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# Constructing silicon nanotubes by assembling hydrogenated silicon clusters
Lingju Guo Xiaohong Zheng Chunsheng Liu Wanghuai Zhou Zhi
Zeng111Corresponding author. E-mail: zzeng@theory.issp.ac.cn Key Laboratory of
Materials Physics, Institute of Solid State Physics, Chinese Academy of
Sciences, Hefei 230031, P.R. China
###### Abstract
The search or design of silicon nanostructures similar to their carbon
analogues has attracted great interest recently. In this work, density
functional calculations are performed to systematically study a series of
finite and infinite hydrogenated cluster-assembled silicon nanotubes (SiNTs).
It is found that stable one-dimensional SiNTs with formula
$Si_{m(3k+1)}H_{2m(k+1)}$ can be constructed by proper assembly of
hydrogenated fullerene-like silicon clusters $Si_{4m}H_{4m}$. The stability is
first demonstrated by the large cohesive energies and HOMO-LUMO gaps. Among
all such silicon nanotubes, the ones built from $Si_{20}H_{20}$($m=5$) and
$Si_{24}H_{24}$($m=6$) are the most stable due to the silicon bond angles that
are most close to the bulk $sp^{3}$ type in these structures. Thermostability
analysis further verifies that such tubes may well exist at room temperature.
Finally, both finite nanotubes and infinite nanotubes show a large energy gap.
A direct-indirect-direct band gap transition has been revealed with the
increase of the tube radius. The existence of direct band gap may make them
potential building blocks for electronic and optoelectronic devices.
Keywords: hydrogenation, silicon clusters, cluster-assembled nanotubes
###### pacs:
61.46.Bc, 61.46.Np, 73.22.-f
## I INTRODUCTION
Since the discovery and application of carbon fullerenes and carbon
nanotubes(CNTs) fullerene , stable cage and tube-like structures have
attracted a great deal of attention. Silicon and carbon are members of the
same group in the periodic table, suggesting a potential probability to form
similar structures. Furthermore, due to the fundamental importance of silicon
in present-day integrated circuits, substantial efforts have focused on
investigating nano-scale forms of silicon, both for the purpose of further
miniaturizing the current microelectronic devices and in the hopes of
unveiling new properties that often arise at the nano-scale levelJPCM_16_1373
. However, it is difficult to form cages or tubes like carbon fullerenes or
nanotubes purely with Si atoms because silicon does not favor the $sp$2
hybridization that carbon does. Carbon normally forms strong $\pi$ bonds
through $sp^{2}$ hybridization, which can facilitate the formation of two-
dimensional spherical cages (or planar structures such as benzene and
graphene). Silicon, on the other hand, usually forms covalent $\sigma$ bonds
through $sp^{3}$ hybridization, which favors a three dimensional diamond-like
structure.
Interestingly, it has been reported that Si cage clusters can be synthesized
by encapsulating suitable foreign atoms to terminate the dangling Si bonds
that inherently arise in cage-like networks. Many researchersjctn257 ; cms1 ;
prb77195417 reported that transition metal(TM) atoms are the most suitable
elements for cage formation due to their $d$ band features. In addition, rare
earth atoms have also been doped into silicon cagesjcp084711 ; epjd343 ;
prb125411 ; prb115429 . Another way to stabilize the Si cages is to terminate
the cluster surface by hydrogenprb075402 ; prb155425 ; prb80195417 ; handbook
, which is similar to the dodecahedral C20H20.
Meanwhile, tube-like silicon nanostructures have also attracted great
attention. Thus far, a few hollow and nonhollow silicon nanotube structures
have been proposed based on intuition or the behavior of similar materials and
theoretically characterized in recent years pnas2664 ; handbook ; prb80195417
; jmst127 ; prl265502 ; nl301 ; nl1243 ; nano109 ; jmc555 ; njp78 ; prl146802
; cpl81 ; prb075420 ; prb195426 ; jpcb7577 ; jpcb8605 ; prb9994 ; jpcc5598 ;
prb11593 ; prb205315 ; prb075328 ; ss257 ; prb193409 ; jpcc16840 ; prl1958 ;
pssr7 ; prb155432 ; prl792 ; jpcc1234 . Nevertheless, most of these structures
have the instability problem arising from the unsaturated dangling bonds.
Among all the present schemes for constructing silicon nanotubes, one is of
great interest in which metal atoms are encapsulated in the tubes. This scheme
has two advantages. On one hand, the metal atom is able to support the tube
wall so as not to collapse. On the other hand, it can saturate the dangling
bonds and further stabilize the structure. However, this scheme is limited to
tubes with very small radii($R$$\leq$1nm) since for larger tubes, the tube
becomes metallic, then the above two advantages will disappear.
In comparison, hydrogen saturation outside the tubes should also be a very
good way for the construction of silicon tubes for three reasons. At first,
each dangling bond can be saturated by one hydrogen atom. Thus the instability
caused by the dangling bonds would be removed and this scheme is not limited
to small size tubes. Moreover, compared with the metal encapsulation scheme,
the intrinsic features of silicon dominate in the properties of the tubes with
no interference of metal atoms. More importantly, in geometry, it is a “real”
empty tube, instead of a filled one. In fact, the effect of outside saturation
of dangling bonds has been already demonstrated in experiments, where the
surfaces of silicon nanowires(SiNWs) are always passivated by hydrogen atoms
science1874 ; jpcb8605 or by silicon oxide layers am1219 ; am1172 ; am564 ;
prl116102 .
Therefore, hydrogen terminated silicon cages will be perfect building blocks
for Si nanotubes and in this work, we present our design of hydrogenated
cluster assembled single wall silicon nanotubes and systematically investigate
their stabilities and electronic properties using density functional
theory(DFT) calculations. We find that stable one-dimensional silicon
nanotubes (SiNTs) with formula Sim(3k+1)H2m(k+1) can be constructed by proper
assembly of hydrogenated fullerene-like silicon clusters Si4mH4m and these
tubes can even exist at room temperatures.
## II COMPUTATIONAL DETAILS AND MODEL DESIGN
All theoretical computations are performed with the DFT approach implemented
in the Dmol3 package jcp92 ; jcp113 , using all electron treatment and the
double numerical basis including the $d$-polarization function(DNP)jcp92 . The
exchange-correlation interaction is treated within the generalized gradient
approximation(GGA) using BLYP functional. Self-consistent field calculations
are performed with a convergence criterion of 2$\times$10-5 Hartree on total
energy. The converge threholds are set to 0.002 Hartree/Å for forces and
0.005Å for the displacement.
The single Si4mH4m ($m=4,5,6,7,8$) cage-like clusters are optimized first, and
some of the initial structures are based on the results reported in Refs.
prb075402, & prb155425, . Then the optimized stable single clusters are taken
as basic units (keep them as original) and stacked together along the axis of
symmetry to construct finite nanotubes, with the adjacent two units sharing
the same bottom surface. One dimensional infinite nanotubes are also
investigated by including the smallest repeated unit cell in the supercell,
with the size chosen as 25Å$\times$25Å$\times$$L$z, where the direction of $z$
is defined as the axial direction of the nanotubes, and $L_{z}$ is the length
of the supercell in the z direction. Meanwhile, in order to avoid interaction
from the adjacent tubes, a sufficiently large vacuum region is introduced
along the radial directions. The Brillouin zone was sampled with a $1\times
1\times 20$ irreducible Monkhorst-Pack k-point grid for structural relaxation
and band structure calculations.
Thermal stability of the hydrogenated silicon nanotubes is studied within $ab$
$initio$ quantum molecular dynamics framework performed by heating at 400K by
using NVT(constant volume and temperature) dynamics with a massive Noseé-
Hoover thermostat. Time step is set as 1.0 fs, total simulation time was set
as 4.0 ps.
## III RESULTS AND DISCUSSIONS
### III.1 Structures of Si4mH4m clusters and finite nanotubes
The fully optimized structures of Si4mH4m ($m$=4, 5, 6, 7, 8) clusters are
shown in Fig. 1. All these structures share the following common
characteristics: 1. All of them are fullerene-like hollow structures; 2. Each
Si atom has three Si neighbors, with one H atom saturating the dangling bond
outside the cage and thus an $sp^{3}$ type hybridization is satisfied; 3. All
these structures consist of $2m$ polygons and two other polygons at the two
ends, with the edge number of these two polygons as $m$. Meanwhile, these two
polygons are parallel to each other, but with a relative angle of
$\frac{\pi}{m}$ between them. Thus each vertex atom of one polygon falls
exactly on the perpendicular bisector of one edge in the other polygon.
Specifically, for $Si_{20}H_{20}$($m$=5), the cage is composed of 12
pentagons, which is very similar to the structure of carbon fullerene
$C_{20}H_{20}$. In addition, structures of $Si_{16}H_{16}$($m$=4),
$Si_{24}H_{24}$($m$=6) and $Si_{28}H_{28}$($m$=7) have been widely discussed
in very recent years prb075402 ; prb155425 ; handbook and the structural
information we obtained is consistent with these reports.
Taking these original clusters as basic units, we stack them along the central
axis of the cage to form finite nanotubes. The two adjacent cages share the
same bottom polygon. We note that for the shared polygon, there is no need for
hydrogen saturation because each Si atom already has four Si neighbors and
thus the $sp^{3}$ hybridization bond type is fulfilled. The molecular formula
of the finite tube can be written as $Si_{m(3k+1)}H_{2m(k+1)}$, where the
number of units $k$ defines the length of the nanotube, while the number of
atoms $m$ in the bottom polygon defines the size in the directions vertical to
the axis and can be taken as a measurement of the radius. Consequently, each
group of $m$ and $k$ uniquely defines a nanotube with different radius and
length therefore such a nanotube can be denoted as NT($m,k$). After full
optimization, for one certain value $m$($m=4,5,6,7$), and for $k$ ranging from
2 to 4 concerned in the present work, the tubes are always straight and
stable. Furthermore, if the number of repeated units $k$ is fixed, the length
of the tubes decreases with the increasing $m$. The angles of H-Si-Si and Si-
Si-Si inside the repeated units are all about 109∘, which is very close to the
109.5∘ of $sp^{3}$, but the Si-Si-Si angle between two units becomes smaller
and smaller with the increase of tube radius (from 127.8∘ of $m$=4 to 99.0∘ of
$m$=8).
### III.2 Electronic structures
In order to measure the relative stability of the tubes as well as the
influence of the length and width, we have calculated the cohesive energy
($E_{coh}$). The $E_{coh}$ are defined by the following formulaapl203112 ;
prl157405 ; prb1419 :
$\ E_{coh}=\\{BE[NT(m,k)]+\mu_{H}N_{H}\\}/N_{Si}$ (1)
where BE[NT(m, k)] is the binding energy of finite
nanotube($Si_{m(3k+1)}H_{2m(k+1)}$), $N_{Si}$ and $N_{H}$ are the number of Si
and H atoms, respectively, $\mu_{H}$ is the chemical potential of H, and thus
the comparison of relative stability of different systems becomes
straightforward. This effectively removes the energy contribution of all Si-H
bonds in every system.
As illustrated in Fig. 2(a), the cohesive energy $E_{coh}$ of the finite tubes
is above 2.95eV and increases gradually as the length $k$ increases for each
fixed $m$, which indicates that the tube becomes increasingly stable as it
gets longer. On the other hand, the stability of the tube does not depend
monotonously on the tube radius as measured by $m$. Particularly, the
$E_{coh}$ curves of $m$=5 or $m$=6 are very close and are obviously higher
than the others, which means that, for any length $k$, the tubes with radius
$m$=5 or $m$=6 are the most stable.
In addition, Fig. 2(b) shows the variation of energy gaps between the highest
occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital
(LUMO) of the finite tubes. When $m$ (radius of the tube) is fixed, the HOMO-
LUMO gap decreases as the length ($k$) of the nanotube increases. Comparing
the gaps of the tubes with different radius (or $m$, see Fig. 2(b)), we find
that the tubes NT($4,k$), NT($5,k$) and NT($6,k$) always have similar HOMO-
LUMO gaps due to the analogous structural parameters characterized by bond
angle and length. With the further increase of the tube radius($m$= 7, 8), the
HOMO-LUMO gap decreases rapidly.
The spatial resolved local density of states (LDOS) of HOMO and LUMO states of
cage clusters and finite nanotubes are plotted in Fig. 3 for the understanding
of the bonding properties. For the convenience of descriptions, we define the
atoms of two opposite polyhedron as bottom atoms, and the other atoms lying
between the two polyhedrons as side atoms. For small size clusters ($m=4,5$),
both bottom atoms and side atoms have contributions on HOMOs, but for large
size clusters ($m=6,7,8$), HOMOs are almost localized on the side silicon
atoms. On the other hand, The LUMO states of small size clusters ($m=4,5,6$)
are mainly localized inside the cage (Fig. 3(a)) while for bigger size
clusters $m=7,8$, the LUMO states are localized around the bottom atoms. For
finite nanotubes NT($m,4$) with small size ($m=4,5$), both the HOMO and LUMO
are distributed on the side atoms and the shared bottom atoms (see (Fig.
3(b)). However, for finite tubes with large sizes ($m=6,7,8$), the HOMO is
localized on the side atoms while the LUMO is localized on the shared bottom
atoms, which are very similar to their building units, namely, the cage
clusters. Furthermore, with the increase of the radius, both the HOMO and LUMO
tend to be localized on the middle atoms while the contributions from the end
atoms become negligible.
### III.3 Infinite nanotubes
The increased stability of finite nanotubes with increasing length leads to
our interest in examining further the stability of infinite nanotubes. The
smallest repeated unit cell of the infinite nanotubes can be obtained by
removing the Si and H atoms of one bottom polygon and the H atoms of the other
bottom polygon of a finite tube NT$(m,2)$ or $Si_{7m}H_{6m}$, and thus it can
be described by a formula $Si_{6m}H_{4m}$, where $m$ is again the number of
atoms in the bottom or connecting polygon. Note that the repeated unit cell
should not be based on NT$(m,1)$ but on NT$(m,2)$ to produce a periodic
system. Therefore we can define these infinite tubes as NT(m, $\infty$)
($m=4,5,6,7,8$). Two repeated cells of the infinite tubes are shown in Fig. 4
with different radii concerned in this work.
Full structure relaxation indicates that the infinite nanotubes have similar
geometric structures to finite ones, but the length of the smallest repeated
cell is slightly changed. The lengths of the smallest repeated cell of the
$m=4$ tube and the $m=5$ tube become 0.08Å and 0.11Å longer than those of the
finite ones for producing these repeated cells. But for $m=6$, $m=7$ and
$m=8$, the lengths become 0.075Å, 0.212Å and 0.365Å shorter. The diameter of
tubes and Si-H bond lengths are almost the same as those of the finite ones.
The average Si-Si bond length is 2.401Å, and the Si-H bond length is 1.497Å,
which are similar to the 2.36Å and 1.50Å in Ref. pssr7, , and the
exohydrogenated carbon nanotube like structures (2.34Å and 1.51Å in Ref.
prb193409, , 2.335Å and 1.521Å in Ref. ss257, ).
In order to study the stability of the infinite tubesNT(m, $\infty$), their
cohesive energies are calculated and included in Fig. 2(a). By comparison, we
find that the cohesive energies of them are larger than those of finite ones,
which means that it is possible to synthesize long tubes. Further, thermal
stability of the hydrogenated silicon tubes has been checked within $ab$
$initio$ quantum molecular dynamics framework performed by heating at 400K for
4.0 ps with the time step of 1.0 fs using NVT(constant volume and temperature)
dynamics with a massive Noseé-Hoover thermostat. The cluster model is
implemented for these tubes without any symmetry constraints so that all atoms
are allowed to move freely. No collapse is found at this time scale,
indicating that these hydrogenated silicon nanotubes may survive at room
temperature.
Another concern about the structural stability of the proposed hydrogenated
silicon tubes comes with whether multiple tubes will be collapsed and merged
into a large cluster when they are put together. Our calculations on NT(5,
$\infty$) and NT(7, $\infty$) show that when two, three or four nanotubes are
placed together in parallel, with very small initial distance between them,
after full optimization, all of these tubes separate away from each other with
no collapse or distortion. The separation is largely due to the mutual
repulsion of the surface hydrogen atoms. It gives another proof that single
hollow silicon nanotubes can well exist and they will not be combined together
with the “protection” from the surface hydrogen atoms between the tubes.
Finally, the band gaps($\Delta_{g}$) of the infinite tubes are analyzed and
exhibit large values from 2.3eV of NT(8, $\infty$) to 2.8eV of NT(4,
$\infty$), which implies that they are wide gap semiconductors. Particularly,
the band gaps of NT(5, $\infty$) and NT(6, $\infty$) are obtained as 2.69 eV
and 2.71 eV, respectively, which agree very well with the 2.65 eV and 2.70 eV
reported in Ref. pssr7, for the same structures. The band gap of the tube is
inversely proportional to the radius, which is similar to the gap changes in
exohydrogenated single-wall carbon nanotubes(SWCNT)prb075404 and
exohydrogenated single-wall silicon nanotubesprb193409 . Meanwhile, seen from
Fig. 5, the type of the band gaps can be controlled by tuning the tube radius.
The smallest tube NT(4, $\infty$) (Fig. 5(a)) has a direct band gap at
Z-point, while NT(5,$\infty$)(Fig. 5(b)) has an indirect band gap. When $m$ in
NT($m$, $\infty$) goes to $m=6,7,8$, these tubes all display a direct band gap
at $\Gamma$-point. From Fig. 5, we see that for $m=4$, both the bottom of the
conduction band (BC) and the top of the valence band (VT) lie at Z point, thus
the $m=4$ tube is a semiconductor with a direct gap at Z point. Interestingly,
with the increase of the radius, both the BC and TV move toward the $\Gamma$
point. However, the BC moves much faster than TV, thus an indirect gap is
observed with $m=5$. After $m=6$, both BC and TV arrive at $\Gamma$ point and
a direct gap is always observed . We believe such an band evolution is related
to the structure changes with the radius, and the nanotubes have a one-
dimension-like to three-dimension-like transition as the increasing radius.
For the thinnest nanotube NT(4, $\infty$), the confinement along Z direction
is smaller than that along vertical directions because of the one dimension
feature. Consequently the band gap at $\Gamma$ is larger than that at Z, then
the effective gap opens at Z point. On the other hand, for the largest radius
tube NT(8, $\infty$), three dimensional feature is apparently shown and the
confinement along vertical directions is relatively smaller than that of small
tubes. This makes the band gap at $\Gamma$ be small, so that the effective gap
opens at $\Gamma$ point. Accordingly, as the radius of tube changes from NT(4,
$\infty$) to NT(8, $\infty$), the confinement along vertical directions
changes evidently. This variation leads to a transition between direct band
gap and indirect band gap. Larger tubes NT (9, $\infty$) and NT (10, $\infty$
) are tested, and the band structures of the tubes are shown in supplementary
materials. We can find that they are also both semiconductors with a direct
band gap at $\Gamma$ point. Our results also show that band gap open at
$\Gamma$ is more sensitive to the change of diameter than that at Z point. The
reason for this is that the size of basic repeated unit cell keeps almost
unchanged in building nanotube, hence the physical phenomena close to the size
of basic repeated unit cell are not sensitive to the size change of nanotube.
In fact, such a size induced change is also observed in the HOMO-LUMO gap and
the cohesive energy in the finite tubes (see Fig. 2). In Fig. 2, we see that
$m=5$ is a special size, since after $m=5$, both the HOMO-LUMO gap and the
cohesive energy changes monotonously with the radius. The existence of direct
band gap in hydrogenated silicon nanotubes is quite important for the
utilization of these nanosturctures in building nanoscale optoelectronic
devices.
## IV CONCLUSIONS
A series of finite and infinite hydrogenated silicon nanotubes are
systematically studied by performing first-principles calculations. Our
results reveal that one-dimensional stable SiNTs $Si_{m(3k+1)}H_{2m(k+1)}$ can
be built by stacking $Si_{4m}H_{4m}$ cagelike clusters along the central axis
of the cage. These tubes have large cohesive energies and HOMO-LUMO gaps.
Among all the tubes, those with the sizes of $m$=5 and 6 are the most stable,
because their $sp^{3}$ bond angle is most close to 109.5∘. Thermodynamics
analysis shows that these tubes may exist at room temperature, which further
confirms the stability of the proposed silicon nanotubes. The infinite silicon
nanotubes have also been investigated and it is found that there is a direct-
indirect-direct band gap change with the increasing radius. For $m=4$, the
direct gap opens at Z point, while for $m\geq 6$, the direct gap opens at
$\Gamma$ point. In the $m=5$ case, an indirect gap is observed.
We want to note that, although single wall silicon nanotubes have attracted
great attention recently, such silicon nanotubes have not been synthesized yet
experimentally due to the $sp^{3}$ hybridization of silicon and the subsequent
unsaturated dangling bonds. Our study indicates that hydrogen passivation may
be a good way to stabilize hollow single wall silicon nanotubes. Particularly,
necklace-like hollow structures built from single cage-like clusters are
proposed in our work. These findings may provide some useful suggestions for
experimental fabrications of hollow single wall silicon nanotubes.
## ACKNOWLEDGMENTS
This work was supported by the National Science Foundation of China under
Grants No. 10774148 and No. 10904148, the special Funds for Major State Basic
Research Project of China (973) under grant No. 2007CB925004. Some of the
calculations were performed at the Center for Computational Science of
CASHIPS.
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* (49) A. Puzder, A. J. Williamson, F. A. Reboredo, and G. Galli, Phys. Rev. Lett. 91, 157405 (2003),
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* (51) T. Yildirim, O. Gülseren, and S. Ciraci, Phys. Rev. B 64, 075404(2001).
Figure captions $Fig$. 1. (Color online) Top view of the optimized
$Si_{4m}H_{4m}$ ($m$=4, 5, 6, 7, 8) clusters (the 1st row) and side view of
cluster-assembled nanotubes $Si_{m(3k+1)}H_{2m(k+1)}$ ($m$=4, 5, 6, 7, 8;
$k$=1, 2, 3, 4) (all the rest). $Fig$. 2. (Color online) (a) The cohesive
energy of finite $Si_{m(3k+1)}H_{2m(k+1)}$ ($m$=4, 5, 6, 7, 8; $k$=1, 2, 3,
4,), (b) HOMO-LUMO gaps of finite $Si_{m(3k+1)}H_{2m(k+1)}$ ($m$=4, 5, 6, 7,
8; $k$=1, 2, 3, 4) tubes. For comparison, the binding energies of infinite
nanotube NT(m, $\infty$) are also put at the ending of (a). $Fig$. 3. (Color
online) The spatial resolved local density of states (LDOS) with the level of
0.015e/Å3 of highest-occupied molecular orbital (HOMO) and lowest-unoccupied
molecular orbital (LUMO) of (a) $Si_{4m}H_{4m}$ ($m$=4, 5, 6, 7, 8) clusters
and (b) $Si_{m(3k+1)}H_{2m(k+1)}$ ($m$=4, 5, 6, 8; $k$=4) tubes. $Fig$. 4.
(Color online) Side views of two unit cells of $Si_{6m}H_{4m}$ infinite
nanotubes NT($m$, $\infty$) ($m$=5, 6, 7, 8). $Fig$. 5. Electronic band
structures of the infinite nanotubes (a) NT(4, $\infty$), (b) NT(5, $\infty$),
(c) NT(6, $\infty$), (d) NT(7, $\infty$), (e) NT(8, $\infty$).
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
|
arxiv-papers
| 2011-07-17T09:17:50 |
2024-09-04T02:49:20.663918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lingju Guo, Xiaohong Zheng, Chunsheng Liu, Wanghuai Zhou, and Zhi Zeng",
"submitter": "Lingju Guo",
"url": "https://arxiv.org/abs/1107.3286"
}
|
1107.3356
|
# Self-adjoint commuting differential operators and commutative subalgebras of
the Weyl algebra
Andrey E. Mironov Sobolev Institute of Mathematics, Novosibirsk, Russia, _
_ Novosibirsk State University, _and_
Laboratory of Geometric Methods in Mathematical Physics, Moscow State
University mironov@math.nsc.ru
This paper was finished in the Hausdorff Research Institute for Mathematics
(Bonn). The author is grateful to the Institute for hospitality. This work was
also partially supported by the Presidium of the Russian Academy of Sciences
(under the program ”Fundamental Problems of Nonlinear Dynamics”); grant
MD-5134.2012.1 from the President of Russia; and a grant from Dmitri Zimin’s
”Dynasty” foundation.
> Abstract. In this paper we study self-adjoint commuting ordinary
> differential operators. We find sufficient conditions when an operator of
> fourth order commuting with an operator of order $4g+2$ is self-adjoint. We
> introduce an equation on potentials $V(x),W(x)$ of the self-adjoint operator
> $L=(\partial_{x}^{2}+V)^{2}+W$ and some additional data. With the help of
> this equation we find the first example of commuting differential operators
> of rank two corresponding to a spectral curve of arbitrary genus. These
> operators have polynomial coefficients and define commutative subalgebras of
> the first Weyl algebra.
## 1\. Introduction
The problem of finding commuting differential operators is a classical problem
of differential equations (for the first results see [1]–[3]). In the case of
operators of rank greater than one, this problem has not been solved until
now. In this paper we study self-adjoint commuting ordinary differential
operators. One of the main results of this paper is the following. We find an
example of commuting differential operators of rank two corresponding to
spectral curves of arbitrary genus.
If two differential operators
$L_{n}=\partial_{x}^{n}+\sum_{i=0}^{n-2}u_{i}(x)\partial_{x}^{i},\ \
L_{m}=\partial_{x}^{m}+\sum_{i=0}^{m-2}v_{i}(x)\partial_{x}^{i}\ $
commute, then there is a nonzero polynomial $R(z,w)$ such that
$R(L_{n},L_{m})=0$ (see [3]). The curve $\Gamma$ defined by $R(z,w)=0$ is
called the spectral curve. This curve parametrizes common eigenvalues of the
operators. If
$L_{n}\psi=z\psi,\qquad L_{m}\psi=w\psi,$
then $(z,w)\in\Gamma$. For almost all $(z,w)\in\Gamma$ the dimension of the
space of common eigenfunctions $\psi$ is the same. The dimension is called the
rank. The rank equals the greatest common divisor of $m$ and $n$.
In this paper we consider only commuting ordinary differential operators whose
spectral curves are smooth. Commutative rings of such operators were
classified by Krichever [4], [5]. The ring is determined by the spectral curve
and some additional spectral data. If the rank is one, then the spectral data
define commuting operators by explicit formulas (see [4]). In the case of
operators of rank greater than one there are the following results. Krichever
and Novikov [6], [7] using the method of deformation of Tyurin parameters
found operators of rank two corresponding to an elliptic spectral curves.
These operators were studied in the papers [8]–[16]. Mokhov [17], using the
same method found operators of rank three also corresponding to elliptic
spectral curves. Besides this there are examples of operators of rank grater
than one corresponding to spectral curves of genus $2,3$ and $4$ (see
[18]–[21]).
The main results of this paper are the following. We consider a pair
$L_{4},L_{4g+2}$ of commuting differential operators of rank two whose
spectral curve is a hyperelliptic curve $\Gamma$ of genus $g$
(1) $w^{2}=F_{g}(z)=z^{2g+1}+c_{2g}z^{2g}+\dots+c_{0}.$
Operators $L_{4}$ and $L_{4g+2}$ satisfy the equation
$(L_{4g+2})^{2}=F_{g}(L_{4})$. The curve $\Gamma$ has a holomorphic involution
$\sigma:\Gamma\rightarrow\Gamma,\qquad\sigma(z,w)=(z,-w).$
Common eigenfunctions of $L_{4}$ and $L_{4g+2}$ satisfy the second order
differential equation [5]
(2)
$\psi^{\prime\prime}(x,P)=\chi_{1}(x,P)\psi^{\prime}(x,P)+\chi_{0}(x,P)\psi(x,P).$
The coefficients $\chi_{0}(x,P),\chi_{1}(x,P)$ are rational functions on
$\Gamma$ with $2g$ simple poles depending on $x$, $\chi_{0}$ has also an
additional simple pole at infinity. These functions satisfy Krichever’s
equations (see below). To find operators $L_{4},L_{4g+2}$ it is enough to find
$\chi_{0},\chi_{1}$.
It is not difficult to prove that if $\chi_{1}$ is invariant under the
involution $\sigma$, then the operator $L_{4}$ is self-adjoint. S.P. Novikov
has proposed the conjecture that the inverse is also true. In this paper we
prove this conjecture.
Theorem 1 The operator $L_{4}$ is self-adjoint if and only if
(3) $\chi_{1}(x,P)=\chi_{1}(x,\sigma(P)).$
At $g=1$ Theorem 1 was proved by Grinevich and Novikov [8].
Let us assume that the operator $L_{4}$ is self-adjoint
$L_{4}=(\partial_{x}^{2}+V(x))^{2}+W(x),$
then the functions $\chi_{0},\chi_{1}$ have simple poles at some points
$(\gamma_{i}(x),\pm\sqrt{F_{g}(\gamma_{i}(x))}),\ 1\leq i\leq g.$
In the next theorem we find the form of $\chi_{0}(x,P),\chi_{1}(x,P).$
Theorem 2 If operator $L_{4}$ is self-adjoint, then
$\chi_{0}=-\frac{1}{2}\frac{Q^{\prime\prime}}{Q}+\frac{w}{Q}-V,\qquad\chi_{1}=\frac{Q^{\prime}}{Q},$
where $Q=(z-\gamma_{1}(x))\dots(z-\gamma_{g}(x))$. Functions $Q,V,W$ satisfy
the equation
(4)
$4F_{g}(z)=4(z-W)Q^{2}-4V(Q^{\prime})^{2}+(Q^{\prime\prime})^{2}-2Q^{\prime}Q^{(3)}+2Q(2V^{\prime}Q^{\prime}+4VQ^{\prime\prime}+Q^{(4)}),$
where $Q^{\prime},Q^{\prime\prime},Q^{(k)}$ mean
$\partial_{x}Q,\partial_{x}^{2}Q,\partial_{x}^{k}Q.$
To find self-adjoint operators $L_{4},L_{4g+2}$ it is enough to solve the
equation (4).
In this paper we find partial solutions of the equation for arbitrary $g$.
These solutions correspond to operators with polynomial coefficients.
Theorem 3 The operator
$L^{{}^{\sharp}}_{4}=(\partial_{x}^{2}+\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\alpha_{0})^{2}+g(g+1)\alpha_{3}x,\qquad\alpha_{3}\neq
0$
commutes with a differential operator $L_{4g+2}^{{}^{\sharp}}$ of order
$4g+2$. The operators $L_{4}^{{}^{\sharp}},$ $L_{4g+2}^{{}^{\sharp}}$ are
operators of rank two. For generic values of parameters
$(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})$ the spectral curve is a
nonsingular hyperelliptic curve of genus $g$.
If $g=1$, $\alpha_{1}=\alpha_{2}=0,\ \alpha_{3}=1$, then the operators
$L^{{}^{\sharp}}_{4},L_{4g+2}^{{}^{\sharp}}$ coincide with the famous Dixmier
operators [22] whose spectral curve is an elliptic curve. Operators
$L^{{}^{\sharp}}_{4},L_{4g+2}^{{}^{\sharp}}$ define commutative subalgebras in
the first Weyl algebra $A_{1}$. Theorem 3 means that the equation
$Y^{2}=X^{2g+1}+c_{2g}X^{2g}+\dots+c_{0}$ has nonconstant solutions $X,Y\in
A_{1}$ for some $c_{i}$. It is easy to see that the group $Aut(A_{1})$
preserves the space of all such solutions. It would be very interesting to
describe the orbits of $Aut(A_{1})$ in the space of solutions under the action
of $Aut(A_{1})$. This gives a chance to compare $End(A_{1})$ and $Aut(A_{1})$
(the Dixmier conjecture is: $End(A_{1})=Aut(A_{1})$).
In Section 2 we recall the method of deformations of Tyurin parameters. In
Sections 3–5 we prove Theorems 1–3.
The author is grateful to I.M. Krichever, O.I. Mokhov, S.P. Novikov and V.V.
Sokolov for valuable discussions and stimulating interest.
## 2\. Operators of rank $l>1$
Common eigenfunctions of commuting differential operators are Baker–Akhiezer
functions. Let me recall the definition of the Baker–Akhiezer function at
$l>1$ [5]. We take the spectral data
$\\{\Gamma,q,k^{-1},\gamma,v,\omega(x)\\},$
where $\Gamma$ is a Riemann surface of genus $g$, $q$ is a fixed point on
$\Gamma$, $k^{-1}$ is a local parameter near $q$,
$\omega(x)=(\omega_{0}(x),\dots,\omega_{l-2}(x))$
is a set of smooth functions, $\gamma=\gamma_{1}+\dots+\gamma_{lg}$ is a
divisor on $\Gamma$, $v$ is a set of vectors
$v_{1},\dots,v_{lg},\qquad v_{i}=(v_{i,1},\dots,v_{i,l-1}).$
The pair $(\gamma,v)$ is called the Tyurin parameters. The Tyurin parameters
define a stable holomorphic vector bundle on $\Gamma$ of rank $l$ and degree
$lg$ with holomorphic sections $\eta_{1},\dots,\eta_{l}$. The points
$\gamma_{1},\dots,\gamma_{lg}$ are the points of the linear dependence
$\eta_{l}(\gamma_{i})=\sum_{i=1}^{l-1}v_{j,i}\eta_{j}(\gamma_{i}).$
The vector-function $\psi=(\psi_{1},\dots,\psi_{l})$ is defined by the
following properties.
1\. In the neighbourhood of $q$ the vector-function $\psi$ has the form
$\psi(x,P)=\left(\sum_{s=0}^{\infty}\xi_{s}(x)k^{-s}\right)\Psi_{0}(x,k),$
where
$\xi_{0}=(1,0,\dots,0),\xi_{i}(x)=(\xi_{i}^{1}(x),\dots,\xi_{i}^{l}(x))$, the
matrix $\Psi_{0}$ satisfies the equation
$\frac{d\Psi_{0}}{dx}=A\Psi_{0},\
A=\left(\begin{array}[]{cccccc}0&1&0&\dots&0&0\\\ 0&0&1&\dots&0&0\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&0&0&\dots&0&1\\\
k+\omega_{0}&\omega_{1}&\omega_{2}&\dots&\omega_{l-2}&0\end{array}\right).$
2\. The components of $\psi$ are meromorphic functions on
$\Gamma\backslash\\{q\\}$ with the simple poles
$\gamma_{1},\dots,\gamma_{lg}$, and
${\rm Res}_{\gamma_{i}}\psi_{j}=v_{i,j}{\rm Res}_{\gamma_{i}}\psi_{l},\quad
1\leq i\leq lg,\ 1\leq j\leq l-1.$
For the rational function $f(P)$ on $\Gamma$ with the unique pole of order $n$
at $q$ there is a linear differential operator $L(f)$ of order $ln$ such that
$L(f)\psi(x,P)=f(P)\psi(x,P).$
For two such functions $f(P),g(P)$ operators $L(f)$, $L(g)$ commute.
The main difficulty to construct operators of rank $l>1$ is the fact that the
Baker–Akhiezer function is not found explicitly. But the operators can be
found by the method of deformation of Tyurin parameters.
The common eigenfunctions of commuting differential operators of rank $l$
satisfy the linear differential equation of order $l$
$\psi^{(l)}(x,P)=\chi_{0}(x,P)\psi(x,P)+\dots+\chi_{l-1}(x,P)\psi^{(l-1)}(x,P).$
Coefficients $\chi_{i}$ are rational functions on $\Gamma$ [5] with simple
poles $P_{1}(x),\dots,P_{lg}(x)\in\Gamma$, and with the following expansions
in the neighbourhood of $q$
$\chi_{0}(x,P)=k+g_{0}(x)+O(k^{-1}),$ $\chi_{j}(x,P)=g_{j}(x)+O(k^{-1}),\ \
0<j<l-1,$ $\chi_{l-1}(x,P)=O(k^{-1}).$
Let $k-\gamma_{i}(x)$ be a local parameter near $P_{i}(x)$. Then
$\chi_{j}=\frac{c_{i,j}(x)}{k-\gamma_{i}(x)}+d_{i,j}(x)+O(k-\gamma_{i}(x)).$
Functions $c_{ij}(x),d_{ij}(x)$ satisfy the following equations [5].
Theorem 4
(5) $c_{i,l-1}(x)=-\gamma^{\prime}_{i}(x),$ (6)
$d_{i,0}(x)=v_{i,0}(x)v_{i,l-2}(x)+v_{i,0}(x)d_{i,l-1}(x)-v^{\prime}_{i,0}(x),$
(7)
$d_{i,j}(x)=v_{i,j}(x)v_{i,l-2}(x)-v_{i,j-1}(x)+v_{i,j}(x)d_{i,l-1}(x)-v^{\prime}_{i,j}(x),j\geq
1,$
where
$v_{i,j}(x)=\frac{c_{i,j}(x)}{c_{i,l-1}(x)},\ \ 0\leq j\leq l-1,\ 1\leq i\leq
lg.$
To find $\chi_{i}$ one should solve equations (5)–(7).
## 3\. Proof of Theorem 1
In the case of operators of rank two the common eigenfunctions of $L_{4}$ and
$L_{4g+2}$ satisfy equation (2). In the neighbourhood of $q$ we have the
expansions
(8)
$\chi_{0}=\frac{1}{k}+a_{0}(x)+a_{1}(x)k+O(k^{2}),\quad\chi_{1}=b_{1}(x)k+b_{2}(x)k^{2}+O(k^{3}).$
Functions $\chi_{0},\chi_{1}$ have $2g$ simple poles
$P_{1}(x),\dots,P_{2g}(x)$, and by Theorem 4
(9)
$\chi_{0}(x,P)=\frac{-v_{i,0}(x)\gamma^{\prime}_{i}(x)}{k-\gamma_{i}(x)}+d_{i,0}(x)+O(k-\gamma_{i}(x)),$
(10)
$\chi_{1}(x,P)=\frac{-\gamma^{\prime}_{i}(x)}{k-\gamma_{i}(x)}+d_{i,1}(x)+O(k-\gamma_{i}(x)),$
(11) $d_{i,0}(x)=v_{i,0}^{2}(x)+v_{i,0}(x)d_{i,1}(x)-v^{\prime}_{i,0}(x).$
Let $\Gamma$ be the hyperelliptic spectral curve (1), $q=\infty\in\Gamma,$
$k=\frac{1}{\sqrt{z}}$.
Let us find coefficients of the operator of order 4 corresponding to $z$,
$L_{4}\psi=z\psi.$
Lemma 1 The operator
$L_{4}=\partial_{x}^{4}+f_{2}(x)\partial_{x}^{2}+f_{1}(x)\partial_{x}+f_{0}(x)$
has the following coefficients:
$f_{0}=a_{0}^{2}-2a_{1}-2b_{1}^{\prime}-a_{0}^{\prime\prime},\qquad
f_{1}=-2(b_{1}+a_{0}^{\prime}),\qquad f_{2}=-2a_{0}.$
Operator $L_{4}$ is self-adjoint if and only if $b_{1}=0$, herewith
$L_{4}=(\partial_{x}^{2}+V(x))^{2}+W(x),$ where $V(x)=-a_{0}(x),$
$W=-2a_{1}(x)$.
Proof. From (2) it follows that the fourth derivative of $\psi$ is
$\psi^{(4)}=(\chi_{0}^{2}+\chi_{1}\chi_{0}^{\prime}+\chi_{0}(\chi_{1}^{2}+2\chi_{1}^{\prime})+\chi_{0}^{\prime\prime})\psi+(\chi_{1}^{3}+2\chi_{0}^{\prime}+\chi_{1}(2\chi_{0}+3\chi_{1}^{\prime})+\chi_{1}^{\prime\prime})\psi^{\prime}.$
With the help of (2) and the last equality we rewrite $L_{4}\psi=z\psi$ in the
form
$P_{1}\psi+P_{2}\psi^{\prime}=z\psi,$
where
$P_{1}=f_{0}+f_{2}\chi_{0}+\chi_{0}^{2}+\chi_{1}\chi_{0}^{\prime}+\chi_{0}(\chi_{1}^{2}+2\chi_{1}^{\prime})+\chi_{0}^{\prime\prime},$
$P_{2}=f_{1}+f_{2}\chi_{1}+\chi_{1}^{3}+2\chi_{0}^{\prime}+\chi_{1}(2\chi_{0}+3\chi_{1}^{\prime})+\chi_{1}^{\prime\prime}.$
This gives
(12) $P_{1}=z=\frac{1}{k^{2}},\qquad\ P_{2}=0.$
From (8) we have
$P_{1}-\frac{1}{k^{2}}=\frac{f_{2}+2a_{0}}{k}+(f_{0}+a_{0}(f_{2}+a_{0})+2(a_{1}+b_{1}^{\prime})+a_{0}^{\prime\prime})+O(k)=0,$
$P_{2}=(f_{1}+2(b_{1}+a_{0}^{\prime}))+O(k)=0.$
From here we find the coefficients of $L_{4}$.
Operator $L_{4}$ is self-adjoint if $f_{1}=f_{2}^{\prime}$, i.e. at $b_{1}=0$.
Lemma 1 is proved.
If $\chi_{1}$ satisfies (3) then $\chi_{1}=\sum_{s>1}b_{2s}k^{2s},$ hence, by
Lemma 1 $L_{4}$ is self-adjoint.
Let us prove the inverse part of Theorem 1. We assume that $L_{4}$ is self-
adjoint
$L_{4}=L_{4}^{*}=\partial_{x}^{4}+f_{2}(x)\partial_{x}^{2}+f_{2}^{\prime}(x)\partial_{x}+f_{0}(x).$
If $\psi_{1},\psi_{2}\in Ker(L_{4}-z)$, then
$\psi_{1}L_{4}\psi_{2}-\psi_{2}L_{4}\psi_{1}=\partial_{x}(\psi_{1}\psi_{2}^{\prime\prime\prime}-\psi_{2}\psi_{1}^{\prime\prime\prime}-(\psi_{1}^{\prime}\psi_{2}^{\prime\prime}-\psi_{2}^{\prime}\psi_{1}^{\prime\prime})+f_{2}(\psi_{1}\psi_{2}^{\prime}-\psi_{2}\psi_{1}^{\prime}))=0.$
Hence, on the space $Ker(L_{4}-z)$ the following skew-symmetric bilinear form
$(.,.):Ker(L_{4}-z)\times Ker(L_{4}-z)\rightarrow{\mathbb{C}},$
$(\psi_{1},\psi_{2})=\psi_{1}\psi_{2}^{\prime\prime\prime}-\psi_{2}\psi_{1}^{\prime\prime\prime}-(\psi_{1}^{\prime}\psi_{2}^{\prime\prime}-\psi_{2}^{\prime}\psi_{1}^{\prime\prime})+f_{2}(\psi_{1}\psi_{2}^{\prime}-\psi_{2}\psi_{1}^{\prime})$
is defined. Let $\psi_{1}(x,P),\psi_{2}(x,P)$ satisfy the equation (2). Using
$\psi_{i}^{\prime\prime\prime}=(\chi_{0}+\chi_{1}^{2}+\chi_{1}^{\prime})\psi_{i}^{\prime}+(\chi_{0}\chi_{1}+\chi_{0}^{\prime})\psi_{i}$
we get
$(\psi_{1},\psi_{2})=(\psi_{1}\psi_{2}^{\prime}-\psi_{2}\psi_{1}^{\prime})(f_{2}+2\chi_{0}+\chi_{2}^{2}+\chi_{1}^{\prime}).$
Since $\psi_{1},\psi_{2}$ satisfy the second order differential equation (2)
we have,
$(\psi_{1},\psi_{2})=e^{\int\chi_{1}(x,z,w)dx}g_{1}(z,w)\left(f_{2}(x)+2\chi_{0}(x,z,w)+\chi_{1}^{2}(x,z,w)+\chi_{1}^{\prime}(x,z,w)\right)$
$=g_{2}(z,w),$
where $g_{1}(z,w),g_{2}(z,w)$ are some functions on $\Gamma$. Let us represent
$\chi_{1}$ in the form
$\chi_{1}(x,z,w)=G_{1}(x,z)+wG_{2}(x,z),$
where $G_{1},G_{2}$ are rational functions on $\Gamma$. Let
$\tilde{G}_{1}(x,z)=\int G_{1}(x,z)dx,\qquad\tilde{G}_{2}(x,z)=\int
G_{2}(x,z)dx,$
then
$e^{\tilde{G}_{1}(x,z)}\left(e^{\tilde{G}_{2}(x,z)}\right)^{w}\frac{g_{1}(z,w)}{g_{2}(z,w)}=\frac{1}{f_{2}+2\chi_{0}+\chi_{2}^{2}+\chi_{1}^{\prime}}.$
From the last identity it follows that for arbitrary $x=x_{1},x=x_{2}$ the
function
$e^{\tilde{G}_{1}(x_{1},z)-\tilde{G}_{1}(x_{2},z)}\left(e^{\tilde{G}_{2}(x_{1},z)-\tilde{G}_{2}(x_{2},z)}\right)^{w}$
is a rational function on $\Gamma$. This is possible only if
$\tilde{G}_{2}(x_{1},z)-\tilde{G}_{2}(x_{2},z)=0,$
or equivalent $G_{2}=0$. Hence, $\chi_{1}=G_{1}(x,z)$. This means that
$\chi_{1}$ is invariant under the involution $\sigma$. Thus, Theorem 1 is
proved.
## 4\. Proof of Theorem 2
Assume that $\chi_{1}$ is invariant under $\sigma$, then by (8)–(10) we have
$\chi_{0}=\frac{H_{1}(x)}{z-\gamma_{1}(x)}+\dots+\frac{H_{g}(x)}{z-\gamma_{g}(x)}+\frac{w(z)}{(z-\gamma_{1}(x))\dots(z-\gamma_{g}(x))}+\kappa(x),$
$\chi_{1}(x,P)=-\frac{\gamma^{\prime}_{1}(x)}{z-\gamma_{1}(x)}-\dots-\frac{\gamma^{\prime}_{g}(x)}{z-\gamma_{g}(x)},$
where $H_{i}(x),\kappa(x)$ are some functions. In the neighbourhood of $q$ the
function $\chi_{0}$ has the expansion
$\chi_{0}=\frac{1}{k}+\kappa+\left(\gamma_{1}+\dots+\gamma_{g}+\frac{c_{2g}}{2}\right)k+O(k^{2}).$
Hence, by Lemma 1
(13) $V=-\kappa,\qquad W=-2(\gamma_{1}+\dots+\gamma_{g})-c_{2g}.$
Thus
$\chi_{0}=\frac{Q_{1}}{Q}+\frac{w}{Q}-V(x),\qquad\chi_{1}(x,P)=\frac{Q^{\prime}}{Q}.$
Let us substitute $\chi_{0},\chi_{1}$ into (12). From $P_{2}=0$ we get
$Q_{1}=-\frac{Q^{\prime\prime}+s}{2}$, where $s$ is a constant. From $P_{1}=z$
we get
$s^{2}-4sw+4w^{2}-4(z-W)Q^{2}+4V(Q^{\prime})^{2}-(Q^{\prime\prime})^{2}+2Q^{\prime}Q^{(3)}$
$-2Q(2V^{\prime}Q^{\prime}+4VQ^{\prime\prime}+Q^{(4)})=0.$
The last identity is possible only if $s=0$ because $Q$ is a polynomial in
$z$. Theorem 2 is proved.
Let us differentiate (4) in $x$ and divide the result by $Q$. We get the
following equation.
Corollary 1 The functions $Q,W,V$ satisfy the equation
$Q^{(5)}+4VQ^{3}+2Q^{\prime}(2z-2W-V^{\prime\prime})+6V^{\prime}Q^{\prime\prime}-2QW^{\prime}=0.$
Let us substitute $z=\gamma_{j}$ in (4). It gives
$V(x)=\left(\frac{(Q^{\prime\prime})^{2}-2Q^{\prime}Q^{(3)}-4F_{g}(z)}{4(Q^{\prime})^{2}}\right)\mid_{z=\gamma_{j}}.$
We get $g-1$ equations on $\gamma_{1}(x),\dots,\gamma_{g}(x)$.
Corollary 2 The functions $\gamma_{1}(x),\dots,\gamma_{g}(x)$ satisfy the
equations
$\left(\frac{(Q^{\prime\prime})^{2}-2Q^{\prime}Q^{(3)}-4F_{g}(z)}{4(Q^{\prime})^{2}}\right)\mid_{z=\gamma_{j}}=\left(\frac{(Q^{\prime\prime})^{2}-2Q^{\prime}Q^{(3)}-4F_{g}(z)}{4(Q^{\prime})^{2}}\right)\mid_{z=\gamma_{k}}.$
## 5\. Proof of Theorem 3
Let
(14)
$\chi_{0}=-\frac{1}{2}\frac{Q^{\prime\prime}}{Q}+\frac{\sqrt{F_{g}(z)}}{Q}-(\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\alpha_{0}),$
(15) $\chi_{1}=\frac{Q^{\prime}}{Q}.$
Let us consider the equations (4) where $V,W$ are potentials of the operator
$L^{{}^{\sharp}}_{4}$
$4F_{g}(z)=4(z-g(g+1)\alpha_{3}x)Q^{2}-4(\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\alpha_{0})(Q^{\prime})^{2}+(Q^{\prime\prime})^{2}-2Q^{\prime}Q^{(3)}$
(16)
$+2Q(2(3\alpha_{3}x^{2}+2\alpha_{2}x+\alpha_{1})Q^{\prime}+4(\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\alpha_{0})Q^{\prime\prime}+Q^{(4)}).$
We prove that the nonlinear equation (16) has a polynomial solution $Q(x,z)$
of degree $g$ in $z$ and degree $g$ in $x$ for some polynomial $F_{g}(z)$.
After that we prove that $\chi_{0},\chi_{1}$ satisfy (11) for the curve
$w^{2}=F_{g}(z)$. The functions $\chi_{0},\chi_{1}$ have required asymptotic
(8) in $q=\infty$. From here it follows that $L_{4}^{{}^{\sharp}}$ commutes
with an operator of order $4g+2$ corresponding to the rational function $w$ on
$\Gamma$ with the unique pole of order $2g+1$ at $q$.
Lemma 2 Equation (16) has a solution of the form
(17) $Q=(z-\gamma_{1}(x))\dots(z-\gamma_{g}(x)),$
for some polynomial $F_{g}(z)$ of degree $2g+1$.
Proof. Let us differentiate both sides of (16) with respect to $x$ and divide
the result by $Q$
$Q^{(5)}+4(\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\alpha_{0})Q^{(3)}+4(\alpha_{2}-(g^{2}+g-3)\alpha_{3}x+z)Q^{\prime}$
(18)
$+6(3\alpha_{3}x^{2}+2\alpha_{2}x+\alpha_{1})Q^{\prime\prime}-2g(g+1)\alpha_{3}Q=0.$
We find a solution of (18) as a polynomial in $x$
(19)
$Q=\delta_{g}x^{g}+\dots+\delta_{1}x+\delta_{0},\qquad\delta_{i}=\delta_{i}(z).$
From (18) we have
$\delta_{s}=\frac{(s+1)}{\alpha_{3}(g-s)(s+g+1)(2s+1)}\left(2(\alpha_{2}(s+1)^{2}+z)\delta_{s+1}+\alpha_{1}(s+2)(2s+3)\delta_{s+2}\right.$
(20)
$\left.+2\alpha_{0}(s+2)(s+3)\delta_{s+3}+1/2(s+2)(s+3)(s+4)(s+5)\delta_{s+5}\right),$
where $0\leq s<g-1$, $\delta_{g}$ is a constant, and $\delta_{s}=0$ at $s>g$.
In particular
(21) $\delta_{g-1}=\frac{\delta_{g}(\alpha_{2}g^{2}+z)}{\alpha_{3}(2g-1)}.$
From (20) it follows that $Q$ is a polynomial of degree $g$ in $z$, and up to
the multiplication by a constant, the polynomial $Q$ has the form (17). The
right-hand side of (16) has degree $2g+1$. Lemma 2 is proved.
Lemma 3 The polynomial $Q$ has no multiple root in $z$
$\gamma_{i}\neq\gamma_{j}\ \mbox{at}\ i\neq j.$
Proof. Let us represent $Q$ in the form
$Q=Q_{H}+\tilde{Q},$
where $Q_{H}$ is a homogeneous polynomial in $x,z$
$Q_{H}=\tilde{\delta}_{g}x^{g}+\tilde{\delta}_{g-1}x^{g-1}z+\tilde{\delta}_{g-2}x^{g-2}z^{2}+\dots+\tilde{\delta}_{0}z^{g},\qquad\tilde{\delta}_{0},\tilde{\delta}_{g}\neq
0$
and ${\rm deg}\tilde{Q}<g$. Since $\tilde{\delta}_{g}\neq 0$, the polynomial
$Q$ has no constant roots (i.e. $\gamma_{i}\neq const$).
Let us note that $Q$ has no multiple roots of order higher than 2. Indeed, if
$Q=(z-\gamma_{i}(x))^{p}\hat{Q},\ p>2$, then from (16)
$F_{g}(\gamma_{i}(x))=0$, but this is impossible.
If $Q$ has multiple roots, then $Q_{H}$ also has multiple roots. This follows
from the following fact. The discriminant of $Q$ is a polynomial
$b_{N}x^{N}+b_{N-1}x^{N-1}+\dots+b_{0}$ in $x$. The discriminant of $Q_{H}$ is
$b_{N}x^{N}$, so if the discriminant of $Q$ is equal to zero, then the
discriminant of $Q_{H}$ is also zero.
From (20) it follows that
$\tilde{\delta}_{s}=\frac{2(s+1)\tilde{\delta}_{s+1}}{\alpha_{3}(g-s)(s+g+1)(2s+1)},\qquad
0\leq s\leq g-1,$
and that $Q_{H}$ satisfies the equation
$2\alpha_{3}x^{3}Q_{H}^{(3)}+2((3-g-g^{2})\alpha_{3}x+z)Q_{H}^{\prime}+9\alpha_{3}x^{2}Q_{H}^{\prime\prime}-g(g+1)\alpha_{3}Q_{H}=0.$
Let us multiply this equation by $Q_{H}$ and integrate in $x$. We get
$\tilde{F}_{g}(z)+(g(g+1)\alpha_{3}x-z)Q_{H}^{2}+\alpha_{3}x^{3}(Q_{H}^{\prime})^{2}-\alpha_{3}x^{2}Q_{H}(3Q_{H}^{\prime}+2xQ_{H}^{\prime\prime}))=0,$
where $\tilde{F}_{g}(z)$ is a polynomial of degree $2g+1$ in $z$.
From the last equation it follows that if $Q_{H}$ has multiple roots, then the
polynomial $\tilde{F}_{g}(z)$ has the same roots. However, this is impossible,
because all roots of $\tilde{F}_{g}(z)$ are constant, but $Q_{H}$ has not
constant roots. Lemma 3 is proved.
Lemma 4 If $(\alpha_{0},\dots,\alpha_{3})\in U$, the curve $w^{2}=F_{g}(z)$ is
nonsingular, where $U\subset{\mathbb{C}}^{4}$ is some Zariski open set.
Proof. The idea of the proof is the following. We represent $F_{g}$ in the
form
$F_{g}(z)=F_{g}^{0}(z)+\alpha_{3}F_{g}^{1}(z)+O(\alpha_{3}^{2}),$
and prove that $F_{g}^{0}(z)+\alpha_{3}F_{g}^{1}(z)$ has not multiple roots.
Therefore, $F_{g}(z)$ has not multiple roots for small $\alpha_{3}$, and
consequently for $(\alpha_{0},\dots,\alpha_{3})\in U$.
Let us consider (19)–(21). We put $\delta_{g}=\alpha_{3}^{g}$, then
$\delta_{g-1}=\alpha_{3}^{g-1}\frac{\alpha_{2}g^{2}+z}{2g-1}.$
Moreover, from (19) it follows that $Q$ has the form
(22)
$Q=\alpha_{3}^{g}x^{g}+\dots+\alpha_{3}^{s}x^{s}(p_{s}(z)+\alpha_{3}q_{s}(z)+O(\alpha_{3}^{2}))+\dots+(p_{0}(z)+\alpha_{3}q_{0}(z)+O(\alpha_{3}^{2})).$
Let us note that from (21) it follows that
$p_{g}=1,\qquad p_{g-1}=\frac{\alpha_{2}g^{2}+z}{2g-1},\qquad q_{g}=0,\qquad
q_{g-1}=0.$
Let us substitute (22) into (16). We get
$F_{g}(z)=p_{0}^{2}(z)z+\alpha_{3}p_{0}(z)(\alpha_{1}p_{1}(z)+2q_{0}(z)z)+O(\alpha_{3}^{2}),$
so,
$F_{g}^{0}(z)=p_{0}^{2}(z)z,\qquad
F_{g}^{1}(z)=p_{0}(z)(\alpha_{1}p_{1}(z)+2q_{0}(z)z).$
To prove Lemma 4 it is enough to prove that $p_{0}(z)z$ and
$\alpha_{1}p_{1}(z)+2q_{0}(z)z$ have no common roots.
Let us find $p_{i}$ and $q_{i}$. For this we again substitute (22) into (18)
and find the coefficients at $\alpha_{3}^{i+1}x^{i}$ and
$\alpha_{3}^{i+2}x^{i}$. These coefficients must be equal to zero. It gives us
(23)
$p_{i}=\frac{2(i+1)(\alpha_{2}(i+1)^{2}+z)}{(2i+1)(g^{2}+g-i^{2}-i)}p_{i+1},\qquad
0\leq i\leq g-1,$ (24)
$q_{i}=\frac{2(i+1)(\alpha_{2}(i+1)^{2}+z)}{(g-i)(g+i+1)(2i+1)}q_{i+1}+\frac{\alpha_{1}(i+1)(i+2)(2i+3)}{(g-i)(g+i+1)(2i+1)}p_{i+2},$
where $0\leq i\leq g-2.$ Hence
$p_{i}(z)=(\alpha_{2}(i+1)^{2}+z)\dots(\alpha_{2}g^{2}+z)A_{i},\qquad 0\leq
i\leq g-1,$
where $A_{i}$ is a constant. Thus to prove that $p_{0}(z)z$ and
$\alpha_{1}p_{1}(z)+2q_{0}(z)z$ have no common roots we should prove that
$z=-\alpha_{2}2^{2},\dots,z=-\alpha_{2}g^{2}$ are not roots of $q_{0}(z)$.
Assume that $q_{0}(-\alpha_{2}s^{2})=0$ for some $s$, $2\leq s\leq g.$ From
(23) it follows that $p_{k}(-\alpha_{2}s^{2})=0$ at $0\leq k<s$,
$p_{k}(-\alpha_{2}s^{2})\neq 0$ at $k\geq s$, and from (24) it follows that
$q_{k}(-\alpha_{2}s^{2})=0$ at $0\leq k\leq s-2$.
First of all we consider the case $s=g$. If $i=g-2$, then (24) yields
$q_{g-2}(z)=\frac{\alpha_{1}(g-1)g(2s-1)}{2(2g-1)(2g-3)}p_{g}(z).$
Hence, if $q_{0}(-\alpha_{2}g^{2})=0$, then $q_{g-2}(-\alpha_{2}g^{2})=0$, but
this is impossible, since $p_{g}=1$, so $s<g.$
Formulas (23), (24) at $i=s-2,i=s-1$ give us
$q_{s-2}-\frac{2(s-1)(\alpha_{2}(s-1)^{2}+z)}{(g-s+2)(g+s-1)(2s-3)}\frac{2s(\alpha_{2}s^{2}+z)q_{s}+\alpha_{1}s(s+1)(2s+1)p_{s+1}}{(g-s+1)(g+s)(2s-1)}-$
$\frac{\alpha_{1}(s-1)s(2s-1)}{(g-s+2)(g+s-1)(2s-3)}\frac{2(s+1)(\alpha_{2}(s+1)^{2}+z)}{(2s+1)(g^{2}+g-s^{2}-s)}p_{s+1}=0.$
Let $z$ be $-\alpha_{2}s^{2}$. After the simplification we have
$g^{2}+g-3s^{2}=0.$
This is impossible, hence $q_{0}(-\alpha_{2}s^{2})\neq 0$ and
$F_{g}^{0}(z)+\alpha_{3}F_{g}^{2}(z)$ has no multiple roots. Lemma 4 is
proved.
Functions $\chi_{0},\chi_{1}$ are rational functions on the curve
$w^{2}=F_{g}(z)$. Let $k=\frac{1}{\sqrt{z}}$ be a local parameter near
$q=\infty$. Functions $\chi_{0},\chi_{1}$ have asymptotic (3). By Lemma, 3
$\chi_{0}$ and $\chi_{1}$ have simple poles
$P_{i}^{\pm}=(\gamma_{i},\pm\sqrt{F_{g}(\gamma_{i})}).$ Let us choose in the
neighbourhood of $P_{i}^{\pm}$ the local parameter $z-\gamma_{i}(x)$.
Lemma 5 Functions $\chi_{0},\chi_{1}$ satisfy the equation (11).
Proof. From (15) we have
$\chi_{1}(x,P)=\frac{-\gamma^{\prime}_{i}(x)}{z-\gamma_{i}(x)}+d_{i,1}(x)+O(z-\gamma_{i}(x))$
for some $d_{i,1}(x).$ Function $\chi_{1}$ has simple poles at
$\gamma_{i}(x)$, thus
$\chi_{0}(x,P)=\frac{-v_{i,0}(x)\gamma^{\prime}_{i}(x)}{z-\gamma_{i}(x)}+d_{i,0}(x)+O(z-\gamma_{i}(x)),$
for some $v_{i,0}(x),d_{i,0}(x)$. By our construction $\chi_{0}$, $\chi_{1}$
satisfy (12). Let us substitute $\chi_{0}$, $\chi_{1}$ in (12). We get
$\frac{(v_{i,0}^{2}(x)-d_{i,0}(x)+d_{i,1}(x)v_{i}(x)-v_{i}^{\prime}(x))(\gamma_{i}^{\prime}(x))^{2}}{(z-\gamma_{i}(x))^{2}}+O\left(\frac{1}{z-\gamma_{i}(x)}\right)=0.$
Hence $d_{i,0}(x),d_{i,1}(x),v_{i,0}(x)$ satisfy (11). Lemma 5 and Theorem 3
are proved.
Operator $L^{{}^{\sharp}}_{4g+2}$ commuting with $L_{4}^{{}^{\sharp}}$ can be
found from
$L_{4}^{{}^{\sharp}}L^{{}^{\sharp}}_{4g+2}=L^{{}^{\sharp}}_{4g+2}L_{4}^{{}^{\sharp}}$.
For the simplicity of the formulas we restrict ourselves to the case
$\alpha_{1}=\alpha_{2}=0,\alpha_{3}=1$. Let us introduce the notations:
$H=\partial_{x}^{2}+x^{3}+\alpha_{0},$ $\langle A,B\rangle=AB+BA.$
Examples.
a) $g=2:$
$L^{{}^{\sharp}}_{10}=H^{5}+\frac{15}{2}\langle x,H^{3}\rangle+45\langle
x^{2},H\rangle,$ $F_{2}(z)=z^{5}+27\alpha_{0}z^{2}+81.$
b) $g=3:$
$L^{{}^{\sharp}}_{14}=H^{7}+21\langle x,H^{5}\rangle+\frac{945}{2}\langle
x^{2},H^{3}\rangle-5418H^{2}+\frac{45}{2}\langle
113\alpha_{0}+287x^{3},H\rangle-486x,$
$F_{3}=z^{7}+594\alpha_{0}z^{4}-2025z^{2}+91125\alpha_{0}^{2}z.$
## References
* [1] G. Wallenberg, Über die Vertauschbarkeit homogener linearer Differentialausdrücke. Arch. Math. Phys. 4 (1903), 252–268.
* [2] J. Schur, Über vertauschbare lineare Differentialausdrücke. Sitzungsber. der Berliner Math. Gesell. 4 (1905), 2–8.
* [3] J.L. Burchnall, I.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Society. Ser. 2. 21 (1923), 420- 440.
* [4] I.M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Functional Anal. Appl., 11: 1 (1977), 12–26.
* [5] I.M. Krichever, Commutative rings of ordinary linear differential operators, Functional Anal. Appl., 12:3 (1978), 175–185.
* [6] I.M. Krichever, S.P. Novikov, Holomorphic bundles over Riemann surfaces and the Kadomtsev Petviashvili equation. I, Functional Anal. Appl., 12:4 (1978), 276–286.
* [7] I.M. Krichever, S.P. Novikov, Holomorphic bundles over algebraic curves and nonlinear equations, Russian Math. Surveys, 35:6 (1980), 47–68.
* [8] P.G. Grinevich, S.P. Novikov, Spectral theory of commuting operators of rank two with periodic coefficients, Functional Anal. Appl., 16:1 (1982), 19–20.
* [9] P.G. Grinevich, Rational solutions for the equation of commutation of differential operators, Functional Anal. Appl., 16:1 (1982), 15–19.
* [10] F. Grunbaum, Commuting pairs of linear ordinary differential operators of orders four and six, Phys. D, 31:3 (1988), 424- 433.
* [11] G. Latham, Rank $2$ commuting ordinary differential operators and Darboux conjugates of KdV, Appl. Math. Lett. 8:6 (1995), 73- 78.
* [12] G. Latham, E. Previato, Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations, Acta Appl. Math. 39 (1995), 405–433.
* [13] O.I.Mokhov, On commutative subalgebras of Weyl algebra, which are associated with an elliptic curve. International Conference on Algebra in Memory of A.I. Shirshov (1921- 1981). Barnaul, USSR, 20 -25 August 1991. Reports on theory of rings, algebras and modules. 1991. P. 85.
* [14] O.I.Mokhov, On the commutative subalgebras of Weyl algebra, which are generated by the Chebyshev polynomials. Third International Conference on Algebra in Memory of M.I.Kargapolov (1928 -1976). Krasnoyarsk, Russia, 23- 28 August 1993. Krasnoyarsk: Inoprof, 1993. P. 421.
* [15] E. Previato, G. Wilson, Differential operators and rank $2$ bundles over elliptic curves, Compositio Math. 81:1 (1992), 107- 119.
* [16] P. Dehornoy. Operateurs differentiels et courbes elliptiques, Compositio Math. 43:1 (1981), 71- 99
* [17] O.I. Mokhov, Commuting differential operators of rank 3 and nonlinear differential equations, Mathematics of the USSR-Izvestiya, 35:3 (1990), 629–655.
* [18] A.E. Mironov, A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2, Sbornik: Math., 195:5 (2004), 711 -722.
* [19] A.E. Mironov, On commuting differential operators of rank 2, Siberian Electronic Math. Reports. 6 (2009), 533–536.
* [20] A.E. Mironov, Commuting rank 2 differential operators corresponding to a curve of genus 2, Functional Anal. Appl., 39:3 (2005), 240 -243.
* [21] D. Zuo, Commuting differential operators of rank 3 associated to a curve of genus 2, arxiv: 1105.5774.
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|
arxiv-papers
| 2011-07-18T05:29:53 |
2024-09-04T02:49:20.673196
|
{
"license": "Public Domain",
"authors": "Andrey E. Mironov",
"submitter": "Andrey Mironov",
"url": "https://arxiv.org/abs/1107.3356"
}
|
1107.3581
|
# Global systematics of octupole excitations in even-even nuclei
L.M. Robledo luis.robledo@uam.es http://gamma.ft.uam.es/robledo Departamento
de Física Teórica, Módulo 15, Universidad Autónoma de Madrid, E-28049 Madrid,
Spain G.F. Bertsch Institute for Nuclear Theory and Dept. of Physics, Box
351560, University of Washington, Seattle, Washington 98915, USA
###### Abstract
We present a computational methodology for a theory of the lowest octupole
excitations applicable to all even-even nuclei beyond the lightest. The theory
is the well-known generator-coordinate extension (GCM) of the Hartree-Fock-
Bogoliubov self-consistent mean field theory (HFB). We use the discrete-basis
Hill-Wheeler method (HW) to compute the wave functions with an interaction
from the Gogny family of Hamiltonians. Comparing to the compiled experimental
data on octupole excitations, we find that the performance of the theory
depends on the deformation characteristics of the nucleus. For nondeformed
nuclei, the theory reproduces the energies to about $\pm 20$ % apart from an
overall scale factor of $\approx 1.6$. The performance is somewhat poorer for
(quadrupole) deformed nuclei, and for both together the dispersion of the
scaled energies about the experimental values is about $\pm 25$ %. This
compares favorably with the performance of similar theories of the quadrupole
excitations. Nuclei having static octupole deformations in HFB form a special
category. These nuclei have the smallest measured octupole excitation energies
as well as the smallest predicted energies. However, in these cases the
energies are seriously underpredicted by the theory. We find that a simple
two-configuration approximation, the Minimization After Projection method,
(MAP) is almost as accurate as the full HW treatment, provided that the
octupole-deformed nuclei are omitted from the comparison. This article is
accompanied by a tabulation of the predicted octupole excitations for 818
nuclei extending from dripline to dripline, computed with several variants of
the Gogny interaction.
## I Introduction
The octupole excitations of nuclei have been well-studied theoretically on a
case-by-case basis but there has never been a global study for a fixed
Hamiltonian and well-defined computational methodology. Such studies are
important for several reasons. Seeing the systematic trends, one can better
assess the deficiencies in the Hamiltonian or the underlying theory, which
could hopefully lead to improvements on both sides. Also, the predictive power
of the theory with the given Hamiltonians can be measured by the comparison to
a large body of nuclear data. In this work we carry out a study of this kind
using the Hartree-Fock-Bogoliubov (HFB) approximation extended by Generator
Coordinate Method (GCM). Earlier studies of the octupole degree of freedom
using this and similar methods are in Refs. ma83 ; he01 ; eg91 ; sk93b ; he94
; ga98 . A competing methodology is based on the quasiparticle random phase
approximation; recent application to octupole modes may be found in Refs. se02
; co03 ; an09 . For a general review of the theory of octupole deformations
and collective excitations, see Ref. bu96 .
A global theory not only needs to treat the consequences of static octupole
deformations in HFB ground states but also to treat the more ordinary
situation where the degree of freedom appears more as a collective vibration
of a symmetric HFB ground state. The latter is typically treated by RPA or
QRPAse02 ; co03 ; an09 , but the most of the studies consider a small body of
nuclei chosen by considerations emphasizing one characteristic or another, for
example semi-magic isotope chains. Our study is the first to encompass not
only magic and semimagic ordinary nuclei, but the quadrupole- and octupole-
deformed nuclei as well. This follows in spirit the studies of the nuclear
quadrupole degrees of freedom in Refs. sa07 ; de10 . We mention that our GCM
coordinate is a one-dimensional variable labeled by the mass octupole moment.
A two-dimensional treatment of the octupole deformations treating the
quadrupole deformation as a separate degree of freedom is important in theory
of fissiongo05 , and is likely to play a role in spectroscopy as wellme95 .
The HFB fields and quasiparticle wave functions are assumed to have the
following symmetries: time reversal, axial symmetry, and the $z$-component of
isospin. We can only consider even-even nuclei under these restrictions. The
restriction to axial symmetry is harmless in spherical nuclei, but for
deformed nuclei it causes two problems. The first is that theory only treats
the $K=0$ excitations of deformed nuclei. As we will see, some of the
identified octupole excitations very likely have nonzero $K$ quantum number.
The second difficulty that arises with deformed nuclei is that angular
momentum is not a good quantum number of the HFB/GCM wave function. On a
practical level, we shall compare the calculated excitation energies with the
spectroscopic $0^{+}\rightarrow 3^{-}$ transitions, assuming that the
rotational inertias can be neglected.
The calculations are carried Gogny’s form of the interaction in the
Hamiltonian. In particular, the D1S Gogny interaction has been well-tested in
many HFB calculations and also gives good results in (Q)RPA pe08 and GCM
extensions of HFB ro02b . Specific results for that interaction will be
presented in the text, and results for other Gogny interactions are provided
in the supplementary material accompanying this article.
## II Implementing the GCM
### II.1 GCM
In the GCM, an external field is added to the Hamiltonian to generate a set of
mean-field configurations to be taken as a basis for the HW minimization. We
take for the generating field the mass octupole operator,
$\hat{Q}_{3}=\sqrt{\frac{4\pi}{7}}r^{3}Y^{3}_{0}(\hat{r})=z^{3}-\frac{3}{2}z(x^{2}+y^{2}).$.
We label the solutions of the HFB equations in the presence of the field
$\lambda\hat{Q}_{3}$ by the expectation value of $\hat{Q}_{3}$,
$\langle q|\hat{Q}_{3}|q\rangle=q.$ (1)
For convenience, we will use the nominal value of $\beta_{3}$ instead of $q$
in discussing the wave functions. These are related by the formula
$q=\sqrt{9/28\pi}(1.2)^{3}A^{2}\beta_{3}$. We also fix the (average) center-
of-mass of the nucleus at the origin with the constraint
$\langle|\hat{z}|\rangle=0$ to avoid a spurious octupole moment associated
with the position of the nucleus.
The GCM wave function is constructed by combining the configurations
$|q\rangle$ to build a correlated wave function $|\sigma\rangle$. This is
expressed formally in the GCM as an integral over configurations
$|\sigma\rangle=\int dq\,f_{\sigma}(q)|q\rangle.$ (2)
The function $f$ in Eq. (2) is to be determined by applying the variational
principle to the expression
$E=\frac{\langle\sigma|H|\sigma\rangle}{\langle\sigma|\sigma\rangle}.$ (3)
While Eq. (3) and (4) define the GCM formally, further approximations are
required to arrive at a well-defined computational methodology. One way common
in the literature is to keep the formal integral Eq. (3) and use the Gaussian
overlap approximation to calculate the matrix elements in Eq. (4), as was done
in Ref. de10 to map the quadrupole deformation onto a collective Hamiltonian,
and in Ref. ro10b for the octupole degree of freedom. A quite different way
is the discrete basis Hill-Wheeler method, first carried out for the octupole
excitations in Ref. ma83 . This method, which we will follow here,
approximates the integral using a discrete set of configurations. The
minimization is equivalent to solving the matrix eigenvalue equation
$\sum_{j}\langle q_{i}|H|q_{j}\rangle c_{j}=E\langle q_{i}|q_{j}\rangle
c_{j}.$ (4)
The states will have good parity if the basis is reflection symmetric, i.e. if
$|-q_{i}\rangle$ is in the basis if it contains $|q_{i}\rangle$.
For either method one needs the overlap integrals between configurations
$\langle q|q^{\prime}\rangle$, the matrix elements of Hamiltonian $\langle
q|H|q^{\prime}\rangle$ and the matrix elements of one-body operators such as
$\langle q|\hat{Q}_{3}|q^{\prime}\rangle$. The basic overlap integral is
computed with the Onishi formulaonishi . The matrix elements of one-body
operators and products of one-body operators are then evaluated using the
generalized Wick’s theorembalian . Unfortunately, the Gogny interaction cannot
be expressed in this way due to its $\rho^{1/3}(\vec{r})$ density dependence.
This gives rise to well-known ambiguities in treating the interaction as a
Hamiltonian in a multiconfiguration space. Of the various prescriptions
available, we use the "mixed density" method. Here the $\rho$ in the
$\rho^{1/3}$ factor is replaced by $\rho_{BB}(\vec{r})$ given by
$\rho_{BB}(\vec{r})=\frac{\langle
q|\hat{\rho}(\vec{r})|q^{\prime}\rangle}{\langle q|q^{\prime}\rangle}$ (5)
and the resulting $\vec{r}$-dependent interaction is evaluated in the usual
way. The mixed-density prescription was introduced in Ref bo90 and first
applied to parity-projected HFB as "Prescription 2" in Ref eg91 . It is
consistent with the mean field limit and is a scalar under symmetry
transformations ro07 . Another prescription which seems plausible at first
sight is to use the projected density for $\rho^{1/3}$. However, this gives
unphysical results for octupole deformationsro10a .
While the configurations $|q\rangle$ constructed with the octupole constraint
have mixed parity, the HW solutions restore the parity quantum number,
provided that we use a basis that contains the both signs of $q$ in the
included configurations. In effect, the parity projection needed to calculate
spectroscopic properties can be obtained from the HW minimization without any
extra effort. However, as a practical matter, it is easier to define the
parity operator in the harmonic oscillator basis and use it to construct
$|-q\rangle$ from $|q\rangle$ thus avoiding a separate HFB minimization for
the $-q$ configuration.
The HW states of interest are the lowest lying even- and odd-parity states of
spectrum, which we call $|e\rangle$ and $|o\rangle$. Taking them to be
normalized, the energies of ground state $E_{e}$, the odd parity state
$E_{o}$, and the excitation energy difference $E_{3}$ are given by
$E_{e}=\langle e|H|e\rangle;\,\,\,\,E_{o}=\langle
o|H|o\rangle;\,\,\,\,E_{3}=E_{o}-E_{e}$ (6)
We follow the usual procedure to solve the matrix equation Eq. (4), using if
necessary the singular value decomposition to avoid difficulties with an
overcomplete space.
One first diagonalizes the overlap matrix and transforms all of the matrices
to the diagonalized basis. Often there will be vectors which very small norms
and the basis is truncated to exclude vectors whose norms are less than a
certain value $n_{min}$. The Hamiltonian is diagonalized in this basis, called
the collective space, to give the HW energies. The eigenvectors are used to
calculate matrix elements of other operators between energy eigenstates.
The main problem with the discrete Hill-Wheeler method is that the calculated
values cannot be considered reliable unless both the range of deformations has
been fully covered and that the singular value decomposition has been set to a
robust truncation. For most of the nuclei, we shall take as a basis the set of
$\beta_{3}$ from -0.5 to +0.5 in steps of 0.025. For lighter nuclei, the range
is extended from -1.2 to +1.2. The calculations are carried out as a function
of the dimension $N_{basis}$ of the singular-value truncation. There is
generally a broad range of $N_{basis}$ for which the excitation energies have
converged to some value; we take the value on this plateau as the HW result.
An example is shown in detail in the next section.
The computation of the HW starting matrices is not trivial, requiring $N^{2}$
Hamiltonian matrix elements for a basis size $N$. While this is not an
important issue here, if one were to attempt GCM calculations in more than one
variable, the number of states $N_{basis}$ could be large. It is therefore of
interest to investigate the accuracy of simpler approximations using fewer
configurations. One of the simplest treatments is to take two configurations,
$|q_{e}\rangle$ and $|q_{o}\rangle$, for the even-parity and odd-parity state,
respectively. The values of $q$ are chosen to minimize the projected energies
of the configuration. We follow Ref. sa07 calling this the Minimization After
Projection (MAP) procedure. The deformations and energies at the minima
denoted $\beta_{3p},E_{p}$ and $\beta_{3m},E_{m}$ for the two projected
states. The MAP excitation energy is defined as
$E_{3}^{MAP}=E_{m}-E_{p}$ (7)
One last general point of the computational procedure needs to be mentioned.
While the individual HFB configurations are constructed with the desired
proton and neutron particle numbers, the mixed configurations in the HW wave
function may have slightly different expectation values of $N$ and $Z$. The
energy depends strongly on $\langle N\rangle$ and $\langle Z\rangle$, and
changes must be corrected for. We do this by adding to the HW Hamiltonian the
term $\lambda_{p}(\hat{Z}-Z)+\lambda_{n}(\hat{N}-N)$, where $\lambda_{p,n}$
are the nucleon chemical potentials at $\beta_{3p}$ bo90 .
### II.2 HFB
The constrained HFB calculations were carried out using the code HFBaxial
written by one of us (L.M.R.). It uses a harmonic oscillator basis specified
by the length parameters $b_{z}$ and $b_{t}$ of the oscillator potential and
the number of shells $N_{osc}$ in the basis. For the calculations reported
here we have taken a fixed spherical basis for all nuclei with oscillator
length parameters $b_{z}=b_{t}=2.1$ fm. The number of oscillator shells
included in the basis is 10,12, and 14 for nuclei in the ranges
$Z=[8,50],[52,82]$, and $[84,100]$ respectively. This is more than enough to
provide converged results for energy differences. We report on the results for
the D1S Gogny interaction in sections below. More detailed results for the D1S
as well as for other interactions of the Gogny form are given in the
supplemental material epaps .
## III Examples
In this section we will go through the details for four examples illustrating
the application to a spherical nucleus, 208Pb, a well-deformed nucleus, 158Gd,
the nucleus 226Ra whose HFB ground state has a static octupole deformation,
and a light nucleus having a very large transitional octupole moment, 20Ne. A
summary of the results for these nuclei is given in Table 2 at the end of this
Section.
### III.1 208Pb
The nucleus 208Pb is a paradigm for a doubly magic nucleus. It is one of the
very few nuclei whose first excited state has $J^{\pi}=3^{-}$ quantum numbers.
The excitation energy is 2.62 MeV and the transition rate is strongly
collective with strength of $B(E3,\uparrow)=0.611$ e2b3 or 34 Weisskopf
unitswu . For the theory, we first shown HFB and projected energies of the GCM
configurations in Fig. 1.
Figure 1: Energy of 208Pb as a function of octupole deformation $\beta_{3}$.
Open circles: HFB energy of constrained configurations ; Solid squares: energy
$E_{e}$ of the even-parity projected wave function; Solid circles: the odd-
parity projected energy $E_{o}$. See the Appendix for explanation of the
fitted lines.
The minimum energy projected configurations, ie. the MAP states, are at
$\beta_{3p}\approx 0.0375$ and $\beta_{3m}\approx 0.075$. One sees that the
energy of the ground state is lower by projecting from a nonzero $\beta_{3}$;
the associated correlation energy has the order of magnitude of one MeV. The
MAP approximation to the excitation energy $E_{3}$ is given by the difference
of the minima of the plus- and minus-projected energy curves, which is about
4.2 MeV.
To see how the calculated $E_{3}$ depends on the basis, we show it in Fig. 2
as a function of $N_{basis}$. The difference of MAP energies is the open
square, and solid circles show the results with various truncations. The full
basis set is comprised of the 41 configurations between $\beta_{3}=-0.5$ to
$\beta_{3}=+0.5$ in steps of 0.025. The truncation is carried out by the
singular-value decomposition.
Figure 2: Excitation energy $E_{3}$ in 208Pb as a function of the
configuration space choice. Solid circles: HW using the singular value
decomposition to keep $N_{basis}$ states; solid square: HW with the two MAP
states; open square: Energy difference of the two MAP states.
One sees that the energy has converged at about $N_{basis}\approx 14$ and the
numerics remain stable up to much larger values. The converged energy, 4.0
MeV, is fairly close to the difference of MAP energies. In fact, one can do
even better in the 4-dimensional space allowing the MAP configurations to mix.
This is shown as the solid square in the figure. We note that our excitation
energy of 4.0 MeV is close to the value found in Ref. he01 using the GCM/HW
method but with the Skyrme SLy4 interaction.
We see here that the MAP could be a very useful simplification, but its
validity depends on the circumstances. It is instructive to examine the GCM/HW
wave function and compare it with MAP. These are shown in Fig. 3, for both the
ground state and the odd-parity excited state. The wave function amplitudes
are formally defined by the integral
$g_{\sigma}(\beta_{3})=\int d\beta_{3}^{\prime}{\cal
N}^{1/2}(\beta_{3},\beta_{3}^{\prime})f_{\sigma}(\beta_{3}^{\prime})$ (8)
where $f$ is normalized $1=\int d\beta_{3}\,d\beta_{3}^{\prime}{\cal
N}(\beta_{3},\beta_{3}^{\prime})f_{\sigma}(\beta_{3}^{\prime})f_{\sigma}(\beta_{3})$.
The above relation establishes the connection between the standard GCM
amplitudes $f$ with the amplitudes $g$ entering the expansion of the GCM wave
functions in terms of orthogonal states $|q\rangle_{\textrm{orth}}=\int
dq^{\prime}{\cal N}^{-1/2}(q,q^{\prime})|q^{\prime}\rangle$. The square root
of the norm overlap has to be understood in terms of the relation $\int
dq^{\prime\prime}{\cal N}^{1/2}(q,q^{\prime\prime}){\cal
N}^{1/2}(q^{\prime\prime},q^{\prime})={\cal N}(q,q^{\prime})$. The ground and
excited state wave functions can be distinguished by the amplitude at
$\beta_{3}=0$, which is finite for the even-parity ground state and zero for
the odd-parity excited state. The HW wave function and the MAP approximation
are shown as solid and dashed lines, respectively. It is clear that the MAP
configuration is a good approximation to the full wave function of both the
ground and excited states, for this particular nucleus.
Figure 3: Wave function amplitudes. See text for explanation.
More insight into the collective physics of the octupole degree of freedom can
be obtained comparing with simple models of the excitation (See Appendix). If
the configuration energies and interactions can be treated as quadratic
functions of the deformation coordinate, and the matrix elements between
different configurations can be treated by the GOA, the GCM/HW reduces to the
RPA and is exact. The line through the HFB energy curve in Fig. 1 is a
quadratic fit. It appears to be well satisfied. Also, the energy of the even-
parity projected configuration follows well the predicted dependence according
to the GOA, Eq. (16). This shown as the line through the even-parity projected
energies in the figure. Thus two of the conditions are met to reduce the
GCM/HW theory to an RPA of a single collective state.
### III.2 158Gd
Our example of a strongly deformed nucleus is 158Gd. It has a $3^{-}$
excitation at 1.04 MeV with a transition strength $B(E3\uparrow)=0.12$ e2b3.
The energies from the GCM calculation are shown in Fig. 4. Overall, the energy
curves look quite similar to those for 208Pb. The HFB curve is also well fit
by a quadratic dependence on $\beta_{3}$ but the curvature here is much
shallower. The projected energy function $E_{e}(\beta_{3})$ also has a similar
shape to the curve for 208Pb, and can be fitted by the same functional form,
Eq. (14). The ratio of MAP minimum points is found to be
$\beta_{3p}/\beta_{3m}\approx 2$, similar to the situation for 208Pb. The
excitation energy $E_{3}$ comes out to about 1.7 MeV, much smaller than the
208Pb value. This is to be expected in view of the softer HFB curve. The
correlation energy of the ground state, $E_{0}-E_{e}$, is similar to the 208Pb
value, about one MeV.
Figure 4: Energy of 158Gd as a function of octupole deformation $\beta_{3}$.
Open circles: HFB energy of constrained configurations ; Solid squares: energy
$E_{e}$ of the even-parity projected wave function; Solid circles: the odd-
parity projected energy $E_{o}$. The line along the HFB values is the function
$E_{q}=E_{0}+K_{1}\beta_{3}^{2}$ with $K_{1}=48.8$ MeV fitted to the values
$\beta_{3}\leq 0.05$. The line along the $E_{e}$ values is the fit motivated
by the Gaussian overlap approximation,
$E_{e}=E_{q}-K_{2}\beta^{2}/(1.0+\exp(\alpha\beta^{2}))$, with $K_{2}$ and
$\alpha$ fitted.
Experimentally, the situation is complicated by the deformation and the
splitting of the octupole strength into different $K$-bands. There are three
negative parity bands known experimentally at low energy. There is a $K=1^{-}$
with an $1^{-}$ state at 977 keV, a $K=0^{-}$ with the $1^{-}$ state at 1263
keV and finally a $K=2^{-}$ with a $2^{-}$ state at 1793 keV. Our excitation
energy of 1.7 MeV should be compared with the 1263 keV of the $1^{-}$ state of
the $K=0^{-}$ band. The theoretical value is stretched by a factor 1.4 with
respect to the experimental value (see discussion below). Note that the
measured octupole transition at 1.04 MeV is not relevant for the comparison
because it corresponds to a different $K$ value.
### III.3 226Ra
226Ra has the lowest $3^{-}$ excitation energy of any in the compilation ki02
, $E_{3}=320$ keV. It also has the highest transition strength in the
compilation, $W(E3)=54$ Weisskopf unitswu . On the theory side, the nucleus is
predicted to deformed both in the quadrupole ($\beta_{2}\approx 0.3$) and the
octupole degrees of freedom. The HFB/GCM energy curve, shown in Fig. 5, has a
minimum at $\beta_{3}\approx 0.13$.
This nucleus is very interesting for our survey, not only because of the
static octupole deformation, but because the theory is seen to fail badly if
the large amplitude fluctuations are not properly accounted for. The predicted
excitation energies for different treatments of the GCM configurations are
shown in Table 1. The most naive theory (top line) would ignore the GCM
construction and simply take the HFB minimum and project from that. The
overlap $\langle-q|q\rangle$ at the HFB minimum is essentially zero and the
$E_{3}$ comes out less than 1 keV. In the next approximation we consider
(second line), we take the single configuration that gives the MAP ground
state. Here the deformation is much closer to zero. However, the $E_{3}$
calculated as the difference between the even and odd projected states is now
far too large, 1.7 MeV. Of course in the full MAP approximation we should take
the configurations at different $\beta_{3}$ for odd and even projections. This
is done in line 3 of the Table, and now the $E_{3}$ has the correct order of
magnitude. Adding more configurations, the valued do not change much on an
absolute MeV scale, but on a relative scale there is a considerable change.
The most complete HW treatment, on the bottom line, underpredicts the energy
by a factor of $\approx 2$.
Figure 5: Energy of 226Ra as a function of octupole deformation $\beta_{3}$
as in Figs. 1,4g. A very similar plot is shown in Fig. 3 of Ref. eg91 .
We also show the HW and MAP wave functions in Fig. 3. It is clear that the
full wave functions are far from harmonic and that the MAP approximation fails
badly.
$N_{q}$ | $\beta_{3}$ | $E_{e}$ | $E_{o}$ | $E_{3}$
---|---|---|---|---
1 | 0.15 | -1722.63 | -1722.63 | 0.00
1 | 0.05 | -1722.71 MeV | -1721.01 | 1.7 MeV
2 | 0.05,0.15 | -1723.43 | | 0.37
3 | 0.05,0.1,0.15 | -1723.45 | | 0.31
4 | 0.025,0.075,0.125,0.175 | -1723.53 | | 0.22
12 | [-0.5,0.5] | | | 0.16
Table 1: Calculated energies of 226Ra with various choices of the
configuration set. Figure 6: Energy of 20Ne as a function of octupole
deformation $\beta_{3}$ as in Figs. 1,4,5. Figure 7: Nucleon density
distribution in 20Ne at $\beta_{3p}$ (left) and $\beta_{3m}$ (right).
### III.4 20Ne
20Ne illustrates some differences that one sees in treating light nuclei by
the GCM/HW, first studied by this method in Ref. ma83 . Due to the incipient
alpha clustering, the equilibrium octupole deformation of the projected
configurations can be very large. The HFB and projected energies are shown in
Fig. 6. Note that the HFB energy deviates from a quadratic dependence on the
deformation, and looks almost linear at large $\beta_{3}$. Fig. 7 shows the
density distribution at the two projected minima. One sees a compact localized
density, suggestive of an alpha particle, outside a nearly spherical core.
Since the alpha emission threshold is rather low in this nucleus, one should
expect a softness in with respect to the generator coordinate corresponding to
alpha cluster separation. In a multipole representation, this requires
changing both the quadrupole and the octupole deformation. This is in fact
what occurs in our GCM wave functions. Fig. 8 shows their deformations in the
two multipolarities. The coupling of the multipolarities can cause problems,
however. We will come back to this in the Appendix, referring to the coupling
in 16O, also shown on the figure.
Figure 8: Deformation of the octupole-constrained HFB configurations for 16O and 20Ne. Nucleus | $E_{3}$ (MeV) | $W(E3)$
---|---|---
| Exp. | Present | Other | Theory | Eq. | Exp.
20Ne | 5.6 | 6.7 | 5.2a | 12. | (11) | 13.
208Pb | 2.6 | 4.0 | 4.0b | 53. | (12) | 34.
158Gd | 1.04 | 1.93 | | 11.6 | (11) | 12.
226Ra | 0.32 | 0.16 | | 43. | (11) | 54.
Table 2: Summary of results for the four examples discussed in the text.
References for column 4, other theory: a) ma83 ; b) he01 .
## IV Systematics
We have applied the HFB/GCM/HW theory across the chart of nuclides including
818 nuclei between $8\leq Z\leq 110$. About 6% of them are octupole deformed
in the HFB ground state. The nuclei are shown in Fig. 9. Favorable conditions
for static octupole deformation occur when a high-$j$ intruder orbital is
close to an opposite-parity orbital with three units less of orbital angular
momentum near the Fermi energybu96 , which happens for $Z$ and $N$ values
around 36, 56, 88, and 134. The regions around Ba and Ra are well-known in
earlier studies. We also find static deformations near 80Zr and near $Z\approx
N\approx 56$ (for this region, see also Ref. he94 . There are also
calculations in the literature reporting static octupole deformations in other
regions as wellmo08 ; zh10 . In any case, the HFB deformation is not an
observable. Physically, one can only measure excitation energies and
transitions strength. These are compared with experiment in the two
subsections following.
Figure 9: Chart of the nuclides shown those calculated in the present study.
Those in black have static octupole deformations in HFB. Except for the nuclei
near $N\sim Z\sim 40$, the nucleon numbers correspond well to the numbers 56,
88, and 136 listed in Ref. bu96 as especially favorable for octupole
deformation.
### IV.1 Excitation energies
We now compare theory with the experimental data from the review by Kibédi and
Spear ki02 . The excitation energies of the 284 tabulated nuclei with $Z\geq
8$ are shown in Fig. 10, plotted as a function of $A$. The data show a strong
overall $A$-dependence as well as shell-related fluctuations. The line shows a
fit to the smooth trend in $A$ with the phenomenological parameterization
$E(A)=103/A^{0.85}$ MeV. The most pronounced fluctuation about the trend is
the rise and sudden drop near $A=208$; the drop is to low values is due to the
extreme softness in the octupole mode. The theoretical energies, shown as
triangles, replicate the overall trend with $A$ and the dramatic fluctuation
at $A\sim 208$. However, overall the theoretical energies are too high,
particularly in the light nuclei.
Figure 10: Octupole excitation energies as a function of mass number $A$.
Circles: experiment; triangles: theory.
A more detailed comparison of theory and experiment may be seen on the scatter
plot Fig. 11. For excitation energies above 1 MeV, the theoretical values
track the experimental but scaled by a factor. Around 1 MeV and below the
theoretical values become closer to experiment. The lowest energy measured
excitations are in the Ra isotopes, where the theoretical HFB wave functions
have static octupole deformations. The theory reproduces the low energies to
several hundred keV on an absolute energy scale, but does not do well on the
logarithmic energy scale shown in the figure.
Figure 11: Octupole excitation energies, comparing the theory with
experiment. Filled circles are excitations with measured $B(E3)$ strengths;
open circles are other identified octupole transitionski02 .
We also make some quantitative assessment of the performance of the theory,
which should be useful in the future for comparing with other theories. We use
the same performance measures as was used to assess theories of quadrupole
excitationssa07 ; de10 , namely to compare ratios of theoretical to
experimental quantities on a logarithmic scale. In terms of
$R_{E}=\log(E(th)/E(exp))$ we determine the average value
$\bar{R}_{E}=\langle R_{E}\rangle$ (9)
and the dispersion about the average,
$\sigma_{E}=\langle(R_{E}-\bar{R}_{E})^{2}\rangle^{1/2}.$ (10)
The results are shown in Table 3. The first line shows the comparison taking
the full HW treatment on the theoretical side and the full data set on the
experimental side. One sees that the predicted energy is systematically too
high, by a factor of $e^{0.44}\approx 1.6$. This is similar to the situation
with the quadrupole excitations. There the understanding is that the wave
function is missing components that would be included in collective theories
using Thouless-Valatin inertial parameters. There may be other reasons for the
systematic overprediction here that we will come back to in Sect. V. The
dispersion in the values is $\sigma_{E}\approx 0.4$, corresponding to errors
in the ratio of theory to experiment of $-30\%$ to $+50\%$. This is larger
than the global dispersion found for the GCM-based theories of quadrupole
excitations. However, we saw in Fig. 11 that there are differences in the
nuclear structure that are responsible for the variable performance of the
theory. Most importantly, the nuclei with calculated static octupole
deformations should be treated separately. Taking out these nuclei, the
dispersion decreases dramatically, as shown on the second line of the Table. A
further distinction can be made between well-deformed and other nuclei,
spherical and soft, respect to ordinary quadrupole deformations. A good
theoretical indicator for deformed nuclei is the ratio of $4^{+}$ to $2^{+}$
excitation energies, called $R_{42}$. The values are available for the Gogny
D1S interaction from the global study de10 , and we use them to set the
condition $R_{42}>2.9$ to define the set of well-deformed nuclei. The results
are shown in the third and fourth rows of the table. One sees that the
dispersion becomes even narrower for the nuclei in the nondeformed set. Thus,
we can claim that the HFB/GCM/HW methodology is quite successful for
nondeformed nuclei, when allowing for the overall scale factor. On the other
hand, the deformed set is significantly poorer, with the average predicted
energies higher and a larger dispersion. A possible cause of this poorer
performance could be the misidentification of transitions in deformed nuclei.
We have assumed here that all transitions are associated with the axially
symmetric octupole operator ($K=0$). As discussed in the next section, it is
clear that some of the measured energies are for transitions with $K\neq 0$
(see also the 158Gd example). Since all the $K$ values in spherical nuclei are
degenerate, this would explain the better overall agreement there.
| | HW | MAP
---|---|---|---
Selection | Number | $\bar{R}_{e}$ | $\sigma_{e}$ | $\bar{R}_{e}$ | $\sigma_{e}$
all | 284 | 0.45 | 0.40 | |
$\beta_{3}=0$ | 277 | 0.55 | 0.23 | 0.59 | 0.22
$\beta_{3}=0$, def. | 59 | 0.62 | 0.32 | 0.75 | 0.26
$\beta_{3}=0$, sph. | 196 | 0.52 | 0.19 | 0.53 | 0.17
Table 3: Performance of the HW theory for excitation energies compared to the
experimental data tabulated in Ref. ki02 . The performance measures $r_{E}$
and $\sigma_{E}$ are given in Eq. (9) and (10) of the text. The performance of
MAP is shown as well on lines 2-4 for subsets of nuclei selected by
deformation criteria.
### IV.2 Transition strengths
The octupole transition strength is computed from the proton octupole
transition matrix element $\langle o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle$.
In a strongly deformed nucleus, the excitation is in a $K=0$ odd-parity band
and the spectroscopic matrix element from the $3^{-}$ state in the band is
given by
$B(E3,3^{-}\rightarrow 0^{+})=\frac{e^{2}}{4\pi}\langle
o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle^{2}.$ (11)
This formula was used in Ref. eg91 to estimate the octupole transition
strengths in Ra isotopes and other possible octupole-deformed nuclei. On the
other hand, if the state $|e\rangle$ is spherical, then the excitation induced
by $Q_{3}$ gives a state $|o\rangle$ that has good angular momentum and the
transition strength can be calculated directly as
$B(E3,3^{-}\rightarrow 0^{+})=\frac{7e^{2}}{4\pi}\langle
o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle^{2}.$ (12)
Notice that this is a factor of 7 larger than Eq. (11). The reason for the
difference is that Eq. (12) gives a total octupole transition strength, while
Eq.(11) only gives the transition strength for the $K=0$ components.
Besides these limiting cases, there are soft nuclei which should fall in
between. Thus, it is imperative to restore good angular for the theory to have
a global applicability. While angular momentum projection has been carried out
in the pasteg93 ; su94 ; eg96 , it is beyond the scope of this article.
Instead, we examine here the range of predicted values using a theoretical
marker of the deformation to distinguish nuclei falling in the different
categories. Fig. 12 shows the ratios of theoretical to experimental $B(E3)$
values, using the experimental data set from Ref. ki02 and Eq. (11) for the
theory. The data is plotted as a function of the quantity $R_{42}$, the ratio
of the lowest $4^{+}$ to $2^{+}$ excitation energies. Values around 2 or less
are characteristic of spherical nuclei, while strongly deformed nuclei have
$R_{42}\geq 3$. We take the values for $R_{42}$ from the spectroscopic
calculations of Ref. de10 , based on HFB/GCM with the same Gogny D1S
interaction used for the theory here. The plot show a lot of scatter, but one
can see two groups of nuclei, the lefthand representing deformed nuclei. There
is a trend visible in the $B(E3)$ ratios consistent with the above discussion.
Figure 12: Ratio of theoretical octupole transition strength to experimental,
with the theoretical strength obtained using Eq. (11). The horizontal axis is
the ratio $R_{42}$ from the theory of Ref. de10 . Experimental $B(E3)$ values
are from Ref. ki02 .
To make the analysis more quantitative, we examine the logarithmic averages
$\bar{R}$ dividing the nuclei into two group according to $R_{42}$. The
results are shown in Table 4. Since we use Eq. (11) to determine $R$, we
should find $\bar{R}=0$ for the first row of the Table. In fact, the average
is about 40 % high. For the second row, if all the nuclei were spherical, the
strength should be a factor of 7 larger. This implies that the $\bar{R}$
calculated with the deformed formula should give a value $0.33-\log(7)=-1.6$.
The value found, -0.99, shows that there is an important effect of the
deformation but that it to simplistic to assume that these nuclei are all
spherical.
Selection | Number | $\bar{R}$ | $\sigma$
---|---|---|---
Deformed, $R_{42}>2.9$ | 41 | 0.34 | 0.5
Other, $R_{42}<2.9$ | 112 | -0.99 | 0.7
Table 4: Ratio of theoretical to experimental $B(E3)$ strengths. The second
column is the number of nuclei in the data set.
We note that the enhancement of the $B(E3)$ for the less deformed nuclei is
evident in the projected calculations for 16O (eg93 ) and Pb isotopes near
$A=208$ (eg96 ). Also, in Ref. sk93a the authors remark on a strong
disagreement between theory and experiment for 96Zr. This is the case if one
uses Eq. (11), but that nucleus is spherical according to the $R_{42}$
criterion and Eq. (12) gives a satisfactory agreement. agreement is satisfying
It is of interest to examine the nuclei that deviate most strongly from the
theory. In Fig. 12 there is a group of three outlier nuclei in the upper
right-hand corner. The nuclei are 170Er and its neighbors. In these cases, the
experimental transitions are likely to be to excited states with $K\neq 0$.
The lowest $1^{-}$ excitation in 170Er at 1.26 MeV has a $K=1^{-}$ character,
and the first $K=0^{-}$ is higher by 0.6 MeV. There are some studies in the
literature in which the $K$-dependence of the octupole excitation is
examinedsk93b ; ya01 ; zb06 . In Ref. ya01 ; zb06 the $K=0^{-}$ bands were
found to be higher in energy than other $K$ values.
The other glaring anomaly is the nucleus 64Zn at $R_{42}\approx 2.4$, which
has a grossly underpredicted $B(E_{3})$. It turns out that the quadrupole
deformation of this nucleus changes sign as $\beta_{3}$ is increased. The
ground state at $\beta_{3}=0$ is oblate, but it switches to another minimum
with a prolate shape at moderate values of $\beta_{3}$. The very small
predicted $B(E3)$ is due to the small overlap between the oblate and prolate
configurations. Clearly, the GCM must include explicitly both quadrupole and
octupole degrees of freedom to properly treat this nucleus. A few other nuclei
with similar Z values show the same behavior. We note that the $B(E3)$ comes
out much closer to experiment if both even and odd states are taken from
configurations having the same sign of quadrupole moment.
## V Discussion
We have demonstrated that a global theory of the octupole degree of freedom
can be constructed using the HFB/GCM/HW methodology. The theory reproduces the
secular trend of the excitations, the effects of an incipient static octupole
deformation, and the most visible shell effects. However, the theory has
obvious deficiencies. Most notably, we require a overall scaling factor of 1.6
to make quantitative comparison with experiment. It is urgent to understand
what physics is needed to make predictions on an absolute energy scale. There
are several possible reasons for the absolute errors. One is the Hamiltonian
itself. Besides the Gogny interaction, there have been calculations with the
BCP interaction, interactions from the Skyrme family and from relativistic
mean-field theory. Ref. ro10b found that the D1S Gogny interaction and the
BCP interaction gave significant differences in the odd-parity excitations of
Ra isotopes. The Gogny interaction is guided by nuclear Hartree-Fock theory,
and one of the characteristics is a nucleon effective mass less than the
physical mass. This implies that single-particle excitation energies will be
higher than for a non-interacting system, and these effects could carry over
to the collective excitations as well. We note that the calculation of the
208Pb in Ref. he01 using a Skyrme interaction with a similar effective mass
to D1S agrees with our results. However, the Relativistic Mean Field
Hamiltonian also has a small effective mass, but excellent agreement was
obtained for $E_{3}$ in an isotone chain by (Q)RPA an09 .
This brings up another source of systematic error in the GCM/HW, the
restriction of the degrees of freedom in the excitation to a single variable.
It is well-known in the theory of quadrupole excitations that time-odd
components must be included in the wave function to obtain good moments of
inertia ro00 . For large amplitude deformations, this can be achieved by self-
consistent cranking. When no time-odd components are allowed in the angular
momentum projected (AMP) GCM calculation the excitation energy is stretched
with respect to standard cranking calculations by a factor of around 1.4. This
correction factor is compatible with the discrepancies observed between our
results and the experiment in the case of 158Gd as well as with the overall
1.6 factor for the negative parity excitation energies discussed previously.
More generally, one can introduce methods that would reduce to (Q)RPA in the
small amplitude limit. The raises the question of how well (Q)RPA would
perform in a global context. As shown in the Appendix, for a large fraction of
nuclei the GCM/HW methodology is essentially equivalent to (Q)RPA in a single
collective variable. For these nuclei, the (Q)RPA is justified and is very
likely to give lower excitation energies.
The interaction of the octupole with the quadrupole degree of freedom is an
interesting problem that appears in several contexts in our study. First, the
HFB static quadrupole deformation of many nuclei invalidates a spectroscopic
interpretation of the observables for the physical angular momentum
eigenstates of the system. We saw this most directly in the discussion of the
$B(E3)$ transition strengths. The solution is to carry out angular momentum
projection. Another aspect missing from our study is the inclusion of $K\neq
0$ excitations in deformed nuclei. This has been done in HFB-BCS in Ref. sk93b
; da99 ; zb06 and in HFB in Ref. ya01 . Since $K\neq 0$ bands can fall below
the $K=0$ octupole excitation band, it is essential for a complete theory of
the octupole excitations in deformed nuclei.
Some aspects of the quadrupole-octupole mixing may require a two-dimensional
GCM to describe properly. It was clear in the light nuclei that octupole and
quadrupole deformations are strongly coupled in forming alpha-clusters. Also
we found that the severe problem describing the $B(E3)$ in 64Zn could be
traced to the coupling. We note that the two-dimensional GCM has been
implemented in the past. In Refs. me95 the coupled GCM was applied to the
complex spectroscopy of the nucleus 194Pb. Also, the microscopic theory of
asymmetric fissiongo05 requires at least a two-dimensional GCM.
One last aspect of the theory should be mentioned. We have seen in the
examples that the correlation energy of the ground state associated with the
$K=0$ octupole excitation is of the order of one MeV. This can have an
important influence on the theory of the nuclear masses. We plan to
investigate the systematics of the correlation energy in a future publication.
The approximation of a single degree of freedom can break down in different
ways. One is if the coupling between different multipoles is important in
determining the configurations. This is the case for light alpha-particle
nuclei. Fig. 8 shows the $\beta_{2}$ and $\beta_{3}$ deformations of the GCM
configurations for 16O and 20Ne. It may be seen that the $\beta_{3}$
deformation carries a $\beta_{2}$ deformation along with it for all but the
smallest values of $\beta_{3}$. Whether this is physical or not depends on the
matrix elements of the interaction connecting the different configurations.
For 16O, the admixtures are perturbative and thus should not change
$\beta_{2}$. However, if the configurations are sampled on coarse mesh, there
will be significant admixture of quadrupole excitations and the Gaussian
overlap approximation will fail. As mentioned in the Introduction, a two-
dimensional treatment of the GCM taking the quadrupole and octupole
deformations as independent variables is important in fissiongo05 . It has
also been carried out for the 194Pb nucleusme95 which is very soft with
respect to quadrupole deformations.
The single-operator approximation is also problematic due to the fragmentation
of octupole strength in the full spectrum. Roughly speaking, the octupole
strength has two important branches: the low collective excitation that is
under study here, and the high-lying excitation characterized as
$3\hbar\omega$ in the harmonic oscillator model. Our generating field
introduces amplitudes of both into the constrained wave function.
## VI Acknowledgments
We thank H. Goutte, P.-H. Heenen and W. Nazarewicz for reading and comments on
the manuscript. This work was supported in part by the U.S. Department of
Energy under Grant DE-FG02-00ER41132, and by the National Science Foundation
under Grant PHY-0835543. The work of LMR was supported by MICINN (Spain) under
grants Nos. FPA2009-08958, and FIS2009-07277, as well as by Consolider-Ingenio
2010 Programs CPAN CSD2007-00042 and MULTIDARK CSD2009-00064.
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## Appendix: Simplified approximations and limits
It is important to understand the limiting behavior of any computationally
demanding theory, both to check the reliability of the calculations as well as
to see whether approximations are justified that would simplify the
calculations. For the GCM/HW methodology, the theory becomes analytic or
nearly so if a few conditions are met. One requirement is that there be only a
single degree of freedom necessary to describe the excitation of the system.
There are simple Hamiltonians that satisfy this condition. Examples are the
Lipkin modelRS , ro92 , where the degree of freedom is the number of particles
in the excited orbital, and the two-particle problem treated in Ref. ha00
where the degree of freedom is the center-of-mass displacement. In the last
model and other like it the theory becomes analytic and reduces to RPA the if
overlap integrals satisfy the Gaussian Overlap Approximation and the matrix
elements of the Hamiltonian reduce to a quadratic functions times the overlap.
In fact the relation to RPA remains even if there are many degrees of freedom
in the GCM br68 ; ja64 .
To make the discussion concrete, let us assume that there is a single
continuous degree of freedom $q$ and we can write the overlap integral and the
Hamiltonian matrix element as
$\langle q^{\prime}|q\rangle=e^{-(q-q^{\prime})^{2}/q_{s}^{2}}$ (13)
$\frac{\langle q^{\prime}|H|q\rangle}{\langle
q^{\prime}|q\rangle}=E_{0}+\frac{1}{2}v(q+q^{\prime})^{2}-\frac{1}{2}w(q-q^{\prime})^{2}$
(14)
The solution obtained by the Hill-Wheeler construction is identical to the
solution of the RPA equation for the operator $\hat{Q}$ that generates the GCM
states $|q\rangle$. The HW wave functions have the form of Gaussians in the
variable $q$ and the excitation energy is given by
$\hbar\omega_{RPA}=q_{s}^{2}\sqrt{vw}.$ (15)
Let us now compare with the MAP approximation. Here one first calculates
projected energies as a function of $q$,
$\frac{\langle e|H|e\rangle}{\langle
e|e\rangle}=2vq^{2}\frac{v-we^{-4(q/q_{s})^{2}}}{1+e^{-4(q/q_{s})^{2}}}$ (16)
and
$\frac{\langle o|H|o\rangle}{\langle
o|o\rangle}=2vq^{2}\frac{v+we^{-4(q/q_{s})^{2}}}{1-e^{-4(q/q_{s})^{2}}}$ (17)
The energies are then minimized with respect to $q$. The results for a range
of values of the ratio $w/v$ are given in Table 5. The ratios $q_{0}/q_{e}$
are close to $\sqrt{3}$, which may reflect the harmonic oscillator character
of the exact HW wave functions. In the last columns we compare the MAP
excitations energy with the RPA values. They are remarkably close.
$w/v$ | $q_{e}$ | $E_{e}$ | $q_{o}$ | $E_{o}$ | $E_{o}-E_{e}$ | $\hbar\omega_{RPA}$
---|---|---|---|---|---|---
1.5 | 0.226 | -0.0125 | 0.390 | 1.212 | 1.225 | 1.225
2. | 0.292 | -0.0421 | 0.509 | 1.373 | 1.415 | 1.414
4. | 0.400 | -0.232 | 0.716 | 1.782 | 2.01 | 2.00
8. | 0.469 | -0.721 | 0.870 | 2.207 | 2.93 | 2.83
Table 5: The MAP solution in the harmonic limit. Deformations are in units of
$q_{s}$ and energies are in units of $vq_{s}^{2}$. The last column shows the
(Q)RPA excitation energy, Eq (15).
As a general conclusion, we find that if the MAP conditions are satisfied, the
energies are close to the RPA performed with a single collective variable. For
those nuclei, it would better to extend the space for the calculation using
more RPA degrees of freedom than by going to large amplitudes in a single
collective variable.
It would be nice to find a criterion to test for validity of the simplified
treatment. The first condition we can check is the ratio $q_{o}/q_{e}$. This
is graphed in Fig. 13 for the 284 nuclei tabulated in Ref. ki02 .
Figure 13: Ratio of MAP deformations $\beta_{3p}/\beta_{3m}$ for nuclei with
measured $E_{3}$ ki02 .
There is a strong peak at $\beta_{3m}/\beta_{3p}\approx 1.9$. This is slightly
higher than the single-mode (Q)RPA, but still close enough to make a further
investigation of the quadratic Hamiltonian approximation. There are also wings
on the distribution extending from 0.9 (16O) to 3.2 (230U). Excluding the
wings below 1.7 and above 2.2, the peak contains 80 % of the measured nuclei.
To examine the validity of the quadratic approximation, we compared the
extracted coefficients $vq_{s}^{2}$ and $wq_{s}^{2}$ at the two deformations
$\beta_{3p}$ and $\beta_{3m}$. If the quadratic approximation is valid, they
should be equal. For example, the values of $\beta_{3p}$ and $\beta_{3m}$ at
the closest mesh points are 0.0375 and 0.075, respectively. The values of
$v\beta_{3p}^{2}$ and $w\beta_{3p}^{2}$ extracted at that mesh point are 0.23
MeV and 1.72 MeV, respectively. The corresponding numbers for $\beta_{3m}$ are
0.94 MeV and 7.20 MeV, very close to 4 times the values at $\beta_{3p}$. This
is just what is expected given $\beta_{3m}/\beta_{3p}=2$, showing that 208Pb
satisfies the conditions for the quadratic Hamiltonian. With these values for
$v$ and $w$, the RPA energy formula Eq. (15) gives 3.9 MeV, close to the
GCM/HW value of $4.0$ MeV. The results for the nuclei within the peak of Fig.
14 is shown as a scatter plot of the ratios. In general, the $w$ term follows
a quadratic dependence very well. The $v$ term can have large deviations,
particularly for nuclei that are soft to octupole deformations. However, for
most of the nuclei, the quadratic approximation is valid to an accuracy far
better than needed, given the overall performance of the theory in non-
octupole deformed nuclei at the 25% level in the scaled energies.
Figure 14: Ratio of MAP deformations $\beta_{3m}/\beta_{3p}$ for nuclei with
measured $E_{3}$ ki02 .
|
arxiv-papers
| 2011-07-18T21:13:32 |
2024-09-04T02:49:20.685427
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L.M. Robledo and G.F. Bertsch",
"submitter": "George F. Bertsch",
"url": "https://arxiv.org/abs/1107.3581"
}
|
1107.3614
|
# New construction APN quadratic functions
Zahid Mounir Université de Paris 8
###### Abstract
Le but de cet exposé est de détailler l’article de Mr Carlet. Au passage je
ferais un rappel sur quelques résultats intéressants en théorie des corps
finis, puis je donnerais des preuves (nouvelles) de quelques résultats connus,
ensuite je généraliserais la construction d’une famille de fonction APN. La
référence du résultat précédera ce dernier, en cas d’absence de référence, la
preuve sera de l’auteur.
## 1 Corps finies
Certains résultats ne seront pas prouvés nous renvoyons le lecteur curieux à
[1]. Certains résultats ne requièrent pas la finitude du corps, nous renvoyons
le lecteur à cette référence [2], dans la suite $\mathbf{K}$ désigne un corps
commutatif quelconque pas forcément fini.
###### Proposition 1.1.
[2] Étant donné un corps $\mathbf{K}$.$P\in\mathbf{K}[X]$. Le polynôme $P$ n’a
pas de facteur carrés si et seulement si $\gcd(P,P^{\prime})=1$
###### Proposition 1.2.
[2] $\mathbf{K}$ corps, $n\in\mathbb{N}_{>1}$, $s\in\mathbb{N\up{*}}$, dans
$\mathbf{K}[X]$ on a:
$\gcd(X^{s}-1,X^{n}-1)=X^{\gcd(s,n)}-1$
###### Corollaire 1.1.
[2] $\mathbf{K}$ corps , $n\in\mathbb{N}_{>1}$, $s\in\mathbb{N\up{*}}$:
$X^{s}-1|X^{n}-1\Leftrightarrow s|n\;\mathrm{dans}\;\mathbb{N\up{*}}$
###### Proposition 1.3.
[1] $\mathbb{F}$ un corps fini t.q $\\#\mathbb{F}=q\;\mathrm{alors}\;\;\forall
x\in\mathbb{F}\;$
$x^{q}=x.$ (1)
et inversement les solution de l’equation (1) sont exactement les éléments de
$\mathbb{F}$ .
###### Corollaire 1.2.
[1] $p$ un nombre premier $s,n\in\mathbb{N\up{*}}$. alors :
$\mathbb{F}_{p^{s}}\bigcap\mathbb{F}_{p^{n}}=\mathbb{F}_{p^{\gcd(s,n)}}$
###### Corollaire 1.3.
$\mathbb{F}_{2^{n}}$ un corps fini $\forall\beta\in\mathbb{F}_{2^{n}}$
$\forall a\in\mathbb{N}$:
${\beta}^{2^{a}}={\beta}^{2^{a\bmod n}}$
###### Proof.
Par division euclidienne de $a$ par $n$, $\exists(q,r)\in\mathbb{N}^{2}$ tel
que : $a=qn+r\quad\mathrm{avec}\;0\leq r<n.$ Donc
$n|nq\Rightarrow\mathbb{F}_{2^{n}}\subset\mathbb{F}_{2^{nq}}\Rightarrow\forall\beta\in\mathbb{F}_{2^{n}},\;{\beta}^{2^{qn}}=\beta$
on conclut avec :
${\beta}^{2^{a}}={\beta}^{2^{qn+r}}=({\beta}^{2^{qn}})^{2^{r}}={\beta}^{2^{r}}={\beta}^{2^{a\bmod
n}}$
∎
###### Corollaire 1.4.
Soit $n$ un entier pair non nul.
1. (i)
$\;\forall
x\in\mathbf{F}_{2^{n}},\;x^{2^{\frac{n}{2}}+1}\in\mathbf{F}_{2^{n/2}}$
2. (ii)
Si $n/2$ impair, alors
$\mathbb{F}_{2^{n/2}}\cap\mathbb{F}_{2^{2}}=\mathbb{F}_{2}$.
###### Proof.
1. (i)
Il suffit d’appliquer le corollaire 1.2.
2. (ii)
En effet
${(x^{2^{\frac{n}{2}}+1})}^{2^{\frac{n}{2}}}=x^{2^{n}+2^{\frac{n}{2}}}=x^{2^{\frac{n}{2}}+1}$
∎
###### Proposition 1.4.
[2] $p$ premier, $n\in\mathbb{N\up{*}}$ et $q=p^{n}$. Les
$\mathbb{F}_{p}$-sous espaces vectorielles de $\mathbb{F}_{q}$ sont au nombre
de :
$\sum_{s=0}^{n}\frac{(p^{n}-1)(p^{n-1}-1)\ldots(p^{n-s+1}-1)}{(p^{s}-1)(p^{s-1}-1)\ldots(p-1)}$
###### Proof.
Pour $s\in\\{1,\ldots,n\\}$ , dénombrons les $\mathbb{F}_{p}$-sous espaces
vectorielles de dimension $s$ de $\mathbb{F}_{q}$ .
* .
Le premier vecteur étant choisi non nul : $p^{n}-1$ possibilités.
* .
Le second vecteur,non colinéaire au premiers : $p^{n}-p$ possibilités.
* .
Le troisième vecteur non lié aux deux premiers : $p^{n}-p^{2}$ possibilités.
* .
$\ldots$
* .
Le $s$-ième vecteur non lié aux précédents : $p^{n}-p^{s-1}$ possibilités.
Il y’a donc $(p^{n}-1)(p^{n}-p)\ldots(p^{n}-p^{s-1})$ systèmes libres à $s$
éléments. Le même raisonnement montre qu’un $\mathbb{F}_{p}$-sous espace
vectorielle de dimension $s$ de $\mathbb{F}_{q}$ admet:
$(p^{s}-1)(p^{s}-p)\ldots(p^{s}-p^{s-1})$ bases. Le nombre de
$\mathbb{F}_{p}$-espace vectorielle de dimension de dim $s$ est donc:
$\frac{(p^{n}-1)(p^{n}-p)\ldots(p^{n}-p^{s-1})}{(p^{s}-1)(p^{s}-p)\ldots(p^{s}-p^{s-1}}=\frac{(p^{n}-1)(p^{n-1}-1)\ldots(p^{n-s+1}-1)}{(p^{s}-1)(p^{s-1}-1)\ldots(p-1)}$
∎
### 1.1 Critère d’irréductibilité
###### Proposition 1.5.
[2] Soit $P\in\mathbf{K}[X]\;\mathrm{tel\,que}\;D\up{\textdegree}(P)\leq 3$.
$P$ est irréductible sur $\mathbf{K}$ si et seulement si $P$ n’a pas de racine
dans $\mathbf{K}$
###### Proposition 1.6.
[2] Soit $P\in\mathbf{K[X]}\;\text{avec}\;D\up{\textdegree}(P)=n$.
$P$ est irréductible si et seulement si $P$ n’a pas de racines dans toutes
extension $\mathbf{L}/\mathbf{K}$ tel que: $[\mathbf{L}:\mathbf{K}]\leq n/2$.
###### Proof.
###### Condition nécéssaire.
$P$ irréductible sur $\mathbf{K}$. Soit $\alpha\in\mathbf{L}$, racine de $P$
alors $\mathbf{K(\alpha)}$ est un corps de rupture de $P$.
Donc $[\mathbf{K(\alpha)}:\mathbf{K}]=n\Rightarrow[\mathbf{L}:\mathbf{K}]\geq
n>n/2$.
∎
###### Condition suffisante.
Par Contraposition. Si $P$ n’est pas irréductible, il existe
$(Q,R)\in{\mathbf{K[X]}}^{2}$. tel que : $P=QR\;\mathrm{et}\;1\leq
D\up{\textdegree}(R),D\up{\textdegree}(Q)<n$, sans perte de généralité on peut
supposer que : $D\up{\textdegree}(Q)\leq\frac{n}{2}$. Soit $f$ un facteur
irréductible de $Q$, et $\mathbf{L}=\mathbf{K(\alpha)}$ un corps de rupture de
$f$ alors $\alpha\in\mathbf{L}$ est une racine de $P(X)$ et
$[\mathbf{L}:\mathbf{K}]=D\up{\textdegree}(f)\leq\frac{n}{2}$. ∎
∎
###### Proposition 1.7.
[2] Soit $P\in\mathbf{K[X]}$ irréductible, $D\up{\textdegree}(P)=n$ et
$\mathbf{L}/\mathbf{K}$ extension de degré $m$ de $\mathbf{K}$ avec
$\gcd(m,n)=1$ alors $P$ est irréductible dans $\mathbf{L[X]}$.
###### Proof.
Supposons $P$ est réductible dans $\mathbf{L[X]}$, soit $f$ un facteur
irréductible de $P\;\mathrm{dans}\;\mathbf{L[X]}$ alors
$0<D\up{\textdegree}(f)<n$. Soit $M=\mathbf{L(\alpha)}$ un corps de rupture de
$f$.
$P$ étant irréductible dans $\mathbf{K[X]}$, donc $\mathbf{K(\alpha)}$ corps
de rupture de $P$ sur $\mathbf{K}$. Alors $[\mathbf{K(\alpha)}:\mathbf{K}]=n$
, donc
$[\mathbf{M}:\mathbf{K}]=[\mathbf{M}:\mathbf{K(\alpha)}][\mathbf{K(\alpha)}:\mathbf{K}]$
est divisible par $n$. Or
$[\mathbf{M}:\mathbf{K}]=[\mathbf{M}:\mathbf{L}].[\mathbf{L}:\mathbf{K}]=D\up{\textdegree}(f)\times
m$. Comme $\gcd(m,n)=1$, il vient $n$ divise $D\up{\textdegree}(f)$,
contradiction. ∎
Remarque: La proposition 1.7, peut être déduite de la proposition qui va
suivre, si nous avons éviter, c’est pour insister sur son caractère générique.
###### Proposition 1.8.
[1] Soit $P\in\mathbb{F}_{q}[X]$ irréductible de degré $n$ et soit
$k\in\mathbb{N}^{*}$, alors $P$ se factorise en $d$ polynômes irréductibles
sur $\mathbb{F}_{q^{k}}[X]$ de degré $n/d$ avec $d=\gcd(n,k)$
### 1.2 Trace sur un corps
###### Définition 1.1.
Soient $\mathbf{K}=\mathbb{F}_{q}$ et $\mathbf{F}=\mathbb{F}_{q^{m}}$. Pour
tout $\alpha\in\mathbf{F}$, la trace
$\textbf{Tr}_{\mathbf{F}/\mathbf{K}}(\alpha)$ de $\alpha$ sur $\mathbf{K}$ est
définie par:
$\textbf{Tr}_{\mathbf{F}/\mathbf{K}}(\alpha)=\alpha+\alpha^{q}+\ldots+\alpha^{q^{m-1}}$
Si $\mathbf{K}$ est un corps premier la trace est dite absolue et on la note
seulement $\textbf{Tr}_{\mathbf{F}}.$
La trace a des propriétés intéressantes que nous énoncerons sous forme d’un
théorème:
###### Théorème 1.1.
[1] Soient $\mathbf{F}=\mathbb{F}_{q^{m}}$ et $\mathbf{K}=\mathbb{F}_{q}$
alors:
1. (i)
$\textbf{Tr}_{\mathbf{F}/\mathbf{K}}$ est une forme $\mathbf{K}$-linéaire non
nulle surjective.
2. (ii)
Pour tout $a\in\mathbf{K}\;,\;\textbf{Tr}_{\mathbf{F}/\mathbf{K}}(a)=ma$.
3. (iii)
Pour tout
$\alpha\in\mathbf{F},\;\textbf{Tr}_{\mathbf{F}/\mathbf{K}}(\alpha^{q})=\textbf{Tr}_{\mathbf{F}/\mathbf{K}}(\alpha)$
(La trace est stable par le Frobenius).
###### Proposition 1.9 (Transitivité de la trace).
[1]
Soient $\mathbf{K}$ un corps fini, $\mathbf{F}$ une extension finie de
$\mathbf{K}$ et $\mathbf{E}$ une extension finie de $\mathbf{F}$. Alors
$\textbf{Tr}_{\mathbf{E}/\mathbf{K}}=\textbf{Tr}_{\mathbf{F}/\mathbf{K}}o\textbf{Tr}_{\mathbf{E}/\mathbf{F}}$
###### Corollaire 1.5.
Soit $n$ un entier pair non nul. Alors:
1. 1.
$\forall
x\in\mathbb{F}_{2^{n/2}},\;\textbf{Tr}_{\mathbb{F}_{2^{n}}}(x)=0.\,(\;\textit{i.e}\;\mathbb{F}_{2^{n/2}}\subset\mathrm{\mathcal{K}er}(\textbf{Tr}_{\mathbb{F}_{2^{n}}}))$
2. 2.
Il existe $\omega\in\mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{n/2}}\;\text{tel
que}\;\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)=1$
###### Proof.
1. 1.
Par définition : Pour tout
$x\in\mathbb{F}_{2^{n}}\;,\;\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(x)=x+x^{2^{\frac{n}{2}}}$
et donc
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}$ est
nulle sur $\mathbb{F}_{2^{n/2}}$ on conclut avec
$\textbf{Tr}_{\mathbb{F}_{2^{n}}}=\textbf{Tr}_{\mathbb{F}_{2^{n/2}}}o\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}$(
cf. proposition 1.9)
2. 2.
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}{}$
est une forme $\mathbb{F}_{2^{n/2}}$-linéaire non nulle.Donc,il existe
$x_{0}\in\mathbb{F}_{2^{n}}$ tel que
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(x_{0})\neq
0$. Il suffit de choisir
$\omega=\frac{x_{0}}{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(x_{0})}$
est conclure par la $\mathbb{F}_{2^{n/2}}$ -linéarité de la
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}{}$.
∎
Remarque: le point $(2)$ peut-être directement prouvé on utilisant la
surjectivité de la trace, seulement je voulais donner une construction
effective.
###### Proposition 1.10.
Soient $n$ un entier pair non nul, $q=2^{n/2}$ et $c\in\mathbb{F}_{2^{n}}$
vérifiant : $c^{q+1}=1$ alors
$(\frac{1}{c^{2^{n-1}}})^{q}=\frac{c}{c^{2^{n-1}}}$ de plus on a:
$\forall\omega\in\mathbb{F}_{2^{n}}\quad\frac{\omega+c{\omega}^{q}}{c^{2^{n-1}}}\in\mathbb{F}_{2^{n/2}}.$
###### Proof.
$c^{q+1}=1\Leftrightarrow c=\frac{1}{c^{q}}$ $\Rightarrow$
$\displaystyle\left(\frac{1}{c^{2^{n-1}}}\right)^{q}$
$\displaystyle=c^{2^{n-1}}$ $\displaystyle=c^{2^{n}-2^{n-1}}$
$\displaystyle=\frac{c^{2^{n}}}{c^{2^{n-1}}}$
$\displaystyle=\frac{c}{c^{2^{n-1}}}$
Soit $\omega\in\mathbb{F}_{2^{n}}$.
$\frac{\omega+c{\omega}^{q}}{c^{2^{n-1}}}=\frac{w}{c^{2^{n-1}}}+{\omega}^{q}\frac{c}{c^{2^{n-1}}}=\frac{\omega}{c^{2^{n-1}}}+\left(\frac{\omega}{c^{2^{n-1}}}\right)^{q}\\\
=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{\omega}{c^{2^{n-1}}}\right)\in\mathbb{F}_{2^{n/2}}$
∎
### 1.3 Permutation particulière sur un corps fini
###### Lemme 1.3.1.
Soit $G=<x>$ un groupe cyclique d’ordre $n$. Alors $\forall d,\;d\mid
n\;\mathrm{dans}\;\mathbb{N^{*}}$
il existe un unique sous groupe d’ordre $d$ de $G$. Il est engendré par
$x^{k}$, $k=\frac{n}{d}$
###### Proposition 1.11.
Soient $\mathbf{F}_{q}$ un corps fini, $i\in\mathbb{N}^{*}$.
1. (i)
$X^{i}$ permute $\mathbf{F}_{q}$ si et seulement si $\gcd(i,q-1)=1$.
2. (ii)
Le nombre de i puissance non nulle dans $\mathbf{F}_{q}$(
i.e$\;\\#{{\mathbb{F^{*}}_{q}}^{i}}\,$) est $(q-1)/\gcd(i,q-1)$.
###### Proof.
Remarquons d’abord que $0$ est la seule solution de l’équation $X^{i}=0$.
Considérons le morphisme de groupe $\mathfrak{F_{i}}\colon
x\in{\mathbb{F}_{q}}^{*}\to x^{i}\in\mathbb{F^{*}}_{q}$. Calculons son noyau:
$\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})=\\{\,x\in{\mathbb{F}_{q}}^{*}\mid
x^{i}=1\,\\}$
On a
$x\in\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})\Rightarrow\;\mathrm{ord}(x)\mid
i\Rightarrow\mathrm{ord}(x)\mid d\,,\,\mathrm{ou}\;d=\gcd(i,q-1)\Rightarrow
x\in\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})$ l’inverse est évidente. Donc:
$\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})=\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})\;\mathrm{avec}\;d=\\\
gcd(i,q-1).$
Le polynôme $P(x)=X^{d}-1$ n’a que des racines simples. ( Voir Proposition 1.1
). Étant un polynôme de degré $d$, $P$ a donc $d$ racine distinct.
$d\mid(q-1)$ implique $X^{d}-1\mid X^{(q-1)}-1$. Donc tous les racines de $P$
sont dans ${\mathbb{F}_{q}}^{*}$.
Ceci entraîne $\\#{\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})}=d$.
$\mathfrak{F_{i}}$ est un isomorphisme si et seulement si $d=1$, c-à-d
$\gcd(i,q-1)=1$.
D’après le 1er théorème d’isomorphisme
${\mathbb{F}_{q}}^{*}/\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})\cong\mathcal{I}m(\mathfrak{F_{i}})$
or $\mathcal{I}m(\mathfrak{F_{i}}):=$ les i puissances non nulles dans
$\mathbf{F}_{q}$, ce qui conclut la preuve. ∎
Remarque:
1. 1.
$\mathcal{I}m(\mathfrak{F_{i}})=\mathcal{I}m(\mathfrak{F}_{d})$; c-à-d les i
puissances non nulles dans $\mathbf{F}_{q}$ sont exactement les d puissances
non nulles dans $\mathbf{F}_{q}$.
2. 2.
$\forall(x,y)\in{\mathbf{F}_{q}}^{2}\quad x^{i}=y^{i}\Leftrightarrow
x^{d}=y^{d}$ ( Découle de
$\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})=\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})$
).
3. 3.
Soit $\alpha$ un générateur de ${\mathbb{F}_{q}}^{*}$
$\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})$ est un sous-groupe de
${\mathbb{F}_{q}}^{*}$ d’ordre $d$, il est donc engendré par
$\xi={\alpha}^{(q-1)/d}$ ( cf. lemme 1.3.1 ).
$\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})=\left\\{\xi^{k};k=0,\ldots,d-1\right\\}$
4. 4.
Définissons une relation d’équivalence sur ${\mathbb{F}_{q}}^{*}$ par:
$\forall(x,y)\in{{\mathbb{F}_{q}}^{*}}^{2},\quad
x\mathfrak{R}y\;\mathrm{sietseulementsi}\;y\in
x\mathrm{\mathcal{K}er}(\mathfrak{F_{i}})$
C’est bien une relation d’équivalence et : $\forall
x\in{\mathbb{F}_{q}}^{*}\quad\mathcal{C}l(x)=x\mathrm{\mathcal{K}er}(\mathfrak{F}_{d})=\left\\{x\xi^{k};k=0,\ldots,d-1\right\\}$.
Les classes forment une partition de l’ensemble en question:
${\mathbb{F}_{q}}^{*}=\bigcup_{x\in\mathcal{I}}\mathcal{C}l(x)\quad\mathrm{avec}\;\\#\mathcal{I}=\frac{q-1}{d}\;\mathrm{et}\;\\#\mathcal{C}l(x)=d.$
### 1.4
$\mathbb{F}_{2^{n}}\;\mathrm{versus}\;\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}$
Certaines fonctions sont définies sur
$\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}$, on aimerait bien expliciter
leur représentation univariée et pour cela il faut les définir sur
$\mathbb{F}_{2^{n}}$, nous allons voir comment:
Soit $\omega\in\mathbb{F}_{2^{n}}/\mathbb{F}_{2^{n/2}}$. $(1,\omega)$ est une
base du $\mathbb{F}_{2^{n/2}}$-espace vectorielle $\mathbb{F}_{2^{n}}$. Pour
tout $X\in\mathbb{F}_{2^{n}}$, il existe un unique couple $(x,y)$ dans
$\mathbb{F}_{2^{n/2}}$ tel que
$X=x+\omega y$ (2)
(
i.e$\;\mathbb{F}_{2^{n}}=\mathbb{F}_{2^{n/2}}\oplus\omega\mathbb{F}_{2^{n/2}}).$
Nous allons expliciter $x$ et $y$ en fonction de $X$. Appliquons
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}$ a
l’equation (2), on obtient
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(X)=y\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)$
de même d’après (2):
$X\omega^{2^{n/2}}+X^{2^{n/2}}\omega=x(\omega^{2^{n/2}}+\omega)$ c-à-d
$x=\frac{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(X\omega^{2^{n/2}}\right)}{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)}$
Nous avons le $\mathbb{F}_{2^{n/2}}$-isomorphisme d’espace vectorielle
suivant:
$X\in\mathbb{F}_{2^{n}}\rightarrow(x,y)\in\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}$
avec
$x=\frac{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(X\omega^{2^{n/2}}\right)}{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)}\;\mathrm{et}\;y=\frac{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(X)}{\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)}$
Remarque:
1. 1.
D’après le corollaire 1.5 $\omega$ peut-être choisi tel que
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)=1$
ce qui simplifie considérablement le calcul.
2. 2.
Dans le cas ou $n/2$ est impair, on a davantage de simplification, il suffit
de choisir $\omega$ élément primitif de $\mathbb{F}_{4}$ on a d’après la
corollaire 1.3: $\omega^{2^{n/2}}=w^{2}$
3. 3.
Rien n’empêche de prendre $\omega=\alpha^{2^{n/2}-1}$ et dans ce
cas:$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(\omega)=\omega+\omega^{-1}$
## 2 Fonctions Courbes
Nous allons donner quelques résultats intéressants, pour une étude plus
approfondie, nous envoyons à [3].
### 2.1 Transformée de Walsh
La Transformée de Walsh d’une fonction booléenne $f$ et la transformée de
Fourier de sa fonction signe. Son expression est donc:
$\forall
u\in\mathbb{F}_{2}^{n}\quad\hat{\chi}_{f}(u)=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{f(x)+u\cdot
x}\;\textrm{ o\\`{u} $u\cdot x$ d\'{e}signe le produit scalaire dans
$\mathbb{F}_{2}^{n}$ }$
Si $\hat{\chi}_{f}(0)=0$ alors $f$ est équilibrée.
Si $\forall u\in\mathbb{F}_{2}^{n}\;\hat{\chi}_{f}(u)=\pm 2^{n/2}$ alors $f$
est dite courbe.
### 2.2 Étude d’une fonction booléenne particulière
Dans la suite nous identifions $\mathbb{F}_{2^{n}}$ a $\mathbb{F}_{2}^{n}$ et
nous posons $u\cdot x=\textbf{Tr}_{\mathbb{F}_{2^{n}}}(xu)$
Étude de la fonction
$f(x)=\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})\;\mathrm{avec}\;a\neq 0$. On
peu déjà remarquer que $\gcd(i,2^{n}-1)\neq 1$ sinon $f$ serait équilibrée,
une telle fonction n’est jamais courbe.
#### 2.2.1 Cas $a\in{\mathbb{F}}^{*i}_{2^{n}}$
###### Proposition 2.1.
$f$ est courbe si et seulement si
$g(x)=\textbf{Tr}_{\mathbb{F}_{2^{n}}}(x^{i})$ est courbe.
###### Proof.
$a\in\mathbb{F}^{*i}_{2^{n}}\Leftrightarrow\exists
b\in{\mathbb{F}}^{*}_{2^{n}}\mid\,a=b^{i}.$ Soit
$\beta\in\mathbb{F}_{2^{n}}\;$
$\displaystyle\hat{\chi}_{f}(\beta)$
$\displaystyle=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
x)}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}((bx)^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
x)}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(x^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\frac{\beta}{b}x)}$
$\displaystyle=\hat{\chi}_{g}\left(\frac{\beta}{b}\right)$
∎
Remarque: L’étude que j’ai menée sur le caractère courbe de $g$ sur
$\mathbb{F}_{2^{k}}\,\mathrm{o\\`{u}}\;k=4,\ldots,22$, a montré que $g$ est
courbe uniquement sur $\mathbb{F}_{2^{8}}$ avec l’exposant
$i=(1+j)15,\;j=0,\ldots,15$. Ce qui correspond a un exposant de Dillon,
étrangement $\mathbb{F}_{2^{8}}$ c’est le corps où est définit l’A.E.S, y’a
t-il une causalité??.
#### 2.2.2 Cas $a\notin{\mathbb{F}}^{*i}_{2^{n}}$
Dans cette partie nous allons tirer profit de l’étude que nous avons réalisé
dans la Proposition 1.11. Les notations sont celles de la-dite Proposition et
de la remarque qui l’a suivie avec $q=2^{n}$. Soit
$\beta\in\mathbb{F}_{2^{n}}$
$\xi={\alpha}^{(2^{n}-1)/d},\mathrm{ou}\;d=\gcd(2^{n}-1,i)$
$\displaystyle\hat{\chi}_{f}(\beta\xi)$
$\displaystyle=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta\xi
x)}$ $\displaystyle=\hat{\chi}_{f}(\beta).\left(\,\mathrm{car}\,x\longmapsto
x\xi\,\text{est une permutation sur}\;\mathbb{F}_{2^{n}}\,\right)$
Pour résumer :
$\boxed{\forall y\in\mathcal{C}l(x)\quad\hat{\chi}_{f}(x)=\hat{\chi}_{f}(y)}$
La transformée de Walsh est constante sur les classes,ceci peut conduire à un
algorithme plus rapide(?). On peut aussi réécrire la TW autrement:
$\displaystyle\hat{\chi}_{f}(\beta)$
$\displaystyle=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
x)}$ (3)
$\displaystyle=1+\sum_{x\in\mathbb{F}^{*}_{2^{n}}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})+\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
x)}$ (4)
$\displaystyle=1+\sum_{x\in\mathcal{I}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})}\sum_{y\in\mathcal{C}l(x)}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
y)}$ (5)
$\displaystyle=1+\sum_{x\in\mathcal{I}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})}\sum_{k=0}^{d-1}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(\beta
x\xi^{k})}$ (6)
###### Proposition 2.2.
Si $f(x)=\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})$ est courbe, alors:
$\hat{\chi}_{f}(0)=\begin{cases}2^{n/2},&\text{ssi}\;\gcd(i,2^{n/2}+1)=1\\\
-2^{n/2},&\text{ssi}\;\gcd(i,2^{n/2}-1)=1\end{cases}$
###### Proof.
$d=\gcd(i,2^{n}-1)=\gcd(i,2^{n/2}-1).\gcd(i,2^{n/2}+1).$
$k=\gcd(i,2^{n/2}-1)$ et $l=\gcd(i,2^{n/2}+1).$En posant $\beta=0$ dans
l’égalité (6)
$\hat{\chi}_{f}(0)-1=kl\sum_{x\in\mathcal{I}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})}$
1er cas:
$\hat{\chi}_{f}(0)=2^{n/2}$ ceci entraîne
$2^{n/2}-1=kl\sum_{x\in\mathcal{I}}(-1)^{\textbf{Tr}_{\mathbb{F}_{2^{n}}}(ax^{i})}$
Alors $l|2^{n/2}-1\;\mathrm{comme}\;\gcd(l,2^{n/2}-1)=1$, cela implique $l=1$
2me cas:
$\hat{\chi}_{f}(0)=-2^{n/2}$ le même raisonnement conduit à $k=1$.
∎
## 3 Construction d’une classe APN a partir d’une fonction Bent
###### Définition 3.1.
Une $(n,n)-$fonction $F$ est dite APN si:
$\forall a\in\mathbb{F^{*}}_{2^{n}},\forall b\in\mathbb{F}_{2^{n}}$.
L’équation : $F(x)+F(x+a)=b$ à au plus $0$ ou $2$ solutions .
Notation:
$\forall a\in\mathbb{F^{*}}_{2^{n}},D_{a}F(x)=F(x+a)+F(x).$ (7)
###### Proposition 3.1.
[3]
1. (i)
$B$ est courbe si et seulement si $D_{a}B$ est équilibré.
2. (ii)
$B$ quadratique alors $D_{a}B$ est affine.
###### Proof.
voir livre [3]. ∎
Remarque: Si $B$ une $(n,n/2)-$courbe quadratique, alors les solutions de
$D_{a}B(x)=b$ avec $(a,b)\in\mathbb{F^{*}}_{2^{n}}\times\mathbb{F}_{2^{n/2}}$
est un sous-espaces affine de $\mathbb{F}_{2^{n}}$ affinement isomorphe a
$\mathbb{F}_{2^{n/2}}$. Seulement c’est isomorphisme il n’est pas simple de
l’explicité, d’autant plus qu’il depend de $a$ et $b$. On va voir que dans le
cas de la fonction simple de Mairona Mac Farland ce n’est pas le cas. L’
auteur de l’article [4] a exploité cette idée, pour construire une classe de
fonction APN.
Posons : $B(x)=X^{2^{n/2}+1}$ et soit $G$ une $(n,n/2)-$fonction.
Et définissons
$F:\,x\in\mathbb{F}_{2^{n}}\rightarrow(B(x),G(x))\in\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}.$
Probleme: Donner une condition nécessaire et suffisante portant sur $G$ pour
que $F$ soit APN.
$F$ APN ssi $\forall
a\in\mathbb{F^{*}}_{2^{n}},\forall(c,d)\in\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}\;D_{a}F(X)=(c,d)$
à au plus $0$ ou deux solutions dans $\mathbb{F}_{2^{n}}$.
$\begin{cases}B(X)+B(X+a)&=c\\\ G(X)+G(X+a)&=d\end{cases}$ (8)
or
$D_{a}B(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(a^{2^{n/2}}X)+a^{2^{n/2}+1}$
et donc:
$D_{a}B(X)=c\Leftrightarrow\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(a^{2^{n/2}}X)+a^{2^{n/2}+1}=c\Leftrightarrow\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(X)=1+\frac{c}{a^{2^{n/2}+1}}\quad\textrm{chgement
de variable $X\to aX$ }$
Soit $b\in\mathbb{F}_{2^{n}}\;\textrm{tel
que}\,\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(b)=1+\frac{c}{a^{2^{n/2}+1}}\;\textrm{
Voir la surj\'{e}ctivit\'{e} de la trace }$ et donc:
$D_{a}B(X)=c\Leftrightarrow\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(X+b)=0\Leftrightarrow\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(X)=0\quad\textrm{changement
de variable $X\to X+b$ }\Leftrightarrow X\in\mathbb{F}_{2^{n/2}}$
L’îsomorphisme affine est
$\varphi:X\in\mathbb{F}_{2^{n/2}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}aX+b\in(D_{a}B)^{-1}(c)\;\textrm{avec}\;\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}(b)=1+\frac{c}{a^{2^{n/2}+1}}$
En reportant dans l’equation (8)
$F$ APN ssi
$\forall a\in\mathbb{F^{*}}_{2^{n}},\forall b\in\mathbb{F}_{2^{n}},\forall
d\in\mathbb{F}_{2^{n/2}}.G(aX+b)+G(aX+b+a)=d$ (9)
à au plus $0$ ou deux solutions sur $\mathbb{F}_{2^{n/2}}$.
On a le théorème suivant:
###### Théorème 3.1.
Soit $B(x)=X^{2^{n/2}+1}$ et soit $G$ une $(n,n/2)-$fonction,
$L:\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}\rightarrow\mathbb{F}_{2^{n}}$
un isomorphisme linéaire et $F:X\in\mathbb{F}_{2^{n}}\rightarrow
L(B(x),G(x))\in\mathbb{F}_{2^{n}}$ alors $F$ est APN si et seulement si
$\forall a\in\mathbb{F^{*}}_{2^{n}},\forall b\in\mathbb{F}_{2^{n}},\forall
d\in\mathbb{F}_{2^{n/2}}\;G(aX+b)+G(aX+b+a)=d$ (10)
à $0$ ou $2$ solutions au plus dans $\mathbb{F}_{2^{n/2}}$.
###### Lemme 3.0.1.
1. (i)
$\forall
a,b\in\mathbb{F}_{2^{r}},(aX+b)^{2^{k}+2^{j}}+(aX+a+b)^{2^{k}+2^{j}}=a^{2^{k}+2^{j}}(X^{2^{k}}+X^{2^{j}}+1)+b^{2^{k}}a^{2^{i}}+b^{2^{i}}a^{2^{k}}$
2. (ii)
Soient $i$ et $r$ premiers entre eux, $c$ élément de $\mathbb{F}_{2^{r}}$,
alors l’équation : $X^{2^{i}}+X+c=0$ à $0$ ou $2$ solutions au plus dans
$\mathbb{F}_{2^{r}}$
###### Proof.
Le point (i) est calculatoire, le point (ii) découle du fait que,
$X\rightarrow X^{2^{i}+1}$ est APN (Gold). ∎
#### 3.0.1 Familles de fonctions connues
Je commencerais par trois lemmes que j’ai jugé fort utile.
###### Lemme 3.0.2.
Pour tout entier $i$
1. (i)
$2^{i}+1$ est divisible par $3$ ssi $i$ est impair.
2. (ii)
$2^{i}-1$ est divisible par $3$ ssi $i$ est pair.
###### Proof.
Tout entier $i$ peut s’écrire :
$i=2k+\epsilon\;\textrm{avec}\,(k,\epsilon)\in\mathbb{N}\times\left\\{0,1\right\\}$
1. (i)
$2^{i}+1=2^{2k+\epsilon}+1=4^{k}2^{\epsilon}+1\equiv(2^{\epsilon}+1)\mod 3$
donc $2^{i}+1\equiv 0\mod 3\;\textrm{ssi}\;\epsilon=1$
2. (ii)
$2^{i}-1=2^{2k+\epsilon}-1=4^{k}2^{\epsilon}-1\equiv(2^{\epsilon}-1)\mod 3$
donc $2^{i}-1\equiv 0\mod 3\;\textrm{ssi}\;\epsilon=0$
∎
###### Lemme 3.0.3.
Soient $n$ un entier pair et $i$ un entier vérifiant $\gcd(i,n/2)=1$ alors
$\gcd(2^{i}+1,2^{n/2}+1)=\begin{cases}1,&\text{ssi $i$ pair}\\\ 1,&\text{ssi
$i$ impair et $n/2$ pair}\\\ 3,&\text{ssi $i$ impair et $n/2$
impair}\end{cases}$
###### Proof.
On a d’un coté
$\gcd(2^{2i}-1,2^{n}-1)=\gcd(2^{i}+1,2^{n}-1).\gcd(2^{i}-1,2^{n}-1)=\gcd(2^{i}+1,2^{n/2}-1)\gcd(2^{i}+1,2^{n/2}+1)(2^{\gcd(i,n)}-1)$.
de l’autre $\gcd(2^{2i}-1,2^{n}-1)=2^{2.\gcd(i,n/2)}-1=3$. Soit
$\boxed{3=\gcd(2^{i}+1,2^{n/2}-1)\gcd(2^{i}+1,2^{n/2}+1)(2^{\gcd(i,n)}-1)}$
on conclut on traitons selon la parité de $i$ et on utilisant le lemme 3.0.2.
∎
###### Lemme 3.0.4.
Soit $q=2^{n/2}$, les solutions de l’équation
$X^{q+1}+1=0$ (11)
sont exactement les éléments de $\mathbb{F}_{2^{n}}^{*(q-1)}$
###### Proof.
En effet, soit $x\in\mathbb{F}_{2^{n}}^{*(q-1)}$, donc il existe
$y\in\mathbb{F}_{2^{n}}^{*}$ vérifiant $x=y^{q-1}$, soit
$x^{q+1}=y^{q^{2}-1}=1$ donc $x$ est solution de l’équation 11. Or d’après la
proposition 1.11, $\,\\#{\mathbb{F}_{2^{n}}^{*(q-1)}}=q+1$.
D’un autre coté les solutions de l’équation (11) sont simples voir proposition
1.1, ils sont au nombre de $q+1$, vue que son degré est $q+1$. Ce qui achève
la preuve. ∎
Dorénavant et dans toutes la suite, nous prendrons pas en compte, les termes
de $\mathbb{F}_{2^{n/2}}$ indépendants de $X$ qui apparaissent dans
$G(aX+b^{\prime})+G(aX+b^{\prime}+a)$, puisqu’on peut toujours les affectés à
$d$ dans l’égalité (10).
###### Corollaire 3.1.
La fonction $F(X)=X^{2^{2i}+2^{i}}+bX^{q+1}+cX^{q(2^{2i}+2^{i})}$ où
$\gcd(i,n/2)=1,\;q=2^{n/2},\;(c,b)\in{\mathbb{F}_{2^{n}}}^{2},\;\textrm{tel
que :
}\;c^{q+1}=1,\;c\notin\left\\{\lambda^{(2^{i}+1)(q-1)},\;\lambda\in\mathbb{F}_{2^{n}}\right\\}\,\textrm{et
}\;cb^{q}+b\neq 0$ est APN
###### Proof.
Nous allons commencer par quelques remarques simples:
1. (i)
$c\notin\mathbb{F}_{2^{n/2}}$
En effet: Si $c\in\mathbb{F}_{2^{n/2}}$ alors $c^{q+1}=c^{2}=1\Rightarrow
c=1\Rightarrow
c\in\left\\{\lambda^{(2^{i}+1)(q-1)},\lambda\in\mathbb{F}_{2^{n}}\right\\}$,
contradiction.
2. (ii)
$\frac{b}{c^{2^{(n-1)}}}\notin\mathbb{F}_{2^{n/2}}$
En effet, sinon :
$\left(\frac{b}{c^{2^{(n-1)}}}\right)^{q}=\frac{b}{c^{2^{(n-1)}}}$ or
$\left(\frac{1}{c^{2^{(n-1)}}}\right)^{q}=\frac{c}{c^{2^{(n-1)}}}$ (cf.
proposition 1.10)
$\Leftrightarrow
b^{q}\frac{c}{c^{2^{(n-1)}}}=\frac{b}{c^{2^{(n-1)}}}\Leftrightarrow
cb^{q}+b=0$, contradiction.
et donc $(1,\frac{b}{c^{2^{(n-1)}}})$ forme une base de $\mathbb{F}_{2^{n}}$
sur $\mathbb{F}_{2^{n/2}}$.
3. (iii)
$F$ est APN si et seulement si $\frac{F}{c^{2^{(n-1)}}}$ est APN ( evident ).
Sans perte de généralité nous pouvons identifier $F$ à
$\frac{F}{c^{2^{(n-1)}}}$ et donc
$F=\frac{X^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}+\frac{b}{c^{2^{(n-1)}}}X^{q+1}+\frac{c}{c^{2^{(n-1)}}}X^{q(2^{2i}+2^{i})}=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{X^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}\right)+\frac{b}{c^{2^{(n-1)}}}X^{q+1}$
Posons $L(u,v)=u+v\frac{b}{c^{2^{(n-1)}}}$ c’est bien un isomorphisme, et
$G(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{X^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}\right).$
Montrons que $G$ vérifie l’équation (10).
$\displaystyle G(aX+b^{\prime})+G(aX+a+b^{\prime})$
$\displaystyle=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}(X^{2^{2i}}+X^{2^{i}}+1)\right)$
$\displaystyle=(X^{2^{2i}}+X^{2^{i}}+1)\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}\right)$
$\displaystyle=(X^{2^{i}}+X+1)^{2^{i}}\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}\right)$
$\displaystyle=d$
Or $x\rightarrow x^{2^{i}}$ est une permutation sur $\mathbb{F}_{2^{n/2}}$,
d’après le lemme 3.0.1, et le théorème 3.1, $F$ est APN si et seulement si
$\forall
a\in\mathbb{F^{*}}_{2^{n}}\quad\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}\right)=0$
n’a pas de solution.
Supposons que c’est le cas, ceci équivaut à
$\displaystyle\frac{a^{2^{2i}+2^{i}}}{c^{2^{(n-1)}}}$
$\displaystyle=\frac{a^{q(2^{2i}+2^{i})}}{c^{q2^{(n-1)}}}$
$\displaystyle=\frac{c}{c^{2^{(n-1)}}}a^{q(2^{2i}+2^{i})}\quad\textrm{( cf.
proposition\leavevmode\nobreak\ \ref{clef} ) }$
Ceci implique
$c=a^{-(q-1)(2^{2i}+2^{i})}=a^{-(q-1)2^{i}(2^{i}+1)}\in\mathbb{F}_{2^{n}}^{(q-1)2^{i}(2^{i}+1)}$
Comme $x\rightarrow x^{2^{i}}$ est une permutation, ce qui entraîne
$c\in\mathbb{F}_{2^{n}}^{(q-1)(2^{i}+1)}$ contradiction, ce qui achève la
preuve. ∎
Remarque:
1. (i)
Les lemmes que j’ai donné 3.0.2, 3.0.4 et surtout 3.0.3, sont très puissantes,
ils trouveront leur application dans ce qui va suivre, mais peuvent être
appliqués en dehors de l’article.
