]. Suppose that
$s<1+q$. Then there exists a $c\in C$ which does not appear in an interval
$[b,c]$ for some $b\in(B\setminus\\{x_{1}x_{t}\\})$. Note that $c$ cannot be a
$c_{j}$ for $j\in[p]$ and so $c\in(E^{\prime})$, let us say $c\in(a)$ for some
$a\in E^{\prime}$. Let $j$ be such that $x_{j}|a$. We have $x_{1}x_{j}\in B$
and there exists as above a partition ${\mathcal{P}}_{j}$ with sdepth $3$. Let
$I_{a}=(B\setminus\\{a\\})$, $J_{a}=I_{a}\cap J$. We see that
${\mathcal{P}}_{j}$ induces a partition ${\mathcal{P}}_{a}$ of $I_{a}/J_{a}$
with sdepth $3$ replacing the interval $[a,x_{1}a]$ from ${\mathcal{P}}_{j}$
with $[x_{1}x_{j},x_{1}a]$.
In ${\mathcal{P}}_{a}$ there is an interval
$[x_{1}x_{t},x_{1}a^{\prime\prime}_{1}]$ for some
$a^{\prime\prime}_{1}=x_{t}x_{i}\in E^{\prime}$. We have
$a^{\prime\prime}_{1}\not=a^{\prime}$ because otherwise we may change in
${\mathcal{P}}_{t}$ the interval
$[a^{\prime\prime}_{1},x_{1}a^{\prime\prime}_{1}]$ by
$[a^{\prime\prime}_{1},c]$, which is false. Then there is in
${\mathcal{P}}_{a}$ an interval $[a^{\prime\prime}_{1},c^{\prime\prime}_{1}]$.
If $c^{\prime\prime}_{1}$ is not a $c_{b}$ as above then we may replace in
${\mathcal{P}}_{t}$ the interval
$[a^{\prime\prime}_{1},x_{1}a^{\prime\prime}_{1}]$ by
$[a^{\prime\prime}_{1},c^{\prime\prime}_{1}]$, which is again false. Thus
$c^{\prime\prime}_{1}=c_{b_{1}}$ for some
$b_{1}\in(B\setminus\\{x_{1}x_{t}\\})$. If $b_{1}=a$ we may replace in
${\mathcal{P}}_{t}$ the intervals
$[a^{\prime\prime}_{1},x_{1}a^{\prime\prime}_{1}]$,
$[b_{1},c^{\prime\prime}_{1}]$ by
$[a^{\prime\prime}_{1},c^{\prime\prime}_{1}]$, $[b_{1},c]$, which is false.
Then there is in ${\mathcal{P}}_{a}$ an interval
$[b_{1},c^{\prime\prime}_{2}]$. By recurrence we find in ${\mathcal{P}}_{a}$
the intervals $[x_{1}x_{t},x_{1}a^{\prime\prime}_{1}]$,
$[a^{\prime\prime}_{1},c^{\prime\prime}_{1}]$,
$[a^{\prime\prime}_{2},c^{\prime\prime}_{2}],\ldots$ which define a partition
${\mathcal{P}}_{a}$, where $c$ is not present in an interval $[b,c]$,
$b\in(B\setminus\\{a\\})$. Adding the interval $[a,c]$ to ${\mathcal{P}}_{a}$
we get a partition ${\mathcal{P}}^{\prime}$ with sdepth $3$ of $I_{B}/J_{B}$,
where $I_{B}=(B)$, $J_{B}=I_{B}\cap J$. But then we replace in
${\mathcal{P}}^{\prime}$ the intervals
$[x_{1}x_{t},x_{1}a^{\prime\prime}_{1}]$,
$[x_{1}x_{i},x_{1}a^{\prime\prime}_{1}]$ by
$[x_{1},x_{1}a^{\prime\prime}_{1}]$ and we get a partition of $I/J$ with
sdepth $3$. Contradiction!
###### Theorem 1.10.
Suppose that $I\subset S$ is generated by $x_{1}$ and a nonempty set $E$ of
square free monomials of degrees $2$ in $x_{2},\ldots,x_{n}$ and
$\operatorname{sdepth}_{S}I/J=2$. Let $E^{\prime}=\\{a\in E:x_{1}a\in C\\}$
and $E^{\prime\prime}=E\setminus E^{\prime}$. Assume that one of the following
conditions holds:
1. (1)
$|E^{\prime\prime}|\leq|C\setminus(x_{1},E^{\prime})|$
2. (2)
$|E^{\prime\prime}|>|C\setminus(x_{1},E^{\prime})|$ and $|B|\not=|C|+1$.
Then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
We may assume $n>2$ and there exists $c=x_{1}x_{n-1}x_{n}\not\in J$ after
renumbering the variables $x$, otherwise we apply Proposition 1.3. Then
$z=x_{n-1}x_{n}\not\in J$.
First suppose that we may find $c$ with $z\not\in I$. Set
$I^{\prime}=(B\setminus\\{x_{1}x_{n-1},x_{1}x_{n}\\})$ and
$J^{\prime}=I^{\prime}\cap J$. Then necessary
$B\supsetneq\\{x_{1}x_{n-1},x_{1}x_{n}\\}$ and so $I^{\prime}\not=J^{\prime}$
because otherwise $\operatorname{sdepth}_{S}I/J=3$. Note that no $b$ dividing
$c$ belongs to $I^{\prime}$ and so $c\not\in(J+I^{\prime})$. In the following
exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(I^{\prime}+J)\rightarrow 0$
the last term has sdepth $\geq 3$ since $[x_{1},c]$ is the whole poset of
$(x_{1})/(x_{1})\cap(I^{\prime}+J)$ except some monomials of degrees $\geq 3$.
It has also depth $\geq 3$ because
$x_{n-1}x_{n}\not\in((J+I^{\prime}):x_{1})$. The first term has sdepth
$\leq\operatorname{sdepth}_{S}I/J=2$ by [14, Lemma 2.2] and so it has depth
$\leq 2$ by [12, Theorem 4.3]. It follows $\operatorname{depth}_{S}I/J\leq 2$.
Next suppose that there exist no such $c$, that is any square free monomial
$u\in S$ of degree $2$, which is not in $I$ satisfies $x_{1}u\in J$. We may
assume that $C\subset(x_{1},B)$ by Lemma 1.6. Now it is enough to apply
Proposition 1.9.
###### Example 1.11.
Let $n=3$, $r=1$, $I=(x_{1},x_{2}x_{3})$, $J=0$. We have
$c=x_{1}x_{2}x_{3}\not\in J$ and $x_{2}x_{3}\in I$. Note also that
$\operatorname{sdepth}_{S}I=\operatorname{depth}_{S}I=2.$
## References
* [1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition. Cambridge University Press (1998).
* [2] W. Bruns, C. Krattenthaler, J. Uliczka, Stanley decompositions and Hilbert depth in the Koszul complex, J. Commutative Alg., 2 (2010), 327-357.
* [3] M. Cimpoeas, The Stanley conjecture on monomial almost complete intersection ideals, Bull. Math. Soc. Sci. Math. Roumanie, 55(103), (2012), 35-39.
* [4] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [5] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc., 140 (2012), 493-504, arXiv:AC/1011.6462v1.
* [6] B. Ichim, J. J. Moyano-Fernandez, How to compute the multigraded Hilbert depth of a module, (2012), arXiv:AC/1209.0084.
* [7] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra 40(2012), no. 1, 87 97.
* [8] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010),363-372, arXiv:AC/1008.3680.
* [9] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
* [10] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, to appear in Communications in Alg.,arXiv:AC/1009.5646.
* [11] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19 (2), (2011), 187-194.
* [12] D. Popescu, Depth of factors of square free monomial ideals, to appear in Proceedings AMS, arXiv:AC/1110.1963.
* [13] D. Popescu, Upper bounds of depth of monomial ideals, to appear in J. Commutative Alg., (2012),arXiv:AC/1206.3977.
* [14] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784.
* [15] Y.H. Shen, Lexsegment ideals of Hilbert depth 1, (2012), arxiv:AC/1208.1822v1.
* [16] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
* [17] J. Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscripta Math., 132 (2010), 159-168.
* [18] A. Zarojanu, Stanley Conjecture on three monomial primary ideals, Bull. Math. Soc. Sc. Math. Roumanie, 55(103),(2012), 335-338, arXiv:AC/11073211.
|
arxiv-papers
| 2012-11-05T12:24:05 |
2024-09-04T02:49:37.552483
|
{
"license": "Public Domain",
"authors": "Dorin Popescu and Andrei Zarojanu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1211.0842"
}
|
1211.0899
|
# A Helly-type problem
Nguyen Luu Danh
###### Abstract.
We raise a question related to Helly’s theorem with the added elements of
geometric transformations.
The classic theorem of Helly tells us the following fact: if any $d+1$ among
$n$ convex bodies in $\mathbb{R}^{d}$ have non-empty intersection, then all
$n$ of them have non empty-intersection. $d+1$ is the best constant for a
general family of $d$-dimensional convex body, and so in a way the theorem
gives another interpretation for the concept of dimensions. This simple
looking result has served to inspire many great developments in discrete
geometry.
A general format that applies to all Helly-related theorems is as follows: for
a collection geometric objects with sufficiently generic properties, if any
$k$-subcollection satisfies some special property, then so does the whole
collection. Usually, such a minimal threshold $k$ exists and depends on the
particular class of objects. For convex bodies, it is the number of dimensions
plus $1$. For non–convex bodies with sufficiently nice intersections, there is
still a corresponding threshold, although not solely dependent upon the
dimension. The “special property” mentioned can vary from non-empty
intersection to common transversal line, etc. For a good reference, please
look at [Mat] or [VGG].
We look at another situation where certain transformations are allowed to act
on the objects, in this paper it is rotations. First, let us state a
straightforward result following from Helly’s theorem:
###### Corollary 1.
Given $n$ points in $\mathbb{R}^{d}$, if any $d+1$ among them can be covered
by a certain translate of a convex body $K$ then all $n$ points can also be
covered in that way.
###### Proof.
A translate $t+K$ contains a point $v$ if and only if $t\in v+(-K)$. Forming a
new family $\\{v_{i}+(-K)\\}$, the given condition is now equivalent to any
$d+1$ family members having non-empty intersection. Helly’s theorem guarantees
a point belonging to all the objects and this is the translation vector that
makes $K$ cover each $v_{i}$. ∎
In this corollary’s setting, what if we allow both translations and rotations
of the body $K$, is there still a similar result? This of course does not say
anything new if $K$ is a round ball, but if $K$ is any other convex body, is
there still another $k$ in place of $d+1$. Should we expect that such
statement is true then $k$ must really depends on the particular shape of $K$
and can be very large for special bodies. Ironically, we are going to show
that such a result would never exist, at least for a large family of convex
sets. Call the largest circle inscribable in $K$ its incircle, we have:
###### Lemma 2.
Let $O$ be $K$’s incircle (possibly one among many), if $\partial K\cup O$ is
a discrete set then no such number k exists.
###### Proof.
$O$$r$$K$$O^{\prime}$$P$$r+\epsilon$$R$
In the above two figures, the left one shows a two dimensional convex body $K$
with the incircle $O$ having radius $r$. On the right we have a slightly
larger circle $O^{\prime}$ with radius $r+\epsilon$, also there is a regular
polygon $P$ containing $O^{\prime}$ as its incircle. $P$ has $n$ edges
touching $O^{\prime}$ and the distance from its vertices to the center of
$O^{\prime}$ is $R$. We have $\lim_{\begin{subarray}{c}\epsilon\to 0\\\
n\to\infty\end{subarray}}R=r$ but clearly any such $P$ cannot be contained in
$K$. Assume $n$ and $\epsilon$ are respectively big and small enough, we mark
on the boundary of $K$ any point that has distance less than $R$ away from the
center of $O$. Let $\alpha$ be the total measure of the angles subtended
radically by these marked points, then $\lim_{\begin{subarray}{c}\epsilon\to
0\\\ n\to\infty\end{subarray}}\alpha=0$. We first translate the two figures so
that $O$ and $O^{\prime}$ are concentric. Now for any vertex of $P$, we can
properly rotate $P$ to make sure that it avoids all marked portions and thus
lies in side $K$. The possible set of rotations for this vertex measures
$2\pi-\alpha$. Hence, we choose any simultaneously rotate any $k$ among $n$
vertices in this way provided that $k\alpha<2\pi$. As $\alpha$ approaches $0$,
$k$ can be as large as possible but still the whole polygon $K$ is not
inscribable in $K$ after any rotation. ∎
The above proof is also adaptable to all higher dimensional convex bodies
whose intersections with boundaries of inner circles have zero measure. This
rules out all convex polytopes as well as simple cases like a sphere with
pieces sliced away by hyperplanes. We are left with cases when $\partial K$
shares a positive-measured intersection with its incircle. So far the author
knows of no effective method to construct counterexamples for these remaining
cases. Thinking from the opposite perspective, if we denote $\alpha$ as the
$(d-1)$-dimensional measure of $\partial K\cap O$ and $\beta$ as that of $O$,
then such a $k$ if exists should be bounded below by $\frac{\beta}{\alpha}$.
## References
* [VGG] E. C. de Verdière, G. Ginot, X. Goaoc. Helly numbers of acyclic families. ArXiv:1101.6006, [math.CO], 2011.
* [Mat] J. Matoušek. Lectures on Discrete Geometry. Springer-Verlag, New York, 2002.
|
arxiv-papers
| 2012-11-02T06:15:21 |
2024-09-04T02:49:37.562787
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nguyen Luu Danh",
"submitter": "Danny Nguyen",
"url": "https://arxiv.org/abs/1211.0899"
}
|
1211.1112
|
KU-PH-013
TTP12-042
SFB/CPP-12-85
Full $\mathcal{O}(\alpha)$ electroweak radiative corrections to
$e^{+}e^{-}\rightarrow t\bar{t}\gamma$ with GRACE-Loop
P.H. KhiemA,B, J. FujimotoA, T. IshikawaA, T. KanekoA,
K. KatoC, Y. KuriharaA,
Y. ShimizuA, T. UedaD, J.A.M. VermaserenE, Y. YasuiF
A)KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan.
B)SOKENDAI University, Shonan Village, Hayama, Kanagawa 240-0193 Japan.
C)Kogakuin University, Shinjuku, Tokyo 163-8677, Japan.
D)Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany.
E)NIKHEF, Science Park 105, 1098 XG Amsterdam, The Netherlands.
F)Tokyo Management College, Ichikawa, Chiba 272-0001, Japan.
## 1 Introduction
The experimental results of CDF [1] and D$0$ [2] on the measurement of top
pair production at the Tevatron show an unexpected large top quark forward-
backward asymmetry. The precise theoretical calculations of the top pair
production play an important role in explaining the experimental data. QCD
radiative corrections to top pair production from proton-proton collisions
were calculated by several authors [3], [4], [5], [6], [7]. However, the
measurement is affected by a huge background from QCD. A good example is the
$gg\rightarrow t\bar{t}$ reaction. In the future, the measurement will be
performed at the ILC without QCD background. Therefore, we consider the
precise calculations of top pair production and top pair with photon
production in $e^{+}e^{-}$ collisions. A completed full one-loop electroweak
correction calculation to the process $e^{+}e^{-}\rightarrow t\bar{t}$ has
already been presented in refs [8], [9], [10]. In this paper, we calculate the
full $\mathcal{O}(\alpha)$ electroweak radiative corrections to both the
process $e^{+}e^{-}\rightarrow t\bar{t}$ and $e^{+}e^{-}\rightarrow
t\bar{t}\gamma$ at the ILC.
The data of the ATLAS [11] and CMS [12] experiments prove the existence of a
new boson with mass around $126$GeV. It is assumed to be the standard-model
Higgs particle. Once the discovery of the Higgs boson is confirmed, the next
important task is to measure its properties. However, it is clear that such a
measurement is much easier at the cleaner environment of the ILC than at the
LHC with its large QCD backgrounds. To measure the properties of the new boson
it is important that also the radiative correction calculations for the ILC
take the complete standard model into account.
The experiments at the ILC require a precise determination of the luminosity
which will be based on higher order theoretical calculations of the Bhabha
scattering cross-section. Thus, the computation of electroweak radiative
corrections to the process $e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$ is
mandatory. This will be our eventual target. However, as a first step, we are
going to calculate the process $e^{+}e^{-}\rightarrow t\bar{t}\gamma$ which is
easier in several respects: there are fewer diagrams and the numerical
cancellations between the diagrams are less severe. It will provide a
framework for our target calculation.
In the scope of this paper, we discuss the full $\mathcal{O}(\alpha)$
electroweak radiative corrections to the process $e^{+}e^{-}\rightarrow
t\bar{t}\gamma$ at ILC. We then examine the numerical results of the top quark
forward-backward asymmetry as well as the genuine weak corrections in both the
$\alpha$ scheme and the $G_{\mu}$ scheme [13] as compared to the process
$e^{+}e^{-}\rightarrow t\bar{t}$.
The paper is organized as follows. In section 2 we introduce the GRACE-Loop
system and set up the calculation. In section 3 we discuss the numerical
results of the calculation. Future plans and conclusions of our paper are
presented in section 4.
## 2 GRACE-Loop and the process $e^{-}e^{+}\rightarrow t\bar{t}\gamma$
### 2.1 GRACE-Loop
The computation is performed with the help of the GRACE-Loop system which is a
generic program for the automatic calculation of scattering processes in High
Energy Physics. Of course, with a system this complicated still under
development, it is important to have as many tests as possible of the
correctness of the answer. Hence the GRACE-Loop system has been equipped with
non-linear gauge fixing terms in the Lagrangian which will be described in
some of the next paragraphs. The renormalization has been carried out with the
on-shell renormalization condition of the Kyoto scheme, as described in ref
[14]. The program was presented and checked carefully with a variety of
$2\rightarrow 2$-body electroweak processes in ref [15]. The GRACE-Loop system
has also been used to calculate $2\rightarrow 3$-body processes such as
$e^{+}e^{-}\rightarrow ZHH$ [16], $e^{+}e^{-}\rightarrow t\bar{t}H$ [17],
$e^{+}e^{-}\rightarrow\nu\bar{\nu}H$ [18]. The above calculations have been
done independently by other groups, for example the process
$e^{+}e^{-}\rightarrow ZHH$ [19], $e^{+}e^{-}\rightarrow t\bar{t}H$ [20],
[21], [22] and $e^{+}e^{-}\rightarrow\nu\bar{\nu}H$ [23], [24]. Moreover, the
$2\rightarrow 4$-body process as
$e^{+}e^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}HH$ [25] was calculated
successfully by GRACE-loop system.
The steps of calculating a process in the GRACE system are as follows. First
the system requires input files that describe the Feynman rules of the model.
In this case, we use the standard model. These files are considered part of
the system but for different models the user would have to provide them. Next
a (small) file is needed that selects the model, the names of the incoming and
outgoing particles, and one of a set of predefined kinematic configurations
for the phase space integration. In the intermediate stage symbolic
manipulation handles all Dirac and tensor algebra in $n$-dimensions, reduces
the formulas to coefficients of tensor one-loop integrals and writes the
formulas in terms of FORTRAN subroutines on a diagram by diagram basis. For
this manipulation either FORM [26] or REDUCE [27] is used. The FORTRAN
routines will be combined with libraries which contain the routines that
reduce the tensor one-loop integrals into scalar one-loop integrals. The
scalar one-loop integrals will be numerically evaluated by one of the FF [28]
or LoopTools [29] packages. The ultraviolet divergences (UV-divergences) are
regulated by dimensional regularization and the infrared divergences (IR-
divergences) will be regulated by giving the photon an infinitesimal mass
$\lambda$. Eventually all FORTRAN routines are compiled and linked with the
GRACE libraries which include the kinematic libraries and the Monte Carlo
integration program BASES [30]. The resulting executable program can then
calculate cross-sections and generate events.
Ref [15] describes the method used by the GRACE-Loop system to reduce the
tensor one-loop five- and six-point functions into one-loop four-point
functions.
The GRACE-Loop system allows the use of non-linear gauge fixing conditions
[31] which are defined by:
$\displaystyle{{\cal L}}_{GF}$ $\displaystyle=$
$\displaystyle-\frac{1}{\xi_{W}}|(\partial_{\mu}\;-\;ie\tilde{\alpha}A_{\mu}\;-\;igc_{W}\tilde{\beta}Z_{\mu})W^{\mu+}+\xi_{W}\frac{g}{2}(v+\tilde{\delta}H+i\tilde{\kappa}\chi_{3})\chi^{+}|^{2}$
(1) $\displaystyle\;-\frac{1}{2\xi_{Z}}(\partial\cdot
Z+\xi_{Z}\frac{g}{2c_{W}}(v+\tilde{\varepsilon}H)\chi_{3})^{2}\;-\frac{1}{2\xi_{A}}(\partial\cdot
A)^{2}\;.$
We are working in the $R_{\xi}$-type gauges with condition
$\xi_{W}=\xi_{Z}=\xi_{A}=1$ (with so-called the ’t Hooft-Feynman gauge), there
is no contribution of the longitudinal term in the gauge propagator. This
choice has not only the advantage of making the expressions much simpler. It
also avoids unnecessary large cancellations, high tensor ranks in the one-loop
integrals and extra powers of momenta in the denominators which cannot be
handled by the FF package.
The GRACE-Loop system can also use an axial gauge for external photons. This
has two advantages.
1. 1.
It cures a problem with large numerical cancellations. This is very useful
when calculating the process at small angle and energy cuts of the final state
particles.
2. 2.
It provides a useful tool to check the consistency of the results which, due
to the Ward identities, must be independent of the choice of the gauge.
### 2.2 The numerical test of the process $e^{-}e^{+}\rightarrow
t\bar{t}\gamma$
The full set of Feynman diagrams with the non-linear gauge fixing as described
before consists of $16$ tree diagrams and $1704$ one-loop diagrams (of which
168 are pentagon diagrams). In Fig 1 we show some selected diagrams.
Figure 1: Typical Feynman diagrams as generated by the GRACE-Loop system.
The results are checked carefully by three kinds of consistency tests. These
tests are performed with quadruple precision at a few random points in the
phase space. The first test is ultraviolet finiteness of the results. This
test is done on all virtual one-loop diagrams and their counter terms and we
treat $C_{UV}=1/\epsilon-\gamma_{E}+\log 4\pi$ as a parameter. In order to
regularize the infrared divergences, we give the virtual photon a fictitious
mass, $\lambda=10^{-17}$GeV. In the table 1 we present the numerical results
of the test at one random point in the phase space. The result is stable over
more than 30 digits for various values of the ultraviolet parameter.
$C_{UV}$ | $2\mathcal{\Re}(\mathcal{T}_{Tree}^{+}\mathcal{T}_{Loop})$
---|---
$0$ | $-5.3131630854021768119477116628317605E^{-3}$
$10$ | $-5.3131630854021768119477116628317726E^{-3}$
$100$ | $-5.3131630854021768119477116628320404E^{-3}$
Table 1: Test of $C_{UV}$ independence of the amplitude. In this table, we
take the non-linear gauge parameters to be $0$, $\lambda=10^{-17}$GeV and we
use $1$ TeV for the center-of-mass energy.
The second test is the independence of the result on the fictitious photon
mass $\lambda$. In this case, we take $C_{UV}=0$. This test will be performed
by including as well the virtual loop diagrams as the soft bremsstrahlung
contribution. In table 2 the numerical results of the test are presented. We
find that the result is stable over more than 15 digits when varying the
parameter $\lambda$ over a wide range.
$\lambda$ [GeV] | $2\mathcal{\Re}(\mathcal{T}_{Tree}^{+}\mathcal{T}_{Loop})$+soft contribution
---|---
$10^{-17}$ | $-1.6743892369492021873805611201763810E^{-3}$
$10^{-19}$ | $-1.6743892369492020397654354220438766E^{-3}$
$10^{-21}$ | $-1.6743892369492020382892402083349623E^{-3}$
Table 2: Test of the IR finiteness of the amplitude. In this table we take
the non-linear gauge parameters to be $0$, $C_{UV}=0$ and the center-of-mass
energy is $1$ TeV.
The independence of the result on the five parameters
$\tilde{\alpha},\tilde{\beta},\tilde{\delta},\tilde{\kappa},\tilde{\varepsilon}$
is also checked. The result is presented in table 3. We find that the result
is stable over more than 26 digits while varying the non-linear gauge
parameters.
$(\tilde{\alpha},\tilde{\beta},\tilde{\kappa},\tilde{\delta},\tilde{\epsilon})$ | $2\mathcal{\Re}(\mathcal{T}_{Tree}^{+}\mathcal{T}_{Loop})$
---|---
$(0,0,0,0,0)$ | $-5.3131630854021768119477116628317605E^{-3}$
$(1,2,3,4,5)$ | $-5.3131630854021768119477116637537265E^{-3}$
$(10,20,30,40,50)$ | $-5.3131630854021768119477116582762373E^{-3}$
Table 3: Gauge invariance of the amplitude. In this table, we set $C_{UV}=0$,
the photon mass is $10^{-17}$GeV and a $1$ TeV center-of-mass energy.
Finally, we check the stability of the result versus the soft photon cut
parameter ($k_{c}$). This test includes both the soft photon and the hard
photon contributions. The hard photon bremsstrahlung part is the process
$e^{+}e^{-}\rightarrow t\bar{t}\gamma\gamma$. It is important to note that we
have two photons at the final state. One of them has to be applied an energy
cut of $E_{\gamma}^{cut}\geq 10$ GeV and an angle cut of
$10^{\circ}\leq\theta_{\gamma}^{cut}\leq 170^{\circ}$. Another one is a hard
photon with energy is greater than $k_{c}$ and smaller than the first photon’s
energy. This part will be generated by the tree level version of GRACE [32]
with the phase space integration by BASES. The result is tested by changing
the value of $k_{c}$ from $10^{-5}$ GeV to $0.1$ GeV. In table 4 we find that
the results are in agreement with an accuracy which is better than $0.1\%$
when we vary $k_{c}$.
$k_{c}$[GeV] | $\sigma_{H}$ | $\sigma_{S}$ | $\sigma_{S+H}$
---|---|---|---
$10^{-5}$ | $4.172723E^{-02}$ | $5.885469E^{-02}$ | $0.10058192$
$10^{-3}$ | $2.926684E^{-02}$ | $7.131737E^{-02}$ | $0.10058421$
$10^{-1}$ | $1.678994E^{-02}$ | $8.377319E^{-02}$ | $0.10056313$
Table 4: Test of the $k_{c}$-stability of the result. We choose the photon
mass to be $10^{-17}$ GeV and the center-of-mass energy is $1$ TeV. The second
column presents the hard photon cross-section and the third column presents
the soft photon cross-section. The final column is the sum of both.
We found that the numerical results are in good agreement when varying
$C_{UV}$, the gauge parameters, photon mass, and $k_{c}$. Hereafter, we set
$\lambda=10^{-17}$ GeV, $C_{UV}=0$ and
$\tilde{\alpha}=\tilde{\beta}=\tilde{\delta}=\tilde{\kappa}=\tilde{\varepsilon}=0$.
## 3 Results
Our input parameters for the calculation are as follows. The fine structure
constant in the Thomson limit is $\alpha^{-1}=137.0359895$. The mass of the Z
boson is $M_{Z}=91.187$ GeV. In the on-shell renormalization scheme we take
the mass of the $W$ boson ($M_{W}$) as an input parameter. It will be derived
through the electroweak radiative corrections to the muon decay width ($\Delta
r$) [33] with $G_{\mu}=1.16639\times 10^{-5}$ GeV-2. Therefore, $M_{W}$ is a
function of $M_{H}$. In this calculation, we take $M_{H}=120$ GeV and the
numerical value of $M_{W}$ is $80.3759$ GeV. For the lepton masses we take
$m_{e}=0.51099891$ MeV, $m_{\tau}=1776.82$ MeV and $m_{\mu}=105.658367$ MeV.
For the quark masses we take $m_{u}=1.7$ MeV, $m_{d}=4.1$ MeV, $m_{c}=1.27$
GeV, $m_{s}=101$ MeV, $m_{t}=172.0$ GeV and $m_{b}=4.19$ GeV. We apply an
energy cut of $E_{\gamma}^{cut}\geq 10$ GeV and an angle cut of
$10^{\circ}\leq\theta_{\gamma}^{cut}\leq 170^{\circ}$ on the photon.
All numerical results are generated by the GRACE-Loop system. For $t\bar{t}$
production the results were first checked with the results in refs [8], [9],
[10]. Then we use the values of the parameters above to produce the results of
$t\bar{t}$ production in this paper and compare them with $t\bar{t}\gamma$
production.
In Fig 2 the total cross-section is a function of the center-of-mass energy
$\sqrt{s}$. We vary the value of $\sqrt{s}$ from $360$ GeV to $1$ TeV. We find
that the cross-section is largest near the threshold, $\sqrt{s}$ around $550$
GeV for $t\bar{t}\gamma$ production and $410$ GeV for $t\bar{t}$ production.
The total cross-section of $t\bar{t}\gamma$ production is considerably less
than $10\%$ of the total cross-section for the $t\bar{t}$ reaction. In
addition we find a negative correction for $t\bar{t}\gamma$ production in
contrast to the positive correction for $t\bar{t}$ production.
$\begin{array}[]{cc}\vspace*{-0.5cm}e^{-}e^{+}\rightarrow
t\bar{t}&e^{-}e^{+}\rightarrow t\bar{t}\gamma\\\
\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{eett.eps}&\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{eeytt.eps}\end{array}$
Figure 2: The total cross-section as a function of center-of-mass energy. The
left figure is the result of $t\bar{t}$ production and the right figure shows
the result of the $t\bar{t}\gamma$ reaction. The triangle points are the
result of the tree level calculation while the rectangular points are the sum
of the tree level calculation combined with the full one-loop electroweak
radiative corrections. Lines are only guide for the eyes.
The full $\mathcal{O}(\alpha)$ electroweak corrections take into account the
tree graphs and the full one-loop virtual corrections as well as the soft and
hard bremsstrahlung contributions. The relative correction is defined as
$\displaystyle\delta_{EW}=\frac{\sigma(\alpha)}{\sigma_{Tree}}-1.$ (2)
In order to extract the genuine weak correction in the $G_{\mu}$ scheme, we
first evaluate the QED initial radiative correction ($\delta_{QED}$). Applying
the structure function method described in ref [34], $\delta_{QED}$ is defined
as
$\displaystyle\delta_{QED}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{QED}-\sigma_{Tree}}{\sigma_{Tree}},$ (3)
with
$\displaystyle\sigma_{QED}(s)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}dx\;\mathcal{H}(x,s)\;\sigma_{0}(s(1-x)),$
(4)
here $\mathcal{H}(x,s)$ is a radiator which is defined by formula (11.213) in
ref [34]:
$\displaystyle\mathcal{H}(x,s)$ $\displaystyle=$ $\displaystyle\Delta\beta
x^{\beta-1}-\beta(1-\frac{x}{2})$ (5) $\displaystyle+$
$\displaystyle\frac{\beta^{2}}{8}\Big{[}-4(2-x)\ln
x-\frac{1+3(1-x)^{2}}{x}\ln(1-x)-6+x\Big{]}$
with $\beta=\frac{2\alpha}{\pi}\Big{(}\ln(\frac{s}{m_{e}^{2}})-1\Big{)}$ and
$\Delta=1+\frac{\alpha}{\pi}\Big{(}\frac{3}{2}\ln(\frac{s}{m_{e}^{2}})+\frac{\pi^{2}}{3}-2\Big{)}$.
After obtaining the QED correction, we define the genuine weak correction in
the $\alpha$ scheme:
$\displaystyle\delta_{W}=\delta_{EW}-\delta_{QED}.$ (6)
Having subtracted the genuine weak correction in the $\alpha$ scheme, one can
express the correction in the $G_{\mu}$ scheme. Next we subtract the universal
weak correction which is obtained from $\Delta r$. The genuine weak correction
in the $G_{\mu}$ scheme is defined by
$\displaystyle\delta^{G_{\mu}}_{W}=\delta_{W}-n\Delta r,$ (7)
with $\Delta r=2.55\%$ for $M_{H}=120$ GeV and $n=3(2)$ for $t\bar{t}\gamma$
(for $t\bar{t}$) production respectively.
In Fig 3, we present the full electroweak correction and the genuine weak
correction in both the $\alpha$ and the $G_{\mu}$ schemes for $t\bar{t}\gamma$
production as compared to $t\bar{t}$ production. These corrections are shown
as a function of the center-of-mass energy, $\sqrt{s}$. We vary $\sqrt{s}$
from $360$ GeV to $1$ TeV. The figures show clearly that the QED correction is
dominant in the low energy region. In the high energy region it is much
smaller ($\sim-5\%$ at $1$ TeV). In contrast to the QED correction the weak
correction in the $\alpha$ scheme is less than $10\%$ for low energies but
reaches $-16\%$ at $1$ TeV center-of-mass energy. For $t\bar{t}\gamma$
production, we find that the value of the genuine weak correction in the
$G_{\mu}$ scheme varies from $2\%$ to $-24\%$ over $\sqrt{s}$ from $360$ GeV
to $1$ TeV.
$\begin{array}[]{cc}\vspace*{-0.5cm}e^{-}e^{+}\rightarrow
t\bar{t}&e^{-}e^{+}\rightarrow t\bar{t}\gamma\\\
\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{eettdelat.eps}&\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{eeyttdelat.eps}\end{array}$
Figure 3: The full electroweak correction and the genuine weak correction as a
function of the center-of-mass energy. The left figure shows the results for
$t\bar{t}$ production while the right figure shows the results for
$t\bar{t}\gamma$ production. The circle points represent the QED correction,
the empty rectangle points are the results for the full electroweak correction
while the triangle points are the results for the genuine weak correction in
the $\alpha$ scheme. The filled rectangle points represent the results of the
genuine weak correction in the $G_{\mu}$ scheme. Lines are only guide for the
eyes.
Now we turn our attention to the forward-backward asymmetry $A_{FB}$. This
quantity is defined as
$\displaystyle A_{FB}$ $\displaystyle=$
$\displaystyle\frac{\sigma(0^{\circ}\leq\theta_{t}\leq
90^{\circ})-\sigma(90^{\circ}\leq\theta_{t}\leq
180^{\circ})}{\sigma(0^{\circ}\leq\theta_{t}\leq
90^{\circ})+\sigma(90^{\circ}\leq\theta_{t}\leq 180^{\circ})},$ (8)
with $\theta_{t}$ the angle of the top quark.
Fig 4 shows the results for $A_{FB}$ as a function of the center-of-mass
energy. The figures show clearly that the top quark asymmetry in the full
results is smaller than the asymmetry at the tree level results only.
$\begin{array}[]{cc}\vspace*{-0.5cm}e^{-}e^{+}\rightarrow
t\bar{t}&e^{-}e^{+}\rightarrow t\bar{t}\gamma\\\
\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{AFB.eps}&\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{Afby.eps}\end{array}$
Figure 4: The top quark forward-backward asymmetry as a function of the
center-of-mass energy. Left figure is the results for $t\bar{t}$ production
and right one is the results for $t\bar{t}\gamma$ production. The triangle
points represent the tree level results and the rectangle points are the
results including the full radiative corrections. Lines are only guide for the
eyes.
In Fig 5 we compare the values of $A_{FB}$ in $t\bar{t}\gamma$ production
directly with its value for $t\bar{t}$ production. From the figures, we find
that $A_{FB}$ in $t\bar{t}\gamma$ production is larger than $A_{FB}$ in
$t\bar{t}$ production. This is the most important result of the paper. The
effect should be clearly observable at the ILC.
$\begin{array}[]{cc}\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{AfTree.eps}&\includegraphics[width=216.81pt,height=213.39566pt,angle={-90}]{AfLoop.eps}\end{array}$
Figure 5: The value of the top quark asymmetry in $t\bar{t}\gamma$ production
as compared to $t\bar{t}$ production. The rectangular points (circle points)
represent the result for $t\bar{t}\gamma$ production ($t\bar{t}$ production)
respectively. Lines are only guide for the eyes.
## 4 Conclusions
We have presented the full $\mathcal{O}(\alpha)$ electroweak radiative
corrections to the process $e^{+}e^{-}\rightarrow t\bar{t}\gamma$ and
$e^{+}e^{-}\rightarrow t\bar{t}$ at ILC. The calculations were done with the
help of the GRACE-Loop system.
GRACE-Loop have implemented a generalised non-linear gauge fixing condition
which includes five gauge parameters. With the UV, IR finiteness and gauge
parameters independence checks, the system provide a powerful tool to test the
results in the consistency way. In the numerical checks of this calculation,
we find that the results are numerically stable when quadruple precision is
used.
We find that the numerical value of the genuine weak corrections in $G_{\mu}$
scheme varies from $2\%$ to $-24\%$ in the range of center-of-mass energy from
$360$ GeV to $1$TeV. We also obtain a large value for the top quark forward-
backward asymmetry in the $t\bar{t}\gamma$ process as compared with the one in
$t\bar{t}$ production.
We also introduce the axial gauge for the external photon in the GRACE-Loop
system. It helps to avoid a large numerical cancellation problem. This is very
useful when calculating Bhabha scattering at small angle and energy cuts of
the final state particles. Bhabha scattering and related processes are not
only used as luminosity monitor, but also play an important role as
backgrounds for the process
$e^{-}e^{+}\rightarrow\tilde{\chi}^{-}\tilde{\chi}^{+}\gamma$, which is a very
interesting reaction for the search for dark matter. We will address it in a
future publication.
In addition, this calculation will provide a framework for calculating the
full $\mathcal{O}(\alpha)$ electroweak radiative corrections for the process
$e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$. This reaction and these corrections
will play an important role at future $e^{+}e^{-}$ colliders like the ILC.
## Acknowledgments
We wish to thank Dr. F. Yuasa and Dr. N. Watanabe for valuable discussions and
comments. This work was supported by JSPS KAKENHI Grant Number 20340063. The
work of T.U. was supported by the DFG through SFB/TR 9 “Computational Particle
Physics”.
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|
arxiv-papers
| 2012-11-06T05:12:28 |
2024-09-04T02:49:37.577543
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P.H. Khiem, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato, Y. Kurihara,\n Y. Shimizu, T. Ueda, J.A.M. Vermaseren, Y. Yasui",
"submitter": "Khiem Hong Phan",
"url": "https://arxiv.org/abs/1211.1112"
}
|
1211.1113
|
# THE BALDWIN EFFECT IN THE NARROW EMISSION LINES OF AGNS
Kai Zhang11affiliation: Key Laboratory for Research in Galaxies and Cosmology,
The University of Sciences and Technology of China, Chinese Academy of
Sciences, Hefei, Anhui 230026, China;
zkdtc@mail.ustc.edu.cn; twang@ustc.edu.cn; xbdong@ustc.edu.cn 22affiliation:
Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,
China , Ting-Gui Wang11affiliation: Key Laboratory for Research in Galaxies
and Cosmology, The University of Sciences and Technology of China, Chinese
Academy of Sciences, Hefei, Anhui 230026, China;
zkdtc@mail.ustc.edu.cn; twang@ustc.edu.cn; xbdong@ustc.edu.cn , C. Martin
Gaskell33affiliation: Centro de Astrofísica de Valparaíso y Departamento de
Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Av. Gran
Bretaña 1111, Valparaíso, Chile. martin.gaskell@uv.cl , and Xiao-Bo
Dong11affiliation: Key Laboratory for Research in Galaxies and Cosmology, The
University of Sciences and Technology of China, Chinese Academy of Sciences,
Hefei, Anhui 230026, China;
zkdtc@mail.ustc.edu.cn; twang@ustc.edu.cn; xbdong@ustc.edu.cn
zkdtc@mail.ustc.edu.cn
###### Abstract
The anti-correlations between the equivalent widths of emission lines and the
continuum luminosity in AGNs, known as the Baldwin effect are well established
for broad lines, but are less well studied for narrow lines. In this paper we
explore the Baldwin effect of narrow emission lines over a wide range of
ionization levels and critical densities using a large sample of broad-line,
radio-quiet AGNs taken from Sloan Digital Sky Survey (SDSS) Data Release 4.
These type1 AGNs span three orders of magnitude in continuum luminosity. We
show that most narrow lines show a similar Baldwin effect slope of about -0.2
while the significant deviations of the slopes for [N II] $\lambda$6583, [O
II] $\lambda$3727, [Ne V] $\lambda$3425 , and the narrow component of
H$\alpha$ can be explained by the influence of metallicity, star-formation
contamination and possibly by difference in the shape of the UV-optical
continuum. The slopes do not show any correlation with either the ionization
potential or the critical density. We show that a combination of 50%
variations in continuum near 5100Å and a log-normal distribution of observed
luminosity can naturally reproduce a constant Baldwin effect slope of -0.2 for
all narrow lines. The variations of the continuum could be due to variability,
intrinsic anisotropic emission, or an inclination effect.
###### Subject headings:
galaxies: active–galaxies:Seyfert–(galaxies:) quasars: emission lines
## 1\. Introduction
The anti-correlation between the equivalent widths of broad emission lines and
AGN luminosity (the “Baldwin effect”; hereinafter BE) was first discovered by
Baldwin (1977) for the C IV $\lambda$1549 broad emission line in high redshift
AGNs. It was initially hoped to be able to use the effect to calibrate the AGN
luminosity to be able to use AGNs as cosmological standard candles (Baldwin et
al. 1978), but the large dispersion of this relationship rendered this
impossible (Baldwin et al. 1989; Zamorani et al. 1992). The BE is now well
established for nearly all broad emission lines and the slope of the BE
steepens with increasing ionization potential (Zheng & Malkan 1993; Dietrich
et al. 2002). Several mechanisms have been proposed to explain this effect
(See Shields 2007 for a review). Among these perhaps the most widely accepted
one is that the ionizing continuum softens with increasing luminosity so there
are relatively fewer ionizing photons for broad emission line formation in
high-luminosity AGNs. This model can reproduce the ionization energy–BE slope
relationship fairly well (Korista et al. 1998) and the assumption has
observational support (Binette et al. 1989; Zheng & Malkan 1993; Wang & Lu
1998; Korista et al. 1998). Some theoretical models (Netzer 1985, 1987;
Netzer, Laor, & Gondhalekar 1992; Wandel 1999a,b) could produce a softer
ionizing spectrum in high-luminosity sources, but the standard thin disc model
(Shakura & Sunyaev 1973) they adopt suffers from many problems (Antonucci
2002; Gaskell & Klimek 2003; Gaskell 2008; Lawrence 2012; Antonucci 2012). In
this sense, the explanation of broad line BE is still elusive and
controversial. An outlier of the ionization energy–BE slope relationship is N
V $\lambda$1240, which has an ionization energy of 97.7eV but shows no BE at
all (Dietrich et al. 2002). It has been proposed that this can be explained by
a dependence of metallicity on AGN luminosity, which in turn is a combination
of Eddington ratio ($L/L\mathrm{{}_{Edd}}$) and black hole mass (Korista et
al. 1998; Hamann & Ferland 1993, 1999; Dietrich et al. 1999; Dietrich &
Wilhelm-Erkens 2000). In local galaxies, gas metallicity correlates well with
the mass of galaxies (Tremonti et al. 2004), and more massive galaxies have
larger black hole masses ($M_{\mathrm{BH}}$) according to the
$M_{\mathrm{BH}}$-$M_{*}$ relationship. For Type1 AGN population, the brighter
AGNs would have higher $M_{\mathrm{BH}}$ on average (Kollmeier et al. 2006;
Steinhardt & Elvis 2010a,b; Lusso et al. 2012) and thus higher metallicities.
Recent studies of the BE for C IV, Mg II and Fe II, however, challenge this
picture by showing that the BE might instead be driven by the EW’s correlation
with Eddington ratio (Baskin & Laor 2004; Bachev et al. 2004; Warner, Hamann &
Dietrich 2004; Zhou et al. 2006; Dong et al. 2009a,b) or with
$M_{\mathrm{BH}}$(Netzer, Laor & Gondhalekar 1992; Wandel, Peterson & Malkan
1999; Shields 2007; Kovacevic et al. 2011).
The BE of narrow emission lines is much less well studied, and some results
are still controversial. Steiner (1981) discovered a strong BE for [O III]
$\lambda$5007 (especially in AGNS with strong optical Fe II) and Wills et al.
(1993) found the narrow lines in high-luminosity AGNs are very weak.
Meanwhile, Croom et al. (2002) found a significant BE for [Ne V] $\lambda$3425
and [O II] $\lambda$3727 but obtained a null result for [Ne III] $\lambda$3870
and [O III] $\lambda$5007 using the 2dF sample. Subsequent work shows,
however, that [O III] $\lambda$5007 does show a BE (Dietrich et al. 2002,
Netzer et al. 2004). Hönig et al. (2008) and Keremedjiev et al. (2009), used
Spitzer data to find that the BEs of different mid-IR narrow emission lines
have a nearly constant slope. Several mechanisms have been proposed to explain
the NLR BE. NLRs follow a size–luminosity relationship (Schmitt et al. 2003b;
Bennert et al. 2002; Greene et al. 2011). The size of the NLR may grow beyond
the size of the host galaxy in high-luminosity AGNs, thus turning from
ionization-bounded to matter-bounded, and so producing the BE (Croom et al.
2002). As the EW is proportional to the covering factor (CF) and it is found
that the CF contributes to much of the variance of the EW (Baskin & Laor
2005), a luminosity- dependent CF is also a possible cause of the BE (Shields
et al. 1995; Stern & Laor 2012b; Stern & Laor 2012c). It has recently been
proposed that the EW of narrow emission lines is dependent on the inclination
to the accretion disc (Risaliti et al. 2011), at least in the highest EW
sources. In principle, this is a potential cause of the BE too.
To make progress in understanding the NLR BE, we need to measure narrow
emission lines of a wide range of ionization potentials and critical
densities, and determine the nature of their BEs more accurately. In this
paper, we use a well-defined sample drawn from SDSS DR4 and employ the
technique of composite spectra to investigate the BE for prominent narrow
emission lines in the optical band and to study the origin of the NLR BE. We
use a cosmology with $H_{\rm 0}$ = 70 km s-1 Mpc-1, $\Omega_{\rm m}$ = 0.3,
and $\Omega_{\rm\Lambda}$ = 0.7 throughout this paper.
## 2\. Sample and Measurements
### 2.1. Sample
We need accurate emission line and continuum measurements to ensure reliable
determinations of equivalent widths and continuum luminosities. From the
spectral data set of the Sloan Digital Sky Survey Fourth Data Release
(Adelman-McCarthy et al. 2006), we have selected 4178 Seyfert 1 galaxies and
quasars (i.e., type 1 AGNs) as described in Dong et al. (2011). We apply a
redshift cutoff of 0.8 so that the redshift of the spectrum can be accurately
determined using [O III] $\lambda$5007\. To ensure high-quality spectra we
require a median signal-to-noise ratio (S/N) of $\geq 10$ per pixel in the
optical. To minimize the host-galaxy contamination (see the Appendix of Dong
et al. 2011), we restrict the weak stellar absorption features, such that the
rest-frame EWs of Ca K (3934 Å), Ca H + H$\epsilon$ (3970 Å), and H$\delta$
(4102 Å) absorption features are undetected at $<2\,\sigma$ significance. This
criterion ensures that the host contamination is less than 10% around
4200Å(Dong et al. 2011). One may note that Dong et al. (2011) do not consider
very young stellar population (i.e., emission line galaxies). But previous
analysis have shown that in massive galaxies, the optical continuum is not
dominated by very young stellar population although the UV continuum may be
(Schawinski et al. 2007). After removing duplications and sources with too
many bad pixels in the H$\beta$ \+ [O III] region, we obtain 4178 type 1 AGNs.
Including radio-loud AGNs may influence the measurement of EW of emission thus
produce false effect for two reasons: firstly, the jet may interact with the
ISM to enhance the narrow line emission (Labiano et al. 2007) and secondly, it
might also enhance the continuum through beaming if the jet points close to
our line of sight. By matching with the FIRST catalog (Becker et al. 1995)
using the method of Lu et al. (2010), we reject 499 radio-loud AGNs so that
our final sample consists of 3677 sources.
### 2.2. Spectral fitting and measurements
We give a brief description of our spectrum fitting process here; the details
can be found in Dong et al. (2011). To model the spectrum, we used a code
based on the MPFIT package (Markwardt 2009) and fit the AGN featureless
continuum, the Fe II multiplets, and other emission lines simultaneously. The
AGN continuum is represented locally by a power-law, for the region of
4200–5600 Å and for the H$\alpha$ region (if present). The Fe II template by
Véron-Cetty et al. (2004) that we use, is constructed using the identification
and measurement of Fe II lines in I Zw 1. It has two separate sets of
templates in _analytical_ forms, one for the broad-line system and the other
for the narrow-line system. Within each system, the relative velocity shifts
and relative strength are assumed to be the same as those in I Zw 1. Broad Fe
II lines share the same profile as broad H$\beta$, while each narrow Fe II
lines is modeled with a Gaussian. During the fitting, the normalization and
redshift of each system are taken as free parameters. The broad Balmer lines
are fitted with as many Gaussians as is statistically justified. All narrow
emission lines, except for the [O III] $\lambda\lambda$4959, 5007 doublet
lines, are fitted with a single Gaussian. Each line of the [O III] doublet is
modeled with two Gaussians, one accounting for the line core and the other for
a possible blue wing, as seen in many objects. Since the sources in our sample
do not suffer from significant host galaxy contamination, we do not apply
starlight corrections to individual spectra. For each source, we use $\lambda
L_{\lambda}(5100)$ and FWHM of broad H$\beta$ line to obtain the
$M_{\mathrm{BH}}$ using the virial mass estimates by the Dibai method (Dibai
1977) using the formalism of Wang et al. (2009). The typical statistical
scatter about the $M_{\mathrm{BH}}$ obtained by reverberation-mapping is about
0.4 dex, and it may also be subject to more systematic errors. (Krolik 2001;
Collin et al. 2006; Shen et al. 2008; Fine et al. 2008; Marconi et al. 2008;
Denney et al. 2009; Rafiee & Hall 2011a; Steinhardt 2011). The error of
$L/L\mathrm{{}_{Edd}}$ is of similar magnitude as that of $M_{\mathrm{BH}}$.
### 2.3. Composite Spectra Generating and Fitting
A convenient way to explore the correlations between EWs and other parameters
like luminosity is to make composite spectra for different parameter bins. We
normalize individual spectrum to the mean flux around 4200Å and then construct
the geometric composite as Vanden Berk et al. (2001). To get an accurate
continuum measurement, we need to subtract the broad emission lines,
especially, H$\alpha$ and H$\beta$ and Fe II emission from the spectrum. For
the broad component of H$\beta$ and Fe II we subtract them from the original
spectrum; but for the broad component of H$\alpha$, we leave it un-subtracted
when making composite spectra. This is because the broad H$\alpha$ is highly
blended with [N II] $\lambda$6583 and [N II] $\lambda$6548, so deblending in
individual spectrum is not reliable while deconvolving the blends in the high
SN composite spectrum is easier. The fitting algorithm used to model the
composite spectrum is the same as described in Section 2.2. More specifically,
we fit the blends using one gaussian for the narrow component of H$\alpha$, [N
II] $\lambda$6583 and [N II] $\lambda$6548, two gaussians for [S II]
$\lambda\lambda$6717, 6731 and three for H$\alpha^{b}$. The broad-line-
subtracted spectra are shown in Fig. 1. The embedded panels show the fitting
result for the H$\alpha^{b}$\+ [N II] blends.
## 3\. Results
### 3.1. The Baldwin effect for different lines
First, we want to explore if the BE exists in the prominent narrow lines [Ne
V] $\lambda$3425 , [O II] $\lambda$3727, [Ne III] $\lambda$3870 ,
H$\beta^{n}$, [O III] $\lambda$5007, [O I]$\lambda$6300, H$\alpha^{n}$, [N II]
$\lambda$6583 and [S II] $\lambda\lambda$6717, 6731. Measuring narrow lines in
individual spectrum are subject to the S$/$N limit and the EW-$L_{5100}$
relation of all our lines shows a large dispersion (typical 0.2 dex, this can
be seen clearly in the middle panel of Fig. 4), so we turn to composite
spectra for reliable measurements. We divide our sample into intervals of 0.3
dex in $5100\AA$ luminosity starting at log $L_{5100}$ = 43.2 to 45.9
$ergs^{-1}$. and make a composite spectrum in each luminosity bin as described
in Section 2.3. From Fig.1 we can see two clear and remarkable results:
* •
With increasing luminosity, the narrow lines vanish.
* •
The slope of the observed continuum become bluer with increasing luminosity.
To see the dependencies of the EWs on the luminosity more clearly, in Fig. 2
we plot the log EW that derived from composite spectra for each luminosity bin
for each of the lines listed above against log $L_{5100}$ in order to get more
qualitative results. For each line, we show a weighted linear regression of
the logarithm of the EW on the logarithm of the luminosity and we list all the
BE slopes of the narrow emission lines together with their ionization energies
and critical densities in Table.1. When fitting the relationship, we add an
error to each data point that would help to reduce the normalized $\chi^{2}$
to $\sim$1\. These added errors are the same for individual narrow line, and
they account for the potential other systematic errors. The BE slopes of the
high-ionization lines [Ne III] $\lambda$3870 and [Ne V] $\lambda$3425 are
$-0.26\pm 0.02$, $-0.31\pm 0.015$ respectively which are consistent with the
results of Keremedjiev et al. (2009) who obtained $-0.22\pm 0.06$, and
$-0.19\pm 0.06$ respectively for the Ne III $\lambda$ 15.56$\mu$m, and Ne V
$\lambda$ 14.32$\mu$m lines. Our [O III] $\lambda$5007 BE slope of $-0.21\pm
0.016$ is similar to the values of $\sim-0.2$ found by Kovacevic et al. (2011)
and steeper than the $-0.1\pm 0.02$ found by Dietrich et al. (2002). The
difference might be due to different sample we employ. The error of slopes we
give here include only the statistic error of the fitting, but do not include
the measuring error and intrinsic dispersion of the EW, which may reach 0.2
dex typically.
In addition to finding the BE in high-ionization lines, we also find the BE in
the low-ionization lines of [O I]$\lambda$6300 [S II] $\lambda\lambda$6717,
6731 and [O II] $\lambda$3727\. The narrow recombination line: H$\beta^{n}$
also shows a BE with a similar slope as for the forbidden lines. For
comparison, the broad H$\beta$ line shows an inverse BE with a slope of 0.16
for our sample. This is consistent with Greene & Ho (2005) and Croom et al.
(2002) who reported $EW(H\beta^{b})\propto L_{5100}^{0.13}$ and
$EW(H\beta^{b})\propto L_{5100}^{0.19}$ respectively. We can see in Fig. 2
that the EW of the narrow component of H$\alpha$ increases with luminosity
too. These indicate that the behaviors of broad lines and narrow lines are
different.
The correlation coefficients between EW([O III]) and $\lambda
L_{\lambda}(5100)$, z, $M_{\mathrm{BH}}$, $L/L\mathrm{{}_{Edd}}$ are -0.12,
-0.05, -0.01, -0.21, respectively. These are similar to the results in Zhang
et al. (2011). For our flux-limited sample, the redshift, luminosity, black
hole mass, and Eddington ratio are correlated with each other. The correlation
coefficients between $\lambda L_{\lambda}(5100)$ and $z$, $M_{\mathrm{BH}}$,
$L/L\mathrm{{}_{Edd}}$ are 0.81, 0.67, 0.15 respectively. After controlling
for $z$ or $L/L\mathrm{{}_{Edd}}$ in a partial correlation analysis, EW([O
III]) correlates with $\lambda L_{\lambda}(5100)$ with
$r_{\rm\scriptscriptstyle S}$=-0.14 and -0.16. This indicates that the BE of
narrow lines is not a secondary effect of correlations between the EW and
redshift or $L/L\mathrm{{}_{Edd}}$ but an independent phenomenon. We note that
as the correlation strength is not strong, one possibility is that there are
other factors that regulate the EW as discussed in Zhang et al. (2011). Also,
measurement error in EW([O III]) may act to smear the correlation. The
measurement error could be introduced by limited S$/$N and fitting process. In
spite of these uncertainties, the BE of narrow lines does exist and may shed
light on physical process in AGNs as explored in detail below.
### 3.2. The ionization energy – BE slope relationship: no correlation
A dependence of the BE slope on ionization energy is found for broad emission
lines in AGNs (see introduction) and this is the most compelling evidence for
a luminosity-dependent ionizing spectrum. The NLR lies far from the nucleus,
has a complex geometry and may contain dust (Netzer & Laor 1993; Tomono et al.
2001; Radomski et al. 2003; Schweitzer et al. 2008). This makes the response
of the narrow-line flux to changes in the SED complicated. We plot the slope
of BE against the ionization energies of the narrow lines in the left panel of
Fig. 3. We also plot the Keremedjiev et al. (2009) data for [SIV]
10.51$\mu$m:$-0.29\pm 0.05$, [Ne II] 12.81$\mu$m:$-0.25\pm 0.06$, [Ne III]
15.56$\mu$m:$-0.22\pm 0.06$ and [Ne V] 14.32$\mu$m:$-0.19\pm 0.06$ in purple
crosses and the broad-line BE from Dietrich et al. (2002) in blue rectangles
for comparison. We can see that the narrow-line BE slopes do not correlate
with the ionization-energy ( $P_{null}=0.78$ meaning the two-sided probability
that a correlation is not present is 78%.) but cluster around -0.2 with a
dispersion of $\pm 0.1$. [N II] $\lambda$6583, H$\alpha^{n}$ and [O II]
$\lambda$3727 have slopes of $-0.10\pm 0.014$, $-0.29\pm 0.033$, and $-0.37\pm
0.011$ so they deviate from -0.2 significantly. Possible reasons for this are
discussed in Section 4.1.
### 3.3. Critical Density – Slope relationship: No correlation
It is well known that the NLR is stratified that the high-ionization lines
rise from the inner part of the NLR while low-ionization lines rise further
away (see, for example, Veilleux et al. 1991; Robinson et al. 1994; Bennert et
al. 2006a,b; Kraemer et al. 2009). The central electron temperature, density,
and ionization parameter are, in general, higher in Seyfert 1s than in Seyfert
2s (Gaskell 1984; Schmitt 1998; Bennert et al. 2006b). The lines of lower
critical density are not as strong near the nuclei as the higher critical
density lines. A possible explanation of this is that the change in critical
density marks a transition between difference types of clouds. To see if this
is a factor in the NLR BE we therefore plot the EWs against critical density
in the right panel of Fig. 2 using the same symbols of panel (a). We again
fail to find any correlation in this plot ($P_{null}=0.82$ using a Spearman
rank correlation analysis).
## 4\. Discussions
Using our sample of 3677 radio-quiet AGNs from SDSS DR4 that have little
contamination by host galaxy, we find that, in contrast with the BE of broad
emission lines, the narrow lines show a nearly constant BE slope irrespective
of ionization potential or critical density. The [O II] $\lambda$3727, [N II]
$\lambda$6583 and H$\alpha^{n}$ lines show significantly different BE slopes
from other lines.
### 4.1. Deviation from constant BE slope
Although the slope of different lines cluster around -0.2, we do find some
lines that show large deviations.
Firstly, [N II] $\lambda$6583 show a much flatter slope ($-0.10\pm 0.014$)
than other lines ($P<$ 5% in two-sided KS test, meaning if the two samples are
drawn from the same distribution, we expect to see a difference as large or
larger than we see here only 5 in 100 times.). This flattening was also found
for N V $\lambda$1240, which has an ionization potential of 77.7 eV but no BE.
One plausible explanation of a weaker or absent BE for the nitrogen lines is
that the increase of metallicity in brighter AGN will enhance nitrogen
abundance relative to carbon, oxygen etc., because nitrogen is a secondary
element whose abundance scale as $Z^{2}$. This trend could compensate for the
decrease of EW in higher-luminosity sources thus making the BE slope flatter.
This explanation is supported by a number of recent studies. The metallicities
of the BLR (Hamann & Ferland 1993; Nagao et al. 2006a; Juarez et al. 2009) and
NLR (e.g., Nagao et al. 2006b; Matsuoka et al. 2009) are both found to
correlate with the luminosity of the AGN. So in principal, the $Z_{NLR}$–L
relationship could produce the flattening of the BE of [N II] $\lambda$6583 we
observe here. However, this explanation has a major problem because N IV] and
N III], which would be expected to deviate from the ionization energy–slope
relationship for the same reason, lie on it (Dietrich et al. 2002). Despite
this serious problem to be resolved, Occam’s razor suggests that an enhanced
abundance in high-luminosity AGNs is the simplest and most plausible
explanation of the flatter BE slope of [N II] $\lambda$6583.
The second significant deviation from a BE slope of -0.2 is the steep slopes
of H$\alpha^{n}$ and [O II] $\lambda$3727\. Both of these are star formation
(SF) indicators in star-forming galaxies (Kenicutt et al. 1998, Ho 2005) and
their EW may reach several hundred Å in starburst galaxies. So even though the
AGNs we selected show no absorption feature in the continuum, the emission
lines are still possible subject to SF contamination because of their large
EW. In low-luminosity sources whose continuum is low, the emission line from
SF may contribute significantly to the total line flux, and thus enhance the
EW. In type-2 AGNs, the SF contribution of H$\alpha^{n}$ is estimated to be
more than 60% (Brinchmann et al. 2004). The H$\alpha$ emitting region of
type-1 AGNs, however, may have a higher fraction of the emission originating
with the AGN (Zhang et al. 2008) but still be heavily influenced by SF.
Furthermore, a variety of studies have established a correlation between the
strength of AGN activity and star formation in the local universe ( e.g.,
Rowan-Robinson 1995; Croom et al. 2002; Netzer et al. 2007, 2009; Shao et al.
2010). The steepest correlation is $L_{SF}\propto L_{AGN}^{0.8}$ (Netzer 2009)
meaning $L_{SF}/L_{AGN}\propto L_{AGN}^{-0.2}$, a BE slope steeper than -0.2.
So it makes sense that the SF in host galaxies would steepen the BE slopes of
H$\alpha^{n}$ and [O II] $\lambda$3727\. However, the [O II] $\lambda$3727 and
H$\alpha^{n}$ BEs cannot be attributed entirely to SF–AGN relationship. For a
plausible range of ionization parameters, densities, and ionizing spectra, the
intensity of [O II] $\lambda$3727 is proportional to that of [O III]
$\lambda$5007 (10% - 30%, Ferland & Osterbrock 1986; Ho et al. 1993a, 1993b).
Since [O III] $\lambda$5007 shows a significant BE, [O II] $\lambda$3727 is
unlikely to show a radically different trend. So it is safe to conclude that
the [O II] $\lambda$3727 and H$\alpha^{n}$ BE could be partly (but not
totally) produced by SF contamination.
A third effect we can see in Fig. 1 is the optical continuum getting bluer
towards higher luminosity. This would leverage the continuum and lower the EW
if the line flux remains unchanged. This could arise if the NLR sees a
filtered spectral-energy distribution (SED) (Kraemer et al. 1998; Groves et
al. 2004a,b) so that a change in the ionizing spectrum of an AGN would not
change the SED the NLR sees much. The [Ne V] $\lambda$3425 , [O II]
$\lambda$3727 and [Ne III] $\lambda$3870 lines are most likely to be
influenced by a continuum shape difference effect because of their shorter
wavelength. A possible explanation of the bluer color of the UV-optical SED is
the anisotropy of the continuum emission. Because the accretion disc is
optically thick, we will see a dimmer continuum when viewing it edge-on. (This
is the combined result of the $\cos i$ projection effect and the disk
equivalent of “limb darkening”.) We could also preferentially be seeing the
inner, high-temperature part of the disc when viewing face-on. These effects
will combine to give a higher observed luminosity with a face-on viewing
angle. The reddening of spectrum depends on viewing angle too (Keel 1980; de
Zotti & Gaskell 1985; Zhang et al. 2008). Gaskell et al. (2004) used radio
orientations to get AGN reddening curves and found the continuum shape is
profoundly affected by reddening for all but the bluest AGNs. Because of these
effects sources with small inclination would both show large luminosity and
have low reddening. An alternative explanation of continuum shape difference
is host galaxy contamination. Shen et al. (2011) made composite spectra of
different $\lambda L_{\lambda}(5100)$ luminosity bins and found that the UV
part of all the composite spectra are similar while the optical part flatten
with decreasing luminosity. They interpreted this trend as due to host galaxy
contamination in the optical region of the spectrum in low-luminosity AGNs.
Similar argument is given by Stern & Laor et al. (2012a). Even though we have
rejected objects with significant stellar light contributions, weak absorption
lines can be spotted on the final composite spectrum of the lowest luminosity
bin because of its extremely high signal to noise ratio. Our data cannot
distinguish different mechanisms that give rise to the continuum shape
difference, and this is beyond the scope of this paper. After correcting the
deviations listed above, our conclusion that the slopes of BE for different
narrow lines are nearly constant is further strengthened.
### 4.2. Possible causes of the NLR Baldwin effect
#### 4.2.1 Softening of the ionizing continuum
It has been argued on both observational and theoretical grounds that there is
a softening of the ionizing continuum with increasing luminosity (see, for
example, Binette et al. 1989; Netzer, Laor, & Gondhalekar 1992; Zheng & Malkan
1993; Wang & Lu 1998; Korista et al. 1998). If this is indeed the case, an
important prediction of this is the ionization potential – BE slope
relationship (Korista et al. 1998). It is successful in explaining the broad
line BE, but a major failing is that it fails to explain the constant BE slope
of narrow lines. It is already known that the NLRs of different AGNs are
similar in the sense that the line ratios show less than 0.5 dex difference
from object to object. (Koski 1978; Veilleux & Osterbrock 1987; Veilleux
1991a, 1991b, 1991c; Veron-Cetty & Veron 2000; Dopita et al. 2002; Gorjian et
al. 2007). Kraemer et al. (2000), Dopita et al. (2002) and Groves et al.
(2004a,b) proposed a NLR model where dust regulates the incident ionizing
spectrum so as to keep the ionization parameter in the NLR constant. In this
model, variation of the SED is filtered by dust in the NLR and thus the
continuum shape seen by the NLR is dominated by the effect of the dust rather
than by intrinsic changes in the SED. This leads to more stable conditions in
the NLR. This could be the cause of the lack of a dependence of the slope of
the BE on the ionization energy.
#### 4.2.2 Luminosity-dependence Covering Factor
A luminosity-dependent covering factor is a natural explanation of both a BLR
and NLR BE. However, to explain the ionization dependence, the variation in
covering factor must have a different dependence for lines of different
ionizations. This would not be a surprise for the BLR since there is strong
radial ionization stratification and the highest ionization lines are an order
of magnitude closer to the black hole than the lowest ionization lines (see
Gaskell 2009 for a review). Shields, Ferland, & Peterson (1995) suggested that
the broad-line BE could be caused by a luminosity-dependent covering factor
for clouds that are optically-thin to photons with wavelength less than 912Å.
The optically-thin clouds which have small column densities, preferentially
emit high-ionization lines. However, comparison of line profiles shows that
the BLR BE is due to changes in the low velocity BLR gas (Francis et al. 1992)
rather than changes in the very broad component. Furthermore, Snedden &
Gaskell (2007) argued that optically-thin gas does not make a substantial
contribution to the BLR. Nevertheless, given the strong radial ionization
stratification of the BLR, and the decreasing covering factor with ionization
(e.g., Francis et al. 1992), an additional luminosity-dependence of the
covering factor could explain the BE and why the high-ionization lines should
show a steeper slope than low-ionization lines. While this could be consistent
with the BLR, it is not obvious how such an explanation can be reconciled with
the lack of an ionization-dependence we find here for the BE.
#### 4.2.3 A Disappearing NLR?
The NLR size is correlated with the luminosity of an AGN as $R_{NLR}\propto
L^{0.5}$ (Bennert et al. 2002; Schmitt et al. 2003b; Bennert et al. 2006a,b;
Greene et al. 2011). So in high-luminosity AGNs, the NLR may possibly turn
from ionization-bounded to matter-bounded, and the luminosity of narrow lines
would cease increasing with AGN luminosity. This model could in principle
explain part of the BE of NLR (Croom et al. 2002) but it would be too rash a
conclusion that it is the origin of the BE because the bulk of the NLR
emission originates from within the central few tens or hundreds of pc
(Schmitt et al. 2003a) so is unlikely to exceed the scale of the galactic
bulge. Besides, the NLR has no natural size because it has no definite edge,
so some cut in surface brightness or line-ratio is needed to define the size
(Schmitt et al. 2003b; Bennert et al. 2006a,b; Greene et al. 2011). This makes
the interpretation of size complicated. Netzer et al. (2004) argue that that
$R_{NLR}\propto L^{0.5}$ is theoretically sound yet this relationship must
break down for $R_{NLR}$ exceeding a few kpc. They found that high-$z$ AGNs
have NLR sizes no larger than 10kpc. It is also well known that the the NLR is
stratified so that the high-ionization lines rise from the inner part of NLR
while low-ionization lines from further away (Veilleux et al. 1995; Bennert et
al. 2006a,b). So the “disappearing NLR effect”, if exists, could
preferentially influence low-ionization and low-critical density lines. This
again stands against the constant slope we find.
#### 4.2.4 Continuum Variation
Because different lines all seem to share a similar BE slope, it is natural to
think that it might be the continuum, rather than the emission lines, which is
the cause. Jiang et al. (2006) proposed that the continuum variation can
produce a weak BE ($\beta=-0.05\pm 0.05$) if the light-crossing time for the
region emitting narrow $FeK\alpha$ exceeds the variability timescale for the
X-ray continuum and the amplitude of variability anti-correlates with the
luminosity. Shu et al. (2012) found a strong anti-correlation between the EW
of the narrow $FeK\alpha$ line and $L_{X}$ ($EW/\langle
EW\rangle\propto(L/\langle L\rangle)^{-0.82\pm 0.10}$ (where $\langle\rangle$
means time-averaged values) consistent with the X-ray BE expected in an
individual AGN if the narrow-line flux remains constant while the continuum
varies. For a NLR whose light-crossing time is about $10^{3}$ yr, the narrow-
line luminosity can safely be assumed to be constant. The amplitude of
variability of the ionizing continuum can be obtained from the monitoring of
AGNs after correction for the constant host galaxy light contribution. As is
well known, the amplitude of variability of an AGN increases with time. This
can be most readily seen from the structure functions of AGNs (the variance as
a function of time interval between observations). We are interested in
variability on the longest timescales (timescales similar to the NLR light-
crossing time). We do not, of course, have monitoring on such long timescales,
but we can estimate the amplitude from structure functions. Cid Fernandez et
al. (2000) have presented structure functions for bright AGNs. These are all
consistent with the amplitude of variability rising as the time interval
increases to a characteristic time of a year or a few years and then remaining
constant. Collier & Peterson (2001) got a similar result for lower-luminosity
AGNs (but with a shorter characteristic time) and also find that the forms of
the UV structure functions are similar to the optical ones. There are
indications that the structure function increases gradually on much longer
timescales, but data on this part of the structure function are limited.
Observed variability thus gives a lower limit to variability on the light-
crossing timescale of the NLR.
In order to minimize the host galaxy light contribution one needs to go to as
short a wavelength as possible. This means using $U$-band observations or
space ultraviolet observations. For example, for NGC 4151 Merkulova (2006)
found an observed peak-to-peak $U$-band amplitude of a factor of 7.5. UV
variability can be a lot larger. For example, the peak-to-peak amplitude at
1300Å for Fairall 9 is a factor of 25 (Koratkar & Gaskell 1989). Inspection of
the results of long-term UV monitoring of a number of AGNs with the IUE
satellite (Koratkar & Gaskell 1989, Koratkar & Gaskell 1991a,b, Clavel et al.
1991, and O’Brien et al. 1998) give a median UV peak-to-peak variability of a
factor of 7, but the upper quartile is a factor of 13. After correction for
host-galaxy light, the amplitudes of optical variability are similarly large.
For example, the NGC 4151 photometry of Lyuty & Doroshenko (1999) gives a
peak-to-peak amplitude of a factor of about 25 to 30.
Continuum anisotropy (Wang & Lu 1999) can produce an effect similar to that
caused by the actual variability of the continuum. It was recently found by
Risaliti et al. (2011) that the optically-thick disc emission responsible for
the continuum and isotropic [O III] emission will produce the EW([O III])
distribution very well. Variability and continuum anisotropy are, in practice,
indistinguishable, so we can consider them together.
We made Monte-Carlo simulations to explore whether continuum variability can
explain the BE slope. In a flux-limited survey, the luminosity will show an
approximately log-normal distribution. We generated an artificial sample of
sources with a similar distribution of $\lambda L_{\lambda}(5100)$ as the
observed AGN sample (0.35 dex here). We set the EW of the artificial-sources
to be the mean EW of the whole sample, and we add a Gaussian of 0.22 dex to
the EW to account for the intrinsic dispersion. UV and optical AGN variability
is approximately log-normal – i.e., it looks normal when plotted in magnitudes
(see Gaskell 2004 for a discussion of log-normal variation of AGNs). The
continuum variation was simulated by adding a Gaussian with $\sigma=50\%$
(0.18 dex) to the continuum while the emission line flux was kept unchanged.
This is an over-simplification, but it can help us to gain some insight into
the effect that continuum variability brings. We generated 4000 sources in
each round and measured the slope of BE using the 4000 sources. An example of
the simulation is shown in Fig. 4. With a peak-to-peak continuum variation of
a factor of three, the simulation can produce a BE slope of $-0.2\pm 0.01$
while a factor of six variation produces a BE slope of $-0.3\pm 0.01$. The
simulated distribution of EW-$\lambda L_{\lambda}(5100)$ can be seen to be
very similar to what is observed. Obviously, this model will produce a similar
slope for every narrow line if the continuum change with similar amplitude in
the wavelength range we concern. It should be noted that due to variation in
continuum slope, not all lines show the same BE slope, but depends on
wavelength. We assume a constant narrow line flux during the continuum
variability. This is an approximation for the continuum variations on time
scales of much shorter than the light travel time over the NLR because at such
short time scale, the NLR has little response to the continuum variations. For
variations on longer time scales, one needs to properly convolve the continuum
variations with the response of emission lines (transfer function). By
considering the latter response, the variations of equivalent width will be
somewhat smaller, but this is equivalent to requiring a somewhat larger
continuum variability amplitude.
Conversely, since continuum variability or the equivalent of continuum
variability will inevitably produce a BE, the BE slope we observe could give
an upper-limit on the few $10^{2}$yrs to $10^{3}$yrs variability of AGN
optical continuum. If the variability amplitudes exceed the factor of three a
steeper BE will emerge. This might seem at first sight to be at variance with
the amplitudes of UV variability and structure functions observed, but it must
be remembered that the equivalent width is the ratio of the line flux to the
observed optical continuum and the observed continuum has a substantial
starlight contamination. An example of how the apparent continuum variability
is substantially less than the real variability is shown in Fig. 4 of Gaskell
et al (2008). Host galaxy light limits the apparent peak-to-peak continuum
variability to about a factor of three (i.e., a rms variability of a few
tenths of a magnitude). This is just what is observed for the PG AGNs studied
by Cid Fernandez et al. Since the structure functions are relatively flat
after a year or so (Cid Fernandez et al. 2000), the variability effect on the
BE should be apparent in a few years. An obvious test of this is to reobserve
AGNs after a few years.
### 4.3. Drawbacks
In summary, the combination of a 50% variation of $\lambda L_{\lambda}(5100)$
and a log-normal distribution of luminosity will naturally produce a -0.20
slope of BE for every narrow line, as we observe. The model we employ is
obviously an oversimplification. To make a more realistic model, we need a
dedicated treatment of sample selection effects and to make a more realistic
assumption of the variability including amplitude, variation form, and their
dependence on wavelength etc, about the light curve. Despite these
simplifications, our results do show that continuum variation will inevitably
produce a similar BE for each narrow line. Meanwhile, while the model predicts
the same BE slope for each narrow line, the differences in slope are still
significant due to the small error bars. There must therefore be some
additional factor at work. It has been shown in Fig. 3 that the differences do
not correlate with the ionization potential or critical density, so there must
be other factors taking effect. A deeper exploration of other factors like
ionization slope, NLR geometry and a more realistic model are needed to make
progress, but these are beyond the scope of this paper.
## 5\. Conclusion
We have constructed a sample of 3677 $z<0.8$ radio-quiet AGNs from Sloan
Digital Survey Data Release 4 that span three orders of magnitude in
luminosity to explore the relationship between equivalent width of narrow
lines and $\lambda L_{\lambda}(5100)$. We have computed composite spectra for
each $\lambda L_{\lambda}(5100)$ bin to enhance the S/N ratio. H$\alpha^{b}$
and H$\beta^{b}$ as well as Fe II were subtracted to get an accurate
measurement of narrow emission lines and continuum. We find that most narrow
lines show a similar BE slope of about -0.2 while the large deviation of [N
II] $\lambda$6583, [O II] $\lambda$3727, H$\alpha^{n}$ and [Ne V]
$\lambda$3425 might be explained by a metallicity effect, SF contamination, or
the UV-optical continuum difference. The slope does not show any correlation
with ionization energy and critical density. We propose that the combination
of a 50% variation of the continuum near 5100Å and a log-normal distribution
of observed luminosity distribution will naturally produce a -0.2 slope of BE
for every narrow line.
We thank our referee: Robert Antonucci for inspiring comments that help to
improve the paper significantly. And we thank Brent Groves for helpful
discussion on NLR model. This work is supported by Chinese NSF grants
NSF-10703006, NSF-10973013, NSF-11073019, and NSF-11233002, the GEMINI-CONICYT
Fund of Chile through project N°32070017 and FONDECYT of Chile through project
N° 1120957. Funding for the Sloan Digital Sky Survey (SDSS) has been provided
by the Alfred P. Sloan Foundation, the Participating Institutions, the
National Aeronautics and Space Administration, the National Science
Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and
the Max Planck Society. The SDSS is managed by the Astrophysical Research
Consortium (ARC) for the Participating Institutions. The SDSS web site is
http://www.sdss.org/.
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Table 1Ionization energy, Critical density and Slope $\beta$ of log EW – log
$\lambda L_{\lambda}(5100)$ Table
Line | $\chi_{ion}$ | log $n_{c}$ | $\lambda L_{\lambda}(5100)$ slope
---|---|---|---
(1) | (2) | (3) | (4)
[Ne V] $\lambda$3425 | 97.12 | 7.3 | -0.31 $\pm$ 0.015
[O II] $\lambda$3727 | 13.62 | 3.5 | -0.37 $\pm$ 0.011
[Ne III] $\lambda$3870 | 40.96 | 5.5 | -0.26 $\pm$ 0.020
H$\beta^{n}$ | 13.6 | Infinity | -0.28 $\pm$ 0.015
[O III] $\lambda$5007 | 35.11 | 5.8 | -0.21 $\pm$ 0.016
[O I]$\lambda$6300 | 0 | 6.3 | -0.16 $\pm$ 0.030
H$\alpha^{n}$ | 13.6 | Infinity | -0.29 $\pm$ 0.033
[N II] $\lambda$6583 | 14.5 | 4.82 | -0.10 $\pm$ 0.014
[S II] $\lambda\lambda$6717, 6731 | 10.36 | 2.3 | -0.20 $\pm$ 0.016
Columns from left to right: (1)The name of the line. (2) The ionization energy
needed to create the ion. (3) The critical density of specific line. (4) The
log EW – log $\lambda L_{\lambda}(5100)$ slope.
Figure 1.— Normalized composite spectra with the broad H$\alpha$, the broad
H$\beta$ and Fe II subtracted are shown for luminosity bins of $\Delta
log\lambda L_{\lambda}(5100)=0.3$ dex, starting from the top with log $\lambda
L_{\lambda}(5100)$=43.35 [$ergs^{-1}$]. The spectra are normalized to the
[4200Å,4300Å] window and shifted vertically to show the weakening of lines
with luminosity more clearly. The lines we concern with are marked with dashed
lines labeled at the top. The embedded plots show the fitting results for the
H$\alpha$+[N II]+[S II] region. The left embedded panel is a composite
spectrum for log $\lambda L_{\lambda}(5100)$$\in$ [43.2,43.5] [$ergs^{-1}$]
and the right one is for log $\lambda L_{\lambda}(5100)$$\in$ [44.4,44.7]
[$ergs^{-1}$]. Figure 2.— Line equivalent widths: $W_{\lambda}$, against 5100Å
continuum luminosity: $\lambda L_{\lambda}(5100)$. We show weighted linear
regressions (green lines) for each line and give the BE slopes as well as
their errors in the upper right corners in each panel. In the narrow H$\beta$
panel we also include the narrow component of H$\alpha$ (dashed line) for
comparison. Figure 3.— Panel (a) BE slopes of different lines against the
ionization energy: $\chi_{ion}$ needed to create the ion. The diamonds and
triangles are our data; the red triangles are lines that are contaminated by
star formation. The data from Dietrich et al. (2002) and Keremedjiev et al.
(2009) are shown with solid blue rectangles and purple crosses respectively,
and the typical error bars are shown in left up corner. Panel (b) BE slope
against critical density($n_{c}$) for the same lines in as in panel (a). The
recombination lines are marked with right arrows. Figure 4.— The variability-
driven BE simulation result. Panel(a):The simulated BE, the slope of BE is
shown on the right up corner. Panel(b): The observed EW-log $\lambda
L_{\lambda}(5100)$ distribution of [O III] $\lambda$5007 from our SDSS DR4
sample. The slope of BE is shown on the right up corner too.
|
arxiv-papers
| 2012-11-06T05:27:16 |
2024-09-04T02:49:37.584802
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kai Zhang (1,2), Ting-Gui Wang (1), C. Martin Gaskell (3), Xiao-Bo\n Dong (1) ((1) Key Laboratory for Research in Galaxies and Cosmology, The\n University of Sciences and Technology of China, (2) Key Laboratory for\n Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, (3)\n Centro de Astrof\\'isica de Valpara\\'iso y Departamento de F\\'isica y\n Astronom\\'ia, Facultad de Ciencias, Universidad de Valpara\\'iso)",
"submitter": "Kai Zhang",
"url": "https://arxiv.org/abs/1211.1113"
}
|
1211.1230
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-333 LHCb-PAPER-2012-038 March 5, 2013
Observation of $D^{0}-\kern 4.14793pt\overline{\kern-4.14793ptD}{}^{0}$
oscillations
The LHCb collaboration†††Authors are listed on the following pages.
We report a measurement of the time-dependent ratio of $D^{0}\rightarrow
K^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$ decay rates in
$D^{*+}$-tagged events using $1.0\mbox{\,fb}^{-1}$ of integrated luminosity
recorded by the LHCb experiment. We measure the mixing parameters $x^{\prime
2}=(-0.9\pm 1.3)\times 10^{-4}$, $y^{\prime}=(7.2\pm 2.4)\times 10^{-3}$ and
the ratio of doubly-Cabibbo-suppressed to Cabibbo-favored decay rates
$R_{D}=(3.52\pm 0.15)\times 10^{-3}$, where the uncertainties include
statistical and systematic sources. The result excludes the no-mixing
hypothesis with a probability corresponding to $9.1$ standard deviations and
represents the first observation of $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations from a single
measurement.
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, L. Anderlini17,f, J. Anderson37, R. Andreassen57, R.B.
Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M.
Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C.
Baesso54, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44,
A. Bates48, 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, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, 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.
Borghi51,48, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, 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,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, 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. Contu15,52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, 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, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46,35, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R.
Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, 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. Garofoli53,
P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E.
Gersabeck11, M. Gersabeck51, 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, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C.
Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, 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, Y. Li3, L. Li Gioi5, 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, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, M. Maino20, S. Malde52, 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 Santos34, D. Martins Tostes2, A. Massafferri1, R.
Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A.
Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57, M. Meissner11,
M. Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5,
D. Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37,
R. Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
S. Nisar56, 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, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, 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, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51,48, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, 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. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26,
F. Teubert35, C. Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M.
Tobin37, S. Tolk39, D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6,
P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, D. Vieira2, X. Vilasis-Cardona33,n, 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,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, 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
25National Center for Nuclear Research (NCBJ), 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
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
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
pMassachusetts Institute of Technology, Cambridge, MA, United States
Meson-antimeson oscillations are a manifestation of flavor changing neutral
currents that occur because the flavor eigenstates differ from the physical
mass eigenstates of the meson-antimeson system. Short-range quark-level
transitions as well as long-range processes contribute to this phenomenon. The
former are governed by loops in which virtual heavy particles are exchanged
making the study of flavor oscillations an attractive area to search for
physics beyond the standard model (SM). Oscillations have been observed in the
$K^{0}-\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ [1], $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ [2] and $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ [3] systems, all with rates in
agreement with SM expectations. Evidence of $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations has been reported by
three experiments using different $D^{0}$ decay channels [4, 5, 6, 7, 8]. Only
the combination of these measurements provides confirmation of $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations, also referred to as
charm mixing, with more than $5\sigma$ significance [9]. While it is accepted
that charm mixing occurs, a clear observation of the phenomenon from a single
measurement is needed to establish it conclusively.
Charm mixing is characterized by two parameters: the mass and decay width
differences, $\Delta m$ and $\Delta\Gamma$, between the two mass eigenstates
expressed in terms of the dimensionless quantities $x=\Delta m/\Gamma$ and
$y=\Delta\Gamma/2\Gamma$, where $\Gamma$ is the average $D^{0}$ decay width.
The charm mixing rate is expected to be small, with predicted values of
$|x|,|y|\lesssim\mathcal{O}(10^{-2})$, including significant contributions
from non-perturbative long-range processes that compete with the short-range
electroweak loops [10, 11, 12, 13]. This makes the mixing parameters difficult
to calculate and complicates the unambiguous identification of potential non-
SM contributions in the experimental measurements [14, 15, 16].
In the analysis described in this Letter, $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations are observed by
studying the time-dependent ratio of $D^{0}\rightarrow K^{+}\pi^{-}$ to
$D^{0}\rightarrow K^{-}\pi^{+}$ decay rates.111The inclusion of charge-
conjugated modes is implied throughout this Letter. The $D^{0}$ flavor at
production time is determined using the charge of the soft (low-momentum)
pion, $\pi_{\rm s}^{+}$, in the strong $D^{*+}\rightarrow D^{0}\pi_{\rm
s}^{+}$ decay. The $D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi_{\rm
s}^{+}$ process is referred to as right-sign (RS), whereas the
$D^{*+}\rightarrow D^{0}(\rightarrow K^{+}\pi^{-})\pi_{\rm s}^{+}$ is
designated as wrong-sign (WS). The RS process is dominated by a Cabibbo-
favored (CF) decay amplitude, whereas the WS amplitude includes contributions
from both the doubly-Cabibbo-suppressed (DCS) $D^{0}\rightarrow K^{+}\pi^{-}$
decay, as well as $D^{0}-\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
mixing followed by the favored $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decay. In
the limit of small mixing ($|x|,|y|\ll 1$), and assuming negligible $C\\!P$
violation, the time-dependent ratio, $R$, of WS to RS decay rates is
approximated by [10]
$R(t)\approx R_{D}+\sqrt{R_{D}}\ y^{\prime}\ \frac{t}{\tau}+\frac{x^{\prime
2}+y^{\prime 2}}{4}\left(\frac{t}{\tau}\right)^{2},$ (1)
where $t/\tau$ is the decay time expressed in units of the average $D^{0}$
lifetime $\tau$, $R_{D}$ is the ratio of DCS to CF decay rates,
$x^{\prime}=x\cos\delta+y\sin\delta$, $y^{\prime}=y\cos\delta-x\sin\delta$,
and $\delta$ is the strong phase difference between the DCS and CF amplitudes.
The analysis is based on a data sample corresponding to $1.0\mbox{\,fb}^{-1}$
of $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions recorded by
LHCb during 2011. The LHCb detector [17] is a single-arm forward spectrometer
covering the pseudorapidity range $2<\eta<5$, designed for the study of
particles containing $b$ or $c$ quarks. Detector components particularly
relevant for this analysis are the silicon Vertex Locator, which provides
identification of displaced, secondary vertices of $b$\- and $c$-hadron
decays; the tracking system, which measures charged particles with momentum
resolution $\Delta p/p$ that varies from $0.4\%$ at
$5\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c$}$ to $0.6\%$ at
$100\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c$}$, corresponding to a typical
mass resolution of approximately
$8\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c^{2}$}$ for a two-body charm-meson
decay; and the ring imaging Cherenkov detectors, which provide kaon-pion
discrimination.
Events are triggered by signatures consistent with a hadronic charm decay. The
hardware trigger demands a hadronic energy deposition with a transverse
component of at least $3\,\mathrm{\,Ge\kern-1.00006ptV}$. Subsequent software-
based triggers require two oppositely-charged tracks to form a $D^{0}$
candidate with a decay vertex well separated from the associated primary $pp$
collision vertex (PV). Additional requirements on the quality of the online-
reconstructed tracks, their transverse momenta ($p_{\rm T}$) and their impact
parameters (IP), defined as the distance of closest approach of the
reconstructed trajectory to the PV, are applied in the final stage of the
software trigger. For the offline analysis only $D^{0}$ candidates selected by
this trigger algorithm are considered.
The $D^{0}$ daughter particles are both required to have $p_{\rm
T}>800\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c$}$,
$p>5\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c$}$ and $\chi^{2}(\text{IP})>9$.
The $\chi^{2}(\text{IP})$ is defined as the difference between the $\chi^{2}$
of the PV reconstructed with and without the considered particle, and is a
measure of consistency with the hypothesis that the particle originates from
the PV. Selected $D^{0}$ candidates are required to have $p_{\rm
T}>3.5\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c$}$ and are combined with a
track with $p_{\rm T}>300\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c$}$ and
$p>1.5\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c$}$ to form a $D^{*+}$
candidate. Contamination from $D$ mesons originating from $b$-hadron decays
(secondary $D$) is reduced by requiring the $\chi^{2}(\text{IP})$ of the
$D^{0}$ and of $\pi_{\rm s}^{+}$ candidates to be smaller than $9$ and $25$,
respectively. In addition, the ring imaging Cherenkov system is used to
distinguish between pions and kaons and to suppress the contamination from
misidentified two-body charm decays in the sample. Backgrounds from
misidentified singly Cabibbo-suppressed decays are specifically removed by
requiring the $D^{0}$ candidate mass reconstructed under the $K^{+}K^{-}$ and
$\pi^{+}\pi^{-}$ hypotheses to differ by more than
$40\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c^{2}$}$ from the known $D^{0}$
mass [18]. Contamination from electrons to the soft pion sample is also
suppressed using particle identification information. Finally, it is required
that the $D^{0}$ and the $\pi_{\rm s}^{+}$ form a vertex, which is constrained
to the measured PV. Only candidates with reconstructed $K\pi$ mass within
$24\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c^{2}$}$ of the known $D^{0}$ mass
and with reconstructed $D^{0}\pi_{\rm s}^{+}$ mass below
$2.02\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}$/$c^{2}$}$ are considered further.
The $D^{0}\pi_{\rm s}^{+}$ mass, $M(D^{0}\pi_{\rm s}^{+})$, is calculated
using the vector sum of the momenta of the three charged particles and the
known $D^{0}$ and $\pi^{+}$ masses [18]; no mass hypotheses for the $D^{0}$
daughters enter the calculation, ensuring that all two-body signal decays have
the same $M(D^{0}\pi_{\rm s}^{+})$ distribution [19]. Events with multiple RS
or WS $D^{*+}$ candidates occur about $2.5\%$ of the time, and all candidates
are kept.
Figure 1: Time-integrated $D^{0}\pi_{\rm s}^{+}$ mass distributions for the
selected RS $D^{0}\rightarrow K^{-}\pi^{+}$ (left) and WS $D^{0}\rightarrow
K^{+}\pi^{-}$ (right) candidates with fit projections overlaid. The bottom
plots show the normalized residuals between the data points and the fits.
Figure 1 shows the $M(D^{0}\pi_{\rm s}^{+})$ distribution for the selected RS
and WS candidates. Overlaid is the result of a binned $\chi^{2}$ fit used to
separate the $D^{*+}$ signal component, with a mass resolution of about
$0.3\,\mbox{$\mathrm{\,Me\kern-1.00006ptV}$/$c^{2}$}$, from the background
component, which is dominated by associations of real $D^{0}$ decays and
random pions. The signal mass shape is modeled as the sum of one Johnson
$S_{U}$ [20] and three Gaussian distributions, which account for the
asymmetric tails and the central core of the distribution, respectively. The
background is described by an empirical function of the form
$\left[M(D^{0}\pi_{\rm s}^{+})-m_{0}\right]^{a}e^{-b\left[M(D^{0}\pi_{\rm
s}^{+})-m_{0}\right]}$, where the threshold $m_{0}$ is fixed to the sum of the
known $D^{0}$ and $\pi^{+}$ masses [18]. We reconstruct approximately
$3.6\times 10^{4}$ WS and $8.4\times 10^{6}$ RS decays. To determine the time-
dependent WS/RS ratio the data are divided into thirteen $D^{0}$ decay time
bins, chosen to have a similar number of candidates in each bin. The decay
time is estimated from the distance $L$ between the PV and the $D^{0}$ decay
vertex and from the $D^{0}$ momentum as $t/\tau=m_{D^{0}}L/p\tau$, where
$m_{D^{0}}$ and $\tau$ are the known $D^{0}$ mass and lifetime [18],
respectively. The typical decay-time resolution is $\sim 0.1\tau$. The signal
yields for the RS and WS samples are determined in each decay time bin using
fits to the $M(D^{0}\pi_{\rm s}^{+})$ distribution. The shape parameters and
the yields of the two components, signal and random pion background, are left
free to vary in the different decay time bins. We further assume that the
$M(D^{0}\pi_{\rm s}^{+})$ signal shape for RS and WS decay are the same.
Hence, we first perform a fit to the abundant and cleaner RS sample to
determine the signal shape and yield, and then, use those shape parameters
with fixed values when fitting for the WS signal yield. The signal yields from
the thirteen bins are used to calculate the WS/RS ratios, shown in Fig. 2, and
the mixing parameters are determined in a binned $\chi^{2}$ fit to the time-
dependence according to Eq. (1).
Since WS and RS events are expected to have the same decay-time acceptance and
$M(D^{0}\pi_{\rm s}^{+})$ distributions, most systematic uncertainties
affecting the determination of the signal yields as a function of decay time
cancel in the ratio between WS and RS events. Residual biases from
noncanceling instrumental and production effects, such as asymmetries in
detection efficiencies or in production, are found to modify the WS/RS ratio
only by a relative fraction of ${\cal O}(10^{-4})$ and are neglected.
Uncertainties in the distance between Vertex Locator sensors can lead to a
bias of the decay-time scale. The effect has been estimated to be less than
0.1% of the measured time [21] and translates into relative systematic biases
of $0.1\%$ and $0.2\%$ on $y^{\prime}$ and $x^{\prime 2}$, respectively. At
the current level of statistical precision, such small effects are negligible.
The main sources of systematic uncertainty are those which could alter the
observed decay-time dependence of the WS/RS ratio. Two such sources have been
identified: $(1)$ secondary $D$ mesons, and $(2)$ backgrounds from charm
decays reconstructed with the wrong particle identification assignments, which
peak in $M(D^{0}\pi_{\rm s}^{+})$ and are not accounted for in our mass fit.
These effects, discussed below, are expected to depend on the true value of
the mixing parameters and are accounted for in the time-dependent fit.
The contamination of charm mesons produced in $b$-hadron decays could bias the
time-dependent measurement, as the reconstructed decay time is calculated with
respect to the PV, which, in this case, does not coincide with the $D^{0}$
production vertex. When this secondary component is not subtracted, the
measured WS/RS ratio can be written as $R(t)\left[1-\Delta_{B}(t)\right]$,
where $R(t)$ is the ratio of promptly-produced candidates according to Eq.
(1), and $\Delta_{B}(t)$ is a time-dependent bias due to the secondary
contamination. Since $R(t)$ is measured to be monotonically nondecreasing [9]
and the reconstructed decay time for secondary decays overestimates the true
decay time of the $D^{0}$ meson, it is possible to bound $\Delta_{B}(t)$, for
all decay times, as
$0\leqslant\Delta_{B}(t)\leqslant f_{B}^{\rm
RS}(t)\left[1-\frac{R_{D}}{R(t)}\right],$ (2)
where $f_{B}^{\rm RS}(t)$ is the fraction of secondary decays in the RS sample
at decay time $t$. The lower bound in Eq. (2) corresponds to the case when the
parent $b$-hadron decays instantaneously and the reconstructed $D^{0}$ decay
time is the true decay time. The upper bound corresponds to the case when the
$D^{0}$ decays instantaneously and the reconstructed decay time $t$ is
entirely due to the $b$-hadron lifetime. Since $\Delta_{B}\geqslant 0$, it
follows that the background from secondary $D$ decays decreases the observable
mixing effect. To include the corresponding systematic uncertainty, we modify
the fitting function for the mixing hypothesis assuming the largest possible
bias from Eq. (2). The value of $f_{B}^{\rm RS}(t)$ is constrained to the
measured value, obtained by fitting the $\chi^{2}(\text{IP})$ distribution of
the RS $D^{0}$ candidates in bins of decay time. In this fit, the promptly-
produced component is described by a time-independent $\chi^{2}(\text{IP})$
shape, which is derived from data using the candidates with $t<0.8\tau$. The
$\chi^{2}(\text{IP})$ shape of the secondary component, and its dependence on
decay time, is also determined from data by studying the sub-sample of
candidates that are reconstructed, in combination with other tracks in the
events, as $B\rightarrow D^{*}(3)\pi$, $B\rightarrow D^{*}\mu X$ or
$B\rightarrow D^{0}\mu X$. The measured value of $f_{B}^{\rm RS}(t)$ increases
almost linearly with decay time from $(0.0\pm 0.5)\%$ up to $(14\pm 5)\%$, for
a time-integrated value of $(2.7\pm 0.2)\%$. We checked on pseudoexperiments,
before fitting the data, and then also on data that such a small contamination
results in a shift on the measured mixing parameters that is much smaller than
the increase in the uncertainty when the secondary bias is included in the
fit.
Background from incorrectly reconstructed $D$ meson decays, peaking in the
$M(D^{0}\pi_{\rm s}^{+})$ distribution, arises from $D^{*+}$ decays for which
the correct soft pion is found but the $D^{0}$ is partially reconstructed or
misidentified. This background is suppressed by the use of tight particle
identification and two-body mass requirements. From studies of the events in
the $D^{0}$ mass sidebands, we find that the dominant peaking background is
from RS events that survive the requirements of the WS selection; they are
estimated to constitute $(0.4\pm 0.2)\%$ of the WS signal. This contamination
is expected to have the same decay time dependence of RS decays and, if
neglected, would lead to a small increase in the measured value of $R_{D}$.
From the events in the $D^{0}$ mass sidebands, we derive a bound on the
possible time dependence of this background, which is included in the fit in a
similar manner to the secondary background. Contamination from peaking
background due to partially reconstructed $D^{0}$ decays is found to be much
smaller than $0.1\%$ of the WS signal and neglected in the fit.
The $\chi^{2}$ that is minimized in the fit to the WS/RS decay-time dependence
is
$\chi^{2}(r_{i},t_{i},\sigma_{i}|\bm{\theta})=\sum_{i}\left(\frac{r_{i}-R(t_{i}|\bm{\theta})[1-\Delta_{B}(t_{i}|\bm{\theta})]-\Delta_{p}(t_{i}|\bm{\theta})}{\sigma_{i}}\right)^{2}+\chi^{2}_{B}(\bm{\theta})+\chi^{2}_{p}(\bm{\theta}),$
(3)
where $r_{i}$ and $\sigma_{i}$ are the measured WS/RS ratio and its
statistical uncertainty in the decay time bin $i$, respectively. The decay
time $t_{i}$ is the average value in each bin of the RS sample. The fit
parameters, $\bm{\theta}$, include the three mixing parameters ($R_{D}$,
$y^{\prime}$, $x^{\prime 2}$) and five nuisance parameters used to describe
the decay time evolution of the secondary $D$ fraction ($\Delta_{B}$) and of
the peaking background ($\Delta_{p}$). The nuisance parameters are constrained
to the measured values by the additional $\chi_{B}^{2}$ and $\chi_{p}^{2}$
terms, which account for their uncertainties including correlations.
The analysis procedure is defined prior to fitting the data for the mixing
parameters. Measurements on pseudoexperiments that mimic the experimental
conditions of the data, and where $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations are simulated,
indicate that the fit procedure is stable and free of any bias.
Figure 2: Decay-time evolution of the ratio, $R$, of WS $D^{0}\rightarrow K^{+}\pi^{-}$ to RS $D^{0}\rightarrow K^{-}\pi^{+}$ yields (points) with the projection of the mixing allowed (solid line) and no-mixing (dashed line) fits overlaid. Table 1: Results of the time-dependent fit to the data. The uncertainties include statistical and systematic sources; ndf indicates the number of degrees of freedom. Fit type | Parameter | Fit result | | Correlation coefficient
---|---|---|---|---
($\chi^{2}$/ndf) | | ($10^{-3}$) | | $R_{D}$ | $y^{\prime}$ | $x^{\prime 2}$
Mixing | $R_{D}$ | $\quad 3.52\pm 0.15$ | | $1$ | $-0.954$ | $+0.882$
($9.5/10$) | $y^{\prime}$ | $\quad 7.2\pm 2.4$ | | | $1$ | $-0.973$
| $x^{\prime 2}$ | $\,-0.09\pm 0.13$ | | | | 1
No mixing | $R_{D}$ | $\quad 4.25\pm 0.04$ | | | |
($98.1/12$) | | | | | |
The fit to the decay-time evolution of the WS/RS ratio is shown in Fig. 2
(solid line), with the values and uncertainties of the parameters $R_{D}$,
$y^{\prime}$ and $x^{\prime 2}$ listed in Table 1. The value of $x^{\prime 2}$
is found to be negative, but consistent with zero. As the dominant systematic
uncertainties are treated within the fit procedure (all other systematic
effects are negligible), the quoted errors account for systematic as well as
statistical uncertainties. When the systematic biases are not included in the
fit, the estimated uncertainties on $R_{D}$, $y^{\prime}$ and $x^{\prime 2}$
become respectively $6\%$, $10\%$ and $11\%$ smaller, showing that the quoted
uncertainties are dominated by their statistical component. To evaluate the
significance of this mixing result we determine the change in the fit
$\chi^{2}$ when the data are described under the assumption of the no-mixing
hypothesis (dashed line in Fig. 2). Under the assumption that the $\chi^{2}$
difference, $\Delta\chi^{2}$, follows a $\chi^{2}$ distribution for two
degrees of freedom, $\Delta\chi^{2}=88.6$ corresponds to a $p$-value of
$5.7\times 10^{-20}$, which excludes the no-mixing hypothesis at $9.1$
standard deviations. This is illustrated in Fig. 3 where the $1\sigma$,
$3\sigma$ and $5\sigma$ confidence regions for $x^{\prime 2}$ and $y^{\prime}$
are shown.
Figure 3: Estimated confidence-level (CL) regions in the $(x^{\prime
2},y^{\prime})$ plane for $1-\text{CL}=0.317$ ($1\sigma$), $2.7\times 10^{-3}$
($3\sigma$) and $5.73\times 10^{-7}$ ($5\sigma$). Systematic uncertainties are
included. The cross indicates the no-mixing point.
As additional cross-checks, we perform the measurement in statistically
independent sub-samples of the data, selected according to different data-
taking periods, and find compatible results. We also use alternative decay-
time binning schemes, selection criteria or fit methods to separate signal and
background, and find no significant variations in the estimated parameters.
Finally, to assess the impact of events where more than one candidate is
reconstructed, we repeat the time-dependent fit on data after randomly
removing the additional candidates and selecting only one per event; the
change in the measured value of $R_{D}$, $y^{\prime}$ and $x^{\prime 2}$ is
$2\%$, $6\%$ and $7\%$ of their uncertainty, respectively.
In conclusion, we measure the decay time dependence of the ratio between
$D^{0}\rightarrow K^{+}\pi^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$ decays
using $1.0\mbox{\,fb}^{-1}$ of data and exclude the no-mixing hypothesis at
$9.1$ standard deviations. This is the first observation of $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations in a single
measurement. The measured values of the mixing parameters are compatible with
and have substantially better precision than those from previous measurements
[4, 6, 22].
## Acknowledgements
This Letter is dedicated to the memory of our friend and colleague Javier
Magnin. 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 the LHCb institutes. We acknowledge support from
CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2012-11-06T14:16:18 |
2024-09-04T02:49:37.600619
|
{
"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, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, 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, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.A.\n Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L.\n De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, J.\n Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo Bonal, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M.\n John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson,\n T.M. Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, V. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, 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, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n H. Luo, A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev,\n M. Maino, S. Malde, 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, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A.\n Mazurov, J. McCarthy, R. McNulty, B. Meadows, M. Meissner, M. Merk, D.A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, T. Nikodem, S. Nisar, 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, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani,\n C. 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, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, 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, M. Smith, K. Sobczak,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, 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, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson,\n A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, 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, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Angelo Di Canto",
"url": "https://arxiv.org/abs/1211.1230"
}
|
1211.1233
|
# Extended $q$-Dedekind-type Daehee-Changhee sums associated with Extended
$q$-Euler polynomials
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com;
saraci88@yahoo.com.tr; mtsrkn@gmail.com and Mehmet Acikgoz University of
Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310
Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
In the present paper, our goal is to introduce a $p$-adic continuous function
for an odd prime to inside a $p$-adic $q$-analogue of the higher order
Dedekind-type sums with weight $\alpha$ related to Extended $q$-Euler
polynomials by using $p$-adic $q$-integral in the $p$-adic integer ring.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Dedekind Sums, $q$-Dedekind-type Sums, $p$-adic
$q$-integral, Extended $q$-Euler numbers and polynomials.
## 1\. Introduction
Assume that $p$ be a fixed odd prime number. Throughout this paper
$\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$, $\mathbb{C}$ and $\mathbb{C}_{p}$ will,
respectively, denote the ring of $p$-adic rational integers, the field of
$p$-adic rational numbers, the complex numbers and the completion of algebraic
closure of $\mathbb{Q}_{p}$.
Let $v_{p}$ be normalized exponential valuation of $\mathbb{C}_{p}$ by
$\left|p\right|_{p}=p^{-v_{p}\left(p\right)}=\frac{1}{p}\text{.}$
When one talks of $q$-extension, $q$ is variously considered as an
indeterminate, a complex number $q\in\mathbb{C}$ or $p$-adic number
$q\in\mathbb{C}_{p}$. If $q\in\mathbb{C}$, we assume that $\left|q\right|<1$.
If $q\in\mathbb{C}_{p}$, we assume that $\left|1-q\right|_{p}<1$ (see,
[1-15]).
A $q$-analogue of $p$-adic Haar distribution is defined by Kim as follows: for
any postive integer $n$,
$\mu_{q}\left(a+p^{n}\mathbb{Z}_{p}\right)=\left(-q\right)^{a}\frac{\left(1+q\right)}{1+q^{p^{n}}}$
for $0\leq a<p^{n}$ and this can be extended to a measure on $\mathbb{Z}_{p}$
(for details, see [1-7]).
Extended $q$-Euler polynomials are defined as
(1)
$\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(x\right)=\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{n}d\mu_{q}\left(\xi\right)$
for $n\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,\cdots\right\\}$. We note that
$\lim_{q\rightarrow
1}\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(x\right)=E_{n}\left(x\right)$
where $E_{n}\left(x\right)$ are $n$-th Euler polynomials, which are defined by
the rule:
$\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}=e^{tx}\frac{2}{e^{t}+1},\text{
}\left|t\right|<\pi$
(for details, see [12]). Taking $x=0$ into (1), then we have
$\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(0\right):=\widetilde{E}_{n,q}^{\left(\alpha\right)}$
are called extended $q$-Euler numbers.
Extended $q$-Euler numbers and polynomials have the following equalities
(2) $\displaystyle\widetilde{E}_{n,q}^{\left(\alpha\right)}$ $\displaystyle=$
$\displaystyle\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l+1}}\text{,}$ (3)
$\displaystyle\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{q^{\alpha
lx}}{1+q^{\alpha l+1}}\text{,}$ (4)
$\displaystyle\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
lx}\widetilde{E}_{l,q}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
x}}{1-q^{\alpha}}\right)^{n-l}\text{.}$
Also, for $d\in\mathbb{N}$ with $d\equiv 1\left(\mathop{\mathrm{m}od}2\right)$
(5)
$\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(x\right)=\left(\frac{1+q}{1+q^{d}}\right)\left(\frac{1-q^{\alpha
d}}{1-q^{\alpha}}\right)^{n}\sum_{a=0}^{d-1}\left(-1\right)^{a}\widetilde{E}_{n,q}^{\left(\alpha\right)}\left(\frac{x+a}{d}\right)\text{,}$
(for more information, see [12]).
For any positive integer $h,k$ and $m$, Dedekind-type DC sums are given by Kim
in [1], [2] and [3] as follows:
$S_{m}\left(h,k\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\frac{M}{k}\overline{E}_{m}\left(\frac{hM}{k}\right)$
where $\overline{E}_{m}\left(x\right)$ are the $m$-th periodic Euler function.
Recently, weighted $q$-Bernoulli numbers and polynomials was firstly defined
by Taekyun Kim in [8]. After, many mathematicians, by utilizing from Kim’s
paper [8], have introduced a new concept in Analytic numbers theory as
weighted $q$-Bernoulli, weighted $q$-Euler, weighted $q$-Genocchi polynomials
in [11], [12], [14] and [15]. Also, Kim derived some interesting properties
for Dedekind-type DC sums. He considered a $p$-adic continuous function for an
odd prime number to contain a $p$-adic $q$-analogue of the higher order
Dedekind-type DC sums $k^{m}S_{m+1}\left(h,k\right)$ in [2]. In [9], Simsek
also studied to $q$-analogue of Dedekind-type sums. He also derived their
interesting properties.
By the same motivation, we, by using $p$-adic $q$-integral on
$\mathbb{Z}_{p}$, will construct weighted $p$-adic $q$-analogue of the higher
order Dedekind-type DC sums $k^{m}S_{m+1}\left(h,k\right)$.
## 2\. Extended $q$-Dedekind-type Sums associated with Extended $q$-Euler
polynomials
Let $w$ be the $Teichm\ddot{u}ller$ character ($\mathop{\mathrm{m}od}p$). For
$x\in\mathbb{Z}_{p}^{\ast}$ $:=\mathbb{Z}_{p}/p\mathbb{Z}_{p}$, set
$\left\langle
x:q\right\rangle=w^{-1}\left(x\right)\left(\frac{1-q^{x}}{1-q}\right)\text{.}$
Let $a$ and $N$ be positive integers with $\left(p,a\right)=1$ and $p\mid N$.
We now consider the following
$\widetilde{C}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=w^{-1}\left(a\right)\left\langle
x:q^{\alpha}\right\rangle^{s}\sum_{j=0}^{\infty}\binom{s}{j}q^{\alpha
aj}\left(\frac{1-q^{\alpha N}}{1-q^{\alpha
a}}\right)^{j}\widetilde{E}_{j,q^{N}}^{\left(\alpha\right)}\text{.}$
In particular, if $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$, then
$\displaystyle\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{1-q^{\alpha
a}}{1-q^{\alpha}}\right)^{m}\sum_{j=0}^{m}\binom{m}{j}q^{\alpha
aj}\widetilde{E}_{j,q^{N}}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha a}}\right)^{j}$ $\displaystyle=$
$\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)\text{.}$
Thus, $\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$ is a
continuous $p$-adic extension of
$\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\widetilde{E}_{m,q^{N}}^{\left(\alpha\right)}\left(\frac{a}{N}\right)\text{.}$
Let $\left[.\right]$ be the Gauss’ symbol and let
$\left\\{x\right\\}=x-\left[x\right]$. Thus, we are now ready to introduce
$q$-analogue of the higher order Dedekind-type DC sums
$\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)$ by the rule:
$\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha
k}}\right)\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(l\xi+l\left\\{\frac{hM}{k}\right\\}\right)}}{1-q^{\alpha
l}}\right)^{m}d\mu_{q^{l}}\left(\xi\right)\text{.}$
If $m+1\equiv 0\left(\mathop{\mathrm{m}od}p-1\right)$
$\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha k}}\right)\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$ $\displaystyle=$
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
where $p\mid k$, $\left(hM,p\right)=1$ for each $M$. Thanks to (1), we easily
state the following
(6) $\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\left(-1\right)^{M-1}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,\left(hM\right)_{k}:q^{k}\right)$
where $(hM)_{k}$ denotes the integer $x$ such that $0\leq x<n$ and
$x\equiv\alpha\left(\mathop{\mathrm{m}od}k\right)$.
It is not difficult to indicate the following
(7)
$\displaystyle\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{k}d\mu_{q}\left(\xi\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
m}}{1-q^{\alpha}}\right)^{k}\frac{1+q}{1+q^{m}}\sum_{i=0}^{m-1}\left(-1\right)^{i}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
m\left(\xi+\frac{x+i}{m}\right)}}{1-q^{\alpha
m}}\right)^{k}d\mu_{q^{m}}\left(\xi\right)\text{.}$
On account of (6) and (7), we easily see that
(8) $\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle=\frac{1+q^{N}}{1+q^{Np}}\sum_{i=0}^{p-1}\left(-1\right)^{i}\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)\text{.}$
Because of (6), (7) and (8), we develop the $p$-adic integration as follows:
$\widetilde{C}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=\frac{1+q^{N}}{1+q^{Np}}\sum_{\underset{a+iN\neq
0(\mathop{\mathrm{m}od}p)}{0\leq i\leq
p-1}}\left(-1\right)^{i}\widetilde{C}_{q}^{\left(\alpha\right)}\left(s,\left(a+iN\right)_{pN},p^{N}:q^{pN}\right)\text{.}$
So,
$\displaystyle\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)=\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)$
where $\left(p^{-1}a\right)_{N}$ denotes the integer $x$ with $0\leq x<N$,
$px\equiv a\left(\mathop{\mathrm{m}od}N\right)$ and $m$ is integer with
$m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$. Therefore, we procure the following
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)$
where $p\nmid k$ and $p\nmid hm$ for each $M$. Thus, we give the following
definition, which seems interesting for further studying in theory of Dedekind
sums.
###### Definition 1.
Let $h,k$ be positive integer with $\left(h,k\right)=1$, $p\nmid k$. For
$s\in\mathbb{Z}_{p},$ we define $p$-adic Dedekind-type DC sums as follows:
$\widetilde{J}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{C}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)\text{.}$
As a result of the above definition, we state the following theorem.
###### Theorem 2.1.
For $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$ and $\left(p^{-1}a\right)_{N}$
denotes the integer $x$ with $0\leq x<N$, $px\equiv
a\left(\mathop{\mathrm{m}od}N\right)$, then, we have
$\displaystyle\widetilde{J}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{J}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)\text{.}$
In the special case $\alpha=1$, our applications in theory of Dedekind sums
resemble Kim’s results in [2]. These results seem to be interesting for
further studies in [1], [3] and [9].
## References
* [1] T. Kim, A note on $p$-adic $q$-Dedekind sums, C. R. Acad. Bulgare Sci. 54 (2001), 37–42.
* [2] T. Kim, Note on $q$-Dedekind-type sums related to $q$-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125.
* [3] T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contemp. Math. 18 (2009), 249–260.
* [4] T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161–170.
* [5] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299.
* [6] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008), 598–608.
* [7] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [8] T. Kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
* [9] Y. Simsek, $q$-Dedekind type sums related to $q$-zeta function and basic $L$-series, J. Math. Anal. Appl. 318 (2006), 333-351.
* [10] M. Acikgoz, Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
* [11] S. H. Rim and J. Jeong, A note on the Modified $q$-Euler Numbers and polynomials with weight $\alpha$, International Mathematical Forum, Vol. 6, 2011, no. 65, 3245-3250.
* [12] C. S. Ryoo, A note on the weighted $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), page 47-54
* [13] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, accepted in Bulletin of the Malaysian Mathematical Sciences and Society.
* [14] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, accepted in Bulletin of the Korean Mathematical Society.
* [15] S. Araci, D. Erdal and 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.
* [16] S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 298531, 11 pages (Article in press).
|
arxiv-papers
| 2012-11-02T23:40:07 |
2024-09-04T02:49:37.610059
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1211.1233"
}
|
1211.1249
|
# New Existence Theorems about the Solutions of Some Stochastic Integral
Equations
Xuemei Chen and Yingying Qi and Chunyan Yang
Department of Mathematics, Sichuan University,
Chengdu 610064, P.R.China
> Abstract Picard’s iteration has been used to prove the existence and
> uniqueness of the solution for stochastic integral equations, here we use
> Schauder’s fixed point theorem to give a new existence theorem about the
> solution of a stochastic integral equation, our theorem can weak some
> conditions gotten by applying Banach’s fixed point theorem.
>
> Keywords Stochastic integral equation; Schauder’s fixed point theorem;
> bounded closed convex subset; compact operator
>
> AMS Subject Classification 47H10, 60H05, 60H10
## 1 Introduction and Main Results
Many mathematical models of phenomena occuring in sociology, physics, biology
and engineering involve random differential and integral equations.
Theoretical and applied treatments of problems concerning random differential
and integral equations can be found in many papers and monographs: Bharucha-
Reid ([4]); Doob ([5]); Padgeett and Tsokos ([6]); Tsokos and Padgett ([7]);
Rao and Tsokos ([8]).
For stochastic differential equations, firstly we should prove the existence
for the solutions. In filtering problem we know the system $X_{t}$ satisfying
$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dB(t),$ the observations $Z_{s}$
satisfying $dZ_{s}=c(s,X_{s})ds+d(s,X_{s})dV(s),Z_{0}=0$ for $0\leq s\leq t$,
where V(s) is Brownian motion, we want to find the best estimate $\hat{X}_{t}$
of the state $X_{t}$ of the system based on these observations, before we find
the estimate $\hat{X}_{t}$, we should give some assumptions for the existence
of the corresponding stochastic integral equations.
Some mathematicians have used Picard’s iteration or Banach’s contraction
mapping principle to prove the existence and uniqueness of some stochastic
integral equations. The goal of this paper is to give a new existence theorem
about stochastic integral equations using Schauder’s fixed point theorem, in
order to apply Schauder’s fixed point theorem we need to construct a compact
operator A and a convex bounded and closed nonempty subset M. Furthermore,
comparing with Banach’s fixed point theorem we weak some conditions.
Theorem 1.1([1])(Schauder’s fixed point theorem). The compact operator
$A:M\longrightarrow M$
has at least one fixed point when M is a bounded, closed, convex, nonempty
subset of a Banach space X over real field.
Theorem 1.2([1])(Banach’s fixed point theorem).We assume that:
(a) M is a closed nonempty subset in the Banach space X over field K, and
(b) the operator $A:M\longrightarrow M$ is k-contractive, i.e., there is
$0\leq k<1$ such that:
${\|Au-Av\|\leq k\|u-v\|\ \ for\ all\ u,v\in M.}$
Then, the operator A has exactly one fixed point $u$ on the set M.
Schauder’s Fixed Point Theorem can be applied to many fields in mathematics,
especially to the integral equation:
$\displaystyle{u(x)=\lambda\int_{a}^{b}F(x,y,u(y))dy,\ \ \ a\leq x\leq b,}$
(1)
where $-\infty<a<b<+\infty$ and $\lambda\in R$. Let
$Q={\\{(x,y,u)\in R^{3};\ x,y\in[a,b],\ |u|\leq r\\}}\ \ for\ fixed\ r>0.$
Theorem 1.3([1])Assume the following conditions:
(a) The function $F:Q\longrightarrow R$ is continuous.
(b) We define $(b-a)$$\mathcal{M}$ $:=\max_{(x,y,u)\in Q}$ $|F(x,y,u)|$, let
the real number $\lambda$ be given such that
$\displaystyle|\lambda|{\mathcal{M}}\leq r$ (2)
(c) We set $X:=C[a,b]$ and $M:=\\{u\in X;\|u\|\leq r\\}$.
Then the original integral equation (1) has at least one solution $u\in M$.
It well known that Banach’s Fixed Point Theorem can be used to prove the
existence and uniqueness of the solution for the integral equation(1).
Theorem 1.4([1])Assume the following conditions:
(a) The function $F:[a,b]\times[a,b]\times R\longrightarrow R$ is continuous,
and the partial derivative
$F_{u}:[a,b]\times[a,b]\times R\longrightarrow R$
is also continuous.
(b) There is a number $\mathcal{L}$ such that
$|F_{u}(x,y,u)|\leq{\mathcal{L}}\ \ for\ all\ x,y\in[a,b],u\in R.$
(c) Let the real number $\lambda$ be given such that
$\displaystyle(b-a)|\lambda|{\mathcal{L}}<1.$ (3)
(d) Set $X:=C[a,b]$ and $\|u\|:=max_{a\leq x\leq b}|u(x)|$.
Then the original equation (1) has a unique solution $u\in M$.
From Theorem 1.1 and Theorem 1.2 we know that Schauder’s fixed point theorem
is a existence principle, while Banach’s fixed point theorem is a existence
and uniqueness theorem. It seems that the conditions in Theorem 1.3 are weaker
comparing with Theorem 1.4, that is, (2) is easier to reach than (3).
Here, a nature question is whether Schauder’s and Banach’s fixed point
theorems can be applied to stochastic integral equations. Furthermore, whether
the conditions coming from Banach’s fixed point theorem are stronger than the
conditions from Schauder’s fixed point theorem.
Notation 2.1([2]) For convenience, we will use
$X:=L_{ad}^{2}([a,b]\times\Omega)$ to denote the space of all stochastic
process $f(t,\omega),a\leq t\leq b,\omega\in\Omega$ satisfying the following
conditions:
(1)$f(t,\omega)$ is adapted to the filtration $\mathcal{F}$t;
(2)$\int_{a}^{b}E(|f(t)|^{2})dt<+\infty.$
Now we want to solve the stochastic integral equation:
$\displaystyle{x(t;w)=h(t_{0};w)+\int_{a}^{t}\sigma(s,x(s;w))dB(s)+\int_{a}^{t}f(s,x(s;w))ds,\
a\leq t\leq b,}$ (4)
where:
(1)$\omega\in\Omega$ , where $\Omega$ is the supporting set of the probability
measure space $(\Omega,{\mathcal{F}},P)$ with $\mathcal{F}$ being the $\sigma$
-algebra and P the probability measure;
(2)$x(t;w)$ is the unknown random variable for each $t\in[a,b]$;
(3)$h(t_{0};w)$ is the knowm random variable and $E|h|^{2}<+\infty$;
(4)$B(t)$ be a Brownian motion and $\\{{\mathcal{F}}_{t};a\leq t\leq b\\}$ be
a filtration so there $B(t)$ is ${\mathcal{F}}_{t}$ -measurable for each $t$
and $B(t)-B(s)$ is independent of ${\mathcal{F}}_{s}$ for any $s<t$.
Let
$Q={\\{(t,X_{t})\in R^{2};\ t\in[a,b],\ and\ \|X_{t}\|\leq r\ for\ fixed\
r>0\\}}.$
Theorem 1.5 Assume the following conditions:
(a) $f(s,X_{s}),\ \sigma(s,X_{s})$ are measurable on $[a,b]\times\Omega$;
$f(s,X_{s}):Q\longrightarrow R$ is continuous;
$\sigma(s,X_{s}):Q\longrightarrow R$ is continuous;
(b) We define
$\displaystyle d=\sup_{(s,X_{s})\in
Q}\\{\|f(s,X_{s})\|,\|\sigma(s,X_{s})\|\\},$ (5)
and let the real number $a,\ b,\ d$ and random variable $h$ be given that
$\displaystyle 3E[h^{2}]+3(1+b-a)(b-a)d^{2}\leq r^{2}.$ (6)
(c) We set $X:=L_{ad}^{2}([a,b]\times\Omega)$ (see Notation 2.1) and
$M:=\\{X_{t}\in X;\|X_{t}\|\leq r\\}$.
Then the stochastic integral equation (4) has at least one solution $X_{t}\in
M$.
Theorem 1.6 Assume the following conditions:
(a) $f(s,X_{s}),\ \sigma(s,X_{s})$ are measurable on $[a,b]\times\Omega$;
(b)
$\displaystyle\ |f(s,X_{s})-f(s,Y_{s})|$ $\displaystyle\leq$ $\displaystyle
k_{1}|X_{s}-Y_{s}|;$ (7) $\displaystyle|\sigma(s,X_{s})-\sigma(s,Y_{s})|$
$\displaystyle\leq$ $\displaystyle k_{2}|X_{s}-Y_{s}|;$ (8)
(c) Let the real number $a,\ b,\ c=\\{k_{1},k_{2}\\}$ be given such that
$\displaystyle 0\leq 2c^{2}(1+b-a)(b-a)<1.$ (9)
(d) We set $X:=L_{ad}^{2}([a,b]\times\Omega)$ (see Notation 2.1) and
$M:=\\{X_{t}\in X;\|X_{t}\|\leq r\\}.$
Then the stochastic integral equation (4) has a unique solution $X_{t}\in M$.
## 2 Some Lemmas
We require the following Lemmas for proving the existence of the stochastic
integral equation.
Lemma 2.1([1]) Let X and Y be normed spaces over field K, and let
$A:M\subseteq X\longrightarrow Y$
be a continuous operator on the compact nonempty subset M of X. Then, A is
uniformly continuous on M.
Lemma 2.2([1]) Let $X:=L_{ad}^{2}([a,b]\times\Omega)$ with
$\|X_{t}\|:=(E|X_{t}|^{2})^{\frac{1}{2}}$ and $-\infty<a<b<+\infty$. Suppose
that we are given a set M in X such that
(1)M is bounded , i.e., $\|X_{t}\|\leq r$ for all $X_{t}\in M$ and fixed
$r\geq 0$.
(2)M is equicontinuous, i.e., for each $\varepsilon>0$, there is a $\delta>0$
such that
$|t_{1}-t_{2}|<\delta\ \ \ and\ \ X_{t}\in M\ \ \ imply\ \
|X_{t_{1}}-X_{t_{2}}|<\varepsilon.$
Then, M is a relatively compact subset of X.
Definition 2.1([1]) Let X and Y be normed spaces over field K. The operator
$A:M\subseteq X\longrightarrow Y$
is called compact iff
(1) A is continuous, and
(2) A transforms bounded sets into relatively compact sets.
Lemma 2.3([3]) (It$\hat{o}$ Isometry)
For each $X_{t},Y_{t}\in L_{ad}^{2}([a,b]\times\Omega)$, we have
$E[(\int_{a}^{b}f(t,w)dB(t))^{2}]=E[\int_{a}^{b}f^{2}(t,w)dt]$
## 3 The Proof of Theorem 1.5
Proof: We divide the proof into three steps:
$Step$ 1: We prove that $M=\\{X_{t}\in X;\|X_{t}\|\leq r\\}$ is closed, convex
subset of $L_{ad}^{2}([a,b]\times\Omega)$ (see Notation 2.1).
(A) We prove M is closed.
Let $X_{t}^{(n)}\in M$ for all $n$, i.e.,
$\|X_{t}^{(n)}\|\leq r\ \ \ for\ all\ n.$
If $X_{t}^{(n)}\longrightarrow X_{t}$ as $n\longrightarrow+\infty$, then
$\|X_{t}\|\leq r$, and hence $X_{t}\in M$.
(B) We prove M is convex.
If $X_{t},Y_{t}\in M$ and $0\leq\alpha\leq 1$, then
$\displaystyle\ \|\alpha X_{t}+(1-\alpha)Y_{t}\|$ $\displaystyle\leq$
$\displaystyle\|\alpha X_{t}\|+\|(1-\alpha)Y_{t}\|$ $\displaystyle\leq$
$\displaystyle\mbox{}\alpha r+(1-\alpha)r$ $\displaystyle=$
$\displaystyle\mbox{}r$
Hence
$\alpha X_{t}+(1-\alpha)Y_{t}\in M.$
$Step$ 2: We prove that $A:M\longrightarrow M$ is a compact operator.
Define $A:M\longrightarrow M$
$\displaystyle{A(X_{t})=h(t_{0};w)+\int_{a}^{t}\sigma(s,x(s;w))dB(s)+\int_{a}^{t}f(s,x(s;w))ds,\
a\leq t\leq b,}$ (10)
Then
(a) $A:M\longrightarrow M$ is a continuous operator.
By Lemma 2.1, we know $f(s,X_{s}),\sigma(s,X_{s})$ are uniformly continuous on
the compact set Q. This implies that, for each $\varepsilon>0$, there is a
number $\delta>0$ such that
$\|\sigma(s,X_{s})-\sigma(s,Y_{s})\|<\varepsilon_{1}$
$\|f(s,X_{s})-f(s,Y_{s})\|<\varepsilon_{2}$
for all$(s,X_{s}),(s,Y_{s})\in Q$ with $\|X_{s}-Y_{s}\|<\delta.$
For each $X_{t},Y_{t}\in M$, we have
$\displaystyle\ \|AX_{t}-AY_{t}\|^{2}$ $\displaystyle=$ $\displaystyle
E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s)$ $\displaystyle+$
$\displaystyle\mbox{}\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$
Using the inequality $(a+b)^{2}\leq 2(a^{2}+b^{2})$ to get
$\displaystyle\ \|AX_{t}-AY_{t}\|^{2}$ $\displaystyle\leq$ $\displaystyle
2E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s))^{2}{}$ (11)
$\displaystyle+$
$\displaystyle\mbox{}2E(\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$
Applying the It$\hat{o}$ Isometry to
$E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s))^{2}$ , we get:
$\displaystyle\ E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s))^{2}$
$\displaystyle=$ $\displaystyle
E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))^{2}ds){}$ (12)
$\displaystyle=$
$\displaystyle\mbox{}\int_{a}^{t}E(\sigma(s,X_{s})-\sigma(s,Y_{s}))^{2}ds{}$
$\displaystyle=$
$\displaystyle\mbox{}\int_{a}^{t}\|\sigma(s,X_{s})-\sigma(s,Y_{s})\|^{2}ds{}$
$\displaystyle<$ $\displaystyle\mbox{}(b-a)\varepsilon_{1}^{2}$
For $E(\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$, we use Schwarz’s
inequality to get
$\displaystyle\ E(\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$
$\displaystyle\leq$ $\displaystyle
E((t-a)\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))^{2}ds){}$ (13) $\displaystyle\leq$
$\displaystyle\mbox{}(b-a)\int_{a}^{t}E(f(s,X_{s})-f(s,Y_{s}))^{2}ds{}$
$\displaystyle=$
$\displaystyle\mbox{}(b-a)\int_{a}^{t}\|f(s,X_{s})-f(s,Y_{s})\|^{2}ds{}$
$\displaystyle<$ $\displaystyle\mbox{}(b-a)^{2}\varepsilon_{2}^{2}$
Put equations (12) and (13) into equation (11) to get
$\displaystyle\ \|AX_{t}-AY_{t}\|^{2}$ $\displaystyle<$ $\displaystyle
2(b-a)[(b-a)\varepsilon_{2}^{2}+\varepsilon_{1}^{2}]$ $\displaystyle=$
$\displaystyle\mbox{}\varepsilon^{2}$
Therefore for each $X_{t},Y_{t}\in M$, there exists $\delta>0$, when
$\|X_{t}-Y_{t}\|\leq\delta$, we have
$\displaystyle\|AX_{t}-AY_{t}\|<\varepsilon.$
That is: A is a continuous operator.
(b) A(M) is bounded.
For each $X_{t}\in M$
$\displaystyle\ \|AX_{t}\|^{2}$ $\displaystyle=$ $\displaystyle
E(h(t_{0};\omega)+\int_{a}^{t}\sigma(s,X_{s})dB(s)+\int_{a}^{t}f(s,X_{s})ds)^{2}{}$
(14) $\displaystyle\leq$
$\displaystyle\mbox{}3E[h^{2}]+3E(\int_{a}^{t}\sigma(s,X_{s})dB(s))^{2}+3E(\int_{a}^{t}f(s,X_{s})ds)^{2}{}$
$\displaystyle\leq$
$\displaystyle\mbox{}3E[h^{2}]+3\int_{a}^{t}E|\sigma(s,X_{s})|^{2}ds+3(b-a)\int_{a}^{t}E|f(s,X_{s})|^{2}ds{}$
$\displaystyle=$
$\displaystyle\mbox{}3E[h^{2}]+3\int_{a}^{t}\|\sigma(s,X_{s})\|^{2}ds+3(b-a)\int_{a}^{t}\|f(s,X_{s})\|^{2}ds{}$
$\displaystyle\leq$ $\displaystyle\mbox{}3E[h^{2}]+3(b-a)(1+b-a)d^{2}{}$
$\displaystyle\leq$ $\displaystyle\mbox{}r^{2}$
Thus A(M) is bounded.
(c) A(M) is equicontinuous.
For each $X_{t}\in M$, we have
$\displaystyle\ \|AX_{t_{1}}-AX_{t_{2}}\|^{2}$ $\displaystyle=$ $\displaystyle
E(\int_{t_{2}}^{t_{1}}\sigma(s,X_{s})dB(s)+\int_{t_{2}}^{t_{1}}f(s,X_{s})ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}2E(\int_{t_{2}}^{t_{1}}\sigma(s,X_{s})dB(s))^{2}+2E(\int_{t_{2}}^{t_{1}}f(s,X_{s})ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}2\int_{t_{2}}^{t_{1}}E|\sigma(s,X_{s})|^{2}ds+2(b-a)\int_{t_{2}}^{t_{1}}E|f(s,X_{s})|^{2}ds$
$\displaystyle\leq$ $\displaystyle\mbox{}2(1+b-a)|t_{1}-t_{2}|d^{2}$
Take
$\delta=\frac{\varepsilon^{2}}{2(1+b-a)d^{2}},$
then for each $\varepsilon>0$, there exists
$\delta=\frac{\varepsilon^{2}}{2(1+b-a)d^{2}},$
when $|t_{1}-t_{2}|<\delta$, we have
$\|AX_{t_{1}}-AX_{t_{2}}\|<\varepsilon.$
Hence A(M) is equicontinuous.
Then by Lemma 2.2 and Definition 2.1, we know $A:M\longrightarrow M$ is a
compact operator.
$Step$ 3: we prove that $A(M)\subseteq M$.
For each $X_{t}\in M$, we have
$\displaystyle\ \|AX_{t}\|^{2}$ $\displaystyle=$ $\displaystyle
E(h+\int_{a}^{t}\sigma(s,X_{s})dB(s)+\int_{a}^{t}f(s,X_{s})ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}3E[h^{2}]+3E(\int_{a}^{t}\sigma(s,X_{s})dB(s))^{2}+3E(\int_{a}^{t}f(s,X_{s})ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}3E[h^{2}]+3\int_{a}^{t}E|\sigma(s,X_{s})|^{2}ds+3(b-a)\int_{a}^{t}E|f(s,X_{s})|^{2}ds$
where $f(s,X_{s}),\sigma(s,X_{s})\in L_{ad}^{2}([a,b]\times\Omega)$.
So
$\int_{a}^{b}\|AX_{t}\|^{2}dt<+\infty,$
that is
$AX_{t}\in L_{ad}^{2}([a,b]\times\Omega),$
meanwhile, we have proved $\|AX_{t}\|\leq r$ in (14), therefore $AX_{t}\in M$,
that is $A(M)\subseteq M$.
Thus the Schauder’s Fixed Point Theorem tells us that equation (4) has at
least one solution $X_{t}\in M$.
## 4 The Proof of the Theorem 1.6
Proof: In the proof of Theorem 1.5, we have proved that M is closed.
We now show that A is a contractive mapping:
For each $X_{t},Y_{t}\in M$, we have
$\displaystyle\ \|AX_{t}-AY_{t}\|^{2}$ $\displaystyle=$ $\displaystyle
E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s)+\mbox{}\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}2E(\int_{a}^{t}(\sigma(s,X_{s})-\sigma(s,Y_{s}))dB(s))^{2}+2E(\int_{a}^{t}(f(s,X_{s})-f(s,Y_{s}))ds)^{2}$
$\displaystyle\leq$
$\displaystyle\mbox{}2\int_{a}^{t}E(\sigma(s,X_{s})-\sigma(s,Y_{s}))^{2}ds+2(b-a)\int_{a}^{t}E(f(s,X_{s})-f(s,Y_{s}))^{2}ds$
$\displaystyle\leq$
$\displaystyle\mbox{}2k_{2}^{2}\int_{a}^{t}E|X_{s}-Y_{s}|^{2}ds+2k_{1}^{2}(b-a)\int_{a}^{t}E|X_{s}-Y_{s}|^{2}ds$
$\displaystyle\leq$
$\displaystyle\mbox{}2c^{2}(1+b-a)(b-a)\|X_{t}-Y_{t}\|^{2}$
Let
$k^{2}=2c^{2}(1+b-a)(b-a)<1,$
then
$\|AX_{t}-AY_{t}\|\leq k\|X_{t}-Y_{t}\|,\ \ 0\leq k<1,$
Therefore A is k-contractive.
Then the Banach’s fixed point theorem tells us that the stochastic integral
equation (4) has a unique solution $X_{t}\in M$.
## 5 Comparing Theorem 1.5 with Theorem 1.6
Comparing Theorem 1.5 with Theorem 1.6 we know when we use Schauder’s fixed
point theorem to prove the existence of the solution for the integral
equation, we need conditions (5) and (6).
But when we use Banach’s fixed point theorem to prove the existence of the
solution for the stochastic integral equation, we need conditions (7), (8) and
(9).
Obviously, the condition (6) is weaker than the condition (9).
## 6 An Example for Theorem 1.5
We apply the above Theorem 1.5 to the linear stochastic integral equation:
$\displaystyle X_{t}=\int_{0}^{t}f(s)X_{s}ds+\int_{0}^{t}g(s)X_{s}dB(s),\ \
0\leq t\leq 1,$ (15)
Proof: Define the operator:
$A(X_{t})=\int_{0}^{t}f(s)X_{s}ds+\int_{0}^{t}g(s)X_{s}dB(s),\ \ 0\leq t\leq
1,$
Obviously, $f(s)X_{s}$ and $g(s)X_{s}$ are measurable and continuous. We
define that $d=\sup E|X_{s}|^{2}$, $c=\max\\{f(s)^{2},g(s)^{2}\\}$,
$6cd=r^{2}$ and set $X:=L_{ad}^{2}([a,b]\times\Omega)$ and $M=\\{X_{t}\in
X;\|X_{t}\|\leq r\\}$.
Then all conditions of Theorem 1.5 hold, so (15) has at least one solution.
Especially, if we let $f(s)=u,g(s)=\sigma$, then the equation is the geometric
Brownian motion equation
$\displaystyle X_{t}=\int_{0}^{t}uX_{s}ds+\int_{0}^{t}\sigma X_{s}dB(s),\ \
0\leq t\leq 1,$ (16)
where $u$ is the expected return rate(constant), $\sigma$ is
volatility(constant), $B(t)$ is standard Brown motion.
We can easily to get that the equation (16) has at least one solution. It well
known that the existence of the solution of the equation (16) is important to
the financial Black-Scholes model .
Acknowledgements. We would like to thank our teacher Processor Zhang Shiqing
for his lectures on Functional Analysis, meanwhile We would like to thank him
for organizing the seminar on financial mathematics and his many helpful
discussions, suggestions and corrections about this paper.
## References
* [1] E. Zeidler, Applied Functional Analysics: Applications to Mathematical Physics, Spring-Verlag, New York-Berlin, 1995, P. 18-64.
* [2] Hui-Hsiung Kuo, Introduction to Stochastic Integration, Springer, 2006, P. 43-61.
* [3] Bernt $\phi$ Ksendl,Stochastic Differential Equations: An introduction with Applications, Springer-Verlag, 2005, P.29.
* [4] Bharucha-Reid, A. P., Random Integral Equations, Academic Press, New York, 1972.
* [5] Doob, J. Stochastic Processes, Wiley, New York, 1953.
* [6] Padgett, W. J. and Tsokos, C. P., On a stochastic integral of the Volterra type in telephone traffic theory, J. Appl. Prob. 8, 1971, 269-275.
* [7] Tsokos, C. P. and Padgett, W. J., Random Integral Equations with Application to Life Sciences and Engineering, Academic Press, New York, 1974.
* [8] Rao, A. N. V. and Tsokos, C. P., On a class of stochastic integral equation, Coll. Math. 35, 1976, 141-146.
|
arxiv-papers
| 2012-11-06T15:01:12 |
2024-09-04T02:49:37.615994
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xuemei Chen and Yingying Qi and Chunyan Yang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1211.1249"
}
|
1211.1451
|
arxiv-papers
| 2012-11-07T05:17:28 |
2024-09-04T02:49:37.625569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Pavan Kumar Reddy, Y. Ganesh Kumar Reddy, K. Chandra Mouli, U.\n Seshadri",
"submitter": "Pallavali Radha Krishna Reddy",
"url": "https://arxiv.org/abs/1211.1451"
}
|
|
1211.1487
|
Mass spectra in $\mathbf{{\cal N}=1}$ SQCD with additional fields. II
Victor L. Chernyak
(e-mail: v.l.chernyak@inp.nsk.su)
Budker Institut of Nuclear Physics SB RAS and Novosibirsk State University,
630090 Novosibirsk, Russia
Abstract
This article continues arXiV: 1205.0410 [hep-th]. Considered is the ${\cal
N}=1$ SQCD-like theory with $SU(N_{c})$ colors and $3N_{c}/2<N_{F}<2N_{c}$
flavors of light quarks $\,Q_{i},{\overline{Q}}_{j}$, and with the additional
$N^{2}_{F}$ colorless flavored fields $\Phi_{ij}$ with the large mass
parameter $\mu_{\Phi}\gg\Lambda_{Q}$. The mass spectra of this $\Phi$ \-
theory (and its dual variant, the $d\Phi$ \- theory) are calculated at
different values of $\mu_{\Phi}/\Lambda_{Q}\gg 1$ within the dynamical
scenario which implies that quarks can be in two different phases only :
either this is the HQ (heavy quark) phase where they are confined, or they are
higgsed at appropriate values of the lagrangian parameters. It is shown that
at the left end of the conformal window, i.e. at $0<(2N_{F}-3N_{c})/N_{F}\ll
1$, the mass spectra of the direct and dual theories are parametrically
different.
Contents
1 Introduction
1.1 Direct $\mathbf{\Phi}$ \- theory 3
1.2 Dual $\,\mathbf{d\Phi}$ \- theory 6
2 Direct theory. Unbroken flavor symmetry
2.1 $\mathbf{L}$ \- vacua 8
2.2 $\mathbf{S}$ \- vacua 9
3 Dual theory. Unbroken flavor symmetry
3.1 $\mathbf{L}$ \- vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm o}}/N_{F}\ll 1$
11
3.2 $\mathbf{S}$ \- vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm o}}/N_{F}\ll 1$
13
4 Direct theory. Broken flavor symmetry. The region
$\mathbf{\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}}$
4.1 $\mathbf{L}$ \- type vacua 15
4.2 ${\rm\bf br2}$ vacua 15
4.3 Special vacua, $\mathbf{n_{1}={\overline{N}}_{c},\,n_{2}=N_{c}}$ 17
5 Dual theory. Broken flavor symmetry. The region
$\mathbf{\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}}$
5.1 $\mathbf{L}$ \- type vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm
o}}/N_{F}\ll 1$ 17
5.2 ${\rm\bf br2}$ vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm o}}/N_{F}=O(1)$
17
5.3 ${\rm\bf br2}$ vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm o}}/N_{F}\ll 1$
19
5.4 Special vacua, $\mathbf{n_{1}={\overline{N}}_{c},\,n_{2}=N_{c}}$ 20
6 Direct theory. Broken flavor symmetry. The region $\mathbf{\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}}$
6.1 $\rm\bf br1$ vacua, 20
6.2 $\rm\bf br2$ and special vacua 23
7 Dual theory. Broken flavor symmetry. The region $\mathbf{\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}}$
7.1 $\rm\bf br1$ vacua, $\,\,\mathbf{{\rm\overline{b}}_{\rm o}}/N_{F}\ll 1$ 23
7.2 $\rm\bf br2$ and special vacua 25
8 Conclusions 26
References 26
## 1 Introduction
We continue in this paper our study of mass spectra in the $\mathbf{\Phi}$ \-
theory (and in its dual variant $d\mathbf{\Phi}$) that was started in [1]. The
difference with [1] is that we calculate here mass spectra at
$3N_{c}/2<N_{F}<2N_{c}$ within the dynamical scenario $\\#2$. 111 The mass
spectra of the direct $\mathbf{\Phi}$ \- theory at $0<N_{F}<N_{c}$ are the
same in both scenarios $\\#1$ and $\\#2$, see section 2 in [1]. We recall,
see [2], that this scenario implies that quarks can be in the two different
phases only. They are either in the heavy quark (HQ) phase where they are
confined or they are higgsed (the Higgs phase), under appropriate conditions.
For a reader convenience, we recall below in this section the definitions of
$\mathbf{\Phi}$ and $d\mathbf{\Phi}$ \- theories and some their most general
properties (see sections 1 and 3 and the appendix A in [1] for more details).
1.1. Direct $\mathbf{\Phi}$ \- theory
The field content of this direct ${\cal N}=1\,\,$ $\mathbf{\Phi}$ \- theory
includes $SU(N_{c})$ gluons and $3N_{c}/2<N_{F}<2N_{c}$ flavors of light
quarks ${\overline{Q}}_{j},Q_{i}$. Besides, there is $N^{2}_{F}$ colorless but
flavored fields $\Phi_{ji}$ (fions).
The Lagrangian at scales $\mu\leq\Lambda_{Q}$ is taken in the form
$\displaystyle K={\rm Tr}\,\Bigl{(}\Phi^{\dagger}\Phi\Bigr{)}+{\rm
Tr}\Bigl{(}\,Q^{\dagger}Q+(Q\rightarrow{\overline{Q}})\,\Bigr{)}\,,\quad
W=-\frac{2\pi}{\alpha(\mu,\Lambda_{Q})}S+W_{\Phi}+{\rm
Tr}\,{\overline{Q}}(m_{Q}-\Phi)Q\,,$ $\displaystyle
W_{\Phi}=\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]}\,.$ (1.1)
Here : $\mu_{\Phi}$ and $m_{Q}$ are the mass parameters,
$S=-W^{a}_{\beta}W^{a,\,\beta}/32\pi^{2}$ where $W^{a}_{\beta}$ is the gauge
field strength, $a=1...N_{c}^{2}-1,\,\beta=1,2$,
$\alpha(\mu,\Lambda_{Q})=g^{2}(\mu,\Lambda_{Q})/4\pi$ is the gauge coupling
with its scale factor $\Lambda_{Q}$, ${\overline{N}}_{c}=N_{F}-N_{c}$ , the
exponents with gluons in the quark Kahler term are implied here and below.
This normalization of fields at $\mu=\Lambda_{Q}$ will be used everywhere
below. Besides, the perturbative NSVZ $\beta$-function for massless SUSY
theories [3, 4] will be used everywhere in this paper.
Therefore, the $\mathbf{\Phi}$-theory we deal with in this paper has the
parameters : $N_{c}\,,\,3N_{c}/2<N_{F}<2N_{c}\,,\,\mu_{\Phi}$,
$\Lambda_{Q},\,m_{Q}$, with the strong hierarchies
$\mu_{\Phi}\gg\Lambda_{Q}\gg m_{Q}$. The mass parameter $\mu_{\Phi}$ will be
varied while $m_{Q}$ and $\Lambda_{Q}$ will stay intact.
The Konishi anomalies [5] for the $i$-th flavor look as (${\it
i}=1\,...\,N_{F}$)
$\displaystyle\langle\Phi_{ij}\rangle=\delta_{ij}\langle\Phi_{i}\rangle\,,\quad\langle\Phi_{i}\rangle\langle\frac{\partial
W_{\Phi}}{\partial\Phi_{i}}\rangle=0\,,\quad\langle m_{Q,i}^{\rm
tot}\rangle\langle{\overline{Q}}_{i}Q_{i}\rangle=\langle
S\rangle\,,\quad\langle m_{Q,\,i}^{\rm
tot}\rangle=m_{Q}-\langle\Phi_{i}\rangle\,,$
$\displaystyle\langle\Phi_{ij}\rangle=\frac{1}{\mu_{\Phi}}\Biggl{(}\langle{\overline{Q}}_{j}Q_{i}\rangle-\delta_{ji}\frac{1}{N_{c}}{\rm
Tr}\,\langle{\overline{Q}}Q\rangle\Biggr{)}\,,\quad\langle{\overline{Q}}_{j}Q_{i}\rangle=\delta_{ji}\langle{\overline{Q}}_{i}Q_{i}\rangle\,,$
(1.2)
and $\langle m_{Q,i}^{\rm tot}\rangle$ is the value of the quark running mass
at $\mu=\Lambda_{Q}$.
At all scales $\mu<\Lambda_{Q}$ until the field $\Phi$ remains too heavy and
non-dynamical, i.e. until its perturbative running mass $\mu_{\Phi}^{\rm
pert}(\mu)>\mu$, it can be integrated out and the Lagrangian takes the form
$\displaystyle K=z_{Q}(\Lambda_{Q},\mu){\rm
Tr}\Bigl{(}Q^{\dagger}Q+Q\rightarrow{\overline{Q}}\Bigr{)},\quad
W=-\frac{2\pi}{\alpha(\mu,\Lambda_{Q})}S+W_{Q}\,,$ $\displaystyle
W_{Q}=m_{Q}{\rm Tr}({\overline{Q}}Q)-\frac{1}{2\mu_{\Phi}}\Biggl{(}{\rm
Tr}\,({\overline{Q}}Q)^{2}-\frac{1}{N_{c}}\Bigl{(}{\rm
Tr}\,{\overline{Q}}Q\Bigr{)}^{2}\Biggr{)}.$ (1.3)
The Konishi anomalies for the Lagrangian (1.3) look as
$\displaystyle\langle S\rangle=\langle{\overline{Q}}_{i}\frac{\partial
W_{Q}}{\partial{\overline{Q}}_{i}}\rangle=m_{Q}\langle{\overline{Q}}_{i}Q_{i}\rangle-\frac{1}{\mu_{\Phi}}\Biggl{(}\sum_{j}\langle{\overline{Q}}_{i}Q_{j}\rangle\langle{\overline{Q}}_{j}Q_{i}\rangle-\frac{1}{N_{c}}\langle{\overline{Q}}_{i}Q_{i}\rangle\langle{\rm
Tr}\,{\overline{Q}}Q\rangle\Biggr{)}=$
$\displaystyle=\langle{\overline{Q}}_{i}Q_{i}\rangle\Biggl{[}\,m_{Q}-\frac{1}{\mu_{\Phi}}\Biggl{(}\langle{\overline{Q}}_{i}Q_{i}\rangle-\frac{1}{N_{c}}\langle{\rm
Tr}\,{\overline{Q}}Q\rangle\Biggr{)}\Biggr{]}\,,\quad
i=1\,...\,N_{F}\,,\quad\langle
S\rangle=\langle\frac{\lambda\lambda}{32\pi^{2}}\rangle\,,$ (1.4)
$\displaystyle 0=\langle{\overline{Q}}_{i}\frac{\partial
W_{Q}}{\partial{\overline{Q}}_{i}}-{\overline{Q}}_{j}\frac{\partial
W_{Q}}{\partial{\overline{Q}}_{j}}\rangle=\langle{\overline{Q}}_{i}Q_{i}-{\overline{Q}}_{j}Q_{j}\rangle\Biggl{[}\,m_{Q}-\frac{1}{\mu_{\Phi}}\Biggl{(}\langle{\overline{Q}}_{i}Q_{i}+{\overline{Q}}_{j}Q_{j}\rangle-\frac{1}{N_{c}}\langle{\rm
Tr}\,{\overline{Q}}Q\rangle\Biggr{)}\Biggr{]}\,.$
It is most easily seen from (1.4) that there are only two types of vacua : a)
the vacua with the unbroken flavor symmetry, b) the vacua with the
spontaneously broken flavor symmetry, and the breaking is of the type
$U(N_{F})\rightarrow U(n_{1})\times U(n_{2})$ only. In these vacua
$\displaystyle\langle{\overline{Q}}_{1}Q_{1}+{\overline{Q}}_{2}Q_{2}\rangle-\frac{1}{N_{c}}{\rm
Tr}\,\langle{\overline{Q}}Q\rangle=m_{Q}\mu_{\Phi},\quad\langle
S\rangle=\frac{1}{\mu_{\Phi}}\langle{\overline{Q}}_{1}Q_{1}\rangle\langle{\overline{Q}}_{2}Q_{2}\rangle,\quad\langle{\overline{Q}}_{1}Q_{1}\rangle\neq\langle{\overline{Q}}_{2}Q_{2}\rangle\,,$
$\displaystyle\langle m^{\rm
tot}_{Q,1}\rangle=m_{Q}-\langle\Phi_{1}\rangle=\frac{\langle{\overline{Q}}_{2}Q_{2}\rangle}{\mu_{\Phi}},\quad\langle
m^{\rm
tot}_{Q,2}\rangle=m_{Q}-\langle\Phi_{2}\rangle=\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle}{\mu_{\Phi}}\,.$
(1.5)
The values of quark condensates
$\langle{\overline{Q}}_{j}Q_{i}\rangle=\delta_{ij}\langle{\overline{Q}}_{i}Q_{i}\rangle$
in different vacua look as follows, see section 3 in [1].
1.1.1 The region $\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm
o}\,,\,\,\,\mu_{\Phi,\rm
o}=\Lambda_{Q}\Bigl{(}\Lambda_{Q}/m_{Q}\Bigr{)}^{(2N_{c}-N_{F})/N_{c}}$
a) Unbroken flavor symmetry, $(2N_{c}-N_{F})$ L - vacua
$\displaystyle\langle{\overline{Q}}Q\rangle_{L}\sim\Lambda_{Q}^{2}\Biggl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Biggr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\ll\Lambda_{Q}^{2}\,.$
(1.6)
b) Unbroken flavor symmetry, ${\overline{N}}_{c}=(N_{F}-N_{c})$ S - vacua
$\displaystyle\langle{\overline{Q}}Q\rangle_{S}\simeq-\frac{{\overline{N}}_{c}}{N_{c}}\,m_{Q}\mu_{\Phi}\ll\Lambda_{Q}^{2}\,.$
(1.7)
c) Broken flavor symmetry, $U(N_{F})\rightarrow U(n_{1})\times
U(n_{2}),\,n_{1}\leq[N_{F}/2]$. In these, there are $n_{1}$ equal condensates
$\langle{\overline{Q}}_{1}Q_{1}\rangle$ and $n_{2}\geq n_{1}$ equal
condensates
$\langle{\overline{Q}}_{2}Q_{2}\rangle\neq\langle{\overline{Q}}_{1}Q_{1}\rangle$.
At $n_{2}\lessgtr N_{c}$, including $n_{1}=n_{2}=N_{F}/2$ for even $N_{F}$ but
excluding $n_{2}=N_{c}$ , there are
$(2N_{c}-N_{F}){\overline{C}}^{\,n_{1}}_{N_{F}}$ L - type vacua with
$\displaystyle(1-\frac{n_{1}}{N_{c}})\langle{\overline{Q}}_{1}Q_{1}\rangle_{Lt}\simeq-(1-\frac{n_{2}}{N_{c}})\langle{\overline{Q}}_{2}Q_{2}\rangle_{Lt}\sim\Lambda_{Q}^{2}\Biggl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Biggr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}},$
(1.8) $\displaystyle N_{F}=n_{1}+n_{2}\,,\quad n_{1}\leq[N_{F}/2]\,,\quad
n_{2}\geq[N_{F}/2]\,,\quad
C^{\,n_{1}}_{N_{F}}=\frac{N_{F}\,!}{n_{1}\,!\,n_{2}\,!}\,,$
${\overline{C}}^{\,n_{1}}_{N_{F}}$ differ from the standard
$C^{\,n_{1}}_{N_{F}}$ only by ${\overline{C}}^{\,n_{1}={\rm k}}_{N_{F}=2{\rm
k}}=C^{\,n_{1}={\rm k}}_{N_{F}=2{\rm k}}/2$.
d) Broken flavor symmetry. At $n_{2}>N_{c}$ there are also
$(n_{2}-N_{c})C^{\,n_{1}}_{N_{F}}$ $\rm br2$ \- vacua with
$\displaystyle\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}\simeq\frac{N_{c}}{N_{c}-n_{2}}\,m_{Q}\mu_{\Phi}\,,\quad\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{n_{2}-N_{c}}}\,,$
(1.9) $\displaystyle\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}}{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{c}}{n_{2}-N_{c}}}\ll 1.$
e) Broken flavor symmetry. At $n_{1}={\overline{N}}_{c},\,n_{2}=N_{c}$ there
are $(2N_{c}-N_{F})C^{\,n_{1}={\overline{N}}_{c}}_{N_{F}}$ ”special” vacua
with
$\displaystyle\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
spec}=\frac{N_{c}}{2N_{c}-N_{F}}\,m_{Q}\mu_{\Phi}\,,\quad\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
spec}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\,,$
(1.10) $\displaystyle\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}}{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{c}}{2N_{c}-N_{F}}}\ll 1\,.$
1.1.2 The region $\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}\,,\,\,\,\mu_{\Phi,\rm
o}=\Lambda_{Q}\Bigl{(}\Lambda_{Q}/m_{Q}\Bigr{)}^{(2N_{c}-N_{F})/N_{c}}$
a) Unbroken flavor symmetry, $N_{c}$ SQCD - vacua
$\displaystyle\langle{\overline{Q}}Q\rangle_{SQCD}\sim\Lambda_{Q}^{2}\Biggl{(}\frac{m_{Q}}{\Lambda_{Q}}\Biggr{)}^{{\overline{N}}_{c}/N_{c}}\,.$
(1.11)
b) Broken flavor symmetry. At all values of $n_{2}\lessgtr N_{c}$, including
$n_{1}=n_{2}=N_{F}/2$ at even $N_{F}$ and the ”special” vacua with
$n_{1}={\overline{N}}_{c},\,n_{2}=N_{c}$, there are
$(N_{c}-n_{1}){\overline{C}}^{\,n_{1}}_{N_{F}}$ $\rm br1$ \- vacua with
$\displaystyle\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br1}\simeq\frac{N_{c}}{N_{c}-n_{1}}\,m_{Q}\mu_{\Phi}\,,\quad\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br1}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{1}}{N_{c}-n_{1}}}\Bigl{(}\frac{\Lambda_{Q}}{m_{Q}}\Bigr{)}^{\frac{N_{c}-n_{2}}{N_{c}-n_{1}}}\,,$
(1.12) $\displaystyle\frac{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br1}}{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br1}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{N_{c}-n_{1}}}\ll 1\,.$
c) Broken flavor symmetry. At $n_{2}<N_{c}$, including $n_{1}=n_{2}=N_{F}/2$
at even $N_{F}$, there are
$(N_{c}-n_{2}){\overline{C}}^{\,n_{2}}_{N_{F}}=(N_{c}-n_{2}){\overline{C}}^{\,n_{1}}_{N_{F}}$
$\rm br2$ \- vacua with
$\displaystyle\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}\simeq\frac{N_{c}}{N_{c}-n_{2}}\,m_{Q}\mu_{\Phi}\,,\quad\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{2}}{N_{c}-n_{2}}}\Bigl{(}\frac{\Lambda_{Q}}{m_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{N_{c}-n_{2}}}\,,$
(1.13) $\displaystyle\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}}{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{N_{c}-n_{2}}}\ll 1\,.$
The perturbative running mass of fions $\mu^{\rm
pert}_{\Phi}(\mu)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu\ll\Lambda_{Q})\ll\mu_{\Phi}$
decreases with the diminishing scale $\mu\ll\Lambda_{Q}$ because
$z_{\Phi}(\Lambda_{Q},\mu\ll\Lambda_{Q})\gg 1$. Hence, if nothing prevents,
the fions become effectively massless and dynamically relevant at scales
$\mu<\mu^{\rm conf}_{o}$, see section 4 in [1],
$\displaystyle\mu^{\rm
conf}_{o}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3(2N_{c}-N_{F})}\,.$
(1.14)
The RG evolution of $\mu^{\rm
pert}_{\Phi}(\mu)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu)$ becomes frozen at
$\mu<\mu_{H}$, where $\mu_{H}<\Lambda_{Q}$ is the largest mass in the quark-
gluon sector, $\mu^{\rm
pert}_{\Phi}(\mu<\mu_{H})=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu=\mu_{H})$
(within the dynamical scenario $\\#2$ used in this paper, this $\mu_{H}$ may
be either the quark pole mass $m_{Q}^{\rm pole}$ or the gluon mass $\mu_{\rm
gl}$ if quarks are higgsed). Hence, if $\mu_{H}\gg\mu^{\rm conf}_{o}$, then
fions remain too heavy and dynamically irrelevant at all scales
$\mu<\Lambda_{Q}$. But if $\mu_{H}\ll\mu^{\rm conf}_{o}$, then there is a pole
in the fion propagator at the momentum $p=\mu^{\rm conf}_{o}$, so that there
appears the second generation of fions with the pole masses $\mu_{2}^{\rm
pole}(\Phi)=\mu^{\rm conf}_{o}$. Moreover, because the limiting low energy
value of the perturbative running mass of fions is $\mu^{\rm
pert}(\Phi)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu_{H})\ll\mu_{H}$ at
$\mu_{H}\ll\mu^{\rm conf}_{o}$, and if this $\mu^{\rm pert}(\Phi)$ is the
largest contribution to the fion mass, then there is also a pole in the fion
propagator at the momentum $p=\mu^{\rm pert}(\Phi)$, so that there appears the
third generation of fions with the pole masses $\mu_{3}^{\rm
pole}(\Phi)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu_{H})\ll\mu_{H}$.
In vacua with broken flavor symmetry the above reasonings have to be applied
separately to $\Phi_{11},\,\Phi_{22}$ and $\Phi_{12}+\Phi_{21}$.
1.2. Dual $\mathbf{d\Phi}$ \- theory
In parallel with the direct $\mathbf{\Phi}$ \- theory we consider at
$3N_{c}/2<N_{F}<2N_{c}$ also the Seiberg dual variant [6, 7] (the
$d\mathbf{\Phi}$ \- theory), with the dual Lagrangian at $\mu=\Lambda_{q}$ [1]
$\displaystyle K={\rm Tr}\,\Bigl{(}\Phi^{\dagger}\Phi\Bigr{)}+{\rm
Tr}\Bigl{(}q^{\dagger}q+(q\rightarrow\overline{q})\,\Bigr{)}+{\rm
Tr}\,\Bigl{(}\frac{M^{\dagger}M}{\mu_{2}^{2}}\Bigr{)}\,,\quad
W=\,-\,\frac{2\pi}{\overline{\alpha}(\mu=\Lambda_{q})}\,{\overline{s}}+{\overline{W}}_{M}+W_{q}\,,$
$\displaystyle{\overline{W}}_{M}=\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]}+{\rm Tr}\,M(m_{Q}-\Phi),\quad
W_{q}=-\,\frac{1}{\mu_{1}}\,\rm{Tr}\Bigl{(}{\overline{q}}\,M\,q\Bigr{)}\,.$
(1.15)
Here : the number of dual colors is
${\overline{N}}_{c}=(N_{F}-N_{c}),\,{\rm\overline{b}}_{\rm
o}=3{\overline{N}}_{c}-N_{F}$, and $M_{ij}$ are the $N_{F}^{2}$ elementary
mion fields,
${\overline{a}}(\mu)={\overline{N}}_{c}{\overline{\alpha}}(\mu)/2\pi={\overline{N}}_{c}{\overline{g}}^{2}(\mu)/8\pi^{2}$
is the dual running gauge coupling,
${\overline{s}}=-{\rm\overline{w}}^{b}_{\beta}{\rm\overline{w}}^{b,\,\beta}/32\pi^{2}$,
${\rm\overline{w}}^{b}_{\beta}$ is the dual gluon field strength. The gluino
condensates of the direct and dual theories are matched,
$\langle{-\,\overline{s}}\rangle=\langle S\rangle=\Lambda_{YM}^{3}$.
By definition, $\mu=\Lambda_{q}$ is such a scale that, at
$0<{\rm\overline{b}}_{\rm o}/{\overline{N}}_{c}\ll 1$, the dual theory already
entered sufficiently deep into the conformal regime, i.e. both the gauge and
Yukawa couplings, ${\overline{a}}(\mu=\Lambda_{q})$ and
$a_{f}(\mu=\Lambda_{q})={\overline{N}}_{c}f^{2}(\mu=\Lambda_{q})/8\pi^{2}$,
are already close to their small fixed point values,
$\overline{\delta}=({\overline{a}}_{*}-{\overline{a}}(\mu=\Lambda_{q})/{\overline{a}}_{*}\ll
1$, and similarly for $a_{f}(\mu=\Lambda_{q})$. We take
$|\Lambda_{q}|=\Lambda_{Q}$ for simplicity (because this does not matter
finally, see the appendix in [2] for more details). The condensates $\langle
M_{ji}(\mu=\Lambda_{Q})\rangle=\langle{\overline{Q}}_{j}Q_{i}(\mu=\Lambda_{Q})\rangle$
can always be matched at $\mu=|\Lambda_{q}|=\Lambda_{Q}$, at the appropriate
choice of $\mu_{1}$ in (1.15).
At $3/2<N_{F}/N_{c}<2$ this dual theory can be taken as UV free at
$\mu\gg\Lambda_{Q}$ and this requires that its Yukawa coupling at
$\mu=\Lambda_{Q},\,f(\mu=\Lambda_{Q})=\mu_{2}/\mu_{1}$, cannot be larger than
its gauge coupling ${\overline{g}}(\mu=\Lambda_{Q})$, i.e.
$\mu_{2}/\mu_{1}\lesssim 1$. The same requirement to the value of the Yukawa
coupling follows from the conformal behavior of this theory at
$3/2<N_{F}/N_{c}<2$ and $\mu<\Lambda_{Q}$, i.e. $a_{f}(\mu=\Lambda_{Q})\simeq
a^{*}_{f}\sim{\overline{a}}_{*}=O({\rm\overline{b}}_{\rm o}/N_{F})$. We
consider below this dual theory at $\mu\leq\Lambda_{Q}$ only where it claims
to be equivalent to the direct $\Phi$ \- theory.
As was explained in [2] (see section 4 and appendix therein, the fixed point
value of the dual gauge coupling at ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ is
${\overline{a}}_{*}\simeq 7{\rm\overline{b}}_{\rm o}/3{\overline{N}}_{c}$
[11]), one has to take
$\displaystyle\mu_{1}\equiv Z_{q}\Lambda_{Q}\,,\quad
Z_{q}\sim\exp\Bigl{\\{}-\frac{1}{3{\overline{a}}_{*}}\Bigr{\\}}\sim\exp\Bigl{\\{}-\frac{{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\ll 1\,,\quad\frac{{\rm\overline{b}}_{\rm o}}{N_{F}}\ll 1$ (1.16)
to match the gluino condensates in the direct and dual theories at
$0<{\rm\overline{b}}_{\rm o}/N_{F}\ll 1$. The value of $Z_{q}$ in (1.16) is
valid with the exponential accuracy only, i.e. powers of
${\rm\overline{b}}_{\rm o}/{\overline{N}}_{c}\ll 1$ are ignored and only
powers of $Z_{q}$ are tracked explicitly. This exponential accuracy is
sufficient for our purposes. Besides, the small number ${\rm\overline{b}}_{\rm
o}/{\overline{N}}_{c}\ll 1$ does not compete in any way with the much smaller
parameter $m_{Q}/\Lambda_{Q}\ll 1$, see [2]. At ${\rm\overline{b}}_{\rm
o}/N_{F}=O(1)$ one has to replace $Z_{q}\rightarrow 1$ in (1.16). Then, from
$\mu_{2}\lesssim\mu_{1}$ and (1.16), we take
$\mu_{2}=\mu_{1}=Z_{q}\Lambda_{Q}$ in (1.15) everywhere below.
The fields $\Phi$ remain always too heavy and dynamically irrelevant in this
$d\mathbf{\Phi}$ \- theory, so that they can be integrated out once and
forever and, finally, we write the Lagrangian of the dual $d\mathbf{\Phi}$
theory at $\mu=\Lambda_{Q}$ in the form
$\displaystyle K={\rm
Tr}\Bigl{(}q^{\dagger}q+(q\rightarrow\overline{q})\Bigr{)}+{\rm
Tr}\,\frac{M^{\dagger}M}{Z_{q}^{2}\Lambda_{Q}^{2}}\,,\quad
W=\,-\,\frac{2\pi}{\overline{\alpha}(\mu=\Lambda_{Q})}\,{\overline{s}}+W_{M}+W_{q}\,,$
$\displaystyle W_{M}=m_{Q}{\rm Tr}\,M-\frac{1}{2\mu_{\Phi}}\Biggl{[}{\rm
Tr}\,(M^{2})-\frac{1}{N_{c}}({\rm Tr}\,M)^{2}\Biggr{]}\,,\quad
W_{q}=-\,\frac{1}{Z_{q}\Lambda_{Q}}\,\rm{Tr}\Bigl{(}{\overline{q}}\,M\,q\Bigr{)}\,.$
(1.17)
From (1.17), the Konishi anomalies for the $i$-th flavor, ${\it
i}=1\,...\,N_{F}$, look here as
$\displaystyle\langle M_{i}\rangle\langle N_{i}\rangle=Z_{q}\Lambda_{Q}\langle
S\rangle\,,\quad\frac{\langle
N_{i}\rangle}{Z_{q}\Lambda_{Q}}=m_{Q}-\frac{1}{\mu_{\Phi}}\Bigl{(}\langle
M_{i}-\frac{1}{N_{c}}{\rm Tr}\,M\rangle\Bigr{)}=\langle m_{Q,i}^{\rm
tot}\rangle\,,$ (1.18) $\displaystyle\langle
N_{i}\rangle\equiv\langle{\overline{q}}_{i}q_{i}(\mu=\Lambda_{Q})\rangle\,.$
In vacua with the broken flavor symmetry these can be rewritten as
$\displaystyle\langle M_{1}+M_{2}\rangle-\frac{1}{N_{c}}{\rm Tr}\,\langle
M\rangle=m_{Q}\mu_{\Phi},\quad\langle S\rangle=\frac{1}{\mu_{\Phi}}\langle
M_{1}\rangle\langle M_{2}\rangle,\quad\langle M_{1}\rangle\neq\langle
M_{2}\rangle\,,$ $\displaystyle\frac{\langle
N_{1}\rangle}{Z_{q}\Lambda_{Q}}=\frac{\langle S\rangle}{\langle
M_{1}\rangle}=\frac{\langle
M_{2}\rangle}{\mu_{\Phi}}=m_{Q}-\frac{1}{\mu_{\Phi}}\Bigl{(}\langle
M_{1}\rangle-\frac{1}{N_{c}}{\rm Tr}\,\langle M\rangle\Bigr{)}=\langle m^{\rm
tot}_{Q,1}\rangle\,,$ (1.19) $\displaystyle\frac{\langle
N_{2}\rangle}{Z_{q}\Lambda_{Q}}=\frac{\langle S\rangle}{\langle
M_{2}\rangle}=\frac{\langle
M_{1}\rangle}{\mu_{\Phi}}=m_{Q}-\frac{1}{\mu_{\Phi}}\Bigl{(}\langle
M_{2}\rangle-\frac{1}{N_{c}}{\rm Tr}\,\langle M\rangle\Bigr{)}=\langle m^{\rm
tot}_{Q,2}\rangle\,.$
The multiplicities of vacua are the same in the direct and dual theories [1].
Our purpose in this paper is to calculate mass spectra in the two above
theories, $\mathbf{\Phi}$ and $\mathbf{d\Phi}$. At present, unfortunately, to
calculate the mass spectra in ${\cal N}=1$ SQCD-like theories one has to rely
on a definite dynamical scenario. Two different scenarios, $\\#1$ and $\\#2$,
have been considered in [8, 9] and [2] and the mass spectra were calculated in
the standard direct ${\cal N}=1$ SQCD with the superpotential $W={\rm
Tr}\,(\,{\overline{Q}}m_{Q}Q)$ and in its dual variant [6, 7]. It was shown
that the direct theory and its Seiberg dual variant are not equivalent in both
scenarios. In this paper we calculate the mass spectra in the $\mathbf{\Phi}$
and $\mathbf{d\Phi}$ theories within the scenario $\\#2$. As will be shown
below, similarly to the standard SQCD, the use of the small parameter
${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ allows to trace explicitly the
parametric differences in mass spectra of the direct and dual theories.
## 2 Direct theory. Unbroken flavor symmetry
2.1 L - vacua
In these $(2N_{c}-N_{F})$ vacua the current quark mass at
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$ looks as, see (1.5),
$\displaystyle\langle m^{\rm tot}_{Q}\rangle_{L}\equiv\langle m^{\rm
tot}_{Q}(\mu=\Lambda_{Q})\rangle_{L}=m_{Q}-\langle\Phi\rangle_{L}=m_{Q}+\frac{{\overline{N}}_{c}}{N_{c}}\frac{\langle{\overline{Q}}Q\rangle_{L}}{\mu_{\Phi}}\,,$
(2.1)
$\displaystyle\langle{\overline{Q}}Q\rangle_{L}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\gg
m_{Q}\mu_{\Phi},\quad\langle m^{\rm
tot}_{Q}\rangle_{L}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{2N_{c}-N_{F}}}\,,$
$\displaystyle m^{\rm pole}_{Q,\,L}=\frac{\langle m^{\rm
tot}_{Q}\rangle_{L}}{z_{Q}(\Lambda_{Q},m^{\rm
pole}_{Q,\,L})}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\sim\Lambda_{YM}^{(L)}\,,\quad
z_{Q}(\Lambda_{Q},\mu\ll\Lambda_{Q})\sim\Bigl{(}\frac{\mu}{\Lambda_{Q}}\Bigr{)}^{{\rm
b_{o}}/N_{F}}\ll 1\,.$
We compare $m^{\rm pole}_{Q,\,L}$ with the gluon mass due to possible higgsing
of quarks. This last looks as
$\displaystyle\mu_{{\rm gl},\,L}^{2}\sim\Bigl{(}a_{*}\sim
1\Bigr{)}z_{Q}(\Lambda_{Q},\mu_{{\rm
gl},\,L})\langle{\overline{Q}}Q\rangle_{L}\quad\rightarrow\quad\mu_{{\rm
gl},\,L}\sim m^{\rm pole}_{Q,\,L}\sim\Lambda_{YM}^{(L)}=\langle
S\rangle^{1/3}_{L}\,.$ (2.2)
Hence, qualitatively, the situation is the same as in the standard SQCD [2].
And one can use here the same reasonings, see the footnote 3 in [2]. In the
case considered, there are only $(2N_{c}-N_{F})$ these isolated L vacua with
unbroken flavor symmetry. If quarks were higgsed in these L vacua, then the
flavor symmetry will be necessary broken spontaneously due to the rank
restriction because $N_{F}>N_{c}$, and there will appear the genuine exactly
massless Nambu-Goldstone fields $\Pi$ (pions), so that there will be a
continuous family of non-isolated vacua. This is ”the standard point of
tension” in the dynamical scenario $\\#2$, see [2]. Therefore, as in [2], to
save this scenario $\\#2$, here and everywhere below in a similar situations
we will assume that $\mu_{\rm gl}=m^{\rm pole}_{Q,\,L}/(\rm several)$, so that
quarks are not higgsed but are in the HQ (heavy quark) phase and are confined.
Otherwise, this scenario $\\#2$ should be rejected at all.
Therefore (see sections 3, 4 in [2]), after integrating out all quarks as
heavy ones at $\mu<m^{\rm pole}_{Q,\,L}$ and then all $SU(N_{c})$ gluons at
$\mu<\Lambda_{YM}^{(L)}=m^{\rm pole}_{Q,\,L}/(\rm several)$ via the Veneziano-
Yankielowicz (VY) procedure [10], we obtain the Lagrangian of fions
$\displaystyle K=z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q,\,L}){\rm
Tr}\,(\Phi^{\dagger}\Phi)\,,\quad z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,\,L})\sim\frac{1}{z^{2}_{Q}(\Lambda_{Q},m^{\rm
pole}_{Q,\,L})}\sim\Bigl{(}\frac{\Lambda_{Q}}{m^{\rm
pole}_{Q,\,L}}\Bigr{)}^{2{\rm b_{o}}/N_{F}}\gg 1\,,$ (2.3) $\displaystyle
W=N_{c}S+\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]}\,,\quad S=\Bigl{(}\Lambda_{Q}^{{\rm b_{o}}}\det
m^{\rm tot}_{Q,L}\Bigr{)}^{1/N_{c}}\,,\quad m^{\rm tot}_{Q,L}=(m_{Q}-\Phi)\,,$
and one has to choose the L - vacua in (2.3).
There are two contributions to the mass of fions in (2.3), the perturbative
one from the term $\sim\mu_{\Phi}\Phi^{2}$ in $W$ and the non-perturbative one
from $\sim S$, and both are parametrically the same,
$\sim\Lambda_{YM}^{(L)}\gg m_{Q}$. Therefore,
$\displaystyle\mu(\Phi)\sim\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,\,L})}\sim m^{\rm pole}_{Q,\,L}\sim\Lambda_{YM}^{(L)}\,.$ (2.4)
Besides, see (1.14), because in these L - vacua
$\displaystyle\mu^{\rm
conf}_{o}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\sim
m^{\rm pole}_{Q,\,L}\sim\Lambda_{YM}^{(L)}\,,$ (2.5)
and fions are dynamically irrelevant at $\mu^{\rm conf}_{o}<\mu<\Lambda_{Q}$
and can become relevant only at the scale $\mu<\mu^{\rm conf}_{o}$, it remains
unclear whether there is a pole in the fion propagators at $p\sim\mu^{\rm
conf}_{o}\sim m^{\rm pole}_{Q,\,L}$. May be yes but maybe not, see section 4
in [2].
On the whole for the mass spectrum in these L - vacua. The quarks
${\overline{Q}},Q$ are confined and strongly coupled here, the coupling being
$a_{*}\sim 1$. Parametrically, there is only one scale
$\sim\Lambda_{YM}^{(L)}$ in the mass spectrum at
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$. And there is no parametrical
guaranty that there is the second generation of fions with the pole masses
$\mu_{2}^{\rm pole}(\Phi)\sim\Lambda_{YM}^{(L)}$.
The condensate $\langle{\overline{Q}}Q\rangle_{L}$ and the quark pole mass
$m^{\rm pole}_{Q,\,L}$ become frozen at their SQCD values at
$\mu_{\Phi}\gg\mu_{\Phi,\rm
o},\,\langle{\overline{Q}}Q\rangle_{SQCD}\sim\Lambda_{Q}^{2}(m_{Q}/\Lambda_{Q})^{{\overline{N}}_{c}/N_{c}},\,\,m^{\rm
pole}_{SQCD}\sim\Lambda_{YM}^{(SQCD)}\sim\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{N_{F}/3N_{c}}$
[2], while $\mu_{\Phi}$ increases and $\mu^{\rm conf}_{o}\ll m^{\rm
pole}_{Q,SQCD}$ decreases, see (1.14). Hence, the perturbative contribution
$\sim\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q,\,L})\gg m^{\rm
pole}_{Q,SQCD}$ to the fion mass becomes dominant at
$\mu_{\Phi}\gg\mu_{\Phi,\rm o}$ and the fion fields will be dynamically
irrelevant at $\mu<\Lambda_{Q}$.
Finally, it is worth noting for what follows that, unlike the dual theory, in
all vacua of the direct theory the mass spectra remain parametrically the same
at $\,{\rm\overline{b}}_{\rm o}/N_{F}=O(1)$ or $\,{\rm\overline{b}}_{\rm
o}/N_{F}\ll 1$.
2.2 S - vacua
In these ${\overline{N}}_{c}$ vacua the quark mass at
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$ looks as, see (1.7),
$\displaystyle\frac{\langle m^{\rm
tot}_{Q}(\mu=\Lambda_{Q})\rangle_{S}}{\Lambda_{Q}}\sim\frac{\langle
S\rangle_{S}}{\Lambda_{Q}\langle{\overline{Q}}Q\rangle_{S}}\sim\Bigl{(}\frac{\langle{\overline{Q}}Q\rangle_{S}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\sim\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\,,$
$\displaystyle m^{\rm
pole}_{Q,\,S}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\sim\Lambda_{YM}^{(S)}=\langle
S\rangle^{1/3}_{S},\quad\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}\,.$ (2.6)
This has to be compared with the gluon mass due to possible higgsing of quarks
$\displaystyle\mu_{{\rm gl},\,S}^{2}\sim(a_{*}\sim
1)z_{Q}(\Lambda_{Q},\mu_{{\rm
gl},\,S})\langle{\overline{Q}}Q\rangle_{S}\,\rightarrow\,\mu_{{\rm
gl},\,S}\sim m^{\rm
pole}_{Q,\,S}\sim\Lambda_{YM}^{(S)},\,z_{Q}(\Lambda_{Q},\mu_{{\rm
gl},\,S})\sim\Bigl{(}\frac{\mu_{{\rm
gl},\,S}}{\Lambda_{Q}}\Bigr{)}^{\frac{{\rm b_{o}}}{N_{F}}}.$ (2.7)
For the same reasons as in previous section, it is clear that quarks will not
be higgsed in these vacua at $N_{F}>N_{c}$ (as otherwise the flavor symmetry
will be broken spontaneously). Hence, as in [2], we assume here also that the
pole mass of quarks is the largest physical mass, $\mu_{H}=m^{\rm
pole}_{Q,\,S}=(\rm several)\mu_{{\rm gl},\,S}$.
But, in contrast with the L - vacua, the fion fields become dynamically
relevant in these S - vacua at scales $\mu<\mu^{\rm conf}_{o}$, see (1.14) and
section 4 in [1], if
$\displaystyle\mu^{\rm
conf}_{o}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{f}}{3(2N_{c}-N_{F})}}\gg
m^{\rm pole}_{Q,\,S}\quad\rightarrow\quad{\rm
i.e.\,\,at}\quad\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}\,.$ (2.8)
Therefore, there is the second generation of $N_{F}^{2}$ fions with the pole
masses
$\displaystyle\mu_{2}^{\rm pole}(\Phi)\sim\mu_{o}^{\rm conf}\gg m^{\rm
pole}_{Q,\,S}\sim\Lambda_{YM}^{(S)}\,.$ (2.9)
Nevertheless [1], the theory remains in the conformal regime and the quark and
fion anomalous dimensions remain the same in the whole range of $m^{\rm
pole}_{Q,\,S}<\mu<\Lambda_{Q}$ of scales, but fions become effectively
massless at $\mu<\mu^{\rm conf}_{o}$ and begin to contribute to the ’t Hooft
triangles.
The RG evolution of the quark and fion fields becomes frozen at scales
$\mu<m^{\rm pole}_{Q,\,S}$ because the heavy quarks decouple. Proceeding as
before, i.e. integrating out first all quarks as heavy ones at $\mu<m^{\rm
pole}_{Q,\,S}=(\rm several)\Lambda_{YM}^{(S)}$ and then all $SU(N_{c})$ gluons
at $\mu<\Lambda_{YM}^{(S)}$, one obtains the Lagrangian of fions as in (2.3),
with a replacement $z_{Q}(\Lambda_{Q},m^{\rm pole}_{Q,\,L})\rightarrow
z_{Q}(\Lambda_{Q},m^{\rm pole}_{Q,\,S})$ (and the S-vacua have to be chosen
therein).
Because fions became relevant at $m^{\rm pole}_{Q,\,S}\ll\mu\ll\mu^{\rm
conf}_{o}$, one could expect that their running mass will be much smaller than
$m^{\rm pole}_{Q,\,S}$. This is right, but only for $\mu^{\rm
pert}_{\Phi}\sim\mu_{\Phi}/z_{Q}(\Lambda_{Q},m^{\rm pole}_{Q,\,S})\ll m^{\rm
pole}_{Q,\,S}$. But there is also additional non-perturbative contribution to
the fion mass originating from the region of scales $\mu\sim m^{\rm
pole}_{Q,\,S}$ and it is dominant in these S - vacua,
$\displaystyle\mu(\Phi)\sim\frac{1}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,\,S})}\,\frac{\langle S\rangle_{S}}{\langle m^{\rm
tot}_{Q}\rangle^{2}_{S}}\sim m^{\rm pole}_{Q,\,S}\,,\quad
z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,\,S})\sim\Bigl{(}\frac{\Lambda_{Q}}{m^{\rm
pole}_{Q,\,S}}\Bigr{)}^{2{\rm b_{o}}/N_{F}}\,.$ (2.10)
Therefore, despite the fact that the fions are definitely dynamically relevant
in the range of scales $m^{\rm pole}_{Q,\,S}\ll\mu\ll\mu^{\rm
conf}_{o}\ll\Lambda_{Q}$ at $\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$,
whether there is the third generation of fions, i.e. whether there is a pole
in the fion propagator at $p=\mu_{3}^{\rm pole}(\Phi)\sim m^{\rm
pole}_{Q,\,S}\sim\Lambda_{YM}^{(S)}$ remains unclear.
On the whole for the mass spectra in these S - vacua. The largest are the
masses of the second generation fions, $\mu_{2}^{\rm
pole}(\Phi)\sim\Lambda_{Q}\Bigl{(}\Lambda_{Q}/\mu_{\Phi}\Bigr{)}^{N_{F}/3(2N_{c}-N_{F})}\gg
m^{\rm pole}_{Q,\,S}$. The scale of all other masses is $\sim m^{\rm
pole}_{Q,\,S}\sim\Lambda_{YM}^{(S)}$, see (2.6). There is no parametrical
guaranty that there is the third generation of fions with the pole masses
$\mu_{3}^{\rm pole}(\Phi)\sim\Lambda_{YM}^{(S)}$. May be yes, but maybe not.
The vacuum condensates $\langle{\overline{Q}}Q\rangle_{S}$ and $m^{\rm
pole}_{Q,\,S}$ evolve into their independent of $\mu_{\Phi}$ SQCD-values at
$\mu_{\Phi}\gg\mu_{\Phi,\rm o}$,
$\displaystyle\langle{\overline{Q}}Q\rangle_{SQCD}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{{\overline{N}}_{c}/N_{c}}\,,\quad
m^{\rm
pole}_{Q,SQCD}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\,,$
(2.11)
and the perturbative contribution $\sim\mu_{\Phi}/z_{Q}(\Lambda_{Q},m^{\rm
pole}_{Q,SQCD})$ to the fion mass becomes dominant. Hence, because $m^{\rm
pole}_{Q,SQCD}\gg\mu^{\rm conf}_{o}$, the fions fields become dynamically
irrelevant at all scales $\mu<\Lambda_{Q}$ when $\mu_{\Phi}\gg\mu_{\Phi,\rm
o}$.
## 3 Dual theory. Unbroken flavor symmetry
3.1 $\bf L$ \- vacua, ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$
Because $\Lambda_{Q}^{2}/\mu_{\Phi}\ll\Lambda_{Q}$, the mions are effectively
massless and dynamically relevant at $\mu\sim\Lambda_{Q}$ (and so in some
range of scales below $\Lambda_{Q}$). By definition, $\mu\sim\Lambda_{Q}$ is
such a scale that the dual theory already entered sufficiently deep the
conformal regime, i.e. the dual gauge coupling
${\overline{a}}(\mu<\Lambda_{Q})={\overline{N}}_{c}{\overline{\alpha}}(\mu<\Lambda_{Q})/2\pi$
is sufficiently close to its small frozen value,
$\overline{\delta}=[{\overline{a}}_{*}-{\overline{a}}(\mu\sim\Lambda_{Q})]/{\overline{a}}_{*}\ll
1$, and $\overline{\delta}$ is neglected everywhere below in comparison with 1
(and the same for the Yukawa coupling
${\overline{a}}_{f}={\overline{N}}_{c}{\overline{\alpha}}_{f}/2\pi$), see [2]
and appendix therein. The fixed point value of the dual gauge coupling is
${\overline{a}}_{*}\simeq 7{\rm\overline{b}}_{\rm o}/3{\overline{N}}_{c}\ll 1$
[11].
We recall also that the mion condensates are matched to the condensates of
direct quarks in all vacua, $\langle
M_{ij}(\mu=\Lambda_{Q})\rangle=\langle{\overline{Q}}_{j}Q_{i}(\mu=\Lambda_{Q})\rangle$
. Hence, in these L - vacua
$\displaystyle\langle
M\rangle_{L}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\,,\quad\langle
N\rangle_{L}=\frac{Z_{q}\Lambda_{Q}\langle S\rangle_{L}}{\langle
M\rangle_{L}}\sim
Z_{q}\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{2N_{c}-N_{F}}}\,,$
(3.1) $\displaystyle
Z_{q}\sim\exp\Bigl{\\{}-\frac{1}{3{\overline{a}}_{*}}\Bigr{\\}}\sim\exp\Bigl{\\{}-\frac{{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\ll 1\,.$
The current mass $\mu_{q,L}\equiv\mu_{q,L}(\mu=\Lambda_{Q})$ of dual quarks
${\overline{q}},\,q$ and their pole mass in these $(2N_{c}-N_{F})$ L - vacua
are, see (1.17),(3.1),
$\displaystyle\frac{\mu_{q,L}}{\Lambda_{Q}}=\frac{\langle
M\rangle_{L}}{Z_{q}\Lambda_{Q}^{2}}\sim\frac{1}{Z_{q}}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\,,\quad\mu^{\rm
pole}_{q,L}=\frac{\mu_{q,L}}{z_{q}(\Lambda_{Q},\mu^{\rm pole}_{q,L})}\,,\quad
z_{q}(\Lambda_{Q},\mu^{\rm pole}_{q,L})\sim\Bigl{(}\frac{\mu^{\rm
pole}_{q,L}}{\Lambda_{Q}}\Bigr{)}^{{\rm\overline{b}}_{\rm o}/N_{F}}\,,$
$\displaystyle\mu^{\rm
pole}_{q,L}\sim\Lambda_{Q}\Bigl{(}\frac{\mu_{q,L}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\sim\frac{\Lambda_{Q}}{Z_{q}}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\sim\frac{1}{Z_{q}}\Lambda_{YM}^{(L)}\gg\Lambda_{YM}^{(L)}\,,\quad\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm
o}\,,$ (3.2)
$\displaystyle\frac{\mu_{q,L}}{\Lambda_{Q}}\sim\frac{1}{Z_{q}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{N_{c}}}\,,\quad\mu^{\rm
pole}_{q,L}\sim\frac{\Lambda_{Q}}{Z_{q}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{F}}{3N_{c}}}\sim\frac{1}{Z_{q}}\Lambda_{YM}^{(SQCD)}\gg\Lambda_{YM}^{(SQCD)}\,,\quad\mu_{\Phi}\gg\mu_{\Phi,\rm
o}\,,$
while the gluon mass due to possible higgsing of dual quarks looks at
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$ as
$\displaystyle{\overline{\mu}}_{\rm gl,L}\sim\Bigl{[}{\overline{a}}_{*}\langle
N\rangle_{L}\,z_{q}(\Lambda_{Q},{\overline{\mu}}_{\rm gl})\Bigr{]}^{1/2}\sim
Z_{q}^{1/2}\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3(2N_{c}-N_{F})}\sim
Z_{q}^{3/2}\mu^{\rm pole}_{q,L}\ll\mu^{\rm pole}_{q,L}\,.$ (3.3)
Therefore, the dual quarks are definitely in the HQ phase in these L - vacua
at ${\rm\overline{b}}_{\rm o}/{\overline{N}}_{c}\ll 1$.
With decreasing scale the perturbative current mass $\mu_{M}(\mu)$ of mions
$\displaystyle\mu_{M}(\mu)\sim\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{\mu_{\Phi}z_{M}(\mu)}=\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{\mu_{\Phi}}\Bigl{(}\frac{\mu}{\Lambda_{Q}}\Bigr{)}^{2{\rm\overline{b}}_{\rm
o}/N_{F}}$ (3.4)
decreases but more slowly than the scale $\mu$ itself, because
$\gamma_{M}=-(2{\rm{\rm\overline{b}}_{\rm o}}/N_{F}),\,\,|\gamma_{M}|<1$ at
$3/2<N_{F}/N_{c}<2$ , and becomes frozen at $\mu<\mu^{\rm pole}_{q,L}$,
$\mu_{M}(\mu<\mu^{\rm pole}_{q,L})=\mu_{M}(\mu=\mu^{\rm pole}_{q,L})$.
After integrating out all dual quarks as heavy ones at $\mu<\mu^{\rm
pole}_{q,L}$ and then all $SU({\overline{N}}_{c})$ gluons at
$\mu<\Lambda_{YM}^{(L)}$ via the Veneziano-Yankielowicz (VY) procedure [10],
the Lagrangian of mions looks as
$\displaystyle K=\frac{z^{(L)}_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,L})}{Z_{q}^{2}\Lambda_{Q}^{2}}{\rm Tr}\,(M^{\dagger}M)\,,\quad
z^{(L)}_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,L})\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu^{\rm
pole}_{q,L}}\Bigr{)}^{2{\rm\overline{b}}_{\rm o}/N_{F}}\,,\quad
S=\Biggl{(}\,\frac{\det M}{\Lambda_{Q}^{{\rm
b_{o}}}}\,\Biggr{)}^{1/{\overline{N}}_{c}},$ $\displaystyle
W=-{\overline{N}}_{c}S+m_{Q}{\rm Tr}\,M-\frac{1}{2\mu_{\Phi}}\Biggl{[}{\rm
Tr}\,(M^{2})-\frac{1}{N_{c}}({\rm Tr}\,M)^{2}\Biggr{]}\,.$ (3.5)
There are two contributions to the mass of mions in (3.5), the perturbative
one from the term $\sim M^{2}/\mu_{\Phi}$ in $W$ and non-perturbative one from
$\sim S$. Both are parametrically the same and the total contribution looks as
$\displaystyle\mu^{\rm
pole}(M)\sim\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,L})\mu_{\Phi}}\sim
Z_{q}^{2}\Lambda_{YM}^{(L)}\ll\Lambda_{YM}^{(L)}\ll\mu^{\rm pole}_{q,L}\,,$
(3.6)
and this parametrical hierarchy guarantees that the mass $\mu^{\rm pole}(M)$
in (3.6) is indeed the pole mass of mions.
On the whole, the mass spectrum in these dual L - vacua looks as follows at
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$. a) There is a large number of
heaviest flavored hadrons made of weakly interacting and weakly confined (the
string tension being $\sqrt{\sigma}\sim\Lambda_{YM}^{(L)}\ll\mu^{\rm
pole}_{q,L}$) non-relativistic quarks ${\overline{q}},q$ with the pole masses
$\mu^{\rm
pole}_{q,L}/\Lambda_{YM}^{(L)}\sim\exp({\overline{N}}_{c}/7{\rm\overline{b}}_{\rm
o})\gg 1$. The mass spectrum of low-lying flavored mesons is Coulomb-like with
parametrically small mass differences
$\Delta\mu_{H}/\mu_{H}=O({\rm\overline{b}}_{\rm
o}^{2}/{\overline{N}}_{c}^{2})\ll 1$. b) A large number of gluonia made of
$SU({\overline{N}}_{c})$ gluons with the mass scale
$\sim\Lambda_{YM}^{(L)}\sim\Lambda_{Q}(\Lambda_{Q}/\mu_{\Phi})^{N_{F}/3(2N_{c}-N_{F})}$.
c) $N_{F}^{2}$ lightest mions with parametrically smaller masses $\mu^{\rm
pole}(M)/\Lambda_{YM}^{(L)}\sim\exp(-2{\overline{N}}_{c}/7{\rm\overline{b}}_{\rm
o})\ll 1$.
At $\mu_{\Phi}\gg\mu_{\Phi,\rm o}$ these L - vacua evolve into the vacua of
the dual SQCD theory (dSQCD), see section 4 in [2].
3.2 $\bf S$ \- vacua, ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$
The current mass $\mu_{q,S}\equiv\mu_{q,S}(\mu=\Lambda_{Q})$ of dual quarks
${\overline{q}},\,q$ at the scale $\mu=\Lambda_{Q}$ in these $(N_{F}-N_{c})$
dual S-vacua is, see (1.17),(1.7),
$\displaystyle\mu_{q,S}=\frac{\langle
M\rangle_{S}}{Z_{q}\Lambda_{Q}}\sim\frac{m_{Q}\mu_{\Phi}}{Z_{q}\Lambda_{Q}}\,,\quad
Z_{q}\sim\exp\Bigl{\\{}-\frac{{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\ll 1\,.$ (3.7)
In comparison with the L - vacua in section 3.1, a qualitatively new element
here is that $\mu^{\rm pole}(M)$ is the largest mass, $\mu^{\rm
pole}(M)\gg\mu^{\rm pole}_{q,S}$, in the wide region
$\Lambda_{Q}\ll\mu_{\Phi}\ll Z_{q}^{\,3/2}\mu_{\Phi,\rm o}$ (see (3.14)
below). In this region: a) the mions are effectively massless and dynamically
relevant at scales $\mu^{\rm pole}(M)\ll\mu\ll\Lambda_{Q}$, b) there is a pole
in the mion propagator at the momentum $p=\mu^{\rm pole}(M)$,
$\displaystyle\mu^{\rm
pole}(M)=\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))\mu_{\Phi}}\,,\quad z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu^{\rm
pole}(M)}\Bigr{)}^{2{\rm\overline{b}}_{\rm o}/N_{F}}\,,$
$\displaystyle\mu^{\rm pole}(M)\sim
Z_{q}^{2}\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\,,\quad
Z_{q}^{2}\sim\exp\\{-\frac{2{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm
o}}\\}\ll 1\,.$ (3.8)
The mions then become too heavy and dynamically irrelevant at $\mu\ll\mu^{\rm
pole}(M)$. Due to this, they decouple from the RG evolution of dual quarks and
gluons and from the ’t Hooft triangles, and (at $\mu_{\Phi}$ not too close to
$\mu_{\Phi,\rm o}$ to have enough ”time” to evolve, see (3.13) ) the remained
dual theory of $N_{F}$ quarks ${\overline{q}},q$ and $SU({\overline{N}}_{c})$
gluons evolves into a new conformal regime with a new smaller value of the
frozen gauge coupling,
${\overline{a}}^{\,\prime}_{*}\simeq{\rm\overline{b}}_{\rm
o}/3{\overline{N}}_{c}={\overline{a}}_{*}/7\ll 1$. It is worth noting that, in
spite of that mions are dynamically irrelevant at $\mu<\mu^{\rm pole}(M)$,
their renormalization factor $z_{M}(\mu<\mu^{\rm pole}(M))$ still runs in the
range of scales $\mu^{\rm pole}_{q,S}<\mu<\mu^{\rm pole}(M)$ being induced by
loops of still effectively massless quarks and gluons.
The next physical scale is the perturbative pole mass of dual quarks
$\displaystyle\mu^{\rm pole}_{q,S}=\frac{\langle
M\rangle_{S}}{Z_{q}\Lambda_{Q}}\,\frac{1}{z_{q}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})}\,,\quad z_{q}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})=\Bigl{(}\frac{\mu^{\rm
pole}_{q,S}}{\Lambda_{Q}}\Bigr{)}^{{\rm\overline{b}}_{\rm
o}/N_{F}}{\overline{\rho}}_{S}\,,$
$\displaystyle{\overline{\rho}}_{S}=\Bigl{(}\frac{{\overline{a}}_{*}}{{\overline{a}}^{\,\prime}_{*}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{N_{F}}}\exp\Bigl{\\{}\frac{{\overline{N}}_{c}}{N_{F}}\Bigl{(}\frac{1}{{\overline{a}}_{*}}-\frac{1}{{\overline{a}}^{\,\prime}_{*}}\Bigr{)}\Bigr{\\}}\sim\frac{{\overline{Z}}_{q}}{Z_{q}}\ll
1\,,\quad{\overline{Z}}_{q}\sim\exp\Bigl{\\{}-\frac{{\overline{N}}_{c}}{{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\sim\Bigl{(}Z_{q}\Bigr{)}^{7}\ll Z_{q}\,,$ $\displaystyle\mu^{\rm
pole}_{q,S}\sim\frac{1}{{\overline{Z}}_{q}}\Lambda_{YM}^{(S)}\gg\Lambda_{YM}^{(S)}\,,\quad\Lambda_{YM}^{(S)}=\Lambda_{Q}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\,.$
(3.9)
This has to be compared with the gluon mass due to possible higgsing of
${\overline{q}},q$
$\displaystyle{\overline{\mu}}^{\,2}_{\rm gl,S}\sim\langle
N\rangle_{S}\,z_{q}(\Lambda_{Q},{\overline{\mu}}_{\rm
gl,S})\,\,\rightarrow\,\,{\overline{\mu}}_{\rm
gl,S}\sim{\overline{Z}}_{q}^{\,1/2}\Lambda_{YM}^{(S)}\ll\Lambda_{YM}^{(S)}\ll\mu^{\rm
pole}_{q,S}.$ (3.10)
The parametrical hierarchy in (3.10) guarantees that the dual quarks are in
the HQ phase in these S - vacua.
Hence, after integrating out all quarks at $\mu<\mu^{\rm pole}_{q,S}$ and,
finally, $SU({\overline{N}}_{c})$ gluons at $\mu<\Lambda_{YM}^{(S)}$, the
Lagrangian looks as in (3.5) but with a replacement
$z^{(L)}_{M}(\Lambda_{Q},\mu^{\rm pole}_{q,L})\rightarrow
z^{(S)}_{M}(\Lambda_{Q},\mu^{\rm pole}_{q,S})$,
$\displaystyle z^{(S)}_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})=\frac{a_{f}(\mu=\Lambda_{Q})}{a_{f}(\mu=\mu^{\rm
pole}_{q,S})}\frac{1}{z^{2}_{q}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})}\sim\frac{1}{z^{2}_{q}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})}\sim\frac{Z_{q}^{\,2}}{{\overline{Z}}_{q}^{\,2}}\Bigl{(}\frac{\Lambda_{Q}}{\mu^{\rm
pole}_{q,S}}\Bigr{)}^{2{\rm\overline{b}}_{\rm o}/N_{F}}\,.$ (3.11)
The contribution of the term $\sim M^{2}/\mu_{\Phi}$ in the superpotential
(3.5) to the frozen low energy value $\mu(M)$ of the running mion mass is
dominant at $\mu_{\Phi}/\mu_{\Phi,\rm o}\ll 1$ and is
$\displaystyle\mu(M)=\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{z^{(S)}_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,S})\mu_{\Phi}}\sim\frac{{\overline{Z}}_{q}^{\,2}\Lambda_{Q}^{2}}{\mu_{\Phi}}\Bigl{(}\frac{\mu^{\rm
pole}_{q,S}}{\Lambda_{Q}}\Bigr{)}^{\frac{2{\rm\overline{b}}_{\rm
o}}{N_{F}}}\ll\mu^{\rm pole}(M)\,.$ (3.12)
The requirement of self-consistency looks in this case as
$\displaystyle\frac{\mu(M)}{\mu^{\rm
pole}_{q,S}}\sim{\overline{Z}}_{q}^{\,3}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\gg
1\quad\rightarrow\quad\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\ll{\overline{Z}}^{\,3/2}_{q}\sim\exp\Bigl{\\{}-\frac{3{\overline{N}}_{c}}{2{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\ll Z_{q}^{3/2}\,,$ (3.13)
the meaning of (3.13) is that only at this condition the range of scales
between $\mu^{\rm pole}(M)$ in (3.8) and $\mu^{\rm pole}_{q,S}\ll\mu^{\rm
pole}(M)$ in (3.9) is sufficiently large that theory has enough ”time” to
evolve from ${\overline{a}}_{*}=7{\rm\overline{b}}_{\rm
o}/3{\overline{N}}_{c}$ to
${\overline{a}}^{\,\prime}_{*}={\rm\overline{b}}_{\rm o}/3{\overline{N}}_{c}$.
There is no pole in the mion propagator at the momentum $p=\mu(M)\gg\mu^{\rm
pole}_{q,S}$.
The opposite case with $\mu^{\rm pole}_{q,S}\gg\mu^{\rm pole}(M)$ is realized
if the ratio $\mu_{\Phi}/\mu_{\Phi,\rm o}$ is still $\ll 1$ but is much larger
than $Z_{q}^{\,3/2}\gg{\overline{Z}}_{q}^{\,3/2}$, see (3.14) below. In this
case the theory at $\mu^{\rm pole}_{q,S}<\mu<\Lambda_{Q}$ remains in the
conformal regime with ${\overline{a}}_{*}=7{\rm\overline{b}}_{\rm
o}/3{\overline{N}}_{c}$ and the largest mass is $\mu^{\rm pole}_{q,S}$. One
has in this case instead of (3.8),(3.9),(3.13)
$\displaystyle{\overline{\rho}}_{S}\sim 1\,,\quad\mu^{\rm
pole}_{q,S}\sim\frac{1}{Z_{q}}\Lambda_{YM}^{(S)}\,,\quad\frac{\mu^{\rm
pole}(M)}{\mu^{\rm pole}_{q,S}}\sim Z^{3}_{q}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\,,$ (3.14)
$\displaystyle\frac{\mu^{\rm pole}(M)}{\mu^{\rm pole}_{q,S}}\ll
1\quad\rightarrow\quad Z_{q}^{\,3/2}\ll\frac{\mu_{\Phi}}{\mu_{\Phi,\rm o}}\ll
1\,.$
On the whole, the mass spectrum in these ${\overline{N}}_{c}$ dual S - vacua
looks as follows at
$\Lambda_{Q}\ll\mu_{\Phi}\ll{\overline{Z}}_{q}^{\,3/2}\mu_{\Phi,\rm o}$. a)
The heaviest are $N_{F}^{2}$ mions with the pole masses (3.8). b) There is a
large number of flavored hadrons made of weakly interacting and weakly
confined (the string tension being
$\sqrt{\sigma}\sim\Lambda_{YM}^{(S)}\ll\mu^{\rm pole}_{q,S}$) non-relativistic
dual quarks ${\overline{q}},q$ with the perturbative pole masses (3.9). The
mass spectrum of low-lying flavored mesons is Coulomb-like with parametrically
small mass differences $\Delta\mu_{H}/\mu_{H}=O({\rm\overline{b}}_{\rm
o}^{2}/{\overline{N}}_{c}^{2})\ll 1$. b) A large number of gluonia made of
$SU({\overline{N}}_{c})$ gluons with the mass scale
$\sim\Lambda_{YM}^{(S)}\sim\Lambda_{Q}(m_{Q}\mu_{\Phi}/\Lambda_{Q}^{2})^{N_{F}/3{\overline{N}}_{c}}$.
The mions with the pole masses (3.8) remain the heaviest ones, $\mu^{\rm
pole}(M)\gg\mu^{\rm pole}_{q,S}$, at values $\mu_{\Phi}$ in the range
${\overline{Z}}_{q}^{\,3/2}\mu_{\Phi,\rm o}\ll\mu_{\Phi}\ll
Z_{q}^{\,3/2}\mu_{\Phi,\rm o}$, while the value $\mu^{\rm pole}_{q,S}$ varies
in a range $\Lambda_{YM}^{(S)}/Z_{q}\ll\mu^{\rm
pole}_{q,S}\ll\Lambda_{YM}^{(S)}/{\overline{Z}}_{q}$ . Finally, in a close
vicinity of $\mu_{\Phi,\rm o},\,\,Z_{q}^{\,3/2}\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}$, the perturbative pole mass of quarks,
$\mu^{\rm pole}_{q,S}\sim\Lambda_{YM}^{(S)}/Z_{q}\gg\Lambda_{YM}^{(S)}$,
becomes the largest one, while the pole masses of mions $\mu^{\rm
pole}(M)\ll\Lambda_{YM}^{(S)}$ become as in (3.14).
At $\mu_{\Phi}\gg\mu_{\Phi,\rm o}$ these S - vacua evolve into the vacua of
dSQCD, see section 4 in [2].