2. (ii)
L’étude faite à la sous-section 1.3, montre que
$\;\mathbb{F}_{2^{n}}^{(q-1)(2^{i}+1)}=\mathbb{F}_{2^{n}}^{\gcd((q-1)(2^{i}+1),2^{n}-1)}=\mathbb{F}_{2^{n}}^{(q-1)\gcd(2^{i}+1,q+1)}$
On utilisant le lemme 3.0.3, on a:
$\mathbb{F}_{2^{n}}^{(q-1)(2^{i}+1)}=\begin{cases}\mathbb{F}_{2^{n}}^{(q-1)},&\text{ssi
$i$ pair}\\\ \mathbb{F}_{2^{n}}^{(q-1)},&\text{ssi $i$ impair et $n/2$
pair}\\\ \mathbb{F}_{2^{n}}^{3(q-1)},&\text{ssi $i$ impair et $n/2$
impair}\end{cases}$
3. (iii)
Le corollaire 3.1, n’est pas tout à fait correcte, car ils y’a des cas où les
hypotheses ne seront jamais satisfaites, et ça c’est très important quand on
implémente, de chercher la où on peut trouver. En effet le lemme 3.0.4,
supprime les deux cas : $i$ pair et $i$ impair avec $n/2$ pair, nous allons
donné une version corrigée est optimale de ce résultat.
###### Corollaire 3.2 (version optimale).
Soient $q=2^{n/2}$, $i$ et $n/2$ impairs, vérifiant $\gcd(i,n/2)=1$. Alors:
La fonction $F(X)=X^{2^{2i}+2^{i}}+bX^{q+1}+cX^{q(2^{2i}+2^{i})}$ où
$\;c,b\in\mathbb{F}_{2^{n}},\textrm{ tel que :
}\;c^{q+1}=1,\,c\notin\mathbb{F}_{2^{n}}^{*3(q-1)},\textrm{et }\;cb^{q}+b\neq
0$ est APN
###### Corollaire 3.3.
Soient $q=2^{n/2}$, $s$ et $n/2$ impairs, vérifiant $\gcd(s,n/2)=1$,
$b\in\mathbb{F}_{2^{n}}$ non cube, et
$c\in\mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{n/2}}$,
$r_{i}\in\mathbb{F}_{2^{n/2}}$. Alors la fonction $F$ définit par:
$F(X)=bX^{2^{s}+1}+b^{q}X^{q(2^{s}+1)}+cX^{q+1}+\sum_{i=1}^{n/2-1}r_{i}X^{2^{i}(q+1)}$
est APN.
###### Proof.
On a l’isomorphisme suivant
$L(u,v)=cu+\sum_{i=1}^{n/2-1}r_{i}u^{2^{i}}+v\;(\textrm{cf :c
$\in\mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{n/2}}$})$
et donc
$G(X)=bX^{2^{s}+1}+b^{q}X^{q(2^{s}+1)}=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(bX^{2^{s}+1}\right)$
Montrons que $G$ vérifie l’équation (10)
$\displaystyle G(aX+b^{\prime})+G(aX+a+b^{\prime})$
$\displaystyle=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(ba^{2^{s}+1}(X^{2^{s}}+X+1)\right)$
$\displaystyle=(X^{2^{s}}+X+1)\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(ba^{2^{s}+1}\right)$
$\displaystyle=d$
D’après le lemme 3.0.1, et le théorème 3.1, $F$ est APN si et seulement si
$\forall
a\in\mathbb{F^{*}}_{2^{n}}\quad\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(ba^{2^{s}+1}\right)=0$
n’a pas de solutions. C’est le cas, sinon :
$\;ba^{2^{s}+1}\in\mathbb{F}_{2^{n/2}}$ comme
$\mathbb{F}_{2^{n}}^{(2^{s}+1)}=\mathbb{F}_{2^{n}}^{\gcd(2^{s}+1,2^{n}-1)}=\mathbb{F}_{2^{n}}^{3}$
et que les élément de $\mathbb{F}_{2^{n/2}}$ sont tous des cubes, ceci conduit
à $b$ est un cube, contradiction. Donc $F$ est APN. ∎
###### Corollaire 3.4.
Soient
$\gcd(i,n/2)=1\;,\;c\in\mathbb{F}_{2^{n}}\;,\;s\in\mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{n/2}}\;,\;q=2^{n/2}$
$F(X)=X(X^{2^{i}}+X^{q}+cX^{2^{i}q})+X^{2^{i}}(c^{q}X^{q}+sX^{q2^{i}})+X^{(2^{i}+1)q}$.
où : $X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1$ est irréductible sur
$\mathbb{F}_{2^{n}}$. Alors $F$ est APN.
###### Proof.
$\displaystyle F(X)$
$\displaystyle=X^{2^{i}+1}+X^{q+1}+cX^{2^{i}q+1}+c^{q}X^{q+2^{i}}+sX^{2^{i}(q+1)}+X^{(2^{i}+1)q}.$
$\displaystyle=X^{q+1}+sX^{2^{i}(q+1)}+X^{2^{i}+1}+cX^{2^{i}q+1}+c^{q}X^{q+2^{i}}+X^{(2^{i}+1)q}$
$\displaystyle=L(B(X),G(X))$
où : $L(u,v)=u+su^{2^{i}}+v$ est un isomorphisme;
$G(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(X^{2^{i}+1}+cX^{2^{i}q+1}\right)$
Montrons que $G$ vérifie l’équation (10)
$\displaystyle G(aX+b^{\prime})+G(aX+a+b^{\prime})$
$\displaystyle=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}(X^{2^{i}}+X+1)+ca^{2^{i}q+1}(X^{2^{i}q}+X+1)\right)\quad(\textrm{
comme $\,X\in\mathbb{F}_{2^{n/2}}$})$
$\displaystyle=(X^{2^{i}}+X+1)\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}+ca^{2^{i}q+1}\right)$
$\displaystyle=d.$
Supposons que
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}+ca^{2^{i}q+1}\right)=0$
a une solution dans $\mathbb{F}_{2^{n/2}}$, ceci équivaut a :
$a^{2^{i}+1}+ca^{2^{i}q+1}=a^{q(2^{i}+1)}+c^{q}a^{2^{i}+q}$, en divisant par
$a^{2^{i}+1}$, et en notant :
$P(X)=X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1$, ceci donne $P(a^{q-1})=0$,
contradiction. ∎
On voit que le théorème 3.1, permet de construire une multitude de fonction
APN, moi même j’en donne deux, assez générales.
posons
$G(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(X^{2^{i}+1}+cX^{2^{i}q+1}+tX^{2^{i}+q}\right)$
et trouvons les conditions nécéssaire et suffisante pour que ca conduit a une
fonction APN.
$\displaystyle G(aX+b^{\prime})+G(aX+a+b^{\prime})$
$\displaystyle=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}(X^{2^{i}}+X+1)+ca^{2^{i}q+1}(X^{2^{i}q}+X+1)+ta^{q+2^{i}}(X^{2^{i}}+X+1)\right)\quad(\textrm{
comme $\,X\in\mathbb{F}_{2^{n/2}}$})$
$\displaystyle=(X^{2^{i}}+X+1)\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}+ca^{2^{i}q+1}+ta^{q+2^{i}}\right)$
$\displaystyle=d.$
Supposons que
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(a^{2^{i}+1}+ca^{2^{i}q+1}+ta^{q+2^{i}}\right)=0$
a une solution dans $\mathbb{F}_{2^{n}}$. Ceci équivaut a :
$P(a^{q-1})=0$ où $P(X)=X^{2^{i}+1}+(t^{q}+c)X^{2^{i}}+(c^{q}+t)X+1$, il
suffit de choisir $P$ irréductible sur $\mathbb{F}_{2^{n}}$
###### Corollaire 3.5.
Soient $q=2^{n/2}$, $i$ tel que $\gcd(i,n/2)=1$, $B(X)=X^{q+1}$.
$G(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(X^{2^{i}+1}+cX^{2^{i}q+1}+tX^{2^{i}+q}\right)$,
et
$L:\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}\rightarrow\mathbb{F}_{2^{n}}$
un isomorphisme quelconque.
Alors $F=L(B,G)$ est APN si et seulement si
$P(X)=X^{2^{i}+1}+(t^{q}+c)X^{2^{i}}+(c^{q}+t)X+1$ n’a pas de racines dans
$\mathbb{F}_{2^{n}}$.
###### Corollaire 3.6.
Soient $n/2$ impair, et $i,j$ vérifiant $(j-i)$ impairs et $\gcd(j-i,n/2)=1$,
$c$ élément de
$\mathbb{F}_{2^{n}}^{*(q-1)}\setminus\mathbb{F}_{2^{n}}^{*3(q-1)}$,
$B(X)=X^{q+1}$.
$G(X)=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{X^{2^{j}+2^{i}}}{c^{2^{n-1}}}\right)$,
et
$L:\mathbb{F}_{2^{n/2}}\times\mathbb{F}_{2^{n/2}}\rightarrow\mathbb{F}_{2^{n}}$
un isomorphisme quelconque. Alors
$F=L(B,G)$ est APN.
###### Proof.
Vérifiant que $G$ satisfait l’équation (10).
$\displaystyle G(aX+b^{\prime})+G(aX+a+b^{\prime})$
$\displaystyle=\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{j}+2^{i}}}{c^{2^{n-1}}}(X^{2^{j}}+X^{2^{i}}+1)\right)$
$\displaystyle=(X^{2^{j-i}}+X+1)^{2^{i}}\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{j}+2^{i}}}{c^{2^{n-1}}}\right)$
$\displaystyle=d$
Les mêmes arguments utiliser jusqu’ici, montre que $F$ est APN ssi
$\textbf{Tr}_{\mathbf{\mathbb{F}_{2^{n}}}/\mathbf{\mathbb{F}_{2^{n/2}}}}\left(\frac{a^{2^{j}+2^{i}}}{c^{2^{n-1}}}\right)=0$
n’a pas de solutions pour tout $a$ dans ${\mathbb{F}^{*}}_{2^{n}}$. Supposons
que c’est le cas alors $c\in\mathbb{F}_{2^{n}}^{*(q-1)(2^{j}+2^{i})}$ soit
d’apres le lemme 3.0.3, $c\in\mathbb{F}_{2^{n}}^{*3(q-1)}$ contradiction, donc
$F$ est APN.
∎
## References
* [1] R.Lidl,H.Niederreiter ,_Finite Fields_.
* [2] Y. Gozard,_Theorie de Galois_.
* [3] C.Carlet,_Boolean Functions for Cryptography and Error Correcting Codes_.
* [4] C.Carlet,_Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions_
|
arxiv-papers
| 2011-07-19T02:54:51 |
2024-09-04T02:49:20.696902
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zahid Mounir",
"submitter": "Zahid Mounir",
"url": "https://arxiv.org/abs/1107.3614"
}
|
1107.3674
|
arxiv-papers
| 2011-07-19T10:31:20 |
2024-09-04T02:49:20.707536
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Venkatesh. M, K. Kumar and Srinivas. V",
"submitter": "Dimple Juneja Dr.",
"url": "https://arxiv.org/abs/1107.3674"
}
|
|
1107.3732
|
LMU-ASC 31/11
ROM2F/2011/09
Magnetized E3-brane instantons
in F-theory
Massimo Bianchi$\,{}^{\heartsuit}$111massimo.bianchi@roma2.infn.it, Andrés
Collinucci$\,{}^{\diamondsuit}$222andres.collinucci@physik.uni-muenchen.de &
Luca Martucci$\,{}^{\heartsuit}$333luca.martucci@roma2.infn.it
♡I.N.F.N. Sezione di Roma “TorVergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”,
Via della Ricerca Scienti ca, 00133 Roma, Italy
♢Arnold Sommerfeld Center for Theoretical Physics,
LMU München, Theresienstraße 37, D-80333 München, Germany
Abstract
We discuss E3-brane instantons in ${\cal N}=1$ F-theory compactifications to
four dimensions and clarify the structure of E3-E3 zero modes for general
world-volume fluxes. We consistently incorporate SL$(2,\mathbb{Z})$
monodromies and highlight the relation between F-theory and perturbative IIB
results. We explicitly show that world-volume fluxes can lift certain
fermionic zero-modes, whose presence would prevent the generation of non-
perturbative superpotential terms, and we discuss in detail the geometric
interpretation of the zero-mode lifting mechanism. We provide a IIB derivation
of the index for generation of superpotential terms and of its modification to
include world-volume fluxes, which reproduces and generalizes available
results. We apply our general analysis to the explicit, though very simple,
example of compactification on $\mathbb{P}^{3}$ and its orientifold weak-
coupling limit. In particular, we provide an example in which a non-rigid
divisor with fluxes contributes to the superpotential.
###### Contents
1. 1 Introduction and summary of the results
2. 2 A short review on F-theory vacua
1. 2.1 Generalities on F-theory compactifications
2. 2.2 Supersymmetric structures
3. 2.3 Orientifold limit
4. 2.4 Relation with the M-theory viewpoint
5. 2.5 A working example
3. 3 E3-instantons without fluxes: the IIB perspective
1. 3.1 Fluxless E3-fermionic action
2. 3.2 Zero-modes
3. 3.3 M5-brane index from IIB perspective
4. 3.4 Comparison with the dual M-theory result
5. 3.5 $\chi_{\rm E3}$ as the perturbative Ext index
6. 3.6 Implementation in the working example
4. 4 Magnetized E3-instantons
1. 4.1 World-volume fluxes and duality
2. 4.2 Fermions and fluxes: the simplest case of constant $\tau$
3. 4.3 Fermions and fluxes with generic $\tau$
4. 4.4 Flux-modified zero modes
5. 4.5 Index for magnetized E3-branes
6. 4.6 On flux-induced zero-modes lifting
5. 5 Lifting zero-modes in a one-modulus example
6. A U(1)-bundles and holomorphic line bundles
7. B Some useful properties of the extrinsic curvature
## 1 Introduction and summary of the results
F-theory provides a large class of non-perturbative vacua of Type IIB whereby
the complexified axio-dilaton develops a non trivial profile due to the
presence of 7-branes [1, 3, 2]. In recent times a resurgence of interest in
this class of models has been triggered by the observation that local
configurations of 7-branes can accommodate interesting aspects of particle
physics phenomenology beyond the standard model, such as gauge coupling
unification and textures in the Yukawa couplings, starting with [4, 5, 6] –
see e.g. [7, 8] for reviews and more complete lists of references.
Although it is still unclear which local F-theory solutions admit a global
embedding, non-perturbative effects generated by Euclidean D3-branes
(E3-branes) are widely recognized as crucial ingredients in IIB/F-theory model
building, see for instance [9, 10, 11, 12, 13, 14]. Most of the applications
involving M5/E3 non-perturbative effects are built on the results of [15],
which in turn relies on the study of the fermionic zero modes derived from the
standard Dirac action for a Euclidean M5-brane, dual to the E3-brane. However,
even in the absence of background fluxes, world-volume fluxes are known to
deform the standard Dirac action on D-branes [16] and M5-branes [17]. Since
the effect of world-volume fluxes on the fermionic zero-mode structure is not
taken into account in [15], understanding this and related issues remains an
open problem. This constitutes the main motivation of the present
investigation, that aims at tackling the effect of world-volume fluxes on
E3-brane instantons in general terms. Henceforth we will dub these as
magnetized E3-branes.
This is not just an academic problem. On the one hand, consistency actually
requires that for every four-cycle wrapped by an E3-brane, one must sum over
the infinite (discrete) family of possible world-volume fluxes. On the other
hand, fluxes can in fact have important physical effects. For instance, there
is evidence that they can modify the zero-mode counting [18] and thus the
nature of E3-brane instanton corrections to the effective theory. In
particular [14], fluxes can play a key role in alleviating the tension between
chirality and moduli stabilization [19].
While [15] and many other papers on F-theory are largely based on the dual
M-theory point of view, that has of course several advantages, in this paper
we will mostly work in the IIB picture. One practical motivation behind this
choice is that it allows one to use the (Wick-rotated) well-understood
effective action for D3-branes, hence avoiding the M5-brane effective action
and in particular its subtle chiral three-form flux. Furthermore, our study of
the effect of world-volume fluxes on E3-branes will have as byproduct a better
comprehension of the relation between the IIB and M-theory pictures even in
absence of fluxes – see [12, 20, 21, 13] for previous works in this direction.
An improved understanding of the IIB/M-theory dictionary can be important
since the IIB approach has the unquestionable advantage of admitting – in some
cases – local or global weak coupling descriptions, wherein one can use
perturbative string theory techniques, not available on the M-theory side.
Indeed, in the past few years, a lot of progress has been achieved in the
understanding of non-perturbative effects generated by unoriented D-brane
instantons using their open string description [22, 23]111See also [24, 25,
26, 27, 28, 29, 30] for early work and [31, 32] for recent reviews..
Although in this paper we will not consider background type IIB three-form
fluxes (or, dually, background M-theory four-form fluxes), the understanding
of E3-branes in their presence actually constitutes another motivation for our
work. Indeed, background fluxes play a crucial role in several applications
and their effect on M5/E3 brane instantons in F-theory/IIB orientifold
backgrounds has been already considered in [33, 34, 35, 36, 37, 38, 18, 39,
40], see also [41, 42] for studies on the effect of bulk fluxes on D-brane
instantons by using the string world-sheet techniques. However, the papers
[34, 35, 36, 38] work within the simplifying assumption that the world-volume
flux vanishes and [18, 39, 40] provide just partial results on its possible
effects. In this context, although legitimate in some cases, the assumption
that there is no world-volume flux constitutes a very non-generic condition,
since the world-volume Bianchi identities relate world-volume and background
fluxes. Take for instance a IIB three-form flux $H_{\it 3}$. The world-volume
Bianchi identity reads $\text{d}{\cal F}=-\iota^{*}H_{\it 3}$, where
$\iota^{*}H_{\it 3}$ denotes the pull-back of $H_{\it 3}$ onto the E3-brane
world-volume. Clearly, if $H_{\it 3}\neq 0$, then generically the world-volume
flux ${\cal F}$ will be non vanishing as well. Hence, a proper understanding
of the effect of world-volume fluxes alone on E3/M5-branes constitutes an
important step that necessarily precedes a consistent incorporation of
background fluxes.
Let us summarize our findings:
* •
We clarify the structure of E3-E3 zero modes for general world-volume fluxes
and propose a modification of the M5-brane index [43] that reproduces and
generalizes all available results.
* •
We consistently incorporate SL$(2,\mathbb{Z})$ non-trivial monodromies and
highlight the relation between F-theory and perturbative IIB results, that was
not so manifest even in the flux-less case [12, 20, 21].
* •
We explicitly show that world-volume fluxes can lift certain fermionic zero-
modes. We discuss in detail the associated geometrical interpretation of the
flux-induced fermionic zero-mode lifting mechanism. We eventually apply our
general analysis to the explicit, though very simple, example with
$\mathbb{P}^{3}$ as compactification space and its orientifold weak-coupling
limit.
* •
In particular, we discuss a concrete example that provides evidence that non-
rigid divisors can generically contribute to the superpotential. This
drastically broadens the possibilities for non-perturbatively generated
superpotential terms. For instance, once properly combined with bulk fluxes,
E3-brane instantons with world-volume fluxes can lead to significant
improvement of the moduli-lifting problem.
The plan of the paper is as follows. In Section 2, we will briefly review some
basic properties of F-theory vacua, including monodromy, S-duality and,
whenever possible, orientifold limit. We then pass to discuss in Section 3 the
fermionic parts of the E3 action and the index counting the fermionic zero-
modes and present a very useful though simple working example. Section 4 is
devoted to study the effect of turning on world-volume fluxes. In Section 5 we
present an explicit example whereby a non-rigid divisor with fluxes
contributes to the superpotential.
We have collected some useful formulae on holomorphic line bundles and
properties of the extrinsic curvature in two Appendices.
Note added: While this paper was being typed a couple of interesting papers
appeared [44, 45] that discuss instantons in F-theory and provide
complementary results to the ones in the present work.
## 2 A short review on F-theory vacua
The aim of this section is to review the structure of F-theory backgrounds. We
will emphasize the IIB viewpoint, that is somewhat less ‘standard’, being
often rephrased in terms of the dual M-theory description. The material
covered in this section is of course not new at all, but we have decided to
include it for self-completeness of the paper, in order to fix notation,
definitions and main properties of these vacua and facilitate the reading of
the subsequent sections. For more details, see for instance the recent reviews
[46, 8].
F-theory vacua are type IIB backgrounds that include full back-reaction to
7-brane induced fluxes. The 7-branes can be either D7-branes or more general
$(p,q)$ 7-branes, obtained by acting on a D7-brane – i.e. a (1,0) 7-brane – by
an SL(2,$\mathbb{Z}$) duality transformation. A D7-brane sources one unit of
Ramond-Ramond (RR) flux $F_{{\it 1}}=\text{d}C_{\it 0}$ and is then
characterized by a monodromy $\tau\rightarrow\tau+1$ of the axion-dilaton
$\tau:=C_{{\it 0}}+{\rm i}e^{-\phi}$ on a closed loop linking the D7-brane.
Analogously, a $(p,q)$ 7-brane is characterized by a more general
SL(2,$\mathbb{Z}$) monodromy of $\tau$.
Local and global tadpole constraints strongly restrict the consistent
configurations and often require the presence of mutually non-local 7-branes,
i.e. 7-branes that cannot contemporarily be seen as a set of D7-branes or an
SL(2,$\mathbb{Z}$) transform thereof. In other words, these 7-brane
configurations are intrinsically non-perturbative, in the sense that the
axion-dilaton $\tau$ generically undergoes SL(2,$\mathbb{Z})$ duality
transformations when going from one patch of the internal manifold to another.
A simple example of such a configuration is provided by the O7-planes of
perturbative string theory, more appropriately described in F-theory as a
bound states of two mutually non-local 7-branes [47, 48].
In this paper we are interested in (minimally) supersymmetric F-theory
compactification to four-dimensions. They are dual to M-theory
compactifications to three-dimensions on an elliptically fibered Calabi-Yau
four-fold, in which the complex structure of the elliptic fiber corresponds to
the axion-dilaton $\tau$. This dual description nicely geometrizes the non-
trivial features of the F-theory backgrounds and turns out to be very
convenient to study several aspects that are harder to handle within a direct
IIB framework. However, the direct type IIB viewpoint can be important for
other purposes, as we will show in this paper.
Hence, in sections 2.1, 2.2 and 2.3 we will describe some basic features of
the F-theory backgrounds within the purely IIB description, without making any
use of the dual M-theory picture. The latter will be considered only in
subsection 2.4. We will also try to make very explicit the relation between
the two descriptions. This will be important in the subsequent sections. In
section 2.5 we provide a simple example of F-theory compactification and its
weak coupling orientifold limit. This example will constitute the playground
on which we will apply the subsequent general discussions.
### 2.1 Generalities on F-theory compactifications
F-theory backgrounds are more conveniently described in the Einstein frame.
Focusing on compactifications to flat four-dimensional Minkowski space and
demanding supersymmetry, the ten-dimensional space is the direct product
$\mathbb{R}^{1,3}\times{X}$, where ${X}$ is a complex three-fold, i.e. a real
six-dimensional manifold, that is also complex and Kähler. The Einstein frame
ten-dimensional metric splits as
$\text{d}s^{2}_{10}=\text{d}x^{\mu}\text{d}x_{\mu}+\text{d}s^{2}_{{X}}$ (2.1)
Here $\text{d}s^{2}_{{X}}=g_{mn}\text{d}y^{m}\text{d}y^{n}$ is the Kähler
metric on ${{X}}$. In this paper we denote the complex structure by
$I=I^{m}{}_{n}\text{d}y^{n}\otimes\partial_{m}$ and the associated Kähler form
by $J=\frac{1}{2}J_{mn}\text{d}y^{m}\wedge\text{d}y^{n}$, where
$J_{mn}=g_{mk}I^{k}{}_{n}$. One can also include a non-trivial warping, that
can be sourced by possible D3-brane charge and three-form flux present on
${X}$, but in the present paper we will focus on the cases in which such
warping can be consistently considered constant.
As recalled at the beginning of this section, the physically non-trivial
feature characterizing the F-theory backgrounds is provided by the presence of
7-branes. In the present setting, they fill $\mathbb{R}^{1,3}$ and wrap
internal four-cycles that are required to be holomorphically embedded by
supersymmetry. Hence, by adopting the algebraic geometry terminology, the
7-branes wrap divisors of the internal space. The presence of 7-branes is
signaled by the presence of a non-trivial axion-dilaton $\tau$, that
supersymmetry requires to depend holomorphically on the internal coordinates
$\bar{\partial}\tau=0$ (2.2)
where
$\bar{\partial}:=\text{d}\bar{z}^{\bar{I}}\wedge\partial/\partial\bar{z}^{\bar{I}}$,
where $(z^{I},\bar{z}^{\bar{I}})$ are complex coordinates on ${X}$.
In general, when passing from one local patch on ${X}$ to another, the axion-
dilaton is allowed to experience non-trivial SL(2,$\mathbb{Z}$) duality
transformations
$\tau\quad\rightarrow\quad\tau^{\prime}=\frac{a\tau+b}{c\tau+d}$ (2.3)
with $a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$. In particular, non-trivial
monodromies signal the presence of 7-branes. A D7-branes is associated to a
so-called $T$-monodromy
$M_{\rm D7}\equiv M_{1,0}=\left(\begin{array}[]{cc}1&1\\\
0&1\end{array}\right)\equiv T$ (2.4)
This implies that close to a D7-brane $\tau$ takes the form
$\tau\simeq\frac{1}{2\pi{\rm i}}\log(z-z_{\rm D7})$, where $z$ denotes some
local coordinate transverse to the D7-brane. Indeed, close to a D7-brane
$\tau$ must satisfy the second-order equation
$\partial\bar{\partial}\tau=-\delta^{\it 2}({\rm D7})$. More generically, the
monodromy around a $(p,q)$ 7-brane is given by
$M_{p,q}=\left(\begin{array}[]{cc}1-pq&p^{2}\\\ -q^{2}&1+pq\end{array}\right)$
(2.5)
The non-trivial axion-dilaton contributes to the energy-momentum tensor. Hence
the internal metric is not Ricci-flat and indeed supersymmetry implies the
following Einstein equations
$R^{{X}}_{{I}\bar{J}}=\nabla_{I}\bar{\nabla}_{\bar{J}}\phi$ (2.6)
It is useful to introduce the following composite one-form
$Q_{\it 1}=\frac{{\rm
i}}{2}\frac{\text{d}(\tau+\bar{\tau})}{\tau-\bar{\tau}}=\frac{1}{2}\,e^{\phi}\,F_{\it
1}=\frac{{\rm i}}{2}(\bar{\partial}\phi-\partial\phi)$ (2.7)
where, in the last equality, we have made use of (2.2). Now, $Q_{\it 1}$ can
be seen as a connection for a U(1) bundle, that we call U(1)Q. It is defined
as follows: If the background undergoes an SL(2,$\mathbb{Z})$ duality
transformation
$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ (2.8)
when going from one patch to another, the corresponding U(1)Q transition
function is given by $e^{{\rm i}\arg(c\tau+d)}$. This U(1)Q bundle will play a
crucial role in the following, in that several fields can be seen to transform
as sections of the associated complex line $L_{Q}$ and powers thereof under
SL(2,$\mathbb{Z}$). Hence, $Q_{\it 1}$ can be used to construct
SL(2,$\mathbb{Z}$)-covariant derivatives, that allow one to obtain manifestly
SL(2,$\mathbb{Z}$)-covariant quantities. This will be very important in the
following discussions.
From (2.7), one can easily compute the curvature associated with $Q_{\it 1}$:
${\cal R}_{Q}=\text{d}Q={\rm i}\partial\bar{\partial}\phi$ (2.9)
Then, by (2.6) it is immediate to check that it is equal to the Ricci form
${\cal
R}_{{X}}:=R^{{X}}_{{I}\bar{J}}\,\text{d}z^{I}\wedge\text{d}\bar{z}^{\bar{J}}$:
${\cal R}_{{X}}\equiv{\cal R}_{Q}$ (2.10)
The curvature ${\cal R}_{Q}$ computes the first Chern Class of the line bundle
$L_{Q}$, $c_{1}(L_{Q})=\frac{1}{2\pi}\left[{\cal R}_{Q}\right]$, while the
Ricci form computes the first Chern Class of the tangent bundle of ${X}$,
$c_{1}({X})=\frac{1}{2\pi}\left[{\cal R}_{{X}}\right]$. Hence, one can
explicitly see that the non-triviality of the line bundle $L_{Q}$ is directly
related to the failure of the Calabi-Yau condition for ${X}$. Furthermore, by
Yau’s theorem, given a certain holomorphic axion-dilaton $\tau$ with
associated bundle $L_{Q}$ on ${X}$, the topological condition
$c_{1}(L_{Q})=c_{1}({X})$ is actually sufficient for an F-theory metric to
exist for every Kähler class $[J]\in H^{1,1}({X})$.
Notice that, by standard arguments in algebraic geometry, since ${\cal R}_{Q}$
has vanishing (2,0) and (0,2) parts, one can associate to $L_{Q}$ a
holomorphic line bundle ${\cal L}_{Q}$, whose sections $f$ satisfy holomorphic
gluing conditions $f\rightarrow(c\tau+d)f$ – see appendix A for more details.
As we will explicitly see, the possibility to trade $L_{Q}$ for ${\cal L}_{Q}$
will allow to translate several geometrical quantities in terms of purely
holomorphic data.
For instance, it is well known that $c_{1}({X})=-c_{1}(K_{{X}})$, where
$K_{{X}}$ is the canonical bundle on ${X}$, i.e. the holomorphic bundle of
$(3,0)$ forms on ${X}$. Hence, (2.10) implies that $c_{1}({\cal
L}_{Q})=-c_{1}(K_{{X}})$ and one can conclude that the holomorphic line bundle
${\cal L}_{Q}$ is isomorphic to the inverse of the canonical bundle on ${X}$:
${\cal L}_{Q}\simeq K^{-1}_{{X}}$ (2.11)
### 2.2 Supersymmetric structures
We are interested in minimally supersymmetric F-theory compactifications, i.e.
preserving four-supersymmetries. The supersymmetry generators are described by
the two type IIB Majorana-Weyl spinors $\epsilon_{1,2}$. They both have
positive chirality $\Gamma_{11}\epsilon_{1,2}=\epsilon_{1,2}$, where
$\Gamma_{11}=\Gamma^{\underline{0123456789}}$ is the ten-dimensional chirality
operator.222In this paper, we distinguish flat indices by underlying them. We
work in the Majorana representation in which the charge conjugation matrix is
$\Gamma^{\underline{0}}$ and all ten-dimensional gamma matrices are real (in
Minkowskian signature). Hence, one can take $\epsilon_{1,2}$ to be real. One
can group them in a bi-spinor $\epsilon$
$\epsilon=\left(\begin{array}[]{c}\epsilon_{1}\\\
\epsilon_{2}\end{array}\right)$ (2.12)
In order to describe the supersymmetry generators, one can first introduce the
following split of the ten-dimensional gamma matrices
$\Gamma^{\underline{\mu}}=\hat{\gamma}^{\underline{\mu}}\otimes\mathbbm{1}\,,\quad\Gamma^{\underline{m}}=\gamma_{5}\otimes\gamma^{\underline{m}}$
(2.13)
where $\hat{\gamma}^{\underline{\mu}}$ and $\gamma^{\underline{m}}$ are four-
and six-dimensional gamma matrices respectively, and furthermore
$\Gamma_{11}=\gamma_{5}\otimes\gamma_{7}\,,\quad\hat{\gamma}_{5}=-{\rm
i}\hat{\gamma}^{\underline{0123}}\,,\quad\gamma_{7}={\rm
i}\gamma^{\underline{123456}}$ (2.14)
with $\hat{\gamma}_{5}$ and $\gamma_{7}$ the four- and six-dimensional
chirality operators respectively.
In the Einstein frame, the supersymmetry generators in double-notation have
the structure
$\epsilon=\epsilon^{\rm R}+\epsilon^{\rm
L}=\left(\begin{array}[]{c}\epsilon^{\rm R}_{1}\\\ \epsilon^{\rm
R}_{2}\end{array}\right)+\left(\begin{array}[]{c}\epsilon^{\rm L}_{1}\\\
\epsilon^{\rm L}_{2}\end{array}\right)$ (2.15)
where
$\epsilon^{\rm R}_{1}={\rm i}\epsilon_{2}^{\rm R}=\zeta_{{\rm
R}}\otimes\eta\quad,\quad\epsilon_{1}^{\rm L}=-{\rm i}\epsilon_{2}^{\rm
L}=\zeta_{\rm L}\otimes\eta^{*}$ (2.16)
with $\zeta_{{\rm R},{\rm L}}$ and $\eta$ external and internal chiral
spinors: $\hat{\gamma}_{5}\zeta_{\rm R}=\zeta_{\rm R}$,
$\hat{\gamma}_{5}\zeta_{\rm L}=-\zeta_{\rm L}$ and $\gamma_{7}\eta=\eta$. In
Minkowskian signature, one has $\zeta_{\rm L}=\zeta^{*}_{{\rm R}}$. However,
in order to consider Euclidean D-branes, one needs to analytically continue
$\zeta_{{\rm R}}$ and $\zeta_{\rm L}$ to two independent ‘holomorphic’ 4D
spinors of opposite chirality.
Notice that the complex combination $\epsilon_{1}+{\rm i}\epsilon_{2}$
transforms with U(1)Q-charge $+1/2$ under the SL(2,$\mathbb{Z}$) duality
group. Hence, the same is true for $\eta$, that can be then considered a
section of $S_{+}\otimes L_{Q}^{1/2}$ or, more precisely, of the associated
Spin${}^{c}_{+}$ bundle over ${X}$, that reduces to $S_{+}\otimes L_{Q}^{1/2}$
only if $S_{+}$ and $L_{Q}^{1/2}$ separately exist. Indeed, consistently with
the discussion of the previous subsection, $\eta$ must satisfy the following
SL(2,$\mathbb{Z}$)-covariant Killing spinor equations
$(\nabla_{m}-\frac{{\rm i}}{2}\,Q_{m})\eta=0$ (2.17)
where $\nabla_{m}$ is the Levi-Civita spin connection. In the following we
will choose the normalization condition $\eta^{\dagger}\eta=1$.
The Kähler form $J$ on ${X}$ can be constructed from $\eta$ as
$J_{mn}={\rm i}\eta^{\dagger}\gamma_{mn}\eta\quad$ (2.18)
It is closed and in fact covariantly constant since it does not transform
under the structure group of $L_{Q}$. On the other hand, one can use $\eta$ to
construct also a holomorphic (3,0) form $\Omega$ as follows
$\Omega_{mnp}=e^{\phi/2}\eta^{T}\gamma_{mnk}\eta$ (2.19)
Notice that
$\Omega\wedge\bar{\Omega}=\frac{4{\rm i}}{3}e^{\phi}J\wedge J\wedge J$ (2.20)
This explicitly shows that the metric is not Calabi-Yau if the dilaton is not
constant.
We would like to emphasize that $\Omega$ should not be considered as a section
of the canonical bundle, since one can easily check that it transforms non-
trivially under SL(2,$\mathbb{Z}$)-duality:
$\Omega\rightarrow(c\tau+d)\Omega$ (2.21)
Hence, $\Omega$ must be rather considered as a section of $K_{Q}\otimes{\cal
L}_{Q}$. Because of the isomorphism (2.11), one can see that $\Omega$ is a
holomorphic section of a trivial line-bundle, which guarantees that $\Omega$
is globally well-defined and never vanishing.
Notice that $\Omega$ is not covariantly constant. On the other hand, according
to the general discussion of appendix A, one can associate to $\Omega$ the
$L_{Q}$-valued three-form $e^{-\phi/2}\Omega$, that is covariantly constant
under the correspondingly U(1)Q-covariant derivative:
$\nabla_{m}^{Q}(e^{-\phi/2}\Omega)\equiv(\nabla_{m}-{\rm
i}Q_{m})(e^{-\phi/2}\Omega)=0$ (2.22)
### 2.3 Orientifold limit
Backgrounds with O7-planes and D7-branes can be properly described as
particular limits of the more general F-theory backgrounds considered in the
previous sections [47, 48], in which the string coupling $g_{s}$ can be tuned
to be small. In this limit, the O7-plane can be seen as a bound state of two
mutually non-local 7-branes separated by a perturbatively invisible distance
suppressed by $e^{-1/g_{s}}$. Neglecting such a non-perturbatively resolution,
the O7-plane appears as a singular non-dynamical object characterized by an
SL(2,$\mathbb{Z}$) monodromy
$M_{\rm O7}=\left(\begin{array}[]{cc}-1&-4\\\ 0&-1\end{array}\right)$ (2.23)
When four D7-branes coincide with one O7-plane, there is a net cancellation of
the resulting backreaction, with a residual monodromy
$M_{\rm O7}M^{4}_{\rm D7}=\left(\begin{array}[]{cc}-1&0\\\
0&-1\end{array}\right)$ (2.24)
This monodromy does not act on the axion-dilaton but reverses the sign of the
fields transforming as doublets under SL(2,$\mathbb{Z}$). The standard way to
describe this effect is to construct a double cover $\tilde{X}$ and describe
the F-theory space as a $\mathbb{Z}_{2}$-quotient ${X}=\tilde{X}/\sigma$,
where $\tilde{X}$ is a double-cover of ${X}$ branched over the locus of the
O7-plane, and $\sigma$ is the orientifold involution. At the level of complex
structure, $\tilde{X}$ is a Calabi-Yau, in the sense that the holomorphic
(3,0) form $\Omega$ defined in (2.19) pulls-back to a globally defined
nowhere-vanishing section of the canonical bundle of $\tilde{X}$. When all
7-branes organize in groups of O7+4 D7, then $\tau$ is globally constant and
$\Omega$ is covariantly constant too, so that $\tilde{X}$ has Ricci-flat
Calabi-Yau metric.
One may also relax this limit where the axion-dilaton is constant by allowing
a configuration with an O7-plane and one or several D7-branes that do not lie
on it. Then, the only SL(2,$\mathbb{Z}$) monodromies will be those that shift
$\tau$ by real constants, and the sign reversing orientifold action on
doublets. In such a limit, $g_{s}$ can be kept small everywhere except for an
exponentially small region around the O7-plane. Hence, one can still proceed
within perturbative string theory as long as one is aware of this subtlety.
### 2.4 Relation with the M-theory viewpoint
The backgrounds described in the previous subsections are dual to M-theory
compactifications on elliptically fibered Calabi-Yau four-folds ${Y}$ – see
for instance [46] for a detailed discussion of this duality. In this section
we would like to provide the concrete prescription on how to construct ${Y}$
starting from the type IIB data.
Given a holomorphic axion-dilaton on a Kähler three-fold ${X}$, one defines an
elliptic fibration over it by first creating a fiber bundle over it, whose
fibers are the weighted projective space $\mathbb{CP}^{2}_{231}$ with
homogeneous coordinates $[x:y:z]$. One way to make this bundle non-trivial
over ${X}$ is by allowing the projective coordinates to transform as sections
of line bundles ${\cal L}_{x}$, ${\cal L}_{y}$ and ${\cal L}_{z}$ on ${X}$
under transitions between patches in ${X}$. In order to make the elliptically
fibered four-fold, we would like to impose fiber by fiber the Weierstrass
equation:
$y^{2}=x^{3}+f\,x\,z^{4}+g\,z^{6}$ (2.25)
that cuts out an elliptic curve inside the weighted projective space. In
particular, the complex structure of the elliptic fiber must coincide with the
axion-dilaton $\tau$ on ${X}$.
On the other hand, it is a standard fact that elliptic curves with such a
Weierstrass representation have very tractable modular properties. More
precisely, it is known that, under an SL(2,$\mathbb{Z}$) transformation, $f$
and $g$ transform as modular forms of weight 4 and 6 respectiverly, i.e.
$f\rightarrow(c\tau+d)^{4}f\quad,\quad g\rightarrow(c\tau+d)^{6}g$ (2.26)
By identifying the modular transformations of the elliptic fiber with the
S-duality group of IIB string theory, one can clearly see that according to
the definition in 2.1 of the line bundle ${\cal L}_{Q}$, $f$ and $g$ can be
regarded as holomorphic sections of ${\cal L}^{4}_{Q}$ and ${\cal L}^{6}_{Q}$:
$f\in\Gamma({\cal L}_{Q}^{4})\quad{\rm and}\quad g\in\Gamma({\cal L}_{Q}^{6})$
(2.27)
Then, by requiring consistency of (2.25), one can see that ${\cal L}_{x}={\cal
L}_{z}^{2}\otimes{\cal L}^{2}_{Q}$ and ${\cal L}_{y}={\cal
L}_{z}^{3}\otimes{\cal L}^{3}_{Q}$. At this point, one has some freedom in the
choice ${\cal L}_{z}$. One can for instance take ${\cal L}_{z}={\cal O}_{{X}}$
(i.e. the trivial line bundle on ${X}$), and then ${\cal L}_{x}={\cal
L}^{2}_{Q}$ and ${\cal L}_{y}={\cal L}^{3}_{Q}$. This choice facilitates the
description of the open patch where $z\neq 0$, by allowing one to gauge fix
$z\rightarrow 1$, and recovering the familiar Weierstrass equation. Another
useful possibility is ${\cal L}_{z}={\cal L}_{Q}^{-1}$, for which ${\cal
L}_{x}={\cal L}_{y}={\cal O}_{{X}}$. Since ${X}$ is itself holomorphically
embedded in ${Y}$ as the divisor $z=0$, this choice is useful because it
easily allows us to restrict integrals on ${Y}$ to integrals on ${X}$.
In section 2.1 we have also shown that supersymmetry constrains this line
bundle to be isomorphic to the anti-canonical bundle of the three-fold, ${\cal
L}_{Q}\simeq K_{B}^{-1}$. Via standard adjunction formulae, one can compute
the first Chern class of the F-theory elliptically fibered four-fold ${Y}$ to
be
$c_{1}({Y})=c_{1}({X})-c_{1}({\cal L}_{Q})$ (2.28)
Hence, since $c_{1}({X})=-c_{1}(K_{{X}})$, the supersymmetry condition ${\cal
L}_{Q}\simeq K_{B}^{-1}$ is equivalent to imposing the Calabi-Yau condition on
${Y}$.
Having established the basics about the M-theory perspective as well as the
IIB supergravity point of view, let us connect it to the perturbative IIB
string theory description via Sen’s limit [47, 48]. Given a choice for $f$ and
$g$, the corresponding axion-dilaton is determined by the following relation:
$j(\tau)=\frac{4\,(24\,f)^{3}}{4\,f^{3}+27\,g^{2}}\quad{\rm where}\quad
j(\tau)=e^{-2\pi{\rm i}\tau}+744+{\cal O}(e^{2\pi{\rm i}\tau})$ (2.29)
is Klein’s modular function. Let us reparametrize $f$ and $g$ as follows:
$\displaystyle f$ $\displaystyle=$ $\displaystyle-3\,h^{2}+\epsilon\,\eta$
(2.30) $\displaystyle g$ $\displaystyle=$
$\displaystyle-2\,h^{3}+\epsilon\,h\,\eta-\epsilon^{2}\,\chi/12$ (2.31)
where $h,\eta,$ and $\chi$ are sections of ${\cal L}_{Q}^{2},{\cal
L}_{Q}^{4},$ and ${\cal L}_{Q}^{6}$, respectively. Then, one finds that
$g_{s}\sim(\log\epsilon)^{-1}$ everywhere except at the locus $h=0$. A
monodromy analysis reveals that the $(p,q)$ branes have rearranged into the
following perturbative configuration:
$\displaystyle\text{ O7-plane at}:$ $\displaystyle\quad h=0$ (2.32)
$\displaystyle\text{D7-brane at}:$ $\displaystyle\quad\eta^{2}-h\,\chi=0$
(2.33)
Away from this limit, however, the total $C_{{\it 0}}$-tadpole cancelling
7-brane configuration becomes one recombined object wrapping the following
divisor:
$4\,f^{3}+27\,g^{2}=0$ (2.34)
of class ${\cal L}_{Q}^{12}$.
In the non-backreacted probe approximation, where $g_{s}\rightarrow 0$, we
expect the internal space to be Ricci flat, i.e. a Calabi-Yau three-fold.
Sen’s limit also allows one to recover the internal Calabi-Yau three-fold. It
appears as a double-cover of ${X}$ branched over the O7-plane locus $h=0$. It
is defined by tagging a new coordinate $\xi$ on ${X}$, such that $\xi$
transforms as a section of ${\cal L}_{Q}$, and imposing the equation:
$\xi^{2}-h=0$ (2.35)
Once we make the choice ${\cal L}_{Q}\equiv K_{{X}}^{-1}$, this new space
$\tilde{X}$ will be guaranteed to be Calabi-Yau.
### 2.5 A working example
In order the make the generalities of the previous section more palatable, we
will introduce a simple working example. Let the internal three-fold be
${X}=\mathbb{P}^{3}$, with homogeneous coordinates $[z_{1}:\ldots:z_{4}]$.
In general, a section of a holomorphic line bundle vanishes along a complex
codimension one holomorphic submanifold, i.e. a divisor $D$. The line bundle
can be then denoted by ${\cal O}(D)$. It turns out that its Poincaré dual
$[D]$ coincides with the first Chern class of the bundle. On $\mathbb{P}^{3}$
one can define the so-called _hyperplane bundle_ , the line bundle such that
(linear combinations of) $z_{I}$ are sections thereof. The associated divisor
is the so-called hyperplane divisor $H$ and the hyperplane bundle is indicated
with ${\cal O}_{\mathbb{P}^{3}}(1)\equiv{\cal O}_{\mathbb{P}^{3}}(H)$.
Analogously, any homogenous polynomial $P^{(n)}(z_{1},\ldots,z_{4})$ of degree
$n$ is a section of ${\cal O}_{\mathbb{P}^{3}}(n)\equiv{\cal
O}_{\mathbb{P}^{3}}(n\,H)$. Hence, one has the following useful identities
${\rm PD}(\\{P^{(n)}=0\\})=c_{1}({\cal O}_{\mathbb{P}^{3}}(n))=n\,[H]$ (2.36)
The canonical bundle of $\mathbb{P}^{3}$ is $K_{\mathbb{P}^{3}}={\cal
O}_{\mathbb{P}^{3}}(-4)$. Therefore, if $\mathbb{P}^{3}$ is chosen as F-theory
compactification space ${X}$, the supersymmetry of the background requires
that we set ${\cal L}_{Q}={\cal O}_{\mathbb{P}^{3}}(4)$. From these data, one
can easily define the corresponding F-theory Calabi-Yau four-fold ${Y}$ as a
hypersurface in the following ambient space:
$\begin{array}[]{ccccccc|c}z_{1}&z_{2}&z_{3}&z_{4}&x&y&z&\textrm{eq.}\eqref{weierstrass}\\\
\hline\cr 1&1&1&1&0&0&-4&0\\\ 0&0&0&0&2&3&1&6\end{array}\begin{array}[]{r}\\\
\\\ \end{array}$ (2.37)
That is, in terms of the general discussion of section 2.4, one has made the
choice ${\cal L}_{x}={\cal L}_{y}={\cal O}_{\mathbb{P}^{3}}$ and ${\cal
L}_{z}={\cal L}_{Q}^{-1}={\cal O}_{\mathbb{P}^{3}}(-4)$.
This space is a generalization of a weighted projective space, whereby one
takes the quotient w.r.t. two rescalings as follows:
$(z_{1},\ldots,z_{4},x,y,z)\sim(\lambda\,z_{1},\ldots,\lambda\,z_{4},\mu^{2}\,x,\mu^{3}\,y,\,\lambda^{-4}\,\mu\,z)\,,\quad\text{for}\quad\lambda,\mu\in\mathbb{C}^{*}$
(2.38)
The other necessary data to define this space are the so-called exceptional
sets that are excluded from the space, generalizing the usual exclusion of the
origin in a projective space. In the present case, there are two sets of
coordinates that are forbidden from vanishing simultaneously:
$(z_{1}\,,\ldots\,,z_{4})\neq(0,\ldots,0))\,;\quad{\rm
and}\quad(x\,,y\,,z)\neq(0,0,0)$ (2.39)
The Calabi-Yau hypersurface ${Y}$ is given by the Weierstrass equation (2.25),
with $f$ and $g$ sections of ${\cal O}_{\mathbb{P}^{3}}(16)$ and ${\cal
O}_{\mathbb{P}^{3}}(24)$ respectively, that is homogeneous of bi-degree
$(0,6)$ with respect to the two projective rescalings.
Note that one can easily construct the Calabi-Yau three-fold $\tilde{X}$ for
the corresponding perturbative string theory description of the example under
consideration in Sen’s limit. According to the general discussion of section
2.4, $\tilde{X}$ is defined by (2.35), with $h$ section of ${\cal
O}_{\mathbb{P}^{3}}(8)$, that is
$\xi^{2}=h(z_{1},\ldots,z_{4})$ (2.40)
with $h(z_{1},\ldots,z_{4})$ a degree 8 homogenous polynomial. Clearly, (2.40)
can be seen as a degree eight hypersurface on the weighted projective four-
fold $\mathbb{P}^{4}_{11114}$ with homogeneous coordinates
$[z_{1},\ldots,z_{4},\xi]$. This Calabi-Yau three-fold is denoted by
$\mathbb{P}^{4}_{11114}[8]$. On this space, the orientifold involution acts on
the target space by sending $\xi\rightarrow-\xi$.
## 3 E3-instantons without fluxes: the IIB perspective
Let us come back to our main motivation, namely the study of E3-instantons in
F-theory backgrounds. In this section we first focus on the case in which
there are no world-volume fluxes on the E3-brane, i.e.
${\cal F}:=2\pi\alpha^{\prime}F_{\rm E3}-\iota^{*}B_{\it 2}=0$ (3.1)
This is the case considered by Witten in his seminal paper [15] and by most of
the subsequent papers on this subject.
The approach of [15] starts from the M-theory viewpoint and identifies the
conditions under which an M5-brane instanton can contribute to the
superpotential. Here we would like to retrace Witten’s procedure working
directly in IIB. In this way we will set the basis for our discussion on the
inclusion of world-volume fluxes, that will be considered in section 4. As a
byproduct, we will obtain a clear physical picture on how the M-theory results
of [15] should be interpreted from the IIB viewpoint. This can be useful in
other developments based on the IIB picture, in which one can take advantage
of perturbative string theory techniques. See for instance [20, 21, 45] for
other papers exploring this perspective.
Consider an E3-brane instanton wrapping a four-cycle $D$ in the internal space
${X}$. In order to preserve supersymmetry the four-cycle must be
holomorphically embedded or, in other words, must be an effective divisor.
Hence, the on-shell E3-action is just given by the value of the complexified
Kähler modulus
$T_{\rm E3}:=\frac{2\pi}{\ell_{s}^{4}}\int_{D}\left(\frac{1}{2}\,\,J\wedge
J+{\rm i}C_{\it 4}\right)$ (3.2)
where $\ell_{s}:=2\pi\sqrt{\alpha^{\prime}}$, and then the possible correction
to the superpotential looks like
$W_{\rm np}={\cal A}(\ldots)\,e^{-T_{\rm E3}}$ (3.3)
where ${\cal A}(\ldots)$ can depend generically on other moduli in the
compactification [43, 49], see also [50].333 If in addition 7-brane U(1)
fluxes are turned on, ${\cal A}(\ldots)$ acquires a dependence on gauge-
invariant combinations of the charged massless chiral multiplets up to
anomalous U(1)’s under which $T_{\rm E3}$ shifts [24, 26, 27, 29, 28, 30].
In order to understand the precise structure of $W_{\rm np}$ one should in
principle study the complete instantonic path integral. This problem may be
attacked by either using Green-Schwarz-like effective actions, as in [51, 15,
52, 53, 54, 55], or by microscopic string theory techniques, as in [24, 26,
27, 29, 28, 30]. In particular, in order to understand whether an E3-brane can
contribute to the superpotential, one has to study the structure of the
fermionic zero-modes. Here we are going to focus on the fermions associated to
(putative) ‘open strings’ with both ends on the E3-brane, whose dynamics is
described by a Green-Schwarz-like effective action that incorporates the
effect of the non-trivial axion-dilaton in a controlled way. As we will
explicitly check, these fermions correspond to the fermions in the dual
M5-brane effective action. There could be additional chiral fermions localized
at the intersection of the E3-brane with the background 7-branes, associated
with (putative) ‘open strings’ connecting the E3 and the 7-branes. In the dual
M5-brane, they are incorporated in the chiral (self-dual) three-form and their
effect can be reabsorbed in the moduli dependence of the pre-factor ${\cal
A}(\ldots)$ in (3.3) [43].
### 3.1 Fluxless E3-fermionic action
In order to study the E3-brane fermionic zero modes, we start from the
effective action for the E3 fermions on an F-theory IIB background. By some
general arguments [56], the world-volume fermions should naturally experience
a topological twist and then be represented by world-volume forms. Here we
would like to explicitly derive the corresponding topologically twisted theory
from the general E3 action. As we will see, a new key role with respect to the
more standard topological twist on Kähler spaces will be provided by the non-
trivial axion-dilaton. Understanding the role role of the axion-dilaton in
this case will be important later, when one will introduce world-volume
fluxes.
The fermions on a D-brane in a general supergravity background are more
conveniently described in the Green-Schwarz formalism, in which the ordinary
bosonic embedding is substituted by an embedding in the ten-dimensional
superspace. Hence, the world-volume fermions are described by a pair of ten-
dimensional Majorana-Weyl spinors $\theta_{1},\theta_{2}$, that one can
combine into the two-component vector
$\Theta=\left(\begin{array}[]{c}\theta_{1}\\\ \theta_{2}\end{array}\right)$
(3.4)
The apparent mismatch between bosonic and ferminic degrees of freedom is cured
by the presence of a world-volume gauge-symmetry, usually called
$\kappa$-symmetry, that acts in the following way
$\delta_{\kappa}\Theta=(\mathbbm{1}+\Gamma_{\rm E3})\kappa$ (3.5)
where $\kappa=(\kappa_{1},\kappa_{2})$, with $\kappa_{1,2}$ two arbitrary
Majorana-Weyl spinors. In the lowest order expansion in the fermionic fields,
the operator $\Gamma_{\rm E3}$ depends just on the bosonic world-volume
degrees of freedom and one natural way to remove these redundant degrees of
freedom is to impose the $\kappa$-fixing condition
$(\mathbbm{1}+\Gamma_{\rm E3})\Theta=0$ (3.6)
In the present case, since for the moment one is assuming (3.1), $\Gamma_{\rm
E3}$ is given by
$\Gamma_{\rm E3}=\left(\begin{array}[]{cc}0&\hat{\gamma}_{\rm E3}\\\
\hat{\gamma}^{-1}_{\rm
E3}&0\end{array}\right)\qquad\qquad\text{with}\qquad\hat{\gamma}_{\rm
E3}=-\frac{{\rm i}}{4!}\,\frac{\epsilon^{a_{1}\ldots a_{4}}}{\sqrt{\det
h}}\,\Gamma_{a_{1}\ldots a_{4}}$ (3.7)
We have introduced world-volume coordinates $\sigma^{a}$, $a=1,\ldots,4$,
$h\equiv g|_{D}$ denotes the pull-back of the bulk-metric onto the E3 world-
volume $D$
$h_{ab}=g_{mn}(y)\frac{\partial y^{m}}{\partial\sigma^{a}}\frac{\partial
y^{n}}{\partial\sigma^{b}}$ (3.8)
and $\Gamma_{a}$ denotes the pull-back of the ten-dimensional gamma-matrices
The explicit $\kappa$-symmetric quadratic fermionic action for D-branes on
general bosonic backgrounds has been worked out, in the string frame, in [16].
Here one has just to apply the general result of [16] to the F-theory
backgrounds described in detailed in section 2. By taking just a little care
of the passage from string to Einstein frame444Notice that world-volume
fermions are rescaled as $\Theta\rightarrow e^{\phi/8}\Theta$ when passing
from string to Einstein frame. and imposing the $\kappa$-fixing condition
(3.6), one obtains the following Green-Schwarz action for an E3-brane wrapping
a four-cycle $D\subset B$:
$S_{\rm F}=\frac{2\pi{\rm
i}}{\ell^{4}_{s}}\int_{D}\text{d}^{4}\sigma\,\sqrt{\det
h}\,\,\overline{\Theta}\Gamma^{a}(\hat{\nabla}_{a}+\frac{{\rm
i}}{4}\,e^{\phi}F_{a}\sigma_{2})\Theta$ (3.9)
Here $\hat{\nabla}_{a}$ is the pull-back of the bulk covariant derivative and
$F_{a}$ is the pull-back of the R-R one-form $F_{\it 1}$. Furthermore
$\overline{\Theta}\equiv\Theta^{T}\Gamma^{\underline{0}}$.
So far, we have not explicitly used the supersymmetry condition on $D$. As
already mention, this boils down to the requirement that $D$ should be a
holomorphically embedded hypersurface or, in other words, an effective
divisor. More specifically, if $D$ is an effective divisor then the E3-brane
preserves the two $\epsilon_{\rm R}$ supersymmetries as defined in (2.15) and
(2.16), while if $D$ is a negative divisor, that is a anti-holomorphic
hypersurface, then the two $\epsilon_{\rm L}$ are preserved.
One way to derive these results is by imposing the usual supersymmetry
condition
$\Gamma_{\rm E3}\epsilon_{\rm R}=\epsilon_{\rm R}$ (3.10)
for an effective divisor, or alternatively $\Gamma_{\rm E3}\epsilon_{\rm
L}=\epsilon_{\rm L}$ for a negative divisor. Focusing on E3-branes preserving
(3.10), by using the decompositions (2.13) and (2.16), one can more explicitly
write (3.10) as a condition involving only the internal space:
$\gamma_{\rm E3}\eta={\rm i}\eta\quad\quad\text{($\zeta_{\rm R}$ preserved)}$
(3.11)
with
$\gamma_{\rm E3}=-\frac{{\rm i}}{4!}\,\frac{\epsilon^{a_{1}\ldots
a_{4}}}{\sqrt{\det h}}\,\gamma_{a_{1}\ldots a_{4}}$ (3.12)
We are now in a position to introduce a more natural parametrization of the
world-volume fermion $\Theta$, that explicitly uses supersymmetry and
automatically solves the $\kappa$-fixing condition (3.6). Indeed, one can
decompose the two components $\theta_{1}$ and $\theta_{2}$ as follows. Split
$\theta_{1}=\theta_{1}^{\rm R}+\theta_{1}^{\rm L}$ and
$\theta_{2}=\theta_{2}^{\rm R}+\theta_{2}^{\rm L}$ and set:
$\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
R}=\frac{1}{2}\big{(}\tilde{\lambda}\otimes\eta+\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\big{)}\\\
\theta_{2}^{\rm R}=\frac{{\rm
i}}{2}\big{(}\tilde{\lambda}\otimes\eta-\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\big{)}\par\end{array}\right.$
(3.15) $\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
L}=\frac{1}{2}\big{(}\lambda\otimes\eta^{*}+\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\big{)}\\\
\theta_{2}^{\rm L}=\frac{{\rm
i}}{2}\big{(}\lambda\otimes\eta^{*}-\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\big{)}\end{array}\right.$
(3.18)
Hence, one has traded $\Theta$ for a new sets of world-volume fermionic
fields. They are not spinors on $D$ but rather forms, hence explicitly
realizing the expected topological twist [56], with purely holomorphic or
anti-holomorphic indices. More explicitly, if $s^{i}$
($\bar{s}^{\bar{\imath}}$) are (anti-)holomorphic world-volume coordinates,
one can write
$\tilde{\psi}=\tilde{\psi}_{i}\,\text{d}s^{i}\,,\quad\psi=\tilde{\psi}_{\bar{\imath}}\,\text{d}\bar{s}^{\bar{\imath}}\,,\quad\tilde{\rho}=\frac{1}{2}\rho_{\bar{\imath}\bar{\jmath}}\,\text{d}\bar{s}^{\bar{\imath}}\wedge\text{d}\bar{s}^{\bar{\jmath}}\,,\quad\rho=\frac{1}{2}\rho_{ij}\,\text{d}s^{i}\wedge\text{d}s^{j}$
(3.19)
On the other hand, the new world-volume fermions keep their spinor nature
related to four-dimensional external flat space. Namely, if $S_{\pm}$ denote
the (anti-)chiral spin bundles associated to the four flat directions,
$\tilde{\lambda},\tilde{\psi}$ and $\tilde{\rho}$ have spinorial index in
$S_{+}$, while $\lambda,\psi$ and $\rho$ have spinorial index in $S_{-}$.
Finally, since the E3-brane can cover different background patches related by
a possible SL(2,$\mathbb{Z}$)-duality transformations, it is important to
understand how the world-volume fermions transform under the duality. By
consistency with the superspace formulation, the Green-Schwarz fermion (3.4)
must transform as as the bulk supersymmetry generator $\epsilon$. Namely, the
combinations $\theta_{1}\pm{\rm i}\theta_{2}$ must transform with U(1)Q
charges $\pm 1/2$ – see for instance [57]. Since the internal spinor $\eta$
transforms itself with U(1)Q charges $\pm 1/2$, it is easy to recover that
$\lambda,\psi$ and $\rho$ are neutral under the SL(2,$\mathbb{Z}$)-duality,
while $\tilde{\lambda}$, $\tilde{\psi}$ and $\tilde{\rho}$ transform with
U(1)Q charges $-1,+1$ and $-1$ respectively. One can then more synthetically
summarize these properties by saying that
$\displaystyle\begin{array}[]{c|c|c}\text{l.h.\
fermions}&\text{U(1)${}_{Q}$-charge}&\text{associated bundle}\\\
\hline\cr\lambda^{\alpha}&0&S_{-}\otimes\Lambda^{0,0}\\\
\psi^{\alpha}&0&S_{-}\otimes\Lambda^{0,1}\\\
\rho^{\alpha}&0&S_{-}\otimes\Lambda^{2,0}\end{array}$ (3.24) (3.25)
$\displaystyle\begin{array}[]{c|c|c}\text{r.h.\
fermions}&\text{U(1)${}_{Q}$-charge}&\text{associated bundle}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&-1&S_{+}\otimes\Lambda^{0,0}\otimes
L_{Q}^{-1}\\\ \tilde{\psi}^{\dot{\alpha}}&+1&S_{+}\otimes\Lambda^{1,0}\otimes
L_{Q}\\\ \tilde{\rho}^{\dot{\alpha}}&-1&S_{+}\otimes\Lambda^{0,2}\otimes
L_{Q}^{-1}\end{array}$ (3.30)
where the restriction to $D$ of the bundles defined on $X$ is understood.
By plugging the expansion (3.15) into the action (3.9), after some
manipulations, one gets the following effective action for the new fermionic
fields
$S_{F}=\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}(\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}_{Q}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial_{Q}\tilde{\psi})$
(3.31)
where $*$ is the Hodge-star computed using the induced metric $h$ and
$\partial_{Q}$ and $\bar{\partial}_{Q}$ are U(1)Q-covariant Dolbeault
differentials, that is $\partial_{Q}=\partial\mp iQ^{1,0}$ and
$\bar{\partial}_{Q}=\bar{\partial}\mp iQ^{0,1}$ on fields with U(1)Q-charge
$\pm 1$.
A couple of comments are in order. First of all, the axion-dilaton enter in
exactly the right and minimal way to render the fermionic Lagrangian
manifestly invariant under SL(2,$\mathbb{Z}$)-duality transformations. The
preservation of this important property will be a guiding principle when we
will introduce a non-vanishing world-volume flux ${\cal F}\neq 0$. Second, the
action does not depend on the components of background tensorial fields that
are transversal to the E3-brane. As we will see, this will not any longer be
true when ${\cal F}\neq 0$.
### 3.2 Zero-modes
From the action (3.31) one can easily obtain the following fermionic equations
of motion
$\displaystyle\partial\lambda=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}\tilde{\lambda}=0$ (3.32)
$\displaystyle\bar{\partial}^{\dagger}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}^{\dagger}\tilde{\psi}=0$ (3.33)
$\displaystyle\partial^{\dagger}\rho=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}^{\dagger}\tilde{\rho}=0$ (3.34)
$\displaystyle\bar{\partial}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}\tilde{\psi}=0$
where $\bar{\partial}_{Q}^{\dagger}$ and $\partial_{Q}^{\dagger}$ are the
adjoint Dolbeault operator obtained by using the usual hermitian inner product
between forms: $(\chi_{1},\chi_{2})=\int\chi_{1}\wedge*\bar{\chi}_{2}$.
Hence, the zero modes are given by the (twisted) harmonic representatives of
the following cohomology groups
$\displaystyle\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{cohomology
group}\\\ \hline\cr\lambda^{\alpha}_{\rm z.m.}&H_{\partial}^{0,0}(D)\simeq
H^{0}(D,\bar{\cal O}_{D})\\\ \psi^{\alpha}_{\rm
z.m.}&H_{\bar{\partial}}^{0,1}(D)\simeq H^{1}(D,{\cal O}_{D})\\\
\rho^{\alpha}_{\rm z.m.}&H_{\partial}^{2,0}(D)\simeq H^{2}(D,\bar{\cal
O}_{D})\end{array}$ (3.39) (3.40)
$\displaystyle\begin{array}[]{c|c}\text{r.h.\ zero modes}&\text{cohomology
group}\\\ \hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm
z.m.}&H_{\bar{\partial}}^{0,0}(D,{L}_{Q}^{-1})\simeq H^{0}(D,{\cal
L}_{Q}^{-1})\\\ \tilde{\psi}^{\dot{\alpha}}_{\rm
z.m.}&H_{\partial}^{1,0}(D,{L}_{Q})\simeq H^{1}(D,\bar{\cal L}^{-1}_{Q})\\\
\tilde{\rho}^{\dot{\alpha}}_{\rm
z.m.}&H_{\bar{\partial}}^{0,2}(D,{L}_{Q}^{-1})\simeq
H^{2}(D,\mathcal{L}_{Q}^{-1})\end{array}$ (3.45)
We have already used the fact that, as discussed in section 2.1 and more in
detail in appendix A, the $U(1)_{Q}$ connection $Q$ defines the holomorphic
line bundle $\mathcal{L}_{Q}=K_{{X}}^{-1}$, whose holomorphic sections
correspond to the $\bar{\partial}_{Q}$-closed section of $L_{Q}$. This allows
to give an interpretation of the fermionic zero modes in terms of sheaf
cohomology groups $H^{i}(D,{\cal O}_{D}),H^{i}(D,{\cal L}^{-1}_{Q})$ and their
complex conjugated, as explicitly indicated in (3.39).
For later notational convenience, let us define the associated Hodge numbers
as follows:
$\displaystyle h^{i}(D)$ $\displaystyle\equiv$ $\displaystyle{\rm
dim}\,H^{i}(D,{\cal O}_{D})={\rm dim}\,H^{i}(D,\bar{\cal O}_{D})$ (3.46)
$\displaystyle h^{i}_{Q}(D)$ $\displaystyle\equiv$ $\displaystyle{\rm
dim}\,H^{i}(D,\mathcal{L}_{Q}^{-1})={\rm dim}\,H^{i}(D,\bar{\cal L}_{Q}^{-1})$
(3.47)
Hence, since each fermion has two four-dimensional spinorial indices, the
left-handed zero modes are counted by $h^{i}(D)$, while the right-handed zero-
modes are counted by $h^{i}_{Q}(D)$.
The structure of this fermionic zero-mode spectrum can be interpreted as
follows.
First, consider the case in which the restriction of the line bundle ${\cal
L}_{Q}$ onto the E3-brane, ${\cal L}_{Q}|_{\rm E3}$, is trivial. Since, as
recalled in section 2.4, the seven-brane divisor $D_{7\text{-brane}}$ is
associated to a section of ${\cal L}^{12}_{Q}$, whose restriction on $D$ is
also trivial, one is requiring that the intersection two-cycle $D\cap
D_{\text{7-brane}}$ is homologically trivial. In this case, the E3-brane
spectrum is transparent to the background 7-branes and is similar to the
spectrum for an E3-brane on a Calabi-Yau three-fold. Namely, the right- and
left-handed zero-modes are in one-to-one correspondence and have a precise
physical interpretation. First, since $h^{0}(D)=1$ for a connected $D$, there
are two left-handed universal zero-modes $\lambda^{\alpha}_{\rm z.m.}$, often
denoted as $\theta^{\alpha}$ in the literature, that can be seen as goldstini
associated to the supersymmetries $\zeta^{\alpha}_{\rm L}$ broken by the
E3-brane. Analogously, in the present case, one has $h^{0}_{Q}(D)=1$ too, that
implies that there are two additional right-handed zero-modes
$\tilde{\lambda}^{\dot{\alpha}}$, often denoted as $\bar{\tau}^{\dot{\alpha}}$
in the literature – see for instance [31] – associated to the hidden
supersymmetry broken by the 7-branes. The other zero-modes have a clear
geometrical interpretation. $h^{1}(D)=h^{1}_{Q}(D)$ counts left- and right-
handed zero-modes that are supersymmetric partners of the Wilson lines, while
$h^{2}(D)=h^{2}_{Q}(D)$ counts left- and right-handed zero-modes that are
supersymmetric partners to the geometric deformations. The latter statements
uses the fact that the restriction to $D$ of the three-fold canonical bundle
$K_{Q}\simeq{\cal L}_{Q}^{-1}$ is trivial too. This implies that
$h^{2}(D)=h^{2}_{Q}(D)=\dim H^{0}(D,N_{D})$ through the standard adjunction
formula and Serre duality.
Let us now assume that ${\cal L}_{Q}|_{\rm E3}$, is non-trivial, in the sense
that the two-cycle $D\cap D_{\text{7-brane}}$ is now homologically non-
trivial. In this case, generically, there is a mismatch between left- and
right-handed zero modes. For instance, one obviously still has the two
universal zero-modes $\lambda^{\alpha}_{\rm z.m.}\sim\theta^{\alpha}$, but
there could be a different number $2h^{0}_{Q}(D)$ of almost-universal zero
modes $\tilde{\lambda}_{\rm
z.m.}^{\dot{\alpha}}\sim\bar{\tau}^{\dot{\alpha}}$. More precisely,
$h^{0}_{Q}(D)$ counts the number of holomorphic sections of $K_{{X}}|_{\rm
E3}$. Requiring $h^{0}_{Q}(D)=0$ implies that $K_{{X}}|_{\rm E3}$ must be a
negative line-bundle on the E3-brane.
Finally, it can be useful to consider the weak coupling orientifold limit of
the above results, in which the axion-dilaton can be approximated as constant
and each O7-plane, covered by four D7-branes, generate a monodromy (2.24).
This monodromy acts on the world-volume fermions
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$ by reversing their sign, while it
leaves $\lambda,\psi,\rho$ invariant. By uplifting ${X}$ to the double cover
Calabi-Yau $\tilde{X}$, one can see that the $\lambda,\psi,\rho$ are even
under the orientifold involution, while
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$ are odd. Then the fermionic zero
modes are even/odd harmonic representatives of (complex conjugated of the)
even/odd cohomology groups $H^{0,i}_{\pm}(\tilde{D})$, where $\tilde{D}$ is
the double cover of $D$:
$\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{cohomology group}\\\
\hline\cr\lambda^{\alpha}_{\rm z.m.}&H_{+}^{0,0}(\tilde{D})\\\
\psi^{\alpha}_{\rm z.m.}&H_{+}^{0,1}(\tilde{D})\\\ \rho^{\alpha}_{\rm
z.m.}&H_{+}^{2,0}(\tilde{D})\end{array}\qquad\begin{array}[]{c|c}\text{r.h.\
zero modes}&\text{cohomology group}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{0,0}(\tilde{D})\\\
\tilde{\psi}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{1,0}(\tilde{D})\\\
\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}&H_{-}^{0,2}(\tilde{D})\end{array}$
(3.48)
This spectrum is counted by
$h^{i}_{\pm}(D):=\dim H^{0,i}_{\pm}(D)$ (3.49)
and agrees with the results obtained in the literature based on specific
orientifolds models. In particular, in the case in which
$h^{1}_{\pm}(D)=h^{2}_{\pm}(D)=0$ (that in particular implies that the divisor
is rigid), the non-universal zero modes
$\tilde{\lambda}_{\text{z.m.}}^{\dot{\alpha}}\sim\bar{\tau}^{\dot{\alpha}}$
are absent only if $h^{0,0}_{-}=0$, that is possible only if the E3-brane
coincides with its orientifold image, i.e. only if it is a so-called O(1)
E3-brane [27, 28, 29].
### 3.3 M5-brane index from IIB perspective
In [15], Witten provided a necessary condition for an E3-brane to contribute
to the superpotential in terms of the arithmetic genus of the divisor wrapped
by the dual M5-brane instanton. Here we would like to revisit Witten’s
argument from the IIB perspective. Of course, the two approaches must be
equivalent, as we will discuss in the the following section.
Witten’s argument is based on the observation that the M5-brane configuration
is symmetric under a U(1) rotation along the normal bundle to the divisor
wrapped by the M5-brane. In the IIB picture, one has to consider a rotation
along the two directions transverse to the divisor $D\subset{X}$. In
particular, one has to consider the action of this rotation on the world-
volume fermionic fields.
Let us denote with $({\underline{5}},\underline{6})$ the two (flat) directions
normal to the E3-brane. Rotations in these two directions can be seen as
R-symmetry tranformations. We denote this symmetry by U(1)R. The ten-
dimensional spinorial action of the U(1)R rotation is generated by the matrix
$R=\frac{1}{2}\mathbbm{1}\otimes\gamma_{\underline{56}}$, where we have used
the gamma-matrices decomposition (2.13). By using (2.14) and (3.12), one can
alternatively write $R=-\frac{1}{2}\mathbbm{1}\otimes(\gamma_{\rm
E3}\gamma_{7})$. This generator acts in the same way on both components
$\theta_{1}$ and $\theta_{2}$ of (3.4). By looking at the expansion (3.15) and
using (3.11), it is easy to see that the world-volume fields transform with
charges
$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{U(1)${}_{R}$ charge}\\\
\hline\cr\lambda^{\alpha}&-1/2\\\ \psi^{\alpha}&+1/2\\\
\rho^{\alpha}&-1/2\end{array}\quad\quad\begin{array}[]{c|c}\text{r.h.\
fermions}&\text{U(1)${}_{R}$ charge}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&+1/2\\\
\tilde{\psi}^{\dot{\alpha}}&-1/2\\\
\tilde{\rho}^{\dot{\alpha}}&+1/2\end{array}$ (3.50)
Taking into account the two-component spinorial structure of the world-volume
zero-modes, the path integral integration measure produces a violation of the
U(1)R symmetry given by
$\displaystyle\chi_{\rm E3}$ $\displaystyle=$
$\displaystyle(h^{0}(D)-h^{1}(D)+h^{2}(D))-\big{[}h^{0}_{Q}(D)-h^{1}_{Q}(D)+h^{2}_{Q}(D)\big{]}$
(3.51) $\displaystyle\equiv$ $\displaystyle\chi(D,{\cal O}_{D})-\chi(D,{\cal
L}^{-1}_{Q})$ (3.52)
of the U(1)R symmetry. As already written in (3.51), $\chi_{\rm E3}$ can be
seen as the difference of the holomorphic Euler characteristics
$\displaystyle\chi(D,{\cal O}_{D})$ $\displaystyle:=$ $\displaystyle
h^{0}(D)-h^{1}(D)+h^{2}(D)$ (3.53) $\displaystyle\chi(D,{\cal L}^{-1}_{Q})$
$\displaystyle:=$ $\displaystyle h^{0}_{Q}(D)-h^{1}_{Q}(D)+h^{2}_{Q}(D)$
(3.54)
The U(1)R symmetry should not be anomalous, that implies that the U(1)R
violation generated by the fermion zero-modes must be compensated by a shift
$T_{\rm E3}\rightarrow T_{\rm E3}-{\rm i}\chi_{\rm E3}$ in the term
$e^{-T_{\rm E3}}$ appearing in the non-perturbative contribution. In
particular, a superpotential is generated only if in the complete path-
integral there are exactly two zero modes, that implies that a necessary
condition for having a non-trivial superpotential is that
$\chi_{\rm E3}\equiv\chi(D,{\cal O}_{D})-\chi(D,{\cal L}^{-1}_{Q})=1$ (3.55)
In order to evaluate $\chi_{\rm E3}$, it can be useful to express it in terms
of characteristic classes by using the Hirzebruch-Riemann-Roch index theorem:
$\chi(D,{\cal O}_{D})=\int_{D}{\rm
Td}(D)\,,\quad\chi(D,\mathcal{L}_{Q}^{-1})=\int_{D}{\rm ch}({\cal
L}^{-1}_{Q})\wedge{\rm Td}(D)$ (3.56)
By using the isomorphism ${\cal L}_{Q}\simeq K_{{X}}^{-1}$ and the adjunction
formulas, the identities (3.56) can be useful for separately computing
$\chi(D,{\cal O}_{D})$ and $\chi(D,{\cal L}^{-1}_{Q})$. For instance, the
combination that appears in $\chi_{\rm E3}$ simplifies to
$\chi_{\rm E3}=-\tfrac{1}{2}\,\int_{{X}}c_{1}({\cal L}_{Q})\wedge[D]\wedge[D]$
(3.57)
where $[D]$ is the Poincaré dual to $D$ in ${X}$.
By recalling the topology of the total divisor wrapped by the background
7-branes, it is immediate to see from (3.57) that
$\chi_{\rm E3}=-\frac{1}{24}n$ (3.58)
where $n$ is the total number of intersections of the 7-branes with the
divisor self-intersection $D\cap D$, with the appropriate multiplicities. In
particular, the tadpole conditions ensure that $n$ is a multiple of $24$.
Finally, notice that in the weak-coupling orientifold limit $\chi_{\rm E3}$
reduces to the equivariant index known as the holomorphic Lefschetz number
$L(\tilde{D})$ discussed in [20]
$\chi_{\rm
E3}=L(D)=h^{0}_{+}(\tilde{D})-\big{[}h^{0}_{-}(\tilde{D})+h^{1}_{+}(\tilde{D})\big{]}+\big{[}h^{1}_{-}(\tilde{D})+h^{2}_{+}(\tilde{D})\big{]}-h^{2}_{-}(\tilde{D})$
(3.59)
that is computed in the double cover Calabi-Yau $\tilde{X}$, with $\tilde{D}$
being the double cover of the divisor $D$.
In the orientifold limit, the formula (3.58) for $\chi_{\rm E3}$ can be
expressed just in terms of intersections with the O7-planes. Let us continue
working on $X=\tilde{X}/\mathbb{Z}_{2}$. Taking into account that each
O7-plane is the bound-state of two mutually nonlocal 7-branes, one can write
$n=n_{\rm D7}+2n_{\rm O7}$ (3.60)
On the other hand, the tadpole condition restricted on $D\cap D$ gives the
constraint
$n_{\rm D7}=4n_{\rm O7}$ (3.61)
Hence, $n=6n_{\rm O7}$ and one can write the index $\chi_{\rm E3}$ just as
$\displaystyle\chi_{\rm E3}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,n_{\rm O7}=-\frac{1}{4}\,\int_{{X}}[D_{\rm
O7}]\wedge[D]\wedge[D]$ (3.62) $\displaystyle=$
$\displaystyle-\frac{1}{4}\,\int_{\tilde{X}}[D_{\rm
O7}]\wedge[\tilde{D}]\wedge[\tilde{D}]$ (3.63)
where in the second line we have rewritten the result in the double cover
Calabi-Yau $\tilde{X}$.
More directly, in the weak coupling description reviewed in section 2.4, the
total divisor $D_{\rm O7}$ wrapped by the O7-planes is defined by (2.32),
where $h$ is a section of ${\cal L}^{2}_{Q}$. Hence $[D_{\rm O7}]=c_{1}({\cal
L}_{Q}^{2})=2c_{1}({\cal L}_{Q})$ and then (3.57) immediately implies the
first line of (3.62).
### 3.4 Comparison with the dual M-theory result
In the past sections we have learned that the twisted theory on an E3-brane
allows us to define the zero-modes as elements of sheaf cohomologies of the
form $H^{i}(D,{\cal O}_{D})$ and $H^{i}(D,{\cal L}^{-1}_{Q})$. In this
section, we will briefly establish the link between these objects and the
zero-modes of the dual M5-brane considered by Witten in [15]. Most of this
analysis is based on the papers [60, 61, 62], see also [63].
Witten’s analysis translates the fermionic zero-modes on the M5-instanton in
terms of its worldvolume $(0,i)$-forms. Given an E3-brane on a divisor
$D\subset{X}$, its dual M5-brane is simply the restriction onto $D$ of the
elliptic fibration over ${X}$. Hence, if $\pi$ is the projection of the
elliptic fiber $\pi:{Y}\rightarrow{X}$, then the dual M5 wraps a divisor of
the Calabi-Yau four-fold ${Y}$ is defined as
$\hat{D}\equiv\pi^{-1}(D)$ (3.64)
Since the fibration is holomorphic, the cohomologies of $D$ and $\hat{D}$ are
related by the so-called Leray spectral sequence (see [58], and [59] for some
material), that eventually implies that the Euler characteristics be related
as follows:
$\chi(\hat{D},{\cal O}_{\hat{D}})=\chi_{E3}\equiv\chi(D,{\cal
O}_{D})-\chi(D,K_{{X}})$ (3.65)
Therefore, since $K_{{X}}={\cal L}^{-1}_{Q}$, the new IIB index (3.51)
computes exactly the index introduced by Witten in [43] from the M-theory
vantage point.
In order to learn about the zero-modes individually, however, the results of
Kollár [62] are crucial, because they show that the Leray spectral sequence
simplifies drastically555I.e. it degenerates at ‘page’ $E_{2}$. and one
actually has the following relations:
$\displaystyle\text{dim }H^{0}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})$ $\displaystyle\text{dim
}H^{1}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,K_{X})+\text{dim }H^{1}(D,{\cal O}_{D})$ (3.66)
$\displaystyle\text{dim }H^{2}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{1}(D,K_{X})+\text{dim }H^{2}(D,{\cal O}_{D})$
$\displaystyle\text{dim }H^{3}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{2}(D,K_{X})$
that originally appeared in [60, 61]. Clearly, the cohomology groups in the
r.h.s. of (3.4) coincide with the (complex conjugated) cohomologies appearing
in (3.39), that were found from the direct IIB zero mode spectrum.
The relations (3.4) were only understood as cohomological identities with no
particular meaning. The present treatment of the E3-instanton gives them
physical significance and provides an interpretation of the M5 Hodge numbers
in terms of more palatable quantities from the IIB perspective, that can be
summarized by the diagram in Fig. 1. Therefore, given an instanton in a IIB
setup, one does not necessarily need to construct an M/F-theory lift of it, in
order to determine whether a non-perturbative superpotential will be
generated.666It would be interesting to use the IIB/M-theory relation to
explore the geometrization of non-perturbative effects proposed in [64, 65,
66, 67] from the point of view of E3/M5-brane instantons.
$h^{2}_{Q}({\rm E3})$ $h^{2}({\rm E3})$ $h^{1}_{Q}({\rm E3})$ $h^{1}({\rm
E3})$ $h^{0}_{Q}({\rm E3})$ $h^{0}({\rm E3})$ geom. mod. tw. geom. mod. tw.
Wilson lines Wilson lines $\bar{\tau}^{\dot{\alpha}}$ $\theta^{\alpha}$
$h^{3}({\rm M5})$ $h^{2}({\rm M5})$ $h^{1}({\rm M5})$ $h^{0}({\rm M5})$ Figure
1: Schematic description of the reorganization of the E3 fermionic zero modes
in terms of the corresponding zero modes on the dual M5-brane. Here
$h^{i}({\rm E3})\equiv h^{i}(D)$, $h^{i}_{Q}({\rm E3})\equiv h^{i}_{Q}(D)=\dim
H^{i}(D,K_{X})$ and $h^{i}({\rm M5})\equiv\dim H^{i}(\hat{D},{\cal
O}_{\hat{D}})\equiv\dim H^{0,i}_{\bar{\partial}}(\hat{D})$.
### 3.5 $\chi_{\rm E3}$ as the perturbative Ext index
In this section, we will show that the new index $\chi_{\rm E3}$, despite
being defined in a non-perturbative context, can be understood in terms of
objects that appear naturally in string theory, namely Ext groups.
A divisor $D\subset{X}$ is associated to a holomorphic sheaf ${\cal E}$ with
support on $D$, defined by the short exact sequence
$0\rightarrow{\cal O}_{\rm{X}}(-D)\stackrel{{\scriptstyle\cdot
P_{D}}}{{\longrightarrow}}{\cal
O}_{\rm{X}}\stackrel{{\scriptstyle|_{P_{D}=0}}}{{\longrightarrow}}{\cal
E}_{D}\rightarrow 0$ (3.67)
The first object corresponds to the negative bundle whose sections have degree
opposite to $P_{D}$, where $P_{D}$ is the section of ${\cal O}_{\rm{X}}(D)$
that vanishes on $D$. The middle map corresponds to multiplication by $P_{D}$,
and the map into ${\cal E}_{D}$ corresponds to setting $P_{D}=0$, i.e.
restricting onto $D$. In mathematics, this is called a _resolution_ of ${\cal
E}_{D}$ by line-bundles (locally free sheaves).
One can now compute the groups ${\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ –
see for instance [68, 69] for an introduction to Ext-groups and to their
applications to D-brane physics. Using the so called local-to-global spectral
sequence one gets
$\displaystyle\dim{\rm Ext}^{0}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{0}(D,{\cal O}_{D})$ $\displaystyle\dim{\rm
Ext}^{1}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$ $\displaystyle\dim
H^{1}(D,{\cal O}_{D})+\dim H^{0}(D,N_{D})$ (3.68) $\displaystyle=$
$\displaystyle\dim H^{1}(D,{\cal O}_{D})+\dim H^{2}(D,K_{{X}})$ (3.69)
$\displaystyle\dim{\rm Ext}^{2}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{2}(D,{\cal O}_{D})+\dim H^{1}(D,N_{D})$ $\displaystyle=$
$\displaystyle\dim H^{0}(D,K_{{X}})+\dim H^{1}(D,K_{{X}})$
$\displaystyle\dim{\rm Ext}^{3}({\cal E}_{D},{\cal E}_{D})$ $\displaystyle=$
$\displaystyle\dim H^{2}(D,N_{D})=\dim H^{0}(D,K_{{X}})$
where $N_{D}$ is the holomorphic normal bundle to $D$.
By recalling that $K_{{X}}\simeq{\cal L}^{-1}_{Q}$, one can see that the the
${\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ classes coincide with the E3
fermionic zero-mode spectrum (3.39). Furthermore, it is easy to see that
$\chi_{\rm E3}$ reproduces the Ext-index777The Hirzebruch-Riemann-Roch theorem
for Ext-groups reads $\int_{{X}}{\rm ch}({\cal E})\,{\rm ch}({\cal
F})^{*}\,{\rm Td}({X})$, that can be used to further check the equivalence
with $\chi_{\rm E3}$.
$\chi_{\rm E3}\equiv{\rm Ind}_{\rm Ext}({\cal E}_{D},{\cal
E}_{D}):=\sum_{i}(-)^{i}\dim{\rm Ext}^{i}({\cal E}_{D},{\cal E}_{D})$ (3.70)
In general, in the perturbative framework in which there are no
SL$(2,\mathbb{Z})$-monodromies and ${X}$ is a Calabi-Yau, given two
holomorphic sheaves ${\cal E}$ and ${\cal F}$ on ${X}$ associated to two
D-branes, the Ext groups count the perturbative spectrum of open strings
stretching between the two D-branes [70]. However, if ${X}$ is a Calabi-Yau
then Serre-duality (i.e. CPT conjugation) implies that $\dim{\rm
Ext}^{i}({\cal E},{\cal F})=\dim{\rm Ext}^{i}({\cal F},{\cal G})$ and then
${\rm Ind}_{\rm Ext}({\cal E}_{D},{\cal E}_{D})\equiv 0$. In the present case,
one gets in general a non-vanishing (3.70) because $K_{{X}}\simeq{\cal
L}_{Q}^{-1}$ is non-trivial, due to the fact that one is actually considering
non-perturbative backgrounds.
In any case, it is somewhat surprising from a physical perspective that the
Ext groups and the associated index, that are usually applied to perturbative
D-brane physics, capture the spectrum of the E3-brane in non-perturbative
backgrounds. It is not completely clear to us whether this just an accidental
correspondence or it hides a deeper motivation.
### 3.6 Implementation in the working example
Let us now put the results of this section into practice by using the example
with ${X}=\mathbb{P}^{3}$ that was introduced in 2.5. We will first show them
from the general IIB point of view on ${X}$, then in Sen’s limit from the
double-cover point of view, and finally in terms of the dual M5-instanton.
Since this is a one-modulus three-fold , all divisors are Poincaré dual to
some multiple of the hyperplane class $[H]\in H^{2}({X},\mathbb{Z})$. Let use
then choose as divisor wrapped by the E3-brane an hypersurface defined by
$D:\\{P^{(n)}(z_{1},\ldots,z_{4})=0\\}$ (3.71)
for any degree $n$ polynomial $P^{(n)}(z_{1},\ldots,z_{4})$. This fixes the
cohomology class of its Poincaré dual to be
$[D]\simeq n[H]$ (3.72)
According to the discussion of section (3.2), in order to compute the spectrum
of E3 fermionic zero-modes, one needs to compute $H^{i}(D,{\cal O}_{D})$ and
$H^{i}(D,{\cal L}^{-1}_{Q})$.
Let us first focus on the cohomology groups $H^{i}(D,{\cal O}_{D})$, that
counts the zero modes associated to the left-handed world-volume fields
$\lambda,\psi,\rho$, see (3.39). In order to compute them, one can use the
short exact sequence:
$0\rightarrow{\cal O}_{\mathbb{P}^{3}}(-n)\rightarrow{\cal
O}_{\mathbb{P}^{3}}\rightarrow{\cal O}_{D}\rightarrow 0$ (3.73)
where the first map is multiplication by the section of ${\cal
O}_{\mathbb{P}^{3}}(n)$ that vanishes on $D$ and the second map is just
restriction.
This induces a long exact sequence of cohomologies. However, it is known that
$H^{i}(\mathbb{P}^{3},{\cal O}_{\mathbb{P}^{3}}(k))=0$ for $i\neq 0,3$ and any
$k$. Hence the long sequence breaks into several short ones and one eventually
gets
$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})$ $\displaystyle=$
$\displaystyle 1$ (3.74) $\displaystyle\text{dim }H^{1}(D,{\cal O}_{D})$
$\displaystyle=$ $\displaystyle 0$ (3.75) $\displaystyle\text{dim
}H^{2}(D,{\cal O}_{D})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,{\cal O}_{\mathbb{P}^{3}}(n-4))$ (3.76)
The dimension of $H^{2}(D,{\cal O}_{D})$ can be more explicitly computed by
recalling that, in general, one can count the sections of a line bundle ${\cal
O}_{\mathbb{P}^{3}}(d)$ by counting monomials of degree $d$, that yields:
$\text{dim }H^{0}(\mathbb{P}^{3},{\cal O}(d))=\binom{d+3}{3}$ (3.77)
Hence, one obtains the following spectrum of left-handed fermions on the
E3-brane
$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{\\#\ zero modes}\\\
\hline\cr\lambda^{\alpha}&2\times 1\\\ \psi^{\alpha}&0\\\
\rho^{\alpha}&2\times\binom{n-1}{3}\end{array}\begin{array}[]{r}\\\ \\\
\end{array}$ (3.78)
As expected, for every $n$, there are the two universal zero modes
$\lambda^{\alpha}_{\rm z.m.}\sim\theta^{\alpha}$. On the other hand, if $n\geq
4$, there are additional $\frac{2(n-1)!}{3!(n-4)!}$ zero-modes
$\rho^{\alpha}_{\rm z.m.}$.
One can now turn to the right-handed fermions
$\tilde{\lambda},\tilde{\psi},\tilde{\rho}$, whose zero modes are counted by
$H^{i}(D,{\cal L}_{Q}^{-1})\equiv H^{i}(D,{\cal O}_{\mathbb{P}^{3}}(-4))$, see
(3.39). In order to compute their dimensions, one can follow the same strategy
as above, using now the short exact sequence
$0\rightarrow{\cal O}_{\mathbb{P}^{3}}(-n)\otimes
K_{\mathbb{P}^{3}}\rightarrow K_{\mathbb{P}^{3}}\rightarrow
K_{\mathbb{P}^{3}}|_{D}\rightarrow 0$ (3.79)
The result is
$\displaystyle\text{dim }H^{0}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$
$\displaystyle 0$ (3.80) $\displaystyle\text{dim }H^{1}(D,K_{\mathbb{P}^{3}})$
$\displaystyle=$ $\displaystyle 0$ (3.81) $\displaystyle\text{dim
}H^{2}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(\mathbb{P}^{3},{\cal O}_{\mathbb{P}^{3}}(n))-1$ (3.82)
By using the general formula (3.77), for the right-handed fermionic spectrum
one then obtains
$\begin{array}[]{c|c}\text{r.h.\ fermions}&\text{\\#\ zero modes}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&0\\\ \tilde{\psi}^{\dot{\alpha}}&0\\\
\tilde{\rho}^{\dot{\alpha}}&2\times\big{[}\binom{n+3}{3}-1\big{]}\end{array}\begin{array}[]{r}\\\
\\\ \end{array}$ (3.83)
One can then see that the right-handed sector contributes with
$\frac{2(n+3)!}{n!3!}-2$ zero modes $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$.
In summary, in addition to the two universal zero modes $\lambda^{\alpha}_{\rm
z.m.}\sim\theta^{\alpha}$ there is always a certain number of zero modes
$\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ and, for $n\geq 4$, additional zero
modes $\rho^{\alpha}_{\rm z.m.}$. The zero modes
$\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ have a clear geometrical
interpretation, as $\text{dim}\,H^{2}(D,K_{\mathbb{P}^{3}})={\rm
dim}\,H^{0}(D,N_{D})$ by Serre duality and then they can be seen as the
supersymmetric partner of the geometrical deformations of the divisor $D$.
Since $N_{D}\simeq{\cal O}_{\mathbb{P}^{3}}(n)|_{D}$, they can be directly
counted as follows:
$\displaystyle\text{dim }H^{2}(D,K_{\mathbb{P}^{3}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}({\cal O}_{\mathbb{P}^{3}}(n))$ $\displaystyle=$
$\displaystyle\text{\\# monomials of degree n in }z_{1},\ldots z_{4}\text{
minus one}$ $\displaystyle=$ $\displaystyle\binom{n+3}{3}-1$ (3.85)
where the minus one corresponds subtracting the defining polynomial $P^{(n)}$,
since we only count deformations. On the other hand, $\rho^{\alpha}_{\rm
z.m.}$ are associated to some kind of twisted geometrical deformations, whose
interpretation will become more transparent in the weak coupling orientifold
limit.
Now, the index $\chi_{\rm E3}$ can be immediately computed from (3.51) to be
$\chi_{\rm E3}=-2\,n^{2}$ (3.86)
that is always different from $1$. Alternatively, one can easily get the same
result from (3.57), by using (3.72), together with the identity $c_{1}({\cal
L}_{Q})=-c_{1}(K_{\mathbb{P}^{3}})=4[H]$ and the fact that
$\int_{\mathbb{P}^{3}}[H]\wedge[H]\wedge[H]=1$.
Let us now revisit these results from the orientifold point of view, as was
done in [20]. The divisor $D\subset\mathbb{P}^{3}$ has a double-cover divisor
$\tilde{D}$ in the double-cover Calabi-Yau three-fold
$\subset\tilde{X}=\mathbb{P}^{4}_{11114}[8]$ discussed in section 2.5. Now,
the canonical bundle of the three-fold is trivial, but the relevant
information is encoded in the $\mathbb{Z}_{2}$-equivariant structure of the
orientifold involution $\xi\rightarrow-\xi$.
The condition (3.72) translates into $[\tilde{D}]=n\,[H]$ in $\tilde{X}$,
where we continue to use the symbol $H$ to denote the double cover of $H$,
hoping that this will not cause confusion. Its Poicaré dual $[H]$ in
$\tilde{X}$ can be thought of as the pullback of the hyperplane class $[H]$ in
$\mathbb{P}^{3}$ under the $\mathbb{Z}_{2}$ projection and we use the notation
${\cal O}_{\tilde{X}}(n)\equiv{\cal O}_{\tilde{X}}(n\,H)$. The most general
effective divisor $\tilde{D}$ of this kind is given by the intersection of
(2.40) and the vanishing locus
$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)=0$ (3.87)
of a homogeneous degree $n$ polynomial
$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)$, where $\xi$ itself has degree four,
that is furthermore even in $\xi$ in order to respect the orientifold
involution $\xi\rightarrow-\xi$. However, we restrict to configurations that
are irreducible double covers of single connected holomorphic hypersurfaces
$D$ in ${X}\equiv\mathbb{P}^{3}$, with (3.87) just given by (3.71), i.e.
independent of $\xi$. This means that the divisor $\tilde{D}$ is transversely
invariant under the orientifold action, i.e. is a so-called O(1) instanton.
In order to understand the fermionic zero-mode spectrum, one needs to compute
the orientifold-even and -odd cohomology groups $H^{i}_{\pm}(\tilde{D},{\cal
O}_{\tilde{D}})$.
First, considering only connected transversely invariant divisors, one finds
$\text{dim }H^{0,0}_{+}(\tilde{D})=1\,,\quad\text{dim
}H^{0,0}_{-}(\tilde{D})=0$ (3.88)
This matches (3.78) and (3.83), cf. with (3.48).
Furthermore, it is useful to note that the line bundle ${\cal
O}_{\tilde{X}}(1)$ is very ample. Then, the Lefschetz hyperplane theorem
guarantees that
$H^{1}(\tilde{D},{\cal O}_{\tilde{D}})=0\quad\Leftrightarrow\quad
H_{\pm}^{0,1}(\tilde{D})=0$ (3.89)
By inspecting (3.48), one can see that this is again in agreement with (3.78)
and (3.83).
Finally, by Serre duality, one knows that
$\text{dim }H^{0,2}_{\pm}(\tilde{D})=\text{dim
}H^{0}_{\mp}(\tilde{D},N_{\tilde{D}})$ (3.90)
where $N_{\tilde{D}}$ is the normal bundle to $\tilde{D}$ in $\tilde{X}$. Note
that, since the holomorphic $(3,0)$ form is orientifold-odd, Serre duality
exchanges the two subgroups of the cohomology with opposite $\mathbb{Z}_{2}$
parity. Therefore, $H^{0,2}_{-}(\tilde{D})$ corresponds to divisor
deformations that respect the orientifold involution, whereas
$H^{0,2}_{+}(\tilde{D})$ corresponds to those that are odd under it. In fact,
one can directly count the even and odd sections. Orientifold _odd_ sections
of the normal bundle correspond to deformations of the divisor’s polynomial
$\tilde{P}^{(n)}(z_{1},\ldots,z_{4},\xi)$ that are linear in $\xi$. The even
ones do not contain $\xi$. Notice that higher powers of $\xi$ can be
eliminated via the hypersurface equation (2.40).
It is then easy to see that the counting of $H^{0,2}_{-}(\tilde{D})$ is
identical to the counting in (3.6) and gives the same result. This is of
course what one was to reproduce: the number of zero modes $\tilde{\rho}_{\rm
z.m.}^{\dot{\alpha}}$ in (3.83).
On the other hand, the odd sections, that are linear in $\xi$, are counted by
$\displaystyle\dim H^{0,2}_{+}(\tilde{D})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}_{-}(\tilde{D},N_{\tilde{D}})$ $\displaystyle=$
$\displaystyle\text{\\# monomials of degree $n-4$ in }z_{1},\ldots z_{4}$
$\displaystyle=$ $\displaystyle\binom{n-4+3}{3}$ (3.92)
This indeed coincides with the dimension of $H^{2}(D,{\cal O}_{D})$ and gives
a clearer interpretation of the zero modes $\rho^{\alpha}_{\rm z.m.}$ found in
(3.78).
Finally, according to the discussion in section 3.4 one can easily lift all
this information to the dual M5-brane. By using (3.4), one can relate the
cohomologies of $D\subset{X}$ to the cohomologies of the M5-instanton divisor
$\hat{D}\equiv\pi^{-1}(D)\subset{Y}$, where $\pi:{Y}\rightarrow{X}$ is the
projection from the elliptically fibered Calabi-Yau four-fold to its base
${X}$, as follows:
$\displaystyle\text{dim }H^{0}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{0}(D,{\cal O}_{D})=1$ $\displaystyle\text{dim
}H^{1}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{0}(D,K_{X})+\text{dim }H^{1}(D,{\cal O}_{D})=0$ (3.93)
$\displaystyle\text{dim }H^{2}(\hat{D},{\cal O}_{\hat{D}})$ $\displaystyle=$
$\displaystyle\text{dim }H^{1}(D,K_{X})+\text{dim }H^{2}(D,{\cal
O}_{D})=\binom{n-1}{3}$ $\displaystyle\text{dim }H^{3}(\hat{D},{\cal
O}_{\hat{D}})$ $\displaystyle=$ $\displaystyle\text{dim
}H^{2}(D,K_{X})=\binom{n+3}{3}-1$
The relevant index for the M5, i.e. the holomorphic characteristic
$\chi(\hat{D},{\cal O})$ [15], does obviously coincide with (3.86).
## 4 Magnetized E3-instantons
From now on, we would like to relax the simplifying assumption (3.1) and study
how a non-trivial world-volume flux
${\cal F}:=2\pi\alpha^{\prime}F_{\rm E3}-\iota^{*}B_{\it 2}\neq 0$ (4.1)
can change the story. The supersymmetry of the E3-brane imposes that it still
wrap an effective divisor $D$ and that the world-volume flux ${\cal F}$
satisfy the anti-self-duality condition
${\cal F}=-*{\cal F}$ (4.2)
that can be alternatively expressed by the pair of conditions
$\displaystyle{\cal F}^{0,2}$ $\displaystyle=$ $\displaystyle
0\qquad(\text{holomorphy})$ (4.3) $\displaystyle J\wedge{\cal F}$
$\displaystyle=$ $\displaystyle 0\qquad(\text{primitivity})$ (4.4)
This conditions can be derived from standard arguments, paying particular
attention to the analytic continuation necessary to extend the fermions to
Euclidean signature, as in section 3.1.888See for instance appendix D of [64]
for a detailed general discussion, that includes the setting considered in
this paper as a particular subcase.
### 4.1 World-volume fluxes and duality
As it is clear form (4.1), the world-volume flux ${\cal F}$ receives two kinds
of contributions. One comes from the pull-back of the bulk NS-NS $B_{\it 2}$
field, and the other from the world-volume flux $F_{\rm E3}$. Both transform
non-trivially under the SL(2,$\mathbb{Z}$)-duality transformation.
Let us first recall that $B_{\it 2}$ pairs up with the R-R two form $C_{\it
2}$ to transform as a doublet under SL(2,$\mathbb{Z}$)-duality
$\left(\begin{array}[]{c}C_{\it 2}\\\ B_{\it
2}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{c}C_{\it 2}\\\ B_{\it
2}\end{array}\right)$ (4.5)
As we are going to discuss, this transformation is consistent with the
transformation of the world-volume flux $F_{\rm E3}$.
Let us then turn to the purely world-volume part $F_{\rm E3}$. First introduce
the dual field-strength $F^{D}_{\rm E3}$ defined as
$F^{D}_{\rm E3}=-2\pi{\rm i}\,\frac{\delta S_{\rm E3}}{\delta F_{\rm E3}}$
(4.6)
In this section, as already stated, we work in Euclidean signature. Thus most
equations should be considered as the analytical continuation of equations in
Minkwskian signature, wherein their meaning would be more transparent. In
(4.6) the functional derivative is defined by $\delta S_{\rm
E3}=\int_{D}\frac{\delta S_{\rm E3}}{\delta F_{\rm E3}}\wedge\delta F_{\rm
E3}$ and $S_{\rm E3}$ is the (Wick rotatated) general off-shell action
$S_{\rm E3}=\frac{1}{(2\pi)^{3}\alpha^{\prime
2}}\int_{D}\text{d}^{4}\sigma\sqrt{\det(h+e^{-\phi/2}{\cal F})}\,+\frac{{\rm
i}}{(2\pi)^{3}\alpha^{\prime 2}}\int_{D}\Big{(}C_{\it 4}+C_{\it 2}\wedge{\cal
F}+\frac{1}{2}C_{\it 0}{\cal F}\wedge{\cal F}\Big{)}$ (4.7)
This action, and actually its full supersymmetric extension, is invariant
under SL(2,$\mathbb{Z}$)-duality, under which $F^{D}_{\rm E3}$ and $F_{\rm
E3}$ transform as a doublet – see for instance the explicit discussion in
[57]. Hence, it is useful to introduce
${\cal F}^{D}:=2\pi\alpha^{\prime}F^{D}_{\rm E3}-\iota^{*}C_{\it 2}$ (4.8)
that pairs with ${\cal F}$ into a doublet under SL(2,$\mathbb{Z}$)-duality:
$\left(\begin{array}[]{c}{\cal F}^{D}\\\ {\cal
F}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{c}{\cal F}^{D}\\\ {\cal
F}\end{array}\right)$ (4.9)
It is then convenient to consider the combinations
$\displaystyle{\cal F}^{-}$ $\displaystyle:=$ $\displaystyle{\rm
i}e^{\phi}(\bar{\tau}{\cal F}-{\cal F}^{D})$ (4.10) $\displaystyle{\cal
F}^{+}$ $\displaystyle:=$ $\displaystyle-{\rm i}e^{\phi}(\tau{\cal F}-{\cal
F}^{D})$ (4.11)
that transform as sections of ${\cal L}_{Q}$ and $\bar{\cal L}_{Q}$
respectively under SL(2,$\mathbb{Z}$)-duality.
Now, the key observation is that in the case we are interested in, namely
E3-branes wrapping a divisor $D$ with anti-self-dual flux (4.2), one has
${\cal F}^{-}=2{\cal F}\,,\qquad{\cal F}^{+}=0\qquad\qquad\text{(for $*{\cal
F}=-{\cal F}$)}$ (4.12)
A quick check of this is obtained by using the fact that the supersymmetric
E3-brane is calibrated, in generalized sense of [71]. For the purpose of
evaluating the on-shell value of $F^{D}_{\rm E3}$ from (4.6), this allows to
use the simplified action obtained by substituting the DBI term in (4.7) with
the value provided by the generalized calibration ${\rm Re\hskip
1.00006pt}(e^{{\rm i}J})$, that is
$S_{\rm E3}=\frac{1}{(2\pi)^{3}\alpha^{\prime
2}}\int_{D}\Big{[}\frac{1}{2}J\wedge J+{\rm i}\Big{(}\frac{1}{2}\,\tau\,{\cal
F}\wedge{\cal F}+C_{\it 4}+C_{\it 2}\wedge{\cal F}\Big{)}\Big{]}+\ldots$
(4.13)
where the $\ldots$ on the right are contributions at least quadratic in terms
that vanish on the supersymmetric configurations [72] and then they do not
effectively contribute to the evaluation of the on-shell $F^{D}_{\rm E3}$.
From (4.13) and (4.6) one readily gets $2\pi\alpha^{\prime}F^{D}_{\rm
E3}=\tau{\cal F}+C_{\it 2}$, that indeed implies (4.12). Equations (4.12) have
the consequence that, in the case we are interested in, one should consider
${\cal F}$ as taking values in ${\cal L}_{Q}$, that is
$e^{-\phi/2}{\cal F}\in\Lambda^{1,1}\otimes L_{Q}$ (4.14)
Furthermore, since $\bar{\partial}{\cal F}=0$, then
$\bar{\partial}_{Q}(e^{-\phi/2}{\cal F})=0$ and ${\cal F}$ identifies the
following cohomology classes
$[e^{-\phi/2}{\cal F}]\in
H^{1,1}_{\bar{\partial}}(D,L_{Q})\quad\Leftrightarrow\quad[{\cal F}]\in
H^{1}(D,T^{*}_{D}\otimes{\cal L}_{Q})$ (4.15)
where $T^{*}_{D}$ is the holomorphic cotangent bundle on $D$.
At this point one has to face a little subtlety. Take the Minkowskian Bianchi
identity $\text{d}{\cal F}=0$ (if $H_{\it 3}=0$) and assume that it is valid
in the Euclidean case we are interested in. Being ${\cal F}$ purely $(1,1)$,
this naturally splits into $\bar{\partial}{\cal F}=0$ and $\partial{\cal
F}=0$. Now, as we saw above, $\bar{\partial}{\cal F}=0$ is
SL$(2,\mathbb{Z})$-covariant, being equivalent to
$\bar{\partial}_{Q}(e^{-\phi/2}{\cal F})=0$. On the other hand, $\partial{\cal
F}=0$ is not SL$(2,\mathbb{Z})$-covariant. This effect is accompanied by a
failure of the expected field equation $\text{d}{\cal
F}^{D}=\text{d}(\tau{\cal F})=0$. Again, this can be split into two parts:
$\bar{\partial}{\cal F}^{D}=0$ and $\partial{\cal F}^{D}=0$. Since ${\cal
F}^{D}=\tau{\cal F}$, the first is indeed automatically implied by the Bianchi
identity $\bar{\partial}{\cal F}=0$ and then it is
SL$(2,\mathbb{Z})$-covariant by itself. On the other hand, the second is
incompatible with the SL$(2,\mathbb{Z})$ non-covariant Bianchi Identity
$\partial{\cal F}=0$, and furthermore it is not SL$(2,\mathbb{Z})$-covariant
by itself. Hence, by invoking the principle of SL$(2,\mathbb{Z})$-covariance,
it is natural to consider the possibility that $\partial{\cal F}=0$ and
$\partial(\tau{\cal F})=0$ are both incorrect and should be substituted by a
single SL$(2,\mathbb{Z})$-covariant equation, that should reduce to
$\partial{\cal F}=0$ when $\tau$ is constant. The most natural choice is
$\partial_{Q}(e^{-\phi/2}{\cal F})=0$ (4.16)
We may interpret this possibility as follows. In Minkowski signature Bianchi
identities and equations of motion are usually put on different footings and
only the second ones are influenced by the background axion-dilaton. On the
other hand, switching to Euclidean signature, the supersymmetric anti-self-
duality condition mixes Bianchi identities and equations of motion into an
anti-holomorphic ($\bar{\partial}$) and holomorphic ($\partial$) part. Now it
would be the anti-holomorphic part that is insensitive to the non-trivial
$\tau$, while the holomorphic one would be modified by it.999This unbalancing
between holomorphic and anti-holomorphic sectors generated by the non-trivial
$\tau$, that is an ubiquitous ingredient in the present discussion, has a
simpler manifestation if one considers E($-1$)-instantons. Indeed, the action
would simply be given by $\tau(z_{\rm E(-1)})$, and so it would automatically
be extremized wrt the anti-holomorphic variables ($\bar{z}_{\rm E(-1)}$) but
not wrt the holomorphic ones ($z_{\rm E(-1)}$). In fact, we will never
explicitly need (4.16), with the only exception of section 4.6, where one will
be naturally led to consider it.
The present situation is reminiscent of the fate of YM instantons in the
presence of non-trivial VEV’s for charged scalars. Instanton solutions no
longer satisfy the classical coupled field equations and they only extremize
the Euclidean action if some ‘constraint’ is added by including additional
terms in the action, whence the name ‘constrained instantons’. Although one
can feel uneasy by expanding around a configuration that is not a solution of
the ‘naive’ field equations, the success of supersymmetric instanton
calculus101010See e.g.[73] for a recent review., that relies on elegant and
sound localization techniques, should be taken as an encouraging analogy for
the validity of our analysis. At any rate, whenever a perturbative orientifold
limit with constant $\tau$ is possible, these subtleties seems to disappear.
Furthermore, even for weakly coupled orientifold limits with non-constant
$\tau$, one could check the validity of the analysis by directly using open-
string techniques.
### 4.2 Fermions and fluxes: the simplest case of constant $\tau$
We now come back to our main problem, the study of the effect of a non-trivial
world-volume flux (4.1) on the fermionic zero-modes spectrum. In this
subsection we restrict to the simpler subcase in which the axion-dilaton is
(or is approximated to be) constant and the internal metric is Ricci-flat.
Clearly the weak-coupling orientifold backgrounds fit into this class.
What one needs to do is to repeat the steps of section 3.1, so one can proceed
quite speedily, mutatis mutandis for taking into account the non-trivial flux.
First of all, the operator $\Gamma_{\rm E3}$ gets modified. Namely, by using
the gamma-matrix decomposition (2.13), in (3.7) one should take
$\hat{\gamma}_{\rm E3}=\mathbbm{1}\otimes\gamma_{\rm E3}({\cal F})$, with
$\gamma_{\rm E3}({\cal F})=\frac{-{\rm i}\,\epsilon^{a_{1}\ldots
a_{4}}}{\sqrt{\det(h+e^{-\phi/2}{\cal
F})}}\Big{(}\frac{1}{4!}\gamma_{a_{1}\ldots
a_{4}}+\frac{1}{4}\,e^{-\phi/2}{\cal
F}_{a_{1}a_{2}}\gamma_{a_{3}a_{4}}+\frac{1}{8}\,e^{-\phi}{\cal
F}_{a_{1}a_{2}}{\cal F}_{a_{3}a_{4}}\Big{)}$ (4.17)
We continue imposing $\kappa$-fixing condition in the form (3.6) and the
supersymmetry condition still reads (3.12).
The quadratic action for the $\kappa$-fixed Green-Schwarz fermions is now
given by [16]
$S_{\rm F}=\frac{2\pi{\rm
i}}{\ell^{4}_{s}}\int_{D}\text{d}^{4}\sigma\,\sqrt{\det
M}\,\,\overline{\Theta}({\cal M}^{-1})^{ab}\Gamma_{a}\hat{\nabla}_{b}\Theta$
(4.18)
where $M_{ab}:=h_{ab}+e^{-\phi/2}{\cal F}_{ab}$ and ${\cal
M}_{ab}:=h_{ab}\mathbbm{1}+e^{-\phi/2}{\cal F}_{ab}\,\sigma_{3}$.
One also needs to decompose $\Theta$ in terms of new (topologically twisted)
world-volume fields. Because of the modification of $\Gamma_{\rm E3}$, the
decomposition (3.15) does not fulfill the $\kappa$-fixing condition (3.6)
anymore and must be deformed into a new one. A new decomposition that we will
find convenient is
$\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
R}=\frac{1}{2}\Big{(}\tilde{\lambda}\otimes\eta+\sqrt{\frac{\det h}{\det
M}}M^{a}{}_{b}\tilde{\psi}_{a}\otimes\gamma^{b}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\Big{)}\\\
\theta_{2}^{\rm R}=\frac{{\rm
i}}{2}\Big{(}\tilde{\lambda}\otimes\eta-\sqrt{\frac{\det h}{\det
M}}(M^{T})^{a}{}_{b}\tilde{\psi}_{a}\otimes\gamma^{a}\eta^{*}+\frac{1}{2}\tilde{\rho}_{ab}\otimes\gamma^{ab}\eta\Big{)}\end{array}\right.$
(4.21) $\displaystyle\left\\{\begin{array}[]{l}\theta_{1}^{\rm
L}=\frac{1}{2}\Big{(}\lambda\otimes\eta^{*}+\sqrt{\frac{\det h}{\det
M}}M^{a}{}_{b}\psi_{a}\otimes\gamma^{b}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\Big{)}\\\
\theta_{2}^{\rm L}=\frac{{\rm
i}}{2}\Big{(}\lambda\otimes\eta^{*}-\sqrt{\frac{\det h}{\det
M}}(M^{T})^{a}{}_{b}\psi_{a}\otimes\gamma^{a}\eta+\frac{1}{2}\rho_{ab}\otimes\gamma^{ab}\eta^{*}\Big{)}\end{array}\right.$
(4.24)
The decomposition (4.21) clearly reduces to (3.15) in the limit ${\cal
F}\rightarrow 0$, in which $M\rightarrow h$.
By pugging (4.21) into the action (4.18), after some manipulations, one gets
$\displaystyle S_{\rm F}$ $\displaystyle=$ $\displaystyle\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\big{(}\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial\tilde{\psi}\big{)}$
(4.26) $\displaystyle\quad\qquad+\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\sqrt{\det h}\,\big{(}\rho\cdot{\cal S}_{\cal
F}\cdot\rho-\tilde{\rho}\cdot\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}\big{)}$
We have introduced the short-handed notation
$\rho\cdot{\cal S}_{\cal F}\cdot\rho:=\frac{1}{4}\rho^{ab}({\cal S}_{{\cal
F}})_{ab}{}^{cd}\rho_{cd}\,,\quad\tilde{\rho}\cdot\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}:=\frac{1}{4}\tilde{\rho}^{ab}(\tilde{\cal S}_{{\cal
F}})_{ab}{}^{cd}\tilde{\rho}_{cd}$ (4.27)
where ${\cal S}_{\cal F}$ and $\tilde{\cal S}_{{\cal F}}$ are tensors that are
anti-symmetric and purely (anti-)holomorphic in $(ab)$ and $(cd)$, explicitly
given in (anti-)holomorphic indices by
$\displaystyle({\cal S}_{\cal F})_{\bar{\imath}\bar{\jmath}}{}^{uv}$
$\displaystyle=$ $\displaystyle-e^{-\phi}{\cal
K}^{m}{}_{\bar{t}[\bar{\imath}}{\cal
F}_{\bar{\jmath}]}{}^{\bar{t}}\bar{\Omega}_{m}{}^{uv}$ (4.28)
$\displaystyle(\tilde{\cal S}_{\cal F})_{ij}{}^{\bar{u}\bar{v}}$
$\displaystyle=$ $\displaystyle-e^{-\phi}{\cal K}^{m}{}_{t[i}{\cal
F}_{j]}{}^{t}\Omega_{m}{}^{\bar{u}\bar{v}}\ $ (4.29)
The first line in (4.26) is just (3.31) in the case of constant axion-dilaton,
$Q_{\it 1}=0$. Hence, one can see that the new decomposition (4.21) allows us
to get the same kinetic terms for the topologically twisted world-volume
fields. The possibility to reach this result is not a priory obvious at all.
The second line of (4.26) provides a new mass-like term for $\rho$ and
$\tilde{\rho}$, induced by the world-volume flux through the appearance of the
extrinsic curvature ${\cal K}^{m}{}_{ab}$. The definition and some useful
properties of the extrinsic curvature are reviewed in appendix B.
In the case of an orientifold background, the action of the E3-brane wrapping
the double-cover divisor $\tilde{D}$ in the double cover Calabi-Yau
$\tilde{X}$ take the very same form as (4.26). Both $\Omega$ and ${\cal F}$
are odd under the orientifold involution and then $S_{\cal F}$ and
$\tilde{S}_{\cal F}$ stay invariant. Furthermore, since the monodromy (2.24)
acts on $\Theta$ as $-{\rm i}\sigma_{2}$ and on $\eta$ as $\eta\rightarrow{\rm
i}\eta$, it is easy to see from (4.21) that $\lambda$, $\psi$ and $\rho$ are
even under the O7-involution, while $\tilde{\lambda}$, $\tilde{\psi}$ and
$\tilde{\rho}$ are odd. Hence the action (4.26) is manifestly invariant under
the orientifold involution, as required.
We stress that the above transformation rules under the orientifold involution
for the topologically twisted world-volume fermions are the same as those
found in the flux-less case discussed in section 3.2. This is probably one of
the reasons why our choice (4.21) leads to the particularly simple action
(4.26). Crucially, as we will see in the next subsection, this observation
continue to hold for the complete SL(2,$\mathbb{Z}$)-duality group.
### 4.3 Fermions and fluxes with generic $\tau$
We can now address the generalization of the above results to the case in
which $\tau$ has a more general non-constant (and holomorphic) profile. In
principle, one could repeat the steps of the previous subsection, starting
back again from the general action in [16] and simply allowing for a non-
constant $\tau$. Although straightforward, this procedure turns out to be
technically quite intricate. Then, here we follow an alternative strategy,
that allows to obtain the desired result without much effort.
The important ingredient is that the action should be invariant under
SL(2,$\mathbb{Z}$) duality. This has been demonstrated in full generality in
[57] and in this section we would like to explore the consequences of this
general result for the specific problem at hand. In the flux-less case of
section 3 we found that the topologically twisted world-volume fermions
transform nicely under the SL(2,$\mathbb{Z}$) duality and in section 4.2 we
have already seen that, for constant $\tau$, this property is preserved for
the $\mathbb{Z}_{2}$ subgroup corresponding to the orientifold involution.
What we would like to argue here is that, even for non-constant $\tau$, the
decomposition (4.21) provides topologically twisted world-volume fermions that
continue to transform under the SL(2,$\mathbb{Z}$) duality in exactly the same
simple way that was found in the flux-less case.
The understanding of the transformation properties of the topologically
twisted world-volume fermions is not as straightforward as in the fluxless
case for the following reason. As already mentioned, in general the
combinations $\theta_{1}\pm{\rm i}\theta_{2}$ of the Majorana-Weyl fermions
appearing in (3.4) transform with U(1)Q charge $\pm 1/2$ under
SL(2,$\mathbb{Z}$) duality. If one tries to naively apply this transformation
to (4.21), taking into account that $\eta$ and ${\cal F}$ have U(1)Q charge
$+1/2$ and $+1$, one seems to encounters an horrible non-linear propagation of
this U(1)Q action on the topologically twisted world-volume fermions, that
originates exactly in the (non-linear) presence of the non-vanishing ${\cal
F}$ in (4.21). However, one has to be careful since, in presence of world-
volume fluxes, this U(1)Q action generically breaks the $\kappa$-fixing
condition (3.6). Let us try to be very explicit on this point.
In bi-spinor formalism, the U(1)Q action is given by
$\Theta\rightarrow\Theta^{\prime}=e^{-\frac{{\rm
i}}{2}\alpha\,\sigma_{2}}\Theta$ (4.30)
where $\alpha=\arg(c\tau+d)$. On the other hand, the infinitesimal
$\kappa$-symmetry gauge transformation around a bosonic background can be
written as
$\delta_{\kappa}\Theta=P^{+}_{\kappa}({\cal F})\kappa$ (4.31)
for an arbitrary spinor $\kappa$, where use of the projectors has been made
$P^{\pm}_{\kappa}({\cal F})=\frac{1}{2}\big{(}\mathbbm{1}\pm\Gamma_{\rm
E3}\big{)}$ (4.32)
that, as explicitly indicated, depend on ${\cal F}$. In this paper we have
chosen to gauge-fix (4.31) by imposing that
$P^{+}_{\kappa}({\cal F})\Theta=0$ (4.33)
Now, it should be evident that the SL(2,$\mathbb{Z}$) action (4.30)
generically breaks (4.33), that must then be re-established by acting with a
compensating $\kappa$-symmetry transformation.
In order to find the $\kappa$-fixing preserving transformation, it is
convenient to work with an infinitesimal SL(2,$\mathbb{R}$)-trasformation of
the form (4.30), that should still be preserved at the supergravity level.
Hence
$\delta_{S}\Theta=-\frac{{\rm i}}{2}\,\alpha\,\sigma_{2}\Theta$ (4.34)
Now, the deformed bispinor $\Theta^{\prime}\simeq\Theta+\delta_{S}\Theta$
should be corrected by a $\kappa$-symmetry transformation in such a way that
it preserves the $\kappa$-fixing condition $P^{+}_{\kappa}({\cal
F}^{\prime})\Theta^{\prime}=0$, where $e^{-\phi^{\prime}/2}{\cal
F}^{\prime}=e^{{\rm i}\alpha}e^{-\phi/2}{\cal F}\simeq(1+{\rm
i}\alpha)e^{-\phi/2}{\cal F}$. Hence, by defining
$\delta_{S}P^{-}_{\kappa}({\cal F})=P^{-}_{\kappa}({\cal
F}^{\prime})-P^{-}_{\kappa}({\cal F})$, the $\kappa$-fixing preserving first
order deformation is given by
$\hat{\delta}_{S}\Theta=P^{-}_{\kappa}({\cal F})\,\delta_{S}\Theta+\delta
P^{-}_{\kappa}({\cal F})\,\Theta$ (4.35)
At this point, all one has to do is to plug the decomposition (4.21) into
(4.35), taking into account that $\eta$ transform with U(1)Q charge $+1/2$ and
extract the resulting transformations for the world-volume fermions. After
some tedious algebra, one eventually finds that all nasty terms drop out and
one is left with a simple linear action of the topologically twisted world-
volume fermions
$\displaystyle\hat{\delta}_{S}\lambda=\hat{\delta}_{S}\psi=\hat{\delta}_{S}\rho=0$
(4.36) $\displaystyle\hat{\delta}_{S}\tilde{\lambda}=-{\rm
i}\alpha\tilde{\lambda}\,,\quad\hat{\delta}_{S}\tilde{\psi}={\rm
i}\alpha\tilde{\psi}\,,\quad\hat{\delta}_{S}\tilde{\rho}=-{\rm
i}\alpha\tilde{\rho}$ (4.37)
Hence, on the twisted world-volume fermions the apparently complicated action
of the infinitesimal SL(2,$\mathbb{R}$) duality simplifies drastically and
reduces to a simple linear action. In particular, this can be readily
exponentiated into a simple finite SL(2,$\mathbb{R}$), and hence
SL(2,$\mathbb{Z}$), action of the form
$\lambda,\psi,\rho\rightarrow\lambda,\psi,\rho\quad,\quad\tilde{\lambda}\rightarrow
e^{-{\rm i}\alpha}\tilde{\lambda}\quad,\quad\tilde{\psi}\rightarrow e^{{\rm
i}\alpha}\tilde{\psi}\quad,\quad\tilde{\rho}\rightarrow e^{-{\rm
i}\alpha}\tilde{\rho}$ (4.38)
Hence, even in presence of fluxes, the topologically twisted world-volume
fermions continue to transform under SL(2,$\mathbb{Z}$)-duality with the same
U(1)Q charges as in the fluxless case. In particular, their properties are
still summarized by (3.24). One can then see the naturalness of the
decomposition (4.21), even in the case of non-constant $\tau$.
Furthermore, notice that the tensors ${\cal S}_{\cal F}$ and $\tilde{S}_{\cal
F}$ transform nicely under SL(2,$\mathbb{Z}$)-transformation, namely with
charge $0$ and $+2$. In particular, they can be seen as operators
${\cal S}_{\cal F}:\Lambda^{2,0}\rightarrow\Lambda^{0,2}\,,\quad\tilde{\cal
S}_{\cal F}:\Lambda^{0,2}\otimes L_{Q}^{-1}\rightarrow\Lambda^{2,0}\otimes
L_{Q}$ (4.39)
One can now make the following observation. Take $\tau$-constant. With the
above rules, it is clear that the action (4.26) is invariant under a global
SL(2,$\mathbb{Z}$)-duality. This action must extend to the case of a more
general F-theory background with non-constant $\tau$ in such a way as to be
invariant under the SL(2,$\mathbb{Z}$)-transformations that characterize it.
Furthermore this generalization must reduce to (3.31) in the case of constant
flux. All these ingredients mashed together practically uniquely fix the
complete effective action to be the simple ‘covariantization’ of (4.26)
obtained by replacing the ordinary derivatives on U(1)Q-charged fields with
the U(1)Q-covariant derivatives.
Hence, we conclude that, in presence of a non-trivial world-volume flux, the
fermionic E3-brane action is given by
$\displaystyle S_{\rm F}$ $\displaystyle=$ $\displaystyle\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\big{(}\psi\wedge*\partial\lambda-\tilde{\psi}\wedge*\bar{\partial}_{Q}\tilde{\lambda}-\rho\wedge*\bar{\partial}\psi+\tilde{\rho}\wedge*\partial_{Q}\tilde{\psi}\big{)}$
(4.41) $\displaystyle\quad\qquad+\frac{4\pi{\rm
i}}{\ell_{s}^{4}}\int_{D}\sqrt{\det h}\,\big{(}\rho\cdot{\cal S}_{\cal
F}\cdot\rho-\tilde{\rho}\cdot\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}\big{)}$
where $\partial_{Q}:=\partial-{\rm i}qQ^{1,0}$,
$\bar{\partial}_{Q}:=\bar{\partial}-{\rm i}qQ^{0,1}$, with $q$ being the
U(1)Q-charge given in (3.24).
### 4.4 Flux-modified zero modes
The action (3.31) leads to the following flux-modified fermionic equations
$\displaystyle\partial\lambda=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}\tilde{\lambda}=0$ (4.42)
$\displaystyle\bar{\partial}^{\dagger}\psi=0\quad$ ,
$\displaystyle\quad\partial_{Q}^{\dagger}\tilde{\psi}=0$ (4.43)
$\displaystyle\partial^{\dagger}\rho=0\quad$ ,
$\displaystyle\quad\bar{\partial}_{Q}^{\dagger}\tilde{\rho}=0$ (4.44)
$\displaystyle\bar{\partial}\psi={\cal S}_{\cal F}\cdot\rho\quad$ ,
$\displaystyle\quad\partial_{Q}\tilde{\psi}=\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}$ (4.45)
Clearly, comparing these equations with the flux-less ones (3.32), one can see
that the only effect of the flux is to slightly mix $\psi$ and $\rho$, and
analogously for $\tilde{\psi}$ and $\tilde{\rho}$. However, in order to count
the zero modes associated to $\psi$ and $\tilde{\psi}$, one can consistently
set $\rho=\tilde{\rho}=0$. So one obtains that $\psi_{\rm z.m.}$ and
$\tilde{\psi}_{\rm z.m.}$ are still given by harmonic representatives of the
cohomology groups $H^{0,1}_{\bar{\partial}}(D)$ and
$H^{1,0}_{\partial}(D,L_{Q})$ respectively. Nothing changes for $\lambda_{\rm
z.m.}$ and $\tilde{\lambda}_{\rm z.m.}$ as well, that are still given by the
harmonic representatives of $H^{0,0}_{\partial}(D)$ and
$H^{0,0}_{\bar{\partial}}(D,L^{-1}_{Q})$.
The story changes for $\rho$ and $\tilde{\rho}$. On the one hand, the third
line of (4.42) implies that $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm z.m.}$
must still be a harmonic representatives of $H^{2,0}_{\partial}(D)$ and
$H^{0,2}_{\bar{\partial}}(D,L^{-1}_{Q})$ respectively, as in the flux-less
case. However, for $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm z.m.}$ to be true
zero-modes, the last line in (4.42) requires ${\cal S}_{\cal F}\cdot\rho_{\rm
z.m.}$ to be $\bar{\partial}$-exact and $\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}_{\rm z.m.}$ must be $\partial_{Q}$-exact.
One can rewrite this condition as follows. First, restricting the action of
${\cal S}_{\cal F}$ and $\tilde{S}_{\cal F}$ onto harmonic forms, one can see
them as maps between cohomology classes
$\displaystyle{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2,0}_{\partial}(D)\rightarrow H^{0,2}_{\bar{\partial}}(D)$ (4.46)
$\displaystyle\tilde{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{0,2}_{\bar{\partial}}(D,L_{Q}^{-1})\rightarrow H^{2,0}_{\partial}(D,L_{Q})$
(4.47)
Hence, the condition for harmonic $\rho_{\rm z.m.}$ and $\tilde{\rho}_{\rm
z.m.}$ to be true zero-modes is that
$\displaystyle[{\cal S}_{\cal F}\cdot\rho]=0$ $\displaystyle\in$
$\displaystyle H_{\bar{\partial}}^{0,2}(D)$ (4.48) $\displaystyle[\tilde{\cal
S}_{\cal F}\cdot\tilde{\rho}]=0$ $\displaystyle\in$ $\displaystyle
H_{\partial}^{2,0}(D,L_{Q})$ (4.49)
Once (4.48) is guaranteed, the last equation in (4.42) can be integrated by
using the standard Hodge decomposition
$\displaystyle\psi$ $\displaystyle=$ $\displaystyle\psi_{\rm
z.m.}+\bar{\partial}^{\dagger}\Delta_{\bar{\partial}}^{-1}({\cal S}_{\cal
F}\cdot\rho_{\rm z.m.})$ (4.50) $\displaystyle\tilde{\psi}$ $\displaystyle=$
$\displaystyle\tilde{\psi}_{\rm
z.m.}+\partial^{\dagger}_{Q}\Delta^{-1}_{\partial_{Q}}(\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}_{\rm z.m.})$ (4.51)
Here $\psi_{\rm z.m.}$ and $\tilde{\psi}_{\rm z.m.}$ are the harmonic forms
introduced above, that count as independent zero modes. On the other hand, one
can see that if $\rho_{\rm z.m.}$ ($\tilde{\rho}_{\rm z.m.}$) is non-vanishing
and satisfies the condition (4.48), then $\psi$ ($\tilde{\psi}$) acquires an
additional corresponding non-vanishing but uniquely determined profile.
One could re-express the above results in terms of sheaf cohomologies. Namely,
one can see ${\cal S}_{\cal F}$ and $\tilde{S}_{\cal F}$ as maps
$\displaystyle{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2}(D,\bar{\cal O})\rightarrow H^{2}(D,{\cal O})$ (4.52)
$\displaystyle\tilde{\cal S}_{\cal F}$ $\displaystyle:$ $\displaystyle
H^{2}(D,{\cal L}_{Q}^{-1})\rightarrow H^{2}(D,\bar{\cal L}^{-1}_{Q})$ (4.53)
Then, in summary, one finds that the in presence of fluxes the fermionic zero
modes are associated with
$\displaystyle\begin{array}[]{c|c}\text{l.h.\ zero modes}&\text{vector
space}\\\ \hline\cr\lambda^{\alpha}_{\rm z.m.}&H^{0}(D,\bar{\cal O}_{D})\\\
\psi^{\alpha}_{\rm z.m.}&H^{1}(D,{\cal O}_{D})\\\ \rho^{\alpha}_{\rm
z.m.}&\ker{\cal S}_{\cal F}\subset H^{2}(D,\bar{\cal
O}_{D})\end{array}\qquad\begin{array}[]{c|c}\text{r.h.\ zero
modes}&\text{vector space}\\\ \hline\cr\tilde{\lambda}^{\dot{\alpha}}_{\rm
z.m.}&H^{0}(D,{\cal L}_{Q}^{-1})\\\ \tilde{\psi}^{\dot{\alpha}}_{\rm
z.m.}&H^{1}(D,\bar{\cal L}^{-1}_{Q})\\\ \tilde{\rho}^{\dot{\alpha}}_{\rm
z.m.}&\ker\tilde{\cal S}_{\cal F}\subset H^{2}(D,{\cal
L}_{Q}^{-1})\end{array}$ (4.62)
These results clearly show that a world-volume flux can potentiallly lift part
of the zero-modes of $\rho$ and $\tilde{\rho}$. According to the discussion of
section 3.2, these zero-modes are the ($Q$-twisted) supersymmetric partners of
the bosonic zero modes describing the infinitesimal deformations of the
divisor. In fact, the above flux-induced fermionic moduli lifting has a
geometrical counterpart – see for instance [70, 68, 74]. We will come back to
this point in subsection 4.6
On the other hand, one can also see very explicitly that world-volume fluxes
cannot lift the $h^{0}(D)$, $h^{1}(D)$, $h^{0}_{Q}(D)$ and $h^{1}_{Q}(D)$ zero
modes. This observation fits in well with the results scattered in the
literature, mostly based on specific examples – see for instance [31] and
references therein.
### 4.5 Index for magnetized E3-branes
In section 3.3 we have provided a direct IIB derivation of the index in [15].
The derivation is based on the U(1)R symmetry correponding to a rotation of
the complex coordinate orthogonal to the divisor. Notice that, even in
presence of fluxes, the U(1)R charges of the topologically twisted world-
volume fermions are still given by (3.50). Hence by looking at the quadratic
fermionic action (4.41) it is immediate to realize that, contrary to the naive
expectation, the presence of world-volume flux explicitly breaks the U(1)R
symmetry. The breaking has its origin in the appearance of the extrinsic
curvature. However, even though the U(1)R symmetry acting only on the word-
volume fermions is broken, the extrinsic curvature appears contracted with
three-form $\Omega$ or its complex conjugated. As it is evident from (2.19),
$\Omega$ transforms with U(1)R charge $-1$. Hence, from the world-volume
perspective, $\Omega$ can be thought of as a spurion restoring the U(1)R that,
regarded as a local Lorentz symmetry, should not be broken in the complete
theory.
Such spurionic couplings give a mass to the $\rho$ and $\tilde{\rho}$ zero
modes that lie in the kernel of the operators ${\cal S}_{\cal F}$ and
$\tilde{\cal S}_{\cal F}$ defined above. Hence, in the path-integral, the
integration of these massive would-be zero-modes will pull-down such spurionic
mass terms that, combined with the factor $e^{-T_{\rm E3}}$, will modify the
anomaly argument.
Hence by defining, in addition to (3.46), the flux-modified Hodge numbers
$h^{2}({\cal F}):={\rm dim}\,[\ker{\cal S}_{\cal F}\subset H^{2}(D,\bar{\cal
O})]\,,\quad h^{2}_{Q}({\cal F}):=\dim[\ker\tilde{\cal S}_{\cal F}\subset
H^{2}(D,{\cal L}_{Q}^{-1})]$ (4.63)
one is led to consider the modified index
$\displaystyle\chi_{\rm E3}({\cal F})$ $\displaystyle:=$ $\displaystyle
h^{0}-(h^{0}_{Q}+h^{1})+[h^{1}_{Q}+h^{2}({\cal F})]-h^{2}_{Q}({\cal F})$
(4.64) $\displaystyle\equiv$ $\displaystyle\chi^{0}_{\rm E3}-(\dim{\rm
Im\hskip 1.00006pt}{\cal S}_{\cal F}-\dim{\rm Im\hskip 1.00006pt}\tilde{\cal
S}_{\cal F})$ (4.65)
where we have defined $\chi^{0}_{\rm E3}:=\chi_{\rm E3}({\cal F}=0)$.
Clearly $\chi_{\rm E3}({\cal F})$ counts the amount of surviving zero-modes
weighted by the sign of their U(1)R charge. The natural generalization of
Witten’s criterion, necessary but by no means sufficient, is then given by
$\chi_{\rm E3}({\cal F})=1$ (4.66)
So far, we have only considered the quadratic fermionic action. Hence, it
could appear that in principle there could be other U(1)R violating higher
order terms that could spoil (4.66). However, standard non-rinormalization
arguments imply that, up to the overall factor $e^{-T_{\rm E3}}$, the
resulting superpotential can be computed in the limit $T_{\rm
E3}\rightarrow\infty$, in which the metric of the bulk three-fold $X$ is
infinitely scaled up. In this limit the higher order terms vanish and hence we
do not expect them to modify the above arguments. Under this reasonable
assumption, (4.66) appears as the most natural necessary condition to be
considered.
Finally, in section 3.5 we have pointed out the identity (3.70) between
$\chi_{\rm E3}$ in the absence of fluxes and the Ext-index. It is not clear to
us whether this correspondence can be in some way extended to the case in
which world-volume fluxes are turned on. One of the reasons is that, in an
F-theory background, the flux cannot be easily associated to standard
perturbative data underlying the definition of Ext-groups.
It is conceivable that one might be able to calculate the index (4.66) in the
orientifold weak coupling limit via some generalization of the
$\mathbb{Z}_{2}$-equivariant Hirzebruch-Riemann-Roch index theorems exploited
in [75], but for Ext-groups.
### 4.6 On flux-induced zero-modes lifting
We would like to understand better the mechanism that regulates the flux-
induced moduli-lifting. As already mentioned, this mechanism has a geometric
counterpart.
The infinitesimal deformations of the divisor are generated by the holomorphic
sections of the normal bundle $H^{0}(D,N_{D})$. Even though by definition
these deformations preserve the holomorphy of the embedding, it is important
to realize that part of them can also produce a deformation of the complex
structure induced on $D$ by the bulk. This effect originates from the non-
holomorphic split of the short exact sequence
$0\rightarrow T_{D}\rightarrow T_{X}|_{D}\rightarrow N_{D}\rightarrow 0$
(4.67)
Namely, not every holomorphic section of $N_{D}$ can be uplifted to a
holomorphic section of $T_{X}|_{D}$. Take the long exact sequence
$\ldots\rightarrow H^{0}(D,T_{X}|_{D})\stackrel{{\scriptstyle
m}}{{\rightarrow}}H^{0}(D,N_{D})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(D,T_{D})\rightarrow\ldots$
(4.68)
Since Im$\,m={\rm Ker}\,\delta$, one can see that the holomorphic sections of
$N_{D}$ which cannot be uplifted to holomorphic sections of $T_{X}|_{D}$ are
the ones whose $\delta$-image in $H^{1}(D,T_{D})$ does not vanish.
In appendix B it is shown how the extrinsic curvature provides an explicit
realization of the $\delta$-map. Namely, working in terms of the associated
smooth complex bundles, given a covariantly holomorphic section $V$ of $N_{D}$
(regarded as a U(1) line bundle), one finds that a representative of
$H^{1}(D,T_{D})\simeq H^{0,1}_{\bar{\partial}}(D,T_{D})$ is provided by
$\Upsilon_{m}V^{m}$, where $\Upsilon_{m}$ is defined in (B.12).
Before proceeding, notice that one can analogously consider the twisted short
exact sequence
$0\rightarrow T_{D}\otimes{\cal L}_{Q}|_{D}\rightarrow(T_{X}\otimes{\cal
L}_{Q})|_{D}\rightarrow N_{D}\otimes{\cal L}_{Q}|_{D}\rightarrow 0$ (4.69)
and repeat all the arguments above. Take a covariantly holomorphic section $V$
of $N_{D}\otimes L_{Q}$, which corresponds to a holomorphic section of
$N_{D}\otimes L_{Q}$. Then the obstruction for the latter to become a
holomorphic section of $(T_{X}\otimes{\cal L}_{Q})|_{D}$ is given by a non-
vanishing cohomology class of $\Upsilon_{m}V^{m}$ in
$H^{0,1}_{\bar{\partial}}(D,T_{D}\otimes L_{Q})\simeq
H^{1}(D,T_{D}\otimes{\cal L}_{Q})$.
The above arguments are clearly valid for the complex-conjugated anti-
holomorphic quantities or if one replaces ${\cal L}_{Q}$ and $L_{Q}$ by any
other line bundles on ${X}$.
Take now the world-volume fermions $\rho$ and $\tilde{\rho}$. We have seen
that, for them to be zero-modes, they must define harmonic representatives of
$H_{\partial}^{2,0}(D)$ and $H_{\bar{\partial}}^{0,2}(D,L^{-1}_{Q})$
respectively. By using $\Omega$, one can associate with them the vectors $V\in
H_{\bar{\partial}}^{0,0}(D,N_{D}\otimes L^{-1}_{Q})$ and $\tilde{V}\in
H_{\partial}^{0,0}(D,\bar{N}_{D})$ given by
$V^{m}=\frac{1}{2}e^{-\phi/2}\bar{\Omega}^{mab}\rho_{ab}\quad,\quad\tilde{V}^{m}=\frac{1}{2}e^{-\phi/2}\Omega^{mab}\tilde{\rho}_{ab}$
(4.70)
In holomorphic indices, one then has
$(\Upsilon\cdot V)^{i}{}_{\bar{\jmath}}=-\frac{1}{2}e^{-\phi/2}{\cal
K}^{mi}{}_{\bar{\jmath}}\bar{\Omega}_{m}{}^{uv}\rho_{uv}\,,\quad(\Upsilon\cdot\tilde{V})^{\bar{\imath}}{}_{j}=-\frac{1}{2}e^{-\phi/2}{\cal
K}^{m\bar{\imath}}{}_{j}\Omega_{m}{}^{\bar{u}\bar{v}}\tilde{\rho}_{\bar{u}\bar{v}}$
(4.71)
so that $\Upsilon\cdot V\in H^{0,1}_{\bar{\partial}}(D,T_{D}\otimes
L^{-1}_{Q})$ and $\Upsilon\cdot\tilde{V}\in H^{1,0}_{\partial}(D,\bar{T}_{D})$
are exactly the combinations which appear in the fermionic equations of motion
(4.42).
We can now turn our attention onto the world-volume flux. Supersymmetry
demands it to be $(1,1)$. Clearly, any deformation of the complex structure of
$D$ generated by $v\in H^{0,1}_{\bar{\partial}}(D,T_{D})\simeq H^{1}(D,T_{D})$
can break supersymmetry since it could generate a non-vanishing
$\delta_{v}{\cal F}^{0,2}\equiv v\cdot{\cal F}:=v^{i}{}_{\bar{\jmath}}{\cal
F}_{i\bar{u}}\text{d}\bar{s}^{\bar{\jmath}}\wedge\text{d}s^{\bar{u}}$ (4.72)
More precisely, supersymmetry is actually broken only if $\delta_{v}{\cal
F}^{0,2}$ cannot be reabsorbed by deforming the world-volume gauge-fields,
that is only if $\delta_{v}{\cal F}^{0,2}$ is cohomologically non-trivial.
In the fermionic equations of motion, one can write
$\displaystyle{\cal S}_{\cal F}\cdot\rho$ $\displaystyle\equiv$ $\displaystyle
e^{-\phi/2}(\Upsilon\cdot V)\cdot{\cal F}\equiv e^{-\phi/2}\delta_{V}{\cal
F}^{0,2}$ (4.73) $\displaystyle\tilde{\cal S}_{\cal F}\cdot\tilde{\rho}$
$\displaystyle\equiv$ $\displaystyle
e^{-\phi/2}(\Upsilon\cdot\tilde{V})\cdot{\cal F}\equiv
e^{-\phi/2}\delta_{\tilde{V}}{\cal F}^{2,0}$ (4.74)
Hence ${\cal S}_{\cal F}\cdot\rho$ and $\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}$ exactly account for the generation of $(0,2)$ and $(2,0)$
components produced by the (twisted) complex structure deformations associated
to $\Upsilon\cdot V$ and $\Upsilon\cdot\tilde{V}$ respectively. From this
point of view, the moduli lifting conditions (4.48) are exactly demanding that
the good zero modes should not be associated to (twisted) geometrical
deformations which generate cohomologically non-trivial components
$\delta_{V}{\cal F}^{0,2}$ or $\delta_{\tilde{V}}{\cal F}^{2,0}$, i.e.
deformations which break supersymmetry.
The question is now: are there general circumstances under which zero-mode
lifting is for sure impossible? First of all, this happens if $\Upsilon\cdot
V$ and $\Upsilon\cdot\tilde{V}$ are all trivial in cohomology, that is, the
short exact sequence (4.67) splits holomorphically. Indeed, in this case one
can write
$\Upsilon\cdot
V=\bar{\partial}_{Q}U\,,\quad\Upsilon\cdot\tilde{V}=\partial\tilde{U}$ (4.75)
with $U\in T_{D}\otimes L^{-1}_{Q}$ and $\tilde{U}\in\bar{T}_{D}$. Then,
assuming (4.16), one can deduce that
${\cal S}_{\cal F}\cdot\rho=\bar{\partial}(e^{-\phi/2}U\cdot{\cal
F})\,,\quad\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}=\partial_{Q}(e^{-\phi/2}\tilde{U}\cdot{\cal F})$ (4.76)
and one can see that the conditions (4.48) are always satisfied. If on the
other hand (4.16) is not assumed, then one cannot reach any definitive
conclusion about $\tilde{\rho}$.
The other special possibility is that the cohomological non-triviality of
${\cal F}$ is inherited from the bulk. Remember that $e^{-\phi/2}{\cal F}$
defines a cohomology class in $H^{1,1}_{\bar{\partial}}(D,L_{Q})$ and,
assuming (4.16), also in $H^{1,1}_{\partial}(D,L_{Q})$. On the other hand,
through the associated pull-back, the embedding $\iota:D\rightarrow{X}$ allows
to see part of these cohomology groups as inherited from the bulk. In other
words, ${\rm Im}\,\iota^{*}$ contains the classes which can be seen as the
pull-back of a non-trivial class in ${X}$.
Now, if $e^{-\phi/2}{\cal F}\in{\rm Im}\,\iota^{*}$ in $\partial_{Q}$ and
$\bar{\partial}_{Q}$ cohomology, then it cannot generate any zero-mode
lifting. Indeed, one can always write $\Upsilon\cdot V=\bar{\partial}_{Q}{\cal
U}$ and $\Upsilon\cdot\tilde{V}=\partial\tilde{\cal U}$, where ${\cal U}$ and
$\tilde{\cal U}$ are smooth sections of the (anti-)holomorphic bundles
$T_{X}|_{D}\otimes L^{-1}_{Q}$ and $\bar{T}_{X}|_{D}$ respectively. Hence, if
${\cal F}$ is the pull-back of a bulk form $\hat{\cal F}$, one can write
${\cal S}_{\cal F}\cdot\rho=\bar{\partial}[\iota^{*}(e^{-\phi/2}{\cal
U}\cdot\hat{\cal F})]\,,\quad\tilde{\cal S}_{\cal
F}\cdot\tilde{\rho}=\partial_{Q}[\iota^{*}(e^{-\phi/2}\tilde{\cal
U}\cdot\hat{\cal F})]$ (4.77)
which implies that the $\rho$ and $\tilde{\rho}$ zero modes cannot be lifted
by the flux. Again, we stress that we need (4.16) to arrive to such a
conceivable conclusion on $\tilde{\rho}$.
Finally, notice that the subtleties related to (4.16) disappear if one only
considers F-theory vacua admitting a weak coupling orientifold limit and works
on the double cover Calabi-Yau three-fold $\tilde{X}$. In this case ${\cal F}$
defines an odd cohomology class in $H^{1,1}_{-}(\tilde{D})$, where $\tilde{D}$
is the double cover of $D$. The above arguments run smoothly in this case and
then one can conclude that there is no flux-induced lifting of the
$\rho^{\alpha}_{\rm z.m.}$ and $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ zero
modes when either the short exact sequence (4.67) splits holomorphically or
${\cal F}$ can be written as the pull-back of a two-form in
$H^{1,1}_{-}(\tilde{X})$.
## 5 Lifting zero-modes in a one-modulus example
In this section we will use the model developed in sections 2.5 and 3.6. The
setting is simply ${X}=\mathbb{P}^{3}$, and the Calabi-Yau double-cover three-
fold $\tilde{X}$ is the octic hypersurface $\mathbb{P}^{4}_{11114}[8]$ defined
by (2.40).
This provides a very simple example that shows that even in a one-modulus
case, non-perturbative superpotentials can be generated. We choose as divisor
wrapped by the E3-brane the hyperplane divisor, $D\simeq H$. This corresponds
to the choice $n=1$ in the discussion of section 3.6. Hence, from the general
results (3.78) and (3.83), we see that in absence of world-volume fluxes the
fermionic spectrum for this specific case is
$\begin{array}[]{c|c}\text{l.h.\ fermions}&\text{\\#\ zero m.\ (${\cal
F}=0$)}\\\ \hline\cr\lambda^{\alpha}&2\times 1\\\ \psi^{\alpha}&0\\\
\rho^{\alpha}&0\end{array}\quad\quad\begin{array}[]{c|c}\text{r.h.\
fermions}&\text{\\#\ zero m.\ (${\cal F}=0$)}\\\
\hline\cr\tilde{\lambda}^{\dot{\alpha}}&0\\\ \tilde{\psi}^{\dot{\alpha}}&0\\\
\tilde{\rho}^{\dot{\alpha}}&2\times 3\end{array}$ (5.1)
Hence, in addition to the two universal zero modes $\lambda_{\rm
z.m.}^{\alpha}\sim\theta^{\alpha}$, there are $2\times 3$ (the factor $2$
counts the two Weyl indices) zero modes $\tilde{\rho}_{\rm
z.m.}^{\dot{\alpha}}$. In absence of fluxes, one has $\chi_{E3}=-2$ and then,
a priori, such an instanton cannot contribute to a superpotential.
We will now show how the addition of world-volume fluxes on $D$ can lead to
the removal of the six zero modes $\tilde{\rho}_{\rm z.m.}^{\dot{\alpha}}$, so
that the divisor can contribute to the superpotential. Actually, as it should
be clear from the general discussion of section 4.1, describing a general
world-volume flux ${\cal F}$ is difficult, as it can undergo non-trivial
monodromies, and hence cannot be treated as curvatures of ordinary line
bundles or as ordinary two-forms. This difficulty is analogous to the
difficulty in describing worldvolume self-dual 3-form fluxes on the dual
M5-instanton explicitly in a concrete situation.
On the other hand, via the weak coupling orientifold limit, these difficulties
are greatly alleviated, basically because they can be addressed in a
perturbative IIB string theory picture. In that case, the purely worldvolume
fluxes $F_{\rm E3}$ on the double-cover divisor $\tilde{D}$ are closed,
orientifold-odd two-forms of $(1,1)$-type, i.e. elements of
$H^{1,1}_{-}(\tilde{D})$. Hence, via Poincaré duality they can be understood
as divisor classes on $\tilde{D}$, i.e. holomorphic curves:111111There is a
subtlety, related to the Freed-Witten quantization condition, that reads
${\cal F}+\iota^{*}B_{\it 2}+\frac{1}{2}c_{1}(K_{\tilde{D}})\in
H^{2}(\tilde{D},\mathbb{Z})$ [76, 77]. In our case
$c_{1}(K_{\tilde{D}})=\iota^{*}[H]$. Since $[H]\in
H^{2}_{+}(\tilde{X};\mathbb{Z})$, one must turn on an even $[B^{+}_{\it
2}]=\frac{1}{2}[H]$ to cancel it, which is consistent with the orientifold
action because of the periodic identification $[B^{+}_{\it 2}]\sim[B^{+}_{\it
2}]+[\omega_{\it 2}]$ for any $[\omega_{\it 2}]\in
H^{2}(\tilde{X},\mathbb{Z})$, and then in particular for $[\omega_{\it
2}]=-[H]$.
$\frac{F_{\rm
E3}}{2\pi}=\tfrac{1}{2}[\tilde{D}]+\sum_{i}n_{i}\big{(}[C_{i}]-[C_{i}^{\prime}]\big{)}\,,\quad
n_{i}\in\mathbb{Z}$ (5.2)
where the $[C_{i}]$ and $[C_{i}^{\prime}]$ are the Poincaré duals of
holomorphic curves $C_{i}\subset\tilde{D}$ and their orientifold images,
respectively. We suppress pullback symbols for simplicity. In order for the
flux to survive the orientifold projection $\sigma^{*}({\cal F}_{E3})=-{\cal
F}_{E3}$, where $\sigma$ is the involution, we turn on a B-field $B_{\it
2}=\tfrac{1}{2}\,[H]$, such that
${\cal
F}=(2\pi\ell_{s})^{2}\sum_{i}n_{i}\big{(}[C_{i}]-[C_{i}^{\prime}]\big{)}\,,\quad
n_{i}\in\mathbb{Z}$ (5.3)
Furthermore, the primitivity condition ${\cal F}\wedge J=0$ translates into
the integrated condition
$\sum_{i}n_{i}\int_{C_{i}}J=0$ (5.4)
Such curves can be explicitly constructed and hence provide with a very hands-
on way of introducing fluxes. This approach was pioneered in the context of
black hole microstate counting in [78, 79, 80], and further developed and
exploited in [81, 82, 83].
The key point is that, as explained in section 4.6, the flux-induced lifting
of the zero modes $\tilde{\rho}^{\dot{\alpha}}_{\rm z.m.}$ have a clear
geometric counterpart in terms of lifting of geometric moduli. In particular,
only world-volume fluxes that cannot be seen as the pull-back of some $(1,1)$
flux on the bulk can contribute. Hence, only the purely world-volume flux
$F_{\rm E3}$ really matters. Then, instead of directly calculating the action
of the operator $\tilde{\cal S}_{\cal F}$, one can Poincaré-dualize the
problem and look for world-volume fluxes (5.2) that ‘rigidify’ the divisor
$D$. Geometrically, this will boil down to requiring that the holomorphic
curves $C_{i}$ cannot be deformed together with $D$ while preserving their
holomorphy .
Let us first discuss the geometric deformations more in detail. In the
orientifold limit, they are associated to the $H^{2}_{-}(\tilde{D})$
cohomology. They correspond via Serre duality to $H^{0}_{+}(H,{\cal
O}_{\tilde{X}}(1))$, i.e. to the sections of the normal bundle
$N_{\tilde{D}}\simeq{\cal O}_{\tilde{X}}(1)|_{\tilde{D}}$ that are _even_
under the orientifold involution. These can be seen quite directly by writing
down the equation for a generic hyperplane divisor $H$ in
$\tilde{X}=\mathbb{P}^{4}_{11114}[8]$. Let the coordinates of the three-fold
be as defined earlier, $[z_{1}:\ldots:z_{4}:\xi]$, where $\xi\rightarrow-\xi$
under the involution, and $\xi$ has projective weight four. Then the E3
divisor $\tilde{D}$ is given by:
$P_{D}=a_{1}\,z_{1}+\ldots+a_{4}\,z_{4}=0$ (5.5)
The vanishing locus of this equation is invariant under rescalings, hence the
moduli space of this divisor is a $\mathbb{P}^{3}$. Hence, $\text{dim
}H_{+}^{0}(H,{\cal O}_{\tilde{X}}(1))=3$, that indeed gives the $2\times 3$
zero modes $\tilde{\rho}^{\dot{\alpha}}$. Notice however that, because $\xi$
cannot appear at degree one, all deformations will keep the divisor invariant,
i.e. as an O$(1)$ instanton. This explains why $H^{2}_{+}(\tilde{D},{\cal
O}_{\tilde{D}})$ is trivial and there are no zero modes $\rho^{\alpha}_{\rm
z.m.}$.
In order to create fluxes that can rigidify the divisor, one needs to identify
holomorphically embedded curves in $\tilde{X}$ that are rigid, i.e. that do
not admit holomorphic deformations. The reason is the following. If one
identifies fluxes on $\tilde{D}$ with their Poincaré dual by imposing that the
divisor _contain_ assigned curves, then the condition ${\cal F}^{0,2}=0$
translates into requiring that the curves remain holomorphic when the divisor
moves. If the curves are rigid, then some or all of the divisor moduli will
become obstructed.
Take the double-cover $\tilde{X}$ defining equation (2.40) in
$\mathbb{P}^{4}_{11114}$ to be given by:
$z_{1}^{8}+\ldots+z_{4}^{8}+\xi^{2}+\psi^{2}\,\big{(}P_{4}(z)\big{)}^{2}=0\,,$
(5.6)
for $\psi\in\mathbb{C}$, and $P_{4}(z)$ some generic polynomial of homogeneous
degree four in the $z_{I}$. At a generic locus in complex structure moduli
space, this Calabi-Yau is known to contain $29504$ isolated holomorphic degree
one curves of genus zero, i.e. $\mathbb{P}^{1}$’s [84, 85]. We turned on the
$\psi^{2}$ deformation in order to ensure genericity.
The curves we are interested in cannot be written as the complete intersection
of the Calabi-Yau hypersurface (5.6) and other two equations, regardless of
whether one of these is the E3-divisor hypersurface (5.5). Indeed, such curves
would be necessarily not rigid, as they would be holomorphic regardless of the
position of the E3-brane. Instead, they must be defined as complete
intersections of three equations in the ambient weighted projective space.
Take the following rational (genus zero) degree one curve:
$C_{1}:\quad z_{1}=\eta\,z_{2}\quad\cap\quad
z_{3}=\tilde{\eta}\,z_{4}\quad\cap\quad\xi=\psi\,P_{4}\qquad\subset\quad\mathbb{P}^{4}_{11114}\,,$
(5.7)
where $\eta^{8}=\tilde{\eta}^{8}=-1$ are eighth roots of minus one. This curve
is a $\mathbb{P}^{1}$, and it clearly lies inside $\tilde{X}$ since it
automatically satisfies (5.6). It can be shown via exact sequences that its
normal bundle in $\tilde{X}$ has no sections, i.e. that
$N_{C_{1}/\tilde{X}}={\cal O}_{\mathbb{P}^{1}}(-1)\oplus{\cal
O}_{\mathbb{P}^{1}}(-1)$ (5.8)
In other words, this curve is fully obstructed already at first order. Its
orientifold image is simply given by
$C_{1}^{\prime}:\quad z_{1}=\eta\,z_{2}\quad\cap\quad
z_{3}=\tilde{\eta}\,z_{4}\quad\cap\quad\xi=-\psi\,P_{4}\qquad\subset\quad\mathbb{P}^{4}_{11114}\,,$
(5.9)
and is clearly also obstructed at first order. If one now defines the flux on
the E3 as follows:
$\frac{F_{\rm E3}}{2\pi}=\tfrac{1}{2}\,[H]+[C_{1}]-[C_{1}^{\prime}]$ (5.10)
then $(2\pi\ell_{s})^{-2}{\cal F}\in H^{1,1}_{-}\cap H^{2}(H,\mathbb{Z})$.
Requiring that this flux remain of type $(1,1)$ as the E3 deforms amounts to
requiring that these two rigid curves be contained in the divisor. This
imposes the following linear constraints on the $a_{i}$ coefficients of the
divisor (5.5):
$a_{1}\,\eta+a_{2}=0\quad\cap\quad a_{3}\,\tilde{\eta}+a_{4}=0$ (5.11)
The hyperplane now looks like:
$P_{D}(C_{1})=a_{1}\,(z_{1}-\eta\,z_{2})+a_{3}\,(z_{3}-\tilde{\eta}\,z_{4})=0$
(5.12)
where by $P_{D}(C_{1})$ we mean the most generic polynomial for a divisor $D$
subject to the constraint of containing $C_{1}$. Hence, of the three moduli,
two have been lifted: $h^{0,2}_{-}=3\mapsto 1$.
Since the $\tilde{X}$ is a one-modulus CY, $C_{1}$ and $C_{1}^{\prime}$ are
homologous in $\tilde{X}$, hence their difference is trivial in
$H_{2}(\tilde{X},\mathbb{Z})$ and then in particular (5.4) is automatically
satisfied. However, the difference $C_{1}-C_{1}^{\prime}$ is not trivial on
the divisor. To check this, it is sufficient to find a single non-zero
intersection number of this difference with some curve on the E3. Let us
compute the following intersection number
$I=\left([C_{1}]-[C_{1}^{\prime}]\right)\cdot[C_{1}]$ (5.13)
The first term can be computed via the adjunction formula for the tangent
bundle of the curve in terms of $[\tilde{D}]$ and the Poincaré dual $[C_{1}]$:
$\chi(C_{1})=-\int_{\tilde{D}}\left([\tilde{D}]\wedge[C_{1}]+[C_{1}]\wedge[C_{1}]\right)$
(5.14)
The first term is simply the intersection number $\tilde{D}\cdot C_{1}=H\cdot
C_{1}=1$ since the curve has degree one. The fact that it is a
$\mathbb{P}^{1}$ tells us that $\chi=2$, hence one deduces that
$C_{1}\cdot C_{1}=-3$ (5.15)
In order to get the intersection number $C_{1}\cdot C_{1}^{\prime}$, one first
notices that both curves have two linear equations in common. Then, one
imposes two more equations of degree $4$ to intersect the curves. Hence, on
this weighted projective space, one needs
$\displaystyle C_{1}\cdot C_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{P}^{4}_{11114}}[H]\wedge[H]\wedge[4\,H]\wedge[4\,H]=4$
(5.16)
Therefore
$I=-7$ (5.17)
Hence, we conclude that
$\displaystyle i_{*}(C_{1}-C_{1}^{\prime})$ $\displaystyle=$ $\displaystyle
0\in H_{2}(\tilde{X},\mathbb{Z})\quad\text{but}$ (5.18) $\displaystyle
C_{1}-C_{1}^{\prime}$ $\displaystyle\neq$ $\displaystyle 0\in
H_{2}(\tilde{D},\mathbb{Z})$ (5.19)
In Poincaré dual terms, this implies that the flux cannot be seen as the pull-
back of a (closed) flux on $\tilde{X}$. This is exactly what is required for
the zero-mode lifting mechanism to work, as discussed in general in 4.6.
Since, the class $[C_{1}]-[C_{1}^{\prime}]$ is non-trivial in the divisor,
these two curves will not somehow recombine and annihilate.
Now that two zero-modes have been eliminated, the remaining one can be lifted
by adding another rigid curve to the [Poincaré dual to the] flux. Take the
following curve and its orientifold image:
$C_{2},C_{2}^{\prime}:\quad z_{1}=\eta\,z_{4}\quad\cap\quad
z_{3}=\tilde{\eta}\,z_{2}\quad\cap\quad\xi=\pm\psi\,P_{4}\qquad\subset\quad\mathbb{P}^{4}_{11114}$
(5.20)
Requiring that the E3-brane contain these curves will impose further
restrictions on its divisor moduli that read
$a_{1}\,\eta+a_{4}=0\quad\cap\quad a_{3}\,\tilde{\eta}+a_{2}=0$ (5.21)
The divisor under consideration is described by
$P_{D}(C_{1},C_{2})=a(z_{1}-\eta\,z_{2}+\tfrac{\eta}{\tilde{\eta}}\,z_{3}-\eta\,z_{4})=0$
(5.22)
In conclusion, the flux
$\frac{F_{\rm
E3}}{2\pi}=\tfrac{1}{2}\,[H]+[C_{1}]+[C_{2}]-[C_{1}^{\prime}]-[C_{2}^{\prime}]$
(5.23)
fully freezes the E3-instanton, thereby lifting all of its $2\times 3$
$\tilde{\rho}_{\rm z.m.}^{\dot{\alpha}}$ zero-modes.121212One may wonder,
whether the intersecting curves $C_{1}$ and $C_{2}$ might recombine into a
degree two $\mathbb{P}^{1}$. However, for a generic hypersurface equation,
such degree two curves are known to come in finite discrete amounts. In this
case there are $128834912$ of them [86], meaning they must be rigid. One can
again check that the resulting linear combination of curves is non-trivial in
the homology of the divisor.
Let us address one subtlety that is fortunately automatically absent in one-
modulus cases. E3-instantons with fluxes could in principle have the so-called
chiral zero-modes localized at intersections with D7-branes, as pointed out in
[19]. However, the index that counts the next chirality of such zero-modes
$\int_{D7\,\cap\,E3}\left(F_{E3}-F_{D7}\right)$ (5.24)
gets no additional contribution from the fluxes we have seen here. The fact
that the differences of curves considered are trivial in the $\tilde{X}$
implies that the Poincaré-dual flux integrates to zero along any curve made by
intersecting two divisors in $\tilde{X}$. Hence, the ‘magnetic’ fluxes under
consideration do not alter the chiral zero-mode problem. They simply lift
neutral zero-modes.
Finally, note that, although we used the specific form (5.6) of the Calabi-Yau
double-cover $\tilde{X}$ to make life easier, we could add a host of terms to
make it more generic, such as
$(z_{1}-\eta\,z_{2})\,(z_{1}-\eta\,z_{4})\,P_{6}$. It is known that Calabi-Yau
threefolds generically contain finite but large amounts of degree one
$\mathbb{P}^{1}$’s (in this case $290540$). Hence, this zero-mode lifting
effect is a generic feature, and it means that this simple one-modulus Calabi-
Yau admits E3-induced non-perturbative superpotentials. Since Calabi-Yau
three-folds typically contain rigid curves, this phenomenon can be applied to
many more situations. This opens up a new possibility for model-building. One
no longer needs to construct Calabi-Yau four-fold containing exceptional
divisors to put their M5-instantons on. Instead, one can study simple spaces,
and discover that they already contain divisors that get frozen by such easily
described fluxes.
Acknowledgments
We would like to thank L. Anderson, I. Brunner, V. Braun, F. Fucito, J. Gray,
S. Kachru, F. Morales, R. Richter, T. Weigand for useful discussions. L. M.
would like to thank P. Koerber for collaboration on a related project. The
work of M. B. and L. M. was partially supported by the ERC Advanced Grant
n.226455 Superfields , by the Italian MIUR-PRIN contract 20075ATT78, by the
NATO grant PST.CLG.978785. The work of A. C. is supported in part by the
Cluster of Excellence “Origin and Structure of the Universe” in München,
Germany, and by a EURYI award of the European Science Foundation. A.C. would
like to thank the Kavli Institute for Theoretical Physics for its hospitality
during early stages of this project. L. M. would like to thank the Laboratoire
de Physique Théorique et Hautes Energies for its hospitality during the course
of this work.
Appendix
## Appendix A U(1)-bundles and holomorphic line bundles
A U$(1)$-connection with (1,1) field-strength associated to a smooth complex
line bundle $L$ allows to reinterpret $L$ as a holomorphic line bundle, that
we denote as ${\cal L}$, and viceversa. Let us explicitly apply this well
known result to the present case. The following discussion is completely
standard.
Let us define a complex line bundle $L^{q}_{Q}$ with charge $q$ under the
$U(1)_{Q}$ symmetry. Clearly, $L_{Q}\equiv L^{q=1}_{Q}$ and
$L^{q}_{Q}=(L_{Q})^{q}$. Hence, if $f$ is a section of $L^{q}_{Q}$, going from
one patch to the other it undergoes a transformation
$f\rightarrow e^{{\rm i}q\arg(c\tau+d)}f$ (A.1)
Its covariant derivative is given by
$\nabla_{Q}f:=(\text{d}-{\rm i}qQ)f=:\partial_{Q}f+\bar{\partial}_{Q}f$ (A.2)
Now, since $(\text{d}Q)^{0,2}=0$, $Q$ defines a holomorphic line bundle.
Indeed, define
$\hat{f}=({\rm Im\hskip 1.00006pt}\tau)^{-q/2}f$ (A.3)
Since, going from one patch to the other we have
${\rm Im\hskip 1.00006pt}\tau\rightarrow|c\tau+d|^{-2}{\rm Im\hskip
1.00006pt}\tau$ (A.4)
it is easy to see that $\hat{f}$ has the following gluing conditions
$\hat{f}\rightarrow(c\tau+d)^{q}\hat{f}$ (A.5)
Hence, we see the transition functions of $\hat{f}$ are holomorphic and then
define an associated holomorphic line bundle, that we denote by ${\cal
L}_{Q}^{q}\equiv({\cal L}_{Q})^{q}$. Clearly, the covariant anti-holomorphic
derivative $\bar{\partial}_{Q}f$ is mapped into the anti-holomorphic
derivative $\bar{\partial}\hat{f}$ and then
$\bar{\partial}_{Q}f=0\quad\Leftrightarrow\quad\bar{\partial}\hat{f}=0$ (A.6)
For instance, using this terminology, we immediately recognize that $\Omega$
as defined in (2.19) can be seen as a holomorphic section of ${\cal
L}_{Q}\otimes K_{{X}}$, whereas $e^{-\phi/2}\Omega$ is a covariantly constant
section of $L_{Q}\otimes K_{{X}}$.
Finally, notice that from $L^{q}_{Q}$ we can alternatively construct an anti-
holomorphic line bundle $\bar{\cal L}^{-q}_{Q}$, whose sections can be defined
as $\tilde{f}=({\rm Im\hskip 1.00006pt}\tau)^{q/2}f$. The sections $\tilde{f}$
of $\bar{\cal L}^{q}_{Q}$ have transition functions of the form
$\tilde{f}\rightarrow(c\bar{\tau}+d)^{q}\tilde{f}$ (A.7)
In this case $\partial_{Q}f\rightarrow\partial\tilde{f}$ and covariantly anti-
holomorphic sections of $L^{q}_{Q}$ are mapped into anti-holomorphic sections
of $\bar{\cal L}^{-q}_{Q}$.
Clearly, we have the isomorphisms
$L^{-q}_{Q}\simeq(L^{q}_{Q})^{-1}\simeq(L^{q}_{Q})^{*}$, ${\cal
L}^{-q}_{Q}\simeq({\cal L}_{Q}^{q})^{-1}$ and $\bar{\cal
L}^{q}_{Q}\simeq({\cal L}_{Q}^{q})^{*}$.
## Appendix B Some useful properties of the extrinsic curvature
Here we review some useful properties of the extrinsic curvature.
Consider a space $M$ with metric $g$ and a submanifold $D$ with induced metric
$h=\iota^{*}g$. We can accordingly split $T_{M}|_{D}=T_{D}\oplus
T^{\perp}_{D}$. Then, the extrinsic curvature ${\cal K}$ can be seen as a map
${\cal K}:T_{D}\otimes T_{D}\rightarrow T^{\perp}_{D}$ (B.1)
defined by
${\cal K}(v,w)={\cal K}(w,v)=(\hat{\nabla}_{v}w)^{\perp}\qquad\forall v,w\in
T_{D}$ (B.2)
where $\hat{\nabla}$ is computed by using the bulk metric $g$. Alternatively
$\langle X,{\cal K}\rangle\equiv X_{m}{\cal K}^{m}=-\frac{1}{2}({\cal
L}_{X}g)|_{D}\qquad\forall X\in T^{\perp}_{D}$ (B.3)
that can be easily proved by starting from the definition (B.2).
Assume now that $M$ is Kähler, with complex structure $I$ and Kähler form
$J=g\cdot I$. We have in particular that $I$ and $J$ are covariantly constant
with respect to the metric $g$. Furthermore, let us restrict to holomorphic
submanifolds $D$. This latter condition can be written by saying that
$I\cdot v\subset T_{D}\qquad\forall v\in T_{D}$ (B.4)
Then,
${\cal K}(Iv,w)={\cal K}(v,Iw)=I{\cal K}(v,w)$ (B.5)
which means that ${\cal K}$ has pure holomophic or anti-holomorphic indices.
Notice also that, since ${\cal K}$ has pure indices, the definition (B.3) is
telling us that ${\cal K}$ measures the deformation of the complex structure
on $D$ induced by a deformation of $D$ preserving the holomorphy of the
embedding. Namely, suppose that $V$ is a section of the ordinary normal bundle
that generates such a deformation.
Then, the variation of the induced metric generated by $V$ is given by
$\delta_{V}h=({\cal L}_{V}g)|_{D}=-2V_{m}{\cal K}^{m}$ (B.6)
so that
$(\delta_{V}h)_{i\bar{\jmath}}=0\quad,\quad(\delta_{V}h)_{\bar{\imath}\bar{\jmath}}=-2V_{m}{\cal
K}^{m}{}_{\bar{\imath}\bar{\jmath}}$ (B.7)
Notice that the geometric deformation does not induce any deformation of the
Kähler structure on $D$. One can alternatively parametrize the change in the
complex structure on $D$ by a section of
$(\delta_{V}I_{D})^{i}{}_{\bar{\jmath}}\in\Gamma(T_{D}\otimes\bar{T}^{*}_{D})$,
where $T_{D}$ and is the holomorphic tangent bundle. $I_{D}+\delta_{V}I_{D}$
denotes the new complex structure131313The condition
$(I_{D}+\delta_{V}I_{D})^{2}=-\mathbbm{1}$ implies that $\delta I_{D}$ has no
pure indices. in which $h+\delta h$ is hermitian. Hence
$h_{ac}(\delta_{V}I_{D})^{c}{}_{b}+h_{bc}(\delta_{V}I_{D})^{c}{}_{a}={\rm
i}\delta h_{ab}=-2iV_{m}{\cal K}^{m}{}_{ab}$ (B.8)
Since $X$ preserves the holomorphy of the embedding, in complex coordinates
$(s^{i},\bar{s}^{\bar{\imath}})$ we have
$\bar{\partial}_{[\bar{\imath}}(\delta_{V}I_{D})^{j}{}_{\bar{u}]}=0$ (B.9)
and hence we can locally write
$(\delta_{V}I_{D})^{i}{}_{\bar{\jmath}}=\bar{\partial}_{\bar{\jmath}}\,\delta_{V}\zeta^{i}$
(B.10)
where $\delta_{V}\zeta^{i}$ defines the shift in the new complex coordinates
$\tilde{\zeta}^{i}=\zeta^{i}+\delta_{V}\zeta^{i}$. Using this one can show
that $h_{ac}(\delta_{V}I_{D})^{c}{}_{b}$ is symmetric in $(ab)$. Hence from
(B.8) we get
$(\delta_{V}I_{D})^{i}{}_{\bar{\jmath}}=-{\rm i}h^{i\bar{u}}V_{m}{\cal
K}^{m}{}_{\bar{u}\bar{\jmath}}$ (B.11)
We can use the extrinsic curvature to define the operator
${\Upsilon}:T^{\perp}_{D}\rightarrow T^{1,0}_{D}\otimes
T^{*0,1}_{D}\qquad\qquad\text{with}\quad(\Upsilon_{m})^{i}{}_{\bar{\jmath}}:=-{\rm
i}h^{i\bar{u}}g_{mn}{\cal K}^{n}{}_{\bar{u}\bar{\jmath}}$ (B.12)
In particular, the action of $\Upsilon$ descends to an action at the level of
sheaf cohomology classes
$\Upsilon:H^{0}(D,N_{D})\rightarrow H^{1}(D,T_{D})\ $ (B.13)
where $N_{D}$ is the holomorphic normal bundle to $D$. In the form (B.13),
$\Upsilon$ gives precisely the map that associates any infinitesimal
deformation of the divisor with the corresponding infinitesimal deformation of
complex structure on $D$.
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|
arxiv-papers
| 2011-07-19T14:57:19 |
2024-09-04T02:49:20.717841
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Massimo Bianchi, Andres Collinucci and Luca Martucci",
"submitter": "Luca Martucci",
"url": "https://arxiv.org/abs/1107.3732"
}
|
1107.3805
|
# Baryon number violation at the LHC: the top option
Zhe Donga, Gauthier Durieuxb, Jean-Marc Gérardb, Tao Hana, Fabio Maltonib
aDepartment of Physics, University of Wisconsin, Madison, WI 53706, USA
bCentre for Cosmology, Particle Physics and Phenomenology (CP3), Université
catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
(19 July 2011)
###### Abstract
Subject to strong experimental constraints at low energies, baryon number
violation is nonetheless well motivated from a theoretical point of view. We
examine the possibility of observing baryon-number-violating top-quark
production or decay at hadron colliders. We adopt a model independent
effective approach and focus on operators with minimal mass-dimension.
Corresponding effective coefficients could be directly probed at the Large
Hadron Collider (LHC) already with an integrated luminosity of 1 fb-1 at 7
TeV, and further constrained with 30 (100) fb-1 at 7 (14) TeV.
###### pacs:
12.38.Bx,12.60.-i,14.65.Ha,11.30.Fs
††preprint: MADPH-11-xxxx,CP3-11-xxxx
## .1 Introduction
In the Standard Model (SM) for fundamental interactions, baryon ($B$) and
lepton ($L$) numbers, associated with accidental global symmetries, are
classically conserved quantities; a (tiny) violation is, however, induced by
non-perturbative effects ’t Hooft (1976). Baryon number violation (BNV) also
naturally occurs in Supersymmetry Weinberg (1982), in Grand Unified Theories
Georgi and Glashow (1974), where BNV is notably mediated by new gauge bosons,
and in black hole physics Bekenstein (1972). The cosmological production of
matter from a matter–anti-matter symmetric initial condition moreover requires
$B$ to have been violated in the early Universe Sakharov (1967).
On the other hand, experimental constraints on several BNV processes have
reached impressive heights. Nucleon decay channels provide the best examples,
though baryon-number-violating decays of the $\tau$ lepton or, much more
recently, of heavy mesons have also been investigated Nakamura et al. (2010).
These latter measurements opened the way for direct experimental tests of the
baryon number conservation law within the second and third generations of
quarks and leptons but have not really extended the range of energy scales.
The only direct experimental constraints on BNV beyond the GeV scale are
bounds on the $Z\to p\,e^{-},\>p\,\mu^{-}$ branching ratios obtained at LEP
Nakamura et al. (2010). The LHC comes as a natural step forward in probing the
baryon number conservation law beyond the TeV scale and the first generation.
In particular, with a very large production rate and unique experimental
signatures, top quarks are an interesting option: the top flavor can be
clearly identified, the $t$ and $\bar{t}$ are distinguishable via the charged
lepton in their decay, and the hadronization effects unimportant.
Consequently, BNV could be probed at the quark level.
We choose to consider interactions involving one single top quark and a
charged lepton. The presence of a single final state lepton produced from the
proton-proton initial state implies a total change in lepton number $\Delta
L=\pm 1$, and, by conservation of angular momentum $\Delta(L+3B)\in
2\mathbb{Z}$, requires a simultaneous violation of $B$. Thus a single charged
lepton without missing energy points toward BNV. In the presence of neutrinos
in the production process, though, the lepton number is intractable.
Consequently, baryon-number-violating processes would be more difficult to
identify unambiguously, and could, for instance, be confused with flavor-
changing neutral currents Andrea et al. (2011); Kamenik and Zupan (2011).
## .2 Effective operators
The effective BNV Lagrangian can easily be built out of five lowest
dimensional effective operators Weinberg (1979); Wilczek and Zee (1979);
Abbott and Wise (1980) that preserve Lorentz invariance and $SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{Y}$ gauge symmetries, along with an accidental global
$B-L$ symmetry. Following the notation of Ref. Weinberg (1979), we write
${\cal L}^{\rm dim=6}_{\rm
BNV}=\frac{1}{\Lambda^{2}}\sum_{i=1}^{5}c_{i}\,O^{(i)}\,,$ (1)
where $c_{i}$ are the effective (dimensionless) coefficients of the
corresponding operators $O^{(i)}$ and $\Lambda$ is the mass scale associated
with physics responsible for BNV beyond the Standard Model.
Expanding $SU(2)_{L}$ indices in operators $O^{(1-5)}$ and identifying one up-
type quark as the top, the effective terms that do not contain neutrinos can
be parametrized as linear combinations of only two operators (and their
Hermitian conjugates),
$\displaystyle
O^{(s)}\equiv\epsilon^{\alpha\beta\gamma}[\overline{t^{c}_{\alpha}}(aP_{L}+bP_{R})D_{\gamma}][\overline{U^{c}_{\beta}}(cP_{L}+dP_{R})E],$
(2) $\displaystyle
O^{(t)}\equiv\epsilon^{\alpha\beta\gamma}[\overline{t^{c}_{\alpha}}(a^{\prime}P_{L}+b^{\prime}P_{R})E][\overline{U^{c}_{\beta}}(c^{\prime}P_{L}+d^{\prime}P_{R})D_{\gamma}],$
where $D,\,U,\,E$ respectively denote generic down-, up-type quarks and
charged leptons. We emphasize that fermions in Eq. (2) are taken as mass
eigenstates. Charge conjugated fields are defined as $\psi^{c}\equiv
C\overline{\psi}^{T}$ with $C$, the charge conjugation matrix; $2P_{L/R}\equiv
1\mp\gamma^{5}$; colors are labeled by Greek indices; $a,\ a^{\prime},\ldots$
are fermion-flavor-dependent effective parameters.
Three comments are in order. First, the $(s)$, $(t)$ labeling in Eq. (2)
reminds that the scale $\Lambda$ in Eq. (1) may be linked to the mass of a
heavy mediator (with electric charge $1/3$) exchanged in $s$ or $t$ channels,
respectively. If so identified, then the coupling parameters $a,\
a^{\prime},\ldots$ could be naturally of the order of unity. We stress that
the operator
$O^{(u)}\\!\equiv\epsilon^{\alpha\beta\gamma}\;[\overline{t^{c}_{\alpha}}P_{L}U_{\beta}][\overline{D^{c}_{\gamma}}P_{L}E$]
arising from $O^{(4)}$ and possibly associated with a mediator of electric
charge $4/3$ exchanged in the $u$ channel does not need to be introduced at
the effective level. The reason is that the Schouten identity,
$\displaystyle[CP_{L}]_{ij}[CP_{L}]_{kl}-[CP_{L}]_{ik}[CP_{L}]_{jl}+[CP_{L}]_{il}[CP_{L}]_{jk}=0,$
can be used to express $O^{(u)}$ in terms of $O^{(s)}$ and $O^{(t)}$, i.e.,
$O^{(u)}=-O^{(s)}(a=c=1,\,b=d=0)-O^{(t)}(a^{\prime}=c^{\prime}=1,\,b^{\prime}=d^{\prime}=0)$.
Second, heavy gauge mediators (vectors) give rise to $O^{(1,2)}$ only Weinberg
(1979); Dorsner and Fileviez Perez (2005), which in our basis, Eq. (2),
entails $a=0=d$ or $b=0=c$ (or primed analogs). Third, operators involving two
top quarks can also be obtained from Eq. (2) by substituting $t$ for $U$.
Note, however, that in this case $O^{(s)}$ and $O^{(t)}$ are no longer
independent and considering only one of the two is then sufficient. Such
operators could, for example, mediate processes like
$e^{-}d\to\bar{t}\,\bar{t}$ in future $e^{-}p$ colliders, or
$gd\to\bar{t}\,\bar{t}e^{+}$ at the LHC.
## .3 Processes
At the LHC, possibly relevant BNV processes involving a top quark are
$\begin{array}[]{ll}t\xrightarrow{\text{BNV}}\overline{U}\,\overline{D}\,E^{+}&({\rm
decay})\\\ U\,D\xrightarrow{\text{BNV}}\bar{t}\,E^{+}&({\rm
production})\end{array}$ (3)
(and their charge conjugate analogs) where, in the first case, top quarks are
produced through SM processes. Since, as mentioned above, a single charged
lepton without any missing transverse energy ($\cancel{E}_{T}$) in the final
state is a clear signal for BNV, it is simpler to avoid signatures that lead
to neutrinos in the final state. Fully reconstructed top leptonic decays could
be considered in more refined analyses. We also note that the flavor
assignments can be very relevant from the phenomenological point of view. In
decay, heavy flavors such as charm and bottom could be tagged in jets. In
production, the relevance of initial quark flavors is determined also by
proton parton distribution functions (PDFs).
Neglecting all fermion masses but the top one, $m_{t}$, and using the
algebraic rules introduced in Ref. Denner et al. (1992), the squared amplitude
for the processes in Eq. (3) induced by the operators of Eq. (2) reads
$\displaystyle\sum_{\text{\scriptsize\parbox{28.45274pt}{\centering spins,
colors\@add@centering}}}|\mathcal{M}|^{2}=\frac{24}{\Lambda^{4}}\Big{[}$
$\displaystyle(p_{t}\cdot p_{D})\>(p_{U}\cdot p_{E})\>(A+C)$ $\displaystyle-$
$\displaystyle(p_{t}\cdot p_{U})\>(p_{D}\cdot p_{E})\>C$ $\displaystyle+$
$\displaystyle(p_{t}\cdot p_{E})\>(p_{D}\cdot p_{U})\>(B+C)\Big{]}$ (4)
where
$\displaystyle A$
$\displaystyle\equiv\left(|a|^{2}+|b|^{2}\right)\left(|c|^{2}+|d|^{2}\right),$
(5) $\displaystyle B$
$\displaystyle\equiv\left(|a^{\prime}|^{2}+|b^{\prime}|^{2}\right)\left(|c^{\prime}|^{2}+|d^{\prime}|^{2}\right),$
$\displaystyle C$
$\displaystyle\equiv\mathfrak{Re}\big{\\{}a^{*}c^{*}a^{\prime}c^{\prime}+b^{*}d^{*}b^{\prime}d^{\prime}\big{\\}},$
arise respectively from the square of $O^{(s)}$, of $O^{(t)}$ and from the
interference between these two operators.
Figure 1: Energy spectrum of the charged lepton in the SM $t\to bE^{+}\nu_{E}$
(blue curve) and in BNV $t\to\overline{U}\,\overline{D}\,E^{+}$ top decays.
For a BNV decay, we obtain the following partial width:
$\displaystyle\Gamma^{\text{BNV}}_{t}\\!$
$\displaystyle=\int\limits_{0}^{m_{t}/2}\text{d}E_{\scriptscriptstyle
E}\>\frac{m_{t}^{2}E_{\scriptscriptstyle
E}^{2}}{32\pi^{3}\Lambda^{4}}\left[\\!\left(\frac{A}{3}+B+C\right)\left(1-\frac{2E_{\scriptscriptstyle
E}}{m_{t}}\right)+\frac{A}{6}\\!\right]$
$\displaystyle=\frac{m_{t}^{5}}{192\pi^{3}}\frac{1}{16\Lambda^{4}}\left[A+B+C\right]\,,$
where $E_{\scriptscriptstyle E}$ is the lepton energy in the top rest frame.
In Fig. 1, we compare the charged lepton energy spectrum in a SM decay to that
in a BNV decay for three different representative choices of $A,B$ and $C$.
Inputing the SM width for the top quark ($1.4$ GeV), the BNV branching ratio
can be conveniently written as
$\displaystyle{\rm Br}^{\text{BNV}}_{t}\\!=1.2\times
10^{-6}\left(\frac{m_{t}}{173~{}\text{GeV}}\right)^{5}\left(\frac{1\text{TeV}}{\Lambda}\right)^{4}\left[A+B+C\right].$
Taking the $t\bar{t}$ production cross section at the 7 (14) TeV LHC to be
$150$ (950) pb, we can expect 0.35 (2.2)$/$fb-1 BNV top decays if $A+B+C=1$,
for each allowed flavor combination.
For BNV production, the partonic cross section reads
$\displaystyle\hat{\sigma}^{\text{BNV}}_{t}\\!$
$\displaystyle=\frac{1}{96\pi\Lambda^{4}}\int\limits_{m_{t}^{2}-\hat{s}}^{0}\text{d}\hat{t}\>\left[A\frac{\hat{t}\left(\hat{t}-m_{t}^{2}\right)}{\hat{s}^{2}}+B\frac{\left(\hat{s}-m_{t}^{2}\right)}{\hat{s}}+2C\frac{\hat{t}}{\hat{s}}\,\right]$
$\displaystyle=\frac{\hat{s}}{96\pi\Lambda^{4}}\left(1-\frac{m_{t}^{2}}{\hat{s}}\right)^{2}\left[\left(\frac{A}{3}+B+C\right)+\frac{m_{t}^{2}}{\hat{s}}\frac{A}{6}\right]\,,$
(6)
with the Mandelstam variables $\hat{s}\equiv(p_{U}+p_{D})^{2}$ and
$\hat{t}\equiv(p_{U}-p_{E})^{2}$. As expected from dimensional arguments the
cross section induced by the operators in Eq. (2) grows as
$\hat{s}/\Lambda^{4}$. However, in setting lower bounds on the scale of new
physics, it is important to always keep in mind that the validity (and
unitarity) of the effective field theory itself assumes
$\hat{s}\ll\Lambda^{2}$.
Out of the six possible initial quark flavor assignments, (namely,
$ud,\,us,\,ub,\,cd,\,cs$ and $cb$), we consider
$\begin{array}[]{l@{\hspace{4mm}-\hspace{1.5mm}}p{4.8cm}}u\,d\to\bar{t}\,E^{+}\hfil\hskip
11.38109pt-\hskip 4.2679pt&the most PDF-favored,\\\
u\,b\to\bar{t}\,e^{+}\hfil\hskip 11.38109pt-\hskip 4.2679pt&possibly flavor-
unsuppressed,\\\ c\,b\to\bar{t}\,\mu^{+}\hfil\hskip 11.38109pt-\hskip
4.2679pt&the most PDF-suppressed, yet,\\\\[-5.69054pt] \hfil\hskip
11.38109pt-\hskip 4.2679pt&possibly flavor-unsuppressed,\end{array}$
as well as their charge conjugate analogs. Operators with two pairs of
fermions in the same generation could be favored by the flavor structure of
the underlying theory. In Table 1, we collect the cross sections for the
different processes at the LHC with $\sqrt{s}=7$ (14) TeV. To enforce
unitarity and not to artificially overestimate cross sections, we impose
$\sqrt{\hat{s}}<\Lambda$. This cut has an important effect on valence quark
initiated processes but a very mild one on processes initiated by sea quarks.
$\sigma\\!\,$[fb] | $ud\to\bar{t}E^{+}$ | $ub\to\bar{t}e^{+}$ | $cb\to\bar{t}\mu^{+}$
---|---|---|---
$A\;B\;C$ | $\bar{u}\bar{d}\to tE^{-}$ | $\bar{u}\bar{b}\to te^{-}$ | $\bar{c}\bar{b}\to t\mu^{-}$
$1\;\;0\;\;0$ | 250 (690) | 30 (150) | 1.2 (10)
14 (74) | 3.1 (21) | 1.2 (10)
$0\;\;1\;\;0$ | 910 (1 900) | 110 (440) | 3.7 (28)
45 (220) | 9.1 (60) | 3.7 (28)
$1\;\;1\;\;1$ | 2 100 (4 600) | 240 (980) | 9.1 (66)
110 (500) | 22 (140) | 9.1 (66)
Table 1: Cross sections (fb) for representative BNV production processes at
the LHC, with three different choices of $A,B$ and $C$,
$\sqrt{\hat{s}}<\Lambda=1$ TeV, $\sqrt{s}=7$ TeV (14 TeV in parentheses) and
CTEQ6L1 PDF Pumplin et al. (2002) (renormalization and factorization scales
set at $m_{t}=173$ GeV).
## .4 LHC searches
We now briefly discuss BNV signatures at the LHC. For the sake of illustration
we make a definite choice for the fermion flavors in Eq. (3) and consider
I) BNV decay: $pp\xrightarrow{\text{SM}}t\,\bar{t}$ with the top decaying via
a BNV interaction $t\xrightarrow{\text{BNV}}\bar{b}\,\bar{c}\,\mu^{+}$ and the
anti-top decaying fully hadronically, which leads to the $\mu^{+}$+5-jet final
state; II) BNV production: $p\,p\xrightarrow{\text{BNV}}\bar{t}\,\mu^{+}$ with
$u,d$ flavors in the initial state, the anti-top decaying fully hadronically,
leading to $\mu^{+}+3$ jets.
The first interesting observation is that there are no irreducible backgrounds
to such signatures as both of them have no $\cancel{E}_{T}$. On the other
hand, processes resulting from a leptonically decaying $W$ with a small
reconstructed $\cancel{E}_{T}$ could mimic the signal. A proper investigation
of such backgrounds requires not only parton showering, hadronization and
realistic detector simulation but also data driven methods. However, a few
relevant observations can already be made with a simple parton-level
simulation. To this aim, we have implemented BNV interactions in MadGraph 5
Alwall et al. (2011) via FeynRules Christensen and Duhr (2009) and generated
events for both signal and representative backgrounds in the same simulation
framework.
The search for BNV decays proceeds through the selection of $\mu^{+}+5$ jets
with an upper cut on the $\cancel{E}_{T}$. The presence of two tops, one
hadronically decaying $W$ and possibly two $b$-tagged jets can be efficiently
used to better reconstruct the event kinematics. In addition, note that the
BNV decay of a top quark gives $\mu^{+}\bar{b}$ at variance with the SM semi-
leptonic decay which gives $\mu^{+}b$. Determining the bottom quark charge
(e.g., via a lepton tag) in the BNV decay could therefore offer crucial
discrimination power. The main SM backgrounds to this signature come from
$t\,\bar{t}+1$ jet and $W^{+}+5$ jets, the former being dominant after $b$
tagging.
Figure 2: Transverse momentum for the charged lepton in the BNV production
signal $\bar{t}\,\mu^{+}$ (from $ud$ initial state) and in the $W^{+}+$3-jet
and $\bar{t}W^{+}$ backgrounds. Top quarks are decayed hadronically. Selection
cuts on the three jets and the muon are given in the text.
The search for BNV production proceeds through the selection of $\mu^{+}+3$
jets with an upper cut on the $\cancel{E}_{T}$. The reconstruction is simpler
than in the BNV decay search, as there is no combinatorial background and the
top and $W$ mass constraints can be used to improve the resolution on the
signal kinematics. In Fig. 2, we compare the $p_{T}$ of the charged lepton in
the signal to that of the $W^{+}$+3-jet and $\bar{t}\>W^{+}$ (with
$W^{+}\to\mu^{+}\nu_{\mu}$) backgrounds. We require three central jets
($p_{T}>40$ GeV$,|\eta|<2.5,\Delta R_{jj}>0.5$), a central isolated lepton
($|\eta|<2.5,\Delta R_{j\mu}>0.5$) and $\cancel{E}_{T}<30$ GeV. In the
$W^{+}$+3-jet background we also demand $|m_{jjj}-m_{t}|<40$ GeV and a $b$
tag. As expected from the $\hat{s}$ enhancement in the cross section,
eventually tamed by requiring $\sqrt{\hat{s}}<\Lambda=1$ TeV, the $p_{T}$
distribution of the lepton in the signal is much harder than in the
backgrounds.
The LHC reach for the processes in Table 1 can be expressed in terms of the
minimal value for the parameters defined in Eq. (5) leading to a sensitivity
$S/\sqrt{S+B}\geq 5$. For the sake of illustration we consider 30 (100) fb-1
of collected luminosity at the LHC for $\sqrt{s}=7$ (14) TeV, the event
selection described in the above paragraph with the additional requirement
$p_{T}>150$ GeV for the charged lepton (one flavor), both $t$ and $\bar{t}$
production, only the $tW$ background, and $A=B=C$. In so doing, we find
$\begin{array}[]{l@{\hspace{1.5mm}:\hspace{4mm}}p{3cm}p{2cm}}u\,d\to
t\,E\hfil\hskip 4.2679pt:\hskip 11.38109pt&$A,\,B,\,C\geq 0.0076$&(0.0046)\\\
u\,b\to t\,e\hfil\hskip 4.2679pt:\hskip 11.38109pt&$A,\,B,\,C\geq
0.084$&(0.026)\\\ c\,b\to t\,\mu\hfil\hskip 4.2679pt:\hskip
11.38109pt&$A,\,B,\,C\geq 1.6$&(0.21)\end{array}$
which point to a sensitivity at the $10^{-1}-10^{-2}$ level for the effective
coefficients $c_{i}$ of Eq. (1) at the TeV scale.
Finally, we stress that, for both BNV production and decay signatures,
selecting high-$p_{T}$ tops could be advantageous. In this limit, the BNV
production signal is enhanced with respect to the backgrounds, while for the
BNV decay search, top decay products might cluster into one jet, curbing, for
instance, the combinatorial problems in the $\mu^{+}$+5-jet signature and also
controlling better $\cancel{E}_{T}$ uncertainties. To this aim, efficient
boosted reconstruction techniques for the top quark should be employed Plehn
et al. (2010).
## .5 Indirect constraints
In principle, the operators considered in Eq. (2) also contribute indirectly
to nucleon decays Hou et al. (2005) through tree and/or loop diagrams. Tree-
level diagrams with one $W$-emission, such as that in Fig. 3a provide
formidable high lower bounds on $\Lambda$ (or equivalently upper bounds on the
effective parameters) if the lepton is not a $\tau$. In fact, two
$W$-emissions are needed for a $udt\tau$-operator to be relevant in nucleon
decays and the constraints become weaker. Moreover, if the dominant BNV
dimension-six operators only involve the third and second generations of
quarks and leptons, three $W$-emissions are required, and the rate is
suppressed to a level consistent with the data. In BNV production, these
theoretical considerations tend to favor the PDF-suppressed processes of Table
1. By considering a single operator contribution at a time, with fixed flavors
in the two-loop diagram of Fig. 3b, extremely small upper bounds on the
effective parameters can also be obtained Hou et al. (2005). Yet, strong
cancellations may occur when summing over all possible $UDUE$ virtual
contributions and allow effective parameters to be large (say, of order one).
Mechanisms that could lead to such _GIM-like_ cancellations at one- and two-
loop level remain to be examined within a complete theory for flavor, starting
from dimension-six BNV operators expressed in terms of weak eigenstates.
(a) (b)
Figure 3: Representative (a) tree-level and (b) two-loop-level diagrams
involving the BNV operators given in Eq. (2) and leading, in principle, to
nucleon decay.
## .6 Conclusions
We have studied lowest dimensional BNV operators and their consequences for
top physics at the LHC. Corresponding effective coefficients could be probed
directly at the TeV scale up to the $10^{-1}-10^{-2}$ level. In this prospect,
possible flavor models (e.g., warped extra-dimensions or horizontal
symmetries) leading to nucleon decay rates consistent with the present
experimental bounds should be examined.
## .7 Acknowledgements
We thank Giacomo Bruno, Michele Gabusi and Davide Pagano for very enjoyable
and stimulating discussions on promising BNV signatures at the LHC,
Christopher Smith and Pavel Fileviez Perez for insights on the flavor aspects
of nucleon decays. This research has been supported in part by the U.S.
Department of Energy under grant No. DE-FG02-95ER40896, by the Belgian IAP
Program BELSPO P6/11-P and by the IISN.
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|
arxiv-papers
| 2011-07-19T19:07:22 |
2024-09-04T02:49:20.738789
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Zhe Dong, Gauthier Durieux, Jean-Marc Gerard, Tao Han, Fabio Maltoni",
"submitter": "Gauthier Durieux",
"url": "https://arxiv.org/abs/1107.3805"
}
|
1107.3935
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-007, CERN-PH-EP-2011-106 (submitted to PLB)
Measurement of the inclusive $\phi$ cross-section in $p{\kern-1.03706pt}p$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$
The LHCb Collaboration 111Authors are listed on the following pages.
The cross-section for inclusive $\phi$ meson production in
$p{\kern-0.50003pt}p$ collisions at a centre-of-mass energy of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ has been measured with the LHCb
detector at the Large Hadron Collider. The differential cross-section is
measured as a function of the $\phi$ transverse momentum $p_{T}$ and rapidity
$y$ in the region $0.6<p_{T}<5.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\text{ and
}2.44<y<4.06$. The cross-section for inclusive $\phi$ production in this
kinematic range is $\sigma(pp\rightarrow\phi X)=1758\pm
19\mathrm{(stat)}\,^{+43}_{-14}\mathrm{(syst)}\pm
182\mathrm{(scale)}\rm\,\upmu b$, where the first systematic uncertainty
depends on the $p_{T}$ and $y$ region and the second is related to the overall
scale. Predictions based on the Pythia 6.4 generator underestimate the cross-
section.
## LHCb Collaboration
R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z.
Ajaltouni5, J. Albrecht37, F. Alessio6,37, M. Alexander47, G. Alkhazov29, P.
Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,
R.B. Appleby50, O. Aquines Gutierrez10, L. Arrabito53, A. Artamonov 34, M.
Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S.
Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S.
Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I.
Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17,37, M.
van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T.
Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V.
Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52,
T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J.
Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H.
Brook42, A. Büchler-Germann39, A. Bursche39, J. Buytaert37, S. Cadeddu15, J.M.
Caicedo Carvajal37, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i, A.
Cardini15, L. Carson36, K. Carvalho Akiba23, G. Casse48, M. Cattaneo37, M.
Charles51, Ph. Charpentier37, N. Chiapolini39, X. Cid Vidal36, P.E.L.
Clarke46, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J.
Cogan6, P. Collins37, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M.
Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C.
D’Ambrosio37, P. David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C.
Deplano15, O. Deschamps5, F. Dettori15,d, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34,
C. Eames49, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D.G. d’Enterria35,o, D. Esperante Pereira36, L. Estève43, A.
Falabella16,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23, S.
Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C.
Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,
Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, S.
Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C.
Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R. Harji49, N.
Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K. Hennessy48,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, W. Hofmann10, K.
Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12,
D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R.
Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23, P.
Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B.
Jost37, S. Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, S.
Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25, S.
Kukulak25, R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J.
van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O.
Leroy6, T. Lesiak25, L. Li3, Y.Y. Li43, L. Li Gioi5, M. Lieng9, R. Lindner37,
C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, J. Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O.
Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G.
Mancinelli6, N. Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti22, A. Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez
Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E.
Maurice6, B. Maynard52, A. Mazurov32,16,37, G. McGregor50, R. McNulty12, C.
Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P.
Morawski25, J.V. Morris45, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39,
R. Muresan28,38, B. Muryn26, M. Musy35, P. Naik42, T. Nakada38, R.
Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M. Needham46, N.
Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N. Nikitin31, A.
Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41,
R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, B. Pal52, J.
Palacios39, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C.
Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du
Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H.
Rademacker42, B. Rakotomiaramanana38, I. Raniuk40, G. Raven24, S. Redford51,
M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43,
V. Romanovsky34, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J.
Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M.
Sannino19,i, R. Santacesaria22, R. Santinelli37, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, A.
Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, N. Serra39, J.
Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P.
Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V.
Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P. Skottowe43, T.
Skwarnicki52, A.C. Smith37, N.A. Smith48, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, U. Straumann39, N. Styles46, S. Swientek9, M.
Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F.
Teubert37, C. Thomas51,45, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin39, S. Topp-Joergensen51, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23,
A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g,
K. Vervink37, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36,
A. Vollhardt39, D. Voong42, A. Vorobyev29, H. Voss10, K. Wacker9, S.
Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Yang3, R. Young46, O.
Yushchenko34, M. Zavertyaev10,a, L. Zhang52, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
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
oInstitució Catalana de Recerca i Estudis Avan$c$cats (ICREA), Barcelona,
Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Two specific regimes can be distinguished in hadron production in
$p{\kern-0.50003pt}p$ collisions: the so called hard regime at high transverse
momenta, which can be described by perturbative QCD; and the soft regime,
which is described by phenomenological models. The underlying event in
$p{\kern-0.50003pt}p$ processes falls into the second category. Therefore soft
QCD interactions need careful study to enable tuning of the models at a new
centre-of-mass energy. Strangeness production is an important ingredient of
this effort. Measurements of $\phi$ production have been reported by various
experiments [1, 2, 3, 4, 5, 6, 7] in different collision types, for different
centre-of-mass energies and different kinematic coverage. LHCb is fully
instrumented in the forward region and thus yields unique results
complementary to previous experiments and to the other LHC experiments.
A measurement of the inclusive differential $\phi$ cross-section in
$p{\kern-0.50003pt}p$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$
is presented in this paper. The analysis uses as kinematic variables the
$\phi$ meson transverse momentum $p_{T}$ and the rapidity
$y=\frac{1}{2}\ln\left[(E+p_{z})/(E-p_{z})\right]$ measured in the
$p{\kern-0.50003pt}p$ centre-of-mass system222The detector reference frame is
a right handed coordinate system with $+z$ pointing downstream from the
interaction point in the direction of the spectrometer and the $+y$ axis
pointing upwards.. $\phi$ mesons are reconstructed using the $K^{+}K^{-}$
decay mode and thus the selection relies strongly on LHCb’s RICH (Ring Imaging
Cherenkov) detectors for particle identification (PID) purposes. Their
performance is determined from data with a tag-and-probe approach. The
measured cross-section is compared to two different Monte Carlo (MC)
predictions based on Pythia 6.4 [8].
## 2 LHCb detector and data set
Designed for precise measurements of $B$ meson decays, the LHCb detector is a
forward spectrometer with a polar angle coverage with respect to the beam line
of approximately $15$ to $300\;\text{mrad}$ in the horizontal bending plane,
and $15$ to $250\;\text{mrad}$ in the vertical non-bending plane. The tracking
system consists of the Vertex Locator (VELO) surrounding the
$p{\kern-0.50003pt}p$ interaction region, a tracking station upstream of the
dipole magnet, and three tracking stations downstream of the magnet.
Particles travelling from the interaction region to the downstream tracking
stations are deflected by a dipole field of around $4\,\rm{Tm}$, whose
polarity can be switched. For this study, roughly the same amount of data was
taken with both magnet polarities.
The detector has a dedicated PID system that includes two Ring Imaging
Cherenkov detectors. RICH1 is installed in front of the magnet and uses two
radiators (Aerogel and $\text{C}_{4}\text{F}_{10}$), and RICH2 is installed
beyond the magnet, with a $\text{CF}_{4}$ radiator. Combining all radiators,
the RICH system provides pion-kaon separation in a momentum range up to
$100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Downstream of the tracking stations
the detector has a calorimeter system, consisting of the Scintillating Pad
Detector (SPD), a preshower, the electromagnetic and the hadronic calorimeter,
and five muon stations. Details of the LHCb detector can be found in Ref. [9].
The study described in this note is based on an integrated luminosity of
14.7$\mbox{\,nb}^{-1}$ of $p{\kern-0.50003pt}p$ collisions collected in May
2010, where the instantaneous luminosity was low.
The trigger system consists of a hardware based first level trigger and a high
level trigger (HLT) implemented in software. The first level trigger was in
pass-through mode, whereas at least one track, reconstructed with VELO
information, was required to be found by the HLT. On Monte Carlo simulated
events, this trigger configuration is found to be 100% efficient for
reconstructed $\phi$ candidates. However, to limit the acquisition rate, a
prescaling was applied.
The luminosity was measured by two van der Meer scans [10] and a novel method
measuring the beam geometry with the VELO, as described in Ref. [11]. Both
methods rely on the measurement of the beam currents as well as the beam
profile determination. Using these results, the absolute luminosity scale is
determined, using the method described in Ref. [12], with a 3.5% uncertainty,
dominated by the knowledge of the beam currents. The instantaneous luminosity
determination is then based on a continuous recording of the hit rate in the
SPD, which has been normalized to the absolute luminosity scale. The
probability for multiple $p{\kern-0.50003pt}p$ collisions per bunch-crossing
was negligibly low in the data taking period considered here.
As the RICH detectors are calibrated separately for the two magnet polarities,
the measurement is carried out separately for each sample before combining
them for the final result.
Trigger and reconstruction efficiencies are determined using a sample of
$1.25\cdot 10^{8}$ simulated minimum bias events. These have been produced in
the LHCb MC setting, which is based on a custom Pythia tune for the
description of $p{\kern-0.50003pt}p$ collisions, while particle decays are
generally handled by EvtGen [13]. The total minimum bias cross-section in LHCb
MC simulation is $91.05\rm\,mb$, composed of the following Pythia process
types: $48.80\rm\,mb$ inelastic-non-diffractive, $2\times 6.84\rm\,mb$ single
diffractive, $9.19\rm\,mb$ double diffractive and $19.28\rm\,mb$ elastic.
Details on the LHCb MC setting can be found in Ref. [14].
## 3 Data selection and efficiencies
Two oppositely charged tracks, each of which are required to have hits in both
the VELO and the main tracking system, are combined to form $\phi\rightarrow
K^{+}K^{-}$ candidates. The RICH system provides kaon-pion separation for
reconstructed tracks, which is crucial for the inclusive $\phi$ production
analysis. As a first step, at least one kaon is required to pass a tight cut
based on the RICH response during the selection. In a second step, both kaons
have to pass this criterion. The samples of $\phi$ candidates passing the cuts
of the first and second steps are referred to as “tag” and “probe” samples,
respectively. They are used to measure the PID efficiency in the selection as
explained below. The reconstructed $K^{+}K^{-}$ mass is required to be between
995${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
1045${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in both samples.
No cut designed to discriminate prompt and non-prompt $\phi$ mesons is applied
in the selection, so the measurement includes both. However, due to the high
minimum bias cross-section compared to charm or beauty production, the non-
prompt contribution is small; in MC simulation it is found to be 1.6%.
The region of interest $0.6<p_{T}<5.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.44<y<4.06$ is divided into 9 bins in $y$ and 12 bins in $p_{T}$. The
differential cross-section per bin in $p_{T}$ and $y$ is determined by the
equation:
$\frac{\text{d}^{2}\sigma}{\text{d}y\,\text{d}p_{T}}=\frac{1}{\Delta
y\,\Delta\mbox{$p_{T}$}}\cdot\frac{N_{\mathrm{tag}}}{{\cal
L}\cdot\varepsilon_{\rm reco}\cdot\varepsilon_{\rm pid}\cdot{\cal
B}(\phi\rightarrow K^{+}K^{-})},$ (1)
where $N_{\mathrm{tag}}$ is the number of reconstructed $\phi$ candidates in
the tag sample, ${\cal L}$ the integrated luminosity and ${\cal
B}(\phi\rightarrow K^{+}K^{-})=(49.2\pm 0.6)\%$ the branching fraction taken
from Ref. [15]. The selection efficiency is split up into two parts in Eq. 1:
reconstruction $\varepsilon_{\rm reco}$, including the geometrical acceptance,
and the PID efficiency $\varepsilon_{\rm pid}$. Both efficiencies are a
function of the $p_{T}$ and $y$ values of the $\phi$ meson and thus determined
separately for each bin.
In the centre of the kinematic region, the reconstruction efficiency is of the
order of 65-70%. It drops to 30-40% with low transverse momenta and high or
low rapidity values. The PID efficiency is above 95% in the centre of the
kinematic region and drops to 60-70% at the edges of the considered kinematic
region.
The reconstruction efficiency is determined from simulation. To limit the MC
dependence, the PID efficiency is determined from data using the tag-and-probe
method: in the $\phi$ selection, at least one of the two kaons is required to
pass the PID criterion. The number of $\phi$ candidates passing this
requirement is given by $N_{\text{tag}}$. In a subsequent step, both kaons
must pass the PID criterion. The number of $\phi$ candidates passing this step
is given by $N_{\text{probe}}$. The efficiency $\varepsilon_{\rm pid}$ that at
least one of the two kaons from a $\phi$ candidate fulfils the kaon PID
requirement for each bin is thus given by:
$\varepsilon_{\rm pid}=1-\left(\frac{N_{\rm tag}-N_{\rm probe}}{N_{\rm
tag}+N_{\rm probe}}\right)^{2}.$ (2)
This formula is valid only if the efficiencies that the two kaons satisfy the
requirements are independent. However, owing to the variation of the RICH
efficiency with track multiplicity, correlations between the values of the
discriminant variable of the RICH are observed and are accounted for in the
systematic uncertainty.
## 4 Signal extraction
Simultaneous maximum likelihood fits to the $\phi$ candidate mass
distributions on the tag and the probe samples are performed in each bin of
$p_{T}$ and $y$ to extract the signal yields. The number of reconstructed
candidates without PID requirements
$N_{\mathrm{reco}}=N_{\mathrm{tag}}/\varepsilon_{\rm pid}$ is a free parameter
in the fit. A Breit-Wigner distribution convolved with a Gaussian resolution
function is used to describe the signal shape
$f_{\text{sig}}=\frac{1}{\left(m-m_{0}\right)^{2}+\frac{1}{4}\Gamma^{2}/c^{4}}\
\otimes\ \exp{\left(-\frac{1}{2}\frac{m^{\prime 2}}{\sigma^{2}}\right)}$ (3)
while the background shape is described by
$f_{\text{bkg}}=1-\exp{(c_{1}\cdot(m-c_{2}))}$ (4)
containing two free parameters.
The fitted $\phi$ mass and the Gaussian width $\sigma$ are common parameters
for both tag and probe sample, while the Breit-Wigner width $\Gamma$ is fixed
to the value 4.26 $\mathrm{\,Me\kern-1.00006ptV}$ taken from Ref. [15]. In
Fig. 1, fit results to the two samples in a given $p_{T}$/$y$ bin are shown
for illustration purposes.
Figure 1: Fit to the tag (left) and the probe (right) sample in the bin
$0.6<p_{T}<0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $3.34<y<3.52$ for one of
the two magnet polarities. Shown are the data points, the fit result (thick
solid line) as well as the signal (thin solid line) and the background
component (dash-dotted line).
## 5 Systematic uncertainties
The uncertainties in this analysis are dominated by systematic contributions,
divided into the ones which are common to all bins and the ones which vary
from bin to bin. The former are summarized in Table 1, whereas the latter are
plotted with the data in Figure 2 and listed in Table 2. The bin-dependent
uncertainties consist of the reconstruction efficiency uncertainty due to the
limited simulation sample size and to the modelling of a diffractive
contribution, as well as the uncertainty of the tag-and-probe PID
determination due to correlations. The combined uncertainties contribute 3–7%
for the statistically dominant bins.
Table 1: Summary of relative systematic uncertainties that are common to all bins. Source | | (%) |
---|---|---|---
Tracking efficiency | | 8 |
Luminosity (normalization) | | 4 |
Track multiplicity | | 3 |
Fit systematics | | 3 |
MC association | | 2 |
Doubly identified candidates | | 2 |
Branching fraction | | 1 |
Bin migration | | 1 |
Material interactions | | 1 |
Total | | 10 |
The largest shared systematics are the uncertainty on the tracking
efficiencies, which have been discussed in Ref. [16], and the luminosity
normalization. The track multiplicity in data is higher than in the
simulation. Studies of the track multiplicity dependence of the reconstruction
efficiency result in an uncertainty of 3% due to this multiplicity difference.
Two major effects contribute to the uncertainty due to the fit procedure.
Fixing the Gaussian width to the same value on tag-and-probe sample introduces
only a 1% systematic uncertainty, since the distribution is dominated by the
Breit-Wigner width. A larger systematic effect (2-3%) is observed when varying
the mass range of the fit, which results in a total uncertainty of 3%.
In the simulation, the reconstructed track is required to match the true
generated track to determine the reconstruction efficiency. A 2% uncertainty
is assigned due to this procedure. A small fraction of doubly identified
candidates is found: it is possible that the detector hits from one particle
are reconstructed as more than one track. The rate difference of these doubly
identified candidates between data and simulation is found to be 2%, which is
the systematic uncertainty assigned due to this effect. The $\phi\rightarrow
K^{+}K^{-}$ branching fraction contributes a 1% systematic uncertainty.
Migration of candidates between different bins due to resolution effects is
found to be small, and is accounted for by assigning a 1% uncertainty.
Uncertainties from the modelling of the material budget and the material
interaction cross-section are estimated to be 1%.
## 6 Results
The cross-sections determined with the two magnet polarities agree within
their statistical uncertainties. All results given here are unweighted
averages of the two samples. Comparisons to simulation samples generated with
two different Pythia tunings are made, namely Perugia 0 [17] and the LHCb
default Monte Carlo tuning.
The integrated cross-section in the region
$0.6<p_{T}<5.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\text{ and }2.44<y<4.06$ is
$\sigma(pp\rightarrow\phi X)=1758\pm
19\mathrm{(stat)}\,^{+43}_{-14}\mathrm{(syst)}\pm
182\mathrm{(scale)}\rm\,\upmu b,$
where the first systematic uncertainty arises from the bin-dependent
contribution, while the second one is the common systematic uncertainty, as
described in Section 5. The differential cross-section values are given in
Table 2 and projections on the $y$ and $p_{T}$ axes within the same kinematic
region are shown in Figure 2.
The simulations underestimate the measured $\phi$ production in the considered
kinematic region by a factor $1.43\pm 0.15$ (LHCb MC) and $2.06\pm 0.22$
(Perugia 0). Additionally, the shape of the $p_{T}$ spectrum and the slope in
the $y$ spectrum differ between the data and the simulation (see Fig. 2).
Fitting a straight line $\frac{\text{d}\sigma}{\text{d}y}=a\cdot y+b$ to the
$y$ spectrum, the slope is $a=-44\pm 27\rm\,\upmu b$ on data, but $a=-181\pm
2\rm\,\upmu b$ for the default LHCb MC tuning and $a=-149\pm 3\rm\,\upmu b$
for the Perugia 0 tuning. Uncertainties given on $a$ are statistical only.
The mean $p_{T}$ in the range
$0.6<p_{T}<5.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is $1.24\pm
0.01{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (data, stat. error only),
$1.077{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (LHCb MC) and
$1.238{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (Perugia 0 MC).
Figure 2: Inclusive differential $\phi$ production cross-section as a function
of $p_{T}$ (top) and $y$ (bottom), measured with data (points), and compared
to the LHCb default MC tuning (solid line) and Perugia 0 tuning (dashed line).
The error bars represent the statistical uncertainty, the braces show the bin
dependent systematic errors, the overall scale uncertainty from Table 1 is not
plotted. The lower parts of the plots show the ratio data cross-section over
Monte Carlo cross-section. Error bars in the ratio plots show statistical
uncertainties only. Table 2: Binned differential cross-section, in
${\rm\,\upmu b}/{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, as function of $p_{T}$
(GeV/c) and $y$. The statistical and the bin-dependent systematic
uncertainties are quoted. There is an additional bin-independent uncertainty
of 10% related to the normalization (Table 1).
$p_{T}$/ $y$ | 2.44-2.62 | 2.62-2.80 | 2.80-2.98
---|---|---|---
0.6-0.8 | $1.001\pm 0.140\,^{+0.076}_{-0.026}$ | $0.853\pm 0.114\,^{+0.081}_{-0.022}$ | $1.069\pm 0.108\,^{+0.093}_{-0.027}$
0.8-1.0 | $0.959\pm 0.112\,^{+0.129}_{-0.015}$ | $0.797\pm 0.084\,^{+0.074}_{-0.012}$ | $0.819\pm 0.079\,^{+0.053}_{-0.012}$
1.0-1.2 | $0.758\pm 0.043\,^{+0.089}_{-0.009}$ | $0.776\pm 0.038\,^{+0.063}_{-0.009}$ | $0.795\pm 0.026\,^{+0.042}_{-0.009}$
1.2-1.4 | $0.648\pm 0.033\,^{+0.067}_{-0.009}$ | $0.627\pm 0.028\,^{+0.049}_{-0.008}$ | $0.604\pm 0.026\,^{+0.024}_{-0.008}$
1.4-1.6 | $0.469\pm 0.023\,^{+0.037}_{-0.008}$ | $0.511\pm 0.022\,^{+0.033}_{-0.008}$ | $0.521\pm 0.022\,^{+0.023}_{-0.008}$
1.6-1.8 | $0.422\pm 0.020\,^{+0.039}_{-0.008}$ | $0.381\pm 0.017\,^{+0.021}_{-0.007}$ | $0.409\pm 0.018\,^{+0.015}_{-0.007}$
1.8-2.0 | $0.334\pm 0.016\,^{+0.027}_{-0.007}$ | $0.323\pm 0.015\,^{+0.014}_{-0.007}$ | $0.276\pm 0.012\,^{+0.009}_{-0.005}$
2.0-2.4 | $0.209\pm 0.008\,^{+0.010}_{-0.004}$ | $0.192\pm 0.007\,^{+0.006}_{-0.003}$ | $0.201\pm 0.007\,^{+0.003}_{-0.003}$
2.4-2.8 | $0.127\pm 0.005\,^{+0.003}_{-0.003}$ | $0.112\pm 0.005\,^{+0.002}_{-0.003}$ | $0.111\pm 0.004\,^{+0.002}_{-0.002}$
2.8-3.2 | $0.078\pm 0.004\,^{+0.002}_{-0.002}$ | $0.069\pm 0.003\,^{+0.002}_{-0.002}$ | $0.063\pm 0.003\,^{+0.002}_{-0.002}$
3.2-4.0 | $0.040\pm 0.002\,^{+0.001}_{-0.001}$ | $0.038\pm 0.002\,^{+0.001}_{-0.001}$ | $0.034\pm 0.001\,^{+0.001}_{-0.001}$
4.0-5.0 | $0.014\pm 0.001\,^{+0.001}_{-0.001}$ | $0.014\pm 0.001\,^{+0.001}_{-0.000}$ | $0.011\pm 0.001\,^{+0.000}_{-0.000}$
$p_{T}$/ $y$ | 2.98-3.16 | 3.16-3.34 | 3.34-3.52
---|---|---|---
0.6-0.8 | $1.171\pm 0.100\,^{+0.058}_{-0.029}$ | $1.060\pm 0.092\,^{+0.027}_{-0.043}$ | $1.131\pm 0.146\,^{+0.029}_{-0.176}$
0.8-1.0 | $1.032\pm 0.080\,^{+0.049}_{-0.015}$ | $0.862\pm 0.080\,^{+0.014}_{-0.013}$ | $1.170\pm 0.082\,^{+0.018}_{-0.058}$
1.0-1.2 | $0.818\pm 0.034\,^{+0.031}_{-0.009}$ | $0.851\pm 0.033\,^{+0.010}_{-0.010}$ | $0.781\pm 0.031\,^{+0.009}_{-0.009}$
1.2-1.4 | $0.648\pm 0.026\,^{+0.016}_{-0.008}$ | $0.693\pm 0.026\,^{+0.009}_{-0.008}$ | $0.661\pm 0.023\,^{+0.011}_{-0.008}$
1.4-1.6 | $0.484\pm 0.019\,^{+0.013}_{-0.006}$ | $0.499\pm 0.018\,^{+0.009}_{-0.007}$ | $0.470\pm 0.017\,^{+0.013}_{-0.006}$
1.6-1.8 | $0.408\pm 0.016\,^{+0.008}_{-0.007}$ | $0.382\pm 0.015\,^{+0.008}_{-0.006}$ | $0.348\pm 0.013\,^{+0.009}_{-0.005}$
1.8-2.0 | $0.320\pm 0.014\,^{+0.006}_{-0.007}$ | $0.308\pm 0.008\,^{+0.009}_{-0.006}$ | $0.255\pm 0.010\,^{+0.009}_{-0.004}$
2.0-2.4 | $0.206\pm 0.006\,^{+0.004}_{-0.004}$ | $0.194\pm 0.006\,^{+0.006}_{-0.003}$ | $0.169\pm 0.005\,^{+0.005}_{-0.003}$
2.4-2.8 | $0.109\pm 0.004\,^{+0.003}_{-0.002}$ | $0.106\pm 0.004\,^{+0.003}_{-0.002}$ | $0.106\pm 0.004\,^{+0.005}_{-0.002}$
2.8-3.2 | $0.065\pm 0.003\,^{+0.002}_{-0.002}$ | $0.057\pm 0.003\,^{+0.002}_{-0.001}$ | $0.053\pm 0.003\,^{+0.003}_{-0.001}$
3.2-4.0 | $0.031\pm 0.001\,^{+0.001}_{-0.001}$ | $0.029\pm 0.001\,^{+0.001}_{-0.001}$ | $0.025\pm 0.002\,^{+0.001}_{-0.001}$
4.0-5.0 | $0.010\pm 0.001\,^{+0.001}_{-0.000}$ | $0.010\pm 0.001\,^{+0.000}_{-0.000}$ | $0.009\pm 0.001\,^{+0.000}_{-0.000}$
$p_{T}$/ $y$ | 3.52-3.70 | 3.70-3.88 | 3.88-4.06
---|---|---|---
0.6-0.8 | $1.341\pm 0.158\,^{+0.034}_{-0.207}$ | $1.164\pm 0.157\,^{+0.030}_{-0.065}$ | $1.341\pm 0.193\,^{+0.120}_{-0.036}$
0.8-1.0 | $0.816\pm 0.075\,^{+0.013}_{-0.035}$ | $1.065\pm 0.075\,^{+0.018}_{-0.059}$ | $0.975\pm 0.115\,^{+0.018}_{-0.070}$
1.0-1.2 | $0.785\pm 0.032\,^{+0.010}_{-0.012}$ | $0.690\pm 0.031\,^{+0.010}_{-0.011}$ | $0.760\pm 0.039\,^{+0.013}_{-0.039}$
1.2-1.4 | $0.609\pm 0.023\,^{+0.012}_{-0.008}$ | $0.561\pm 0.022\,^{+0.010}_{-0.008}$ | $0.531\pm 0.027\,^{+0.012}_{-0.010}$
1.4-1.6 | $0.484\pm 0.018\,^{+0.016}_{-0.007}$ | $0.433\pm 0.017\,^{+0.011}_{-0.007}$ | $0.409\pm 0.021\,^{+0.016}_{-0.008}$
1.6-1.8 | $0.336\pm 0.013\,^{+0.008}_{-0.006}$ | $0.315\pm 0.014\,^{+0.011}_{-0.006}$ | $0.279\pm 0.014\,^{+0.011}_{-0.006}$
1.8-2.0 | $0.231\pm 0.010\,^{+0.006}_{-0.004}$ | $0.228\pm 0.011\,^{+0.009}_{-0.005}$ | $0.213\pm 0.011\,^{+0.007}_{-0.005}$
2.0-2.4 | $0.164\pm 0.005\,^{+0.007}_{-0.003}$ | $0.140\pm 0.005\,^{+0.006}_{-0.002}$ | $0.131\pm 0.006\,^{+0.003}_{-0.003}$
2.4-2.8 | $0.082\pm 0.002\,^{+0.004}_{-0.002}$ | $0.078\pm 0.004\,^{+0.003}_{-0.002}$ | $0.070\pm 0.004\,^{+0.004}_{-0.002}$
2.8-3.2 | $0.059\pm 0.003\,^{+0.004}_{-0.002}$ | $0.049\pm 0.003\,^{+0.002}_{-0.001}$ | $0.039\pm 0.003\,^{+0.006}_{-0.001}$
3.2-4.0 | $0.022\pm 0.001\,^{+0.001}_{-0.001}$ | $0.019\pm 0.001\,^{+0.002}_{-0.000}$ | $0.022\pm 0.002\,^{+0.003}_{-0.001}$
4.0-5.0 | $0.008\pm 0.001\,^{+0.001}_{-0.000}$ | $0.007\pm 0.001\,^{+0.001}_{-0.000}$ | $0.007\pm 0.002\,^{+0.000}_{-0.002}$
## 7 Conclusions
A study of inclusive $\phi$ production in $p{\kern-0.50003pt}p$ collisions at
a centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$ at the Large
Hadron Collider is reported. The differential cross-section as a function of
$p_{T}$ and $y$ measured in the range
$0.6<p_{T}<5.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\text{ and }2.44<y<4.06$ is
$\sigma(pp\rightarrow\phi X)=1758\pm
19\mathrm{(stat)}\,^{+43}_{-14}\mathrm{(syst)}\pm
182\mathrm{(scale)}\rm\,\upmu b$, where the first systematic uncertainty
depends on the $p_{T}$ and $y$ scale and the second is related to the overall
scale. Predictions based on the Pythia 6.4 generator underestimate the cross-
section.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Région Auvergne.
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|
arxiv-papers
| 2011-07-20T10:34:01 |
2024-09-04T02:49:20.750972
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Anderson, R.B.\n Appleby, O. Aquines Gutierrez, L. Arrabito, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer,\n Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, 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, S. Brisbane, M. Britsch,\n T. Britton, N.H. Brook, A. B\\\"uchler-Germann, A. Bursche, J. Buytaert, S.\n Cadeddu, J.M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, X. Cid Vidal, P.E.L. Clarke, M. Clemencic, H.V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, F. Constantin, G.\n Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G.A. Cowan, R. Currie, B.\n D'Almagne, C. D'Ambrosio, P. David, I. De Bonis, S. De Capua, M. De Cian, F.\n De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del Buono, C. Deplano, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, D. Dossett,\n A. Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L. Est\\`eve, A.\n Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, N. Gauvin, 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.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, W.\n Hofmann, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S.\n Huston, 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, S. Kandybei,\n M. Karacson, T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, Y.M. Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, S. Kukulak, R. Kumar, 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, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.Y.\n Li, L. Li Gioi, M. Lieng, R. Lindner, C. Linn, B. Liu, G. Liu, J.H. Lopes, E.\n Lopez Asamar, N. Lopez-March, J. Luisier, F. Machefert, I.V. Machikhiliyan,\n F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel,\n R. Messi, S. Miglioranzi, D.A. Milanes, M.-N. Minard, S. Monteil, D. Moran,\n P. Morawski, J.V. Morris, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R.\n Muresan, B. Muryn, M. Musy, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli,\n I. Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S.\n Nies, V. Niess, N. Nikitin, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, B.\n Pal, J. Palacios, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n A. Petrella, A. Petrolini, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, J. Prisciandaro,\n V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n I. Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, K.\n Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G.J. Rogers, V. Romanovsky, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti,\n C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, N. Serra,\n J. Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, H.P. Skottowe, T. Skwarnicki, A.C. Smith, N.A. Smith, K.\n Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, U. Straumann, N. Styles, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, M.T.\n Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja, P. Urquijo, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, K. Vervink, B. Viaud, I. Videau, X. Vilasis-Cardona, J.\n Visniakov, A. Vollhardt, D. Voong, A. Vorobyev, H. Voss, K. Wacker, S.\n Wandernoth, J. Wang, D.R. Ward, A.D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Yang, R. Young, O. Yushchenko, M. Zavertyaev, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Sebastian Schleich",
"url": "https://arxiv.org/abs/1107.3935"
}
|
1107.4017
|
# Phase transitions of H2 adsorbed on the surface of single carbon nanotubes
M.C. Gordillo Departamento de Sistemas Físicos, Químicos y Naturales,
Facultad de Ciencias Experimentales, Universidad Pablo de Olavide, Carretera
de Utrera, km 1. 41013 Sevilla, Spain J. Boronat Departament de Física i
Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5,
08034 Barcelona, Spain
###### Abstract
By means of Diffusion Monte Carlo calculations, we obtained the complete phase
diagrams of H2 adsorbed on the outer surface of isolated armchair carbon
nanotubes of radii ranging from 3.42 to 10.85 Å. We only considered density
ranges corresponding to the filling of the first adsorption layer in these
curved structures. In all cases, the zero-temperature ground state was found
to be an incommensurate solid, except in the widest tube, in which the
structure with lowest energy is an analogous of the $\sqrt{3}\times\sqrt{3}$
phase found in planar substrates. Those incommensurate solids result form the
arrangement of the hydrogen molecules in circumferences whose plane is
perpendicular to the main axis of the carbon nanotube. For each tube, there is
only one of such phases stable in the density range considered, except in the
case of the (5,5) and (6,6) tubes, in which two of these incommensurate solids
are separated by novel first order phase transitions.
###### pacs:
67.25.dp,05.30.Jp,87.80.bd,68.90.+g
## I Introduction
A carbon nanotube is a cylindrical structure Iijima that can be thought as
the result of the wrapping up of a single graphene sheet science2004 ;
pnas2005 over itself. In the same way that , in the past , graphene sheets
were always found stacked to form graphite, carbon nanotubes were always found
forming nanotube bundles, structures in which the only surfaces available for
adsorption were the exposed parts of the tubes in the fringes of the bundle
talapatra ; prl2006 ; prb2007 . The adsorbed phases in a planar structure such
as graphite are quite different from the ones observed on a highly patterned
substrate like the external surface of a nanotube bundle. This can be seen by
comparing the results on a bundle of carbon nanotubes and graphite for 4He
glyde ; prl2008 (bundles) grey2 ; prl2009 (graphite), H2 vilchesjltp1 ;
vilchesjltp2 (bundles) frei1 ; frei3 ; prb2010 (graphite), and Ne talapatra2
; vilchesne (bundles) and tiby ; colebook (graphite). Similar to the
opportunities for adsorption that the achievement of a single graphene sheet
offers, a recent experimental work shows the proper technique to suspend a
single carbon nanotube and study the phases of different gases (Ar,Kr)
adsorbed on its external surface vilchesscience . In this latter work, the
results indicated that the phase diagrams were similar to those of the same
gases on graphite, but with phase transitions shifted to higher pressures. In
the present work, we have studied the adsorption behavior of H2 on a series of
armchair ((n,n)) carbon nanotubes with different diameters in order to see if
a quantum gas would exhibit novel phases when adsorbed on curved surfaces. The
carbon nanotubes considered, together with their indexes and radii, are given
in Table I. The most noticeable results driven from our calculations are: i),
the existence of a solid-solid phase transition for (5,5) and (6,6) nanotubes,
and ii), the stability of the curved version of the $\sqrt{3}\times\sqrt{3}$
commensurate phase only for the widest nanotube studied (16,16).
## II Method
Our calculations were performed with the diffusion Monte Carlo (DMC) method.
This well-established technique allows us to solve exactly the Schroedinger
equation within some statistical errors to obtain the ground state energy of a
system of interacting bosons boro94 . This is exactly our case, since in its
lowest energy state ($para$-H2) the hydrogen molecule behaves as a boson. To
apply the DMC algorithm, we need the potentials to describe the H2-H2
interaction (we chose the Silvera and Goldman expression silvera , a standard
model for calculations involving $para$-H2), and the C-H2 one. For the latter,
we used a Lennard-Jones potential with parameters taken from Ref. coleh2, , as
used previously to describe the phase diagram of H2 adsorbed on graphene
prb2010 . All the individual C-H2 pairs were considered, i.e., we took into
consideration the corrugation effects due to the real structure of the
nanotube.
The last ingredient needed to perform a calculation in a DMC scheme is a trial
wave function. It regulates the Monte Carlo importance sampling, and can be
considered as a variational approximation to the exact description of the
system. In this work, we used a trial wave function formed by the product of
two terms. The first one is
$\displaystyle\Phi({\bf r}_{1},{\bf r}_{2},\ldots,{\bf
r}_{N})=\prod_{i<j}\exp\left[-\frac{1}{2}\left(\frac{b_{{\text{H}}_{2}{\text{-}}{\text{H}}_{2}}}{r_{ij}}\right)^{5}\right]$
$\displaystyle\times\prod_{i}\prod_{J}\exp\left[-\frac{1}{2}\left(\frac{b_{{\text{C}}{\text{-}}{\text{H}}_{2}}}{r_{iJ}}\right)^{5}\right]\prod_{i}\exp(-a(r_{i}-r_{0})^{2})\
,$
where ${\bf r}_{1}$, ${\bf r}_{2}$, …, ${\bf r}_{N}$ are the coordinates of
the H2 molecules and ${\bf r}_{J}$ the position of the carbon atoms in the
nanotube. The first term in Eq. (II) is a two-body Jastrow function depending
on the H2 intermolecular distances $r_{ij}$, while the second one has the same
meaning but for each $r_{iJ}$ (C-H2). Finally, the third term is a product of
one-body Gaussians with $r_{i}=\sqrt{x_{i}^{2}+y_{i}^{2}}$, that depend on the
distance of each H2 to the center of the tube ($r_{0}$).
Table 1: Armchair carbon nanotubes considered in this work, together with their tube radii ($r_{t}$) and the most probable distance for an adsorbed hydrogen molecule to the center of the cylinder ($r_{0}$). In the last column, the adsorption energy of a single H2 molecule ($e_{0}$) is reported. Tube | $r_{t}$ (Å) | $r_{0}$ (Å) | e0 (K)
---|---|---|---
(5,5) | 3.42 | 6.36 | -320.6 $\pm$ 0.1
(6,6) | 4.10 | 7.05 | -329.0 $\pm$ 0.1
(7,7) | 4.75 | 7.70 | -335.5 $\pm$ 0.1
(8,8) | 5.45 | 8.39 | -341.0 $\pm$ 0.1
(10,10) | 6.80 | 9.75 | -349.2 $\pm$ 0.2
(12,12) | 8.14 | 11.10 | -353.7 $\pm$ 0.2
(14,14) | 9.49 | 12.46 | -356.5 $\pm$ 0.1
(16,16) | 10.85 | 13.80 | -357.2 $\pm$ 0.2
Equation II is appropriate for describing liquid phases, but if the system is
a solid, we need to multiply the trial wave function above (II) by
$\prod_{i}\exp\left\\{-c\left[(x_{i}-x_{\text{site}})^{2}+(y_{i}-y_{\text{site}})^{2}+(z_{i}-z_{\text{site}})^{2}\right]\right\\}\
,$ (2)
whose purpose is to limit the location of the hydrogen molecules to regions
close to the $x_{\text{site}},y_{\text{site}},z_{\text{site}}$ regularly
distributed coordinates of a solid arrangement. The variational parameters
appearing in the whole trial wave function ($b_{{\text{H}}_{2}\text{-H}_{2}}$,
$b_{\text{C-H}_{2}}$, $a$ and $c$) were fixed to be the same values than the
ones for H2 adsorbed on graphene prb2010 , after having checked that there
were no appreciable differences when variationally optimized for the (5,5)
tube, i.e. $b_{{\text{H}}_{2}\text{-H}_{2}}=3.195$ Å, $b_{\text{C-H}_{2}}$ =
2.3 Åand $a$ = 3.06 Å-2. Parameter $c$ was varied with density in the same way
than for graphene. The only remaining parameter, $r_{0}$, was optimized
independently for each tube; the results are listed in Table I.
## III Results
Table I contains also the binding energies ($e_{0}$) of a single H2 molecule
on top of the series of nanotubes considered in this work. We observe, quite
reasonably, that this energy increases with the radius of the cylinder.
However, even the value for the (16,16) tube is quite far from the result on a
flat graphene sheet (-431.79 $\pm$ 0.06 K prb2010 ), due to the surface
curvature, that distorts the C-H2 distances with respect to those of a flat
graphene sheet.
Our aim in the present work is to describe the possible phases of H2 adsorbed
on the surface of different (n,n) nanotubes. To do so, we considered the
curved counterparts of all the commensurate solid phases found experimentally
for most of the quantum gases (4He, H2 and D2) on graphite grey2 ; frei1 ;
frei3 ; w , which included the curved version of a $\sqrt{3}\times\sqrt{3}$
phase, perfectly possible in these (n,n) tube substrates vilchesscience . For
the incommensurate phases, we tried structures similar to the triangular
phases found in graphene, but wrapped up around to form hydrogen cylinders of
radius $r_{0}$ (see Table I). One of these structures is shown in Fig. 1,
which displays the projections on a plane of the H2 site locations for an
incommensurate solid wrapped around a (5,5) tube. Here, the $r$ coordinate
represents the hydrogen positions on a circumference of radius $r_{0}$ = 6.36
Å, while the $z$ axis is chosen parallel to the main axis of the tube. One can
see that the solid is built by locating five H2 molecules in circumferences on
planes perpendicular to $z$, and rotating the molecules in neighboring
circumferences half the distance between H2 molecules. We defined a phase by
the number of hydrogen molecules in one of such circumferences, and varied
their density by changing the distance between them. Alternatively, this
structure can be thought as the result of having five helices of pitch 17
Åwrapped around the tube, each one of them including 10 molecules per turn of
the helix.
Figure 1: Projection on a flat surface of an incommensurate structure
corresponding to an areal density of 0.076 Å-2 to be wrapped around a (5,5)
tube. It can be described as an set of five intertwined helices whose pitch
(indicated by an arrow) is 17 Å. Every turn of the helix contains 10 H2
molecules. Figure 2: Energies per H2 molecule for different adsorbed phases
on a (5,5) tube. We display here the liquid (open squares), the different
incommensurate solids: five-in-a-row (solid circles), six-in-a row (open
circles), four-in-a-row (open triangles) and seven-in-a-row (full triangles).
DMC results for the equations of state corresponding to different phases of H2
adsorbed on a (5,5) tube are shown in Fig. 2. Open squares indicate a liquid
phase (obtained by considering as a trial function only Eq. (II)), while the
circles represent solid incommensurate phases with five (full circles) and six
(open circles) H2 molecules per row. We display with triangles results for a
four molecules per row (open), and seven molecules per row (full)
arrangements. For all the areal densities considered (up to 0.0937 Å-2, the
experimental density for a second layer promotion in planar graphite colebook
), these last two phases are unstable (of higher energy) with respect to the
first two. All the alternative commensurate structures are also unstable, as
can be seen in Table II and III. In that table, we also display their areal
densities, different for different tubes, even though the pattern of the
adsorbed hydrogen molecules is the same. This is due to the fact that the
areal densities depend on $r_{0}$, while these phases are registered with
respect to structures whose dimensions depend on $r_{t}$. Other solid
incommensurate structures, such as helices with different number of molecules
per turn but with the same pitch, i.e., more or less H2 molecules on top of
the dashed lines in Fig. 1, have energies greater than the phases represented
by the circles in Fig. 2. For instance, a structure with eleven molecules per
turn, instead of the ten displayed in Fig. 1, has a minimum energy of -336.0
$\pm$ 0.1 K, for a density of 0.062 $\pm$ 0.003 Å-2. Therefore, the ground
state of H2 adsorbed in the outer surface of a (5,5) carbon nanotube is an
incommensurate solid with five H2’s per row. Upon a density increase, there is
a first-order solid-solid phase transition to another incommensurate solid,
similar to the first one but with six atoms per row. The equilibrium densities
for both structures at the transition are obtained from a double tangent
Maxwell construction: 0.0685 $\pm$ 0.0001 Å-2 (five molecules per row, energy
per hydrogen molecule -345.3 $\pm$ 0.1 K) and 0.0795 $\pm$ 0.0001 Å-2 (six
intertwined helices, with energy per hydrogen molecule -334.7 $\pm$ 0.1 K).
Table 2: Energies per particle ($e_{b}$) and equilibrium densities ($\rho$) for different phases proposed in the literature grey2 for quantum gases on graphite, when adsorbed in tubes of different radii. The liquid and incommensurate helical densities are the values corresponding to the minimum energies obtained by means of third-degree polynomial fits to curves of the type displayed in Fig. 1. The rest correspond to exact densities. Error bars within parenthesis represent the uncertainty of the last figure shown. Phase | | liquid | | 2/5 | | 3/7 | | $\sqrt{3}\times\sqrt{3}$ | | helical incommensurate
---|---|---|---|---|---|---|---|---|---|---
Tube | $\rho$ (Å-2) | $e_{b}$ (K) | $\rho$ (Å-2) | $e_{b}$ (K) | $\rho$ (Å-2) | $e_{b}$ (K) | $\rho$ (Å-2) | $e_{b}$ (K) | $\rho$ (Å-2) | $e_{b}$ (K)
(5,5) | 0.0574(4) | -340.3(1) | 0.0407 | -333.7(2) | 0.0436 | -325.4(2) | 0.0339 | -329.03(4) | 0.0621(3) | -349.0(2)
(6,6) | 0.0586(3) | -349.70(9) | 0.0441 | -344.4(1) | 0.0472 | -334.3(2) | 0.0367 | -339.52(6) | 0.0673(1) | -360.34(4)
(7,7) | 0.0615(6) | -356.2(2) | 0.0471 | -352.28(7) | 0.0504 | -339.7(2) | 0.0392 | -347.66(3) | 0.0716(1) | -367.89(5)
(8,8) | 0.0611(7) | -360.3(2) | 0.0494 | -358.9(1) | 0.0529 | -343.4(2) | 0.0411 | -354.44(3) | 0.0742(1) | -372.43(3)
(10,10) | 0.0578(3) | -365.95(6) | 0.0531 | -367.11(8) | 0.0569 | -347.5(2) | 0.0443 | -364.12(5) | 0.0758(1) | -375.6(1)
(12,12) | 0.0565(4) | -369.1(1) | 0.0559 | -371.8(1) | 0.0600 | -346.3(3) | 0.0466 | -369.93(5) | 0.0703(2) | -375.96(5)
(14,14) | 0.0530(4) | -370.4(1) | 0.0581 | -373.5(1) | 0.0623 | -342.5(3) | 0.0485 | -373.10(8) | 0.0681(2) | -375.3(1)
(16,16) | 0.0511(8) | -367.57(9) | 0.0600 | -371.17(5) | 0.0643 | -333.8(3) | 0.0500 | -371.7(1) | 0.0660(1) | -371.4(1)
Figure 3: Equation of state for the incommensurate solids adsorbed in carbon
nanotubes of increasing radii: (5,5), (open squares, five-in-a-row solid; full
squares, six-in-a-row arrangement); (6,6), (open circles, six-in-a-row
structure; full circles, seven-in-a-row solid); (7,7), (open triangles);
(8,8), (full triangles); (10,10), (inverted triangles). See further
explanation in the text.
The stable phases of H2 on other (n,n) tubes, up to n = 14, are similar to the
(5,5) one. In all cases, the ground states are the same type of incommensurate
solids already described, the only difference being the number of intertwined
helices (or molecules in the same circumference) forming the structure. This
number was found to be always equal to the index $n$ of the nanotube. In Fig.
3, we show the equation of state for some of such solids, from the already
displayed (5,5) case (open circles; five molecules in a row), to the (10,10)
one (inverted open triangles; ten molecules in a row), going through the (6,6)
(open circles), (7,7) (open triangles) and (8,8) (full triangles) tubes. In
all cases the dashed lines are mere guides-to-the-eye. Between the two
narrowest cylinders and the rest there is an important difference, though: the
(6,6) tube exhibits a first order phase transition between two incommensurate
solids with six and seven molecules per row while, while for the other tubes,
there is only one stable incommensurate structure in the areal density range
corresponding to an adsorbed monolayer The zero-pressure densities and
energies corresponding to the ground state for all the tubes considered are
given in Table II. There, and in Table III, we can see that those
incommensurate solids are the truly ground states for the systems under
consideration at zero pressure, since their energies are lower than the
corresponding to any of the commensurate structures on the same tubes. The
only exception is the (16,16) nanotube, in which the ground state is the same
$\sqrt{3}\times\sqrt{3}$ structure than in a flat surface. The stability
limits for the two solids in the (6,6) tube are 0.0755 $\pm$ 0.0001 Å-2
($e_{b}$ = -338.2 $\pm$ 0.1 K) and 0.0855 $\pm$ 0.0001 Å-2 ($e_{b}$ = -354.1
$\pm$ 0.1 K), also obtained by a Maxwell construction. The $1\times\sqrt{3}$
structure defined in Ref. colesolido, for (n,0) nanotubes was found also to
be unstable for all the tubes considered. For instance, for the (16,16) tube,
the density of this phase is 0.0750 Å-2, with a binding energy of -303.9 $\pm$
0.6 K.
Table 3: Same than in Table II, but for two other commensurate structures found experimentally for D2 on graphite. The dimensions of their unit cells make them only possible for the tubes shown. In all cases, they are unstable with respect to the incommensurate structures (n$\leq$ 14) or to the $\sqrt{3}\times\sqrt{3}$ one (n= 16). | 7/16corboz ; w |
---|---|---
Tube | $\rho$ (Å-2) | $e_{b}$ (K)
(8,8) | 0.0540 | -359.4 $\pm$ 0.1
(12,12) | 0.0612 | -371.6 $\pm$ 0.1
(16,16) | 0.0656 | -369.9 $\pm$ 0.1
| 31/75 ($\delta$ phase in H2 on graphite w ) |
Tube | $\rho$ | $e_{b}$
(5,5) | 0.0421 | -334.8 $\pm$ 0.1
(10,10) | 0.0549 | -367.0 $\pm$ 0.1
## IV Concluding remarks
Summarizing, we studied all the possible H2 phases adsorbed on the surface of
armchair carbon nanotubes ranging from (5,5) to (16,16). In all cases, but the
last one, we have found that the stable phases are incommensurate solids
formed by molecules adsorbed on circumferences perpendicular to the main tube
axis. Where the curvature of the surface is more relevant, i.e., in the
narrowest tubes [(5,5),(6,6)], our results show the existence of a solid-solid
zero-temperature phase transition between two incommensurate structures. The
first commensurate solid phase, curved version of the well-known
$\sqrt{3}\times\sqrt{3}$ phase, appears only when the radius of the nanotube
is big enough: a (16,16) tube for H2.
###### Acknowledgements.
We acknowledge partial financial support from the Junta de Andalucía group
PAI-205 and grant FQM-5985, DGI (Spain) Grants No. FIS2010-18356 and
FIS2008-04403, and Generalitat de Catalunya Grant No. 2009SGR-1003.
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|
arxiv-papers
| 2011-07-20T16:04:00 |
2024-09-04T02:49:20.761622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. C. Gordillo, J. Boronat",
"submitter": "Jordi Boronat",
"url": "https://arxiv.org/abs/1107.4017"
}
|
1107.4018
|
# Supremum of Perelman’s entropy and Kähler-Ricci flow on a Fano manifold
Gang Tian , Shijin Zhang , Zhenlei $\text{Zhang}^{*}$ and Xiaohua
$\text{Zhu}^{**}$ Gang Tian
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
and Department of Mathematics, Princeton University, New Jersey, NJ 02139, USA
tian@math.mit.edu Shijin Zhang, School of Mathematical Sciences and BICMR,
Peking University, Beijing, 100871, P.R.China, shijin_zhang@yahoo.com Zhenlei
Zhang
Department of Mathematics, Beijing Capital Normal University, Beijing, China
zhleigo@yahoo.com.cn Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
xhzhu@math.pku.edu.cn
###### Abstract.
In this paper, we extend the method in [TZhu5] to study the energy level
$L(\cdot)$ of Perelman’s entropy $\lambda(\cdot)$ for Kähler-Ricci flow on a
Fano manifold. Consequently, we first compute the supremum of $\lambda(\cdot)$
in Kähler class $2\pi c_{1}(M)$ under an assumption that the modified
Mabuchi’s K-energy $\mu(\cdot)$ defined in [TZhu2] is bounded from below.
Secondly, we give an alternative proof to the main theorem about the
convergence of Kähler-Ricci flow in [TZhu3].
###### Key words and phrases:
Kähler-Ricci flow, Kähler-Ricci solitons, Perelman’s entropy
###### 1991 Mathematics Subject Classification:
Primary: 53C25; Secondary: 53C55, 58E11
* Partially supported by a grant of BMCE 11224010007 in China
** Partially supported by the NSFC Grant 10990013
## Introduction
In this paper, we extend the method in [TZhu5] to study the energy level
$L(\cdot)$ of Perelman’s entropy $\lambda(\cdot)$ for Kähler-Ricci flow on an
$n$-dimensional compact Kähler manifold $(M,J)$ with positive first Chern
class $c_{1}(M)>0$ (namely called a Fano manifold). We will show that
$L(\cdot)$ is independent of choice of initial Kähler metrics in $2\pi
c_{1}(M)$ under an assumption that the modified Mabuchi’s K-energy
$\mu(\cdot)$ is bounded from below (cf. Proposition 3.1 in Section 3). The
modified Mabuchi’s K-energy $\mu(\cdot)$ is a generalization of Mabuchi’s
K-energy. It was showed in [TZhu2] that $\mu(\cdot)$ is bounded from below if
$M$ admits a Kähler-Ricci soliton.
As an application of Proposition 3.1, we first compute the supremum of
Perelman’s entropy $\lambda(\cdot)$ in Kähler class $2\pi c_{1}(M)$ [Pe]. More
precisely, we prove that
###### Theorem 0.1.
Suppose that the modified Mabuchi’s K-energy is bounded from below. Then
(0.1)
$\sup\\{\lambda({g^{\prime}})|~{}g^{\prime}\in\mathcal{K}_{X}\\}=(2\pi)^{-n}[nV-
N_{X}(c_{1}(M))].$
Here the quantity $N_{X}(c_{1}(M))$ is a nonnegative invariance in
$\mathcal{K}_{X}$ and it is zero iff the Futaki-invariant vanishes [Fu]. We
denote $\mathcal{K}_{X}$ to be a class of $K_{X}$-invariant Kähler metrics in
$2\pi c_{1}(M)$, where $K_{X}$ is an one-parameter compact subgroup of
holomorphisms transformation group generated by an extremal holomorphic vector
field $X$ for Kähler-Ricci solitons on $M$ [TZhu2]. We note that we do not
need to assume an existence of Kähler-Ricci solitons in Theorem 0.1. In fact,
if we assume the existence of Kähler-Ricci solitons then we can use a more
direct way to prove Theorem 0.1 and that the supremum of $\lambda(\cdot)$ can
be achieved in $\mathcal{K}_{X}$ (cf. Section 1). It seems that the supremum
of $\lambda(\cdot)$ can be achieved in the total space of Kähler potentials in
$2\pi c_{1}(M)$ if $M$ admits a Kähler-Ricci soliton. In a special case of
small neighborhood of a Kähler-Ricci soliton the positivity has been verified
by computing the second variation of $\lambda(\cdot)$ in [TZhu4].
As another application of Proposition 3.1, we prove the following convergence
result for Kähler-Ricci flow.
###### Theorem 0.2.
Let $(M,J)$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Then Kähler-Ricci flow with any initial Kähler metric in
$\mathcal{K}_{X}$ will converge to a Kähler-Ricci soliton in $C^{\infty}$ in
the sense of Kähler potentials. Moreover, the convergence can be made
exponentially.
We note that that without loss of generality we may assume that a Kähler-Ricci
soliton $g_{KS}$ on $M$ is corresponding to the above $X$ (cf. [TZhu1],
[TZhu2]). Theorem 0.2 was first proved by Tian and Zhu in [TZhu3] by using an
inequality of Moser-Trudinger type established in [CTZ]111We need to add more
details about how to use the Moser-Trudinger typed inequality in general
case.. Here we will modify arguments in [TZhu5] in our general case that
$(M,J)$ admits a Kähler-Ricci soliton to give an alternative proof of this
theorem. This new proof does not use such an inequality of Moser-Trudinger
type. Moreover, in particular, in case that $(M,J)$ admits a Kähler-Einstein
metric this new proof allows us to avoid to use a deep result recentlly proved
by Chen and Sun in [CS] for the uniqueness of Kähler-Einsteins in the sense of
orbit space to give a self-contained proof to the main theorem in [TZhu5].
The organization of paper is as follows. In Section 1, we discuss an upper
bound of $\lambda(\cdot)$ in general case-without any condition for
$\mu(\cdot)$ and show that the quantity $(2\pi)^{-n}[nV-N_{X}(c_{1}(M))]$ is
an upper bound of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ (cf. Proposition 1.4).
In Section 2, we will summarize to give some estimates for modified Ricci
potentials of evolved Kähler metrics along Kähler-Ricci flow (cf. Proposition
2.3). In Section 3, we prove Proposition 3.1 and so do Theorem 0.1. Theorem
0.2 will be proved in Section 6. In Section 4, we improve our key Lemma 3.2 in
Section 3 independent of time $t$ (cf. Proposition 4.2). Section 5 is a
discussion about an upper bound of $\lambda(\cdot)$ in $\mathcal{K}_{Y}$ for a
general holomorphic vector field $Y\in\eta_{r}(M)$. Section 7 is an appendix
where we discuss the gradient estimate and Laplace estimate for the minimizers
of Perelman’s $W$-functional along the Kähler-Ricci flow.
## 1\. An upper bound of $\lambda(\cdot)$
In this section, we first review Perelman’s $W$-functional for triples
$(g,f,\tau)$ on a closed $m$-demensional Riemannian manifold $M$ (cf. [Pe],
[TZhu5]). Here $g$ is a Riemannian metric, $f$ is a smooth function and $\tau$
is a constant. In our situation, we will normalize volume of $g$ by
(1.1) $\int_{M}dV_{g}\equiv\,V$
and so we can fix $\tau$ by $\frac{1}{2}$. Then the $W$-functional depends
only on a pair $(g,f)$ and it can be reexpressed as follows:
(1.2) $\displaystyle W(g,f)=(2\pi)^{-m/2}\int_{M}[\frac{1}{2}(R(g)+|\nabla
f|^{2})+f]e^{-f}dV_{g},$
where $R(g)$ is a scalar curvature of $g$ and $(g,f)$ satisfies a
normalization condition
(1.3) $\int_{M}e^{-f}dV_{g}\,=V.$
Then Perelman’s entropy $\lambda(g)$ is defined by
$\lambda(g)=\inf_{f}\\{W(g,f)|~{}(g,f)~{}\text{satisfies}~{}(\ref{norm-1})\\}.$
It is well known that $\lambda(g)$ can be attained by some smooth function $f$
(cf. [Ro]). In fact, such a $f$ satisfies the Euler-Lagrange equation of
$W(g,\cdot)$,
(1.4) $\triangle f+f+\frac{1}{2}(R-|\nabla
f|^{2})=(2\pi)^{m/2}V^{-1}\lambda(g).$
Following Perelman’s computation in [Pe], we can deduce the first variation of
$\lambda(g)$,
(1.5) $\delta\lambda(g)=-(2\pi)^{-m/2}\int_{M}<\delta
g,\mathrm{Ric}(g)-g+\nabla^{2}f>e^{-f}dV_{g},$
where $\mathrm{Ric}(g)$ denotes the Ricci tensor of $g$ and $\nabla^{2}f$ is
the Hessian of $f$. Hence, $g$ is a critical point of $\lambda(\cdot)$ if and
only if $g$ is a gradient shrinking Ricci-soliton which satisfies
(1.6) $\mathrm{Ric}(g)+\nabla^{2}f=g,$
where $f$ is a minimizer of $W(g,\cdot)$. The following lemma was proved in
[TZhu5] for the uniqueness of solutions (1.4) when $g$ is a gradient shrinking
Ricci soliton.
###### Lemma 1.1.
If $g$ satisfies (1.6) for some $f$, then any solution of (1.4) is equal to
$f$ modulo a constant. Consequently, a minimizer of $W(g,\cdot)$ is unique if
the metric $g$ is a gradient shrinking Ricci-soliton. Conversely, if $f$ is a
function in (1.6) for $g$, then $f$ satisfies (1.4).
In case that $(M,J)$ is an $n$-dimensional Fano manifold, for any Kähler
metric $g$ in $2\pi c_{1}(M)$, (1.1) is equal to
(1.7)
$\int_{M}dV_{g}=\int_{M}\omega_{g}^{n}\,=\,(2\pi)^{n}\int_{M}c_{1}(M)^{n}\,\equiv\,V.$
Moreover, (1.6) becomes an equation for Kähler-Ricci solitons,
$Ric(\omega_{g})-\omega_{g}=\mathrm{L}_{X}\omega_{g},$
where $Ric(\omega_{g})$ is a Ricci form of $g$ and $L_{X}$ denotes the Lie
derivative along a holomorphic vector field $X$ on $M$. By the uniqueness of
Kähler-Ricci solitons [TZhu1], [TZhu2], we may assume that $X$ lies in a
reductive Lie subalgebra $\eta_{r}(M)$ of $\eta(M)$ after a holomorphism
transformation, where $\eta(M)$ consists of all holomorphic vector fields on
$M$. Such a $X$ ( we call it an extremal holomorphic vector field for Kähler-
Ricci solitons ) can be determined as follows.
Let $\text{Aut}_{r}(M)$ be a connected Lie subgroup of automorphisms group of
$M$ generated by $\eta_{r}(M)$. Let $K$ be a maximal compact subgroup of
$\text{Aut}_{r}(M)$. Without loss of generality, we may choose a $K$-invariant
background metric $g$ with its Kähler form $\omega_{g}$ in $2\pi c_{1}(M)$. In
[TZhu2], as an obstruction to Kähler-Ricci solitons, Tian and Zhu introduced a
modified Futaki-invariant $F_{X}(v)$ for any $X,v~{}\in~{}\eta(M)$ by
(1.8)
$F_{X}(Z)=\int_{M}Z(h_{g}-\hat{\theta}_{X,\omega_{g}})e^{\hat{\theta}_{X,\omega_{g}}}\omega_{g}^{n},\quad\forall~{}Z\in\eta(M),$
where $h_{g}$ is a Ricci potential of $g$ and $\hat{\theta}_{X,\omega_{g}}$ is
a real-valued potential of $X$ associated to $g$ defined by
$L_{X}\omega_{g}=\sqrt{-1}\partial\bar{\partial}\hat{\theta}_{X,\omega_{g}}$
with a normalization condition
(1.9) $\int_{M}\hat{\theta}_{X,\omega_{g}}e^{h_{g}}\omega_{g}^{n}=0.$
It was showed that there exists a unique $X~{}\in~{}\eta_{r}(M)$ such that
$F_{X}(v)\equiv 0,~{}\forall~{}v\in~{}\eta_{r}(M).$
Moreover, $F_{X}(v)\equiv 0$, for any $v\in~{}\eta(M)$ if $(M,J)$ admits a
Kähler-Ricci soliton.
Let $K_{X}$ be an one-parameter compact subgroup of holomorphisms
transformation group generated by $X$. We denote $\mathcal{K}_{X}$ to be a
class of $K_{X}$-invariant Kähler metrics in $2\pi c_{1}(M)$. Let
$\theta_{X,\omega_{g}}$ be a real-valued potential of $X$ associated to $g$
with a normalization condition
(1.10)
$\int_{M}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}=\int_{M}\omega_{g}^{n}=V.$
Clearly, $\theta_{X,\omega_{g}}=\hat{\theta}_{X,\omega_{g}}-c_{X}$ for some
constant $c_{X}$ which is independent of $g\in\mathcal{K}_{X}$ .
###### Definition 1.2.
For $g\in\mathcal{K}_{X}$, define $N_{X}(\omega_{g})$ by
$N_{X}(\omega_{g})=\int_{M}\theta_{X,\omega_{g}}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}.$
By Jensen’s inequality, it is easy to see
$\displaystyle\begin{aligned}
&\frac{1}{V}\int_{M}(-\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&\leq\text{log}\\{\frac{1}{V}\int_{M}e^{-\theta_{X,\omega_{g}}}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\}=0.\end{aligned}$
The equality holds if and only if $\theta_{X,\omega_{g}}=0$. This shows that
$N_{X}(\omega_{g})$ is nonnegative and it is zero if and only if the Futaki-
invariant vanishes [Fu]. Moreover, we have
###### Lemma 1.3.
$N_{X}(\omega_{g})$ is independent of choice of $g$ in $\mathcal{K}_{X}$.
###### Proof.
Choose a $K$-invariant Kähler form $\omega$ in $2\pi c_{1}(M)$. Then for any
Kähler metric $g$ in $\mathcal{K}_{X}$ there exists a Kähler potential
$\varphi$ such that the imaginary part of $X(\varphi)$ vanishes and Kähler
form of $g$ satisfies
$\omega_{g}=\omega_{\varphi}=\omega+\sqrt{-1}\partial\bar{\partial}\varphi.$
Thus we suffice to prove
$N_{X}(\omega_{\varphi})=N_{X}(\omega_{t\varphi}),~{}~{}\forall~{}t\in~{}[0,1],$
where
$\omega_{t\varphi}=\omega+t\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\varphi.$
This follows from
$\displaystyle\begin{aligned}
\frac{dN_{X}(\omega_{t\varphi})}{dt}&=\int_{M}X(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}+\int_{M}\theta_{X,\omega_{t\varphi}}(X+\triangle)(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}\\\
&=\int_{M}X(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}-\int_{M}\nabla^{i}\varphi\nabla_{\bar{i}}\theta_{X,\omega_{t\varphi}}\omega_{t\varphi}^{n}\\\
&=0.\end{aligned}$
Here we have used the fact
$\theta_{X,\omega_{t\varphi}}=\theta_{X,\omega_{0}}+tX(\varphi).$
∎
By the above lemma, $N_{X}(\cdot)$ is an invariance on $\mathcal{K}_{X}$,
which is independent of choice of $g$. For simplicity, we denote this
invariance by $N_{X}(c_{1}(M))$. The following proposition gives an upper
bound of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ related to $N_{X}(c_{1}(M))$.
###### Proposition 1.4.
$\sup_{g\in\mathcal{K}_{X}}\lambda(g)\leq(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$
###### Proof.
Since $\lambda(g)\leq W(g,-\theta_{X,\omega_{g}})$, we suffice to prove
(1.11) $W(g,-\theta_{X,\omega_{g}})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$
In fact, by using the facts $R(g)=2n+\Delta h_{g}$ and
$\int_{M}(\Delta\theta_{X,\omega_{g}}+|\nabla\theta_{X,\omega_{g}}|^{2})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}=0,$
we have
$\displaystyle\begin{aligned}
&\int_{M}(R(g)+|\nabla\theta_{X,\omega_{g}}|^{2})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV+\int_{M}(\Delta
h_{g}-\Delta\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-\int_{M}<\nabla(h_{g}-\theta_{X,\omega_{g}}),\nabla\theta_{X,\omega_{g}}>e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-2\int_{M}X(h_{g}-\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-2e^{-c_{X}}F_{X}(X).\end{aligned}$
In the last equality above, we used the relation (1.8). Since $X$ is extremal
, we have
$F_{X}(X)=0.$
Thus by (1.2) for $f=-\theta_{X,\omega}$ together with Lemma 1.3, one will get
(1.11). ∎
In case that $M$ admits a Kähler-Ricci soliton $g_{KS}$, by Lemma 1.1, a
minimizer $f$ of $W(g_{KS},\cdot)$ in $\mathcal{K}_{X}$ must be $-\theta_{X}$.
Thus for any $g\in\mathcal{K}_{X}$, by Proposition 1.4, we have
$\displaystyle\begin{aligned} \lambda(g_{KS})&=W(g_{KS},-\theta_{X})\\\
&=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))]\geq\lambda(g).\end{aligned}$
Therefore we get the following corollary.
###### Corollary 1.5.
Suppose that $(M,J)$ admits a Kähler-Ricci soliton $g_{KS}$. Then $g_{KS}$ is
a global maximizer of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ and
(1.12) $\lambda(g_{KS})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$
###### Remark 1.6.
Corollary 1.5 implies that a Kähler-Einstein metric is a global maximizer of
$\lambda(\cdot)$ in $2\pi c_{1}(M)$ even with varying complex structures and
supremum of $\lambda(\cdot)$ is $(2\pi)^{-n}nV$ since $N_{X}(c_{1}(M))=0$.
Note that $N_{X}(c_{1}(M))>0$ if the Futaki-invariant does not vanish. Thus
Corollary 1.5 also implies that the supremum of $\lambda(\cdot)$ in case that
$(M,J)$ admits a Kähler-Ricci soliton is strictly less than one in case that
$(M,J)$ admits a Kähler-Einstein metric.
## 2\. Estimates for modified Ricci potentials
In this section, we summarize some apriori estimates for modified Ricci
potentials of evolved Kähler metrics along Kähler-Ricci flow. Some similar
estimates have been also discussed in [TZhu3] and [PSSW], we refer the readers
to those two papers. We consider the following (normalized) Kähler-Ricci flow:
(2.1) $\frac{\partial g(t,\cdot)}{\partial t}\,=\,-{\rm
Ric}(g(t,\cdot))+g(t,\cdot),~{}~{}~{}g(0)=g,$
where Kähler form of $g$ is in $2\pi c_{1}(M)$. It was proved in [Ca] that
(2.1) has a global solution $g_{t}=g(t,\cdot)$ for all time $t>0$. For
simplicity, we denote by $(g_{t};g)$ a solution of (2.1) with initial metric
$g$. Since the flow preserves the Kähler class, we may write Kähler form of
$g_{t}$ as
$\omega_{\phi}\,=\,\omega_{g}+\sqrt{-1}\partial\overline{\partial}\phi$
for some Käher potential $\phi=\phi_{t}$.
Let $X\in\eta_{r}(M)$ be the extremal holomorphic vector field on $M$ as in
Section 1 and $\sigma_{t}=\exp\\{tX\\}$ an one-parameter subgroup generated by
$X$. Let $\phi^{\prime}=\phi_{\sigma_{t}}$ be corresponding Kähler potentials
of $\sigma_{t}^{\star}\omega_{\phi_{t}}$. Then $\omega_{\phi^{\prime}}$ will
satisfy a modified Kähler-Ricci flow,
(2.2) $\frac{\partial}{\partial
t}\omega_{\phi^{\prime}}=-\text{Ric}(\omega_{\phi^{\prime}})+\omega_{\phi^{\prime}}+L_{X}\omega_{\phi^{\prime}}.$
Equation (2.2) is equivalent to the following Monge-Ampére flow for
$\phi^{\prime}$ (modulo a constant),
(2.3) $\frac{\partial\phi^{\prime}}{\partial
t}=\log\frac{\omega_{\phi^{\prime}}^{n}}{\omega_{g}^{n}}+\phi^{\prime}+\theta_{X,\omega_{\phi^{\prime}}}-h_{g},~{}~{}~{}~{}\phi^{\prime}(0,\cdot)=c,$
where $c$ is a constant and all Kähler potentials
$\phi^{\prime}=\phi_{t}^{\prime}=\phi^{\prime}(t,\cdot)$ are in a space given
by
$\mathcal{P}_{X}(M,\omega)=\\{\varphi\in
C^{\infty}(M)|~{}\omega_{\varphi}=\omega+\sqrt{-1}\partial\overline{\partial}\varphi>0,\quad\mathrm{Im}(X(\varphi))=0\\}.$
By using the maximum principle to (2.2) or (2.3), we get
(2.4)
$h_{\phi^{\prime}}-\theta_{X,\omega_{\phi^{\prime}}}=-\frac{\partial}{\partial
t}\phi^{\prime}+c_{t},$
for some constants $c_{t}$. Here $h_{\phi^{\prime}}$ are Ricci potentials of
$\omega_{\phi^{\prime}}$ which are normalized by
(2.5) $\int_{M}e^{h_{\phi^{\prime}}}\omega_{\phi^{\prime}}^{n}=V.$
The following estimates are due to G. Perelman. We refer the readers to [ST]
for their proof.
###### Lemma 2.1.
There are constants $c$ and $C$ depending only on the initial metric $g$ such
that (a) $\text{diam}(M,\omega_{\phi^{\prime}})\leq C$; (b)
$\text{vol}(B_{r}(p),\omega_{\phi^{\prime}})\geq cr^{2n}$; (c)
$\|h_{\phi^{\prime}}\|_{C^{0}(M)}\leq C$; (d) $\|\nabla
h_{\phi^{\prime}}\|_{\omega_{\phi^{\prime}}}\leq C$; (e) $\|\Delta
h_{\phi^{\prime}}\|_{C^{0}(M)}\leq C$.
Recall that the modified Mabuchi’s K-energy $\mu(\cdot)$ is defined in
$\mathcal{P}_{X}(M,\omega)$ by
$\displaystyle\begin{aligned}
\mu(\varphi)=~{}&-\frac{n}{V}\int_{0}^{1}\int_{M}\dot{\psi}[\text{Ric}(\omega_{\psi})-\omega_{\psi}-\sqrt{-1}\partial\overline{\partial}\theta_{X,\omega_{\psi}}\\\
&+\sqrt{-}\overline{\partial}(h_{\omega_{\psi}}-\theta_{X,\omega_{\psi}})\wedge\partial\theta_{X,\omega_{\psi}})]\wedge
e^{\theta_{X,\omega_{\psi}}}\omega_{\psi}^{n-1}\wedge dt,\end{aligned}$
where $\psi=\psi_{t}$ $(0\leq t\leq 1)$ is a path connecting $0$ to $\varphi$
in $\mathcal{P}_{X}(M,\omega)$. If $X=0$, then $\mu_{\omega_{g}}(\phi)$ is
nothing but Mabuchi’s $K$-energy [Ma]. Then by (2.2), we have
(2.6)
$\frac{d\mu(\phi^{\prime})}{dt}=-\frac{1}{V}\int_{M}\|\overline{\partial}\frac{\partial\phi^{\prime}}{\partial
t}\|_{\omega_{\phi^{\prime}}}^{2}e^{\theta_{X,\omega_{\phi^{\prime}}}}(\omega_{\phi^{\prime}})^{n}\leq
0.$
This implies that $\mu(\phi^{\prime})$ is uniformly bounded if $\mu(\cdot)$ is
bounded from below in $\mathcal{P}_{X}(M,\omega)$.
Let
$u_{X,\phi^{\prime}}=u_{X,\omega_{g_{t}^{\prime}}}=h_{\phi^{\prime}}-\theta_{X,\omega_{\phi^{\prime}}}$.
Then
###### Lemma 2.2.
There exists a uniform $C$ such that
$\|\nabla u_{X,\phi^{\prime}}\|_{\omega_{\phi^{\prime}}}\leq C.$
###### Proof.
First we note that $\theta_{X,\omega_{\phi^{\prime}}}$ is uniformly bounded in
$\mathcal{P}_{X}(M,\omega)$ (cf. [Zhu1], [ZZ]). Then by (c) of Lemma 2.1, we
have
$\|u_{X,\phi^{\prime}}\|_{C^{0}}=\|u_{X,\omega_{g_{s}^{\prime}}}\|_{C^{0}}\leq
C,~{}\forall~{}s>0$
for some uniform constant $C$. Now we consider the flow (2.3) with zero as an
initial Kähler potential and the background Kähler form $\omega_{g}$ replaced
by $\omega_{g_{s}^{\prime}}$. By an estimate in Lemma 4.3 in [CTZ], we see
$t\|\nabla
u_{X,\omega_{g_{s+t}^{\prime}}}\|_{\omega_{g_{s+t}^{\prime}}}^{2}\leq
e^{2t}\|u_{X,\omega_{g_{s}^{\prime}}}\|_{C^{0}},~{}~{}\forall~{}t>0.$
In particular, we get
$\|\nabla
u_{X,\omega_{g_{s+t}^{\prime}}}\|_{\omega_{g_{s+t}^{\prime}}}^{2}\leq
C^{\prime},~{}~{}\forall~{}t\in[1,2].$
Since the above estimate is independent of $s$, we conclude that the lemma is
true.
∎
Now we begin to prove the main result in this section.
###### Proposition 2.3.
Suppose that $\mu(\cdot)$ is bounded from below in
$\mathcal{P}_{X}(M,\omega)$. Then we have:
(a) $\lim_{t\rightarrow\infty}||u_{X,\phi^{\prime}}||_{C^{0}}=0$;
(b) $\lim_{t\rightarrow\infty}||\nabla
u_{X,\phi^{\prime}}||_{\omega_{\phi^{\prime}}}=0$;
(c) $\lim_{t\rightarrow\infty}||\triangle u_{X,\phi^{\prime}}||_{C^{0}}=0$.
###### Proof.
Let $H(t)=\int_{M}|\nabla
u_{X,\omega_{g_{t}}^{\prime}}|^{2}e^{\theta_{X,\omega_{g_{t}}^{\prime}}}\omega_{g_{t}^{\prime}}^{n}$.
Then by (2.6), one sees that there exists a sequence of $t_{i}\in[i,i+1]$ such
that
$\lim_{i\rightarrow\infty}H(t_{i})=0.$
Thus by using a differential inequality
$\frac{dH(t)}{dt}\leq CH(t),$
where $C$ is a uniform constant (cf. [PSSW]), we get
(2.7) $\lim_{t\rightarrow\infty}\int_{M}|\nabla
u_{X,\omega_{g_{t}^{\prime}}}|_{g_{t}^{\prime}}^{2}e^{\theta_{X,\omega_{g_{t}^{\prime}}}}\omega_{g_{t}^{\prime}}^{n}=0.$
Let
$\tilde{u}_{t}=u_{X,\omega_{g_{t}^{\prime}}}-\frac{1}{V}\int_{M}u_{X,\omega_{g_{t}^{\prime}}}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n},$
where $h_{t}^{\prime}=h_{\phi^{\prime}(t,\cdot)}$. Then by using the weighted
Poincaré inequality in [TZhu3] together with (c) of Lemma 2.1, we obtain from
(2.7),
$\int_{M}\tilde{u}_{t}^{2}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}\leq\int_{M}|\nabla
u_{X,\omega_{g_{t}^{\prime}}}|_{g_{t}^{\prime}}^{2}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}\to
0,~{}~{}\text{as }~{}t\to\infty.$
Consequently, we derive
(2.8)
$\lim_{t\rightarrow\infty}\int_{M}\tilde{u}_{t}^{2}\omega_{g_{t}^{\prime}}^{n}=0.$
We claim
$\lim_{t\rightarrow\infty}\|\tilde{u}_{t}\|_{C^{0}}=0.$
The claim immediately implies (a) of Proposition 2.3 by the normalization
conditions
$\int_{M}e^{\theta_{X,\omega_{g_{t}^{\prime}}}}\omega_{g_{t}^{\prime}}^{n}=\int_{M}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}=V.$
To prove the claim, we need to use an inequality
(2.9) $\|\tilde{u}_{t}\|_{C^{0}}^{n+1}\leq C\|\nabla
u_{X,\omega_{g_{t}^{\prime}}}\|^{n}_{g_{t}^{\prime}}[\int_{M}\tilde{u}_{t}^{2}\omega_{g_{t}^{\prime}}^{n}]^{\frac{1}{2}}.$
(2.9) can be proved by using the non-collapsing estimate (b) in Lemma 2.1 (cf.
[PSSW], [Zhu2]). Thus by Lemma 2.2 and (2.8), the claim is proved.
By (a) we can show that after a suitable choice of constant $c$ in the flow
(2.3) it holds
$\lim_{t\to\infty}\|\frac{\partial}{\partial t}\phi^{\prime}\|_{C^{0}}=0.$
In fact under the assumption of lower bound of modified $K$-energy, one can
choose such a $c$ (cf. [TZhu3]) such that
$\lim_{t\to\infty}\int_{M}\frac{\partial}{\partial
t}\phi^{\prime}e^{\theta_{X,\omega_{\phi^{\prime}}}}\omega_{\phi^{\prime}}^{n}=0.$
Then by (2.4), we will get the conclusion. On the other hand, by Lemma 2.2 and
(d) of Lemma 2.1, we have
$\sup_{t\in[0,\infty)}\|X\|_{g_{t}^{\prime}}<C$
for some uniform constant $C$. Therefore, by using the following lemma we
prove (b) and (c).
∎
###### Lemma 2.4.
([PSSW]) There exist $\delta,K>0$ depending only on $n$ and the constant
$C_{X}=\sup_{t\in[0,\infty)}\|X\|_{g_{t}^{\prime}}$ with the following
property. For $\epsilon$ with $0<\epsilon\leq\delta$ and any $t_{0}>0$, if
$\|\frac{\partial\phi^{\prime}}{\partial t}\|_{C^{0}}(t_{0})\leq\epsilon,$
then
$\|\nabla
u_{X,\omega_{g_{t_{0}+2}^{\prime}}}\|^{2}_{g_{t_{0}+2}^{\prime}}+\|\Delta
u_{X,\omega_{g_{t_{0}+2}^{\prime}}}\|_{C^{0}}\leq K\epsilon.$
## 3\. Proof of Theorem 0.1
According to [TZhu5], an energy level $L(g)$ of entropy $\lambda(\cdot)$ along
Kähler-Ricci flow $(g_{t};g)$ is defined by
$L(g)=\lim_{t\rightarrow\infty}\lambda(g_{t}).$
By the monotonicity of $\lambda(g_{t})$, we see that $L(g)$ exists and it is
finite. In this section, our goal is to prove
###### Proposition 3.1.
Suppose that the modified Mabuchi’s K-energy is bounded from below in
$\mathcal{K}_{X}$. Then for any $g\in\mathcal{K}_{X}$.
(3.1) $L(g)=(2\pi)^{-n}(nV-N_{X}(c_{1}(M)).$
The above proposition shows that the energy level $L(g)$ of entropy
$\lambda(\cdot)$ does not depend on the initial Kähler metric
$g\in\mathcal{K}_{X}$. Thus by using the Kähler-Ricci flow $(g_{t};g)$ for any
Kähler metric $g\in\mathcal{K}_{X}$ and the monotonicity of $\lambda(g_{t})$,
we will get Theorem 0.1.
To prove Proposition 3.1, we need the following key lemma.
###### Lemma 3.2.
Let $f_{t}$ be a minimizer of $W(g_{t},\cdot)$-functional associated evolved
Kähler metric $g_{t}$ of (2.1) at time $t$ and $h_{t}$ a Ricci potential of
$g_{t}$ which satisfying the normalization (2.5) . Then there exists a
sequence of $t_{i}\in[i,i+1]$ such that
(a)
$\lim_{t_{i}\to\infty}\|\Delta(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\,=\,0$;
(b)
$\lim_{t_{i}\to\infty}\|\nabla(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\,=\,0$;
(c) $\lim_{t_{i}\to\infty}\|f_{t_{i}}+h_{t_{i}}\|_{C^{0}}\,=\,0$.
###### Proof.
Lemma 3.2 is a generalization of Proposition 4.4 in [TZhu5]. We will follow
the argument there. First by (1.5), it is easy to see that
$\displaystyle\begin{aligned}
\frac{d}{dt}\lambda(g_{t})=(2\pi)^{-n}\int_{M}|\mathrm{Ric}(g_{t})-g_{t}+\nabla^{2}f_{t}|_{g_{t}}^{2}e^{-f_{t}}\omega_{g_{t}}^{n}.\end{aligned}$
It follows that
$\frac{d}{dt}\lambda(g_{t}))\geq(2\pi)^{-n}\frac{1}{2n}\int_{M}|\triangle(h_{t}+f_{t})|^{2}e^{-f(t)}\omega_{g_{t}}^{n}.$
Since $\lambda(g_{t})\leq W(g_{t},0)=(2\pi)^{-n}nV$ are uniformly bounded, we
see that there exists a sequence of $t_{i}\in[i,i+1]$ such that
$\lim_{i\rightarrow\infty}\int_{M}|\triangle(h_{t_{i}}+f_{t_{i}})|^{2}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}=0.$
Note that $f_{t}$ is uniformly bounded [TZhu5]. Hence we see that that (a) of
the lemma is true. By (a), we also get
(3.2) $\displaystyle\begin{aligned}
&\lim_{t_{i}\to\infty}\|\nabla(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\\\
&\leq\lim_{t_{i}\to\infty}\int_{M}|f_{t_{i}}+h_{t_{i}}||\triangle(f_{t_{i}}+h_{t_{i}})|\omega_{g_{t_{i}}}^{n}\\\
&\leq
C\lim_{t_{i}\to\infty}\|\Delta(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}=0.\end{aligned}$
This proves (b) of the lemma. It remains to prove (c).
Let $q_{t}=f_{t}+h_{t}$. Then
(3.3) $\displaystyle\begin{aligned} -\Delta{q_{t}}&=-\Delta f_{t}-\Delta
h_{t}\\\ &=f_{t}+\frac{1}{2}(R-|\nabla
f_{t}|^{2})-(2\pi)^{2n}V^{-1}\lambda(g_{t})-\Delta h_{t}\leq C.\end{aligned}$
Define
$\tilde{q_{t}}=q_{t}-\frac{1}{V}\int_{M}q_{t}e^{h_{t}}\omega_{g_{t}}^{n}.$
By using the weighted Poincaré inequality (cf. [TZhu3]), we have
$\int_{M}\tilde{q_{t}}^{2}e^{h_{t}}\omega_{g_{t}}^{n}\leq\int_{M}|\nabla
q_{t}|^{2}e^{h_{t}}\omega_{g_{t}}^{n}.$
It follows by (b),
(3.4)
$\lim_{i\rightarrow\infty}\int_{M}\tilde{q}_{t_{i}}^{2}\omega_{g_{t_{i}}}^{n}=0.$
Hence, following an argument in the proof of Proposition 4.4 in [TZhu5], we
will get estimates
(3.5) $\|\tilde{q_{t_{i}}}^{+}\|_{C^{0}}\leq
C\|\tilde{q}_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}\to
0,~{}~{}\text{as}~{}i\to\infty,$
and
(3.6)
$\lim_{i\rightarrow\infty}\int_{M}\tilde{q_{t_{i}}}^{-}\omega_{g_{t_{i}}}^{n}=0,$
where $q_{t}^{+}=\max\\{q_{t},0\\}$ and $q_{t}^{-}=\min\\{q_{t},0\\}$.
Consequently, we derive
$\int_{M}\tilde{q_{t_{i}}}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}=0.$
This implies
(3.7)
$\lim_{i\rightarrow\infty}\int_{M}q_{t_{i}}e^{-f_{t_{i}}}\omega^{n}_{g_{t_{i}}}=0$
according to the normalization
$\int_{M}e^{-f_{t}}\omega^{n}_{g_{t}}=\int_{M}e^{h_{t}}\omega^{n}_{g_{t}}=V$.
Next we improve that
(3.8) $\lim_{i\rightarrow\infty}|q_{t_{i}}|=0.$
Let $u_{t}=e^{-\frac{f_{t}}{2}}-e^{\frac{h_{t}}{2}}$. We claim
(3.9)
$\lim_{i\rightarrow\infty}\|u_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}=0.$
In fact, by Jensen’s inequality and (3.7), one sees
$\displaystyle\begin{aligned}
\frac{1}{V}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}&=\frac{1}{V}\int_{M}e^{\frac{f_{t_{i}}+h_{t_{i}}}{2}}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}\\\
&\geq
e^{\frac{1}{2V}\int_{M}(f_{t_{i}}+h_{t_{i})}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}}\to
1,~{}\text{as}~{}i\to\infty.\end{aligned}$
On the other hand,
$\int_{M}e^{-\frac{f_{t}}{2}}e^{\frac{h_{t}}{2}}\omega_{g_{t}}^{n}\leq(\int_{M}e^{-f_{t}}\omega_{g_{t}}^{n})^{\frac{1}{2}}(\int_{M}e^{h_{t}}\omega_{g_{t}}^{n})^{\frac{1}{2}}=V.$
Hence
$\lim_{i\rightarrow\infty}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}=V.$
It follows
$\lim_{i\rightarrow\infty}\int_{M}u_{t_{i}}^{2}\omega_{g_{t_{i}}}^{n}=2V-2\lim_{i\rightarrow\infty}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}=0.$
This completes the proof of claim.
Since equation (1.4) is equivalent to
(3.10) $\Delta
v_{t}-\frac{1}{2}f_{t}v_{t}-\frac{1}{4}R(g_{t})v_{t}=\frac{1}{2V}(2\pi)^{n}\lambda(g_{t})v_{t},$
where $v_{t}=e^{\frac{-f_{t}}{2}}$, by Lemma 2.1, it is easy to see
$|\Delta u_{t}|\leq C.$
Then by the standard Moser’s iteration, we get from (3.9),
$\|u_{t_{i}}\|_{C^{0}}\leq C\|u_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}\to
0,~{}\text{as}~{}i\to\infty.$
This implies (3.8), so we obtain (c) of the lemma.
∎
###### Proof of Proposition 3.1.
Note that $\frac{R(g_{t})}{2}=n+\frac{1}{2}\Delta h_{t}$, where $\Delta$ is
the Beltrima-Laplacian operator associated to the Riemannian metric $g_{t}$.
Then
$\int_{M}\frac{1}{2}(R(g_{t})+|\nabla
f_{t}|^{2})e^{-f_{t}}dV_{g_{t}}=nV+\frac{1}{2}\int_{M}\Delta(f_{t}+h_{t})e^{-f_{t}}dV_{g_{t}}.$
Thus by (a) of Lemma 3.2, one sees that there exists a sequence of time
$t_{i}$ such that
(3.11) $\lim_{i\rightarrow\infty}\int_{M}\frac{1}{2}(R(g_{t_{i}})+|\nabla
f_{t_{i}}|^{2})e^{-f_{t_{i}}}dV_{g_{t_{i}}}=nV.$
On the other hand, since the modified Mabuchi’s K-energy is bounded from
below, we see that (a) of Proposition 2.3 is true. Then by (c) of Lemma 3.2,
it follows
(3.12)
$\lim_{i\to\infty}\|f_{t_{i}}+\theta_{X,\omega_{g_{t_{i}}}}\|_{C^{0}}=0.$
Here we used a fact
$\sigma_{t}^{\star}\theta_{X,\omega_{g_{t}}}=\theta_{X,\omega_{g_{t}^{\prime}}}$
since $X$ lies in the center of $\eta_{r}(M)$ [TZhu1]. Hence
(3.13) $\displaystyle\begin{aligned}
&\lim_{i\to\infty}\int_{M}f_{t_{i}}e^{-f_{t_{i}}}dV_{g_{t_{i}}}\\\
&=-\lim_{i\to\infty}\int_{M}\theta_{X,\omega_{g_{t_{i}}}}e^{\theta_{X,\omega_{g_{t_{i}}}}}\omega^{n}_{g_{t_{i}}}=-N_{X}(c_{1}(M)).\end{aligned}$
By combining (3.11) and (3.13), we get
$\displaystyle\begin{aligned}
&\lim_{i\to\infty}\lambda(g_{t_{i}})=\lim_{i\to\infty}\int_{M}[\frac{1}{2}(R(g_{t_{i}})+|\nabla
f_{t_{i}}|^{2})+f_{t_{i}}]e^{-f_{t_{i}}}dV_{g_{t_{i}}}\\\ &=nV-
N_{X}(c_{1}(M))\end{aligned}$
Therefore, by using the monotonicity of $\lambda(g_{t})$ along the flow
$(g_{t};g)$, we obtain (3.1).
∎
It was showed in [TZhu4] that a Kähler-Ricci soliton is a local maximizer of
$\lambda(\cdot)$ in the Kähler class $2\pi c_{1}(M)$. Together with Corollary
1.5, one may guess that a Kähler-Ricci soliton is a global maximizer of
$\lambda(\cdot)$. More general, according to Theorem 0.1 , we propose the
following conjecture.
###### Conjecture 3.3.
Suppose that the modified Mabuchi’s K-energy is bounded from below. Then
$\sup_{\omega_{g^{\prime}}\in 2\pi
c_{1}(M)}\lambda({g^{\prime}})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$
## 4\. Improvement of Lemma 3.2
In this section, we use Perelman’s backward heat flow to improve estimate (c)
in Lemma 3.2 independent of $t$. Moreover, we show the gradient estimate of
$f_{t}+h_{t}$ also holds. Although Lemma 3.2 is sufficient to be applied to
prove Theorem 0.1 and Theorem 0.2, results of this section are independent of
interests. We hope that these results will have applications in the future.
Fix any $t_{0}\geq 1$. We consider a backward heat equation in
$t\in[t_{0}-1,t_{0}]$,
(4.1) $\frac{\partial}{\partial t}f_{t_{0}}(t)=-\triangle f_{t_{0}}(t)+|\nabla
f_{t_{0}}(t)|^{2}-\triangle h_{t}$
with an initial $f_{t_{0}}(t_{0})=f_{t_{0}}$. Clearly, the equation preserves
the normalizing condition
$\frac{1}{V}\int_{M}e^{-f_{t_{0}}(t)}\omega_{g_{t}}^{n}=1$. Moreover, by the
maximum principle, we have
(4.2) $\|f_{t_{0}}(t)\|_{C^{0}}\leq C(g),\hskip
8.5359pt\forall~{}t\in[t_{0}-1,t_{0}],$
since $\triangle h_{t}$ are uniformly bounded. Here the constant $C(g)$
depends only on the initial metric $g$ of (2.1).
Similarly to (1.5), we can compute
(4.3) $\displaystyle\begin{aligned} &\frac{d}{dt}W(g_{t},f_{t_{0}})\\\
&=(2\pi)^{-n}\int_{M}(\|\partial\overline{\partial}(h_{t}+f_{t_{0}}(t))\|^{2}+\|\partial\partial
f_{t_{0}}(t)\|^{2})e^{-f_{t_{0}}(t)}\omega_{{}_{g_{t}}}^{n}.\end{aligned}$
By using (4.3), we want to prove
###### Lemma 4.1.
(4.4) $\|f_{t}+h_{t}-c_{t}\|_{L^{2}(M,g_{t})}\rightarrow
0,~{}\text{as}~{}t\rightarrow\infty,$
where $c_{t}=\frac{1}{V}\int_{M}(f_{t}+h_{t})e^{h_{t}}\omega_{g_{t}}^{n}$.
###### Proof.
First by (4.3), one sees
$\displaystyle\lambda(g_{t_{0}})-\lambda(g_{t_{0}-1})$ $\displaystyle\geq$
$\displaystyle
W(g_{t_{0}},f_{t_{0}}(t_{0}))-W(g_{t_{0}-1},f_{t_{0}}(t_{0}-1))$
$\displaystyle\geq$
$\displaystyle(2\pi)^{-n}\frac{1}{2n}\int_{t_{0}-1}^{t_{0}}\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}e^{-f_{t_{0}}(t)}\omega_{g_{t}}^{n}dt.$
It follows
$\int_{t_{0}-1}^{t_{0}}\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}dt\rightarrow
0,~{}\text{as }~{}t_{0}\rightarrow\infty.$
Thus by using the weighted Poincaré inequality as in (3.4) in last section ,
we will get
(4.5) $\displaystyle\begin{aligned}
&\int_{t_{0}-1}^{t_{0}}dt\int_{M}(f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t))^{2}\omega^{n}_{g_{t}}\\\
&\leq
C(g_{0})[\int_{t_{0}-1}^{t_{0}}dt\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}]^{1/2}\to
0,~{}\text{as}~{}t_{0}\to\infty,\end{aligned}$
where
$c_{t_{0}}(t)=\frac{1}{V}\int_{M}(f_{t_{0}}(t)+h_{t})e^{h_{t}}\omega_{g_{t}}^{n}.$
Next, since $\frac{dh_{t}}{dt}=\Delta h_{t}+h_{t}-a_{t}$, where
$a_{t}=\frac{1}{V}\int_{M}h_{t}e^{h_{t}}\omega^{n}_{g_{t}}$, by a
straightforward calculation, we see
$\displaystyle\frac{d}{dt}\int_{M}(h_{t}+f_{t_{0}}-c_{t_{0}}(t))^{2}\omega^{n}_{g_{t}}$
$\displaystyle=\int_{M}[2(h_{t}+f_{t_{0}}(t)-c_{t_{0}}(t))(\triangle
f_{t_{0}}-|\nabla f_{t_{0}}(t)|^{2}+h_{t}-a_{t}-\frac{dc_{t_{0}}}{dt})$
$\displaystyle-(h_{t}+f_{t_{0}}(t)-c_{t_{0}}(t))^{2}\triangle
h_{t}]\omega^{n}_{g_{t}}$
Then by Lemma 3.2, we get
$\displaystyle|\frac{d}{dt}\int_{M}(h_{t}+f_{t_{0}}(t)-c_{t_{0}})^{2}\omega^{n}_{g_{t}}|$
$\displaystyle\leq C+C\int_{M}\big{(}|\triangle f_{t_{0}}(t)|+|\nabla
f_{t_{0}}(t)|^{2}+|\frac{dc_{t_{0}}(t)}{dt}|\big{)}\omega^{n}_{g_{t}}$
$\displaystyle\leq
C+C\int_{M}\big{(}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}+|\frac{dc_{t_{0}}(t)}{dt}|\big{)}\omega^{n}_{g_{t}}.$
Notice that
$\displaystyle\frac{dc_{t_{0}}}{dt}=\frac{1}{V}\int_{M}[\triangle
f_{t_{0}}(t)-|\nabla
f_{t_{0}}(t)|^{2}-(h_{t}+f_{t_{0}}(t))(h_{t}+a_{t})]e^{h_{t}}\omega^{n}_{g_{t}}.$
We can also estimate
$|\frac{dc_{t_{0}}}{dt}|\leq
C+C\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}.$
Hence we derive
(4.6)
$\big{|}\frac{d}{dt}\int_{M}(f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t))^{2}dv\big{|}\leq
C+C\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}.$
Therefore, according to
$\displaystyle\int_{t_{0}-1}^{t_{0}}dt\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}e^{-f_{t_{0}}(t)}\omega^{n}_{g_{t}}$
$\displaystyle\leq(2\pi)^{n}(\lambda(g_{t_{0}})-\lambda(g_{t_{0}-1}))\to
0,~{}\text{as}~{}t_{0}\rightarrow\infty,$
(4.5) and (4.6) will implies
$\|f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t)\|_{L^{2}(g_{t},M)}\rightarrow
0,~{}\text{as}~{}t_{0}\rightarrow\infty,~{}\forall~{}t\in[t_{0}-1,t_{0}].$
Consequently, we get (4.4). ∎
###### Proposition 4.2.
(4.7) $\|f_{t}+h_{t}\|_{C^{0}}+\|\nabla(f_{t}+h_{t})\|_{g_{t}}\rightarrow
0,~{}\text{ as}~{}t\rightarrow\infty.$
###### Proof.
With the help of Lemma 4.1, by using same argument in the proof of (c) in
Lemma 3.2, we can prove that
(4.8) $\|f_{t}+h_{t}\|_{C^{0}}\rightarrow 0,~{}\text{
as}~{}t\rightarrow\infty.$
So we suffice to prove
(4.9) $\|\nabla(f_{t}+h_{t})\|_{g_{t}}\rightarrow 0,~{}\text{
as}~{}t\rightarrow\infty.$
We will use the Moser’s iteration to obtain (4.9 ) as in lemma 7.2 in
Appendix. We note by (4.8) and Theorem 7.1 that
(4.10) $\int_{M}|\nabla
q_{t}|^{2}\omega_{g_{t}}^{n}=|\int_{M}-\Delta(f_{t}+h_{t})(f_{t}+h_{t})\omega_{g_{t}}^{n}|\leq
C\|f_{t}+h_{t}\|_{C^{0}}\to 0,$
where $q_{t}=f_{t}+h_{t}$ satisfies an equation
$\triangle q_{t}=\frac{1}{2}(|\nabla f_{t}|^{2}-2f_{t}+\triangle
h_{t})+(2\pi)^{n}V^{-1}\lambda(g_{t})-n.$
Let $w_{t}=|\nabla q_{t}|^{2}$. Then by the Bochner formula, we have
$\displaystyle\triangle w_{t}$ $\displaystyle=$ $\displaystyle|\nabla\nabla
q_{t}|^{2}+|\nabla\bar{\nabla}q_{t}|^{2}+\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}q_{t}+\nabla_{\bar{i}}\triangle
q_{t}\nabla_{i}q_{t}+R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}{q_{t}}.$
Hence for any $p\geq 2$, it follows
(4.11) $\displaystyle\begin{aligned} &\frac{4(p-1)}{p^{2}}\int_{M}|\nabla
w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}=-\int_{M}w_{t}^{p-1}\triangle
q_{t}\omega_{g_{t}}^{n}\\\ &=-\int_{M}w_{t}^{p-1}(|\nabla\nabla
q_{t}|^{2}+|\nabla\bar{\nabla}q_{t}|^{2})\omega_{g_{t}}^{n}\\\
&-2\text{Re}\int_{M}q_{t}^{p-1}\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}q_{t}\omega_{g_{t}}^{n}-\int_{M}q_{t}^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}.\end{aligned}$
On the other hand, by Lemma 2.1 and Theorem 7.1, we estimate
$\displaystyle\begin{aligned}
&\quad-2\text{Re}\int_{M}w_{t}^{p-1}\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}w_{t}\omega_{g_{t}}^{n}\\\
&=-\text{Re}\int_{M}w_{t}^{q-1}\nabla_{i}(|\nabla f_{t}|^{2}-2f+\triangle
h_{t})\nabla_{\bar{i}}q_{t}\omega_{g_{t}}^{n}\\\ &=-\text{Re}\int_{M}(|\nabla
f_{t}|^{2}-2f_{t}+\triangle
h_{t})(\frac{2(p-1)}{p}w_{t}^{\frac{p}{2}-1}\nabla_{i}w_{t}^{p/2}\nabla_{\bar{i}}q_{t}+w_{t}^{q-1}\triangle
q_{t})\omega_{g_{t}}^{n}\\\ &\leq
C(g)[\int_{M}\frac{2(p-1)}{p}w_{t}^{\frac{p-1}{2}}|\nabla
w_{t}^{p/2}|\omega_{g_{t}}^{n}+\int_{M}w_{t}^{p-1}|\triangle
q_{t}|]\omega_{g_{t}}^{n}\\\ &\leq\frac{p-1}{p^{2}}\int_{M}|\nabla
w^{p/2}|^{2}\omega_{g_{t}}^{n}+C(g)^{\prime}p\int_{M}w^{p-1}\omega_{g_{t}}^{n}\end{aligned}$
and
$\displaystyle\begin{aligned}
&\quad-\int_{M}w_{t}^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}\\\
&=-\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}-\int_{M}w_{t}^{p-1}\nabla_{i}\nabla_{\bar{j}}h_{t}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}\\\
&=-\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}+\int_{M}w_{t}^{p-1}\nabla_{\bar{j}}h_{t}(\nabla_{\bar{i}}q_{t}\nabla_{i}\nabla_{j}q_{t}\omega_{g_{t}}^{n}+\triangle
q_{t}\nabla_{j}q_{t})\omega_{g_{t}}^{n}\\\
&\quad+\frac{2(p-1)}{p}\int_{M}w_{t}^{\frac{p}{2}-1}\nabla_{i}w_{t}^{p/2}\nabla_{\bar{j}}h_{t}\nabla_{\bar{i}}q_{t}\nabla_{j}q\omega_{g_{t}}^{n}\\\
&\leq\int_{M}w_{t}^{p-1}(|\nabla\nabla
q_{t}|^{2}+\frac{1}{2}|\nabla\bar{\nabla}q_{t}|^{2})\omega_{g_{t}}^{n}+\frac{p-1}{p^{2}}\int_{M}|\nabla
w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}\\\
&+C(g)(p-1)\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}.\end{aligned}$
Then substituting the above two inequalities into (4.11), we get
$\displaystyle\int_{M}|\nabla w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}\leq
C(g)(p-1)^{2}\int_{M}w_{t}^{p-1}\omega_{g_{t}}^{n},\hskip
5.69046pt\forall~{}p\geq 2.$
By using Zhang’s Sobolev inequality [Zha], we deduce
(4.12)
$\displaystyle\big{(}\int_{M}w_{t}^{p\nu}\big{)}^{1/\nu}\omega_{g_{t}}^{n}\leq
C(g)C_{s}(q-1)^{2}\int_{M}w_{t}^{p-1}\omega_{g_{t}}^{n},\quad\forall~{}p\geq
2,$
where $\nu=\frac{n}{n-1}$. To run the iteration we put $p_{0}=1$ and
$p_{k+1}=p_{k}\nu+\nu$, $k\geq 0$. Hence
$\displaystyle\|w_{t}\|_{L^{p_{k+1}}}$ $\displaystyle\leq$
$\displaystyle(CC_{s})^{\frac{1}{p_{k}+1}}p_{k}^{\frac{2}{p_{k}+1}}\|w\|_{L^{p_{k}}}^{\frac{p_{k}}{p_{k}+1}}$
$\displaystyle\leq$
$\displaystyle(CC_{s})^{\sum_{i=0}^{i=k}\frac{\nu^{k-i}}{p_{k}+1}}\prod_{i=0}^{i=k}p_{i}^{\frac{2\nu^{k-i}}{p_{k}+1}}\|w_{t}\|_{L^{1}}^{\prod\frac{p_{i}}{p_{i}+1}}$
$\displaystyle\leq$ $\displaystyle
C(n,g)C_{s}^{\frac{n}{2}}\|w_{t}\|_{L^{1}}^{\gamma(n)}$
for a constant $\gamma(n)$ depending only on $n$, where we have used the fact
$p_{k}\leq 2\nu^{k}$ for $k\geq 1$. Therefore by (4.10), we prove
$\|w_{t}\|_{C^{0}}\leq
C(n,g)C_{s}^{\frac{n}{2}}\|w_{t}\|_{L^{1}}^{\gamma(n)}\to
0,~{}\text{as}~{}t\to\infty.$
∎
## 5\. Another version of the invariance $N_{X}(\omega_{g})$
Let $Y\in\eta_{r}(M)$ so that $\text{Im}(Y)$ generates an one-parameter
compact subgroup of $K$. Denote $\mathcal{K}_{Y}$ to be a class of
$K_{Y}$-invariant Kähler metrics in $2\pi c_{1}(M)$. Then according to the
proof of Proposition 1.4, we actually prove
(5.1) $\displaystyle\begin{aligned}
\sup_{g\in\mathcal{K}_{Y}}\lambda(g)\leq(2\pi)^{-n}[nV-\tilde{F}_{Y}(Y)-N_{Y}(c_{1}(M))].\end{aligned}$
Note that
$\tilde{F}_{Y}(Y)=\int_{M}Y(h_{g}-\theta_{Y,\omega})e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}$
and
$N_{Y}(c_{1}(M))=\int_{M}\theta_{Y,\omega_{g}}e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}$
are both holomorphic invariances of $M$. In this section, we want to show
###### Proposition 5.1.
Let $H(Y)=\tilde{F}_{Y}(Y)+N_{Y}(c_{1}(M))$. Then
$\sup_{Y\in\eta_{r}(M)}H(Y)=N_{X}(c_{1}(M)),$
where $X$ is the extremal vector field as in Section 1.
###### Proof.
Choose a constant $c_{Y}$ so that
$\hat{\theta}_{Y,\omega_{g}}=\theta_{Y,\omega_{g}}+c_{Y}$ satisfies a
normalization condition
(5.2) $\displaystyle\begin{aligned}
\int_{M}\hat{\theta}_{Y,\omega_{g}}e^{h_{g}}\omega_{g}^{n}=0.\end{aligned}$
Then $\hat{\theta}_{Y,\omega_{g}}$ satisfies an equation
$\Delta\hat{\theta}_{X,\omega_{g}}+X(h_{g})+\hat{\theta}_{X,\omega_{g}}=0$
Thus using the integration by part, we have
$\tilde{F}_{Y}(Y)+\int_{M}\hat{\theta}_{Y,\omega_{g}}e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}=0$
It follows
(5.3) $H(Y)=-c_{Y}V=\int_{M}\theta_{Y,\omega_{g}}e^{h_{g}}\omega_{g}^{n}.$
We compute the first variation of $H(Y)$ in $\eta_{r}(M)$. By the definition
of $\theta_{Y+tY^{\prime}}$, we see that there exist constants $b(t)$ such
that $\theta_{Y+tY^{\prime}}=\theta_{Y}+t\theta_{Y^{\prime}}+b(t)$. Since
$\int_{M}e^{\theta_{Y+tY^{\prime}}}\omega_{g}^{n}=V$, we have
$e^{-b(t)}=\frac{1}{V}\int_{M}e^{\theta_{Y}+t\theta_{Y^{\prime}}}\omega_{g}^{n}.$
Thus we get
(5.4)
$\frac{dH(Y+tY^{\prime})}{dt}|_{t=0}=\int_{M}\theta_{Y^{\prime}}e^{h_{g}}\omega_{g}^{n}-\int_{M}\theta_{Y^{\prime}}e^{\theta_{Y}}\omega_{g}^{n}=\tilde{F}_{Y}(Y^{\prime}).$
Therefore, by [TZhu2], we see that there exists a unique critical
$X\in\eta_{r}(M)$ of $H(\cdot)$ such that
(5.5) $\tilde{F}_{X}(Y^{\prime})=F_{X}(Y^{\prime})\equiv
0,~{}~{}\forall~{}Y^{\prime}\in\eta_{r}(M).$
Similarly, we have
$\theta_{tY+(1-t)Y^{\prime}}=t\theta_{Y}+(1-t)\theta_{Y^{\prime}}+b(t)^{\prime},~{}\forall~{}t\in[0,1]$
for some constants $b(t)^{\prime}$. Then
$\displaystyle\begin{aligned}
V=\int_{M}e^{\theta_{tY+(1-t)Y^{\prime}}}\omega_{g}^{n}&=e^{b(t)^{\prime}}\int_{M}e^{t\theta_{Y}+(1-t)\theta_{Y^{\prime}}}\omega_{g}^{n}\\\
&\leq
e^{b(t)^{\prime}}[t\int_{M}e^{\theta_{Y}}\omega_{g}^{n}+(1-t)\int_{M}e^{\theta_{Y^{\prime}}}\omega_{g}^{n}]\\\
&=e^{b(t)^{\prime}}V.\end{aligned}$
Thus $b(t)^{\prime}\geq 0$. Consequently
$H(tX+(1-t)Y)\geq tH(X)+(1-t)H(Y).$
This means that $H(\cdot)$ is a concave functional on $\eta_{r}(M)$. It
follows that $X$ is a global maximizer of $H(\cdot)$. Therefore we prove the
proposition by using the fact $H(X)=N_{X}(c_{1}(M))$.
∎
###### Corollary 5.2.
Let $\mathcal{K}_{K}$ be a class of $K$-invariant Kähler metrics in $2\pi
c_{1}(M)$. Suppose that
(5.6) $\displaystyle\begin{aligned}
\sup_{g\in\mathcal{K}_{K}}\lambda(g)<\inf_{Y\in\eta_{r}(M)}(2\pi)^{-n}[nV-
F_{Y}(Y)-N_{Y}(c_{1}(M))].\end{aligned}$
Then $(M,J)$ could not admit any Kähler-Ricci soliton. Furthermore, the
modified Mabuchi’s K-energy could not be bounded from below.
###### Proof.
The first part of corollary follows from Proposition 5.1 and Corollary 1.5.
The second part follows from Proposition 5.1 and Theorem 0.1.
∎
The above corollary gives a new obstruction to the existence of Kähler-Ricci
solitons.
## 6\. Proof of Theorem 0.2
In this section, we will modify the proof of Main Theorem in [TZhu5] to prove
Theorem 0.2. The proof in [TZhu5] depends on a generalized uniqueness theorem
for Kähler-Einsteins recently proved by Chen and Sun in [CS]. Here we avoid to
use their theorem so that we can generalize the proof to the case of Kähler-
Ricci solitons by applying Proposition 3.1. As in [TZhu5], we write an initial
Kähler form $\omega_{g}$ of Kähler-Ricci flow (2.1) by
$\omega_{g}=\omega_{\varphi}=\omega_{g_{KS}}+\sqrt{-1}\partial\overline{\partial}\varphi\in
2\pi c_{1}(M)$
for a Kähler potential $\varphi$ on $M$. We define a path of Kähler forms
$\omega_{g^{s}}=\omega_{g_{KS}}+s\sqrt{-1}\partial\overline{\partial}\varphi$
and set
$\displaystyle\begin{aligned}
I=\\{s\in~{}[0,1]|~{}&(g_{t}^{s};g^{s})~{}\text{converges to a K\"{a}hler-
Ricci soliton in }~{}C^{\infty}~{}~{}\\\ &\text{in sense of K\"{a}hler
potentials}\\}.\end{aligned}$
Clearly, $I$ is not empty by the assumption of existence of Kähler-Ricci
solitons on $M$. We want to show that $I$ is in fact both open and closed.