## 4 Direct theory. Broken flavor symmetry.
The region $\mathbf{\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}}$
4.1 L - type vacua
The quark condensates are parametrically the same as in the L - vacua with
unbroken flavor symmetry in section 3.1,
$\displaystyle(1-\frac{n_{1}}{N_{c}})\langle{\overline{Q}}_{1}Q_{1}\rangle_{Lt}\simeq-(1-\frac{n_{2}}{N_{c}})\langle{\overline{Q}}_{2}Q_{2}\rangle_{Lt},\quad\langle
S\rangle=\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle\langle{\overline{Q}}_{2}Q_{2}\rangle}{\mu_{\Phi}}\,,$
(4.1)
$\displaystyle\langle{\overline{Q}}_{1}Q_{1}\rangle_{Lt}\sim\langle{\overline{Q}}_{2}Q_{2}\rangle_{Lt}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{2N_{c}-N_{F}}}\,.$
All quarks are in the HQ phase and are confined and the Lagrangian of fions
looks as in (2.3), but one has to choose the L - type vacua with the broken
flavor symmetry in (2.3). Due to this, see (1.5), the masses of hybrid fions
$\Phi_{12},\Phi_{21}$ are qualitatively different, they are the Nambu-
Goldstone particles here and are massless. The ”masses” of $\Phi_{11}$ and
$\Phi_{22}$ are parametrically as in (2.4),
$\displaystyle\mu(\Phi_{11})\sim\mu(\Phi_{22})\sim\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q})}\sim m^{\rm pole}_{Q,1}\sim m^{\rm
pole}_{Q,2}\sim\Lambda_{YM}^{(L)}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\,,$
(4.2)
and hence there is no guaranty that these are the pole masses of fions, see
section 4 in [1]. May be yes, but maybe not.
On the whole, there are only two characteristic scales in the mass spectra in
these L - type vacua. The hybrid fions $\Phi_{12},\Phi_{21}$ are massless
while all other masses are $\sim\Lambda_{YM}^{(L)}$.
4.2 $\rm\bf br2$ vacua
The condensates of quarks look as
$\displaystyle\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}\simeq\Bigl{(}\rho_{2}=-\frac{n_{2}-N_{c}}{N_{c}}\Bigr{)}m_{Q}\mu_{\Phi},\,\,\,\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{n_{2}-N_{c}}},$
(4.3) $\displaystyle\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}}{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{c}}{n_{2}-N_{c}}}\ll 1$
in these vacua with $n_{2}>N_{c}\,,n_{1}<{\overline{N}}_{c}$ . Hence, the
largest among the masses smaller than $\Lambda_{Q}$ are the masses of the
$N_{F}^{2}$ second generation fions, see (1.14),
$\displaystyle\mu^{\rm pole}_{2}(\Phi_{ij})=\mu_{o}^{\rm
conf}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\,,$
(4.4)
while some other possible characteristic masses look here as
$\displaystyle\langle m^{\rm tot}_{Q,2}\rangle_{\rm
br2}=\frac{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm br2}}{\mu_{\Phi}}\sim
m_{Q}\,,\quad m^{\rm
pole}_{Q,1}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\gg{\tilde{m}}^{\rm
pole}_{Q,2}\,,$ (4.5) $\displaystyle\mu_{\rm gl,2}^{2}\sim
z_{Q}(\Lambda_{Q},\mu_{\rm gl,2})\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br2},\quad z_{Q}(\Lambda_{Q},\mu_{\rm
gl,2})\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\rm gl,2}}\Bigr{)}^{\frac{{\rm
b_{o}}}{N_{F}}}\ll 1\,,$ $\displaystyle\mu_{\rm
gl,2}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu_{\rm
gl,1}\,,\quad\frac{\mu_{\rm gl,2}}{m^{\rm
pole}_{Q,1}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{F}}{3{\overline{N}}_{c}}}\ll 1\,,$ (4.6)
where $m^{\rm pole}_{Q,1}$ and ${\tilde{m}}^{\rm pole}_{Q,2}$ are the pole
masses of quarks ${\overline{Q}}_{1},Q_{1}$ and ${\overline{Q}}_{2},Q_{2}$ and
$\mu_{\rm gl,1},\,\mu_{\rm gl,2}$ are the gluon masses due to possible
higgsing of these quarks. Hence, the largest mass is $m^{\rm pole}_{Q,1}$ and
the overall phase is $HQ_{1}-HQ_{2}$.
The lower energy theory at $\mu<m^{\rm pole}_{Q,1}$ has $N_{c}$ colors and
$N_{F}^{\prime}=n_{2}>N_{c}$ flavors of quarks ${\overline{Q}}_{2},Q_{2}$. In
the range of scales $m^{\rm pole}_{Q,2}<\mu<m^{\rm pole}_{Q,1}$, it will
remain in the conformal regime at $2n_{1}<{\rm\overline{b}}_{\rm o}$, while it
will be in the strong coupling regime at $2n_{1}>{\rm\overline{b}}_{\rm o}$,
with the gauge coupling $a(\mu\ll m^{\rm pole}_{Q,1})\gg 1$. We do not
consider the strong coupling regime in this paper and for this reason we take
${\rm\overline{b}}_{\rm o}/{\overline{N}}_{c}=O(1)$ in this subsection.
After the heaviest quarks ${\overline{Q}}_{1},Q_{1}$ decouple at $\mu<m^{\rm
pole}_{Q,1}$, the pole mass of quarks ${\overline{Q}}_{2},Q_{2}$ in the lower
energy theory looks as
$\displaystyle m^{\rm pole}_{Q,2}=\frac{1}{z^{\,\prime}_{Q}(m^{\rm
pole}_{Q,1},m^{\rm
pole}_{Q,2})}\Biggl{(}\,\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br2}}{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm br2}}\,m^{\rm
pole}_{Q,1}\,\Biggr{)}\sim\Lambda_{YM}^{(\rm br2)}\,,$ (4.7) $\displaystyle
z^{\,\prime}_{Q}(m^{\rm pole}_{Q,1},m^{\rm
pole}_{Q,2})\sim\Bigl{(}\frac{m^{\rm pole}_{Q,2}}{m^{\rm
pole}_{Q,1}}\Bigr{)}^{\frac{3N_{c}-n_{2}}{N_{F}}}\ll 1\,.$
Hence, after integrating out quarks ${\overline{Q}}_{1},Q_{1}$ at $\mu<m^{\rm
pole}_{Q,1}$ and then quarks ${\overline{Q}}_{2},Q_{2}$ and $SU(N_{c})$ gluons
at $\mu<\Lambda_{YM}^{(\rm br2)}$, the Lagrangian of fions looks as
$\displaystyle K=z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q,1})\,{\rm
Tr}\,\Bigl{[}\,\Phi_{11}^{\dagger}\Phi_{11}+\Phi_{12}^{\dagger}\Phi_{12}+\Phi_{21}^{\dagger}\Phi_{21}+z^{\,\prime}_{\Phi}(m^{\rm
pole}_{Q,1},m^{\rm pole}_{Q,2})\Phi_{22}^{\dagger}\Phi_{22}\,\Bigr{]}\,,$
(4.8) $\displaystyle z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})\sim\Bigl{(}\frac{\Lambda_{Q}}{m^{\rm
pole}_{Q,1}}\Bigr{)}^{\frac{2(3N_{c}-N_{F})}{N_{F}}}\gg 1\,,\quad
z^{\,\prime}_{\Phi}(m^{\rm pole}_{Q,1},m^{\rm
pole}_{Q,2})\sim\Bigl{(}\frac{m^{\rm pole}_{Q,1}}{m^{\rm
pole}_{Q,2}}\Bigr{)}^{\frac{2(3N_{c}-n_{2})}{N_{F}}}\gg 1\,,$ $\displaystyle
W=N_{c}S+W_{\Phi}\,,\quad m^{\rm tot}_{Q}=(m_{Q}-\Phi)\,,\quad$ (4.9)
$\displaystyle S=\Bigl{(}\Lambda_{Q}^{{\rm b_{o}}}\det m^{\rm
tot}_{Q}\Bigr{)}^{1/N_{c}}\,,\quad W_{\Phi}=\frac{\mu_{\Phi}}{2}\Bigl{(}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}({\rm Tr}\,\Phi)^{2}\,\Bigr{)}.$
From (4.8),(4.9), the main contribution to the mass of the third generation
fions $\Phi_{11}$ gives the term $\sim\mu_{\Phi}\Phi^{2}_{11}$,
$\displaystyle\mu^{\rm
pole}_{3}(\Phi_{11})\sim\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm o}}\Bigr{)}m^{\rm
pole}_{Q,1}\,,$ (4.10)
while the third generation hybrid fions $\Phi_{12},\Phi_{21}$ are massless,
$\mu^{\rm pole}_{3}(\Phi_{12})=\mu^{\rm pole}_{3}(\Phi_{21})=0$. As for the
third generation fions $\Phi_{22}$, the main contribution to their masses
comes from the non-perturbative term $\sim S$ in the superpotential
$\displaystyle\mu_{3}(\Phi_{22})\sim\frac{\langle S\rangle}{\langle m^{\rm
tot}_{Q,2}\rangle^{2}}\frac{1}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})z^{\,\prime}_{\Phi}(m^{\rm pole}_{Q,1},m^{\rm pole}_{Q,2})}\sim
m^{\rm pole}_{Q,2}\sim\Lambda_{YM}^{(\rm br2)}.$ (4.11)
In such a situation there is no guaranty that there is a pole in the
propagator of $\Phi_{22}$ at the momentum $p\sim m^{\rm pole}_{Q,2}$. May be
yes but maybe not, see section 4 in [1].
4.3 Special vacua, $n_{1}={\overline{N}}_{c},\,\,n_{2}=N_{c}$
Nothing specific for the dynamical scenario $\\#1$ with the diquark condensate
(DC) was used in a description of these vacua in section 7.3 of [1]. Hence,
all remains the same in the dynamical scenario $\\#2$ here.
## 5 Dual theory. Broken flavor symmetry.
The region $\mathbf{\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,\rm o}}$
5.1 L - type vacua, $\,\,{\rm\overline{b}}_{\rm o}/N_{F}\ll 1$
The condensates of mions and dual quarks look here as
$\displaystyle\langle M_{1}+M_{2}-\frac{1}{N_{c}}{\rm
Tr}\,M\rangle_{Lt}=m_{Q}\mu_{\Phi}\quad\rightarrow\quad\frac{\langle
M_{1}\rangle_{Lt}}{\langle
M_{2}\rangle_{Lt}}\simeq-\,\frac{N_{c}-n_{1}}{N_{c}-n_{2}}\,,$
$\displaystyle\langle M_{1}\rangle_{Lt}\langle N_{1}\rangle_{Lt}=\langle
M_{2}\rangle_{Lt}\langle N_{2}\rangle_{Lt}=Z_{q}\Lambda_{Q}\langle
S\rangle_{Lt},\quad\langle S\rangle_{Lt}=\frac{\langle
M_{1}\rangle_{Lt}\langle M_{2}\rangle_{Lt}}{\mu_{\Phi}}\,.$
I.e., all condensates are parametrically the same as in the L - vacua with
unbroken flavor symmetry in section 3.1 and the overall phase is also
$HQ_{1}-HQ_{2}$. The pole masses of dual quarks are as in (3.2), the
Lagrangian of mions is as in (3.5) and the pole masses of mions $M_{11}$ and
$M_{22}$ are as in (3.6). But the masses of hybrid mions $M_{12}$ and $M_{21}$
are qualitatively different here. They are the Nambu-Goldstone particles now
and are exactly massless, $\mu(M_{12})=\mu(M_{21})=0$.
5.2 $\rm\bf br2$ vacua, $\,\,{\rm\overline{b}}_{\rm o}/N_{F}=O(1)$
In these vacua with $n_{2}>N_{c}\,,n_{1}<{\overline{N}}_{c}$ the condensates
of mions and dual quarks look as
$\displaystyle\langle M_{2}\rangle_{\rm
br2}\simeq-\,\frac{n_{2}-N_{c}}{N_{c}}\,m_{Q}\mu_{\Phi}\,,\quad\langle
M_{1}\rangle_{\rm
br2}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{n_{2}-N_{c}}},\,\,$
(5.1) $\displaystyle\frac{\langle M_{1}\rangle_{\rm br2}}{\langle
M_{2}\rangle_{\rm br2}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{c}}{n_{2}-N_{c}}}\ll 1\,,$ $\displaystyle\langle
N_{1}\rangle_{\rm
br2}=\langle{\overline{q}}_{1}q_{1}(\mu=\Lambda_{Q})\rangle_{\rm
br2}=\frac{\Lambda_{Q}\langle S\rangle_{\rm br2}}{\langle M_{1}\rangle_{\rm
br2}}=\frac{\Lambda_{Q}\langle M_{2}\rangle_{\rm br2}}{\mu_{\Phi}}\sim
m_{Q}\Lambda_{Q}\gg\langle N_{2}\rangle_{\rm br2}\,.$
From these, the heaviest are $N_{F}^{2}$ mions $M_{ij}$ with the pole masses
$\displaystyle\mu^{\rm
pole}(M)=\frac{\Lambda_{Q}^{2}/\mu_{\Phi}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}},\,z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu^{\rm
pole}(M)}\Bigr{)}^{\frac{2{\rm\overline{b}}_{\rm o}}{N_{F}}}\gg 1\,\,$ (5.2)
while some other possible characteristic masses look as
$\displaystyle\mu_{q,2}=\frac{\langle
M_{2}\rangle}{\Lambda_{Q}}\sim\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}},\quad{\tilde{\mu}}^{\rm
pole}_{q,2}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu^{\rm
pole}_{q,1}\,,$ (5.3) $\displaystyle{\overline{\mu}}_{\rm
gl,1}\sim\Lambda_{Q}\Bigl{(}\frac{\langle
N_{1}\rangle}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3N_{c}}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\gg{\overline{\mu}}_{\rm
gl,2}\,,\quad\frac{{\overline{\mu}}_{\rm gl,1}}{{\tilde{\mu}}^{\rm
pole}_{q,2}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg 1\,,$
where $\mu^{\rm pole}_{q,1}$ and ${\tilde{\mu}}^{\rm pole}_{q,2}$ are the pole
masses of quarks ${\overline{q}}_{1},q_{1}$ and ${\overline{q}}_{2},q_{2}$ and
${\overline{\mu}}_{\rm gl,1},\,{\overline{\mu}}_{\rm gl,2}$ are the gluon
masses due to possible higgsing of these quarks. Hence, the largest mass is
${\overline{\mu}}_{\rm gl,1}$ and the overall phase is $Higgs_{1}-HQ_{2}$.
After integrating out all higgsed gluons and quarks
${\overline{q}}_{1},q_{1}$, we write the dual Lagrangian at
$\mu={\overline{\mu}}_{{\rm gl},\,1}$ as
$\displaystyle K=z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}){\rm
Tr}\,\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1}){\rm Tr}\,\Bigl{[}\,2\sqrt{N_{11}^{\dagger}N_{11}}+K_{\rm
hybr}+\Bigl{(}{\textsf{q}}^{\dagger}_{2}{\textsf{q}}_{2}+({\textsf{q}}_{2}\rightarrow{\overline{\textsf{q}}}_{2})\Bigr{)}\,\Bigr{]}\,,$
$\displaystyle K_{\rm
hybr}=\Biggl{(}N^{\dagger}_{12}\frac{1}{\sqrt{N_{11}N^{\dagger}_{11}}}N_{12}+N_{21}\frac{1}{\sqrt{N^{\dagger}_{11}N_{11}}}N^{\dagger}_{21}\Biggr{)},\quad
z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})=\Bigl{(}\frac{{\overline{\mu}}_{{\rm
gl},\,1}}{\Lambda_{Q}}\Bigr{)}^{{\rm\overline{b}}_{\rm o}/N_{F}}\,,$ (5.4)
$\displaystyle z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})=1/z^{2}_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}),\quad
W=\Bigl{[}-\frac{2\pi}{{\overline{\alpha}}(\mu)}{\overline{\textsf{s}}}\Bigr{]}-\frac{1}{\Lambda_{Q}}{\rm
Tr}\,\Bigl{(}{\overline{\textsf{q}}}_{2}M_{22}\textsf{q}_{2}\Bigr{)}-W_{MN}+W_{M},$
$\displaystyle W_{MN}=\frac{1}{\Lambda_{Q}}{\rm
Tr}\,\Bigl{(}M_{11}N_{11}+M_{21}N_{12}+N_{21}M_{12}+M_{22}N_{21}\frac{1}{N_{11}}N_{12}\Bigr{)}\,,$
where $\overline{\textsf{q}}_{2},\textsf{q}_{2}$ are the quarks
${\overline{q}}_{2},q_{2}$ with unhiggsed colors, $\overline{\textsf{s}}$ is
the field strength of unhiggsed dual gluons and the hybrid nions $N_{12}$ and
$N_{21}$ are, in essence, the quarks ${\overline{q}}_{2},q_{2}$ with higgsed
colors, $W_{M}$ is given in (1.17). The lower energy theory at
$\mu<{\overline{\mu}}_{{\rm gl},\,1}$ has
${\overline{N}}_{c}^{\,\prime}={\overline{N}}_{c}-n_{1}$ colors and
$n_{2}>N_{c}$ flavors, ${\rm\overline{b}}_{\rm
o}^{\,\prime}={\rm\overline{b}}_{\rm o}-2n_{1}<{\rm\overline{b}}_{\rm o}$. We
consider here only the case ${\rm\overline{b}}_{\rm o}^{\,\prime}>0$ when it
remains in the conformal window. In this case the value of the pole mass
$\mu^{\rm pole}_{q,2}$ in this lower energy theory is
$\displaystyle\mu^{\rm pole}_{q,2}\sim\frac{\langle
M_{2}\rangle}{\Lambda_{Q}}\frac{1}{z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})z^{\,\prime}_{q}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{q,2})}\sim\Lambda_{YM}^{(\rm br2)}\,,\quad
z^{\,\prime}_{q}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{q,2})\sim\Bigl{(}\frac{\mu^{\rm pole}_{q,2}}{{\overline{\mu}}_{{\rm
gl},\,1}}\Bigr{)}^{{\rm\overline{b}}_{\rm o}^{\,\prime}/n_{2}}\ll 1\,.$ (5.5)
The fields $N_{11},N_{12},N_{21}$ and $M_{11},M_{12},M_{21}$ are frozen and do
not evolve at $\mu<{\overline{\mu}}_{{\rm gl},\,1}$. After integrating out
remained unhiggsed quarks $\overline{\textsf{q}}_{2},\textsf{q}_{2}$ as heavy
ones and unhiggsed gluons at $\mu<\Lambda_{YM}^{(\rm br2)}$ the Lagrangian of
mions and nions looks as, see (5.4),
$\displaystyle K=z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}){\rm
Tr}\,K_{M}+z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\Bigl{[}\,2\sqrt{N_{11}^{\dagger}N_{11}}+K_{\rm
hybr}\,\Bigr{]},\,\,z^{\,\prime}_{M}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{q,2})\sim\Bigl{(}\frac{{\overline{\mu}}_{{\rm gl},\,1}}{\mu^{\rm
pole}_{q,2}}\Bigr{)}^{\frac{2{\rm\overline{b}}_{\rm o}^{\,\prime}}{n_{2}}}\gg
1,$ $\displaystyle
K_{M}=\frac{1}{\Lambda_{Q}^{2}}\Bigl{(}M_{11}^{\dagger}M_{11}+M_{12}^{\dagger}M_{12}+M_{21}^{\dagger}M_{21}+z^{\,\prime}_{M}({\overline{\mu}}_{{\rm
gl},\,1},\mu^{\rm pole}_{q,2})M_{22}^{\dagger}M_{22}\Bigr{)}\,,$ (5.6)
$\displaystyle W=-{\overline{N}}_{c}^{\,\prime}S-W_{MN}+W_{M}\,,\quad
S=\Bigl{(}\Lambda_{YM}^{(\rm br2)}\Bigr{)}^{3}\Biggl{(}\det\frac{\langle
N_{1}\rangle}{N_{11}}\det\frac{M_{22}}{\langle
M_{2}\rangle}\Biggr{)}^{1/{\overline{N}}_{c}^{\,\prime}},\quad\Lambda_{YM}^{(\rm
br2)}\sim\Bigl{(}m_{Q}\langle M_{1}\rangle\Bigr{)}^{1/3}.$
From (5.6), the ”masses” of mions look as
$\displaystyle\mu(M_{11})\sim\mu(M_{12})\sim\mu(M_{21})\sim\frac{\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\mu_{\Phi}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm gl},\,1}\,,$ (5.7)
$\displaystyle\mu(M_{22})\sim\frac{\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})z^{\prime}_{M}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{q,2})\mu_{\Phi}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{3N_{c}-n_{2}}{3(n_{2}-N_{c})}}\,{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm gl},\,1}\,,$ (5.8)
while the pole masses of nions $N_{11}$ are
$\displaystyle\mu^{\rm pole}(N_{11})\sim\frac{\mu_{\Phi}\langle
N_{1}\rangle_{\rm br2}}{z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\Lambda_{Q}^{2}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}{\overline{\mu}}_{{\rm gl},\,1}\,,$ (5.9)
and the hybrid nions $N_{12},N_{21}$ are massless,
$\mu(N_{12})=\mu(N_{21})=0$. The mion ”masses” (5.7),(5.8) are not the pole
masses but simply the low energy values of mass terms in their propagators,
the only pole masses are given in (5.2).
5.3 $\rm\bf br2$ vacua, ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$
Instead of (5.2), the pole mass of mions is parametrically smaller now, see
(1.17),(4.4),
$\displaystyle\mu^{\rm
pole}(M)=\frac{Z_{q}^{\,2}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))}\sim
Z_{q}^{\,2}\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}},\quad\frac{\mu^{\rm
pole}(M)}{\mu^{\rm pole}_{2}(\Phi)}\sim Z_{q}^{\,2}\ll 1\,,$ (5.10)
while instead of (5.3) we have now, see (4.5),
$\displaystyle\mu_{q,2}=\frac{\langle
M_{2}\rangle}{Z_{q}\Lambda_{Q}}\sim\frac{m_{Q}\mu_{\Phi}}{Z_{q}\Lambda_{Q}},\quad{\tilde{\mu}}^{\rm
pole}_{q,2}\sim\frac{\Lambda_{Q}}{Z_{q}}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu^{\rm
pole}_{q,1}\,,$ (5.11) $\displaystyle{\overline{\mu}}_{\rm
gl,1}\sim\Lambda_{Q}\Bigl{(}\frac{\langle
N_{1}\rangle}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3N_{c}}\sim
Z_{q}^{1/2}\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\gg{\overline{\mu}}_{\rm
gl,2}\,,\quad\frac{{\overline{\mu}}_{\rm gl,1}}{m^{\rm pole}_{Q,1}}\sim
Z_{q}^{1/2}\ll 1\,,$ (5.12) $\displaystyle\quad\frac{{\overline{\mu}}_{\rm
gl,1}}{{\tilde{\mu}}^{\rm pole}_{q,2}}\sim
Z_{q}^{3/2}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg
1\,,\quad\Lambda_{Q}\ll{\mu_{\Phi}}\ll Z_{q}^{\,3/2}\mu_{\Phi,\rm o}\,,\quad
Z_{q}\sim\exp\\{-\frac{{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm o}}\\}\ll
1\,.$ (5.13)
Hence, at the condition (5.13), the largest mass is ${\overline{\mu}}_{\rm
gl,1}$ and the overall phase is also $Higgs_{1}-HQ_{2}$. But now, at
${\rm\overline{b}}_{\rm o}/{\overline{N}}_{c}\ll 1$, it looks unnatural to
require ${\rm\overline{b}}_{\rm o}^{\,\prime}=({\rm\overline{b}}_{\rm
o}-2n_{1})>0$. Therefore, with $n_{1}/{\overline{N}}_{c}=O(1)$, the lower
energy theory at $\mu<{\overline{\mu}}_{\rm gl,1}$ has ${\rm\overline{b}}_{\rm
o}^{\,\prime}<0$ and is in the logarithmic IR free regime in the range of
scales $\mu^{\rm pole}_{q,2}<\mu<{\overline{\mu}}_{\rm gl,1}$. Then instead of
(5.5) (ignoring all logarithmic renormalization factors),
$\displaystyle\Lambda_{YM}^{(\rm br2)}\ll\mu^{\rm pole}_{q,2}\sim\frac{\langle
M_{2}\rangle_{\rm
br2}}{\Lambda_{Q}}\frac{1}{z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})}\sim\frac{\mu_{\Phi}}{Z_{q}^{3/2}\mu_{\Phi,\rm
o}}\,{\overline{\mu}}_{\rm gl,1}\ll{\overline{\mu}}_{\rm gl,1}\,.$ (5.14)
The Lagrangian of mions and nions has now the form (5.6) with a replacement
$z^{\,\prime}_{M}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm pole}_{q,2})\sim 1$
and so $\mu(M_{22})\sim\mu(M_{11})\sim\mu(M_{12})\sim\mu(M_{21})$ now, see
(5.7),(5.8),(5.13),
$\displaystyle\mu(M_{ij})\sim\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\mu_{\Phi}}\sim Z_{q}^{3/2}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm gl},\,1}\,,$ (5.15)
while, instead of (5.9), the mass of nions looks now as
$\displaystyle\mu^{\rm pole}(N_{11})\sim\frac{\mu_{\Phi}\langle
N_{1}\rangle_{\rm br2}}{z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\Lambda_{Q}^{2}}\sim
Z_{q}^{1/2}\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}\,{\overline{\mu}}_{{\rm gl},\,1}\,.$ (5.16)
On the whole for the mass spectra in this case. a) The heaviest are
$N_{F}^{2}$ mions with the pole masses (5.10) (the ”masses” (5.15) are not the
pole masses but simply the low energy values of mass terms in the mion
propagators). b) The next are the masses (5.12) of
$n_{1}(2{\overline{N}}_{c}-n_{1})$ higgsed gluons and their superpartners. c)
There is a large number of flavored hadrons, mesons and baryons, made of non-
relativistic and weakly confined (the string tension being
$\sqrt{\sigma}\sim\Lambda_{YM}^{(\rm br2)}\ll\mu^{\rm pole}_{q,2}$ ) quarks
$\overline{\textsf{q}}_{2},\textsf{q}_{2}$ with unhiggsed colors. The mass
spectrum of low-lying flavored mesons is Coulomb-like with parametrically
small mass differences, $\Delta\mu_{H}/\mu_{H}=O({\rm\overline{b}}_{\rm
o}^{\,2}/N^{2}_{F})\ll 1$. d) A large number of gluonia made of
$SU({\overline{N}}_{c}-n_{1})$ gluons with the mass scale
$\sim\Lambda_{YM}^{(\rm br2)}$. e) $n_{1}^{2}$ nions $N_{11}$ with the masses
(5.16). f) The hybrid nions $N_{12},N_{21}$ are massless.
5.4 Special vacua, $n_{1}={\overline{N}}_{c},\,\,n_{2}=N_{c}$
Nothing specific for the dynamical scenario $\\#1$ with the diquark condensate
(DC) was used in a description of these vacua in section 9.3 of [1]. Hence,
all remains the same in the dynamical scenario $\\#2$ here.
## 6 Direct theory. Broken flavor symmetry.
The region $\mathbf{\mu_{\Phi,\rm o}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}}$
6.1 $\rm\bf br1$ vacua
The values of quark condensates are here
$\displaystyle\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br1}\simeq\frac{N_{c}}{N_{c}-n_{1}}\,m_{Q}\mu_{\Phi}\,,\quad\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br1}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{1}}{N_{c}-n_{1}}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}-N_{c}}{N_{c}-n_{1}}}\,,$
(6.1) $\displaystyle\frac{\langle{\overline{Q}}_{2}Q_{2}\rangle_{\rm
br1}}{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br1}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{N_{c}-n_{1}}}\ll 1\,.$
From these, the values of some potentially relevant masses look as
$\displaystyle\mu_{\rm gl,1}^{2}\sim\Bigl{(}a_{*}\sim
1\Bigr{)}z_{Q}(\Lambda_{Q},\mu_{\rm
gl,1})\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm br1}\,,\quad
z_{Q}(\Lambda_{Q},\mu_{\rm gl,1})\sim\Bigl{(}\frac{\mu_{\rm
gl,1}}{\Lambda_{Q}}\Bigr{)}^{{\rm b_{o}}/N_{F}}\,,$ $\displaystyle\mu_{\rm
gl,1}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu_{\rm
gl,2}\,,$ (6.2) $\displaystyle\langle m^{\rm
tot}_{Q,2}\rangle=\frac{\langle{\overline{Q}}_{1}Q_{1}\rangle_{\rm
br1}}{\mu_{\Phi}}\sim m_{Q}\,,\quad{\tilde{m}}^{\rm pole}_{Q,2}=\frac{\langle
m^{\rm tot}_{Q,2}\rangle_{\rm br1}}{z_{Q}(\Lambda_{Q},{\tilde{m}}^{\rm
pole}_{Q,2})}\,,$ $\displaystyle{\tilde{m}}^{\rm
pole}_{Q,2}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\gg
m^{\rm pole}_{Q,1},\quad\,\,\,\frac{{\tilde{m}}^{\rm pole}_{Q,2}}{\mu_{\rm
gl,1}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\ll 1.$ (6.3)
Hence, the largest mass is $\mu_{\rm gl,1}$ due to higgsing of
${\overline{Q}}_{1},Q_{1}$ quarks and the overall phase is $Higgs_{1}-HQ_{2}$.
The lower energy theory at $\mu<\mu_{\rm gl,1}$ has
$N_{c}^{\prime}=N_{c}-n_{1}$ colors and $n_{2}\geq N_{f}/2$ flavors. At
$2n_{1}<{\rm b_{o}}$ it remains in the conformal window with ${\rm
b}_{o}^{\prime}>0$, while at $2n_{1}>{\rm b_{o}},\,\,{\rm b}_{o}^{\prime}<0$
it enters the logarithmic IR free perturbative regime.
We start with ${\rm b_{o}}^{\prime}>0$. Then the value of the pole mass of
quarks $\overline{\textsf{Q}}_{2},\,\textsf{Q}_{2}$ with unhiggsed colors
looks as
$\displaystyle m^{\rm pole}_{\textsf{Q}_{2}}=\frac{\langle m^{\rm
tot}_{Q,2}\rangle_{\rm br1}}{z_{Q}(\Lambda_{Q},\mu_{\rm
gl,1})z_{Q}^{\,\prime}(\mu_{\rm gl,1},m^{\rm pole}_{\textsf{Q}_{2}})}\,,\quad
z_{Q}^{\,\prime}(\mu_{\rm gl,1},m^{\rm
pole}_{\textsf{Q}_{2}})\sim\Bigl{(}\frac{m^{\rm
pole}_{\textsf{Q}_{2}}}{\mu_{\rm gl,1}}\Bigr{)}^{{\rm
b}_{o}^{\prime}/n_{2}}\,,$ $\displaystyle m^{\rm
pole}_{\textsf{Q}_{2}}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{1}}{3(N_{c}-n_{1})}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}-n_{1}}{3(N_{c}-n_{1})}}\sim\Lambda_{YM}^{(\rm
br1)}\,.$ (6.4)
It is technically convenient to retain all fion fields $\Phi$ although, in
essence, they are too heavy and dynamically irrelevant at
$\mu_{\Phi}\gg\mu_{\Phi,\rm o}$. After integrating out all heavy higgsed
gluons and quarks ${\overline{Q}}_{1},Q_{1}$, we write the Lagrangian at
$\mu=\mu_{\rm gl,1}$ in the form
$\displaystyle K=\Bigl{[}\,z_{\Phi}(\Lambda_{Q},\mu_{\rm gl,1}){\rm
Tr}(\Phi^{\dagger}\Phi)+z_{Q}(\Lambda_{Q},\mu^{2}_{\rm
gl,1})\Bigl{(}K_{{\textsf{Q}}_{2}}+K_{\Pi}\Bigr{)}\,\Bigr{]},\quad
z_{\Phi}(\Lambda_{Q},\mu_{\rm gl,1})=1/z^{2}_{Q}(\Lambda_{Q},\mu_{\rm
gl,1})\,,$ $\displaystyle K_{{\textsf{Q}}_{2}}={\rm
Tr}\Bigl{(}{\textsf{Q}}^{\dagger}_{2}{\textsf{Q}}_{2}+({\textsf{Q}}_{2}\rightarrow{\overline{\textsf{Q}}}_{2})\Bigr{)}\,,\quad
K_{\Pi}=2{\rm Tr}\sqrt{\Pi^{\dagger}_{11}\Pi_{11}}+K_{\rm hybr},$ (6.5)
$\displaystyle K_{\rm hybr}={\rm
Tr}\Biggl{(}\Pi^{\dagger}_{12}\frac{1}{\sqrt{\Pi_{11}\Pi^{\dagger}_{11}}}\Pi_{12}+\Pi_{21}\frac{1}{\sqrt{\Pi^{\dagger}_{11}\Pi_{11}}}\Pi^{\dagger}_{21}\Biggr{)},$
$\displaystyle W=\Bigl{[}-\frac{2\pi}{\alpha(\mu_{\rm
gl,1})}{\textsf{S}}\Bigr{]}+\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]}+{\rm
Tr}\Bigl{(}{\overline{\textsf{Q}}_{2}}m^{\rm
tot}_{{\textsf{Q}}_{2}}{\textsf{Q}}_{2}\Bigr{)}+W_{\Pi},$ $\displaystyle
W_{\Pi}={\rm Tr}\Bigl{(}m_{Q}\Pi_{11}+m^{\rm
tot}_{{\textsf{Q}}_{2}}\,\Pi_{21}\frac{1}{\Pi_{11}}\Pi_{12}\Bigr{)}-{\rm
Tr}\Bigl{(}\Phi_{11}\Pi_{11}+\Phi_{12}\Pi_{21}+\Phi_{21}\Pi_{12}\Bigr{)},\quad
m^{\rm tot}_{{\textsf{Q}}_{2}}=(m_{Q}-\Phi_{22}).$
In (6.5): $\overline{\textsf{Q}}_{2},\,\textsf{Q}_{2}$ and V are the active
${\overline{Q}}_{2},Q_{2}$ guarks and gluons with unhiggsed colors (S is their
field strength squared), $\Pi_{12},\Pi_{21}$ are the hybrid pions (in essence,
these are the quarks ${\overline{Q}}_{2},Q_{2}$ with higgsed colors),
$z_{Q}(\Lambda_{Q},\mu^{2}_{\rm gl,1})$ is the corresponding perturbative
renormalization factor of massless quarks, see (6.2), while
$z_{\Phi}(\Lambda_{Q},\mu_{\rm gl,1})$ is that of fions. Evolving now down in
the scale and integrating out at $\mu<\Lambda_{YM}^{(\rm br1)}$ quarks
$\overline{\textsf{Q}}_{2},\,\textsf{Q}_{2}$ as heavy ones and unhiggsed
gluons, we obtain the Lagrangian of pions and fions
$\displaystyle K=\Bigl{[}z_{\Phi}(\Lambda_{Q},\mu_{\rm gl,1}){\rm
Tr}\Bigl{(}\Phi^{\dagger}_{11}\Phi_{11}+\Phi^{\dagger}_{12}\Phi_{12}+\Phi^{\dagger}_{21}\Phi_{21}+z^{\,\prime}_{\Phi}(\mu_{\rm
gl,1},m^{\rm
pole}_{\textsf{Q}_{2}})\Phi^{\dagger}_{22}\Phi_{22}\Bigr{)}+z_{Q}(\Lambda_{Q},\mu^{2}_{\rm
gl,1})K_{\Pi}\Bigr{]},\,$ $\displaystyle
W=(N_{c}-n_{1})S+W_{\Phi}+W_{\Pi}\,,\quad S=\Biggl{[}\frac{\Lambda_{Q}^{{\rm
b_{o}}}\det m^{\rm
tot}_{{\textsf{Q}}_{2}}}{\det\Pi_{11}}\Biggr{]}^{\frac{1}{N_{c}-n_{1}}}\,,$
(6.6) $\displaystyle W_{\Phi}=\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]},\quad z^{\,\prime}_{\Phi}(\mu_{\rm gl,1},m^{\rm
pole}_{\textsf{Q}_{2}})\sim\Bigl{(}\frac{\mu_{\rm gl,1}}{m^{\rm
pole}_{\textsf{Q}_{2}}}\Bigr{)}^{2{\rm b}_{o}^{\prime}/n_{2}}\,.$
We obtain from (6.6) that all fions are heavy with the ”masses”
$\displaystyle\mu(\Phi_{11})\sim\mu(\Phi_{12})\sim\mu(\Phi_{21})\sim\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},\mu_{\rm
gl,1})}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\mu_{\rm gl,1}\gg\mu_{\rm gl,1}\,,$
(6.7)
$\displaystyle\mu(\Phi_{22})\sim\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},\mu_{\rm
gl,1})z^{\,\prime}_{\Phi}(\mu_{\rm gl,1},m^{\rm
pole}_{\textsf{Q}_{2}})}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{N_{c}}{N_{c}-n_{1}}}\,m^{\rm pole}_{\textsf{Q}_{2}}\gg
m^{\rm pole}_{\textsf{Q}_{2}}\,.$ (6.8)
These are not the pole masses but simply the low energy values of mass terms
in their propagators. All fions are dynamically irrelevant at all scales
$\mu<\Lambda_{Q}$. The mixings of
$\Phi_{12}\leftrightarrow\Pi_{12},\,\Phi_{21}\leftrightarrow\Pi_{21}$ and
$\Phi_{11}\leftrightarrow\Pi_{11}$ are parametrically small and are neglected.
We obtain then for the masses of pions $\Pi_{11}$
$\displaystyle\mu(\Pi_{11})\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}({\rm
b_{o}}-2n_{1})}{3{\overline{N}}_{c}(N_{c}-n_{1})}}\,\Lambda_{YM}^{(\rm
br1)}\sim\Bigl{(}\frac{\mu_{\Phi,\rm o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}({\rm
b_{o}}-2n_{1})}{3{\overline{N}}_{c}(N_{c}-n_{1})}}\,m^{\rm
pole}_{\textsf{Q}_{2}}\ll m^{\rm pole}_{\textsf{Q}_{2}}\,,$ (6.9)
and, finally, the hybrids $\Pi_{12},\Pi_{21}$ are massless,
$\mu(\Pi_{12})=\mu(\Pi_{21})=0$.
At $2n_{1}>{\rm b_{o}}$ the RG evolution at $m^{\rm
pole}_{\textsf{Q}_{2}}<\mu<\mu_{\rm gl,1}$ is only slow logarithmic (and is
neglected). We replace then $z^{\,\prime}_{Q}(\mu_{\rm gl,1},m^{\rm
pole}_{\textsf{Q}_{2}})\sim 1$ in (6.4) and $z^{\,\prime}_{\Phi}(\mu_{\rm
gl,1},m^{\rm pole}_{\textsf{Q}_{2}})\sim 1$ in (6.8) and obtain
$\displaystyle\mu(\Phi_{22})\sim\mu(\Phi_{11})\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\mu_{\rm gl,1}\gg\mu_{\rm gl,1}\,,$
(6.10) $\displaystyle\mu(\Pi_{11})\sim m^{\rm
pole}_{\textsf{Q}_{2}}\sim\frac{m_{Q}}{z_{Q}(\Lambda_{Q},\mu^{2}_{\rm
gl,1})}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{{\rm
b_{o}}/3{\overline{N}}_{c}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{2\,{\rm\overline{b}}_{\rm
o}/3{\overline{N}}_{c}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{c}/{\overline{N}}_{c}}\mu_{\rm gl,1}\ll\mu_{\rm
gl,1}.$ $\displaystyle\frac{\Lambda_{YM}^{(\rm br1)}}{m^{\rm
pole}_{\textsf{Q}_{2}}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\Delta}\ll 1,\quad\Delta=\frac{N_{c}(2n_{1}-{\rm
b_{o}})}{3{\overline{N}}_{c}(N_{c}-n_{1})}>0\,.$ (6.11)
6.2 $\rm\bf br2$ and $\rm\bf special$ vacua
At $n_{2}<N_{c}$ there are also $\rm br2$ \- vacua. All their properties can
be obtained by a replacement $n_{1}\leftrightarrow n_{2}$ in formulas of the
preceding section 6.1. The only difference is that, because $n_{2}\geq
N_{F}/2$ and so $2n_{2}>{\rm b_{o}}$, there is no analog of the conformal
regime at $\mu<\mu_{\rm gl,1}$ with $2n_{1}<{\rm b_{o}}$. I.e. at
$\mu<\mu_{\rm gl,2}$ the lower energy theory will be always in the
perturbative IR free logarithmic regime and the overall phase will be
$Higgs_{2}-HQ_{1}$.
As for the special vacua, all their properties can also be obtained with
$n_{1}={\overline{N}}_{c},\,n_{2}=N_{c}$ in formulas of the preceding section
6.1.
## 7 Dual theory. Broken flavor symmetry.
The region $\mathbf{\mu_{\Phi,\rm o}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}}$
7.1 $\rm\bf br1$ vacua, $\,\,{\rm\overline{b}}_{\rm o}/N_{F}\ll 1$
We recall, see (1.12), that condensates of mions and dual quarks in these
vacua are
$\displaystyle\langle M_{1}\rangle_{\rm
br1}\simeq\frac{N_{c}}{N_{c}-n_{1}}\,m_{Q}\mu_{\Phi}\,,\quad\langle
M_{2}\rangle_{\rm
br1}\sim\Lambda_{Q}^{2}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{1}}{N_{c}-n_{1}}}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}-N_{c}}{N_{c}-n_{1}}}\,,$
(7.1) $\displaystyle\frac{\langle M_{2}\rangle_{\rm br1}}{\langle
M_{1}\rangle_{\rm br1}}\sim\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}}{N_{c}-n_{1}}}\ll 1\,,$
$\displaystyle\langle N_{2}\rangle_{\rm
br1}\equiv\langle{\overline{q}}_{2}q_{2}(\mu=\Lambda_{Q})\rangle_{\rm
br1}=Z_{q}\frac{\langle M_{1}\rangle_{\rm br1}\Lambda_{Q}}{\mu_{\Phi}}\sim
Z_{q}m_{Q}\Lambda_{Q}\gg\langle N_{1}\rangle_{\rm br1}\,,$
and some potentially relevant masses look here as
$\displaystyle\mu_{q,1}=\mu_{q,1}(\mu=\Lambda_{Q})=\frac{\langle
M_{1}\rangle_{\rm
br1}}{Z_{q}\Lambda_{Q}}\sim\frac{m_{Q}\mu_{\Phi}}{Z_{q}\Lambda_{Q}}\,,\quad\frac{\mu_{q,2}}{\mu_{q,1}}=\frac{\langle
M_{2}\rangle_{\rm br1}}{\langle M_{1}\rangle_{\rm br1}}\ll 1\,,$ (7.2)
$\displaystyle
Z_{q}\sim\exp\Bigl{\\{}-\frac{1}{3{\overline{a}}_{*}}\Bigr{\\}}\sim\exp\Bigl{\\{}-\frac{{\overline{N}}_{c}}{7{\rm\overline{b}}_{\rm
o}}\Bigr{\\}}\ll 1\,,$ $\displaystyle\mu^{\rm
pole}_{q,1}\sim\frac{\Lambda_{Q}}{Z_{q}}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu^{\rm
pole}_{q,2}\,,\quad\frac{\Lambda_{YM}^{(\rm br1)}}{\mu^{\rm pole}_{q,1}}\sim
Z_{q}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{2}N_{c}}{3{\overline{N}}_{c}(N_{c}-n_{1})}}\ll
1\,,$ (7.3) $\displaystyle{\overline{\mu}}_{{\rm
gl},\,2}\sim\Lambda_{Q}\Bigl{(}\frac{\langle
N_{2}\rangle}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3N_{c}}\sim
Z_{q}^{1/2}\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\gg{\overline{\mu}}_{{\rm
gl},\,1}\,,$ $\displaystyle\frac{{\overline{\mu}}_{{\rm gl},\,2}}{\mu^{\rm
pole}_{q,1}}\sim Z_{q}^{3/2}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\ll 1\,.$ (7.4)
Hence, the largest mass is $\mu^{\rm pole}_{q,1}$ while the overall phase is
$HQ_{1}-HQ_{2}$. We consider below only the case $n_{1}<{\rm b_{o}}/2$, so
that the lower energy theory with ${\overline{N}}_{c}$ colors and
$N^{\prime}_{F}=n_{2}$ flavors at $\mu<\mu^{\rm pole}_{q,1}$ remains in the
conformal window.
After integrating out the heaviest quarks ${\overline{q}}_{1},q_{1}$ at
$\mu<\mu^{\rm pole}_{q,1}$ and ${\overline{q}}_{2},q_{2}$ quarks at
$\mu<\mu^{\rm pole}_{q,2}$ and, finally, all $SU({\overline{N}}_{c})$ dual
gluons at $\mu<\Lambda_{YM}^{(\rm br1)}$, the Lagrangian of mions looks as
$\displaystyle K=\frac{z_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,1})}{Z^{2}_{q}\Lambda_{Q}^{2}}\,{\rm
Tr}\Bigl{[}\,M_{11}^{\dagger}M_{11}+M_{12}^{\dagger}M_{12}+M_{21}^{\dagger}M_{21}+z^{\,\prime}_{M}(\mu^{\rm
pole}_{q,1},\mu^{\rm pole}_{q,2})M_{22}^{\dagger}M_{22}\Bigr{]}\,,$ (7.5)
$\displaystyle W=-{\overline{N}}_{c}S+W_{M}\,,\quad\quad S=\Bigl{(}\frac{\det
M}{\Lambda_{Q}^{{\rm
b_{o}}}}\Bigr{)}^{1/{\overline{N}}_{c}}\,,\quad\Lambda_{YM}^{(\rm
br1)}=\langle S\rangle^{1/3}\sim\Bigl{(}m_{Q}\langle
M_{2}\rangle\Bigr{)}^{1/3}\,.$ $\displaystyle W_{M}=m_{Q}{\rm
Tr}M-\frac{1}{2\mu_{\Phi}}\Bigl{[}\,{\rm Tr}(M^{2})-\frac{1}{N_{c}}({\rm
Tr}M)^{2}\Bigr{]}\,,\quad z_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,1})\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu^{\rm
pole}_{q,1}}\Bigr{)}^{2\,{\rm\overline{b}}_{\rm o}/N_{F}}\gg 1\,.$
From (7.5) : the hybrids $M_{12}$ and $M_{21}$ are massless,
$\mu(M_{12})=\mu(M_{21})=0$, while the pole mass of $M_{11}$ is (compare with
(6.9) )
$\displaystyle\mu^{\rm
pole}(M_{11})\sim\frac{Z^{2}_{q}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,1})\mu_{\Phi}}\,,\quad\frac{\mu^{\rm
pole}(M_{11})}{\Lambda_{YM}^{(\rm br1)}}\sim
Z^{2}_{q}\Bigl{(}\frac{\mu_{\Phi,\rm o}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{c}({\rm
b_{o}}-2n_{1})}{3{\overline{N}}_{c}(N_{c}-n_{1})}}\ll 1\,.$ (7.6)
The parametric behavior of $\mu^{\rm pole}_{q,2}$ and
$z^{\,\prime}_{M}(\mu^{\rm pole}_{q,1},\mu^{\rm pole}_{q,2})$ depends on the
value $\mu_{\Phi}\lessgtr{\tilde{\mu}}_{\Phi,1}$ (see below). We consider
first the case $\mu_{\Phi}\gg{\tilde{\mu}}_{\Phi,1}$ so that, by definition,
the lower energy theory with ${\overline{N}}_{c}$ colors and $n_{2}$ flavors
had enough ”time” to evolve and entered already the new conformal regime at
$\mu^{\rm pole}_{q,2}<\mu\ll\mu^{\rm pole}_{q,1}$, with
${\rm\overline{b}\,}^{\prime}_{o}/{\overline{N}}_{c}=(3{\overline{N}}_{c}-n_{2})/{\overline{N}}_{c}=O(1)$
and ${\overline{a}\,}^{\prime}_{*}=O(1)$. Hence, when the quarks
${\overline{q}}_{2},q_{2}$ decouple as heavy ones at $\mu<\mu^{\rm
pole}_{q,2}$, the coupling ${\overline{a}}_{YM}$ of the remained
$SU({\overline{N}}_{c})$ Yang-Mills theory is
${\overline{a}}_{YM}\sim{\overline{a}\,}^{\prime}_{*}=O(1)$ and this means
that $\mu^{\rm pole}_{q,2}\sim\Lambda_{YM}^{(\rm br1)}$. This can be obtained
also in a direct way. The running mass of quarks ${\overline{q}}_{2},q_{2}$ at
$\mu=\mu^{\rm pole}_{q,1}$ is, see (7.1)-(7.3),
$\displaystyle\mu_{q,2}(\mu=\mu^{\rm pole}_{q,1})=\frac{\langle
M_{2}\rangle_{\rm br1}}{\langle M_{1}\rangle_{\rm br1}}\,\mu^{\rm
pole}_{q,1}\,,\quad\mu^{\rm pole}_{q,2}=\frac{\mu_{q,2}(\mu=\mu^{\rm
pole}_{q,1})}{z^{\,\prime}_{q}(\mu^{\rm pole}_{q,1},\mu^{\rm
pole}_{q,2})}\sim\Lambda_{YM}^{(\rm br1)}\sim\Bigl{(}m_{Q}\langle
M_{2}\rangle\Bigr{)}^{1/3}\,,$ (7.7) $\displaystyle z^{\,\prime}_{q}(\mu^{\rm
pole}_{q,1},\mu^{\rm pole}_{q,2})=\Bigl{(}\frac{\mu^{\rm pole}_{q,2}}{\mu^{\rm
pole}_{q,1}}\Bigr{)}^{\frac{{\rm\overline{b}\,}^{\prime}_{o}}{n_{2}}}\rho\,,\quad\rho=\Bigl{(}\frac{{\overline{a}\,}_{*}}{{\overline{a}\,}^{\prime}_{*}}\Bigr{)}^{\frac{{\overline{N}}_{c}}{n_{2}}}\exp\Bigl{\\{}\frac{{\overline{N}}_{c}}{n_{2}}\Bigl{(}\frac{1}{{\overline{a}\,}_{*}}-\frac{1}{{\overline{a}\,}^{\prime}_{*}}\Bigr{)}\Bigr{\\}}\sim\exp\Bigl{\\{}\frac{{\overline{N}}_{c}}{n_{2}}\frac{1}{{\overline{a}\,}_{*}}\Bigr{\\}}\gg
1\,.$
We obtain from (7.5) that the main contribution to the mass of mions $M_{22}$
originates from the non-perturbative term $\sim S$ in the superpotential and,
using (7.5),(7.7),
$\displaystyle z^{\,\prime}_{M}(\mu^{\rm pole}_{q,1},\mu^{\rm
pole}_{q,2})=\frac{{\overline{a}}_{f}(\mu=\mu^{\rm
pole}_{q,1})}{{\overline{a}}_{f}(\mu=\mu^{\rm
pole}_{q,2})}\Bigl{(}\frac{1}{z^{\,\prime}_{q}(\mu^{\rm pole}_{q,1},\mu^{\rm
pole}_{q,2})}\Bigr{)}^{2}\sim\Bigl{(}\frac{1}{z^{\,\prime}_{q}(\mu^{\rm
pole}_{q,1},\mu^{\rm pole}_{q,2})}\Bigr{)}^{2}\,,$ (7.8)
$\displaystyle\mu(M_{22})\sim\frac{Z_{q}^{2}\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}_{q,1})z^{\,\prime}_{M}(\mu^{\rm pole}_{q,1},\mu^{\rm
pole}_{q,2})}\Biggl{(}\frac{\langle S\rangle}{\langle
M_{2}\rangle^{2}}=\frac{\langle M_{1}\rangle}{\langle
M_{2}\rangle}\frac{1}{\mu_{\Phi}}\Biggr{)}_{\rm br1}\,\,\sim\Lambda_{YM}^{(\rm
br1)}\sim\mu^{\rm pole}_{q,2}\,.$ (7.9)
We consider now the region $\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll{\tilde{\mu}}_{\Phi,1},\,2n_{1}\lessgtr{\rm b_{o}}$ where,
by definition, $\mu^{\rm pole}_{q,2}$ is too close to $\mu^{\rm pole}_{q,1}$,
so that the range of scales $\mu^{\rm pole}_{q,2}<\mu<\mu^{\rm pole}_{q,1}$ is
too small and the lower energy theory at $\mu<\mu^{\rm pole}_{q,1}$ has no
enough ”time” to enter a new regime (conformal at $2n_{1}<{\rm b_{o}}$ or
strong coupling one at $2n_{1}>{\rm b_{o}}$) and remains in the weak coupling
logarithmic regime. Then, ignoring logarithmic effects in renormalization
factors, $z^{\,\prime}_{q}(\mu^{\rm pole}_{q,1},\mu^{\rm pole}_{q,2})\sim
z^{\,\prime}_{M}(\mu^{\rm pole}_{q,1},\mu^{\rm pole}_{q,2})\sim 1$, and
keeping as always only the exponential dependence on
${\overline{N}}_{c}/{\rm\overline{b}}_{\rm o}$ :
$\displaystyle\mu^{\rm pole}_{q,2}\sim\frac{\langle M_{2}\rangle_{\rm
br1}}{\langle M_{1}\rangle_{\rm br1}}\,\mu^{\rm
pole}_{q,1}\,,\quad\quad\frac{\Lambda_{YM}^{(\rm br1)}}{\mu^{\rm
pole}_{q,2}}\ll 1\quad\rightarrow\quad\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll{\tilde{\mu}}_{\Phi,1}\,,$
$\displaystyle{\tilde{\mu}}_{\Phi,1}\sim\exp\Bigl{\\{}\frac{(N_{c}-n_{1})}{2n_{1}}\frac{1}{{\overline{a}\,}_{*}}\Bigr{\\}}\mu_{\Phi,\rm
o}\gg\mu_{\Phi,\rm o}\,.$ (7.10)
The pole mass of mions $M_{22}$ looks in this case as
$\displaystyle\frac{\mu^{\rm pole}(M_{22})}{\mu^{\rm
pole}(M_{11})}\sim\frac{\langle M_{1}\rangle_{\rm br1}}{\langle
M_{2}\rangle_{\rm br1}}\gg 1,\quad\frac{\mu^{\rm
pole}(M_{22})}{\Lambda_{YM}^{(\rm br1)}}\sim
Z^{2}_{q}\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,\rm
o}}\Bigr{)}^{\frac{2n_{1}N_{c}}{3{\overline{N}}_{c}(N_{c}-n_{1})}}\ll 1\,.$
(7.11)
On the whole, see (7.10), the mass spectrum at $\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll{\tilde{\mu}}_{\Phi,1}$ and $2n_{1}\lessgtr{\rm b_{o}}$
looks as follows. a) There is a large number of heaviest hadrons made of
weakly coupled (and weakly confined, the string tension being
$\sqrt{\sigma}\sim\Lambda_{YM}^{(\rm br1)}\ll\mu^{\rm pole}_{q,1})$
nonrelativistic quarks ${\overline{q}}_{1},q_{1}$, the scale of their masses
is $\mu^{\rm pole}_{q,1}$, see (7.3). b) The next physical mass scale is due
to $\mu^{\rm pole}_{q,2},\,\,\Lambda_{YM}^{(\rm br1)}\ll\mu^{\rm
pole}_{q,2}\ll\mu^{\rm pole}_{q,1}$. Hence, there is also a large number of
hadrons made of weakly coupled and weakly confined nonrelativistic quarks
${\overline{q}}_{2},q_{2}$, the scale of their masses is $\mu^{\rm
pole}_{q,2}$, see (7.10), and a large number of heavy hybrid hadrons with the
masses $\sim(\mu^{\rm pole}_{q,1}+\mu^{\rm pole}_{q,2})$. Because all quarks
are weakly coupled and non-relativistic in all three flavor sectors,
$"11",\,"22"$ and $"12+21"$, the mass spectrum of low-lying flavored mesons is
Coulomb-like with parametrically small mass differences
$\Delta\mu_{H}/\mu_{H}=O({\rm\overline{b}}_{\rm
o}^{2}/{\overline{N}}_{c}^{2})\ll 1$. c) A large number of gluonia made of
$SU({\overline{N}}_{c})$ gluons, with the mass scale $\sim\Lambda_{YM}^{(\rm
br1)}\sim\Bigl{(}m_{Q}\langle M_{2}\rangle\Bigr{)}^{1/3}$, see (7.5),(7.1). d)
$n^{2}_{2}$ mions $M_{22}$ with the pole masses $\mu^{\rm
pole}(M_{22})\ll\Lambda_{YM}^{(\rm br1)}$, see (7.11). e) $n^{2}_{1}$ mions
$M_{11}$ with the pole masses $\mu^{\rm pole}(M_{11})\ll\mu^{\rm
pole}(M_{22})$, see (7.6),(7.11). f) $2n_{1}n_{2}$ hybrids $M_{12},M_{21}$ are
massless, $\mu(M_{12})=\mu(M_{21})=0$.
The pole mass of quarks ${\overline{q}}_{2},q_{2}$ is smaller at
${\tilde{\mu}}_{\Phi,1}\ll\mu_{\Phi}\ll\Lambda_{Q}^{2}/m_{Q}$ and $2n_{1}<{\rm
b_{o}}$, and stays at $\mu^{\rm pole}_{q,2}\sim\Lambda_{YM}^{(\rm br1)}$,
while the mass of mions $M_{22}$ is larger and also stays at
$\mu(M_{22})\sim\Lambda_{YM}^{(\rm br1)}$.
7.2 $\rm\bf br2$ and $\rm\bf special$ vacua, ${\rm\overline{b}}_{\rm
o}/N_{F}\ll 1$
The condensates of mions look in these br2 - vacua as in (7.1) with the
exchange $1\leftrightarrow 2$. The largest mass is $\mu^{\rm pole}_{q,2}$,
$\displaystyle\mu^{\rm
pole}_{q,2}\sim\frac{\Lambda_{Q}}{Z_{q}}\Bigl{(}\frac{m_{Q}\mu_{\Phi}}{\Lambda_{Q}^{2}}\Bigr{)}^{N_{F}/3{\overline{N}}_{c}}\gg\mu^{\rm
pole}_{q,1}\,,\quad\frac{\Lambda_{YM}^{(\rm br2)}}{\mu^{\rm pole}_{q,2}}\sim
Z_{q}\Bigl{(}\frac{\mu_{\Phi,\rm
o}}{\mu_{\Phi}}\Bigr{)}^{\frac{n_{1}N_{c}}{3{\overline{N}}_{c}(N_{c}-n_{2})}}\ll
1\,,$ (7.12)
and the overall phase is $HQ_{1}-HQ_{2}$. After decoupling the heaviest quarks
${\overline{q}}_{2},q_{2}$ at $\mu<\mu^{\rm pole}_{q,2}$ the lower energy
theory remains in the weak coupling logarithmic regime at, see (7.10),
$\displaystyle\frac{\Lambda_{YM}^{(\rm br2)}}{\mu^{\rm pole}_{q,1}}\ll
1\quad\rightarrow\quad\mu_{\Phi,\rm
o}\ll\mu_{\Phi}\ll{\tilde{\mu}}_{\Phi,2}\,,\quad\frac{{\tilde{\mu}}_{\Phi,2}}{\mu_{\Phi,\rm
o}}\sim\exp\Bigl{\\{}\frac{(N_{c}-n_{2})}{2n_{2}}\frac{1}{{\overline{a}\,}_{*}}\Bigr{\\}}\gg
1\,.$ (7.13)
Hence, the mass spectra in this range $\mu_{\Phi,\rm
o}<\mu_{\Phi}\ll{\tilde{\mu}}_{\Phi,2}$ can be obtained from corresponding
formulas in section 7.1 by the replacements $n_{1}\leftrightarrow n_{2}$.
But because $n_{2}\geq N_{F}/2$, the lower energy theory with
$1<n_{1}/{\overline{N}}_{c}<3/2$ is in the strong coupling regime at
$\mu_{\Phi}\gg{\tilde{\mu}}_{\Phi,2}$, with ${\overline{a}}(\mu)\gg 1$ at
$\Lambda_{YM}^{(\rm br2)}\ll\mu\ll\mu^{\rm pole}_{q,2}$. We do not consider
the strong coupling regime in this paper.
As for the special vacua, the overall phase is also $HQ_{1}-HQ_{2}$ therein.
The mass spectra are obtained by substituting $n_{1}={\overline{N}}_{c}$ into
the formulas of section 7.1. At $5/3<N_{F}/N_{c}<2$ and
$\mu_{\Phi}\gg{\tilde{\mu}}_{\Phi,1}$ the lower energy theory in these special
vacua enters the strong coupling regime at $\Lambda_{YM}^{(\rm
spec)}\ll\mu\ll\mu^{\rm pole}_{q,1}$.
## 8 Conclusions
The mass spectra of the direct $\bf\Phi$ -theory and its dual variant, the
$\bf d\Phi$ -theory, were calculated in [1] within the dynamical scenario
$\\#1$ which implies a possibility of (quasi)spontaneous breaking of chiral
flavor symmetry. I.e., the phase is formed with a large coherent diquark
condensate (DC), $\langle{\overline{Q}}Q\rangle$, and without higgsing of
quarks, $\langle{\overline{Q}}\rangle=\langle Q\rangle=0$. As a result, the
light quarks acquire the large dynamical constituent mass
$\mu_{C}\sim\langle{\overline{Q}}Q\rangle^{1/2}$ and much lighter (pseudo)
Nambu-Goldstone pions appear. This paper continues [1]. The mass spectra of
the $\bf\Phi$ and $\bf d\Phi$ theories are calculated here within the
dynamical scenario $\\#2$ which implies that the DC phase is not formed and
the quarks may be in the two different phases only : either they are in the HQ
(heavy quark) phase where they are confined, or they are higgsed at
appropriate values of the lagrangian parameters.
As was shown above in the main text, the use of the additional small parameter
${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ allows to trace explicitly parametrical
differences in the mass spectra of the direct $\bf\Phi$ and dual $\bf d\Phi$
theories. This shows that these two theories are not equivalent. A similar
situation takes place when comparing the mass spectra of the ordinary ${\cal
N}=1$ SQCD and its dual variant [2].
At present, unfortunately, to calculate the mass spectra of ${\cal N}=1$ SQCD-
like theories one is forced to assume a definite dynamical scenario. But
nevertheless, from our standpoint, a most important thing at present is a very
ability to calculate the mass spectra of various ${\cal N}=1$ SQCD-like
theories, even within a given dynamical scenario. It seems clear that further
developments will allow to find a unique right scenario in each such theory.
This work is supported in part by Ministry of Education and Science of the
Russian Federation and RFBR grant 12-02-00106-a.
## References
* [1] V.L. Chernyak, Mass spectrum in SQCD with additional fields. I, arXiv: 1205.0410[hep-th]
* [2] V.L. Chernyak, On mass spectrum in SQCD and problems with the Seiberg duality.
Another scenario, JETP 114 (2012) 61, arXiv: 0811.4283 [hep-th]
* [3] V. Novikov, M. Shifman, A. Vainshtein, V. Zakharov, Exact Gell-Mann-Low function of
supersymmetric Yang-Mills theories from instanton calculus, Nucl. Phys. B 229
(1983) 381
* [4] M. Shifman, A. Vainshtein, Solution of the anomaly puzzle in SUSY gauge theories and the Wilson operator expansion, Nucl. Phys. B 277 (1986) 456
* [5] K. Konishi, Anomalous supersymmetry transformation of some composite operators in SQCD, Phys. Lett. B 135 (1984) 439
* [6] N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857, hep-th/9402044
* [7] N. Seiberg, Electric - magnetic duality in supersymmetric nonabelian gauge theories,
Nucl. Phys. B 435 (1995) 129, hep-th/9411149
* [8] V.L. Chernyak, On mass spectrum in SQCD and problems with the Seiberg duality.
Equal quark masses, JETP 110 (2010) 383, arXiv : 0712.3167 [hep-th]
* [9] V.L. Chernyak, On mass spectrum in SQCD. Unequal quark masses, JETP 111 (2010) 949, arXiv : 0805.2299 [hep-th]
* [10] G. Veneziano, S. Yankielowicz, An effective Lagrangian for the pure ${\cal N}=1$ supersymmetric Yang-Mills theory, Phys. Lett. B 113 (1982) 231
* [11] I. Kogan, M. Shifman, A. Vainshtein, Phys. Rev. D 53 (1996) 4526,
hep-th/9507170 ; Err: Phys. Rev. D 59 (1999) 109903
|
arxiv-papers
| 2012-11-07T09:04:30 |
2024-09-04T02:49:37.631716
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Victor L. Chernyak",
"submitter": "Victor Chernyak",
"url": "https://arxiv.org/abs/1211.1487"
}
|
1211.1497
|
# Mobility edge phenomenon in a Hubbard chain: A mean field study
Santanu K. Maiti santanu.maiti@isical.ac.in Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India Abraham Nitzan School of Chemistry, Tel Aviv University, Ramat-
Aviv, Tel Aviv-69978, Israel
###### Abstract
We show that a tight-binding one-dimensional chain composed of interacting and
non-interacting atomic sites can exhibit multiple mobility edges at different
values of carrier energy in presence of external electric field. Within a mean
field Hartree-Fock approximation we numerically calculate two-terminal
transport by using Green’s function formalism. Several cases are analyzed
depending on the arrangements of interacting and non-interacting atoms in the
chain. The analysis may be helpful in designing mesoscale switching devices.
###### pacs:
73.63.Nm, 72.20.Ee, 73.21.-b
## I Introduction
Electronic localization phenomena in one-dimensional ($1$D) quantum systems
have long been a central problem in condensed matter physics. It is well
established that in infinite $1$D systems with random site potentials,
irrespective of the strength of randomness, all the energy eigenstates are
exponentially localized anderson . Apart from this Anderson type localization
another kind of localization known as Wannier-Stark localization is also
observed in $1$D materials, even in absence of any disorder, when the system
is subjected to an external electric field wan . For both cases i.e., infinite
$1$D systems with random site potentials and $1$D chains in presence of
external electric field, one never encounters mobility edges separating the
localized energy eigenstates from the extended ones, since all eigenstates are
localized. However, there are some classes of $1$D systems such as, correlated
disordered models, quasi-periodic Aubry-Andre model where several classic
features of mobility edges at some specific values of energy are obtained dun
; sanch ; fa ; fm ; dom ; aubry ; san6 ; eco ; das ; rolf . Although the
existence of such mobility edges in one- or two-dimensional systems has been
described by several groups eco ; das ; rolf ; sch ; san1 ; san2 , a
comprehensive study of this phenomenon is still lacking, particularly in the
presence of electron-electron interaction. Still open fundamental questions
are whether some special features exist in disordered $1$D systems, or in the
response of $1$D systems to an externally applied electric field even in the
presence of electron-electron interaction.
In the present article we investigate two-terminal electron transport through
a $1$D mesoscopic chain composed of interacting and non-interacting atomic
sites in presence of external electric field. Although some works have been
done in such superlattice structures pai1 ; pai2 ; pai3 ; pai4 ; wang , the
analogous representation of metallic multilayered structures which exhibit
several novel features multi1 ; multi2 ; multi3 , no rigorous effort has been
made so far, to the best of our knowledge, to unravel the effect of the
interplay of electron-electron interaction and an imposed external electric
field on electron transport in such systems. Here we show that a traditional
$1$D lattice with electron-electron interaction, evaluated at the Hartree-Fock
(HF) mean field (MF) level, is characterized by a mobility edge behavior at
finite bias voltage. Furthermore, a superlattice structure comprising sites on
which electron-electron interactions are expressed differently (some sites are
interacting and some sites are non-interacting) is characterized by multiple
occurrence of mobility edges at several values of the carrier energy. The
applicability of mean field approximation in such superlattice geometries has
already been reported in a recent work meanfield .
## II Model and Calculation
We adopt a tight-binding (TB) framework to describe the model quantum system
and numerically calculate two-terminal transport within a mean field
Figure 1: (Color online). A $1$D mesoscopic chain, composed of interacting
(filled black circle) and non-interacting (filled green circle) atomic sites,
is attached to two semi-infinite $1$D metallic electrodes, representing source
and drain.
Hartree-Fock approximation using a Green’s function formalism. Several cases
characterized by different arrangements of interacting and non-interacting
atomic sites in the chain, are analyzed. For these models we calculate the
average density of states (ADOS) and the two-terminal transmission
probability, and find that sharp crossovers from completely opaque to fully or
partly transmitting zones take place at one or more specific electron
energies. This observation suggests the possibility of controlling the
transmission characteristics by gating the transmission zone, and using such
superlattice structures as switching devices.
Let us refer to Fig. 1 where a $1$D mesoscopic chain, composed of non-
interacting and interacting atomic sites, is attached to two semi-infinite
$1$D non-interacting source and drain electrodes. In the arrangement of the
two different atomic sites shown in Fig. 1, $M$ ($M\geq 1$) non-interacting
sites are placed between two interacting sites. Here and in what follows we
make a restriction that interacting atoms are not placed successively. In a
Wannier basis, the TB Hamiltonian for a $N$-site chain reads,
$\displaystyle H_{C}$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}+\sum_{\langle
ij\rangle,\sigma}t\left[c_{i\sigma}^{\dagger}c_{j\sigma}+c_{j\sigma}^{\dagger}c_{i\sigma}\right]$
(1) $\displaystyle+$
$\displaystyle\sum_{i}U_{i}c_{i\uparrow}^{\dagger}c_{i\uparrow}c_{i\downarrow}^{\dagger}c_{i\downarrow}$
where, $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$) is the creation (annihilation)
operator of an electron at the site $i$ with spin $\sigma$
($=\uparrow,\downarrow$), $t$ is the nearest-neighbor hopping element,
$\epsilon_{i\sigma}$ is the on-site energy of an electron at the site $i$ of
spin $\sigma$ and $U_{i}$ is the strength of on-site Coulomb interaction where
$U_{i}=0$ for the non-interacting sites. In presence of bias voltage $V$
between the two electrodes an electric field is developed and the site
energies become voltage dependent,
$\epsilon_{i\sigma}=\epsilon_{i}^{0}+\epsilon_{i}(V)$, where
$\epsilon_{i}^{0}$ is a voltage independent term. For the ordered chain
$\epsilon_{i}^{0}$ is a
Figure 2: (Color online). Variation of voltage dependent site energies in a
$1$D chain with $300$ lattice sites for three different electrostatic
potential profiles when the bias voltage $V$ is fixed at $0.2$.
constant independent of $i$ that can be chosen zero without loss of
generality, while for the disordered case we select it randomly from a uniform
“Box” distribution function in the range $-W/2$ to $W/2$.
The voltage dependence of $\epsilon_{i}(V)$ reflects the bare electric field
in the bias junction as well as screening due to longer range e-e interaction
not explicitly accounted for in Eq. 1. In the absence of such screening the
electric field is uniform along the chain and $\epsilon_{i}(V)=V/2-iV/(N+1)$.
Below we consider this as well as screened electric field profiles, examples
of which are shown in Fig. 2. We will see that the appearance of multiple
mobility edges in superlattice geometries strongly depends on the existence of
finite bias and on the profile of the bias drop along the chain.
The Hamiltonian for the non-interacting ($U_{i}=0$) electrodes can be
expressed as,
$H_{\mbox{lead}}=\sum_{p}\epsilon_{0}c_{p}^{\dagger}c_{p}+\sum_{<pq>}t_{0}\left(c_{p}^{\dagger}c_{q}+c_{q}^{\dagger}c_{p}\right)$
(2)
with site energy and nearest-neighbor intersite coupling $\epsilon_{0}$ and
$t_{0}$, respectively. These electrodes are directly coupled to the $1$D chain
through the lattice sites $1$ and $N$. The hopping integrals between the
source and chain and between the chain and drain are denoted by $\tau_{S}$ and
$\tau_{D}$, respectively.
In the generalized HF approach san3 ; san4 ; san5 ; kato ; kam , the full
Hamiltonian is decoupled into its up-spin and down-spin components by
replacing the interaction terms by their mean field (MF) counterparts. This
redefines the on-site energies as
$\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 MF approximation in the decoupled form
$\displaystyle H_{MF}$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i\uparrow}^{\prime}n_{i\uparrow}+\sum_{\langle
ij\rangle}t\left[c_{i\uparrow}^{\dagger}c_{j\uparrow}+c_{j\uparrow}^{\dagger}c_{i\uparrow}\right]$
(3) $\displaystyle+$
$\displaystyle\sum_{i}\epsilon_{i\downarrow}^{\prime}n_{i\downarrow}+\sum_{\langle
ij\rangle}t\left[c_{i\downarrow}^{\dagger}c_{j\downarrow}+c_{j\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle-$ $\displaystyle\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ $\displaystyle=$
$\displaystyle H_{C,\uparrow}+H_{C,\downarrow}-\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$
where, $H_{C,\uparrow}$ and $H_{C,\downarrow}$ correspond to the effective
Figure 3: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D non-interacting
($U_{i}=0$ $\forall$ $i$) ordered ($W=0$) chain with $N=300$ sites. The
electrostatic potential profile varies linearly (red curve in Fig. 2), with
the total potential drop across the chain to be (a) $V=0$ and (b) $V=0.2$.
TB Hamiltonians for the up and down spin electrons, respectively. The last
term provides a shift in the total energy that depends on the mean populations
of the up and down spin states.
Figure 4: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for an ordered ($W=0$) $1$D chain.
The left column corresponds to the case where all sites are interacting
($U_{i}=2$), while the right column represents the results for a $1$D
superlattice geometry where four non-interacting ($U_{i}=0$) atoms are placed
between two interacting ($U_{i}=2$) atoms. The 1st, 2nd and 3rd rows
correspond to $V=0$, $0.1$ and $0.2$, respectively. All these results are
shown for a linear bias drop along the chain.
With these decoupled Hamiltonians ($H_{C,\uparrow}$ and $H_{C,\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.
The converged mean field Hamiltonian is a sum of single electron up and down
spin Hamiltonians. The transmission function is therefore a sum
$T(E)=\sum_{\sigma}T_{\sigma}(E)$ where datta
$T_{\sigma}={\mbox{Tr}}\left[\Gamma_{S}\,G_{C,\sigma}^{r}\,\Gamma_{D}\,G_{C,\sigma}^{a}\right]$.
Here, $G_{C,\sigma}^{r}$ and $G_{C,\sigma}^{a}$ are the retarded and advanced
Green’s functions, respectively, of the chain including the effects of the
electrodes.
$G_{C,\sigma}=\left(E-H_{C,\sigma}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$, where
$\Sigma_{S}$ and $\Sigma_{D}$ are the self-energies due to coupling of the
chain to the source and drain, respectively, while $\Gamma_{S}$ and
$\Gamma_{D}$ are their imaginary parts.
## III Results and Discussion
In what follows we limit ourselves to absolute zero temperature and use the
units where $c=h=e=1$. For the numerical calculations we choose $t=1$,
$\epsilon_{0}=0$, $t_{0}=3$ and $\tau_{S}=\tau_{D}=1$. The energy scale is
measured in unit of $t$.
Before addressing the central problem i.e., the possibility of getting
multiple mobility edges in $1$D superlattice geometries, first we explore
Figure 5: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) in
absence of disorder ($W=0$) with on-site interaction $U_{i}=2$ and bias
voltage $V=0.2$ that varies linearly along the chain. Here we set $M=5$ i.e,
five non-interacting atoms are placed between two interacting atoms.
the effect of finite bias on electron transport in two simple systems, one for
a standard non-interacting chain and the other for a conventional Hubbard
chain where all sites are interacting.
In Fig. 3 we show the variation of total transmission probability (T) together
with the average density of states as a function of energy $E$ for an ordered
($\epsilon_{i}^{0}=0$ for all atomic sites $i$ in the chain) non-interacting
chain for two different magnitudes of the voltage bias, assuming a linear bias
drop (uniform electric field) across the chain. In the absence of electric
field electron transmission takes place throughout the energy band as clearly
seen from the spectrum Fig. 3(a), since in this case all the energy
eigenstates are extended. On the other hand, when a finite bias drop takes
Figure 6: (Color online). Same as Fig. 5, with $M=6$.
place along the chain, several energy eigenstates appear in the energy regions
around the band edges for which the transmission probability is exactly zero
(Fig. 3(b)). Therefore, the chain appears insulating when Fermi energy is
within the zone of zero transmission, while finite transmission, $T\neq 0$, is
seen more towards the band centre. The sharp transition between these regimes
illustrates the existence of a mobility edge phenomenon under finite bias
Figure 7: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) in
presence of disorder ($W=0.5$) for the same parameter values used in Fig.
4(f): $U_{i}=2$, $V=0.2$ (linear potential profile) and $M=4$.
condition. For a finite bias, the localization of energy levels always starts
from the band edges and the width of the localized energy zones can be
controlled by the imposed electric field. Obviously, for strong enough
electric field almost all the energy levels are localized and the extended
energy regions disappear, so that in this particular case metal-insulator (MI)
transition will no longer be observed. This localization phenomenon in
presence of an external electric field has already been established in the
literature, but the central issue of our present investigation - the interplay
between the Hubbard interaction strength, the superlattice configuration and
the electric field has not been addressed earlier.
To explore it, we present in Fig. 4 the results of a traditional Hubbard chain
where all sites are interacting (left column) together with the results of a
superlattice geometry where four non-interacting atoms are placed between two
interacting atoms (right column). The results are shown for three different
values of the voltage bias, taking a linear bias drop along the $1$D chain.
For the chain where all sites are interacting a single energy gap only appears
at the band centre, while in the superlattice geometry, depending on the unit
cell configuration, multiple energy gaps are generated which are clearly
visible from the ADOS spectra. Therefore, in a superlattice geometry, in
presence of external electric field associated with bias voltage $V$ between
two electrodes, zero transmission ($T=0$) energy regions exist, and are
separated by regions of extended states compared to the traditional Hubbard
chain, and, it leads to the possibility of getting an MI-like transition at
multiple energies.
The total number of energy sub-bands in a superlattice geometry for a
Figure 8: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) with no
disorder ($W=0$). The model parameters are $U=2$, $M=4$ and $V=0.2$, with the
bias potential profile taken as the green curve given of Fig. 2.
particular energy range generated in the ADOS profile strongly depends on
structural details i.e., the number $M$ of non-interacting atoms between two
interacting lattice sites. This is shown in Figs. 5 and 6 which show the ADOS
and the transmission probability for two models that are identical in all
details (see caption to Fig. 5) except that $M=5$ in Fig. 5 and $M=6$ in Fig.
6. These structures show more mobility edge phenomena, that is crossovers
between fully opaque and a transmitting zone, than in the corresponding case
of Fig. 4(f), suggesting a design concept based on such superlattice
structures as a switching devices at multiple energies.
The robustness of the observed behavior can be examined by its sensitivity to
the presence of disorder. Figure 7 displays the ADOS spectrum and the total
transmission probability for a $1$D chain in presence of diagonal disorder
affected by choosing $\epsilon_{i}^{0}$ from a uniform distribution of width
$W=0.5$ ($-0.25$ to $+0.5$). An average over $50$ disorder configurations is
presented. The resulting ADOS and transmission show similar qualitative
features, with sharp transitions between localized and extended spectral
regions as seen above for the ordered cases. Note that the presence of
disorder alone can cause state localization. For strong enough disorder almost
all energy levels get localized, even for such a finite size $1$D chain. In
this limit such crossover behavior will no longer be observed.
In the calculations presented so far we have assumed a linear drop of the
electrostatic potential along the chain. Figures 8 and 9 show results obtained
for an identical chain length with other potential profiles that are
characteristics of stronger screening.
Figure 9: (Color online). Same as Fig. 8, with the electrostatic potential
profile given by the blue curve in Fig. 2.
We see that the localized region gradually decreases with increasing flatness
of the potential profile in the interior of the conducting bridge. If the
potential drop takes place only at the chain-to-electrode interfaces i.e.,
when the potential profile becomes almost flat along the chain the width of
the localized region almost vanishes and the metal-insulator transition is not
observed, as was the case for the zero bias limit.
Finally, we point out that by locating the Fermi energy in appropriate places
of the sub-bands, the system can be used as a p-type or an n-type
semiconductor. For example, let us imagine, at absolute zero temperature, the
Fermi level is fixed in the localized region which is very close to the fully
transmitting zone (right hand side). In this case, the left sub-bands up to
the Fermi level are completely filled with electrons. Now, if the energy gap
between the Fermi level, pinned in the localized region, and the bottom of the
transmitting region (right hand side) is small enough for electrons to hop,
then the system will behave as an n-type semiconductor. On the other hand, by
reverting the situation we can generate a p-type semiconductor where electrons
hop from a filled transmitting zone (valence band) to unoccupied localized
zone (conduction band) generating holes in the valence band.
## IV Conclusion
To summarize, we have investigated in detail the two-terminal finite bias
electron transport in a $1$D superlattice structure composed of interacting
and non-interacting atoms. The electron-electron interaction is considered in
the Hubbard form, and the Hamiltonian is solved within a generalized HF
scheme. We numerically calculate two-terminal transport by using a Green’s
function formalism and analyze the results for some specific chain structures
characterized by different arrangements of the atomic sites in the chain. Our
analysis may be utilized in designing a tailor made switching device for
multiple values of Fermi energy (or, more practically, for different values of
a gating potential). The sensitivity of this switching action i.e., metal-to-
insulator transition and vice versa on the electric field variation has also
been discussed. Though the results presented in this article are worked out at
absolute zero temperature limit, the results should remain valid even at
finite temperatures ($\sim 300\,$K) since the broadening of the energy levels
of the superlattice structure due to its coupling with the metal electrodes is
much higher than that of the thermal broadening datta .
## V Acknowledgment
The research of A.N. is supported by the Israel Science Foundation, the
Israel-US Binational Science Foundation, the European Research Council under
the European Union’s Seventh Framework Program (FP7/2007-2013; ERC Grant No.
226628) and the Israel-Niedersachsen Research Fund.
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|
arxiv-papers
| 2012-11-07T09:59:45 |
2024-09-04T02:49:37.646349
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti and Abraham Nitzan",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1211.1497"
}
|
1211.1541
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-327 LHCb-PAPER-2012-033 Dec. 24, 2012
First observation of the decays $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The first observation of the decays $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are reported using an integrated luminosity of
1.0 $\rm fb^{-1}$ recorded by the LHCb experiment. The branching fractions,
normalized with respect to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively, are measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2},$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$
where the first uncertainty is statistical and the second is systematic. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay is of particular interest as it can be
used to measure the weak phase $\gamma$. First observation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decay is also presented, and its branching fraction relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is found to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(4.0\pm 1.0\pm 0.4)\times
10^{-3}.$
Submitted to Physical Review D
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, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
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, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, 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,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, 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. Contu52,15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, 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, O. Deschamps5, F. Dettori39, A. Di
Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F.
Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, 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, P. Garosi51, J. Garra
Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, 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, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, 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. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F.
Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K.
Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M.
Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D.
Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, 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, Y. Li3, L. Li Gioi5, 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, H. Luo47,
A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O.
Maev27,35, J. Magnin1, M. Maino20, S. Malde52, 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, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, 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, T.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, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, 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, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
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Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, 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, 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.
Shears49,35, 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, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, G. Veneziano36, M. Vesterinen35, 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,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, 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
25National Center for Nuclear Research (NCBJ), 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
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
In the Standard Model (SM), the amplitudes associated with flavor-changing
processes depend on four Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] matrix
parameters. Contributions from physics beyond the Standard Model (BSM) add
coherently to these amplitudes, leading to potential deviations in rates and
CP-violating asymmetries when compared to the SM contributions alone. Since
the SM does not predict the CKM parameters, it is important to make precise
measurements of their values in processes that are expected to be insensitive
to BSM contributions. Their values then provide a benchmark to which BSM-
sensitive measurements can be compared.
The least well-determined of the CKM parameters is the weak phase
${\gamma\equiv{\rm arg}\left(-{V_{\rm ub}^{*}V_{\rm ud}\over V_{\rm
cb}^{*}V_{\rm cd}}\right)}$, which, through direct measurements, is known to a
precision of ${\sim 10^{\rm o}-12^{\rm o}}$ [3, 4]. It may be probed using
time-independent rates of decays such as $B^{-}\rightarrow DK^{-}$ [5,
*Dunietz:1992ti, *Atwood:1994zm, *Atwood:1996ci, 9, *Gronau:1991dp, 11], or by
analyzing the time-dependent decay rates of processes such as
$B_{s}^{0}\rightarrow D_{s}^{\mp}K^{\pm}$ [12, 13, 14, 15]. Sensitivity to the
weak phase $\gamma$ results from the interference between $b\rightarrow c$ and
$b\rightarrow u$ transitions, as indicated in Figs. 1(a-c). Such measurements
may be extended to multibody decay modes, such as $B^{-}\rightarrow
DK^{-}\pi^{+}\pi^{-}$ [16] for a time-independent measurement, or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ in the case of a time-dependent analysis.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ decay, while having the same final state as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, receives contributions not only from the
$W$-exchange process (Fig. 1(d)), but also from $b\rightarrow c$ transitions
in association with the production of an extra $s\bar{s}$ pair (Figs. 1(e-f)).
The decay may also proceed through mixing followed by a $b\rightarrow u$,
$W$-exchange process (not shown). However, this amplitude is Cabibbo-,
helicity- and color-suppressed, and is therefore negligible compared to the
$b\rightarrow c$ amplitude.
Figure 1: Diagrams contributing to the $B^{0}_{s},\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ (a-c) and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ (d-f) decays, as described in the text. In
(a-d), the additional ($\pi^{+}\pi^{-}$) indicates that the
$K^{-}\pi^{+}\pi^{-}$ may be produced either through an excited strange kaon
resonance decay, or through fragmentation.
This paper reports the first observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, and measurements of their branching fractions
relative to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively. The data sample is based on an
integrated luminosity of 1.0 $\rm fb^{-1}$ of $pp$ collisions at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb experiment. The same
data sample is also used to observe the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decay for the first time, and measure its branching fraction relative to
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$. The inclusion of charge-conjugated modes is
implied throughout this paper.
## 2 Detector and simulation
The LHCb detector [17] 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 (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) 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 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. The software trigger requires a two-, three- or
four-track secondary vertex with a high $p_{\rm T}$ sum of the tracks and a
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, an IP $\chi^{2}$ greater than
16 with respect to all PVs, and a track fit $\chi^{2}/\rm{ndf}<2$, where ndf
is the number of degrees of freedom. The IP $\chi^{2}$ is defined as the
difference between the $\chi^{2}$ of the PV reconstructed with and without the
considered particle. A multivariate algorithm is used for the identification
of secondary vertices [18].
For the simulation, $pp$ collisions are generated using Pythia 6.4 [19] with a
specific LHCb configuration [20]. Decays of hadronic particles are described
by EvtGen [21] in which final state radiation is generated using Photos [22].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [23, *Agostinelli:2002hh] as
described in Ref. [25].
## 3 Signal selection
Signal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ decay
candidates are formed by pairing a $D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$
candidate with either a $\pi^{-}\pi^{+}\pi^{-}$ (hereafter referred to as
$X_{d}$) or a $K^{-}\pi^{+}\pi^{-}$ combination (hereafter referred to as
$X_{s}$). Tracks used to form the $D^{+}_{s}$ and $X_{d,s}$ are required to be
identified as either a pion or a kaon using information from the RICH
detectors, have $p_{\rm T}$ in excess of 100
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and be significantly detached from any
reconstructed PV in the event.
Signal $D^{+}_{s}$ candidates are required to have good vertex fit quality, be
significantly displaced from the nearest PV, and have invariant mass,
$M(K^{+}K^{-}\pi^{+})$, within 20 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the $D^{+}_{s}$ mass [26]. To suppress combinatorial and charmless
backgrounds, only those $D^{+}_{s}$ candidates that are consistent with
decaying through either the $\phi$
($M(K^{+}K^{-})<1040~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) or $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
($|M(K^{-}\pi^{+})-m_{K^{*0}}|<75~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
resonances are used (here, $m_{K^{*0}}$ is the $K^{*0}$ mass [26]). The
remaining charmless background yields are determined using the $D^{+}_{s}$
mass sidebands. For about 20% of candidates, when the $K^{+}$ is assumed to be
a $\pi^{+}$, the corresponding $K^{-}\pi^{+}\pi^{+}$ invariant mass is
consistent with the $D^{+}$ mass. To suppress cross-feed from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}X$ decays, a
tighter particle identification (PID) requirement is applied to the $K^{+}$ in
the $D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ candidates when
$|M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}|<20~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
($m_{D^{+}}$ is the $D^{+}$ mass [26]). Similarly, if the invariant mass of
the particles forming the $D^{+}_{s}$ candidate, after replacing the $K^{+}$
mass with the proton mass, falls within 15
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\Lambda_{c}^{+}$ mass,
tighter PID selection is applied. The sizes of these mass windows are about
2.5 times the invariant mass resolution, and are sufficient to render these
cross-feed backgrounds negligible.
Candidate $X_{d}$ and $X_{s}$ are formed from $\pi^{-}\pi^{+}\pi^{-}$ or
$K^{-}\pi^{+}\pi^{-}$ combinations, where all invariant mass values up to 3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are accepted. To reduce the level
of combinatorial background, we demand that the $X_{d,s}$ vertex is displaced
from the nearest PV by more than 100 $\,\upmu\rm m$ in the direction
transverse to the beam axis and that at least two of the daughter tracks have
$\mbox{$p_{\rm T}$}>300~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Backgrounds
to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ search from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{(*)+}\pi^{-}\pi^{+}\pi^{-}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}K^{+}\pi^{-}$ decays are suppressed by applying more stringent
PID requirements to the $K^{-}$ and $\pi^{+}$ in $X_{s}$. The PID requirements
have an efficiency of about 65% for selecting $X_{s}$, while rejecting about
97% of the favored three-pion background. To suppress peaking backgrounds from
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}D^{-}_{s}$ decays, where
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+},~{}K^{+}\pi^{-}\pi^{+}$, it is
required that $M(X_{d,s})$ is more than 20
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ away from the $D^{+}_{s}$ mass.
Signal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson candidates are
then formed by combining a $D^{+}_{s}$ with either an $X_{d}$ or $X_{s}$. The
reconstructed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ candidate is
required to be well separated from the nearest PV with a decay time larger
than 0.2 ps and have a good quality vertex fit. To suppress remaining
charmless backgrounds, which appear primarily in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the vertex separation (VS) $\chi^{2}$ between
the $D^{+}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ decay
vertices is required to be greater than 9. Candidates passing all selection
requirements are refit with both $D^{+}_{s}$ mass and vertex constraints to
improve the mass resolution [27].
To further suppress combinatorial background, a boosted decision tree (BDT)
selection [28, *Roe] with the AdaBoost algorithm[30] is employed. The BDT is
trained using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays for the signal distributions, and the
high $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mass sideband in data are
used to model the backgrounds. The following thirteen variables are used:
* •
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ candidate: IP $\chi^{2}$, VS
$\chi^{2}$, vertex fit $\chi^{2}$, and $p_{\rm T}$;
* •
$D^{+}_{s}$ candidate: Flight distance significance from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ vertex;
* •
$X_{d,s}$ candidate: IP $\chi^{2}$, maximum of the distances of closest
approach between any pair of tracks in the decay;
* •
$X_{d,s}$ daughters: min(IP $\chi^{2}$), max(IP $\chi^{2}$), min($p_{\rm T}$);
* •
$D^{+}_{s}$ daughters: min(IP $\chi^{2}$), max(IP $\chi^{2}$), min($p_{\rm
T}$),
where min and max denote the minimum and maximum of the indicated values
amongst the daughter particles. The flight distance significance is the
separation between the $D^{+}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ vertices, normalized by the
uncertainty. The training produces a single variable, $x$, that provides
discrimination between signal decays and background contributions. The cut
value is chosen by optimizing $S(x_{\rm cut})/\sqrt{S(x_{\rm cut})+B(x_{\rm
cut})}$, where $S(x_{\rm cut})$ and $B(x_{\rm cut})$ are the expected signal
and background yields, respectively, after requiring $x>x_{\rm cut}$. At the
optimal point, a signal efficiency of $\sim$90% is expected while rejecting
about $85\%$ of the combinatorial background (after the previously discussed
selections are applied). After all selections, about 3% of events have more
than one signal candidate in both data and simulation. All candidates are kept
for further analysis.
## 4 Fits to data
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ invariant mass spectra are each modeled by the
sum of a signal and several background components. The signal shapes are
obtained from simulation, and are each described by the sum of a Crystal Ball
(CB) [31] shape and a Gaussian function. The CB shape parameter that describes
the tail toward low mass is fixed based on simulated decays. A common, freely
varying scale factor multiplies the width parameters in the CB and Gaussian
functions to account for slightly larger resolution in data than in
simulation. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass fit, the difference between the mean $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ masses is fixed to 87.35
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26].
Several non-signal $b$-hadron decays produce broad peaking structures in the
$D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ invariant
mass spectra. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, the only significant source of peaking
background is from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$, where the photon or $\pi^{0}$ from the
$D^{*+}_{s}$ decay is not included in the reconstructed decay. Since the full
decay amplitude for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is not known, the simulation may not
adequately model the decay. Simulation is therefore used to provide an
estimate for the shape, but the parameters are allowed to vary within one
standard deviation about the fitted values.
For $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, backgrounds from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ and from misidentified $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays are considered. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ shape is fixed to be the same as that obtained
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ component in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass fit. This same shape is assumed for both
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, where for the former, a shift
by the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass difference is included.
For the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ cross-feed, simulated decays and kaon
misidentification rates taken from $D^{*+}$ calibration data are used to
obtain their expected yields and invariant mass shapes. The cross-feed
contribution is about 3% of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ yields; the corresponding cross-feed yields
are fixed in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ fit. The shape is obtained by parameterizing the
invariant mass spectrum obtained from the simulation after replacing the
appropriate $\pi^{-}$ mass in $X_{d}$ with the kaon mass. The combinatorial
background is described by an exponential function whose slope is allowed to
vary independently for both mass fits.
Figure 2 shows the invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates passing all selection criteria. The
fitted number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ signal events is $5683\pm 83$. While it is
expected that most of the low mass background emanates from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays, contributions from other sources such
as $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ are also possibly absorbed into this
background component. Figure 3 shows the invariant mass distribution for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal yields are $402\pm
33$ $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $216\pm 21$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ events.
Figure 2: Invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal probability
disrtibution function (PDF) is indicated by the dashed line and the background
shapes are shown as shaded regions, as described in the text. Figure 3:
Invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal (dashed lines) and
background shapes (shaded/hatched regions) are shown, as described in the
text.
The $D^{+}_{s}$ mass sidebands, defined to be from 35 to 55
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ on either side of the nominal
$D^{+}_{s}$ mass, are used to estimate the residual charmless background that
may contribute to the observed signals. The numbers of $B^{0}_{s}$ decays in
the $D^{+}_{s}$ sidebands are $61\pm 16$ , $0^{+5}_{-0}$, and $9\pm 5$ for the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays, respectively; they are subtracted from
the observed signal yields to obtain the corrected number of signal decays.
The yields in the signal and sideband regions are summarized in Table 1.
Table 1: Summary of event yields from data in the $D^{+}_{s}$ signal and sidebands regions, and the background corrected yield. The signal and sideband regions require $D^{+}_{s}$ candidates to have invariant mass $|M(K^{+}K^{-}\pi^{+})-m_{D^{+}_{s}}|<20$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $35<|M(K^{+}K^{-}\pi^{+})-m_{D^{+}_{s}}|<55$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, where $m_{D^{+}_{s}}$ is the $D^{+}_{s}$ mass [26]. Decay | Signal Region | Sideband Region | Corrected Yield
---|---|---|---
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ | $5683\pm 83$ | $\phantom{1}61\pm 16$ | $5622\pm 85$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\phantom{1}216\pm 21$ | $\phantom{1}0^{+5}_{-0}$ | $\phantom{1}216\pm 22$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\phantom{1}402\pm 33$ | $~{}~{}9\pm 5$ | $\phantom{1}393\pm 33$
## 5 Mass distributions of ${\mathbf{X}_{\boldsymbol{d,s}}}$ and two-body
masses
In order to investigate the properties of these $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ decays, sWeights [32]
obtained from the mass fits are used to determine the underlying $X_{d,s}$
invariant mass spectra as well as the two-body invariant masses amongst the
three daughter particles. Figure 4 shows (a) the $\pi^{-}\pi^{+}\pi^{-}$ mass,
(b) the smaller $\pi^{+}\pi^{-}$ mass and (c) the larger $\pi^{+}\pi^{-}$ mass
in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ data and simulated decays. A prominent peak,
consistent with the $a_{1}(1260)^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}$ is
observed, along with structures consistent with the $\rho^{0}$ in the two-body
masses. There appears to be an offset in the peak position of the
$a_{1}(1260)^{-}$ between data and simulation. Since the mean and width of the
$a_{1}(1260)^{-}$ resonance are not well known, and their values may even be
process dependent, this level of agreement is reasonable. A number of other
spectra have been compared between data and simulation, such as the $p_{\rm
T}$ spectra of the $D^{+}_{s}$, $X_{d}$ and the daughter particles, and
excellent agreement is found.
Figure 4: Invariant mass distributions for (a) $X_{d}$, (b) smaller
$\pi^{+}\pi^{-}$ mass in $X_{d}$ and (c) the larger $\pi^{+}\pi^{-}$ mass in
$X_{d}$, from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays using sWeights. The points are the data
and the solid line is the simulation. The simulated distribution is normalized
to have the same yield as the data.
Figure 5 shows the corresponding distributions for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay. A peaked structure at low
$K^{-}\pi^{+}\pi^{-}$ mass, consistent with contributions from the lower-lying
excited strange mesons, such as the $K_{1}(1270)^{-}$ and $K_{1}(1400)^{-}$,
is observed. As many of these states decay through $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\rho^{0}$ mesons,
significant contributions from these resonances are observed in the
$K^{-}\pi^{+}$ and $\pi^{+}\pi^{-}$ invariant mass spectra, respectively. The
simulation provides a reasonable description of the distributions in the data.
Figure 5: Invariant mass distributions for (a) $X_{s}$, (b) $\pi^{+}\pi^{-}$
in $X_{s}$ and (c) the $K^{-}\pi^{+}$ in $X_{s}$, from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ data using sWeights. The points are data and the
solid line is the simulation. The simulated distribution is normalized to have
the same yield as the data.
Figure 6 shows the same distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. The $K^{-}\pi^{+}\pi^{-}$ invariant mass is
quite broad, with little indication of any narrow structures. There are
indications of $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$\rho^{0}$ contributions in the $K^{-}\pi^{+}$ and $\pi^{-}\pi^{+}$ invariant
mass spectra, respectively, but the contribution from resonances such as the
$K_{1}(1270)^{-}$ or $K_{1}(1400)^{-}$ appear to be small or absent. In the
$K^{-}\pi^{+}$ invariant mass spectrum, there may be an indication of a $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ contribution. The
simulation, which models the $K^{-}\pi^{+}\pi^{-}$ final state as 10%
$K_{1}(1270)^{-}$, 10% $K_{1}(1400)^{-}$, 40% $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\pi^{-}$ and 40% $K^{-}\rho^{0}$,
provides a reasonable description of the data, which suggests that processes
such as those in Figs. 1(e-f) constitute a large portion of the total width
for this decay.
Figure 6: Invariant mass distributions for (a) $X_{s}$, (b) $\pi^{+}\pi^{-}$
in $X_{s}$ and (c) the $K^{-}\pi^{+}$ in $X_{s}$, from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ data using sWeights. The points are data and the
solid line is the simulation. The simulated distribution is normalized to have
the same yield as the data.
## 6 First observation of $\boldsymbol{\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow~{}D_{s1}(2536)^{+}\pi^{-}}$
A search for excited $D^{+}_{s}$ states, such as $D_{sJ}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}$, contributing to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ final state is performed. Signal candidates
within $\pm$40 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal
$B^{0}_{s}$ mass are selected, and from them the invariant mass difference,
$\Delta M=M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s}$) is formed, where both
$\pi^{+}\pi^{-}$ combinations are included. The $\Delta M$ distribution for
candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
signal window is shown in Fig. 7. A peak corresponding to the
$D_{s1}(2536)^{+}$ is observed, whereas no significant structures are observed
in the upper $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass
sideband (5450$-$5590 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The
distribution is fitted to the sum of a signal Breit-Wigner shape convolved
with a Gaussian resolution function, and a second order polynomial to describe
the background contribution. The Breit-Wigner width is set to 0.92
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26], and the Gaussian resolution
is fixed to 3.8 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ based on
simulation. A signal yield of $20.0\pm 5.1$ signal events is observed at a
mass difference of $565.1\pm 1.0~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
which is consistent with the known $D_{s1}(2536)^{+}-D^{+}_{s}$ mass
difference of $566.63\pm 0.35~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
[26]. The significance of the signal is 5.9, obtained by fitting the invariant
mass distribution with the mean mass difference fixed to 566.63
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26], and computing $\sqrt{-2{\rm
ln}({\mathcal{L}_{0}}/{\mathcal{L}_{\rm max}}})$. Here, ${\mathcal{L}_{\rm
max}}$ and ${\mathcal{L}_{0}}$ are the fit likelihoods with the signal yields
left free and fixed to zero, respectively. Several variations in the
background shape were investigated, and in all cases the signal significance
exceeded 5.5. This decay is therefore observed for the first time. To obtain
the yield in the normalization mode ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$), the signal function is integrated from 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ below to 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ above the nominal $B^{0}_{s}$ mass.
A yield of $5505\pm 85$ events is found in this restricted mass interval.
Figure 7: Distribution of the difference in invariant mass,
$M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$, using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass
(points) and in the upper $B^{0}_{s}$ mass sidebands (filled histogram). The
fit to the distribution is shown, as described in the text.
## 7 Selection efficiencies
The ratios of branching fractions can be written as
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle={Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}\times\epsilon_{\rm rel}^{s}$ (1)
and
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle={Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}\times\epsilon_{\rm rel}^{d}\times f_{s}/f_{d},$
(2)
where $Y$ are the measured yields, $\epsilon_{\rm rel}^{s}=\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})/\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ and $\epsilon_{\rm rel}^{d}=\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}))/\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ are the relative selection efficiencies
(including trigger), and $f_{s}/f_{d}=0.267\pm 0.021$ [33] is the $B^{0}_{s}$
fragmentation fraction relative to $B^{0}$. The ratios of selection
efficiencies are obtained from simulation, except for the PID requirements,
which are obtained from a dedicated $D^{*+}$ calibration sample, weighted to
match the momentum spectrum of the particles that form $X_{d}$ and $X_{s}$.
The selection efficiencies for each decay are given in Table 2. The efficiency
of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decay is about 35% larger than the values
obtained in either the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay; the efficiencies of the latter two are
consistent with each other. The lower efficiency is due almost entirely to the
tighter PID requirements on the $K^{-}$ and $\pi^{+}$ in $X_{s}$. Two
additional multiplicative correction factors, also shown in Table 2, are
applied to the measured ratio of branching fractions in Eqs. 1 and 2. The
first is a correction for the $D^{+}_{s}$ mass veto on $M(X_{d,s})$, and the
second is due to the requirement that
$M(X_{s,d})<3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The former, which
represents a small correction, is estimated from the sWeight-ed distributions
of $M(X_{d,s})$ shown previously. For the latter, the fraction of events with
$M(X_{d,s})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is obtained from
simulation, and scaled by the ratio of yields in data relative to simulation
for the mass region
$2.6<M(X_{s,d})<3.0~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. A 50%
uncertainty is assigned to the estimated correction. Based on the qualitative
agreement between data and simulation in the $M(X_{d,s})$ distributions (see
Sect. 5), and the fact that the phase space approaches zero as
$M(X_{d,s})\rightarrow~{}3.5~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, this
uncertainty is conservative.
Table 2: Selection efficiencies and correction factors for decay modes under study. The uncertainties on the selection efficiencies are statistical only, whereas the correction factors show the total uncertainty. Quantity | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$
---|---|---|---
Total $\epsilon$ ($10^{-4}$) | $4.97\pm 0.08$ | $3.67\pm 0.10$ | $3.59\pm 0.10$
$D^{+}_{s}$ veto corr. | $1.013\pm 0.003$ | $1.013\pm 0.003$ | $1.017\pm 0.005$
$M>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ corr. | $1.02\pm 0.01$ | $1.04\pm 0.02$ | $1.14\pm 0.07$
The relative efficiency between $B^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is estimated from simulation, and is found to
be $0.90\pm 0.05$.
## 8 Systematic uncertainties
Several uncertainties contribute to the ratio of branching fractions. The
sources and their values are listed in Table 3. The largest uncertainty, which
applies only to the ratio ${{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$, is from the $b$ hadronization fraction,
$f_{s}/f_{d}=0.267\pm 0.021$ [33], which is 7.9%. Another large uncertainty
results from the required correction factor to account for the signal with
$M(X_{s,d})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Those corrections
are described in Sect. 7.
The selection efficiency depends slightly on the modeling of the $X_{d,s}$
decay. The momentum spectra of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$, $D^{+}_{s}$, $X_{d,s}$ and the
$X_{d,s}$ daughters have been compared to simulation, and excellent agreement
is found. The selection efficiency is consistent with being flat as a function
of $M(X_{d,s})$ at the level of two standard deviations or less. To assess a
potential systematic uncertainty due to a possible $M(X_{d,s})$-dependent
efficiency, the relative differences between the nominal selection
efficiencies and the ones obtained by reweighting the measured efficiencies by
the $X_{d,s}$ mass spectra in data are computed. The relative deviations of
0.5%, 1.1% and 1.2% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively, are the assigned uncertainties.
The systematic uncertainty on the BDT efficiency is determined by fitting the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass distribution in data with and without the
BDT requirement. The efficiency is found to agree with simulation to better
than the 1% uncertainty assigned to this source. In total, the simulated
efficiencies have uncertainties of 1.6% and 1.9% in the two ratios of
branching fractions. The PID efficiency uncertainty is dominated by the usage
of the $D^{*+}$ calibration sample to determine the efficiencies of a given
PID requirement [34]. This uncertainty is assessed by comparing the PID
efficiencies obtained directly from simulated signal decays with the values
obtained using a simulated $D^{*+}$ calibration sample that is re-weighted to
match the kinematics of the signal decay particles. Using this technique, an
uncertainty of 2% each on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ PID efficiencies is obtained, which is 100%
correlated, and a 1% uncertainty for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$. The trigger is fully simulated, and given the
identical number of tracks and the well-modeled $p_{\rm T}$ spectra, the
associated uncertainty cancels to first order. Based on previous studies [16],
a 2% uncertainty is assigned.
The uncertainties in the signal yield determinations have contributions from
both the background and signal modeling. The signal shape uncertainty was
estimated by varying all the fixed signal shape parameters one at a time by
one standard deviation, and adding the changes in yield in quadrature (0.5%).
A double Gaussian signal shape model was also tried, and the difference was
negligible. For the combinatorial background, the shape was modified from a
single exponential to either the sum of two exponentials, or a linear
function. For $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, the difference in yield was 0.4%. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the maximum change was 4%, and for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the maximum shift was 1%. In the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass fit, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D_{s}^{*+}K^{-}\pi^{+}\pi^{-}$ contribution was modeled using the shape from
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass fit. To estimate an uncertainty from this
assumption, the data were fitted with the shape obtained from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ simulation. A deviation of 5.5% in the fitted
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ yield is found, with almost no change in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ yield. The larger sensitivity on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ yield than the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ yield arises because these
background contributions have a rising edge in the vicinity of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak, which is far enough
below the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak to
have negligible impact. These yield uncertainties are added in quadrature to
obtain the values shown in Table 3. The uncertainties due to the finite
simulation sample sizes are 3.0%.
Table 3: Summary of systematic uncertainties (in %) on the measurements of the ratios of branching fractions. Source | ${{\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ | ${{\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$
---|---|---
$f_{s}/f_{d}$ | - | 7.9
$M(X_{s,d})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ | 2.2 | 7.0
Efficiency | 1.6 | 1.9
PID | 2.2 | 0.0
Trigger | 2.0 | 2.0
Signal yields | 4.0 | 6.9
Simulated sample size | 3.0 | 3.0
Total | 6.4 | 13.4
The major source of systematic uncertainty on the branching fraction for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$, is
from the relative efficiency (5%), and on the fraction of events with
$M>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (10%). This 10% uncertainty
is conservatively estimated by assuming a flat distribution in $M(X_{d})$ up
to 3 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and then a linear decrease to
zero at the phase space limit of $\sim$3.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Other systematic uncertainties
related to the fit model are negligible. Thus in total, a systematic
uncertainty of 11% is assigned to the ratio ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})$.
## 9 Results and summary
This paper reports the first observation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decays. The ratios of branching fractions are measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2}$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$
and
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}=(4.0\pm 1.0\pm 0.4)\times 10^{-3},$
where the uncertainties are statistical and systematic, respectively. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is consistent with
expectations from Cabibbo suppression. This decay is particularly interesting
because it can be used in a time-dependent analysis to measure the CKM phase
$\gamma$. Additional studies indicate that this decay mode, with selections
optimized for only $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, can contribute about an additional 35% more
signal events relative to the signal yield in $B_{s}^{0}\rightarrow
D_{s}^{\mp}K^{\pm}$ alone.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is about 50% of that for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. Compared to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}$ decay
that proceeds only via a $W$-exchange diagram, where ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-})\sim
0.1$ [26], the ratio ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ is about 5 times larger. A consistent
explanation of this larger $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is that only about 1/5 of the
rate is from the $W$-exchange process (Fig. 1(d)) and about 4/5 comes from the
diagrams shown in Figs. 1(e-f). The observed $M(X_{s})$, $M(K^{-}\pi^{+})$ and
$M(\pi^{+}\pi^{-})$ distributions in Fig. 6 also support this explanation, as
evidenced by the qualitative agreement with the simulation.
## 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), CIEMAT, IFAE and UAB (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
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|
arxiv-papers
| 2012-11-07T13:25:00 |
2024-09-04T02:49:37.657538
|
{
"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, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, 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, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, 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, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, 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, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n 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,\n S. Kandybei, M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel,\n A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov, R. F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, 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, Y. Li, L. Li Gioi, 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, H. Luo, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n 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, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J.\n Molina 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, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, 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, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, 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, M. Smith, K.\n Sobczak, F. J. P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M. T. Tran, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht,\n D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F.\n Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Steven R. Blusk",
"url": "https://arxiv.org/abs/1211.1541"
}
|
1211.1551
|
# The buckling and invagination process during consolidation of colloidal
droplets111Soft Matter (2012) DOI: 10.1039/c2sm26530c
F. Boulogne,† F. Giorgiutti-Dauphiné,† and L. Pauchard222CNRS, UMR 7608, Lab
FAST, Bat 502, Campus Univ - F-91405, Orsay, France, EU. Fax: +33 1 69 15 80
60; Tel: +33 1 69 15 80 49; E-mail: pauchard@fast.u-psud.fr
###### Abstract
Drying a droplet of colloidal dispersion can result in complex pattern
formation due to both development and deformation of a skin at the drop
surface. The present study focus on the drying process of droplets of
colloidal dispersions in a confined geometry where direct observations of the
skin thickness are allowed. During the drying, a buckling process is followed
by a single depression growth inside the drop. The deformation of the droplet
is found to be generic and is studied for various colloidal dispersions. The
final shape can be partly explained by simple energy analysis based on the
competition between bending and stretching deformations. Particularly, the
final shape enables us to determine precisely the critical thickness of the
shell for buckling. This study allow us to validate theory in 2D droplets and
apply it to the case of 3D droplets where the thickness is not accessible by
direct observation.
## 1 Introduction
Drying droplets of complex fluids as polymer solutions or colloidal
dispersions involves a large number of microscopic phenomena: solvent
diffusion, transfers at the vapor/medium interface, skin formation then skin
deformation. Over the last decade, there has been a great deal of scientific
and technological interests in studying the flow and deposition of materials
in drying droplets [11]. As an example, the fabrication of dried milk requires
the transformation of liquid droplets into a powder form using nozzles; also
liquid food concentrate is atomized into droplets using spray-drying processes
[14, 24, 10]. Particularly functional properties have to be controlled such as
dispersibility, and solubility that are essential for the capability of the
powder to be re-hydrated. Therefore, the droplet morphology, or the droplet
size changes have to be controlled and the effects of various initial or
environmental conditions on a single droplet development have to be
understood.
The concentration at the free surface induced by desiccation can cause skin
formation of the film [17] and is often responsible for strong distortions of
the droplet [18, 21]. Deformation of nearly spheroidal droplets were
considered in the case of polymer solutions[18] or colloidal dispersions [19,
29]: the various shapes have been related to buckling process of a skin formed
at the drop surface. Such deformations are not specifically related to drying
droplets and can be observed at different scales: capsules formation [25, 26]
or lock particles formation when controlled shell buckling that forms
spherical cavities[27]. However in spheroidal structure the thickness of the
skin is not directly accessible generally. Thus, we focus here on a single
drop of a colloidal dispersion confined between two circular glass plates and
left to evaporate (see figure 1). In this configuration, diffusion processes
and drying kinetics have been carefully studied [5, 16, 21, 6, 7, 20].
Moreover this geometry allows us a direct observation and quantification of
the skin growth near the drop/air interface by imaging analysis. When its
structural strength is sufficient, the skin can withstand internal stress and
can be regarded as an elastic shell. By appropriate selection of physical
quantities, constrained shrinkage can result in buckling instability, causing
the circular shell to inversion of curvature as shown in sequence in figure
2a: the resulting single depression grows and is continued by an invagination
tube penetrating inside the drop. The complex pattern formation is studied for
various colloidal dispersions and drop sizes. The generic shape is explained
by simple energy analysis. Particularly, the critical thickness of the shell
for buckling is precisely determined by the final shape of the drop. In
addition, this study allow us to validate theory in 2D droplets and apply it
to the case of 3D droplets where the thickness is not directly accessible. The
case of three-dimensional droplets and large Péclet numbers corresponds to
common industrial processes (spray-drying consisting in rapidly drying
aerosols to manufacture dry powders).
## 2 Experimental
Our experiments were performed with various stable aqueous dispersions (see
Table 1). (i) silica particles Ludox HS-40 (solid volume fraction $\phi=0.22$,
particle diameter $2a=15\pm 2$ nm, polydispersity index $\sim 0.16$) and
diluted dispersion of Ludox TM-50 (solid volume fraction $\phi=0.12$, particle
diameter $2a=22\pm 2$ nm, polydispersity index $\sim 0.2$ [9]) from Sigma-
Aldrich; (ii) Nanolatex particles, polystyrene (solid volume fraction
$\phi=0.25$, particle diameter $2a=25$ nm, glass transition temperature
$=100^{\circ}$C) provided by Rhodia Recherche (Aubervilliers, France); (iii)
Green fluorescent silica particles sicastar-greenF (solid volume fraction
$\phi=0.10$, particle diameter $2a=50\pm 3$ nm, polydispersity index $\sim
0.2$, absorbance: $\lambda_{excitation}=475$ nm and emission:
$\lambda_{emission}=510$ nm) commercially available from Micromod. Most of the
experiments were performed with the last system. The colloidal particles are
adequately polydispersed to avoid cristallization. In addition during the
drying process, the particles density increases resulting in the formation of
a gel phase, defined as a porous matrix saturated with water. Dilution of
dispersion was possibly done by adding deionized water (quality Milli-$\rho$).
The confined geometry consists of a thin cell made of two circular, parallel
and horizontal glass slides of radius $R_{s}$ (figure 3). It has been shown
that the drying kinetics can be controlled by the slide size and is not
strongly affected by the relative humidity of the surrounding[5]; experiments
were conducted at room temperature and relative humidity ($\sim 40\%$). The
slides are lubricated with a thin film of silicon oil $V1000$ (viscosity $1$
Pa.s). This treatment prevents from the pinning process of the contact line
[21]. A drop of the dispersion is placed on the bottom substrate with a
micropipet. Then the upper glass slide is carefully placed to squeeze the drop
(see figure 1a). The gap $\delta$, between the slides is controlled using
three thin spacers of controlled thickness. In all experiments the radius of
the glass slides is kept to $R_{s}=8$ mm and the gap is constant,
$\delta=100\pm 5\mu$m. During the drying process, images of the drop evolution
are recorded at different times and show a dark ring corresponding to the
meniscus at its periphery as shown in figure 2a. The high contrast of the
images allows an accurate detection of the drop edge and provides variations
with time of both the periphery length and surface area of the drop.
Fluorescent microscopy was performed using a DM2500 Leica microscope with
objective $5\times$ and $50\times$ magnification for global view (drops
deformation) and close-up view (envelope thickness measurements) respectively.
The measurement of the size of the densely-packed region at the drop
periphery, envelope thickness, have been investigated by fluorescence
microscopy (see figure 4). Focusing on the air/drop interface, in the middle
of the cell ($z=\frac{\delta}{2}$ in figure 4a), the size of the envelope
thickness, $h$, can be estimated from the fluorescence intensity profile with
an accuracy of $5\mu$m (figure 4b).
Figure 1: (a) Set-up: a droplet of solution is sandwiched between two circular
glass slides in side view (left); the cell is illuminated by transmitted
light. (b) Sketch in top view of a quarter of cell showing the formation of a
densely-packed particles at the liquid-vapour interface during solvent
removal.
## 3 Results
The deformation of the drop during the drying process exhibits different
configurations shown in the sequence in figure 2a and in the superposition of
the drop profiles in figure 2b. The successive configurations adopted by the
drop are driven by the volume decrease that is induced by solvent removal. In
a first stage the drop progressively shrinks with solvent evaporation (figure
2a(1),(2)). Then, contrasting with the case of a pure solvent, the droplet
stops shrinking isotropically: a sudden inversion of curvature occurs (figure
2a(3)). Consequently, the depression is continued by an invagination tip that
deepens with time (figure 2a(4),(5)) until the formation of a bean-shaped
material that cracks in figure 2a(6)). These configurations are detailed in
the following. Most of quantities used are shown in figure 3.
### 3.1 Isotropic shrinkage and shell formation
During the drying process, the gas that escapes from the liquid/air interface
is driven outward from the cell by diffusion [5]. Direct observation of the
drop evolution with time shows a progressive and isotropic shrinkage in a
first stage (figure 2a (from (1) to (2)), figure 2b): this first stage is
named as configuration I. As a result both periphery length and surface area
decrease steadily with time[21] (figure 2d). In the following we suppose that
the radius decreases linearly with time as: $R(t)=R_{0}-Jt$ where $R_{0}$ is
the initial drop radius and $J$ is the evaporation rate. In addition time
variations of dimensionless periphery length are shown in figure 2e for
different drops. In particular, the duration of the shrinkage process
increases when the initial volume fraction of dispersions dereases (this is
observed in the case of pure and diluted Ludox HS-40).
Starting from suspended compounds uniformly distributed within the droplet,
particles are advected to the drop periphery due to solvent removal (figure
1b). The radial flow, induced by the evaporation rate, gives the typical
velocity scale characterizing the transport of particles. A Péclet number can
be defined as the diffusion time $t_{D}=\frac{R_{0}^{2}}{D_{0}}$
(characteristic time for the diffusion of a particle along a distance $R_{0}$)
divided by the evaporation time $t_{E}=\frac{R_{s}}{J}$, as follows:
$Pe=\frac{R_{0}^{2}J}{R_{s}D_{0}}$ with the diffusion coefficient
$D_{0}=\frac{k_{B}T}{6\pi\eta a}=8.5\times 10^{-12}m^{2}.s^{-1}$ using the
Stoke Einstein relation ($\eta$ is the viscosity of the suspending fluid) and
the evaporation deduced from the time variation of the periphery length,
$J\sim 10^{-7}m.s^{-1}$. In our experimental conditions: $10<Pe<10^{2}$,
diffusion is slow compared with the rate of evaporation. As a result, the
transport of particles to the drop-air interface strongly suggests the
formation of a porous envelope during the evaporation process [13, 28]. This
envelope thickens with time as shown by the direct measurements in figure 4d.
Figure 2: (a) Digitized images taken during a typical deformation of a drop of
colloidal dispersion (Ludox HS-40 (diluted), $R_{0}=2.9$mm, $R_{s}=8$mm). (b)
Time variation of the relative volume of the drop (measurements); at the
buckling onset a indentation depth, can be defined and characterize the growth
of a depression. (c) Superposition of drop profiles during the drying process:
duration elapsed between two images is $1$ minute. (d) Time variations of
periphery length (crosses) and surface area (circles) of the drop; the
inversion of curvature takes place at time $t_{B}$, says the buckling time.
The dashed line corresponds to the linear decrease of the drop periphery
length with time. (e) Time variations of the dimensionless periphery length
for various drops.
Due to the drop shrinkage, the structure of the envelope is possibly modified
by a redistribution of particles. In this hypothesis, the particle volume
fraction in the envelope increases until a maximum value close to the random
close packing $\phi_{c}$. Applying mass conservation of particles the relative
thickness $h$, of the shell of radius $R$ can be determined during isotropic
shrinkage[29].
$\frac{h}{R}=1-\left(\frac{\phi_{c}-\phi\left(\frac{R_{0}}{R}\right)^{2}}{\phi_{c}-\phi}\right)^{1/2}$
(1)
Figure 1 allows us to plot the time variation of the envelope thickness from
the time variation of the drop radius. Measurements of the envelope thickness
are well fitted by this expression in the isotropic regime (figure 4d). Due to
the meniscus the thickness is possibly underestimated which could explain the
discrepancy with expresion 1. The thickening of the envelope does not impede
solvent loss since the transport takes place through the envelope in
accordance with the Darcy law:
$J=\frac{k}{\eta}\frac{\Delta P}{h}$ (2)
where $\Delta P$ is the pressure drop across the shell thickness $h$, required
to produce a given fluid flow and $k$ is the permeability of the porous shell
estimated using the Carman-Kozeny relation[4].
Figure 3: Mechanical stress field in cylindrical coordinates (left part of the
sketch) and definition of physical quantities in the drying droplet (right
part of the sketch).
When the structural strength of the envelope is sufficient, the envelope can
withstand internal stress. At this time, particles do not redistribute in the
envelope that will be considered as an elastic shell. The material is assumed
to be homogeneous and characterized by an elastic modulus $E$; $E$ is supposed
to be constant during the deformation since the gel is continuously drained by
the solvent.
Figure 4: (a) Fluorescence images in top view (dispersion of Green fluorescent
silica particles). (b) Corresponding intensity profiles across the air/drop
interface, along the dashed line in (a); in the referential bound to the
drop/air interface, the increase of the envelope thickness $h(t)$, can be
measured with time. (c) Menisci at the drop edge in side view $z=0$ and
$z=\delta$ denote positions with the substrates ($\delta=100\pm 5\mu$m); the
contact angle of the aqueous dispersion on the surface is close to $60^{o}$.
(d) Ratio of the envelope thickness $h$, to the drop radius $R$, as a function
of time: crosses are measurements deduced from the intensity profiles in (b)
($R_{0}=2.9$mm, $R_{s}=8$mm), dashed line is theory in accordance with
expression 1 in the isotropic regime.
In addition the mechanical stress is assumed to be homogeneously distributed
in the elastic shell. Observing that the shell thickness is much smaller than
the other two spatial lengths (drop radius $R$ and cell gap $\delta$), the
following assumptions can be made. Let us consider the three components of the
mechanical stress field in the elastic shell, in cylindrical coordinates: the
ortho-radial, radial and normal components of the stress field are denoted by
$\sigma_{\theta\theta}$, $\sigma_{rr}$ and $\sigma_{zz}$ respectively (see
figure 3). Equilibrium of the corresponding internal forces in the elastic
shell results in the following relations:
$\frac{\sigma_{\theta\theta}}{\sigma_{rr}}\sim\frac{R}{h}>>1$ and
$\frac{\sigma_{\theta\theta}}{\sigma_{zz}}\sim\frac{R}{\delta}>>1$ since
$\frac{h}{R}<<1$ during the elastic deformation of the envelope. Therefore the
main component of the stress in the shell is the ortho-radial one. This
quantity is responsible for the buckling process of the shell. Indeed, above a
critical thickness, deformations occur mainly by bending, which is much less
energetic than stretching. Therefore a buckling process occurs leading to an
inversion of curvature of the shell (figure 5a and figure 5b). In the
following suffix $B$ is related to quantities taken at time $t_{B}$.
Expression 2 gives the relation between the shell thickness $h_{B}$, and the
pressure drop $\Delta P_{B}$ across $h_{B}$ at the onset of buckling:
$h_{B}=\frac{\Delta P_{B}}{J}\frac{k}{\eta}$ (3)
Firstly expression 1 at $t=t_{B}$ gives $R_{B}$ as a function of $R_{0}$ and
$h_{B}$. Replacing $h_{B}$ by the expression 3, $\frac{R_{B}}{R_{0}}$ can be
expressed as a function of $R_{0}$ and the pressure drop across the shell at
the onset of buckling, $\Delta P_{B}$. In this way $\Delta P_{B}$ is a
parameter and can be estimated for each system studied. $\frac{R_{B}}{R_{0}}$
slightly increases with the initial drop radius as shown by the measurements
for different colloidal systems (figure 5a). These results are close to the
linear increase of $R_{B}$ with $R_{0}$ obtained in spherical drops by Tsapis
et al.[29]. Secondly, combining expressions 1 and 3 the critical thickness at
the point of buckling is plotted in figure 5b; measurements of the thickness
using the method described in figure 4. The order of magnitude of $\Delta
P_{B}$ is found to be in agreement with the values obtained by Tsapis et
al.[29]. Values are listed in Table 1 for different systems.
Figure 5: (a) Ratio of the radius of the shell at the onset of buckling to the initial drop radius $\frac{R_{B}}{R_{0}}$, as a function of $R_{0}$. (b) Ratio of the critical envelope thickness to the drop radius at the point of buckling $\frac{h_{B}}{R_{B}}$, as a function of $R_{0}$. Dots are measurements and lines are fits to the theoretical expressions 1 and 3. Table 1: Main characteristics for the samples considered in the experiments: particle diameter $2a$ (and polydispersity is known), particle volume fraction $\phi$, permeability of the gel $k$, and critical pressure drop across the shell at the onset of buckling $\Delta P_{B}$, deduced from the measurements. | $2a(nm)$ | $\phi$ | $k(\times 10^{-18}m^{2})$ | $\Delta P_{B}(kPa)$
---|---|---|---|---
Ludox HS-40 | $15\pm 2$ | $0.22$ | $1.2$ | $9$
Ludox HS-40 (diluted) | $15\pm 2$ | $0.10$ | $1.2$ | $9$
Ludox TM-50 (diluted) | $22\pm 2$ | $0.12$ | $2.4$ | $5$
nanolatex | $25$ | $0.25$ | $4.4$ | $3$
Green fluo silica | $50\pm 3$ | $0.10$ | $12$ | $0.8$
### 3.2 Depression growth
As shown previously, the shell stops shrinking ($R=R_{B}$) and a buckling
process results in the formation of a depression in a region of the shell
(figure 2b and figure 6a). In the following, the existence of the depression
in the shell is referred as configuration II. A part of the elastic energy is
released by the formation of this depression (figure 6a). The depression
growth is characterized by the increase of the indentation length $e$, and
happens in a region of linear size $\ell$. Measurements of $\ell$ as a
function of $e$ are shown in figure 6b. These results are well fitted by the
geometric relation: $\ell=2\sqrt{2R_{B}e-e^{2}}$ approximated by
$2\sqrt{2R_{B}e}$ to first order in $e$, as a consequence of the inversion of
curvature.
Figure 6: (a) Top: superposition of digitized profiles showing the depression
growth in a part of the shell; the whites dots locate the regions of highest
curvature. Down: sketch of the depression formed by inversion of the circular
cap: the indentation length $e$, happens in a region of size $\ell$, between
$A$ and $B$. The angle $\alpha$ between the asymptotes $\Delta$ and
$\Delta^{\prime}$ limit the fold; the quantity $d$ is the lateral extension of
the fold limiting the circular shell and the inverted cap. (b) Ratio of the
lateral size of the depression, $\ell_{exp}$, to the radius of the drop at the
onset of buckling $R_{B}$, as a function of the indentation length $e$, to the
critical thickness $h_{B}$, at the onset of buckling;
$\ell^{\prime}=2\sqrt{2R_{B}e-e^{2}}$, $\ell=2\sqrt{2R_{B}e}$. $h_{B}=12\pm
5\mu m$. Above a critical indentation length, the quantity $\ell$ becomes
constant: $\ell=\ell_{c}$. (c) Sketch of the shell forming at the drop
periphery: the geometrical quantities of the shell are defined taking into
account the shell thickness $h$.
The elastic energy of configuration II is the sum of three contributions:
$U_{II}=U_{0}+U_{fold}+U_{inversion}$ (4)
1. 1.
The term $U_{0}$ is the contribution of the elastic energy due to the part of
the shell not inverted: it is considered as a reference energy.
2. 2.
During the depression growth the elastic energy contains a main contribution
coming from the creation of the folds $A$ and $B$ (see figure 6a): a fold is
limited by asymptotes (asymptotes $\Delta$ and $\Delta^{\prime}$ for $A$) and
extends over a typical distance $d$. Assuming $h/R<<1$, which is actually
realized during the drop deformation, the Föppl-von Kàrmàn theory for elastic
shells [8, 15, 1] allows us to obtain the energy due to the fold $U_{fold}$.
Minimizing the total elastic energy (bending energy and in-plane elastic
energy) with respect to the distance $d$ leads to $d=(hR_{B})^{1/2}$ (this was
described in references [15, 22]). Consequently the relevant curvature radius
is $d/\tan{\alpha}$ where $\alpha$ is the angle between the asymptotes
limiting the fold (figure 6a). Therefore $U_{fold}\sim
E\frac{h^{2}s}{R_{B}}\tan{\alpha/2}^{2}$, where $s=2d\delta$ is the area of
the fold[22, 23]. As the depression grows, the angle $\alpha$ increases, in
accordance with $\tan{\alpha/2}^{2}\sim\frac{e}{R_{B}}$, so does the energy
$U_{fold}$. $U_{fold}$ is due to both the increase of the indentation length
and the increase of the shell thickness. Consequently, at each time denoted by
$t_{i}$, an indentation length $e_{i}$, and a shell thickness $h_{i}$, can be
defined.
$U_{fold}(h_{i},e_{i})=2c_{0}\delta Eh_{i}^{5/2}\frac{1}{R_{B}^{3/2}}e_{i}$
(5)
where $c_{0}$ is a parameter that only depends on the Poisson ratio of the
shell.
3. 3.
The term $U_{inversion}$ corresponds to the change in elastic energy due the
inversion of the circular section $AB$ in accordance with figure 6c. In this
way the arc of external radius $R_{B}+h_{B}/2$ becomes compressed after
inversion; also the arc of internal radius $R_{B}-h_{B}/2$ becomes stretched
after inversion. The length change makes this energy contribution linear in
the deformation [23].
$U_{inversion}(h_{i},e_{i})=\frac{c_{1}\sqrt{2}}{4}E\delta
h_{i}^{3}\frac{1}{R_{B}^{3/2}}e_{i}^{1/2}$ (6)
where $c_{1}$ is a parameter that only depends on the Poisson ratio of the
shell. Note that $U_{inversion}(h_{i},e_{i})$ becomes negligible compared with
$U_{fold}(h_{i},e_{i})$ for $e_{i}>>h$.
The energy $(U_{II}-U_{0})/(Eh^{2}\delta)$ is plotted as a function of the
relative volume variation, $\left|\frac{\Delta V}{V_{0}}\right|\sim\frac{\pi
R_{B}^{2}-(2R_{B})^{1/2}e^{3/2}}{\pi R_{0}^{2}}$ in figure 7. This illustrates
the tendency of the elastic energy to increase during the shell deformation.
Figure 7: Dimensionless elastic energy as a function of the relative volume
variation $\mid\frac{\Delta V}{V_{0}}\mid$ (the lowest-energy state is plotted
as a solid line and the higher-energy one is plotted as a dashed line); the
plotrange starts at the buckling process $\mid\frac{\Delta
V_{B}}{V_{0}}\mid=\frac{R_{B}^{2}}{R_{0}^{2}}$. At a threshold value,
$\mid\frac{\Delta V_{C}}{V_{0}}\mid$, configuration II becomes energetically
more favourable than configuration III. The high stretching process occuring
during the configuration III possibly causes weakening and breakage of the
inverted part of the shell; in a realistic point of view the vertical dashed
line is not reached during the deformation of the elastic shell.
Since the energy concentrated in the folds becomes enormous during the
depression growth, another configuration, denoted in the following by
configuration III, is energetically more favourable. This is realized by
assuming the angle $\alpha$ constant ($\alpha$ defined in figure 6a). Hence,
the lateral size of the depression $\ell$, stops increasing and becomes
constant as shown by our measurements in figure 6b (region named III). Now,
the elastic energy due to the fold only increases through the growth of the
shell thickness.
However, the decrease of the inner volume of the shell need another mode of
deformation of the shell. Assuming $\alpha$ constant and keeping $\ell$
constant, as shown by empirical observations, require an increase of the
indentation length $e$ (figure 6a) and results in a length change of the
inverted part of the shell between $A$ and $B$. Let us consider the stretching
energy due to the length change of the inverted part of the shell. Usually,
when an elastic material is deformed the work is stored in the body as a
strain energy. If the elastic deformation $\epsilon^{\prime}$ (strain), occurs
in an elementary volume $dV\sim\delta h\ell$ ($\ell$ being the length scale of
the stretched region), the elastic energy expresses as:
$U_{stretch}=E\int dV\epsilon^{\prime 2}$
As a result, the energy of the inverted part due to the relative change in
length $u$ expresses as:
$U_{stretch}(h_{i},e_{i})=c_{2}\delta Eh_{i}\ell u(e_{i})^{2}$ (7)
where $c_{2}$ is a parameter that only depends on the Poisson ratio of the
shell. The relative length change $u(e_{i})$ differs from zero only for
$e_{i}>e_{c}$. Then, the length change is related to the indentation length
by333Starting from a circular arc AB, the simplest perturbation corresponding
to an elongational shape is a part of an ellipse. We use the approximation by
Ramanujan [3] for the circumference of an ellipse gives:
$C_{(a,b)}\sim\pi\big{(}3(a+b)-\sqrt{(3a+b)(a+3b)}\big{)}$ where $a$ and $b$
are one-half of the ellipse’s major and minor axes respectively. Taking
$a=R_{B}+e_{i}$ and $b=R_{B}$, the relative length change in length between
such an ellipse and a circle of radius $R_{B}$ expresses as:
$\frac{C_{(R_{B}+e_{i},R_{B})}-C_{(R_{B},R_{B})}}{C_{(R_{B},R_{B})}}\sim\frac{e_{i}}{2R_{B}}$
to first order in $e_{i}$. The relative length change related to the
transition from configuration II to configuration III in 7 is then
$\frac{e_{i}}{4R_{B}}$. $u(e_{i})\sim\frac{1}{2}\frac{e_{i}}{2R_{B}}$.
The contribution $U_{stretch}$ governs the elastic energy of the configuration
III as follows:
$U_{III}=U_{0}+U_{fold}(h_{i},e_{c})+U_{inversion}(h_{i},e_{i})+U_{stretch}(h_{i},e_{i})$
(8)
Here, $U_{fold}(h_{i},e_{c})$ only varies by the thickness variation because
$\ell$ is constant.
The energy $(U_{III}-U_{0})/(Eh^{2}\delta)$ is plotted as a function of the
relative volume variation $\mid\frac{\Delta V}{V_{0}}\mid$ in figure 7: for
small deformations, $U_{II}<U_{III}$ and configuration II is energetically
more favourable than configuration III. Note that this new configuration costs
rapidly a high energy.
Figure 8: Ideal scheme of the transition between different configurations of
the elastic shell. Configuration II corresponds to the inversion of a part of
the elastic shell characterized by an indentation length $e$ and a lateral
size $\ell$. Configuration II can be followed by two possible configurations:
(i) configuration II’ corresponds to the increase of both the lateral size and
the indentation length; (ii) configuration III where the lateral size is the
same than in configuration II and the indentation length is the same than in
configuration II’; consequently a stretching process of the inverted part is
needed.
Let us now predict the critical quantity $e=e_{c}$, and consequently
$\ell=\ell_{c}$, at which the transition between configuration II and III
arises. Starting with a deformed shell characterized by an indentation depth
$e$ (configuration II in the sketch figure 8), an increase of the indentation
length can result in two possible configurations:
* •
both the indentation depth $e$ and the lateral size $\ell$ increase in
accordance with configuration II’ in figure 8);
* •
only the indentation length increases at fixed lateral size $\ell=\ell_{c}$ in
accordance with configuration III in 8: in this case a length change of the
inverted region is needed.
At the threshold, the energy of configuration II’ has to equal the energy of
configuration III. Assuming that the elastic energy due to the inversion,
$U_{inversion}$, is dominated both by the energy due to the folds and the one
due to the stretching process, we have:
$U_{fold}(h,e+e_{c})=U_{fold}(h,e)+U_{stretch}(h,e+e_{c})$ using the notations
in figure 8. Starting from configuration II with $e<<e_{c}$, it comes:
$U_{fold}(h,e_{c})\sim U_{stretch}(h,e_{c})$ (at the threshold). Using
relations 5 and 7, it comes directly that $e_{c}$ is proportional to the
thickness at the onset of buckling, $h_{B}$, with a prefactor only depending
on the Poisson ratio: $e_{c}=8\frac{c_{0}}{c_{2}}h_{B}$. Taking into account
numerical values of the parameters $c_{0}$ and $c_{2}$ with a Poisson ratio
equal $0.3$ we obtain[2]: $e_{c}\sim 4h_{B}$. In addition, since the
transition to lower-energy results in a constant lateral size $\ell_{c}$, this
length can be expressed as a function of the characteristics of the shell at
the buckling onset as: $\ell_{c}\sim
2(2R_{B}e_{c})^{1/2}=4\sqrt{2}(R_{B}h_{B})^{1/2}$.
As a consequence the thickness of the shell at the onset of buckling $h_{B}$,
can be precisely deduced from the measurement of the distorted drop at a
macroscopic scale since:
$h_{B}\sim\frac{l_{c}^{2}}{32R_{B}}$ (9)
Figure 9 shows the dimensionless quantity
$\frac{l_{c}^{2}}{32R_{B}}\times\frac{1}{h_{B}}$ for a range of initial drop
radii. Evaluation of $h_{B}$ using the relation 9 is related to the
measurements of macroscopic quantities ($l_{c}$, $R_{B}$) and is consequently
more accurate than the direct measurement of $h_{B}$. Typically, the error by
direct measurement of $h_{B}$ is $20\%$ while it is equal to $5\%$ in the
measurement of $\frac{l_{c}^{2}}{32R_{B}}$. Note that the discrepancy between
theory and experiment could be due to the systematic discrepancy between
measurements of the enveloppe thickness and prediction as shown in figure 4d.
Figure 9: Measurements of the quantity
$\frac{\ell_{c}^{2}}{32R_{B}}\times\frac{1}{h_{B}}$ for a range of initial
drop radii $R_{0}$, and for different systems.
The continuation of the deformation is shown in figure 10a: the depression
deepens and forms an invaginating tube (configuration IV in figure 10b). As
shown in figure 7, the stretching process of the shell is the main
contribution to the elastic energy of configuration III. However the energy
required to proceed with this deformation is rapidly enormous. Therefore a
weakening or a breakage of the stretched part of the shell is suspected.
Indeed, a relative length change larger than $20\%$ as observed experimentally
is often not feasible in such colloidal gels (figure 10). Let us compare the
tensile stress in the stretched part of the shell with the cohesive stress in
the shell. The critical stress $\sigma_{c}$, needed to separate two particles
of radius $a$ can be estimated from the van der Waals interaction in terms of
equilibrium separation distance $Z_{0}$, and Hamaker constant $A$, whose value
depends on the surface chemistry of the particles[12]:
$\sigma_{c}\sim\frac{Aa}{12Z_{0}^{2}}\times a^{-2}$.
This critical stress is easily balanced by the tensile stress in the inverted
region of the shell during the relative length change $u$: $Eu\sim\sigma_{c}$.
Taking orders of magnitude for the Young modulus, $E\sim 10^{7}$Pa, obtained
by indentation testing, and $A=0.83\times 10^{-20}$J, $Z_{0}\sim 0.2$nm for
Ludox HS-40, the relative length change needed to separate particles is found
to be: $u=0.18$. This value corresponds to a relative volume variation of
$\mid\Delta V/V_{0}\mid\sim 0.52$ (note that the same order of magnitude is
obtained for the different colloidal systems studied). Therefore the strong
stretching of the shell possibly causes rapidly decohesion of the gel as the
indentation depth grows. Then, breakage or weakening of the inverted region is
possibly followed by a healing process of the shell due to particles
accumulation during drying.
Figure 10: The indentation depth $e$ at fixed lateral size $\ell=\ell_{c}$
turns into an invagination tip (superposition of profiles in (a)). (b) In
configuration II, the length of the inverted cap, $C$, increases with the
indentation depth $e$; the black line is the theoretical arc length of
inverted cap of radius $R_{B}$ indented by $e$ (arc length
$=R_{B}cos^{-1}(1-4e/R_{B})$). In addition the lateral size $\ell$ increases
with the indentation depth $e$. The vertical line delimits configuration II
and configuration III. In configuration III, the lateral size keeps a constant
value, $\ell=\ell_{c}$, while the length of the inverted part, $C$, still
increases. When the length change of the inverted part reaches a critical
value, possible decohesion in the inverted part occurs leading to a transition
to configuration IV. This last configuration is characterized by the
progression of an invagination tube that deepens in the droplet.
### 3.3 Case of spheroidal shells
The case of three-dimensional droplets and large Péclet numbers corresponds to
common industrial processes. As in the confined geometry (2D case), drying a
droplet of a colloidal dispersion leads to strong distortions (figure 11a).
However in 3D case, measurement of the shell thickness is not possible by
direct observation. The previous study can easily be applied to three-
dimensional droplets to obtain shell thickness at the buckling instability.
The experiments were carried out with droplets of Nanolatex dispersion. The
nearly spheroidal geometry is obtained by depositing droplets on a hydrophobic
substrate; the surface is sprayed with Lycopodium spores which form a rough
texture in which air remains trapped when a drop is deposited. In the first
stage the droplet shrinks as described in the superposition of profiles in
figure 11b. Then an inverted region grows at the apex of the droplet (figure
11a,b)[19]. As in the 2D case, the depression is continued by an invagination
process. Similar arguments can be applied to the case of spheroidal drops as
explained in references [22, 23]. In particular the energy due to the fold and
the one due to the stretching to the shell express now as:
$U_{fold}^{3D}(h_{i},e{i})=\frac{c^{\prime}_{0}}{4}E\frac{h_{i}^{5/2}}{R_{B}}e_{i}^{3/2}$
(10)
$U_{stretch}^{3D}(h_{i},e_{i})=\frac{\pi
c^{\prime}_{2}}{16}E\frac{h_{i}}{R_{B}}e_{i}^{3}$ (11)
$c^{\prime}_{0}$ and $c^{\prime}_{2}$ only depend on the Poisson ratio. From
10 and 11 it comes that $e_{c}$ is proportional to the thickness at the onset
of buckling $h_{B}$, such as:
$e_{c}\sim\big{(}\frac{4c^{\prime}_{0}}{c^{\prime}_{2}}\big{)}^{2/3}h_{B}$.
Since the geometric relation $\ell_{c}\sim 2\sqrt{2R_{B}e_{c}}$ still holds,
it comes:
$h_{B}\sim\frac{1}{8\times
4^{2/3}}\frac{c^{\prime}_{2}}{c^{\prime}_{0}}^{2/3}\frac{l_{c}^{2}}{R_{B}}\sim\frac{1}{12}\frac{l_{c}^{2}}{R_{B}}$
(12)
Consequently, using relation 12, measurements of the macroscopic quantities
$l_{c}$ and $R_{B}$ allow us to evaluate with a good accuracy the critical
thickness for buckling of spheroidal shells. In the case of a droplet of
nanolatex dispersion (solid volume fraction $0.25$, particle diameter $25$ nm,
glass transition temperature $=100^{\circ}$C) dried at $RH=50\%$ and
$T=20^{\circ}C$, we find $h_{B}=25\pm 2\mu$m.
Figure 11: Droplet of a Nanolatex dispersion deposited on a super-hydrophobic
surface. (a) Distorted droplet: digitized image showing a depression of
lateral size $l_{c}$ at the top of the droplet. (b) Superposition of
dimensionless profiles (apex height $H$, vs. radius $R$) measured at different
times by lateral imaging: the duration between two consecutive profiles is
$300$ s.
## 4 Conclusion
Desiccation of drops of complex fluids can display large shape distortions
related to the development of an elastic skin at the drop surface. We studied
such mechanical instabilities in a confined geometry where direct measurements
of the skin thickness are possible. The drop deformation appears to be generic
and does not depend on the initial drop size: a depression due to an inversion
of curvature in the elastic shell is formed and deepens in the drop. This
deformation can be related to a mismatch between bending and stretching
processes. From the comparison between the corresponding elastic energies, we
obtain a relation between the critical thickness for buckling and the
characteristic lengths of the final drop shape that are easily measurable. The
validation of this method in the confined geometry (2D geometry) is then
applied to spheroidal drops exhibiting identical distortions.
## 5 Acknowledgment
We would like to thank E. Sultan for useful discussions and A. Aubertin, L.
Auffray, C. Borget, R. Pidoux for technical support.
## References
* [1] M. Ben Amar and Y. Pomeau. Proc. Roy. Lond. A, 453:1, 1997.
* [2] B. Audoly and Y. Pomeau. Elasticity and geometry: from hair curls to the nonlinear response of shells. Oxford University Press, 2010.
* [3] R.W. Barnard, K. Pearce, and L. Schovanec. Inequalities for the perimeter of an ellipse. Journal of mathematical analysis and applications, 260(2):295–306, 2001.
* [4] C. Brinker and G.W. Scherer. Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing. Academic Press: New York, 1990.
* [5] F. Clément and J. Leng. Langmuir, 20:6538, 2004.
* [6] L. Daubersies and J.-B. Salmon. Phys. Rev. E, 84:031406, 2011.
* [7] Laure Daubersies, Jacques Leng, and Jean-Baptiste Salmon. Confined drying of a complex fluid drop: phase diagram, activity, and mutual diffusion coefficient. Soft Matter, 8:5923–5932, 2012.
* [8] A. Foppl. Vorlesungen uber technische Mechanik, page 132, 1907.
* [9] W. L. Griffith and A. L. Triolo. Phys.Rev.A, 35:2200 – 2206, 1987.
* [10] Christopher S. Handscomb and Markus Kraft. Simulating the structural evolution of droplets following shell formation. Chemical Engineering Science, 65:713–725, 2010.
* [11] Hua Hu and Ronald G. Larson. Langmuir, 21(9):3963–3971, 2005.
* [12] J.N. Israelachvili. Intermolecular and Surface Forces. Elsevier, third edition edition, 2011.
* [13] J.L. Keddie and A.F. Routh. Fundamental of latex film formation: processes and properties. Springer, 2010.
* [14] S. Kentish, M. Davidson, H. Hassan, and C. Bloore. Milk skin formation during drying. Chemical Engineering Science, 60(3):635 – 646, 2005.
* [15] L. D. Landau and E. M. Lifshitz. Theory of Elasticity (Course of Theoretical Physics), volume 7. Elsevier, 1986.
* [16] J. Leng. Phys. Rev. E, 82:021405, 2010.
* [17] T. Okuzono, K. Ozawa, and M. Doi. Phys. Rev. Lett., 97:136103–136106, 2006.
* [18] L. Pauchard and C. Allain. Europhys. Lett., 62:897–903, 2003.
* [19] L. Pauchard and Y. Couder. Europhys. Lett., 66(5):667, 2004.
* [20] L. Pauchard and F. Giorgiutti-Dauphiné. Journal Colloid Interface Science, submitted.
* [21] L. Pauchard, M. Mermet-Guyennet, and F. Giorgiutti-Dauphiné. Chemical Engineering and Processing, 94:483, 2011.
* [22] L. Pauchard, Y. Pomeau, and S. Rica. C. R. Acad. Sci. Paris, 324:411–418, 1997.
* [23] L. Pauchard and S. Rica. Philosophical Magazine B, 78(2):225–233, 1998.
* [24] Sean Xu Qi Lin and Xiao Dong Chen. Engineering data of diameter change during air drying of milk droplets with 40 wt Drying Technology, 27(10):1028–1032, 2009.
* [25] C. Quilliet. Phys. Rev. E, 74:046608, 2006.
* [26] C. Quilliet. Eur.Phys.J.E, 48:35, 2012.
* [27] S. Sacanna, W. T. M. Irvine, P. M. Chaikin, and D. J. Pine. Lock and key colloids. Nature, 464(7288):575–578, 2010.
* [28] R.W. Style and S.S.L. Peppin. Crust formation in drying colloidal suspensions. Proceedings of the Royal Society of London, Series A, 467:174–193, 2011.
* [29] N. Tsapis, E.R. Dufresne, S.S. Sinha, C.S. Riera, J.W. Hutchinson, L. Mahadevan, and D.A. Weitz. Phys. Rev. Lett., 94:018302, 2005.
|
arxiv-papers
| 2012-11-07T13:50:13 |
2024-09-04T02:49:37.668716
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne, Fr\\'ed\\'erique Giorgiutti-Dauphin\\'e, Ludovic\n Pauchard",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1211.1551"
}
|
1211.1565
|
11institutetext: DERI, NUI Galway, Ireland
11email: firstname.lastname@deri.org 22institutetext: iSOCO, Madrid, Spain
22email: bvillazon@isoco.com
# Data Shapes and Data Transformations
Michael Hausenblas 11 Boris Villazón-Terrazas 22 Richard Cyganiak 11
###### Abstract
Nowadays, information management systems deal with data originating from
different sources including relational databases, NoSQL data stores, and Web
data formats, varying not only in terms of data formats, but also in the
underlying data model. Integrating data from heterogeneous data sources is a
time-consuming and error-prone engineering task; part of this process requires
that the data has to be transformed from its original form to other forms,
repeating all along the life cycle. With this report we provide a principled
overview on the fundamental data shapes _tabular_ , _tree_ , and _graph_ as
well as transformations between them, in order to gain a better understanding
for performing said transformations more efficiently and effectively.
## 1 Motivation
These days, content and information management systems have to deal with data
originating from an array of sources, such as relational databases, NoSQL data
stores, and Web data formats. The data sources vary not only in terms of data
formats, but first and foremost in the underlying data model, be it
implicit—such as with JSON—or explicit, think: RDF.
As recently put forward by Helland [Hel11], data integration of heterogeneous
data sources is a time-consuming, costly, and error-prone engineering task.
Typically, the data has to be transformed from its original form to other
forms, repeating all along the life cycle. For example, let us assume we want
to publish data from an government agency such as spreadsheets containing
statistical information into the Linked Open Data cloud111http://lod-
cloud.net. One task of the Linked Data life cycles would then be to transform
the original tabular spreadsheet data into (graph-shaped) RDF. Once we have
the data transformed into RDF according to, say, the RDF Data Cube
Vocabulary222http://www.w3.org/TR/vocab-data-cube/, we want to visualize it in
an appealing way, so we decide to use the Google Charts
API333https://developers.google.com/chart/ requiring us to provide input as
tabular data. So, again we have to transform a graph into a tabular, and then
the application visualize the information in an appealing way.
Apparently, even in this toy example, we have to transform data from its
original form to potentially many (intermediate) other forms. This was the
motivation for us to compile this report, aiming to provide a principled
overview of possible fundamental data shapes and transformations between them.
In the following we will focus on the data transformation within the context
of the “Extract, Transform and Load” process, a valuable process, growing in
its use and scale. We use the term _data shape_ to refer to how the data is
arranged and structured, closely related to the term data
model444http://en.wikipedia.org/wiki/Structured_data: we have identified three
fundamental _data shapes_ : tabular, tree, and graph and respective
transformations between them.
The remainder of the report is organized as follows: in Section 2 we introduce
and motivate the fundamental data shapes, then in Section 3 we describe
transformations between data shapes, and, finally, in Section 4 we discuss
open issues and challenges concerning the transformations.
## 2 Fundamental data shapes
In the following we motivate and introduce the three fundamental data shapes
_tabular_ , _tree_ , and _graph_ , derived from data
structures555http://en.wikibooks.org/wiki/Data_Structures and as found in the
wild in various datas sources, including but not limited to relational
databases (RDB), NoSQL data stores [Cat11], or Web data formats such as JSON,
OData and RDF serialisations.
### 2.1 Tabular
A tabular data shape organizes data items into a table. A table is a set of
data elements (values) that are organized using a model of vertical columns
(identified by their name), and horizontal rows. A table has a specified
number of columns. Examples of tabular data shapes are:
* •
CSV (Comma Separated Values) files as of RFC
4180666http://tools.ietf.org/html/rfc4180—These files are used to store
tabular data, capable of storing numbers as well as text in a plain-text
format that can be easily written and read by humans and software alike.
* •
RDB (relational databases)—A relational database is essentially a group of
tables (entities). Tables are made up of columns and rows (tuples). Those
tables have constraints, and relationships are defined between them.
Relational databases are queried using SQL, and result sets are produced from
queries that access data from one or more tables.
### 2.2 Tree
A tree is a non-empty set, one element of which is designated the root of the
tree while the remaining elements are partitioned into non-empty sets each of
which is a subtree of the root. Tree nodes have many useful properties. The
depth of a node is the length of the path (or the number of edges) from the
root to that node. The height of a node is the longest path from that node to
its leaves. The height of a tree is the height of the root. A leaf node has no
children, its only path is up to its parent.
A particular case of a tree is the _key-value data shape_ —a linked list of
key-value pairs. Examples of tree data shapes are:
* •
XML (eXtensible Markup Language)—An open and flexible format used to exchange
a wide variety of data on and off the Web. XML is a tree structure of nodes
and nested nodes of information where the user defines the names of the
nodes777http://www.w3.org/XML/.
* •
JSON (JavaScript Object Notation)— A lightweight data-interchange format. It
is easy for humans to read and write as well as straightforward for machines
to parse and generate888http://json.org/.
* •
YAML (YAML Ain’t Markup Language)—A Super-set of JSON and general-purpos data
serialization language designed to be human-friendly and work well with modern
programming languages for common everyday tasks999http://yaml.org/.
### 2.3 Graph
A graph is a mathematical structure consisting of a set of vertexes (also
called nodes), and a set of edges. An edge is a pair of vertexes. The two
vertexes are called edge endpoints. A graph may be either undirected or
directed. Intuitively, an undirected edge models a “two-way” or “duplex”
connection between its endpoints, while a directed edge is a “one-way”
connection, and is typically represented by an arrow.
Examples of graphs are:
* •
RDF (Resource Description Framework)— A family of World Wide Web Consortium
(W3C) specifications originally designed as a metadata model. It has come to
be used as a general method for conceptual description or modeling of
information that is implemented in web resources, using a variety of syntax
formats101010http://www.w3.org/RDF/.
* •
Topic Maps—Topic maps are an ISO standard for describing knowledge structures
and associating them with information
resources111111http://www.isotopicmaps.org/.
## 3 Data shapes transformations
In this section we compare the possible transformations we can perform between
two given data shapes. To this end, we have identified a set of features along
three dimensions—the input, the output, and the transformation process—and
provide motivational usage scenarios per transformation. We acknowledge that
the characterisations and the formats presented in following are neither
exhaustive nor complete, however, serve as a useful starting point.
* •
Dimension 1—concerning the input data shape:
* –
The generic data shape, e.g., tabular, tree or graph.
* –
The specific implementation of the data shape, e.g., XML, JSON, relational
database, ect.
* •
Dimension 2—concerning the output data shape:
* –
The generic data shape, e.g., tabular, tree or graph.
* –
The specific implementation of the data shape, e.g., XML, JSON, relational
database, ect.
* •
Dimension 3—concerning the transformation process:
* –
The transformation process can be declarative or operational.
* *
_Declarative_. There is a transformation description, the transformation is
based on a language that describes the mappings between the input and output
shapes.
* *
_Operational_. The transformation is only based on an _ad-hoc_ transformation
engine.
* –
The transformation process can have an information loss (also known as lossy
transformation) defined by: “all queries that are possible on the original
shape are also possible on the resultant shape”. We have information loss when
we change the abstraction level; this happens typically, when we transform a
“richer” shape into a “less rich shape”, e.g., from graph to tabular.
The Table 3.1 illustrates all possible transformations between two given data
shapes and provides pointers to the respective subsections where we discuss
them in further detail.
Table 3.1: Data shapes transformations overview. _from/to_ | tree | tabular | graph
---|---|---|---
tree | cf. Section 3.1 | cf. Section 3.2 | cf. Section 3.3
tabular | cf. Section 3.4 | cf. Section 3.5 | cf. Section 3.6
graph | cf. Section 3.7 | cf. Section 3.8 | cf. Section 3.9
### 3.1 Tree–Tree
In this case, the transformation takes as input a given tree and outputs
another tree. Let us suppose we have a set of XML documents that contain the
description of the transactions of a company, and we need to submit these tin
JSON files instead of XML, so we need to perform a transformation from tree to
tree. Examples of these transformations are:
* •
XML to XML. An XSLT that turns a DocBook121212http://www.docbook.org/ file
into XHTML.
* •
XML to JSON. A program that turns a XML file into JSON, or, for example via
XSLT.
### 3.2 Tree–Tabular
This transformation takes as input a tree and outputs a tabular. Let us
suppose we have a set of XML that contain the description of the transactions
of a company, and we need to submit these to an entity such as a government
agency that works with CSV files instead of XML, so we need to perform a
transformation from tree to tabular. Examples of these transformations are:
* •
XML to RDB:
* –
In [Fli09] a technique is described to transform XML into a RDB. The thecnique
relies on the XSD of the XML.
* –
The connect xml-2-db
tool131313http://www.skyhawksystems.com/users_guide/runningxml2db.htm relies
on mapping files.
* •
XML to CSV:
* –
XSLT141414http://www.w3.org/TR/xslt.
* –
Scripts151515For example, http://www.ricebridge.com/xml-csv-convert.htm.
### 3.3 Tree–Graph
This transformation takes as input a tree and outputs a graph. Let us suppose
we have a set of XML document that contain the description of the transactions
of a company, and we need to submit these to an entity such as a government
agency that works with RDF for integration purposes.In this setup, we need to
transform from tree to graph. Examples of these transformations are:
* •
For example, with _Gleaning Resource Descriptions from Dialects of Languages_
(GRDDL)161616http://www.w3.org/TR/grddl/ one can turn an OData
document171717http://www.odata.org/ file into a corresponding RDF
representation.
* •
Rhizomik ReDeFer181818http://rhizomik.net/redefer/ that includes XSD2OWL,
XML2RDF.
* •
XSPARQL191919http://www.w3.org/Submission/xsparql-language-specification is a
query language combining XQuery and SPARQL for transformations between RDF and
XML.
### 3.4 Tabular–Tree
This transformation takes as input a tabular shape and outputs a tree shape.
Let us suppose we have a relational database containing transaction data of a
company, and we need to submit these transactions that requires, for
integration purposes, the data in XML, so we need to transform from tabular to
tree. Examples of these transformations are not standardised, but there are
bespoke systems such as:
* •
XML representation of a relational
database202020http://www.w3.org/XML/RDB.html.
* •
XMLSpy Relational Database
Integration212121http://www.altova.com/xmlspy/database-xml.html.
* •
CSV-to-XML222222http://csv2xml.sourceforge.net/.
### 3.5 Tabular–Tabular
This transformation takes as input a tabular and outputs a tabular. Let us
suppose we have a relational database that contain the description of the
transactions in our company. We need to display these transactions in the
company web page. To this end, we have to transform from a tabular (RDB) to a
tabular (web page). Examples of these transformations are:
* •
RDB to RDB: SQL SELECT.
* •
CSV to RDB: relying on a particular DBMS import tool.
### 3.6 Tabular–Graph
This transformation takes as input a tabular and outputs a graph. Let us
suppose we have a relational database that contain the description of the
transactions in our company, and we need to submit these transactions into the
central office in London. For integration purposes the central office is using
RDF, so we need to transform from tabular to graph. Examples of these
transformations are:
* •
RDB to RDF:
* –
W3C’s RDB2RDF activity232323http://www.w3.org/2001/sw/rdb2rdf/: Direct Mapping
and R2RML, a language for expressing customized mappings from relational
databases to RDF datasets.
* •
CSV to RDF:
* –
XLWrap - language
* –
TopBraid - tool
* –
RDF extension of Google Refine - tool - GUI
* •
RDB to Topic maps [NP09].
### 3.7 Graph–Tree
This transformation takes as input a graph and outputs a tree. Let us suppose
we have an RDF dataset for representing the statistical information of a
company and we need to transfer this information to an XML-based format such
as PC-Axis, used by the Irish CSO. Examples of these transformations are:
* •
Turning RDF to XML242424http://www.w3.org/wiki/ConverterFromRdf.
* •
XSPARQL252525http://www.w3.org/Submission/xsparql-language-specification is a
query language combining XQuery and SPARQL for transformations between RDF and
XML.
* •
Geo2KML262626http://graphite.ecs.soton.ac.uk/geo2kml/ web service which
converts RDF to KML suitable for showing on Google Earth & Maps.
### 3.8 Graph–Tabular
The transformation takes as input a graph and outputs a tabular. Let us
suppose we have an RDF dataset for representing the statistical information of
a company and want to use Google Charts for visualising it. This requires a
tabular representation in CSV and therefore we have to perform a
transformation from graph to tabular. Examples of these transformations are:
* •
SPARQL SELECT
* •
_ad-hoc_ conversion scripts.
### 3.9 Graph–Graph
This transformation takes as input a graph and outputs a graph. Let us suppose
we have an RDF dataset for representing the statistical information of a
company, expressed in the W3C RDF Data Cube
vocabulary272727http://www.w3.org/TR/vocab-data-cube/. Now, further assume
that someone is still using the deprecated
SCOVO282828http://purl.org/NET/scovo# vocabulary for representing the
statistical information. Therefore we need to transform our data expressed in
RDF Data Cube to SCOVO. In this case, we have to perform a transformation from
a graph to graph. Examples of these transformations are:
* •
RDF to RDF:
* –
SPARQL CONSTRUCT.
* –
R2R292929http://www4.wiwiss.fu-berlin.de/bizer/r2r/.
* •
JSON to RDF: JSON to RDF web
service303030http://graphite.ecs.soton.ac.uk/rdf2json/
* •
RTM313131http://www.ontopia.net/topicmaps/materials/rdf2tm.html is a
vocabulary that can be used to describe the mapping of an RDF vocabulary to
topic maps in such a way that RDF data using that vocabulary can be converted
automatically to topic maps.
### 3.10 Summary
In Table 3.2 we provide a summary of the data shapes transformation and their
characteristics.
Table 3.2: Data shapes transformations comparison. Input | Output | Nature | Lossy? | Standard
---|---|---|---|---
Tabular (RDB) | Tabular (RDB) | Declarative | No | SQL
Tabular (RDB) | Tree (XML) | Operational | No | No
Tabular (RDB) | Graph (RDF) | Declarative | No | RDB2RDF
Tree (XML) | Tabular (RDB) | Operational | No | No
Tree (XML) | Tree (XML) | Declarative | No | XSLT
Tree (XML) | Tree (XML) | Declarative | No | XSLT
Graph (RDF) | Tabular (RDB) | Declarative | Yes | SPARQL SELECT
Graph (RDF) | Tree (XML) | Declarative | Yes | No
Graph (RDF) | Graph (RDF) | Declarative | No | SPARQL CONSTRUCT
## 4 Discussion
Motivated by our experiences gathered in data integration projects as well as
in standardisation activities within W3C we wanted to provide a principled
overview on the fundamental data shapes and transformations between them.
Summarising, we can state the following:
* •
We can perform (loss-less) data shape transformations between certain shapes.
* •
A number of data shape transformations are already standards or in the process
of being standardised, including:
* –
For RDB2RDF, see R2RML and Direct Mapping.
* –
For XML2XML, see XSLT.
* –
For XML2RDF, see GRDDL.
* •
We found that some data shape transformations are declarative in nature and it
would be interesting to learn if others can and should be expressed
declaratively as well.
* •
To this end, we have not taken provenance information in the transformation
process into account. Again, this is something worthwhile to follow up on.
* •
In certain cases we have to deal with lossy transformations. A more systematic
study of these cases, including an assessment of the implications concerning
the data integration process is subject to future research.
We hope that the report in the current form is useful for both researchers and
practitioners alike and consider it as one contribution in helping to
establish a discussion around data shapes and their transformations in order
to advance the state of the art.
## 5 Acknowledgements
The authors are grateful for the support received through the European
Commission FP7 ICT projects BIG– _Big Data Public Private Forum_ (Grant
Agreement No. 257943) as well as LATC– _LOD Around-The-Clock_ (Grant Agreement
No. 256975).
## References
* [Cat11] Rick Cattell. Scalable SQL and NoSQL data stores. SIGMOD Rec., 39:12–27, May 2011.
* [Fli09] Amy Flik. Transforming XML into a Relational Database Using XML Schema Document Type. Technical Library. Paper 48., Grand Valley State University, 2009\.
* [Hel11] Pat Helland. If You Have Too Much Data, then ’Good Enough’ Is Good Enough. Queue, 9:40:40–40:50, May 2011.
* [NP09] Thomas Neidhart and Rani Pinchuk. Semantic Integration of Relational Data Sources With Topic Maps. In Fifth International Conference on Topic Maps Research and Applications, 2009.
|
arxiv-papers
| 2012-11-07T14:54:46 |
2024-09-04T02:49:37.677494
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Hausenblas and Boris Villazon-Terrazas and Richard Cyganiak",
"submitter": "Michael Hausenblas",
"url": "https://arxiv.org/abs/1211.1565"
}
|
1211.1606
|
# On identities generated by compositions of positive integers
Vladimir Shevelev Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel. e-mail: shevelev@bgu.ac.il
###### Abstract.
We prove astonishing identities generated by compositions of positive
integers. In passing, we obtain two new identities for Stirling numbers of the
first kind. In the two last sections we clarify an algebraic sense of these
identities and obtain several other structural close identities.
###### 1991 Mathematics Subject Classification:
05A19. Key words and phrases: compositions of integer, combinatorial
identities, polynomials, Stirling numbers of the first kind
## 1\. Introduction
Recall (cf.[2]) that a composition of a positive integer $n$ is a way of
writing $n$ as a sum of a sequence of positive integers. These integers are
called parts of a composition. Thus to a composition of $n$ with $r$ parts
corresponds $r$\- fold vector $(k_{1},...,k_{r})$ of positive integer
components with the condition $k_{1}+k_{2}+...+k_{r}=n.$ From the definition
it follows that, in contrast to partitions of $n,$ the order of parts matters.
Note that the set of all solutions of the Diophantine equation
$k_{1}+k_{2}+...+k_{r}=n,\;k_{i}\geq 1$ is the set of all compositions with
$r$ parts. We start with two examples.
###### Example 1.
Let $k=3.$ We have the following compositions of $3:$ $1+1+1=1+2=2+1=3.$
Let us map a composition $k=k_{1}+k_{2}+...+k_{r}$ to the following product of
binomial coefficients:
$\binom{n}{k_{1}}\binom{n}{k_{2}}\cdot...\cdot\binom{n}{k_{r}}$ and all
compositions of $k$ we map to the sum of such products, where the summand are
taken with the sign $(-1)^{k-r}.$ After summing the products with the same
sets of factors, we obtain a liner combinations of such products. In our case
$k=3,$ we have the following linear combination of products of binomial
coefficients:
(1) $c_{3}(n)=\binom{n}{1}^{3}-2\binom{n}{1}\binom{n}{2}+\binom{n}{3}.$
It is easy to verify that
(2) $c_{3}(n)=\binom{n+2}{3}.$
###### Example 2.
We have the following compositions of $k=4:$
$1+1+1+1=2+1+1=1+2+1=1+1+2=1+3=3+1=2+2=4.$
Thus we have the following linear combination of products of binomial
coefficients:
(3)
$c_{4}(n)=\binom{n}{1}^{4}-3\binom{n}{1}^{2}\binom{n}{2}+2\binom{n}{1}\binom{n}{3}+\binom{n}{2}^{2}-\binom{n}{4}$
and it is easy to verify that
(4) $c_{4}(n)=\binom{n+3}{4}.$
In general, we obtain the following.
###### Theorem 3.
(5) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\binom{n}{k_{i}}=\binom{n+k-1}{k}.$
In cases $k=3$ and $k=4,$ formula (5), evidently, leads to Examples 1-2. It is
interesting to note (using a simple induction) that the $k$-th polynomial in
$n$ of the sequence $\\{\binom{n+k-1}{k}\\}$ is the partial sum of values of
the $(k-1)$-th one:
(6) $\sum_{j=1}^{n}\binom{j+k-2}{k-1}=\binom{n+k-1}{k}.$
## 2\. An equivalent form of identity (5)
We calculate the interior sum in (5) in a combinatorial way. First, let us
consider also zero parts in the compositions of $k.$ In this case we have the
sum
(7) $\Sigma_{1}=\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
0}\prod_{i=1}^{r}\binom{n}{k_{i}}.$
To calculate this sum, suppose that we have $rn$ white points and mark $k$
from them. This we can do in $\binom{rn}{k}$ ways. On the other hand, we can
mark $k_{1}$ from $n$ points (since the white points are indistinguishable, we
can choose any $n$ points), $k_{2}$ from another $n$ points, etc. Thus we
immediately obtain the equality
(8) $\Sigma_{1}=\binom{rn}{k}.$
To calculate the required interior sum in (5)
(9) $\Sigma_{2}=\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\binom{n}{k_{i}},$
we should remove zero parts in $\Sigma_{1}$ (7), using ”include-exclude”
formula. Hence, we find
$\Sigma_{2}=\binom{rn}{k}-\binom{r}{1}\binom{(r-1)n}{k}+$ (10)
$\binom{r}{2}\binom{(r-2)n}{k}-...+(-1)^{r-1}\binom{r}{r-1}\binom{n}{k}.$
Now, by (9)-(10), we see that (5) is equivalent to the identity
$\sum_{r=1}^{k}(-1)^{k-r}\sum_{j=0}^{r-1}(-1)^{j}\binom{r}{j}\binom{n(r-j)}{k}=\binom{n+k-1}{k},$
or, putting $i=r-j,$ to the identity
(11)
$\sum_{r=1}^{k}\sum_{i=1}^{r}(-1)^{i}\binom{r}{i}\binom{ni}{k}=(-1)^{k}\binom{n+k-1}{k}.$
Changing here the order of summing, we have
$\sum_{i=1}^{k}\sum_{r=i}^{k}(-1)^{i}\binom{r}{i}\binom{ni}{k}=$ (12)
$\sum_{i=1}^{k}(-1)^{i}\binom{ni}{k}\sum_{r=i}^{k}\binom{r}{i}=(-1)^{k}\binom{n+k-1}{k}.$
As is well known,
$\sum_{r=i}^{k}\binom{r}{i}=\binom{k+1}{i+1}.$
Therefore, (5) is equivalent to the identity:
(13)
$\sum_{i=1}^{k}(-1)^{i}\binom{ni}{k}\binom{k+1}{i+1}=(-1)^{k}\binom{n+k-1}{k}.$
## 3\. (13) as a polynomial identity in $n$
Unfortunately, we are not able to give a direct inductive proof of (13). Note
that (13) means the equality between two polynomials in $n$ of degree $k.$
Therefore, for a justification of (13), it is natural to use Stirling numbers
of the first kind with the generating polynomial for them ([1]):
(14) $x(x-1)\cdot...\cdot(x-n+1)=\sum_{j=1}^{n}s(n,j)x^{j},\;n\geq 1.$
Writing (13) in the form
$\sum_{i=1}^{k}(-1)^{i}in(in-1)\cdot...\cdot(in-k+1)\binom{k+1}{i+1}=$
(15) $(-1)^{k}(n+k-1)(n+k-2)\cdot...\cdot n,$
by (14), we have
$\sum_{i=1}^{k}(-1)^{i}\binom{k+1}{i+1}\sum_{t=1}^{k}s(k,t)(in)^{t}=$ (16)
$(-1)^{k}\sum_{t=1}^{k}s(k,t)(n+k-1)^{t}.$
In the left hand side of (16), the coefficient of $n^{t}$ equals
$s(k,t)\sum_{i=0}^{k}(-1)^{i}\binom{k+1}{i+1}i^{t}=$
$-s(k,t)\sum_{j=1}^{k+1}(-1)^{j}\binom{k+1}{j}(j-1)^{t}=$
$-s(k,t)\sum_{j=0}^{k+1}(-1)^{j}\binom{k+1}{j}(j-1)^{t}+s(k,t)(-1)^{t}.$
Since $t\leq k,$ then the $(k+1)$-th difference
$\Delta^{k+1}[(j-1)^{t}]=\sum_{j=0}^{k+1}(-1)^{j}\binom{k+1}{j}(j-1)^{t}=0$
and we conclude that for $t\geq 1$
(17)
$Coef_{n^{t}}(\sum_{i=1}^{k}(-1)^{i}in(in-1)\cdot...\cdot(in-k+1)\binom{k+1}{i+1})=(-1)^{t}s(k,t).$
In the right hand side of (16), the coefficient of $n^{t}$ equals
$(-1)^{k}\sum_{j=0}^{k}s(k,j)Coef_{n^{t}}(n+k-1)^{j}=$
$(-1)^{k}\sum_{j=t}^{k}s(k,j)\binom{j}{t}(k-1)^{j-t}.$
Thus, comparing with (17), we conclude that identity (13) is equivalent to the
identity
(18) $\sum_{j=t}^{k}\binom{j}{t}s(k,j)(k-1)^{j-t}=(-1)^{k+t}s(k,t).$
Further we need two lemmas.
## 4\. Lemmas
###### Lemma 4.
For $1\leq t\leq k,$ we have
(19) $\sum_{j=t+1}^{k}\binom{j}{t}s(k,j)=ks(k-1,t).$
Proof. We prove the lemma in the form:
(20) $\sum_{i=1}^{k-t}\binom{t+i}{t}s(k,t+i)=ks(k-1,t),\;1\leq t\leq k.$
We use induction over $k.$ Note that (20) is valid for $k=1$ and $t\geq 1.$
Suppose that
(21) $\sum_{i=1}^{k-1-t}\binom{t+i}{t}s(k-1,t+i)=(k-1)s(k-2,t),\;t\geq 1,$
or, the same, changing the summing index $i:=i-1,$
(22) $\sum_{i=2}^{k-t}\binom{t+i-1}{t}s(k-1,t+i-1)=(k-1)s(k-2,t),\;t\geq 1,$
or
$\sum_{i=1}^{k-t}\binom{t+i-1}{t}s(k-1,t+i-1)=$ (23)
$(k-1)s(k-2,t)+s(k-1,t),\;t\geq 1.$
For $t\geq 2,$ put in (21) $t:=t-1.$ Then, for $t\geq 1,$ we have
(24) $\sum_{i=1}^{k-t}\binom{t+i-1}{t-1}s(k-1,t+i-1)=(k-1)s(k-2,t-1).$
This we sum with (23). We find
$\sum_{i=1}^{k-t}\binom{t+i}{t}s(k-1,t+i-1)=$
(25) $(k-1)s(k-2,t-1)+(k-1)s(k-2,t)+s(k-1,t),\;t\geq 1.$
Recall that ([1])
(26) $s(n,t)=s(n-1,t-1)-(n-1)s(n-1,t).$
For $k\neq 1,$ put here $n=k-1$ and multiply by $k-1.$ We have
$(k-1)s(k-1,t)=$ $(k-1)s(k-2,t-1)-(k-1)(k-2)s(k-2,t)=$
$(k-1)s(k-2,t-1)-((k-1)^{2}-(k-1))s(k-2,t),$
whence
$(k-1)^{2}s(k-2,t)=(k-1)s(k-2,t-1)-$ (27) $(k-1)s(k-1,t)+(k-1)s(k-2,t).$
Taking into account the inductive supposition (21), from (27) we find
$(k-1)\sum_{i=1}^{k-1-t}\binom{t+i}{t}s(k-1,t+i)=$ (28)
$(k-1)s(k-2,t-1)-(k-1)s(k-1,t)+(k-1)s(k-2,t).$
Note that, since $s(k-1,k)=0,$ then in (28) we can consider the summing up to
$i=k-t.$ Subtracting (28) from (25), we have
$\sum_{i=1}^{k-t}\binom{t+i}{t}(s(k-1,t+i-1)-(k-1)s(k-1,t+i))=ks(k-1,t).$
Since
$s(k-1,t+i-1)-(k-1)s(k-1,t+i)=s(k,t+i),$
then we find
$\sum_{i=1}^{k-t}\binom{t+i}{t}s(k,t+i)=ks(k-1,t)$
which, comparing with (21), means the step of induction. $\square$
###### Lemma 5.
We have
$\sum_{i=1}^{k}(-1)^{i}(\binom{(n-1)i}{k}-\binom{ni}{k})\binom{k+1}{i+1}=$
(29) $\sum_{i=1}^{k-1}(-1)^{i}\binom{ni}{k-1}\binom{k}{i+1}.$
Proof. We prove (29) in the form
$\sum_{i=1}^{k}(-1)^{i}\binom{(n-1)i}{k}\binom{k+1}{i+1}=$ (30)
$\sum_{i=1}^{k}(-1)^{i}\binom{ni}{k}\binom{k+1}{i+1}+\sum_{i=1}^{k-1}(-1)^{i}\binom{ni}{k-1}\binom{k}{i+1}.$
According to (17) (which not depends on the validity of (13)), the coefficient
of $n^{t}$ of right hand side of (30) equals
$\frac{(-1)^{t}}{k!}s(k,t)+\frac{(-1)^{t}}{(k-1)!}s(k-1,t).$ Thus, by (30), we
should prove that
$Coef_{n^{t}}(\sum_{i=1}^{k}(-1)^{i}\binom{(n-1)i}{k}\binom{k+1}{i+1})=\frac{(-1)^{t}}{k!}(s(k,t)+ks(k-1,t),$
or
$Coef_{n^{t}}(\sum_{i=1}^{k}(-1)^{i}\binom{k+1}{i+1})\sum_{r=0}^{k}s(k,r)((n-1)i)^{r}=$
$\sum_{i=1}^{k}(-1)^{i}\binom{k+1}{i+1})\sum_{r=t}^{k}s(k,r)i^{r}(-1)^{r-t}\binom{r}{t}=$
$(-1)^{t}(s(k,t)+ks(k-1,t),$
or, changing the order of summing, equivalently we should prove that
(31)
$\sum_{r=t}^{k}(-1)^{r}\binom{r}{t}s(k,r)\sum_{i=0}^{k}(-1)^{i}\binom{k+1}{i+1}i^{r}=s(k,t)+ks(k-1,t)$
(we can sum over $i\geq 0,$ since $r\geq t\geq 1$). Note that the interior sum
of (31) is
$\sum_{i=0}^{k}(-1)^{i}\binom{k+1}{i+1}i^{r}=\sum_{j=1}^{k+1}(-1)^{j-1}\binom{k+1}{j}(j-1)^{r}=$
$\sum_{j=0}^{k+1}(-1)^{j-1}\binom{k+1}{j}(j-1)^{r}+(-1)^{r}.$
However, since $r\leq k,$ then
$\Delta^{k+1}[(j-1)^{r}]=\sum_{j=0}^{k+1}(-1)^{j}\binom{k+1}{j}(j-1)^{r}=0$
and thus
$\sum_{i=0}^{k}(-1)^{i}\binom{k+1}{i+1}i^{r}=(-1)^{r}.$
Now the left hand side of (31) is $\sum_{r=t}^{k}\binom{r}{t}s(kr)$ and, by
Lemma 4, is $s(k,t)+ks(k-1,t).\;\;\;\;\;\square$
## 5\. Completion of proof of Theorem 3
In Section 2 we proved that (5) is equivalent to (13). Therefore, our aim is
to prove (13). We use induction over $k.$ Note that (13), evidently, satisfies
in case $k=1$ and every $n.$ Suppose that (13) holds for $k:=k-1$ and every
$n,$ i.e.,
(32)
$\sum_{i=1}^{k-1}(-1)^{i}\binom{ni}{k-1}\binom{k}{i+1}=(-1)^{k-1}\binom{n+k-2}{k-1}.$
By Lemma 5, the inductive supposition (32) is equivalent to the identity
$\sum_{i=1}^{k}(-1)^{i}(\binom{(n-1)i}{k}-\binom{ni}{k})\binom{k+1}{i+1}=$
(33) $(-1)^{k-1}\binom{n+k-2}{k-1}.$
Putting $n:=j,$ and summing (33) over $j$ from $j=1$ up to $j=n,$ according to
(6), we find
$\sum_{i=1}^{k}(-1)^{i}\binom{ni}{k}\binom{k+1}{i+1}=(-1)^{k}\binom{n+k-1}{k}$
which is realized the step of induction. $\square$
Simultaneously, in view of the proved in Section 3 equivalence of (13) and
(18), we proved the identity (18).
## 6\. Remarks on the newness of identities (13), (18) and (19)
Formally, the identities (13), (18) and (19) (and, consequently, (5)) appear
to be new, since they are absent in so fundamental sources as [1],[4],[7].
However, there is a deeper reason. The newness of (13) (and together with it
(18) and (19)) is explained by the fact that there are no known identities
involving $\binom{in}{k}$ with the summing index $i.$ Indeed, the only known
generator of similar sums is Rothe-Hagen coefficient $A_{k}(x,n)$ [4]-[5]. It
is defined alternatively by the following formulas:
(34) $A_{k}(x,n)=\frac{x}{x+kn}\binom{x+kn}{k},$ (35)
$A_{k}(x,n)=\sum_{i=0}^{k-1}(-1)^{i+k+1}\binom{k}{i}\binom{x+in}{k}\frac{x}{x+in},\;k\geq
1.$
The comparison of these formulas leads to the identity of the form
$\sum_{i=1}^{k-1}(-1)^{i+k+1}\binom{k}{i}\binom{x+in}{k}\frac{x}{x+in}=$ (36)
$\frac{x}{x+kn}\binom{x+kn}{k}+(-1)^{k}\binom{x}{k}.$
Unfortunately, the attempt to eliminate from $x$ in $\binom{x+in}{k},$ putting
$x=0,$ lead to the trivial identity $0=0.$ Consider another attempt. For
$k>x\geq 1,$ we have
$\sum_{i=1}^{k-1}(-1)^{i+k+1}\binom{k}{i}\binom{x+in}{k}\frac{1}{x+in}=\frac{1}{x+kn}\binom{x+kn}{k},$
or
(37)
$\sum_{i=1}^{k}(-1)^{i-1}\binom{k}{i}\binom{x+in}{k}\frac{1}{x+in}=0,\;x\geq
1.$
In the ”singular ” case $x=0,$ we obtain the required factor of the form
$\binom{ni}{k}$ and found (quite independently on (37)) a nice identity
(38)
$\sum_{i=1}^{k}\frac{(-1)^{i-1}}{i}\binom{in}{k}\binom{k}{i}=\frac{(-1)^{k-1}n}{k}$
which, most likely, is also new, but different from (13). Indeed, denote the
left hand side of (38) by $a_{n}(k).$ Using (14),we have
$a_{n}(k)=\frac{1}{n!}\sum_{i=1}^{n}\frac{(-1)^{i-1}}{i}\binom{n}{i}(ik)(ik-1)\cdot...\cdot(ik-n+1)=$
$\frac{1}{n!}\sum_{i=1}^{n}\frac{(-1)^{i-1}}{i}\binom{n}{i}\sum_{t=0}^{n}s(n,t)(ik)^{t}.$
Thus, since $s(n,0)=0,$ then
(39) $Coef_{k^{t}}(a_{n}(k))=\begin{cases}0,\;\;if\;\;t=0,\\\
\frac{s(n,\;t)}{n!}\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}i^{t-1},\;\;if\;\;t\geq
1.\end{cases}$
Further, since
$s(n,1)=(-1)^{n-1}(n-1)!,\;\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}=1,$
then
(40) $Coef_{k}(a_{n}(k))=\frac{(-1)^{n-1}}{n}.$
It is left to show that, for $t\geq 2,$ we have
(41) $s(n,\;t)\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}i^{t-1}=0.$
Indeed, if $2\leq t\leq n,$ then we have
$\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}i^{t-1}=(-1)^{n-1}\sum_{i=0}^{n}(-1)^{i}\binom{n}{i}(k-i)^{t-1}.$
The latter is the $n$-th difference $\Delta^{n}[k^{t-1}]$ which, for $t\leq
n,$ equals 0. If $t>n,$ then $s(n,\;t)=0,$ and (41) follows. $\square$
## 7\. Gessel’s short proof of (13)
Gessel [3] proposed a short proof of the identity (13).
Let $P(x)$ be a polynomial of degree $k.$ Then, for the $(k+1)$-th difference
of $P(x),$ we have
$\Delta^{k+1}[P(x)]=\sum_{j=0}^{k+1}(-1)^{k+1-j}\binom{k+1}{j}P(x+j)=0.$
In particular, for $x=0,$
$\sum_{j=0}^{k+1}(-1)^{j}\binom{k+1}{j}P(j)=0.$
Put here
$P(j)=P_{n,\;k}(j)=\binom{n(j-1)}{k}$
which is a polynomial in $j$ of degree $k.$ We have
$\sum_{j=0}^{k+1}(-1)^{j}\binom{k+1}{j}\binom{n(j-1)}{k}=0.$
Putting here $j-1=i,$ we find
$\sum_{i=-1}^{k}(-1)^{i}\binom{k+1}{i+1}\binom{ni}{k}=0,$
or, the same, for $k\geq 1,$ we have
$\sum_{i=1}^{k}(-1)^{i}\binom{k+1}{i+1}\binom{ni}{k}=\binom{-n}{k}=$
$\frac{(-n)(-n-1)\cdot...\cdot(-n-(k-1))}{k!}=$
$(-1)^{k}\frac{(n+k-1)(n+k-2)\cdot...\cdot
n}{k!}=(-1)^{k}\binom{n+k-1}{k}.\;\;\;\;\square$
It is interesting to note that, if the author was successful to find such an
elegant and simple proof, then, most likely, the identities (18), (19) and
(38) were not discovered.
## 8\. Dual case of identity (5)
Note that, together with Example1, we have the following identity
$\binom{n}{1}^{3}-2\binom{n}{1}\binom{n+1}{2}+\binom{n+2}{3}=\binom{n}{3}.$
In general, together with (5), we prove the following dual identity.
###### Theorem 6.
(42) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\binom{n+k_{i}-1}{k_{i}}=\binom{n}{k}.$
###### Proof.
Again we calculate the interior sum in (42) in a combinatorial way with
firstly consideration also zero parts in the compositions of $k.$ In this case
we have the sum
(43) $\Sigma_{3}=\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
0}\prod_{i=1}^{r}\binom{n+k_{i}-1}{k_{i}}.$
To calculate this sum, suppose that we have $rn$ white points and mark $k$
from them, _but now every point could be marked several times._ According to
well known formula for the number of the combination with repetitions (cf.
[8], p.10), this we can do in $\binom{rn+k-1}{k}$ ways. On the other hand, we
can mark (with repetitions) $k_{1}$ from $n$ points, $k_{2}$ from another $n$
points, etc. This leads us to the equality
(44) $\Sigma_{3}=\binom{rn+k-1}{k}.$
To calculate the required interior sum in (42)
(45) $\Sigma_{4}=\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\binom{n+k_{i}-1}{k_{i}},$
we should remove zero parts in $\Sigma_{3}$ (44), using ”include-exclude”
formula. We find
$\Sigma_{4}=\binom{rn+k-1}{k}-\binom{r}{1}\binom{(r-1)n+k-1}{k}+$ (46)
$\binom{r}{2}\binom{(r-2)n+k-1}{k}-...+(-1)^{r-1}\binom{r}{r-1}\binom{n+k-1}{k}.$
Now in an analogous way, as in Section 2, we find that the identity (42) is
equivalent to the identity dual to (13):
(47)
$\sum_{i=1}^{k}(-1)^{i}\binom{ni+k-1}{k}\binom{k+1}{i+1}=(-1)^{k}\binom{n}{k}.$
The latter identity is easily proved as (13) in Section 7. ∎
## 9\. An algebraic approach
L. Tevlin [10] outlined the contours of quite another proof of Theorem 3 in
frameworks of the good advanced theory of symmetric functions. Recall (cf.[6],
[8]) that, for each integer $k\geq 0,$
$i)$ the $k$-th elementary symmetric function $e_{k}$ is the sum of all
products of $k$ distinct variables $x_{i},$ so that $e_{0}=1$ and, for $k\geq
1,$
(48) $e_{k}=\sum_{i_{1}<i_{2}<...<i_{k}}x_{i_{1}}x_{i_{2}}...x_{i_{k}};$
$ii)$ the $k$-th complete symmetric function $h_{k}$ is defined as $h_{0}=1$
and, for $k\geq 1,$
(49) $h_{k}=\sum_{i_{1}\leq i_{2}\leq...\leq
i_{k}}x_{i_{1}}x_{i_{2}}...x_{i_{k}}.$
In particular, $h_{1}=e_{1}.$ It is convenient to define $e_{k}=h_{k}=0$ for
$k<0;$
$iii)$ it is well known that
(50)
$h_{k}=\left|\begin{matrix}e_{1}&e_{2}&e_{3}&\ldots&e_{k-2}&e_{k-1}&e_{k}\\\
1&e_{1}&e_{2}&\ldots&e_{k-3}&e_{k-2}&e_{k-1}\\\
0&1&e_{1}&\ldots&e_{k-4}&e_{k-3}&e_{k-2}\\\
0&0&1&\ldots&e_{k-5}&e_{k-4}&e_{k-3}\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ 0&0&0&\ldots&1&e_{1}&e_{2}\\\
0&0&0&\ldots&0&1&e_{1}\end{matrix}\right|$
and
(51)
$e_{k}=\left|\begin{matrix}h_{1}&h_{2}&h_{3}&\ldots&h_{k-2}&h_{k-1}&h_{k}\\\
1&h_{1}&h_{2}&\ldots&h_{k-3}&h_{k-2}&h_{k-1}\\\
0&1&h_{1}&\ldots&h_{k-4}&h_{k-3}&h_{k-2}\\\
0&0&1&\ldots&h_{k-5}&h_{k-4}&h_{k-3}\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ 0&0&0&\ldots&1&h_{1}&h_{2}\\\
0&0&0&\ldots&0&1&h_{1}\end{matrix}\right|_{.}$
Diagonals of these determinants has very simple cycle structure that allows to
give an explicit formulas for them.
###### Lemma 7.
The following formulas hold
(52) $h_{k}=\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}e_{k_{i}};$ (53)
$e_{k}=\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}h_{k_{i}}.$
###### Proof.
Consider products of nonzero elements of cycles of the Toeplitz matrix (50).
It is easy to see that for every cycle of length 1 this product is $e_{1},$
for every cycle of length 2 this product is $e_{2},$ …, for every cycle of
length $i$ this product is $e_{i}.$ Therefore, a diagonal with cycles of
length $k_{1},k_{2},...,k_{r},$ such that $k_{1}+k_{2}+...+k_{r}=k,$ has the
product of its elements $\prod_{i=1}^{r}e_{k_{i}}$ and in the determinant this
product appears with sign $(-1)^{k-r}.$ Hence, (52) follows. Dually we have
also (53). ∎
However, by Ex.1, p.26 in [6], it follows that, if $e_{k}=\binom{n}{k},$ then
$h_{k}=\binom{n+k-1}{k}.$ Thus, in view of Lemma 7, we obtain new proofs of
identities (5) and (42).
## 10\. Other identities generated by compositions of integers
In [6] we find also other pairs $\\{e_{k},h_{k}\\}$ given by explicit
formulas. So we obtain other interesting identities generated by compositions
of integers. We restrict ourself by the following five pairs of identities.
1) Pair $e_{k}=\frac{a(a-k)^{k-1}}{k!},\;h_{k}=\frac{a(a+k)^{k-1}}{k!},\;k\geq
1,$ leads to identities:
(54) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\frac{a(a-k_{i})^{k_{i}-1}}{k_{i}!}=\frac{a(a+k)^{k-1}}{k!};$
(55) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\frac{a(a+k_{i})^{k_{i}-1}}{k_{i}!}=\frac{a(a-k)^{k-1}}{k!};$
2) Pair
$e_{k}=\frac{(-1)^{k}a^{k}B_{k}}{k!},\;h_{k}=\frac{a^{k}}{(k+1)!},\;k\geq 1,$
where $B_{k}$ is the $k$-th Bernoulli number, leads to identities:
(56) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\frac{(-1)^{k_{i}}a^{k_{i}}B_{k_{i}}}{k_{i}!}=\frac{a^{k}}{(k+1)!};$
(57) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\frac{a^{k_{i}}}{(k_{i}+1)!}=\frac{(-1)^{k}a^{k}B_{k}}{k!};$
3) Pair $e_{k}=q^{\frac{k(k-1)}{2}}\left[\begin{matrix}n\\\
k\end{matrix}\right],\;h_{k}=\left[\begin{matrix}n+k-1\\\
k\end{matrix}\right],$ where $\left[\begin{matrix}n\\\ k\end{matrix}\right]$
denotes the ”$q$-binomial coefficient” or Gaussian polynomial
$\left[\begin{matrix}n\\\
k\end{matrix}\right]=\frac{(1-q^{n})(1-q^{n-1})...(1-q^{n-k+1})}{(1-q)(1-q^{2})...(1-q^{k})},$
leads to identities:
(58) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}q^{\frac{k_{i}(k_{i}-1)}{2}}\left[\begin{matrix}n\\\
k_{i}\end{matrix}\right]=\left[\begin{matrix}n+k-1\\\ k\end{matrix}\right];$
(59) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\left[\begin{matrix}n+k_{i}-1\\\
k_{i}\end{matrix}\right]=q^{\frac{k(k-1)}{2}}\left[\begin{matrix}n\\\
k\end{matrix}\right];$
4) Pair $e_{k}=q^{\frac{k(k-1)}{2}}/\varphi_{k}(q),\;h_{k}=1/\varphi_{k}(q),$
where
$\varphi_{k}(q)=(1-q)(1-q^{2})...(1-q^{k}),$
leads to identities:
(60) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}(q^{\frac{k_{i}(k_{i}-1)}{2}}/\varphi_{k_{i}}(q))=1/\varphi_{k}(q);$
(61) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}1/\varphi_{k_{i}}(q)=q^{\frac{k(k-1)}{2}}/\varphi_{k}(q);$
5) Pair
$e_{k}=\prod_{i=1}^{k}\frac{a-bq^{i-1}}{1-q^{i}},\;h_{k}=\prod_{i=1}^{k}\frac{aq^{i-1}-b}{1-q^{i}},$
leads to identities:
(62) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\prod_{j=1}^{k_{i}}\frac{a-bq^{j-1}}{1-q^{j}}=\prod_{i=1}^{k}\frac{aq^{i-1}-b}{1-q^{i}};$
(63) $\sum_{r=1}^{k}(-1)^{k-r}\sum_{k_{1}+k_{2}+...+k_{r}=k,\;k_{i}\geq
1}\prod_{i=1}^{r}\prod_{j=1}^{k_{i}}\frac{aq^{j-1}-b}{1-q^{j}}=\prod_{i=1}^{k}\frac{a-bq^{i-1}}{1-q^{i}}.$
## 11\. Acknowledgments
The author thanks Ira M. Gessel for private communication [3]. Especially he
is grateful to Lenny Tevlin for very useful discussions which lead to writing
the last two sections of the paper.
## References
* [1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York: Dover, pp. 804-806, 1972.
* [2] G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998.
* [3] I. M. Gessel, Private communication.
* [4] H. W. Gould, Combinatorial identities, Morgantown, 1972.
* [5] J. G. Hagen, Synopsis der Hoeheren Mathematik, V.1 (1891), 64-68.
* [6] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Univ. Press, Second edition, 1995.
* [7] J. Riordan, Combinatorial Identities, Wiley, New-York, 1968.
* [8] J. Riordan, An introduction to combinatorial analysis, Wiley, Fourth printing, 1967.
* [9] D. Salamon, A survey of symmetric functions, Grassmannians, and representations of the unitary group, Preprint, 1996\.
* [10] L. Tevlin, Private communication.
|
arxiv-papers
| 2012-09-23T10:17:58 |
2024-09-04T02:49:37.684956
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir Shevelev",
"submitter": "Vladimir Shevelev",
"url": "https://arxiv.org/abs/1211.1606"
}
|
1211.1783
|
# The arithmetic Grothendieck-Riemann-Roch theorem for general projective
morphisms
José Ignacio Burgos Gil Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3),
Calle Nicolás Cabrera 15 28049 Madrid, Spain burgos@icmat.es , Gerard
Freixas i Montplet Institut de Mathématiques de Jussieu (IMJ), Centre
National de la Recherche Scientifique (CNRS), France freixas@math.jussieu.fr
and Răzvan Liţcanu Faculty of Mathematics, University Al. I. Cuza Iasi,
Romania litcanu@uaic.ro
###### Abstract.
The classical arithmetic Grothendieck-Riemann-Roch theorem can be applied only
to projective morphisms that are smooth over the complex numbers. In this
paper we generalize the arithmetic Grothendieck-Riemann-Roch theorem to the
case of general projective morphisms between regular arithmetic varieties. To
this end we rely on the theory of generalized analytic torsion developed by
the authors.
## 1\. Introduction
The Grothendieck-Riemann-Roch theorem is a fundamental statement in algebraic
geometry. It describes the behavior of the Chern character from algebraic
$K$-theory to suitable cohomology theories (for instance Chow groups), with
respect to the push-forward operation by proper maps. It provides a vast
generalization of the classical Riemann-Roch theorem on Riemann surfaces and
the Hirzebruch-Riemann-Roch theorem on compact complex manifolds. In their
development of arithmetic intersection theory, Gillet and Soulé were lead to
extend the Grothendieck-Riemann-Roch theorem to the context of arithmetic
varieties. In this setting, vector bundles are equipped with additional smooth
hermitian metrics, for which an extension of algebraic $K$-theory can be
defined. There is a theory of characteristic classes for hermitian vector
bundles, with values in the so-called arithmetic Chow groups [16]. In analogy
to the classical algebraic geometric setting, it is natural to ask about the
behavior of the arithmetic Chern character with respect to proper push-
forward. This is the question first addressed by Gillet-Soulé [18] and later
by Gillet-Rössler-Soulé [14]. These works had to restrict to push-forward by
morphisms which are smooth over the complex points of the arithmetic
varieties. This assumption was necessary in defining both push-forward on
arithmetic $K$-theory and the arithmetic Chow groups. While on arithmetic Chow
groups push-forward resides on elementary operations (direct images of cycles
and fiber integrals of differential forms), they used holomorphic analytic
torsion on the arithmetic $K$-theory level [5].
The aim of this article is to extend the work of Gillet-Soulé and Gillet-
Rössler-Soulé to arbitrary projective morphisms of regular arithmetic
varieties. Hence we face to the difficulty of non-smoothness of morphisms at
the level of complex points. To accomplish our program, we have to introduce
generalized arithmetic Chow groups and arithmetic $K$-theory groups which
afford proper push-forward functorialities for possibly non-smooth projective
morphisms. Loosely speaking, this is achieved by replacing smooth differential
forms in the theory of Gillet and Soulé by currents with possibly non-empty
wave front sets. To motivate the introduction of these currents, we remark
that they naturally appear as push-forwards of smooth differential forms by
morphisms whose critical set is non-empty. In this concrete example, the wave
front set of the currents is controlled by the normal directions of the
morphism. The definition of our generalized arithmetic Chow groups is a
variant of the constructions of Burgos-Kramer-Kühn [12], specially their
covariant arithmetic Chow groups. As an advantage with respect to _loc. cit._
, the presentation we give simplifies the definition of proper push-forward,
which is the main operation we have to deal with in the present article. At
the level of Chow groups, this operation relies on push-forward of currents
and keeps track of the wave front sets. For arithmetic $K$-groups, we replace
the analytic torsion forms of Bismut-Köhler by a choice of a generalized
analytic torsion theory as developed in our previous work [9]. While a
generalized analytic torsion theory is not unique, we proved it is uniquely
determined by the choice of a real genus. We establish an arithmetic
Grothendieck-Riemann-Roch theorem for arbitrary projective morphisms, where
this real genus replaces the $R$-genus of Gillet and Soulé. We therefore
obtain the most general possible formulation of the theorem. In particular,
the natural choice of the 0 genus, corresponding to what we called the
homogenous theory of analytic torsion, provides an exact Grothendieck-Riemann-
Roch type formula, which is the formal translation of the classical algebraic
geometric theorem to the setting of Arakelov geometry. The present work is
thus the abutment of the articles [13] (by the first and third named authors)
and [10]–[9].
Let us briefly review the contents of this article. In section 2 we develop
our new generalization of arithmetic Chow groups, and consider as particular
instances the arithmetic Chow groups with currents of fixed wave front set. We
study the main operations, such as pull-back, push-forward and products. In
section 3 we carry a similar program to arithmetic $K$-theory. We also
consider an arithmetic version of our hermitian derived categories [10], that
is specially useful to deal with complexes of coherent sheaves with hermitian
structures. The essentials on arithmetic characteristic classes are treated in
section 4. With the help of our theory of generalized analytic torsion,
section 5 builds push-forward maps on the level of arithmetic derived
categories and arithmetic $K$-theory. The last section, namely section 6, is
devoted to the statement and proof of the arithmetic Grothendieck-Riemann-Roch
theorem for arbitrary projective morphisms of regular arithmetic varieties. As
an application, we compute the main characteristic numbers of the homogenous
theory, a question that was left open in [9].
## 2\. Generalized arithmetic Chow groups
Let $(A,\Sigma,F_{\infty})$ be an arithmetic ring [15]: that is, $A$ is an
excellent regular Noetherian integral domain, together with a finite non-empty
set of embeddings $\Sigma$ of $A$ into ${\mathbb{C}}$ and a linear conjugate
involution $F_{\infty}$ of the product ${\mathbb{C}}^{\Sigma}$ which commutes
with the diagonal embedding of $A$. Let $F$ be the field of fractions of $A$.
An arithmetic variety $\mathcal{X}$ is a flat and quasi-projective scheme over
$A$ such that $\mathcal{X}_{F}=\mathcal{X}\times\operatorname{Spec}F$ is
smooth. Then
$X_{{\mathbb{C}}}:=\coprod_{\sigma\in\Sigma}\mathcal{X}_{\sigma}({\mathbb{C}})$
is a complex algebraic manifold, which is endowed with an anti-holomorphic
automorphism $F_{\infty}$. One also associates to $\mathcal{X}$ the real
variety $X=(X_{{\mathbb{C}}},F_{\infty})$. Whenever we have arithmetic
varieties $\mathcal{X},\ \mathcal{Y},\dots$ we will denote by
$X_{{\mathbb{C}}},\ Y_{{\mathbb{C}}},\dots$ the associated complex manifolds
and by $X,\ Y,\dots$ the associated real manifolds.
To every regular arithmetic variety Gillet and Soulé have associated
arithmetic Chow groups, denoted
$\operatorname{\widehat{CH}}^{\ast}(\mathcal{X})$, and developed an arithmetic
intersection theory [15].
The arithmetic Chow groups defined by Gillet and Soulé are only covariant for
morphism that are smooth on the generic fiber. Moreover they are not suitable
to study the kind of singular metrics that appear naturally when dealing with
non proper modular varieties. In order to have arithmetic Chow groups that are
covariant with respect to arbitrary proper morphism, or that are suitable to
treat certain kind of singular metrics, in [12] different kinds of arithmetic
Chow groups are constructed, depending on the choice of a Gillet sheaf of
algebras $\mathcal{G}$ and a $\mathcal{G}$-complex $\mathcal{C}$. We denote by
$\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{C})$ the arithmetic
Chow groups defined in _op. cit._ Section 4.
The basic example of a Gillet algebra is the Deligne complex of sheaves of
differential forms with logarithmic singularities $\mathcal{D}_{\log}$,
defined in [12, Definition 5.67]; we refer to _op. cit._ for the precise
definition and properties. Therefore, to any $\mathcal{D}_{\log}$-complex we
can associate arithmetic Chow groups. In particular, considering
$\mathcal{D}_{\log}$ itself as a $\mathcal{D}_{\log}$-complex, we obtain
$\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\log})$, the
arithmetic Chow groups defined in [12, Section 6.1]. When $\mathcal{X}_{F}$ is
projective, these groups agree, up to a normalization factor, with the groups
defined by Gillet and Soulé.
The groups $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{C})$
introduced in [12, Section 6.1] have several technical issues: they depend on
the sheaf structure of $\mathcal{C}$ and not only on the complex of global
sections $\mathcal{C}(X)$; moreover, they are not completely satisfactory if
the cohomology determined by $\mathcal{C}$ does not satisfy a weak purity
property; finally the definition of direct images is intricate. To overcome
these difficulties we introduce here a variant of the cohomological arithmetic
Chow groups that only depends on the complex of global sections of a
$\mathcal{D}_{\log}$-complex.
###### Definition 2.1.
Let $\mathcal{X}$ be an arithmetic variety,
$X_{{\mathbb{C}}}=\mathcal{X}_{\Sigma}$ the associated complex manifold and
$X=(X_{{\mathbb{C}}},F_{\infty})$ the associated real manifold. A
_$\mathcal{D}_{\log}(X)$ -complex_ is a graded complex of real vector spaces
$C^{\ast}(\ast)$ provided with a morphism of graded complexes
$\operatorname{c}\colon\mathcal{D}_{\log}^{\ast}(X,\ast)\longrightarrow
C^{\ast}(\ast).$
Given two $\mathcal{D}_{\log}(X)$-complexes $C$ and $C^{\prime}$, we say that
$C^{\prime}$ is a $C$-complex if there is a commutative diagram of morphisms
of graded complexes
$\textstyle{\mathcal{D}_{\log}^{\ast}(X,\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{c^{\prime}}$$\textstyle{C^{\ast}(\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{C^{\prime\ast}(\ast).}$
In this situation, we say that $\varphi$ is a morphism of
$\mathcal{D}_{\log}(X)$-complexes.
We stress the fact that a $\mathcal{D}_{\log}$-complex is a complex of sheaves
while a $\mathcal{D}_{\log}(X)$-complex is a complex of vector spaces. If
$\mathcal{C}$ is a $\mathcal{D}_{\log}$-complex of real vector spaces, then
the complex of global sections $\mathcal{C}^{\ast}(X,\ast)$ is a
$\mathcal{D}_{\log}(X)$-complex. We are mainly interested in the
$\mathcal{D}_{\log}(X)$-complexes of Example 2.2 made out of differential
forms and currents. We will follow the conventions of [12, Section 5.4]
regarding differential forms and currents. In particular, both the current
associated to a differential form and the current associated to a cycle have
implicit a power of the trivial period $2\pi i$.
###### Example 2.2.
1. (i)
The Deligne complex $\mathcal{D}^{\ast}_{{\text{\rm a}}}(X,\ast)$ of
differential forms on $X$ with arbitrary singularities at infinity. Namely, if
$E^{\ast}(X_{{\mathbb{C}}})$ is the Dolbeault complex ([12, Definition 5.7])
of differential forms on $X_{{\mathbb{C}}}$ then
$\mathcal{D}_{{\text{\rm
a}}}^{\ast}(X,\ast)=\mathcal{D}^{\ast}(E^{\ast}(X_{{\mathbb{C}}}),\ast)^{\sigma},$
where $\mathcal{D}^{\ast}(\underline{\ },\ast)$ denotes the Deligne complex
([12, Definition 5.10]) associated to a Dolbeault complex and $\sigma$ is the
involution $\sigma(\eta)=\overline{F_{\infty}^{\ast}\eta}$ as in [12, Notation
5.65].
Note that $\mathcal{D}^{\ast}_{{\text{\rm a}}}(X,\ast)$ is the complex of
global sections of the $\mathcal{D}_{\log}$-complex
$\mathcal{D}^{\ast}_{{\text{\rm l, ll, a}}}$ that appears in [11, Section 3.6]
with empty log-log singular locus. In particular, by [11, Theorem 3.9] it
satisfies the weak purity condition.
2. (ii)
The Deligne complex $\mathcal{D}^{\ast}_{\text{{\rm cur}},{\text{\rm
a}}}(X,\ast)$ of currents on $X$. Namely, if $D^{\ast}(X_{{\mathbb{C}}})$ is
the Dolbeault complex of currents on $X_{{\mathbb{C}}}$ then
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}^{\ast}(X,\ast)=\mathcal{D}^{\ast}(D^{\ast}(X_{{\mathbb{C}}}),\ast)^{\sigma}.$
Note that here we are considering arbitrary currents on $X_{{\mathbb{C}}}$ and
not extendable currents as in [12, Definition 6.30].
3. (iii)
Let $T^{\ast}X_{{\mathbb{C}}}$ be the cotangent bundle of $X_{{\mathbb{C}}}$.
Denote by $T_{0}^{\ast}X_{{\mathbb{C}}}=T^{\ast}X_{{\mathbb{C}}}\setminus
X_{{\mathbb{C}}}$ the cotangent bundle with the zero section deleted and let
$S\subset T_{0}^{\ast}X_{{\mathbb{C}}}$ be a closed conical subset that is
invariant under $F_{\infty}$. Let $D^{\ast}(X_{{\mathbb{C}}},S)$ be the
complex of currents whose wave front set is contained in $S$ [13, Section 4].
The Deligne complex of currents on $X$ having the wave front set included in
the fixed set $S$ is given by
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}^{\ast}(X,S,\ast)=\mathcal{D}^{\ast}(D^{\ast}(X_{{\mathbb{C}}},S),\ast)^{\sigma}.$
The maps of complexes $\mathcal{D}^{\ast}_{{\text{\rm
a}}}(X,\ast)\to\mathcal{D}^{\ast}_{\text{{\rm cur}},{\text{\rm a}}}(X,\ast)$
given by $\eta\mapsto[\eta]$ is injective and makes of
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X,\ast)$ a
$\mathcal{D}^{\ast}_{{\text{\rm a}}}(X,\ast)$-complex. We will use this map to
identify $\mathcal{D}^{\ast}_{{\text{\rm a}}}$ with a subcomplex of
$\mathcal{D}^{\ast}_{\text{{\rm cur}},{\text{\rm a}}}$. Since
$\mathcal{D}^{\ast}_{{\text{\rm a}}}(X,\ast)=\mathcal{D}^{\ast}_{\text{{\rm
cur}},{\text{\rm a}}}(X,\emptyset,\ast)$ and $\mathcal{D}^{\ast}_{D,{\text{\rm
a}}}(X,\ast)=\mathcal{D}^{\ast}_{\text{{\rm cur}},{\text{\rm
a}}}(X,T^{\ast}_{0}X,\ast)$, examples (i) and (ii) are particular cases of
(iii).
###### Remark 2.3.
With these examples at hand, we can specialize the definition of $C$-complex
(Definition 2.1) to $C=\mathcal{D}_{{\text{\rm a}}}(X)$. When dealing with
hermitian structures on sheaves on non-necessarily proper varieties, we will
to work with $\mathcal{D}_{{\text{\rm a}}}(X)$-complexes rather than
$\mathcal{D}_{\log}(X)$-complexes.
We will also follow the notation of [12, Section 3] regarding complexes. In
particular we will write
$\widetilde{C}^{2p-1}(p)=C^{2p-1}(p)/\operatorname{Im}\operatorname{d}_{C},\quad\operatorname{Z}C^{2p}(p)=\operatorname{Ker}\operatorname{d}_{C}\cap
C^{2p}(p).$
###### Definition 2.4.
The _arithmetic Chow groups with $C$ coefficients_ are defined as
(2.5)
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)=\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\log})\times\widetilde{C}^{2p-1}(p)/\sim$
where $\sim$ is the equivalence relation generated by
(2.6) $(a(g),0)\sim(0,c(g)).$
If $\varphi:C\rightarrow C^{\prime}$ is a morphism of
$\mathcal{D}_{\log}(X)$-complexes (so that $C^{\prime}$ is a $C$-complex),
then there is a natural surjective morphism
(2.7)
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\times\widetilde{C^{\prime}}^{2p-1}(p)\longrightarrow\operatorname{\widehat{CH}}^{p}(\mathcal{X},C^{\prime}).$
Introducing the equivalence relation generated by
$((0,c),0)\equiv((0,0),\varphi(c)),$
we see that (2.7) induces an isomorphism
(2.8)
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\times\widetilde{C}^{\prime
2p-1}(p)/\equiv\;\overset{\sim}{\longrightarrow}\operatorname{\widehat{CH}}^{p}(\mathcal{X},C^{\prime}).$
We next unwrap Definition 2.4 in order to get simpler descriptions of the
arithmetic Chow groups associated to the complexes of Example 2.2. We start by
recalling the construction of
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\log})$. The group
of codimension $p$ arithmetic cycles is given by
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})=\left\\{(Z,\widetilde{g})\in\operatorname{Z}^{p}(\mathcal{X})\times\widetilde{\mathcal{D}}_{\log}^{2p-1}(X\setminus\mathcal{Z}^{p},p)\left|\begin{aligned}
\operatorname{d}_{\mathcal{D}}\widetilde{g}&\in\mathcal{D}_{\log}^{2p}(X,p)\\\
\operatorname{cl}(Z)&=[(\operatorname{d}_{\mathcal{D}}\widetilde{g},\widetilde{g})]\end{aligned}\right.\right\\},$
where $\operatorname{Z}^{p}(\mathcal{X})$ is the group of codimension $p$
algebraic cycles of $\mathcal{X}$, $\mathcal{Z}^{p}$ is the ordered system of
codimension at least $p$ closed subsets of $X$,
$\widetilde{\mathcal{D}}_{\log}^{2p-1}(X\setminus\mathcal{Z}^{p},p)=\lim_{\begin{subarray}{c}\longrightarrow\\\
W\in\mathcal{Z}^{p}\end{subarray}}\widetilde{\mathcal{D}}_{\log}^{2p-1}(X\setminus
W,p),$
and $\operatorname{cl}(Z)$ and
$[(\operatorname{d}_{\mathcal{D}}\widetilde{g},\widetilde{g})]$ denote the
class in the real Deligne-Beilinson cohomology group
$H^{2p}_{\mathcal{D},\mathcal{Z}^{p}}(X,{\mathbb{R}}(p))$ with supports on
$\mathcal{Z}^{p}$ of the cycle $Z$ and the pair
$(\operatorname{d}_{\mathcal{D}}\widetilde{g},\widetilde{g})$ respectively.
For each codimension $p-1$ irreducible variety $W$ and each rational function
$f\in K(W)$, there is a class $[f]\in
H^{2p-1}_{\mathcal{D}}(X\setminus|\operatorname{div}f|,{\mathbb{R}}(p))$.
Hence a class
$\operatorname{b}([f])\in\widetilde{\mathcal{D}}_{\log}^{2p-1}(X\setminus\mathcal{Z}^{p},p)$
that is denoted $\mathfrak{g}(f)$. Then
$\operatorname{\widehat{Rat}}^{p}(\mathcal{X},\mathcal{D}_{\log})$ is the
group generated by the elements of the form
$\operatorname{\widehat{div}}(f)=(\operatorname{div}(f),\mathfrak{g}(f))$.
Then
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\log})=\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})\left/\operatorname{\widehat{Rat}}^{p}(\mathcal{X},\mathcal{D}_{\log})\right..$
We will use the following well stablished notation. If $B$ is any subring of
${\mathbb{R}}$ we will denote by
$\operatorname{Z}^{p}_{B}(\mathcal{X})=\operatorname{Z}^{p}(\mathcal{X})\otimes
B$, by $\operatorname{\widehat{Z}}^{p}_{B}(\mathcal{X},\mathcal{D}_{\log})$
the group with the same definition as
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})$ with
$\operatorname{Z}^{p}_{B}(\mathcal{X})$ instead of
$\operatorname{Z}^{p}(\mathcal{X})$, and we write
$\operatorname{\widehat{Rat}}^{p}_{B}(\mathcal{X},\mathcal{D}_{\log})=\operatorname{\widehat{Rat}}^{p}(\mathcal{X},\mathcal{D}_{\log})\otimes
B$. Finaly we write
$\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{\log})=\operatorname{\widehat{Z}}^{p}_{B}(\mathcal{X},\mathcal{D}_{\log})\left/\operatorname{\widehat{Rat}}^{p}_{B}(\mathcal{X},\mathcal{D}_{\log})\right.$
Note that
$\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\log})=\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\log})\otimes{\mathbb{Q}}$
but, in general,
$\operatorname{\widehat{CH}}^{p}_{{\mathbb{R}}}(\mathcal{X},\mathcal{D}_{\log})\not=\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\log})\otimes{\mathbb{R}}$.
We will use the same notation for all variants of the arithmetic Chow groups.
Now, in the definition of $\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)$ we
can first change coefficients and then take rational equivalence. We define
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)=\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})\times\widetilde{C}^{2p-1}(p)/\sim$
where again $\sim$ is the equivalence relation generated by
$(a(g),0)\sim(0,c(g))$.
There are maps
$\displaystyle\zeta_{C}$
$\displaystyle\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{Z}^{p}(\mathcal{X}),$
$\displaystyle\zeta_{C}((Z,\widetilde{g}),\widetilde{c})$ $\displaystyle=Z,$
$\displaystyle\operatorname{a}_{C}$
$\displaystyle\colon\widetilde{C}^{2p-1}(p)\longrightarrow\operatorname{\widehat{Z}}^{p}(\mathcal{X},C),$
$\displaystyle\operatorname{a}_{C}(\widetilde{c})$
$\displaystyle=((0,0),-\widetilde{c}),$ $\displaystyle\omega_{C}$
$\displaystyle\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{Z}C^{2p}(p),$
$\displaystyle\qquad\omega_{C}((Z,\widetilde{g}),\widetilde{c})$
$\displaystyle=\operatorname{c}(\operatorname{d}_{\mathcal{D}}\widetilde{g})+\operatorname{d}_{C}\widetilde{c}.$
We also consider the map
$\operatorname{b}_{C}\colon
H^{2p-1}(C^{\ast}(p))\to\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)$
obtained by composing $\operatorname{a}_{C}$ with the inclusion
$H^{2p-1}(C^{\ast}(p))\to\widetilde{C}^{2p-1}(p)$, and the map
$\rho_{C}\colon\operatorname{CH}^{p,p-1}(\mathcal{X})\to
H^{2p-1}(C^{\ast}(p))$
obtained by composing the regulator map
$\rho\colon\operatorname{CH}^{p,p-1}(\mathcal{X})\to
H^{2p-1}_{\mathcal{D}}(X,{\mathbb{R}}(p))$ in [12, Notation 4.12] with the map
$\operatorname{c}\colon H^{2p-1}_{\mathcal{D}}(X,{\mathbb{R}}(p))\to
H^{2p-1}(C^{\ast}(p))$. We will also denote by $\rho_{C}$ the analogous map
with target $\widetilde{C}^{2p-1}(p)$.
There are induced maps
$\displaystyle\zeta_{C}$
$\displaystyle\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{CH}^{p}(\mathcal{X}),$
$\displaystyle\operatorname{a}_{C}$
$\displaystyle\colon\widetilde{C}^{2p-1}(p)\longrightarrow\operatorname{\widehat{CH}}^{p}(\mathcal{X},C),$
$\displaystyle\omega_{C}$
$\displaystyle\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{Z}C^{2p}(p).$
###### Lemma 2.9.
1. (i)
Let $\operatorname{\widehat{Rat}}^{p}(\mathcal{X},C)$ denote the image of
$\operatorname{\widehat{Rat}}^{p}(\mathcal{X},\mathcal{D}_{\log})$ in the
group $\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)$. Then
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)=\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\left/\operatorname{\widehat{Rat}}^{p}(\mathcal{X},C)\right..$
2. (ii)
There is an exact sequence
(2.10)
$0\to\widetilde{C}^{2p-1}(p)\overset{\operatorname{a}_{C}}{\longrightarrow}\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\overset{\zeta_{C}}{\longrightarrow}\operatorname{Z}^{p}(\mathcal{X})\to
0.$
3. (iii)
There are exact sequences
(2.11)
$\operatorname{CH}^{p,p-1}(\mathcal{X})\overset{\rho_{C}}{\longrightarrow}\widetilde{C}^{2p-1}(p)\overset{\operatorname{a}_{C}}{\longrightarrow}\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\overset{\zeta_{C}}{\longrightarrow}\operatorname{CH}^{p}(\mathcal{X})\to
0,$
and
(2.12)
$\operatorname{CH}^{p,p-1}(\mathcal{X})\overset{\rho_{C}}{\longrightarrow}H^{2p-1}(C^{\ast}(p))\overset{\operatorname{b}_{C}}{\longrightarrow}\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\overset{\zeta_{C}\oplus\omega_{C}}{\longrightarrow}\\\
\operatorname{CH}^{p}(\mathcal{X})\oplus\operatorname{Z}C^{2p}(p)\longrightarrow
H^{2p}(C^{\ast}(p))\to 0.$
###### Proof.
(i) Follows easily from the definition.
(ii) By [8, Proposition 5.5] there is an exact sequence
$0\to\widetilde{\mathcal{D}}_{\log}^{2p-1}(X,p)\overset{\operatorname{a}}{\longrightarrow}\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})\overset{\zeta}{\longrightarrow}\operatorname{Z}^{p}(\mathcal{X})\to
0.$
From it and the definition of
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})$ we derive the
exactness of (2.10).
(iii) Follows from the exact sequences of [8, Theorem 7.3] and the definition
of $\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)$. ∎
The contravariant functoriality of
$\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\log})$ is easily
translated to other coefficients. Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a
morphism of regular arithmetic varieties. Let $C$ be a
$\mathcal{D}_{\log}(X)$-complex and $C^{\prime}$ a
$\mathcal{D}_{\log}(Y)$-complex, such that there exists a map of complexes
$f^{\ast}:C^{\prime\ast}(\ast)\to C^{\ast}(\ast)$ that makes the following
diagram commutative:
$\textstyle{\mathcal{D}_{\log}^{\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\mathcal{D}_{\log}^{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{C^{\ast}(X).}$
Then we define
$f^{\ast}((\mathcal{Z},g),c)=(f^{\ast}(\mathcal{Z},g),f^{\ast}(c)).$
It is easy to see that this map is well defined, because the pull-back map
$f^{\ast}:\operatorname{\widehat{CH}}^{\ast}(\mathcal{Y})\to\operatorname{\widehat{CH}}^{\ast}(\mathcal{X})$
(for the Chow groups corresponding to $\mathcal{D}_{\log}$) is compatible to
the map $\operatorname{a}$.
Before stating concrete examples of this contravariant functoriality we need
some notation ([20, Theorem 8.2.4], see also [13, Section 4]). Let
$f_{{\mathbb{C}}}\colon X_{{\mathbb{C}}}\to Y_{{\mathbb{C}}}$ denote the
induced map of complex manifolds. Let $N_{f}$ be the set of normal directions
of $f_{{\mathbb{C}}}$, that is
$N_{f}=\\{(f(x),\xi)\in
T_{0}^{\ast}Y_{{\mathbb{C}}}\mid\operatorname{d}f_{{\mathbb{C}}}^{t}\xi=0\\}.$
Let $S\subset T^{\ast}_{0}Y_{{\mathbb{C}}}$ be a closed conical subset
invariant under $F_{\infty}$. When $N_{f}\cap S=\emptyset$, the function $f$
is said to be transverse to $S$. In this case we write
$f^{\ast}S=\\{(x,\operatorname{d}f_{{\mathbb{C}}}^{t}\xi)\mid(f(x),\xi)\in
S\\}.$
It is a closed conical subset of $T^{\ast}_{0}X_{{\mathbb{C}}}$ invariant
under $F_{\infty}$.
###### Proposition 2.13.
Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a morphism of regular arithmetic
varieties.
1. (i)
There is a pull-back morphism
$f^{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{Y},\mathcal{D}_{{\text{\rm
a}}}(Y))\to\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$.
2. (ii)
Let $N_{f}$ be the set of normal directions of $f_{{\mathbb{C}}}$ and
$S\subset T^{\ast}_{0}Y_{{\mathbb{C}}}$ a closed conical subset invariant
under $F_{\infty}$. If $N_{f}\cap S=\emptyset$, then there is a pull-back
morphism
$f^{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,S))\to\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,f^{\ast}S)).$
3. (iii)
If $f_{F}$ is smooth (hence $N_{f}=\emptyset$) then there is a pull-back
morphism
$f^{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y))\to\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X)).$
Similarly the multiplicative properties of
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\log})$
can be transferred to other coefficients. Let $C$, $C^{\prime}$ and
$C^{\prime\prime}$ be $\mathcal{D}_{\log}(X)$-complexes such that there is a
commutative diagram of morphisms of complexes
$\textstyle{\mathcal{D}_{\log}(X)\otimes\mathcal{D}_{\log}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathcal{D}_{\log}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\otimes
C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{C^{\prime\prime}.}$
Then we define a product
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\times\operatorname{\widehat{CH}}^{q}(\mathcal{X},C^{\prime})\to\operatorname{\widehat{CH}}^{p+q}_{{\mathbb{Q}}}(\mathcal{X},C^{\prime\prime})$
by
(2.14)
$((\mathcal{Z},g),c)\cdot((\mathcal{Z}^{\prime},g^{\prime}),c^{\prime})=((\mathcal{Z},g)\cdot(\mathcal{Z}^{\prime},g^{\prime}),c\bullet\operatorname{c}(\omega(g^{\prime}))+\operatorname{c}(\omega(g))\bullet
c^{\prime}+\operatorname{d}_{C}c\bullet c^{\prime}).$
As a consequence we obtain the following result.
###### Proposition 2.15.
Let $\mathcal{X}$ be a regular arithmetic variety.
1. (i)
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$ is an associative commutative graded ring.
2. (ii)
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$ is a module over
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$.
3. (iii)
Let $S,S^{\prime}$ be closed conic subsets of $T^{\ast}_{0}X_{{\mathbb{C}}}$
that are invariant under $F_{\infty}$. Then
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S))$ is a module over
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$. Moreover, if $S\cap(-S^{\prime})=\emptyset$, there is a graded
bilinear map
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S))\times\operatorname{\widehat{CH}}^{q}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S^{\prime}))\longrightarrow\\\
\operatorname{\widehat{CH}}^{p+q}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S\cup S^{\prime}\cup(S+S^{\prime}))).$
4. (iv)
The product is compatible with the pull-back of Proposition 2.13.
We now turn our attention towards direct images. The definition of direct
images for general $\mathcal{D}_{\log}$-complexes is quite intricate,
involving the notion of covariant $f$-pseudo-morphisms (see [12, Definition
3.71]). By contrast, we will give a another description of the groups
$\operatorname{\widehat{CH}}^{\ast}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$ for which the definition of push-forward is much
simpler.
By [7, 3.8.2] we know that any $\mathcal{D}_{\log}$-Green form is locally
integrable. Therefore there is a well defined map
$\varphi\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\log})\longrightarrow\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p-1}(X,p)$
given by $(Z,\widetilde{g})\to(Z,\widetilde{[g]})$ for any representative $g$
of $\widetilde{g}$. The previous map can be extended to a map
$\varphi_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)}\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X))\longrightarrow\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p-1}(X,p)$
given by $\varphi_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)}((Z,\widetilde{g}),\widetilde{h})=(Z,\widetilde{[g]}+\widetilde{h})$.
The following result is clear.
###### Lemma 2.16.
The map $\varphi_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X)}$ is an
isomorphism.
This lemma gives us a more concrete description of the group
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$. In fact, we will identify it with the group
$\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p-1}(X,p)$ when necessary. Some care has to be taken
when doing this identification. For instance
(2.17) $\omega_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)}(\varphi_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)}^{-1}(Z,\widetilde{g}))=\operatorname{d}_{\mathcal{D}}g+\delta_{Z}.$
Let now $f\colon\mathcal{X}\to\mathcal{Y}$ be a proper morphism of regular
arithmetic varieties of relative dimension $e$. Using the above
identification, we define
(2.18)
$f_{\ast}\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X))\longrightarrow\operatorname{\widehat{Z}}^{p-e}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y))$
by $f_{\ast}(Z,\widetilde{g})=(f_{\ast}Z,\widetilde{f_{\ast}g})$, where $g$ is
any representative of $\widetilde{g}$, and $f_{\ast}(g)$ is the usual direct
image of currents given by $f_{\ast}(g)(\eta)=g(f^{\ast}\eta)$.
###### Proposition 2.19.
The map $f_{\ast}$ in (2.18) sends the group
$\operatorname{\widehat{Rat}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$ to the group
$\operatorname{\widehat{Rat}}^{p-e}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y))$. Therefore it induces a map
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X))\longrightarrow\operatorname{\widehat{CH}}^{p-e}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y)).$
In order to transfer this push-forward to other coefficients we introduce two
extra properties for a $\mathcal{D}_{\log}(X)$-complex $C$.
:
(H1) There is a commutative diagram of injective morphisms of complexes
$\textstyle{\mathcal{D}_{\log}^{\ast}(X,\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{c}}$$\textstyle{C^{\ast}(\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{c}^{\prime}}$$\textstyle{\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}^{\ast}(X,\ast).}$
Since $\operatorname{c}^{\prime}$ is injective we will usually identify $C$
with its image by $\operatorname{c}^{\prime}$.
:
(H2) The map $\operatorname{c}^{\prime}$ induces isomorphisms
$\displaystyle H^{n}(C^{\ast}(p))\cong H^{n}(\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}^{\ast}(X,p))$
for all $p\geq 0$ and $n=2p-1,2p$.
The conditions (H1) and (H2) have two consequences. First if $\eta\in
D_{\text{{\rm cur}},{\text{\rm a}}}^{2p-1}(X,p)$ is a current such that
$\operatorname{d}_{\mathcal{D}}\eta\in C^{2p}(p)$, there exist
$a\in\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}^{2p-2}(X,p)$ such that
$\operatorname{d}_{\mathcal{D}}a+\eta\in C^{2p-1}(p).$
Second, the induced map
$\widetilde{C}^{2p-1}(p)\to\widetilde{\mathcal{D}}^{2p-1}(X,p)$ is injective.
Let $C$ be a $\mathcal{D}_{\log}(X)$-complex satisfying (H1). Consider the
diagram
$\textstyle{\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{\omega_{C}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Z}C^{2p}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\operatorname{Z}\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p}(p)}$
where $A$ is defined by the cartesian square, $i$ is induced by (H1) and $j$
is induced by $i$ and $\omega_{C}$.
###### Lemma 2.20.
If $C$ also satisfies (H2) then $j$ is an isomorphism.
###### Proof.
By the injectivity of
$\widetilde{C}^{2p-1}(p)\to\widetilde{\mathcal{D}}^{2p-1}(X,p)$ and Lemma 2.9
(iii), the map $i$ is injective. Hence the map $j$ is injective and we only
need to prove that $j$ is surjective.
Let $x=((Z,\widetilde{g}),\eta)\in A$. This means that
$(Z,\widetilde{g})\in\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$, $\eta\in\operatorname{Z}C^{2p}(p)$ and
$\omega_{\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)}(Z,\widetilde{g})=\operatorname{d}_{\mathcal{D}}g+\delta_{Z}=\eta$.
Let $g^{\prime}\in\mathcal{D}^{2p-1}_{\log}(X\setminus|Z|,p)$ be a Green form
for $Z$. Then $g-[g^{\prime}]\in\mathcal{D}^{2p-1}_{\text{{\rm
cur}},{\text{\rm a}}}(X,p)$ satisfies
$\operatorname{d}_{\mathcal{D}}(g-[g^{\prime}])\in C^{2p}(p)$. By (H2) there
is $a\in\mathcal{D}^{2p-2}_{\text{{\rm cur}},{\text{\rm a}}}(X,p)$ such that
$g^{\prime\prime}:=g-[g^{\prime}]+\operatorname{d}_{\mathcal{D}}a\in
C^{2p-1}(p)$. Consider the element
$x^{\prime}=((Z,\widetilde{g}^{\prime}),\widetilde{g}^{\prime\prime})\in\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)$.
To see that $j(x^{\prime})=x$ we have to check that
$\omega_{C}(x^{\prime})=\eta$ and $i(x^{\prime})=(Z,\widetilde{g})$. We
compute
$\displaystyle\omega_{C}(x^{\prime})=\operatorname{c}(\operatorname{d}_{\mathcal{D}}g^{\prime})+\operatorname{d}_{C}g^{\prime\prime}=\operatorname{d}_{\mathcal{D}}[g^{\prime}]+\delta_{Z}+\operatorname{d}_{\mathcal{D}}g-\operatorname{d}_{\mathcal{D}}[g^{\prime}]+\operatorname{d}_{\mathcal{D}}\operatorname{d}_{\mathcal{D}}a=\eta,$
$\displaystyle
i(x^{\prime})=(Z,([g^{\prime}]+g-[g^{\prime}]+\operatorname{d}_{\mathcal{D}}a)^{\sim})=(Z,\widetilde{g}),$
concluding the proof of the lemma. ∎
We can rephrase the lemma as follows.
###### Theorem 2.21.
Let $C$ be a $\mathcal{D}_{\log}(X)$-complex that satisfies (H1) and (H2).
Then the map $\varphi$ can be extended to an injective map
$\varphi_{C}\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{D}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p-1}(X,p).$
Moreover
(2.22)
$\operatorname{Im}(\varphi_{C})=\left\\{(Z,\widetilde{g})\in\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm
a}}}^{2p-1}(X,p)\mid\operatorname{d}_{\mathcal{D}}g+\delta_{Z}\in
C^{2p}(p)\right\\}.$
In view of this theorem, if $C$ satisfies (H1) and (H2), we will identify
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)$ with the right hand side of
equation (2.22).
###### Proposition 2.23.
The $\mathcal{D}_{\log}(X)$-complexes $\mathcal{D}_{{\text{\rm a}}}(X)$,
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X,S)$ and
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X)$ satisfy (H1) and (H2).
Therefore we can identify
$\displaystyle\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$
$\displaystyle\cong\left\\{(Z,\widetilde{g})\in\operatorname{Z}^{p}\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm
a}}}^{2p-1}(X,p)\mid\operatorname{d}_{\mathcal{D}}g+\delta_{Z}\in\mathcal{D}^{2p}_{{\text{\rm
a}}}(X,p)\right\\}$
$\displaystyle\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S))$
$\displaystyle\cong\left\\{(Z,\widetilde{g})\in\operatorname{Z}^{p}\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm
a}}}^{2p-1}(X,p)\mid\operatorname{d}_{\mathcal{D}}g+\delta_{Z}\in\mathcal{D}^{2p}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S,p)\right\\}$
$\displaystyle\operatorname{\widehat{Z}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$
$\displaystyle\cong\operatorname{Z}^{p}(\mathcal{X})\oplus\widetilde{\mathcal{D}}_{\text{{\rm
cur}},{\text{\rm a}}}^{2p-1}(X,p).$
###### Proof.
The result for $\mathcal{D}_{{\text{\rm a}}}(X)$ follows from the Poincaré
$\overline{\partial}$-Lemma for currents [19, Pag. 382]. The result for
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X,S)$ follows from the Poincaré
$\overline{\partial}$-Lemma for currents with fixed wave front set [13,
Theorem 4.5]. The statement for $\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X)$ is Lemma 2.16. ∎
###### Remark 2.24.
The previous proposition gives us a more concrete description of the groups
$\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)$ for $C=\mathcal{D}_{{\text{\rm
a}}}(X)$, $\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X,S)$ and
$\mathcal{D}_{\text{{\rm cur}},{\text{\rm a}}}(X)$ and also a more concrete
description of the groups $\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)$. In
particular it easily implies that the groups
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$, $p\geq 0$, agree (up to a normalization factor) with the arithmetic
Chow groups of Gillet-Soulé $\operatorname{\widehat{CH}}^{p}(\mathcal{X})$ and
that the groups
$\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X))$, $p\geq 0$, agree (again up to a normalization
factor) with the arithmetic Chow groups of Kawaguchi-Moriwaki
$\operatorname{\widehat{CH}}^{p}_{D}(\mathcal{X})$, introduced in [21].
We want to transfer the push-forward of Proposition 2.19 to complexes
satisfying (H1) and (H2). Let again $f\colon\mathcal{X}\to\mathcal{Y}$ be a
proper morphism of regular arithmetic varieties of relative dimension $e$. Let
$C$ be a $\mathcal{D}_{\log}(X)$-complex and $C^{\prime}$ a
$\mathcal{D}_{\log}(Y)$-complex, both satisfying (H1) and (H2). We assume
there exists a map of complexes $f_{\ast}\colon C^{\ast}(\ast)\to
C^{\prime\ast-2e}(\ast-e)$ that makes the following diagram commutative:
(2.25)
$\textstyle{C^{\ast}(\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{C^{\prime\ast-2e}(\ast-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{\text{{\rm
cur}}}^{\ast}(X,\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{\mathcal{D}_{\text{{\rm
cur}}}^{\ast-2e}(Y,\ast-e).}$
Then, for $p\geq 0$, we define maps
$f_{\ast}\colon\operatorname{\widehat{Z}}^{p}(\mathcal{X},C)\to\operatorname{\widehat{Z}}^{p-e}(\mathcal{Y},C^{\prime})$
that, using the identification of Theorem 2.21 are given by
(2.26) $f_{\ast}(Z,\widetilde{g})=(f_{\ast}Z,\widetilde{f_{\ast}(g)}).$
By Proposition 2.19, these maps send
$\operatorname{\widehat{Rat}}^{p}(\mathcal{X},C)$ to
$\operatorname{\widehat{Rat}}^{p-e}(\mathcal{Y},C^{\prime})$. Therefore they
induce morphisms
$f_{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},C)\longrightarrow\operatorname{\widehat{CH}}^{p-e}(\mathcal{Y},C^{\prime}).$
Before stating concrete examples of this covariant functoriality we need some
notation. Let $f_{{\mathbb{C}}}\colon X_{{\mathbb{C}}}\to Y_{{\mathbb{C}}}$
denote the induced proper map of complex manifolds. If $S\subset
T^{\ast}_{0}X_{{\mathbb{C}}}$ is a closed conical subset invariant under
$F_{\infty}$, then we write
$f_{\ast}S=\\{(f(x),\xi)\in
T^{\ast}_{0}Y_{{\mathbb{C}}}\mid(x,\operatorname{d}f^{t}\xi)\in S\\}\cup
N_{f}.$
It is a closed conical subset of $T^{\ast}_{0}Y_{{\mathbb{C}}}$ invariant
under $F_{\infty}$.
###### Proposition 2.27.
Let $f\colon\mathcal{X}\to\mathcal{Y}$, $g\colon\mathcal{Y}\to\mathcal{Z}$ be
proper morphism of regular arithmetic varieties of relative dimension $e$ and
$e^{\prime}$, and $S\subset T^{\ast}_{0}X_{{\mathbb{C}}}$ a closed conical
subset invariant under $F_{\infty}$. Then there are maps
$f_{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S))\to\operatorname{\widehat{CH}}^{p-e}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,f_{\ast}S)),$
and similarly
$g_{\ast}\colon\operatorname{\widehat{CH}}^{p^{\prime}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,f_{\ast}S))\to\operatorname{\widehat{CH}}^{p^{\prime}-e^{\prime}}(\mathcal{Z},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Z,g_{\ast}f_{\ast}S)).$
Furthermore, the relation $(g\circ f)_{\ast}=g_{\ast}\circ f_{\ast}$ is
satisfied.
As a particular case of the above proposition we obtain the following cases.
###### Corollary 2.28.
Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a proper morphism of regular
arithmetic varieties of relative dimension $e$.
1. (i)
There are maps
$f_{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))\to\operatorname{\widehat{CH}}^{p-e}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,N_{f})).$
2. (ii)
If $N_{f}=\emptyset$, then there are maps
$f_{\ast}\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))\to\operatorname{\widehat{CH}}^{p-e}(\mathcal{Y},\mathcal{D}_{{\text{\rm
a}}}(Y)).$
The functoriality and the multiplicative structures satisfy the following
compatibility properties.
###### Theorem 2.29.
Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a morphism of regular arithmetic
varieties. Let $N_{f}$ be the set of normal directions of $f_{{\mathbb{C}}}$
and $S,\,S^{\prime}\subset T^{\ast}_{0}Y_{{\mathbb{C}}}$ closed conical
subsets invariant under $F_{\infty}$. Then
$f^{\ast}(S\cup S^{\prime}\cup(S+S^{\prime}))=f^{\ast}(S)\cup
f^{\ast}(S^{\prime})\cup(f^{\ast}(S)+f^{\ast}(S^{\prime}))).$
If $N_{f}\cap(S\cup S^{\prime}\cup S+S^{\prime})=\emptyset$ and
$S\cap(-S^{\prime})=\emptyset$ then
$f^{\ast}(S)\cap(-f^{\ast}(S^{\prime}))=\emptyset$. In this case, if
$\alpha\in\operatorname{\widehat{CH}}^{p}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,S))$ and
$\beta\in\operatorname{\widehat{CH}}^{q}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,S^{\prime}))$ then
$f^{\ast}(\alpha\cdot\beta)=f^{\ast}(\alpha)\cdot
f^{\ast}(\beta)\in\operatorname{\widehat{CH}}^{p+q}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,f^{\ast}(S\cup S^{\prime}\cup(S+S^{\prime})))).$
###### Theorem 2.30.
Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a proper morphism of regular
arithmetic varieties of relative dimension $e$. Let $N_{f}$ be the set of
normal directions of $f_{{\mathbb{C}}}$, $S\subset
T^{\ast}_{0}X_{{\mathbb{C}}}$ and $S^{\prime}\subset
T^{\ast}_{0}Y_{{\mathbb{C}}}$ closed conical subsets invariant under
$F_{\infty}$. Then
$f_{\ast}(S\cup f^{\ast}(S^{\prime})\cup(S+f^{\ast}(S^{\prime})))\subset
f_{\ast}(S)\cup S^{\prime}\cup(f_{\ast}(S)+S^{\prime})).$
If $f_{\ast}(S)\cap(-S^{\prime})=\emptyset$ then $N_{f}\cap
S^{\prime}=\emptyset$ and $S\cap(-f^{\ast}(S^{\prime}))=\emptyset$. In this
case, if
$\alpha\in\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S))$ and
$\beta\in\operatorname{\widehat{CH}}^{q}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,S^{\prime}))$ then
$f_{\ast}(\alpha\cdot
f^{\ast}(\beta))=f_{\ast}(\alpha)\cdot\beta\in\operatorname{\widehat{CH}}^{p+q-e}_{{\mathbb{Q}}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,f_{\ast}(S)\cup
S^{\prime}\cup(f_{\ast}(S)+S^{\prime}))).$
## 3\. Arithmetic $K$-theory and derived categories
### 3.1. Arithmetic $K$-theory
As for arithmetic Chow groups, the arithmetic $K$-groups of Gillet-Soulé can
be generalized to include more general coefficients at the archimedean places.
Because the definition of arithmetic $K$-groups involves hermitian vector
bundles whose metrics have arbitrary singularities at infinity, we are
actually forced to consider $\mathcal{D}_{{\text{\rm a}}}(X)$-complex
coefficients. The reader is referred to [11, Sec. 4.2] for the construction of
the groups $\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm a}}})$, which
provide the base of our extension (see also Definition 3.1 below). An
arithmetic Chern character allows to compare these generalized arithmetic
$K$-groups and the generalized arithmetic Chow groups. The arithmetic Chern
character is automatically compatible with pull-back and products whenever
defined.
###### Definition 3.1.
Let $\mathcal{X}$ be an arithmetic variety and $C^{\ast}(\ast)$ a
$\mathcal{D}_{{\text{\rm a}}}(X)$-complex with structure morphism
$\operatorname{c}:\mathcal{D}_{{\text{\rm a}}}^{\ast}(X,\ast)\rightarrow
C^{\ast}(\ast)$. The arithmetic $K$ group of $\mathcal{X}$ with $C$
coefficients is the abelian group $\widehat{K}_{0}(\mathcal{X},C)$ generated
by pairs $(\overline{\mathcal{E}},\eta)$, where $\overline{\mathcal{E}}$ is a
smooth hermitian vector bundle on $\mathcal{X}$ and $\eta\in\bigoplus_{p\geq
0}\widetilde{C}^{2p-1}(p)$, modulo the relations
$(\overline{\mathcal{E}}_{1},\eta_{1})+(\overline{\mathcal{E}}_{2},\eta_{2})=(\overline{\mathcal{E}},\operatorname{c}(\widetilde{\operatorname{ch}}(\overline{\varepsilon}))+\eta_{1}+\eta_{2}),$
for every exact sequence
$\overline{\varepsilon}\colon\quad
0\longrightarrow\overline{\mathcal{E}}_{1}\longrightarrow\overline{\mathcal{E}}\longrightarrow\overline{\mathcal{E}}_{2}\longrightarrow
0$
with Bott-Chern secondary class
$\widetilde{\operatorname{ch}}(\overline{\varepsilon})$.
An equivalent construction can be given in the same lines as for the
generalized arithmetic Chow groups. For this, observe there is a natural
morphism
(3.2) $\begin{split}\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}})&\longrightarrow\widehat{K}_{0}(\mathcal{X},C)\\\
[\overline{\mathcal{E}},\eta]&\longmapsto[\overline{\mathcal{E}},\operatorname{c}(\eta)]\end{split}$
induced by $\operatorname{c}:\mathcal{D}_{{\text{\rm
a}}}^{\ast}(X,\ast)\rightarrow C^{\ast}(\ast)$, and also
(3.3) $\begin{split}\bigoplus_{p\geq
0}\widetilde{C}^{2p-1}(p)&\longrightarrow\widehat{K}_{0}(\mathcal{X},C)\\\
\eta&\longmapsto[0,\eta].\end{split}$
One easily sees the maps (3.2)–(3.3) induce a natural isomorphism of groups
(3.4) $\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}})\times\bigoplus_{p\geq
0}\widetilde{C}^{2p-1}(p)/\equiv\;\overset{\cong}{\longrightarrow}\widehat{K}_{0}(\mathcal{X},C),$
where $\equiv$ is the equivalence relation generated by
$((\overline{\mathcal{E}},\eta),0)\equiv((\overline{\mathcal{E}},0),\operatorname{c}(\eta)).$
Generalized arithmetic $K$-groups for suitable complexes have pull-backs and
products. Let $f:\mathcal{X}\to\mathcal{Y}$ be a morphism of arithmetic
varieties, and suppose given $\mathcal{D}_{{\text{\rm a}}}(X)$ and
$\mathcal{D}_{{\text{\rm a}}}(Y)$ complexes $C$ and $C^{\prime}$ respectively,
for which there is a commutative diagram
(3.5) $\textstyle{\mathcal{D}_{{\text{\rm
a}}}^{\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\mathcal{D}_{{\text{\rm
a}}}^{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{C^{\ast}.}$
By description (3.4) and the contravariant functoriality of
$\widehat{K}_{0}(\underline{\ },\mathcal{D}_{{\text{\rm a}}})$, we see there
is an induced morphism of groups
$f^{\ast}:\widehat{K}_{0}(\mathcal{Y},C^{\prime})\longrightarrow\widehat{K}_{0}(\mathcal{X},C).$
For this kind of functoriality, an analog statement to Proposition 2.13 holds,
and we leave to the reader the task of stating it. As for products, let $C$,
$C^{\prime}$ and $C^{\prime\prime}$ be $\mathcal{D}_{{\text{\rm
a}}}(X)$-complexes with a product $C\otimes
C^{\prime}\overset{\bullet}{\rightarrow}C^{\prime\prime}$ and a commutative
diagram
(3.6) $\textstyle{\mathcal{D}_{{\text{\rm
a}}}(X)\otimes\mathcal{D}_{{\text{\rm
a}}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathcal{D}_{{\text{\rm
a}}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\otimes
C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{C^{\prime\prime}.}$
Then there is an induced product at the level of $\widehat{K}_{0}$
$\widehat{K}_{0}(\mathcal{X},C)\times\widehat{K}_{0}(\mathcal{X},C^{\prime})\longrightarrow\widehat{K}_{0}(\mathcal{X},C^{\prime\prime})$
described by the rule
$[\overline{\mathcal{E}},\eta]\cdot[\overline{\mathcal{E}}^{\prime},\eta^{\prime}]=[\overline{\mathcal{E}}\otimes\overline{\mathcal{E}}^{\prime},c(\operatorname{ch}(\overline{\mathcal{E}}))\bullet\eta^{\prime}+c^{\prime}(\operatorname{ch}(\overline{\mathcal{E}}^{\prime}))\bullet\eta+\operatorname{d}_{C}\eta\bullet\eta^{\prime}].$
With respect to this laws, the groups $\widehat{K}_{0}$ enjoy of the analogue
properties to Proposition 2.15.
Finally, we discuss on the arithmetic Chern character. If $\mathcal{X}$ is a
regular arithmetic variety, we recall there is an isomorphism of rings [11,
Thm. 4.5], namely
$\operatorname{\widehat{ch}}:\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}})_{{\mathbb{Q}}}\longrightarrow\bigoplus_{p\geq
0}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}).$
If $C$ is a $\mathcal{D}_{{\text{\rm a}}}(X)$-complex, by the presentations
(2.8) of $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},C)$ and (3.4) of
$\widehat{K}_{0}$ it is clear that $\operatorname{\widehat{ch}}$ extends to an
isomorphism of groups
$\operatorname{\widehat{ch}}:\widehat{K}_{0}(\mathcal{X},C)_{{\mathbb{Q}}}\longrightarrow\bigoplus_{p\geq
0}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},C).$
Suppose now that $C,C^{\prime},C^{\prime\prime}$ are $\mathcal{D}_{{\text{\rm
a}}}(X)$-complexes with a product $C\otimes C^{\prime}\rightarrow
C^{\prime\prime}$ as above. Then, it is easily seen that there is a
commutative diagram of morphisms of groups
$\textstyle{\widehat{K}_{0}(\mathcal{X},C)_{{\mathbb{Q}}}\times\widehat{K}_{0}(\mathcal{X},C^{\prime})_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\cdot}$$\scriptstyle{(\operatorname{\widehat{ch}},\operatorname{\widehat{ch}})}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C^{\prime\prime})_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widehat{ch}}}$$\textstyle{\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},C)\times\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},C^{\prime\prime}).}$
This is in particular true for complexes of currents with controlled wave
front set, a result that we next record.
###### Proposition 3.7.
Let $S,S^{\prime}$ be closed conical subsets of $T_{0}^{\ast}X_{{\mathbb{C}}}$
invariant under the action of complex conjugation, with
$S\cap(-S^{\prime})=\emptyset$. Define $T=S\cup S^{\prime}\cup(S+S^{\prime})$.
If $\alpha\in\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,S))_{{\mathbb{Q}}}$ and
$\beta\in\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X,S^{\prime}))_{{\mathbb{Q}}}$, then
$\operatorname{\widehat{ch}}(\alpha)\cdot\operatorname{\widehat{ch}}(\beta)=\operatorname{\widehat{ch}}(\alpha\cdot\beta)\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,T)).$
Another feature of the Chern character is its compatibility with pull-back
functoriality. Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of
regular arithmetic varieties. Let $C,C^{\prime}$ be $\mathcal{D}_{{\text{\rm
a}}}(X)$ and $\mathcal{D}_{{\text{\rm a}}}(Y)$ complexes, respectively,
together with a morphism of complexes $f^{\ast}:C^{\prime}\rightarrow C$
satifying the commutativity (3.5). Then there is a commutative diagram
$\textstyle{\widehat{K}_{0}(\mathcal{Y},C^{\prime})_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\scriptstyle{\operatorname{\widehat{ch}}}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C)_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widehat{ch}}}$$\textstyle{\operatorname{\widehat{CH}}_{{\mathbb{Q}}}^{\ast}(\mathcal{Y},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\operatorname{\widehat{CH}}_{{\mathbb{Q}}}^{\ast}(\mathcal{X},C).}$
Again, the proof is a simple consequence for the known compatibility in the
case of the $\mathcal{D}_{{\text{\rm a}}}$ arithmetic Chow groups. In
particular we have the following proposition.
###### Proposition 3.8.
Let $f\colon\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of regular
arithmetic varieties, and $S\subset T_{0}^{\ast}Y_{{\mathbb{C}}}$ a closed
conical subset, invariant under complex conjugation and disjoint with the
normal directions $N_{f}$ of $f_{{\mathbb{C}}}$. Then there is a commutative
diagram
$\textstyle{\widehat{K}_{0}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,S))_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\scriptstyle{\operatorname{\widehat{ch}}}$$\textstyle{\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,f^{\ast}(S))_{{\mathbb{Q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widehat{ch}}}$$\textstyle{\operatorname{\widehat{CH}}_{{\mathbb{Q}}}^{\ast}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,S))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\operatorname{\widehat{CH}}_{{\mathbb{Q}}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,f^{\ast}(S)).}$
### 3.2. Arithmetic derived categories
For the problem of defining direct images on arithmetic $K$-theory it is
useful to deal with arbitrary complexes of coherent sheaves instead of locally
free sheaves. We therefore introduce an arithmetic counterpart of our theory
of hermitian structures on derived categories of coherent sheaves, developed
in [10], and the $\operatorname{\mathbf{\widehat{D}}^{b}}$ categories in [9].
We then compare this construction to the arithmetic $K$ groups.
###### Definition 3.9.
Let $S$ be a scheme. We denote by $\operatorname{\mathbf{D}^{b}}(S)$ the
derived category of cohomological complexes of quasi-coherent sheaves with
bounded coherent cohomology.
If $\mathcal{X}$ is a regular arithmetic variety, every object of
$\operatorname{\mathbf{D}^{b}}(\mathcal{X})$ is quasi-isomorphic to a bounded
cohomological complex of locally free sheaves. One checks there is a well
defined map
$\operatorname{Ob}\operatorname{\mathbf{D}^{b}}(\mathcal{X})\longrightarrow
K_{0}(\mathcal{X})$
that sends an object $\mathcal{F}^{\ast}$ to the class
$\sum_{i}(-1)^{i}[\mathcal{E}^{i}]$, where $\mathcal{E}^{\ast}$ is quasi-
isomorphic to $\mathcal{F}^{\ast}$, and that is compatible with derived tensor
products on $\operatorname{\mathbf{D}^{b}}(\mathcal{X})$ and the ring
structure on $K_{0}(\mathcal{X})$. Our aim is thus to extend this picture and
incorporate hermitian structures.
Let us consider the complex quasi-projective manifold $X_{{\mathbb{C}}}$
associated to an arithmetic variety $\mathcal{X}$. It comes equipped with the
conjugate-linear involution $F_{\infty}$. Recall the data
$X=(X_{{\mathbb{C}}},F_{\infty})$ uniquely determines a smooth quasi-
projective scheme $X_{{\mathbb{R}}}$ over ${\mathbb{R}}$, whose base change to
${\mathbb{C}}$ is isomorphic to $X_{{\mathbb{C}}}$, and such that its natural
automorphism given by complex conjugation gets identified to $F_{\infty}$. The
abelian category of quasi-coherent (resp. coherent) sheaves over
$X_{{\mathbb{R}}}$ is equivalent to the category of quasi-coherent (resp.
coherent) sheaves over $X_{{\mathbb{C}}}$, equivariant with respect to the
action of $F_{\infty}$. Namely, giving a quasi-coherent (resp. coherent) sheaf
on $X_{{\mathbb{R}}}$ is equivalent to giving a quasi-coherent (resp.
coherent) sheaf $\mathcal{F}$ on $X_{{\mathbb{C}}}$, together with a morphism
of sheaves
$\mathcal{F}\longrightarrow F_{\infty\ast}\mathcal{F}$
compatible with the conjugate-linear morphism
$F_{\infty}^{\sharp}:{\mathcal{O}}_{X_{{\mathbb{C}}}}\longrightarrow
F_{\infty\ast}{\mathcal{O}}_{X_{{\mathbb{C}}}}$
induced by the morphism of ${\mathbb{R}}$-schemes $F_{\infty}\colon
X_{{\mathbb{C}}}\rightarrow X_{{\mathbb{C}}}$. A similar condition
characterizes morphisms of quasi-coherent (resp. coherent) sheaves. Therefore,
we denote the bounded derived category of coherent sheaves on
$X_{{\mathbb{R}}}$ just by $\operatorname{\mathbf{D}^{b}}(X)$.
The theory of hermitian structures on the bounded derived category of coherent
sheaves on a complex algebraic manifold developed in [10] can be adapted to
the real situation of $\operatorname{\mathbf{D}^{b}}(X)$, by considering
hermitian structures invariant under the action of complex conjugation. All
the results in _loc. cit._ carry over to the real case. We denote by
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$ the category whose objects are
objects of $\operatorname{\mathbf{D}^{b}}(X)$ endowed with a hermitian
structure (_loc. cit._ , Def. 3.10) invariant under complex conjugation, and
whose morphisms are just morphisms in $\operatorname{\mathbf{D}^{b}}(X)$.
Thus, every object $\overline{\mathcal{F}}^{\ast}$ in
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$ is represented by a quasi-
isomorphism $\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast}$,
where $\overline{\mathcal{E}}^{\ast}$ is a bounded complex of hermitian
locally free sheaves on $X_{{\mathbb{C}}}$, equivariant under $F_{\infty}$.
There is an obvious forgetful functor
$\mathfrak{F}:\operatorname{\overline{\mathbf{D}}^{b}}(X)\rightarrow\operatorname{\mathbf{D}^{b}}(X)$,
that makes of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ a principal
fibered category over $\operatorname{\mathbf{D}^{b}}(X)$, with structural
group $\overline{\operatorname{\mathbf{KA}}}(X)$, the group of hermitian
structures over the 0 object [10, Def. 2.34, Thm. 3.13].
Base change to ${\mathbb{R}}$ induces a covariant functor
$\operatorname{\mathbf{D}^{b}}(\mathcal{X})\rightarrow\operatorname{\mathbf{D}^{b}}(X)$.
###### Definition 3.10.
We define the category $\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$
as the fiber product category
$\textstyle{\operatorname{\mathbf{D}^{b}}(\mathcal{X})\times_{\operatorname{\mathbf{D}^{b}}(X)}\operatorname{\overline{\mathbf{D}}^{b}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\overline{\mathbf{D}}^{b}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{F}}$$\textstyle{\operatorname{\mathbf{D}^{b}}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathbf{D}^{b}}(X).}$
We still denote by $\mathfrak{F}$ the forgetful functor
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})\rightarrow\operatorname{\mathbf{D}^{b}}(\mathcal{X})$.
By construction, $\mathfrak{F}$ makes of
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$ a principal fibered
category over $\operatorname{\mathbf{D}^{b}}(\mathcal{X})$, with structure
group $\overline{\operatorname{\mathbf{KA}}}(X)$. In particular,
$\overline{\operatorname{\mathbf{KA}}}(X)$ acts on
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$.
The Bott-Chern secondary character $\operatorname{\widetilde{ch}}$ can be
defined at the level of $\overline{\operatorname{\mathbf{KA}}}$ groups [10,
Sec. 4, Def. 4.6]. In our situation, we actually have a morphism of groups
(3.11)
$\operatorname{\widetilde{ch}}:\overline{\operatorname{\mathbf{KA}}}(X)\longrightarrow\bigoplus_{p}\widetilde{\mathcal{D}}_{{\text{\rm
a}}}^{2p-1}(X,p).$
More generally, if $C$ is a $\mathcal{D}_{{\text{\rm a}}}(X)$-complex, we may
consider a secondary Chern character with values in
$\bigoplus_{p}\widetilde{C}^{2p-1}(p)$, that we denote
$\operatorname{\widetilde{ch}}_{C}$. In particular,
$\overline{\operatorname{\mathbf{KA}}}(X)$ acts on
$\bigoplus_{p}\widetilde{C}^{2p-1}(p)$ through
$\operatorname{\widetilde{ch}}_{C}$.
###### Definition 3.12.
The arithmetic derived category
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$ is defined as the
cartesian product
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})\times_{\overline{\operatorname{\mathbf{KA}}}(X),\operatorname{\widetilde{ch}}_{C}}\bigoplus_{p}\widetilde{C}^{2p-1}(p).$
###### Remark 3.13.
If $\mathcal{X}$ is regular, then every object of
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$ can be represented by
$(\mathcal{F}^{\ast},\overline{\mathcal{E}}_{{\mathbb{C}}}\dashrightarrow\mathcal{F}^{\ast}_{{\mathbb{C}}},\widetilde{\eta})$
where $\mathcal{E}^{\ast}\dashrightarrow\mathcal{F}^{\ast}$ is any quasi-
isomorphism from a bounded complex of locally free sheaves over $\mathcal{X}$.
Indeed, $\mathcal{X}$ is assumed to be regular, so that $\mathcal{F}^{\ast}$
is quasi-isomorphic to a bounded complex of locally free sheaves
$\mathcal{E}^{\ast}$. One then endows the $\mathcal{E}^{i}$ with smooth
hermitian metrics invariant under complex conjugation, and takes into account
that $\overline{\operatorname{\mathbf{KA}}}(X)$ acts transitively on the
hermitian structures on $\mathcal{F}^{\ast}_{{\mathbb{C}}}$. We introduce the
simplified notation
$(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F},\widetilde{\eta})$
for such representatives.
The categories $\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$ are
the arithmetic analogues to those we introduced in [9, Sec. 4], and have
similar properties. We are only going to review some of them.
For the next proposition, we recall that any group can be considered as a
category, with morphisms given by the group law. This in particular the case
of $\widehat{K}_{0}(\mathcal{X},C)$.
###### Theorem 3.14.
If $\mathcal{X}$ is regular, there is a natural functor
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\longrightarrow\widehat{K}_{0}(\mathcal{X},C),$
that, at the level of objects is given by
$(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F},\widetilde{\eta})\longmapsto\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{i},\widetilde{\eta}\right],$
and at the level of morphism sends any
$f\in\operatorname{Hom}_{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)}(A,B)$
to $[B]-[A]$.
###### Proof.
It is enough to see that the defining assignment does not depend on the
representatives. For this, take two objects
$(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta})$
and
$(\overline{\mathcal{E}}^{\prime\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta}^{\prime})$
giving raise to the same class. We have an equivalence of objects
$\begin{split}(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta})\sim&(\overline{\mathcal{E}}^{\prime\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta}^{\prime})\\\
&\sim(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta}^{\prime}+c(\widetilde{\operatorname{ch}}(\operatorname{id}\colon\overline{\mathcal{F}}^{\ast}\rightarrow\overline{\mathcal{F}}^{\prime\ast}))).\end{split}$
Consequently
$\widetilde{\eta}=\widetilde{\eta}^{\prime}+c(\widetilde{\operatorname{ch}}(\operatorname{id}\colon\overline{\mathcal{F}}^{\ast}\rightarrow\overline{\mathcal{F}}^{\prime\ast})).$
Hence, the image of
$(\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast},\widetilde{\eta})$
is equivalently written
$\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{i},\widetilde{\eta}^{\prime}+c(\widetilde{\operatorname{ch}}(\operatorname{id}\colon\overline{\mathcal{F}}^{\ast}\rightarrow\overline{\mathcal{F}}^{\prime\ast}))\right].$
We thus have to show the equality in $\widehat{K}_{0}(\mathcal{X},C)$
$[0,c(\widetilde{\operatorname{ch}}(\operatorname{id}\colon\overline{\mathcal{F}}^{\ast}\rightarrow\overline{\mathcal{F}}^{\prime\ast}))]\overset{?}{=}\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{\prime
i},0\right]-\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{i},0\right].$
By [10, Lemma 3.5], the quasi-isomorphism
$\overline{\mathcal{E}}^{\ast}\dashrightarrow\overline{\mathcal{E}}^{\prime\ast}$
inducing the identity on $\mathcal{F}^{\ast}$ can be lifted to a diagram
$\textstyle{\overline{\mathcal{E}}^{{}^{\prime\prime}\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{\overline{\mathcal{E}}^{\ast}}$$\textstyle{\overline{\mathcal{E}}^{\prime\ast},}$
where $a$ and $b$ are quasi-isomorphisms and
$\operatorname{\overline{cone}}(a)$ is meager [10, Def. 2.9]. On the one hand,
by the characterization [10, Thm. 2.13] of meager complexes, one can show
$\left[\sum_{i}(-1)^{i}\operatorname{\overline{cone}}(a)^{i},0\right]=0.$
On the other hand, we have an exact sequence of complexes
$0\rightarrow\overline{\mathcal{E}}^{{}^{\prime\prime}\ast}\rightarrow\operatorname{\overline{cone}}(a)\rightarrow\overline{\mathcal{E}}^{\ast}[1]\rightarrow
0,$
whose constituent rows are orthogonally split. This shows
(3.15)
$\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{{}^{\prime\prime}i},0\right]-\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{i},0\right]=\left[\sum_{i}(-1)^{i}\operatorname{\overline{cone}}(a)^{i},0\right]=0.$
Similarly we have
(3.16)
$\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{{}^{\prime}i},0\right]-\left[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{{}^{\prime\prime}i},0\right]=\left[\sum_{i}(-1)^{i}\operatorname{\overline{cone}}(b)^{i},0\right].$
But the complex underlying $\operatorname{\overline{cone}}(b)$ is acyclic, so
that
(3.17)
$\left[\sum_{i}(-1)^{i}\operatorname{\overline{cone}}(b)^{i},0\right]=[0,c(\widetilde{\operatorname{ch}}(\operatorname{\overline{cone}}(b))].$
Finally, by [10, Def. 3.14, Thm. 4.11] we have
(3.18)
$\widetilde{\operatorname{ch}}(\operatorname{id}:\overline{\mathcal{F}}^{\ast}\rightarrow\overline{\mathcal{F}}^{{}^{\prime}\ast})=\widetilde{\operatorname{ch}}(\operatorname{\overline{cone}}(b)).$
Putting (3.15)–(3.18) together allows to conclude. ∎
###### Notation 3.19.
Let $\mathcal{X}$ be a regular arithmetic variety. Then we still denote the
image of an object
$[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$
by the morphism of the proposition by
$[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]$. We call this image the
class of $[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]$ in arithmetic
$K$-theory.
###### Remark 3.20.
Two tightly isomorphic objects in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$ have the same class
in arithmetic $K$-theory.
Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of regular arithmetic
varieties and let $C$, $C^{\prime}$ be $\mathcal{D}_{{\text{\rm a}}}(X)$ and
$\mathcal{D}_{{\text{\rm a}}}(Y)$ complexes respectively, with a commutative
diagram as in (3.5). Then there is a commutative diagram of functors
$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{Y},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C).}$
The following statement is a particular case.
###### Proposition 3.21.
Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of regular arithmetic
varieties, and $S\subset T_{0}^{\ast}Y_{{\mathbb{C}}}$ a closed conical subset
invariant under complex conjugation, disjoint with the normal directions
$N_{f}$ of $f_{{\mathbb{C}}}$. Then there is a commutative diagram
$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,S))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,f^{\ast}(S)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(Y,S))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,f^{\ast}(S))).}$
If $C,C^{\prime},C^{\prime\prime}$ are $\mathcal{D}_{{\text{\rm
a}}}(X)$-complexes with a product $C\otimes C^{\prime}\rightarrow
C^{\prime\prime}$ compatible with the product of $\mathcal{D}_{{\text{\rm
a}}}(X)$ and $\mathcal{X}$ is regular, then there is a commutative diagram of
functors
$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\times\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C)\times\widehat{K}_{0}(\mathcal{X},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C^{\prime\prime}),}$
where the derived tensor product $\otimes$ is defined by
$[\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu]=[\overline{\mathcal{F}}\otimes\overline{\mathcal{G}},c(\operatorname{ch}(\overline{\mathcal{F}}))\bullet\nu+\eta\bullet
c^{\prime}(\operatorname{ch}(\overline{\mathcal{G}}))+\operatorname{d}_{C}\eta\bullet\nu]$
###### Proposition 3.22.
Let $S,S^{\prime}$ be closed conical subsets of $T_{0}^{\ast}X_{{\mathbb{C}}}$
invariant under the action of complex conjugation, with
$S\cap(-S^{\prime})=\emptyset$. Define $T=S\cup S^{\prime}\cup(S+S^{\prime})$.
If $\mathcal{X}$ is regular, then there is a commutative diagram of functors
$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S))\times\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,T))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S))\times\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,S^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,T)).}$
Furthermore, it is compatible with pull-back $f^{\ast}$ whenever defined.
Finally, the class functor and the arithmetic Chern character on arithmetic
$K$ groups, allow to extend it to arithmetic derived categories.
###### Notation 3.23.
Let $\mathcal{X}$ be a regular arithmetic variety and $C$ a
$\mathcal{D}_{{\text{\rm a}}}(X)$ complex. We denote by
$\operatorname{\widehat{ch}}:\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\longrightarrow\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},C)$
the arithmetic Chern character on
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$, obtained as the
composition of the class functor and $\operatorname{\widehat{ch}}$ on
$\widehat{K}_{0}(\mathcal{X},C)$.
## 4\. Arithmetic characteristic classes
Let $\mathcal{X}$ be a regular arithmetic variety. From the previous sections,
there exists a natural functor
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})\longrightarrow\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$
that factors through $\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$,
and there is a ring isomorphism
$\operatorname{\widehat{ch}}:\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))_{{\mathbb{Q}}}\longrightarrow\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X)).$
We therefore obtain a functor
$\operatorname{\widehat{ch}}:\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})\longrightarrow\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$
automatically satisfying several compatibilities with the operations in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$ and distinguished
triangles. More generally, in this section we construct arithmetic
characteristic classes attached to real additive or multiplicative genera. The
case of the arithmetic Todd class will be specially relevant, since it is
involved in the arithmetic Riemann-Roch theorem. Our construction relies on
the one given by Gillet-Soulé [16].
Let $B$ be a subring of ${\mathbb{R}}$ and $\varphi\in B[[x]]$ a real power
series, defining an additive genus. For each hermitian vector bundle
$\overline{\mathcal{E}}$, in [16], there is attached a class
$\widehat{\varphi}(\overline{\mathcal{E}})\in\operatorname{\widehat{CH}}_{B}^{\ast}(\mathcal{X})$.
By the isomorphism [11, Theorem 3.33] we obtain a class
$\widehat{\varphi}(\overline{\mathcal{E}})\in\operatorname{\widehat{CH}}_{B}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$.
For every finite complex of smooth hermitian vector bundles
$\overline{\mathcal{E}}^{\ast}$, we put
$\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})=\sum_{i}(-1)^{i}\widehat{\varphi}(\overline{\mathcal{E}}^{i})\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X)).$
Let us now consider an object $\overline{\mathcal{F}}^{\ast}$ in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$. We choose an
auxiliary quasi-isomorphism
$\psi:\mathcal{E}^{\ast}\dashrightarrow\mathcal{F}^{\ast}$, where
$\mathcal{E}^{\ast}$ is a bounded complex of locally free sheaves on
$\mathcal{X}$. This is possible since $\mathcal{X}$ is regular by assumption.
We also fix auxiliary smooth hermitian metrics on the individual terms
$\mathcal{E}^{i}$. We thus obtain an isomorphism
$\overline{\psi}\colon\overline{\mathcal{E}}^{\ast}\dashrightarrow\overline{\mathcal{F}}^{\ast}$
that in general is not tight. The lack of tightness is measured by a class
$[\overline{\psi}_{{\mathbb{C}}}]\in\overline{\operatorname{\mathbf{KA}}}(X)$,
that we simply denote $[\overline{\psi}]$ [10, Sec. 3]. Recall that Bott-Chern
secondary classes can be defined at the level of
$\overline{\operatorname{\mathbf{KA}}}(X)$ (see _loc. cit._ Sec. 4, and
especially the characterization given in Prop. 4.6). In particular we have a
class
$\widetilde{\varphi}(\overline{\psi}):=\widetilde{\varphi}([\overline{\psi}])\in\bigoplus_{p}\widetilde{\mathcal{D}}_{{\text{\rm
a}}}^{2p-1}(X,p).$
###### Lemma 4.1.
The class
(4.2)
$\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})+\operatorname{a}(\widetilde{\varphi}(\overline{\psi}))\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$
depends only on $\overline{\mathcal{F}}^{\ast}$.
###### Proof.
Let
$\psi^{\prime}\colon\mathcal{E}^{\prime\ast}\dashrightarrow\mathcal{F}^{\ast}$
be another finite locally free resolution, and choose arbitrary metrics on the
$\mathcal{E}^{\prime i}$. We can construct a commutative diagram of complexes
in $\operatorname{\mathbf{D}^{b}}(\mathcal{X})$
(4.3)
$\textstyle{\mathcal{E}^{\prime\prime\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{\mathcal{E}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathcal{E}^{\prime\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$$\textstyle{\mathcal{F}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{\mathcal{F}^{\ast},}$
where $\mathcal{E}^{\prime\prime}$ is also a finite complex of locally free
sheaves, that we endow with smooth hermitian metrics, and $\alpha$, $\beta$
are quasi-isomorphisms. Because the exact sequences
$\displaystyle
0\rightarrow\overline{\mathcal{E}}^{\prime\ast}\rightarrow\operatorname{\overline{cone}}(\alpha)\rightarrow\overline{\mathcal{E}}^{\prime\prime\ast}[1]\rightarrow
0,$ $\displaystyle
0\rightarrow\overline{\mathcal{E}}^{\ast}\rightarrow\operatorname{\overline{cone}}(\beta)\rightarrow\overline{\mathcal{E}}^{\prime\prime\ast}[1]\rightarrow
0$
have orhtogonally split constituent rows, we find
(4.4)
$\displaystyle\widehat{\varphi}(\operatorname{\overline{cone}}(\alpha))=\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\prime\ast}),$
(4.5)
$\displaystyle\widehat{\varphi}(\operatorname{\overline{cone}}(\beta))=\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\prime\ast}).$
Also, the complexes $\operatorname{cone}(\alpha)$ and
$\operatorname{cone}(\beta)$ are acyclic, so that
(4.6)
$\displaystyle\widehat{\varphi}(\operatorname{\overline{cone}}(\alpha))=\operatorname{a}(\widetilde{\varphi}(\operatorname{\overline{cone}}(\alpha)))=\operatorname{a}(\widetilde{\varphi}(\overline{\alpha})),$
(4.7)
$\displaystyle\widehat{\varphi}(\operatorname{\overline{cone}}(\beta))=\operatorname{a}(\widetilde{\varphi}(\operatorname{\overline{cone}}(\beta)))=\operatorname{a}(\widetilde{\varphi}(\overline{\beta})),$
where we took into account the very definition of the class of an isomorphism
in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. From the relations
(4.5)–(4.7) we derive
(4.8)
$\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\ast})=\operatorname{a}(\widetilde{\varphi}(\overline{\alpha})-\widetilde{\varphi}(\overline{\beta}))=\operatorname{a}(\widetilde{\varphi}(\overline{\alpha}\circ\overline{\beta}^{-1})),$
where we plugged
$\widetilde{\varphi}(\overline{\alpha}\circ\overline{\beta}^{-1})=\widetilde{\varphi}(\overline{\alpha})-\widetilde{\varphi}(\overline{\beta})$
[10, Prop. 4.13]. But by diagram (4.3) we have
$\overline{\alpha}\circ\overline{\beta}^{-1}=\overline{\psi}^{-1}\circ\overline{\psi}^{\prime}$.
This fact combined with (4.8) implies
$\begin{split}\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\ast})=&\operatorname{a}(\widetilde{\varphi}(\overline{\alpha})-\widetilde{\varphi}(\overline{\beta}))\\\
&=\operatorname{a}(\widetilde{\varphi}(\overline{\psi}^{-1}\circ\overline{\psi}^{\prime}))\\\
&\hskip
11.38092pt=\operatorname{a}(\widetilde{\varphi}(\overline{\psi}^{\prime}))-\operatorname{a}(\widetilde{\varphi}(\overline{\psi})).\end{split}$
This completes the proof of the lemma. ∎
###### Definition 4.9.
The notations being as above, we define
$\widehat{\varphi}(\overline{\mathcal{F}}^{\ast}):=\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})+\operatorname{a}(\widetilde{\varphi}(\overline{\psi}))\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X)),$
The additive arithmetic characteristic classes are indeed additive with
respect to direct sum, and are compatible with pull-back by morphisms of
arithmetic varieties. However, the most important property of additive
arithmetic characteristic classes is the behavior with respect to
distinguished triangles. The reader may review [10, Def. 3.29, Thm. 3.33, Def.
4.17, Thm. 4.18] for definitions and main properties, especially for the class
in $\overline{\operatorname{\mathbf{KA}}}(X)$ and secondary class of a
distinguished triangle.
###### Theorem 4.10.
Let us consider a distinguished triangle in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$:
$\overline{\tau}\colon\quad\overline{\mathcal{F}}^{\ast}\dashrightarrow\overline{\mathcal{G}}^{\ast}\dashrightarrow\overline{\mathcal{H}}^{\ast}\dashrightarrow\overline{\mathcal{F}}^{\ast}_{0}[1].$
Then we have
$\widehat{\varphi}(\mathcal{F}^{\ast})-\widehat{\varphi}(\mathcal{G}^{\ast})+\widehat{\varphi}(\mathcal{H}^{\ast})=\operatorname{a}(\widetilde{\varphi}(\overline{\tau}))$
in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$. In particular, if $\overline{\tau}$ is tightly distinguished, we
have
$\widehat{\varphi}(\overline{\mathcal{G}}^{\ast})=\widehat{\varphi}(\overline{\mathcal{F}}^{\ast})+\widehat{\varphi}(\overline{\mathcal{H}}^{\ast}).$
###### Proof.
It is possible to find a diagram
$\textstyle{\overline{\eta}\colon}$$\textstyle{\mathcal{E}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{E}^{\prime\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\operatorname{cone}(\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\mathcal{E}^{\ast}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f[1]}$$\textstyle{\overline{\tau}\colon}$$\textstyle{\mathcal{F}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\ast}[1],}$
where $\mathcal{E}^{\ast}$, $\mathcal{E}^{\prime\ast}$ are bounded complexes
of locally free sheaves and the vertical arrows are isomorphisms in
$\operatorname{\mathbf{D}^{b}}(\mathcal{X})$. We choose arbitrary smooth
hermitian metrics on the $\mathcal{E}^{i}$, $\mathcal{E}^{\prime j}$, and put
the orthogonal sum metric on $\operatorname{cone}(\alpha)$. Then, by
construction of the arithmetic characteristic classes, we have
$\begin{split}\widehat{\varphi}(\overline{\mathcal{F}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{G}}^{\ast})+\widehat{\varphi}(\overline{\mathcal{H}}^{\ast})=&\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\ast})+\widehat{\varphi}(\operatorname{\overline{cone}}(\alpha))\\\
&+a(\widetilde{\varphi}(\overline{f})-\widetilde{\varphi}(\overline{g})+\widetilde{\varphi}(\overline{h})).\end{split}$
Because the exact sequence
$0\rightarrow\overline{\mathcal{E}}^{\prime\ast}\rightarrow\operatorname{\overline{cone}}(\alpha)\rightarrow\overline{\mathcal{E}}^{\ast}[1]\rightarrow
0$
has orthogonally split constituent rows, we observe
$\widehat{\varphi}(\overline{\mathcal{E}}^{\ast})-\widehat{\varphi}(\overline{\mathcal{E}}^{\prime\ast})+\widehat{\varphi}(\operatorname{\overline{cone}}(\alpha))=0.$
Moreover, by [10, Thm 3.33 (vii)] the equality
$\widetilde{\varphi}(\overline{f})-\widetilde{\varphi}(\overline{g})+\widetilde{\varphi}(\overline{h})=\widetilde{\varphi}(\overline{\tau})-\widetilde{\varphi}(\overline{\eta}),$
holds, and $\widetilde{\varphi}(\overline{\eta})=0$ since $\overline{\eta}$ is
tightly distinguished. The theorem now follows. ∎
We may also say a few words on multiplicative arithmetic characteristic
classes. We follow the discussion in [10, Sec. 5]. Assume from now on that
${\mathbb{Q}}\subset B$. Let $\psi\in B[[x]]$ be a formal power series with
$\psi^{0}=1$. We denote also by $\psi$ the associated multiplicative genus.
Then
$\varphi=\log(\psi)\in B[[x]]$
defines an additive genus, to which we can associate an addtive arithmetic
characteristic genus $\widehat{\varphi}$. Then, given an object
$\overline{\mathcal{F}}^{\ast}$ in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$, we have a well
defined class
$\widehat{\psi}_{m}(\overline{\mathcal{F}}^{\ast}):=\exp(\widehat{\varphi}(\overline{\mathcal{F}}^{\ast}))\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X)).$
Observe this construction uses the ring structure of
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$, hence the regularity of $\mathcal{X}$ and the product structure of
$\mathcal{D}_{{\text{\rm a}}}(X)$. In case $\psi$ has rational coefficients,
$\widehat{\psi}(\overline{\mathcal{F}}^{\ast})$ takes values in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$. We also recall that there is a multiplicative secondary class
associated to $\psi$, denoted $\widetilde{\psi}_{m}$, that can be expressed in
terms of $\widetilde{\varphi}$:
$\widetilde{\psi}_{m}(\theta)=\frac{\exp(\varphi(\theta))-1}{\varphi(\theta)}\widetilde{\varphi}(\theta),$
where $\theta$ is any class in $\overline{\operatorname{\mathbf{KA}}}(X)$. If
$\overline{\tau}$ is a distinguished triangle in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$, we will simply write
$\widetilde{\psi}_{m}(\overline{\tau})$ instead of
$\widetilde{\psi}_{m}([\overline{\tau}])$.
###### Theorem 4.11.
Let $\psi$ be a multiplicative genus, with degree 0 component $\psi^{0}=1$.
Then, for every distinguished triangle in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$
$\overline{\tau}\colon\overline{\mathcal{F}}^{\ast}\dashrightarrow\overline{\mathcal{G}}^{\ast}\dashrightarrow\overline{\mathcal{H}}^{\ast}\dashrightarrow\overline{\mathcal{F}}[1]$
we have the relation
$\widehat{\psi}(\overline{\mathcal{F}}^{\ast})^{-1}\widehat{\psi}(\overline{\mathcal{G}})\widehat{\psi}(\overline{\mathcal{H}})^{-1}-1=\operatorname{a}(\widetilde{\psi}_{m}(\overline{\tau})).$
In particular, if $\overline{\tau}$ is tightly distinguished, the equality
$\widehat{\psi}(\overline{\mathcal{G}}^{\ast})=\widehat{\psi}(\overline{\mathcal{F}}^{\ast})\widehat{\psi}(\overline{\mathcal{H}}^{\ast})$
holds.
###### Proof.
It is enough to exponentiate the relation provided by Theorem 4.10 and observe
that, due to the mutliplicative law in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$, one has
$\exp(\operatorname{a}(\widetilde{\varphi}(\overline{\tau})))=1+\operatorname{a}\left(\frac{\exp(\varphi(\overline{\tau}))-1}{\exp(\varphi(\overline{\tau}))}\widetilde{\varphi}(\overline{\tau})\right).$
∎
###### Example 4.12.
1. (i)
The arithmetic Chern class, is the additive class attached to the additive
genus $\operatorname{ch}(x)=e^{x}$. It coincides with the character
$\operatorname{\widehat{ch}}$ of the previous section.
2. (ii)
The arithmetic Todd class, is the multiplicative class attached to the genus
$\operatorname{Td}(x)=\frac{x}{1-e^{-x}}.$
Observe that the formal series of $\operatorname{Td}(x)$ has constant
coefficient 1.
###### Remark 4.13.
If $C$ is a $\mathcal{D}_{{\text{\rm a}}}(X)$-complex, we can define
arithmetic characteristic classes with values in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},C)$, just taking
their image by the natural morphism
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))\rightarrow\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{B}(\mathcal{X},C)$.
We will use the same notations to refer to these classes.
## 5\. Direct images and generalized analytic torsion
In the preceding section we introduced the arithmetic $K$-groups and the
arithmetic derived categories. They satisfy elementary functoriality
properties and are related by a class functor. A missing functoriality is the
push-forward by _arbitrary_ projectives morphisms of arithmetic varieties.
Similarly to the work of Gillet-Rössler-Soulé [14], we will define direct
images after choosing a generalized analytic torsion theory in the sense of
[9]. Our theory is more general in that we don’t require our morphisms to be
smooth over the generic fiber. At the archimedean places, we are thus forced
to work with complexes of currents with controlled wave front sets. In this
level of generality, the theory of arithmetic Chow groups, arithmetic
$K$-theory and arithmetic derived categories has already been discussed. Let
us recall that a generalized analytic torsion theory is not unique, but
according to [9, Thm. 7.7 and Thm. 7.14] it is classified by a real additive
genus.
Our theory of generalized analytic torsion classes involves the notion of
relative metrized complex [9, Def. 2.5]. In the sequel we will need a variant
on real smooth quasi-projective schemes $X=(X_{{\mathbb{C}}},F_{\infty})$.
With respect to _loc. cit._ , this amounts to imposing an additional
invariance under the action of $F_{\infty}$.
###### Definition 5.1.
A real relative metrized complex is a triple
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}}^{\ast},\overline{f_{\ast}\mathcal{F}^{\ast}})$,
where
* •
$\overline{f}:X\rightarrow Y$ is a projective morphism of real smooth quasi-
projective varieties, together with a hermitian structure on the tangent
complex $T_{f}$, invariant under the action of complex conjugation;
* •
$\overline{\mathcal{F}}^{\ast}$ is an object in
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$;
* •
$\overline{f_{\ast}\mathcal{F}}^{\ast}$ is an object in
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ lying over
$f_{\ast}\mathcal{F}^{\ast}$.
The following lemma is checked by a careful proof reading of the construction
of generalized analytic torsion classes in [9].
###### Lemma 5.2.
Let $T$ be a theory of generalized analytic torsion classes. Then, for every
real relative metrized complex $\overline{\xi}=(\overline{f}\colon X\to
Y,\overline{\mathcal{F}}^{\ast},\overline{f_{\ast}\mathcal{F}}^{\ast})$,
$T(\overline{\xi})$ is a class of real currents,
$T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{\text{{\rm
cur}},{\text{\rm a}}}(X,N_{f},p),$
where $N_{f}$ is the cone of normal directions to $f$.
It will be useful to have an adaptation of the category
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ [10, Sec. 5] to
real quasi-projective schemes, that we denote
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{R}}}$. In this category,
the objects are real smooth quasi-projective schemes, and the morphisms are
projective morphisms with a hermitian structure invariant under complex
conjugation. The composition law is then described in _loc. cit._ , Def. 5.7,
with the help of the hermitian cone construction. We will follow the notation
introduced in [9, Def. 2.12].
###### Notation 5.3.
Let $\overline{f}\colon X\to Y$ be in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{R}}}$, of pure relative
dimension $e$, $C$ a $\mathcal{D}_{{\text{\rm a}}}(X)$-complex, $C^{\prime}$ a
$\mathcal{D}_{{\text{\rm a}}}(Y)$-complex and $f_{\ast}:C\rightarrow
C^{\prime}$ a morphism of fitting into the commutative diagram (2.25). Assume
furthermore that $C$ is a $\mathcal{D}_{{\text{\rm a}}}(X)$-module with a
product law $\bullet$ as in (3.6). Then we put
$\begin{split}\overline{f}_{\flat}:C^{\ast}(\ast)&\longrightarrow
C^{\prime\ast-2e}(p-e)\\\ \eta&\longmapsto
f_{\ast}(\eta\bullet\operatorname{Td}(T_{\overline{f}})).\end{split}$
This morphism induces a corresponding morphism on
$\widetilde{C}^{\ast}(\ast)$, for which we use the same notation.
For an arithmetic ring $A$, we introduce
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$ the category of quasi-
projective regular arithmetic varieties over $A$, with projective morphisms
endowed (at the archimedean places) with a hermitian structure invariant under
complex conjugation. By construction, there is a natural base change functor
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}\longrightarrow\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{R}}}.$
Given $\overline{f}\colon\mathcal{X}\to\mathcal{Y}$ a morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$ and objects
$\overline{\mathcal{F}}^{\ast}$, $\overline{f_{\ast}\mathcal{F}}^{\ast}$ in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$ and
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{Y})$, respectively, we may
consider the corresponding real relative metrized complex, that we will
abusively write
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\ast})$,
and its analytic torsion class $T(\overline{\xi})$. We may also write
$\overline{f}_{\flat}$ instead of $\overline{f}_{{\mathbb{C}}\,\flat}$, etc.
We are now in position to construct the arithmetic counterpart of [9, Eq.
(10.6)], namely the direct image functor on arithmetic derived categories, as
well as a similar push-forward on arithmetic $K$-theory.
###### Definition 5.4.
Let $\overline{f}\colon\mathcal{X}\to\mathcal{Y}$ be a morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$, $C$ a
$\mathcal{D}_{{\text{\rm a}}}(X)$-complex, $C^{\prime}$ a
$\mathcal{D}_{{\text{\rm a}}}(Y)$-complex, both satisfying the hypothesis (H1)
and (H2), and $f_{\ast}:C\rightarrow C^{\prime}$ a morphism fitting into a
commutative diagram like (2.25).
1. (i)
We define the functor
$\overline{f}_{\ast}\colon\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\longrightarrow\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},C^{\prime}),$
acting on objects by the assignment
$[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]\longmapsto[\overline{f_{\ast}\mathcal{F}}^{\ast},\overline{f}_{\flat}(\widetilde{\eta})-c^{\prime}(T(\overline{f},\overline{\mathcal{F}}^{\ast},\overline{f_{\ast}\mathcal{F}}^{\ast}))].$
Here $\overline{f_{\ast}\mathcal{F}}^{\ast}$ carries an arbitrary choice of
hermitian structure. The action on morphisms of $\overline{f}_{\ast}$ is just
the usual action $f_{\ast}$ on morphisms of
$\operatorname{\mathbf{D}^{b}}(\mathcal{X})$.
2. (ii)
We define a morphism of groups
$\begin{split}f_{\ast}:\widehat{K}_{0}(\mathcal{X},C)&\longrightarrow\widehat{K}_{0}(\mathcal{X},C^{\prime})\\\
[\overline{\mathcal{E}},\widetilde{\eta}]&\longmapsto[\sum_{i}(-1)^{i}\overline{\mathcal{E}}^{\prime
i},\overline{f}_{\flat}(\widetilde{\eta})-T(\overline{f},\overline{\mathcal{E}},\overline{f_{\ast}\mathcal{E}}^{\ast})],\end{split}$
where we choose an arbitrary quasi-isomorphism
$\mathcal{E}^{\prime\ast}\dashrightarrow f_{\ast}\mathcal{E}$ and arbitrary
smooth hermitian metrics on the $\mathcal{E}^{\prime i}$. Here
$f_{\ast}\mathcal{E}$ denotes the derived direct image of the single locally
free sheaf $\mathcal{E}$.
Notice the previous definition makes sense by the anomaly formulas satisfied
by analytic torsion theories [9, Prop. 7.4]. Both push-forwards are compatible
through the class map from arithmetic derived categories to arithmetic
$K$-theory.
###### Theorem 5.5.
There is a commutative diagram of functors
$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}_{0}(\mathcal{X},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\textstyle{\widehat{K}_{0}(\mathcal{Y},C^{\prime}).}$
This is in particular true for $C=\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(X,S)$ and $C^{\prime}=\mathcal{D}_{\text{{\rm cur}},{\text{\rm
a}}}(Y,f_{\ast}(S))$, where $S\subset T_{0}^{\ast}X$ is a closed conical
subset invariant under the action of complex conjugation.
###### Proof.
Let $[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]$ be an object in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$. By Remark 3.13, we
can suppose the hermitian structure on $\overline{\mathcal{F}}^{\ast}$ is
given by a quasi-isomorphism
$\overline{\mathcal{E}}^{\ast}\dashrightarrow\mathcal{F}^{\ast}$, where
$\overline{\mathcal{E}}^{\ast}$ is a finite complex of locally free sheaves,
each one endowed with a smooth hermitian metric. For every $i$, we have to
endow $f_{\ast}\mathcal{E}^{i}$ with a hermitian structure given by a finite
locally free resolution $\overline{\mathcal{E}}^{i\ast}\dashrightarrow
f_{\ast}\mathcal{E}^{i}$ and a choice of arbitrary smooth hermitian metric on
every piece $\mathcal{E}^{ij}$. From the data
$\overline{\mathcal{E}}^{i\ast}\dashrightarrow f_{\ast}\mathcal{E}^{i}$, every
$i$, the procedure of [10, Def. 3.39] produces a hermitian structure on
$f_{\ast}\mathcal{E}^{\ast}$, via the hermitian cone construction. Observe the
construction of _loc. cit._ can be done in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{Y})$. Combined with the
fixed quasi-isomorphism $\mathcal{E}^{\ast}\dashrightarrow\mathcal{F}^{\ast}$,
we thus obtain a hermitian structure on $f_{\ast}\mathcal{F}^{\ast}$, that we
denote by $\overline{f_{\ast}\mathcal{F}}^{\ast}$.
The class of $[\overline{\mathcal{F}},\widetilde{\eta}]$ in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},C)$ is thus
$\sum_{i}(-1)^{i}[\overline{\mathcal{E}}^{i},0]+[0,\widetilde{\eta}].$
Its image under $\overline{f}_{\ast}$ is
(5.6)
$\sum_{i}(-1)^{i}\left((\sum_{j}(-1)^{j}[\overline{\mathcal{E}}^{ij},0])-[0,c^{\prime}(T(\overline{f},\overline{\mathcal{E}}^{i},\overline{f_{\ast}\mathcal{E}}^{i}))]\right)+[0,\overline{f}_{\flat}(\widetilde{\eta})].$
The class of $\overline{f}_{\ast}[\overline{\mathcal{F}},\widetilde{\eta}]$ in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},C^{\prime})$ is
(5.7)
$[\overline{f_{\ast}\mathcal{F}}^{\ast},\overline{f}_{\flat}(\widetilde{\eta})-c^{\prime}(T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\ast}))].$
By the choice of the hermitian structures on $\overline{\mathcal{F}}^{\ast}$
and $\overline{f_{\ast}\mathcal{F}}^{\ast}$ and by Theorem [9, Prop. 7.6], we
have
$T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\ast})=\sum_{i}(-1)^{i}T(\overline{f},\overline{\mathcal{E}}^{i},\overline{f_{\ast}\mathcal{E}}^{i}).$
Therefore the class in arithmetic $K$-theory of (5.7) equals (5.6). ∎
There are several compatibilities between direct images, inverse images and
derived tensor product. We now state them without proof, referring the reader
to [9, Thm. 10.7] for the details.
###### Proposition 5.8.
Let $\overline{f}\colon\mathcal{X}\to\mathcal{Y}$ and
$\overline{g}\colon\mathcal{Y}\to\mathcal{Z}$ be morphisms in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$. Let $S\subset
T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed conical subsets.
1. (i)
(Functoriality of push-forward) We have the relation
$(\overline{g}\circ\overline{f})_{\ast}=\overline{g}_{\ast}\circ\overline{f}_{\ast},$
as functors
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},S)\rightarrow\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Z},g_{\ast}f_{\ast}S)$.
2. (ii)
(Projection formula) Assume that $T\cap N_{f}=\emptyset$ and that
$T+f_{\ast}S$ does not cross the zero section of $T^{\ast}Y$. Let
$[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]$ be in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},S)$ and
$[\overline{\mathcal{F}}^{\prime\ast},\widetilde{\eta}^{\prime}]$ in
$\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},T)$. Then
$\overline{f}_{\ast}([\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]\otimes
f^{\ast}[\overline{\mathcal{F}}^{\prime\ast},\widetilde{\eta}^{\prime}])=\overline{f}_{\ast}[\overline{\mathcal{F}},\widetilde{\eta}]\otimes[\overline{\mathcal{F}}^{\prime},\widetilde{\eta}^{\prime}]$
in $\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},W)$, where
$W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$
3. (iii)
There are analogous relations on the level of arithmetic $K$-theory,
compatible with the class functor from arithmetic derived categories.
###### Remark 5.9.
Strictly speaking, the equalities provided by the previous statement should be
canonical isomorphisms, but as usual we abuse the notations and pretend they
are equalities.
## 6\. Arithmetic Grothendieck-Riemann-Roch
### 6.1. Statement and reductions
#### Hermitian tangent complexes.
Let $\overline{f}\colon\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$. We explain how to construct
the associated hermitian tangent complex $T_{\overline{f}}$. This is an object
in $\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$, well defined up to
tight isomorphism.
Because $\mathcal{X}$, $\mathcal{Y}$ are regular schemes and $f$ is
projective, it is automatically a l.c.i. morphism. The tangent complex of $f$
is an object in $\operatorname{\mathbf{D}^{b}}(\mathcal{X})$, well defined up
to isomorphism. Consider a factorization
$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{Y},}$
with $i$ being a closed regular immersion and $\pi$ a smooth morphism. For
instance, one may choose $\mathcal{Z}={\mathbb{P}}^{n}_{\mathcal{Y}}$, for
some $n$. Let us denote by $\mathcal{I}$ the ideal defining the closed
immersion $i$. Then $\mathcal{I}/\mathcal{I}^{2}$ is a locally free sheaf on
$\mathcal{X}$ and, as customary, we define the normal bundle
$N_{\mathcal{X}/\mathcal{Z}}=(\mathcal{I}/\mathcal{I}^{2})^{\vee}$. There is a
morphism of coherent sheaves
$\varphi\colon i^{\ast}T_{\mathcal{Z}/\mathcal{Y}}\longrightarrow
N_{\mathcal{X}/\mathcal{Z}},$
namely the dual of the differential map $d:\mathcal{I}/\mathcal{I}^{2}\to
i^{\ast}\Omega_{\mathcal{Z}/\mathcal{Y}}$. We consider
$T_{\mathcal{Z}/\mathcal{Y}}$ as a complex concentrated in degree 0, and
$N_{\mathcal{X}/\mathcal{Z}}$ as a complex concentrated in degree one. We then
put
$T_{f}:=\operatorname{cone}(\varphi)[-1].$
The isomorphism class of $T_{f}$ in
$\operatorname{\mathbf{D}^{b}}(\mathcal{X})$ is independent of the
factorization.
The base change to ${\mathbb{C}}$ of $T_{f}$ is naturally isomorphic to the
tangent complex $T_{f_{{\mathbb{C}}}}:TX_{{\mathbb{C}}}\rightarrow
f_{{\mathbb{C}}}^{\ast}Y_{{\mathbb{C}}}$, which is equipped with a hermitian
structure by assumption. Therefore, the data provided by the constructed
complex $T_{f}$ and the hermitian structure on $\overline{f}$ determine an
object $T_{\overline{f}}$ in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$, which is well defined
up to tight isomorphism. By Theorem 4.11, the arithmetic Todd class of
$T_{\overline{f}}$ is unambiguously defined.
###### Definition 6.1.
The arithmetic Todd class of $\overline{f}$ is
$\widehat{\operatorname{Td}}(\overline{f}):=\widehat{\operatorname{Td}}(T_{\overline{f}})\in\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X)).$
###### Theorem 6.2.
Let $\overline{f}\colon\mathcal{X}\rightarrow\mathcal{Y}$,
$\overline{g}\colon\mathcal{Y}\rightarrow\mathcal{Z}$ be morphisms in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$. Then we have an equality
$\widehat{\operatorname{Td}}(\overline{g}\circ\overline{f})=f^{\ast}\widehat{\operatorname{Td}}(\overline{g})\cdot\widehat{\operatorname{Td}}(\overline{f})$
in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{X},\mathcal{D}_{{\text{\rm
a}}}(X))$.
###### Proof.
By construction of $T_{\overline{f}}$ and definition of the composition rule
of morphisms in $\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$, there a is
tightly distinguished triangle in
$\operatorname{\overline{\mathbf{D}}^{b}}(\mathcal{X})$
$T_{\overline{f}}\dashrightarrow
T_{\overline{g}\circ\overline{f}}\dashrightarrow
f^{\ast}T_{\overline{g}}\dashrightarrow T_{\overline{f}}[1].$
We conclude by an application of Theorem 4.11. ∎
#### Statement.
The arithmetic Grothendieck-Riemann-Roch theorem describes the behavior of the
arithmetic Chern character with respect to the push-forward functor. Recall
that the definition of the push-forward functor depends on the choice of a
theory of generalized analytic torsion classes. In its turn, such a theory
corresponds to a real additive genus.
###### Theorem 6.3.
Let $\overline{f}\colon\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$. Fix a closed conical subset
$W$ of $T_{0}^{\ast}X$ and a theory of generalized analytic torsion classes
$T$, whose associated real additive genus is $S$. Then, the derived direct
image functor
$\overline{f}_{\ast}\colon\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm
a}}}(X,W))\longrightarrow\operatorname{\mathbf{\widehat{D}}^{b}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,f_{\ast}(W)))$
attached to $T$ satisfies the equation
(6.4)
$\operatorname{\widehat{ch}}(\overline{f}_{\ast}\alpha)=f_{\ast}(\operatorname{\widehat{ch}}(\alpha)\widehat{\operatorname{Td}}(\overline{f}))-\operatorname{a}(f_{\ast}(\operatorname{ch}(\mathcal{F}^{\ast}_{{\mathbb{C}}})\operatorname{Td}(T_{f_{{\mathbb{C}}}})S(T_{f_{{\mathbb{C}}}})),$
for every object $\alpha=[\overline{\mathcal{F}}^{\ast},\widetilde{\eta}]$.
The equality takes place in the arithmetic Chow group
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,f_{\ast}(W)))$.
Recall that the genus $S$=0 corresponds to the homogeneous theory $T^{h}$,
which is characterized by satisfying an additional homogeneity property [13,
Sec. 9]. Roughly speaking, this condition is exactly the one guaranteeing an
arithmetic Grothendieck-Riemann-Roch theorem without correction term.
###### Corollary 6.5.
The direct image functor $\overline{f}^{h}_{\ast}$ attached to the homogeneous
generalized analytic torsion theory $T^{h}$ satisfies an exact Grothendieck-
Riemann-Roch type formula:
$\operatorname{\widehat{ch}}(\overline{f}^{h}_{\ast}\alpha)=f_{\ast}(\operatorname{\widehat{ch}}(\alpha)\widehat{\operatorname{Td}}(\overline{f})).$
A Grothendieck-Riemann-Roch type theorem in Arakelov geometry was first proven
by Gillet-Soulé [18], for the degree 1 part of the Chern character (namely the
determinan of the cohomology) and under the restriction on the morphism $f$ to
be smooth over ${\mathbb{C}}$. Also they can only deal with hermitian vector
bundles. They used the holomorphic analytic torsion [2], [3], [4] and deep
results of Bismut-Lebeau [6] on the compatibility of analytic torsion with
closed immersions. The holomorphic analytic torsion was later generalized by
Bismut-Köhler [5], to the holomorphic analytic torsion forms, that transgress
the whole Grothendieck-Riemann-Roch theorem for Kähler submersions, at the
level of differential forms. The extension of the arithmetic Grothendieck-
Riemann-Roch theorem to the full Chern character and generically smooth
morphisms was finally proven by Gillet-Rössler-Soulé [14]. They applied the
analogue to the Bismut-Lebeau immersion theorem, for analytic torsion forms,
established in the monograph by Bismut [1].
Theorem 6.3 provides an extension of the previous results in several
directions. First, we allow the morphism to be an arbitrary projective
morphism, non necessarily generically smooth. Second, we can deal with
metrized objects in the bounded derived category of coherent sheaves. Third,
we provide all the possible forms of such a theorem, by introducing our theory
of generalized analytic torsion classes, thus explaining the topological
correction term.
#### Reductions.
The proof of our version of the arithmetic Grothendieck-Riemann-Roch theorem
follows the pattern of the classical approach in algebraic geometry. Namely,
it proceeds by factorization of the morphism $\overline{f}$ into a regular
closed immersion and a trivial projective bundle projection. The advantage of
working with the formalism of hermitian structures on the derived category of
coherent sheaves makes the whole procedure more transparent and direct, in
particular avoiding the appearance of several secondary classes. Also the
cocyle type relation expressing the behaviour of generalized analytic torsion
with respect to composition of morphisms in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{R}}}$ (see the axioms
[9, Sec. 7]) is well suited to this factorization argument.
By the classification of generalized analytic torsion theories, it is enough
to prove Theorem 6.3 for the homogenous theory $T^{h}$. From now on we fix
this choice, and hence all derived direct image functors will be with respect
to this theory.
Let $\overline{f}$ be a morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$, and consider a
factorization
$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{\mathcal{{\mathbb{P}}}^{n}_{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{Y}.}$
The tangent complex of $\pi$ is canonically isomorphic to
$p^{\prime\ast}T_{{\mathbb{P}}^{n}_{A}/A}$, where
$\textstyle{{\mathbb{P}}^{n}_{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\operatorname{Spec}A.}$
We may thus endow $\pi$ with the pull-back by $p^{\prime}$ of the Fubini-Study
metric on $T_{{\mathbb{P}}^{n}_{A}/A}$ [9, Sec. 5]. We write $\overline{\pi}$
for the resulting morphism in
$\overline{\operatorname{\mathbf{Reg}}}_{\ast/A}$. By [10, Lemma 5.3], there
exists a unique hermitian structure on $i$ such that
$\overline{f}=\overline{\pi}\circ\overline{i}$. Then we recall that we have
the equality of functors
$f_{\ast}=\overline{\pi}_{\ast}\circ\overline{i}_{\ast}$
and the equality of arithmetic Todd genera
$\widehat{\operatorname{Td}}(\overline{f})=i^{\ast}\widehat{\operatorname{Td}}(\overline{\pi})\widehat{\operatorname{Td}}(\overline{i}).$
###### Lemma 6.6.
It is enough to prove Theorem 6.3 for $\overline{i}$ and $\overline{\pi}$
individually.
###### Proof.
Let us assume the theorem known for $\overline{i}$ and for $\overline{\pi}$.
Because the theory is known for $\overline{\pi}$, we may apply it to the
object $\overline{i}_{\ast}\alpha$ in
$\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{\mathcal{Y}})$, to
obtain
(6.7)
$\operatorname{\widehat{ch}}(\overline{f}_{\ast}\alpha)=\operatorname{\widehat{ch}}(\overline{\pi}_{\ast}(\overline{i}_{\ast}\alpha))=\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{i}_{\ast}\alpha)\widehat{\operatorname{Td}}(\overline{\pi})).$
Because the theorem is known for $\overline{i}$, we also have
(6.8)
$\begin{split}\operatorname{\widehat{ch}}(\overline{i}_{\ast}\alpha)\widehat{\operatorname{Td}}(\overline{\pi})=&i_{\ast}(\operatorname{\widehat{ch}}(\alpha)\widehat{\operatorname{Td}}(\overline{i}))\widehat{\operatorname{Td}}(\overline{\pi})\\\
&=i_{\ast}(\operatorname{\widehat{ch}}(\alpha)\widehat{\operatorname{Td}}(\overline{i})i^{\ast}\widehat{\operatorname{Td}}(\overline{\pi}))\\\
&\hskip
8.5359pt=i_{\ast}(\operatorname{\widehat{ch}}(\alpha)\widehat{\operatorname{Td}}(\overline{f})).\end{split}$
Observe we used the projection formula in arithmetic Chow groups (Theorem
2.30), and that this equality holds in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y,f_{\ast}(W)))$, because the definition of direct image
of wave front sets and the fact that $\pi$ is smooth yield
$f_{\ast}(W)=\pi_{\ast}(i_{\ast}(W))$. We conclude by putting (6.7)–(6.8)
together and using the fact $f_{\ast}=\pi_{\ast}i_{\ast}$ on arithmetic Chow
groups (Proposition 2.27). ∎
###### Lemma 6.9.
Theorem 6.3 holds for $\overline{i}$.
###### Proof.
In [13, Thm 10.28], the authors prove the theorem for direct images by closed
immersions, defined on arithmetic $K$ groups. They suppose as well that the
hermitian structure on $T_{i}$ is given by a smooth hermitian metric on
$N_{\mathcal{X}/{\mathbb{P}}^{n}_{\mathcal{Y}}}$. Finally, the result in
_loc,. cit._ holds in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}(\mathcal{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(Y))$.
By the anomaly formula provided by Theorem 4.11 and the anomaly formula of
generalized analytic torsion theories [9, Prop. 7.4] for change of hermitian
structure on the tangent complex, one can extend [13, Thm 10.28] to arbitrary
hermitian structures on $i$, in particular the one we fixed. Also, a careful
proof reading of the proof in _loc. cit._ shows that the theorem can be
adapted to allow $\alpha\in\widehat{K}_{0}(\mathcal{X},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}(X,W))$, taking values in
$\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{{\mathbb{Q}}}({\mathbb{P}}^{n}_{Y},\mathcal{D}_{\text{{\rm
cur}},{\text{\rm a}}}({\mathbb{P}}^{n}_{Y},i_{\ast}W))$. Finally, to get the
result for $\overline{i}_{\ast}$ on the arithmetic derived category, we use
the class functor to arithmetic $K$-theory, the commutativity Theorem 5.5 and
that the Chern character on arithmetic derived category factors, by
construction, through arithmetic $K$ groups (see Notation 3.23). ∎
###### Lemma 6.10.
To prove Theorem 6.3 for $\overline{\pi}$, it is enough to prove it when
$\mathcal{Y}=\operatorname{Spec}A$ and $\alpha=\overline{{\mathcal{O}}(k)}$,
$-n\leq k\leq 0$, where the chosen hermitian structure on ${\mathcal{O}}(k)$
is the Fubini-Study metric.
###### Proof.
The derived category
$\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{\mathcal{Y}})$ is generated
by coherent sheaves of the form $\pi^{\ast}\mathcal{G}\otimes
p^{\prime\ast}{\mathcal{O}}(k)$ [9, Cor. 4.11]. By the anomaly formulas [9,
Prop. 7.4, Prop. 7.6], we thus reduce to prove the theorem for $\alpha$ of the
form $\pi^{\ast}\overline{\mathcal{G}}\otimes
p^{\prime\ast}\overline{{\mathcal{O}}(k)}$, for any metric on $\mathcal{G}$.
On the one hand, by the formulas in Prop 5.8, the multiplicativity and pull-
back functoriality (Prop. 3.8) of the Chern character, we have
$\begin{split}\operatorname{\widehat{ch}}(\overline{\pi}_{\ast}\alpha)=&\operatorname{\widehat{ch}}(\overline{\mathcal{G}}\otimes\overline{\pi}_{\ast}p^{\prime\ast}\overline{{\mathcal{O}}(k)})\\\
=&\operatorname{\widehat{ch}}(\overline{\mathcal{G}})\operatorname{\widehat{ch}}(\overline{\pi}_{\ast}p^{\prime\ast}\overline{{\mathcal{O}}(k)})\\\
&\hskip
8.5359pt=\operatorname{\widehat{ch}}(\overline{\mathcal{G}})\operatorname{\widehat{ch}}(p^{\ast}\overline{\pi}^{\prime}_{\ast}\overline{{\mathcal{O}}(k)})\\\
&\hskip
17.07182pt=\operatorname{\widehat{ch}}(\overline{\mathcal{G}})p^{\ast}\operatorname{\widehat{ch}}(\overline{\pi}^{\prime}_{\ast}\overline{{\mathcal{O}}(k)}).\end{split}$
Here we recall
$\pi^{\prime}:{\mathbb{P}}^{n}_{A}\rightarrow\operatorname{Spec}A$ is the
structure morphism, that we endow with the Fubini-Study metric. On the other
hand, we similarly prove
$\begin{split}\pi_{\ast}(\operatorname{\widehat{ch}}(\pi^{\ast}\overline{\mathcal{G}}\otimes
p^{\prime\ast}\overline{{\mathcal{O}}(k)})\widehat{\operatorname{Td}}(\overline{\pi}))=&\operatorname{\widehat{ch}}(\overline{\mathcal{G}})\pi_{\ast}(p^{\prime\ast}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(k)})p^{\prime\ast}\widehat{\operatorname{Td}}(\overline{\pi}^{\prime}))\\\
&=\operatorname{\widehat{ch}}(\overline{\mathcal{G}})p^{\ast}\pi^{\prime}_{\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(k)})\widehat{\operatorname{Td}}(\overline{\pi^{\prime}})).\end{split}$
We thus reduce to prove the theorem for $\overline{\pi}^{\prime}$ and
$\alpha=\overline{{\mathcal{O}}(k)}$, as was to be shown. ∎
#### The case of projective spaces.
To prove Theorem 6.3 in full generality, it remains to treat the case of the
projection
$\overline{\pi}\colon{\mathbb{P}}^{n}_{A}\rightarrow\operatorname{Spec}A$,
where we endow $\pi$ with the Fubini-Study metric. Since this projection is
the pull-back of the projection
$\overline{\pi}\colon{\mathbb{P}}^{n}_{{\mathbb{Z}}}\rightarrow\operatorname{Spec}{\mathbb{Z}}$
it is enough to treat the case when $A={\mathbb{Z}}$.
Furthermore, we showed that it is enough to consider $\alpha$ of the form
$\overline{{\mathcal{O}}(k)}$, $-n\leq k\leq 0$, with the Fubini-Study metric
as well and any metric of the direct image $\pi_{\ast}{\mathcal{O}}(k)$. In
[9, Def. 5.7], we introduced the main characteristic numbers of $T^{h}$,
$t_{n,k}^{h}=T^{h}(\overline{\pi},\overline{{\mathcal{O}}(k)},\overline{\pi_{\ast}{\mathcal{O}}(k)}),\
-n\leq k\leq 0,$
where $\pi_{\ast}{\mathcal{O}}(k)$ was endowed with its $L^{2}$ metric. Since
we will only consider the homogeneous analytic torsion we will shorthand
$t^{h}_{n,k}=t_{n,k}$.
We will denote by $\overline{0}$ the trivial vector bundle with trivial
hermitian structure and, for any $X$, we denote by
$\overline{\mathcal{O}}_{X}$ the structural sheaf with the metric $\|1\|=1$.
For $-n\leq k<0$, the complex $\overline{\pi_{\ast}{\mathcal{O}}(k)}$ can be
represented by $\overline{0}$, while the complex
$\overline{\pi_{\ast}{\mathcal{O}}(0)}$ can be represented by the structural
sheaf ${\mathcal{O}}_{{\mathbb{Z}}}$ with the metric $\|1\|=1/n!$. Since this
metric depends on $n$ it will be simpler to consider the complexes
$\overline{\pi_{\ast}{\mathcal{O}}(k)}^{\prime}=\overline{\pi_{\ast}{\mathcal{O}}(k)}$
for $-n\leq k<0$ and
$\overline{\pi_{\ast}{\mathcal{O}}(k)}^{\prime}=\overline{\mathcal{O}}_{{\mathbb{Z}}}$.
Then we write
$t^{\prime}_{n,k}=T^{h}(\overline{\pi},\overline{{\mathcal{O}}(k)},\overline{\pi_{\ast}{\mathcal{O}}(k)}^{\prime}),\
-n\leq k\leq 0.$
These characteristic numbers satify
$t^{\prime}_{n,k}=\begin{cases}t_{n,k},&\text{ if }-n\leq k<0,\\\
t_{n,0}-(1/2)\log(n!),&\text{ if }k=0.\end{cases}$
Clearly it is equivalent to work with the characteristic numbers $t_{n,k}$ or
with $t^{\prime}_{n,k}$.
In order to finish the the proof of Theorem 6.3 it only remains to show the
following particular cases.
###### Theorem 6.11.
For every $0\leq k\leq n$, we have the equality in
$\operatorname{\widehat{CH}}^{1}(\operatorname{Spec}{\mathbb{Z}})={\mathbb{R}}$
(6.12)
$\operatorname{a}(t^{\prime}_{n,-k})=\operatorname{\widehat{ch}}(\overline{\pi_{\ast}{\mathcal{O}}(-k)}^{\prime})-\
\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})\widehat{\operatorname{Td}}(\overline{\pi})).$
The proof will proceed by induction. The next proposition treats the first
case.
###### Proposition 6.13.
Equation (6.12) holds for $k=0$.
###### Proof.
The proof exploits the behavior of generalized analytic torsion with respect
to composition of morphisms. Let us consider the diagram
$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\Delta\
\ \
}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{Z}}}\underset{{\mathbb{Z}}}{\times}{\mathbb{P}}^{n}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\operatorname{Spec}{\mathbb{Z}}.}$
The Fubini-Studi metric on the tangent space
$T_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}$ is invariant under complex conjugation.
It induces a metric on the tangent space of the product of projective spaces.
On each morphism on the above diagram we consider the relative hermitian
structure deduced by the hermitian metrics on each tangent space. With this
choice
(6.14) $\overline{\operatorname{id}}=\overline{p}_{2}\circ\overline{\Delta},$
where the composition of relative hermitian structures is defined in [10,
Definition 5.7]. On the relative tangent bundle $T_{p_{2}}$ we consider the
metric induced by the metric on $T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}$.
Since the short exact sequence
$0\longrightarrow\overline{T}_{p_{2}}\longrightarrow\overline{T}_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\longrightarrow
p_{2}^{\ast}\overline{T}_{{\mathbb{P}}^{2}}\longrightarrow 0$
is orthogonaly split, we deduce that the metric we consider on
$\overline{p}_{2}$ agrees with the metric on the vector bundle
$\overline{T}_{p_{2}}$ and this, in turn agrees with the metric on
$p_{1}^{\ast}T_{{\mathbb{P}}^{n}}$.
Let $\overline{Q}$ be the tautological quotient bundle on
${\mathbb{P}}^{n}_{{\mathbb{Z}}}$ with any hermitian metric invariant under
complex conjugation. We denote by $K$ the Koszul resolution of the diagonal
and $\overline{K}$ the same resolution with the induced metrics. Namely
$\overline{K}$ is the complex
$0\to p_{2}^{\ast}\Lambda^{n}\overline{Q}^{\vee}\otimes
p_{1}^{\ast}\overline{{\mathcal{O}}(-n)}\to\dots\to
p_{2}^{\ast}\overline{Q}^{\vee}\otimes
p_{1}^{\ast}\overline{{\mathcal{O}}(-1)}\to\overline{\mathcal{O}}_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\to
0,$
and there is an isomorphism in the derived category
$K\to\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}$. Since the
theories of homogeneous torsion classes for closed immersions and for
projective spaces are compatible [9, Definition 6.2] the equation
(6.15)
$T(\overline{p}_{2},\overline{K},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+\overline{p}_{2\flat}(T(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}))=0$
holds. Then
$\displaystyle
T(\overline{p}_{2},\overline{K},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})$
$\displaystyle=T(\overline{p}_{2},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+\sum_{i=1}^{n}(-1)^{i}T(\overline{p}_{2},p_{2}^{\ast}\Lambda^{i}\overline{Q}^{\vee}\otimes
p_{1}^{\ast}\overline{{\mathcal{O}}(-i)},\overline{0})$
$\displaystyle=\pi_{1}^{\ast}T(\overline{\pi},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{Z}}})\bullet\operatorname{ch}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+$
$\displaystyle\qquad\qquad\sum_{i=1}^{n}(-1)^{i}\pi_{1}^{\ast}T(\overline{\pi},\overline{{\mathcal{O}}(-i)},\overline{0})\bullet\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\bullet\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee}).$
Using that
$\pi_{1\ast}(\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})\operatorname{Td}(\overline{\pi}_{1}))=\begin{cases}1,&\text{
if }i=0,\\\ 0,&\text{ otherwise,}\end{cases}$
we deduce
(6.16)
$\overline{\pi}_{1\flat}(T(\overline{p}_{2},\overline{K},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}}))=t_{n,0}.$
By the arithmetic Riemann-Roch theorem for closed immersions
$\operatorname{a}(T(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}))=\sum_{i=0}^{n}(-1)^{i}p_{2}^{\ast}\operatorname{\widehat{ch}}(\Lambda^{i}\overline{Q}^{\vee})\cdot
p_{1}^{\ast}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-i)})-\Delta_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\Delta})).$
For $i>0$
$\pi_{1\ast}\left(p_{2\ast}\left(p_{2}^{\ast}\operatorname{\widehat{ch}}(\Lambda^{i}\overline{Q}^{\vee})p_{1}^{\ast}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-i)})\widehat{\operatorname{Td}}(\overline{p_{2}})\right)\widehat{\operatorname{Td}}(\overline{\pi}_{1})\right)\\\
=\pi_{1\ast}(\operatorname{\widehat{ch}}(\Lambda^{i}\overline{Q}^{\vee})\widehat{\operatorname{Td}}(\overline{\pi}_{1}))\cdot\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-i)})\widehat{\operatorname{Td}}(\overline{\pi})).$
Since
$\zeta(\pi_{1\ast}(\operatorname{\widehat{ch}}(\Lambda^{i}\overline{Q}^{\vee})\widehat{\operatorname{Td}}(\overline{\pi}_{1})))=\zeta(\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-i)})\widehat{\operatorname{Td}}(\overline{\pi})))=0$
and $\operatorname{Ker}\zeta$ is a square zero ideal of the arithmetic Chow
ring, we deduce that, for $i>0$
(6.17)
$\pi_{1\ast}\left(p_{2\ast}\left(p_{2}^{\ast}\operatorname{\widehat{ch}}(\Lambda^{i}\overline{Q}^{\vee})p_{1}^{\ast}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-i)})\widehat{\operatorname{Td}}(\overline{p_{2}})\right)\widehat{\operatorname{Td}}(\overline{\pi}_{1})\right)=0.$
For $i=0$ we compute
$\pi_{1\ast}\left(p_{2\ast}\left(p_{2}^{\ast}\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})p_{1}^{\ast}\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{p_{2}})\right)\widehat{\operatorname{Td}}(\overline{\pi}_{1})\right)=\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\pi}))^{2}.$
Using that
$\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})=1$ and
that
$\pi_{\ast}(\widehat{\operatorname{Td}}(\overline{\pi}))-1\in\operatorname{Ker}\zeta$
we obtain that
(6.18)
$\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\pi}))^{2}=-1+2\pi_{\ast}(\widehat{\operatorname{Td}}(\overline{\pi})).$
Furthermore, by the choice of metrics on the relative tangent complexes
(6.19)
$\pi_{1\ast}p_{2\ast}\Delta_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\Delta})\widehat{\operatorname{Td}}(\overline{p}_{2})\widehat{\operatorname{Td}}(\overline{\pi}_{1}))=\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\pi})).$
Using equations (6.17), (6.18) and (6.19) we deduce that
(6.20)
$\operatorname{a}(\overline{\pi}_{1\flat}(\overline{p}_{2\flat}(T(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}))))=\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\pi}))-\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{Z}}}).$
By equations (6.15), (6.16) and (6.20) we conclude
$\operatorname{a}(t^{\prime}_{n,0})=\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{Z}}})-\pi_{\ast}(\operatorname{\widehat{ch}}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\widehat{\operatorname{Td}}(\overline{\pi}))$
proving the proposition. ∎
###### Proof of Theorem 6.11.
We now proceed with the induction step. We assume that equation (6.12) holds
for some $k\geq 0$ and all $n\geq k$. Fix now $n\geq k+1$. Consider the
diagram
$\textstyle{{\mathbb{P}}^{n-1}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-1}}$$\scriptstyle{s}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{\operatorname{Spec}{\mathbb{Z}},}$
where $s$ is the closed immersion induced by a section $s$ of
${\mathcal{O}}_{{\mathbb{P}}^{n}}(1)$. As in the proof of Proposition 6.13, we
consider the relative hermitian structures defined by the Fubini-Study metric
on the tangent bundles. In this way
$\overline{\pi}_{n-1}=\overline{\pi}_{n}\circ\overline{s}$.
We consider the Koszul complex
$K_{n,k}\colon{\mathcal{O}}_{{\mathbb{P}}^{n}}(-k-1)\overset{s}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^{n}}(-k)$
and we denote by $\overline{K}_{n,k}$ the same complex provided with the
Fubini-Study metrics.
By the transitivity [9, Definition 7.1] of the homogeneous theory of analytic
torsion, the equation
$T^{h}(\overline{\pi}_{n-1},\overline{{\mathcal{O}}(-k)},\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime})=\\\
T^{h}(\overline{\pi}_{n},\overline{K}_{n,k},\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime})+\overline{\pi}_{n\flat}(T^{h}(\overline{s},\overline{{\mathcal{O}}(-k)},\overline{K}_{n,k})).$
By definition
$T^{h}(\overline{\pi}_{n-1},\overline{{\mathcal{O}}(-k)},\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime})=t^{\prime}_{n-1,-k}$.
By the choice of metrics, the exact sequence
(6.21)
$0\to\overline{\pi_{n\ast}{\mathcal{O}}(-k-1)}^{\prime}\to\overline{\pi_{n\ast}{\mathcal{O}}(-k)}^{\prime}\to\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime}\to
0$
is orthogonal split (all the terms appearing in this short exact sequence are
either $\overline{0}$ or $\overline{\mathcal{O}}_{{\mathbb{Z}}}$). This
implies that
$T^{h}(\overline{\pi}_{n},\overline{K}_{n,k},\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime})=\\\
T^{h}(\overline{\pi}_{n},\overline{{\mathcal{O}}(-k)},\overline{\pi_{n\ast}{\mathcal{O}}(-k)}^{\prime})-T^{h}(\overline{\pi}_{n},\overline{{\mathcal{O}}(-k-1)},\overline{\pi_{n\ast}{\mathcal{O}}(-k-1)}^{\prime})=\\\
t^{\prime}_{n,-k}-t^{\prime}_{n,-k-1}.$
By Lemma 6.9 (the case of closed immersions)
$T^{h}(\overline{s},\overline{{\mathcal{O}}(-k)},\overline{K}_{n,k})=\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})-\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}}(-k-1))-\overline{s}_{\flat}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})).$
This implies that
$\overline{\pi}_{n\flat}(T^{h}(\overline{s},\overline{{\mathcal{O}}(-k)},\overline{K}_{n,k}))=\\\
\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})-\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}}(-k-1))-\overline{\pi}_{n-1\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)}).$
Summing up, we deduce
(6.22)
$\operatorname{a}(t^{\prime}_{n-1,-k}-t^{\prime}_{n,-k}+t^{\prime}_{n,-k-1})=\\\
-\overline{\pi}_{n-1\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})+\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k)})+-\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k-1)}).$
Applying the induction hypothesis we get
$t^{\prime}_{n,-k-1}=-\operatorname{\widehat{ch}}(\overline{\pi_{n-1\ast}{\mathcal{O}}(-k)}^{\prime})+\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}(-k)}^{\prime})-\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k-1)}).$
Using again that the exact sequence (6.21) is orthogonally split, we deduce
that
$t^{\prime}_{n,-k-1}=\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}(-k-1)}^{\prime})-\overline{\pi}_{n\flat}\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}(-k-1)}).$
completing the inductive step and proving the Theorem 6.11 and therefore
Theorem 6.3. ∎
Once we have proved that the formal properties of an analytic torsion theory
imply the arithmetic Riemann-Roch theorem we can compute easily the
characteristic numbers $t_{n,k}$ for the homogeneous analytic torsion and all
$n\geq 0$ and $k\in{\mathbb{Z}}$. By the Theorem 6.3 they satisfy
$\operatorname{a}(t_{n,k})=\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})-\
\pi_{n\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})\widehat{\operatorname{Td}}(\overline{\pi}_{n})).$
Observe it is enough to compute $t_{n,k}$ for $k\geq-n$, since the self-
duality of the homogenous analytic torsion [9, Thm. 9.12] immediately yields
the relation
$t_{n,k}=(-1)^{n}t_{n,-k-n-1}.$
Therefore, from now on we restrict to this range of values of $k$.
We first compute
$\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})^{(1)}$.
For $-n\leq k\leq-1$ this quantity vanishes:
$\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})^{(1)}=0,\quad-n\leq
k\leq-1.$
Suppose now $k\geq 0$. Using that the volume form $1/n!\omega_{FS}^{n}$ is
given, in a coordinate patch, by
$\mu=\left(\frac{i}{2\pi}\right)^{n}\frac{\operatorname{d}z_{1}\land\operatorname{d}\overline{z}_{1}\land\dots\land\operatorname{d}z_{n}\land\operatorname{d}\overline{z}_{n}}{(1+\sum_{i=1}^{n}z_{i}\overline{z}_{i})^{n+1}},$
it is easy to see that the basis $\\{x_{0}^{a_{0}}\dots
x_{n}^{a_{n}}\\}_{a_{0}+\dots+a_{n}=k}$ is orthonormal and satisfies
$\|x_{0}^{a_{0}}\dots x_{n}^{a_{n}}\|^{2}_{L^{2}}=\frac{a_{0}!\dots
a_{n}!}{(k+n)!}.$
Therefore
$\operatorname{\widehat{ch}}(\overline{\pi_{n\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})^{(1)}=\sum_{a_{0}+\dots+a_{n}=k}-\left(\frac{1}{2}\right)\log\left(\frac{a_{0}!\dots
a_{n}!}{(k+n)!}\right).$
To compute
$\pi_{n\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})\widehat{\operatorname{Td}}(\overline{\pi}_{n}))$
we follow [17], where the case $k=0$ is considered. Let $\alpha_{n,k}$ be the
coefficient of $x^{n+1}$ in the power series
$e^{kx}\left(\frac{x}{1-e^{-x}}\right)^{n+1}$ and let $\beta_{n,k}$ be the
coefficient of $x^{n}$ in the power series
$\int_{0}^{1}\frac{\phi(t)-\phi(0)}{t}\operatorname{d}t,$ where
$\phi(t)=e^{kx}\left(\frac{1}{tx}-\frac{e^{-tx}}{1-e^{-tx}}\right)\left(\frac{x}{1-e^{-x}}\right)^{n+1}.$
Then, by a slight modification of the argument in [17, Proposition 2.2.2 &
2.2.3], we derive
$\pi_{n\ast}(\operatorname{\widehat{ch}}(\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{Z}}}}(k)})\widehat{\operatorname{Td}}(\overline{\pi}_{n}))^{(1)}=\frac{1}{2}\operatorname{a}\left(\alpha_{n,k}\sum_{p=1}^{n}\sum_{j=1}^{p}\frac{1}{j}+\beta_{n,k}\right).$
The factor $(1/2)$ appears from the different normalization used here (see
[11, Theorem 3.33]). We thus have to determine the coefficients $\alpha_{n,k}$
and $\beta_{n,k}$. The numbers $\alpha_{n,k}$ can be obtained similarly to
[17, Eq. (27)]. We obtain
$\alpha_{n,k}=\begin{cases}0&\text{for }-n\leq k\leq-1,\\\
\binom{k+n}{n}&\text{for }k\geq 0.\end{cases}$
The values of $\beta_{n,k}$ are expressed in terms of some secondary Todd
numbers, that in turn are determined by a generating series:
$\beta_{n,k}=\sum_{j=0}^{n}\widetilde{\operatorname{Td}}_{n-j}\frac{k^{j}}{j!},$
where the $\widetilde{\operatorname{Td}}_{m}$ are given by the equality of
generating series
(6.23) $\sum_{m\geq
0}\frac{\widetilde{\operatorname{Td}}_{m}}{m+1}T^{m+1}=\sum_{m\geq
1}\frac{\zeta(-(2m-1))}{2m-1}\frac{y^{2m}}{(2m)!},\quad T=1-e^{-y}.$
Here $\zeta$ stands for the Riemann zeta function. The result is a direct
consequence of the definition of $\beta_{n,k}$ and the computation of the
numbers $\beta_{n,0}$ in [17, Prop. 2.2.3 & Lemma 2.4.3].
We summarize these computations for the principal characteristic numbers
$t_{n,k}$, $-n\leq k\leq 0$, since it is particularly pleasant.
###### Proposition 6.24.
The principal characteristic numbers $t_{n,k}$ are given by
$t_{n,k}=\begin{cases}-\frac{1}{2}\sum_{p=1}^{n}\sum_{j=1}^{p}\frac{1}{j},&\text{for
}k=0,\\\ \\\
-\frac{1}{2}\sum_{j=0}^{n}\widetilde{\operatorname{Td}}_{n-j}\frac{k^{j}}{j!},&\text{for
}-n\leq k\leq-1,\end{cases}$
where the sequence of numbers $\widetilde{\operatorname{Td}}_{m}$, $m\geq 0$,
is determined by (6.23).
## References
* [1] J.-M. Bismut, _Holomorphic families of immersions and higher analytic torsion forms_ , Astérisque, vol. 244, SMF, 1997.
* [2] J.-M. Bismut, H. Gillet, and C. Soulé, _Analytic torsion and holomorphic determinant bundles I_ , Comm. Math. Phys. 115 (1988), 49–78.
* [3] by same author, _Analytic torsion and holomorphic determinant bundles II_ , Comm. Math. Phys. 115 (1988), 79–126.
* [4] by same author, _Analytic torsion and holomorphic determinant bundles III_ , Comm. Math. Phys. 115 (1988), 301–351.
* [5] J.-M. Bismut and K. Köhler, _Higher analytic torsion forms for direct images and anomaly formulas_ , J. Alg. Geom. 1 (1992), 647–684.
* [6] J.-M. Bismut and G. Lebeau, _Complex immersions and Quillen metrics_ , Publ. Math. IHES 74 (1991), 1–298.
* [7] J. I. Burgos Gil, _Green forms and their product_ , Duke Math. J. 75 (1994), 529–574.
* [8] by same author, _Arithmetic Chow rings and Deligne-Beilinson cohomology_ , J. Alg. Geom. 6 (1997), 335–377.
* [9] J. I. Burgos Gil, G. Freixas i Montplet, and R. Liţcanu, _Generalized holomorphic analytic torsion_ , arxiv:1011.3702.
* [10] by same author, _Hermitian structures on the derived category of coherent sheaves_ , arxiv:1102.2063.
* [11] J. I. Burgos Gil, J. Kramer, and U. Kühn, _Arithmetic characteristic classes of automorphic vector bundles_ , Documenta Math. 10 (2005), 619–716.
* [12] by same author, _Cohomological arithmetic Chow rings_ , J. Inst. Math. Jussieu 6 (2007), no. 1, 1–172.
* [13] J. I. Burgos Gil and R. Liţcanu, _Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions_ , Doc. Math. 15 (2010), 73–176.
* [14] H. Gillet, D. Rössler, and C. Soulé, _An arithmetic Riemann-Roch theorem in higher degrees_ , Ann. Inst. Fourier 58 (2008), 2169–2189.
* [15] H. Gillet and C. Soulé, _Arithmetic intersection theory_ , Publ. Math. IHES 72 (1990), 94–174.
* [16] by same author, _Characteristic classes for algebraic vector bundles with hermitian metric I, II_ , Annals of Math. 131 (1990), 163–203,205–238.
* [17] by same author, _Analytic torsion and the arithmetic Todd genus_ , Topology 30 (1991), no. 1, 21–54, With an appendix by D. Zagier.
* [18] by same author, _An arithmetic Riemann-Roch theorem_ , Invent. Math. 110 (1992), 473–543.
* [19] P. Griffiths and J. Harris, _Principles of algebraic geometry_ , John Wiley & Sons, Inc., 1994.
* [20] L. Hörmander, _The analysis of linear partial differential operators. I_ , second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis.
* [21] S. Kawaguchi and A. Moriwaki, _Inequalities for semistable families of arithmetic varieties_ , J. Math. Kyoto Univ. 41 (2001), no. 1, 97–182.
|
arxiv-papers
| 2012-11-08T07:59:13 |
2024-09-04T02:49:37.702981
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jos\\'e Ignacio Burgos Gil, Gerard Freixas i Montplet, Razvan Litcanu",
"submitter": "Jos\\'e Ignacio Burgos Gil",
"url": "https://arxiv.org/abs/1211.1783"
}
|
1211.1862
|
# Three methods to detect the predicted $D\bar{D}$ scalar meson $X(3700)$
C. W. Xiao 1 and E. Oset 1 1Departamento de Física Teórica and IFIC, Centro
Mixto Universidad
de Valencia-CSIC, Institutos de Investigación de Paterna, Apartado 22085,
46071 Valencia, Spain
###### Abstract
In analogy to the $f_{0}(500)$, which appears as a $\pi\pi$ resonance in
chiral unitary theory, and the $f_{0}(980)$, which appears as a quasibound
$K\bar{K}$ state, the extension of this approach to the charm sector also
predicts a quasibound $D\bar{D}$ state with mass around 3720 MeV, named as
$X(3700)$, for which some experimental support is seen in the $e^{+}e^{-}\to
J/\psi D\bar{D}$ reaction close to the $D\bar{D}$ threshold. In the present
work we propose three different experiments to observe it as a clear peak. The
first one is the radiative decay of the $\psi(3770)$, $\psi(3770)\to\gamma
X(3700)\to\gamma\eta\eta^{\prime}$. The second one proposes the analogous
reaction $\psi(4040)\to\gamma X(3700)\to\gamma\eta\eta^{\prime}$ and the third
reaction is the $e^{+}e^{-}\to J/\psi X(3700)\to J/\psi\eta\eta^{\prime}$.
Neat peaks are predicted for all the reactions and the calculated rates are
found within measurable range in present facilities.
## I Introduction
The use of the chiral unitary approach in the meson-meson interaction gives
rise to the $f_{0}(500)(or\ \sigma),\ f_{0}(980),\ a_{0}(980)$ scalar mesons
npa ; norbert ; markushin ; rios1 ; rios2 ; juanclassic ; juanenrique from
the unitarization in coupled channels of the meson-meson interaction provided
by the chiral Lagrangians weinberg ; gasser . The $f_{0}(500)$ appears
basically as a $\pi\pi$ resonance and the $f_{0}(980),\ a_{0}(980)$ as
basically $K\bar{K}$ quasibound states that decay into $\pi\pi$ and $\pi\eta$
respectively. The similarity between $\bar{K}$ and $D$ ($K$ and $\bar{D}$)
suggest that there could be also a $D\bar{D}$ quasibound state around
$3700\textrm{ MeV}$, that we shall call $X(3700)$, decaying into pairs of
light pseudoscalars, $\pi\pi,\ \eta\eta,\ \eta\eta^{\prime},\ K\bar{K}$. In an
extrapolation of the chiral unitary approach to the SU(4) sector danione it
was found that, indeed, a quasibound scalar $D\bar{D}$ state with $I=0$
emerged with a small width, since transition matrix elements from $D\bar{D}$
to the light sector were strongly suppressed. This finding has been
corroborated recently in manolito ; carlos using models that incorporate
heavy quark symmetry.
Later on it was found in daniee that the bump in the $D\bar{D}$ spectrum
close to the $D\bar{D}$ threshold observed at Belle in the $e^{+}e^{-}\to
J/\psi D\bar{D}$ reaction eebelle was better interpreted in terms of the
bound state below threshold, with $M_{X}\simeq 3723\textrm{ MeV}$ than with a
new resonance as suggested in eebelle . So far, this is the strongest
experimental support for this state, in spite of the fact that some other
reaction has been suggested to observe it. Indeed, in danirad a suggestion
was made to detect the state in the radiative decay of the $\psi(3770)$. The
idea is based in the fact that the $\psi(3770)$ couples strongly to $D\bar{D}$
and with the emission of a photon one can bring the $D\bar{D}$ state below the
threshold into the region of the resonance. A width of $\Gamma_{\psi\to\gamma
X}=(1.05\pm 0.41)\textrm{ KeV}$ was found which would be in the measurable
range. However, a problem of this suggestion is that this peak would have to
be seen over a background of $\psi\to\gamma\ +$ anything, which is estimated
to have a branding ratio of the order of $10^{-2}$, judging by the rate of
some measured channels reported in the Particle Date Group (PDG) pdg , while
$\Gamma_{\psi\to\gamma X}/\Gamma_{\psi}\simeq 4\times 10^{-5}$. The signal
would be of the order of 1% of smaller on top of a background and the
prospects to see it there would be dim. There is another problem since the
peak for the decay appears at small photon momentum where there would be
radiative decays displaying Bremsstrahlung of the photons, with accumulated
strength at low photon energies, precisely where the peak of the $X$ would
appear. The selection of a particular decay channel where the background would
be much reduced would be then much welcome and this is what we do here,
suggesting the $\eta\eta^{\prime}$ channel for reasons that would be clear
later on.
On the other hand, with the advent of BESIII the production of the
$\psi(4040)$ state is being undertaken and in this case the photon has more
energy in the radiative decay, removing the peak from the Bremsstrahlung
region, with obvious advantages.
We have also investigated another method, taking the same reaction as
performed in eebelle but looking for $e^{+}e^{-}\to J/\psi\eta\eta^{\prime}$.
We predict a peak for $\eta\eta^{\prime}$ production and compare the strength
of the peak with the cross sections measured in eebelle for the $J/\psi
D\bar{D}$ production.
With all these studies we find out three methods which would allow to see the
neat peak for that state and the widths or cross sections are found within
present measuring range, such that devoted experiments would be most
opportune.
## II Formalism
In danirad the radiative decay of $\psi(3770)$ into $\gamma X(3700)$ was
studied. The work of danione was redone including the channels $D^{+}D^{-},\
D^{0}\bar{D}^{0},\ D^{+}_{s}D^{-}_{s},\ \pi^{+}\pi^{-},\ K^{+}K^{-},\
\pi^{0}\pi^{0},\ K^{0}\bar{K}^{0},\ \eta\eta,\ \eta\eta^{\prime},\
\eta^{\prime}\eta^{\prime},\ \eta_{c}\eta,\\\ \ \eta_{c}\eta^{\prime}$. Using
a potential derived from an SU(4) extension of the SU(3) chiral Lagrangians
weinberg ; gasser with an explicit SU(4) breaking for terms exchanging charm,
the Bethe-Salpeter equations were solved to obtain the scattering matrix
$T=[1-VG]^{-1}V,$ (1)
with $V$ the potential and $G$ the loop function for the integral of
intermediate two meson propagators. With this formalism a pole was obtained
for the $T$ matrix around $3722\textrm{ MeV}$ below the $D\bar{D}$ threshold.
What is of relevance for the present work is the coupling of this state to the
different channels. In Table 1 we show the results obtained in danirad :
Table 1: Coupling of the pole at $(3722-i18)\textrm{ MeV}$ to the channels. channel | Re($g_{X}$) [MeV] | Im($g_{X}$) [MeV] | $|g_{X}|$ [MeV]
---|---|---|---
$\pi^{+}\pi^{-}$ | 9 | 83 | 84
$K^{+}K^{-}$ | 5 | 22 | 22
$D^{+}D^{-}$ | 5962 | 1695 | 6198
$\pi^{0}\pi^{0}$ | 6 | 83 | 84
$K^{0}\bar{K}^{0}$ | 5 | 22 | 22
$\eta\eta$ | 1023 | 242 | 1051
$\eta\eta^{\prime}$ | 1680 | 368 | 1720
$\eta^{\prime}\eta^{\prime}$ | 922 | -417 | 1012
$D^{0}\bar{D}^{0}$ | 5962 | 1695 | 6198
$D^{+}_{s}D^{-}_{s}$ | 5901 | -869 | 5965
$\eta_{c}\eta$ | 518 | 659 | 838
$\eta_{c}\eta^{\prime}$ | 405 | 9 | 405
As we can see, the largest couplings are for $D\bar{D}$ and
$D_{s}\bar{D}_{s}$. However, the separation in energy of the
$D_{s}\bar{D}_{s}$ component makes the $D\bar{D}$ component to stand as the
more relevant meson-meson component of this state, which qualifies
approximately as a $D\bar{D}$ quasibound state. The width obtained from the
decay of this state in all the allowed channels is $36\textrm{ MeV}$. Note
that the transition to light, open, channels is suppressed and this determines
the large lifetime of the state. We observe from the Table that the largest
coupling to the light channels is to $\eta\eta^{\prime}$ which also contains
two different particles, hence, this will be the channel that we will adopt to
have the $X(3700)$ state detected.
In danirad the decay of $\psi(3770)$ to $\gamma X$ was evaluated recalling
that the $\psi(3770)$ decays basically with $D\bar{D}$. This allows one to
obtain the coupling of $\psi(3770)$ to $D^{+}D^{-}$ and then, from the
triangular diagram of Fig. 1, the $\psi(3770)\to\gamma X$ transition
Figure 1: Diagram for $\psi(3770)\to\gamma X$ that contains the $d$ term.
amplitude was evaluated. We do not repeat here the steps of the calculations
in danirad and quote the final results. The transition amplitude for the
diagram of Fig. 1 is given by
$i{\cal M}=i\epsilon_{\psi}^{\mu}(P)\epsilon_{\gamma}^{\nu}(K){\cal
T}_{\mu\nu},$ (2)
and since the problem has two independent four momenta, by Lorentz invariance
one may write
${\cal
T}_{\mu\nu}=ag_{\mu\nu}+bP_{\mu}P_{\nu}+cP_{\mu}K_{\nu}+dP_{\nu}K_{\mu}+eK_{\mu}K_{\nu}.$
(3)
Due to gauge invariance only the structures $ag_{\mu\nu}$ and
$dP_{\nu}K_{\mu}$ (with $P,\ K$ the $\psi$ and $\gamma$ momentum), which leads
to a convergent integral, survive, and, in addition, one has $a=-d\,K\cdot P$.
The $d$ coefficient is evaluated using the Feynman parameterization of the
loop function corresponding to the diagram of Fig. 1 and one finds
$d=-\sum_{j}\frac{g_{\psi}g_{X,j}e}{2\pi^{2}}\int_{0}^{1}dx\int_{0}^{x}dy\frac{y(1-x)}{s+i\epsilon},$
(4)
with $s$ given by
$s=(1-x)(xM_{\psi}^{2}-m_{2}^{2}-2yP\cdot K)-xm_{1}^{2},$ (5)
with $e$ the electron charge ($e^{2}/4\pi=\alpha=1/137$), $g_{\psi}$ the
coupling of $\psi(3770)$ to $D^{+}D^{-},\ g_{\psi}=11.7$, and $j$ summing over
the two relevant channels $D^{+}D^{-}$ and $D^{+}_{s}D^{-}_{s}$. The partial
decay width for $\psi(3770)\to\gamma X$ is given by
$\Gamma_{\psi\rightarrow\gamma X}=\frac{|\vec{K}|}{12\pi
M_{\psi}^{2}}\;(P\cdot K)^{2}\;|d|^{2}.$ (6)
The result obtained in danirad is 111Within uncertainties it turned out to be
$\Gamma_{\psi\rightarrow\gamma X}=(1.05\pm 0.41)\textrm{ KeV}$. We refer to
the value $0.65\textrm{ KeV}$ here obtained with the standard parameters for
comparison reasons.
$\Gamma_{\psi\rightarrow\gamma X}=0.65\textrm{ KeV}.$ (7)
As mentioned in the Introduction, determining the peak corresponding to this
process over a background of $\gamma X$ events is problematic and thus we
choose the $\eta\eta^{\prime}$ to detect the $X$ peak. For this the diagram of
Fig. 1 has to be changed to the one of Fig. 2.
Figure 2: Diagram for $\psi(3770)\to\gamma X\to\gamma\eta\eta^{\prime}$.
Technically all we have to do is substitute $d$ by $d\,^{\prime}$ where
$d\,^{\prime}=d\frac{1}{M_{inv}^{2}-M_{X}^{2}+iM_{X}\Gamma_{X}}g_{X,\eta\eta^{\prime}},$
(8)
with $d$ defined in Eq. (4),
$M_{inv}^{2}=(p_{\eta}+p_{\eta^{\prime}})^{2},$ (9)
and $g_{X,\eta\eta^{\prime}}$ the coupling of the $X$ to the
$\eta\eta^{\prime}$ channel given in Table 1.
The relevant magnitude now is
$\frac{d\Gamma}{dM_{inv}}=\frac{1}{4(2\pi)^{3}}\frac{1}{M_{\psi}^{2}}\,p_{\gamma}\,\tilde{p}_{\eta}\,\overline{\sum}\sum|T|^{2},$
(10)
which provides the invariant mass distribution, where
$\displaystyle p_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M_{\psi}^{2},0,M_{inv}^{2})}{2M_{\psi}},$
(11) $\displaystyle\tilde{p}_{\eta}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M_{inv}^{2},m_{\eta}^{2},m_{\eta^{\prime}}^{2})}{2M_{inv}},$
(12) $\displaystyle\overline{\sum}\sum|T|^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\,|d\,^{\prime}|^{2}(K\cdot P)^{2},$ (13)
with $p_{\gamma}$, $\tilde{p}_{\eta}$ the $\gamma$ momentum in the
$\psi(3770)$ rest frame and the $\eta$ momentum in the $\eta\eta^{\prime}$
rest frame respectively.
In Fig. 3 we show this distribution,
Figure 3: The mass distribution of the $\eta\eta^{\prime}$ in the decay of
$\psi(3770)$ to $\gamma X(3700)\to\gamma\eta\eta^{\prime}$.
and we see a clear peak at $M_{inv}\simeq 3722\textrm{ MeV}$ with a narrow
width. The peak is still around the upper threshold for the invariant mass.
However, the fact that we have chosen a neutral channel to identify the $X$
state prevents Bremsstrahlung to occur and the identification of a peak there
would be a clear signal of a state. The integrated width around the peak
($3600<M_{inv}<3770\textrm{ MeV}$) gives
$\Gamma=\int_{3600}^{3770}\frac{d\Gamma}{dM_{inv}}dM_{inv}=0.293\textrm{
KeV},$ (14)
which is smaller than the total width of Eq. (7) which integrates over the
whole range of $M_{inv}$.
The largest contribution comes from the $D^{+}D^{-}$ channel which by itself
provides 73% of the rate. The coherent sum with the $D_{s}^{+}D_{s}^{-}$
contribution makes up for the rest of the rate.
The width of Eq. (14) represents a branching ratio of $10^{-5}$. In this sense
one should note that CLEO has set thresholds of the order of magnitude of
$10^{-4}$ and at BESIII one can get a production $\psi(3770)$ of about a
factor one hundred times bigger, which would make this measurement feasible in
that Lab.
## III Radiative decay of the $\psi(4040)$
The $\psi(4040)$ shares the same quantum numbers as the $\psi(3770)$, however
the largest branching ratio is not to $D\bar{D}$ but to $D^{*}\bar{D}+cc$.
From the data in the PDG we find that
$\displaystyle\frac{\Gamma(D\bar{D})}{\Gamma(D^{*}\bar{D}+cc)}$
$\displaystyle=$ $\displaystyle 0.24\pm 0.05\pm 0.12,$ (15)
$\displaystyle\frac{\Gamma(D^{*}\bar{D}^{*})}{\Gamma(D^{*}\bar{D}+cc)}$
$\displaystyle=$ $\displaystyle 0.18\pm 0.14\pm 0.03,$ (16)
Assuming that the $D^{*}\bar{D}+cc$, $D\bar{D}$ and $D^{*}\bar{D}^{*}$ provide
most of the contribution, this allows us to get the coupling
$g_{\psi(4040),D^{+}D^{-}}=2.15,$ (17)
and then we can recalculate the invariant mass distribution and width for
$\psi(4040)\to\gamma X(3700)$. In Fig. 4 we show the results for the invariant
mass distribution.
Figure 4: The mass distribution of $\eta\eta^{\prime}$ in the $\psi(4040)$
decay to $\gamma X(3700)\to\gamma\eta\eta^{\prime}$.
We find now a neat peak around the mass of the $X$. The novelty here is that
the peak is far away from all thresholds which could eventually be seen in the
spectrum of inclusive $d\Gamma/dE_{\gamma}$ without the risk to confuse the
peak with Bremsstrahlung like in the case of the $\psi(3770)\to\gamma X$. In
any case, as advocated here, the direct measurement of the $\eta\eta^{\prime}$
channel should drastically reduce the background and allow a clear peak to be
identified. The integrated width around this peak ($3600<M_{inv}<3800\textrm{
MeV}$) gives
$\Gamma=\int_{3600}^{3800}\frac{d\Gamma}{dM_{inv}}dM_{inv}=0.496\textrm{
KeV},$ (18)
which is about double than in the case of the $\psi(3770)$. In this case, the
larger phase space for decay has overcome the reduction due to the reduced
coupling of Eq. (17).
We should note that the largest contribution comes from the $D^{+}D^{-}$
channel, this channel alone providing about half the rate of Eq. (18) while
$D_{s}^{+}D_{s}^{-}$ alone only given 19% of this rate.
The width of Eq. (18) is a bit bigger than the one obtained for the
$\psi(3770)$, yet, the rate of production at BESIII is smaller. Present plans
are to produce 2.8 million $\psi(4040)$ events and no plans are made for the
future yet 222We would like to thank Cheng-Ping Shen for providing us the
information.. With this statistics and the width of Eq. (18), which
corresponds to a branching ratio of $6.2\times 10^{-6}$, one could get about
17 events of this radiative decay. It is clear that more statistics would be
needed to see a clear peak.
## IV The $e^{+}e^{-}\to J/\psi X\to J/\psi\eta\eta^{\prime}$ reaction
In daniee the $e^{+}e^{-}\to J/\psi X\to J/\psi D\bar{D}$ reaction was
studied and it was concluded that the data on the $D\bar{D}$ invariant mass
distribution was better described in terms of the $X(3700)$ resonance that in
terms of a new state suggested in eebelle . The mechanism for this reaction is
given in Fig. 5.
Figure 5: Feynman diagram of the reaction $e^{+}e^{-}\to J/\psi X\to J/\psi
D\bar{D}$.
The differental cross section is given by daniee
$\frac{d\sigma}{dM_{inv}(D\bar{D})}=\frac{1}{(2\pi)^{3}}\frac{m_{e}^{2}}{s\sqrt{s}}|\vec{k}|\;|\vec{p}|\;|T|^{2},$
(19)
with
$\displaystyle|\vec{k}|$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M_{inv}^{2}(D\bar{D}),m_{D}^{2},m_{D}^{2})}{2M_{inv}(D\bar{D})},$
(20) $\displaystyle|\vec{p}|$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(s,M_{J/\psi}^{2},M_{inv}^{2}(D\bar{D}))}{2\sqrt{s}},$
(21)
where $T$ is given by
$T=C\frac{1}{M_{inv}^{2}(D\bar{D})-M_{X}^{2}+i\Gamma_{X}M_{X}}.$ (22)
As in daniee we restrain from giving absolute values but we can give relative
values with respect to $D\bar{D}$ production simply multiply $T$ of Eq. (22)
by $g_{X,\eta\eta^{\prime}}/\sqrt{2}g_{X,D^{+}D^{-}}$, where the factor
$\sqrt{2}$ will take into account in $|T|^{2}$ that we compare
$\eta\eta^{\prime}$ production versus $D^{+}D^{-}+D^{0}\bar{D}^{0}$
production.
Figure 6: The mass distribution of the final states $J/\psi\eta\eta^{\prime}$
compared to $J/\psi D\bar{D}$.
In Fig. 6 we show the results for $\eta\eta^{\prime}$ production with the same
scale as for $D\bar{D}$ production. We can see that the strength of the peak
is bigger for $\eta\eta^{\prime}$ production than for $D\bar{D}$, in spite of
having a smaller coupling to $X(3700)$. The reason is that the
$\eta\eta^{\prime}$ production is not suppressed by the threshold factors that
inhibit $D\bar{D}$ production. The peak seen in the $\eta\eta^{\prime}$ mass
spectrum is neat and the strength larger than for $D\bar{D}$ production. Since
$D\bar{D}$ has been observed in eebelle , this guarantees that the
$\eta\eta^{\prime}$ peak is within present measurable range.
## V Conclusions
In the present work, we have investigated some reactions by means of which one
could observe the predicted scalar meson formed as a quasibound state of
$D\bar{D}$. This state appears in analogy to the $f_{0}(500)$ and $f_{0}(980)$
states which are described within the chiral unitary approach as a $\pi\pi$
resonance and a quasibound $K\bar{K}$ state respectively. Some suggestion had
been made before to observe this state in the $\psi(3770)\to\gamma X(3700)$
decay by looking at the $\gamma$ energy distribution. Yet, this has the
inconvenience of having to observe a small peak in a large background. In
order to suppress the background we have chosen one of the main decay channels
of the $X(3700)$ state, the $\eta\eta^{\prime}$ channel, and suggest to look
at the $\eta\eta^{\prime}$ invariant mass distribution in the reaction
$\psi(3770)\to\gamma X(3700)\to\gamma\eta\eta^{\prime}$. Since BESIII already
can produce the $\psi(4040)$, we also suggest to look at the
$\psi(4040)\to\gamma X(3700)\to\gamma\eta\eta^{\prime}$ decay channel. A third
reaction was motivated by the only indirect experimental “evidence” of this
state. Indeed, in the BELLE reaction $e^{+}e^{-}\to J/\psi D\bar{D}$ eebelle ,
a peak was observed in the $D\bar{D}$ invariant mass distribution close to the
$D\bar{D}$ threshold, which was interpreted in daniee as a signal of a
$D\bar{D}$ resonance below the $D\bar{D}$ threshold. In the present work we
have suggested to look at the reaction $e^{+}e^{-}\to
J/\psi\eta\eta^{\prime}$, allowing the $X(3700)$ to be produced and decay into
$\eta\eta^{\prime}$.
We find clear peaks in all the invariant mass distributions of
$\eta\eta^{\prime}$. In the two radiative decays, the rates are within present
measurable range at BESIII, although in the case of $\psi(4040)$ radiative
decay the statistics with presently planned $\psi(4040)$ production would be
very low. In the case of the $e^{+}e^{-}$ reaction we do not evaluate absolute
cross sections and we find more instructive to compare the cross section of
the $e^{+}e^{-}\to J/\psi\eta\eta^{\prime}$ reaction with the one of
$e^{+}e^{-}\to J/\psi D\bar{D}$ already measured. We observe that the cross
section for the former reaction is bigger than for the latter one and produces
a clear peak that does not have the ambiguity of a threshold enhancement as in
the $e^{+}e^{-}\to J/\psi D\bar{D}$ reaction. This is the best guarantee that
the reaction is within measurable range.
The experimental search for this state is timely and its observation would
clarify issues concerning the interaction of hadrons in the charm sector,
which is not so well known as the non charmed one, and which would be much
welcome.
## Acknowledgements
This work is partly supported by DGICYT contract number FIS2011-28853-C02-01,
and the Generalitat Valenciana in the program Prometeo, 2009/090. We
acknowledge the support of the European Community-Research Infrastructure
Integrating Activity Study of Strongly Interacting Matter (acronym
HadronPhysics3, Grant Agreement n. 283286) under the Seventh Framework
Programme of the EU.
## References
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* (4) J. R. Pelaez and G. Rios, Phys. Rev. Lett. 97, 242002 (2006) [hep-ph/0610397].
* (5) C. Hanhart, J. R. Pelaez and G. Rios, Phys. Rev. Lett. 100, 152001 (2008) [arXiv:0801.2871 [hep-ph]].
* (6) J. Nieves and E. Ruiz Arriola, Phys. Lett. B 455, 30 (1999) [nucl-th/9807035].
* (7) J. Nieves and E. Ruiz Arriola, Phys. Rev. D 80, 045023 (2009) [arXiv:0904.4344 [hep-ph]].
* (8) S. Weinberg, Physica A 96, 327 (1979).
* (9) J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984).
* (10) D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, Phys. Rev. D 76, 074016 (2007) [hep-ph/0612179].
* (11) J. Nieves and M. P. Valderrama, Phys. Rev. D 86, 056004 (2012) [arXiv:1204.2790 [hep-ph]].
* (12) C. Hidalgo-Duque, J. Nieves and M. P. Valderrama, arXiv:1210.5431 [hep-ph].
* (13) D. Gamermann and E. Oset, Eur. Phys. J. A 36, 189 (2008) [arXiv:0712.1758 [hep-ph]].
* (14) P. Pakhlov et al. [Belle Collaboration], Phys. Rev. Lett. 100 (2008) 202001 [arXiv:0708.3812 [hep-ex]].
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* (16) J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012).
|
arxiv-papers
| 2012-11-08T14:22:50 |
2024-09-04T02:49:37.725499
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. W. Xiao and E. Oset",
"submitter": "Chuwen Xiao",
"url": "https://arxiv.org/abs/1211.1862"
}
|
1211.1935
|
# Electromagnetic momentum in a dielectric and the energy–momentum tensor
Michael E. Crenshaw US Army Aviation and Missile Research, Development, and
Engineering Center, Redstone Arsenal, AL 35898, USA
###### Abstract
The Abraham–Minkowski momentum controversy is the outwardly visible symptom of
an inconsistency in the use of the energy–momentum tensor in the case of a
plane quasimonochromatic field in a simple linear dielectric. We show that the
Gordon form of the electromagnetic momentum is conserved in a
thermodynamically closed system. We regard conservation of the components of
the four-momentum in a thermodynamically closed system as a fundamental
property of the energy–momentum tensor. Then the first row and column of the
energy–momentum tensor is populated by the electromagnetic energy density and
the Gordon momentum density. We derive new electromagnetic continuity
equations for the electromagnetic energy and momentum that are based on the
Gordon momentum density. These continuity equations can be represented in the
energy–momentum tensor using a material four-divergence operator in which
temporal differentiation is performed with respect to $ct/n$.
## I Introduction
The energy–momentum tensor is a concise way to represent the conservation
properties of a flow field. Simple in concept, the form of the energy–momentum
tensor for the field in a dielectric has been at the center of the century-
long Abraham-Minkowski momentum controversy BIPfeifer ; BIRL . The tensor that
was proposed by Minkowski BIMin in 1908 is not symmetric and consequently
does not conserve angular momentum BILL . Abraham BIAbr subsequently proposed
a symmetric tensor at the expense of a phenomenological force. The original
point of contention of the Abraham–Minkowski momentum controversy was whether
the Abraham momentum density
${\bf g}_{A}=\left({\bf E}\times{\bf H}\right)/c$ (1)
or the Minkowski momentum density
${\bf g}_{M}=\left({\bf D}\times{\bf B}\right)/c$ (2)
provides the correct description of the momentum of light in a linear medium.
The absence of an experimental decision allowed the debate to persist until
the 1960s, when resolution of the Abraham–Minkowski dilemma was provided by
Penfield and Haus BIPenHau , based on earlier work by Møller BIMol , who
showed that the issue is undecidable because neither the spatial integral of
the Abraham momentum density nor the spatial integral of the Minkowski
momentum density is the total momentum of the closed system. Likewise, the
total energy–momentum tensor of the closed system is not the Abraham
energy–momentum tensor and it is not the Minkowski energy–momentum tensor.
What, then, is the total momentum and the total energy–momentum tensor? Here,
we adopt the definition of the total momentum as the momentum quantity that is
conserved in a thermodynamically closed system. This approach has the
advantage of obtaining the total momentum directly from global conservation
principles and neatly avoids any issues regarding the ill-defined roles of the
Abraham and Minkowski momentums. We construct a thermodynamically closed
system consisting of a homogeneous dielectric block illuminated by a
quasimonochromatic field at normal incidence through an antireflection-
coating. In this closed system, the total energy and total momentum are
conserved quantities and are well-defined by virtue of being conserved. We
find that the total energy is the spatial integral of the electromagnetic
energy density
$\rho_{e}=\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf B}^{2}\right)$ (3)
and that the total momentum is the spatial integral of the Gordon BIGord
momentum density
${\bf g}_{G}=\left(n{\bf E}\times{\bf B}\right)/c.$ (4)
The four-momentum density is therefore $g_{\alpha}=(\rho_{e}/c,{\bf g}_{G})$.
We then show that the elements of the first row of the energy–momentum tensor
can be chosen to be energy and momentum densities that satisfy the requirement
for conservation of the components of the four-momentum. By doing so, the
four-divergence of the energy–momentum tensor produces an energy continuity
equation that is incompatible with Poynting’s theorem. Retaining the
conservation properties of the total energy–momentum tensor, we derive new
electromagnetic continuity equations. We define a material four-divergence
operator in which the temporal differentiation is performed with respect to
$ct/n$ and obtain a traceless symmetric energy–momentum tensor. The first-row
and first-column components of the tensor are the densities of conserved
energy and momentum quantities and the material four-divergence of the tensor
generates the new electromagnetic continuity equations in terms of the
electromagnetic energy density and the Gordon total momentum density.
## II The Total Energy and Total Momentum
The first step toward obtaining the total energy and total momentum is to
define a thermodynamically closed free-body system. We consider a stationary
dielectric block illuminated at normal incidence from vacuum by a plane
quasimonochromatic field. The dielectric has a gradient-index antireflection
coating that allows radiation pressure on the vacuum/dielectric interface to
be neglected. Then the refractive index $n$ for a linear, isotropic,
transparent dispersion-negligible dielectric is real and time-independent. The
dielectric is homogeneous and occupies a finite region of 3-dimensional space
so spatial variation of the refractive index, $n=n({\bf r})$, is limited to
the transition region of the coating. The quasimonochromatic electromagnetic
field is represented by the vector potential
${\bf A}({\bf r},t)=\frac{1}{2}\left({\bf\tilde{A}}({\bf
r},t)e^{-i(\omega_{d}t-{\bf k}_{d}\cdot{\bf r})}+{\bf\tilde{A}}^{*}({\bf
r},t)e^{i(\omega_{d}t-{\bf k}_{d}\cdot{\bf r})}\right)$ (5)
where $\tilde{A}$ is a slowly varying function of ${\bf r}$ and $t$,
$\omega_{d}$ is the center frequency of the field, ${\bf
k}_{d}=(n\omega_{d}/c){\bf e}_{\bf k}$, and ${\bf e}_{\bf k}$ is a unit vector
in the direction of propagation. Figure 1 shows the envelope of the incident
field
$\tilde{A}_{i}(z)=({\bf\tilde{A}}(z,t_{0})\cdot{\bf\tilde{A}}^{*}(z,t_{0}))^{1/2}$
about to enter the dielectric through a gradient-index antireflection coating.
Figure 2 shows a time-domain numerical solution of the wave equation at a
later time when the refracted field
$\tilde{A}_{r}(z)=({\bf\tilde{A}}(z,t_{1})\cdot{\bf\tilde{A}}^{*}(z,t_{1}))^{1/2}$
is entirely inside the dielectric. As shown in Fig.2, the amplitude of the
refracted field is ${\tilde{A}}_{r}={\tilde{A}}_{i}/\sqrt{n}$ and the spatial
extent of the refracted pulse is $w_{r}=w_{i}/n$ in terms of the width $w_{i}$
of the incident pulse. It can be shown that this linear result is quite
general in terms of the refractive index, as well as the width and amplitude
of the field.
For a stationary dielectric, the electromagnetic energy is
$U=\int_{\sigma}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf B}^{2}\right)dv,$ (6)
where the integration extends over all space $\sigma$. For the example shown
in Figs. 1 and 2, there is no significant field outside the rectangular pulse.
Then, the incident energy at $t=t_{0}$,
$U(t_{0})=\frac{\omega^{2}_{d}w_{i}}{2c^{2}}A_{i}^{2},$ (7)
is equal to the refracted energy at $t=t_{1}$,
$U(t_{1})=\frac{n^{2}\omega^{2}_{d}w_{r}}{2c^{2}}A_{r}^{2}=\frac{n^{2}\omega^{2}_{d}(w_{i}/n)}{2c^{2}}\left(\frac{A_{i}}{\sqrt{n}}\right)^{2}=U(t_{0}).$
(8)
The field has been averaged on a scale that is long compared to an optical
period, but short compared to $t_{1}-t_{0}$, accounting for a factor of one-
half. It can be demonstrated, in a similar manner, that the electromagnetic
energy at any time $t>t_{0}$ is equal to the incident energy. The temporal
invariance of the electromagnetic energy makes $U$, Eq. (6), the conserved
energy of the closed system.
Conservation of the electromagnetic energy is all that is needed to show that
the Gordon momentum
${\bf G}_{G}=\int_{\sigma}\frac{n{\bf E}\times{\bf B}}{c}dv,$ (9)
obtained by spatially integrating the Gordon momentum density BIGord , Eq.
(4), is the total momentum of our closed system. For monochromatic radiation,
where ${\bf B}=n{\bf E}$, we can write the momentum as
${\bf G}_{G}=\int_{\sigma}\frac{n^{2}{\bf E}^{2}}{c}{\bf\hat{e}}_{\bf
k}dv=\int_{\sigma}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf
B}^{2}\right)\frac{{\bf\hat{e}}_{\bf k}}{c}dv=\frac{U}{c}{\bf\hat{e}}_{\bf k}$
(10)
in the direction of the propagation unit vector ${\bf\hat{e}}_{\bf k}$. If the
total energy $U$ is temporally invariant, and therefore conserved, then so is
${\bf G}_{G}=({U}/{c}){\bf\hat{e}}_{\bf k}$. Alternately, we can show that the
momentum balance
${\bf
G}_{G}(t_{1})=\frac{n^{2}\omega^{2}_{d}w_{r}}{2c^{3}}A_{r}^{2}{\bf\hat{e}}_{\bf
k}=\frac{n^{2}\omega^{2}_{d}(w_{i}/n)}{2c^{3}}\left(\frac{A_{i}}{\sqrt{n}}\right)^{2}{\bf\hat{e}}_{\bf
k}={\bf G}_{G}(t_{0})$ (11)
is temporally invariant BICB . The momentum formula, Eq. (9), was originally
derived in 1973 by Gordon BIGord . Although there are some issues with
Gordon’s derivation, temporal invariance is decisive. Hence, the total
momentum in our closed system is given by Gordon’s formula, Eq. (9), and the
total momentum density is the Gordon momentum density ${\bf g}_{G}$, Eq. (4)
BICB .
## III The Energy–Momentum Tensor
The energy and momentum continuity equations of a thermodynamically closed
system can be combined to form a tensor differential equation and the
energy–momentum tensor is the central element of this formalism. While the
energy–momentum tensor formalism is straightforward, it has not been
successful in application to classical continuum electrodynamics. Now that we
have identified the total momentum by global conservation principles BICB , we
can use the apparatus of energy–momentum tensor theory to derive the
energy–momentum tensor. However, it is not that simple. The Abraham–Minkowski
momentum controversy arises from an incompatibility between two of the
properties of the energy–momentum tensor.
The energy–momentum tensor has a number of important properties. Here, we
employ the summation convention where identical indices on the same side of
the equation are summed over, Greek indices belong to $(0,1,2,3)$, and Roman
indices run from 1 to 3. Then, the four main properties of the energy–momentum
tensor $T^{\alpha\beta}$ are: i) The four-divergence operator
$\partial_{\alpha}=(c^{-1}\partial_{t},\partial_{x},\partial_{y},\partial_{z})$
applied to the rows of the tensor generates continuity equations
$\partial_{\alpha}T^{\alpha\beta}=0$ (12)
for the electromagnetic energy and the components of the momentum. ii)
Conservation of angular momentum requires diagonal symmetry
$T^{\alpha\beta}=T^{\beta\alpha}.$ (13)
iii) The array has a vanishing trace
$T^{\alpha}_{\alpha}=0$ (14)
corresponding to massless particles BIJackson ; BILL . iv) The components of
the four-momentum
$U=\int_{\sigma}dvT^{00}$ (15a) $G^{i}=\frac{1}{c}\int_{\sigma}dvT^{0i}$ (15b)
are conserved BILL in an unimpeded flow. Conditions iv) and ii) dictate the
elements of the first row and first column of the energy-momentum tensor, such
that
$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],$ (16)
where the elements of $W$ are yet to be specified. Applying condition i) to
the first row of the array in Eq. (16), we have
$\frac{1}{c}\frac{\partial\rho_{e}}{\partial t}=\nabla\cdot\left(n{\bf
E}\times{\bf B}\right),$ (17)
which is incompatible with the Poynting theorem
$\frac{\partial\rho_{e}}{\partial t}=\nabla\cdot c\left({\bf E}\times{\bf
H}\right).$ (18)
Alternatively, we can use Poynting’s theorem and the momentum continuity
equation
$\frac{\partial}{\partial t}\left({\bf D}\times{\bf
B}\right)/c+\nabla\cdot{\bf W}=0$ (19)
to populate the tensor
$T^{\alpha\beta}_{M}=\left[\begin{matrix}\rho_{e}&{c{g}_{\rm
A}}_{1}&{c{g}_{\rm A}}_{2}&{c{g}_{\rm A}}_{3}\cr{c{g}_{\rm
M}}_{1}&W_{11}&W_{12}&W_{13}\cr{c{g}_{\rm
M}}_{2}&W_{21}&W_{22}&W_{23}\cr{c{g}_{\rm
M}}_{3}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right],$ (20)
where
$W_{ij}=-E_{i}D_{j}-B_{i}B_{j}+\frac{1}{2}\left({\bf E}\cdot{\bf D}+{\bf
B}\cdot{\bf B}\right)\delta_{ij}$ (21)
is the Maxwell stress tensor. The resulting Minkowski energy–momentum tensor
violates condition iv), in addition to condition ii). Because we are in a
regime of negligible dispersion, we can rewrite the momentum continuity
equation, Eq. (19), as
$\frac{\partial}{\partial t}\left({\bf E}\times{\bf
H}\right)/c+\nabla\cdot{\bf W}=(1-n^{2})\frac{\partial}{\partial t}\left({\bf
E}\times{\bf B}\right)/c$ (22)
and obtain the Abraham energy–momentum tensor,
$T^{\alpha\beta}_{A}=\left[\begin{matrix}\rho_{e}&{c{g}_{\rm
A}}_{1}&{c{g}_{\rm A}}_{2}&{c{g}_{\rm A}}_{3}\cr{c{g}_{\rm
A}}_{1}&W_{11}&W_{12}&W_{13}\cr{c{g}_{\rm
A}}_{2}&W_{21}&W_{22}&W_{23}\cr{c{g}_{\rm
A}}_{3}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right].$ (23)
Our condition i) becomes
$\partial_{\alpha}T^{\alpha\beta}=f^{A}_{\alpha}.$ (24)
However, the Abraham force $f^{A}_{\alpha}=(0,c(1-n^{2})\partial_{t}{\bf
g}_{A})$ is a source or sink of momentum such that momentum is not conserved
in a closed system, also in violation of condition iv).
We have demonstrated an inconsistency in the energy–momentum tensor
formulation of classical continuum electrodynamics. Specifically we have
demonstrated that all of the properties of the energy–momentum tensor cannot
be simultaneously satisfied for electrodynamic fields in a dielectric, even if
the system is thermodynamically closed. The elements of the first row of the
energy–momentum tensor can be chosen such that the four-divergence of the
first row is the Poynting theorem or they can be chosen to satisfy
conservation of the components of the four-momentum, but not both.
In order to fully resolve the Abraham–Minkowski controversy, we must make the
properties of the energy–momentum tensor self-consistent when applied to
electromagnetic radiation in matter. Conservation of the momentum four-vector
is paramount for the total energy–momentum tensor of a thermodynamically
closed system. Likewise, conservation of angular momentum is required and we
affirm our tensor properties iv) and ii). Then the first row and column of the
tensor are populated by the densities of the electromagnetic energy density
and the Gordon total momentum density as shown in Eq. (16). These densities,
integrated over space, correspond to the conserved total quantities of energy
and momentum as shown in Eq. (15). Property iii) is not in question.
The problem that remains is a situation in which the four-divergence of the
total energy–momentum tensor, property i), is not consistent with the
electromagnetic continuity equations, and Poynting’s theorem in particular. We
can multiply Poynting’s theorem by $n({\bf r})$ and use a vector identity to
commute the refractive index with the divergence operator to obtain a
continuity equation
$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left(n{\bf
E}\times{\bf B}\right)=\frac{\nabla n}{n}\cdot\left(n{\bf E}\times{\bf
B}\right).$ (25)
The second-order energy continuity equation, Eq. (25), can be written as two
first-order equations
$\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}=\left(\nabla\times{\bf
B}\right)$ (26a) $\frac{n}{c}\frac{\partial{\bf B}}{\partial
t}=-\left(\nabla\times(n{\bf E})\right)+\frac{\nabla n}{n}\times(n{\bf E}).$
(26b)
Then, for a homogeneous dielectric, we can drop the term $\nabla n\times{\bf
E}$ and combine Eqs. (26) to obtain the energy and momentum continuity
equations
$\frac{n}{c}\frac{\partial}{\partial t}\left[\frac{1}{2}\left(n^{2}{\bf
E}^{2}+{\bf B}^{2}\right)\right]+\nabla\cdot\left(n{\bf E}\times{\bf
B}\right)=0$ (27a) $\frac{n}{c}\frac{\partial}{\partial t}\left(n{\bf
E}\times{\bf B}\right)+\nabla\cdot{\bf W}=0.$ (27b)
The elements of ${\bf W}$ are the elements of the stress tensor
$W_{ij}=-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}.$ (28)
We define a material four-divergence operator
$\bar{\partial}_{\alpha}=\left(\frac{n}{c}\frac{\partial}{\partial
t},\partial_{x},\partial_{y},\partial_{z}\right)$ (29)
and replace property i) with
$\bar{\partial}_{\alpha}T^{\alpha\beta}=0$ (30)
that generates Eqs. (27). Then the tensor, Eq. (16) is a traceless, symmetric
energy momentum tensor whose material four-divergence generates the new
electromagnetic energy and momentum continuity equations. It should be noted
that the new energy continuity equation, Eq. (27a), is mathematically
equivalent to Poynting’s theorem, Eq. (18) because the two are related by a
vector identity. Likewise the new momentum continuity equation, Eq. (27b), is
mathematically equivalent to the momentum continuity equation, Eq. (19) if the
radiation is sufficiently monochromatic and sufficiently far from any material
resonances that dispersion can be neglected.
## IV Summary
A transparent linear dielectric block in free space illuminated by a
quasimonochromatic field can be configured as an isolated system. However, the
partial reflection of the field at the surface causes a change in the momentum
of the field and the resulting radiation pressure accelerates the block. In
principle, we could write a complete set of equations of motion at the
microscopic level, but the effects of radiation pressure are not treated in a
complete way at the level of the macroscopic Maxwell equations. In order to
derive a theory of minimum complexity, we assumed a simple dielectric defined
to be linear, isotropic, homogeneous, transparent, and dispersionless with
negligible electrostrictive and magnetostrictive effects. The block is
stationary in free space and radiation pressure on the antireflection coating
is negligible. For this system, the elements of the energy–momentum tensor
$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]$ (31)
are the electromagnetic energy density $\rho_{e}=(n^{2}{\bf E}^{2}+{\bf
B}^{2})/2$, the Gordon momentum density ${\bf g}_{G}=(n{\bf E}\times{\bf
B})/c$ and the Maxwell stress tensor, Eq. (21), in the form
$W_{ij}=-nE_{i}nE_{j}-B_{i}B_{j}+\frac{1}{2}\left(n^{2}{\bf E}\cdot{\bf
E}+{\bf B}\cdot{\bf B}\right)\delta_{ij}$. The properties of the
energy–momentum tensor are: i) The material four-divergence operator
$\bar{\partial}_{\alpha}=(\frac{n}{c}\frac{\partial}{\partial_{t}},\frac{\partial}{\partial_{x}},\frac{\partial}{\partial_{y}},\frac{\partial}{\partial_{z}})$
(32)
applied to the rows of the tensor generates continuity equations
$\bar{\partial}_{\beta}T^{\alpha\beta}=0$ (33)
for the electromagnetic energy and momentum. The energy continuity equation
and the momentum continuity equation are mathematically equivalent to
Poynting’s theorem and the momentum continuity equation for a linear,
isotropic, homogeneous, transparent, dispersionless dielectric. ii)
Conservation of angular momentum requires diagonal symmetry
$T^{\alpha\beta}=T^{\beta\alpha}.$ (34)
iii) The array has a vanishing trace
$T^{\alpha}_{\alpha}=0$ (35)
corresponding to massless particles BIJackson ; BILL . iv) The components of
the four-momentum
$U=\int_{\sigma}dvT^{00}$ (36a) $G^{i}=\frac{1}{c}\int_{\sigma}dvT^{0i}$ (36b)
are conserved BILL in an unimpeded flow.
## References
* (1) For a recent review, see: Pfeifer, R. N. C., Nieminen, T. A., Heckenberg, N. R., and Rubinsztein-Dunlop, H., “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
* (2) Griffiths, D. J., “Resource Letter EM-1: Electromagnetic Momentum,” Am. J. Phys. 80, 7–18 (2012).
* (3) Minkowski, H., Natches. Ges. Wiss. Gättaining, 53, (1908).
* (4) Landau, L. D. and Lifshitz, E. M. The Classical Theory of Fields, 4th. ed., Elsevier (2006).
* (5) Abraham, M. Rend. Circ. Mat. Palermo, 28, 1 (1909).
* (6) Penfield, Jr., P., and Haus, H. A., Electrodynamics of Moving Media, MIT Press (1967).
* (7) Møller, C., The Theory of Relativity, 2nd ed., Oxford University Press (1972).
* (8) Gordon, J. P. “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14-21 (1973).
* (9) Crenshaw, M. E. and Bahder, T. B. “Energy–momentum tensor for the electromagnetic field in a dielectric,” Opt. Comm. 284, 2460–2465 (2011).
* (10) Jackson, J. D., Classical Electrodynamics, 2nd ed., Wiley, (1975).
|
arxiv-papers
| 2012-11-08T18:31:58 |
2024-09-04T02:49:37.733645
|
{
"license": "Public Domain",
"authors": "Michael E. Crenshaw",
"submitter": "Michael Crenshaw",
"url": "https://arxiv.org/abs/1211.1935"
}
|
1211.1993
|
# Quasiconvexity and Relatively Hyperbolic Groups that Split
Hadi Bigdely Dept. of Math & Stats.
McGill University
Montreal, Quebec, Canada H3A 2K6 bigdely@math.mcgill.ca and Daniel T. Wise
wise@math.mcgill.ca
###### Abstract.
We explore the combination theorem for a group $G$ splitting as a graph of
relatively hyperbolic groups. Using the fine graph approach to relative
hyperbolicity, we find short proofs of the relative hyperbolicity of $G$ under
certain conditions. We then provide a criterion for the relative
quasiconvexity of a subgroup $H$ depending on the relative quasiconvexity of
the intersection of $H$ with the vertex groups of $G$. We give an application
towards local relative quasiconvexity.
###### Key words and phrases:
relatively hyperbolic, relatively quasiconvex, graphs of groups
###### 2000 Mathematics Subject Classification:
20F06,
Research supported by NSERC
The goal of this paper is to examine relative hyperbolicity and quasiconvexity
in graphs of relatively hyperbolic vertex groups with almost malnormal
quasiconvex edge groups. The paper hinges upon the observation that if $G$
splits as a graph of relatively hyperbolic groups with malnormal relatively
quasiconvex edge groups, then a fine hyperbolic graph for $G$ can be built
from fine hyperbolic graphs for the vertex groups. This leads to short proofs
of the relative hyperbolicity of $G$ as well as to a concise criterion for the
relative quasiconvexity of a subgroup $H$ of $G$.
Bestvina and Feighn proved a combination theorem that characterized the
hyperbolicity of groups splitting as graphs of hyperbolic groups [2]. Their
geometric characterization is akin to the flat plane theorem characterization
of hyperbolicity for actions on CAT(0) spaces, and leads to explicit positive
results, especially in an “acylindrical” scenario where some form of
malnormality is imposed on the edge groups. The Bestvina-Feighn combination
theorem has been revisited multiple times in a hyperbolic setting, and more
recently in a relatively hyperbolic context but through diverse methods.
Dahmani proved a combination theorem for relatively hyperbolic groups using
the convergence group approach [4]. Later Alibegović proved similar results in
[1] using a method generalizing parts of the Bestvina-Feighn approach. Osin
reproved Dahman’s result in the general context of relative Dehn functions
[19]. Most recently, Mj and Reeves gave a generalization of the Bestvina-
Feighn combination theorem that follows Farb’s approach but uses a generalized
“partial electrocution” [17]. Their result appears to be a far-reaching
generalization at the expense of complex geometric language.
Our own results revisit these relatively hyperbolic generalizations, and we
offer a very concrete approach employing Bowditch’s fine hyperbolic graphs.
The most natural formulation of our main combination theorem (proven as
Theorem 1.4) is as follows:
###### Theorem A (Combining Relatively Hyperbolic Groups Along Parabolics).
Let $G$ split as a finite graph of groups. Suppose each vertex group is
relatively hyperbolic and each edge group is parabolic in its vertex groups.
Then $G$ is hyperbolic relative to $\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$ where
each $Q_{i}$ is the stabilizer of a “parabolic tree”. (See Definition 1.3.)
A simplistic example illustrating Theorem A is an amalgamated product
$G=G_{1}*_{C}G_{2}$ where each $G_{i}=\pi_{1}M_{i}$ and $M_{i}$ is a cusped
hyperbolic manifold with a single boundary torus $T_{i}$. And $C$ is an
arbitrary common subgroup of $\pi_{1}T_{1}$ and $\pi_{1}T_{2}$. Then $G$ is
hyperbolic relative to $\pi_{1}T_{1}*_{C}\pi_{1}T_{2}$.
We note that Theorem A is more general than results in the same spirit that
were obtained by Dahmani, Alibegović, and Osin. In particular, they require
that edge groups be maximal parabolic on at least one side, but we do not. We
believe that Theorem A could be deduced from the results of Mj-Reeves.
In Section 4, we employ work of Yang [22] on extended peripheral structures,
to obtain the following seemingly more natural corollary of Theorem A which is
proven as Corollary 4.6:
###### Corollary B.
Let $G$ split as a finite graph of groups. Suppose
* (a)
Each $G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$;
* (b)
Each $G_{e}$ is total and relatively quasiconvex in $G_{\nu}$;
* (c)
$\\{G_{e}:e~{}\textrm{is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.
Then $G$ is hyperbolic relative to
$\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$.
The “omitted repeats” in the conclusion of Corollary B refer to (some of) the
parabolic subgroups of vertex groups that are identified through an edge
group.
It is not clear whether Corollary B could be obtained using the method of
Dahmani, Alibegović, or Osin. However, we suspect it could be extracted from
the result of Mj-Reeves.
###### Definition 0.1.
(Tamely generated) Let $G$ split as a graph of groups with relatively
hyperbolic vertex groups. A subgroup $H$ is _tamely generated_ if the induced
graph of groups $\Gamma_{H}$ has a $\pi_{1}$-isomorphic subgraph of groups
$\Gamma^{\prime}_{H}$ that is a finite graph of groups each of whose vertex
groups is relatively quasiconvex in the corresponding vertex group of $G$.
Note that $H$ is tamely generated when $H$ is finitely generated and there are
finitely many $H$-orbits of vertices $v$ in $T$ with $H_{v}$ nontrivial, and
each such $H_{v}$ is relatively quasiconvex in $G_{v}$. However the above
condition is not necessary. For instance, let $G=F_{2}\times\mathbb{Z}_{2}$,
and consider a splitting where $\Gamma$ is a bouquet of two circles, and each
vertex and edge group is isomorphic to $\mathbb{Z}_{2}$. Then every f.g.
subgroup $H$ of $F_{2}\times\mathbb{Z}_{2}$ is tamely generated, but no
subgroup containing $\mathbb{Z}_{2}$ satisfies the condition that there are
finitely many $H$-orbits of vertices $\omega$ with $H_{\omega}$ nontrivial.
The geometric construction proving Theorem A allows us to give a simple
criterion for quasiconvexity of a subgroup $H$ relative to $\mathbb{Q}$.
Again, coupling this with Yang’s work, we obtain (as Theorem 4.13) the
following criterion for quasiconvexity relative to $\mathbb{P}$:
###### Main Theorem C (Quasiconvexity Criterion).
Let $G$ be hyperbolic relative to $\mathbb{P}$ where each $P\in\mathbb{P}$ is
finitely generated. Suppose $G$ splits as a finite graph of groups. Suppose
1. (a)
Each $G_{e}$ is total in $G$;
2. (b)
Each $G_{e}$ is relatively quasiconvex in $G$;
3. (c)
Each $G_{e}$ is almost malnormal in $G$.
Let $H\leq G$ be tamely generated. Then $H$ is relatively quasiconvex in $G$.
Recall that $G$ is _locally relatively quasiconvex_ if each finitely generated
subgroup $H$ of $G$ is quasiconvex relative to the peripheral structure of
$G$. I. Kapovich first recognized that hyperbolic limit groups are locally
relatively quasiconvex [12], and subsequently Dahmani proved that all limit
groups are locally relatively quasiconvex [4].
A group $P$ is _small_ if there is no embedding $F_{2}\hookrightarrow P$, and
$G$ has a _small hierarchy_ if it can be built from small subgroups by a
sequence of AFP’s and HNN’s along small subgroups (see Definition 3.4). When
$\mathbb{P}$ is a collection of free-abelian groups, the following inductive
consequence of Corollary 3.3 generalizes Dahmani’s result.
###### Theorem D.
Let $G$ be hyperbolic relative to a collection of Noetherian subgroups
$\mathbb{P}$ and suppose $G$ has a small hierarchy. Then $G$ is locally
relatively quasiconvex.
Although Theorem D is implicit in Dahmani’s work, we believe Theorem C is new.
Figure 1. A fine graph $K_{G}$ for $G=A*_{C}B$ is built from copies of fine
graphs $K_{A}$ and $K_{B}$ for $A$ and $B$ by gluing new edges together along
vertices stabilized by $C$. The parabolic trees of $T$ are images of trees
formed from the new edges in $K_{G}$. We obtain a fine hyperbolic graph
$\bar{K}_{G}$ with finite edge stabilizers as a quotient
$K_{G}\rightarrow\bar{K}_{G}$.
The main construction and its application: Although we work in somewhat
greater generality, let us focus on the simple case of an amalgamated product
$G=A*_{C}B$ where $A,B$ are relatively hyperbolic and $C$ is parabolic on each
side. The central theme of this paper is a construction that builds a fine
hyperbolic graph $\bar{K}_{G}$ for $G$ from fine hyperbolic graphs $K_{A}$ and
$K_{B}$ for $A,B$. (See Figure 1.) This is done in two steps: Guided by the
Bass-Serre tree, we first construct a graph $K_{G}$ which is a tree of spaces
whose vertex spaces are copies of $K_{A}$ and $K_{B}$, and whose edge spaces
are ordinary edges. Though $K_{G}$ is fine and hyperbolic, its edges have
infinite stabilizers. We remedy this by quotienting these edge spaces to form
the fine hyperbolic graph $\bar{K}_{G}$. The vertices of $\bar{K}_{G}$ are
quotients of “parabolic trees” in $K_{G}$. The fine hyperbolic graph
$\bar{K}_{G}$ quickly proves that $G$ is hyperbolic relative to the collection
$\mathbb{Q}$ of subgroups stabilizing parabolic trees. Variations on the
construction, hypotheses on the edge groups, and interplay with previous work
on peripheral structures, leads to a variety of relatively hyperbolic
conclusions. The simplest and most immediate in the case above, is that $G$ is
hyperbolic relative to
$\mathbb{P}_{G}=\mathbb{P}_{A}\cup\mathbb{P}_{B}-\\{C\\}$ when $C$ is maximal
parabolic on each side and $A,B$ are hyperbolic relative to $\mathbb{P}_{A}$
and $\mathbb{P}_{B}$.
Our primary application is to give an easy criterion to recognize
quasiconvexity. A subgroup $H$ is relatively quasiconvex in $G$ if there is an
$H$-cocompact quasiconvex subgraph $\bar{L}\subset\bar{K}_{G}$ of the fine
hyperbolic $G$-graph. The tree-like nature of our graph $\bar{K}_{G}$, permits
us to naturally build the quasiconvex $H$-graph $\bar{L}$. When $H$ is
relatively quasiconvex, there are finitely many $H$-orbits of nontrivially
$H$-stabilized vertices in the Bass-Serre tree $T$, and each of these
stabilizers is relatively quasiconvex in its vertex group. Choosing finitely
many quasiconvex subgraphs in the corresponding copies of $K_{A}$ and $K_{B}$,
we are able to combine these together to form $L$ in $K_{G}$ and then to form
a quasiconvex $H$-subgraph $\bar{L}$ in $\bar{K}_{G}$.
We conclude by mentioning the following consequence of Corollary 1.5 that is a
natural consequence of the viewpoint developed in this paper.
###### Corollary E.
Let $M$ be a compact irreducible 3-manifold. And let $M_{1},\ldots,M_{r}$
denote the graph manifolds obtained by removing each (open) hyperbolic piece
in the geometric decomposition of $M$. Then $\pi_{1}M$ is hyperbolic relative
to $\\{\pi_{1}M_{1},\ldots,\pi_{1}M_{r}\\}$.
As explained to us by the referee, the relative hyperbolicity of $\pi_{1}(M)$
was previously proved by Drutu-Sapir using work of Kapovich-Leeb. This
previous proof is deep as it uses the structure of the asymptotic cone due to
Kapovich-Leeb together with the technical proof of Drutu-Sapir that
asymptotically tree graded groups are relatively hyperbolic [13, 5].
## 1\. Combining Relatively Hyperbolic Groups along Parabolics
The class of relatively hyperbolic groups was introduced by Gromov [8] as a
generalization of the class of fundamental groups of complete finite-volume
manifolds of pinched negative sectional curvature. Various approaches to
relative hyperbolicity were developed by Farb [6], Bowditch [3] and Osin [20],
and as surveyed by Hruska [10], these notions are equivalent for finitely
generated groups. We follow Bowditch’s approach:
###### Definition 1.1 (Relatively Hyperbolic).
A circuit in a graph is an embedded cycle. A graph $\Gamma$ is fine if each
edge of $\Gamma$ lies in finitely many circuits of length $n$ for each $n$.
A group G is _hyperbolic relative to a finite collection of subgroups_
$\mathbb{P}$ if $G$ acts cocompactly(without inversions) on a connected, fine,
hyperbolic graph $\Gamma$ with finite edge stabilizers, such that each element
of $\mathbb{P}$ equals the stabilizer of a vertex of $\Gamma$, and moreover,
each infinite vertex stabilizer is conjugate to a unique element of
$\mathbb{P}$. We refer to a connected, fine, hyperbolic graph $\Gamma$
equipped with such an action as a _$(G;\mathbb{P})$ -graph_. Subgroups of G
that are conjugate into subgroups in $\mathbb{P}$ are _parabolic_.
###### Technical Remark 1.2.
Given a finite collection of parabolic subgroups $\\{A_{1},\dots,A_{r}\\}$, we
choose $\mathbb{P}$ so that there is a prescribed choice of parabolic subgroup
$P_{i}\in\mathbb{P}$ so that $A_{i}$ is _“declared”_ to be conjugate into
$P_{i}$. This is automatic for an infinite parabolic subgroup $A$ but for
finite subgroups there could be ambiguity. One way to resolve this is to
revise the choice of $\mathbb{P}$ as follows: For any finite collection of
parabolic subgroups $\\{A_{1},\dots,A_{r}\\}$ in $G$, we moreover assume each
$A_{i}$ is conjugate to a subgroup of $\mathbb{P}$ and we assume that no two
(finite) subgroups in $\mathbb{P}$ are conjugate. We note that finite
subgroups can be freely added to or omitted from the peripheral structure of
$G$ (see e.g. [16]).
###### Definition 1.3 (Parabolic tree).
Let $G$ split as a finite graph of groups where each vertex group $G_{\nu}$ is
hyperbolic relative to $\mathbb{P}_{\nu}$, and where each edge group $G_{e}$
embeds as a parabolic subgroup of its two vertex groups. Let $T$ be the Bass-
Serre tree. Define the _parabolic forest_ $F$ by:
1. (1)
A _vertex_ in $F$ is a pair $(u,P)$ where $u\in T^{0}$ and $P$ is a
$G_{u}$-conjugate of an element of $\mathbb{P}_{u}$.
2. (2)
An _edge_ in $F$ is a pair $(e,G_{e})$ where $e$ is an edge of $T$ and $G_{e}$
is its stabilizer.
3. (3)
The edge $(e,G_{e})$ is _attached_ to $(\iota(e),\iota(P_{e}))$ and
$(\tau(e),\tau(P_{e}))$ where $\iota(e)$ and $\tau(e)$ are the initial and
terminal vertex of $e$ and $\iota(P_{e})$ is the $G_{\iota(e)}$-conjugate of
an element of $\mathbb{P}$ that is declared to contain $G_{e}$. Likewise for
$(\tau(e),\tau(P_{e}))$. We arranged for this unique determination in
Technical Remark 1.2.
Each component of $F$ is a _parabolic tree_ and the map $F\rightarrow T$ is
injective on the set of edges, and in particular each parabolic tree embeds in
$T$. Let $S_{1},\dots,S_{j}$ be representatives of the finitely many orbits of
parabolic trees under the $G$ action on $F$. Let $Q_{i}=stab(S_{i})$, for each
$i$.
###### Theorem 1.4 (Combining Relatively Hyperbolic Groups Along Parabolics).
Let $G$ split as a finite graph $\Gamma$ of groups. Suppose each vertex group
is relatively hyperbolic and each edge group is parabolic in its vertex
groups. Then $G$ is hyperbolic relative to
$\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$.
###### Proof.
For $u\in\Gamma^{0}$, let $G_{u}$ be hyperbolic relative to $\mathbb{P}_{u}$
and let $K_{u}$ be a $(G_{u};\mathbb{P}_{u})$-graph. For each
$P\in\mathbb{P}_{u}$, following the Technical Remark 1.2, we choose a specific
vertex of $K_{u}$ whose stabilizer equals $P$. Note that, in general there
could be more than one possible choice when $|P|<\infty$, but by Technical
Remark 1.2 we have a unique choice. Translating determines a “choice” of
vertex for conjugates.
We now construct a $(G;\mathbb{Q})$-graph $\bar{K}$. Let $K$ be the tree of
spaces whose underlying tree is the Bass-Serre tree $T$ with the following
properties:
1. (1)
Vertex spaces of $K$ are copies of appropriate elements in
$\\{K_{u}:u\in\Gamma^{0}\\}$. Specifically, $K_{\nu}$ is a copy of $K_{u}$
where $u$ is the image of $\nu$ under $T\rightarrow\Gamma$.
2. (2)
Each edge space $K_{e}$ is an ordinary edge, denoted as an ordered pair
$(e,G_{e})$ that is attached to the vertices in $K_{\iota(e)}$ and
$K_{\tau(e)}$ that were chosen to contain $G_{e}$.
Note that each $G_{\nu}$ acts on $K_{\nu}$ and there is a $G$-equivariant map
$K\rightarrow T$. Let $\bar{K}$ be the quotient of $K$ obtained by contracting
each edge space. Observe that $G$ acts on $\bar{K}$ and there is a
$G$-equivariant map $K\rightarrow\bar{K}$. Moreover the preimage of each open
edge of $\bar{K}$ is a single open edge of $K$.
We now show that $\bar{K}$ is a $(G;\mathbb{Q})$-graph. Since any embedded
cycle lies in some vertex space, the graph $\bar{K}$ is fine and hyperbolic.
There are finitely many orbits of vertices in $K$ and therefore finitely many
orbits of vertices in $\bar{K}$. Likewise, there are finitely many orbits of
edges in $\bar{K}$. The stabilizer of an (open) edge of $\bar{K}$ equals the
stabilizer of the corresponding (open) edge in $K$, and is thus finite. By
construction, there is a $G$-equivariant embedding $F\hookrightarrow K$ where
$F$ is the parabolic forest associated to $G$ and $T$. Finally, the preimage
in $K$ of a vertex of $\bar{K}$ is precisely a parabolic tree and thus the
stabilizer of a vertex of $\bar{K}$ is a conjugate of some $Q_{j}$. ∎
We now examine some conclusions that arise when the parabolic trees are small.
An extreme case arises when the edge groups are isolated from each other as
follows:
###### Corollary 1.5.
Let $G$ split as a finite directed graph of groups where each vertex group
$G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$. Suppose that:
1. (1)
Each edge group is parabolic in its vertex groups.
2. (2)
Each outgoing infinite edge group $G_{\vec{e}}$ is maximal parabolic in its
initial vertex group $G_{\nu}$ and for each other incoming and outgoing
infinite edge group $G_{{}\reflectbox{$\vec{\reflectbox{$e$}}$}}$ or
$G_{\vec{d}}$ or $G_{\reflectbox{$\vec{\reflectbox{$d$}}$}}$, none of its
conjugates lie in $G_{\vec{e}}$.
Then $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{outgoing edge groups}\\}$.
###### Proof.
We can arrange for finitely stabilized edges of $F$ to be attached to distinct
chosen vertices when they correspond to distinct edges of $T$. Thus, parabolic
trees are singletons and/or $i$-pods consisting of edges that all terminate at
the same vertex $\\{(\nu,P^{g})\\}$ where $P\in\mathbb{P}_{\nu}$ and $g\in
G_{\nu}$. Recall that an _$i$ -pod_ is a tree consisting of $i$ edges glued to
a central vertex. ∎
###### Corollary 1.6.
Let $G$ split as a finite graph of groups. Suppose each vertex group $G_{\nu}$
is hyperbolic relative to $\mathbb{P}_{\nu}$. For each $G_{\nu}$ assume that
the collection $\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is a collection of
maximal parabolic subgroups of $G_{\nu}$. Then $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$. Specifically,
we remove an element of $\bigcup_{\nu}\mathbb{P}_{\nu}$ if it is conjugate to
another one.
The first two of the following cases were treated by Dahmani, Alibegović, and
Osin [1, 4, 19]:
###### Corollary 1.7.
1. (1)
Let $G_{1}$ and $G_{2}$ be hyperbolic relative to $\mathbb{P}_{1}$ and
$\mathbb{P}_{2}$. Let $G=G_{1}\ast_{P_{1}={P_{2}^{\prime}}}G_{2}$ where each
$P_{i}\in\mathbb{P}_{i}$ and $P_{1}$ is identified with the subgroup
${P_{2}}^{\prime}$ of $P_{2}$. Then $G$ is hyperbolic relative
$\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$.
2. (2)
Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $P_{1}\in\mathbb{P}$
be isomorphic to a subgroup ${P_{2}}^{\prime}$ of a maximal parabolic subgroup
$P_{2}$ not conjugate to $P_{1}$. Let
$G=G_{1}*_{{P_{1}}^{t}={P_{2}}^{\prime}}$ where ${P_{1}}^{t}=t^{-1}P_{1}t$.
Then $G$ is hyperbolic relative to $\mathbb{P}-\\{P_{1}\\}$.
3. (3)
Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $P\in\mathbb{P}$ be
isomorphic to ${P}^{\prime}\leq P$. Let $G=G_{1}*_{{P}^{t}={P}^{\prime}}$.
Then $G$ is hyperbolic relative to $\mathbb{P}\cup\langle P,t\rangle-\\{P\\}$.
###### Remark 1.8.
Note that in this Corollary and some similar results when we say
$P_{i}\in\mathbb{P}_{i}$, we mean if $P^{g}_{i}\in\mathbb{P}_{i}$ then replace
$P^{g}_{i}$ by $P_{i}$ in $\mathbb{P}_{i}$.
###### Proof.
(1): In this case, the parabolic trees are either singletons stabilized by a
conjugate of an element of $\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$, or
parabolic trees are $i$-pods stabilized by conjugates of $P_{2}$.
(2): The proof is similar.
(3): All parabolic trees are singletons except for those that are translates
of a copy of the Bass-Serre tree for $P*_{P^{t}=P^{\prime}}$. Following the
proof of Theorem 1.4, let $\nu\in\bar{K}$, if the preimage of $\nu$ in $K$ is
not attached to an edge space, then $G_{\nu}$ is conjugate to an element of
$\mathbb{P}-\\{P\\}$, otherwise $G_{\nu}$ is conjugate to $\langle
P,t\rangle$. ∎
###### Example 1.9.
We encourage the reader to consider the case of Theorem 1.4 and Corollaries
1.6 and 1.7, in the scenario where $G$ splits as a graph of free groups with
cyclic edge groups. A very simple case is: Let $G=\langle
a,b,t~{}|~{}(W^{n})^{t}=W^{m}\rangle$ where $W\in\langle a,b\rangle$ and
$m,n\geq 1$. Then $G$ is hyperbolic relative to $\langle W,t\rangle$.
## 2\. Relative Quasiconvexity
Dahmani introduced the notion of relatively quasiconvex subgroup in [4]. This
notion was further developed by Osin in [20], and later Hruska investigated
several equivalent definitions of relatively quasiconvex subgroups [10].
Martinez-Pedroza and the second author introduced a definition of relative
quasiconvexity in the context of fine hyperbolic graphs and showed this
definition is equivalent to Osin’s definition [16]. We will study relatively
quasiconvexity using this fine hyperbolic viewpoint. Our aim is to examine the
relative quasiconvexity of a certain subgroup which are themselves amalgams,
and we note that powerful results in this direction are given in [15].
###### Definition 2.1 (Relatively Quasiconvex).
Let $G$ be hyperbolic relative to $\mathbb{P}$. A subgroup $H$ of $G$ is
_quasiconvex relative to $\mathbb{P}$_ if for some (and hence any)
$(G;\mathbb{P})$-graph $K$, there is a nonempty connected and quasi-
isometrically embedded, $H$-cocompact subgraph $L$ of $K$. In the sequel, we
sometimes refer to $L$ as a _quasiconvex $H$-cocompact subgraph_ of $K$.
###### Remark 2.2.
It is immediate from the Definition 2.1 that in a relatively hyperbolic group,
any parabolic subgroup is relatively quasiconvex, and any relatively
quasiconvex subgroup is also relatively hyperbolic. In particular, the
relatively quasiconvex subgroup $H$ is hyperbolic relative to the collection
$\mathbb{P}_{H}$ consisting of representatives of $H$-stabilizers of vertices
of $L\subseteq K$. Note that a conjugate of a relatively quasiconvex subgroup
is also relatively quasiconvex. And the intersection of two relatively
quasiconvex subgroups is relatively quasiconvex. Specifically, this last
statement was proven when $G$ is f.g. in [15], and when $G$ is countable in
[10].
Relative quasiconvexity has the following transitive property proven by Hruska
for countable relatively hyperbolic groups in [10]:
###### Lemma 2.3.
Let $G$ be hyperbolic relative to $\mathbb{P}_{G}$. Suppose that $B$ is
relatively quasiconvex in $G$, and note that $B$ is then hyperbolic relative
to $\mathbb{P}_{B}$ as in Remark 2.2. Then $A\leq B$ is quasiconvex relative
to $\mathbb{P}_{B}$ if and only if $A$ is quasiconvex relative to
$\mathbb{P}_{G}$.
###### Proof.
Let $K$ be a $(G;\mathbb{P}_{G})$-graph. As $B$ is quasiconvex relative to
$\mathbb{P}_{G}$, there is a $B$-cocompact and quasiconvex subgraph $L\subset
K$. Note that $L$ is a $(B;\mathbb{P}_{B})$-graph. Let $A\leq B$.
If $A$ is quasiconvex in $B$ relative to $\mathbb{P}_{B}$, there is an
$A$-cocompact quasiconvex subgraph $M\subset L$. Since the composition
$L_{A}\rightarrow L_{B}\rightarrow K$ is a quasi-isometric embedding, $A$ is
quasiconvex relative to $\mathbb{P}_{G}$. Conversely, if $A$ is quasiconvex in
$G$ relative to $\mathbb{P}_{G}$, then there is an $A$-cocompact quasiconvex
subgraph $M\subset K$. Let $L^{\prime}=L\cup BM$ and note that $L^{\prime}$ is
$B$-cocompact and hence also quasiconvex, and thus $L^{\prime}$ also serves as
a fine hyperbolic graph for $B$. Now $M\subset L^{\prime}$ is quasiconvex
since $M\subset L$ is quasiconvex so $A$ is relatively quasiconvex in $B$. ∎
###### Remark 2.4.
One consequence of Theorem 1.4 and its various Corollaries, is that when $G$
splits as a graph of relatively hyperbolic groups with parabolic subgroups,
then each of the vertex groups is quasiconvex relative to the peripheral
structure of $G$. (For Theorem 1.4 this is $\mathbb{Q}$, and for Corollary 1.6
this is $\mathbb{P}-\\{\text{repeats}\\}$.) Indeed, $K_{v}$ is a
$G_{v}$-cocompact quasiconvex subgraph in the fine graph $K$ constructed in
the proof.
###### Lemma 2.5.
Let $G$ be a f.g. group that split as a finite graph of groups $\Gamma$. If
each edge group is f.g. then each vertex group is f.g.
###### Proof.
Let $G=\langle g_{1},\dots,g_{n}\rangle$. We regard $G$ as $\pi_{1}$ of a
$2$-complex corresponding to $\Gamma$. We show that each vertex group $G_{v}$
equals $\langle\\{G_{e}\\}_{\text{$e$ attached to~{}$v$}}\cup\\{g\in
G_{v}:g~{}\text{in normal form of some}~{}g_{i}\\}\rangle$. Let $a\in G_{v}$
and consider an expression of $a$ as a product of normal forms of the
$g_{i}^{\pm 1}$. Then $a$ equals some product
$a_{1}{t_{1}}^{\epsilon_{1}}b_{1}{t_{2}}^{\epsilon_{2}}a_{2}\cdots
a_{n}{t_{m}}^{\epsilon_{m}}b_{k}$. There is a disc diagram $D$ whose boundary
path is
$a^{-1}a_{1}{t_{1}}^{\epsilon_{1}}b_{1}{t_{2}}^{\epsilon_{2}}a_{2}\cdots
a_{n}{t_{m}}^{\epsilon_{m}}b_{k}$. See Figure 2. The region of $D$ that lies
along $a$ shows that $a$ equals the product of elements in edge groups
adjacent to $G_{v}$, together with elements of $G_{v}$ that lie in the normal
forms of $g_{1},\dots,g_{n}$. ∎
Figure 2.
###### Theorem 2.6 (Quasiconvexity of a Subgroup in Parabolic Splitting).
Let $G$ split as a finite graph $\Gamma$ of relatively hyperbolic groups such
that each edge group is parabolic in its vertex groups. (Note that $G$ is
hyperbolic relative to $\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$ by Theorem 1.4.)
Let $H\leq G$ be tamely generated. Then $H$ is quasiconvex relative to
$\mathbb{Q}$. Moreover if each $H_{v}$ in the Bass-Serre tree $T$ is finitely
generated then $H$ is finitely generated.
###### Proof.
Since there are finitely many orbits of vertices whose stabilizers are
finitely generated, $H$ is finitely generated. For each $u\in\Gamma^{0}$, let
$G_{u}$ be hyperbolic relative to $\mathbb{P}_{u}$ and let $K_{u}$ be a
$(G_{u};\mathbb{P}_{u})$-graph. Let $K$ be the $(G;\mathbb{Q})$-graph
constructed in the proof of Theorem 1.4 and let $\bar{K}$ be its quotient. We
will construct an $H$-cocompact quasiconvex, connected subgraph $\bar{L}$ of
$\bar{K}$.
Let $T_{H}$ be the minimal $H$-invariant subgraph of $T$. Recall that each
edge of $T$ (and hence $T_{H}$) corresponds to an edge of $K$. Let $F_{H}$
denote the subgraph of $K$ that is the union of all edges correspond to edges
of $T_{H}$. Let $\\{\nu_{1},\dots,\nu_{n}\\}$ be a representatives of
$H$-orbits of vertices of $T_{H}$. For each $i$, let $L_{i}\hookrightarrow
K_{\nu_{i}}$ be a $(H\cap G_{\nu_{i}}^{g_{i}})$-cocompact quasiconvex subgraph
such that $L_{i}$ contains $F_{H}\cap K_{\nu_{i}}$. (There are finitely many
$(H\cap G_{\nu_{i}}^{g_{i}})$-orbits of such endpoints of edges in
$K_{\nu_{i}}$.) Let $L=F_{H}\cup\bigcup_{i=1}^{n}HL_{i}$ and let $\bar{L}$ be
the image of $L$ under $K\rightarrow\bar{K}$. Observe that $L$ is quasiconvex
in $K$ since $K$ is a “tree union” and each such $L_{i}$ of $L$ is quasiconvex
in $K_{\nu_{i}}$. And likewise, $\bar{L}$ is quasiconvex in $\bar{K}$. ∎
###### Corollary 2.7 (Characterizing Quasiconvexity in Maximal Parabolic
Splitting).
Let $G$ split as a finite graph of countable groups. For each $\nu$, let
$G_{\nu}$ be hyperbolic relative to $\mathbb{P}_{\nu}$ and let the collection
$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ be a collection of maximal
parabolic subgroups of $G_{\nu}$. $($Note that $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$ by Corollary
1.6.$)$ Let $T$ be the Bass-Serre tree and let $H$ be a subgroup of $G$. The
following are equivalent:
1. (1)
$H$ is tamely generated and each $H_{v}$ in the Bass-Serre tree $T$ is f.g.
2. (2)
$H$ is f.g. and quasiconvex relative to $\mathbb{P}$.
###### Proof.
(1 $\Rightarrow$ 2): Follows from Theorem 1.4 and Theorem 2.6.
(2 $\Rightarrow$ 1): Since $H$ is f.g., the minimal $H$-subtree $T_{H}$ is
$H$-cocompact, and so $H$ splits as a finite graph of groups $\Gamma_{H}$.
Since $H$ is quasiconvex in $\mathbb{P}$, it is hyperbolic relative to
intersections with conjugates of $\mathbb{P}$. In particular, the infinite
edge groups in the induced splitting of $H$ are maximal parabolic, and are
thus f.g. since the maximal parabolic subgroups of a f.g. relatively
hyperbolic group are f.g. [20]. Each vertex group of $\Gamma_{H}$ is f.g. by
Lemma 2.5.
By Remark 2.4, each vertex group of $G$ is quasiconvex relative to
$\mathbb{P}$, and hence each $G_{\nu}$ is relatively quasiconvex by Remark 2.2
since it is a conjugate of a vertex group. Thus $H_{\nu}=H\cap G_{\nu}$ is
quasiconvex relative to $\mathbb{P}$ by Remark 2.2. Finally, $H_{\nu}$ is
quasiconvex in $G_{\nu}$ by Lemma 2.3. ∎
## 3\. Local Relative Quasiconvexity
A relatively hyperbolic group $G$ is _locally relatively quasiconvex_ if each
f.g. subgroup of $G$ is relatively quasiconvex. The focus of this section is
the following criterion for showing that the combination of locally relatively
quasiconvex groups is again locally relatively quasiconvex.
Recall that $N$ is _Noetherian_ if each subgroup of $N$ is f.g. We now give a
criterion for local quasiconvexity of a group that splits along parabolic
subgroups.
###### Theorem 3.1 (A Criterion for Locally Relatively Quasiconvexity).
1. (1)
Let $G_{1}$ and $G_{2}$ be locally relatively quasiconvex relative to
$\mathbb{P}_{1}$ and $\mathbb{P}_{2}$. Let
$G=G_{1}\ast_{P_{1}={P_{2}^{\prime}}}G_{2}$ where each
$P_{i}\in\mathbb{P}_{i}$ and $P_{1}$ is identified with the subgroup
${P_{2}}^{\prime}$ of $P_{2}$. Suppose $P_{1}$ is Noetherian. Then $G$ is
locally quasiconvex relative to
$\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$.
2. (2)
Let $G_{1}$ be a locally relatively quasiconvex relative to $\mathbb{P}$. Let
$P_{1}\in\mathbb{P}$ be isomorphic to a subgroup ${P_{2}}^{\prime}$ of a
maximal parabolic subgroup $P_{2}$ not conjugate to $P_{1}$. Let
$G=G_{1}*_{{P_{1}}^{t}={P_{2}}^{\prime}}$. Suppose $P_{1}$ is Noetherian. Then
$G$ is locally quasiconvex relative to $\mathbb{P}-\\{P_{1}\\}$.
3. (3)
Let $G_{1}$ be a locally quasiconvex relative to $\mathbb{P}$. Let $P$ be a
maximal parabolic subgroup of $G_{1}$, isomorphic to ${P}^{\prime}\leq P$. Let
$G=G_{1}*_{{P}^{t}={P}^{\prime}}$ and suppose $P$ is Noetherian. Then $G$ is
also locally quasiconvex relative to $\mathbb{P}\cup\langle
P,t\rangle-\\{P\\}$.
###### Proof.
(1): By Corollary 1.7, $G$ is hyperbolic relative to
$\mathbb{P}=\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$. Let $H$ be a
finitely generated subgroup of $G$. We show that $H$ is quasiconvex relative
to $\mathbb{P}$. Let $T$ be the Bass-Serre tree of $G$. Since $H$ is f.g., the
minimal $H$-subtree $T_{H}$ is $H$-cocompact, and so $H$ splits as a finite
graph of groups $\Gamma_{H}$. Moreover, the edge groups of this splitting are
f.g. since the edge groups of $G$ are Noetherian by hypothesis. Thus each
vertex group of $\Gamma_{H}$ is f.g. by Lemma 2.5. Since $G_{1}$ and $G_{2}$
are locally relatively quasiconvex, each vertex group of $T_{H}$ is relatively
quasiconvex in its “image vertex group” under the map $T_{H}\rightarrow T$.
Now by Theorem 2.6, $H$ is quasiconvex relative to $\mathbb{P}$. The proof of
(2) and (3) are similar. ∎
###### Definition 3.2 (Almost Malnormal).
A subgroup $H$ is _malnormal_ in $G$ if $H\cap H^{g}=\\{1\\}$ for $g\notin H$,
and similarly $H$ is _almost malnormal_ if this intersection $H\cap H^{g}$ is
always finite. Likewise, a collection of subgroups $\\{H_{i}\\}$ is _almost
malnormal_ if $H_{i}^{g}\cap H_{j}^{h}$ is finite unless $i=j$ and $gh^{-1}\in
H_{i}$.