Then it follows that $I=[0,1]$. This will finish the proof Theorem 0.2.
The openness of $I$ is related to the following stability theorem of Kähler-
Ricc flow, which was proved in [Zhu2].
###### Lemma 6.1.
Let $(M,J)$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Let $\psi$ be a Kähler potential of a $K_{X}$-invariant initial
metric $g$ of (2.1). Then there exists a small $\epsilon$ such that if
$\|\psi\|_{C^{3}}\leq\epsilon,$
the solution $g(t,\cdot)$ of (2.1) will converge to a Kähler-Ricci soliton
with respect to $X$ in $C^{\infty}$ in the sense of Kähler potentials.
Moreover, the convergence can be made exponentially.
###### Remark 6.2.
Lemma 6.1 is still true if the $K_{X}$-invariant condition is removed for the
initial metric $g$ (cf. [Zhu2]). But we do not know whether the convergence is
exponentially fast or not.
###### Proof of openness of $I$.
Suppose that $s_{0}\in I$. Then by the uniqueness of Kähler-Ricci solitons
[TZhu1], the flow $(g_{t}^{s_{0}};\omega_{s_{0}})$ converges to $g_{KS}$ after
a holomorphism transformation in $\text{Aut}_{r}(M)$. Namely, there exists a
$\sigma\in\text{Aut}_{r}(M)$ such that
$\sigma^{\star}\omega_{g_{t}^{s_{0}}}=\omega_{KS}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}^{s_{0}})_{\sigma}$
with property
$\|(\varphi_{t}^{s_{0}})_{\sigma}\|_{C^{k}}\leq C_{k}e^{-\alpha_{k}t},$
where $C_{k},\alpha_{k}>0$ are two uniform constants. Then we can choose $T$
sufficiently large such that
$\|(\varphi_{t}^{s_{0}})_{\sigma}\|_{C^{3}(M)}<\frac{\delta}{2},$
where $\delta$ is a small number determined in Lemma 6.1. Since the Kähler-
Ricci flow is stable for any fixed finite time, there is a small $\epsilon>0$
such that
$\|\varphi_{T}^{s}-(\varphi_{T}^{s_{0}})_{\sigma}\|_{C^{3}(M)}\,<\,\frac{\delta}{2},~{}\forall~{}s\in[s_{0},s_{0}+\epsilon],$
where $\varphi_{T}^{s}$ is a Kähler potential of evolved Kähler metric
$g_{T}^{s}$ of Kähler-Ricci flow $(g_{t}^{s};\sigma^{\star}\omega_{s})$ at
time $T$. Hence, we have
(6.1)
$\displaystyle\|\varphi_{T}^{s}\|_{C^{3}(M)}\,<\,\delta,~{}\forall~{}s\in[s_{0},s_{0}+\epsilon].$
Then the flow $(g_{t};g_{T}^{s})$ with initial $g_{T}^{s}$ will converge to a
Kähler-Ricci soliton in $C^{\infty}$ according to Lemma 6.1. This shows $s\in
I$ for any $s\in[s_{0},s_{0}+\epsilon]$
∎
Let $\varphi_{t}^{s}$ be a family of Kähler potentials of evolved Kähler
metric $g_{t}^{s}$ of Kähler-Ricci flow $(g_{t}^{s};\omega_{s})$. To make
potentials $\varphi^{s}_{t}$ more smaller to control, we need the following
lemma, which was proved in [TZhu1].
###### Lemma 6.3.
Let $M$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Let $\varphi$ be a $K_{X}$-invariant Kähler potential. Then
there exists a unique holomorphism transformation $\sigma\in\text{Aut}_{r}(M)$
such that $\varphi_{\sigma}\in\Lambda^{\perp}(\omega_{KS})$ with property
$J(\varphi_{\sigma})=\inf_{\tau\in\text{Aut}_{r}(M)}J(\varphi_{\tau}),$
where $\Lambda^{\perp}(\omega_{KS})$ is an orthogonal space to kernel space of
linear operator $(\Delta_{g_{KS}}+X+\text{Id})(\psi)$ and
$J(\varphi)=-\int_{M}\varphi
e^{\theta_{X,\omega_{\varphi}}}\omega_{\varphi}^{n}+\int_{0}^{1}\int_{M}\varphi
e^{\theta_{X,\omega_{\lambda\varphi}}}\omega_{\lambda\varphi}^{n}\wedge
d\lambda\geq 0.$
Moreover,
$\|\sigma-\text{Id}\|\leq C(\|\varphi\|_{C^{5}}),$
where $\|\sigma-\text{Id}\|$ denotes the distance norm in Lie group
$\text{Aut}_{r}(M)$ .
###### Proof of closedness of $I$.
By the openness of $I$, we see that there exists a $\tau_{0}\leq 1$ with
$[0,\tau_{0})\subset I$. We need to show that $\tau_{0}\in I$. In fact we want
to prove that for any $\delta>0$ there exists a large $T$ such that
(6.2)
$\displaystyle\|(\phi^{s}_{t})_{\sigma_{s,t}}\|_{C^{5}}\leq\delta,~{}\forall~{}t\geq
T~{}~{}\text{and}~{}s<\tau_{0},$
where $\sigma_{s,t}$ are some holomorphisms in $\text{Aut}_{r}(M)$. We will
use an argument by contradiction as in [TZhu5]. On contrary, we can find a
sequence of evolved Kähler metrics $g_{t_{i}}^{s_{i}}$ of Kähler-Ricci flows
$(g_{t}^{s_{i}};g^{s_{i}})$, where $s_{i}\to\tau_{0}$ and $t_{i}\to\infty$,
and a sequence of unique holomorphisms
$\sigma_{s_{i},t_{i}}\in\text{Aut}_{r}(M)$ for pairs $(s_{i},t_{i})$ such that
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\in\Lambda^{\perp}(\omega_{KS})$
and
(6.3)
$\displaystyle\|(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\|_{C^{5}}\geq\delta_{0}>0,$
for some constant $\delta_{0}$. Since the Kähler-Ricci flow
$(g_{t}^{s_{i}};g^{s_{i}})$ converges to some Kähler-Ricci soliton, by the
uniqueness of Kähler-Ricci solitons [TZhu1], the flow after a holomorphism
transformation in $\text{Aut}_{r}(M)$ converges to $g_{KS}$. So we may further
assume that $\phi^{s_{i}}_{t_{i}}$ satisfy
(6.4)
$\displaystyle\|(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\|_{C^{5}}\leq
2\delta_{0}.$
Then there exists a subsequence
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ (still used by
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ ) of
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ converging to a potential
$\phi_{\infty}\in\Lambda^{\perp}(\omega_{KS})$ with property
(6.5) $\displaystyle 2\delta_{0}\geq\|\phi_{\infty}\|_{C^{5}}\geq\delta_{0}.$
We want to show that
(6.6)
$\displaystyle\lambda(\omega_{\phi_{\infty})}=\lambda(g_{KS})=(2\pi)^{-n}(nV-
N_{X}(c_{1}(M)).$
First we note that the modified $K$-energy is bounded from below since $M$
admits a Kähler-Ricci soliton [TZhu2]. Then by Proposition 3.1 and the
monotonicity of $\lambda(g_{t}^{\tau_{0}})$, we see that for any $\epsilon>0$,
there exists a large $T>0$ such that
$\lambda(g^{\tau_{0}}_{t})\geq(2\pi)^{-n}(nV-
N_{X}(c_{1}(M))-\frac{\epsilon}{2},~{}~{}\forall~{}t\geq T.$
Since Kähler-Ricci flow is stable in finite time and $\lambda(g_{t}^{s})$ is
monotonic in $t$, there is a small $\delta>0$ such that for any
$s\geq\tau_{0}-\delta$, we have
(6.7) $\displaystyle\lambda(g_{t}^{s})\geq(2\pi)^{-n}(nV-
N_{X}(c_{1}(M))-\epsilon,~{}~{}\forall~{}t\geq T.$
Since $s_{i}\to\tau_{0}$ and $t_{i}\to\infty$, we conclude that
$\lim_{s_{i}\to\tau_{0},t_{i}\to\infty}\lambda(\sigma_{s_{i},t_{i}}^{\star}g^{s_{i}}_{t_{i}})=\lim_{s_{i}\to\tau_{0},t_{i}\to\infty}\lambda(g^{s_{i}}_{t_{i}})=(2\pi)^{-n}(nV-
N_{X}(c_{1}(M)).$
By the continuity of $\lambda(\cdot)$, we will get $(\ref{identity-3})$.
Now by Corollary 1.4 together with $(\ref{identity-3})$, we see that
$\omega_{\phi_{\infty}}$ is a global maximizer of $\lambda(\cdot)$ in
$\mathcal{K}_{X}$, so it is a critical point of $\lambda(\cdot)$. Then it is
easy to show that $\omega_{\phi_{\infty}}$ a Kähler-Ricci soliton with respect
to $X$ by computing the first variation of $\lambda(\cdot)$ as done in (1.5).
Thus by the uniqueness result for Kähler-Ricci solitons in [TZhu1], we get
$\omega_{\phi_{\infty}}=\sigma^{\star}\omega_{KS},$
where $\sigma\in\text{Aut}_{r}(M)$. Since
$\phi_{\infty}\in\Lambda^{\perp}(\omega_{KS})$, by Lemma 6.3, $\phi_{\infty}$
must be zero. This is a contradiction to $(\ref{contradiction-2})$. The
contradiction implies that (6.2) is true.
By $(\ref{claim-5})$, we see that for any $\delta>0$ there exists a large
$T_{0}$ and $\sigma_{0}\in\text{Aut}_{r}(M)$ such that
(6.8)
$\displaystyle\|(\phi_{T_{0}}^{\tau_{0}})_{\sigma_{0}}\|_{C^{5}}\leq\delta.$
Then by Lemma 6.1, the Kähler-Ricci flow
$(g_{t};\omega_{(\phi_{T_{0}}^{\tau_{0}})_{\sigma_{0}}})$ converge to a
Kähler-Ricci soliton. Thus, we prove that $\tau_{0}\in I$.
∎
###### Remark 6.4.
According to the proof of Theorem 0.2 and Remark 6.2, Theorem 0.2 will be
still true if the $K_{X}$-invariant condition is removed for the initial
metric $g$ of (2.1) assuming that Conjecture (3.3) is true.
## 7\. Appendix
In [TZhu5], it was proved the minimizer $f_{t}$ of $W(g_{t},\cdot)$-functional
associated to evolved Kähler metric $g_{t}$ of Kähler-Ricci flow (2.1) is
uniformly bounded (see also [TZha]). In this appendix, we show that the
gradients of $f_{t}$ are also uniformly bounded, and so are $\triangle f_{t}$
by (1.4). Namely, we prove
###### Theorem 7.1.
There is a uniform constant $C$ such that
$\|f_{t}\|+\|\nabla f_{t}\|+\|\triangle f_{t}\|\leq C,~{}~{}\forall~{}t>0.$
We will derive $\|\nabla f_{t}\|$ in Theorem 7.1 by studying a general
nonlinear elliptic equation as follows:
(7.1) $\triangle w(x)=w(x)F(x,w(x))$
where the Laplace operator $\triangle$ is associated to a Kähler metric $g$ in
$2\pi c_{1}(M)$ and $F$ is a smooth function on $M\times\mathbb{R}^{+}$, which
satisfies a structure condition:
(7.2) $-A-Bt^{\alpha}\leq F(\cdot,t)\leq H(t).$
Here $0\leq A,B\leq\infty,0\leq\alpha<\frac{2}{n}$ are constants, and $H$ is a
proper function on $\mathbb{R}^{+}$ which satisfies a growth control at $0$:
(7.3) $\limsup_{t\rightarrow 0}\big{(}tH(t)\big{)}<\infty.$
###### Lemma 7.2.
Let $w$ is a positive solution of (7.1). Then
(7.4) $\|\nabla w\|_{C^{0}}\leq C(n)C_{s}^{\frac{n}{2}}\big{(}\|\nabla
h\|_{C^{0}}+\|wF\|_{C^{0}}\big{)}^{n}\big{(}\int_{M}(1+|\nabla
w|^{2})dV_{g}\big{)}^{1/2},$
where $C_{s}$ is a Sobelev constant of $g$ and $h$ is a Ricci potential of
$g$.
###### Proof.
We will use the Moser’s iteration to $L^{p}$-estimate of $|\nabla w|$. By the
Bochner formula, we have
$\displaystyle\begin{aligned} \triangle|\nabla w|^{2}&=|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}+\nabla_{i}\triangle
w\nabla_{\bar{i}}w+\nabla_{i}w\nabla_{\bar{i}}\triangle
w+R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}w\\\ &=|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}+\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{i}w\nabla_{\bar{i}}(wF)+R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}w.\end{aligned}$
Put $\eta=|\nabla w|^{2}+1$ . Then for $p\geq 2$, it follows
(7.5) $\displaystyle\begin{aligned}
&\frac{4(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\\\
&=-\int_{M}\eta^{p-1}\triangle\eta dV_{g}\\\
&=-\int_{M}\eta^{p-1}\big{(}|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}\big{)}dV_{g}-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}.\end{aligned}$
The last term on the right hand side can be estimate as follows.
$\displaystyle-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}$
$\displaystyle=\int_{M}wF\big{(}\nabla_{i}\eta^{p-1}\nabla_{\bar{i}}w+\nabla_{\bar{i}}\eta^{p-1}\nabla_{i}w+2\eta^{p-1}\triangle
w\big{)}dV_{g}$
$\displaystyle=\frac{2(p-1)}{p}\int_{M}wF\eta^{\frac{p}{2}-1}\big{(}\nabla_{i}\eta^{p/2}\nabla_{\bar{i}}w+\nabla_{\bar{i}}\eta^{p/2}\nabla_{i}w\big{)}dV_{g}$
$\displaystyle+2\int_{M}wF\eta^{p-1}\triangle wdV_{g}.$
Then
(7.6)
$\displaystyle-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}wdV_{g}+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}$
$\displaystyle\leq\frac{2(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+2(p-1)p\int_{M}(wF)^{2}\eta^{p-2}(\eta-1)dV_{g}$
$\displaystyle+\int_{M}\eta^{p-1}\big{(}\frac{(\triangle
w)^{2}}{2n}+2n(wF)^{2}\big{)}dV_{g}$
$\displaystyle\leq\frac{2(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}$
$\displaystyle+2[p(p-1)+n]\|wF\|_{C^{0}}^{2}\int_{M}\eta^{p}dV_{g}.$
For the second term on the right hand side, we note
$R_{i\overline{j}}=g_{i\overline{j}}+h_{i\overline{j}}.$
Then
$\displaystyle\begin{aligned}
&-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&=\int_{M}\eta^{p-1}\nabla_{i}\nabla_{\bar{j}}h\nabla_{\bar{i}}w\nabla_{j}wdV_{g}-\int_{M}\eta^{p}dV_{g}\\\
&=\frac{2(p-1)}{p}\int_{M}\eta^{p/2-1}\nabla_{\bar{j}}h\nabla_{i}\eta^{p/2}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&+\int_{M}\eta^{p-1}\nabla_{\bar{j}}h\big{(}\triangle
w\nabla_{j}w+\nabla_{\bar{i}}w\nabla_{i}\nabla_{j}w\big{)}dV_{g}-\int_{M}\eta^{p}dV_{g}.\end{aligned}$
Thus
(7.7)
$\displaystyle-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}$
$\displaystyle\leq\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+p(p-1)\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-2}(\eta-1)^{2}dV_{g}$
$\displaystyle+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}+\frac{n}{2}\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-1}(\eta-1)dV_{g}$
$\displaystyle+\frac{1}{2n}\int_{M}\eta^{p-1}|\nabla\nabla
w|^{2}dV_{g}+\frac{n}{2}\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-1}(\eta-1)dV_{g}$
$\displaystyle\leq\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}$ $\displaystyle+\frac{1}{2n}\int\eta^{p-1}|\nabla\nabla
w|^{2}dV_{g}+[p(p-1)+n]\|\nabla h\|_{C^{0}}^{2}\int_{M}\eta^{p}dV_{g}.$
Substituting (7.6) and (7.7) into (7.5), we get
$\displaystyle\begin{aligned}
&\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\\\
&\leq-\int_{M}\eta^{p-1}\big{(}|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}\big{)}dV_{g}\\\
&+\frac{1}{n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}|\nabla\nabla w|^{2}dV_{g}\\\
&+[2(p-1)p+n]\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g}\\\ &\leq
C(n)p^{2}\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g}.\end{aligned}$
It follows
$\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\leq C(n)p^{3}\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g},\hskip
8.5359pt\forall~{}p\geq 2.$
Therefore, by iteration, we derive
$\sup\eta\leq C(n)D^{n/2}\big{(}\int_{M}\eta^{2}dV_{g}\big{)}^{1/2},$
where $D=C_{s}\big{(}\|\nabla h\|_{C_{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}$. This
implies (7.4) . ∎
###### Proposition 7.3.
(7.8) $\|\nabla w\|_{C^{0}}\leq C(\|w\|_{L^{2}}),$
where the constant $C$ depends only on $n,C_{s},A,B,\alpha,H$,
$\text{Vol}(g)$, $\|\nabla h\|_{C^{0}}$ and $\|w\|_{L^{2}}$.
###### Proof.
First we note that by using the standard Moser’s iteration to equation
$\triangle w(x)\geq-A-Bw^{\alpha},$
it is easy to see
$\sup w\leq C(1+\|w\|_{L^{2}}^{\gamma})$
for some constants $C$ and $\gamma$ which depend only on
$n,C_{s},A,B,\alpha,H$ and $\rm{Vol}(g)$. On the other hand, by (7.1), we have
$\int_{M}|\nabla w|^{2}dV_{g}=-\int_{M}w\triangle wdV_{g}=-\int_{M}wFdV_{g}.$
Then we see that $\|\nabla w\|_{L^{2}}$ is bounded by $\|w\|_{L^{2}}$. Thus
the proposition follows from Lemma 7.2.
∎
Since $v_{t}=e^{-\frac{f_{t}}{2}}$ satisfies (3.10) which is a type of
equation (7.1), then by Perelman’s estimates (d) in Lemma 2.1 and Zhang’s
estimate for Sobelev constants associated to $g_{t}$ in [Zha] together with
$C^{0}$-estimate for $f_{t}$ in [TZhu5], we obtain a uniform gradient estimate
for $v_{t}$ from Proposition 7.3, and so for $f_{t}$ . By equation (1.4), we
also derive a a uniform Laplacian estimate for $f_{t}$. Thus Theorem 7.1 is
true. Theorem 7.1 will be used in Section 4.
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* [18] [Zhu2] X.H. Zhu, Stability on Kähler-Ricci flow on a compact Kähler manifold with positive first Chern class; I, Arxiv/0908.1488v1.
* [19] [ZZ] B. Zhou and X.H. Zhu, Relative K–stability and modified K-energy on toric manifolds, Advances in Math., Vol 219 (2008), 1327-1362.
|
arxiv-papers
| 2011-07-20T16:05:58 |
2024-09-04T02:49:20.767967
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gang Tian, Shijin Zhang, Zhenlei Zhang, and Xiaohua Zhu",
"submitter": "Xiaohua Zhu",
"url": "https://arxiv.org/abs/1107.4018"
}
|
1107.4194
|
Also at ]Dokuz Eylül University, Graduate School of Natural and Applied
Sciences, Turkey
# Critical behavior and phase diagrams of a spin-1 Blume-Capel model with
random crystal field interactions: An effective field theory analysis
Yusuf Yüksel [ Ümit Akıncı Department of Physics, Dokuz Eylül University,
TR-35160 Izmir, Turkey Hamza Polat hamza.polat@deu.edu.tr Department of
Physics, Dokuz Eylül University, TR-35160 Izmir, Turkey
###### Abstract
A spin-1 Blume-Capel model with dilute and random crystal fields is examined
for honeycomb and square lattices by introducing an effective-field
approximation that takes into account the correlations between different spins
that emerge when expanding the identities. For dilute crystal fields, we have
given a detailed exploration of the global phase diagrams of the system in
$k_{B}T_{c}/J-D/J$ plane with the second and first order transitions, as well
as tricritical points. We have also investigated the effect of the random
crystal field distribution characterized by two crystal field parameters $D/J$
and $\triangle/J$ on the phase diagrams of the system. The system exhibits
clear distinctions in qualitative manner with coordination number $q$ for
random crystal fields with $\triangle/J,D/J\neq 0$. We have also found that,
under certain conditions, the system may exhibit a number of interesting and
unusual phenomena, such as reentrant behavior of first and second order, as
well as a double reentrance with three successive phase transitions.
PACS numbers
75.10.Dg, 75.10.Hk, 75.30.Kz
††preprint: APS/123-QED
###### Contents
1. I Introduction
2. II Formulation
3. III Results and Discussion
1. III.1 Phase diagrams of the system with dilute crystal field
2. III.2 Phase diagrams of the system with random crystal field
4. IV Conclusions
5. A Fundamental correlation functions of the system for a square lattice
6. B The complete set of twenty one linear equations of a honeycomb lattice
## I Introduction
Spin-1 Blume-Capel (BC) model blume ; capel is one of the most extensively
studied models in statistical mechanics and condensed matter physics. The
model exhibits a variety of multicritical phenomena such as a phase diagram
with ordered ferromagnetic and disordered paramagnetic phases separated by a
transition line that changes from a continuous phase transition to a first-
order transition at a tricritical point. On the other hand, as an extension of
the model, BC model with a random crystal field represents the critical
behavior of $\mathrm{{}^{3}He}-\mathrm{{}^{4}He}$ mixtures in a random media,
i.e., aerogel where $S=0$ and $S=\pm 1$ states represent $\mathrm{{}^{3}He}$
and $\mathrm{{}^{4}He}$ atoms, respectively aerogel1 ; aerogel2 . From the
theoretical point of view, BC model with a random crystal field (RCF) has been
studied by a variety of techniques such as cluster variational method (CVM)
aerogel2 , Bethe lattice approximation (BLA) albayrak , effective field theory
(EFT) kaneyoshi1 ; kaneyoshi2 ; kaneyoshi3 ; kaneyoshi4 ; yan , finite cluster
approximation (FCA) benyoussef1 ; ilkovic , mean field theory (MFT)
benyoussef2 ; borelli ; carneiro2 ; boccara ; carneiro1 ; bahmad , Monte Carlo
(MC) simulations puha , pair approximation (PA) lara , and renormalization
group (RG) method branco . Among these studies, EFT and MFT have been widely
used to investigate the thermal and magnetic properties of BC model with a RCF
distribution. For example, Kaneyoshi and Mielnicki kaneyoshi3 investigated
the phase diagram of the system for a honeycomb lattice by using EFT with
correlations and they found some important differences from the results
obtained by the standard MFT. Similarly, in a recent paper, Yan and Deng yan
considered the same model within the framework of EFT, and they derived the
expressions of magnetizations for honeycomb and square lattices. On the other
hand, in several studies based on MFT, benyoussef2 ; carneiro2 the authors
paid attention to the effects of crystal field dilution on the phase diagrams
of the system and they observed that the system may exhibit a reentrant
behavior, as well as first order phase transitions.
However, EFT and MFT studies mentioned above have some unsatisfactory results.
Namely, the results obtained by EFT are limited to second-order phase
transitions and tricritical points, and a detailed description of first-order
transitions has not been reported. Other than this, it is well known that
magnetic systems with dilute crystal fields exhibit qualitatively similar
characteristics when compared to site dilution problem of magnetic atoms. From
this point of view, for a BC model with diluted crystal fields, MFT predicts
that the phase transition temperature of the system will remain at a finite
value until zero concentration is reached. In this context, we believe that BC
model with RCF still deserves particular attention for investigating the
proper phase diagrams, especially the first-order transition lines that
include reentrant phase transition regions. Conventional EFT approximations
include spin-spin correlations resulting from the usage of the Van der Waerden
identities, and provide results that are superior to those obtained within the
traditional MFT. However, these conventional EFT approximations are not
sufficient enough to improve the results, due to the usage of a decoupling
approximation (DA) that neglects the correlations between different spins that
emerge when expanding the identities. Therefore, taking these correlations
into consideration will improve the results of conventional EFT
approximations. In order to overcome this point, recently we proposed an
approximation that takes into account the correlations between different spins
in the cluster of a considered lattice ak nc . Namely, an advantage of the
approximation method proposed by this study is that no decoupling procedure is
used for the higher-order correlation functions.
In this paper, we intent to investigate the effects of RCF distributions on
the phase diagrams of spin-1 BC model on 2D lattices, namely honeycomb $(q=3)$
and square $(q=4)$ lattices. For this purpose, the paper is organized as
follows: In Sec. II we briefly present the formulations. The results and
discussions are presented in Sec. III, and finally Sec. IV contains our
conclusions.
## II Formulation
In this section, we give the formulation of the present study for a 2D lattice
which has $N$ identical spins arranged. We define a cluster on the lattice
which consists of a central spin labeled $S_{0}$, and $q$ perimeter spins
being the nearest neighbors of the central spin. The cluster consists of
$(q+1)$ spins being independent from the spin operator $\hat{S}$. The nearest-
neighbor spins are in an effective field produced by the outer spins, which
can be determined by the condition that the thermal average of the central
spin is equal to that of its nearest-neighbor spins. The Hamiltonian
describing our model is
$H=-J\sum_{\langle
i,j\rangle}S_{i}^{z}S_{j}^{z}-\sum_{i}D_{i}(S_{i}^{z})^{2},$ (1)
where the first term is a summation over the nearest-neighbor spins with
$S_{i}^{z}=\pm 1,0$ and the term $D_{i}$ on the second summation represents a
random crystal field, distributed according to a given probability
distribution. In this paper, we primarily deal with two kinds of probability
distributions, namely, a quenched diluted crystal field distribution and a
double peaked delta distribution which are given by Eqs. (2) and (3),
respectively as follows
$\displaystyle P(D_{i})$ $\displaystyle=$ $\displaystyle
p\delta(D_{i}-D)+(1-p)\delta(D_{i}),$ (2) $\displaystyle P(D_{i})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\delta[D_{i}-(D-\triangle)]+\delta[D_{i}-(D+\triangle)]\right\\}.$
(3)
where $p$ denotes the concentration of the spins on the lattice which are
influenced by a crystal field $D$.
We can construct the mathematical background of our model by using the
approximated spin correlation identities sa_barreto_et_al by taking into
account random configurational averages
$\langle\langle\\{f_{i}\\}S_{i}^{z}\rangle\rangle_{r}=\left\langle\left\langle\\{f_{i}\\}\frac{\mathrm{Tr}_{i}\left(S_{i}^{z}\right)\exp{(-\beta
H_{i})}}{\mathrm{Tr}_{i}\exp{(-\beta H_{i})}}\right\rangle\right\rangle_{r},$
(4)
$\langle\langle\\{f_{i}\\}(S_{i}^{z})^{2}\rangle\rangle_{r}=\left\langle\left\langle\\{f_{i}\\}\frac{\mathrm{Tr}_{i}(S_{i}^{z})^{2}\exp{(-\beta
H_{i})}}{\mathrm{Tr}_{i}\exp{(-\beta H_{i})}}\right\rangle\right\rangle_{r},$
(5)
where $\beta=1/k_{B}T$, $\\{f_{i}\\}$ is an arbitrary function which is
independent of the spin variable $S_{i}$ and the inner $\langle...\rangle$ and
the outer $\langle...\rangle_{r}$ brackets represents the thermal and random
configurational averages, respectively.
In order to apply the differential operator technique kaneyoshi_honmura ;
kaneyoshi5 , we should separate the Hamiltonian (1) into two parts as
$H=H_{i}+H^{{}^{\prime}}$. Here, the effective Hamiltonian $H_{i}$ includes
all the contributions associated with the site $i$, and the other part
$H^{{}^{\prime}}$ does not depend on the site $i$.
$-H_{i}=ES_{i}^{z}+D_{i}\left(S_{i}^{z}\right)^{2},$ (6)
where $E=J\sum_{j}S_{j}^{z}$ is the local field on the site $i$. If we use the
matrix representations of the operators $S_{i}^{z}$ and $(S_{i}^{z})^{2}$ for
the spin-1 system then we can obtain the matrix form of Eq. (6)
$-H_{i}=\left(\begin{array}[]{ccc}E+D&0&0\\\ 0&0&0\\\ 0&0&-E+D\\\
\end{array}\right).$ (7)
Hereafter, we apply the differential operator technique in Eqs. (4) and (5)
with $\\{f_{i}\\}=1$. From Eq. (4) we obtain the following spin identity with
thermal and configurational averages of a central spin for a lattice with a
coordination number $q$ as
$\langle\langle
S_{0}^{z}\rangle\rangle_{r}=\left\langle\left\langle\prod_{j=1}^{q}\left[1+S_{j}^{z}\mathrm{sinh}(J\nabla)+(S_{j}^{z})^{2}\\{\mathrm{cosh}(J\nabla)-1\\}\right]\right\rangle\right\rangle_{r}F(x)|_{x=0}.$
(8)
The function $F(x)$ in Eq. (8) is defined by
$F(x)=\int dD_{i}P(D_{i})f(x,D_{i}),$ (9)
where
$\displaystyle f(x,D_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{n=1}^{3}\exp(\beta\lambda_{n})}\sum_{n=1}^{3}\langle\varphi_{n}|S_{i}^{z}|\varphi_{n}\rangle\exp(\beta\lambda_{n}),$
$\displaystyle=$ $\displaystyle\frac{2\sinh(\beta x)}{2\cosh(\beta
x)+e^{-\beta D_{i}}}.$
In Eq. (II), $\lambda_{n}$ denotes the eigenvalues of $-H_{i}$ matrix in Eq.
(7), and $\varphi_{n}$ represents the eigenvectors corresponding to the
eigenvalues $\lambda_{n}$ of $-H_{i}$ matrix. With the help of Eq. (II), and
by using the distribution functions defined in Eqs. (2) and (3), the function
$F(x)$ in Eq. (9) can be easily calculated by numerical integration.
Hereafter, we will focus our attention on the construction of the correlation
functions, as well as magnetization and quadrupole moment identities of a
honeycomb lattice with $q=3$. A brief formulation of the fundamental spin
identities for a square lattice with $q=4$ can be found in Appendix A.
By expanding the right-hand side of Eq. (8) for a honeycomb lattice with
$q=3$, we get the longitudinal magnetization as
$\displaystyle m_{z}=\langle\langle S_{0}^{z}\rangle\rangle_{r}$
$\displaystyle=$ $\displaystyle l_{0}+3k_{1}\langle\langle
S_{1}\rangle\rangle_{r}+3(l_{1}-l_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3l_{2}\langle\langle S_{1}S_{2}\rangle\rangle_{r}$
(11) $\displaystyle+6(k_{2}-k_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+3(l_{0}-2l_{1}+l_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle+k_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+3(l_{4}-l_{2})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+3(k_{1}-2k_{2}+k_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-l_{0}+3l_{1}-3l_{3}+l_{5})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r},$
We note that, for the sake of simplicity, the superscript $z$ is omitted from
the correlation functions on the right-hand side of Eq. (11). The coefficients
in Eq. (11) are defined as follows:
$\displaystyle l_{0}=F(0),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
l_{1}=\mathrm{cosh}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $ $\displaystyle k_{1}=\mathrm{sinh}(J\nabla)F(x)|_{x=0},$
$\displaystyle l_{2}=\mathrm{sinh}^{2}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
k_{2}=\mathrm{cosh}(J\nabla)\mathrm{sinh}(J\nabla)F(x)|_{x=0},$ $\displaystyle
l_{3}=\mathrm{cosh}^{2}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $ $\displaystyle k_{3}=\mathrm{sinh}^{3}(J\nabla)F(x)|_{x=0},$
$\displaystyle
l_{4}=\mathrm{cosh}(J\nabla)\mathrm{sinh}^{2}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \
\ \ \ \ $ $\displaystyle
k_{4}=\mathrm{cosh}^{2}(J\nabla)\mathrm{sinh}(J\nabla)F(x)|_{x=0},$
$\displaystyle l_{5}=\mathrm{cosh}^{3}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ $ (12)
Next, the average value of the perimeter spin in the system can be written as
follows, and it is found as
$m_{1}=\langle\langle S_{\delta}^{z}\rangle\rangle_{r}=\langle\langle
1+S_{0}^{z}\mathrm{sinh}(J\nabla)+(S_{0}^{z})^{2}\\{\mathrm{cosh}(J\nabla)-1\\}\rangle\rangle_{r}F(x+\gamma)|_{x=0},$
(13) $\langle\langle
S_{1}\rangle\rangle_{r}=a_{1}\left(1-\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}\right)+a_{2}\langle\langle
S_{0}^{z}\rangle\rangle_{r}+a_{3}\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r},$
(14)
with the coefficients
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle F(\gamma),$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\mathrm{sinh}(J\nabla)F(x+\gamma)|_{x=0},$ $\displaystyle a_{3}$
$\displaystyle=$ $\displaystyle\mathrm{cosh}(J\nabla)F(x+\gamma)|_{x=0},$ (15)
where $\gamma=(q-1)A$ is the effective field produced by the $(q-1)$ spins
outside the system and $A$ is an unknown parameter to be determined self-
consistently. In the effective-field approximation, the number of independent
spin variables describes the considered system. This number is given by the
relation $\nu=\langle\langle(S_{i}^{z})^{2S}\rangle\rangle_{r}$. As an example
for the spin-1 system, $2S=2$ which means that we have to introduce the
additional parameters $\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}$ and
$\langle\langle(S_{\delta}^{z})^{2}\rangle\rangle_{r}$ resulting from the
usage of the Van der Waerden identity for the spin-1 Ising system. With the
help of Eq. (5), quadrupolar moment of the central spin can be obtained as
follows
$\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}=\left\langle\left\langle\prod_{j=1}^{q}\left[1+S_{j}^{z}\mathrm{sinh}(J\nabla)+(S_{j}^{z})^{2}\\{\mathrm{cosh}(J\nabla)-1\\}\right]\right\rangle\right\rangle_{r}G(x)|_{x=0},$
(16)
where the function $G(x)$ is defined as
$G(x)=\int dD_{i}P(D_{i})g(x,D_{i}).$ (17)
Definition of the function $g(x,D_{i})$ in Eq. (17) is given as follows and
the expression in Eq. (II) can be evaluated by using the eigenvalues and
corresponding eigenvectors of the effective Hamiltonian matrix in Eq. (7).
$\displaystyle g(x,D_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{n=1}^{3}\exp(\beta\lambda_{n})}\sum_{n=1}^{3}\langle\varphi_{n}|\left(S_{i}^{z}\right)^{2}|\varphi_{n}\rangle\exp(\beta\lambda_{n}),$
$\displaystyle=$ $\displaystyle\frac{2\cosh(\beta x)}{2\cosh(\beta
x)+e^{-\beta D_{i}}}.$
Hence, we get the quadrupolar moment by expanding the right-hand side of Eq.
(16)
$\displaystyle\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}$
$\displaystyle=$ $\displaystyle r_{0}+3n_{1}\langle\langle
S_{1}\rangle\rangle_{r}+3(r_{1}-r_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3r_{2}\langle\langle S_{1}S_{2}\rangle\rangle_{r}$
(19) $\displaystyle+6(n_{2}-n_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+3(r_{0}-2r_{1}+r_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+n_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$ $\displaystyle+3(r_{4}-r_{2})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}+3(n_{1}-2n_{2}+n_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-r_{0}+3r_{1}-3r_{3}+r_{5})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r},$
with
$\displaystyle r_{0}=G(0),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
r_{1}=\mathrm{cosh}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ $ $\displaystyle n_{1}=\mathrm{sinh}(J\nabla)G(x)|_{x=0},$
$\displaystyle r_{2}=\mathrm{sinh}^{2}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
n_{2}=\mathrm{cosh}(J\nabla)\mathrm{sinh}(J\nabla)G(x)|_{x=0},$ $\displaystyle
r_{3}=\mathrm{cosh}^{2}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $ $\displaystyle n_{3}=\mathrm{sinh}^{3}(J\nabla)G(x)|_{x=0},$
$\displaystyle
r_{4}=\mathrm{cosh}(J\nabla)\mathrm{sinh}^{2}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \
\ \ \ \ $ $\displaystyle
n_{4}=\mathrm{cosh}^{2}(J\nabla)\mathrm{sinh}(J\nabla)G(x)|_{x=0},$
$\displaystyle r_{5}=\mathrm{cosh}^{3}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ $ (20)
Corresponding to Eq. (13),
$\langle\langle(S_{\delta}^{z})^{2}\rangle\rangle_{r}=\langle\langle
1+S_{0}^{z}\mathrm{sinh}(J\nabla)+(S_{0}^{z})^{2}\\{\mathrm{cosh}(J\nabla)-1\\}\rangle\rangle_{r}G(x+\gamma),$
(21) $\langle\langle
S_{1}^{2}\rangle\rangle_{r}=b_{1}\left(1-\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}\right)+b_{2}\langle\langle
S_{0}^{z}\rangle\rangle_{r}+b_{3}\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}.$
(22)
where
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle G(\gamma),$
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\mathrm{sinh}(J\nabla)G(x+\gamma)|_{x=0},$ $\displaystyle b_{3}$
$\displaystyle=$ $\displaystyle\mathrm{cosh}(J\nabla)\rangle
G(x+\gamma)|_{x=0}.$ (23)
Eqs. (11), (14), (19) and (22) are the fundamental spin identities of the
system. When the right-hand sides of Eqs. (8) and (16) are expanded, the
multispin correlation functions appear. The simplest approximation, and one of
the most frequently adopted is to decouple these correlations according to
$\left\langle\left\langle
S_{i}^{z}(S_{j}^{z})^{2}...S_{l}^{z}\right\rangle\right\rangle_{r}\cong\left\langle\left\langle
S_{i}^{z}\right\rangle\right\rangle_{r}\left\langle\left\langle(S_{j}^{z})^{2}\right\rangle\right\rangle_{r}...\left\langle\left\langle
S_{l}^{z}\right\rangle\right\rangle_{r},$ (24)
for $i\neq j\neq...\neq l$ tamura_kaneyoshi . The main difference of the
method used in this study from the other approximations in the literature
emerges in comparison with any decoupling approximation (DA) when expanding
the right-hand sides of Eqs. (8) and (16). In other words, one advantage of
the approximation method used in this study is that no uncontrolled decoupling
procedure is used for the higher-order correlation functions.
For spin-1 Ising system with $q=3$, taking Eqs. (11), (14), (19) and (22) as a
basis, we derive a set of linear equations of the spin identities. At this
point, we assume that (i) the correlations depend only on the distance between
the spins, (ii) the average values of a central spin and its nearest-neighbor
spin (it is labeled as the perimeter spin) are equal to each other with the
fact that, in the matrix representations of spin operator $\hat{S}$, the
spin-1 system has the properties $(S_{\delta}^{z})^{3}=S_{\delta}^{z}$ and
$(S_{\delta}^{z})^{4}=(S_{\delta}^{z})^{2}$. Thus, the number of the set of
linear equations obtained for the spin-1 Ising system with $q=3$ reduces to
twenty one, and the complete set is given in Appendix B.
If Eq. (B) is written in the form of a $21\times 21$ matrix and solved in
terms of the variables $x_{i}[(i=1,2,...,21)(e.g.,x_{1}=\langle\langle
S_{0}^{z}\rangle\rangle_{r},x_{2}=\langle\langle
S_{1}S_{0}\rangle\rangle_{r},...)]$ of the linear equations, all of the spin
correlation functions, as well as magnetizations and quadrupolar moments can
be easily determined as functions of the temperature and Hamiltonian
parameters. Since the thermal and configurational average of the central spin
is equal to that of its nearest-neighbor spins within the present method then
the unknown parameter $A$ can be numerically determined by the relation
$\langle\langle S_{0}^{z}\rangle\rangle_{r}=\langle\langle
S_{1}\rangle\rangle_{r}\qquad{\rm{or}}\qquad x_{1}=x_{4}.$ (25)
By solving Eq. (25) numerically at a given fixed set of Hamiltonian parameters
we obtain the parameter $A$. Then we use the numerical values of $A$ to obtain
the spin correlation functions which can be found from Eq. (B). Note that
$A=0$ is always the root of Eq. (25) corresponding to the disordered state of
the system. The nonzero root of $A$ in Eq. (25) corresponds to the long-range
ordered state of the system. Once the spin identities have been evaluated then
we can give the numerical results for the thermal and magnetic properties of
the system. Since the effective field $\gamma$ is very small in the vicinity
of $k_{B}T_{c}/J$, we can obtain the critical temperature for the fixed set of
Hamiltonian parameters by solving Eq. (25) in the limit of $\gamma\rightarrow
0$ then we can construct the whole phase diagrams of the system. Depending on
the values of Hamiltonian and crystal field distribution parameters, there may
be two solutions [i.e., two critical temperature values which satisfy Eq.
(25)] corresponding to the first (or second) and second-order phase-transition
points, respectively. We determine the type of the transition by looking at
the temperature dependence of magnetization for selected values of system
parameters.
## III Results and Discussion
In this section, we will discuss the effect of the crystal field distributions
defined in Eqs. (2) and (3) on the global phase diagrams of the system where
the second and first order transitions are shown by solid and dashed curves,
respectively with tricritical points (shown by hollow circles) for honeycomb
$(q=3)$ and square $(q=4)$ lattices. Also, in order to clarify the type of the
transitions in the system, we will give the temperature dependence of the
order parameter.
### III.1 Phase diagrams of the system with dilute crystal field
In this section, we illustrate the phase diagrams and magnetization curves of
the system with a dilute crystal field distribution defined in Eq.(2) where
crystal field $D$ is turned on, or turned off with probabilities $p$ and
$(1-p)$ on the lattice sites, respectively. In Figs. (1a) and (1c), we plot
the phase diagrams of the system in $(k_{B}T_{c}/J-D/J)$ plane for honeycomb
and square lattices with coordination numbers $q=3$ and $q=4$, respectively.
As seen in Figs. (1a) and (1c), phase diagrams of the system can be divided
into three parts with different concentration values $p$. For the curves in
the first group with $p<p^{*}$, the system always exhibits a second order
phase transition with a finite critical temperature $k_{B}T_{c}/J$ which
extent to $D/J\rightarrow-\infty$. If the concentration $p$ reaches its
critical value $p^{*}$ then the critical temperature depresses to zero.
Physical reason underlying this behavior can be explained as follows; when we
select sufficiently large negative crystal field values
$(\mathrm{i.e.}D/J\rightarrow-\infty)$, all of the spins in the system tend to
align in $S=0$ state. As $p$ increases starting from zero, the ratio of spins
which aligned in $S=0$ state increases, and therefore, magnetization weakens,
and accordingly, critical temperature of the system decreases. According to
our numerical results, the critical concentration value is obtained as
$p^{*}=0.3795$ for $q=3$ and $p^{*}=0.5875$ for $q=4$. In the second group of
the phase diagrams in Figs. (1a) and (1c), the system exhibits a reentrant
behavior of second order, whereas the curves in the third group, exhibit a
reentrant behavior of first order with a tricritical point at which a first
order transition line turns into a second order transition line. Besides, the
curves which exhibit a reentrant behavior of first (or second) order, depress
to zero at three successive values of crystal field $D/J=-3.0,-2.0,-1.0$.
Moreover, in $D/J\rightarrow\infty$ limit, the system behaves like spin-$1/2$
for $p=1.0$. In the case of $p\neq 0$, the ratio of spins that behave like
$S=\pm 1$ increases as $p$ increases. Therefore, for $0\leq p\leq 1.0$, all
transition lines have finite critical temperatures which increase with
increasing $p$ values for $D/J\rightarrow\infty$. At this point, we also note
that if we select $D/J=0$ in Eq. (2), all lattice sites expose to a crystal
field $D_{i}/J=0$ independent from $p$. Hence, all transition lines intersect
each other on the point $(D/J,k_{B}T_{c}/J)=(0,1.3022)$ for $q=3$, and
$(D/J,k_{B}T_{c}/J)=(0,1.9643)$ for $q=4$. Meanwhile, previous studies based
on EFT are not capable of obtaining first order transition lines of the
system. From this point of view, we see that our method improves the results
of the other EFT works and we take the conventional EFT method one step
forward by investigating the global phase diagrams, especially the first-order
transition lines that include reentrant phase transition regions.
Table 1: Critical concentration $p^{*}$ obtained by several methods and the
present work for honeycomb $(q=3)$ and square $(q=4)$ lattices.
---
Lattice | EFT-I kaneyoshi3 ; kaneyoshi4 | EFT-II yan | MFTbenyoussef2 ; carneiro2 | PA lara | Present Work
$q=3$ | 0.484 | 0.492 | 1.0 | 0.5 | 0.3795
$q=4$ | 0.604 | 0.610 | 1.0 | 0.667 | 0.5875
On the other hand, Figs. (1b) and (1d) shows the phase boundary in
$(k_{B}T_{c}/J-p)$ plane which separates the ferromagnetic and paramagnetic
phases with $D/J\rightarrow-\infty$. According to this figure, critical
temperature $k_{B}T_{c}/J$ of system decreases gradually, and ferromagnetic
region gets narrower as $p$ increases, and $k_{B}T_{c}/J$ value depresses to
zero at $p=p^{*}$. Such a behavior is an expected fact in dilution problems.
Numerical value of critical concentration $p^{*}$ for honeycomb $(q=3)$ and
square $(q=4)$ lattices is given in Table 1, and compared with the other works
in the literature. As seen in Table 1, numerical values of $p_{c}$ for $q=3$
and $q=4$ are new results in literature. Furthermore, MFT benyoussef2 ;
carneiro2 predicts that the system always has a finite critical temperature
and exists in a ferromagnetic state at lower temperatures in
$D/J\rightarrow-\infty$ limit, except that $p=1.0$. This artificial result can
be regarded as a failure of the MFT.
In Fig.(2), we plot the temperature dependencies of magnetization curves
corresponding to the phase diagrams depicted in Fig. (1) for $q=3$. As seen in
Fig. (2), as $p$ increases then critical temperature $k_{B}T_{c}/J$ values
decrease for $D/J<0$, except the reentrant phase transition temperatures which
occur at low temperatures. On the other hand, effect of increasing $p$ values
on the shape of magnetization curves depends on value of $D/J$. Namely, in
Figs. (2a) and (2b) we see that ground state saturation values of
magnetization curves decreases as $p$ increases for $D/J=-10.0$ and $-3.1$.
Moreover, for $D/J=-3.1$, magnetization curves of the system exhibit a broad
maximum at low temperatures for $p=0.37$, and a reentrant behavior of second
order for $p=0.4$. If we select $D/J=-2.5$ as in Fig. (2c), saturation values
of magnetization curves remain unchanged for $p=0,0.2,0.3$ and tend to
decrease for $p>0.3$. If $p$ increases further, a reentrant behavior of second
order appears for $p=0.53$, and we see a broad maximum at low temperatures for
$p=0.517$ and $0.5$ which tends to depress as $p$ decreases. This broad
maximum behavior of magnetization curves originates from the increase in the
number of spins directed parallel to the z-direction with increasing
temperature, due to the thermal agitation. For $D/J=-2.0$ in Fig. (2d),
magnetization curves saturates at $m=1$ at the ground state and reentrant
behavior disappears. If we increase $D/J$ further, for example for $D/J=-1.5$
(Fig. (2e)), another type of reentrant behavior occurs in the system in which
a first order transition is followed by a second order transition. Finally,
for sufficiently large positive values of crystal field, magnetization curves
always saturate at $m=1$ and the system always undergoes a second order phase
transition from a ferromagnetic phase to a paramagnetic phase with increasing
temperature, which can be seen in Fig. (2f) with $D/J=10.0$. As a common
property of the curves in Fig. (2), we see that effect of $p$ on the
saturation values, as well as temperature dependence of magnetization curves
strictly depend on the strength of $D/J$. Hence, according to us, the presence
of dilute crystal fields on the system should produce a competition effect on
the phase diagrams of the system. We also note that, although it has not been
shown in the present work, magnetization curves for $q=4$ corresponding to the
phase diagrams depicted in Fig. (1c) exhibit qualitatively similar behavior
with those of Fig. (2) with $q=3$.
As seen in Fig. (1), for a dilute crystal field distribution defined in Eq.
(2), the global phase diagrams which are plotted in $(k_{B}T_{c}/J-D/J)$
plane, as well as the phase boundaries in $(k_{B}T_{c}/J-p)$ plane for $q=3$
exhibit qualitatively similar characteristics when compared with those for
$q=4$. Hence, in order to examine the phase diagrams which are plotted in
$(k_{B}T_{c}/J-D/J)$ plane in Figs. (1a) and (1c) in detail, we plot the
evolution of the global phase diagrams in Fig. (3) only for $q=4$. From this
point of view, Fig. (3a) shows how the phase diagrams in Fig. (1c) evolve when
the concentration $p$ changes from 0.5 to 0.6. As seen in Fig. (3a), we
observe a second order phase transition line with a finite critical
temperature $k_{B}T_{c}/J$ which extent to $D/J\rightarrow-\infty$ for
$p=0.575$. If $p$ increases, namely for $p=0.580$ and $0.583$, we see that a
low temperature transition line arises between $-4.0<D/J<-3.0$, as well as a
high temperature phase boundary which extents to $D/J\rightarrow-\infty$. If
$p$ increases further, such as for $p=0.584$, $0.585$ and $0.587$, high
temperature phase boundary is gradually connected to the transition line which
arises between $-4.0<D/J<-3.0$, and the phase diagrams exhibit a bulge on the
right hand side of $(k_{B}T_{c}/J-D/J)$ plane, whereas another transition line
emerges within the range of $-\infty<D/J<-4.0$, which disappears for
$p>0.587$. Similarly, evolution of the phase diagrams in Fig. (1c) when the
concentration $p$ changes from 0.6 to 0.7 can be seen in Fig. (3b). As seen in
this figure, the curves for $p=0.60$, $0.62$, $0.64$, $0.66$ exhibit a
reentrant behavior of second order, while for $p=0.68$ reentrance disappears
and for $p=0.70$ and $0.71$ double reentrance with three successive second
order phase transitions occurs in a very narrow region of $D/J$. On the other
hand, increasing values of $p$ generates first order phase transitions with
tricritical points, as well as reentrant behavior of first order. This
phenomena is illustrated in Fig. (3c). From Fig. (3c), we see that, the second
order transition temperatures decrease as absolute value of $D/J$ increases,
and turn into first order transition lines at tricritical points. Evidently,
the phase diagrams change abruptly for $p\geq 0.7164$. Hence, the behavior of
the $p=0.7163$ curve is completely different from that of $p=0.7164$. Namely,
first order transition temperatures of the system for $p\leq 0.7163$ and
$p\geq 0.7164$ depress to zero at $D/J=-3.0$ and $D/J=-2.0$, respectively. In
order to investigate the phase transition features of the system further, we
should continue increasing the value of $p$. In Fig. (3d), we see that the
curves for $p=0.72$, $0.74$, $0.76$ and $0.78$ exhibit a reentrant behavior of
first order, whereas the curves with $p=0.80$, $0.82$, and $0.84$ exhibit
double reentrance with two first order and a second order transition
temperature.
### III.2 Phase diagrams of the system with random crystal field
Next, in order to investigate the effect of the random crystal fields defined
in Eq.(3) on the thermal and magnetic properties of the system, we represent
the phase diagrams and corresponding magnetization curves for honeycomb
$(q=3)$ and square lattices $(q=4)$ throughout Figs. (4) and (8).
We note that random crystal field distribution given in Eq. (3) with $D/J=0$
corresponds to a bimodal distribution function, while for $\triangle/J=0$, we
obtain pure BC model with homogenous crystal field $D/J$. In Fig. (4), phase
diagrams of the system corresponding to the bimodal distribution function are
shown in $(k_{B}T_{c}/J-\triangle/J)$ plane. For a bimodal distribution, the
phase diagrams have symmetric shape with respect to $\triangle/J$ which comes
from the fact that $p=1/2$, and as seen in Fig. (4), transition temperatures
are second order, and it is clear that the system exhibit different
characteristic features depending on the coordination number $q$. Namely, for
$q=3$, transition temperature decreases with increasing $\triangle/J$ and
exhibits double reentrance with three second order phase transition
temperatures, then falls to zero at $\triangle/J=3.0$ (left panel in Fig.
(4)). On the other hand, as seen on the right panel in Fig. (4), as
$\triangle/J$ increases then the transition temperature of the system for
$q=4$ decreases and remains at a finite value for
$\triangle/J\rightarrow\infty$ which means that ferromagnetic exchange
interactions for $q=3$ are insufficient for the system to keep its
ferromagnetic order for $\triangle/J>3.0$, while for $q=4$ these interactions
are dominant in the system, and the presence of a disorder in the crystal
fields cannot destruct the ferromagnetic order.
At the same time, in order to see the effect of the random crystal fields with
$\triangle/J,D/J\neq 0$ on the phase diagrams and magnetization curves of the
system for $q=3$ and $4$, we plot the phase diagrams in $(k_{B}T_{c}/J-D/J)$
plane in Fig. (5) and variation of the corresponding magnetization curves with
temperature in Figs. (6) and (7), respectively. At first sight, it is obvious
that the phase diagrams in Fig.(5) represent evident differences in
qualitative manner with coordination number $q$. That is, as seen in Fig.
(5a), the curve corresponding to $\triangle/J=0$ represents the phase diagram
of pure BC model for a honeycomb lattice which exhibits a reentrant behavior
of first order with first and second order transition lines, as well as a
tricritical point. From Fig. (5a), we see that as $\triangle/J$ increases then
the tricritical point and first order transitions disappear, and the first
order reentrance turns into double reentrance with three transition
temperatures of second order, and phase transition lines shift to positive
crystal field direction without changing their shapes. On the other hand, the
situation is very different for a square lattice. Namely, as seen in Figs.
(5a) and (5b), $\triangle/J=0$ curves for $q=3$ and $q=4$ are qualitatively
identical to each other. However, as seen in Fig. (5b), for $\triangle/J\neq
0$, first order transition lines and tricritical points do not disappear from
the system for $q=4$, but shift to negative crystal field values. Besides, the
system does not exhibit double reentrance for $q=4$. Furthermore, for
$\triangle/J\geq 1.5$ in Fig. (5a), and $\triangle/J\geq 0$ in Fig. (5b), all
phase diagrams exhibit similar behavior as $D/J$ varies. Namely, critical
temperature $k_{B}T_{c}/J$ in Fig. (5a) reduces to zero at
$D/J=\triangle/J-3.0$. On the other hand, first order transition temperatures
in Fig. (5b), reduce to zero at $D/J=-\triangle/J$.
It is important to note that these observations are consistent with the
results shown in Figs. (1a) and (1c). In other words, the distribution
function given in Eq. (2) with $p=0.5$ and $D/J=2D_{0}/J$ is identical to Eq.
(3) for $\triangle/J=\pm D_{0}/J$ and $D/J=D_{0}/J$. For example, according to
Eq. (2), if we select $D_{0}/J=4.0$ with $p=0.5$, it means that half of the
spins on the lattice sites expose to a crystal field $D/J=0$, while a crystal
field given by $D/J=8.0$ acts on the other half of the spins. On the other
hand, if we select $\triangle/J=\pm D_{0}/J$ and $D/J=D_{0}/J$ by using Eq.
(3), we generate the same distribution again. Hence, we expect to get the same
results in Figs. (1) and (5) for these system parameters. For instance, for
$D_{0}/J=4.0$ in Fig. (1a), we get $D/J=8.0$, and the system exhibits a
ferromagnetic order in the ground state, which can also be seen in Fig. (5a)
with a critical temperature $k_{B}T_{c}/J=1.4395$. These conditions are also
valid for $q=4$, and for the whole temperature region on the phase diagrams.
Therefore, the state (para-or ferro), as well as thermal and magnetic
properties of a selected $(k_{B}T/J,D/J)$ point with respect to $p=0.5$ curves
in Figs. (1a) and (1c) is identical to the state of a point $(k_{B}T/J,D/2J)$
in Figs. (5a) and (5b) with respect to the curve $\triangle/J=\pm D/J$,
respectively. Moreover, the qualitative differences between Figs. (5a) and
(5b) mentioned above are strongly related to the percolation threshold value
of the lattice. Namely, distribution function Eq.(3) is valid only for
$p=0.5$. However, as seen in Table 1, we obtain $p_{c}<0.5$ for $q=3$, and
$p_{c}>0.5$ for $q=4$.
In Fig. (6), we examine the temperature dependence of magnetization curves for
$q=3$, corresponding to the phase diagrams shown in Fig. (5a) with $D/J=-1.0$.
Fig. (6a), shows how the temperature dependence of magnetization curves evolve
when $\triangle/J$ changes. According to Fig. (6a), magnetization curves
saturate at a partially ordered state at low temperatures. Besides, for
$\triangle/J=1.68,1.74$ and $1.80$, the system undergoes three successive
phase transitions of second order, which confirms the existence of double
reentrance. Similarly, Fig. (6b) shows how the shape of the magnetization
curves change as $D/J$ changes for constant $\triangle/J=6.0$. As seen in Fig.
(6b), magnetization curves exhibit a second order phase transition from a
ferromagnetic (fully ordered) to a paramagnetic phase at certain values of
crystal field, namely at $D/J=4.0,4.1$ and $4.4$, whereas for
$D/J=3.3,3.5,3.7$ and $3.9$ the system can only achieve a partially ordered
phase. In addition, the curves corresponding to $D/J=3.5,3.7$ and $3.9$
exhibit a broad maximum at low temperatures, and then decrease as the
temperature increases, whereas for $D/J=3.3$, we observe double reentrance.
Fig. (7) shows the magnetization curves for $q=4$, corresponding to the phase
diagrams shown in Fig. (5b). In Fig. (7a), we see that magnetization curves
exhibit a second order phase transition from a paramagnetic phase to a fully
ordered ferromagnetic phase for $\triangle/J=0$ and $5.0$. On the other hand,
the curves corresponding to $D/J=5.5$, $5.8$ and $6.0$, saturate at a
partially ordered state at low temperatures, and exhibit a broad maximum with
increasing temperature which depresses gradually as $\triangle/J$ increases,
then fall rapidly at a second order phase transition temperature. The broad
maximum behavior observed in these curves disappears for $\triangle/J=8.0$.
Additionally, Fig. (7b) represents the magnetization versus temperature curves
for $q=4$ with with $D/J=-4.0$. In Fig. (7b), it is clearly evident that, at
low temperatures, the system saturates at a partially ordered phase for
$\triangle/J=4.0$, $5.0$, $6.0$ and $7.0$, while for $\triangle/J=3.7$ and
$3.8$, a reentrant behavior of first order occurs. Again we see that, there is
a competition between ferromagnetic exchange interactions and disorder effects
in crystal fields which determines the saturation values and temperature
dependence of magnetization curves of the system.
Finally, dependence of magnetization of the system on the crystal field
$\triangle/J$ for fixed temperature values $k_{B}T/J=0.01,0.05,0.1$ and $0.2$
with $D/J=2.0$ is shown in Fig. (8) for $q=3$ and $4$, respectively. We see
that at sufficiently low temperatures such as $k_{B}T/J=0.01$, magnetization
curves exhibit three phases for $q=3$. A first order transition is
characterized by a gap in this figure. On the left panel in Fig. (8), which is
plotted for $q=3$, we observe two successive first order phase transitions for
$k_{B}T/J=0.01$. The first one is from the fully ordered ferromagnetic phase
($m=1.0$) to the partly ordered phase ($m=0.47$), and the other is from partly
ordered phase to disordered phase ($m=0.0$). On the other hand, according to
the right panel in Fig. (8), the system can not reach a paramagnetic phase at
the ground state for $q=4$.Hence, we observe two phases. Namely, for
$k_{B}T/J=0.01$, a first order phase transition from a fully ordered phase
($m=1.0$) to a partly ordered phase $(m=0.59)$. Then, saturation magnetization
of the partly ordered phase reduces continuously to ($m=0.527$). Moreover, the
first order transitions disappear with increasing temperatures, both for $q=3$
and $4$. Since in Figs. (5a) and (5b), all phase diagrams exhibit similar
behavior as $D/J$ varies, behavior of magnetization curves in Fig. (8) should
be the same for different $D/J$ values. Namely, the left panel of Fig. (8) can
be regarded as the variation of magnetization with $\triangle/J$ within the
range $D/J+1.0<\triangle/J<D/J+4.0$ for $q=3$. for a selected value of $D/J$.
It is possible to observe the similar behavior on the right panel in Fig. (8).
This general behavior is an expected result, since the behavior of the system
depends on the relation between $\triangle/J$ and $D/J$ parameters for a given
temperature, not their values independently.
## IV Conclusions
In this work, we have studied the phase diagrams of a spin-1 Blume-Capel model
with diluted and random crystal field interactions on two dimensional
lattices. We have introduced an effective-field approximation that takes into
account the correlations between different spins in the cluster of a
considered lattice and examined the phase diagrams as well as magnetization
curves of the system for different types of crystal field distributions,
namely, dilute crystal fields and a double peaked delta distribution, given by
Eqs. (2) and (3), respectively.
For dilute crystal fields, we have given a detailed exploration of the global
phase diagrams of the system in $k_{B}T_{c}/J-D/J$ plane with the second and
first order transitions, as well as tricritical points. We have also shown
that the system with dilute crystal fields exhibits a percolation threshold
value $p_{c}$ which can not be predicted by standard MFA. In addition, we have
observed multi-reentrant phase transitions for specific set of system
parameters.
On the other hand, we have investigated the effect of the random crystal field
distribution characterized by two crystal field parameters $D/J$ and
$\triangle/J$ on the phase diagrams of the system. As a limited case, we have
also focused on a bimodal distribution with $D/J=0$. Particulary, we have
reported the following observations for a bimodal distribution: It has been
found that the phase diagrams have symmetric shape with respect to
$\triangle/J$ which comes from the fact that $p=1/2$. The transition
temperatures are of second order, and the system exhibit different
characteristic features depending on the coordination number $q$. Besides, we
have realized that the system may exhibit clear distinctions in qualitative
manner with coordination number $q$ for random crystal fields with
$\triangle/J,D/J\neq 0$. Moreover, we have discussed a competition effect
which arises from the presence of dilution, as well as random crystal fields,
and we have observed that saturation values of the magnetization curves are
strongly related to these effects.
As a result, we can conclude that all of the points mentioned above show that
our method improves the conventional EFT methods based on decoupling
approximation. Therefore, we hope that the results obtained in this work may
be beneficial from both theoretical and experimental points of view.
## Acknowledgements
One of the authors (Y.Y.) would like to thank the Scientific and Technological
Research Council of Turkey (TÜBİTAK) for partial financial support. This work
has been completed at Dokuz Eylül University, Graduate School of Natural and
Applied Sciences. Partial financial support from SRF (Scientific Research
Fund) of Dokuz Eylül University (2009.KB.FEN.077) (H.P.) is also acknowledged.
## Appendix A Fundamental correlation functions of the system for a square
lattice
Magnetization of the central spin for a square lattice is given as follows
$\displaystyle\left\langle\langle S_{0}^{z}\right\rangle\rangle$
$\displaystyle=$ $\displaystyle\mu_{0}+4c_{1}\langle\langle
S_{1}\rangle\rangle_{r}+4(\mu_{2}-\mu_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}$ (26) $\displaystyle+6\mu_{1}\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+12(c_{2}-c_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle+6(\mu_{0}-2\mu_{2}+\mu_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+4c_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$
$\displaystyle+12(\mu_{4}-\mu_{1})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}+12(c_{4}-2c_{2}+c_{1})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+4(\mu_{5}-3\mu_{3}+3\mu_{2}-\mu_{0})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}+\mu_{8}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}$
$\displaystyle+4(c_{5}-c_{3})\langle\langle
S_{1}S_{2}S_{3}S_{4}^{2}\rangle\rangle_{r}+6(\mu_{1}-2\mu_{4}+\mu_{6})\langle\langle
S_{1}S_{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r}$
$\displaystyle+4(c_{6}-3c_{4}+3c_{2}-c_{1})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r}$
$\displaystyle+(\mu_{0}-4\mu_{2}+6\mu_{3}-4\mu_{5}+\mu_{7})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r},$
where the coefficients are given by
$\displaystyle\mu_{0}=F(0),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ $ $\displaystyle\mu_{1}=\sinh^{2}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \
\ \ \ \ \ \ \ $ $\displaystyle c_{1}=\sinh(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{2}=\cosh(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle c_{2}=\sinh(J\nabla)\cosh(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{3}=\cosh^{2}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle c_{3}=\sinh^{3}(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{4}=\sinh^{2}(J\nabla)\cosh(J\nabla)F(x)|_{x=0},$
$\displaystyle c_{4}=\cosh^{2}(J\nabla)\sinh(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{5}=\cosh^{3}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle c_{5}=\sinh^{3}(J\nabla)\cosh(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{6}=\sinh^{2}(J\nabla)\cosh^{2}(J\nabla)F(x)_{x=0},$
$\displaystyle c_{6}=\cosh^{3}(J\nabla)\sinh(J\nabla)F(x)_{x=0},$
$\displaystyle\mu_{7}=\cosh^{4}(J\nabla)F(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle\mu_{8}=\sinh^{4}(J\nabla)F(x)_{x=0}.\ \ \ \ \ \ \ \ \ \ \ \ \
\ $
Quadrupolar moment corresponding to equation (B) defined as
$\displaystyle\left\langle\langle(S_{0}^{z})^{2}\right\rangle\rangle$
$\displaystyle=$ $\displaystyle\rho_{0}+4\eta_{1}\langle\langle
S_{1}\rangle\rangle_{r}+4(\rho_{2}-\rho_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}$ (27) $\displaystyle+6\rho_{1}\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+12(\eta_{2}-\eta_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle+6(\rho_{0}-2\rho_{2}+\rho_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+4\eta_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$
$\displaystyle+12(\rho_{4}-\rho_{1})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}+12(\eta_{4}-2\eta_{2}+\eta_{1})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+4(\rho_{5}-3\rho_{3}+3\rho_{2}-\rho_{0})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}+\rho_{8}\langle\langle
S_{1}S_{2}S_{3}S_{4}\rangle\rangle_{r}$
$\displaystyle+4(\eta_{5}-\eta_{3})\langle\langle
S_{1}S_{2}S_{3}S_{4}^{2}\rangle\rangle_{r}+6(\rho_{1}-2\rho_{4}+\rho_{6})\langle\langle
S_{1}S_{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r}$
$\displaystyle+4(\eta_{6}-3\eta_{4}+3\eta_{2}-\eta_{1})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r}$
$\displaystyle+(\rho_{0}-4\rho_{2}+6\rho_{3}-4\rho_{5}+\rho_{7})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}S_{4}^{2}\rangle\rangle_{r},$
where
$\displaystyle\rho_{0}=G(0),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ $ $\displaystyle\rho_{1}=\sinh^{2}(J\nabla)G(x)|_{x=0},\ \ \ \ \
\ \ \ \ \ \ \ \ $ $\displaystyle\eta_{1}=\sinh(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{2}=\cosh(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\eta_{2}=\sinh(J\nabla)\cosh(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{3}=\cosh^{2}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle\eta_{3}=\sinh^{3}(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{4}=\sinh^{2}(J\nabla)\cosh(J\nabla)G(x)|_{x=0},$
$\displaystyle\eta_{4}=\cosh^{2}(J\nabla)\sinh(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{5}=\cosh^{3}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle\eta_{5}=\sinh^{3}(J\nabla)\cosh(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{6}=\sinh^{2}(J\nabla)\cosh^{2}(J\nabla)G(x)_{x=0},$
$\displaystyle\eta_{6}=\cosh^{3}(J\nabla)\sinh(J\nabla)G(x)_{x=0},$
$\displaystyle\rho_{7}=\cosh^{4}(J\nabla)G(x)|_{x=0},\ \ \ \ \ \ \ \ \ \ \ \ \
$ $\displaystyle\rho_{8}=\sinh^{4}(J\nabla)G(x)_{x=0}.\ \ \ \ \ \ \ \ \ \ \ \
\ \ $
Finally, perimeter spin identities are as follows
$\displaystyle\langle\langle S_{1}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle\alpha_{1}(1-\langle\langle(S_{0})^{2}\rangle\rangle_{r})+\alpha_{2}\langle\langle
S_{0}\rangle\rangle_{r}+\alpha_{3}\langle\langle(S_{0})^{2}\rangle\rangle_{r},$
(28) $\displaystyle\langle\langle S_{1}^{2}\rangle\rangle_{r}$
$\displaystyle=$ $\displaystyle\omega_{1}+\omega_{2}\langle\langle
S_{0}\rangle\rangle_{r}+(\omega_{3}-\omega_{1})\langle\langle
S_{0}^{2}\rangle\rangle_{r}$ (29)
with the coefficients
$\displaystyle\alpha_{1}=F(\gamma)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\omega_{1}=G(\gamma)|_{x=0}$
$\displaystyle\alpha_{2}=\sinh(J\nabla)F(x+\gamma)$
$\displaystyle\omega_{2}=\sinh(J\nabla)G(x+\gamma)|_{x=0}$
$\displaystyle\alpha_{3}=\cosh(J\nabla)F(x+\gamma)$
$\displaystyle\omega_{3}=\cosh(J\nabla)G(x+\gamma)|_{x=0}$
where $\gamma=(q-1)A$ with $q=4$, and the functions $F(x)$ and $G(x)$ are
defined in equations (9) and (17).
## Appendix B The complete set of twenty one linear equations of a honeycomb
lattice
$\displaystyle\langle\langle S_{0}^{z}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle l_{0}+3k_{1}\langle\langle
S_{1}\rangle\rangle_{r}+3(l_{1}-l_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3l_{2}\langle\langle S_{1}S_{2}\rangle\rangle_{r}$
$\displaystyle+6(k_{2}-k_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+3(l_{0}-2l_{1}+l_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle+k_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}+3(l_{4}-l_{2})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+3(k_{1}-2k_{2}+k_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-l_{0}+3l_{1}-3l_{3}+l_{5})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{0}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle(3l_{1}-2l_{0})\langle\langle
S_{1}\rangle\rangle_{r}+3k_{1}\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3(l_{0}-2l_{1}+l_{2}+l_{3})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle+6(k_{2}-k_{1})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+k_{3}\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-l_{0}+3l_{1}-3l_{2}-3l_{3}+3l_{4}+l_{5})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+3(k_{1}-2k_{2}+k_{4})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}S_{0}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle({l_{0}}-3{l_{1}}+3{l_{2}}+3{l_{3}})\langle\langle{S_{1}}{S_{2}}\rangle\rangle_{r}+(6{k_{2}}-3{k_{1}})\langle\langle{S_{1}}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle+(-{l_{0}}+3{l_{1}}-3{l_{2}}-3{l_{3}}+3{l_{4}}+{l_{5}})\langle\langle{S_{1}}{S_{2}}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(3{k_{1}}-6{k_{2}}+{k_{3}}+3{k_{4}})\langle\langle{S_{1}}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle\langle\langle S_{1}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle
a_{1}(1-\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r})+a_{2}\langle\langle
S_{0}^{z}\rangle\rangle_{r}+a_{3}\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}$
$\displaystyle\langle\langle S_{1}S_{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle a_{1}\langle\langle S_{1}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}\rangle\rangle_{r}+(a_{3}-a_{1})\langle\langle
S_{1}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
a_{1}\langle\langle S_{1}S_{2}\rangle\rangle_{r}+a_{2}\langle\langle
S_{0}S_{1}S_{2}\rangle\rangle_{r}+(a_{3}-a_{1})\langle\langle
S_{1}S_{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{1}(1-\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r})+b_{2}\langle\langle
S_{0}^{z}\rangle\rangle_{r}+b_{3}\langle\langle(S_{0}^{z})^{2}\rangle\rangle_{r}$
$\displaystyle\langle\langle S_{1}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle=$ $\displaystyle b_{1}\langle\langle
S_{1}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}S_{1}\rangle\rangle_{r}+(b_{3}-b_{1})\langle\langle
S_{1}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{1}\langle\langle S_{1}^{2}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}S_{1}^{2}\rangle\rangle_{r}+(b_{3}-b_{1})\langle\langle
S_{1}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{0}S_{1}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{3}\langle\langle S_{0}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{0}S_{1}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{3}\langle\langle S_{0}S_{1}\rangle\rangle_{r}+b_{2}\langle\langle
S_{1}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{0}S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{3}\langle\langle S_{0}S_{1}^{2}\rangle\rangle_{r}+b_{2}\langle\langle
S_{1}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{1}\langle\langle S_{1}S_{2}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}S_{1}S_{2}\rangle\rangle_{r}+(b_{3}-b_{1})\langle\langle
S_{1}S_{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{1}\langle\langle S_{1}S_{2}^{2}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}S_{1}S_{2}^{2}\rangle\rangle_{r}+(b_{3}-b_{1})\langle\langle
S_{1}S_{2}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
b_{1}\langle\langle S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+b_{2}\langle\langle
S_{0}S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+(b_{3}-b_{1})\langle\langle
S_{1}^{2}S_{2}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$ $\displaystyle
r_{0}+3n_{1}\langle\langle
S_{1}\rangle\rangle_{r}+3(r_{1}-r_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3r_{2}\langle\langle S_{1}S_{2}\rangle\rangle_{r}$
$\displaystyle+6(n_{2}-n_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+3(r_{0}-2r_{1}+r_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+n_{3}\langle\langle
S_{1}S_{2}S_{3}\rangle\rangle_{r}$ $\displaystyle+3(r_{4}-r_{2})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}+3(n_{1}-2n_{2}+n_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-r_{0}+3r_{1}-3r_{3}+r_{5})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle(3r_{1}-2r_{0})\langle\langle
S_{1}\rangle\rangle_{r}+3n_{1}\langle\langle
S_{1}^{2}\rangle\rangle_{r}+(3r_{2}+3r_{0}-6r_{1}+3r_{3})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}$ $\displaystyle+6(n_{2}-n_{1})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+n_{3}\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-r_{0}+3r_{1}-3r_{2}-3r_{3}+3r_{4}+r_{5})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+3(n_{1}-2n_{2}+n_{4})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle(3r_{1}-2r_{0})\langle\langle
S_{1}^{2}\rangle\rangle_{r}+3n_{1}\langle\langle
S_{1}\rangle\rangle_{r}+(3r_{2}+3r_{0}-6r_{1}+3r_{3})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle+6(n_{2}-n_{1})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+(3n_{1}-6n_{2}+n_{3}+3n_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-r_{0}+3r_{1}-3r_{2}-3r_{3}+3r_{4}+r_{5})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle(r_{0}-3r_{1}+3r_{2}+3r_{3})\langle\langle
S_{1}S_{2}\rangle\rangle_{r}+(-3n_{1}+6n_{2})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle(-r_{0}+3r_{1}-3r_{2}-3r_{3}+3r_{4}+r_{5})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(3n_{1}-6n_{2}+n_{3}+3n_{4})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}S_{2}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle(r_{0}-3r_{1}+3r_{2}+3r_{3})\langle\langle
S_{1}S_{2}^{2}\rangle\rangle_{r}+(-3n_{1}+6n_{2})\langle\langle
S_{1}S_{2}\rangle\rangle_{r}$
$\displaystyle(-r_{0}+3r_{1}-3r_{2}-3r_{3}+3r_{4}+r_{5})\langle\langle
S_{1}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(3n_{1}-6n_{2}+n_{3}+3n_{4})\langle\langle
S_{1}S_{2}S_{3}^{2}\rangle\rangle_{r}$ $\displaystyle\langle\langle
S_{1}^{2}S_{2}^{2}S_{0}^{2}\rangle\rangle_{r}$ $\displaystyle=$
$\displaystyle({r_{0}}-3{r_{1}}+3{r_{2}}+3{r_{3}})\langle\langle
S_{1}^{2}S_{2}^{2}\rangle\rangle_{r}+(-3{n_{1}}+6{n_{2}})\langle\langle{S_{1}}S_{2}^{2}\rangle\rangle_{r}$
$\displaystyle+(3{n_{1}}-6{n_{2}}+{n_{3}}+3{n_{4}})\langle\langle{S_{1}}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}$
$\displaystyle+(-{r_{0}}+3{r_{1}}-3{r_{2}}-3{r_{3}}+3{r_{4}}+{r_{5}})\langle\langle
S_{1}^{2}S_{2}^{2}S_{3}^{2}\rangle\rangle_{r}.$
## References
* (1)
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* (15) M. E. S. Borelli and C. E. I Carneiro, Physica A 230, 249 (1996).
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* (17) N. Boccara, A. El Kenz, and M. Saber, J. Phys. Condens. Matter 1, 5721 (1989).
* (18) C. E. I Carneiro, V. B. Henriques, and S. R. Salinas, J. Phys. A Math. Gen. 23, 3383 (1990).
* (19) L. Bahmad, A. Benyoussef, and A. El Kenz, J. Magn. Magn. Mater 320, 397 (2008).
* (20) I. Puha and H. T. Diep, J. Magn. Magn. Mater. 224, 85 (2001).
* (21) D. P Lara and J. A. Plascak, Physica A 260, 443 (1998).
* (22) N. S. Branco and B. M. Boechat, Phys. Rev. B 56, 11673 (1997).
* (23) Ü. Akıncı, Y. Yüksel, and H. Polat, Phys. Rev. E 83, 061103 (2011).
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Figure 1: (a) Phase diagrams of the system for $q=3$ in a $(k_{B}T_{c}/J-D/J)$
plane corresponding to dilute crystal field distribution defined in Eq. (2).
The solid and dashed lines correspond to second- and first-order phase
transitions, respectively. The open circles denote the tricritical points, and
the numbers on each curve represent the value of concentration $p$. (b) Phase
diagrams of the system for $q=3$ in a $(k_{B}T_{c}/J-p)$ plane with a selected
value of the crystal field $D/J=-10.0$. (c) Phase diagrams of the system for
$q=4$ in a $(k_{B}T_{c}/J-D/J)$ plane corresponding to dilute crystal field
distribution defined in Eq. (2). The solid and dashed lines correspond to
second- and first-order phase transition, respectively. The open circles refer
to the tricritical points, and the numbers on each curve represent the value
of concentration $p$. (d) Phase diagrams of the system for $q=4$ in a
$(k_{B}T_{c}/J-p)$ plane with a selected value of the crystal field
$D/J=-10.0$.
Figure 2: (Color online) Temperature dependence of magnetization corresponding
to Fig. (1a) with some selected values of crystal field. (a) $D/J=-10.0$, (b)
$D/J=-3.1$, (c) $D/J=-2.5$, (d) $D/J=-2.0$, (e) $D/J=-1.5$, and (f)
$D/J=10.0$. The numbers on each curve denote the value of concentration $p$.
The solid and dashed lines correspond to second- and first-order phase
transitions, respectively.
Figure 3: (Color online) Evolution of the phase diagrams corresponding to Fig.
(1c). The numbers on each curve denote the value of concentration $p$. The
solid and dashed lines correspond to second- and first-order phase
transitions, respectively. The open circles indicate the tricritical points.
Figure 4: Phase diagrams of the system in a $(k_{B}T_{c}/J-\triangle/J)$ plane
for a bimodal crystal field distribution corresponding to Eq. (3) with
$D/J=0.0$. Left and right-hand side panels are plotted for $q=3$ and $q=4$,
respectively.
Figure 5: (Color online) Phase diagrams of the system in a
$(k_{B}T_{c}/J-D/J)$ plane corresponding to random crystal field distribution
defined in Eq. (3) for (a) $q=3$, (b) $q=4$. The numbers on each curve denote
the value of $\triangle/J$. The open circles represent the tricritical points,
and the solid and dashed lines correspond to second- and first-order phase
transitions, respectively.
Figure 6: (Color online) (a) Temperature dependence of magnetization curves
corresponding to Fig. (5a) for $q=3$ (a) with $D/J=-1.0$ and for some selected
values of $\triangle/J$, and (b) with $\triangle/J=6.0$ and for some selected
values of $D/J$.
Figure 7: (Color online) (a) Temperature dependence of magnetization curves
for $q=4$ corresponding to Fig. (5b) for (a) $D/J=2.0$ and (b) $D/J=-4.0$ with
some selected values of $\triangle/J$.
Figure 8: (Color online) Variation of the magnetization curves as a function
of $\triangle/J$ with $D/J=2.0$. Left and right panels are plotted for $q=3$
and $4$, respectively. Each curve is plotted for different temperature values,
namely $k_{B}T/J=0.01,0.05,0.1$ and $0.2$.
|
arxiv-papers
| 2011-07-21T08:14:47 |
2024-09-04T02:49:20.786781
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yusuf Y\\\"uksel, \\\"Umit Ak{\\i}nc{\\i}, and Hamza Polat",
"submitter": "Yusuf Yuksel",
"url": "https://arxiv.org/abs/1107.4194"
}
|
1107.4214
|
# Competitive Brownian and Lévy walkers
E. Heinsalu IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos
(CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de
Mallorca, Spain National Institute of Chemical Physics and Biophysics, Rävala
10, Tallinn 15042, Estonia E. Hernández-García IFISC, Instituto de Física
Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat de les
Illes Balears, E-07122 Palma de Mallorca, Spain C. López IFISC, Instituto de
Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat de
les Illes Balears, E-07122 Palma de Mallorca, Spain
(November 18, 2011)
###### Abstract
Population dynamics of individuals undergoing birth and death and diffusing by
short or long ranged twodimensional spatial excursions (Gaussian jumps or Lévy
flights) is studied. Competitive interactions are considered in a global case,
in which birth and death rates are influenced by all individuals in the
system, and in a nonlocal but finite-range case in which interaction affects
individuals in a neighborhood (we also address the noninteracting case). In
the global case one single or few-cluster configurations are achieved with the
spatial distribution of the bugs tied to the type of diffusion. In the Lévy
case long tails appear for some properties characterizing the shape and
dynamics of clusters. Under non-local finite-range interactions periodic
patterns appear with periodicity set by the interaction range. This length
acts as a cut-off limiting the influence of the long Lévy jumps, so that
spatial configurations under the two types of diffusion become more similar.
By dividing initially everyone into different families and following their
descent it is possible to show that mixing of families and their competition
is greatly influenced by the spatial dynamics.
###### pacs:
05.40.-a, 05.40.Fb, 87.18.Hf, 87.23.Cc
## I Introduction
Birth and death are the most relevant processes in determining the dynamics of
biological populations which in the context of statistical physics can be
modeled using interacting particle models where particle number is changing in
time. As it is understood by now, birth and death processes are also
responsible for clustering mechanisms in systems where random-walking
individuals undergo reproduction and death. As a result, aggregation of
organisms can occur even in simple models where birth and death processes are
combined with spatial diffusion. In fact, in the most simple Brownian bug
model, where particles reproduce and die with the same probability and undergo
Brownian motion Zhang et al. (1990); Young et al. (2001); Felsenstein (1975),
clustering of particles was observed. In this model the clustering is produced
simply by the reproductive correlations (the offspring is born at the same
location of the parent) and by the irreversibility of the death process. Birth
and death models of moving individuals are the pertinent framework to capture
properties of biological systems such as planktonic populations Young et al.
(2001), or patterns in amoebae Houchmandzadeh (2008) and bacteria Ramos et al.
(2008).
Taking into account another central ingredient that is present in ecological
systems, namely, the competition with other individuals in the neighborhood
for resources, the formation of periodic spatial structures was observed in
Refs. Hernández-García and López (2004); Hernández-Garcia and López (2005);
López and Hernández-Garcia (2004). In these nonlocally interacting Brownian
bug models it was assumed that the reproduction probability depends on the
number of other organisms in the neighborhood. In Ref. Heinsalu et al. (2010)
nonlocally interacting Lévy bugs, i.e., reproducing and dying organisms that
undergo Lévy flights, were studied. This type of motion is relevant to model
cell migration Dieterich et al. (2008), biological searching strategies Sims
et al. (2008); Bartumeus et al. (2005), bacteria dynamics Levandowsky et al.
(1997), or pattern formation of mussels de Jager et al. (2011). In Ref.
Heinsalu et al. (2010) it was shown that the formation of a periodic pattern
is robust with respect to the type of spatial motion that the particles
perform. The periodic arrangement of clusters in these nonlocally interacting
bug models is a consequence of the competitive interaction and has a spatial
scale determined by the interaction range Hernández-García and López (2004).
However, a deeper analysis of the differences and similarities between the
Brownian and Lévy cases is still missing. In particular, as shown in
Hernández-Garcia and López (2005); Brigatti et al. (2008), this analysis can
be very conveniently performed by considering the limit of the interaction
distance reaching the system size (global interaction), since a unique cluster
appears which helps to understand and characterize the cluster properties, and
the fluctuations of the population size.
In the present paper we report on differences between the systems of Brownian
and Lévy bugs, in the situations of global and non-local interactions, as well
as in the noninteracting case. In addition, results on the dependence of
population on diffusion, and mixing of families of particles are presented for
the finite-range interaction case. The paper is organized as follows: in Sec.
II we describe the models to be analyzed. In Sec. III the noninteracting bug
systems are studied. The infinite competition range where each particle is
competing with all the others is analyzed in Sec. IV. Finally, the nonlocally
interacting (i.e. with a finite interaction range) models are investigated in
Sec. V.
## II Model and numerical algorithm
We consider a system consisting initially of $N_{0}$ point-like particles,
which could be thought as being biological organisms or bugs, placed randomly
in a two-dimensional $L\times L$ square domain with periodic boundary
conditions. Except when explicitly stated, we take $L=1$, so that lengths are
measured in units of system size. Bugs diffuse, reproduce at rate $r^{i}_{b}$,
and die at rate $r^{i}_{d}$; $i=1,\ldots,N$, and $N\equiv N(t)$ is the number
of bugs in the system at time $t$. The numerical algorithm used to evolve the
system follows the one suggested in Ref. Birch and Young (2006). The following
sequence of steps is repeated until the final simulation time is reached:
We first compute the random time $\tau$ after which the next demographic event
(birth or death) will occur. For this we need to determine the total birth and
death rates,
$B_{\mathrm{tot}}=\sum_{i=1}^{N}r^{i}_{b}\,,\qquad
D_{\mathrm{tot}}=\sum_{i=1}^{N}r^{i}_{d}\,,$ (1)
and compute also the total rate
$R_{\mathrm{tot}}=B_{\mathrm{tot}}+D_{\mathrm{tot}}=\sum_{i=1}^{N}(r^{i}_{b}+r^{i}_{d})\,.$
(2)
For the random times $\tau$ we choose an exponential probability density with
the complementary cumulative distribution
$p(\tau)=\exp(-\tau/\tilde{\tau})$ (3)
so that values of $\tau$ could be generated from
$\tau=-\tilde{\tau}\ln(\xi_{0})$, where $\xi_{0}$ is a uniform random number
on $(0,1)$ Press et al. (1992). The characteristic time or time-scale
parameter $\tilde{\tau}=\langle\tau\rangle$ is determined by the total rate:
$\tilde{\tau}=R_{\mathrm{tot}}^{-1}\,.$ (4)
After the random time $\tau$, an individual $i$, chosen among all the $N(t)$
bugs, either reproduces or disappears. With probability
$B_{\mathrm{tot}}/R_{\mathrm{tot}}$ the event is reproduction and with
probability $D_{\mathrm{tot}}/R_{\mathrm{tot}}$ it is death. The probability
of choosing a particular individual $i$ is weighted proportionally to its
contribution to the corresponding total rate. In the case of reproduction, the
new bug is located at the same position $(x_{i},y_{i})$ as the parent
individual $i$. Finally, all the bugs perform a jump of random length $\ell$
in a random direction characterized by an angle uniformly distributed on
$(0,2\pi)$ ($\ell$ and the direction of the jump are independent for each
particle). The new present time is $t+\tau$, bugs are relabeled with indices
$i=1,2,...,N(t+\tau)$, and the process is repeated.
When bugs undergo normal diffusion (Brownian bugs), a Gaussian jump-length
probability density function is used,
$\varphi(\ell)=\frac{2}{\tilde{\ell}\sqrt{2\pi}}\exp\left(-\frac{\ell^{2}}{2{\tilde{\ell}}^{2}}\right)\,,\,l\geq
0$ (5)
with second moment $\langle\ell^{2}\rangle={\tilde{\ell}}^{2}$; $\tilde{\ell}$
is the space-scale parameter. Since we draw the angle specifying the direction
of the jump from the interval $(0,2\pi)$, we restrict $\ell$ in Eq. (5) to
have positive sign. The random jump length $\ell$ can be computed from
$\ell=\tilde{\ell}\,\xi_{G}$, where $\xi_{G}$ is sampled from the standard
Gaussian distribution with average $0$ and standard deviation $1$, and
neglecting the sign. Note that the random walk defined in this way is not
exactly the same as the one in which the walker performs jumps extracted from
a two-dimensional Gaussian distribution, but it also leads to normal diffusion
and allows a more direct comparison with the Lévy case. The corresponding
diffusion coefficient can be estimated as
$\kappa=\langle\ell^{2}\rangle/(2\langle\tau\rangle)={\tilde{\ell}}^{2}/(2\tilde{\tau})\,.$
(6)
As we choose to fix the value of $\kappa$, and the demographic rates, then the
space-scale parameter is determined by
$\tilde{\ell}=\sqrt{2\kappa\tilde{\tau}}=\sqrt{2\kappa/R_{\mathrm{tot}}}\,.$
(7)
In order to simulate the system where the bugs undergo superdiffusive Lévy
flights (Lévy bugs) one can use a symmetric Lévy-type probability density
function for the jump size ($\ell\geq 0$), behaving asymptotically as Klages
et al. (2008); Metzler and Klafter (2000)
$\varphi_{\mu}(\ell)\approx{\tilde{\ell}}^{\mu}|\ell|^{-\mu-1}\,,\quad\ell\to\infty\quad(\ell\gg\tilde{\ell})$
(8)
with the Lévy index $0<\mu<2$. For all Lévy-type probability density functions
with $\mu<2$ the second moment diverges, $\langle\ell^{2}\rangle=\infty$,
leading to the occurrence of extremely long jumps, and typical trajectories
are self-similar, showing at all scales clusters of shorter jumps interspersed
with long excursions. For $0<a<\mu<2$ fractional moments
$\langle\ell^{a}\rangle$ are finite. For the Lévy index in the range $1<\mu<2$
the value of $\langle\ell\rangle$ is finite. The complementary cumulative
distribution corresponding to (8) behaves as
$P_{\mu}(\ell)\approx{\mu}^{-1}(\ell/\tilde{\ell})^{-\mu}\,,\quad\ell\to\infty\,.$
(9)
As a simple form of complementary cumulative distribution function which
behaves asymptotically as (9), we use
$P_{\mu}(\ell)=(1+b^{1/\mu}\ell/\tilde{\ell})^{-\mu}\,,$ (10)
with $b=[\Gamma(1-\mu/2)\Gamma(\mu/2)]/\Gamma(\mu)$, and $\ell\geq 0$. As
before, the direction of the jump is assigned by drawing an random angle on
$(0,2\pi)$. The particular expression for $b$ is chosen for consistency with
previous work (Heinsalu et al., 2010). It gives to the tail of the jump
distribution the same prefactor as for the Lévy-stable distribution (Nolan,
2012), but any other positive value of $b$ should lead to the same results
reported here. One can generate a random step-length $\ell$ by inverting (10):
$\ell=\tilde{\ell}\frac{(\xi_{0}^{-1/\mu}-1)}{b^{1/\mu}}\,.$ (11)
with $\xi_{0}$ being a uniform random variable on the unit interval. Now the
diffusion coefficient (6) is infinite, but one can define a generalized
diffusion coefficient in terms of the scales $\tilde{\ell}$ and $\tilde{\tau}$
as Metzler and Klafter (2000); Klages et al. (2008)
$\kappa_{\mu}={\tilde{\ell}}^{\mu}/(2\tilde{\tau})\,.$ (12)
Therefore, in the case of the Lévy flights, when fixing the value of
$\kappa_{\mu}$, the space-scale parameter is:
$\tilde{\ell}=(2\kappa_{\mu}\tilde{\tau})^{1/\mu}=(2\kappa_{\mu}/R_{\mathrm{tot}})^{1/\mu}\,.$
(13)
As we consider the bugs to be point-like, the spatial dynamics does not
include any interaction between them. The interaction is instead taken into
account through reproduction and death rates, which we assume to be affected
by competitive interactions.
If the birth and death rates of a bug are influenced by the number of other
bugs within a certain radius $R$, one talks about a nonlocal interaction of
finite range. In the present paper we assume that the birth and death rates of
the $i$-th individual depend linearly on the number of neighbors in the
interaction range Hernández-García and López (2004),
$\displaystyle r^{i}_{b}$ $\displaystyle=$
$\displaystyle\mathrm{max}\left(0,r_{b0}-\alpha N_{R}^{i}\right)\,,$ (14)
$\displaystyle r^{i}_{d}$ $\displaystyle=$
$\displaystyle\mathrm{max}\left(0,r_{d0}+\beta N_{R}^{i}\right)\,.$ (15)
Here $N_{R}^{i}$ is the number of bugs which are at a distance smaller than
$R$ from bug $i$, the parameters $r_{b0}$ and $r_{d0}$ are the zero-neighbor
birth and death rates, and the parameters $\alpha$ and $\beta$ determine how
$r^{i}_{b}$ and $r^{i}_{d}$ depend on the neighborhood. For positive values of
$\alpha$ and $\beta$, the more neighbors an individual has within the radius
$R$, the smaller is the probability for reproduction and the larger is the
probability that the bug does not survive, which could arise from competition
for resources. The function $\mathrm{max}()$ enforces the positivity of the
rates. Since we take $R<L/2$ (in fact $R\ll L$), the periodic boundary
conditions are straightforwardly implemented and bugs are never counted twice.
If the birth and death rates of a bug are instead influenced by all the other
individuals in the system, i.e.,
$\displaystyle r^{i}_{b}$ $\displaystyle\equiv$ $\displaystyle
r_{b}=\mathrm{max}\left\\{0,r_{b0}-\alpha[N(t)-1]\right\\}\,,$ (16)
$\displaystyle r^{i}_{d}$ $\displaystyle\equiv$ $\displaystyle
r_{d}=\mathrm{max}\left\\{0,r_{d0}+\beta[N(t)-1]\right\\}\,,$ (17)
then one talks about global interaction. This is formally equivalent to Eqs.
(14) and (15) with $R$ sufficiently large for the interaction domain to
include the whole system, but taking care of counting each bug only once, so
that $N^{i}_{R}=N(t)-1$.
In the case the rates of the demographic events are the same for all the bugs
and assume constant values,
$\displaystyle r^{i}_{b}\equiv r_{b}=r_{b0}\,,\qquad r^{i}_{d}\equiv
r_{d}=r_{d0}\,,$ (18)
which is equivalent to $\alpha=\beta=0$ in Eqs. (14), (15), bugs are
noninteracting.
In the following we discuss the Brownian and Lévy bug systems when individuals
do not influence each other and when inter-particle interaction occurs, either
global or nonlocal. Although we formally maintain the parameter $\beta$ in
Eqs. (15) and (17), in our numerical examples we restrict to $\beta=0$.
Figure 1: (Color) Simple bug models with no interaction; spatial configuration
of bugs at different times $t$: (a) Brownian bugs with $\kappa=10^{-6}$ and
(b) Lévy bugs with $\kappa_{\mu}=10^{-5}$ and $\mu=1$. Reproduction and death
occur with equal probability, $r_{b}=r_{d}=0.1$, and the initial number of
individuals is $N_{0}=1000$. Bugs are colored with the color their ancestors
bear in the panel at $t=0$.
## III Simple bug models with no interaction
### III.1 Noninteracting Brownian bugs
The simple Brownian bug model with no interaction, i.e., when the birth and
death rates of the individuals are given by Eq. (18), has been studied and
discussed in various works Zhang et al. (1990); Young et al. (2001). The
ensemble average of the total population size follows
$\langle N(t)\rangle=N_{0}\exp{[\Delta(t-t_{0})]}\,,$ (19)
independently of the diffusivity of the bugs; it only depends on the
difference $\Delta=r_{b}-r_{d}$. If the birth rate is larger than the death
rate, $\Delta>0$, the total population generally explodes exponentially,
though there is a finite probability for extinction that depends on the
initial size of the population and decreases with increasing $\Delta$. If the
death rate is larger than the birth rate, $\Delta<0$ the extinction of the
population takes place with probability $1$. If birth and death are equally
probable, $\Delta=0$, then the average over many realizations is $\langle
N(t)\rangle=N_{0}$ and the average lifetime is infinite. However, in single
realizations the fluctuations in the number of individuals are huge leading to
fast extinction in some runs. In fact, there exists a typical lifetime
proportional to $N_{0}$, defined as the time for which the fluctuations become
as large as the mean value, beyond which the population is extinct with
probability close to $1$ Zhang et al. (1990).
As a surprising effect, in the systems where the noninteracting Brownian bugs
undergo death and reproduction with equal probabilities, spatial clustering of
the bugs was observed in single realizations Zhang et al. (1990); Young et al.
(2001); Felsenstein (1975). A typical time evolution of such a system is
illustrated in Fig. 1a. We note that in all figures presenting the spatial
configurations of the bugs, we have divided the individuals according to their
initial position into nine groups characterized by different colors as in Fig.
1 at time $t=0$; if reproduction takes place, the newborn bug assumes the same
color as the parent. From Fig. 1a one can see that many small clusters form
some time after starting from a uniform initial distribution. The occurrence
of the clustering is related to the fact that in the case of reproduction the
new bug is located at the same position as the parent. Due to the fluctuations
and irreversibility of death the number of clusters decreases in time, until
there will be a single cluster consisting of individuals coming from a single
ancestor. There are constant, spontaneous, short-lived break-offs from the
main cluster, which are always located near it. The center of mass of such a
cluster undergoes a motion similar to that of a single bug Zhang et al. (1990)
and its linear width fluctuates with a typical value proportional to
$\sqrt{N_{0}}$ Zhang et al. (1990). Furthermore, the larger the diffusion
coefficient $\kappa$, the wider is the cluster (notice that when simulating
the system numerically, if the diffusivity becomes so large that the jump
lengths become comparable to the system size, one needs to take a larger
simulation box). Finally, due to the fluctuations also the last cluster
disappears.
### III.2 Noninteracting Lévy bugs
In the case of noninteracting Lévy bugs, the number of individuals still
follows Eq. (19), independently of the Lévy index $\mu$, and also the cluster
formation observed in the case of Brownian bugs takes place. Now, however, as
bugs can perform long jumps, there are also small clusters continuously
appearing and disappearing far from the main clusters (Fig. 1b). The smaller
the value of $\mu$ the more anomalous the system, i.e., the larger is the
probability for long jumps and therefore there are more flash-clusters. When
the number of clusters has already decreased to one, due to the long jumps and
fluctuation of the number of individuals, new clusters that are placed far
from the central cluster can appear in the system also for some time and often
the disappearance of the main cluster takes place whereas another new central
cluster appears somewhere else. As a result the center of mass undergoes
anomalous diffusion as single bugs do. The value of the Lévy index $\mu$
influences also the linear size of the main clusters: the smaller is $\mu$,
the more compact are the clusters, although also more particle jumps to long
distances occur. The influence of the value of $\kappa_{\mu}$ is similar as in
the case of Brownian bugs, i.e., a larger value of the anomalous diffusion
coefficient results in a larger linear size of the clusters.
Figure 2: (Color) Globally interacting bug models; spatial configuration of
bugs at different times $t$: (a) Brownian bugs with $\kappa=10^{-5}$ and (b)
Lévy bugs with $\kappa_{\mu}=10^{-4}$ and $\mu=1$. The parameters in the
reproduction and death rates are: $r_{b0}=1$, $r_{d0}=0.1$, $\alpha=0.02$,
$\beta=0$. Bugs are colored with the color their ancestors bear in the panel
at $t=0$.
## IV Global interaction
### IV.1 Formation of a cluster
Let us now investigate the behavior of the Brownian and Lévy bug systems in
the case of global interaction, i.e., birth and death rates of the individuals
are given by Eqs. (16), (17). The time evolutions of the globally interacting
Brownian and Lévy bug systems are illustrated by Fig. 2a and 2b, respectively.
In both systems we start from $N_{0}=500$ bugs uniformly distributed in the
simulation area and choose for the parameters characterizing death and birth
rates the following values: $r_{b0}=1$, $r_{d0}=0.1$, $\alpha=0.02$,
$\beta=0$. As in the noninteracting case, the final state of the dynamics is
complete extinction, since there is always a nonvanishing probability for a
fluctuation strong enough to produce that. However, if the number of bugs in
the system is large this happens at very long times Doering et al. (2005).
Then, there is a long-lived quasistable state for which the average number of
individuals $\langle N(t)\rangle$ can be estimated from the condition that
death and birth are equally probable, $r_{b}^{i}=r_{d}^{i}$. From there,
$\langle N(t)\rangle=\frac{\Delta_{0}}{\alpha+\beta}+1\,,$ (20)
where $\Delta_{0}=r_{b0}-r_{d0}$. We have restricted to parameter values so
that the $\mathrm{max}$ functions in Eqs. (16-17) do not operate. Since we
have chosen $N_{0}>\langle N(t)\rangle=46$ in Fig. 2, death is more probable
at small times and the number of bugs decreases rapidly. Approximately at time
$t=30$ the number of individuals has reached the value at which death and
birth become in average equally probable and after this time particle number
fluctuates around that value; parameters of the birth and the death rates can
be chosen so that these fluctuations are weak. At this time small clusters
start to form due to the reproductive pair correlations. As in the case of
noninteracting bugs, fluctuations and irreversibility of death makes the
number of clusters to decrease in time, although now the fluctuations of the
particle content of the different clusters are correlated to keep the total
number close to the value given by Eq. (20) and the process is faster. Finally
a single cluster consisting of bugs coming from the same ancestor remains (as
stated before, it will also disappear at very long times due to finite-size
fluctuation effects) though there are also now spontaneous short-lived break-
offs from the central cluster as in the case of noninteracting bugs. The
center of mass of such a cluster is moving in space and its linear size is a
fluctuating quantity. The clustering of the globally interacting bugs was
quantitatively discussed in Ref. Hernández-Garcia and López (2005) for the
one-dimensional Brownian bug system.
### IV.2 Fluctuations of the number of bugs
As indicated by Eq. (20), for given values of $\alpha$ and $\beta$, the
average number of individuals in the system with global interaction depends
solely on the difference $\Delta_{0}=r_{b0}-r_{d0}$. It is independent of the
concrete values of $r_{b0}$ and $r_{d0}$, as well as of the value of $\kappa$
or $\kappa_{\mu}$ and $\mu$; in fact, it does not even depend on whether the
system consists of Brownian or Lévy bugs. Nevertheless, fluctuations of the
number of bugs do indeed depend on the values of $r_{b0}$ and $r_{d0}$, even
if the difference $\Delta_{0}$, and thus the average number of bugs, has the
same value. To illustrate this, let us calculate from the simulations time
series the probability distribution of the number of individuals in the
globally interacting Brownian and Lévy bug systems. As can be seen from Fig.
3, for a given value of $\Delta_{0}$, larger values of $r_{b0}$ and $r_{d0}$
lead to larger fluctuations. This is a simple consequence of the Poisson
character of the birth and death events for which fluctuations in each of the
instantaneous rates are proportional to the mean rates. When the distributions
are narrow, they are close to Gaussian. For larger rates particle number
distribution gets broader implying that there is an enhanced probability that
particle number becomes zero at some moment, after which bugs become extinct
(remember that what is in fact plot in Fig. 3 is the numerical particle number
distribution in the long-lived metastable state before extinction). For the
present case with $\Delta_{0}=0.9$ and $\alpha=0.02$, $\beta=0$, rate values
above the ones shown in Fig. 3 (i.e. $r_{b0}>2$, $r_{d0}>1.1$) lead to
observable extinction after some tenths of thousands of steps. An ecological
implication of this could be the following: one can think of two colonies of
organisms of the same type, having both the same equilibrium size determined
for example by the size of the living area. Now if in one of the systems the
population has no enemies and the natural death rate is low, but in the other
the death rate is higher due to the presence of a predator, then the latter
system will more probably go to extinction sooner due to the presence of
larger fluctuations.
Figure 3: (Color online) Probability distribution of the number of bugs in
globally interacting bug systems. The results are numerically obtained from
the time series of the particle number in the very long-lived state before the
fluctuations lead the system to the extinction. For all the curves
$\alpha=0.02$, $\beta=0$, and the rate difference is
$\Delta_{0}=r_{b0}-r_{d0}=0.9$, but the rates $r_{b0}$, $r_{d0}$ assume
different values: (a) $r_{b0}=1$, $r_{d0}=0.1$; (b) $r_{b0}=1.5$,
$r_{d0}=0.6$; (c) $r_{b0}=2$, $r_{d0}=1.1$. The overlapping curves correspond
to Brownian and Lévy bug systems; the distributions do not depend on the type
of diffusion nor on the values of $\kappa$, $\kappa_{\mu}$ or $\mu$ in this
globally interacting case.
### IV.3 The average cluster shape, cluster width, and center of mass motion
Let us keep in the following $\alpha=0.02$, $\beta=0$ and $r_{b0}=1$,
$r_{d0}=0.1$ [the same parameter values as in Fig. 2 and in Fig. 3 for curve
(a)] and study the behavior of the cluster formed in the case of a system with
global interaction defined by Eqs. (16)-(17). As mentioned, even after the
transition from an uniform distribution of bugs to a single cluster (and
before eventual extinction at large times), at some moment the system can
consist actually of more than one cluster. In such cases we define that all
the individuals in the system belong to the same cluster, even though in the
Lévy case the distance between the bugs (subclusters) can be rather large. In
order to avoid the boundary effects as much as possible, in Figs. 4-7 the
linear size of the simulation area was taken as $L=1000$ and to have enough
statistics simulations were run until $t=5\times 10^{8}$.
Let us start by analyzing the average shape of the cluster. The average
cluster, $\rho(x,y)$, is obtained setting at each time the origin in the
center of mass of the cluster (distances under the periodic boundary
conditions are computed under a minimum distance convention) and averaging
over a long time (after the transition from uniform distribution to one single
cluster but before long-time extinction). A one-dimensional cut of it (say
across $x$ for $y=0$, i.e., $\rho(x)\equiv\rho(x,y=0)$) is shown in Figs. 4
and 5. For the case of Brownian bugs, the tail of the average cluster is
approximately exponential. A pair distribution function, which should be
related but not identical to the average cluster discussed here, was
analytically calculated in Ref. Birch and Young (2006) for a globally
interacting Brownian bug model of our type but in which total extinction was
forbidden. This quantity also displayed a fast decaying tail. In the case of
Lévy bugs the tail of $\rho(x)$ follows instead a power law, $\rho(x)\sim
x^{-(2+\mu)}$, see Fig. 4b, arising from the long jumps. Note that, in the
present case of circular symmetry, the relation
$\rho(x,y)dx\,dy=R(r)(2\pi)^{-1}dr\,d\theta$ of $\rho(x,y)$ with the radial
density of the average cluster, $R(r)$, where $r$ and $\theta$ are the polar
coordinates centered at the cluster center, implies
$\rho(x)=R(r=|x|)(2\pi|x|)^{-1}$, so that the asymptotic behavior of the
radial density is $R(r)\sim r^{-(1+\mu)}$. This is the same asymptotic
behavior as the individual radial jumps in (8) and it is also the asymptotic
tail of the probability of displacement from the original position of
nonreproducing bugs moving by Lévy flights Metzler and Klafter (2000). We note
also that, for $\kappa=\kappa_{\mu}$, the central part of $\rho(x)$ is
narrower and higher in the Lévy than in the Brownian bug system, and it is
narrower and higher the smaller the value of $\mu$ (see Fig. 4a). This is a
somehow counterintuitive effect of the Lévy motion on clusters, already
commented in the noninteracting case: increasing anomalous diffusion (smaller
$\mu$) induces larger jumps and longer tails, but at small scales it acts as
making the cluster more compact.
Figure 4: (Color online) (a) $\rho(x)$, the cross-section of the two-
dimensional particle density of the average cluster in semi-log scale;
comparison between the Brownian and Lévy bug systems; $\kappa=10^{-5}$,
$\kappa_{\mu}=10^{-5}$, $r_{b0}=1$, $r_{d0}=0.1$, $\alpha=0.02$, $\beta=0$.
(b) The tails of $\rho(x)$ in log-log scale in the case of the Lévy bug
systems for different values of $\mu$. Solid lines correspond to fitting
curves $\propto x^{-(2+\mu)}$.
The influence of the diffusivity is similar in both systems: the larger is the
value of $\kappa$ or $\kappa_{\mu}$ the more spread is the average cluster
(see Fig. 5). For the Brownian one-dimensional case it was shown in Ref.
Hernández-Garcia and López (2005) that cluster width is essentially the
distance associated to the Brownian walk during the lifetime of a bug and
their descendants. Thus, the width increases as $\kappa^{1/2}$. In the Lévy
case, defining the distance associated to the walk is more subtle, since
higher moments of displacements diverge. But the behavior of lower ones and
dimensional analysis indicate that typical displacements during a lifetime
scale as $\kappa_{\mu}^{1/\mu}$, and then this should determine the width of
$\rho(x)$ or $R(r)$ (i.e., the spatial dependence should occur only through
the combinations [$x\kappa_{\mu}^{-1/\mu}$] or [$r\kappa_{\mu}^{-1/\mu}$]).
Imposing additionally that the total number of bugs in the average cluster in
this global interaction case does not depend on particle motion or
distribution, and it is thus independent on the value of $\kappa_{\mu}$ we
have $R(r)=\kappa_{\mu}^{-1/\mu}F(r\kappa_{\mu}^{-1/\mu})$, or
$\rho(x)=\frac{1}{\kappa_{\mu}^{2/\mu}}G\left(\frac{x}{\kappa_{\mu}^{1/\mu}}\right)\
,$ (21)
with $G(u)=F(u)/u$. The analogous scaling form for the average cluster in the
Gaussian diffusion case is
$\rho(x)=\frac{1}{\kappa}G\left(\frac{x}{\kappa^{1/2}}\right)\ .$ (22)
The insets in Fig. 5 show the validity of these scaling forms.
Figure 5: (Color online) $\rho(x)$, the cross-section of the two-dimensional
particle density of the average cluster in semi-log scale for different values
of diffusivity: (a) Brownian bugs and (b) Lévy bugs with $\mu=1$. Other
parameters are as in Figs. 2 and 4. The insets check the correctness of the
scaling forms (22) (with $G(u)=\kappa\rho$ and $u=x/\kappa^{1/2}$) and (21)
(with $G(u)=\kappa_{\mu}^{2/\mu}\rho$ and $u=x/\kappa^{1/\mu}$). Figure 6:
(Color online) Probability density $\pi(\sigma)$ of the standard deviation
$\sigma$ of the bug positions with respect to the center of mass of the
cluster in the Brownian and Lévy bug systems; $\kappa=10^{-5}$,
$\kappa_{\mu}=10^{-5}$. The distribution is obtained averaging over a long
time. The curves corresponding to the Lévy bug systems are well fitted by
$\propto\sigma^{-(1+\mu)}$ (not shown). Other parameters are as in Figs. 2, 4,
and 5. Figure 7: (Color online) Probability density $p(\Delta_{CM})$ of the
jump lengths of the center of mass in the Brownian and Lévy bug systems. Same
parameter values as in Fig. 6. The tails of the curves corresponding to Lévy
bugs are well fitted by $\propto\Delta_{CM}^{-(1+\mu)}$ (not shown).
In addition to the average cluster shape, which gives information on the
cluster width, it is also interesting to study the fluctuations of the cluster
width in time. We characterize cluster width at each instant of time by the
standard deviation of the bug positions with respect to the center of mass of
the cluster at that time. Then, a probability density $\pi(\sigma)$ is
constructed from the values of $\sigma$ at different times. Figure 6 shows
that in the case of globally interacting Brownian bugs the distribution of
$\sigma$ is short-tailed. In the case of globally interacting Lévy bugs, in
contrast, the distribution for $\sigma$ is characterized by tails with a power
law decay with exponent $-(1+\mu)$. This means that in the latter case the
cluster width can undergo arbitrarily large fluctuations in time. We note that
the tails in $\pi(\sigma)$ decay with the same exponent as the radial density
$R(r)$ of the average cluster, thus suggesting that the tails of the average
cluster are produced by the large fluctuations in the width of the
instantaneous clusters (which in fact include splitting events).
The individual motion of bugs drives the behavior of the center of mass of the
system. Figure 7 depicts the probability density $p(\Delta_{CM})$ of the jump
lengths $\Delta_{CM}$ performed by the center of mass each time the bug motion
step is executed in the globally interacting Brownian and Lévy bug systems.
For Brownian bugs it is short-tailed. In fact, from the arguments in Ref.
Hernández-Garcia and López (2005), the center of mass motion of the cluster
for globally interacting Brownian bugs is characterized by a Brownian process
with the same diffusion coefficient as the individual bugs. In the case of
Lévy bugs the probability density of the jump lengths of the center of mass is
described by a distribution with a power-law tail with exponent $-(1+\mu)$,
i.e., the center of mass of the cluster formed in the case of globally
interacting Lévy bugs undergoes jumps that follow asymptotically the same law
as the single bugs, Eq. (8), and as the radial tails of the average cluster.
This reflects the fact commented previously that, due to the long jumps of the
Lévy bugs, additional clusters far from the main one appear from time to time,
strongly displacing the center of mass of the system. Due to the fluctuations
it is even possible that the cluster that used to be the main cluster
disappears and a new main cluster forms somewhere else. As a result the center
of mass motion undergoes the same type of superdiffusion as the individual
bugs of the system.
Extending the arguments for the Brownian bugs Hernández-Garcia and López
(2005) (which were themselves adapted from the ones in Zhang et al. (1990)) to
the Lévy case one can heuristically show that the distributions of $\sigma$
and $\Delta_{CM}$ are related. To this aim one makes the approximation that
the number of bugs in the system is constant, say $N$, instead of being
constant on average. The center of mass receives a positive contribution from
the new bugs appearing (at location $\vec{x}_{i}$) due to the reproduction
between diffusion steps (say at time $t_{i}$), a negative contribution from
the bugs disappearing during that time (say from position $\vec{x}_{j}$ at
time $t_{j}$), and the direct contribution from the Lévy jumps
$\vec{\ell}_{k}$ of all bugs present at the diffusion step:
$\vec{\Delta}_{CM}=\frac{1}{N}\sum_{i\in
B}\vec{x}_{i}(t_{i})-\frac{1}{N}\sum_{j\in
D}\vec{x}_{j}(t_{j})+\frac{1}{N}\sum_{k=1}^{N}\vec{\ell}_{k}\ .$ (23)
$B$ and $D$ denote the sets of bugs that have been born or dead, respectively,
between diffusion steps. The two first terms can combined in a single one
$\vec{S}\approx N^{-1}\sum_{p=1}^{n}\vec{\sigma}_{p}$ by considering that the
two sets have approximately the same number of individuals, $n$.
$\vec{\sigma}_{p}=\vec{x}_{i}-\vec{x}_{j}$ is the displacement between a pair
of these bugs, one just born and the other just disappeared, sampled inside
the same cluster. Then the modulus of each $\sigma_{p}$ should be of the order
of the cluster width $\sigma$, which fluctuates in time with probability tails
ruled by an exponent $-(1+\mu)$. This contribution in Eq. (23) gives the
motion of the center of mass due to the birth and death processes. The
contribution from the individual particle jumps is in the last term in (23),
which is a sum of Lévy jumps of parameter $\mu$ so that the tails of the
probability density are characterized by a decay with the same exponent
$-(1+\mu)$. These heuristic arguments imply that the modulus $\Delta_{CM}$
will also be distributed with long tails characterized by an exponent
$-(1+\mu)$, as observed.
## V Nonlocal interaction
### V.1 Formation of a periodic pattern
Figure 8: (Color) Interacting Brownian bug model with $R=0.1$, $r_{b0}=1$,
$r_{d0}=0.1$ and $\alpha=0.02$, $\beta=0$. Spatial configuration of bugs at
time $45000$ in systems with different diffusion coefficients: (a)
$\kappa=10^{-5}$, (b) $\kappa=2\times 10^{-5}$, (c) $\kappa=4\times 10^{-5}$,
(d) $\kappa=10^{-4}$. The initial configuration of bugs is the same as in
Figs. 1 and 2 at time $t=0$.
In Refs. Hernández-García and López (2004); López and Hernández-Garcia (2004);
Heinsalu et al. (2010) on the nonlocally interacting Brownian and Lévy bugs it
was assumed that the birth and death rates of the $i$-th individual are given
by Eqs. (14), (15). In the case of Brownian bugs, for small enough diffusion
coefficient and large enough $\Delta_{0}$, the occurrence of a periodic
pattern consisting of clusters that are arranged in a hexagonal lattice was
observed (see Fig. 8a-c) Hernández-García and López (2004); López and
Hernández-Garcia (2004). For large values of the diffusion coefficient such
periodic pattern is replaced by a more homogeneous distribution of bugs (Fig.
8d). In the case of Lévy bugs, since the diffusion coefficient (6) is
infinite, one could expect that the spatial distribution will not reveal a
periodic pattern; however, as shown in Ref. Heinsalu et al. (2010), for proper
parameters periodic cluster arrangements do indeed occur (see Fig. 9). The
reason for the divergence of the diffusion coefficient in the Lévy case is in
the statistical weight of large jumps. These large jumps have some influence
on the characteristics of the pattern formed, but the relevant structure is
ruled mainly by the interactions between individuals. In the Lévy bug system
however, at variance with the Brownian case, even at small values of
$\kappa_{\mu}$ there are many solitary bugs appearing for short time periods
in the space between the periodically arranged clusters due to the large jumps
Heinsalu et al. (2010), c.f. Figs. 8a and 9a. However, the periodicity of the
pattern is of the order of $R$ (the interaction range) in both systems, being
only slightly influenced by $\kappa$ or $\kappa_{\mu}$ and $\mu$, as
demonstrated in Refs. Hernández-García and López (2004); Heinsalu et al.
(2010) through a mean-field theory calculation.
Figure 9: (Color) Interacting Lévy bug model with $R=0.1$, $r_{b0}=1$,
$r_{d0}=0.1$ and $\alpha=0.02$, $\beta=0$ (same parameters as in Fig. 8 for
Brownian bugs). The spatial configuration of bugs at time $45000$ in systems
with different generalized diffusion coefficients and anomalous exponent: (a)
$\kappa_{\mu}=10^{-4}$, $\mu=1$; (b) $\kappa_{\mu}=10^{-3}$, $\mu=1$; (c)
$\kappa_{\mu}=10^{-4}$, $\mu=1.5$; (d) $\kappa_{\mu}=5\times 10^{-5}$,
$\mu=1.5$. The initial configuration of bugs is the same as in Figs. 1 and 2
at time $t=0$.
In Ref. Heinsalu et al. (2010) also the two-dimensional particle density of
the average cluster, obtained by setting the origin at the center of mass of
each cluster forming the periodic pattern and averaging over all the clusters
in the simulation area and over time, was studied. In both, Brownian and Lévy
bug systems the central part of the average cluster, where most of the
individuals are concentrated, was well fitted by a Gaussian function, but the
way the particle density decreases when moving away from the center of mass of
the cluster is rather different. In the Brownian case a Gaussian decay
provides a good approximation, whereas in the Lévy case it is close to
exponential. The comparison with the systems with global interaction,
discussed in Sec. IV.3, reveals therefore that the interaction range $R$ turns
the exponential decay into Gaussian and the power law decay into exponential.
For a given value of diffusion coefficient, there exists a critical value of
$\Delta_{0}$ below which the system gets extinct, independently of $\alpha$
Hernández-García and López (2004). Above this critical value, for every
$\alpha$ the increase of $\Delta_{0}$ results in the increase of the average
number of bugs, but the pattern formation is not much influenced. The latter
is, however, true solely if $\Delta_{0}$ increases through the increase of
$r_{b0}$ and the death rate is low. Namely, as in the case of global
interaction discussed in Sec. IV.2, an increase of the death rate, though
accompanied by a compensating increase of birth rate, leads to larger
fluctuations in the particle number. In numerical simulations we have observed
that the larger are the fluctuations in the number of bugs, the more difficult
is the formation of the periodic pattern, and finally the individuals do not
arrange in the periodic pattern but in random clusters (see also Ref. Birch
and Young (2006)). This effect may in fact make difficult to observe the
periodic clustering phenomenon in real competitive biological systems.
In the following we keep for the parameters in the birth and death rate the
same values as in the case of global interaction, i.e., $r_{b0}=1$,
$r_{d0}=0.1$, $\alpha=0.02$, $\beta=0$. For these parameter values the number
of bugs fluctuates only weakly around the mean value. Differently from the
case of global interaction, now the average number of bugs in the system is
influenced not only by the birth and death rates, but also by the diffusion,
i.e., in the case of Brownian bugs by $\kappa$ and in the case of Lévy bugs by
$\kappa_{\mu}$ and $\mu$, see Fig. 10. The smaller is $\kappa$, $\kappa_{\mu}$
or $\mu$, the larger the particle number. At the same time Figs. 8 and 9
indicate that by decreasing $\kappa$ or $\kappa_{\mu}$ the linear width of the
clusters becomes smaller, the particle density in the clusters higher, and the
density between the clusters lower (c.f. Sec. IV.3 and see also Ref. Heinsalu
et al. (2010)). Somehow counterintuitively, the effect of decreasing $\mu$
seems to have the same effects, as commented above for the noninteracting and
global cases. Furthermore, the value of $\kappa$ or $\kappa_{\mu}$ and $\mu$
seems to weakly influence the number of clusters in the system: In Figs. and 8
and 9 smaller values lead to larger number of clusters. This observation is
not explained by the linear instability analysis of Hernández-García and López
(2004); Heinsalu et al. (2010).
Figure 10: (Color online) a): Average number of bugs versus diffusion
coefficient in the system with Gaussian jumps. b): Average number of bugs
versus anomalous exponent $\mu$ in the system with Lévy jumps for various
values of the anomalous diffusion coefficient. Other parameters as in Figs. 8
and 9.
### V.2 Mixing of different families
It is interesting to study the evolution of the system also regarding the
disappearance or survival of the different groups, by dividing initially the
bugs into different families and following their descent. In the case of
nonlocally interacting Brownian bugs, a very low diffusion coefficient leads
to the situation in which after cluster formation the inter-cluster travel is
very rare because the individuals are not capable to make the jumps from one
cluster to another one, and it is also very unlikely to arrive to the next
cluster doing a multistep random walk because death is very probable between
the clusters. Therefore, in the case of very low diffusion different families
would remain inside their initial clusters. If one assumes that initially each
individual represents a different family, then only inter-cluster competition
occurs and the final number of families is equal to the number of clusters. If
instead initially individuals are assigned to families according to large
areas of initial positions (larger than typical cluster size as done in Figs.
1 and 2 at time $t=0$), there is no family competition internal to the
clusters, most families survive and the clusters coming from different
families occupy approximately the territory of the ancestors even after a long
time, as can be seen from Fig. 8a. In that case, the travel of a cluster to a
new territory away from the other clusters of the same family can take place
due to the diffusion of the cluster as a whole during the clusters arrangement
into the periodic pattern. For larger values of $\kappa$ the inter-cluster
travel is possible which leads to the conquering of new territories, i.e.,
bugs can be found in a region where their ancestors were not from, Fig. 8b.
The effect is larger for larger $\kappa$ and leads to the disappearance of
some families, as can be seen from Fig. 8c. Finally, for increased diffusion,
intra-cluster competition will force all surviving bugs to be from a single
family (in fact, from a single ancestor); which family (ancestor) wins is a
random event. The process is faster for larger diffusion. Increasing the
diffusivity further even the periodic pattern disappears, Fig. 8d.
Figure 9 illustrates the family mixing for nonlocally interacting Lévy bugs.
In this case the inter-cluster traveling is supported by the long jumps.
Differently from the case of Brownian bugs, now the individuals can reach not
only the next neighboring cluster but also clusters far away. Consequently, a
cluster originally consisting of bugs coming from one ancestor can after some
time turn into a cluster consisting of bugs coming from different families
placed initially far away. Thus, intra-cluster bug competition becomes soon
competition between families, and even if the diffusivity of the bugs is very
low, at the end the Lévy bug system would consist of individuals coming from
one or just a few ancestors. As in the case of Brownian bugs, the process of
the disappearance of families is faster the greater is the generalized
diffusion coefficient.
Besides the diffusion of a cluster as a whole during the formation of the
periodic pattern and the conquering of new territories through the migration
to and survival in another cluster, the mixing of clusters from different
families can take place also due to the appearance of a new cluster if in the
periodic pattern there is a dislocation. In the case of Brownian bugs the new
cluster is formed through the splitting of an old cluster. In the case of Lévy
bugs, instead, the new cluster can appear also far from the original
territory.
## VI Conclusions and outlook
We have presented some detailed properties of interacting particle systems in
which the individuals are Brownian or Lévy random walkers which interact in a
competitive manner. We have seen strong differences between the globally and
the finite-range nonlocally interacting systems. In the systems with global
interaction the spatial distribution of the bugs becomes tied to the type of
diffusion, Brownian or Lévy. Typical configurations consist of a single or a
few clusters for both types of motion. For the Lévy bug systems long tails
appear in the mean cluster shape and in probability distributions of cluster
width and of jumps of the center of mass. For Brownian bug systems these
quantities appear to be much shorter ranged. This is qualitatively also the
situation in the noninteracting case, although then the effects of the
particle-number fluctuations are much stronger. Under non-local finite-range
interactions the situation is rather different. First, single cluster
configurations are generally replaced by periodic patterns with periodicity
set by the interaction range $R$. Motion of individual clusters is severely
restricted by the presence of the neighboring clusters. In addition, the
natural spatial cut-off introduced by the interaction range $R$ seems to limit
the influence of the long Lévy jumps, so that measures of spatial cluster
shape do not generally exhibit power laws, making spatial configurations under
both types of diffusion more similar. Mixing of families and their competition
is nevertheless greatly influenced by the type of motion. This suggests that
it would be interesting to consider the influence of different types of
diffusion into competitive genetic mixing processes Sayama et al. (2002).
Obtaining analytic understanding in this type of interacting systems is
difficult, but at least the nature of the instability leading to pattern
formation and its relevant spatial scale have been clarified by using partial
integro-differential equation descriptions of the mean field type Hernández-
García and López (2004); Heinsalu et al. (2010), which are useful in broader
contexts Fuentes et al. (2003); Clerc et al. (2005); Maruvka and Shnerb
(2006). However, from previous work in the Gaussian case López and Hernández-
Garcia (2004); Hernández-García and López (2005); Hernández-Garcia and López
(2005), it is known that quantities such as cluster width and structure or
transition thresholds strongly depend on particle-number fluctuations. Thus,
obtaining additional results from differential equation approaches would need
the inclusion of effective multiplicative noise terms Ramos et al. (2008) or
focus on statistical quantities such as pair correlation functions Young et
al. (2001); Birch and Young (2006); Houchmandzadeh (2009).
###### Acknowledgements.
This work has been supported by the targeted financing project SF0690030s09,
Estonian Science Foundation through grant no. 7466, by the Balearic Government
(E.H.), and by Spanish MICINN and FEDER through project FISICOS
(FIS2007-60327).
## References
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* Heinsalu et al. (2010) E. Heinsalu, E. Hernández-Garcia, and C. López, Europhys. Lett. 92, 40011 (2010), Erratum: 95, 69902 (2011).
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|
arxiv-papers
| 2011-07-21T09:48:44 |
2024-09-04T02:49:20.796513
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Heinsalu, E. Hernandez-Garcia and C. Lopez (IFISC, CSIC-UIB)",
"submitter": "Emilio Hernandez-Garcia",
"url": "https://arxiv.org/abs/1107.4214"
}
|
1107.4234
|
# Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors
Jian-Feng Wu ***wujf@itp.ac.cn, Ying-Ying Xu †††yyxu@itp.ac.cn , and Ming
Yu‡‡‡yum@itp.ac.cn
_Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing, 100190, China_
## 1 Introduction and Conclusion
The Calogero-Sutherland (CS) Model, which is an integrable 1d many-body
system, plays important roles in many different research areas in physics and
mathematics. Among them are the 2D conformal field theories(CFTs)[1, 16], the
generalized matrix models[2, 4], the fractional quantum hall effects(FQHE) [6,
7, 11, 12], and there is an even more surprising correspondence related to the
$N=2^{\ast}$ 4D supersymmetric gauge systems[23, 24, 25, 26, 27, 28, 29]. The
spectrum of the CS model can be generated from the Jack polynomials[5, 17, 18,
19, 32]. From the CFT point of view, Jack symmetric functions are naturally
the building blocks for the conformal towers, the characters of which encode
the (extended) conformal symmetries. For instance, the Jack functions related
to a given Young tableaux are believed to be in one to one correspondence with
the singular vectors of the $W$-algebra[1], which reflects the hidden
$W_{1+\infty}$ symmetry of the CS model. Singular vectors in 2d CFT are the
keys to the calculation of the correlation functions in CFT and may also
reveal important physical properties of the CS model. Unfortunately, on the
CFT side, it is not clear how to relate the construction of the secondary
states in the conformal tower to that of the Jack functions, except for some
simple cases, i.e. the Virasoro singular vectors [10]. On the 2d CFT side, the
calculation of conformal blocks is based on the conformal Ward-identities,
$[L_{n},V_{h}(z)]=(z^{n+1}\partial_{z}+(n+1)hz^{n})V_{h}(z).$
And the calculation is carried out perturbatively level by level [8, 9]. In
some special cases, the decoupling of the Virasoro null vectors can be
implemented as differential equations for the conformal blocks. For the
general case, recursion relations have been proposed by Zamolodchikov on the
meromorphic structures of the conformal blocks either in complex $c$-plane or
$h$-plane. However, it remains unclear (to us) how to construct the basis
vectors in the given conformal tower by making use of the Zamolodchikov’s
recursion formulae explicitly. In contrast, there are various ways in the
explicit constructions of the Jack polynomials, each follows different
strategies. For example, there are two integral representations. The one given
in [3] is based on the $W_{1+\infty}$ symmetry hidden in the CS model, the
other, by the authors of [13], starts from the so called shift Jack
polynomials. There is also an operator formalism [14, 15] based on the Dunkel
(exchange) operators. However, here we follow a different strategy in
constructing the Jack polynomials which are intrinsically related to the
singular vectors in the Virasoro algebra, and present a new recursion relation
derived from the construction. We also do not need to invoke the hidden
$W$-algebra. In fact there is a hidden Virasoro structure in the Hilbert space
of the CS model which can be used recursively in our construction. To be more
specific, the ket states in the Fock space realization of the CS model is
mapped a la Feigin-Fuks-Dotsenko-Fateev Coulomb gas formalism [20][21] to the
singular vectors of the Virasoro algebra and its skew hierarchical
descendants. The construction of the singular vectors defines a new recursion
relation for the Jack functions and finally leads to a new integral
representation which differs from the one appeared in [3, 13]. Hence our
approach may supply new insights in dealing with CS model or more general
integrable systems.
The structure of this article is organized as following. In section 2, we
review some useful properties of the Jack polynomials and the CS model. In
section 3, we review the construction of the Virasoro singular vectors which
are related to the Jack polynomials with rectangular Young tableaux. It should
be stressed again that the Virasoro symmetry is hidden in the ket (or bra)
Hilbert space only, and is not the true symmetry in the CS model. The reason
is that the prescribed Hermiticity in the CS model is not respected by the
conjugation of the hidden Virasoro algebra. I.E., the Virasoro structure in
the bra and ket Hilbert spaces, respectively, are not related by the Hermitian
conjugation in the CS model. Our main results are given in sections 4 and 5.
In section 4, we propose a skew (recursion) formula for generating new Jack
functions stating from the simple ones which inevitably involve the Virasoro
singular vectors, or equivalently, the Jack functions of the rectangular
graph. Our proof of the skew (recursion) formula can be considered as an
operator formalism generalization of its counterpart for the Jack symmetric
polynomials found by Kadell in [18]. The basic skew relation is further
developed recursively in section 5. This can be made explicit first in the
operator formalism, based on which we develop a new integral representation
for the Jack symmetric functions associated with any generic Young tableaux.
One immediately sees the advantages of our operator formalism of the skew
(recursion) formula over the one proposed by Kadell. Since our formalism does
not depend on explicitly the number of argument variables $\\{z_{i}\\}$ for
$i=1,\cdots,N$, so the recursion is done without worrying the change of the
total number of arguments. Finally, the integral representation of the Jack
symmetric polynomials which depend explicitly on finite number of variables
$\\{z_{i}\\}$, for $i=1,\cdots,N$, are presented as a by-product.
## 2 Jack Polynomials and Calogero-Sutherland Model
Now we review the Jack polynomials and the Calogero-Sutherland model. The Jack
polynomials can be viewed as a special one parameter generalization of the
Schur polynomials[32] and the Jack symmetric function is the large N limit of
the Jack polynomials. For physicists, the most familiar integrable system
which involves Jack symmetric functions as its spectrum functions is the
Calogero-Sutherland model. This model is an integrable system and shows a
great deal of interesting aspects, such as duality, conformal invariance, and
even the combinatorial property of the spectrum etc. An elementary
introduction can be found, for instance, in [11] and further studies in [19]
and [32]. Here we only review some basics of the Jack polynomials.
### 2.1 Partitions and Jack Polynomials
Given a partition: $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{n})$,
$\lambda_{1}\geq\lambda\geq\cdots\geq\lambda_{n}$ ,$n\equiv l(\lambda)$, one
defines the related Jack polynomial as the basis function for the symmetric
homogeneous polynomials in $N$ variables $\\{z_{i}\\},i=1,2,...,N$, of degree
$|\lambda|=\sum\lambda_{i}$
$J_{\lambda}(\\{z_{i}\\})=\sum_{\lambda^{\prime}\leq\lambda,\,l(\lambda^{\prime})\leq
N}C_{\lambda}^{\lambda^{\prime}}z^{\lambda^{\prime}}\,,$ (1)
which should satisfy the second order differential equation:
$\displaystyle H_{J}J_{\lambda}$ $\displaystyle=$ $\displaystyle
E_{\lambda}J_{\lambda},\,\,\,\,\,H_{J}=H_{J}^{0}+\beta H_{J}^{I}$ (2)
$\displaystyle H_{J}^{0}$ $\displaystyle=$
$\displaystyle\sum_{i}^{N}\left(z_{i}\partial_{z_{i}}\right)^{2},\,\,\,\,H_{J}^{I}=\sum_{i<j}\dfrac{z_{i}+z_{j}}{z_{i}-z_{j}}(z_{i}\partial_{z_{i}}-z_{j}\partial_{z_{j}})\,,$
(3)
here
$z^{\lambda}:=\sum_{P}z_{P(1)}^{\lambda_{1}}\cdots z_{P(N)}^{\lambda_{N}}\,,$
(4)
$\lambda^{\prime}\leq\lambda\Rightarrow\sum_{i=1}^{j}\lambda^{\prime}_{i}\leq\sum_{i=1}^{j}\lambda_{i},\,for\,j=1,2,\cdots
l(\lambda^{\prime})$
$P$ means the permutations of $N$ objects. We have also defined
$\lambda_{i}=0$ for $i>l(\lambda)$ and
$\lambda_{1}+\lambda_{2}+\cdots+\lambda_{l(\lambda)}=|\lambda|$, $|\lambda|$
is the level of the partition. $\lambda$ can be represented graphically as a
Young tableau $\lambda=\\{(i,j)|1\leq i\leq l(\lambda)\\},1\leq
j\leq\lambda_{i}$. And the corresponding transposed Young tableau is
represented as
$\lambda^{t}=(\lambda^{t}_{1},\lambda^{t}_{2},\dots,\lambda^{t}_{\lambda_{1}})\Rightarrow\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{\lambda^{t}_{1}}).$
It is clear that $l(\lambda)\equiv\lambda^{t}_{1}$. We shall see later that
the defining differential equation can be derived from the CS Hamiltonian.
The Jack polynomials can be generated by the power sum symmetric polynomials
as well: $p_{l}=\sum_{i=1}^{N}z_{i}^{l}$
$\displaystyle
J_{\lambda}(p)=\sum_{|\lambda^{\prime}|=|\lambda|}d_{\lambda}^{\lambda^{\prime}}p_{\lambda^{\prime}},\
\ \
p_{\lambda^{\prime}}=p_{\lambda^{\prime}_{1}}p_{\lambda^{\prime}_{2}}\cdots
p_{\lambda^{\prime}_{m}},\ \ \ d_{\lambda}^{1^{|\lambda|}}=1$
when $N$ large, $J_{\lambda}$ spans the Hilbert space of free oscillators, and
each power sum symmetric polynomial behaves as a single oscillator. By using
the conjugacy class representation of the Young tableau
$\lambda=\\{i^{m_{i}}\\},~{}~{}i=1,2,\dots$, where $m_{i}$ means the
multiplicity of the rows of $i$ squares in Young tableau $\lambda$, the
normalization of the power sum symmetric polynomial is derived from that of
the oscillators,
$\displaystyle\langle p_{\lambda},p_{\lambda^{\prime}}\rangle$
$\displaystyle=$
$\displaystyle\langle\frac{a_{\lambda}a_{-\lambda^{\prime}}}{\beta^{\frac{1}{2}l(\lambda)}\beta^{\frac{1}{2}l(\lambda^{\prime})}}\rangle$
$\displaystyle=$
$\displaystyle\delta_{\lambda\lambda^{\prime}}i^{m_{i}}m_{i}!\beta^{-l(\lambda)}$
$\displaystyle\langle a_{n}a_{-m}\rangle$ $\displaystyle=$
$\displaystyle\delta_{n,m}n,\,\,\,\beta=k^{2}\,,$
In fact, the above normalization is consistent with that of Jack symmetric
functions[11, 19]. The normalization of the Jack polynomials is derived from
that of the wave function in the CS model:
$\displaystyle\left(\prod_{i}^{N}\int_{0}^{\pi}dx_{i}\right)J_{\lambda^{\prime}}(p^{\ast})J_{\lambda}(p)\prod_{i<j}|z_{i}-z_{j}|^{2\beta}=\Gamma_{N}^{2}\delta_{\lambda,\lambda^{\prime}}j_{\lambda}\dfrac{\bar{A}_{\lambda,N}}{\bar{B}_{\lambda,N}}\,,$
(6)
here
$\displaystyle j_{\lambda}$ $\displaystyle=$ $\displaystyle
A_{\lambda}^{1/{\beta}}B_{\lambda}^{1/\beta},\,\,\,z_{i}=e^{2ix_{i}}$ (7)
$\displaystyle A_{\lambda}^{1/\beta}$ $\displaystyle=$
$\displaystyle\prod_{s\in\lambda}\left(a_{\lambda}(s)\beta^{-1}+l_{\lambda}(s)+1\right),\,\,\,B_{\lambda}^{1/\beta}=\prod_{s\in\lambda}\left((a_{\lambda}(s)+1)\beta^{-1}+l_{\lambda}(s)\right)\,$
$a_{\lambda}(s)$ and $l_{\lambda}(s)$ are called arm-length and leg-length of
the box $s$ in the Young tableau $\lambda$:
$a_{\lambda}(s)=\lambda_{i}-j,\,\,\,\,\,l_{\lambda}(s)=\lambda^{t}_{j}-i,$
$\lambda^{t}_{j}$ is the $j$-th part of the partition related to the
transposed Young tableau $\lambda$.
$\bar{A}_{\lambda,N}=\prod_{s\in\lambda}\left(N+a^{\prime}_{\lambda}(s)/\beta-l^{\prime}_{\lambda}(s)\right),\,\,\,\,\bar{B}_{\lambda,N}=\prod_{s\in\lambda}\left(N+(a^{\prime}_{\lambda}(s)+1)/\beta-(l^{\prime}_{\lambda}(s)+1)\right),$
$a^{\prime}_{\lambda}(i,j)=j-1$, $l^{\prime}_{\lambda}(i,j)=i-1$ denote the
co-arm-length and co-leg-length for the box $s=(i,j)$ in Young tableau
$\lambda$.
$\Gamma_{N}^{2}\equiv\pi^{N}\dfrac{\Gamma(1+N\beta)}{\Gamma^{N}(1+\beta)}$ is
the normalization of the ground state. When $N\to\infty$, and after dividing
out the ground state normalization $\Gamma_{N}^{2}$, we obtain the
normalization for the Jack functions:
$\displaystyle\langle J_{\lambda},J_{\lambda^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle\int
J_{\lambda^{\prime}}(p^{\ast})J_{\lambda}(p)\dfrac{\prod_{i<j}|z_{i}-z_{j}|^{2\beta}}{\Gamma_{N}^{2}}\prod_{i=1}^{N}dx_{i}\mid_{N\to\infty}=\delta_{\lambda,\lambda^{\prime}}j_{\lambda}$
(8)
We shall see in the next section that eq.(6) implies the following integral
formula
$J_{\lambda}(\dfrac{a_{-}}{k})=\int
e^{k\sum_{n>0}\frac{a_{-n}}{n}p_{n}}J_{\lambda}(p^{\ast})\dfrac{\prod_{i<j}|z_{i}-z_{j}|^{2\beta}}{\Gamma_{N}^{2}}\prod_{i=1}^{N}dx_{i}\,.$
(9)
Now we shall clarify some notations used in this work. $J_{\lambda}(p)$ means
Jack polynomials in the power sum polynomial basis, $J_{\lambda}(\frac{a}{k})$
the annihilation operator valued Jack symmetric function, i.e. with the
substitution $p_{n}\rightarrow\dfrac{a_{n}}{k}$, and $J_{-\lambda}\equiv
J_{\lambda}(\frac{a_{-}}{k})$ the creation operator valued Jack symmetric
function, i.e. with the substitution
$p^{\ast}_{n}\rightarrow\dfrac{a_{-n}}{k}$.
### 2.2 Calogero-Sutherland Model
The CS model is introduced for studying Coulomb interacting electrons
distributed on a circle. The Hamiltonian for this system can be written
as111For convenience, we set the circumference of the circle as $L=\pi$:
$\displaystyle H_{CS}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{N}\frac{1}{2}\partial_{i}^{2}+\sum_{i<j}\frac{\beta(\beta-1)}{\sin^{2}(x_{ij})}$
(10)
here $\partial_{i}=\partial_{x_{i}},\,\hbar^{2}/8m=1$, $\beta=k^{2}$, $k$ is
the charge of the N identical electrons. This is an exact solvable system.
However, let’s consider another auxiliary Hamiltonian which is positive
definite and differs from the original one only by a shift of the constant
ground state energy
$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{N}(\partial_{i}+\partial_{i}\ln\prod_{l<j}\sin^{\beta}x_{lj})(\partial_{i}-\partial_{i}\ln\prod_{l<j}\sin^{\beta}x_{lj})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{N}\partial_{i}^{2}+\beta(\beta-1)\sum_{i<j}\frac{1}{\sin^{2}x_{ij}}-\frac{1}{6}\beta^{2}N(N+1)(N-1)$
$\displaystyle=$ $\displaystyle H_{CS}-E_{0}\,,$ $\displaystyle E_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{6}\beta^{2}N(N+1)(N-1)\,.$
In going from the first line to the second line of eq. (2.2), we have used the
identity
$\displaystyle\sum_{i,j\neq k}\cot x_{ij}\cot x_{ik}+\sum_{j,i\neq k}\cot
x_{ji}\cot x_{jk}+\sum_{k,i\neq j}\cot x_{ki}\cot x_{kj}$ $\displaystyle=$
$\displaystyle\sum_{i,j\neq k}\frac{-\cos x_{ij}-\cos x_{ik}\cos x_{jk}}{\sin
x_{ik}\sin x_{jk}}$ $\displaystyle=$ $\displaystyle\sum_{i,j\neq
k}(-1)=-N(N-1)(N-2)\,.$
where $x_{ij}\equiv x_{i}-x_{j}$. It is also convenient to define
$z_{i}=e^{2ix_{i}}$ for later use. The ground state should be a solution to
the eigen-equation:
$H_{CS}\psi_{0}=E_{0}\psi_{0},$
and can be easily read out:
$\psi_{0}=\prod_{i<j}(2\sin^{\beta}x_{ij})\,,$
where the factor of 2 is included for normalization reason. If one defines the
excited state as
$\psi_{\lambda}=J_{\lambda}\psi_{0}\,,$
then it can be shown that this state actually satisfies the energy eigen-
equation:
$\displaystyle H\psi_{\lambda}$ $\displaystyle=$ $\displaystyle
H\psi_{0}J_{\lambda}(p)=\psi_{0}(\psi_{0}^{-1}H\psi_{0})J_{\lambda}(p)$
$\displaystyle=$ $\displaystyle
2\psi_{0}H_{J}J_{\lambda}(p)=2\psi_{0}E_{\lambda}J_{\lambda}(p)\,,$
$\displaystyle H_{J}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\psi_{0}^{-1}H\psi_{0}=-\frac{1}{4}\sum_{i=1}^{N}(\partial_{i}+2\beta\sum_{j\neq
i}\cot x_{ij})\partial_{i}\,,$ (14)
thus the eigen-equation can be rewritten as
$\displaystyle H_{J}J_{\lambda}$ $\displaystyle=$
$\displaystyle\left[-\dfrac{1}{4}\sum_{i=1}^{N}\partial_{i}^{2}-\dfrac{1}{2}\beta\sum_{i<j}\cot
x_{ij}(\partial_{i}-\partial_{j})\right]J_{\lambda}=E_{\lambda}J_{\lambda}\,.$
(15)
We see that this coincides with the defining differential equation eq.(2) for
the Jack polynomials. The eigenstates of $H_{J}$ in the form of eq.(14)
and(15), means that $H_{J}$ is triangular with respect to the symmetric
monomials.
$J_{\lambda}\sim\left(\prod_{i=1}^{l(\lambda)}z_{i}^{\lambda_{i}}+symmetrization\right)+daughter\,\,\,terms\,.$
Here the daughter terms are the symmetrized monomials associated with Young
tableau $\lambda^{\prime}<\lambda$. That is to say, given a Young tableau
$\lambda$, one can squeeze the partition $\\{\lambda\\}$ to other partitions
by moving squares in $\lambda$ downwards to get new Young tableaux. These
terms actually reflect the triangular property of the interaction $H_{J}^{I}$
as in eq.(2). fig.1 gives an example of squeezing.
Figure 1: An example for squeezing Young tableau, where the square 3 has been
squeezed downward to form a different Young tableau.
It is easy to read out the eigenvalue of $H_{J}$ which can be read off from
the diagonal value of the leading term of the eigenstate
$\displaystyle E_{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\sum_{i}^{N}(-4\lambda_{i}^{2})-constant\,\,term\,\,part\,\,\left(\frac{1}{2}\sum_{i<j}i\beta\frac{(z_{i}+z_{j})}{(z_{i}-z_{j})}\frac{(z_{i}^{\lambda_{i}}z_{j}^{\lambda_{j}}-z_{j}^{\lambda_{i}}z_{i}^{\lambda_{j}})}{(z_{i}^{\lambda_{i}}z_{j}^{\lambda_{j}}+z_{j}^{\lambda_{i}}z_{i}^{\lambda_{j}})}(2i\lambda_{i}-2i\lambda_{j})\right)$
(16) $\displaystyle=$
$\displaystyle\sum_{i}^{N}\lambda_{i}^{2}+\beta\sum_{i<j}(\lambda_{i}-\lambda_{j})$
$\displaystyle=$
$\displaystyle\sum_{i}^{N}\lambda_{i}^{2}+\beta\sum_{i}^{N}(N-2i+1)\lambda_{i}\,.$
Here in the last step of eq.(16), we have used the fact that
$\displaystyle\sum_{i<j}^{N}(\lambda_{i}-\lambda_{j})$ $\displaystyle=$
$\displaystyle(N-1)\lambda_{1}+(N-2-1)\lambda_{2}+\cdots+(N-2i+1)\lambda_{i}\cdots$
$\displaystyle=$ $\displaystyle\sum_{i}^{N}(N-2i+1)\lambda_{i}\,.$
Here, since we are concerning ourselves to all the Young tableaux
$\lambda^{\prime}\leq\lambda$ with the restriction $l(\lambda^{\prime})\leq
N$, so we redefine $\lambda$ as well as $\lambda^{\prime}$ to include the
trailing null parts such that $l(\lambda^{\prime})=l(\lambda)=N$. Actually,
eq.(16) can be written as the following more compact formula:
$\displaystyle E_{\lambda}$ $\displaystyle=$ $\displaystyle
k\left(k^{-1}||\lambda||^{2}-k||\lambda^{t}||^{2}+kN|\lambda|\right)$ (17)
$\displaystyle||\lambda||^{2}$ $\displaystyle\equiv$
$\displaystyle\sum_{i}^{N}\lambda_{i}^{2},\,\,\,\,||\lambda^{t}||^{2}\equiv\sum_{i}^{N}(\lambda_{i}^{t})^{2},\,\,\,\,|\lambda|\equiv\sum_{i}^{N}\lambda_{i}.$
### 2.3 Second Quantized Form
In fact, the second quantized form of the CS model can be realized as a theory
of 2D scalar field $\varphi(z)$[4]. In the corresponding CFT, $\varphi(z)$ is
a scalar defined on the unit circle but can be analytically continued to
complex plane. The vertex operator for CS model reads
$\displaystyle V_{k}(z)$ $\displaystyle=$ $\displaystyle:e^{k\varphi(z)}:$
(18) $\displaystyle\varphi(z)$ $\displaystyle=$ $\displaystyle
q+plnz+\sum_{n\in z,n\neq 0}\frac{a_{-m}}{m}z^{m}$
$\displaystyle\langle\varphi(z)\varphi(w)\rangle$ $\displaystyle=$
$\displaystyle\log(z-w)\,,$
here
$[a_{n},a_{m}]=n\delta_{n+m,0},\,\,\,\,[p,q]=1,\,\,\,\,\,\varphi(z)^{\dagger}=-\varphi(z)\,.$
It is easy to show that the ground state of the CS model can be written as the
holomorphic part of the correlation function in conformal field theory
$\displaystyle\langle k_{f}|V_{k}(z_{1})\cdots V_{k}(z_{n})|k_{i}\rangle$
$\displaystyle=$
$\displaystyle\prod_{i<j}^{N}(z_{i}-z_{j})^{k^{2}}\prod_{j=1}^{N}z_{j}^{k_{i}\cdot
k}.$ (19)
If one choose $a_{0}|k_{i}\rangle=k_{i}|k_{i}\rangle$,
$k_{i}=\frac{k}{2}(1-N)$, the correlation function reproduces the ground state
of CS model up to a phase factor 222For simplicity, we drop this phase factor
in the following context.:
$\prod_{i<j}^{N}(z_{i}-z_{j})^{k^{2}}\prod_{j=1}^{N}z_{j}^{k_{i}\cdot
k}=(i)^{k^{2}\frac{N(N-1)}{2}}\prod_{i<j}(2\sin^{\beta}x_{ij}).$ (20)
Noticing that the ground state actually comes from the contraction of the
vertex operators, then we can define the state $|\psi\rangle$ as
$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{n}V_{k}(z_{j})|k_{i}\rangle\sim\prod_{i<j}(2\sin^{\beta}x_{ij}):\prod_{j=1}^{N}V_{k}(z_{j}):|k_{i}\rangle$
$\displaystyle=$
$\displaystyle\psi_{0}(x_{i})e^{k\sum_{n>0}a_{-n}p_{n}/n}|\frac{N+1}{2}k\rangle\equiv\psi_{0}(x_{i})V_{k}^{(-)}(p)|\frac{N+1}{2}k\rangle\
\,,$
with the action of the CS Hamiltonian, one obtains:
$\displaystyle\frac{1}{2}H|\psi\rangle$ $\displaystyle\sim$
$\displaystyle-\frac{1}{2}\psi_{0}\sum_{i}(\partial_{i}+2\beta\sum_{j\neq
i}\cot x_{ij})\partial_{i}V_{k}^{(-)}(p)|k_{i}\rangle$ $\displaystyle=$
$\displaystyle\psi_{0}\left(\sum_{n>0}a_{-n}a_{n}(\beta
N+n(1-\beta))+k\sum_{n,m>0}(a_{-n-m}a_{n}a_{m}+a_{-n}a_{-m}a_{n+m})\right)V_{k}^{(-)}(p)|k_{i}\rangle.$
Then we get the second quantized form of the Hamiltonian,
$\displaystyle\hat{H}:=\sum_{n>0}a_{-n}a_{n}(\beta
N+n(1-\beta))+k\sum_{n,m>0}(a_{-n-m}a_{n}a_{m}+a_{-n}a_{-m}a_{n+m}),$ (22)
and the wave function in coordinate space:
$\psi_{\lambda}(\\{z_{i}\\})=\langle
k_{f}|J_{\lambda}(a/k)\psi_{0}V_{k}^{(-)}(p)|k_{i}\rangle$ (23)
does satisfy the eigen-equation
$\displaystyle H\psi_{\lambda}$ $\displaystyle=$ $\displaystyle\langle
k_{f}|J_{\lambda}(a/k)H\psi_{0}V_{k}^{(-)}(p)|k_{i}\rangle$ $\displaystyle=$
$\displaystyle 2\langle
k_{f}|J_{\lambda}(a/k)\psi_{0}\hat{H}V_{k}^{(-)}(p)|k_{i}\rangle$
$\displaystyle=$ $\displaystyle 2E_{\lambda}\psi_{\lambda}\,,$
provided the following defining operator equation for the Jack functions is
satisfied.
$\langle 0|J_{\lambda}(a/k)\hat{H}=\langle 0|J_{\lambda}(a/k)E_{\lambda}\,.$
(25)
### 2.4 Duality Relation
Since for CS system $a_{0}=\dfrac{1}{2}(N+1)k$, the Hamiltonian, eq.(22) can
be written in a more compact form as333For convenience, we neglect the
summation symbols, one can recover them whenever one needs.
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{n,m>0}k(a_{-n}a_{-m}a_{n+m}+a_{-n-m}a_{n}a_{m})+\sum_{n>0}(2a_{0}a_{-n}a_{n}+(1-\beta)na_{-n}a_{n}-\beta
a_{-n}a_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{3}k\oint(z\partial_{z}\varphi(z)-a_{0})^{3}\dfrac{dz}{2\pi
iz}+\sum_{n>0}2a_{0}a_{-n}a_{n}+\sum_{n>0}(1-\beta)na_{-n}a_{n}-\beta
a_{-n}a_{n}$ $\displaystyle\equiv$ $\displaystyle
k\left(\hat{H^{\prime}}(k)+(2a_{0}-k)a_{-n}a_{n}\right)\,.$
There exists an explicit duality relation which can be read off as follows.
First, we have the non-zero mode part of $\hat{H}$ as
$\displaystyle\hat{H^{\prime}}(k)=\sum_{n,m>0}(a_{-n}a_{-m}a_{n+m}+a_{-n-m}a_{n}a_{m})+\sum_{n>0}(k^{-1}-k)na_{-n}a_{n}\,,$
(27)
it has an apparent symmetry
$\displaystyle k^{-1}\leftrightarrow-k\,,$ (28)
namely, let $\tilde{k}=-k^{-1}$
$\displaystyle\hat{H^{\prime}}(\tilde{k})=\hat{H^{\prime}}(k)\,.$ (29)
Now we shall show that $k\rightarrow\tilde{k}$ sends Young tableau $\lambda$
to its dual diagram (transposed diagram)
$\lambda^{t}=\\{\lambda_{1}^{t},\lambda_{2}^{t},\dots,\lambda_{N}^{t}\\}$.