###### Corollary 3.3.
Let $G$ split as a finite graph of groups. Suppose
* a)
Each $G_{\nu}$ is locally relatively quasiconvex;
* b)
Each $G_{e}$ is Noetherian and maximal parabolic in its vertex groups;
* c)
$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$,
for any vertex $\nu$.
Then $G$ is locally relatively quasiconvex relative to $\mathbb{P}$, see
Corollary 1.6.
### 3.1. Small-hierarchies and local quasiconvexity
The main result in this subsection is a consequence of Theorem 3.1 that
employs results of Yang [22] stated in Theorems 4.7 and 4.2, and also depends
on Lemma 4.9 which is independent of Section 4. The reader may choose to read
this subsection and refer ahead to those results, or return to this subsection
after reading Section 4.
###### Definition 3.4 (Small-Hierarchy).
A group is _small_ if it has no rank 2 free subgroup. Any small group has a
length 0 small-hierarchy. $G$ has a length $n$ small-hierarchy if $G\cong
A*_{C}B$ or $G\cong A*_{C^{t}=C^{\prime}}$, where $A$ and $B$ have length
$(n-1)$ small-hierarchies, and $C$ is small and f.g. We say $G$ has a _small-
hierarchy_ if it has a length $n$ small-hierarchy for some $n$.
We can define $\mathcal{F}$-hierarchy by replacing “small” by a class of
groups $\mathcal{F}$ closed under subgroups and isomorphisms. For instance,
when $\mathcal{F}$ is the class of finite groups, the class of groups with an
$\mathcal{F}$-hierarchy is precisely the class of virtually free groups.
###### Remark 3.5.
The _Tits alternative_ for relatively hyperbolic groups states that every f.g.
subgroup is either: elementary, parabolic, or contains a subgroup isomorphic
to $F_{2}$. The Tits alternative is proven for countable relatively hyperbolic
groups in [8, Thm 8.2.F]. A proof is given for convergence groups in [21]. It
is shown in [20] that every cyclic subgroup $H$ of a f.g. relatively
hyperbolic group $G$ is relatively quasiconvex.
###### Theorem 3.6.
Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$ where each element of
$\mathbb{P}$ is Noetherian. Suppose $G$ has a small-hierarchy. Then $G$ is
locally relatively quasiconvex.
###### Proof.
The proof is by induction on the length of the hierarchy. Since edge groups
are f.g., the Tits alternative shows that there are three cases according to
whether the edge group is finite, virtually cyclic, or infinite parabolic, and
we note that the edge group is relatively quasiconvex in each case. These
three cases are each divided into two subcases according to whether
$G=A*_{C_{1}}B$ or $G=A*_{C_{1}^{t}=C_{2}}$.
Since $C_{1}$ and $G$ are f.g. the vertex groups are f.g. by Lemma 2.5. Thus,
since $C_{1}$ is relatively quasiconvex the vertex groups are relatively
quasiconvex by Lemma 4.9.
When $C_{1}$ is finite the conclusion follows in each subcase from Theorem
3.1.
When $C_{1}$ is virtually cyclic but not parabolic, then $C_{1}$ lies in a
unique maximal virtually cyclic subgroup $Z$ that is almost malnormal and
relatively quasiconvex by [18]. Thus $G$ is hyperbolic relative to
$\mathbb{P}^{\prime}=\mathbb{P}\cup\\{Z\\}$ by Theorem 4.2.
Observe that $C_{1}$ is maximal infinite cyclic on at least one side, since
otherwise there would be a nontrivial splitting of $Z$ as an amalgamated free
product over $C_{1}$. We equip the (relatively quasiconvex) vertex groups with
their induced peripheral structures. Note that $C_{1}$ is maximal parabolic on
at least one side and so $G$ is locally relatively quasiconvex relative to
$\mathbb{P}^{\prime}$ by Theorem 3.1. Finally, by Theorem 4.7, any subgroup
$H$ is quasiconvex relative to the original peripheral structure $\mathbb{P}$
since intersections between $H$ and conjugates of $Z$ are quasiconvex relative
to $\mathbb{P}$.
When $C_{1}$ is infinite parabolic, we will first produce a new splitting
before verifying local relative quasiconvexity.
When $G=A*_{C_{1}}B$. Let $D_{a},D_{b}$ be the maximal parabolic subgroups of
$A,B$ containing $C_{1}$, and refine the splitting to:
$A*_{D_{a}}(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B$
The two outer splittings are along a parabolic that is maximal on the outside
vertex group. The inner vertex group $D_{a}*_{C_{1}}D_{b}$ is a single
parabolic subgroup of $G$. Indeed, as $C_{1}$ is infinite,
$D_{a}\supset{C_{1}}\subset D_{b}$ must all lie in the same parabolic subgroup
of $G$. It is obvious that $D_{a}*_{C_{1}}D_{b}$ is locally relatively
quasiconvex with respect to its induced peripheral structure since it is
itself parabolic in $G$. Consequently $(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B$ is
locally relatively quasiconvex by Theorem 3.1, therefore
$G=A*_{D_{a}}\big{(}(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B\big{)}$ is locally
relatively quasiconvex by Theorem 3.1.
When $G\cong A*_{{C_{1}}^{t}=C_{2}}$, let $M_{i}$ be the maximal parabolic
subgroup of $G$ containing $C_{i}$. There are two subsubcases:
[$t\in M_{1}$] Then $C_{2}\leq M_{1}$ and we revise the splitting to $G\cong
A*_{D_{1}}M_{1}$ where $D_{1}=M_{1}\cap A$. And in this splitting the edge
group is maximal parabolic at $D_{1}\subset A$, and $M_{1}$ is parabolic.
[$t\notin M_{1}$] Let $D_{i}$ denote the maximal parabolic subgroup of $A$
containing $C_{i}$. Observe that $\\{D_{1},D_{2}\\}$ is almost malnormal since
$D_{i}=M_{i}\cap A$. We revise the HNN extension to the following:
$\bigg{(}D_{1}^{t}*_{C_{1}^{t}=C_{2}}A\bigg{)}*_{{D_{1}}^{t}=D_{1}}$
where the conjugated copies of $D_{1}$ in the HNN extension embed in the first
and second factor of the AFP.
In both cases, the local relative quasiconvexity of $G$ now holds by Theorem
3.1 as before. ∎
## 4\. Relative Quasiconvexity in Graphs of Groups
Gersten [7] and then Bowditch [3] showed that a hyperbolic group $G$ is
hyperbolic relative to an almost malnormal quasiconvex subgroup. Generalizing
work of Martinez-Pedroza [14], Yang introduced and characterized a class of
parabolically extended structures for countable relatively hyperbolic groups
[22]. We use his results to generalize our previous results. The following was
defined in [22] for countable groups.
###### Definition 4.1 (Extended Peripheral Structure).
A peripheral structure consists of a finite collection $\mathbb{P}$ of
subgroups of a group $G$. Each element $P\in\mathbb{P}$ is a peripheral
subgroup of $G$. The peripheral structure $\mathbb{E}=\\{E_{j}\\}_{j\in J}$
_extends_ $\mathbb{P}=\\{P_{i}\\}_{i\in I}$ if for each $i\in I$, there exists
$j\in J$ such that $P_{i}\subseteq E_{j}$. For $E\in\mathbb{E}$, we let
$\mathbb{P}_{E}=\\{P_{i}\,:\,P_{i}\subseteq E,P_{i}\in\mathbb{P},i\in I\\}$.
We will use the following result of Yang [22].
###### Theorem 4.2 (Hyperbolicity of Extended Peripheral Structure).
Let $G$ be hyperbolic relative to $\mathbb{P}$ and let the peripheral
structure $\mathbb{E}$ extend $\mathbb{P}$. Then $G$ is hyperbolic relative to
$\mathbb{E}$ if and only if the following hold:
1. (1)
$\mathbb{E}$ is almost malnormal;
2. (2)
Each $E\in\mathbb{E}$ is quasiconvex in $G$ relative to $\mathbb{P}$.
###### Definition 4.3 (Total).
Let $G$ be hyperbolic relative to $\mathbb{P}$. The subgroup $H$ of $G$ is
total relative to $\mathbb{P}$ if: either $H\cap P^{g}=P^{g}$ or $H\cap P^{g}$
is finite for each $P\in\mathbb{P}$ and $g\in G$.
The following is proven in [5]:
###### Lemma 4.4.
If $G$ is f.g. and hyperbolic relative to $\mathbb{P}=\\{P_{1},\dots,P_{n}\\}$
and each $P_{i}$ is hyperbolic relative to
$\mathbb{H}_{i}=\\{H_{i1},\dots,H_{i{m_{i}}}\\}$, then $G$ is hyperbolic
relative to $\bigcup_{1\leq i\leq n}\mathbb{H}_{i}$.
As an application of Theorem 4.2, we now generalize Corollary 1.7 to handle
the case where edge groups are quasiconvex and not merely parabolic.
###### Theorem 4.5 (Combination along Total, Malnormal and Quasiconvex
Subgroups).
1. (1)
Let $G_{i}$ be hyperbolic relative to $\mathbb{P}_{i}$ for $i=1,2$. Let
$C_{i}\leq G_{i}$ be almost malnormal, total and relatively quasiconvex. Let
${C_{1}}^{\prime}\leq C_{1}$. Then $G=G_{1}*_{{C_{1}}^{\prime}=C_{2}}G_{2}$ is
hyperbolic relative to
$\mathbb{P}=\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{2}\in\mathbb{P}_{2}~{}:~{}P^{g}_{2}\subseteq
C_{2},~{}\text{for some}~{}g\in G_{2}\\}$.
2. (2)
Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $\\{C_{1},C_{2}\\}$ be
almost malnormal and assume each $C_{i}$ is total and relatively quasiconvex.
Let ${C_{1}}^{\prime}\leq C_{1}$. Then
$G=G_{1}*_{{{C_{1}}^{\prime}={C_{2}}^{t}}}$ is hyperbolic relative to
$\mathbb{P}=\mathbb{P}-\\{P_{2}\in\mathbb{P}_{2}~{}:~{}P^{g}_{2}\subseteq
C_{2},~{}\text{for some}~{}g\in G_{2}\\}$.
###### Proof.
(1): For each $i$, let
$\mathbb{E}_{i}=\mathbb{P}_{i}-\\{P\in\mathbb{P}_{i}\colon
P^{g}\leq{C_{i}},~{}\text{for some}~{}g\in G_{i}\\}\cup\\{C_{i}\\}$
Without loss of generality, we can assume that $\mathbb{E}_{i}$ extends
$\mathbb{P}_{i}$, since we can replace an element of $\mathbb{P}_{i}$ by its
conjugate. We now show that $G_{i}$ is hyperbolic relative to $\mathbb{E}_{i}$
by verifying the two conditions of Theorem 4.2: $\mathbb{E}_{i}$ is malnormal
in $G_{i}$, since $\mathbb{P}_{i}$ is almost malnormal and $C_{i}$ is total
and almost malnormal. Each element of $\mathbb{E}_{i}$ is relatively
quasiconvex, since $C_{i}$ is relatively quasiconvex by hypothesis and each
element of $\mathbb{P}_{i}$ is relatively quasiconvex by Remark 2.2.
We now regard each $G_{i}$ as hyperbolic relative to $\mathbb{E}_{i}$.
Therefore since the edge group $C_{2}={C_{1}}^{\prime}$ is maximal on one
side, by Corollary 1.7, $G$ is hyperbolic relative to
$\mathbb{E}=\mathbb{E}_{1}\cup\mathbb{E}_{2}-\\{C_{2}\\}$.
We now apply Lemma 4.4 to show that $G$ is hyperbolic relative to
$\mathbb{P}$. We showed that $G$ is hyperbolic relative to $\mathbb{E}$. But
each element of $\mathbb{E}$ is hyperbolic relative to $\mathbb{P}$ that it
contains. Thus by Lemma 4.4, we obtain the result.
(2): The proof is analogous to the proof of $(1)$. ∎
The following can be obtained by induction using Theorem 4.5 or can be proven
directly using the same mode of proof.
###### Corollary 4.6.
Let $G$ split as a finite graph of groups. Suppose
* (a)
Each $G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$;
* (b)
Each $G_{e}$ is total and relatively quasiconvex in $G_{\nu}$;
* (c)
$\\{G_{e}:e~{}\textrm{is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.
Then $G$ is hyperbolic relative to
$\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$.
Yang characterized relative quasiconvexity with respect to extensions in [22]
as follows:
###### Theorem 4.7 (Quasiconvexity in Extended Peripheral Structure).
Let $G$ be hyperbolic relative to $\mathbb{P}$ and relative to $\mathbb{E}$.
Suppose that $\mathbb{E}$ extends $\mathbb{P}$. Then
1. (1)
If $H\leq G$ is quasiconvex relative to $\mathbb{P}$, then $H$ is quasiconvex
relative to $\mathbb{E}$.
2. (2)
Conversely, if $H\leq G$ is quasiconvex relative to $\mathbb{E}$, then $H$ is
quasiconvex relative to $\mathbb{P}$ if and only if $H\cap{E}^{g}$ is
quasiconvex relative to $\mathbb{P}$ for all $g\in G$ and $E\in\mathbb{E}$.
We recall the following observation of Bowditch (see [16, Lem 2.7 and 2.9]).
###### Lemma 4.8 ($G$-attachment).
Let $G$ act on a graph $K$. Let $p,q\in K^{0}$ and $e$ be a new edge whose
endpoints are $p$ and $q$. The _$G$ -attachment_ of $e$ is the new graph
$K^{\prime}=K\cup Ge$ which consists of the union of $K$ and copies $ge$ of
$e$ attached at $gp$ and $gq$ for any $g\in G$. Note that $K^{\prime}$ is
$G$-cocompact/fine/hyperbolic if $K$ is.
In the following lemma, we prove that when a relatively hyperbolic group $G$
splits then relative quasiconvexity of vertex groups is equivalent to relative
quasiconvexity of the edge groups.
###### Lemma 4.9 (Quasiconvex Edges $\Longleftrightarrow$ Quasiconvex
Vertices).
Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a finite
graph of groups whose vertex groups and edge groups are finitely generated.
Then the edge groups are quasiconvex relative to $\mathbb{P}$ if and only if
the vertex groups are quasiconvex relative to $\mathbb{P}$.
###### Proof.
If the vertex groups are quasiconvex relative to $\mathbb{P}$ then so are the
edge groups, since relative quasiconvexity is preserved by intersection (see
[10, 15]) in the f.g. group $G$. Assume the edge groups are quasiconvex
relative to $\mathbb{P}$. Let $K$ be a $(G;\mathbb{P})$ graph and let $T$ be
the Bass-Serre tree for $G$. Let $f\colon K\rightarrow T$ be a $G$-equivariant
map that sends vertices to vertices and edges to geodesics. Subdivide $K$ and
$T$, so that each edge is the union of two length $\frac{1}{2}$ _halfedges_.
Let $\nu$ be a vertex in $T$. It suffices to find a $G_{\nu}$-cocompact
quasiconvex subgraph $L$ of $K$.
Let $\\{e_{1},\dots,e_{m}\\}$ be representatives of the $G_{\nu}$-orbits of
halfedges attached to $\nu$. Let $\omega_{i}$ be the other vertex of $e_{i}$
for $1\leq i\leq m$. Since each $G_{\omega_{i}}=G_{e_{i}}$ is f.g. by
hypothesis, we can perform finitely many $G_{\omega_{i}}$-attachments of arcs
so that the preimage of $\omega_{i}$ is connected for each $i$. This leads to
finitely many $G$-attachments to $K$ to obtain a new fine hyperbolic graph
$K^{\prime}$. By mapping the newly attached edges to their associated vertices
in $T$, we thus obtain a $G$-equivariant map $f^{\prime}\colon
K^{\prime}\rightarrow T$ such that $M^{\prime}_{i}=f^{\prime-1}(\omega_{i})$
is connected and $G_{\omega_{i}}$-cocompact for each $i$.
Consider $L^{\prime}=f^{\prime-1}(N_{\frac{1}{2}}(\nu))$ where
$N_{\frac{1}{2}}(\nu)$ is the closed $\frac{1}{2}$-neighborhood of $\nu$. To
see that $L^{\prime}$ is connected, consider a path $\sigma$ in $K^{\prime}$
between distinct components of $L^{\prime}$. Moreover choose $\sigma$ so that
its image in $T$ is minimal among all such choices. Then $\sigma$ must leave
and enter $L^{\prime}$ through the same $g_{\nu}M^{\prime}_{i}$ which is
connected by construction.
We now show that $L^{\prime}$ is quasiconvex. Consider a geodesic $\gamma$
that intersects $L^{\prime}$ exactly at its endpoints. As before the endpoints
of $\gamma$ lie in the same $g_{\nu}M^{\prime}_{i}$. Since
$g_{\nu}M^{\prime}_{i}$ is $\kappa_{i}$-quasiconvex for some $\kappa_{i}$, we
see that $\gamma$ lies in $\kappa$-neighborhood of $g_{\nu}M^{\prime}_{i}$ and
hence in the $\kappa$-neighborhood of $L^{\prime}$. ∎
###### Lemma 4.10 (Total Edges $\Longleftrightarrow$ Total Vertices).
Let $G$ be hyperbolic relative to $\mathbb{P}$. Let $G$ act on a tree $T$. For
each $P\in\mathbb{P}$ let $T_{P}$ be a minimal $P$-subtree. Assume that no
$T_{P}$ has a finite edge stabilizer in the $P$-action. Then edge groups of
$T$ are total in $G$ iff vertex groups are total in $G$.
###### Proof.
Since the intersection of two total subgroups is total, if the vertex groups
are total then the edge groups are also total. We now assume that the edge
groups are total. Let $G_{\nu}$ be a vertex group and $P\in\mathbb{P}$ such
that $P^{g}\cap G_{\nu}$ is infinite for some $g\in G$. If $|P^{g}\cap
G_{e}|=\infty$ for some edge $e$ attached to $\nu$, then $P\subseteq G_{e}$,
thus $P\subseteq G_{e}\subseteq G_{\nu}$. Now suppose that $|P^{g}\cap
G_{e}|<\infty$ for each $e$ attached to $\nu$. If $P^{g}\nleq G_{\nu}$ then
the action of $P^{g}$ on $gT$ violates our hypothesis. ∎
###### Remark 4.11.
Suppose $G$ is f.g. and $G$ is hyperbolic relative to $\mathbb{P}$. Let
$P\in\mathbb{P}$ such that $P=A*_{C}B$ [$P=A*_{C={C^{\prime}}^{t}}$] where $C$
is a finite group. Since $P$ is hyperbolic relative to $\\{A,B\\}$
[$\\{A\\}$], by Lemma 4.4, $G$ is hyperbolic relative to
$\mathbb{P}^{\prime}=\mathbb{P}-\\{P\\}\cup\\{A,B\\}$
[$\mathbb{P}^{\prime}=\mathbb{P}-\\{P\\}\cup\\{A\\}$].
We now describe a more general criterion for relative quasiconvexity which is
proven by combining Corollary 2.7 with Theorem 4.7.
###### Theorem 4.12.
Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
a finite graph of groups. Suppose
1. (a)
Each $G_{e}$ is total in $G$;
2. (b)
Each $G_{e}$ is relatively quasiconvex in $G$;
3. (c)
$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.
Let $H\leq G$ be tamely generated subgroup of $G$. Then $H$ is relatively
quasiconvex in $G$.
###### Proof.
_Technical Point:_ By splitting certain elements of $\mathbb{P}$ to obtain
$\mathbb{P}^{\prime}$ as in Remark 4.11, we can assume that $G$ is hyperbolic
relative to $\mathbb{P}^{\prime}$ and each $G_{\nu}$ is hyperbolic relative to
the conjugates of elements of $\mathbb{P}^{\prime}$ that it contains.
Indeed for any $P\in\mathbb{P}$, if the action of $P$ on a minimal subtree
$T_{P}$ of the Bass-Serre tree $T$, yields a finite graph $\Gamma$ of groups
some of whose edge groups are finite, then following Remark 4.11, we can
replace $\mathbb{P}$ by the groups that complement these finite edge groups,
(i.e. the fundamental groups of the subgraphs obtained by deleting these edges
from $\Gamma$.) Therefore $G$ is hyperbolic relative to $\mathbb{P}^{\prime}$.
No $P\in\mathbb{P}^{\prime}$ has a nontrivial induced splitting as a graph of
groups with a finite edge group.The edge groups are total relative to
$\mathbb{P}^{\prime}$ since they are total relative to $\mathbb{P}$. Therefore
by Lemma 4.10 the vertex groups are total in $G$ relative to
$\mathbb{P}^{\prime}$. By Lemma 4.9, each vertex group $G_{\nu}$ is relatively
quasiconvex in $G$ relative to $\mathbb{P}$, therefore by Theorem 4.7 each
$G_{\nu}$ is quasiconvex in $G$ relative to $\mathbb{P}^{\prime}$. Thus
$G_{\nu}$ has an induced relatively hyperbolic structure
$\mathbb{P}^{\prime}_{\nu}$ as in Remark 2.2. By totality of $G_{\nu}$, we can
assume each element of $\mathbb{P}^{\prime}_{\nu}$ is a conjugate of an
element of $\mathbb{P}^{\prime}$. And as usual we may omit the finite
subgroups in $\mathbb{P}^{\prime}_{\nu}$.
_Step 1:_ We now extend the peripheral structure of each $G_{\nu}$ from
$\mathbb{P}^{\prime}_{\nu}$ to $\mathbb{E}_{\nu}$ where
$\mathbb{E}_{\nu}=\\{G_{e}:e\textrm{ is attached to
}\nu\\}\cup\\{P\in\mathbb{P}^{\prime}_{\nu}:P^{g}\nleq G_{e}\textrm{ for any
$g\in G_{\nu}$}\\}$
Almost malnormality of $\mathbb{E}_{\nu}$ follows from Condition (c) and the
totality of the edge groups in their vertex groups which follows by the
totality of the edge groups in $G$, also relative quasiconvexity of the new
elements $G_{e}$ is Condition (b). Thus by $G_{\nu}$ is hyperbolic relative to
$\mathbb{E}_{\nu}$ by Theorem 4.7.
_Step 2:_ For each $\tilde{\nu}$ in the Bass-Serre tree, its $H$-stabilizer
$H_{\tilde{\nu}}$ lies in $G_{\tilde{\nu}}$ which we identify (by a conjugacy
isomorphism) with the chosen vertex stabilizer $G_{\nu}$ in the graph of group
decomposition. Then $H_{\tilde{\nu}}$ is quasiconvex in $G_{\nu}$ relative to
$\mathbb{E}_{\nu}$ for each $\nu$ by Theorem 4.7, since $\mathbb{E}_{\nu}$
extends $\mathbb{P}^{\prime}_{\nu}$ and each $H_{\tilde{\nu}}$ is quasiconvex
in $G_{\nu}$ relative to $\mathbb{P}^{\prime}_{\nu}$. Therefore $H$ is
quasiconvex relative to $\bigcup\mathbb{E}_{\nu}$ by Corollary 2.7.
_Step 3:_ $H$ is quasiconvex relative to
$\mathbb{P}^{\prime}=\bigcup\mathbb{P}^{\prime}_{\nu}$. Since
$\bigcup\mathbb{E}_{\nu}$ extends
$\mathbb{P}=\bigcup\mathbb{P}^{\prime}_{\nu}$, by Theorem 4.7, it suffices to
show that $H\cap K^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$ for
all $K\in\bigcup\mathbb{E}_{\nu}$ and $g\in G$. There are two cases:
Case 1: $K\in\mathbb{P}^{\prime}_{\nu}$ for some $\nu$. Now $H\cap K^{g}$ is a
parabolic subgroup of $G$ relative to $\mathbb{P}^{\prime}$ and is thus
quasiconvex relative to $\mathbb{P}^{\prime}$.
Case 2: $K=G_{e}$ for some $e$ attached to some $\nu$. The group $K$ is
relatively quasiconvex in $G_{\nu}$, therefore by Remark 2.2, $K^{g}$ is also
relatively quasiconvex but in $G_{g\nu}$. Now since $K^{g}\cap H=K^{g}\cap
H_{g\nu}$ and $K^{g}$ and $H_{g\nu}$ are both relatively quasiconvex in
$G_{g\nu}$, the group $K^{g}\cap H$ is relatively quasiconvex in $G_{g\nu}$.
Since by Lemma 4.9, $G_{g\nu}$ is quasiconvex relative to
$\mathbb{P}^{\prime}$, Lemma 2.3 implies that $K^{g}\cap H$ is quasiconvex
relative to $\mathbb{P}^{\prime}$.
Now $H$ is quasiconvex relative to $\mathbb{P}$ by Theorem 4.7, since
$\mathbb{P}$ extends $\mathbb{P}^{\prime}$. ∎
The following result strengthens Theorem 4.12, by relaxing Condition (c).
###### Theorem 4.13 (Quasiconvexity Criterion for Relatively Hyperbolic
Groups that Split).
Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$ such that $G$ splits
as a finite graph of groups. Suppose
1. (a)
Each $G_{e}$ is total in $G$;
2. (b)
Each $G_{e}$ is relatively quasiconvex in $G$;
3. (c)
Each $G_{e}$ is almost malnormal in $G$.
Let $H\leq G$ be tamely generated. Then $H$ is relatively quasiconvex in $G$.
###### Remark 4.14.
By Lemma 2.3 and Remark 2.4, Condition (b) is equivalent to requiring that
each $G_{e}$ is quasiconvex in $G_{\nu}$. Also we can replace Condition (a) by
requiring $G_{e}$ to be total in $G_{\nu}$.
###### Proof.
We prove the result by induction on the number of edges of the graph of groups
$\Gamma$. The base case where $\Gamma$ has no edge is contained in the
hypothesis. Suppose that $\Gamma$ has at least one edge $e$ (regarded as an
open edge). If $e$ is nonseparating, then $G=A*_{C^{t}=D}$ where $A$ is the
graph of groups over $\Gamma-e$, and $C,D$ are the two images of $G_{e}$.
Condition (c) ensures that $\\{C,D\\}$ is almost malnormal in $A$, and by
induction, the various nontrivial intersections $H\cap A^{g}$ are relatively
quasiconvex in $A^{g}$, and thus $H$ is relatively quasiconvex in $G$ by
Theorem 4.12. A similar argument concludes the separating case. ∎
###### Corollary 4.15.
Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
a finite graph of groups. Assume:
* (a)
Each $G_{\nu}$ is locally relatively quasiconvex;
* (b)
Each $G_{e}$ is Noetherian, total and relatively quasiconvex in $G$;
* (c)
Each $G_{e}$ is almost malnormal in $G$.
Then $G$ is locally relatively quasiconvex relative to $\mathbb{P}$.
###### Theorem 4.16.
Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a graph
$\Gamma$ of groups with relatively quasiconvex edge groups. Suppose $\Gamma$
is bipartite with $\Gamma^{0}=V\sqcup U$ and each edge joins vertices of $V$
and $U$. Suppose each $G_{v}$ is maximal parabolic for $v\in V$, and for each
$P\in\mathbb{P}$ there is at most one $v$ with $P$ conjugate to $G_{v}$. Let
$H\leq G$ be tamely generated. Then $H$ is quasiconvex relative to
$\mathbb{P}$.
The scenario of Theorem 4.16 arises when $M$ is a compact aspherical
3-manifold, from its JSJ decomposition. The manifold $M$ decomposes as a
bipartite graph $\Gamma$ of spaces with $\Gamma^{0}=U\sqcup V$. The
submanifold $M_{v}$ is hyperbolic for each $v\in V$, and $M_{u}$ is a graph
manifold for each $u\in U$. The edges of $\Gamma$ correspond to the
“transitional tori” between these hyperbolic and complementary graph manifold
parts. Some of the graph manifolds are complex but others are simpler Seifert
fibered spaces; in the simplest cases, thickened tori between adjacent
hyperbolic parts or $I$-bundles over Klein bottles where a hyperbolic part
terminates. Hence $\pi_{1}M$ decomposes accordingly as a graph $\Gamma$ of
groups, and $\pi_{1}M$ is hyperbolic relative to $\\{\pi_{1}M_{u}:u\in U\\}$
by Theorem 1.4 or indeed, Corollary 1.5.
###### Proof.
Let $K_{o}$ be a fine hyperbolic graph for $G$. Each vertex group is
quasiconvex in $G$ by Lemma 4.9, and so for each $u\in U$ let $K_{u}$ be a
$G_{u}$-quasiconvex subgraph, and in this way we obtain finite hyperbolic
$G_{u}$-graphs, and for $v\in V$, we let $K_{v}$ be a singleton. We apply the
Construction in the proof of Theorem 1.4 to obtain a fine hyperbolic $G$-graph
$K$ and quotient $\bar{K}$. Note that the parabolic trees are $i$-pods. We
form the $H$-cocompact quasiconvex subgraph $L$ by combining
$H_{\omega}$-cocompact quasiconvex subgraphs $K_{\omega}$ as in the proof of
Theorem 2.6. ∎
###### Theorem 4.17.
Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
graph $\Gamma$ of groups with relatively quasiconvex edge groups. Suppose
$\Gamma$ is bipartite with $\Gamma^{0}=V\sqcup U$ and each edge joins vertices
of $V$ and $U$. Suppose each $G_{v}$ is almost malnormal and total in $G$ for
$v\in V$. Let $H\leq G$ be tamely generated. Then $H$ is quasiconvex relative
to $\mathbb{P}$.
Theorem 4.17 covers the case where edge groups are almost malnormal on both
sides since we can subdivide to put barycenters of edges in $V$.
Another special case where Theorem 4.17 applies is where
$G=G_{1}*_{{C_{1}}^{\prime}=C_{2}}G_{2}$ is hyperbolic relative to
$\mathbb{P}$, and $C_{2}\leq G_{2}$ is total and relatively quasiconvex in $G$
and almost malnormal in $G_{2}$.
###### Proof.
Following the Technical Point in the proof of Theorem 4.12, by splitting
certain elements of $\mathbb{P}$ to obtain $\mathbb{P}^{\prime}$ as in Remark
4.11, we can assume that $G$ is hyperbolic relative to $\mathbb{P}^{\prime}$
where each $P^{\prime}\in\mathbb{P}^{\prime}$ is elliptic with respect to the
action of $G$ on the Bass-Serre tree $T$. Since $\mathbb{P}$ extends
$\mathbb{P}^{\prime}$ and each $G_{v}\cap P^{g}$ is conjugate to an element of
$\mathbb{P}^{\prime}$, we see that each $G_{v}$ is quasiconvex in $G$ relative
to $\mathbb{P}^{\prime}$ by Theorem 4.7, and moreover, since elements of
$\mathbb{P}^{\prime}$ are vertex groups of elements of $\mathbb{P}$, each
$G_{v}$ is total relative to $\mathbb{P}^{\prime}$. Therefore each $G_{v}$ is
hyperbolic relative to a collection $\mathbb{P}^{\prime}_{v}$ of conjugates of
elements of $\mathbb{P}^{\prime}$.
We argue by induction on the number of edges of $\Gamma$. If $\Gamma$ has no
edge the result is contained in the hypothesis. Suppose $\Gamma$ has at least
one edge $e$. If $e$ is separating and $\Gamma=\Gamma_{1}\sqcup
e\sqcup\Gamma_{2}$ where $e$ attaches $v\in\Gamma^{0}_{1}$ to
$u\in\Gamma^{0}_{2}$ then $G=G_{1}*_{G_{e}}G_{2}$ where
$G_{i}=\pi_{1}(\Gamma_{i})$. Each $G_{e}$ is the intersection of vertex groups
and hence quasiconvex relative to $\mathbb{P}^{\prime}$. By Lemma 4.9, the
groups $G_{1}$ and $G_{2}$ are quasiconvex in $G$ relative to
$\mathbb{P}^{\prime}$. Thus $G_{i}$ is hyperbolic relative to
$\mathbb{P}^{\prime}_{i}$ by Remark 2.2.
Observe that $T$ contains subtrees $T_{1}$ and $T_{2}$ that are the Bass-Serre
trees of $\Gamma_{1}$ and $\Gamma_{2}$, and $T-G\tilde{e}=\\{gT_{1}\cup
gT_{2}\ :\ g\in G\\}$. The Bass-Serre tree $\bar{T}$ of $G_{1}*_{G_{e}}G_{2}$
is the quotient of $T$ obtained by identifying each $gT_{i}$ to a vertex.
Since $H$ is relatively finitely generated, there is a finite graph of groups
$\Gamma_{H}$ for $H$, and a map $\Gamma_{H}\rightarrow\Gamma$. Removing the
edges mapping to $e$ from $\Gamma_{H}$, we obtain a collection of finitely
many graphs of groups - some over $\Gamma_{1}$ and some over $\Gamma_{2}$.
Each component of $\Gamma_{H}$ corresponds to the stabilizer of some $gT_{i}$
and is denoted by $H_{gT_{i}}$, and since that component is a finite graph
with relatively quasiconvex vertex stabilizers, we see that each $H_{gT_{i}}$
is relatively quasiconvex in $G_{i}$ relative to $\mathbb{P}^{\prime}_{i}$ by
induction on the number of edges of $\Gamma_{H}$.
We extend the peripheral structure $\mathbb{P}^{\prime}_{1}$ of $G_{1}$ to
$\mathbb{E}_{1}=\\{G_{1}\\}$. Note that now each $H_{gT_{1}}$ is quasiconvex
in $G_{1}$ relative to $\mathbb{E}_{1}$ by Theorem 4.7. Let
$\mathbb{E}=\mathbb{E}_{1}\cup\mathbb{P}^{\prime}_{2}-\\{P\in\mathbb{P}^{\prime}_{2}\colon
P^{g}\leq{G_{e}},~{}\text{for some}~{}g\in G_{2}\\}.$
Observe that $\mathbb{E}$ extends $\mathbb{P}^{\prime}$. Since $G_{v}$ is
total and quasiconvex in $G$ relative to $\mathbb{P}^{\prime}$ and
$\mathbb{E}$ extends $\mathbb{P}^{\prime}$, the group $G_{1}$ is total and
quasiconvex in $G$ relative to $\mathbb{E}$ by Theorem 4.7. Therefore $G$ is
hyperbolic relative to $\mathbb{E}$ by Theorem 4.2.
Since $G_{1}$ is maximal parabolic in $G$, by Theorem 4.16 $H$ is quasiconvex
in $G$ relative to $\mathbb{E}$. The graph $\Gamma_{H}$ shows that $H$ is
generated by finitely many hyperbolic elements and vertex stabilizers
$H_{g\bar{T}_{i}}$ and each $H_{g\bar{T}_{i}}=H_{gT_{i}}$ which we explained
above is relatively quasiconvex in $G_{i}$.
We now show that $H$ is quasiconvex relative to $\mathbb{P}^{\prime}$ and
therefore relative to $\mathbb{P}$ by Theorem 4.7. Since $\mathbb{E}$ extends
$\mathbb{P}^{\prime}$, by Theorem 4.7, it suffices to show that $H\cap E^{g}$
is quasiconvex relative to $\mathbb{P}^{\prime}$ for all $E\in\mathbb{E}$ and
$g\in G$. There are two cases:
Case 1: $E\in\mathbb{P}^{\prime}_{2}$. Now $H\cap E^{g}$ is a parabolic
subgroup of $G$ relative to $\mathbb{P}^{\prime}$ and is thus quasiconvex
relative to $\mathbb{P}^{\prime}$.
Case 2: $E=G_{1}$. Then $H\cap E^{g}$ is quasiconvex relative to
$\mathbb{P}^{\prime}_{1}$ since $(H\cap E^{g})=H_{gT_{1}}$ is quasiconvex in
$G_{1}^{g}$ relative to $\mathbb{E}_{1}^{g}=\\{G_{1}^{g}\\}$. Since
$E^{g}=G_{1}^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$, Lemma 2.3
implies that $H\cap E^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$.
Now assume that $e$ is nonseparating. Let $u\in U$ and $v\in V$ be the
endpoints of $e$. Then $G=G_{1}*_{C^{t}=D}$ where $G_{1}$ is the graph of
groups over $\Gamma-e$, and $C$ and $D$ are the images of $G_{e}$ in $G_{v}$
and $G_{u}$ respectively. We first reduce the peripheral structure of $G$ from
$\mathbb{P}$ to $\mathbb{P}^{\prime}$, and we then extend from
$\mathbb{P}^{\prime}$ to $\mathbb{E}$ with:
$\mathbb{E}=\\{G_{v}\\}\cup\mathbb{P}^{\prime}-\\{P\in\mathbb{P}^{\prime}\colon
P^{g}\leq{G_{v}},~{}\text{for some}~{}g\in G\\}.$
$G$ is hyperbolic relative to $\mathbb{E}$ by Theorem 4.2 as $G_{v}$ is almost
malnormal, total, and quasiconvex relative to $\mathbb{P}$ . The argument
follows by induction and Theorem 4.16 as in the separating case. ∎
Theorem 4.13 suggests the following criterion for relative quasiconvexity:
###### Conjecture 4.18.
Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a finite
graph of groups with f.g. relatively quasiconvex edge groups. Suppose $H\leq
G$ is tamely generated such that each $H_{v}$ is f.g. for each $v$ in the
Bass-Serre tree. Then $H$ is relatively quasiconvex in $G$.
When the edge groups are separable in $G$, there is a finite index subgroup
$G^{\prime}$ whose splitting has relatively malnormal edge groups (see e.g.
[11, 9]). Consequently, if moreover, the edge groups of $G$ are total, then
the induced splitting of $G^{\prime}$ satisfies the criterion of Theorem 4.13,
and we see that Conjecture 4.18 holds in this case. In particular, Conjecture
4.18 holds when $G$ is virtually special and hyperbolic relative to virtually
abelian subgroups, provided that edge groups are also total. We suspect the
totalness assumption can be dropped totally.
As a closing thought, consider a hyperbolic $3$-manifold $M$ virtually having
a malnormal quasiconvex hierarchy (conjecturally all closed $M$). Theorem 4.13
suggests an alternate approach to the tameness theorem, which could be
reproven by verifying:
_If the intersection of a f.g. $H$ with a malnormal quasiconvex edge group is
infinitely generated then $H$ is a virtual fiber._
Acknowledgement: We are extremely grateful to the anonymous f.g. referee whose
very helpful corrections and adjustments improved the results and exposition
of this paper.
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|
arxiv-papers
| 2012-11-08T21:28:59 |
2024-09-04T02:49:37.741850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hadi Bigdely, Daniel T. Wise",
"submitter": "Hadi Bigdely",
"url": "https://arxiv.org/abs/1211.1993"
}
|
1211.2188
|
# Torsion of rational elliptic curves over cubic fields and sporadic points on
$X_{1}(n)$
Filip Najman Department of Mathematics
University of Zagreb
Bijenička cesta 30
10000 Zagreb
Croatia fnajman@math.hr
###### Abstract.
We classify the possible torsion structures of rational elliptic curves over
cubic fields. Along the way we find a previously unknown torsion structure
over a cubic field, $\mathbb{Z}/21\mathbb{Z}$, which corresponds to a sporadic
point on $X_{1}(21)$ of degree 3, which is the lowest possible degree of a
sporadic point on a modular curve $X_{1}(n)$.
###### Key words and phrases:
Elliptic curves, torsion subgroups, cubic fields, modular curves
###### 2000 Mathematics Subject Classification:
11G05, 11G18, 14G25
The author was supported by the Ministry of Science, Education, and Sports,
Republic of Croatia, grant 037-0372781-2821.
## 1\. Introduction
When trying to understand elliptic curves over number fields, an important
problem is to classify the possible torsion structures. The first such
classification was done by Mazur [26, 27], proving that the torsion of an
elliptic curve over $\mathbb{Q}$ has to be isomorphic to one of the following
15 groups:
(1) $\mathbb{Z}/m\mathbb{Z},1\leq m\leq 12,\ m\neq 11,$
---
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2m\mathbb{Z},\ 1\leq m\leq 4.$
After this result, attention shifted toward number fields. Kamienny [16,
Theorem 3.1] proved that if a torsion point of an elliptic curve over a
quadratic field has prime order $p$, then $p\leq 13$. This, when combined with
a theorem of Kenku and Momose [22, Theorem (0.1).], gave a complete list of
possible torsion structures over quadratic fields:
(2) $\mathbb{Z}/m\mathbb{Z},1\leq m\leq 18,\ m\neq 17,$
---
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2m\mathbb{Z},\ 1\leq m\leq 6,$
$\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/3m\mathbb{Z},\ m=1,2,$
$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}.$
The author gave a similar complete list for the fields $\mathbb{Q}(i)$ and
$\mathbb{Q}(\sqrt{-3})$ (see [33, Theorem 2.]), and a procedure how to make
such a list was developed by Kamienny and the author [17].
As one can see, much is known about the possible torsion structures of
elliptic curves over $\mathbb{Q}$ and over quadratic fields. Unfortunately,
already over cubic fields a classification of possible torsion structures of
elliptic curves is not known. However, it is known that if an elliptic curve
over a cubic field has a point of prime order $p$, then $p\leq 13$ (see [36,
37]). Jeon, Kim and Schweizer [13, Theorem 3.4.] found all the torsion
structures that appear infinitely often as one runs through all elliptic
curves over all cubic fields:
(3) $\mathbb{Z}/m\mathbb{Z},1\leq m\leq 20,\ m\neq 17,19,$
---
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2m\mathbb{Z},\ 1\leq m\leq 7.$
Jeon, Kim and Lee [11] constructed infinite families having each of the
torsion structures from the list (3).
However, it was unknown whether the list (3) is complete, i.e. if one runs
through all elliptic curves over all cubic fields, do there exist torsion
structures that appear only finitely many times? We show, by finding an
elliptic curve with torsion $\mathbb{Z}/21\mathbb{Z}$ over a cubic field, that
the answer is yes and that the list (3) is not the complete list of possible
torsion structures over cubic fields. Note that this, in contrast with what
happens over $\mathbb{Q}$ and over quadratic fields, where each group that can
appear at all, is the torsion group of infinitely many non-isomorphic elliptic
curves.
The main purpose of this paper is to study the possible torsion structures of
all rational elliptic curves (meaning that all their coefficients are
$\mathbb{Q}$-rational) over all cubic fields. This is a natural question to
consider as, apart from being interesting in itself, it is often important to
study rational elliptic curves over extensions of $\mathbb{Q}$ when solving
Diophantine equations (see for example [2]). Somewhat similar problems were
studied by Fujita [8], who studied the possible torsion groups of rational
elliptic curves over the compositum of all quadratic fields, by Lozano-Robledo
[25], who studied the minimal degree of the field of definition of points of
order $p$ on rational elliptic curves and by González-Jiménez and Tornero
[14], who studied how can the torsion of a rational elliptic curve grow upon
base changing to a quadratic field.
The main result of this paper is the following theorem.
###### Theorem 1.
Let $E/\mathbb{Q}$ be a rational elliptic curve, and let $K/\mathbb{Q}$ be a
cubic extension. Then $E(K)_{tors}$ is one of the following groups
(4) $\mathbb{Z}/m\mathbb{Z},\text{ }m=1,\ldots,10,12,13,14,18,21,$
---
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2m\mathbb{Z},\text{ }m=1,2,3,4,7.$
The elliptic curve $162B1$ over $\mathbb{Q}(\zeta_{9})^{+}$ is the unique
rational elliptic curve over a cubic field with torsion
$\mathbb{Z}/21\mathbb{Z}$. For all the other groups $T$ in the list (4), there
exists infinitely many rational elliptic curves that have torsion $T$ over
some cubic field.
To prove Theorem 1, we will first need to solve the analogous problem for
quadratic fields, which we do in Section 3.
We prove Theorem 1 by studying the action of the Galois group on the torsion
points, division polynomials, and by finding the rational points on certain
(modular) curves.
Let us mention that a somewhat similar problem to the one considered in this
paper is the problem of finding the possible torsion structures of rational
elliptic curves with integral $j$-invariant [32, 38] and with complex
multiplication over number fields [5, 3].
Recall that the gonality $\gamma(X)$ of an algebraic curve $X$ is the lowest
degree of a nonconstant rational map from $X$ to the projective line. We call
points of degree $d$ on the modular curves $Y_{1}(m,n)$ (see Section 2 for
definitions of modular curves and note that we only consider modular curves
with $m=1,2$ in this paper), when $d<\gamma(Y_{1}(m,n))$ _sporadic_. Since all
the modular curves $Y_{1}(m,n)$ that correspond to the torsion structures in
the list(1) are of genus 0 and have (infinitely many) rational points (since
some of the cusps of $X_{1}(m,n)$ are rational) and hence are of gonality 1.
Similarly, all the modular curves $Y_{1}(m,n)$ that correspond to the torsion
structures in the list (2) are of genus $\leq 2$ (and hence have gonality 1 or
2), so it follows that there are no sporadic points of degree 1 or 2. Van
Hoeij [40] found sporadic points of degree $6$ on $X_{1}(37)$ (of gonality
18), and of degree 9 on $X_{1}(29)$ and $X_{1}(31)$ (of gonality 11 and 12,
respectively).
Since the modular curve $X_{1}(21)$ has gonality $4$, the unique rational
elliptic curve with $21$-torsion over a cubic field gives us a degree 3
sporadic point, which is the lowest degree possible. By the method used to
construct this point we rediscover van Hoeij’s degree 6 point on $X_{1}(37)$.
## 2\. Conventions and notation
Throughout this paper, $K$ will be a cubic field and $L$ will be its normal
closure over $\mathbb{Q}$. This means that when $K/\mathbb{Q}$ is normal, then
$L=K$, and otherwise $L$ is a degree 6 extension of $\mathbb{Q}$ such that
$\operatorname{Gal}(L/\mathbb{Q})\simeq S_{3}$. We denote by $M$ the unique
field with the property that $M$ is a subfield of $L$ such that $[L:M]=3$
(from which it follows that
$\operatorname{Gal}(L/M)\simeq\mathbb{Z}/3\mathbb{Z}$). If $K$ is normal over
$\mathbb{Q}$, this will mean that $M=\mathbb{Q}$.
Let $E[n]=\\{P\in E(\overline{\mathbb{Q}})|nP=0\\}$ denote the $n$-th division
group of $E$ over $\overline{\mathbb{Q}}$ and let $\mathbb{Q}(E[n])$ be the
$n$-th division field of $E$.
We will denote by $E^{d}$ a quadratic twist of $E$ by
$d\in\mathbb{Q}^{*}/(\mathbb{Q}^{*})^{2}$. By $\zeta_{n}$ we will denote a
$n$th primitive root of unity and by $\mathbb{Q}(\zeta_{n})^{+}$ the maximal
real subfield of $\mathbb{Q}(\zeta_{n})$.
For $n$ an odd positive integer, we denote by $\psi_{n}$ the $n$-th division
polynomial of an elliptic curve $E$ (see [41, Section 3.2] for details), which
satisfies that, for a point $P\in E$, $\psi_{n}(x(P))=0$ if and only if
$nP=0$. As before, although the division polynomial depends on the curve $E$,
we will leave $E$ out of the index as it will be clear what elliptic curve we
are referring to.
Let $E/\mathbb{Q}$ be an elliptic curve and $d>1$ be an integer. Factor
$\psi_{n}$ and then consider all factors of degree $l$, where $l|d$. Each of
these factors generates a field $F$ over which the $x$-coordinate of a point
$P$ such that $nP=0$ is defined. The torsion point $P$ is then defined over
$F^{\prime}$ which is either $F$ itself or a quadratic extension of $F$
obtained by adjoining the $y$-coordinate of $P$. By considering all such
fields $F^{\prime}$ of degree dividing $d$, we can check whether a fixed
rational elliptic curve $E$ obtains $n$-torsion over some extension of degree
$d$. We call this method the _division polynomial method_.
The division polynomial method can also be effectively used in determining
which, if any, twists of a given curve $E/F$ have non-trivial $n$-torsion over
the base field. This is done by finding all the quadratic extensions
$F(\sqrt{d})$ in which the $n$-torsion grows and then computing for which of
the finitely many $d$ obtained does the curve $E^{d}(F)$ have non-trivial
$n$-torsion.
If there exists a $K$-rational cyclic isogeny $\phi:E\rightarrow E^{\prime}$
of degree $n$, this implies that $\ker\phi$ is
$\operatorname{Gal}(\overline{K}/K)$-invariant cyclic group of order $n$ and
we will say that $E/K$ has an $n$-isogeny.
When counting rational elliptic curves, unless stated otherwise, we will count
up to $\mathbb{Q}$-isomorphism. When referring to specific elliptic curves we
will list them as they appear in Cremona’s tables [4], as we already did in
Theorem 1.
Let $m|n$ and denote by $Y_{1}(m,n)$ the affine modular curve whose
$K$-rational points classify isomorphism classes of the triples
$(E,P_{m},P_{n})$, where $E$ is an elliptic curve (over $K$) and $P_{m}$ and
$P_{n}$ are torsion points (over $K$) which generate a subgroup isomorphic to
$\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}/n\mathbb{Z}$. For simplicity, we write
$Y_{1}(n)$ instead of $Y_{1}(1,n)$. Let $X_{1}(m,n)$ be the compactification
of the curve $Y_{1}(m,n)$ obtained by adjoining its cusps.
Denote by $Y_{0}(n)$ the affine curve whose $K$-rational points classify
isomorphism classes of pairs $(E,C)$, where $E/K$ is an elliptic curve and $C$
is a cyclic (Gal$(\overline{K}/K)$-invariant) subgroup of $E$. Let $X_{0}(n)$
be the compactification of $Y_{0}(n)$.
Recall that a _$\mathbb{Q}$ -curve_ is an elliptic curve $E/K$ over a number
field, such that it is $\overline{\mathbb{Q}}$-isogenuos to all of its
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-conjugates.
For computations we use Magma [1].
## 3\. Torsion of rational elliptic curves over quadratic fields
The results of this short section will be needed in the proof of Theorem 1,
but are also interesting in their own right. They also provide a nice
introductory exercise for the much harder cubic fields case.
###### Theorem 2.
Let $E$ be a rational elliptic curve and $F$ a quadratic field.
* a)
The torsion of $E(F)$ is isomorphic to one of the following groups
(5) $\mathbb{Z}/m\mathbb{Z},\text{ }m=1,\ldots,10,12,15,16$
---
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2m\mathbb{Z},\text{ }1\leq m\leq 6.$
$\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/3m\mathbb{Z},\text{ }m=1,2,$
$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}.$
* b)
Each of these groups, except for $\mathbb{Z}/15\mathbb{Z}$, appears as the
torsion structure over a quadratic field for infinitely many rational elliptic
curves $E$.
* c)
The elliptic curves $50B1$ and $50A3$ have $15$-torsion over
$\mathbb{Q}(\sqrt{5})$, $50B2$ and $450B4$ have $15$-torsion over
$\mathbb{Q}(\sqrt{-15})$. These are the only rational curves having non-
trivial $15$-torsion over any quadratic field.
###### Remark 3.
* a)
The elliptic curves 50B1 and 50A3 are twists by 5 of each other, and hence
become isomorphic over $\mathbb{Q}(\sqrt{5})$. Similarly, 50B2 and 450B4 are
twists of each other by $-15$ and become isomorphic over
$\mathbb{Q}(\sqrt{-15})$.
* b)
These elliptic curves are "exceptional curves", in the sense that they are the
only elliptic curves (not just rational) over the respective quadratic fields
with $15$-torsion (see [17, 34] for details).
Before we proceed with the proof of Theorem 2, we will prove the following
lemma.
###### Lemma 4.
Let $E$ be a rational elliptic curve. Then in the family of all quadratic
twists $E^{d}$ of $E$ (including $E$ itself) there is at most one elliptic
curve with nontrivial $n$-torsion for $n=5,7$, and at most $2$ curves with
$3$-torsion.
###### Proof.
We will often use the fact (see for example [14, Theorem 3 and Corollary 4])
that if $F=\mathbb{Q}(\sqrt{d})$, and $n$ an odd integer $>1$ then
(6) $E(F)[n]=E(\mathbb{Q})[n]\oplus E^{d}(\mathbb{Q})[n].$
Suppose that two twists $E^{d}$ and $E^{d^{\prime}}$ of $E$ have non-trivial
$n$-torsion, for $n=5$ or $7$. Then $E^{d^{\prime}}$ is a twist of $E^{d}$ by
$d^{\prime}/d$ and now (6) implies that
$E^{d}(\mathbb{Q}(\sqrt{d/d^{\prime}}))$ has full $n$-torsion, which is
impossible.
It is impossible that $E$ has 3 twists with $3$-torsion, as this would imply
that, by [8, Lemma 9], $E^{d}(F_{2})$ would contain
$(\mathbb{Z}/3\mathbb{Z})^{3}$ for some twist $E^{d}$ of $E$ and some
biquadratic extension $F_{2}$ of $\mathbb{Q}$. ∎
###### Proof of Theorem 2.
The possible torsion structures of a rational elliptic curve over a quadratic
field is obviously a subset of the list (2). The equality (6) rules out the
possibility of $n$-torsion for $n=11,13$. Note that the number of points of
order $2$ over $F$ on an elliptic curve
$E:y^{2}=f(x)=x^{3}+ax+b$
is equal to the number of roots of $f$ over $F$. It follows that if
$E(\mathbb{Q})$ has no $2$-torsion, then neither does $E(F)$.
Suppose $E(F)$ had $2n$-torsion, for $n=7$ or $9$. Then, as noted above, both
$E(\mathbb{Q})$ and $E^{d}(\mathbb{Q})$ would have a $2$-torsion point, and by
(6) it would follow that either $E(\mathbb{Q})$ or $E^{d}(\mathbb{Q})$ has a
point of order $n$ and therefore also a point of order $2n$, which is
impossible by Mazur’s theorem.
It can be seen from [12, Theorems 3.2., 3.3., 3.4., 3.5. and 3.6.] that there
exist infinitely many rational elliptic curves with torsion
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/10\mathbb{Z},\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/12\mathbb{Z},\ $
$\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z},\
\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/6\mathbb{Z}$ and
$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$.
In [24, Remark 2.6. (d)] one is given a construction which can be used to
construct infinitely many rational elliptic curves with $16$-torsion over
quadratic fields.
Finally, we wish to find all rational elliptic curves with $15$-torsion over
quadratic fields. Let $E/\mathbb{Q}$ be an elliptic curve which attains
$15$-torsion over a quadratic field. By (6) this implies that,
(7) $E(\mathbb{Q})_{tors}\simeq\mathbb{Z}/3\mathbb{Z}\text{ and
}E^{d}(\mathbb{Q})_{tors}\simeq\mathbb{Z}/5\mathbb{Z}$
or vice versa. Suppose without loss of generality that it is as in (7). It
also follows, since if $E$ has a $p$-isogeny, so do all the quadratic twists
$E^{d}$ of $E$, that $E$ has to have a $15$-isogeny. But there are only 4
families of twists of rational elliptic curves with a $15$-isogeny [30,
p.78–80], those with $j$-invariants
$-25/2,\ -349938025/8,\ -121945/32,\ 46969655/32768,$
which are the twists of the elliptic curves 50A1, 50A2, 50B1 and 50B2,
respectively.
By the division polynomial method we find that 50B1 has $15$-torsion only over
one quadratic field, namely $\mathbb{Q}(\sqrt{5})$, that 50B2 has $15$-torsion
only over one quadratic field, namely $\mathbb{Q}(\sqrt{-15})$, and that 50A1
and 50A2 have no twists with $5$-torsion, completing the proof of the theorem.
∎
## 4\. Auxiliary results
In this section we prove a series of results that we will need for the proof
of Theorem 1.
###### Lemma 5.
Let $F/\mathbb{Q}$ be a quadratic extension, $n$ an odd positive integer, and
$E/\mathbb{Q}$ an elliptic curve such that $E(F)$ contains
$\mathbb{Z}/n\mathbb{Z}$. Then $E/\mathbb{Q}$ has an $n$-isogeny.
###### Proof.
Factor $n$ as $n=\prod_{i=1}^{k}p_{i}^{e_{i}}$, where $p_{i}$ and $p_{j}$ are
distinct primes for $i\neq j$. As already mentioned in (6), if
$F=\mathbb{Q}(\sqrt{d})$, then for each $i=1,\ldots,k$
$E(F)[p_{i}^{e_{i}}]=E(\mathbb{Q})[p_{i}^{e_{i}}]\oplus
E^{d}(\mathbb{Q})[p_{i}^{e_{i}}].$
Thus either $E(\mathbb{Q})$ or $E^{d}(\mathbb{Q})$ has a point of order
$p_{i}^{e_{i}}$ and thus a
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant subgroup
generated by this point. If an elliptic curve has a
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant cyclic
subgroup of order $p_{i}^{e_{i}}$, then so does every twist of $E$ and hence
we conclude that $E$ has a
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant cyclic
subgroup of order $n$, proving the lemma. ∎
We will also extensively use the well-known classification of possible degrees
of cyclic isogenies over $\mathbb{Q}$.
###### Theorem 6 ([27, 19, 20, 21]).
Let $E/\mathbb{Q}$ be an elliptic curve with an $n$-isogeny. Then $n\leq 19$
or $n\in\\{21,25,27,37,43,67,163\\}$. If $E$ does not have complex
multiplication, then $n\leq 18$ or $n\in\\{21,25,37\\}$.
Next we show that from 2 independent isogenies of degrees $m$ and $n$ on an
elliptic curve $E$, one can deduce the existence of a $mn$-isogeny on an
isogenous curve $E^{\prime}$.
###### Lemma 7.
Let $E/F$ be an elliptic curve over a number field with $2$ independent
isogenies (the intersection of their kernels is trivial) of degrees $m$ and
$n$ (over $F$). Then $E$ is isogenous (over $F$) to an elliptic curve
$E^{\prime}/F$ with an $mn$-isogeny.
###### Proof.
Suppose $E$ has an $m$-isogeny $f:E\rightarrow E^{\prime}$ and an $n$-isogeny
$g:E\rightarrow E^{\prime\prime}$. We claim that
$h=g\circ\hat{f}:E^{\prime}\rightarrow E^{\prime\prime}$, where $\hat{f}$ is
the dual isogeny of $f$, is a cyclic $mn$-isogeny.
Suppose the opposite, that $h$ is not cyclic. Then for some integer $l\geq 2$
which divides both $m$ and $n$, $E^{\prime}[l]\subset\ker h$. Note that
$\hat{f}(E^{\prime}[l])\simeq\mathbb{Z}/l\mathbb{Z}\text{ and
}\hat{f}(E^{\prime}[l])\subset\ker f$
since $f\circ\hat{f}=[n]_{E^{\prime}}$. But since $\hat{f}(E^{\prime}[l])$
contains non-zero points and $\ker g$ and $\ker f$ have trivial intersection,
this means that $g$ cannot send $\hat{f}(E^{\prime}[l])$ to $0$. Therefore
$h(E^{\prime}[l])\neq 0$, which is a contradiction. ∎
The following four results will be useful in controlling the 2-primary torsion
of rational elliptic curves over cubic fields.
###### Lemma 8 ([34, Lemma 1]).
If $E(\mathbb{Q})$ has a nontrivial $2$-Sylow subgroup, $E(K)$ has the same
$2$-Sylow subgroup as $E(\mathbb{Q})$.
###### Proposition 9.
Let $M\neq\mathbb{Q}(i)$, and let $E/\mathbb{Q}$ be an elliptic curve such
that $E(M)[2]=0$. Then the $2$-Sylow subgroup of $E(L)$ is either trivial or
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
###### Proof.
Suppose $E(L)[2]\neq 0$. Write
$E:y^{2}=f(x)=x^{3}+ax+b,$
where $f(x)$ is irreducible over $M$. As $L/M$ is Galois, it follows that
since $f$ has one root over $L$, all the roots of $f$ are defined over $L$.
Hence $E(L)[2]\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
Suppose that $E(L)$ has a point of order 4. As $L$ does not contain $i$, it
follows that the only possibility for $E(L)$ to have a $4$-torsion point is
that $E(L)[4]\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$.
We now prove that a group of order 3 has to fix a line of
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. Let
$G:=\operatorname{Gal}(L/M)$. We have the short exact sequence
(8) $0\rightarrow E(L)[2]\rightarrow E(L)[4]\rightarrow
E(L)[4]/E(L)[2]\rightarrow 0$
and it follows that
(9) $0\rightarrow E(L)[2]^{G}\rightarrow
E(L)[4]^{G}\rightarrow(E(L)[4]/E(L)[2])^{G}\rightarrow H^{1}(G,E(L)[2])$
It is easy to compute that $H^{1}(G,E(L)[2])=0$, and since $E(L)[4]/E(L)[2]$
is a group of order $2$, it follows that
$(E(L)[4]/E(L)[2])^{G}\simeq\mathbb{Z}/2\mathbb{Z},$
from which we conclude that $E(L)[4]^{G}\neq 0$. We conclude that $E(M)$ has a
$2$-torsion point, which is a contradiction. ∎
###### Proposition 10.
Let $M=\mathbb{Q}(i)$. Let $E/\mathbb{Q}$ be an elliptic curve with no
$\mathbb{Q}$-rational points of order $2$. Then
* a)
If $E(K)$ has a point of order $4$, then
$\Delta(E)\in-1\cdot(\mathbb{Q}^{*})^{2},$ $j(E)=-4t^{3}(t+8)$ for some
$t\in\mathbb{Q}$ and
$E(L)[4]\simeq\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$.
* b)
$E(K)$ has no points of order $8$.
###### Proof.
First note that from the assumption that $E(\mathbb{Q})[2]=0$, it follows that
$E(\mathbb{Q}(i))[2]=0$. Suppose $E(L)[2]\neq 0$. Since $L/\mathbb{Q}(i)$ is
Galois, it follows that
$E(L)[2]\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
If $E(L)$ has a point of order 4, by the same argument as in the proof of
Proposition 9 it follows that $E(L)[4]$ cannot be
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. Thus
$E(L)[4]\simeq\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z},$
and $\mathbb{Q}(E[2])=\mathbb{Q}(E[4]).$ Note that for any elliptic curve
$E^{\prime}$, the field $\mathbb{Q}(E^{\prime}[2])$ contains
$\mathbb{Q}(\sqrt{\Delta})$ and $\mathbb{Q}(E^{\prime}[4])$ contains
$\mathbb{Q}(i)$, and since $\mathbb{Q}(E[2])$ is a $S_{3}$ extension of $E$,
it follows that $\Delta$ is a square in $\mathbb{Q}(i)$, but not in
$\mathbb{Q}$, i.e. $\Delta(E)\in-1\cdot(\mathbb{Q}^{*})^{2}.$
By [6, Lemma], since $\operatorname{Gal}(\mathbb{Q}(E[4])/\mathbb{Q})\simeq
S_{3}$, which is isomorphic to a subgroup of
$H_{24}=\mathbb{Z}/3\mathbb{Z}\rtimes D_{8}$, it follows that
$j(E)=-4t^{3}(t+8)$ for some $t\in\mathbb{Q}$. This concludes the proof of a).
Suppose now $E(K)$ has a point of order 8. It follows that
$E(L)[8]\simeq\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}$. Let
$G:=\operatorname{Gal}(L/M)$ and take the short exact sequence
(10) $0\rightarrow E(L)[2]\rightarrow E(L)[8]\rightarrow
E(L)[8]/E(L)[2]\rightarrow 0.$
It follows that
(11) $0\rightarrow E(L)[2]^{G}\rightarrow
E(L)[8]^{G}\rightarrow(E(L)[8]/E(L)[2])^{G}\rightarrow H^{1}(G,E(L)[2]).$
Note that
$E(L)[8]/E(L)[2]\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ and
since we have shown in the proof of Proposition 9 that
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ has a $G$-invariant line,
it follows that $(E(L)[8]/E(L)[2])^{G}\neq 0$. Now from the fact that
$H^{1}(G,E(L)[2])=0$ it follows that $E(L)[8]^{G}\neq 0$, from which it
follows $E(M)[2]\neq 0$, which is a contradiction. ∎
###### Remark 11.
Note that there exist elliptic curves such that
$\mathbb{Q}(E[2])=\mathbb{Q}(E[4])$ and
$\operatorname{Gal}(\mathbb{Q}(E[2])/\mathbb{Q})\simeq\operatorname{Gal}(\mathbb{Q}(E[4])/\mathbb{Q})\simeq
S_{3}.$
The elliptic curve 1936D1 is such a curve.
We can combine Propositions 9 and 10 into the following corollary.
###### Corollary 12.
Let $E/\mathbb{Q}$ be an elliptic curve such that $E(\mathbb{Q})$ has no
points of order $4$. Then $E(K)$ has no $8$-torsion and has a point of order
$4$ only if $E(\mathbb{Q})[2]=0$, $M=\mathbb{Q}(i)$,
$\Delta(E)\in-1\cdot(\mathbb{Q}^{*})^{2},$ and $j(E)=-4t^{3}(t+8)$ for some
$t\in\mathbb{Q}$.
###### Proof.
If $E(\mathbb{Q})$ has a $2$-torsion point, the statement follows from Lemma
8.
If $E(\mathbb{Q})$ has no $2$-torsion, the statement follows from Propositions
9 and 10. ∎
The next step towards the proof of Theorem 1 is to control the growth of the
$3$-torsion, for which the following two propositions will be useful.
###### Lemma 13.
If $E(M)$ does not have a point of order $3$, neither does $E(L)$.
###### Proof.
Let $G=\operatorname{Gal}(L/M)$. Then $E(L)[3]$ is $\mathbb{F}_{3}$-linear
representation of $G$. Thus if $E(L)[3]\neq 0$, then $E(L)[3]^{G}=E(M)[3]\neq
0$ by [39, Proposition 26, p.64.]. ∎
###### Proposition 14.
Suppose $E(K)$ has a point of order $9$. Then $E/\mathbb{Q}$ has an isogeny of
degree $9$ or $2$ independent isogenies of degree $3$.
###### Proof.
Suppose the opposite, that $E/\mathbb{Q}$ does not have an isogeny of degree 9
nor 2 independent isogenies of degree 3. Let
$\langle\sigma\rangle=\operatorname{Gal}(L/M)$,
$\langle\tau\rangle=\operatorname{Gal}(L/K)$ and $P\in E(K)$ be a point of
order 9. It is easy to see that $E(L)$ has a $9$-torsion point, and that
$E(M)$ has a point of order 3 by Lemma 13. Also, $E(M)$ cannot have a point of
order 9, since this would imply that $E/\mathbb{Q}$ has a 9-isogeny by Lemma
5, which would contradict our assumption.
We examine how $\sigma$ acts on $P$. Let $E[9]=\langle P,Q\rangle$ and
$P^{\sigma}=\alpha P+\beta Q$.
Suppose $\beta=0$. Then $\sigma$ fixes $\langle P\rangle$ and since $P$ is a
$K$-rational point, $\tau$ also fixes $\langle P\rangle$. Since $\sigma$ and
$\tau$ generate $\operatorname{Gal}(L/\mathbb{Q})$, it follows that that
$E/\mathbb{Q}$ has a 9-isogeny, contradicting our assumption.
Note that $P^{\sigma}\in E(L)$ so $(9-\alpha)P+P^{\sigma}=\beta Q\in E(L)$,
and it follows that $\beta$ has to be $3$ or $6$, otherwise the full
$9$-torsion would be defined over $L$, which would further imply that
$L=\mathbb{Q}(\zeta_{9})$, which is impossible since
$\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/3\mathbb{Z}$ or $S_{3}$ and
$\operatorname{Gal}(\mathbb{Q}(\zeta_{9})/\mathbb{Q})\simeq\mathbb{Z}/6\mathbb{Z}$.
Furthermore, it follows that
$E(L)[9]\simeq\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/9\mathbb{Z}$ and from
this $M=\mathbb{Q}(\sqrt{-3})$.
Let $E[3]=\langle P^{\prime},Q^{\prime}\rangle$. By Lemma 13, $E(M)$ has non-
trivial $3$-torsion; suppose $P^{\prime}\in E(M)$. Now
$G=\operatorname{Gal}(L/M)$ acts on $\langle Q^{\prime}\rangle$, and since $G$
is a group of order 3, it follows that $\langle Q^{\prime}\rangle^{G}=\langle
Q^{\prime}\rangle$ and hence $E(M)$ has full $3$-torsion. Since
$M=\mathbb{Q}(\sqrt{-3})$, it holds that $E(M)[3]=E(\mathbb{Q})[3]\oplus
E^{-3}(\mathbb{Q})[3]$ by (6), and since an elliptic curve over $\mathbb{Q}$
cannot have full $3$-torsion, it follows that
$E(\mathbb{Q})[3]\simeq\mathbb{Z}/3\mathbb{Z}$. Suppose without loss of
generality that $P^{\prime}\in E(\mathbb{Q})$. Then $\langle
P^{\prime}\rangle$ is obviously a
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant subgroup and
we will show that there exists another orthogonal to it. Let
$\langle\mu\rangle=\operatorname{Gal}(M/\mathbb{Q})$. Then
$Q^{\prime\mu}=\alpha P^{\prime}+\beta Q^{\prime},$
$Q^{\prime\mu^{2}}=Q^{\prime}=\alpha(1+\beta)P^{\prime}+\beta^{2}Q^{\prime},$
where $\alpha,\beta\in\\{0,1,2\\}$. It follows that $\beta=1$ or $2$. If
$\beta=1$ then $\alpha=0$, which would imply that $E(\mathbb{Q})$ has full
$3$-torsion, which is impossible. Hence $\beta=2$ and $\langle
Q^{\prime}+2\alpha P^{\prime}\rangle$ is a
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant subgroup
orthogonal to $\langle P^{\prime}\rangle$, which shows that $E$ has 2
independent 3-isogenies, giving a contradiction. ∎
We now move to the study of the $p$-torsion of $E(K)$ for $p>3$.
###### Lemma 15.
For $p>3$ prime, $E(L)[p]=0\text{ or }\mathbb{Z}/p\mathbb{Z}$.
###### Proof.
This follows from the fact that
$E(L)[p]\simeq\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}$ would, by
the existence of the Weil pairing, require the $p$-th cyclotomic field
$\mathbb{Q}(\zeta_{p})$ to be contained in $L$, which is impossible. ∎
Next we study for which $p$, $E(M)[p]=0$ implies $E(L)[p]=0$. We prove a more
general statement, that applies beyond the case of cubic extensions.
###### Lemma 16.
Let $p,q$ be odd distinct primes, $F_{2}/F_{1}$ a Galois extension of number
fields such that
$\operatorname{Gal}(F_{2}/F_{1})\simeq\mathbb{Z}/q\mathbb{Z}$, and $E/F_{1}$
an elliptic curve with no $p$-torsion over $F_{1}$. Then if $q$ does not
divide $p-1$ and $\mathbb{Q}(\zeta_{p})\not\subset F_{2}$, then
$E(F_{2})[p]=0$.
###### Proof.
Since if one point of order $p$ is defined over $F_{2}$, then so are all its
multiples, it follows that either $p-1$ or $p^{2}-1$ points of order $p$ are
defined over $F_{2}$, but not over $F_{1}$. But it is impossible that all
$p^{2}-1$ appear because of $\mathbb{Q}(\zeta_{p})\not\subset F_{2}$.
Let $\langle\sigma\rangle=\operatorname{Gal}(F_{2}/F_{1})$, $P$ be a point of
order $p$ in $E(F_{2})$ and then note that $P^{\sigma}\neq P$, and that
$\operatorname{Gal}(F_{2}/F_{1})$ acts on $\langle P\rangle$. We have a
homomorphism
$\mathbb{Z}/q\mathbb{Z}\simeq\operatorname{Gal}(F_{2}/F_{1})\rightarrow\operatorname{Aut}(\langle
P\rangle)\simeq\operatorname{Aut}(\mathbb{Z}/p\mathbb{Z})\simeq(\mathbb{Z}/p\mathbb{Z})^{\times},$
so either $\operatorname{Gal}(F_{2}/F_{1})$ acts trivially on $P$ and
therefore $P\in E(F_{1})[p]$ or $q$ divides $p-1$.
∎
In this paper we will use only the special case $q=3$, $F_{1}=M$ and $F_{2}=L$
of Lemma 16. We also need to show the non-existence of points of order $p^{2}$
in $E(L)$. Again, we prove a more general statement.
###### Lemma 17.
Let $p$ be an odd prime number, $q$ a prime not dividing $p$, $F_{2}/F_{1}$ a
Galois extension of number fields such that
$\operatorname{Gal}(F_{2}/F_{1})\simeq\mathbb{Z}/q\mathbb{Z}$, $E/F_{1}$ an
elliptic curve, and suppose $E(F_{1})\supset\mathbb{Z}/p\mathbb{Z}$,
$E(F_{1})\not\supset\mathbb{Z}/p^{2}\mathbb{Z}$ and $\zeta_{p}\not\in F_{2}$.
Then $E(F_{2})\not\supset\mathbb{Z}/p^{2}\mathbb{Z}$.
###### Proof.
Suppose $\mathbb{Z}/p^{2}\mathbb{Z}\subset E(F_{2})$. By the assumption
$\zeta_{p}\not\in F_{2}$ and the existence of the Weil pairing, it follows
that $E(F_{2})[p^{2}]\simeq\mathbb{Z}/p^{2}\mathbb{Z}$. Let $P\in E(F_{1})$ be
of order $p$ and $\langle\sigma\rangle=\operatorname{Gal}(F_{2}/F_{1})$. Let
$S=\\{Q\in E(F_{2})|pQ=P\\}.$
The set $S$ has $p$ elements, on which $\operatorname{Gal}(F_{2}/F_{1})$ acts.
By the Orbit Stabilizer Theorem, the orbits under the action of
$\operatorname{Gal}(F_{2}/F_{1})$ have to have length $q$, since if a point in
$S$ was left fixed, it would mean that it is $F_{1}$-rational. This implies
that $S$ decomposes into orbits of $q$ elements each, which is a contradiction
with our assumption that $q$ does not divide $p$. ∎
We will again use Lemma 17 only in the special case $q=3$, $F_{1}=M$ and
$F_{2}=L$. For $n$ coprime to 6, the existence of a point of order $n$ in
$E(K)$ will imply the existence of an $n$-isogeny over $\mathbb{Q}$, as the
following lemma shows.
###### Lemma 18.
Let $n$ be an odd integer not divisible by $3$ and suppose $E(K)$ has a point
of order $n$. Then $E/\mathbb{Q}$ has an isogeny of degree $n$.
###### Proof.
First note that $E(L)$ has a point $P$ of order $n$. Let
$\langle\sigma\rangle=\operatorname{Gal}(L/M),\
\langle\tau\rangle=\operatorname{Gal}(L/K),\
\langle\sigma,\tau\rangle=\operatorname{Gal}(L/\mathbb{Q}).$
As $P$ is $K$-rational, it follows that $P^{\tau}=P$. Let $E[n]=\langle
P,Q\rangle$ and $P^{\sigma}=\alpha P+\beta Q\in E(L)$. We have
$(n-\alpha)P+P^{\sigma}=\beta Q\in E(L)$. If $\beta\not\equiv 0\pmod{n}$ then
$\beta Q$ is a point of order $l|n$ not contained in $\langle P\rangle$, from
which it follows that that $E(L)$ has full $l$-torsion, which is impossible by
Lemma 15. We conclude that $\beta\equiv 0\pmod{n}$.
Hence
$P^{\mu}=kP\text{ for all }\mu\in\operatorname{Gal}(L/\mathbb{Q}),$
and since the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
on $\langle P\rangle$ factors through $\operatorname{Gal}(L/\mathbb{Q})$, it
follows that
$P^{\mu}=kP\text{ for all
}\mu\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),$
which means that $E/\mathbb{Q}$ has an $n$-isogeny. ∎
In the special case when $K=L$, the conclusion of Lemma 18 follows when $n$ is
a multiple of 3.
###### Lemma 19.
Suppose $K=L$, i.e. $K/\mathbb{Q}$ is a Galois extension. Let $n$ be an odd
integer and suppose $E(K)$ has a point of order $n$. Then $E/\mathbb{Q}$ has
an isogeny of degree $n$.
###### Proof.
The proof follows by a similar argument as in Lemma 18 and by using the fact
that $\mathbb{Q}(\zeta_{k})$ is not contained in $K$, for any divisor $k\geq
3$ of $n$. ∎
## 5\. Proof of Theorem 1
We are now ready to prove Theorem 1, which we will do in a series of Lemmas
and Propositions. Recall that if a point in $E(K)$ has prime order $p$, then
$p\leq 13$ (see [36, 37]).
###### Lemma 20.
The $3$-Sylow subgroup of $E(K)$ is isomorphic to a subgroup of
$\mathbb{Z}/9\mathbb{Z}$.
###### Proof.
This follows by [31, Theorem (4.1)] and by the existence of the Weil pairing.
∎
###### Lemma 21.
The $5$-Sylow groups of $E(\mathbb{Q})$ and $E(K)$ are equal.
###### Proof.
First note that $E(L)$ cannot have full $5$-torsion by Lemma 15.
Suppose $E(\mathbb{Q})[5]\simeq\mathbb{Z}/5\mathbb{Z}$. Then
$E(M)\simeq\mathbb{Z}/5\mathbb{Z}$ and Lemma 17 shows that $E(L)$ cannot have
a point of order 25, and hence the $5$-Sylow groups of $E(\mathbb{Q})$ and
$E(L)$ (and hence also $E(K)$) are equal and all isomorphic to
$\mathbb{Z}/5\mathbb{Z}$.
Suppose $E(\mathbb{Q})[5]=0$. If $E(M)[5]=0$, then $E(L)[5]=0$ (and hence
$E(K)[5]=0$) by Lemma 16. If $E(M)[5]\simeq\mathbb{Z}/5\mathbb{Z}$, then
$0=E(\mathbb{Q})[5]=E(M)[5]\cap E(K)[5]$, from which it follows that
$E(K)[5]=0$ (note that $E(K)$ here cannot have a $5$-torsion point that is not
in $E(M)$ because then $E(L)$ would have full $5$-torsion). ∎
###### Lemma 22.
There are no points of order $11$ in $E(K)$.
###### Proof.
By Lemma 16, $E(L)$ has $11$-torsion only if $E(M)$ has $11$-torsion, which is
never true, as rational elliptic curves cannot have $11$-torsion over
$\mathbb{Q}$ or over a quadratic field, by Theorem 2. Hence $E(K)$ has no
$11$-torsion. ∎
###### Lemma 23.
$E(K)$ has no points of order $35,$ $49,$ $65,$ $91$ or $169$.
###### Proof.
This follows by Lemma 18 and Theorem 6. ∎
###### Lemma 24.
There exists no rational elliptic curves with points of order $15$ or $16$
over a cubic field.
###### Proof.
As $E(L)[5]=E(M)[5]$ and $E(M)[3]=0\Longrightarrow E(L)[3]=0$ by Lemmas 16 and
13, it follows that the only way for $E(L)$ to have $15$-torsion is for $E(M)$
to have $15$-torsion.
It follows that, by Theorem 2 c), $E$ is 50B1, 50B2, 50A3 or 450B4. By the
division polynomial method, we find that none of these curves have points of
order $15$ over any cubic field.
If $E(\mathbb{Q})$ has a non-trivial $2$-Sylow group, then the $2$-Sylow
subgroups of $E(\mathbb{Q})$ and $E(K)$ are equal by Lemma 8 and hence $E(K)$
has no points of order $16$. If the $2$-Sylow subgroup of $E(\mathbb{Q})$ is
trivial, then by Corollary 12, $E(K)$ has no points of order $8$. ∎
###### Proposition 25.
The elliptic curve $162B1$ has torsion isomorphic to $\mathbb{Z}/21\mathbb{Z}$
over $\mathbb{Q}(\zeta_{9})^{+}$. This is the unique pair $(E,K)$ of a
rational elliptic curve $E$ and a cubic field such that $E(K)$ has a point of
order $21$.
###### Proof.
By Lemma 13, if $E(M)_{tors}=0,\text{ or }\mathbb{Z}/7\mathbb{Z}$, then
$E(L)[3]=0$. Suppose now that $E(K)[21]\supset\mathbb{Z}/21\mathbb{Z}$. By
Lemma 18, it follows that $E/\mathbb{Q}$ has an isogeny of degree 7, and since
it has a $3$-torsion point, it also has a $21$-isogeny. There are 4 curves (up
to $\overline{\mathbb{Q}}$-isomorphism) with a 21-isogeny [30, p.78–80]. These
are the curves in the 162B or 162C isogeny classes. Note that the 162B isogeny
class is a $-3$ twist of the 162C class.
By the division polynomial method we find that the only twists of the curves
in the 162B and 162C isogeny classes with non-trivial $3$-torsion are the
curves 162C1, 162C3, 162B1 and 162B3. By the division polynomial method, we
find that only 162B1 has a $7$-torsion point over a cubic field, the field
$\mathbb{Q}(\zeta_{9})^{+}$, which is generated by $x^{3}-3x^{2}+3$. ∎
Note that since 162B1 is the unique curve with $21$-torsion over any cubic
field and since it has torsion exactly $\mathbb{Z}/21\mathbb{Z}$, this means
that there exist no points of order $21n$ on rational elliptic curves over
cubic fields, for any integer $n\geq 2$.
###### Remark 26.
Note that in [34, Remark 2] we misstated that from the fact that
(12) $X_{0}(21)(\mathbb{Q}(\zeta_{9})^{+})=X_{0}(21)(\mathbb{Q})$
(note that $K_{7}=\mathbb{Q}(\zeta_{9})^{+},$ using the notation of [34]) one
can conclude that there are no elliptic curves with $21$-torsion over
$\mathbb{Q}(\zeta_{9})^{+}$, while what should have been written is that from
(12) we can determine whether $21$-torsion appears over
$\mathbb{Q}(\zeta_{9})^{+}$ by checking whether twists of rational elliptic
curves with rational $21$-isogeny have $21$-torsion over
$\mathbb{Q}(\zeta_{9})^{+}$. The point of the remark, that if we could find
out whether $Y_{1}(25)(\mathbb{Q}(\zeta_{9})^{+})=\emptyset$, then we could
completely classify the possible torsion groups of elliptic curves over
$\mathbb{Q}(\zeta_{9})^{+}$, remains true.
###### Lemma 27.
$E(K)$ does not have points of order $20$, $24$, $26$, $28$, $36$ or $39$.
###### Proof.
Suppose $E(K)$ has a $20$-torsion point. Then $E(M)$ has to have a $5$-torsion
point by Lemma 16, and by (6) either $E^{d}(\mathbb{Q})$ or $E(\mathbb{Q})$
has a $5$-torsion point, where $M=\mathbb{Q}(\sqrt{d})$. We can suppose
without loss of generality that $E(\mathbb{Q})$ has a $5$-torsion point,
otherwise we proceed with the proof using $E^{d}$ instead of $E$.
Thus, by [23, Table 3.], $E/\mathbb{Q}$ has a model
(13) $E:y^{2}+(1-t)xy-ty=x^{3}-tx^{2}\text{ for some }t\in\mathbb{Q}^{*}.$
If $E(\mathbb{Q})$ had a $4$-torsion point, it would follow that it has a
$20$-torsion point which is impossible. Thus by Corollary 12, it follows that
$E(\mathbb{Q})[2]=0$ and $\Delta(E)=-k^{2}$ for some $k\in\mathbb{Q}^{*}$.
Hence
$-k^{2}=\Delta(E)=t^{5}(t^{2}-11t-1)$
for some $t\in\mathbb{Q}^{*}$, which is equivalent to
$X:y^{2}=t^{3}+11t^{2}-t$
having rational points with $t\in\mathbb{Q}^{*}$. But
$X(\mathbb{Q})=\\{0,(0,0)\\}$.
If $E(K)$ had a $24$-torsion point, then Corollary 12 would imply that
$E(\mathbb{Q})$ has a non-trivial $2$-Sylow group, and from this fact and
Lemma 8 it follows that $E(\mathbb{Q})$ has a point of order 8. By Lemma 13,
$E(M)$ has $3$-torsion point and hence a $24$-torsion point which is in
contradiction with Theorem 2, since $M/\mathbb{Q}$ is quadratic.
Suppose $E(K)\supset\mathbb{Z}/26\mathbb{Z}$. It follows by Lemma 18 that
$E/\mathbb{Q}$ has an isogeny of degree 13, and this implies that
$E(\mathbb{Q})[2]=0$, since by Theorem 6 there are no elliptic curves with
26-isogenies over $\mathbb{Q}$. Because $E$ has a 13-isogeny, but no
$13$-torsion over $\mathbb{Q}$, it follows that $\mathbb{Q}(\langle
P\rangle)/\mathbb{Q}$ is Galois and non-trivial. Since this is contained in
$K$ and $K$ is of prime degree, we must have $\mathbb{Q}(\langle P\rangle)=K$
and therefore $K/\mathbb{Q}$ is Galois.
From the facts that $K/\mathbb{Q}$ is cubic and Galois, $E(\mathbb{Q})[2]=0$
and $E(K)[2]\neq 0$, it follows that $E(K)$ has full $2$-torsion and hence
$E(K)\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/26\mathbb{Z}$.
By [18, 29] an elliptic curve with a $13$-isogeny over $\mathbb{Q}$ has
$j$-invariant
$j=\frac{(t^{2}+5t+13)(t^{4}+7t^{3}+20t^{2}+19t+1)^{3}}{t},\
t\in\mathbb{Q}^{\times}.$
It follows that $E$ is a quadratic twist of an elliptic curve $E_{0}$ with
discriminant
$\Delta(E_{0})=t(t^{2}+5t+13)^{2}(t^{4}+7t^{3}+20t^{2}+19t+1)^{6}(t^{2}+6t+13)^{9}$
$\times(t^{6}+10t^{5}+46t^{4}+108t^{3}+122t^{2}+38t-1)^{6},$
and thus $\Delta(E)=u^{12}\Delta(E_{0})$ for some $u\in\mathbb{Q}^{\times}$.
The curve $E$ gains full $2$-torsion over a cubic field only if $\Delta(E)$ is
a square which happens only if
$X:y^{2}=x(x^{2}+6x+13)\text{ for some }x,y\in\mathbb{Q}^{\times}$
has solutions. But $X(\mathbb{Q})=\\{0,(0,0)\\}$, and hence that is
impossible.
Suppose $E(K)$ has a $28$-torsion point. It follows that $E/\mathbb{Q}$ has a
7-isogeny by Lemma 18. If $E(\mathbb{Q})$ had a $4$-torsion point, this would
imply that $E/\mathbb{Q}$ has a 28-isogeny, which is impossible by Theorem 6.
Thus $E(\mathbb{Q})$ does not have a $4$-torsion point and by Corollary 12 it
follows that $K$ is not a Galois extension of $\mathbb{Q}$ and
$\Delta(E)=-k^{2}$, for some $k\in\mathbb{Q}^{*}$.
Note that since $E$ cannot have 2 independent rational 7-isogenies (see [23,
Proposition III.2.1.]), it follows that the kernel of the rational 7-isogeny
is equal to $E(K)[7]$. As the points in the kernel of the rational 7-isogeny
are defined over a Galois extension of $\mathbb{Q}$, it follows that they must
be defined already over $\mathbb{Q}$, since they cannot be $K$-rational but
not $\mathbb{Q}$-rational (because $K/\mathbb{Q}$ is not Galois).
Hence $E(\mathbb{Q})$ has points of order 7, and by [23, Table 3.], it follows
that $E$ is of the form
(14) $E:y^{2}+(-t^{2}+t+1)xy+(-t^{3}+t^{2})y=x^{3}+(-t^{3}+t^{2})x^{2},\text{
for some }t\in\mathbb{Q},t\not\in\\{0,1\\}.$
and that
$-k^{2}=\Delta(E)=t^{7}(t-1)^{7}(t^{3}-8t^{2}+5t-1),$
which is equivalent to
$X:y^{2}=t(t+1)(t^{3}+8t^{2}+5t+1),$
having rational points such that $t\not\in\\{0,-1\\}$. But the Jacobian $J$ of
$X$ has rank 0 over $\mathbb{Q}$ and it is an easy computation in Magma (using
the Chabauty0 function) to show that
$X(\mathbb{Q})=\\{\infty,(0,0),(-1,0)\\},$
and thus completing the proof that there are no rational elliptic curves with
$28$-torsion over a cubic field.
Suppose $E(K)$ has a $36$-torsion point. This implies that, by Corollary 12,
$E(\mathbb{Q})$ has either a point of order $4$ or no $2$-torsion. Also, by
Proposition 14, $E/\mathbb{Q}$ has to have a $9$-isogeny or $2$ independent
isogenies of degree $3$.
Suppose first that $E(\mathbb{Q})$ has a $4$-torsion point. If $E/\mathbb{Q}$
had a $9$-isogeny, this would imply, by Lemma 7 that $E/\mathbb{Q}$ is
isogenous to a rational curve which has a 36-isogeny, which is impossible by
Theorem 6. On the other hand, by [23, Main Result 2.], an elliptic curve with
2 independent $3$-isogenies cannot have a $4$-torsion point.
Suppose now that $E(\mathbb{Q})$ has no $2$-torsion. We split this case into
$2$ subcases: when $E(\mathbb{Q})$ has a rational $9$-isogeny, and when $E$
has $2$ independent rational isogenies of degree $3$.
Suppose $E$ has a $9$-isogeny. Then, by [23, 15], $E$ is a twist of an
elliptic curve $E_{0}$ with $j$-invariant
(15) $j=\frac{t^{12}-72t^{9}+1728t^{6}-13824t^{3}}{t^{3}-27},\
t\in\mathbb{Q}\backslash\\{0,3\\}$
and
$\Delta(E_{0})=2^{12}3^{6}(t^{3}-27),$
and since $E$ has to be a twist of $E_{0}$, it follows that
$\Delta(E)=u^{12}\Delta(E_{0})$, for some $u\in\mathbb{Q}^{\times}$. By
Corollary 12, the curve $E$ gains a $4$-torsion point over a cubic field only
if $-y^{2}=\Delta(E)\neq 0$ has solutions, which is equivalent to
$X:y^{2}=t^{3}+27,\ t\in\mathbb{Q}\backslash\\{0,-3\\},y\in\mathbb{Q}$
having a solution. But $X(\mathbb{Q})=\\{0,(-3,0)\\}$.
Suppose now that $E$ has 2 independent rational 3-isogenies and that $E$ gains
a point of order 9 over $K$. By the same type of argument as in the proof of
Proposition 14, it follows that $M=\mathbb{Q}(\sqrt{-3})$. Since $E$ has no
rational $4$-torsion, but has a $4$-torsion point over $K$, by Corollary 12,
it follows that $M=\mathbb{Q}(i)$, which is a contradiction.
If $E(K)$ had a $39$-torsion point, this would imply that $E(M)$ has a
$3$-torsion point by Lemma 13, from which it would follow, by Lemma 5, that
$E/\mathbb{Q}$ has a 3-isogeny. Also, $E/\mathbb{Q}$ would have a $13$-isogeny
by Lemma 18. But this would imply that there is an elliptic curve over
$\mathbb{Q}$ with a 39-isogeny by Lemma 7, which is impossible by Theorem 6. ∎
###### Lemma 28.
$E(K)$ cannot have subgroups isomorphic to
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/10\mathbb{Z}$,
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/12\mathbb{Z}$ or
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/18\mathbb{Z}$.
###### Proof.
Suppose $E(K)\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/10\mathbb{Z}$. By
Lemma 16, this implies that $E(\mathbb{Q})$ has a $5$-torsion point. This
implies, by [23, Table 3.], that $E$ has a model as in (13). Since
$E(\mathbb{Q})$ cannot contain
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/10\mathbb{Z}$, either
$E(\mathbb{Q})[2]=0$ or $\mathbb{Z}/2\mathbb{Z}$. But if
$E(\mathbb{Q})[2]=\mathbb{Z}/2\mathbb{Z}$, then $E$ would gain full
$2$-torsion over a quadratic instead of a cubic field. We conclude that
$E(\mathbb{Q})[2]=0$, and that $E$ gains full $2$-torsion over the cubic field
$K$. This happens if and only if the discriminant $\Delta(E)$ is a square in
$\mathbb{Q}^{\times}$, i.e. the equation
$\Delta(E)=y^{2}=t^{7}-11t^{6}-t^{5},\text{ for some
}y,t\in\mathbb{Q}^{\times}$
has solutions. Dividing out by $t^{4}$ and by change of variables we get
$X:A^{2}=t^{3}-11t^{2}-t,\text{ for some }A,t\in\mathbb{Q}^{\times}.$
The curve $X$ is an elliptic curve and
$X(\mathbb{Q})\simeq\mathbb{Z}/2\mathbb{Z}$, where the rational points are $0$
and $(0,0)$. Thus
$E(K)\not\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/10\mathbb{Z}$.
Suppose $E(K)\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/12\mathbb{Z}$.
Also, $E(\mathbb{Q})$ has to either have a $4$-torsion point or no $2$-torsion
by Lemma 8 and Corollary 12.
If $E(\mathbb{Q})$ had a $4$-torsion point, then by Lemma 8,
$E(\mathbb{Q})\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. We
see that $E(M)$ has a $3$-torsion point by Lemma 13 and hence by Lemma 5 it
follows that $E/\mathbb{Q}$ has a 3-isogeny and moreover a $4$-torsion point
is in the kernel of a rational 12-isogeny. It follows that $E/\mathbb{Q}$ has
a 12-isogeny and an independent 2-isogeny and now from Lemma 7 it follows that
there exists an elliptic curve over with a 24-isogeny over $\mathbb{Q}$, which
is in contradiction with Theorem 6.
If $E(\mathbb{Q})$ had trivial $2$-torsion, then $K/\mathbb{Q}$ would have to
be a Galois extension for $E(K)$ to have full $2$-torsion. But then by
Corollary 12, $E(K)$ cannot have points of order $4$, which is a
contradiction.
Suppose $E(K)\supset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/18\mathbb{Z}$. By
Lemma 14, $E/\mathbb{Q}$ has either a 9-isogeny or two independent isogenies
of degree 3.
Suppose now that $E(\mathbb{Q})$ has a $2$-torsion point. Then it follows, by
Lemma 8, that $E(\mathbb{Q})$ has full $2$-torsion. But an elliptic curve with
full $2$-torsion cannot have a 9-isogeny [23, Table 2.] nor 2 independent
3-isogenies [23, Proposition III.2.3]. Thus it follows that
$E(\mathbb{Q})[2]=0$ and from this that $K/\mathbb{Q}$ is a Galois extension.
Now it follows, by Lemma 19, that $E/\mathbb{Q}$ in fact has a 9-isogeny.
By [15, Appendix], since $E$ has a 9-isogeny, it is a twist of an elliptic
curve $E_{0}$ with $j$-invariant as given in (15) and
$\Delta(E_{0})=2^{12}3^{6}(t^{3}-27),$ and since $E$ has to be a twist of
$E_{0}$, it follows that $\Delta(E)=u^{12}\Delta(E_{0})$, for some
$u\in\mathbb{Q}^{\times}$. The curve $E$ gains full $2$-torsion over a cubic
field only if $\Delta(E)$ is a square, which is equivalent to
$X:y^{2}=t^{3}-27,\ t\in\mathbb{Q}\backslash\\{0,3\\},y\in\mathbb{Q}$
having a solution. But $X(\mathbb{Q})=\\{0,(3,0)\\}$, so there exist no such
curves. ∎
It is shown in Lemmas 8 and Proposition 10 that $2$-Sylow subgroup of
$E(K)_{tors}$ is contained in
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}$, in Lemma 20 that the
largest power of $3$ that can divide $|E(K)_{tors}|$ is $9$, and in Lemmas 21
and 23 that no powers of $5,$ $7$ and $13$ (apart from $5,$ $7$ and $13$
themselves) divide $|E(K)_{tors}|$. By [36, 37] and Lemma 22 the
aforementioned primes are the only ones that can divide $|E(K)_{tors}|$.
Now Lemma 23, 24 and 27 and Proposition 25 show that $|E(K)_{tors}|$ is
divisible by more than one element from the set $\\{3,5,7,13\\}$ only for the
pair $(E,K)$ from Proposition 25. The possible combinations of $2$-Sylow
subgroups and $p$-Sylow subgroups for $p=3,5,7,13$ are dealt with in Lemmas 27
and 28 and Proposition 29.
This completes the proof that the groups that appear as torsion groups of
rational elliptic curves over cubic fields are contained in the list (4). Note
first that by [13, Lemma 3.2 a)], all of the groups from the list (1) appear
infinitely often and the group $\mathbb{Z}/21\mathbb{Z}$ has already been
dealt with in Proposition 25.
Any elliptic curve $E/\mathbb{Q}$ with torsion isomorphic to
$\mathbb{Z}/9\mathbb{Z}$ over $\mathbb{Q}$ gains a $2$-torsion point over a
cubic field $K$ defined by the cubic polynomial $f(x)$, when $E$ is written in
short Weierstrass form $E:y^{2}=f(x)$. Then by Lemmas 27 and 28 it follows
that $E(K)_{tors}$ does not contain $\mathbb{Z}/36\mathbb{Z}$ or
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/18\mathbb{Z}$, respectively, and
therefore it follows that $E(K)_{tors}\simeq\mathbb{Z}/18\mathbb{Z}$.
It remains to prove, for each of the groups $T=\mathbb{Z}/13\mathbb{Z}$,
$\mathbb{Z}/14\mathbb{Z}$ and
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$, that there exist
infinitely many elliptic curves $E$ and cubic fields $K$ such that
$E(K)_{tors}\simeq T$.
We will deal with the groups $\mathbb{Z}/14\mathbb{Z}$ and
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$ simultaneously in the
following proposition.
###### Proposition 29.
There exists infinitely many elliptic curves $E/\mathbb{Q}$ such that there
exists a cubic field $K$ over which $E(K)_{tors}\simeq\mathbb{Z}/14\mathbb{Z}$
and there exists infinitely many elliptic curves $E/\mathbb{Q}$ such that
there exists a cubic field $K$ over which
$E(K)_{tors}\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$.
###### Proof.
Let $E/\mathbb{Q}$ be an elliptic curve such that
$E(\mathbb{Q})_{tors}\simeq\mathbb{Z}/7\mathbb{Z}$. By, [23, Table 3.], this
curve has the model as given in (14). If $E$ is written in short Weierstrass
form $y^{2}=f(x)$, then $E$ gains a point of order 2 over the cubic field $K$
generated by $f$. We will show that
$E(K)_{tors}\simeq\mathbb{Z}/14\mathbb{Z}$. By Lemma 27, $E(K)$ cannot have a
point of order 28, so it only remains to show that
$E(K)_{tors}\not\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$
If $E$ gained full $2$-torsion over $K$, this would imply that
$\Delta(E)=t^{7}(t-1)^{7}(t^{3}-8t^{2}+5t-1)$
is a square in $\mathbb{Q}$, which is equivalent to
$X:y^{2}=t(t-1)(t^{3}-8t^{2}+5t-1),$
having rational points such that $t\not\in\\{0,1\\}$. But the Jacobian $J$ of
$X$ has rank 0 over $\mathbb{Q}$ and it is an easy computation in Magma to
show that
$X(\mathbb{Q})=\\{\infty,(0,0),(1,0)\\},$
proving the claim.
To prove that there exist infinitely many curves $E/\mathbb{Q}$ such that for
each curve there exists a cubic field $K$ such that
$E(K)_{tors}\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$, we
note that every elliptic curve from the infinite family of elliptic curves
having torsion $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$ over a
cubic field from [11, Theorem 4.2] has rational $j$-invariant and the cubic
field over which it has $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/14\mathbb{Z}$
torsion has discriminant $(t^{6}+4t^{5}+13t^{4}-40t^{3}+19t^{2}+36t+31)^{2}$,
which implies that it is Galois. These are lengthy but completely
straightforward calculations, and hence we leave them out.
The fact that these curves have rational $j$-invariant does not yet prove that
the curves are rational, but just that they are quadratic twists by $\delta\in
O_{K}$ of some rational elliptic curve. We need to prove that in fact $\delta$
is rational.
Let $E_{1}$ be one of the curves from the family [11, Theorem 4.2] and let $E$
be a rational elliptic curve with the same $j$-invariant and denote
$\langle\sigma\rangle=\operatorname{Gal}(K/\mathbb{Q})$. As already noted
$E_{1}=E^{\delta}$. Let $P\in E^{\delta}(K)$ be a point of order 7. It follows
that $P^{\sigma}$ is a point of order 7 in $E^{\sigma(\delta)}$ and that
$P^{\sigma^{2}}$ is a point of order 7 in $E^{\sigma^{2}(\delta)}$.
Now we will show that in a family of quadratic twists over a cubic field $K$
there can be only one elliptic curve with a point of order 7. Suppose that
$E_{2}/K$ and $E_{2}^{d}/K$ are quadratic twists by a $d\in K^{\times}$ which
are not $K$-isomorphic, and that both have a point of order 7. Then it follows
that
$E_{2}(K(\sqrt{d}))[7]\simeq E_{2}(K)[7]\oplus
E_{2}^{d}(K)[7]\supset\mathbb{Z}/7\mathbb{Z}\oplus\mathbb{Z}/7\mathbb{Z},$
or in other words $E$ has full $7$-torsion over $K(\sqrt{d})$. Since
$K(\sqrt{d})$ has to contain $\zeta_{7}$ it follows that
$K(\sqrt{d})=\mathbb{Q}(\zeta_{7})$. But elliptic curves over
$\mathbb{Q}(\zeta_{7})$ cannot have full $7$-torsion (see [28, Theorem.]).
Thus it follows that $E^{\delta}$, $E^{\sigma(\delta)}$ and
$E^{\sigma^{2}(\delta)}$ are all $K$-isomorphic which means that $E^{\delta}$
is a $\mathbb{Q}$-curve. It is known [7, Section 2.] that $\mathbb{Q}$-curves
are either rational or defined over a $(2,\ldots,2)$ extension of
$\mathbb{Q}$. Hence $E^{\delta}$ is defined over $\mathbb{Q}$, completing the
proof. ∎
###### Remark 30.
In the proof of Proposition 29, once we have proven that $E^{\delta}$,
$E^{\sigma(\delta)}$ and $E^{\sigma^{2}(\delta)}$ are all $K$-isomorphic, an
alternative way of proving that $E^{\delta}$ is rational, without using
$\mathbb{Q}$-curves can be done in the following way.
We can see that $E^{\delta}$, $E^{\sigma(\delta)}$ and
$E^{\sigma^{2}(\delta)}$ are $K$-isomorphic if and only if
$\delta\sigma(\delta),\sigma(\delta)\sigma^{2}(\delta)\text{ and
}\delta\sigma^{2}(\delta)$
are all squares in $K$. But since
$N_{K/\mathbb{Q}}(\delta)=\delta\sigma(\delta)\sigma^{2}(\delta)=k\in\mathbb{Q},$
it follows that
$\delta=ka^{2}\text{ for some }k\in\mathbb{Q}\text{ and }a\in K,$
or in other words that $E_{1}$ is a rational twist of $E/\mathbb{Q}$, and
hence can be defined over $\mathbb{Q}$.
Note that it is not hard to prove that there exist rational elliptic curves
with non-trivial $13$-torsion over cubic fields; a short search in Cremona’s
tables shows that 147B1 is such a curve. The hard part is proving that there
are infinitely many such curves. In fact, 147B1 is the only curve with this
property that we found in our (short) search.
Let $\\{\pm 1\\}\leq\Delta\leq(\mathbb{Z}/N\mathbb{Z})^{\times}$ and define
the congruence subgroup
$\Gamma_{\Delta}=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z})\lvert a\text{ mod
}N\in\Delta,N|c\right\\}.$
For $N$ prime, the modular curve $Y_{\Delta}(N)$ corresponding to $\Delta$ has
the following moduli space interpretation: a $F$-rational point on
$Y_{\Delta}(N)$ corresponds to an isomorphism class of pairs $(E,\langle
P\rangle)$ of an elliptic curve $E/F$ and a subgroup $\langle P\rangle\in
E(\overline{\mathbb{Q}})$ of order $N$ such that every
$\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/F)$ acts on $\langle
P\rangle$ as multiplication by some $\alpha(\sigma)\in\Delta$. Let
$X_{\Delta}(N)$ be the compactification of $Y_{\Delta}(N)$. Note that if
$\Delta=(\mathbb{Z}/N\mathbb{Z})^{\times}$ then $X_{\Delta}(N)=X_{0}(N)$ and
if $\Delta=\\{\pm 1\\}$ then $X_{\Delta}(N)=X_{1}(N)$ ($X_{\\{\pm 1\\}}$ is
then defined as the quotient of $\mathbb{H}$ by $\pm\Gamma_{1}(N)$, but since
the action of $\pm 1$ is trivial on $\mathbb{H}$, it follows that
$X_{1}(N)=X_{\\{\pm 1\\}}(N))$. All intermediate curves between $X_{1}(N)$ and
$X_{0}(N)$ are of the form $X_{\Delta}(N)$, for some $\Delta$.
We now prove that there are in fact infinitely many rational elliptic curves
with non-trivial $13$-torsion over some cubic field.
###### Proposition 31.
There exists infinitely many elliptic curves $E/\mathbb{Q}$ such that there
exists a cubic field $K$ such that $E(K)$ has a $13$-torsion point.
###### Proof.
Let $\Delta=\\{\pm 1,\pm 3,\pm 4\\}\subset(\mathbb{Z}/13\mathbb{Z})^{\times}$.
Then by [10, Theorem 1.1. and Table 1.], $X_{\Delta}(13)$ has genus 0. The
existence of a $\mathbb{Q}$\- rational cusp on $X_{\Delta}(13)$ now
immediately shows that $X_{\Delta}(13)(\mathbb{Q})$ (and hence
$Y_{\Delta}(13)(\mathbb{Q})$) has infinitely many points.
Now let $(E,\langle R\rangle)$, where $E/\mathbb{Q}$ and $\langle R\rangle$ is
a $13$-cycle of $E$, be a point on $X_{\Delta}(13)(\mathbb{Q})$. If every
$\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on
$\langle R\rangle$ by multiplication by an element of
$(\mathbb{Z}/13\mathbb{Z})^{\times}$ of order 3, then it would follow that $R$
is defined over a cubic field and we are done.
Suppose the opposite, that $\sigma$ acts on $\langle R\rangle$ by
multiplication as an element of $(\mathbb{Z}/13\mathbb{Z})^{\times}$ of order
6. It can be seen, however that when $E$ is written in short Weierstrass form,
$\sigma$ actually permutes the three $x$-coordinates of $\pm R,\pm 3R$ and
$\pm 4R$ and since $x(T)=x(-T)$ for any $T\in E(\overline{\mathbb{Q}})$, this
implies that the $x$ coordinates of the points in $\langle R\rangle$ are
defined over a cubic field $K$. Let $F\supset K$ be the field of definition of
$R$. If $F=K$ we are done so suppose $F=K(\sqrt{\delta})$, for some $\delta\in
K$. Then $E(F)$ has a point of order 13 and since
$E(F)[13]=E(K)[13]+E^{\delta}(K)[13],$
it follows that either $E$ or $E^{\delta}$ have a $K$-rational point of order
13. If $E(K)$ has a point of order $13$ we are done. Suppose that
$E^{\delta}(K)$ has a point of order 13. Since $F=\mathbb{Q}(\langle
R\rangle)$, the field $F$ is a Galois extension of $\mathbb{Q}$ with
$\operatorname{Gal}(F/\mathbb{Q})\simeq\mathbb{Z}/6\mathbb{Z}$ and it follows
that $K$ is Galois over $\mathbb{Q}$ with
$\operatorname{Gal}(K/\mathbb{Q})\simeq\mathbb{Z}/3\mathbb{Z}$. Let
$\langle\tau\rangle=\operatorname{Gal}(K/\mathbb{Q})$.
Using the same argument as in the proof of Proposition 29, one can prove that
$E^{\delta}$, $E^{\tau(\delta)}$ and $E^{\tau^{2}(\delta)}$ all have to be
$K$-isomorphic and hence it follows that $E$ is a $\mathbb{Q}$-curve and it
follows that $E^{\delta}$ has to be defined over $\mathbb{Q}$.
Thus for every rational elliptic curve $E$ represented by a point on
$X_{\Delta}(13)$, there exists a rational twist $E^{\prime}$ such that there
exists a cubic field $K$ with the property that $E^{\prime}(K)$ has a point of
order 13. ∎
## 6\. Sporadic points on $X_{1}(n)$
As we have seen in Proposition 25, there exists a sporadic point of degree 3
on $X_{1}(21)$, which is a curve of gonality 4 (the gonality of $X_{1}(21)$
can be deduced from [13, Theorem 2.3.] and [9, Theorem 0.1.]). This point was
essentially constructed by starting with an elliptic curve $E/\mathbb{Q}$ with
a $21$-isogeny and then using the division polynomial method to determine the
minimal degree of a field $K$ over which the points in the
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant subgroup of
order 21 of some twist of $E$ becomes $K$-rational.
It is a natural question to ask whether the same procedure can be used to find
other sporadic points by starting with other rational elliptic curves with
isogenies. We have tried this and this method gives us (only) a degree 6 point
on $X_{1}(37)$; we describe the procedure used to find it below.
There are 2 families of twists of elliptic curves with $37$-isogenies. We
start with the elliptic curve $E=1225H1$, take a short Weierstrass model
$y^{2}=x^{3}-10395x+444150$
of it and factor (over $\mathbb{Q}$) its $37$-division polynomial $\psi_{37}$
finding a degree 6 factor
$f_{6}=x^{6}-3150x^{5}+796635x^{4}-75770100x^{3}+3111596775x^{2}$
$-44606598750x-85333003875.$
This implies that the $x$-coordinate of a point of order $37$ of $E$ is
defined over a sextic field $F$ and since twisting does not change the roots
of division polynomials, it follows that there exists an unique twist (over
$F$) of $E$ such that it has a point of order $37$ over $F$. This can be found
simply by finding over which quadratic extension $F(\sqrt{\delta})$ the points
of order $37$ become defined, and then the quadratic twist we are looking for
is $E^{\delta}$. Let $w$ be a root of $f_{6}$. We compute that
$\delta=w^{3}-10395w+444150$
and that $E^{\delta}$ indeed has a point of order $37$. Thus $E^{\delta}$,
together with a point of order $37$ represents a sporadic point of degree 6 on
$X_{1}(37)$, which has gonality 18. Note that this is the same curve which has
been previously found by van Hoeij using computational methods [40].
Acknowledgements.
We thank Kęstutis Česnavičius for finding a mistake in Proposition 10 in an
earlier version and for helping simplify the proofs of Propositions 9 and 10
and the anonymous referee for many suggestions that greatly improved this
paper. We are also grateful to Andew Sutherland for providing useful
computational data, and to Andrej Dujella, Enrique González-Jiménez and Petra
Tadić for helpful comments.
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|
arxiv-papers
| 2012-11-09T16:59:14 |
2024-09-04T02:49:37.757276
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Filip Najman",
"submitter": "Filip Najman",
"url": "https://arxiv.org/abs/1211.2188"
}
|
1211.2240
|
# Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories
Nikita Nekrasov IHES, France & ITEP, Russia & SCGP, USA
nikitastring@gmail.com and Vasily Pestun IAS, USA pestun@ias.edu
###### Abstract.
Seiberg-Witten geometry of mass deformed ${\mathcal{N}}=2$ superconformal ADE
quiver gauge theories in four dimensions is determined. We solve the limit
shape equations derived from the gauge theory and identify the space
${{\mathfrak{M}}}$ of vacua of the theory with the moduli space of the genus
zero holomorphic (quasi)maps to the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$G^{\mathbb{C}}$-bundles on a (possibly degenerate) elliptic curve
$\mathscr{E}$ defined in terms of the microscopic gauge couplings, for the
corresponding simple ADE Lie group $G$. The integrable systems
${\mathfrak{P}}$ underlying, or, rather, overlooking the special geometry of
${{\mathfrak{M}}}$ are identified. The moduli spaces of framed $G$-instantons
on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, of $G$-monopoles with
singularities on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$, the Hitchin systems
on curves with punctures, as well as various spin chains play an important
rôle in our story. We also comment on the higher dimensional theories. In the
companion paper the quantum integrable systems and their connections to the
representation theory of quantum affine algebras will be discussed.
### Acknowledgments
We are grateful to S. Cherkis, R. Donagi, E. Frenkel, A. Kapustin, J. Morgan,
H. Nakajima, A. Okounkov, S. Shatashvili, and E. Witten for discussions and
useful remarks.
Research of NN was supported in part by RFBR grants 12-02-00594, 12-01-00525,
by Agence Nationale de Recherche via the grant ANR 12 BS05 003 02, by Simons
Foundation, and by Stony Brook Foundation.
VP gratefully acknowledges support from Institute for Advanced Study, NFS
grant PHY-0969448, Rogen Dashen Membership, MSERF 14.740.11.0081, NSh
3349.2012.2 and RFBR grant 10-02-00499
The work was reported at the conferences and invited talks [conferences]. We
thank the organizers of these conferences and the institutions for the
invitations and the participants for the questions and the comments.
###### Contents
1. 1 Introduction
1. 1.1 Organization of the material
2. 1.2 Notations
2. 2 Supersymmetric quiver theories
1. 2.1 Gauge group, matter fields, couplings, parameters
3. 3 The ADE classifications of superconformal ${\mathcal{N}}=2$ theories
1. 3.1 Beta functions and Cartan matrix
2. 3.2 Class I theories
3. 3.3 Class II theories
4. 3.4 Class II* theories
4. 4 Low-energy effective theory
1. 4.1 Special Kähler geometry
2. 4.2 Extended moduli space
3. 4.3 Finite size effects
4. 4.4 The appearance of an integrable system
5. 4.5 Integrable systems from classical gauge theories
6. 4.6 Extended moduli space as a complex integrable system
5. 5 The limit shape equations
1. 5.1 The amplitude functions
2. 5.2 The limit shape prepotential
3. 5.3 The iWeyl group.
4. 5.4 Moduli of vacua and mass parameters
6. 6 Solution of the limit shape equations
1. 6.1 From invariants to the curve
2. 6.2 The cameral curve as a modular object
3. 6.3 Spectral curves
4. 6.4 Obscured curve
7. 7 The Seiberg-Witten curves in some detail
1. 7.1 Class I theories of $A$ type
2. 7.2 Class I theories of $D$ type
3. 7.3 Class I theories of $E$ type
4. 7.4 Class II theories of $A$ type and class II* theories
5. 7.5 Class II theories of $D$ type
6. 7.6 Class II theories of $E$ type
8. 8 The integrable systems of monopoles and instantons
1. 8.1 Periodic Monopoles and the phase space of Class I theories
2. 8.2 Double-periodic instantons and Class II theories
3. 8.3 Noncommutative instantons and Class II* theories
9. 9 Higher dimensional theories
10. A McKay correspondence, D-branes, and M-theory
1. A.1 From finite groups to Lie groups
2. A.2 Platonic symmetries
3. A.3 D-branes at singularities
11. B Partitions and free fermions
12. C Lie groups and Lie algebras
1. C.1 Finite dimensional Lie algebras
2. C.2 Affine Lie algebras
3. C.3 ADE Cartan matrices, roots and weights
4. C.4 Affine Weyl group
5. C.5 Conjugacy classes and moduli of bundles
6. C.6 Infinite matrices and their Weyl group
7. C.7 The representation theory
## Chapter 1 Introduction
In this work a class of quiver ${\mathcal{N}}=2$ supersymmetric theories in
four dimensions is analyzed. The first problem of this sort was solved in
[Seiberg:1994rs, Seiberg:1994aj] for the $SU(2)$ gauge theory in four
dimensions with eight supercharges.
We study mass perturbed ${\mathcal{N}}=2$ superconformal theories, and compute
the exact metric
$ds_{{\mathfrak{M}}}^{2}=G_{{\mathcal{I}}{\bar{\mathcal{J}}}}du^{\mathcal{I}}d{\bar{u}}^{\bar{\mathcal{J}}}$
on the moduli space ${{\mathfrak{M}}}$ of vacua of the low-energy effective
theory. We also compute the vacuum expectation values
$\langle{\mathcal{O}}_{i,n}\rangle_{u}$
of all gauge invariant ${\mathcal{N}}=2$ chiral operators.
Our theories have the gauge group $G_{\text{g}}$ which is a product of a
finite number of special unitary groups. The technique we use is the saddle
point approach to the calculation of the supersymmetric partition function of
the theory in $\Omega$-background [Nekrasov:2002qd]. The partition function is
given by the sum over special instanton configurations. In the limit, where
the $\Omega$-deformation is removed so that the theory approaches the original
flat space theory, the sum over the special instantons is dominated by the
contribution of one particular special instanton configuration, of a very
large instanton charge (with the expected small effective density of instanton
charge). This configuration, the so-called limit shape, is found in this work
using a novel approach, built on the analytic techniques of [Nekrasov:2003rj].
Namely, we interpret the limit shape equations as the conditions defining the
analytic continuations of the generating functions
${\mathscr{Y}}_{i}(x)={\exp}\ \langle\,{\operatorname{tr}}\,{\rm
log}\left(x-{\Phi}_{i}\right)\,\rangle_{u},$ (1.1)
where $i$ labels the simple factors in the gauge group $G_{\text{g}}$, and
${\Phi}_{i}$ is the corresponding complex adjoint Higgs scalar field. We get
the system of (algebraic) equations determining these functions by fixing the
set of basic invariants of the monodromy of the analytic continuation.
Recall that a complex Lie group ${\mathbf{G}}_{\text{q}}$ is naturally
associated with the quiver gauge theory. This group is different from the
original gauge group $G_{\text{g}}$ of the theory. Roughly speaking the Dynkin
diagram of ${\mathbf{G}}_{\text{q}}$ is the universal cover of the quiver of
the gauge theory. The group ${\mathbf{G}}_{\text{q}}$ may be infinite-
dimensional. In fact, for the ${\mathcal{N}}=2$ superconformal theories the
corresponding Lie algebra $\mathfrak{g_{\text{q}}}$ is the finite dimensional
simple Lie algebra of the ADE type $\mathfrak{g}$, or its affine version
$\widehat{\mathfrak{g}}$, or the algebra $\widehat{\mathfrak{gl}}_{\infty}$.
Our main construct is the $x$-dependent element $g(x)$ of the maximal torus
${\mathbf{T}}^{\text{ad}}_{\text{q}}$ of
${\mathbf{G}}^{\text{ad}}_{\text{q}}$, which can also be viewed as the multi-
valued ${\bf T}_{\text{q}}$-valued function. The group element is locally
analytic in $x$:
$g(x)=\prod_{i\in\widetilde{\mathrm{Vert}_{\gamma}}}{\mathscr{P}}_{i}(x+{\mu}_{i})^{-{\lambda}^{\vee}_{i}}{\mathscr{Y}}_{i}(x+{\mu}_{i})^{{\alpha}_{i}^{\vee}}$
(1.2)
where $i$ runs over the set $\widetilde{\mathrm{Vert}_{\gamma}}$ of vertices
of the universal cover of the quiver graph $\gamma$, the polynomials
${\mathscr{P}}_{i}(x)$ and the complex parameters ${\mu}_{i}$ are determined
by the gauge couplings and the masses of matter hypermultiplets, and
${\alpha}_{i}^{\vee}$ and ${\lambda}_{i}^{\vee}$ are the simple coroots and
the fundamental coweights of $\mathfrak{g_{\text{q}}}$. It is also convenient
to introduce another group element:
$g_{\infty}(x)=\prod_{i\in\widetilde{\mathrm{Vert}_{\gamma}}}{\mathscr{P}}_{i}(x+{\mu}_{i})^{-{\lambda}^{\vee}_{i}}$
(1.3)
Our main claim is that the conjugacy class
$[g(x)]\in{{\mathbf{T}}^{\text{ad}}_{\text{q}}}/W({\mathfrak{g_{\text{q}}}})$
is holomorphic in $x$, so that the basic adjoint invariants of
${\mathbf{G}}_{\text{q}}$, evaluated on $[g(x)]$ (up to some twist discussed
further) are polynomials of $x$, leading to a system of equations relating
${\mathscr{Y}}$ and $x$:
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=T_{i}(x)=T_{i,0}x^{{\mathbf{v}}_{i}}+T_{i,1}x^{{\mathbf{v}}_{i}-1}+\ldots+T_{i,{\mathbf{v}}_{i}}$
(1.4)
which define what we call the _cameral curve_
${\mathcal{C}}_{u}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times\left({\mathbb{C}}^{\times}\right)^{\mathrm{Vert}_{\gamma}}$
(1.5)
The invariants ${{\mathscr{X}}}_{i}$ are normalized characters of $g(x)$ in
the fundamental representations ${\mathbf{R}}_{i}$ of
${\mathbf{G}}_{\text{q}}$, of the highest weight $\lambda_{i}$:
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=g_{\infty}(x)^{-\lambda_{i}}\,{\operatorname{Tr}}_{R_{i}}g(x)$
Moreover, from the work of Steinberg [Steinberg:1965] (see also
[Morgan:2002math]) we know that for the finite-dimensional
${\mathbf{G}}_{\text{q}}$ one can conjugate $g(x)$ in
${\mathbf{G}}_{\text{q}}$ to obtain a smooth ${\mathbf{G}}_{\text{q}}$-valued
function ${\bf g}(x)$ of $x$. Further inspection shows that ${\bf g}(x)$ is a
quasi-classical limit of an element of the Yangian algebra
$Y({\mathfrak{g_{\text{q}}}})$, built on the Lie algebra
$\mathfrak{g_{\text{q}}}$ of ${\mathbf{G}}_{\text{q}}$. Hopefully the
analogous statements hold for all ${\mathbf{G}}_{\text{q}}$’s we encounter.
In this way one recovers all known results about the Seiberg-Witten geometries
of the ${\mathcal{N}}=2$ theories in four dimensions (we do not review all of
them in this work) as well as finds new results. In particular, we find the
families of curves describing the geometry of the moduli space of vacua for
the theories which were previously believed not to have such description. We
also find that the special geometry of the quiver theories is captured in
general by a polylogarithmic system of differentials on these curves.
##### Higher dimensions
The gauge theories we discuss can be also lifted to five dimensional theories
compactified on a circle $\mathbb{S}^{1}_{\left\langle\beta\right\rangle}$ of
circumference $\beta$, or even to the six dimensional theories, compactified
on a two-torus, of the area $\beta^{2}$. In the limit $\beta\to 0$ one
recovers the original four dimensional theory. In the five dimensional case
the polynomials $T_{i}(x)$ in Eq. (1.4) are replaced by Laurent polynomials in
$e^{\mathrm{i}\beta x}$, while in the six dimensional case the functions
$T_{i}(x)$ become elliptic.
##### Defreezing
One of the initial questions which led us to the subject of this work was the
following. Consider the $SU(2)$ theory with $N_{f}=4$ hypermultiplets in the
fundamental representation, with the coupling $\mathfrak{q}$. By now there is
an overwhelming evidence for the connection of this theory to Liouville
conformal blocks on a sphere with four punctures. The momenta of Liouville
vertex operators at the punctures are related to the masses of the
hypermultiplets, the locations of the vertex operators are, e.g.
$0,1,{\mathfrak{q}},{\infty}$, and the momentum at the intermediate channel is
the Coulomb parameter $a$.
Let us view this theory as a $U(2)$ theory, and let us single out the maximal
torus $U(1)^{4}$ of the $Spin(8)$ flavor symmetry group. Let us gauge these
$U(1)$ groups. This gauging is possible in the noncommutative geometry setup.
One acquires four additional coupling constants. What will happen to the
Liouville theory?
Upon some reflection one concludes that the resulting theory is a particular
case of the ${\widehat{D}}_{4}$ theory, with
${\mathbf{v}}_{0}={\mathbf{v}}_{1}={\mathbf{v}}_{3}={\mathbf{v}}_{4}=1$, and
${\mathbf{v}}_{2}=2$. We then decided to solve the general quiver
superconformal theory which led us to discover many other interesting things.
Figure 1.1. The three major ways to construct $\mathcal{N}=2$ theories
##### Classification
Another motivation was the question whether Hitchin’s system exhaust the list
of all reasonable Seiberg-Witten integrable systems. From the early discovery
[Donagi:1995cf] that the ${\mathcal{N}}=2^{*}$ theory with
${G_{\text{g}}}=SU(N)$ is governed by the $SU(N)$ Hitchin system on a one-
punctured torus (which is nothing but the elliptic Calogero-Moser system, as
shown previously in [Gorsky:1994dj]), proposals in [Martinec:1996by], and
subsequent developments culminating in the introduction of the ‘‘S-class’’
theories [Witten:1997sc, Gaiotto:2009we, Gaiotto:2009hg, Alday:2009aq] there
was a lot of activity with experimental evidence suggesting that
${\mathcal{N}}=2$ theories can be described by some version of Hitchin’s
system. The underlying construction in these approaches is the
compactification of the six dimensional superconformal $(0,2)$-theory on some
Riemann surface embedded as a supersymmetric cycle in some ambient geometry,
and it is believed that the global features of the embedding should play
virtually no rôle in the effective gauge theory dynamics.
Another way of engineering ${\mathcal{N}}=2$ theories, using string theory, is
the so-called geometric engineering [Katz:1996fh, Lerche:1996xu], which is the
study of the gravity-decoupled limit of the IIA compactification on a Calabi-
Yau threefold, with the Calabi-Yau becoming effectively non-compact. A large
class of models comes from toric Calabi-Yau’s. One then employs the local
mirror symmetry to generate curves with differentials, whose periods capture
the special geometry of the ${\mathcal{N}}=2$ theory.
In our work we presented another characterization of the integrable systems
underlying the special geometry of the ${\mathcal{N}}=2$ theories with the
superconformal ultraviolet limit. Namely, we identify these systems with the
moduli spaces of some gauge/Higgs configurations, such as monopoles or
instantons, with the gauge group ${\mathbf{G}}_{\text{q}}$ corresponding to
the quiver diagram encoding, among other things, the matter sector of the
theory. Unlike all previous approaches, see fig. 1.1, which involved some
reference to the non-perturbative dualities, or even embedding of the gauge
theory to string theory and M-theory, we derive these statements within the
quantum field theory, by analyzing the instanton contributions to the low-
energy effective action.
In some cases (e.g. in a simple fashion for the $A_{r}$ type theories, in a
more subtle way for the $D_{r}$ type theories) our phase spaces can be
identified with the phase spaces of Hitchin systems on the low genus curves
with punctures, using some version of Nahm-Fourier-Mukai transform, but in
general we don’t have such a duality. Provided a complete description of the
Seiberg-Witten curves and algebraic integrable systems for the $\mathcal{N}=2$
ADE quiver theories it would be interesting to further investigate this ADE
quiver class, along the lines of [Gaiotto:2009hg, Gaiotto:2010be] or
[Alim:2011ae] for the ‘‘S-class’’.
In the other cases, e.g. the class II $E_{r}$ type theories we can use the
relation between the moduli of del Pezzo surfaces and the moduli of
$E$-bundles on elliptic curve to assign to our version of the Seiberg-Witten
curve a one-parametric family of del Pezzo surfaces, which can be viewed as an
example of the mirror noncompact threefold of [Katz:1997eq].
##### Outlook
| | | | classical | quantum | double quantum
---|---|---|---|---|---|---
| | | | $\begin{array}[]{c}\epsilon_{1}=0\\\ \epsilon_{2}=0\end{array}$ | $\begin{array}[]{c}\epsilon_{1}\neq 0\\\ \epsilon_{2}=0\end{array}$ | $\begin{array}[]{c}\epsilon_{1}\neq 0\\\ \epsilon_{2}\neq 0\end{array}$
4d | $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}$ | XXX | rational | ${\mathbf{G}}_{\text{q}}(\mathbb{C})$ | $Y(\mathfrak{g_{\text{q}}})$ | …
5d | $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}^{\times}$ | XXZ | trigonometric | ${\mathbf{G}}_{\text{q}}(\mathbb{C}^{\times})$ | $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ | …
6d | $\mathbf{C}_{\left\langle x\right\rangle}=E_{T}$ | XYZ | elliptic | ${\mathbf{G}}_{\text{q}}(E)$ | $E_{T,\eta}(\mathfrak{g_{\text{q}}})$ | …
Table 1.1. The (rational/trigonometric/elliptic) by (classical/quantum/double
quantum)
In the companion paper [NP2012b] we study in details the connection between
the class of ADE quiver gauge theories and _quantum_ integrable ADE
spinchains. In particular we explain there that the five dimensional version
of the ADE quiver gauge theory on the twisted bundle
$\mathbb{R}^{4}{\tilde{\times}}\mathbb{S}^{1}_{\left\langle\epsilon_{1},\epsilon_{2};\beta\right\rangle}$
[Nekrasov:2003af] with the equivariant parameters set to
$\epsilon_{1}=\epsilon$, $\epsilon_{2}=0$ as in [Nekrasov:2009ui] is
associated with the XXZ spin chain $\mathfrak{g_{\text{q}}}$. The theory is
solved by the quantum version of the master equation (1.4): the group
${\mathbf{G}}_{\text{q}}$ is replaced by the _quantum affine algebra_
$U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ with the quantum parameter
$q=e^{\mathrm{i}\beta\epsilon}$, while the characters ${\mathscr{X}}_{i}$ are
promoted to the $q$-characters of Frenkel-Reshetikhin [Frenkel:1998]. (If
${\mathbf{G}}_{\text{q}}$ is itself affine Kac-Moody group
${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ then
$U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ is naturally _quantum toroidal
algebra_). In the four dimensional limit the XXZ $\mathfrak{g_{\text{q}}}$
spin chain turns into the XXX $\mathfrak{g_{\text{q}}}$ spin chain, the
quantum affine algebra $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ degenerates
into Yangian $Y(\mathfrak{g_{\text{q}}})$, and the gauge theory on the twisted
bundle becomes the four dimensional theory subject to a two dimensional
$\Omega$-background. Finally, the six dimensional theory compactified on a
torus $E_{T}$ corresponds to the XYZ $\mathfrak{g_{\text{q}}}$ spinchain, with
the quantum affine group $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ elevated
to the quantum elliptic group $E_{T,{\eta}}(\mathfrak{g_{\text{q}}})$
[Drinfeld_1986, Felder_1995, Felder_1996], with
${\eta}={\beta}{\epsilon}/2\pi$.
It is clear that there is an even larger picture in which the algebraic
integrable systems we encountered in this work are in a additionally
quantized, or deformed, this time with the with parameter $\epsilon_{2}$, c.f.
table 1.1, with the rational/trigonometric/elliptic trichotomy in the vertical
direction established in [Faddeev:1982, Sklyanin:1979, Belavin:1982] and
connected with the gauge theories in [Nekrasov:1996cz]. It would be exciting
to explore the connection with H. Nakajima’s work [Nakajima:2001q] on quiver
varieties and quantum affine algebras as well as the connection with elliptic
cohomology [Ginzburg_1995, Grojnowski_2007, Lurie_2009] of moduli spaces.
Notice that the quantum or double quantum exploration of the ADE quiver world
is in a sense orthogonal to the approach of [Alday:2009aq] dealing with the
‘‘S-class’’ world in the Fig. 1.1. Classically, on the overlap, the relation
between the corresponding algebraic integrable systems comes from the Nahm-
Fourier-Mukai/Corrigan-Goddard/ADHM reciprocity relating the moduli space of
$\mathbf{G}$-bundles and Hitchin systems. The (doubly) quantum version of this
Nahm transform, if it exists, seems to cover the ‘‘quantum’’ geometric
Langlands duality, Separation of variables for quantum systems
[Sklyanin:1987ih, Feigin:1994in, Frenkel:1995zp, Sklyanin:1995bm,
Enriquez:1996xc] and the quasi-particle or ‘‘free field’’ Dotsenko-Fateev
[Dotsenko:1984nm, Gerasimov:1990fi, Feigin:1990qn] representation of conformal
blocks. The new ingredient [Nekrasov:2009rc] in this relatively classic field
of research are the supersymmetric gauge theories in four dimensions.
The task to fill the table 1.1 with all glory (double) quantum details is left
for future work.
### 1.1. Organization of the material
Chapter 2 introduces the quiver supersymmetric gauge theories which we shall
study.
Chapter 3 presents the classification of the gauge theories which are
${\mathcal{N}}=2$ superconformal in the ultraviolet. We distinguish three
classes of such theories, I, II, and II*. The I and II classes have an ADE
classification so that for class I ${\mathbf{G}}_{\text{q}}=\mathbf{G}$ and
for class II ${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ where $\mathbf{G}$
is ADE group, the II* theories correspond to $\widehat{GL}_{\infty}$ group.
Chapter 4 reviews the special Kähler geometry of the vectormultiplet moduli
spaces ${\mathfrak{M}}$ of vacua of ${\mathcal{N}}=2$ theories. We also recall
the relation of ${\mathfrak{M}}$ to the algebraic integrable systems and the
hyperKähler manifolds. We give some examples to be used later.
Chapter 5 introduces our main tool: the limit shape equations, which summarize
the microscopic gauge theory calculation leading to the effective low-energy
action, i.e. the prepotential ${{\mathscr{F}}}$.
Chapter 6 presents the solution of the limit shape equation. We reformulate
the equations as the Riemann-Hilbert problem for the set of functions
${\mathscr{Y}}(x)$ and solve it by equating the invariants
${{\mathscr{X}}}({\mathscr{Y}}(x))$ of the monodromy group, the _iWeyl group_
which we attach to every ${\mathcal{N}}=2$ gauge theory, to some polynomials
$T(x)$. In this manner we find an (algebraic) curve ${\mathcal{C}}$ and a
system of differentials, whose periods give the special coordinates
$\mathfrak{a}$ and the derivatives
${\partial}{{\mathscr{F}}}/{\partial}{\mathfrak{a}}$ of the prepotential
${\mathscr{F}}$.
Chapter 7 analyzes the solution in some detail. We interpret the data for the
solution of the class I theories as describing a holomorphic map with
prescribed singularities of ${\mathbb{C}\mathbb{P}}^{1}$ to the space of
conjugacy classes ${\mathbf{T}}/W({\mathbf{g}})$ in a complex Lie group
$\mathbf{G}$, which can be also viewed as the moduli space of holomorphic
$\mathbf{G}$-bundles on a degenerate elliptic curve. For the class II theories
the analogous data parametrizes (quasi)maps to the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$\mathbf{G}$-bundles on elliptic curve. In some cases we relate the curve
$\mathcal{C}$ to the more familiar Seiberg-Witten curves. For the theories
corresponding to the $A$ series we manage to relate our curves $\mathcal{C}$
to the spectral curves of rational and elliptic Gaudin models (the Hitchin
system on the genus zero and one curves with punctures), and also reproduce
the results of [Witten:1997sc], [Shadchin:2005cc]. For the class II $D$ type
theories we reproduce the results of [Kapustin:1998fa]. For the class II $E$
type theories we find yet another interpretation of our solution, in terms of
families of del Pezzo surfaces. In this way we get a field theory
understanding of some of the local mirror symmetry predictions [Katz:1997eq]
and brane construction [Kapustin:1998pb, Kapustin:1998xn].
Chapter 8 discusses the moduli spaces ${\mathfrak{P}}$ of vacua of the gauge
theory compactified on a circle ${\mathbb{S}}^{1}$. We don’t present the full
analysis of the hyperKähler metric on ${{\mathfrak{P}}}$ in this work.
Instead, we focus on the geometry of ${{\mathfrak{P}}}$ in the complex
structure inherited from four dimensions (this complex structure is sometimes
called the complex structure $\bf I$), in which it presents itself as an
(algebraic) integrable system. Our solution of the four dimensional theory
comes in a form which leads to a natural guess for the phase space
${\mathfrak{P}}$ of the integrable systems corresponding to our theories. For
the class I theories it is the moduli space of $G$-monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with singularities, for the class II
theories it is the moduli space of $G$-instantons on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, and for the class II*
${\widehat{A}}_{r}$ theories it is the moduli space of noncommutative $U(r+1)$
instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$. Of course these spaces
have a natural hyperkähler structure which depends in the expected fashion on
all the parameters of the theory and its compactification. Although our
motivation comes from the field theory analysis in the previous chapters, our
results confirm the conjectures of [Chalmers:1996xh], [Hanany:1996ie],
[Kapustin:1998fa],[Kapustin:1998pb], [Kapustin:1998xn], [Cherkis:1997aa],
[Cherkis:2000cj], [Cherkis:2001gm] which are motivated by the string theory
analysis, and in particular by the brane constructions.
Chapter 9 discusses the modifications of our solutions in the five and six
dimensional cases.
In the Appendix A we review the affine ADE graphs, the McKay correspondence
and the M-theory/D-brane picture for the present work; in the Appendix B we
put our conventions on the partitions and representations by free fermions; in
the Appendix C we review some standard material on Lie groups and Lie algebras
which we use in solving our theories. We recall the notions of the (co)root
and the (co)weight lattices, Weyl groups, and the integrable highest weight
representations; in the Appendix LABEL:se:elliptic we collect our conventions
for elliptic functions; in the Appendix LABEL:se:appendixE we give some
technical details on spectral curves of affine E-series.
### 1.2. Notations
###### Quivers
chapter 2
$\displaystyle{\mathrm{Vert}_{\gamma}}$ $\displaystyle{\rm set\ of\ vertices}$
$\displaystyle{\mathrm{Edge}_{\gamma}}$ $\displaystyle{\rm set\ of\ edges}$
$\displaystyle s(e)\in\mathrm{Vert}_{\gamma}$ $\displaystyle{\rm the\ source\
of\ the\ edge\ }e\in\mathrm{Edge}_{\gamma}$ $\displaystyle
t(e)\in\mathrm{Vert}_{\gamma}$ $\displaystyle{\rm the\ target\ of\ the\ edge\
}e\in\mathrm{Edge}_{\gamma}$
$\displaystyle\mathfrak{q}_{i}=e^{2\pi\mathrm{i}\tau_{i}}$ gauge coupling
constants $\displaystyle{\tilde{\mathfrak{q}}}$
$\displaystyle({\mathfrak{q}}_{i})_{i\in\mathrm{Vert}_{\gamma}}$
$\displaystyle\mathbf{v}_{i}$ number of colors for $i$-th node gauge group
$SU(\mathbf{v}_{i})$ $\displaystyle\mathbf{w}_{i}$ number of flavors for
$i$-th node fundamental matter $\displaystyle\mathfrak{a}_{i,\mathbf{a}}$
eigenvalues of the complex scalars
$\displaystyle\mathfrak{a}_{i,\mathbf{a}}\leftrightarrow\mathfrak{a}^{\mathcal{I}}$
the special coordinates on Coulomb moduli space $\displaystyle C_{ij}$ Cartan
matrix associated to the quiver by its Dynkin graph $\displaystyle
r=\operatorname{rk}(C)$ $|\mathrm{Vert}_{\gamma}|$ if $\gamma$ is finite ADE
or $|\mathrm{Vert}_{\gamma}|-1$ if $\gamma$ is affine ADE
###### Lie groups
$\displaystyle\mathrm{i}$ $\displaystyle\sqrt{-1}$ $\displaystyle
G_{\text{g}}$ Gauge group of the four-dimensional theory
$\displaystyle{\mathbf{G}}_{\text{q}}$ Kac-Moody group associated with quiver
Dynkin diagram $\displaystyle{G_{M}}$ the flavor group
$\displaystyle{\mathbf{G}}=G^{\mathbb{C}}$ $\displaystyle{\rm finite\
dimensional\ complex\ Lie\ group}\quad$ $\displaystyle{\widehat{\mathbf{G}}}$
affine Kac-Moody group for $\mathbf{G}$ $\displaystyle{G}$ maximal compact
subgroup of $\mathbf{G}$ $\displaystyle{\mathbf{T}}$ $\displaystyle{\rm
maximal\ torus\ of}\ {\mathbf{G}}\quad$ $\displaystyle T$ $\displaystyle{\rm
maximal\ torus\ of}\ G$ $\displaystyle Z$ the center of both $G$ and
$\mathbf{G}$ $\displaystyle{{\mathbf{G}}^{\text{ad}}}={\mathbf{G}}/Z$
$\displaystyle{\rm adjoint\ form\ of\ the\ complex\ Lie\ group}\
{\mathbf{G}}\quad$ $\displaystyle{{\mathbf{T}}^{\text{ad}}}={\mathbf{T}}/Z$
the maximal torus of ${\mathbf{G}}^{\text{ad}}$
$\displaystyle{{\mathbf{G}}^{\text{ad}}_{\text{q}}}={{\mathbf{G}}_{\text{q}}}/Z$
$\displaystyle{\rm adjoint\ form\ of\ the\ complex\ Lie\ group}\
{{\mathbf{G}}_{\text{q}}}\quad$
$\displaystyle{{\mathbf{T}}^{\text{ad}}_{\text{q}}}={{\bf T}_{\text{q}}}/Z$
$\displaystyle{\rm its\ maximal\ torus}\quad$
###### Lie algebras
$\displaystyle\mathfrak{g_{\text{q}}}$ Kac-Moody Lie algebra associated with
quiver Dynkin diagram $\displaystyle{\mathfrak{g}}$ $\displaystyle\ {\rm
Lie}({\mathbf{G}})$ $\displaystyle{\mathfrak{h}}$ $\displaystyle\ {\rm
Lie}({\mathbf{T}})$
###### Representation theory
appendix C
$\displaystyle a_{i}$ Kac-Dynkin marks $\displaystyle{{\rm
Q}},{\Lambda}^{\vee}$ $\displaystyle{\rm root\ lattice},\ {\rm coweight\
lattice}$ $\displaystyle{\Lambda},{{\rm Q}}^{\vee}$ $\displaystyle{\rm weight\
lattice},\ {\rm coroot\ lattice}$ $\displaystyle R_{i}$ $\displaystyle
i^{\prime}{\rm th\ fundamental\ representation\ of}\ {\mathbf{G}}$
$\displaystyle{\widehat{R}}_{i}$ $\displaystyle i^{\prime}{\rm th\
fundamental\ representation\ of}\ {\widehat{\mathbf{G}}}$
$\displaystyle{\mathcal{R}}_{i}$ $\displaystyle i^{\prime}{\rm th\
fundamental\ representation\ of}\ {\widehat{GL}_{\infty}}$
###### Spaces
$\displaystyle{B({\mathfrak{g}})}$
$\displaystyle{\mathbf{T}}/W({\mathfrak{g}})\text{ the space of conjugacy
classes in $\mathbf{G}$ }$ $\displaystyle{B({\mathfrak{g_{\text{q}}}})}$
$\displaystyle{{\bf T}_{\text{q}}}/W({\mathfrak{g_{\text{q}}}})\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}_{\text{q}}}$
$\displaystyle{B^{\text{ad}}({\mathfrak{g}})}$
$\displaystyle{\mathbf{T}}/\left(Z\times W({\mathfrak{g}})\right)\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}^{\text{ad}}}$
$\displaystyle{B^{\text{ad}}({\mathfrak{g_{\text{q}}}})}$ $\displaystyle{{\bf
T}_{\text{q}}}/\left(Z\times W({\mathfrak{g_{\text{q}}}})\right)\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}^{\text{ad}}_{\text{q}}}$
$\displaystyle{\mathbf{C}_{\left\langle x\right\rangle}}$ complex plane
$\mathbb{C}$ in 4d, cylinder $\mathbb{C}^{\times}$ in 5d, torus $E$ in 6d
$\displaystyle{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$
$\displaystyle={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\,\infty\,\\}$
$\displaystyle{\mathscr{E}}(\mathfrak{q})$ elliptic curve
$\mathbb{C}^{\times}/\mathfrak{q}^{\mathbb{Z}}$ $\displaystyle\mathfrak{q}$
$\prod_{i\in\mathrm{Vert}_{\gamma}}\mathfrak{q}_{i}^{a_{i}}$ for class II
theories $\displaystyle\mathrm{Bun}_{\mathbf{G}}(\mathscr{E})$ coarse moduli
space of semistable holomorphic $\mathbf{G}$-bundles on $\mathscr{E}$
$\displaystyle{\mathfrak{M}}$ the Couloumb moduli space of the 4d $\times
SU(\mathbf{v}_{i})$ gauge theory $\displaystyle{\mathfrak{M}}^{\mathrm{ext}}$
the Couloumb moduli space of the 4d $\times U(\mathbf{v}_{i})$ gauge theory
$\displaystyle{\mathfrak{P}}\to{\mathfrak{M}}$ the algebraic integrable system
$\dim_{\mathbb{C}}{\mathfrak{P}}=2\dim_{\mathbb{C}}{\mathfrak{M}}$
$\displaystyle{\mathfrak{P}}^{\mathrm{ext}}\to{\mathfrak{M}}^{\mathrm{ext}}$
the complex integrable system
$\dim_{\mathbb{C}}{\mathfrak{P}}^{\mathrm{ext}}=2\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}$
###### Seiberg-Witten curves
$\displaystyle\mathcal{C}_{u}$ cameral curve: section 6.2 $\displaystyle
C_{u}$ spectral curve: section 6.3 $\displaystyle\mathscr{C}$ obscure curve:
section 6.4 $\displaystyle x$ flat coordinate on $\mathbf{C}_{\left\langle
x\right\rangle}$ $\displaystyle\mathscr{Y}_{i}(x)$ amplitudes (the solution of
the theory): section 5.1 $\displaystyle T_{i}(x)$ gauge polynomials of degree
$\mathbf{v}_{i}$ $\displaystyle\mathscr{P}_{i}(x)$ matter polynomials of
degree $\mathbf{w}_{i}$ $\displaystyle g(x)$
${\mathbf{T}}^{\text{ad}}_{\text{q}}$ valued analytic function on
$\mathbf{C}_{\left\langle x\right\rangle}$ $\displaystyle\mathbf{g}(x)$
${\mathbf{G}}_{\text{q}}$ valued analytic function on
$\mathbf{C}_{\left\langle x\right\rangle}$ $\displaystyle{\mathscr{X}}_{j}$
${\mathbf{G}}_{\text{q}}$ character (or Weyl invariant) for $i$-th fundamental
weight of ${\mathbf{G}}_{\text{q}}$
###### Partitions
$\displaystyle{\lambda}$ $\displaystyle{\rm partition}\
({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots{\lambda}_{{\ell}({\lambda})}>0),\qquad{\lambda}_{i}\in{\mathbb{Z}}_{\geq
0}$ $\displaystyle{\ell}({\lambda})$ $\displaystyle{\rm the\ length\ of\ the\
partition\ }{\lambda}$ $\displaystyle|{\lambda}|$ the size of the partition
$|\lambda|=\sum_{i=1}^{{\ell}({\lambda})}{\lambda}_{i}$
Let $x\in R^{\mathbb{Z}}$ be a sequence $(x_{i})_{i\in\mathbb{Z}}$, with
$x_{i}$ in some ring $R$.
###### Consecutive products
$\displaystyle x^{[i]}_{j}\equiv$
$\displaystyle\,\frac{\prod_{k=-\infty}^{k=j+i-1}x_{k}}{\prod_{k=-\infty}^{j-1}x_{k}}\,,$
$\displaystyle x^{[i]}=$ $\displaystyle\,x^{[i]}_{1},$ $\displaystyle
x^{[i+1]}=$ $\displaystyle\,x^{[i]}x_{i+1}$
for example
$\displaystyle x^{[0]}=$ $\displaystyle\,1,$ $\displaystyle x^{[2]}_{4}=$
$\displaystyle\,x_{4}x_{5},$ $\displaystyle x^{[-1]}=$
$\displaystyle\,x_{0}^{-1},$ $\displaystyle x_{5}^{[-4]}=$
$\displaystyle\,x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}$
###### Consecutive sums
$\displaystyle x_{(i)}=$
$\displaystyle\,\sum_{j=-\infty}^{i}x_{j}-\sum_{j=-\infty}^{0}x_{j}$
$\displaystyle x_{(i+1)}=$ $\displaystyle\,x_{(i)}+x_{i+1}$
for example
$\displaystyle x_{(0)}=$ $\displaystyle\,0,$ $\displaystyle x_{(3)}=$
$\displaystyle\,x_{1}+x_{2}+x_{3},$ $\displaystyle x_{(-2)}=$
$\displaystyle\,-x_{0}-x_{-1},$
$\displaystyle{\lambda}_{({\ell}({\lambda}))}=$ $\displaystyle\,|{\lambda}|$
## Chapter 2 Supersymmetric quiver theories
Consider any ${\mathcal{N}}=2$ supersymmetric field theory in four dimensions
whose gauge group is a product of special unitary groups, while the matter
hypermultiplets are in the fundamental, bi-fundamental, and adjoint
representations. The field content, the parameters of the Lagrangian, and the
choice of the vacuum are conveniently encoded in the quiver data, which is:
1. (1)
An oriented graph $\gamma$ with the set $\mathrm{Vert}_{\gamma}$ of vertices
and the set
$\mathrm{Edge}_{\gamma}\subset\mathrm{Vert}_{\gamma}\times\mathrm{Vert}_{\gamma}$
of oriented edges. Let
$s,t:\mathrm{Vert}_{\gamma}\times\mathrm{Vert}_{\gamma}\to\mathrm{Vert}_{\gamma}$
by the projections onto the first and the second factors. They define the two
maps $s,t:\mathrm{Edge}_{\gamma}\to\mathrm{Vert}_{\gamma}$ which assign to an
oriented edge its source and the target, respectively. In what follows we
shall use the notation
$|\mathrm{Vert}_{\gamma}|=\\#\mathrm{Vert}_{\gamma}$
for the number of vertices in the quiver.
2. (2)
An assignment of polynomials to the vertices:
${\mathscr{T}},{\mathscr{P}}:\mathrm{Vert}_{\gamma}\to{\mathbb{C}}[x],\qquad
i\mapsto{\mathscr{T}}_{i}(x),{\mathscr{P}}_{i}(x)$ (2.1)
and ${\mathbf{v}},{\mathbf{w}}:\mathrm{Vert}_{\gamma}\to{\mathbb{Z}}_{+}$,
where
${\mathbf{v}}_{i}={\deg}\,{\mathscr{T}}_{i},\
{\mathbf{w}}_{i}={\deg}\,{\mathscr{P}}_{i},\qquad i\in\mathrm{Vert}_{\gamma}$
The polynomials ${\mathscr{T}}_{i}(x)$ are monic, the highest order term
coefficients ${\mathfrak{q}}_{i}$ of the polynomials ${\mathscr{P}}_{i}(x)$
are required to obey: $|{\mathfrak{q}}_{i}|<1$.
3. (3)
A $1$-cocycle ${\bf m}\in{\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$, in other
words an assignment
$e\in\mathrm{Edge}_{\gamma}\mapsto m_{e}\in{\mathbb{C}}\ .$ (2.2)
We now proceed with the explanation of the rôles of the polynomials
${\mathscr{T}},{\mathscr{P}}$, as well as that of the cocycle $\bf m$.
### 2.1. Gauge group, matter fields, couplings, parameters
#### 2.1.1. The gauge group
We denote the gauge group by $G_{\text{g}}$. It is the product:
${G_{\text{g}}}=\prod_{i\in\mathrm{Vert}_{\gamma}}SU({\mathbf{v}}_{i})$ (2.3)
The vector multiplet therefore splits into a collection of vector multiplets
for the $SU({\mathbf{v}}_{i})$ gauge factors:
${\Phi}_{\text{g}}=\left({\Phi}_{i}\right)_{i\in\mathrm{Vert}_{\gamma}}$
We have a gauge coupling $e_{i}$ and the theta angle $\vartheta_{i}$ for each
$i\in\mathrm{Vert}_{\gamma}$. As usual, we combine them into the complexified
gauge couplings,
${\tau}_{i}=\frac{1}{2\pi\mathrm{i}}{\log}({\mathfrak{q}}_{i})=\frac{{\vartheta}_{i}}{2\pi}+\frac{4\pi\mathrm{i}}{e_{i}^{2}}$
(2.4)
The bosonic part of the action for gauge fields is given by:
${\mathcal{L}}_{\text{YM}}=\sum_{i\in{\mathrm{Vert}_{\gamma}}}\left(\frac{1}{e_{i}^{2}}\int{\operatorname{tr}}_{{\mathbf{v}}_{i}}F_{A_{i}}\wedge\star
F_{A_{i}}+\frac{{\mathrm{i}}{\vartheta}_{i}}{8\pi^{2}}\int{\operatorname{tr}}_{{\mathbf{v}}_{i}}F_{A_{i}}\wedge
F_{A_{i}}\right)$ (2.5)
where ${\operatorname{tr}}_{v}$ denotes the trace of a $v\times v$ matrix. The
exponentiated coupling
${\mathfrak{q}}_{i}=e^{2\pi\mathrm{i}\tau_{i}}$
enters the path integral measure. The perturbative effects do not depend on
$\vartheta_{i}$, while the non-perturbative effects, which are the
contributions of the gauge fields with non-trivial instanton charge, depend on
${\mathfrak{q}}_{i},{\bar{\mathfrak{q}}}_{i}$. In other words, the partition
function is expected to be invariant under the shifts
${\tau}_{i}\longrightarrow{\tau}_{i}+1\ .$ (2.6)
For $i\in\mathrm{Vert}_{\gamma}$ let ${\Phi}_{i}$ denote the corresponding
complex scalar in the adjoint representation. The bosonic potential of the
vector multiplet field $\Phi_{i}$ contains a universal term
${\operatorname{tr}}_{{\mathbf{v}}_{i}}\,[{\Phi}_{i},{\Phi}_{i}^{\dagger}]^{2}$
plus some possible non-negative terms coming from interactions with matter
fields. If the matter fields are massive then this term alone forces
${\Phi}_{i}$ to commute with its conjugate at low energies. Therefore, at low
energy the field $\Phi_{i}$ can be diagonalized:
${\Phi}_{i}\longrightarrow{\rm
diag}(\mathfrak{a}_{i,{\alpha}})_{{\alpha}=1}^{{\mathbf{v}}_{i}},$ (2.7)
The gauge invariant order parameters are the vacuum expectation values of the
coefficients of the characteristic polynomial ${\mathscr{T}}_{i}$ of
${\Phi}_{i}$:
${\mathscr{T}}_{i}(x)=\left\langle{\det}_{{\mathbf{v}}_{i}}(x-{\Phi}_{i})\right\rangle_{u}$
(2.8)
where we assume the normalization $\langle 1\rangle_{u}=1$, and ${\det}_{v}$
denotes the determinant of a $v\times v$-matrix. Therefore the polynomials
${\mathscr{T}}_{i}(x)$ in (2.8) are monic.
Thus, a collection of polynomials ${\mathscr{T}}_{i}(x)$,
$i\in{\mathrm{Vert}_{\gamma}}$ fixes the choice of the vacuum
$u=(u_{i,{\mathbf{a}}})\in{{\mathfrak{M}}}$:
${\mathscr{T}}_{i}(x)=x^{{\mathbf{v}}_{i}}+\sum_{{\mathbf{a}}=2}^{{\mathbf{v}}_{i}}u_{i,{\mathbf{a}}}x^{{\mathbf{v}}_{i}-{\mathbf{a}}}$
(2.9)
Because of the non-perturbative (instanton) effects the relation between
$u_{i,{\mathbf{a}}}$ and $\mathfrak{a}_{i,{\mathbf{a}}}$ is not polynomial,
and for the same reason ${\mathscr{T}}_{i}(x)\neq{\mathscr{Y}}_{i}(x)$.
#### 2.1.2. The hypermultiplets in the bi-fundamental, or adjoint
representations
The bifundamental or adjoint hypermultiplet $H_{e}$,
$e\in{\mathrm{Edge}_{\gamma}}$ transforms in the following ${G_{\text{g}}}$
representation:
$\left({\overline{\mathbf{v}}_{s(e)}},{\mathbf{v}}_{t(e)}\right)\oplus\left({\overline{\mathbf{v}}_{t(e)}},{\mathbf{v}}_{s(e)}\right),\qquad{\rm
for}\quad s(e)\neq t(e)$ ${\rm Adj}(su({\mathbf{v}}_{i})),\qquad{\rm for}\quad
s(e)=t(e)=i$
The masses of the bi-fundamental hypermultiplets are conveniently represented
by the $1$-cocycle: ${\bf m}\in{\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$:
$e\mapsto{\bf m}_{e}$
Let $[{\bf m}]\in H^{1}({\gamma},{\mathbb{C}})$ be the corresponding
cohomology class. If we denote by $m^{*}$ a particular representative of
$[{\bf m}]$ in ${\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$, then
${\bf
m}=m^{*}+{\delta}{\mu}\,,\qquad{\mu}\in{\mathcal{C}}^{0}({\gamma},{\mathbb{C}})$
(2.10)
or, in components,
${\bf m}_{e}=m^{*}_{e}+{\mu}_{t(e)}-{\mu}_{s(e)}$
#### 2.1.3. The hypermultiplets in the fundamental representations
These are assigned to the vertices $i\in\mathrm{Vert}_{\gamma}$. We have
${\mathbf{w}}_{i}$ such multiplets. Write
${\mathscr{P}}_{i}(x)={\mathfrak{q}}_{i}\,\prod_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}(x-m_{i,{\mathfrak{f}}})$
(2.11)
Then $m_{i,{\mathfrak{f}}}$ are the masses of the fundamental hypermultiplets,
charged under $SU({\mathbf{v}}_{i})$. A ${\mathbf{w}}_{i}$-tuplet of
fundamental hypermultiplets can be thought as a bifundamental
$(\mathbf{v}_{i},\bar{\mathbf{w}}_{i})$ for $SU({\mathbf{v}}_{i})$ and an
auxiliary frozen $U({\mathbf{w}}_{i})$, so that $m_{i,{\mathfrak{f}}}$ can be
interpreted as the values of the frozen scalar field in the vector multiplet
of $U({\mathbf{w}}_{i})$.
## Chapter 3 The ADE classifications of superconformal ${\mathcal{N}}=2$
theories
In quantum gauge theory the coupling constants $e_{i}$ are subject to the
renormalization which leads to their dependence on the energy scale at which
one measures the interaction between the charged particles. The consistent
theories have the gauge couplings which tend to zero as the scale approaches
ultraviolet, or approach some fixed values. These theories are called
asymptotically free and asymptotically conformal, respectively. Moreover,
starting with the asymptotically conformal theory, one can perturb it by the
mass terms. Then, by tuning the masses and the bare couplings, one arrives at
the asymptotically free theory. All asymptotically free quiver theories arise
in this way. Therefore it suffices to solve the asymptotically conformal
theories.
From the perspective of geometrical engineering the ADE quiver theories were
studied in [Katz:1997eq], and three dimensional ADE quiver theories were
studied in [Gaiotto:2008ak].
### 3.1. Beta functions and Cartan matrix
The running of the gauge coupling ${\tau}_{i}$ is described by the Gell-Mann-
Low equations which are one-loop exact for the ${\mathcal{N}}=2$
supersymmetric theories, the result of [Novikov:1983uc]. The actual
contributions of the matter and gauge multiplets to the gauge couplings are:
${\beta}_{i}={2\pi\mathrm{i}}\frac{d{\tau}_{i}}{d{\log}{\Lambda}}={\mathbf{w}}_{i}-2{\mathbf{v}}_{i}+\sum_{e:\
t(e)=i}{\mathbf{v}}_{s(e)}+\sum_{e:\ s(e)=i}{\mathbf{v}}_{t(e)}$ (3.1)
where ${\Lambda}$ is the energy scale. The theory is asymptotically conformal
if ${\beta}_{i}=0$ vanishes for all $i\in\mathrm{Vert}_{\gamma}$. The theory
is asymptotically free if ${\beta}_{i}<0$.
Let us define the incidence of the pair of vertices $I_{ij}$ to be the number
of edges $e$ connecting the vertices $i$ and $j$:
$I_{ij}=\\#\\{e\,|\,s(e)=i,\,t(e)=j,\ {\rm or}\ s(e)=j,\,t(e)=i\\}$ (3.2)
with the understanding that if the vertex $i\in\mathrm{Vert}_{\gamma}$ is
connected to itself by a loop, then the corresponding edge contributes $2$ to
the incidence matrix element $I_{ii}$. Define, for all quivers, the Cartan
matrix of size $|\mathrm{Vert}_{\gamma}|\times|\mathrm{Vert}_{\gamma}|$
$C_{ij}=2{\delta}_{ij}-I_{ij}$ (3.3)
Then
${\beta}_{i}\propto({\mathbf{w}}-C{\mathbf{v}})_{i}$
where
$(C{\mathbf{v}})_{i}=\sum_{j\in\mathrm{Vert}_{\gamma}}C_{ij}{\mathbf{v}}_{j}\
.$
Let us solve the ${\beta}_{i}=0$ conditions (cf. [Howe:1983wj], [Katz:1997eq],
[Lawrence:1998ja]). It is convenient to separate the solutions into three
cases, which we shall call the theories of Class I, the theories of Class II
and the theories of Class II*. By $r$ we shall denote the rank of the Cartan
matrix $C$
$r=\operatorname{rk}(C)$ (3.4)
The main difference between the Class I and Class II,II* theories is that the
Cartan matrix of Class I theories has the maximal rank
$r(C_{\text{$\gamma$ of Class I}})=|\mathrm{Vert}_{\gamma}|$ (3.5)
while for the theories of Class II and Class II* the Cartan matrix has one-
dimensional kernel,
$r(C_{\text{$\gamma$ of Class II}})=|\mathrm{Vert}_{\gamma}|-1$ (3.6)
### 3.2. Class I theories
The solutions to the equations ${\beta}_{i}=0$ with ${\mathbf{w}}\neq 0$ are
the theories of Class I. It is well-known that the graph $\gamma$ is in this
case a Dynkin diagram of a finite dimensional simple simply-laced Lie algebra
$\mathfrak{g_{\text{q}}}$, of the ADE type, with $\mathrm{Vert}_{\gamma}$
labeling the simple roots of $\mathfrak{g_{\text{q}}}$:
$i\in\mathrm{Vert}_{\gamma}\mapsto{\alpha}_{i}$
To solve the equation ${\beta}_{i}=0$ is equivalent to finding two vectors
${\mathbf{v}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{v}}_{i}{\alpha}_{i}^{\vee},\qquad{\mathbf{w}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{w}}_{i}{\alpha}_{i}^{\vee}$
(3.7)
with non-negative components
${\mathbf{v}}_{i},{\mathbf{w}}_{i}\in{\mathbb{Z}}_{\geq 0}$, such that (cf.
(3.3)
${\mathbf{w}}=C^{\mathfrak{g_{\text{q}}}}{\mathbf{v}}$ (3.8)
where $C^{\mathfrak{g_{\text{q}}}}$ is the Cartan matrix of the corresponding
finite dimensional Lie algebra $\mathfrak{g_{\text{q}}}$ of the ADE type.
Equivalently
${\mathbf{v}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{w}}_{i}{\lambda}_{i}^{\vee}$
where ${\lambda}_{i}^{\vee}$ are the fundamental coweights of
$\mathfrak{g_{\text{q}}}$.
For class I theories we set ${\mathbf{G}}_{\text{q}}=\mathbf{G}$ where
$\mathbf{G}$ is finite dimensional complex ADE group.
###### Remark.
In the case of $\mathfrak{g_{\text{q}}}$ of the $A_{r}$ type the dimensions
${\mathbf{v}}_{i}$ must be a convex function of $i$. In particular, they grow
with $i$, for $i=1,\ldots,i_{*}$, and then decrease:
${\mathbf{v}}_{1}\leq{\mathbf{v}}_{2}\leq\ldots\leq{\mathbf{v}}_{i_{*}-1}\leq{\mathbf{v}}_{i_{*}}\geq{\mathbf{v}}_{i_{*}+1}\geq\ldots\geq{\mathbf{v}}_{r}$
(3.9)
###### Remark.
The graphs of the $D_{r}$ and $E_{r}$ Dynkin type have a single tri-valent
vertex, let us call it $i_{*}$. One can easily show using the ${\beta}_{i}=0$
equations that ${\mathbf{v}}_{i_{*}}$ is the maximal value of
${\mathbf{v}}_{i}$ on $i\in\mathrm{Vert}_{\gamma}$, and that
${\mathbf{v}}_{i}$ decrease along each leg emanating from the tri-valent
vertex $i_{*}$.
### 3.3. Class II theories
The Class II theories have ${\mathbf{w}}\equiv 0$, and $[{\bf m}]=0$. It is
well-known that the graphs $\gamma$, such that the corresponding Cartan matrix
$C$ has a zero eigenvector with positive integer entries are in one-to-one
correspondence with the simply laced affine Dynkin diagrams (see Appendix A
for our conventions on ADE graphs and McKay correspondence):
1. (1)
${\widehat{A}}_{r}$, $r\geq 2$,
2. (2)
${\widehat{D}}_{r}$, $r\geq 5$,
3. (3)
${\widehat{E}}_{r}$, $r=7,8,9$
These Dynkin diagrams correspond to the affine Lie algebras
$\widehat{\mathfrak{g}}$ associated to finite dimensional Lie algebras
$\mathfrak{g}$ of rank $r$. We set
${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ and
$\mathfrak{g_{\text{q}}}=\widehat{\mathfrak{g}}$. We discuss the relevant
aspects of the theory of affine Kac-Moody algebras in the following
subsections.
Note that the ${\widehat{A}}_{0}$ case (its quiver has one vertex and one edge
connecting it to itself), given our constraint $[{\bf m}]=0$ for the Class II
theory, corresponds to the ${\mathcal{N}}=4$ superconformal theory. It is well
known that the classical moduli space of vacua gets no quantum corrections in
this theory.
The dimensions ${\mathbf{v}}$ are uniquely specified, up to a single multiple:
${\mathbf{v}}_{i}=Na_{i}$ (3.10)
where $a_{i}$ are the so-called Dynkin labels. We shall recall several
interpretations of these numbers below.
### 3.4. Class II* theories
The Class II* theories have ${\mathbf{w}}=0$ and $[{\bf m}]\neq 0$. The first
condition reduces our choice of $\gamma$ to the affine Dynkin diagrams
(including the ${\widehat{A}}_{0}$ case of the quiver with one vertex and one
loop connecting this vertex with itself). The second condition implies that
$\gamma$ is the Dynkin diagram of the ${\widehat{A}}_{r}$ type for some $r\geq
1$. Indeed, only in this the affine Dynkin diagram has
$H^{1}({\gamma},{\mathbb{C}})={\mathbb{C}}$, the diagram being a regular
$r+1$-gon. The dimensions ${\mathbf{v}}_{i}$ are all equal to $N$, a non-
negative integer.
In particular, the Class II* $r=0$, ${\widehat{A}}_{0}$-theory with $[{\bf
m}]\neq 0$ is the celebrated ${\mathcal{N}}=2^{*}$ theory, the $SU(N)$ theory
with massive adjoint hypermultiplet.
We shall see that the Kac-Moody Lie algebra which corresponds to the theories
of Class II* is the $\widehat{{\mathfrak{g}\mathfrak{l}}({\infty})}$ algebra,
which contains $\widehat{\mathfrak{u}(r)}$ as a subalgebra of $r$-periodic
matrices.
## Chapter 4 Low-energy effective theory
We now can proceed with the main subject of our study. Our goal is to
determine the two-derivative/four-fermions terms in the low-energy effective
action of our theory.
The low-energy effective theory of the ${\mathcal{N}}=2$ supersymmetric quiver
theory with generic masses $(m_{e}),(m_{i,{\mathfrak{f}}})$ is the abelian
${\mathcal{N}}=2$ theory of ${\mathbf{r}}$ vector multiplets,
${\mathbf{r}}=\sum_{i\in{\mathrm{Vert}_{\gamma}}}({\mathbf{v}}_{i}-1)$ (4.1)
For generic masses the theory has the manifold ${\mathfrak{M}}$ of vacua,
which is a complex variety of complex dimension $\mathbf{r}$:
${\rm dim}_{\mathbb{C}}\,{{\mathfrak{M}}}={\mathbf{r}}\,,$
The effective theory is a sigma model on ${\mathfrak{M}}$, interacting with
$\mathbf{r}$ abelian gauge fields $A_{\mu}^{\mathcal{I}}$,
${\mathcal{I}}=1,\ldots,{\mathbf{r}}$, and some fermionic fields. Our goal is
to determine the metric on ${\mathfrak{M}}$, the effective gauge couplings
${\rm Im}{\tau}_{\mathcal{I}\mathcal{J}}$ and the effective theta-angles ${\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}$ of these gauge fields.
### 4.1. Special Kähler geometry
One can interpret the eigen-values (2.7) obeying
$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}=0$
(4.2)
as the _special coordinates_ on the moduli space ${\mathfrak{M}}$ of vacua. As
is well known, ${\mathfrak{M}}$ is a Kähler manifold, with a peculiar metric,
and a rigid system of local coordinate systems. The corresponding geometry is
called the _rigid special geometry_ , and it is a limit [Seiberg:1994rs] of
the _special geometry_ of ${\mathcal{N}}=2$ supergravity, studied in
[Cremmer:1984hj].
Let us label the effective abelian vector multiplets
${\mathcal{A}}^{\mathcal{I}}$ by ${\mathcal{I}}=(i,{\mathbf{a}})$,
$i\in\mathrm{Vert}_{\gamma}$, ${\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}$. In
components:
${\mathcal{A}}^{\mathcal{I}}={\mathfrak{a}}^{\mathcal{I}}+{\vartheta}{\psi}^{\mathcal{I}}+{\vartheta}{\vartheta}(F_{A}^{\mathcal{I}})^{-}+\ldots$
where
$(F_{A}^{\mathcal{I}})^{\pm}=\frac{1}{2}\left(F_{A}^{\mathcal{I}}\pm\star
F_{A}^{\mathcal{I}}\right)$ (4.3)
The scalar components
${\mathfrak{a}}^{\mathcal{I}}\leftrightarrow{\mathfrak{a}}_{i,{\mathbf{a}}}$,
${\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}$, more precisely their vacuum
expectation values are the local _special coordinates_. Globally they are
subject to monodromy transformations, unlike the global coordinates
$(u_{i,{\mathbf{a}}})$ in Eq. (2.9), which are defined via the expectation
values of the gauge invariant local operators of the microscopic theory.
The monodromy transformations act by symplectic transformations mixing the
special coordinates ${\mathfrak{a}}^{\mathcal{I}}$ and their duals
${\mathfrak{a}}_{\mathcal{I}}^{D}$ together with the masses $(m_{e})$,
$e\in\mathrm{Edge}_{\gamma}$ and $(m_{i,{\mathfrak{f}}})$,
$i\in\mathrm{Vert}_{\gamma}$, ${\mathfrak{f}}=1,\ldots,{\mathbf{w}}_{i}$. The
dual coordinates are the derivatives of the prepotential ${{\mathscr{F}}}$,
${\mathfrak{a}}^{D}_{\mathcal{I}}=\frac{{\partial}{{\mathscr{F}}}}{{\partial}{\mathfrak{a}}^{\mathcal{I}}}$
(4.4)
The prepotential is a multi-valued analytic function of
${\mathfrak{a}}^{\mathcal{I}}$, it is the superspace action which determines
the low-energy effective action in the approximation we are working:
${\mathcal{L}}^{\rm
eff}=\frac{\mathrm{i}}{4\pi}\int{\tau}_{\mathcal{I}\mathcal{J}}(F_{A}^{\mathcal{I}})^{-}\wedge(F_{A}^{\mathcal{J}})^{-}+{\bar{\tau}}_{\mathcal{I}\mathcal{J}}(F_{A}^{\mathcal{I}})^{+}\wedge(F_{A}^{\mathcal{J}})^{+}-\mathrm{i}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\star
d{\bar{\mathfrak{a}}}^{\mathcal{J}}$ (4.5)
where
${\tau}_{\mathcal{I}\mathcal{J}}=\frac{{\partial}^{2}{{\mathscr{F}}}}{{\partial}{\mathfrak{a}}^{\mathcal{I}}{\partial}{\mathfrak{a}}^{\mathcal{J}}}$
(4.6)
The invariant formulation of Eq. (4.4) is that the two-form
$\sum_{\mathcal{I}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}^{D}_{\mathcal{I}}$
identically vanishes on ${\mathfrak{M}}$. The proper formulation of this
condition uses the additional structure which we review below.
### 4.2. Extended moduli space
In our solution of the theories of class I and class II it would be sometimes
convenient to trade the bifundamental masses formally with the $U(1)$ factors
as explained in (2.10) if one considers
$\times_{i\in\mathrm{Vert}_{\gamma}}U(\mathbf{v}_{i})$ gauge group instead of
$\times_{i\in\mathrm{Vert}_{\gamma}}SU(\mathbf{v}_{i})$. The
$|\mathrm{Vert}_{\gamma}|-1$ bifundamental masses111Recall that in the class
II $\widehat{A}_{r}$ theory there are $r$ mass parameters that can be traded
for the $U(1)$ scalars and $1$ additional “twist” mass parameter $m^{*}$
promoting the class II to class II* and one overall $U(1)$ factor add
$|\mathrm{Vert}_{\gamma}|$ parameters to ${\mathfrak{M}}$. We set
${\mathfrak{M}}^{\mathrm{ext}}={\mathfrak{M}}\times\mathbb{C}^{\mathrm{Vert}_{\gamma}}$
(4.7)
with
$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=\dim_{\mathbb{C}}{\mathfrak{M}}+|\mathrm{Vert}_{\gamma}|=\sum_{i\in\mathrm{Vert}_{\gamma}}\mathbf{v}_{i}$
(4.8)
For the class II theories
$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=\sum_{i\in\mathrm{Vert}_{\gamma}}\mathbf{v}_{i}=Nh$
(4.9)
where $h=\sum_{i\in\mathrm{Vert}_{\gamma}}a_{i}$ is the Coxeter number of
$\mathbf{G}$. Recall that we only encounter simply-laced Lie algebras for
which $h=h^{\vee}$.
We should emphasize that only the true moduli space ${\mathfrak{M}}$ of vacua
has the special geometry with (4.6) defining a positive (outside the loci of
singularities which signal the appearance of massless BPS particles) metric in
the appropriate duality frame. On the extended moduli space
${\mathfrak{M}}^{\mathrm{ext}}$ the prepotential ${\mathscr{F}}$ still defines
some kind of metric, but it cannot be positive everywhere throughout the
variety of masses. This is because the dependence on masses is purely
perturbative. Once we gauge the flavor symmetry (an example of such gauging,
promoting the $A_{1}$ class I theory to the $\widehat{D}_{4}$ class II theory,
will be discussed in section 7.5.1), we correct the metric by the instanton
contributions.
### 4.3. Finite size effects
Subjecting the gauge theory to some boundary conditions reveals more
structure.
For example, we can compactify the four dimensional ${\mathcal{N}}=2$ gauge
theory on a circle ${\mathbb{S}}^{1}$ of radius $R$. The resulting theory
looks like a three dimensional sigma model with the target space
${{\mathfrak{P}}}$ which is a hyperkähler manifold of real dimension
$4{\mathbf{r}}$. The hyperkähler metric on ${\mathfrak{P}}$ contains a lot of
interesting information about the particle content of the original four
dimensional theory.
The hyperkähler structure on ${\mathfrak{P}}$ is a triplet of integrable
complex structures, ${\bf I},{\bf J},{\bf K}$, such that every linear
combination $(a{\bf I}+b{\bf J}+c{\bf K})$ for $a^{2}+b^{2}+c^{2}=1$ is also
an integrable complex structure, and a triplet of the corresponding symplectic
forms ${\omega}_{\bf I},{\omega}_{\bf J},{\omega}_{\bf K}$ which are the
Kähler forms for the metric $g$ on ${\mathfrak{P}}$ in the corresponding
complex structures.
Among the two-sphere of complex structures, one complex structure, which is
usually called $\bf I$, plays a special rôle. This complex structure and the
corresponding $(2,0)$ symplectic form ${\Omega}_{\bf I}={\omega}_{\bf
J}+\mathrm{i}{\omega}_{\bf K}$ are visible in the limit $R\to\infty$, where
${\mathfrak{P}}$ as a metric space collapses to ${\mathfrak{M}}$. For very
large but finite $R$ the manifold ${\mathfrak{P}}$ looks like a fibration over
${\mathfrak{M}}$ whose fibers ${{\mathscr{A}}_{u}}$, $u\in{{\mathfrak{M}}}$
are the abelian varieties (complex tori, which we describe in more detail
momentarily) of diameter which scales like $R^{-1}$.
These fibers ${{\mathscr{A}}_{u}}$ parametrize the ${\mathbb{S}}^{1}$ holonomy
of the abelian gauge fields $A^{\mathcal{I}}$ and their duals
$A^{D}_{\mathcal{I}}$. The reduction of the action (4.5) on ${\mathbb{S}}^{1}$
gives:
$\displaystyle{\mathcal{L}}^{\rm eff3d}=\int\mathrm{i}{\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\alpha}^{\mathcal{I}}\wedge
B^{\mathcal{J}}+$ (4.10) $\displaystyle\qquad\frac{1}{2}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,\left(Rd{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+R^{-1}d{\alpha}^{\mathcal{I}}\wedge\ast
d{\alpha}^{\mathcal{J}}+RB^{\mathcal{I}}\wedge\ast B^{\mathcal{J}}\right)$
where we denote by $\ast$ the three dimensional Hodge star, and by
$B^{\mathcal{I}}$ the curvature of the three dimensional gauge field
$B^{\mathcal{I}}=dA^{\mathcal{I}}_{\rm 3d}$ which is obtained by decomposing
$A^{\mathcal{I}}_{\rm 4d}={\alpha}^{\mathcal{I}}d{\theta}+A^{\mathcal{I}}_{\rm
3d}$. The scalar ${\alpha}^{\mathcal{I}}$ is actually circle-valued, since the
gauge transformations $e^{2\pi\mathrm{i}n_{\mathcal{I}}{\theta}}$ shift it by
$2\pi\mathrm{i}n^{\mathcal{I}}$, $n^{\mathcal{I}}\in{\mathbb{Z}}$. Next we
dualize the three dimensional abelian gauge field, by promoting
$B^{\mathcal{I}}$ to the independent $2$-form, and coupling it to the dual
scalar ${\beta}_{\mathcal{I}}$, which is also circle-valued, in order to
ensure the flux quantization of the original gauge curvature
$B_{\mathcal{I}}$:
$\displaystyle{\mathcal{L}}^{\rm
eff3dd}=\int\mathrm{i}\left(d{\beta}_{\mathcal{J}}+{\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\alpha}^{\mathcal{I}}\right)\wedge
B^{\mathcal{J}}+$ (4.11) $\displaystyle\qquad\frac{1}{2}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,\left(R\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+R^{-1}d{\alpha}^{\mathcal{I}}\wedge\ast
d{\alpha}^{\mathcal{J}}+RB^{\mathcal{I}}\wedge\ast B^{\mathcal{J}}\right)$
$\displaystyle\qquad\longrightarrow\frac{R}{2}\int{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+\frac{1}{2R}\int({\rm
Im}{\tau}^{-1})^{\mathcal{I}\mathcal{J}}dz_{\mathcal{I}}\wedge\ast
d{\bar{z}}_{\mathcal{J}}$
In the last line we have integrated out the unconstrained Gaussian field
$B_{\mathcal{I}}$. We also introduced the holomorphic coordinates
$z_{\mathcal{I}}={\beta}_{\mathcal{I}}+{\tau}_{\mathcal{I}\mathcal{J}}{\alpha}^{\mathcal{J}},\qquad
I=1,\ldots,{\mathbf{r}}$ (4.12)
on the fibers ${{\mathscr{A}}_{u}}$ of the fibration
${{\mathfrak{P}}}\to{{\mathfrak{M}}}$. Both ${\mathfrak{a}}_{\mathcal{I}}$ and
$z_{\mathcal{I}}$ are the ${\bf I}$-holomorphic coordinates on
${\mathfrak{P}}$. By construction, the coordinates $z_{\mathcal{I}}$ are
subject to the periodic identifications:
$z_{\mathcal{I}}\to
z_{\mathcal{I}}+2\pi\mathrm{i}\left(n_{\mathcal{I}}+{\tau}_{\mathcal{I}\mathcal{J}}m^{\mathcal{J}}\right),\qquad
n_{\mathcal{I}},m^{\mathcal{I}}\in{\mathbb{Z}}$ (4.13)
which confirm our assertion that the fibers ${{\mathscr{A}}_{u}}$ of the map
${\mathfrak{P}}\to{\mathfrak{M}}$ are abelian varieties (recall that the
metric ${\rm Im}{\tau}d{\mathfrak{a}}\otimes d{\bar{\mathfrak{a}}}$ is
positive definite, the unitarity requirement). The coordinates
${\mathfrak{a}}_{\mathcal{I}},z_{\mathcal{I}}$ are the Darboux coordinates for
the $(2,0)$ form ${\Omega}_{\bf I}$:
${\Omega}_{\bf
I}=\sum_{{\mathcal{I}}=1}^{{\mathbf{r}}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
dz_{\mathcal{I}}=\sum_{{\mathcal{I}}=1}^{{\mathbf{r}}}d{\mathfrak{a}}^{D}_{\mathcal{I}}\wedge
dz_{D}^{\mathcal{I}}$ (4.14)
as well as the electric-magnetic duals ${\mathfrak{a}}^{D}_{\mathcal{I}}$ and
$z_{D}^{\mathcal{I}}=({\tau}^{-1})^{\mathcal{I}\mathcal{J}}{\beta}_{\mathcal{J}}+{\alpha}^{\mathcal{I}}$.
The fibers ${{\mathscr{A}}_{u}}$ are Lagrangian with respect to $\Omega_{\bf
I}$.
The metric on ${\mathfrak{P}}$, which enters the kinetic term in the Eq.
(4.11) is actually not the correct hyperkähler metric on ${\mathfrak{P}}$ for
finite $R$. It receives corrections which are exponentially small with $R$,
$\sim e^{-M({\mathfrak{a}})R}$ (4.15)
where $M({\mathfrak{a}})$ is the mass of a BPS particle in the Hilbert space
of the theory in four dimensions built over the vacuum $u\in{\mathfrak{M}}$.
As is well-known, the masses of some BPS particles vanish along some loci in
${\mathfrak{M}}$, where the corrections (4.15) become significant. One can
show, however, that $\Omega_{\bf I}$ does not get corrected by the finite size
effects of these BPS particles.
One can also compactify the theory on a two-dimensional Riemann surface
$\Sigma$ (with a partial twist along $\Sigma$, to preserve some
supersymmetry). For $\Sigma$ other then two-torus this leads to the two
dimensional theory with ${\mathcal{N}}=2$ supersymmetry. One has various
sectors labeled by the electric and magnetic fluxes ${\bf
e}=(e_{\mathcal{I}}),{\bf m}=(m^{\mathcal{I}})$ through $\Sigma$. In the
sector where $({\bf e},{\bf m})\neq(0,0)$ one gets an effective superpotential
[Losev:1997tp]:
$W_{({\bf e},{\bf
m})}=\sum_{\mathcal{I}=1}^{\mathbf{r}}e_{\mathcal{I}}\mathfrak{a}^{\mathcal{I}}+m^{\mathcal{I}}{\mathfrak{a}}^{D}_{\mathcal{I}}$
which in four dimensional theory is the central charge of the
${\mathcal{N}}=2$ superalgebra. It is also equal to one of the _action
variables_ of the Seiberg-Witten integrable system [Gorsky:1995zq],
[Donagi:1995cf], [Donagi:1997sr].
If one compactifies on a two-torus, then the resulting two dimensional theory
is the ${\mathcal{N}}=4$ supersymmetric sigma model whose target space
${\mathfrak{P}}$ is the hyperkähler manifold.
It turns out to be quite useful to interpret the ${\mathcal{N}}=2$ theory on a
four dimensional manifold $X$ which can be viewed as a two-torus fibration
over some base $B$, as an effective sigma model with $B$ as a world sheet. In
case where the fibration has singularities of real codimension one (for
example, if $X$ is a product of a disk and a cylinder), then $B$ has a
boundary, and the smoothness of the four dimensional field configurations
translates to particular boundary conditions in the two dimensional sigma
model [Nekrasov:2010ka]. An interesting class of such boundary conditions come
from the so-called canonical coisotropic branes [Kapustin:2001ij,
Kapustin:2006pk, Nekrasov:2010ka]. The algebra of the open string vertex
operators corresponding to such a brane turns out to be the deformation
quantization [MR2062626] of the algebra of holomorphic (in the appropriate
complex structure) functions on ${\mathfrak{P}}$. Remarkably, when
${\mathfrak{P}}$ is an algebraic integrable system in one of the complex
structures, one can apply the fiberwise T-duality along the Liouville fibers,
leading to the mirror perspective on the quantization procedure. First of all,
in the case of the Hitchin system the mirror manifold turns out to be the
Hitchin system for the Langlands dual group. In the general case the mirror
${{\mathfrak{P}}}^{\vee}$ of the original hyperkähler manifold
${\mathfrak{P}}$ is also expected to be an integrable system. The mirror of
the canonical coisotropic brane is believed to to be a holomorphic (in an
appropriate complex structure) Lagrangian brane. In the case of Hitchin system
this brane is the so-called brane ${\mathcal{B}}_{\mathcal{O}}$ of opers
[Kapustin:2006pk].
This approach to quantization of the integrable systems from gauge theories
will be elaborated upon in [NP2012b].
### 4.4. The appearance of an integrable system
The complex symplectic manifold $({{\mathfrak{P}}},{\Omega}_{\bf I})$, its
projection ${\pi}:{\mathfrak{P}}\to{\mathfrak{M}}$ with Lagrangian fibers
${{\mathscr{A}}_{u}}={\pi}^{-1}(u)$, $u\in{\mathfrak{M}}$, which are
principally polarized abelian varieties (the principal polarization comes from
the restriction of ${\omega}_{\bf I}$ onto the fibers) define what is known as
the _algebraic integrable system_ [Donagi:1995am], [Donagi:1995cf],
[Donagi:1997sr]. It is one of the possible complexifications of the familiar
notion of the completely integrable system in the classical mechanics.
The other possibility, namely a complex symplectic manifold with the
Lagrangian fibration whose fibers are the complex tori
$\left({\mathbb{C}}^{\times}\right)^{\mathbf{r}}$, is also realized in the
context of gauge theories. However, the base of such a system typically
parametrizes the space of mass parameters of the gauge theory.
The fibers ${{\mathscr{A}}_{u}}$ are the Liouville tori, while
$({\mathfrak{a}}_{\mathcal{I}},z_{\mathcal{I}})$ are the action-angle
variables. The novelty of the complex case is the doubling of the possible
choices of the action-angle variables with fixed Liouville fibration. Indeed,
the fibers ${{\mathscr{A}}_{u}}$ are the $2{\mathbf{r}}$-real dimensional
tori, therefore in producing the action variables as in the Arnol’d-Liouville
theorem one has a choice of $\mathbf{r}$ out of $2\mathbf{r}$ cycles in
$H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$. The lattice
$H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$ has a symplectic form $\varpi$,
which comes from the polarization, i.e. a properly normalized class of the
restriction ${\omega}_{\bf I}|_{{{\mathscr{A}}_{u}}}$. It turns out that any
Lagrangian sublattice $L$ in $H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$ defines
a system of local coordinates $({\mathfrak{a}}_{I})$ on the base
${\mathfrak{M}}$ near the point $u\in{\mathfrak{M}}$, as well as the conjugate
angle-like coordinates $(z_{\mathcal{I}})$ on the fiber ${{\mathscr{A}}_{u}}$
itself. Let ${\mathcal{A}}_{\mathcal{I}}$ be the integral basis of this
sublattice $L\subset H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$. Then
$d{\mathfrak{a}}^{\mathcal{I}}=\oint_{{\mathcal{A}}_{\mathcal{I}}}{\Omega}_{\bf
I}$ (4.16)
One can also define
$d{\mathfrak{a}}^{D}_{\mathcal{I}}=\oint_{{\mathcal{B}}^{\mathcal{I}}}{\Omega}_{\bf
I}$ (4.17)
where ${\mathcal{B}}^{\mathcal{I}}$ is the basis in the dual sublattice
$L^{\vee}\subset H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$, such that
${\varpi}({\mathcal{A}}_{\mathcal{I}},{\mathcal{A}}_{\mathcal{J}})={\varpi}({\mathcal{B}}^{\mathcal{I}},{\mathcal{B}}^{\mathcal{J}})=0,\qquad{\varpi}({\mathcal{A}}_{\mathcal{I}},{\mathcal{B}}^{\mathcal{J}})={\delta}_{\mathcal{I}}^{\mathcal{J}}$
(4.18)
One then shows that
$\sum_{{\mathcal{I}}=1}^{\mathbf{r}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}_{\mathcal{I}}^{D}\equiv 0$ (4.19)
on ${\mathfrak{M}}$, which, in turn, implies (4.4). The coordinates
$z_{\mathcal{I}}$ along ${{\mathscr{A}}_{u}}$ are defined using (4.14) with
the normalization (4.13) that half of the periods of $z_{\mathcal{I}}$ are in
$2\pi\mathrm{i}{\mathbb{Z}}$.
The integrable systems which one encounters in the classical mechanics are
rarely given in the form of the action-angle variables. Usually one has the
phase space ${\mathfrak{P}}$, the symplectic form ${\Omega}_{\bf I}$, perhaps
some Darboux coordinates
${\Omega}_{\bf I}=\sum_{{\mathcal{I}}=1}^{\mathbf{r}}dp_{\mathcal{I}}\wedge
dq^{\mathcal{I}}$
and the collection of Poisson-commuting functionally independent Hamiltonians
$U_{1}(p,q),\ldots,U_{\mathbf{r}}(p,q)$. One then looks for the action-angle
coordinates, i.e. the Darboux coordinates $({\mathfrak{a}},z)$, such that the
Hamiltonians $U_{\mathcal{I}}(p,q)=u_{\mathcal{I}}({\mathfrak{a}})$ depend
only on ${\mathfrak{a}}$, the action variables. The Hamiltonian evolution then
linearizes on the fibers ${{\mathscr{A}}_{u}}$, which are the level sets of
the Hamiltonians. The motion is a constant velocity motion in the $z$
coordinates:
$z_{\mathcal{I}}(t)=z_{\mathcal{I}}(0)+\sum_{\mathcal{J}=1}^{\bf
r}\,t_{\mathcal{J}}\,\frac{{\partial}u_{\mathcal{J}}}{{\partial}{\mathfrak{a}}_{\mathcal{I}}}$
It is interesting to study the level sets ${{\mathscr{A}}_{u}}$ of the
Hamiltonians, the Liouville tori. The algebraic integrable systems are such,
that the fibers can be compactified to become the polarized abelian varieties.
Where do the polarized abelian varieties come from?
### 4.5. Integrable systems from classical gauge theories
One source of the polarized abelian varieties are the Jacobians of the
algebraic curves. The Liouville tori of algebraic integrable systems can be
often found inside the Jacobians of the algebraic curves, constructed while
solving some classical gauge field equations.
#### 4.5.1. Hitchin system
There is an interesting class of algebraic integrable systems for which the
Liouville tori are precisely these Jacobians. Take the $U(N)$ Hitchin system
on a genus $g$ Riemann surface. The phase space ${\mathfrak{P}}$ is the
cotangent bundle (up to a birational transformation) to the moduli space
$M_{N,c}$ of holomorphic rank $N$ vector bundles $E$ over $\Sigma$ with fixed
first Chern class $c=c_{1}(E)$. It is convenient to take $(c,N)=1$ to avoid
complications coming from the reducible connections.
In the complex structure $\bf I$ the holomorphic coordinates on
${\mathfrak{P}}$ are $({\bar{A}},{\Phi})$, where ${\bar{\partial}}+{\bar{A}}$
is the $(0,1)$-connection on the smooth vector bundle $E$ which endows it with
the complex structure, and $\Phi\in End(E)\otimes{\Omega}^{1,0}({\Sigma})$ is
the holomorphic Higgs field:
${\bar{\partial}}{\Phi}+[{\bar{A}},{\Phi}]=0$ (4.20)
The symplectic form on ${\mathfrak{P}}$ comes from the $(2,0)$ symplectic form
on the space of all smooth pairs $({\bar{A}},{\Phi})$
${\Omega}_{\bf
I}=\int_{\Sigma}{\operatorname{tr}}\,{\delta}{\Phi}\wedge{\delta}{\bar{A}}$
(4.21)
by the symplectic reduction with respect to the action of the gauge group:
$g:\,({\bar{A}},{\Phi})\longrightarrow(g^{-1}{\bar{A}}g+g^{-1}{\bar{\partial}}g,g^{-1}{\Phi}g)$
The set of Poisson-commuting Hamiltonians is given by:
$U_{i,{\mathbf{a}}}=\int_{\Sigma}{\nu}_{i,{\mathbf{a}}}{\operatorname{tr}}\,{\Phi}^{i},\qquad
i=1,\ldots,N$ (4.22)
where ${\nu}_{i,{\mathbf{a}}}\in H^{0,1}({\Sigma},K_{\Sigma}^{\otimes(1-i)})$,
${\mathbf{a}}=1,\ldots,(2i-1)(g-1)+{\delta}_{i,1}$ form a basis in the space
of holomorphic $(1-i,1)$-differentials. Fixing the values $u_{i,{\mathbf{a}}}$
of all the Hamiltonians $U_{i,{\mathbf{a}}}$ gives us a point
$u\in{{\mathfrak{M}}}$ in the vector space
${{\mathfrak{M}}}=\bigoplus_{i}H^{0,1}({\Sigma},K_{\Sigma}^{\otimes(1-i)})$
(4.23)
One defines the _spectral curve_ $C_{u}\subset T^{*}{\Sigma}$ as the zero
locus of the characteristic polynomial of $\Phi$:
${\rm Det}({\Phi}-{\lambda})=0$ (4.24)
It is a holomorphic curve thanks to (4.20), which is invariant under the
Hamiltonian flows generated by the Hamiltonians (4.22). The curve $C$ is an
$N$-sheeted cover of $\Sigma$
${\pi}:C_{u}\to\Sigma$ (4.25)
Its genus can be computed using the Riemann-Hurwitz formula:
$2-2g_{C_{u}}=N(2-2g_{\Sigma})-{\delta}$
where ${\delta}=2N(N-1)(g_{\Sigma}-1)$ is the number of branch points. The
latter is the number of zeroes of the discriminant of the polynomial (4.24),
which is a holomorphic $N(N-1)$-differential on $\Sigma$. Thus:
$g_{C_{u}}=N^{2}(g_{\Sigma}-1)+1$
The Jacobian of $C$ is thus an abelian variety of dimension
$g_{C_{u}}=g_{\Sigma}+\sum_{j=2}^{N}(2j-1)(g_{\Sigma}-1)$ (4.26)
which is equal to the dimension of the base ${\mathfrak{M}}$ of the Hitchin
fibration. The fibers ${\mathscr{A}}_{u}$ of the Hitchin fibration are thus
the Jacobians of the corresponding spectral curves.
One generalization is to study the $SL(N)$ Hitchin system. In this case the
corresponding rank $N$ vector bundles have the trivial determinant, and the
corresponding Higgs field is traceless. The base of the Hitchin fibration now
has the dimension $(N^{2}-1)(g_{\Sigma}-1)$, the equation (4.24) has vanishing
$\propto{\lambda}^{N-1}$ term, and the fibers ${\mathscr{A}}_{u}$ are not the
full Jacobians of the spectral curve $C_{u}$, which still has the genus
$g_{C_{u}}$ (4.26) but the kernel $J_{0}$ of the map ${\pi}_{*}:Jac(C_{u})\to
Jac({\Sigma})$, which sends the degree zero line bundle $L$ on $C_{u}$ to the
line bundle ${\mathscr{L}}={\rm Det}{\pi}_{*}L$ on $\Sigma$, whose fiber
${\mathscr{L}}_{z}$ over the point $z\in\Sigma$ is the tensor product of the
fibers $L_{y}$ of $L$ over all preimages of $z$:
${\mathscr{L}}_{z}=\bigotimes_{y\in{\pi}^{-1}(z)}L_{y}$ (4.27)
The Hitchin system can be defined [Hitchin:1987mz] for any algebraic Lie group
$G$, with the maximal torus $T$. Let $\mathfrak{g}=Lie(G)$,
$\mathfrak{h}=Lie(T)$. The Hitchin space is the moduli space of stable pairs
$({\mathcal{P}},{\Phi})$, where ${\mathcal{P}}$ is a holomorphic $G$-bundle
over $\Sigma$, and $\Phi$ is a holomorphic $(1,0)$-form on $\Sigma$, valued in
the bundle of Lie algebras $\mathfrak{g}$, associated with $\mathcal{P}$ via
the adjoint representation:
${\Phi}\in H^{0}({\Sigma},K_{\Sigma}\otimes{\rm ad}({\mathcal{P}}))\ .$
The Hitchin fibration is again defined by fixing the gauge-invariant
polynomials $P_{j}({\Phi})\in H^{0}\left({\Sigma},K_{\Sigma}^{\otimes
d_{j}}\right)$ of the $\mathfrak{g}$-valued Higgs field $\Phi$:
$u=\left(P_{j}({\Phi})\right)_{j=1}^{r}\in\bigoplus_{j=1}^{r}{\mathbb{C}}^{(2d_{j}-1)(g_{\Sigma}-1)}={{\mathfrak{M}}}$
where $d_{j}$’s are the degrees of basic Ad-invariant polynomials on
$\mathfrak{g}$.
The fibers of the Hitchin fibration are now trickier to define. First of all,
there is no preferred notion of the spectral curve. For some gauge groups one
can use the minuscule representation, but this is not always available.
One option is to consider the so-called _cameral curve_ ${\mathcal{C}}_{u}$,
which is a $W({\mathfrak{g}})$-cover of the base curve $\Sigma$. The points of
the cameral curve ${\mathcal{C}}_{u}$ are, over generic $z$, the pairs
$({\varphi},z)$, where $z\in\Sigma$ and $\varphi\in\mathfrak{h}$ is the
element of the fixed Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ which
is conjugate to the Higgs field $\Phi(z)$. This definition makes sense for the
points $z\in\Sigma$ for which $\Phi(z)$ is semi-simple, i.e. belongs to the
$ad(G)$-orbit of an element in $\mathfrak{h}$. If this is not the case (e.g.
$\Phi(z)$ is conjugate to a Jordan block in the $GL(N)$ case), one can find an
appropriate representative in $\mathfrak{h}$ by modifying the equivalence
relation (e.g. two matrices are equivalent if their characteristic polynomials
coincide). To stress the fact that ${\mathcal{C}}_{u}$ depends on $u$ which is
the set of holomorphic $d_{j}$-differentials $P_{j}({\Phi})$ .
Over ${\mathcal{C}}_{u}$ so defined one has $r$ line bundles,
${\mathcal{L}}_{i}$, $i=1,\ldots,r$, which correspond to the fundamental
weights $\lambda_{i}\in\mathfrak{h}^{*}$. The line bundle ${\mathcal{L}}_{i}$
is a subbundle in the holomorphic vector bundle ${\bf
R}_{i}=R_{i}\times_{G}{\mathcal{P}}$, associated with $\mathcal{P}$ via the
$i$’th fundamental representation $R_{i}$ of $G$. The fiber of
${\mathcal{L}}_{i}\subset{\bf R}_{i}$ over $({\varphi},z)$ is the eigenspace
corresponding to the eigenvalue ${\lambda}_{i}({\varphi})$.
To any weight vector ${\lambda}\in\Lambda$ a line bundle
${\mathcal{L}}_{\lambda}$ over $\mathcal{C}_{u}$ can be associated:
${\lambda}=\sum_{i=1}^{r}n_{i}{\lambda}_{i}\
\mapsto{\mathcal{L}}_{\lambda}=\bigotimes_{i=1}^{r}{\mathcal{L}}_{i}^{\otimes\,n_{i}}$
(4.28)
In a more physical language, the Hitchin moduli space is the quotient of the
space of pairs $({\bar{A}},{\Phi})$, where ${\bar{A}}$ is a $(0,1)$-connection
on smooth principal $G$-bundle ${\mathcal{P}}$ over $\Sigma$, and $\Phi$ is a
$(1,0)$ $\mathfrak{g}$-valued form, which are compatible, i.e. solve the Eq.
(4.20), and are considered up to the $G$-gauge transformations:
$g:({\bar{A}},{\Phi})\mapsto(g^{-1}{\bar{\partial}}g+Ad_{g}{\bar{A}},Ad_{g}{\Phi})$
By fixing the partial gauge ${\Phi}={\varphi}\in\mathfrak{h}$ for fixed
$\mathfrak{h}\subset\mathfrak{g}$, one reduces the gauge invariance from $G$
to $N(T)$. The Eq. (4.20) imply that in this gauge $\bar{A}$ is a
$T$-connection $\bar{A}=\bar{a}$, with the $T$ subgroup of $N(T)$ acting by
the $T$-gauge transformations $\bar{a}\mapsto\bar{a}+{\bar{\partial}}{\chi}$,
$e^{\chi}\in T$. On $\Sigma$ the $T$-valued gauge field and the
$\mathfrak{h}$-valued Higgs field $\varphi$ are not well-defined, since there
are the $W({\mathfrak{g}})=N(T)/T$ remaining gauge transformations. On
$\mathcal{C}$, however, both $\varphi$ and $\bar{a}$ are well-defined. In
fact, ${\bar{a}}$ defines on $\mathcal{C}_{u}$ a holomorphic principal
$T$-bundle $\mathcal{T}$, so that $L_{i}={\mathcal{T}}^{{\lambda}_{i}}$. The
$T$-bundle $\mathcal{T}$ is $W({\mathfrak{g}})$-equivariant. This is the
translation of the fact that the Weyl group $W({\mathfrak{g}})$ acts
simultaneously on $\varphi$ and $\bar{a}$. The isomorphism (properly
understood at the ramification points)
Holomorphic principal $G$-bundles $\mathcal{P}$ on $\Sigma$, holomorphic Higgs
fields $\Phi\in H^{0}({\Sigma},K_{\Sigma}\otimes{\rm ad}({\mathcal{P}}))$
$\Leftrightarrow$ $W({\mathfrak{g}})$-covers $\mathcal{C}$ of $\Sigma$,
$\mathcal{C}\subset T^{*}{\Sigma}\otimes\mathfrak{h}$, holomorphic
$W({\mathfrak{g}})$-equivariant principal $T$-bundles on $\mathcal{C}$
allows to represent the Hitchin moduli space as a fibration over the vector
space ${\mathfrak{M}}$, whose points are the $W({\mathfrak{g}})$-invariant
curves $\mathcal{C}_{u}$ sitting in the tensor product
$T^{*}\Sigma\otimes\mathfrak{h}$ (this is almost a tautology: a
$W({\mathfrak{g}})$-invariant curve in $T^{*}\Sigma\otimes\mathfrak{h}$ is a
curve in $T^{*}\Sigma\otimes\mathfrak{h}/W({\mathfrak{g}})$, i.e. a
holomorphic section of the vector bundle
$T^{*}{\Sigma}\otimes{\mathbb{C}}[{\mathfrak{h}}]^{W({\mathfrak{g}})}$).
The fiber ${\mathscr{A}}_{u}$ of the Liouville fibration (which is called
Hitchin’s fibration in this case) is a generalized Prym variety, which is,
roughly speaking,
${{\mathscr{A}}_{u}}\approx{\operatorname{Hom}}_{W({\mathfrak{g}})}\left({\Lambda},{\rm
Pic}({\mathcal{C}}_{u})\right)={\mathrm{Bun}}_{T}({\mathcal{C}}_{u})^{W({\mathfrak{g}})}$
(4.29)
The papers [Donagi:1995alg], [Donagi:2000dr], [Donagi:1998vx] correct the Eq.
( 4.29 ) in a couple of subtle points as well as provide the additional
theory.
#### 4.5.2. Instanton moduli spaces as integrable systems
Hitchin’s equations (4.20), for flat $\Sigma$, are the dimensional reduction
of the instanton (or anti-self-duality) equations from four dimensions. It
turns out that one can get an integrable system directly from the moduli
spaces of four dimensional instantons, or three dimensional monopoles.
We only briefly sketch the constructions here.
Let $S$ be an elliptic K3 manifold, i.e. an algebraic surface, with the
holomorphic ${\omega}_{S}^{2,0}$ form, and with the projection
${\pi}:S\to{\mathbb{C}\mathbb{P}}^{1}$ whose fibers ${\pi}^{-1}(z)$,
$z\in{\mathbb{C}\mathbb{P}}^{1}$ are the elliptic curves ${\mathscr{E}}_{z}$
(generically nonsingular). One can endow $S$ with the hyperkähler metric.
Consider the moduli space ${{\mathfrak{P}}}={\mathcal{M}}_{N}(G)$ of charge
$N$ $G$-instantons on $S$, i.e. the solutions to the system of partial
differential equations
$\displaystyle F_{A}\wedge{\omega}_{\bf I}=F_{A}$
$\displaystyle\wedge{\omega}_{\bf J}=F_{A}\wedge{\omega}_{\bf K}=0$ (4.30)
$\displaystyle F^{0,2}_{A}=0$
(the last equation is a linear combination of the $\bf J$ and $\bf K$
equations from the first line) of fixed instanton charge $N\geq 0$:
$-\frac{1}{8\pi^{2}}\int_{S}{\operatorname{tr}}F_{A}\wedge F_{A}=N$ (4.31)
Here $G$ is some compact simply-connected simple Lie group, which has a
simply-laced Lie algebra $\mathfrak{g}$. The moduli space
${{\mathfrak{P}}}={\mathcal{M}}_{N}(G)$ is also hyperkähler, in particular it
is holomorphic symplectic, with the $(2,0)$-form given by:
${\Omega}_{\bf
I}^{N,G}=\int_{S}{\omega}_{S}^{2,0}\wedge\,{\operatorname{tr}}\,{\delta}{\bar{A}}\wedge{\delta}{\bar{A}}$
(4.32)
The integrable system structure is obtained by studying the restriction of the
instanton gauge field on the elliptic fibers, where generically they define a
point in the coarse moduli space $\mathrm{Bun}_{\bf G}({\mathscr{E}}_{z})$ of
semi-stable principal holomorphic $\bf G$-bundles on the fiber, see C.5.
Thanks to E. Loojienga’s theorem, this moduli space is a weighted projective
space, which can be identified for different non-singular fibers. One gets
thus a section of the locally trivial bundle of
${\Pi}:{\mathscr{P}}=\bigcup_{z\in{\mathbb{C}\mathbb{P}}^{1}}\mathrm{Bun}_{\bf
G}({\mathscr{E}}_{z})\longrightarrow{\mathbb{C}\mathbb{P}}^{1}$
One has to be careful at the singular fibers. The base ${\mathfrak{M}}$ of the
integrable systems is the properly compactified moduli space of the
holomorphic sections
${\sigma}:{\mathbb{C}\mathbb{P}}^{1}\longrightarrow{\mathscr{P}}$ of
appropriate degree with some ramification conditions at the discriminant locus
of the original elliptic fibration $\pi$.
In this work we shall not encounter these difficulties.
In fact, as we shall explain in more detail in the section 8, the moduli
spaces of vacua of the quiver gauge theories we study lead to the integrable
systems which arise from the the moduli spaces of $G$-monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ for class I theories with
${\mathbf{G}}_{\text{q}}=\mathbf{G}$, or from the moduli spaces of
$G$-instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ for class II
theories with ${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$. Here $G$ is a
compact Lie group, whose complexification is the complex simple Lie group
$\mathbf{G}$.
The moduli space ${\mathfrak{P}}$ of $G$-instantons, viewed in the complex
structure where
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}={\mathbb{C}}^{1}\times{\mathscr{E}}$,
is birational to the moduli space of semi-stable holomorphic
$\mathbf{G}$-bundles on ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$, with fixed trivialization at
${\infty}\times{\mathscr{E}}$. The moduli space ${{\mathfrak{P}}}$ projects
down to the moduli space ${\mathfrak{M}}$ of quasimaps from
${\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}$ to the moduli space
of semi-stable holomorphic bundles $\mathrm{Bun}_{\mathbf{G}}(\mathscr{E})$ on
a fixed elliptic curve $\mathscr{E}$. The moduli space of monopoles maps to
the moduli space of quasimaps with prescribed singularities on
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$ to
$B({\mathfrak{g}})={\mathbf{G}}/Ad({\mathbf{G}})={\mathbf{T}}/W({\mathfrak{g}})$.
### 4.6. Extended moduli space as a complex integrable system
The extended moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ is a base of a
complex, but not algebraic, integrable system
${{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$. The
Liouville tori of this integrable system are acted on by an algebraic torus
$({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$, so that the quotients are
the compact abelian varieties, the Liouville tori fibered over
${\mathfrak{M}}$. The symplectic quotient of ${\mathfrak{P}}^{\mathrm{ext}}$
with respect to $({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$ at some
level of the moment map, which is linearly determined by the values of the bi-
fundamental masses, gives ${\mathfrak{P}}$. Recall that Duistermaat-Heckmann
theorem [Duistermaat:1982vw] then implies that the cohomology class
$[{\Omega}_{\bf I}]$ of the $(2,0)$-symplectic form ${\Omega}_{\bf I}$ on
${\mathfrak{M}}$ is linear with masses.