Since $\hat{H^{\prime}}(k)$ acts on Jack function gives
$\displaystyle\hat{H^{\prime}}(k)|J_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle E^{(k)}_{Y}|J_{\lambda}\rangle$ (30) $\displaystyle
E^{(k)}_{\lambda}$ $\displaystyle=$
$\displaystyle\sum_{i}\left(\lambda_{i}^{2}k^{-1}-(2i-1)\lambda_{i}k\right)$
$\displaystyle=$
$\displaystyle\sum_{i}\left({\lambda^{t}}_{i}^{2}\tilde{k}^{-1}-(2i-1)\lambda^{t}_{i}\tilde{k}\right)=E^{(\tilde{k})}_{\lambda^{t}}$
Here we have used the following identity
$\sum_{i=1}^{l(\lambda)}(2i-1)\lambda_{i}=\sum_{j=1}^{l(\lambda^{t})}(\lambda^{t}_{j})^{2}.$
We now conclude that
$J_{\lambda}^{1/\beta}(\dfrac{a_{-}}{k})=(\dfrac{-1}{\beta})^{|\lambda|}J^{\beta}_{\lambda^{t}}(-ka_{-})\,.$
(32)
Finally, notice that the inclusion of of the $a_{0}$ part of $\hat{H}$,
$(2a_{0}-k)a_{n}a_{n}=Nk|\lambda|$, will not change eq.(32), only the
eigenvalue of $\hat{H}$ is different on the two sides of eq.(32).
### 2.5 Generating Function and Skew-Fusion Coefficient
The bra state $|\psi\rangle$ defined in eq.(2.3), is actually the second
quantized form of the wave packet, which is a linear superposition of the
incoming energy eigenfunctions defined on the unit circle. The superposition
coefficients are understood as the creation operators creating incoming energy
eigenstates. To see this, from eq.(32) and orthogonality condition, eq.(8), we
can expand ,
$\displaystyle\exp(k\sum_{n>0}\dfrac{a_{-n}}{n}p_{n})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}\dfrac{J_{\lambda}^{1/\beta}(\frac{a_{-}}{k})}{j_{\lambda}^{1/\beta}}J_{\lambda}^{1/\beta}(p)\Rightarrow$
(33) $\displaystyle\exp(-\dfrac{1}{k}\sum_{n>0}\dfrac{a_{-n}}{n}p_{n})$
$\displaystyle=$
$\displaystyle\sum_{\lambda}\dfrac{J_{\lambda}^{1/\beta}(\dfrac{a_{-}}{k})(-)^{|\lambda|}P_{\lambda^{t}}^{\beta}(p)}{A_{\lambda}^{1/\beta}}\,.$
Here, the last equality in eq.(33) comes from the duality relation, eq.(32),
and we have also defined
$P_{\lambda}^{\beta}(p)=\frac{J_{\lambda}^{\beta}(p)}{\prod_{s\in\lambda}(a_{\lambda}(s)\beta+l_{\lambda}(s)+1)}=\frac{J_{\lambda}^{\beta}(p)}{A_{\lambda}^{\beta}}\,,$
which is proportional to the Jack polynomials but normalized differently,
$P_{\lambda}^{1/\beta}(p)=z^{\lambda}+\sum_{\lambda^{\prime}<\lambda}m_{\lambda}^{\lambda^{\prime}}z^{\lambda^{\prime}}\,,$
(34)
Similarly $\psi^{\dagger}$ creates outgoing states,
$\displaystyle\exp(\dfrac{1}{k}\sum_{n>0}\frac{a_{n}}{-n}p_{-n})$
$\displaystyle=$
$\displaystyle\sum_{\lambda}\dfrac{J_{\lambda}^{1/\beta}(\dfrac{a}{k})(-)^{|\lambda|}{P_{\lambda^{t}}^{\beta}}(p^{\ast})}{A_{\lambda}^{1/\beta}}\,;$
(35) $\displaystyle\exp(-k\sum_{n>0}\dfrac{a_{n}}{-n}p_{-n})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}\dfrac{J_{\lambda}^{1/\beta}(\dfrac{a}{k})J^{1/\beta}_{\lambda}(p^{\ast})}{j_{\lambda}^{1/\beta}}\,.$
(36)
Besides being a wave packet, the state $|\psi\rangle$ is also a coherent state
which is the eigenstate for all the annihilation operators
${J_{\lambda}}(k^{-1}a)$ with eigenvalue $J_{\lambda}(p)$ for each Young
tableau $\lambda$. To show that the r.h.s of eq.(33) is actually a coherent
state, we need define a 3-point function[19] $\langle
J_{\mu}J_{\nu}J_{-\lambda}\rangle\equiv g_{\mu\nu}^{\lambda}\Rightarrow
J_{\mu}(p)J_{\nu}(p)=\sum_{\lambda}g_{\mu\nu}^{\lambda}j_{\lambda}^{-1}J_{\lambda}(p)\,,$
and
$J_{\mu}J_{-\nu}|\rangle=\sum_{\lambda}g_{\mu\lambda}^{\nu}j_{\lambda}^{-1}J_{-\lambda}|\rangle:=J_{-\nu/\mu}|\rangle\,$
is called the skew Jack symmetric function. Hence we have the following
equation.
$\displaystyle
J_{\mu}|\psi\rangle=\sum_{\nu}J_{\mu}\dfrac{J_{-\nu}J_{\nu}(p)}{j_{\nu}}|k_{i}\rangle=\sum_{\lambda,\nu}g_{\lambda,\mu}^{\nu}\dfrac{J_{-\lambda}}{j_{\lambda}}\dfrac{J_{\nu}(p)}{j_{\nu}}|k_{i}\rangle=\sum_{\lambda}\dfrac{J_{-\lambda}J_{\lambda}(p)}{j_{\lambda}}J_{\mu}(p)|k_{i}\rangle$
(37) $\displaystyle=J_{\mu}(p)|\psi\rangle$
In general, the fusion coefficient $g_{\mu\nu}^{\lambda}$ is not a simple
expression. However, if a rectangular Young tableau is involved, then
$g_{\lambda,s^{r}/\lambda}^{s^{r}}$ can be derived from the generating
function, eq.(33) and the normalization condition, eq.(6). One gets444In this
particular case, $s^{r}/\lambda$ is taken to represent the Young tableau as in
fig.3 and $J_{s^{r}/\lambda}$ the corresponding Jack function associated with
it.
$\displaystyle J_{\lambda}J_{-s^{r}}|0\rangle$ $\displaystyle=$ $\displaystyle
A_{\lambda}\oiint
V_{k}^{(-)}(p)P_{\lambda}(p)z_{i}^{-s-1}dz_{i}\dfrac{\prod_{i<j}|z_{i}-z_{j}|^{2\beta}}{\Gamma_{r}^{2}}B_{s^{r}}|{0}\rangle$
$\displaystyle=$ $\displaystyle
A_{\lambda}J^{1/\beta}_{s^{r}/\lambda}(\dfrac{a_{-}}{k})\dfrac{\bar{A}_{s^{r}/\lambda},r}{A_{s^{r}/\lambda}\bar{B}_{s^{r}/\lambda,r}}B_{s^{r}}|{0}\rangle\,,$
In deriving this555We drop the superscript $1/{\beta}$ of
$A_{\lambda}^{1/\beta}$ etc and add it explicitly when necessary., use has
been made of the following identity [18],
$P_{\lambda}(p)z_{i}^{-s}=P_{s^{r}/\lambda}(p^{\ast})\,.$ (39)
Thus
$\displaystyle g_{\lambda,s^{r}/\lambda}^{s^{r}}j^{-1}_{s^{r}/\lambda}$
$\displaystyle=$ $\displaystyle
A_{\lambda}\dfrac{\bar{A}_{s^{r}/\lambda,r}}{A_{s^{r}/\lambda}\bar{B}_{s^{r}/\lambda,r}}B_{s^{r},r}$
$\displaystyle=$
$\displaystyle\dfrac{B_{s^{r},r}\bar{A}_{\lambda,r}}{\bar{B}_{s^{r}/\lambda,r}}.$
In reaching the last line in the above equation, we have used the following
interesting identity:
$\displaystyle
A_{\lambda}\bar{A}_{s^{r}/\lambda,r}=A_{s^{r}/\lambda}\bar{A}_{\lambda,r}.$
(41)
This identity can be proven diagrammatically by moving squares in the Young
tableaux. The detailed presentation on this diagrammatic proof will appear
elsewhere [31].
Another example which involves the skew Jack function is of two sets of
oscillators. Let’s consider the following expansion:
$\exp(\sum_{n>0}k\dfrac{(a_{-n}+\tilde{a}_{-n})p_{n}}{n})=\sum_{l(\lambda)\leq
N}\dfrac{J_{\lambda}(\dfrac{a_{-}+\tilde{a}_{-}}{k})J_{\lambda}(p)}{j_{\lambda}}\,.$
(42)
One can also expand $\exp(\sum_{n>0}k\frac{(a_{-n}+\tilde{a}_{-n})p_{n}}{n})$
in another way,
$\displaystyle\exp(\sum_{n>0}k\dfrac{a_{-n}p_{n}}{n})\exp(\sum_{n>0}k\dfrac{\tilde{a}_{-n}p_{n}}{n})$
$\displaystyle=$
$\displaystyle\sum_{\mu,\nu}\dfrac{J_{\mu}(\dfrac{a_{-}}{k})}{j_{\mu}}J_{\mu}(p)\dfrac{J_{\nu}(\dfrac{\tilde{a}_{-}}{k})}{j_{\nu}}J_{\nu}(p)$
$\displaystyle=$
$\displaystyle\sum_{|\mu|+|\nu|=|\lambda|}\dfrac{J_{\mu}(\dfrac{a_{-}}{k})}{j_{\mu}}\dfrac{J_{\nu}(\dfrac{\tilde{a}_{-}}{k})}{j_{\nu}}\dfrac{g_{\mu\nu}^{\lambda}J_{\lambda}(p)}{j_{\lambda}}$
$\displaystyle=$
$\displaystyle\sum_{\mu,\lambda}\dfrac{J_{\mu}(\dfrac{a_{-}}{k})}{j_{\mu}}J_{\lambda/\mu}(\dfrac{\tilde{a}_{-}}{k})\dfrac{J_{\lambda}(p)}{j_{\lambda}}\,.$
Comparing eq.(42) and eq.(2.5), we find
$\displaystyle
J_{\lambda}(\dfrac{a_{-}+\tilde{a}_{-}}{k})=\sum_{\mu}\dfrac{J_{\mu}(\dfrac{a_{-}}{k})}{j_{\mu}}J_{\lambda/\mu}(\dfrac{\tilde{a}_{-}}{k})\,.$
(44)
Such that the skew Jack function can be obtained from the inner product
$J_{\lambda/\mu}(\dfrac{\tilde{a}_{-}}{k})=\langle{0}|J_{\mu}(\dfrac{a}{k})J_{\lambda}(\dfrac{a_{-}+\tilde{a}_{-}}{k})|{0}\rangle\,.$
(45)
Here, $\langle{0}|$ and $|{0}\rangle$ are the bra and ket vacuum states for
the $a_{n}$’s only. Eq.(45) turns out to be very useful when we develop a
skew-recursive integral for the construction of Jack states in section 5.
## 3 Virasoro Singular Vectors in Calogero-Sutherland Model
From the discussions in the previous sections, we can see that there exist
apparent similarities between the CS model and the Coulomb gas picture. The
Coulomb gas picture endowed with screening charges originated in [21] and
[20]. This method plays an important role in the calculations of the
correlation functions in 2D conformal field theories. The conformal blocks are
calculated with the insertions of the primary vertex operators which usually
ends up with a charge deficit. In Coulomb gas picture, such kind of charge
deficit can be compensated by sandwiching a number of conformally invariant
screening charges to make charge balanced while keeping conformal invariance
intact. To see the similarities between the CS model and the Coulomb gas
picture, notice the following:
1) For one scalar theory, we have two kinds of vertex operators which may be
interpreted as screening vertex operators with the charges $\alpha_{\pm}$ in
CFT and $\pm k^{\mp}$ in CS model.
2) In both cases there are zero norm states.
3) In CFT the descendant states are generated by the Virasoro algebra, while
in the CS model the Jack symmetric functions. Both expands a complete set of
basis.
However, despite all of those similarities, we have to address some apparent
dis-similarities:
1) $\alpha_{+}\alpha_{-}=-2$ while $k^{\pm}(-k)^{\mp}=-1$
2) In CFT, zero norm state exists for generic $\alpha_{\pm}$, while in CS
model only for $k^{2}\leq 0$, see eq.(6)
3) In CFT the conjugate state is defined by $L_{-n}^{\dagger}=L_{n}$, while in
CS model $a_{-n}^{\dagger}=a_{n}$. The two conjugations coincide only in the
case when $k^{2}\leq 0\Rightarrow c\geq 25$.
Combining the above comparison 2) and 3) we see that there is an chance to map
the two systems into each other in the case of Liouville type CFT, provided we
can solve the problem 1), i.e. mapping between $\alpha_{\pm}$ and $\pm
k^{\mp}$. It turns out that it can be solved by introducing an additional
scalar field. For example, in AGT conjecture, an additional $U(1)$ scalar is
needed to make the comparison between Nekrasov instanton counting and the
conformal blocks of the Liouville type, where the Virasoro structure is
explicitly shown ([23, 24]). In that case, Jack functions are the essential
ingredients in building up the desired conformal blocks. We shall postpone our
discussion on this point until our next paper[30] which is finishing soon.
However, in the present paper, we shall restrict ourselves to the case of one
set of oscillators in the operator formalism and to the case of generic $k$.
In this case, we shall see that the Virasoro structure is implicit.
### 3.1 Hidden Virasoro Structure
The existence of the Virasoro structure in the Jack symmetric function has
been investigated by the authors of [1, 2]. In particular, it has been found
there is a direct map between Virasoro singular vectors and the Jack functions
of the rectangular Young tableau. Although it was suggested in [1] that such
relationship should lead to an integral representations for the Jack
functions, only in some simple cases, the explicit construction was found.
Starting from the next section, we shall present a complete construction for
the Jack functions based on the Virasoro null vectors and their skew
hierarchies. Here, to see how it works, we shall make some preparations. Let’s
rewrite the Hamiltonian $\hat{H}$ as666This redefinition is not unitary but it
makes the following computation simpler.
$\displaystyle\hat{H}=\sum_{n>0}\left(\alpha_{+}\tilde{a}_{-n}\tilde{L}_{n}+(N\beta+\beta-1-\alpha_{+}\tilde{a}_{0})\tilde{a}_{-n}\tilde{a}_{n}\right),$
(46)
here we have redefined
$\tilde{a}_{n}=\sqrt{2}a_{n},\tilde{a}_{-n}=\frac{a_{-n}}{\sqrt{2}},n>0,\tilde{a}_{0}=\sqrt{2}a_{0},\alpha_{\pm}=\pm\sqrt{2}k^{\pm
1},\alpha_{+}+\alpha_{-}=2\alpha_{0}$, and the Virasoro generator
$\displaystyle\tilde{L}_{n}=\frac{1}{2}\sum_{m\in\mathbb{Z}}:\tilde{a}_{m}\tilde{a}_{n-m}:-(n+1)\alpha_{0}\tilde{a}_{n}.$
(47)
Notice that in this convention, the Hamiltonian separates into two parts, one
for the ”Virasoro part” which is proportional to $\tilde{L}_{n}$ and the other
part is in fact the conserved charge and is always diagonal on Jack functions
and its eigenvalue proportional to the norm of the Young tableau. It is clear
that any “Virasoro” singular vector $|\chi_{r,s}\rangle$ is an eigenstate of
$\hat{H}$ whose eigenvalue suggests that $|\chi_{r,s}\rangle$ is proportional
to the Jack state $J_{\\{s^{r}\\}}$. Of course, The singular vector in the
“Virasoro” sector is not singular on the CS model side, since for generic $k$,
Jack functions has non-zero norm. This is because the redefinition, eq.(47),is
not unitary and the conjugation in the “Virasoro” sector is not hermitian.
While in the CS model, the conjugation is always Hermitian for real $k$.
Figure 2: Felder’s integration contour
To make the comparison more clear, we shall assume that the $a_{0}$
eigenvalues differ from $k_{i}$ defined in eq.(19). Consider a general vacuum
state $|p\rangle$ in the CS model, which is mapped to a highest weight state
with conformal dimension $h_{p}=\frac{1}{2}p(p-2\alpha_{0})$ in the Virasoro
sector. The singular vectors appears when its descendant states combine
themselves into a highest weight state again. This can happen for quantized
$p$
$p=p_{r,s}\equiv\frac{1}{2}(1-r)\alpha_{+}+\frac{1}{2}(1-s)\alpha_{-}\,.$
and at the level $rs$. And this null vector can be constructed explicitly by
making use of the fact that $h_{p_{r,s}}=h_{p_{-r,-s}}$,
$|\chi_{r,s}\rangle=S^{r}|p_{r,-s}\rangle$ which satisfies
$\displaystyle\tilde{L}_{n}|\chi_{r,s}\rangle$ $\displaystyle=$
$\displaystyle\delta_{n,0}(h_{p_{r,s}}+rs)|\chi_{r,s}\rangle\,,n\geq 0$ (48)
$\displaystyle\tilde{a}_{0}|\chi_{r,s}\rangle$ $\displaystyle=$ $\displaystyle
p_{-r,-s}|\chi_{r,s}\rangle.$
Here $S\equiv S^{+}=\oint
V^{+}(z)dz\,,V^{\pm}(z)=:\exp(\alpha_{\pm}\tilde{\varphi}(z)):\,,\alpha_{\pm}=\pm\sqrt{2}k^{\pm
1}$ are called the screening charges in the Virasoro sector. When multiple
$S$’ act together, we take Felder’s contour [22] for $S^{r}$ (see fig.2). to
get
$|\chi_{r,s}\rangle=S^{r}|p_{r,-s}\rangle=\oiint\prod_{i<j}^{r}|z_{i}-z_{j}|^{2\beta}e^{k\sum_{n>0}a_{-n}p_{n}}\prod_{i=1}^{r}z_{i}^{-s-1}dz_{i}|p_{-r,-s}\rangle\propto
J_{-s^{r}}|p_{-r,-s}\rangle\,.$ (49)
Notice that in the equation above we have used $a_{-n}$ instead of
$\tilde{a}_{-n}$ to make the comparison with eq.(2.3).
### 3.2 An Example of Single Screening Charge
The construction of the Jack states for the rectangular diagrams, as well as
the null vectors of the Virasoro algebra hidden in the CS model, thus reduces
to the evaluation of the multi-integrals of the Selberg type in eq.(49). Since
there is no closed formula for such type of operator valued multi-integrals,
we choose to discuss some simple cases here. The simplest one is the case of
one screening charge for the Young tableau $\\{1^{n}\\}$. From eq.(49) and
duality relation eq.(32), one can verify that the state777We have dropped the
factor $\frac{1}{2\pi i}$ for convenience. We also use the label $J_{s^{r}}$
instead of $J_{\\{s^{r}\\}}$ for the same reason.
$\displaystyle|J_{1^{n}}\rangle$ $\displaystyle=$ $\displaystyle\oint
e^{-\frac{1}{k}\sum_{m>0}\frac{a_{-m}z^{m}}{m}}(-1)^{n}n!z^{-n-1}dz|p_{-n,-1}\rangle$
$\displaystyle=$ $\displaystyle\oint
e^{\alpha_{-}\tilde{\varphi}(z)}(-1)^{n}n!dz|p_{-n,1}\rangle$ $\displaystyle=$
$\displaystyle\oint
e^{p_{-n,1}\tilde{q}}e^{z\tilde{L}_{-1}}(-1)^{n}n!z^{-n-1}dz|p_{1,-1}\rangle$
$\displaystyle=$ $\displaystyle
e^{p_{-n,1}\tilde{q}}(-\tilde{L}_{-1})^{n}|p_{1,-1}\rangle$
reproduces the Jack polynomial $J_{\\{1^{n}\\}}$. To take its conjugate state
we have to be careful to take its Hermitian conjugation. Now let’s workout the
Hermitian conjugate of $\tilde{L}_{-1}$. $\tilde{L}_{-1}=\sum_{n\geq
0}\tilde{a}_{-n-1}\tilde{a}_{n}=\sum_{n\geq 0}a_{-n-1}a_{n}\equiv L_{-1}$.
Here we have defined $L_{n}=\frac{1}{2}\sum_{m\in Z}:a_{m}a_{n-m}:$. It can be
checked that $L_{-n}^{\dagger}=L_{n}$ and
$L_{0}|p_{1,-1}\rangle=\frac{k^{2}}{2}|p_{1,-1}\rangle$ Thus the normalization
of $J_{1^{n}}$ reads
$\displaystyle\langle p_{1,-1}|(L_{1})^{n}(L_{-1})^{n}|p_{1,-1}\rangle$
$\displaystyle=$ $\displaystyle(2h+n-1)(n)(2h+n-2)(n-1)\cdots(2h)\cdot 1$
$\displaystyle=$ $\displaystyle\prod_{s\in
1^{n}}(l(s)+1+a(s)\frac{1}{\beta})(l(s)+(a(s)+1)\frac{1}{\beta})$
which coincides with the Stanley’s normalization for the Jack polynomials
[19]. Since there is a natural duality in CS model which states that if one
change $k\rightarrow-1/k$ and meanwhile transpose the partition(Young
tableau), the theory doesn’t change. This implies one can define the Jack
polynomial with Young tableau $\\{n\\}$ as:
$\langle k|(L_{1})^{n}$
up to a normalization factor $k^{-2n}$,
$J_{n}=k^{-2n}\langle k|(L_{1})^{n}=n!k^{-2n}\langle 0|\oint
e^{-k\varphi(w)}w^{n-1}dw.$
## 4 Skew-Recursion Formula for Jack States
In the previous section we have shown that any “Virasoro” null vector,
represented by a multiple integral of the Selberg type, is a Jack state of the
rectangular graph up to normalization. One may naturally ask how the other
Jack states be represented. Our answer to this question is positive. In this
and the following sections we shall show that any “Virasoro” null vector, or
equivalently, the Jack state of the rectangular graph, skewed by another Jack
state is again a Jack state. In this way we can build any desired Jack state
recursively either in operator or multiple integral formalism. There are
already two kinds of integral representations of the Jack symmetric
polynomials[1][13]. Both are based on the method that the number of arguments
$N$ in $J_{\lambda}(p)$ are increased recursively. The method we have
developed is, however, in a different manner. While other methods are based on
adding blocks of squares to the Young tableau, we are trying to subtract a
block of squares from a given rectangular one. And the other difference is
that we first build an operator formalism, and later an integral formalism
based on it (in contrast to the pure operator formalism, [14]. The way to
subtract a block of squares from a given Young tableau is described in
mathematical language as ”skewing”. We have already seen this method in
section 2.5. The skewing of $\lambda$ by $\mu$ when $\lambda$ is a rectangular
one is, however, simpler. In this case, the summation only contains one term.
This fact is proven by Kadell in [18] and is presented as
$P_{\lambda}(p)\prod_{i=1}^{N}z_{i}^{-n}=P_{n^{N}/\lambda}(p^{\ast})$ with the
Young tableau
$n^{N}/\lambda:=\\{n,\cdots,n,n-\lambda_{l},n-\lambda_{l-1},\cdots,n-\lambda_{1}\\}\Rightarrow\lambda_{1}\leq
n$ and $l\leq N$. In fact, in eq.(2.5), we have made use of this identity in
the calculation of the fusion coefficients. Here, however, we shall show that
this particular skew relation has profound meaning related to the Virasoro
singular vectors. One can also view our method as an alternative proof on
Kadell’s formula, eq.(39).
### 4.1 Proposition and Examples
Figure 3: The Young tableau for $n^{N}/\lambda$, the shadowed part labeled as
$\lambda^{R}$ has been cut out from $n^{N}$.
###### Proposition 1.
Given a Jack bra state of the rectangular graph,
$|p_{-N,-n}\rangle_{\\{n^{N}\\}}=J_{-n^{N}}|p_{-N,-n}\rangle,$
if it is acted from the left by a Jack annihilation operator $J_{\lambda}$,
$\lambda\prec n^{N}$,
$J_{n^{N}/\lambda}|p_{-N,-n}\rangle:=J_{\lambda}(\frac{a}{k})|p_{-N,-n}\rangle_{\\{n^{N}\\}}$,
then $J_{n^{N}/\lambda}|p_{-N,-n}\rangle$ is again a Jack bra state up to a
normalization constant.
$J_{\lambda}(\frac{a}{k})|p_{-N,-n}\rangle_{\\{n^{N}\\}}\propto|p_{-N,-n}\rangle_{\\{n^{N}/\lambda\\}}.$
Here, the introducing of $p_{-N,-n}$ for the oscillator vacuum state is
artificial. It just make the comparison with the “Virasoro” null vector
easier. The Young tableau $\\{n^{N}\\}/\lambda$ is shown in fig.3. Before
rushing to the proof of the proposition, we start from some simple examples
according to the level of the graphs being cut.
#### 4.1.1 Example 0: level 0
In order to show that
$|p_{-N,-n}\rangle_{\\{n^{N}\\}}\propto|\chi_{N,n}\rangle\,,$ (52)
we just have to calculate
$\displaystyle\hat{H}|\chi_{N,n}\rangle$ $\displaystyle=$
$\displaystyle(N\beta+\beta-1-\alpha_{+}\tilde{a}_{0})\tilde{a}_{-n}\tilde{a}_{n}|\chi_{N,n}\rangle$
$\displaystyle=$ $\displaystyle Nn^{2}|\chi_{N,n}\rangle$
Notice that $E_{n^{N}}=Nn^{2}$, eq.(52) is proved.
#### 4.1.2 Example 1: level 1
Level one graph is just a single square. If we cut a square in the SE corner
of the rectangular graph, the resulting state is proportional to
$\tilde{a}_{1}J_{n^{N}}|p_{-N,-n}\rangle.$
It is easy to show that the ”Virasoro part” of the Hamiltonian have eigenvalue
on the resulting state
$\alpha_{+}\tilde{a}_{-n}\tilde{L}_{n}\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}}=((N-1)\beta-(n-1))\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}},$
And the diagonal part of $\hat{H}$ has the eigenvalue
$(\beta
N+\beta-1-\alpha_{+}\tilde{a}_{0})(nN-1)\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}}=n(nN-1)\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}}.$
Combining the two parts together, we have
$\hat{H}\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}}=E_{n^{N}/\Box}\tilde{a}_{1}|p_{-N,-n}\rangle_{\\{n^{N}\\}},$
Again this state has the correct property corresponding to the skew Young
tableau $\\{n^{N}/\Box\\}$.
#### 4.1.3 Example 2: level 2
There are two different Young tableaux $\lambda^{(1)}$ and $\lambda^{(2)}$ at
level 2. If we cut these Young tableaux from a rectangular one $s^{r}$, the
resulting states will span a two dimensional Hilbert space. Let us denote them
as
$|\chi\rangle=(\frac{\tilde{a}_{1}^{2}}{2k^{2}}+A\tilde{a}_{2})|\psi\rangle_{\\{n^{N}\\}},$
here $A$ is an undetermined parameter. Note that the “diagonal part” of the
Hamiltonian only shift the eigenvalue by a global constant. So for the eigen-
equation
$\hat{H}|\chi\rangle=E_{\chi}|\chi\rangle.$
we can drop this diagonal term and consider only the ”Virasoro part” of the
Hamiltonian. After a simple computation, one finds
$A=\frac{1}{\sqrt{2}k^{3}}\,\,\,\text{or}\,\,\,\frac{-1}{\sqrt{2}k}$
corresponding to $\lambda^{(1)}=\\{2\\}$ and $\lambda^{(2)}=\\{1^{2}\\}$
respectively.
#### 4.1.4 Example 3: level 3
It is straightforward to continue on to level 3 graphs being cut. The
resulting state is denoted as
$|\chi\rangle=(\frac{\tilde{a}_{1}^{3}}{2\sqrt{2}k^{3}})+A\tilde{a}_{1}\tilde{a}_{2}+B\tilde{a}_{3}|\psi\rangle_{\\{n^{N}\\}},$
here $A,B$ are undetermined parameters. There are three independent solutions
for the eigen-equation corresponds to the three Young tableau at level 3.
For the horizontal Young tableau $\\{3,0\\}$, one gets :
$A=3/2k^{4},\,\,\,\,B=\sqrt{2}/k^{5}.$
For the vertical Young tableau $\\{1,1,1\\}$,
$A=-3/2k^{4},\,\,\,\,B=\sqrt{2}/k^{2}.$
For the symmetric Young tableau $\\{2,1\\}$,
$A=-\frac{1}{2k^{2}}(1-\frac{1}{k^{2}}),\,\,\,\,B=\frac{-1}{\sqrt{2}k^{3}}.$
These results reproduce the level 3 Jack polynomials.
### 4.2 Proof by Brute Force Operator Formalism
Having checked for the low level skew Jack states, we are encouraged to find a
more general proof for the proposition 1. In this section, we shall show that
if $\langle J_{\lambda}|$ is a Jack symmetric function related to Young
tableau $\lambda$, then $J_{\lambda}|\chi_{r,s}\rangle$ is proportional to a
Jack symmetric function related to a Young tableau $s^{r}/\lambda$, with
$\lambda\prec s^{r}$. Here, $|\chi_{r,s}\rangle$ is a Virasoro singular vector
descendant from $|p_{-r,-s}\rangle$, see eq.(49). We can prove this in
operator formalism first by “brute force”. Later in the next section we shall
present it in a more compact manner. To proceed, we need to write the operator
valued Jack function as follows:
$J_{\lambda}=\sum_{\lambda^{\prime},|\lambda^{\prime}|=|\lambda|}C_{\lambda}^{\lambda^{\prime}}a_{\lambda^{\prime}_{1}}\cdots
a_{\lambda_{s}^{\prime}}\,.$ (54)
Then consider the commutator of $J_{\lambda}$ and $\hat{H}$ defined in
eq.(22),
$\displaystyle[J_{\lambda},\hat{H}]$ $\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l}C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots[a_{\lambda^{\prime}_{l}},\hat{H}]\cdots
a_{\lambda^{\prime}_{s}}$ $\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l}C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots[(1-\beta)(\lambda^{\prime}_{l})^{2}a_{\lambda^{\prime}_{l}}+2k\lambda^{\prime}_{l}L_{\lambda^{\prime}_{l}}^{\prime}+\beta
Nla_{\lambda^{\prime}_{l}}]\cdots a_{\lambda_{s}^{\prime}}\,,$
here
$L^{\prime}_{l}=\frac{1}{2}\sum_{m\in Z}(:a_{m}a_{l-m}:)-a_{0}a_{l}\,.$
In deriving this, we have used the commutation between $a_{l}$ and $\hat{H}$:
$\displaystyle[a_{l},\hat{H}]$ $\displaystyle=$
$\displaystyle(1-\beta)l^{2}a_{l}+\sum_{m>0}2kla_{-m}a_{l+m}+\sum_{l>m>0}kla_{l-m}a_{m}+\beta
Nla_{l}$ $\displaystyle=$
$\displaystyle(1-\beta)l^{2}a_{l}+2klL^{\prime}_{l}+\beta Nla_{l}\,.$
In moving $L_{\lambda^{\prime}_{l}}^{\prime}$ to the most left by the
commutation relation $[L^{\prime}_{n},a_{m}]=-ma_{m+n}$ for $n,m>0$, more
terms are generated,
$\displaystyle[J_{\lambda},\hat{H}]$ $\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l}C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{l-1}}[(1-\beta)(\lambda^{\prime}_{l})^{2}a_{\lambda^{\prime}_{l}}+\beta
N\lambda^{\prime}_{l}a_{\lambda^{\prime}_{l}}]\cdots a_{\lambda^{\prime}_{s}}$
$\displaystyle+$
$\displaystyle\sum_{\lambda^{\prime},n<l}C_{\lambda}^{\lambda^{\prime}}(2k\lambda^{\prime}_{l}\lambda^{\prime}_{n})a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{n}+\lambda^{\prime}_{l}}\cdots
a_{\lambda^{\prime}_{l-1}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda_{s}^{\prime}}$ $\displaystyle+$
$\displaystyle\sum_{\lambda^{\prime},l}C_{\lambda}^{\lambda^{\prime}}(2k\lambda^{\prime}_{l})L_{\lambda_{l}^{\prime}}^{\prime}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda_{l-1}^{\prime}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}\,.$
Let us define some notations to simplify our calculation. We denote the first
line on the r.h.s. of eq.(4.2) as $A_{0}$ since this term retain the same
number of $a_{n}$’s comparing to the original terms in $J_{\lambda}$, the
second line is named as $A_{-}$ since it contains one less $a_{n}$ comparing
to the original term in $J_{\lambda}$, the third line separates into two terms
$A_{+}+\not{A}$, which are defined as
$\displaystyle A_{+}$ $\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l,\lambda^{\prime}_{l}>m>0}C_{\lambda}^{\lambda^{\prime}}k\lambda^{\prime}_{l}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda_{l-1}^{\prime}}(a_{\lambda^{\prime}_{l}-m}a_{m})a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}$ $\displaystyle\not{A}$ $\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l,m>0}C_{\lambda}^{\lambda^{\prime}}(2k\lambda^{\prime}_{l})(a_{-m}a_{\lambda^{\prime}_{l}+m})a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda_{l-1}^{\prime}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda_{s}^{\prime}}\,.$
If we apply eq.(4.2) to a bra vacuum sate $\langle{0}|$, the contribution of
$\not{A}$ vanishes. Since $\langle{J_{\lambda}}|$ is an eigenstate of
$\hat{H}$, we conclude
$\langle{0}|J_{\lambda}\hat{H}=\langle{0}|J_{\lambda}E_{\lambda}\Rightarrow$
$\left[J_{\lambda},\hat{H}\right]=E_{\lambda}J_{\lambda}+\not{A}$ (57)
$E_{\lambda}J_{\lambda}=A_{+}+A_{-}+A_{0}$ (58)
Now we calculate the action of $\hat{H}$ on the ket state
$J_{\lambda}|{\chi_{r,s}}\rangle$
$\hat{H}J_{\lambda}|{\chi_{r,s}}\rangle=[\hat{H},J_{\lambda}]|{\chi_{r,s}}\rangle+J_{\lambda}\hat{H}|{\chi_{r,s}}\rangle=[\hat{H},J_{\lambda}]|{\chi_{r,s}}\rangle+E_{rs}J_{\lambda}|{\chi_{r,s}}\rangle\,.$
(59)
By moving $L_{\lambda_{l}}^{\prime}$ in eq.(4.2) to the most right, we get
$\displaystyle[\hat{H},J_{\lambda}]$ $\displaystyle=$ $\displaystyle
E_{\lambda}J_{\lambda}-2A_{+}-2A_{0}$ $\displaystyle-$
$\displaystyle\sum_{\lambda^{\prime},l}(2k\lambda^{\prime}_{l})C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{l-1}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}\left[L_{\lambda^{\prime}_{l}}^{\prime}-\frac{1}{2}\sum_{\lambda^{\prime}_{l}>m>0}a_{\lambda^{\prime}_{l}-m}a_{m}\right]\,.$
and
$\displaystyle
2A_{+}+2A_{0}+\sum_{\lambda^{\prime},l}(2k\lambda^{\prime}_{l})C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{l-1}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}\left[L_{\lambda^{\prime}_{l}}^{\prime}-\frac{1}{2}\sum_{\lambda^{\prime}_{l}>m>0}a_{\lambda^{\prime}_{l}-m}a_{m}\right]$
$\displaystyle=$
$\displaystyle\sum_{\lambda^{\prime},l}2k\lambda^{\prime}_{l}C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{l-1}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}$ $\displaystyle\times$
$\displaystyle[\sum_{m>0}(a_{-m}a_{\lambda^{\prime}_{l}+m})+\sum_{\lambda^{\prime}_{l}>m>0}(a_{\lambda^{\prime}_{l}-m}a_{m})+[-(k-\frac{1}{k})\lambda_{l}^{\prime}+Nk]a_{\lambda^{\prime}_{l}}]$
$\displaystyle=$ $\displaystyle
2k(\sqrt{2}\alpha_{0}-\sqrt{2}\tilde{a}_{0})+Nk)|\lambda|J_{\lambda}+\sum_{\lambda^{\prime},l}2k\lambda^{\prime}_{l}C_{\lambda}^{\lambda^{\prime}}a_{\lambda_{1}^{\prime}}\cdots
a_{\lambda^{\prime}_{l-1}}a_{\lambda^{\prime}_{l+1}}\cdots
a_{\lambda^{\prime}_{s}}\tilde{L}_{\lambda^{\prime}_{l}}\,,$
In deriving these, use has been made of eq.(58) and eq.(47). Substituting the
results in eqs.(4.2-4.2) to eq.(59) and using the property of the Virasoro
singular vector, $\tilde{L}_{l}|{\chi_{r,s}}\rangle=0,\,\,l>0$, we conclude
$\displaystyle\hat{H}J_{\lambda}|{\chi_{r,s}}\rangle=\left[E_{\lambda}+E_{r,s}-|\lambda|\hat{M}\right]J_{\lambda}|{\chi_{r,s}}\rangle\,,$
(62)
here
$\hat{M}=2k(\sqrt{2}\alpha_{0}-\sqrt{2}\tilde{a_{0}}+Nk),$
on $|{\chi_{r,s}}\rangle$, $\tilde{a_{0}}$ gives
$\tilde{a}_{0}|{\chi_{r,s}}\rangle=\left(\dfrac{1+r}{2}\alpha_{+}+\dfrac{1+s}{2}\alpha_{-}\right)|{\chi_{r,s}}\rangle\,.$
(63)
The establishment of eq.(62) finishes the proof of the proposition 1 we
proposed before, that is, Jack polynomials for rectangular Young tableaux,
skewed by an Jack state is again a Jack state.
### 4.3 More Compact Proof
The proposition 1 is proven in the previous subsection by making use of the
eigen-equation for $\hat{H}$. However, we know that the eigenstate of
$\hat{H}$ can always been written as an integral transformation,
$\displaystyle\langle{J_{\lambda}}|$ $\displaystyle\propto$
$\displaystyle\langle{0}|\oint
e^{k\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}\prod_{i<j}|z_{i}-z_{j}|^{2\beta}J_{\lambda}(\\{z_{i}\\})\prod_{i}\dfrac{dz_{i}}{z_{i}}$
$\displaystyle\equiv$ $\displaystyle\langle{0}|\oint
F_{\lambda}(a,z)\prod_{i}\dfrac{dz_{i}}{z_{i}}\,.$
Here $a\equiv\\{a_{n}\\}$, $z\equiv\\{z_{i}\\}$, and in the following the
integration measure $\prod_{i}\dfrac{dz_{i}}{z_{i}}$ will be implied without
written out explicitly. With $J_{\lambda}$ realized in this way, we found that
the brute force proof can be rewritten in a more compact form with less
indices involved. Using:
$\displaystyle
a_{-m}e^{k\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}=e^{k\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}(a_{-m}+kp_{-m})$
(65) $\displaystyle
e^{k\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}a_{-m}=(a_{-m}-kp_{-m})e^{k\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}\,,$
(66)
we have
$\displaystyle\int F_{\lambda}(a,z)\hat{H}$ $\displaystyle=$
$\displaystyle\int\left[\sum_{n,m=1}^{\infty}k\left((a_{-n}-kp_{-n})(a_{-m}-kp_{-m})a_{n+m}+(a_{-n-m}-kp_{-n-m})a_{n}a_{m}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{n=1}^{\infty}(a_{-n}-kp_{-n})a_{n}(\beta
N+n(1-\beta))\right]F_{\lambda}(a,z)$ $\displaystyle=$
$\displaystyle\int\hat{H}F_{\lambda}(a,z)+\int\left[\sum_{n,m=1}^{\infty}(k^{3}p_{-n}p_{-m}a_{n+m}-2k^{2}p_{-m}a_{-n}a_{n+m}-k^{2}p_{-n-m}a_{n}a_{m})\right.$
$\displaystyle-$ $\displaystyle\left.\sum_{n=1}^{\infty}kp_{-n}a_{n}(\beta
N+n(1-\beta))\right]F_{\lambda}(a,z)\,.$
Since the terms containing $a_{-n}$’s on the most left will annihilate the bra
vacuum $\langle{0}|$, we conclude the following identity
$\displaystyle\left[J_{\lambda},\hat{H}\right]$ $\displaystyle=$
$\displaystyle E_{\lambda}J_{\lambda}-\int
2k^{2}\sum_{n,m=1}^{\infty}p_{-m}a_{-n}a_{n+m}F_{\lambda}(a,z)$ (68)
will be true. Comparing eq.(4.3) and eq.(68), we have
$\displaystyle\int\left[\sum_{n,m=1}^{\infty}(k^{3}p_{-n}p_{-m}a_{n+m}-k^{2}p_{-n-m}a_{n}a_{m})\right.$
$\displaystyle-$ $\displaystyle\left.\sum_{n=1}^{\infty}kp_{-n}a_{n}(\beta
N+n(1-\beta))\right]F_{\lambda}(a,z)=E_{\lambda}J_{\lambda}\,.$
Now we move $a_{-n}$’s in the last term in eq. (68) to the most right to get:
$\left[J_{\lambda},\hat{H}\right]=E_{\lambda}J_{\lambda}-\int
2k^{2}\sum_{n,m=1}^{\infty}F_{\lambda}(a,z)p_{-m}(a_{-n}+kp_{-n})a_{n+m}\,.$
(70)
Using eq.(4.3), the last term in the above eqation can be rewritten as
$\displaystyle-2E_{\lambda}J_{\lambda}-\int
F_{\lambda}(a,z)\left(\sum_{n,m=1}^{\infty}2k^{2}(p_{-m}a_{-n}a_{n+m}+p_{-n-m}a_{n}a_{m})\right.$
$\displaystyle+$ $\displaystyle\left.\sum_{n=1}^{\infty}2kp_{-n}a_{n}(\beta
N+n(1-\beta))\right)\,,$
Substituting this result into eq.(70), we have
$\displaystyle\left[J_{\lambda},\hat{H}\right]$ $\displaystyle=$
$\displaystyle-E_{\lambda}J_{\lambda}-2k^{2}\int
F_{\lambda}(a,z)\left\\{\sum_{n,m=1}^{\infty}p_{-m}\left(\sum_{n=1}^{\infty}(a_{-n}a_{n+m})+\sum_{n=1}^{m-1}a_{n}a_{m-n}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{m=1}^{\infty}p_{-m}a_{m}(kN+m(k^{-1}-k))\right\\}$
$\displaystyle=$ $\displaystyle-E_{\lambda}J_{\lambda}-2k^{2}\int
F_{\lambda}(a,z)\left\\{\sum_{m=1}^{\infty}p_{-m}(\tilde{L}_{m}+a_{m}(kN-(k^{-1}-k))-2a_{0})\right\\}\,,$
where $\tilde{L}_{m}$ is the same as what we defined in eq.(47). When we apply
eq.(4.3) to a Virasoro singular vector $|{\chi_{rs}}\rangle$,
$\tilde{L}_{n}|{\chi_{rs}}\rangle=0$ implies:
$\displaystyle\left[\hat{H},J_{\lambda}\right]|{\chi_{rs}}\rangle=\left(E_{\lambda}J_{\lambda}+2k^{2}\int
F_{\lambda}(a,z)\sum_{m>0}p_{-m}a_{m}(kN+(k-k^{-1})-2a_{0})\right)|{\chi_{rs}}\rangle\,.$
(73)
Now we can check, using eqs.(4.3-65),
$-k\int
F_{\lambda}(a,z)\sum_{m>0}p_{-m}a_{m}=[J_{\lambda},\sum_{m>0}a_{-m}a_{m}]=|\lambda|J_{\lambda}\,,$
which leads to
$\hat{H}J_{\lambda}|{\chi_{rs}}\rangle=\left(E_{\lambda}+E_{\chi_{rs}}-2k|\lambda|(kN+k-k^{-1}-2a_{0})\right)J_{\lambda}|{\chi_{rs}}\rangle\,.$
(74)
Here and before we have assumed that Virasoro $\tilde{L}_{n}$ singular state
$|{\chi_{rs}}\rangle$ is an eigenstate for CS Hamiltonian $\hat{H}$ with
eigenvalue $E_{\chi_{rs}}$. This can be checked as follows. From the formula,
eq.(46)
$H=k\sum_{n=1}^{\infty}a_{-n}\tilde{L}_{n}+\sum_{n=1}^{\infty}(\beta
N+\beta-1-2ka_{0})a_{-n}a_{n}\,,$
we arrive at: $E_{\chi_{rs}}=(\beta N+\beta-1-\sqrt{2}kp_{-r,-s})l$, here $l$
is the level of the decendant states. By the construction of Virasoro singular
vectors, we know $l=rs,\,\,\sqrt{2}p_{-r,-s}=(1+r)k-(1+s)k^{-1}$, hence
$\displaystyle E_{\chi_{rs}}=\left[\beta
N+\beta-1-k((1+r)k-(1+s)\frac{1}{k})\right]|\lambda|$ (75)
$\displaystyle=rs^{2}+\beta(N-r)rs=E_{\\{s^{r}\\}}\,.$
Thus eq.(73) implies that $J_{\lambda}|{\chi}\rangle$ is an eigenstate of
$\hat{H}$ with the eigenvalue
$\displaystyle
E_{\lambda}+E_{s^{r}}-2k|\lambda|(kN+k-k^{-1}-(1+r)k+(1+s)k^{-1})$
$\displaystyle=$ $\displaystyle
E_{\lambda}+E_{s^{r}}-2|\lambda|((N-r)\beta+s))=E_{s^{r}/\lambda}\,.$
This concludes our proof of proposition 1.
## 5 Skew-Recursive Construction of Jack States
In the previous sections we have shown that if we cut, inside a rectangular
Young tableau of size $r\times s$, any sub-Young tableau in a skew way, the
resulting Young tableau is unique and hence the corresponding Jack function,
which is named as $J_{s^{r}/\lambda}$. This Jack function,
$J_{s^{r}/\lambda}$, can be used again to cut another bigger rectangular Young
tableau of size $r_{1}\times s_{1}$ to get $J_{s_{1}^{r_{1}}/(s^{r}/\lambda)}$
and so forth. If we know the construction of the Jack function for a definite
Young tableau, we can build a tower of Jack functions upon it in such a skew
way. Of course, if we start with a trivial Young tableau (empty), then the
tower of Jack functions is built upon the constructions of the Jack function
for rectangular Young tableau only, which are in fact Virasoro singular
vectors. Following is the precise procedure which leads to the recursive
construction of the Jack functions.
### 5.1 Operator Formalism
First, $J_{-\lambda}$ acts on the left vacuum to create a bra state
${}_{\lambda}\langle{0}|\equiv\langle{0}|J_{\lambda},$
$J_{\lambda}$ acts to the right will produce a skew ket state
$\displaystyle J_{\lambda}|{0}\rangle_{\\{s_{1}^{r_{1}}\\}}$
$\displaystyle\equiv$ $\displaystyle
J_{\lambda}J_{-s_{1}^{r_{1}}}|{0}\rangle\equiv
J_{-s_{1}^{r_{1}}/\lambda}|{0}\rangle$ $\displaystyle=$ $\displaystyle
g_{\lambda,s_{1}^{r_{1}}/\lambda}^{s_{1}^{r_{1}}}j^{-1}_{[\lambda,s_{1}^{r_{1}}]}J_{-[\lambda,s_{1}^{r_{1}}]}|{0}\rangle$
$\displaystyle=$ $\displaystyle
g_{\lambda,s_{1}^{r_{1}}/\lambda}^{s_{1}^{r_{1}}}j^{-1}_{[\lambda,s_{1}^{r_{1}}]}|{0}\rangle_{\\{s_{1}^{r_{1}}/\lambda\\}}\,.$
Here we use the symbol $[\lambda,s^{r}]$ to represent the unique Young tableau
$s^{r}/\lambda$, see fig.3, where $\lambda^{R}$ means $\lambda$ rotated by
$\pi$ angle. Such type of Young tableau, i.e., a rectangular one cut in the SE
corner by a rotated $\lambda$, will be frequently used recursively. For
example, $[[\lambda,s_{1}^{r_{1}}],s_{2}^{r_{2}}]$ will define another Jack
function associated with the Young tableau $s_{2}^{r_{2}}$ cut in the SE
corner by $[\lambda,s_{1}^{r_{1}}]$ rotated.
To facilitate such recursive procedure, we shall define the following
abbreviation
$\displaystyle[r,s]_{\lambda,n}$ $\displaystyle\equiv$
$\displaystyle\left[\cdots[[\lambda,s_{1}^{r_{1}}],s_{2}^{r_{2}}],\cdots,s_{n}^{r_{n}}\right]$
(78) $\displaystyle\langle J_{(r,s)_{\lambda,n+1}}$ $\displaystyle\equiv$
$\displaystyle\langle J_{s_{n+1}^{r_{n+1}}}J_{-(r,s)_{\lambda,n}}\,,$
$\displaystyle J_{(r,s)_{\lambda,0}}$ $\displaystyle=$ $\displaystyle
J_{\lambda}\,.$
Here and after, however, we shall take $\lambda$ to be the empty Young
tableau, so we shall use the abbreviation
$\displaystyle[r,s]_{n}$ $\displaystyle\equiv$
$\displaystyle\left[\cdots[[s_{1}^{r_{1}},s_{2}^{r_{2}}],s_{3}^{r_{3}}],\cdots,s_{n}^{r_{n}}\right]$
(79) $\displaystyle\langle{J_{(r,s)_{n+1}}}|$ $\displaystyle\equiv$
$\displaystyle\langle{J_{s_{n+1}^{r_{n+1}}}J_{-(r,s)_{n}}}|$ $\displaystyle
J_{(r,s)_{0}}$ $\displaystyle=$ $\displaystyle 1\,.$
It is clear that any regular Young tableau can be represented uniquely by two
integer vectors of dimension $n$ each, $[r,s]_{n}$, where $n-1$ is the total
number of skews for the Young tableau considered according to our convention.
From the definition eq. (5.1), we know that $J_{(r,s)_{n}}$ differ from the
standard Jack symmetric function $J_{[r,s]_{n}}$ only by a normalization
constant. For example,
$\displaystyle|J_{-(r,s)_{1}}\rangle$ $\displaystyle=$ $\displaystyle
J_{-s_{1}^{r_{1}}}|{0}\rangle$ (80) $\displaystyle\langle{J_{(r,s)_{2}}}|$
$\displaystyle=$
$\displaystyle\langle{0}|J_{s_{2}^{r_{2}}}J_{-s_{1}^{r_{1}}}\equiv\langle{0}|J_{s_{2}^{r_{2}}/s_{1}^{r_{1}}}$
$\displaystyle=$ $\displaystyle
g_{s_{1}^{r_{1}},s_{2}^{r_{2}}/s_{1}^{r_{1}}}^{s_{2}^{r_{2}}}j^{-1}_{s_{2}^{r_{2}}/s_{1}^{r_{1}}}\langle{0}|J_{[s_{1}^{r_{1}},s_{2}^{r_{2}}]}$
$\displaystyle J_{-(r,s)_{3}}|{0}\rangle$ $\displaystyle=$ $\displaystyle
J_{(r,s)_{2}}J_{-s_{3}^{r_{3}}}|{0}\rangle$ $\displaystyle=$ $\displaystyle
g_{s_{1}^{r_{1}},s_{2}^{r_{2}}/s_{1}^{r_{1}}}^{s_{2}^{r_{2}}}j^{-1}_{[s_{1}^{r_{1}},s_{2}^{r_{2}}]}J_{[s_{1}^{r_{1}},s_{2}^{r_{2}}]}J_{-s_{3}^{r_{3}}}|{0}\rangle$
$\displaystyle=$ $\displaystyle
g_{s_{1}^{r_{1}},s_{2}^{r_{2}}/s_{1}^{r_{1}}}^{s_{2}^{r_{2}}}j^{-1}_{[s_{1}^{r_{1}},s_{2}^{r_{2}}]}g_{[s_{1}^{r_{1}},s_{2}^{r_{2}}],[[s_{1}^{r_{1}},s_{2}^{r_{2}}],s_{3}^{r_{3}}]}^{s_{3}^{r_{3}}}$
$\displaystyle\times$ $\displaystyle
j^{-1}_{[[s_{1}^{r_{1}},s_{2}^{r_{2}}],s_{3}^{r_{3}}]}J_{-[[s_{1}^{r_{1}},s_{2}^{r_{2}}],s_{3}^{r_{3}}]}|{0}\rangle$
In general, the normalization constant can be determined as following. Suppose
$J_{(r,s)_{n}}=C_{[r,s]_{n}}J_{[r,s]_{n}}\,,$
then
$\displaystyle\langle{J_{(r,s)_{n+1}}}|$ $\displaystyle=$
$\displaystyle\langle{0}|J_{s_{n+1}^{r_{n+1}}}J_{-(r,s)_{n}}=\langle{0}|J_{s_{n+1}^{r_{n+1}}}J_{-[r,s]_{n}}C_{[r,s]_{n}}$
$\displaystyle=$ $\displaystyle
g_{[r,s]_{n},[[r,s]_{n},s_{n+1}^{r_{n+1}}]}^{s_{n+1}^{r_{n+1}}}j^{-1}_{[[r,s]_{n},s_{n+1}^{r_{n+1}}]}\langle{J_{[r,s]_{n+1}}}|C_{[r,s]_{n+1}}\,,$
so $C_{\lambda}$ can be defined recursively:
$C_{[r,s]_{n+1}}=C_{[r,s]_{n}}g_{[r,s]_{n},[r,s]_{n+1}}^{s_{n+1}^{r_{n+1}}}j_{[r,s]_{n+1}}^{-1},$
(84)
where the fusion coefficient $g_{[r,s]_{n},[r,s]_{n+1}}^{s_{n+1}^{r_{n+1}}}$
is calculated in eq.(2.5).
### 5.2 Integral Representation
In practice, an integral formalism is more useful in analysis. Based on the
operator formalism, we derive the following integrals for building the Jack
symmetric functions.
#### 5.2.1 Auxiliary Scalar Fields
Since $J_{s^{r}}$’s are essentially the building blocks for any generic Jack
function $J_{[r,s]_{n}}$, we come back to the construction of $J_{s^{r}}$ by
the following integral,
$\displaystyle J_{-s^{r}}|{p}\rangle$ $\displaystyle=$
$\displaystyle\int[dz]_{r}^{+}\prod_{i=1}^{r}z_{i}^{-s-1}e^{\sum_{n>0}\frac{a_{-n}p_{n}}{n}}|{p}\rangle\,,$
(85)
here we have defined
$\displaystyle[dz]_{r}^{+}$ $\displaystyle\equiv$
$\displaystyle\dfrac{B_{s^{r}}}{\Gamma_{r}^{2}}\prod_{i<j}|z_{i}-z_{j}|^{2\beta}\prod_{i=1}^{r}dz_{i}$
$\displaystyle\,[dz]_{s}^{-}$ $\displaystyle\equiv$
$\displaystyle\dfrac{(-1)^{sr}A_{s^{r}}}{\Gamma_{r}^{2}}\prod_{i<j}|z_{i}-z_{j}|^{2/\beta}\prod_{i=1}^{s}dz_{i}\,.$
To relate $J_{-s^{r}}|{p}\rangle$ to a Virasoro singular vector, we introduce
two scalar field, $\varphi^{(0)}$ and $\varphi^{(1)}$ to provide the right
integration measure $[dz]$,
$\langle\varphi^{(i)}(z)\varphi^{(j)}(z^{\prime})\rangle=\delta_{ij}\log(z-z^{\prime})\,,$
and define the vertex operator integral
$V_{01}^{\pm}=\oint:e^{k^{\pm 1}(\varphi^{(0)}+\varphi^{(1)})(z)}:\,\,\,\,.$
Clearly, $V_{01}^{\pm}$ is the screening charge for the Virasoro algebra
$L_{n}^{\pm}$ respectively. Here
$\displaystyle T^{01,\pm}(z)$ $\displaystyle\equiv$
$\displaystyle\sum_{n\in\mathbb{Z}}L_{n}^{\pm}z^{-n-2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\partial_{z}(\varphi^{(0)}+\varphi^{(1)}))^{2}\pm\frac{1}{2}(k-\frac{1}{k})\partial^{2}_{z}(\varphi^{(0)}+\varphi^{(1)})\,.$
Define
$\displaystyle|{\chi_{rs}^{+}}\rangle$ $\displaystyle=$
$\displaystyle(V_{01}^{+})^{r}|{p_{r,-s}}\rangle$
$\displaystyle\langle{\chi^{-}_{rs}}|$ $\displaystyle=$
$\displaystyle\langle{p_{-r,s}}|(V_{01}^{-})^{s}\,,$
clearly we have
$\displaystyle L_{n}^{+}|{\chi_{rs}^{+}}\rangle=0,\,\,\,n>0$
$\displaystyle\langle{\chi_{rs}^{-}}|L_{-n}^{-}=0,\,\,\,n>0\,.$
However, to get $J_{s^{r}}$, we have to project out one of the two scalar
fields, say, $\varphi^{(0)}$ and from eq.(45) we get,
$\displaystyle
J_{s^{r}}(\dfrac{a^{(1)}_{-}}{k})|{p}\rangle_{1}\propto_{0}\langle{p}|\chi_{rs}^{+}\rangle$
(86)
${}_{1}\langle{p}|J_{s^{r}}(\dfrac{a^{(1)}}{k})\propto\langle{\chi_{rs}^{-}}|p\rangle_{0}\,,$
(87)
so that now $J_{\pm s^{r}}$ contains only $a^{(1)}_{\pm n}$’s.
Now the Jack states read
$\displaystyle|J_{-s^{r}}\rangle$ $\displaystyle=$
$\displaystyle\int\prod_{i=1}^{r}z_{i}^{-s-1}[dz]_{r}^{+}e^{k\sum_{n>0}\frac{a_{-n}^{(1)}p_{n}}{n}}|{p}\rangle_{1}\equiv
J_{-(r,s)_{1}}\rangle$ (88) $\displaystyle\langle J_{s^{r}}|$ $\displaystyle=$
${}_{1}\langle{p}|\int
e^{\frac{1}{k}\sum_{n>0}\frac{a_{n}^{(1)}p_{-n}}{-n}}\prod_{i=1}^{s}z_{i}^{r-1}[dz]_{s}^{-}\equiv\langle
J_{(r,s)_{1}}\,,$ (89)
here $|{p}\rangle_{i}$ is the vacuum state (no oscillator excitations) for the
$\varphi^{(i)}$ scalar
$a_{n}^{(i)}|{p}\rangle_{i}=\delta_{n,0}p^{(i)}|{p}\rangle_{i}\,\,\,\,n\geq 0$
(90)
Notice that since $a_{-n}^{(0)}$ has been projected out, $J_{-s^{r}}$ is no
longer a null vector for $L_{n}^{+}$. However, $J_{-s^{r}}$ is still a null
vector for the modified Virasoro generator $\tilde{L}_{n}$ constructed with
$\varphi^{(1)}$ only, see, eq. (47).
#### 5.2.2 Bra and Ket States
Now we shall specify how the bra state $\langle{p^{+}_{p_{r,s}}}|$ and the ket
state $|{p^{-}_{p_{r,s}}}\rangle$ are labeled.
Since we have $L_{n}^{\pm}$ acts on ket-state and bra-state respectively, so
we have different screening charges for $L_{n}^{\pm}$ respectively.
$\alpha^{++}=\sqrt{2}k,\,\,\,\alpha^{+-}=-\sqrt{2}k^{-1}$
for $L_{n}^{+}$, and
$\alpha^{-+}=-\sqrt{2}k,\alpha^{--}=\sqrt{2}k^{-1}$
for $L_{n}^{-}$. If we combine $\varphi^{(i)}+\varphi^{(i+1)}$ into a single
scalar,
$\varphi=\dfrac{1}{\sqrt{2}}(\varphi^{(i)}+\varphi^{(i+1)}),$
and
$\displaystyle a_{0}|{p_{r,s}}\rangle$ $\displaystyle=$ $\displaystyle
p^{+}_{r,s}|{p_{r,s}}\rangle$ $\displaystyle\langle{p_{r,s}}|a_{0}$
$\displaystyle=$ $\displaystyle\langle{p_{r,s}}|p^{-}_{rs}\,,$
then we define
$\displaystyle p^{+}_{rs}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-r)\alpha^{++}+\frac{1}{2}(1-s)\alpha^{+-}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(1-r)\sqrt{2}k-\frac{1}{2}(1-s)\frac{\sqrt{2}}{k}$
$\displaystyle p_{rs}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+r)\alpha^{-+}+\frac{1}{2}(1+s)\alpha^{--}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(1+r)\sqrt{2}k+\frac{1}{2}(1+s)\frac{\sqrt{2}}{k}$
Now consider
$a_{0}|{p^{+}_{r,s}}\rangle_{i,i+1}=p^{+}_{r,s}|{p_{r,s}}\rangle_{i,i+1}\,.$
However, when, say $\varphi^{(i)}$ is projected out, then
$a_{0}^{(i+1)}|{p^{+}_{r,s}}\rangle_{i+1}=\dfrac{1}{\sqrt{2}}p^{+}_{r,s}|{p^{+}_{r,s}}\rangle_{i+1}\,.$
For $\langle{p^{-}_{r,s}}|$, the projection is similar. To see this notation
will provide the correct integration measure, one could check:
$\displaystyle\langle{p^{-}_{-r,s}}|(V^{-})^{s}/\Gamma_{s}^{2}|{p^{+}_{-r,-s}}\rangle_{i}$
$\displaystyle=$
$\displaystyle\langle{p^{-}_{-r,-s}}|\int\dfrac{\prod_{i<j}(z_{i}-z_{j})^{\frac{2}{k^{2}}}\prod_{i=1}^{s}z_{i}^{\frac{\sqrt{2}}{k}a_{0}}}{\Gamma_{s}^{2}}e^{\frac{\sqrt{2}}{k}\sum_{n>0}\frac{a_{n}}{-n}p_{-n}}\prod
dz_{i}|{p^{+}_{-r,-s}}\rangle_{i}$ $\displaystyle=$
${}_{i+1}\langle{p^{-}_{-r,-s}}|\int\dfrac{\prod_{i<j}(z_{i}-z_{j})^{\frac{2}{k^{2}}}}{\Gamma_{s}^{2}}\prod_{i=1}^{s}z_{i}^{\frac{\sqrt{2}}{k}(\frac{1}{2}(1-r)(-\sqrt{2}k)+\frac{1}{2}(1-s)\frac{\sqrt{2}}{k})}e^{\frac{1}{k}\sum_{n>0}\frac{a^{(i+1)}_{n}}{-n}p_{-n}}\prod_{i}dz_{i}$
$\displaystyle=$
${}_{i+1}\langle{p^{-}_{-r,-s}}|\int\prod_{i<j}\left[\dfrac{(z_{i}-z_{j})^{2}}{z_{i}z_{j}}\right]^{\frac{1}{k^{2}}}e^{\frac{1}{k}\sum_{n>0}\frac{a^{(i+1)}_{n}}{-n}p_{-n}}\prod_{i=1}^{s}z_{i}^{r-1}dz_{i}/\Gamma_{s}^{2}$
$\displaystyle=$
${}_{i+1}\langle{\chi_{rs}}|\propto\langle{p^{-}_{-r,-s}}|\int
e^{\frac{1}{k}\sum_{n>0}\frac{a^{(i+1)}_{n}}{-n}p_{-n}}\prod_{i=1}^{s}(z_{i})^{r-1}[dz]^{-}_{s}$
$\displaystyle=$
${}_{i+1}\langle{p^{-}_{-r,-s}}|J_{s^{r}}(\frac{a^{(i+1)}}{k})\,,$
produces the Jack states of rectangular graph.
#### 5.2.3 Integral Recursion
Now we have
$\displaystyle|{J_{-s_{1}^{r_{1}}}}\rangle$ $\displaystyle=$
$\displaystyle|{J_{-(r,s)_{1}}}\rangle=\,\,_{0}\langle{p_{0}}|{\chi_{r_{1},s_{1}}}\rangle_{01}$
$\displaystyle=$ $\displaystyle\int
e^{\sum_{n>0}\frac{a_{-n}^{(1)}p_{n}}{n}k}\prod_{i=1}^{r_{1}}(z_{1,i})^{-s-1}[dz_{1}]_{r_{1}}^{+}|{p^{+}_{-r_{1},-s_{1}}}\rangle_{1}$
$\displaystyle p_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}p^{+}_{-r_{1},-s_{1}}=\frac{1}{2}(1+r_{1})k-\frac{1}{2}(1+s_{1})\frac{1}{k}\,.$
Figure 4: a. Young tableau $\\{s_{2}^{r_{2}}\\}/\\{s_{1}^{r_{1}}\\}$ b. Young
tableau $\\{s_{3}^{r_{3}}\\}/(\\{s_{2}^{r_{2}}\\}/\\{s_{1}^{r_{1}}\\})$, this
is a three-ladder Young tableau.
For one skew Young tableau of the type as in fig.4.a, we have to introduce
$\varphi^{(2)}$ scalar and project out $\varphi^{(1)}$ scalar. The resulting
state is actually the skew Jack state, as what has been shown in eq.(45); We
proceed to construct
$\displaystyle\langle{J_{(r,s)_{2}}}|$ $\displaystyle=$
${}_{12}\langle{\chi_{r_{2},s_{2}}}|e^{\delta
k_{21}q}J_{-(r,s)_{1}}|p^{+}_{-r_{1},-s_{1}}\rangle_{1}$ $\displaystyle=$
${}_{12}\langle{p^{-}_{-r_{2},-s_{2}}}|(V_{12}^{-})^{s_{2}}e^{\delta
k_{21}q^{(1)}}J_{-{(r,s)_{1}}}|p^{+}_{-r_{1},-s_{1}}\rangle_{1}$
$\displaystyle=$ ${}_{2}\langle{p^{-}_{-r_{2},-s_{2}}}|\iint
e^{\sum{n>0}\frac{1}{k}\frac{a_{n_{2}}^{(2)}\sum
z_{2,i}^{-n_{2}}}{-n_{2}}}\prod_{i=1}^{s_{2}}(z_{2,i})^{r_{2}-1}\prod_{s_{2},r_{1}}(1-\frac{z_{1}}{z_{2}})\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz_{2}]_{s_{2}}^{-}[dz_{1}]^{+}_{r_{1}}\,.$
Here we have defined
$\prod^{s_{m},r_{n}}(1-\frac{z_{n}}{z_{m}})\equiv\prod_{i=1}^{s_{m}}\prod_{j=1}^{r_{n}}(1-\frac{z_{n,j}}{z_{m,i}})\,.$
and $e^{\delta k_{21}q}$ is introduced to eliminate the charge deficit in
$\varphi^{(1)}$ sector, that is
${}_{1}\langle{p^{-}_{-r_{2},-s_{2}}}|e^{\delta
k_{21}q^{(1)}}|{p^{+}_{-r_{1},-s_{1}}}\rangle_{1}\neq 0\,.$ (96)
will give the following equation,
$\displaystyle\dfrac{1}{\sqrt{2}}p^{-}_{-r_{2},-s_{2}}$ $\displaystyle=$
$\displaystyle\delta k_{21}+\frac{1}{\sqrt{2}}p^{+}_{-r_{1},-s_{1}}$ (97)
$\displaystyle\delta k_{21}$ $\displaystyle=$
$\displaystyle(p^{-}_{-r_{2},-s_{2}}-p^{+}_{-r_{1},-s_{1}})\dfrac{1}{\sqrt{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{1}{k}-k)-\frac{k}{2}(1+r_{1}-r_{2})+\frac{1}{2k}(1+s_{1}-s_{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{2}}\left\\{\alpha_{0}^{-}+p^{+}_{r_{1}-r_{2},s_{1}-s_{2}}\right\\}$
$\displaystyle 2\alpha_{0}^{\pm}$ $\displaystyle=$
$\displaystyle\alpha^{\pm+}+\alpha^{\pm-}\,.$
For two skew Young tableau,fig.4.b, $\varphi^{(3)}$ is introduced and
$\varphi^{(2)}$ eliminated.
$\displaystyle|{J_{-(r,s)_{3}}}\rangle$ $\displaystyle=$ ${}_{2}\langle
p^{-}_{-r_{2},-s_{2}}|J_{(r,s)_{2}}e^{\delta
k_{23}q^{(2)}}|{\chi_{r_{3},s_{3}}}\rangle_{23}$ $\displaystyle=$
${}_{2}\langle p^{-}_{-r_{2},-s_{2}}|J_{(r,s)_{2}}|e^{\delta
k_{23}q^{(2)}}(V_{2}^{+})^{r_{3}}|{p^{+}_{-r_{3},-s_{3}}}\rangle_{23}$
$\displaystyle=$
$\displaystyle\int\prod_{i=1}^{r_{3}}(z_{3,i})^{-s_{3}-1}[dz_{3}]_{r_{3}}^{+}\prod^{s_{2},r_{3}}\left(1-\frac{z_{3}}{z_{2}}\right)\prod_{i=1}^{s_{2}}(z_{2,i})^{r_{2}-1}[dz_{2}]_{s_{2}}^{-}$
$\displaystyle\times$
$\displaystyle\prod^{s_{2},r_{1}}\left(1-\frac{z_{1}}{z_{2}}\right)\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz_{1}]_{r_{1}}^{+}\exp\left(k\sum_{n>0}\frac{a_{-n}^{(3)}}{n}\sum_{i}z_{3,i}^{n}\right)|{p^{+}_{-r_{3},-s_{3}}}\rangle_{3}$
Similarly, we have
$\displaystyle\delta k_{23}+\frac{1}{\sqrt{2}}p^{+}_{-r_{3},-s_{3}}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}p^{-}_{-r_{2},-s_{2}}$ (100)
$\displaystyle\delta k_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-\alpha_{0}^{+}-p^{+}_{r_{2}-r_{3},s_{2}-s_{3}})=\frac{1}{\sqrt{2}}(\alpha_{0}^{-}+p^{-}_{r_{3}-r_{2},s_{3}-s_{2}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{1}{k}-k)-\frac{k}{2}(1+r_{3}-r_{2})+\frac{1}{2k}(1+s_{3}-s_{2})\,.$
In general, proceed recursively, we have, for $n$ odd
$\displaystyle J_{-(r,s)_{n}}|{p^{+}_{-r_{n},-s_{n}}}\rangle_{n}$
$\displaystyle=$ ${}_{n-1}\langle
p^{-}_{-r_{n-1},-s_{n-1}}|J_{(r,s)_{n-1}}e^{\delta
k_{n-1,n}q^{(n-1)}}|{\chi_{r_{n},s_{n}}}\rangle_{n-1,n}$ $\displaystyle=$
$\displaystyle\iiint\exp\left({k\sum_{m>0}\dfrac{a_{-m}^{(n)}\sum_{i=1}^{r_{n}}z_{n,i}^{m}}{m}}\right)\prod_{i=1}^{r_{n}}(z_{n,i})^{-s_{n}-1}\prod^{s_{n-1},r_{n}}(1-\frac{z_{n}}{z_{n-1}})$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{s_{n-1}}(z_{n-1,i})^{r_{n-1}-1}\prod^{s_{n-1},r_{n-2}}(1-\frac{z_{n-2}}{z_{n-1}})\cdots\prod_{i=1}^{s_{2}}(z_{2,i})^{r_{2}-1}$
$\displaystyle\times$
$\displaystyle\prod^{s_{2},r_{1}}(1-\frac{z_{1}}{z_{2}})\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz]^{o}_{[n]!}|{p^{+}_{-r_{n},-s_{n}}}\rangle_{n}\,.$
Here
$\displaystyle\delta k_{n-1,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-\alpha_{0}^{+}-p^{+}_{r_{n-1}-r_{n},s_{n-1}-s_{n}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\alpha_{0}^{-}+p^{-}_{r_{n}-r_{n-1},s_{n}-s_{n-1}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{1}{k}-k)-\frac{k}{2}(1+r_{n}-r_{n-1})+\frac{1}{2k}(1+s_{n}-s_{n-1})\,.$
For $n$ even,
${}_{n}\langle{p^{-}_{-r_{n},-s_{n}}}|J_{(r,s)_{n}}$ $\displaystyle=$
$\displaystyle\langle{\chi_{r_{n},s_{n}}}|e^{\delta
k_{n,n-1}q^{(n-1)}}J_{-{(r,s)_{n-1}}}|p^{+}_{-r_{n-1},-s_{n-1}}\rangle_{n-1}$
$\displaystyle=$
${}_{n}\langle{p^{-}_{-r_{n},-s_{n}}}|\iiint\exp\left({\dfrac{1}{k}\sum_{m>0}\dfrac{a_{m}^{(n)}\sum_{i=1}^{s_{n}}z_{n,i}^{-m}}{-m}}\right)\prod_{i=1}^{s_{n}}(z_{n,i})^{r_{n}-1}\prod^{s_{n},r_{n-1}}(1-\frac{z_{n-1}}{z_{n}})$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{r_{n-1}}(z_{n-1,i})^{-s_{n-1}-1}\prod^{s_{n-2},r_{n-1}}(1-\frac{z_{n-2}}{z_{n-1}})\cdots\prod_{i=1}^{s_{2}}(z_{2,i})^{r_{2}-1}$
$\displaystyle\times$
$\displaystyle\prod^{s_{2},r_{1}}(1-\frac{z_{1}}{z_{2}})\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz]^{e}_{[n]!}\,.$
Here,
$\displaystyle\delta k_{n,n-1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-\alpha_{0}^{+}-p^{+}_{r_{n}-r_{n-1},s_{n}-s_{n-1}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\alpha_{0}^{-}+p^{-}_{r_{n-1}-r_{n},s_{n-1}-s_{n}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{1}{k}-k)-\frac{k}{2}(1+r_{n-1}-r_{n})+\frac{1}{2k}(1+s_{n-1}-s_{n})\,.$
The integration measures are defined as following: for $n$ odd,
$[dz]_{[n]!}^{o}\equiv[dz_{1}]_{r_{1}}^{+}[dz_{2}]_{s_{2}}^{-}\cdots[dz_{n}]_{r_{n}}^{+}\,.$
For $n$ even,
$[dz]_{[n]!}^{e}\equiv[dz_{1}]_{r_{1}}^{+}[dz_{2}]_{s_{2}}^{-}\cdots[dz_{n}]_{s_{n}}^{-}\,.$
Eq.(5.2.3) and eq.(5.2.3) are the main results of our present work. 888In
fact, one can easily see that the distinguishment between even and odd skews
is artificial. It provides an integral representation for any Jack symmetric
function which, in our formalism, is labeled by two integer vectors of
dimension $n$ each, $(r,s)_{n}$.
The integral representation not only provide a useful tool in analyzing
problems involving Jack symmetric functions, but also give an explicit
construction of the Jack symmetric functions in terms of free bosons. It is
also desirable to work out explicitly the Selberg type multi-integrals
appearing in eq.(5.2.3) and eq.(5.2.3).
### 5.3 Integral Representation for Jack Symmetric Polynomials
Having got the integral representation for a general Jack symmetric function,
it is then straightforward to get the Jack symmetric polynomials in any number
$N$ of arguments $z_{i}$. Notice that in the following we shall present the
unnormalized Jack polynomials. However, the normalization constants can be
easily worked out.
First, let us consider $n$ even, thus
$\displaystyle J_{(r,s)_{n}}^{1/k^{2}}(\\{z_{i}\\})$ $\displaystyle\equiv$
$\displaystyle\langle
J_{(r,s)_{n}}\exp\left(k\sum_{m>0}\frac{a_{-m}^{(n)}}{m}\sum_{i=1}^{N}z_{i}^{m}\right)|{p^{+}_{n}}\rangle_{n}$
$\displaystyle=$
$\displaystyle\iiint\prod^{s_{n},N}(1-\frac{z}{z_{n}})\prod_{i=1}^{s_{n}}(z_{n,i})^{r_{n}-1}[dz_{n}]_{s_{n}}^{-}\prod^{s_{n},r_{n-1}}(1-\frac{z_{n-1}}{z_{n}})\prod_{i=1}^{r_{n-1}}(z_{n-1,i})^{-s_{n-1}-1}[dz_{n-1}]_{r_{n-1}}^{+}$
$\displaystyle\times$
$\displaystyle\prod^{s_{n-2},r_{n-1}}(1-\frac{z_{n-1}}{z_{n-2}})\prod_{i=1}^{s_{n-2}}(z_{n-2,i})^{r_{n-2}-1}[dz_{n-2}]_{s_{n-2}}^{-}\cdots$
$\displaystyle\times$
$\displaystyle\prod^{s_{2},r_{1}}(1-\frac{z_{1}}{z_{2}})\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz_{i}]_{r_{1}}^{+}\,.$
And for $n$ odd,
$\displaystyle J^{1/k^{2}}_{(r,s)_{n}}(\\{z_{i}^{-1}\\})$
$\displaystyle\equiv$
${}_{n}\langle{p_{n}^{-}}|\exp\left(\frac{1}{k}\sum_{m>0}\frac{a_{m}^{(n)}}{-m}\sum_{i=1}^{N}z_{i}^{-m}\right)J_{-(r,s)_{n}}\rangle$
$\displaystyle=$
$\displaystyle\iiint\prod^{N,r_{n}}(1-\frac{z_{n}}{z})\prod_{i=1}^{r_{n}}(z_{n,i})^{-s_{n}-1}[dz_{n}]_{r_{n}}^{+}\prod^{s_{n-1},r_{n}}(1-\frac{z_{n}}{z_{n-1}})\prod_{i=1}^{s_{n-1}}(z_{n-1,i})^{r_{n-1}-1}[dz_{n-1}]^{-}_{s_{n-1}}$
$\displaystyle\times$
$\displaystyle\prod^{s_{n-1},r_{n-2}}(1-\frac{z_{n-2}}{z_{n-1}})\prod_{i=1}^{r_{n-2}}(z_{n-2,i})^{-s_{n-2}-1}[dz_{n-2}]^{+}_{s_{n-2}}\cdots$
$\displaystyle\times$
$\displaystyle\prod^{s_{2},r_{1}}(1-\frac{z_{1}}{z_{2}})\prod_{i=1}^{r_{1}}(z_{1,i})^{-s_{1}-1}[dz_{1}]_{r_{1}}^{+}\,.$
Now $p_{n}^{\pm}$ can be easily worked out,
$\displaystyle p^{+}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}p^{-}_{-r_{n},-s_{n}}=\frac{1}{\sqrt{2}}\left(\frac{1}{2}(1-r_{n})\alpha^{-+}+\frac{1}{2}(1-s_{n})\alpha^{--}\right)$
$\displaystyle=$ $\displaystyle-\frac{k}{2}(1-r_{n})+\frac{1}{2k}(1-s_{n})$
$\displaystyle p^{-}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}p^{+}_{-r_{n},-s_{n}}=\frac{k}{2}(1+r_{n})-\frac{1}{2k}(1+s_{n})\,.$
(108)
## 6 Acknowledgement
This work is part of the CAS program ”Frontier Topics in Mathematical Physics”
(KJCX3-SYW-S03) and is supported partially by a national grant NSFC(11035008).
## References
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|
arxiv-papers
| 2011-07-21T11:16:53 |
2024-09-04T02:49:20.808218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-Feng Wu, Ying-Ying Xu, and Ming Yu",
"submitter": "Ming Yu",
"url": "https://arxiv.org/abs/1107.4234"
}
|
1107.4470
|
# Symmetry Breaking in Neuroevolution: A Technical Report
Onay Urfalioglu and Orhan Arikan
###### Abstract
Artificial Neural Networks (ANN) comprise important symmetry properties, which
can influence the performance of Monte Carlo methods in Neuroevolution. The
problem of the symmetries is also known as the competing conventions problem
or simply as the permutation problem. In the literature, symmetries are mainly
addressed in Genetic Algoritm based approaches. However, investigations in
this direction based on other Evolutionary Algorithms (EA) are rare or
missing. Furthermore, there are different and contradictionary reports on the
efficacy of symmetry breaking. By using a novel viewpoint, we offer a possible
explanation for this issue. As a result, we show that a strategy which is
invariant to the global optimum can only be successfull on certain problems,
whereas it must fail to improve the global convergence on others. We introduce
the _Minimum Global Optimum Proximity_ principle as a generalized and adaptive
strategy to symmetry breaking, which depends on the location of the global
optimum. We apply the proposed principle to Differential Evolution (DE) and
Covariance Matrix Adaptation Evolution Strategies (CMA-ES), which are two
popular and conceptually different global optimization methods. Using a wide
range of feedforward ANN problems, we experimentally illustrate significant
improvements in the global search efficiency by the proposed symmetry breaking
technique.
## 1 Introduction
Artificial Neural Networks (ANN) are general function approximators [13] and
can be used to find a functional representation of a data set. Another point
of view is that ANN’s represent a way of data compression [2]. The compression
ratio depends on the number of neurons used in the ANN which encodes the data:
the less neurons at the same representation quality, the better the
compression.
Given a problem, there are generally two kinds of optimization tasks for the
learning process of ANN’s. The first one is to find a network topology, i.e.,
the optimal number of layers and the optimal number of neurons per layer. The
second task is to find the parameters of the network, given a topology. In
this paper, we focus on the second task and assume a predefined topology.
The estimation of the ANN-parameters is generally a computationally demanding
task [28]. The corresponding Maximum-Likelihood derived error function
comprises many local optima. Therefore, local search techniques to find an
optimal solution generally fail and typically converge to a suboptimal
solution [13]. In addition, local search techniques are mainly sequential
methods and parallel implementations are limited. On the other hand, global
optimization techniques based on Monte Carlo methods such as the Genetic
Algorithm (GA) [7, 21], Covariance Matrix Adaptation Evolution Strategies
(CMA-ES) [12, 11] or Differential Evolution (DE) [29, 22, 34] are generally
very well parallelizable. Differential Evolution is one of the most popular
and robust Monte Carlo global search methods, which outperforms many other
evolutionary algorithms on a wide range of problems [3, 33, 36]. DE is
successfully used in various engineering problems such as multiprocessor
synthesis [23], optimization of radio network designs [20], training Radial
Basis Function networks [18], training multi layer neural networks [15] and
many others [5]. On the other hand, CMA-ES is a state-of-the-art evolutionary
algorithm, which is also used for ANN-learning [27, 26, 8] and other
engineering tasks [24, 16, 25].
Due to inherent symmetries in the parametric representation of ANN’s, there
are multiple _global_ optima in the parameter space. The multiple global
optima result from point symmetries and permutation symmetries [30, 31]. In
the literature, this problem is also known as the _competing conventions
problem_ , or simply the _permutation problem_. In [32, 31], significant
improvements are reported by different approaches to symmetry breaking for
GA’s. However, in both publications, the improvement is shown using only one
single test-case, respectively. On the other hand, in [10, 9] contradictionary
results are presented, where the effect of removing these symmetries on GA’s
is reported to be minimal and negligable, and even leading to reduced
performance.
Furthermore, crossover operators used in GA’s are reported to be a source of
the problems caused by symmetries [6]. Therefore, some researchers disable
crossover or apply EA’s which do not have crossover at all [37].
To our best knowledge, there are no reports on the impact of the ANN-
symmetries regarding the performance of the DE and CMA-ES methods. In this
paper, we show that the performance of DE and CMA-ES are highly sensitive to
the presence of multiple global optima, and that symmetries are also an issue
on the performance of EA’s without crossover operators. We show that there are
infinitely many ways of symmetry breaking, which differ in the way they
partititon the parameter space. Furthermore, we argue that an effective way of
partitioning should depend on the location of the global optimum and its
symmetric replicas. Therefore, we derive a symmetry breaking operator based on
considerations about the partitioning of the ANN-parameter space, which is
optimal according to a _Minimum Global Optimum Proximity_ condition. By
theoretical considerations and numerous experimental studies on offline
supervised learning problems, we show that typical approaches to symmetry
breaking, which are invariant to the global optimum, may lead to superior or
inferior results, depending on the ANN-problem.
On the other hand, we show that the proposed global optimum variant approach
for symmetry breaking leads to consistent and significant improvements in the
estimation of ANN-parameters.
The paper is organized as follows. In the following Section, we briefly review
Artificial Feedforward Neural Networks (ANN). Section 3 defines the term
’symmetry’ and introduces the types of symmetries found in the optimization of
ANN-parameters. In Section 4, we discuss existing approaches to symmetry
breaking. In this Section, we also reformulate the rules applied by existing
approaches to prepare a more general view to the topic. In Section 5, we
introduce the ’Minimum global optimum proximity’ principle and propose
symmetry breaking methods based on this principle. In Section 6, we present
the conducted experiments and obtained results, followed by the Section of
Conclusions, where the main contributions are emphasized.
## 2 Brief review of Artificial Feedforward Neural Networks
Artificial (Feedforward) Neural Networks (ANN) are used for approximation of
functions $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{q}$. ANN’s typically have
multiple layers of artificial neurons. Assuming that an ANN has $L$ layers,
the first and the last layer are called as the input and the output layer,
respectively. Remaining $L-2$ layers are called as hidden layers.
For the $n$-th neuron $(l,n)$ in layer $l$, we denote a parameter vector by
$\bm{\eta}^{l}_{n}=(\bm{w}^{l}_{n},\tau^{l}_{n}),\ n=1,...,N_{l},$ (1)
where $\bm{w}^{l}_{n}$ is the weight vector of dimension equal to the number
of inputs available to the neuron and $\tau^{l}_{n}$ is the shift scalar. The
output of a tanh-type sigmoid neuron $(l,n)$ is given by
$x^{l}_{n}=\tanh({\bm{w}^{l}_{n}}^{\top}\bm{x}^{l-1}+\tau^{l}_{n}),$ (2)
where $\bm{x}^{l}=(x^{l}_{1},\ldots,x^{l}_{N_{l}})$ is the output vector of
layer $l$. After all hidden layers $l=2,3,...,L-1$ are evaluated, the output
layer component $\hat{y}_{n}$ of the output vector $\hat{\bm{y}}$ is typically
obtained by the following two alternative ways:
$\displaystyle\hat{y}_{n}={\bm{w}^{L}_{n}}^{\top}\bm{x}^{L-1},\ $
$\displaystyle n=1,...,q\text{ (regression)},$ (3)
$\displaystyle\hat{y}_{n}=\tanh({\bm{w}^{L}_{n}}^{\top}\bm{x}^{L-1}),\ $
$\displaystyle n=1,...,q\text{ (classification).}$ (4)
We denote the parameter vector of all neurons in a layer $l$ by
$\bm{\lambda}^{l}$, where
$\bm{\lambda}^{l}=(\bm{\eta}^{l}_{1},\ldots,\bm{\eta}^{l}_{N_{l}}).$ (5)
The vector of all the parameters in the network is given by
$\bm{\theta}_{a}=(\bm{\lambda}^{2},\ldots,\bm{\lambda}^{L-1},\bm{w}^{L}_{1},\ldots,\bm{w}^{L}_{q}),$
(6)
where $\bm{w}_{n}^{L}=(w^{L}_{n,1},\ldots,w^{L}_{n,N_{L-1}}),\ n=1,...,q$, is
the vector of the output layer weights for output $\hat{y}_{n}$. The function
defined by the network is denoted by
$\hat{\bm{y}}=\Omega(\bm{\theta}_{a};\bm{x}),$ (7)
where $\bm{x}$ is the input vector, which is notationwise equal to the output
of the input layer, so that $\bm{x}^{1}\equiv\bm{x}$.
Assuming additive normal i.i.d. noise on the available data
$(\bm{x}_{k},\bm{y}_{k}),k=1,...,K$, the ML-estimate $\hat{\bm{\theta}}_{a}$
of the parameters $\bm{\theta}_{a}$ can be obtained by the minimizer to the
following least squares optimization problem:
$\hat{\bm{\theta}}_{a}=\arg\min_{\bm{\theta}_{a}}\sum_{k=1}^{K}(\bm{y}_{k}-\Omega(\bm{\theta}_{a};\bm{x}_{k}))^{\top}(\bm{y}_{k}-\Omega(\bm{\theta}_{a};\bm{x}_{k})).$
(8)
For regression problems, the output layer is linear as shown in Eqn. (3).
Thus, the corresponding weights $\bm{w}^{L}$ can be determined by a least
squares method, as described in [19], which we adopt in this paper. This has
the advantage that global search is applied only to the non-linear part of the
parameter space, which generally speeds up convergence. For classification
problems, we assume that an output vector $\bm{y}$ of a data-sample
designating class $i$ has the following format
$y_{j}=\left\\{\begin{matrix}1&\mathrm{for}\ j=i\\\
0&\mathrm{else.}\end{matrix}\right.$ (9)
Although the output layer is non-linear as shown in Eqn. (4), corresponding
weights $\bm{w}_{n}^{L}$ can still be determined linearly in the training
phase. For this, the output vectors of the training data are rescaled by
factor 20, such that $\tanh(20)\approx 1$ and $\tanh(0)=0$. The weights of the
output layer are determined by a least squares method using the rescaled data.
Given the remaining parameters, Eqn. (8) is applied by using the non-rescaled
data.
Consequently, the parameter vector $\bm{\theta}$ for the global optimization
can be reduced to
$\bm{\theta}=(\bm{\lambda}^{2},\ldots,\bm{\lambda}^{L-1}).$ (10)
The important problem of how to choose the net topology is not considered in
this paper. For a given net-topology, we focus on the effect of symmetry
breaking on the efficiency of the optimization of the parameters in (10). In
the following Section, we investigate the symmetries in the ANN-parameter
space.
## 3 Symmetries in ANN’s
A _symmetry_ is an operator $\Phi$ which does not change the output of an ANN
when applied to the parameter vector $\bm{\theta}$:
$\Omega(\bm{\theta};\bm{x})=\Omega(\Phi(\bm{\theta});\bm{x}),\
\forall\bm{\theta},\bm{x}.$ (11)
Non reducable ANN’s comprise two types of symmetries [30]. The first type is a
_point symmetry_ on the neuron parameter level, since
$w\tanh(x)=-w\tanh(-x),\ \forall w,x.$ (12)
The following definition of a point symmetry operator $O^{l}_{n}$
$O^{l}_{n}(\bm{\theta}):\left\\{\begin{array}[]{lcl}\bm{\eta}^{l}_{n}&\rightarrow&-\bm{\eta}^{l}_{n}\\\
w^{l+1}_{i,n}&\rightarrow&-w^{l+1}_{i,n},\
i=1,\ldots,N_{l+1}\end{array}\right.$ (13)
changes the sign of the parameters of neuron $(l,n)$ and the $n$-th weight
component $w^{l+1}_{i,n}$ of all neurons $(l+1,i)$ in the following layer
$l+1$. It satisfies the symmetry condition because of Eqn. (12). In Fig. 1, an
example for the application of $O^{2}_{1}$ is shown. For each layer $l$, the
point symmetry yields $2^{N_{l}}$ symmetric replicas of the parameter vector
$\bm{\theta}$.
Figure 1: Application of the point symmetry operator $O^{2}_{1}$, which
changes the signs of $\bm{\eta}^{2}_{1}$-parameters in layer two and
$w^{3}_{i,1}$-parameters in layer three, respectively.
The second type of symmetry is a _permutation symmetry_ by the neuron
parameters $\bm{\eta}$ and the corresponding weight parameters in the next
layer. A permutation operator $P^{l}_{j,k}$ defined by
$P^{l}_{j,k}(\bm{\theta}):\left\\{\begin{array}[]{lcl}\bm{\eta}^{l}_{j}\leftrightarrow\bm{\eta}^{l}_{k}\\\
w^{l+1}_{i,j}\leftrightarrow w^{l+1}_{i,k},\
i=1,\ldots,N_{l+1}\end{array}\right.$ (14)
leaves the output invariant. Note that $P^{l}_{j,k}=P^{l}_{k,j}$. In Fig. 2,
the application of $P^{2}_{1,2}=P^{2}_{2,1}$ is illustrated. In each layer
$l$, there are $N_{l}!$ symmetric replicas of the parameter vector
$\bm{\theta}$ due to permutation symmetries. Combining both symmetries, the
total count of symmetric replicas per layer $l$ is $2^{N_{l}}N_{l}!$. Another
important property is that the length of the vector $\bm{\theta}$ is invariant
under such symmetry operators,
$||\Phi(\bm{\theta})||=||\bm{\theta}||,\ \forall\bm{\theta},$ (15)
since the point symmetry operator only changes the sign of some components of
the parameter vector, whereas the permutation symmetry operator only swaps
some components.
Figure 2: Application of the permutation symmetry operator
$P^{2}_{1,2}=P^{2}_{2,1}$, which exchanges the parameters
$\bm{\eta}^{2}_{1}\leftrightarrow\bm{\eta}^{2}_{2}$ in layer two and the
parameters $w^{3}_{1,1}\leftrightarrow w^{3}_{1,2}$,
$w^{3}_{2,1}\leftrightarrow w^{3}_{2,2}$ in layer three.
###### Lemma 3.1.
Symmetry operators are linear and orthogonal operators.
###### Proof.
The proof for the linearity of these operators is trivial and therefore
omitted in this paper. The orthogonality follows from Eqn. (15):
$\displaystyle||\Phi\bm{\theta}||=||\bm{\theta}||,\forall\bm{\theta}$
$\displaystyle\Rightarrow||\Phi\bm{\theta}||^{2}=||\bm{\theta}||^{2},\forall\bm{\theta}$
(16)
$\displaystyle\Rightarrow(\Phi\bm{\theta})^{\top}(\Phi\bm{\theta})=\bm{\theta}^{\top}\bm{\theta},\forall\bm{\theta}$
(17)
$\displaystyle\Rightarrow\bm{\theta}^{\top}\Phi^{\top}\Phi\bm{\theta}=\bm{\theta}^{\top}\bm{\theta},\forall\bm{\theta}\Rightarrow\Phi^{-1}=\Phi^{\top}.$
(18)
∎
Furthermore, applying the same point symmetry operator two times subsequently
does not change the parameter vector, since switching the signs of selected
components a second time reverts the first sign-change. The same holds also
for the permutation symmetry operator: swapping the selected components a
second time reverts the first swapping. Therefore, we can write
$O^{l}_{i}O^{l}_{i}=\mathcal{I},\ P^{l}_{i,j}P^{l}_{i,j}=\mathcal{I},$ (19)
where $\mathcal{I}$ is the identity operator. As a result, point symmetry,
permutation symmetry as well as joint symmetry operators correspond to
rotations and all symmetric replicas of a global optimum lie on a hypersphere.
Since such symmetries multiply the local and global optima count in the
parameter space, the ultimate goal of symmetry breaking is to reduce the total
number of local optima in the parameter space by avoiding all but one
symmetrically equivalent space partitions.
There are infinitely many ways for symmetry breaking by using the operators
$O^{l}_{n}$ and $P^{l}_{j,k}$, which depend on the _condition_ upon which
these operators are applied. As an example, consider a 2-D point symmetry as
illustrated in Fig. 3. Limiting the search space to the upper half plane
($y>0$) is one possibility to break the symmetry, where only one global
optimum remains and the space is separated into two partitions. In this case,
the point symmetry operator is to be applied only for $y<0$. Another
possibility is to reduce the space to the right half plane ($x>0$). This is
realized by applying the point symmetry operator only on the condition $x<0$.
By rotating the coordinate system, we obtain infinitely many other ways to
separate and reduce the space. As a result, there is a degree of freedom on
the choice of a specific condition or separation. We derive similar results
also for the permutation symmetry. In Section 5, we argue that there is an
optimal choice for a specific symmetry breaking condition (separation) based
on considerations about the location of the global optimum. We exploit the
degree of freedom on the choice of a specific condition by choosing a
condition such that the distance of the global optimum to the separating
region is maximal. In other words, we demand that the proximity of the global
optimum to the separating region is minimal. This way, the influence of
neighboring global optima is minimized and the symmetry breaking can be
realized most effectively.
A detailed discussion about an optimal separation follows in Section 5.
## 4 Existing approaches to deal with symmetries
A commonly used method is to reduce the parameter space to one single
symmetrically equivalent region, also called partition. To achieve this, the
following rules can be applied [31]:
rule-1
The shift parameter of _all_ neurons is ensured to be positive by flipping the
signs of the parameters when required, for each neuron.
rule-2
In each hidden layer, neurons are sorted according to the shift parameter.
This method and all other similar methods can be realized by applying a chain
of the operators $O^{l}_{n}$ and $P^{l}_{j,k}$. In the following, we show that
these rules are suboptimal, and in some cases may even cause inferior
performance. We show that rules for symmetry breaking should take the position
of the global optimum into account in order to be effective. Therefore, we
denote rule-1 and rule-2 as _global optimum invariant_ , and rules which
depend on the global optimum as _global optimum variant_.
### 4.1 Global optimum invariant point symmetry breaking
Assuming a point symmetric function $f(x,y)=f(-x,-y),\ \forall x,y$, Fig. 3
shows two cases where rule-1 is applied such that all $y$-coordinates are
forced to be positive. As a consequence, all solution candidates are located
in the upper half plane and the parameter space is effecively reduced. There
is only one remaining global optimum $\bar{\bm{\eta}}$. In the left plot, the
global optima $\bar{\bm{\eta}}$ and $-\bar{\bm{\eta}}$ are relatively far away
from the $x$-axis, whereas in the right plot, the global optima are close to
the $x$-axis, although they have the same distance to the origin in both
plots. In case of the right plot, there exists an ’artificial’ local optimum
due to the proximity of the hidden global optimum $-\bar{\bm{\eta}}$, where
some solution candidates may be attracted to. The main problem is that after
applying symmetry breaking, some solution candidates may still be closer to
the hidden global optimum $-\bar{\bm{\eta}}$ than to $\bar{\bm{\eta}}$. As a
result, the goal of reducing the influence of other global optima is not fully
achieved. Furthermore, the introduced artificial local optimum may trap some
solution candidates without having a chance to ever reach the corresponding
’hidden’ global optimum $-\bar{\bm{\eta}}$. We believe that this is the main
reason why an inferior performance is reported by some symmetry breaking
approaches. Note that this situation depends on the location of the global
optimum, which in turn depends on the problem at hand. Therefore, this issue
arises on some problems, whereas on others, a symmetry breaking with increased
performance can be achieved by these rules.
Figure 3: Example for a point symmetry in 2-D, where $f(x,y)=f(-x,-y)\
\forall x,y$. It is assumed that rule-1 is applied to force all solution
candidates to be in the upper half plane ($y\geq 0$). As a result, the
parameter space is effecively reduced and there is only one remaining global
optimum $\bar{\bm{\eta}}$. In the left plot, the global optima
$\bar{\bm{\eta}}$ and $-\bar{\bm{\eta}}$ are relatively far away from the
$x$-axis, whereas in the right plot, the global optima are close to the
$x$-axis, although they have the same distance to the origin in both cases. In
case of the right plot, there exists an ’artificial’ local optimum due to the
proximity of the hidden global optimum $-\bar{\bm{\eta}}$, where some solution
candidates may be attracted to. The main problem is that after applying
symmetry breaking, some solution candidates may still be closer to the hidden
global optimum $-\bar{\bm{\eta}}$ than to $\bar{\bm{\eta}}$.
In Fig. 3, the $x$-axis is the region $\mathcal{S}$ of separation
$\mathcal{S}=\\{\lambda:(\lambda,0)\\}.$ (20)
The separating region depends on the rule and divides the parameter space into
partitions. As an example, an alternative rule, which would force all $x$
coordintates to be positive, would have the $y$-axis as the separating region.
We repeat that the distance of the global optimum to the separating region is
crucial for effective symmetry breaking, and that it should be arranged to
have this distance as large as possible. Another equivalent goal is to apply
symmetry breaking such that no solution candidate is closer to the hidden
global optimum than to the global optimum of the selected partition.
### 4.2 Global optimum invariant permutation symmetry breaking
Similar problems caused by rule-1 also arise by the application of rule-2.
This is shown in the following example. We use a 2x2 parameter structure,
i.e., two neurons with two parameters $(a_{i},b_{i})$ per neuron $i$:
$\bm{\theta}=(a_{1},b_{1},a_{2},b_{2})$. From the permutation symmetry follows
that
$f((a_{1},b_{1},a_{2},b_{2}))=f((a_{2},b_{2},a_{1},b_{1})),\ \forall
a_{1},b_{1},a_{2},b_{2},$ (21)
where $f$ shall be the error function. Let the global optimum be at
$\bar{\bm{\theta}}=(2,1,-2,3)$. There are two possibilities to apply rule-2:
sorting by parameter $a$ or sorting by parameter $b$, respectively. The
separating region varies for each choice. Choosing to sort by parameter $a$
yields $\mathcal{S}_{a}$, whereas sorting by parameter $b$ yields
$\mathcal{S}_{b}$:
$\mathcal{S}_{a}=\\{\alpha,\beta,\lambda:(\lambda,\alpha,\lambda,\beta)\\},\
\mathcal{S}_{b}=\\{\alpha,\beta,\lambda:(\alpha,\lambda,\beta,\lambda)\\}.$
(22)
We show that each separation region has a different distance to the global
optimum $\bar{\bm{\theta}}$. The closest point on $\mathcal{S}_{a}$ to
$\bar{\bm{\theta}}$ is at $\lambda=0,\alpha=1,\beta=3$, which yields the
distance $\sqrt{8}$. On the other hand, the closest point on $\mathcal{S}_{b}$
to $\bar{\bm{\theta}}$ is at $\lambda=2,\alpha=2,\beta=-2$, which yields the
distance $\sqrt{2}$. In this example, applying rule-2 by ordering the
$a$-coordinates results in a better sparation of the partitions. Would the
global optimum be at $\bar{\bm{\theta}}=(1,2,3,-2)$, the opposite case would
apply. Consequently, similar to rule-1 in the previous Section 4.1, rule-2 can
only be effective on some problems.
## 5 Minimum global optimum proximity principle
In this Section we propose new methods for symmetry breaking to avoid the
problems described in Section 4. Here, we assume that the _basin_ , or the
region of influence of the global optimum is isotropic. Although this
assumption does not apply in general, it is introduced to simplify the
discussion. Also, this simplification enables us to easily derive
theoretically motivated methods, which prove to be very effective in a wide
range of problems. In the presentation, we first consider the point symmetry,
then the permutation symmetry and finally the general joint symmetry as a
combination of both point and permutation symmetries.
### 5.1 Minimum global optimum proximity principle for point symmetry
The differences between possible rules to apply the point symmetry operator
arise from the _condition_ on which the operator is to be applied. Fig. 4
shows different rules with corresponding separation regions for breaking a
point symmetry in relation to the global optimum.
Figure 4: Example for a point symmetry in 2-D, where
$f(\bm{\eta})=f(-\bm{\eta})\ \forall\bm{\eta}$. The plots show worst case
(left), suboptimal (middle) and optimal separation lines (right) for point
symmetry breaking. The separating line divides the parameter space in two
parts, where each partition contains a global optimum ($\bar{\bm{\eta}}$ and
$-\bar{\bm{\eta}}$).
It can be seen that the separating region which has maximum distances to the
global optima, which means that the according _proximity_ is minimal, enables
the optimal separation or partitioning. This way, an optimal isolation between
all symmetric replicas of the global optimum is achieved. As a result, the
disturbing influence of other neighboring global optima is decreased to a
minimum, which in turn effectively maximizes the attraction of the global
optimum of the selected partition.
The following Lemma provides a more general perspective for rule-1 presented
in Section 4. Note that the shift parameter is the last entry in the parameter
vector.
###### Lemma 5.1.
Rule-1 from Section 4 modifies a parameter vector $\bm{\eta}$ as:
$\bm{\eta}^{\prime}=\left\\{\begin{array}[]{lcl}\bm{\eta}&\mathrm{if}&||\bm{\eta}-(0,...,0,1)||^{2}\leq||-\bm{\eta}-(0,...,0,1)||^{2}\\\
-\bm{\eta}&\mathrm{if}&||\bm{\eta}-(0,...,0,1)||^{2}>||-\bm{\eta}-(0,...,0,1)||^{2}.\end{array}\right.$
(23)
###### Proof.
From the first line in Eqn. (23) follows with $\bm{\eta}=(x_{1},...,x_{D})$
and a reference vector $\bm{r}=(0,...,0,1)$
$\displaystyle||\bm{\eta}-\bm{r}||^{2}$
$\displaystyle\leq||-\bm{\eta}-\bm{r}||^{2}$ (24)
$\displaystyle||\bm{\eta}-(0,...,0,1)||^{2}$
$\displaystyle\leq||-\bm{\eta}-(0,...,0,1)||^{2}$ (25)
$\displaystyle\Leftrightarrow\left(\sum_{i=1}^{D-1}x_{i}^{2}\right)+(x_{D}-1)^{2}$
$\displaystyle\leq\left(\sum_{i=1}^{D-1}x_{i}^{2}\right)+(-x_{D}-1)^{2}.$ (26)
Further simplifying both sides of the equation yields
$-x_{D}\leq x_{D}\ \Leftrightarrow x_{D}\geq 0.$ (27)
This means that the conditional Equation (23) is equivalent to rule-1 which
demands that the shift parameters shall be positive. ∎
The rule-structure introduced by Lemma 5.1 can be used to formulate the
following strategy to maximize the distance of the global optimum
$\bar{\bm{\eta}}$ to the separating region.
$\bm{\eta}^{\prime}=\left\\{\begin{array}[]{lcl}\bm{\eta}&\mathrm{for}&||\bm{\eta}-\bar{\bm{\eta}}||^{2}\leq||-\bm{\eta}-\bar{\bm{\eta}}||^{2}\\\
-\bm{\eta}&\lx@intercol\hfil\mathrm{otherwise}\hfil\lx@intercol\end{array}\right.$
(28)
###### Theorem 5.2.
The solution candidate $\bm{\eta}^{\prime}$ determined by rule (28) is always
closer to $\bar{\bm{\eta}}$ than to $-\bar{\bm{\eta}}$.
We will prove Theorem 5.2 in a more general setting in Section 5.3.
### 5.2 Minimum global optimum proximity principle for permutation symmetry
In this Section we introduce an optimal rule for breaking a permutation
symmetry for parameter spaces with two blocks of permutation-invariant
parameters. We define a parameter vector $\bm{\theta}$ as
$\bm{\theta}=(\bm{\eta}_{1},\bm{\eta}_{2})\equiv(\bm{\eta}_{1}|\bm{\eta}_{2}),$
(29)
where the notation $(\bm{\eta}_{1}|\bm{\eta}_{2})$ is used to emphasize the
block structure. The permutation symmetry is given by
$f(\bm{\theta})=f((\bm{\eta}_{1},\bm{\eta}_{2}))=f(P\bm{\theta})=f((\bm{\eta}_{2},\bm{\eta}_{1})),\
\forall\bm{\theta},$ (30)
where $f$ is the error function and $P$ is a permutation operator defined by
$P(\bm{\eta}_{1},\bm{\eta}_{2})=(\bm{\eta}_{2},\bm{\eta}_{1}),\
\forall\bm{\eta}_{1},\bm{\eta}_{2}.$ (31)
The following Lemma restates rule-2 as a distance dependent rule.
###### Lemma 5.3.
Assuming the shift parameter is the last parameter in the parameter block
$\bm{\eta}$, rule-2, presented in Section 4, can alternatively be described in
a more general form by the following rule:
$\bm{\theta}^{\prime}=\left\\{\begin{array}[]{lcl}(\bm{\eta}_{1}|\bm{\eta}_{2})&\mathrm{for}&||(\bm{\eta}_{1}|\bm{\eta}_{2})-(0,...,0|0,...,1)||^{2}\leq||(\bm{\eta}_{2}|\bm{\eta}_{1})-(0,...,0|0,...,1)||^{2}\\\
(\bm{\eta}_{2}|\bm{\eta}_{1})&\lx@intercol\hfil\mathrm{otherwise}\hfil\lx@intercol\end{array}\right.$
(32)
###### Proof.
From Eqn. (32) follows with $\bm{\eta}_{i}=(x_{i,1},...,x_{i,D})$
$\displaystyle||(x_{1,1},...,x_{1,D}|x_{2,1},...,x_{2,D})-(0,...,0|0,...,1)||^{2}$
$\displaystyle\leq$
$\displaystyle||(x_{2,1},...,x_{2,D}|x_{1,1},...,x_{1,D})-(0,...,0|0,...,1)||^{2}$
(33) $\displaystyle\Leftrightarrow$ $\displaystyle
x_{1,D}^{2}+(x_{2,D}-1)^{2}\leq(x_{1,D}-1)^{2}+x_{2,D}^{2}$ (34)
$\displaystyle\Leftrightarrow$ $\displaystyle x_{2,D}\leq x_{1,D}$ (35)
∎
We state the following proposal in order to maximize the distance of the
global optimum $\bar{\bm{\theta}}$ to the separating region, according to the
rule-structure introduced by Lemma 5.3
$\bm{\theta}^{\prime}=\left\\{\begin{array}[]{lcl}(\bm{\eta}_{1}|\bm{\eta}_{2})&\mathrm{for}&||(\bm{\eta}_{1}|\bm{\eta}_{2})-\bar{\bm{\theta}}||^{2}\leq||(\bm{\eta}_{2}|\bm{\eta}_{1})-\bar{\bm{\theta}}||^{2}\\\
(\bm{\eta}_{2}|\bm{\eta}_{1})&\lx@intercol\hfil\mathrm{otherwise}\hfil\lx@intercol\end{array}\right.$
(36)
###### Theorem 5.4.
The solution candidate $\bm{\theta}^{\prime}$ determined by rule (36) is
always closer to $\bar{\bm{\theta}}$ than to $P\bar{\bm{\theta}}$.
Theorem 5.4 will be proved in a more general setting in Section 5.3.
### 5.3 Ideal symmetry breaking
For a given ANN-optimization problem, let $\mathcal{P}$ be the set of all
possible symmetry operators. Note that a symmetry operator
$\Phi\in\mathcal{P}$ may be a point symmetry, a permutation symmetry or a
joint symmetry operator. A joint symmetry operator is generally composed of a
chain of point symmetry and permutation symmetry operators. As an example,
$\Phi=O^{2}_{2}\circ P^{3}_{2,4}$ applies a permutation symmetry followed by a
point symmetry operator. The following properties of symmetry operators are
relevant in the following discussion. According to Eqn. (11), a symmetry
operator does not change the output of the ANN when applied to the parameter
vector $\bm{\theta}$. According to Eqn. (15) a symmetry operator does not
change the length of a parameter vector. Furthermore, according to Eqn. (18),
symmetry operators are orthogonal.
Given a parameter vector $\bm{\theta}$, the set $\mathcal{R}_{\bm{\theta}}$ of
all symmetric replicas of $\bm{\theta}$ is defined by
$\mathcal{R}_{\bm{\theta}}=\\{\Phi\in\mathcal{P}:\Phi\bm{\theta}\\}=\mathcal{P}\bm{\theta}.$
(37)
Recall that the ultimate goal of symmetry breaking is to minimize the
influence of all symmetric replicas of the selected global optimum and to
concentrate the global search to the partition where the selected global
optimum is located. To achieve this, we propose the following joint separation
condition:
$\bm{\theta}^{\prime}=\arg\min_{\tilde{\bm{\theta}}\in\mathcal{R}_{\bm{\theta}}}||\tilde{\bm{\theta}}-\bar{\bm{\theta}}||^{2}.$
(38)
In other words, this optimization selects the closest symmetric replica of
$\bm{\theta}$ to the selected global optimum $\bar{\bm{\theta}}$. Finding the
closest symmetric replica of $\bm{\theta}$ means finding the corresponding
symmetry operator $\Phi^{\prime}$, where
$\bm{\theta}^{\prime}=\Phi^{\prime}\bm{\theta}.$ (39)
In case the parameter vector $\bm{\theta}$ is already close to
$\bar{\bm{\theta}}$, i.e., it is in the corresponding partition, the solution
for $\Phi^{\prime}$ is the identity operator $\mathcal{I}$. Note that,
according to Eqn. (19), the identity operator $\mathcal{I}$ is in
$\mathcal{P}$. In Fig. 5, ideal symmetry breaking according to Eqn. (38) is
illustrated on a hypothetical 2-D space.
$\bar{\bm{\theta}}$$\bm{\theta}^{\prime}$$\bm{\theta}$Symmetric replica of
global optimum
Figure 5: Ideal symmetry breaking according to Eqn. (38) shown on a
hypothetical 2-D space. In this example, applying a point symmetry operator
followed by a permutation symmetry operator maps $\bm{\theta}$ to
$\bm{\theta}^{\prime}$, which is located at the partition of the selected
global optimum $\bar{\bm{\theta}}$, marked by a star. Note that a symmetry
operator corresponds to a rotation, which preserves lengths as well as angles.
###### Theorem 5.5.
The solution $\bm{\theta}^{\prime}$ determined by Equation (38) ensures that
no other symmetric replica of the selected global optimum $\bar{\bm{\theta}}$
is closer to $\bm{\theta}^{\prime}$ than $\bar{\bm{\theta}}$. In other words,
it minimizes the influence of the symmetric replicas of the selected global
optimum.
###### Proof.
We prove this by contradiction. According to Eqn. (38),
$||\bm{\theta}^{\prime}-\bar{\bm{\theta}}||^{2}$ is minimal. Assume that there
exists a global optimum replica
$\bar{\bm{\theta}}^{\prime}\neq\bar{\bm{\theta}}$ with
$||\bm{\theta}^{\prime}-\bar{\bm{\theta}^{\prime}}||^{2}<||\bm{\theta}^{\prime}-\bar{\bm{\theta}}||^{2}.$
(40)
Due to the underlying symmetry properties, each global optimum replica can be
mapped to another replica by a symmetry operator, i.e., there exists a
symmetry operator $\Phi\in\mathcal{P}/\\{\mathcal{I}\\}$ which satisfies
$\bar{\bm{\theta}}^{\prime}=\Phi\bar{\bm{\theta}}\Rightarrow\bar{\bm{\theta}}=\Phi^{-1}\bar{\bm{\theta}}^{\prime}.$
(41)
Due to length-preserving property of symmetry operators, using Eqn. (39), the
left-hand side of the Relation (40) can be written as
$||\bm{\theta}^{\prime}-\bar{\bm{\theta}^{\prime}}||^{2}=||\Phi^{-1}(\bm{\theta}^{\prime}-\bar{\bm{\theta}^{\prime}})||^{2}=||\Phi^{-1}\bm{\theta}^{\prime}-\Phi^{-1}\bar{\bm{\theta}^{\prime}}||^{2}=||\Phi^{-1}\bm{\theta}^{\prime}-\bar{\bm{\theta}}||^{2}$
(42)
Since $\Phi\neq\mathcal{I}$ and therefore $\Phi^{-1}\neq\mathcal{I}$, it
follows that $\Phi^{-1}\bm{\theta}^{\prime}\neq\bm{\theta}^{\prime}$. But this
means that $\bm{\theta}^{\prime}$ does not minimize the distance to
$\bar{\bm{\theta}}$, which contradicts Eqn. (38). ∎
### 5.4 Approximations of the ideal separation
In order to take advantage of these results, we have to address two issues.
First, the global optimum is not known a priori. Second, the brute force
method for finding an optimal solution to (38) has exponential complexity, but
a low-complexity algorithm is desired. In order to circumvent the first
problem, we propose to use an estimate for the global optimum, which can be
determined by the population of solution candidates at each iteration of the
applied Monte Carlo method. Naturally, this estimate improves with increasing
iteration number. The second problem can be addressed by using an
approximation for the ideal separation achieved by (38).
To describe the proposed method, for each neuron $(l,n)$, we define a symmetry
relevant parameter block $\bm{\beta}^{l}_{n}$ as
$\displaystyle\bm{\beta}^{l}_{n}$ $\displaystyle=$
$\displaystyle(\bm{\eta}^{l}_{n},w^{l+1}_{1,n},\ldots,w^{l+1}_{N_{l+1},n}),\
l=2,...,L-2,$ (43) $\displaystyle\bm{\beta}^{L-1}_{n}$ $\displaystyle=$
$\displaystyle\bm{\eta}^{L-1}_{n},$ (44)
which includes also some corresponding parameters from the next layer $l+1$.
Given a parameter vector $\bm{\theta}$ and an estimate of the global optimum
$\hat{\bm{\theta}}$ with corresponding parameter blocks $\bm{\beta}^{l}_{n}$
and $\hat{\bm{\beta}}^{l}_{n}$, the pseudocode 1 describes the proposed
approximation for ideal symmetry breaking.
Algorithm 1 Proposed symmetry breaking method. A symmetry operator $\Phi$ is
only applied to the parameter vector $\bm{\theta}$ when it decreases the
distance to the global optimum $\hat{\bm{\theta}}$, i.e.,
$||\Phi\bm{\theta}-\hat{\bm{\theta}}||<||\bm{\theta}-\hat{\bm{\theta}}||$.
Algorithm input: $\bm{\theta}$ and $\hat{\bm{\theta}}$. Effect: modification
of the parameter vector $\bm{\theta}$ when appropriate.
[breaking point symmetry]
for all hidden layers $l=2,\ldots,L-1$ do
for all neurons $(l,n),\ n=1,\ldots,N_{l}$ per layer $l$ do
// would the point symmetry operator $O^{l}_{n}$ decrease the distance?
($||O^{l}_{n}\bm{\theta}-\hat{\bm{\theta}}||\stackrel{{\scriptstyle?}}{{<}}||\bm{\theta}-\hat{\bm{\theta}}||$)
calculate distance-square for NOT applying $O^{l}_{n}$:
$D_{1}=||\bm{\beta}^{l}_{n}-\hat{\bm{\beta}}^{l}_{n}||^{2}$
calculate distance-square for applying $O^{l}_{n}$:
$D_{2}=||-\bm{\beta}^{l}_{n}-\hat{\bm{\beta}}^{l}_{n}||^{2}$
if $D_{1}>D_{2}$ then
apply point symmetry operator $O^{l}_{n}$: set
$\bm{\beta}^{l}_{n}=-\bm{\beta}^{l}_{n}$
end if
end for
end for
[breaking permutation symmetry]
for all hidden layers $l=2,\ldots,L-1$ do
randomly choose two neurons $(l,m),\ (l,n)\in\\{1,\ldots,N_{l}\\}$ in hidden
layer $l$ with $m\neq n$
// would the permutation operator $P^{l}_{m,n}$ decrease the distance?
($||P^{l}_{m,n}\bm{\theta}-\hat{\bm{\theta}}||\stackrel{{\scriptstyle?}}{{<}}||\bm{\theta}-\hat{\bm{\theta}}||$)
calculate distance-square for NOT applying $P^{l}_{m,n}$:
$D_{1}=||\bm{\beta}^{l}_{n}-\hat{\bm{\beta}}^{l}_{n}||^{2}+||\bm{\beta}^{l}_{m}-\hat{\bm{\beta}}^{l}_{m}||^{2}$
calculate distance-square for applying $P^{l}_{m,n}$:
$D_{2}=||\bm{\beta}^{l}_{n}-\hat{\bm{\beta}}^{l}_{m}||^{2}+||\bm{\beta}^{l}_{m}-\hat{\bm{\beta}}^{l}_{n}||^{2}$
if $D_{1}>D_{2}$ then
apply permutation symmetry operator $P^{l}_{m,n}$: swap
$\bm{\beta}^{l}_{m}\leftrightarrow\bm{\beta}^{l}_{n}$
end if
end for
In Fig. 6, the effect of the several symmetry breaking approaches is
demonstrated on a hypothetical 2-D parameter space.
Figure 6: Examples of symmetry breaking methods. Given a distribution of
solution candidates as shown in the upper circle, typical outcomes of three
different symmetry breaking methods are shown. In the left-bottom case, all
solution candidates are mapped into the selected partition, but the global
optimum is not necessarily centered within the partition. As a downside, there
is a relatively strong influence of the global optimum from the neighbor
partition. In the center-bottom case, the selected partition is chosen such
that the distance to other symmetric replica of the global optimum are
maximized, and all solution candidates are mapped into the selected partition.
The right-bottom case shows the proposed approximate global optimum variant
symmetry breaking. It equals the center-bottom case, except that the solution
candidates are not necessarily mapped into the selected partition, but also to
other partitions close to the selected one.
#### 5.4.1 DE with symmetry breaking
The DE method [29, 22] comprises a population of solution candidates
$\bm{\theta}_{i}$, which are iteratively updated and moved towards an optimal
solution. We propose to choose the centroid of the population at each
iteration as an estimate for the global optimum $\hat{\bm{\theta}}$.
The DE method extended by the global optimum invariant symmetry breaking [31]
is denoted by DE-INV-SB, DE extended by the proposed global optimum variant
symmetry breaking, described by Algorithm 1, is denoted by DE-SB and DE with
global optimum variant ideal symmetry breaking using brute force search is
denoted by DE-SB-BF. As shown in Fig. 7, in DE-based symmetry breaking
approaches, symmetry breaking is always applied on each solution candidate
$\bm{\theta}_{i}$ right after it has been updated for the next iteration. Only
in DE-SB, we apply an additional step by increasing the error yield of some
solution candidates which are not in the same partition as the selected
partition holding $\hat{\bm{\theta}}$. This increases the probability that
these solution candidates are updated and moved closer to the selected
partition. This is not required for symmetry breaking approaches which map
each solution cadidate exactly to the selected partition, such as DE-INV-SB or
DE-SB-BF. The DE-SB method is described in Algorithm 2.
Algorithm 2 DE-SB. Algorithm input: population of candidate vectors
$\bm{\theta}_{j},\ j=1,...,N_{p}$ and the centroid of the population as the
estimate for the global optimum $\hat{\bm{\theta}}$. Effect: modify candidate
vectors $\bm{\theta}_{j},\ j=1,...,N_{p}$ when appropriate.
for all candidate vectors $\bm{\theta}_{j},\ j=1,...,N_{p}$ do
apply symmetry breaking on $\bm{\theta}_{j}$, see Algorithm 1
if $\bm{\theta}_{j}$ modified (a symmetry operator was applied) and
$j<N_{p}/2$ then
multiply the stored error yield of $\bm{\theta}_{j}$ by factor $100$
end if
end for
Figure 7: Flowgraph for DE with symmetry breaking.
#### 5.4.2 CMA-ES with symmetry breaking
The CMA-ES method [12, 11] adapts a global step size $\sigma$, the mean
$\bf{m}$ and a covariance matrix $C$ at each iteration. According to the
Gaussian distribution $\mathcal{N}(\bf{m},\sigma C)$ with mean $\bf{m}$ and
covariance matrix $\sigma C$, $N_{p}$ solution candidate vectors are drawn.
After sorting the population by the error each candidate vector yields, the
best $N_{p}/2$ samples are used to update the mean, covariance matrix and the
step size for the next iteration.
In the following discussion, the CMA-ES method extended by the global optimum
invariant symmetry breaking [31] is denoted by CM-ES-INV-SB, CMA-ES extended
by the proposed global optimum variant symmetry breaking, described by
Algorithm 1, is denoted by CMA-ES-SB and CMA-ES with global optimum variant
ideal symmetry breaking using brute force search is denoted by CMA-ES-SB-BF.
In CMA-ES-INV-SB, CMA-ES-SB and CMA-ES-SB-SF, symmetry breaking is applied
right after the evaluation of all candidate vectors and prior to updating the
parameters of the Gaussian distribution. In CMA-ES-SB, we propose to use the
best candidate vector (yielding the smallest error) so far as the estimate for
the global optimum, denoted by $\hat{\bm{\theta}}$. In Fig. 8, the flowgraph
for CMA-ES-based symmetry breaking approaches is shown. For CMA-ES-SB, the
update of the mean is described in Algorithm 3. In all other CMA-ES-based
methods, the original update formula for the mean is applied.
In CMA-ES, applying symmetry breaking introduces a bias in the mean, which can
lead to an excessive increase of the global step size and negatively affect
the performance. This bias results from the rotations caused by the symmetry
operators. These rotations move solution candidates to the vicinity of one
partiton, which typically increases the radius of the population mean, as
shown in Fig. 6. In order to prevent such an increase, in all CMA-ES-based
symmetry breaking methods, we modify the damping term for the update of the
global step size $\sigma$. Let $\bm{s}$ be the shift vector of the centroid of
the best $N_{p}/2$ solution candidates induced by applying symmetry breaking.
The regular update formula for $\sigma$
$\sigma_{k+1}=\sigma_{k}\exp(\chi)$ (45)
is changed to
$\sigma_{k+1}=\sigma_{k}\exp\left[\chi\exp(-0.05D^{2}||\bm{s}||)\right],$ (46)
where $k$ is the iteration number and $\chi$ is a term depending on the
difference of the previous mean and the current mean, and several other
parameters.
Algorithm 3 CMA-ES-SB. Algorithm input: population of candidate vectors
$\bm{\theta}_{j},\ j=1,...,N_{p}$, the estimate for the global optimum
$\hat{\bm{\theta}}$ and weights $w_{j},\ j=1,...,N_{p}$. Effect: modify
candidate vectors $\bm{\theta}_{j},\ j=1,...,N_{p}$ when appropriate.
for all candidate vectors $\bm{\theta}_{j},\ j=1,...,N_{p}$ do
set mean vector $\bm{m}:=\bm{0}$
apply symmetry breaking on $\bm{\theta}_{j}$, see Algorithm 1
if $\bm{\theta}_{j}$ modified (a symmetry operator was applied) then
add weighted global optimum estimate to mean vector:
$\bm{m}:=\bm{m}+w_{j}\hat{\bm{\theta}}$
else
add weighted candidate vector to mean vector:
$\bm{m}:=\bm{m}+w_{j}\bm{\theta}$
end if
end for
Figure 8: Flowgraph for CMA-ES with symmetry breaking.
## 6 Experiments
In this section, we introduce results of experiments to demonstrate the
performance improvements by symmetry breaking. The following methods are
compared using regression and classification tests. From the DE-family:
Differential Evolution (DE), DE with global optimum invariant symmetry
breaking (DE-INV-SB), DE with global optimum variant symmetry breaking (DE-SB)
and DE with global optimum variant ideal symmetry breaking using brute force
search (DE-SB-BF). From the CMA-ES-family: Covariance Matrix Adaptation
Evolution Strategies (CMA-ES), CMA-ES with global optimum invariant symmetry
breaking (CMA-ES-INV-SB), CMA-ES with global optimum variant symmetry breaking
(CMA-ES-SB) and CMA-ES with global optimum variant ideal symmetry breaking
using brute force search (CMA-ES-SB-BF). It should be noted that the purpose
of this investigation is not to present the best global optimization method
for ANN-learning, but to demonstrate the benefits of symmetry breaking.
With a $D$-dimensional parameter space, all tests are performed with following
settings:
* •
DE, DE-SB, DE-INV-SB and DE-SB-BF settings: $F=0.5$, $C_{r}=0.9$, initial
population is randomly generated in $D$-dim. hypercube $[-1,1]^{D}$
(uniformly),
* •
CMA-ES, CMA-ES-SB, CMA-ES-INV-SB and CMA-ES-SB-BF settings: we used suggested
settings for enhanced global search abilities, mentioned in the C-code
reference implementation.
* •
in all experiments, the optimization is finished when a maximum number of ANN-
function-evaluations is reached.
Given a parameter $\bm{\theta}$ and a data set $(\bm{x}_{i},\bm{y}_{i})$, we
define the Mean Squared Error (MSE) $\epsilon$ according to Eqn. (8):
$\epsilon=\frac{1}{K\cdot
q}\sum_{k=1}^{K}(\bm{y}_{k}-\Omega(\bm{\theta};\bm{x}_{k}))^{\top}(\bm{y}_{k}-\Omega(\bm{\theta};\bm{x}_{k})).$
(47)
In order to limit the $D$-dimensional parameter space to a feasible region, we
apply a penalty approach. Due to the length-invariance by the symmetry
operators as shown in Eqn. (15), the feasible region is defined by a
hypersphere. In case of $||\bm{\theta}||>\sqrt{D}$, the error function (47) is
evaluated at a rescaled parameter vector $\frac{\bm{\theta}}{||\bm{\theta}||}$
and a penalty term $50(||\bm{\theta}||-\sqrt{D})$ is added to the error
$\epsilon$.
In self-generated data sets, we add normal distributed noise with zero mean
and variance $\sigma^{2}$ to the function values $y_{i}$
$y_{i}=f(\bm{x}_{i})+\mu,\ \mu\sim\mathcal{N}(0,\sigma^{2}),\
\sigma=5\\!\\!\times\\!\\!10^{-3}.$ (48)
### 6.1 Experimental setup
In all experiments, data is normalized such that mean is zero and variance is
one. The population size $N_{p}$ used in DE and CMA-ES depends on the problem
and the choice of the optimization method. Therefore, it is manually adapted
accordingly. For each problem and each optimization method, we conduct 50
independent repetitions of the optimization process and record the error over
the number of ANN-evaluations. To test for statistical significance of the
obtained results, first the Kruskal-Wallis test [14] for the hypothesis that
all performance means are equal is applied. In case this hypothesis is
rejected, the Wilcoxon rank sum test [35] is applied to all pairs of means to
identify significantly different results. All tests are based on a
significance level of $0.05$. In Table 1, normalized training set errors for
the regression and the autoencoding problems, and normalized test set errors
for the classification problems are shown.
Table 1: Normalized training set errors for the regression and the autoencoding problems, and normalized test set errors for the classification problems. The best results are printed in boldface. For each problem and method, errors are normalized by the maximum error from within the corresponding regular method, its extension by global optimization invariant symmetry breaking and its extension by global optimization variant symmetry breaking. | DE | DE-INV-SB | DE-SB | CMA-ES | CMA-ES-INV-SB | CMA-ES-SB
---|---|---|---|---|---|---
syn5 | 0.958 $\pm$ 0.079 | 1.000 $\pm$ 0.186 | 0.949 $\pm$ 0.039 | 1.000 $\pm$ 4.446 | 0.386 $\pm$ 0.469 | 0.093 $\pm$ 0.007
sinc | 1.000 $\pm$ 0.859 | 0.412 $\pm$ 0.166 | 0.114 $\pm$ 0.008 | 0.459 $\pm$ 0.271 | 1.000 $\pm$ 0.784 | 0.139 $\pm$ 0.051
inc-sinc | 1.000 $\pm$ 0.963 | 0.337 $\pm$ 0.155 | 0.089 $\pm$ 0.016 | 0.287 $\pm$ 0.336 | 1.000 $\pm$ 0.707 | 0.082 $\pm$ 0.035
sinc2d | 1.000 $\pm$ 0.387 | 0.995 $\pm$ 0.094 | 0.875 $\pm$ 0.029 | 0.975 $\pm$ 0.139 | 1.000 $\pm$ 0.241 | 0.089 $\pm$ 0.253
sinc3d | 0.622 $\pm$ 0.029 | 1.000 $\pm$ 0.572 | 0.603 $\pm$ 0.033 | 1.000 $\pm$ 1.401 | 0.090 $\pm$ 0.013 | 0.043 $\pm$ 0.021
autoenc-circle | 0.057 $\pm$ 0.082 | 1.000 $\pm$ 1.850 | 0.020 $\pm$ 0.030 | 1.000 $\pm$ 0.295 | 0.626 $\pm$ 0.548 | 0.077 $\pm$ 0.164
autoenc-spiral | 0.341 $\pm$ 0.545 | 1.000 $\pm$ 0.932 | 0.116 $\pm$ 0.308 | 0.248 $\pm$ 0.232 | 1.000 $\pm$ 0.882 | 0.030 $\pm$ 0.024
autoenc-sphere | 0.554 $\pm$ 0.321 | 1.000 $\pm$ 0.064 | 0.022 $\pm$ 0.012 | 0.050 $\pm$ 0.012 | 1.000 $\pm$ 0.416 | 0.032 $\pm$ 0.008
two-circles | 0.450 $\pm$ 0.225 | 1.000 $\pm$ 0.182 | 0.269 $\pm$ 0.074 | 0.635 $\pm$ 0.368 | 1.000 $\pm$ 0.284 | 0.326 $\pm$ 0.169
two-spirals | 0.918 $\pm$ 0.260 | 1.000 $\pm$ 0.213 | 0.426 $\pm$ 0.197 | 1.000 $\pm$ 0.228 | 0.930 $\pm$ 0.201 | 0.683 $\pm$ 0.293
digits | 0.325 $\pm$ 0.087 | 1.000 $\pm$ 0.111 | 0.272 $\pm$ 0.062 | 1.000 $\pm$ 0.352 | 0.805 $\pm$ 0.113 | 0.668 $\pm$ 0.099
### 6.2 Regression problems
As in [4], we apply learning only on a training set to compare the performance
of the introduced methods. In the following, the regression problems are
introduced and corresponding results are shown.
#### 6.2.1 Dataset syn5
The syn5 dataset is generated by the fourth-degree polynome
$(x-0.5)^{2}\cdot(0.1+(x+0.65)^{2})$ with uniformly distributed random input
values $x_{i}\in(-1,1)$. We use a 1-3-1 net and 200 data samples. The
population size for all DE-based methods is $N_{p}=80$, and $N_{p}=48$ for all
CMA-ES-based methods. Fig. 9 shows the resulting convergence curves and box
plots for the learning process.
Figure 9: Convergence curves for regression by DE (left) and CMA-ES (right)
using the syn5 dataset.
For the DE-family, the Kruskal-Wallis test showed no significant difference in
means. In contrast, according to the Wilcoxon tests, the inequality of means
of CMA-ES and CMA-ES-SB is rejected by a narrow margin, with a corresponding
p-value of $0.08$. The other means are significantly different. All DE
variants reach the same low-error, where DE-SB shows the fastest decrease in
error. As for the CMA-ES variants, CMA-ES fails to reach a low error in a few
runs, which leads to a larger mean error in average. In contrast, CMA-ES-SB
proves to be more robust and reaches a relatively low error in all runs.
#### 6.2.2 Dataset sinc
The sinc dataset is generated by the function $\frac{\sin(10x)}{10x}$ with
uniformly distributed random input values $x_{i}\in(-1,1)$. We use a 1-5-1 net
and 200 data samples. The population size for all DE-based methods is
$N_{p}=120$, and $N_{p}=400$ for all CMA-ES-based methods. Fig. 10 shows the
resulting convergence curves and box plots for the learning process.
Figure 10: Convergence curves for regression by DE (left) and CMA-ES (right)
using the sinc dataset.
According to the Wilcoxon tests, all pairwise differences are significant. DE-
SB clearly outperforms DE and DE-INV-SB. Similarly, CMA-ES-SB is the fastest
among the CMA-ES-based methods.
#### 6.2.3 Dataset inc-sinc
The inc-sinc dataset is generated by the function
$\frac{x}{2}+\frac{\sin(10x)}{10x}$ with uniformly distributed random input
values $x_{i}\in(-1,1)$. We use a 1-5-1 net and 200 data samples. The
population size for all DE-based methods is $N_{p}=144$, and $N_{p}=400$ for
all CMA-ES-based methods. Fig. 11 shows the resulting convergence curves and
box plots for the learning process.
Figure 11: Convergence curves for regression by DE (left) and CMA-ES (right)
using the inc-sinc dataset.
According to the Wilcoxon tests, all pairwise differences are significant.
Interestingly, the global optimum invariant symmetry breaking approach leads
to in improvement for DE (DE-INV-SB), but shows inferior performance on CMA-ES
(CMA-ES-INV-SB). This proves that symmetry breaking approaches should be
specific to the selected global optimization method. Again, DE-SB and CMA-ES-
SB are the fastest methods.
#### 6.2.4 Dataset sinc2d
The sinc2d dataset is generated by the function $\frac{\sin(5||x||)}{15||x||}$
with uniformly distributed random input values $x_{i}\in(-1,1)^{2}$. We use a
2-3-1-3-1 net and 1000 data samples. The population size for all DE-based
methods is $N_{p}=96$, and $N_{p}=1000$ for all CMA-ES-based methods. Fig. 12
shows the resulting convergence curves and box plots for the learning process.
Figure 12: Convergence curves for regression by DE (left) and CMA-ES (right)
using the sinc2d dataset.
According to the Wilcoxon tests, all pairwise differences are significant,
except the difference between CMA-ES and CMA-ES-INV-SB. The proposed symmetry
breaking approach shows a very clear impact on the CMA-ES-variants. While CMA-
ES and CMA-ES-INV-SB fail to solve this problem completely, CMA-ES-SB
successfully trains the ANN in the majority of the 50 runs.
#### 6.2.5 Dataset sinc3d
The sinc3d dataset is generated by the function $\frac{\sin(5||x||)}{15||x||}$
with uniformly distributed random input values $x_{i}\in(-1,1)^{3}$. We use a
3-4-1-4-1 net and 1000 data samples. The population size for all DE-based
methods is $N_{p}=120$, and $N_{p}=1000$ for all CMA-ES-based methods. Fig. 13
shows the resulting convergence curves and box plots for the learning process.
Figure 13: Convergence curves for regression by DE (left) and CMA-ES (right)
using the sinc3d dataset.
According to the Wilcoxon tests, all pairwise differences are significant.
Again, DE-SB and CMA-ES-SB are the fastest methods. This time, in contrast to
previous experiments, CMA-ES-INV-SB clearly outperforms CMA-ES.
### 6.3 Autoencoding problems
In this section, all $d$-dimensional data samples lie on a $s$-dimensional
set, where $s<d$. As a result, the data can be described, or ’encoded’ by an
$s$-dimensional subset. On the other hand, there is also a $s$-D to $d$-D
mapping to ’decode’ the data. The task is to approximate both the encoding and
decoding mapping by an ANN. As in the case of the regression problems, the
performance is compared only on the training using a training set.
#### 6.3.1 Dataset autoenc-circle
In this problem, the data samples lie on a 2-D circle centered at the origin
with radius one. We use a 2-5-3-2-1-2-3-5-2 net and 200 data samples to encode
from 2-D to 1-D and decode back to 2-D. The population size for all DE-based
methods is $N_{p}=64$, and $N_{p}=4000$ for all CMA-ES-based methods. Fig. 14
shows the resulting convergence curves and box plots for the learning process.
Figure 14: Convergence curves for regression by DE (left) and CMA-ES (right)
using the sinc3d dataset.
All pairwise differences prove to be statistically significant. The proposed
symmetry beraking approach improves the training in both methods. On CMA-ES-
SB, the difference turns out to be quite significant.
#### 6.3.2 Dataset autoenc-spiral
In this problem, the data samples lie on a 3-D spiral with radius one, defined
by
$(\cos(\phi),\sin(\phi),\phi),\phi\in[0,6\pi].$
We use a 3-1-3-4-7-3 net and 1000 data samples to encode from 3-D to 1-D and
decode back to 3-D. The population size for all DE-based methods is
$N_{p}=80$, and $N_{p}=400$ for all CMA-ES-based methods. Fig. 15 shows the
resulting convergence curves and box plots for the learning process.
Figure 15: Convergence curves for regression by DE (left) and CMA-ES (right)
using the sinc3d dataset.
All pairwise differences prove to be statistically significant.
#### 6.3.3 Dataset autoenc-sphere
In this problem, the data samples lie on a 3-D sphere centered at the origin
with radius one. We use a 3-8-5-2-5-8-3 net and 1000 data samples to encode
from 3-D to 2-D and decode back to 3-D. The population size for all DE-based
methods is $N_{p}=96$, and $N_{p}=1000$ for all CMA-ES-based methods. Fig. 16
shows the resulting convergence curves and box plots for the learning process.
Figure 16: Convergence curves for regression by DE (left) and CMA-ES (right)
using the autoenc-sphere dataset.
All pairwise differences prove to be statistically significant. Clearly, DE-SB
and CMA-ES-SB are significantly faster then the other methods.
### 6.4 Classification problems
In classification problems, data samples are divided into a training set, a
validation set and a test set. All three sets are generated by random
selection of samples. A winner-takes-all scheme is applied to distinguish
different classes, i.e., given an input, the ANN-output component with the
greatest value determines the class. In order to improve generalization,
classification performance measures on the training and test set are updated
only on each improvement of the validation set classification performance.
#### 6.4.1 Dataset: Two-Circles
In this problem, the 2-D data domain $[-1,1]^{2}$ is divided into two parts,
where one part is given by the union area of two circles and the remaining
part is the disjunct space. Hence, there are two classes: samples which lie
inside any circle and samples which lie outside of both circles. One circle is
specified by center $(0.5,0.5)$ and radius $r_{1}=0.39894$, and the other
circle by center $(-0.5,-0.5)$ and same radius $r_{2}=0.39894$. We use a
2-4-2-4-2 net with 400 samples for each training, validation and test set,
having a total of 1200 samples. The population size for all DE-based methods
is $N_{p}=80$, and $N_{p}=400$ for all CMA-ES-based methods. Fig. 15 shows the
resulting convergence curves and box plots for the learning process.
Figure 17: Classification error rates over ANN-evaluations on the Two-Circles
dataset using the DE-variants.
Figure 18: Classification error rates over ANN-evaluations on the Two-Circles
dataset using the CMA-ES-variants.
All pairwise differences prove to be statistically significant. It can be seen
that again DE-SB and CMA-ES-SB dominate the performances.
#### 6.4.2 Dataset: Two-Spirals
This problem [17] contains 2-D data-samples from two spirals on the plane,
both starting at the origin and going around each other. The task is to
classify each data sample by deciding to which spiral it belongs to. We use a
2-8-3-1-3-8-2 net, 114 samples for the training set, 40 samples for the
validation set and 40 samples for the test set. The population size for all
DE-based methods is $N_{p}=120$, and $N_{p}=1000$ for all CMA-ES-based
methods. Fig. 19 and 20 show the resulting convergence curves and box plots
for the learning process.
Figure 19: Classification error rates over ANN-evaluations on the Two-Spirals
dataset using the DE-variants.
Figure 20: Classification error rates over ANN-evaluations on the Two-Spirals
dataset using the CMA-ES-variants.
On the training and test set, DE and DE-INV-SB mean results are not
statistical significantly different. Furthermore, on the test set, CMA-ES and
CMA-ES-INV-SB mean results are not significantly different. Otherwise, all
other pairwise differences prove to be statistically significant. DE-SB and
CMA-ES-SB continue to show the best results.
#### 6.4.3 Dataset: Digits
This problem [1] deals with the recognition of handwritten digits, which
results in a classification problem with 10 classes. The data is generated by
asking several writers to write 250 digits in random order inside boxes of 500
by 500 tablet pixel resolution. There are 16 features extracted from the
digitized data. We use a 16-8-3-10-10 net and 1000 data samples each for the
training set, the validation set as well as the test set.
Figure 21: Classification error rates over ANN-evaluations on the Digits
dataset using the DE-variants.
Figure 22: Classification error rates over ANN-evaluations on the Digits
dataset using the CMA-ES-variants.
All pairwise differences prove to be statistically significant. Again, DE-SB
and CMA-ES-SB are the fastest methods.
### 6.5 Ideal separation
In this Section, we compare the ideal separation to the proposed
approximations. Since the complexity of the brute force method for the ideal
separation is exponential, we restrict the experiments to small networks as
used in the problems syn5, sinc and inc-sinc. It can be seen that the results
are almost identical.
Figure 23: Comparing DE-SB with DE using ideal separation by brute force
symmetry breaking (DE-SB-BF) on the syn5, sinc and inc-sinc datasets.
Figure 24: Comparing CMA-ES-SB with CMA-ES using ideal separation by brute
force symmetry breaking (CMA-ES-SB-BF) on the syn5, sinc and inc-sinc
datasets.
## 7 Conclusions
The problem of symmetries in ANN-parameter space is a well known problem
resulting in important complication in the training of ANN’s. However, a
detailed investigation of this problem for Evolutionary Algorithms other than
Genetic Algorithms is missing in the literature. Furthermore, there are
contradictionary results about the efficacy of symmetry breaking methods in
the performance of the global search. We show that a possible explanation for
this situation is the use of symmetry breaking methods which are invariant to
the global optimum and therefore can only be effective on a limited number of
problems. Furthermore, we show theoretically and illustrate experimentally,
that the application of global optimum invariant symmetry breaking may even
lead to inferior performance. To circumvent these problems, we propose methods
for global optimum variant symmetry breaking approaches for Differential
Evolution (DE) and Covariance Matrix Adaptation Evolution Strategies (CMA-ES),
which are two popular, robust and state-of-the-art global optimization
methods.
Experimental studies conducted on fixed topology feedforward neural networks
indicate a significant improvement over standard DE and CMA-ES techniques in
terms of global convergence speed. Further comparisons of the proposed
approach with a common global optimum invariant symmetry breaking approach
support our hypotheses.
Based on the obtained results, we conclude that other global optimization
based methods may also benefit from the use of the proposed global optimum
variant symmetry breaking. Further research is required to adapt the proposed
approach to other techniques to improve their performance.
The proposed method can be tested and verified using the open source C++ Monte
Carlo Machine Learning Library (MCMLL), which is available under the GNU GPLv2
license. The website of the library can be found on: mcmll.sourceforge.net.
The project website is available at sourceforge.net/projects/mcmll.
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|
arxiv-papers
| 2011-07-22T10:08:58 |
2024-09-04T02:49:20.825151
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Onay Urfalioglu and Orhan Arikan",
"submitter": "Onay Urfalioglu",
"url": "https://arxiv.org/abs/1107.4470"
}
|
1107.4631
|
# A More General Model for the Intrinsic Scatter in Type Ia Supernova Distance
Moduli
John Marriner,11affiliation: Center for Particle Astrophysics, Fermi National
Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA;
marriner@fnal.gov J.P. Bernstein,22affiliation: Argonne National Laboratory,
9700 South Cass Avenue, Lemont, IL 60439, USA. Richard Kessler,33affiliation:
Department of Astronomy and Astrophysics, The University of Chicago, 5640
South Ellis Avenue, Chicago, IL 60637, USA. 44affiliation: Kavli Institute for
Cosmological Physics, The University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA. Hubert Lampeitl,55affiliation: Institute of Cosmology
and Gravitation, Dennis Sciama Building, Burnaby Road, University of
Portsmouth, Portsmouth, PO1 3FX, UK. Ramon Miquel,66affiliation: Institució
Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain.
77affiliation: Institut de Física d’Altes Energies, E-08193 Bellaterra
(Barcelona), Spain. Jennifer Mosher,88affiliation: Department of Physics and
Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA
19104, USA Robert C. Nichol,55affiliation: Institute of Cosmology and
Gravitation, Dennis Sciama Building, Burnaby Road, University of Portsmouth,
Portsmouth, PO1 3FX, UK. Masao Sako, 88affiliation: Department of Physics and
Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA
19104, USA Donald P. Schneider,99affiliation: Department of Astronomy and
Astrophysics, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA. and Mathew Smith1010affiliation: Astrophysics,
Cosmology and Gravitation Centre (ACGC) University of Cape Town, Rondebosch,
Cape Town, Republic of South Africa.
###### Abstract
We describe a new formalism to fit the parameters $\alpha$ and $\beta$ that
are used in the SALT2 model to determine the standard magnitudes of Type Ia
supernovae. The new formalism describes the intrinsic scatter in Type Ia
supernovae by a covariance matrix in place of the single parameter normally
used. We have applied this formalism to the Sloan Digital Sky Survey Supernova
Survey (SDSS-II) data and conclude that the data are best described by
$\alpha=0.135_{-.017}^{+.033}$ and $\beta=3.19_{-0.24}^{+0.14}$, where the
error is dominated by the uncertainty in the form of the intrinsic scatter
matrix. Our result depends on the introduction of a more general form for the
intrinsic scatter of the distance moduli of Type Ia supernovae than is
conventional, resulting in a larger value of $\beta$ and a larger uncertainty
than the conventional approach. Although this analysis results in a larger
value of $\beta$ and a larger error, the SDSS data differ (at a 98% confidence
level) with $\beta=4.1$, the value expected for extinction by the type of dust
found in the Milky Way. We have modeled the distribution of supernovae Ia in
terms of their color and conclude that there is strong evidence that variation
in color is a significant contributor to the scatter of supernovae Ia around
their standard candle magnitude.
dark energy, supernovae: general
††slugcomment: July 22, 2011
## 1 Introduction
Type Ia supernovae (SN) have been extensively used as “standard candles” to
measure astronomical distances. However, they are often called “standardizable
candles” to emphasize the fact that the light curve shape and color can be
used to correct the apparent magnitudes, resulting in a more accurate
measurement of the distance. The correlation between light curve shape and
magnitude was demonstrated by Phillips et al. (1993), who parameterized the
light curve shape with the parameter ${\Delta}m_{15}$, the decay in $B$-band
magnitude 15 days after the peak. Different parameterizations of the light
curve shape parameter have been used: $\Delta$ for MLCS2k2 (Jha et al., 2007),
$x_{1}$ for SALT2 (Guy et al., 2007, 2010), and $s_{B}$ for SiFTO (Conley et
al., 2008), but the parameter is essentially determined empirically to produce
the tightest correlation between light curve shape and distance. This work
will use the SuperNova ANAlysis (SNANA) software package (Kessler et al.,
2009a) implementation of the SALT2 model that uses $x_{1}$ as the light curve
shape parameter.
Type Ia SN color depends on the light curve shape parameter, but SN also
exhibit variations in color for fixed light curve shape. The source of color
variation is not well understood, but it is widely suspected that variations
in the explosion process, host galaxy dust, and the circumstellar environment
all play a role. The difference in measured color and the mean color for a
given $x_{1}$ is denoted by the parameter $c$. Regardless of the process that
produces different colors, it may be expected that the apparent $B$-band
magnitude ($m_{B}$) will show some dependence on color, and that an
appropriate color correction should improve the precision of SN as standard
candles. These considerations have led to the widespread practice of fitting
SN light curves with a one parameter family of templates plus a color law to
describe the color.
The determination of color is straight-forward given multi-band photometric
measurements and a color law, and the result is traditionally given as
$E(B-V)$. However, the amount of correction to apply to a given SN has been
less clear. The correction has been often expressed as an extinction
correction, which is related to the color through the parameter
$R_{V}=A_{V}/E(B-V)$, where $A_{V}$ is the extinction in $V$-band. Many
efforts to determine this correction have relied on the parameterization of
Cardelli, Clayton, and Mathis (Cardelli et al., 1989), which we will call the
CCM model, where the parameter $R_{V}$ changes both the shape of the
extinction curve as a function of wavelength and the amount of extinction for
a given color excess $E(B-V)$. The simplest approach (Jha et al., 2007) is to
fix $R_{V}=3.1$, the average value determined from the studies of extinction
in the Milky Way (Sneden et al., 1978). However, many studies of SN light
curve data have indicated that smaller values of $R_{V}$ provide a better
description of the SN data. In the CCM model the single parameter $R_{V}$ can
be measured by estimating $A_{V}$ or by comparing the relative extinction in
different filters (by considering $E(B-V)-E(V-R)$, for example).
The SALT2 model parameterizes the SN rest-frame spectrum $S(\lambda,p)$ as a
function of wavelength ($\lambda$) and phase ($p$) according to
$S(\lambda,p)=x_{0}\,[S_{0}(\lambda,p)+x_{1}S_{1}(\lambda,p)]\,e^{[-cCL(\lambda)]},$
(1)
where the fitted parameters $x_{0}$, $x_{1}$, and $c$ are the overall scale,
the light curve shape parameter, and the color respectively. The functions
$S_{0}$ and $S_{1}$ are fixed by a training sample of SN light curves and
spectra. The SALT2 light curve fit does not rely on the CCM parameterization,
but uses a color law $CL(\lambda)$ determined empirically from SN data. The
SALT2 parameters are determined by fitting the measured light curve data for
each SN to the synthetic magnitudes calculated from the redshifted spectrum
given by Equation (1).
The light curve parameters $x_{0}$, $x_{1}$, and $c$ specify a model SN
spectrum, but an additional step is needed to use the light curve parameters
to calculate a standard SN magnitude. One can calculate the apparent magnitude
$m_{B}$, where $m_{B}$ is the apparent $B$-band synthesized from the rest-
frame spectrum given by Equation (1), and a standardized SN brightness
$m_{B}^{s}$ is computed with parameters $\alpha$ and $\beta$:
$m_{B}^{s}=m_{B}+\alpha x_{1}-\beta c.$ (2)
Guy et al. (2007) describe a technique that simultaneously determines the
cosmological parameters and the coefficients ($\alpha$ and $\beta$) that
minimize the residuals on the Hubble diagram. The SALT2 approach to correction
for color is formally equivalent to the description in terms of $R_{V}$ and
$E(B-V)$, except for the interpretation that the value of $\beta$ may result
from some effect other than extinction by host galaxy dust. Regardless of the
interpretation, $\beta$, which gives the magnitude correction for $B$-band,
and $R_{V}$, which gives the magnitude correction for $V$-band, may be
compared via the relation $\beta=R_{B}=R_{V}+1$.
In this work, we present a new method for fitting the parameters $\alpha$ and
$\beta$ from the data. The method has been implemented in the program SALT2mu,
which has been added to the SNANA software package. The motivations for a new
program are:
* •
The fit for $\alpha$ and $\beta$ is decoupled from the cosmology. If the mean
values of $x_{1}$ and $c$ are a function of redshift (a magnitude limited
survey will have such selection effects), the average residual depends on both
the cosmology and the $\alpha$ and $\beta$ parameters. We prefer to determine
the $\alpha$ and $\beta$ solely by minimizing the rms of the distribution and
to let the mean of the distribution determine the cosmology.
* •
A new description of intrinsic scatter is used as discussed below.
* •
The SALT2mu output can be used interchangeably with different cosmology
fitters since SALT2mu produces distance moduli separately from the cosmology
fit. This feature allows a direct comparison of SN distance moduli without
reference to a particular cosmology.
* •
The sample used to determine $\alpha$ and $\beta$ can be different from the
sample used to determine the cosmology.
A significant additional benefit to writing a new program is that SALT2mu
reads files from the SALT2 light curve fits and outputs files that can be read
by any of the SNANA cosmology fitters. The fitting model resides in a single
function making it easy to modify the $\chi^{2}$ function by adding new
parameters, for example.
The inclusion of a more general description of the intrinsic scatter of the
fitted SALT2 parameters is a major feature, as is the determination of
$\alpha$ and $\beta$ in a way that is independent of the cosmology. The
details of the incorporation of the intrinsic scatter are given in the
mathematical description (§2) below, and most of the paper is devoted to the
issues associated with the more general description of the intrinsic scatter.
The program is then applied to the Sloan Digital Sky Survey supernova (SDSS-II
SN) data, which is used to illustrate some of the features of the program and
issues with its use.
The SDSS-II SN data (Frieman et al., 2008) were obtained during 2004-2007 as
part of extension of the original SDSS (York et al., 2000). The SDSS telescope
(Gunn et al., 2006) and imaging camera (Gunn et al., 1998) produce photometric
measurements in each of the 5 SDSS filters (Fukugita et al., 1996) spanning
the range of 350 to 1000 nm. The most useful filters for observing SDSS SN,
however, are g, r, and i because the SN are too faint in u and z to be well
measured by the SDSS SN survey except at low redshifts. Further details of the
survey, data processing and selections are given in §3.
## 2 Mathematical Description
The standardized 10 pc supernova magnitude is determined by the expression
$M_{0}^{(B)}=m_{B}-\mu(z)+\alpha(z)x_{1}-\beta(z)c,$ (3)
where $\mu(z)$ is the distance modulus, and $M_{0}^{(B)}$ is the standardized
SN magnitude for $B$-band. We have generalized the unknown parameters $\alpha$
and $\beta$ to functions of redshift to accommodate possible effects of
evolution of supernovae or their environment. These functions are not known
from first principles, but must be determined from the data. We use a nearly
equivalent formulation
$M_{0}(z)=m_{x}-\mu(z)+\alpha(z)x_{1}-\beta(z)c,$ (4)
where
$m_{x}=-2.5\log_{10}(x_{0}).$ (5)
The definition shown in Equation (5) has the advantage that the magnitude
$m_{x}$ does not depend on the shape of the $B$-band filter. It differs from
the $B$-band standard magnitude by about 10 magnitudes. The functions
$M_{0}(z)$, $\alpha(z)$ and $\beta(z)$ are to be found by the minimization of
the expression for $\chi^{2}$
$\chi^{2}=\displaystyle\sum_{n=1}^{N}[m_{xn}-\mu(z_{n})+\alpha(z_{n})x_{1n}-\beta(z_{n})c_{n}\\\
-M_{0}(z_{n})]^{2}/(\sigma_{n}^{2}+\sigma_{\rm{int}})^{2}$ (6)
where $m_{xn}$, $x_{1n}$ and $c_{n}$ are the fitted SALT2 parameters for the
$n^{\rm{th}}$ SN, and $N$ is the number of SN. The quantity $\sigma_{n}$ is
the error on the quantity in the numerator due to measurement uncertainty for
the $n^{\textrm{th}}$ SN. The quantity $\sigma_{\rm{int}}$ is discussed below
and its form is displayed in Equation (7). The function $M_{0}(z)$ is, by
construction, degenerate with cosmological information contained in the
distance modulus $\mu(z)$. The introduction of a nuisance function $M_{0}(z)$
allows our fit to minimize the scatter in the Hubble diagram while being
insensitive to the cosmology. In order to reduce this function to a few
parameters we assume that $M_{0}(z)$ is an arbitrary constant in each of
several redshift bins centered at $z=z_{b}$, and we use the exact expression
for the distance modulus $\mu(z)$ for some assumed cosmology. Effectively we
are using an assumed cosmology to extrapolate the residual to the redshift at
the center of the bin and minimizing the Hubble diagram scatter at those
reference points. Since our method introduces several new parameters, the
statistical accuracy of the fit is reduced by the additional parameters. And
since our method relies on the minimization of the scatter of distance moduli
in a single bin, it requires a bin population of at least two SN—preferably
many more. These considerations are of no consequence for the SDSS sample, but
may be of some concern in applying our method to samples with sparse data or
unconventional cosmologies. We show in §4 that the fit results are insensitive
to the choice of cosmological parameters if the number of redshift bins is
sufficiently large.