## Chapter 5 The limit shape equations
In this section we return to the microscopic analysis of our gauge theory.
Recall that the ${\mathcal{N}}=2$ supersymmetry algebra is generated by four
supercharges ${\mathcal{Q}}_{{\alpha}i}$, ${\alpha}=1,2$, $i=1,2$ of the left
and by four supercharges ${\bar{\mathcal{Q}}}_{\dot{\alpha}}^{i}$,
${\dot{\alpha}}=1,2$ of the right chirality. The prepotential
${\mathcal{F}}({\mathcal{A}})$ of the theory is a function of the superfield
${\mathcal{A}}$ which is annihilated by ${\mathcal{Q}}_{{\alpha}i}$’s. We
shall now focus on the observables which are in the cohomology of one of the
${\mathcal{Q}}_{{\alpha}i}$ supercharges, which we shall call simply
$\mathcal{Q}$.
### 5.1. The amplitude functions
The basic such observable is the scalar ${\Phi}_{i}$ in the vector multiplet.
More precisely, any gauge invariant functional, in particular the local
operator $P({\Phi}_{i}({\bf x}))$, where $P$ is some invariant polynomial on
the Lie algebra of $SU(\mathbb{B}_{i})$, and $\bf x$ is a point in space-time,
is annihilated by $\mathcal{Q}$. Moreover, the observables $P({\Phi}_{i}({\bf
x}))$ and $P({\Phi}_{i}({\bf x}^{\prime}))$ for two different points ${\bf x}$
and ${\bf x}^{\prime}$ are in the same $\mathcal{Q}$-cohomology class.
Therefore, one may talk about the vacuum expectation value of $P({\Phi}_{i})$
without specifying the point $\bf x$.
Consider the observables
${\mathcal{O}}_{n,i}={\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\Phi}_{i}^{n}$.
Form the generating function:
${\mathscr{Y}}_{i}(x)=x^{{\mathbf{v}}_{i}}\,{\exp}\,\left(-\sum_{n=1}^{\infty}\frac{\langle{\mathcal{O}}_{n,i}\rangle_{u}}{n}\,x^{-n}\right)$
(5.1)
which turns out to be well-behaved for sufficiently large $x$. We shall denote
the $x$-plane where ${\mathscr{Y}}_{i}(x)$ are defined, by
$\mathbf{C}_{\left\langle x\right\rangle}$. Actually, the analytic
continuation in the $x$ variable gives us the set ${\bf\mathscr{Y}}(x)$ of
multi-valued analytic functions on $\mathbf{C}_{\left\langle x\right\rangle}$,
${\bf\mathscr{Y}}(x)=({\mathscr{Y}}_{i}(x))_{i\in\mathrm{Vert}_{\gamma}},$
This set of multi-valued functions captures the vacuum expectation values of
all the local gauge invariant observables commuting with the supercharge
${\mathcal{Q}}$.
###### Remark.
The general relation between the amplitude functions ${\mathscr{Y}}_{i}(x)$
and the polynomials ${\mathscr{T}}_{i}(x)$ generating the first non-trivial
Casimirs of the gauge group $G_{\text{g}}$ is
${\mathscr{T}}_{i}(x)=\left({\mathscr{Y}}_{i}(x)\right)_{+}$
where $(\ldots)_{+}$ denotes the polynomial part.
In what follows we shall use another set of
$(T_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ polynomials, ${\rm
deg}T_{i}(x)={\mathbf{v}}_{i}$, which are not monic. The coefficients of
$T_{i}(x)$ are related to the coefficients of ${\mathscr{T}}_{i}(x)$ by a
‘‘mirror map’’ change of variables, which will become clear in the course of
our exposition.
###### Remark.
In the way we defined these functions, the information about the vacuum
expectation values of these observables is contained in the expansion of
${\mathscr{Y}}_{i}(x)$ near $x=\infty$ on the physical sheet of these
functions. It would be interesting to see whether the expansion at $x=\infty$
of the branches of ${\mathscr{Y}}_{i}(x)$ contains information about the vevs
of the chiral observables of the theories, related to the one we started with
via some version of $S$-duality.
The functions ${\mathscr{Y}}_{i}(x)$ are the integral transforms of the
densities
${\rho}_{i}({\mathrm{x}}),\qquad{\mathrm{x}}\in{\mathbb{R}},i\in\mathrm{Vert}_{\gamma},$
which describe the combinatorics of the set of fixed points of the symmetry
group action on the instanton moduli space used in the localization approach
to the calculation of the supersymmetric partition function of the gauge
theory. We do not review the formalism [Nekrasov:2003rj] in this work. For the
prior work on the subject see [Moore:1997dj, Lossev:1997bz, Losev:1997wp,
Moore:1998et, Losev:1999nt, Losev:1999tu, Losev:2000bf, Nekrasov:2002dd,
Nekrasov:2002qd, Nekrasov:2003zj, Losev:2003py] and for the novel applications
and refinements see [Pestun:2007rz, Alday:2009aq].
Instead we write down the equations obeyed by the amplitude functions, the so-
called _limit shape equations_ , generalizing the limit shape equations
studied in [Nekrasov:2003rj, Nekrasov:2004vw, Shadchin:2005mx,
Shadchin:2005cc, Shadchin:2005hp]. We shall solve the limit shape equations
using the analytic properties of the amplitude functions. One finds that the
analytic continuation of these functions is governed by the monodromy group,
which we shall call the iWeyl group (the _instanton Weyl group_).
The iWeyl group is the Weyl group $W(\mathfrak{g_{\text{q}}})$. For the Class
I theories $W(\mathfrak{g_{\text{q}}})$ is the finite Weyl group
$W({\mathfrak{g}})$ of the corresponding ADE simple Lie algebra
$\mathfrak{g}$, for the Class II theories the iWeyl group turns out to be the
affine Weyl group $W(\widehat{\mathfrak{g}})$ of the corresponding affine Lie
algebra $\widehat{\mathfrak{g}}=\mathfrak{g_{\text{q}}}$. The Weyl group of
$\widehat{GL}_{\infty}$ shows up in the Class II* theories.
We solve the limit shape equations by constructing the iWeyl invariants
${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))$ of ${\mathscr{Y}}_{i}(x+{\mu}_{i})$,
for the appropriate shifts ${\mu}_{i}$, and showing that these invariants are
polynomials in $x$,
${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))=T_{j}(x),\qquad
j\in\mathrm{Vert}_{\gamma}$ (5.2)
For the Class II theories the invariants ${{\mathscr{X}}}_{j}$ are convergent
power series in ${\mathfrak{q}}_{j}$. Moreover, in each order in expansion in
${\mathfrak{q}}=\prod_{i}{\mathfrak{q}}_{i}^{a_{i}}$ they are finite Laurent
polynomials in ${\mathscr{Y}}_{j}$’s. For the Class II* theories the
invariants ${{\mathscr{X}}}_{j}$ are convergent power series in
${\mathfrak{q}}_{j}$, and finite Laurent polynomials in
${\mathscr{Y}}_{j}(x+{\mu}_{j}+lm^{*})$, for a finite collection of integers
$l\in{\mathbb{Z}}$, again in every order in $\mathfrak{q}$ expansion. For the
Class I theories the functions ${{\mathscr{X}}}_{j}$ are polynomials in
${\mathscr{P}}_{j}(x)$ and Laurent polynomials in ${\mathscr{Y}}_{j}(x)$.
#### 5.1.1. The densities and the amplitude functions
The amplitude functions ${\mathscr{Y}}_{i}(x)$ are the multi-valued analytic
functions, which we defined, for large $x$, via Eq. (5.1):
${\mathscr{Y}}_{i}(x)={\exp}\
\langle{\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\rm
log}\left(x-{\Phi}_{i}\right)\rangle_{u}$ (5.3)
One shows, using the fixed point techniques that
${\mathscr{Y}}_{i}(x)={\exp}\
\int_{\mathbb{R}}\,d{\mathrm{x}}\,{\rho}_{i}({\mathrm{x}}){\log}(x-{\mathrm{x}})$
(5.4)
where the _density_ function ${\rho}_{i}(x)$ has compact support which
consists of ${\mathbf{v}}_{i}$ intervals
${\rm supp}\,{\rho}_{i}=\bigcup_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\qquad
I_{i,{\mathbf{a}}}$
The intervals $I_{i,{\mathbf{a}}}$ should be thought of as the ‘‘fattened’’
versions of the eigen-values ${\mathfrak{a}}_{i,{\mathbf{a}}}$. More
precisely:
$1=\int_{I_{i,{\mathbf{a}}}}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}},\qquad{\mathfrak{a}}_{i,{\mathbf{a}}}=\int_{I_{i,{\mathbf{a}}}}{\mathrm{x}}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}}$
(5.5)
The functions ${\mathscr{Y}}_{i}(x)$ have, therefore, the cuts at the
intervals $I_{i,{\mathbf{a}}}$, with the limit values
${\mathscr{Y}}^{\pm}_{i}$ of the ${\mathscr{Y}}_{i}$ function at the top and
the bottom banks of the interval $I_{i,{\mathbf{a}}}$ being related via:
${\mathscr{Y}}_{i}^{+}(x)/{\mathscr{Y}}_{i}^{-}(x)={\exp}\
2\pi\mathrm{i}\int^{x}_{-\infty}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}}$
One then analytically continues ${\mathscr{Y}}_{i}(x)$ across the cuts, which
leads to the set of the multi-valued analytic functions. We shall describe
this analytic continuation in detail in the coming section.
#### 5.1.2. The special coordinates
From the Eqs. (5.3) and (5.5) one derives:
${\mathfrak{a}}_{i,{\mathbf{a}}}=\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d\,{\log}{\mathscr{Y}}_{i}(x)$
(5.6)
where $A_{i{\mathbf{a}}}$ is a small loop surrounding the cut
$I_{i,{\mathbf{a}}}$, see Fig. 5.1.
Figure 5.1. Support intervals $I_{i,\mathbf{a}}$ of the densities $\rho_{i}$
and the cycles $A_{i\mathbf{a}}$
### 5.2. The limit shape prepotential
The prepotential of the low-energy theory is expressed in terms of the
densities as follows:
$\displaystyle{\mathcal{F}}({\mathfrak{a}};m;{\tau})$ $\displaystyle\ =\
-\int\int_{{\mathbb{R}}^{2}}d{\mathrm{x}}^{\prime}d{\mathrm{x}}^{\prime\prime}\sum_{i\in\mathrm{Vert}_{\gamma}}{\rho}_{i}({\mathrm{x}}^{\prime}){\rho}_{i}({\mathrm{x}}^{\prime\prime}){\mathcal{K}}({\mathrm{x}}^{\prime}-{\mathrm{x}}^{\prime\prime})$
$\displaystyle+\int\int_{{\mathbb{R}}^{2}}d{\mathrm{x}}^{\prime}d{\mathrm{x}}^{\prime\prime}\sum_{e\in\mathrm{Edge}_{\gamma}}{\rho}_{t(e)}({\mathrm{x}}^{\prime}){\rho}_{s(e)}({\mathrm{x}}^{\prime\prime}){\mathcal{K}}({\mathrm{x}}^{\prime}-{\mathrm{x}}^{\prime\prime}+m_{e})$
$\displaystyle+\sum_{i\in\mathrm{Vert}_{\gamma}}\int_{{\mathbb{R}}}d{\mathrm{x}}\,{\rho}_{i}({\mathrm{x}})\left(({\log}{\mathfrak{q}}_{i})\frac{{\mathrm{x}}^{2}}{2}+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}({\mathrm{x}}-m_{i,{\mathfrak{f}}})\right)$
$\displaystyle\qquad\qquad\qquad+\sum_{i,\mathbf{a}}\mathfrak{b}_{i,\mathbf{a}}\left(1-\int_{I_{i,\mathbf{a}}}\rho_{i}({\mathrm{x}})d{\mathrm{x}}\right)$
$\displaystyle+\sum_{i,\mathbf{a}}\mathfrak{a}_{i,\mathbf{a}}^{D}\left(\mathfrak{a}_{i,\mathbf{a}}-\int_{I_{i,\mathbf{a}}}{\mathrm{x}}\rho_{i}({\mathrm{x}})\,d{\mathrm{x}}\right)$
where the constraints (5.5) are incorporated by the last two lines via
Lagrangian multipliers
$\mathfrak{b}_{i,\mathbf{a}},\mathfrak{a}_{i,\mathbf{a}}^{D}$,
${\mathcal{K}}(x)=\frac{x^{2}}{2}\left(\log\left(\frac{x}{\Lambda_{UV}}\right)-\frac{3}{2}\right)$
(5.7)
and $\Lambda_{UV}$ is the UV cutoff scale. In fact, the
$\Lambda_{UV}$-dependence drops out for the theories solving the
${\beta}_{i}=0$ equations. However, in the intermediate formulae we keep the
explicit $\Lambda_{UV}$-dependence.
#### 5.2.1. The limit shape equations.
Originally the limit shape equations were derived as the variational equations
on $\mathcal{F}$ (see [Nekrasov:2003rj], [Nekrasov:2004vw]). These are linear
integral equations on the densities ${\rho}_{i}(x)$: for any $x\in
I_{i,{\mathbf{a}}}$ the following should hold
$\displaystyle-2\int_{\mathbb{R}}{\rho}_{i}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}})d{\mathrm{x}}+$
(5.8) $\displaystyle\qquad+\sum_{e:\
t(e)=i}\int_{\mathbb{R}}{\rho}_{s(e)}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}}+m_{e})d{\mathrm{x}}$
$\displaystyle\qquad\quad+\sum_{e:\
s(e)=i}\int_{\mathbb{R}}{\rho}_{t(e)}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}}-m_{e})d{\mathrm{x}}$
$\displaystyle\qquad\qquad\qquad+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}(x-m_{i,\mathfrak{f}})+\frac{x^{2}}{2}{\log}({\mathfrak{q}}_{i})=x\,{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}+{\mathfrak{b}}_{i,{\mathbf{a}}}$
where ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$, ${\mathfrak{b}}_{i,{\mathbf{a}}}$
are some constants, the Lagrange multipliers for the conditions (5.5) which
are determined from the solution. Actually,
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ _is_ the dual special coordinate, cf.
(4.4)
$\frac{{\partial}{\mathcal{F}}}{{\partial}{\mathfrak{a}}_{i,{\mathbf{a}}}}={\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$
(5.9)
We find it useful to rewrite the second derivative with respect to $x$ of the
linear integral equations (5.8) on $\rho_{i}({\mathrm{x}})$ as the non-linear
polynomial difference equations on the amplitudes ${\mathscr{Y}}_{i}(x)$:
${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathscr{P}}_{i}(x)\prod_{e:\
t(e)=i}{\mathscr{Y}}_{s(e)}(x+m_{e})\prod_{e:\
s(e)=i}{\mathscr{Y}}_{t(e)}(x-m_{e})$ (5.10)
for $x\in I_{i,{\mathbf{a}}}$, ${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}$, where
we used the notation:
${\mathscr{Y}}^{\pm}_{i}(x)={\mathscr{Y}}_{i}(x\pm\mathrm{i}0),\qquad x\in
I_{i,{\mathbf{a}}}$ (5.11)
#### 5.2.2. The mass cocycles.
In what follows we shall redefine the amplitude functions and the
${\mathscr{P}}$-polynomials
${\mathscr{Y}}_{i}(x)\to{\mathscr{Y}}_{i}(x+{\mu}_{i}),\qquad{\mathscr{P}}_{i}(x)\to{\mathscr{P}}_{i}(x+{\mu}_{i})$
(5.12)
so as to simplify the shifts of the arguments by the masses $m_{e}$ of the bi-
fundamental hypermultiplets:
$m_{e}\longrightarrow m_{e}+{\mu}_{t(e)}-{\mu}_{s(e)}$
The equations (5.10) are the main equations which determine the low-energy
effective action as well as the expectation values of all gauge invariant
chiral observables. One can view the Eqs. (5.10) as a Riemann-Hilbert problem.
They are also similar, but not identical, to the so-called Y-systems and
discrete Hirota equations.
For the class I and class II theories the shift (5.12) maps the equations
(5.10) to
${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathscr{P}}_{i}(x)\prod_{j\in\mathrm{Vert}_{\gamma},j\neq
i}{\mathscr{Y}}_{j}(x)^{I_{ij}}$ (5.13)
where for the class II theories ${\mathscr{P}}_{i}(x)={\mathfrak{q}}_{i}$. For
the class II* theory $\widehat{A}_{r}^{*}$, with the clockwise, say,
orientation of the quiver, we can make all masses $m_{e}$ to be equal,
$m_{e}=\frac{1}{r+1}{\mathfrak{m}}$, by using the shift (5.12). More
precisely, in writing (5.14) we chose the representative $m^{*}$ such that if
all the edges are oriented so that $t(e)=(s(e)+1)\,{\rm mod}\,(r+1)$, then
$m^{*}_{e}=\frac{1}{r+1}{\mathfrak{m}}$. Then,
${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathfrak{q}}_{i}\,{\mathscr{Y}}_{i-1}\left(x-\frac{\mathfrak{m}}{r+1}\right){\mathscr{Y}}_{i+1}\left(x+\frac{\mathfrak{m}}{r+1}\right)$
(5.14)
where ${\mathscr{Y}}_{i+r+1}(x)={\mathscr{Y}}_{i}(x)$.
#### 5.2.3. Analytic continuation
We use Eq. (5.3) to analytically continue the functions ${\mathscr{Y}}_{i}(x)$
through the cuts $I_{i,{\mathbf{a}}}$: for the Class I and II theories, after
the redefinition (5.12) which eliminates $m_{e}$:
$r_{i}:{\mathscr{Y}}_{i}(x)\longrightarrow{\mathscr{Y}}_{i}(x){\mathscr{P}}_{i}(x)\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathscr{Y}}_{j}(x)^{-C_{ij}}$
(5.15)
For the Class II* theories after the redefinition (5.12) we have the following
analogue of (5.15):
$r_{i}:{\mathscr{Y}}_{i}(x)\longrightarrow{\mathfrak{q}}_{i}\frac{{\mathscr{Y}}_{i-1}\left(x-\frac{\mathfrak{m}}{r+1}\right){\mathscr{Y}}_{i+1}\left(x+\frac{\mathfrak{m}}{r+1}\right)}{{\mathscr{Y}}_{i}(x)}$
(5.16)
### 5.3. The iWeyl group.
The transformations (5.15, 5.16) generate a group, which we shall call the
_instanton Weyl group_ , or _iWeyl group_ , ${{}^{i}{\mathcal{W}}}$, for
short. This group can be defined for a much larger class of ${\mathcal{N}}=2$
theories, not necessarily of the superconformal quiver type we study in this
work.
It is clear that the transformations $r_{i}$ are reflections $r_{i}\circ
r_{i}=Id$, so the iWeyl group is the group, generated by reflections.
Now, by comparing Eq. (5.15) and Eqs. (C.31), (C.58) we see that for the Class
I theories the iWeyl group is the finite Weyl group $W({\mathfrak{g}})$.
Similarly, for the Class II theories the iWeyl group coincides with the affine
Weyl group $W(\widehat{\mathfrak{g}})$. For the Class II* theory the iWeyl
group is the Weyl group $W({\widehat{\mathfrak{gl}}_{\infty}})$ of the group
$\widehat{GL}_{\infty}$.
The groups $\mathbf{G}$, $\widehat{\mathbf{G}}$, $\widehat{GL}_{\infty}$ and
their Weyl groups are discussed in Appendix.
### 5.4. Moduli of vacua and mass parameters
After all the redefinitions (5.12) the original mass parameters $m_{e}$, the
moduli $({\mathfrak{a}}^{\mathcal{I}})_{{\mathcal{I}}=1}^{\bf r}$ of the vacua
$u\in{\mathfrak{M}}$, and the derivatives of the prepotential
$({\partial}{\mathcal{F}}/{\partial}{\mathfrak{a}}^{\mathcal{I}})$ are
recovered from the study of periods of certain differentials on the curve
${\mathcal{C}}_{u}$ defined as follows.
#### 5.4.1. The first glimpses of the cameral curve
The functions ${\mathscr{Y}}_{i}(x)$, $i\in\mathrm{Vert}_{\gamma}$, after the
maximal analytic continuation through the cuts $I_{i,{\mathbf{a}}}$ form a
local ${{}^{i}{\mathcal{W}}}$-system. It is easy to see that, as long as
$|{\mathfrak{q}}_{j}|\ll 1$ for all $j\in\mathrm{Vert}_{\gamma}$, there is
exactly one branch of ${\mathscr{Y}}_{i}(x)$ as $x\to\infty$ which behaves as:
${\mathscr{Y}}_{i}^{\rm phys}(x)\sim x^{{\mathbf{v}}_{i}}+\ldots$ (5.17)
The other branches behave as
${\mathscr{Y}}_{i}^{\rm
unphys}(x)\sim\left(\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathfrak{q}}_{j}^{n_{ji}}\right)\,x^{{\mathbf{v}}_{i}}+\ldots$
(5.18)
with $n_{ji}\geq 0$, and
$\sum_{j}n_{ji}>0\ .$
Now, the branches meet at the cuts
$\bigcup_{i\in\mathrm{Vert}_{\gamma};\,{\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}}I_{i,{\mathbf{a}}}$
(5.19)
for the Class I and II theories, and at the cuts
$\bigcup_{i\in\mathrm{Vert}_{\gamma};\,{\mathbf{a}}=1,\ldots,N}\,I_{i,{\mathbf{a}}}+\frac{\mathfrak{m}}{r+1}\,{\mathbb{Z}}$
(5.20)
for the Class II* theories.
The collection of these branches defines a curve ${\mathcal{C}}_{u}$ which we
shall describe explicitly in the next section. The curve ${\mathcal{C}}_{u}$
is a ${{}^{i}{\mathcal{W}}}$-cover of the $x$-plane $\mathbf{C}_{\left\langle
x\right\rangle}$, with the branch points at the ends of the cuts
$I_{i,{\mathbf{a}}}$. Because of the ${{}^{i}{\mathcal{W}}}$-action on
${\mathcal{C}}_{u}$ and the relation to the Weyl groups which permute Weyl
cameras, the curve ${\mathcal{C}}_{u}$ will be called the _cameral curve_ ,
following [Donagi:1995alg]
#### 5.4.2. The special coordinates and the mass parameters
Figure 5.2. The cycle at $x=\infty$ surrounding the branch cuts of
$\mathscr{Y}_{i}^{\text{phys}}(x)$
Take the physical branch and expand it at $x=\infty$. Then the next-to-leading
term gives ${\mu}_{i}$, the mass shift which determines (or partially
determines, in the II* case) the bi-fundamental masses:
${\mathscr{Y}}^{\rm phys}_{i}(x)\sim
x^{{\mathbf{v}}_{i}}+{\mathbf{v}}_{i}{\mu}_{i}x^{{\mathbf{v}}_{i}-1}+\ldots$
(5.21)
The Eq. (5.6) is modified by the ${\mu}_{i}$-shift:
${\mathfrak{a}}_{i,{\mathbf{a}}}+{\mu}_{i}=\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d{\log}{\mathscr{Y}}_{i}(x)$
(5.22)
where $A_{i{\mathbf{a}}}$ is a loop on the physical sheet of
${\mathcal{C}}_{u}$ which surrounds the cut $I_{i,{\mathbf{a}}}$. Note that
the only singularities of ${\mathscr{Y}}_{i}(x)$ on the physical sheet are at
$x=\infty$ and at the cuts. Therefore (see fig. 5.2):
$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d{\log}{\mathscr{Y}}_{i}(x)=\frac{1}{2\pi\mathrm{i}}\oint_{\infty}\,x\,d{\log}{\mathscr{Y}}_{i}(x)={\mathbf{v}}_{i}{\mu}_{i}$
(5.23)
which is consistent with Eqs. (5.21), (5.22) thanks to (2.3), i.e.
$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}=0$
(5.24)
###### Remark.
The residues
$U_{ji}=\frac{1}{2\pi\mathrm{i}}\oint_{\infty}\,x^{j}\,d{\log}{\mathscr{Y}}_{i}(x)$
(5.25)
determine the ‘‘mirror map’’, the change of variables
$\left(T_{i}(x)\right)_{i\in\mathrm{Vert}_{\gamma}}\longrightarrow\left({\mathscr{T}}_{i}(x)\right)_{i\in\mathrm{Vert}_{\gamma}}$
we talked about earlier.
#### 5.4.3. The dual special coordinates
Now let us discuss the dual coordinates ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$.
First of all, the Eq. (5.9) does not quite make sense in view of (5.24).
Suppose we relax (5.24) by absorbing ${\mu}_{i}$ into the definition of
${\mathfrak{a}}_{i,{\mathbf{a}}}$. We can then analyze the limit shape problem
in the usual fashion. We should keep in mind, however, that only the
$SU({\mathbf{v}}_{i})$ part of the gauge group is dynamical. The trace part of
the dual special coordinates ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$,
${\mu}^{D}_{i}=\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$
is ambiguous. The traceless part, i.e.
$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\lambda_{\mathbf{a}}{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$
for any weight vector $({\lambda}_{\mathbf{a}})$,
$\sum_{\mathbf{a}}{\lambda}_{\mathbf{a}}=0\ ,$
should be well-defined. Let us now see how this works in detail.
By differentiating (5.8) with respect to $x$ we find
$\displaystyle-2\int_{\mathbb{R}}{\rho}_{i}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}})d{\tilde{x}}+$
(5.26)
$\displaystyle\qquad+\sum_{e:t(e)=i}\int_{\mathbb{R}}{\rho}_{s(e)}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}}+m_{e})d{\tilde{x}}+\sum_{e:s(e)=i}\int_{\mathbb{R}}{\rho}_{t(e)}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}}-m_{e})d{\tilde{x}}$
$\displaystyle\qquad\qquad\qquad+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}^{\prime}(x-m_{i,\mathfrak{f}})+x\,{\log}{\mathfrak{q}}_{i}={\mathfrak{a}}_{i,{\mathbf{a}}}^{D},\qquad\qquad
x\in I_{i,{\mathbf{a}}}$
where
${\mathcal{K}}^{\prime}(x)=x\,\log\left(\frac{x}{\tilde{\Lambda}_{UV}}\right)\equiv\int_{\tilde{\Lambda}_{UV}}^{x}\mathcal{K}^{\prime\prime}({\tilde{x}})d{\tilde{x}}$
with
$\mathcal{K}^{\prime\prime}(x)=\log\left(\frac{x}{\Lambda_{UV}}\right)$ (5.27)
and $\tilde{\Lambda}_{UV}=\exp(1)\Lambda_{UV}$. Using the definition (5.4) of
$\mathscr{Y}_{i}(x)$ functions we find
$\displaystyle{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\tilde{\Lambda}_{UV}\log\mathfrak{q}_{i}-\int_{\tilde{\Lambda}_{UV}}^{x}\,d{\tilde{x}}\,\log\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,+$
(5.28)
$\displaystyle\qquad\int_{\tilde{\Lambda}_{UV}}^{x}d{\tilde{x}}\left(-\log\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})+\sum_{e:\,t(e)=i}\log\mathscr{Y}^{\mathrm{phys}}_{s(e)}({\tilde{x}}+m_{e})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\sum_{e:\,s(e)=i}\log\mathscr{Y}^{\mathrm{phys}}_{t(e)}({\tilde{x}}-m_{e})+\log{\mathscr{P}}_{i}({\tilde{x}})\right)$
The integration contour in the above formula runs over a physical sheet from a
marked point $p_{*}\in{\mathcal{C}}_{u}$ which sits over the point
$\tilde{\Lambda}_{UV}\in\mathbf{C}_{\left\langle x\right\rangle}$ to a point
$x\in I_{i,\mathbf{a}}$ which we view as sitting on $\mathcal{C}_{u}$. The
choice of $x$ is is irrelevant111Physically the meaning of the integrand in
the effective electrostatic problem is the force acting on elementary charge,
and the integral is the chemical potential for the charge, or the energy
required to move an elementary charge from the density support to infinity.
The force vanishes on the support of the charge in the stationary charge
distribution. as long as $x\in I_{i,\mathbf{a}}$ precisely due to the critical
point equations (5.10). The above expression for
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ can be converted into much nicer form by
noting that the integral on the second line in (5.28) is in fact
$\int_{\tilde{\Lambda}_{UV}}^{x}\log\left(\,{}^{r_{i}}\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)d{\tilde{x}}$
(5.29)
where $r_{i}$ is the $i$’th reflection (5.15). In other words, (5.29) is the
integral of the analytic continuation of the function
$\mathscr{Y}_{i}({\tilde{x}})$ onto the mirror sheet of ${\mathcal{C}}_{u}$
obtained from the physical sheet by the ${{}^{i}{\mathcal{W}}}$-reflection
$r_{i}$, i.e. by continuing across any of the cuts $I_{i,\mathbf{a}}$,
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}$, supporting the density
$\rho_{i}(x)$. Thus the integral of the expression in the brackets on the last
two lines in (5.28) is equal to the integral
$\int_{r_{i}(p_{*})}^{x}\,\log{\mathscr{Y}}_{i}({\tilde{x}})d{\tilde{x}}$
(5.30)
Thus we conclude
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\tilde{\Lambda}_{UV}\log\mathfrak{q}_{i}-\int_{B_{i\mathbf{a}}}\,\log\mathscr{Y}_{i}({\tilde{x}})d{\tilde{x}}$
(5.31)
where the contour $B_{i\mathbf{a}}$ starts at the point $p_{*}$ which sits
over $x=\tilde{\Lambda}_{UV}$ on the physical sheet, runs through the cut
$I_{i,\mathbf{a}}$ to the mirror sheet $r_{i}(\mathrm{phys})$ and terminates
at the point $r_{i}(p_{*})$, which sits over $x=\tilde{\Lambda}_{UV}$ on the
mirror sheet.
It is tempting to send $\Lambda_{UV}$ to infinity. However, there is a
subtlety which we already discussed in the beginning of this section. The
integral (5.31) diverges for ${\Lambda}_{UV}\to\infty$. The linear divergence
is canceled by the constant term
${\tilde{\Lambda}}_{UV}{\log}{\mathfrak{q}}_{i}$ due to
$\log\left(\,{}^{r_{i}}\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)-\log\left(\,\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)=\log\mathfrak{q}_{i},\quad{\tilde{x}}\to\infty$
(5.32)
However the subleading logarithmic divergence does not, in general, cancel.
The simplest way to calculate it is to compute the logarithmic derivative
${\Lambda}_{UV}d{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}/d{\Lambda}_{UV}$ and then
send $\Lambda_{UV}\to\infty$:
${\Lambda}_{UV}\frac{d{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}}{d{\Lambda}_{UV}}=-\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}m_{i,{\mathfrak{f}}}+\sum_{e:\,t(e)=i}{\mathbf{v}}_{s(e)}m_{e}-\sum_{e:\,s(e)=i}{\mathbf{v}}_{t(e)}m_{e}$
(5.33)
Luckily the right hand side of (5.33) does not depend on ${\mathbf{a}}$.
We can use the formal expression
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\int_{B_{i\mathbf{a}}}\,x\,d\,{\log}\,\mathscr{Y}_{i}$
(5.34)
where the contour
$B_{i\mathbf{a}}:\infty_{\mathrm{phys}}\to\ \stackrel{{\scriptstyle
I_{i,\mathbf{a}}}}{{\qquad}}\to r_{i}(\infty_{\mathrm{phys}})$ (5.35)
starts at the point $x=\infty_{\mathrm{phys}}$ on the physical sheet, then
runs through the cut $I_{i,\mathbf{a}}$ to the mirror sheet
$r_{i}(\mathrm{phys})$ and finishes at the point
$r_{i}(\infty_{\mathrm{phys}})$ on this mirror sheet.
The canonical contour $B_{i\mathbf{a}}$ computing
$\mathfrak{a}_{i\mathbf{a}}^{D}$ is an open contour, and, as we said above,
the integral of $xd{\log}{\mathscr{Y}}_{i}$ is divergent. However the
variation of Coulomb parameters in $SU({\mathbf{v}}_{i})$ concerns only the
differences
$\mathfrak{a}_{i,\mathbf{a}^{\prime}}^{D}-\mathfrak{a}_{i,\mathbf{a}^{\prime\prime}}^{D}=\int_{B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}}\,x\,d\,\log\mathscr{Y}_{i}$
(5.36)
computed by the closed contour
$B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}=B_{i\mathbf{a}^{\prime}}-B_{i\mathbf{a}^{\prime\prime}}$
running on physical sheet through the cut $I_{i\mathbf{a}^{\prime}}$ to the
mirror sheet $r_{i}(\mathrm{phys})$ and then through the cut
$I_{i\mathbf{a}^{\prime\prime}}$ back to the physical sheet. The divergence
(5.33) cancels in the integration over
$B_{i{\mathbf{a}}^{\prime}}-B_{i{\mathbf{a}}^{\prime\prime}}$.
In fact, the difference (5.36) is represented as the integral over the closed
contour $B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}$ without any
divergent quantities, as follows immediately from the (5.26), by connecting
two points $x^{\prime}\in I_{i,\mathbf{a}^{\prime}}$ and $x^{\prime\prime}\in
I_{i,\mathbf{a}^{\prime\prime}}$ on the physical sheet and replacing the
integrand as in the second integral of (5.28) over the physical sheet by an
integral of $-\log\mathscr{Y}(x^{\prime})dx^{\prime}$ over the return segment
from $x^{\prime\prime}\in I_{i,\mathbf{a}^{\prime\prime}}$ to $x^{\prime}\in
I_{i,\mathbf{a}^{\prime}}$ on the mirror sheet $r_{i}(\mathrm{phys})$, see
Fig. 5.3
Figure 5.3. The cycles $B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}$
In the weakly coupled regime we have the following BPS particles in the gauge
theory: for each gauge group factor $i$ the $W$-bosons associated with the
breaking $SU({\mathbf{v}}_{i})\to U(1)^{{\mathbf{v}}_{i}-1}$, which correspond
to the roots of the $SU({\mathbf{v}}_{i})$, and magnetic monopoles, which
correspond to the fundamental weights. Accordingly, it seems natural to define
the following cycles on the cameral curve: the $\mathcal{A}$\- and
$\mathcal{B}$-cycles, more precisely ${\mathcal{A}}_{\mathcal{I}}$,
${\mathcal{B}}^{\mathcal{I}}$, labelled by ${\mathcal{I}}=(i,{\mathbf{a}})$,
with $i\in\mathrm{Vert}_{\gamma}$, and
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}-1$:
${\mathcal{A}}_{\mathcal{I}}=A_{i{\mathbf{a}}}-A_{i({\mathbf{a}}+1)}\,,\qquad{\mathcal{B}}^{\mathcal{I}}=\sum_{{\mathbf{a}}^{\prime}=1}^{\mathbf{a}}B_{i{\mathbf{a}}^{\prime}}-\frac{\mathbf{a}}{{\mathbf{v}}_{i}}\sum_{{\mathbf{a}}^{\prime\prime}=1}^{{\mathbf{v}}_{i}}B_{i{\mathbf{a}}^{\prime\prime}}$
(5.37)
The cycles ${\mathcal{A}}_{\mathcal{I}},{\mathcal{B}}^{\mathcal{I}}$ determine
the special coordinates
${\mathfrak{a}}^{\mathcal{I}}=\oint_{{\mathcal{A}}_{\mathcal{I}}}\,x\,d{\log}\,{\mathscr{Y}}_{i},\qquad{\mathfrak{a}}_{\mathcal{I}}^{D}=\oint_{{\mathcal{B}}^{\mathcal{I}}}\,x\,d{\log}\,{\mathscr{Y}}_{i}$
(5.38)
In the weak coupling regime the pairing (5.38) between the cycles and the
differentials is non-zero only for ${\mathcal{I}}=(i,{\mathbf{a}})$, for some
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}-1$. The main property of these cycles
is the vanishing of the following two-form on the space ${\mathfrak{M}}$ of
$u$-parameters:
$0=\sum_{\mathcal{I}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}_{\mathcal{I}}^{D}$ (5.39)
which follows simply from the Eq. (5.9).
## Chapter 6 Solution of the limit shape equations
In this section we solve the Eqs (5.10), (5.13), (5.14), and get an explicit
formula for the curve ${\mathcal{C}}_{u}$.
### 6.1. From invariants to the curve
Our strategy is to define a set of basic invariants
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))$ of the ${{}^{i}{\mathcal{W}}}$ group.
We shall find the basic invariants which are power series in
${\mathfrak{q}}_{j}$’s, and which are normalized in such a way that, for the
class I and II theories:
$\displaystyle{{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}(x)+\sum_{{\nu},\,|{\nu}|>0}\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathscr{P}}_{j}^{{\nu}_{j}}(x)\
{\Psi}_{i\nu}({\mathscr{Y}}_{1}(x),\ldots,{\mathscr{Y}}_{r}(x))$ (6.1)
$\displaystyle\qquad{\nu}=({\nu}_{1},\ldots,{\nu}_{r}),\,\qquad|{\nu}|=\sum_{j\in\mathrm{Vert}_{\gamma}}{\nu}_{j}$
where
${\Psi}_{\nu}({\mathscr{Y}}(x))\in{\mathbb{C}}[{\mathscr{Y}}_{j}(x),{\mathscr{Y}}_{j}(x)^{-1}]$
are quasi-homogeneous Laurent polynomials:
$\sum_{j\in\mathrm{Vert}_{\gamma}}\left({\mathbf{v}}_{j}{\mathscr{Y}}_{j}\frac{\partial}{{\partial}{\mathscr{Y}}_{j}}+{\mathbf{w}}_{j}{\nu}_{j}\right){\Psi}_{i\nu}={\mathbf{v}}_{i}{\Psi}_{i\nu}$
(6.2)
For the class II* theories there is one modification:
$\displaystyle{{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}(x)+\sum_{{\nu},\,|{\nu}|>0}\
{\tilde{\mathfrak{q}}}^{\nu}\,{\tilde{\Psi}}_{i\nu}({\mathscr{Y}}(x))$ (6.3)
$\displaystyle\qquad{\nu}=({\nu}_{0},\ldots,{\nu}_{r}),\,\qquad|{\nu}|=\sum_{j=0}^{r}{\nu}_{j}$
where
${\tilde{\mathfrak{q}}}^{\nu}=\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{{\nu}_{j}}$
and
${\tilde{\Psi}}_{i\nu}({\mathscr{Y}}(x))\in{\mathbb{C}}\biggl{[}{\mathscr{Y}}_{j}(x+\frac{n}{r+1}{\mathfrak{m}}),{\mathscr{Y}}_{j}^{-1}(x+\frac{n}{r+1}{\mathfrak{m}})\biggr{]}$
with $-|{\nu}|\leq n\leq|{\nu}|$. The functions $\tilde{\Psi}_{i\nu}$ are
quasi-homogeneous:
$\sum_{j\in\mathrm{Vert}_{\gamma}}a_{j}{\mathscr{Y}}_{j}\frac{\partial}{{\partial}{\mathscr{Y}}_{j}}{\tilde{\Psi}}_{i\nu}=a_{i}{\tilde{\Psi}}_{i\nu}$
(6.4)
#### 6.1.1. Master equations
Now, the ${{}^{i}{\mathcal{W}}}$-invariance of
${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))$ implies that they are continuous
across all the cuts, that is they are single-valued analytic functions of $x$.
Given their large $x$ asymptotics, they are polynomials in $x$:
$\boxed{{{\mathscr{X}}}_{j}({\mathscr{Y}}(x))=T_{j}(x)}$ (6.5)
where
$T_{j}(x)=T_{j,0}({\tilde{\mathfrak{q}}})x^{{\mathbf{v}}_{j}}+T_{j,1}({\tilde{\mathfrak{q}}};m)x^{{\mathbf{v}}_{j}-1}+\ldots+T_{j,{\mathbf{v}}_{j}}$
(6.6)
The coefficients $T_{j,0}({\tilde{\mathfrak{q}}})$ are determined by the gauge
couplings ${\mathfrak{q}}_{k}$
$T_{j,0}({\tilde{\mathfrak{q}}})=1+\sum_{{\nu},\,|{\nu}|>0}{\tilde{\mathfrak{q}}}^{\nu}{\Psi}_{j\nu}(1,\ldots,1)$
The coefficients $T_{j,1}({\tilde{\mathfrak{q}}};m)$ are determined by the
masses $m_{e}$ from (6.1),(6.3),(5.21).
The rest of the coefficients
$\left(T_{j,{\mathbf{a}}}({\tilde{\mathfrak{q}}};m,u)\right)^{j\in\mathrm{Vert}_{\gamma}}_{{\mathbf{a}}=2,\ldots,{\mathbf{v}}_{j}}\
$ (6.7)
is determined by
$u=(u_{i,{\mathbf{a}}})_{i\in\mathrm{Vert}_{\gamma}}^{{\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}}$.
The Eqs. (6.5,6.6) define an analytic curve
${\mathcal{C}}_{u}^{\circ}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}\,,$
which can be compactified to the _cameral curve_ ${\mathcal{C}}_{u}$ which is
the ${{}^{i}{\mathcal{W}}}$-cover of
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\infty\\}$:
${\mathcal{C}}_{u}\longrightarrow{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$
unramified (for generic $\tilde{\mathfrak{q}}$) over $x=\infty$. The details
of the compactification of ${\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$ and
${\mathcal{C}}_{u}^{\circ}\subset{\mathcal{C}}_{u}$ will be discussed
elsewhere. In what follows we drop the superscript $\circ$ in the definition
of $\mathcal{C}_{u}$.
#### 6.1.2. The periods
The cameral curve ${\mathcal{C}}_{u}$ depends on $u$, forming a family
${\mathcal{C}}$ of curves parametrized by the ‘‘u-plane’’ ${\mathfrak{M}}$.
The family $\mathcal{C}$ depends on the microscopic couplings
${\mathfrak{q}}_{i}$ and on the mass parameters. Let us keep the masses fixed.
When $|{\mathfrak{q}}_{i}|\ll 1$ for all $i\in\mathrm{Vert}_{\gamma}$ we have
a well-separated system of cycles ${\mathcal{A}}_{\mathcal{I}}$ and
${\mathcal{B}}^{\mathcal{I}}$, which we defined in the section 5.4. We
transport this system of cycles throughout the moduli space of gauge couplings
via Gauß-Manin connection.
#### 6.1.3. Vector-valued Seiberg-Witten differential
Let us introduce the following vector-valued $1$-differential which is
schematically given by:
$d{\mathbb{S}}=x\sum_{i\in\mathrm{Vert}_{\gamma}}\left(d\,{\log}\,{\mathscr{Y}}_{i}{\alpha}^{\vee}_{i}-d\,{\log}\,{\mathscr{P}}_{i}(x){\lambda}^{\vee}_{i}\right)$
(6.8)
where ${\alpha}_{i}^{\vee}$ and ${\lambda}_{i}^{\vee}$ are the simple coroots
and the fundamental coweights of $\mathfrak{g_{\text{q}}}$. We shall have more
specific notations for each class.
The differential $d{\mathbb{S}}$ takes values in the vector space
${\mathbb{C}}^{\mathrm{Vert}_{\gamma}}$ which is acted upon by the
${{}^{i}{\mathcal{W}}}$-group. The group ${{}^{i}{\mathcal{W}}}$ also acts on
the curve $\mathcal{C}_{u}$. It is clear from our construction that
$d{\mathbb{S}}$ is ${{}^{i}{\mathcal{W}}}$-equivariant:
$w^{*}d{\mathbb{S}}=w\cdot d{\mathbb{S}}$ (6.9)
for any $w\in{{}^{i}{\mathcal{W}}}$.
#### 6.1.4. Degeneration and filtration
In this section we consider the theories of class $I$ and class $II$, and the
extended Coulomb moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ (which includes
the masses of bifundamental hypermultiplets, recall eq. 4.7). Consider
conformal quiver with assigned dimensions $(\gamma,\mathbf{v},\mathbf{w})$ at
vertices and recall that they satisfy $\mathbf{w}=C\mathbf{v}$,
$\mathbf{w}_{i}\geq 0,\mathbf{v}_{i}\geq 0$ where $C$ is the Cartan matrix of
$\gamma$. We say that the theory $(\gamma,\mathbf{v},\mathbf{w})$ strictly
contains the theory $(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ if
$0\leq\mathbf{v}^{\prime}\leq\mathbf{v},0\leq\mathbf{w}^{\prime}\leq\mathbf{w}$
and $C\mathbf{v}^{\prime}=\mathbf{w}^{\prime}$, and
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})\neq(\gamma,\mathbf{v},\mathbf{w})$.
The extended Coulomb moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ of theory
$(\gamma,\mathbf{v},\mathbf{w})$ contains a locus related to the Coulomb
moduli ${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}$ space of
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ as follows. Suppose that
given a point $\mathfrak{m}_{f}\in\mathbf{C}_{\left\langle x\right\rangle}$
the polynomials $T_{i}(x),\mathscr{P}_{i}(x),i\in\mathrm{Vert}_{\gamma}$
factorize as follows
$\displaystyle T_{i}(x)$
$\displaystyle=T^{\prime}_{i}(x)(x-\mathfrak{m}_{f})^{\mathbf{v}_{i}-\mathbf{v}^{\prime}_{i}}$
(6.10) $\displaystyle\mathscr{P}_{i}(x)$
$\displaystyle=\mathscr{P}^{\prime}_{i}(x)(x-\mathfrak{m}_{f})^{\mathbf{w}_{i}-\mathbf{w}^{\prime}_{i}}$
Then it is clear that the character equations (6.5) factorize as well, and the
$\mathscr{Y}_{i}(x)$ functions solving the theory
$(\gamma,\mathbf{v},\mathbf{w})$ are expressed in terms of
$\mathscr{Y}_{i}^{\prime}(x)$ functions solving the theory
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ as
$\mathscr{Y}_{i}(x)=(x-\mathfrak{m}_{f})^{(\mathbf{v}_{i}-\mathbf{v}^{\prime}_{i})}\mathscr{Y}_{i}^{\prime}(x)$
(6.11)
Suppose that the degeneration eq. 6.10 is minimal, i.e. there is no
intermediate and different
$(\gamma,\mathbf{v}^{\prime\prime},\mathbf{w}^{\prime\prime})$ such that
$\mathbf{v}^{\prime}\leq\mathbf{v}^{\prime\prime}\leq\mathbf{v}$ and
$\mathbf{w}^{\prime}\leq\mathbf{w}^{\prime\prime}\leq\mathbf{w}$. Then we see
that ${\mathfrak{M}}^{\mathrm{ext}}$ includes the loci
${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}\times\mathbf{C}_{\left\langle
x\right\rangle}$
${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}\times\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathfrak{M}}^{\mathrm{ext}}$ (6.12)
where $\mathbf{C}_{\left\langle x\right\rangle}$ parameterizes the location of
$\mathfrak{m}_{f}$.
In the monopole picture of 8.1 such degeneration corresponds to the complete
screening of the point-like non-abelian monopoles by several Dirac monopoles.
Recall that dimensions $\mathbf{v}$ for the theories of class II parametrized
by a single integer $N$ such that $\mathbf{v}_{i}=Na_{i}$ where $a_{i}$ are
Dynkin marks. Therefore, the above inclusion is
${\mathfrak{M}}^{\mathrm{ext}}_{N-1}\times\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathfrak{M}}^{\mathrm{ext}}_{N}$ (6.13)
Geometrically, such inclusion for theories of class II is associated with
freckled (point) instantons described in more details after eq. 6.30.
### 6.2. The cameral curve as a modular object
In this section we give the modular interpretation of the curve
$\mathcal{C}_{u}$.
#### 6.2.1. The Class I theories
Let $\mathbf{G}={\mathbf{G}}_{\text{q}}$ be the simple complex Lie group
corresponding to the quiver of the class I theory. Let $Z$ be its center, and
let C$\mathbf{G}$ be the conformal extension of $\mathbf{G}$. Let
${\check{\lambda}}_{i}^{\vee},{\check{\alpha}}_{i}^{\vee}$, $i=1,\ldots,r$ be
the fundamental coweights and the simple coroots in C$\mathfrak{h}\subset$
C$\mathfrak{g}$. Let
$g(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\check{\lambda}}^{\vee}_{i}}{\mathscr{Y}}_{i}(x)^{{\check{\alpha}}_{i}^{\vee}}\in{\rm
C}{\mathbf{T}}\subset{\rm C}{\mathbf{G}}$ (6.14)
In the notations of (C.56):
$g(x)={\mathbf{g}}_{{\mathscr{P}}(x),{\mathscr{Y}}_{1}(x),\ldots,{\mathscr{Y}}_{r}(x)}$
We also use
$g_{\infty}(x)={\mathbf{g}}_{{\mathscr{P}}(x),1,\ldots 1}$
The importance of $g(x)$ is that it transforms by the reflection in the Weyl
group $W(\mathfrak{g})$ when crossing the cuts $I_{i,{\mathbf{a}}}$, cf.
(C.58),
$g_{+}(x)\longrightarrow g_{-}(x)=\ ^{r_{i}}g_{+}(x)$
which implies that for the class I theories the iWeyl group
${{}^{i}{\mathcal{W}}}$ is the Weyl group $W(\mathfrak{g})$ of the
corresponding simple Lie group $\mathbf{G}$. In order to construct the
${{}^{i}{\mathcal{W}}}$-invariants one could take any C$\mathbf{G}$-invariant
function on C$\mathbf{G}$. In fact, cf. (C.56)
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=g_{\infty}(x)^{-{\lambda}_{i}}\,{\chi}_{i}\left(g(x)\right)$
(6.15)
Using the formulae (LABEL:eq:wmult),(LABEL:eq:poswe) we write:
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}\sum_{{\nu}=({\nu}_{1},\ldots,{\nu}_{r})}c_{\nu}^{i}\,\prod_{j=1}^{r}\left({\mathscr{P}}_{j}\prod_{k=1}^{r}{\mathscr{Y}}_{k}^{-C_{kj}^{\mathfrak{g_{\text{q}}}}}\right)^{{\nu}_{j}}$
(6.16)
where
$c_{\nu}^{i}={\chi}_{R_{i},{\lambda}_{i}-\sum_{j=1}^{r}{\nu}_{j}{\alpha}_{j}},\qquad{\nu}\in{\mathbb{Z}}_{+}^{r}$
(6.17)
In fact, the sum in (6.16) is finite, i.e. only for a finite number of vectors
$\nu$’s the multiplicity $c_{\nu}^{i}$ is non-zero.
We thus obtain the following geometric picture. The solution of the Class I
theory is a $\mathbf{C}_{\left\langle x\right\rangle}$-parametrized family
$[g(x)]$ of conjugacy classes in C${\mathbf{G}}$, which vary with $x$
polynomially, in the appropriate sense, and such that the value of the
$D$-homomorphism on $[g(x)]$ is fixed for the theory:
$D(g(x))=\left(\prod_{j=1}^{r}{\mathscr{P}}_{j}(x)^{-l_{j{\xi}}}\right)_{{\xi}=1}^{z_{\mathfrak{g}}}$
(6.18)
Actually, as we explain in the section C.1.8, the coweights
${\check{\lambda}}_{i}^{\vee}$ are not uniquely specified. The group element
$g(x)$ in (6.14) defines a well-defined conjugacy class
$[g(x)]\in{B^{\text{ad}}({\mathfrak{g_{\text{q}}}})}$ in
${{\mathbf{G}}_{\text{q}}}/Z$. Its lift to $C\mathbf{G}$ can be twisted by any
C-valued (meromorphic) function of $x$. We shall use this freedom in our
manipulations with spectral curves.
The cameral curve ${\mathcal{C}}_{u}$ can be viewed, geometrically, as the
lift to $C\mathbf{T}$ of the parametrized rational curve in
$C\mathbf{T}/W({\mathfrak{g}})\approx({\mathbb{C}}^{\times})^{z_{\mathfrak{g}}}\times{\mathbb{C}}^{r}$:
$x\mapsto D(g(x))\times\left(T_{1}(x),\ldots,T_{r}(x)\right)$ (6.19)
#### 6.2.2. The class II theories
As we mentioned above, the quivers of the class II theories correspond to the
simply laced affine Kac-Moody algebras, i.e.
$\mathfrak{g_{\text{q}}}=\widehat{\mathfrak{g}}$. Let $\widehat{\mathbf{G}}$
be the corresponding Kac-Moody group. Let ${\widehat{\lambda}}_{i}^{\vee}$,
${\widehat{\alpha}}_{i}^{\vee}$ be the corresponding affine coweights and
coroots, $i=0,1,\ldots,r$ (see Appendix C.2). Define:
$g(x)=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}(x)^{{\widehat{\alpha}}_{i}^{\vee}}\in{\widehat{\mathbf{T}}}\subset{\widehat{\mathbf{G}}}$
(6.20)
Again, strictly speaking $g(x)$ takes values in ${\widehat{\mathbf{G}}}/Z$ and
so we should consider the modification of $\widehat{\mathbf{G}}$ corresponding
to the conformal extension $C\widehat{\mathbf{G}}$, but since the subtlety
with the center $Z\subset\mathbf{G}$ only involves the $x$-independent factor
${\widehat{g}}_{\infty}=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}$
(6.21)
it will not affect the $x$-dependence of the invariants. The limit shape
equations, as in the class I case, translate to the jump conditions
$g_{+}(x)\longrightarrow g_{-}(x)=\,^{r_{i}}g_{+}(x)$
for $x\in I_{i,{\mathbf{a}}}$, with $r_{i}$ being the simple reflections
generating the affine Weyl group $W({\widehat{\mathfrak{g}}})$, which is the
${{}^{i}{\mathcal{W}}}$ group for the class II theories.
The invariants of $W({\widehat{\mathfrak{g}}})$ are constructed using the
characters $\widehat{\chi}_{i}$ of the fundamental representations
${\widehat{R}}_{i}$ of $\widehat{\mathbf{G}}$:
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=\left({\widehat{g}}_{\infty}^{{\widehat{\lambda}}_{i}}\right)^{-1}\,{\widehat{\chi}}_{i}(g(x))$
(6.22)
They can also be obtained by starting with ${\mathscr{Y}}_{i}(x)$ and
averaging with respect to the $W({\widehat{\mathfrak{g}}})$-action. The
$W({\widehat{\mathfrak{g}}})$-action consists of the translations by the
coroot lattice ${{\rm Q}}^{\vee}$ and the $W({\mathfrak{g}})$-transformations.
The ${{\rm Q}}^{\vee}$-averaging produces the lattice theta-functions of
various characteristics, of the schematic form (the details are given in the
Appendix LABEL:se:lattice-theta):
${\Theta}({\xi},{\mathfrak{q}})=\sum_{{\nu}\in{{\rm
Q}}^{\vee}}{\mathfrak{q}}^{\frac{1}{2}\langle{\nu},{\nu}\rangle}e^{i\langle{\nu},{\xi}\rangle}$
(6.23)
where
${\mathfrak{q}}=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{a_{i}}$ (6.24)
The affine analogue of the formula (6.16) is an infinite sum, however, it is a
power series in ${\mathfrak{q}}$. Using the fact that the weights
$\widehat{\lambda}$ of the fundamental representation ${\widehat{R}}_{i}$
differ from the highest weight $\widehat{\lambda}_{i}$ by a positive linear
combination of simple roots,
${\widehat{\lambda}}={\widehat{\lambda}}_{i}-{\widehat{\nu}}$,
${\widehat{\nu}}=\sum_{j=0}^{r}{\nu}_{j}{\widehat{\alpha}}_{j},\qquad{\nu}_{j}\in{\mathbb{Z}}_{+}$
we can write, with
${\tilde{\mathfrak{q}}}^{\widehat{\nu}}=\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{\nu_{j}}$
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x);{\tilde{\mathfrak{q}}})={\mathscr{Y}}_{i}\sum_{\widehat{\nu}}{\widehat{c}}_{\widehat{\nu}}^{i}{\tilde{\mathfrak{q}}}^{\widehat{\nu}}\,\prod_{k,j=0}^{r}{\mathscr{Y}}_{k}(x)^{-C_{kj}^{\widehat{\mathfrak{g}}}{\nu}_{j}}$
(6.25)
where we made the
${\tilde{\mathfrak{q}}}=({\mathfrak{q}}_{0},\ldots,{\mathfrak{q}}_{r})$
dependence explicit, and
${\widehat{c}}_{\widehat{\nu}}^{i}={\chi}_{{\widehat{R}}_{i},{\widehat{\lambda}}_{i}-{\widehat{\nu}}},\qquad$
(6.26)
Write ${\widehat{\nu}}=n{\delta}+{\nu}$, where $n\in{\mathbb{Z}}_{+}$, and
${\nu}\in{{\rm Q}}$ belongs to the root lattice of $\mathfrak{g}$. Notice that
the factor ${\tilde{\mathfrak{q}}}^{\widehat{\nu}}$ in (6.25) depends on $n$
only via the ${\mathfrak{q}}^{n}$ factor. For fixed $n$ the number of
${\nu}\in{{\rm Q}}$ such that ${\widehat{c}}_{n{\delta}+{\nu}}^{i}\neq 0$ is
finite.
The characters of $\widehat{\mathbf{G}}$ are well-studied [Kac:1984].
Physically they are the torus
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$ conformal
blocks of the WZW theories with the group $G$, and levels $k=a_{i}$,
$i=0,1,\ldots,r$ (see [Dolan:2007eh] for recent developments). The argument of
the characters can be viewed as the background $\mathbf{G}$ $(0,1)$-gauge
field $\bar{\bf A}$, which couples to the holomorphic current ${\bf
J}=g^{-1}{\partial}g$:
$Z_{k}({\tau};{\bar{\bf A}})=\int Dg\,{\exp}\,k\left(S_{\rm
WZW}(g)+\int_{\mathscr{E}}\left\langle{\bf J},{\bar{\bf
A}}\right\rangle\right)=\sum_{\widehat{\lambda}\text{ at level
$k$}}c_{\widehat{\lambda}}\cdot{\widehat{\chi}}_{\widehat{\lambda}}({\widehat{t}};{\mathfrak{q}})$
(6.27)
The background gauge field has only $r$ moduli. In practice, one chooses the
gauge ${\bar{\bf A}}=\frac{\pi}{{\rm Im}{\tau}}{\xi}$, where ${\xi}={\rm
const}\in{\mathfrak{h}}$.
Technically, it is more convenient to built the characters using the free
fermion theory, at least for the $A_{r}$, $D_{r}$ cases, and for the groups
$E_{6},E_{7},E_{8}$ at level $1$. We review this approach in the appendix.
The master equations (6.5)
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x);{\tilde{\mathfrak{q}}})=T_{i}(x)$
describe a curve ${\mathcal{C}}_{u}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{r+1}$ which is a
$W({\widehat{\mathfrak{g}}})$-cover of the $x$-parametrized rational curve
$\Sigma_{u}$ in ${\mathbb{C}}^{r+1}={\rm
Spec}{\mathbb{C}}[{\widehat{\chi}}_{0},\ldots,{\widehat{\chi}}_{r}]$, cf.
(6.22):
${\widehat{\chi}}_{i}=\prod_{j}{\mathfrak{q}}_{j}^{-{\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee})}\
T_{i}(x),\qquad i=0,\ldots,r$ (6.28)
Now, as we recall in the section C.5, the characters ${\widehat{\chi}}_{i}$,
$i=0,\ldots,r$ are the sections of the line (orbi)bundle ${\mathcal{O}}(1)$
over the coarse moduli space ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of
holomorphic principal semi-stable $\mathbf{G}$-bundles over the elliptic curve
$\mathscr{E}$. Therefore (6.28),(6.5) define for each $u$ a quasimap $U$ of
the compactified $x$-plane ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ to ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$, which is
actually an honest map near $x=\infty$, whose image approaches the fixed
$\mathbf{G}$-bundle ${\mathcal{P}}_{\tilde{\mathfrak{q}}}$. This bundle can be
described, e.g. by the transition function $g_{\infty}$, which is one of the
$\mathbf{T}$ lifts of
${\tilde{g}}_{\infty}=\prod_{i=1}^{r}{\mathfrak{q}}_{i}^{-{\lambda}^{\vee}_{i}}\in{\mathbf{T}}/Z$
(6.29)
By definition, the local holomorphic sections of
${\mathcal{P}}_{\tilde{\mathfrak{q}}}$ are the ${\mathbf{G}}$-valued functions
${\Psi}(z)$, defined in some domain in ${\mathbb{C}}^{\times}$ such that
${\Psi}({\mathfrak{q}}z)=g_{\infty}{\Psi}(z)$
The complex dimension of the space of quasimaps $U$ with fixed $U({\infty})$
is the number of coefficients in the polynomials
$(T_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ excluding the highest coefficients,
that is (cf. Eq.(4.7)),
$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=\sum_{i\in\mathrm{Vert}_{\gamma}}\mathbf{v}_{i}=Nh\
.$
We say that $U$ is a quasimap, and not just a holomorphic map
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$ for two
reasons. Technically, a collection of ${\widehat{\chi}}_{i}$ in (6.28) defines
a point in $\mathbb{W}\mathbb{P}^{a_{0},a_{1},\ldots,a_{r}}$ only if the
polynomials $T_{i}(x)$ don’t have common weighted factors. If, however, for
some ${\mathfrak{m}}_{f}\in{\mathbf{C}_{\left\langle x\right\rangle}}$:
$T_{i}(x)={\tilde{T}}_{i}(x)(x-{\mathfrak{m}}_{f})^{a_{i}},{\rm for\ all\
}i=0,\ldots,r$ (6.30)
then the map ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}\to{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ is not well-
defined at $x={\mathfrak{m}}_{f}$. It is trivial to extend the map there by
removing the $(x-{\mathfrak{m}}_{f})^{a_{i}}$ factors. This operation lowers
$N\to N-1$. In a way, the point ${\mathfrak{m}}_{f}$ carries a unit of the
instanton charge. Such a configuration is called a freckled instanton
[Losev:1999tu]. Thus, the extended moduli space eq. 4.7 of vacua
${{\mathfrak{M}}^{\mathrm{ext}}}_{N}$ of the gauge theory with
${G_{\text{g}}}=\times_{i}SU(Na_{i})$, contains the locus
${{\mathfrak{M}}^{\mathrm{ext}}}_{N-1}\times{\mathbf{C}_{\left\langle
x\right\rangle}}$. Allowing for several freckles at the unordered points
${\mathfrak{m}}_{f1},{\mathfrak{m}}_{f2},\ldots,{\mathfrak{m}}_{fi}$ we arrive
at the hierarchy of embeddings of the moduli spaces of vacua of the gauge
theories with different gauge groups $G_{\text{g}}$:
$\displaystyle{{\mathfrak{M}}^{\mathrm{ext}}}_{N}=$
$\displaystyle\,\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-1}\times{\mathbf{C}_{\left\langle
x\right\rangle}}\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-2}\times
Sym^{2}{\mathbf{C}_{\left\langle x\right\rangle}}$ (6.31)
$\displaystyle\qquad\qquad\cup\ldots\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-i}\times
Sym^{i}{\mathbf{C}_{\left\langle x\right\rangle}}\cup\ldots\cup
Sym^{N}{\mathbf{C}_{\left\langle x\right\rangle}}$
where $\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$ stands for the space of
degree $N$ rational maps $U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$.
This hierarchy of gauge theories is more familiar in the context of class I
theories. Presently, the freckled instantons to
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ correspond to the loci in
${\mathfrak{M}}$ where a Higgs branch of the gauge theory can open. Indeed, if
(6.30) holds, then we can solve the master equation (6.5) by writing
${\mathscr{Y}}_{j}(x)=(x-{\mathfrak{m}}_{f})^{a_{j}}\
{\tilde{\mathscr{Y}}}_{j}(x)$ (6.32)
with ${\tilde{\mathscr{Y}}}_{j}(x)$ solving the master equation (6.5) of the
$\times_{i\in\mathrm{Vert}_{\gamma}}\ SU\left(\left({N-1}\right)a_{i}\right)$
gauge theory. In the IIB string theory picture A.3 the full collection of
fractional branes in the amount of $a_{i}$ for the $i$’th type recombine, and
detach themselves from the fixed locus, moving away at the position
${\mathfrak{m}}_{f}$ on the transverse
${\mathbb{R}}^{2}={\mathbf{C}_{\left\langle x\right\rangle}}$.
Now let us take $u\in\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$. The
corresponding map $U$ defines a rational curve $\Sigma_{u}$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of degree $N$.
###### Remark.
Actually, there is another compactification of
$\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$, via genus zero Kontsevich
stable maps of bi-degree $(1,N)$ to
${\mathbb{C}\mathbb{P}}^{1}\times{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$
(see [Givental:1997] where the space of quasimaps is called the toric map
spaces). It would be interesting to study its gauge theoretic meaning.
###### Remark.
The highest order coefficients $T_{i,0}({\tilde{\mathfrak{q}}})$ of the
polynomials $T_{i}(x)$ depend only on the gauge coupling constants, and
determine the limit $U(x)$, $x\to\infty$
$U({\infty})=[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]\in{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$
(6.33)
The next-to-leading terms $T_{i,1}({\tilde{\mathfrak{q}}},m)$ depend only on
the gauge couplings and the bi-fundamental masses. These define the first jet
${\mathscr{T}}_{[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]}{\Sigma}_{u}$ of the
rational curve $\Sigma_{u}$ at $x=\infty$.
Summarizing, _the moduli space ${{\mathfrak{M}}}_{N}$ of vacua of the class II
theory with the gauge group_
${G_{\text{g}}}=\times_{i\in\mathrm{Vert}_{\gamma}}\ SU(Na_{i})$
_is the moduli space of degree $N$ finely framed at infinity quasimaps_
$U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}\approx{\mathbb{W}\mathbb{P}}^{a_{0},a_{1},\ldots,a_{r}}$
(6.34)
_where the fine framing is the condition that $U$ is the honest map near
$x=\infty$, and the first jet (the value and the tangent vector) at $x=\infty$
are fixed:_
$\left(U({\infty}),U^{\prime}({\infty})\right)\leftrightarrow\left({\tilde{\mathfrak{q}}},m\right)$
(6.35)
_We also have the identification of the extended moduli space
${{\mathfrak{M}}^{\mathrm{ext}}}$ with the space of framed quasimaps_
#### 6.2.3. The class II* theories
The theories with the affine quiver of the ${\widehat{A}}_{r}$ type can be
solved uniformly in both class II and class II* cases. This is related to the
fact that the current algebra ${\widehat{u(r+1)}}$, the affine Kac-Moody
algebra based on $U(r+1)$ is a subalagebra of
$\widehat{\mathfrak{gl}}_{\infty}$, consisting of the $(r+1)$-periodic
infinite matrices.
Let $\gamma$ be the affine Dynkin graph of the ${\widehat{A}}_{r}$ algebra. We
have, $\mathrm{Vert}_{\gamma}=\mathrm{Edge}_{\gamma}=\\{0,1,\ldots,r\\}$.
Choose such an orientation of the graph $\gamma$ that for any
$e\in\mathrm{Edge}_{\gamma}$: $s(e)=e$, $t(e)=(e+1)$ mod $r+1$. Let $m_{e}$,
$e=0,\ldots,r$ be the corresponding bi-fundamental multiplet masses, and
${\mathfrak{m}}=\sum_{e=0}^{r}m_{e}$ (6.36)
We are in the class II* theory iff $\mathfrak{m}\neq 0$.
It is convenient to extend the definition of $m_{e}$ to the universal cover of
$\gamma$. Thus, we define
$\displaystyle{\mathfrak{m}}_{i}=m_{i\,\mathrm{mod}\,(r+1)},$ (6.37)
$\displaystyle
Y_{i}(x)={\mathscr{Y}}_{i\,\mathrm{mod}\,(r+1)}(x-{\mathfrak{m}}_{(i)})\
,\qquad i\in\mathbb{Z}$
The extended amplitudes $Y_{i}(x)$ obey
$Y_{i+r+1}(x)=Y_{i}\left(x-{\mathfrak{m}}\right)$ (6.38)
Define
$t_{j}(x)={\check{t}}_{j}\,\frac{Y_{j}(x)}{Y_{j-1}(x)}$ (6.39)
where
$\displaystyle{\check{t}}_{j+1}={\mathfrak{q}}_{j\,\mathrm{mod}\,(r+1)}\,{\check{t}}_{j}$
(6.40) $\displaystyle\prod_{j=0}^{r}{\check{t}}_{j}=1\,,\ $
$\displaystyle{\check{t}}_{j+r+1}={\mathfrak{q}}\,{\check{t}}_{j}$
Then
$t_{j+r+1}(x)={\mathfrak{q}}\,t_{j}(x-{\mathfrak{m}})$
where for the $\widehat{A}_{r}$-series,
${\mathfrak{q}}=\prod_{j=0}^{r}{\mathfrak{q}}_{j}$
Now, consider the following element of $\widehat{GL}_{\infty}$:
$g(x)={\mathscr{Y}}_{0}(x)^{K}\times\prod_{i\in\mathbb{Z}}t_{i}(x)^{E_{i,i}}$
(6.41)
with $t_{i}(x)$ from (6.39), and $E_{i,j}$ denoting the matrix with all
entries zero except $1$ at the $i$’th row and $j$’th column. A closer
inspection shows (6.41) is the direct generalization of (6.20) with the
$r+1$-periodic matrix $g_{\infty}$, and
$({\mathscr{Y}}_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ replaced by the infinite
array $(Y_{i}(x))_{i\in\mathbb{Z}}$. Indeed, the simple coroots of
$\widehat{GL}_{\infty}$ are the diagonal matrices, shifted in the central
direction
${\alpha}_{i}^{\vee}=K{\delta}_{i,0}+E_{i,i}-E_{i+1,i+1},\qquad
i\in\mathbb{Z}$ (6.42)
so that the analogue of (C.64) holds
$K=\sum_{i\in\mathbb{Z}}{\alpha}_{i}^{\vee}$
if we drop the telescopic sum $\sum_{i\in\mathbb{Z}}E_{i,i}-E_{i+1,i+1}\sim
0$.
We do not need to deal with all the coweights of $\widehat{GL}_{\infty}$, only
with the $r+1$-periodic ones, defined via:
$\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{-{\tilde{\lambda}}_{j}^{\vee}}=\prod_{b\in\mathbb{Z}}\prod_{j=1}^{r+1}\left({\mathfrak{q}}^{b}{\check{t}}_{j}\right)^{E_{i+b(r+1),i+b(r+1)}}$
These coweights are the coweights of the ${\widehat{A}}_{r}$ Kac-Moody
algebra, embedded into $\mathfrak{gl}_{\infty}$ as the subalgebra of
$r+1$-periodic matrices
$\sum_{i,j\in\mathbb{Z}}a_{i,j}E_{i,j},\qquad a_{i+r+1,j+r+1}=a_{i,j}$
We shall describe the solution of this theory in detail in the next section.
### 6.3. Spectral curves
The cameral curve captures all the information about the limit shape, the
special coordinates, the vevs of the chiral operators, and the prepotential.
Its definition is universal.
However, the cameral curve is not very convenient to work with. In many cases
one can extract the same information from a ‘‘smaller’’ curve, the so-called
_spectral curve_. In fact, there are several notions of the spectral curve in
the literature.
Suppose ${\lambda}\in{\rm
Hom}(({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}},{\mathbb{C}}^{\times})$
is a dominant weight, i.e. ${\lambda}({\alpha}_{i}^{\vee})\geq 0$ for all
$i\in\mathrm{Vert}_{\gamma}$. Let $R_{\lambda}$ be the irreducible highest
weight module of ${\mathbf{G}}_{\text{q}}$ with the highest weight $\lambda$,
and ${\pi}_{\lambda}:{{\mathbf{G}}_{\text{q}}}\longrightarrow{\rm
End}(R_{\lambda})$ the corresponding homomorphism. Then the spectral curve
$C^{R_{\lambda}}_{u}$ in $\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbf{C}_{\left\langle t\right\rangle}$ is
${\det}_{R_{\lambda}}\left(1-t^{-1}{\zeta}(x)^{-1}{\pi}_{\lambda}(g(x))\right)=0$
(6.43)
where
1. (1)
for the class I theories we introduce the factor
$\zeta(x)=g_{\infty}(x)^{\lambda}\times{\rm\ a\ rational\ function\ of\ }x\,,$
having to do with the lift of the conjugacy class $[g(x)]$ from
${{\mathbf{G}}^{\text{ad}}}$ to ${\rm C}{\mathbf{G}}$. The rational function
is chosen so as to minimize the degree of the curve $C^{R_{\lambda}}_{u}$, as
we explain in the examples below.
2. (2)
for the class II, II* theories the factor $\zeta(x)$ is a constant.
Generally, the curve $C^{R_{\lambda}}_{u}$ defined by (6.43) is not
irreducible. The equation (6.43) factorizes into a product of components, one
component for each Weyl orbit in the set of weights $\Lambda_{R_{\lambda}}$
for the module $R_{\lambda}$. Each Weyl orbit intersects dominant chamber at
one point and therefore can be parametrized by dominant weights $\mu$.
Therefore
$C^{R_{\lambda}}_{u}=\bigcup_{\mu\in\Lambda_{R_{\lambda}}\cap\Lambda^{+}}{\mathrm{mult}(\lambda:\mu)}\cdot\left(C_{u}^{\mu}\right)$
(6.44)
where $\mathrm{mult}(\lambda:\mu)$ denotes multiplicity of weight $\mu$ in the
module $R_{\lambda}$. If $R_{\lambda}$ is minuscule module, then, by
definition, the curve $C^{R_{\lambda}}_{u}$ is irreducible.
###### Example.
Consider the $A_{1}$ theory and take $\lambda=3\lambda_{1}$, i.e. the spin
$\frac{3}{2}$ representation. If
$T_{1}(x)=\operatorname{tr}_{R_{1}}g(x)=t(x)+t(x)^{-1}$ one finds that
$\displaystyle C^{R_{\lambda_{1}}}:$ $\displaystyle\ 0=1-T_{1}(x)t+t^{2}$
(6.45) $\displaystyle C^{R_{3\lambda_{1}}}:$ $\displaystyle\
0=(1-T_{1}(x)t+t^{2})(1+3T_{1}(x)t-T_{1}(x)^{3}t+t^{2})$
Let ${{}^{i}{\mathcal{W}}}_{\mu}\subset{{}^{i}{\mathcal{W}}}$ be the
stabilizer of $\mu$ in ${{}^{i}{\mathcal{W}}}$, a subgroup of
${{}^{i}{\mathcal{W}}}$. Consider the map:
$p_{\mu}:{\mathbf{C}_{\left\langle
x\right\rangle}}\times\left({\mathbb{C}}^{\times}\right)^{\mathrm{Vert}_{\gamma}}\longrightarrow{\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}$
given by:
$\displaystyle
p_{\mu}:(x,({\mathscr{Y}}_{i})_{i\in\mathrm{Vert}_{\gamma}})\mapsto$
$\displaystyle\,(x,t(x)),$ (6.46) $\displaystyle\qquad
t(x)=g(x)^{\mu}/g_{\infty}(x)^{\mu}=\prod_{i\in\mathrm{Vert}_{\gamma}}{\mathscr{Y}}_{i}^{{\mu}({\alpha}_{i}^{\vee})}$
Under the map $p_{\mu}$ the curve ${\mathcal{C}}_{u}$ maps to
$C_{u}^{\mu}={\mathcal{C}}_{u}/{{{}^{i}{\mathcal{W}}}}_{\mu}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}$, the
irreducible $\mu$-component of the spectral curve. This curve comes with the
canonical differential, which is the restriction of the differential on
${\mathbf{C}_{\left\langle x\right\rangle}}\times{\mathbf{C}_{\left\langle
t\right\rangle}}^{\times}$:
$dS=x\frac{dt}{t}$ (6.47)
Actually, in the case of the class II, II* theories the commonly used notion
of the spectral curve differs from the one in (6.43).
Although we suspect the study of spectral curves associated with the
integrable highest weight representations of affine Kac-Moody algebras may be
quite interesting, in this paper for the analysis of the class II and II*
theories we use the conventional notion of the spectral curve used for the
study of families of $\mathbf{G}$-bundles.
To define it, let us fix an irreducible representation $R$ of $\mathbf{G}$,
${\pi}_{R}:{\mathbf{G}}\to{\rm End}(R)$, and let us study the theory of a
complex chiral fermion valued in $R$, more precisely, an $(1,0)$ $bc$ system
in the representations $(R^{*},R)$:
${\mathscr{L}}_{bc}=\sum_{i=1}^{{\rm dim}R}\int b_{i}{\bar{\partial}}c^{i}$
(6.48)
coupled to a background ${\mathbf{G}}\times{\mathbb{C}}^{\times}$ gauge field
$\bar{\bf A}\oplus\bar{A}$, and compute its partition function on the torus
$\mathscr{E}$:
$Z({\bf t},t,q)={\operatorname{Tr}}_{{\mathcal{H}}_{R}}\left((-t)^{J}_{0}{\bf
t}^{{\bf J}_{0}}q^{L_{0}}\right)$ (6.49)
Mathematically, we consider the space
$H_{R}=R[z,z^{-1}]=H_{R}^{+}\oplus H_{R}^{-}$ (6.50)
of $R$-valued functions on the circle ${\mathbb{S}}^{1}$. In (6.50) we took
Laurent polynomials in $z\in{\mathbb{C}}^{\times}$, which correspond to
Fourier polynomials on the circle. We may take some completion of $H_{R}$ but
we shall not do this in the definition of the spectral determinant below.
Consider an element ${\widehat{g}}\in{\widehat{\mathbf{G}}}$ of the affine
Kac-Moody group, i.e. the central extension of
${\widetilde{L\mathbf{G}}}=L{\mathbf{G}}\ltimes{\mathbb{C}}^{\times}$, the
loop group $L{\mathbf{G}}$ extended by the $\mathbb{C}^{\times}$ acting by
loop rotations. We have the canonical homomorphism-projection
$f:{\widehat{\mathbf{G}}}\longrightarrow{\widetilde{L\mathbf{G}}}$ with the
fiber ${\mathbb{C}}^{\times}$, the center of the central extension:
$f:{\widehat{g}}\mapsto g(z)q^{z{\partial}_{z}}$ (6.51)
The projection is topologically non-trivial.
Now, ${\widetilde{L\mathbf{G}}}$ acts in $H_{R}$ via rotation and evaluation,
and so does $\widehat{\mathbf{G}}$ thanks to (6.51) : for ${\Psi}\in H_{R}$:
$\left(f({\widehat{g}})\cdot{\Psi}\right)(z)={\pi}_{R}(g(z))\cdot{\Psi}(qz)$
(6.52)
We would like to define the spectral determinant of $f({\widehat{g}})$ in the
representation $H_{R}$. The eigenvalues of $f({\widehat{g}})$ are easy to
compute
${\rm Eigen}(f({\widehat{g}}))=\\{\,{\bf t}^{\mu}q^{n}\,|\
{\mu}\in{\Lambda}_{R},\,n\in\mathbb{Z}\,\\}$ (6.53)
where we transformed $g(z)$ to a constant ${\bf t}\in\mathbf{T}$ by means of a
$z$-dependent $\mathbf{G}$-gauge transformation:
$g(z)\mapsto h^{-1}(z)g(z)h(qz)={\bf t}$ (6.54)
The fibration $f:{\widehat{\mathbf{G}}}\to{\widetilde{L{\mathbf{G}}}}$,
restricted onto
${\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}\subset{\widetilde{L{\mathbf{G}}}}$
becomes trivial,
$f^{-1}\left({\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}\right)\approx{\mathbb{C}}^{\times}_{c}\times{\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}$.
Let us denote by $c$ the coordinate on the first factor.
The eigenvalues (6.53) concentrate both near $0$ and $\infty$, so we define:
$\displaystyle{\det}_{H_{R}}\,\left(1-t^{-1}{\widehat{g}}\right):={\det}_{H_{R}^{+}}(1-t^{-1}{\widehat{g}}){\det}_{H_{R}^{-}}(1-t{\widehat{g}}^{-1})=$
(6.55) $\displaystyle\qquad\qquad
c^{{\kappa}_{R}}\prod_{{\mu}\in{\Lambda}_{R}}\prod_{n=0}^{\infty}\left(1-q^{n}t^{-1}{\bf
t}^{\mu}\right)\left(1-q^{n+1}t\,{\bf t}^{-{\mu}}\right)$
The expression (6.55) is $W({\widehat{\mathfrak{g}}})$-invariant. The shifts
by ${\rm Q}$ act as follows, cf. (C.104):
$({\bf t},c)\mapsto\left(q^{\beta}\cdot{\bf t},\,{\bf
t}^{\beta}q^{\frac{1}{2}\left\langle\beta,\beta\right\rangle}\cdot c\right)$
(6.56)
where we view $\beta\in{\rm Q}$ both as a vector in the root lattice and as a
vector in the coroot lattice, and $\left\langle,\right\rangle$ is the Killing
metric. The level ${\kappa}_{R}$ in (6.55) is defined as follows:
$\sum_{{\mu}\in{\Lambda}_{R}}{\mu}\left\langle\mu,\beta\right\rangle={\kappa}_{R}{\beta}$
(6.57)
for any vector $\beta\in{\rm Q}$. Geometrically the spectral curve
corresponding to $R$ is obtained as follows: consider the universal principal
$\mathbf{G}$-bundle ${\mathcal{U}}$ over
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})\times\mathscr{E}$, and associate
the vector bundle $\mathscr{R}$ with the fiber $R$:
${\mathscr{R}}={\mathcal{U}}\times_{\mathbf{G}}R$
Now restrict it onto the rational curve
$\Sigma_{u}\subset{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$. We get the
$R$-bundle over $\Sigma_{u}\times{\mathscr{E}}$.
For generic point $x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ over the corresponding point $U(x)\in\Sigma_{u}$ we get the
vector bundle ${\mathscr{R}}_{x}$ over $\mathscr{E}$, which is semi-stable,
and splits as a direct sum of line bundles
${\mathscr{R}}_{x}=\bigoplus_{{\mu}\in{\Lambda}_{R}}{\mathscr{L}}_{{\mu},x}$
(6.58)
where the summands are the degree zero line bundles on $\mathscr{E}$. Under
the identification $Pic_{0}({\mathscr{E}})$ with $\mathscr{E}$ the line bundle
${\mathscr{L}}_{{\mu},x}$ corresponds to the point ${\bf t}(x)^{\mu}$
mod$\,{\mathfrak{q}}^{\mathbb{Z}}$ for some ${\bf
t}(x)\in{\mathbf{T}}/{\mathfrak{q}}^{{{\rm Q}}^{\vee}}$. The closure of the
union
$\bigcup_{x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}}\,\left\\{\ {\bf t}(x)^{\mu}\ |\ {\mu}\in{\Lambda}_{R}\
\right\\}\ \subset{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$ (6.59)
is the spectral curve
$C^{R}_{u}\subset{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$. It is given by the vanishing locus of
the regularized determinant (6.55):
$c(x)^{{\kappa}_{R}}\prod_{{\mu}\in{\Lambda}_{R}}{\theta}(t^{-1}{\bf
t}(x)^{\mu};q)=0$ (6.60)
the choice of the $x$-dependence of $c(x)$ seems immaterial at this point, as
long as $c(x)\in{\mathbb{C}}^{\times}$.
##### Degree of the spectral curve
The $x$-degree of the spectral curve for class II theories in representation
$R$ is $N\kappa_{R}$ where $\kappa_{R}$ is given by (6.57). The $\kappa_{R}$
is the proportionality constant for the second Casimir in representation $R$
$\operatorname{tr}_{R}(\cdot,\cdot)=\kappa_{R}(\cdot,\cdot)_{2}$ where the
$(\cdot,\cdot)_{2}$ is the canonical Killing form in which the long roots have
length square equal to $2$. The standard computations leads to
$\kappa_{R}=\frac{\dim_{R}}{\dim_{\mathfrak{g}}}(\lambda_{R},\lambda_{R}+2\rho)_{2}$
(6.61)
where $\rho=\frac{1}{2}\sum_{\alpha>0}\alpha$ is the Weyl vector. For
fundamental representations $R_{1}$ we find for all cases
$\displaystyle\kappa_{R_{1}}(A_{r})=1$ (6.62)
$\displaystyle\kappa_{R_{1}}(D_{r})=2$ $\displaystyle\kappa_{R_{1}}(E_{6})=6$
$\displaystyle\kappa_{R_{1}}(E_{7})=12$
$\displaystyle\kappa_{R_{1}}(E_{8})=60$
### 6.4. Obscured curve
In the previous construction, in view of the identification
${\mathscr{L}}_{{\mu},x}\leftrightarrow{\bf t}(x)^{\mu}$ we can decompose, for
each weight
${\mu}=\sum_{i=1}^{r}{\mu}_{i}{\lambda}_{i}\in{\Lambda}_{R}$
${\mathscr{L}}_{{\mu},x}=\bigotimes_{i=1}^{r}{\mathbb{L}}_{i,x}^{\otimes{\mu}_{i}}$
for some "basic" line bundles ${\mathbb{L}}_{i,x}$ corresponding to the
fundamental weights. These basic line bundles are ordered, so they define a
point
$\\{\ {\mathbb{L}}_{1,x},\ldots,{\mathbb{L}}_{r,x}\ \\}\in
Pic_{0}({\mathscr{E}})^{r}\approx{\mathscr{E}}^{r}\ ,$
the Cartesian product of $r$ copies of the elliptic curve. Taking the whole
family and including the parametrization we obtain the _obscured curve_
${\mathscr{C}}_{u}$:
${\mathscr{C}}_{u}=\\{\ (x;{\mathbb{L}}_{1,x},\ldots,{\mathbb{L}}_{r,x})\ |\
x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}\
\\}\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}^{r}$ (6.63)
Let us present another simple construction of ${\mathscr{C}}_{u}$. Namely, let
us use the fact [Friedman:1997yq, Friedman:1997ih, Donagi:1997dp,
Friedman:1998si, Friedman:2000ze], that
${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}=({\mathscr{E}}\otimes{{\rm
Q}})/W({\mathfrak{g}})$ (6.64)
where the tensor product is understood in the category of abelian groups. At
the level of manifolds, (6.64) simply says that
${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}={\mathscr{E}}^{r}/W({\mathfrak{g}})$
(6.65)
for some natural action of the Weyl group $W({\mathfrak{g}})$ on the Cartesian
product of $r$ copies of $\mathscr{E}$. Let us denote by ${\pi}_{W}$ the
projection
${\pi}_{W}:{\mathscr{E}}^{r}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}={\mathscr{E}}^{r}/W({\mathfrak{g}})$
(6.66)
The rational curve $\Sigma_{u}$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ lifts to a curve in
${\mathscr{E}}^{r}$, and the graph of the parametrized curve
$\Sigma_{u}\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$ lifts to
the graph in ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}^{r}$ which is our friend _obscured curve_
${\mathscr{C}}_{u}$. It is the quotient of the cameral curve by the lattice
${\rm Q}^{\vee}$:
${\mathscr{C}}_{u}={\mathcal{C}}_{u}/{{\rm Q}}^{\vee}$ (6.67)
In the section 8.2 we shall present yet another construction of
${\mathbb{L}}_{i,x}$’s, using gauge theory.
There is the so-called determinant line bundle $L$ over the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$, whose sections are the
fundamental characters $\widehat{\chi}_{i}$, $i=0,1,\ldots,r$. In the E.
Loojienga’s identification
${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}\approx{\mathbb{W}\mathbb{P}}^{a_{0},a_{1},\ldots,a_{r}}$
this line bundle is just the ${\mathcal{O}}(1)$ orbibundle over the weighted
projective space.
We have then the line bundle ${\mathscr{L}}$ over ${\mathscr{E}}^{r}$:
${\mathscr{L}}={\pi}_{W}^{*}L$ (6.68)
Let us call this line bundle _the abelianized determinant line bundle_.
## Chapter 7 The Seiberg-Witten curves in some detail
In this section we shall discuss the geometry of curves describing the limit
shape configurations and the special geometry of the gauge theories under
consideration. When possible we identify the cameral or the spectral curves
with the analogous curves of some algebraic integrable systems, namely the
Hitchin systems on the genus zero (i.e. Gaudin model) or genus one (i.e. spin
elliptic Calogero-Moser system) curves with punctures. These identifications
are less universal then the identification with the spectral curves of the
spin chains based on the Yangian algebra built on $\mathfrak{g}$,
$\widehat{\mathfrak{g}}$, or $\widehat{GL}_{\infty}$, respectively. The latter
identification is a subject of a separate venue of research which touches upon
various advances in geometric representation theory, study of the symplectic
geometry of moduli spaces of instantons and monopoles, quantum cohomology of
quiver varieties, to name just a few. We shall only mention the relation to
spin chains in a few examples, in this work.
Throughout this section we shall use the notation
$g_{\lambda}(x)={\zeta}(x)^{-1}{\pi}_{\lambda}(g(x))$ (7.1)
for the projectively modified operator in the representation
$(R_{\lambda},{\pi}_{\lambda})$ of ${\mathbf{G}}_{\text{q}}$, corresponding to
the group element $g(x)\in{\mathbf{G}}_{\text{q}}$.
### 7.1. Class I theories of $A$ type
This is the so-called linear quiver theory. The set of vertices
$\mathrm{Vert}_{\gamma}=\\{1,\ldots,r\\}$, the set of edges
$\mathrm{Edge}_{\gamma}=\\{1,\ldots,r-1\\}$, the maps $s,t$ for a particular
orientation are given by: $s(e)=e$, $t(e)=e+1$. The bi-fundamental masses are
a trivial cocycle:
$m_{e}={\mu}_{e+1}-{\mu}_{e}$ (7.2)
The corresponding conformal group $C\mathbf{G}=GL(r+1,{\mathbb{C}})$, the
fundamental characters ${\chi}_{i}$ are the characters of the representations
${\Lambda}^{i}{\mathbb{C}}^{r+1}$. We shall now describe the spectral curve in
the representation $R_{\lambda_{1}}\approx{\mathbb{C}}^{r+1}$. The
corresponding group element $g_{\lambda_{1}}(x)$ in (6.14) is the diagonal
matrix
$g_{\lambda_{1}}(x)={\rm diag}(t_{1}(x),\ldots,t_{r+1}(x))$
with
$\displaystyle t_{1}(x)={\zeta}(x){\mathscr{Y}}_{1}(x),\quad
t_{r+1}(x)={\zeta}(x){\mathscr{P}}^{[r]}(x)\,{\mathscr{Y}}_{r}(x)^{-1}$ (7.3)
$\displaystyle\quad
t_{i}(x)={\zeta}(x){\mathscr{P}}^{[i-1]}(x)\,{\mathscr{Y}}_{i}(x){\mathscr{Y}}_{i-1}(x)^{-1},\qquad
i=2,\ldots,r$
with some normalization factor ${\zeta}(x)$ which we choose shortly, and the
explicit formula for the invariants ${\mathcal{X}}_{i}({\mathscr{Y}}(x))$ is
(we omit the $x$-dependence in the right hand side):
$\displaystyle{\mathcal{X}}_{i}({\mathscr{Y}}(x))=\prod_{j=1}^{i-1}{\mathscr{P}}_{j}^{j-i}\times$
(7.4) $\displaystyle\qquad
e_{i}\left({\mathscr{Y}}_{1},{\mathscr{Y}}_{2}{\mathscr{Y}}_{1}^{-1}{\mathscr{P}}^{[1]},\ldots,{\mathscr{Y}}_{i}{\mathscr{Y}}_{i-1}^{-1}{\mathscr{P}}^{[i-1]},\ldots,{\mathscr{Y}}_{r}^{-1}{\mathscr{P}}^{[r]}\right)$
where $e_{i}$ are the elementary symmetric polynomials in $r+1$ variables. Our
master equations (6.5) equate the right hand side of (7.4) with the degree
${\mathbf{v}}_{i}$ polynomial $T_{i}(x)$ in $x$, cf. (6.6).
It is convenient to organize the invariants (7.4) into a generating
polynomial, which is nothing but the characteristic polynomial of the group
element $g(x)$ in some representation of $C{\mathbf{G}}$. The most economical
is, of course, the defining fundamental representation ${\mathbb{C}}^{r+1}$
with the highest weight ${\lambda}_{1}$:
$\displaystyle{\det}\left(t\cdot 1_{r+1}-g_{\lambda_{1}}(x)\right)=$ (7.5)
$\displaystyle\qquad
t^{r+1}+\sum_{i=1}^{r}(-1)^{i}t^{r+1-i}{\zeta}(x)^{i}\prod_{j=1}^{i-1}{\mathscr{P}}_{j}^{i-j}(x)\,{\mathcal{X}}_{i}({\mathscr{Y}}(x))$
$\displaystyle\qquad\qquad\qquad+(-{\zeta}(x))^{r+1}\prod_{j=1}^{r}{\mathscr{P}}_{j}^{r+1-j}(x)$
The group ${{}^{i}{\mathcal{W}}}$ is the symmetric group
${\mathcal{S}}_{r+1}$, which acts by permuting the eigenvalues of $g(x)$ in
(7.3). The cameral curve ${\mathcal{C}}_{u}$ is the $(r+1)!$-fold ramified
cover of the compactified $x$-plane ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}$. The points in the fiber are the _ordered_ sets of roots
$(t_{1}(x),\ldots,t_{r+1}(x))$ of the polynomial (7.5).
The curve ${\mathcal{C}}_{u}$ covers the _spectral curve_ $C_{u}$. The latter
is defined as the zero locus of the characteristic polynomial (7.5). The cover
${\mathcal{C}}_{u}\to C_{u}$ is $r!:1$, it sends the ordered $r+1$-tuple of
roots $(t_{1},\ldots,t_{r+1})$ to the first root $t_{1}$. The cover
$C_{u}\to\mathbf{C}_{\left\langle x\right\rangle}$ is $(r+1):1$.
Explicitly, the curve $C_{u}$ is given by:
$0={\mathcal{P}}(t,x)=\sum_{i=0}^{r+1}(-1)^{i}t^{r+1-i}{\zeta}(x)^{i}\,{\prod_{j=1}^{i-1}{\mathscr{P}}_{j}(x)^{i-j}}\
T_{i}(x)$ (7.6)
#### 7.1.1. Relation to Gaudin model
Figure 7.1. Degree profile example for $A_{4}$ theory and
$(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4})=(4,7,8,5)$. For
convenience one can set boundary conditions
$\mathbf{v}_{0}=\mathbf{v}_{r+1}=\mathbf{w}_{0}=\mathbf{w}_{r+1}=0$.
It is easy to see, using the Eqs. (3.8), (3.9) and Fig. 7.1 that
${\mathbf{w}}_{i_{*}}=w_{+}+w_{-}$, where
$\displaystyle w_{+}={\mathbf{v}}_{i_{*}}-{\mathbf{v}}_{i_{*}+1}\geq 0$ (7.7)
$\displaystyle w_{-}={\mathbf{v}}_{i_{*}}-{\mathbf{v}}_{i_{*}-1}\geq 0$
and it useful to record
$\displaystyle\mathbf{v}_{1}$
$\displaystyle=\mathbf{w}_{1}+\dots+\mathbf{w}_{i_{*}-1}+\mathbf{w}_{-}$ (7.8)
$\displaystyle\mathbf{v}_{r}$
$\displaystyle=\mathbf{w}_{r}+\dots+\mathbf{w}_{i_{*}+1}+\mathbf{w}_{+}$
$\displaystyle\mathbf{v}_{*}$
$\displaystyle=\sum_{i=1}^{i_{*}-1}i\mathbf{w}_{i}+i_{*}\mathbf{w}_{-}$
$\displaystyle\mathbf{v}_{*}$
$\displaystyle=\sum_{i=i_{*}+1}^{r}(r+1-i)\mathbf{w}_{i}+(r+1-i_{*})\mathbf{w}_{+}$
Accordingly, we can factorize the polynomial ${\mathscr{P}}_{i_{*}}(x)$ as:
${\mathscr{P}}_{i_{*}}(x)={\mathfrak{q}}_{i_{*}}{\mathscr{P}}^{+}(x){\mathscr{P}}^{-}(x)$
(7.9)
where ${\mathscr{P}}^{\pm}$ are monic polynomials of degrees
${\rm deg}{\mathscr{P}}^{\pm}=w_{\pm}$
We can actually transform (7.6) into something nice, by adjusting
${\zeta}(x)$:
${\zeta}(x)^{-1}={\mathscr{P}}^{-}(x){\mathscr{P}}^{[i_{*}-1]}(x)$ (7.10)
Then $D(g(x))$ is given by:
$\displaystyle D(g(x))=\frac{P_{0}(x)}{P_{\infty}(x)}$ (7.11) $\displaystyle
P_{0}(x)={\mathscr{P}}^{+}(x)^{r+1-i_{*}}\prod_{j=i_{*}+1}^{r}{\mathscr{P}}_{j}(x)^{r+1-j}$
$\displaystyle
P_{\infty}(x)={\mathscr{P}}^{-}(x)^{i_{*}}\prod_{j=1}^{i_{*}-1}{\mathscr{P}}_{j}(x)^{j}$
Then ${\mathcal{P}}(t,x)$ can be written as:
${\mathcal{P}}(t,x)=\prod_{j=1}^{i_{*}-1}{\mathfrak{q}}_{j}^{j}\cdot\frac{P(t,x)}{P_{\infty}(x)}$
where $P(t,x)$ is a degree $N={\mathbf{v}}_{i_{*}}$ polynomial in $x$, and the
degree $r+1$ polynomial in $t$, which is straightforward to calculate:
$\displaystyle(-1)^{i_{*}}\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}^{j}\ P(t,x)=$
(7.12)
$\displaystyle\qquad(-{\mathfrak{q}}_{i_{*}})^{i_{*}}t^{r+1}P_{\infty}(x)\,+$
$\displaystyle\qquad\qquad+\sum_{i=1}^{i_{*}}t^{r+1-i}\,T_{i}(x){\mathfrak{q}}_{i_{*}}^{i_{*}-i}(-{\mathscr{P}}_{*}^{-}(x))^{i_{*}-i}\prod_{j=i}^{i_{*}-1}{\mathscr{P}}_{j}^{j-i}(x)$
$\displaystyle\qquad\qquad\qquad+\sum_{i=i_{*}+1}^{r}t^{r+1-i}\,T_{i}(x)(-{\mathscr{P}}_{*}^{+}(x))^{i-i_{*}}\prod_{j=i_{*}+1}^{i-1}{\mathscr{P}}_{j}^{i-j}(x)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(-1)^{r+1-i_{*}}P_{0}(x)$
Now, recall that $T_{j,0}$ is fixed by the couplings ${\mathfrak{q}}$:
$T_{j,0}({\mathfrak{q}})=\prod_{j=1}^{i-1}{\mathfrak{q}}_{j}^{j-i}\,e_{i}(1,{\mathfrak{q}}_{1},{\mathfrak{q}}_{1}{\mathfrak{q}}_{2},\ldots,{\mathfrak{q}}_{1}{\mathfrak{q}}_{2}\ldots{\mathfrak{q}}_{i},\ldots{\mathfrak{q}}_{1}\ldots{\mathfrak{q}}_{r})$
(7.13)
and the coefficient $T_{j,1}$ is fixed by the masses $m_{i,{\mathfrak{f}}}$
and $m_{e}$.
Therefore, the coefficient of $x^{N}$ in $P(t,x)$ can be computed explicitly:
$\displaystyle\sum_{i=0}^{r+1}(-1)^{i}t^{r+1-i}\,\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}^{-i}\,e_{i}(1,{\mathfrak{q}}_{1},{\mathfrak{q}}_{1}{\mathfrak{q}}_{2},\ldots,{\mathfrak{q}}_{1}{\mathfrak{q}}_{2}\ldots{\mathfrak{q}}_{i},\ldots,{\mathfrak{q}}_{1}\ldots{\mathfrak{q}}_{r})=$
(7.14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)$
where
${\check{t}}_{i}=\frac{\prod_{j=1}^{i}{\mathfrak{q}}_{j}}{\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}},\qquad\qquad
i=0,\ldots,r$ (7.15)
We thus rewrite the curve $C_{u}$ in the $(x,t)$-space, defined by the
equation
$\displaystyle
0={\mathcal{R}}_{A_{r}}(t,x)=\frac{P(t,x)}{\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)}=\prod_{l=1}^{N}\left(x-x_{l}(t)\right)=$
(7.16)
$\displaystyle\qquad\qquad\qquad\qquad=x^{N}+\frac{1}{\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)}\sum_{j=1}^{N}p_{j}(t)x^{N-j}$
where
$N={\mathbf{v}}_{i_{*}}$ (7.17)
It is clear from the Eq. (7.16) that as $t\to{\check{t}}_{i}$ one of the roots
$x_{l}(t)$ has a pole, while the other $N-1$ roots are finite. Near $t=0$ the
polynomial $P(t,x)$ approaches:
$P(0,x)=(-1)^{r+1-i_{*}}P_{0}(x)$ (7.18)
while near $t=\infty$
$P(t,x)t^{-r-1}\to(-{\mathfrak{q}}_{i_{*}})^{i_{*}}P_{\infty}(x)$ (7.19)
Let
$dS=x\frac{dt}{t}$ (7.20)
Then our discussion above implies that the differential $dS$ has the first
order poles on $C_{u}$: at one of the $N$ preimages of the points
${\check{t}}_{i}$, $i=0,1,\ldots,r$, and at all preimages of the points $t=0$
and $t=\infty$. The residues of $dS$ are linear combinations of the masses of
the hypermultiplets, in agreement with the observations in
[Seiberg:1994aj],[Donagi:1995cf].
Remarkably, we can identify $C_{u}$ with the spectral curve of the meromorphic
Higgs field ${\Phi}$:
${\Phi}={\Phi}(t)dt=\sum_{j=-1}^{r+1}\,{\Phi}_{j}\frac{dt}{t-{\check{t}}_{j}}$
(7.21)
where ${\check{t}}_{-1}=0$, ${\check{t}}_{r+1}={\infty}$, and ${\Phi}_{j}$ are
$N\times N$ matrices, which have rank one for $j=0,1,\ldots,r$, and have the
maximal rank for $j=-1,r+1$. Moreover, the eigenvalues of ${\Phi}_{j}$ are all
fixed in terms of the masses. The spectra of ${\Phi}_{j}$, $j=-1,\ldots,r+1$
have specified multiplicity:
1. (1)
the matrix ${\Phi}_{-1}$ has $w_{+}$ eigenvalues of multiplicity $r+1-i_{*}$,
and ${\mathbf{w}}_{r+1-j}$ eigenvalues of multiplicity $j$, for
$j=1,\ldots,r-i_{*}$;
the eigenvalues are fixed by the masses
2. (2)
the matrices ${\Phi}_{j}$, $j=0,1,\ldots,r$ has one non-vanishing eigenvalue
each, and $N-1$ vanishing eigenvalues; We can write
$({\Phi}_{j})_{a}^{b}=u_{a}^{j}v_{j}^{b},\qquad a,b=1,\ldots,N$
for some vectors $u^{j}$, $v_{j}\in{\mathbb{C}}^{N}$, obeying
$\sum_{a=1}^{N}u^{j}_{a}v_{j}^{a}=M_{j}$ (7.22)
and considered up to an obvious ${\mathbb{C}}^{\times}$-action, for some
$M_{j}$ which is linear in the bi-fundamental and fundamental masses.
3. (3)
the matrix ${\Phi}_{r+1}$ has $w_{-}$ eigenvalues of multiplicity $i_{*}$, and
${\mathbf{w}}_{j}$ eigenvalues of multiplicity $j$, for $j=1,\ldots,i_{*}-1$.
Then:
$\left(\frac{dt}{t}\right)^{N}\,{\mathcal{R}}_{A_{r}}(t,x)={\det}\left(\,x\frac{dt}{t}-{\Phi}\right)$
(7.23)
We can make an $SL(N)$ Higgs field out of $\Phi$ by shifting it by the scalar
meromorphic one-form $\frac{1}{N}{\operatorname{Tr}}_{N}{\Phi}$, which is
independent of the moduli $u$ of the curve $C_{u}$.
The moduli space of $r+3$-ples of matrices ${\Phi}_{j}$, obeying
$\sum_{j=-1}^{r+1}{\Phi}_{j}=0$ (7.24)
with fixed eigenvalues of the above mentioned multiplicity, considered up to
the simultaneous $SL(N)$-similarity transformation, is the phase space
${{\mathfrak{P}}}^{H}_{0,r+3}$ of the genus zero version of $SL(N)$ Hitchin
system, the classical Gaudin model on $r+3$ sites. The general Gaudin model
has the residues ${\Phi}_{j}$ belonging to arbitrary conjugacy classes.
See [Kronheimer:1990a, Kronheimer:1990] for the geometry of complex coadjoint
orbits. The Hitchin system with singularities was studied in [Gorsky:1994dj,
Nekrasov:1995nq, Donagi:1995am, Gukov:2006jk, Witten:2007td, Gukov:2008sn]. In
[Gaiotto:2009we, Nanopoulos:2009uw, Nanopoulos:2010zb, Nanopoulos:2010ga,
Nanopoulos:2009xe] this Hitchin system with singularities was discussed from
the point of view of brane constructions such as [Witten:1997sc,
Gaiotto:2009we].
###### Remark.
The curve $C_{u}$ is much more economical then ${\mathcal{C}}_{u}$. However,
the price we pay is the complexity of the relation between the special
coordinates ${\mathfrak{a}}_{i{\mathbf{a}}}$,
${\mathfrak{a}}_{i{\mathbf{a}}}^{D}$ and the moduli $u$ of the curve $C_{u}$.
Roughly speaking all special coordinates are linear combinations of the
periods of the differential
$x\frac{dt}{t}$
and the masses. The coordinates ${\mathfrak{a}}_{1{\mathbf{a}}}$ come from the
periods
$\oint x\,d{\log}g_{1}(x)\sim\oint xdt/t$
the coordinates ${\mathfrak{a}}_{2{\mathbf{a}}}$ come from the periods
$\oint x\,d{\log}(g_{1}(x)g_{2}(x))\sim\oint xdt/t+\oint xdt/t$
the coordinates ${\mathfrak{a}}_{i{\mathbf{a}}}$ come from the periods
$\oint x\,d{\log}(g_{1}(x)\ldots g_{i}(x))\sim\oint xdt/t+\ldots+\oint xdt/t$
etc.
###### Remark.
In the $A_{2}$ case our solution matches the one found in [Shadchin:2005cc].
###### Remark.
We can connect the cameral curve ${\mathcal{C}}_{u}$ to the spectral curve
$C_{u}$ via a tower of ramified covers:
${\mathcal{C}}_{u}\to C_{u}^{(r)}\to C_{u}^{(r-1)}\to\dots\to
C_{u}^{(1)}=C_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ (7.25)
which we can call the _Gelfand-Zeitlin_ tower of curves. The curve
$C_{u}^{(i)}$ is the quotient of ${\mathcal{C}}_{u}$ by the subgroup
$W(A_{r-i})$ of the Weyl group $W(A_{r})$, which acts on the amplitudes
$({\mathscr{Y}}_{i+1},\ldots,{\mathscr{Y}}_{r})$ while preserving
$({\mathscr{Y}}_{1},\ldots,{\mathscr{Y}}_{i})$.
###### Remark.
We should warn the reader that our cameral curves need not be the cameral
curves of Hitchin systems [Donagi:1995alg]. We mapped the spectral curve of
the family of conjugacy classes $[g(x)]$ corresponding to the fundamental
representation $R_{1}$ to the spectral curve of the $GL(N)$-Gaudin system,
i.e. the genus zero Hitchin system, corresponding to the $N$-dimensional
representation. One could then build the cameral curve for the $GL(N)$-Gaudin
system. This curve has all the reasons to differ from our cameral curve
${\mathcal{C}}_{u}$.
However, the identification of ${\mathfrak{M}}$ with the moduli spaces of
curves describing the spectrum of the transfer matrix in the quasi classical
limit of the $Y(A_{r})$ spin chain is more natural, and carries over to the
level of cameral curves. This statement will be elaborated upon below and in
[NP2012b].
###### Remark.
In view of [Gaiotto:2009we] it is natural to identify the space of couplings
${\tilde{\mathfrak{q}}}=({\mathfrak{q}}_{1},\ldots,{\mathfrak{q}}_{r})$ with a
coordinate patch in the moduli space $\overline{\mathcal{M}}_{0,r+3}$ of
stable genus zero curves with $r+3$ punctures. In this fashion the linear
quiver theories (the class I type $A_{r}$ theories) can be analytically
continued to other weakly coupled regions (weak coupling corresponds to the
maximal degeneration of the stable curve). Most of these regions do not have a
satisfactory Lagrangian description. Nevertheless, it would be interesting to
try to generalize the limit shape equations even without knowing their
microscopic origin. What would the iWeyl group look like in this case?
#### 7.1.2. Quiver description
We have thus found that a particular subset of Gaudin-Hitchin models, with all
but two residues of the minimal type, are the Seiberg-Witten integrable
systems of the class I $A_{r}$ type theories. As a check, let us compute the
dimension of the moduli space ${{\mathfrak{P}}}^{H}_{0,r+3}$ of solutions to
the (traceless part of the) moment map equation (7.24) divided by the
$SL(N,{\mathbb{C}})$-action is equal to:
$\displaystyle 2(r+1)(N-1)-2(N^{2}-1)+$ (7.26)
$\displaystyle\qquad+\left(N^{2}-\sum_{j=1}^{i_{*}-1}j^{2}{\mathbf{w}}_{j}-i_{*}^{2}w_{-}\right)+$
$\displaystyle\qquad+\left(N^{2}-\sum_{j=i_{*}+1}^{r}(r+1-j)^{2}{\mathbf{w}}_{j}-(r+1-i_{*})^{2}w_{+}\right)=$
$\displaystyle\qquad\qquad\qquad=2\sum_{i=1}^{r}({\mathbf{v}}_{i}-1)=2\,{\dim}{{\mathfrak{M}}}$
Actually, the moduli space ${{\mathfrak{P}}}^{H}_{0,r+3}$ can be described as
a quiver variety. Its graph is an $r+3$-pointed star, with $r+1$ legs of
length $1$, and two long legs, of the lengths $l_{-1}={\mathbf{v}}_{r}-1$ and
$l_{r+1}={\mathbf{v}}_{1}-1$, respectively. The dimensions of the vector
spaces assigned to vertices are: the $r+3$-valent vertex (the star) has
dimension $N$, the tails of the short legs all have dimension $1$, the
dimensions along the long legs start at $1$ at the tails, then grow with the
step $1$ for the first ${\mathbf{w}}_{1}$ (respectively, ${\mathbf{w}}_{r}$)
vertices, then grow with the step $2$ for the next ${\mathbf{w}}_{2}$
(respectively, ${\mathbf{w}}_{r-1}$) and so on. (See example in figure 7.2).
Figure 7.2. The example quiver variety for $A_{4}$ quiver at
$\mathbf{v}=(7,10,8,5)$ and $\mathbf{w}=(4,5,1,2)$ with $i_{*}=2$ and
$\mathbf{w}_{*}=\mathbf{w}_{-}+\mathbf{w}_{+}$ with $\mathbf{w}_{-}=3$ and
$\mathbf{w}_{+}=2$. The labels at vertices denote the dimensions in the
pattern as explained
The extended phase space ${\mathfrak{P}}^{\mathrm{ext}}$ for the class I
$A_{r}$ type theories is easy to describe. One just need to relax the
${\mathbb{C}}^{\times}$ moment map constraints (7.22) as well as the analogous
${\mathbb{C}}^{\times}$ constraints for the ${\Phi}_{-1}$, ${\Phi}_{r+1}$
residues. In the quiver description we make the quiver gauge group the product
of the special unitary groups as opposed to the product of unitary groups.
#### 7.1.3. Reduction to the spin chain
The simplest example of the Class I theory of the $A$ type is, of course, the
$A_{1}$ theory. This is the celebrated $N_{f}=2N_{c}$ theory, with
${\mathbf{w}}_{1}=N_{f}$, ${\mathbf{v}}_{1}=N_{c}=N$, in our notation. Let
${\mathfrak{q}}={\mathfrak{q}}_{1}$ and let $T(x)=T_{1,0}^{-1}T_{1}(x)$ denote
the monic degree $N$ polynomial.
The reduced curve (7.12) assumes a very simple form:
${\mathfrak{q}}{\mathscr{P}}^{-}(x)t+{\mathscr{P}}^{+}(x)t^{-1}=(1+{\mathfrak{q}})T(x)$
(7.27)
It is not difficult to recognize in this formula the quasiclassical limit of
Baxter’s $T-Q$ equation [Baxter:1985] for the $XXX$ $sl_{2}$ spin chain. In
fact, it was observed already in [Gorsky:1996hs, Gorsky:1996qp] that the
Seiberg-Witten curve of the ${\mathcal{N}}=2$ supersymmetric QCD can be
interpreted using the integrable spin chain, albeit in a somewhat different
fashion. Note that a possible lift of $[g(x)]$ to
$C{\mathbf{G}}=GL(2,{\mathbb{C}})$ in this case is given by the diagonal
matrix:
$g(x)=\left(\begin{matrix}{\mathfrak{q}}t{\mathscr{P}}^{-}(x)&0\\\
0&t^{-1}{\mathscr{P}}^{+}(x)\end{matrix}\right)$ (7.28)
where $t$ solves (7.27). However this choice of $g(x)$ is not continuous in
$x$. As we cross the cuts $I_{1,{\mathbf{a}}}$ the matrix $g(x)$ will have its
diagonal entries exchanged. We can conjugate $g(x)\to h^{-1}(x)g(x)h(x)$ into
a form, e.g.
$\mathbf{g}(x)=\left(\begin{matrix}{\mathfrak{q}}T(x)&1\\\
{\mathfrak{q}}\left(T^{2}(x)-{\mathscr{P}}^{+}(x){\mathscr{P}}^{-}(x)\right)&T(x)\end{matrix}\right)$
(7.29)
whose entries are polynomials. This is a particular case of a general
statement [Steinberg:1965], lifting a family of conjugacy classes in
${\mathbf{G}}_{\text{q}}$ to ${\mathbf{G}}_{\text{q}}$ itself (slightly
adapted for the conformal extension $C\mathbf{G}$). The lift (7.29) does not
depend on the split ${\mathscr{P}}(x)$ into the product of
${\mathscr{P}}^{\pm}$ factors.
There is yet another lift of $[g(x)]$ to $C\mathbf{G}$, which does depend on
the factorization, and makes closer contact with spin chains. We shall discuss
it in the section devoted to the study of the phase spaces of the integrable
systems corresponding to our gauge theories.
#### 7.1.4. Duality
In the mapping to the Gaudin-Hitchin system we employed a particular lift
$g(x)$ of the conjugacy class $[g(x)]$ in
$SL(r+1,{\mathbb{Z}})/{\mathbb{Z}}_{r+1}$ to the conjugacy class in
$GL(r+1,{\mathbb{C}})$ by a judicious choice of the normalization factor
${\zeta}(x)$. More importantly, the spectral one-form describing the
eigenvalues of the Higgs field, is equal to $xdt/t$ where $x$ is the argument
of the amplitude function, and $t$ is the spectral variable describing the
eigenvalues of $g(x)$. For the group $GL(r+1,{\mathbb{C}})$ the eigenvalues of
$g(x)$ in some representation take values in ${\mathbf{C}_{\left\langle
t\right\rangle}}={\mathbb{C}}^{\times}$ which gets naturally compactified to
${\mathbb{C}\mathbb{P}}^{1}$ to allow the degenerations.
To summarize, the Lax operator of Gaudin-Hitchin system, the Higgs field
${\Phi}(t)dt/dt$ lives on the curve ${\mathbf{C}_{\left\langle
t\right\rangle}}$ of the eigenvalues of the ‘‘Lax operator’’ $g(x)$ of the
gauge theory. Vice versa, the ‘‘Lax operator’’ $g(x)$ of the gauge theory
lives on the curve ${\mathbf{C}_{\left\langle x\right\rangle}}$ of the
eigenvalues of the Higgs field of Hitchin system.
We shall encounter some versions of this ‘‘eigenvalue – spectral parameter’’
duality in other sections of this work.
### 7.2. Class I theories of $D$ type
These are the $SU({\mathbf{v}}_{1})\times\ldots\times SU({\mathbf{v}}_{r})$
theories whose quiver contains a trivalent vertex which connects two one-
vertex legs to a leg of the length $r-3$. The corresponding group
${\mathbf{G}}_{\text{q}}$ is $Spin(2r,{\mathbb{C}})$, its conformal version
$C\mathbf{G}$ is the extension of $\mathbf{G}$ by ${\mathbb{C}}^{\times}$ or
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$, depending on the parity of
$r$.
Passing from the $A$ type theories to the $D$ type theories we encounter new
phenomenon. In addition to the exterior powers $\wedge^{i}V$ of the vector
representation $V={\mathbb{C}}^{2r}$ of $Spin(2r)$ the fundamental
representations of the group ${\mathbf{G}}_{\text{q}}$ come also from spin
representations $S_{\pm}$. We should use the cameral curve ${\mathcal{C}}_{u}$
to get the special coordinates and the prepotential, however a lot of
information is contained in the spectral curve $C_{u}^{R}$ in some fundamental
representation $R$, which we shall take to be the vector $2r$ dimensional
representation $V=R_{\lambda_{1}}={\mathbb{C}}^{2r}$. In order to describe the
spectral curve we need to know the characters of the group element $g(x)$
(6.14) in the representations $\wedge^{i}V$, for $i=1,\ldots,2r$. When we deal
with $V$ and its exterior powers only, we do not see the full conformal
version of $\mathbf{G}$, only its one-dimensional extension (which we shall
denote simply by C$\mathbf{G}$) which consists of the matrices $g\in
GL(2r,{\mathbb{C}})$, such that $gg^{t}=D(g)\cdot{\bf 1}_{2r}$, with
$D(g)\in{\mathbb{C}}^{\times}$ a scalar.
The spectral curve $C_{u}=C^{V}_{u}$ in the vector representation can be
modified by the transformation similar to (7.10) to get the curve of minimal
degree in $x$. Let us label the vertices of the $D_{r}$ Dynkin diagram in such
a way, that the trivalent vertex is $r-2$, the tails are $r-1$, $r$, and the
end vertex of the ‘‘long leg’’ has the label $1$, see Appendix A. Then the
product of the matter polynomials ${\mathscr{P}}_{r-1}$ and
${\mathscr{P}}_{r}$ has degree
${\rm
deg}({\mathscr{P}}_{r-1}{\mathscr{P}}_{r})=2({\mathbf{v}}_{r-1}+{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2})$
Now we shall factorize ${\mathscr{P}}_{r-1}{\mathscr{P}}_{r}$ into a product
of two factors of equal degrees
${\mathscr{P}}_{r-1}{\mathscr{P}}_{r}={\mathscr{P}}^{+}{\mathscr{P}}^{-},\qquad{\rm
deg}{\mathscr{P}}^{+}={\rm
deg}{\mathscr{P}}^{-}={\mathbf{v}}_{r-1}+{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2}$
(7.30)
There are many possible factorizations. For example, if
${\mathbf{w}}_{r-1}\leq{\mathbf{w}}_{r}$, then we can take:
${\mathscr{P}}_{r}(x)={\mathscr{P}}^{+}(x)S(x)$,
${\mathscr{P}}^{-}(x)=S(x){\mathscr{P}}_{r-1}(x)$ for any degree
${\mathbf{v}}_{r}+{\mathbf{v}}_{r-1}-{\mathbf{v}}_{r-2}\leq{\mathbf{w}}_{r}=2{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2}$
subfactor ${\mathscr{P}}^{+}(x)$ in ${\mathscr{P}}_{r}(x)$. We shall normalize
${\mathscr{P}}^{\pm}(x)$ so that the highest coefficient in both polynomials
equals
$\sqrt{{\mathfrak{q}}_{r-1}{\mathfrak{q}}_{r}}$
That there exist different decompositions (7.30) is a generalization of
$S$-duality of the ${\mathcal{S}}$-class ${\mathcal{N}}=2$ theories of the
$A_{r}$ type studied in [Gaiotto:2009we].
The spectral curve $C_{u}$ corresponding to the $2r$-dimensional vector
representation of $CSpin(2r,{\mathbb{C}})$ is mapped to the curve
$P_{D_{r}}^{C}(t,x)=0$ in the $(t,x)$-space, where
$P_{D_{r}}^{C}(t,x)=t^{-r}P_{\infty}(x){\det}_{R_{1}}(t\cdot 1_{2r}-g(x))$
(7.31)
with some polynomial $P_{\infty}(x)$ to be determined below. The group element
$g(x)$ in the vector representation ${\mathbb{C}}^{2r}$ of
$C{\mathbf{G}}_{\text{q}}$ is given by
$g(x)=E^{-1}\operatorname{diag}(g_{1}(x),\dots,g_{2r}(x))E$ (7.32)
with $E$ being any matrix such that
$(EE^{t})_{ij}={\delta}_{i,2r+1-j}$
represents the symmetric bilinear form on ${\mathbb{C}}^{2r}$ and
$\displaystyle g_{1}(x)=\zeta(x)\mathscr{Y}_{1}(x)$ (7.33) $\displaystyle
g_{i}(x)=\zeta(x)\mathscr{P}^{[i-1]}(x)\frac{\mathscr{Y}_{i}(x)}{\mathscr{Y}_{i-1}(x)},\quad
i=2,\dots,r-2$ $\displaystyle
g_{r-1}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\frac{\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x)}{\mathscr{Y}_{r-2}(x)}$
$\displaystyle
g_{r}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\mathscr{P}_{r-1}(x)\frac{\mathscr{Y}_{r}(x)}{\mathscr{Y}_{r-1}(x)}$
$\displaystyle
g_{r+1}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\mathscr{P}_{r}(x)\mathscr{Y}_{r-1}(x)/\mathscr{Y}_{r}(x)$
$\displaystyle
g_{r+2}(x)=\zeta(x)\mathscr{P}^{[r]}(x)\mathscr{Y}_{r-2}(x)/(\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x))$
$\displaystyle
g_{2r+1-i}(x)=\zeta(x)\frac{\mathscr{P}^{[r]}(x)\mathscr{P}^{[r-2]}(x)}{{\mathscr{P}}^{[i-1]}(x)}\frac{\mathscr{Y}_{i-1}(x)}{\mathscr{Y}_{i}(x)},\quad
i=2,\dots,r-2$ $\displaystyle
g_{2r}=\zeta(x)\mathscr{P}^{[r]}(x)\mathscr{P}^{[r-2]}(x)\frac{1}{\mathscr{Y}_{1}(x)}$
The factor $\zeta(x)$ which likely gives the minimal degree curve is
$\zeta(x)^{-1}=\mathscr{P}^{+}(x)\mathscr{P}^{[r-2]}(x)$ (7.34)
Thus, the scalar $D(g(x))$ is equal to:
$D(g(x))=\frac{{\mathscr{P}}^{-}(x)}{{\mathscr{P}}^{+}(x)}$ (7.35)
and the prefactor in (7.31) is:
$P_{\infty}(x)=\mathscr{P}^{+}(x)^{r}\prod_{j=1}^{r-2}\mathscr{P}_{j}(x)^{j}.$
(7.36)
After some manipulations we find
$\displaystyle(-1)^{r}P_{D_{r}}^{C}(t,x)=T_{r}^{2}{\mathscr{P}}_{r-1}+T_{r-1}^{2}{\mathscr{P}}_{r}-{\eta}T_{r-1}T_{r}+$
(7.37)
$\displaystyle\qquad\sum_{l=1}^{\left[\frac{r}{2}\right]}T_{r-2l}\left(\prod_{j=r+1-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l}\right)\,{\xi}_{l}^{2}-$
$\displaystyle\qquad\qquad-\sum_{l=1}^{\left[\frac{r-1}{2}\right]}T_{r-2l-1}\left(\prod_{j=r-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l+1}\right)\,{\xi}_{l}{\xi}_{l+1}\,,$
$\displaystyle{\xi}_{l}=({\mathscr{P}}^{+}t)^{l}-({\mathscr{P}}^{-}t^{-1})^{l},\
{\eta}={\mathscr{P}}^{+}t+{\mathscr{P}}^{-}t^{-1}$
This equation has degree
$N=2({\mathbf{v}}_{r}+{\mathbf{v}}_{r-1})-{\mathbf{v}}_{r-2}$ in the $x$
variable. Note
${\mathbf{v}}_{r-2}\leq N\leq 2{\mathbf{v}}_{r-2}$ (7.38)
As in the $A_{r}$ case, the curve $C_{u}$ has branches going off to infinity
in the $x$-direction, over $2r$ points ${\check{t}}_{i},{\check{t}}_{i}^{-1}$,
$i=1,\ldots,r$ in the $t$-line ${\mathbb{C}\mathbb{P}}^{1}_{t}$ which
correspond to the weights of $R_{1}$
${\check{t}}_{i}=\frac{1}{\sqrt{{\mathfrak{q}}_{r-1}{\mathfrak{q}}_{r}}}\frac{{\mathfrak{q}}^{[i-1]}}{{\mathfrak{q}}^{[r-2]}}$
(7.39)
In addition, there are special points $t=0,\infty$. Over these points the
curve $C_{u}$ has $N$ branches, where $x$ approaches one of the roots of the
polynomial $P_{0}(x)$
$P_{0}(x)={\mathscr{P}}^{-}(x)^{r}\prod_{j=1}^{r-2}{\mathscr{P}}_{j}(x)^{j}\
.$ (7.40)
and $P_{\infty}(x)$, cf. (7.36), respectively.
The curve $C_{u}$ is invariant under the involution
$t\mapsto\frac{{\mathscr{P}}_{-}(x)}{{\mathscr{P}}_{+}(x)}t^{-1}$ (7.41)
The fixed points of (7.41) are the points of intersection of the curve $C_{u}$
and the curve
${\mathscr{P}}^{+}(x)t-{\mathscr{P}}^{-}(x)t^{-1}=0$ (7.42)
The equations ${\mathcal{R}}_{D_{r}}(t,x)=0$ (7.37) and (7.42) imply
$\displaystyle
T_{r}^{2}(x){\mathscr{P}}_{r-1}(x)+T_{r-1}^{2}(x){\mathscr{P}}_{r}(x)=$ (7.43)
$\displaystyle\qquad\qquad\qquad
T_{r-1}(x)T_{r}(x)({\mathscr{P}}^{+}(x)t+{\mathscr{P}}^{-}(x)t^{-1})$
and
$T_{r}^{2}(x){\mathscr{P}}_{r-1}(x)=T_{r-1}^{2}(x){\mathscr{P}}_{r}(x)$ (7.44)
Again, the curve $C_{u}$ is more economical then the full cameral curve
${\mathcal{C}}_{u}$. Again, the special coordinates
${\mathfrak{a}}_{i,{\mathbf{a}}}$ and the duals
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ are the linear combinations of the
periods of the differential $xdt/t$ and the masses.
Let us map the curve $C_{u}$ to the curve $\Sigma_{u}$ in the space $S$ which
is a ${\mathbb{Z}}_{2}$-quotient of the (blowup of the)
${\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbb{C}\mathbb{P}}^{1}_{t}$ space, parametrized by
$(x,s)$ where
$s=\frac{{\mathscr{P}}^{+}(x)}{{\mathscr{P}}^{-}(x)}t^{2}$
The curve $\Sigma_{u}$ is described by the equations $s+s^{-1}=2c$ and:
$P_{D_{r}}^{\Sigma}(x,c)\equiv{\bf
A}(x,c)^{2}-2{\mathscr{P}}_{r}(x){\mathscr{P}}_{r-1}(x)(c+1){\bf
B}(x,c)^{2}=0$ (7.45)
where ${\bf A},{\bf B}$ are the polynomials in $x$ and $c$ of bi-degrees
$(N,\left[\frac{r}{2}\right])$ and
$({\mathbf{v}}_{r-1}+{\mathbf{v}}_{r},\left[\frac{r-1}{2}\right])$,
respectively:
$\displaystyle{\bf
A}(x,c)=T_{r}^{2}{\mathscr{P}}_{r-1}+T_{r-1}^{2}{\mathscr{P}}_{r}+$ (7.46)
$\displaystyle\qquad\qquad+2\sum_{l=1}^{\left[\frac{r}{2}\right]}\,{\bf
C}_{l}(c)\,T_{r-2l}{\mathscr{P}}_{r-1}^{l}{\mathscr{P}}_{r}^{l}\prod_{j=r+1-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l},$
$\displaystyle{\bf
B}(x,c)=T_{r-1}T_{r}+2\sum_{l=1}^{\left[\frac{r-1}{2}\right]}\,{\bf
D}_{l}(c)\,T_{r-2l-1}{\mathscr{P}}_{r-1}^{l}{\mathscr{P}}_{r}^{l}\prod_{j=r-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l+1},$
where the degree $l$ polynomials ${\bf C}_{l}(c)$, ${\bf D}_{l}(c)$ are
defined as follows:
$\displaystyle{\bf C}_{l}(c)=\frac{1}{2}(s^{l}+s^{-l})-1,\qquad s+s^{-1}=2c$
(7.47) $\displaystyle{\bf
D}_{l}(c)=\frac{(s^{l}-1)(s^{l+1}-1)}{2s^{l}(s+1)}=\sum_{j=0}^{l-1}(-1)^{j}{\bf
C}_{l-j}(c)$
Over the points $c=1$ and $c=-1$ the equation for $\Sigma_{u}$ becomes
reducible: at $c=1$:
$P_{D_{r}}^{\Sigma}(x,1)=({\mathscr{P}}_{r-1}T_{r}^{2}-{\mathscr{P}}_{r}T_{r-1}^{2})^{2}$
(7.48)
and at $c=-1$:
$P_{D_{r}}^{\Sigma}(x,-1)={\bf A}(x,-1)^{2}$ (7.49)
It is easy to see that the curve $\Sigma_{u}$ has double points at $(x,s)$
where either $s=1$ and $x$ being any of the $N$ roots of (7.48) or $s=-1$ and
$x$ is any of the $N$ roots of (7.49). The locations of these roots are not
fixed by the masses of the matter fields.
Let us normalize the equation of $\Sigma_{u}$ by dividing $P_{D_{r}}^{\Sigma}$
by the coefficient at $x^{2N}$:
${\mathcal{R}}_{D_{r}}(x,c)=\frac{P_{D_{r}}^{\Sigma}(x,c)}{\prod_{i=1}^{r}(s-{\check{t}}_{i}^{2})(1-s^{-1}{\check{t}}_{i}^{-2})}$
(7.50)
times a constant such that $\mathcal{R}_{D_{r}}(x,c)$ is monic in $x$ and a
rational function of $c$.
We thus arrive at the following interpretation of the curve $\Sigma_{u}$. It
is the spectral curve
${\mathcal{R}}_{D_{r}}\left(x,\frac{s^{2}+1}{2s}\right)\left(\frac{ds}{s}\right)^{2N}={\rm
Det}_{2N}\left(x\frac{ds}{s}-{\Phi}(s)\right)$ (7.51)
of the genus zero Higgs field
${\Phi}(s)=\sum_{s_{j}\in J}{\Phi}_{j}\frac{ds}{s-s_{j}}$ (7.52)
where $J\subset{\mathbb{C}\mathbb{P}}^{1}_{s}$ is the set of $2r+2$
singularities:
$J=\\{0,\infty\,\\}\cup\\{\,{\check{t}}_{i}^{2},\
{\check{t}}_{i}^{-2}\,|\,i=1,\ldots,r\\}$
Let ${\sigma}:{\mathbb{C}\mathbb{P}}^{1}_{s}\to{\mathbb{C}\mathbb{P}}^{1}_{s}$
be the involution ${\sigma}(s)=s^{-1}$. The Higgs field must obey:
${\sigma}^{*}{\Phi}={\Omega}{\Phi}^{t}{\Omega}^{-1}$ (7.53)
where ${\Omega}$ is a constant anti-symmetric matrix (cf. [Kapustin:1998fa]),
which defines the symplectic structure on $V={\mathbb{C}}^{2N}$. If we expand:
$\displaystyle{\Phi}(s)={\Phi}_{0}\frac{ds}{s}+\sum_{i=1}^{r}{\Phi}_{i}^{+}\frac{ds}{s-{\check{t}}_{i}^{2}}+\sum_{i=1}^{r}{\Phi}_{i}^{-}\frac{ds}{s-{\check{t}}_{i}^{-2}}\,,$
(7.54)
$\displaystyle\qquad\qquad{\Phi}_{\infty}=-{\Phi}_{0}-\sum_{i=1}^{r}({\Phi}_{i}^{+}+{\Phi}_{i}^{-})$
Then (7.53) implies:
$\displaystyle{\Phi}_{\infty}={\Omega}{\Phi}_{0}^{t}{\Omega}^{-1},\qquad{\Phi}_{i}^{+}={\Omega}({\Phi}_{i}^{-})^{t}{\Omega}^{-1},\qquad
i=1,\ldots,r$ (7.55)
Also, the matrices ${\Phi}^{+}_{i},{\Phi}^{-}_{i}$, $i=1,\ldots,r$, must have
rank one, while the matrices ${\Phi}_{0,\infty.\pm 1}$ have rank $2N$. We can
interpret
$\displaystyle{\mu}={\Phi}_{0}+{\Phi}_{\infty}+\sum_{i=1}^{r}({\Phi}_{i}^{+}+{\Phi}_{i}^{-})\,,$
(7.56)
$\displaystyle\qquad\qquad\qquad\qquad{\mu}^{t}={\Omega}^{-1}{\mu}{\Omega}$
as the moment map for the $Sp(2N)$ group action on the product of some orbits
${\mathcal{O}}_{0}\times{\mathcal{O}}_{-1}\times{\mathcal{O}}_{1}\times_{i=1}^{r}{\mathcal{O}}_{i}$
which generates the action ${\Phi}_{j}\mapsto g^{-1}{\Phi}_{j}g$ of $g\in
Sp(2N)$, such that:
$g{\Omega}g^{t}={\Omega}\ .$ (7.57)
It would be nice to develop further the theory of these orbifold Hitchin-
Gaudin systems. We shall encounter a genus one version of such theory in the
Class II $D_{r}$ section below.
The differential whose periods determine the special coordinates is equal to
$dS=x\frac{ds}{s}$ (7.58)
#### 7.2.1. Freezing example
Here we will illustrate how the $D_{4}$ theory with
$v_{1}=v_{3}=v_{4}=v,v_{2}=2v$ and $w_{1}=w_{3}=w_{4}=0,w_{2}=v$ reduces to
$A_{3}$ with $v_{1}=v_{3}=v,v_{2}=2v$ and $w_{2}=2v$ when the node 4 freezes
under $\mathfrak{q}_{4}\to 0$. Keeping in mind unfreezing to the affine
$\widehat{D}_{4}$, let polynomial $Y_{0}$ of degree $v$ denote the fundamental
matter polynomial attached to the node ‘‘2’’.
The $D_{4}$ spectral curve for the node ‘‘1’’ from (7.37) in terms of variable
$\eta$
$\eta=t+\frac{\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}}{t}$
(7.59)
where $\mathscr{Y}_{2}=Y_{0}t$ is
$\mathcal{R}_{D_{4}}(\eta,x)=\eta^{4}Y_{0}^{2}-\eta^{3}T_{1}Y_{0}+\eta^{2}\left(\mathfrak{q}_{1}T_{2}-4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}Y_{0}^{2}\right)+\\\
\eta\left(-\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}T_{3}T_{4}+4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}T_{1}Y_{0}\right)-4\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}T_{2}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{4}T_{3}^{2}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}T_{4}^{2}$
(7.60)
Notice that the curve is polynomial of degree $4$ in $\eta$ with polynomial
coefficients in $x$ of degree $2v$. In the limit $x\to\infty$ we find the
limiting values of $\eta$ are
$1+\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}+\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}+\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}+\mathfrak{q}_{1}\mathfrak{q}_{3}\mathfrak{q}_{4}$
(7.61)
Notice that the differential is
$\lambda=x\frac{dt}{t}=x\frac{d\eta}{(\eta^{2}-4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4})^{\frac{1}{2}}}$
(7.62)
Also notice that at $\eta=\pm
2\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}}$
the curve factorizes as
$\mathcal{R}_{D_{4}}(\pm
2\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}},x)=\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}(\mathfrak{q}_{3}T_{4}(x)\mp\mathfrak{q}_{4}T_{3}(x))^{2}$
(7.63)
as well as it factorizes at $\eta=\infty$
$\mathcal{R}_{D_{4}}(\eta=\infty,x)=Y_{0}(x)^{2}$ (7.64)
We can interpret the multi-valued nature of $\lambda$ on the $\eta$-plane as
the deformation of the punctured sphere underlying the $A_{r}$-type theories
to the curve describing the $D_{r}$-type theories, by opening punctures into
cuts. Perhaps one can elevate this observation to the corresponding
deformation of the Liouville theory coupled to some conformal matter, along
the lines of [Knizhnik:1987xp, Gerasimov:1988gy].
We see that in the decoupling limit $\mathfrak{q}_{4}=0$ the above curve
reduces to
$\mathcal{R}_{A_{3}}(\eta,x)=\eta^{4}Y_{0}^{2}-\eta^{3}T_{1}Y_{0}+\eta^{2}\mathfrak{q}_{1}T_{2}-\eta\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}T_{3}Y_{4}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}Y_{4}^{2}$
(7.65)
where we just set that $\mathscr{Y}_{4}$ freezes and converts to a factor of
degree $v$ contributing to the fundamental matter polynomial for the node
‘‘2’’; we denote this factor by $Y_{4}\equiv\mathscr{Y}_{4}=T_{4}$. The curve
(7.65) is precisely the $A_{3}$ curve for the node ‘‘1’’ (7.12) in terms of
the variable $\mathscr{Y}_{1}=Y_{0}\eta$. This curve corresponds to the
$GL(2)$ Hitchin system with punctures at four punctures
$1,\quad\mathfrak{q}_{1},\quad\mathfrak{q}_{1}\mathfrak{q}_{2},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}$
(7.66)
Moreover, from the discussion after (7.21) (we have
$\mathbf{w}_{0}=0,\mathbf{w}_{2}=2,\mathbf{w}_{3}=0$ and $i_{*}=2$ and
$\mathbf{w}_{+}=\mathbf{w}_{-1}=1$) it is clear the the eigenvalues of the
Higgs field residues at $\eta=0$ and at $\eta=\infty$ are doubly degenerate
which effectively means that $SL(2,\mathbb{C})$ part of the Higgs field does
not have punctures at $\eta=0$ and $\eta=\infty$. We can continue the freezing
reduction and now we shall set $\mathfrak{q}_{3}=0$ declaring the function
$\mathscr{Y}_{3}$ as contributing to the fundamental matter at the node ‘‘2’’,
we denote $\mathscr{Y}_{3}=T_{3}=Y_{3}$. After factoring out $\eta$, the curve
(7.65) reduces to the $A_{2}$ curve
$\mathcal{R}_{A_{2}}(\eta,x)=\eta^{3}Y_{0}^{2}-\eta^{2}T_{1}Y_{0}+\eta\mathfrak{q}_{1}T_{2}-\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}Y_{3}Y_{4}$
(7.67)
The corresponding Gaudin system has punctures at $\eta=0$ and $\eta=\infty$
and at
$1,\quad\mathfrak{q}_{1},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}$ (7.68)
Finally, we can freeze the node ‘‘1’’ by sending $\mathfrak{q}_{1}$ to zero
and rescaling $\eta=\tilde{\eta}\mathfrak{q}_{1}$ so that the former punctures
$\mathfrak{q}_{1},\mathfrak{q}_{1}\mathfrak{q}_{2}$ on the
$\tilde{\eta}$-plane in terms of $\tilde{\eta}$ become
$1,\quad\mathfrak{q}_{2}$ (7.69)
while the puncture $\eta=1$ is send away to $\tilde{\eta}=\infty$. We set
$Y_{1}\equiv\mathscr{Y}_{1}=T_{1}$ and find that (7.67) reduces to the
familiar $A_{1}$ curve with gauge polynomial $T_{2}$ of degree $2v$ and four
factors $(Y_{0},Y_{1},Y_{3},Y_{4})$ of degree $v$ which make fundamental
polynomial of degree $4v$
$\mathcal{R}_{A_{2}}(\eta,x)=-\tilde{\eta}^{2}Y_{1}Y_{0}+\tilde{\eta}T_{2}-\mathfrak{q}_{2}Y_{3}Y_{4}$
(7.70)
The punctures of the corresponding Gaudin model in $\tilde{\eta}$ plane are at
$(0,\mathfrak{q}_{2},1,\infty)$.
### 7.3. Class I theories of $E$ type
We are using Bourbaki conventions to label the nodes on the Dynkin graph of
$E_{r}$ series, see figures in the Appendix A. One can construct the analogues
of the spectral curves $C_{u}$ or $\Sigma_{u}$ using the minuscule
representations in the $E_{6}$ and $E_{7}$ cases. For $E_{8}$ one can
construct the spectral curve using the adjoint representation $\bf 248$.
However it seems more advantageous to use the degenerate version of del
Pezzo/$E$-bundle correspondence, which we review below in the discussion of
Class II theories of $E$ type. For the standard conformal $E_{r}$ quivers,
which are obtained by freezing of the node ‘‘0’’ in the affine $E_{r}$ quivers
with ranks ${\mathbf{v}}_{i}=Na_{i}$ where $a_{i}$ are Dynkin marks, we find
spectral curves of $(t,x)$-degree equal to $(27,6N)$ for $E_{6}$, $(56,12N)$
for $E_{7}$ and $(240,60N)$ for $E_{8}$. These degrees can be understood from
the degeneration of $\widehat{E}_{r}$ spectral curves computed in section 6.3.
#### 7.3.1. The $E_{6}$ theory
The spectral curve in the fundamental representation $R_{6}=\mathbf{27}$
associated with the node ‘‘6’’, in which the group element of the conformal
extension of $E_{6}$ is $g(x)=(\mathscr{Y}_{6}(x),\dots)$ has the form
$\mathcal{R}_{E_{6}}(t,x)=0$ (7.71)
where the explicit expression is of the form111The explicit expression, which
we do not list here, is available upon a request; it is computed by the
straightforward expansion of the exterior powers $\bigwedge^{\bullet}R_{6}$ in
the representation ring $\mathrm{Rep}(E_{6})$ over the fundamental
representations $R_{1},\dots,R_{6}$.
$\mathcal{R}_{E_{6}}(t,x)={\det}_{R_{6}}(t\cdot
1_{27}-g(x))=t^{27}-t^{26}T_{6}+t^{25}\mathscr{P}_{6}T_{5}-t^{24}\mathscr{P}_{5}\mathscr{P}_{6}^{2}T_{4}+t^{23}\left(-\mathscr{P}_{2}^{2}\mathscr{P}_{3}^{2}\mathscr{P}_{4}^{4}\mathscr{P}_{5}^{4}\mathscr{P}_{6}^{4}T_{1}^{2}+\mathscr{P}_{1}\mathscr{P}_{2}^{2}\mathscr{P}_{3}^{2}\mathscr{P}_{4}^{4}\mathscr{P}_{5}^{4}\mathscr{P}_{6}^{4}T_{3}+\mathscr{P}_{4}\mathscr{P}_{5}^{2}\mathscr{P}_{6}^{3}T_{2}T_{3}-\mathscr{P}_{2}\mathscr{P}_{3}\mathscr{P}_{4}^{2}\mathscr{P}_{5}^{2}\mathscr{P}_{6}^{3}T_{1}T_{5}+\mathscr{P}_{1}^{2}\mathscr{P}_{2}^{3}\mathscr{P}_{3}^{4}\mathscr{P}_{4}^{6}\mathscr{P}_{5}^{5}\mathscr{P}_{6}^{4}T_{6}\right)+\dots-\mathscr{P}_{1}^{18}\mathscr{P}_{2}^{27}\mathscr{P}_{3}^{36}\mathscr{P}_{4}^{54}\mathscr{P}_{5}^{45}\mathscr{P}_{6}^{36}$
(7.72)
where we have omitted the explicit expressions for the terms from $t^{24}$ to
$t^{1}$, and we omitted the dependence on $x$ in the notations for the
polynomial coefficients so that $\mathcal{P}_{i}\equiv\mathcal{P}_{i}(x)$ and
$T_{i}\equiv T_{i}(x)$. The curve 7.72 has $x$-degree $27v_{6}$, and, of
course, is not the most economical. By rescaling $g(x)\to\zeta(x)g(x)$ with a
suitably chosen $\zeta(x)$ of degree $-v_{6}$ made of some powers of the
factors in fundamental polynomials we can reduce the degree of (7.72).
The most standard conformal $E_{6}$ quiver, which arises from the degenerate
limit $\mathfrak{q}_{0}\to 0$ in the node ‘‘0’’ of the affine
$\widehat{E}_{6}$ quiver, has matter polynomial
$\mathscr{P}_{2}=\mathfrak{q}_{2}Y_{0}$ of degree $N$ only at the node ‘‘2’’
to which the affine node ‘‘0’’ was attached, while the degrees of the gauge
polynomials are fixed by the Dynkin marks ${\mathbf{v}}_{i}=Na_{i}$, that is
$(\mathbf{v}_{1},\dots,\mathbf{v}_{6})=(N,2N,2N,3N,2N,N)$. For such conformal
$E_{6}$ quiver, the curve 7.72 has canonical reduced form under the choice
$\zeta^{-1}(x)=Y_{0}(x)$ and the degree of the reduced curve is
$6N=2\mathbf{v}_{*}$ where $\mathbf{v}_{*}\equiv\mathbf{v}_{4}=3N$ denotes the
rank in the trivalent node ‘‘4’’. The reduced curve of such special conformal
$E_{6}$ quiver is $\mathcal{R}_{E_{6}}(t,x)$, with
$\mathscr{P}_{i}=\mathfrak{q}_{i},i\neq
2;\mathscr{P}_{2}=\mathfrak{q}_{2}Y_{0}$ we find
$\mathcal{R}_{E_{6}}(t,x)=t^{27}Y_{0}^{6}-t^{26}Y_{0}^{5}T_{6}+t^{25}\mathfrak{q}_{6}Y_{0}^{4}T_{5}-t^{24}\mathfrak{q}_{5}\mathfrak{q}_{6}^{2}Y_{0}^{3}T_{4}+t^{23}\left(-\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{4}\mathfrak{q}_{5}^{4}\mathfrak{q}_{6}^{4}Y_{0}^{4}T_{1}^{2}+\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{4}\mathfrak{q}_{5}^{4}\mathfrak{q}_{6}^{4}Y_{0}^{4}T_{3}+\mathfrak{q}_{4}\mathfrak{q}_{5}^{2}\mathfrak{q}_{6}^{3}Y_{0}^{2}T_{2}T_{3}-\mathfrak{q}_{2}\mathfrak{q}_{3}\mathfrak{q}_{4}^{2}\mathfrak{q}_{5}^{2}\mathfrak{q}_{6}^{3}Y_{0}^{3}T_{1}T_{5}+\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{3}\mathfrak{q}_{3}^{4}\mathfrak{q}_{4}^{6}\mathfrak{q}_{5}^{5}\mathfrak{q}_{6}^{4}Y_{0}^{5}T_{6}\right)+\dots-t^{2}\mathfrak{q}_{1}^{15}\mathfrak{q}_{2}^{23}\mathfrak{q}_{3}^{30}\mathfrak{q}_{4}^{46}\mathfrak{q}_{5}^{39}\mathfrak{q}_{6}^{32}Y_{0}^{4}T_{3}+t\mathfrak{q}_{1}^{16}\mathfrak{q}_{2}^{25}\mathfrak{q}_{3}^{33}\mathfrak{q}_{4}^{50}\mathfrak{q}_{5}^{42}\mathfrak{q}_{6}^{34}Y_{0}^{5}T_{1}-\mathfrak{q}_{1}^{18}\mathfrak{q}_{2}^{27}\mathfrak{q}_{3}^{36}\mathfrak{q}_{4}^{54}\mathfrak{q}_{5}^{45}\mathfrak{q}_{6}^{36}Y_{0}^{6}$
(7.73)
where again we only indicated the middle terms but skipped the explicit
expressions. Indeed, one sees that the curve 7.73 of the $E_{6}$ quiver with
the standard rank assignments $\mathbf{v}_{i}=Na_{i}$ has degree $6N$. At the
limit $x\to\infty$ the 27 roots of $\mathcal{R}_{E_{6}}(t,x)$ in 7.73 approach
the set of points in the $t$-plane labeled by the weights $\lambda$ in the
$\mathbf{27}$ representation of $E_{6}$ and given explicitly by
$\prod_{i=1}^{6}\mathfrak{q}_{i}^{(\lambda_{i},\lambda-\lambda_{i})}$, or
$\left\\{\ \prod_{i=1}^{6}\mathfrak{q}_{i}^{n_{i}}\ |\
\sum_{i=1}^{6}n_{i}\alpha_{i}\
=\lambda_{6}-\lambda,\quad\lambda\in\mathrm{weights}(R_{6})\right\\}$ (7.74)
where $n_{i}$ are the coefficients of the expansion in the basis of simple
roots of the difference between a given weight in $\mathbf{27}$ and the
highest weight. One can associate a Higgs field to the spectral curve 7.73
with poles in the 27 punctures (7.74) with certain relations. In other words,
the curve 7.73 realizes a certain embedding of the standard conformal $E_{6}$
quiver theory with gauge group ranks $\mathbf{v}_{i}=(N,2N,2N,3N,2N,N)$ to
some specialization of the $A_{26}$ theory with ranks $(6N,6N,\dots,6N)$, and
this embedding can be lifted to the Higgs field spectral curve representation
of (7.73).
For non-standard assignments of $\mathbf{w}_{i}$ and $\mathbf{v}_{i}$ for the
conformal $E_{6}$ quiver we did not find a simple choice of $\zeta(x)$
reducing the curve 7.72 to the minimal degree. For small ranks
$\mathbf{v}_{i},\mathbf{w}_{i}$ we can find the reduced curve using the brute
search minimization problem on the total degree of the reduced curve under
$g(x)\to\zeta(x)g(x)$. We have found different chambers in the space of
parameters $\mathbf{w}_{i},\mathbf{v}_{i}$ with piece-wise linear dependence
of the reduced degree of $\mathbf{w}_{i}$ or $\mathbf{v}_{i}$’s but not a
simple expression. For example, in several examples we find
$(w_{i})$ | $(v_{i})$ | reduced curve $x$-degree
---|---|---
$(0,4,0,0,0,0)$ | $(4,8,8,12,8,4)$ | 24
$(3,0,0,0,0,3)$ | $(6,6,9,12,9,6)$ | 33
$(6,0,0,0,0,0)$ | $(8,6,10,12,8,4)$ | 40
$(4,0,0,0,0,1)$ | $(6,5,8,10,7,4)$ | 31
$(6,0,0,0,0,3)$ | $(10,9,14,18,13,8)$ | 53
where the first three lines list different conformal $E_{6}$ quivers sharing
the same $\mathbf{v}_{*}=12$, and one can see that the curve of the minimal
degree $2\mathbf{v}_{*}$ is obtained in the standard assignment
$\mathbf{w}_{i}=0,i\neq 2$ associated to the degenerate limit of the affine
$E_{6}$.
#### 7.3.2. $E_{7}$ theory
We write the spectral curve in, for example, the $\mathbf{56}$ representation
of $E_{7}$ similar to the $E_{6}$ case. If
$(\mathbf{v}_{0},\dots,\mathbf{v}_{7})=Na_{i}$ where $a_{i}$ are Dynkin marks
of $E_{7}$ quiver, again, similar to $E_{6}$ quiver we find that the reduced
curve of the standard conformal $E_{7}$ quiver obtained from the degenerate
limit of the affine theory has $x$-degree $12N=3\mathbf{v}_{*}$ where
$\mathbf{v}_{*}=\mathbf{v}_{4}=4N$ is rank at the trivalent node. The standard
$E_{7}$ quiver spectral curve hence is realized as a specialization of the
spectral curve for $A_{55}$ quiver with ranks $(12N,12N,\dots,12N)$, or
Hitchin system with $56$ punctures on $t$-plane associated to the weights in
$\mathbf{56}$.
#### 7.3.3. $E_{8}$ theory
For $E_{8}$ the minimal representation is adjoint $\mathbf{248}$. The reduced
curve in the adjoint representation for the standard conformal $E_{8}$ quiver
obtained from the degenerate limit of the affine theory has $x$-degree
$60N=10\mathbf{v}_{*}$ where $\mathbf{v}_{*}=6N$ is rank at the trivalent
node. Hence the standard conformal $E_{8}$ quiver spectral curve is realized
as a specialization of the spectral curve for $A_{247}$ quiver with ranks
$60(N,N,\dots,N)$, or Hitchin system with $240$ punctures on $t$-plane
associated to the non-zero adjoint weights in $\mathbf{248}$.
### 7.4. Class II theories of $A$ type and class II* theories
Let us start with the simplest nontrivial examples, and then pass onto a
general case.
#### 7.4.1. Class II $\widehat{A}_{1}$ theory
For the class II theory we shift the arguments of ${\mathscr{Y}}_{i}(x)$ by
${\mu}_{i}$ to get rid of the bi-fundamental masses.
Let $g(x)\in\widehat{SL_{2}}$:
$g(x)={\mathfrak{q}}_{0}^{-{\widehat{\lambda}}_{0}^{\vee}}{\mathfrak{q}}_{1}^{-{\widehat{\lambda}}_{1}^{\vee}}{\mathscr{Y}}_{0}(x)^{{\widehat{\alpha}}_{0}^{\vee}}{\mathscr{Y}}_{1}(x)^{\widehat{\alpha}_{1}^{\vee}}$
(7.75)
We have: ${\mathfrak{q}}={\mathfrak{q}}_{0}{\mathfrak{q}}_{1}$,
$g(x)^{\alpha_{1}}=\frac{{\mathscr{Y}}_{1}^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}^{2}},\quad
g(x)^{-\delta}={\mathfrak{q}},\quad
g(x)^{\widehat{\lambda}_{0}}={\mathscr{Y}}_{0}(x)$ (7.76)
The normalized $\widehat{sl_{2}}$ characters (6.22) of the fundamental
representations ${\widehat{R}}_{0},{\widehat{R}}_{1}$ are equal to
$\displaystyle{{\mathscr{X}}}_{0}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=\,\frac{{\mathscr{Y}}_{0}(x)}{\phi({\mathfrak{q}})}\theta_{3}\left(\frac{{\mathscr{Y}}_{1}(x)^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}(x)^{2}};{\mathfrak{q}}^{2}\right)$
(7.77)
$\displaystyle{{\mathscr{X}}}_{1}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=\,\left(\frac{{\mathfrak{q}}_{1}}{\mathfrak{q}_{0}}\right)^{\frac{1}{4}}\frac{{\mathscr{Y}}_{0}(x)}{\phi({\mathfrak{q}})}\theta_{2}\left(\frac{{\mathscr{Y}}_{1}(x)^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}(x)^{2}};{\mathfrak{q}}^{2}\right)$
(see the appendix for our conventions on elliptic functions). The characters
(7.77) are invariant under the Weyl transformations
$\displaystyle{\mathscr{Y}}_{0}\to{\mathfrak{q}}_{0}{\mathscr{Y}}_{0}^{-1}{\mathscr{Y}}_{1}^{2}$
(7.78)
$\displaystyle{\mathscr{Y}}_{1}\to{\mathfrak{q}}_{1}{\mathscr{Y}}_{1}^{-1}{\mathscr{Y}}_{0}^{2}$
and therefore we can equate them to the polynomials:
$\displaystyle{{\mathscr{X}}}_{0}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=T_{0}(x)$
(7.79) $\displaystyle\qquad\qquad
T_{0,0}=\frac{\theta_{3}\left({\mathfrak{q}}_{1}^{-1};{\mathfrak{q}}^{2}\right)}{\phi({\mathfrak{q}})}$
$\displaystyle{{\mathscr{X}}}_{1}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=T_{1}(x)$
$\displaystyle\qquad\qquad
T_{1,0}=\left(\frac{{\mathfrak{q}}_{1}}{\mathfrak{q}_{0}}\right)^{\frac{1}{4}}\frac{\theta_{2}\left({\mathfrak{q}}_{1}^{-1};{\mathfrak{q}}^{2}\right)}{\phi({\mathfrak{q}})}$
The values of characters (7.77) and ${\mathfrak{q}}_{0},{\mathfrak{q}}_{1}$
define ${\mathscr{Y}}_{0}$ and ${\mathscr{Y}}_{1}$ up to an affine Weyl
transformation. To recover ${\mathscr{Y}}_{0}$ and ${\mathscr{Y}}_{1}$ we
invert the relations (7.77):
$\displaystyle{\mathscr{Y}}_{1}(x)={\mathfrak{q}}_{1}^{\frac{1}{2}}{\mathscr{Y}}_{0}(x)t$
(7.80)
$\displaystyle{\mathscr{Y}}_{0}(x)=\frac{\phi({\mathfrak{q}})}{\theta_{3}(t^{2};{\mathfrak{q}}^{2})}T_{0}(x)$
and express
$\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{{\theta}_{3}(t^{2};{\mathfrak{q}}^{2})}{{\theta}_{2}(t^{2};{\mathfrak{q}}^{2})}=\frac{T_{0}(x)}{T_{1}(x)}$
(7.81)
Actually, the ratio
${\xi}=\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{{\theta}_{3}(t^{2};{\mathfrak{q}}^{2})}{{\theta}_{2}(t^{2};{\mathfrak{q}}^{2})}$
is a meromorphic function on $\mathscr{E}$ with two first order poles at
$t=\pm\mathrm{i}$ and two simple zeroes at $t=\pm\mathrm{i}\mathfrak{q}$.
Therefore
$\xi={\xi}_{\infty}\frac{X(t,\mathfrak{q})-X_{0}}{X(t,\mathfrak{q})-X_{1}},\quad
X_{0}:=X(\mathrm{i}\mathfrak{q},\mathfrak{q}),\quad
X_{1}:=X(\mathrm{i},\mathfrak{q})$ (7.82)
with
${\xi}_{\infty}=\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{\theta_{3}(1,\mathfrak{q}^{2})}{\theta_{2}(1,\mathfrak{q}^{2})}$
(7.83)
and the explicit $\mathfrak{q}$-series for $X(t,\mathfrak{q})$ is given in
(LABEL:eq:weierx1),(LABEL:eq:weierx2). Hence, the algebraic Seiberg-Witten
curve $C_{u}$ describing the $\widehat{A}_{1}$ theory is a two-fold cover of
the rational curve ${\Sigma}_{u}$
$({\xi}_{\infty}T_{1}(x)-T_{0}(x))X-({\xi}_{\infty}T_{1}(x)X_{0}-T_{0}(x)X_{1})=0$
(7.84)
defined by the Weierstraß cubic (LABEL:eq:wxy). There are $4N$ branch points
of the $2:1$ cover $C_{u}\to{\Sigma}_{u}$:
$\displaystyle{\xi}_{\infty}T_{1}(x_{\infty,{\mathbf{a}}})-T_{0}(x_{\infty,{\mathbf{a}}})=0$
(7.85)
$\displaystyle({\xi}_{\infty}T_{1}(x_{{\alpha},{\mathbf{a}}})-T_{0}(x_{{\alpha},{\mathbf{a}}}))e_{\alpha}-({\xi}_{\infty}T_{1}(x_{{\alpha},{\mathbf{a}}})X_{0}-T_{0}(x_{{\alpha},{\mathbf{a}}})X_{1})=0$
$\displaystyle{\alpha}=1,2,3,\qquad{\alpha}=1,\ldots,N$
which can be split into $2$ groups of $N$ pairs, corresponding to the cycles
$A_{i{\mathbf{a}}}$ with $i=0,1$, e.g. $A_{0,{\mathbf{a}}}$ is a small circle
around the cut which connects $x_{1,{\mathbf{a}}}$ to $x_{2,{\mathbf{a}}}$,
while $A_{1,{\mathbf{a}}}$ is a small circle around the cut which connects
$x_{3,{\mathbf{a}}}$ to $x_{\infty,{\mathbf{a}}}$. The special coordinates are
computed by the periods of
$dS_{-}=x\,d{\log}\,(t)=x\,\frac{dX}{Y}$
The curve $C_{u}$ is the spectral curve. The cameral curve ${\mathcal{C}}_{u}$
is a $\mathbb{Z}$-cover of spectral curve $C_{u}$, which is given by the same
equations but now with $t\in{\mathbb{C}}^{\times}$ as opposed to
$t\in{\mathscr{E}}$. On cameral curve ${\mathcal{C}}_{u}$ we have the second
differential
$dS_{+}=x\,d{\log}\,{\theta}_{3}(t^{2};{\mathfrak{q}})$
which would be a multi-valued differential on spectral curve $C_{u}$ whose
periods are defined up to the periods of $dS_{-}$, similar to the
polylogarithm motives [Cartier:1987].
#### 7.4.2. Class II* $\widehat{A}_{0}$ theory
This is a (noncommutative) $U(1)$ ${\mathcal{N}}=2^{*}$ theory. This theory
was solved in [Nekrasov:2003rj] by the similar method. There is only one
amplitude ${\mathscr{Y}}(x)={\mathscr{Y}}_{0}(x)$, with the single interval
$I$ as its branch cut, the single function
$t(x)\equiv
t_{0}(x)=\frac{{\mathscr{Y}}(x)}{{\mathscr{Y}}(x+{\mathfrak{m}})}.$
with two branch cuts $I$ and $I-{\mathfrak{m}}$. Crossing the $I$ cut maps
$t(x)\mapsto{\mathfrak{q}}t(x-{\mathfrak{m}})$. Crossing the cut
$I-{\mathfrak{m}}$ has the opposite effect:
$t(x)\mapsto{\mathfrak{q}}^{-1}t(x+{\mathfrak{m}})$. The extended functions
$t_{j}(x)={\mathfrak{q}}^{j}t(x-j{\mathfrak{m}})$
The analytically continued function $t(x)$ has cuts at
$I+{\mathfrak{m}}\mathbb{Z}$. The sheets of the Riemann surface of $t(x)$ are
labeled by $j\in\mathbb{Z}$, so that on the sheet $j$ the cuts are at
$I-j{\mathfrak{m}}$, and $I-(j+1){\mathfrak{m}}$. Upon crossing
$I+j{\mathfrak{m}}$ the $t_{j}(x)$ function transforms to $t_{j+1}(x)$
function. As $x\to\infty$ on this sheet the corresponding branch of $t(x)$
approaches ${\mathfrak{q}}^{j}$. These conditions uniquely fix the inverse
function to be the logarithmic derivative of $\theta_{1}$:
$x=a+{\mathfrak{m}}\,t\frac{d}{dt}{\rm log}\,{\theta}_{1}(t;{\mathfrak{q}})$
(7.86)
#### 7.4.3. Class II $A_{r}$ theories
In order to solve the general rank $r$ theory, it is convenient to form a
linear combination of fundamental characters of ${\widehat{A}}_{r}$.
Ultimately we would like to define a regularized version of the characteristic
polynomial of $g(x)$, where, as in the general case, after the shift of the
arguments of ${\mathscr{Y}}_{i}(x)\to{\mathscr{Y}}_{i}(x+{\mu}_{i})$:
$g(x)=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}(x)^{{\widehat{\alpha}}_{i}^{\vee}}$
(7.87)
Using $t_{i}(x)=g(x)^{e_{i}}$ (see the appendix), we compute:
$t_{i}(x)={\check{t}}_{i}\,\frac{{Y}_{i}(x)}{{Y}_{i-1}(x)},\quad
i=1,\dots,r+1$ (7.88)
where we extended the amplitude functions ${\mathscr{Y}}_{j}(x)$ defined for
$j=0,\ldots,r$ to be defined for all $j\in\mathbb{Z}$ by periodicity
${Y}_{j}(x)={\mathscr{Y}}_{j+(r+1)}(x)$ and where
${\bf t}(x)=(t_{1}(x),t_{2}(x),\ldots,t_{r+1}(x))$
represents an element of the maximal torus of $SL(r+1,{\mathbb{C}})$, i.e.
$\prod_{i=1}^{r+1}t_{i}(x)=1.$
The $\check{t}_{i}$ are the asymptotic values at $x\to\infty$ of $t_{i}(x)$
and are given by
$\check{t}_{i}=(\mathfrak{q}_{i}\dots\mathfrak{q}_{r})^{-1}(\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\dots\mathfrak{q}_{r}^{r})^{\frac{1}{r+1}},\quad
i=1\dots r+1$ (7.89)
and
$g(x)^{-\delta}={\mathfrak{q}},\quad
g(x)^{\widehat{\lambda}_{0}}={\mathscr{Y}}_{0}(x).$ (7.90)
Now we shall explore the relation between the conjugacy classes in Kac-Moody
group and the holomorphic bundles on elliptic curve $\mathscr{E}$. We will
consider a family of bundles on $\mathscr{E}$ parametrized by the
$\mathbf{C}_{\left\langle x\right\rangle}$-plane, as e.g. in
[Friedman:1997ih]. We start with individual bundles.
Let V be a rank $r+1$ polystable vector bundle of degree zero over the
elliptic curve ${\mathscr{E}}=\mathbb{C}^{\times}/\mathfrak{q}^{\mathbb{Z}}$,
with trivial determinant,
${\rm det}V\approx{\mathcal{O}}_{\mathscr{E}}$
Such bundle always splits as a direct sum of line bundles
$V=\bigoplus_{i=1}^{r+1}L_{i}\ .$
Each summand is a degree zero line bundle $L_{i}$ which can be represented as
$L_{i}={\mathcal{O}}(p_{0})^{-1}{\mathcal{O}}(t_{i})$ where ${\mathcal{O}}(p)$
is the degree one line bundle whose divisor is a single point $p\in E$ and
$p_{0}$ denotes the point $t=1$ corresponding to the identity in the abelian
group law on the elliptic curve $\mathscr{E}$. A meromorphic section $s_{i}$
of $L_{i}$ with a simple pole at $t=1$ and zero at $t=t_{i}$ can be written
explicitly using the theta-functions:
$s_{i}(t)=\frac{\theta(t/t_{i};\mathfrak{q})}{\theta(t;\mathfrak{q})}$ (7.91)
and is unique up to a multiplicative constant. To each degree zero vector
bundle $V$ with the divisor
$D_{V}=-(r+1)p_{0}+t_{1}+\dots+t_{r+1}$
of ${\rm det}V$ we associate a projectively unique section $s$ of its
determinant ${\det}V$ which has zeroes at $t_{1},\dots,t_{r+1}$ and a pole of
the order not greater than $r+1$ at $t=1$:
$s(t;{\bf
t})=\prod_{i=1}^{r+1}\frac{\theta(t/t_{i};\mathfrak{q})}{\theta(t;\mathfrak{q})}$
(7.92)
where we explicitly indicate the $\bf t$ dependence of the section $s$. Now
set $t_{i}=t_{i}(x)$ given by (7.88). The meromorphic sections
$s(t;{\mathbf{t}}(x);\mathfrak{q})$ can be expanded in terms of the theta-
functions $\Theta_{j}(\mathscr{Y}_{0}(x);{\mathbf{t}};\mathfrak{q})$ and
characters of $\widehat{A}_{r}$ (see (LABEL:eq:Archar)(LABEL:eq:Ar-theta)) as
follows
$\mathscr{Y}_{0}(x)\prod_{i=1}^{r+1}\frac{\theta(t/t_{i}(x);\mathfrak{q})}{\theta(t,{\mathfrak{q}})}=\\\
\sum_{i=0}^{r}\mathfrak{q}^{-\frac{i}{2}}\mathfrak{q}^{\frac{i^{2}}{2(r+1)}}\Theta_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})\phi_{i}(t;\mathfrak{q})=\\\
=\phi(\mathfrak{q})^{r}\sum_{i=0}^{r}\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})\phi_{i}(t;\mathfrak{q})$
(7.93)
where the functions $\phi_{i}(t;\mathfrak{q})$ are normalized meromorphic
elliptic functions defined in the appendix LABEL:subsubsec:phi. Hence we find
from (6.39) and (LABEL:eq:T-matrix) that the section $s(t,x)$ (7.92) obeys
${\mathscr{Y}}_{0}(x)s(t,x)=\phi(\mathfrak{q})^{r}\sum_{i=0}^{r}\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})M_{i{\tilde{j}}}(\mathfrak{q})\tilde{\phi}_{{\tilde{j}}}(t;\mathfrak{q})$
(7.94)
where ${\tilde{\phi}}_{\tilde{j}}(t;{\mathfrak{q}})$ denotes the Weierstraß
monomials of Weierstraß elliptic functions $X(t,\mathfrak{q})$ and
$Y(t,\mathfrak{q})$; and $M_{i{\tilde{j}}}$ is a certain modular matrix as
defined in LABEL:subsubsec:phi. Recalling (6.5) that the characters
$(\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q}))$ evaluated on the
solutions $(\mathscr{Y}_{i}(x))$ are polynomials in $x$, from (6.22)(6.28) we
get
$\frac{{\mathscr{Y}}_{0}(x)s(t,x)}{\phi(\mathfrak{q})^{r}}=\sum_{i=0}^{r}\left(\prod_{j=0}^{r}\mathfrak{q}_{j}^{-\widehat{\lambda}_{i}(\widehat{\lambda}_{j}^{\vee})}\right)T_{i}(x)\sum_{{\tilde{j}}}M_{i\tilde{j}}(\mathfrak{q})\tilde{\phi}_{{\tilde{j}}}(t;\mathfrak{q})$
(7.95)
The section $s(t,x)$ vanishes at the $r+1$ points $t_{1}(x),\dots,t_{r+1}(x)$
for each $x\in\mathbf{C}_{\left\langle x\right\rangle}$, and hence defines the
$r+1$-folded spectral cover of $\mathbf{C}_{\left\langle x\right\rangle}$
plane by the equation
$R(t,x)=0$ (7.96)
where $R(t,x)$ is the right-hand side of (7.95). The curve (7.96) coincides
with the curve in [Witten:1997sc] constructed from by lifting to $M$-theory
the IIA brane arrangement realizing the elliptic model with $\mathfrak{m}=0$.
#### 7.4.4. Class II* theory
Recall that in (6.37) we defined an infinite set of functions
$Y_{i}(x),i\in\mathbb{Z}$ . The analogue of the formula (1.2) is the matrix
$g(x)\in{\widehat{GL}_{\infty}}$, (cf. (6.39)):
$g(x)=Y_{0}(x)^{K}\times{\rm
diag}\,\left(\,t_{i}(x)\,\right)_{i\in{\mathbb{Z}}},\qquad
t_{i}(x)={\check{t}}_{i}\,\frac{Y_{i}(x)}{Y_{i-1}(x)}$ (7.97)
where ${\check{t}}_{i}$, $i\in\mathbb{Z}$ solve
${\check{t}}_{i+1}={\mathfrak{q}}_{i\,{\rm mod}\,(r+1)}{\check{t}}_{i},$
and are normalized as in (6.40)
$\prod_{j=1}^{r+1}{\check{t}}_{j}=1$
so that for $i=1,\ldots,r+1$ the ${\check{t}}_{i}$ coincide with those in
(6.40), and
${\check{t}}_{i+b(r+1)}={\check{t}}_{i}{\mathfrak{q}}^{b},\qquad{\check{t}}^{[i+b(r+1)]}={\check{t}}^{[i]}\left({\mathfrak{q}}^{r+1}\right)^{\frac{b(b-1)}{2}}$
(7.98)
The fundamental characters of $\widehat{GL}_{\infty}$ evaluated on $g(x)$,
${\chi}_{i}(g(x))$ are associated with representations ${\mathcal{R}}_{i}$ of
$\widehat{GL}_{\infty}$ with the highest weight taking value (cf.
(LABEL:eq:hwgli)):
$g(x)^{\tilde{\lambda}_{i}}=\ Y_{i}(x)\ {\check{t}}^{[i]}=\ Y_{0}(x)\
t(x)^{[i]}$ (7.99)
The characters are given by the infinite sums over all partitions
${\lambda}=({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots\geq{\lambda}_{{\ell}({\lambda})}>0)$
and so are the normalized invariants
$\displaystyle{\mathscr{X}}_{i}(\\{Y_{j}(x)\\},\mathfrak{q})=$
$\displaystyle\,\frac{1}{{\check{t}}^{[i]}}{\chi}_{i}(g(x))=$ (7.100)
$\displaystyle\sum_{{\lambda}}\prod_{j=1}^{{\ell}({\lambda})}\left({\mathfrak{q}}_{i-j+1}^{[{\lambda}_{j}]}\frac{Y_{i+{\lambda}_{j}-j+1}(x)}{Y_{i+\lambda_{j}-j}(x)}\right)Y_{i-{\ell}({\lambda})}(x)$
$\displaystyle\qquad=Y_{i}(x)+{\mathfrak{q}}_{i}\frac{Y_{i+1}(x)Y_{i-1}(x)}{Y_{i}(x)}+\ldots$
where we use the notation section 1.2.
The invariant ${\mathscr{X}}_{i}$ in (7.100) is a convergent series for
$|{\mathfrak{q}}_{i}|<1$ like the theta-series, if $t_{i}(x)$ is uniformly
bounded. In fact, for the periodic chain of arguments, i.e. for
$Y_{i}(x)=Y_{i+r+1}(x)$ the $\mathfrak{gl}_{\infty}$ character (7.100) reduces
to the usual affine character of $\widehat{\mathfrak{g}\mathfrak{l}}_{r}$. The
convergence of ${\mathscr{X}}_{i}$ in the class II* case is more subtle. We
shall comment on this below. For the moment let us view the invariants as the
formal power series in $\mathfrak{q}$ with coefficients in Laurent polynomials
in $Y_{i}(x)$.
For the class II* theory the extended amplitudes $Y_{i}(x)$ are quasi-periodic
in $i$, cf. (6.38), so
${{\mathscr{X}}}_{i+r+1}\left(\\{Y_{j}(x)\\},\mathfrak{q}\right)={{\mathscr{X}}}_{i}\left(\\{Y_{j}(x-(r+1){\mathfrak{m}})\\},\mathfrak{q}\right)$
(7.101)
The cameral curve ${\mathcal{C}}_{u}$ for the class II* $A_{r}$ theory is
defined by the system of $r+1$ functional equations
${\mathscr{X}}_{i}(\,\\{\,Y_{j}(x)\\},\mathfrak{q})=T_{i}(x),\quad
i=0,\dots,r$ (7.102)
with
$\displaystyle
T_{i}(x)=T_{i,0}x^{N}+T_{i,1}x^{N-1}+\sum_{{\mathbf{a}}=2}^{N}u_{i,{\mathbf{a}}}x^{N-\mathbf{a}}\,,$
(7.103) $\displaystyle\qquad\qquad
T_{i,0}=\sum_{\lambda}\prod_{j=1}^{{\ell}({\lambda})}{\mathfrak{q}}_{i-j+1}^{[\lambda_{j}]}$
Let us now describe the II* analogue of the spectral curve, and find its
realization in terms of some version of the Hitchin’s system. Along the way we
shall get an alternative derivation of (7.95) with the benefit of getting its
Hitchin’s form as well.
We form the generating function of ${\mathscr{X}}_{i}$’s and study its
automorphic properties. The idea is to regularize the infinite product
$\prod_{i\in\mathbb{Z}}(1-t_{i}(x)/t)/(1-{\check{t}}_{i}/t)$
while keeping the same set of zeroes and poles. Thus, we define
$R(t,x)=\frac{Y_{0}(x)}{D_{0}(t;{\bf\mathfrak{q}})}\prod_{k=1}^{\infty}(1-t_{k}(x)t^{-1})(1-t\,t_{1-k}(x)^{-1})$
(7.104)
where
$\displaystyle
D_{0}(t;{\bf\mathfrak{q}})=\prod_{k=1}^{\infty}(1-{\check{t}}_{k}t^{-1})(1-t\,{\check{t}}_{1-k}^{-1})$
(7.105)
$\displaystyle\qquad\qquad\qquad=\prod_{i=1}^{r+1}\frac{{\theta}(t/{\check{t}}_{i};{\mathfrak{q}})}{{\phi}({\mathfrak{q}})}$
First of all, given that at large $x$ the eigenvalues $t_{k}(x)$ approach
${\check{t}}_{k}$ which, in turn, behave as ${\mathfrak{q}}^{\frac{k}{r+1}}$,
we expect (7.104) to define the converging product, at least for large enough
$x$.
Secondly, let us check that (7.104) is ${}^{i}{\mathcal{W}}$-invariant. Let
$i=0,\ldots,r$, ${\mathbf{a}}=1,\ldots,N$. While crossing the
$I_{i,{\mathbf{a}}}$ cut the ‘‘eigen-value’’ $t_{i}(x)$ maps to $t_{i+1}(x)$,
which, in case $i\geq 1$ or $i<0$, leaves (7.104) manifestly invariant. For
$i=0$ several factors in $\Delta(t,x)$ transform, altogether conspiring to
make it invariant:
$\displaystyle
Y_{0}(x)\mapsto{\mathfrak{q}}_{0}Y_{-1}(x)Y_{1}(x)/Y_{0}(x)=t_{1}(x)/t_{0}(x),$
(7.106) $\displaystyle(1-t_{1}(x)t^{-1})(1-t\,t_{0}(x)^{-1})\mapsto$
$\displaystyle\qquad(1-t_{0}(x)t^{-1})(1-t\,t_{1}(x)^{-1})=\frac{t_{0}(x)}{t_{1}(x)}(1-t_{1}(x)t^{-1})(1-t\,t_{0}(x)^{-1})$
Thirdly, let us introduce the analogues of the spectral determinants for all
fundamental representations ${\mathcal{R}}_{i}$:
$\displaystyle{\Delta}_{i}(t,x)=\frac{Y_{i}(x)}{D_{i}(t;{\bf\mathfrak{q}})}\prod_{k=i+1}^{\infty}(1-t_{k}(x)t^{-1})(1-t\,t_{2i+1-k}(x)^{-1})$
(7.107) $\displaystyle\qquad\qquad
D_{i}(t;{\bf\mathfrak{q}})=\prod_{k=i+1}^{\infty}(1-{\check{t}}_{k}t^{-1})(1-t\,{\check{t}}_{2i+1-k}^{-1})$
Using
$D_{i+1}(t;{\mathfrak{q}})=-t{\check{t}}_{i+1}^{-1}D_{i}(t;{\mathfrak{q}})$,
$Y_{i+1}(x)=t_{i+1}(x){\check{t}}_{i+1}^{-1}Y_{i}(x)$ we derive:
${\Delta}_{i}(t,x)=R(t,x)$ for all $i\in\mathbb{Z}$.
Then, the quasi-periodicity (7.101), (7.98) implies
$R({\mathfrak{q}}t,x+{\mathfrak{m}})={\Delta}_{r+1}(t,x)=R(t,x)$ (7.108)
Given the large $x$ asymptotics of $Y_{0}(x)$ and $t_{i}(x)$, we conclude:
$R(t,x)=x^{N}+\sum_{k=1}^{N}{\delta}_{k}(t)x^{N-k}$ (7.109)
where ${\delta}_{k}(t)$ are the quasi-elliptic functions, which have the first
order poles at $t={\check{t}}_{i}$, $i=0,\ldots r$ on the elliptic curve
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$. Indeed, the
poles come from the $D_{0}(t;{\mathfrak{q}})$ denominator, while the quasi-
ellipticity of $\delta_{k}(t)$ follows from (7.108):
${\delta}_{i}({\mathfrak{q}}t)-{\delta}_{i}(t)={\mathfrak{m}}^{i}+{\rm
polynomial\ in}\ {\mathfrak{m}}\ {\rm linear\ in}\
{\delta}_{k}({\mathfrak{q}}t),\qquad k<i$ (7.110)
Now use (LABEL:eq:fermch), (LABEL:eq:chari) to rewrite $R(t,x)$ as:
$\displaystyle R(t,x)\,=$
$\displaystyle\frac{\sum_{i\in\mathbb{Z}}(-t)^{i}{\check{t}}^{[i]}{{\mathscr{X}}}_{i}\left(\\{Y_{j}(x)\\},\mathfrak{q}\right)}{D_{0}(t;{\bf\mathfrak{q}})}$
(7.111)
$\displaystyle\qquad\qquad=\frac{1}{D_{0}(t;{\bf\mathfrak{q}})}\sum_{i\in\mathbb{Z}}(-t)^{i}{\check{t}}^{[i]}T_{i}(x)$
where we extended the definition of gauge polynomials $T_{i}(x)$ to
$i\in\mathbb{Z}$ by quasi-periodicity implied by (7.101):
$T_{i+r+1}(x)=T_{i}(x-{\mathfrak{m}})$ (7.112)
Armed with (7.112), (7.98) we reduce (7.111) to a finite sum: let
$r(t,x)=\sum_{i=0}^{r}(-t)^{i}{\check{t}}^{[i]}T_{i}(x)\,,$
then (cf. (LABEL:eq:phi_p))
$\displaystyle
R(t,x)=\frac{1}{D_{0}(t;{\mathfrak{q}})}\sum_{b\in{\mathbb{Z}}}r(t,x-b{\mathfrak{m}})\left((-t)^{b}{\mathfrak{q}}^{\frac{b(b-1)}{2}}\right)^{r+1}$
(7.113)
$\displaystyle\qquad\qquad=\frac{1}{D_{0}(t;{\mathfrak{q}})}\left({\theta}\left(-(-t)^{r+1};{\mathfrak{q}}^{r+1}\right)\ast_{{\mathfrak{m}}}r(t,x)\right)$
where the $\ast_{\hbar}$-product is defined by the usual Moyal formula:
$\left(f\ast_{\hbar}g\right)(t,x)=e^{\hbar\frac{\partial^{2}}{{\partial}{\xi}_{1}{\partial}{\eta}_{2}}-\hbar\frac{\partial^{2}}{{\partial}{\xi}_{2}{\partial}{\eta}_{1}}}|_{{\xi}={\eta}=0}\,f(t+{\eta}_{1},x+{\xi}_{2})g(t+{\eta}_{2},x+{\xi}_{2})$
(7.114)
The appearance of the $\ast$-product is the first hint that the class II*
theory has something to do with the noncommutative geometry. We shall indeed
soon see that a natural interpretation of the solution to the limit shape
equations of the class II* theory involves instantons on the noncommutative
four-manifold ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, where the
noncommutativity is ‘‘between’’ the $\mathbb{R}^{2}$ and the $\mathbb{T}^{2}$
components.
#### 7.4.5. Hitchin system on $T^{2}$
The above solution can be represented by the affine $GL(N)$ Hitchin system on
$\mathscr{E}$:
${\Phi}({\mathfrak{q}}t)={\Phi}(t)+N{\mathfrak{m}}\cdot{\bf 1}_{N}$ (7.115)
with $r+1$ rank $1$ punctures ${\check{t}}_{j}$:
$\displaystyle{\Phi}(t)\sim{\Phi}_{j}\frac{dt}{t-{\check{t}}_{j}},\qquad
j=1,\ldots,r+1$ (7.116)
$\displaystyle\qquad\qquad{\Phi}_{j}={\mathbf{u}}_{j}\otimes{\mathbf{v}}_{j}^{t},\qquad{\mathbf{u}}_{j},{\mathbf{v}}_{j}\in{\mathbb{C}}^{N}$
whose eigenvalues are fixed in terms of masses:
${\mathbf{v}}_{j}^{t}{\mathbf{u}}_{j}={\operatorname{tr}}{\Phi}_{j}=Nm_{j}\ .$
(7.117)
Actually, the vectors and covectors ${\mathbf{v}}_{j},{\mathbf{u}}_{j}$ are
defined up to the ${\mathbb{C}}^{\times}$-action
$({\mathbf{v}}_{j},{\mathbf{u}}_{j})\mapsto(z_{j}{\mathbf{v}}_{j},z_{j}^{-1}{\mathbf{u}}_{j}),\qquad
z_{j}\in{\mathbb{C}}^{\times}$ (7.118)
and (7.117) is the corresponding moment map equation, defining the coadjoint
orbit ${\mathcal{O}}_{j}$ of $SL(N,{\mathbb{C}})$. We can shift ${\Phi}(t)$ by
the meromorphic scalar matrix
${\bf\Phi}(t)={\Phi}(t)-\sum_{j=1}^{r+1}m_{j}{\xi}(t/{\check{t}}_{j})\frac{dt}{t}\,{\bf
1}_{N}\ ,$
which gives the following traceless meromorphic Higgs field (see
[Nekrasov:1995nq]):
${\bf\Phi}({t})=\left\|p_{a}{\delta}_{a}^{b}+\sum_{j=0}^{r}u_{j}^{b}v^{j}_{a}(1-{\delta}_{a}^{b})\frac{{\theta}_{1}({t}/{t}_{j}w_{b}/w_{a}){\theta}_{1}^{\prime}(1)}{{\theta}_{1}({t}/{t}_{j}){\theta}_{1}(w_{b}/w_{a})}\right\|_{a,b=1}^{N}$
(7.119)
which depends, in addition to the $SL(N,{\mathbb{C}})$-orbits
${\mathcal{O}}_{1},\ldots,{\mathcal{O}}_{r+1}$ on the choice
$(w_{1},\ldots,w_{N})$ of a holomorphic $SL(N,{\mathbb{C}})$ bundle on
$\mathscr{E}$, and the dual variables $(p_{1},\ldots,p_{N})$, subject to
$\sum_{a=1}^{N}p_{a}=0,\qquad\prod_{a=1}^{N}w_{a}=1$
There are additional constraints:
$\sum_{j=1}^{r+1}u_{j}^{a}v_{a}^{j}={\mathfrak{m}}$ (7.120)
which generate the action of the residual gauge transformations in the maximal
torus ${\bf T}=({\mathbb{C}}^{\times})^{N-1}$ of $SL(N,{\mathbb{C}})$. The
dimension of the corresponding phase space ${\mathfrak{P}}$, whose open subset
${{\mathfrak{P}}}^{\circ}$ is isomorphic to
${{\mathfrak{P}}}^{\circ}\approx\left(T^{*}\mathrm{Bun}_{SL(N,{\mathbb{C}})}({\mathscr{E}})\times\times_{j=1}^{r+1}{\mathcal{O}}_{j}\right)//{\bf
T}$ (7.121)
is equal to
${\rm dim}{{\mathfrak{P}}}=2(N-1)+(r+1)(2(N-1))-2(N-1)=2(r+1)(N-1)=2{\bf r}$
(7.122)
which is twice the dimension of the moduli space ${\mathfrak{M}}$ of vacua of
the class II* $A_{r}$ theory with the gauge group
${G_{\text{g}}}=SU(N)^{r+1}$. The remaining $r+1$ mass parameters are encoded
in the symplectic moduli of the coadjoint orbits ${\mathcal{O}}_{j}$, as
expected.
The relation to our solution is in the equality of two spectral determinants:
$R(t,x)={\rm
Det}_{N}\,\left[\left(x-\sum_{j}m_{j}{\xi}(t/t_{j})\right)\cdot{\bf
1}_{N}-{\bf\Phi}({t})\right]=0$ (7.123)
which is established by comparing the modular properties and the residues of
the left and the right hand sides.
Note the duality of the twisted periodicities of the gauge theory and
Hitchin’s system Lax operators:
$\displaystyle{\Phi}({\mathfrak{q}}t)=w^{-1}{\Phi}(t)w+{\mathfrak{m}}\cdot{\bf
1}_{N}\in{\mathfrak{sl}}(N,{\mathbb{C}})$ (7.124)
$\displaystyle{\mathfrak{q}}\cdot
g(x-{\mathfrak{m}})=S^{-1}g(x)S\in\widehat{GL}_{\infty}$
where $S$ is the shift operator $S=\sum_{i\in\mathbb{Z}}E_{i,i+r+1}$, and
$w={\rm diag}(w_{1},\ldots,w_{N})$. The Eq. (7.123) can be suggestively
written as:
${\rm Det}_{N}(x-{\Phi}(t))\approx{\rm Det}_{H}(t-g(x))$ (7.125)
where $H$ is the single-particle Hilbert space of a free fermion $\psi$.
#### 7.4.6. Relation to many-body systems and spin chains
The parameters of the spectral curve (7.123) are holomorphic functions on
${{\mathfrak{P}}}^{\circ}$, which Poisson-commute, and define the integrable
system. One way of enumerating the Hamiltonians of the integrable system is to
mimic the construction of Hamiltonians (4.22) of the higher genus Hitchin
system. For example, the quadratic Casimir is a meromorphic $2$-differential
on $\mathscr{E}$ with the fixed second order poles at $t={\check{t}}_{j}$
${\operatorname{tr}}{\bf\Phi}(t)^{2}=\sum_{j=1}^{r+1}N^{2}m_{j}^{2}{\wp}(t/{\check{t}}_{j})dt^{2}+\sum_{j=1}^{r+1}U_{2,1,j}{\xi}(t/{\check{t}}_{j})+U_{2,0}$
The Hamiltonians $U_{2,0}$, $U_{2,1,j}$ are computed explicitly in
[Nekrasov:1995nq]. They describe the motion of $N$ particles on $\mathscr{E}$
with the coordinates $w_{1},\ldots,w_{N}$, which have additional
$GL(r+1,{\mathbb{C}})$-spin degrees of freedom. However, in view of our gauge
theory analysis, it seems more natural to view this system as the
$\widehat{GL}_{\infty}$-spin chain. We conjecture that the deformation
quantization of the properly compactified phase space ${\mathfrak{P}}$ will
contain the subalgebra ${\mathcal{A}}_{\mathfrak{m}}$ of the Yangian
$Y(\widehat{GL}_{\infty})$ algebra, which is a deformation of the Yangian of
the affine ${\widehat{A}}_{r}$.
The relation of many-body systems and spin chains based on finite dimensional
symmetry groups was discussed in the context of Hecke symplectic
correspondences in [Levin:2001nm, Olshanetsky:2008uu]. One can also interpret
the results of [Felder:1995iv] as the quantum version of this correspondence.
### 7.5. Class II theories of $D$ type
In this section $\mathfrak{g_{\text{q}}}=\widehat{D}_{r}$.
The fundamental weights of $\widehat{D}_{r}$ are
$\lambda_{0},\widehat{\lambda}_{i}=a_{i}^{\vee}\lambda_{0}+\lambda_{i},i=1,\dots,r$
where $\lambda_{i}$ are fundamental weights of $D_{r}$, and Dynkin labels are
$(a_{0},\dots,a_{r})=(1,1,2,\dots,2,1,1)$ (see Appendix C.3.2).
Correspondingly,
$\displaystyle t_{1}(x)=$
$\displaystyle\,\check{t}_{1}\mathscr{Y}_{1}(x)/\mathscr{Y}_{0}(x),$ (7.126)
$\displaystyle t_{2}(x)=$
$\displaystyle\,\check{t}_{2}\mathscr{Y}_{2}(x)/(\mathscr{Y}_{1}(x)\mathscr{Y}_{0}(x)),$
$\displaystyle t_{i}(x)=$
$\displaystyle\,\check{t}_{i}\mathscr{Y}_{i}(x)/\mathscr{Y}_{i-1}(x),\quad
i=3,\ldots,r-2$ $\displaystyle t_{r-1}(x)=$
$\displaystyle\,\check{t}_{r-1}\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x)/\mathscr{Y}_{r-2}(x),$
$\displaystyle t_{r}(x)=$
$\displaystyle\,\check{t}_{r}\mathscr{Y}_{r}(x)/\mathscr{Y}_{r-1}(x)$
with
$\displaystyle\check{t}_{i}$
$\displaystyle=\left(\mathfrak{q}_{i}\mathfrak{q}_{i+1}\dots\mathfrak{q}_{r-2}\right)^{-1}\left(\mathfrak{q}_{r-1}\mathfrak{q}_{r}\right)^{-\frac{1}{2}}$
(7.127) $\displaystyle\qquad i=1,\ldots,r-2$
$\displaystyle\check{t}_{r-1}=(\mathfrak{q}_{r-1}\mathfrak{q}_{r})^{-\frac{1}{2}}\,,\quad\check{t}_{r}=(\mathfrak{q}_{r-1}/\mathfrak{q}_{r})^{\frac{1}{2}}$
There are $4$ irreducible $\widehat{D}_{r}$ highest weight modules
$\widehat{R}_{0},\widehat{R}_{1},\widehat{R}_{r-1},\widehat{R}_{r}$ at level
$1$, and $r-3$ irreducible $\widehat{D}_{r}$ highest weight modules
$\widehat{R}_{2},\dots,\widehat{R}_{r-2}$ at level $2$. In this section, to
shorten formulae, we are using not the characters of $\widehat{R}_{i}$
themselves but the closely related affine Weyl invariant functions
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ at level $2$ and
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ at level $1$ expressed terms
of theta-functions explicitly as given in the appendix (LABEL:eq:Dr-inv). Such
functions ${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ and
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ differ from the actual
characters by a simple power of Euler function $\phi(\mathfrak{q})$ and some
$\mathfrak{q}$-dependent constant, also
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{0},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{1}$
appear as a linear combination of $\widehat{R}_{0}$ and $\widehat{R}_{1}$
characters, while
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r-1}$
appear as linear combination of $\widehat{R}_{r-1}$ and $\widehat{R}_{r}$
characters (see appendix (LABEL:eq:Dr-inv)).
$\begin{cases}{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}(\mathscr{Y}_{0}(x),{\mathbf{t}(x)};\mathfrak{q})=T_{j}(x)\\\
{}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}(\mathscr{Y}_{0}(x),{\mathbf{t}(x)};\mathfrak{q})=T_{j}(x)\\\
\end{cases}$ (7.128)
where polynomials $T_{j}(x)$ are of degree $N$ for $j=0,1,r-1,r$ and of degree
$2N$ for $j=2,\dots,r-2$. so that ${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}$
is of degree $1$ in $\mathscr{Y}_{0}$ for $j=0,1,r-1,r$ and
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}$ is of degree $2$ in
$\mathscr{Y}_{0}$ for $j=2,\dots,r-2$. Also, in this section the highest
coefficient of the polynomial $T_{j}(x)$ is normalized differently then in
6.22; one can find it as theta-series evaluating (LABEL:eq:Dr-inv) on
$\check{t}_{i}$.
Using the standard embedding $so(2r)\subset sl(2r)$ we construct the algebraic
equation of the spectral curve of the $\widehat{D}_{r}$ theory as the
specialization of the spectral curve for $\widehat{A}_{2r-1}$ theory. Indeed,
a vector bundle V associated to the vector representation of $SO(2r)$ splits
as the sum of $r$ pairs of line bundles
$L_{t_{i}}\oplus L_{t_{i}^{-1}}$
with the degree zero line bundle $L_{t}$ being
$L_{t}=\mathcal{O}(p_{0})^{-1}\mathcal{O}(t)$ (7.129)
and $p_{0}\in{\mathscr{E}}$ is our friend $t=1$. Then we proceed as in
(LABEL:eq:phi_p)(LABEL:eq:s-ThetaD)(7.92) by considering a meromorphic section
of the determinant bundle ${\rm det}$V$\approx{\mathcal{O}}_{\mathscr{E}}$
$s(t,x)=\prod_{i=1}^{r}\frac{\theta(t/t_{i}(x);\mathfrak{q})}{\theta(t;\mathfrak{q})}\frac{\theta(t/t_{i}(x)^{-1};\mathfrak{q})}{\theta(t;\mathfrak{q})}$
(7.130)
From LABEL:se:phiD we find
$\mathscr{Y}_{0}^{2}s(t,x)=\sum_{i=0}^{r}\Xi_{i}(\mathscr{Y}_{0};\mathbf{t}(x);\mathfrak{q})M_{ij}(\mathfrak{q})X^{j}(t;\mathfrak{q})$
(7.131)
where $X^{j}(t,\mathfrak{q})$ are powers of Weierstraß monomials forming a
basis in the space $H^{0}_{\text{even}}({\mathscr{E}},\mathcal{O}(2rp_{0}))$
of meromorphic functions on elliptic curve symmetric under the reflection
$t\to t^{-1}$ and with a pole of order no greater then $2r$ at the origin, and
$M_{ij}(\mathfrak{q})$ is a certain modular matrix.
The linear relations (LABEL:eq:D-theta-relations) allow to express $\Xi_{i}$
in terms of
$\displaystyle\tilde{\Xi}_{i}={}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{i}\quad
i=2,\dots,r-2$ (7.132)
$\displaystyle\tilde{\Xi}_{i}=({}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{i})^{2}\quad
i=0,1,r-1,r$
as
$\Xi_{i}=\sum_{{\tilde{i}}=0}^{r}{\tilde{\Xi}}_{\tilde{i}}{\tilde{M}}_{{\tilde{i}}i}(\mathfrak{q})$
(7.133)
where $\tilde{M}_{i\tilde{i}}(\mathfrak{q})$ is a certain modular
transformation matrix. Using the character equations (7.128) the spectral
curve (7.131) turns into
$\mathscr{Y}_{0}^{2}s(t,x)=\sum_{\tilde{i},j}\tilde{T}_{\tilde{i}}(x)\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})X^{j}(t,\mathfrak{q})$
(7.134)
where
$\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})=\tilde{M}_{\tilde{i}i}(\mathfrak{q})M_{ij}(\mathfrak{q})$
and
$\displaystyle\tilde{T}_{\tilde{i}}(x)=T_{\tilde{i}}(x)\quad\tilde{i}=2,\dots,r-1$
(7.135)
$\displaystyle\tilde{T}_{\tilde{i}}(x)=(T_{\tilde{i}}(x))^{2}\quad\tilde{i}=0,1,r-1,r$
The spectral curve of the $\widehat{D}_{r}$ theory is the algebraic equation
$R(t,x)=0$ where $R(t,x)$ is the right hand side of 7.134 combined with the
Weierstraß cubic equation (LABEL:eq:wxy). The $\widehat{D}_{r}$ curve is the
specialization of the $\widehat{A}_{2r-1}$ curve in two ways. First, there are
no odd in $Y$ monomials in (7.134), and, second, the polynomial coefficients
$\tilde{T}_{\tilde{i}}(x)$ of degree $2N$ in $x$ satisfy factorization
condition: they are full squares for $\tilde{i}=0,1,r-1,r$.
To interpret the curve in Hitchin-Gaudin formalism we will rewrite it in a
slightly different form. First, notice that222Indeed, the LHS and RHS is the
meromorphic elliptic function with $2r$ zeroes at points $X_{i},Y$ and
$X_{i},-Y$ and the pole of order $2r$ at $t=1$, or $X=\infty$. Such function
is unique up to a normalization which is fixed by the asymptotics at $t=1$.
$\prod_{i=1}^{r}\frac{\theta_{1}(t/\check{t}_{i};\mathfrak{q})}{\theta_{1}(t;\mathfrak{q})}\frac{\theta_{1}(t/\check{t}_{i}^{-1};\mathfrak{q})}{\theta_{1}(t;\mathfrak{q})}=\prod_{i=1}^{r}\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})(X-\check{X}_{i})$
(7.136)
We used here the notations (LABEL:eq:weierx2), (LABEL:eq:eal) for the
Weierstraß functions and
$\displaystyle\check{X}_{i}=X(\check{t}_{i};{\mathfrak{q}}),\quad\check{Y}_{i}^{2}=4\prod_{{\alpha}=1}^{3}(\check{X}_{i}-e_{\alpha})$
Then, if we divide (7.130) by (7.136) we find333And use
$\theta_{1}(t,\mathfrak{q})$ in lieu of $\theta(t,\mathfrak{q})$ as the basic
function, so that strictly speaking there is slightly different transformation
matrix $\tilde{\tilde{M}}_{\tilde{i}j}$ compared to (7.130)(LABEL:eq:s-ThetaD)
$\mathscr{Y}_{0}^{2}(x)\prod_{i=1}^{r}\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})\times\\\
\prod_{i=1}^{r}\frac{{\theta}_{1}(t/t_{i}(x);\mathfrak{q})}{{\theta}_{1}(t/\check{t}_{i};\mathfrak{q})}\frac{{\theta}_{1}(t/t_{i}^{-1}(x);\mathfrak{q})}{{\theta}_{1}(t/\check{t}_{i}^{-1};\mathfrak{q})}=R(x,X(t,{\mathfrak{q}}))\\\
R(x,X):=\frac{\sum_{\tilde{i},j=0}^{r}\tilde{T}_{\tilde{i}}(x)\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})X^{j}}{\prod_{i=1}^{r}(X-\check{X}_{i})}$
(7.137)
Now, at the order two points on $\mathscr{E}$444e.g. the points
$(1,-1,-\mathfrak{q}^{-1/2},\mathfrak{q}^{1/2})$ in the $t$-parametrization,
where vanish the respective theta functions
$\theta_{1}(t;\mathfrak{q}),\theta_{2}(t;\mathfrak{q}),\theta_{3}(t;\mathfrak{q}),\theta_{4}(t;\mathfrak{q})$,
or, equivalently, at the four branch points in the $X$ plane:
$(\infty,e_{1},e_{2},e_{3})$, the value of the section $R(x,X)$ can be
expressed in terms of the weight $1$ invariants
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r-1},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{0},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{1}$
(compare with (LABEL:eq:Dr-inv) and (LABEL:eq:D1-theta)(LABEL:eq:D1-theta-
jacobi)), and it factorizes as
$\displaystyle R(x,X)|_{X\to\infty}=(T_{r-1}(x))^{2}$ (7.138) $\displaystyle
R(x,X)|_{X\to e_{1}}=c_{2}({\tilde{\mathfrak{q}}})\,(T_{r}(x))^{2}$
$\displaystyle R(x,X)|_{X\to
e_{2}}=c_{3}({\tilde{\mathfrak{q}}})\,(T_{0}(x))^{2}$ $\displaystyle
R(x,X)|_{X\to e_{3}}=c_{4}({\tilde{\mathfrak{q}}})\,(T_{1}(x))^{2}$
where
$c_{k}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{r}\frac{\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})}{\theta_{k}(\check{t}_{i};\mathfrak{q})\theta_{k}(\check{t}_{i}^{-1};\mathfrak{q})},\quad
k=2,3,4$ (7.139)
The Seiberg-Witten differential is given by:
${\lambda}=x\frac{dX}{Y}$ (7.140)
It is defined on the two fold cover $C_{u}$ of the curve $R(x,X)=0$, which is
a curve in the product
${\mathbb{C}\mathbb{P}}^{2}_{(X:Y:Z)}\times\mathbf{C}_{\left\langle
x\right\rangle}$, given by the equations:
$\displaystyle Y^{2}Z=4(X-e_{1}Z)(X-e_{2}Z)(X-e_{3}Z)$ (7.141) $\displaystyle
F(x,Z,X)=Z^{r}R(x,X/Z)=0$
The curve $C_{u}$ can be interpreted at the spectral curve of $GL(2N)$
Hitchin-Gaudin system on the orbifold ${\mathscr{E}}/\mathbb{Z}_{2}$, such
that at the fixed point $X=\infty,e_{1},e_{2},e_{3}$ the $GL(2N)$ system
reduces to the $Sp(2N)$ system. For more details on the Hitchin system, Nahm
transform and the brane construction of the the spectral curve for the
${\widehat{D}}_{r}$ quiver see [Kapustin:1998fa, Kapustin:1998xn]. Our main
result is the rigorous derivation of the spectral curve and its periods from
the gauge theory considerations.
#### 7.5.1. Deforming the $N_{f}=4$ SU(2) theory
The $\widehat{D}_{4}$ theory can be interpreted as the theory obtained from
gauging the flavor group of the $D_{4}$ theory with
$({\mathbf{v}}_{1},{\mathbf{v}}_{2},{\mathbf{v}}_{3},{\mathbf{v}}_{4})=(N,2N,N,N)$
theory, and with
$({\mathbf{w}}_{1},{\mathbf{w}}_{2},{\mathbf{w}}_{3},{\mathbf{w}}_{4})=(0,N,0,0)$
matter multiplets. In the limit $\mathfrak{q}_{0}\to 0$ the elliptic curve
$\mathscr{E}$ degenerates to the cylinder $\mathbb{C}_{\left\langle
t\right\rangle}^{\times}$, while Seiberg-Witten curve (7.141) degenerates to
the Seiberg-Witten curve of the $D_{4}$ theory (7.50).
Let us consider the case $N=1$. Let us parametrize the polynomials
$T_{0},T_{1},T_{3},T_{4}$ as:
$T_{i}(x)=T_{i,0}({\tilde{\mathfrak{q}}})(x-m_{i}),\qquad i=0,1,3,4$ (7.142)
and
$T_{2}(x)=T_{2,0}({\tilde{\mathfrak{q}}})(x^{2}-m_{2}x+u)$
where parameters $q_{i},m_{i}$ and $u$ are related to the microscopic
couplings ${\mathfrak{q}}_{i}$ and the $U(1)^{4}\times SU(2)$ Coulomb moduli
$\displaystyle
T_{3,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{1}(\check{t}_{i}),\quad$
$\displaystyle
T_{4,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{2}(\check{t}_{i}),$
(7.143) $\displaystyle
T_{0,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{3}(\check{t}_{i}),\quad$
$\displaystyle
T_{1,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{4}(\check{t}_{i}),$
$\displaystyle\qquad\qquad\qquad\qquad
T_{2,0}({\tilde{\mathfrak{q}}})=\Xi_{2}(1,\mathbf{\check{t}},\mathfrak{q})$
where $\check{t}_{i}$ are defined in (7.127). Then the spectral curve of the
${\widehat{D}}_{4}$ theory (7.137)(LABEL:eq:consD) has the generic form:
$R(x,X)=T_{3}^{2}(x)+\sum_{i=1}^{4}\frac{b_{i}(x)}{X-\check{X}_{i}}$ (7.144)
where $b_{i}(x)$ are some polynomials of degree $2$ that we want to relate to
the coupling constants and Coulomb parameters. The first thing to notice is
that $R(x,X)$ in (7.144) obtained from (7.137) does not have poles at
$X=\check{X}_{i}$ at $x\to\infty$ in the leading order $x^{2}$. Therefore, the
polynomials $b_{i}(x)$ are actually degree 1 polynomials containing $8$
coefficients. There are 6 linear equations on these coefficients coming from 3
factorization equations (LABEL:eq:consD) viewed as coefficients at $x^{1}$ and
$x^{0}$ (and notice that the equations at $x^{2}$ are identically satisfied
because of (7.143) and (7.139))
$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{1}-\check{X}_{i}}=c_{2}T_{4}^{2}(x)-T_{3}^{2}(x)$
(7.145)
$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{2}-\check{X}_{i}}=c_{3}T_{0}^{2}(x)-T_{3}^{2}(x)$
$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{3}-\check{X}_{i}}=c_{4}T_{1}^{2}(x)-T_{3}^{2}(x)$
The above three equations determine four linear functions $b_{i}(x)$ up to a
single linear function, which depends on two parameters
${\tilde{m}}_{2},{\tilde{u}}$:
$\tilde{b}_{j}(x)=(-1)^{j}(-\tilde{m}_{2}x+\tilde{u})\,{\rm
Det}\left\|\begin{matrix}\frac{1}{e_{a}-\check{X}_{b}}\end{matrix}\right\|_{a=1,\ldots
3}^{b=1,\ldots 4,\,b\neq j}$ (7.146)
From (7.137) it is clear that $\tilde{m}_{2},\tilde{u}$ are proportional to
$m_{2},u_{2}$. To summarize, we can describe the spectral curve (7.144) of
$\widehat{D}_{4}$ theory by the coupling constants
$\mathfrak{q}_{i},i=0,\dots,4$, which define the elliptic curve
$\mathscr{E}(\mathfrak{q})$ with modulus
$\mathfrak{q}=\mathfrak{q}_{0}\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}$
and positions of 4 punctures $\check{X}_{i}$ in the $\mathbb{C}_{\left\langle
X\right\rangle}$ plane for Weierstraß cubic using (7.127), the 4 parameters
$m_{i},i=0,1,3,4$ entering into relations (7.145) through (7.142) and 2
parameters $\tilde{m}_{2},\tilde{u}$ in (7.146).
Now consider the limit $\mathfrak{q}_{0}\to 0$ which turns the
$\widehat{D}_{4}$ class II quiver theory to the $D_{4}$ class I quiver theory.