Although this formalism allows fitting an arbitrary redshift
dependence111SALT2mu includes the possibility of fitting a linear dependence
for $\alpha$ and $\beta$. for $\alpha$ and $\beta$, we will assume that
$\alpha$ and $\beta$ are constants, independent of redshift, in this paper.
Recently, a number of papers (Kelly et al., 2010; Lampeitl et al., 2010;
Sullivan et al., 2010) have found that the value of $M_{0}$ shows a dependence
on host galaxy properties. Other papers have found correlations between line
velocities in SN spectra (Wang et al., 2009; Foley & Kasen, 2011) and their
standardized magnitudes and other correlations with spectral features (Nordin
et al., 2011). Those aspects will not be explored in this paper, but it is
straightforward to generalize Equation (4) by adding a new term $\gamma y_{n}$
to the numerator of Equation (6), where $y_{n}$ is some measured property of
the $n^{\textrm{th}}$ SN and $\gamma$ is a parameter to be determined by the
minimization. In addition, the measurement uncertainty and intrinsic scatter
in $y_{n}$ should be incorporated into $\sigma_{n}$ and $\sigma_{\rm{int}}$,
respectively, following the formalism described below.
When the SALT2 light curve parameters are used according to Equation (4) to
construct a SN Hubble diagram, the dispersion exceeds that expected from the
light curve fit parameter errors. A simple way to parameterize the Hubble
diagram scatter is to assert that the distribution of light curve parameters
can be described by an arbitrary population in the plane described by Equation
(4) plus a Gaussian spread that is described by a covariance matrix, which we
will refer to as the intrinsic scatter. The quanitity $\sigma_{\rm{int}}$ in
the denominator of Equation (6) describes the expected error in the numerator
due to the intrinsic scatter. We further assume that the covariance matrix is
constant, independent of the SN light curve parameters. These considerations
result in an expression for the intrinsic scatter
$\sigma_{\rm{int}}^{2}=\Sigma_{00}+\alpha(z)^{2}\Sigma_{11}+\beta(z)^{2}\Sigma_{cc}+2\alpha(z)\Sigma_{01}\\\
-2\beta(z)\Sigma_{0c}-2\alpha(z)\beta(z)\Sigma_{1c},$ (7)
where $\Sigma$ is the intrinsic covariance matrix of the parameters and
$\Sigma_{00}$, $\Sigma_{11}$, and $\Sigma_{cc}$ are the diagonal elements that
correspond to the SALT2 parameters $m_{x}$, $x_{1}$, and $c$. Our introduction
of a covariance matrix to describe the intrinsic scatter differs from the
conventional approach where the intrinsic scatter is assumed to be a single
number. Equation (7) shows that the conventional approach is equivalent to our
model if $\Sigma_{00}$ is the only non-zero term, leading to the conclusion
that the conventional approach applies the intrinsic scatter to $m_{x}$ but
not to $x_{1}$ or $c$. We introduce the more general approach because the
additional terms have not been excluded on either experimental or theoretical
grounds and, in fact, there is good reason to believe that there is a
significant scatter that can be attributed to the SALT2 color parameter. The
measurement error term ($\sigma_{n}^{2}$) in Equation (6) is dominated by
photometric errors and is calculated by the SALT2 light curve fit for each SN
and is given by a covariance matrix that is of the same form as the intrinsic
scatter covariance displayed in Equation (7). The denominator in Equation (6)
is therefore the sum of a constant intrinsic scatter covariance matrix, which
we have introduced above, and the measurement covariance, which has
traditionally been included. The fact that the measurement error and the
intrinsic error are both of the form shown in Equation (7) means that each of
the SALT2 parameters has both an intrinsic scatter and a measurement error
that are taken into account in computing $\chi^{2}$. The formulation of the
minimization problem defined by Equations (6) and (7) is an extension of the
formula given in Press et al. (2007) for the case of fitting a line to two
variables with known errors. A more extensive treatment of the linear
regression problem from an astronomical perspective has been given, for
example, by Kelly (2007).
After the parameters $\alpha$ and $\beta$ are determined by the minimization
of $\chi^{2}$ the distance moduli are calculated according to
$\mu_{n}=m_{xn}+\alpha(z_{n})x_{1n}-\beta(z_{n})c_{n}-M_{0}(z_{b})$ (8)
These distance moduli can then be used for input to cosmological fits.
## 3 Application to the SDSS data
We now apply this method to the SDSS 4-year sample of 529 spectroscopically
confirmed SN. The SDSS SN survey is a so-called “rolling search,” where a
portion of the sky is repeatedly scanned to discover new SN and to measure the
light curves of ones already discovered. The survey measured an equatorial
stripe about 2.5 degrees wide in declination between a right ascension of 20 h
and 04 h. Full coverage of the stripe can be obtained in two nights, but the
average cadence was about four nights because of inclement weather and
interference from moonlight. The data were obtained in the Fall of 2005, 2006,
and 2007 and a smaller amount of data was taken in a test run in 2004. The
survey is sensitive to SN beyond a redshift of 0.4, but beyond a redshift of
0.2 the completeness and the ability to obtain accurate photometry
deteriorates. The SDSS camera images were processed by the SDSS imaging
software and SN were identified via a frame subtraction technique. More
details and references can be found in Frieman et al. (2008). The candidate
selection and spectroscopic identification has been described by Sako et al.
(2008). The supernova photometry used in this analysis was obtained using the
scene modeling technique of Holtzman et al. (2008), where the sequence of SN
observations is modeled as a variable point source and a host galaxy
background that is constant in time.
We select a sample of well-measured light curves for this analysis based on
the criteria used by Kessler et al. (2009b). We require
* •
Spectroscopically confirmed as a Type Ia SN using techniques similar to those
used for the first year sample (Zheng et al., 2008)
* •
Redshift in the range $0.02<z<0.42$
* •
A minimum number of measurements without any requirement on the measurement
error and a minimum number of well-measured points, where well-measured means
an error of 20% or less (i.e., $S/N>5$)
* –
At least 5 well-measured points
* –
At least 2 well-measured points in different filters
* –
At least 1 point measured before peak light in $B$-band
* –
At least 1 point measured later than 9.5 days in the rest frame after peak
light in $B$-band
* •
An acceptable fit to the SALT2 model (confidence level greater than 0.001)
These selections result in a total sample of 343 SNe with negligible
contamination from sources that are not Type Ia SN. The redshift range spans
0.036 to 0.419, with a mean redshift of 0.22. The SDSS sample thus defined
does not have uniform selection criteria as a function of redshift and,
therefore, there is a significant change in the population as a function of
redshift. In particular, the spectroscopically-selected sample of SDSS SN is
significantly biased towards bluer, brighter SN as the redshift increases; the
selection bias is discussed in detail in Kessler et al. (2009b). To the extent
that the data are described by the model of Equation (4) with an accurate
parameterization of $\alpha(z)$ and $\beta(z)$ and the SALT2 fit parameters
are unbiased, we expect to be insensitive to having a biased population.
An application of SALT2mu to the SDSS data using four redshift bins yields the
results shown in Table 1, which shows the fit $\chi^{2}$ for 337 degrees of
freedom (343 SN), the fitted values of $\alpha$ and $\beta$, and their errors
$\sigma_{\alpha}$ and $\sigma_{\beta}$. In section §4 we show that four bins
is adequate to decouple the SDSS data from the determination of the
cosmological parameters. The first row in Table 1 shows the fit when the
intrinsic scatter is assumed to be zero resulting in $\chi^{2}=1305$,
reproducing the well-known result that the scatter in the Hubble diagram is
larger than what is expected from the measurement error alone. Subsequent rows
illustrate how the fitted parameters change with different assumptions on the
nature of the intrinsic scatter. Each of the fit variables is assumed, in
turn, to be totally responsible for the intrinsic scatter with the magnitude
of the scatter being adjusted to achieve $\chi^{2}/dof\approx 1$. In Table 1
we have introduced new variables to indicate the size of the intrinsic
scatter: $\sigma_{0}=\sqrt{\Sigma_{00}}$, $\sigma_{1}=\sqrt{\Sigma_{11}}$, and
$\sigma_{c}=\sqrt{\Sigma_{cc}}$. From Table 1 we see that the different
assumptions on the nature of the intrinsic scatter result in rather different
estimates of the $\alpha$ and $\beta$ parameters compared to the reported
statistical error. An assumed scatter in $m_{x}$ ($\sigma_{0}\neq 0$)
decreases both $\alpha$ and $\beta$ compared to the case where the intrinsic
scatter is taken to be zero. An assumed scatter in $x_{1}$ ($\sigma_{1}\neq
0$) increases $\alpha$ by $15.7\sigma$ while decreasing $\beta$ by $1.4\sigma$
relative to the fit where the intrinsic scatter is assumed to be entirely due
to $m_{x}$. The opposite effect is seen when the scatter is assumed to be due
to $c$ ($\sigma_{c}\neq 0$): $\beta$ increases by $7.1\sigma$ and $\alpha$
decreases by $3.1\sigma$. Even more variation could be exhibited by a more
general intrinsic scatter covariance matrix. However, the variations in
$\alpha$ and $\beta$ are not without limit. The values are all the same sign
and approximately the same magnitude. Roughly speaking, we are changing our
estimate of how much to correct the observed slope for the flattening caused
by the scatter described by the intrinsic covariance matrix (Equation (7)).
Table 1: SALT2mu fits to the SDSS Data
Intrinsic Error | $\chi^{2}$ | $\alpha$ | $\sigma_{\alpha}$ | $\beta$ | $\sigma_{\beta}$
---|---|---|---|---|---
$\sigma_{0}=\sigma_{1}=\sigma_{c}=0$ | 1305 | 0.1891 | 0.0064 | 2.820 | 0.050
$\sigma_{0}=0.14$,$\sigma_{1}=\sigma_{c}=0$ | 350 | 0.1465 | 0.0098 | 2.650 | 0.078
$\sigma_{1}=0.70$,$\sigma_{0}=\sigma_{c}=0$ | 330 | 0.3002 | 0.0200 | 2.405 | 0.116
$\sigma_{c}=0.05$,$\sigma_{0}=\sigma_{1}=0$ | 333 | 0.1330 | 0.0110 | 3.208 | 0.102
Note. — The SALT2mu program is applied to the 343 selected SDSS SN. The values
of $\alpha$ and $\beta$ and their calculated errors ($\sigma_{\alpha}$ and
$\sigma_{\beta}$) are shown. There are 6 fit parameters ($\alpha$, $\beta$,
and the 4 $M_{0}$ values for the 4 redshift bins) so the number of degrees of
freedom is 337. The variable $\sigma_{0}$, $\sigma_{1}$, and $\sigma_{c}$ give
the assumed rms scatter in the SALT2 parameters $m_{x}$, $x_{1}$, and $c$,
respectively. The first row in the table shows the fit assuming an intrinsic
scatter of zero; the following 3 rows assume that the intrinsic scatter
results, in turn, from a single SALT2 fit parameter.
If we are only interested in SN as distance indicators, $\alpha$ and $\beta$
are nuisance parameters of no particular interest. The average SN brightness
at a given redshift will be given by the mean of Equation (4): if the mean
values of $x_{1}$ and $c$ do not evolve with redshift, then different values
of $\alpha$ and $\beta$ will result in a redshift-independent offset that is
not significant in determining the cosmology. The mean values of $x_{1}$ and
$c$ for the SDSS data are plotted in Figure 1 showing, however, a substantial
increase in $x_{1}$ and a decrease in $c$ with redshift. These trends are
primarily caused by the spectroscopic-selection bias towards brighter SN Ia.
The different values listed in Table 1 would result in differences of $\sim
0.1$ in the average values of $\mu(z)$, a rather large error for cosmological
studies. In practice, the effect can be mitigated by limiting the sample to
lower redshifts, where the selection efficiency is higher, correcting for
biases, and combining high and low redshift experiments that have different
efficiencies. Nonetheless, the possibility of an uncorrected bias is a
concern.
Figure 1: The average value of (a) SALT2 shape ($x_{1}$) and (b) color ($c$)
as a function of redshift for the SDSS sample of 343 SN. The general trends of
the data can be explained by selection effects.
Our results can be compared to the results of Amanullah et al. (2010) who
combined data from several sources to obtain a value of $\beta=2.51\pm 0.07$
with an analysis most similar to our case with $\sigma_{0}=0.14$. Guy et al.
(2007) found $\beta=1.77\pm 0.16$ for the first year data from the SNLS and
Kessler et al. (2009b) found $\beta=2.65\pm 0.08$ for the first year of SDSS
data. Amanullah et al. (2010) also quote numbers for their four largest
samples: $\beta=2.38\pm 0.14$ (SDSS), $2.73\pm 0.13$ (CfA), $2.50\pm 0.17$
(ESSENCE), and $1.72\pm 0.18$ (SNLS). More recently Conley et al. (2011) found
$\beta=3.18\pm 0.10$ for the SNLS 3-year sample using a chi-squared
minimization technique. The study of Nobili and Goobar (2008) found the
variations in SN colors were most consistent with a CCM parameterization when
$R_{V}=1.75\pm 0.27$. However, the Nobili and Goobar (2008) study differs
significantly from this analysis in that their value of $R_{V}$ is based on
their measurement of the wavelength dependence of the reddening using the
Cardelli parameterization. Kessler et al. (2009b) also investigated SN color
and found $R_{V}=2.18\pm 0.14(stat)\pm 0.48(syst)$, but that result relies on
modeling the color changes due to the magnitude limited sample. All these
results are broadly consistent in finding $\beta<4.1$, but they exhibit more
scatter than one would expect from the quoted errors.
## 4 Tests with simulated SN samples
We will return to the SDSS SN data in §5, but it is useful to first examine
the performance of SALT2mu on simulated SN data. We first determine the number
of data bins that are required to decouple our fits from the cosmology over a
wide range of parameters and conclude that 4 bins are adequate for the SDSS
data. We then verify the formalism with a simplified simulation and show that
we recover the input values of $\alpha$ and $\beta$ without bias. We next look
at a realistic simulation and show that the fitted $\alpha$ and $\beta$
dependence on the intrinsic covariance matrix $\Sigma$ follow the trends seen
in the data. We compute the intrinsic scatter covariance matrix from the
simulation (which requires knowing the simulation parameters), and show that
the input values of $\alpha$ and $\beta$ are recovered when the correct
intrinsic scatter covariance matrix is used.
We generate the simulated SNe by using SNANA to generate SN light curve data
according to the SALT2 model. The SN are generated with a redshift
distribution appropriate for the SDSS survey and with $\alpha=0.18$ and
$\beta=3.4$. Realistic cadence, measurement errors, and selection effects are
applied to the simulated SN, and they are processed with the same light curve
fitting program and selection criteria as the SDSS SN, resulting in SALT2 fit
parameters for the simulated SN. In addition to the photometric measurement
errors, the simulation applies color smearing to model the intrinsic scatter.
Color smearing consists of random magnitude offsets that are generated for
each SN and observed filter and added to each light curve point measured in
that filter. The magnitude of the smearing is adjusted to 0.08 in each
observed filter to give approximately the same Hubble diagram scatter as is
seen in the real data.
The first test is a relatively trivial one to verify that the fit results for
$\alpha$ and $\beta$ are, indeed, independent of cosmology. We expect that a
large number of parameters $M_{0}(z_{b})$, will decouple the determination of
$\alpha$ and $\beta$ from the cosmological parameters, but wish to minimize
the number of nuisance parameters that are introduced. We illustrate the fit
dependence on the number of redshift bins in Figure 2 using the simulation of
the SDSS dataset, which has over ten times the number of SN (4417) as the SDSS
data. Figure 2 shows the fit results for $\alpha$ and $\beta$ assuming three
different cosmologies: $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$, $w=-1$
(red), $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.0$ (green), and
$\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$, $w=0$ (blue). The difference in
the fitted parameters is noticeable when only 1 bin is used, but becomes
negligible as the number of bins becomes greater than about 5. We conclude
that using 4 redshift bins is adequate for the SDSS data, given the lower
statistical precision of the data and the established constraints on the
cosmological parameters. All results in this paper are based on dividing the
SDSS data (real or simulated) into 4 equal sized bins spanning the redshift
range $0.02<z<0.42$.
Figure 2: The dependence of the fitted parameters $\alpha$ (a) and $\beta$ (b)
is displayed as a function of the number of redshift bins for three
assumptions about cosmological parameters and a particular choice of the
intrinsic scatter matrix ($\sigma_{0}=0.14$, $\sigma_{1}=\sigma_{c}=0$). As
the number of bins increases, the fits converge to a common value
$\alpha=0.184$ and $\beta=2.86$ shown as dotted lines on the respective
figures.
While the fit results in Figure 2 show that we have successfully decoupled the
fit for $\alpha$ and $\beta$ from the cosmology, the dependence on the
assumptions about the intrinsic scatter matrix remains. For the fit results
shown in Figure 2 we took $\sigma_{0}=0.14$ and $\sigma_{1}=\sigma_{c}=0$, and
this choice results in fitted values of $\alpha=0.184$ and $\beta=2.86$ which
differ from those used to generate the simulated SNe, namely $\alpha=0.18$ and
$\beta=3.4$. The discrepancy is not surprising since our simulation includes
significant color smearing that we have not taken into account. We will
determine more accurate parameters for the intrinsic scatter covariance below,
but first we check the formalism with a simplified simulation. We use the same
simulated SN but replace the simulated values of $m_{x}$, $x_{1}$, and $c$
with the exact values that were used to generate the light curves but smeared
according to the $3\times 3$ error matrix including any assumed intrinsic
scatter. This procedure tests the self-consistency of the SALT2mu model and
eliminates concerns about biases in the SALT2 parameter fitting and
inaccuracies in the calculation of the parameter errors. The results of the
fits are shown in Table 2 with the same four assumptions about the intrinsic
scatter that were used in Table 1. We have also tested the formalism when all
the elements of the intrinsic scatter covariance matrix are non-zero; the last
line in Table 2 is an example. For each case we recover the input values that
are consistent within the statistical error of about 1%. The $\chi^{2}/dof$
consistent with 1. We do not observe any variation in the parameter values
since the parameters of each SN are generated according to a Gaussian error
distribution that is the combination of the measurement error and whatever
intrinsic scatter was assumed.
Table 2: SALT2mu fits to the simplified, simulated SDSS Data
Intrinsic Scatter | $\chi^{2}$ | $\alpha$ | $\sigma_{\alpha}$ | $\beta$ | $\sigma_{\beta}$
---|---|---|---|---|---
$\sigma_{0}=\sigma_{1}=\sigma_{c}=0$ | 4445 | 0.1802 | 0.0009 | 3.391 | 0.010
$\sigma_{0}=0.14$ | 4487 | 0.1774 | 0.0019 | 3.395 | 0.024
$\sigma_{1}=0.70$ | 4319 | 0.1815 | 0.0018 | 3.423 | 0.023
$\sigma_{c}=0.05$ | 4277 | 0.1818 | 0.0022 | 3.379 | 0.028
Full covarianceaaThe full covariance matrix used for this example was $\sigma_{0}=0.05$, $\sigma_{1}=0.35$, $\sigma_{c}=0.025$, $\xi_{01}=-0.2$, $\xi_{0c}=0.5$, and $\xi_{1c}=0.4$, where the $\xi$ are the correlation coefficients defined in the text following Equation (9). | 4564 | 0.1811 | 0.0015 | 3.356 | 0.019
Note. — In the SALT2mu simplified simulation, the SALT2 fit parameters
($m_{x}$, $x_{1}$, and $c$) are replaced with the true parameters plus an
random error determined by the combination of the measurement and intrinsic
scatter. The number of simulated SN is 4417 and the number of degrees of
freedom for this fit is 4411.
Using the parameters of the SALT2 fits to the simulated data provides a more
complicated test since the color smearing model is different than applying an
intrinsic scatter to the simulated SALT2 parameters used to generate the light
curves. We apply SALT2mu to the simulated SN using the same procedure used for
the SDSS data, and display the results in Table 3. The first line has the
intrinsic scatter set to zero, and the succeeding lines have the intrinsic
scatter that results in $\chi^{2}/dof\approx 1$. The general trends shown in
Table 3 are very similar to those seen in Table 1: assuming an intrinsic
scatter in $x_{1}$ and $c$ increases the fitted value of $\alpha$ and $\beta$,
respectively. The magnitudes of the intrinsic scatter required to achieve
$\chi^{2}/dof\approx 1$ are similar to those required by the SDSS data. But we
can also compare the results with the “true” values for the simulated data.
While the assumption that the intrinsic scatter is entirely due to color
errors (last line in Table 3) produces values of $\alpha$ and $\beta$ that are
close to those used to generate the simulation ($\alpha$ differs by
$2.6\sigma$), it is not a very accurate representation of the intrinsic
scattering covariance matrix that describes the simulated SN as shown below.
Table 3: SALT2mu fits to the simulated SDSS Data
Intrinsic Error | $\chi^{2}$ | $\alpha$ | $\sigma_{\alpha}$ | $\beta$ | $\sigma_{\beta}$
---|---|---|---|---|---
$\sigma_{0}=\sigma_{1}=\sigma_{c}=0$ | 19500 | 0.2143 | 0.0011 | 3.104 | 0.011
$\sigma_{0}=0.14$ | 4340 | 0.1846 | 0.0019 | 2.868 | 0.021
$\sigma_{1}=0.65$ | 4375 | 0.2482 | 0.0025 | 2.882 | 0.026
$\sigma_{c}=0.045$ | 4343 | 0.1855 | 0.0021 | 3.370 | 0.025
Note. — The first row in the table shows the fit assuming an intrinsic scatter
of zero; the following 3 rows assume that the intrinsic scatter results, in
turn, from a single SALT2 fit parameter. The number of simulated SN is 4417
and the number of degrees of freedom for this fit is 4411.
In principle, we know the smearing formulae applied in the simulation and can
calculate the intrinsic scatter covariance matrix $\Sigma$. But the
calculation of the intrinsic smearing matrix is complicated because the error
model used in the simulation smears individual light curve points in a way
that is not easily expressed in terms of an intrinsic smearing matrix in
$m_{x}$, $x_{1}$, and c. However, it is relatively straightforward to
calculate the error a posteriori. The full error matrix is determined by the
variance of the simulated measurements of $m_{x}$, $x_{1}$, and $c$ relative
to the true values (which, of course, are known for the simulated data). The
light curve fitter calculates covariance matrix of measurement errors, which,
when added to the intrinsic scatter covariance matrix, should yield the total
error matrix. The intrinsic scatter covariance matrix can therefore be
calculated for the simulated data by subtracting the observed variance from
the prediction for the measurement error. The total error covariance matrix
calculated from the variance of the reconstructed SALT2 parameters from the
simulated ones is
$\begin{array}[]{lll}\sigma_{0}=0.0781,&\xi_{01}=0.253,&\xi_{0c}=0.612,\\\
\sigma_{1}=0.351,&\xi_{1c}=0.183,&\textrm{and}\\\
\sigma_{c}=0.0634.\end{array}$ (9)
where the various $\xi$ are the correlation coefficients
($\xi_{01}=\Sigma_{01}/(\sigma_{0}\sigma_{1})$,
$\xi_{0c}=\Sigma_{0c}/(\sigma_{0}\sigma_{c})$, and
$\xi_{1c}=\Sigma_{1c}/(\sigma_{1}\sigma_{c})$). Using the same notation to
represent the elements of the measurement error matrix we find
$\begin{array}[]{lll}\sigma_{0}=0.0365,&\xi_{01}=0.581,&\xi_{0c}=0.694,\\\
\sigma_{1}=0.347,&\xi_{1c}=0.307,&\textrm{and}\\\
\sigma_{c}=0.0286.\end{array}$ (10)
Subtracting (10) from (9) gives an excess error, which we attribute to the
intrinsic scatter. The result is
$\begin{array}[]{lll}\sigma_{0}=0.0690,&\xi_{01}=-0.119,&\xi_{0c}=0.590,\\\
\sigma_{1}=0.0506&\xi_{1c}=0.362,&\textrm{and}\\\
\sigma_{c}=0.0566.\end{array}$ (11)
We are now in position to apply the intrinsic scatter from Equation (11) above
to the fit from SALT2mu. The result is shown in the first row in Table 4. The
result is consistent with the generated values ($\alpha=0.18$ and
$\beta=3.4$). The small discrepancy in $\beta$ ($2.6\sigma$) not a serious
concern because it is quite small compared to the accuracy of the SDSS
dataset. However, we suspect that uncertainties in the calculation of the
measurement errors, including treatment of non-linearities and a consistent
handling of model errors, may contribute a systematic bias at this level of
precision. We have performed additional fits with other intrinsic scatter
matrices. The second row is a fit with all the non-diagonal elements of the
intrinsic scatter matrix set to zero. The fitted parameters are only slightly
changed although there is a decrease in $\chi^{2}$. The next row shows a fit
that has been selected to give a larger change in $\chi^{2}$ by tuning the
correlations for maximum effect. The last row in Table 4 displays a fit, which
may be counter-intuitive: the error in $x_{1}$ is decreased, but the
$\chi^{2}$ increases. These results illustrate how complicated the fit
behavior can be for the five free parameters that live in the intrinsic
scatter matrix.
Table 4: SALT2mu fits to the simulated SDSS data with non-zero, off-diagonal
elements of the intrinsic scatter matrix
Intrinsic Error | $\chi^{2}$ | $\alpha$ | $\sigma_{\alpha}$ | $\beta$ | $\sigma_{\beta}$
---|---|---|---|---|---
As computed | 4161 | 0.1832 | 0.0022 | 3.469 | 0.027
As computed but $\xi_{01}=\xi_{0c}=\xi_{1c}=0.0$ | 2752 | 0.1822 | 0.0026 | 3.320 | 0.031
As computed but $\xi_{01}=0.847,\xi_{0c}=-0.913,\xi_{1c}=-0.831$ | 1758 | 0.1820 | 0.0031 | 3.227 | 0.036
As computed but $\sigma_{1}=0$ | 3130 | 0.1817 | 0.0024 | 3.400 | 0.030
Note. — The first row shows the intrinsic scatter matrix computed as described
in the text and succeeding rows illustrate various changes to the intrinsic
scatter matrix. The simulation is the same as was used to produce Table 3.
## 5 Determination of the intrinsic scatter matrix
We do not have a rigorous approach to estimating the intrinsic scatter matrix
for actual SN data. Since we don’t know the underlying, unsmeared distribution
or the actual form of the smearing, the observed distribution is difficult to
interpret. The scatter in the Hubble diagram sets the overall scale, but more
information is needed to determine the individual elements of the intrinsic
scatter matrix.
We can derive some information from the color distribution with the help of
some plausible assumptions, which were first used by Jha et al. (2007). We
assume that the color distribution arises from random color smearing that we
parameterize as a Gaussian of unknown width ($\sigma_{c}$) and host galaxy
extinction that we characterize as an exponential distribution of unknown
slope ($\tau$). We do not expect to be sensitive to the exact shape assumed
for galaxy extinction: the important point is that it is one-sided (extinction
can only make objects redder, not bluer) and that the most probable value of
extinction is zero. In addition, we do not know the position of the edge of
the extinction distribution, so we must introduce an additional parameter
($c_{0}$) to be determined from the data. Assuming that the intrinsic scatter
and host galaxy extinction are independent, the convolution of the two
processes results in the following distribution for $c$:
$\rho(c)=A\int_{c-c_{o}}^{\infty}\exp\left(-\frac{c-c_{o}-c^{\prime}}{\tau}\right)\\\
\times\exp\left(-\frac{{c^{\prime}}^{2}}{2\sigma_{c}^{2}}\right)dc^{\prime}$
(12)
In order to minimize the contribution of measurement errors to the width of
the Gaussian component, we include only SN where the estimated measurement
error for $c$ is less than 0.04, reducing our sample of 343 SN to 247. Fitting
Equation (12) to the data, we find $\sigma_{c}=0.061\pm 0.007$, $\tau=0.076\pm
0.010$, and $c_{0}=-0.073\pm 0.009$. The data and resulting fit are shown in
Figure 3. As can be seen, the functional form of Equation (12) provides a good
description of the data. The fitted value for $\tau$ is substantially smaller
than the value $\tau=0.334\pm 0.088$ quoted by Kessler et al. (2009b), but our
result is an effective $\tau$ for the spectroscopically-selected sample with
the additional requirement of a well-measured color and is therefore biased
against large extinctions. Measurement errors are calculated by the SALT2
light-curve fit and contribute, on average,
$\sqrt{\langle\sigma_{c}^{2}\rangle}=0.026$ to the width of the Gaussian
component, resulting in an estimate of $0.055\pm 0.007$ for the color
smearing. We do not expect our values of $c_{0}$ and $\sigma_{c}$ to be
significantly affected by the selection bias, and we do not make a correction
for the bias. This result is comparable to the amount of color smearing
($\sigma_{c}=0.05$) that was required in Table 1 to explain, by itself, the
scatter in the Hubble diagram.
Figure 3: The distribution in SALT2 color of 247 SDSS SN selected to have a
calculated measurement error $\sigma_{c}<0.04$ is shown as a histogram, and
the dashed curve is the best fit to the data. Individual points on the curve
are shown at the center of the histogram bins with errors corresponding to the
expected fluctuations from Poisson statistics.
Unfortunately, we do not have any useful a priori model for the $x_{1}$
distribution. We can, however, make the somewhat trivial observation that the
intrinsic scatter cannot be larger than the width of the distribution. It is
known that passive and star forming galaxies have different distributions in
$x_{1}$ and, if we assume that the intrinsic scatter is the same for both, we
can obtain an upper limit for the intrinsic scatter from the distribution that
excludes passive galaxies. The distributions in $x_{1}$ were shown by Lampeitl
et al. (2010) for the SDSS data, and we have used that analysis for the 296 SN
this paper has in common with that analysis. The resulting distribution is
shown in Figure 4 along with a Gaussian of width 1.0. The Gaussian is not a
good fit to the tails of the distribution, but it is a good match to the core
and we conclude that an upper limit for the intrinsic scatter in $x_{1}$ is
1.0, a limit that is greater than the amount required to explain the scatter
in the Hubble diagram.
Figure 4: The distribution in SALT2 shape parameter ($x_{1}$) of 296 SDSS SN
from our sample of 343 SN that were also determined to be hosted by star-
forming galaxies according to the analysis of (Lampeitl et al., 2010) is shown
as a histogram and the dashed curve is Gaussian of width 1. The Gaussian is a
good description of the core of the distribution but does not describe the
tail towards low $x_{1}$.
The simulated data exhibits a significant correlation between $m_{x}$ and $c$,
and it is interesting to ask whether that correlation should be expected in
the data. In the simulation, the correlation arises because the SALT2 model
defines $B$-band to be the reference point for the color and because the
measured data is largely redward of $B$-band. The assumed independent
variation of each measured band results in the correlation between peak
magnitude and color. While the model of color smearing is arbitrarily
constructed to reproduce the scatter in the Hubble diagram, there is some
evidence to support independent fluctuations in each of the filters. As
pointed out in (Kessler et al., 2010), if the redshift is fit from the light
curve data, the dispersion of the results around the spectroscopically
determined redshifts is larger than can be accounted for by photometric errors
alone, but the color smearing model of SNANA gives approximately the right
dispersion. Based on these considerations, we conclude that the correlation
between $m_{x}$ and $c$ is likely to be positive (as seen in the simulations)
for the SDSS data.
While we have been able to make some observations on the elements of the
intrinsic scatter matrix, our picture is incomplete and rests on assumptions
of uncertain validity. An alternative approach is to marginalize over all
possible intrinsic scatter matices. We implement this marginalization by
running SALT2mu repeatedly with randomly generated intrinsic scatter matrices.
We start with a vector randomly generated on the unit sphere subject to the
restriction that all components of the vector are positive. We then multiply
the three components of the vector by 0.2, 1.0, and 0.07 respectively and set
$\sigma_{0}$, $\sigma_{1}$, and $\sigma_{c}$ to the scaled vector components.
The constants are chosen to be about 50% larger than the values (see Table 1)
that were required to explain the intrinsic scatter. The normalized
correlations are chosen randomly between $-1$ and $+1$.
A total of 10000 intrinsic scatter matrices were generated, but the procedure
does not guarantee that the matrices are positive. A total of 6380 of the
matrices were found by SALT2mu to be positive definite and were used to
determine values for $\alpha$ and $\beta$. Each intrinsic scatter matrix is
scaled by an arbitrary factor to achieve a $\chi^{2}/dof=1$. The red (dotted)
curve in Figure 5 shows the results of the fits for (a) $\alpha$ and (b)
$\beta$. The median value is $\alpha=0.150_{-0.021}^{+0.083}$ and
$\beta=2.91_{-0.35}^{+0.30}$, where the error is the range that includes 68%
of the results. Our intrinsic scatter covariance matrix defines an intrinsic
error in 3 dimensions, but the chi-squared is only sensitive to the error in
one direction. The intrinsic errors in the other two directions just broaden
the distribution of points in the plane defined by Equation (8). If the
intrinsic errors are large enough to explain, by themselves, the observed
population, the slopes in Equation (8) become undefined and produce the
outliers that are observed in Figure 5. The data show a secondary peak near
$\beta=2.5$, but this should not be interpreted as evidence for 2 populations
with different slopes. The shape of the plot in Figure 5, of course, depends
on the distribution of covariance matrices, which we chose to be uniform. The
lower values of $\beta$ result primarily from intrinsic scatter matrices with
a small $\sigma_{c}$. The fit becomes insensitive to $\sigma_{c}$ as
$\sigma_{c}\to 0$ and the peak at $\beta=2.5$ is an artifact of our choice of
distribution for the intrinsic scatter matrices and the insensitivity of the
fit parameter $\beta$ in the vicinity of $\sigma_{c}=0$.
The blue (solid) curve is the result of restricting the intrinsic scatter
matrix to a range favored by the arguments in this section. Specifically we
require the intrinsic scatter matrix color element to be within 3 standard
deviations of the value determined by the fit to Equation (12), namely
$0.040<\sigma_{c}<0.082$. We also require $\sigma_{1}<1$ since the intrinsic
scatter cannot exceed the measured width of the distribution, and $\xi_{0c}>0$
since we expect a positive correlation when the color smearing includes a
component that is uncorrelated between the observed filter bands. While the
exact values of these cuts are somewhat arbitrary, we have chosen a wide range
to account for model uncertainties. Only 1650 of the 6380 intrinsic scatter
matrices satisfy these three criteria, but even with the restrictions a wide
range of values are found. However, the median values and the ranges that
contains 68% of the data are $\alpha=0.135_{-0.013}^{+0.031}$ and
$\beta=3.19_{-0.22}^{+0.10}$. In addition to the range of values that result
from different assumptions about the intrinsic scatter matrix, the statistical
errors reported by SALT2mu are $\sigma_{\alpha}=0.011$ and
$\sigma_{\beta}=0.10$. Adding these errors in quadrature, the final result is
$\alpha=0.135_{-0.017}^{+0.033}$ and $\beta=3.26_{-0.24}^{+0.14}$. For each
entry in Figure 5, we compute the statistical probability that the entry is
consistent with $\beta\geq 4.1$ using the computed statistical error and
assuming a normal distribution. The uncertainty in the intrinsic scatter
matrix is computed by averaging the statistical probability for all the
entries in Figure 5. From this calculation, we conclude that the probability
that our result is consistent with $\beta\geq 4.1$ is 2%. The probability is
higher than would be naively estimated from the value
$\beta=3.26_{-0.24}^{+0.14}$ because our calculation includes the effect of
the non-Gaussian tail of the distribution shown in Figure 5b.
Figure 5: The distribution of SALT2mu fit results for $\alpha$ (a) and $\beta$
(b) for a random distribution of intrinsic scatter matrices that yield
$\chi^{2}$ per degree of freedom equal to 1 is shown as the dotted red curve..
The solid blue curve shows the more restricted subset of error matrices that
satisfy $\sigma_{1}<1$, $0.040<\sigma_{c}<0.082$, and $\xi_{0c}>0$. The left
axis shows the vertical scale for the red curve; the right axis is the scale
for the blue curve. Of the 1650 entries in the restricted subset of intrinsic
scatter matrices, there are 32 with $\alpha<0$, 15 with $\alpha>0.4$, 1 with
$\beta<0$ and 15 with $\beta>5$, which are not shown in the figures. The
underflow and overflow entries result from cases where the entire distribution
can be explained by the assumed intrinsic scatter of the data and the
parameters $\alpha$ and $\beta$ become indeterminate, resulting in large
values and large estimated errors for the parameters.
## 6 Redshift Dependence of $\alpha$ and $\beta$ in the SDSS data
The parameters $\alpha$ and $\beta$ are usually assumed to be constants,
independent of redshift, but the possibility of a redshift dependence has been
considered previously by, for example, Kessler et al. (2009b). We might
observe a redshift dependence of the fits for $\alpha$ and $\beta$ for a
number of reasons:
* •
The properties of the SN, themselves, may evolve with redshift.
* •
The model of Equation (4) may not be an exact model and the resulting fit
parameters may be in tension between different SN populations. If the
population is a function of redshift, the amount of tension may vary,
resulting in different results for $\alpha$ and $\beta$.
* •
Our SN selection has biases, particularly at higher redshift where we tend to
select the brighter SN.
While we could directly fit for $\alpha(z)$ and $\beta(z)$, the likelihood
that selection effects will play a significant role suggests that a better
approach might be to split the SDSS sample into 4 redshift bins. The results
of the fits are shown in Figure 6. We have used the the same ensemble of
intrinsic scatter matrices with $\sigma_{1}<1$, $0.040<\sigma_{c}<0.082$, and
$\xi_{0c}>0$ that was used to construct Figure 5. The errors shown are the
combined statistical and systematic error that arise from the uncertainty in
the intrinsic scatter matrix. The results shown for $\alpha$ in Figure 6(a)
are consistent with the sample average although there is an indication that
the value of $\alpha$ may be rising as a function of redshift. Figure 6(b)
show a value of $\beta$ that is $2.1\sigma$ lower in the lowest redshift bin,
$2.2\sigma$ higher in the range $0.22<z<0.32$, and $3.7\sigma$ lower in the
highest redshift bin than the result for the complete sample. The accuracy in
the determination of $\beta$ as a function of redshift varies not only because
of the statistics but because highly reddened SN are too dim to meet our
selection criteria at high redshift. The number of SN in each redshift bin and
the number of highly reddened ($c>0.3$) SN are: 36 and 4 ($0.02<z<0.12$), 149
and 2 ($0.12<z<0.22$), 116 and 0 ($0.22<z<0.32$), and 41 and 0
($0.32<z<0.42$), respectively. The reduced range of $c$ at the higher
redshifts reduces the accuracy of the fit and makes the fit considerably more
sensitive to the amount of intrinsic scatter in the data. The highest-redshift
sample consists of 41 SN with $-0.319<c<0.126$ that show very little
correlation between the distance modulus and $c$, resulting in a low value of
$\beta$. The range of observed values of $c$ is most likely primarily due to
measurement error and intrinsic scatter resulting in the weak correlation
between distance modulus and observed $c$. In principle, our formalism should
recover the correct value of $\beta$ when the correct measurement errors and
intrinsic scattering are included, but selection effects have significantly
distorted the natural spread. In addition to the restricted color range, the
absolute magnitudes of SN with $0.32<z<0.42$ are $0.069\pm 0.042$ magnitudes
brighter on average, assuming standard cosmological parameters
($\Omega_{\Lambda}=0.7$ and $w=-1$). As a consequence, our fit to the last
redshift bin should not be considered to be a good representation of what
would be obtained with an unbiased sample. A more robust result would be
possible with an improved simulation of the selection effects and augmenting
the spectroscopically confirmed sample with those identified photometrically
(Sako et al., 2011) to obtain a more complete sample. Selection effects should
be much less important for the midrange ($0.22<z<0.32$), where is $\beta$
larger than the sample average by $2.2\sigma$. While the lack of highly
reddened SN ($c>0.3$) degrades our ability to determine $\beta$ in this
redshift region, we can see if their absence biases $\beta$ to higher values
by excluding SN with $c>0.3$ in the low redshift bins. The $\beta$ values for
the two lowest redshift regions change by less than 0.1 when the highly
reddened SN are removed from the sample.
Figure 6: The SALT2mu fit results for $\alpha$ (a) and $\beta$ (b) in 4
redshift bins using the same a distribution of intrinsic scatter matrices that
yield chi-squared per degree of freedom equal to 1 and satisfy $\sigma_{1}<1$,
$0.040<\sigma_{c}<0.082$, and $\xi_{0c}>0$. The results for the full sample
are indicated by the dashed lines.
## 7 Discussion
The histograms in Figure 5 show the same trend that was observed in Table 1:
when the intrinsic scatter in color is large, the best fit value of $\beta$
will increase while $\alpha$ decreases slightly. The fact that our value for
$\beta$ is the highest value of those reported in §3 depends on our
incorporation of a large intrinsic scatter in supernova color. We have not
made any correction for the selection bias of our sample based on the results
of the simulation, which is subject to the same selection criteria, and shows
no significant bias in the determination of $\alpha$ and $\beta$.
The error on our result is dominated by an uncertainty in the intrinsic
scatter matrix, but there are other potential problems that have not been
taken into account. Certainly, our assumption that the intrinsic scatter is a
constant, independent of any SN properties, is suspect given our lack of
knowledge of the processes involved. In addition, our model assumes that the
data follow the relationship of Equation (4) with a fixed value of $\beta$.
However, we know that the value of $R_{V}$ varies significantly in our own
galaxy (Schlafly et al., 2010), and we should expect to see at least some
variation in SN host galaxies. Variations in $\beta$ produce effects similar
to the intrinsic scatter in color, but the effect can be magnified in a
magnitude limited sample like SDSS where there are few highly reddened SN,
which have higher weight in determining $\beta$. Variations in $\beta$ could
explain the small differences seen between the values of $\beta$ in Figure 6
in the two lowest redshift bins.
We have seen that neglecting intrinsic scatter in the SALT2 $c$ parameter
results in a value of $\beta$ that is biased low. The amount of bias depends
on the amount of intrinsic scatter relative to the range in the variable $c$.
Since the range in $c$ decreases as the redshift increases, neglecting or
underestimating the amount of intrinsic scatter in $c$ would be expected to
result in a decreasing value of $\beta$ as a function of redshift. The fits to
the SDSS data as a function of redshift include intrinsic scatter in color so
this effect should not be the explanation for the SDSS data—unless the scatter
was underestimated. However, an overestimate of the intrinsic scatter in
$m_{x}$ leads to similar biases and is more likely explanation for the lower
value of $\beta$ that is observed in the highest redshift bin.
A more problematic type of variation in $\beta$ could arise from the color
smearing. We have argued that color variations of SN must have some component
of host galaxy extinction and also some intrinsic color variation, which could
be due to variations in the explosion process or local environment or both. We
have modeled the intrinsic color variation as color smearing with an effective
value $\beta=0$, but it is possible that each mechanism that contributes to
the observed color requires a different correction to the distance modulus.
Our model can only produce an effective value of $\beta$ that could be a
function of color, redshift, host galaxy type or other parameters and could,
therefore, differ depending on the characteristics of the particular SN
sample.
Although we have not included correlations with galaxy types or spectral line
features, the introduction of additional dependent variables into Equation (4)
is straightforward. In fact, having a more complete description of the
supernova explosion will not only provide a smaller dispersion for the
numerator in Equation (6), it will decrease the size of the intrinsic scatter
that is required and decrease the importance of the uncertainty as to its
form.
Recently a number of papers (Kim, 2011; Lago et al., 2011; March et al., 2011)
have explored the technique of determining the intrinsic scatter from the
data. The approach of (March et al., 2011) is similar to ours in that it
included our more general form for the intrinsic error matrix but assumed that
the off-diagonal terms involving $M_{B}$ were zero. We have not considered
including the intrinsic scattering error as a fit parameter partly for
simplicity and partly because the underlying physics of the intrinsic scatter
is still uncertain. However, the more rigorous statistical approach described
by these authors is a possible direction for future work.
We argue from the SDSS SN photometric data that there is likely some intrinsic
scatter in SN color that is responsible for some of the scatter in the Hubble
diagram. A similar conclusion is reached by Foley & Kasen (2011), based on
much more detailed information involving differences in the line velocities
measured from the SN spectra, which are modeled as arising from differences in
viewing angle. While this type of detailed spectral information may not be
available for all future SN surveys, an understanding of the underlying
physical processes will be important for the most accurate treatment of the
data.
## 8 Summary
We have described a new formalism to fit the parameters $\alpha$ and $\beta$
that are used to determine the standard magnitudes of Type Ia supernovae. The
determination of these parameters is accomplished in a way that is independent
of cosmology by introducing a few nuisance parameters to accommodate any
reasonable cosmology. The formalism also introduces a more general form for
the intrinsic scatter which introduces a large uncertainty into the fit if the
parameters describing the intrinsic scattering are not known accurately. We
have shown by simulation that the mathematical formalism is self-consistent,
but have pointed out ways in which the model may fail to be a complete
description of SN.
We find that the SDSS data, when fit to the form of Equation (4) are described
by $\alpha=0.135_{-0.017}^{+0.033}$ and $\beta=3.19_{-0.24}^{+0.14}$ by
marginalizing over the uncertainty in the intrinsic scatter matrix. Our result
relies on our conclusion that SNe are subject to a substantial color smearing
in addition to reddening from host galaxy dust, as indicated by our fit to the
color distribution where we determined that $\sigma_{c}=0.055\pm 0.007$. The
uncertainty in the parameters of the intrinsic scatter matrix results in a
much larger error than is obtained if the parameters are assumed to be known.
However, even with the larger value of $\beta$ and the larger error, we find
that SDSS data differ at the 98% confidence level from $\beta=4.1$, the value
expected for extinction by dust in the Milky Way.
## 9 Acknowledgements
The SDSS-II SN survey was managed by the Astrophysical Research Consortium for
the Participating Institutions. The Participating Institutions were the
American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, Cambridge University, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute
for Astronomy (MPA), the Max-Planck-Institute for Astrophysics (MPiA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory, and the University of Washington.
This work is based in part on observations made at the following telescopes.
The Hobby- Eberly Telescope (HET) is a joint project of the University of
Texas at Austin, the Pennsylvania State University, Stanford University,
Ludwig-Maximillians-Universität München, and Georg-August-Universität
Göttingen. The HET is named in honor of its principal benefactors, William P.
Hobby and Robert E. Eberly. The Marcario Low-Resolution Spectrograph is named
for Mike Marcario of High Lonesome Optics, who fabricated several optical
elements for the instrument but died before its completion; it is a joint
project of the Hobby-Eberly Telescope partnership and the Instituto de
Astronomía de la Universidad Nacional Autónoma de México. The Apache Point
Observatory 3.5 m telescope is owned and operated by the Astrophysical
Research Consortium. We thank the observatory director, Suzanne Hawley, and
former site manager, Bruce Gillespie, for their support of this project. The
Subaru Telescope is operated by the National Astronomical Observatory of
Japan. The William Herschel Telescope (WHT) is operated by the Isaac Newton
Group, the Nordic Optical Telescope (NOT) is operated jointly by Denmark,
Finland, Iceland, Norway, and Sweden, and the Telescopio Nazionale Galileo
(TNG) is operated by the Fundación Galileo Galilei of the Italian INAF
(Istituto Nazionale di Astrofisica) all on the island of La Palma in the
Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias. Observations at the ESO New Technology Telescope at
La Silla Observatory were made under programme IDs 77.A-0437, 78.A-0325, and
79.A-0715. Kitt Peak National Observatory, National Optical Astronomy
Observatories (NOAO), is operated by the Association of Universities for
Research in Astronomy, Inc. (AURA) under cooperative agreement with the NSF.
The South African Large Telescope (SALT) of the South African Astronomical
Observatory is operated by a partnership between the National Research
Foundation of South Africa, Nicolaus Copernicus Astronomical Center of the
Polish Academy of Sciences, the Hobby-Eberly Telescope Board, Rutgers
University, Georg- August-Universität Göttingen, University of Wisconsin-
Madison, University of Canterbury, University of North Carolina-Chapel Hill,
Dartmouth College, Carnegie Mellon University, and the United Kingdom SALT
consortium. The WIYN Observatory is a joint facility of the University of
Wisconsin- Madison, Indiana University, Yale University, and NOAO. The W.M.
Keck Observatory is operated as a scientific partnership among the California
Institute of Technology, the University of California, and the National
Aeronautics and Space Administration. The Observatory was made possible by the
generous financial support of the W. M. Keck Foundation.
This work was supported in part by the U.S. Department of Energy under
contract number DE-AC0276SF00515.
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|
arxiv-papers
| 2011-07-22T21:40:41 |
2024-09-04T02:49:20.838742
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "John Marriner, J.P. Bernstein, Richard Kessler, Hubert Lampeitl, Ramon\n Miquel, Jennifer Mosher, Robert C. Nichol, Masao Sako, Donald P. Schneider,\n and Mathew Smith",
"submitter": "John Marriner",
"url": "https://arxiv.org/abs/1107.4631"
}
|
1107.4661
|
# MediaWiki Grammar Recovery
Vadim Zaytsev, vadim@grammarware.net
SWAT, CWI, NL
(August 27, 2024)
## 1 Introduction
Wiki is the simplest online database that could possibly work [41]. It usually
takes a form of a website or a webpage where the presentation is predefined to
some extent, but the content can be edited by a subset of users. The editing
ideally does not require any additional software nor extra knowledge, takes
place in a browser and utilises a simple notation for markup. Currently there
are more than a hundred of such notations, varying slightly in concrete syntax
but mostly providing the same set of features for emphasizing fragments of
text, making tables, inserting images, etc [10]. The most popular notation of
all is the one of MediaWiki engine, it is used on Wikipedia, Wikia and
numerous Wikimedia Foundation projects.
In order to facilitate development of new wikiware and to simplify maintenance
of existing wikiware, one can rely on methods and tools from software language
engineering. It is a field that emerged in recent years, generalising
theoretical and practical aspects of programming languages, markup languages,
modelling languages, data definition languages, transformation languages,
query languages, application programming interfaces, software libraries, etc
[15, 23, 25, 70] and believed to be the successor for the object-oriented
paradigm [14]. The main instrument of software language engineering is on
disciplined creation of new domain specific languages with emphasis on
extensive automation. Practice shows that automated software maintenance,
analysis, migration and renovation deliver considerable benefits in terms of
costs and human effort compared to alternatives (manual changes, legacy
rebuild, etc), especially on large scale [11, 61, 65]. However, automated
methods do require special foundation for their successful usage.
Wikiware (wiki engines, parsers, bots, etc) is a specific case of grammarware
(parsers, compilers, browsers, pretty-printers, analysis and manipulation
tools, etc) [25, 75]. The most straightforward definition of grammarware can
be of software which input and/or output must belong to a certain language
(i.e., can be specified implicitly or explicitly by a formal grammar). An
operational grammar is needed to parse the code, to get it from a textual form
that the programmers created into a specialised generational and
transformational infrastructure that usually utilises a tree-like internal
format. In spite of the fact that the formal grammar theory is quite an
established area since 1956 [9], the grammars of mainstream programming
languages are rarely freely obtainable, they are complex artefacts that are
seen as valuable IT assets, require considerable effort and expertise to
compose and therefore are not always readily disclosed to public by those who
develop, maintain and reverse engineer them. A syntactic grammar is basically
a mere formal description of what can and what cannot be considered valid in a
language. The most obvious sources for this kind of information are: language
documentation, grammarware source code, international standards, protocol
definitions, etc.
However, documentation and specifications are neither ever complete nor error-
free [79]. To obtain correct grammars and ensure their quality level, special
techniques are needed: grammar adaptation [32], grammar recovery [36], grammar
engineering [25], grammar derivation [27], grammar reverse engineering,
grammar re-engineering, grammar archaeology [34], grammar extraction [75,
§5.4], grammar convergence [37], grammar relationship recovery [39], grammar
testing [33], grammar inference [64], grammar correction [75, §5.7],
programmable grammar transformation [74], and so on. The current document is
mainly a demonstration of application of such techniques to the MediaWiki BNF
grammar that was published as [47, 46, 51, 52, 49, 50, 48, 44].
### 1.1 Objectives
The project reported in this document aims at extraction and initial recovery
of the MediaWiki grammar. However, the extracted grammar is not the final
goal, but rather a stepping stone to enable the following activities:
Parse wiki pages.
The current state of Wikipedia is based on a PHP rewriting system that
transforms wiki layout directly into HTML [53]. However, it can not always be
utilised in other external wikiware: for example, future plans of Wikimedia
Foundation include having an in-browser editor with a WYSIWYG front-end in
JavaScript [69]. Having an operational grammar means anyone can parse wiki
pages more freely with their own technology of choice, either directly or by
deriving tolerant grammars from the baseline grammar [27].
Aid wiki migration.
The ability to easily parse and transform wiki pages can deliver considerable
benefits when migrating wiki content from one platform to another [78].
Validate existing wiki pages.
The current state of MediaWiki parser [53] allows users to submit wiki pages
that are essentially incorrect: they may combine wiki notation with bare HTML,
contain unbalanced markup, refer to nonexistent templates. This positively
affects the user-friendliness of the wiki, but makes some wiki pages possibly
problematic. Such pages can be identified and repaired with static code
analysis techniques [7].
Test existing wiki parsers.
There is considerable prior research in the field of grammar-based testing,
both stochastic [43, 60] and combinatorial [19, 33, 42, 56, 72], with
important recent advances in formulating coverage criteria and achieving
automation [16, 35]. These results can be easily reproduced to provide an
extensive test data suite containing different wiki text fragments to explore
every detail specified by the grammar in a fully automated fashion. Such test
data suites can be used to determine existing parsers’ conformance, can help
in developing new parsers, find problematic combinations that are treated
differently by different parsers, etc.
Improve grammar readability.
It is known that the grammar is meant to both define the language for the
computer to parse, and describe it for the language engineers to understand.
However, these two goals are usually conflicting, and more often than not, one
opts for an executable grammar that is harder to read, than for a perfectly
readable one that cannot be used in constructing grammarware. Unfortunately,
the effort and expertise needed to fully achieve either of them, and most
language documents contain non-operational grammars [28, 72, 79]. The practice
of using two grammars: the ‘‘more readable’’ one and the ‘‘more
implementable’’, adopted in the Java specification [18], has also proven to be
very ineffective and error-prone [38, 39].
Perform automated adaptation.
Grammars commonly need to be adapted in order to be useful and efficient in
wide range of circumstances [32]. Grammar transformation frameworks such as
GDK [30], GRK [34] or XBGF [74] can be used to apply adapting transformations
in a safe disciplined way with validation of applicability preconditions and
full control over the language delta. In fact, some of the transformations can
even be generated automatically and applied afterwards.
Establish inter-grammar relationships.
As of today, several MediaWiki notation grammars exist and are available in
one form or another: in EBNF [76], in ANTLR [5], etc (none of them are fully
operational). Furthermore, there exist various other wiki notations: Creole
[22], Wikidot [68], etc. Relationships among all these notations are unknown:
they are implicit even when formal grammars actually exist, and are totally
obscured when the notation is only documented in a manual. A special technique
called language convergence can help to reengineer such relationships in order
to make stronger claims about compatibility and expressivity [37, 75, 77].
### 1.2 Related work: grammar recovery initiatives
Most of operational grammars for mainstream software languages are
handcrafted, many are not publicly disclosed, few are documented. The first
case reported in detail in 1998 was PLEX (Programming Language for EXchanges),
a proprietary DSL for real time embedded software systems by Ericsson [59], a
successful application of the same technology to COBOL followed [62]. Grammar
recovery technique is not only needed for legacy languages, examples of more
modern and presumably more accurately engineered grammars being nontrivially
extracted include C# in [73] and [75, §3] and Java in [38] and [39]. The whole
process of MediaWiki grammar extraction is documented by this report, all
corrections and refactorings are available online, as is the end result (under
CC-BY-SA license).
### 1.3 Related work: Wiki Creole
Wiki Creole 1.0111http://wikicreole.org is an attempt for engineering an ideal
wiki syntax and a formal grammar for it. While the goal of specifying the wiki
syntax with a grammar is not foreign to us, but the benefits listed in [22,
p.3] are highly questionable:
1\. Trivial parser construction.
In the paper cited above it is claimed that applying a parser generator is
trivial. However, the main prerequisite for it is successful grammar
adaptation for the particular parsing technology [32]. A Wiki Creole grammar
was specifically geared toward ANTLR, and it is a highly sophisticated task to
migrate it anywhere if at some point ANTLR use is deemed to be undesirable.
Hence, the result is not reproducible without considerable effort and
expertise.
2\. Foundation for subsequent semantics specification.
The grammar can certainly serve as a basis for specifying semantics. However,
the choice of a suitable calculus for such semantics specification is of even
more importance. Furthermore, syntax definition does not guarantee the absence
of ambiguities in semantics, or even changes of semantics as a part of
language evolution (cf., evolutionary changes of HTML elements).
3\. Improved communication between wikiware developers.
The paper claimed that if wiki syntax is specified with a grammar, there can
be no different interpretations of it. However, it is quite common to have
different interpretations (dialects) of even mainstream programming languages,
plus wiki technology in its current state heavily relies on fault tolerance
(somewhat less so in the future when no bare text editing should be taking
place).
4\. Same rendering behaviour that users rely on.
Depending on the browser or the particular gadget that the end user deploys to
access the wiki, rendering behaviour can be vastly different, and this has
nothing to do with the syntax specification.
5\. Simplified syntax extension.
It is a very known fact in formal grammar theory [1] that grammar classes are
not compositional: that is, the result of combining two LL(*) grammars (which
ANTLR uses) does not necessarily belong to the LL(*) class; we can only prove
that it will still be context-free [9]. In other words, it is indeed easy to
specify a syntax extension, but such the extended grammar sometimes will not
be operational. Modular grammars can be deployed in frameworks which use
different parsing technologies, such as in Meta-Environment [24] or in Rascal
[26] or in MPS [66], but not in ANTLR.
6\. Performance predictions.
The paper claims that it is easier to predict performance of a parser made
with ‘‘well-understood language theory’’ than with a parser based on regular
expressions. However, there are implementation algorithms of regular
expressions that demonstrate quadratic behaviour [12], and ANTLR uses the same
technology for matching lookahead anyway, which immediately means that their
performance is the same.
7\. Discovering ambiguities.
It is true that ambiguity analysis is easier on a formal grammar than on the
prose, but it is not achieved by ‘‘more rigorous specification mechanism’’ and
even the most advanced techniques of today do not always succeed [4].
8\. Well-defined interchange format.
A well designed interchange format between different types of wikiware is a
separate effort that should be based on appropriate generalisations of many
previously existing wiki notations, not on one artificially created one, even
if that one is better designed.
In general, Wiki Creole initiative is relevant for us because it can serve as
a common grammar denominator later to converge several wiki grammars [37, 77],
but is neither contributing nor conflicting directly with our grammar recovery
project.
## 2 Grammar notation
One of the first steps in grammar extraction is understanding the grammar
definition formalism (i.e., the notation) used in the original artefact to
describe the language. In the case of MediaWiki, Backus-Naur form is claimed
to be used [45]. Manual cursory examination of the grammar text [47, 46, 51,
52, 49, 50, 48, 44] allows us to identify the following metasymbols in the
spirit of [20] and [75]:
Name | Value
---|---
Start grammar symbol | <source lang=bnf>
End grammar symbol | </source>
Start comment symbol | /*
End comment symbol | */
Defining symbol | ::=
Definition separator symbol | |
Start nonterminal symbol | <
End nonterminal symbol | >
Start terminal symbol | "
End terminal symbol | "
Start option symbol | [
End option symbol | ]
Start group symbol | (
End group symbol | )
Start repetition star symbol | {
End repetition star symbol | }
Start repetition plus symbol | {
End repetition plus symbol | }+
As we know from [3] and its research in [75, §6.3], BNF was originally defined
as follows:
Name | Value
---|---
Defining symbol | $:\equiv$
Definition separator symbol | $\overline{\texttt{or}}$
Terminator symbol | $\hookleftarrow$
Start nonterminal symbol | <
End nonterminal symbol | >
While the difference in the appearances of defining symbols is minor and is
commonly overlooked, there are several properties of the notation used for
MediaWiki grammar definition that place it well outside BNF, namely:
* •
Using delimiters to explicitly denote terminal symbols (instead of using
underlined decoration for keywords and relying on implicit assumptions for
non-alphanumeric characters).
* •
Presence of comments in the grammar (not in the text around it).
* •
Allowing inconsistent terminator symbol (i.e., a newline or a double newline,
sometimes a semicolon).
* •
Having metalanguage symbols for marking optional parts of productions.
* •
Having metalanguage symbols for marking repeated parts of productions.
* •
Having metalanguage symbols for grouping parts of productions.
Hence, it is not BNF. For the sake of completeness, let us compare it to the
classic EBNF, originally proposed in [71] (sometimes that dialect is referred
to as Wirth Syntax Notation) and standardised much later by ISO as [20]:
Name | Value in WSN | Value in ISO EBNF
---|---|---
Concatenate symbol | | ,
Start comment symbol | | (*
End comment symbol | | *)
Defining symbol | = | =
Definition separator symbol | | | |
Terminator symbol | . | ;
Start terminal symbol | " | "
End terminal symbol | " | "
Start option symbol | [ | [
End option symbol | ] | ]
Start group symbol | ( | (
End group symbol | ) | )
Start repetition star symbol | { | {
End repetition star symbol | } | }
Exception symbol | | -
Postfix repetition symbol | | *
We notice again a list of differences of MediaWiki grammar notation versus WSN
and ISO EBNF:
* •
Allowing inconsistent terminator symbol (i.e., a newline or a double newline).
* •
Presence of comments (consistent only with ISO EBNF).
* •
Lack of concatenate metasymbol (consistent only with WSN).
* •
Having metalanguage symbol for exceptions (consistent only with WSN).
* •
Not having a specially designated postfix symbol for denoting repetition
(consistent only with WSN).
Hence, the notation adopted by MediaWiki grammar, is neither BNF nor EBNF, but
an extension of a subset of EBNF. Since we cannot reuse any previously
existing automated grammar extractor, we define this particular notation with
EDD (EBNF Dialect Definition), a part of SLPS (Software Language Processing
Suite) [80] — and use Grammar Hunter, a universal configurable grammar
extraction tool, for extracting the first version. The definition itself is a
straightforward XML-ification of the first table of this section, so we leave
it out of this document. The only addition is switching on the options of
disregarding extra spaces and extra newlines that are left after tokenising
the grammar. The EDD is freely available for re-use in the subversion
repository of SLPS222Available as config.edd..
## 3 Guided grammar extraction
Since the grammar extraction process is performed for this particular notation
for the first time, we use _guided_ extraction, when the results of the
extraction are visually compared to the original text by an expert in grammar
engineering. This document is a detailed explanation of observations collected
in that process and actions undertaken to resolve the spotted issues.
Given previous experience, it is safe to assume that once the grammar is
extracted, we would like to change some parts of it (for grammar adaptation
[32], deyaccification [58] and other activities common for grammar recovery
[34]). In order for those changes to stay fully traceable and transparent, we
will take the approach of _programmable grammar transformation_. In this
methodology, we take a baseline grammar and an operator suite and by choosing
the right operators and parametrising them, we _program_ the desired changes
in the same way mainstream programmers use programming languages to create
software. These transformation scripts are executable with the grammar
transformation engine: any meta-programming facility would suffice, for this
particular work we use XBGF [74] which was shown in [39] to be the best and
the most versatile grammar transformation infrastructure at this moment. The
tools of SLPS that surround XBGF also allow for easy publishing by providing
immediate possibilities to transform XBGF scripts to LaTeX or XHTML.
### 3.1 Source for extraction
The grammar of MediaWiki is available on subpages of [45]. Striving for more
automation, we can use the ‘‘raw’’ action to download the content from the
same makefile that performs the extraction333Available as Makefile.. For
example, the wiki source of Article Title [47] is
http://www.mediawiki.org/w/index.php?title=Markup_spec/BNF/Article_title&action=raw.
In order to make our setup stable for the future when the contents of the wiki
page may change (in fact, changing them is one of the main objectives of this
work), we can add the revision number to that command, making it
http://www.mediawiki.org/w/index.php?title=Markup_spec/BNF/Article_title&action=raw&oldid=295042.
### 3.2 Article title
Parsing Article Title [47] with Grammar Hunter is not hard and does not report
many problems. One particular peculiarity that we notice when comparing the
resulting grammar with the original, is the ‘‘...?’’ symbol:
<canonical-page-first-char> ::= <ucase-letter> | <digit> | <underscore> | ...?
<canonical-page-char> ::= <letter> | <digit> | <underscore> | ...?
The ‘‘...?’’ symbol is not explained anywhere, but the intuitive meaning is
that it is a metasymbol for a possible future extension point. For example, if
in the future one decides to allow a hash symbol (#) in an article title
(currently not allowed for technical reasons), it will be added as an
alternative to the production defining canonical-page-char. The very notion of
such extension points contradicts the contemporary view on language evolution.
It is commonly assumed that a grammar engineer cannot predict in advance all
the places in the grammar that will need change in the future: hence, it is
better to not mark any of such places explicitly and assume that any place can
be extended, replaced, adapted, transformed, etc. Modern grammar
transformation engines such as XBGF [74], Rascal [26] or TXL [13] all have
means of extending a grammar in almost any desired place. Since it seems
reasonable to remove these extension points at all, we can do it with XBGF
after the extraction444Part of remove-extension-points.xbgf.:
⬇
vertical( in canonical-page-first-char );
removeV(
canonical-page-first-char:
"." "." "." "?"
);
horizontal( in canonical-page-first-char );
vertical( in canonical-page-char );
removeV(
canonical-page-char:
"." "." "." "?"
);
horizontal( in canonical-page-char );
vertical( in page-first-char );
removeV(
page-first-char:
"." "." "." "?"
);
horizontal( in page-first-char );
vertical( in page-char );
removeV(
page-char:
"." "." "." "?"
);
horizontal( in page-char );
By looking at the grammar where this transformation chain does not apply, one
can notice productions in this style:
<canonical-article-title> ::= <canonical-page> [<canonical-sub-pages>]
<canonical-sub-pages> ::= <canonical-sub-page> [<canonical-sub-pages>]
<canonical-sub-page> ::= <sub-page-separator> <canonical-page-chars>
In simple words, what we see here is an optional occurrence of a nonterminal
called canonical-sub-pages, which is defined as a list of one or more
nonterminals called canonical-sub-page. So, in fact, that optional occurrence
consists of zero or more canonical-sub-page nonterminals. However, these
observations are not immediate when looking at the definition, because the
production is written with explicit right recursion. This style of writing
productions belong to very early versions of compiler compilers like YACC
[21], which required manual optimisation of each grammar before parser
generation was possible. It has been reported later on multiple occasions [25,
58, etc] that it is highly undesirable to perform premature optimisation of a
general purpose grammar for a specific parsing technology that may or may not
be used with it at some point in the future. The classic construct of a list
of zero or more nonterminal occurrences is called a Kleene closure [1] or
Kleene star (since it is commonly denoted as a postfix star) and is
omnipresent in modern grammarware practice.
Using the Kleene star makes the grammars much more concise and readable. Most
parser generators that require right-recursive (or left-recursive) expansions
of a Kleene star, can do them automatically on the fly. Another possible
reason for not using a star repetition could have been to stay within limits
of pure BNF, but since we have already noted earlier that this goal was not
reached anyway, we see no reason to pretend to seek it. A well-known grammar
beautification technique known as ‘‘deyaccification’’ [58] is performed by the
following grammar refactoring chain555Part of deyaccify.xbgf.:
⬇
massage(
canonical-sub-pages$?$,
(canonical-sub-pages | $\varepsilon$));
distribute( in canonical-sub-pages );
vertical( in canonical-sub-pages );
deyaccify(canonical-sub-pages);
inline(canonical-sub-pages);
massage(
(canonical-sub-page+ | $\varepsilon$),
canonical-sub-page⋆);
massage(
canonical-page-chars$?$,
(canonical-page-chars | $\varepsilon$));
distribute( in canonical-page-chars );
vertical( in canonical-page-chars );
deyaccify(canonical-page-chars);
inline(canonical-page-chars);
massage(
(canonical-page-char+ | $\varepsilon$),
canonical-page-char⋆);
massage(
sub-pages$?$,
(sub-pages | $\varepsilon$));
distribute( in sub-pages );
vertical( in sub-pages );
deyaccify(sub-pages);
inline(sub-pages);
massage(
(sub-page+ | $\varepsilon$),
sub-page⋆);
massage(
page-chars$?$,
(page-chars | $\varepsilon$));
distribute( in page-chars );
vertical( in page-chars );
deyaccify(page-chars);
inline(page-chars);
massage(
(page-char+ | $\varepsilon$),
page-char⋆);
Even the simplest metrics can show us that these refactorings have simplified
the grammar, reducing it from 15 VAR and 25 PROD to 11 VAR and 17 PROD [55],
without any fallback in functionality. They have also removed technological
idiosyncrasies and improved properties that are somewhat harder to measure,
like readability and understandability.
### 3.3 Article
Article [46] contains seven grammar fragments, out of which only the first
three conform to the chosen grammar notation. The last four were copy-pasted
from elsewhere and use a different EBNF dialect, which we luckily can also
analyse and identify:
Name | Value
---|---
Defining symbol | =
Definition separator symbol | |
Start special symbol | ?
End special symbol | ?
Start terminal symbol | "
End terminal symbol | "
Start option symbol | [
End option symbol | ]
Start group symbol | (
End group symbol | )
Start repetition star symbol | {
End repetition star symbol | }
Exception symbol | -
We will not lay out its step by step comparison with the notation used in the
rest of the MediaWiki grammar, but it suffices to say that the presence of the
exception symbol in the metalanguage is enough to make some grammars
inexpressible in a metalanguage without it. BGF does not have a metasymbol for
exception, but we still could express the dialect in EDD666Available at
metawiki.edd. and extract these parts of the grammar with it. Judging by the
presence of the Kleene star in the metalanguage, the grammar engineers who
developed those parts did not intend to stay within BNF limits. Thus, we can
also advise to add the use of a plus repetition for denoting a sequence of one
or more nonterminal occurrences, in order to improve readability of
productions like these:
Line = PlainText { PlainText } { " " { " " } PlainText { PlainText } } ;
Text = Line { Line } { NewLine { NewLine } Line { Line } } ;
Or, in postfix-oriented BNF that we use within SLPS:
⬇ Line: PlainText PlainText⋆ (" " " "⋆ PlainText PlainText⋆)⋆ Text: Line Line⋆
(NewLine NewLine⋆ Line Line⋆)⋆
Compare with the version that we claim to be more readable:
⬇ Line: PlainText+ (" "+ PlainText+)⋆ Text: Line+ (NewLine+ Line+)⋆
In fact, many modern grammar definition formalisms have a metaconstruct called
‘‘separator list’’, because Text above is nothing more than a (multiple)
Newline-separated list of Lines. We do not enforce this kind of metaconstructs
here, but we do emphasize the fact that the very understanding of Text being a
separated list of Lines was not clear before our proposed refactoring. In the
case if MediaWiki still wants the grammar representation to have only one type
of repetition or even no repetition at all, such a view can be automatically
derived from the baseline grammar preserved in a more expressive metalanguage.
The refactorings that utilise the plus notation are rather
straightforward777Part of utilise-repetition.xbgf.:
⬇
massage(
PlainText PlainText⋆,
PlainText+);
massage(
Line Line⋆,
Line+);
massage(
NewLine NewLine⋆,
NewLine+);
massage(
" " " "⋆,
" "+);
Further investigation draws our attention to these productions:
PageName = TitleCharacter , { [ " " ] TitleCharacter } ;
PageNameLink = TitleCharacter , { [ " " | "_" ] TitleCharacter } ;
The comma used in both productions is not a terminal symbol ‘‘,’’: in fact, it
is a concatenate symbol from ISO EBNF [20]. Since ISO EBNF is not the notation
used, the commas must have been left out unintentionally—this is what usually
happens when grammars are transformed manually and not in a disciplined way.
Grammar Hunter assumed that the quotes were forgotten in this place (since a
comma is not a good name for a nonterminal), so we need to project it away
(the corresponding operator is called abstractize because it shifts a grammar
from concrete syntax to abstract syntax). These are the transformations that
we write down888Complete listing of remove-concatenation.xbgf.:
⬇
abstractize(
PageName:
TitleCharacter $\langle$","$\rangle$ (" "$?$ TitleCharacter)⋆
);
abstractize(
PageNameLink:
TitleCharacter $\langle$","$\rangle$ ((" " | "_")$?$ TitleCharacter)⋆
);
The following fragment uses excessive bracketing: parenthesis are used to
group symbols together, which is usually necessary for inner choices and
similar cases when one needs to override natural priorities. However, in this
case it is unnecessary:
SectionTitle = ( SectionLinkCharacter - "=" )
{ [ " " ] ( SectionLinkCharacter - "=" ) } ;
LinkTitle = { UnicodeCharacter { " " } } ( UnicodeCharacter - "]" ) ;
Excessive bracketing is not a problem for SLPS toolset since all BGF grammars
are normalised before serialisation, and it includes a step of refactoring
trivial subsequences, but we still report it for the sake of reproducibility
within a different environment.
The following grammar production uses a strange-looking construction that is
explained in the text to be the ‘‘non-greedy’’ variant of the optional
newline:
<special-block-and-more> ::=
<special-block> ( EOF | [<newline>] <special-block-and-more>
| (<newline> | "") <paragraph-and-more> )
The purpose of a syntax definition such as a BNF is to define syntax of a
language. Thus, any references to the semantics of the parsing process should
be avoided. The definition of ‘‘greediness’’ as ordered alternatives, given at
the first page of [45], contradicts the classic definition based on token
consumption, and contradicts the basics of EBNF. Approaches alternative to
context-free grammars such as PEG [17] should be considered if ordered
alternatives are really required. For EBNF (or BGF), we refactor the
singularity as follows999Part of utilise-question.xbgf.:
⬇
massage(
(newline | $\varepsilon$),
newline$?$);
Since at this point the subgrammar of this part must be rather consistent, we
can execute some simple grammar analyses to help assess the grammar quality.
One of them is based on a well-known notion of bottom and top nonterminals
[58, 59]: a top is one that is defined but never used; a bottom is one that is
used but never defined. We were surprised to see WhiteSpaces in the list of
top nonterminals, while Whitespaces was in the list of bottom nonterminals.
Apparently, a renaming is needed101010Part of unify-whitespace.xbgf.:
⬇
unite(WhiteSpaces, Whitespaces);
The definition of nonterminal BlockHTML contains textual annotation claiming
that it is not yet referred to. We decided to parse it anyway and validate
that assertion afterwards. Indeed, it showed up as an unconnected grammar
fragment, which we can then safely remove111111Part of connect-grammar.xbgf.:
⬇
eliminate(BlockHTML);
### 3.4 Noparse block
Apart from quoting the language name in the source tags, which makes the start
grammar symbol change from <source lang=bnf> to <source lang="bnf">, the
Noparse Block [51] uses the same EBNF dialect that we derived as the starting
step of our extraction. However, there are two major exceptions:
* •
Round brackets and square brackets have swapped their meaning.
* •
A lookahead assertion metasymbol is used, borrowed from Perl Compatible
Regular Expressions library.
The first impression given by cursory examination of the extracted grammar is
that it uses excessive bracketing (mentioned in the previous section):
<pre-block> ::= <pre-opening-tag> (<whitespace>) <pre-body>
(<whitespace>) [<pre-closing-tag> | (?=EOF) ]
<pre-opening-tag> ::= "<pre" (<whitespace> (<characters>)) ">"
<pre-closing-tag> ::= "</pre" (<whitespace>) ">"
<pre-body> ::= <characters>
However, if we assume this to be true, the meaning of the grammar will become inadequate: for example, it will have mandatory whitespace in many places. On the other hand, making the last part of the grammar (<nowiki-closing-tag> | (?=EOF)) optional is also inadequate, because optional assertion will never make sense. This particular lookahead assertion is displayed as (?=EOF) and means basically an $\varepsilon$ that must be followed by EOF (even that definition is not that apparent from the low-level description saying _‘‘It asserts that an EOF follows, but does not consume the EOF.’’_). The presence or absence of lookahead based facilities is heavily dependent on the parsing technology, and therefore should be avoided as much as possible, as noted by multiple sources [25, 36, 58]. More straightforward and high level assertions like ‘‘should be followed by’’ and ‘‘should not be followed by’’ are available in modern metaprogramming languages like Rascal [26] instead.
Since the general problem of leaving opened tags at the end of the article
text is much bigger than the tags described in this part of the grammar, we
opt for removing these assertions altogether and solving the problem later
with suitable technology. EBNF has never been intended for and has never been
good at defining tolerant parsers [27]. Since we have to construct another
EBNF dialect in order to parse the Noparse Block fragment correctly anyway, we
specify ‘‘(?=EOF)’’ as a notation for $\varepsilon$ (otherwise we would have
to fix the problem later with a horizontal remove operator from XBGF). Those
explicit empty sequence metasymbols need to be refactored into proper optional
symbols121212Part of remove-lookahead.xbgf.:
⬇
massage(
(nowiki-closing-tag | $\varepsilon$),
nowiki-closing-tag$?$);
massage(
(pre-closing-tag | $\varepsilon$),
pre-closing-tag$?$);
massage(
(html-closing-tag | $\varepsilon$),
html-closing-tag$?$);
In every notation that comprises similar looking symbols and metasymbols that
can be encountered within the same context, there is need for _escaping_ some
special characters. In this part of the MediaWiki grammar escaping is done in
HTML entities, which is not explainable with grammar-based arguments. However,
we recall that our extraction source is a handcrafted grammar that was meant
to reproduce the behaviour of the MediaWiki PHP parse—so, in a sense, it was
(manually) extracted, and what we have just encountered is in fact a legacy
artefact randomly inherited from its source. Such legacy should be removed by
following transformation steps131313Complete listing of dehtmlify.xbgf.:
⬇
renameT("<nowiki", "<nowiki");
renameT("</nowiki", "</nowiki");
renameT("<pre", "<pre");
renameT("</pre", "</pre");
renameT("<html", "<html");
renameT("</html", "</html");
renameT("<!--", "<!--");
replace(">", ">");
There are two more problems in the Noparse Block part that concern the
nonterminal characters. First, it is undefined (bottom). As we will see in
§3.8, there is a nonterminal called character—issues like these with
‘‘forgetting’’ to define some nonterminals with readable names are quite
common in handcrafted grammars, as noted by [28] and other sources. A
trivially guessed definition for characters is either ‘‘one-or-more’’ or
‘‘zero-or-more’’ repetition of character. Since characters is mostly used as
an optional nonterminal, we assume that it is one or more141414Part of
connect-grammar.xbgf.:
⬇
define(
characters:
character+
);
The second problem is its usage in html-comment (remember that round brackets
mean optionality here):
<html-comment> ::= "<!--" ({ characters }) "-->"
Since we do not need to make a Kleene repetition optional, we can refactor it
as follows151515Part of refactor-repetition.xbgf.:
⬇
unfold(characters in html-comment);
massage(
character+⋆,
character⋆);
massage(
character⋆$?$,
character⋆);
massage(
character⋆,
character+$?$);
fold(characters in html-comment);
More detailed information about leaving combinations of various kinds of
repetition and optionality in the deployed grammar will be given in the next
section.
### 3.5 Links
Figure 1: A syntax that even MediaWiki cannot colour-code properly [52].
Links definitions [52] exhibit bits of yet another notation, namely the one
where a set of possible values is given, assuming that only one should be
picked. In the MediaWiki grammar it is erroneously called a ‘‘regex
format’’—regular expressions do use this notation in some places, but not
everywhere and it is not exclusive to them. This notation is very much akin to
‘‘one-of’’ metaconstructs also encountered in definitions of other software
languages such as C# [75, §3.2.4]. In the MediaWiki grammar, it looks like
this:
/* Specified using regex format, obviously... */
<title-legal-chars> ::= " %!\"$&’()*,\\-.\\/0-9:;=?@A-Z\\\\^_‘a-z~\\x80-\\xFF+"
The unobviousness of the notation is perfectly simplified by the fact that
even the MediaWiki engine itself fails to parse and colour-code it correctly,
as seen on Figure 1. In fact, when we look at the expression more closely, we
can notice that it is even incorrect in itself, since it uses double-escaping
for most backslashes (ruining them) and does not escape the dot (which denotes
any character when unescaped). Some other characters like * or + should
arguably also be escaped, but it is impossible to decide firmly on escaping
rules when we have no engine to process this string. However, the correct
expression should have looked similar to this:
<title-legal-chars> ::= " %!\"$&’()*,\-\.\/0-9:;=?@A-Z\\^_‘a-z~\x80-\xFF+"
Which we rewrite as (some invisible characters are omitted for readability):
This refactored version with all alternatives given explicitly was created
automatically by a trivial Python one-liner and can be parsed without any
trouble by Grammar Hunter. We should also note that the name for this
nonterminal is misleading, since it represents only one character. This is not
a technical mistake, but we can improve _learnability_ of the grammar by
fixing it161616Part of fix-names.xbgf.:
⬇
renameN(title-legal-chars, title-legal-char);
Grammar Hunter displays an error message but is capable of dealing with this
fragment:
<article-link> ::=
[<interwiki-prefix> | ":" ] [<namespace-prefix] <article-title>
The problem in this grammar production is in ‘‘[<namespace-prefix]’’ (note the
unbalanced angle brackets). The start nonterminal symbol here is followed by
the name of the nonterminal and then by the end option symbol without the end
nonterminal symbol. This kind of problems are rather common in grammars that
have been created manually and have never been tested in any environment that
would make them executable or validate consistency otherwise. Grammar Hunter
can resolve this problem by using the heuristic of next best guess, which is
to assume that the nonterminal name ended at the first alphanumeric/non-
alphanumeric border that happened after the unbalanced start nonterminal
symbol.
Next, consider the following two grammar productions that lead to several
problems simultaneously:
<article-title> ::= { [<title-legal-chars> | "%" ] } +
<section-id> ::= { [<title-legal-chars> | "%" | "#" ] } +
As we have explained above, the grammar notation used for the MediaWiki
grammar was never defined explicitly in any formal or informal way, so we had
to infer it in §2. When inferring its semantics, we had two options: to treat
the plus as a postfix metasymbol or to treat ‘‘{’’ and ‘‘}+’’ as bracket
metasymbols. Both variants are possible and feasible, since Grammar Hunter is
capable of dealing with ambiguous starting metasymbols (i.e., ‘‘{’’ as both a
start repetition star symbol and a start repetition plus symbol). We obviously
opt for the latter variant because from the formal language theory we all know
that for any $x$ it is always true that $(x^{*})^{+}\equiv x^{*}$, so a
postfix plus operation on a star repetition is useless and we tend to assume
good faith of grammar engineers who made use of it. But even if we assume it
to be a transitive closure (a plus repetition), which is a common notation for
a sequence of one or more occurrences of a subexpression, the productions
become parseable, but they are bound to deliver problems with ambiguities [4]
on later stages of grammar deployment, since in these particular grammar
fragments optional symbols are iterated.
To give a simple example, suppose we have a nonterminal $x$ defined as
$a^{+}$, and $a$ itself is defined as $``a"?$ (either $``a"$ or
$\varepsilon$). Then the following are two distinct possibilities to parse
$``aa"$ with such a grammar:
$\begin{array}[]{ccccccc}a^{+}&&&&&a^{+}&\\\
\nnearrow\>\>\>\nnwarrow&&&&\phantom{xx}\nearrow&\uparrow&\nwarrow\phantom{xx}\\\
a\qquad a&&&&a&a&a\\\ \uparrow\qquad\uparrow&&&&\uparrow&\uparrow&\uparrow\\\
``a"\quad``a"&&&&``a"&\varepsilon&``a"\\\ \end{array}$
The number of such ways to parse even the simplest of expressions is infinite,
and special algorithms need to be utilised to detect such problems at the
parser generator level. Thus, to prevent this trouble from happening, we
massage the productions above to use a simple star repetition instead, which
is an equivalent unambiguous construct171717Part of utilise-repetition.xbgf.:
⬇
massage(
(title-legal-chars | "%")$?$+,
(title-legal-chars | "%")⋆);
massage(
(title-legal-chars | "%" | "#")$?$+,
(title-legal-chars | "%" | "#")⋆);
Reading further, we notice one of the nonterminals being defined with explicit
right recursion:
<extra-description> ::= <letter> [<extra-description>]
The problem is known and has been discussed above, all we need here is proper
deyaccification181818Part of deyaccify.xbgf.:
⬇
massage(
extra-description$?$,
(extra-description | $\varepsilon$)
in extra-description);
distribute( in extra-description );
vertical( in extra-description );
deyaccify(extra-description);
The last problem with the Links part of the grammar is the use of natural
language inside a BNF production:
<protocol> ::= ALLOWED_PROTOCOL_FROM_CONFIG (e.g. "http://", "mailto:")
Examples are never a part of a syntax definition: the alternatives are either
listed exhaustively (like we will do later when we make the grammar complete)
or belong in the comments (like it was undoubtedly intended here). A
projection is needed to remove them from the raw extracted
grammar191919Complete listing of remove-comments.xbgf.:
⬇
project(
protocol:
ALLOWED_PROTOCOL_FROM_CONFIG $\langle$(e "." g "." "http://" ","
"mailto:")$\rangle$
);
### 3.6 Magic links
Just like Noparse Block discussed above in §3.4, Magic Links [49] also uses
<source lang="bnf"> as the start grammar symbol, but this is the least problem
encountered in this fragment. Consider the following productions:
<isbn> ::= "ISBN" (" "+) <isbn-number> ?(non-word-character /\b/)
<isbn-number> ::= ("97" ("8" | "9") (" " | "-")?) (DIGIT (" " | "-")?)
{9} (DIGIT | "X" | "x")
We see a notation where:
* •
A postfix plus repetition metasymbol is used, which is not encountered
anywhere else in the MediaWiki.
* •
The character used as the postfix repetition metasymbol clashes with end
repetition plus metasymbol from Inline Text [48] and Links [52]202020Indirect
clash of “}+” being an end repetition plus symbol as well as a sequence of an
end repetition star symbol and a postfix repetition metasymbol..
* •
A postfix optionality metasymbol is used, which is not encountered anywhere
else in the MediaWiki.
* •
The character used as the postfix optionality metasymbol clashes with start
special metasymbol and end special metasymbol from Article [46], Inline Text
[48] and Special Block [50].
* •
The same character used as the postfix optionality metasymbol is used as in a
prefix notation that relies on lookahead.
* •
A regular expression is used inside the lookahead assertion.
* •
A terminal symbol (‘‘9’’) is not explicitly marked as such.
* •
A nonterminal symbol (‘‘DIGIT’’) is not explicitly marked as such.
Along with the discussion from §3.4, we first remove the lookahead assertions.
They (arguably) do not belong in EBNF at all, and definitely do not belong in
such a form212121Part of remove-lookahead.xbgf.:
⬇
project(
isbn:
"ISBN" " " "+" isbn-number $\langle$("$?$" non-word-character "/" "\" b
"/")$\rangle$
);
We do not even try to add the postfix plus repetition metasymbol to the
notation definition, since it is used only once, since it clashes with
something else, and since there is a special nonterminal spaces that should be
used instead anyway222222Part of unify-whitespace.xbgf.:
⬇
replace(
" " "+",
spaces);
Then we adjust the grammar for the untreated postfix question
metasymbol232323Part of utilise-question.xbgf.:
⬇
abstractize(
isbn-number:
"97" ("8" | "9") (" " | "-") $\langle$"?"$\rangle$ DIGIT (" " | "-") $\langle$"?"$\rangle$ "9"⋆
(DIGIT | "X" | "x")
);
widen(
(" " | "-"),
(" " | "-")$?$
in isbn-number);
### 3.7 Special block
Just as in [47], the Special Block uses a special metasymbol for omitted
grammar fragments [50]. This case is subtly different from the one discussed
in §3.2 in a sense that it explicitly says in the accompanying text that ‘‘The
dots need to be filled in’’. This information is undoubtedly useful, but
considering the fact that its very presence renders the grammar non-
executable, we decide to remove it from the grammar and let the documentation
tell the story about how much of the intended language does the grammar
cover242424Part of remove-extension-points.xbgf.:
⬇
vertical( in special-block );
removeV(
special-block:
"." "." "."
);
horizontal( in special-block );
In the same first production there is an alternative that reads
<nowiki><table></nowiki>, which seems like either a leftover after manually
cleaning up the markup, or a legacy escaping trick. Either way, nowiki
wrapping is not necessary for displaying this fragment and is generally
misleading: the chevrons around ‘‘table’’ mean to denote it explicitly as a
nonterminal, not as an HTML tag. We project away the unnecessary
parts252525Part of fix-markup.xbgf.:
⬇
vertical( in special-block );
project(
special-block:
$\langle$nowiki$\rangle$ table $\langle$/ nowiki$\rangle$
);
horizontal( in special-block );
There are also more cases of excessive bracketing which are fixed
automatically by Grammar Hunter:
<defined-term> ::= ";" <text> [ (<definition>)]
A nonterminal symbol called dashes is arguably superfluous and can be replaced
by a Kleene star of a dash terminal:
<horizontal-rule> ::= "----" [<dashes>] [<inline-text>] <newline>
<dashes> ::= "-" [<dashes>]
Still, we can keep it in the grammar for the sake of possible future BNF-
ification, but refactor the idiosyncrasy (the right recursion)262626Part of
deyaccify.xbgf.:
⬇
massage(
dashes$?$,
(dashes | $\varepsilon$)
in dashes);
distribute( in dashes );
vertical( in dashes );
deyaccify(dashes);
The worst part of the Special Block part is the section titled ‘‘Tables’’: it
contains eight productions in a different notation, with a comment ‘‘From
meta…minor reformatting’’. This reformatting has obviously been performed
manually, since it does not utilise the standard notation of the rest of the
grammar, nor is it compatible with the MetaWiki notation that we have
encountered in §3.3: the defining symbol is from the MediaWiki notation, the
terminator symbol is from the MetaWiki notation, etc:
Name | Value
---|---
Defining symbol | ::=
Terminator symbol | ;
Definition separator symbol | |
Start special symbol | ?
End special symbol | ?
Start terminal symbol | "
End terminal symbol | "
Start nonterminal symbol | <
End nonterminal symbol | >
Start option symbol | [
End option symbol | ]
To save the trouble of post-extraction fixing, we used this configuration as a
yet another EDD file to extract this grammar fragment and merge it with the
rest of the grammar. The naming convention of the fragment is still not
synchronised with the rest (i.e., camel case vs. dash-separated lowercase),
but we will deal with it later in §5.
We also see a problem similar to the one discussed above in §3.4, namely an
optional zero-or-more repetition:
<space-block> ::= " " <inline-text> <newline> [ {<space-block-2} ]
The solution is also already known to us272727Part of refactor-
repetition.xbgf.:
⬇
massage(
space-block-2⋆$?$,
space-block-2⋆);
When comparing the list of top nonterminals with the list of bottom ones, we
notice TableCellParameters being used while TableCellParameter being defined.
Judging by its clone named TableParameters, the intention was to name it
plural, so we perform unification282828Part of fix-names.xbgf.:
⬇
unite(TableCellParameter, TableCellParameters);
### 3.8 Inline text
Suddenly, [48] uses bulleted-list notation for listing alternatives in a
grammar:
<text-with-formatting> ::=
| <formatting>
| <inline-html>
| <noparseblock>
| <behaviour-switch>
| <open-guillemet> | <close-guillemet>
| <html-entity>
| <html-unsafe-symbol>
| <text>
| <random-character>
| (more missing?)...
This is almost never encountered in grammar engineering, but not completely
unknown to computer science—for example, TLA+ uses this notation [40]. In our
case it is confusing for Grammar Hunter since newlines are also used in the
notation to separate production rules, and since it only happens in two
productions, we decide to manually remove the first bar there. The last line
of the sample above also shows an extension point discussed earlier in §3.2
and §3.7, which we remove292929Part of remove-extension-points.xbgf.:
⬇
vertical( in text-with-formatting );
removeV(
text-with-formatting:
(more missing "?") "." "." "."
);
horizontal( in text-with-formatting );
Nonterminal noparseblock is referenced in the same grammar fragment, but never
encountered elsewhere in the grammar, later we will unite it with noparse-
block when specifically considering enforcing consistent naming convention in
§5.5.
The next problematic fragment is the following:
<html-entity-name> ::= Sanitizer::$wgHtmlEntities (case sensitive)
(* "Aacute" | "aacute" | ... *)
It has three problems:
* •
Referencing PHP variables from the grammar is unheard of.
* •
Static semantics within postfix parenthesis in plain English is not helpful.
* •
A comment that uses ‘‘(*’’ and ‘‘*)’’ as delimiters instead of ‘‘/*’’ and
‘‘*/’’ used in the rest of the grammar.
These identified problems can be solved with projecting excessive symbols,
leaving only one nonterminal reference, which will remain undefined for
now303030Complete listing of remove-php-legacy.xbgf.:
⬇
project(
html-entity-name:
$\langle$(Sanitizer ":" ":" "$")$\rangle$ wgHtmlEntities $\langle$(case
sensitive (("⋆" "Aacute")
| "aacute" | ("." "." "." "⋆")))$\rangle$
);
Later in §5.6 we will reuse the source code of Sanitizer class to formally
complete the grammar by defining wgHtmlEntities nonterminal.
The following fragment combines two double problems that have already been
encountered before. The first problem is akin to the one we have noticed in
§3.5, namely having a nonterminal with ‘‘-characters’’ in its name, which is
supposed to denote only one character taken from a character class; the second
part of that problem is the usage of the regular expression notation. The
second problem is an omission/extension point (cf. §3.2 and §3.7), which is
expressed in Latin:
<harmless-characters> ::= /[A-Za-z0-9] etc
We rewrite it as follows:
<harmless-characters> ::=
"A" | "B" | "C" | "D" | "E" | "F" | "G" | "H" | "I" | "J" | "K" | "L" | "M"
| "N" | "O" | "P" | "Q" | "R" | "S" | "T" | "U" | "V" | "W" | "X" | "Y" | "Z"
| "a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i" | "j" | "k" | "l" | "m"
| "n" | "o" | "p" | "q" | "r" | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z"
| "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
The name of the nonterminal symbol harmless-characters is misleading, since it
represents only one character. In fact, simple investigation into top and
bottom nonterminals [36] shows that it is not referenced anywhere in the
grammar, but a nonterminal harmless-character is used in the definition of
text. Hence, we want to unite those two nonterminals313131Part of fix-
names.xbgf.:
⬇
unite(harmless-characters, harmless-character);
The immediately following production contains a special symbol written in the
style of ISO EBNF and MetaWiki:
<random-character> ::= ? any character ... ?
Instead of adjusting the assumed notation definition, we choose to let Grammar
Hunter parse it as it is, and to subsequently transform the result to a
special BGF metasymbol with the same semantics (i.e., ‘‘any
character’’)323232Part of define-lexicals.xbgf.:
⬇
redefine(
random-character:
ANY
);
The next problematic fragment once again contains omission/extension points:
<ucase-letter> ::= "A" | "B" | ... | "Y" | "Z"
<lcase-letter> ::= "a" | "b" | ... | "y" | "z"
<decimal-digit> ::= "0" | "1" | ... | "8" | "9"
Since in fact they represent all possible alternatives from the given range,
we rewrite them as follows:
<ucase-letter> ::=
"A" | "B" | "C" | "D" | "E" | "F" | "G" | "H" | "I" | "J" | "K" | "L" | "M"
| "N" | "O" | "P" | "Q" | "R" | "S" | "T" | "U" | "V" | "W" | "X" | "Y" | "Z"
<lcase-letter> ::=
"a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i" | "j" | "k" | "l" | "m"
| "n" | "o" | "p" | "q" | "r" | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z"
<decimal-digit> ::=
"0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
The same looking metasymbol is used later as a pure extension point:
<symbol> ::= <html-unsafe-symbol> | <underscore> | "." | "," | ...
The crucial difference in the semantics of these two metasymbols both denoted as ‘‘...’’ lies in the fact that in the former one (i.e., "A" | ... | "Z") it is basically a macro definition that can be expanded by any human reader, but in the latter one (i.e., "." | ...) the only thing the reader learns from looking at it is that something can or should be added. Hence, following the conclusions we drew above, we expand the former omission metasymbol right in the grammar source, but we remove the latter omission metasymbol with grammar transformation333333Part of remove-extension-points.xbgf.:
⬇
vertical( in symbol );
removeV(
symbol:
"." "." "."
);
horizontal( in symbol );
Finally, we notice some of the productions using explicit right recursion:
<newlines> ::= <newline> [<newlines>]
<space-tabs> ::= <space-tab> [<space-tabs>]
<spaces> ::= <space> [<spaces>]
<decimal-number> ::= <decimal-digit> [<decimal-number>]
<hex-number> ::= <hex-digit> [<hex-number>]
The deyaccifying transformation steps are straightforward343434Part of
deyaccify.xbgf.:
⬇
massage(
newlines$?$,
(newlines | $\varepsilon$)
in newlines);
distribute( in newlines );
vertical( in newlines );
deyaccify(newlines);
massage(
space-tabs$?$,
(space-tabs | $\varepsilon$)
in space-tabs);
distribute( in space-tabs );
vertical( in space-tabs );
deyaccify(space-tabs);
massage(
spaces$?$,
(spaces | $\varepsilon$)
in spaces);
distribute( in spaces );
vertical( in spaces );
deyaccify(spaces);
massage(
decimal-number$?$,
(decimal-number | $\varepsilon$)
in decimal-number);
distribute( in decimal-number );
vertical( in decimal-number );
deyaccify(decimal-number);
massage(
hex-number$?$,
(hex-number | $\varepsilon$)
in hex-number);
distribute( in hex-number );
vertical( in hex-number );
deyaccify(hex-number);
We should specially note here that the form used to define spaces prevented us
earlier in §3.6 from using less invasive grammar transformation operators.
What we ideally want is a transformation that is as semantics preserving as
possible353535Part of unify-whitespace.xbgf.:
⬇
fold(space);
fold(spaces);
It is intentional that these two steps affect the whole grammar. We will
return to this issue later in §5.2.
There are two views given on formatting: an optimistic one and a realistic
one. Since the grammar needs to define the allowed syntax in a structured way,
we scrap the the latter363636Complete listing of remove-duplicates.xbgf.:
⬇
removeV(
formatting:
apostrophe-jungle
);
eliminate(apostrophe-jungle);
The whole section describing Inline HTML was removed from [48] prior to
extraction because it combines two aspects that are not intended to be defined
with (E)BNF: it defines a different language embedded inside the current one
(this can be done in a clean way by using modules in advanced practical
frameworks like Rascal [26]) and it tries to define rules for automated error
fixing (cf. fault-tolerant parsing, tolerant parsing, etc). It suffices to
note here that the metalanguage used in the parts of that section that were
formulated not in plain English, is fascinatingly different from the parts of
the MediaWiki grammar that we have already processed: it uses attributed
(parametrised) nonterminals and postfix modifiers for case (in)sensitivity.
The same metasyntax is used in the next section about images, so we do need to
find a way to process chunks like this:
ImageModeManualThumb ::= mw("img_manualthumb");
ImageModeAutoThumb ::= mw("img_thumbnail");
ImageModeFrame ::= mw("img_frame");
ImageModeFrameless ::= mw("img_frameless");
/* Default settings: */
mw("img_manualthumb") ::= "thumbnail=", ImageName | "thumb=", ImageName
mw("img_thumbnail") ::= "thumbnail" | "thumb";
mw("img_frame") ::= "framed" | "enframed" | "frame";
mw("img_frameless") ::= "frameless";
ImageOtherParameter ::= ImageParamPage | ImageParamUpright | ImageParamBorder
ImageParamPage ::= mw("img_page")
ImageParamUpgright ::= mw("img_upright")
ImageParamBorder ::= mw("img_border")
/* Default settings: */
mw("img_page") ::= "page=$1" | "page $1" ??? (where is this used?)
mw("img_upright") ::= "upright" [, ["=",] PositiveInteger]
mw("img_border") ::= "border"
We try to list the problems within that grammar fragment:
* •
Parametrised nonterminals are used in a style of function calls. This is not
completely uncommon to grammarware since the invention of van Wijngaarden
grammars [63] and attribute grammars [29], but unnecessary here.
* •
Some productions end with a terminator symbol ‘‘;’’, others don’t.
* •
Concatenate metasymbol ‘‘,’’ is used rather inconsistently (occurs between
some metasymbols, doesn’t occur between some nonterminal symbols).
* •
Inline comments are given in English without consistent explicit separation
from the BNF formulae.
The shortest way to overcome these difficulties is to reformat them lexically,
unchaining parametrised nonterminals and appending terminator symbols to
productions that did not have them. The result looks like this:
ImageModeManualThumb ::= "thumbnail=", ImageName | "thumb=", ImageName ;
ImageModeAutoThumb ::= "thumbnail" | "thumb";
ImageModeFrame ::= "framed" | "enframed" | "frame";
ImageModeFrameless ::= "frameless";
ImageOtherParameter ::= ImageParamPage | ImageParamUpright | ImageParamBorder
ImageParamPage ::= "page=$1" | "page $1"; /* ??? (where is this used?) */
ImageParamUpgright ::= "upright" [, ["=",] PositiveInteger]
ImageParamBorder ::= "border"
One of the fragments fixed in this way contains postfix metasymbols for case
insensitivity:
<behaviourswitch-toc> ::= "__TOC__"i
<behaviourswitch-forcetoc> ::= "__FORCETOC__"i
<behaviourswitch-notoc> ::= "__NOTOC__"i
<behaviourswitch-noeditsection> ::= "__NOEDITSECTION__"i
<behaviourswitch-nogallery> ::= "__NOGALLERY__"i
These untypical metasymbols are parsed by Grammar Hunter as separate
nonterminals, which we remove by projection373737Complete listing of remove-
postfix-case.xbgf.:
⬇
project(
behaviourswitch-toc:
"__TOC__" $\langle$i$\rangle$
);
project(
behaviourswitch-forcetoc:
"__FORCETOC__" $\langle$i$\rangle$
);
project(
behaviourswitch-notoc:
"__NOTOC__" $\langle$i$\rangle$
);
project(
behaviourswitch-noeditsection:
"__NOEDITSECTION__" $\langle$i$\rangle$
);
project(
behaviourswitch-nogallery:
"__NOGALLERY__" $\langle$i$\rangle$
);
There is also a mistake that is easily overlooked unless you analyse top and
bottom nonterminals (look at the second option):
ImageAlignParameter ::= ImageAlignLeft | ImageAlign|Center |
ImageAlignRight | ImageAlignNone
This extra unnecessary bar is parsed as a regular choice separator, so we need
to fix it this way383838Part of fix-names.xbgf.:
⬇
replace(
(ImageAlign | Center),
(ImageAlignCenter));
The same analysis shows us a fragment in the resulting grammar, which is
unconnected because ImageOption does not list it with the others393939Part of
connect-grammar.xbgf.:
⬇
vertical( in image-option );
addV(
image-option:
image-other-parameter
);
horizontal( in image-option );
The first and the last productions of the Images subsection contain an
explicitly marked nonterminal symbol:
ImageInline ::= "[[" , "Image:" , PageName, ".",
ImageExtension, ( { <Pipe>, ImageOption, } ) "]]" ;
Caption ::= <inline-text>
A production in the middle of the Images subsection and the first production
of the Media subsection make inconsistent use of a concatenate symbol:
ImageSizeParameter ::= PositiveNumber "px" ;
MediaInline ::= "[[" , "Media:" , PageName "." MediaExtension "]]" ;
And, finally, the last production of the Media subsection contains a wrong
defining symbol:
MediaExtension = "ogg" | "wav" ;
These three problems were reported and overcome by Grammar Hunter but not
solved automatically, because usually there is more that one way to resolve
such issues, and a human intervention is needed to make a choice. After the
unified notation is enforced everywhere, we can extract the grammar and
continue recovering it with grammar transformation steps. It should be noted
that Grammar Hunter could not resolve the lack of concatenate symbols, since
it starts assuming that the following symbol is a part of the current one
(originally the concatenate symbol was proposed in [20] in order to allow
nonterminal names contain spaces), but it easily dealt with excessive
concatenate symbols because they just virtually insert $\varepsilon$ here and
there, which gets easily normalised.
Back to the rest of the section, we have a fragment with essentially an
extension point specified in plain English as the right hand side of a
production:
GalleryImage ::= (to be defined: essentially foo.jpg[|caption] )
We can easily decide to disregard this definition in favour of a really
working one404040Part of remove-extension-points.xbgf.:
⬇
redefine(
GalleryImage:
ImageName ("|" Caption)$?$
);
After analysing top and bottom nonterminals, we easily spot unespaced-less-
than being bottom and unescaped-less-than being top—apparently, they were
meant to be one, and the other one is a misspelled variation typically found
in big handcrafted grammars. The same issue arises with some other
nonterminals, apparently this grammar fragment was typed by someone rather
careless at spelling414141Part of fix-names.xbgf.:
⬇
unite(unespaced-less-than, unescaped-less-than);
unite(ImageParamUpgright, ImageParamUpright);
unite(ImageValignParameter, ImageVAlignParameter);
### 3.9 Fundamental elements
Surprisingly for those who did not look at the text of the Inline Text part,
the Fundamental Elements [44] does not contain any new grammar productions for
us, because all of them were encountered within the Inline Text, slightly
reordered.
## 4 Conclusion
This section contains the list of imperfections found in the MediaWiki grammar
definition. In the parenthesis we refer to the section in the text that
unveils the problem or explains it.
* •
Non-extended Backus-Naur form was claimed to be used (§2)
* •
Three different metalanguages used for parts of the grammar (§3.3, §3.4, §3.7)
* •
Bulleted-list notation for alternatives is used, both untraditional and
inconsistent with other grammar fragments (§3.8)
* •
Atypical metasymbols used:
* •
‘‘...?’’ (§3.2) — not defined, assumed to be an extension point
* •
‘‘(?=EOF)’’ (§3.4) — defined in terms of lookahead symbols
* •
‘‘(’’ and ‘‘)’’ (§3.4) — unexpectedly used to denote optionality
* •
‘‘[’’ and ‘‘]’’ (§3.4) — unexpectedly used for grouping
* •
‘‘+’’ (§3.6) — not defined, assumed to be a plus repetition
* •
‘‘?’’ (§3.6) — not defined, assumed to be a postfix optionality
* •
‘‘?()’’ (§3.6) — not defined, assumed to be a lookahead assertion
* •
‘‘...’’ (§3.7, §3.8) — omissions due to the lack of knowledge
* •
‘‘...’’ (§3.8) — omissions to denote values from the range of alternatives
* •
‘‘(*’’ and ‘‘*)’’ (§3.8) — start and end comment symbols
* •
An undesirable omission/extension point metasymbol was used (§3.2, §3.7, §3.8)
* •
An undesirable exception metasymbol was used (§3.3)
* •
An attempt to use metasyntax to distinguish between two choice semantics
(§3.3)
* •
‘‘Yaccified’’ productions with explicit right recursion (§3.2, §3.8)
* •
Underused metalanguage functionality: obfuscated ‘‘plus’’ repetitions and
separator lists (§3.3)
* •
Misspelled nonterminal names w.r.t. case: WhiteSpaces vs. Whitespaces (§3.3),
InlineText vs. inline-text (§3.7, §3.8), etc
* •
Mistyped nonterminal names: unespaced-less-than vs. unescaped-less-than and
ImageParamUpgright vs. ImageParamUpright (§3.8)
* •
Varying grammar fragment delimiters (§3.4, §3.6)
* •
Not marking terminals explicitly with the chosen notation (§3.6)
* •
Not marking nonterminals explicitly with the chosen notation (§3.6)
* •
Escaping special characters with HTML entities (§3.4)
* •
Usage of ‘‘regexp format’’ to specify title legal characters (§3.5)
* •
Insufficient and excessive escaping within ‘‘regexp format’’ (§3.5)
* •
Misleading nonterminal symbol name: plural name for a single character (§3.5,
§3.8)
* •
Improper omission of the end nonterminal metasymbol (§3.5)
* •
Natural language (examples given in parenthesis) as a part of a BNF production
(§3.5)
* •
Inherently ambiguous constructs like $a?^{+}$ and $a*?$ (§3.4, §3.5, §3.7)
* •
Excessive bracketing (§3.3, §3.7)
* •
Unintentionally undefined nonterminals (§3.4)
* •
Referencing PHP variables like Sanitizer::$wgHtmlEntities and configuration
functions like `mw("img_thumbnail")` (§3.8, §5.6)
## 5 Finishing touches
| TERM | VAR | PROD | Bottom | Top
---|---|---|---|---|---
After extraction | 304 | 188 | 691 | 78 | 29
After utilise-repetition.xbgf | 304 | 188 | 691 | 78 | 29
After remove-concatenation.xbgf | 304 | 188 | 691 | 78 | 29
After remove-extension-points.xbgf | 304 | 188 | 684 | 73 | 29
After remove-php-legacy.xbgf | 302 | 188 | 684 | 70 | 29
After deyaccify.xbgf | 302 | 187 | 680 | 70 | 29
After remove-comments.xbgf | 300 | 187 | 680 | 68 | 29
After remove-lookahead.xbgf | 300 | 184 | 680 | 66 | 29
After remove-duplicates.xbgf | 300 | 183 | 678 | 66 | 29
After dehtmlify.xbgf | 299 | 183 | 678 | 66 | 29
After utilise-question.xbgf | 299 | 183 | 678 | 66 | 29
After fix-markup.xbgf | 299 | 183 | 678 | 64 | 29
After define-special-symbols.xbgf | 299 | 183 | 678 | 62 | 29
After fake-exclusion.xbgf | 299 | 183 | 678 | 58 | 26
After remove-postfix-case.xbgf | 299 | 183 | 678 | 57 | 26
After fix-names.xbgf | 307 | 182 | 681 | 37 | 14
After unify-whitespace.xbgf | 307 | 181 | 681 | 31 | 13
After connect-grammar.xbgf | 307 | 181 | 671 | 16 | 7
After refactor-repetition.xbgf | 307 | 181 | 671 | 16 | 7
After define-lexicals.xbgf | 310 | 187 | 671 | 9 | 7
After subgrammar | 310 | 177 | 664 | 8 | 1
Table 1: Simple metrics computed on grammars during transformation.
Table 1 shows the progress of several grammar metrics during recovery: TERM is
the number of unique terminal symbols used in the grammar, VAR is the number
of nonterminals defined or referenced there, PROD is the number of grammar
production rules (counting each top alternative in them) [31]. We have already
discussed bottom and top nonterminals from [36, 58, 59] earlier in §3.3. It is
known and intuitively understood that high numbers of top and bottom
nonterminals indicate unconnected grammar. In the ideal grammar, only few top
nonterminals exist (preferably just one, which is the start symbol) and only
few bottoms (only those that need to be defined elsewhere—lexically or in
another language) [36]. Thus, our finishing touches mostly involved inspection
of the tops and bottoms and their elimination. The very last step called
‘‘subgrammar’’ in Table 1 extracted only the desired start symbol (wiki-page)
and all nonterminals reachable from its definition.
Using the terminology of [36], in this section we move from a level 1 grammar
(i.e., raw extracted one) to a level 2 grammar (i.e., maximally connected
one).
### 5.1 Defining special nonterminals
There is a range of nonterminals used in the MediaWiki grammar that have
noticeably specific names (starting and ending with a question sign or being
uppercased): they are not defined by the grammar, but usually the text around
their definition is enough for a human reader to derive the intended semantics
and then to specify lacking grammar productions. We also unify the naming
convention while doing so (the final steps of that unification will be present
in §5.5) and leave some nonterminals undefined (bottom) to serve connection
points to other languages (more of that in §5.6)424242Complete listing of
define-special-symbols.xbgf.:
⬇
vertical( in TableCellParameter );
removeV(
TableCellParameter:
? HTML cell attributes ?
);
addV(
TableCellParameter:
html-cell-attributes
);
horizontal( in TableCellParameter );
vertical( in TableParameters );
removeV(
TableParameters:
? HTML table attributes ?
);
addV(
TableParameters:
html-table-attributes
);
horizontal( in TableParameters );
define(
FROM_LANGUAGE_FILE:
"#redirect"
);
inline(FROM_LANGUAGE_FILE);
define(
STRING_FROM_DB:
"Wikipedia"
);
inline(STRING_FROM_DB);
define(
STRING_FROM_CONFIG:
STR
);
inline(STRING_FROM_CONFIG);
define(
NS_CATEGORY:
"Category"
);
inline(NS_CATEGORY);
define(
ALLOWED_PROTOCOL_FROM_CONFIG:
"http://"
"https://"
"ftp://"
"ftps://"
"mailto:"
);
inline(ALLOWED_PROTOCOL_FROM_CONFIG);
unite(LEGAL_ARTICLE_ENTITY, article-title);
### 5.2 Unification of whitespace and lexicals
Another big metacategory of nonterminal symbols represent the lexical part,
which is not always properly specified by a syntactic grammar. In the
MediaWiki grammar case, there were several attempts to cover all lexical
peculiarities including problems arising from using Unicode (i.e., different
types of spaces and newlines), so the least we can do is to unify those
attempts. Future work on deriving a level 3 grammar from the result of this
project, will use test-driven correction to complete the lexical part
correctly [36]. Our current goal is to provide a high quality level 2 grammar
without destroying too much information that can be reused later434343Part of
unify-whitespace.xbgf.:
⬇
unite(?_variants_of_spaces_?, space);
unite(?_carriage_return_and_line_feed_?, newline);
unite(?_carriage_return_?, CR);
unite(?_line_feed_?, LF);
inline(NewLine);
unfold(newline in Whitespaces);
fold(newline in Whitespaces);
unite(?_tab_?, TAB);
Another specificity is only referenced but not defined directly by the
grammar. According to the text of Inline Text section [48], this is a patch
for dealing with French punctuation. It is highly debatable whether such
specificity should be found in the baseline grammar, but since it is not
defined properly anyway, we decide to root it out444444Part of unify-
whitespace.xbgf.:
⬇
vertical( in text-with-formatting );
removeV(
text-with-formatting:
open-guillemet
);
removeV(
text-with-formatting:
close-guillemet
);
horizontal( in text-with-formatting );
Some bottom lexical nonterminals are trivially defined in BGF454545Part of
define-lexicals.xbgf.:
⬇
define(
TAB:
"\t"
);
define(
CR:
"\r"
);
define(
LF:
"\n"
);
define(
any-text:
unicode-character⋆
);
define(
sort-key:
any-text
);
define(
any-supported-unicode-character:
ANY
);
### 5.3 Connecting the grammar
The Magic Links part (see 3.6) apparently referenced some nonterminals that
were never used. We can easily pinpoint them with a simple grammar analysis
showing bottom nonterminals, and after that program the appropriate
transformations464646Part of connect-grammar.xbgf.:
⬇
define(
digits:
digit+
);
unite(digit, decimal-digit);
unite(DIGIT, decimal-digit);
Undefined nonterminals PositiveInteger and PositiveNumber both can be merged
with this new nonterminal474747Part of connect-grammar.xbgf.:
⬇
unite(PositiveInteger, digits);
unite(PositiveNumber, digits);
Nonterminal newlines defined at [48] and [44], is also never used and can be
eliminated484848Part of connect-grammar.xbgf.:
⬇
eliminate(newlines);
Last connecting steps are easy since there are not that many top and bottom
nonterminals left, and a simple human inspection can show that some of them
are actually misspelled pairs like this one494949Part of connect-
grammar.xbgf.:
⬇
unite(ImageModeThumb, image-mode-auto-thumb);
unite(category, category-link);
In Links section [52] there is a discussion on whether there should be a
syntactic category for all links (i.e., internal and external). The discussion
seems to be unfinished, with the nonterminal link specified, but unused (i.e.,
top). Since the definition is already available, we decided to use it by
folding wherever possible505050Part of connect-grammar.xbgf.:
⬇
fold(link);
### 5.4 Mark exclusion
BGF does not have a metaconstruct for exclusion (‘‘a should be parseable as b
but not as c’’, mostly specified as ‘‘<a> ::= <b> \- <c>’’ within the
MediaWiki grammar), but we still want to preserve the information for further
refactoring. One of the ways to do so is to used a marking construct usually
found in parameters to transformation operators such as project or
addH515151Part of fake-exclusion.xbgf.:
⬇
replace(
?_all_supported_Unicode_characters_?_-_Whitespaces,
$\langle$(any-supported-unicode-character Whitespaces)$\rangle$);
replace(
UnicodeCharacter_-_WikiMarkupCharacters,
$\langle$(UnicodeCharacter WikiMarkupCharacters)$\rangle$);
replace(
SectionLinkCharacter_- "=",
$\langle$(SectionLinkCharacter "=")$\rangle$);
replace(
UnicodeCharacter_- "]",
$\langle$(UnicodeCharacter "]")$\rangle$);
replace(
UnicodeCharacter_-_BadTitleCharacters,
$\langle$(UnicodeCharacter BadTitleCharacters)$\rangle$);
replace(
UnicodeCharacter_-_BadSectionLinkCharacters,
$\langle$(UnicodeCharacter BadSectionLinkCharacters)$\rangle$);
### 5.5 Naming convention
There are three basic problems with the naming convention if we look at the
whole extracted grammar, namely:
Unintelligible nonterminal names.
When looking at a particular grammar production rule situated close to a piece
of text explaining all kinds of details that did not fit in the BNF, it is
easy to overlook non-informative names. In the case of MediaWiki, in the final
grammar we have bottom nonterminals with the names like `FROM_LANGUAGE_FILE`,
`STRING_FROM_CONFIG`, `STRING_FROM_DB`. Such names do not belong in the
grammar, because they obfuscate it, and the main reason for having a grammar
printed out in an EBNF-like form in the first place is to make it readable for
a human.
Letters capitalisation.
Nonterminal names can be always written in lowercase, or in uppercase, or in
any mixture of them. The choice of parsing technology can influence that
choice: for instance, Rascal [26] can only process capitalised nonterminal
names and ANTLR [54] treats uppercase nonterminals and non-uppercase ones
differently. These implicit semantic details need to be acknowledged and
accounted for, in a consistent manner, which was not the case in the MediaWiki
grammar.
Word separation.
Most of the nonterminals have names that consist of several natural words
(e.g., ‘‘wiki’’ and ‘‘page’’). There are several ways to separate them: by
straightforward concatenating (‘‘wikipage’’), by camelcasing (‘‘WikiPage’’ or
‘‘wikiPage’’), by hyphenating (‘‘wiki-page’’), by allowing spaces in
nonterminal names (‘‘wiki page’’), etc. It does not matter too much which
convention is used, as long as it is the same throughout the whole grammar. In
the case of MediaWiki there is no consistency, which leads to not only
decreased readability, but also to problems like noparse-block being defined
in [51] and noparseblock being used in [48] (they were obviously meant to be
one nonterminal).
The complete transformation script enforcing a consistent naming convention
and fixing related problems on the way, looks like this525252Part of fix-
names.xbgf.:
⬇
unite(noparseblock, noparse-block);
unite(GalleryBlock, gallery-block);
unite(ImageInline, image-inline);
unite(MediaInline, media-inline);
unite(Table, table);
unite(Text, text);
unite(InlineText, inline-text);
unite(Pipe, pipe);
renameN(AnyText, any-text);
renameN(BadSectionLinkCharacters, bad-section-link-characters);
renameN(BadTitleCharacters, bad-title-characters);
renameN(Caption, caption);
renameN(GalleryImage, gallery-image);
renameN(ImageAlignCenter, image-align-center);
renameN(ImageAlignLeft, image-align-left);
renameN(ImageAlignNone, image-align-none);
renameN(ImageAlignParameter, image-align-parameter);
renameN(ImageAlignRight, image-align-right);
renameN(ImageExtension, image-extension);
renameN(ImageModeAutoThumb, image-mode-auto-thumb);
renameN(ImageModeFrame, image-mode-frame);
renameN(ImageModeFrameless, image-mode-frameless);
renameN(ImageModeManualThumb, image-mode-manual-thumb);
renameN(ImageModeParameter, image-mode-parameter);
renameN(ImageName, image-name);
renameN(ImageOption, image-option);
renameN(ImageOtherParameter, image-other-parameter);
renameN(ImageParamBorder, image-param-border);
renameN(ImageParamPage, image-param-page);
renameN(ImageParamUpright, image-param-upright);
renameN(ImageSizeParameter, image-size-parameter);
renameN(ImageValignBaseline, image-valign-baseline);
renameN(ImageValignBottom, image-valign-bottom);
renameN(ImageValignMiddle, image-valign-middle);
renameN(ImageVAlignParameter, image-valign-parameter);
renameN(ImageValignSub, image-valign-sub);
renameN(ImageValignSuper, image-valign-super);
renameN(ImageValignTextBottom, image-valign-text-bottom);
renameN(ImageValignTextTop, image-valign-text-top);
renameN(ImageValignTop, image-valign-top);
renameN(Line, line);
renameN(LinkTitle, link-title);
renameN(MediaExtension, media-extension);
renameN(PageName, page-name);
renameN(PageNameLink, page-name-link);
renameN(PlainText, plain-text);
renameN(SectionLink, section-link);
renameN(SectionLinkCharacter, section-link-character);
renameN(SectionTitle, section-title);
renameN(TableCellParameters, table-cell-parameters);
renameN(TableColumn, table-column);
renameN(TableColumnLine, table-column-line);
renameN(TableColumnMultiLine, table-column-multiline);
renameN(TableFirstRow, table-first-row);
renameN(TableParameters, table-parameters);
renameN(TableRow, table-row);
renameN(TitleCharacter, title-character);
renameN(UnicodeCharacter, unicode-character);
renameN(UnicodeWiki, unicode-wiki);
renameN(WikiMarkupCharacters, wiki-markup-characters);
As one can see, we reinforce hyphenation in almost all places, except for
nonterminals inherited from other languages (e.g., blockquote from HTML). The
list of plain renamings was derived automatically by a Python one-liner that
transformed CamelCase to dash-separated names. The XBGF engine always checks
preconditions for renaming a nonterminal (i.e., the target name must be
fresh), so then it was trivial to turn the non-working renameN calls into
unite calls.
### 5.6 Embedded languages
We may recall seeing wgHtmlEntities undefined nonterminal being referenced in
§3.8. There are more like it—in fact, at the end of our recovery project there
are 8 bottom nonterminals in the grammar:
* •
`LEGAL_URL_ENTITY`: designates a character that is allowed in a URL; defined
by the corresponding RFC [6].
* •
inline-html: was removed deliberately due to incompleteness and questionable
representation; defined partially by the accompanying English text, partially
by the HTML standard [57].
* •
math-block: the syntax used by the math extension to MediaWiki [67].
* •
CSS: cascading style sheets used to specify layout of tables and table cells
[8].
* •
html-table-attributes and html-cell-attributes: also layout of tables and
table cells, but in pure HTML.
* •
wgHtmlEntities: one of the HTML entities (‘‘quot’’, ‘‘dagger’’, ‘‘auml’’,
etc).
They are all essentially different languages that are reused here, but are not
exactly a part of wiki syntax. Some wiki engines may allow for different
subsets of HTML and CSS features to be used within their pages, but
conceptually these limitations are import parameters, not complete
definitions. For instance, we could derive a lacking grammar fragment for
wgHtmlEntities by looking at the file `mw_sanitizer.inc` from MediaWiki
distribution535353Available as mediawiki.config.wiki.:
<wgHtmlEntities> ::= "Aacute" | "aacute" | "Acirc" | "acirc" | "acute" | "AElig"
| "aelig" | "Agrave" | "agrave" | "alefsym" | "Alpha" | "alpha" | "amp" | "and"
| "ang" | "Aring" | "aring" | "asymp" | "Atilde" | "atilde" | "Auml" | "auml"
| "bdquo" | "Beta" | "beta" | "brvbar" | "bull" | "cap" | "Ccedil" | "ccedil"
| "cedil" | "cent" | "Chi" | "chi" | "circ" | "clubs" | "cong" | "copy"
| "crarr" | "cup" | "curren" | "dagger" | "Dagger" | "darr" | "dArr" | "deg"
| "Delta" | "delta" | "diams" | "divide" | "Eacute" | "eacute" | "Ecirc"
| "ecirc" | "Egrave" | "egrave" | "empty" | "emsp" | "ensp" | "Epsilon"
| "epsilon" | "equiv" | "Eta" | "eta" | "ETH" | "eth" | "Euml" | "euml" | "euro"
| "exist" | "fnof" | "forall" | "frac12" | "frac14" | "frac34" | "frasl"
| "Gamma" | "gamma" | "ge" | "gt" | "harr" | "hArr" | "hearts" | "hellip"
| "Iacute" | "iacute" | "Icirc" | "icirc" | "iexcl" | "Igrave" | "igrave"
| "image" | "infin" | "int" | "Iota" | "iota" | "iquest" | "isin" | "Iuml"
| "iuml" | "Kappa" | "kappa" | "Lambda" | "lambda" | "lang" | "laquo" | "larr"
| "lArr" | "lceil" | "ldquo" | "le" | "lfloor" | "lowast" | "loz" | "lrm"
| "lsaquo" | "lsquo" | "lt" | "macr" | "mdash" | "micro" | "middot" | "minus"
| "Mu" | "mu" | "nabla" | "nbsp" | "ndash" | "ne" | "ni" | "not" | "notin"
| "nsub" | "Ntilde" | "ntilde" | "Nu" | "nu" | "Oacute" | "oacute" | "Ocirc"
| "ocirc" | "OElig" | "oelig" | "Ograve" | "ograve" | "oline" | "Omega"
| "omega" | "Omicron" | "omicron" | "oplus" | "or" | "ordf" | "ordm" | "Oslash"
| "oslash" | "Otilde" | "otilde" | "otimes" | "Ouml" | "ouml" | "para" | "part"
| "permil" | "perp" | "Phi" | "phi" | "Pi" | "pi" | "piv" | "plusmn" | "pound"
| "prime" | "Prime" | "prod" | "prop" | "Psi" | "psi" | "quot" | "radic"
| "rang" | "raquo" | "rarr" | "rArr" | "rceil" | "rdquo" | "real" | "reg"
| "rfloor" | "Rho" | "rho" | "rlm" | "rsaquo" | "rsquo" | "sbquo" | "Scaron"
| "scaron" | "sdot" | "sect" | "shy" | "Sigma" | "sigma" | "sigmaf" | "sim"
| "spades" | "sub" | "sube" | "sum" | "sup" | "sup1" | "sup2" | "sup3" | "supe"
| "szlig" | "Tau" | "tau" | "there4" | "Theta" | "theta" | "thetasym" | "thinsp"
| "THORN" | "thorn" | "tilde" | "times" | "trade" | "Uacute" | "uacute" | "uarr"
| "uArr" | "Ucirc" | "ucirc" | "Ugrave" | "ugrave" | "uml" | "upsih" | "Upsilon"
| "upsilon" | "Uuml" | "uuml" | "weierp" | "Xi" | "xi" | "Yacute" | "yacute"
| "yen" | "Yuml" | "yuml" | "Zeta" | "zeta" | "zwj" | "zwnj"
These are 252 entities taken from the DTD of HTML 4.0 [57]. XHTML 1.0 defines
an additional entity called ‘‘apos’’ [2], which, technically speaking, can be
handled by MediaWiki since in its current state it rewrites wikitext to XHTML
1.0 Transitional. Whether it is the grammar’s role to report an error when it
is used, remains an open question. Furthermore, suppose we are developing
wikiware which is not a WYSIWYG editor, but a migration tool or an analysis
tool: this would mean that the details about all particular entities are of
little importance, and one could define an entity name to be just any
alphanumeric word. Questions like these arise when languages are combined, and
for this particular project we leave the bottom nonterminals that represent
import points, undefined.
## 6 Results and future work
This document has reported on a successful grammar recovery effort. The input
for this project was a community-created MediaWiki grammar manually extracted
from the PHP tool that is used to transform wiki text to HTML. This grammar
contained unconnected fragments in at least five different notations, bearing
various kinds of errors from conceptual underuse of base notation to simple
misspellings, rendering the grammar fairly useless. As an output we provide a
level 2 grammar, ready to be connected to adjacent modules (grammars of HTML,
CSS, etc) and made into a higher level grammar (e.g., test it on a real wiki
code). Naturally, this effort is one step in a long way, and we take the rest
of the report to sketch the next milestones and planned deliverables:
Fix grammar fragments.
The first thing we can do is regenerate the original grammar fragments in the
same notation. One one hand, this would help to not alienate the grammar from
its creators; on the other hand, the fragments will use a consistent notation
throughout the grammar and be validated as not having any misspellings,
metasymbol omissions, etc.
Derive several versions.
Just in case the same MediaWiki grammar is needed in several different
notations (e.g., BNF and EBNF), we can derive them from the baseline grammar
with either inferred or programmable grammar transformation.
Propose a better notation.
Whether or not the pure BNF grammar is delivered to Wikimedia Foundation, it
will be of limited use to most people. ANTLR notation that Wiki Creole used,
is more useful, but even less easy to comprehend. Both more readable and more
expressive variants of grammar definition formalisms exist and can be advised
for use based on the required functionality.
Find ambiguities and other problems.
Various grammar analysis techniques referenced in the text above can be used
to perform deeper analyses on the grammar in order to make it fully
operational in Rascal, resolve existing ambiguities, and perhaps even spot
problems that are unavoidable with the current notation.
Complete the lexical part.
Some lexical definitions were already found in the source grammar, and were
mostly preserved through the recovery process. A level 3 grammar can be
derived from our current result by reinspecting these definitions together
with textual annotations found on MediaWiki.org.
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|
arxiv-papers
| 2011-07-23T06:30:44 |
2024-09-04T02:49:20.852004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vadim Zaytsev",
"submitter": "Vadim Zaytsev",
"url": "https://arxiv.org/abs/1107.4661"
}
|
1107.4731
|
# A New Formula for the Natural Logarithm of a Natural Number
Shahar Nevo Bar-Ilan University, Department of Mathematics, Ramat-Gan 52900,
Israel nevosh@macs.biu.ac.il
###### Abstract.
For every natural number $T,$ we write $\operatorname{Ln}T$ as a series,
generalizing the known series for $\operatorname{Ln}2.$
###### 2010 Mathematics Subject Classification:
26A09, 40A05, 40A30
This research is part of the European Science Foundation Networking Programme
HCAA and was supported by Israel Science Foundation Grant 395/07.
## 1\. Introduction
The Euler-Mascheroni constant $\gamma$, [1], is given by the limit
(1) $\gamma=\lim_{n\to\infty}A_{n},$
where for every $n\geq 1,$
$A_{n}:=1+\frac{1}{2}+\dots+\frac{1}{n}-\operatorname{Ln}n.$ An elementary way
to show the convergence of $\\{A_{n}\\}_{n=1}^{\infty}$ is to consider the
series $\sum_{n=0}^{\infty}(A_{n+1}-A_{n})$. (Here $A_{0}:=0.)$ Indeed, by
Lagrange’s Mean Value Theorem, there exists for every $n\geq 1$ a number
$\theta_{n},$ $0<\theta_{n}<1$ such that
$A_{n+1}-A_{n}=\frac{1}{n+1}-\operatorname{Ln}(n+1)+\operatorname{Ln}n=\frac{1}{n+1}-\frac{1}{n+\theta_{n}}=\frac{\theta_{n}-1}{(n+1)(n+\theta_{n})},$
and thus $0>A_{n+1}-A_{n}>\frac{-1}{n(n+1)}$ and the series converges to some
limit $\gamma.$
## 2\. The new formula
Let $T\geq 2$ be an integer. We have
(2)
$A_{nT}=\sum_{k=0}^{n-1}\sum_{j=1}^{T}\frac{1}{kT+j}-\operatorname{Ln}(nT)\underset{n\to\infty}{\rightarrow}\gamma.$
By subtracting (1) from (2) and using
$\operatorname{Ln}(nT)=\operatorname{Ln}n+\operatorname{Ln}T,$ we get
$\sum_{k=0}^{n-1}\left(\sum_{j=1}^{T}\frac{1}{kT+j}-\frac{1}{k+1}\right)\underset{n\to\infty}{\rightarrow}\operatorname{Ln}T,$
that is,
(3)
$\operatorname{Ln}T=\sum_{k=0}^{\infty}\left(\frac{1}{kT+1}+\frac{1}{kT+2}+\dots+\frac{1}{kT+(T-1)}-\frac{(T-1)}{kT+T}\right).$
We observe that (3) generalizes the formula
$\operatorname{Ln}2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots\,.$
We can write (3) also as
(4)
$\operatorname{Ln}T=\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{T}-1\right)+\left(\frac{1}{T+1}+\frac{1}{T+2}+\dots+\frac{1}{2T}-\frac{1}{2}\right)+\dots$
and this gives $\operatorname{Ln}T$ as a rearrangement of the conditionally
convergent series $1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3}+\dots.$
The formula (4) holds also for $T=1.$ Formulas (3) and (4) can be applied also
to introduce $\operatorname{Ln}Q$ as a series for any positive rational
$Q=\frac{M}{L}$ since
$\operatorname{Ln}\frac{M}{L}=\operatorname{Ln}M-\operatorname{Ln}L.$
Now, for any $k\geq 0,$ the nominators of the $k$-th element in (3) are the
same and their sum is 0. This fact is not random. For every constant
$a_{1},a_{2},\dots a_{T}$, the sum
(5)
$S_{T}(a_{1},\dots,a_{T}):=\sum_{k=0}^{\infty}\left(\frac{a_{1}}{kT+1}+\frac{a_{2}}{kT+2}+\dots+\frac{a_{T}}{(k+1)T}\right)$
converges if and only $a_{1}+a_{2}+\dots+a_{T}=0.$ This follows by comparison
to the series $\sum_{k=1}^{\infty}\frac{1}{k^{2}}<\infty.$ By (3) and the
notation (5), $\operatorname{Ln}T=S_{T}(1,1,\dots,1,T-1).$
For $T\geq 2,$ let us denote by $\Sigma(T)$ the collection of all sums of
rational series of type (5), i.e.,
$\Sigma(T)=\big{\\{}S_{T}(a_{1},\dots,a_{T}):a_{i}\in Q,\,1\leq i\leq
T,a_{1}+\dots+a_{T}=0\big{\\}}.$
The collection $\Sigma(T)$ is a linear space of real numbers over $\mathbb{Q}$
(or over the field of algebraic numbers if we would define $\Sigma(T)$ to be
with algebraic coefficients instead of rational coefficients), and
$\dim\Sigma(T)\leq T-1.$ A spanning set of $T-1$ elements of $\Sigma(T)$ is
$\big{\\{}S_{T}(1,-1,0,0,\dots,0),S_{T}(0,1,-1,0,0,\dots,0),\dots,S_{T}(0,\dots,0,1,-1)\big{\\}}.$
Also, if $T$ is not a prime number, then $\dim\Sigma(T)<T-1.$ If
$Q=\frac{M}{L}$ is a positive rational number and $P_{1},P_{2},\dots,P_{k}$
are all the prime factors of $M$ and $L$ together, then
$\operatorname{Ln}Q\in\Sigma(P_{1}P_{2}\dots P_{k}).$
We can get a non-trivial series for $x=0$:
$\operatorname{Ln}4=2\operatorname{Ln}2=S_{2}(2,-2)=S_{4}(2,-2,2,-2)$, and
also $\operatorname{Ln}(4)=S_{4}(1,1,1,-3).$ Hence
$\displaystyle 0$
$\displaystyle=S_{4}(2,-2,2,-2)-S_{4}(1,1,1-3)=S_{4}(1,-3,1,1)$
$\displaystyle=\left(\frac{1}{1}-\frac{3}{2}+\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}-\frac{3}{6}+\frac{1}{7}+\frac{1}{8}\right)+\dots\,.$
## 3\. The integral approach
The formula (3) can as well be deduced in the following way.
$\displaystyle\operatorname{Ln}T$ $\displaystyle=\lim_{x\to
1^{-}}\operatorname{Ln}(1+x+\dots+x^{T-1})=\lim_{x\to
1^{-}}\operatorname{Ln}\left(\frac{1-x^{T}}{1-x}\right)$
$\displaystyle=\lim_{x\to
1^{-}}(\operatorname{Ln}(1-x^{T})-\operatorname{Ln}(1-x))=\lim_{x\to
1^{-}}\int_{0}^{x}\left[\frac{Tu^{T-1}}{u^{T}-1}+\frac{1}{1-u}\right]du$
$\displaystyle=\lim_{x\to
1^{-}}\int_{0}^{x}\frac{Tu^{T-1}-(1+u+\dots+u^{T-1})}{u^{T}-1}du$
$\displaystyle=\lim_{x\to
1^{-}}\int_{0}^{x}\frac{-1-u-u^{2}-\dots-u^{T-2}+(T-1)u^{T-1}}{u^{T}-1}du$
$\displaystyle=\lim_{x\to
1^{-}}\bigg{[}1\cdot\int_{0}^{x}\frac{u-1}{u^{T}-1}du+2\cdot\int_{0}^{x}\frac{u^{2}-u}{u^{T}-1}du+3\int_{0}^{x}\frac{u^{3}-u^{2}}{u^{T}-1}du+\dots$
(6)
$\displaystyle\quad+(T-2)\int_{0}^{x}\frac{u^{T-2}-u^{T-3}}{u^{T}-1}du+(T-1)\int_{0}^{x}\frac{u^{T-1}-u^{T-2}}{u^{T}-1}du\bigg{]}.$
For every $1\leq j\leq T-1,$
$\displaystyle\lim_{x\to 1^{-}}$
$\displaystyle\int_{0}^{x}\frac{u^{j}-u^{j-1}}{u^{T}-1}du=\lim_{x\to
1^{-}}\int_{0}^{x}\bigg{[}u^{j-1}\sum_{k=0}^{\infty}u^{kT}-u^{j}\sum_{k=0}^{\infty}u^{kT}\bigg{]}du$
$\displaystyle=\lim_{x\to
1^{-}}\int_{0}^{x}\bigg{(}\sum_{k=0}^{\infty}u^{kT+j-1}-\sum_{k=0}^{\infty}u^{kT+j}\bigg{)}du=\lim_{x\to
1^{-}}\sum_{k=0}^{\infty}\bigg{(}\frac{x^{kT+j}}{kT+j}-\frac{x^{kT+j+1}}{kT+j+1}\bigg{)}.$
The series in the last expression converges at $x=1,$ and thus it defines a
continuous function in $[0,1]$ and so the limit is
(7)
$\int_{0}^{1}\frac{u^{j}-u^{j-1}}{u^{T}-1}du=\sum_{k=0}^{\infty}\left(\frac{1}{kT+j}-\frac{1}{kT+j+1}\right).$
By (6), we now get that
$\displaystyle\operatorname{Ln}T$
$\displaystyle=\sum_{k=0}^{\infty}\left(\frac{1}{kT+1}-\frac{1}{kT+2}\right)+2\sum_{k=0}^{\infty}\left(\frac{1}{kT+2}-\frac{1}{kT+3}\right)+\dots$
$\displaystyle\quad+(T-2)\sum_{k=0}^{\infty}\left(\frac{1}{kT+T-1}-\frac{1}{kT+T-1}\right)+(T-1)\sum_{k=0}^{\infty}\left(\frac{1}{kT+T-1}-\frac{1}{(k+1)T}\right)$
$\displaystyle=\sum_{k=0}^{\infty}\left(\frac{1}{kT+1}+\frac{1}{kT+2}+\dots+\frac{1}{kT+T-1}-\frac{(T-1)}{(k+1)T}\right),$
and this is formula (3).
If we put $T=3,$ $j=1$ into (7), we get that
(8)
$\int_{0}^{1}\frac{u-1}{u^{3}-1}du=\sum_{k=0}^{\infty}\left(\frac{1}{3k+1}-\frac{1}{3k+2}\right)=S_{3}(1,-1,0).$
On the other hand,
$\int\frac{u-1}{u^{3}-1}du=\frac{2}{\sqrt{3}}\arctan\left(\frac{2u+1}{\sqrt{3}}\right),$
and together with (8), this gives
$\frac{2}{\sqrt{3}}\left(\arctan\sqrt{3}-\arctan\frac{1}{\sqrt{3}}\right)=S_{3}(1,-1,0)$
or
$\pi=3\sqrt{3}\cdot
S_{3}(1,-1,0)=3\sqrt{3}\left[\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{7}-\frac{1}{8}\right)+\dots\right].$
## References
* [1] L. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.
|
arxiv-papers
| 2011-07-24T08:03:51 |
2024-09-04T02:49:20.869943
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shahar Nevo",
"submitter": "Shahar Nevo",
"url": "https://arxiv.org/abs/1107.4731"
}
|
1107.4740
|
# Baryon production and net-proton distributions in relativistic heavy ion
collisions
C. B. Yang and Xin Wang Institute of Particle Physics, Central China Normal
University,Wuhan 430079, People’s Republic of China Key Laboratory of Quark
and Lepton Physics (CCNU), Ministry of Education, People’s Republic of China
###### Abstract
The higher order moments of the net-baryon distributions in relativistic heavy
ion collisions are useful probes for the QCD critical point and fluctuations.
We study the net-proton distributions and their moments in a simple model
which considers the baryon stopping and pair production effects in the
processes. It is shown that a single emission source model can explain the
experimental data well. Centrality and energy dependence of the distributions
and higher moments is discussed.
###### pacs:
25.75.Gz, 21.65.Qr
## I Introduction
The investigation of QCD phase diagram is fundamental to our understanding of
strong interactions. At vanishing baryon chemical potential, lattice QCD
calculations predict the occurrence of a cross-over from hadronic phase to the
quark-gluon plasma phase above a critical temperature of about 170-190 MeV YA
; JB . A distinct singular feature of the phase diagram is the QCD critical
point MAS which is located at the end of the transition boundary. A
characteristic feature of the critical point is the divergence of the
correlation length $\xi$ and extremely large critical fluctuations. In ultra-
relativistic heavy ion collisions, however, because of finite size and rapid
expansion of the system, those divergence may be washed out. As estimated in
MAS , the critical correlation length in heavy ion collisions is not divergent
but about 2-3 fm. Remnants of those critical large fluctuations may become
accessible in heavy ion collisions through an event-by-event analysis of
fluctuations in various channels of conservative hadron quantum numbers, for
example, baryon number, electric charge, and strangeness FLUC . In an energy
scan there would be a non-monotonic behavior of non-Gaussian multiplicity
fluctuations, which would be a clear signature for the existence of a critical
point. In fact, at vanishing chemical potential it has been shown that moments
of conservative charge fluctuations are sensitive indicators for the
occurrence of a transition from hadronic to partonic matter SE .
Recently, the higher order moments of net-baryon distributions in heavy ion
collisions at RHIC energies have aroused great interest both experimentally
STAR and theoretically THEO1 ; THEO2 ; THEO3 ; THEO4 . Experimentally,
neutrons can not be detected easily and the reconstruction efficiency is very
low for strange hadrons. Fortunately, theoretical calculations confirmed that
the net-proton distribution can be a meaningful observable for the purpose of
detecting the critical fluctuations of net baryons in heavy ion collisions NP
. The theoretical interest on these higher order moments comes from the
discovery of the relation between the moments and the thermal fluctuations
near the critical points. It is shown that the higher order moments have
stronger dependence on the correlation length $\xi$ and are therefore more
sensitive to the critical fluctuations. If some memory of large correlation
length persists in the thermal medium in hadronization process, this must be
reflected in higher order moments of the distributions. It has been predicted
MAS2 that the third moment, called skewness, is proportional to $\xi^{4.5}$
and fourth moment, or kurtosis, proportional to $\xi^{7}$ while the second
moment proportional to $\xi^{2}$. More importantly, the moments are closely
related to the susceptibilities of the thermal medium. It has been argued that
information of QCD phase diagram and the critical point can be obtained from
the energy dependence of those moments THEO1 . The moments of net-proton
distributions are studied with different theoretical models such as AMPT and
UrQMD THEO3 , HIJING LUO , and hadron resonance gas model FK etc. All those
theoretical models are quite complicated and many microscopic processes are
involved, and as a result many parameters can be tuned in the investigations.
Therefore, the underlying physics behind the experimental results on the
higher order moments of the net-proton distribution is not very transparent
from the model studies. In addition, those studies focused on the moments only
and made no direct comparison with the experimentally obtained distributions.
In this paper, we will investigate the net-proton distributions in Au+Au
collisions at $\sqrt{s_{NN}}=200{\rm GeV}$ from very simple physics
considerations: baryon stopping and baryon pair production. These physics
effects are well known from studying heavy ion collisions in the past decades.
We will show that such simple physics can be used to reproduce the
experimentally observed net-proton distributions at different colliding
centralities with parameters chosen properly. Then higher moments can be
calculated numerically from the distributions. In this way the centrality
dependence of those moments can be predicted.
This paper is organized as follows. In next section, we will address the
physics points in our considerations for an emission source. Analytical
expressions for the net-proton distribution will be given. The model will be
used to fit the experimental data on the distribution. The model can fit the
data nearly perfectly. The centrality dependence of the moments will then be
predicted. Also the moments at LHC energies are discussed. The last section
will be for a brief summary.
## II Model consideration for an emission source
It was well established that the net-baryon number would be zero in heavy ion
collisions in central rapidity region if there were no nuclear stopping in the
processes. Because the nuclear stopping effect depends on the collision energy
and the size of the system, the baryon number stopped in a rapidity region is
closely related to the number of participant nucleons $N_{\rm part}$. In more
central collisions the net baryon number will be larger. We consider a case in
which all final state baryons are assumed being produced from one emission
source. The initial mean nucleon number in the source is denoted as $B$ which
may be different for different colliding centralities. Considering the
randomness and independence of the nucleon-nucleon collisions in a heavy ion
collision, the probability of finding $N_{0}$ baryons stopped in the kinematic
region under investigation can be assumed, with the given mean number $B$, as
$P_{0}(N_{0},B)=\frac{B^{N_{0}}}{N_{0}!}\exp(-B)\ .$ (1)
Out of those $N_{0}$ stopped nucleons, some of them are proton, others
neutrons. The probability of finding $N_{p}$ protons from $N_{0}$ nucleons is
$Q_{0}(N_{0},N_{p})=C^{N_{p}}_{N_{0}}\rho^{N_{p}}(1-\rho)^{N_{0}-N_{p}}\ $ (2)
with $\rho=Z/A$ the fraction of proton in the nucleus. The above formulas
determine the distribution of net proton number in the source in the initial
state of the collisions.
Then one can consider the baryon production in the collisions. Baryons can be
produced from various channels. The baryon number is a conserved observable,
thus baryons must be produced in baryon-antibaryon pairs. The pair production
may be independent, and as a result of the independence, the probability for
producing $M$ baryon pairs must be a Poissonian with the given mean number of
produced pairs $\mu$ as a parameter
$P_{0}(M,\mu)=\frac{\mu^{M}}{M!}\exp(-\mu)\ .$ (3)
The parameter $\mu$ depends on the colliding centrality. For more central
collisions, the colliding system is larger, therefore $\mu$ should be larger,
and more nucleon-antinucleon pairs can be produced in the process.
It is the right place to compare the number distributions used in this paper
and in others. We use a Poisson distribution for the baryon pair distribution,
supposing the independent production of the pairs. In chen the distributions
for both proton and anti-proton are assumed Poissonian, implying that protons
and anti-protons are produced completely independently. Therefore the baryon
number conservation may be violated in any event. In begun , a canonical
ensemble is employed to derive the number distribution for $\pi$ systems. This
is reasonable because there are a lot of $\pi$ particles in the final state of
heavy ion collisions. But a simple transportation of the method to the case
for baryon production may be problematic, because the relevant baryon particle
number may be not large enough for an equilibrium statistical description.
In the strong production of nucleon-antinucleon pairs, isospin is conserved.
Suppose that $N_{1}$ protons, $N_{2}$ anti-protons, $N_{3}$ neutrons and
$N_{4}$ anti-neutrons are produced in the process, the conservation of isospin
reads $N_{1}-N_{2}=N_{3}-N_{4}$ if the effect from the presence of mesons is
neglected. Thus we have $N_{1}+N_{4}=N_{2}+N_{3}=M$. The probability of
finding $N_{1}$ protons can be assumed as
$Q_{1}(N_{1},M)=2^{-M}C_{M}^{N_{1}}\ .$ (4)
In writing this equality, we assume that all the produced pairs are within the
kinematic range detected experimentally. Of course, this is a rather rough
approximation. In fact, some of the produced nucleons can go out of that range
and cannot be included in measuring the net-protons in the event. The effect
from limited kinematical acceptance can be taken into account by introducing
one more parameter for the probability of the produced baryon in the detected
region. To avoid this complexity, in this paper, such an effect is effectively
treated as having the number of pairs $M$ a little smaller in the event.
Therefore, the value of the parameter $\mu$ obtained from the fitting in this
paper should be a little bit smaller than the real one. In the same way, the
probability of finding $N_{2}$ anti-proton is $Q_{1}(N_{2},M)$. Then the
distribution of net-proton $\Delta p$ from an emission source can be expressed
as
$\displaystyle P(\Delta
p)=\sum_{N_{0},N_{p},M,N_{1},N_{2}}P_{0}(N_{0},B)Q_{0}(N_{0},N_{p})$
$\displaystyle Q_{1}(N_{1},M)Q_{1}(N_{2},M)P_{0}(M,\mu)\delta_{\Delta
p,N_{p}+N_{1}-N_{2}}\ .$ (5)
By inserting an identity expression
$\delta_{m,n}=\int_{0}^{2\pi}dxe^{i(m-n)x}/2\pi$, the above equation can be
rewritten as
$\displaystyle P(\Delta p)$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\frac{dx}{\pi}e^{-(2B\rho+\mu)\sin^{2}\frac{x}{2}}\cos(x\Delta
p-B\rho\sin x)\ .$ (6)
As can be seen from the above expression, the net-baryon distribution depends
on two combined parameters, $B\rho$ and $\mu$ instead of $B,\rho$ and $\mu$
separately. One can check easily that $B\rho$ is the mean value of the
distribution $P(\Delta p)$.
## III Comparison with the experimental data
The expression Eq. (6) enables us to compare the calculated net-proton
distributions from an emission source to the experimental data from STAR STAR
, as shown in Fig. 1. The parameters used are tabulated in TABLE 1. The
parameters show the expected behaviors from central to peripheral collisions.
As one can see from the figure, the agreement with the data is very good over
five orders of magnitude. Almost all calculated points for $F(\Delta p)$ lie
within the experimental error bars.
Figure 1: The net-proton distributions from single emission source. The points are from Ref. STAR , and the curves are calculated from Eq. 6. centrality | $N_{\rm part}$ | $\mu$ | $B\rho$
---|---|---|---
0-5% | 351.4 | 14.5 | 1.65
30-40% | 114.2 | 5.5 | 0.632
70-80% | 13.4 | 0.83 | 0.075
Table 1: Fitted parameters for Fig. 1
To make predictions for the net-proton distributions at other centralities,
one can parameterize the values of parameters $\mu$ and $B\rho$ tabulated in
TABLE 1 by polynomials of the number of participants $N_{p}$ as
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle 0.171+0.0495N_{p}-2.5\times
10^{-5}N_{p}^{2},$ (7) $\displaystyle B\rho$ $\displaystyle=$
$\displaystyle(-4.63+5.99N_{p}-3.7\times 10^{-3})/1000.$ (8)
From this parameterization, one can calculate the moments for the
distributions easily. The obtained moments are shown, as functions of the
number of participants $N_{\rm part}$, in Figs. 2-5. The corresponding
experimental data from STAR are shown in the figures for comparison. The
calculated mean, variance, skewness and kurtosis are well in agreement with
the data. The good agreement shows that the basic merits for the baryon
production mechanism have been exhibited in our model consideration.
Figure 2: The mean net-proton from multiple emission sources. The points are
from Ref. STAR , and the curve is from our model calculation. Figure 3: The
variance for the net-proton distributions from multiple emission sources. The
points are from Ref. STAR , and the curve is from our model calculation.
Figure 4: The skewness for the net-proton distributions from multiple emission
sources. The points are from Ref. STAR , and the curve is from our model
calculation. Figure 5: The kurtosis for the net-proton distributions from
multiple emission sources. The points are from Ref. STAR , and the curve is
from our model calculation.
In Ref. THEO4 , the observed moments are related to the ones from one emission
source by using the central limit theorem. This expectation is deduced from
the independent emission of particles from each source. In fact, if the
baryons are produced from $N_{S}$ identical sources, one can get relations of
the moments for the measured distributions and those for the emission sources
as
$\displaystyle M$ $\displaystyle=$ $\displaystyle M_{i}N_{S},$
$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\sigma_{i}\sqrt{N_{S}},$
$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{i}/\sqrt{N_{S}},$
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\kappa_{i}/N_{S},$
where quantities with subscript $i$ are for moments from one emission source.
From the above expressions on centrality dependence, one can expect constant
$S\sigma$ and $\kappa\sigma^{2}$ for all $N_{\rm part}$. In our fitting with a
single emission source, both $\mu$ changes strongly with centrality. As a
result of the centrality dependence of $B\rho$ and $\mu$, the centrality
dependence of the moments are well reproduced, as can be seen from Figs. 2-5.
One can also calculate moment products $S\sigma$ and $\kappa\sigma^{2}$. The
centrality dependence of the products are shown in Fig. 6. In the $N_{\rm
part}$ range shown, $S\sigma$ increases a few percent, while
$\kappa\sigma^{2}$ is almost exactly 1, as expected from the hadron resonance
gas model HRG .
Figure 6: The moment products, $S\sigma$ and $\kappa\sigma^{2}$, for the net-
proton distributions from multiple emission sources as functions of $N_{\rm
part}$. The dotted line is for $\kappa\sigma^{2}=1$ expected from the hadron
resonance gas model.
With the good agreement with STAR data at RHIC energy at hand, one can go one
step further to predict the higher order moments of the net-proton
distributions at LHC energies. For specific, let the center-of-mass energy of
the colliding nucleon-nucleon pair be 2.76TeV. From fitting data in STAR one
can find the center-of-mass energy dependence of the parameter $B\rho$ and
extrapolate to LHC/ALICE energy. In this way, one gets $B\rho\simeq 0$ at
$\sqrt{s_{NN}}=2.76$ TeV. To get the parameter $\mu$ at LHC/ALICE, one can
write $\mu\propto\exp(m_{p}/T)$, with $T$ being the effective emission
temperature of the sources and $m_{p}$ the mass of proton. In Tem the
effective temperature of the medium is given as a function of the center-of-
mass energy $\sqrt{s}$ as
$T=T_{\rm Lim}/[1+\exp(2.6-\ln(\sqrt{s})/0.45)]$
with $T_{\rm Lim}=164$ MeV. At LHC, the center of mass energy is much higher
than at RHIC, so the value of parameter $\mu$ is much smaller than obtained in
the above. The smaller value of $\mu$ will allow more baryon pairs to be
produced from a source. The predicted moments are shown in Fig. 7. Now the
mean value of the distribution is zero. Because of zero mean value, the net-
proton distribution at LHC/ALICE is symmetric about 0 and one gets zero value
for the skewness. Since the kurtosis is zero for Gaussian distributions,
extremely small $\kappa$ value at large $N_{\rm part}$ at LHC energies means
that the net-proton distributions at LHC energies can be well parameterized by
Gaussian, but not for small $N_{\rm part}$.