In this limit the Weierstraß cubic degenerates: $e_{1}=-2e_{3},\quad
e_{2}=e_{3}=1/12$,
$Y^{2}=4\left(X-e_{3}\right)^{2}\left(X+2e_{3}\right)^{2}$ (7.147)
with
$X=\frac{t}{(1-t)^{2}}+\frac{1}{12},\quad Y=\frac{t(1+t)}{(1-t)^{3}}$ (7.148)
The Seiberg-Witten differential $x\frac{dX}{Y}$ becomes $x\frac{dt}{t}$. The
elliptic curve $\mathscr{E}$ degenerates to the rational curve which is the
double cover $t\mapsto X$ of the complex projective line
$\mathbb{C}\mathbb{P}^{1}_{X}$. To make contact with the Seiberg-Witten curve
of the the $D_{4}$ quiver theory it is convenient to work in the coordinate
which is related to the coordinate $X$ by rational transformation
$\eta=2+\frac{1}{X-e_{3}}=t+t^{-1}$ (7.149)
The function $\eta(X)$ is degree two meromorphic function on $\mathscr{E}$
with values at the four $\mathbb{Z}_{2}$ invariant points given by
$\eta(e_{2})=\eta(e_{3})=\infty,\quad\eta(e_{1})=-2,\quad\eta(\infty)=2$
(7.150)
Rewriting (7.137) in terms of $\eta(x)$ we find the equation of spectral curve
$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)=0$ for
$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)=\sum_{i=0}^{4}\eta^{i}p_{i}(x)$ (7.151)
where $p_{i}(x)$ are some polynomials of degree $2$ in $x$. Moreover, the
factorization conditions (LABEL:eq:consD) translates to the statement that
$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)$ is full square at $\eta=\infty$ and at
$\eta=\pm 2$ in the polynomial ring of $x$. Notice that this is precisely the
factorization conditions (7.63)(7.64) of the curve (7.60) for the $D_{4}$
quiver. (The variables $t$ and $\eta$ in the equations (7.60), (7.59)
correspond to $t$ and $\eta$ of this section multiplied by a factor
$\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}}$.)
Given the above discussion and the section 7.2.1 let us summarize the freezing
hierarchy $\widehat{D}_{4}\to D_{4}\to A_{3}\to A_{2}\to A_{1}$. For
$\widehat{D}_{4}$ theory we start with elliptic curve
$\mathscr{E}(\mathfrak{q})$ with $\mathbb{Z}_{2}$ reflection symmetry $t\to
t^{-1}$ (or $Y\to-Y$) and $8$ $\mathbb{Z}_{2}$-symmetrically located punctures
in 4 pairs $(\check{t}_{i},\check{t}_{i}^{-1})$. As we freeze
$\mathfrak{q}_{0}\to 0$, the elliptic curve $\mathscr{E}(\mathfrak{q})$
degenerates to a $\mathbb{Z}_{2}$-symmetrical cylinder
$\mathbb{C}^{\times}_{t}$ with 4 old pairs
$(\check{t}_{i},\check{t}_{i}^{-1})$ of punctures. The cylinder
$\mathbb{C}^{\times}_{t}$ double covers its $\mathbb{Z}_{2}$-quotient
$\mathbb{C}\mathbb{P}^{\times}_{\eta}$. This is the situation of $D_{4}$
quiver theory (7.60). As we freeze $\mathfrak{q}_{4}\to 0$ the second sheet of
the double cover $\mathbb{C}^{\times}_{t}\to\mathbb{C}^{\times}_{\eta}$ is
removed to infinity and we are left with $4$ punctures of $A_{3}$ quiver at
$(\mathfrak{q}_{1}^{-1},1,\mathfrak{q}_{2},\mathfrak{q}_{2}\mathfrak{q}_{3})$.555Keeping
in mind the ultimate configuration of the $A_{1}$ quiver dynamical at node “2”
we have rescaled the position of punctures by a factor of $\mathfrak{q}_{1}$.
Notice, that as discussed after (7.65) the $SL(2,\mathbb{C})$ residues of the
Higgs field vanish at the punctures in $0$ and $\infty$. As we freeze
$\mathfrak{q}_{3}\to 0$ the puncture at $\mathfrak{q}_{2}\mathfrak{q}_{3}$
(with non-trivial $SL(2,\mathbb{C})$ residue of Higgs field) merges with the
puncture $0$ and we are in the situation of the $A_{2}$ quiver with
$SL(2,\mathbb{C})$ punctures at $(\mathfrak{q}_{1}^{-1},1,\mathfrak{q}_{2},0)$
and $GL(1,\mathbb{C})$ puncture at $\infty$. Finally as we freeze
$\mathfrak{q}_{1}\to 0$ the puncture at $\mathfrak{q}_{1}^{-1}$ (with non-
trivial residue of the $SL(2,\mathbb{C})$ Higgs field) is merged with the
puncture at $\infty$ and we are left with $\mathbb{C}\mathbb{P}^{1}$ with
$SL(2,\mathbb{C})$ punctures at $(\infty,1,\mathfrak{q}_{2},0)$ for the
$A_{1}$ quiver theory defined at the dynamical node ‘‘2’’. See figure 7.3.
Figure 7.3. The freezing $\widehat{D}_{4}\to D_{4}\to A_{3}\to A_{2}\to
A_{1}$. The live nodes are denoted by red, the frozen nodes are denoted by
blue. The nodes are labeled as $i_{\mathbf{v}_{i}}$
### 7.6. Class II theories of $E$ type
The main technical tool is the natural isomorphism between the moduli space of
the $E_{k}$-bundles on elliptic curve $\mathscr{E}$ and the moduli space of
del Pezzo surfaces $\mathcal{S}_{k}$, which are obtained by blowing up $k$
points in ${\mathbb{C}}{\mathbb{P}}^{2}$, and have the fixed elliptic curve
$\mathscr{E}$ as the anticanonical divisor. The spectral curve is found using
the ‘‘cylinder map’’ [Kanev], and see [Donagi:2008kj, Donagi:2008ca,
Curio:1998bva] for applications.
Another way of encoding the geometry of the moduli space the $E_{k}$-bundles
is in the unfolding of the parabolic unimodular singularities [Arnold:1985]
${\widehat{T}}_{a,b,c}$ with
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$
which are:
$\displaystyle{\tilde{E}}_{6}=P_{8}={\widehat{T}}_{3,3,3}\,:$
$\displaystyle\,x^{3}+y^{3}+z^{3}+mxyz,\qquad\qquad m^{3}+27\neq 0,$ (7.152)
$\displaystyle{\tilde{E}}_{7}=X_{9}={\widehat{T}}_{2,4,4}\,:$
$\displaystyle\,x^{4}+y^{4}+z^{2}+mxyz,\qquad\qquad m^{4}-64\neq 0,$
$\displaystyle{\tilde{E}}_{8}=J_{10}={\widehat{T}}_{2,3,6}\,:$
$\displaystyle\,x^{6}+y^{3}+z^{2}+mxyz,\qquad\qquad 4m^{6}-432\neq 0$
We shall not pursue this direction in this work.
###### Remark.
Another important question left for future work is the connection between our
description of the special geometry via the periods of $d{\mathbb{S}}$ and the
periods of non-compact Calabi-Yau threefolds of [Katz:1997eq].
#### 7.6.1. Del Pezzo and $E_{6}$ bundles
The Del Pezzo surface
$\mathcal{S}_{6}\subset\mathbb{W}\mathbb{P}^{1,1,1,1}={\mathbb{C}\mathbb{P}}^{3}$
is a zero locus of a homogeneous degree $3$ polynomial:
${\Gamma}_{3}(X_{0},X_{1},X_{2},X_{3})=\sum_{i=0}^{3}X_{0}^{3-i}\,{\mathcal{G}}_{i}(X_{1},X_{2},X_{3})$
(7.153)
where ${\mathcal{G}}_{i}$ is the degree $i$ homogeneous polynomial in
$X_{1},X_{2},X_{3}$. In particular,
${\mathcal{G}}_{3}(X_{1},X_{2},X_{3})=-X_{1}X_{3}^{2}+4X_{2}^{3}-g_{2}X_{1}^{2}X_{2}-g_{3}X_{1}^{3}$
(7.154)
defines the elliptic curve ${\mathscr{E}}$, which determines the gauge
coupling $\mathfrak{q}={\exp}\,2\pi\mathrm{i}\tau$, cf. (LABEL:eq:wxy):
${\tau}=\frac{\oint_{B}dX/Y}{\oint_{A}dX/Y}\,,\qquad
X=X_{2}/X_{1},\,Y=X_{3}/X_{1}$ (7.155)
The rest of the coefficient functions ${\mathcal{G}}_{0,1,2}$ is parametrized
as follows:
$\displaystyle{\mathcal{G}}_{2}(X_{1},X_{2},X_{3})=p_{0}X_{1}^{2}+p_{1}X_{1}X_{2}+p_{6}X_{2}X_{3}$
(7.156)
$\displaystyle{\mathcal{G}}_{1}(X_{1},X_{2},X_{3})=p_{2}X_{1}+p_{3}X_{2}+p_{5}X_{3}$
(7.157) $\displaystyle{\mathcal{G}}_{0}(X_{1},X_{2},X_{3})=p_{4}$ (7.158)
The isomorphism classes of $\mathcal{S}_{6}$ surfaces containing the fixed
elliptic curve ${\mathscr{E}}$ are in one-to-one correspondence with the
points
$[p]=(p_{0}:p_{1}:p_{6}:p_{2}:p_{3}:p_{5}:p_{4})\in{\mathcal{M}}$ (7.159)
in the weighted projective space
${\mathcal{M}}={\mathbb{W}\mathbb{P}}^{1,1,1,2,2,2,3}$, which is also
isomorphic, by E.Loojienga’s theorem [Loojienga:1976], to the moduli space
$\mathrm{Bun}_{E_{6}({\mathbb{C}})}^{ss}(\mathscr{E})$ of holomorphic semi-
stable principal $E_{6}$-bundles on $\mathscr{E}$. We label the projective
coordinates $p_{i}$ in such a way that the projective weight of $p_{i}$ equals
Dynkin mark $a_{i}$ in our conventions Appendix A. The correspondence between
the $E_{6}$-bundles on $\mathscr{E}$ and the del Pezzo surfaces
$\mathcal{S}_{6}$ is geometric: there are precisely $27$ degree $1$ rational
curves (‘‘the $(-1)$-lines’’) $C_{a}$ on $\mathcal{S}_{6}$, $a=1,\ldots,27$,
which are the divisors of the line bundles ${\mathscr{L}}_{a}$ on
$\mathcal{S}_{6}$. The direct sum
${\mathcal{U}}=\bigoplus_{a=1}^{27}{\mathscr{L}}_{a}\,,$ (7.160)
has no infinitesimal deformations, as a bundle on $\mathcal{S}_{6}$. The
mapping class group of $\mathcal{S}_{6}$ acts on the $(-1)$-lines by the
$E_{6}$ Weyl transformations. As a result, the bundle ${\mathcal{U}}$ is a
vector bundle associated to a canonical principal $E_{6}({\mathbb{C}})$-bundle
${\mathcal{P}}_{\mathcal{S}_{6}}$ over $\mathcal{S}_{6}$ with the help of a
${\bf 27}$ representation:
${\mathcal{U}}={\mathcal{P}}_{\mathcal{S}_{6}}\times_{E_{6}({\mathbb{C}})}{\bf
27}$ (7.161)
The restriction of ${\mathcal{P}}_{\mathcal{S}_{6}}|_{E}$ is the holomorphic
principal $E_{6}({\mathbb{C}})$ bundle over $E$ which corresponds to the point
$[s]$ in Loojienga’s theorem. Again, the associated rank $27$ vector bundle
${\mathcal{U}}|_{E}$ splits
${\mathcal{U}}|_{\mathscr{E}}=\bigoplus_{a=1}^{27}{\mathscr{L}}_{a}$ (7.162)
The line subbundles ${\mathscr{L}}_{a}$ can be expressed as:
${\mathscr{L}}_{a}=\bigotimes_{i=1}^{6}\,{\mathbb{L}}_{i}^{w_{a,i}}$ (7.163)
where $w_{a,i}$, $i=1,\ldots,6$, $a=1,\ldots,27$ are the components of the
weight vector. The line bundles ${\mathbb{L}}_{i}$, $i=1,\ldots,6$ are defined
up to the action of the $E_{6}$ Weyl group. Let us now compute the
${\mathscr{L}}_{a}$’s. The rational curve of degree one in $\mathcal{S}_{6}$
is a rational curve of degree one in ${\mathbb{C}\mathbb{P}}^{3}$ which is
contained in $\mathcal{S}_{6}$. A parametrized rational curve of degree one in
${\mathbb{C}\mathbb{P}}^{3}$ is a collection of $4$ linear functions:
${\zeta}\mapsto{\mathbf{X}}({\zeta})$,
${\mathbf{X}}({\zeta})=\left(X_{0}+{\zeta}v_{0},X_{1}+{\zeta}v_{1},X_{2}+{\zeta}v_{2},X_{3}+{\zeta}v_{3}\right)$
(7.164)
The two quadruples
${\mathbf{X}}({\zeta})\qquad{\rm
and}\qquad(c{\zeta}+d)\,{\mathbf{X}}\left(\frac{a{\zeta}+b}{c{\zeta}+d}\right)$
for
$\left(\begin{matrix}a&b\\\ c&d\\\ \end{matrix}\right)\in{\rm
GL}_{2}({\mathbb{C}})$ (7.165)
define identical curves in ${\mathbb{C}\mathbb{P}}^{3}$. We can fix the
GL${}_{2}({\mathbb{C}})$ gauge by choosing the parameter $\zeta$ so that:
${\mathbf{X}}({\zeta})=\left({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}\right)$
(7.166)
The requirement that the curve lands in
$\mathcal{S}_{6}\subset{\mathbb{C}\mathbb{P}}^{3}$ reads as
${\Gamma}_{3}\left({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}\right)=\sum_{i=0}^{3}{\zeta}^{i}{\Xi}_{i}(X,Y;v_{X},v_{Y})\equiv
0$ (7.167)
which is a system of $4$ equations
${\Xi}_{i}(X,Y;v_{X},v_{Y})=0,\qquad i=0,\ldots,3$
on $4$ unknowns $X,Y,v_{X},v_{Y}$:
$\displaystyle\Xi_{0}$ $\displaystyle=-Y^{2}+4X^{3}-g_{2}X-g_{3}$
$\displaystyle\Xi_{1}$
$\displaystyle=-g_{2}v_{X}+p_{6}XY+p_{1}X+p_{0}+12X^{2}v_{X}-2Yv_{Y}$
$\displaystyle\Xi_{2}$
$\displaystyle=p_{6}Yv_{X}+p_{1}v_{X}+p_{6}Xv_{Y}+p_{3}X+p_{5}Y+p_{2}+12Xv_{X}^{2}-v_{Y}^{2}$
$\displaystyle\Xi_{3}$
$\displaystyle=p_{6}v_{X}v_{Y}+p_{3}v_{X}+p_{5}v_{Y}+p_{4}+4v_{X}^{3}$
The equation $\Xi_{0}=0$ in the above system is the equation of the elliptic
curve $\mathscr{E}$. To find the equation of the spectral cover associated
with the vector bundle $\mathcal{U}|_{\mathscr{E}}$ in the $\mathbf{27}$
representation we can express $v_{Y}$ from the equation $\Xi_{1}=0$, then plug
it into the remaining equations $\Xi_{2}=0$ and $\Xi_{3}=0$, compute the
resultant of these two polynomials with respect to the variable $v_{X}$,
reduce modulo the equation $\Xi_{0}=0$ defining the elliptic curve
$\mathscr{E}$, arriving at:
$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})=-4Y^{4}\mathrm{res}_{v_{X}}(\Xi_{2}|_{v_{Y}:\Xi_{1}=0},\Xi_{3}|_{v_{Y}:\Xi_{1}=0})\mod\Xi_{0}$
(7.168)
The resultant $C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})$ is a polynomial
in $X,Y$ with polynomial coefficients in $g_{2},g_{3},p_{0},\dots,p_{6}$ of
the form
$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})=\\\
(p_{0}^{6}+\dots)+(6p_{0}^{5}p_{1}+\dots)X+\dots+(-256p_{3}^{3}+\dots)X^{12}\\\
+(12g_{3}p_{0}^{4}p_{5}+\dots)Y+(32g_{3}p_{0}^{4}p_{5}+\dots)XY+\dots+(-256p_{5}^{3}+\dots)X^{12}Y$
(7.169)
(A short `Mathematica` version of this formula is given in appendix
LABEL:sec:_E6-delPezzo-code.) Now let us imagine having a family
${\mathcal{U}}$ of the $E_{6}$-bundles on $E$.
In our solution the vacuum $u$ of the gauge theory is identified with the
degree $N$ quasimap:
$\displaystyle p:{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}\to\mathrm{Bun}_{E_{6}(\mathbb{C})}(\mathscr{E})\simeq\mathbb{W}\mathbb{P}^{1,1,1,2,2,2,3}$
given by the polynomials $p_{i}(x)$ of degree $Na_{i}$
$p_{i}=p_{i}(x),\quad i=0,\dots,6$ (7.170)
Together with the equation of the Weierstraß cubic
$\Xi_{0}(X,Y,g_{2},g_{3})=0$, the equation
$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0}(x),\dots,p_{6}(x))=0$ (7.171)
defines the Seiberg-Witten curve of the affine $E_{6}$ theory as an algebraic
curve. Given that the degree of $X$ on $\mathscr{E}$ is $2$ and the degree $Y$
on $\mathscr{E}$ is $3$ the polynomial $C^{E_{6}}$ is of degree $27$, i.e. the
equation $C^{E_{6}}=0$ defines $27$ points on the elliptic curve
$\mathscr{E}$.
The top degree coefficients of the polynomials $p_{i}(x)$ are determined
explicitly in terms of the coupling constants $\mathfrak{q}_{i}$. Indeed, the
$E_{6}$ characters, or more conveniently in the present case, the $E_{6}$
theta-functions, $c_{i}(\mathscr{Y}_{0},\mathbf{t},\mathfrak{q})$ as set in
(LABEL:eq:c-def), define a set of projective coordinates on
$\mathbb{W}\mathbb{P}^{1,1,1,2,2,2,3}$ (which differs slightly from the set
$({\widehat{\chi}}_{i})_{i=0}^{6}$):
$(c_{0}:c_{1}:c_{6}:c_{2}:c_{3}:c_{5}:c_{4})$ (7.172)
In these coordinates the solution of the theory has the canonical form (6.28)
$c_{i}(\mathscr{Y}_{0},\mathbf{t},\mathfrak{q})=\left(\prod_{j=0}^{r}\mathfrak{q}_{j}^{-\lambda_{i}(\lambda_{j}^{\vee})}\right)\,T_{i}(x)$
(7.173)
The ‘‘del Pezzo projective coordinates’’ $(p_{i})_{i=0}^{6}$ are related to
the theta-function coordinates $(c_{i})$ on
$\mathrm{Bun}_{E_{6}(\mathbb{C})}^{ss}(\mathscr{E})$ by a polynomial map of
the form
$p_{i}=\sum_{j_{1}\leq j_{2}\leq
j_{3}\dots}M_{i,\\{j_{1},j_{2},j_{3},\dots\\}}(\mathfrak{q})c_{j_{1}}c_{j_{2}}c_{j_{3}}\dots$
(7.174)
where $M_{i,\\{j_{1},j_{2},\dots,\\}}(\mathfrak{q})$ is certain modular
transformation matrix. This matrix can be explicitly computed by comparing the
spectral curve (7.168) and the $\widehat{A}_{26}$ spectral curve (7.95)
specialized to the the embedding $\widehat{E}_{6}\subset\widehat{A}_{26}$ by
fundamental representation. The coefficients
$M_{i,\\{j_{1},j_{2},j_{3},\dots\\}}$ are modular forms for modular group
$\Gamma(6)$ with a certain modular weights that can computed by observation
that the weights of variables $(X_{0},X_{1},X_{2},X_{3})$ under the modular
transformation $\tau\to-\tau^{-1}$ on $\mathscr{E}(\mathfrak{q})$ for
$\mathfrak{q}=\exp(2\pi\mathrm{i}\tau)$ are
$\begin{pmatrix}X_{0}&X_{1}&X_{2}&X_{3}\\\ 6&1&2&3\end{pmatrix}$ (7.175)
This implies that the modular weights of $p_{i}$ are
$\begin{pmatrix}&p_{0}&p_{1}&p_{6}&p_{2}&p_{3}&p_{5}&p_{4}\\\
&0&-2&-5&-6&-8&-9&-12\end{pmatrix}$ (7.176)
The $(c_{i})$ have modular weight $3$ because they are rank $6$ lattice theta-
functions. From this assignment of weights one finds the modular weights of
all coefficients $M_{i,\\{j_{1},\dots,j_{k}\\}}(\mathfrak{q})$; for example
$M_{4,\\{4\\}}$ has modular weight $15$. The space of modular forms for
$\Gamma(N)$ of a given weight $k$ is a finite dimensional vector space. (For
any integer $k>0$ the dimension is $k+1$ for $\Gamma(3)$ and $6k$ for
$\Gamma(6)$). Matching a finite number of the coefficients in the
$\mathfrak{q}$ expansion one finds explicitly the modular coefficients
$M_{i,\\{j_{1},\dots,\\}}(\mathfrak{q})$, see appendix LABEL:sec:E6-modular-
matrix.
#### 7.6.2. Del Pezzo and $E_{7}$ bundles
The story for the $E_{7}$-bundles is similar. There is a family of del Pezzo
surfaces $\mathcal{S}_{7}\subset{\mathbb{W}}{{\mathbb{P}}}^{1,1,1,2}$,
described by the degree $4$ equation
${\Gamma}_{4}(X_{0},X_{1},X_{2},X_{3})=\sum_{i=0}^{4}X_{0}^{i}\,{\mathcal{G}}_{4-i}(X_{1},X_{2},X_{3})$
(7.177)
with
${\mathcal{G}}_{4}(X_{1},X_{2},X_{3})=-X_{3}^{2}+4X_{1}X_{2}^{3}-g_{2}X_{1}^{3}X_{2}-g_{3}X_{1}^{4}(X_{3}^{2}-4X_{1}X_{2}^{3}+g_{2}X_{2}X_{1}^{3}+g_{3}X_{1}^{4})$
(7.178)
and
$\displaystyle{\mathcal{G}}_{3}(X_{1},X_{2},X_{3})=p_{0}X_{1}^{3}+p_{7}X_{1}^{2}X_{2}(a_{1}X_{1}^{3}+a_{2}X_{2}X_{1}^{2})$
$\displaystyle{\mathcal{G}}_{2}(X_{1},X_{2},X_{3})=p_{1}X_{1}^{2}+p_{2}X_{1}X_{2}+p_{6}X_{2}^{2}(b_{1}X_{1}^{2}+b_{2}X_{1}X_{2}+b_{3}X_{2}^{2})$
$\displaystyle{\mathcal{G}}_{1}(X_{1},X_{2},X_{3})=p_{3}X_{1}+p_{5}X_{2}(c_{1}X_{1}+c_{2}X_{2})$
$\displaystyle{\mathcal{G}}_{0}=p_{4}$
Again, the divisor $X_{0}=0$ is the elliptic curve ${\mathscr{E}}$, which is
realized as a zero locus of the degree $4$ polynomial equation
${\mathcal{G}}_{4}(X_{1},X_{2},X_{3})=0$ in
${\mathbb{W}}{\mathbb{P}}^{1,1,2}$.
The isomorphism classes of $\mathcal{S}_{7}$, containing the fixed elliptic
curve ${\mathscr{E}}$, are in one-to-one correspondence with the points
$[p]=(p_{0}:p_{7}:p_{1}:p_{2}:p_{5}:p_{3}:p_{5}:p_{4})\in{\mathbb{W}\mathbb{P}}^{1,1,2,2,2,3,3,4}$
(7.179)
Again, we study the ‘‘(-1)-curves’’, which in a particular gauge look like:
${\mathbf{x}}({\zeta})=({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y})$
(7.180)
where $(X,Y;v_{X},v_{Y},w_{Y})$ obey a system of $5$ equations
${\Xi}_{i}(X,Y;v_{X},v_{Y},w_{Y})=0$, $i=0,\ldots,4$:
${\Gamma}_{4}({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y})=\sum_{i=0}^{4}{\zeta}^{i}\,{\Xi}_{i}(X,Y;v_{X},v_{Y},w_{Y})$
(7.181)
where
$\displaystyle\Xi_{0}=-Y^{2}+4X^{3}-g_{2}Y-g_{3}$ (7.182)
$\displaystyle\Xi_{1}=p_{0}+Xp_{7}+12X^{2}v_{X}-g_{2}v_{X}-2Yv_{Y}$
$\displaystyle\Xi_{2}=p_{1}+Xp_{2}+X^{2}p_{6}+p_{7}v_{X}+12Xv_{X}^{2}-v_{Y}^{2}-Yw_{Y}$
$\displaystyle\Xi_{3}=p_{3}+Xp_{5}+p_{2}v_{X}+2Xp_{6}v_{X}+4v_{X}^{3}-v_{Y}w_{Y}$
$\displaystyle\Xi_{4}=p_{4}+p_{5}v_{X}+p_{6}v_{X}^{2}-\frac{w_{Y}^{2}}{4}$
We proceed similarly to the $E_{6}$ case: we solve for $v_{Y}$ and $w_{Y}$
from the equations $\Xi_{1}$ and $\Xi_{2}$, plug the solution into the
polynomial $\Xi_{3}$ and $\Xi_{4}$ and compute the resultant
$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{7})=\\\
-2^{10}Y^{18}\mathrm{res}_{v_{X}}(\Xi_{3}|_{v_{Y},w_{Y}:\Xi_{1,2}=0},\Xi_{4}|_{v_{Y},w_{Y}:\Xi_{1,2}=0})\mod\Xi_{0}$
(7.183)
which has the structure of the degree $28$ polynomial in $X$ with the
coefficients polynomial in $(p_{0},\dots,p_{7})$ of total degree 12:
$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{7})=(p_{0}^{12}+24g_{3}p_{0}^{10}p_{1}+\dots)+(24g_{2}p_{0}^{10}p_{1}+\dots)X+\\\
+\dots+(2^{16}p_{5}^{4}-2^{19}p_{4}p_{5}^{2}p_{6}+2^{20}p_{4}^{2}p_{6}^{2})X^{28}$
(7.184)
(A short `Mathematica` version of this formula is given in appendix
LABEL:sec:_E7-delPezzo-code.) The polynomial
$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0}(x),\dots,p_{7}(x))$ together with the
Weierstraß cubic $\Xi_{0}(X,Y;g_{2},g_{3})$ defines the algebraic Seiberg-
Witten curve for $E_{7}$ quiver theory. Since on the elliptic curve
$\mathscr{E}$ the degree of $X$ is $2$, at each $x\in\mathbf{C}_{\left\langle
x\right\rangle}$ the spectral curve (7.184) defines $58$ points on
$\mathscr{E}$ encoding the vector bundle in the $\mathbf{58}$ representation
of $E_{7}$.
The relation between the del Pezzo parametrization $(p_{i})$ and the theta-
function parametrization $(c_{i})$ of
$\mathrm{Bun}_{E_{7}(\mathbb{C})}^{ss}(\mathscr{E})$ can be in principle
written in terms of a certain modular matrix $M(\mathfrak{q})$, as in the the
$E_{6}$ theory LABEL:sec:E6-modular-matrix. We do not record this
transformation in this work.
#### 7.6.3. Del Pezzo and $E_{8}$ bundles
To get an effective description of $E_{8}$-bundles we study the family of del
Pezzo surfaces ${\mathcal{S}}_{8}\subset{\mathbb{W}}{\mathbb{P}}^{1,1,2,3}$:
$\displaystyle{\Gamma}_{6}(X_{0},X_{1},X_{2},X_{3})=$ $\displaystyle\ =\
-X_{3}^{2}+4X_{2}^{3}-X_{2}G_{2}(X_{0},X_{1})-G_{3}(X_{0},X_{1})=0$
$\displaystyle\qquad\qquad
G_{2}(X_{0},X_{1})=g_{2}X_{1}^{4}+\sum_{j=1}^{4}a_{j}X_{0}^{j}X_{1}^{4-j}$
$\displaystyle\qquad\qquad
G_{3}(X_{0},X_{1})=g_{3}X_{1}^{6}+\sum_{j=2}^{6}b_{j}X_{0}^{j}X_{1}^{6-j}$
The isomorphism classes of the del Pezzo surfaces ${\mathcal{S}}_{8}$
containing the fixed elliptic curve
$Y^{2}=4X^{3}-g_{2}X-g_{3}$
are parametrized by:
$[s]=(a_{1}:a_{2}:b_{2}:a_{3}:b_{3}:a_{4}:b_{4}:b_{5}:b_{6})\in{\mathbb{W}\mathbb{P}}^{1,2,2,3,3,4,4,5,6}$
(7.185)
The ‘‘-1’’-curves in ${\mathcal{S}}_{8}$ are described by the parametrizations
${\mathbf{x}}({\zeta})=({\zeta},1,X+{\zeta}v_{X}+\frac{1}{2}{\zeta}^{2}w_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y}+\frac{1}{6}{\zeta}^{3}u_{Y})$,
where $(X,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})$ to be found from the equations
${\Xi}_{i}(Y,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})=0,\qquad i=0,\ldots,6$ (7.186)
where
${\Gamma}_{6}({\zeta},1,X+{\zeta}v_{X}+\frac{1}{2}{\zeta}^{2}w_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y}+\frac{1}{6}{\zeta}^{3}u_{Y})=$
$\qquad=\sum_{i=0}^{6}{\zeta}^{i}{\Xi}_{i}(X,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})$
To find explicitly the equation of affine $E_{8}$ spectral curve, one shall
proceed in spirit similarly to the $E_{6},E_{7}$ cases considered above.
However the explicit computation becomes much more tedious as the minimal
representation of $E_{8}$ is $\bf{248}$, and the expected $x$-degree of the
curve is $60N$ (see below). We leave this task for future investigation.
## Chapter 8 The integrable systems of monopoles and instantons
As we reviewed above, the ${\mathcal{N}}=2$ gauge theory compactified on a
circle ${\mathbb{S}}^{1}$ becomes, at low energy, the ${\mathcal{N}}=4$
supersymmetric sigma model with the hyperkäbler target space ${\mathfrak{P}}$.
The triplet of complex structures on ${\mathfrak{P}}$ is in correspondence
with the choices of a supercharge ${\mathcal{Q}}$, which is nilpotent up to an
infinitesimal translation along ${\mathbb{S}}^{1}$. The one supercharge which
is nilpotent even in the decompactified theory (it corresponds to the
topological supercharge of the Donaldson theory, for pure ${\mathcal{N}}=2$
super-Yang-Mills theory) corresponds to the complex structure $\bf I$. In this
complex structure ${{\mathfrak{P}}}$ has the structure of an algebraic
integrable system $({{\mathfrak{P}}},{\Omega},h)$:
$h:{{\mathfrak{P}}}\longrightarrow{{\mathfrak{M}}},\qquad{\Omega}|_{h^{-1}(u)}=0,\
u\in{{\mathfrak{M}}}$ (8.1)
The $\bf I$-holomorphic $(2,0)$ form ${\Omega}$ is the form which we
previously denoted by ${\Omega}_{\bf I}$.
We shall now describe these systems for the class I, II and II* theories we
studied so far. For some theories several presentations of the same integrable
system are possible.
In all cases we study the phase spaces ${\mathfrak{P}}$ have parameters
corresponding to the masses $m$ of the matter field in the gauge theory. The
cohomology class of $[{\Omega}]$ is linear in $m$. An explanation of this fact
in the symplectic geometry is the existence of a ‘‘larger’’ symplectic
manifold ${\mathfrak{P}}^{\mathrm{ext}}$ with the holomorphic Hamiltonian
torus ${\mathbb{T}}=({\mathbb{C}}^{\times})^{M}$ action, whose holomorphic
symplectic quotient of ${\mathfrak{P}}^{\mathrm{ext}}$ at some level $m$ of
the moment map produces ${\mathfrak{P}}$.
The explanation in gauge theory is the three dimensional mirror symmetry. Our
phase space ${\mathfrak{P}}$ is the Coulomb branch of the three dimensional
${\mathcal{N}}=4$ gauge theory, obtained by the ${\mathbb{S}}^{1}$
compactification of the four dimensional ${\mathcal{N}}=2$ theory. The masses
of the matter fields are the vacuum expectation values of the scalars in the
vector multiplet. Under the three dimensional mirror symmetry
[Intriligator:1996ex] these are exchanged with the Fayet-Illiopoulos terms,
which are the levels of the three moment maps in the hyperkähler quotient
construction [Hitchin:1986ea] of the Higgs branch of the mirror theory.
It is amusing to identify ${\mathfrak{P}}^{\mathrm{ext}}$ and the action of
the torus in the examples below. We shall treat the case of the class II
theories in some detail, leaving other examples to the interested reader.
### 8.1. Periodic Monopoles and the phase space of Class I theories
We shall now demonstrate that for the class I theories the phase space
${\mathfrak{P}}$ is the moduli space of the charge ${\mathbf{v}}$
$G$-monopoles on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with Dirac
singularities, whose location and the embedding of the Dirac $U(1)$-monopoles
into $G$ is parametrized by ${\mathbf{w}}$ and the masses
$m_{i,{\mathfrak{f}}}$.
Let us discuss the monopole moduli space in more detail.
The ordinary $G$-monopoles are the solutions of Bogomolny equation on
${\mathbb{R}}^{3}$:
$D_{A}{\phi}+\star F_{A}=0$ (8.2)
with finite $L^{2}$-energy:
${\mathcal{E}}(A,{\phi})=\int_{{\mathbb{R}}^{3}}\langle F_{A},\star
F_{A}\rangle+\langle D_{A}{\phi},\star D_{A}{\phi}\rangle$ (8.3)
One shows that as ${\vec{x}}\to\infty$, the conjugacy class of
$\phi({\vec{x}})$ approaches a fixed value. Equivalently,
${\phi}({\vec{x}})\longrightarrow
g^{-1}({\vec{x}})\,{\phi}_{\infty}\,g({\vec{x}})$, for some fixed
${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}$, the Cartan subalgebra of the
maximal compact subgroup $G$. Actually,
${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}/W({\mathfrak{g}})$, but, since
${\mathbb{S}}^{2}_{\infty}$ is simply connected, one can choose a uniform
representative ${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}$. This lift from
$\mathfrak{h}_{\mathbb{R}}/W({\mathfrak{g}})$ to $\mathfrak{h}_{\mathbb{R}}$
is going to be trickier in the case of periodic monopoles we shall study
below.
Suppose $\phi_{\infty}$ is generic, i.e. the only gauge transformations which
commute with it belong to the normalizer $N(T)$ of a maximal torus $T$. The
restriction of $\phi$ onto a two-sphere ${\mathbb{S}}^{2}_{\infty}$ of a very
large radius defines a map:
${\varphi}:{\mathbb{S}}^{2}_{\infty}\longrightarrow G/T$ (8.4)
and a $T$-subbundle $\mathbb{T}$ of the trivial $G$-bundle
$P=G\times{\mathbb{S}}^{2}_{\infty}$. The latter is characterized by its Chern
classes, which can be also identified with the class $[{\varphi}]$ of
${\varphi}$ in
${\pi}_{2}(G/T)\approx{\pi}_{1}(T)\approx{{\rm Q}}^{\vee}\,,$
also known as the magnetic charges of the monopole solution. The magnetic
charge can also be read off the solution of (8.2) by projecting the curvature
$F_{A}$ to the Cartan subalgebra $\mathfrak{h}$ defined by
$\phi\biggr{|}_{{\mathbb{S}}^{2}_{\infty}}$, and by taking the corresponding
integrals:
${\bf
m}_{\varphi}=\frac{1}{2\pi\mathrm{i}}\int_{{\mathbb{S}}^{2}_{\infty}}F_{A}^{\mathfrak{h}}$
(8.5)
Now let us compactify one of the spatial directions, i.e. replace
${\mathbb{R}}^{3}$ by $M^{3}={\mathbb{S}}^{1}\times{\mathbb{R}}^{2}$. Let
$\psi\in[0,2\pi)$ be the angular coordinate on $\mathbb{S}^{1}$ and let
$x=x_{1}+\mathrm{i}x_{2}$ be the complex coordinate on $\mathbb{R}^{2}$. Let
us normalize the metric on $M^{3}$ so that the circumference of
$\mathbb{S}^{1}$ is equal to one.
Consider the complex connection in the $\mathbb{S}^{1}$ direction (we use the
physical convention where $A$ is represented by Hermitian matrices, i.e. by a
real-valued one-form for the $U(1)$ gauge group):
${\nabla}={\partial}_{\psi}+\mathrm{i}A_{\psi}-{\phi}$ (8.6)
The Eq. (8.2) implies that the ${{\bar{x}}}$-variation of $\nabla$ is an
infinitesimal gauge transformation:
${\bar{\partial}}_{{\bar{x}}}{\nabla}+[A_{{\bar{x}}},{\nabla}]=0$ (8.7)
Therefore, the conjugacy class of the holonomy $g(x,{{\bar{x}}})$ of $\nabla$
around $\mathbb{S}^{1}$ varies holomorphically with $x$, and, in the gauge
where $g(x,{{\bar{x}}})\in\mathbf{T}$, it is locally holomorphic:
$[g(x)]=\left[P{\exp}\,\oint_{0}^{2\pi}\,\mathrm{i}d\psi\left(A_{\psi}({\psi},x,{{\bar{x}}})+\mathrm{i}{\phi}({\psi},x,{{\bar{x}}})\right)\right]\in{B({\mathfrak{g}})}$
(8.8)
As we shall clarify later, when $x\to\infty$,
$[g(x)]\to\left[\prod_{i=1}^{r}{\mathfrak{q}}_{i}^{-{\lambda}_{i}^{\vee}}\right]=b_{\infty}\in{B^{\text{ad}}({\mathfrak{g}})}\
.$
One is left with the quasimap $u:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{B^{\text{ad}}({\mathfrak{g}})}$. It is instructive to
calculate (8.8) for the Dirac monopole on $M^{3}$.
Recall that the Dirac monopole at ${\vec{p}}\in{\mathbb{R}}^{3}$ is the
connection in the $U(1)$ bundle over ${\mathbb{R}}^{3}\backslash{\vec{p}}$,
which is a pullback of the constant curvature connection on the Hopf bundle
over ${\mathbb{S}}^{2}$ via the projection map
${\pi}_{\vec{p}}:{\mathbb{R}}^{3}\backslash{\vec{p}}\longrightarrow{\mathbb{S}}^{2},\qquad{\pi}_{\vec{p}}({\vec{r}})=\frac{{\vec{r}}-{\vec{p}}}{|{\vec{r}}-{\vec{p}}|}$
The corresponding curvature two-form $F$ is given by:
$F=2\pi\mathrm{i}\,{\pi}_{\vec{p}}^{*}{\varpi}_{2}=\frac{\mathrm{i}}{2}\frac{({\vec{x}}-{\vec{p}})\cdot
d{\vec{x}}\times d{\vec{x}}}{|{\vec{x}}-{\vec{p}}|^{3}}$
where
$\int_{{\mathbb{S}}^{2}}{\varpi}_{2}=1$
The fact that up to the $|{\vec{r}}-{\vec{p}}|^{2}$ rescaling the two-form
${\varpi}_{2}$ coincides with the volume form on $\mathbb{S}^{2}$ obtained
from the flat metric on ${\mathbb{R}}^{3}$ implies that $F$ solves Maxwell
equations in ${\mathbb{R}}^{3}\backslash{\vec{p}}$ and moreover there is a
magnetic potential $\phi$, such that
$d{\phi}+\star_{3}F=0$
Moreover, if $\phi$ is normalized to approach zero at infinity, then
${\phi}=-\frac{1}{2|{\vec{r}}-{\vec{p}}|}$ (8.9)
The periodic Dirac monopole, i.e. the solution of Maxwell equations on
$M^{3}\backslash p=({\psi}_{0},x_{0},{{\bar{x}}}_{0})$ can be obtained from
the basic monopole in $\mathbb{R}^{3}$ by taking the superposition of the
fields of an infinite periodic array of monopoles, living on the universal
cover $\widetilde{M^{3}\backslash
p}={\mathbb{R}}^{3}\backslash({\psi}_{0}+2\pi{\mathbb{Z}},x_{0},{{\bar{x}}}_{0})$.
The magnetic potential is given by the regularized sum of potentials (8.9):
$\displaystyle{\phi}({\psi},x,{{\bar{x}}};p)={\phi}_{\infty}+\frac{{\rm
log}(\pi)-{\gamma}}{2\pi}$ (8.10)
$\displaystyle\qquad\qquad+\sum_{n\in{\mathbb{Z}}}\left(\varphi({\psi}-{\psi}_{0}-2\pi
n,x-x_{0})+\frac{1-{\delta}_{n,0}}{4\pi|n|}\right)$
$\displaystyle\qquad\qquad\qquad\qquad{\varphi}({\psi},x)=-\frac{1}{2\sqrt{{\psi}^{2}+x{{\bar{x}}}}}$
We calculate:
$\frac{1}{2\pi}\int_{0}^{2\pi}{\phi}({\psi},x,{{\bar{x}}};p)d{\psi}={\phi}_{\infty}+\frac{1}{2\pi}{\rm
log}\,|x-x_{0}|$ (8.11)
The calculation of
$\int_{0}^{2\pi}A_{\psi}({\psi},x,{{\bar{x}}};p)d{\psi}$
is a bit tricky. Fortunately its derivative is easy to compute:
$\displaystyle
d\int_{0}^{2\pi}A_{\psi}({\psi},x,{{\bar{x}}};p)d{\psi}=\oint_{{\mathbb{S}}^{1}}F=$
(8.12)
$\displaystyle\qquad\frac{\mathrm{i}}{4}\sum_{n\in{\mathbb{Z}}}\int_{0}^{2\pi}\,d\psi\wedge\,\frac{(x-x_{0})d{{\bar{x}}}-({{\bar{x}}}-{{\bar{x}}}_{0})dx}{\left(\,|x-x_{0}|^{2}+\left({\psi}-{\psi}_{0}+2\pi
n\right)^{2}\,\right)^{3/2}}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\frac{\mathrm{i}}{2}d\,{\rm
log}\left(\frac{x-x_{0}}{{{\bar{x}}}-{{\bar{x}}}_{0}}\right)$
Thus
$d\oint(A_{\psi}+\mathrm{i}{\phi})d\psi=\mathrm{i}d\,{\rm log}(x-x_{0})$
and the monodromy is equal to
$g(x)=(x-x_{0})^{-1}$ (8.13)
up to some multiplicative constant. Now, if we have a superposition of several
Dirac monopoles, in the theory with the gauge group $T$, with the monopoles of
the type $i$, i.e. corresponding to the coweight
${\lambda}_{i}^{\vee}\in{\operatorname{Hom}}(U(1),T)$ located at the points
$({\psi}_{i,{\mathfrak{f}}},m_{i,\mathfrak{f}},{\bar{m}}_{i,\mathfrak{f}})$,
then the monodromy of the corresponding complexified connection
$A+\mathrm{i}{\phi}d\psi$ is equal to:
$g(x)\propto\prod_{i=1}^{r}\prod_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}(x-m_{i,\mathfrak{f}})^{-{\lambda}^{\vee}_{i}}$
(8.14)
Now let us consider the nonabelian Bogomolny equation on $M^{3}$. Instead of
solving the Eq. (8.2) modulo $G$-gauge transformations, let us solve two out
of three equations in (8.2), namely the equation
$[D_{{\bar{x}}},{\nabla}]=0$ (8.15)
modulo the action of the group ${\mathcal{G}}^{\mathbb{C}}$ of
$\mathbf{G}$-valued (complex) gauge transformations. In fact, (8.15) can be
viewed as the complex moment map for ${\mathcal{G}}^{\mathbb{C}}$, acting on
the space of $(A,{\Phi})$, endowed with the
${\mathcal{G}}^{\mathbb{C}}$-invariant holomorphic symplectic form:
${\Omega}=\frac{1}{2\pi}\int_{M^{3}}\langle{\delta}{\nabla}\wedge{\delta}A_{{\bar{x}}}\rangle\
d\psi dxd{{\bar{x}}}$ (8.16)
Let us now try to analyze the solutions to (8.15) in some domain
$D\times{\mathbb{S}}^{1}\subset M^{3}$ over $D\subset\mathbf{C}_{\left\langle
x\right\rangle}$. We fix the $\mathbf{G}$-gauge where
$A_{\psi}+\mathrm{i}\phi={\xi}(x,{{\bar{x}}})$, ${\partial}_{\psi}{\xi}=0$.
This gauge leaves some residual gauge freedom. Passing to
$g(x)={\exp}\,2\pi\mathrm{i}\xi(x)$
partially reduces the residual gauge invariance. The Eq. (8.15) implies that
${\bar{\partial}}_{{\bar{x}}}{\xi}=0$, and
$A_{{\bar{x}}}=a_{{\bar{x}}}\in\mathfrak{h}$. We then proceed with
constructing the cameral curve ${\mathcal{C}}_{u}$ which is the union of the
$W({\mathfrak{g}})$-orbits of $g(x)$ over all
$x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\infty\\}$.
The fiber ${\mathscr{A}}_{u}$ of the projection
$h:{{\mathfrak{P}}}\to{{\mathfrak{M}}}$ is the space of
$W({\mathfrak{g}})$-equivariant $\mathbf{T}$-bundles over $\mathcal{C}_{u}$ of
fixed multi-degree. We shall discuss in more detail the analogous situation
for the class II theories in the next section.
The asymptotics of the solution to (8.2) is characterized by a vector of
magnetic charges. Namely, over
${\mathbb{T}}^{2}_{\infty}={\mathbb{S}}^{1}\times{\mathbb{S}}^{1}_{\infty}$
where ${\mathbb{S}}^{1}_{\infty}$ is a large radius circle in
${\mathbb{R}}^{2}$, the gauge group $G$ is broken down to $T$. The gauge
bundle is therefore characterized by the vector of the first Chern classes:
${\bf m}\in H^{2}({\mathbb{T}}^{2}_{\infty},{\pi}_{1}({\mathbf{T}}))={{\rm
Q}}^{\vee}$ (8.17)
One can compute ${\bf m}$ by analyzing the behavior of the conjugacy class of
the holonomy $g(x)$ of the complexified connection ${\nabla}$. One can show
that the finite energy (8.3) solutions with non-trivial magnetic charge do not
exist. Indeed, macroscopically the system looks two dimensional, and
asymptotically it looks like a charged vortex, whose energy diverges
logarithmically at large distances.
However, infinite $3$-dimensional energy solutions may correspond to the
finite tension higher dimensional objects. For example, the noncommutative
$U(1)$ monopole describes a finite tension string, which is attached to the
worldvolume of the gauge theory [Gross:2000wc]. Similarly, the infinite energy
periodic monopole solutions have interpretation in the higher dimensional
theory, e.g. in the brane realization [Witten:1997sc] of the pure
${\mathcal{N}}=2$ $SU(N)$ gauge theory in four dimensions [Cherkis:2000cj].
One can actually make the energy finite in the infrared by allowing point-like
singularities in $M^{3}$. The idea is to screen the asymptotic magnetic charge
of the non-abelian solution by the opposite charge of the Dirac monopole
singularities.
Let us study the general situation. Suppose
${\gamma}\subset{\mathbf{C}_{\left\langle x\right\rangle}}$ is a closed
contour. For each point $x\in\gamma$ compute the holonomy $g(x,{{\bar{x}}})$
of the complexified connection $\nabla$, e.g. starting at the point ${\psi}=0$
on the fiber ${\mathbb{S}}^{1}$. It is a functional of the gauge field $A$ and
the Higgs field $\phi$: $(A,{\phi})\mapsto g(x,{{\bar{x}}})$. The gauge
transformed $(A,\phi)$ leads to the similarity-transformed function
$g(x,{{\bar{x}}})$: $(A^{h},{\phi}^{h})\mapsto
h^{-1}(0,x,{{\bar{x}}})g(x,{{\bar{x}}})h(0,x,{{\bar{x}}})$. We have a well-
defined map:
$[g_{\gamma}]:{\gamma}\longrightarrow B({\mathfrak{g}})$ (8.18)
which is a restriction on $\gamma$ of the holomorphic (cf. (8.15)) map
$U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{B^{\text{ad}}({\mathfrak{g}})}$, $U:x\mapsto[g(x)]$.
Now let ${\Xi}({\mathfrak{g}})\subset{B^{\text{ad}}({\mathfrak{g}})}$ denote
the set of irregular orbits of $W({\mathfrak{g}})$ in ${\mathbf{T}}/Z$.
Generically the image $\Sigma_{u}$ of $[g(x)]$ crosses ${\Xi}({\mathfrak{g}})$
at some isolated points, which are the branch points of the cameral curve
$\mathcal{C}_{u}$:
${\Xi}_{x}=U^{-1}\left({\Sigma}_{u}\cap{\Xi}({\mathfrak{g}})\right)$
Let us consider the subvariety
$B({\mathfrak{g}})^{reg}={B^{\text{ad}}({\mathfrak{g}})}\backslash{\Xi}({\mathfrak{g}})$.
The fundamental group ${\pi}_{1}(B({\mathfrak{g}})^{reg})$ is related to Artin
braid group associated with the $\mathfrak{g}$ root system. Let us also define
${\mathbf{T}}^{reg}$ to be the subvariety in $\mathbf{T}$ consisting of the
regular elements, i.e. the elements of the maximal torus whose stabilizer in
$\mathbf{G}$ is the maximal torus $\mathbf{T}$. It is invariant under the
translations by $Z$. We have a map:
${\pi}_{*}:{\pi}_{1}({\mathbf{T}}^{reg})\to{\pi}_{1}(B({\mathfrak{g}})^{reg})$,
induced by the projection
${\pi}:{\mathbf{T}}^{reg}\to{\mathbf{T}}^{reg}/\left(Z\times
W({\mathfrak{g}})\right)$.
Now, let us go back to the loop $\gamma$. We would like to define a
$\mathbf{T}$-bundle over ${\mathbb{S}}^{1}\times{\gamma}$, by choosing a gauge
where $A_{\psi}+\mathrm{i}\phi$ is $\psi$-independent element of the Lie
algebra $\mathfrak{h}\subset\mathfrak{g}$. Then Bogomolny equations imply that
$A_{{\bar{x}}}$ also belongs to $\mathfrak{h}$, assuming that
$A_{\psi}+\mathrm{i}\phi$ is regular, i.e. its stabilizer in $\mathbf{G}$ is
the maximal torus $\mathbf{T}$.
There is an obstruction for such a gauge being possible throughout $\gamma$.
Namely the class of $\gamma$ in ${\pi}_{1}(B({\mathfrak{g}})^{reg})$ should
lie in the image of ${\pi}_{*}$. This is related to our solution of the four
dimensional ${\mathcal{N}}=2$ theory of the class I as follows.
For the asymptotically conformal theories, with the assignments of dimensions
${\mathbf{v}}$, ${\mathbf{w}}$, we have defined a ${\mathbf{G}}/Z$-valued
function on $\mathbf{C}_{\left\langle x\right\rangle}$ minus a finite number
of points:
$g(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\lambda}^{\vee}_{i}}{\mathscr{Y}}_{i}(x)^{{\alpha}_{i}^{\vee}}$
(8.19)
We identify $g(x)$ in (8.19) with the holonomy in (8.8). The factor
$g_{\infty}(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\lambda}^{\vee}_{i}}$
clearly corresponds to the Dirac monopoles sitting at some points
$({\psi}_{i,{\mathfrak{f}}},m_{i,{\mathfrak{f}}},{\bar{m}}_{i,\mathfrak{f}})$
with the charges ${\lambda}^{\vee}_{i}$. The remaining factor has to do with
the nonabelian monopoles.
Recall that the map $U:\mathbf{C}_{\left\langle
x\right\rangle}\backslash{\Xi}_{x}\to B({\mathfrak{g}})$ is determined by the
collection of gauge polynomials $T_{i}(x)$, $i=1,\ldots,r$. The singular locus
${\Xi}_{x}$ of $U$ are at the zeroes and poles of the discriminant
${\Delta}(x)=g(x)^{-2{\rho}}\prod_{{\alpha}\in
R_{+}}\left(g(x)^{\alpha}-1\right)^{2}$ (8.20)
where $R_{+}$ is the set of positive roots of $\mathfrak{g}$, and
${\rho}=\frac{1}{2}\sum_{{\alpha}\in R_{+}}{\alpha}\ .$
The discriminant ${\Delta}(x)$ is a rational function in $T_{i}(x)$’s and
${\mathscr{P}}_{i}(x)$’s. Now, given a loop ${\gamma}$ in
${\mathbf{C}_{\left\langle x\right\rangle}}\backslash{\Xi}_{x}$, when can we
lift $[g(x)]|_{\gamma}$ to the $\mathbf{T}$-valued loop?
For the simple root ${\alpha}_{i}$ let us denote by ${\Xi}_{x,i}$ the set of
solutions to the equation $g(x)^{{\alpha}_{i}}=1$ on the physical sheet of
$\mathcal{C}_{u}$.
${\Xi}_{x,i}=\left\\{\,x\,|\ g\left(x\right)^{{\alpha}_{i}}=1\,\right\\}$
(8.21)
so that
${\Xi}_{x}=\bigcup_{i=1}^{r}\ {\Xi}_{x,i}$ (8.22)
Actually the points of $\Xi_{x,i}$ are the endpoints of the cuts
$I_{i,{\mathbf{a}}}$.
Our claim is that for the loops ${\gamma}=A_{i{\mathbf{a}}}$ which encircle
the individual cuts $I_{i,\mathbf{a}}$ the class $[g(x)]\in
B({\mathfrak{g}})^{reg}$ lifts to ${\mathbf{T}}^{reg}$, and defines a
${\mathbf{T}}$-bundle over ${\mathbb{S}}^{1}\times{\gamma}$. Its
characteristic class is equal to ${\alpha}_{i}^{\vee}\in{{\rm Q}}^{\vee}$.
Thus the magnetic monopoles which correspond to the limit shape of the
${\mathcal{N}}=2$ theory have the Dirac monopole charges ${\bf q}_{\rm
Dir}=-\sum_{i=1}^{r}{\mathbf{w}}_{i}{\lambda}_{i}^{\vee}$ which are
distributed at the points
$({\psi}_{i,{\mathfrak{f}}},m_{i,\mathfrak{f}},{\bar{m}}_{i,{\mathfrak{f}}})$,
and the nonabelian monopole charges ${\bf
q}_{\rm{}^{\prime}tHP}=\sum_{i=1}^{r}{\mathbf{v}}_{i}{\alpha}_{i}^{\vee}$
which are located over the cuts $I_{i,{\mathbf{a}}}$. The net charge at
infinity is equal to
${\bf q}_{\rm Dir}+{\bf q}_{\rm{}^{\prime}tHP}=0$ (8.23)
for the asymptotically conformal theories.
For the asymptotically free theories the net charge at infinity is equal to
(cf. (3.1)):
${\bf q}_{\rm Dir}+{\bf
q}_{\rm{}^{\prime}tHP}=-\sum_{i=1}^{r}{\beta}_{i}{\lambda}_{i}{{}^{\vee}}$
(8.24)
The fact that ${\beta}_{i}\leq 0$ should follow from the positivity of energy
(as it does in the $A_{1}$ case) but we couldn’t find a simple proof for
general $\mathbf{G}$.
The relation of the monopole picture of ${\mathfrak{P}}$ to the Hitchin system
picture we had in the section 7.1.1 goes via the Nahm transform, or, since we
are ultimately working only in a particular complex structure of the moduli
space, via a version of Fourier-Mukai transform. The $U(k)$ monopoles on
$\mathbb{R}^{3}$ are mapped via Nahm’s transform to the solutions of a one-
dimensional system of Nahm’s equations. The $U(k)$ monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ are mapped via Nahm’s transform to
the solutions of a two-dimensional system of Hitchin’s equations, with
singularities. Indeed, our spectral curve in the form (7.123) captures the
solutions to two, ${\bar{\partial}}_{A}{\Phi}=0$, ${\partial}_{A}{\bar{\Phi}}$
out of three Hitchin’s equations. The remaining equation
$F_{A}+[{\Phi},{\bar{\Phi}}]=0$ away from the punctures would fix the
hyperkähler metric on ${\mathfrak{P}}$. Unfortunately we are not in the
position to discuss the metric on ${\mathfrak{P}}$ as long as we stay within
the realm of the four dimensional gauge theory. See [Nahm:1981nb,
Corrigan:1983sv, Hitchin:1983ay, Braam:1988qk].
We do need to discuss the holomorphic symplectic geometry of ${\mathfrak{P}}$.
The symplectic form on ${\mathfrak{P}}$ descends from the two-form (8.16) via
the Hamiltonian reduction with (8.15) being the moment map. The textbook
construction of the action-angle variables of the integrable system produces
the special coordinates
${\mathfrak{a}}^{\mathcal{I}},{\mathfrak{a}}_{\mathcal{I}}^{D}$ of the gauge
theory. We claim this construction is equivalent to the one using the periods
of the differentials $x\,d{\log}{\mathscr{Y}}_{i}$ on the cameral curve. The
essential points of the demonstration are identical for the class I and for
the class II theories.
We thus return to this question in the section 8.2.
Now let us study the $A_{1}$ case in some more detail. We wish to present yet
another perspective on the phase space ${\mathfrak{P}}$.
Consider the product of $N$ $A_{1}$ surfaces ${\mathcal{O}}_{a}$,
$a=1,\ldots,N$, the complex coadjoint orbits of $SL(2,{\mathbb{C}})$. Each
surface ${\mathcal{O}}_{i}$ is a quadric in ${\mathbb{C}}^{3}$, given by the
equation:
${\xi}^{2}_{a}+{\eta}_{a}^{2}+{\zeta}_{a}^{2}=s_{a}^{2}$ (8.25)
with some fixed constant $s_{a}\in\mathbb{C}$. The surface $\mathcal{O}_{a}$
has the holomorphic symplectic form:
${\varpi}_{a}=\frac{d{\xi}_{a}\wedge d{\eta}_{a}}{\zeta_{a}}$
which has the period $s_{a}$ along a non-contractible two-sphere in
$\mathcal{O}_{a}$. The moment map for the action of $SL(2,{\mathbb{C}})$ on
${\mathcal{O}}_{a}$ is a $2\times 2$ traceless matrix. Let us extend it into a
general $2\times 2$ matrix (which corresponds to the conformal extension of
the group):
$L_{a}(x)=\left(\begin{matrix}x+{\xi}_{a}&{\eta}_{a}+\mathrm{i}{\zeta}_{a}\\\
-{\eta}_{a}+\mathrm{i}{\zeta}_{a}&x-{\xi}_{a}\end{matrix}\right)$ (8.26)
and define the ‘‘monodromy matrix’’
$L(x)=\left(\begin{matrix}{\mathfrak{q}}&0\\\ 0&1\end{matrix}\right)\times
L_{N}(x-{\mu}_{N})L_{N-1}(x-{\mu}_{N-1})\ldots L_{1}(x-{\mu}_{1})$ (8.27)
It is actually better to work with the somewhat differently normalized ‘‘local
Lax operators’’:
$g_{a}(x)={\bf
1}_{2}+\frac{1}{x-s_{a}}\left(\begin{matrix}u_{a}^{+}&v_{a}^{+}\\\
v_{a}^{-}&u_{a}^{-}\end{matrix}\right)$ (8.28)
where
$u_{a}^{\pm}=s_{a}\pm{\xi}_{a},\qquad
v_{a}^{\pm}=\mathrm{i}{\zeta}_{a}\pm{\eta}_{a}$
obey
$u_{a}^{+}u_{a}^{-}-v_{a}^{+}v_{a}^{-}=0$ (8.29)
and define
$g(x)=\left(\begin{matrix}{\mathfrak{q}}&0\\\ 0&1\end{matrix}\right)\times
g_{N}(x-{\mu}_{N})g_{N-1}(x-{\mu}_{N-1})\ldots g_{1}(x-{\mu}_{1})$ (8.30)
Then, cf. [Gorsky:1996hs]:
${\rm
Det}(g(x))={\mathfrak{q}}\frac{{\mathscr{P}}^{+}(x)}{{\mathscr{P}}^{-}(x)}$
(8.31)
where we defined
${\mathscr{P}}^{\pm}(x)=\prod_{a=1}^{N}(x-{\mu}_{a}\pm s_{a})$
Now, define the phase space to be the complex symplectic quotient of the
product of the $A_{1}$ surfaces by the diagonal action of the
${\mathbb{C}}^{\times}$,
${{\mathfrak{P}}}=\times_{a=1}^{N}{\mathcal{O}}_{a}//{\mathbb{C}}^{\times}$
(8.32)
generated by
$H_{1}=\sum_{a=1}^{N}{\xi}_{a}$
The variety (8.32), defined by fixing the level ${\xi}$ of $H_{1}$ and
dividing the corresponding level set $H_{1}^{-1}({\xi})$ by
${\mathbb{C}}^{\times}$, carries the induced holomorphic symplectic structure.
The functions $h_{k}$ defined as:
${\operatorname{tr}}_{2}\,g(x)=(1+{\mathfrak{q}})\left(1+\sum_{k=1}^{\infty}x^{-k}h_{k}\right)$
(8.33)
Poisson-commute with respect to the induced Poisson structure. Moreover,
$h_{1}=H_{1}+\frac{{\mathfrak{q}}-1}{{\mathfrak{q}}+1}\sum_{a}s_{a}$
and the next $N-1$ $h_{k}$’s are independent. The rest of the expansion
coefficients can be expressed in terms of the first $N$.
We argue that the normalized Lax operator $g(x)$ is the complexified monodromy
field $g(x)$ in the corresponding periodic singular monopole problem, with the
monopole group $U(2)$ (the compact form of C$\mathbf{G}$) and ${\mu}_{a}\pm
s_{a}$ are the locations of $2N$ Dirac monopoles.
It is not difficult to convince oneself that the ‘‘local Lax operator’’ (8.28)
is indeed the complexified monodromy of a single $U(2)$ monopole screened by
two Dirac monopoles of the opposite $U(1)$ charges, located at $\pm s_{a}$.
What is amusing is that the Eq. (8.30) suggests that the complexified
monodromy of the charge $N$ $U(2)$ monopole screened by $2N$ Dirac monopoles
factorizes as the product of $N$ elementary monodromy matrices.
Note in passing that if we do not perform the reduction with respect to
${\mathbb{C}}^{\times}$ generated by $H_{1}$, i.e. work with the
$2N$-dimensional phase space $\tilde{\mathfrak{P}}$ (this is a first step
towards the extended phase space ${\mathfrak{P}}^{\mathrm{ext}}$), then
Sklyanin’s separation of variables [Sklyanin:1987ih, Sklyanin:1995bm]
identifies its open subset ${\tilde{\mathfrak{P}}}^{\circ}$ with the $N$’th
symmetric product of (which is most likely [Gorsky:1999rb] resolved into the
Hilbert scheme of $N$ points on) ${\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}^{\times}$.
Incidentally [Donaldson:1985id], this manifold is symplectomorphic to the
moduli space of regular charge $N$ $SU(2)$ monopoles on ${\mathbb{R}}^{3}$.
In recent years the connection between the gauge theories and the spin chains,
and the inspired duality between the Gaudin-like integrable systems and the
Heisenberg spin chain was discussed in [Chen:2011sj, Dorey:2011pa,
Muneyuki:2011qu, Mironov:2012ba], building on the earlier work in
[Moore:1997dj, Gerasimov:2006zt, Gerasimov:2007ap, Nekrasov:2009zz,
Nekrasov:2009ui].
Before concluding this section, note that the masses of the bi-fundamental
matter hypermultiplets are encoded in the next-to-leading terms in the
asymptotics of the complexified connection $A_{\psi}+i{\phi}$ near
$x\to\infty$. We shall explain this in more detail in the similar context in
the section 8.2.
The moduli space of singular $\mathbf{G}$-monopoles with fixed conjugacy class
of the monodromy of $A_{\psi}+i{\phi}$ at $x\to\infty$, with unspecified
location of Dirac singularities of specified charges defines our extended
phase space ${\mathfrak{P}}^{\mathrm{ext}}$. It is acted upon by the torus
$T\times T_{M}$, where $T_{M}\subset{G_{M}}$ is the maximal torus of the
flavor group. The action of $T$ is via the constant gauge transformations
preserving the gauge field and the Higgs field at infinity, while $T_{M}$ acts
by changing the glueing data for the grafted Dirac monopoles. Fixing the level
of the corresponding moment maps and reducing with respect to the
complexification of the $T\times T_{M}$ action produces ${\mathfrak{P}}$.
Summarizing, we conclude with the conjecture: _the moduli space of singular
$\mathbf{G}$-monopoles on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ is acted
upon by the Poisson-Lie group, which is a quasi-classical limit of the Yangian
$Y({\mathfrak{g_{\text{q}}}})$. The deformation quantization of
${\mathfrak{P}}^{\mathrm{ext}}$ (in the complex structure $I$) produces the
Yangian $Y({\mathfrak{g}})$. Fixing the asymptotics of the complexified gauge
field at infinity as well as the locations of the Dirac singularities would
specify._ It should be interesting to explore the holomorphic symplectic
geometry of ${\mathfrak{P}}$ in all of its complex structures and find the
analogue of the variety of opers (cf. [Nekrasov:2010ka, Nekrasov:2011bc]),
which in the $A_{r}$ case is the variety of local systems on the genus zero
curve coming from the $N$’th order differential operators with regular
singularities at $r+3$ punctures with the monodromies whose eigenvalues
coincide with our description of the residues of the Higgs field just above
the Eq. (7.23) (cf. [Frenkel:2003qx]).
Note that in [Gerasimov:2005qz] a connection between the representation theory
of Yangians and the moduli spaces of monopoles on ${\mathbb{R}}^{3}$ was
found. Also, in [Atiyah:1991gd, Atiyah:1996ij] the relations between the
monopole solutions and the solutions to the Yang-Baxter equations were
discussed. It remains to be seen, whether any of these connections carries
over to the ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ case and whether it is
the one we need.
What happens if one tries to study periodic $\widehat{\mathbf{G}}$-monopoles
where $\widehat{\mathbf{G}}$ is Kac-Moody affine group? One naturally finds
double periodic instantons similar to relation between
$\widehat{\mathbf{G}}$-monopoles and periodic $\mathbf{G}$ instantons
[Garland_1988].
### 8.2. Double-periodic instantons and Class II theories
Let us now discuss the class II theories. Recall from the section 6.2.2 that
an elliptic curve
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$, with
$\mathfrak{q}$ given by 6.24, is associated with the class II gauge theory.
Also recall that the gauge group is the product of special unitary groups
${G_{\text{g}}}=\times_{i=0}^{r}\ SU(Na_{i})$
for some number $N$. Using (6.28) we have identified the extended moduli space
${\mathfrak{M}}^{\mathrm{ext}}_{N}$ of vacua of the theory (4.7),
$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}_{N}=Nh^{\vee}$, with the
moduli space of degree $N$ framed quasimaps of
$({\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle},{\infty})$ to the
moduli space ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$\mathbf{G}$-bundles on $\mathscr{E}$, sending ${\infty}$ to a particular
bundle $[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]$. This space of quasimaps is a
natural base of the projection from the moduli space
${{\mathfrak{P}}^{\mathrm{ext}}}_{N}\approx\mathrm{Bun}^{ss}_{\mathbf{G};N}(S_{\mathscr{E}})^{{\mathcal{P}}_{\tilde{\mathfrak{q}}}}_{\infty}$
(8.34)
of framed semi-stable holomorphic principal $\mathbf{G}$-bundles
$\mathcal{P}$, $c_{2}({\mathcal{P}})=N$, on the surface
$S_{\mathscr{E}}\equiv{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$, where the framing is the identification
of the restriction of $\mathcal{P}$ at the fiber at infinity:
${\mathcal{P}}\biggr{|}_{\\{\infty\\}\times{\mathscr{E}}}\approx{\mathcal{P}}_{\tilde{\mathfrak{q}}}$
(8.35)
In what follows we drop the subscript $N$.
The projection
${{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$ is defined
on the open dense subset of ${\mathfrak{P}}^{\mathrm{ext}}$ by restricting the
bundle ${\mathcal{P}}\in{\mathfrak{P}}^{\mathrm{ext}}$ to the fiber
${\mathscr{E}}_{x}$ and taking its equivalence class ${\bf
t}(x)=[{\mathcal{P}}|_{{\mathscr{E}}_{x}}]$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$. This gives the desired quasimap
$U:x\mapsto{\bf t}(x)$. This quasimap is a map near $x=\infty$, approaching a
particular holomorphic bundle
$[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]\in{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$
over $\mathscr{E}_{\infty}=\\{{\infty}\\}\times{\mathscr{E}}$. The reason $U$
is not, in general, a map, has to do with the usual difference between the
stability condition in two complex dimensions and the fiberwise stability
condition, cf. [Losev:1999tu].
The semi-stable framed holomorphic principal bundles $\mathcal{P}$ on
$S_{\mathscr{E}}$, are in one-to-one correspondence with the $G$-instantons on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ with the flat metric (cf.
[Donaldson:1985zz]), i.e. connections on a principal $G$-bundle $P$ over
${\mathbb{S}}^{2}\times{\mathbb{T}}^{2}$, endowed with some metric, which is
conformally flat on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, which obey
$F_{A}=-\star F_{A}$
and have finite action,
$\int_{{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}}\left\langle F_{A}\wedge\star
F_{A}\right\rangle<\infty$
(as we explain later, these instantons correspond to the M2 branes and the
instanton action is the tension of the stack of M2 branes) and this forces the
curvature $F_{A}$ of the $G$-gauge field tend to zero as
$|{\vec{x}}|\to\infty$, ${\vec{x}}\in{\mathbb{R}}^{2}$. We fix the instanton
charge:
$N=-\frac{1}{8\pi^{2}h}\int_{{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}}\left\langle
F_{A}\wedge F_{A}\right\rangle$ (8.36)
(remember that $\left\langle,\right\rangle$ is the Killing form, which is the
trace in the adjoint representation). The real dimension of the moduli space
${\mathfrak{P}}^{\mathrm{ext}}$ of $G$-instantons of finite action on
$\mathbb{R}^{2}\times\mathbb{T}^{2}$ with fixed framing at infinity
$\mathbb{T}^{2}_{\infty}$ is equal to $4Nh$.
Actually, there is a subtlety here. The moduli space of charge $N$
$G$-instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ may have several
components. This has to do with the fact that the moduli space of flat
$G$-connections on
${\mathbb{T}}^{3}={\mathbb{S}}^{1}_{\infty}\times{\mathbb{T}}^{2}$ may have
several components [Witten:1997bs]. We shall assume we are always in the
component of the trivial connection.
Note that ${\mathfrak{P}}^{\mathrm{ext}}$ is acted upon by the maximal torus
$T$ of $G$, the symmetry group of the flat connection at infinity. This action
lifts to the action of the algebraic torus ${\mathbf{T}}$ on the moduli space
of framed holomorphic bundles ${{\mathfrak{P}}^{\mathrm{ext}}}$. This is
entirely parallel to the action of the group $\mathbf{G}$ on the moduli space
${{\mathfrak{M}}}^{\mathrm{framed}}_{G}({\mathbb{R}}^{4})$ of framed
$G$-instantons on ${\mathbb{R}}^{4}$.
A holomorphic principal $\mathbf{G}$-bundle ${\mathcal{P}}$ on
$S_{\mathscr{E}}$ can be described using the transition functions
$g_{\alpha\beta}:U_{\alpha\beta}\to{\mathbf{G}}$ defined on the overlaps
$U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$ of the open sets in the appropriate
open cover $(U_{\alpha})_{{\alpha}\in A}$ of $S_{\mathscr{E}}$. The transition
functions are holomorphic ${\bar{\partial}}g_{\alpha\beta}=0$, must obey the
cocycle condition $g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=1$ on
$U_{\alpha\beta\gamma}=U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$, and the
cocycles differing by the holomorphic coboundaries define equivalent bundles:
$g_{\alpha\beta}\sim g_{\alpha}g_{\alpha\beta}g_{\beta}^{-1}$
where $g_{\alpha}:U_{\alpha}\to{\mathbf{G}}$ are holomorphic
$\mathbf{G}$-valued functions on the open sets $U_{\alpha}$ themselves.
Another way to describe the holomorphic bundle is to introduce a connection
$\nabla=d+A$ on a smooth bundle, such that its $(0,1)$-part is $(0,2)$-flat:
${\nabla}_{\bar{\partial}}^{2}=0$ (8.37)
The local holomorphic sections of
${\mathcal{P}}_{\alpha}={\mathcal{P}}|_{U_{\alpha}}$ are the solutions to
${\nabla}_{\bar{\partial}}s_{\alpha}=0$. Over an intersection
$U_{\alpha\beta}$ these solutions must differ by a holomorphic
$\mathbf{G}$-valued gauge transformation:
$s_{\alpha}({\zeta},{\bar{\zeta}})=g_{\alpha\beta}({\zeta})s_{\beta}({\zeta},{\bar{\zeta}})$
where $\zeta$ stand for local holomorphic coordinates. This is sometimes
expressed by saying that locally ${\bar{A}}$ is a pure $\mathbf{G}$-gauge
${\bar{A}}\biggr{|}_{U_{\alpha}}=-s_{\alpha}^{-1}{\bar{\partial}}s_{\alpha}$
In the case at hand ${\zeta}=(x,z)$, where $x\in\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}$ and $z$ is the additive coordinate on $\mathscr{E}$. The Eq.
(8.37) reads:
${\bar{\partial}}_{{\bar{x}}}A_{{\bar{z}}}-{\bar{\partial}}_{{\bar{z}}}A_{{\bar{x}}}+[A_{{\bar{x}}},A_{{\bar{z}}}]=0$
(8.38)
This equation can be viewed as the complex moment map equation for the action
of the group $\mathcal{G}$ of the $\mathbf{G}$-gauge transformations on the
space $\mathcal{A}$ of connections on a given smooth principal
$\mathbf{G}$-bundle over $S_{\mathscr{E}}$, endowed with the holomorphic
symplectic form:
${\Omega}=\int_{S_{\mathscr{E}}}dx\wedge
dz\wedge\langle{\delta}{\bar{A}}\wedge{\delta}{\bar{A}}\rangle$ (8.39)
We have a little subtlety here. The two-form $dx\wedge dz$ was perfectly good
on $S_{\mathscr{E}}^{\circ}$ but on $S_{\mathscr{E}}$ is has a pole along the
divisor ${\mathscr{E}}_{\infty}=\\{{\infty}\\}\times{\mathscr{E}}$. We impose
the condition that $A_{{\bar{z}}}$ approaches a specific value as
$x\to\infty$. More specifically, recall (C.113) that generic $A_{{\bar{z}}}$
can be $\mathcal{G}$-transformed to the normal form
$A_{{\bar{z}}}\to\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi\in\mathfrak{h}$,
${\partial}_{z}{\xi}={\partial}_{{\bar{z}}}{\xi}=0$. We impose the boundary
conditions:
$A_{{\bar{z}}}(x,{{\bar{x}}})\to\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi_{\infty}+o(|x|^{-2}),\qquad|x|\to\infty$
(8.40)
for fixed $\xi_{\infty}$, which we relate to the gauge couplings
${\mathfrak{q}}_{i}$, $i=0,1,\ldots,r$ via:
${\xi}_{\infty}=-\frac{1}{2\pi\mathrm{i}}\sum_{i=1}^{r}{\log}{\mathfrak{q}}_{i}\,{\lambda}_{i}^{\vee}$
(8.41)
The limiting gauge field
${\bar{\partial}}_{{\bar{z}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi_{\infty}$
corresponds to the holomorphic bundle ${\mathcal{P}}_{\tilde{\mathfrak{q}}}$
on $\mathscr{E}$. The decay rate (8.40) makes the integral (8.39) convergent.
One can impose weaker conditions, allowing even the $O(x^{-1})$ (but no
$x^{-1}{{\bar{x}}}^{-1}$ terms!) decay, which would make (8.39) convergent
with the principal value prescription. In fact, these subleading terms
correspond to the bi-fundamental masses of the $G_{\text{g}}$-theory. Let us
interpret them using the $\mathbf{T}$-action on
${{\mathfrak{P}}^{\mathrm{ext}}}$.
The constant $\mathbf{T}$-gauge transformation with the parameter
${\varepsilon}\in\mathfrak{h}$ acts on the $(0,1)$-gauge field ${\bar{A}}$ as
follows:
${\delta}{\bar{A}}=[{\varepsilon},{\bar{A}}]+{\nabla}_{\bar{\partial}}{\eta}_{\varepsilon}$
(8.42)
where ${\eta}_{\varepsilon}(x,{{\bar{x}}},z,{{\bar{z}}})$ is the compensating
gauge transformation which sufficiently fast decays at $x\to\infty$.
Contracting the vector field (8.42) with $\Omega$ in (8.39) gives us a closed
one-form ${\delta}{\bf m}_{\varepsilon}$ on the space of connections obeying
(8.38), where ${\bf m}_{\varepsilon}$ is linear in $\varepsilon$, ${\bf
m}_{\varepsilon}=\left\langle{\varepsilon},{\bf m}\right\rangle$ for some
${\bf m}\in\mathfrak{h}$:
$\displaystyle{\delta}{\bf m}_{\varepsilon}\,=$
$\displaystyle\,\int_{S_{\mathscr{E}}}\,dx\wedge
dz\wedge\left\langle\left([{\varepsilon},{\bar{A}}]+{\nabla}_{\bar{\partial}}{\eta}_{\varepsilon}\right){\delta}{\bar{A}}\right\rangle=\int_{S_{\mathscr{E}}}dx\wedge
dz\wedge{\bar{\partial}}\left\langle{\varepsilon},{\delta}{\bar{A}}\right\rangle$
(8.43) $\displaystyle\qquad\qquad{\delta}\oint_{x=\infty}dx\
\left\langle{\varepsilon},\int_{{\mathscr{E}}_{x}}d^{2}z\,A_{{\bar{z}}}\right\rangle$
Thus the moment map ${\bf m}$ for the $\mathbf{T}$ action is the residue at
$x=\infty$ of the zero mode of the $\mathfrak{h}$-projection of the
$A_{{\bar{z}}}$ gauge field. The analogous statement holds for the monopoles
of the section 8.1.
Now let us solve (8.38) locally over some domain $D$ in
$\mathbf{C}_{\left\langle x\right\rangle}$. We choose the gauge (C.113) over
each point $x\in D$:
$A_{{\bar{z}}}(z,{{\bar{z}}};x,{{\bar{x}}})\longrightarrow\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}{\xi}(x,{{\bar{x}}})\in\mathfrak{h}$
(8.44)
If $\infty\in D$, then, using (8.43), in the gauge (8.44):
${\xi}(x,{{\bar{x}}})={\xi}_{\infty}+\frac{{\bf m}}{x}+\ldots\,,\qquad
x\to\infty$ (8.45)
where ${\bf m}\in\mathfrak{h}$ is a linear function of the bi-fundamental
masses $({\bf m}_{e})_{e\in\mathrm{Edge}_{\gamma}}$.
Recall that $\mathbf{G}$ is a simple simply-connected Lie group. Therefore,
the restriction ${\mathcal{P}}|_{{\mathscr{E}}_{x}}$ is trivial as a smooth
$\mathbf{G}$-bundle. Therefore ${\bar{A}}$ is just a $\mathfrak{g}$-valued
$(0,1)$-form on $D\times\mathscr{E}$. Now let us decompose
$A_{{\bar{x}}}=a_{{\bar{x}}}+W_{{\bar{x}}}$, with
$a_{{\bar{x}}}\in\mathfrak{h}$ and
$W_{{\bar{x}}}\in{\mathfrak{h}}^{\perp}\subset\mathfrak{g}$, the orthogonal
decomposition being provided by the Killing form $\langle\cdot,\cdot\rangle$.
Then (8.38) implies: ${\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})=0$,
${\bar{\partial}}_{{\bar{z}}}a_{{\bar{x}}}=0$, $W_{{\bar{x}}}=0$, the latter
equation being valid for generic ${\xi}(x)$, corresponding to the irreducible
bundles on $\mathscr{E}$. Indeed, the $F^{0,2}=0$ equation (8.38) splits in
$\mathfrak{g}={\mathfrak{h}}\oplus{\mathfrak{h}}^{\perp}$ as follows:
$\displaystyle F^{0,2}|_{{\mathfrak{h}}}\ =$ $\displaystyle\
{\bar{\partial}}_{{\bar{x}}}{\xi}-{\bar{\partial}}_{{\bar{z}}}a_{{\bar{x}}}=0$
(8.46) $\displaystyle F^{0,2}|_{{\mathfrak{h}}^{\perp}}\ =$ $\displaystyle\
{\bar{\partial}}_{{\bar{z}}}W_{{\bar{x}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}[{\xi},W_{{\bar{x}}}]=0$
For generic $\xi\in\mathfrak{h}$ (irreducible ${\rm
ad}({\mathcal{P}})^{\perp}$) the $\mathfrak{h}^{\perp}$-equation (8.46) does
not have non-zero solutions, hence $W_{{\bar{x}}}=0$. As for the
$\mathfrak{h}$-equation (8.46) in our gauge its solution is:
$a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})=a_{{\bar{x}}}^{(0)}(x,{{\bar{x}}},z)+{{\bar{z}}}{\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})$
Since on $D\times{\mathscr{E}}$ the connection form
$A_{{\bar{z}}}d{{\bar{z}}}+a_{{\bar{x}}}d{{\bar{x}}}$ is simply an
$\mathfrak{h}$-valued $(0,1)$-form the component
$a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})$ must be $z$-periodic. This is
only possible if ${\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})=0$, which
also implies $a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})$ is $z$-independent.
Of course, (8.44) is not the complete gauge fixing: there remain the
$z$-independent $\mathbf{T}$-valued gauge transformations and the
$W({\mathfrak{g}})$-transformations (C.114), which combine into the locally
$z$-constant $N({\mathbf{T}})$-gauge transformations. There are also the
shifts (C.115) by the lattice ${{\rm Q}}^{\vee}\oplus{\tau}{{\rm Q}}^{\vee}$,
generated by the $(z,{{\bar{z}}})$-dependent $\mathbf{T}$-valued gauge
transformations with discrete, as far as the $(x,{\bar{x}})$-dependence is
concerned, parameters ${\beta}_{1},{\beta}_{2}\in{{\rm Q}}^{\vee}$. Let us
expand:
${\xi}(x)=\sum_{i=1}^{r}{\xi}_{i}(x){\alpha}^{\vee}_{i}$ (8.47)
The residual shifts act on the components ${\xi}_{i}(x)$ by
${\xi}_{i}(x)\mapsto{\xi}_{i}(x)+{\beta}_{1,i}+{\beta}_{2,i}{\tau},\qquad
i=1,\ldots,r+1$
where ${\beta}_{1,i},{\beta}_{2,i}\in\mathbb{Z}$, for $i=1,\ldots,r$ are the
expansion coefficients:
${\beta}_{A}=\sum_{i=1}^{r}{\beta}_{A,i}{\alpha}_{i}^{\vee},\qquad A=1,2$
(8.48)
Let ${\bf
t}(x)=\left(t_{i}(x)=e^{2\pi\mathrm{i}\xi_{i}(x)}\right)_{i=1}^{r}\in({\mathbb{C}}^{\times})^{r}$
and $[{\bf t}(x)]\in{\mathscr{E}}^{r}$ be the equivalence classes for the
actions of the lattices ${{\rm Q}}^{\vee}$ and ${{\rm
Q}}^{\vee}\oplus{\tau}{{\rm Q}}^{\vee}$, respectively.
Thus, dividing by all but the locally $z$-independent $N({\mathbf{T}})$-gauge
transformations we arrive at the collection of $r$ points on $\mathscr{E}$,
or, in a more sophisticated fashion, a point $[{\bf t}(x)]$ in
${\mathscr{E}}\otimes{{\rm Q}}$, in addition to the $\mathfrak{h}$-valued
gauge field $a_{{\bar{x}}}(x,{{\bar{x}}})d{{\bar{x}}}$. This is all done over
the generic point $x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$. We haven’t completely fixed the gauge, though.
Let us forget for a moment about the gauge field
${\bar{a}}=a_{{\bar{x}}}(x,{{\bar{x}}})d{{\bar{x}}}$ (8.49)
Then we’d divide by the action $W({\mathfrak{g}})$, giving as back the point
$[{\mathcal{P}}|_{{\mathscr{E}}_{x}}]$ in the orbispace
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})={\mathscr{E}}\otimes{{\rm
Q}}/W({\mathfrak{g}})$, the holomorphic $\mathbf{G}$-bundle on $\mathscr{E}$
whose holomorphic structure is given by
${\bar{\partial}}_{{\bar{z}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}{\xi}(x)$.
The (quasi)map $U:x\mapsto U(x)=[{\mathcal{P}}_{x}]$ is point
$u\in{{\mathfrak{M}}^{\mathrm{ext}}}$ in the extended moduli space of the four
dimensional class II gauge theory.
The considerations similar to those in [Friedman:1997ih] show that the
instanton charge $c_{2}({\mathcal{P}})=N$ bundles on $S_{\mathscr{E}}$
correspond to the degree $N$ quasimaps
$u:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$. This is of
course in line with the original observations on the relations between the
sigma model and gauge instantons [Atiyah:1984tk].
By following the whole $W({\mathfrak{g}})$-orbit of $[{\bf t}(x)]$ in
${\mathscr{E}}^{r}$ as $x$ varies we obtain the curve ${\mathscr{C}}_{u}$,
which is a ramified $W({\mathfrak{g}})$-cover of
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. The fiber of the
projection ${\mathscr{C}}_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ over a point $x$ is the orbit
$W({\mathfrak{g}})\cdot[{\bf t}(x)]\subset{\mathscr{E}}\otimes{{\rm Q}}$
This leads us to the _obscured curve_ which we have encountered earlier in our
solution of the gauge theory using the limit shape equations.
Now, as in our prior discussion of Hitchin systems, let us recall the gauge
field ${\bar{a}}$ in (8.49). It looks like the $\mathfrak{h}$-valued one-form
on ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}$, since it is
locally $z$-independent, and therefore might be a $(0,1)$ part of a
$\mathbf{T}$-connection, defining a holomorpic $\mathbf{T}$-bundle over
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. However, we have
$|W({\mathfrak{g}})|$ worth of choices for ${\bar{\partial}}+{\bar{a}}$ at any
particular point $x$ since the residual $W({\mathfrak{g}})$-symmetry acts both
on $[{\bf t}(x)]$ and ${\bar{a}}$. This means that $\bar{a}$ becomes well-
defined when lifted to ${\mathscr{C}}_{u}$. The way it transforms under the
$W({\mathfrak{g}})$-action permuting the sheets of the cover
${\mathscr{C}}_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ makes it into the $W({\mathfrak{g}})$-equivariant gauge
field on $\mathscr{C}_{u}$, defining a $W({\mathfrak{g}})$-equivariant
holomorphic $\mathbf{T}$-bundle ${\mathscr{T}}$ over $\mathscr{C}_{u}$:
${\mathscr{T}}\in{\mathscr{A}}_{u}={\mathrm{Bun}}_{\mathbf{T}}({\mathscr{C}}_{u})^{W({\mathfrak{g}})}\approx{\operatorname{Hom}}_{W({\mathfrak{g}})}\left({\Lambda},{\rm
Pic}({\mathscr{C}}_{u})\right)\,,$ (8.50)
the idea of the last equality is that every weight
${\lambda}\in\Lambda=\operatorname{Hom}({\mathbf{T}},{\mathbb{C}}^{\times})$
defines a $\mathbb{C}^{\times}$-bundle ${\mathscr{T}}^{\lambda}$ on
$\mathscr{C}_{u}$, in the $W({\mathfrak{g}})$-compatible fashion.
Of course this discussion is not adequate at the branch points, yet hopefully
it can be extended similarly to other constructions of spectral covers
[Hitchin:1987mz], [Donagi:1995alg]. In particular, the analysis near the
branch points should demonstrate that the bundle ${\mathscr{T}}$ has a fixed
topological type, which we shall not attempt to determine in this paper.
We conjecture that (8.50) is the fiber of the projection
${\pi}:{{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$,
sending the extended moduli space of vacua of the class II gauge theory on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ to that of the infinite volume four
dimensional gauge theory. Since the projection ${\mathcal{P}}\mapsto U$ is
roughly of the form $(A_{{\bar{x}}},A_{{\bar{z}}})\mapsto A_{{\bar{z}}}$, it
is Lagrangian with respect to (8.39). This is in agreement with
$\dim_{\mathbb{C}}{\mathfrak{P}}^{\mathrm{ext}}=2\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=2Nh$.
From now on we fix the masses, divide by the global $\mathbf{T}$-action and
work with ${\mathfrak{P}}$ and ${\mathfrak{M}}$.
Thus the study of holomorphic bundles on $S_{\mathscr{E}}$ brought us the
following picture: over each point
$x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$ we hang the
$W({\mathfrak{g}})$-orbit of $[{\bf t}(x)]$ in ${\mathscr{E}}\otimes{{\rm
Q}}$. As $x$-varies, so does the orbit, spanning a curve ${\mathscr{C}}_{u}$
in ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}\otimes{{\rm Q}}$, which projects down to
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. However
${\mathscr{C}}_{u}$ is not yet the cameral curve $\mathcal{C}_{u}$. It is
however relatively straightforward to lift $\mathscr{C}_{u}$ to
$\mathcal{C}_{u}$ using the abelianized determinant line bundle over
${\mathscr{E}}^{r}$ we discussed in the section 6.4.
Now let us discuss the period map. The moduli space ${\mathfrak{P}}$ carries
the holomorphic symplectic structure $\Omega$ which descends from (8.39). From
the complexified textbooks on classical mechanics we learn, as we reviewed in
section 4.4, that choosing some basis $A_{\mathcal{I}},B^{\mathcal{I}}$, in
the integral homology $H_{1}({\mathscr{A}}_{u},{\mathbb{Z}})$ lattice, such
that the cup product of the basis vectors obeys (4.18) and then defining
${\mathfrak{a}}^{\mathcal{I}}$, ${\mathfrak{a}}_{\mathcal{I}}^{D}$ using
(4.16), (4.17), one verifies (4.19). It remains to compare this definition of
the special coordinates with the periods of the differentials
$x\,d{\log}{\mathscr{Y}}_{i}(x)$.
Let us work in the domain $|{\mathfrak{q}}_{i}|\ll 1$. The cycles
${\mathcal{A}}_{\mathcal{I}}$ and ${\mathcal{B}}^{\mathcal{I}}$ which we
defined (cf. Figs. 5.1, 5.3) on the cameral curve ${\mathcal{C}}_{u}$ define
the corresponding one-cycles on ${\mathscr{C}}_{u}={\mathcal{C}}_{u}/{{\rm
Q}}^{\vee}$ and consequently on ${\mathscr{A}}_{u}$.
Now let us compute the symplectic form $\Omega$ on the reduced phase space
${\mathfrak{P}}$. Using that in our gauge
$\int_{\mathscr{E}}d^{2}zA_{{\bar{z}}}(x,{{\bar{x}}})={\xi}(x)$
we obtain
${\Omega}=\int\,d^{2}x\left\langle{\delta}a_{{\bar{x}}}\wedge{\delta}{\xi}\right\rangle=\frac{1}{|W({\mathfrak{g}}|}\int_{{\mathscr{C}}_{u}}\left\langle{\delta}{\bar{a}}\wedge{\delta}{\varphi}\right\rangle$
(8.51)
where we denoted by $\bar{a}$ the $W({\mathfrak{g}}$-equivariant
$\mathbf{T}$-gauge field on $\mathscr{C}_{u}$ and by ${\delta}{\varphi}$ an
$\mathfrak{h}$-valued, $W({\mathfrak{g}})$-equivariant $(1,0)$-form on
$\mathscr{C}_{u}$, given by
${\delta}{\varphi}=x{\delta}{\xi}$
It is now easy to pin down the periods of $d{\mathbb{S}}$ among the periods of
${\delta}{\varphi}$. One uses the cycles in ${\mathscr{A}}_{u}$ where the
monodromy of the gauge field $\bar{a}$ along the cycle
${\mathcal{A}}_{\mathcal{I}}$ or ${\mathcal{B}}^{\mathcal{I}}$ changes by
$2\pi$.
To conclude this section, in parallel with the class I story we conjecture,
that the theories of class II lead to the representation theory of the
Yangians built on the Kac-Moody algebras, i.e. the toroidal algebras. It would
be nice to see whether the Kac-Moody symmetry of instanton moduli spaces
[Dolan:1982dc, Licata_2007] leads to the quasi-classical limit of the Yangian
action, at the level of the phase space ${\mathfrak{P}}^{\mathrm{ext}}$
symmetry.
### 8.3. Noncommutative instantons and Class II* theories
In this subsection we show that the data $t_{i}(x)$ with the cross-cut
transformations (5.16) correspond naturally to the charge $N$ instantons on
the noncommutative space ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, with the
gauge group $U(r+1)$. We shall use the constructions analogous to the
constructions [Nekrasov:2000zz] of instantons on the noncommutative
${\mathbb{R}}^{4}$ adapted to the periodic case. The noncommutative solitons
on a cylinder were studied in [Demidov:2003xq].
The idea of the construction is the following. Consider first the commutative
situation. Let us denote the coordinates on ${\mathbb{R}}^{2}$ by
$x_{1},x_{2}$. Let
$(u_{1},u_{2})\in{\mathbb{T}}^{2}\subset{\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$,
$|u_{1}|=|u_{2}|=1$. Let us denote by $z,x$ the holomorphic local coordinates
on ${\mathbf{C}_{\left\langle x\right\rangle}}\times{\mathscr{E}}$,
$x=x_{1}+{\mathrm{i}}x_{2},\ z=\frac{1}{2\pi\mathrm{i}}\left({\rm
log}(u_{1})+{\tau}{\rm log}(u_{2})\right)$ (8.52)
Consider the $(0,1)$-component of $U(r+1)$ gauge field
${\bar{A}}=A_{{\bar{z}}}d{{\bar{z}}}+A_{{\bar{x}}}d{{\bar{x}}}$, where
$A_{{\bar{z}}}$, $A_{{\bar{x}}}$ are the $(r+1)\times(r+1)$ complex matrices.
They obey (8.38) and in the gauge (8.53) we have
$A_{{\bar{z}}}=\frac{2\pi i}{{\bar{\tau}}-{\tau}}{\rm
diag}\left({\xi}_{i}(x,{{\bar{x}}})\right)_{i=1}^{r+1}$ (8.53)
where, by the considerations analogous to those around (8.46) we conclude that
${\bar{\partial}}_{{\bar{x}}}{\xi}_{i}=0$. Again, there are the residual gauge
transformations
$A_{{\bar{z}}}\mapsto G^{-1}A_{{\bar{z}}}G+G^{-1}{\partial}_{{\bar{z}}}G$
which permute the eigenvalues ${\xi}_{i}(x)$:
$\\{{\xi}_{i}(x)\\}\mapsto\\{{\xi}_{{\sigma}(i)}(x)\\}$, for some
${\sigma}\in{\mathcal{S}}_{r+1}$, and shift them
${\xi}_{i}(x)\mapsto{\xi}_{i}(x)+l_{i}-{\tau}n_{i}$ (8.54)
The latter are generated by the diagonal gauge transformations:
$G={\rm diag}\left(u_{1}^{n_{i}}u_{2}^{l_{i}}\right)_{i=1}^{r+1}$ (8.55)
We can partially reduce the ambiguity (8.54) by passing to the exponential
variables:
$t_{j}(x)=e^{2\pi\mathrm{i}{\xi}_{j}(x)}\,\qquad j=1,\ldots,r+1$ (8.56)
The residual gauge transformations (8.54) become
$t_{j}(x)\mapsto{\mathfrak{q}}^{n_{j}}t_{j}(x),\qquad t_{j}(x)\mapsto
t_{{\sigma}(j)}(x)$ (8.57)
for some integers $n_{i}\in{\mathbb{Z}}$ and permutations
${\sigma}\in{\mathcal{S}}_{r+1}$.
Let us now consider the noncommutative case. Let us replace the algebra of
functions of $u_{1},u_{2},x_{1},x_{2}$ by the noncommutative algebra, with the
generators
${\widehat{u}}_{1},{\widehat{u}}_{2},{\widehat{x}}_{1},{\widehat{x}}_{2}$,
obeying:
$\displaystyle{\widehat{u}}_{1}^{-1}{\widehat{x}}_{i}{\widehat{u}}_{1}={\widehat{x}}_{i}+{\hbar}_{i}\cdot
1$ (8.58)
$\displaystyle{\widehat{u}}_{2}^{-1}{\widehat{x}}_{i}{\widehat{u}}_{2}={\widehat{x}}_{i},$
$\displaystyle{\widehat{u}}_{1}{\widehat{u}}_{2}={\widehat{u}}_{2}{\widehat{u}}_{1},$
$\displaystyle[{\widehat{x}}_{1},{\widehat{x}}_{2}]=0$
where
$\frac{\mathfrak{m}}{r+1}={\hbar}={\hbar}_{1}+{\mathrm{i}}{\hbar}_{2}$
Then the analogue of (8.54) for
$G={\rm
diag}\left({\widehat{u}}_{1}^{n_{i}}{\widehat{u}}_{2}^{l_{i}}\right)_{i=1}^{r+1}$
(8.59)
gives:
$t_{i}(x)\mapsto q^{n_{i}}t_{i}(x-\frac{n_{i}}{r+1}{\mathfrak{m}}),\qquad
t_{i}(x)\mapsto t_{{\sigma}(i)}(x)$ (8.60)
These are precisely the ${{}^{i}{\mathcal{W}}}$-transformations of the class
II* type ${\widehat{A}}_{r}$ gauge theory, with $t_{i}(x)$ given in (5.14),
(7.97).
Instead of giving more systematic discussion along the lines of
[Connes:1997cr, Nekrasov:1998ss, Astashkevich:1998uc, Nekrasov:2000zz,
Schwarz:2001ru, Douglas:2001ba] let us comment on the relation to the group
$\widehat{GL}_{\infty}$. We claim that the $U(r+1)$ instantons on the
noncommutative ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ with both
${\mathbb{R}}^{2}$ and ${\mathbb{T}}^{2}$ separately commutative, can be
interpreted as the commutative periodic monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with the gauge group
$\widehat{GL}_{\infty}$.
The idea is to interpret the noncommutative $U(r+1)$ gauge fields on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ as the $\widehat{GL}_{\infty}$-gauge
fields on ${\mathbb{R}}^{2}_{x_{1},x_{2}}\times{\mathbb{S}}^{1}_{\psi}$, using
the relation of the group $\widehat{GL}_{\infty}$ to the quantization of the
volume-preserving diffeomorphisms of a cylinder
${\mathbb{R}}^{1}_{p}\times{\mathbb{S}}^{1}_{q}$, i.e. the pseudo-differential
operators ${\Psi}DO$ on ${\mathbb{S}}^{1}_{q}$. Here the generators
${\widehat{u}}_{1,2},{\widehat{x}}_{1,2}$ of the algebra (8.58) are related to
the commutative coordinates $(x_{1},x_{2},\psi)$ of the monopole theory and
the generators $(p,e^{\mathrm{i}q})$ of $\widehat{GL}_{\infty}$ in the via
$\displaystyle{\widehat{u}}_{2}=e^{\mathrm{i}\psi}$ (8.61)
$\displaystyle{\widehat{u}}_{1}=e^{\mathrm{i}q}$
$\displaystyle{\widehat{x}}_{1}=x_{1}-{\mathrm{i}}{\hbar}_{1}{\partial}_{q}$
$\displaystyle{\widehat{x}}_{2}=x_{2}-{\mathrm{i}}{\hbar}_{2}{\partial}_{q}$
## Chapter 9 Higher dimensional theories
In this section we briefly go over the higher dimensional generalizations. We
shall consider the lifts of all our theories to five dimensions, and the lifts
of some of our theories to six dimensions. The latter restriction comes from
the possibility of encountering gauge and mixed anomalies in six dimensions,
which would prohibit the decoupling of the gauge sector from the supergravity
and ultimately string theory modes.
The classical ${\mathcal{N}}=2$ theories in four dimensions admit a canonical
lift to the ${\mathcal{N}}=1$ theories in six dimensions. Under this lift the
vector multiplets become the vector multiplets, and the hypermultiplets become
the hypermultiplets. The structure of the hypermultiplet does not change,
while the structure of the vector multiplet does change, for the complex
scalar $\Phi$ in the adjoint representation becomes the remaining component
$A_{5}+\mathrm{i}A_{6}$ of the gauge field.
We then compactify the theory on a two-torus ${\mathbb{T}}^{2}$. In addition
to the metric on the torus we shall also fix a background $B$-field, a
constant two-form. The combined metric and the two-form moduli are described
by a $2\times 2$ matrix $G$, with the positive definite symmetric part $g$. It
is convenient to parametrize $G$ by two complex numbers $T,U$, with ${\rm
Im}U,{\rm Im}T\geq 0$. The parameter $T$ encodes the complex structure of
$\mathbb{T}^{2}$, while the parameter $U$ is the complexified Kähler class:
$\displaystyle
U=\int_{\mathbb{T}^{2}}B+\mathrm{i}\,\int_{\mathbb{T}^{2}}\sqrt{{\rm
det}(g)},$ (9.1) $\displaystyle T=\frac{g_{12}+\mathrm{i}\sqrt{{\rm
det}(g)}}{g_{11}}$
Five (six) dimensional supersymmetric gauge theory compactified on a circle
(two-torus) would look four dimensional at low energy. The microscopic gauge
coupling ${\tau}$ in four dimensions is proportional to $U$ while $T$
determines the complex geometry of the $x$-plane $\mathbf{C}_{\left\langle
x\right\rangle}$. One can study the corresponding Seiberg-Witten geometry. Its
key feature compared to special geometry of more traditional four dimensional
models is the periodicity in the $x$-variable [Nekrasov:1996cz]. In the five
dimensional theory the $x$-plane becomes the cylinder
${\mathbf{C}_{\left\langle x\right\rangle}}^{\times}$, while in the six
dimensional theory the $x$-plane becomes the two-torus
${\mathbf{C}_{\left\langle
x\right\rangle}}/2{\pi}{\mathrm{i}}({\mathbb{Z}}\oplus T{\mathbb{Z}})$. It has
to do with the large gauge transformations. The result of these additional
symmetries is the relativistic nature of the corresponding integrable systems.
For example, the periodic Toda chain describing the pure ${\mathcal{N}}=2$
theory in four dimensions becomes the relativistic Toda chain. The
Hamiltonians of the relativistic systems have periodic dependence on momenta,
which are the rapidities of the particles. The resulting quantized
Hamiltonians are the difference operators.
Our discussion modifies in the case of five dimensional gauge theories in two
aspects. First, the notion of the amplitude function accommodates the large
gauge transformations:
${\mathscr{Y}}_{i}(x)={\exp}\,\langle\,{\operatorname{tr}}_{{\mathbf{v}}_{i}}\
{\log}\left(e^{{{\mathrm{i}\beta}}(x-{\Phi}_{i})}-1\right)\,\rangle_{u}$ (9.2)
where the dimension length $\beta$ characterizes the circumference of the
compactification circle. The limit shape integral equations (5.8) generalize
straightforwardly, with the kernel:
${\mathcal{K}}_{\beta}({\mathrm{x}})=\frac{{\mathrm{i}\beta}}{12}{\mathrm{x}}^{3}-\frac{{\rm
log}({{\mathrm{i}\beta}}{\Lambda}_{\rm
UV})}{2}{\mathrm{x}}^{2}-\frac{1}{{\mathrm{i}\beta}^{2}}\,{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}\right)$ (9.3)
Secondly, there are additional couplings in five dimensions: the levels
$k_{i}$ of the Chern-Simons interactions CS${}_{5}(A^{i})$,
$i\in\mathrm{Vert}_{\gamma}$. Effectively the Chern-Simons term changes the
gauge coupling ${\mathfrak{q}}_{i}$ to the $x$-dependent quantity:
${\mathfrak{q}}_{i}\longrightarrow{\mathfrak{q}}_{i}\,e^{{{\mathrm{i}\beta}}k_{i}x}$
In the six dimensional case the amplitude and the kernel (9.3) modify to:
${\mathscr{Y}}_{i}(x)={\exp}\,\langle\,{\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\log}\,{\theta}\left(e^{{{\mathrm{i}\beta}}(x-{\Phi}_{i})};Q\right)\,\rangle_{u}$
(9.4)
$\displaystyle{\mathcal{K}}_{\beta}({\mathrm{x}})=\frac{{\mathrm{i}\beta}}{12}{\mathrm{x}}^{3}-\frac{{\rm
log}({{\mathrm{i}\beta}}{\Lambda}_{\rm
UV})}{2}{\mathrm{x}}^{2}-\frac{1}{{\mathrm{i}\beta}^{2}}\,{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}\right)-$ (9.5)
$\displaystyle\qquad\qquad\qquad\sum_{n=1}^{\infty}\left({\rm
Li}_{3}\left(e^{{{\mathrm{i}\beta}}{\mathrm{x}}}Q^{n}\right)+{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}Q^{n}\right)\right)$
where $Q={\exp}\,2\pi\mathrm{i}T$, and now ${\beta}^{2}$ is the scale of the
area of the compactification torus ${\mathbb{T}}^{2}$. We will provide further
details on five and six dimensional theories in the companion paper [NP2012b].
## Appendix A McKay correspondence, D-branes, and M-theory
### A.1. From finite groups to Lie groups
McKay correspondence states [McKay:1980] that the affine ADE graphs $\gamma$
can be constructed from the representation theory of finite subgroups $\Gamma$
of $SU(2,{\mathbb{C}})$.
The vertices $i\in\mathrm{Vert}_{\gamma}$ in this case are the irreducible
representations of $\Gamma$, $i\mapsto{\mathcal{R}}_{i}$. One usually
enumerates them, $\mathrm{Vert}_{\gamma}=\\{0,1,\ldots,r\\}$, so that $0$
corresponds to the trivial representation ${\mathcal{R}}_{0}={\mathbb{C}}$. By
$r$ in this section we denote the number of nodes in the finite graph
$\gamma_{\text{fin}}$ obtained by discarding the node ‘‘0’’ from affine
$\gamma$. In all cases $r$ is the rank of the Cartan matrix associated to
Dynkin graph. Moreover, the Dynkin marks $a_{i}$ are the dimensions of
$V_{i}$, and the numbers of colors in affine quivers is
${\mathbf{v}}_{i}=Na_{i},\ \qquad$ (A.1)
where $a_{i}={\rm dim}{\mathcal{R}}_{i}$, and $N$ is some non-negative
integer. As we said above, $i=0$ corresponds to the trivial representation, so
$a_{0}=1$. In the table of McKay correspondence we write ‘‘$i(a_{i})$’’ by
each node to denote its label $i$ and Dynkin mark $a_{i}$. We label the nodes
in Bourbaki conventions.
The number of edges $I_{ij}$ in the McKay graph between the node $i$ and the
node $j$ is the multiplicity of representation $\mathcal{R}_{j}$ in the tensor
product of $\mathcal{R}_{i}$ with the defining representation $\mathbb{C}^{2}$
${\mathbb{C}}^{2}\otimes{\mathcal{R}}_{i}=\bigoplus_{j}{\mathbb{C}}^{I_{ij}}\otimes{\mathcal{R}}_{j}.$
(A.2)
The equation ${\beta}_{i}=0$ is verified by computing the dimensions of the
left and the right hand sides.
The order of $\Gamma$ agrees with the dimensions of the irreducible
representations / Dynkin marks of the McKay/Dynkin affine graph computed using
the standard relation from the orthogonality of characters
$|\Gamma|=\sum_{i}\dim\mathcal{R}_{i}^{2}=\sum_{i}a_{i}^{2}$ (A.3)
Polyhedron | $\Gamma$ | $T_{a,b,c}$ | Affine Dynkin graph $\gamma$ with labels $i(a_{i})$ | Lie $\widehat{\mathfrak{g}}$
---|---|---|---|---
| $\mathbb{Z}_{r+1}$ (r=5) | $T_{r,1,1}$ | | $\widehat{A}_{r}$ (r=5)
| $\mathbb{B}\mathbb{D}_{r-2}$ (r=7) | $T_{r-2,2,2}$ | | $\widehat{D}_{r}$ (r=7)
| $\mathbb{B}\mathbb{T}$ | $T_{3,3,2}$ $(\widehat{T}_{3,3,3})$ | | $\widehat{E}_{6}$
| $\mathbb{B}\mathbb{O}$ | $T_{4,3,2}$ $(\widehat{T}_{4,4,2})$ | | $\widehat{E}_{7}$
| $\mathbb{B}\mathbb{I}$ | $T_{5,3,2}$ $(\widehat{T}_{6,3,2})$ | | $\widehat{E}_{8}$
McKay’s observation is that $\gamma$ is affine ADE Dynkin diagram, with the
trivial representation $\mathcal{R}_{0}$ associated with the affine node
‘‘0’’. The finite Dynkin graph $\gamma_{\text{fin}}$ is always a tri-star
graph $T_{a,b,c}$ with one trivalent vertex and three legs containing $a,b,c$
vertices (where in the counting we included the center trivalent vertex).
Hence the rank of the finite quiver $\gamma_{\mathrm{fin}}$ is $a+b+c-2$. In
fact $a,b,c$ have simple interpretation: the group $\Gamma$ is always a
Coxeter group $\mathrm{Cox}(a,b,c)$ defined on three generators $(x,y,z)$
subject to relation
$x^{a}=y^{b}=z^{c}=xyz$ (A.4)
The affine graph of $\widehat{E}_{r}$ series is also a trivalent graph denoted
by $(\widehat{T}_{a,b,c})$ in the table of McKay correspondence. Note that for
each of the three cases $\widehat{E}_{6},\widehat{E}_{7},\widehat{E}_{8}$ the
identity
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ (A.5)
holds, which makes contact with the unimodular parabolic singularities.
Now we shall list explicitly the discrete subgroups of $SU(2)$, the
classification going back to Plato and Klein [Klein:1884].
### A.2. Platonic symmetries
#### A.2.1. Cyclic group
Let us present cyclic group $\mathbb{Z}_{r+1}$ in the $(a,b,c)$ notation:
consider group generated by $x,y,z$ subject to
$x^{r}=y=z=xyz$ (A.6)
Then $z=y$ and $xy=1$ and $y=x^{r}$ hence
$x^{r+1}=1$ (A.7)
hence $\mathbb{Z}_{r+1}=\mathrm{Cox}(r,1,1)=\mathrm{McKay}(A_{r})$. The order
of $\Gamma$ is
$|{\mathbb{Z}}_{r+1}|=r+1=\sum_{i\in\gamma}a_{i}^{2}=(r+1)\cdot 1$ (A.8)
#### A.2.2. Dihedral group
The dihedral group
$\mathbb{D}_{r-2}\simeq\mathbb{Z}_{r-2}\rtimes\mathbb{Z}_{2}$ of order
$2(r-2)$ is the symmetry group of regular $r-2$-gon considered as a subgroup
of $SO(3)$. The lift of dihedral group
$\mathbb{Z}_{2}\to\mathbb{B}\mathbb{D}_{r-2}\to\mathbb{D}_{r-2}$ to a subgroup
of $SU(2)$ is called bidihedral group $\mathbb{B}\mathbb{D}$, which has order
$4(r-2)$. The $\mathbb{B}\mathbb{D}_{r}$ is generated by $(x,y,z)$ subject to
relations
$x^{r-2}=y^{2}=z^{2}=xyz$ (A.9)
hence $\mathbb{B}\mathbb{D}_{r-2}=\mathrm{Cox}(r-2,2,2)=\mathrm{McKay}(D_{r})$
The order of $\Gamma$ is
$|\mathbb{B}\mathbb{D}_{r-2}|=4(r-2)=\sum_{i\in\gamma}a_{i}^{2}=(r-3)\cdot
4+4\cdot 1$ (A.10)
#### A.2.3. Tetrahedral group
The tetrahedral group $\mathbb{T}\simeq A_{4}$, of order 12, is the subgroup
of $SO(3)$ realizing the symmetries of tetrahedron. The lift of tetrahedral
group $\mathbb{Z}_{2}\to\mathbb{B}\mathbb{T}\to\mathbb{T}$ to a subgroup of
$SU(2)$ is called bitetrahedral group $\mathbb{B}\mathbb{T}$, which has order
$24$ and is generated by $(x,y,z)$ subject to the relations
$x^{3}=y^{3}=z^{2}=xyz$ (A.11)
Hence $\mathbb{B}\mathbb{T}=\mathrm{Cox}(3,3,2)=\mathrm{McKay}(E_{6})$. The
order of $\Gamma$ is
$|\mathbb{B}\mathbb{T}|=24=\sum_{i\in\gamma}a_{i}^{2}=3\cdot(1^{2}+2^{2})+3^{2}=24$
(A.12)
#### A.2.4. Octahedral group
The octahedral group $\mathbb{O}\simeq S_{4}$, of order $24$, is the subgroup
of $SO(3)$ realizing the symmetries of the cube/octahedron. The lift to
$SU(2)$ is called bioctahedral group $\mathbb{B}\mathbb{O}$, of order $48$;
the $\mathbb{B}\mathbb{O}$ can be generated by $(x,y,z)$ subject to relations
$x^{4}=y^{3}=z^{2}=xyz$ (A.13)
associated to the symmetry axes of the cube/octahedron of the order $4,3$ and
$2$. Hence $\mathbb{B}\mathbb{O}=\mathrm{Cox}(4,3,2)=\mathrm{McKay}(E_{7})$.
The order of $\Gamma$ is
$|\mathbb{B}\mathbb{O}|=48=\sum_{i\in\gamma}a_{i}^{2}=2\cdot(1^{2}+2^{2}+3^{2})+4^{2}+2^{2}$
(A.14)
#### A.2.5. Icosahedral group
The icosahedral group $\mathbb{I}\simeq A_{5}$, of order $60$, is the subgroup
of $SO(3)$ realizing the symmetries of the icosahedron/dodecahedron The lift
to $SU(2)$ is called biicosahedral group $\mathbb{B}\mathbb{O}$, of order
$120$; the biicosahedral group $\mathbb{B}\mathbb{I}$ can be generated by
$(x,y,z)$ subject to relations
$x^{5}=y^{3}=z^{2}=xyz$ (A.15)
associated to the symmetry axes of the icosahedron/dodecahedron of the order
$5,3$ and $2$. Hence
$\mathbb{B}\mathbb{O}=\mathrm{Cox}(5,3,2)=\mathrm{McKay}(E_{8})$. The order of
$\Gamma$ is
$|\mathbb{B}\mathbb{I}|=120=\sum_{i\in\gamma}a_{i}^{2}=(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2})+2^{2}+4^{2}+3^{2}$
(A.16)
### A.3. D-branes at singularities
The physical explanation of the relation between the
${\mathbb{C}}^{2}/{\Gamma}$-singularities [Kronheimer:1989] and the
superconformal theories with ${\mathcal{N}}=2$ supersymmetry is the following.
Consider the IIB string in the background
${\mathbb{C}}^{2}/{\Gamma}\times{\mathbb{C}}^{1}\times{\mathbb{R}}^{1,3}$,
where ${\Gamma}\subset SU(2)\subset SO(4)$ acts on
${\mathbb{C}}^{2}\approx{\mathbb{R}}^{4}$ by the hyperkähler rotations. This
background preserves half of the ten dimensional IIB supersymmetry. Now add a
stack of $N$ regular D3 branes. At the orbifold singularity the regular D3
brane splits as a collection of $r+1$ types of fractional branes, which
correspond to the irreducible representations of $\Gamma$ [Douglas:1996sw,
Johnson:1996py]. Moreover, the regular brane contains $a_{i}$ fractional
branes of the ${\mathcal{R}}_{i}$ type. The stack of $N$ regular brane splits,
therefore, as a collection of $Na_{i}$ fractional branes of $\mathcal{R}_{i}$
type, for all $i=0,\ldots,r$. Each cluster of fractional branes can be
classically moved anywhere in the two-plane, transverse to the singularity and
the D3 brane worldvolume. To summarize, the D3 branes are located at:
$0\,\times\,\\{{\mu}_{0}^{Na_{0}},{\mu}_{1}^{Na_{1}},\ldots,{\mu}_{r}^{Na_{r}}\\}\times{\mathbb{R}}^{1,3}$
Here $\\{{\mu}_{0}^{v_{0}},\ldots,{\mu}_{r}^{v_{r}}\\}$ represents the
positions of the $v_{i}$ copies of the $\mathcal{R}_{i}$ type of the
fractional branes in the two dimensional plane
${\mathbb{C}}^{1}\approx{\mathbb{R}}^{2}$ transverse to the singular ALE space
${\mathbb{C}}^{2}/{\Gamma}$ and the worldvolume ${\mathbb{R}}^{1,3}$ of the
branes. The positions ${\mu}_{i}$ are the ones which enter the Eq. (2.10). At
the low energy the worldvolume theory on the D3 branes is the quiver gauge
theory we are studying. If all ${\mu}_{i}$ parameters are scaled to zero (or
at least all coincide), the theory has no scale except for the string scale
which is absent in the low energy description. Therefore the theory has the
scaling invariance, which is promoted to the full ${\mathcal{N}}=2$
superconformal invariance. The couplings ${\tau}_{i}$, except for
$\tau=\sum_{i=0}^{r}a_{i}{\tau}_{i}$ (A.17)
which corresponds to the IIB dilaton-axion field, are not of geometric origin.
The gauge couplings $\tau_{i}$, $i=1,\ldots,r$, come from the twisted sector
fields:
$\tau_{i}=\int_{{\Sigma}_{i}}B_{RR}+{\tau}B_{NS}$ (A.18)
where $B_{NS}$, $B_{RR}$ are the Neveu-Schwarz(NSNS) and the Ramond-Ramond
two-form fields of the IIB supergravity, and $\Sigma_{i}$ stand for the non-
trivial two-cycles in the resolved $\widetilde{{\mathbb{C}}^{2}/{\Gamma}}$
geometry. These exceptional cycles are well-known to correspond to the simple
roots of $\mathfrak{g_{\text{q}}}$.
The 4d gauge theory on the stack of D3 branes in the above geometry is
$\mathcal{N}=2$ supersymmetric theory of the affine ADE quiver type. To find
the algebraic integrable system associated to the IR solution of such theory
we compactify the worldvolume of the D3 branes $(x_{0},x_{1},x_{2},x_{3})$ on
a circle $S^{1}$, say, along $x_{3}$. Hence, our setup is $N$ D3 branes along
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}$ inside the IIB string theory on
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma$. Here $\mathbf{C}_{\left\langle
x\right\rangle}$ is the real-two dimensional space associated with the scalars
of $\mathcal{N}=2$ vector multiplet in 4 dimensions. For 4d theories this is
affine complex line $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}$, for
5d theories on $S^{1}$ this is a cylinder $\mathbf{C}_{\left\langle
x\right\rangle}=\mathbb{C}^{\times}$ and for 6d theories on $T^{2}$ this is a
torus $\mathbf{C}_{\left\langle x\right\rangle}=\mathscr{E}_{\left\langle
x\right\rangle}$.
Given this type IIB string theory realization of the affine ADE quiver theory,
first we perform T-duality along $\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}$. The stack of $N$ D3-branes converts to the stack of $N$
D2-branes on $\mathbb{R}^{2,1}$ in type IIA string theory on
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma$. Next we lift the type IIA string
to M-theory on $\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma\times\mathbb{S}^{1}_{\left\langle
x_{10}\right\rangle}$ where $\mathbb{S}^{1}_{\left\langle
x_{10}\right\rangle}$ is the M-theory circle. The radius of M-theory circle is
determined by IIB coupling constant (A.17). We treat the product of two
circles along $x_{3}$ and along $x_{10}$ as the elliptic curve
$\mathscr{E}=\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbb{S}^{1}_{\left\langle x_{10}\right\rangle}$
which has elliptic modulus $\mathfrak{q}=e^{2\pi\mathrm{i}\tau}$. The stack of
$N$ D2 branes converts to the stack of $N$ M2 branes along $\mathbb{R}^{1,2}$.
So finally we arrive to the M-theory picture.
That is, we consider the following configuration in M-theory: the space-time
background is
${\mathbb{R}}^{1,2}\times X_{4}\times{\mathbb{C}}^{2}/{\Gamma}$ (A.19)
to which we add a stack of $N$ M2 branes whose worldvolume is
${\mathbb{R}}^{1,2}$ in (A.19) localized at the orbifold singularity in
${\mathbb{C}}^{2}/{\Gamma}$, and anywhere in $X_{4}=\mathbb{C}_{\left\langle
x\right\rangle}\times\mathscr{E}_{\mathfrak{q}}$. The M-theory on
$\mathbb{C}^{2}/\Gamma$ is believed to contain the seven dimensional $G$-gauge
theory with sixteen supersymmetries, localized at the singular locus. The
maximal torus $T$ part of the gauge bosons and their superpartners descend
from the supergravity modes associated with the hyperkähler deformations of
the metric on ${\mathbb{R}}^{4}/{\Gamma}$. In particular, the gauge fields are
the modes of the three-form field $C_{3}$ integrated along the collapsed two-
cycles $\Sigma_{i}$, $i=1,\ldots,r$ in the deformed geometry
${\widetilde{{\mathbb{R}}^{4}/{\Gamma}}}$. The remaining $W$-bosons come from
the M2-branes wrapped on various two-cycles in
$H_{2}({\widetilde{{\mathbb{R}}^{4}/{\Gamma}}},{\mathbb{Z}})\approx{{\rm Q}}$.
In the IR, the stack of $N$ M2 branes stretched along $\mathbb{R}^{1,2}$ in
this 7d $G$-gauge theory on $\mathbb{R}^{1,2}\times X_{4}$ dissolve into the
charge $N$ $G$-instantons on $X_{4}=\mathbf{C}_{\left\langle
x\right\rangle}\times\mathscr{E}_{\mathfrak{q}}$ [Intriligator:1996ex,
deBoer:1996ck, Witten:1997kz, Blum:1997mm, Kapustin:1998fa, Kapustin:1998xn,
Katz:1997eq]. The moduli space of framed charge $N$ $G$-instantons on $X_{4}$
is the phase space of the algebraic integrable system (see chapter 4) we have
found from the microscopic four dimensional instanton counting in the quiver
gauge theory.
## Appendix B Partitions and free fermions
Recall that a partition $\lambda$ is a non-increasing sequence of integers,
stabilizing at zero,
${\lambda}=({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots\geq{\lambda}_{{\ell}({\lambda})}>0=0=0\ldots)$
(B.1)
The number ${\ell}({\lambda})\geq 0$ is called the _length_ of the partition
$\lambda$, while
$|{\lambda}|=\sum_{i=1}^{\ell({\lambda})}{\lambda}_{i}$ (B.2)
is called the _size_ of the partition $\lambda$.
Consider the theory of a single chiral complex fermion, which is a
$(1/2,0)$-differential, in two dimensions
${\mathcal{L}}=\int\,{\tilde{\psi}}{\bar{\partial}}{\psi}$ (B.3)
The theory (B.3) has a $U(1)$-symmetry
$({\psi},{\tilde{\psi}})\mapsto(e^{i{\alpha}}{\psi},e^{-i{\alpha}}{\tilde{\psi}})$
(B.4)
One can couple the fermion to the background abelian gauge field corresponding
to this symmetry:
${\bar{\partial}}\mapsto{\bar{\partial}}+{\bar{A}}$ (B.5)
so that the Lagrangian (B.3) deforms to
${\mathcal{L}}\to{\mathcal{L}}+\int{\bar{A}}J$
where $J={\tilde{\psi}}{\psi}$ is the $U(1)$ current, which we define as an
operator below.
In studying the space of states ${\mathcal{H}}$ corresponding to the
quantization of the theory (B.4) one can distinguish various sectors,
corresponding to twisted boundary conditions: Expand:
$\displaystyle{\psi}(z)=\left(\frac{dz}{z}\right)^{\frac{1}{2}}\sum_{r\in{\mathbb{Z}}+{\alpha}}{\psi}_{r}z^{-r}$
(B.6)
$\displaystyle{\tilde{\psi}}(z)=\left(\frac{dz}{z}\right)^{\frac{1}{2}}\sum_{r\in{\mathbb{Z}}-{\alpha}}{\tilde{\psi}}_{r}z^{-r}$
where ${\alpha}=\int_{{\mathbb{S}}^{1}}{\bar{A}}$. The fermion modes
${\psi}_{r},{\tilde{\psi}}_{s}$ form the Clifford algebra:
$\\{\,{\psi}_{r},\,{\tilde{\psi}}_{s}\,\\}={\delta}_{r+s,0}\,\qquad\\{\,{\psi}_{r},\,{\psi}_{s}\,\\}=\\{\,{\tilde{\psi}}_{r},\,{\tilde{\psi}}_{s}\,\\}=0$
(B.7)
The vacuum state $|{\emptyset};{\alpha}\rangle$ is annihilated by all the
${\alpha}$-positive harmonics
$\displaystyle{\psi}_{r}\,|{\emptyset};{\alpha}\rangle=0,\qquad r>{\alpha}$
(B.8) $\displaystyle{\tilde{\psi}}_{r}\,|{\emptyset};{\alpha}\rangle=0,\qquad
r>-{\alpha}$
The space of states is built by acting on $|{\emptyset};{\alpha}\rangle$ with
creation operators ${\psi}_{r}$, $r\leq{\alpha}$, and ${\tilde{\psi}}_{s}$,
$s\leq-{\alpha}$. The resulting Hilbert space ${\mathcal{H}}_{\alpha}$ has a
basis labeled by partitions:
$\displaystyle\qquad\qquad\qquad{\mathcal{H}}_{\alpha}=\bigoplus_{\lambda}{\mathbb{C}}|{\lambda};{\alpha}\rangle$
(B.9) $\displaystyle\qquad|{\lambda};{\alpha}\rangle=$
$\displaystyle{\psi}_{-{\lambda}_{1}+{\alpha}}{\psi}_{-{\lambda}_{2}+1+{\alpha}}\ldots{\psi}_{-{\lambda}_{{\ell}({\lambda})}+{\ell}({\lambda})-1+{\alpha}}\,{\tilde{\psi}}_{-{\alpha}}{\tilde{\psi}}_{-{\alpha}-1}\ldots{\tilde{\psi}}_{-{\alpha}+1-{\ell}({\lambda})}\,|{\emptyset};{\alpha}\rangle$
Define the normal ordering with respect to the ${\alpha}=0$ vacuum:
$:{\psi}_{i}{\tilde{\psi}}_{j}:\
=\left\\{\begin{matrix}{\psi}_{i}{\tilde{\psi}}_{j}\,,&j>0\\\ &\\\
-{\tilde{\psi}}_{j}{\psi}_{i}\,,&j\leq 0\end{matrix}\right.$ (B.10)
The $U(1)$ symmetry of the Lagrangian (B.3) is promoted to the
${\widehat{u}(1)}$ current algebra symmetry. It is generated by the operator
$\displaystyle J(z)=$ $\displaystyle\ :{\tilde{\psi}}(z){\psi}(z):\
=\sum_{n\in\mathbb{Z}}J_{n}z^{-n-1}dz$ (B.11) $\displaystyle
J_{n}=\sum_{r\in{\mathbb{Z}}+{\alpha}}:{\tilde{\psi}}_{r}{\psi}_{n-r}:$
The generating function
$\sum_{{\lambda}}{\mathfrak{q}}^{|{\lambda}|}t^{{\ell}({\lambda})}=\prod_{n=1}^{\infty}\frac{1}{1-t{\mathfrak{q}}^{n}}$
(B.12)
is a character of the fundamental $\widehat{u(1)}$ module,
${\mathcal{H}}_{0}$.
## Appendix C Lie groups and Lie algebras
In this section we fix our notations for the notions from the Lie group and
Lie algebra theory we are using in the work. In this section we work over the
field $\mathbb{C}$ of complex numbers.
### C.1. Finite dimensional Lie algebras
Let $\mathfrak{g}$ be a finite-dimensional simply-laced simple Lie algebra,
$\mathfrak{h}$ its Cartan subalgebra. Let $\mathbf{G}$ be the corresponding
_simply-connected_ simple Lie group, and $\mathbf{T}\subset\mathbf{G}$ the
corresponding to $\mathfrak{h}$ maximal torus. We have the exponential map
${\exp}_{\mathfrak{h}}:{\mathfrak{h}}\to\mathbf{T}$, which is a restriction on
$\mathfrak{h}$ of the exponential map
${\exp}_{\mathfrak{g}}:{\mathfrak{g}}\to\mathbf{G}$. We shall only use the
${\exp}_{\mathfrak{h}}$ map in this work and will omit the $\mathfrak{h}$
subscript in what follows. We also use the notation:
${\mathbf{e}}({\bf x})={\exp}\,(2\pi\mathrm{i}{\bf x})=e^{2\pi\mathrm{i}{\bf
x}}\in\mathbf{T},\qquad{\rm for}\qquad{\bf x}\in\mathfrak{h}$ (C.1)
#### C.1.1. The coroots
The kernel of the map (C.1), i.e. the set ${{\rm
Q}}^{\vee}\subset\mathfrak{h}$ which is mapped to the identity element in
$\mathbf{T}$, is called the coroot lattice (it is obviously an abelian group).
The coroot lattice ${{\rm Q}}^{\vee}$ has the rank $r={\rm rk}\mathbf{G}={\rm
dim}{\mathfrak{h}}$. Let us denote by ${\alpha}_{i}^{\vee}$, $i=1,\ldots,r$
its integral basis:
${{\rm
Q}}^{\vee}={\mathbb{Z}}{\alpha}_{1}^{\vee}\oplus{\mathbb{Z}}{\alpha}_{2}^{\vee}\oplus\ldots\oplus{\mathbb{Z}}{\alpha}_{r}^{\vee}$
The generators ${\alpha}_{i}^{\vee}$ are called the simple coroots.
Now, let $z\in{\mathbb{C}}^{\times}$, and ${\alpha}^{\vee}\in{{\rm
Q}}^{\vee}$, then
$z^{{\alpha}^{\vee}}\equiv{\mathbf{e}}\,\left(\frac{{\log}(z)}{2\pi\mathrm{i}}{\alpha}^{\vee}\right)\in\mathbf{T}$
(C.2)
is independent of the choice of the branch of the ${\log}(z)$. This identifies
the coroot lattice with the lattice of homomorphisms:
${{\rm Q}}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},\mathbf{T})$
(C.3)
Using Eq. (C.2) we can parametrize $\mathbf{T}$ by
${\mathbf{z}}=\left(z_{1},\ldots,z_{r}\right)\mapsto
t_{\mathbf{z}}=\prod_{i=1}^{r}z_{i}^{{\alpha}_{i}^{\vee}}$ (C.4)
where $z_{i}\in{\mathbb{C}}^{\times}$.
#### C.1.2. The weights
The dual lattice is called the lattice of weights:
${\Lambda}={\operatorname{Hom}}(\mathbf{T},{\mathbb{C}}^{\times})$ (C.5)
Let us represent element $t\in\mathbf{T}$ as
$t={\mathbf{e}}({\mathbf{x}})$ (C.6)
for some $\bf x\in\mathfrak{h}$, which is defined, as we recall, up to an a
shift by an element of the coroot lattice ${\rm Q}^{\vee}$. Then for any
$\lambda\in\Lambda$, the value of the homomorphism ${\lambda}$ on $t$, which
we denote by $t^{\lambda}\in{\mathbb{C}}^{\times}$, can be computed as:
$t^{\lambda}=e^{2\pi\mathrm{i}\lambda({\mathbf{x}})}$ (C.7)
where ${\lambda}({\bf x})$ is a linear function of $\bf x$. In this way we
view ${\lambda}$ as an element of $\mathfrak{h}^{*}$, so that
$\Lambda\subset\mathfrak{h}^{*}$. In order for (C.7) be independent of the
choice of $\bf x$ in (C.6), the value of $\lambda$ on any element of the
coroot lattice must be an integer:
${\lambda}({{\rm Q}}^{\vee})\subset{\mathbb{Z}}\ .$ (C.8)
which is another definition of the dual lattices. Let us fix the basis
$({\lambda}_{i})_{i=1}^{r}$ of $\Lambda$:
${\Lambda}={\mathbb{Z}}{\lambda}_{1}\oplus{\mathbb{Z}}{\lambda}_{2}\oplus\ldots\oplus{\mathbb{Z}}{\lambda}_{r}\,,\
$
dual to the basis $({\alpha}_{i}^{\vee})_{i=1}^{r}$ of simple coroots, so that
${\lambda}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$ (C.9)
#### C.1.3. The roots
The torus $\mathbf{T}$ acts on $\mathbf{G}$ via the adjoint action:
$Ad:\mathbf{T}\times\mathbf{G}\to\mathbf{G},\qquad(t,g)\mapsto t^{-1}gt$
Infinitesimally, it acts linearly on the Lie algebra
$Ad:\mathbf{T}\longrightarrow{\rm Aut}({\mathfrak{g}}),\qquad
Ad_{t}({\xi})=\frac{d}{ds}\biggr{|}_{s=0}t^{-1}{\exp}(s{\xi})t\in\mathfrak{g}$
(C.10)
for $\xi\in\mathfrak{g}$, and finally, this defines an action of
$\mathfrak{h}$ on $\mathfrak{g}$:
${\rm ad}:{\mathfrak{h}}\to{\rm End}({\mathfrak{g}}),\qquad{\rm ad}_{\bf
x}({\xi})=[{\xi},{\bf x}]\in\mathfrak{g}$ (C.11)
for ${\bf x}\in\mathfrak{h}$, ${\xi}\in\mathfrak{g}$.
This action gives us several structures: the root decomposition of
$\mathfrak{g}$:
${\mathfrak{g}}={\mathfrak{h}}\oplus\bigoplus_{{\alpha}\in
R}{\mathbb{C}}e_{\alpha}$ (C.12)
where $R\subset\Lambda$ is a set of non-vanishing weights of the adjoint
representation:
$ad_{t}(e_{\alpha})=t^{\alpha}e_{\alpha}$
These weights are called roots, and the lattice ${\rm Q}\subset\Lambda$ they
generate is a sublattice of $\Lambda$, called the _root lattice_. It has a
basis $({\alpha}_{i})_{i=1}^{r}$ of _positive simple roots_ :
${{\rm
Q}}={\mathbb{Z}}{\alpha}_{1}\oplus{\mathbb{Z}}{\alpha}_{2}\oplus\ldots\oplus{\mathbb{Z}}{\alpha}_{r}$
(C.13)
which allows us to define the _Cartan matrix_ $C^{\mathfrak{g}}$ of
$\mathfrak{g}$:
$C_{ij}^{\mathfrak{g}}={\alpha}_{i}({\alpha}_{j}^{\vee})$ (C.14)
which is non-degenerate. The additional requirement we impose on
$\mathfrak{g}$ is that it is _simply-laced_ , i.e. by an appropriate choice of
integral bases one can make $C^{\mathfrak{g}}$ symmetric:
$C_{ij}^{\mathfrak{g}}=C_{ji}^{\mathfrak{g}}$
#### C.1.4. The center
The quotient
$Z={\Lambda}/{{\rm Q}}$ (C.15)
is an abelian group, which is a subgroup both of $\mathbf{T}$ and
$\mathbf{G}$. In fact, it is the center of $\mathbf{G}$. Clearly, the center
does not act in the adjoint representation, so that in the Eq. (C.10) it is
the quotient $\mathbf{T}/Z$ which acts faithfully. Hence the root lattice
${{\rm Q}}\subset{\Lambda}\subset{\mathfrak{h}}^{*}$
can be also identified with the lattice of $\mathbf{T}/Z$ characters
${{\rm Q}}={\operatorname{Hom}}(\mathbf{T}/Z,{\mathbb{C}}^{\times})$ (C.16)
Finally, the coweight lattice ${\Lambda}^{\vee}\subset\mathfrak{h}$ is both
the integral dual to ${\rm Q}$, and the lattice of $\mathbf{T}/Z$
cocharacters:
${\Lambda}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},\mathbf{T}/Z)$
(C.17)
with the basis $({\lambda}_{i}^{\vee})_{i=1}^{r}$, dual to that of ${\rm Q}$:
${\alpha}_{i}({\lambda}_{j}^{\vee})={\delta}_{ij}$ (C.18)
The expression
$w^{{\lambda}^{\vee}}={\mathbf{e}}\,\left(\frac{{\log}(w)}{2\pi\mathrm{i}}{\lambda}^{\vee}\right)\in\mathbf{T}$
_does_ depend on the choice of the branch of the logarithm ${\log}(w)$,
however, the ambiguity is in the multiplicative $Z$-valued factor, since for
any ${\lambda}^{\vee}\in{{\rm Q}}^{\vee}$:
${\mathbf{e}}({\lambda}^{\vee})\in Z\subset\mathbf{T}$
Thus, for any $w\in{\mathbb{C}}^{\times}$, $w^{{\lambda}^{\vee}}$ is well-
defined as an element of $\mathbf{T}/Z$, as is claimed by (C.17).
The center $Z$, being a finite abelian group, is isomorphic to a product of
cyclic groups
$Z\approx\bigotimes_{{\xi}=1}^{z_{\mathfrak{g}}}\
{\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$ (C.19)
for some $z_{\mathfrak{g}}$, which is equal to $0$, $1$, or $2$.
$\bf G$ | $z_{\mathfrak{g}}$ | ${\ell}_{\xi}$, ${\xi}=1,\ldots,a_{\mathfrak{g}}$
---|---|---
$A_{r}$ | $1$ | $r+1$
$D_{2s}$ | $2$ | $2,2$
$D_{2s+1}$ | $1$ | $4$
$E_{6}$ | $1$ | $3$
$E_{7}$ | $1$ | $2$
$E_{8}$ | $0$ |
Table C.1. The number of cyclic factors in $Z$ with their orders
The table (C.1) shows the values of $z_{\mathfrak{g}}$’s and ${\ell}_{\xi}$’s
for all simple simply-laced Lie groups.
Let ${\varpi}_{\xi}\in\mathbf{T}$ be the generator of the
${\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$ factor in $Z\subset\mathbf{T}$. In
other words
$\displaystyle{\varpi}_{\xi}^{k}\neq 1\ {\rm for}\
k=1,2,\ldots,{\ell}_{\xi}-1,$ (C.20) $\displaystyle\qquad{\rm and}\
{\varpi}_{\xi}^{{\ell}_{\xi}}=1\in T\ .$
Of course, the Eq. (C.20) does not characterize ${\varpi}_{\xi}$ uniquely.
Indeed, for any integer $s_{\xi}$, which is mutually prime with
${\ell}_{\xi}$, i.e. $(s_{\xi},{\ell}_{\xi})=1$, the $Z$ element
$\widetilde{{\varpi}_{\xi}}={\varpi}_{\xi}^{s_{\xi}}$ also generates
${\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$. We write,
$\displaystyle{\mathbf{e}}({\lambda}_{i}^{\vee})=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\varpi}_{\xi}^{l_{i{\xi}}},\qquad
i=1,\ldots,r$ (C.21)
$\displaystyle{\varpi}_{\xi}=\prod_{j=1}^{r}{\mathbf{e}}\left(\frac{w_{{\xi}j}}{\ell_{\xi}}{\alpha}_{j}^{\vee}\right)$
for some integers $l_{i{\xi}},w_{{\xi}j}\in\mathbb{Z}$, which are normalized
$0\leq l_{i{\xi}}<{\ell}_{{\xi}},\qquad 0\leq w_{{\xi}j}<{\ell}_{\xi}$ (C.22)
and
$(w_{{\xi}j},{\ell}_{{\xi}})=1\qquad.$
Note that
${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{\mathfrak{g}})^{-1}_{ij}\in{\mathbb{Q}}$
(C.23)
By combining Eqs. (C.23) with (C.21), we derive:
$(C^{\mathfrak{g}})^{-1}_{ij}=\sum_{{\xi}=1}^{z_{\mathfrak{g}}}\frac{l_{i{\xi}}w_{{\xi}j}}{{\ell}_{{\xi}}}+{\mathcal{L}}^{\mathfrak{g}}_{ij}$
(C.24)
where ${\mathcal{L}}^{\mathfrak{g}}$ is some integral matrix. Note in passing
that if we were to study the group $G$ over the field ${\mathbb{F}}_{p^{n}}$
where $p$ divides ${\ell}_{y}$ then $C^{\mathfrak{g}}_{ij}$ would correspond
to an affine (or even double affine) root system.
#### C.1.5. Killing metric
Another structure we gain from the adjoint action of $T$ on $\mathfrak{g}$ is
the Killing metric on $\mathfrak{h}$,
$\langle
x,x\rangle=\frac{1}{h_{\mathfrak{g}}}\,{\operatorname{tr}}_{\mathfrak{g}}\,{\rm
ad}_{x}^{2}$ (C.25)
which identifies ${\mathfrak{h}}$ with ${\mathfrak{h}}^{*}$, ${{\rm
Q}}^{\vee}$ with ${{\rm Q}}$, and ${\Lambda}^{\vee}$ with $\Lambda$, and the
constant $h_{\mathfrak{g}}$ is chosen so that
$\langle{\lambda}_{i},{\alpha}_{j}\rangle={\delta}_{ij},\quad\langle{\alpha}_{i},{\alpha}_{j}\rangle=C_{ij}^{\mathfrak{g}},\quad\langle{\lambda}_{i},{\lambda}_{j}\rangle=(C^{\mathfrak{g}}_{ij})^{-1}$
(C.26)
#### C.1.6. The Weyl group
The torus $\mathbf{T}$ has some symmetries within $\mathbf{G}$. Namely, for
any $t\in\mathbf{T}$ there are transformations of the form:
$t\mapsto g^{-1}tg$ (C.27)
for some $g\in\mathbf{G}$, for which $g^{-1}tg\in\mathbf{T}$ also. Such
transformations form a group, which is called the _normalizer $N(\mathbf{T})$
of $\mathbf{T}$_. This group obviously contains $\mathbf{T}$, since the
transformation (C.27) with $g\in\mathbf{T}$ doesn’t even move $t$. It turns
out that there are additional nontrivial transformations. These additional
transformations form _the Weyl group_
$W({\mathfrak{g}})=N(\mathbf{T})/\mathbf{T}$.
For $t\in\mathbf{T}$ and $w\in W({\mathfrak{g}})$ let us denote by ${}^{w}t$
the result of the application of $g_{w}$ representing $w$ in
$N(\mathbf{T})\subset\mathbf{G}$:
${}^{w}t=g_{w}^{-1}tg_{w}$ (C.28)
By taking the limit $t\to 1$ we get the action of $W({\mathfrak{g}})$ on
$\mathfrak{h}$: for $\xi\in\mathfrak{h}$
${\xi}\mapsto\
^{w}{\xi}=\frac{d}{ds}\biggr{|}_{s=0}\,^{w}\left({\exp}\,s{\xi}\right)$ (C.29)
It is clear that $W({\mathfrak{g}})$ acts on $\mathfrak{h}$ by the orthogonal
transformations preserving the metric (C.25). The action of
$W({\mathfrak{g}})$ on $\mathfrak{h}^{*}$ ($\mathfrak{h}$) preserves both the
(co)root lattice and the (co)weight lattice.
The less trivial result is that the group $W({\mathfrak{g}})$ is generated by
reflections $r_{i}$ at the simple roots. The corresponding transformations on
$\mathfrak{h}$ and $\mathbf{T}$ are
${}^{r_{i}}x=x-{\alpha}_{i}(x){\alpha}_{i}^{\vee},\qquad^{r_{i}}t=t\left(t^{-{\alpha}_{i}}\right)^{{\alpha}_{i}^{\vee}}$
(C.30)
The $W({\mathfrak{g}})$-action on $\mathbf{T}$ can be also described in the
$z$-coordinates:
${}^{r_{i}}t_{\bf z}=t_{\bf z}(t_{\bf
z}^{-{\alpha}_{i}})^{{\alpha}_{i}^{\vee}}=t_{\tilde{\bf z}}\ ,$
where
$\displaystyle{\tilde{z}}_{j}=$ $\displaystyle z_{j},\qquad j\neq i$ (C.31)
$\displaystyle{\tilde{z}}_{i}=$ $\displaystyle
z_{i}\,\prod_{k=1}^{r}z_{k}^{-C^{\mathfrak{g}}_{ik}}$
#### C.1.7. Weyl group and the center
Since $W({\mathfrak{g}})$ acts on $\mathbf{T}$ by similarity transformations
in $\mathbf{G}$, the center $Z\subset\mathbf{T}\subset\mathbf{G}$ is fixed by
any $w\in W({\mathfrak{g}})$. In particular, $r_{i}$ preserves
${\varpi}_{\xi}$ for any $i$ and $\xi$, which is equivalent to
${\varpi}_{\xi}^{{\alpha}_{i}}=1$ (C.32)
Substituting into Eq. (C.32) the representation (C.22) and recalling the
definition (C.14) of $C^{\mathfrak{g}}$ we get:
$\frac{1}{{\ell}_{\xi}}\sum_{j=1}^{r}w_{{\xi}j}C_{ji}^{\mathfrak{g}}={\mathfrak{C}}_{{\xi}i}^{\mathfrak{g}}\in{\mathbb{Z}},\qquad{\xi}=1,\ldots,z_{\mathfrak{g}},\
i=1,\ldots,r\ .$ (C.33)
Combining this relation with Eq. (C.24) we obtain:
${\delta}_{ij}=\sum_{{\xi}=1}^{z_{\mathfrak{g}}}l_{i{\xi}}{\mathfrak{C}}_{{\xi}j}^{\mathfrak{g}}+\sum_{k=1}^{r}{\mathcal{L}}^{\mathfrak{g}}_{ik}C^{\mathfrak{g}}_{kj}$
(C.34)
On the other hand, using the Eq. (C.21), and the relations
$\sum_{j=1}^{r}C^{\mathfrak{g}}_{ij}{\lambda}_{j}^{\vee}={\alpha}_{i}^{\vee}$,
and ${\mathbf{e}}({\alpha}_{i}^{\vee})=1$ for any $i$, we derive:
$\prod_{j=1}^{r}\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\varpi}_{\xi}^{C_{ij}^{\mathfrak{g}}l_{j\xi}}=1$
which implies
$\frac{1}{{\ell}_{\xi}}\sum_{j=1}^{z_{\mathfrak{g}}}C_{ij}^{\mathfrak{g}}l_{j\xi}={\mathfrak{c}}_{i\xi}^{\mathfrak{g}}\in{\mathbb{Z}}$
(C.35)
#### C.1.8. Langlands dual, adjoint, and conformal groups
The simple Lie group $\mathbf{G}/Z$ has a trivial center, but it is not
simply-connected. This group is also denoted by $\mathbf{G}^{\text{ad}}$,
since it is represented faithfully in the adjoint representation
$\mathfrak{g}$. The maximal torus $\mathbf{T}^{\text{ad}}$ of
$\mathbf{G}^{\text{ad}}$, is the quotient $\mathbf{T}/Z={\mathfrak{h}}/{{\rm
Q}}$. Also, since the lattices of weights and roots of $\mathbf{G}$ and
$\mathbf{G}^{\text{ad}}$ are dual to each other, these groups are Langlands
duals, ${}^{L}\mathbf{G}=\mathbf{G}^{\text{ad}}$.
The group $\mathbf{G}^{\text{ad}}$ is not very convenient to work with. For
one thing, the center $Z$ looks differently for different groups. One defines
the _conformal extension_ $C\mathbf{G}$ of $\mathbf{G}$ as the group (see
[Levin:2010ve, Levin:2010mz, Morgan:2000math] for recent applications in a
related context),
${\mathrm{C}}\mathbf{G}=({\rm C}\times\,\mathbf{G})/Z$ (C.36)
where
${\rm C}=({\mathbb{C}}^{\times})^{z_{\mathfrak{g}}}$ (C.37)
where the center $Z$ acts on $\mathbf{G}$ in the usual way, and on $\rm C$ via
some character
${\chi}\in{\operatorname{Hom}}(Z,{\rm C})$ (C.38)
For example, we can choose
${\chi}({\varpi}_{x})=({\zeta}_{{\ell}_{y}}^{{\delta}_{x,y}})_{y=1}^{z_{\mathfrak{g}}}$
where ${\zeta}_{l}$ is the primitive $l$’th root of unity:
${\zeta}_{l}=e^{2\pi\frac{\mathrm{i}}{l}}$ (C.39)
The elements of the group $C\mathbf{G}$ are the classes of pairs
$[({\mathfrak{b}};g)]$, where ${\mathfrak{b}}\in\rm C$ is the
$z_{\mathfrak{g}}$-tuple of non-zero complex numbers,
${\mathfrak{b}}=({\mathfrak{b}}_{1},\ldots,{\mathfrak{b}}_{z_{\mathfrak{g}}})$,
and $g\in\mathbf{G}$, under the equivalence
$({\mathfrak{b}};g)\sim(\,^{x}{\mathfrak{b}};{\varpi}_{x}g\,)$, where
${}^{x}{\mathfrak{b}}=({\mathfrak{b}}_{y}{\zeta}_{{\ell}_{x}}^{-{\delta}_{x,y}})_{y=1}^{z_{\mathfrak{g}}}$
with the multiplication law
$[({\mathfrak{b}}_{1};g_{1})]\cdot[({\mathfrak{b}}_{2};g_{2})]=[({\mathfrak{b}}_{1}{\mathfrak{b}}_{2};g_{1}g_{2})]$
The maximal torus $C\mathbf{T}$ of $C\mathbf{G}$ has rank
$r+z_{\mathfrak{g}}$. The center $CZ$ of $C\mathbf{G}$ is the subgroup of
$C\mathbf{G}$ which consists of the equivalence classes containing
$({\mathfrak{b}};1)$, with $\mathfrak{b}\in{\rm C}$. Clearly, $CZ\approx{\rm
C}$. In this way, the centers of the conformal extensions look the same for
all groups of equal $z_{\mathfrak{g}}$.
The lattice
$C{{\rm Q}}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},C\mathbf{T})$
(C.40)
is an extension of ${{\rm Q}}^{\vee}$ by the rank $z_{\mathfrak{g}}$ lattice
with the generators ${\beta}_{\xi}^{\vee}$, ${\xi}=1,\ldots,z_{\mathfrak{g}}$:
$C{{\rm
Q}}^{\vee}=\bigoplus_{i=1}^{r}{\mathbb{Z}}{\alpha}_{i}^{\vee}\oplus\bigoplus_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathbb{Z}}{\beta}_{\xi}^{\vee}$
(C.41)
The root subspace decomposition of ${\mathfrak{c}\mathfrak{g}}={\rm
Lie}(C\mathbf{G})={\mathfrak{g}}\oplus{\mathbb{C}}^{z_{\mathfrak{g}}}$ is easy
to compute. There are $r$ simple roots which we denote by ${\alpha}_{i}$. They
act on ${\mathfrak{c}}{\mathfrak{h}}={\rm Lie}(C\mathbf{T})$ as follows:
$\displaystyle{\alpha}_{i}({\alpha}_{j}^{\vee})=C_{ji}^{\mathfrak{g}}$ (C.42)
$\displaystyle{\alpha}_{i}({\beta}_{\xi}^{\vee})={\mathfrak{C}}_{{\xi}i}^{\mathfrak{g}}$
so that if we define
$K_{\xi}=-{\ell}_{\xi}{\beta}_{\xi}^{\vee}+\sum_{j=1}^{r}w_{{\xi}j}{\alpha}_{j}^{\vee}$
(C.43)
then
${\alpha}_{i}(K_{\xi})=0$ (C.44)
for any $i$, $\xi$. The meaning of (C.43) is the following. The lattice of
cocharacters of the maximal torus of $\mathrm{C}\times\mathbf{G}$ is the
direct sum of the lattice ${{\rm Q}}^{\vee}$ of coroots of $G$ and the lattice
$\displaystyle{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})=L\oplus{{\rm Q}}^{\vee}$ (C.45) $\displaystyle
L={\pi}_{1}({\rm
C})=\bigoplus_{{\eta}=1}^{z_{\mathfrak{g}}}\,{\mathbb{Z}}K_{\eta},$
so that a generic element of $\mathrm{C}\times\mathbf{T}$ can be represented
by:
${\widehat{t}}_{\mathfrak{b},{\bf
g}}=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{b}}_{\xi}^{K_{\xi}}\times\prod_{i=1}^{r}g_{i}^{{\alpha}_{i}^{\vee}}$
(C.46)
(again, in writing (C.46) we assume normalization ${\mathbf{e}}(K_{\xi})=1\in$
C.) The action of $Z$ on $\mathrm{C}\times\mathbf{G}$ translates to the action
on ${\mathfrak{b}},{\bf g}$:
${\varpi}_{\xi}:{\widehat{t}}_{{\mathfrak{b}},{\bf
g}}\mapsto{\widehat{t}}_{{\mathfrak{b}},{\bf
g}}\,{\zeta}_{{\ell}_{\xi}}^{-K_{\xi}+\sum_{i=1}^{r}w_{{\xi}i}{\alpha}_{i}^{\vee}}$
(C.47)
The quotient $C\mathbf{T}=({\rm C}\times\mathbf{T})/Z$ is coordinatized by
$\displaystyle[({\mathfrak{b}};t_{\mathbf{z}})]={\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{u}}_{\xi}^{{\beta}_{\xi}^{\vee}}\prod_{i=1}^{r}z_{i}^{{\check{\alpha}}_{i}^{\vee}}\in
CT,$ (C.48)
$\displaystyle{\mathfrak{u}}_{\xi}={\mathfrak{b}}_{\xi}^{-{\ell}_{\xi}},\qquad
z_{i}=g_{i}\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{b}}_{\xi}^{w_{{\xi}i}}$
in agreement with (C.40). Simply put,
${\beta}_{\xi}^{\vee}\in{\operatorname{Hom}}({\mathbb{C}}^{\times},C\mathbf{T})$,
${\ell}_{\xi}{\beta}_{\xi}^{\vee}\in{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})$, and
${\beta}^{\vee}_{\xi}\notin{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})$.
Clearly, there are other choices of the $Z$-invariant coordinates, which
differ by multiplication of $z_{i}$ by any function of
${\mathfrak{u}}_{\xi}$’s.
Comparing the Eqs. (C.48) and (C.46) we arrive at (C.43). The Eq. (C.44)
simply reflects the fact that C acts trivially in the adjoint representation.
The lattice $C{{\rm Q}}$ generated by ${\alpha}_{i}$ is isomorphic to the root
lattice ${{\rm Q}}$.
The weight lattice $\widetilde{C\Lambda}$ of $\mathrm{C}\times\mathbf{G}$ is
the direct sum of the weight lattice of C and that of $\mathbf{G}$:
$\widetilde{C\Lambda}={\operatorname{Hom}}({\rm
C}\times\mathbf{T},{\mathbb{C}}^{\times})=L^{\vee}\oplus{\Lambda}$ (C.49)
The weight lattice $C\Lambda$ of $C\mathbf{G}$ is a sublattice of
$\widetilde{C\Lambda}$ which consists of the weights which are trivial on the
elements of $\mathrm{C}\times\mathbf{T}$ of the form
${\zeta}_{{\ell}_{\xi}}^{-K_{\xi}+\sum_{i=1}^{r}w_{{\xi}i}{\alpha}_{i}^{\vee}}$
for any $\xi$. In the $\widetilde{C\Lambda}$-basis
$({\mu}_{\xi};{\lambda}_{i})$, ${\xi}=1,\ldots,z_{\mathfrak{g}}$,
$i=1,\ldots,r$ such that
${\mu}_{\xi}(K_{\eta})={\delta}_{{\xi},{\eta}},\
{\mu}_{\xi}({\alpha}_{i}^{\vee})=0$ ${\lambda}_{i}(K_{\eta})=0,\
{\lambda}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$
the lattice $C\Lambda$ is spanned by
${\check{\mu}}_{\xi}=-{\ell}_{\xi}{\mu}_{\xi},\qquad{\check{\lambda}}_{i}={\lambda}_{i}+\sum_{\xi}w_{{\xi}i}{\mu}_{\xi}\
.$ (C.50)
An easy computation gives:
$\displaystyle{\check{\mu}}_{\xi}({\beta}_{\eta}^{\vee})={\delta}_{\xi\eta},\quad{\check{\lambda}}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$
(C.51)
$\displaystyle{\check{\mu}}_{\xi}({\alpha}_{i}^{\vee})=0,\quad{\check{\lambda}}_{i}({\beta}_{\eta}^{\vee})=0$
Finally, the lattice $C\Lambda^{\vee}$ of coweights of $C\mathbf{G}$, which is
dual to the root lattice $C{{\rm Q}}$, is generated by the fundamental
coweights ${\check{\lambda}}_{i}^{\vee}$, which obey
${\alpha}_{i}({\check{\lambda}}_{j}^{\vee})={\delta}_{ij}$ (C.52)
The equations (C.52) define ${\check{\lambda}}_{i}^{\vee}$ up to the shifts by
the integer multiples of $K_{\xi}$. We choose the representative
${\check{\lambda}}_{i}^{\vee}=\sum_{j=1}^{r}{\mathcal{L}}_{ij}^{\mathfrak{g}}{\alpha}_{j}^{\vee}+\sum_{{\xi}=1}^{z_{\mathfrak{g}}}l_{i{\xi}}{\beta}_{\xi}^{\vee}$
(C.53)
Shifting ${\check{\lambda}}_{i}^{\vee}$ by $d_{\xi}K_{\xi}$ would change the
coefficient $l_{i{\xi}}\mapsto l_{i{\xi}}+{\ell}_{\xi}d_{\xi}$.
As we see, the coweights ${\check{\lambda}}_{i}^{\vee}$ are the integral
linear combinations of the coroots ${\alpha}^{\vee}$ and ${\beta}^{\vee}$, as
if $C\mathbf{G}$ were the simple simply-connected group with the trivial
center.
#### C.1.9. The D-homomorphism
By construction of the conformal group, there is a homomorphism
$\displaystyle D:C\mathbf{G}\longrightarrow{\rm C}/Z\approx{\rm C}$ (C.54)
$\displaystyle
D[({\mathfrak{b}};g)]=({\mathfrak{b}}_{\xi}^{{\ell}_{\xi}})_{{\xi}=1}^{z_{\mathfrak{g}}}$
In terms of the weights, the restriction of the homomorphism $D$ onto
$C\mathbf{T}$ is given by:
$D({\check{t}}_{{\mathfrak{u}},{\mathbf{z}}})=({\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}^{{\check{\mu}}_{\xi}})_{{\xi}=1}^{z_{\mathfrak{g}}}=({\mathfrak{u}}_{\xi})_{{\xi}=1}^{z_{\mathfrak{g}}}$
(C.55)
#### C.1.10. Twisted Weyl group action
Now we can define the twisted action of Weyl group $W({\mathfrak{g}})$. It is
nothing but the natural action of $W({\mathfrak{g}})$ on $C\mathbf{T}$.
However, we shall encounter a somewhat redundant parametrization of
$C\mathbf{T}$, by $\mathbf{T}\times\mathbf{T}$:
${\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}=\prod_{i=1}^{r}{\mathscr{P}}_{i}^{-\,{\check{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}^{{\check{\alpha}}_{i}^{\vee}}$
(C.56)
which is equal to ${\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}$ with
${\mathfrak{u}}_{\xi}=\prod_{j}{\mathscr{P}}_{j}^{-l_{j\xi}},\qquad
z_{i}={\mathscr{Y}}_{i}\prod_{j}{\mathscr{P}}_{j}^{-{\mathcal{L}}_{ji}}$
(C.57)
Using (C.42) we compute the Weyl group $W({\mathfrak{g}})$ action on
$C\mathbf{T}$: under the reflection $r_{i}$ the group element
${\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}$ maps to
${}^{r_{i}}{\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}={\mathbf{g}}_{{\mathscr{P}},{\tilde{\mathscr{Y}}}}$,
where
$\displaystyle{\tilde{\mathscr{Y}}}_{j}={\mathscr{Y}}_{j},\qquad j\neq i$
(C.58)
$\displaystyle{\tilde{\mathscr{Y}}}_{i}={\mathscr{P}}_{i}{\mathscr{Y}}_{i}\prod_{j}{\mathscr{Y}}_{j}^{-C_{ij}^{\mathfrak{g}}}$
The homomorphism $D$ is $W({\mathfrak{g}})$-invariant:
$D({\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}})=D({\mathbf{g}}_{{\mathscr{P}},{\tilde{\mathscr{Y}}}})=(\prod_{j=1}^{r}{\mathscr{P}}_{j}^{-l_{j{\xi}}})_{{\xi}=1}^{z_{\mathfrak{g}}}$
### C.2. Affine Lie algebras
The affine Lie algebras $\widehat{\mathfrak{g}}$ show up in the solution of
the theories of the class II. In preparing this section we consulted with
[Bardakci:1970nb], [Kac:1984]. Given a simple Lie group $\mathbf{G}$ with its
Lie algebra $\mathfrak{g}$ one defines the loop group $L\mathbf{G}$ and the
loop algebra $L\mathfrak{g}$ of (suitably defined, analytic, formal,
polynomial) maps of the neighborhood of a circle in $\mathbb{C}^{\times}$ into
$\mathbf{G}$ and $\mathfrak{g}$, respectively:
$L\mathbf{G}={\rm Maps}({\mathbb{C}}^{\times},\mathbf{G}),\quad
L{\mathfrak{g}}={\rm Maps}({\mathbb{C}}^{\times},{\mathfrak{g}}),$ (C.59)
with the point-wise product and Lie brackets, respectively. Then one defines
the central extension
$\widetilde{L{\mathfrak{g}}}=L{\mathfrak{g}}\oplus{\mathbb{C}}K$ by
$[f_{1}(t)\oplus a_{1}\,K,f_{2}(t)\oplus
a_{2}\,K]=[f_{1}(t),f_{2}(t)]\oplus\oint_{{\mathbb{S}}^{1}}\langle
df_{1}(t),f_{2}(t)\rangle\,K$
and the additional (non-central) extension $\widehat{\mathfrak{g}}$ of
$\widetilde{L{\mathfrak{g}}}$ by $\mathbb{C}$, which acts on $L\mathfrak{g}$
by the infinitesimal rotation of ${\mathbb{C}}^{\times}$ in (C.59). The
elements of $\widehat{\mathfrak{g}}$ can be represented as the
$\mathfrak{g}$-valued first order differential operators on $\mathbb{S}^{1}$
plus a constant:
${\tau}d+f(t)\oplus
aK\in\widehat{\mathfrak{g}},\qquad{\tau},a\in\mathbb{C},\,f(t)\in{\mathfrak{g}}$
with the commutation relations:
$\displaystyle[{\tau}_{1}d+f_{1}(t)\oplus a_{1}K,{\tau}_{2}d+f_{2}(t)\oplus
a_{2}K]=$ (C.60) $\displaystyle\quad 0\cdot
d+t\left({\tau}_{1}f_{2}^{\prime}(t)-{\tau}_{2}f_{1}^{\prime}(t)\right)+[f_{1}(t),f_{2}(t)]\,\oplus\oint_{{\mathbb{S}}^{1}}\langle
f_{1}(t),df_{2}(t)\rangle\,K$
Analogously, one defines the central extension $\widetilde{L\mathbf{G}}$ which
is a non-trivial ${\mathbb{C}}^{\times}$-bundle over $L\mathbf{G}$ (this is
analogous to the construction of the conformal group $C\mathbf{G}$ in the
previous section), and then the extension $\widehat{\mathbf{G}}$ of
$\widetilde{L\mathbf{G}}$ by ${\mathbb{C}}^{\times}$ which acts by rotation of
${\mathbb{C}}^{\times}$ in (C.59).
The algebra $\widehat{\mathfrak{g}}$ can also be defined by the general
construction of the generalized Kac-Moody algebras associated with the Cartan
matrix $C_{ij}^{\widehat{\mathfrak{g}}}$. The affine Cartan matrix has exactly
one eigenvector with the eigenvalue zero:
$\sum_{j=0}^{r}C_{ij}^{\widehat{\mathfrak{g}}}a_{j}=0$ (C.61)
The Cartan subalgebra $\widehat{\mathfrak{h}}$ of the corresponding affine Lie
algebra $\widehat{\mathfrak{g}}$ with the underlying finite dimensional
simply-laced Lie algebra $\mathfrak{g}$ of rank $r$ is the complex vector
space of dimension $r+2$. The dual space $\widehat{\mathfrak{h}}^{*}$ contains
the root lattice ${\widehat{\rm Q}}\subset\widehat{\mathfrak{h}}^{*}$, which
is generated by the simple roots ${\widehat{\alpha}}_{i}$, $i=0,\ldots,r$, of
which the simple roots ${\widehat{\alpha}}_{i}$ with $i>0$ generate the root
lattice of $\mathfrak{g}$. Likewise, the Cartan subalgebra
$\widehat{\mathfrak{h}}$ contains the coroot lattice ${\widehat{\rm
Q}}^{\vee}\subset\widehat{\mathfrak{h}}$, generated by the simple coroots
${\alpha}_{i}^{\vee}$, which obey:
${\widehat{\alpha}}_{i}({\widehat{\alpha}}_{j}^{\vee})=C_{ij}^{\widehat{\mathfrak{g}}}$
(C.62)
The following linear combination of the simple roots:
${\delta}=\sum_{i=0}^{r-1}a_{i}{\widehat{\alpha}}_{i}\in\widehat{\mathfrak{h}}^{*}$
(C.63)
is called the imaginary root. It annihilates the simple coroots, cf. (C.61):
${\delta}({\widehat{\alpha}}_{i}^{\vee})=0\,,\ i=0,\ldots,r-1$
The analogous linear combination of coroots
$K=\sum_{i=0}^{r-1}a_{i}{\widehat{\alpha}}_{i}^{\vee}\in\widehat{\mathfrak{h}}$
(C.64)
obeys
${\widehat{\alpha}}_{i}(K)=0\,,\ i=0,\ldots,r$ (C.65)
and generates the center of the affine Kac-Moody algebra since (C.65) implies
it commutes with everything in $\widehat{\mathfrak{g}}$. Then
${\widehat{\rm Q}}^{\vee}_{\mathbb{C}}={\mathfrak{h}}\oplus{\mathbb{C}}K\,,\
{\widehat{\rm Q}}_{\mathbb{C}}={\mathfrak{h}}^{*}\oplus{\mathbb{C}}{\delta}$
(C.66)
where $\mathfrak{h}$ and $\mathfrak{h}^{*}$ are the Cartan subalgebra and its
dual space of the corresponding finite dimensional simply-laced Lie algebra
$\mathfrak{g}$, respectively. In order to generate $\widehat{\mathfrak{h}}$
and $\widehat{\mathfrak{h}}^{*}$ as the vector spaces over $\mathbb{C}$ we
need to add one more generator in addition to the simple roots and the simple
coroots, respectively:
${\widehat{\mathfrak{h}}}^{*}={\widehat{\rm
Q}}_{\mathbb{C}}\oplus{\mathbb{C}}{\lambda}_{0}\,,\
{\widehat{\mathfrak{h}}}={\widehat{\rm
Q}}^{\vee}_{\mathbb{C}}\oplus{\mathbb{C}}{\delta}^{\vee}$ (C.67)
which obey:
${\widehat{\lambda}}_{0}({\widehat{\alpha}}_{i}^{\vee})={\widehat{\alpha}}_{i}({\delta}^{\vee})={\delta}_{i0}$
(C.68)
The generator ${\delta}^{\vee}$ is equal to the generator of the infinitesimal
loop rotation $d$ we used in (C.60).
The weight lattice ${\widehat{\Lambda}}$ of $\widehat{\mathfrak{g}}$ is
generated by the fundamental weights
${\lambda}_{i}\in\widehat{\mathfrak{h}}^{*}$, $i=0,1,\ldots,r$:
${\widehat{\Lambda}}={\Lambda}\oplus{\mathbb{Z}}{\lambda}_{0}\subset\widehat{\mathfrak{h}}^{*}$
(C.69)
which obey the following basic relations:
${\widehat{\lambda}}_{i}({\widehat{\alpha}}_{j}^{\vee})={\delta}_{ij},\
{\widehat{\lambda}}_{i}({\delta}^{\vee})=0,\
{\delta}({\delta}^{\vee})=a_{0}=1$ (C.70)
The fundamental coweights
${\widehat{\lambda}}_{i}^{\vee}\in\widehat{\mathfrak{h}}$ obey:
${\widehat{\alpha}}_{j}({\widehat{\lambda}}_{i}^{\vee})={\delta}_{ij},$
and form the coweight lattice
${\widehat{\Lambda}}^{\vee}=\bigoplus_{i=0}^{r}{\mathbb{Z}}{\lambda}_{i}^{\vee}\subset{\mathfrak{h}}$
The level of a weight ${\widehat{\lambda}}\in\widehat{\Lambda}$ is defined as
$k={\widehat{\lambda}}(K)$ so that the level of the $i$’th fundamental weight
is equal to
${\widehat{\lambda}}_{i}(K)=a_{i}={\delta}({\widehat{\lambda}}^{\vee}_{i})\ .$
(C.71)
Note that ${\widehat{\rm Q}}^{\vee}_{\mathbb{C}}$ is the Cartan subalgebra of
the central extension $\widehat{L{\mathfrak{g}}}$ of the loop algebra
$L{\mathfrak{g}}$. Adding ${\delta}^{\vee}$ makes up the Cartan of the affine
Kac-Moody algebra of $\mathfrak{g}$, which is an extension of
$\widehat{L{\mathfrak{g}}}$ by the operator of the infinitesimal loop
rotation, the zero mode of the Virasoro generator. In physics literature the
more common notation for ${\delta}^{\vee}$ is $L_{0}$.
Also we notice that
${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{\mathfrak{g}})^{-1}_{ij},\qquad
i,j=1,\ldots,r$ (C.72)
hence
${\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee})=\begin{pmatrix}0&0\\\
0&(C^{\mathfrak{g}})^{-1}\end{pmatrix}_{ij}.$ (C.73)
A useful relation obtained from identity operator
$|\Lambda_{i}\rangle\langle\alpha_{i}^{\vee}|+|\delta\rangle\langle\delta^{\vee}|$
is
$\sum_{i=0}^{r}{\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee}){\widehat{\alpha}}_{k}({\widehat{\alpha}}_{i}^{\vee})={\widehat{\alpha}}_{k}({\widehat{\lambda}}_{j}^{\vee})-{\delta}({\widehat{\lambda}}_{j}^{\vee}){\alpha}_{k}({\delta}^{\vee})=\delta_{jk}-a_{j}\delta_{k0}$
(C.74)
Let $\lambda\in\mathfrak{h}^{*}$ for
$\widehat{\lambda}\in\widehat{\mathfrak{h}}^{*}$ denote the image of the
projection $\widehat{\mathfrak{h}}^{*}\to\mathfrak{h}^{*}$, i.e. forgetting
components spanned by $\delta$ and $\widehat{\lambda}_{0}$. Then
$\displaystyle\widehat{\lambda}_{i}^{\vee}=a_{i}\delta^{\vee}+\lambda_{i}^{\vee}$
(C.75)
$\displaystyle\widehat{\lambda}_{j}=a_{j}\widehat{\lambda}_{0}+\lambda_{j}$
### C.3. ADE Cartan matrices, roots and weights
Our ADE conventions are summarized in the table A.
#### C.3.1. $A_{r}$ series
The Cartan matrix $C_{ij}^{\widehat{A}_{r}}$ with $i,j=0,\dots,r$
$C^{\widehat{A}_{r}}=\begin{pmatrix}2&-1&0&\dots&\dots&-1\\\
-1&2&-1&\dots&\dots&0\\\ 0&-1&2&-1&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&0&\dots&\dots&2&-1\\\
-1&0&\dots&\dots&-1&2\end{pmatrix}$ (C.76)
The affine simple roots are
${\widehat{\alpha}}_{0}=\delta-\theta,\widehat{\alpha}_{1}=e_{1}-e_{2},\widehat{\alpha}_{2}=e_{2}-e_{3},\dots,\widehat{\alpha}_{r}=e_{r}-e_{r+1}$
where $\theta=e_{1}-e_{r+1}$ is the highest root. The Dynkin marks are
$a_{i}=1,i=0,\dots,r$. The fundamental weights are
$\displaystyle\widehat{\lambda}_{0},$ (C.77)
$\displaystyle\widehat{\lambda}_{1}=\widehat{\lambda}_{0}+e_{1}-\frac{1}{r}{\bf
e}$
$\displaystyle\widehat{\lambda}_{2}=\widehat{\lambda}_{0}+e_{1}+e_{2}-\frac{2}{r}{\bf
e}$ $\displaystyle\dots$
$\displaystyle\widehat{\lambda}_{r}=\widehat{\lambda}_{0}+e_{1}+e_{2}+\dots+e_{r}-\frac{r}{r+1}{\bf
e}$
where
${\bf e}=e_{1}+\dots+e_{r}$
The inverse Cartan matrix of the block $i,j=1\dots r$, i.e. the inverse Cartan
matrix $(C^{A_{r}})^{-1}$ of ${\mathfrak{g}}=A_{r}$ is
${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{A_{r}})^{-1}_{ij}={\rm
max}(i,j)-\frac{ij}{r+1},\qquad i,j=1,\ldots,r$ (C.78)
#### C.3.2. $D_{r}$ series
In the standard basis $\\{e_{i}\,|\,i=1,\ldots,r\\}$ the simple roots of
$D_{r}$ are
$\displaystyle\alpha_{i}$ $\displaystyle=e_{i}-e_{i+1},\qquad i=1,\ldots,r-1$
(C.79) $\displaystyle\alpha_{r}$ $\displaystyle=e_{r-1}+e_{r}$
and the fundamental weights are
$\displaystyle\lambda_{i}$ $\displaystyle=e_{1}+e_{2}+\dots+e_{i},\quad
i=1\dots r-2$ (C.80) $\displaystyle\lambda_{r-1}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+\dots+e_{r-1}-e_{r})$
$\displaystyle\lambda_{r}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+\dots+e_{r-1}+e_{r})$
The highest root is $\theta=\sum_{i=1}^{r}\alpha_{i}a_{i}=e_{1}+e_{2}$.
In the basis $(e_{i})$ the root (coroot) lattice of $D_{r}$ is given by
$Q=Q^{\vee}=\\{\mathbf{n}\in\mathbb{Z}^{r}||{\bf n}|\in 2\mathbb{Z}\\}$ (C.81)
The weight (coweight) lattice of $D_{r}$ is
$\oplus_{i=1}^{r}\mathbb{Z}\lambda_{i}=\bigcup_{{\varepsilon}=0,\frac{1}{2}}\left({\mathbb{Z}}+{\varepsilon}\right)^{r}\
.$ (C.82)
The inverse Cartan matrix is
$\displaystyle(C^{D_{r}})^{-1}=$ (C.83)
$\displaystyle\qquad\qquad\|\lambda_{i}(\lambda_{j}^{\vee})\|_{i,j}^{r}=$
$\displaystyle\qquad\qquad\begin{pmatrix}1&1&1&1&\cdots&1&\frac{1}{2}\quad&\frac{1}{2}\quad\\\
1&2&2&2&\cdots&2&\quad\frac{2}{2}&\quad\frac{2}{2}\\\
1&2&3&3&\cdots&3&\frac{3}{2}\quad&\frac{3}{2}\quad\\\
1&2&3&4&\cdots&4&\quad\frac{4}{2}&\quad\frac{4}{2}\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&2&3&4&\cdots&r-2&\quad\frac{r-2}{2}&\quad\frac{r-2}{2}\\\
\frac{1}{2}&\frac{2}{2}&\frac{3}{2}&\frac{4}{2}&\cdots&\frac{r-2}{2}&\frac{r}{4}\quad&\frac{r-2}{4}\quad\\\
\quad\frac{1}{2}&\quad\frac{2}{2}&\quad\frac{3}{2}&\quad\frac{4}{2}&\cdots&\quad\frac{r-2}{2}&\quad\frac{r-2}{4}&\quad\frac{r}{4}\end{pmatrix}$
The basis of the affine Cartan dual is
$\\{\lambda_{0},\delta,e_{1},\dots,e_{r}\\}$. The simple roots are
$\widehat{\alpha}_{0}=\delta-\theta$ and the roots $\alpha_{i}$ of $D_{r}$.
The fundamental weights are
$\displaystyle\widehat{\lambda}_{0},$ (C.84)
$\displaystyle\widehat{\lambda}_{i}=a_{i}\widehat{\lambda}_{0}+{\lambda}_{i},\quad
i=1,\ldots,r$
where $\lambda_{i}$ denote the fundamental weights of $D_{r}$, and $a_{i}$ are
the Dynkin labels:
$(a_{0},\dots,a_{r})=(1,1,2,\dots,2,1,1)$ (C.85)
#### C.3.3. $E_{6}$
The Cartan matrix of $E_{6}$ is
$C^{E_{6}}=\begin{pmatrix}2&0&-1&0&0&0\\\ 0&2&0&-1&0&0\\\ -1&0&2&-1&0&0\\\
0&-1&-1&2&-1&0\\\ 0&0&0&-1&2&-1\\\ 0&0&0&0&-1&2\end{pmatrix}$ (C.86)
and the inverse is
$(C^{E_{6}})^{-1}=\begin{pmatrix}\frac{4}{3}&1&\frac{5}{3}&2&\frac{4}{3}&\frac{2}{3}\\\
1&2&2&3&2&1\\\ \frac{5}{3}&2&\frac{10}{3}&4&\frac{8}{3}&\frac{4}{3}\\\
2&3&4&6&4&2\\\ \frac{4}{3}&2&\frac{8}{3}&4&\frac{10}{3}&\frac{5}{3}\\\
\frac{2}{3}&1&\frac{4}{3}&2&\frac{5}{3}&\frac{4}{3}\end{pmatrix}$ (C.87)
In the affine Cartan matrix the affine node ‘‘0’’ of $\widehat{E}_{6}$
connects to the node ‘‘2’’ of $E_{6}$. The Dynkin marks
$(a_{0},\dots,a_{6})=(1,1,2,2,3,2,1).$ (C.88)
#### C.3.4. $E_{7}$
The Cartan matrix of $E_{7}$ is
$C^{E_{7}}=\begin{pmatrix}2&0&-1&0&0&0&0\\\ 0&2&0&-1&0&0&0\\\
-1&0&2&-1&0&0&0\\\ 0&-1&-1&2&-1&0&0\\\ 0&0&0&-1&2&-1&0\\\ 0&0&0&0&-1&2&-1\\\
0&0&0&0&0&-1&2\end{pmatrix}$ (C.89)
and the inverse is
$(C^{E_{7}})^{-1}=\begin{pmatrix}2&2&3&4&3&2&1\\\
2&\frac{7}{2}&4&6&\frac{9}{2}&3&\frac{3}{2}\\\ 3&4&6&8&6&4&2\\\
4&6&8&12&9&6&3\\\ 3&\frac{9}{2}&6&9&\frac{15}{2}&5&\frac{5}{2}\\\
2&3&4&6&5&4&2\\\ 1&\frac{3}{2}&2&3&\frac{5}{2}&2&\frac{3}{2}\end{pmatrix}$
(C.90)
In the affine Cartan matrix the affine node ‘‘0’’ of $\widehat{E}_{7}$
connects to the node ‘‘1’’ of $E_{7}$. The Dynkin marks
$(a_{0},\dots,a_{7})=(1,2,2,3,4,3,2,1).$ (C.91)
#### C.3.5. $E_{8}$
The Cartan matrix of $E_{8}$ is
$C^{E_{8}}=\begin{pmatrix}2&0&-1&0&0&0&0&0\\\ 0&2&0&-1&0&0&0&0\\\
-1&0&2&-1&0&0&0&0\\\ 0&-1&-1&2&-1&0&0&0\\\ 0&0&0&-1&2&-1&0&0\\\
0&0&0&0&-1&2&-1&0\\\ 0&0&0&0&0&-1&2&-1\\\ 0&0&0&0&0&0&-1&2\end{pmatrix}$
(C.92)
and the inverse is
$(C^{E_{8}})^{-1}=\begin{pmatrix}4&5&7&10&8&6&4&2\\\ 5&8&10&15&12&9&6&3\\\
7&10&14&20&16&12&8&4\\\ 10&15&20&30&24&18&12&6\\\ 8&12&16&24&20&15&10&5\\\
6&9&12&18&15&12&8&4\\\ 4&6&8&12&10&8&6&3\\\ 2&3&4&6&5&4&3&2\end{pmatrix}$
(C.93)
In the affine Cartan matrix the node ‘‘0’’ of $\widehat{E}_{8}$ connects to
the node ‘‘8’’ of $E_{8}$. The Dynkin marks
$(a_{0},\dots,a_{8})=(1,3,2,4,6,5,4,3,2).$ (C.94)
### C.4. Affine Weyl group
For the Class II theories the relevant reflection group turns out to be the
affine Weyl group $W({\widehat{\mathfrak{g}}})$. As a group, it is a semi-
direct product of the finite Weyl group $W({\mathfrak{g}})$ and the root
lattice ${\rm Q}$ of $\mathfrak{g}$ (recall that for $\mathfrak{g}$ the root
and the coroot lattices are identified):
$W({\widehat{\mathfrak{g}}})=W({\mathfrak{g}})\ltimes{\rm Q}$ (C.95)
We can view $W({\widehat{\mathfrak{g}}})$ as the group acting in
$\widehat{\mathfrak{h}}$, preserving the non-degenerate scalar product:
$(\cdot,\cdot)$ which extends the Killing form on $\mathfrak{h}$ by the
pairing between $K$ and ${\delta}^{\vee}$, as follows. For
$x={\tau}{\delta}^{\vee}+{\sigma}K+{\xi},\qquad{\tau},{\sigma}\in{\mathbb{C}},\
{\xi}\in\mathfrak{h}$ (C.96) $(x,x)=\langle{\xi},{\xi}\rangle+2{\tau}{\sigma}$
(C.97)
The group $W({\widehat{\mathfrak{g}}})$ is generated by simple reflections
$r_{i}$, $i=0,1,\ldots,r$. The action of $r_{i}$ on $\widehat{\mathfrak{h}}$
is given by:
$r_{i}:x\mapsto x-{\alpha}_{i}(x){\alpha}_{i}^{\vee}\,,\qquad i=0,\ldots,r$
(C.98)
for $x\in{\widehat{\mathfrak{h}}}$. Similarly, the action of $r_{i}$ on
$\widehat{\mathfrak{h}}^{*}$ is given by
$r_{i}:p\mapsto p-p({\alpha}_{i}^{\vee}){\alpha}_{i}\,,\qquad i=0,\ldots,r$
(C.99)
Note that $K$ is invariant under the reflections (C.98), while ${\delta}$ is
invariant under the reflections (C.99), cf. (C.63).
On the hyperplane
$H_{\tau}=\\{\,x\,|\,{\delta}(x)={\tau}\,\\}\subset\widehat{\mathfrak{h}}\,$
(C.100)
the group $W({\widehat{\mathfrak{g}}})$ acts by orthogonal transformations,
generated by the ordinary reflections, and by translations. In the
decomposition (C.96) we have:
$r_{0}({\tau}{\delta}^{\vee}+{\sigma}K+{\xi})={\tau}{\delta}^{\vee}+{\sigma}K+{\xi}-({\tau}-{\theta}({\xi}))(K-{\theta}^{\vee})$
(C.101)
where we introduced the highest root $\theta\in\mathfrak{h}^{*}$, and the
highest coroot $\theta^{\vee}\in\mathfrak{h}$:
$n\theta={\delta}-{\alpha}_{0}=\sum_{i=1}^{r}a_{i}{\alpha}_{i},\qquad\theta^{\vee}=K-{\alpha}_{0}^{\vee}=\sum_{i=1}^{r}a_{i}{\alpha}_{i}^{\vee}$
(C.102)
which obey:
${\theta}({\theta}^{\vee})=\langle{\theta}^{\vee},{\theta}^{\vee}\rangle=2$,
and also ${\theta}({\xi})=\langle\xi,{\theta}^{\vee}\rangle$ for any
$\xi\in\mathfrak{h}$.
Now, to make the translational part of the $W({\widehat{\mathfrak{g}}})$
action explicit, let us perform an additional reflection at $\theta$:
$\displaystyle
r_{\theta}r_{0}(x)=r_{0}(x)-{\theta}({\xi}-({\tau}-{\theta}({\xi}))(-{\theta}^{\vee})){\theta}^{\vee}=$
(C.103)
$\displaystyle\qquad{\tau}{\delta}^{\vee}+({\sigma}-{\tau}+{\theta}({\xi}))K+{\xi}-{\tau}{\theta}^{\vee}$
Finally, the general element of $W({\widehat{\mathfrak{g}}})$ can be
represented as a pair $(w,{\beta})$, where $w\in W({\mathfrak{g}})$ and
$\beta\in{\rm Q}$, with the composition rule:
$(w_{1},{\beta}_{1})\cdot(w_{2},{\beta}_{2})=(w_{1}\cdot
w_{2},{\beta}_{1}+{\beta}_{2}^{w_{1}})$
The element ${\widehat{w}}=(w,{\beta})$ acts on $x\in H_{\tau}$ as follows:
$\displaystyle({\tau}{\delta}^{\vee}+{\sigma}K+{\xi})^{\widehat{w}}=$ (C.104)
$\displaystyle{\tau}{\delta}^{\vee}+\left({\sigma}-\langle\xi^{w},\beta\rangle-\frac{\tau}{2}\langle\beta,\beta\rangle\right)K+\left({\xi}^{w}+{\tau}{\beta}\right)$
The dual picture is the action of $W({\widehat{\mathfrak{g}}})$ on the fixed
level hyperplane
$H^{*}_{k}=\\{\,p\,|\,p(K)=k\\}\subset\widehat{\mathfrak{h}}^{*}$. Write
$p=k{\lambda}_{0}+{\mu}{\delta}+{\eta},\qquad{\eta}\in\mathfrak{h}^{*},k,{\mu}\in{\mathbb{C}}$
(C.105)
then
$p(x)=k{\sigma}+{\tau}{\mu}+{\eta}({\xi})$
Then the affine Weyl transformation $w_{\beta}$ generated by $\beta\in{\rm Q}$
acts on $p$ as follows:
$w_{\beta}(p)=p+\left(-\langle{\eta},{\beta}\rangle+\frac{k}{2}\langle{\beta},{\beta}\rangle\right){\delta}+k{\beta}$
(C.106)
We need the affine Weyl lattice action (C.106) to construct lattice theta-
functions in section LABEL:se:lattice-theta.
### C.5. Conjugacy classes and moduli of bundles
In this section we recall the relation of the moduli space of holomorphic
$G$-bundles on elliptic curve and the space of conjugacy classes in
$\widehat{\mathfrak{g}}$, see e.g [Baranovsky-Ginzburg_1996].
Let $H_{\tau}$ be the hyperplane (C.100) and let $B_{\tau}$ be the quotient
$H_{\tau}/{\mathbb{C}}K$, i.e. we forget about the ${\sigma}K$ part in the
expansion
$x={\tau}{\delta}^{\vee}+{\sigma}K+\sum_{i=1}^{r}{\xi}_{i}{\alpha}_{i}^{\vee}$
(C.107)
We can identify the quotient with the Cartan subalgebra $\mathfrak{h}$ of the
finite-dimensional Lie algebra $\mathfrak{g}$,
$B_{\tau}\approx{\mathfrak{h}}$. In this way we arrive at the
${\tau}$-dependent action of $W({\widehat{\mathfrak{g}}})$ on $\mathfrak{h}$:
${\xi}\mapsto{\xi}^{\widehat{w}}={\xi}^{w}+{\tau}{\beta}$ (C.108)
Note that this action does not preserve the coroot lattice ${\rm
Q}^{\vee}\subset\mathfrak{h}$. However, it preserves the ‘‘doubled’’ coroot
lattice ${{\rm Q}}^{\vee}+{\tau}{{\rm Q}}^{\vee}\subset\mathfrak{h}$.
Recall that _Bernstein-Schwarzman group_ $W_{\tau}({\widehat{\mathfrak{g}}})$
[Bernstein_1978] is the semi-direct product of $W({\mathfrak{g}})$ and the
lattice ${{\rm Q}}^{\vee}+{\tau}{{\rm Q}}^{\vee}\subset\mathfrak{h}$. The
quotient
${\mathfrak{h}}/W_{\tau}({\widehat{\mathfrak{g}}})$ (C.109)
is identified with the coarse moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic principal semi-
stable $\mathbf{G}$-bundles on the elliptic curve
${\mathscr{E}}={\mathbb{C}}/({\mathbb{Z}}+{\tau}{\mathbb{Z}})$
[Narasimhan_1965, Ramanathan_1975]. Let us denote by $z$ the additive
coordinate on $\mathscr{E}$:
$z\sim z+m+n{\tau},\qquad m,n\in{\mathbb{Z}}$
We can perform the quotient (C.109) in two steps: first, divide by ${{\rm
Q}}^{\vee}$, and then, divide by the action of $W({\widehat{\mathfrak{g}}})$.
The first step is accomplished by the exponential map from $\mathfrak{h}$ to
${\mathfrak{h}}/{{\rm Q}}^{\vee}\approx\mathbf{T}$. In the second step we
divide by the lattice $\tau{\rm Q}^{\vee}$, giving us
${\mathscr{E}}\otimes{{\rm Q}}^{\vee}$ (as an abelian group), and finally we
divide by the Weyl group $W({\mathfrak{g}})$:
$\mathbf{T}/W({\widehat{\mathfrak{g}}})=(\mathbf{T}/{\tau}{{\rm
Q}}^{\vee})/W({\mathfrak{g}})=({\mathscr{E}}\otimes{{\rm
Q}}^{\vee})/W({\mathfrak{g}})$ (C.110)
It is instructive to recall ‘‘Dolbeault’’ realization of the moduli space of
bundles, since it will be useful in our further analysis. A holomorphic
structure in the $G$-bundle is a $(0,1)$-connection
${\bar{\nabla}}_{{\bar{z}}}={\partial}_{{\bar{z}}}+A_{{\bar{z}}}$ on a
principal smooth $G$-bundle $P$, which is topologically trivial since we
assume $G$ simply connected, and therefore $A_{{\bar{z}}}$ is a
$\mathfrak{g}$-valued $(0,1)$-form on $\mathscr{E}$. The gauge equivalent
connections ${\nabla}_{{\bar{z}}}$ correspond to the isomorphic bundles:
$A_{{\bar{z}}}\mapsto g^{-1}A_{{\bar{z}}}g+g^{-1}{\partial}_{{\bar{z}}}g,\
g\in C^{\infty}({\mathscr{E}},G)$ (C.111)
Recall that the holomorphic structure on $P$ is determined by the condition
that the local holomorphic sections of $P$ are annihilated by
$\nabla_{{\bar{z}}}$, e.g. in an open neighborhood
$U_{\alpha}\subset{\mathscr{E}}$, the holomorphic sections $s_{\alpha}$ obey:
${\nabla}_{{\bar{z}}}s_{\alpha}=0\ \Leftrightarrow\
A_{{\bar{z}}}|_{U_{\alpha}}=-({\partial}_{{\bar{z}}}s_{\alpha})s_{\alpha}^{-1}$
(C.112)
One can find a gauge for $A_{{\bar{z}}}$, where it is a given by a constant
$\mathfrak{h}$-valued $(0,1)$-form:
${\nabla}_{{\bar{z}}}={\partial}_{{\bar{z}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}{\xi}\
,$ (C.113)
which still leaves a room for the residual gauge transformations, which
preserve the fact that $\xi\in\mathfrak{h}$ and
${\partial}_{z}{\xi}={\partial}_{{\bar{z}}}{\xi}=0$. There are two kinds of
transformations (the general such transformation is a composition of these
two):
${\xi}\mapsto{\xi}^{w}=g_{w}^{-1}{\xi}g_{w},\qquad g_{w}\in
N(\mathbf{T})/\mathbf{T}=W({\mathfrak{g}})$ (C.114)
and
${\xi}\mapsto{\xi}+\frac{{\tau}-{\bar{\tau}}}{2\pi\mathrm{i}}{\partial}_{{\bar{z}}}{\log}g={\xi}-{\beta}_{1}-{\tau}{\beta}_{2}$
(C.115)
where
$g(z,{{\bar{z}}})={\exp}\,2\pi\mathrm{i}\,\left(\frac{z-{{\bar{z}}}}{{\tau}-{\bar{\tau}}}{\beta}_{1}+\frac{z{\bar{\tau}}-{{\bar{z}}}{\tau}}{{\tau}-{\bar{\tau}}}{\beta}_{2}\right)$
(C.116)
with ${\beta}_{1},{\beta}_{2}\in{\rm Q}^{\vee}$, cf. (C.3). Thus the space of
all the $(0,1)$-connections ${\nabla}_{{\bar{z}}}$ modulo the smooth gauge
transformations is isomorphic to the quotient $({\mathscr{E}}\otimes{{\rm
Q}})/W({\mathfrak{g}})$, as we claimed.
E.Loojienga’s theorem [Loojienga:1976] identifies the moduli space of
$\mathbf{G}$-bundles on $\mathscr{E}$ with the weighted projective space:
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})\approx{\mathbb{W}\mathbb{P}}^{a_{0},a_{1},\ldots,a_{r}}$
(C.117)
There are several mathematical interpretations of this theorem
[Friedman:1997yq, Friedman:1997ih, Donagi:1997dp, Friedman:1998si,
Friedman:2000ze]. We shall give yet another, more physical explanation of this
result, using $\widehat{\mathfrak{g}}$ representation theory. This realization
of (C.117) is closer related to our problem.
### C.6. Infinite matrices and their Weyl group
We shall also encounter the group $\widehat{GL}_{\infty}$. It is the group
whose Lie algebra is the central extension of the Lie algebra
${\mathfrak{g}\mathfrak{l}}({\infty})$ of infinite matrices which have only a
finite number of non-vanishing elements away from the main diagonal:
$A=\sum_{i,j\in\mathbb{Z}}a_{ij}E_{ij}\,,\qquad a_{ij}=0,\ {\rm for}\ |i-j|\gg
0$
where $E_{ij}$ is the matrix with all the entries vanishing except for the
$(i,j)$ entry. The central extension is given by the cocycle:
$[A\oplus{\alpha}K,B\oplus{\beta}K]=[A,B]\oplus{\gamma}(A,B)K,\qquad{\gamma}(A,B)={\operatorname{tr}}A[J,B]$
(C.118)
where
$J=\frac{1}{2}\sum_{i\in\mathbb{Z}}\,{\rm
sign}\left(i-\frac{1}{2}\right)\,E_{ii}.$
The Cartan subalgebra
$\widehat{\mathfrak{h}}_{\infty}\subset\widehat{{\mathfrak{g}\mathfrak{l}}({\infty})}$
consists of the diagonal matrices plus the span of the central generator:
$\widehat{\mathfrak{h}}_{\infty}=\bigoplus_{i\in\mathbb{Z}}{\mathbb{C}}E_{ii}\oplus{\mathbb{C}}K$
(C.119)
The Weyl group $W(\widehat{{\mathfrak{g}\mathfrak{l}}({\infty})})$ is the
group of finite permutations of the eigenvalues of the infinite diagonal
matrix.
### C.7. The representation theory
Let us start over with the finite dimensional simple Lie group $\mathbf{G}$.
Recall that the weight ${\mu}\in{\Lambda}$ is _dominant_ ,
${\mu}\in{\Lambda}_{+}$ iff ${\mu}({\alpha}_{i}^{\vee})\geq 0$ for all
$i=1,\ldots,r$.
Recall that to every dominant weight ${\mu}\in{\Lambda}_{+}$ an irreducible
highest weight finite dimensional representation ${\mathcal{V}}_{\mu}$ of the
group $\mathbf{G}$ is associated. In particular, the fundamental
representations $R_{i}={\mathcal{V}}_{{\lambda}_{i}}$ of the group
$\mathbf{G}$ correspond to the fundamental weights ${\lambda}_{i}$. Moreover,
the Grothendieck ring ${\rm Rep}(\mathbf{G})$ of finite dimensional
representations of $\mathbf{G}$ is generated by $R_{i}$.
###### Example.
For the $A_{r}$ series, the group $\mathbf{G}=SL(r+1,{\mathbb{C}})$, the
fundamental representations $R_{i}={\bigwedge}^{i}{\mathbb{C}}^{r+1}$,
$i=1,\ldots,r$, are the exterior powers of the defining $r+1$-dimensional
representation. The center $Z={\mathbb{Z}}_{r+1}$, the adjoint group
$G^{\text{ad}}=PGL(r+1,{\mathbb{C}})=SL(r+1,{\mathbb{C}})/{\mathbb{Z}}_{r+1}$.
|
arxiv-papers
| 2012-11-09T21:05:26 |
2024-09-04T02:49:37.776358
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Nikita Nekrasov and Vasily Pestun",
"submitter": "Nikita Nekrasov",
"url": "https://arxiv.org/abs/1211.2240"
}
|
1211.2312
|
# Threshold energy for sub-barrier fusion hindrance phenomenon
V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, W. Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, 0025
Yerevan, Armenia
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, 102413 Beijing, China
###### Abstract
The relationship between the threshold energy for a deep sub-barrier fusion
hindrance phenomenon and the energy at which the regime of interaction changes
(the turning-off of the nuclear forces and friction) in the sub-barrier
capture process, is studied within the quantum diffusion approach. The
quasielastic barrier distribution is shown to be a useful tool to clarify
whether the slope of capture cross section changes at sub-barrier energies.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: threshold energy, sub-barrier capture, fusion hindrance,
quasifission, quasielastic barrier distribution
The experiments with various medium-light and heavy nuclei have shown that the
experimental slopes of the complete fusion excitation function keep increasing
at low sub-barrier energies and may become much larger than the predictions of
standard coupled-channel calculations. This was identified as the fusion
hindrance with the threshold energy $E_{s}$ Jiang ; Gomes ; Bertulani . More
experimental and theoretical studies of sub-barrier fusion hindrance are
required to improve our understanding of its physical reason, which may be
especially important in astrophysical fusion reactions Zvezda .
As shown within the quantum diffusion approach EPJSub1 ; EPJSub2 ; EPJSub3 ;
EPJSub4 ; EPJSub5 , due to a change of the regime of interaction (the turning-
off of the nuclear forces and friction) at deep sub-barrier energies, the
curve related to the capture cross section as a function of bombarding energy
has smaller slope. In the present paper we try to demonstrate the relationship
between the threshold energy $E_{s}$ for a deep sub-barrier fusion hindrance
phenomenon and the energy $E_{ch}$ at which the regime of interaction changes
in the sub-barrier capture process.
In the quantum diffusion approach the capture of nuclei is treated in terms of
a single collective variable: the relative distance between the colliding
nuclei. The neutron transfer and nuclear deformation effects are taken into
consideration through the dependence of the nucleus-nucleus potential on the
isotopic compositions, deformations and orientations of interacting nuclei.
Our approach takes into consideration the fluctuation and dissipation effects
in collisions of heavy ions which model the coupling with various channels
(for example, coupling of the relative motion with low-lying collective modes
such as dynamical quadrupole and octupole modes of target and projectile
Ayik333 ). We have to mention that many quantum-mechanical and non-Markovian
effects Hofman ; Ayik ; Hupin accompanying the passage through the potential
barrier are taken into consideration in our formalism EPJSub1 ; EPJSub2 ;
EPJSub3 ; EPJSub4 ; EPJSub5 . The details of used formalism are presented in
our previous articles EPJSub1 ; EPJSub2 . With this approach many heavy-ion
capture reactions at energies above and well below the Coulomb barrier have
been successfully described.
Within the quantum diffusion model EPJSub1 ; EPJSub2 ; EPJSub3 ; EPJSub4 ;
EPJSub5 the nuclear forces start to play a role at relative distance
$R_{int}=R_{b}+1.1$ fm ($R_{b}$ is the position of the Coulomb barrier at
given angular momentum and orientations of the interacting nuclei) where the
nucleon density of colliding nuclei approximately reaches 10% of saturation
density. If the colliding nuclei approach the distance $R_{int}$ between their
centers, the nuclear forces start to act in addition to the Coulomb
interaction. Thus, at $R<R_{int}$ the relative motion may be more coupled with
other degrees of freedom. At $R>R_{int}$ the relative motion is almost
independent of the internal degrees of freedom. Depending on whether the value
of external turning point $R_{ex}$ is larger or smaller than interaction
radius $R_{int}$, the impact of coupling with other degrees of freedom upon
the barrier passage seems to be different. So, two regimes of interaction at
sub-barrier energies differ by the action of nuclear forces and, respectively,
of nuclear friction. Due to the switching-off the nuclear interaction at
external turning point $R_{ex}$, the cross sections falls with the smaller
rate at a deep sub-barrier energies.
Figure 1: The experimental threshold energy $E_{s}$ for a deep sub-barrier
fusion hindrance phenomenon Jiang and the calculated energy $E_{ch}$ at which
the regime of interaction changes in the indicated sub-barrier capture
reactions as a function of $Z_{1}Z_{2}[A_{1}A_{2}/(A_{1}+A_{2})]^{1/2}$.
As seen in Fig. 1, for the reactions 4He + 208Pb, 58Ni + 54Fe, 48Ca +
48Ca,90,96Zr, 40Ca + 90,96Zr, 58Ni + 58,60,64Ni, 60Ni + 89Y, 64Ni +
64Ni,100Mo, 90Zr+90Zr, and 16O + 208Pb,238U, there is a good agreement between
the threshold energy $E_{s}$ for a deep sub-barrier fusion hindrance
phenomenon and the energy $E_{ch}$ at which the regime of interaction changes
in the sub-barrier capture process. The values $E_{s}$ and $E_{ch}$ almost
coincide and linearly increase with
$Z_{1}Z_{2}[A_{1}A_{2}/(A_{1}+A_{2})]^{1/2}$.
and the capture cross section is the sum of the fusion and quasifission cross
sections, from the comparison of calculated capture cross sections and
measured fusion cross sections one can extract the hindrance factor and the
threshold incident energy for a deep sub-barrier fusion hindrance phenomenon.
The small fusion cross section at energies well below the Coulomb barrier may
indicate that the quasifission channel is preferable and the system goes to
this channel after the capture EPJSub1 ; EPJSub2 ; EPJSub3 ; EPJSub4 ; EPJSub5
. So, the observed hindrance factor may be understood in term of quasifission.
At deep sub-barrier energies, the quasifission event corresponds to the
formation of a nuclear-molecular state or dinuclear system with small
excitation energy that separates (in the competition with the compound nucleus
formation process) by the quantum tunneling through the Coulomb barrier in a
binary event with mass and charge close to the colliding nuclei. In this sense
the quasifission is the general phenomenon which takes place in the reactions
with the massive GSI ; Volkov ; nasha ; Avaz , and medium-mass nuclei EPJSub2
.
Figure 2: The calculated (solid line) $D_{qe}(E_{\rm c.m.})=dP_{cap}(E_{\rm
c.m.},J=0)/dE_{\rm c.m.}$ for the 16O + 120Sn reaction. The experimental data
(symbols) are from Ref. Sinha . The values of $D_{qe}(E_{\rm c.m.})$ are shown
in the linear (a) and logarithmic (b) scales.
Since the quasielastic measurements are usually not as complex as the capture
(fusion) measurements, and they are well suited to survey the decreasing rate
of fall of the sub-barrier capture cross section. There is a direct
relationship between the capture and the quasielastic scattering processes,
because any loss from the quasielastic channel contributes directly to the
capture (the conservation of the reaction flux):
$P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)=1$
and
$dP_{cap}/dE_{\rm c.m.}=-dP_{qe}/dE_{\rm c.m.},$
where $P_{qe}$ is the reflection probability and $P_{cap}$ is the capture
(transmission) probability. The quasielastic scattering is the sum of elastic,
inelastic, and transfer processes. The reflection probability
$P_{qe}(E_{\rm c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$
for angular momentum $J=0$ is given by the ratio of the quasielastic
differential cross section and Rutherford differential cross section at 180
degrees Timmers ; Timmers2 ; Zhang ; Sonzogni ; Sinha ; Piasecki .
Figure 3: The same as in Fig. 2 but for the 16O + 208Pb reaction. The
experimental data (symbols) are from Ref. Timmers2 . Figure 4: The calculated
(solid line) $D_{qe}(E_{\rm c.m.})=dP_{cap}(E_{\rm c.m.},J=0)/dE_{\rm c.m.}$
for the 4He + 208Pb reaction.
The barrier distribution is extracted by taking the first derivative of the
$P_{qe}$ with respect to $E_{\rm c.m.}$, that is,
$D_{qe}(E_{\rm c.m.})=-dP_{qe}(E_{\rm c.m.},J=0)/dE_{\rm c.m.}=$
$=dP_{cap}(E_{\rm c.m.},J=0)/dE_{\rm c.m.}.$
Thus, one can observe the change of the fall rate of $P_{cap}(E_{c.m.},J=0)$
at sub-barrier energies by measuring the barrier distribution $D_{qe}$. By
employing the quantum diffusion approach and calculating $dP_{cap}(E_{\rm
c.m.},J=0)/dE_{\rm c.m.}$, one can obtain $D_{qe}(E_{\rm c.m.})$. In addition
to the mean peak position of the $D_{qe}$ around the barrier height, we
predict the sharp change of the slope of $D_{qe}$ below the threshold energy
because of a change of the regime of interaction in the sub-barrier capture
process (Figs. 2–4). The effect seems to be more pronounced in the collisions
of spherical nuclei (Figs. 3 and 4). The collisions of deformed nuclei occurs
at various mutual orientations on which the value of $R_{int}$ depends. Thus,
the deformation and neutron transfer effects can smear out this effect. The
reactions 4He,16O + ASn,144Sm,208Pb and 48,40Ca$,^{36}$S+90Zr with the
spherical nuclei are preferable for the experimental study of $D_{qe}(E_{\rm
c.m.})$.
In conclusions, employing the quantum diffusion approach, we demonstrated the
relationship between the threshold energy for a deep sub-barrier fusion
hindrance phenomenon and the energy at which the regime of interaction changes
in the sub-barrier capture process. We predicted the sharp change of the slope
of the quasielastic barrier distribution below the threshold energy. This is
expected to be the experimental indication of a change of the regime of
interaction in the sub-barrier capture. One concludes that the quasielastic
technique could be an important tool in capture (fusion) research.
This work was supported by DFG, NSFC, and RFBR. The IN2P3(France)-JINR(Dubna)
and Polish - JINR(Dubna) Cooperation Programmes are gratefully acknowledged.
## References
* (1) C.L. Jiang et al., Phys. Rev. Lett. 89, 052701 (2002); C.L. Jiang et al., Phys. Rev. C 71, 044613 (2005); C.L. Jiang, B.B. Back, H. Esbensen, R.V.F. Janssens, and K.E. Rehm, Phys. Rev. C 73, 014613 (2006); H. Esbensen and C.L. Jiang, Phys. Rev. C 79, 064619 (2009).
* (2) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, (2006) 1.
* (3) C.A. Bertulani, EPJ Web Conf. 17, 15001 (2011).
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* (5) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Eur. Phys. J. A 47, 38 (2011); J. of Phys.: Conf. Ser. 282, 012001 (2011); EPJ Web Conf. 17, 04003 (2011).
* (6) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Phys. C 84, 064614 (2011); Phys. Rev. C 85, 024616 (2012); Phys. Rev. C 85, 069903 (2012).
* (7) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 45, 125 (2010).
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* (9) R.A. Kuzyakin, V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, E.E. Saperstein, and S.V. Tolokonnikov, Phys. Rev. C 85, 034612 (2012).
* (10) S. Ayik, B. Yilmaz, and D. Lacroix, Phys. Rev. C 81, 034605 (2010).
* (11) H. Hofmann, Phys. Rep. 284, 137 (1997); C. Rummel and H. Hofmann, Nucl. Phys. A 727, 24 (2003).
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* (13) G. Hupin and D. Lacroix, Phys. Rev. C 81, 014609 (2010).
* (14) J.G. Keller et al., Nucl. Phys. A452, 173 (1986).
* (15) V.V. Volkov, Particles and Nuclei, 35, 797 (2004).
* (16) G.G. Adamian, N.V. Antonenko, and W.Scheid, Phys. Rev. C 68, 034601 (2003); Lecture Notes in Physics 848, Clusters in Nuclei, Vol. 2, ed. by C. Beck (Springer-Verlag, Berlin, 2012) p. 165.
* (17) G. Giardina et al., Nucl. Phys. A671, 165 (2000); A. Nasirov et al., Nucl. Phys. A759, 342 (2005); Z.-Q. Feng, G.-M. Jin, J.-Q. Li, and W. Scheid, Phys. Rev. C 76, 044606 (2007); H.Q. Zhang, C.L. Zhang, C.J. Lin, Z.H. Liu, F. Yang, A.K. Nasirov, G. Mandaglio, M. Manganaro, and G. Giardina, Phys. Rev. C 81, 034611 (2010).
* (18) H. Timmers, J.R. Leigh, M. Dasgupta, D.J. Hinde, R.C. Lemmon, J.C. Mein, C.R. Morton, J.O. Newton, and N. Rowley, Nucl. Phys. A584, 190 (1995).
* (19) H. Timmers, Ph. D. thesis, Australian National University (1996).
* (20) H.Q. Zhang, F. Yang, C. Lin, Z. Liu, F. Yang, and Y. Hu, Phys. Rev. C 57, R1047 (1998); Inter. workshop on Nuclear reactions and beyond, Wold Scientific (2000), p.95; F. Yang, C.J. Lin, X.K. Wu, H.Q. Zhang, C.L. Zhang, P.Zhou, and Z.H. Liu, Phys. Rev. C 77, 014601 (2008).
* (21) A.A. Sonzogni, J.D. Bierman, M.P. Kelly, J.P. Lestone, J.F. Liang, and R. Vandenbosch, Phys. Rev. C 57, 722 (1998).
* (22) S. Sinha, M.R. Pahlavani, R. Varma, R.K. Choudhury, B.K. Nayak, and A. Saxena, Phys. Rev. C 64, 024607 (2001).
* (23) E. Piasecki et al., Phys. Rev. C 65, 054611 (2002); E. Piasecki et al., Phys. Rev. C 80, 054613 (2009); E. Piasecki et al., Phys. Rev. C 85, 054604 (2012); ibid 85, 054608 (2012).
|
arxiv-papers
| 2012-11-10T10:11:07 |
2024-09-04T02:49:37.829031
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid, and H. Q.\n Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1211.2312"
}
|
1211.2320
|
# Search of systematic behavior of breakup probability in reactions with
weakly bound projectiles at energies around Coulomb barrier
V.V. Sargsyan1,2, G.G. Adamian1, N.V.Antonenko 1, W. Scheid3, and H.Q. Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, 0025
Yerevan, Armenia
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
Comparing the capture cross sections calculated without the breakup effect and
experimental complete fusion cross sections, the breakup was analyzed in
reactions with weakly bound projectiles 6,7,9Li, 9,11Be, and 6,8He. A trend of
a systematic behavior for the complete fusion suppression as a function of the
target charge and bombarding energy is not achieved. The quasielastic
backscattering is suggested to be an useful tool to study the behavior of the
breakup probability in reactions with weakly bound projectiles.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: capture, breakup effects, weakly bound nuclei, quantum diffusion
approach
## I Introduction
In recent years, many efforts have been made to understand the effect of
breakup of weakly bound nuclei during the fusion reaction in very asymmetric
reactions where the capture cross section is equal to the complete fusion
cross section Bertulani ; PRSGomes1 ; Alamanos ; PRSGomes2 ; PRSGomes3 ;
PRSGomes4 ; Rafi ; Alinka ; PRSGomes5 . The light radioactive nuclei,
especially halo nuclei, such as 6He, 8B, 11Be, and the stable nuclei 6,7Li and
9Be are weakly bounded, hence there is a chance of the breakup in the
colliding process. By performing a comparison of fusion data with theoretical
predictions which do not take into account the dynamic breakup plus transfer
channel effects, it has been shown PRSGomes2 ; PRSGomes3 ; PRSGomes4 ;
PRSGomes5 , that for energies from about $1.1V_{b}$ to $1.5V_{b}$ ($V_{b}$ is
the height of the Coulomb barrier) complete fusion in the reactions
6,7Li+208Pb,209Bi and 9Be+89Y,124Sn,208Pb,209Bi is suppressed by about 30%.
However, the 9Be+144Sm data is out of the systematics, showing a much smaller
suppression of about 15%. The total fusion (incomplete fusion + sequential
complete fusion + complete fusion) cross section for the same projectiles on
targets of any mass, including 9Be + 27Al,64Zn, does not seem to be affected
by the dynamic breakup and transfer effects PRSGomes4 ; PRSGomes5 . As the
charge of the target decreases, one expects that the Coulomb breakup becomes
weaker, and consequently the complete fusion suppression and incomplete fusion
probability decrease. The lack of a clear systematic behavior of the complete
fusion suppression as a function of the target charge was explained in Ref.
PRSGomes5 by different effects of the transfer channels on the complete
fusion and by some problems with the experimental data analysis.
In the present article we try to reveal a systematic behavior of the complete
fusion suppression as a function of the target charge $Z_{T}$ and colliding
energy $E_{\rm c.m.}$ by using the quantum diffusion approach EPJSub ; EPJSub1
and by comparing the calculated capture cross sections in the absence of
breakup with the experimental complete and total fusion cross sections. The
effects of deformation and neutron transfer on the complete fusion are taken
into consideration.
## II Model
In the quantum diffusion approach EPJSub ; EPJSub1 the collision of nuclei is
described with a single relevant collective variable: the relative distance
between the colliding nuclei. This approach takes into account the fluctuation
and dissipation effects in collisions of heavy ions which model the coupling
with various channels (for example, coupling of the relative motion with low-
lying collective modes such as dynamical quadrupole and octupole modes of the
target and projectile nuclei Ayik333 ). We like to mention that many quantum-
mechanical and non-Markovian effects accompanying the passage through the
potential barrier are considered in our formalism EPJSub ; EPJSub1 ; PRCPOP .
The nuclear deformation effects are taken into account through the dependence
of the nucleus-nucleus potential on the deformations and mutual orientations
of the colliding nuclei. To calculate the nucleus-nucleus interaction
potential $V(R)$, we use the procedure presented in Refs. EPJSub ; EPJSub1 .
For the nuclear part of the nucleus-nucleus potential, a double-folding
formalism with a Skyrme-type density-dependent effective nucleon-nucleon
interaction is used. Within this approach many heavy-ion capture reactions
with stable and radioactive beams at energies above and well below the Coulomb
barrier have been successfully described EPJSub ; EPJSub1 ; PRCPOP . One
should note that other diffusion models, which include the quantum statistical
effects, were also proposed in Hofman .
We assume that the sub-barrier capture mainly depends on the optimal one-
neutron ($Q_{1n}>Q_{2n}$) or two-neutron ($Q_{2n}>Q_{1n}$) transfer with a
positive $Q$-value. Our assumption is that, just before the projectile is
captured by the target-nucleus (just before the crossing of the Coulomb
barrier) which is a slow process, the transfer occurs that can lead to the
population of the first excited collective state in the recipient nucleus
SSzilner . So, the motion to the $N/Z$ equilibrium starts in the system before
the capture because it is energetically favorable in the dinuclear system in
the vicinity of the Coulomb barrier. For the reactions under consideration,
the average change of mass asymmetry is connected to the one- or two-neutron
transfer. Since after the transfer the mass numbers, the isotopic composition
and the deformation parameters of the interacting nuclei, and,
correspondingly, the height $V_{b}=V(R_{b})$ and shape of the Coulomb barrier
are changed, one can expect an enhancement or suppression of the capture. When
the isotopic dependence of the nucleus-nucleus interaction potential is weak
and the deformations of the interacting nuclei after the transfer have not
changed, there is no effect of the neutron transfer on the capture cross
section. This scenario was verified in the description of many reactions in
Ref. EPJSub1 .
## III Results of calculations
All calculated results are obtained with the same set of parameters as in Ref.
EPJSub . We use the friction coefficient in the relative distance coordinate
which is close to that calculated within the mean field approaches obzor . The
absolute values of the quadrupole deformation parameters $\beta_{2}$ of even-
even deformed nuclei are taken from Ref. Ram . For the nuclei deformed in the
ground state, the $\beta_{2}$ in the first excited collective state is similar
to the $\beta_{2}$ in the ground state. For the quadruple deformation
parameter of an odd nucleus, we choose the maximal value of the deformation
parameters of neighboring even-even nuclei. For the double magic and
neighboring nuclei, in the ground state we set $\beta_{2}=0$. There are
uncertainties in the definition of the values of $\beta_{2}$ in light-mass
nuclei. However, these uncertainties weakly influence the capture cross
sections in the asymmetric reactions treated. In the calculations for light
nuclei we use $\beta_{2}$ from Ref. Kanada .
### III.1 Breakup probabilities
In Figs. 1-13 we compare the calculated $\sigma_{c}^{th}$ capture cross
sections with the experimental $\sigma_{fus}^{exp}$ complete and total fusion
cross sections in the reactions induced by projectiles 9Be, 10,11B, 6,7,9Li,
4,6,8He B11Bi209 ; B11Tb159 ; Be9Bi209 ; Be9Pb208 ; Be9Sm144 ; Be9Sn124 ;
Be9Y89 ; Be9Zn64 ; Be9Al27 ; Li7Al27 ; Li7Au197 ; Li6Bi209 ; Li6Pb208 ;
Li6Pt198 ; Li6Sm144 ; Li6Zn64 ; Li7Ho165 ; He4Bi211 ; He4Zn64 ; Li9Zn70 ;
He4Au197 ; Li9Pb208 . The difference between the capture cross section and the
complete fusion cross section can be ascribed to the breakup effect. Comparing
$\sigma_{c}^{th}$ and $\sigma_{fus}^{exp}$, one can estimate the breakup
probability
$\displaystyle P_{\rm BU}=1-\sigma_{fus}^{exp}/\sigma_{c}^{th}.$ (1)
If at some energy $\sigma_{fus}^{exp}>\sigma_{c}^{th}$, the values of
$\sigma_{c}^{th}$ was normalized so to have $P_{\rm BU}\geq 0$ at any energy.
Note that $\sigma_{fus}^{exp}=\sigma_{fus}^{noBU}+\sigma_{fus}^{BU}$ contains
the contribution from two processes: the direct fusion of the projectile with
the target ($\sigma_{fus}^{noBU}$), and the breakup of the projectile followed
by the fusion of the two projectile fragments with the target
($\sigma_{fus}^{BU}$). A more adequate estimate of the breakup probability
would then be:
$\displaystyle P_{\rm BU}=1-\sigma_{fus}^{noBU}/\sigma_{c}^{th},$ (2)
which leads to larger values of $P_{\rm BU}$ than the expression employed by
us. However, the ratio between $\sigma_{fus}^{noBU}$ and $\sigma_{fus}^{BU}$
cannot be measured experimentally, but can be estimated with the approach
suggested in Refs. Maximka ; PLATYPUS . The parameters of the potential are
taken to fit the height of the Coulomb barrier obtained in our calculations.
The parameters of the breakup function Maximka are set to describe the value
of $\sigma_{fus}^{exp}$. As shown in Ref. Maximka and in our calculations, in
the 8Be+208Pb reaction the fraction of $\sigma_{fus}^{BU}$ in
$\sigma_{fus}^{exp}$ does not exceed few percents at $E_{\rm c.m.}-V_{b}<$4
MeV. This fraction rapidly increases and reaches about 12–20%, depending on
the reaction, at $E_{\rm c.m.}-V_{b}\approx$10 MeV. Because we are mainly
interested in the energies near and below the barrier, the estimated
$\sigma_{fus}^{BU}$ does not exceed 20% of $\sigma_{fus}^{exp}$ at $E_{\rm
c.m.}-V_{b}<$10 MeV. The results for $P_{\rm BU}$ are presented taking
$\sigma_{fus}^{noBU}$ into account in Eq. (2).
As seen in Figs. 14 and 15, at energies above the Coulomb barriers the values
of $P_{\rm BU}$ vary from 0 to 84%. In the reactions 9Be+144Sm,208Pb,209Bi the
value of $P_{\rm BU}$ increases with charge number of the target at $E_{\rm
c.m.}-V_{b}>3$ MeV. This was also noted in Ref. PRSGomes5 . However, the
reactions 9Be+89Y,124Sn are out of this systematics. In the reactions
6Li+144Sm,198Pt,209Bi the value of $P_{\rm BU}$ decreases with increasing
charge number of the target at $E_{\rm c.m.}-V_{b}>3$ MeV. While in the
reactions 9Be+89Y,144Sm,208Pb,209Bi the value of $P_{\rm BU}$ has a minimum at
$E_{\rm c.m.}-V_{b}\approx 0$ and a maximum at $E_{\rm
c.m.}-V_{b}\approx-(1-3)$ MeV, in the 9Be+124Sn reaction the value of $P_{\rm
BU}$ steady decreases with energy. In the reactions 6Li+144Sm,198Pt,209Bi,
7Li+208Pb,209Bi, and 9Li+208Pb there is maximum of $P_{\rm BU}$ at $E_{\rm
c.m.}-V_{b}\approx-(0-1)$ MeV. However, in the reactions 6Li+208Pb and
7Li+165Ho $P_{\rm BU}$ has a minima $E_{\rm c.m.}-V_{b}\approx 2$ MeV and no
maxima at $E_{\rm c.m.}-V_{b}\approx 0$. For 9Be, the breakup threshold is
slightly larger than for 6Li. Therefore, we can not explain a larger breakup
probability at smaller $E_{\rm c.m.}-V_{b}$ in the case of 9Be.
In Figs. 1-13 we also show the calculated capture cross sections normalized by
some factors to obtain a rather good agreement between the experimental and
theoretical results. These average normalization factors are 0.7, 0.75, 0.9,
0.64, 0.7, 1, 0.9 for the reactions 9Be+209Bi,208Pb,144Sm,124Sn,89Y,64Zn,27Al,
respectively, 0.52, 0.5, 0.5, 0.42, 0.65 for the reactions
6Li+209Bi,208Pb,198Pt,144Sm,64Zn, respectively, 0.6, 0.7, 0.65, 0.6, 0.65,
0.75 for the reactions 7Li+209Bi,197Au,165Ho,159Tb,64Zn,27Al, respectively.
For the reactions 9Li+208Pb (Fig. 10), 6He+209Bi (Fig. 11), 6He+64Zn (Fig.
11), 6He+197Au (Fig. 12), 8He+197Au (Fig. 12), 11B+209Bi (Fig. 13) and
11B+159Tb (Fig. 13), a satisfactory agreement between experimental fusion data
and capture cross sections can be reached with average normalization factors
0.6, 0.68, 0.4, 0.8, 0.7, 0.82, and 0.95, respectively. Note that these
average normalization factors do not depend on $E_{\rm c.m.}$. With the 9Be
projectile we obtain the complete fusion suppressions similar to those
reported in Refs. PRSGomes4 ; PRSGomes5 . For lighter targets, when the
Coulomb breakup becomes weaker, one expects that the suppression of complete
fusion becomes smaller than for heavy targets.
An expected behavior for complete fusion suppression is that fusion
probability increases with decreasing $Z_{T}$. However, one can observe
deviations from this rule. In the reactions 9Be+124Sn,89Y the data show quite
larger complete fusion suppression (30–36)%. For the reactions induced by a
6Li projectile, one can see that the fusion suppression is nearly independent
of $Z_{T}$. The replacement of 7Li by 6Li in the reactions in Fig. 9 almost
does not change the experimental Li7Au197 ; Li7Al27 and calculated data. The
Coulomb fields for very light systems 9Be+27Al, and 6He,7Li+64Zn are not
strong enough to produce an appreciable breakup. It is not realistic that the
fusion suppression in the 9Be+64Zn reaction is smaller than the one in the
9Be+27Al reaction or the suppressions of fusion coincide in the reactions
4,6He+64Zn (Fig. 11) with stable and exotic projectiles. Note that the
experimental data for the reactions 6,7Li+27Al,64Zn and 9Be+27Al,64,70Zn are
for the total fusion. In general, the total fusion does not seem to be
affected by breakup PRSGomes4 ; PRSGomes5 .
So, there is a lack of a systematic behavior of the complete fusion
suppression for the systems treated. The possible explanation of it is that
there are probably some problems with the data analysis which were earlier
noted in Refs. PRSGomes4 ; PRSGomes5 from the point of view of a universal
fusion function representation. It could be also that at energies near the
Coulomb barrier the characteristic time of the breakup is larger than the
characteristic time of the capture process and influences the complete fusion.
For the reactions 6,7Li+208Pb, the characteristic times of the prompt and
delayed breakup were studied recently in Ref. Luong .
The large positive $Q_{2n}$-value in the 9Li+208Pb reaction Li9Pb208 gives a
possibility of a two-neutron transfer before the capture. However, the capture
cross sections calculated with and without neutron transfer are very close to
each other because the effect of neutron transfer is rather weak in asymmetric
reactions EPJSub ; EPJSub1 . The calculated capture cross sections normalized
by a factor of 0.6 are shown by the dotted line in the lower part of Fig. 10.
In the upper part of Fig. 10, the predicted capture cross sections for the
reaction 11Li+208Pb are shown.
### III.2 Quasielastic backscattering - tool for search of breakup process in
reactions with weakly bound projectiles
The lack of a clear systematic behavior of the complete fusion suppression as
a function of the target charge requires new additional experimental and
theoretical studies. The quasielastic backscattering has been used Timmers ;
Zhang ; Sinha ; Piasecki as an alternative to investigate fusion (capture)
barrier distributions, since this process is complementary to fusion. Since
the quasielastic experiment is usually not as complex as the capture (fusion)
and breakup measurements, they are well suited to survey the breakup
probability. There is a direct relationship between the capture, the
quasielastic scattering and the breakup processes, since any loss from the
quasielastic and breakup channel contributes directly to capture (the
conservation of the reaction flux):
$\displaystyle P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)+P_{BU}(E_{\rm
c.m.},J)=1,$ (3)
where $P_{qe}$ is the reflection quasielastic probability, $P_{BU}$ is the
breakup (reflection) probability, and $P_{cap}$ is the capture (transmission)
probability. The quasielastic scattering is the sum of all direct reactions,
which include elastic, inelastic, and transfer processes. Equation (3) can be
rewritten as
$\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}+\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}=P_{qe}^{nBU}(E_{\rm c.m.},J)+P_{cap}^{nBU}(E_{\rm c.m.},J)=1,$ (4)
where
$P_{qe}^{nBU}(E_{\rm c.m.},J)=\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}$
and
$P_{cap}^{nBU}(E_{\rm c.m.},J)=\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}$
are the quasielastic and capture probabilities, respectively, in the absence
of the breakup process. From these expressions we obtain the useful formulas
$\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{P_{cap}(E_{\rm
c.m.},J)}=\frac{P_{qe}^{nBU}(E_{\rm c.m.},J)}{P_{cap}^{nBU}(E_{\rm
c.m.},J)}=\frac{P_{qe}^{nBU}(E_{\rm c.m.},J)}{1-P_{qe}^{nBU}(E_{\rm
c.m.},J)}=a.$ (5)
Using Eqs. (3) and (5), we obtain the relationship between breakup and
quasielastic processes:
$\displaystyle P_{BU}(E_{\rm c.m.},J)$ $\displaystyle=$ $\displaystyle
1-[P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)]=1-P_{qe}(E_{\rm
c.m.},J)[1+1/a]$ (6) $\displaystyle=$ $\displaystyle 1-P_{qe}(E_{\rm
c.m.},J)/P_{qe}^{nBU}(E_{\rm c.m.},J).$
The last equation is one of important results of the present paper.
Analogously one can find other expression
$\displaystyle P_{BU}(E_{\rm c.m.},J)=1-P_{cap}(E_{\rm
c.m.},J)/P_{cap}^{nBU}(E_{\rm c.m.},J),$ (7)
which relates the breakup and capture processes.
The reflection quasielastic probability
$\displaystyle P_{qe}(E_{\rm c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$ (8)
for bombarding energy $E_{\rm c.m.}$ and angular momentum $J=0$ is given by
the ratio of the quasielastic differential cross section $\sigma_{qe}$ and
Rutherford differential cross section $\sigma_{Ru}$ at 180 degrees Timmers ;
Zhang ; Sonzogni ; Sinha ; Piasecki . Employing Eqs. (6), (8), and the
experimental quasielastic backscattering data with toughly and weakly bound
isotopes-projectiles and the same compound nucleus, one can extract the
breakup probability of the exotic nucleus. For example, using Eq. (6) at $J=0$
and the experimental $P_{qe}^{nBU}$[4He+208Pb] of the 4He+208Pb reaction with
toughly bound nuclei (without breakup) and $P_{qe}$[6He+206Pb] of the
6He+206Pb reaction with weakly bound projectile (with breakup), and taking
into consideration $V_{b}$(4He+208Pb)$\approx V_{b}$(6He+206Pb) for the very
asymmetric systems, one can extract the breakup probability of the 6He:
$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm
c.m.},J=0)[^{6}He+^{206}Pb]}{P_{qe}^{nBU}(E_{\rm
c.m.},J=0)[^{4}He+^{208}Pb]}.$ (9)
Comparing the experimental quasielastic backscattering cross sections in the
presence and absence of breakup data in the reaction pairs 6He+68Zn and
4He+70Zn, 6He+122Sn and 4He+124Sn, 6He+236U and 4He+238U, 8He+204Pb and
4He+208Pb, 9Be+208Pb and 10Be+207Pb, 11Be+206Pb and 10Be+207Pb, 8B+208Pb and
10B+206Pb, 8B+207Pb and 11B+204Pb, 9B+208Pb and 11B+206Pb, 15C+207Pb and
14C+208Pb, 17F+206Pb and 19F+208Pb leading to the same corresponding compound
nuclei, one can analyze the role of the breakup channels in the reactions with
the light weakly bound projectiles 6,8He, 9,11Be, 8,9B, 15C, and 17F at near
and below barrier energies. One concludes that the quasielastic technique
could be a very important tool in breakup research. We propose to extract the
breakup probability directly from the quasielastic cross sections of systems
mentioned above.
## IV Summary
Comparing the calculated capture cross sections in the absence of breakup data
and experimental complete fusion data, we analyzed the role of the breakup
channels in the reactions with the light projectiles 9Be, 6,7,9Li and 6,8He at
near-barrier energies. Within the quantum diffusion approach the neutron
transfer and deformation effects were taken into account. Analyzing the
extracted breakup probabilities, we showed that there are no systematic trends
of breakup in the reactions studied. Moreover, for some system with larger
(smaller) $Z_{T}$ we found the contribution of breakup to be smaller (larger).
Almost for all reactions considered we obtained a satisfactory agreement
between calculated capture cross section and experimental fusion data, if the
calculated capture cross section or the experimental fusion data are
renormalized by some average factor which does not depend on the bombarding
energy. Note that our conclusions coincide with those of Refs. PRSGomes4 ;
PRSGomes5 , where the universal fusion function formalism was applied for the
analysis of experimental data. One needs to measure directly the breakup
process in different systems, especially light ones, to understand the role of
the Coulomb breakup in the complete fusion process. The other important
subject to be investigated both experimental and theoretically is the
characteristic time of the breakup. The first steps in these directions were
done in Refs. Rafi ; Gomesnew ; Luong .
As shown, one no needs to measure directly the breakup process in different
systems, especially light ones, to understand the role of the breakup in the
capture (complete fusion) process. Employing the experimental quasielastic
backscattering data with weakly and toughly bound isotopes of light nucleus
and Eq. (6), the dependence of breakup probability on $E_{\rm c.m.}$ can be
extracted for the systems suggested. Analyzing the extracted breakup
probabilities, one can indirectly study the trends of breakup in the different
reactions at energies near and below Coulomb barrier.
## V acknowledgements
We thank P.R.S. Gomes for fruitful discussions and useful suggestions.
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Figure 1: The calculated capture cross sections vs $E_{\rm c.m.}$ for the
reactions 9Be+209Bi and 9Be+208Pb (solid lines). The experimental data
(squares) are from Refs. Be9Bi209 ; Be9Pb208 . The calculated capture cross
sections normalized by factors 0.7 and 0.75 for the reactions 9Be+209Bi and
9Be+208Pb, respectively, are presented by dotted lines. Figure 2: (Color
online) The calculated capture cross sections vs $E_{\rm c.m.}$ for the
reactions 9Be+144Sm and 9Be+124Sn (solid lines). The experimental data
(squares) are from Refs. Be9Sm144 ; Be9Sn124 . The experimental total fusion
data Be9Sm144 for the 9Be+144Sm reaction are shown by open circles. The
calculated capture cross sections normalized by factors 0.9 and 0.64 for the
reactions 9Be+144Sm and 9Be+124Sn, respectively, are presented by dotted
lines. Figure 3: The calculated capture cross sections vs $E_{\rm c.m.}$ for
the reactions 9Be+89Y and 9Be+64Zn (solid lines). The experimental data
(squares) are from Refs. Be9Y89 ; Be9Zn64 . The calculated capture cross
sections normalized by a factor 0.7 for the 9Be+89Y reaction are presented by
a dotted line. Figure 4: The calculated capture cross sections vs $E_{\rm
c.m.}$ for the reaction 9Be+27Al (solid line). The experimental data (squares)
are from Ref. Be9Al27 . The calculated capture cross sections normalized by a
factor 0.9 for the 9Be+27Al reaction are presented by a dotted line. Figure
5: The calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions
6Li+209Bi and 6Li+208Pb (solid lines). The experimental data (squares) are
from Refs. Li6Bi209 ; Li6Pb208 . The calculated capture cross sections
normalized by factors 0.52 and 0.5 for the reactions 6Li+209Bi and 6Li+208Pb,
respectively, are presented by dotted lines. Figure 6: The calculated capture
cross sections vs $E_{\rm c.m.}$ for the reactions 6Li+198Pt and 6Li+144Sm
(solid lines). The experimental data (squares) are from Refs. Li6Pt198 ;
Li6Sm144 . The calculated capture cross sections normalized by factors 0.5 and
0.42 for the reactions 6Li+198Pt and 6Li+144Sm, respectively, are presented by
dotted lines. Figure 7: The calculated capture cross sections vs $E_{\rm
c.m.}$ for the reactions 7Li+209Bi and 7Li+64Zn (solid lines). The calculated
results for the reactions 6,7Li+64Zn almost coincide. The experimental data
(squares) are from Refs. Li6Bi209 ; Be9Zn64 . The experimental data for the
reactions 7Li+64Zn (squares) and 6Li+64Zn (circles and stars) are from Refs.
Be9Zn64 ; Li6Zn64 . The calculated capture cross sections normalized by
factors 0.6 and 0.65 for the reactions 7Li+209Bi and 7Li+64Zn, respectively,
are presented by dotted lines. Figure 8: The calculated capture cross
sections vs $E_{\rm c.m.}$ for the reactions 7Li+165Ho and 7Li+159Tb (solid
lines). The experimental data (squares) are from Refs. Li7Ho165 ; B11Tb159 .
The experimental total fusion data Li7Ho165 ; B11Tb159 are shown by the solid
triangles. The calculated capture cross sections normalized by factors 0.65
and 0.6 for the reactions 7Li+165Ho and 7Li+159Tb, respectively, are presented
by dotted lines. Figure 9: The calculated capture cross sections vs $E_{\rm
c.m.}$ for the reactions 7Li+197Au and 7Li+27Al (solid lines). The
experimental data (squares) are from Refs. Li7Au197 ; Li7Al27 . The calculated
capture cross sections normalized by factors 0.7 and 0.75 for the reactions
7Li+197Au and 7Li+27Al, respectively, are presented by dotted lines. Figure
10: The calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions
11Li+208Pb and 9Li+208Pb (solid lines). The experimental data (squares) are
from Ref. Li9Pb208 . The calculated capture cross sections normalized by a
factor 0.6 for the 9Li+208Pb reaction are presented by a dotted line. Figure
11: The calculated capture cross sections vs $E_{\rm c.m.}$ for the indicated
reactions 6He+209Bi (solid line), 6He+64Zn (solid line), and 4He+64Zn (dashed
line). The experimental data for the reactions 4He+64Zn (solid squares) and
6He+64Zn (open squares) are from Refs. He4Bi211 ; He4Zn64 . The calculated
capture cross sections normalized by factors 0.68, 0.4, and 0.4 for the
reactions 6He+209Bi (dotted line), 6He+64Zn (dotted line), and 4He+64Zn (dash-
dotted line), respectively, are shown. Figure 12: (Color online) The
calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions 9Li+70Zn
(solid line), 4,6He+197Au (solid lines) and 8He+197Au (dashed line). The
results for the reactions 4,6He+197Au almost coincide. The experimental data
for the reactions with 9Li, 4He (solid squares), 6He (open squares), and 8He
(solid triangles) are from Refs. Li9Zn70 ; He4Au197 . The calculated capture
cross sections normalized by factors 0.8 and 0.6 for the reactions 4,6He+197Au
(dotted lines) and 8He+197Au (dash-dotted line), respectively, are shown.
Figure 13: (Color online) The calculated capture cross sections vs $E_{\rm
c.m.}$ for the reactions 10,11B+209Bi and 10,11B+159Tb (solid lines). The
experimental data B11Bi209 ; B11Tb159 for the reactions 10B+159Tb,209Bi and
11B+159Tb,209Bi are marked by circles and squares, respectively. The
experimental total fusion data B11Bi209 are shown by the solid stars. The
calculated capture cross sections normalized by factors 0.82 and 0.95 for the
reactions 11B+209Bi and 11B+159Tb, respectively, are presented by dotted
lines. Figure 14: (Color online) The dependence of the extracted breakup
probability $P_{BU}$ vs $E_{c.m.}-V_{b}$ for the indicated reactions with 9Be-
projectiles in %. Formula (2) was used. Figure 15: (Color online) The same as
in Fig. 14, but for the indicated reactions with 6,7,9Li-projectiles.
|
arxiv-papers
| 2012-11-10T11:14:16 |
2024-09-04T02:49:37.834393
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Sargsyan, G.G. Adamian, N.V.Antonenko, W. Scheid, and H.Q. Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1211.2320"
}
|
1211.2350
|
# A note on the $q$-Dedekind-type Daehee-Changhee sums with weight $\alpha$
arising from modified $q$-Genocchi polynomials with weight $\alpha$
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com;
saraci88@yahoo.com.tr; mtsrkn@gmail.com , Mehmet Acikgoz University of
Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310
Gaziantep, TURKEY acikgoz@gantep.edu.tr and Ayhan Esi University of Adiyaman,
Faculty of Science and Arts, Department of Mathematics, 02040 Adiyaman, TURKEY
aesi23@hotmail.com
###### Abstract.
In the present paper, our objective is to treat a $p$-adic continuous function
for an odd prime to inside a $p$-adic $q$-analogue of the higher order
Dedekind-type sums with weight $\alpha$ in connection with modified
$q$-Genocchi polynomials with weight $\alpha$ by using $p$-adic invariant
$q$-integral on $\mathbb{Z}_{p}$.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Dedekind sums, $q$-Dedekind-type sums, $p$-adic
$q$-integral on $\mathbb{Z}_{p}$, modified $q$-Genocchi polynomials with
weight $\alpha.$
## 1\. Introduction
Assume that $p$ be a fixed odd prime number. We now begin with the definition
of the following notations. Let $\mathbb{Q}_{p}$ be the field $p$-adic
rational numbers and let $\mathbb{C}_{p}$ be the completion of algebraic
closure of $\mathbb{Q}_{p}$.
Thus,
$\boldsymbol{\mathbb{Q}}_{p}=\left\\{x=\sum_{n=-k}^{\infty}a_{n}p^{n}:0\leq
a_{n}<p\right\\}\text{.}$
Then $\mathbb{Z}_{p}$ is integral domain, which is given by
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x=\sum_{n=0}^{\infty}a_{n}p^{n}:0\leq
a_{n}<p\right\\}$
or
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq
1\right\\}\text{.}$
We suppose that $q\in\mathbb{C}_{p}$ with $\left|1-q\right|_{p}<1$ as an
indeterminate. The $p$-adic absolute value $\left|.\right|_{p}$, is normally
introduced by
$\left|x\right|_{p}=\frac{1}{p^{n}}$
where $x=p^{n}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and $n\in\mathbb{Q}$
(see [1-15]).
The $p$-adic $q$-Haar distribution was originally introduced by Kim as
follows: For each postive integer $n$,
$\mu_{q}\left(a+p^{n}\mathbb{Z}_{p}\right)=\left(-q\right)^{a}\frac{\left(1+q\right)}{1+q^{p^{n}}}$
for $0\leq a<p^{n}$ and this can be extended to a measure on $\mathbb{Z}_{p}$
(for details, see [1-7]).
In [12], modified $q$-Genocchi polynomials with weight
$\left(\alpha,\beta\right)$ are defined by Araci et al. as follows:
(1)
$\widetilde{G}_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)=n\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{n-1}d\mu_{q^{\beta}}\left(\xi\right)$
for $n\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,\cdots\right\\}$. We easily see that
$\lim_{q\rightarrow
1}\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)=G_{n}\left(x\right)$
where $G_{n}\left(x\right)$ are Genocchi polynomials, which are given in the
form:
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=e^{tx}\frac{2t}{e^{t}+1},\text{
}\left|t\right|<\pi$
(for details, see [14]). Taking $x=0$ into (1), then we have
$\widetilde{G}_{n,q}^{\left(\alpha,\beta\right)}\left(0\right):=\widetilde{G}_{n,q}^{\left(\alpha,\beta\right)}$
are called modified $q$-Genocchi numbers with weight
$\left(\alpha,\beta\right)$.
It seems to be interesting for studying equation (1) at $\beta=1$. Then, we
can state the following:
(2)
$\widetilde{G}_{n,q}^{\left(\alpha,1\right)}\left(x\right):=\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)=n\int_{\mathbb{Z}_{p}}q^{-\xi}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{n-1}d\mu_{q}\left(\xi\right)\text{.}$
where $\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)$ are called
modified $q$-Genocchi polynomials with weight $\alpha$.
Modified $q$-Genocchi numbers and polynomials with weight $\alpha$ have the
following identities:
(3) $\displaystyle\widetilde{G}_{n+1,q}^{\left(\alpha\right)}$
$\displaystyle=$
$\displaystyle\left(n+1\right)\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l}}\text{,}$ (4)
$\displaystyle\widetilde{G}_{n+1,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\left(n+1\right)\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{q^{\alpha
lx}}{1+q^{\alpha l}}\text{,}$ (5)
$\displaystyle\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$ $\displaystyle q^{-\alpha
x}\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
lx}\widetilde{G}_{l,q}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
x}}{1-q^{\alpha}}\right)^{n-l}\text{.}$
Additionally, for $d$ odd natural number, we have
(6)
$\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(dx\right)=\left(\frac{1+q}{1+q^{d}}\right)\left(\frac{1-q^{\alpha
d}}{1-q^{\alpha}}\right)^{n-1}\sum_{a=0}^{d-1}\left(-1\right)^{a}\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x+\frac{a}{d}\right)\text{,}$
(for details about this subject, see [12]).
For any positive integer $h,k$ and $m$, Dedekind-type D-C sums are given by
Kim in [1], [2] and [3] as follows:
$S_{m}\left(h,k\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\frac{M}{k}\overline{E}_{m}\left(\frac{hM}{k}\right)$
where $\overline{E}_{m}\left(x\right)$ are the $m$-th periodic Euler function.
In 2011, Taekyun Kim introduced weighted $q$-Bernoulli numbers and polynomials
in [8]. He derived not only new but also ineteresting properties for weighted
$q$-Bernoulli numbers and polynomials. In [12], Araci et al. extended Kim’s
method for $q$-Genocchi polynomials and also they defined $q$-Genocchi numbers
and polynomials with weight ($\alpha,\beta$).
In [2], Kim has given some fascinating properties for Dedekind-type D-C sums.
He firstly considered a $p$-adic continuous function for an odd prime number
to contain a $p$-adic $q$-analogue of the higher order Dedekind-type D-C sums
$k^{m}S_{m+1}\left(h,k\right)$. In previous paper [17], Araci and Acikgoz also
introduced the definition of the extended $q$-Dedekind-type sums and given the
relation between extended $q$-Euler polynomials.
By using $p$-adic invariant $q$-integral on $\mathbb{Z}_{p}$, in this paper,
we shall give the definition of $q$-Dedekind-type sums with weight $\alpha$.
Also, we shall derive interesting property for $q$-Dedekind sums with weight
$\alpha$ arising from modified $q$-Genocchi polynomials with weight $\alpha$.
## 2\. $q$-Dedekind-type D-C sums with weight $\alpha$ related to modified
$q$-Genocchi polynomials with weight $\alpha$
If $x$ is a $p$-adic integer, then $w\left(x\right)$ is the unique solution of
$w\left(x\right)=w\left(x\right)^{p}$ that is congruent to
$x\mathop{\mathrm{m}od}p$. It can also be defined by
$w\left(x\right)=\lim_{n\rightarrow\infty}x^{p^{n}}\text{.}$
The multiplicative group of $p$-adic units is a product of the finite group of
roots of unity, and a group isomorphic to the $p$-adic integers. The finite
group is cylic of order $p-1$ or $2$, as $p$ is odd or even, respectively, and
so it is isomorphic. Actually, the teichmüller character gives a canonical
isomorphism between these two groups.
Let $w$ be the $Teichm\ddot{u}ller$ character ($\mathop{\mathrm{m}od}p$). For
$x\in\mathbb{Z}_{p}^{\ast}$ $:=\mathbb{Z}_{p}/p\mathbb{Z}_{p}$, set
$\left\langle
x:q\right\rangle=w^{-1}\left(x\right)\left(\frac{1-q^{x}}{1-q}\right)\text{.}$
Let $a$ and $N$ be positive integers with $\left(p,a\right)=1$ and $p\mid N$.
We now introduce the following
$\widetilde{A}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=w^{-1}\left(a\right)\left\langle
a:q^{\alpha}\right\rangle^{s}q^{-\alpha
a}\sum_{j=0}^{\infty}\binom{s}{j}q^{\alpha aj}\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha
a}}\right)^{j}\widetilde{G}_{j,q^{N}}^{\left(\alpha\right)}\text{.}$
Obviously, if $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$, then
$\displaystyle\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{1-q^{\alpha
a}}{1-q^{\alpha}}\right)^{m}q^{-\alpha a}\sum_{j=0}^{m}\binom{m}{j}q^{\alpha
aj}\widetilde{G}_{j,q^{N}}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha a}}\right)^{j}$ $\displaystyle=$
$\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-N\xi}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)\text{.}$
Then, $\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$ is a
continuous $p$-adic analogue of
$\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\frac{\widetilde{G}_{m+1,q^{N}}^{\left(\alpha\right)}\left(\frac{a}{N}\right)}{m+1}\text{.}$
Let $\left[.\right]$ be the Gauss’ symbol and let
$\left\\{x\right\\}=x-\left[x\right]$. Thus, we are now ready to treat
$q$-analogue of the higher order Dedekind-type D-C sums
$\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)$ in the form:
$\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha
k}}\right)\int_{\mathbb{Z}_{p}}q^{-l\xi}\left(\frac{1-q^{\alpha\left(l\xi+l\left\\{\frac{hM}{k}\right\\}\right)}}{1-q^{\alpha
l}}\right)^{m}d\mu_{q^{l}}\left(\xi\right)\text{.}$
If $m+1\equiv 0\left(\mathop{\mathrm{m}od}p-1\right)$
$\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha k}}\right)\int_{\mathbb{Z}_{p}}q^{-\xi
k}\left(\frac{1-q^{\alpha k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$ $\displaystyle=$
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-\xi
k}\left(\frac{1-q^{\alpha k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
where $p\mid k$, $\left(hM,p\right)=1$ for each $M$. By means of the equation
(1), we easily derive the following:
(7) $\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\left(-1\right)^{M-1}\int_{\mathbb{Z}_{p}}q^{-\xi
k}\left(\frac{1-q^{\alpha k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,\left(hM\right)_{k}:q^{k}\right)$
where $(hM)_{k}$ denotes the integer $x$ such that $0\leq x<n$ and
$x\equiv\alpha\left(\mathop{\mathrm{m}od}k\right)$.
It is easy to indicate the following:
(8)
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\xi}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{k}d\mu_{q}\left(\xi\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
m}}{1-q^{\alpha}}\right)^{k}\frac{1+q}{1+q^{m}}\sum_{i=0}^{m-1}\left(-1\right)^{i}\int_{\mathbb{Z}_{p}}q^{-m\xi}\left(\frac{1-q^{\alpha
m\left(\xi+\frac{x+i}{m}\right)}}{1-q^{\alpha
m}}\right)^{k}d\mu_{q^{m}}\left(\xi\right)\text{.}$
Due to (7) and (8), we get
(9) $\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-\xi
N}\left(\frac{1-q^{\alpha N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle=\frac{1+q^{N}}{1+q^{Np}}\sum_{i=0}^{p-1}\left(-1\right)^{i}\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-\xi
pN}\left(\frac{1-q^{\alpha pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)\text{.}$
Via the (7), (8) and (9), we discover the following $p$-adic integration:
$\widetilde{A}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=\frac{1+q^{N}}{1+q^{Np}}\sum_{\underset{a+iN\neq
0(\mathop{\mathrm{m}od}p)}{0\leq i\leq
p-1}}\left(-1\right)^{i}\widetilde{A}_{q}^{\left(\alpha\right)}\left(s,\left(a+iN\right)_{pN},p^{N}:q^{pN}\right)\text{.}$
In the other words,
$\displaystyle\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)=\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-\xi
N}\left(\frac{1-q^{\alpha N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}q^{-\xi
N}\left(\frac{1-q^{\alpha pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)$
where $\left(p^{-1}a\right)_{N}$ denotes the integer $x$ with $0\leq x<N$,
$px\equiv a\left(\mathop{\mathrm{m}od}N\right)$ and $m$ is integer with
$m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$. Then we derive the following
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)$
where $p\nmid k$ and $p\nmid hm$ for each $M$. Thus, we get the following
definition.
###### Definition 1.
Let $h,k$ be positive integer with $\left(h,k\right)=1$, $p\nmid k$. For
$s\in\mathbb{Z}_{p},$ we define $p$-adic Dedekind-type DC sums as follows:
$\widetilde{Y}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{A}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)\text{.}$
As a result of the above definition, we derive the following theorem.
###### Theorem 2.1.
For $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$ and $\left(p^{-1}a\right)_{N}$
denotes the integer $x$ with $0\leq x<N$, $px\equiv
a\left(\mathop{\mathrm{m}od}N\right)$, then we have
$\displaystyle\widetilde{Y}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{Y}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)\text{.}$
## References
* [1] T. Kim, A note on $p$-adic $q$-Dedekind sums, C. R. Acad. Bulgare Sci. 54 (2001), 37–42.
* [2] T. Kim, Note on $q$-Dedekind-type sums related to $q$-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125.
* [3] T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contemp. Math. 18 (2009), 249–260.
* [4] T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161–170.
* [5] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299.
* [6] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008), 598–608.
* [7] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [8] T. Kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
* [9] T. Kim and J. Choi, On the $q$-Euler Numbers and Polynomials with Weight $0$, Abstract and Applied Analysis, vol. 2012, Article ID 795304, 7 pages.
* [10] Y. Simsek, $q$-Dedekind type sums related to $q$-zeta function and basic $L$-series, J. Math. Anal. Appl. 318 (2006), 333-351.
* [11] M. Acikgoz, Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
* [12] S. Araci, M. Acikgoz, Feng Qi and H. Jolany, A note on the modified $q$-Genocchi numbers and polynomials with weight $\left(\alpha,\beta\right)$ and their interpolation function at negative integers, Fasc. Math. Journal (accepted for publication).
* [13] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, accepted in Bulletin of the Korean Mathematical Society.
* [14] S. Araci, M. Acikgoz and J. J. Seo, A study on the weighted $q$-Genocchi numbers and polynomials with their interpolation function, Honam Mathematical J. 34 (2012), No. 1, pp. 11-18.
* [15] S. Araci, D. Erdal and 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.
* [16] S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 298531, 11 pages.
* [17] S. Araci, and M. Acikgoz, Extended $q$-Dedekind-type sums Daehee-Changhee sums associated with extended $q$-Euler polynomials, submitted.
|
arxiv-papers
| 2012-11-10T21:03:45 |
2024-09-04T02:49:37.840252
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz, and Ayhan Esi",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1211.2350"
}
|
1211.2411
|
22affiliationtext: Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, China33affiliationtext: University of Chinese Academy of Sciences,
No. 19A, Yuquan Road, Beijing, 100049, China
# A search for 95 GHz class I methanol masers in molecular outflows
Cong-Gui Gan11affiliation: Key Laboratory for Research in Galaxies and
Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80
Nandan RD, Shanghai, 200030, China 2 2affiliationmark: 3 3affiliationmark: ,
Xi Chen11affiliationmark: 2 2affiliationmark: , Zhi-Qiang,
Shen11affiliationmark: 2 2affiliationmark: , Ye Xu44affiliation: Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China 2
2affiliationmark: , Bing-Gang Ju44affiliationmark: 2 2affiliationmark:
cggan@shao.ac.cn
###### Abstract
We have observed a sample of 288 molecular outflow sources including 123 high-
mass and 165 low-mass sources to search for class I methanol masers at 95 GHz
transition and to investigate relationship between outflow characteristics and
class I methanol maser emission with the PMO-13.7m radio telescope. Our survey
detected 62 sources with 95 GHz methanol masers above 3$\sigma$ detection
limit, which include 47 high-mass sources and 15 low-mass sources. Therefore
the detection rate is 38% for high-mass outflow sources and 9% for low-mass
outflow sources, suggesting that class I methanol maser is relatively easily
excited in high-mass sources. There are 37 newly detected 95 GHz methanol
masers (including 27 high-mass and 10 low-mass sources), 19 of which are newly
identified (i.e. first identification) class I methanol masers (including 13
high-mass and 6 low-mass sources). Statistical analysis for the distributions
of maser detections with the outflow parameters reveals that the maser
detection efficiency increases with outflow properties (e.g. mass, momentum,
kinetic energy and mechanical luminosity of outflows etc.). Systematic
investigations of relationships between the intrinsic luminosity of methanol
maser and the outflow properties (including mass, momentum, kinetic energy,
bolometric luminosity and mass loss rate of central stellar sources) indicate
a positive correlations. This further supports that class I methanol masers
are collisionally pumped and associated with shocks, where outflows interact
with the surrounding ambient medium.
Stars: formation - ISM: jets and outflows - ISM:masers -Radio lines: ISM
## 1 Introduction
Methanol masers are widespread in our Galaxy, with more than 20 transitions in
a wide frequency range from centimeter to millimeter discovered to date (Cragg
et al., 2005). Their observed connections with other star formation activities
(e.g., infrared dark clouds, millimetre and sub-millimetre dust continuum
emissions and ultracompact (UC) Hii regions) made them one of the most
effective tools to investigate star forming regions (e.g., Ellingsen, 2006).
Their trigonometric parallaxes provide a direct and accurate measurement of
distances to star formation regions wherein methanol masers reside (e.g., Xu
et al., 2006; Rygl et al., 2010). Their multiple frequency transitions enable
us to investigate the physical and chemical conditions of star forming regions
(e.g., Leurini et al., 2004, 2007; Purcell et al., 2009), and evolutionary
stages of star formation (e.g., Ellingsen et al., 2007; Chen et al., 2011,
2012).
Methanol masers can be divided into two classes (class I and II) according to
the empirical classification on the basis of their different exciting
locations (Batrla et al., 1987; Menten, 1991). Class I methanol masers are
found usually offset ($\sim 1^{\prime}$, nearly 1 pc at a distance of 4 Kpc)
from the presumed origin of excitation, and can be further categorized to
widespread class I methanol masers (e.g., 44 and 95 GHz) and rare or weak
class I methanol masers (e.g., 9.9 and 104 GHz) (Voronkov et al., 2012). The
rare or weak masers trace stronger shock regions which have higher
temperatures and densities with regard to widespread masers (Sobolev et al.,
2005; Voronkov et al., 2012). In contrast, class II methanol masers are often
found to reside close to (within 1$\arcsec$) high-mass young stellar objects
(YSOs) (e.g., Caswell et al. 2010) and are frequently associated with UC Hii
regions, infrared sources and OH masers. They also can be further categorized
to widespread (e.g., 6.7 and 12.2 GHz) and rare (e.g., 19.9, 23.1 and 37.7
GHz) class II methanol masers (Ellingsen et al., 2011; Bartkiewicz & van
Langevelde, 2012). Recently Ellingsen et al. (2011) found 37.7 GHz methanol
masers (rare class II) related with most luminous 6.7 and 12.2 GHz sources and
thus they suggested that the rare 37.7 GHz methanol masers are tracing more
evolved sources and arise prior to the cessation of widespread class II
methanol maser activity. The excitations of these two classes of masers depend
on two different pumping mechanisms: the pumping mechanism of class I masers
is dominated by collisions with molecular hydrogen, whereas class II masers
are pumped by external far-infrared radiation (e.g., Cragg et al., 1992;
Voronkov, 1999; Voronkov et al., 2005). There is a competition between the two
mechanisms since strong radiation from a nearby infrared source suppresses
class I masers but strengthens class II masers (see Voronkov et al. (2005) for
details). Surveys of class II methanol masers found that they only exist in
high-mass star forming regions (Minier et al., 2003; Ellingsen, 2006; Xu et
al., 2008); while class I methanol masers have been detected not only in high-
mass star forming regions, but also in low-mass star forming regions
(Kalenskii et al., 2006, 2010).
Many surveys of methanol masers have been carried out in last four decades.
The surveys of class II methanol masers (mainly at 6.7 GHz transition) have
detected nearly 900 sources to date (e.g., Pestalozzi et al., 2005; Pandian et
al., 2007; Xu et al., 2008, 2009; Cyganowski et al., 2009; Green et al., 2009,
2010, 2012; Caswell et al., 2010, 2011). While studies and surveys of class I
methanol masers are rare compared to class II methanol masers surveys. There
are only a few single-dish surveys (e.g., Haschick et al., 1990; Slysh et al.,
1994; Val’tts et al., 2000; Ellingsen, 2005) as well as interferometric
searches (e.g., Kurtz et al., 2004; Cyganowski et al., 2009). However class I
masers have recently become the focus of more intensive research (e.g., Sarma
& Momjian, 2009, 2011; Fontani et al., 2010; Kalenskii et al., 2010; Voronkov
et al., 2010a, b, 2011; Chambers et al., 2011; Chen et al., 2011, 2012; Fish
et al., 2011; Pihlström et al., 2011). Some surveys have been carried out at
the rare maser transitions, e.g., at 9.9 GHz by Voronkov et al. (2010a) and
23.4 GHz by Voronkov et al. (2011). To date altogether $\sim$300 class I
methanol maser sources have been detected in our Galaxy (see Chen et al.,
2011, 2012, for details).
Earlier observations of class I methanol masers (e.g., Plambeck & Menten,
1990; Kurtz et al., 2004) had found that class I methanol masers located at
the interface regions between outflows and interstellar medium, suggesting
that locations associated with outflows may be one of the best target sites
for class I maser search. Statistical analysis by Chen et al. (2009) found
that 67% of outflow sources including millimeter line molecular outflows
(cataloged by Wu et al. 2004) and EGOs, are associated with the class I
methanol maser (at 95 GHz and 44 GHz) within 1′. The EGOs are identified from
$Spitzer$ Infra Red Array Camera (IRAC) images in the 4.5 $\mu$m band, which
is thought to be a powerful outflow tracer and produced by shock-excited of H2
and CO (Cyganowski et al., 2008). A follow-up systematic survey towards a
nearly complete EGO sample (192 sources) with the Australia Telescope National
Facility (ATNF) Mopra 22-m radio telescope done by Chen et al. (2011) has
detected 105 new 95 GHz class I methanol masers, thus supporting a high
detection rate (55%) of 95 GHz methanol masers towards EGOs. In this paper we
mainly focus on 95 GHz class I methanol maser searches in another outflow
cataloged sample included in the statistical analysis of Chen et al. (2009) –
the outflow sources identified from millimeter molecular spectral lines
cataloged by Wu et al. (2004) to check whether these millimeter molecular
outflows have also indeed a high detection rate of class I methanol maser as
expected in Chen et al. (2009). In addition, although the spatial distribution
relationship between class I methanol masers and outflows has been
investigated by a series of mapping observations (e.g., Johnston et al., 1997;
Sandell et al., 2003, 2005), most of these observations only confirmed their
spatial connections. The statistical studies of their physical relationships
(e.g., methanol luminosity and outflow properties) are still absent. So it is
also necessary to perform a systematical search for class I methanol masers
(e.g., 95 GHz) in outflow sources to investigate the physical dependences
between outflows and the masers.
In this paper we report our result from the survey of 95 GHz class I methanol
maser toward the outflow sources selected from Wu et al. (2004) outflow
catalog. We describe our sample selection and observation in $\S$ 2\. In $\S$
3, we present our results of class I methanol maser detections. We discuss
methanol maser detections with outflow parameters and the relationships
between outflow parameters and maser luminosity in $\S$ 4\. The conclusion is
summarized in $\S$ 5.
## 2 OBSERVATION
### 2.1 Sample Selection
Our sample sources are selected from Wu et al. (2004), which cataloged a list
of molecular outflow sources ($\sim$400), identified from millimeters
molecular lines, along with their outflow parameters. These sources are
compiled mainly on the basis of mapping observations of CO at low transitions
($J=1-0$ and $J=2-1$), showing evidence of large scale red- and blue-lobes. We
choose the sources with Dec.$>$ -10 degrees, which can be accessible to the
Purple Mountain Observatory (PMO) 13.7-m telescope, and exclude sources with
95 GHz class I methanol maser observed before the observing epoch, which
included in the statistical study of Chen et al. (2009). A total of 288
molecular outflow sources were selected for our survey. The sample includes
123 high-mass and 165 low-mass sources according to their available bolometric
luminosity or outflow mass. Wu et al. (2004) pointed out that the high-mass
sources have bolometric luminosity of larger than $10^{3}$ $L_{\odot}$(for
sources with bolometric luminosity calculated) or outflow mass of larger than
3 $M_{\odot}$(for sources without bolometric luminosity calculated), the
others below these limits are classified as low-mass sources. Their different
mass ranges are very helpful for comparing the 95 GHz class I methanol maser
detections and the relationships between outflow properties and methanol
masers. However, such classification for high- and low-mass sources may not be
reliable for some cases. We will discuss this in Section 4.4.
### 2.2 Observation and Data Reduction
Single-point observations of 95 GHz class I methanol maser toward the selected
288 sources were made in a period from 2010 June to 2010 September with the
PMO 13.7-m telescope in Delingha, China. The rest frequency of the observed
$8_{0}-7_{1}A^{+}$ transition is set to 95.169463 GHz. The half-power
beamwidth of the telescope is about $\sim$$55^{\prime\prime}$, and the
pointing rms is better than $5^{\prime}$ during the observations. A cooled SIS
receiver working in the 80$-$115 GHz band was used and the system temperature
was about 180$-$250 K, depending on weather conditions. The spectra were
recorded with an Acousto-Optical Spectrometer (AOS) backend which has 1024
channels, 42 KHz for each channel and a total bandwidth of 42.7 MHz, resulting
in a velocity resolution of 0.13 $\mbox{km~{}s}^{-1}$and total velocity
coverage of 135 $\mbox{km~{}s}^{-1}$. The observations were performed in
position-switch mode with off positions offset $15^{\prime}$ from targeted
point (no emission was found in each off position). The aperture efficiency of
the telescope is 60%, which implies that 1 K of antenna temperature
($T^{\star}_{A}$) corresponds to 31 Jy in flux density scale. The observation
was firstly carried out with an integrated time of 30 minutes for each source
(achieving a typical rms noise level of 1.2 Jy), and then extend integration
time (typical integration time is 60 minutes, resulting in a typical rms of
0.9 Jy) for the potential weak sources.
The spectral data were reduced and analyzed with the GILDAS/CLASS package. A
first-order polynomial baseline subtraction was performed for the majority of
the observed sources, but for sources with no good solutions from the first-
order polynomial fits, we carried out a second (or third)-order polynomial
baseline subtraction. After such a baseline removed, Hanning smoothing was
applied to obtain the spectra with a velocity resolution of 0.22 km s-1.
Frequently the detected 95 GHz methanol spectra do not exactly show a
particularly Gaussian profile, possibly due to that multiple maser features
within a similar velocity range confuse the spectra. However, each maser
feature contributing to the complex spectra often shows a single Gaussian
profile. Thus, to characterize the spectral characteristics of the total
emission, we have performed Gaussian fitting to each feature for each detected
source.
## 3 RESULT
We have detected a total of 62 sources with 95 GHz methanol maser emission
flux above 3$\sigma$. A summary is given in Table 1. The references which were
used to catalog newly-identified class I methanol maser and 95 GHz methanol
maser are presented in Table 2 & 3. These references include almost all the
known class I methanol maser surveys (including 36 GHz, 44 GHz and 95 GHz) to
date. By cross-matching our detections with the previously known-detected
methanol masers from above references within a spatial scale of 1′, we found
37 newly-detected 95 GHz methanol maser sources. And among them 19 are newly
identified as class I methanol maser sources, i.e. the first identification of
a class I maser transition associated with these objects. This further
increases the sample of class I methanol maser, adding up to nearly 300 class
I methanol maser sources (see Chen et al., 2011, 2012). Appendix A gives the
undetected sources along with their 1 rms noise, which have a range from 0.3
to 2.8 Jy, depending on integration time and weather condition (a typical rms
is 1 Jy).
### 3.1 High-mass sources
Our observations find 47 high-mass sources with 95 GHz methanol masers above a
detection limit of 3$\sigma$, which include 27 newly-detected 95 GHz methanol
masers and 13 of them are newly-identified class I methanol masers. A list of
the detected 95 GHz methanol masers along with their Gaussian fitting
parameters in high-mass sources are presented in Table 2, which includes three
sub-tables: (a) sources had 95 GHz class I methanol masers detected
previously; (b) sources detected at 95 GHz class I methanol masers in the
first time; (c) sources detected only at 95 GHz class I methanol masers so far
(i.e. Newly-identified class I methanol masers). We also listed information as
to whether the detected class I methanol masers are associated with class II
methanol masers or not in the table. Note that we only use the catalog of 6.7
GHz class II methanol masers for which accurate positions (better than 1′′)
have been published (Xu et al., 2009; Caswell, 2009; Caswell et al., 2011;
Green et al., 2010, 2012) in the cross-match. There are 13 sources which are
found to be associated with class II methanol masers within the measured
figure of the outflow region in the 47 detected high-mass sources. Figure 1
shows the detected 95 GHz methanol maser spectra and the fitted Gaussian
profiles with different color lines representing different fitted components
for these sources. The figure also shown in three sub-figures according to
whether sources had 95 GHz or other class I methanol masers detected
previously. The total integrated intensities of 95 GHz methanol maser range
from 3.5 to 1070 $\mbox{Jy~{}km~{}s}^{-1}$ with a mean of 47
$\mbox{Jy~{}km~{}s}^{-1}$ for high-mass sources.
### 3.2 Low-mass sources
There are 15 sources which have 95 GHz methanol maser emission detected with
flux density above 3$\sigma$ in the low-mass sample. Among them, 10 sources
are newly-detected 95 GHz methanol masers and 6 are newly identified as class
I methanol masers. The detected 95 GHz methanol masers along with their
Gaussian fitting parameters for low-mass sources are listed in Table 3, which
is also subdivided into three sub-tables according to whether sources had 95
GHz or other class I methanol masers detected previously. We also listed
information as to whether the detected 95 GHz methanol maser sources are
associated with class II methanol masers or not in the table in the same
approach as high-mass sources (see section 3.1). There is only one source
(G206.54-16.36, we will discuss this source in Section 4.4) which is found to
be associated with high accurate position class II methanol masers in the 15
detected sources (see Table 3). The detected 95 GHz spectra and fitted
Gaussian profiles for these low-mass sources are presented in Figure 2 and
also including three sub-figures as high-mass counterpart. The total
integrated intensities of 95 GHz methanol maser range from 1.3
$\mbox{Jy~{}km~{}s}^{-1}$ to 45.5 $\mbox{Jy~{}km~{}s}^{-1}$ with a mean of 12
$\mbox{Jy~{}km~{}s}^{-1}$ for the low-mass sources.
Therefore the maximal methanol intensity in high-mass sources is nearly 24
times than that in low-mass sources, while the minimal methanol intensity in
high-mass source is 3 times that in low-mass sources. The average intensity in
high-mass sources is nearly 4 times larger than that in low-mass sources.
## 4 DISCUSSION
### 4.1 Detection Rates
A total of 62 sources have been detected 95 GHz methanol masers toward a
sample of 288 outflow sources, giving a detection rate of 22% (62/288) for our
survey. Out of 62 detections, 47 sources are high-mass sources, thus a
detection rate of 38% (47/123) for high-mass sources. The remaining 15 belong
to low-mass sources, thus a detection rate of 9% (15/165) for low-mass
sources. However the actual detections/detection rates of methanol masers may
be affected by the following factors: 1) it can be clearly seen from Figures 1
and 2 that only one single broad (and weak) Gaussian profile was detected
toward a number of sources (including G70.29+1.60, G79.88+2.55, G105.37+9.84,
G111.25-0.77, G173.58+2.44 and G213.70-12.60 in high mass source sample, and
G65.78-2.61, G183.72-3.66 and G208.77-19.24 in low mass source sample). We can
not determine whether they are thermal emission, or one or more maser spectral
features appending together from our current single dish observations,
although earlier high resolution observations to the similar broad emission
profiles showed that they are usually masers, e.g, at 95 GHz (Voronkov et al.,
2006), and 44 GHz (Cyganowski et al., 2009; Voronkov et al., 2010b); 2) it
should be noted that some detected methanol masers are located within the
PMO-13.7 m beam (e.g., high-mass sources G111.53+0.76, G111.54+0.75 and
G111.55+0.75 have angular separations within 30′′; low-mass sources
G205.10-14.39 and G205.12-14.38 have angular separations of $\sim
40^{\prime\prime}$; low-mass source G206.56-16.36 and high-mass source
G206.57-16.36 have angular separations of $\sim 25^{\prime\prime}$). If these
nearby masers are only excited by the same one source, the total detection
rate would be 20% (58/284; out of these 58 detections, 45 sources are high-
mass sources, resulting in a detection of 45/121=37%; the remaining 13 sources
belong to low-mass group, thus a detection rate of 13/163=8%). However, these
nearby detected masers show different spectral profiles, suggesting that they
may be excited by different driving sources; 3) the detection rate may be
affected by the possible extended spatial distribution of class I methanol
masers arising from sources with larger scale outflows (some sources listed in
Wu et al. (2004) catalog show larger scale outflows extending to several arc
minutes). Therefore there is a possibility that the detected maser emissions
in one source may actually originate from nearby sources with extended
outflows to several arc minutes along the line-of-sight. All above factors
would only be clarified with further higher resolution observations, but these
factors would not bring too much changes to the actual detection rates. So we
keep the methanol maser detection rates derived from the current single dish
observations in the following subsequent discussions.
Interestingly, the detection rate of 22% for the full observing sample is in
number nearly consistent with the finding of Val’tts & Larionov (2007), which
derived that 25% of mm molecular line outflows were associated with class I
masers including 36 GHz, 44 GHz and 95 GHz within 2′ from a statistical
analysis. However it is significantly lower than that expected from the
statistical analysis of Chen et al. (2009) for the Wu et al. (2004) outflow
catalog. They have analyzed 34 outflow sources from the Wu et al. (2004)
catalog which have been included in previous four class I methanol maser
surveys (including the 44 GHz transition by Slysh et al. (1994) and Kurtz et
al. (2004); and the 95 GHz transition by Val’tts et al. (2000) and Ellingsen
(2005)). They found that 23 sources are associated with one or both of the 95
and 44 GHz class I methanol masers within 1′, thus the expected detection rate
of class I masers in Wu et al. (2004) outflow catalog is 67% at this
resolution. An actual lower detection rate was achieved in the majority
($\sim$288/400) of the cataloged outflow sources may be due to that previous
statistical study is subject to influences produced by the target selection
effects as followings: 1) only a small size sample which includes 34 outflow
sources was used in the statistical study; 2) in previous four class I
methanol maser surveys used in the study most of target samples were pointed
to UC Hii regions, class II masers and known class I maser sources; 3) the
majority (29/34) of sample used in previous analysis is high-mass sources
which usually show high detection rate of class I maser with regard to low-
mass sources (for example 38% vs. 9% in our observations, we will discuss them
in more details later); 4) previous statistical analysis combined 44 and 95
GHz class I maser searches, and emission from the 44 GHz transition is
generally 3 times stronger than that at 95 GHz (Val’tts et al., 2000), thus
the search for methanol masers only at 95 GHz transition is likely to have a
lower detection rate than at 44 GHz or both 44 and 95 GHz transitions under
comparable sensitivity. Combining these it is not surprising for an
overestimated detection rate of class I methanol maser from previous
statistical study.
Comparing the detection rate of 95 GHz class I methanol maser (55%) achieved
toward nearly complete EGOs ($\sim$ 200) with the Mopra telescope by Chen et
al. (2011), we found that the actual detection rate (of 22%) of our 95 GHz
class I masers in the full observing outflow sample is also lower than that in
the EGO sample. However, EGOs trace a population of high-mass young stellar
objects with ongoing outflow activities. If only considering high-mass outflow
sources in our sample, the detection rate of 95 GHz class I maser of 38% for
them is still smaller than that of 55% for EGOs. The difference between them
is mainly due to the different detection sensitivities in the two surveys: the
typical detection sensitivity of $\sim$ 3 Jy (3$\sigma$) in our observations
is about two times that of $\sim$ 1.6 Jy (3$\sigma$) in the Chen et al. (2011)
EGO surveys. To check this, we re-examined the detected maser sources in the
Chen et al. (2011) EGO surveys. When we excluded the sources (24 in total)
with maser peak flux density of less than $\sim$3 Jy from the detected sources
in the EGO survey, we found the detection rate is 81/192$=$42%, which is
nearly consistent with that achieved for Wu et al. (2004) outflow catalog in
this work at same sensitivity. This also suggests that the EGOs have similar
properties to outflow sources traced by millimeter molecular lines from the
view of their correlation with 95 GHz class I methanol masers.
Note that our detection rate in low-mass sources is consistent with that
reported by Kalenskii et al. (2010), which detected four sources (NGC 1333I2A,
NGC 1333I4A, HH25MMS, and L1157) at 44 GHz, and one source (NGC 2023) at 36
GHz in a total of 44 low-mass outflow sources, resulting in a detection rate
of 11%. Although the low-mass sources are closer to us than high-mass sources
on average, the detection rate and flux of methanol maser in low-mass sources
are lower than that in high-mass counterparts on average in our observations.
This suggests that the high-mass sources have a higher outflow power than low-
mass sources, which could cause a higher collisional efficiency between
methanol molecule and surrounding clouds than low-mass sources (see below).
### 4.2 Detection rates with outflow parameters
There are a series of outflow parameters including bolometric luminosity of
driving sources ($L_{bol}$), outflow mass ($M$), momentum of outflows ($P$),
kinetic energy of outflows ($E_{k}$), force derived from outflow ($F$),
mechanical luminosity of outflows ($L_{m}$), mass loss rate of central stellar
sources ($\dot{M}_{loss}$) and dynamic time ($\tau$) associated with outflows
etc, presented in Wu et al. (2004). We performed a series of investigations on
maser detections with each of the outflow parameters. Histogram showing the
detection rate of class I methanol masers as a function of the bolometric
luminosity of driving sources ($L_{bol}$), outflow mass ($M$), momentum of
outflows ($P$), kinetic energy of outflows ($E_{k}$), force derived from
outflows ($F$), mechanical luminosity of outflows ($L_{m}$), are presented in
(a)–(f) of Figure 3, respectively. We adopt the mean value for parameters
which have been given more than one values in Wu et al. (2004) in the
analysis. The total sources and detected sources are presented with different
shapes in top panel of each diagram. The bottom panel in each diagram denotes
the corresponding detection rate with the outflow properties. The
corresponding detection rate in each bin is represented by black dot and a low
order polynomial fit for the detection rates is marked with solid line (only
fitted for the data points with total observed source number larger than 5).
All panels presented in Figure 3 show a clear tendency that the detection
rates of 95 GHz methanol maser increase with the increment of outflow
properties. With the increments of outflow properties (e.g., outflow mass,
momentum of outflows, kinetic energy of outflows), much more materials would
be ejected in form of outflows, which would compress parent clouds and
increase methanol abundance, resulting in increasing collision between
methanol molecule and surrounding medium (mainly H2) and stimulating methanol
molecule to higher energy levels. These processes would ultimately cause a
brighter maser excitation (we will discuss this further in Section 4.3),
making it more easily to be detected in sources with higher outflow parameter
values.
The methanol maser detection rates are related with outflow properties from
above analysis. We plot box plot to show the significance of each outflow
properties associated with methanol maser presence, similar to the approach
used in Breen et al. (2007, 2011). Figure 4 shows result of box plot of the
methanol maser presence in consideration of outflow properties, which can be
divided into two categories of ‘n’ and ‘y’, corresponding to those not related
with methanol maser, and those related with methanol maser. It can be clearly
seen that the sources with maser detected have a higher (larger) range of
outflow properties than the sources without maser detected. It suggests that
these outflow properties could play important roles in predicting methanol
maser presence. Methanol masers can be more easily detected in sources with
higher outflow properties than in those with lower outflow properties
relatively. This is consistent with above detection rate analysis.
### 4.3 Class I methanol maser emission with outflow properties
The statistical studies of correlations between outflow properties and class I
methanol maser emission based on a large sample are crucial for investigating
the physical relationship between class I methanol masers and outflows. It
could be an important complement to mapping observations (e.g., Plambeck &
Menten, 1990; Kurtz et al., 2004; Voronkov et al., 2006; Cyganowski et al.,
2009) – the mapping observations were only made for limited size sample, and
they only present the spatial associations between class I methanol masers and
outflows at present. We have detected a large number of 95 GHz methanol masers
(62 in total) toward molecular outflow sources in our observations. Most of
the detected sources were provided with the outflow properties in Wu et al.
(2004) catalog. All of them decide that our observations are very suitable for
such a statistical study. We performed a series of analysis for the
correlations between intrinsic luminosities of detected methanol maser and
outflow properties. The luminosities of methanol maser can be calculated with
$L=F_{m}\cdot 4{\pi}\cdot d^{2}$, where $L$ is the intrinsic luminosity of
methanol maser, $F_{m}$ is the total integrated intensity of 95 GHz methanol
masers estimated from Gaussian fitting, $d$ is the distance to outflow source.
The result indicates that there is significant correlation between methanol
maser intrinsic luminosity and outflow properties including bolometric
luminosity of central source ($L_{bol}$), outflow mass ($M$), momentum of
outflows ($P$), mechanical luminosity of outflows ($E_{k}$). We show the
log–log distributions of methanol maser luminosity versus these four outflow
properties in Figure 5 (a) – (d). The green squares and red triangles in each
panel of this figure represent high-mass sources and low-mass sources,
respectively. The black solid line in each panel denotes the best linear fit
for each distribution. We give the best fitting results in Table 4. From this
figure, we can clearly see that most of high-mass sources reside at top right
place in each panel, with higher outflow properties and more luminous methanol
masers, whereas low-mass sources locate at the bottom left place, with lower
outflow properties and less luminous methanol masers. The correlation
coefficients for all these four relationships are larger than 0.66, suggesting
that strong correlations exist between 95 GHz methanol maser luminosity and
these outflow properties. This is consistent with the theoretical expectation.
The protostar ejects materials in form of outflows, which squeeze clouds
surrounding the protostar. The generated shock propagating through high
density medium would stimulate methanol formation (Wirström et al., 2011) and
enhance methanol abundance (e.g., Gibb & Davis, 1998; Garay et al., 2002;
Voronkov et al., 2010b). These effects combined would increase the collision
efficiency of methanol molecule with surrounding clouds and raise up the
pumping efficiency of class I methanol masers. As high-mass sources have
higher outflow power than low-mass sources, a brighter methanol maser would
not be unexpected to excite in high-mass sources. Our result for the
correlation between maser luminosity and bolometric luminosity of outflow
driving source is also comparable to the finding of Bae et al. (2011). They
also demonstrated that there is a correlation between bolometric luminosity of
outflow driving source and isotropic luminosity of only twelve 44 GHz methanol
maser sources detected in 180 intermediate-mass star forming regions, with a
correlation coefficient of 0.72.
The mass loss rate of central stellar sources directly reflect the ejected
materials from the central objects per unit time. So the relationship between
mass loss rate and class I methanol maser luminosity is essential to interpret
the dependence between methanol masers and outflows. However, there are only a
few sources in our sample with mass loss rate of central stellar source
estimates presented by Wu et al. (2004), so there shows a poor correlation
between them. We plot relationship between intrinsic luminosity of 95 GHz
methanol maser and mass loss rate of central stellar sources in panel (e) of
Figure 5. The best linear fitting result for this dependence is also listed in
Table 4. Its correlation coefficient is 0.33 due to the small size of the
sample. But it also shows the similar tendency to the other four outflow
properties discussed above, that the intrinsic luminosity of methanol maser
increases with the increment of mass loss rate of central stellar sources
(i.e., the flux of methanol maser is proportional to mass loss rate of central
stars in logarithm). This may further support that class I methanol masers are
collisionally excited, under which with increment of outflow efficiency (e.g.,
outflow properties), the phenomenon (e.g., shock) triggering methanol
population inversion is boosted up and hence more methanol excitations appear.
### 4.4 Low-mass sources
Studies of methanol maser in low-mass sources are an effective and direct
method in explaining the properties of methanol masers, because the majority
of detected low mass sources are closer to us than high-mass sources. To date
a total of 14 low-mass sources have class I methanol masers detected including
in one or more transitions from 36 GHz, 44 GHz or 95 GHz. We list the
previously known class I methanol maser detections in low-mass sources in
Table 5. Among them, 8 sources have also been detected in our 95 GHz class I
methanol maser survey (see Table 5). Note that the previously detected source
G205.11-14.38 is close to the two sources G205.10-14.39 and G205.12-14.38
detected in our survey, with a separation of $\sim$20′′ to each of the two
sources respectively. It means the previous detected source is located within
the PMO-13.7 m beam of our detected two sources. Thus we suggest that the two
sources detected in our survey had class I methanol masers detected
previously. Our observations have found another 6 new class I methanol masers
at 95 GHz in low-mass sources, a significant increase in the low-mass sample
size. This also confirms the existence of class I methanol maser in low-mass
star formation regions.
However we should note that the classifications of the high-mass and low-mass
sources on the basis of bolometric luminosity of the centeral source or
outflow mass proposed by Wu et al. (2004) may not be exact for some cases. For
example, the source G206.54-16.36 was classified as low-mass sources due to
its low outflow mass (0.04 M⊙) according to Wu et al. (2004). But a 6.7 GHz
class II methanol maser which is exclusive tracer of high-mass star-formation
has been detected in this source, suggesting that it should be a high-mass
rather than low-mass star forming region. Therefore part of low-mass sources
classified by Wu et al. (2004) may not truly correspond to the regions wherein
only low-mass star forms. On the other hand, some theories have proposed that
high-mass star formation regions may evolve from low-mass star formation
regions (e.g., Arce et al., 2007). If considering the above possible
evolutionary effects, our observed different mass type sources can be seen to
locate at different evolutionary stages. Our results shows that the detection
rate of 95 GHz class I methanol maser in high-mass sources are 4 times larger
than that in low-mass sources, meaning that the detection rate in more evolved
sources (i.e. high-mass regions) are also 4 times larger than that in less
evolved sources (i.e. low-mass regions). This conclusion is consistent with
Fontani et al. (2010) which cataloged a total of 88 sources and classified
them into two groups including Low sources and High sources according to their
IRAS colours. The Low sources are younger than the High sources according to
their criteria. Their result shows the detection rate of class I methanol
masers in High sources are nearly 3 times (2.9 times for 44 GHz and 3.3 times
for 95 GHz methanol masers) than that in Low sources. This also supports that
more evolved sources are more easily detected class I methanol maser than less
evolved sources during evolutionary stage of star formation. That is, with the
source evolving from low-mass to high-mass, the pumping efficiency also
increases and hence a brighter class I methanol maser is excited. Therefore as
to whether the class I masers detected in low-mass sources in our observations
are truly associated low-mass forming stars, we can not completely exclude
effects from the inaccurate mass-type classifications and possible
evolutionary effects from low-mass to high-mass.
## 5 SUMMARY
A systematic survey of 95 GHz class I methanol masers was performed towards
288 molecular outflow sources including 123 high- and 165 low-mass sources
selected from Wu et al. (2004) outflow catalog with the PMO-13.7 m telescope.
We detected 62 sources with 95 GHz class I maser above a detection limit of 3
$\sigma$, which include 47 high-mass sources and 15 low-mass sources. This
suggests that the detection rate of high-mass sources is 38% and low-mass
sources is 9%. The detection rate in high-mass sources is nearly 4 times that
in low-mass sources, suggesting that 95 GHz class I methanol masers are easily
excited in high-mass sources. There are 37 newly detected 95 GHz methanol
maser sources (including 27 high-mass sources and 10 low-mass sources), and 19
of them are newly detected class I methanol maser sources (including 13 high-
mass sources and 6 low-mass sources). This further increases the number of the
known class I methanol masers (adding on top of the previous $\sim$300 class I
maser sources) in our Galaxy. We performed statistical analysis for the
distribution of detection rates with outflow properties. It shows a clear
tendency that the distributions of methanol maser detection rates increase
with the increment of outflow properties including outflow mass, momentum of
outflows, kinetic energy of outflows, bolometric luminosity of central source,
mechanical luminosity of outflows and force derived from outflows. Analysis of
the relationship between intrinsic luminosity of methanol masers and outflow
properties show that intrinsic luminosity of methanol masers is
logarithmically proportional to outflow mass, momentum of outflows, kinetic
energy, and, bolometric luminosity and mass loss rate from central stellar
sources. This is in accord with the pumping mechanism (collisionally excited)
of class I methanol masers and confirms the physical connections of methanol
masers and outflows.
## Acknowledgments
We are grateful to the staff of Qinghai Station of Purple Mountain Observatory
for their help during the observations. This work is partly supported by China
Ministry of Science and Technology under State Key Development Program for
Basic Research (2012CB821800), the National Natural Science Foundation of
China (grants 10621303, 10625314, 10803017, 11121062, 10921063, 11073041,
11073054, 11133008, 11173046 and 11273043), the CAS/SAFEA International
Partnership Program for Creative Research Teams, and Key Laboratory for Radio
Astronomy, CAS.
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Table 1: Summary of the 95 GHz class I methanol maser detected toward outflow
sources
Summary | High-mass | Low-mass | Total
---|---|---|---
Outflow Targets | 123 | 165 | 288
Detections | 47 | 15 | 62
Detection rate | 38% | 9% | 22%
aClass I (new detections) | 27 (13) | 10 (6) | 37 (19)
* •
aNumber of class I methanol masers newly-identified at 95 GHz transition in
the first time. The value in bracket present the number of newly-identified
class I methanol masers in our work.
Table 2: Detected 95 GHz methanol maser in high-mass sources
Source Other name RA DEC $V_{LSR}$ $\Delta$$v$ $P$ $S$ $S_{int}$ $RMS$ Class I
Class II Name (J2000) (J2000) ( $\mbox{km~{}s}^{-1}$) ( $\mbox{km~{}s}^{-1}$)
(Jy) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) (Jy) 36 GHza
44 GHzb 95 GHzc 6.7 GHzd (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
(13) (14) (a) Sources had 95 GHz class I methanol masers detected previously.
G10.84-2.59 GGD27 18:19:12.1 -20:47:26 13.15(0.04) 0.70(0.06) 72.98
54.12(8.56) 69.20 2.55 - Y Y - 12.26(0.19) 0.92(0.75) 15.39 15.08(10.11)
G17.02-2.40 L379IRAS(2) 18:30:34.9 -15:14:38 14.80(0.11) 0.88(0.23) 3.94
3.67(0.87) 30.68 1.16 - - Y - 18.49(0.21) 2.16(0.56) 6.43 14.75(3.41)
20.11(0.08) 1.09(0.20) 7.74 8.96(2.88) 22.65(0.27) 1.46(0.46) 2.13 3.30(1.02)
G19.88-0.54 18264-1152 18:29:14.7 -11:50:25 44.37(0.07) 0.82(0.14) 8.94
7.75(2.02) 59.19 0.85 - - Y Y 43.42(0.19) 4.53(0.55) 4.41 21.27(2.54)
43.32(0.03) 1.14(0.08) 24.19 29.22(2.48) 40.98(0.05) 0.26(6.03) 3.40
0.95(0.40) G25.65+1.05 18316-0602 18:34:20.8 -5:59:42 41.62(0.02) 0.54(0.04)
26.47 15.33(1.30) 58.54 2.44 - Y Y - 43.85(0.04) 0.30(0.10) 8.60 2.77(0.87)
42.51(0.07) 0.37(0.13) 6.29 2.49(0.96) 42.95(0.21) 4.56(0.47) 7.83 37.95(3.66)
G35.20-0.74 G35.2-0.74 18:58:12.9 1:40:37 36.71(0.67) 2.56(1.00) 3.09
8.43(6.88) 50.31 1.30 - Y Y Y 34.52(0.06) 1.92(0.26) 12.23 25.05(4.25)
33.35(0.63) 3.87(1.73) 4.08 16.83(3.19) G43.17+0.00 W49 19:10:15.6 9:06:09
15.37(0.68) 4.68(1.06) 1.10 5.46(1.61) 8.41 0.53 Y Y Y Y 14.31(0.08)
0.85(0.31) 2.22 1.99(0.84) 12.46(0.15) 0.71(0.30) 1.26 0.96(0.56) G59.78+0.07
19410+2336(L) 19:43:11.3 23:44:06 22.98(0.29) 3.74(0.67) 1.97 7.82(1.64) 18.70
0.78 - - Y - 22.31(0.05) 1.30(0.16) 6.33 8.74(1.58) 22.44(0.02) 0.35(0.06)
5.70 2.15(0.59) G75.78+0.34 G75C 20:21:44.1 37:26:42 3.50(0.06) 1.08(0.14)
5.20 5.98(0.65) 14.70 0.79 Y Y Y - 0.58(0.07) 0.93(0.15) 4.94 4.91(0.81)
-1.10(0.19) 1.55(0.44) 2.32 3.82(0.96) G81.68+0.54 DR21 20:39:00.0 42:19:28
-3.68(0.01) 0.51(0.04) 20.47 11.00(0.87) 48.29 1.30 - Y Y - -2.00(0.08)
0.97(0.15) 5.14 5.32(1.27) -3.12(0.10) 4.13(0.23) 7.28 31.96(2.17) G81.88+0.78
W75-N 20:38:37.4 42:37:57 8.87(0.01) 0.37(0.03) 15.30 5.99(0.43) 62.22 1.06 Y
Y Y Y 11.92(0.09) 0.73(0.19) 2.54 1.98(0.59) 8.52(0.05) 4.25(0.13) 11.98
54.25(1.33) G105.37+9.84 NGC7129 FIR 21:43:01.3 66:03:37 -8.53(0.14)
3.84(0.36) 1.61 6.58(0.50) 6.58 0.26 - Y Y - G106.80+5.31 S140 22:19:18.1
63:18:54 -7.15(0.13) 2.58(0.22) 2.95 8.09(0.81) 8.69 0.75 Y Y Y - -6.63(0.10)
0.43(0.19) 1.31 0.60(0.40) G108.59+0.49 22506+5944 22:52:36.9 60:00:48
-51.11(0.26) 2.41(0.26) 2.86 7.32(0.34) 14.14 0.70 - Y Y - -49.63(0.26)
0.48(0.26) 1.17 0.59(0.34) -51.90(0.26) 1.29(0.26) 2.18 2.99(0.34)
-46.45(0.26) 1.77(0.26) 1.72 3.23(0.34) G111.53+0.76 NGC7538 IRAS11 23:13:44.7
61:26:54 -56.91(0.13) 1.13(0.13) 8.17 9.79(0.81) 89.82 1.74 - Y Y -
-53.67(0.13) 3.03(0.13) 5.85 18.89(0.81) -50.16(0.13) 1.38(0.13) 4.49
6.60(0.81) -54.77(0.13) 2.72(0.13) 11.19 32.43(0.81) -57.55(0.13) 2.37(0.13)
8.75 22.11(0.81) G111.54+0.75 NGC7538 A 23:13:47.8 61:26:39 -50.80(0.18)
1.69(0.34) 2.19 3.94(0.84) 55.53 1.18 - Y Y - -53.24(0.03) 0.78(0.09) 8.62
7.11(1.30) -54.95(0.08) 2.86(0.25) 12.04 36.62(2.98) -57.06(0.06) 1.09(0.14)
6.80 7.85(1.64) G111.54+0.78 NGC7538 23:13:44.7 61:28:10 -57.39(0.04)
1.00(0.18) 12.11 12.88(4.19) 31.76 1.25 Y Y Y - -57.02(0.20) 2.86(0.60) 6.20
18.88(3.97) G173.49+2.44 05358+3543 5:39:10.6 35:45:19 -16.05(0.11) 1.26(0.27)
2.18 2.94(0.90) 13.84 0.50 Y Y Y Y -18.84(0.40) 5.58(0.85) 1.84 10.90(1.59)
Table 2: – _continued_
Source Other name RA DEC $V_{LSR}$ $\Delta$$v$ $P$ $S$ $S_{int}$ $RMS$ Class I
Class II Name (J2000) (J2000) ( $\mbox{km~{}s}^{-1}$) ( $\mbox{km~{}s}^{-1}$)
(Jy) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) (Jy) 36 GHza
44 GHzb 95 GHzc 6.7 GHzd (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
(13) (14) G192.60-0.05 S255-IRS1 6:12:54.4 17:59:25 4.98(0.03) 0.30(0.08) 5.25
1.69(0.34) 19.35 1.10 - Y Y Y 7.29(0.06) 0.94(0.17) 6.48 6.50(1.92) 8.45(0.50)
2.46(0.60) 2.94 7.70(2.02) 11.33(0.07) 0.73(0.16) 3.56 2.75(0.59) 9.62(0.14)
0.42(0.34) 1.57 0.71(0.78) G208.99-19.38 ORION-A 5:35:14.5 -5:22:21 7.85(0.09)
1.54(0.16) 34.60 56.83(10.23) 1070.60 2.88 Y Y Y Y 16.58(0.01) 12.39(0.34)
31.00 408.74(10.48) 6.53(0.06) 8.69(0.26) 34.62 320.42(9.55) 8.82(0.01)
3.62(0.09) 66.15 255.23(9.21) 8.60(0.02) 0.54(0.05) 51.66 29.39(4.74)
G209.01-19.41 ORION-S 5:35:12.4 -5:24:11 6.57(0.14) 3.05(0.16) 25.03
81.21(4.22) 143.69 1.92 Y - Y Y 6.10(0.22) 2.10(0.38) 8.18 18.33(6.98)
8.35(0.24) 5.84(0.54) 7.10 44.15(5.18) (b) Sources detected at 95 GHz class I
methanol masers in the first time. G9.62+0.19 G9.62+0.19F 18:06:14.8 -20:31:39
5.82(0.58) 6.20(0.82) 1.50 9.90(7.75) 28.47 0.75 - Y - Y 3.74(0.08) 2.55(0.21)
5.33 14.45(0.96) 0.41(0.20) 1.87(0.33) 2.07 4.12(0.87) G45.07+0.13 G45.07+0.13
19:13:21.7 10:50:53 59.29(0.14) 0.95(0.41) 0.86 0.87(0.39) 5.79 0.25 - Y - -
58.44(0.42) 6.36(1.01) 0.73 4.92(0.68) G53.03+0.12 19266+1745 19:28:53.9
17:51:56 5.05(0.18) 2.32(0.50) 1.47 3.62(0.62) 5.02 0.41 - Y - - 6.71(0.09)
0.46(0.21) 1.20 0.59(0.31) 8.51(0.06) 0.46(0.11) 1.67 0.82(0.22) G77.46+1.76
20188+3928 20:20:39.6 39:37:52 -2.87(0.19) 1.01(0.32) 1.43 1.53(0.50) 10.55
0.69 - Y - - -0.72(0.10) 0.85(0.20) 2.41 2.17(0.50) 1.72(0.10) 1.89(0.24) 3.39
6.84(0.74) G78.12+3.63 20126+4104 20:14:26.0 41:13:32 -0.59(0.14) 0.61(0.35)
1.38 0.90(0.47) 24.18 0.75 - Y - - -4.61(0.06) 0.67(0.18) 3.12 2.23(0.87)
-3.29(0.02) 0.37(0.05) 6.01 2.36(0.40) -3.01(0.09) 2.47(0.21) 7.11 18.69(1.43)
G78.97+0.36 20293+3952 20:31:10.4 40:03:10 6.59(0.09) 1.45(0.25) 3.67
5.66(0.78) 11.17 1.04 - Y - - 4.98(0.06) 0.84(0.14) 4.99 4.46(0.68) 3.63(0.24)
0.89(0.38) 1.12 1.05(0.53) G110.09-0.07 23033+5951 23:05:25.0 60:08:12
-54.14(0.15) 0.96(0.25) 6.00 6.15(2.26) 18.57 1.19 - Y - - -54.55(0.03)
0.35(0.08) 7.34 2.72(1.40) -53.81(0.47) 5.52(1.31) 1.65 9.70(1.86)
G111.24-1.24 23151+5912 23:17:21.4 59:28:49 -51.73(0.13) 0.38(0.62) 1.32
0.54(0.37) 5.76 0.58 - Y - - -52.74(0.08) 0.85(0.17) 2.86 2.58(0.43)
-55.64(0.03) 0.26(0.95) 2.37 0.66(0.25) -54.73(0.07) 0.73(0.17) 2.56
1.98(0.37) G111.55+0.75 NGC7538 IRAS9 23:13:53.8 61:27:09 -57.32(0.12)
1.10(0.14) 1.63 1.91(0.34) 14.33 0.35 - Y - - -55.55(0.16) 0.90(0.40) 1.11
1.06(0.64) -55.62(0.17) 4.92(0.49) 2.17 11.36(0.87) G121.30+0.66 00338+6312
0:36:47.2 63:29:02 -17.80(0.20) 2.74(0.48) 2.57 7.49(1.21) 8.95 0.85 - Y - Y
-17.56(0.06) 0.41(0.28) 3.29 1.45(0.74) G122.01-7.07 00420+5530 0:44:57.2
55:47:18 -51.10(0.00) 1.09(0.22) 1.36 1.58(0.31) 3.49 0.34 - Y - -
-48.50(0.00) 1.56(0.30) 1.15 1.91(0.37) G173.72+2.69 S235B 5:40:52.5 35:41:26
-16.40(0.01) 0.95(0.03) 15.22 15.31(0.37) 17.65 0.61 - Y - - -17.37(0.03)
0.43(0.08) 3.24 1.48(0.28) -18.43(0.08) 0.49(0.21) 1.66 0.87(0.28)
G174.20-0.08 AFGL5142 5:30:46.0 33:47:52 -0.38(0.15) 1.00(0.25) 3.33
3.56(0.96) 23.73 1.54 - Y - - -3.15(0.03) 0.88(0.20) 7.92 7.38(3.16)
-4.62(0.50) 2.18(1.04) 2.60 6.05(2.79) -1.90(0.13) 1.27(0.33) 5.00 6.75(1.36)
G188.95+0.89 AFGL 5180 6:08:54.2 21:38:25 2.27(0.45) 2.98(1.12) 3.49
11.08(2.08) 23.01 1.35 Y - - Y 3.89(0.49) 2.74(0.93) 4.09 11.93(2.23)
Table 2: – _continued_
Source Other name RA DEC $V_{LSR}$ $\Delta$$v$ $P$ $S$ $S_{int}$ $RMS$ Class I
Class II Name (J2000) (J2000) ( $\mbox{km~{}s}^{-1}$) ( $\mbox{km~{}s}^{-1}$)
(Jy) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) (Jy) 36 GHza
44 GHzb 95 GHzc 6.7 GHzd (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
(13) (14) (c) Sources detected only at 95 GHz class I methanol masers so far
(newly-identified class I methanol masers). G37.55+0.20 18566+0408 18:59:10.1
4:12:14 85.65(0.20) 4.33(0.54) 2.00 9.22(0.90) 9.22 0.53 - - - - G51.68+0.72
19217+1651 19:23:58.8 16:57:37 2.66(0.28) 2.93(0.90) 1.30 4.04(0.96) 8.23 0.48
- - - - -0.23(0.21) 1.00(0.38) 1.17 1.24(0.50) 4.66(0.11) 0.48(0.23) 1.29
0.66(0.37) 5.59(0.18) 0.73(0.36) 0.93 0.73(0.37) 7.55(0.18) 1.29(0.30) 1.13
1.56(0.40) G70.29+1.60 19598+3324 20:01:45.6 33:32:43 -24.63(0.54) 7.24(1.03)
0.66 5.07(0.66) 5.07 0.19 - - - - G79.87+1.18 20286+4105 20:30:28.4 41:15:48
-3.12(0.23) 3.80(0.43) 2.78 11.24(1.36) 15.03 0.65 - - - - -4.09(0.08)
0.76(0.25) 3.01 2.42(0.87) -6.79(0.11) 0.71(0.28) 1.81 1.38(0.47) G79.88+2.55
20227+4154 20:24:31.1 42:04:17 4.84(0.58) 10.41(1.52) 1.34 14.82(1.67) 14.82
0.64 - - - - G109.99-0.28 23032+5937 23:05:23.2 59:53:53 -51.99(0.30)
0.43(0.49) 0.74 0.34(0.84) 5.11 0.52 - - - - -50.55(0.19) 1.19(0.44) 1.07
1.36(0.43) -54.13(0.29) 1.00(1.61) 0.95 1.01(1.02) -52.71(0.15) 0.93(0.62)
2.44 2.41(1.46) G111.25-0.77 23139+5939 23:16:09.4 59:55:23 -44.98(0.12)
2.67(0.27) 2.54 7.23(0.62) 7.23 0.40 - - - - G123.07-6.31 00494+5617 0:52:23.9
56:33:45 -30.73(0.13) 5.88(0.36) 5.43 34.00(1.61) 45.28 0.81 - - - Y
-40.31(0.28) 4.35(0.57) 2.43 11.27(1.33) G138.30+1.56 AFGL4029 3:01:32.7
60:29:12 -33.98(0.10) 0.85(0.19) 1.66 1.50(0.34) 6.50 0.51 - - - -
-36.67(0.14) 1.25(0.34) 1.99 2.65(0.62) -38.05(0.11) 0.86(0.22) 2.08
1.91(0.53) -39.12(0.05) 0.27(0.73) 1.53 0.44(0.19) G173.58+2.44 05361+3539
5:39:27.5 35:40:43 -16.79(0.14) 1.58(0.32) 2.18 3.68(0.64) 3.68 0.54 - - - -
G206.57-16.36 NGC2024 FIR 6 5:41:45.5 -1:56:02 11.77(0.10) 0.74(0.18) 1.96
1.54(0.40) 7.84 0.70 - - - - 10.26(0.13) 1.88(0.20) 2.30 4.61(0.03) 7.16(0.07)
0.72(0.16) 2.20 1.69(0.31) G207.27-1.81 AFGL961 6:34:37.6 04:12:44 12.09(0.13)
0.84(0.38) 2.84 2.54(0.76) 5.39 0.74 - - - - 14.10(0.14) 0.68(0.33) 2.03
1.48(0.59) 17.27(0.13) 0.62(0.26) 2.09 1.37(0.56) G213.70-12.60 MON R2
6:07:48.3 -06:22:54 10.14(0.17) 2.34(0.39) 1.88 4.68(0.67) 4.68 0.47 - - - Y
Columns (1) present the source name sorted by the galactic coordinates.
Columns (2) - (4) list other source name from Wu et al. (2004) and
corresponding equatorial coordinates. Columns (5) - (9) present Gaussian
fitting parameters of detected methanol emission: the velocity at peak
$V_{LSR}$, the line FWHM $\Delta$$v$, the integrated intensity $S$, the total
integrated intensity $S_{int}$. Value in brackets is the fitting error.
Columns (10) list 1-${\sigma}_{rms}$ noise of observations. Columns (11) -
(14) list the 36 GHz, 44 GHz and 95 GHz class I and 6.7 GHz class II methanol
maser associations: Y = Yes, N = No, ”-” = No information. References: a
Liechti & Wilson (1996) b Val’tts & Larionov (2007); Kurtz et al. (2004);
Larionov & Val’tts (2007); Fontani et al. (2010); Kalenskii et al. (2006,
2010); Litovchenko et al. (2011); Bae et al. (2011) c Val’tts et al. (1995);
Val’tts & Larionov (2007); Kalenskii et al. (1994); Kalenskii et al. (2006);
Larionov et al. (1999); Fontani et al. (2010); Chen et al. (2011, 2012) d Xu
et al. (2009); Caswell (2009); Caswell et al. (2010, 2011); Green et al.
(2010, 2012)
Table 3: Detected 95 GHz methanol maser low-mass sources
Name Other name RA DEC $V_{LSR}$ $\Delta$$v$ $P$ $S$ $S_{int}$ $RMS$ Class I
Class II Name (J2000) (J2000) ( $\mbox{km~{}s}^{-1}$) ( $\mbox{km~{}s}^{-1}$)
(Jy) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) ( $\mbox{Jy~{}km~{}s}^{-1}$ ) (Jy) 36 GHza
44 GHzb 95 GHzc 6.7 GHzd (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
(13) (14) (a) Sources had 95 GHz class I methanol masers detected previously.
G99.98+4.17 IC1396-N 21:40:42.1 58:16:10 0.00(0.04) 0.39(0.13) 3.21 1.32(0.40)
13.75 0.94 - Y Y - -0.84(0.03) 0.47(0.07) 5.44 2.75(0.53) -0.76(0.19)
3.61(0.41) 2.52 9.68(1.02) G158.40-20.57 NGC1333/IRAS4A 3:29:10.5 31:13:32
7.18(0.08) 0.30(2.67) 6.86 2.19(0.90) 17.39 1.42 - Y Y - 6.81(0.12) 1.92(0.28)
7.44 15.20(1.92) G205.10-14.39 HH26IR 5:46:05.5 -00:14:17 10.42(0.06)
0.83(0.16) 3.69 3.25(0.53) 6.17 0.53 - Y Y - 9.17(0.13) 0.90(0.29) 1.83
1.74(0.53) 12.14(0.16) 0.43(0.77) 2.57 1.18(0.35) G205.11-14.11 NGC2071
5:47:04.1 0:21:42 8.43(0.19) 1.34(0.37) 3.69 5.26(1.58) 14.64 0.86 Y - Y -
12.96(0.41) 1.58(0.83) 1.35 2.27(0.99) 10.22(0.16) 1.59(0.46) 4.20 7.10(1.77)
G206.54-16.36 NGC2024 FIR 4 5:41:43.5 -1:54:45 9.64(0.39) 1.50(0.47) 2.40
3.83(0.65) 8.75 1.22 Y - Y Y 10.31(0.51) 0.72(0.77) 2.19 1.68(0.84)
10.86(0.29) 0.91(0.30) 3.36 3.24(0.99) (b) Sources detected at 95 GHz class I
methanol masers in the first time. G65.78-2.61 20050+2720 20:07:06.2 27:28:53
6.32(0.17) 2.76(0.47) 1.12 3.29(0.43) 3.29 0.46 - Y - - G110.48+1.48
23011+6126 23:03:13.0 61:42:26 -11.09(0.26) 1.26(0.26) 3.15 4.21(0.31) 15.87
0.69 - Y - - -9.46(0.26) 2.02(0.26) 1.45 3.13(0.31) -13.82(0.26) 2.08(0.26)
2.18 4.81(0.31) -17.17(0.26) 0.46(0.26) 2.01 0.98(0.31) -18.53(0.26)
2.04(0.26) 1.26 2.74(0.31) G119.80-6.03 00259+5625 0:28:44.8 56:42:07
-38.55(0.14) 2.20(0.38) 8.37 19.62(3.94) 45.54 1.37 - Y - - -35.66(0.30)
2.98(0.71) 5.27 16.70(4.15) -31.95(0.15) 2.14(0.36) 4.05 9.23(1.46)
G205.12-14.38 HH25 MMS 5:46:07.5 -0:13:36 12.45(0.16) 1.18(0.30) 1.79
2.25(0.56) 11.30 1.00 - Y - - 11.02(0.09) 0.45(0.18) 2.04 0.98(0.40)
7.89(0.31) 2.54(0.66) 1.53 4.14(0.93) 10.18(0.04) 0.65(0.11) 5.72 3.93(0.62)
(c) Sources detected only at 95 GHz class I methanol masers so far (newly-
identified class I methanol masers). G75.79+0.33 G75E 20:21:47.1 37:26:30
-0.41(0.23) 3.83(0.60) 1.58 6.44(0.80) 8.75 0.41 - - - - 3.82(0.10) 1.09(0.21)
1.99 2.31(0.41) G87.06-4.19 L944 SMM1 21:17:43.8 43:18:47 0.06(0.17)
1.57(0.48) 1.44 2.39(0.62) 8.00 0.52 - - - - 4.60(0.14) 1.83(0.30) 1.96
3.80(0.56) 2.12(0.08) 0.88(0.30) 1.93 1.80(0.47) G183.72-3.66 GGD4 5:40:24.1
23:50:54 2.64(0.26) 2.79(0.67) 1.17 3.47(0.67) 3.47 0.41 - - - - G206.56-16.36
NGC2024 FIR 5 5:41:44.5 -1:55:38 11.20(0.00) 1.05(0.41) 2.44 2.73(1.60) 9.78
0.46 - - - - 11.00(0.00) 4.11(1.38) 1.61 7.05(1.38) G208.77-19.24 FIR 1
5:35:21.8 -05:07:37 10.90(0.18) 1.56(0.35) 0.75 1.25(0.27) 1.25 0.25 - - - -
G208.97-19.37 ORION-KLN 5:35:15.5 -5:20:41 9.84(0.06) 1.39(0.17) 6.60
9.73(0.90) 11.79 0.82 - - - - 11.52(0.16) 0.98(0.37) 1.99 2.06(0.74) Columns
(1) present the source name sorted by the galactic coordinates. Columns (2) -
(4) list other source name from Wu et al. (2004) and corresponding equatorial
coordinates. Columns (5) - (9) present Gaussian fitting parameters of detected
methanol emission: the velocity at peak $V_{LSR}$, the line FWHM $\Delta$$v$,
the integrated intensity $S$, the total integrated intensity $S_{int}$. Value
in brackets is the fitting error. Columns (10) list 1-${\sigma}_{rms}$ noise
of observations. Columns (11) - (14) list the 36 GHz, 44 GHz and 95 GHz class
I and 6.7 GHz class II methanol maser associations: Y = Yes, N = No, ”-” = No
information. References: a Liechti & Wilson (1996) b Val’tts & Larionov
(2007); Kurtz et al. (2004); Larionov & Val’tts (2007); Fontani et al. (2010);
Kalenskii et al. (2006, 2010); Litovchenko et al. (2011); Bae et al. (2011) c
Val’tts et al. (1995); Val’tts & Larionov (2007); Kalenskii et al. (1994);
Kalenskii et al. (2006); Larionov et al. (1999); Fontani et al. (2010); Chen
et al. (2011, 2012) d Xu et al. (2009); Caswell (2009); Caswell et al. (2010,
2011); Green et al. (2010, 2012)
Table 4: Linear fitting results of Log-Log relationships between methanol maser luminosity and outflow properties. Outflow properties | Slope | Intercept | correlation coefficient
---|---|---|---
$L_{bol}$ | 0.514(0.081) | -7.769(0.326) | 0.66
$M_{outflow}$ | 0.581(0.074) | -6.361(0.110) | 0.73
$P$ | 0.616(0.079) | -7.036(0.179) | 0.74
$E_{k}$ | 0.448(0.072) | -6.270(0.123) | 0.67
$\dot{M}_{loss}$ | 0.334(0.229) | -6.089(0.260) | 0.33
Table 5: A list of previously known class I methanol maser low-mass sources.
Source Other name R.A. Dec. Frequency (GHz) References Our name (1) (2) (3)
(4) (5) (6) (7) G65.78-2.61 20050+2720 20:07:06.7 27:28:53 44 9 G65.78-2.61
G99.98+4.17 Mol 138 21:40:41.0 58:16:16 44,95 5,6,7 G99.98+4.17 G102.64+15.78
L1157 B1 20:39:08.1 68:01:13 44,95 2,3 G102.65+15.80 L1157-mm 20:39:06.2
68:02:15 95 1 G110.48+1.48 IRAS 23011+6126 23:03:13.0 61:42:26 44 8
G110.48+1.48 G119.80-6.03 CB 3 0:28:42.7 56:42:07 44 8 G119.80-6.03
G158.36-20.58 NGC 1333I2A 3:29:01.0 31:14:20 44 3 G158.39-20.57 NGC 1333I4A
3:29:10.3 31:03:13 44 3 G158.40-20.58 NGC 1333IRAS4A 3:29:12.0 31:13:09 44,95
2 G158.40-20.57 G205.07-14.36 HH 25MMS 5:46:08.0 0:16:26 44,95 2 G205.11-14.11
NGC 2071 5:47:04.1 0:21:42 36,95 4,5 G205.11-14.11 G205.11-14.38 HH25MMS
5:46:06.5 -0:13:54 44,95 3 G205.10-14.39, G205.12-14.38 G206.54-16.36 NGC 2024
5:41:42.9 -1:54:34 36,95 4,5 G206.54-16.36 G206.89-16.60 NGC2023 5:41:28.5
-2:19:19 36 3
* •
Column (1) list source name sorted by the galactic coordinates.
* •
Columns (2)–(4) present other source name and corresponding equatorial
coordinates from references.
* •
Columns (5)–(6) present frequency of previous detections and corresponding
references.
* •
Column (7) list corresponding low-mass source names with 95 GHz methanol maser
detections in this work.
* •
References: (1) Kalenskii et al. (2001); (2) Kalenskii et al. (2006);
(3)Kalenskii et al. (2010); (4)Liechti & Wilson (1996); (5)Val’tts et al.
(1995); (6) Kurtz et al. (2004); (7)Val’tts & Larionov (2007); (8)Bae et al.
(2011); (9)Fontani et al. (2010).
(a)
Figure 1: A set of spectra of the detected 95 GHz methanol masers in high-mass
sources: (a) sources had 95 GHz class I methanol masers detected previously;
(b) sources detected at 95 GHz class I methanol masers in the first time; (c)
sources detected only at 95 GHz class I methanol masers so far (newly-
identified class I methanol masers). The dotted lines with different colors
represent Gaussian fits to different velocity components (A color version of
this figure is available in the online journal).
(b)
Figure 1: – _continued_
(c)
Figure 1: – _continued_
(a)
(b)
(c)
Figure 2: A set of spectra of the detected 95 GHz methanol masers in low-mass
sources: (a) sources had 95 GHz class I methanol masers detected previously;
(b) sources detected at 95 GHz class I methanol masers in the first time; (c)
sources detected only at 95 GHz class I methanol masers so far (newly-
identified class I methanol masers). The dotted lines with different colors
represent Gaussian fits to different velocity components (A color version of
this figure is available in the online journal). Figure 3: The detection
statistics of outflow properties including bolometric luminosity of central
source (a), outflow mass (b), momentum of outflows (c), kinetic energy of
outflows (d), driving force of outflows (e) and mechanical luminosity of
outflows (f). The boxes shown with white boxes and shaded boxes in the top of
each diagram represent total observed and detected sources. The dots in the
bottom of each diagram represent the corresponding detection rates in each bin
and solid line marks the lower-order polynomial fits for the detection rates.
Note that the polynomial fits are only for the data points where the total
source number is larger than 5. Figure 4: Statistical box plots of the
molecular outflow properties presented in the categories of yes and no,
according to the presence of methanol maser emission. There is distinct
detection difference in outflow properties including bolometric luminosity of
central source (a), outflow mass (b), momentum of outflows (c), kinetic energy
of outflows (d), driving force of outflows (e), mechanical luminosity of
outflows (f). The box contains data from 25% to 75%, the line within each box
represents the median of the data. The vertical lines from the top of the box
and from the bottom to the box represent from 75% to maximum value and from
25% to minimum value,respectively. The open circles represent the outliers,
which were excluded from the statistics. Figure 5: The log–log distributions
between class I methanol maser luminosity and outflow properties including
bolometric luminosity of central sources (a), outflow mass (b), momentum of
outflows (c), kinetic energy of outflows (d) and mass loss rate of central
stars (e). The green squares and red triangles represent high-mass and low-
mass sources, respetively. The solid black line is the best linear fit of the
plot (A color version of this figure is available in the online journal).
## Appendix A Undetected 95 GHz methanol maser sources
Name Other name RA DEC RMS Mass (J2000) (J2000) (Jy) (1) (2) (3) (4) (5)
(6) G1.36+20.97 L43 16:34:29.3 -15:47:01.0 1.62 L G8.66+22.18 16442-0930
16:46:58.4 -09:35:22.0 0.54 L G11.30-2.02 18150-2016 18:17:59.2 -20:06:58.0
1.54 H G11.42-1.68 18139-1952 18:16:56.9 -19:51:08.0 1.28 H G11.42-1.37
18128-1943 18:15:47.7 -19:42:18.0 0.74 H G11.50-1.48 18134-1942 18:16:21.7
-19:41:31.0 0.85 H G17.64+0.15 18196-1331 18:22:26.8 -13:30:15.0 1.35 H
G23.57+1.58 18258-0737 18:28:34.6 -07:35:31.0 1.12 H G24.89+5.38 L483
18:17:30.2 -04:39:38.0 0.6 L G28.58-1.73 18470-0044 18:49:41.1 -04:39:35.0
0.79 H G28.75+3.52 W40 18:31:16.2 -02:06:49.0 1.43 H G30.64+3.25 18331-0035
18:35:42.4 -00:33:18.0 0.69 L G31.59+5.35 S68 Firs1 18:29:57.4 01:14:45 0.99 L
G31.59+5.38 Serpens SMM1-11 18:29:50.4 01:15:19 1.32 L G39.39-0.14 19012+0536
19:03:45.4 05:40:40 1.35 H G39.82-11.92 S87 19:46:20.5 00:35:24 1.59 H
G40.62-0.14 19035+0641 19:06:01.2 06:46:35 0.67 H G42.16-11.49 19471+2641
19:49:09.8 02:48:52 1.44 H G42.64-10.17 L810 19:45:24.2 03:51:01 1.38 H
G43.23-12.57 19529+2704 19:54:59.7 03:12:52 0.54 H G43.83-14.82 CB214
20:03:59.1 02:38:14 1.3 L G43.94-11.94 19520+2759 19:54:06.4 04:07:25 1.08 H
G44.93-6.55 B335 19:36:59.7 07:34:07 1.42 L G45.12+0.13 G45.12+0.13 19:13:28.6
10:53:22 1.33 H G46.17-1.55 AS353 19:21:30.6 11:02:14 1.41 L G46.31-1.19 L673
19:20:29.2 11:19:40 1.27 L G46.34-1.15 L673 SMM1 19:20:25.2 11:22:17 1.16 L
G46.53-1.02 CB188 19:20:17.9 11:35:57 1.31 H G48.57-10.19 19550+3248
19:56:55.0 08:56:32 0.72 L G50.84-11.92 20056+3350 20:07:31.5 09:59:39 0.65 H
G50.92-12.07 20062+3350 20:08:12.6 09:59:20 1.2 L G51.25-27.69 L988-a
21:02:23.0 02:03:06 0.65 L G51.36-27.26 V1331Cyg 21:01:09.0 02:21:46 0.93 L
G51.58-28.00 L988-f 21:04:03.2 02:07:49 1.14 L G51.67-27.92 L988-e 21:03:58.0
02:14:38 1.22 L G52.10+1.04 19213+1723 19:23:37.1 17:28:59 1.63 H G52.98+3.05
L723 2 Flows 19:17:53.9 19:12:19 1.2 L G53.16-11.98 20106+3545 20:12:31.3
11:54:46 1.31 H G53.60-10.36 CB217 20:07:45.9 13:07:01 1.27 L G53.63+0.02
19282+1814 19:30:28.4 18:20:53 0.66 H G53.78-24.68 20582+7724 20:57:10.6
05:35:46 1.11 L G53.88-15.15 20231+3440 SMM1 20:25:00.3 10:50:05 0.7 L
G53.93-15.13 20231+3440 SMM2 20:25:01.2 10:53:05 0.94 L G55.72-13.06 G75N
20:21:42.1 13:27:20 1.36 L G55.86-12.99 G75.78NE 20:21:44.8 13:37:00 1.03 H
G57.08-33.39 21307+5049 21:32:31.2 03:02:22 0.94 H G57.39-34.01 21334+5039
21:35:09.2 02:53:09 0.63 H G57.61-35.37 V645Cyg 21:39:58.6 02:14:22 0.7 H
G58.15+3.51 L778 19:26:32.2 23:58:42 1.36 L G59.11-13.13 AFGL2591 20:29:24.9
16:11:19 1.15 H G59.13-20.12 20520+6003 20:53:13.9 12:14:44 1.27 L
G59.15-20.29 20526+5958 20:53:50.4 12:09:46 0.59 L G59.24-11.32 20216+4107
20:23:24.2 17:17:40 1.28 H G59.27-13.20 20281+4006 20:30:00.9 16:16:36 1.15 L
G59.36-12.88 20272+4021 20:29:05.3 16:31:58 1.1 H G59.36-0.21 19411+2306
19:43:18.0 23:13:59 0.34 H G59.47-0.05 1548C27 19:42:55.7 23:24:20 1.23 L
G59.60+0.92 19374+2352 19:39:32.8 23:59:55 1.44 H G60.35-10.94 20228+4215
20:24:34.5 18:25:01 1.19 H G61.30-13.63 20343+4129 d 20:36:07.6 17:40:01 0.85
H G62.22-4.54 CB216 20:05:54.1 23:27:04 1.57 L G62.68-33.11 21413+5442
21:43:01.2 06:56:18 0.59 H G63.11-12.29 20353+6742 20:35:45.4 19:52:59 0.56 L
G63.71-12.84 L1157 20:39:06.9 20:02:13 0.77 L Name Other name RA DEC RMS Mass
(J2000) (J2000) (Jy) (1) (2) (3) (4) (5) (6) G63.73-31.59 GN21.38.9 21:40:29.8
08:35:13 1.17 L G64.60-14.18 Pvcep 20:45:53.6 19:57:39 1.46 L G65.55-32.10
IC1396-E 21:46:07.0 09:26:23 1.05 L G66.12-34.15 21519+5613 21:53:38.9
08:27:46 0.96 H G66.76-16.43 V1057Cyg 20:58:53.5 20:15:29 0.46 L G66.95-41.06
22142+5206 22:16:10.7 04:21:25 0.93 H G67.01-17.30 L1172D 21:02:24.0 19:54:27
0.67 L G67.17-17.10 21015+6757 21:02:09.5 20:09:09 1.54 L G69.72-19.82 CB230
21:17:40.0 20:17:32 0.73 L G70.43-45.27 22343+7501 22:35:24.3 03:17:06 1.25 L
G71.14-39.02 22172+5549 22:19:09.3 08:04:45 1.15 H G71.19-45.93 22376+7455
22:38:47.3 03:11:29 1.11 L G71.38-16.87 B361 21:12:26.2 23:24:24 1.14 L
G71.94-35.92 22103+5828 22:12:07.8 10:43:33 1.05 L G72.26-23.87 CB232
21:37:10.9 19:20:36 1.1 L G72.34-25.73 LKHa234 21:43:06.1 18:06:52 0.71 H
G72.67-36.37 22134+5834 22:15:08.9 10:49:09 0.55 H G76.20-39.62 22305+5803
22:32:24.2 10:18:58 1.01 H G76.94-22.25 21432+4719 21:45:10.2 23:33:21 0.79 L
G76.98-22.12 21429+4726 21:44:51.9 23:40:31 0.85 L G77.03-22.02 21428+4732
21:44:43.9 23:46:45 0.73 L G77.12-22.35 21441+4722 21:45:59.3 23:36:04 1.52 L
G77.31-22.61 EL1-12 21:47:20.8 23:32:05 0.82 L G77.49-22.66 21461+4722
21:48:00.8 23:36:38 1.2 L G77.53-33.61 S140-N 22:19:28.6 15:32:56 0.81 L
G78.03-23.66 BD+46 3471 21:52:34.6 23:13:43 1.07 L G78.21-33.68 L1204-A
22:21:27.9 15:51:42 1.3 H G80.15-34.54 L1206 22:28:52.0 16:13:43 1.35 H
G81.79-41.02 22475+5939 22:49:29.6 11:54:57 0.97 H G83.48-30.65 L1221
22:28:02.7 21:01:13 1 L G84.01-42.75 22570+5912 22:59:06.7 11:28:28 1.22 H
G84.50-54.06 CB244 23:25:45.8 02:17:38 1.38 L G85.13-31.50 22336+6855
22:35:06.1 21:10:53 0.68 L G85.19-40.26 Cep A 22:56:17.9 14:01:46 1.27 H
G89.19-40.94 MBM 55 23:08:23.7 15:05:16 1.19 H G90.23-43.99 MWC1080 23:17:27.2
12:50:16 0.59 H G90.40-43.15 23140+6121 23:16:11.8 13:37:45 1.02 H
G94.25-42.40 L1246 SMM1 23:25:04.8 15:36:40 0.58 L G95.21-45.81 23314+6033
23:33:44.6 12:50:30 0.49 H G97.69-46.21 23385+6053 23:40:51.1 13:10:29 1.13 H
G104.92-43.54 23545+6508 23:57:05.1 17:25:11 1.39 H G107.26-50.82 LKHA 198
00:11:25.6 10:49:47 0.91 H G110.24-45.48 00117+6412 00:14:26.7 16:28:44 0.48 H
G113.80-44.60 00213+6530 00:24:10.4 17:47:02 0.98 L G115.28-45.09 00259+6510
00:28:49.1 17:26:47 0.91 L G117.95-46.66 00342+6374 00:37:13.0 16:04:15 1.21 H
G131.56-45.61 01133+6434 01:16:37.2 16:50:39 1.24 H G154.86-42.86 W3-IRS5
02:25:40.5 14:05:52 1.29 H G156.19-42.92 IC1805-W 02:29:02.3 13:33:32 1.16 H
G157.66-42.02 02310+6133 02:34:46.4 13:46:22 1.44 H G161.63-39.79 02461+6147
02:50:09.5 13:59:58 0.97 H G168.75-39.96 AFGL437 03:07:24.4 10:31:08 0.74 H
G173.06-36.36 AFGL490-iki 03:27:28.0 10:54:10 1.65 H G173.20-36.41 AFGL490
03:27:38.6 10:47:04 1.5 H G176.26-39.68 RNO13 03:25:09.5 06:46:22 1.23 L
G176.37-39.62 L1448 03:25:36.5 06:45:19 1.02 L G176.40-39.62 L1448 U-star
03:25:38.5 06:44:04 1.13 L G176.61-38.74 NGC1333 03:28:39.5 07:13:34 0.75 L
G176.64-38.69 IRAS2NNE-SSW 03:28:53.5 07:14:53 0.71 L G176.65-38.60 HH6
03:29:10.8 07:18:19 1.13 L G176.66-38.64 HH7-11 SSV 03:29:03.7 07:16:04 1.4 L
G177.24-39.63 L1455NW 03:27:26.1 06:15:48 1.26 L G177.34-39.62 RNO15FIR
03:27:40.1 06:13:03 0.74 L G177.38-39.60 L1455M 03:27:48.1 06:12:06 1.67 L
Name Other name RA DEC RMS Mass (J2000) (J2000) (Jy) (1) (2) (3) (4) (5) (6)
G177.63-38.58 03282+3035 03:31:20.2 06:45:25 1.32 L G177.73-37.96 03301+3057
03:33:22.8 07:07:30 0.7 L G177.73-39.03 03271+3013 03:30:14.6 06:23:49 0.84 L
G178.78-26.46 PP 13S 04:10:41.1 14:07:54 0.64 L G178.86-19.97 HL/XZ Tau
04:31:38.0 18:13:59 1.41 L G178.92-34.02 B5-IRS4 03:47:45.9 09:03:45 1.17 L
G178.93-20.05 L1551-IRS5 04:31:33.9 18:08:05 0.69 L G178.93-20.03 L1551NE
04:31:36.9 18:08:35 1.39 L G179.08-34.17 B5-IRS1 03:47:41.6 08:51:43 0.7 L
G179.09-34.38 B5-IRS3 03:47:05.4 08:43:09 1.41 L G179.11-35.42 HH211
03:43:57.1 08:00:50 1.29 L G179.19-34.13 B5-IRS2 03:48:03.6 08:49:28 1.3 L
G179.56-23.49 04191+1523 04:21:59.5 15:30:17 1.29 L G186.95-3.84 CB34
05:47:02.3 21:00:10 1 L G187.96-12.83 05137+3919 05:17:13.5 15:22:23 0.85 H
G188.51-34.95 L1489 04:04:43.5 02:18:57 1.21 L G189.03+0.78 AFGL 6366S
06:08:41.1 21:31:01 1.01 H G190.20-31.37 04166+2706 04:19:42.7 03:13:40 1.21 L
G190.63-31.18 04181+2655 04:21:10.5 03:02:06 0.69 L G190.75-13.70 05168+3634
05:20:16.5 12:37:21 1.72 H G192.07-11.00 RNO43N 05:32:27.0 12:57:06 1.44 L
G192.13-11.06 RNO43S 05:32:23.9 12:52:07 1.23 L G192.16-11.11 L1582B
05:32:15.8 12:49:20 1.09 L G192.16-3.82 05553+1631 05:58:13.6 16:32:00 0.63 H
G192.16-11.08 RNO43 05:32:21.8 12:49:40 0.8 L G192.88-3.18 HD250550 06:02:00.2
16:13:04 1.05 L G192.99+0.15 06114+1745 06:14:24.1 17:44:36 0.96 H
G193.81-31.25 04239+2436 04:26:56.9 00:43:36 1.16 L G193.86-10.32 05358+3543
A+B 05:38:34.6 11:47:35 1.01 H G194.37-30.83 L1524 04:29:23.8 00:32:58 1 L
G194.45-30.44 ZZ Tau 04:30:53.0 00:41:50 1.08 L G194.52-27.82 L1527 04:39:53.3
02:03:06 1.19 L G194.58-28.04 04361+2547 04:39:14.0 01:53:22 1.34 L
G194.81-30.30 TMC2A 04:31:59.8 00:30:49 1.1 L G194.82-28.07 TMC1A 04:39:34.8
01:41:46 1.62 L G194.82-27.96 IC2087 04:39:58.9 01:45:06 1.88 L G194.99-27.68
04381+2540 04:41:13.0 01:46:37 0.74 L G195.03-11.71 05327+3404 05:36:05.7
10:06:12 1.26 L G195.05-30.21 L1529 04:32:44.7 00:23:13 0.88 L G195.27-16.98
HH114/115 05:18:17.0 07:11:01 1.44 H G195.72-29.75 04325+2402 04:35:33.5
00:08:15 1.34 L G196.93-10.42 B35 05:44:26.4 09:08:14 0.91 L G197.11-12.41
AFGL5157 05:37:48.2 07:59:24 0.7 H G198.58-9.14 L1598NW 05:52:11.5 08:21:28
1.16 L G201.87-7.62 S241 06:03:53.7 06:14:44 0.84 H G202.12+2.65 Mon OB1I
06:41:05.4 10:49:46 1.07 L G202.30+2.54 Mon OB1H 06:41:03.1 10:37:03 0.97 L
G202.93+2.26 Mon OB1G 06:41:12.3 09:55:35 1.78 L G203.23+2.07 Mon OB1D
06:41:03.9 09:34:39 1.1 L Name Other name RA DEC RMS Mass (J2000) (J2000) (Jy)
(1) (2) (3) (4) (5) (6) G203.32-11.94 05487+0255 05:51:23.1 02:55:45 1.03 L
G203.36-11.73 05490+2658 05:52:12.7 02:59:33 1.08 H G203.47-11.92 05491+0247
05:51:46.0 02:48:35 1.26 L G203.76+1.27 NGC2261 06:39:09.9 08:44:12 1.02 L
G204.88-13.85 NGC2071N 05:47:34.5 00:41:00 2.75 L G205.38-14.41 NGC2068/LBS17
05:46:29.7 -00:00:37.0 1.02 L G205.42-14.42 NGC2068 05:46:31.7 -00:02:56.0
1.18 L G205.52-14.57 HH24 05:46:11.5 -00:12:17.0 0.93 L G205.95-17.09 Ori-I-2
05:38:04.7 -01:45:09.0 1.36 L G206.01-15.48 HH212 05:43:51.5 -01:02:52.0 0.61
L G206.56-16.34 NGC2024 Ori B 05:41:49.5 -01:55:17.0 0.83 H G206.84-2.38
06291+0421 06:31:47.8 04:19:31 0.93 H G206.86-16.55 NGC 2023 05:41:37.1
-02:15:58.0 1.19 L G206.86-16.61 NGC2023-MM1 05:41:25.0 -02:18:09.0 1.51 L
G207.33-2.15 06308+0402 06:33:31.4 04:00:07 1.05 H G207.60-23.03 L1634
05:19:49.0 -05:52:05.0 1.33 L G208.63-19.21 CSO 2 05:35:13.9 -04:59:22.0 1.48
L G208.66-19.21 AC 3 05:35:18.3 -05:00:41.0 1.06 L G208.75-19.22 OMC-2/3(MMS
9) 05:35:25.8 -05:05:37.0 0.85 L G208.77-19.19 MMS 10 05:35:33.8 -05:05:38.0
2.02 L G208.90-20.05 Ori A-W 05:32:42.2 -05:35:48.0 1.56 L G209.25-19.12
05341-0539 05:36:38.4 -05:28:16.0 1.46 L G209.85-20.27 HH83 05:33:32.2
-06:29:44.0 1.03 L G210.04-19.81 HH 34 05:35:30.2 -06:26:57.0 0.72 L
G210.35-19.69 V380/OriNE 05:36:26.0 -06:39:12.0 1.36 L G210.40-19.72 V380 Ori
05:36:25.9 -06:42:38.0 1.3 L G210.43-19.74 05339-0646 05:36:23.9 -06:44:45.0
0.88 L G210.44-19.76 CS-star HH1 05:36:20.8 -06:45:35.0 1.55 L G210.44-19.75
VLA3 05:36:22.9 -06:45:22.0 1.55 L G210.45-19.76 HH 1-2 05:36:22.8 -06:46:07.0
1.96 L G210.58-19.81 V380/OriS 05:36:25.7 -06:54:12.0 1.49 L G210.82-36.61
L1642 04:34:49.9 -14:13:09.0 0.94 L G210.96-19.34 05363-0702 05:38:44.5
-07:01:03.0 1.04 L G211.44-19.39 Haro 4-255 05:39:22.0 -07:26:45.0 0.99 L
G211.57-19.29 L1641S3 05:39:56.0 -07:30:26.0 0.45 L G211.58-19.15 L1641S
05:40:28.0 -07:27:28.0 0.99 L G212.25-19.36 L1641S4 05:40:49.2 -08:06:51.0
1.36 L G212.63-19.00 L1641S2 05:42:47.0 -08:17:06.0 0.98 L G213.88-11.83 GGD
12-15 06:10:51.5 -06:11:27.0 2.52 H G217.30-0.05 BFS 56 06:59:14.4 -03:54:52.0
0.53 L G217.38-0.08 BiP 14 06:59:16.3 -03:59:39.0 0.62 H G218.02-0.32 S287-C
06:59:36.5 -04:40:22.0 1.72 L G218.06-0.11 S287-A 07:00:23.6 -04:36:38.0 0.54
L G218.10-0.37 S287-B 06:59:34.4 -04:46:00.0 1.77 L G218.15-15.00 06047-1117
06:06:41.4 -11:18:40.0 1.05 L G224.35-2.01 07028 1100 07:05:13.2 -11:04:41.0
0.73 L G224.61-2.56 Z Cma 07:03:43.6 -11:33:06.0 1.02 L G228.99-4.62 CB54
07:04:20.9 -16:23:20.0 0.9 L G356.09+20.75 16191 1936 16:22:04.4 -19:43:26.0
1.57 L
---
* •
Columns (1)–(6) list the source name, other source name from Wu et al. (2004)
catalog, corresponding equatorial coordinates, rms noise and mass type (L
represents low-mass sources, H represents high-mass sources), respectively
|
arxiv-papers
| 2012-11-11T12:34:36 |
2024-09-04T02:49:37.848209
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cong-Gui Gan, Xi Chen, Zhi-Qiang Shen, Ye Xu, Bing-Gang Ju",
"submitter": "Conggui Gan",
"url": "https://arxiv.org/abs/1211.2411"
}
|
1211.2414
|
# Resonances for activity waves in spherical mean field dynamos
D. Moss moss@maths.manchester.ac.uk School of Mathematics, University of
Manchester, Manchester M13 9PL, UK D.D. Sokoloff${}^{\,}$
sokoloff.dd@gmail.com Department of Physics, Moscow State University, 119999,
Moscow, Russia
###### Abstract
We study activity waves of the kind that determine cyclic magnetic activity of
various stars, including the Sun, as a more general physical rather than a
purely astronomical problem. We try to identify resonances which are expected
to occur when a mean-field dynamo excites waves of quasi-stationary magnetic
field in two distinct spherical layers. We isolate some features that can be
associated with resonances in the profiles of energy or frequency plotted
versus a dynamo governing parameter. Rather unexpectedly however the
resonances in spherical dynamos take a much less spectacular form than
resonances in many more familiar branches of physics. In particular, we find
that the magnitudes of resonant phenomena are much smaller than seem
detectable by astronomical observations, and plausibly any related effects in
laboratory dynamo experiments (which of course are not in gravitating
spheres!) are also small. We discuss specific features relevant to resonant
phenomena in spherical dynamos, and find parametric resonance to be the most
pronounced type of resonance phenomena. Resonance conditions for these dynamo
wave resonances are rather different from those for more conventional branches
of physics. We suggest that the relative insignificance of the phenomenon in
this case is because the phenomena of excitation and propagation of the
activity waves are not well-separated from each other and this, together with
the nonlinear nature of more-or-less realistic dynamos, suppress the
resonances and makes them much less pronounced than resonant effects, for
example in optics,.
###### pacs:
95.30.Qd, 97.10.Jb
## I Introduction
The well known solar activity cycle is associated with propagation of a wave
of quasistationary magnetic field somewhere within the solar convective
envelope. This wave is believed to be excited and supported by electromagnetic
induction effects driven by the joint action of differential rotation and
mirror-asymmetric flows. This is known as dynamo action, e.g. S04 .
Correspondingly, the wave is known as a dynamo wave. More specifically, there
are at least two, more or less independent, dynamo waves, one propagating
equatorwards in each solar hemisphere.
For quite a long time, solar activity waves provided the unique example of
dynamo wave propagation, and so the phenomenon was not investigated in a wider
context. Even such a limited statement of the problem provides a quite rich
range of instructive physical phenomena. From time to time the dynamo engine
”misfires”, and solar Grand Minima, including the Maunder Minimum in the
XVIIth - early XVIIIth centuries, occur (see for review e.g. SY03 ). At the
end of the Maunder Minimum, the dynamo wave propagated in one solar hemisphere
only, and a so-called ”mixed-parity” magnetic configuration appeared SNR94 .
Analysis of XVIIIth century sunspot data A08 gives a hint Ietal10 that from
time to time magnetic field becomes concentrated near to the solar equator,
rather than in two propagating belts in middle latitudes, and a configuration
of quadrupole-like (instead of the usual dipole-like) symmetry then appears.
Recently the variety of physical phenomenon associated with dynamo wave
propagation has enlarged substantially. The first attempts to isolate dynamo
wave propagation from stellar activity data were undertaken in BH07 ; K10 ,
and generation of oscillating magnetic field, probably indicating dynamo wave
propagation, was obtained in the VKS dynamo experiment Metal07 . This opens
new perspectives in the topic, in particular the opportunity to study dynamo
waves as a physical, rather than a purely astronomical, phenomenon.
Figure 1: Periods of oscillation in energy when dynamo action occurs in a
single layer. Crosses: $C^{I}_{\alpha}=0$, $C^{II}_{\alpha}$ varies;
asterisks: $C^{II}_{\alpha}=0$, $C^{I}_{\alpha}$ varies .
This change of viewpoint in the investigation of dynamo waves has attracted
attention to these problems, which had previously remained underinvestigated.
In particular, it was stressed DG12 that resonance, which is a prominent and
sometimes spectacular phenomenon in the conventional theory of various
oscillations and wave propagations, must also play an important role in dynamo
wave propagation. We share their view that the topic was not adequately
addressed in previous studies, and only a few papers (including KS93 ; Moss96
) have addressed the topic. The point is that the natural statement of the
problem relevant to astronomical contexts did not focus attention on this
issue.
Here we present our findings associated with the problem of resonance
phenomena in dynamo wave propagation. Our general conclusion can be formulated
as follows. Resonances in dynamo systems do occur, and sometimes they can be
isolated in the solutions of the governing equations (e.g. KS93 ; Moss96 ).
However in general dynamo wave propagation as a physical phenomenon differs
quite substantially from, say, the propagation of acoustic or electromagnetic
waves. The point is that conventional wave propagation can be easily separated
from the wave excitation, whereas dynamo wave propagation is usually
intimately associated with dynamo wave excitation. As a result, solutions of
the dynamo equations usually fail to separate in any explicit way resonant
phenomena from the more general phenomena of dynamo wave excitation.
Figure 2: Time variations of magnetic energy for
$C_{\alpha}^{I}=6,C_{\alpha}^{II}=-5$. Upper panel - toroidal field energy in
lower (solid curve) and upper (broken) layers separately, lower panel - total
magnetic energy.
We illustrate the problem by numerical investigation of a dynamo model in
which dynamo waves are excited in two adjacent spherical shells with different
distributions of dynamo governing parameters. The magnetic field is continuous
across the boundary between the two domains. A related model was discussed
previously in MS07 ; MSL11 in a more specifically astronomical context.
## II The dynamo model
We consider a standard mean field dynamo equation ($\alpha^{2}\omega$-dynamo)
${\frac{\partial{\bf B}}{\partial t}}={\rm curl}\,({\bf V\times B}+\alpha{\bf
B}-\eta\,{\rm curl}\,{\bf B}),$ (1)
where $\eta$ is the turbulent diffusivity and $\alpha$ represents the usual
isotropic alpha-effect (for details of the model see MS07 ; the codes used
there are in turn based on that of MB00 ). In Eq. (1) $\bf V$ is the velocity
from the (differential) rotation and we only look for axisymmetric solutions.
A simple algebraic $\alpha$-quenching (in which $\alpha$ is reduced by a
factor $1+(B/B_{0})^{2}$, where $B_{0}$ is a field strength at which dynamo
saturation is expected) is used to suppress dynamo action as the magnetic
field grows.
Figure 3: The case $C_{\alpha}^{II}=-5$,
$C_{\omega}^{I}=C_{\omega}^{II}=10^{5}$. Top panel: toroidal field energies in
the lower and upper layers (asterisks and crosses respectively). The lower
panel shows the total energy and the middle panel shows the variation of the
period of oscillation of the energy. In each panel, quantities are plotted as
a function of $C_{\alpha}^{I}$.
We solve the equation in spherical geometry for two adjacent dynamo active
layers, the ’deep’ layer I with $0.7\leq r\leq 0.85$ and the ’shallow’ layer
II with $0.85\leq r\leq 1$ ($r$ is the fractional radius). We consider a very
simple rotation profile with rotation velocity linear in $r$, and
$\alpha$-profiles in each layer are proportional to $\cos\theta$ where
$\theta$ is the polar angle, so that $\theta=90^{\circ}$ is the equator.
$C^{I}_{\alpha}$ and $C^{II}_{\alpha}$ are dimensionless amplitudes of
$\alpha$ in layers I and II correspondingly, and $C^{I}_{\omega}$ and
$C^{II}_{\omega}$ are dimensionless amplitudes of the differential rotation.
We take $C^{I}_{\omega}=C^{II}_{\omega}$ in the examples discussed below.
$\alpha$ is continuous across the interface at $r=0.85$. All solutions
discussed have odd (”dipole-like”) parity.
Our investigation is focussed on the more physical aspects of the problem, but
it would be straightforward to rescale the results for particular celestial
bodies, or to take a more realistic rotation curve.
## III Looking for resonance in dynamo solutions
Figure 4: Extracts from the time series of total energy for solutions outside
of the resonant domain (upper panel, $C^{I}_{\alpha}=6.5$), and inside this
domain (lower panel, $C^{I}_{\alpha}=4.5$).
### III.1 Background
We obtained solutions for our model for various governing parameters. We first
determined the periods when dynamo action occurs in one layer only, i.e.
either $C_{\alpha}^{I}$ or $C_{\alpha}^{II}$ is zero. Note that the magnetic
field does penetrate the passive layer, although its strength is substantially
reduced there. Here we either take $C_{\alpha}^{I}>0$ so that with
$C_{\alpha}^{II}=0$ we obtain poleward migration, or $C_{\alpha}^{II}<0$ so
that with $C_{\alpha}^{I}=0$ there is equatorward migration. The variations of
period with $|C_{\alpha}^{I,II}|$ for these cases are shown in Fig. 1.
More generally, the model includes two interacting dynamo regions, which are
physically separate but adjacent, which possess specific individual
eigenfrequencies, so quite naturally the resulting time-series for the total
magnetic energy and magnetic energy for the upper and lower layer can be more
complicated than just a harmonic oscillation; see Fig. 2 where the result for
a typical solution ($C_{\alpha}^{I}=6,C_{\alpha}^{II}=-5$) is shown.
### III.2 Resonant solutions
We then set $C_{\alpha}^{II}=-5$ and allowed $C_{\alpha}^{I}$ to vary, and
plotted the period of the total energy (defined by a mean over $O(1000)$
oscillations) The variations of toroidal field energies in the upper and lower
layers, the (common) period and the total magnetic energy are shown in Fig. 3
as a function of $C_{\alpha}^{I}$. We see a feature, in the form of a local
peak of period, in the region $C^{I}_{\alpha}=5\pm 1$. A related feature
appears in the plots of all three quantities (although it is maybe less
pronounced in the lower panel). Examination of Fig. 2 shows that this feature
does not correspond to $C_{\alpha}^{I}$ values for which the periods of the
layers, taken individually, are equal. For $C_{\alpha}^{II}=-5$, periods are
approximately equal when $C_{\alpha}^{I}\approx 7$.
We show in Fig. 4 short samples of the time series of the total energy for
solutions outside of the resonant domain ($C_{\alpha}^{I}=6.5$, upper panel),
and inside the domaon ($C_{\alpha}^{I}=4.5$, lower panel).
### III.3 Resonance with a subcritical dynamo layer
Figure 5: The case $C_{\alpha}^{I}=-5$,
$C_{\omega}^{I}=C_{\omega}^{II}=10^{5}$, with thin lower layer, $0.7\leq r\leq
0.775$. Top panel: toroidal field energies in the lower and upper layers
(asterisks and crosses respectively). The lower panel shows the total energy
and the middle panel shows the variation of period.
We recognize that the resonant behaviour was not very well pronounced in the
previous results and thus try to explore and explain this situation. We do not
doubt that resonance in its purely mathematical sense occurs in dynamo
solutions. More specifically, if the governing equations for a kinematic
dynamo problem have a degenerate eigenvalue
$\Gamma_{n}=\gamma_{n}+i\omega_{n}$ ($\gamma_{n}>0$), then it is more than
plausible that the corresponding set of eigenvectors becomes insufficient to
provide a basis, and the desired solution, apart from terms proportional to
$\exp(\gamma_{n}+i\omega_{n})t$, may contain resonant terms proportional to
$t\exp(\gamma_{n}+i\omega_{n})t$. The point however is that the exponential
growth of the solution as $\exp\gamma_{n}t$ or $t\exp\gamma_{n}t$ occurs just
at the initial stage of the dynamo wave excitation and is a matter of more or
less purely theoretical interest. A more practical issue is the amplitude of
the steady-state oscillation obtained after the exponential growth is somehow
saturated – here by the term $1+(B/B_{0})^{2}$. The saturation reduces the
growth rate $\gamma_{n}$ to zero, and this is a much stronger effect than the
additional power-low growth due to the resonance. The exceptional situation
discussed by, say, KS93 ; Moss96 ) addresses the cases where excitation of a
particular solution without a resonance does not occur, and it is the
existence of a resonance that allows this solution to develop. In contrast,
resonances in acoustic or electromagnetic wave propagation occur when a wave
can marginally propagate and does not grow ($\gamma=0$), and the resonance is
clearly recognisable as a growing (and then saturated) solution.
In another investigation, the lower layer was made much thinner $(0.7\leq
r\leq 0.775)$, so a dynamo in this layer alone is subcritical (i.e. with
$C_{\alpha}^{I}=\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$
}}\;\hskip-2.0pt10,C_{\alpha}^{II}=0$, dynamo action does not occur) and the
upper layer correspondingly thicker, $0.775\leq r\leq 1.0$. We keep
$C_{\alpha}^{II}=-5$, $C_{\omega}^{I}=C_{\omega}^{II}=10^{5}$. There is again
a (smaller) feature attributable to a resonance near $C_{\alpha}^{I}=4.25$.
See Fig. 5.
We conclude from Fig. 5 that the resonance here is of a similar nature to that
in the previous case.
### III.4 Parametric resonance
Now we seek the other kind of resonance, i.e. a parametric resonance in the
dynamo wave driven by harmonic variations of the dynamo governing parameters.
Our preliminary expectations here are twofold. From one hand, the resonances
recognized in KS93 ; Moss96 for galactic dynamos are parametric. On the other
hand, the parametric resonance suggested in MPS02 to explain some details in
the magnetic activity of stellar binary systems does not seem very
spectacular.
We imposed a modulation $\alpha=\alpha_{0}(1+f\cos\omega_{P}t)$ in the region
I, with $500\leq\omega_{P}\leq 1500$, $f=0.2$, keeping $C_{\alpha}^{II}=-5$.
We started from the unperturbed model which has $C_{\alpha}^{I}=7.5$ (i.e.
outside of the resonant region discussed in Sect. III.2). The corresponding
period is $P(E)=0.00539$, so $\omega(E)=1166$. Results are shown in Fig. 6,
where resonant peaks are clearly recognizable.
The feature at $\omega_{P}\approx 1150$ is quite expected, as it satisfies the
standard resonance condition $\omega_{P}\approx\omega(E)=2\omega(B)$ (where
$\omega(B)$ is the frequency of the magnetic field oscillations). The features
near $\omega_{p}\approx 700$ and $\omega_{p}\approx 900$ have to be associated
with rather unusual excitation conditions. Note that these resonant solutions
do not settle to a completely steady (maybe modulated) oscillation, even when
run for about 2000 oscillations. In the two anomalous cases the period is not
quite stable, but the averaging process seems robust, in that averages over
different temporal sub-ranges give the same values. Note that the energy in
the upper and lower regions separately (top panel in Fig. 6) is the toroidal
field energy only, whereas the ”total energy” includes also the poloidal. Over
long enough time intervals the average period in region II is slightly longer
than that of the total energy (see middle panel of Fig. 6).
Figure 6: Parametric resonance: notation as in Fig. 3. The middle panel now
includes the periods of oscillations in the upper layer II (shown by small
solid circles). The dashed horizontal lines are values for the unperturbed
model. The period of the total energy is plotted by solid dots in the middle
panel. In each panel, quantities are plotted a functions of $\omega({\rm P})$.
In general, parametric resonance effects seem to be much clearer and more
pronounced than the resonances analysed in previous models (Sects. III.2 and
III.3). However the peaks in energy profiles for parametric resonances are
quite broad.
## IV Discussion
The concept of resonance belongs to the basic concepts of contemporary
physics, and it looks a priori implausible that it should not be applicable to
dynamo wave propagation. We have verified this natural expectation in the
framework of activity waves generated by dynamo waves, in a quite traditional
model of a two-layer mean-field spherical mean-field dynamo of the type
conventionally considered as a model for periodic stellar magnetic activity.
Indeed, we found features in the profiles for energies or periods
(frequencies) plotted versus a dynamo governing parameter which can be
identified with a resonance. However, it looks noteworthy that the details
resemble more discontinuities in the profiles of the behaviour of period and
energy as $C_{\alpha}^{I}$ varies, rather than the conventional resonant peaks
shown in physics textbooks. It is even difficult to claim that the resonance
makes dynamo excitation more efficient and enhances, say, the total magnetic
energy. The lower panel in Fig. 5 is instructive in this respect: the
interaction here appears as a fall in total energy, rather than an increase.
Of course, resonances are far from being the only effects that determine the
relevant behaviour in, say, optics, and there are many cases where other
effects make recognition of resonance peaks problematic. The point however is
that it is not difficult to provide examples of pronounced resonant peaks in
optics. When presenting our results, we have of course chosen the more
pronounced examples of resonances for dynamo waves, admittedly from a rather
small sample.
Correspondingly, we cannot suggest that the sort of resonances discussed here
will leave observable signatures. Our intention is to draw attention to novel
behaviour. Our results are not so marked as those in DG12 , arguably because a
linear problem was considered there. The inclusion of a nonlinear saturation
mechanism reduces these effects.
The parametric resonances present in the spherical dynamo models investigated
here (Sect. III.4) are much clearer than the other resonance features (Sects.
III.2 and III.3). Perhaps, this is a consequence of controlling by varying the
frequency $\omega_{P}$, which governs the parametric excitation but, leaves
unchanged the ”nominal” parameters of the dynamo model for the vanishing
modulation amplitude $f=0$.
Our numerical experiment illuminate the unusual excitation conditions for
parametric resonances. Perhaps, it can be said that the Mathieu equation is
not a fully adequate model for parametric resonance in spherical dynamos.
Isolation of the resonance condition here appears an attractive topic for
dynamo theory, but it is however obviously beyond the scope of this paper. The
transition time required for the dynamo system to achieve a resonant type
solution can be long and a qualitative explanation in terms of an appropriate
model equation also looks desirable.
We note also the existence of a quasi-resonant phenomenon in a case where one
of the dynamo layers is passive, i.e. with the chosen parameters the lower
layer taken in solution does not support dynamo action.
Perhaps the basic differences between resonant phenomena for dynamo waves and
those in more conventional branches of physics can be summarized as follows.
It is difficult to separate dynamo wave propagation from dynamo wave
excitation, while such processes are well separated in many other branches of
physics. Correspondingly, eigenvalues of the linear (kinematic) dynamo
problems take complex values. Coincidence of two complex quantities is a more
severe requirement rather than equality of two real valued quantities. Dynamo
self-excitation is in practice saturated by some back-reaction of the dynamo
generated magnetic field on the hydrodynamics. This saturation determines the
resulting magnetic energy and affects the periods of dynamo waves. The
corresponding effects specific to dynamo action are more important than any
potential resonant behaviour. In additional, the two layers of dynamo action
are separated hydrodynamically but magnetic field penetrates from one activity
layer to the other, even if the dynamo action in one layer remains
subcritical. Thus the very concept of the frequency of dynamo waves in one
activity layer taken alone is not fully applicable.
In general, it seems that the situation under discussion reflects the every-
day wisdom that small problems (here resonances) can only spoil your life in
the absence of larger ones (in this case other uncertainties associated with
mean field dynamos are much larger).
###### Acknowledgements.
DS is grateful to financial support from RFBR under grant 12-02-00170-a.
## References
* (1) Stix, M., 2004, The Sun: An Introduction, B., Springer
* (2) Soon, W.W.-H., Yaskell, S.H., 2003, The Maunder Minimum: The Variable Sun-Earth Connection, Singapore, World Sci.
* (3) Sokoloff, D., Nesme-Ribes, E., 1994, Astron. Astrophys., 288, 293
* (4) Arlt, R., 2008, Solar Phys., 247, 399
* (5) Illarionov, E., Sokoloff, D., Arlt, R., Khlystova, A., 2011, Astronomische Nachrichten, 332, 590
* (6) Berdyugina, S.V., Henry, G.W., 2007, Astrophys. J., 659, L157
* (7) Katsova, M.M., Livshits, M.A., Soon, W., Baliunas, S.L., Sokoloff, D.D., 2010, New Astron., 15, 274
* (8) Monchaux, R., Berhanu, M., Bourgoin, M., Moulin, M., Odier, Ph., Pinton, J.-F., Volk, R., Fauve, S., Mordant, N., Petrelis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Gasquet, C., Marié, L., Ravelet, F., 2007, PRL, 98, 044502
* (9) Gilman, P.A., Dikpati, M., 2011, Astrophys. J., 738, 108
* (10) Kuzanyan, K.M., Sokoloff, D.D., 1993, Astrophys. Sp. Sci., 208, 245
* (11) Moss, D., 1996, Astron. Astrophys., 308, 381
* (12) Moss, D., Sokoloff, D., 2007, Mon. Not. Roy. Astron. Soc., 377, 1597
* (13) Moss, D., Sokoloff, D.. Lanza, A. F., 2011, Astron. Astrophys., 531, A43
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* (15) Moss, D., Piskunov, N., Sokoloff, D., 2002. Astron. Astrophys., 396, 885
|
arxiv-papers
| 2012-11-11T12:42:36 |
2024-09-04T02:49:37.855938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Moss and Dmitri Sokoloff",
"submitter": "David Moss Dr",
"url": "https://arxiv.org/abs/1211.2414"
}
|
1211.2517
|
# A SVD accelerated kernel-independent fast multipole method and its
application to BEM
Yanchuang Cao caoyanch@126.com Lihua Wen lhwen@nwpu.edu.cn Junjie Rong
rxrjj@126.com College of Astronautics, Northwestern Polytechnical University,
Xi’an 710072, P. R. China
###### Abstract
The kernel-independent fast multipole method (KIFMM) proposed in [1] is of
almost linear complexity. In the original KIFMM the time-consuming M2L
translations are accelerated by FFT. However, when more equivalent points are
used to achieve higher accuracy, the efficiency of the FFT approach tends to
be lower because more auxiliary volume grid points have to be added. In this
paper, all the translations of the KIFMM are accelerated by using the singular
value decomposition (SVD) based on the low-rank property of the translating
matrices. The acceleration of M2L is realized by first transforming the
associated translating matrices into more compact form, and then using low-
rank approximations. By using the transform matrices for M2L, the orders of
the translating matrices in upward and downward passes are also reduced. The
improved KIFMM is then applied to accelerate BEM. The performance of the
proposed algorithms are demonstrated by three examples. Numerical results show
that, compared with the original KIFMM, the present method can reduce about
40% of the iterating time and 25% of the memory requirement.
###### keywords:
boundary element method; kernel-independent fast multipole method; singular
value decomposition; matrix compression
## 1 Introduction
The boundary element method (BEM) has become a promising numerical method in
computational science and engineering. Despite many unique advantages, like
the dimension reduction, high accuracy and suitability for treating infinite
domain problems, a major disadvantage of the BEM is its dense system matrix
which solution cost is prohibitive in large-scale problems. During the past
three decades, several acceleration methods have been proposed to circumvent
this disadvantage. Representative examples are the fast multipole method
(FMM)[2], wavelet compression method[3], $\mathcal{H}$-matrix[4], adaptive
cross approximation (ACA)[5], pre-corrected FFT [6], etc. Among them the FMM
is no doubt the most outstanding one.
The conventional FMM is originally proposed to accelerate the $N$-body
simulations, which requires the analytical expansions of the kernel functions.
This poses a severe limitation on its applications to many problems where the
analytical expansions are hard to be obtained. Besides, the kernel-tailored
expansion makes it difficult to develop a universal FMM code for real-world
applications. To overcome this drawback, the kernel-independent FMM (KIFMM)
has been proposed in the past decade [1, 7, 8]. A salient feature of the KIFMM
is that the expansion of the kernel function is no longer required. Instead,
only a user-defined function for kernel value evaluation is needed; the
structure of the FMM acceleration algorithm is in common for many typical
problems.
In this paper, the KIFMM proposed by Ying et al [1] is concerned. This method
uses equivalent densities in lieu of the analytical expansions. It provides a
unified framework for fast summations with the Laplace, Stokes, Navier and
similar kernel functions. Due to its ease-of-use and high efficiency, it has
attracted the attention of many researchers [9, 10, 11].
The moment-to-local (M2L) translation is the most time-consuming part of the
FMM [12, 13, 8, 14, 7, 15]. In the KIFMM [1] the M2L translation is
accelerated by the fast Fourier transform (FFT), leading to
$\mathcal{O}(p^{3}\log p)$ computational complexity, where $p$ is the number
of equivalent points along the cube side. However, one should note that the
efficiency of the FFT approach tends to become lower when $p$ increases. This
is because the equivalent points lie only on the boundary of each box, while
to use the FFT Cartesian grid points interior the box must be considered as
well. In this paper, the M2L in KIFMM is compressed and accelerated using the
singular value decomposition (SVD); see Section 3. This method is built on the
fact that the M2L matrices are typically of very low numerical ranks. Our
numerical experiments, including those in Section 5, show that the proposed
method is more efficient than the FFT approach. Another advantage of the SVD
accelerating approach is that it is more flexible than the FFT approach,
because the later requires the equivalent and check points to be equally
spaced while this is not needed in the SVD approach. Moreover, as a byproduct,
the orders of the translating matrices in the upward and downward passes can
also be reduced by using the transform matrices of M2L, leading to further
reduction of the CPU time and memory usage for the upward and downward passes.
The original KIFMM in [1] is designed to accelerate the potential evaluation
for particle simulations. Recently, this original method was applied to solve
boundary integral equations (BIEs) in, e.g., blood flow, molecular
electrostatic problems [16, 17, 18]. It is noticed that the central idea of
all those works is to translate the far-field interactions to a particle
summation formulation so that the original KIFMM can be used in a
_straightforward_ manner. For example, Ref. [16] deals with large-scale blood
flow problem. The velocity of each red blood cell is divided into two
components, namely the velocity of a reference point and the relative velocity
reflecting the self deformation of the cell. By doing this, the interactions
between red blood cells can be formulated into “particle summations”
corresponding to the reference points for all the blood cells, and thus the
KIFMM can be used. In [17], the Nyström method is used to discretize the BIE
in order to obtain the particle summation form.
As is well known, the BEM is an dominate numerical method for solving BIEs and
has profound applications in engineer. In this paper, the KIFMM is used to
accelerate the BEM. This work is nontrivial since the KIFMM can not be
straightly used in BEM due to the presence of elements, let alone to maintain
the accuracy and efficiency. For example, the equivalent and check surfaces
are crucial components of the KIFMM. In the original KIFMM these surfaces can
be set as the surfaces of each cube. However, in BEM setting this choice would
deteriorate the accuracy, because the boundary elements belonging to a cube
can often extrude from the cube; see Section 4.1 for the details in choosing
those surfaces.
## 2 Basic idea of the KIFMM
The KIFMM was proposed in [1] to solve the potential problems for particles.
Here its framework is briefly reviewed.
### 2.1 Setup
Assume that there are $N$ source densities $\\{q_{i}\\}$ located at $N$ points
$\\{\mathbf{y}_{i}\\}$. Then the induced field potential $\\{p_{i}\\}$ at
points $\\{\mathbf{x}_{i}\\}$ is given by
$p_{i}=p(\mathbf{x}_{i})=\sum_{j=1}^{N}G(\mathbf{x}_{i},\mathbf{y}_{j})q(\mathbf{y}_{j})=\sum_{j=1}^{N}G_{ij}q_{j},\quad
i=1,2,\cdots,N,$ (1)
where, $G(\mathbf{x},\mathbf{y})$ is the kernel function, which can be of the
single layer, double layer or other layers. The complexity is obviously
$\mathcal{O}(N^{2})$ if the potentials are evaluated naively by a order $N$
matrix-vector multiplication. By using the FMMs this complexity can be reduced
to $\mathcal{O}(N)$.
The central to all the FMMs lies in the low-rank approximation (LRA) of the
submatrices representing the far-field interactions. The efficient realization
of the LRA relies on a spatial tree structure. To construct the tree, all the
particles are first included into a root level cube. Then the cube is equally
divided into 8 cubes, generating the cubes in the next level. This subdivision
is continued until the particles contained in each leaf cube is no more than a
predetermined number $s$.
For each cube $C$, let $\mathscr{N}^{C}$ denote its near field which consists
of cubes in the same level that share at least one vertex with $C$; the union
of the other cubes is defined to be its far field $\mathscr{F}^{C}$. Let $B$
denote the parent cube of $C$, then the interaction field of $C$ is defined as
$\mathscr{I}^{C}=\mathscr{F}^{C}\backslash\mathscr{F}^{B}$. Let
$\mathbf{y}^{C,\mathrm{u}}$ denote the _upward equivalent surface_
corresponding to cube $C$, $\mathbf{x}^{C,\mathrm{u}}$ denote the _upward
check surface_ , $\mathbf{y}^{C,\mathrm{d}}$ denote the _downward equivalent
surface_ and $\mathbf{x}^{C,\mathrm{d}}$ denote the _downward check surface_
[1]. To guarantee the existence of the equivalent densities and check
potentials, these surfaces must satisfy the following conditions:
1. 1.
$\mathbf{y}^{C,\mathrm{u}}$ and $\mathbf{x}^{C,\mathrm{u}}$ lie between $C$
and $\mathscr{F}^{C}$; $\mathbf{x}^{C,\mathrm{u}}$ encloses
$\mathbf{y}^{C,\mathrm{u}}$;
2. 2.
$\mathbf{y}^{C,\mathrm{d}}$ and $\mathbf{x}^{C,\mathrm{d}}$ lie between $C$
and $\mathscr{F}^{C}$; $\mathbf{y}^{C,\mathrm{d}}$ encloses
$\mathbf{x}^{C,\mathrm{d}}$;
3. 3.
$\mathbf{y}^{C,\mathrm{u}}$ encloses $\mathbf{y}^{B,\mathrm{u}}$ where $B$ is
$C$’s child;
4. 4.
$\mathbf{y}^{C,\mathrm{u}}$ is disjoint from $\mathbf{y}^{B,\mathrm{d}}$ for
all $B$ in $\mathscr{F}^{B}$;
5. 5.
$\mathbf{y}^{C,\mathrm{d}}$ lies inside $\mathbf{y}^{B,\mathrm{d}}$ where $B$
is $C$’s parent.
The above conditions can be satisfied by choosing all the related surfaces be
the boundaries of cubes that are concentric with the cube. For each cube $C$
with side length $2r$, $\mathbf{y}^{C,\mathrm{u}}$ and
$\mathbf{x}^{C,\mathrm{d}}$ can be chosen as the boundary of the cube with
halfwidth $(1+d)r$, $\mathbf{x}^{C,\mathrm{u}}$ and
$\mathbf{y}^{C,\mathrm{d}}$ as the boundary of the cube with halfwidth
$(3-2d)r$, where $0\leq d\leq\frac{2}{3}$. Therefore, the distance between the
equivalent surface and the check surfaces involved in each translation is no
less than $(2-3d)r$. This relation is used in the original KIFMM [1], with $d$
being of a small value. In this way the equivalent surface and the check
surfaces are well-separated, and high accuracy can be obtained. However, when
the KIFMM is applied to BEM, $d$ has to be set larger, or a large part of the
source densities on elements belonging to $C$ may extrude from its upward
equivalent surface $\mathbf{y}^{C,\mathrm{u}}$, and the sources belonging to
cube $C$ can not be well represented by its equivalent densities. Thus the
size of the elements should be considered in defining these surfaces for BEM.
See Section 4.1.
### 2.2 Far field translations
Generally, in a FMM, the potentials induced by the sources in the near field
are computed directly by (1), which is named as _source-to-target_ (S2T)
translation. The potentials induced by the sources in the far field are
efficiently evaluated by a series of translations, named as _source-to-
multipole_ (S2M), _multipole-to-multipole_ (M2M), _multipole-to-local_ (M2L),
_local-to-local_ (L2L) and _local-to-target_ (L2T) translations. The main
feature of the KIFMM lies in that the above translations are performed using
equivalent densities and check potentials, while in the conventional FMM the
translations are performed using the multipole expansions and local
expansions. The algorithm for evaluating the potential contribution of far-
field sources in KIFMM is as follows.
1. 1.
_S2M_. The source densities $\mathbf{q}$ in a leaf cube $B$ are translated
into its upward equivalent densities $\mathbf{q}^{B,\text{u}}$; that is,
$\mathbf{q}^{B,\text{u}}=\mathbf{Sq},$ (2)
with $\mathbf{S}$ being the translating matrix [1].
2. 2.
_M2M_. The upward equivalent densities $\mathbf{q}^{B,\text{u}}$ of a cube $B$
are transformed to the upward equivalent densities $\mathbf{q}^{C,\text{u}}$
of its parent $C$,
$\mathbf{q}^{C,\text{u}}=\mathbf{M}\mathbf{q}^{B,\text{u}},$ (3)
with $\mathbf{M}$ being the translating matrix.
3. 3.
_M2L_. The upward equivalent densities $\mathbf{q}^{C,\text{u}}$ of cube $C$
are translated to the downward check potentials $\mathbf{p}^{D,\text{d}}$ of
cube $D\in\mathscr{I}^{C}$ in its interaction field
$\mathbf{p}^{D,\text{d}}=\mathbf{K}\mathbf{q}^{C,\text{u}},$ (4)
where, the translating matrix $\mathbf{K}$ is computed as
$K_{ij}=G(\mathbf{x}_{i},\mathbf{y}_{j}),$
with $\mathbf{x}_{i}$ being the $i$-th downward check point of $D$ and
$\mathbf{y}_{j}$ being the $j$-th upward equivalent point of $C$.
4. 4.
_L2L_. The downward check potentials $\mathbf{p}^{D,\text{d}}$ of cube $D$ are
translated to the downward check potentials of its child cube $E$,
$\mathbf{p}^{E,\text{d}}=\mathbf{L}\mathbf{p}^{D,\text{d}},$ (5)
with $\mathbf{L}$ being the translating matrix.
5. 5.
_L2T_. The downward check potentials $\mathbf{p}^{E,\text{d}}$ of leaf cube
$E$ are translated to the potentials $\mathbf{p}$ on the target points in $E$,
$\mathbf{p}=\mathbf{T}\mathbf{p}^{E,\text{d}},$ (6)
with $\mathbf{T}$ being the translating matrix.
Combining equations (2)–(6), the potential $\mathbf{p}$ on the target points
in a leaf cube induced by the source densities $\mathbf{q}$ in another leaf
cube in its far field can be computed as
$\mathbf{p=TLKMSq}.$ (7)
The M2L translation (4) is the most time-consuming step in the KIFMM. It is
accelerated by FFT in the original KIFMM [1]. In its implementation auxiliary
points must be added inside the upward equivalent surface and the downward
check surface, although one only needs the upward equivalent points and the
downward check points on the corresponding surfaces. This makes the FFT
approach less efficient when the number of the equivalent points and check
points are large because the auxiliary points will account for a large
proportion. In the next section, a SVD approach is proposed to accelerate the
M2L translations as well as other translations.
## 3 SVD-based acceleration for translations
In this section, a new SVD-based accelerating technique is proposed, which can
compress all the transform matrices in KIFMM, thus both the M2L translation
and the upward and downward passes are greatly accelerated.
### 3.1 Matrix dimension reduction for M2L
In the acceleration for M2L, SVD is applied in two stages. In the first stage,
the M2L translating matrices are compressed into more compact forms.
Suppose that the kernel function is translational invariant. The union of
unique translating matrices over all cubes in each level forms a set of 316
matrices. To compress these matrices, first collect them into a fat matrix in
which they are aligned in a single row and a thin matrix in which they are
aligned in a single column
$\mathbf{K}_{\text{fat}}=\left[\mathbf{K}^{(1)}\quad\mathbf{K}^{(2)}\quad\dots\quad\mathbf{K}^{(316)}\right],$
(8a)
$\mathbf{K}_{\text{thin}}=\left[\mathbf{K}^{(1)};\quad\mathbf{K}^{(2)};\quad\dots;\quad\mathbf{K}^{(316)}\right],$
(8b)
where $\mathbf{K}^{(i)}$ is the $i$-th translating matrix. Perform SVD for
these two matrices
$\mathbf{K}_{\text{fat}}=\mathbf{U}\mathbf{\Sigma}\left[{\mathbf{V}^{(1)}}^{\mathtt{T}}\quad{\mathbf{V}^{(2)}}^{\mathtt{T}}\quad\dots\quad{\mathbf{V}^{(316)}}^{\mathtt{T}}\right],$
(9a)
$\mathbf{K}_{\text{thin}}=\left[\mathbf{Q}^{(1)};\quad\mathbf{Q}^{(2)};\quad\dots;\quad\mathbf{Q}^{(316)}\right]\mathbf{\Lambda}\mathbf{R}^{\mathtt{T}}.$
(9b)
Notice that in our algorithm, the entities of M2L matrices $\mathbf{K}^{(i)}$
are the evaluations of single-layer kernel function. In most cases, they are
symmetric, ie., ${\mathbf{K}^{(i)}}^{\mathtt{T}}=\mathbf{K}^{(i)}$, so (9a)
and (9b) are just transposes of each other, and the SVD has to be performed
only once.
Consider the translating matrix $\mathbf{K}^{(i)}$ for one translation
$\mathbf{U}^{\mathtt{T}}\mathbf{K}^{(i)}\mathbf{R}=\mathbf{\Sigma}{\mathbf{V}^{(i)}}^{\mathtt{T}}\mathbf{R}=\mathbf{U}^{\mathtt{T}}\mathbf{Q}^{(i)}\mathbf{\Lambda}.$
(10)
Obviously, $\mathbf{U}^{\mathtt{T}}\mathbf{K}^{(i)}\mathbf{R}$ decays both
along the rows and columns as quickly as the singular values in
$\mathbf{\Sigma}$ and $\mathbf{\Lambda}$, thus it can be approximated by its
submatrix $\mathbf{\tilde{U}}^{\mathtt{T}}\mathbf{K}^{(i)}\mathbf{\tilde{R}}$,
therefore
$\mathbf{K=U(U^{\mathtt{T}}KR)R^{\mathtt{T}}\approx\tilde{U}(\tilde{U}^{\mathtt{T}}K\tilde{R})\tilde{R}^{\mathtt{T}}=\tilde{U}\tilde{K}\tilde{R}},$
(11)
where, $\mathbf{\tilde{U}}$ and $\mathbf{\tilde{R}}$ are the tailored matrices
consisted by columns corresponding with dominant singular values that are not
less than
$\varepsilon_{1}\|\mathbf{K}_{\text{fat}}\|_{2}=\varepsilon_{1}\mathbf{\Sigma}_{0,0}=\varepsilon_{1}\mathbf{\Lambda}_{0,0}$,
and $\mathbf{\tilde{K}}$ is the compressed translating matrix. Substituting
(11) into (7) yeilds
$\mathbf{p}=\mathbf{TL\tilde{U}\tilde{K}{\tilde{R}}^{\mathtt{T}}MSq}.$ (12)
Similar compression scheme was also used in [7].
The compression (11) is performed for M2L translating matrices at all levels.
Let $L$ denote the number of levels, then the computational complexity is
$\mathcal{O}(L)\sim\mathcal{O}(\log N)$. It can be reduced for the cases of
homogeneous kernels. Assume that kernel function $G(\mathbf{x},\mathbf{y})$ is
homogeneous of degree of $m$, that is,
$G(\alpha\mathbf{x},\alpha\mathbf{y})=\alpha^{m}G(\mathbf{x},\mathbf{y})$ for
any nonzero real $\alpha$. Let
$\mathbf{\tilde{K}}_{0}^{(i)}\,(i=1,2,\dots,316)$ be the compressed
translating matrices constructed from the interacting cubes that are scaled to
have unit halfwidth. Then, the compressed translating matrices on the $l$-th
level can be computed efficiently by scaling
$\mathbf{\tilde{K}}_{l}^{(i)}=\left(\frac{r_{0}}{2^{l}}\right)^{m}\mathbf{\tilde{K}}_{0}^{(i)},$
(13)
where, $r_{0}$ is the halfwidth of the root cube in the octree. Therefore only
$\mathbf{\tilde{K}}_{0}^{(i)}$ has to be computed in the compressing
procedure, and the computational complexity can be reduced into
$\mathcal{O}(1)$.
The threshold $\varepsilon_{1}$ affects the balance between the computational
cost and the accuracy of the algorithm. The induced error in each M2L
translation is of order $\varepsilon_{1}$, and the total error is
approximately $L\varepsilon_{1}$ [1]. In order to maintain the error
decreasing rate of BEM with piecewise constant element, $L\varepsilon_{1}$
should decrease by a factor of 2 with each mesh refinement
$L\varepsilon_{1}\sim 2^{-L}.$
In this paper, $\varepsilon_{1}$ is chosen by
$\varepsilon_{1}=C_{1}\frac{2^{-L}}{L},$ (14)
where, $C_{1}$ is a constant coefficient.
### 3.2 Further acceleration for M2L
After the dimension reduction, it is found that most of the compressed M2L
matrices $\mathbf{\tilde{K}}$ are still of low numerical ranks. For example,
figure 1 illustrates the rank distribution of the interaction field of a cube
$C$ used in numerical example 5.1 with $N=2097152,p=8,C_{1}=0.1,C_{2}=100$.
The dimension of the original translating matrix is 296. After compression
using $\mathbf{\tilde{U}}$ and $\mathbf{\tilde{R}}$ the dimension is reduced
to 84. However, the figure clearly shows that the actual numerical ranks of
the matrices are still much lower than 84. This fact indicates that the
computational cost of M2L can be further reduced by using the low rank
decomposition of matrices $\tilde{\mathbf{K}}$. Here the low rank
decomposition is computed by SVD, so that optimal rank can be obtained. Since
the number of the translating matrices is $\mathcal{O}(1)$, this computational
overhead is small.
Figure 1: Numerical rank distribution of M2L matrices
$\mathbf{\tilde{K}}_{84\times 84}$ in numerical example 5.1 with
$N=2097152,p=8,C_{1}=0.1,C_{2}=100$.
Consider the low rank approximations of the scaled matrices
$\mathbf{\tilde{K}}_{0}$ for translational invariant and homogeneous kernels.
Compute the SVD for each M2L matrix $\mathbf{\tilde{K}}_{0}^{(i)}$,
$\mathbf{\tilde{K}}_{0}^{(i)}=\mathbf{U}_{0}^{(i)}\mathbf{S}_{0}^{(i)}\left(\mathbf{Q}_{0}^{(i)}\right)^{\mathtt{T}}.$
Truncate the singular values smaller than
$\varepsilon_{2}\|\mathbf{K}_{0,\text{fat}}\|_{2}$, and discard the
corresponding columns in $\mathbf{U}_{0}^{(i)}$ and $\mathbf{Q}_{0}^{(i)}$.
Then the M2L translation can be approximated by
$\begin{split}\mathbf{\tilde{p}}^{D,\text{d}}_{l}&=\sum_{C\in\mathscr{I}^{D}}\left(\frac{r_{0}}{2^{l}}\right)^{m}\mathbf{\hat{U}}_{0}^{(i)}(\mathbf{\hat{S}}_{0}^{(i)}\mathbf{\hat{Q}}_{0}^{(i)})\mathbf{\tilde{q}}^{C,\text{u}}_{l}\\\
&=\sum_{C\in\mathscr{I}^{D}}\left(\frac{r_{0}}{2^{l}}\right)^{m}\mathbf{\hat{U}}_{0}^{(i)}\mathbf{\hat{V}}_{0}^{(i)}\mathbf{\tilde{q}}^{C,\text{u}}_{l},\end{split}$
(15)
where, $\mathbf{\hat{S}}_{0}^{(i)}$ is the submatrix of $\mathbf{S}_{0}^{(i)}$
containing the dominant singular values that are no smaller than
$\varepsilon_{2}\|\mathbf{K}_{0,\text{fat}}\|_{2}$;
$\mathbf{\hat{U}}_{0}^{(i)}$ and $\mathbf{\hat{Q}}_{0}^{(i)}$ are the matrices
consisted by the corresponding columns of $\mathbf{U}_{0}^{(i)}$ and
$\mathbf{Q}_{0}^{(i)}$, respectively; and
$\mathbf{\hat{V}}_{0}^{(i)}=\mathbf{\hat{S}}_{0}^{(i)}\mathbf{\hat{Q}}_{0}^{(i)}$.
The error of approximation (15) is determined by $\varepsilon_{2}$. Denote
$\mathbf{\hat{K}}_{0}^{(i)}=\mathbf{\hat{U}}_{0}^{(i)}\mathbf{\hat{V}}_{0}^{(i)}$.
From the truncating scheme, there exists
$\|\mathbf{\hat{K}}_{0}^{(i)}-\mathbf{\tilde{K}}_{0}^{(i)}\|_{2}\leq\varepsilon_{2}\|\mathbf{K}_{0,\text{fat}}\|_{2}.$
For arbitrary $m\times n$ matrix $\mathbf{A}$, one has
$\|\mathbf{A}\|_{\text{max}}\leq\|\mathbf{A}\|_{2}\leq\sqrt{mn}\|\mathbf{A}\|_{\text{max}}$.
Thus,
$\|\mathbf{\hat{K}}_{0}^{(i)}-\mathbf{\tilde{K}}_{0}^{(i)}\|_{\text{max}}\leq\varepsilon_{2}\|\mathbf{K}_{\text{fat}}\|_{2}.$
Let $\mathbf{\hat{K}}_{0,\text{fat}}$ and $\mathbf{\tilde{K}}_{0,\text{fat}}$
denote the fat matrices for $\mathbf{\hat{K}}_{0}$ and
$\mathbf{\tilde{K}}_{0}$, respectively, which are constructed similarly as
$\mathbf{K}_{\text{fat}}$ in (8a). It is easy to know that
$\|\mathbf{\hat{K}}_{0,\text{fat}}-\mathbf{\tilde{K}}_{0,\text{fat}}\|_{\text{max}}\leq\varepsilon_{2}\|\mathbf{K}_{0,\text{fat}}\|_{2}.$
Since the dimension of $\mathbf{\tilde{K}}_{0,\text{fat}}$ is $\tilde{p}\times
316\tilde{p}$, where $\tilde{p}$ is the dimension of $\mathbf{\tilde{K}}_{0}$,
and
$\|\mathbf{\hat{K}}_{0,\text{fat}}-\mathbf{\tilde{K}}_{0,\text{fat}}\|_{\text{max}}\geq\frac{1}{\sqrt{316\tilde{p}^{2}}}\|\mathbf{\hat{K}}_{0,\text{fat}}-\mathbf{\tilde{K}}_{0,\text{fat}}\|_{2},$
one has
$\|\mathbf{\hat{K}}_{0,\text{fat}}-\mathbf{\tilde{K}}_{0,\text{fat}}\|_{2}\leq\varepsilon_{2}\sqrt{316\tilde{p}^{2}}\|\mathbf{K}_{0,\text{fat}}\|_{2}.$
Therefore, the error introduced by the low rank approximation is ensured to be
of same order as $\varepsilon_{1}$ by letting
$\varepsilon_{2}\sim\frac{\varepsilon_{1}}{\sqrt{316\tilde{p}^{2}}}\sim\frac{\varepsilon_{1}}{\tilde{p}}.$
In our scheme, it is defined by
$\varepsilon_{2}=C_{2}\frac{\varepsilon_{1}}{\tilde{p}},$ (16)
where $C_{2}$ is a constant coefficient.
### 3.3 Acceleration for upward and downward passes
The transformation matrices $\mathbf{\tilde{U}}$ and $\mathbf{\tilde{R}}$ can
also be used to compress the matrices for upward and downward passes. Since
the columns of $\mathbf{\tilde{R}}$ are orthonormal, thus
$\mathbf{\tilde{R}^{\mathtt{T}}\tilde{R}=I}$. The potentials in
$\mathscr{I}^{B}$ generated by the upward equivalent densities
$\mathbf{q}^{B,\text{u}}$ can be written as follows
$\mathbf{K}\mathbf{q}^{B,\text{u}}\approx\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}}\mathbf{q}^{B,\text{u}}=\mathbf{\tilde{U}\tilde{K}(\tilde{R}^{\mathtt{T}}\tilde{R})\tilde{R}^{\mathtt{T}}}\mathbf{q}^{B,\text{u}}=\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}}\mathbf{q}_{1}^{B,\text{u}},$
(17)
where
$\mathbf{q}_{1}^{B,\text{u}}=\mathbf{\tilde{R}\tilde{R}^{\mathtt{T}}}\mathbf{q}^{B,\text{u}}$
is the projection of $\mathbf{q}^{B,\text{u}}$ to the space spanned by the
columns of $\mathbf{\tilde{R}}$. This suggests that
$\mathbf{q}_{1}^{B,\text{u}}$ can approximately reproduce the potential field
in $\mathscr{I}^{C}$ excited by $\mathbf{q}^{B,\text{u}}$. In other words,
$\mathbf{q}_{1}^{B,\text{u}}$ can be taken as the new upward equivalent
densities for the potential field in $\mathscr{I}^{C}$.
Now consider $B$’s parent cube $C$ and its interacting field
$\mathscr{I}^{C}$. Since $\mathscr{I}^{C}$ lies outside $\mathscr{I}^{B}$,
from potential theory we know that $\mathbf{q}_{1}^{B,\text{u}}$ can also be
used to reproduce the potential field in $\mathscr{I}^{C}$, ie.,
$\begin{split}\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}M}\mathbf{q}^{B,\text{u}}\approx&\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}M}\mathbf{q}_{1}^{B,\text{u}}\\\
=&\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}M\tilde{R}\tilde{R}^{\mathtt{T}}}\mathbf{q}^{B,\text{u}}\\\
=&\mathbf{\tilde{U}\tilde{K}\tilde{R}^{\mathtt{T}}M\tilde{R}\tilde{R}^{\mathtt{T}}S}\mathbf{q}\\\
=&\mathbf{\tilde{U}\tilde{K}\tilde{M}\tilde{S}}\mathbf{q},\end{split}$ (18)
where,
$\mathbf{\tilde{M}=\tilde{R}^{\mathtt{T}}M\tilde{R}}$
is the new translating matrix for M2M;
$\mathbf{\tilde{S}=\tilde{R}^{\mathtt{T}}S}$
is the new translating matrix for S2M.
From the symmetry of the algorithm, ie., the upward pass and the downward pass
playing the same role in the algorithm, we know that the downward pass can be
transformed by $\mathbf{\tilde{U}}$ similarly. Thus,
$\mathbf{\tilde{L}=\tilde{U}^{\mathtt{T}}L\tilde{U}}$
is the new translating matrix for L2L;
$\mathbf{\tilde{T}=\tilde{T}\tilde{U}}$
is the new translating matrix for L2T.
Since both the transformation matrices $\mathbf{\tilde{U}}$ and
$\mathbf{\tilde{R}}$ are thin matrices, the new translating matrices
$\mathbf{\tilde{S}}$, $\mathbf{\tilde{M}}$, $\mathbf{\tilde{L}}$ and
$\mathbf{\tilde{T}}$ are smaller than their original forms, and thus the
computational cost of the upward and downward passes can be reduced.
## 4 KIFMM for BEM
In this section, the KIFMM is applied to accelerate the BEM. One should note
that in BEM the sources distribute continuously on the boundary elements
instead of on a group of discrete points in the original KIFMM [1]. Therefore,
in the BEM the potential evaluation in S2T and S2M operations must be
performed by integration rather than summation. More importantly, since the
continuous sources are represented by nodal basis functions, the sources has
to be grouped based on the supports of nodal functions, which would make the
definition of equivalent and check surfaces different from original KIFMM.
For clarification in explanation, the single-layer BIE for Laplace problem is
considered. Let $\Omega$ be a bounded domain with boundary $\Gamma$. Given a
known potential $f(\mathbf{x})$ on the boundary $\Gamma$, the source density
distribution $q(\mathbf{x})$ satisfies
$\int_{\Gamma}G(\mathbf{x},\mathbf{y})q(\mathbf{y})\mathrm{d}\mathbf{y}=f(\mathbf{x}),\quad\mathbf{x}\in\Gamma,$
(19)
where, $G(\mathbf{x},\mathbf{y})={1/(4\pi|\mathbf{x}-\mathbf{y}|)}$ is the
fundamental solution of the Laplace equation. By partitioning the boundary
$\Gamma$ into triangular elements and using the piecewise constant basis
functions with the nodal points on element centroids, the collocation BEM
leads to a linear system
$\mathbf{Aq=b}$
with $\mathbf{q}$ consisting of the source densities on each triangles,
$\mathbf{b}$ consisting of the known potentials on each collocation points,
and
$A_{i,j}=\int_{\triangle_{j}}G(\mathbf{x}_{i},\mathbf{y})\chi_{j}(\mathbf{y})\mathrm{d}\mathbf{y}.$
(20)
where, $\mathbf{x}_{i}$ is the $i$-th collocation point, $\triangle_{j}$ is
the $j$-th triangle, and $\chi_{j}(\mathbf{y})$ is the basis function on
$\triangle_{j}$. When the system solved by iterative methods, the main
computational cost is spent on the matrix-vector multiplication (MVM) of which
the complexity is $\mathcal{O}(N^{2})$. This complexity can be reduced to
$\mathcal{O}(N)$ by the KIFMM.
### 4.1 The equivalent and check surfaces
As mentioned above, the definition of the equivalent and check surfaces for
BEM has to be different with the original KIFMM for particle simulations,
because the equivalent surface must enclose all the sources according to the
potential theory [1].
In constructing the octree for BEM, the centroids of elements are used as
reference points, and the subdivision process is similar to that in Section
2.1. Figure 2 illustrates a leaf cube $C$ in the octree. The union of all the
elements whose centroids lying in $C$ is denoted by $\Gamma(C)$. One should
note that $\Gamma(C)$ may extrude from $C$. As a result, for BEM the first two
restrictions of surface definition in Section 2.1 have to be modified as
follows:
1. 1.
$\mathbf{y}^{C,\mathrm{u}}$ and $\mathbf{x}^{C,\mathrm{u}}$ lie between
$\Gamma(C)$ and $\mathscr{F}^{C}$; $\mathbf{x}^{C,\mathrm{u}}$ encloses
$\mathbf{y}^{C,\mathrm{u}}$;
2. 2.
$\mathbf{y}^{C,\mathrm{d}}$ and $\mathbf{x}^{C,\mathrm{d}}$ lie between $C$
and $\Gamma(\mathscr{F}^{C})$, with $\Gamma(\mathscr{F}^{C})$ being the union
of all elements that belongs to $\mathscr{F}^{C}$; $\mathbf{y}^{C,d}$ encloses
$\mathbf{x}^{C,\mathrm{d}}$.
Figure 2: The elements and the upward equivalent surface related to a leaf
cube.
The equivalent and check surfaces for BEM are defined similarly with Section
2.1. However, in order to satisfy the above restrictions, the relative
distance between a cube and its upward equivalent surface $d$ has to be chosen
large enough so that for each leaf cube, the triangles “belonging to” it are
enclosed in its upward equivalent surface. For a quasi-uniform element
partition, assume that the size of the element is $h$ and each leaf cube
contains at most $s$ elements, then the halfwidth of the leaf cubes in the
finest level is proportional with $\sqrt{s}h$. The distance between the
outmost vertex and the cube surface is no larger than $h$, thus $d$ is of
order
$d\sim\mathcal{O}\left(\frac{(\sqrt{s}+1)h}{\sqrt{s}h}-1\right)=\mathcal{O}\left(\frac{1}{\sqrt{s}}\right).$
So, in this paper $d$ is evaluated as
$d=C_{d}\frac{1}{\sqrt{s}},$ (21)
where, $C_{d}$ is user-defined constant. Our numerical experience indicates
that $C_{d}=0.5$ is proper for most problems.
### 4.2 S2T and S2M translations
The equivalent points and check points are sampled on the equivalent and check
surfaces, respectively. Then the potentials on the collocation points can be
evaluated efficiently by KIFMM, in which the contribution of the near-field
sources are evaluated by S2T, and the contribution of the far-field sources
are evaluated efficiently using the equivalent densities and check potentials.
The potentials produced by near-field sources are evaluated by S2T
translation. In the original KIFMM, it is performed by direct evaluation (1),
since the sources distribute on discrete points. However, in BEM the sources
distribute on elements, thus these potentials should be evaluated by
integration
$\begin{split}p(\mathbf{x}_{i})&=\int_{\Gamma(\mathscr{N}^{C})}G(\mathbf{x}_{i},\mathbf{y})q(\mathbf{y})\mathrm{d}\mathbf{y}\\\
&=\sum_{j}q_{j}\int_{\triangle_{j}}G(\mathbf{x}_{i},\mathbf{y})\chi_{j}(\mathbf{y})\mathrm{d}\mathbf{y},\quad\triangle_{j}\in\Gamma(\mathscr{N}^{C}),\end{split}$
(22)
where $p(\mathbf{x}_{i})$ is the check potential on the $i$-th collocation
point.
The potentials produced by far-field sources are evaluated by a series of
translations, namely S2M, M2M, M2L, L2L and L2T. Among these translations, S2M
need to evaluate the upward check potentials produced by the sources belonging
to the leaf cube. Similar to S2T translation, this must be implemented by
integration as well
$\begin{split}p^{C,\mathrm{u}}(\mathbf{x})&=\int_{\Gamma(C)}G(\mathbf{x},\mathbf{y})q(\mathbf{y})\mathrm{d}\mathbf{y}\\\
&=\sum_{j}q_{j}\int_{\triangle_{j}}G(\mathbf{x},\mathbf{y})\chi_{j}(\mathbf{y})\mathrm{d}\mathbf{y},\quad\triangle_{j}\in\Gamma(C),\end{split}$
(23)
where, $p^{C,\mathrm{u}}(\mathbf{x})$ is the upward check potential for leaf
cube $C$.
In the above sections, the accelerating algorithm for single layer type
boundary integral is introduced. With slight modifications it can be used to
accelerate double layer boundary integral. That is, only the integral kernel
function in S2T translation and the first step in S2M translation should be
replaced into double layer kernel. Therefore, upward equivalent densities
should be solved by
$\int_{\mathbf{y}^{C,\mathrm{u}}}G(\mathbf{x},\mathbf{y})q^{C,\mathrm{u}}(\mathbf{y})\mathrm{d}\mathbf{y}=\int_{\Gamma(C)}\frac{\partial
G(\mathbf{x},\mathbf{y})}{\partial\mathbf{n(y)}}q(\mathbf{y})\mathrm{d}\mathbf{y},\quad\text{for
any }\mathbf{x}\in\mathbf{x}^{C,\mathrm{u}}.$ (24)
Discretized with upward check points and upward equivalent points, a linear
system can be achieved, and the upward equivalent densities can be solved. The
other steps of the algorithm remains the same with that dealing with single
layer boundary integral.
### 4.3 The complete algorithm
In KIFMM for BEM, the discretized sources are grouped into cubes in a octree,
then the potentials on collocation points are divided into two parts, namely
the contribution of the near-field sources and the contribution of the far-
field sources. The former is evaluated by S2T, while the later is evaluated by
a series of translations. The complete algorithm for BEM is implemented by the
following steps:
* Algorithm SVD accelerated KIFMM for BEM
* Step 1 Setup
* 1 Construct the octree by subdividing the leaf cube recursively.
* 2 For each cube $C$, find the cubes in its near field $\mathscr{N}^{C}$ and interaction field $\mathscr{I}^{C}$.
* 3 Define the equivalent and check surfaces by the method described in Section 4.1.
* 4 Compute and compress the translating matrices by the compressing approach in Section 3.
* Step 2 Upward pass
* 5 for each leaf cube $C$ in _postorder_ traversal of the tree do
* 6 Compute the upward equivalent densities (S2M).
* 7 end for
* 8 for each non-leaf cube $C$ in _postorder_ traversal of the tree do
* 9 Compute the upward equivalent densities (M2M).
* 10 end for
* Step 3 Downward pass
* 11 for each non-leaf cube $C$ in _preorder_ traversal of the tree do
* 12 Add to the downward check potentials produced by the sources in its interaction list (M2L)
* 13 Add to the downward check potentials of its child cubes (L2L)
* 14 end for
* 15 for each leaf cube $C$ in _preorder_ traversal of the tree do
* 16 Evaluate the potentials on the collocation points (L2T)
* 17 end for
* Step 4 Near-field interaction
* 18 for each leaf cube $C$ in _preorder_ traversal of the tree do
* 19 Add to the potential the contribution of near field source densities (S2T), which should be evaluated by Eq. (22)
* 20 end for
In our method, the definition of the equivalent and check surfaces are
different with the original KIFMM. However, this does not affect the
computational cost. The total computational complexity of our new KIFMM for
BEM remains $\mathcal{O}(N)$.
## 5 Numerical Examples
The performance of our SVD-based accelerating technique and the kernel-
independent fast multipole BEM for Laplace BIEs is demonstrated by three
numerical examples. The resulting linear systems are solved by GMRES solver.
The algorithms are implemented based on the public kifmm3d code available from
[19]. All simulations are carried out on a computer with a Xeon 5440 (3.00
GHz) CPU and 28 GB RAM.
### 5.1 Electrostatic problem
In this subsection, the electric charge density on an ellipsoidal conductor is
computed by solving Eq. (19). The ellipsoid can be described by
$(x_{1}/2)^{2}+x_{2}^{2}+(x_{3}/3)^{2}=1$. The analytic solution can be
expressed analytically using ellipsoidal coordinates. The convergence
tolerance for GMRES solver is set to be $10^{-6}$. The surface of the
ellipsoid is first discretized into $N=512$ triangular elements, then the mesh
is refined 6 times. The finest mesh has $N=2097152$ elements.
Table 1: Errors obtained with different $C_{1}$ and $C_{2}$ $N$ | Relative error
---|---
FFT | $C_{1}=0.1$ | $C_{1}=0.5$ | $C_{1}=0.1$ | $C_{1}=0.1$ | $C_{1}=0.1$
| $C_{2}=0$ | $C_{2}=0$ | $C_{2}=10$ | $C_{2}=100$ | $C_{2}=500$
$p=4$ |
512 | 0.069 936 | 0.070 235 | 0.086 715 | 0.071 639 | 0.074 210 | 0.074 210
2 048 | 0.032 898 | 0.033 050 | 0.033 227 | 0.033 080 | 0.039 421 | 0.060 588
8 192 | 0.014 010 | 0.014 047 | 0.016 236 | 0.014 055 | 0.015 502 | 0.062 169
32 768 | 0.006 697 | 0.006 704 | 0.007 845 | 0.006 702 | 0.007 191 | 0.028 167
131 072 | 0.003 517 | 0.003 518 | 0.004 236 | 0.003 521 | 0.003 720 | 0.009 063
524 288 | 0.002 960 | 0.002 962 | 0.003 100 | 0.002 966 | 0.003 098 | 0.004 419
2 097 152 | 0.004 880 | 0.004 880 | 0.004 887 | 0.004 883 | 0.004 896 | 0.005 296
$p=6$ |
512 | 0.069 923 | 0.070 111 | 0.086 805 | 0.071 593 | 0.074 541 | 0.074 541
2 048 | 0.032 901 | 0.033 038 | 0.033 270 | 0.033 082 | 0.043 255 | 0.062 104
8 192 | 0.014 001 | 0.014 076 | 0.016 217 | 0.014 081 | 0.016 608 | 0.063 492
32 768 | 0.006 641 | 0.006 679 | 0.008 849 | 0.006 681 | 0.007 300 | 0.028 298
131 072 | 0.003 182 | 0.003 219 | 0.003 983 | 0.003 220 | 0.003 351 | 0.007 635
524 288 | 0.001 579 | 0.001 587 | 0.002 905 | 0.001 587 | 0.001 607 | 0.002 321
2 097 152 | 0.000 790 | 0.000 791 | 0.001 227 | 0.000 791 | 0.000 793 | 0.000 932
$p=8$ |
512 | 0.069 923 | 0.070 122 | 0.086 831 | 0.071 622 | 0.074 681 | 0.074 681
2 048 | 0.032 901 | 0.033 035 | 0.033 299 | 0.033 075 | 0.046 320 | 0.062 675
8 192 | 0.014 001 | 0.014 065 | 0.016 208 | 0.014 071 | 0.016 880 | 0.064 226
32 768 | 0.006 641 | 0.006 677 | 0.008 673 | 0.006 670 | 0.007 231 | 0.030 980
131 072 | 0.003 182 | 0.003 216 | 0.004 141 | 0.003 217 | 0.003 344 | 0.009 128
524 288 | 0.001 579 | 0.001 585 | 0.002 467 | 0.001 585 | 0.001 602 | 0.002 632
2 097 152 | 0.000 789 | 0.000 790 | 0.001 014 | 0.000 790 | 0.000 793 | 0.000 988
Table 2: CPU times in each iteration $T_{\text{it}}$ and the total memory usage with different $C_{1}$ and $C_{2}$ $N$ | $T_{\text{it}}$ (s) | Memory usage (MB)
---|---|---
FFT | $C_{1}=0.1$ | $C_{1}=0.1$ | FFT | $C_{1}=0.1$ | $C_{1}=0.1$
| $C_{2}=10$ | $C_{2}=100$ | $C_{2}=10$ | $C_{2}=100$
$p=4$
512 | $\sim$ 0 | $\sim$ 0 | $\sim$ 0 | 2.7 | 1.8 | 1.8
2 048 | 0.02 | 0.01 | 0.01 | 9.3 | 7.1 | 6.7
8 192 | 0.10 | 0.06 | 0.05 | 31.4 | 27.8 | 27.4
32 768 | 0.37 | 0.31 | 0.24 | 121.3 | 116.9 | 116.3
131 072 | 1.57 | 1.50 | 1.10 | 471.1 | 466.4 | 465.3
524 288 | 6.30 | 7.34 | 5.25 | 1 879.6 | 1 892.8 | 1 891.6
2 097 152 | 18.25 | 30.12 | 24.32 | 7 504.3 | 7 598.9 | 7 597.7
$p=6$
512 | 0.01 | $\sim$ 0 | $\sim$ 0 | 6.1 | 3.0 | 3.0
2 048 | 0.05 | 0.01 | 0.01 | 19.8 | 8.1 | 7.8
8 192 | 0.31 | 0.06 | 0.05 | 55.8 | 28.3 | 27.9
32 768 | 1.11 | 0.30 | 0.23 | 186.4 | 119.5 | 118.9
131 072 | 4.75 | 1.78 | 1.31 | 736.5 | 493.4 | 492.1
524 288 | 13.26 | 8.56 | 6.46 | 2 917.5 | 2 062.5 | 2 060.8
2 097 152 | 76.16 | 45.51 | 34.79 | 11 669.0 | 8 861.7 | 8 859.8
$p=8$
512 | 0.01 | $\sim$ 0 | $\sim$ 0 | 14.4 | 7.7 | 7.7
2 048 | 0.17 | 0.01 | 0.01 | 44.0 | 12.8 | 12.5
8 192 | 0.70 | 0.06 | 0.04 | 100.8 | 33.0 | 32.6
32 768 | 2.98 | 0.30 | 0.17 | 292.3 | 124.2 | 123.6
131 072 | 12.50 | 1.63 | 1.21 | 1 143.1 | 489.5 | 488.3
524 288 | 49.09 | 8.37 | 4.57 | 4 482.6 | 2 067.1 | 2 065.5
2 097 152 | 198.03 | 44.89 | 34.73 | 17 924.3 | 8 866.4 | 8 864.6
The accuracy and efficiency of the present KIFMM BEM are mainly determined by
parameters $C_{1}$ in (14) and $C_{2}$ in (16). The translating matrices are
independent with the boundary since they are only determined by the position
of the equivalent and check points which are defined in the same manner as
discussed in section 4.1. The SVD accelerating approach truncates small
singular values of these translating matrices, therefore the induced error by
SVD acceleration only depends on $C_{1}$ and $C_{2}$ when $p$ is sufficiently
large. Consequently, $C_{1}$ and $C_{2}$ should keep the same values for
various boundary element analyses. From equation (14) we know that with larger
$C_{1}$, more singular values are discarded, and the translating matrices for
M2L and upward and downward passes would be compressed into more compact form,
thus the computing time could be reduced lower. While on the other hand the
error would become larger. Similar conclusions could be made for $C_{2}$.
Consequently, the choices of $C_{1}$ and $C_{2}$ are determined by the
tradeoff between the accuracy and the efficiency.
First the influence of $C_{1}$ on the accuracy of the algorithm is tested.
Three cases with $C_{1}$ being $0$, $0.1$ and $0.5$ are computed. The results
corresponding to $C_{1}=0$ are computed using the original FFT-accelerating
scheme in [1]. In all the three cases, $C_{2}$ is set to be 0.
The resulting errors are listed in the second to fourth columns of Table 1.
One can see that when $C_{1}=0.1$ the errors are almost the same as that
computed by FFT-accelerating scheme. However, when $C_{1}=0.5$ errors for
$p=6,\,8$ are increased. This indicates that $C_{1}=0.1$ is nearly optimal for
retaining the accuracy. It is noticed that when $p=4$ the error tends to be
larger when the DOF is high. This is because the error of the algorithm is
also relevant with $p$ and the depth $L$ of the octree. To get higher
accuracy, $p$ has to be increased to reduce the error in each translation; see
[1] for the details. The errors with $p=6$ and $p=8$ are almost the same, this
is because the error is bounded by the discretization precision for the BIE.
This also indicates that, for this numerical example, $p=6$ is sufficient to
get the same accuracy with conventional BEM.
The influence of $C_{2}$ is studied by setting $C_{2}=10,\,100,\,500$ while
$C_{1}=0.1$. In Table 1 it can be seen that for $C_{2}=10$ and $C_{2}=100$ the
results keep almost the same errors; while for $C_{2}=500$ the errors
increase. Although the errors raise with the increase of $C_{2}$, the test
case indicates that a choice of $C_{2}$ between 10 and 100 can maintains
almost the same accuracy. The CPU times $T_{\text{it}}$ in each iteration and
the total memory usage of the two methods, FFT-accelerating approach and the
SVD accelerating approach, are listed in Table 2. In Table 2, $T_{\text{it}}$
can reduce considerably with larger $C_{2}$, but the memory cost only reduce
slightly. The reason is that $C_{2}$ only controls the accuracy and efficiency
of the low-rank approximation for M2L matrices, as discussed in section 3.2.
In this problem, since the kernel is translational invariant and homogeneous,
the memory cost for M2L matrices is only of $\mathcal{O}(1)$. Therefore, the
memory reduction is negligible.
From Table 2 one can see that the CPU time in the SVD approach can be reduced
significantly for large $p$, comparing with the FFT approach, since the CPU
time for each iteration in the SVD approach is not sensitive to $p$. For
example, the $T_{\text{it}}$s of the SVD approach for the cases $p=6$ are
almost the same as that for $p=8$. This is because when $p$ is large, the size
of the compressed translating matrices are mainly determined by the
compressing threshold $\varepsilon_{1}$, and the numerical rank of M2L
matrices are only determined by $\varepsilon_{2}$. Both $\varepsilon_{1}$ and
$\varepsilon_{2}$ are independent with $p$. However, in the FFT approach more
auxiliary points has to be added, which makes the FFT approach less efficient.
Besides the CPU time, the memory usage can also be considerably reduced in the
SVD approach, since the translating matrices used in S2M and L2T are
compressed into more condensed form by the scheme in section 3.3.
This example shows that the accuracy reduces with the increase of $C_{1}$ and
$C_{2}$. When $C_{1}=0.1$ and $C_{2}=10$ the SVD accelerating approach is much
more efficient than FFT-accelerating approach without significantly affecting
the accuracy. It is also showed that $p=6$ is sufficient to maintain the
accuracy of BEM in this case. With $p=6$, $C_{1}=0.1$ and $C_{2}=10$, the CPU
time cost in each iteration can be reduced about 40% and the memory cost can
be reduced about 25% by SVD approach compared with the original FFT approach
while maintaining the accuracy of BEM. These parameters will be used in the
next numerical examples.
### 5.2 A mixed boundary condition problem
To demonstrate the performance of the SVD accelerating approach for more
complicated geometry and boundary condition problems, Laplace equation with
mixed boundary conditions on a shaft model illustrated in Figure 3 is
simulated. The analytical solution is set be to
$u=1/|\mathbf{x_{0}}-\mathbf{x}|$, with $\mathbf{x_{0}}$ being outside the
computational domain. The potential $u$ is given on the two end surfaces (red
surfaces in Figure 3), and the flux $q$ is given on the remainder (gray)
surfaces. The converging tolerance for GMRES solver is set to be $10^{-6}$.
Figure 3: A shaft model with mixed boundary condition.
The problem is solved by using the KIFMM BEM with the FFT approach and the SVD
approach, respectively. The results are reported in Table 3. It is showed
again that the SVD approach can save about 40% of the iterating time cost and
25% of the memory cost. The $L^{2}$-error of $u$ decays linearly with
$\mathcal{O}(h)$ and the $L^{2}$-error of $q$ decays as
$\mathcal{O}(\sqrt{h})$. The time consuming in each iteration and the memory
consuming increase almost linearly, which indicate that the computational
complexity of the method is almost $\mathcal{O}(N)$.
Table 3: Performance of SVD accelerating approach and FFT-accelerating approach $N$ | Error of $u$ | Error of $q$ | $T_{\text{it}}$ (s) | Memory (MB)
---|---|---|---|---
FFT | SVD | FFT | SVD | FFT | SVD | FFT | SVD
18 048 | 0.021 521 | 0.020 222 | 0.062 718 | 0.066 147 | 0.73 | 0.20 | 115.1 | 77.6
72 518 | 0.012 367 | 0.011 263 | 0.052 828 | 0.053 375 | 2.97 | 0.83 | 396.9 | 281.6
288 768 | 0.004 648 | 0.004 505 | 0.003 173 | 0.003 580 | 7.15 | 4.52 | 1 532.3 | 1 105.4
1 156 042 | 0.001 862 | 0.001 866 | 0.002 001 | 0.002 415 | 25.00 | 14.85 | 5 899.9 | 4 533.4
### 5.3 Heat conduction problem
To demonstrate the ability of the present KIFMM BEM for solving real-world
problems, a steady-state heat conduction analysis of a engine block is solved
here; see figure 4(b). The temperature field is governed by the Laplace
equation. The conductivity of the engine block is
$\lambda=80\mathrm{W/(m\cdot^{\circ}C)}$. The temperature of the inner surface
of the oblique tube and the temperature of the bottom surface are set to be
$75\mathrm{{}^{\circ}C}$ and $100\mathrm{{}^{\circ}C}$, respectively.
Convective condition with constant film coefficient
$h=10\mathrm{W/(m^{2}\cdot^{\circ}C)}$ and constant bulk temperature
$T_{0}=22^{\circ}\mathrm{C}$ are applied to the other surfaces. Simulations
are performed using a series of meshes with number of elements ranging from 85
680 to nearly 5 million. For comparison, this problem is also solved by finite
element method (FEM) with 698317 tetrahedral elements, 1015653 nodes. The
converging tolerance for GMRES solver is $10^{-4}$.
The CPU times and memory usage for different meshes are listed in Table 4,
where $N_{\mathrm{it}}$ and $T_{\mathrm{it}}$ stand for the number of
iterations and the CPU time for each iteration, respectively. Again one can
see linear behavior of the CPU time and memory requirement. The computed
temperature distribution using mesh with 325 774 elements is exhibited in
Figure 4(b). It can be seen that the temperature distribution obtained by the
KIFMM BEM agrees very well with that by FEM in figure 4(a).
It is noticed that, with the KIFMM BEM in this paper, the largest model with
nearly 5 million DOFs is successfully solved within 5 hours.
Table 4: CPU times (s) and memory usage (MB) for engine-block heat conduction analysis $N$ | $N_{\mathrm{it}}$ | $T_{\mathrm{total}}$ | $T_{\mathrm{it}}$ | Memory
---|---|---|---|---
85 680 | 90 | 582.4 | 3.8 | 502.5
325 774 | 96 | 2 111.1 | 13.3 | 1 735.5
900 420 | 100 | 4 251.0 | 27.5 | 4 589.0
1 370 880 | 103 | 5 946.3 | 34.7 | 7 374.8
4 754 670 | 97 | 18 330.4 | 108.2 | 25 021.5
(a) FEM result
(b) KIFMM result
Figure 4: Temperature distributions of the engine-block model computed by FEM
and KIFMM BEM.
## 6 Conclusion
The FMM is one of the most successful fast algorithms for BEM acceleration.
But it requires the analytical expansion of the kernel function, which makes
it difficult to be applied to some complicated problems. Recently, various
kernel-independent FMMs were developed to overcome this drawback. Among them
the KIFMM proposed in [1] has high efficiency and accuracy, and thus has been
extensively used [16, 17, 18]. The KIFMM uses equivalent densities and check
potentials instead of analytical expansions to construct the fast algorithm.
The time consuming M2L translations are accelerated by using the FFT. However,
it is noticed that when more equivalent and check points are sampled to get
higher accuracy, the efficiency of the FFT approach tends to be lower because
more auxiliary volume grid points have to be added in order to do FFT.
In this paper, the low rank property of the translating matrices in KIFMM is
sufficiently exploited by SVD (called SVD approach in this paper) to
accelerate all the translations, including the most time-consuming M2L. The
acceleration of the M2L translations is carried out in two stages. First the
translating matrix is compressed into more compact form, and then it is
approximated by low-rank decomposition. By using the compression matrices for
M2L, the translating matrices in upward and downward passes can also be
compressed into more compact form. Finally, the above improved KIFMM is
applied to accelerate BEM, leading to a highly efficient KIFMM BEM for solving
large-scale problems.
The accuracy and efficiency of the SVD approach and the KIFMM BEM are
demonstrated by three numerical examples. It is shown that, when compared with
the FFT-accelerated KIFMM, the SVD approach can reduce about 40% of the
iterating time and 25% of the total memory requirement. The presented KIFMM
BEM is of $\mathcal{O}(N)$ complexity. By using this method Laplace problem
with nearly 5 million unknowns can be successfully solved within 5 hours on a
Xeon-5440 2.83 GHz CPU and 28 GB RAM.
## Acknowledgements
This work was supported by the Doctorate Foundation of Northwestern
Polytechnical University under Grant No. CX201220, National Science
Foundations of China under Grants 11074201 and 11102154, and Funds for Doctor
Station from the Chinese Ministry of Education under Grants 20106102120009 and
20116102110006.
## References
## References
* [1] Lexing Ying, George Biros, and Denis Zorin. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. Journal of Computational Physics, 196:591–626, 2004.
* [2] L. Greengard and V. Rokhlin. A fast algorithm for particle simulation. Journal of Computational Physics, 73:325–348, 1987.
* [3] G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and numerical algorithms. Pure Appl. Math., 37:141–183, 1991.
* [4] W Hackbusch. A sparse matrix arithmetic based on h-matrices. part i: Introduction to h-matrices. Computing, 62:89–108, 1999.
* [5] M Bebendorf and S Rjasanow. Adaptive low-rank approximation of collocation matrices. Computing, 70:1–24, 2003.
* [6] Joel R. Phillips and Jocob K. White. A precorrected-fft method for electrostatic analysis of complicated 3-d structures. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 16(10):1059–1072, 1997.
* [7] William Fong and Eric Darve. The black-box fast multipole method. Journal of Computational Physics, 228(23):8712–8725, 2009.
* [8] Pierre-David Létourneau, Cristopher Cecka, and Eric Darve. Generalized fast multipole method. IOP Conference Series: Materials Science and Engineering, 10, 2010\.
* [9] Yuchun Lin, Andrij Baumketner, Shaozhong Deng, Zhenli Xu, Donald Jacobs, and Wei Cai. An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions. The Journal of chemical physics, 131(15):154103, 2009.
* [10] Ming Xiang, Shaozhong Deng, and Wei Cai. A sixth-order image approximation to the ionic solvent induced reaction field. Journal of scientific computing, 41(3):411–435, 2009.
* [11] S. N. Razavi, N. Gaud, N. Mozayani, and A. Koukam. Multi-agent based simulations using fast multipole method: application to large scale simulations of flocking dynamical systems. Artificial Intelligence Review, 35(1), 2011.
* [12] L. Greengard and V. Rokhlin. A new version of the fast multipole method for the laplace equation in three dimensions. Acta Numerica, pages 229–269, 1997.
* [13] W. Elliott and J. Board. Fast fourier transform accelerated fast multipole algorithm. SIAM J. Sci. Comput., 17(2):398–415, 1996.
* [14] Junichiro Makino. Yet another fast multipole method without multipoles — pseudoparticle multipole method. Journal of Computational Physics, 151:910–920, 1999.
* [15] R. Yokota. An fmm based on dual tree traversal for many-core architectures. arXiv: 1209.3516, 2012.
* [16] Abtin Rahimian, Ilya Lashuk, Shravan K. Veerapaneni, Aparna Chandramowlishwaran, Dhairya Malhotra, Logan Moon, Rahul Sampath, Aashay Shringarpure, Jeffrey Vetter, Richard Vuduc, Denis Zorin, and George Biros. Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures. In Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, New Orleans, Louisiana, USA, November 2010. IEEE.
* [17] Lexing Ying, George Biros, and Denis Zorin. A high-order 3d boundary integral equation solver for elliptic pdes in smooth domains. Journal of Comutational Physics, 219:247–275, 2006.
* [18] Chandrajit Bajaj, Shun-Chuan Chen, and Alexander Rand. An efficient higher-order fast multipole boundary element solution for poisson-boltzmann-based molecular electrostatics. SIAM J. Sci. Comput., 33(2):826–848, 2011.