Figure 7: The moments for the net-proton distribution from multi-emission
sources at $\sqrt{s_{NN}}=2.76$ TeV.
## IV Conclusion
The higher order moments of the net-proton distributions in relativistic Au+Au
collisions at $\sqrt{s_{NN}}=200{\rm GeV}$ are studied from a simple model
with effects from initial baryon stopping and final baryon pair emission taken
into account. We have demonstrated that by employing a single emission source
model, the distributions at different collision centralities can be well
reproduced. Then the higher order moments for the distributions can be
calculated without new free parameters. The calculated moments agree well with
the experimental results. The predicted moments for LHC Pb+Pb collisions need
to be verified experimentally.
It should be mentioned that nothing else is assumed in this model except an
initial net-proton and a finite probability for producing baryon pairs from
sources. Therefore, our model has nothing to do with thermal equilibrium
and/or critical fluctuations. Because our model consideration is based on
normal physics effects, our results can be used as a baseline for detecting
novel physics in the processes.
###### Acknowledgements.
This work was supported in part by the National Natural Science Foundation of
China under Grant No. 11075061 and by the Programme of Introducing Talents of
Discipline to Universities under No. B08033. The authors thank Dr. X.F. Luo
for sending us the experimental data. We are grateful to N. Xu and X.F. Luo
for valuable discussions.
## References
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* (3) M. Stephanov, K. Rajagopal and E. Shuryak, Phys. Rev. D 60, 114028 (1999).
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* (5) S. Ejiri, F. Karsch and K. Redlich, Phys. Lett. B 633, 275 (2006).
* (6) M.M. Aggarwal et al., (STAR Collaboration), Phys. Rev. Lett. 105, 022302 (2010).
* (7) S. Gupta et al., Science 332, 1525 (2011); X.F. Luo, B. Mohanty, H.G. Ritter and N. Xu, arXiv:1105.5049.
* (8) F. Karsh and K. Redlich, Phys. Lett. B 693, 136 (2011); M. A. Stephanov, arXiv:1104.1627v1; B. Friman et al., arXiv:1103.3511.
* (9) Y. Zhou et al., Phys. Rev. C 82, 014905 (2010); K. Xiao et al., Chin. Phys. C 35, 467 (2011).
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* (18) A. Andronic et al., arXiv:1106.6321v1.
|
arxiv-papers
| 2011-07-24T09:32:32 |
2024-09-04T02:49:20.874796
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. B. Yang and X. Wang",
"submitter": "Chunbin Yang",
"url": "https://arxiv.org/abs/1107.4740"
}
|
1107.4784
|
# AGT conjecture and AFLT states: a complete construction
###### Abstract
A complete construction of the AFLT states is proposed. With this construction
and for all the cases we have checked, the AGT conjecture on the equivalence
of Nekrasov Instanton Counting (NIC) to the $Vir\oplus u(1)$ conformal block
has been verified to be true.
Bao Shou ***bsoul@itp.ac.cn, Jian-Feng Wu †††wujf@itp.ac.cn , and Ming
Yu‡‡‡yum@itp.ac.cn
_Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing, 100190, China_
PACS: 11.25.Hf, 12.60.Jv, 12.40.Nn, 02.30.Ik
Keywords: Liouville Theory, Conformal Blocks, AGT Conjecture, Supersymmetric
Gauge Theory,
S-duality
## 1 Introduction
Conformal blocks, which are defined on the (punctured) Riemann surfaces,
holomorphic in each $z_{i}$ coordinate except when they meet each other, play
an essential role in building correlation functions in two dimensional
(Euclidean) conformal field theories[1]. They can be best understood as sewing
together chiral vertex operators[2, 3, 4], which by definition, are not local
objects, but the correlation functions are. The later combine both holomorphic
and anti-holomorphic conformal blocks in a consistent way to make modular
covariant objects. On the sphere, the $n$-point conformal block is represented
graphically as in fig.1, where $h_{i}$ is the conformal dimension of the
primary field inserted at coordinate $z_{i}$, and $\tilde{h_{i}}$ labels the
contribution arising from the conformal family descending from a primary field
with the conformal dimension $\tilde{h_{i}}$. The global conformal invariance
is $SL(2)\times SL(2)$, which may be used to fix three coordinates $z_{1}=0$,
$z_{n-1}=1$ and $z_{n}=\infty$. So the independent variables are $z_{i}$,
$i=2,...,n-2$, with the degrees of freedom $n-3$ for the $n$-point conformal
blocks on the sphere.
The calculation of conformal blocks is based on the conformal Ward-identities,
$[L_{n},V_{h}(z)]=(z^{n+1}\partial_{z}+(n+1)hz^{n})V_{h}(z).$
and carried out perturbatively level by level [1, 5, 6]. In some special
cases, the decoupling of the Virasoro null vectors can be implemented as
differential equations for the conformal blocks. For the general case,
recursion relations have been proposed by Zamolodchikov[5, 6] on the
meromorphic structures of the conformal blocks either in complex $c$-plane or
$h$-plane. However, in general, the global perspective of the sewing procedure
for the conformal blocks was still not fully understood until recently when
the AGT duality [13] had been proposed.
AGT conjecture relates 2d Liouville conformal field theories to 4d $N=2$
supersymmetric gauge theories of the $A_{1}$ type. The main idea is coupling
to the Liouville field a $u(1)$ field444In fact, the zero mode of the $u(1)$
field is a gauge symmetry and can be fixed to any desired value., then this
system is dual to a $U(2)=SU(2)\times U(1)$ superconformal 4d theory. In this
case, the partition function by Nekrasov instanton counting(NIC)[15, 16] of
the 4d $U(2)$ theory is to be identified with the conformal blocks of the
$u(1)$ coupled Liouville type. The Liouville CFT is characterized by a 2d one
boson theory with center charge $c\geq 25$. Finally, one can decouple the
$U(1)$ factor and obtain the instanton partition function of the $SU(2)$
theory which duals to Liouville conformal blocks. Liouville interaction breaks
down the charge conservation explicitly and leads to the introduction of the
screening charges. Because of the existence of the screening charges, the
conformal blocks of the Liouville type is much more complicated than its
counterpart of the $u(1)$ free boson theory. However, the AGT conjecture, if
proven true, means that there exists an orthogonal basis upon which the
$Liouville\times u(1)$ conformal blocks are built. From the above reasoning,
there exists a tree-like structure which describes the duality in coupling
space of the $N=2$ 4d superconformal linear quiver gauge theory. The primary
objects for this tree-like structure is the “bifundmental” matter coupling,
which, if translated correctly, should be represented by the inner products of
the bra and ket descendant fields in 2d conformal families sandwiched by a
“primary” vertex operator at position, say, $z$. Such kind of pants-like
diagram can be sewed together to form a linear quiver diagram, which, on the
2d CFT side, is just the $n$-point functions on the sphere for our
consideration. Of course, in the present context, we mean the $Vir\oplus u(1)$
2d CFT.
At first sight, it seems that such duality does not bring in any conveniences.
However, the Nekrasov instanton counting on the 4d field theory shows a rather
compact form for the summands which are completely factorized in “momentum”
$P$. And the summation is well organized into the combinatorial enumeration of
the Young tableuax. This simple structure implies Liouville theory, in
particular, the evaluation of the Liouville conformal blocks, could be
resolved by embedding it into a bigger system. So one may expect a new
construction for the Liouville conformal blocks from the corresponding NIC.
Figure 1: $n$-point conformal block on $S^{2}$
As pointed out by Nakajima[23, 24, 25], the instanton counting for $N=2$ gauge
theory is equivalent to the Hilbert scheme of points on the corresponding
Seiberg-Witten curve (blow-up Riemann surface)[11, 12, 21]. This can be
translated into a topological string description from physicists’ point of
view. By invoking the D4-D0 brane setup[20, 19] for ADHM construction[18] of
the instanton moduli space and the resolving process for ALE
singularities[22], these indicate that the instanton counting is a counting
for D0 branes in a toric Calabi-Yau 3-fold. Actually, there are two kinds of
D0 branes in the Calabi-Yau 3-fold, one is the regular D0 brane, which is in
regular representation of $\Gamma$, the center of the corresponding ADE group.
It carries no flux and can move freely on the Riemann surface. The other is
the fractional D0 brane, which is a D2 brane wrapping on a zero-sized two
sphere. It is always attached to the ALE singularity since it has a nontrivial
monodromy while moving around the singularity. It is these fractional D0
branes that resolve the ALE singularity, and leave fluxes on the blow-up
Riemann surface. This property ensures that one can identify these fractional
instantons as “anyons” on the Riemann surface. On the other hand, the regular
ones are “electric charged” particles on the Riemann surface. So the total
counting is equvalent to solving the problem of “electron gas” system with
insertions of anyons at the blow-up singularity on the Riemann surface. This
point of view is partialy included in Dijkgraaf and Vafa’s article[17]. For
each pants of the pants decomposition for the (punctured) Riemann surface, one
can guess that the instanton partition function can be rewritten as summation
over all the intermediate states passing through the sewn holes[4]. For the
interests of the present paper, we concern ourselves only with the special
pants diagram that one of the tubes is replaced by the blow-up singularity.
Then the summand in the instanton partition function represents itself as an
inner product of the bra and ket states, sandwiched by the anyonic vertex
operator. These bra and ket states should come from the interacting
“electronic”555For each simple root of an ADE group, one should introduce a
kind of “electronic” field. particles. A candidate description of the
“electronic gas” system is the integrable system of multiple Calogero-
Sutherland model, each living on a cycle. The whole (punctured) Riemann
surface, can be obtained by sewing together these pants on nonintersecting
cycles.
There are many efforts on relating the conformal blocks to the NIC[14, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35] from various points of views, and these works
confirm the validity of the AGT duality. However, the explicit construction
for the Liouville conformal blocks has remained largely unclear until the
recent work [7] by Alba, Fateev, Litvinov and Tarnopolsky. In [7], they have
put forward the AGT duality in a more explicit form
$\displaystyle\dfrac{{}_{\vec{Y}^{\prime}}\langle{P^{\prime}}|V_{\alpha}|{P}\rangle_{\vec{Y}}}{\langle{P^{\prime}}|V_{\alpha}|{P}\rangle}$
$\displaystyle=$ $\displaystyle
Z_{bif}(\alpha|P^{\prime},\vec{Y}^{\prime};P,\vec{Y})\,\,,$ (1)
here specifically for a free field realization,
$V_{\alpha}(z)=e^{2i(Q-\alpha)\tilde{\varphi}_{-}(z)}e^{-2i\alpha\tilde{\varphi}_{+}(z)}S^{n}:e^{2i\alpha\varphi(z)}:\,,$
with $P+P^{\prime}+\alpha+nb=0$ , and $S=\oint e^{2ib\varphi(z)}\rm dz$ is the
screening charge in the Virasoro sector. The l.h.s. of eq(1) is the pants-like
(with one of the tubes labeled by $\alpha$ shrinks to a line) conformal block.
The r.h.s. of (1) reproduces $Z_{bif}$ for the instanton counting, which is
given by
$\displaystyle
Z_{bif}(\alpha|P^{\prime},\vec{Y}^{\prime};P,\vec{Y})=\prod_{i,j=1}^{2}\prod_{s\in
Y_{i}}\left(Q-E_{Y_{i},Y_{j}^{\prime}}(P_{i}-P_{j}^{\prime}|s)-\alpha\right)\prod_{t\in
Y_{j}^{\prime}}\left(E_{Y_{j}^{\prime},Y_{i}}(P^{\prime}_{j}-P_{i}|t)-\alpha\right)\,,$
(2)
where $\vec{P}=(P,-P)$, $\vec{P^{\prime}}=(P^{\prime},-P)$ and
$\displaystyle E_{Y,Y^{\prime}}(P|s)\equiv
P+b^{-1}(a_{Y}(s)+1)-bl_{Y^{\prime}}(s)\,.$ (3)
Here $a_{Y}(s)$ and $l_{Y}(s)$ resp. are the arm length and the leg length
resp. of the box $s$ in the Young tableau $Y$, defined as
$a_{Y}(s)|_{s=(i,j)}:=\lambda_{i}-j,\ \ \ \
l_{Y}(s)|_{s=(i,j)}:=\lambda_{j}^{t}-i\,,$
$\lambda_{i}$ and $\lambda_{j}^{t}$ resp. are the $i$-th part of the partition
$\lambda=(\lambda_{1},\lambda_{2},\cdots),\,\,\lambda_{i}\geq\lambda_{i+1}$
and the $j$-th part of the transpose partition $\lambda^{t}$ respectively .
(1) means that the matrix elements of a special “chiral vertex operator”
$V_{\alpha}$ in a suitably chosen basis, can be translated into a 4d theory as
an instanton contribution for a special bifundamental contribution of the NIC.
By sewing together pants-like diagrams one gets any desired duality diagrams
in the coupling space of the linear quiver gauge theory. So, the checking of
the AGT duality reduces to the construction of the states
$|{P}\rangle_{\vec{Y}}$, which we shall call the AFLT states[7], with
$\vec{Y}\equiv(Y_{1},Y_{2})$ the Young tableaux. Here the $Y$’s, the
partitions of natural numbers, or equivalently represented by Young tableaux,
are labels for the orthogonal basis for the descendant fields (Verma modules)
in a $Vir\oplus u(1)$ conformal family from the 2d CFT point of view. By
definition, the AFLT states form a complete set of states for the family
members in a given $Vir\oplus u(1)$ conformal family and the inner products
between them, sandwiched by a vertex operator of the particular form,
$V_{\alpha}(z)$, at position,say, $z=1$, is factorized exactly as the NIC
$Z_{bif}$ presented on the r.h.s. of (1). The explicit formula, (1), puts
strong constraints on the possible forms of the AFLT states and make a
systematic construction of them unaccessible at first glance. In [7], only the
explicit form of the state $|{P}\rangle_{Y,\varnothing}$ has been found,
$|{P}\rangle_{Y,\varnothing}=J^{+}_{-Y}|{P}\rangle\Omega_{Y}(P),$
with $J^{+}_{-Y}$ the creator $(-ib)^{-1}a^{+}_{-n}$’s valued Jack symmetric
function, and $\Omega_{Y}(P)$ the normalization constant.
In our opinion, the AGT conjecture, written in the form of (1), strongly
suggests that the $Vir\oplus u(1)$ conformal family is a Hamiltonian system
with $|{P}\rangle_{\vec{Y}}$ the Hamiltonian eigenstates. So the construction
of the AFLT states becomes a quantum mechanical problem of solving the
Schrodinger equation. Put things in this way, we propose a possible form of
the Hamiltonian $H$ and construct its eigenstates explicitly. We shall
identify those eigenstates as the AFLT states desired. For in all the cases we
have checked, (1) is verified to be true, using the AFLT states we have
constructed. We shall present now as the main results of our present paper the
explicit form of the Hamiltonian $H$ along with the complete construction of
the AFLT states, $|{P}\rangle_{\vec{Y}}$. More elaborated exposition will come
in the subsequent sections.
$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{I}$ (4)
$\displaystyle|{P}\rangle_{\vec{Y}}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\frac{1}{E_{\vec{Y}}(P)-H_{0}}H_{I}}J_{-\vec{Y}}|P\rangle\Omega_{\vec{Y}}(P).$
Here, $J^{\pm}_{\pm Y}$ are the Jack states constructed in terms of the
oscillators $a^{\pm}_{n}$’s or $a^{\pm}_{-n}$’s ($n>0$) solely, $H^{\pm}$ the
corresponding Hamiltonian for the Jack symmetric functions, $H_{0}\equiv
H^{+}+H^{-}$. Thus the eigenstate of $H_{0}$ is just
$J_{-\vec{Y}}|{P}\rangle\equiv J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$ with
the eigenvalue $E_{\vec{Y}}(P)$. $H^{\pm}$ in our formalism is defined to
include zero modes $a^{\pm}_{0}$ also, $-ia^{\pm}_{0}|{P}\rangle=\pm
P|{P}\rangle$. It is important that $H_{I}$ is strictly triangular with
respect to the basis vectors of the $H_{0}$ eigenstates. By “strictly
triangular” we mean the (upper or lower) triangular matrix with zero diagonal
entries. It is easy to see that if the interaction term $H_{I}$ is strictly
triangular, then the eigenvalue spectrum of $H_{0}$ remains unperturbed and
$|{P}\rangle_{\vec{Y}}$ in (4) well defined for non-degenerate $H_{0}$
spectrum descending from a mother state $J_{-\vec{Y}}|{P}\rangle$ for generic
values of $P$’s. Putting things all together, we have
$\displaystyle H=H_{0}+H_{I},\ \ \ H_{0}=H^{+}+H^{-},\ \ \
H_{I}=\sum_{n=1}^{\infty}2Qna_{-n}^{+}a_{n}^{-},$ (5) $\displaystyle
H^{\pm}=\dfrac{-i}{3}\oint\left(z\partial_{z}\varphi^{\pm}\right)^{3}\frac{dz}{2\pi
iz}+\sum_{n=1}^{\infty}Qna_{-n}^{\pm}a_{n}^{\pm},$ $\displaystyle
E_{\vec{Y}}(P)=E_{Y_{1}}+E_{Y_{2}}+2P(|Y_{1}|-|Y_{2}|),\ \ \
E_{Y}=\sum_{i}(y^{2}_{i}b^{-1}+(2i-1)y_{i}b),$
$\displaystyle\Omega_{\vec{Y}}(P)=(-)^{|Y_{1}|}b^{|Y_{1}|+|Y_{2}|}\prod_{Y_{1}}\left(2P+(a_{Y_{1}}+1)b^{-1}-l_{Y_{2}}b\right)\prod_{Y_{2}}\left(2P-a_{Y_{2}}b^{-1}+(l_{Y_{1}}+1)b\right),$
$\displaystyle|{P}\rangle_{\vec{Y}}=\frac{1}{1-\frac{1}{E_{\vec{Y}}(P)-H_{0}}H_{I}}J_{-\vec{Y}}|P\rangle\Omega_{\vec{Y}}(P),$
$\displaystyle
H_{0}J_{-\vec{Y}}|{P}\rangle=E_{\vec{Y}}(P)J_{-\vec{Y}}|{P}\rangle,\ \ \
H|{P}\rangle_{\vec{Y}}=E_{\vec{Y}}(P)|{P}\rangle_{\vec{Y}},\ \ \
-ia^{\pm}_{0}|{P}\rangle=\pm P|{P}\rangle$
Notice that
1) $|{P}\rangle_{Y,\varnothing}$ constructed in [7] are included in our
construction as subcases.
2) The Hamiltonian $H$ constructed by us, albeit in a disguised form, turns
out to coincide up to some trivial factor with $I_{3}$, one of the integrals
of motion found in a different context in appendix C of [7]. $I_{3}$ in [7],
written in the form of $Vir\oplus u(1)$, makes the Virasoro symmetry manifest,
but is not suitable for solving a perturbation theory with perturbation
parameter $Q=b+b^{-1}$. The Hamiltonian $H$ written in terms of the
interacting bi-Jack polynomial system as in (5), shows Virasoro symmetry only
implicitly, but makes the perturbation theory exactly solvable as we shall see
soon after.
The procedure is outlined as follows. On the 2d CFT side, the $Vir\oplus u(1)$
theory can be represented as a theory of two independent scalars
$\tilde{\varphi}(z)$ and $\varphi(z)$. $\tilde{\varphi}(z)$ part is
essentially a free theory of timelike oscillators, while the scalar field
$-i\varphi(z)$ is spacelike but engaged in a Liouville type interaction. The
two scalars can be linearly combined to form the “light-cone” scalars.
$\varphi^{+}(z)$ and $\varphi^{-}(z)$. The labeling $\vec{Y}$ of the basis
vectors strongly suggests that there exist a bi-Jack polynomial structure,
plus possibly some interactions between these two sectors. That is, the “free”
$H^{\pm}$ spectrums should be described by $J_{Y_{1}}^{+}$ and $J_{Y_{2}}^{-}$
respectively, here $J_{Y}$ denotes Jack states related to Young tableau $Y$.
First we construct the “unperturbed” energy operator $H_{0}$ which just sums
up the “energies” in $J^{\pm}_{Y}$ sectors, $H_{0}=H^{+}+H^{-}$. The next
thing is to specify the interaction between these two sectors. Strictly
speaking, $H_{0}$ does not describe a free theory, since it also contains the
interaction terms proportional to $Q$. But the new interaction term $H_{I}$
further mixes the $J^{\pm}_{Y}$’s and the coupling is also a first order in
$Q$. It is good to see that $H_{I}$ is strictly triangular with respect to the
basis vectors of $H_{0}$ eigenstates. Our method can be easily generalized to
wider classes of integrable models, in which the interacting Hamiltonian
splits into two parts, $H^{//}$ and $H^{\perp}$, representing respectively the
shift of energies and the rotations (mixings) of states. The later keeps the
eigenvalue spectrum untouched[37].
Besides being triangular, the form of the interaction term is however much
restricted, also by the Virasoro symmetry. Since the total Hamiltonian is of
the form $Vir\oplus u(1)$, an “interaction energy operator” $H_{I}$ is needed
to make the “full Energy operator” $H=H_{0}+H_{I}$ the combination of
$a_{n}$’s and $L_{n}$’s only. Once the Hamiltonian structure is determined,
then the construction of the Hamiltonian eigenstate $|{P}\rangle_{\vec{Y}}$ is
just a quantum mechanical problem. $H_{0}$ and $H$ share the same eigenvalue
spectrum, but only the eigenstates of $H$, represented by
$|{P}\rangle_{\vec{Y}}$’s, form a complete set of basis vectors for the
$Vir\oplus u(1)$ conformal family.
We have checked by examples the corresponding AGT duality formula, (1) up to
level 4, and have found that indeed Nekrasov instanton counting can be
reproduced with this construction, (5). In fact, we have also checked more
general cases and all get positive answers. But those more general results
will appear elsewhere due to lacking of space to include them in this paper.
The insertions of the screening charges play an important role in checking the
AGT duality. However, in the present work we concern ourselves only with the
cases in which the screening charges can be detached away from the vertex
operator $V_{\alpha}$ and moved on to act on the AFLT states (similar to the
Felder’s BRST operators)[8, 9]. The more general cases in which screening
charges can not be moved away from $V_{\alpha}$ will be under our future
studies.
It is well known that it is possible to map the Liouville theory to the
analytic continuation of the Calogero-Sutherland(CS) model, which was
originally and in most cases considered to be a theory with the parameter
$\beta>0$, while in the Liouville case $\beta<0$ is required. Some explanation
is given in [36]. The physical space of the CS model are created by Jack
polynomials, which are symmetric functions studied in great detail in
mathematics and physics literatures[38, 39]. The integrability of the CS model
may be derived in different ways, e.g., from the knowledge of the hidden
$W_{1+\infty}$ symmetry of the model. A recursion relation related to the
Virasoro singular vectors and an integral representation based on it has
appeared recently in [36], in which more references can be found on the
subjects of the CS model and the Jack symmetric functions. It should be
stressed again that for $\beta>0$, there is no null vectors in the CS model.
So the “null” vectors are not the true null vectors of the CS model, since the
Virasoro algebra based on which the null vectors are constructed is not the
true conformal algebra of the CS model in that case. But for $\beta<0$, yes,
there are null vectors in the CS model. It is possible to describe the
$Liouville\times u(1)$ theory in terms of the Jack polynomials considered as
analytic continuation from $\beta>0$ to $\beta<0$.
There is another hint that the $Liouville\times u(1)$ theory has something to
do with $\beta<0$ CS model. It can be found from the Nekrasov partition
function, in which each term in the summation can be written in the form of
the Carlsson-Okounkov formula[10], for the special cases when no screening
charges are inserted. Carlsson-Okounkov formula is a formula for the inner
products between the bra Jack states and the ket Jack states sandwiched with a
modified vertex operator. This extraordinary formula is of great help in
checking the AGT duality with our construction for the orthogonal basis
vectors $|{P}\rangle_{\vec{Y}}$’s defined in (1).
We notice that the construction we found shares many similarities with the
construction of the Jack functions themselves. Namely, we take the state
$J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$ as the mother state and its
descendants are constructed in such a manner that two partitions are
“squeezed” into other pairs. The squeezing does not change the total level of
the two partitions, but does make the inner products of the descendants a
triangular form.
Although the 4d to 2d duality has just begun to be understood, it has been
known for sometime that 2d conformal blocks can be equivalently described as
insertions of Wilson lines in 3d pure Chern-Simons topological gauge theory.
In fact, we can interpret the $n$-point conformal block represented by fig.1
as a Wilson line insertion inside a three-ball. The path integral in Chern-
Simons-Witten gauge theory thus creates a state living on the boundary of the
three ball, which is punctured $S^{2}$. So it should not be a too big surprise
that 2d conformal field theory has something to do with higher dimensional
quantum field theories. Taking into account that Jack symmetric polynomials
can be taken as some special limit of the two parameter Macdonald symmetric
polynomials, one natural guess is that our construction can be generalized to
the case of Macdonald symmetric polynomials. In that case there should be a 5d
to 3d duality.
This paper is organized in the following way. Our general formalism on the
construction of the AFLT states is presented in the introduction. In section
2, we explore the general structure of the $Vir\oplus u(1)$ conformal family.
We found in some cases it is more convenient to work with the bi-Jack function
basis. Section 3 contains the major derivation of our construction. Section 4
is the conclusion. And in appendix A the explicit construction of the AFLT
states up to level 3 is presented.
## 2 Exploring the $Vir\oplus u(1)$ Structure
We are dealing with a 4d N=2 $U(2)$ linear quiver gauge theory coupled to
special bi-fundmental matter in a superconformal way. According to the
standard AGT duality dictionary, the corresponding 2d conformal block is of
the $Vir\oplus u(1)$ type, which reproduces the instanton part of the Nekrasov
partition function for the $U(2)$ theory. There are two sets of Young diagrams
which measure the partitions of the instantons. If one wants to extract the
Virasoro basis of the conformal blocks, one need to factor out the $u(1)$
factor.
In this section we shall mainly explore the Hilbert space for the $Vir\oplus
u(1)$ theory and find the requirements that the energy operator $H$ should
meet. Our procedure depends heavily on the Nekrasov instanton counting formula
written more suitably for the construction of the conformal blocks, (1).
First, the 2d $u(1)$ conformal block, realized in terms of the oscillators of
the scaler field $\tilde{\varphi}$, is essentially of free theory with center
charge $c=1$. The zero modes can be integrated out trivially and does not play
any significant role here. The vertex operators for $\tilde{\varphi}$, take a
peculiar form
$e^{2i(Q-\alpha)\tilde{\varphi}_{(-)}(z)}e^{-2i\alpha\tilde{\varphi}_{(+)}(z)}$
Here, $\tilde{\varphi}_{(\pm)}$ means the positive (negative) mode part of the
scalar field $\tilde{\varphi}$ . Although the above vertex operator is not the
standard one in 2d CFT, its contribution to the conformal block can be easily
read off and factored out. Second, the $Vir$ part is a Liouville conformal
field theory of the $\varphi(z)$ scalar field and is more complicated because
of the existence of the screening charges.
We have the following mode expansion for the scalar fields $\varphi(z)$ and
$\tilde{\varphi}(z)$,
$\displaystyle\varphi(z)$ $\displaystyle=$ $\displaystyle
q+c_{0}\log(z)+\sum_{n\in Z,n\neq 0}\frac{c_{-n}}{n}z^{n}\,,$ (6)
$\displaystyle\tilde{\varphi}(z)$ $\displaystyle=$
$\displaystyle\tilde{q}+a_{0}\log(z)+\sum_{n\in Z,n\neq
0}\frac{a_{-n}}{n}z^{n}\,,$ $\displaystyle\left[c_{n},c_{m}\right]$
$\displaystyle=$ $\displaystyle\frac{n}{2}\delta_{n+m,0}\,,\ \ \
\,(c_{-n})^{\dagger}=c_{n}\,,\ \ \ \left[c_{0},q\right]=\frac{1}{2}$
$\displaystyle\left[a_{n},a_{m}\right]$ $\displaystyle=$
$\displaystyle\frac{n}{2}\delta_{n+m,0}\,,\ \ \
\,(a_{-n})^{\dagger}=-a_{n}\,,\ \ \
\left[a_{0},\tilde{q}\right]=\frac{1}{2}\,.$
Virasoro generators in the $Vir$ part, $L_{n}$, thus reads
$\displaystyle L_{n}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}c_{k}c_{n-k}-inQc_{n}=\sum_{k\neq
0,n}c_{k}c_{n-k}+i(2\hat{P}-nQ)c_{n},$ (7) $\displaystyle L_{0}$
$\displaystyle=$
$\displaystyle\frac{Q^{2}}{4}-\hat{P}^{2}+2\sum_{k>0}c_{-k}c_{k}\,,$ (8)
here, $c_{0}=i\hat{P}$,
$-ic_{0}|{P}\rangle=\hat{P}|{P}\rangle=P|{P}\rangle,\,\,\,\langle{P}|(-ic_{0})=\langle{P}|\hat{P}=-P\langle{P}|.$
By this construction, $L_{n}$ defined in (7-8) is obviously unitary,
$L_{-n}=L_{n}^{\dagger}.$
In 2d CFT, one frequently meets another (more conventional) definition of the
Virasoro generators,
$L^{0}_{n}=\sum_{k\in\mathbb{Z}}c_{k}c_{n-k}-iQ(n+1)c_{n}\,.$ (9)
If (7) and (8) are combined in this way,
$T(z)=\partial\varphi\partial\varphi+iQ\partial^{2}\varphi+iQz^{-1}\partial\varphi-\frac{Q^{2}}{4}\frac{1}{z^{2}}=\sum_{n}L_{n}z^{-n-2}\,,$
(10)
then $T(z)$ differs from the more conventional one
$T^{0}(z)=\partial\varphi\partial\varphi+iQ\partial^{2}\varphi$ by a
similarity transformation
$\displaystyle T(z)$ $\displaystyle=$ $\displaystyle
e^{-iQq}(\partial\varphi\partial\varphi+iQ\partial^{2}\varphi)e^{iQq}=\sum_{n}\tilde{L}_{n}z^{-n-2}$
(11) $\displaystyle\tilde{L}_{n}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}\tilde{c}_{k}\tilde{c}_{n-k}-iQ(n+1)\tilde{c}_{n}\,\,.$
Comparing (10) and (11), we have
$\displaystyle\tilde{c_{n}}=\left\\{\begin{array}[]{l}c_{n},\,\,\,n\neq 0\\\
c_{0}+\frac{i}{2}Q,\,\,\,n=0\end{array}\right.\,.$ (14)
Viewing the $Vir\oplus u(1)$ model as a 2d sigma model, since under
conjugation $c_{n}$ and $a_{n}$ transform differently we recognize that
$-i\varphi$ is spacelike and $\tilde{\varphi}$ timelike when they are
considered as coordinates in target space. So the target space of the sigma
model under consideration is curved in space and flat in time direction. The
two scalars can also be linearly combined to form the “light-cone” scalars
$\varphi^{\pm}(z)$,
$\displaystyle\varphi^{\pm}(z)=\tilde{\varphi}(z)\pm\varphi(z)\,,$ (15)
$\displaystyle\varphi^{\pm}(z)\varphi^{\pm}(z^{\prime})=\log(z-z^{\prime})\,,$
$\displaystyle\varphi^{\pm}(z)\varphi^{\mp}(z^{\prime})=0\,,$
$\displaystyle\varphi^{\pm\dagger}(z)=\varphi^{\mp}(z)\,.$
The descendant states in the conformal family split into sub-spaces of
different levels, which are measured by
$I_{2}=L_{0}+\sum_{n=1}^{\infty}a_{-n}a_{n}$. Within the sub-space of given
level $N\equiv\sum_{n>0}(a_{-n}a_{n}+c_{-n}c_{n})$, states can be labeled
either by linear conbinations of $a_{-X}L_{-Y}$’s or $J^{+}_{-X}J^{-}_{-Y}$’s,
with $X$, $Y$ the Young tableau, $|X|+|Y|=N$, $J^{\pm}_{\pm Y}$ the
annihilator $(-ib)^{-1}a_{Y}^{\pm}$’s (or creator $(-ib)^{-1}a_{-Y}^{\pm}$’s)
valued Jack symmetric functions. In either case, one can infer from AGT
duality that there exist a Hermitian operator $H$, which commutes with $I_{2}$
and diagonizes this subspace. Hence the eigenstates of $H$ form an orthogonal
basis. We know $I_{2}$ acts on this subspace trivially like an identity
operator. So in order to eliminate the degeneracy, the next candidate $H$ we
are looking for should be at least cubic in the oscillators $a_{n}$’s and
$c_{n}$’s. Once $H$ is introduced, the descendant states will organize
themselves into an orthogonal basis labeled by two sets of Young tableaux
$\\{Y_{1},Y_{2}\\}$. In our opinion, it is better to start with the
$J^{\pm}_{Y}$ system, since there is already a Hamiltonian structure $H^{\pm}$
acting separately on them. But $H_{0}=H^{+}+H^{-}$ does not commute with the
screening charges $S^{\pm}$ pertaining to $L_{n}$,
$S^{\pm}=\oint e^{2ib^{\pm}\varphi(z)}\rm dz\,,$
here $b^{+}\equiv b,\,b^{-}\equiv b^{-1}$ . We then add a new term $H_{I}$ to
$H_{0}$, $H=H_{0}+H_{I}$ and require that $\left[S^{\pm},H\right]=0$. If
$H_{I}$ are chosen correctly, the eigenstates of $H$ will coincide with the
unique orthogonal basis $|{P}\rangle_{\vec{Y}}$, which we call AFLT states and
are defined to satisfy (1), in which the matrix elements
$\frac{{}_{\vec{Y^{\prime}}}\langle{P^{\prime}}|V_{\alpha}|{P}\rangle_{\vec{Y}}}{\langle{P^{\prime}}|V_{\alpha}|{P}\rangle}$
is factorized in a consistent way.
On the 4d theory side, one can decouple a single massless bifundamental
matter666The massless condition implies
$|Y^{\prime}_{1}|+|Y^{\prime}_{2}|=|Y_{1}|+|Y_{2}|$. $(\vec{a}=(P,P),m=0)$. We
shall show that under this condition the contributions can be written as the
orthogonality condition for the $|{P}\rangle_{\vec{Y}}$’s, provided (1) is
satisfied.
###### Proposition 1.
777This is actually Proposition 2.4 in [7], but here we give more details.
If AGT conjecture is true, then the AFLT states, $|{P}\rangle_{\vec{Y}}$’s
defined in (1), form an orthogonal basis.
$\\!\\!{}_{Y^{\prime}_{1},Y^{\prime}_{2}}\langle P|{P}\rangle_{Y_{1},Y_{2}}$
$\displaystyle=$ $\displaystyle
Z_{bifund}^{U(2)inst}(\vec{a},\vec{Y},\vec{a},\vec{Y^{\prime}};0)\propto\delta_{\vec{Y},\vec{Y^{\prime}}}.$
(16)
Proof: We proceed, from (1),
${}_{Y^{\prime}_{1},Y^{\prime}_{2}}\langle
P|{P}\rangle_{Y_{1},Y_{2}}=\,\,\\!\\!_{Y^{\prime}_{1},Y^{\prime}_{2}}\langle{P}|V_{\alpha=0}|{P}\rangle_{Y_{1},Y_{2}}$
$\displaystyle=$
$\displaystyle\prod_{Y_{1}}\\{Q-[(a_{Y_{1}}+1)b^{-1}-l_{Y^{\prime}_{1}}b]\\}\prod_{Y^{\prime}_{1}}\\{(a_{Y^{\prime}_{1}}+1)b^{-1}-l_{Y_{1}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{1}}\\{Q-[2P+(a_{Y_{1}}+1)b^{-1}-l_{Y^{\prime}_{2}}b]\\}\prod_{Y^{\prime}_{2}}\\{-2P+(a_{Y^{\prime}_{2}}+1)b^{-1}-l_{Y_{1}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{2}}\\{Q-[-2P+(a_{Y_{2}}+1)b^{-1}-l_{Y^{\prime}_{1}}b]\\}\prod_{Y^{\prime}_{1}}\\{2P+(a_{Y^{\prime}_{1}}+1)b^{-1}-l_{Y_{2}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{2}}\\{Q-[(a_{Y_{2}}+1)b^{-1}-l_{Y^{\prime}_{2}}b]\\}\prod_{Y^{\prime}_{2}}\\{(a_{Y^{\prime}_{2}}+1)b^{-1}-l_{Y_{2}}b\\}\,.$
We shall prove now that under this situation, if the result is non-zero, one
can conclude
$\vec{Y}=\vec{Y^{\prime}}.$
If
${}_{Y^{\prime}_{1},Y^{\prime}_{2}}\langle P|{P}\rangle_{Y_{1},Y_{2}}\neq 0,$
one gets
$y_{1,1}\leq y^{\prime}_{1,1}.$ 888Here we use the notation $y_{1,r}$ to label
the $r$-th part of the partition $Y_{1}$.
Since otherwise there must exist a box in the tableau $Y_{1}$ satisfying
$a_{Y_{1}}=0,\,\,l_{Y_{1}^{\prime}}=-1\,\,.$
This will lead to
$Q-[(a_{Y_{1}}+1)b^{-1}-l_{Y^{\prime}_{1}}b]=0,$ (18)
This argument cycles and one finally conclude:
$y_{1,i}\leq y^{\prime}_{1,i}\,\,,i=1,2,\dots\,.$
For $Y_{2}$, similarly, the argument follows, and gives:
$y_{2,i}\leq y^{\prime}_{2,i}\,\,,i=1,2,\dots\,.$
However, the original condition
$|Y^{\prime}_{1}|+|Y^{\prime}_{2}|=|Y_{1}|+|Y_{2}|$
then forces $Y_{1}=Y^{\prime}_{1},Y_{2}=Y^{\prime}_{2}$. Q.E.D.
The orthogonality condition, (16), strongly suggests the existence of mutually
commuting Hermitian operators, whose common eigenstates form a complete
orthogonal basis of the Hilbert space. One of the operators, called the energy
operator, probably cubic in $a_{n}$’s and $c_{n}$’s (since this is most likely
the case beyond $I_{2}$), is the first object we are going to construct.
However, hermiticity alone is not enough to constrain the possible forms of
the construction. For example, $H_{0}=H^{+}+H^{-}$ is Hermitian, but does not
belong to $Vir\oplus u(1)$. In fact, we shall show that altogether there
should be at least 3 conditions $H$ are to meet in order the orthogonality of
the $H_{0}$ eigenstates play an important role here.
$\displaystyle i)$ $\displaystyle\hskip 42.67912ptHermiticity$ $\displaystyle
ii)$ $\displaystyle\hskip 42.67912ptTriangularity$ $\displaystyle iii)$
$\displaystyle\hskip 42.67912ptReflection-invariance\,.$
Now we explain what the other two conditions means. Condition ii),
triangularity, means that $H$, in its matrix form,
$H_{\vec{Y^{\prime}},\vec{Y}}(P)=\langle{\vec{Y^{\prime}}}|H(P)|{\vec{Y}}\rangle$,
where $|{\vec{Y}}\rangle$’s are the eigenstates of $H_{0}=H^{+}+H^{-}$(the
“unperturbed” energy operator)999Here we have fixed $\hat{P}$ eigenvalue
equals $P$ ., is lower(or upper)-triangular with $H_{I}\equiv H-H_{0}$
strictly triangular (with zero diagonal entries). Under such circumstances,
the spectrum of $H$ coincides with that of $H_{0}$, and the eigenstates of $H$
can be expressed as
$|{P}\rangle_{Y_{1},Y_{2}}=\Omega_{Y_{1},Y_{2}}(P)R(E)|{\vec{Y},P}\rangle$,
here the normalization constant $\Omega_{Y_{1},Y_{2}}(P)$ will be specified
later on. $R(E)=1+\tilde{R}(E)$, a unitriangular matrix, is again triangular
with identity diagonal entries following the triangularity of $H$. $H^{\pm}$
is the collective mode Hamiltonian for the Calogero-Sutherland model in terms
of the oscillators ${a^{\pm}_{n}}$’s. Thus the eigenstate of $H_{0}$ is just
$J^{+}_{-Y_{1}}J^{-}_{-Y_{2}}|{P}\rangle$. $H^{\pm}$ in our formalism
(including the zero modes $a^{\pm}_{0}$) is defined as
$H^{\pm}=-i\frac{1}{3}\oint(z\partial_{z}\varphi^{\pm}(z))^{3}dz/z+\sum_{n=1}^{\infty}Qna^{\pm}_{-n}a^{\pm}_{n}$
Since $\varphi^{\pm}\dagger=\varphi^{\mp}$, we have
$J^{+}_{Y}\dagger=J^{-}_{-Y}$. There is a natural question on how to define
the inner products between $J^{\pm}_{Y}$’s. The answer is that we need the
condition iii) Reflection-invariance. Notice that
$\langle{P^{\prime}}|P\rangle$ not zero means $P+P^{\prime}=0$. In order to
get a non-vanishing result, we need to shift $\langle{P}|$ to $\langle{-P}|$.
We thus expect that there exists an operation which changes
${}_{Y_{1},Y_{2}}\langle{P}|$ to ${}_{Y_{2},Y_{1}}\langle{-P}|$. We call this
operation reflection following the terminology in a similar situation in [29].
Actually, by looking closer to the NIC formula i.e. the r.h.s of (1), one can
find that there exist an apparent symmetry
${}_{Y_{1},Y_{2}}\langle{P}|\leftrightarrow_{Y_{2},Y_{1}}\langle{-P}|.$ (19)
If we change either bra state $\langle{P}|_{Y_{1},Y_{2}}$ to
${}_{Y_{2},Y_{1}}\langle{-P}|$, or ket state $|{P}\rangle_{Y_{1},Y_{2}}$ to
$|{-P}\rangle_{Y_{2},Y_{1}}$, on the l.h.s. of (1), the factors on the r.h.s.
of eq(1) get reshuffling but the final result keep invariant. We may name this
symmetry “reflection” or “flipping” symmetry. On the 2d CFT side, from general
reasoning that such an operation should be conformally invariant, it is
natural to identify the insertions of the screening charges as this
“reflection” operation. For Liouville theory (or Coulomb gas model), we can
attach to $V_{\alpha=0}$ some screening charges101010We suppose originally
there is no screening charge attached to $V_{\alpha=0}$ for simplicity., such
that
${}_{Y_{1}^{\prime},Y_{2}^{\prime}}\langle{P}|V_{0}S^{n}|{P}\rangle_{Y_{1},Y_{2}}\neq
0$ (20) $\displaystyle S=\oint e^{2ib\varphi(z)}\rm dz\,.$ (21)
Now the neutrality condition forces $2P+nb=0\,\,\,.$ If this is satisfied,
then Felder’s contour for the integration of the screening charges actually
closes and $S^{n}$ becomes a floating charge[8, 9]. Now $S^{n}$ can move away
from $V_{\alpha}$ and communing through $L_{n}$’s and finally acts on the
vacuum state $\langle{P}|$. Since $S^{n}$ acts by not changing the conformal
weight, we deduce
$\langle{P}|S^{n}=\langle{-P}|$
for a suitable normalization of $S^{n}$. Similar arguments apply to the case
of $V_{\alpha},\ \alpha\neq 0$, and one can always move a subset of screening
charges, $S^{\frac{-2P}{b}}$ away from $V_{\alpha}(z)$. Since AGT duality
formula is valid for any $P$, we may assume that $n$ can take arbitrary real
value, as analytical continuation away from integer $n$. This flipping is due
to the fact that $S^{n}$ can be detached from $V_{\alpha}$, and act on the
vacuum directly. Similar operation exists in Felder’s BRST cohomology [8].
We are going to identify the reflection symmetry in NIC as the Hamiltonian
symmetry in 2d CFT for the insertions of the screening charges $S^{n}$ with
$2P=-nb$. Since Hamiltonian $H\in Vir\oplus u(1)$, satisfies
$\left[H,S^{n}\right]=0$ it has the property of double degeneracy. So $S^{n}$
with $2P=-nb$ should map one AFLT state to its partner state. If we require
${}_{Y_{1},Y_{2}}\langle{P}|H$ $\displaystyle=$
$\\!\\!{}_{Y_{1},Y_{2}}\langle{P}|E_{Y_{1},Y_{2}}$ (22)
${}_{Y_{1},Y_{2}}\langle{P}|S^{n}H$ $\displaystyle=$
${}_{Y_{1},Y_{2}}\langle{P}|S^{n}E_{Y_{1},Y_{2}}\,,$ (23)
then we can identify
${}_{Y_{1},Y_{2}}\langle{P}|S^{n}=_{Y_{2},Y_{1}}\langle{-P}|\,,$
since reflection symmetry means $E_{Y_{1},Y_{2}}(P)=E_{Y_{2},Y_{1}}(-P)$.
Notice that nothing has changed for the $u(1)$ part. Define
$P^{\pm}=-ia_{0}\mp ic_{0},$ then we have111111We have set
$a_{0}|{P}\rangle=0$ throughout this paper.:
$P^{\pm}|{P}\rangle=\pm
P|{P}\rangle,\,\,\,\langle{-P}|P^{\pm}=\langle{-P}|(\pm P)\,,$
which obviously shows that $\langle-P|{P}\rangle\neq 0$. So reflection
invariance means that we can identify the inner product
${}_{Y_{1}^{\prime},Y_{2}^{\prime}}\langle{P}|P\rangle_{Y_{1},Y_{2}}$ with
either ${}_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|P\rangle_{Y_{1},Y_{2}}$
or ${}_{Y_{1}^{\prime},Y_{2}^{\prime}}\langle{P}|-P\rangle_{Y_{2},Y_{1}}$ by
the incertions of screening charges satisfying $n=-2Pb^{-1}$.
Having determined that $|{P}\rangle_{\vec{Y}}$ form a normalizable orthogonal
basis, the next step is the determination of their normalization. Before doing
this, let’s review the so-called Carlsson-Okounkov formula[10] which is useful
for our formulation. First, define
$\displaystyle
E=1+e_{1}+e_{2}+\dots=e^{-\sum_{n}\frac{(-)^{n}}{n}p_{n}}=e^{-\frac{1}{k}\varphi_{(-)}(-1)}$
(24) 121212For infinitely many arguments $z_{i}$’s, $i=1,2,\cdots,\infty$, one
may identify $p_{n}\equiv\sum_{i}z_{i}^{n}$ with $\frac{a_{-n}}{k}$,
$k^{2}=\beta$ and $J_{Y}^{1/\beta}(\\{p_{n}\\})$ with
$J_{Y}^{1/\beta}(\\{\frac{a_{-n}}{k}\\})$. Here our convention is that
$\frac{a_{-n}}{k}|{0}\rangle$ creates a state $|{p_{n}}\rangle$. As a
consequence, $e_{n}$ is to be identified with
$P_{-1^{n}}\equiv\dfrac{J_{1^{n}}^{1/\beta}(\\{\frac{a_{-n}}{k}\\})}{n!}\equiv
J_{-1^{n}}/n!$ . Such kind of identification is justified because they share
the same values of their inner products. For more details see [36].
which is a vertex operator, and also a generating function for $J_{-1^{n}}$
$e^{-\frac{1}{k}\varphi_{(-)}(z)}|{0}\rangle=\sum_{n}(-)^{n}\dfrac{J_{-1^{n}}}{n!}z^{n}|{0}\rangle\,,$
here $e_{i}$ are elementary symmetric functions, $p_{n}$ is the power sum
symmetric function. Then
$e^{-\frac{1}{k}\varphi_{(-)}(-1)}|{0}\rangle=\sum_{n}\dfrac{J_{-1^{n}}}{n!}|{0}\rangle\equiv\sum_{n}P_{-1^{n}}|{0}\rangle.$
(25)
The conjugation of $E$ reads
$E^{\ast}=e^{\frac{1}{k}\varphi_{(+)}(-1)}\,,$ (26)
and we have
$\langle{0}|E^{\ast}=\langle{0}|\sum_{n}P_{1^{n}}\,.$
Now the Carlsson-Okounkov formula reads
$\displaystyle\langle E^{m}(E^{\ast})^{\beta-m-1}J_{-Y_{1}},J_{-Y_{2}}\rangle$
$\displaystyle=$
$\displaystyle(-)^{|Y_{1}|}\beta^{-|Y_{1}|-|Y_{2}|}\prod_{Y_{1}}(m+(a_{Y_{1}}+1)+\beta
l_{Y_{2}})\prod_{Y_{2}}(m-a_{Y_{2}}-\beta(l_{Y_{1}}+1))$ $\displaystyle=$
$\displaystyle\langle J_{Y_{1}}E^{\beta-m-1}(E^{\ast})^{m}J_{-Y_{2}}\rangle$
$\displaystyle=$ $\displaystyle\langle
J_{Y_{1}}e^{(-k+k^{-1}+\frac{m}{k})\varphi_{(-)}(-1)}e^{\frac{m}{k}\varphi_{(+)}(-1)}J_{-Y_{2}}\rangle\,.$
For Liouville theory, $k=-ib$. If we set $\frac{m}{-ib}=-2i\alpha$, then the
Carlsson-Okounkov formula reads
$\displaystyle\langle
J_{Y_{1}}e^{i(Q-2\alpha)\varphi_{(-)}(-1)}e^{-2i\alpha\varphi_{(+)}(-1)}J_{-Y_{2}}\rangle$
$\displaystyle=$
$\displaystyle(-)^{|Y_{2}|}b^{-|Y_{1}|-|Y_{2}|}\prod_{Y_{1}}(-2\alpha+(a_{Y_{1}}+1)b^{-1}-l_{-Y_{2}}b)\prod_{Y_{2}}(-2\alpha-
a_{Y_{2}}b^{-1}+(l_{Y_{1}}+1)b)\,.$
The normalization of the AFLT states is inherited from AFLT’s version of the
AGT duality formula, (1) and the orthogonality condition, (2),
${}_{Y_{1},Y_{2}}\langle
P|{P}\rangle_{Y_{1},Y_{2}}=_{Y_{2},Y_{1}}\langle-P|{P}\rangle_{Y_{1},Y_{2}}$
$\displaystyle=$
$\displaystyle\prod_{Y_{1}}\\{-a_{Y_{1}}b^{-1}+(l_{Y_{1}}+1)b\\}\\{(a_{Y_{1}}+1)b^{-1}-l_{Y_{1}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{2}}\\{-a_{Y_{2}}b^{-1}+(l_{Y_{2}}+1)b\\}\\{(a_{Y_{2}}+1)b^{-1}-l_{Y_{2}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{1}}\\{-2P-a_{Y_{1}}b^{-1}+(l_{Y_{2}}+1)b)\\}\prod_{Y_{2}}\\{-2P+(a_{Y_{2}}+1)b^{-1}-l_{Y_{1}}b\\}$
$\displaystyle\times$
$\displaystyle\prod_{Y_{2}}\\{2P-a_{Y_{2}}b^{-1}+(l_{Y_{1}}+1)b)\\}\prod_{Y_{1}}\\{2P+(a_{Y_{1}}+1)b^{-1}-l_{Y_{2}}b\\}$
$\displaystyle=$ $\displaystyle(-)^{|Y_{1}|+|Y_{2}|}j_{Y_{1}}j_{Y_{2}}$
$\displaystyle\times$ $\displaystyle\prod_{Y_{1}}\\{-2Pb-
a_{Y_{1}}+(l_{Y_{2}}+1)b^{2}\\}\prod_{Y_{2}}\\{-2Pb+(a_{Y_{2}}+1)-l_{Y_{1}}b^{2}\\}$
$\displaystyle\times$ $\displaystyle\prod_{Y_{2}}\\{2Pb-
a_{Y_{2}}+(l_{Y_{1}}+1)b^{2}\\}\prod_{Y_{1}}\\{2Pb+(a_{Y_{1}}+1)-l_{Y_{2}}b^{2}\\}$
$\displaystyle\equiv$ $\displaystyle
j_{Y_{1}}j_{Y_{2}}\Omega_{Y_{2},Y_{1}}(-P)\Omega_{Y_{1},Y_{2}}(P)$
$\displaystyle=$ $\displaystyle j_{Y_{1}}j_{Y_{2}}\langle
J_{Y_{2}}e^{i(Q-2P)\varphi_{(-)}(-1)}e^{-i2P\varphi_{(+)}(-1)}J_{-Y_{1}}\rangle$
$\displaystyle\times$ $\displaystyle(b^{4})^{|Y_{1}|+|Y_{2}|}\langle
J_{Y_{1}}e^{i(Q+2P)\varphi_{(-)}(-1)}e^{i2P\varphi_{(+)}(-1)}J_{-Y_{2}}\rangle\,.$
In reaching the last line in the above eqation, Carlsson-Okounkov formula has
been applied, and we have defined
$\displaystyle\Omega_{Y_{1},Y_{2}}(P)$ $\displaystyle=$
$\displaystyle(-)^{|Y_{1}|}b^{|Y_{1}|+|Y_{2}|}\prod_{Y_{1}}\left(2P+(a_{Y_{1}}+1)b^{-1}-l_{Y_{2}}b\right)\prod_{Y_{2}}\left(2P-a_{Y_{2}}b^{-1}+(l_{Y_{1}}+1)b\right)$
$\displaystyle=$ $\displaystyle(-b^{2})^{(|Y_{1}|+|Y_{2}|)}\langle
J_{Y_{1}}e^{i(Q+2P)\varphi_{(-)}(-1)}e^{i2P\varphi_{(+)}(-1)}J_{-Y_{2}}\rangle$
$\displaystyle=$ $\displaystyle b^{2(|Y_{1}|+|Y_{2}|)}\langle
J_{Y_{1}}e^{i(Q+2P)\varphi_{(-)}(1)}e^{i2P\varphi_{(+)}(1)}J_{-Y_{2}}\rangle$
$\displaystyle\equiv$ $\displaystyle\Omega_{\vec{Y}}(P)\,.$
Notice that $\Omega_{\vec{Y}}(P)$ is just a generalization of $\Omega_{Y}(P)$
defined in [7].
## 3 The Construction of the AFLT States
Now we come to our main problem of the construction of the Hamiltonian $H$
with the requirement that its eigenstates be identified with ALFT states
satisfying (1). We prefer to work first on the basis of Jack symmetric
functions $J_{\vec{Y}}$, which already form an orthogonal basis. We found that
if $H_{I}$ matrix elements are strictly triangular on this basis, then the
orthogonality of the $H=H_{0}+H_{I}$ eigenstates follows immediately from the
orthogonality of the $H_{0}$ eigenstates. This is just the simplest way to go
from one orthogonal basis to another one. To see this, let’s introduce an
operator $R(E)$ which map $H_{0}$ eigenstates to $H$ eigenstates, with
$\Omega_{Y_{1},Y_{2}}(P)$ the normalization constant
$\displaystyle|{P}\rangle_{Y_{1},Y_{2}}$ $\displaystyle=$ $\displaystyle
R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle\Omega_{Y_{1},Y_{2}}(P)\,$ (31)
${}_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|$ $\displaystyle=$
$\displaystyle\langle{-P}|J_{Y_{2}^{\prime}}^{-}J_{Y_{1}^{\prime}}^{+}R(E^{\prime})^{\dagger}\Omega_{Y_{2}^{\prime}Y_{1}^{\prime}}(-P)$
$\displaystyle R(E)$ $\displaystyle=$ $\displaystyle
1+\cdots=1+\tilde{R}(E)\,,$
where the reflection symmetry has been applied to the AFLT states
${}_{Y_{1}^{\prime},Y_{2}^{\prime}}\langle{P}|S^{n}=\langle{P}|S^{n}J_{Y_{1}^{\prime}}^{-}J_{Y_{2}^{\prime}}^{+}R(E^{\prime})^{\dagger}\Omega_{Y_{1}^{\prime}Y_{2}^{\prime}}(P)=\\\
_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|=\langle{-P}|J_{Y_{2}^{\prime}}^{-}J_{Y_{1}^{\prime}}^{+}R(E^{\prime})^{\dagger}\Omega_{Y_{2}^{\prime}Y_{1}^{\prime}}(-P),$
and $\tilde{R}(E)$ is strictly lower(or upper)-triangular $\Rightarrow$
$\tilde{R}_{\vec{Y},\vec{Y}}(E)=0$. The way $R(E)$ is expanded in (31) follows
from the normalization condition, (2-2) as we shall see in (3).
The Hermitian operator $H$ should satisfy:
$\displaystyle H|{P}\rangle_{Y_{1},Y_{2}}$ $\displaystyle=$ $\displaystyle
E_{Y_{1},Y_{2}}(P)|{P}\rangle_{Y_{1},Y_{2}}$ (32)
${}_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|H$ $\displaystyle=$
${}_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|E_{Y_{2}^{\prime},Y_{1}^{\prime}}(-P),$
where the energy eigenvalue has the double degeneracy:
$E_{Y_{2},Y_{1}}(-P)=E_{Y_{1},Y_{2}}(P)\,,$ (33)
due to the orthogonality condition,
${}_{Y_{1}^{\prime},Y_{2}^{\prime}}\langle
P|{P}\rangle_{Y_{1},Y_{2}}=_{Y_{2}^{\prime},Y_{1}^{\prime}}\langle{-P}|P\rangle_{Y_{1},Y_{2}}\propto\delta_{\vec{Y},\vec{Y^{\prime}}}\,.$
(34)
Then the next step is to determine if we get the right normalization for
$|{P}\rangle_{Y_{1},Y_{2}}$
${}_{Y_{2},Y_{1}}\langle{-P}|{P}\rangle_{Y_{1},Y_{2}}$ $\displaystyle=$
$\displaystyle\langle{-P}|J_{Y_{2}}^{-}J_{Y_{2}}^{+}R(E)^{\dagger}R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle\Omega_{Y_{1},Y_{2}}(P)\Omega_{Y_{2},Y_{1}}(-P)$
$\displaystyle=$
$\displaystyle\langle{-P}|J_{Y_{2}}^{-}J_{Y_{1}}^{+}(1+\tilde{R}(E)^{\dagger})(1+\tilde{R}(E))J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle\Omega_{Y_{1},Y_{2}}(P)\Omega_{Y_{2},Y_{1}}(-P)$
$\displaystyle=$
$\displaystyle\Omega_{Y_{1},Y_{2}}(P)\Omega_{Y_{2},Y_{1}}(-P)$
$\displaystyle\times$
$\displaystyle\left[j_{Y_{1}}j_{Y_{2}}+\langle{-P}|J_{Y_{2}}^{-}J_{Y_{1}}^{+}(\tilde{R}(E)^{\dagger}+\tilde{R}(E)+\tilde{R}(E)^{\dagger}\tilde{R}(E))J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle\right]$
$\displaystyle=$
$\displaystyle\Omega_{Y_{1},Y_{2}}(P)\Omega_{Y_{2},Y_{1}}(-P)j_{Y_{1}}j_{Y_{2}}\,.$
It is in agreement with (2). In deriving this we have used the fact that if
$R(E)$ is a unitriangular matrix131313A unitriangular matrix is a triangular
matrix with the diagonal entries equal to 1 .,
$R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle=J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle+\sum_{\begin{subarray}{c}|Y_{1}^{\prime}|>|Y_{1}|\\\
|Y_{2}^{\prime}|<|Y_{2}|\\\
|Y_{1}^{\prime}|+|Y_{2}^{\prime}|=|Y_{1}|+|Y_{2}|\end{subarray}}R_{Y_{1},Y_{2}}^{Y_{1}^{\prime}Y_{2}^{\prime}}(E)J_{-Y_{1}^{\prime}}^{+}J_{-Y_{2}^{\prime}}^{-}|{P}\rangle\,,$
(36)
then it is easy to check that
$\displaystyle\langle{-P}|J_{Y_{2}}^{-}J_{Y_{1}}^{+}\tilde{R}(E)^{\dagger}J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$
$\displaystyle=\langle{-P}|J_{Y_{2}}^{-}J_{Y_{1}}^{+}\tilde{R}(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$
$\displaystyle=\langle{-P}|J_{Y_{2}}^{-}J_{Y_{1}}^{+}\tilde{R}(E)^{\dagger}\tilde{R}(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$
$\displaystyle=0\,.$
Now we summarize the requirements for $R(E)$
$\displaystyle i)\hskip 28.45274pt\text{$R(E)$ is unitriangular}$
$\displaystyle ii)\hskip 28.45274pt\text{$R(E)$ creates the eigenstate for
$H$}$ $\displaystyle\hskip
28.45274ptHR(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle=E_{Y_{1},Y_{2}}(P)R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$
$\displaystyle iii)\hskip 28.45274pt\text{Reflection invariant}$
$\displaystyle\hskip
28.45274ptS^{n}R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle\Omega_{Y_{1},Y_{2}}(P)=R(E)J_{-Y_{2}}^{+}J_{-Y_{1}}^{-}|{-P}\rangle\Omega_{Y_{2},Y_{1}}(-P)$
$\displaystyle\hskip 28.45274pt\left[S^{n},H\right]=0,\ \ \ \
E_{Y_{1},Y_{2}}(P)=E_{Y_{2},Y_{1}}(-P)$
This means that the Hamiltonian $H$ should also be triangular, and $H_{I}$
strictly triangular,
$\displaystyle H$ $\displaystyle=$ $\displaystyle H^{+}+H^{-}+H_{I}$ (37)
$\displaystyle H^{\pm}$ $\displaystyle=$
$\displaystyle\dfrac{-i}{3}\oint\left(z\partial_{z}\varphi^{\pm}\right)^{3}+\sum_{n>0}Qna_{-n}^{\pm}a_{n}^{\pm}\,,$
(38)
here $H_{I}$ is to be determined later on.
$\displaystyle H^{+}+H^{-}$ $\displaystyle=$
$\displaystyle\dfrac{-i}{3}\oint\left[\left(z\partial_{z}(\varphi+\tilde{\varphi})\right)^{3}+\left(z\partial_{z}(\varphi-\tilde{\varphi})\right)^{3}\right]\dfrac{\rm
dz}{z}+$
$\displaystyle\sum_{n>0}Qna_{-n}^{+}a_{n}^{+}+\sum_{n>0}Qna_{-n}^{-}a_{n}^{-}$
$\displaystyle=$
$\displaystyle\dfrac{-i}{3}\oint\left[2\left(z\partial_{z}\tilde{\varphi}\right)^{3}+6(z\partial_{z}\tilde{\varphi})(z\partial_{z}\varphi)^{2}\right]\dfrac{\rm
dz}{z}$ $\displaystyle+$ $\displaystyle\sum_{n>0}2Qn(a_{-n}a_{n}+c_{-n}c_{n})$
$\displaystyle=$
$\displaystyle\dfrac{-2i}{3}\oint\left(z\partial_{z}\tilde{\varphi}\right)^{3}\frac{\rm
dz}{z}+\sum_{n>0}2Qna_{-n}a_{n}-2i\oint\left(z\partial_{z}\tilde{\varphi}\right)\left(z\partial_{z}\varphi\right)^{2}\frac{\rm
dz}{z}+2\sum_{n>0}Qnc_{-n}c_{n}\,.$
Now the requirement that $H$ commute with $S^{n}$ is equivalent to say that
$H$ can be written in terms of $L_{n}$’s and $a_{n}$’s. To make $H$
triangular, we may try
$\displaystyle H_{I}$ $\displaystyle\propto$
$\displaystyle\sum_{n}Qna_{-n}^{+}a_{n}^{-}$ $\displaystyle=$
$\displaystyle\sum_{n}Qn(a_{-n}a_{n}-c_{-n}c_{n}-a_{-n}c_{n}+c_{-n}a_{n})$
$\displaystyle=$ $\displaystyle\sum_{n}Qn(a_{-n}a_{n}-c_{-n}c_{n})+Q\oint
z\partial_{z}\tilde{\varphi}(z\partial_{z})^{2}\varphi\frac{\rm dz}{z}\,.$
If we now make use of (10) and choose
$H_{I}=\sum_{n}2Qna_{-n}^{+}a_{n}^{-},\,$
then we get
$\displaystyle H$ $\displaystyle=$
$\displaystyle-\dfrac{2i}{3}\oint(z\partial_{z}\tilde{\varphi})^{3}\dfrac{\rm
dz}{z}+4Q\sum_{n\in\mathbb{N}^{+}}na_{-n}a_{n}-2i\oint(z\partial_{z}\tilde{\varphi})z^{2}T(z)\dfrac{\rm
dz}{z}+2ia_{0}\dfrac{Q^{2}}{4}$ $\displaystyle=$
$\displaystyle-\dfrac{2i}{3}\oint(z\partial_{z}\tilde{\varphi})^{3}\dfrac{\rm
dz}{z}+4Q\sum_{n\in\mathbb{N}^{+}}na_{-n}a_{n}-2i\sum_{n\in\mathbb{Z}}a_{-n}L_{n}+2ia_{0}\dfrac{Q^{2}}{4}$
$\displaystyle=$
$\displaystyle-i\left\\{\sum_{n,m\in\mathbb{N}^{+}}\left(a_{-n-m}^{+}a_{n}^{+}a_{m}^{+}+a_{-n-m}^{-}a_{n}^{-}a_{m}^{-}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{n,m\in\mathbb{N}^{+}}\left(a_{-n}^{+}a_{-m}^{+}a_{n+m}^{+}+a_{-n}^{-}a_{-n}^{-}a_{n+m}^{-}\right)\right\\}$
$\displaystyle+$
$\displaystyle\sum_{n\in\mathbb{N}^{+}}Qn(a_{-n}^{+}a_{n}^{+}+a_{-n}^{-}a_{n}^{-}+2a_{-n}^{+}a_{n}^{-})$
(40) $\displaystyle+$
$\displaystyle\sum_{n\in\mathbb{N}^{+}}-2ia_{0}^{+}a_{-n}^{+}a_{n}^{+}-2ia_{0}^{-}a_{-n}^{-}a_{n}^{-}-\dfrac{i}{3}((a_{0}^{+})^{3}+(a_{0}^{-})^{3})\,.$
(41)
Clearly, $H$ indeed satisfies the three requirements proposed in the previous
section.
$\displaystyle i)$ $\displaystyle\hskip 42.67912ptHermitian$ $\displaystyle
ii)$ $\displaystyle\hskip 42.67912ptTriangular$ $\displaystyle iii)$
$\displaystyle\hskip 42.67912ptReflection-invarint\,.$
Besides, we found that $H\propto I_{3}$, where $I_{3}$ is defined in Appendix
C of [7] as one of the infinitely many commuting operators which may makes the
system integrable. The authors of [7] have checked for the first a few levels
that the AFLT states, $|{P}\rangle_{Y_{1},Y_{2}}$, which satisfies AGT duality
formula, (1), are also the eigenstates of $I_{3}$. But a general formula for
the $I_{3}$ eigenstates is missing in [7].
Now the next question is : how to find all the eigenstates of $H$? First,
let’s consider $H^{+}$
$\displaystyle H^{+}$ $\displaystyle=$
$\displaystyle-i\sum_{n,m\in\mathbb{N}^{+}}\left\\{a_{-n-m}^{+}a_{n}^{+}a_{m}^{+}+a_{-n}^{+}a_{-m}^{+}a_{n+m}^{+}\right\\}$
$\displaystyle+$
$\displaystyle\sum_{n\in\mathbb{N}^{+}}\left\\{nQa_{-n}^{+}a_{n}^{+}+2a^{+}_{0}(-i)a_{-n}^{+}a_{n}^{+}\right\\}-\frac{i(a_{0}^{+})^{3}}{3}\,,$
Its eigenvalue
$H^{+}J^{+}_{-Y}|P^{+}\rangle=E^{+}_{Y}(P^{+})J^{+}_{-Y}|P^{+}\rangle$
$E^{+}_{Y}(P^{+})=\sum_{i}\left\\{y^{2}_{i}b^{-1}+(2i-1)y_{i}b\right\\}+2P^{+}|Y|-\frac{(P^{+})^{3}}{3}.$
Here we have assumed the zero modes take the following eigenvalues,
$a_{0}=iP^{a}\,,\,c_{0}=iP^{c}\,,\,a^{\pm}_{0}=iP^{\pm}=i(P^{a}\pm P^{c})$
(43)
For the bi-Jack system, we have the following eigenequation,
$HR(E)J^{+}_{-Y_{1}}J^{-}_{-Y_{2}}|P^{+},P^{-}\rangle=E_{Y_{1},Y_{2}}(P^{+},P^{-})R(E)J^{+}_{-Y_{1}}J^{-}_{-Y_{2}}|P^{+},P^{-}\rangle\,.$
Triangularity means
$\displaystyle E_{Y_{1},Y_{2}}(P^{+},P^{-})=$ $\displaystyle
E^{+}_{Y_{1}}(P^{+})+E^{-}_{Y_{2}}(P^{-})$ $\displaystyle=$
$\displaystyle\sum_{i}\left\\{y_{1,i}^{2}b^{-1}+(2i-1)y_{1,i}^{2}b\right\\}+\sum_{i}\left\\{y_{2,i}^{2}b^{-1}+(2i-1)y_{2,i}^{2}b\right\\}$
$\displaystyle+2P^{+}|Y_{1}|+2P^{-}|Y_{2}|-\frac{(P^{+})^{3}+(P^{-})^{3}}{3}$
Since $H$ can be constructed in terms of $L_{n}$’s and $a_{n}$’s, so
$S^{n}|P^{+},P^{-}\rangle_{Y_{1},Y_{2}}$ dose not change the eigenvalue. But
$S^{n}$ changes $P^{c}\rightarrow-P^{c}$ and $P^{+}\leftrightarrow P^{-}$ and
from $E_{Y_{1},Y_{2}}(P^{+},P^{-})=E_{Y_{2},Y_{1}}(P^{-},P^{+})$. We conclude
$S^{n}|P^{+},P^{-}\rangle_{Y_{1},Y_{2}}\propto|P^{-},P^{+}\rangle_{Y_{2},Y_{1}}.$
(44)
Next, since $P^{a}$ does not play any important role, we may consider it as a
gauge symmetry and can be fixed to any desired value. For convenience, we fix
$P^{a}=0$, hence, $P^{+}=P^{c}\equiv P$, $P^{-}=-P^{c}\equiv-P$ and
$\displaystyle E_{Y_{1},Y_{2}}(P,-P)$ $\displaystyle\equiv E_{Y_{1},Y_{2}}(P)$
$\displaystyle=E_{Y_{1}}+E_{Y_{2}}+2P(|Y_{1}|-|Y_{2}|)$
Here
$E_{Y}=\sum_{i}\left\\{y^{2}_{i}b^{-1}+(2i-1)y_{i}b\right\\}$
If we define
$\displaystyle|P\rangle$ $\displaystyle\equiv$ $\displaystyle|P,-P\rangle$
$\displaystyle|P\rangle_{Y_{1},Y_{2}}$ $\displaystyle\equiv$ $\displaystyle
R(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle\Omega_{Y_{1},Y_{2}}(P)$
Then we infer from from (44)
$S^{n}|P\rangle_{Y_{1},Y_{2}}=|-P\rangle_{Y_{2},Y_{1}}$
with the proper normalization for $S^{n}$. Now we are going to determine
$R(E)$ which satisfies
$HR(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle=E_{Y_{1},Y_{2}}(P)R(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle.$
###### Proposition 2.
$R(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle=\frac{1}{1-\frac{1}{E-H_{0}}H_{I}}J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle,$
here $H_{0}=H^{+}+H^{-},E=E_{Y_{1},Y_{2}}(P)$. $R(E)$ defined in such a way
should be understood as
$\displaystyle R(E)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\frac{1}{E-H_{0}}H_{I}}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(\frac{1}{E_{Y_{1},Y_{2}}(P)-H_{0}}H_{I})^{n}$
Proof: First, we rewrite $H$ as
$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{I}$
$\displaystyle=$ $\displaystyle E+H_{0}+H_{I}-E$ $\displaystyle=$
$\displaystyle E+(H_{0}-E)(1+\frac{1}{H_{0}-E}H_{I})\,.$
Then from
$\displaystyle HR(E)$ $\displaystyle=$
$\displaystyle(E+(H_{0}-E)(1+\frac{1}{H_{0}-E}H_{I}))\frac{1}{1-\frac{1}{E-H_{0}}H_{I}}$
$\displaystyle=$ $\displaystyle
E\frac{1}{1-\frac{1}{E-H_{0}}H_{I}}+H_{0}-E\,,$ $\displaystyle=$
$\displaystyle ER(E)+H_{0}-E\,,$
one gets
$HR(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle=ER(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle+(H_{0}-E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle\,.$
Since $J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle$ is an eigenstate of $H_{0}$
with eigenvalue $E=E_{Y_{1},Y_{2}}(P)$, we have
$(H_{0}-E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle=0.$
Hence, we conclude that $R(E)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}|{P}\rangle$ is an
eigenstate of $H$ with eigenvalue $E$,
$E\equiv
E_{Y_{1},Y_{2}}(P)=\left(\sum_{i=1,l=1}^{i=2,l=y_{i,1}^{t}}(y_{i,l})^{2}+(2i-1)y_{i,l}\right)+2P(|Y_{1}|-|Y_{2}|)\,,$
Q.E.D.
Now we shall address the question raised in [7] on the possible degeneracy of
$H$. The authors of [7], argued that $I_{3}$ has some degeneracy at level 4
and higher. We have analyzed what causes such kind of degeneracy. After
analyzing the spectrum of $H$, we believe that such degeneracy happens when
$|Y_{1}|=|Y_{2}|$, and we have $2P(|Y_{1}|-|Y_{2}|)=0$,
$E_{Y_{1},Y_{2}}(P)=E_{Y_{1}}+E_{Y_{2}}=E_{Y_{2}}+E_{Y_{1}}=E_{Y_{2},Y_{1}}(P)$
This can happen, for $Y_{1}\neq Y_{2}$, first at level 4,
$|Y_{1}|+|Y_{2}|\equiv|\vec{Y}|=4$, and $Y_{1}=2$, $Y_{2}=1^{2}$. Such
degeneracy can happen at any even level higher or equal to 4. For example at
$\textrm{level}=6$:
$Y_{1}=3,Y_{2}=1^{3},\textrm{or}~{}Y_{1}=3,Y_{2}=\\{2,1\\},\textrm{or}~{}Y_{1}=1^{3},Y_{2}=\\{2,1\\},$
or simply, we have $(Y_{1},Y_{2})$ pair
$(3,1^{3}),(3,\\{2,1\\}),(1^{3},\\{2,1\\})$
Such degeneracy does not cause any problem in constructing the eigenstate of
$H$ for the following reasons.
i) The mother state $J^{+}_{-Y_{1}}J^{-}_{-Y_{2}}|P\rangle$ is uniquely
determined by the Young diagram, even for the degenerate $E$.
ii) Consider power expansion
$R(E)=\sum_{n=0}^{\infty}(\frac{1}{E_{Y_{1},Y_{2}}(P)-H_{0}}H_{I})^{n}.$
For an intermediate state.
$\displaystyle E_{Y_{1},Y_{2}}(P)-H_{0}$ $\displaystyle\sim
E_{Y_{1},Y_{2}}(P)-E_{Y^{{}^{\prime}}_{1},Y^{{}^{\prime}}_{2}}(P)$
$\displaystyle=E_{Y_{1}}+E_{Y_{2}}-E_{Y^{{}^{\prime}}_{1}}-E_{Y^{{}^{\prime}}_{2}}+2P(|Y_{1}|-|Y_{2}|-|Y^{{}^{\prime}}_{1}|+|Y^{{}^{\prime}}_{2}|)$
Since $|Y^{{}^{\prime}}_{1}|>|Y_{1}|$, $|Y^{{}^{\prime}}_{2}|<|Y_{2}|$ and
$|Y_{1}|-|Y^{{}^{\prime}}_{1}|+|Y^{{}^{\prime}}_{2}|-|Y_{2}|<0$ because of
strictly triangularity of $H_{I}$, so for a general value of $P$,
$\frac{1}{E_{Y_{1},Y_{2}}(P)-H_{0}}$ is not singular, and
$R(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle$ is well defined.
iii) The construction given above leads to the orthogonality of the state
$|P\rangle_{Y_{1},Y_{2}}$ for distinct $Y_{1},Y_{2}$ even for the degenerate
values of $E$, cf. eqs.(2,2,3).
iv) It can be proven that the eigenstate of $H$, constructed as in proposition
2, is actually the common eigenstate for all the conseved charges which
commute with $H$, with the mild assumption that all the conserved charges are
triangular in a similar way as $H$ is. Due to lack of space for the present
paper, we shall give a proof on this statement elsewhere.
Finally, we shall make a comment on the possible poles of $R(E)$ in the
complex $p-$plane. The $R(E)$ matrix elements is calculated based on the
following formula,
$\displaystyle|P\rangle_{Y_{1},Y_{2}}$
$\displaystyle=R(E)J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle\Omega_{Y_{1},Y_{2}}(P)$
$\displaystyle=\sum_{n=0}^{\infty}(\frac{1}{E_{Y_{1},Y_{2}}(P)-H_{0}}H_{I})^{n}J^{+}_{-Y_{1}}{J^{-}_{-Y_{2}}}|P\rangle\Omega_{Y_{1},Y_{2}}(P)\,.$
which always ends up with finite order perturbation because $H_{I}$ is
strictly triangular. We found, by the explicit calculations carried out so
far, that there is no pole in the finite $P$ complex plane. The poles in
$R(E)$ either cancels the zeros in $\Omega_{Y_{1},Y_{2}}(P)$ or simply cancels
by summing over all the relevant terms. Of course, this property is also the
necessary condition if $|{P}\rangle_{Y_{1},Y_{2}}$’s satisfy (1). Now the
general AFLT state can be written as
$\displaystyle|{P}\rangle_{Y_{1},Y_{2}}$ $\displaystyle=$
$\displaystyle\left\\{\Omega_{Y_{1},Y_{2}}(P)J_{-Y_{1}}^{+}J_{-Y_{2}}^{-}+\sum_{\begin{subarray}{|}|Y_{1}^{\prime}|=|Y_{1}|+1\\\
|Y_{2}^{\prime}|=|Y_{2}|-1\end{subarray}}C_{Y_{1},Y_{2}}^{Y_{1}^{\prime},Y_{2}^{\prime}}J_{-Y_{1}^{\prime}}^{+}J_{-Y_{2}^{\prime}}^{-}\right.$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{|}|Y_{1}^{\prime\prime}|=|Y_{1}|+2\\\
|Y_{2}^{\prime\prime}|=|Y_{2}|-2\end{subarray}}C_{Y_{1},Y_{2}}^{Y_{1}^{\prime\prime},Y_{2}^{\prime\prime}}J_{-Y_{1}^{\prime\prime}}^{+}J_{-Y_{2}^{\prime\prime}}^{-}$
$\displaystyle+$
$\displaystyle\left.\cdots+\sum_{|Y|=|Y_{1}|+|Y_{2}|}C_{Y_{1},Y_{2}}^{Y,\varnothing}J_{-Y}^{+}\right\\}|{P}\rangle,$
here $C_{Y_{1},Y_{2}}^{Y_{3},Y_{4}}$ is the transition coefficient which
measures the changing from the Young tableau vector $(Y_{1},Y_{2})$ to
$(Y_{3},Y_{4})$.
We have calculated those coefficients up to level 4, the explicit results(up
to level 3) are included in Appdix A. With the coefficients we calculated, one
can check that:
$\displaystyle Z_{bif}(\alpha|P^{\prime},\vec{X};P,\vec{Y})=$ (46)
$\displaystyle\sum_{(X_{1}^{\prime},X_{2}^{\prime}),(Y_{1}^{\prime},Y_{2}^{\prime})}\langle{P^{\prime}}|J^{-}_{X_{1}^{\prime}}J^{+}_{X_{2}^{\prime}}C_{X_{1},X_{2}}^{X_{1}^{\prime},X_{2}^{\prime}}V_{\alpha}C_{Y_{1},Y_{2}}^{Y_{1}^{\prime},Y_{2}^{\prime}}J_{-Y_{1}^{\prime}}J_{-Y_{2}^{\prime}}|{P}\rangle\,,$
holds true, thus (1) is verified. Here for simplicity, we have only verified
the cases without the incertions of the screening charges, i.e.
$P+P^{\prime}+\alpha=0$.
## 4 Conclusion and Perspective
The present work can be generalized in different ways. First, since the one
parameter Jack symmetric function is a special limit of the two parameter
Macdonald symmetric function, we expect that much of our work can be
generalized to the cases where Macdonald symmetric function plays a role. In
that case, we expect a similar relation to the NIC for 5d theory. Second, the
Calogero-Sutherland model is an integrable system. And consequently, Jack
symmetric function is the common eigenstate of the infinitely many commuting
charges which are deformed $W^{\infty}$ charges. And for the construction of
the AFLT states, the conserved charges are further deformed from those for the
Jack symmetric functions. The final construction should give the same results
as $I_{n}$ proposed in [7], which are constructed from integrable KdV
equations. We find in this case, the AFLT states remain to be the eigenstates
for all the conserved charges. However, it is desirable to have infinitely
many conserved charges constructed explicitly. Third, the reflection symmetry
studied in this paper is actually powerful enough to give a closed form for
the construction of the AFLT states. We shall present this result in our
future work. Another interesting idea related to our work is to consider the
Jack function as a perturbation away from the Schur function, we have found
that similar formalism applies [37]. Finally, it is very interesting to see
how we present the full pants diagram for the conformal blocks, comparing to
the one we have considered with one external leg.
## 5 Acknowledgments
We are grateful to Yi-hong Gao for detailed explanations on the the AGT
conjecture and the related topics.
This work is part of the CAS program “Frontier Topics in Mathematical Physics”
(KJCX3-SYW-S03) and is supported partially by a national grant NSFC(11035008).
## Appendix A Coefficients for AFLT States(up to level 3)
Now we give the explicit construction of the AFLT states up to level 3. The
transition coefficients $C_{Y_{1},Y_{2}}^{Y_{1}^{\prime},Y_{2}^{\prime}}$ are
defined as,
$C_{Y_{1},Y_{2}}^{Y_{1}^{\prime},Y_{2}^{\prime}}\equiv
R_{Y_{1},Y_{2}}^{Y_{1}^{\prime},Y_{2}^{\prime}}(E)\Omega_{Y_{1},Y_{2}}(P),\ \
\ \ C_{Y_{1},Y_{2}}^{Y_{1},Y_{2}}\equiv\Omega_{Y_{1},Y_{2}}(P)\,.$
Level 2 coefficients:
$C_{0,1^{2}}^{1,1}=C_{0,2}^{1,1}=-4b\left(1+b^{2}\right)P,\\\
C_{1,1}^{1^{2},0}=\frac{\left(1+b^{2}\right)(1+2bP)}{-1+b^{2}},\\\
C_{1,1}^{2,0}=-\frac{b^{2}\left(1+b^{2}\right)(1+2bP)}{-1+b^{2}},\\\
C_{0,1^{2}}^{12,0}=1+b^{2}\left(3+2b\left(b-\frac{2\left(1+b^{2}\right)P}{-1+b^{2}}\right)\right),\\\
C_{0,1^{2}}^{2,0}=\frac{4b^{3}\left(1+b^{2}\right)P}{-1+b^{2}},\\\
C_{0,2}^{2,0}=\frac{\left(1+b^{2}\right)P^{2}\left(-2+b^{2}+b^{4}+4bP\right)}{-1+b^{2}},\\\
C_{0,2}^{1^{2},0}=\frac{4b\left(1+b^{2}\right)P}{-1+b^{2}}$
Level 3 coefficients:
$C_{2,1}^{3,0}=\left\\{-\frac{b^{3}\left(1+b^{2}\right)(b+2P)\left(1+b^{2}+2bP\right)}{-2+b^{2}}\right\\},\\\
C_{2,1}^{\\{2,1\\},0}=\left\\{\frac{4\left(1+b^{2}\right)(1+bP)\left(1+b^{2}+2bP\right)}{-2+b^{2}}\right\\},\\\
C_{1^{2},1}^{1^{3},0}=\left\\{\frac{\left(1+b^{2}\right)(1+2bP)\left(1+b^{2}+2bP\right)}{-1+2b^{2}}\right\\},\\\
C_{1^{2},1}^{\\{2,1\\},0}=\left\\{-\frac{4b^{3}\left(1+b^{2}\right)(b+P)\left(1+b^{2}+2bP\right)}{-1+2b^{2}}\right\\},\\\
C_{1,2}^{2,1}=\left\\{-\frac{2b^{2}\left(1+b^{2}\right)(1+2bP)\left(-1+b^{2}+2bP\right)}{-1+b^{2}}\right\\},\\\
C_{1,2}^{1^{2},1}=\left\\{\frac{4b\left(1+b^{2}\right)P(1+2bP)}{-1+b^{2}}\right\\},\\\
C_{1,2}^{3,0}=\left\\{\frac{b^{3}\left(1+b^{2}\right)\left(b+b^{3}+4P\right)\left(-1+b^{2}+2bP\right)}{2-3b^{2}+b^{4}}\right\\},\\\
C_{1,2}^{\\{2,1\\},0}=\left\\{-\frac{4\left(1+b^{2}\right)^{2}(1+2bP)(-1+2b(b+P))}{2-5b^{2}+2b^{4}}\right\\},\\\
C_{1,2}^{1^{3},0}=\left\\{\frac{4b\left(1+b^{2}\right)P(1+2bP)}{1-3b^{2}+2b^{4}}\right\\},\\\
C_{1,1^{2}}^{2,1}=\left\\{-\frac{4b^{4}\left(1+b^{2}\right)P(b+2P)}{-1+b^{2}}\right\\},\\\
C_{1,1^{2}}^{1^{2},1}=\left\\{-\frac{2b\left(1+b^{2}\right)(b+2P)\left(-1+b^{2}-2bP\right)}{-1+b^{2}}\right\\},\\\
C_{1,1^{2}}^{3,0}=\left\\{\frac{4b^{6}\left(1+b^{2}\right)P(b+2P)}{2-3b^{2}+b^{4}}\right\\},\\\
C_{1,1^{2}}^{\\{2,1\\},0}=\left\\{\frac{4b^{3}\left(1+b^{2}\right)^{2}(b+2P)\left(-2+b^{2}-2bP\right)}{2-5b^{2}+2b^{4}}\right\\},\\\
C_{1,1^{2}}^{1^{3},0}=\left\\{-\frac{\left(1+b^{2}\right)\left(-1+b^{2}-2bP\right)\left(1+b^{2}+4b^{3}P\right)}{1-3b^{2}+2b^{4}}\right\\},\\\
C_{0,3}^{1,2}=\left\\{-6b\left(1+b^{2}\right)P(-1+2bP)\right\\},\\\
C_{0,3}^{2,1}=\left\\{\frac{6b\left(1+b^{2}\right)P\left(-4+b^{2}+b^{4}+4bP\right)}{-1+b^{2}}\right\\},\\\
C_{0,3}^{1^{2},1}=\left\\{-\frac{12b\left(1+b^{2}\right)P(-1+2bP)}{-1+b^{2}}\right\\},\\\
C_{0,3}^{3,0}=\left\\{-\frac{\left(1+b^{2}\right)\left(12+b\left(b\left(1+b^{2}\right)\left(-8+b^{2}+b^{4}\right)+12\left(-3+b^{2}+b^{4}\right)P+24bP^{2}\right)\right)}{2-3b^{2}+b^{4}}\right\\},\\\
C_{0,3}^{1^{3},0}=\left\\{-\frac{12b\left(1+b^{2}\right)P(-1+2bP)}{1-3b^{2}+2b^{4}}\right\\},\\\
C_{0,3}^{\\{2,1\\},0}=\left\\{\frac{12b\left(1+b^{2}\right)P\left(-5+3b^{2}+2b^{4}+6bP\right)}{2-5b^{2}+2b^{4}}\right\\},\\\
C_{0,1^{3}}^{1,1^{2}}=\left\\{6b^{2}\left(1+b^{2}\right)(b-2P)P\right\\},\\\
C_{0,1^{3}}^{2,1}=\left\\{-\frac{12b^{4}\left(1+b^{2}\right)(b-2P)P}{-1+b^{2}}\right\\},\\\
C_{0,1^{3}}^{1^{2},1}=\left\\{\frac{6b\left(1+b^{2}\right)\left(-1+b^{2}(-1+4b(b-P))\right)P}{-1+b^{2}}\right\\},\\\
C_{0,1^{3}}^{3,0}=\left\\{\frac{12b^{6}\left(1+b^{2}\right)(b-2P)P}{2-3b^{2}+b^{4}}\right\\},\\\
C_{0,1^{3}}^{1^{3},0}=\left\\{-\frac{1}{1-3b^{2}+2b^{4}}\left(1+b^{2}\right)\left(1+b^{2}(2+b(12P+b(-7+4b(b(-2+3(b-2P)(b-P))+3P))))\right)\right\\},\\\
C_{0,1^{3}}^{\\{2,1\\},0}=\left\\{-\frac{12b^{3}\left(1+b^{2}\right)P\left(-2+b^{2}\left(-3+5b^{2}-6bP\right)\right)}{2-5b^{2}+2b^{4}}\right\\},\\\
C_{0,\\{2,1\\}}^{1,1^{2}}=\left\\{\frac{2b^{2}\left(-2+b^{2}\right)\left(1+b^{2}\right)(b-2P)P}{-1+b^{2}}\right\\},\\\
C_{0,\\{2,1\\}}^{1,2}=\left\\{-\frac{2b\left(-1+b^{2}+2b^{4}\right)P(-1+2bP)}{-1+b^{2}}\right\\},\\\
C_{0,\\{2,1\\}}^{2,1}=\left\\{\frac{4b\left(1+b^{2}\right)^{2}P\left(-1+b^{2}+2bP\right)}{-1+b^{2}}\right\\},\\\
C_{0,\\{2,1\\}}^{1^{2},1}=\left\\{\frac{4b\left(1+b^{2}\right)^{2}P\left(-1+b^{2}-2bP\right)}{-1+b^{2}}\right\\},\\\
C_{0,\\{2,1\\}}^{3,0}=\left\\{-\frac{2b^{3}\left(1+b^{2}\right)P\left(-3+2b\left(b+b^{3}+3P\right)\right)}{2-3b^{2}+b^{4}}\right\\},\\\
C_{0,\\{2,1\\}}^{\\{2,1\\},0}=\left\\{-\frac{\left(1+b^{2}\right)\left(4-17b^{4}+4b^{8}-2b\left(4+3b^{2}+3b^{4}+4b^{6}\right)P-36b^{4}P^{2}\right)}{2-5b^{2}+2b^{4}}\right\\},\\\
C_{0,\\{2,1\\}}^{1^{3},0}=\left\\{\frac{2b\left(1+b^{2}\right)\left(-2+b^{2}(-2+3b(b-2P))\right)P}{1-3b^{2}+2b^{4}}\right\\}$
In the above expressions, 0 labels $\\{\varnothing\\}$.
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* [28] H. Itoyama and T. Oota, “Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model,” Nucl. Phys. B 838, 298 (2010) [arXiv:1003.2929 [hep-th]].
* [29] A. Belavin and V. Belavin, “AGT conjecture and Integrable structure of Conformal field theory for c=1,” Nucl. Phys. B 850, 199 (2011) [arXiv:1102.0343 [hep-th]].
* [30] V. Belavin and B. Feigin, “Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories,” arXiv:1105.5800 [hep-th].
* [31] N. Dorey, S. Lee and T. J. Hollowood, “Quantization of Integrable Systems and a 2d/4d Duality,” arXiv:1103.5726 [hep-th].
* [32] H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, “Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String,” arXiv:1008.0574 [hep-th].
* [33] A. Mironov, A. Morozov and S. Shakirov, “On ’Dotsenko-Fateev’ representation of the toric conformal blocks,” J. Phys. A 44, 085401 (2011) [arXiv:1010.1734 [hep-th]].
* [34] A. Mironov, A. Morozov and S. Shakirov, “Conformal blocks as Dotsenko-Fateev Integral Discriminants,” Int. J. Mod. Phys. A 25, 3173 (2010) [arXiv:1001.0563 [hep-th]].
* [35] J. f. Wu, “Note on refined topological vertex, Jack polynomials and instanton counting,” arXiv:1012.2147 [hep-th].
* [36] J. F. Wu, Y. Y. Xu and M. Yu, “Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors,” arXiv:1107.4234 [hep-th].
* [37] J. F. Wu and M. Yu, work in progress.
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|
arxiv-papers
| 2011-07-24T18:26:03 |
2024-09-04T02:49:20.882718
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bao Shou, Jian-Feng Wu, Ming Yu",
"submitter": "Ming Yu",
"url": "https://arxiv.org/abs/1107.4784"
}
|
1107.4820
|
# Role of water in Protein Aggregation and Amyloid Polymorphism
D. Thirumalai thirum@umd.edu Govardhan Reddy [ John E. Straub [
###### Abstract
A variety of neurodegenerative diseases are associated with amyloid plaques,
which begin as soluble protein oligomers but develop into amyloid fibrils. Our
incomplete understanding of this process underscores the need to decipher the
principles governing protein aggregation. Mechanisms of in vivo amyloid
formation involve a number of co-conspirators and complex interactions with
membranes. Nevertheless, understanding the biophysical basis of simpler in
vitro amyloid formation is considered important for discovering ligands that
preferentially bind regions harboring amyloidogenic tendencies. The
determination of the fibril structure of many peptides has set the stage for
probing the dynamics of oligomer formation and amyloid growth through computer
simulations. Most experimental and simulation studies, however, have been
interpreted largely from the perspective of proteins: the role of solvent has
been relatively overlooked in the formation of oligomer assembly to
protofilaments and amyloid fibrils.
In this Account, we provide a perspective on how interactions with water
affect folding landscapes of amyloid beta (A$\beta$) monomers, oligomer
formation in the A$\beta_{16-22}$ fragment, and protofilament formation in a
peptide from yeast prion Sup35. Explicit molecular dynamics simulations of
these systems illustrate how water controls the self-assembly of higher order
structures, providing a structural basis for understanding the kinetics of
oligomer and fibril growth. Simulations show that monomers of A$\beta$
peptides sample a number of compact conformations. The formation of
aggregation-prone structures (N∗) with a salt bridge, strikingly similar to
the structure in the fibril requires overcoming high desolvation barrier. In
general, sequences for which N∗ structures are not significantly populated are
unlikely to aggregate.
Oligomers and fibrils generally form in two steps. First, water is expelled
from the region between peptides rich in hydrophobic residues (for example,
A$\beta_{16-22}$), resulting in disordered oligomers. Then the peptides align
along a preferred axis to form ordered structures with anti-parallel
$\beta$-strand arrangement. The rate-limiting step in the ordered assembly is
the rearrangement of the peptides within a confining volume.
The mechanism of protofilament formation in a polar peptide fragment from the
yeast prion, in which the two sheets are packed against each other and create
a dry interface, illustrates that water dramatically slows down self-assembly.
As the sheets approach each other, two perfectly ordered one-dimensional water
wires form. They are stabilized by hydrogen bonds to the amide groups of the
polar side chains, resulting in the formation of long-lived metastable
structures. Release of trapped water from the pore creates a helically twisted
protofilament with a dry interface. Similarly, the driving force for addition
of a solvated monomer to a preformed fibril is the release of water; the
entropy gain and favorable interpeptide hydrogen bond formation compensate for
loss in entropy in the peptides.
We conclude by offering evidence that a two-step model, similar to that
postulated for protein crystallization, must also hold for higher order
amyloid structure formation starting from N∗. Multiple N∗ structures with
varying water content results in a number of distinct water-laden polymorphic
structures. Water plays multifarious roles in all of these protein
aggregations. In predominantly hydrophobic sequences, water accelerates fibril
formation. In contrast, water-stabilized metastable intermediates dramatically
slow down fibril growth rates in hydrophilic sequences.
University of Maryland]Biophysics Program, Institute for Physical Science and
Technology, Department of Chemistry and Biochemistry, University of Maryland,
College Park, MD 20742 Boston University]Department of Chemistry, Boston
University, Boston, MA 02215
## 1 Introduction
Protein aggregation leading to amyloid fibril formation is linked to a number
of neurodegenerative diseases 1, 2 although in some instances their formation
is also beneficial 3. Understanding how misfolded proteins polymerize into
ordered fibrils, which universally have a characteristic cross
$\beta$-structure 4, may be important in our ability to intervene and prevent
their formation. The physical basis of protein aggregation involving a cascade
of events that drive a monomer to a fibrillar structure is complicated because
of interplay of a number of energy and time scales governing amyloid
formation. In addition, a number of other factors, such as protein
concentration, sequence of proteins, and environmental conditions (pH,
presence of osmolytes, temperature) affect various kinetic steps in distinct
ways, thus making it difficult to describe even in vitro protein aggregation
in molecular terms. Despite these complexities significant advances have been
made, especially in getting structures of peptide amyloids and models for
amyloid fibrils from A$\beta$ and fungal prion proteins. The availability of
structures have made it possible to undertake molecular dynamics simulations,
which have given insights into the role water plays in oligomer formation as
well as assembly and growth of amyloid fibrils.
It has long been appreciated that water plays a major role in the self-
assembly of proteins 5 in ensuring that hydrophobic residues are
(predominantly) sequestered in protein interior. In contrast, the effects of
water on protein aggregation is poorly understood. Indeed, almost all studies
(experimental and computer simulations) on amlyoid assembly mechanisms have
been largely analyzed using a protein centric perspective. The situation is
further exacerbated by experimental difficulties in directly monitoring water
activity during the growth process. Here, we provide a perspective on the role
water plays in protein aggregation by synthesizing results from molecular
dynamics (MD) simulation studies. Briefly our goals are: (i) Describe how
water-mediated interactions affect the energy landscape of monomers and drive
oligomer formation in A$\beta$ peptides. (ii) The key role water plays in late
stages of fibril growth is described by large variations in the sequence
dependent mechanism of self-assembly to $\beta$-sheet rich amyloids. (iii) We
use results of recent MD simulations and concepts in protein crystallization
to provide scenarios for the role water plays in polymorphic amyloid
structures.
## 2 Water influences the energy landscape of A$\beta$ monomers
Although there are several plausible scenarios for the fate of monomer in the
conversion to fibrils the process invariably commences by populating misfolded
conformations (an ensemble of $N^{*}$ structures in Fig. 1) by denaturation
stress or thermal fluctuations. Thus, the pathways to soluble and mobile
oligomer formation and subsequent polymerization depend on the nature of
$N^{*}$, and hence the folding landscape of monomers. Ensemble of $N^{*}$ (or
toplogically related) conformations can collide to populate low order
oligomers with differing molecular structures that contain varying number of
water molecules. Once the oligomers exceed a critical size they nucleate and
form protofilaments and eventually mature fibrils with differing morphologies
(Fig. 1). Thus, the spectra of the states sampled by the monomers can provide
insight into the tendency of specific sequences to form amyloid structures.
The relevance of N∗ in affecting fibril morphology and growth kinetics
suggested in 6 has been confirmed in a number of studies 7, 8, 9, 10.
For A$\beta$ peptides and other sequences for which exhaustive MD simulations
can be performed it is now established that typically the polypeptide chain
samples a large number of conformations belonging to distinct basins, and the
aggregation-prone N∗structures are separated from the lowest free energy
conformations by a free energy barrier. Two extreme scenarios, which follow
from the energy landscape perspective of aggregation 11, 6, 12, can be
envisioned. According to Scenario I, which applies to A$\beta$-peptides and
transthyretin, fibril formation requires partial unfolding of the native state
[30] or partial folding of the unfolded state. Both events, which involve
crossing free energy barriers lead to the transient population of an ensemble
of assembly-competent structures N∗. According to Scenario II, which describes
aggregation of mammalian prions 13, 14 the ensemble of N∗ structures has a
lower free energy than the structures in the native state ensemble thus making
the folded (functional state) state metastable 15. In both scenarios water-
mediated interactions are responsible for erecting free energy barriers
between the ground state and one of the N∗ states. In the case of mammalian
prions (PrP) MD simulations 14 and complementary structural analysis 13 showed
that the structured C-terminus must undergo a conformational transition to the
more stable N∗ structures, which can self-assemble to form self-propagating
PrPsc structures. The need to partially unfold the C-terminal regions results
in a substantial barrier between the cellular form of PrP and the aggregation
prone N∗.
The ensemble of conformations with the lowest free energy in A$\beta_{10-35}$
and the longer A$\beta_{1-40}$ monomer fluctuate16, 17, 18 among a number of
compact structures, whereas in the fibrillar state they adopt a $\beta$-sheet
structure. Solid-state NMR-based structural model of the fibrils of
A$\beta_{1-40}$ is characterized by V24GSN27 turn and intrapeptide salt-bridge
between D23 and K28. Such a structural motif, when stacked in parallel, that
satisfies the amyloid self-organization principle 7, 12 according to which
fibril stability is enhanced by maximizing the number of hydrophobic and
favorable electrostatic interactions (formation of salt-bridges and hydrogen
bonds) 7. Given that the structural precursors of the fibrils manifest
themselves in soluble dynamically fluctuating oligomers, it is natural to
expect that the D23-K28 salt bridge must play an important role in the early
events of self-association of A$\beta$-proteins. Indeed, molecular dynamics
simulations of A$\beta_{9-40}$ fibrils suggest that partially solvated D23-K28
salt bridges appear to be arranged as in a one-dimensional ionic crystal 19.
However, extensive MD simulations, have shown that the formation of a stable
structure with an intact D23-K28 salt bridge and the VGSN turn is highly
improbable in the monomer 7. A natural implication is that overcoming the
large barrier to desolvation of D23 and K28, which can only occur at finite
peptide concentration or by rare flutuations, must be an early event in the
formation of higher order structures.
The folding landscape of A$\beta_{10-35}$ can be partitioned into four basins
of attraction 7. The ensemble of structures with intact salt-bridge, a motif
that resembles the one found in the fibril, is rarely populated. There is a
broad distribution of compact structures stabilized by a variety of
intramolecular interactions. The three most highly populated structures (Fig.
2) are stabilized by solvation of charged residues and by hydrophobic
interactions in a locally dry environment. Snapshots of the
A$\beta_{10-35}$-protein, in which the D23-K28 salt bridge is absent (Fig. 2),
show that the two side chains are separated by three and two solvation shells,
respectively. Clearly, a stable intramolecular salt bridge can only form if
the interning water molecules can be expelled, which involves overcoming a
large desolvation barrier. Large distance separation between D23 and K28
observed in the first structure in Fig. 2 is due to the interposed side chain
of V24 between D23 and K28. This results in a hydrophobic contact between V24
and the aliphatic portion of the K28 side chain. Competition between the
electrostatic D23-K28 and the hydrophobic V24-K28 interactions stabilizes the
turn in the region V24-N27. The last structure in Fig. 2A shows a D23-K28
water-mediated salt-bridge structure in which one water molecule makes
hydrogen bonds with both the D23 and K28 side chains. The lifetime of hydrogen
bonds of the solvated D23 and K28 is nearly three times ($\sim$ 2.4 ps) as
long as bulk water ($\sim$ 0.8 ps). Thus, the side chains of D23 and K28 make
stronger contacts with water than water with itself, indicating that the
desolvation of D23 and K28 is an activated process. The barrier can be reduced
by creating monomers containing preformed D23-K28 salt bridge. Indeed,
experiments show that aggregation of monomers containing a lactam-bridge
between D23-K28 aggregate $\approx 1000$ times faster than the wild type20.
This finding has been rationalized in terms of a reduction in the free energy
barrier between low free energy structures without D23-K28 salt bridge and N∗
structures (ones in which these residues are in proximity) in chemically
linked monomers21.
## 3 Dynamics of Oligomer Formation
The first MD study on interacting peptides 22 focused on the mechanism of
assembly of peptide fragment KLVFFAE [A$\beta_{16-22}$]n ($n$ = 2 and 3),
which contains the central hydrophobic cluster LVFFA (CHC) flanked by the
N-terminal positively charged residue (Lys) and the C-terminal negatively
charged residue (Glu). The peptides form antiparallel $\beta$-sheet structure
in the fibril as assessed by solid state NMR and molecular dynamics
simulations. Somewhat surprisingly, MD simulations showed that even in a
trimer the peptides, which are unstructured as monomers 22, are extended and
arranged in antiparallel fashion. Such an arrangement ensures formation of the
largest number of inter peptide salt-bridges in addition to maximizing the
number of hydrophobic contacts between the peptides, which accords well with
the amyloid-organization principle. Thus, the ordered structure, which
undergoes substantial conformational fluctuations because of finite size,
should be viewed as a ”nematic” droplet in which the strands are aligned along
a common director. Explicit mapping using analysis of MD trajectories showed
that the energy landscape (A$\beta_{16-22}$)2) has nearly six minima 23
including one in which the peptides are antiparallel to each other. However,
the number of minima decrease as $n$ increases and approached a critical size
24.
The mechanism of oligomer formation revealed that the salt-bridges gives rise
to orientational specificity, which renders the antiparallel arrangement
stable. However, the driving force for oligomerization, which initially
produces an ensemble of disordered aggregates, is the hydrophobic interaction
between various residues in the CHC. The early formation of disordered
structures in LVFFAE implies that water must be expelled relatively quickly
upon interaction between the peptides. It was found that at very early stages
the number of water molecules is substantially reduced from the crevices
between the peptides which implies that the ordered nematic droplet, which is
coated on the outside with water, is essentially dry. The first MD studies22
showed that expulsion of water in sequences with a large number of bulky
hydrophobic residues must be an early event, and hence cannot be the rate
limiting step in the ordered assembly of such peptides.
The growth dynamics of oligomers of A$\beta_{16-22}$ peptides further showed
that water is not present in the interior. Simulations of the reaction
(A$\beta_{16-22}$)n-1 \+ A$\beta_{16-22}\leftrightarrow$ (A$\beta_{16-22}$)n
done by adding an unstructured solvated monomer to a preformed oligomer it was
shown that the monomer adds onto the larger particle by a dock-lock mechanism
24. In the first docking step the solvated monomer attaches to the oligomer
rapidly by essentially a diffusive process. In the much slower lock step the
peptide undergoes conformational transitions from a random coil to a
$\beta$-strand conformation, and adopts a conformation that is commensurate
with the structures in the nematic droplet. Interestingly, the interactions
that stabilize the larger oligomers also do not involve water molecules. In
addition, there are very few stable hydrogen bonds that persist between the
peptides, which implies that the higher order oligomer structures are
stabilized largely by interpeptide side-chain contacts. As is the case for
trimers, the antiparallel orientation is guaranteed by the formation of the
salt bridge between K16 from one peptide and E22 from another. Taken together,
these results show that the driving force for oligomerization is the favorable
interpeptide association between residues belonging to the CHC. Although the
role of side chains is a major determining factor in oligomer formation in all
peptides it should be stressed that expulsion of water from the interior of
oligomers in the early stages is highly sequence dependent.
## 4 Water release promotes protofilament formation and amyloid fibril growth
Studies of protein crystallization25 remind us that a major driving force for
crystal formation is the release of water molecules from the hydration layer
upon formation of contacts between protein molecules. A number of
experimental, simulation, and theoretical studies of proteins, which
crystallize with intact folded structures have shown that even in cases when
enthalpy gain upon crystallization is small, it is more than compensated by
depletion of water molecules around the proteins. Crystallization results in a
loss in translational and rotational entropy and the vibrational degrees of
freedom associated with the ordered structure only partly compensate for the
loss. However, the total entropy change, $S_{T}$, ( = $\Delta
S_{protein}+\Delta S_{water}$ where $\Delta S_{protein}$ and $\Delta
S_{water}$ are the changes in protein and water entropy, respectively) is
positive, and is explained by water release mechanism. Water is structured
around the surface of folded proteins with the thickness of hydration layer
being $\sim$ 7Å (Fig. 1). Water molecules in the structured layer are in
dynamic equilibrium with the bulk water. Upon crystallization, the structured
water (typically $\sim$ (5-30)) around the protein is released into the bulk,
which contributes to an increase in $S_{T}$, and has been suggested as a major
thermodynamic driving force for protein crystallization 25. Thus, even if the
polymerization process is endothermic, increase in $S_{T}$ can drive gelation
and crystallization. Self-assembly of Tobacco Mosaic Virus (TMV) 26, 27
provided an early example of water-release mechanism. The endothermic
polymerization reaction28 involving TMV goes to completion by the water
release mechanism leading to an increase in $S_{T}$ of the system. Using the
quartz spring balance experiments27, it is demonstrated that 96 moles of
hydrated water is released per mole of the TMV protein trimer which is shown
to be sufficient to increase the overall entropy of the system and drive the
polymerization reaction. There are a number of experimental studies involving
protein crystallization, which are nearly quantitatively explained by increase
in $\Delta S_{water}$.
Although not discussed explicity, expulsion of water leading to increase in
solvent entropy, which has been observed in oligomer formation of
A$\beta$-peptides 22, 24, 29, 30, is also a major driving force for fibril
formation. Here, it is important to consider both sequence effects and account
for conformational changes that occur. For example, in the case of
A$\beta_{16-22}$ the random coil structure (small size) expands to form
$\beta$-strand (larger size), which is unfavorable not only because it
involves solvent exposure of hydrophobic residues but also results in
reduction in conformational entropy. In this case both release of water and
favorable side chain contacts stabilize the oligomers. Aggregation of Sup35
yeast prion protein 31, the N terminal region of the protein rich in polar
side groups (glutamines and asparagines) participate in the formation of
collapsed disordered aggregates. Simulations have shown, somewhat
surprisingly, that water is a poor solvent (we adopt the terminology used in
polymer physics) for the polypeptide backbone and collapsed disordered
polyglutamine chains are thermodynamically favored 32, 33, 34. To decrease the
interfacial tension between the water and the backbone secondary amides,
polyglutamine becomes compact with the side chain amides solvating the
backbone amides. Thus, different driving forces are involved in the initial
collapse of protein molecules leading to disordered oligomers and the
mechanism depends on the protein sequence. The common universal driving force
is predominantly the release of the structured water 35 around the protein
into the bulk as oligomers, protofilament, and fibrils form. Because the
strength of interpeptide interactions and solvent mediated forces are
sequence-dependent, time scales for fibril formation also can vary greatly
depending on the sequence even under identical external conditions 36.
Two recent simulations on the growth of fibrils (assumed to occur by
incorporating one monomer at a time 30) and self assembly of protofilament 36
vividly illustrate water release as a key factor. Addition of a Sup35 peptide
(GNNQQNY) to an amyloid fibril reveal 30 that the release of the hydrating
water molecules into the bulk and peptide addition to the fibril occur
simultaneously (Fig. 3). The number of water molecules, $N_{W}(t)$ decreases
as the solvated monomer interacts with the underlying fibril lattice (Fig.
3A). As the locking reaction progresses, water molecules in the vicinity of
the monomer in the fibril that are closest to the solvated monomer are
released (Fig. 3B). Comparison of the growth dynamics associated with the
A$\beta$ peptides and Sup35 shows that the dehydration process is dynamically
more cooperative for the polar sequence 30, which emphasizes the role of
sequence discussed above. Fluctuations in the number of water molecules
coincide with the locking events (Fig. 3). The largest fluctuations in the
number of water molecules near the locking monomer, $N_{W}^{L}(t)$ and the
solvent-exposed monomer in the fibril, $N_{W}^{F}(t)$ occur precisely when the
monomer completely locks onto the crystal cooperatively (Fig. 3 A). The
coincidence of the locking step and dehydration is also reflected in the
sharp, decrease in the water content in the zipper region of the Sup35 crystal
(Fig. 3B), which occurs in two well-separated stages. The number of water
molecules, $N_{W}^{Z}(t)$ decreases abruptly from 8 to 2 as the docking is
initiated, and finally goes to zero as the locking process is complete (Fig.
3B). These observations show that dehydration leading to release of ”bound”
water, resulting in the formation of the dry zipper region must be taken into
account in estimating free energy changes that occur upon amyloid fibril
growth.
In a recent study we predicted that there must be large variations (exceeding
a factor of over one thousand) in the time needed for self-assembly of
protofilaments between hydrophobic and polar sequences because of the entirely
different roles water plays in their formation 36. The barrier to the release
of bound water around the polar residues should be high due to the favored
interactions between the polar side chains and water compared to hydrophobic
side chains. As a consequence, protofilaments comprised of polar sequences
must take much longer to form than ones made of hydrophopbic residues. These
expectations were borne out in MD simulations 36 contrasting the role of water
in the protofilament formation from peptides with polar and non-polar
residues. Water forms spontaneously meta-stable ordered one-dimensional wires
in the pores of the protofilaments during the assembly of the $\beta$-sheets
of GNNQQNY preventing the sheets to associate completely. The water wires are
stabilized by the hydrogen bonds with the amide groups in the side chains of
asparagines and glutamines and delay the protofilament formation. The gain in
entropy due to water release can be obtained by comparing the difference in
the energies (obtained from MD simulations36) between the metastable and
stable structures. We find that entropy gains per water molecule released is
$\approx$ 6 cal/mole.K, which is similar to the value estimated from protein
crystallization experiments25.
There are predominantly two major routes to assembly of $\beta$-sheets (Fig.
4). In one of them spontaneously formed nearly perfectly ordered one-
dimensional water from the pore is released into the bulk resulting in
$\beta$-sheet association and protofilament formation. Alternatively, when
fluctuations lead to misalignment in the orientation of the $\beta$-sheets,
water release occurs by leakage through the sides. In such a pathway the
sheets are packed against each other with orientational defect, and could
represent one of the polymorphic structures. In contrast, the assembly of the
$\beta$-sheets of GGVVIA, rich in hydrophobic groups occurs rapidly, and the
water in between the sheets is eliminated concurrently as the $\beta$-sheets
associate with one another 36. The contrasting behavior observed in the
protofilament assembly observed in hydrophobic and polar sequences illustrates
the distinct role of water. In the former case the driving force forming a
protofilament with a dry interior is the hydrophobic interactions. However, if
the amyloid forming sequence is hydrophilic then water release serves as a
substantial driving force. In this case water is a surrogate hydrogen bond
former, upon release of the trapped water, protofilament assembly is
completed.
Similarly, simulations 29 of association between preformed $\beta$-sheets in
A$\beta_{16-22}$ showed that in some of the trajectories water is expelled
early before assembly. In other trajectories, the two processes are observed
to be coincident. The predominant interactions that mediate protofilament
formation are hydrophobic with interactions involving Phe playing a major
role, as was previously shown in the context of oligomer formation. In both
cases water release provides the needed impetus for self-assembly. The
simulations also rationalize experiments37, which showed that the rate of
fibril formation increases significantly on reducing the hydration of
aggregating peptide molecules. It was found that aggregation rate of
A$\beta_{16-22}$ is largest when stabilized in reverse AOT micells containing
the least amount of free water molecules.
## 5 Is Water part of polymorphic structures?
The consequences of misfolding to multiple conformations with subsequent
aggregation into distinct infectious states with differing phenotypes (the so
called strain phenomenon) has been established in prion disorders and A$\beta$
peptides. Originally found in the context of wasting diseases and mammalian
prions, strain phenotypes, which grow from the same protein but lead to
different heritable states, are found even in peptide fibrils and amyloids
grown from A$\beta$-peptides. In general, amyloid fibrils show polymorphism
both in the mature structure38, 39, 40, 41, and is also manifested in
protofilaments 40. Various structures differ in side chain packing, water
content, hydrogen bond networks (Fig. 1) or in the quaternary structure 39,
41, 38. Polymorphism in amyloid fibrils also forms the basis of strain
phenomena in prion protein42. A single prion protein with multiple infectious
conformations, one for each strain, gives rise to distinct phenotypes and is
also inheritable 43. Although polymorphism is widely observed in amyloid
fibrils, the biophysical basis for their formation is lacking.
It is likely that trapped water molecules is part of the observed polymorphic
structures. The rationale for such a suggestion is based on the energy
landscape perspective of protein aggregation (Fig. 1), which provides a
plausible connection to the strain phenotypes that have been extensively
studied especially in yeast prion biology. At what stage of the growth of
fibrils is a particular strain ‘encoded’ in the structure? The suggestion that
the N∗ structures are aggregation prone implies that the strain phenotypes may
be encoded in the monomer structures or low order oligomers. We speculate that
the various N∗ structures can form oligomers with different structures, which
can subsequently lead to structurally distinct fibrils.
It is also clear from MD simulations 22, 24, 7 that the pre-nucleus
structures, which we propose are candidates for encoding polymorphism, are
water-laden. Hence, it follows that the distinct mature fibrils must contain
discrete number of water molecules. A number of studies provide evidence for
our proposal. Formation of water channels near the salt-bridge (D23-K28) has
been observed in simulations 19 of a solid-state NMR-derived structural model.
In the resulting structure, which is a variant of the one proposed using
experimental constraints, the buried salt bridge between D23 and K28 are
arranged in a periodic manner along the fibril axis. Confined water molecules
solvate the salt bridge, which is interestingly reminiscent of high free
energy conformations sampled by A$\beta_{10-35}$ monomer (see the last
structure in Fig. 2A). More recently, a different morphology for
A$\beta_{1-40}$ has been proposed using 2D IR spectroscopy 44. It was found
that water molecules (roughly 1.2 per monomer) are trapped in A$\beta_{1-40}$
fibrils. However, the finding that water molecules are trapped in the
hydrophobic pocket (L17, V18, L34, and V36) that interact with the amide
backbone of L17 and L34 is a surprise. There are two possible explanations for
these findings. If mobile water molecules are not part of the fully mature
fibrils, it is likely that the fibril structures with trapped waters are
metastable. If this were the case then we could argue that on much longer time
scales the trapped water molecules would migrate closer to the charged
residues and populate a structure similar to that found in MD simulations 19.
Alternatively, it is possible that these structures represent a distinct
polymorphic fibril structure. We surmise that other proposed structures for
A$\beta$-peptides must contain discrete number of water molecules trapped in
the fibril interior, and hence must be part of amyloid polymorphism.
The scenarios of fibril formation (Fig. 1) and lessons from protein
crystallization provide a physical picture for polymorphism. It is firmly
believed that crystallization generally and protein crystallization in
particular occurs in two steps 25, 45. In the first step, fluctuations produce
droplets that are rich in proteins leading to structures that are disordered.
In the second stage rearrangement of the structures within the droplet
produces ordered structures which grow by incorporating one monomer at a time.
Globally a similar mechanism qualitatively explains amyloid formation (Fig. 1)
46, 47, 48, 49, 50, 51. The first step involves formation of disordered
oligomers which produces regions that are protein rich droplets. In contrast
to protein crystallization in which proteins are folded, the N∗ structures in
the droplet could contain varying number of water molecules that may be
embedded in the mature fibrils. Once the droplet size becomes large enough (by
collisions with smaller droplets or by monomer addition) they produce distinct
fibrillar structures, which differ not only in inter protein interactions but
also in the content of water. The two-step growth mechanism, which is
reminiscent of nucleated conformational conversion picture 52, differs from
the traditional nucleation mechanism because growth occurs within the liquid-
like disordered droplets that are protein rich. As a result, morphologies
which nucleate more frequently dominate fibril formation rather than ones
which are thermodynamically more stable. Thus, the dominant fibril morphology
emerges from those N∗ structures that minimize surface energies in the protein
rich droplets. It also follows that distinct strain formation might be under
kinetic control 53 .
## 6 Conclusions
Naturally occurring peptides and proteins that form $\beta$-amyloids are
wonderful systems that can be used to study self-assembly of higher order
structures and hydration dynamics. Extrapolation of such biophysical studies
to what transpires physiologically is often fraught with difficulties. For
example, damage to synapses in Alzheimer’s disease is not caused solely by
oligomers of A$\beta$-peptides whose production is a complicated process
involving other enzymes. There are other culprits (one or more kinases) whose
interaction with A$\beta$-oligomers apparently play a significant role in
synapse impairment. Thus, bridging the gap between in vitro and in vivo
studies involve protein-protein recognition, which could also be mediated by
water in different ways.
From a biophysical perspective characterizing the nature of fluctuations that
promote regions that are rich in N∗ peptides (Fig. 1), which is a precursor to
growth of ordered $\beta$-amyloids, as the protein concentration is lowered
from $\sim$ mM (used in computer simulations) to $\sim\mu$M (needed to grow
fibrils in the laboratory) to $\sim$ nM (found in in vivo) conditions is a
challenging problem. Molecular dynamics simulations that probe water-mediated
interactions and the associated dynamics in mesoscale droplets containing a
number of amyloidogenic species will go a long way in our ability to describe
self-assembly of $\beta$-amyloids. In such confined spaces water alters both
hydrophobic and electrostatic interactions substantially 54, 55. Indeed,
amyloid fibrils, which can be can be pictured as water filled nanotubes 35,
are great systems to probe the properties of confined water. In these systems
hydrophobic or hydrophilic environment for the confined water can be
controlled using mutations, which naturally changes water density in the
pores. It is clear from this brief perspective that many facets of amyloid
growth, driven by context dependent interactions involving water and by the
peculiarities of confined water, remain to be explored using carefully planned
simulations and experiments. Finally, it should be noted that lessons from
such simulations can and should be incorporated into simpler models and
theories as summarized in recent reviews 12, 56, 57.
Acknowledgements This work was supported by a generous grant from the National
Institutes of Health (GM076688-08).
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Figure Captions:
Fig. 1: Schematic of protein aggregation mechanisms leading to polymorphic
fibrils. On the left are solvated peptides. Water in the hydration layer is in
red and the bulk water in blue. Even isolated monomers sample aggregation
prone conformations, N∗, which are coated with varying number of water
molecules. The peptides with N∗ conformations aggregate to form disordered
protein rich droplets. A major driving force for aggregation is the release of
water molecules in the hydration layer into the bulk, which facilitates fibril
formation entropically favorable. The structured protein aggregates nucleate
from the protein rich droplet to form protofilaments, which further self-
assembles to form a variety of mature amyloid fibrils. In some of the
polymorphic structures discrete number of water molecules are confined in the
fibril.
Fig. 2: Folding landscape of A$\beta_{10-35}$ monomers. (A) Low free energy
conformations in the which D23 and K28 amino acids, which forms a salt bridge
in the fibril, are separated by three, two and one water solvation shells
respectively (from top to bottom). The backbone oxygen and nitrogen atoms are
in red and blue, respectively. The positively and negatively charged, polar,
and hydrophobic residues are colored blue, red, purple, and green,
respectively. Water molecules around D23 and K28 are in cyan, while water
molecules which separate the two residues are shown in yellow. Hydrogen bonds
are shown as black dashed lines. (B) Hairpin-like conformation of the
A$\beta_{10-35}$ monomer which has a topologically similar structure as the
peptide structure in the A$\beta$ fibrils. The D23-K28 salt bridge is solvated
by the water molecules. The driving force for the formation of hairpin-like
conformation is the interaction between the hydrophobic residues in the N and
C termini shown in red and green respectively.
Fig. 3: Water release in fibril growth. (A) Variation in the number of water
molecules (red) within 3.5Å of the peptide from Sup35, which docks and locks
onto the fibril as a function of time. Time-dependent changes (green) in the
number of water molecules in the neighborhood of the fibril monomer onto which
the solvated peptide docks. (B) Release of water molecules in the zipper
region of the fibril occurs in two stages. In the first stage, water is
eliminated rapidly as the peptide docks onto the fibril, while in the second
stage, the last two water molecules are squeezed out with the concurrent
formation of the protofilament with a dry interior (structure on the right).
Fig. 4: Water molecules play a central role in the association kinetics of two
sheets formed from peptides rich in amino acids with polar side chains. In the
association process starting from a fully solvated pore (structure on the
left) trapped water molecules between the protofilaments form ordered water
wires (top middle structure). If the sheets misalign confined water molecules
are disordered. Release of trapped water molecules results in protofilament
formation (structure on the right). In the upper pathway the water molecules
in the wire file out in orderly fashion whereas in the bottom pathway water
escapes from the crevice on the sides of the protofilament.
|
arxiv-papers
| 2011-07-25T00:52:14 |
2024-09-04T02:49:20.898474
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Thirumalai, Govardhan Reddy, and John E. Straub",
"submitter": "Govardhan Reddy",
"url": "https://arxiv.org/abs/1107.4820"
}
|
1107.4941
|
# On the OUMD property for the column Hilbert space $C$
Yanqi QIU Equipe d’Analyse, Université Paris VI, Paris Cedex 05, France
yanqi-qiu@math.jussieu.fr
###### Abstract.
The operator space OUMD property was introduced by Pisier in the context of
vector-valued noncommutative $L_{p}$-spaces. It is an open problem whether the
column Hilbert space has this property. Based on some complex interpolation
techniques, we are able to relate this problem to the study of the
$\text{OUMD}_{q}$ property for non-commutative $L_{p}$-spaces.
###### Key words and phrases:
column Hilbert space, operator space OUMD property, noncommutative $L_{p}$
spaces
## 1\. Introduction
In Banach space valued martingale theory, the UMD property plays an important
role. Let us recall briefly the definition of the UMD property. Let
$1<p<\infty$. A Banach space $B$ is $\text{UMD}_{p}$, if there exists a
positive constant which depends only on $p$ and the Banach space $B$ (the best
one is usually denoted by $\beta_{p}(B)$), such that for all positive integers
$n$, all sequences $\varepsilon=(\varepsilon_{k})_{k=1}^{n}$ of numbers in
$\\{-1,1\\}$ and all $B$-valued martingale difference sequences
$dx=(dx_{k})_{k=1}^{n}$, we have
$\|\sum_{k=1}^{n}\varepsilon_{k}dx_{k}\|_{L_{p}(B)}\leq\beta_{p}(B)\|\sum_{k=1}^{n}dx_{k}\|_{L_{p}(B)}.$
The UMD property has very deep connections with the boundedness of certain
singular integral operators such as the Hilbert transform, see e.g.
Burkholder’s article [5]. Burkholder and McConnell [3] proved that if a Banach
space $B$ is $\text{UMD}_{p}$, then the Hilbert transform is bounded on the
Bochner space $L_{p}(\mathbb{T},m;B)$. Bourgain [2] showed that if the Hilbert
transform is bounded on $L_{p}(\mathbb{T},m;B)$, then $B$ is $\text{UMD}_{p}$.
The fact that the UMD property is independent of $p$ was first proved by
Pisier (using the Burkholder-Gundy extrapolation techniques). More precisely,
he proved that the finiteness of $\beta_{p}(B)$ for some $1<p<\infty$ implies
its finiteness for all $1<p<\infty$. Examples of UMD spaces include all the
finite dimensional Banach spaces, the Schatten $p$-classes $S_{p}$ and more
generally the noncommutative $L_{p}$-spaces associated to a von Neumann
algebra $\mathcal{M}$, for all $1<p<\infty$. The readers are referred to
Burkholder [4, 5] for information on UMD spaces.
In his monograph [11], Pisier developed a theory of vector-valued
noncommutative $L_{p}$-spaces. For a given hyperfinite von Neumann algebra
$\mathcal{M}$ equipped with an normal, semifinite, faithful trace $\tau$ and a
given operator space $E$, he defined the space $L_{p}(\tau;E)$. In the case
where $\mathcal{M}=B(\ell_{2})$ equipped with the usual trace, this space is
denoted by $S_{p}[E]$. The readers are referred to Pisier’s book [12] for the
details on operator space theory. Noncommutative conditional expectations and
martingales arise naturally in this setting. Following Pisier, we say that an
operator space $E$ is $\text{OUMD}_{p}$ for some $1<p<\infty$, if there exists
a constant (as before, the best one is usually denoted by $\beta_{p}^{os}(E)$,
which depends on $p$ and the operator space structure on $E$) such that for
any interger $n\geq 1$, any
$\varepsilon=(\varepsilon_{k})_{k=1}^{n}\in\\{-1,1\\}^{n}$, and any $E$-valued
noncommutative martingale $(f_{k})_{k=0}^{n}$ in $L_{p}(\tau;E)$ associated to
any increasing filtration $(\mathcal{M}_{k})_{k=1}^{n}$, we have
$\|f_{0}+\sum_{k=1}^{n}\varepsilon_{k}(f_{k}-f_{k-1})\|_{L_{p}(\tau;E)}\leq\beta_{p}^{os}(E)\|f_{n}\|_{L_{p}(\tau;E)}.$
The first remarkable result concerning the OUMD property was proved in [13]
and [14] by Pisier and Xu, where they proved that all the noncommutative
$L_{p}$-spaces are $\text{OUMD}_{p}$. In particular, the Schatten $p$-class
$S_{p}$ is $\text{OUMD}_{p}$. By a well known (but still unpublished) result
of Musat, an operator space $E$ is $\text{OUMD}_{p}$ if and only if $S_{p}[E]$
is UMD.
The following question is well-known to the experts, it is due to Z.-J. Ruan.
###### Question 1.1.
Does the column Hilbert space $C$ have $\text{OUMD}_{p}$ property for some or
all $1<p<\infty$?
The main theorem of this paper is:
###### Theorem 1.2.
Let $1<p<\infty$, then there exist $1<u,v<\infty$, such that we have an
isometric (Banach) embedding
$S_{p}[C]\hookrightarrow S_{u}[S_{v}].$
In principle, this embedding theorem should solve the Ruan problem, since in
[8], Musat presented a proof of the following:
###### Statement 1.3.
For any $1<p,q<\infty$, the operator space $S_{q}$ is $\text{OUMD}_{p}$. In
particular, as a Banach space, $S_{p}[S_{q}]$ is UMD.
However, very recently, Javier Parcet found a gap in the proof of the above
result. The main gap is in the proof of Proposition 4.10 in [8], where the
system (4.19) and (4.20) has solution only for $\frac{3}{2}<p<3$. For the
reader’s convenience, we reproduce in the appendix the results in [8] which
are not affected by this gap.
Since Statement 1.3 is in doubt, the answer to Ruan’s question might be
negative. Actually, this would be more interesting, since it would then imply
that the $\text{OUMD}_{p}$ property depends on $p$. Whether the
$\text{OUMD}_{p}$ property depends on $p$ or not is one of the main open
problems in this direction.
We also prove the following non-embeddability result, showing that the
obtained embedding result in Theorem 1.2 can not be extended to the category
of operator spaces.
###### Theorem 1.4.
Let $1<p_{1},p_{2},\cdots,p_{n}<\infty$. The operator space $C$ can not be
embedded completely isomorphically into any quotient of
$S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]$.
## 2\. Preliminaries
By an operator space we mean a closed subspace of $B(H)$ for some complex
Hilbert space $H$. When $E\subset B(H)$ is an operator space, we denote by
$M_{n}(E)$ the space of all $n\times n$ matrices with entries in $E$, equipped
with the norm induced by $B(\ell_{2}^{n}\otimes_{2}H)$. Let $e_{ij}$ be the
element of $B(\ell_{2})$ corresponding to the matrix whose coefficients equal
to one at the $(i,j)$ entry and zero elsewhere. The column Hilbert space $C$
is defined as
$C=\overline{\textrm{span}}\\{e_{i1}|i\geq 1\\}$
and the row Hilbert space $R$ is defined as
$R=\overline{\textrm{span}}\\{e_{1j}|j\geq 1\\}.$
Their operator space structures are given by the embeddings $C\subset
B(\ell_{2})$ and $R\subset B(\ell_{2})$.
Z.-J. Ruan in [16] gave an abstract characterization of operator spaces in
terms of matrix norms. An abstract operator space is a vector space $E$
equipped with matrix norms $\|\cdot\|_{m}$ on $M_{m}(E)$ for each positive
integer $m$, satisfying the axioms: for all $x\in M_{m}(E),y\in M_{n}(E)$ and
$\alpha,\beta\in M_{m}(\mathbb{C})$, we have
$\left\|\left(\begin{array}[]{cc}x&0\\\
0&y\end{array}\right)\right\|_{m+n}=\max\\{\|x\|_{m},\|y\|_{n}\\},\qquad\|\alpha
x\beta\|_{m}\leq\|\alpha\|\|x\|_{m}\|\beta\|.$
This abstract characterization can be used to define various important
constructions of new operator spaces from given ones. Among these are the
projective tensor product, the quotient, the dual etc. The two constructions
used frequently in this paper are the Haagerup tensor product and the complex
interpolation for operator spaces. Let us recall briefly their definitions and
main properties.
Let $E,F$ be two operator spaces, the Haagerup tensor product $E\otimes_{h}F$
of $E$ and $F$ is defined as the completion of $E\otimes F$ with respect to
the matrix norms
$\|u\|_{h,m}=\inf\\{\|v\|\|w\|:u=v\odot w,v\in M_{m,r}(E),w\in
M_{r,m}(F),r\in\mathbb{N}\\},$
where the element $v\odot w\in M_{m}(E\otimes F)$ is defined by $(v\odot
w)_{ij}=\sum_{k=1}^{m}v_{ik}\otimes w_{kj}$, for all $1\leq i,j\leq m$.
We refer to [1] for details about interpolations of Banach spaces. Now let
$E_{0},E_{1}$ be two operator spaces, such that $(E_{0},E_{1})$ is a
compatible couple in the sense of [1]. Following Pisier, we endow the
interpolation space $E_{\theta}=(E_{0},E_{1})_{\theta}$ with a canonical
operator space structure by defining for all positive integers $m$,
$M_{m}(E_{\theta})=(M_{m}(E_{0}),M_{m}(E_{1}))_{\theta}.$
The Haagerup tensor product is injective, projective, self-dual in the finite
dimensional case, however, it is not commutative, that is we do not have
$E\otimes_{h}F=F\otimes_{h}E$ in general.
We state the following Kouba’s interpolation theorem, which mainly says that
the Haagerup tensor product behaves nicely with respect to the complex
interpolation, for its proof, see e.g. [10].
###### Theorem 2.1 (Kouba).
Let $(E_{0},E_{1})$ and $(F_{0},F_{1})$ be two compatible couples of operator
spaces. Then $(E_{0}\otimes_{h}F_{0},E_{1}\otimes_{h}F_{1})$ is a compatible
couple. Moreover, for all $0<\theta<1$ we have complete isometry
$(E_{0}\otimes_{h}F_{0},E_{1}\otimes_{h}F_{1})_{\theta}=(E_{0},E_{1})_{\theta}\otimes_{h}(F_{0},F_{1})_{\theta}.$
Let us now turn to some basic definitions of vector-valued noncommutative
$L_{p}$-spaces. Let $S_{\infty}$ be the space of compact operators on
$\ell_{2}$. It is viewed as an operator space by the natural embedding
$S_{\infty}\subset B(\ell_{2})$. The finite dimensional version is
$S_{\infty}^{n}=B(\ell_{2}^{n})$. It is well known that the trace class
$S_{1}$ (resp. $S_{1}^{n}$) is the dual space of $S_{\infty}$ (resp.
$S_{\infty}^{n}$). By this duality, we equip $S_{1}$ (resp. $S_{1}^{n}$) with
the dual operator space structure.
Let $E$ be an operator space. Following Pisier, the noncommutative vector-
valued $L_{p}$-spaces in the discrete case are defined by
$S_{\infty}^{n}[E]=S_{\infty}^{n}\otimes_{min}E,\qquad
S_{\infty}[E]=S_{\infty}\otimes_{min}E,$
$S_{1}^{n}[E]=S_{1}^{n}\otimes^{\wedge}E,\qquad
S_{1}[E]=S_{1}\otimes^{\wedge}E.$
It turns out that $(S_{\infty}[E],S_{1}[E])$ (resp.
$(S_{\infty}^{n}[E],S_{1}^{n}[E])$) is a compatible couple. For $1<p<\infty$,
the definitions are
$S_{p}^{n}[E]=(S_{\infty}^{n}[E],S_{1}^{n}[E])_{\frac{1}{p}},\qquad
S_{p}[E]=(S_{\infty}[E],S_{1}[E])_{\frac{1}{p}}.$
Let $C_{p}$ and $R_{p}$ denote respectively the column and the row subspace of
$S_{p}$, for $1\leq p\leq\infty$. In particular, $C_{\infty}=C$ and
$R_{\infty}=R$ are the column and row Hilbert space defined as above. By well
known results, we have
$S_{p}=(S_{\infty},S_{1})_{\frac{1}{p}}.$
By this identity, $S_{p}$ is equipped with an operator space structure, which
is called the canonical operator space structure on $S_{p}$. We endow $C_{p}$
and $R_{p}$ with the induced operator space structure by inclusions
$C_{p}\subset S_{p}$ and $R_{p}\subset S_{p}$. Let $p^{\prime}$ denote the
conjugate exponent of $p$, i.e. $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Then the
natural identification is a complete isometry
$C_{p}\simeq R_{p^{\prime}}.$
This identification will be used freely in the sequel. Note also we have
complete isometry
$C_{p}^{*}\simeq C_{p^{\prime}}\simeq R_{p}.$
The following complete isometries from [11] will also be used:
$C_{p}=(C_{\infty},C_{1})_{\frac{1}{p}}.$
More generally, if
$\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$, then
$C_{p_{\theta}}=(C_{p_{0}},C_{p_{1}})_{\theta}.$
With these notations, we have complete isometry
$S_{p}[E]=C_{p}\otimes_{h}E\otimes_{h}R_{p}.$
The following proposition from [11] is useful for us.
###### Proposition 2.2.
Let $E$ be an operator space. Then we have the following complete isometric
isomorphism
$S_{p}[E]=C_{p}\otimes_{h}E\otimes_{h}R_{p}.$
By a well known (but still unpublished) result of Musat, an operator space $E$
is $\text{OUMD}_{p}$ if and only if $S_{p}[E]$ is UMD, hence Question 1.1 is
equivalent to the following
###### Question 2.3.
Let $1<p<\infty$. Does the Banach space $S_{p}[C]$ have UMD property?
## 3\. The Banach space $S_{p}[C]$
We now turn to the proof of Theorem 1.2. Let us begin by a simple proposition.
###### Proposition 3.1.
Let $1\leq u,v\leq\infty$. Then $C_{u}\otimes_{h}C_{v}$ is isometric to a
Schatten $p$-class $S_{p}$ for certain $1\leq p\leq\infty$. In particular,
either $1\leq u,v<\infty$ or $1<u,v\leq\infty$, the space
$C_{u}\otimes_{h}C_{v}$ is a UMD Banach space.
###### Proof.
Define $\theta=\frac{1}{v},\eta=\frac{1}{u}\in[0,1]$, then we have
$\frac{1}{v}=\frac{1-\theta}{\infty}+\frac{\theta}{1},$
$\frac{1}{u}=\frac{1-\eta}{\infty}+\frac{\eta}{1}.$
Then by Kouba’s interpolation result,
$C_{u}\otimes_{h}C_{\infty}=(C_{\infty}\otimes_{h}C_{\infty},C_{1}\otimes_{h}C_{\infty})_{\eta}.$
By applying the isometric identities
$C_{\infty}\otimes_{h}C_{\infty}=S_{2},C_{1}\otimes_{h}C_{\infty}=S_{1},$
we have
$C_{u}\otimes_{h}C_{\infty}=(S_{2},S_{1})_{\eta}=S_{\frac{2}{1+\eta}}.$
Similarly, we have isometric identity
$C_{u}\otimes_{h}C_{1}=S_{\frac{2}{\eta}}.$
Finally, we obtain that
$C_{u}\otimes_{h}C_{v}=(C_{u}\otimes_{h}C_{\infty},C_{u}\otimes_{h}C_{1})_{\theta}=(S_{\frac{2}{1+\eta}},S_{\frac{2}{\eta}})_{\theta}=S_{\frac{2}{1-1/v+1/u}}.$
The second assertion now follows easily. ∎
The following simple observation will be useful for us.
###### Remark 3.2.
We have complete isometries
$C\otimes_{h}C=C\otimes_{min}C\simeq C,\quad
R\otimes_{h}R=R\otimes_{min}R\simeq R.$
An application of Kouba’s interpolation result yields complete isometry
(1) $C_{p}\otimes_{h}C_{p}\simeq C_{p}$
for all $1\leq p\leq\infty$.
More generally, for any integer $n\geq 1$ we have the following complete
isometry
$\underbrace{C_{p}\otimes_{h}C_{p}\otimes_{h}\cdots\otimes_{h}C_{p}}_{n\,\,\textit{times}}\simeq
C_{p}.$
In particular, we have the following isometry (in the Banach space category)
$\underbrace{C_{1}\otimes_{h}C_{1}\otimes_{h}\cdots\otimes_{h}C_{1}}_{n\,\,\textit{times}}\simeq\underbrace{C_{\infty}\otimes_{h}C_{\infty}\otimes_{h}\cdots\otimes_{h}C_{\infty}}_{n\,\,\textit{times}}\simeq\ell_{2}.$
###### Proof of Theorem 1.2.
Let us first assume that $1<p\leq 2$. Define $\theta=\frac{1+1/p}{2}\in(0,1),$
then $q=\theta p=\frac{p+1}{2}\in(1,\infty)$ and $r=\theta
p^{\prime}=\frac{p+1}{2(p-1)}\in(1,\infty)$. That is
$\frac{1}{p}=\frac{1-\theta}{\infty}+\frac{\theta}{q},$
$\frac{1}{p^{\prime}}=\frac{1-\theta}{\infty}+\frac{\theta}{r}.$
By Proposition 2.2 and Kouba’s interpolation result,
$\displaystyle S_{p}[C]$ $\displaystyle=$ $\displaystyle
C_{p}\otimes_{h}C_{\infty}\otimes_{h}C_{p^{\prime}}$ $\displaystyle=$
$\displaystyle(C_{\infty}\otimes_{h}C_{\infty}\otimes_{h}C_{\infty},C_{q}\otimes_{h}C_{\infty}\otimes_{h}C_{r})_{\theta}$
By Remark 3.2 $\displaystyle\stackrel{{\scriptstyle isometric}}{{=}}$
$\displaystyle(C_{1}\otimes_{h}C_{1}\otimes_{h}C_{1},C_{q}\otimes_{h}C_{\infty}\otimes_{h}C_{r})_{\theta}$
$\displaystyle=$ $\displaystyle
C_{\frac{2p}{p+1}}\otimes_{h}C_{\frac{2p}{p-1}}\otimes_{h}C_{\frac{2p}{3p-3}}$
$\displaystyle=$ $\displaystyle
C_{\frac{2p}{p+1}}\otimes_{h}R_{\frac{2p}{p+1}}\otimes_{h}R_{\frac{2p}{3-p}}$
$\displaystyle=$ $\displaystyle
S_{\frac{2p}{p+1}}\otimes_{h}R_{\frac{2p}{3-p}}.$
Hence we get the desired isometric embedding
$S_{p}[C]\hookrightarrow
C_{\frac{2p}{3-p}}\otimes_{h}S_{\frac{2p}{p+1}}\otimes_{h}R_{\frac{2p}{3-p}}=S_{\frac{2p}{3-p}}[S_{\frac{2p}{p+1}}].$
Similar argument shows that if $1<p\leq 2$, then isometrically, we have
$S_{p^{\prime}}[R]=R_{p}\otimes_{h}R_{\infty}\otimes_{h}R_{p^{\prime}}=R_{\frac{2p}{p+1}}\otimes_{h}R_{\frac{2p}{p-1}}\otimes_{h}R_{\frac{2p}{3p-3}}=S_{\frac{2p}{p-1}}\otimes_{h}R_{\frac{2p}{3p-3}}\hookrightarrow
S_{\frac{2p}{3p-3}}[S_{\frac{2p}{p-1}}].$
By taking the opposite operator space, we have
$(S_{p^{\prime}}[R])^{op}=(R_{p}\otimes_{h}R_{\infty}\otimes_{h}R_{p^{\prime}})^{op}=R_{p^{\prime}}^{op}\otimes_{h}R_{\infty}^{op}\otimes_{h}R_{p}^{op}=C_{p^{\prime}}\otimes_{h}C_{\infty}\otimes_{h}C_{p}=S_{p^{\prime}}[C].$
It follows that the Banach space $S_{p^{\prime}}[C]$ embeds isometrically in
$S_{\frac{2p}{3p-3}}[S_{\frac{2p}{p-1}}]$ whenever $1<p\leq 2$. In other
words, if $2\leq p<\infty$, then
$S_{p}[C]\hookrightarrow S_{2p/3}[S_{2p}].$
∎
###### Corollary 3.3.
If Statement 1.3 holds, then $S_{p}[C]$ is a UMD space.
We will use the following lemma to prove Theorem 1.4.
###### Lemma 3.4.
If $1\leq p_{1},p_{2},\cdots,p_{n}<\infty$ or
$1<p_{1},p_{2},\cdots,p_{n}\leq\infty$. Then there exist
$1<q_{1},q_{2},\cdots,q_{n}<\infty$, such that we have
$C_{p_{1}}\otimes_{h}C_{p_{2}}\otimes_{h}\cdots\otimes_{h}C_{p_{n}}\stackrel{{\scriptstyle
isometric}}{{=}}C_{q_{1}}\otimes_{h}C_{q_{2}}\otimes_{h}\cdots\otimes_{h}C_{q_{n}}.$
Moreover, $C_{p_{1}}\otimes_{h}C_{p_{2}}\otimes_{h}\cdots\otimes_{h}C_{p_{n}}$
is a super-reflexive Banach space.
###### Proof.
Assume first that $1<p_{1},p_{2},\cdots,p_{n}\leq\infty$, then by choosing
$\theta\in(0,1)$ such that $\theta>\max(1/p_{1},1/p_{2},\cdots,1/p_{n})$, we
can define
$\widetilde{p}_{1},\widetilde{p}_{2},\cdots,\widetilde{p}_{n}\in(1,\infty]$ by
$\widetilde{p}_{1}=\theta p_{1},\widetilde{p}_{2}=\theta
p_{2},\cdots,\widetilde{p}_{n}=\theta p_{n}$ such that for $k=1,2,\cdots,n$,
$\frac{1}{p_{k}}=\frac{1-\theta}{\infty}+\frac{\theta}{\widetilde{p}_{k}}.$
It follows that
$\displaystyle
C_{p_{1}}\otimes_{h}C_{p_{2}}\otimes_{h}\cdots\otimes_{h}C_{p_{n}}$
$\displaystyle=$
$\displaystyle(C_{\infty}\otimes_{h}C_{\infty}\otimes_{h}\cdots\otimes_{h}C_{\infty},C_{\widetilde{p}_{1}}\otimes_{h}C_{\widetilde{p}_{2}}\otimes_{h}\cdots\otimes_{h}C_{\widetilde{p}_{n}})_{\theta}$
$\displaystyle\stackrel{{\scriptstyle isometric}}{{=}}$
$\displaystyle(C_{1}\otimes_{h}C_{1}\otimes_{h}\cdots\otimes_{h}C_{1},C_{\widetilde{p}_{1}}\otimes_{h}C_{\widetilde{p}_{2}}\otimes_{h}\cdots\otimes_{h}C_{\widetilde{p}_{n}})_{\theta}$
$\displaystyle=$ $\displaystyle
C_{q_{1}}\otimes_{h}C_{q_{2}}\otimes_{h}\cdots\otimes_{h}C_{q_{n}},$
where
$\frac{1}{q_{k}}=\frac{1-\theta}{1}+\frac{\theta}{\widetilde{p}_{k}}\in(0,1)$,
and hence $1<q_{k}<\infty$ for all $1\leq k\leq n$.
Super-reflexivity of
$C_{p_{1}}\otimes_{h}C_{p_{2}}\otimes_{h}\cdots\otimes_{h}C_{p_{n}}$ follows
from the above first identity, which shows that it is a
$(1-\theta)$-Hilbertian space.
The case when $1<p_{1},p_{2},\cdots,p_{n}\leq\infty$ can be treated similarly
or can be obtained by duality. ∎
###### Proof of Theorem 1.4.
Assume that we have a complete isomorphic embedding
$j:C\rightarrow S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]].$
By the injectivity of the Haagerup tensor product, we have a complete
isomorphic embedding
$j\otimes Id_{R}:C\otimes_{h}R\rightarrow
S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]\otimes_{h}R.$
Since $1<p_{1},p_{2},\cdots,p_{n}<\infty$ and $R=C_{1}$, hence by Lemma 3.4,
$S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]\otimes_{h}R$ is a super-
reflexive Banach space. This implies that $S_{\infty}=C\otimes_{h}R$ is also
super-reflexive, which is a contradiction.
For any closed subspace $F\subset
S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]$, we have
$\frac{S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]}{F}\otimes_{h}R\simeq\frac{S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]\otimes_{h}R}{F\otimes_{h}R}.$
Indeed, by [11], for any $1\leq p\leq\infty$, if $E_{2}\subset E_{1}$ is a
closed subspace, then we have complete isometry
$S_{p}[E_{1}/E_{2}]=S_{p}[E_{1}]/S_{p}[E_{2}].$
Using the above fact, it is easy to see that
$(E_{1}/E_{2})\otimes_{h}R_{p}=\frac{E_{1}\otimes_{h}R_{p}}{E_{2}\otimes_{h}R_{p}}.$
Note that the super-reflexive property is stable under taking the quotient,
hence $\frac{S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]}{F}\otimes_{h}R$ is
super-reflexive. Hence by using the same idea as above, assume that there is a
completely isomorphic embedding:
$i:C\rightarrow S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]/F,$
then we have completely isomorphic embedding:
$i\otimes
Id_{R}:C\otimes_{h}R\rightarrow\frac{S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]}{F}\otimes_{h}R,$
which leads to a contradiction. Hence $C$ can not be embedded completely
isomorphically into $S_{p_{1}}[S_{p_{2}}[\cdots[S_{p_{n}}]\cdots]]/F$. ∎
###### Open problem.
Let $n\geq 3$. Consider the underlying Banach space structure of the operator
space $C_{p_{1}}\otimes_{h}C_{p_{2}}\otimes_{h}\cdots\otimes_{h}C_{p_{n}}$. Is
it always a UMD space whenever $1<p_{1},p_{2},\cdots,p_{n}<\infty$?
Let us state the following result of the author from [15].
Fix $1\leq p,q\leq\infty$. We define by induction:
$E_{0}=\mathbb{C}\text{ \,and\, }E_{n+1}=\ell_{p}(\ell_{q}(E_{n})).$
###### Theorem 3.5.
Let $1\leq p\neq q\leq\infty$. Then there exists $c=c(p,q)>1$ depending only
on $p$ and $q$, such that the $\text{UMD}_{2}$ constants of the above defined
spaces $E_{n}$ satisfy
$\beta_{2}(E_{n})\geq c^{n}.$
Although we can not solve the above open problem, we have the following:
###### Proposition 3.6.
Let $1<p\neq q<\infty$. Define
$X_{n}(p,q)=\underbrace{C_{p}\otimes_{h}C_{q}\otimes_{h}\cdots\otimes_{h}C_{p}\otimes_{h}C_{q}}_{\text{$n$
times $C_{p}\otimes_{h}C_{q}$}}.$
Then we have
$\lim_{n\to\infty}\beta_{2}(X_{n}(p,q))=\infty.$
###### Proof.
To simplify, if there is no risk of confusion, we will use the notation
$X_{n}$ instead of $X_{n}(p,q)$ in the proof. Let us first assume that
$1<p\neq q<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$ (i.e. $p=q^{\prime}$ and
$p\neq 2$). Then $C_{p}=R_{q}$ and $C_{q}=R_{p}$. Hence we have
$\displaystyle X_{2n}$ $\displaystyle=$ $\displaystyle
C_{p}\otimes_{h}C_{q}\otimes_{h}\underbrace{C_{p}\otimes_{h}C_{q}\otimes_{h}\cdots\otimes_{h}C_{p}\otimes_{h}C_{q}}_{\text{$2(n-1)$
times $C_{p}\otimes_{h}C_{q}$}}\otimes_{h}C_{p}\otimes_{h}C_{q}$
$\displaystyle=$ $\displaystyle
C_{p}\otimes_{h}C_{q}\otimes_{h}X_{2(n-1)}\otimes_{h}R_{q}\otimes_{h}R_{p}$
$\displaystyle=$ $\displaystyle S_{p}[S_{q}[X_{2(n-1)}]].$
It follows that
$X_{2n}=\underbrace{S_{p}[S_{q}[\cdots[S_{p}[S_{q}]]\cdots]]}_{\text{$n$ times
$S_{p}[S_{q}]$}}.$
In particular, $E_{n}\subset X_{2n}$ (completely) isometrically. Hence
$\lim_{n\to\infty}\beta_{2}(X_{n})\geq\lim_{n\to\infty}\beta_{2}(E_{n})=\infty.$
Then we treat the general case. It is easy to see (for example, one can draw
the picture of the points
$(\frac{1}{p},\frac{1}{q}),(\frac{1}{r},\frac{1}{r^{\prime}}),(\frac{1}{s},\frac{1}{s})$
in the unit square $(0,1)\times(0,1)$ and use the obvious geometric meaning of
the following equation system) that for every pair $(p,q)$ with $1<p\neq
q<\infty$, there exist $0<\theta<1$ and $1<r,s<\infty$, $r\neq 2$ such that
$\frac{1}{r}=\frac{1-\theta}{s}+\frac{\theta}{p}$
$\frac{1}{r^{\prime}}=\frac{1-\theta}{s}+\frac{\theta}{q}.$
By Kouba’s interpolation theorem, we have
$X_{n}(r,r^{\prime})=(X_{n}(s,s),X_{n}(p,q))_{\theta}.$
Hence by the interpolation property of UMD constant, we have
$\beta_{2}(X_{n}(r,r^{\prime}))\leq\beta_{2}(X_{n}(s,s))^{1-\theta}\beta_{2}(X_{n}(p,q))^{\theta}.$
By definition, we have $X_{n}(s,s)\simeq C_{s},$ and hence
$\beta_{2}(X_{n}(s,s))=1$. Combining with the first step, we have
$\lim_{n\to\infty}\beta_{2}(X_{n}(p,q))\geq\lim_{n\to\infty}\beta_{2}(X_{n}(r,r^{\prime}))^{1/\theta}=\infty.$
∎
## 4\. Further results
In this section, we give some equivalent conditions for $S_{p}[E]$ to be UMD,
or equivalently, for $E$ to be $\text{OUMD}_{p}$. We give the equivalence
between the UMD property and the boundedness of the triangular projection on
$S_{p}[E]$. Applying this equivalence, we prove that $E$ is $\text{OUMD}_{p}$
if and only if $E$ is $\text{OUMD}_{p}$ with respect to the so-called
canonical filtration of matrix algebras.
We first give the following simple observation.
###### Proposition 4.1.
Let $1<p<\infty$, if we denote by $\mathcal{R}$ the Riesz projection
$\mathcal{R}:L_{p}(\mathbb{T},m)\rightarrow L_{p}(\mathbb{T},m)$ defined by
$\sum_{\text{finite}}x_{n}z^{n}\mapsto\sum_{n\geq 0}x_{n}z^{n}.$
Then $S_{p}[E]$ is UMD if and only if
$\mathcal{R}_{E}:=Id_{E}\otimes\mathcal{R}:L_{p}(\mathbb{T},m;E)\rightarrow
L_{p}(\mathbb{T},m;E)$
is completely bounded.
###### Proof.
By the classical results on UMD property, $S_{p}[E]$ is UMD if and only if the
corresponding Riesz projection
$\mathcal{R}_{S_{p}[E]}:L_{p}(\mathbb{T},m;S_{p}[E])\rightarrow
L_{p}(\mathbb{T},m;S_{p}[E])$
is bounded. By the noncommutative Fubini theorem, the natural identification
gives complete isometry
$L_{p}(\mathbb{T},m;S_{p}[E])\simeq S_{p}[L_{p}(\mathbb{T},m;E)].$
In this identification, $\mathcal{R}_{S_{p}[E]}$ corresponds to
$Id_{S_{p}}\otimes\mathcal{R}_{E}:S_{p}[L_{p}(\mathbb{T},m;E)]\rightarrow
S_{p}[L_{p}(\mathbb{T},m;E)].$
A very useful result in [11] tell us that
$\|\mathcal{R}_{E}\|_{cb}=\|Id_{S_{p}}\otimes\mathcal{R}_{E}\|$. Hence we have
$\|\mathcal{R}_{E}\|_{cb}=\|\mathcal{R}_{S_{p}[E]}\|.$
This ends our proof. ∎
The next theorem can be viewed as a special case of a result in [9].
###### Theorem 4.2.
Let $T_{E}$ be the triangular projection on $S_{p}[E]$ defined by
$(x_{ij})\mapsto(x_{ij}1_{j\geq i}).$
Then $\|T_{E}\|_{cb}=\|T_{E}\|=\|\mathcal{R}_{E}\|_{cb}$.