* [19] Lexing Ying. kifmm3d. http://www.math.utexas.edu/users/lexing/software/index.html.
|
arxiv-papers
| 2012-11-12T06:27:22 |
2024-09-04T02:49:37.862369
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanchuang Cao, Lihua Wen, Junjie Rong",
"submitter": "Yanchuang Cao",
"url": "https://arxiv.org/abs/1211.2517"
}
|
1211.2633
|
S. F. Lukomskii
Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups
111This research was carried out with the financial support the Russian
Foundation for Basic Research (grant no. 10-01-00097).
(Russia, Saratov)
lukomskiisf@info.sgu.ru
Abstract. We find the necessary and sufficient conditions for refinable step
function under which this function generates an orthogonal MRA in the
$L_{2}(\mathfrak{G})$ -spaces on Vilenkin groups $\mathfrak{G}$. We consider a
class of refinable step functions for which the mask $m_{0}(\chi)$ is constant
on cosets $\mathfrak{G}_{-1}^{\bot}$ and its modulus $|m_{0}(\chi)|$ takes two
values only: 0 and 1. We will prove that any refinable step function $\varphi$
from this class that generates an orthogonal MRA on $p$-adic Vilenkin group
$\mathfrak{G}$ has Fourier transform with condition ${\rm
supp}\,\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$. We show the
sharpness of this result too.
2000 Math. subject classification.
Primary 65T60; Secondary 42C10, 43A75
Keywords: zero-dimensional groups, MRA, Vilenkin groups, refinable functions,
wavelet bases.
## Introduction
Foundations for wavelet analysis theory on locally compact groups have been
lay in the monograph [1]. In articles [2-4] first examples of orthogonal
wavelets on the dyadic Cantor group are constructed and their properties are
studied. The general scheme for the construction of wavelets is based on the
notion of multiresolution analysis (MRA in the sequel) introduced by Y. Meyer
and S. Mallat [5, 6]. Yu.Farkov [7-12] found necessary and sufficient
conditions for a refinable function under which this function generates an
orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on the Vilenkin group
$\mathfrak{G}$. These conditions use the Strang-Fix and the modified Cohen
properties. In [10] this construction to the p = 3 case in a concrete fashion
are given. In [11], some algorithms for constructing orthogonal and
biorthogonal compactly supported wavelets on Vilenkin groups are suggested. In
[7-11] two types of orthogonal wavelets examples are constructed: step
functions and sums of Vilenkin series.
In these examples all step refinable functions have a support ${\rm
supp}\,\hat{\varphi}(\chi)\subset\mathfrak{G}_{1}^{\bot}=\mathfrak{G}_{0}^{\bot}{\cal
A}$ where $\mathfrak{G}^{\perp}_{0}$ is the unit ball in the character group
and ${\cal A}$ is a dilation operator. Therefore there is an assumption that a
step refinable function which generates an orthogonal MRA on Vilenkin group
$\mathfrak{G}$ has a Fourier transform with support ${\rm
supp}\,\hat{\varphi}(\chi)\subset\mathfrak{G}_{1}^{\bot}$ . We will prove that
it is not true. We consider a class of refinable step functions for which the
mask $m_{0}(\chi)$ is constant on cosets $\mathfrak{G}_{-1}^{\bot}$ and its
modulus $|m_{0}(\chi)|$ takes two values only: 0 and 1. We will prove that any
refinable step function $\varphi$ from this class that generates an orthogonal
MRA on $p$-adic Vilenkin group $\mathfrak{G}$ has Fourier transform with
condition ${\rm supp}\,\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$.
We show the sharpness of this result too.
We should note that in the p-adic analysis, the situation is different. S.
Albeverio, S. Evdokimov, M. Skopina [12] proved, that if a refinable step
function $\varphi$ generates an orthogonal $p$-adic MRA, then
$\hat{\varphi}(\chi)\subset\mathfrak{G}_{0}^{\bot}$.
## 1 Preliminaries
We will consider the Velenkin group as a locally compact zero-dimensional
Abelian group with additional condition $p_{n}g_{n}=0$. Therefore we start
with some basic notions and facts related to analysis on zero-dimensional
groups. A topological group in which the connected component of 0 is 0 is
usually referred to as a zero-dimensional group. If a separable locally
compact group $(G,\dot{+})$ is zero-dimensional, then the topology on it can
be generated by means of a descending sequence of subgroups.
The converse statement holds for all topological groups (see [13, Ch. 1, §
3]). So, for a locally compact group, we are going to say ‘zero-dimensional
group’ instead of saying ‘a group with topology generated by a sequence
subgroups’.
Let $(G,\dot{+})$ be a locally compact zero-dimensional Abelian group with the
topology generated by a countable system of open subgroups
$\cdots\supset G_{-n}\supset\cdots\supset G_{-1}\supset G_{0}\supset
G_{1}\supset\cdots\supset G_{n}\supset\cdots$
where
$\bigcup_{n=-\infty}^{+\infty}G_{n}=G,\quad\quad\bigcap_{n=-\infty}^{+\infty}G_{n}=\\{0\\}$
(0 is the null element in the group $G$). Given any fixed $N\in\mathbb{Z}$,
the subgroup $G_{N}$ is a compact Abelian group with respect to the same
operation $\dot{+}$ under the topology generated by the system of subgroups
$G_{N}\supset G_{N+1}\supset\cdots\supset G_{n}\supset\cdots.$
As each subgroup $G_{n}$ is compact, it follows that each quotient group
$G_{n}/G_{n+1}$ is finite (say, of order $p_{n}$). We may always assume that
all $p_{n}$ are prime numbers. We will name such chain as basic chain. In this
case, a base of the topology is formed by all possible cosets $G_{n}\dot{+}g$,
$g\in G$.
We further define the numbers $(\mathfrak{m}_{n})_{n=-\infty}^{+\infty}$ as
follows:
$\mathfrak{m}_{0}=1,\qquad\mathfrak{m}_{n+1}=\mathfrak{m}_{n}\cdot p_{n}.$
Clearly, for $n\geq 1$,
$\mathfrak{m}_{n}=p_{0}p_{1}\cdots
p_{n-1},\qquad\mathfrak{m}_{-n}=\frac{1}{p_{-1}p_{-2}\cdots p_{-n}}.$
The collection of all such cosets $G_{n}\dot{+}g$, $n\in\mathbb{Z}$, along
with the empty set form the semiring ${\mathscr{K}}$. On each coset
$G_{n}\dot{+}g$ we define the measure $\mu$ by $\mu(G_{n}\dot{+}g)=\mu
G_{n}=1/{m_{n}}$. So, if $n\in\mathbb{Z}$ and $p_{n}=p$, we have $\mu
G_{n}\cdot\mu G_{-n}=1$. The measure $\mu$ can be extended from the semiring
${\mathscr{K}}$ onto the $\sigma$-algebra (for example, by using
Caratheodory’s extension). This gives the translation invariant measure $\mu$,
which agrees on the Borel sets with the Haar measure on $G$. Further, let
$\smash[b]{\displaystyle\int_{G}f(x)\,d\mu(x)}$ be the absolutely convergent
integral of the measure $\mu$.
Given an $n\in\mathbb{Z}$, take an element $g_{n}\in G_{n}\setminus G_{n+1}$
and fix it. Then any $x\in G$ has a unique representation of the form
$x=\sum_{n=-\infty}^{+\infty}a_{n}g_{n},\qquad a_{n}=\overline{0,p_{n}-1}.$
(1.1)
The sum (1.1) contain finite number of terms with negative subscripts, that
is,
$x=\sum_{n=m}^{+\infty}a_{n}g_{n},\qquad a_{n}=\overline{0,p_{n}-1},\quad
a_{m}\neq 0.$ (1.2)
We will name system $(g_{n})_{n\in\mathbb{Z}}$ as a basic system.
Classical examples of zero-dimensional groups are Vilenkin groups and groups
of $p$-adic numbers (see [13, Ch. 1, § 2]). A direct sum of cyclic groups
$Z(p_{k})$ of order $p_{k}$, $k\in\mathbb{Z}$, is called a Vilenkin group.
This means that the elements of a Vilenkin group are infinite sequences
$x=(x_{k})_{k=-\infty}^{+\infty}$ such that:
* 1)
$x_{k}=\overline{0,p_{k}-1}$;
* 2)
only a finite number of $x_{k}$ with negative subscripts are different from
zero;
* 3)
the group operation $\dot{+}$ is the coordinate-wise addition modulo $p_{k}$,
that is,
$x\dot{+}y=(x_{k}\dot{+}y_{k}),\qquad x_{k}\dot{+}y_{k}=(x_{k}+y_{k})\ \
\operatorname{mod}p_{k}.$
A topology on such group is generated by the chain of subgroups
$G_{n}=\bigl{\\{}x\in G:x=(\dots,0,0,\dots,0,x_{n},x_{n+1},\dots),\
x_{\nu}=\overline{0,p_{\nu}-1},\ \nu\geq n\bigr{\\}}.$
The elements $g_{n}=(\dots,0,0,1,0,0,\dots)$ form a basic system. From
definition of the operation $\dot{+}$ we have $p_{n}g_{n}=0$. Therefore we
will name a zero-dimensional group $(G,\dot{+})$ with the condition
$p_{n}g_{n}=0$ as Vilenkin group.
The group $\mathbb{Q}_{p}$ of all $p$-adic numbers ($p$ is a prime number)
also consists of sequences $x=(x_{k})_{k=-\infty}^{+\infty}$,
$x_{k}=\overline{0,p-1}$, only a finite number of $x_{k}$ with negative
subscripts being different from zero. However, the group operation in
$\mathbb{Q}_{p}$ is defined differently. Namely, given elements
$x=(\dots,0,\dots,0,x_{N},x_{N+1},\dots)\;\text{and}\;y=(\dots,0,\dots,0,y_{N},y_{N+1},\dots)\in\mathbb{Q}_{p},$
we again add them coordinate-wise, but whereas in a Vilenkin group
$x_{n}\dot{+}y_{n}=(x_{n}+y_{n})\ \operatorname{mod}p$ (that is, a 1 is not
carried to the next $(n+1)$th position), the corresponding $p$-adic summation
has the property that the 1 occuring as a result of the addition of
$x_{n}+y_{n}$ is carried to the next $(n+1)$th position. We endow the group
$\mathbb{Q}_{p}$ with the topology generated by the same system of subgroups
$G_{n}$ as for a Vilenkin group. Similarly, as a $(g_{n})$, we may again take
the same sequence.
By $X$ denote the collection of the characters of a group $(G,\dot{+})$; it is
a group with respect to multiplication too. Also let $G_{n}^{\bot}=\\{\chi\in
X:\forall\,x\in G_{n}\ ,\chi(x)=1\\}$ be the annihilator of the group $G_{n}$.
Each annihilator $G_{n}^{\bot}$ is a group with respect to multiplication, and
the subgroups $G_{n}^{\bot}$ form an increasing sequence
$\cdots\subset G_{-n}^{\bot}\subset\cdots\subset G_{0}^{\bot}\subset
G_{1}^{\bot}\subset\cdots\subset G_{n}^{\bot}\subset\cdots$ (1.3)
with
$\bigcup_{n=-\infty}^{+\infty}G_{n}^{\bot}=X\quad\text{and}\quad\bigcap_{n=-\infty}^{+\infty}G_{n}^{\bot}=\\{1\\},$
the quotient group $G_{n+1}^{\bot}/G_{n}^{\bot}$ having order $p_{n}$. The
group of characters $X$ may be equipped with the topology using the chain of
subgroups (1.3), the family of the cosets $G_{n}^{\bot}\cdot\chi$, $\chi\in
X$, being taken as a base of the topology. The collection of such cosets,
along with the empty set, forms the semiring ${\mathscr{X}}$. Given a coset
$G_{n}^{\bot}\cdot\chi$, we define a measure $\nu$ on it by
$\nu(G_{n}^{\bot}\cdot\chi)=\nu(G_{n}^{\bot})=m_{n}$ (so that always
$\mu(G_{n})\nu(G_{n}^{\bot})=1$). The measure $\nu$ can be extended onto the
$\sigma$-algebra of measurable sets in the standard way. One then forms the
absolutely convergent integral $\displaystyle\int_{X}F(\chi)\,d\nu(\chi)$ of
this measure.
The value $\chi(g)$ of the character $\chi$ at an element $g\in G$ will be
denoted by $(\chi,g)$. The Fourier transform $\widehat{f}$ of an $f\in
L_{2}(G)$ is defined as follows
$\widehat{f}(\chi)=\int_{G}f(x)\overline{(\chi,x)}\,d\mu(x)=\lim_{n\to+\infty}\int_{G_{-n}}f(x)\overline{(\chi,x)}\,d\mu(x),$
the limit being in the norm of $L_{2}(X)$. For any $f\in L_{2}(G)$, the
inversion formula is valid
$f(x)=\int_{X}\widehat{f}(\chi)(\chi,x)\,d\nu(\chi)=\lim_{n\to+\infty}\int_{G_{n}^{\bot}}\widehat{f}(\chi)(\chi,x)\,d\nu(\chi);$
here the limit also signifies the convergence in the norm of $L_{2}(G)$. If
$f,g\in L_{2}(G)$ then the Plancherel formula is valid
$\int_{G}f(x)\overline{g(x)}\,d\mu(x)=\int_{X}\widehat{f}(\chi)\overline{\widehat{g}(\chi)}\,d\nu(\chi).$
Endowed with this topology, the group of characters $X$ is a zero-dimensional
locally compact group; there is, however, a dual situation: every element
$x\in G$ is a character of the group $X$, and $G_{n}$ is the annihilator of
the group $G_{n}^{\bot}$.
The union of disjoint sets $E_{j}$ we will denote by $\bigsqcup E_{j}$.
## 2 Rademacher functions and dilation operator
In this section we will consider zero-dimensional groups with condition
$p_{n}=p$ for any $n\in\mathbb{Z}$. In this case we define the mapping ${\cal
A}\colon G\to G$ by ${\cal A}x:=\sum_{n=-\infty}^{+\infty}a_{n}g_{n-1}$, where
$x=\sum_{n=-\infty}^{+\infty}a_{n}g_{n}\in G$. The mapping ${\cal A}$ is
called a dilation operator if ${\cal A}(x\dot{+}y)={\cal A}x\dot{+}{\cal A}y$
for all $x,y\in G$. By definition, put $(\chi{\cal A},x)=(\chi,{\cal A}x)$. A
character $r_{n}\in G_{n+1}^{\bot}\backslash G_{n}^{\bot}$ is called the
Rademacher function. Let us denote
$H_{0}=\\{h\in
G:h=a_{-1}g_{-1}\dot{+}a_{-2}g_{-2}\dot{+}\dots\dot{+}a_{-s}g_{-s},s\in\mathbb{N}\\},$
$H_{0}^{(s)}=\\{h\in
G:h=a_{-1}g_{-1}\dot{+}a_{-2}g_{-2}\dot{+}\dots\dot{+}a_{-s}g_{-s}\\},s\in\mathbb{N}.$
The set $H_{0}$ is an analog of the set $\mathbb{N}$.
###### Lemma 2.1
For any zero-dimensional group
1) $\int\limits_{G_{0}^{\bot}}(\chi,x)\,d\nu(\chi)={\bf 1}_{G_{0}}(x)$, 2)
$\int\limits_{G_{0}}(\chi,x)\,d\mu(x)={\bf 1}_{G_{0}^{\bot}}(\chi)$.
The first equation it was proved in [14], the second equation is dual to
first.
###### Lemma 2.2
If $p_{n}=p$ for any $n\in\mathbb{Z}$ and the mapping ${\cal A}$ is additive
then
1) $\int\limits_{G_{n}^{\bot}}(\chi,x)\,d\nu(\chi)=p^{n}{\bf 1}_{G_{n}}(x)$,
2) $\int\limits_{G_{n}}(\chi,x)\,d\mu(x)=\frac{1}{p^{n}}{\bf
1}_{G_{n}^{\bot}}(\chi)$.
Proof. First we prove the equation 1). Using equations
$\int\limits_{X}f(\chi{\cal
A})\,d\nu(\chi)=p\int\limits_{X}f(\chi)\,d\nu(\chi),\;\;\;{\bf
1}_{G_{n}^{\bot}}(x)={\bf 1}_{G_{0}}({\cal A}^{n}x),$
and Lemma 2.1 we have
$\int\limits_{G_{n}^{\bot}}(\chi,x)\,d\nu(\chi)=\int\limits_{X}{\bf
1}_{G_{n}^{\bot}}(\chi)(\chi,x)\,d\nu(\chi)=p^{n}\int\limits_{X}(\chi{\cal
A}^{n},x){\bf 1}_{G_{n}^{\bot}}(\chi{\cal A}^{n})\,d\nu(\chi)=$
$=p^{n}\int\limits_{X}(\chi,{\cal A}^{n}x){\bf
1}_{G_{0}^{\bot}}(\chi)\,d\nu(\chi)=p^{n}{\bf 1}_{G_{0}}({\cal
A}^{n}x)=p^{n}{\bf 1}_{G_{n}}(x).$
The second equation is proved by analogy. $\square$
###### Lemma 2.3
Let $\chi_{n,s}=r_{n}^{\alpha_{n}}r_{n+1}^{\alpha_{n+1}}\dots
r_{n+s}^{\alpha_{n+s}}$ be a character does not belong to $G_{n}^{\bot}$. Then
$\int\limits_{G_{n}^{\bot}\chi_{n,s}}(\chi,x)\,d\nu(\chi)=p^{n}(\chi_{n,s},x){\bf
1}_{G_{n}}(x).$
Proof. By analogy with previously we have
$\int\limits_{G_{n}^{\bot}\chi_{n,s}}(\chi,x)\,d\nu(\chi)=\int\limits_{X}{\bf
1}_{G_{n}^{\bot}}(\chi)(\chi_{n,s}\chi,x)\,d\nu(\chi)=$
$\int\limits_{G_{n}^{\bot}}(\chi_{n,s},x)(\chi,x)\,d\nu(\chi)=p^{n}(\chi_{n,s},x){\bf
1}_{G_{n}}(x).\;\;\square$
###### Lemma 2.4
Let
$h_{n,s}=a_{n-1}g_{n-1}\dot{+}a_{n-2}g_{n-2}\dot{+}\dots\dot{+}a_{n-s}g_{n-s}\notin
G_{n}$. Then
$\int\limits_{G_{n}\dot{+}h_{n,s}}(\chi,x)\,d\mu(x)=\frac{1}{p^{n}}(\chi,h_{n,s}){\bf
1}_{G_{n}^{\bot}}(\chi).$
This lemma is dual to lemma 2.3.
###### Definition 2.1
Let $M,N\in\mathbb{N}$. Denote by ${\mathfrak{D}}_{M}(G_{-N})$ the set of
step-functions $f\in L_{2}(G)$ such that 1)${\rm supp}\,f\subset G_{-N}$, and
2) $f$ is constant on cosets $G_{M}$. Similarly is defined
${\mathfrak{D}}_{-N}(G_{M}^{\bot})$.
###### Lemma 2.5
Let $M,N\in\mathbb{N}$. $f\in\mathfrak{D}_{M}(G_{-N})$ if and only if
$\hat{f}\in\mathfrak{D}_{-N}(G_{M}^{\bot})$.
Proof. 1) Let $f$ be a constant on cosets $G_{M}\dot{+}g$ and ${\rm
supp}\,f\subset G_{-N}$. Let us show that ${\rm supp}\,\hat{f}\subset
G_{M}^{\bot}$. Let $\chi\notin G_{M}^{\bot}$. Then
$\hat{f}(\chi)=\int\limits_{G}f(x)\overline{(\chi,x)}\,d\mu(x)=\int\limits_{G_{-N}}f(x)\overline{(\chi,x)}\,d\mu(x)=$
$=\sum_{h_{M,N}\in
H_{M}^{N}}\int\limits_{G_{M}\dot{+}h_{M,N}}f(x)\overline{(\chi,x)}\,d\mu(x),$
where
$H_{M}^{N}=\\{h_{M,N}=a_{M-1}g_{M-1}\dot{+}a_{M-2}g_{M-2}\dot{+}\dots\dot{+}a_{-N}g_{-N}\\}.$
By lemma 2.4
$\hat{f}(\chi)=\sum
f(G_{M}\dot{+}h_{M,N})\int\limits_{G_{M}\dot{+}h_{M,N}}\overline{(\chi,x)}\,d\mu(x)=$
$=\sum f(G_{M}\dot{+}h_{M,N})\frac{1}{p^{M}}\overline{(\chi,h_{M,N})}{\bf
1}_{G_{M}^{\bot}}(\chi)=0.$
Now we will show that $\hat{f}$ is constant on cosets $G_{-N}^{\bot}\zeta$.
Indeed let $\chi\in G_{-N}^{\bot}\zeta$ and
$\zeta=r_{-N}^{\alpha_{-N}}r_{-N+1}^{\alpha_{-N+1}}\dots
r_{-N+s}^{\alpha_{-N+s}}$. Then $\chi=\chi_{-N}\zeta$ where $\chi_{-N}\in
G_{-N}^{\bot}$. Therefore
$\hat{f}(\chi)=\int\limits_{G_{-N}}f(x)\overline{(\chi,x)}\,d\mu(x)=\int\limits_{G_{-N}}f(x)\overline{(\chi_{-N}\zeta,x)}\,d\mu(x)=\int\limits_{G_{-N}}f(x)\overline{(\zeta,x)}\,d\mu(x).$
It means that $\hat{f}(\chi)$ depends only on $\zeta$. The first part is
proved. The second part is proved similarly. $\square$
###### Lemma 2.6
Let $\varphi\in L_{2}(G)$. The system $(\varphi(x\dot{-}h))_{h\in H_{0}}$ is
orthonormal if and only if the system $\left(p^{\frac{n}{2}}\varphi({\cal
A}^{n}x\dot{-}h)\right)_{h\in H_{0}}$ is orthonormal.
Proof. This lemma follows from the equation
$\int\limits_{G}p^{\frac{n}{2}}\varphi({\cal
A}^{n}x\dot{-}h)p^{\frac{n}{2}}\overline{\varphi({\cal
A}^{n}x\dot{-}g)}\,d\mu=\int\limits_{G}\varphi(x\dot{-}h)\overline{\varphi(x\dot{-}g)}\,d\mu.\;\;\square$
## 3 MRA on Vilenkin groups
In what follows we will consider groups $G$ for which $p_{n}=p$ and $pg_{n}=0$
for any $n\in\mathbb{Z}$. We now that it is a Vilenkin group. We will denote a
Vilenkin group as $\mathfrak{G}$. In this group we can chouse Rademacher
functions in various ways. We define Rademacher functions by the equation
$\left(r_{n},\sum_{k\in\mathbb{Z}}a_{k}g_{k}\right)=\exp\left(\frac{2\pi
i}{p}a_{n}\right).$
In this case
$(r_{n},g_{k})=\exp\left(\frac{2\pi i}{p}\delta_{nk}\right).$
Our main objective is to find a refinable step-function that generates an
orthogonal MRA on Vilenkin group.
###### Definition 3.1
A family of closed subspaces $V_{n}$, $n\in\mathbb{Z}$, is said to be a
multiresolution analysis of $L_{2}(\mathfrak{G})$ if the following axioms are
satisfied:
* A1)
$V_{n}\subset V_{n+1}$;
* A2)
${\vrule
width=0.0pt,depth=0.0pt,height=11.0pt}\overline{\bigcup_{n\in\mathbb{Z}}V_{n}}=L_{2}(\mathfrak{G})$
and $\bigcap_{n\in\mathbb{Z}}V_{n}=\\{0\\}$;
* A3)
$f(x)\in V_{n}$ $\Longleftrightarrow$ $f({\cal A}x)\in V_{n+1}$ (${\cal A}$ is
a dilation operator);
* A4)
$f(x)\in V_{0}$ $\Longrightarrow$ $f(x\dot{-}h)\in V_{0}$ for all $h\in
H_{0}$; ($H_{0}$ is analog of $\mathbb{Z}$).
* A5)
there exists a function $\varphi\in L_{2}(\mathfrak{G})$ such that the system
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ is an orthonormal basis for $V_{0}$.
A function $\varphi$ occurring in axiom A5 is called a scaling function.
Next we will follow the conventional approach. Let
$\varphi(x)\,{\in}\,L_{2}(\mathfrak{G})$, and suppose that
$(\varphi(x\dot{-}\nobreak h))_{h\in H_{0}}$ is an orthonormal system in
$L_{2}(\mathfrak{G})$. With the function $\varphi$ and the dilation operator
${\cal A}$, we define the linear subspaces $L_{j}=(\varphi({\cal
A}^{j}x\dot{-}h))_{h\in H_{0}}$ and closed subspaces $V_{j}=\overline{L_{j}}$.
It is evident that the functions $p^{\frac{n}{2}}\varphi({\cal
A}x\dot{-}h)_{h\in H_{0}}$ form an orthonormal basis for $V_{n}$,
$n\in\mathbb{Z}$. Therefore the axiom A4 is fulfilled. If subspaces $V_{j}$
form a MRA, then the function $\varphi$ is said to generate an MRA in
$L_{2}(\mathfrak{G})$. If a function $\varphi$ generates an MRA, then we
obtain from the axiom A1
$\varphi(x)=\sum_{h\in H_{0}}\beta_{h}\varphi({\cal
A}x\dot{-}h)\;\;\left(\sum|\beta_{h}|^{2}<+\infty\right).$ (3.1)
Therefore we will look up a function $\varphi\in L_{2}(\mathfrak{G})$, which
generates an MRA in $L_{2}(\mathfrak{G})$, as a solution of the refinement
equation (3.1), A solution of refinement equation (3.1) is called a refinable
function.
###### Lemma 3.1
Let $\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$ be a solution of (3.1).
Then
$\varphi(x)=\sum_{h\in H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}h)$
(3.2)
Proof. Let us write $\varphi(x)$ in the form
$\varphi(x)=\sum_{h\in H_{0}^{(N+1)}}\beta_{h}\varphi({\cal
A}x\dot{-}h)+\sum_{h\notin H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}h).$
(3.3)
If $x\in\mathfrak{G}_{-N}$, then ${\cal A}x\in\mathfrak{G}_{-N-1}$. Therefore
${\cal A}x=b_{-N-1}g_{-N-1}\dot{+}b_{-N}g_{-N}\dot{+}\dots$. If $h\notin
H_{0}^{(N+1)}$, then
$h=a_{-1}g_{-1}\dot{+}\dots\dot{+}a_{-N-1}g_{-N-1}\dot{+}a_{-N-2}g_{-N-2}\dot{+}\dots\dot{+}a_{-N-s}g_{-N-s},$
and $a_{-N-2}g_{-N-2}\dot{+}\dots\dot{+}a_{-N-s}g_{-N-s}\neq 0$. Hence ${\cal
A}x\dot{-}h\notin H_{0}^{(N+1)}$ and $\varphi({\cal A}x\dot{-}h)=0$. This
means that
$\sum_{h\notin H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}h)=0$
when $x\in\mathfrak{G}_{-N}$.
Let $x\notin\mathfrak{G}_{-N}$. Then $\varphi(x)=0$ and ${\cal
A}x\notin\mathfrak{G}_{-N-1}$. Hence
${\cal
A}x=\sum_{k=-N-s}^{-N-2}b_{k}g_{k}\dot{+}\sum_{k=-N-1}^{+\infty}b_{k}g_{k}.$
If $h\in H_{0}^{(N+1)}$, then
$h=a_{-1}g_{-1}\dot{+}\dots\dot{+}a_{-N}g_{-N}\dot{+}a_{-N-1}g_{-N-1}$, and
consequently ${\cal A}x\dot{-}h\notin\mathfrak{G}_{-N-1}$. Therefore
$\sum_{h\in H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}h)=0.$
Using equation (3.3) we obtain finally
$\sum_{h\notin H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}h)=0,$
and lemma is proved. $\square$
###### Theorem 3.2
Let $\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$ and let
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ be an orthonormal system. $V_{n}\subset
V_{n+1}$ if and only if the function $\varphi(x)$ is a solution of refinement
equation (3.2).
Proof. First we prove that $V_{n}\subset V_{n+1}$ if and only if $V_{0}\subset
V_{1}$. Indeed, let $V_{0}\subset V_{1}$ and $f\in V_{n}$. Then
$f(x)=\sum_{h}c_{h}\varphi({\cal A}^{n}x\dot{-}h)\Rightarrow f({\cal
A}^{-n}x)=\sum_{h}c_{h}\varphi(x\dot{-}h)\Rightarrow f({\cal A}^{-n}x)\in
V_{0}\Rightarrow$ $\Rightarrow f({\cal A}^{-n}x)\in V_{1}\Rightarrow f({\cal
A}^{-n}x)=\sum_{h}\gamma_{h}\varphi({\cal A}x\dot{-}h)\Rightarrow$
$\Rightarrow f(x)=\sum_{h}\gamma_{h}\varphi({\cal
A}^{n+1}x\dot{-}h)\Rightarrow f\in V_{n+1}.$
So we have, $V_{n}\subset V_{n+1}$. The converse is proved by analogy.
Now we prove that $V_{0}\subset V_{1}$ if and only if the function
$\varphi(x)$ is a solution of the refinement equation (3.2). The necessity is
evident. Let $\varphi$ be a solution of (3.2). We take $f\in{\rm
span}(\varphi(x\dot{-}h))_{h\in H_{0}}$. Then
$f(x)=\sum_{\tilde{h}\in H_{0}^{(m)}}c_{\tilde{h}}\varphi(x\dot{-}\tilde{h})$
for some $m\in\mathbb{N}$.
Since $\varphi$ is a solution of (3.2) then we can write $f$ in the form
$f(x)=\sum_{\tilde{h}\in H_{0}^{(m)}}c_{\tilde{h}}\sum_{h\in
H_{0}^{(N+1)}}\beta_{h}\varphi({\cal A}x\dot{-}({\cal A}\tilde{h}\dot{+}h)).$
Since $\tilde{h}\in H_{0}^{(m)}$ then ${\cal A}\tilde{h}\in H_{0}$. Therefore
${\cal A}\tilde{h}\dot{+}h\in H_{0}$. This means that $f\in{\rm
span}(\varphi({\cal A}x\dot{-}h))_{h\in H_{0}}$. It follows $V_{0}\subset
V_{1}$. $\square$
###### Theorem 3.3
Let $(\varphi(x\dot{-}h))_{h\in H_{0}}$ be an orthonormal basis in $V_{0}$.
Then $\bigcap\limits_{n\in\mathbb{Z}}V_{n}=\\{0\\}$.
Proof. Let $f\in V_{-n}$ for some $n\in\mathbb{N}$. Then $f({\cal A}^{n}x)\in
V_{0}$. Since the system $(\varphi(x\dot{-}h))_{h\in H_{0}}$ is orthonormal we
have the equality
$\frac{1}{p^{n}}\sum_{h\in
H_{0}}\left|\int\limits_{\mathfrak{G}}f(x)\varphi({\cal
A}^{-n}x\dot{-}h)\,d\mu\right|^{2}=\sum_{h\in
H_{0}}\left|\int\limits_{\mathfrak{G}}f({\cal
A}^{n}x)\varphi(x\dot{-}h)\,d\mu\right|^{2}=$ $=||f({\cal
A}^{n}x)||_{2}^{2}=\int\limits_{\mathfrak{G}}|f({\cal
A}^{n}x)|^{2}\,d\mu=\frac{1}{p^{n}}||f||_{2}^{2}.$
It is evident that $(p^{\frac{n}{2}}\varphi({\cal A}^{n}x\dot{-}h))_{h\in
H_{0}}$ is orthonormal basis in $V_{n}$. Therefore
$||f||_{2}^{2}=p^{n}\sum_{h\in
H_{0}}\left|\int\limits_{\mathfrak{G}}f(x)\varphi({\cal
A}^{-n}x\dot{-}h)\,d\mu\right|$
for $f\in V_{n}$. Combining these equations we obtain
$||f||_{2}^{2}=\frac{1}{p^{n}}\sum_{h\in
H_{0}}\left|\int\limits_{\mathfrak{G}}f(x)\varphi({\cal
A}^{-n}x\dot{-}h)\,d\mu\right|^{2}=\frac{1}{p^{n}}||f||_{2}^{2},$
for any $n\in\mathbb{N}$. It follows $f(x)=0$ a.e. $\square$
###### Theorem 3.4
Let $\varphi$ be a solution of the equation (3.2) and
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ an orthonormal basis in $V_{0}$. Then
$\overline{\bigcup\limits_{n\in\mathbb{Z}}V_{n}}=L_{2}(\mathfrak{G})$ if and
only if
$\bigcup\limits_{n\in\mathbb{Z}}{\rm supp}\,\hat{\varphi}(\cdot{\cal
A}^{-n})=X.$
Proof. This theorem is written in [14] for any zero-dimensional group under
the condition $|\hat{\varphi}|={\bf 1}_{\mathfrak{G}_{0}^{\perp}}$. But this
condition was used to get the inclusion $V_{n}\subset V_{n+1}$ only. By
theorems 3.2 the inclusion $V_{n}\subset V_{n+1}$ holds. Therefore the theorem
is true. $\square$
The refinement equation (3.2) may be written in the form
$\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal A}^{-1}),$ (3.4)
where
$m_{0}(\chi)=\frac{1}{p}\sum_{h\in H_{0}^{(N+1)}}\beta_{h}\overline{(\chi{\cal
A}^{-1},h)}$ (3.5)
is a mask of the equation (3.4).
###### Lemma 3.5
Let $\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$. Then the mask
$m_{0}(\chi)$ is constant on cosets $\mathfrak{G}_{-N}^{\bot}\zeta$.
Proof. We will prove that $(\chi,{\cal A}^{-1}h)$ are constant on cosets
$\mathfrak{G}_{-N}^{\bot}\zeta$. Without loss of generality, we can assume
that $\zeta=r_{-N}^{\alpha_{-N}}\dots
r_{-N+s}^{\alpha_{-N}+s}\notin\mathfrak{G}_{-N}^{\bot}$. If
$h=a_{-1}g_{-1}\dot{+}\dots\dot{+}a_{-N-1}g_{-N-1}\in H_{0}^{(N+1)}$
then
${\cal
A}^{-1}h=a_{-1}g_{0}\dot{+}\dots\dot{+}a_{-N-1}g_{-N}\in\mathfrak{G}_{-N}.$
If $\chi\in\mathfrak{G}_{-N}^{\bot}\zeta$ then $\chi=\chi_{-N}\zeta$ where
$\chi_{-N}\in\mathfrak{G}_{-N}^{\bot}$. Therefore $(\chi,{\cal
A}^{-1}h)=(\chi_{-N}\zeta,{\cal A}^{-1}h)=(\zeta,{\cal A}^{-1}h)$. This means
that $(\chi,{\cal A}^{-1}h)$ depends on $\zeta$ only. $\square$
###### Lemma 3.6
The mask $m_{0}(\chi)$ is a periodic function with any period
$r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\dots r_{s}^{\alpha_{s}}$
$(s\in\mathbb{N},\;\alpha_{j}=\overline{0,p-1},\;j=\overline{1,s})$.
Proof. Using the equation $(r_{k},g_{l})=1,(k\neq l)$ we find
$(\chi r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\dots r_{s}^{\alpha_{s}},{\cal
A}^{-1}h)=(\chi r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\dots
r_{s}^{\alpha_{s}},a_{-1}g_{0}\dot{+}a_{-2}g_{-1}\dot{+}\dots\dot{+}a_{-N-1}g_{-N})=$
$=(\chi,a_{-1}g_{0}\dot{+}a_{-2}g_{-1}\dot{+}\dots\dot{+}a_{-N-1}g_{-N})=(\chi{\cal
A}^{-1},h).$
Therefore $m_{0}(\chi r_{1}^{\alpha_{1}}\dots r_{s}^{\alpha_{s}})=m_{0}(\chi)$
and the lemma is proved. $\square$
###### Lemma 3.7
The mask $m_{0}(\chi)$ is defined by its values on cosets
$\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots r_{0}^{\alpha_{0}}$
$(\alpha_{j}=\overline{0,p-1})$.
Proof. Let us denote
$k=\alpha_{0}+\alpha_{-1}p+\dots+\alpha_{-N}p^{N}\in[0,p^{N+1}-1],$
$l=a_{-1}+a_{-2}p+\dots+a_{-N-1}p^{N}\in[0,p^{N+1}-1].$
Then (3.5) can be written as the system
$m_{0}(\chi_{k})=\frac{1}{p}\sum_{l=0}^{p^{N+1}-1}\beta_{l}\overline{(\chi_{k},{\cal
A}^{-1}h_{l})},\;k=\overline{0,p^{N+1}-1}$ (3.6)
in the unknowns $\beta_{l}$. We consider the characters $\chi_{k}$ on the
subgroup $\mathfrak{G}_{-N_{0}}$. Since ${\cal A}^{-1}h_{l}$ lie in
$\mathfrak{G}_{-N}$, it follows that the matrix
$p^{-\frac{N+1}{2}}\overline{(\chi_{k},{\cal A}^{-1}h_{l})}$ is unitary, and
so the system (3.6) has a unique solution for each finite sequence
$(m_{0}(\chi_{k}))_{k=0}^{p^{N+1}-1}$.
$\square$
Remark. The function $m_{0}(\chi)$ constructing in Lemma 3.7 may be not a mask
for $\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$. In the section 4 we find
conditions under which the function $m_{0}(\chi)$ will be a mask.
###### Lemma 3.8
Let $\hat{f}_{0}(\chi)\in{\mathfrak{D}}_{-N}(\mathfrak{G}_{1}^{\bot})$. Then
$\hat{f}_{0}(\chi)=\frac{1}{p}\sum_{h\in
H_{0}^{(N+1)}}\beta_{h}\overline{(\chi,{\cal A}^{-1}h)}.$ (3.7)
Prof. Since
$\int\limits_{\mathfrak{G}_{0}^{\bot}}(\chi,g)\overline{(\chi,h)}\,d\nu(\chi)=\delta_{h,g}$
for $h,g\in H_{0}$ it follows that
$\int\limits_{\mathfrak{G}_{0}^{\bot}}(\chi{\cal
A}^{-1},g)\overline{(\chi{\cal A}^{-1},h)}\,d\nu(\chi)=p\delta_{h,g}$.
Therefore we can consider the set $\left(\frac{{\cal
A}^{-1}h}{\sqrt{p}}\right)_{h\in H_{0}^{(N+1)}}$ as an orthonormal system on
$\mathfrak{G}_{1}^{\bot}$. We know (lemma 3.5) that $(\chi,{\cal A}^{-1}h)$ is
a constant on cosets $\mathfrak{G}_{-N}^{\bot}\zeta$. It is evident the
dimensional of $\mathfrak{D}_{-N}(\mathfrak{G}_{1}^{\bot})$ is equal to
$p^{N+1}$. Therefore the system $\left(\frac{{\cal
A}^{-1}h}{\sqrt{p}}\right)_{h\in H_{0}^{(N+1)}}$ is an orthonormal basis for
$\mathfrak{D}_{-N}(\mathfrak{G}_{1}^{\bot})$ and the equation (3.7) is valid.
$\square$
## 4 The main results. The statements and proofs
In this section we find the necessary and sufficient condition under which a
step function $\varphi(x)\in{\mathfrak{D}}_{M}(\mathfrak{G}_{-N})$ generates
an orthogonal MRA on the $p$-adic Vilenkin group. We will prove also that for
any $n\in\mathbb{N}$ there exists a step function $\varphi$ such that 1)
$\varphi$ generate an orthogonal MRA, 2) ${\rm
supp}\,\hat{\varphi}\subset\mathfrak{G}_{n}^{\perp}$, 3)
$\hat{\varphi}(\mathfrak{G}_{n}^{\bot}\setminus\mathfrak{G}_{n-1}^{\bot})\not\equiv
0$.
First we obtain a test under which the system of shifts
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ is an orthonormal system.
###### Theorem 4.1
Let $\varphi(x)\in{\mathfrak{D}}_{M}(\mathfrak{G}_{-N})$. A shift’s system
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ will be orthonormal if and only if for any
$\alpha_{-N},\alpha_{-N+1},\dots,\alpha_{-1}=\overline{(0,p-1)}$
$\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{M-1}=0}^{p-1}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}})|^{2}=1.$ (4.1)
Proof. First we prove that the system $(\varphi(x\dot{-}h))_{h\in H_{0}}$ will
be orthonormal if and only if
$\sum_{\alpha_{-N},\dots,\alpha_{0},\dots,\alpha_{M-1}}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{M-1}^{\alpha_{M-1}})|^{2}=p^{N}.$ (4.2)
and for any vector $(a_{-1},a_{-2},\dots,a_{-N})\neq(0,0,\dots,0),\
(a_{j}=0,p-1)$
$\sum_{\alpha_{-1},\dots,\alpha_{-N}}\exp\left(\frac{2\pi
i}{p}(a_{-1}\alpha_{-1}+a_{-2}\alpha_{-2}+\dots+a_{-N}\alpha_{-N})\right)\times$
$\times\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{M-1}}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{M-1}^{\alpha_{M-1}})|^{2}=0$ (4.3)
Let $(\varphi(x\dot{-}h))_{h\in H_{0}}$ be an orthonormal system. Using the
Plansherel equality and Lemma 2.3 we have
$\delta_{h_{1}h_{2}}=\int\limits_{\mathfrak{G}}\varphi(x\dot{-}h_{1})\overline{\varphi(x\dot{-}h_{2})}\,d\mu(x)=\int\limits_{\mathfrak{G}_{M}^{\bot}}|\hat{\varphi}(\chi)|^{2}(\chi,h_{2}\dot{-}h_{1})\,d\nu(\chi)=$
$=\sum_{\alpha_{-N},\dots,\alpha_{0},\dots,\alpha_{M-1}}\int\limits_{\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots
r_{M-1}^{\alpha_{M-1}}}|\hat{\varphi}(\chi)|^{2}(\chi,h_{2}\dot{-}h_{1})\,d\nu(\chi)=$
$=\sum_{\alpha_{-N},\dots,\alpha_{M-1}}|\hat{\varphi}(G_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots
r_{M-1}^{\alpha_{M-1}}|^{2}\int\limits_{\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots
r_{M-1}^{\alpha_{M-1}}}(\chi,h_{2}\dot{-}h_{1})\,d\nu(\chi)=$ $=p^{-N}{\bf
1}_{\mathfrak{G}_{-N}}(h_{2}\dot{-}h_{1})\times$
$\times\sum_{\alpha_{-N},\dots,\alpha_{M-1}}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}}|^{2}(r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}},h_{2}\dot{-}h_{1}).$
If $h_{2}=h_{1}$, we obtain the equality (4). If $h_{2}\neq h_{1}$ then
$h_{2}\dot{-}h_{1}=a_{-1}g_{-1}\dot{+}\dots\dot{+}a_{-N}g_{-N}\in\mathfrak{G}_{-N}$
(4.4)
or
$h_{2}\dot{-}h_{1}=a_{-1}g_{-1}\dot{+}\dots\dot{+}a_{-N}g_{-N}\dot{+}\dots\dot{+}a_{-s}g_{-s}\in\mathfrak{G}\backslash\mathfrak{G}_{-N}.$
(4.5)
If the condition (4.5) are fulfilled, then ${\bf
1}_{\mathfrak{G}_{-N}}(h_{2}\dot{-}h_{1})=0$. If the condition (4.4) are
fulfilled, then
${\bf 1}_{\mathfrak{G}_{-N}}(h_{2}\dot{-}h_{1})=1,$
$(r_{-N}^{\alpha_{-N}}\dots r_{0}^{\alpha_{0}}\dots
r_{M-1}^{\alpha_{M-1}},h_{2}\dot{-}h_{1})=(r_{-N},g_{-N})^{a_{-N}\alpha_{-N}}\dots(r_{-1},g_{-1})^{a_{-1}\alpha_{-1}}.$
Using the equality $(r_{n},g_{n})=e^{\frac{2\pi i}{p}}$ we obtain the equality
(4).The conversely may be proved by analogy.
Let as show now if for any vector
$(a_{-1},a_{-2},\dots,a_{-N})\neq(0,0,\dots,0)$ the conditions (4.2) (4) are
fulfilled, then for any
$\alpha_{-N},\alpha_{-N+1},\dots,\alpha_{-1}=\overline{0,p-1}$
$\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{M-1}}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}})|^{2}=1.$ (4.6)
Let us denote
$n=\sum_{j=1}^{N}a_{-j}p^{j-1},\;\;k=\sum_{j=1}^{N}\alpha_{-j}p^{j-1},\;\;C_{n,k}=e^{\frac{2\pi
i}{p}\left(\sum_{j=1}^{N}\alpha_{-j}a_{-j}\right)}.$
and write the equalities (4.2) (4) as the system
$\begin{array}[]{l}C_{0,0}x_{0}+C_{0,1}x_{1}+\dots+C_{0,p^{N}-1}x_{p^{N}-1}=p^{N}\\\
C_{1,0}x_{0}+C_{1,1}x_{1}+\dots+C_{1,p^{N}-1}x_{p^{N}-1}=0\\\
\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\\
C_{p^{N}-1,0}x_{0}+C_{p^{N}-1,1}x_{1}+\dots+C_{p^{N}-1,p^{N}-1}x_{p^{N}-1}=0\\\
\end{array}$ (4.7)
with unknowns
$x_{k}=\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{M-1}}|\hat{\varphi}(G_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}})|^{2}.$
The matrix $(C_{n,k})$ is orthogonal. Indeed, if
$(a_{-1},a_{-2},\dots,a_{-N})\neq(a^{\prime}_{-1},a^{\prime}_{-2},\dots,a^{\prime}_{-N})$,
i.e., $k\neq n^{\prime}$ we obtain
$\sum_{k=0}^{p^{N}-1}C_{n,k}\overline{C_{n^{\prime},k}}=\sum_{\alpha_{-1},\dots,\alpha_{-N}}\exp\left(\frac{2\pi
i}{p}((a_{-1}-a^{\prime}_{-1})\alpha_{-1}+(a_{-N}-a^{\prime}_{-N})\alpha_{-N})\right)=0,$
so at least one of differences $a_{-l}-a^{\prime}_{-l}\neq 0$. So, the system
(4.7) has unique solution. It is evident that $x_{k}=1$ is a solution of this
system. This means that (4.6) is fulfil, and the necessity is proved. The
sufficiency is evident. $\square$
Now we obtain a necessary and sufficient conditions for function $m_{0}(\chi)$
to be a mask on the class $\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$, i.e.
there exists $\hat{\varphi}\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$ for
which
$\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal A}^{-1}).$ (4.8)
If $m_{0}(\chi)$ is a mask of (4.8) then
T1) $m_{0}(\chi)$ is constant on cosets $\mathfrak{G}_{-N}^{\bot}\zeta$,
T2) $m_{0}(\chi)$ is periodic with any period
$r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\dots r_{s}^{\alpha_{s}}$,
$\alpha_{j}=\overline{0,p-1}$,
T3) $m_{0}(\mathfrak{G}_{-N}^{\bot})=1$.
Therefore we will assume that $m_{0}$ satisfies these conditions. Let
$E_{k}\subset\mathfrak{G}_{k}^{\bot}\setminus\mathfrak{G}_{k-1}^{\bot}\;\
,(k=-N+1,-N+2,\dots,0,1,\dots,M,M+1)$
be a set, on which $m_{0}(E_{k})=0$. Since $m_{0}(\chi)$ is constant on cosets
$\mathfrak{G}_{-N}^{\bot}\zeta$, it follows that $E_{k}$ is a union of such
cosets or $E_{k}=\emptyset$.
###### Theorem 4.2
$m_{0}(\chi)$ is a mask of some equation on the class
$\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$ if and only if
$\bigcup\limits_{k=-N+1}^{M+1}E_{k}{\cal
A}^{M+1-k}=\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot}.$ (4.9)
Proof. Since $m_{0}(\chi)=1$ on $\mathfrak{G}_{N}$ it follows that
$m_{0}(\chi{\cal A}^{-M-N})=1$ for $\chi\in\mathfrak{G}_{M}^{\bot}$. Therefore
$m_{0}(\chi)$ will be a mask if and only if
$m_{0}(\chi)m_{0}(\chi{\cal A}^{-1})\dots m_{0}(\chi{\cal A}^{-M-N})=0$ (4.10)
on $\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot}$. Indeed, if
(4.10) is true we set
$\hat{\varphi}(\chi)=\prod\limits_{k=0}^{\infty}m_{0}(\chi{\cal
A}^{-k})\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot}).$
Then $\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal A}^{-1})$ and
$m_{0}(\chi)=\sum_{h\in H_{0}^{(N+1)}}\beta_{h}\overline{(\chi{\cal
A}^{-1},h)}$
for some $\beta_{h}$. Therefore $m_{0}(\chi)$ is a mask. Inversely let
$m_{0}(\chi)$ be a mask, i.e.
$\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal
A}^{-1})\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$. From it we find
$\hat{\varphi}(\chi)=m_{0}(\chi)m_{0}(\chi{\cal A}^{-1})\dots m_{0}(\chi{\cal
A}^{-M-N})\hat{\varphi}(\chi{\cal A}^{-M-N-1}),$
and $\hat{\varphi}(\chi{\cal A}^{-M-N-1})=1$ on $\mathfrak{G}_{M+1}^{\bot}$.
Since $\hat{\varphi}(\chi)=0$ on
$\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot}$, it follows
$m_{0}(\chi)m_{0}(\chi{\cal A}^{-1})\dots m_{0}(\chi{\cal A}^{-M-N})=0$
on $\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot}$.
To conclude the proof, it remains to note that for any $-N+1\leq k\leq M+1$
the inclusion $E_{k}{\cal
A}^{-k+M+1}\subset\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot}$
is true. Therefore the equation (4.9) is fulfil if and only if the equation
(4.10) is true. $\square$
###### Lemma 4.3
Let $\hat{\varphi}\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$ be a solution
of the refinement equation
$\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal A}^{-1}).$
Then for any $\alpha_{-N},\alpha_{-N+1},\dots,\alpha_{-1}=\overline{0,p-1}$
$\sum_{\alpha_{0}=0}^{p-1}|m_{0}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}r_{-N+1}^{\alpha_{-N+1}}\dots
r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})|^{2}=1.$ (4.11)
Proof. Since $\hat{\varphi}\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$, it
follows that
$\hat{\varphi}(\mathfrak{G}_{M+1}^{\bot}\setminus\mathfrak{G}_{M}^{\bot})=0$.
Using theorem 4.1 we have
$1=\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{M-1}=0}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots r_{M-1}^{\alpha_{M-1}})|^{2}=$
$=\sum_{\alpha_{0},\dots,\alpha_{M-1},\alpha_{M}=0}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}\dots
r_{M-1}^{\alpha_{M-1}}r_{M}^{\alpha_{M}})|^{2}=\sum_{\alpha_{0}=0}^{p-1}|m_{0}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}})|^{2}$
$\cdot\sum_{\alpha_{1},\dots,\alpha_{M-1},\alpha_{M}=0}|\hat{\varphi}(\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}+1}\dots
r_{-1}^{\alpha_{0}}r_{0}^{\alpha_{1}}\dots
r_{M-2}^{\alpha_{M-1}}r_{M-1}^{\alpha_{M}})|^{2}=$
$=\sum_{\alpha_{0}=0}^{p-1}|\mathfrak{G}_{-N}^{\bot}r_{-N}^{\alpha_{-N}}\dots
r_{0}^{\alpha_{0}}|^{2}.\;\;\square$
Corollary. If $N=1$ and
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})=\lambda_{\alpha_{-1}+\alpha_{0}p}$
then we can write the equations (4.11) in the form
$\sum_{\alpha_{0}=0}^{p-1}|\lambda_{\alpha_{-1}+\alpha_{0}p}|^{2}=1.$ (4.12)
###### Theorem 4.4
Suppose the function $m_{0}(\chi)$ satisfies the conditions T1,T2,T3, (4.10),
and the function
$\hat{\varphi}(\chi)=\prod\limits_{n=0}^{\infty}m_{0}(\chi{\cal A}^{-n})$
satisfies the condition (4.1). Then
$\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$ generates an orthogonal MRA.
Proof. It is evident that
$\hat{\varphi}\in\mathfrak{D}_{-N}(\mathfrak{G}_{M}^{\bot})$,
$\hat{\varphi}(\chi)=m_{0}(\chi)\hat{\varphi}(\chi{\cal A}^{-1})$ and
$(\varphi(x\dot{-}h))_{h\in H_{0}}$ is an orthonormal system. From theorems
3.4, 3.3, 3.2 we find that the function $\varphi$ generates an orthogonal MRA.
$\square$
###### Definition 4.1
A mask $m_{0}(\chi)$ is called $N$-elementary $(N\in\mathbb{N})$ if it is
constant on cosets $\mathfrak{G}_{-N}^{\bot}\chi$ and its modulus
$|m_{0}(\chi)|$ take two values:0 and 1 only. The refinable function $\varphi$
with Fourier transform
$\hat{\varphi}(\chi)=\prod\limits_{j=0}^{\infty}m_{0}(\chi{\cal A}^{-1})$
is called $N$-elementary too.
###### Theorem 4.5
Let $m_{0}(\chi)$ be an $1$-elementary mask such that
$\sum\limits_{\alpha_{0}=0}^{p-1}|m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})|^{2}=1$
for any $\alpha_{-1}=\overline{0,p-1}$. Let us denote
$E_{0}^{(0)}=\\{\alpha=\overline{0,p-1}:\;m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha})=0\\}$
and $l=\sharp E_{0}^{(0)}$, $0\leq l\leq p-2$. If
$\hat{\varphi}(\chi)=\prod\limits_{j=0}^{\infty}m_{0}(\chi{\cal A}^{-j})$,
then
$\hat{\varphi}(\mathfrak{G}_{l+1}^{\bot}\setminus\mathfrak{G}_{l}^{\bot})=0$.
Proof. Since
$\mathfrak{G}_{l+1}^{\bot}\setminus\mathfrak{G}_{l}^{\bot}=\bigsqcup\limits_{\alpha_{l}=1}^{p-1}\bigsqcup\limits_{\alpha_{l-1},\dots,\alpha_{-1}=0}^{p-1}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})$
we need prove that
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})=0$
for $\alpha_{l}=\overline{1,p-1}$;
$\alpha_{-1},\dots,\alpha_{l-1}=\overline{0,p-1}$. Using a periodicity of
$\varphi$ we can write
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})=$
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})\hat{\varphi}(\mathfrak{G}_{-2}^{\bot}r_{-2}^{\alpha_{-1}}r_{-1}^{\alpha_{0}}\dots
r_{l-2}^{\alpha_{l-1}}r_{l-1}^{\alpha_{l}})=$
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{0}}r_{0}^{\alpha_{1}}\dots
r_{l-2}^{\alpha_{l-1}}r_{l-1}^{\alpha_{l}})=\dots=$
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{0}\alpha_{1}})\dots
m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l-1}}r_{0}^{\alpha_{l}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l}}).$
Let us denote
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{k}r_{0}^{j})=\lambda_{k+jp}$ and write
$\hat{\varphi}$ in the form
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})=\lambda_{\alpha_{-1}+\alpha_{0}p}\cdot\lambda_{\alpha_{0}+\alpha_{1}p}\dots\lambda_{\alpha_{l-1}+\alpha_{l}p}\cdot\lambda_{\alpha_{l}}.$
We will consider numbers $\lambda_{k+jp}$ as elements of the matrix
$\Lambda=(\lambda_{j,k})$, where $j$ is a number of a line, $k$ is a number of
a column. Let us consider the product
$\Pi=\lambda_{\alpha_{-1}+\alpha_{0}p}\cdot\lambda_{\alpha_{0}+\alpha_{1}p}\dots\lambda_{\alpha_{l-2}+\alpha_{l-1}p}\cdot\lambda_{\alpha_{l-1}+\alpha_{l}p}\cdot\lambda_{\alpha_{l}}\;\;(\alpha_{l}\neq
0).$
We need prove that $\Pi=0$ for $\alpha_{j}=\overline{0,p-1}$,
$j=\overline{-1,l-1}$ and for $\alpha_{l}=\overline{1,p-1}$.
If $\alpha_{l}\in E_{0}^{(0)}$, then $\lambda_{\alpha_{l}}=0$ and $\Pi=0$. Let
$\alpha_{l}\in E_{0}^{(1)}$ and $\alpha_{l}\neq 0$.
If $\lambda_{\alpha_{l-1}+\alpha_{l}p}=0$, then $\Pi=0$ and theorem is proved.
Therefore we assume $|\lambda_{\alpha_{l-1}+\alpha_{l}p}|=1$. In this case
$\alpha_{l-1}\in E_{0}^{(0)}$ and $\alpha_{l-1}=0$. Let us consider
$\lambda_{\alpha_{l-2}+\alpha_{l-1}p}$. If
$\lambda_{\alpha_{l-2}+\alpha_{l-1}p}=0$ then $\Pi=0$. Therefore we assume
$|\lambda_{\alpha_{l-2}+\alpha_{l-1}p}|=1$. In this case $\alpha_{l-1}\in
E_{0}^{(0)}$ and $\alpha_{l-1}\neq\alpha_{l-1}$. Let us consider
$\lambda_{\alpha_{l-3}+\alpha_{l-2}p}$. If
$\lambda_{\alpha_{l-3}+\alpha_{l-2}p}=0$ then $\Pi=0$ and the theorem is
proved. Therefore we assume $|\lambda_{\alpha_{l-3}+\alpha_{l-2}p}|=1$. In
this case $\alpha_{l-3}\in E_{0}^{(0)}$ and
$\alpha_{l-3}\notin\\{\alpha_{l-1},\alpha_{l-2}\\}$.
In the general case, if
$|\lambda_{\alpha_{l-s}+\alpha_{l-s+1}p}|\cdot|\lambda_{\alpha_{l-s+1}+\alpha_{l-s+2}p}|\dots|\lambda_{\alpha_{l-1}+\alpha_{l}p}|\cdot|\lambda_{\alpha_{l}}|=1$
and
$\alpha_{l-s}\notin\\{\alpha_{l-s+1},\alpha_{l-s+2},\dots,\alpha_{l-1}\\},\;\;\alpha_{l-s}\in
E_{0}^{(0)}$
then we consider $\lambda_{\alpha_{l-s-1}+\alpha_{l-s}p}$. If
$|\lambda_{\alpha_{l-s-1}+\alpha_{l-s}p}|=0$ then $\Pi=0$ and the theorem is
proved. If $|\lambda_{\alpha_{l-s-1}+\alpha_{l-s}p}|=1$ then
$\alpha_{l-s-1}\notin\\{\alpha_{l-s},\alpha_{l-s+1},\dots,\alpha_{l-1}\\},\;\;\alpha_{l-s-1}\in
E_{0}^{(0)}.$
We have two possible cases.
1) For some $s\leq l$
$\lambda_{\alpha_{l-s}+\alpha_{l-s+1}p}\cdot\lambda_{\alpha_{l-s+1}+\alpha_{l-s+2}p}\dots\lambda_{\alpha_{l-1}+\alpha_{l}p}\cdot\lambda_{\alpha_{l}}=0.$
In this case $\Pi=0$, and the theorem is proved.
2) For $s=l$
$|\lambda_{\alpha_{0}+\alpha_{1}p}|\cdot|\lambda_{\alpha_{1}+\alpha_{2}p}|\dots|\lambda_{\alpha_{l-1}+\alpha_{l}p}|\cdot|\lambda_{\alpha_{l}}|=1.$
In this case $\lambda_{\alpha_{-1}+\alpha_{0}p}=0$ for
$\alpha_{-l}=\overline{0,p-1}$, then $\Pi=0$ and the theorem is proved.
$\square$
Remark. If $l=p-1$, then
$m_{0}(\mathfrak{G}_{0}^{\bot}\setminus\mathfrak{G}_{-1}^{\bot})\equiv 0$. It
follow
$\hat{\varphi}(\mathfrak{G}_{0}^{\bot}\setminus\mathfrak{G}_{-1}^{\bot})$ and
consequently ${\rm supp}\,\hat{\varphi}(\chi)=\mathfrak{G}_{-1}^{\bot}$. In
this case the system of shift $(\varphi(x\dot{-}h))_{h\in H_{0}}$ is not
orthonormal system.
If $l=0$, then $|m_{0}(\mathfrak{G}_{0}^{\bot})|\equiv 1$ and the system of
shifts $(\varphi(x\dot{-}h))_{h\in H_{0}}$ will be orthonormal if and only if
$\hat{\varphi}(\mathfrak{G}_{1}^{\bot}\setminus\mathfrak{G}_{0}^{\bot})\equiv
0$. In this case $\varphi$ generate an orthogonal MRA on any zero-dimensional
group [14].
Corollary. Let $\varphi\in\mathfrak{D}_{M}(\mathfrak{G}_{-N})$ be an
$1$-elementary refinable function and $\varphi$ generate an orthogonal MRA on
$p$-adic Vilenkin group $\mathfrak{G}$ with $p\geq 3$. Then ${\rm
supp}\,\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$.
The next theorem shows the sharpness of this result.
###### Theorem 4.6
Let $\mathfrak{G}$ – be a $p$-adic Vilenkin group, $p\geq 3$. Then for any
$1\leq l\leq p-2$ there exists an $1$-elementary refinable function
$\varphi\in{\mathfrak{D}_{l}(\mathfrak{G}_{-1})}$ that generate an orthogonal
MRA on group $\mathfrak{G}$.
Proof. We will find the Fourier transform $\hat{\varphi}$ as product
$\hat{\varphi}(\chi)=\prod\limits_{j=0}^{\infty}m_{0}(\chi{\cal A}^{-j}),$
where the $1$-elementary mask $m_{0}(\chi)$ is constant on cosets
$\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{s}^{\alpha_{s}}$ $(s\in\mathbb{N}\bigsqcup\\{0\\})$. We will construct the
mask $m_{0}(\chi)$ on the subgroup $\mathfrak{G}_{1}^{\bot}$ only, since
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{s}^{\alpha_{s}})=m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})$.
We will assume also that for any $\alpha_{-1}=\overline{0,p-1}$
$\sum_{\alpha_{0}=0}^{p-1}|m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})|^{2}=1,$
(4.13)
since this condition is necessary for mask $m_{0}(\chi)$.
Choose an arbitrary set $E_{l}^{(0)}\subset\\{1,2,\dots,p-1\\}$ of cardinality
$\sharp E_{l}^{(0)}=l$. Let us denote
$E_{l}^{(1)}=\\{1,2,\dots,p-1\\}\setminus E_{l}^{(0)}$ and
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})=\lambda_{\alpha_{-1}+\alpha_{0}p}$.
First we set
$\lambda_{0}=1,\;\;|\lambda_{\alpha}|=\left\\{\begin{array}[]{ll}0,&\alpha\in
E_{l}^{(0)},\\\ 1,&\alpha\in E_{l}^{(1)}\\\ \end{array}\right..$
Now we will define $\lambda_{\alpha_{-1}+\alpha_{0}p}$ for $\alpha_{0}\geq 1$.
It follow from (4.13) that $\lambda_{\alpha_{-1}+\alpha_{0}p}=0$ for
$\alpha_{-1}\in E_{l}^{(1)}$, $\alpha_{0}\geq 1$. Choose an arbitrary
$\alpha_{l-1}^{(0)}\in E_{l}^{(1)}$ and fix it. Now we choose
$\alpha_{l-2}^{(0)}\in E_{l}^{(0)}$ and set
$|\lambda_{\alpha_{l-2}^{(0)}+\alpha_{l-1}^{(0)}p}|=1,\;\;|\lambda_{\alpha_{l-2}^{(0)}+\alpha
p}|=0\;\mbox{if}\;\alpha\neq\alpha_{l-1}^{(0)}.$
If numbers $\alpha_{l-2}^{(0)},\dots,\alpha_{s}^{(0)}\in E_{l}^{(0)}$
$(s=l-1,l-2,\dots,0)$ have been choosen we choose $\alpha_{s-1}^{(0)}\in
E_{l}^{(0)}\setminus\\{\alpha_{l-2}^{(0)},\dots,\alpha_{s}^{(0)}\\}$ and set
$|\lambda_{\alpha_{s-1}^{(0)}+\alpha_{s}^{(0)}p}|=1,\;\;|\lambda_{\alpha_{s-1}^{(0)}+\alpha
p}|=0\;\mbox{if}\;\alpha\neq\alpha_{s}^{(0)}.$
So the mask $m_{0}(\chi)$ have been defined on the subgroup
$\mathfrak{G}_{1}^{\bot}$ and consequently on the group $\mathfrak{G}$.
It is evident that
$\lambda_{\alpha_{-1}^{(0)}+\alpha_{0}^{(0)}p}\cdot\lambda_{\alpha_{0}^{(0)}+\alpha_{1}^{(0)}p}\dots\lambda_{\alpha_{l-2}^{(0)}+\alpha_{l-1}^{(0)}p}\cdot\lambda_{\alpha_{l-1}^{(0)}}\neq
0.$
Let us show that for any vector
$(\alpha_{-1},\alpha_{0},\dots,\alpha_{l-1})\neq(\alpha_{-1}^{(0)},\alpha_{0}^{(0)},\dots,\alpha_{l-1}^{(0)})$
$\lambda_{\alpha_{-1}+\alpha_{0}p}\cdot\lambda_{\alpha_{0}+\alpha_{1}p}\dots\lambda_{\alpha_{l-2}+\alpha_{l-1}p}\cdot\lambda_{\alpha_{l-1}}=0.$
(4.14)
Indeed, if $\alpha_{l-1}\in E_{l}^{(0)}$ then $\lambda_{\alpha_{l-1}}=0$. If
$\alpha_{l-1}\in E_{l}^{(1)}$ and $\alpha_{l-1}\neq\alpha_{l-1}^{(0)}$ then
$\lambda_{\alpha_{l-2}+\alpha_{l-1}p}=0$. If $\alpha_{l-1}\in E_{l}^{(1)}$ and
$\alpha_{l-1}=\alpha_{l-1}^{(0)}$ then we denote
$s=\min\\{j:\;\alpha_{j}=\alpha_{j}^{(0)}\\}.$
For this $s$ we have $\lambda_{\alpha_{s-1}+\alpha_{s}^{(0)}p}=0$ and the
equality (4.14) is proved. It should be noted that
$\lambda_{\alpha+\alpha_{-1}^{(0)}p}=0$ for $\alpha=\overline{0,p-1}$.
Therefore
$\lambda_{\alpha+\alpha_{-1}^{(0)}p}\cdot\lambda_{\alpha_{-1}^{(0)}+\alpha_{0}^{(0)}p}\dots\lambda_{\alpha_{l-2}^{(0)}+\alpha_{l-1}^{(0)}p}\cdot\lambda_{\alpha_{l-1}^{(0)}}=0.$
(4.15)
Let us show that
$\hat{\varphi}(\mathfrak{G}_{l}^{\bot}\setminus\mathfrak{G}_{l-1}^{\bot})\not\equiv
0$ and
$\hat{\varphi}(\mathfrak{G}_{l+1}^{\bot}\setminus\mathfrak{G}_{l}^{\bot})\equiv
0$. Since $m_{0}(\chi)$ is periodic with any period
$r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\dots r_{s}^{\alpha_{s}}$, it follow that
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}})=$
$m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{0}}r_{0}^{\alpha_{1}}\dots
r_{l-1}^{\alpha_{l-1}})\dots
m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l-2}}r_{0}^{\alpha_{l-1}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l-1}})=$
$=m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{0}}r_{0}^{\alpha_{1}})\dots
m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l-2}}r_{0}^{\alpha_{l-1}})m_{0}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{l-1}})=$
$=\lambda_{\alpha_{-1}+\alpha_{0}p}\cdot\lambda_{\alpha_{0}+\alpha_{1}p}\cdot\dots\cdot\lambda_{\alpha_{l-2}+\alpha_{l-1}p}\cdot\lambda_{\alpha_{l-1}}\neq
0$
for
$(\alpha_{-1},\alpha_{0},\dots,\alpha_{l-2},\alpha_{l-1})=(\alpha_{-1}^{(0)},\alpha_{0}^{(0)},\dots,\alpha_{l-2}^{(0)},\alpha_{l-1}^{(0)})$.
This means that
$\hat{\varphi}(\mathfrak{G}_{l}^{\bot}\setminus\mathfrak{G}_{l-1}^{\bot})\not\equiv
0$.
By analogy
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}}r_{l}^{\alpha_{l}})=\lambda_{\alpha_{-1}+\alpha_{0}p}\cdot\lambda_{\alpha_{0}+\alpha_{1}p}\cdot\dots\cdot\lambda_{\alpha_{l-1}+\alpha_{l}p}\cdot\lambda_{\alpha_{l}}.$
If $\alpha_{l}\in E_{l}^{(0)}$ then $\lambda_{\alpha_{l}}=0$. If
$\alpha_{l}\in E_{l}^{(1)}$ and $\alpha_{l}\neq\alpha_{l-1}^{(0)}$ then
$\lambda_{\alpha_{l-1}+\alpha_{l}p}=0$ for any
$\alpha_{l-1}=\overline{0,p-1}$. If $\alpha_{l}\in E_{l}^{(1)}$ and
$\alpha_{l}=\alpha_{l-1}^{(0)}$ we define the number
$s=\min\\{j:\;\alpha_{j}=\alpha_{j-1}^{(0)}\\}.$
Then
$\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l}^{\alpha_{l}})=\lambda_{\alpha_{-1}+\alpha_{0}p}\dots\lambda_{\alpha_{s-1}+\alpha_{s-1}^{(0)}p}\lambda_{\alpha_{s-1}^{(0)}+\alpha_{s-2}^{(0)}p}\dots\lambda_{\alpha_{l-2}^{(0)}+\alpha_{l-1}^{(0)}p}\cdot\lambda_{\alpha_{l-1}^{(0)}}=0$
since $\lambda_{\alpha_{s-1}+\alpha_{s-1}^{(0)}p}=0$ for any
$\alpha_{s-1}=\overline{0,p-1}$. This means that
$\hat{\varphi}(\mathfrak{G}_{l+1}^{\bot}\setminus\mathfrak{G}_{l}^{\bot})\equiv
0$. Consequently
$\hat{\varphi}\in{\mathfrak{D}}_{-1}(\mathfrak{G}_{l}^{\bot})$.
Let us show that $(\varphi(x\dot{-}h))_{h\in H_{0}}$ is an orthonormal system.
We need show that the sum
$S{(\alpha_{-1})}=\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{l-1}=0}^{p-1}|\hat{\varphi}(\mathfrak{G}_{-1}^{\bot}r_{-1}^{\alpha_{-1}}r_{0}^{\alpha_{0}}\dots
r_{l-1}^{\alpha_{l-1}})|^{2}=$
$=\sum_{\alpha_{0},\alpha_{1},\dots,\alpha_{l-1}=0}^{p-1}|\lambda_{\alpha_{-1}+\alpha_{0}p}|^{2}|\lambda_{\alpha_{0}+\alpha_{1}p}|^{2}\dots|\lambda_{\alpha_{l-2}+\alpha_{l-1}p}|^{2}|\lambda_{\alpha_{l-1}}|^{2}=1$
for any $\alpha_{-1}=\overline{0,p-1}$.
Let us consider next possible cases.
1) If $\alpha_{-1}=0$ then $\lambda_{\alpha_{-1}+\alpha_{0}p}\neq 0$ iff
$\alpha_{0}=0$, $\lambda_{\alpha_{0}+\alpha_{1}p}\neq 0$ iff $\alpha_{1}=0$
and so on.
Consequently $S(\alpha_{-1})\neq 0$ iff
$\alpha_{-1}=\alpha_{0}=\dots=\alpha_{l-1}=0$. It means that
$S(\alpha_{-1})=1$.
2) If $\alpha_{-1}\neq 0$ and $\alpha_{-1}\in E_{l}^{(1)}$ then
$\lambda_{\alpha_{-1}+\alpha_{0}p}\neq 0$ iff $\alpha_{0}=0$ and by analog
$S(\alpha_{-1})=1$.
3) If $\alpha_{-1}\in E_{l}^{(0)}$ and $\alpha_{-1}=\alpha_{-1}^{(0)}$ then
$\lambda_{\alpha_{-1}+\alpha_{0}p}\neq 0$ iff $\alpha_{0}=\alpha_{0}^{(0)}$,
$\lambda_{\alpha_{0}^{(0)}+\alpha_{1}p}\neq 0$ iff
$\alpha_{1}=\alpha_{1}^{(0)}$ and so on. Consequently $S(\alpha_{-1})\neq 0$
iff $\alpha_{0}=\alpha_{0}^{(0)}$, $\alpha_{1}=\alpha_{1}^{(0)},\dots$,
$\alpha_{l-1}=\alpha_{l-1}^{(0)}$. It means that $S(\alpha_{-1})=1$.
4) If $\alpha_{-1}\in E_{l}^{(0)}$ and $\alpha_{-1}=\alpha_{j}^{(0)}$ $(j\geq
0)$ then $\lambda_{\alpha_{-1}+\alpha_{0}p}\neq 0$ iff
$\alpha_{0}=\alpha_{j+1}^{(0)}$,
$\lambda_{\alpha_{0}+\alpha_{1}p}\neq 0$ iff $\alpha_{1}=\alpha_{j+2}^{(0)}$
and so on, $\alpha_{l-j-2}=\alpha_{l-1}^{(0)}$. Then
$\alpha_{l-j-1}=\dots=\alpha_{l-1}=0$. This means that $S(\alpha_{-1})=1$.
$\square$
By theorem 4.4 $\varphi(x)$ generate an orthogonal MRA. $\square$
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|
arxiv-papers
| 2012-11-12T14:29:47 |
2024-09-04T02:49:37.873253
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. F. Lukomskii",
"submitter": "Sergey Lukomskii",
"url": "https://arxiv.org/abs/1211.2633"
}
|
1211.2674
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-335 LHCb-PAPER-2012-043 November 12, 2012
First evidence for the decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$
The 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.
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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
25National Center for Nuclear Research (NCBJ), 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
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
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
pMassachusetts Institute of Technology, Cambridge, MA, United States
A search for the rare decays $B^{0}_{s}\to\mu^{+}\mu^{-}$ and
$B^{0}\to\mu^{+}\mu^{-}$ is performed using data collected in 2011 and 2012
with the LHCb experiment at the Large Hadron Collider. The data samples
comprise 1.1$\mbox{\,fb}^{-1}$ of proton-proton collisions at $\sqrt{s}=8$ TeV
and 1.0$\mbox{\,fb}^{-1}$ at $\sqrt{s}=7$ TeV. We observe an excess of
$B^{0}_{s}\to\mu^{+}\mu^{-}$ candidates with respect to the background
expectation. The probability that the background could produce such an excess
or larger is $5.3\times 10^{-4}$ corresponding to a signal significance of 3.5
standard deviations. A maximum-likelihood fit gives a branching fraction of
${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ = $(3.2^{\,+1.5}_{\,-1.2})\times
10^{-9}$, where the statistical uncertainty is 95 % of the total uncertainty.
This result is in agreement with the Standard Model expectation. The observed
number of $B^{0}\rightarrow\mu^{+}\mu^{-}$ candidates is consistent with the
background expectation, giving an upper limit of ${\cal
B}(B^{0}\to\mu^{+}\mu^{-})$ $<9.4\times 10^{-10}$ at 95 % confidence level.
Submitted to Physical Review Letters
The rare decays $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ are
highly suppressed in the Standard Model (SM). Precise predictions of their
branching fractions, ${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ = $(3.23\pm
0.27)\times 10^{-9}$ and ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ = $(1.07\pm
0.10)\times 10^{-10}$ Buras2012 make these modes powerful probes in the
search for deviations from the SM, especially in models with a non-standard
Higgs sector. Taking the measured finite width difference of the $B^{0}_{s}$
system LHCb-CONF-2012-002 into account deBruyn:2012wk , the time integrated
branching fraction of $B^{0}_{s}\to\mu^{+}\mu^{-}$ that should be compared to
the experimental value is $(3.54\pm 0.30)\times 10^{-9}$.
Previous searches d0_PLB ; cdf_prl ; cms2 ; atlas ; lhcbpaper3 already
constrain possible deviations from the SM predictions. The lowest published
limits are ${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ $<4.5\times 10^{-9}$ and
${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ $<1.0\times 10^{-9}$ at 95 % confidence
level (CL) from the LHCb collaboration using 1.0$\mbox{\,fb}^{-1}$ of data
collected in $pp$ collisions in 2011 at $\sqrt{s}=7$ TeV lhcbpaper3 . This
Letter reports an update of this search with 1.1 fb-1 of data recorded in 2012
at $\sqrt{s}=8$ TeV.
The analysis of 2012 data is similar to that described in Ref. lhcbpaper3
with two main improvements: the use of particle identification to select
$B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ (with $h^{(\prime)}=K,\pi$) decays used to
calibrate the geometrical and kinematic variables, and a refined estimate of
the exclusive backgrounds. To avoid potential bias, the events in the signal
region were not examined until all the analysis choices were finalized. The
updated estimate of the exclusive backgrounds is also applied to the 2011 data
lhcbpaper3 and the results re-evaluated. The results obtained with the
combined 2011 and 2012 datasets supersede those of Ref. lhcbpaper3 .
The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, and is described in detail in Ref.
LHCbdetector . The simulated events used in this analysis are produced using
the software described in Refs. Sjostrand:2006za ; Lange:2001uf ;
Allison:2006ve ; Agostinelli:2002hh ; Golonka:2005pn ; LHCb-PROC-2011-005 ;
LHCb-PROC-2011-006 .
Candidate $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ events are required to be selected by
a hardware and a subsequent software trigger. The candidates are predominantly
selected by single and dimuon triggers LHCb-PUB-2011-017 and, to a smaller
extent, by a generic $b$-hadron trigger LHCb-PUB-2011-016 . Candidate events
in the $B^{+}\to J/\psi K^{+}$ control channel, with $J/\psi\to\mu^{+}\mu^{-}$
(inclusion of charged conjugated processes is implied throughout this Letter),
are selected in a very similar way, the only difference being a different
dimuon mass requirement in the final software trigger. The $B^{0}_{(s)}\to
h^{+}{h}^{\prime-}$ decays are predominantly selected by a hardware trigger
based on the calorimeter transverse energy and subsequently by a generic
$b$-hadron software trigger.
The $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ candidates are selected by requiring two
high quality muon candidates muonid displaced with respect to any $pp$
interaction vertex (primary vertex, PV), and forming a secondary vertex (SV)
with a $\chi^{2}$ per degree of freedom smaller than 9 and separated from the
PV in the downstream direction by a flight distance significance greater than
15. Only candidates with an impact parameter $\chi^{2}$, ${\rm IP}\chi^{2}$
(defined as the difference between the $\chi^{2}$ of the PV formed with and
without the considered tracks) less than 25 are considered. When more than one
PV is reconstructed, that giving the smallest ${\rm IP}\chi^{2}$ for the $B$
candidate is chosen. Tracks from selected candidates are required to have
transverse momentum $p_{\rm T}$ satisfying $0.25<p_{\rm
T}<40$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p<$ 500 GeV/$c$. Only $B$
candidates with decay times smaller than $9\,\tau(B^{0}_{s})$ PDG2012 and
with invariant mass in the range
$[4900,6000]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are kept.
Dimuon candidates from elastic diphoton production are heavily suppressed by
requiring $p_{\rm T}(B)>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
surviving background comprises mainly random combinations of muons from
semileptonic decays of two different $b$ hadrons
($b\bar{b}\to\mu^{+}\mu^{-}X$, where $X$ is any other set of particles).
Two channels, $B^{+}\to J/\psi K^{+}$ and $B^{0}\to K^{+}\pi^{-}$, serve as
normalization modes. The first mode has trigger and muon identification
efficiencies similar to those of the signal, but a different number of tracks
in the final state. The second mode has a similar topology, but is triggered
differently. The selection of these channels is as close as possible to that
of the signal to reduce the impact of potential systematic uncertainties.
The $B^{0}\to K^{+}\pi^{-}$ selection is the same as for
$B^{0}_{(s)}\to\mu^{+}\mu^{-}$ signal except for muon identification. The two
tracks are nevertheless required to be within the muon detector acceptance.
The $J/\psi\to\mu^{+}\mu^{-}$ decay in the $B^{+}\to J/\psi K^{+}$
normalization channel is also selected similarly to the
$B^{0}_{(s)}\to\mu^{+}\mu^{-}$ signals, except for the requirements on the
IP$\chi^{2}$ and mass. Kaon candidates are required to have ${\rm
IP}\chi^{2}>25$.
A two-stage multivariate selection, based on boosted decision trees Breiman ;
AdaBoost is applied to the $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ candidates. A cut
on the first multivariate discriminant, unchanged from Ref. lhcbpaper3 ,
removes 80 % of the background while retaining 92 % of signal. The
efficiencies of this cut for the signal and the normalization samples are
equal within 0.2 % as determined from simulation.
The output of the second multivariate discriminant, called BDT, and the dimuon
invariant mass are used to classify the selected candidates. The nine
variables entering the BDT are the $B$ candidate IP, the minimum IP$\chi^{2}$
of the two muons with respect to any PV, the sum of the degrees of isolation
of the muons (the number of good two-track vertices a muon can make with other
tracks in the event), the $B$ candidate decay time, $p_{\rm T}$, and isolation
cdf_iso , the distance of closest approach between the two muons, the minimum
$p_{\rm T}$ of the muons, and the cosine of the angle between the muon
momentum in the dimuon rest frame and the vector perpendicular to both the $B$
candidate momentum and the beam axis.
The BDT discriminant is trained using simulated samples consisting of
$B^{0}_{s}\to\mu^{+}\mu^{-}$ for signal and $b\bar{b}\to\mu^{+}\mu^{-}X$ for
background. The BDT response is defined such that it is approximately
uniformly distributed between zero and one for signal events and peaks at zero
for the background. The BDT response is independent of the invariant mass for
signal inside the search window. The probability for a
$B^{0}_{(s)}\to\mu^{+}\mu^{-}$ event to have a given BDT value is obtained
from data using $B^{0}\to K^{+}\pi^{-}$, $\pi^{+}\pi^{-}$ and
$B^{0}_{s}\to\pi^{+}K^{-}$, $K^{+}K^{-}$ exclusive decays selected as the
signal events and triggered independently of the tracks from $B^{0}_{(s)}$
candidates.
The invariant mass lineshape of the signal events is described by a Crystal
Ball function crystalball . The peak values for the $B^{0}_{s}$ and $B^{0}$
mesons, $m_{B^{0}_{s}}$ and $m_{B^{0}}$, are obtained from the $B^{0}_{s}\to
K^{+}K^{-}$ and $B^{0}\to K^{+}\pi^{-}$, $B^{0}\to\pi^{+}\pi^{-}$ samples. The
resolutions are determined by combining the results obtained with a power-law
interpolation between the measured resolutions of charmonium and bottomonium
resonances decaying into two muons with those obtained with a fit of the mass
distributions of $B^{0}\to K^{+}\pi^{-}$, $B^{0}\to\pi^{+}\pi^{-}$ and
$B^{0}_{s}\to K^{+}K^{-}$ samples. The results are $\sigma_{B^{0}_{s}}=25.0\pm
0.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\sigma_{B^{0}}=24.6\pm
0.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The transition
point of the radiative tail is obtained from simulated
$B^{0}_{s}\to\mu^{+}\mu^{-}$ events smeared to reproduce the mass resolution
measured in data.
The $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ yields are
translated into branching fractions using
$\displaystyle{\cal B}(B^{0}_{(s)}\to\mu^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle\frac{{\cal B}_{\rm norm}\,{\rm\epsilon_{\rm norm}}\,f_{\rm
norm}}{N_{\rm norm}\,{\rm\epsilon_{sig}}\,f_{d(s)}}\times
N_{B^{0}_{(s)}\to\mu^{+}\mu^{-}}$ (1) $\displaystyle=$
$\displaystyle\alpha^{\rm norm}_{B^{0}_{(s)}\to\mu^{+}\mu^{-}}\times
N_{B^{0}_{(s)}\to\mu^{+}\mu^{-}},$
where ${\cal B}_{\rm norm}$ represents the branching fraction, $N_{\rm norm}$
the number of signal events in the normalization channel obtained from a fit
to the invariant mass distribution, and $N_{B^{0}_{(s)}\to\mu^{+}\mu^{-}}$ is
the number of observed signal events.
The factors $f_{d(s)}$ and $f_{\rm norm}$ indicate the probabilities that a
$b$ quark fragments into a $B^{0}_{(s)}$ meson and into the hadron involved in
the given normalization mode, respectively. We assume $f_{d}=f_{u}$ and use
$f_{s}/f_{d}=0.256\pm 0.020$ measured in $pp$ collision data at $\sqrt{s}=7$
TeV LHCb-PAPER-2012-037 . This value is in agreement within $1.5\,\sigma$ with
that found at $\sqrt{s}=8$ TeV by comparing the ratios of the yields of
$B^{0}_{s}\to J/\psi\phi$ and $B^{+}\to J/\psi K^{+}$ decays. The measured
dependence of $f_{s}/f_{d}$ on $p_{\rm T}(B)$ LHCb-PAPER-2012-037 is found to
be negligible for this analysis.
The efficiency ${\rm\epsilon_{sig(norm)}}$ for the signal (normalization
channel) is the product of the reconstruction efficiency of the final state
particles including the geometric detector acceptance, the selection
efficiency and the trigger efficiency. The ratio of acceptance, reconstruction
and selection efficiencies is computed using simulation. Potential differences
between data and simulation are accounted for as systematic uncertainties.
Reweighting techniques are used for all the distributions in the simulation
that do not match those from data. The trigger efficiency is evaluated with
data-driven techniques tistos . The observed numbers of $B^{+}\to J/\psi
K^{+}$ and $B^{0}\to K^{+}\pi^{-}$ candidates in the 2012 dataset are
$424\,200\pm 1500$ and $14\,600\pm 1100$, respectively. The two normalization
factors $\alpha^{\rm norm}_{B^{0}_{(s)}\to\mu^{+}\mu^{-}}$ are in agreement
within the uncertainties, and their weighted average, taking correlations into
account, gives $\alpha_{B^{0}_{s}\to\mu^{+}\mu^{-}}=(2.52\pm 0.23)\times
10^{-10}$ and $\alpha_{B^{0}\to\mu^{+}\mu^{-}}=(6.45\pm 0.30)\times 10^{-11}$.
In total, 24 044 muon pairs with invariant mass between 4900 and
6000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ pass the trigger and selection
requirements. Given the measured normalization factors and assuming the SM
branching fractions, the data sample is expected to contain about 14.1
$B^{0}_{s}\to\mu^{+}\mu^{-}$ and $1.7$ $B^{0}\to\mu^{+}\mu^{-}$ decays.
The BDT range is divided into eight bins with boundaries
$[0.0,0.25,0.4,0.5,0.6,0.7,0.8,0.9,1.0]$. For the 2012 dataset, only one bin
is considered in the BDT range 0.8–1.0 due to the lack of events in the mass
sidebands for ${\rm BDT}>0.9$. The signal regions are defined by
$m_{B^{0}_{(s)}}\pm 60$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The expected number of combinatorial background events is determined by
interpolating from the invariant mass sideband regions defined as
$[4900{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},m_{B^{0}}-60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$
and
$[m_{B^{0}_{s}}+60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},6000{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$.
The low-mass sideband and the $B^{0}$ and $B^{0}_{s}$ signal regions are
potentially polluted by exclusive backgrounds with or without
misidentification of the muon candidates.
The first category includes $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$, $B^{0}_{(s)}\to
h^{+}{h}^{\prime-}$, $B^{0}_{s}\to K^{-}\mu^{+}\nu_{\mu}$ and
$\Lambda^{0}_{b}\to p\mu^{-}\overline{\nu}_{\mu}$ decays. The
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ and $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$
branching fractions are taken from Ref. PDG2012 . The theoretical estimates of
the $\Lambda^{0}_{b}\to p\mu^{-}\overline{\nu}_{\mu}$ and $B^{0}_{s}\to
K^{-}\mu^{+}\nu_{\mu}$ branching fractions are taken from Refs. datta and
BsKmunu , respectively. The mass and BDT distributions of these modes are
evaluated from simulated samples where the $K\to\mu$, $\pi\to\mu$ and
$p\to\mu$ misidentification probabilities as a function of momentum and
transverse momentum are those determined from $D^{*+}\to D^{0}\pi^{+},D^{0}\to
K^{-}\pi^{+}$ and $\Lambda\to p\pi^{-}$ data samples. We use the
$\Lambda^{0}_{b}$ fragmentation fraction $f_{\Lambda^{0}_{b}}$ measured by
LHCb Aaij:2011jp and account for its $p_{\rm T}$ dependence.
The second category includes $B^{+}_{c}\to
J/\psi(\mu^{+}\mu^{-})\mu^{+}\nu_{\mu}$, $B^{0}_{s}\to\mu^{+}\mu^{-}\gamma$
and $B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$ decays, evaluated assuming branching
fraction values from Refs. Abe:1998wi , Nikitin1 and Bpimumu , respectively.
Apart from $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$, all background modes are
normalized relative to the $B^{+}\to J/\psi K^{+}$ decay. The
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$, $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ and
$B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$ decays are the dominant exclusive modes
in the range ${\rm BDT}>0.8$, which accounts for 70 % of the sensitivity.
In the full BDT range, $8.6\pm 0.7$ doubly misidentified $B^{0}_{(s)}\to
h^{+}{h}^{\prime-}$ decays are expected in the full mass interval,
$4.1^{+1.7}_{-0.8}$ in the $B^{0}$ and $0.76^{+0.26}_{-0.18}$ in the
$B^{0}_{s}$ signal region. The expected yields for
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ and $B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$
are $41.1\pm 0.4$ and $11.9\pm 3.5$, respectively, in the full mass and BDT
ranges. The contributions of these two backgrounds above
$m_{B^{0}}-60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are negligible. The
fractions of these backgrounds with ${\rm BDT}>0.8$, in the full mass range,
are $(19.0\pm 1.4)\,\%$, $(11.1\pm 0.5)\,\%$, and $(12.2\pm 0.3)\,\%$ for
$B^{0}_{(s)}\to h^{+}{h}^{\prime-}$, $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ and
$B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$ decays, respectively.
A simultaneous unbinned maximum-likelihood fit to the mass projections in the
BDT bins is performed on the mass sidebands to determine the number of
expected combinatorial background events in the $B^{0}$ and $B^{0}_{s}$ signal
regions used in the derivation of the branching fraction limit. In this fit
the parameters that describe the mass distributions of the exclusive
backgrounds, their fractional yields in each BDT bin and their overall yields
are limited by Gaussian constraints according to their expected values and
uncertainties. The combinatorial background is parameterized with an
exponential function with slope and normalization allowed to vary. The
systematic uncertainty on the estimated number of combinatorial background
events in the signal regions is determined by fluctuating the number of events
observed in the sidebands according to a Poisson distribution, and by varying
the exponential slope according to its uncertainty. The same fit is then
performed on the full mass range to determine the $B^{0}_{s}\to\mu^{+}\mu^{-}$
and $B^{0}\to\mu^{+}\mu^{-}$ branching fractions, which are free parameters of
the fit. The $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$
fractional yields in BDT bins are constrained to the BDT fractions calibrated
with the $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ sample. The parameters of the
Crystal Ball functions that describe the mass lineshapes and the normalization
factors are restricted by Gaussian constraints according to their expected
values and uncertainties.
The compatibility of the observed distribution of events with that expected
for a given branching fraction hypothesis is computed using the
$\textrm{CL}_{\textrm{s}}$ method Read_02 . The method provides
$\textrm{CL}_{\textrm{s+b}}$, a measure of the compatibility of the observed
distribution with the signal plus background hypothesis,
$\textrm{CL}_{\textrm{b}}$, a measure of the compatibility with the
background-only hypothesis, and
$\textrm{CL}_{\textrm{s}}=\textrm{CL}_{\textrm{s+b}}/\textrm{CL}_{\textrm{b}}$.
The invariant mass signal regions are divided into nine bins with boundaries
$m_{B^{0}_{(s)}}\pm 18,30,36,48,60$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
In each bin of the two-dimensional space formed by the dimuon mass and the BDT
output we count the number of observed candidates, and compute the expected
number of signal and background events.
The comparison of the distributions of observed events and expected background
events in the 2012 dataset results in p-values $(1-\textrm{CL}_{\textrm{b}})$
of $9\times 10^{-4}$ for the $B^{0}_{s}\to\mu^{+}\mu^{-}$ and 0.16 for the
$B^{0}\to\mu^{+}\mu^{-}$ decay, computed at the branching fraction values
corresponding to $\textrm{CL}_{\textrm{s+b}}=0.5$. We observe an excess of
$B^{0}_{s}\to\mu^{+}\mu^{-}$ candidates with respect to background expectation
with a significance of 3.3 standard deviations. The simultaneous unbinned
maximum-likelihood fit gives ${\cal
B}(B^{0}_{s}\to\mu^{+}\mu^{-})=(5.1^{\,+2.3}_{\,-1.9}({\rm
stat})^{\,+0.7}_{\,-0.4}({\rm syst}))\times 10^{-9}$. The statistical
uncertainty reflects the interval corresponding to a change of 0.5 with
respect to the maximum of the likelihood after fixing all the fit parameters
to their expected values except the $B^{0}_{s}\to\mu^{+}\mu^{-}$ and
$B^{0}\to\mu^{+}\mu^{-}$ branching fractions and the slope and normalization
of the combinatorial background. The systematic uncertainty is obtained by
subtracting in quadrature the statistical uncertainty from the total
uncertainty obtained from the likelihood with all nuisance parameters left to
vary according to their uncertainties. An additional systematic uncertainty of
$0.16\times 10^{-9}$ reflects the impact on the result of the change in the
parameterization of the combinatorial background from a single to a double
exponential, and is added in quadrature.
The expected and measured limits on the $B^{0}\to\mu^{+}\mu^{-}$ branching
fraction at 90 % and 95 % CL are shown in Table 1. The expected limits are
computed allowing for the presence of $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ events
according to the SM branching fractions, including cross-feed between the two
modes.
Table 1: Expected and observed limits on the $B^{0}\to\mu^{+}\mu^{-}$ branching fractions for the 2012 and for the combined 2011+2012 datasets. Dataset | Limit at | 90 % CL | 95 % CL
---|---|---|---
2012 | Exp. bkg+SM | $8.5\times 10^{-10}$ | $10.5\times 10^{-10}$
| Exp. bkg | $7.6\times 10^{-10}$ | $9.6\times 10^{-10}$
| Observed | $10.5\times 10^{-10}$ | $12.5\times 10^{-10}$
2011+2012 | Exp. bkg+SM | $5.8\times 10^{-10}$ | $7.1\times 10^{-10}$
| Exp. bkg | $5.0\times 10^{-10}$ | $6.0\times 10^{-10}$
| Observed | $8.0\times 10^{-10}$ | $9.4\times 10^{-10}$
The contribution of the exclusive background components is also evaluated for
the 2011 dataset, modifying the number of expected combinatorial background in
the signal regions. The results for the $B^{0}_{(s)}\to\mu^{+}\mu^{-}$
branching fractions have been updated accordingly. We obtain ${\cal
B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ $<5.1\times 10^{-9}$ and ${\cal
B}(B^{0}\to\mu^{+}\mu^{-})$ $<13\times 10^{-10}$ at 95 % CL to be compared to
the published limits ${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ $<4.5\times
10^{-9}$ and ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ $<10.3\times 10^{-10}$ at 95 %
CL lhcbpaper3 , respectively. The (1-$\textrm{CL}_{\textrm{b}}$) p-value for
$B^{0}_{s}\to\mu^{+}\mu^{-}$ changes from 18 % to 11 % and the
$B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction increases by $\sim
0.3\,\sigma$ from $(0.8^{\,+1.8}_{\,-1.3})\times 10^{-9}$ to
$(1.4^{\,+1.7}_{\,-1.3})\times 10^{-9}$. This shift is compatible with the
systematic uncertainty previously assigned to the background shape lhcbpaper3
. The values of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction obtained
with the 2011 and 2012 datasets are compatible within $1.5\,\sigma$.
The 2011 and 2012 results are combined by computing the
$\textrm{CL}_{\textrm{s}}$ and performing the maximum-likelihood fit
simultaneously to the eight and seven BDT bins of the 2011 and 2012 datasets,
respectively. The parameters that are considered 100 % correlated between the
two datasets are $f_{s}/f_{d}$, ${\cal B}(B^{+}\to J/\psi K^{+})$ and ${\cal
B}(B^{0}\to K^{+}\pi^{-})$, the transition point of the Crystal Ball function
describing the signal mass lineshape, the mass distribution of the
$B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ background, the BDT and mass distributions
of the $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ and
$B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$ backgrounds and the SM predictions of
the $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ branching
fractions. The distribution of the expected and observed events in bins of BDT
in the signal regions obtained from the simultaneous analysis of the 2011 and
2012 datasets, are summarized in Table 2.
The expected and observed upper limits for the $B^{0}\to\mu^{+}\mu^{-}$
channel obtained from the combined 2011+2012 datasets are summarized in Table
1 and the expected and observed $\textrm{CL}_{\textrm{s}}$ values as a
function of the branching fraction are shown in Fig. 1. The observed
$\textrm{CL}_{\textrm{b}}$ value at $\textrm{CL}_{\textrm{s+b}}$ = 0.5 is 89
%.
Figure 1: $\textrm{CL}_{\textrm{s}}$ as a function of the assumed
$B^{0}\to\mu^{+}\mu^{-}$ branching fraction for the combined 2011+2012
dataset. The dashed gray curve is the median of the expected
$\textrm{CL}_{\textrm{s}}$ distribution if background and SM signal were
observed. The shaded yellow area covers, for each branching fraction value, 34
% of the expected $\textrm{CL}_{\textrm{s}}$ distribution on each side of its
median. The solid red curve is the observed $\textrm{CL}_{\textrm{s}}$.
The probability that background processes can produce the observed number of
$B^{0}_{s}\to\mu^{+}\mu^{-}$ candidates or more is $5\times 10^{-4}$ and
corresponds to a statistical significance of $3.5\,\sigma$. The value of the
$B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction obtained from the fit is
${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})=(3.2^{\,+1.4}_{\,-1.2}({\rm
stat})^{\,+0.5}_{\,-0.3}({\rm syst}))\times 10^{-9}$
and is in agreement with the SM expectation. The invariant mass distribution
of the $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ candidates with ${\rm BDT}>0.7$ is shown
in Fig. 2.
Figure 2: Invariant mass distribution of the selected $B^{0}_{s}\to\mu^{+}\mu^{-}$ candidates (black dots) with ${\rm BDT}>0.7$ in the combined 2011+2012 dataset. The result of the fit is overlaid (blue solid line) and the different components detailed: $B^{0}_{s}\to\mu^{+}\mu^{-}$ (red long dashed), $B^{0}\to\mu^{+}\mu^{-}$ (green medium dashed), $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ (pink dotted), $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ (black short dashed) and $B^{0(+)}\to\pi^{0(+)}\mu^{+}\mu^{-}$ (light blue dot dashed), and the combinatorial background (blue medium dashed). Table 2: Expected combinatorial background, $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ peaking background, cross-feed, and signal events assuming the SM prediction, together with the number of observed candidates in the $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ mass signal regions, in bins of BDT for the 2011 (top) and for the 2012 (bottom) data samples. The quoted errors include statistical and systematic uncertainties. Mode | BDT bin | 0.0 – 0.25 | 0.25 – 0.4 | 0.4 – 0.5 | 0.5 – 0.6 | 0.6 – 0.7 | 0.7 – 0.8 | 0.8 – 0.9 | 0.9 – 1.0
---|---|---|---|---|---|---|---|---|---
$B^{0}_{s}\to\mu^{+}\mu^{-}$ | Exp. comb. bkg | $1880^{+33}_{-33}$ | $55.5^{+3.0}_{-2.9}$ | $12.1^{+1.4}_{-1.3}$ | $4.16^{+0.88}_{-0.79}$ | $1.81^{+0.62}_{-0.51}$ | $0.77^{+0.52}_{-0.38}$ | $0.47^{+0.48}_{-0.36}$ | $0.24^{+0.44}_{-0.20}$
(2011) | Exp. peak. bkg | $0.13^{+0.07}_{-0.05}$ | $0.07^{+0.02}_{-0.02}$ | $0.05^{+0.02}_{-0.02}$ | $0.05^{+0.02}_{-0.01}$ | $0.05^{+0.02}_{-0.01}$ | $0.05^{+0.02}_{-0.01}$ | $0.05^{+0.02}_{-0.01}$ | $0.05^{+0.02}_{-0.01}$
| Exp. signal | $2.70^{+0.81}_{-0.80}$ | $1.30^{+0.27}_{-0.23}$ | $1.03^{+0.20}_{-0.17}$ | $0.92^{+0.15}_{-0.13}$ | $1.06^{+0.17}_{-0.15}$ | $1.10^{+0.17}_{-0.15}$ | $1.26^{+0.20}_{-0.17}$ | $1.31^{+0.28}_{-0.25}$
| Observed | $1818$ | $39$ | $12$ | $6$ | $1$ | $2$ | $1$ | $1$
$B^{0}\to\mu^{+}\mu^{-}$ | Exp. comb. bkg | $1995^{+34}_{-34}$ | $59.2^{+3.3}_{-3.2}$ | $12.6^{+1.6}_{-1.5}$ | $4.44^{+0.99}_{-0.86}$ | $1.67^{+0.66}_{-0.54}$ | $0.75^{+0.58}_{-0.40}$ | $0.44^{+0.57}_{-0.38}$ | $0.22^{+0.48}_{-0.20}$
(2011) | Exp. peak. bkg | $0.78^{+0.38}_{-0.29}$ | $0.40^{+0.14}_{-0.10}$ | $0.31^{+0.11}_{-0.08}$ | $0.28^{+0.09}_{-0.07}$ | $0.31^{+0.10}_{-0.08}$ | $0.30^{+0.10}_{-0.07}$ | $0.31^{+0.10}_{-0.08}$ | $0.30^{+0.11}_{-0.08}$
| Exp. cross-feed | $0.43^{+0.13}_{-0.13}$ | $0.21^{+0.04}_{-0.04}$ | $0.16^{+0.03}_{-0.03}$ | $0.15^{+0.03}_{-0.02}$ | $0.17^{+0.03}_{-0.03}$ | $0.17^{+0.03}_{-0.02}$ | $0.20^{+0.03}_{-0.03}$ | $0.21^{+0.05}_{-0.04}$
| Exp. signal | $0.33^{+0.10}_{-0.10}$ | $0.16^{+0.03}_{-0.03}$ | $0.13^{+0.02}_{-0.02}$ | $0.11^{+0.02}_{-0.02}$ | $0.13^{+0.02}_{-0.02}$ | $0.13^{+0.02}_{-0.02}$ | $0.15^{+0.02}_{-0.02}$ | $0.16^{+0.03}_{-0.03}$
| Observed | $1904$ | $50$ | $20$ | $5$ | $2$ | $1$ | $4$ | $1$
Mode | BDT bin | 0.0 – 0.25 | 0.25 – 0.4 | 0.4 – 0.5 | 0.5 – 0.6 | 0.6 – 0.7 | 0.7 – 0.8 | 0.8–1.0
$B^{0}_{s}\to\mu^{+}\mu^{-}$ | Exp. comb. bkg | $2345^{+40}_{-40}$ | $56.7^{+3.0}_{-2.9}$ | $13.1^{+1.5}_{-1.4}$ | $4.42^{+0.91}_{-0.81}$ | $2.10^{+0.67}_{-0.56}$ | $0.35^{+0.42}_{-0.22}$ | $0.39^{+0.33}_{-0.21}$
(2012) | Exp. peak. bkg | $0.250^{+0.08}_{-0.07}$ | $0.15^{+0.05}_{-0.04}$ | $0.08^{+0.03}_{-0.02}$ | $0.08^{+0.02}_{-0.02}$ | $0.07^{+0.02}_{-0.02}$ | $0.06^{+0.02}_{-0.02}$ | $0.10^{+0.03}_{-0.03}$
| Exp. signal | $3.69^{+0.59}_{-0.52}$ | $2.14^{+0.37}_{-0.33}$ | $1.20^{+0.21}_{-0.18}$ | $1.16^{+0.18}_{-0.16}$ | $1.17^{+0.18}_{-0.16}$ | $1.15^{+0.19}_{-0.17}$ | $2.13^{+0.33}_{-0.29}$
| Observed | $2274$ | $65$ | $19$ | $5$ | $3$ | $1$ | $3$
$B^{0}\to\mu^{+}\mu^{-}$ | Exp. comb. bkg | $2491^{+42}_{-42}$ | $59.5^{+3.3}_{-3.2}$ | $13.9^{+1.6}_{-1.5}$ | $4.74^{+1.00}_{-0.89}$ | $2.10^{+0.74}_{-0.61}$ | $0.55^{+0.50}_{-0.31}$ | $0.29^{+0.34}_{-0.19}$
(2012) | Exp. peak. bkg | $1.49^{+0.50}_{-0.36}$ | $0.86^{+0.29}_{-0.22}$ | $0.48^{+0.16}_{-0.12}$ | $0.44^{+0.15}_{-0.11}$ | $0.42^{+0.14}_{-0.10}$ | $0.37^{+0.13}_{-0.09}$ | $0.62^{+0.21}_{-0.15}$
| Exp. cross-feed | $0.63^{+0.10}_{-0.09}$ | $0.36^{+0.07}_{-0.06}$ | $0.20^{+0.04}_{-0.03}$ | $0.20^{+0.03}_{-0.03}$ | $0.20^{+0.03}_{-0.03}$ | $0.20^{+0.03}_{-0.03}$ | $0.36^{+0.06}_{-0.05}$
| Exp. signal | $0.44^{+0.06}_{-0.06}$ | $0.26^{+0.04}_{-0.04}$ | $0.14^{+0.02}_{-0.02}$ | $0.14^{+0.02}_{-0.02}$ | $0.14^{+0.02}_{-0.02}$ | $0.14^{+0.02}_{-0.02}$ | $0.26^{+0.04}_{-0.03}$
| Observed | $2433$ | $59$ | $19$ | $3$ | $2$ | $2$ | $2$
The true value of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction is
contained in the interval $[1.3,5.8]\times 10^{-9}([1.1,6.4]\times 10^{-9})$
at 90 % CL (95 % CL), where the lower and upper limit are the branching
fractions evaluated at $\textrm{CL}_{\textrm{s+b}}$ = 0.95
($\textrm{CL}_{\textrm{s+b}}$ = 0.975) and $\textrm{CL}_{\textrm{s+b}}$ = 0.05
($\textrm{CL}_{\textrm{s+b}}$ = 0.025), respectively. These results are in
good agreement with the lower and upper limits derived from integrating the
profile likelihood obtained from the unbinned fit.
In summary, a search for the rare decays $B^{0}_{s}\to\mu^{+}\mu^{-}$ and
$B^{0}\to\mu^{+}\mu^{-}$ is performed using 1.0$\mbox{\,fb}^{-1}$ and
1.1$\mbox{\,fb}^{-1}$ of $pp$ collision data collected at $\sqrt{s}=7$ TeV and
$\sqrt{s}=8$ TeV, respectively. The data in the $B^{0}$ search window are
consistent with the background expectation and the world’s best upper limit of
${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ $<9.4\times 10^{-10}$ at 95 % CL is
obtained. The data in the $B^{0}_{s}$ search window show an excess of events
with respect to the background-only prediction with a statistical significance
of $3.5\,\sigma$. A fit to the data leads to ${\cal
B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ $=(3.2^{\,+1.5}_{\,-1.2})\times 10^{-9}$ which
is in agreement with the SM prediction. This is the first evidence for the
decay $B^{0}_{s}\to\mu^{+}\mu^{-}$.
## I 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2012-11-12T16:06:32 |
2024-09-04T02:49:37.880867
|
{
"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, L. Anderlini, J. Anderson, R. Andreassen, R. B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, E.\n Bowen, 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, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic,\n H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier,\n G. A. Cowan, D. Craik, S. Cunliffe, 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, A. Di Canto,\n J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo Bonal, S.\n Donleavy, F. Dordei, P. Dornan, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, R. Ekelhof, L. Eklund, I.\n El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C.\n Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F. Ferreira Rodrigues, 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. Garofoli, P. Garosi,\n J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck,\n T. 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.\n A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S. T. 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, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen,\n P. Jaton, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, O. Kochebina, V. Komarov, R. F. Koopman, P. Koppenburg, M.\n Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, 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, Y. Li, L. Li Gioi, 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, H. Luo, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F.\n Maciuc, O. Maev, M. Maino, S. Malde, G. Manca, G. Mancinelli, N. Mangiafave,\n U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A.\n Mazurov, J. McCarthy, R. McNulty, B. Meadows, M. Meissner, M. Merk, D. A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, T. Nikodem, S. Nisar, 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, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C. J.\n Parkinson, G. Passaleva, G. D. Patel, M. Patel, G. N. Patrick, C. Patrignani,\n C. 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, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M.\n S. Rangel, I. Raniuk, N. 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Wilkinson, M.\n P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling,\n S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang,\n R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L.\n Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Gaia Lanfranchi",
"url": "https://arxiv.org/abs/1211.2674"
}
|
1211.2701
|
# Spin polarization and magnetoresistance through a ferromagnetic barrier in
bilayer graphene
Hosein Cheraghchi cheraghchi@du.ac.ir Fatemeh Adinehvand School of Physics,
Damghan University, 6715- 364, Damghan, Iran
###### Abstract
We study spin dependent transport through a magnetic bilayer graphene
nanojunction configured as two dimensional normal/ferromagnetic/normal
structure where the gate-voltage is applied on the layers of ferromagnetic
graphene. Based on the fourband Hamiltonian, conductance is calculated by
using Landauer Butikker formula at zero temperature. For parallel
configuration of the ferromagnetic layers of bilayer graphene, the energy band
structure is metallic and spin polarization reaches to its maximum value close
to the resonant states, while for antiparallel configuration, the nanojunction
behaves as a semiconductor and there is no spin filtering. As a result, a huge
magnetoresistance is achievable by altering the configurations of
ferromagnetic graphene especially around the band gap.
###### pacs:
73.23.-b,73.63.-b
## I Introduction
Since spin-orbit coupling in graphene novoselov is very weak spinorbit and
also there is no nuclear spinhyperfine , spin flip length is so long about
$1\mu m$ in dirty samples and room temperaturespinlength . Clean samples are
expected to have longer spin coherency. This is good opportunity for
spintronic applications based on candidatory of graphene. On the other hand,
graphene has not intrinsically ferromagnetic (FM) properties, however, it is
possible to induce ferromagnetism externally by doping and defectsdefects ,
Coulomb interactions coulomb or by applying an external electric field in the
transverse direction in nanoribbonselectric . Recently, HaugenHaugen proposed
the FM correlations due to strong proximity of magnetic states close to
graphene. The overlap between the wave functions of the localized magnetic
states in the magnetic insulator and the itinerant electrons in graphene
induces an exchange field on itinerant electrons in graphene giving rise spin
splitting of the transport. The exchange splitting which is induced by FM
insulator Euo in graphene was estimated to be of order of $5meV$. This
splitting which is effectively similar to a Zeeman interaction has so large
magnitude that can have important effects. Such spin splitting can be directly
evaluated from the transmission resonances or magnetoresistance of FM graphene
junction. The ferromagnetism leads to a spin splitting effectively similar to
a Zeeman interaction but of much larger magnitude. The induced exchange field
is tunable by an in-plane external electric fieldex-tune . The possibility of
controlling spin conductance in FM monolayer graphene insulator has also been
studied by Yokoyama Yokoyama . It was found that the spin conductance has an
oscillatory behavior in terms of chemical potential and the gate voltage.
Figure 1: Schematic view of normal/ferromagnetic/normal bilayer graphene. Two
gate electrodes can be coated on top of the magnetic insulator strips which
are located on the upper and lower layers.
Bilayer graphene on the other hand has shown to have interesting properties
for application in nanoelectronic devices such as transistors based on
graphene substrate. The new type of integer quantum Hall effects QHE and also
the electronic band gap controllable by vertically applied electric field are
of its unusual properties in compared to monolayer graphene4band ; gap-BG ;
gap-BG1 ; gap-BG2 . Moreover, the parabolic band structure close to the Dirac
points transforms to a Mexican-hat like dispersion when an electric field is
applied on graphene. Optical measurements and theoretical predictions propose
a $200$ meV gap in bilayer graphene. This controllable gap makes bilayer
graphene as an appropriate candidate for spintronic devices. An effective two-
band Hamiltonian can describe the low energy excitations of a graphene bilayer
in the regime of low barrier heightseffective2band . However, the four-band
Hamiltonian is known to give a better agreement with both experimental data
and theoretical tight-binding calculationsbarbier ; 4band . Very recently,
spin splitting of conductance in bilayer graphene has been investigated by
using an effective two-band Hamiltonian emerging from low energy
approximationYu . This approximation is valid when energy of electrons hitting
on the potential barrier is about the barrier height. Application of a
potential difference between upper and lower layers intensifies the failure of
this approximation. On the other hand, magnetoresistance in bilayer graphene
has been studied in Ref.semenov by using $8\times 8$ Hamiltonian when the
induced exchange fields are laid in plane of each layer with a rotation in
their orientations aginst each other. By using Kobo-Greenwood formula, they
have investigated the dependence of conductivity and magnetoresistance on
temperature and induced exchange field.
Motivated by these studies, based on the four-band Hamiltonian and close to
the Dirac points, we study spin current through a magnetic barrier creating by
use of the proximity of a ferromagnetic insulator on bilayer graphene.
Conductance is calculated by use of Landauer-Buttiker formula at zero
temperature. The parameters of the barrier, energy and angle of incident
electrons can affect transport through a magnetic barrier classifying in the
propagating or evanescent modes. The dependnce of resonant peaks in
transmission on system parameters is proposed to follow a resonance condtion.
We have found in some resonant energies and barrier parameters and also around
the gapped region that a remarkable spin polarization and also
magnetoresistance can be achieved.
Model: We consider a normal/ferromagnetic/normal bilayer graphene
nanojunction. The model we have used for a bilayer graphene sandwiched between
two ferromagnetic insulator is shown schematically in Fig. 1. The exchange
fields induced by the ferromagnetic insulators is supposed to be perpendicular
to the graphene plane. Therefore, Hamiltonian of the spin up detaches from the
spin down. Two gate electrodes can be attached to the ferromagnetic graphene
from the upper and lower layers which control the barrier height in each of
layers. This set-up is different from the systems studied by Refs.semenov ;
nguyen . The exchange field splits this potential depending on the spin
parallel $(+)$ or antiparallel ($-$) to the exchange field. So in the
ferromagnetic part of bilayer graphene, we have $V^{\pm}=V_{0}\mp\Delta$ where
$\Delta$ and $V_{0}$ are the exchange field and the potential barrier made by
the gate voltage, respectively. So the two spins are scattered from the
barriers with different heights. It means that energy shift of the top of the
valance band in the barrier is different for parallel and antiparallel spins
to the exchange field. This spin splitting causes to shift conductance as a
function of energy for each spin resulting in magnetoresistance. To
investigate spin polarization and also magnetoresistance, we have considered
two different configurations so that the exchange field inducing by the
magnetic insulators on each layer are parallel or antiparallel with respect to
each other. The configuration prepared for observation of magnetoresistance
differs from the configuration considered by Ref.semenov ; nguyen . The
parallel configuration has a metallic behavior while the antiparallel
configuration induces a potential difference between upper and lower layers
concluding that the system has a semiconductor behavior with the band gap of
$2\Delta$.
This paper is organized as follows: we briefly explain the formalism which is
used for calculating of transmission based on the four-band Hamiltonian.
Before presenting our results, it is so important to have a short review in
section III on transport through a barrier deposited on bilayer graphene and
its dependence on the system parameters such as energy of incident quasi-
particles and their angle hitting into the barrier and also barrier
parameters. The method presented in section II is a detailed analysis
accompanied with some small corrections on the method used by Ref.barbier . We
will present spin polarization in the parallel configuration in section IV.
Magnetoresistance and its dependence on energy of incident particles and also
induced magnetic field will be investigated in section V. Finally, the last
section concludes our results.
## II Formalism
Figure 2: Schematic view of the potential barrier with height $V_{0}$ and
width $w$ and the wave number at normal incidence directed in the x-axis for
three regions.
In the unit cell of bilayer graphene, we suppose that two independent
sublattices A and B related to each monolayer graphene are connected to each
other in the Bernal stacking. Close to the Dirac points and in the nearest
neighbor tight binding approximation, the four band Hamiltonian and also its
eigenfunction is written as the following:
$H=\begin{pmatrix}V_{1}&\pi&t_{\perp}&0\\\ \pi^{\dagger}&V_{1}&0&0\\\
t_{\perp}&0&V_{2}&\pi^{\dagger}\\\
0&0&\pi&V_{2}\end{pmatrix}\,\,\,\,,\,\,\,\,\Psi=\begin{pmatrix}\psi_{A}\\\
\psi_{B}\\\ \psi_{B^{\prime}}\\\ \psi_{A^{\prime}}\end{pmatrix}$ (1)
where
$\pi=\left(p_{x}+ip_{y}\right)v_{f}=-i\hbar
v_{f}\left(\partial_{x}-k_{y}\right)$
and in the above formula $k_{y}=k\sin\theta$ where $\theta$ and $k$ are the
incident angle and wave number of quasi-particles hitting on a barrier which
is created by applying a bias to a metallic strip deposited on bilayer
graphene. Moreover, $V_{1}$ and $V_{2}$ are the gate potentials applied on the
upper and lower layers of bilayer graphene. Such a gate potential can be
manipulated by applying a perpendicular electric field on graphene sheet.
Here, the barrier is approximated by a square potential of barrier with sharp
variation. By solving the eigenvalue equation of $H\Psi=E\Psi$, the four band
spectrum can be concluded as the following:
$(\varepsilon^{\prime})^{2}=k^{2}+\delta^{2}+\frac{(t^{\prime})^{2}}{2}\pm
t^{\prime}\sqrt{4k^{2}\delta^{2}/(t^{\prime})^{2}+(\frac{t^{\prime}}{2})^{2}+k^{2}}$
(2)
where the above parameters are defined as the following:
$\displaystyle\varepsilon^{\prime}=(E-V_{0})/\hbar
v_{f}=\varepsilon-\upsilon_{0},\,\,\,\,\ V_{0}=(V_{1}+V_{2})/2$
$\displaystyle\delta=(V_{1}-V_{2})/2\hbar v_{f},\,\,\,\,\,\ t^{\prime}=t/\hbar
v_{f}.$ (3)
In the case of $\delta=0$ and $k<<t^{\prime}$, in low energy limit, energy
spectrum behaves as $E-V_{0}=\pm\hbar^{2}k^{2}/2m$ where $m=t/(2v_{f}^{2})$ is
an effective mass. This approximation which results in a effective two-band
Hamiltonian is valid when energy of incident electrons is close to the barrier
height. On the other word, in the case of zero potential difference between
two layers, the absolute value of $(E-V_{0})$ should be much smaller than the
interlayer coupling strength (0.4eV). For $\delta\neq 0$, this approximation
may fail for large potential differences. So one should care to choose valid
energy and potential ranges. However, in spite of Ref.Yu , in this paper, we
use four-band Hamiltonian in which the only approximation is the Dirac cone.
If we assume plane wave solution for the Schroedinger equation, the wave
function in each region with a constant potential is written as the following
matrix product.
$\Psi=GM\begin{pmatrix}a\\\ b\\\ c\\\ d\end{pmatrix}$ (4)
where matrix elements of the matrices $G$ and $M$ are defined as the
following:
$G=\begin{pmatrix}1&1&1&1\\\ f_{+}^{+}&f_{-}^{+}&f_{+}^{-}&f_{-}^{-}\\\
h^{+}&h^{+}&h^{-}&h^{-}\\\
g_{+}^{+}h^{+}&g_{-}^{+}h^{+}&g_{+}^{-}h^{-}&g_{-}^{-}h^{-}\end{pmatrix}$
$M(x)=\begin{pmatrix}e^{i\alpha_{+}x}&0&0&0\\\ 0&e^{-i\alpha_{+}x}&0&0\\\
0&0&e^{i\alpha_{-}x}&0\\\ 0&0&0&e^{-i\alpha_{-}x}\end{pmatrix}$ (5)
$\begin{array}[]{c}f_{\pm}^{+}=\dfrac{\pm\alpha_{+}-ik_{y}}{\varepsilon^{\prime}-\delta}\,\,\,\,\,,\,\,\,\,\,f_{\pm}^{-}=\dfrac{\pm\alpha_{-}-ik_{y}}{\varepsilon^{\prime}-\delta}\\\
g_{\pm}^{+}=\dfrac{\pm\alpha_{+}+ik_{y}}{\varepsilon^{\prime}+\delta}\,\,\,\,\,,\,\,\,\,\,g_{\pm}^{-}=\dfrac{\pm\alpha_{-}+ik_{y}}{\varepsilon^{\prime}+\delta}\\\
h^{\pm}=\dfrac{(\varepsilon^{\prime}-\delta)^{2}-\alpha_{\pm}^{2}-k_{y}^{2}}{t^{\prime}(\varepsilon^{\prime}-\delta)}\end{array}$
(6)
Here $\alpha$ is the wave number in the x direction and is defined as:
$\alpha_{\pm}^{2}=[\delta^{2}+(\varepsilon^{\prime})^{2}-k_{y}^{2}\pm\sqrt{4(\varepsilon^{\prime})^{2}\delta^{2}+(t^{\prime})^{2}((\varepsilon^{\prime})^{2}-\delta^{2})}]$
(7)
In the special case of normal incident angle and zero gate potential where
$k_{y}=\delta=0$, the wave number $\alpha_{+}$ is real in the energy range of
$0<\varepsilon^{\prime}<t^{\prime}$ and $\alpha_{-}$ is real if the energy of
incident particles is in the range of
$\varepsilon^{\prime}<0,\varepsilon^{\prime}>-t^{\prime}$. In this paper,
studied system contains a magnetic or electrostatic barrier of potential as
shown in Fig.2. The barrier width is $w$. The electrostatic potential which
plays the role of the gate voltage is set to be $V_{0}$ in the barrier part
and zero in the first and last regions. We suppose that the energy range of
incident particles is limited to the range of
$0<\varepsilon^{\prime}<t^{\prime}$. Consequently in the barrier part we have
$-V_{0}<\varepsilon^{\prime}_{2}<t^{\prime}-V_{0}$. The wave numbers behind
and in front of the barrier $\alpha_{+}^{(1)}$ and $\alpha_{+}^{(3)}$ are real
while $\alpha_{-}^{(1)}$ and $\alpha_{-}^{(3)}$ are imaginary. In the barrier
part, for $\varepsilon^{\prime}_{2}<0$, the wave numbers $\alpha_{+}^{(2)}$
and $\alpha_{-}^{(2)}$ are imaginary and real, respectively, while for
$\varepsilon^{\prime}_{2}>0$ they behave vice versa. A schematic view of the
barrier at normal incidence and wave numbers in each part are shown in Fig.2.
Figure 3: Transmission at normal incidence $\theta=0$ as a function of energy
difference between energy of incident particles and the barrier height
$(E-V_{0})/\hbar v_{F}$ for potential difference of upper layer against lower
layer $\delta=0$ and $10meV$.
By applying continuity of the wave functions on the boundaries of the barrier,
one can connect coefficient matrix of the wave function for the last region
$A_{3}$ to the coefficient matrix for the first region $A_{1}$.
$\begin{array}[]{c}A_{1}=NA_{3}\\\
N=M_{1}^{-1}(0)G_{1}^{-1}G_{2}M_{2}(0)M_{2}^{-1}(w)G_{2}^{-1}G_{3}M_{3}(w)\end{array}$
(8)
where N is called as transfer matrix. Since $\alpha_{-}^{(1)}$ and
$\alpha_{-}^{(3)}$ are imaginary in the interested energy range, that part of
wave function which are associated by such wave numbers are exponentially a
growing or decaying function. So we have to set the coefficient of plane wave
$e^{i\alpha_{-}^{(1)}x}$ ($c$ in Eq. 4) to be zero for the first region,
because this part of wave function grows exponentially when
$x\rightarrow-\infty$. Therefore, coefficients matrix in the first region is
supposed to be as $A_{1}=[1,r,0,e_{g}]^{T}$, where the superscript $T$ refers
to the transpose of a matrix and $e_{g}$ is the coefficient of growing
evanescent state and $r$ is the coefficient of the reflected part of the wave
function. In the last region, we have to set the coefficient ($d$ in Eq. 4) of
$e^{-i\alpha_{-}^{(3)}x}$ to be zero because this part of the wave function
increases exponentially when $x\rightarrow\infty$. Therefore, the coefficients
matrix in the last regions is supposed to be as $A_{3}=[t,0,e_{d},0]^{T}$
where $t$ is the coefficient of the transmitted part of wave function and
$e_{d}$ is the coefficient of decaying evanescent state. In this region, there
is no reflected wave. However, in equation 8 of Ref.barbier , matrices $A_{1}$
and $A_{3}$ have been considered to be completely displaced which leas to
different results.
By rearrangement of the transfer matrix elements of Eq. 8, the coefficient of
transmitted part of wave function is derived in terms of transfer matrix
elements as the following;
$t=[N_{11}-N_{13}N_{31}/N_{33}]^{-1}.$ (9)
Since the first and last regions possess similar wave numbers, transmission
probability is given as $T=|t|^{2}$. Before presenting our results, in the
next section, we will shortly review transport properties through a potential
barrier by using the mentioned formalism.
Figure 4: Contour plot of transmission in plane of the incident angle and
energy difference of $\varepsilon^{\prime}_{2}=(E-V_{0})/\hbar v_{F}$
accompanied with the band structure for potential difference of the upper
layer against the lower layer to be as a) $\delta=0$ and b) $10meV$.
## III Transmission through a Barrier on Bilayer Graphene
The Klein tunneling in monolayer graphene results in a complete transmission
through a barrier potential in normal incident. However, in contrast to
monolayer graphene, as a result of chiral symmetry in bilayer graphene,
transmission is zero for quasiparticles with energies lower than the barrier
height. In the special case of $\delta=k_{y}=0$, transmission through a
potential barrier can be analytically calculated in normal incident and for
two ranges of energy $\varepsilon^{\prime}<0$ and $\varepsilon^{\prime}>0$.
$\begin{array}[]{c}t=e^{i\alpha^{(1)}w}[\cos(\alpha^{(2)}w)-iQ\sin(\alpha^{(2)}w)]^{-1}\\\
\end{array}$ (10)
where
$Q=\frac{1}{2}(\frac{\varepsilon^{\prime}_{1}\alpha^{(2)}}{\varepsilon_{2}^{\prime}\alpha^{(1)}}+\frac{\varepsilon_{2}^{\prime}\alpha^{(1)}}{\varepsilon_{1}^{\prime}\alpha^{(2)}})$
where in the above formula, parameters are defined as
$\varepsilon_{1}^{\prime}=\varepsilon/\hbar
v_{f},\varepsilon_{2}^{\prime}=(\varepsilon-V_{0})/\hbar v_{f}$. So the real
part of the wave numbers inside and outside of the barrier part are defined as
$\alpha^{(1)}=[(\varepsilon_{1}^{\prime})^{2}+\varepsilon_{1}^{\prime}t^{\prime}]^{1/2}$
and
$\alpha^{(2)}=[(\varepsilon_{2}^{\prime})^{2}+\varepsilon^{\prime}_{2}t^{\prime}]^{1/2}$,
respectively. The energy of incident particle is supposed to be always
$\varepsilon^{\prime}_{1}>0$. So $\alpha^{(1)}$ is always real.
For the energy range of $E<V_{0}$, the wave number inside the barrier part
$\alpha^{(2)}$ and consequently $Q$ are imaginary so that
$\alpha^{(2)}=i\kappa$ and $Q=iq$ where $\kappa$ and $q$ are real. As a
result, transmission tends to zero as a function of the system parameters such
as $w$ and $\varepsilon^{\prime}_{2}$.
$T(\theta=0,\varepsilon^{\prime}_{2}<0)=tt^{\ast}=[\cosh^{2}(\kappa
w)+q^{2}\sinh^{2}(\kappa w)]^{-1}$ (11)
This behavior is the trace of chiral symmetry in bilayer graphene. However, if
the incident angle is nonzero, some resonant peaks appear in the transmission
curve (see Fig. 4). For the energy range of $E>V_{0}$, all parameters such as
$\alpha^{(2)}$ and $Q$ are real. Thus transmission has an oscillatory behavior
as a function of $\varepsilon^{\prime}_{2}$ as the following form,
$T(\theta=0,\varepsilon^{\prime}_{2}>0)=[\cos^{2}(\alpha_{+}^{(2)}w)+Q^{2}\sin^{2}(\alpha_{+}^{(2)}w)]^{-1}$
(12)
Figure 5: Contour plot of transmission in plane of the incident angle and the
barrier width for the barrier height $50meV$, the energy of incident angle
$17meV$ and for the case of $\delta=0$.
In the high energies limit $E\gg V_{0}$, we have $Q\longrightarrow 1$ and so
transmission is complete ($T\longrightarrow 1$). By applying a vertically
electric field in the barrier part, a band gap is opened in the band structure
of bilayer graphene which is proportional to the potential difference between
potentials of each layers. In this case, chiral symmetry is failed and
therefore transmission in normal incidence is nonzero for energies lower than
the barrier height ($E<V_{0}$). Transmission at normal incidence is
represented in Fig. 3 as a function of $\varepsilon^{\prime}_{2}$ for
$\delta=0$ and $10meV$. Application of a vertically electric field causes to
emerge some resonant tunneling states for energies of $E<V_{0}$. In this
energy range, resonant states originates from interference of the incident and
scattered waves. For all cases such as nonzero incident angles and $\delta\neq
0$, the resonant peaks are interpreted by the resonance condition relation
proposed as
$\alpha_{b}w=n\pi$ (13)
where $\alpha_{b}$ is the x-component of the wave number inside the barrier
which can be calculated by Eq. 7.
To have more complete view, we prepare a contour plot of transmission in plane
of the incident angle and $\varepsilon^{\prime}_{2}$ which is shown in Fig. 4
for a fixed width of the barrier. For the normal incidence ($\theta=0$),
transmission behavior is compatible with the results shown in Fig. 3.
Figure 6: Energy band structure for parallel and antiparallel configurations.
In the parallel configuration, the direction of the exchange fields inducing
in each layers are parallel with respect to each other. a) for spin parallel
and antiparallel to the exchange field direction in the parallel
configuration. b) for both spins up and down in the antiparallel
configuration.
For energies higher than the barrier height $\varepsilon^{\prime}_{2}>0$,
transmitting channels are opened over all ranges of energies. However,
transmitting window for the incident angles is limited with the condition that
$\alpha_{+}^{(2)}$ (in Eq. 7) is real. In the case of $\delta=0$, the range of
incident angle in which transmission is high can be extracted as
$-\sin^{-1}\frac{\sqrt{(\varepsilon^{\prime}_{2})^{2}+\varepsilon^{\prime}_{2}t^{\prime}}}{k}\leq\theta\leq\sin^{-1}\frac{\sqrt{(\varepsilon^{\prime}_{2})^{2}+\varepsilon^{\prime}_{2}t^{\prime}}}{k}$.
Therefore, by increasing $\varepsilon^{\prime}_{2}$, the range of angles with
high transmission becomes more extended. In the energy range of
$\varepsilon^{\prime}_{2}<0$, independent of the value of $\delta$, resonant
peaks emerge for nonzero incident angles ($\theta\neq 0$) which obey the
resonance condition $\alpha_{b}w=n\pi$. So additional to some resonant energy
states, we have some resonant widths in which transmission is high. Fig. 5
shows transmission in plane of the incident angle and the barrier width for
$\varepsilon^{\prime}<0$ and $\delta=0$. It is shown that based on the
resonance condition (Eqs. 13,7), in large incident angles, $\alpha_{b}$
reduces and so in a fixed resonance order ($n$), the resonance condition is
satisfied for wide barriers. Therefore, the resonance strips with complete
transmission shown in Fig. 5, depend strongly on the incident angle in the
range of wide barriers.
By applying a vertically electric field in the barrier part, a band gap is
opened around $\varepsilon^{\prime}_{2}=0$. This band gap also has a trace in
transmission as a transport gap shown in Fig. 4b.
Figure 7: a) Conductance and b) spin polarization as a function of
$\varepsilon^{\prime}_{2}$ for different induced exchange fields $\Delta$ in
the parallel configuration. Here, the barrier height and width are considered
to be as $50$ meV and $40$ nm, respectively.
## IV Results
By application of an averaged gate voltage $V_{0}$, band structure in the
barrier part is shifted by $V_{0}$ value. Fig. 6 shows band structure of
parallel and antiparallel configuration magnetic insulators when a gate
voltage is applied on the barrier part. In case the exchange fields inducing
in each layers of bilayer graphene are parallel, particles with spin parallel
(spin up) and antiparallel (spin down) to the exchange fields are scattered
from barriers with different heights. In the parallel configuration, spin
splitting of the barrier potential in the ferromagnetic graphene is written as
$V^{-}-V^{+}=2\Delta$. Such spin splitting is also seen in the band structure
that is shown in Fig. 6a. It is seen that the top of valance band are shifted
to lower/higher energies for spins up/down. However, in the antiparallel
configuration, the band structure shown in Fig. 6b is the same for both up and
down spins. A band gap which is proportional to $2\Delta$ appears in the band
structure of the antiparallel configuration.
### IV.1 Spin Polarization
Here, there is a correspondence between the band structure and transmission.
According to the band structure, we expect to emerge spin polarization just
for parallel configuration because energy bands for up and down spins are
shifted by $2\Delta$ with respect to each other. However, since the band
structure for antiparallel configuration is the same for both spins, it is not
expected to have spin polarization for this configuration. The spin
polarization is defined as:
Figure 8: a) Conductance and b) spin polarization as a function of barrier
width for different induced exchange fields $\Delta$ in the parallel
configuration. Here, the barrier height and incident energy are considered to
be as $50$ meV and $17$ meV, respectively.
$P=\dfrac{G_{up}-G_{down}}{G_{up}+G_{down}}$ (14)
where $G_{up}$ and $G_{down}$ are conductance for up and down spins. The
conductance is calculated by using Landauer formalism in the linear regime.
Therefore, conductance is proportional to angularly averaged transmission
projected along the current direction.
$G=\int_{-\pi/2}^{\pi/2}T(E,cos(\theta))cos\theta d\theta$
It is clear that additional to the transmission curves (Fig.4), resonance
peaks also appears in conductance. Since up and down spins in the parallel
configuration see barriers with different heights, resonance peaks in
conductance as a function of Fermi energy $E$ are shifted to higher energies
as $\Delta$ for spin down and to lower energies as $-\Delta$ for spin up. This
mismatching of conductance peaks for two spins causes to a large spin
polarization at resonance states. Fig. 7 displays conductance and spin
polarization as a function of $\varepsilon^{\prime}_{2}$ for the parallel
configuration. It is shown that conductance peaks and consequently spin
polarization appears in the energy range of $\varepsilon^{\prime}_{2}<0$. It
is seen that by inducing an exchange field, conductance peaks in Fig. 7a split
into two peaks which are related to each spin. This spin splitting is about
$2\Delta$. Spin polarization shown in Fig. 7b has an oscillatory behavior with
energy of incident particles for energies lower than the barrier height
$\varepsilon^{\prime}_{2}<0$. The amplitude of spin polarization increases
with the induced exchange field $\Delta$ and reaches to its maximum value.
However, spin polarization tends to zero for energies greater than the
potential height $\varepsilon^{\prime}_{2}>0$ except at $E\sim V_{0}$.
In the parallel configuration and for $\varepsilon_{2}^{\prime}<0$, Fig. 8a
shows that conductance in the resonance widths has a peak. These peaks which
are also seen in the transmission curves of Fig. 5 are explained by the
resonance condition of Eq. 13. It is shown that spin splitting of conductance
peaks also appears in the resonance widths which is originated from different
barrier heights for two spins up and down. It should be noted that the
conductance at resonance widths decreases for wide barriers. In the wide range
of widths, the angularly window for transmitting channels shown in Fig. 5
decreases with the widths.
Figure 9: Conductance in the parallel and antiparallel configurations as a
function of a)$\varepsilon^{\prime}_{2}=(E-V_{0})/\hbar v_{F}$ for a barrier
with the width of 40 nm and, c) the barrier width for a barrier with the
height of 50 meV. Magnetoresistance as a function of b)
$\varepsilon^{\prime}_{2}$ for a barrier with the width of 40 nm, d) the
barrier width for a barrier with the height of 50 meV. The induced exchange
field is considered to be as $\Delta=5$meV.
Fig. 8b shows spin polarization as a function of the barrier width. Again,
spin polarization has an oscillatory behavior with the barrier width. The
amplitude of spin polarization strongly increases by an increase of the
induced exchange field. Therefore, to manifest this spin polarization, we
should manufacture the ferromagnetic graphene part with the special widths in
which spin polarization reaches to the value of unity.
### IV.2 Magnetoresistance
In this section, we will show that by switching between parallel and
antiparallel configurations, one can obtain large magnetoresistance.
Magnetoresistance is defined as the following:
$MR=\dfrac{G^{p}-G^{ap}}{G^{p}+G^{ap}}$ (15)
where $G^{p}=G_{up}^{p}+G_{down}^{p}$ and $G^{ap}=G_{up}^{ap}+G_{down}^{ap}$
are conductance for parallel and antiparallel configurations.
Fig. 9 displays conductance in the parallel and antiparallel configurations
and also magnetoresistance as a function of $\varepsilon^{\prime}_{2}$ and the
barrier width for a fixed exchange field $\Delta=5meV$. As we before
expressed, spin splitting at the resonance states (for
$\varepsilon^{\prime}_{2}<0$) emerges in conductance peaks in the case of the
parallel configuration. This behavior is clear in Fig. 9a and 9c. However,
this splitting will not occur for the case of antiparallel configuration.
Therefore, large magnetoresistance appears around the conductance resonance
peaks. In the parallel configuration, a band gap appears around the barrier
edge in the interval $V_{0}-\Delta<E<V_{0}+\Delta$. This band gap has a trace
in transmission and consequently conductance. Zero conductance region around
the barrier edge $\varepsilon^{\prime}_{2}\sim 0$ which is seen in Fig.9a, is
a result of the band gap. Since there is no such a band gap in the parallel
configuration, magnetoresistance as shown in Fig.9b reaches to its maximum
value in the energy band gap. In the energy range greater than the barrier
height $\varepsilon^{\prime}_{2}>0$, there is no spin splitting and therefore,
magnetoresistance tends to zero.
As we showed before, conductance has peak at resonant widths. Similar to the
previous case, spin splitting occurs just for the parallel configuration. So
magnetoresistance increases around the resonance widths. The oscillatory
behavior of magnetoresistance as a function of the barrier width is
represented in Fig.9d.
Figure 10: a) Conductance in the parallel and antiparallel configuration and
b) magnetoresistance as a function of the induced exchange field $\Delta$ for
a barrier with the height of 50 meV and energy of incident particles as 40
meV. Here the barrier width is 20 nm.
As we showed, there is a large magnetoresistance around the barrier edge
$E\approx V_{0}$. In this range of energies, we investigate the dependence of
magnetoresistance to the induced exchange field. This exchange field of
graphene can be controlled by an in-plane external electric field ex-tune .
Fig. 10b represents that magnetoresistance increases monotonically by
increasing the exchange field $\Delta$. It is interesting by increasing the
exchange field up to $10meV$, magnetoresistance reaches to its maximum value.
To explain this behavior, we investigate the dependence of conductance on the
exchange field in the parallel and antiparallel configurations. In the
antiparallel configuration, the band gap which is limited in the interval of
$V_{0}-\Delta<E<V_{0}+\Delta$, enhances by increasing the exchange field.
Therefore, conductance in the antiparallel configuration goes to zero when the
exchange field is increased. Suppression of the conductance with the exchange
field in the antiparallel configuration is shown in Fig. 10a. However, in the
parallel configuration, conductance increases by enhancement of the exchange
field. The reason of this enhancement comes back to have larger angularly
transmitting windows for larger $\varepsilon^{\prime}_{2}$ (see Fig. 4). In
fact, effective potential for spins up $V^{+}=V_{0}-\Delta$ is decreased by an
increase in $\Delta$. So $\varepsilon^{\prime}_{2}=(E-V_{0})/\hbar v_{F}$ for
a fixed energy is increased and consequently $G_{up}$ and so $G^{p}$ is
increased by $\Delta$. As a conclusion, for the exchange fields up to 10 meV,
suppression of $G^{ap}$ and an increase of $G^{p}$ results in a large
magnetoresistance which is so useful for designing spin memory devices.
## V Conclusion
We have studied spin polarization and magnetoresistance of a
normal/ferromagnetic/normal junction of bilayer graphene by using transfer
matrix method and based on the four-band Hamiltonian. Transport properties
simultaneously is controlled by two gate electrodes ($V_{0}$), which are
applied on the ferromagnetic graphene. Two configurations of the exchange
field is considered perpendicular to the graphene sheet. This exchange field
is induced by the proximity of a localized magnetic orbital in a magnetic
insulator coating on top of each layers of bilayer graphene in the barrier
part. In the parallel configuration which graphene has a metallic behavior, a
spin splitting $2\Delta$ occurs for the conductance at the resonant states
just for energies lower than the barrier height $E<V_{0}$. However, there is
no spin splitting in the antiparallel configuration. A band gap of $2\Delta$
is opened in the antiparallel configuration which makes it a semiconductor. As
a result of spin splitting in the parallel configuration, an oscillating spin
polarization emerges for energies lower than the barrier height. Furthermore,
an oscillatory of magnetoresistance with large amplitude is achievable for
$E<V_{0}$ when we are able to switch between two configurations. There is also
a large magnetoresistance in the energy range around the barrier edge
originating from the band gap which is openned by a vertically electric field.
In this range of energy, magnetoresistance reaches to its maximum value when
the exchange field is increased by an in-plane external electric field.
## References
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* (10) T. Yokoyama, Phys. Rev. B. 77, 073413 (2008).
* (11) K. S. Novoselov, et al., Nature Physics 2, 177 (2006).
* (12) T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Science 313, 951 (2006).
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|
arxiv-papers
| 2012-11-12T17:20:56 |
2024-09-04T02:49:37.888175
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hosein Cheraghchi, Fatemeh Adinehvand",
"submitter": "Hosein Cheraghchi",
"url": "https://arxiv.org/abs/1211.2701"
}
|
1211.2711
|
The following article has been submitted to/accepted by Applied Physics
Letters.
After it is published, it will be found at http://apl.aip.org/.
# Spin lifetime measurements in GaAsBi thin films
Brennan Pursley bpursley@umich.edu Applied Physics Program, University if
Michigan, Ann Arbor, MI 48109 M. Luengo-Kovac Department of Physics,
University of Michigan, Ann Arbor, MI 48109 G. Vardar Department of
Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109
R. S. Goldman Applied Physics Program, University if Michigan, Ann Arbor, MI
48109 Department of Physics, University of Michigan, Ann Arbor, MI 48109
Department of Materials Science and Engineering, University of Michigan, Ann
Arbor, MI 48109 V. Sih Applied Physics Program, University if Michigan, Ann
Arbor, MI 48109 Department of Physics, University of Michigan, Ann Arbor, MI
48109
(November 8, 2012)
###### Abstract
Photoluminescence spectroscopy and Hanle effect measurements are used to
investigate carrier spin dephasing and recombination times in the
semiconductor alloy GaAsBi as a function of temperature and excitation energy.
Hanle effect measurements reveal the product of g-factor and effective spin
dephasing time ($gT_{s}$) ranges from 0.8 ns at 40 K to 0.1 ns at 120 K. The
temperature dependence of $gT_{s}$ provides evidence for a thermally activated
effect, which is attributed to hole localization at single Bi or Bi cluster
sites below 40 K.
The field of spintronics, firmly entrenched in giant magnetoresistive (GMR)
read heads for hard-drive technology, is rapidly progressing to encompass
other elements of computer memory such as magnetic random access memory
(MRAM).Keatley _et al._ (2011) However, a superior spintronic analogue to
silicon compimentary metal-oxide semiconductor (CMOS) logic systems has yet to
be realized. The dilute bismuthides GaAs(1-x)Bix, also referred to as
bismides, are a potentially promising family of semiconductor alloys for both
spintronic and electronic applications. These alloys can be grown on common
GaAs substrates, Bi incorporation has minimal effect on electron mobility,Kini
_et al._ (2009); Cooke _et al._ (2006) and by incorporating nitrogen, the
significant band gap tunability of both bismuthide and nitride alloys might be
achieved while remaining lattice matched to GaAs.Tiedje, Young, and
Mascarenhas (2008) Furthermore, giant spin-orbit bowing in films with low Bi
incorporation has been reportedFluegel _et al._ (2006) which allows tuning of
the spin-orbit splitting. Although bismuthide alloys have intriguing
properties that would be useful for scalable spintronic devices, the carrier
spin dynamics in GaAsBi have not yet been reported.
In this work, we investigate spin dephasing and carrier recombination times in
dilute bismuthide thin films using photoluminescence spectroscopy and Hanle
effect measurements. Photoluminescence (PL) below 1.4 eV is attributed to
carrier recombination in the GaAsBi epilayer and we observe a power dependent
blue shift of the emission peak in agreement with existing
literature.Kudrawiec _et al._ (2009); Imhof _et al._ (2010); Francoeur _et
al._ (2008) In addition, we report excitation dependent broadening of the
spectra between the main bismuthide and GaAs emission. Hanle effect
measurements reveal the product of g-factor and effective spin dephasing time
($gT_{s}$) ranges from 0.8 ns at 40 K to 0.1 ns at 120 K with $gT_{s}$ nearly
constant below and decreasing above 40 K. All observed phenomena are
attributed to hole localization that occurs below 40 K at single Bi or Bi
cluster sites.
GaAsBi epilayers were grown by molecular-beam epitaxy on GaAs substrates at
350º C with a growth rate of 0.1 $\mu$m/hr and As2/Ga and Bi/As2 beam
equivalent pressure ratios of 6 and 0.01, respectively. A 100 nm thick
GaAs0.992Bi0.008 film was grown on a 520 nm GaAs buffer layer on a semi-
insulating (001) GaAs substrate. The GaAsBi film composition was confirmed
using Rutherford back-scattering spectroscopy.
Samples were mounted on the cold-finger of a continuous flow liquid helium
cryostat. PL was performed using a tunable-wavelength mode-locked Ti:Sapphire
laser with pulse duration $\sim$3 ps and a repetition rate of 76 MHz as the
excitation source. Two excitation energies, ExA=1.59eV and ExB=1.45eV, were
used to selectively excite carriers either above (ExA) or below (ExB) the
observed GaAs band gap at all temperatures. The excitation and collection
paths were both normal to the sample surface. Incident light was focused to a
75 $\mu$m diameter spot with intensities ranging from 1 W/cm2 to 250 W/cm2. A
grating spectrometer and liquid nitrogen cooled CCD camera with 0.2 nm
resolution were used for analysis. Using both excitation energies over the
entire intensity range, PL spectra were recorded at temperatures between 10 K
and 200 K. Laser scatter was reduced by orthogonally oriented linear
polarizers and long pass filters.
We measured the Hanle effect using either ExA or ExB excitation, with an
irradiance of 23 W/cm2. A variable retarder, placed in the incident path,
allowed for right ($\sigma_{+}$) or left ($\sigma_{-}$) circularly polarized
excitation. A quarter-wave plate, combined with a linear polarizer and placed
in the collection path, selectively passed $\sigma_{+}$ PL. We recorded PL
within the range from 1.30 eV to 1.38 eV, and in our analysis, we present
Hanle data averaged within a $\pm 2$ meV range of selected energies. An
electromagnet was used to generate a field with maximum magnitude of 250 mT in
the sample plane. For each measurement, the applied magnetic field was varied
from -250 mT to +250 mT. At each field step, the $\sigma_{+}$ PL intensity was
recorded for both $\sigma_{+}$ and $\sigma_{-}$ excitation. The entire process
was repeated several times to average out noise due to fluctuations in the
laser power. The relative polarization
$P_{rel}=(I_{\sigma_{+}}-I_{\sigma_{-}})/(I_{\sigma_{+}}+I_{\sigma_{-}})$,
where $I_{\sigma_{j}}$ is the intensity of $\sigma_{+}$ PL under $\sigma_{j}$
excitation, was calculated using the average intensity of a $\pm$2 meV range
about selected PL energies.
The polarization of PL in direct-gap III-V semiconductors is proportional to
the spin polarization of the photoexcited carriers. Equation 1 describes the
temporal evolution of electron spin polarization where $\Omega$ is the Larmor
precession frequency, $\tau_{D}$ is the ensemble dephasing time, $\tau_{R}$ is
the carrier recombination time, and $S_{0}$ is the initial spin
polarization.Meier and Zakharchenya (1984)
$\frac{d\mathbf{S}}{dt}=\mathbf{\Omega}\times\mathbf{S}-\frac{\mathbf{S}}{\tau_{D}}-\frac{\mathbf{S}-\mathbf{S}_{0}}{\tau_{R}}$
(1)
In bulk GaAs, the degeneracy of the light and heavy hole transitions lead to
$S_{0}$=0.5 for transitions involving circularly-polarized light near k =
0.Žutić and Das Sarma (2004) However, $S_{0}$ could have a different value in
bismuthide samples due to carrier trappingFrancoeur _et al._ (2008) and
strain-induced splitting of the valence bands.Francoeur _et al._ (2003)
Therefore, our only assumption is that $S_{0}$ is a constant.
$\mathbf{\Omega}$, a function of the applied field, is defined by Eq. 2 where
$g$ is the effective electron g-factor, $\mu_{B}$ is the Bohr magneton, and
$\hbar$ is the reduced Planck’s constant.
$\mathbf{\Omega}=\frac{g\mu_{B}\mathbf{B}}{\hbar}$ (2)
It is convenient to define an effective spin lifetime, $T_{S}$, by Eq. 3.
$\frac{1}{T_{S}}=\frac{1}{\tau_{D}}+\frac{1}{\tau_{R}}$ (3)
In steady state, the combination of Eqs. 1 through 3 yields 4.
$0=\frac{g\mu_{B}}{\hbar}\mathbf{B}\times\mathbf{S}-\frac{\mathbf{S}}{T_{S}}+\frac{\mathbf{S_{0}}}{\tau_{R}}$
(4)
The spin polarization, $S_{0}$, of the photo-generated carriers is defined to
be along $+\mathbf{\hat{z}}$, the same direction as the incident light. By
applying a magnetic field $\mathbf{B}$ in the sample plane, which is
perpendicular to $\mathbf{\hat{z}}$, we obtain
$\displaystyle S_{z}(B)$ $\displaystyle=$
$\displaystyle\frac{S_{z}(0)}{1+(\frac{gT_{s}\mu_{B}}{\hbar}B)^{2}}$ (5)
$\displaystyle S_{z}(0)$ $\displaystyle=$
$\displaystyle\frac{S_{0}}{1+\frac{\tau_{R}}{\tau_{D}}}$ (6)
where $S_{z}(0)$ is the zero-field maximum and $S_{z}(B)$ has a Lorentzian
lineshape. Note that $gT_{s}$, the product of the effective spin lifetime and
g-factor, cannot be uncoupled through Hanle measurements. The corresponding
equations for the emitted PL polarization are
$\displaystyle P(B)$ $\displaystyle=$
$\displaystyle\frac{P(0)}{1+(\frac{gT_{s}\mu_{B}}{\hbar}(B-B_{0}))^{2}}+P_{bkg}$
(7) $\displaystyle P(0)$ $\displaystyle=$
$\displaystyle\frac{S_{0}^{2}}{1+\frac{\tau_{R}}{\tau_{D}}}$ (8)
The $S_{0}^{2}$ in Eq. 8 accounts for both the optical generation and
recombination pathways of spin polarized carriers. $B_{0}$ and $P_{bkg}$ are,
respectively, horizontal and vertical offsets added for better fitting.
Figure 1: a) Power dependent photoluminescence (PL) generated by a Ti:Sapphire
laser tuned to 1.45 eV (ExB) for a sample temperature of 10 K. b) Comparison
of PL, normalized to the dilute bismuthide emission peak, for excitations of
1.45 eV and 1.59 eV (ExA) at 10 K and various powers, as labeled. Red lines
are for ExB and blue lines for ExA excitations, respectively. c) Power
dependence of the bismuthide emission peak location at 10 K for both ExB (red
down triangles) and ExA (blue up triangles).
Figure 1(a) shows PL attributed to carrier recombination in the GaAsBi
epilayer measured at 10 K with ExB at intensities of 2.3 W/cm2 to 207 W/cm2.
Increasing power blue shifts the bismuthide emission peak from 1.32 eV to 1.36
eV and increases the low energy tail. PL measured at 10 K, 40 K, 80 K, and 120
K all exhibit similar power dependence even though PL intensity diminishes
with increasing temperature. Figure 1(b) shows a comparison of ExA and ExB
generated PL collected at 10 K and normalized to the bismuthide signal
maximum, which is below 1.40 eV. The three peaks observed above 1.40 eV are
attributed to emission from the underlying GaAs: 1.46 eV, neutral acceptor
exciton; 1.48 eV, valence to conduction band transition; 1.51 eV, free and
neutral donor excitons.Kang, Woo, and Kim (1996); Zemon _et al._ (1986) ExA
at 23 W/cm2 yielded roughly the same bismuthide emission peak location as ExB
at 207 W/cm2, but with ExA there is a broadening of the bismuthide emission at
higher energies. At all temperatures, excitation with ExA compared to ExB
generated a broader high energy tail to the bismuthide signal. Furthermore,
low powers of ExA generated PL with the same bismuthide peak locations as high
powers of ExB. The bismuthide peak locations obtained at 10 K for both
excitations as a function of power is shown in Figure 1(c).
We attribute the differences in PL linewidth and bismuthide emission peak
location between excitation energies to carrier transfer from the underlying
GaAs to the dilute bismuthide layer as opposed to sample heating. Migrating
carriers generated in GaAs by ExA would increase the carrier density of the
bismuthide layer and fill higher energy states, continuing the power dependent
trends for ExB, as observed in 1(b-c). PL measurements show that the
bismuthide peak location exhibits minimal shift when the sample temperature is
varied from 10 K to 120 K. Carrier transfer is supported by the theoretical
prediction,Broderick _et al._ (2011); Usman _et al._ (2011) and likely
observation,Riordan _et al._ (2012) of a type I heterojunction at the
GaAsBi/GaAs interface which energetically favors both electron and hole
migration from GaAs to the bismuthide layer. The power dependence of the
bismuthide signal shift is likely due to the filling of first the defect,
single-Bi, and Bi-cluster bound exciton states, followed by the band edges.
Several recent studies support the interpretation of hole localization at Bi
and Bi-cluster sites through Bi atom distributions,Sales _et al._ (2011);
Ciatto _et al._ (2008) bandgap transition behavior,Kudrawiec _et al._
(2009); Imhof _et al._ (2010); Francoeur _et al._ (2008) hole mobility,Kini
_et al._ (2011) and hole diffusion.Sales _et al._ (2011) The measured spin
behavior, discussed below, provides further support for the effects of carrier
transfer and localization.
Figure 2: Temperature dependent Hanle data for excitation energies of a) 1.59
eV (ExA) and b) 1.45 eV (ExB) evaluated at their respective 10 K bismuthide
photoluminescence peak locations of 1.35 eV and 1.33 eV. Polarization and
linewidth monotonically decrease as temperature decreases.
Figure 2 shows the temperature dependent evolution of Hanle data for PL
energies of 1.35 eV with ExA (Fig. 2a) and 1.33 eV with ExB (Fig. 2b), the
respective PL peak energies for each excitation at 10 K. The data were shifted
to remove the magnetic field independent vertical offset $P_{bkg}$ attributed
to laser scatter. Nonlinear least-squares fits were performed to all data with
Eq. 7 and are shown as black lines in Fig. 2. The fitting parameters were
$P(0)$, $gT_{S}$, $P_{bkg}$, and $B_{0}$. For both excitation energies, the
amplitude and linewidth of the Hanle curves increases with increasing
temperatures. Equation 7 shows that the value of $gT_{S}$ has a direct impact
on the observed Hanle linewidth–large $gT_{S}$ leads to small widths and vice
versa. From Eq. 8, and assuming that $S_{0}$ is constant, an increase in
polarization corresponds to a decrease in the ratio $\tau_{R}/\tau_{D}$. Since
$P(0)$ can obtain at least 7$\%$ at 120 K, this implies that at low
temperature, where the polarization has decreased by an order of magnitude,
$\tau_{R}$ is at least an order of magnitude larger than $\tau_{D}$. According
to Eq. 3, a very large $\tau_{R}/\tau_{D}$ value implies $gT_{S}\simeq
g\tau_{D}$. Therefore $g\tau_{D}$ dominates the observed low temperature Hanle
linewidth.
Figure 3: $P(0)$ as a function of temperature with excitation energies of a)
1.59 eV (ExA) and b) 1.45 eV (ExB) at selected photoluminescence (PL)
energies. $gT_{s}$ as a function of temperature with excitation c) ExA and d)
ExB at selected PL energies and fits (solid black lines) as described in the
text. The blue filled up-triangles and red filled down-triangles respectively
correspond to the ExA and ExB generated Hanle curves shown in Fig. 2. The size
of all data points includes the standard error of the fit values.
Plots of $P(0)$ and $gT_{s}$ as functions of temperature and various PL energy
for both excitations are shown in Fig. 3. Evaluation of ExA generated PL at
1.35 eV reveals an increase in $P(0)$ from 0.1$\%$ to 4$\%$ polarization and a
decrease in $gT_{s}$ from 0.4 ns to 0.1 ns as temperature increases from 10 K
to 120 K. For ExB generated PL, evaluation at 1.33 eV also reveals an increase
of $P(0)$ but from 0.6$\%$ to 7$\%$ and a decrease in $gT_{S}$ from 0.7 ns to
0.2 ns as temperature increases from 10 K to 120 K. The temperature dependence
of $P(0)$ and $gT_{S}$, shown in Fig. 3, appears to show a threshold for
behavior change around 40 K. The change in $gT_{S}$ is more apparent than for
polarization but both occur around 40 K regardless of excitation. As PL energy
increases, P(0) increases in value below 40 K while $gT_{S}$ temperature
dependence remains unchanged. This behavior is attributed to hole localization
which can be achieved by impurities, Bi clusters, or possibly single Bi
atoms.Francoeur _et al._ (2008) The $P(0)$ dependence on PL energy might be
explained by a correlation of 1.37 eV with free carriers and 1.31 eV to bound
carriers. Recently, a strong nonlinear recombination dependence on Bi
composition was reportedNargelas _et al._ (2011) indicating that AsGa defects
play a lesser role in carrier lifetime. Also observed was thermal activation
of hole diffusion which was attributed to strong hole localization.
Furthermore, a report of bismuthide energy gap broadeningsKudrawiec _et al._
(2012) shows that valence to conduction band transitions are much broader than
split-off to conduction band transitions, which is uncommon in III-V alloys
and implies unusually strong perturbations. Bismuth incorporation appears to
strongly perturb the valence band and localize holes at isolated Bi atoms or
clusters.
The role of the D’yakonov-PerelD’yakonov and Perel (1972) and Elliot-
YafetElliott (1954); Yafet (1963) mechanisms in limiting the measured spin
dephasing time should be distinguishable using a temperature dependent power
law model Song and Kim (2002) and an appropriate assumption for the
temperature dependence of the momentum scattering time. However, the sharp
transition in behavior observed at 40 K cannot easily be explained by those
mechanisms, and we have insufficient data at higher temperatures to fit the
measured spin dephasing time to a temperature dependent power law. We believe
that the dominant temperature dependent effect on $gT_{s}$ is due to changes
in carrier localization. Below 40 K, holes are localized and therefore undergo
minimal scattering; above 40 K, the holes are no longer bound and experience
some combination of the above mechanisms. An Arrhenius function (Eq. 9),
analogous to one used in Ref. 20, was used to fit the temperature dependence
of $gT_{s}$ for both excitation energies, and can be seen as black lines in
Fig. 3.
$\frac{1}{gT_{S}}=\alpha e^{-\frac{\Delta E}{kT}}+\frac{1}{g\tau_{0}}$ (9)
$\alpha$ is the pre-exponential factor, T is the sample temperature, k is
Boltzmann’s constant, $\tau_{0}$ is the 0 K limit of the dephasing time, and
both $g$ and $T_{S}$ are the same as previously defined. $\Delta E$ is
interpreted as the thermal activation energy for holes with average fit values
of 33$\pm$8 meV for ExA and 40$\pm$6 meV for ExB. These values are similar to
the value of 46 meV in Ref. 20 when a similar model was applied to the
temperature dependence of hole diffusion in GaAsBi. Future work must isolate
the contribution of hole localization in order to determine the underlying
dephasing mechanisms.
All measurements in this study exhibit an excitation energy dependence which
can be explained by changes in carrier concentration. Spin polarized carriers
optically injected in the underlying GaAs could migrate to the bismuthide
layer, fill the trap states, and substantially increase the carrier population
at the bismuthide band edges. These spins could partially dephase while in
GaAs before crossing the GaAs/GaAsBi interface. Furthermore, spin polarization
of the carriers could scatter as they cross the GaAs/GaAsBi interface, as
reported for GaAs/GaNAs Puttisong _et al._ (2011) and GaAs/ZnSe Malajovich
_et al._ (2000) heterointerfaces. The additional spin dephasing of transferred
carriers would explain the lower measured polarizations and effective
dephasing times for ExA compared to ExB (see Fig. 3c,d). Carrier transfer
could also account for the difference in $\Delta E$ values since a higher
carrier concentration would lead to a larger number of free carriers, and
hence, lower extracted thermal activation energies.
In conclusion, we present measurements of spin dynamics in a dilute bismuthide
semiconductor alloy using excitation energies both above and below the GaAs
band gap and at temperatures ranging from 10 K to 120 K. The values of
$gT_{s}$, extracted from Hanle effect measurements, range from 0.8 ns at 40 K
to 0.1 ns at 120 K, and are modeled by an Arrhenius function with thermal
activation energies of 33$\pm$8 meV for ExA and 40$\pm$6 meV for ExB.
Observations of broad PL spectra below the GaAs bandgap and thermally
activated behavior are consistent with existing literature and both phenomena
are likely due to hole localization caused by single bismuth or bismuth
cluster sites.
This material is based upon work supported by the National Science Foundation
under Grant No. ECCS-0844908, and the Materials Research Science and
Engineering Center program DMR-1120923. BP was supported in part by the
Graduate Student Research Fellowship under Grant No. DGE 1256260. VS
acknowledges support from the Air Force Office of Scientific Research (Award
No. FA9550-12-1-0258) and the Office of Naval Research (Award No.
N00014-12-1-0519). GV was supported in part by a Fulbright Foreign Student
Fellowship. GV and RSG were supported in part by NSF DMR Grant No. 1006835.
## References
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|
arxiv-papers
| 2012-11-12T17:55:05 |
2024-09-04T02:49:37.894198
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Brennan Pursley, M. Luengo-Kovac, G. Vardar, R. S. Goldman, and V. Sih",
"submitter": "Vanessa Sih",
"url": "https://arxiv.org/abs/1211.2711"
}
|
1211.2876
|
# Some results on space-like self-shrinkers
Huaqiao Liu and Y. L. Xin School of Mathematics and Information Sciences,
Henan University, Kaifeng 475004, China. liuhuaqiao@henu.edu.cn Institute of
Mathematics, Fudan University, Shanghai 200433, China. ylxin@fudan.edu.cn
###### Abstract.
We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space
$\mathbb{R}^{m+n}_{m}$with index $m$. We derive drift Laplacian of the basic
geometric quantities and obtain their volume estimates in pseudo-distance
function. Finally, we prove rigidity results under minor growth conditions in
terms of the mean curvature or the image of Gauss maps .
###### 1991 Mathematics Subject Classification:
58E20,53A10.
The authors are supported partially by NSFC
## 1\. Introduction
Let $\mathbb{R}^{m+n}_{m}$ be an $(m+n)$-dimensional pseudo-Euclidean space
with the index $m$. The indefinite flat metric on $\mathbb{R}^{m+n}_{m}$ is
defined by
$ds^{2}=\sum_{i=1}^{n}(dx^{i})^{2}-\sum_{\alpha=n+1}^{m+n}(dx^{\alpha})^{2}.$
In what follows we agree with the following range of indices
$A,\,B,\,C,\cdots=1,\cdots,m+n;\,i,\,j,\,k\cdots=1,\cdots,n;$
$s,t=1,\,\cdots,m;\,\alpha,\,\beta,\cdots=n+1,\cdots,m+n.$
Let $F:M\to\mathbb{R}^{m+n}_{m}$ be a space-like $n$-dimensional submanifold
in $\mathbb{R}^{m+n}_{m}$ with the second fundamental form $B$ defined by
$B_{XY}\mathop{=}\limits^{def.}\left(\bar{\nabla}_{X}Y\right)^{N}$ for
$X,Y\in\Gamma(TM)$. We denote $(\cdots)^{T}$ and $(\cdots)^{N}$ for the
orthogonal projections into the tangent bundle $TM$ and the normal bundle
$NM$, respectively. For $\nu\in\Gamma(NM)$ we define the shape operator
$A^{\nu}:TM\to TM$ by $A^{\nu}(V)=-(\bar{\nabla}_{V}\nu)^{T}.$ Taking the
trace of $B$ gives the mean curvature vector $H$ of $M$ in
$\mathbb{R}^{m+n}_{m}$ and
$H\mathop{=}\limits^{def.}\text{trace}(B)=B_{e_{i}e_{i}},$ where $\\{e_{i}\\}$
is a local orthonormal frame field of $M.$ Here and in the sequel we use the
summation convention. The mean curvature vector is time-like, and a cross-
section of the normal bundle.
We now consider a one-parameter family $F_{t}=F(\cdot,t)$ of immersions
$F_{t}:M\to\mathbb{R}^{m+n}_{m}$ with the corresponding images
$M_{t}=F_{t}(M)$ such that
(1.1) $\begin{split}\frac{d\,}{d\,t}F(x,t)&=H(x,t),\qquad x\in M\\\
F(x,0)&=F(x)\end{split}$
are satisfied, where $H(x,t)$ is the mean curvature vector of $M_{t}$ at
$F(x,t).$ There are many interesting results on mean curvature flow on space-
like hypersurfaces in certain Lorentzian manifolds [10, 11, 12, 13]. For
higher codimension we refer to the previous work of the second author [19].
A special but important class of solutions to (1.1) are self-similar shrinking
solutions, whose profiles, space-like self-shrinkers, satisfy a system of
quasi-linear elliptic PDE of the second order
(1.2) $H=-\frac{X^{N}}{2}.$
Besides the Lagrangian space-like self-shrinkers [2, 14, 9], there is an
interesting paper on curves in the Minkowski plane [15]. The present paper is
devoted to general situation on space-like self-shrinker.
For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ we have the
Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$. The target manifold is a pseudo-
Grassmann manifold, dual space of the Grassmann manifold $\mathbb{G}_{n,m}$.
In the next section we will describe its geometric properties, which will be
used in the paper.
Choose a Lorentzian frame field $\\{e_{i},e_{\alpha}\\}$ in
$\mathbb{R}^{m+n}_{m}$ with space-like $\\{e_{i}\\}\in TM$ and time-like
$\\{e_{\alpha}\\}\in NM$ along the space-like submanifold
$F:M\to\mathbb{R}^{m+n}_{m}$. Define coordinate functions
$x^{i}=\left<F,e_{i}\right>,\;y^{\alpha}=-\left<F,e_{\alpha}\right>.$
We then have
$|F|^{2}=X^{2}-Y^{2},$
where $X=\sqrt{\sum_{i=1}^{n}(x^{i})^{2}},\quad
Y=\sqrt{\sum_{\alpha=n+1}^{m+n}(y^{\alpha})^{2}}.$ We call $|F|^{2}$ the
pseudo-distance function from the origin $0\in M$.
We always put the origin on $M$ in the paper. We see that $|F|^{2}$ is
invariant under the Lorentzian action up to the choice of the origin in
$\mathbb{R}^{m+n}_{m}$. Set $z=|F|^{2}$. It has been proved that $z$ is proper
provided $M$ is closed with the Euclidean topology (see [4] for $m=1$ and [16]
for any codimension $m$).
Following Colding and Minicozzi [6] we can also introduce the drift Laplacian,
(1.3) $\displaystyle\mathcal{L}=\Delta-\frac{1}{2}\langle
F,\nabla(\cdot)\rangle=e^{\frac{z}{4}}div(e^{-\frac{z}{4}}\nabla(\cdot)).$
It can be showed that $\mathcal{L}$ is self-adjoint with respect to the
weighted volume element $e^{-\frac{z}{4}}d\mu,$ where $d\mu$ is the volume
element of $M$ with respect to the induced metric from the ambient space
$\mathbb{R}^{m+n}_{m}$. In the present paper we carry out integrations with
respect to this measure. We denote $\rho=e^{-\frac{z}{4}}$ and the volume form
$d\mu$ might be omitted in the integrations for notational simplicity.
For a space-like submanifold in $\mathbb{R}^{m+n}_{m}$ there are several
geometric quantities. The squared norm of the second fundamental form
$|B|^{2}$, the squared norm of the mean curvature $|H|^{2}$ and the
$w-$function, which is related to the image of the Gauss map. In §3 we will
calculate drift Laplacian $\mathcal{L}$ of those quantities, see Proposition
3.1.
Corresponding to the weighted measure and drift Laplacian there is so-called
the Baker-Emery Ricci tensor. It is noted that in [3]
$\text{Ric}_{f}\geq\frac{z}{4}$ with $f=\frac{z}{4}$. Using the comparison
technique the weighted volume of the geodesic ball can be estimated from above
in terms of the distance function [18].
For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ there are 3
kind global conditions: Closed one with Euclidean topology; entire graph;
complete with induced Riemannian metric. A complete space-like one has to be
entire graph, but the converse claim is not always the case. Closed one with
Euclidean topologg is complete under the parallel mean curvature assumption
(see [4] for codimension one and [16] for higher codimension).
In our case of closed one with Euclidean topology, the pseudo-distance
function $z$ is always proper. It is natural to consider the volume growth in
$z$. For the proper self-shrinkers in Euclidean space Ding-Xin [8] gave the
volume estimates. It has been generalized in [5] for more general situation.
But, the present case does not satisfy the conditions in Theorem 1.1 in [5].
However, the idea in [8] is still applicable for space-like self-shrinkers. In
§4 we will give volume estimates for space-like self-shrinkers, in a similar
manner as in [8], see Theorem 4.1.
Finally, using integral method we can obtain rigidity results as follows.
###### Theorem 1.1.
Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.
###### Theorem 1.2.
Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.
###### Remark 1.1.
In the special situation, for the Lagrangian space-like self-shrinkers, the
rigidity results hold without the growth condition (see [9]). Let ${\tenmsb
R}^{2n}_{n}$ be Euclidean space with null coordinates
$(x,y)=(x_{1},\cdots,x_{n};\,y_{1},\cdots,y_{n})$, which means that the
indefinite metric is defined by $ds^{2}=\sum_{i}dx_{i}dy_{i}.$ If
$M=\\{(x,Du(x))\big{|}\ x\in{\tenmsb R}^{n}\\}$ is a space-like submanifold in
${\tenmsb R}^{2n}_{n}$, then $u$ is convex and the induced metric on $M$ is
given by $ds^{2}=\sum_{i,j}u_{ij}dx_{i}dx_{j}$. $M$ is a space-like Lagrangian
submanifold in $\mathbb{R}^{2n}_{n}$. It is worthy to point out that if $M$ is
entire gradient graph the potential function $u$ is proper, as the following
consideration. On $\mathbb{R}^{n}$ set $\rho=|x|=\sqrt{\sum x_{i}^{2}}$. At
any direction $\theta\in S^{n-1}$
$u_{i}=u_{\rho}\frac{\partial\rho}{\partial x_{i}}=\frac{x_{i}}{\rho}u_{\rho}$
and the pseudo-distance
$z=x_{i}u_{i}=\rho u_{\rho},$
which is positive when the origin is on $M$, since it is space-like. It
implies that $u$ is increasing in $\rho$. Moreover,
$z_{\rho}=u_{\rho}+\rho u_{\rho\rho}>0,$
which means that $z$ is also increasing in $\rho$. Hence,
$u(\rho)-u(\epsilon)=\int_{\epsilon}^{\rho}u_{\rho}d\rho=\int_{\epsilon}^{\rho}\frac{z}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\to\infty$
as $\rho\to\infty$.
###### Remark 1.2.
Rigidity problem for space-like extremal submanifolds was raised by E. Calabi
[1], and solved by Cheng-Yau [4] for codimension $1$. Later, Jost-Xin
generalized the results to higher codimension [16]. The rigidity problem for
space-like submanifolds with parallel mean curvature was studied in [20][22]
and [16] (see also in Chap. 8 of [21]).
## 2\. Geometry of $G_{n,m}^{m}$
In $\mathbb{R}^{n+m}_{m}$ all space-like $n-$subspaces form the pseudo-
Grassmannian $G^{m}_{n,m}.$ It is a specific Cartan-Hadamard manifold which is
the noncompact dual space of the Grassmann manifold $G_{n,m}.$
Let $P$ and $A\in G_{n,m}^{m}$ be two space-like $n-$plane in $R_{m}^{m+n}.$
The angles between $P$ and $A$ are defined by the critical values of angel
$\theta$ between a nonzero vector $x$ in P and its orthogonal projection
$x^{*}$ in $A$ as $x$ runs through $P$.
Assume that $e_{1},\cdots,e_{n}$ are orthonormal vectors which span the space-
like $P$ and $a_{1},\cdots,a_{n}$ for space-like $A.$ For a nonzero vector in
$P$
$x=\sum_{i}x_{i}e_{i},$
its orthonormal projections in $A$ is
$x^{*}=\sum_{i}x_{i}^{*}a_{i}.$
Thus, for any $y\in A,$ we have
$\langle x-x^{*},y\rangle=0.$
Set
$W_{i\,j}=\langle e_{i},a_{j}\rangle,$
We then have
$x_{j}^{*}=\sum_{i}W_{i\,j}x_{i}.$
Since $x$ is a vector in a space-like $n-$plane and its projection $x^{*}$ in
$A$ is also a space-like vector. We then have a Minkowski plane $R_{1}^{2}$
spanned by $x$ and $x^{*}.$ Then angle $\theta$ between $x$ and $x^{*}$ is
defined by
$\cosh\theta=\frac{\langle x,x^{*}\rangle}{|x||x^{*}|}.$
Let
$W=(W_{i\,j})=\left(\begin{array}[]{lll}\langle
e_{1},a_{1}\rangle&\cdots&\langle e_{n},a_{1}\rangle\\\ \quad\vdots&\
\vdots&\quad\vdots\\\ \langle e_{n},a_{1}\rangle&\cdots&\langle
e_{n},a_{n}\rangle\end{array}\right)$
Now define the $w-$function as
$w=\langle e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle=\det W.$
$W^{T}W$ is symmetric, its eigenvalues are $\mu_{1}^{2},\cdots,\mu_{n}^{2},$
then there exist $e_{1},\cdots,e_{n}$ in $P$, such that
$W^{T}W=\left(\begin{array}[]{lll}\mu_{1}^{2}&&0\\\ &\ddots&\\\
0&&\mu_{n}^{2}\end{array}\right),$
in which $\mu_{i}\geq 1$ and $\mu_{i}=\cosh\theta_{i}.$ Then
(2.1)
$w=\prod_{i}\cosh\theta_{i}=\prod_{i}\frac{1}{\sqrt{1-\lambda_{i}^{2}}},\;\lambda_{i}=\tanh\theta_{i}.$
The distance between $P$ and $A$ in the canonical Riemannian metric on
$\mathbb{G}_{n,m}^{m}$ is (see [17] for example)
$d(P,A)=\sqrt{\sum_{i}\theta_{i}^{2}}.$
For the fixed $A\in G_{n,m}^{m},$ which is spanned by $\\{a_{i}\\}$, choose
time-like $\\{a_{n+s}\\}$ such that $\\{a_{i},a_{n+s}\\}$ form an orthonormal
Lorentzian bases of $R^{n+m}_{m}.$
Set
$\displaystyle e_{i}$
$\displaystyle=\cosh\theta_{i}a_{i}+\sinh\theta_{i}a_{n+i}$ $\displaystyle
e_{n+i}$
$\displaystyle=\sinh\theta_{i}a_{i}+\cosh\theta_{i}a_{n+i}\,(\;\text{and}\;e_{n+\alpha}=a_{n+\alpha}\;\text{if}\;m>n).$
Then $e_{i}\in T_{p}M,e_{n+i}\in N_{p}M.$ In this case
(2.2) $\displaystyle w_{i\,\alpha}$ $\displaystyle=$ $\displaystyle\langle
e_{1}\wedge\cdots\wedge e_{i-1}\wedge e_{\alpha}\wedge e_{i+1}\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ (2.3) $\displaystyle=$
$\displaystyle\cosh\theta_{1}\cosh\theta_{i-1}\sinh\theta_{i}\cosh\theta_{i+1}\cosh\theta_{n}=\lambda_{i}w\delta_{n+i\
\alpha},$
which is obtained by replacing $e_{i}$ by $e_{\alpha}$ in $w$. We also have
$w_{i\alpha j\beta}$ by replacing $e_{j}$ by $e_{\beta}$ in $w_{i\,\alpha}.$
We obtain
(2.7) $\displaystyle w_{i\alpha
j\beta}=\left\\{\begin{array}[]{lll}\lambda_{i}\lambda_{j}w&&\alpha=n+i,\beta=n+j\\\
-\lambda_{i}\lambda_{j}w&&\alpha=n+j,\beta=n+i\\\
0&&otherwise.\end{array}\right.$
## 3\. Drift Laplacian of some geometric quantities
The second fundamental form $B$ can be viewed as a cross-section of the vector
bundle Hom($\odot^{2}TM,NM$) over $M.$ A connection on Hom($\odot^{2}TM,NM$)
can be induced from those of $TM$ and $NM$ naturally. There is a natural fiber
metric on Hom($\odot^{2}TM,NM$) induced from the ambient space and it becomes
a Riemannian vector bundle. There is the trace-Laplace operator $\nabla^{2}$
acting on any Riemannian vector bundle.
In [19] we already calculate $\nabla^{2}B$ for general space-like
$n-$submanifolds in $\mathbb{R}^{m+n}_{m}$.
Set
$B_{i\,j}=B_{e_{i}\,e_{j}}=h_{i\,j}^{\alpha}e_{\alpha},\;S_{\alpha\,\beta}=h_{i\,j}^{\alpha}h_{i\,j}^{\beta}.$
From Proposition 2.1 in [19] we have
(3.1)
$\displaystyle\langle\nabla^{2}B,B\rangle=\langle\nabla_{i}\nabla_{j}H,B_{i\,j}\rangle+\langle
B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S_{\alpha\,\beta}^{2},$
where $R^{\perp}$ denotes the curvature of the normal bundle and
$|R^{\perp}|^{2}=-\langle
R_{e_{i}\,e_{j}}\nu_{\alpha},R_{e_{i}\,e_{j}}\nu_{\alpha}\rangle.$
Then from the self-shrinker equation (1.2) we obtain
$\displaystyle\nabla_{i}F^{N}$ $\displaystyle=$
$\displaystyle[\bar{\nabla}_{i}(F-\langle F,e_{j}\rangle e_{j})]^{N}$
$\displaystyle=$ $\displaystyle[e_{i}-\bar{\nabla}_{i}\langle F,e_{j}\rangle
e_{j}-\langle F,e_{j}\rangle\bar{\nabla}_{e_{i}}e_{j}]^{N}$ $\displaystyle=$
$\displaystyle-\langle F,e_{j}\rangle B_{i\,j},$
and
$\displaystyle\nabla_{i}\nabla_{j}F^{N}$ $\displaystyle=$
$\displaystyle-\nabla_{i}[\langle F,e_{k}\rangle B_{k\,j}]$ $\displaystyle=$
$\displaystyle-\delta_{i}^{k}B_{k\,j}-\langle F^{N},B_{k\,i}\rangle
B_{k\,j}-\langle F,e_{k}\rangle\nabla_{i}B_{k\,j}$ $\displaystyle=$
$\displaystyle-B_{i\,j}-\langle F^{N},B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}$ $\displaystyle=$ $\displaystyle-
B_{i\,j}+\langle 2H,B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$
Set $P_{i\,j}=\langle B_{i\,j},H\rangle,$ then
(3.2)
$\displaystyle\nabla_{i}\nabla_{j}H=\frac{1}{2}B_{i\,j}-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$
Substituting (3.2) into (3.1)we obtain
$\displaystyle\langle\nabla^{2}B,B\rangle$ $\displaystyle=$
$\displaystyle\langle\frac{1}{2}B_{i\,j},B_{i\,j}\rangle-\langle
H,B_{k\,i}\rangle\langle B_{k\,j},B_{i\,j}\rangle+\frac{1}{2}\langle
F,e_{k}\rangle\langle\nabla_{k}B_{i\,j},B_{i\,j}\rangle$
$\displaystyle+\langle B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$
This also means that
(3.3) $\displaystyle\langle\nabla^{2}B,B\rangle=\frac{1}{2}\langle
B,B\rangle+\frac{1}{4}\langle F^{T},\nabla\langle
B,B\rangle\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$
Note that $\Delta\langle
B,B\rangle=2\langle\nabla^{2}B,B\rangle+2\langle\nabla B,\nabla B\rangle,$ so
(3.4) $\displaystyle\Delta\langle B,B\rangle$ $\displaystyle=\langle
B,B\rangle+\frac{1}{2}\langle F^{T},\nabla\langle
B,B\rangle\rangle-2|R^{\perp}|^{2}-2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}$
$\displaystyle+2\langle\nabla B,\nabla B\rangle.$
We denote
$|B|^{2}=-\langle B,B\rangle=\sum_{i,j,\alpha}h_{\alpha ij}^{2},\;|\nabla
B|^{2}=-\langle\nabla B,\nabla B\rangle.$ $|H|^{2}=-\langle
H,H\rangle,\;|\nabla H|^{2}=-\langle\nabla H,\nabla H\rangle$
then
(3.5) $\displaystyle\Delta|B|^{2}=|B|^{2}+\frac{1}{2}\langle
F^{T},\nabla|B|^{2}\rangle+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2}$
From (3.2) we also obtain
$\displaystyle\nabla^{2}H=\frac{1}{2}H-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H.$
Since
$\displaystyle\Delta|H|^{2}=-\Delta\langle
H,H\rangle=-2\langle\nabla^{2}H,H\rangle-2\langle\nabla H,\nabla H\rangle,$
we obtain
(3.6) $\displaystyle\Delta|H|^{2}$
$\displaystyle=-2\langle\frac{1}{2}H-P_{k\,i}B_{k\,i}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H,H\rangle-2\langle\nabla H,\nabla H\rangle$
$\displaystyle=|H|^{2}+2|P|^{2}+\frac{1}{2}\langle
F^{T},\nabla|H|^{2}\rangle+2|\nabla H|^{2},$
where $|P|^{2}=\sum_{i,j}P_{ij}^{2}$.
In the pseudo-Grassmann manifold $\mathbb{G}_{n,m}^{m}$ there are
$w-$functions with respect to a fixed point $A\in\mathbb{G}_{n,m}^{m}$, as
shown in §2. For the space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$
we define the Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$, which is obtained
by parallel translation of $T_{p}M$ for any $p\in M$ to the origin in
$\mathbb{R}^{m+n}_{m}$. Then, we have functions $w\circ\gamma$ on $M$, which
is still denoted by $w$ for notational simplicity.
For any point $p\in M$ around $p$ there is a local tangent frame field
$\\{e_{i}\\}$, and which is normal at $p$. We also have a local orthonormal
normal frame field $\\{e_{\alpha}\\}$, and which is normal at $p$. Define a
$w-$function by
$w=\left<e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\right>,$
where $\\{a_{i}\\}$ is a fixed orthonormal vectors which span a fixed space-
like $n-$plane $A$. Denote
$e_{i\,\alpha}=e_{1}\wedge\cdots\wedge e_{\alpha}\wedge\cdots\wedge e_{n},$
which is got by substituting $e_{\alpha}$ for $e_{i}$ in
$e_{1}\wedge\cdots\wedge e_{n}$ and $e_{i\alpha j\beta}$ is obtained by
substituting $e_{\beta}$ for $e_{j}$ in $e_{i\,\alpha}.$ Then
(3.7) $\displaystyle\nabla_{e_{j}}w$ $\displaystyle=\sum_{i=1}^{n}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{j}}e_{i}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}\langle e_{1}\wedge\cdots\wedge B_{i\,j}\cdots
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle e_{1}\cdots\wedge
e_{\alpha}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle
e_{i\,\alpha},a_{1}\wedge\cdots\wedge a_{n}\rangle.$
Furthermore,
(3.10) $\displaystyle\nabla_{e_{i}}\nabla_{e_{j}}w$ $\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}(e_{1}\wedge\cdots\wedge
e_{n}),a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle
e_{1}\wedge\cdots\wedge\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge\bar{\nabla}_{e_{i}}e_{l}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle e_{1}\wedge\cdots\wedge
B_{j\,k}\wedge\cdots\wedge B_{il}\wedge\cdots e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{T}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
Note that
$\displaystyle(\ref{3.13})$ $\displaystyle=$ $\displaystyle\sum_{k\neq
l}h_{j\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle(\ref{3.14})$
$\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k},e_{k}\rangle
w=-\langle\bar{\nabla}_{j}e_{k},\bar{\nabla}_{i}e_{k}\rangle w=-\langle
B_{j\,k},B_{i\,k}\rangle w=h_{j\,k}^{\alpha}h_{i\,k}^{\alpha}w$
$\displaystyle(\ref{3.15})$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}(B_{j\,k}+\nabla_{e_{j}}e_{k}))^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle\nabla_{i}B_{j\,k},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge
a_{n}\rangle=-\langle\nabla_{k}B_{i\,j},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$
where we use the Codazzi equation in the last step. Thus, we obtain
$\Delta w=\sum_{i,k\neq l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{k\beta
l}^{\alpha},a_{1}\wedge\cdots\wedge
a_{n}\rangle+|B|^{2}w-\langle\nabla_{k}H,e_{\alpha}\rangle\langle e_{\alpha
k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$
Since
$\nabla_{i}F^{N}=-\langle F,e_{j}\rangle B_{i\,j},$
from (1.2), we obtain
(3.11) $\displaystyle\nabla_{i}H$ $\displaystyle=\frac{1}{2}\langle
F,e_{j}\rangle B_{i\,j}$ $\displaystyle\langle\nabla_{i}H,e_{\alpha}\rangle$
$\displaystyle=-\frac{1}{2}\langle F,e_{j}\rangle h_{i\,j}^{\alpha},$
so,
(3.12) $\displaystyle\Delta w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,e_{i}\rangle
h_{k\,i}^{\alpha}\langle e_{\alpha k},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,\nabla w\rangle,$
where (3.7) has been used in the last equality.
###### Proposition 3.1.
For a space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
we have
(3.13)
$\mathcal{L}|B|^{2}=|B|^{2}+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2},$ (3.14) $\mathcal{L}|H|^{2}=|H|^{2}+2|P|^{2}+2|\nabla H|^{2},$ (3.15)
$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}.$
###### Proof.
From (1.3,3.5,3.6), we can obtain (3.13) and (3.14) easily.
From (1.3,3.12) we have
(3.16) $\displaystyle\mathcal{L}w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}w_{\alpha k\beta l}$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})w,$
Furthermore, since
$\mathcal{L}(\ln w)=\frac{1}{w}\mathcal{L}w-\frac{|\nabla w|^{2}}{w^{2}},$
we obtain
$\mathcal{L}(\ln w)=|B|^{2}+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})-\frac{|\nabla
w|^{2}}{w^{2}}.$
From (3.7),we obtain
$\displaystyle|\nabla w|^{2}$
$\displaystyle=\sum_{j=1}^{n}|\nabla_{e_{j}}w|^{2}=\sum_{j=1}^{n}(\sum_{i=1}^{n}\sum_{\alpha}h_{ij}^{\alpha}w_{i\,\alpha})^{2}$
$\displaystyle=\sum_{j=1}^{n}(\sum_{i=1}^{n}h_{ij}^{n+i}\lambda_{i}w)^{2}=\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}w^{2}h_{i\,j}^{n+i}h_{k\,j}^{n+k}.$
in the case of $m\geq n$ we rewrite (otherwise, we treat the situation
similarly)
$|B|^{2}=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}.$
So, we obtain
(3.17) $\displaystyle\mathcal{L}(\ln w)$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}(h_{i\,j}^{n+i}h_{j\,k}^{n+k}-h_{i\,j}^{n+k}h_{j\,k}^{n+i})-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+i}h_{j\,k}^{n+k}-\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+k}h_{j\,k}^{n+i}-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{i\,j}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
144.54pt-\sum_{i,j}\lambda_{i}^{2}(h_{ij}^{n+i})^{2}-\sum_{ij,k\neq
i}\lambda_{i}\lambda_{k}h_{ij}^{n+k}h_{jk}^{n+i}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
180.67499pt-2\sum_{j}\sum_{k<i}\lambda_{k}\lambda_{i}h_{jk}^{n+i}h_{ij}^{n+k}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
231.26378pt+\sum_{j}\sum_{i<k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j,k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\prod_{i}(1-\lambda_{i}^{2})\sum_{i,j,k}(h_{ij}^{n+k})^{2}.$
Noting (2.1) the inequality (3.15) has been proved.
###### Remark 3.1.
For a space-like graph $M=(x,f(x))$ with $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$
its induced metric is
$ds^{2}=(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})dx^{i}dx^{j}.$ Set
$g=\det(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})$ then
$w=\frac{1}{\sqrt{g}}$.
###### Remark 3.2.
(3.15) is a generalization of a formula (5.8) for space-like graphical self-
shrinkers in [7] to more general situation.
∎
## 4\. Volume growth
To draw our results we intend to integrate those differential inequalities
obtained in the last section. We need to know the volume growth in the pseudo-
distance function $z$ on the space-like submanifolds. In [16] the following
property has been proved.
###### Proposition 4.1.
(Proposition 3.1 in [16]) Let $M$ be a space-like $n-$submanifold in
$\mathbb{R}^{m+n}_{m}$. If $M$ is closed with respect to the Euclidean
topology, then when $0\in M$, $z=\left<F,F\right>$ is a proper function on
$M$.
we also need a lemma from [8]:
###### Lemma 4.1.
If $f(r)$ is a monotonic increasing nonnegative function on $[0,+\infty)$
satisfying $f(r)\leq C_{1}r^{n}f(\frac{r}{2})$ on $[C_{2},+\infty)$ for some
positive constant $n,C_{1},C_{2}$, here $C_{2}>1$, then $f(r)\leq
C_{3}e^{2n(\log r)^{2}}$ on $[C_{2},+\infty)$ for some positive constant
$C_{3}$ depending only on $n,C_{1},C_{2},f(C_{2})$.
Using the similar method as in [8] we obtain the following volume growth
estimates.
###### Theorem 4.1.
Let $z=\langle F,F\rangle$ be the pseudo-distance of $R_{m}^{n+m},$ where
$F\in R_{m}^{n+m}$ is the position vector with respect to the origin $0\in M$.
Let $M$ be an $n-$dimensional space-like self-shrinker of $R^{n+m}_{m}$.
Assume that $M$ is closed with respect to the Euclidean topology, then for any
$\alpha>0$, $\int_{M}e^{-\alpha z}d\mu$ is finite, in particular $M$ has
finite weighted volume.
###### Proof.
We have
$z_{i}\mathop{=}\limits^{def.}e_{i}(z)=2\left<F,e_{i}\right>,$
$z_{ij}\mathop{=}\limits^{def.}Hess(z)(e_{i},e_{j})=2\left(\delta_{ij}-y^{\alpha}h_{ij}^{\alpha}\right),$
(4.1) $\Delta z=2n-2y^{\alpha}H^{\alpha}=2n+Y^{2},$
where the self-shrinker equation (1.2) has been used in third equality. For
our self-shrinker $M^{n}$ in ${\tenmsb R}^{n+m}_{m}$, we define a functional
$F_{t}$ on any set $\Omega\subset M$ by
$F_{t}(\Omega)=\frac{1}{(4\pi
t)^{n/2}}\int_{\Omega}e^{-\frac{z}{4t}}d\mu,\quad\mathrm{for}\quad t>0.$
Set $B_{r}=\\{p\in\mathbb{R}^{m+n}_{m},z(p)<r^{2}\\}$ and $D_{r}=B_{r}\bigcap
M$. We differential $F_{t}(D_{r})$ with respect to $t$,
$F_{t}^{\prime}(D_{r})=(4\pi)^{-\frac{n}{2}}t^{-(\frac{n}{2}+1)}\int_{D_{r}}(-\frac{n}{2}+\frac{z}{4t})e^{-\frac{z}{4t}}d\mu.$
Noting (4.1)
(4.2) $\displaystyle-e^{\frac{z}{4t}}\mathrm{div}(e^{-\frac{z}{4t}}\nabla z)$
$\displaystyle=-\Delta z+\frac{1}{4t}\nabla z\cdot\nabla z$
$\displaystyle=-2n-Y^{2}+\frac{X^{2}}{t}$
$\displaystyle\geq\frac{z}{t}-2n\quad(\ when\ 0<t\leq 1\ ).$
Since
$\nabla z=2F^{T}$
and the unit normal vector to $\partial D_{r}$ is $\frac{F^{T}}{X}$, then
(4.3) $\displaystyle F_{t}^{\prime}(D_{r})\leq$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{D_{r}}-\mathrm{div}(e^{-\frac{z}{4t}}\nabla
z)d\mu$ $\displaystyle=$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{\partial
D_{r}}-2Xe^{-\frac{z}{4t}}\leq 0.$
We integrate $F_{t}^{\prime}(D_{r})$ over $t$ from $\frac{1}{r}$ to $1,\,r\geq
1$, and get
$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\leq
r^{\frac{n}{2}}\int_{D_{r}}e^{-\frac{zr}{4}}d\mu,$
Since
$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\geq e^{-\frac{r^{2}}{4}}\int_{D_{r}}1d\mu$
and
$\displaystyle\int_{D_{r}}e^{-\frac{zr}{4}}d\mu$ $\displaystyle=$
$\displaystyle\int_{D_{r}\backslash
D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu+\int_{D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu$
$\displaystyle\leq$ $\displaystyle
e^{-\frac{r^{3}}{16}}\int_{D_{r}}1d\mu+\int_{D_{\frac{r}{2}}}1d\mu.$
Set $V(r)=\int_{D_{r}}1d\mu$. Then,
$(e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}})V(r)\leq
r^{\frac{n}{2}}V(\frac{r}{2}).$
Let $g(r)=e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}}$. $g(r)>0$
when $r$ sufficiently large (say $r\geq 8n$) Since
$\displaystyle g^{\prime}(r)$
$\displaystyle=-\frac{r}{2}e^{-\frac{r^{2}}{4}}-\frac{n}{2}r^{\frac{n}{2}-1}e^{-\frac{r^{3}}{16}}+\frac{3r^{2}}{16}r^{\frac{n}{2}}e^{-\frac{r^{3}}{16}}$
$\displaystyle>(-\frac{r}{2}-\frac{n}{2}r^{\frac{n}{2}-1}+\frac{3}{16}r^{\frac{n}{2}+2})e^{-\frac{r^{3}}{16}}>0,$
$g(r)$ is increasing in $r$ and $g^{-1}(r)$ is decreasing in $r$. Therefore,
$g^{-1}(r)\leq\frac{1}{e^{-16n^{2}}-e^{-32n^{3}}(8n)^{\frac{n}{2}}}=C_{1}$
We then have
$V(r)\leq C_{1}r^{n}V(\frac{r}{2})\quad\quad\text{for $r$ sufficiently large
(say, $r\geq 8n$)}.$
By Lemma 4.1, we have
$V(r)\leq C_{4}e^{2n(\log r)^{2}}\quad for\quad r\geq 8n,$
here $C_{4}$ is a constant depending only on $n,V(8n)$. Hence, for any
$\alpha>0$
$\displaystyle\int_{M}e^{-\alpha z}d\mu=$
$\displaystyle\sum_{j=0}^{\infty}\int_{D_{8n(j+1)}\setminus D_{8nj}}e^{-\alpha
z}d\mu\leq\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}V(8n(j+1))$
$\displaystyle\leq$ $\displaystyle
C_{4}\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}e^{2n(\log(8n)+\log(j+1))^{2}}\leq
C_{5},$
where $C_{5}$ is a constant depending only on $n,V(8n)$. So we obtain our
estimates. Certainly, $M$ has weighted finite volume. ∎
###### Corollary 4.1.
Any space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
with closed Euclidean topology has finite fundamental group.
From the Gauss equation we have
$\text{Ric}(e_{i},e_{i})=\left<H,B_{ii}\right>-\sum_{j}\left<B_{ij},B_{ij}\right>,$
and
$\text{Hess}(f)(e_{i},e_{i})=\frac{1}{4}\text{Hess}(z)(e_{i},e_{i})=\frac{1}{2}\delta_{ij}+\frac{1}{2}\left<F,B_{ij}\right>=\frac{1}{2}\delta_{ij}-\left<H,B_{ij}\right>,$
It follows that
$\text{Ric}_{f}(e_{i},e_{i})=\text{Ric}(e_{i},e_{i})+\text{Hess}(f)(e_{i},e_{i})\geq\frac{1}{2}.$
Set $B_{R}(p)\subset M$, a geodesic ball of radius $R$ and centered at $p\in
M$. From Theorem 3.1 in [18] we know that for any $r$ there are constant
$A,\;B$ and $C$ such that
(4.4) $\int_{B_{R}(p)}\rho\leq A+B\int_{r}^{R}e^{-\frac{1}{2}t^{2}+Ct}dt$
## 5\. Rigidity results
Now, we are in a position to prove rigidity results mentioned in the
introduction.
###### Theorem 5.1.
Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.
###### Proof.
Let $\eta$ be a smooth function with compact support in $M,$ then by (3.14) we
obtain
(5.1) $\displaystyle\int_{M}(\frac{1}{2}|H|^{2}+|P|^{2}+|\nabla
H|^{2})\eta^{2}\rho$
$\displaystyle=\frac{1}{2}\int_{M}(\mathcal{L}|H|^{2})\eta^{2}\rho=\frac{1}{2}\int_{M}\text{div}(\rho\nabla|H|^{2})\eta^{2}$
$\displaystyle=-\int_{M}\eta\rho\nabla|H|^{2}\cdot\nabla\eta$
$\displaystyle=2\int_{M}\eta\rho\langle\nabla_{i}H,H\rangle\cdot\nabla_{i}\eta$
$\displaystyle\leq\int_{M}|\nabla
H|^{2}\eta^{2}\rho+\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$
We then have
(5.2)
$\int_{M}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\eta^{2}\rho\leq\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$
Let $\eta=\phi(\frac{|F|}{r})$ for any $r>0,$ where $\phi$ is a nonnegative
function on $[0,+\infty)$ satisfying
$\displaystyle\phi(x)=\left\\{\begin{array}[]{lllll}1&&&if&x\in[0,1)\\\
0&&&if&x\in[2,+\infty),\end{array}\right.$
and $|\phi^{\prime}|\leq C$ for some absolute constant. Since $\nabla
z=2F^{T}$,
$\nabla\eta=\frac{1}{r}\phi^{\prime}\nabla\sqrt{z}=\frac{1}{r}\phi^{\prime}\frac{F^{T}}{\sqrt{z}}.$
By (1.2) we have
$|\nabla\eta|^{2}\leq\frac{C^{2}}{r^{2}}\frac{|F^{T}|^{2}}{z}=\frac{1}{r^{2}z}C^{2}(z+4|H|^{2}).$
It follows that (5.2) becomes
(5.3)
$\int_{D_{r}}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\rho\leq\frac{C^{2}}{r^{2}}\int_{D_{2r}\setminus
D_{r}}|H|^{2}(1+\frac{4|H|^{2}}{z})\rho.$
By Theorem 4.1 then under the condition on $|H|$, we obtain that the right
hand side of (5.3) approaches to zero as $r\rightarrow+\infty.$ This implies
that $H\equiv 0.$
According to Theorem 3.3 in [16] we see that $M$ is complete with respect to
the induced metric from $\mathbb{R}^{m+n}_{m}$. In a geodesic ball $B_{a}(x)$
of radius $a$ and centered at $x\in M$ we can make gradient estimates of
$|B|^{2}$ in terms of the mean curvature. From (2.9) in [16] we have
$|B|^{2}\leq k\frac{2m(n-4)a^{2}}{(a^{2}-r^{2})^{2}}.$
Since $M$ is complete we can fix $x$ and let $a$ go to infinity. Hence,
$|B|^{2}=0$ at any $x\in M$ and $M$ is an $n-$plane. ∎
###### Theorem 5.2.
Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.
###### Proof.
(3.15) tells us
$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}\geq 0.$
As an application to (4.4) (Theorem 3.1 in [18]) the Corollary 4.2 in [18]
tells us that $\ln w$ is constant. This forces $|B|^{2}\equiv 0$. ∎
## References
* [1] E. Calabi: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Global Analysis U. C. Berkeley. (1968).
* [2] A. Chau, J. Chen, and Y. Yuan, Rigidity of Entire self-shrinking solutions to curvature flows, J. reine angew. Math. 664 (2012), 229-239.
* [3] Qun Chen and Hongbing Qiu: Rigidity theorems for self-shrinker in Euclidean space and Pseudo-Eclideanspace. Preprint.
* [4] S. Y. Cheng and S. T. Yau: Maximal spacelike hypersurfaces in the Lorentz- Minkowski spaces. Ann. Math. 104 (1976), 407-419.
* [5] Xu Cheng and Detang Zhou: Volume estimates about shrinkers, arXiv:1106.4950.
* [6] Tobias H. Colding and William P. Minicozzi II, Generic Mean Curvature Flow I; Generic Singularities, Ann. Math. 175 (2012), 755-833.
* [7] Qi Ding and Zhizhang Wang: On the self-shrinking system in arbitrary codimensional spaces. arXiv:1012.0429v2 [math.DG].
* [8] Qi Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, arXiv:1101.1411 [math.DG] (to appear in Asia J. Math.)
* [9] Qi Ding and Y. L. Xin: The rigidity theorems for Lagrangian self-shrinkers. arXiv:1112.2453 [math.DG] (to appear in J. reine angew. Math.)
* [10] K. Ecker: On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime. J. Austral. Math. Soc. Ser A 55 (1993) no. 1, 41-59.
* [11] K. Ecker: Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J. Differential Geom. 46 (1997) no.3, 481-498.
* [12] K. Ecker: Mean curvature flow of of spacelike hypersurfaces near null initial data. Comm. Anal. Geom. 11no. 2(2003), 181-205.
* [13] K. Ecker and G. Huisken: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys. 135(1991), 595-613.
* [14] Rongli Huang and Zhizhang Wang, On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 41 (2011), 321-339.
* [15] Hoeskuldur P. Halldorsson: Self-similar sulutions to the mean curvature flow in the Minkowski plane $\mathbb{R}^{1,1}$. arXiv:1212.0276v1[math.DG].
* [16] J. Jost and Y. L. Xin: Some aspects ofthe global geometry of entire space-like submanifolds. Result Math. 40 (2001), 233-245.
* [17] Yung-Chow Wong: Euclidean n-planes in pseudo-Euclidean spaces and differential geometry of Cartan domain. Bull. A. M. S. 75 (1969), 409-414.
* [18] Guofang Wei and Will Wylie: Comparison Geometry for Bakry-Emery Ricci Tensor, J. Differential Geometry 83(2)(2009), 377-405.
* [19] Y. L. Xin: Mean curvature flow with bounded Gauss image. Results Math. 59(2011), 415-436.
* [20] Y. L. Xin: On the Gauss image of a spacelike hypersurfaces with constant mean curvature in Minkowski space. Comment. Math. Helv.66(1991), 590-598.
* [21] Yuanlong Xin: Minimal submanifolds and related topics. World Scientific Publ. (2003).
* [22] Y. L. Xin and R. G. Ye: Bernstein-type theorems for space-like surfaces with parallel mean curvature. J. rein angew. math. 489(1997), 189-198.
|
arxiv-papers
| 2012-11-13T02:37:31 |
2024-09-04T02:49:37.903754
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huaqiao Liu and Y. L. Xin",
"submitter": "Yuanlong Xin",
"url": "https://arxiv.org/abs/1211.2876"
}
|
1211.2920
|
11institutetext: D. Moździerski 22institutetext: Instytut Astronomiczny,
Uniwersytet Wrocławski, Kopernika 11, 51-622, Wrocław, Poland, 22email:
mozdzierski@astro.uni.wroc.pl
# Variability survey in the young open cluster NGC 457
Dawid Moździerski Andrzej Pigulski Grzegorz Kopacki Marek Stȩślicki and
Zbigniew Kołaczkowski
###### Abstract
We present preliminary results of the photometric variability search in the
field of view of the young open cluster NGC 457. We find over 60 variable
stars in the field, including 25 pulsating or candidate pulsating stars.
## 1 The cluster
NGC 457 is a young open cluster in Cassiopeia, nearby $\phi$ Cas. Its age is
estimated for 10–20 Myr PheJan1994 ; Loktin2001 , distance, for about 2.5–3.0
kpc. It is located in the Perseus arm of the Galaxy. Six variable or suspected
variable stars were known in the observed cluster field prior to our study
(see, e.g., Maciejewski2008 ). We present preliminary results of the
photometric variability survey in this cluster aimed at discovery of B-type
pulsating stars and bright eclipsing binaries.
## 2 Observations and results
The photometric observations of NGC 457 were obtained during three runs: the
run consisting of 4 nights in 1993 carried out in Ostrowik station, University
of Warsaw, by one of us (GK), the second run of 31 nights in the years
1999–2002 (Białków station, University of Wrocław) and the third one
consisting of 24 nights made again in Białków between December 2010 and March
2011. Here, we present results based on the 2010–2011 observations only. Of
the three runs, the last one is of the best quality. During this run we used
60-cm reflecting telescope equipped with the Andor Tech. DW 432-BV CCD camera
covering 13${}^{\prime}\times$ 12′ field of view. All frames were calibrated
in a standard way and reduced with the Daophot II package Stetson1987 .
We have found 64 variable stars in the observed field, including the six
already known. The most interesting is the discovery of pulsating stars,
likely members of the cluster. The sample of pulsating stars includes: a
single $\beta$ Cephei star NGC 457-8 (see Fig. 1), 12 (candidate) SPB stars
and 12 $\delta$ Scuti stars. In addition, nine eclipsing and ellipsoidal
variables were found. The sample includes also stars showing irregular or
(quasi)periodic variations of unknown origin both in the cluster main sequence
and among the reddest stars in the field. Some of B-type stars showing this
type of variability are known as Be stars.
[scale=.65]Mozdzierski_Fig1.eps
Figure 1: Fourier amplitude spectrum for the $V$-filter data of $\beta$ Cephei
star NGC 457-8. The inset shows the light curve phased with the main period.
###### Acknowledgements.
We thank Dominik Drobek, Piotr Śródka and Ewa Zahajkiewicz for making some
observations of NGC 457. The work was supported by the MNiSzW grant No. N N203
302635.
## References
* (1) Loktin, A.V., Gerasimenko, T.P., Malysheva, L.K.: The catalogue of open cluster parameters — second version. Astronomical and Astrophysical Transactions. 20, 607–633 (2001)
* (2) Maciejewski, G., Bukowiecki, L., Brożek, T., Georgiev, Ts., Boeva, S., Kacharov, N., Mihov, B., Latev, G., Ovcharov, E., Valcheva, A.: Variable stars in the field of the open cluster NGC 457. Information Bulletin on Variable Stars. 5864, 1–7 (2008)
* (3) Phelps, R.L., Janes, K.A.: Young open clusters as probes of the star formation process. I. An atlas of open cluster photometry. ApJS. 90, 31–82 (1994)
* (4) Stetson, P.B.: DAOPHOT — A computer program for crowded-field stellar photometry. PASP. 99, 191–222 (1987)
|
arxiv-papers
| 2012-11-13T09:03:13 |
2024-09-04T02:49:37.909496
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Mo\\'zdzierski, A. Pigulski, G. Kopacki, M. St\\c{e}\\'slicki and Z.\n Ko{\\l}aczkowski",
"submitter": "Dawid Mo\\'zdzierski",
"url": "https://arxiv.org/abs/1211.2920"
}
|
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