We refer to [6] and [7] for details on the canonical matrix filtration. As
usual, we regard $M_{n}$ as a non-unital subalgebra of
$M_{\infty}=B(\ell_{2})$ by viewing an $n\times n$ matrix as an infinite one
whose left upper corner of size $n\times n$ is the given $n\times n$ matrix,
and all other entries are zero. The unit of $M_{n}$ is the projection
$e_{n}\in M_{\infty}$ which projects a sequence in $\ell_{2}$ into its first
$n$ coordinates. The canonical matrix filtration is the increasing filtration
$(M_{n})_{n\geq 1}$ of subalgebras of $M_{\infty}$. We denote by
$E_{n}:M_{\infty}\rightarrow M_{n}$ the corresponding conditional expectation.
It is clear that
$E_{n}(a)=e_{n}ae_{n}=\sum_{\max(i,j)\leq n}a_{ij}\otimes e_{ij},\text{ for
all }a=(a_{ij})\in M_{\infty}.$
###### Remark 4.3.
Note that $E_{n}$ is not faithful, thus the noncommutative martingales with
respect to the filtration $(M_{n})_{n\geq 1}$ are different from the usual
ones. But this difference is not essential for what follows.
We can define the $\text{OUMD}_{p}$ property with respect to this canonical
matrix filtration. Let $x\in S_{p}[E]$. Then
$d_{1}x=E_{1}(x),\quad d_{n}x=E_{n}(x)-E_{n-1}(x),\text{ for all }n\geq 2.$
$E$ is said to be $\text{OUMD}_{p}$ with respect to the canonical matrix
filtration, if there exists a constant $K$ depending only on $p$ and $E$, such
that for all positive integers $N$ and all choices of signs
$\varepsilon_{n}=\pm 1$, we have
$\|\sum_{n=1}^{N}\varepsilon_{n}d_{n}x\|_{S_{p}[E]}\leq K\|x\|_{S_{p}[E]}.$
Let $K_{p}(E)$ denote the best such constant.
Every choice of signs $\varepsilon$ generates a transformation
$T_{\varepsilon}$ defined by
$T_{\varepsilon}(x)=\sum_{n}\varepsilon_{n}d_{n}x.$
An element $x\in S_{p}[E]$ is said to have finite support if the support of
$x$ defined by $\text{supp}(x)=\\{(i,j)\in\mathbb{N}^{2}:x_{ij}\neq 0\\}$ is
finite. Note that $T_{\varepsilon}$ is always well-defined on the subspace of
finite supported elements.
An operator space $E$ is $\text{OUMD}_{p}$ with respect to the canonical
matrix filtration if for every choice of signs $\varepsilon$, we have
$\|T_{\varepsilon}(x)\|_{S_{p}[E]}\leq
K_{p}(E)\|x\|_{S_{p}[E]},\quad|\text{supp}(x)|<\infty.$
###### Remark 4.4.
The transformation $T_{\varepsilon}$ is a Schur multiplication associated with
the function $f_{\varepsilon}(i,j)=\varepsilon_{\max(i,j)}$. Indeed, pick up
an arbitrary element $x=(x_{ij})\in S_{p}^{N}[E]$, we have
$d_{n}x=\sum_{\max(i,j)\leq n}x_{ij}\otimes e_{ij}-\sum_{\max(i,j)\leq
n-1}x_{ij}\otimes e_{ij}=\sum_{\max(i,j)=n}x_{ij}\otimes e_{ij},$
thus
$T_{\varepsilon}(x)=\sum_{n=1}^{N}\varepsilon_{n}d_{n}x=\sum_{n=1}^{N}\varepsilon_{n}\sum_{\max(i,j)=n}x_{ij}\otimes
e_{ij}=(\varepsilon_{\max(i,j)}x_{ij}).$
.
###### Remark 4.5.
Let
$D_{\varepsilon}=\text{diag}\\{\varepsilon_{1},\cdots,\varepsilon_{n},\cdots\\}$.
Then $T_{\varepsilon}(x)$ multiplied on the left by the scalar matrix
$D_{\varepsilon}$, we get
$D_{\varepsilon}T_{\varepsilon}(x)=(\varepsilon_{i}\varepsilon_{\max(i,j)}x_{ij})$.
After taking the average according to independent uniformly distributed
choices of signs, we get the lower triangular projection of $x$, i.e, we have
$\int
D_{\varepsilon}T_{\varepsilon}(x)d\varepsilon=\int(\varepsilon_{i}\varepsilon_{\max(i,j)}x_{ij})d\varepsilon=(x_{ij}1_{i\geq
j}).$
The following result is inspired by [6] and [7]
###### Theorem 4.6.
Let $1<p<\infty$. Then $E$ is $\text{OUMD}_{p}$ if and only if it is
$\text{OUMD}_{p}$ with respect to the canonical matrix filtration. Moreover,
we have:
$\frac{1}{2}(K_{p}(E)-1)\leq\|T_{E}\|\leq K_{p}(E).$
###### Proof.
Assume that $E$ is $\text{OUMD}_{p}$. Then $S_{p}[E]$ is UMD and the
triangular projection $T_{E}$ is bounded. Let $T_{E}^{-}$ be the triangular
projection defined by $(x_{ij})\mapsto(x_{ij}1_{j\leq i})$, it is clear that
$\|T_{E}\|=\|T_{E}^{-}\|$. We have
$d_{n}x=d_{n}T_{E}x+d_{n}T_{E}^{-}-D_{n}x,$
where $D_{n}x=e_{nn}xe_{nn}$. Thus
$\displaystyle\|\sum\varepsilon_{n}d_{n}x\|_{S_{p}[E]}$ $\displaystyle\leq$
$\displaystyle\|\sum\varepsilon_{n}d_{n}T_{E}x\|_{S_{p}[E]}+\|\sum\varepsilon_{n}d_{n}T_{E}^{-}x\|_{S_{p}[E]}$
$\displaystyle+\|\sum\varepsilon_{n}D_{n}x\|_{S_{p}[E]}.$
Since $d_{n}T_{E}x$ is the $n$-th column of $T_{E}x$, it is easy to see
$\|\sum\varepsilon_{n}d_{n}T_{E}x\|_{S_{p}[E]}=\|\sum
d_{n}T_{E}x\|_{S_{p}[E]}=\|T_{E}x\|_{S_{p}[E]}\leq\|T_{E}\|\|x\|_{S_{p}[E]}.$
The same reason shows that
$\|\sum\varepsilon_{n}d_{n}T_{E}^{-}x\|_{S_{p}[E]}=\|\sum
d_{n}T_{E}^{-}x\|_{S_{p}[E]}=\|T_{E}^{-}x\|_{S_{p}[E]}\leq\|T_{E}^{-}\|\|x\|_{S_{p}[E]}.$
For the third term, we have obviously that
$\|\sum\varepsilon_{n}D_{n}x\|_{S_{p}[E]}=\|\sum
D_{n}x\|_{S_{p}[E]}\leq\|x\|_{S_{p}[E]}.$
Combining these inequalities, we have
$\|\sum\varepsilon_{n}d_{n}x\|_{S_{p}[E]}\leq(\|T_{E}\|+\|T_{E}^{-}\|+1)\|x\|_{S_{p}[E]}=(2\|T_{E}\|+1)\|x\|_{S_{p}[E]}.$
So $E$ is $\text{OUMD}_{p}$ with respect to the canonical matrix filtration
with $K_{p}(E)\leq 2\|T_{E}\|+1$.
Conversely, assume that $E$ is $\text{OUMD}_{p}$ with respect to the canonical
matrix filtration. We shall show that $E$ is $\text{OUMD}_{p}$. It suffices to
show that the triangular projection $T_{E}$ is bounded. According to the
remark 4.5, we have
$\|(x_{ij}1_{i\geq
j})\|_{S_{p}[E]}\leq\int\|D_{\varepsilon}T_{\varepsilon}(x)\|_{S_{p}[E]}d\varepsilon\leq
K_{p}(E)\|x\|_{S_{p}[E]},$
proving that $\|T_{E}^{-}\|\leq K_{p}(E)$, and hence $\|T_{E}\|\leq K_{p}(E)$.
∎
###### Remark 4.7.
We have a slightly better estimation for $\|T_{E}\|$ as the following
$\frac{1}{2}(K_{p}(E)-1)\leq\|T_{E}\|\leq\frac{1}{2}(K_{p}(E)+1).$
We omit the proof here.
## Appendix
In this appendix, we quickly review the results from [8] that are unaffected
by the already mentioned gap.
We first recall that it follows from [14] that for any $1<p<\infty$, $S_{p}$
has $\text{OUMD}_{p}$.
###### Theorem 4.8 (Musat).
Let $1<p,q<\infty$. Then
$C_{p}\text{ has }\text{OUMD}_{q}.$
###### Proof.
Let
$S=\Big{\\{}(\frac{1}{p},\frac{1}{q})\in(0,1)\times(0,1):C_{p}\text{ has
}\text{OUMD}_{q}\Big{\\}}.$
We need to show that
$S=(0,1)\times(0,1).$
By complex interpolation, it is clear that $S$ is convex. Since $C_{p}$ and
$C_{p^{\prime}}=R_{p}$ are subspaces of $S_{p}$, by [14], both of them have
$\text{OUMD}_{p}$. Hence $(\frac{1}{p},\frac{1}{p})\in S$ and
$(\frac{1}{p^{\prime}},\frac{1}{p})\in S$ for all $1<p<\infty$. The result now
follows since
$(0,1)\times(0,1)=\text{conv}\Big{(}\\{(\frac{1}{p},\frac{1}{p}):1<p<\infty\\}\cup\\{(\frac{1}{p^{\prime}},\frac{1}{p}):1<p<\infty\\}\Big{)}.$
∎
###### Theorem 4.9 (Musat).
Let $L$ be the set of interior points of the closed convex set
$\text{conv}\Big{\\{}(0,0),(1,1),(\frac{1}{2},0),(\frac{1}{2},1)\Big{\\}}$.
Then
$S_{p}\text{ has }\text{OUMD}_{q}\text{ if }(\frac{1}{p},\frac{1}{q})\in L.$
###### Proof.
We have
$L=\text{conv}\Big{(}\\{(\frac{1}{p},\frac{1}{p}):1<p<\infty\\}\cup\\{(\frac{1}{2},\frac{1}{q}):1<q<\infty\\}\Big{)}.$
As operator spaces, we have completely isometrically
$S_{2}\simeq OH\simeq C_{2},$
hence by Theorem 4.8, $S_{2}$ has $\text{OUMD}_{q}$ for all $1<q<\infty$. If
we define
$T=\Big{\\{}(\frac{1}{p},\frac{1}{q})\in(0,1)\times(0,1):S_{p}\text{ has
}\text{OUMD}_{q}\Big{\\}},$
then $T$ is convex and contains all the points
$(\frac{1}{p},\frac{1}{p}),1<p<\infty$ and all the points
$(\frac{1}{2},\frac{1}{q}),1<q<\infty$. Hence
$L\subset T.$
∎
## Acknowledgements
The author is extremely grateful to his advisor Gilles Pisier for helpful
discussions. He also thanks Quanhua Xu for the helpful suggestion during the
preparation of this paper. He would like thank Javier Parcet for explaining
carefully the gap in [8] to him.
## References
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|
arxiv-papers
| 2011-07-25T13:18:48 |
2024-09-04T02:49:20.912072
|
{
"license": "Public Domain",
"authors": "Yanqi Qiu",
"submitter": "Yanqi Qiu",
"url": "https://arxiv.org/abs/1107.4941"
}
|
1107.4999
|
CDF Collaboration222With visitors from aIstituto Nazionale di Fisica Nucleare,
Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy, bUniversity of CA
Irvine, Irvine, CA 92697, USA, cUniversity of CA Santa Barbara, Santa Barbara,
CA 93106, USA, dUniversity of CA Santa Cruz, Santa Cruz, CA 95064, USA,
eCERN,CH-1211 Geneva, Switzerland, fCornell University, Ithaca, NY 14853, USA,
gUniversity of Cyprus, Nicosia CY-1678, Cyprus, hOffice of Science, U.S.
Department of Energy, Washington, DC 20585, USA, iUniversity College Dublin,
Dublin 4, Ireland, jUniversity of Fukui, Fukui City, Fukui Prefecture, Japan
910-0017, kUniversidad Iberoamericana, Mexico D.F., Mexico, lIowa State
University, Ames, IA 50011, USA, mUniversity of Iowa, Iowa City, IA 52242,
USA, nKinki University, Higashi-Osaka City, Japan 577-8502, oKansas State
University, Manhattan, KS 66506, USA, pUniversity of Manchester, Manchester
M13 9PL, United Kingdom, qQueen Mary, University of London, London, E1 4NS,
United Kingdom, rUniversity of Melbourne, Victoria 3010, Australia, sMuons,
Inc., Batavia, IL 60510, USA, tNagasaki Institute of Applied Science,
Nagasaki, Japan, uNational Research Nuclear University, Moscow, Russia,
vUniversity of Notre Dame, Notre Dame, IN 46556, USA, wUniversidad de Oviedo,
E-33007 Oviedo, Spain, xTexas Tech University, Lubbock, TX 79609, USA,
yUniversidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile, zYarmouk
University, Irbid 211-63, Jordan, hhOn leave from J. Stefan Institute,
Ljubljana, Slovenia,
# FERMILAB-PUB-11-345-E
CDF/PUB/BOTTOM/PUBLIC/10383
Measurement of Polarization and Search for $C\\!P$ Violation in
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ Decays
T. Aaltonen Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki,
Finland B. Álvarez Gonzálezw Instituto de Fisica de Cantabria, CSIC-
University of Cantabria, 39005 Santander, Spain S. Amerio Istituto Nazionale
di Fisica Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131
Padova, Italy D. Amidei University of Michigan, Ann Arbor, Michigan 48109,
USA A. Anastassov Northwestern University, Evanston, Illinois 60208, USA A.
Annovi Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica
Nucleare, I-00044 Frascati, Italy J. Antos Comenius University, 842 48
Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice,
Slovakia G. Apollinari Fermi National Accelerator Laboratory, Batavia,
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Artikov Joint Institute for Nuclear Research, RU-141980 Dubna, Russia J.
Asaadi Texas A&M University, College Station, Texas 77843, USA W. Ashmanskas
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of Oxford, Oxford OX1 3RH, United Kingdom W. Badgett Fermi National
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ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy S.
Carrillok University of Florida, Gainesville, Florida 32611, USA S. Carron
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Tufts University, Medford, Massachusetts 02155, USA R.F. Harr Wayne State
University, Detroit, Michigan 48201, USA K. Hatakeyama Baylor University,
Waco, Texas 76798, USA C. Hays University of Oxford, Oxford OX1 3RH, United
Kingdom M. Heck Institut für Experimentelle Kernphysik, Karlsruhe Institute
of Technology, D-76131 Karlsruhe, Germany J. Heinrich University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA M. Herndon University of
Wisconsin, Madison, Wisconsin 53706, USA S. Hewamanage Baylor University,
Waco, Texas 76798, USA D. Hidas Rutgers University, Piscataway, New Jersey
08855, USA A. Hocker Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA W. Hopkinsf Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA D. Horn Institut für Experimentelle Kernphysik,
Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany S. Hou
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
R.E. Hughes The Ohio State University, Columbus, Ohio 43210, USA M. Hurwitz
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
U. Husemann Yale University, New Haven, Connecticut 06520, USA N. Hussain
Institute of Particle Physics: McGill University, Montréal, Québec, Canada H3A
2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6;
University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF,
Vancouver, British Columbia, Canada V6T 2A3 M. Hussein Michigan State
University, East Lansing, Michigan 48824, USA J. Huston Michigan State
University, East Lansing, Michigan 48824, USA G. Introzzi Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy M. Ioriff Istituto Nazionale
di Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di Roma, I-00185
Roma, Italy A. Ivanovo University of California, Davis, Davis, California
95616, USA E. James Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA D. Jang Carnegie Mellon University, Pittsburgh, Pennsylvania
15213, USA B. Jayatilaka Duke University, Durham, North Carolina 27708, USA
E.J. Jeon Center for High Energy Physics: Kyungpook National University,
Daegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea;
Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of Science and
Technology Information, Daejeon 305-806, Korea; Chonnam National University,
Gwangju 500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea
M.K. Jha Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity of
Bologna, I-40127 Bologna, Italy S. Jindariani Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA W. Johnson University of
California, Davis, Davis, California 95616, USA M. Jones Purdue University,
West Lafayette, Indiana 47907, USA K.K. Joo Center for High Energy Physics:
Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea S.Y. Jun Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213, USA T.R. Junk Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA T. Kamon Texas A&M University,
College Station, Texas 77843, USA P.E. Karchin Wayne State University,
Detroit, Michigan 48201, USA A. Kasmi Baylor University, Waco, Texas 76798,
USA Y. Katon Osaka City University, Osaka 588, Japan W. Ketchum Enrico
Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA J. Keung
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA V.
Khotilovich Texas A&M University, College Station, Texas 77843, USA B.
Kilminster Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA D.H. Kim Center for High Energy Physics: Kyungpook National University,
Daegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea;
Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of Science and
Technology Information, Daejeon 305-806, Korea; Chonnam National University,
Gwangju 500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea
H.S. Kim Center for High Energy Physics: Kyungpook National University, Daegu
702-701, Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea H.W. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea J.E. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea M.J. Kim
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy S.B. Kim Center for High Energy Physics: Kyungpook
National University, Daegu 702-701, Korea; Seoul National University, Seoul
151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute
of Science and Technology Information, Daejeon 305-806, Korea; Chonnam
National University, Gwangju 500-757, Korea; Chonbuk National University,
Jeonju 561-756, Korea S.H. Kim University of Tsukuba, Tsukuba, Ibaraki 305,
Japan Y.K. Kim Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637, USA N. Kimura Waseda University, Tokyo 169, Japan M. Kirby
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA S.
Klimenko University of Florida, Gainesville, Florida 32611, USA K.
Kondo††footnotemark: Waseda University, Tokyo 169, Japan D.J. Kong Center
for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University,
Suwon 440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon 305-806, Korea; Chonnam National University, Gwangju 500-757, Korea;
Chonbuk National University, Jeonju 561-756, Korea J. Konigsberg University
of Florida, Gainesville, Florida 32611, USA A.V. Kotwal Duke University,
Durham, North Carolina 27708, USA M. Kreps Institut für Experimentelle
Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany J.
Kroll University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA D.
Krop Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637,
USA N. Krumnackl Baylor University, Waco, Texas 76798, USA M. Kruse Duke
University, Durham, North Carolina 27708, USA V. Krutelyovc Texas A&M
University, College Station, Texas 77843, USA T. Kuhr Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany M. Kurata University of Tsukuba, Tsukuba, Ibaraki 305,
Japan S. Kwang Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637, USA A.T. Laasanen Purdue University, West Lafayette, Indiana
47907, USA S. Lami Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity
of Pisa, ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa,
Italy S. Lammel Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA M. Lancaster University College London, London WC1E 6BT, United
Kingdom R.L. Lander University of California, Davis, Davis, California
95616, USA K. Lannonv The Ohio State University, Columbus, Ohio 43210, USA
A. Lath Rutgers University, Piscataway, New Jersey 08855, USA G. Latinocc
Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity
of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy T. LeCompte
Argonne National Laboratory, Argonne, Illinois 60439, USA E. Lee Texas A&M
University, College Station, Texas 77843, USA H.S. Lee Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637, USA J.S. Lee
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea S.W. Leex
Texas A&M University, College Station, Texas 77843, USA S. Leocc Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy S. Leone Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy J.D. Lewis Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA A. Limosanir
Duke University, Durham, North Carolina 27708, USA C.-J. Lin Ernest Orlando
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA J.
Linacre University of Oxford, Oxford OX1 3RH, United Kingdom M. Lindgren
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA E.
Lipeles University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA A.
Lister University of Geneva, CH-1211 Geneva 4, Switzerland D.O. Litvintsev
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA C. Liu
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA Q. Liu Purdue
University, West Lafayette, Indiana 47907, USA T. Liu Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA S. Lockwitz Yale
University, New Haven, Connecticut 06520, USA A. Loginov Yale University,
New Haven, Connecticut 06520, USA D. Lucchesibb Istituto Nazionale di Fisica
Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131 Padova,
Italy J. Lueck Institut für Experimentelle Kernphysik, Karlsruhe Institute
of Technology, D-76131 Karlsruhe, Germany P. Lujan Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA P. Lukens
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA G. Lungu
The Rockefeller University, New York, New York 10065, USA J. Lys Ernest
Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
R. Lysak Comenius University, 842 48 Bratislava, Slovakia; Institute of
Experimental Physics, 040 01 Kosice, Slovakia R. Madrak Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Maeshima Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA K. Makhoul
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA S.
Malik The Rockefeller University, New York, New York 10065, USA G. Mancaa
University of Liverpool, Liverpool L69 7ZE, United Kingdom A. Manousakis-
Katsikakis University of Athens, 157 71 Athens, Greece F. Margaroli Purdue
University, West Lafayette, Indiana 47907, USA C. Marino Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany M. Martínez Institut de Fisica d’Altes Energies, ICREA,
Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain R.
Martínez-Ballarín Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain P. Mastrandrea Istituto Nazionale di
Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di Roma, I-00185
Roma, Italy M.E. Mattson Wayne State University, Detroit, Michigan 48201,
USA P. Mazzanti Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity
of Bologna, I-40127 Bologna, Italy K.S. McFarland University of Rochester,
Rochester, New York 14627, USA P. McIntyre Texas A&M University, College
Station, Texas 77843, USA R. McNultyi University of Liverpool, Liverpool L69
7ZE, United Kingdom A. Mehta University of Liverpool, Liverpool L69 7ZE,
United Kingdom P. Mehtala Division of High Energy Physics, Department of
Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014,
Helsinki, Finland A. Menzione Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy C. Mesropian The Rockefeller University, New York, New
York 10065, USA T. Miao Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA D. Mietlicki University of Michigan, Ann Arbor, Michigan
48109, USA A. Mitra Institute of Physics, Academia Sinica, Taipei, Taiwan
11529, Republic of China H. Miyake University of Tsukuba, Tsukuba, Ibaraki
305, Japan S. Moed Harvard University, Cambridge, Massachusetts 02138, USA
N. Moggi Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity of
Bologna, I-40127 Bologna, Italy M.N. Mondragonk Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA C.S. Moon Center for High Energy
Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea R. Moore Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA M.J. Morello Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Morlock Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany P. Movilla Fernandez Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A. Mukherjee Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA Th. Muller Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany P. Murat Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA M. Mussiniaa Istituto Nazionale di Fisica Nucleare
Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy J. Nachtmanm Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA Y. Nagai
University of Tsukuba, Tsukuba, Ibaraki 305, Japan J. Naganoma Waseda
University, Tokyo 169, Japan I. Nakano Okayama University, Okayama 700-8530,
Japan A. Napier Tufts University, Medford, Massachusetts 02155, USA J. Nett
Texas A&M University, College Station, Texas 77843, USA C. Neu University of
Virginia, Charlottesville, Virginia 22906, USA M.S. Neubauer University of
Illinois, Urbana, Illinois 61801, USA J. Nielsend Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA L. Nodulman
Argonne National Laboratory, Argonne, Illinois 60439, USA O. Norniella
University of Illinois, Urbana, Illinois 61801, USA E. Nurse University
College London, London WC1E 6BT, United Kingdom L. Oakes University of
Oxford, Oxford OX1 3RH, United Kingdom S.H. Oh Duke University, Durham,
North Carolina 27708, USA Y.D. Oh Center for High Energy Physics: Kyungpook
National University, Daegu 702-701, Korea; Seoul National University, Seoul
151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute
of Science and Technology Information, Daejeon 305-806, Korea; Chonnam
National University, Gwangju 500-757, Korea; Chonbuk National University,
Jeonju 561-756, Korea I. Oksuzian University of Virginia, Charlottesville,
Virginia 22906, USA T. Okusawa Osaka City University, Osaka 588, Japan R.
Orava Division of High Energy Physics, Department of Physics, University of
Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland L.
Ortolan Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de
Barcelona, E-08193, Bellaterra (Barcelona), Spain S. Pagan Grisobb Istituto
Nazionale di Fisica Nucleare, Sezione di Padova-Trento, bbUniversity of
Padova, I-35131 Padova, Italy C. Pagliarone Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine, I-33100 Udine,
Italy E. Palenciae Instituto de Fisica de Cantabria, CSIC-University of
Cantabria, 39005 Santander, Spain V. Papadimitriou Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A.A. Paramonov Argonne
National Laboratory, Argonne, Illinois 60439, USA J. Patrick Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA G. Paulettagg Istituto
Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of
Udine, I-33100 Udine, Italy M. Paulini Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213, USA C. Paus Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA D.E. Pellett University of
California, Davis, Davis, California 95616, USA A. Penzo Istituto Nazionale
di Fisica Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine,
I-33100 Udine, Italy T.J. Phillips Duke University, Durham, North Carolina
27708, USA G. Piacentino Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy E. Pianori University of Pennsylvania, Philadelphia,
Pennsylvania 19104, USA J. Pilot The Ohio State University, Columbus, Ohio
43210, USA K. Pitts University of Illinois, Urbana, Illinois 61801, USA C.
Plager University of California, Los Angeles, Los Angeles, California 90024,
USA L. Pondrom University of Wisconsin, Madison, Wisconsin 53706, USA K.
Potamianos Purdue University, West Lafayette, Indiana 47907, USA O.
Poukhov††footnotemark: Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia F. Prokoshiny Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia A. Pronko Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA F. Ptohosg Laboratori Nazionali di Frascati, Istituto Nazionale di
Fisica Nucleare, I-00044 Frascati, Italy E. Pueschel Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA G. Punzicc Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy J. Pursley University of
Wisconsin, Madison, Wisconsin 53706, USA A. Rahaman University of
Pittsburgh, Pittsburgh, Pennsylvania 15260, USA V. Ramakrishnan University
of Wisconsin, Madison, Wisconsin 53706, USA N. Ranjan Purdue University,
West Lafayette, Indiana 47907, USA I. Redondo Centro de Investigaciones
Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain P. Renton
University of Oxford, Oxford OX1 3RH, United Kingdom M. Rescigno Istituto
Nazionale di Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di
Roma, I-00185 Roma, Italy T. Riddick University College London, London WC1E
6BT, United Kingdom F. Rimondiaa Istituto Nazionale di Fisica Nucleare
Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy L. Ristori44 Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA A. Robson
Glasgow University, Glasgow G12 8QQ, United Kingdom T. Rodrigo Instituto de
Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain T.
Rodriguez University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
E. Rogers University of Illinois, Urbana, Illinois 61801, USA S. Rollih
Tufts University, Medford, Massachusetts 02155, USA R. Roser Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. L. Rosner Enrico
Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA M. Rossi
Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste,
ggUniversity of Udine, I-33100 Udine, Italy F. Rubbo Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA F. Ruffinidd Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy A. Ruiz Instituto de
Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain J.
Russ Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA V. Rusu
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA A.
Safonov Texas A&M University, College Station, Texas 77843, USA W.K.
Sakumoto University of Rochester, Rochester, New York 14627, USA Y. Sakurai
Waseda University, Tokyo 169, Japan L. Santigg Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine, I-33100 Udine,
Italy L. Sartori Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of
Pisa, ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa,
Italy K. Sato University of Tsukuba, Tsukuba, Ibaraki 305, Japan V.
Savelievu LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris,
F-75252 France A. Savoy-Navarro LPNHE, Universite Pierre et Marie
Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France P. Schlabach Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A. Schmidt Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany E.E. Schmidt Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA M.P. Schmidt††footnotemark: Yale University,
New Haven, Connecticut 06520, USA M. Schmitt Northwestern University,
Evanston, Illinois 60208, USA T. Schwarz University of California, Davis,
Davis, California 95616, USA L. Scodellaro Instituto de Fisica de Cantabria,
CSIC-University of Cantabria, 39005 Santander, Spain A. Scribanodd Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy F. Scuri Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy A. Sedov Purdue
University, West Lafayette, Indiana 47907, USA S. Seidel University of New
Mexico, Albuquerque, New Mexico 87131, USA Y. Seiya Osaka City University,
Osaka 588, Japan A. Semenov Joint Institute for Nuclear Research, RU-141980
Dubna, Russia F. Sforzacc Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy A. Sfyrla University of Illinois, Urbana, Illinois
61801, USA S.Z. Shalhout University of California, Davis, Davis, California
95616, USA T. Shears University of Liverpool, Liverpool L69 7ZE, United
Kingdom P.F. Shepard University of Pittsburgh, Pittsburgh, Pennsylvania
15260, USA M. Shimojimat University of Tsukuba, Tsukuba, Ibaraki 305, Japan
S. Shiraishi Enrico Fermi Institute, University of Chicago, Chicago, Illinois
60637, USA M. Shochet Enrico Fermi Institute, University of Chicago,
Chicago, Illinois 60637, USA I. Shreyber Institution for Theoretical and
Experimental Physics, ITEP, Moscow 117259, Russia A. Simonenko Joint
Institute for Nuclear Research, RU-141980 Dubna, Russia P. Sinervo Institute
of Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8;
Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University
of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British
Columbia, Canada V6T 2A3 A. Sissakian††footnotemark: Joint Institute for
Nuclear Research, RU-141980 Dubna, Russia K. Sliwa Tufts University,
Medford, Massachusetts 02155, USA J.R. Smith University of California,
Davis, Davis, California 95616, USA F.D. Snider Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A. Soha Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA S. Somalwar Rutgers University,
Piscataway, New Jersey 08855, USA V. Sorin Institut de Fisica d’Altes
Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra
(Barcelona), Spain P. Squillacioti Istituto Nazionale di Fisica Nucleare
Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale
Superiore, I-56127 Pisa, Italy M. Stancari Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA M. Stanitzki Yale University, New
Haven, Connecticut 06520, USA R. St. Denis Glasgow University, Glasgow G12
8QQ, United Kingdom B. Stelzer Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 O. Stelzer-Chilton Institute of Particle Physics: McGill University,
Montréal, Québec, Canada H3A 2T8; Simon Fraser University, Burnaby, British
Columbia, Canada V5A 1S6; University of Toronto, Toronto, Ontario, Canada M5S
1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3 D. Stentz
Northwestern University, Evanston, Illinois 60208, USA J. Strologas
University of New Mexico, Albuquerque, New Mexico 87131, USA G.L. Strycker
University of Michigan, Ann Arbor, Michigan 48109, USA Y. Sudo University of
Tsukuba, Tsukuba, Ibaraki 305, Japan A. Sukhanov University of Florida,
Gainesville, Florida 32611, USA I. Suslov Joint Institute for Nuclear
Research, RU-141980 Dubna, Russia K. Takemasa University of Tsukuba,
Tsukuba, Ibaraki 305, Japan Y. Takeuchi University of Tsukuba, Tsukuba,
Ibaraki 305, Japan J. Tang Enrico Fermi Institute, University of Chicago,
Chicago, Illinois 60637, USA M. Tecchio University of Michigan, Ann Arbor,
Michigan 48109, USA P.K. Teng Institute of Physics, Academia Sinica, Taipei,
Taiwan 11529, Republic of China J. Thomf Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA J. Thome Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA G.A. Thompson University of
Illinois, Urbana, Illinois 61801, USA E. Thomson University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA P. Ttito-Guzmán Centro de
Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid,
Spain S. Tkaczyk Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA D. Toback Texas A&M University, College Station, Texas 77843, USA
S. Tokar Comenius University, 842 48 Bratislava, Slovakia; Institute of
Experimental Physics, 040 01 Kosice, Slovakia K. Tollefson Michigan State
University, East Lansing, Michigan 48824, USA T. Tomura University of
Tsukuba, Tsukuba, Ibaraki 305, Japan D. Tonelli Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA S. Torre Laboratori Nazionali di
Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy D.
Torretta Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
P. Totaro Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento,
bbUniversity of Padova, I-35131 Padova, Italy M. Trovatoee Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy Y. Tu University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA F. Ukegawa University of
Tsukuba, Tsukuba, Ibaraki 305, Japan S. Uozumi Center for High Energy
Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea A. Varganov University of Michigan, Ann
Arbor, Michigan 48109, USA F. Vázquezk University of Florida, Gainesville,
Florida 32611, USA G. Velev Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA C. Vellidis University of Athens, 157 71 Athens, Greece
M. Vidal Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain I. Vila Instituto de Fisica de
Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain R. Vilar
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain J. Vizán Instituto de Fisica de Cantabria, CSIC-University
of Cantabria, 39005 Santander, Spain M. Vogel University of New Mexico,
Albuquerque, New Mexico 87131, USA G. Volpicc Istituto Nazionale di Fisica
Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola
Normale Superiore, I-56127 Pisa, Italy P. Wagner University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA R.L. Wagner Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA T. Wakisaka Osaka City University,
Osaka 588, Japan R. Wallny University of California, Los Angeles, Los
Angeles, California 90024, USA S.M. Wang Institute of Physics, Academia
Sinica, Taipei, Taiwan 11529, Republic of China A. Warburton Institute of
Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8; Simon
Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University of
Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British
Columbia, Canada V6T 2A3 D. Waters University College London, London WC1E
6BT, United Kingdom M. Weinberger Texas A&M University, College Station,
Texas 77843, USA W.C. Wester III Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA B. Whitehouse Tufts University, Medford,
Massachusetts 02155, USA D. Whitesonb University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA A.B. Wicklund Argonne National
Laboratory, Argonne, Illinois 60439, USA E. Wicklund Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA S. Wilbur Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637, USA F. Wick
Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology,
D-76131 Karlsruhe, Germany H.H. Williams University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA J.S. Wilson The Ohio State University,
Columbus, Ohio 43210, USA P. Wilson Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA B.L. Winer The Ohio State University, Columbus,
Ohio 43210, USA P. Wittichg Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA S. Wolbers Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA H. Wolfe The Ohio State University, Columbus,
Ohio 43210, USA T. Wright University of Michigan, Ann Arbor, Michigan 48109,
USA X. Wu University of Geneva, CH-1211 Geneva 4, Switzerland Z. Wu Baylor
University, Waco, Texas 76798, USA K. Yamamoto Osaka City University, Osaka
588, Japan J. Yamaoka Duke University, Durham, North Carolina 27708, USA T.
Yang Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
U.K. Yangp Enrico Fermi Institute, University of Chicago, Chicago, Illinois
60637, USA Y.C. Yang Center for High Energy Physics: Kyungpook National
University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of
Science and Technology Information, Daejeon 305-806, Korea; Chonnam National
University, Gwangju 500-757, Korea; Chonbuk National University, Jeonju
561-756, Korea W.-M. Yao Ernest Orlando Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA G.P. Yeh Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Yim Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Yoh Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Yorita Waseda
University, Tokyo 169, Japan T. Yoshidaj Osaka City University, Osaka 588,
Japan G.B. Yu Duke University, Durham, North Carolina 27708, USA I. Yu
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea S.S. Yu
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA J.C. Yun
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA A.
Zanetti Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste,
ggUniversity of Udine, I-33100 Udine, Italy Y. Zeng Duke University, Durham,
North Carolina 27708, USA S. Zucchelliaa Istituto Nazionale di Fisica
Nucleare Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy
###### Abstract
We present the first measurement of polarization and $C\\!P$–violating
asymmetries in a $B^{0}_{s}$ decay into two light vector mesons,
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$, and an improved
determination of its branching ratio using $295$ decays reconstructed in a
data sample corresponding to 2.9 $\mathrm{fb^{-1}}$ of integrated luminosity
collected by the CDF experiment at the Fermilab Tevatron collider. The
fraction of longitudinal polarization is determined to be $f_{\textup{L}}\
=0.348\pm 0.041({\rm stat})\pm 0.021({\rm syst})$, and the branching ratio
${\mathcal{B}}(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi)=[2.32\pm
0.18({\rm stat})\pm 0.82({\rm syst})]\times 10^{-5}$. Asymmetries of decay
angle distributions sensitive to $C\\!P$ violation are measured to be
$A_{u}=-0.007\pm 0.064\text{(stat)}\pm 0.018\text{(syst)}$ and
$A_{v}=-0.120\pm 0.064\text{(stat)}\pm 0.016\text{(syst)}$.
###### pacs:
13.25.Hw 11.30.Er 14.40.Nd 12.38.Qk
Several charmless $B^{0}_{s}$ decays were observed at the Tevatron in Run II
prlphix ; BsKK , but a detailed investigation of decay properties and of
$C\\!P$ violation in these decays is still lacking. The
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ process is mediated by a
one–loop flavor–changing neutral current, the
$b\kern-3.00003pt\to\kern-1.99997pts$ penguin, and belongs to the class of
decays where the final state consists of a pair of light spin–1 mesons (V).
Three independent amplitudes govern $B\kern-3.00003pt\to\kern-1.99997ptVV$
decays, corresponding to the polarizations of the final–state vector mesons:
longitudinal polarization, and transverse polarization with spins parallel or
perpendicular to each other. The first two states are $C\\!P$–even, while the
last one is $C\\!P$–odd. Polarization amplitudes can be measured analyzing
angular distributions of final–state particles. Interference between the
$C\\!P$–even and $C\\!P$–odd amplitudes can generate asymmetries in angular
distributions, the triple product (TP) asymmetries, which may signal
unexpected $C\\!P$ violation due to physics beyond the standard model (SM).
The V–A structure of charged weak currents leads to the expectation of a
dominant longitudinal polarization BVV-QCDf ; Ali:2007ff . Approximately equal
longitudinal and transverse polarizations have been measured instead in
$b\kern-3.00003pt\to\kern-1.99997pts$ penguin–dominated $B^{0}$ and $B^{+}$
decay modes phikstar . This is explained in the SM by including either
non–factorizable penguin–annihilation effects puzzle-PA or final state
interactions puzzle-FSI . Recent theoretical predictions BVV-QCDf ; Ali:2007ff
indicate a longitudinal fraction $f_{L}$ in the 40–70 % range, when
phenomenological parameters are adjusted to accommodate present experimental
data. Explanations involving new physics (NP) in the
$b\kern-3.00003pt\to\kern-1.99997pts$ penguin process have also been proposed
NPtheory . Additional experimental information in $B^{0}_{s}$
penguin–dominated decays, such as
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$, may help distinguishing
the various solutions Datta:2008wf , and can be used to derive upper limits
for the mixing–induced $C\\!P$ asymmetries Bartsch:2008ps .
Triple product asymmetries are odd under time-reversal ($T$), and can be
generated either by final–state interactions or $C\\!P$ violation. In
flavor–untagged samples, where the initial $B$ flavor is not identified, TP
asymmetries can be shown to signify genuine $C\\!P$ violation TPtruefake . In
this respect they are very sensitive to the presence of NP in the decay since
they do not require a strong–phase difference between NP and SM amplitudes, as
opposed to direct $C\\!P$ asymmetries TPtheory . The TP asymmetry is defined
as
$\mathcal{A}_{\mathrm{TP}}=\frac{\Gamma(\mathrm{TP}>0)-\Gamma(\mathrm{TP}<0)}{\Gamma(\mathrm{TP}>0)+\Gamma(\mathrm{TP}<0)},$
where $\Gamma$ is the decay width for the given process. In
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decays two TP asymmetries
can be studied, corresponding to the two interference terms between amplitudes
with different $C\\!P$. These asymmetries are predicted to vanish in the SM,
and an observation of a non–zero asymmetry would be an unambigous sign of NP
TPtheory .
In this Letter we present the first measurement of polarization amplitudes and
of TP asymmetries in the $B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$
decay and an updated measurement of its branching ratio using
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ decays
reconstructed in the same dataset as a normalization. Data from an integrated
luminosity of 2.9 $\mathrm{fb^{-1}}$ of $p\bar{p}$ collisions at
$\sqrt{s}=1.96$ TeV are analyzed.
The components of the CDF II detector relevant for this analysis are briefly
described below; a more complete description can be found elsewhere cdf . We
reconstruct charged–particle trajectories (tracks) in the pseudorapidity range
$|\eta|\lesssim 1$ CDFsystem using a silicon microstrip vertex detector
vertex-detectors and a central drift chamber cot , both immersed in a
$1.4\,{\rm T}$ solenoidal magnetic field. The detection of muons in the
pseudorapidity range $|\eta|\lesssim 0.6$ is provided by two sets of drift
chambers located behind the calorimeters (CMU) and behind additional steel
absorbers (CMP), while the CMX detector covers the range
$0.6\lesssim|\eta|\lesssim 1.0$ cmu . A sample enriched with heavy–flavor
particles is selected by the displaced–track trigger trigger , based on the
Silicon Vertex Trigger (SVT) svt . It provides a precise measurement of the
track impact parameter ($d_{0}$), defined as the distance of closest approach
to the beam axis in the transverse plane. Decays of heavy–flavor particles are
identified by requiring two tracks with $120\,\mu{\rm m}\leq d_{0}\leq
1.0\rm\,mm$ and applying a requirement on the two–dimensional decay length,
$L_{\rm xy}>200\rm\,\mu m$ Lxy .
We reconstruct $B^{0}_{s}$ mesons by first forming
$\phi\kern-3.00003pt\to\kern-1.99997ptK^{+}K^{-}$ and
$J\\!/\\!\psi\kern-3.00003pt\to\kern-1.99997pt\mu^{+}\mu^{-}$ candidate decays
from opposite–sign track pairs with mass within 15 and 100
$\mathrm{Me\kern-1.00006ptV\\!/}c^{2}$ of the known pdg2010 $\phi$ and
$J\\!/\\!\psi$ mass, respectively. At least one $J\\!/\\!\psi$ track is
required to match a segment reconstructed in the muon detectors. We form
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$
($B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$) candidates by
fitting to a single vertex the $\phi$ $\phi$ ($J\\!/\\!\psi$ $\phi$) candidate
pairs. In the $B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$
case the fit constrains the mass of the two muons to the $J\\!/\\!\psi$ mass
pdg2010 . At least one pair of tracks in the $B^{0}_{s}$ candidate must
satisfy the trigger requirements. Combinatorial background and partially
reconstructed decays are reduced by exploiting the long lifetime and
relatively hard $p_{T}$ spectrum of $B^{0}_{s}$ mesons. We follow closely the
selection adopted in prlphix , using the vertex fit $\chi^{2}$, the $L_{\rm
xy}$, the reconstructed $B^{0}_{s}$ and $\phi$ meson impact parameters, and
the minimum kaon transverse momentum as discriminating variables. The
selection requirements are set by maximizing the quantity
$\mathrm{S/\sqrt{S+B}}$, where the accepted number of signal events
$\mathrm{S}$ is derived from a Monte Carlo (MC) simulation cdfsim of the CDF
II detector and trigger, while the number of background events $\mathrm{B}$ is
modeled using data in mass sideband regions: $(5.02,5.22)$ and $(5.52,5.72)$
$\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}$. The resulting mass distributions are
shown in Fig. 1.
Figure 1: The invariant mass of the four kaons (left) and of the
$J\\!/\\!\psi$ and two kaons (right) for
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ and
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ candidates,
overlayed with fit projections and separate signal and background components.
The narrower signal peak for the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ is due to the
$J\\!/\\!\psi$ mass constraint applied in the reconstruction.
A binned maximum likelihood (ML) fit to the $m_{B}$ distribution is performed
to determine the $B^{0}_{s}$ yield for both decay modes. The signal is
parameterized by two Gaussian functions with the same mean value, but
different widths. The ratios between the two widths and between the integrals
of the two components are fixed based on MC simulations. The combinatorial
background has a smooth mass distribution near the signal and is modeled with
an exponential function. A reflection from
$B^{0}\kern-3.00003pt\to\kern-1.99997pt\phi K^{\ast}(892)^{0}$
($B^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi K^{\ast}(892)^{0}$) with
misassigned kaon mass to final state pions contaminates the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$
($B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$) signal region.
Parameterizations and efficiencies determined from simulation are used for
these backgrounds. Their normalizations are derived from the known pdg2010
branching ratios, fragmentation fraction ratio $f_{s}/f_{d}$, and the ratio of
the detection efficiencies relative to signal ones. We estimate $(4.19\pm
0.93)\%$ and $(2.7\pm 1.0)\%$ reflection background under the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ and
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ signals, respectively.
Free parameters of the fit are the signal fraction, the $B^{0}_{s}$ mass $M$,
and width $\sigma$, together with the exponential slope $b_{0}$ defining the
combinatorial background mass shape. We estimate the total number of signal
decays as $N_{\phi\phi}=295\pm 20({\rm stat})\pm 12({\rm syst})$ and
$N_{\psi\phi}=1766\pm 48({\rm stat})\pm 41({\rm syst})$, where the systematic
uncertainty is estimated by varying signal and background models.
The $B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decay rate is derived
from the relation
$\displaystyle\frac{{\mathcal{B}}\left(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi\right)}{{\mathcal{B}}\left(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi\right)}=\frac{N_{\phi\phi}}{N_{\psi\phi}}\frac{{\mathcal{B}}\left(J/\psi\to\mu\mu\right)}{{\mathcal{B}}\left(\phi\to
KK\right)}\frac{\epsilon_{\psi\phi}}{\epsilon_{\phi\phi}}\,\epsilon^{\mu}_{\psi\phi}\
,$
where $\epsilon_{\psi\phi}/\epsilon_{\phi\phi}$ is the acceptance times
efficiency ratio for the two decays and $\epsilon^{\mu}_{\psi\phi}$ is the
efficiency for identifying at least one of the two muons. The efficiency ratio
is determined using a MC simulation of the CDF II detector and trigger, whose
reliability in determining relative trigger and reconstruction efficiencies
has been verified for several different decay modes also using data–driven
approaches earlierCDFBR . We estimate
$\epsilon_{\psi\phi}/\epsilon_{\phi\phi}=0.939\pm 0.099$, where the
uncertainty includes systematic effects from polarization uncertainties in the
two decay modes (9%), from the different trigger efficiencies for kaons and
muons (4%), and from the $B^{0}_{s}$ $p_{T}$ spectra (1%). We use inclusive
$J\\!/\\!\psi$ data to derive the single–muon identification efficiency as a
function of muon $p_{T}$. It is determined separately in two pseudorapidity
regions corresponding, respectively, to the CMU/CMP and CMX detectors, and is
described by a turn-on function that depends on a plateau, a slope, and a
threshold parameter. We use simulated
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ decays to
calculate $\epsilon^{\mu}_{\psi\phi}$ treating the efficiencies for the two
muons as uncorrelated: $\epsilon^{\mu}_{\psi\phi}=(86.95\pm 0.44({\rm
stat})\pm 0.75({\rm syst}))\%$. The systematic uncertainty includes the
uncertainty on the background subtraction and effects of residual correlation
between the two muon efficiencies.
We measure
${\mathcal{B}}(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi)/{\mathcal{B}}(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi)=[1.78\pm
0.14({\rm stat})\pm 0.20({\rm syst})]\times 10^{-2}$ and derive
${\mathcal{B}}(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi)=[2.32\pm
0.18({\rm stat})\pm 0.26({\rm syst})\pm 0.78({\rm br})]\times 10^{-5}$, using
the known pdg2010
${\mathcal{B}}(B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi)$,
which contributes the dominant uncertainty, labeled (br). This result is in
agreement and supersedes our previous measurement prlphix with a substantial
reduction of its statistical uncertainty; it is also consistent with recent
theoretical calculations BVV-QCDf ; Ali:2007ff .
We describe the angular distribution of the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decay products using the
helicity variables
$\vec{\omega}=(\cos\vartheta_{1},\cos\vartheta_{2},\varPhi$), where
$\vartheta_{i}$ is the angle between the direction of the $K^{+}$ from each
$\phi$ and the direction opposite the $B^{0}_{s}$ in the vector meson rest
frame, and $\varPhi$ is the angle between the two resonance decay planes in
the $B^{0}_{s}$ rest frame. The three independent complex amplitudes are
$A_{0}$ for the longitudinal polarization and $A_{\parallel}$ ($A_{\perp}$)
for transverse polarization with spins parallel (perpendicular) to each other.
They are related by $|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}=1$. The
differential decay rate is expressed as
$d^{4}\Gamma/(dtd\vec{\omega})\propto\sum_{i=1}^{6}K_{i}(t)f_{i}(\vec{\omega})$,
where the functions $K_{i}(t)$ encode the $B^{0}_{s}$ time evolution including
mixing and depend on the polarization amplitudes, and the
$f_{i}(\vec{\omega})$ are functions of the helicity angles only TPtheory . To
extract the polarization amplitudes we measure the time-integrated angular
distribution assuming no direct $C\\!P$ violation and a negligible weak phase
difference between $B^{0}_{s}$ mixing and
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decay as predicted in the
SM. The time–integrated differential decay rate depends on the polarization
amplitudes at $t=0$ and on the light and heavy $B_{s}^{0}$ mass–eigenstate
lifetimes, $\tau_{\text{L}}$ and $\tau_{\text{H}}$, as follows:
$\displaystyle\frac{d^{3}\Gamma}{d\vec{\omega}}$ $\displaystyle\propto$
$\displaystyle\,\tau_{\textup{L}}\big{(}|A_{0}|^{2}f_{1}(\vec{\omega})+|A_{\parallel}|^{2}f_{2}(\vec{\omega})$
(1)
$\displaystyle+|A_{0}||A_{\parallel}|\cos\delta_{\parallel}f_{5}(\vec{\omega})\big{)}+\tau_{\textup{H}}|A_{\perp}|^{2}f_{3}(\vec{\omega}),$
where $\delta_{\parallel}=\arg(A_{0}^{\star}A_{\parallel})$ and
$\begin{split}f_{1}(\vec{\omega})&=4\cos^{2}\vartheta_{1}\cos^{2}\vartheta_{2},\\\
f_{2}(\vec{\omega})&=\sin^{2}\vartheta_{1}\sin^{2}\vartheta_{2}(1+\cos
2\varPhi),\\\
f_{3}(\vec{\omega})&=\sin^{2}\vartheta_{1}\sin^{2}\vartheta_{2}(1-\cos
2\varPhi),\\\ f_{5}(\vec{\omega})&=\sqrt{2}\sin 2\vartheta_{1}\sin
2\vartheta_{2}\cos\varPhi.\end{split}$
Two triple products are present in $B\kern-3.00003pt\to\kern-1.99997ptVV$
decays: $\mathrm{TP_{2}}\equiv\Im(A_{\parallel}^{\star}A_{\perp})$, and
$\mathrm{TP_{1}}\equiv\Im(A_{0}^{\star}A_{\perp})$. These factors appear,
respectively, in the decay rate terms $K_{4}(t)$ and $K_{6}(t)$ multiplied by
the functions
$\begin{split}f_{4}(\vec{\omega})&=-2\sin^{2}\vartheta_{1}\sin^{2}\vartheta_{2}\sin
2\varPhi,\\\ f_{6}(\vec{\omega})&=-\sqrt{2}\sin 2\vartheta_{1}\sin
2\vartheta_{2}\sin\varPhi.\end{split}$
In flavor–untagged samples the TP terms, that vanish in the absence of NP, are
proportional to the so–called _true_ triple products, and provide two
$C\\!P$–violating observables, $\mathcal{A}^{1}_{\mathrm{TP}}$ and
$\mathcal{A}^{2}_{\mathrm{TP}}$ TPtruefake . We access
$\mathcal{A}^{2}_{\mathrm{TP}}$ through the observable $u=\sin 2\varPhi$. We
measure the $u$ asymmetry, $A_{u}$, by integrating over $\cos\vartheta_{1,2}$
the untagged decay rate and counting events with $u\\!>\\!0$ ($N^{+}_{u}$) and
$u\\!<\\!0$ ($N^{-}_{u}$). Similarly, $\mathcal{A}^{1}_{\mathrm{TP}}$ is
accessed through an asymmetry in $\sin\varPhi$. We define the observable $v$
as $v=\sin\varPhi$ ($v=-\sin\varPhi$) if
$\cos\vartheta_{1}\cos\vartheta_{2}\geq 0$
($\cos\vartheta_{1}\cos\vartheta_{2}<0$) and measure its asymmetry $A_{v}$ by
counting events with $v\\!>\\!0$ ($N^{+}_{v}$) and $v\\!<\\!0$ ($N^{-}_{v}$).
The asymmetries are defined as
$\displaystyle A_{u(v)}$
$\displaystyle=\frac{N^{+}_{u(v)}-N^{-}_{u(v)}}{N^{+}_{u(v)}+N^{-}_{u(v)}}=\mathcal{N}_{u(v)}\times$
(2)
$\displaystyle\big{[}\Im(A^{\star}_{\parallel(0)}A_{\perp})+\Im(\bar{A}^{\star}_{\parallel(0)}\bar{A}_{\perp})\big{]}=\mathcal{N}_{u(v)}\mathcal{A}^{2(1)}_{\rm{TP}},$
where the two normalization factors are $\mathcal{N}_{u}=-2/\pi$ and
$\mathcal{N}_{v}=-\sqrt{2}/\pi$. Both $A_{u}$ and $A_{v}$ are proportional to
$C\\!P$–violating TP asymmetries, and are also sensitive to mixing–induced TP
when considering the decay–width difference of the $B^{0}_{s}$ system.
We perform an unbinned ML fit to the reconstructed mass of the $B^{0}_{s}$
candidates and the helicity angles in order to measure the polarization
amplitudes. The contribution of each candidate to the likelihood is
$\mathcal{L}_{i}=f_{s}\mathcal{P}_{s}(m_{B\,i},\vec{\omega}_{i}|\vec{\xi}_{s})+(1-f_{s})\mathcal{P}_{b}(m_{B\,i},\vec{\omega}_{i}|\vec{\xi}_{b})$,
where $f_{s}$ is the signal fraction and $\mathcal{P}_{j}$ are the probability
density functions (PDFs) for the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ signal ($j=s$) and
background ($j=b$) components, which depend on the fit parameters
$\vec{\xi}_{s}$ and $\vec{\xi}_{b}$, respectively. The effects of neglecting
the reflection background are included in the systematic uncertainties. Both
the signal and the background PDFs are the products of a mass component,
described earlier, and an angular one. The signal angular component is given
by Eq. 1 multiplied by an acceptance factor. The acceptance is computed in
bins of the helicity angles from simulated
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decays averaged over all
possible spin states of the decay products and passed through detector
simulation, full reconstruction, and analysis cuts. We use an empirical
parameterization derived from the observed angular distributions in the mass
sidebands to model the background angular PDF: the product of a flat
distribution for the $\varPhi$ angle and a parabolic function for the other
two, whose single parameter $b_{1}$ is a fit parameter. We fix
$\tau_{\textup{L}}$ and $\tau_{\textup{H}}$ to the world average values
pdg2010 . There are eight free parameters in the fit: $f_{s}$,
$\vec{\xi}_{s}=(M,\sigma,|A_{0}|^{2},|A_{\parallel}|^{2},\cos\delta_{\parallel})$
and $\vec{\xi}_{b}=(b_{0},b_{1})$. The fit has been extensively tested using
simulated samples with a variety of input parameters and shows unbiased
estimates of parameters and their uncertainties. We also perform the
polarization measurement using the sample of $\approx$1700
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997ptJ\\!/\\!\psi\phi$ candidates
described earlier. We find $|A_{0}|^{2}=0.534\pm 0.019({\rm stat})$ and
$|A_{\parallel}|^{2}=0.220\pm 0.025({\rm stat})$, in good agreement with
current measurements psiphiparam .
Figure 2: Angular distribution for
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ events with the fit
projection, signal, and background component superimposed.
The results of the polarization analysis for the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ sample are summarized in
Table 1. In Fig. 2 we show the fit projections onto the helicity angles. The
dominant correlation of the fit parameters is between $|A_{0}|^{2}$ and
$|A_{\parallel}|^{2}$ (-0.447), the others being much smaller.
Table 1: Summary of the $B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ measurements. The first uncertainty quoted is statistical and the second is systematic. Observable | Result
---|---
${\mathcal{B}}$ | $[2.32\pm 0.18\pm 0.82]\times 10^{-5}$
$|A_{0}|^{2}$ | $0.348\pm 0.041\pm 0.021$
$|A_{\parallel}|^{2}$ | $0.287\pm 0.043\pm 0.011$
$|A_{\perp}|^{2}$ | $0.365\pm 0.044\pm 0.027$
$\cos\delta_{\parallel}$ | $-0.91^{+0.15}_{-0.13}\pm 0.09$
$A_{u}$ | $-0.007\pm 0.064\pm 0.018$
$A_{v}$ | $-0.120\pm 0.064\pm 0.016$
Several sources of systematic uncertainty have been studied. We account for
the neglected physics backgrounds considering the
$B^{0}\kern-3.00003pt\to\kern-1.99997pt\phi K^{\ast}(892)^{0}$ decay and two
other possible contaminations: $B^{0}_{s}\kern-3.00003pt\to\kern-1.99997pt\phi
f_{0}(980)$, with $f_{0}\kern-3.00003pt\to\kern-1.99997ptK^{+}K^{-}$, and
$B^{0}_{s}\kern-3.00003pt\to\kern-1.99997pt\phi K^{+}K^{-}$ (non–resonant).
The latter two contributions are normalized to the signal yield in analogy
with similar $B^{0}\kern-3.00003pt\to\kern-1.99997pt\phi X$ decays. We assume
up to 4.6% contamination from $B^{0}_{s}\to\phi f_{0}$ and 0.9% of
$B_{s}^{0}\to\phi K^{+}K^{-}$, and determine a 1.5%(0.4%) shift in the central
value for $|A_{0}|^{2}$($|A_{\parallel}|^{2}$) using simulated experiments.
Biases introduced by the time integration are examined with MC simulation:
they are created by the dependence of the angular acceptance on
$\Delta\Gamma_{s}$ and by a non–uniform acceptance in the $B^{0}_{s}$ proper
decay time introduced by the displaced–track trigger. The assigned systematic
uncertainty (1%) is the full shift expected in the central value, assuming a
value for $\Delta\Gamma_{s}$ equal to the world average plus one standard
deviation pdg2010 . We also consider the propagation of $\tau_{\textup{L(H)}}$
uncertainties to the polarization amplitudes (1%). Other sources of minor
systematic uncertainties are the modeling of the combinatorial background
(0.4%) and of the angular acceptance (0.5%). The impact of $C\\!P$–violating
effects on the measured amplitudes is negligible.
The asymmetries $A_{i}$ ($i=u,v$) are evaluated through an unbinned ML fit to
$m_{B}$ only, using the joint likelihood for the $N^{+}_{i}$ and $N^{-}_{i}$
events with positive and negative $u$($v$). The same $m_{B}$ PDF
parameterization discussed above is used for samples with both $u$($v$) signs.
We multiply the total likelihood by the binomial $f(N^{+}_{i},N^{-}_{i}|p)$,
where the probability $p$ of obtaining $N^{+}_{i}$ and $N^{-}_{i}$ events
depends on the overall signal fraction $f_{s}$, the signal asymmetry $A_{i}$,
and the background asymmetry $A_{b}^{i}$:
$p=\frac{1}{2}[1+A_{i}f_{s}+(1-f_{s})A_{b}^{i}]$. Mass and width for the
$B^{0}_{s}$ signal, as well as signal fraction, are consistent with those
obtained in the polarization analysis, while background asymmetries are
consistent with zero. The measured
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ asymmetries are reported
in Table 1. The systematic uncertainty is evaluated using an alternate
background parameterization as in the polarization analysis and by
conservatively assigning maximal asymmetry to the neglected physics background
peaking in the signal region. Using a large sample of simulated events, we
check that the detector acceptance and resolution introduce a bias in the
asymmetries smaller than 0.2%.
In summary, we measure for the first time the polarization amplitudes and the
triple product asymmetries in the
$B_{s}^{0}\kern-3.00003pt\to\kern-1.99997pt\phi\phi$ decay. We find a
significantly suppressed longitudinal fraction
$f_{\text{L}}=|A_{0}|^{2}=0.348\pm 0.041({\rm stat})\pm 0.021({\rm syst})$,
smaller than in other $b\kern-3.00003pt\to\kern-1.99997pts$ penguin
$B\kern-3.00003pt\to\kern-1.99997ptVV$ decays phikstar . This result agrees
well with predictions BVV-QCDf based on QCD factorization, but only
marginally with perturbative QCD ones Ali:2007ff , and hints at a large
penguin annihilation contribution Datta:2008wf . The two measured asymmetries
are statistically consistent with the no $C\\!P$ violation hypothesis,
although $A_{v}$ is $1.8\,\sigma$ different from zero.
We thank D. London, A. Datta, M. Gronau, and I. Bigi for valuable discussions
on time–integrated TP asymmetries. We thank the Fermilab staff and the
technical staffs of the participating institutions for their vital
contributions. This work was supported by the U.S. Department of Energy and
National Science Foundation; the Italian Istituto Nazionale di Fisica
Nucleare; the Ministry of Education, Culture, Sports, Science and Technology
of Japan; the Natural Sciences and Engineering Research Council of Canada; the
National Science Council of the Republic of China; the Swiss National Science
Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und
Forschung, Germany; the Korean World Class University Program, the National
Research Foundation of Korea; the Science and Technology Facilities Council
and the Royal Society, UK; the Russian Foundation for Basic Research; the
Ministerio de Ciencia e Innovación, and Programa Consolider-Ingenio 2010,
Spain; the Slovak R&D Agency; the Academy of Finland; and the Australian
Research Council (ARC).
## References
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* (11) A. Datta, M. Duraisamy, and D. London, Phys. Lett. B 701, 357 (2011); M. Gronau and J. L. Rosner, arXiv:1107.1232 [hep-ph].
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* (13) D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 032001 (2005).
* (14) CDF uses a cylindrical coordinate system with the $z$ axis along the proton beam axis. Pseudorapidity is $\eta\equiv\ln(\tan(\vartheta/2))$, where $\vartheta$ is the polar angle, and $\phi$ is the azimuthal angle while $p_{T}=|p|\sin(\vartheta)$.
* (15) C. S. Hill, Nucl. Instrum. Methods A530, 1 (2004); A. Sill et al., Nucl. Instrum. Methods A 447, 1 (2000); A. Affolder et al., Nucl. Instrum. Methods A 453, 84 (2000).
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* (17) G. Ascoli et al., Nucl. Instrum. Methods A 268, 33(1988).
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* (19) W. Ashmanskas et al., Nucl. Instrum. Methods A 518, 532 (2004); G. Punzi and L. Ristori, Ann. Rev. Nucl. Part. Sci. 60, 595 (2010).
* (20) $L_{\rm xy}$ is calculated as the displacement of the two-track intersection point with respect to the beam axis, projected onto the total transverse momentum of the track pair.
* (21) K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).
* (22) E. Gerchtein and M. Paulini, eConf C0303241, TUMT005 (2003), arXiv:physics/0306031.
* (23) D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 94, 122001 (2005); T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 103, 031801 (2009); A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 98, 122002 (2007).
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|
arxiv-papers
| 2011-07-25T17:02:58 |
2024-09-04T02:49:20.921269
|
{
"license": "Public Domain",
"authors": "CDF Collaboration",
"submitter": "Marco Rescigno",
"url": "https://arxiv.org/abs/1107.4999"
}
|
1107.5061
|
# Resolving the CO Snow Line in the Disk around HD 163296
Chunhua Qi11affiliation: Harvard–Smithsonian Center for Astrophysics, 60
Garden Street, MS 42, Cambridge, MA 02138, USA; cqi, koberg, dwilner, mhughes,
sandrews@cfa.harvard.edu. , Paola D’Alessio22affiliation: Centro de
Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, 58089
Morelia, Michoacán, México; p.dalessio@crya.unam.mx, sayala2001@gmail.com ,
Karin I. Öberg11affiliation: Harvard–Smithsonian Center for Astrophysics, 60
Garden Street, MS 42, Cambridge, MA 02138, USA; cqi, koberg, dwilner, mhughes,
sandrews@cfa.harvard.edu. 33affiliation: Hubble Fellow , David J.
Wilner11affiliation: Harvard–Smithsonian Center for Astrophysics, 60 Garden
Street, MS 42, Cambridge, MA 02138, USA; cqi, koberg, dwilner, mhughes,
sandrews@cfa.harvard.edu. , A. Meredith Hughes11affiliation:
Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, MS 42,
Cambridge, MA 02138, USA; cqi, koberg, dwilner, mhughes,
sandrews@cfa.harvard.edu. 44affiliation: Department of Astronomy, University
of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA
55affiliation: Miller Fellow , Sean M. Andrews11affiliation:
Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, MS 42,
Cambridge, MA 02138, USA; cqi, koberg, dwilner, mhughes,
sandrews@cfa.harvard.edu. , Sandra Ayala22affiliation: Centro de
Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, 58089
Morelia, Michoacán, México; p.dalessio@crya.unam.mx, sayala2001@gmail.com
###### Abstract
We report Submillimeter Array (SMA) observations of CO (J=2–1, 3–2 and 6–5)
and its isotopologues (13CO J=2–1, C18O J=2–1 and C17O J=3–2) in the disk
around the Herbig Ae star HD 163296 at $\sim 2^{\prime\prime}$ (250 AU)
resolution, and interpret these data in the framework of a model that
constrains the radial and vertical location of the line emission regions.
First, we develop a physically self-consistent accretion disk model with an
exponentially tapered edge that matches the spectral energy distribution and
spatially resolved millimeter dust continuum emission. Then, we refine the
vertical structure of the model using wide range of excitation conditions
sampled by the CO lines, in particular the rarely observed J=6–5 transition.
By fitting 13CO data in this structure, we further constrain the vertical
distribution of CO to lie between a lower boundary below which CO freezes out
onto dust grains (T $\lesssim 19$ K) and an upper boundary above which CO can
be photodissociated (the hydrogen column density from the disk surface is
$\lesssim 10^{21}$ cm-2). The freeze-out at 19 K leads to a significant drop
in the gas-phase CO column density beyond a radius of $\sim$155 AU, a “CO snow
line” that we directly resolve. By fitting the abundances of all CO
isotopologues, we derive isotopic ratios of 12C/13C, 16O/18O and 18O/17O that
are consistent with quiescent interstellar gas-phase values. This detailed
model of the HD 163296 disk demonstrates the potential of a staged, parametric
technique for constructing unified gas and dust structure models and
constraining the distribution of molecular abundances using resolved multi-
transition, multi-isotope observations.
circumstellar matter —techniques: interferometric —planetary systems:
protoplanetary disks —stars: individual(HD 163296) —ISM: abundances —radio
lines: stars
## 1 Introduction
The disks around pre-main sequence stars are the reservoirs of raw material
that represent the initial conditions for the formation of planetary systems.
The spatial distribution of mass in these disks is a fundamental property, as
it sets a constraint on the contents and orbital architectures of planetary
systems. In that sense, measurements of disk densities and temperatures can
provide strong, albeit complex, constraints on planet-forming scenarios.
Because the dominant constituent of these disks is believed to be cold H2 gas
that is generally unobservable, our knowledge of many disk properties
(including mass) relies on the observation and interpretation of minor
constituents. Dust has naturally been the focus of the most work on disk
structure and evolution, as it dominates the opacity and emits bright
continuum radiation (e.g., Andrews et al., 2009, 2010). However, converting
those continuum surface brightnesses to disk (gas) densities is challenging
due to the large uncertainties in the assumed dust properties (composition,
size distribution, etc.) and the dust abundance relative to the gas, which may
deviate substantially from the canonical interstellar value (0.01) and vary
spatially in the disk. In principle, spectral line emission from trace gas
species can provide complementary constraints on the disk structure, but the
interpretation of the observations can also be difficult. These emission lines
depend on the gas abundances that are set by chemical and physical processes,
as well as the excitation conditions set by the local densities, temperatures,
and incident radiation fields. With a dataset of sufficient quality, these
complexities can be leveraged into powerful probes of otherwise inaccessible
disk characteristics (e.g., Kamp et al., 2010).
The chemical property of CO, the most abundant molecule after H2, is thought
to be well understood in disks. The spatial distribution of gas-phase CO is
expected to be shaped by photodissociation both near the star and high in the
disk atmosphere, as well as by a depletion process when it freezes onto dust
grains at low temperatures (below $\sim$20 K) at large radii and deep in the
disk interior (e.g., Aikawa et al., 1996; Aikawa & Nomura, 2006; Gorti &
Hollenbach, 2008). The peak abundance of CO is also predicted to be almost
independent of radius, around 10-4, the typical value in molecular clouds
(Aikawa & Nomura, 2006). However, observational studies of CO emission from
disks have yielded puzzling results. In general, the CO in disks is found to
be under-abundant by factors of 10–100 compared to molecular clouds, a deficit
usually attributed to CO depletion in cold, dense gas (Dutrey et al., 1994,
1996, 1997; van Zadelhoff et al., 2001; Thi et al., 2004). However, multi-
transition and multi-isotope CO studies report a wide range of relative
depletions of CO and its isotopologues. For example, Dartois et al. (2003)
finds that all of the CO isotopologues are depleted by an identical uniform
factor of $\sim$10, but Piétu et al. (2007) finds in a small sample of disks
that the 12CO/13CO ratios in the outer part of disks are much lower than the
standard 12C/13C ratio in the solar neighborhood: these low values were
attributed to significant carbon fractionation. At the same time, both of
these studies suggested that the outer radius derived from 13CO is smaller
than the one derived from 12CO indicating their 12CO/13CO ratios are
abnormally large at the outer disk edge of their models, which leads to
speculations on the importance of selective photo-dissociation due to self-
shielding in the photolysis of gaseous CO (Dutrey et al., 2007). They also
find that a large amount of CO remains gaseous at temperatures as low as 10 K.
Possible explanations for the presence of the inferred cold CO include
vertical mixing (Aikawa, 2007) and photodesorption off dust grains (Hersant et
al., 2009). All above depict a very complex picture of the distribution of CO
gas in the disks, which remains to be explained by detailed chemical models
combined with realistic physical disk structures. However, it is unclear, how
much of these variations depend on the individual sources, and how much, if
any, depends on the modeling approach.
A robust methodology for interpreting CO observations is a high priority for
our efforts to constrain the total gas distribution in disks, and to provide a
basis for interpreting observations of more complex molecules. In this study,
we develop a modeling framework that is able to account for observations of
both the dust continuum and CO line emission from a protoplanetary disk.
Specifically, we combine a previously-established dust emission modeling
formalism with multiple transitions, spatially resolved CO line data (CO
J=2–1, 3–2 and 6–5, 13CO J=2–1, C18O J=2–1 and C17O J=3–2) to construct a
simplified and internally self-consistent model of the disk around the Herbig
Ae star HD 163296. At a distance of $\sim$122 pc (Perryman et al., 1997), this
2.3 M⊙ star (spectral type A1; age $\sim$4 Myr) harbors a large disk with
strong millimeter continuum and molecular line emission (Qi, 2001; Thi et al.,
2004; Natta et al., 2004). It does not seem to be associated with any known
star-formation region, dark clouds, or reflection nebulae. The disk around HD
163296 exhibits a scattered light pattern extending out to a radius of
$\sim$500 AU (Grady et al., 2000), perpendicular to a bipolar microjet traced
by a chain of HH knots visible in coronagraphic observations (Devine et al.,
2000; Grady et al., 2000; Wassell et al., 2006). Millimeter and submillimeter
interferometric observations have explored this large disk and its Keplerian
velocity field in low-J CO lines (Mannings & Sargent, 1997; Isella et al.,
2007). The large disk size and strong molecular emission make this system an
excellent target for resolved millimeter observations and detailed modeling.
This paper is organized as follows. We describe the CO and mm continuum data
in §2, and present the main observational results in §3. Our extensive
modeling effort is detailed in §4. We discuss the modeling results and compare
with previous work on this subject in §5, and provide a brief summary in §6.
## 2 Observations
Observations of HD 163296 (RA: 17h56m21$\fs$279, DEC: $-$21°57′22$\farcs$09;
J2000.0) were conducted between 2005 August and 2010 September using the
8-antenna SMA111The Submillimeter Array is a joint project between the
Smithsonian Astrophysical Observatory and the Academia Sinica Institute of
Astronomy and Astrophysics, and is funded by the Smithsonian Institution and
the Academia Sinica. interferometer located atop Mauna Kea, Hawaii. Table 1
provides a general summary of the observational parameters.
For the observations of the CO 6–5 line, we used the correlator settings
adopted by Qi et al. (2006) in their similar study of the TW Hya disk. At 690
GHz, there is no nearby quasar bright enough to use for phase referencing with
the SMA. However, Callisto was located only 10° away from HD 163296 during our
observation on 2007 March 20, and was monitored every 20 minutes for use in
the gain and absolute flux calibration. At that time, Callisto had a diameter
of $1\farcs 3$ and a zero-spacing flux density of 48.1 Jy at 690 GHz. Based on
the uncertainties of the Callisto emission model, we estimate a 10% systematic
uncertainty in this adopted flux scale. The 690 GHz continuum emission from HD
163296 was sufficiently bright (with a flux density of 7.5 Jy) that we could
correct the gain response of the SMA with a single phase-only self-calibration
iteration.
The observations of the CO 2–1 line took place on 2010 May 17 in the compact
configuration and 2010 September 14 in the extended configuration. The
correlator was configured to include 2048 channels in the 104 MHz segment of
the correlator centered on the rest frequency of the line. The remaining 1.3
GHz of the correlator bandwidth in both sidebands were configured with a
uniform spectral resolution of 256 channels in each 104 MHz correlator segment
to achieve maximum continuum sensitivity. The weather was good on both nights,
with the 225 GHz opacity just below 0.1 and stable atmospheric phase. The
nearby quasar 1733-130 (10.5 degrees from the source) was used to calibrate
the atmospheric and instrumental phase, and the solution was checked using the
quasar 1911-201, which was included for a short period during each loop
between source and calibrator. Uranus and Neptune were used as flux
calibrators on September 14, yielding a 1733-130 flux of 2.71 Jy; Callisto was
used to calibrate the May 17 flux, yielding a 1733-130 flux of 2.04 Jy.
The observations of 13CO 2–1 and C18O 2–1 were carried out simultaneously on
2010 May 15 in the compact configuration and 2010 September 11 in the extended
configuration. The correlator configuration was identical to the CO 2–1 setup,
except that the high resolution correlator chunk was centered on the rest
frequency of 13CO 2–1 with C18O 2–1 in a nearby low-resolution chunk. The
remaining 1.2 GHz were devoted to uniform spectral resolution continuum
observations. The weather was excellent on both nights with stable phase and
225 GHz opacity varying between 0.04 (primarily during the extended track) and
0.06. The same calibrators as for the CO(2-1) observations were used; the
derived flux of 1733-130 was 2.35 on September 11 and 2.04 Jy on May 15.
The observation of the C17O 3–2 line took place on 2005 August 23 in the
compact configuration. The correlator was configured with a uniform spectral
resolution of 128 channels over 104 MHz, which provided 0.8 MHz frequency
resolution. The nearby quasars 1833-210 and 1921-293 were used as the
calibrators. Uranus was used as flux calibrator and the derived fluxes of
1833-210 and 1921-293 were 0.57 and 3.76 Jy, respectively.
The observations of the CO 3–2 observations were summarized in Hughes et al.
(2011). All of the calibration was performed using the MIR software
package222http://www.cfa.harvard.edu/$\sim$cqi/mircook.html. Images of the
continuum and the spectral lines were generated and CLEANed using standard
techniques in the MIRIAD software package.
## 3 Results
Figure 1 shows the disk-averaged spectra of the CO J=2–1, 3–2 and 6–5 lines,
along with the J=2–1 lines of 13CO and C18O and the J=3–2 line of C17O 3–2.
These spectra were extracted from the SMA channel maps in 10′′ square boxes
centered on HD 163296, except for the CO 6–5 line. That fainter spectrum was
produced from a 4′′-wide box chosen to cover the area with statistically
significant emission. Each emission line shows the double-peaked velocity
profile characteristic of disk rotation. The low-J CO lines have an additional
feature in their spectra at $V_{\rm LSR}$ = 13 km s-1, which is distinctly
offset from the disk kinematically and much more obvious in previous single-
dish spectra (Thi et al., 2001, 2004). That feature is apparently spatially
extended, and likely is associated with foreground and/or background molecular
clouds. Table 2 lists the integrated intensities of the observed lines and the
continuum flux densities. Note that the signal-to-noise ratios range from over
200 for the CO 2–1 line to $\sim$10 for the CO 6–5 and C17O 3–2 lines. Figure
2, 3 and 4 show the channel maps of all the transitions except for CO 3–2
which were presented in Hughes et al. (2011).
## 4 Analysis
As a first step in developing a framework for understanding the chemical
structures of protoplanetary disks, we aim to interpret this suite of
observations in the context of an internally self-consistent model for the
physical conditions of the gas and dust in the HD 163296 disk. Our modeling
analysis consists of three distinct stages. First, we construct a template
two-dimensional model for the disk structure based on observations of the
dust, using the broadband spectral energy distribution (SED) and the resolved
millimeter continuum emission as diagnostics (§4.1). Second, we refine the
vertical temperature distribution of that template structure based on the
spatially resolved line ratios of the optically thick CO transitions (§4.2).
And third, we constrain the radial and vertical CO abundance pattern using the
emission from the more optically thin CO isotopologue transitions (§4.3).
### 4.1 Stage 1: Template Disk Structure Model
Our starting point is with a set of disk structures calculated following the
prescription for a steady viscous accretion disk model (see D’Alessio et al.,
1998, 1999, 2001, 2006). In these models, the gas surface density is
determined based on the conservation of angular momentum flux and depends on
the mass flow rate ($\dot{M}$), viscosity coefficient $\alpha$ (Shakura &
Sunyaev, 1973), and midplane temperature ($T_{0}$), such that
$\Sigma\propto\dot{M}/\alpha T_{0}$. We assume that $\dot{M}$ is constant
throughout the disk, but make an important structural modification from
previous generations of these models. Motivated by the similarity solution for
the time evolution of accretion disks (Hartmann et al., 1998) and recent
millimeter-wave observations (Hughes et al., 2008), we allow the viscosity
coefficient to vary radially such that $\alpha=\alpha_{0}\exp{R/R_{c}}$. This
change makes no difference at small radii ($R\ll R_{c}$), but effectively adds
an exponential taper to the surface density profile outside the characteristic
radius, $R_{c}$. Although this modification mimics the $\Sigma$ profile
derived from the similarity solution models, it does not reproduce the
predicted behavior of the mass flow rate as a function of radius. However,
that $\dot{M}$ profile is not expected to have any observable effects on the
disk structure at these large radii, since stellar irradiation is a much more
important heating mechanism than viscous dissipation in those regions.
For a given flow rate ($\dot{M}$), viscosity coefficient ($\alpha_{0}$), and
characteristic radius ($R_{c}$), the density and temperature structure of this
model is determined as described by D’Alessio et al. (1998, 1999). We consider
heating from the mechanical work of viscous dissipation (relevant only in the
midplane of the inner disk), accretion shocks at the stellar surface, and
passive stellar irradiation, and follow the radiative transfer of that energy
with 1+1D calculations using the Eddington approximation and a set of mean
dust opacities (gas opacities are considered negligible). The dust is assumed
to be a mixture of segregated spheres composed of “astronomical” silicates and
graphite, with abundances (relative to the total gas mass) of $\zeta_{\rm
sil}=0.004$ and $\zeta_{\rm gra}=0.0025$ (Draine & Lee, 1984): the “reference”
dust-to-gas mass ratio is $\zeta_{\rm ref}=0.0065$. At any given location in
the disk, the grain size ($a$) distribution of these dust particles is assumed
to be a power-law, $n(a)\propto a^{-3.5}$, between $a_{\rm min}=0.005$ $\mu$m
and a specified $a_{\rm max}$. We assume the disk has two grain populations,
each with a different maximum size. The “small” grains are distributed in the
disk atmosphere and have $a_{\rm max}=0.25$ $\mu$m (as in the interstellar
medium), and the “big” grains are concentrated toward the midplane and have
$a_{\rm max}=1$ mm. Utilizing the dust settling prescription of D’Alessio et
al. (2006), the “small” grain population in the upper disk layers is assumed
to have a dust-to-gas mass ratio $\epsilon\zeta_{\rm ref}$ and the “big” grain
population near the midplane has a dust-to-gas ratio determined so that mass
is conserved at each radius. A smooth vertical transition between the two
grain populations is made at a height $z_{\rm big}$ above the midplane.
The inner boundary of the structure model is taken to be the radius where
silicates are sublimated (where $T\approx 1500$ K). Because this inner rim is
irradiated by the photosphere of the central star (Natta et al., 2000;
Dullemond et al., 2001) and the accretion shock at the stellar surface
(Muzerolle et al., 2004) at a normal incidence angle, the material is heated
to high temperatures and therefore extends (or “puffs” up) to a larger
vertical height, $z_{\rm wall}$. The radial structure of that “wall” feature
(and its emergent spectrum) is calculated as described by D’Alessio et al.
(2005). The structure model is truncated at 540 AU, large enough to have no
effect on our analysis.
In the model calculations, we adopt the HD 163296 stellar parameters advocated
by van den Ancker et al. (1998) – $M_{\ast}=2.3$ M⊙, $R_{\ast}=2$ R⊙, and
$T_{\ast}=9333$ K – and a flow rate equivalent to the accretion rate onto the
star, $\dot{M}=7.6\times 10^{-8}$ M⊙ yr-1 (Garcia Lopez et al., 2006). Based
on the CO observations of Isella et al. (2007), we fixed the disk inclination
angle to $i=44\arcdeg$ and the major axis position angle to 133° east of
north. For this set of fixed parameters, the inner wall is located at a radius
of 0.6 AU. There are five remaining free parameters in the model:
{$\alpha_{0}$, $R_{c}$, $\epsilon$, $z_{\rm wall}$, $z_{\rm big}$}. To
identify a model structure that provides a reasonable match to the HD 163296
observations, we started by comparing synthetic data for a coarse grid of
these parameters with the broadband SED. That parameter search was then
refined by comparing the model predictions with the observed millimeter
continuum visibilities at 271 and 341 GHz. Although not optimized for robust
parameter estimation, this approach yielded a template model that exhibits a
satisfactory match with these diagnostics of the dust disk. The adopted model
has $\alpha_{0}=0.019$, $R_{c}=150$ AU, $\epsilon=0.003$ (the dust-to-gas
ratio in the atmosphere is 0.3% of the reference value, $\zeta_{\rm ref}$),
and $z_{\rm wall}=0.1$ AU. The total mass of the model is 0.089 M⊙. All the
stellar and disk properties are summarized in Table 3. Figure 5 shows how the
value of the $R_{c}$ parameter affects the corresponding continuum visibility
profiles at 271 GHz and 341 GHz, the frequencies with the best resolved
millimeter continuum data. We find the visibility profiles are matched best
with $R_{c}$ at 150 AU, a bit larger than 125 AU derived by Hughes et al.
(2008) with the similarity solutions.
Figure 6 compares the HD 163296 SED with the models with different values of
$z_{\rm big}$, ranging from 0.5$H$ to 2.5$H$, where $H$ is the pressure scale
height calculated from the local temperature value ($H=c_{s}/\Omega$, where
$c_{s}$ is the sound speed and $\Omega$ is the Keplerian angular velocity of
the gas. Note that this is merely a convenient scaling, as the vertical
structure is solved consistently and does not assume a vertically isothermal
profile). The different behaviors of the models are likely caused by the
change in the shape of the irradiation surface, which determines the fraction
of stellar radiative flux intercepted and reprocessed by the disk. Apparently
it is difficult to constrain $z_{\rm big}$ from the SED alone due to the
complexity of the SED modeling in the mid and far-IR wavelengths. The most
substantial discrepancy between the observed SED and the model predictions can
be noted in the strength of the 10 and 18 $\mu$m silicate emission bands. This
might be a reflection that the upper atmosphere layers are hotter than in the
model, or that the optical properties of the grains in the atmosphere are
slightly different (in size distribution and/or composition) than we assume.
The modeling formalism we are using assumes the gas and dust are in thermal
equilibrium at all locations in the disk, even in the upper layers of the
atmosphere. While that is expected to be a good approximation at the height
where most of the stellar radiation is deposited, and where most of the
emission from the bands is produced, the gas at larger heights may be even
hotter (e.g., Glassgold et al., 2004; Kamp & Dullemond, 2004; Jonkheid et al.,
2004; Nomura et al., 2007; Gorti & Hollenbach, 2008; Woitke et al., 2009; Kamp
et al., 2010), which might change the dust density distribution, and hence the
silicate emission.
### 4.2 Stage 2: Refining the Vertical Structure
At the large disk radii probed by our CO observations, the vertical structure
of the disk is determined by irradiation. The small dust grains in the disk
atmosphere absorb energy from the incident stellar radiation field, and some
of that energy is then re-emitted down into the disk interior at longer,
infrared wavelengths. Given the increasing densities near the disk midplane,
this “external” heating of the disk surface naturally produces a structure
with a vertical temperature inversion: the midplane is cooler than the
atmosphere (Calvet et al., 1991; Chiang & Goldreich, 1997; D’Alessio et al.,
1998). In our model prescription for dust settling, the deeper layers in the
disk are populated by big dust grains that have low infrared opacities, and
therefore are heated less efficiently than their smaller counterparts.
Therefore, this concentration of big grains near the disk midplane actually
amplifies the temperature contrast between the surface and interior. In
practice, the location of the transition between the small and big grains,
$z_{\rm big}$, has a pronounced effect on the vertical temperature gradient
(and therefore the vertical density structure). In Figure 7, we demonstrate
how $z_{\rm big}$ impacts the vertical distributions of temperatures and
densities at a fixed radius of 200 AU. A more condensed population of big
grains (lower $z_{\rm big}$) permits more heating at deeper depths into the
disk (i.e., lower $z$), producing a warmer disk interior. Likewise, a more
vertically extended population of big grains (higher $z_{\rm big}$) produces
much lower temperatures in the disk interior.
The effects of $z_{\rm big}$ are difficult to distinguish from the dust
tracers alone: the infrared SED and millimeter visibilities do not effectively
probe the shape of the vertical temperature profile. However, the CO line
emission is expected to be generated in an intermediate disk layer that should
be sensitive to this temperature inversion. The main isotope CO lines are
optically thick, and therefore excellent temperature diagnostics. By measuring
the emission from several CO rotational transitions, their relative strengths
may be used to trace the temperature at different depths in the disk (each
line probing a layer commensurate with its excitation). Previous analysis of
multi-transition CO data have been used to successfully measure the vertical
temperature gradient (e.g., Dartois et al., 2003). In this stage of the
modeling, we utilized the resolved CO 2–1, 3–2, and 6–5 emission lines toward
the HD 163296 disk to determine the value of $z_{\rm big}$ that best
reproduces the inferred temperature gradient present in its intermediate
layer.
To compare the model predictions with the CO data, we first need to fix a set
of geometric and kinematic parameters that affect the observed spatio-
kinematic behavior of the disk. We assume the disk material orbits the central
star in Keplerian motion, and fix the stellar mass and position, and non-
thermal turbulent velocity width ($dV_{turb}$) based on the models of Hughes
et al. (2011) and the references noted in §4.1. The effects of inclination
($i$), position angle of the major axis (PA), and systemic velocity
($V_{lsr}$) are essentially orthogonal to those that depend on the assumed
disk structure, and therefore can be optimized independently. All the disk
geometric and kinematic parameters are summarized in Table 3. We further
assume a simple CO abundance model that depends on the local density and
temperature via the parameters introduced by Qi et al. (2008), described
below. For a given disk structure and CO abundance model, we use the non-local
thermodynamic equilibrium two-dimensional accelerated Monte-Carlo radiative
transfer code RATRAN to calculate the molecular excitation and generate a sky-
projected set of synthetic CO data cubes (Hogerheijde & van der Tak, 2000),
sampled at the same spatial frequencies as each SMA dataset. The collisional
rates are taken from the Leiden Atomic and Molecular Database (Schöier et al.,
2005) for non-LTE line radiative transfer calculations. Specifically, in the
models used in this paper, we have used the new set of the CO collisional rate
coefficients calculated by Yang et al. (2010).
Our adopted abundance model was introduced by Qi et al. (2008), and assumes
that the CO emission originates in a vertical layer of the disk with a
constant abundance. The upper (surface) and lower (midplane) boundaries of
that layer are defined by the parameters $\sigma_{s}$ and $\sigma_{m}$, which
represent vertically-integrated hydrogen column densities from the disk
surface in units of $1.59\times 10^{21}$ cm-2 (the conversion factor of the
hydrogen column to $A_{v}$ for interstellar dust). The CO abundance ($f_{\rm
CO}$) is assumed to be a constant for the hydrogen column densities from the
disk surface $\sigma_{s}\geq N\geq\sigma_{m}$, with respect to the H2 density
specified in the disk structure model.
We compute model CO visibilities for a range of disk structure models with
different $z_{\rm big}$ values, ranging from 0.5–3.0$H$ in 0.5$H$ steps, where
$H$ is the pressure scale height as we described before. For each $z_{\rm
big}$ value, we optimized the CO abundance model ($\sigma_{m}$, $\sigma_{s}$
and fractional abundance $f_{\rm CO}$) to minimize the combined $\chi^{2}$
value in reference to the CO J=2–1 and 3–2 visibility data. Because the
sensitivity of our CO J=6–5 data is relatively modest, this high-excitation
line is better suited to an a posteriori check on the fitting results.
Nevertheless, the J=6–5 line has a sufficiently high excitation that its
luminosity relative to the low-lying J=2–1 and 3–2 lines provides perhaps the
best discriminant between different $z_{\rm big}$ values. Figure 8 shows the
results of this stage of the modeling analysis, marking a direct comparison
between the observed and model spectra of the main CO transitions for 3
different $z_{\rm big}$ values - 1.5, 2.0, and 2.5$H$. While any of these
models provides a suitable match to both the J=2–1 and 3–2 spectra, only the
$z_{\rm big}=2H$ model also matches the J=6–5 line. At R=100 AU of this model,
the abundance of big grains has decreased to 50% of its maximum value at a
height around 16 AU from the midplane. Figure 9 shows our favored two-
dimensional density and temperature distributions for this $z_{\rm big}=2H$
model.
### 4.3 Stage 3: CO Abundance Distributions
We have established a gas+dust structure model that reproduces well the
broadband SED, resolved millimeter continuum images, and a multi-transition
set of CO spectral images. The next step in the modeling analysis is to
constrain the spatial distribution of the CO abundance, relying on the
spatially resolved - and more optically thin - CO isotopologue emission, which
probe much deeper into the midplane. Here we assume CO and its isotopologues
share the same spatial distribution and only differ in fractional abundances.
Instead of $\sigma_{m}$, at this stage we adopt the lower boundary (toward
midplane) governed by the CO freeze-out temperature, TCO, such that $f_{\rm
CO}\rightarrow 0$ when $T<T_{\rm CO}$, which provides a way of interpreting
this lower boundary more physically. For the upper boundary (toward surface),
we still adopt $\sigma_{s}$ as before.
Using the fixed structure model derived in §4.1 and 4.2, we compute a grid of
synthetic 13CO J=2–1 visibility datasets over a range of $\sigma_{s}$, $T_{\rm
CO}$, and $f_{\rm{}^{13}CO}$ values and compare with the observations. Figure
10 shows the $\chi^{2}$ surfaces for the 13CO J=2–1 emission in the space of
these three parameters. We find that the data are best described when
$\sigma_{s}=0.79\pm 0.03$ (i.e., the hydrogen column densities from the disk
surface are below 1.2–1.3$\times 10^{21}$ cm-2), $T_{\rm CO}=19.0\pm 0.3$ K,
and $f_{\rm{}^{13}CO}=9.0(\pm 0.6)\times 10^{-7}$. Figure 9 shows the
locations of the CO emission in gray shade on top of the temperature and
density profiles (top and middle panels) and the two vertical boundaries
($\sigma_{s}=0.79$ and $T_{\rm CO}=19$K) at R = 200 AU (bottom panel).
Figure 11 shows the effect of the CO freeze-out temperatures TCO on the radial
column density of 13CO in the model. Without any freeze-out, the 13CO column
density follows the exponential taper of the H2 density profile. The freeze-
out at 19 K leads to a significant drop in the gas-phase 13CO column density
beyond a radius of $\sim$155 AU (or 310 AU in diameter), which we directly
resolve. Even though the 13CO column densities are different by orders of
magnitude , the line emission difference projected to the line-of-sight from
the outer disk is very subtle and high spectral resolution data is fundamental
for resolving any molecular abundance structure changes at radii beyond
$\sim$155 AU. Indeed a similar analysis of the C18O J=2–1 emission alone
indicates that the data can be fit equally well (or perhaps better) by models
that do not include CO freeze-out (i.e. the lower boundary is the midplane,
$z=0$). However, this apparent inconsistency is likely the result of the
10$\times$ lower velocity resolution of the C18O data. Therefore, we don’t fit
for CO vertical boundaries from the emissions of C18O 2–1 and C17O 3–2 due to
their weaker signals and limited spectral resolutions.
Keeping that resolution effect in mind, we adopt the abundance boundaries
derived from the 13CO 2–1 data and then fit the CO, C18O, and C17O data with
only the fractional abundances as free parameters. For the $\sigma_{s}$ and
$T_{\rm CO}$ derived above, we find the fractional abundances of CO, C18O and
C17O to be $6.0(\pm 0.3)\times 10^{-5}$, $1.35(\pm 0.20)\times 10^{-7}$ and
$3.5(\pm 1.1)\times 10^{-8}$ respectively, corresponding to CO/13CO = $67\pm
8$, CO/C18O = $444\pm 88$ and CO/C17O = $3.8\pm 1.7$. Our derived isotopic
ratios are all consistent with the quiescent interstellar gas-phase values,
which Wilson (1999) find in the local ISM to be CO/13CO = $69\pm 6$, CO/C18O =
$557\pm 30$, and CO/C17O = $3.6\pm 0.2$. Our final model parameters are listed
in Table 4. The models are directly compared with the data channel maps in
Figures 12 and Figure 13 (with the velocities binned in 1 km s-1 channels).
Table 4 also shows the best-fit fractional abundances in a model of HD 163296
that does not include CO freeze-out; the 13CO/C18O ratio is determined to be
$30.6\pm 6.1$, about four times higher than the value $6.7\pm 1.4$ derived
from our best-fit models that consider CO freeze-out at 19 K (Table 4) or the
local ISM value $8.1\pm 1.1$ (Wilson, 1999). This provides an indirect
evidence that we should also take into account CO freeze-out in the C18O 2–1
data analysis.
## 5 Discussion
### 5.1 Modeling Dust Emission and CO Line Emission
We have modeled multiple emission lines of CO and its isotopologues from the
disk around HD 163296 in the context of an accretion disk model structure,
grounded in observations of the broadband SED and resolved millimeter
continuum emission. The goal of this modeling effort is to develop a more
consistent examination of the connection between the gas and dust phases in
the disk. While our modeling framework does not treat the complexity of a
completely self-consistent, simultaneous description of the energy balance and
chemistry between the gas and dust, it effectively employs a set of parameters
that can retrieve molecular abundance information in a way that captures the
essential character of the layered disk structures predicted by those more
sophisticated models. Most importantly, our approach directly addresses two
common issues that are noted in much simpler structure models: (1) a radius
(or size) discrepancy between the dust and CO emission, and (2) the degeneracy
in the vertical temperature structure for models based solely on the dust
emission (i.e., the SED).
A longstanding problem with disk models has been the seemingly different
radial distributions of dust and gas when each is considered independently
(Dutrey et al., 2007). Isella et al. (2007) presented multi-wavelength
millimeter continuum and CO isotopologue observations of the disk around HD
163296 and found a significant discrepancy between the outer radius derived
for the dust continuum ($200\pm 15$ AU) and that derived from CO emission
($540\pm 40$ AU) in their truncated power law models. However, Hughes et al.
(2008) showed that models with tapered outer edges can naturally reconcile the
apparent size discrepancy in dust and gas millimeter imaging. The successful
fitting of CO isotopologues in the HD 163296 disk using the SED-based disk
model with an exponentially tapered outer edge, without invoking an unknown or
unconstrained chemical effect, provides new support for the necessity of
including this feature in outer disk structure.
The SED alone does not provide any direct information on the temperature
structure of the intermediate disk layers where the CO (and other molecular)
emission is generated. In our modeling framework for the HD 163296 disk, the
SED (and millimeter continuum images) can be fit equally well for a wide range
of vertical temperature/density profiles, highlighting the degeneracy of the
dust data with the parameter $z_{\rm big}$, the height marking the transition
between the small grains in the disk atmosphere and the big grains
concentrated toward the midplane. However, we have found that resolved
observations of optically thick CO lines at a range of excitations can be used
to place stringent constraints on the vertical temperature structure. Previous
analysis of the CO J=2–1 and 3–2 lines from the HD 163296 disk suggested that
gas temperatures were always higher than 20 K, ruling out CO freeze-out as a
cause for the depletion of CO abundances (Isella et al., 2007). But as we
discussed in §4.2 (see Figure 8), the small excitation leverage between those
low-lying transitions is not a strong discriminant of the temperature profile.
Here, we make use of the higher-excitation J=6–5 line to better constrain the
vertical structure of the disk, and find a colder midplane that is consistent
with significant CO freeze-out. Although observations of these various CO
transitions is expensive, there are likely other molecules that emit at nearby
frequencies and can be used to trace a sufficient range of excitation
conditions (e.g., HCO+). The essential point is that the temperature structure
in the intermediate layers of a disk can only be constrained well using
several optically thick emission lines that probe a range of excitation.
### 5.2 The “CO Snow Line”
Theoretical models of the molecular abundances in protoplanetary disks predict
three distinct vertical layers (see Bergin et al., 2007). At large heights in
the disk atmosphere, temperatures are high and molecular abundances are
comparable to those in a standard photon-dominated region (PDR). At
intermediate heights, between the midplane and atmosphere, warm temperatures
are suitable for high gas-phase abundances of many molecules. And at the
deepest layers near the midplane, temperatures are low enough that many
molecules are depleted from the gas phase and frozen onto the mantles of dust
grains. Aikawa & Nomura (2006) and others, have used this kind of layered
physical structure in their chemical reaction network models and shown that CO
is expected to be abundant in the warm layer, with $f_{\rm CO}\sim 10^{-4}$
almost independent of radius. The boundaries of that layer are defined by the
CO freeze-out in the midplane and the photodissociation of CO by high-energy
stellar photons in the disk atmosphere. The modeling of the SMA observations
of the HD 163296 disk provides direct support for this basic model structure.
Our estimate of the upper boundary $\sigma_{s}$ is in excellent agreement with
models of the upper PDR layer, where CO is photodissociated into its atomic
constituents by stellar UV and/or X-ray radiation at column densities
$\lesssim 10^{21}$ cm-2 333A total gas column of 1021 cm-2 corresponds to 1
mag of extinction (A${}_{v}=$1) at visible wavelengths when the gas-to-dust
ratio is 100. But in our model the CO photodissociation front is located at
A${}_{v}\ll$1 because of the large dust depletion factor as measured with
$\epsilon=0.003$ in the upper disk layers. (Aikawa & Nomura, 2006; Gorti &
Hollenbach, 2008). The lower boundary that best fits the 13CO data,
corresponding to a temperature of 19 K, is also in excellent agreement with
the laboratory studies of CO freeze-out temperature onto dust grains (Collings
et al., 2003; Bisschop et al., 2006). The main effect of CO photodesorption
(Öberg et al., 2007; Hersant et al., 2009) and other related non-thermal ice
desorption mechanisms is to push the lower boundary to colder temperatures.
Since our best-fit CO lower boundary is consistent with thermal desorption,
there is no evidence for efficient CO photodesorption in the disk of HD
163296.
Since the term “snow line” is often discussed in planetary formation as some
point where the temperature in the midplane would drop below the water ice
sublimation level, e.g. in the “minimum-mass solar nebular” model prescribed
by Hayashi (1981), we propose to use the term “CO snow line” to indicate where
CO freezes out in the disk. It was generally believed that the dust
temperature in the disks of Herbig Ae stars is high enough that even the
temperature close to the disk midplane is above the temperature of freeze-out
of CO (Dutrey et al., 2007; Jonkheid et al., 2007). In this work, the primary
line of evidence for CO freeze-out in the disk of HD 163296 comes from the
detection of sharp reduction of the effective 13CO J=2–1 emission area in the
outer disk. This is demonstrated in Figure 10 where the 13CO visibility is
best fit with the lower boundary at 19 K, indicating no or very few 13CO
emission from the midplane area below this boundary, as shown in Figure 9. The
CO snow line marks drop off at least two orders of magnitude in the column
density of 13CO beyond 155 AU, which is clearly resolved by our
interferometric observations. However, the emission lines from CO and its
isotopologues are still detectable out to large radii, e.g. $>$500 AU, which
indicates that there are still substantial amount of CO toward surface layer
of the outer disk where the temperature is higher than 19 K.
The presence of this CO snow line has a direct impact on some key elements of
chemical networks in a disk, especially related to the ionization fraction.
The CO abundance is expected to be linearly correlated with the HCO+
abundance, but has a strong anti-correlation with N2H+ since the main
destruction pathway for the latter is in reactions with CO (Jørgensen et al.,
2004). In the midplane, the depletion of CO beyond the snow line should
enhance the deuterated molecular ions, H2D+, D2H+ and D${}_{3}^{+}$, such that
they become the most abundant ions in the midplane (Ceccarelli & Dominik,
2005). So, resolving the CO snow line will eventually help constrain the
ionization fraction and the extent of deuterium fractionation in the disk
interior.
### 5.3 CO Isotope Ratios
The model developed here indicates that the gas in the intermediate layer of
the HD 163296 disk has CO isotopologue abundance ratios that are consistent to
those found in the molecular interstellar medium. That finding is in contrast
with some previous studies of this and other disks that instead suggested an
increasing amount of depletion from 12CO to 13CO to C18O (e.g., Isella et al.,
2007; Dutrey et al., 1994, 1996). That difference is likely a manifestation of
the underlying modeling approach, as well as the constraining power now
available to us from the high-excitation J=6–5 line and the rare
isotopologues. When we include CO freeze-out, our models also show no evidence
for radial variations of the CO abundance with respect to the H2 densities
inferred (indirectly) from the dust emission. Previous studies have found
(much) steeper CO surface density profiles compared to dust (e.g., Pietu et
al. 2007, Dutrey et al. 2008). To compare with those findings, we have used an
alternative power-law prescription for the radial abundance fractions,
$f_{i}(r)\propto r^{p_{i}}$, and attempted to re-fit our data. The resulting
best-fit values of $p_{i}$ for 13CO and C18O are $0.0\pm 0.1$ and $0.1\pm
0.2$, respectively: both consistent with a radially constant abundance
profile. So, when taking into account CO freeze-out at low temperatures, there
is no clear evidence that the radial distribution of CO deviates from the
underlying total gas surface density. The steep slopes inferred in previous
work may be a manifestation of the sharp abundance drop beyond the CO snow
line.
This model demonstrates that, given sufficiently strong constraints on the
vertical temperature gradient, a unified model of the gas and dust disk can be
constructed that conforms to standard ISM-based assumptions about the isotopic
abundances and the CO chemistry.
Visser et al. (2009) present a photodissociation model for CO isotopologues
including newly updated depth-dependent and isotope-selective
photodissociation rates. They find grain growth in circumstellar disks can
enhance $N(^{12}{\rm CO})/N({\rm C^{17}O})$ and $N(^{12}{\rm CO})/N({\rm
C^{18}O})$ ratios by a factor of ten relative to the initial isotopic
abundances. We have fit the $\sigma_{s}$ and fractional abundances for C17O
and C18O and found that different pairs of $\sigma_{s}$ and f(C18O) or f(C17O)
can fit the data equally well, i.e. we can not constrain $\sigma_{s}$ due to
the limited signal-to-noise ratios of C18O J=2–1 and C17O 3–2 data. Future
observations with greater sensitivity in the CO isotopologues will be
essential to investigate the isotope-selective photodissociation on the
distributions of CO isotopologues and hence the local isotopic ratios at
different layers of the disks.
### 5.4 Comparison of HD 163296 and TW Hya
In a series of papers, Qi et al. (2004, 2006) modeled the CO J=2–1, 3–2, and
6–5 line emission from the disk around the cooler young star TW Hya. There,
the best fit model to the optically thick CO J=2–1 and 3–2 lines tended to
underestimate the CO 6–5 emission. To fit all three CO lines simultaneously
required additional heating of the disk surface, suggested to be the result of
the intense X-ray irradiation field for that source. No such additional
heating is required to explain the HD 163296 data. On the contrary, the CO
J=6–5 emission required a lower disk interior temperature compared to those
predicted by the typical accretion disk models based on the SED, forcing us to
modify the $z_{\rm big}$ parameter to fit the data.
There are at least two plausible reasons for the different vertical structures
we infer for these disks. First, is the high-energy radiation field from the
two stars. Chandra X-ray observations reveal a point-like object within
$0\farcs 25$ (30 AU) of HD 163296, with an X-ray luminosity of $4\times
10^{29}$ ergs s-1 (Swartz et al., 2005), about 5$\times$ lower than that from
TW Hya (Kastner et al., 1999). Obviously the suggestion that X-ray heating is
dependent on the stellar type will require a much larger sample to confirm, as
well as more detailed radiative transfer modeling of the high-energy heating
of these disks. However, it is an exciting prospect that the interior
structures of these disks may be very different depending on the connection
between the stellar mass and the irradiation environment, with implications
both for disk evolution and chemistry. A second possibility may lie with the
level of turbulence in these disks. In our model, the vertical structure is
governed by the settling of large dust grains. The timescale for gravitational
sedimentation is expected to be short, with big grains reaching the midplane
in less than $\sim$1 Myr (Dullemond & Dominik, 2004) unless the dust is
effectively stirred up by turbulence. The more vertically extended population
of big grains required to model the HD 163296 disk might be commensurate with
more efficient stirring, or rather inhibited settling. Hughes et al. (2011)
suggest a 300 m s-1 turbulent linewidth in the CO layer of the HD 163296 disk,
much larger than the $\lesssim$40 m s-1 they find for the TW Hya disk. Perhaps
turbulent stirring is responsible for lofting the big grains above the
midplane in the HD 163296 disk, enhancing the radiative cooling.
## 6 Summary
We present CO line and dust emission observations of the disk around the
Herbig Ae star HD 163296 and develop a model framework that describes in a
consistent way the spectral energy distribution, resolved millimeter dust
continuum data, and multiple emission lines of CO and its isotopologues. The
fitting results support the general picture of CO vertical distribution
regulated by photodissociation at a surface where the hydrogen column density
is $\lesssim 10^{21}$ cm-2 and by CO freeze-out at depths below 20 K in the
midplane. The main conclusions are summarized here:
1. 1.
We confirm a tapered exponential edge in the surface density distribution, an
outcome of the similarity solution of the time evolution of accretion disks,
can account for the size discrepancy in dust continuum and CO emission (Hughes
et al., 2008). This means that constraints from dust modeling can be applied
to CO gas modeling, significantly reducing the number of free parameters.
2. 2.
We find in the disk model the transition between the “small” and “big” dust
grain populations, $z=z_{\rm big}$ in units of the gas scale height $H$,
regulates the vertical temperature and density profiles between the disk
midplane and surface. Multiple transitions of CO, especially the CO 6-5 line,
which requires much higher excitation, can be used to constrain the location
of $z_{\rm big}$, thus the temperature structure in the disk intermediate
layer. We find the resulting disk model for HD 163296 has a cold midplane
populated by large grains with a large scale height ($z_{\rm big}=2H$). Since
the vertical temperature in the HD 163296 disk is governed mainly by the
settling, or lack of settling, of large grains, a possible explanation for the
relatively cold interior of HD 163296 disk may be a high level of gas
turbulence (see Hughes et al., 2011).
3. 3.
Using the model with temperature structure constrained by the CO multi-
transition analysis, we fit for the location of emission from 13CO and
constrain the vertical distribution of the emission region to lie between a
lower boundary set by the temperature where CO freezes out onto dust grains
(at heights where $T\lesssim 19$ K) and an upper boundary where densities are
low enough that stellar and interstellar radiation can photodissociate the CO
molecule (where the vertically integrated hydrogen column density from the
disk surface is $\lesssim 10^{21}$ cm-2). The CO freeze-out produces a
significant drop in the gas-phase CO column density beyond a radius of about
155 AU, effectively a CO snow line that is resolved directly by the
observations. The CO depletion, generally found in disks, can be successfully
accounted for considering both the CO freeze-out and photodissociation.
4. 4.
Taking the CO freeze-out into consideration in the disk model and assuming CO
isotopologues sharing the same spatial distribution, the isotopic ratios of
12C/13C, 16O/18O and 18O/17O are consistent with the standard quiescent
interstellar gas-phase values and show no evidence for unusual fractionation.
More sensitive data is needed to investigate the distribution differences
among those CO isotopologues and the local isotopic ratios at different layers
of the disks.
This detailed investigation of the HD 163296 disk demonstrates the potential
of a staged, parametric technique for constructing unified gas and dust
structure models and constraining the distribution of molecular abundances
using resolved multi-transition, multi-isotope observations. The analysis
provides the essential framework for more general observational studies of
molecular line emission in protoplanetary disks.
We thank Edwin Bergin, Eugene Chiang and Jeremy Drake for beneficial
conversations, and a referee for constructive comments on the paper. Support
for K. I. O. is provided by NASA through a Hubble Fellowship grant awarded by
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract NAS
5-26555. Support for A. M. H. is provided by a fellowship from the Miller
Institute for Basic Research in Science. We acknowledge NASA Origins of Solar
Systems grant No. NNX11AK63G.
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Table 1: Observational parameters for SMA HD 163296 observations Parameters | CO 2–1 | CO 3–2bb Observational parameters from Hughes et al. (2011) | CO 6–5 | 13CO 2–1 | C18O 2–1 | C17O 3–2
---|---|---|---|---|---|---
Rest Frequency (GHz): | 230.53800 | 345.79599 | 691.47308 | 220.39868 | 219.56036 | 337.06113
Observationsaa C: compact configuration; E: extended configuration. | 2010 May 17 (C) | 2009 May 6 (C) | 2007 Mar 20 (C) | 2010 May 15 (C) | 2010 May 15 (C) | 2005 Aug 23 (C)
| 2010 Sep 14 (E) | 2009 Aug 22(E) | | 2010 Sep 11 (E) | 2010 Sep 11 (E) |
Tsys | 93–207 | 147–273 | 1368–4764 | 82–146 | 82–146 | 210–603
Baselines | 6–151 | 10–146 | 23–158 | 6–145 | 6–145 | 8–80
Spectral resolution (km s-1): | 0.066 | 0.044 | 0.88 | 0.069 | 0.55 | 0.72
Table 2: Continuum and emission line results
$\lambda$(mm) | Beam | PA | Flux (Jy)
---|---|---|---
1.36 | $3\farcs 5\times 2\farcs 2$ | 55.8∘ | 0.615$\pm$0.004
1.33 | $4\farcs 7\times 2\farcs 8$ | 24.2∘ | 0.670$\pm$0.007
1.11 | $3\farcs 0\times 1\farcs 9$ | 57.8∘ | 1.039$\pm$0.007
0.88 | $1\farcs 8\times 1\farcs 3$ | 47.6∘ | 1.74$\pm$0.12
0.44 | $2\farcs 7\times 2\farcs 2$ | -6.9∘ | 7.5$\pm$0.5
Transitions | Beam | PA | Integrated IntensitybbGas scale height. (Jy km-1)
CO 2–1 | $2\farcs 1\times 1\farcs 8$ | 10.5∘ | 54.17$\pm$0.39
CO 3–2 | $1\farcs 7\times 1\farcs 3$ | 46.7∘ | 98.72$\pm$1.69
CO 6–5 | $2\farcs 7\times 1\farcs 4$ | 1.8∘ | 58.66$\pm$6.44
13CO 2–1 | $1\farcs 9\times 1\farcs 8$ | 5.5∘ | 18.76$\pm$0.24
C18O 2–1 | $1\farcs 9\times 1\farcs 8$ | 5.5∘ | 6.30$\pm$0.16
C17O 3–2 | $3\farcs 2\times 2\farcs 2$ | 13.1∘ | 11.64$\pm$0.76
aafootnotetext: Intensity averaged over the whole emission area.
Table 3: Physical model for the disk of HD 163296
Parameters | Values
---|---
Stellar and accretion properties
Spectral type | A1
Effective temperature: T∗(K) | 9333
Visual extinction: Av | 0.3
Estimated distance: d(pc) | 122
Stellar radius: R∗(R⊙) | 2
Stellar mass: M∗(M⊙) | 2.3
Accretion rate: $\dot{M}$($M_{\odot}$ yr-1) | 7.6$\times$10-8
Disk structure properties
Disk mass: Md(M⊙) | 0.089
Characteristic radius: Rc(AU) | 150
Viscosity coefficient: $\alpha_{0}$ | 0.019
Depletion factor of the atmospheric small grains: $\epsilon^{a}$ | 0.003
$z_{\rm big}$a ($H^{b}$) | 2.0
Inner wall radius: Rwall(AU) | 0.6
Inner wall scale height: zwall(AU) | 0.1
Disk geometric and kinematic properties
Inclination: $i$(deg) | 44 $\pm$ 2
Systemic velocity: VLSR(km s-1) | 5.8 $\pm$ 0.2
Turbulent line widthc: vturb(km s-1) | 0.2
Position angle: P.A.(deg) | 133 $\pm$ 3
aafootnotetext: See definition in paper or D’Alessio et al. (2006).
ccfootnotetext: Fixed parameter.
Table 4: Fitting results: fractional abundances and distributions
Parameters | CO 2–1 & 3–2 | 13CO 2–1 | C18O 2–1 | C17O 3–2
---|---|---|---|---
Midplane freeze-out temperature (K) | 19.0∗ | 19.0 $\pm$ 0.3 | 19.0∗ | 19.0∗
$\sigma_{s}$ | 0.79∗ | 0.79 $\pm$ 0.03 | 0.79∗ | 0.79∗
Fractional abundance | (6.0$\pm$0.3) $\times$10-5 | (9.0$\pm$0.6) $\times$10-7 | (1.35$\pm$0.20) $\times$10-7 | (3.5$\pm$1.1) $\times$10-8
Fractional abundance (no freeze-out) | (6.5$\pm$0.3) $\times$10-5 | (4.6$\pm$0.3) $\times$10-7 | (1.5$\pm$0.2) $\times$10-8 | (7.0$\pm$2.2) $\times$10-8
**footnotetext: Parameter values adopted from 13CO 2–1 fitting.
Figure 1: Spatially integrated spectra at the peak continuum (stellar)
position of HD 163296. The fluxes are averaged over the emission areas (4′′
box for CO 6–5; 10′′ box for other lines). The vertical dotted line indicates
the position of the fitted VLSR.
Figure 2: Channel maps of the CO J=2–1 line emission from the disk around HD
163296. The LSR velocity is indicated in the upper right of each channel,
while the synthesized beam size and orientation ($2\farcs 1\times 1\farcs 8$
at a position angle of 10.5∘) is indicated in the upper left panel. The
contours are 0.18 Jy Beam-1 (1$\sigma$) $\times[3,6,9,12,15,18,21,24,27]$. The
star symbol indicates the disk center.
Figure 3: Same as Figure 2, but for 13CO 2–1. The beam size is $1\farcs
9\times 1\farcs 8$ at a position angle of 5.5∘ and the contours are 0.08 Jy
Beam-1 (1$\sigma$) $\times[3,6,9,12,15,18,21,24]$.
Figure 4: Same as Figure 2, but for C18O 2–1, C17O 3–2 and CO 6–5. For C18O
2–1, the beam size is $1\farcs 9\times 1\farcs 8$ at a position angle of 5.5∘
and the contours are 0.025 Jy Beam-1 (1$\sigma$)
$\times[3,6,9,12,15,18,21,24]$; for C17O 3–2, the beam size is $3\farcs
1\times 2\farcs 3$ at a position angle of 12.8∘ and the contours are 0.2 Jy
Beam-1 (1$\sigma$) $\times[2,3,4,5]$; for CO 6–5, the beam size is $2\farcs
7\times 2\farcs 2$ at a position angle of -5.6∘ and the contours are 3.0 Jy
Beam-1 (1$\sigma$) $\times[2,3,4,5,6,7]$.
Figure 5: Top panels (Left): The predicted continuum image vs data at 271
GHz. Contours are shown at 3$\sigma$ intervals. (Right): The predicted 271 GHz
continuum visibility profiles for Models with $R_{c}$=125 AU(dotted line), 150
AU(solid line), 175 AU(dashed line). Bottom panels: The predicted continuum
image and visibilities vs data at 341 GHz.
Figure 6: The HD 163296 SED. Optical and infrared bands: UBVJHKLM from Malfait
et al. (1998); UBVRI from Hillenbrand et al. (1992); BV from Oudmaijer et al.
(2001); BVRIJHK from Tannirkulam et al. (2008); JHKLM and 8–13 $\micron$m
bands from Berrilli et al. (1992); JHK from Eiroa et al. (2001) and 2MASS
(Cutri et al., 2003); ISO spectra (Acke & van den Ancker, 2004; Thi et al.,
2001); the CTIO and Keck telescopes data (Jayawardhana et al., 2001); IRAS
(Beichman et al., 1988); Spitzer-IRS (retrieved from Spitzer archive website:
irsa.ipac.caltech.edu). Millimeter fluxes are from this paper (marked with
circles) and Isella et al. (2007). Overlaid on the SED are the models which
have: $\dot{M}=7.6\times 10^{-8}\ M_{\odot}/yrs$, $\alpha_{0}=0.019$,
$\epsilon=0.003$, $R_{c}=150$ AU, $cosi=0.72$, a vertical wall at the dust
sublimation radius, calculated assuming $T_{sub}=1500$ K, with $R_{wall}=0.6$
AU and a height $z_{wall}=0.17R_{wall}$. The stellar photosphere (the short-
long-dashed line) is from the library for population synthesis from Bruzual A.
& Charlot (1993). The model SED lines are laid out for models with different
values of $z_{\rm big}/H=$ 0.5(long-dashed), 1.0(short-dashed), 1.5(dot-
dashed), 2.0(solid) and 2.5 (dotted).
Figure 7: Upper panel: Temperature vs height at $R=200$ AU for disk models
with $\dot{M}=7.6\times 10^{-8}\ M_{\odot}/yrs$, $\alpha_{0}=0.019$,
$\epsilon=0.003$, $R_{c}=150$ AU, and different values of $z_{\rm big}/H=$
0.5, 1, 1.5, 2.0, 2.5, 3 (from left to right). The fiducial model, with
$z_{\rm big}=2H$ is showed with a solid line. Lower panel: Density vs height
at $R=200$ AU for the same models. The lines are laid out from top to bottom
for models with different values of $z_{\rm big}/H=$ 0.5, 1, 1.5, 2.0, 2.5, 3
at $z>$ 100 AU.
Figure 8: CO J=2–1, 3–2, and 6–5 model spectra (black lines) overlaid with
the HD 163296 spectra in grey shade. The left, middle and right column are the
simulated models with $z_{\rm big}/H=$ 1.5, 2.0, 2.5. Note that all the models
have been sampled at the same spatial frequencies as each SMA dataset.
Figure 9: The irradiated accretion disk structure model for HD 163296 with
$z_{\rm big}/H=$ 2.0 and $R_{c}$=150 AU. Top and middle panels: Temperature
and density profiles are indicated by solid and dashed lines, respectively.
The CO emission area constrained by the best-fit vertical boundaries is shown
in grey shade. Bottom panel: The vertical distributions of temperature and
$\sigma_{s}$ at R = 200 AU are indicated by solid and dashed lines,
respectively. The vertical dotted lines show the best-fit locations of the
lower boundary (TCO = 19 K) and upper boundary ($\sigma_{s}$=0.79) for CO.
Figure 10: Iso-$\chi^{2}$ surfaces of CO freeze-out temperature (TCO), CO
fractional abundance (fCO) and $\sigma_{s}$ for 13CO. Contours correspond to
the 1–5 $\sigma$ errors.
Figure 11: Models of 13CO radial column densities for no freeze-out (dotted),
freeze-out at 18 K (dashed) 19 K (solid) and 20 K (dot-dashed). Each model has
been scaled to fit the 13CO 2–1 emission. Note that the CO snow line is at a
radius of 155 AU for T${}_{\rm CO}=$ 19 K and it increases from 135 to 175 AU
when TCO decreases from 20 to 18 K.
Figure 12: For each panel, the top rows are the velocity channel maps of the
13CO, C18O 2–1 and C17O 3–2 emissions toward HD 163296, respectively
(velocities binned in 1 km s-1. Contours are 0.04 Jy Beam-1 (1$\sigma$)
$\times[3,6,9,12,15,18,24,30,36,42,48,54]$ for 13CO 2–1; 0.03 Jy Beam-1
(1$\sigma$) $\times[2,4,6,8,10,12,14,16,18,20]$ for C18O 2–1; 0.15 Jy Beam-1
(1$\sigma$) $\times[2,3,4,5,6,7]$ for C17O 3–2. The middle rows are the best-
fit models and the bottom rows are the difference between the best-fit models
and data on the same contour scale.
Figure 13: For each panel, the top rows are the velocity channel maps of the
CO 2–1, 3–2 and 6–5 emissions toward HD 163296, respectively (velocities
binned in 1 km s-1). Contours are 0.15 Jy Beam-1 (1$\sigma$)
$\times[3,6,9,12,15,18,21,24]$ for CO 2–1; 0.2 Jy Beam-1 (1$\sigma$)
$\times[3,6,9,12,15,18,21,24,27]$ for CO 3–2; 3.0 Jy Beam-1 (1$\sigma$)
$\times[2,3,4,5,6,7,8]$ for CO 6–5. The middle rows are the best-fit models
and the bottom rows are the difference between the best-fit models and data on
the same contour scale.
|
arxiv-papers
| 2011-07-25T20:09:34 |
2024-09-04T02:49:20.932624
|
{
"license": "Public Domain",
"authors": "Chunhua Qi, Paola D'Alessio, Karin I. Oberg, David J. Wilner, A.\n Meredith Hughes, Sean M. Andrews, Sandra Ayala",
"submitter": "Chunhua Qi",
"url": "https://arxiv.org/abs/1107.5061"
}
|
1107.5228
|
# Non–Uniform Cellular Automata:
classes, dynamics, and decidability111This is an extended and improved version
of the paper [2] presented at LATA2009 conference.
Alberto Dennunzio alberto.dennunzio@unice.fr Enrico Formenti
enrico.formenti@unice.fr Julien Provillard julien.provillard@i3s.unice.fr
Université Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles,
06903 Sophia Antipolis (France)
###### Abstract
The dynamical behavior of non-uniform cellular automata is compared with the
one of classical cellular automata. Several differences and similarities are
pointed out by a series of examples. Decidability of basic properties like
surjectivity and injectivity is also established. The final part studies a
strong form of equicontinuity property specially suited for non-uniform
cellular automata.
###### keywords:
cellular automata , non–uniform cellular automata , decidability , symbolic
dynamics
## 1 Introduction
A complex system is (roughly) defined by a multitude of simple individuals
which cooperate to build a complex (unexpected) global behavior by local
interactions. Cellular automata (CA) are often used to model complex systems
when individuals are embedded in a uniform “universe” in which local
interactions are the same for all. Indeed, a cellular automaton is made of
identical finite automata arranged on a regular lattice. Each automaton
updates its state by a local rule on the basis of its state and the one of a
fixed set of neighbors. At each time-step, the same (here comes uniformity)
local rule is applied to all finite automata in the lattice. For recent
results on CA dynamics and an up-to-date bibliography see for instance [10,
17, 3, 14, 9, 6, 7, 1, 5, 8].
In a number of situations one needs a more general setting. One possibility
consists in relaxing the uniformity constraint. This choice may result winning
for example for
#### Complexity design control
In many phenomena, each individual locally interacts with others but maybe
these interactions depend on the individual itself or on its position in the
space. For example, when studying the formation of hyper-structures in cells,
proteins move in the cellular soup and do not behave just like billiard balls.
They chemically interact each other only when they meet special situations
(see for instance, [18]) or when they are at some special places (in rybosomes
for instance). It is clear that one might simulate all those situations by a
CA but the writing of a single local rule will be an excessive difficult task,
difficult to control. A better option would be to write simpler (but
different) local rules that are applied only at precise positions so that less
constraints are to be taken into account at each time.
#### Structural stability
Assume that we are investigating the robustness of a system w.r.t. some
specific property $P$. If some individuals change their “standard” behavior
does the system still have property $P$? What is the “largest” number of
individuals that can change their default behavior so that the system does not
change its overall evolution?
#### Reliability
CA are more and more used to perform fast parallel computations (beginning
from [4], for example). Each cell of the CA can be implemented by a simple
electronic device (FPGAs for example) [19]. Then, how reliable are
computations w.r.t. failure of some of these devices? (Here failure is
interpreted as a device which behaves differently from its “default” way).
The generalization of CA to non-uniform CA ($\nu\text{-}\mathcal{CA}$) has
some interest in its own since the new model coincides with the set of
continuous functions in Cantor topology. It is clear that the class of
continuous functions is too large to be studied fruitfully. In the present
paper, we present several sub-classes that are also interesting in
applications. First of all, we show that several classical results about the
dynamical behavior of CA are no longer valid in the new setting. Even when the
analysis is restricted to smaller classes of non-uniform CA, the overall
impression is that new stronger techniques will be necessary to study
$\nu\text{-}\mathcal{CA}$. However, by generalizing the notion of De Bruijn
graph, we could prove the decidability of basic set properties like
surjectivity and injectivity. We recall that these property are often
necessary conditions of many classical definitions of deterministic chaos.
Keeping on with surjectivity and injectivity, we give a partial answer about
reliability and structural stability questions issued above. More precisely,
we answer the following question: assuming to perturb some CA in some finite
number of sites, if one knows that the corresponding $\nu\text{-}\mathcal{CA}$
is surjective (resp., injective) does this imply that the original CA was
surjective (resp., injective)?
The final part starts going more in deep with the study of the long-term
dynamical behavior of $\nu\text{-}\mathcal{CA}$. Indeed, we show that under
some conditions, if a $\nu\text{-}\mathcal{CA}$ is a perturbed version of some
equicontinuous or almost equicontinuous CA, then it shares the same dynamics.
Finally, we develop some complex examples showing that even small
perturbations of an almost equicontinuous CA can give raise to sensitive to
initial conditions behavior or to equicontinuous dynamics.
We might conclude that breaking uniformity property in a CA may cause a
dramatic change in the dynamical behavior.
## 2 Background
In this section, we briefly recall standard definitions about CA and dynamical
systems. For introductory matter see [16], for instance. For all
$i,j\in\mathbb{Z}$ with $i\leq j$ (resp., $i<j$), let
$[i,j]=\left\\{i,i+1,\ldots,j\right\\}$ (resp.,
$[i,j)=\left\\{i,\ldots,j-1\right\\}$). Let $\mathbb{N}_{+}$ be the set of
positive integers.
#### Configurations and CA
Let $A$ be an alphabet. A _configuration_ is a function from $\mathbb{Z}$ to
$A$. The _configuration set_ $A^{\mathbb{Z}}$ is usually equipped with the
metric $d$ defined as follows
$\forall x,y\in
A^{\mathbb{Z}},\;d(x,y)=2^{-n},\;\text{where}\;n=\min\left\\{i\geq
0\,:\,x_{i}\neq y_{i}\;\text{or}\;x_{-i}\neq y_{-i}\right\\}\enspace.$
When $A$ is finite, the set $A^{\mathbb{Z}}$ is a compact, totally
disconnected and perfect topological space (i.e., $A^{\mathbb{Z}}$ is a Cantor
space). For any pair $i,j\in\mathbb{Z}$, with $i\leq j$, and any configuration
$x\in A^{\mathbb{Z}}$ we denote by $x_{[i,j]}$ the word $x_{i}\cdots x_{j}\in
A^{j-i+1}$, i.e., the portion of $x$ inside the interval $[i,j]$. Similarly,
$u_{[i,j]}=u_{i}\cdots u_{j}$ is the portion of a word $u\in A^{l}$ inside
$[i,j]$ (here, $i,j\in[0,l)$). In both the previous notations, $[i,j]$ can be
replaced by $[i,j)$ with the obvious meaning. For any word $u\in A^{*}$, $|u|$
denotes its length. A _cylinder_ of block $u\in A^{k}$ and position
$i\in\mathbb{Z}$ is the set $[u]_{i}=\\{x\in A^{\mathbb{Z}}:x_{[i,i+k)}=u\\}$.
Cylinders are clopen sets w.r.t. the metric $d$ and they form a basis for the
topology induced by $d$. A configuration $x$ is said to be _$a$ -finite_ for
some $a\in A$ if there exists $k\in\mathbb{N}$ such that $x_{i}=a$ for any
$i\notin[-k,k]$. In the sequel, the collection of the $a$-finite
configurations for a certain $a$ will be simply called set of finite
configurations.
A (one–dimensional) _CA_ is a structure $\langle A,r,f\rangle$, where $A$ is
the _alphabet_ , $r\in\mathbb{N}$ is the _radius_ and $f:A^{2r+1}\to A$ is the
_local rule_ of the automaton. The local rule $f$ induces a _global rule_
$F:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ defined as follows,
$\forall x\in A^{\mathbb{Z}},\,\forall i\in\mathbb{Z},\quad
F(x)_{i}=f(x_{i-r},\ldots,x_{i+r})\enspace.$ (1)
Recall that $F$ is a uniformly continuous map w.r.t. the metric $d$.
With an abuse of notation, a CA local rule $f$ is extended to the function
$f:A^{*}\to A^{*}$ which map any $u\in A^{*}$ of length $l$ to the word $f(u)$
such that $f(u)=\epsilon$ (empty word), if $l\leq 2r$, and
$f(u)_{i}=f(u_{[i,i+2r]})$ for each $i\in[0,l-2r)$, otherwise.
#### DTDS
A _discrete time dynamical system (DTDS)_ is a pair $(X,G)$ where $X$ is a set
equipped with a distance $d$ and $G:X\mapsto X$ is a map which is continuous
on $X$ with respect to the metric $d$. When $A^{\mathbb{Z}}$ is the
configuration space equipped with the above introduced metric and $F$ is the
global rule of a CA, the pair $(A^{\mathbb{Z}},F)$ is a DTDS. From now on, for
the sake of simplicity, we identify a CA with the dynamical system induced by
itself or even with its global rule $F$.
Given a DTDS $(X,G)$, an element $x\in X$ is an _ultimately periodic point_ if
there exist $p,q\in\mathbb{N}$ such that $G^{p+q}(x)=G^{q}(x)$. If $q=0$, then
$x$ is a _periodic point_ , i.e., $G^{p}(x)=x$. The minimum $p$ for which
$G^{p}(x)=x$ holds is called _period_ of $x$.
Recall that a DTDS $(X,g)$ is _sensitive to the initial conditions_ (or simply
_sensitive_) if there exists a constant $\varepsilon>0$ such that for any
element $x\in X$ and any $\delta>0$ there is a point $y\in X$ such that
$d(y,x)<\delta$ and $d(G^{n}(y),G^{n}(x))>\varepsilon$ for some
$n\in\mathbb{N}$. A DTDS $(X,G)$ is _positively expansive_ if there exists a
constant $\varepsilon>0$ such that for any pair of distinct elements $x,y$ we
have $d(G^{n}(y),G^{n}(x))\geq\varepsilon$ for some $n\in\mathbb{N}$. If $X$
is a perfect set, positive expansivity implies sensitivity. Recall that a DTDS
$(X,g)$ is _(topologically) transitive_ if for any pair of non-empty open sets
$U,V\subseteq X$ there exists an integer $n\in\mathbb{N}$ such that
$g^{n}(U)\cap V\neq\emptyset$. A DTDS $(X,G)$ is _sujective_ (resp.,
_injective_) iff $G$ is surjective (resp., $G$ is injective).
Let $(X,G)$ be a DTDS. An element $x\in X$ is an _equicontinuity point_ for
$G$ if $\forall\varepsilon>0$ there exists $\delta>0$ such that for all $y\in
X$, $d(y,x)<\delta$ implies that $\forall
n\in\mathbb{N},\;d(G^{n}(y),G^{n}(x))<\varepsilon$. For a CA $F$, the
existence of an equicontinuity point is related to the existence of a special
word, called _blocking word_. A word $u\in A^{k}$ is $s$-blocking ($s\leq k$)
for a CA $F$ if there exists an offset $j\in[0,k-s]$ such that for any
$x,y\in[u]_{0}$ and any $n\in\mathbb{N}$, $F(x)_{[j,j+s-1]}=F(y)_{[j,j+s-1]}$
. A word $u\in A^{k}$ is said to be _blocking_ if it is $s$-blocking for some
$s\leq k$. A DTDS is said to be _equicontinuous_ if $\forall\varepsilon>0$
there exists $\delta>0$ such that for all $x,y\in X$, $d(x,y)<\delta$ implies
that $\forall n\in\mathbb{N},\;d(G^{n}(x),G^{n}(y))<\varepsilon$. If $X$ is a
compact set, a DTDS $(X,G)$ is equicontinuous iff the set $E$ of all its
equicontinuity points is the whole $X$. A DTDS is said to be _almost
equicontinuous_ if $E$ is residual (i.e., $E$ contains an infinite
intersection of dense open subsets). In [15], Kůrka proved that a CA is almost
equicontinuous iff it is non-sensitive iff it admits a $r$-blocking word.
Recall that two DTDS $(X,G)$ and $(X^{\prime},G^{\prime})$ are _topologically
conjugated_ if there exists a homeomorphism $\phi:X\mapsto X^{\prime}$ such
that $G^{\prime}\circ\phi=\phi\circ G$. In that case, $(X,G)$ and
$(X^{\prime},G^{\prime})$ share some properties such as surjectivity,
injectivity, transitivity.
## 3 Non–Uniform Cellular Automata
The meaning of (1) is that the same local rule $f$ is applied to each site of
the CA. Relaxing this constraint gives us the definition of a
$\nu\text{-}\mathcal{CA}$. More formally one can give the following notion.
###### Definition 3.1 (Non–Uniform Cellular Automaton
($\nu\text{-}\mathcal{CA}$))
A _Non–Uniform Cellular Automaton_ ($\nu\text{-}\mathcal{CA}$) is a structure
$(A,\\{h_{i},r_{i}\\}_{i\in\mathbb{Z}})$ defined by a family of local rules
$h_{i}:A^{2r_{i}+1}\to A$ of radius $r_{i}$ all based on the same alphabet
$A$.
Similarly to CA, one can define the global rule of a $\nu\text{-}\mathcal{CA}$
as the map $H:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ given by the law
$\forall x\in A^{\mathbb{Z}},\,\forall i\in\mathbb{Z},\quad
H(x)_{i}=h_{i}(x_{i-r_{i}},\dots,x_{i+r_{i}})\enspace.$
From now on, we identify a $\nu\text{-}\mathcal{CA}$ (resp., CA) with the
discrete dynamical system induced by itself or even with its global rule $H$
(resp., $F$).
It is well known that the Hedlund’s Theorem [12] characterizes CA as the class
of continuous functions commuting with the shift map $\sigma:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$, where $\forall x\in A^{\mathbb{Z}},\forall
i\in\mathbb{Z},\sigma(x)_{i}=x_{i+1}$. It is straightforward to prove that a
function $H:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ is the global map of a
$\nu\text{-}\mathcal{CA}$ iff it is continuous. i.e., in other words, iff the
pair $(A^{\mathbb{Z}},H)$ is a DTDS. Remark that the definition of
$\nu\text{-}\mathcal{CA}$ is by far too general to be useful. Therefore, we
are going to focus our attention only over three special subclasses of
$\nu\text{-}\mathcal{CA}$.
###### Definition 3.2 (d$\nu\text{-}\mathcal{CA}$)
A $\nu\text{-}\mathcal{CA}$ $(A,\\{h_{i},r_{i}\\}_{i\in\mathbb{Z}})$ is a
d$\nu\text{-}\mathcal{CA}$ if there exist two naturals $k,r$ and a rule
$h:A^{2r+1}\to A$ such that $h_{i}=h$ for all integers $i$ with $|i|>k$. In
this case, we say that the given $\nu\text{-}\mathcal{CA}$ has $h$ as _default
rule_.
###### Definition 3.3 (p$\nu\text{-}\mathcal{CA}$)
A $\nu\text{-}\mathcal{CA}$ $(A,\\{h_{i},r_{i}\\}_{i\in\mathbb{Z}})$ is a
p$\nu\text{-}\mathcal{CA}$ if there exist two naturals $k$,$r$, a _structural
period_ $p>0$, and two sets $\\{f_{0},\ldots,f_{p-1}\\}$ and
$\\{g_{0},\ldots,g_{p-1}\\}$ of rules of radius $r$ such that for any integer
$i$ with $|i|>k$
$h_{i}=\begin{cases}f_{i\,\text{mod}\,p}&\text{if $i>k$}\\\
g_{i\,\text{mod}\,p}&\text{if $i<-k$}\end{cases}$
If $p=1$, we say that the given $\nu\text{-}\mathcal{CA}$ has $f_{0}$ and
$g_{0}$ as right and left default rules, respectively.
###### Definition 3.4 (r$\nu\text{-}\mathcal{CA}$)
A $\nu\text{-}\mathcal{CA}$ $H=(A,\\{h_{i},r_{i}\\}_{i\in\mathbb{Z}})$ is a
r$\nu\text{-}\mathcal{CA}$ if there exists an integer $r$ such that $\forall
i\in\mathbb{Z}$, $r_{i}=r$. In this case, we say that $H$ has radius $r$.
The first two class restrict the number of positions at which non-default
rules can appear, while the third class restricts the number of different
rules but not the number of occurrences nor it imposes the presence of default
rules. Some simple examples follow.
###### Example 1
Consider the $\nu\text{-}\mathcal{CA}$ $H^{(1)}:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$ defined as
$\forall x\in A^{\mathbb{Z}},\quad H^{(1)}(x)_{i}=\begin{cases}1&\text{if
$i=0$}\\\ 0&\text{otherwise}\end{cases}\enspace.$
Remark that $H^{(1)}$ is a d$\nu\text{-}\mathcal{CA}$ which cannot be a CA
since it does not commute with $\sigma$. This trivially shows that the class
of $\nu\text{-}\mathcal{CA}$ is larger than the one of CA.
###### Example 2
Consider the $\nu\text{-}\mathcal{CA}$ $H^{(2)}:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$ defined as
$\forall x\in A^{\mathbb{Z}},\quad
H^{(2)}(x)_{i}=\begin{cases}x_{i+1}&\text{if $i<0$}\\\ x_{0}&\text{if
$i=0$}\\\ x_{i-1}&\text{if $i>0$}\\\ \end{cases}$
Remark that $H^{(2)}$ is a p$\nu\text{-}\mathcal{CA}$ (with $p=1$) but not a
d$\nu\text{-}\mathcal{CA}$.
###### Example 3
Consider the $\nu\text{-}\mathcal{CA}$ $H^{(3)}:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$ defined as
$\forall x\in A^{\mathbb{Z}},\quad H^{(3)}(x)_{i}=\begin{cases}1&\text{if $i$
is even}\\\ 0&\text{otherwise.}\end{cases}$
Remark that $H^{(3)}$ is a r$\nu\text{-}\mathcal{CA}$ but not a
p$\nu\text{-}\mathcal{CA}$.
###### Example 4
Consider the $\nu\text{-}\mathcal{CA}$ $H^{(4)}:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$ defined as $\forall x\in A^{\mathbb{Z}},H^{(4)}(x)_{i}=x_{0}$.
Remark that $H^{(4)}$ is a $\nu\text{-}\mathcal{CA}$ but not a
r$\nu\text{-}\mathcal{CA}$.
We give some relationships and properties involving the classes of
$\nu\text{-}\mathcal{CA}$ above introduced.
###### Proposition 3.1
$\mathcal{CA}\varsubsetneq d\nu\text{-}\mathcal{CA}\varsubsetneq
p\nu\text{-}\mathcal{CA}\varsubsetneq
r\nu\text{-}\mathcal{CA}\varsubsetneq\nu\text{-}\mathcal{CA}$, where
$\mathcal{CA}$ is the set of all CA.
###### Proof 1
The inclusions $\subseteq$ easily follow from the definitions. For the strict
inclusions refer to Examples 1 to 4.
Similarly to what happens in the context of CA one can prove the following.
###### Proposition 3.2
Any r$\nu\text{-}\mathcal{CA}$ is topologically conjugated to a
r$\nu\text{-}\mathcal{CA}$ of radius 1.
###### Proof 2
Let $H$ be a r$\nu\text{-}\mathcal{CA}$ on the alphabet $A$. If $H$ has radius
$r=1$ then the statement is trivially true. Otherwise, let $B=A^{r}$ and
define $\phi:A^{\mathbb{Z}}\rightarrow B^{\mathbb{Z}}$ as $\forall
i\in\mathbb{Z},\phi(x)_{i}=x_{[ir,(i+1)r)}$. Then, it is not difficult to see
that the r$\nu\text{-}\mathcal{CA}$ $(B^{\mathbb{Z}},H^{\prime})$ of radius
$1$ defined as $\forall x\in A^{\mathbb{Z}},\forall
i\in\mathbb{Z},H^{\prime}(x)_{i}=h^{\prime}_{i}(x_{i-1},x_{i},x_{i+1})$ is
topologically conjugated to $H$ via $\phi$, where $\forall u,v,w\in B,\forall
i\in\mathbb{Z},\forall
j\in\\{0,\dots,r-1\\},(h^{\prime}_{i}(u,v,w))_{j}=h_{ir+j}(u_{[j,r)}vw_{[0,j]})$.
Finally, the following result shows that every r$\nu\text{-}\mathcal{CA}$ is a
subsystem of a suitable CA.
###### Theorem 3.3
Any r$\nu\text{-}\mathcal{CA}$ $H:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ of radius
$r$ is a subsystem of a CA, i.e., there exist a CA $F:B^{\mathbb{Z}}\to
B^{\mathbb{Z}}$ on a suitable alphabet $B$ and a continuous injection
$\phi:A^{\mathbb{Z}}\to B^{\mathbb{Z}}$ such that $\phi\circ H=F\circ\phi$.
###### Proof 3
Consider a r$\nu\text{-}\mathcal{CA}$ $H:A^{\mathbb{Z}}\rightarrow
A^{\mathbb{Z}}$ of radius $r$. Remark that there are only $n=|A|^{|A|^{2r+1}}$
distinct functions $h_{i}:A^{2r+1}\rightarrow A$. Take a numbering
$(f_{j})_{1\leq j\leq n}$ of these functions and let
$B=A\times\\{1,\dots,n\\}$. Define the map $\phi:A^{\mathbb{Z}}\rightarrow
B^{\mathbb{Z}}$ such that $\forall x\in A^{\mathbb{Z}},\forall
i\in\mathbb{Z},\phi(x)_{i}=(x_{i},k)$, where $k$ is the integer for which
$H(x)_{i}=f_{k}(x_{i-r},\dots,x_{i+r})$. Clearly, $\phi$ is injective and
continuous. Now, define a CA $F:B^{\mathbb{Z}}\rightarrow B^{\mathbb{Z}}$
using the local rule $f:B^{2r+1}\rightarrow B$ such that
$f((x_{-r},k_{-r}),\dots,(x_{0},k_{0}),\dots,(x_{r},k_{r}))=(f_{k_{0}}(x_{-r},\dots,x_{r}),k_{0})\enspace.$
It is not difficult to see that $\phi\circ H=F\circ\phi$.
## 4 CA versus $\nu\text{-}\mathcal{CA}$
In this section, we illustrate some differences in dynamical behavior between
CA and $\nu\text{-}\mathcal{CA}$. The following properties which are really
specific for CA are lost in the larger class of $\nu\text{-}\mathcal{CA}$.
1. P1)
_the set of ultimately periodic points is dense in $A^{\mathbb{Z}}$_.
2. P2)
_surjectivity $\Leftrightarrow$ injectivity on finite configurations._
3. P3)
_surjectivity $\Leftrightarrow$ any configuration has a finite number of
pre–images_.
4. P4)
_expansivity $\Rightarrow$ transitivity_
5. P5)
_expansivity $\Rightarrow$ surjectivity_
6. P6)
_injectivity $\Rightarrow$ surjectivity_
Some of the previous properties are not valid for the following
$\nu\text{-}\mathcal{CA}$.
###### Example 5
Let $A=\\{0,1\\}$ and define the d$\nu\text{-}\mathcal{CA}$
$H^{(5)}:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ as
$\forall x\in A^{\mathbb{Z}},\forall i\in\mathbb{Z},\quad
H^{(5)}(x)_{i}=\begin{cases}x_{i}&\text{if $i=0$}\\\
x_{i-1}&\text{otherwise}\enspace.\end{cases}$
P1) is not valid for $H^{(5)}$.
###### Proof 4
Let $H=H^{(5)}$. Denote by $P$ and $U$ the sets of periodic and ultimately
periodic points, respectively. Let $E=\\{x\in A^{\mathbb{Z}}:\forall
i\in\mathbb{N},x_{i}=x_{0}\\}$. Take $x\in P$ with $H^{p}(x)=x$. Remark that
the set $B_{x}=\\{i\in\mathbb{N}:x_{i}\neq x_{0}\\}$ is empty. Indeed, by
contradiction, assume that $B\neq\emptyset$ and let $m=\min B$. It is easy to
check that $\forall y\in A^{\mathbb{Z}},\forall
i\in\mathbb{N},H^{i}(y)_{[0,i]}={y_{0}}^{i+1}$, hence
$x_{m}=H^{pm}(x)_{m}=x_{0}$, contradiction. Thus $x\in E$ and $P\subseteq E$.
Let $y\in H^{-1}(E)$. We show that $B_{y}=\emptyset$. By contradiction, let
$n=\min B_{y}$. Since $H(y)_{n+1}=y_{n}\neq y_{0}=H(y)_{0}$, then $H(y)\notin
E$. Contradiction, then $y\in E$ and $H^{-1}(E)\subseteq E$. So $\forall
n\in\mathbb{N},H^{-n}(E)\subseteq E$. Moreover,
$U=\bigcup_{n\in\mathbb{N}}H^{-n}(P)\subseteq\bigcup_{n\in\mathbb{N}}H^{-n}(E)\subseteq
E$ and $E$ is not dense.
P3) is not valid for $H^{(5)}$
###### Proof 5
We show that $H^{(5)}$ has no configuration with an infinite number of pre-
images although it is not surjective. In particular, any configuration has
either $0$ or $2$ pre-images.
First of all, $H^{(5)}$ is not surjective. Indeed, since $\forall x\in
A^{\mathbb{Z}},H^{(5)}(x)_{0}=H^{(5)}(x)_{1}$, configurations in the set
$B=\\{x\in A^{\mathbb{Z}}:x_{0}\neq x_{1}\\}$ have no pre-image. Furthermore,
any $x\in{A^{\mathbb{Z}}}\setminus B$ has $y$ and $z$ as unique pre–images,
where $y$ and $z$ are configurations such that $\forall
i\notin\\{-1,0\\},y_{i}=z_{i}=x_{i+1},y_{0}=z_{0}=x_{0},y_{-1}=0;z_{-1}=1$.
We stress that $H^{(5)}$ is not surjective, despite it is based on two local
rules each of which generates a surjective CA (namely, the identity CA and the
shift CA).
In order to explore other properties, we introduce an other
$\nu\text{-}\mathcal{CA}$.
###### Example 6
Let $A=\\{0,1\\}$ and define $H^{(6)}:A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ by
$\forall x\in A^{\mathbb{Z}},\forall i\in\mathbb{Z},\quad
H^{(6)}(x)_{i}=\begin{cases}0&\text{if $i=0$}\\\ x_{i-1}\oplus
x_{i+1}&\text{otherwise}\enspace,\end{cases}$
where $\oplus$ is the xor operator.
P2) is not valid for $H^{(6)}$.
###### Proof 6
We prove that $H^{(6)}$ is injective on the $0$-finite configurations but it
is not surjective. It is evident that $H^{(6)}$ is not surjective. Let $x,y$
be two finite configurations such that $H^{(6)}(x)=H^{(6)}(y)$. By
contradiction, assume that $x_{i}\neq y_{i}$, for some $i\in\mathbb{Z}$.
Without loss of generality, assume that $i\in\mathbb{N}$. Since $x_{i}\oplus
x_{i+2}=H^{(6)}(x)_{i+1}=H^{(6)}(y)_{i+1}=y_{i}\oplus y_{i+2}$, it holds that
$x_{i+2}\neq y_{i+2}$ and, by induction, $\forall j\in\mathbb{N},x_{i+2j}\neq
y_{i+2j}$. We conclude that $\forall j\in\mathbb{N},x_{i+2j}=1$ or
$y_{i+2j}=1$ contradicting the assumption that $x$ and $y$ are finite.
P4) and P5) are not valid for $H^{(6)}$.
###### Proof 7
Let $H=H^{(6)}$. $H$ is not transitive since it is not surjective. We show
that $H$ is positively expansive. Let $x$ and $y$ be two distinct
configurations and let $k=\min_{i\in\mathbb{Z}}\\{|i|,x_{i}\neq y_{i}\\}$. If
$k=0$, then $d(H^{0}(x),H^{0}(y))=1\geq\frac{1}{2}$. Without loss of
generality assume $k=n>0$. It is clear that $H(x)_{n-1}=x_{n-2}\oplus
x_{n}\neq y_{n-2}\oplus y_{n}=H(y)_{n-1}$ and $H(x)_{[0,n-2]}=H(y)_{[0,n-2]}$.
Iterating the same reasoning one sees that $H^{n-1}(x)_{1}\neq
H^{n-1}(y)_{1}$. Hence $d(H^{n-1}(x),H^{n-1}(y))\geq\frac{1}{2}$. Thus $H$ is
positively expansive with expansivity constant $\frac{1}{2}$.
Consider now the $\nu\text{-}\mathcal{CA}$ $H^{(2)}$ from Example 2.
P6) is not valid for $H^{(2)}$.
###### Proof 8
Concerning non-surjectivity, just remark that only configurations $x$ such
that $x_{-1}=x_{0}=x_{1}$ have a pre-image. Let $x,y\in A^{\mathbb{Z}}$ with
$H^{(2)}(x)=H^{(2)}(y)$. Then, we have $\forall i>0,x_{i-1}=y_{i-1}$ and
$\forall i<0,x_{i+1}=y_{i+1}$. So $x=y$ and $H$ is injective.
## 5 Basic Properties of $\nu\text{-}\mathcal{CA}$ and Decidability
This section is centered on two fundamental properties, namely surjectivity
and injectivity. Focusing on CA, both properties are strongly related to
peculiar dynamical behaviors. Injectivity coincides with reversibility [12],
while surjectivity is a necessary condition for almost all the widest known
definitions of deterministic chaos (see [11], for instance).
In (1D) CA settings, the notion of De Bruijn graph is very handy to find fast
decision algorithms for surjectivity, injectivity and openness [20]. Here, we
extend this notion to work with p$\nu\text{-}\mathcal{CA}$ having period 1 and
find decision algorithm for surjectivity. We stress that surjectivity is
undecidable for two (or higher) dimensional p$\nu\text{-}\mathcal{CA}$, since
surjectivity is undecidable for 2D CA [13].
###### Definition 5.1
Consider a p$\nu\text{-}\mathcal{CA}$ $H$ of radius $r$ and period $p=1$
having $f$ and $g$ as right and left default rules. Let $k\in\mathbb{N}$ be
the largest natural such that $h_{k}\neq f$ or $h_{-k}\neq g$. The _De Bruijn
graph_ of $H$ is the triple $G=(V,E,\ell_{G})$ where
$V=A^{2r}\times\\{-k,\dots,k+1\\}$ and $E$ is the set of pairs
$((u,\alpha),(v,\beta))\in V^{2}$ with label $\ell_{G}((u,\alpha),(v,\beta))$
in $A\times\\{0,1\\}$ such that $u_{[1,2r)}=v_{[0,2r-1)}$ and one of the
following conditions is verified
1. a)
$\alpha=\beta=-k$; in this case the label is $(g(u_{0}v),0)$
2. b)
$\alpha+1=\beta$; in this case the label is $(h_{\alpha}(u_{0}v),0)$
3. c)
$\alpha=\beta=k+1$; in this case the label is $(f(u_{0}v),1)$
By this graph, a configuration can be seen as a bi-infinite path on vertexes
which passes once from a vertex whose second component is in $[-k+1,k]$ and
infinite times through other vertices. The second component of vertices allows
to single out the positions of local rules different from the default one. The
image of a configuration is the sequence of first components of edge labels.
###### Lemma 5.1
Surjectivity is decidable for p$\nu\text{-}\mathcal{CA}$ with structural
period $p=1$.
###### Proof 9
Let $H$ be a p$\nu\text{-}\mathcal{CA}$ with structural period $p=1$ and let
$G$ be its De Bruijn graph. We prove that $H$ is surjective iff $G$ recognizes
the language $(A\times\\{0\\})^{*}(A\times\\{1\\})^{*}$ when $G$ is considered
as the graph of a finite state automaton.
Let $k$ be as in Definition 5.1 and denote by $(w,s)$ any word of
$(A\times\\{0,1\\})^{*}$ with $w\in A^{l}$, $s\in\\{0,1\\}^{l}$, for some
$l\in\mathbb{N}$.
Assume that $H$ is surjective and take
$(w^{\prime},s)\in(A\times\\{0\\})^{n}(A\times\\{1\\})^{*}$, for any
$n\in\mathbb{N}$. Then, there exists $x\in A^{\mathbb{Z}}$ such that
$H(x)_{[m,m+l)}=w^{\prime}$, where $m=k+1-n$. Set $w=x_{[m-r,m+l+r)}$. Hence,
the word $(w^{\prime},s)$ is the sequence of edge labels of the following
vertex path on $G$:
$(w_{[0,2r)},\alpha_{0}),\dots,(w_{[l,l+2r)},\alpha_{l})$
where
$\alpha_{i}=\left\\{\begin{array}[]{cl}-k&\text{ if }m+i<-k\\\ k+1&\text{ if
}m+i>k\\\ m+i&\text{ otherwise}\end{array}\right.$
For the opposite implication, assume that $G$ recognizes
$(A\times\\{0\\})^{*}(A\times\\{1\\})^{*}$. Take $y\in A^{\mathbb{Z}}$ and let
$n>k$. Since $G$ recognizes $(y_{[-n,n]},0^{n+k+1}1^{n-k})$, there exists
$x\in A^{\mathbb{Z}}$ such that $H(x)_{[-n,n]}=y_{[-n,n]}$. Set
$X_{n}=\left\\{x\in A^{\mathbb{Z}},x_{[n,n]}=y_{[-n,n]}\right\\}$. For any
$n\in\mathbb{N}$, $X_{n}$ is non-empty and compact. Moreover,
$X_{n+1}\subseteq X_{n}$. Therefore,
$X=\bigcap_{n\in\mathbb{N}}X_{n}\neq\emptyset$ and $H(X)=\left\\{y\right\\}$.
Hence, $H$ is surjective.
In order to deal with injectivity, we introduce the following notion.
###### Definition 5.2
Consider a p$\nu\text{-}\mathcal{CA}$ of structural period $1$ and let
$G=(V,E,\ell_{G})$ be its De Bruijn graph. The _product graph_ $P$ of $H$ is a
labeled graph $P=(V\times V,W,\ell_{P})$ where
$(((u,\alpha),(v,\beta)),((w,\gamma),(z,\delta)))\in W$ iff
$\begin{cases}\alpha=\beta\;\text{and}\;\gamma=\delta\\\
((u,\alpha),(w,\gamma))\in E\;\text{and}\;((v,\beta),(z,\delta))\in E\\\
\ell_{G}((u,\alpha),(w,\gamma))=\ell_{G}((v,\beta),(z,\delta))\end{cases}$
and $\ell_{P}:W\to A$ is defined as
$\ell_{P}((((u,\alpha),(v,\beta)),((w,\gamma),(z,\delta))))=\left\\{\begin{array}[]{ll}0,&\text{if}\;u=v\;\text{and}\;w=z\\\
1,&\text{otherwise.}\end{array}\right.$
The _reduced product graph_ $D$ of $P$ is the sub-graph of $P$ made by all the
strongly connected components of $P$ plus all nodes and edges belonging to any
path between two connected components.
Let $k$ be as in Definition 5.1. We can also consider $D$ as the transition
graph of a finite automaton with set of initial states
$\\{((u,-k),(v,-k)):u,v\in A^{2r}\\}$ and set of final states
$\\{((u,k+1),(v,k+1)):u,v\in A^{2r}\\}$. Denote by $L(D)$ the language
recognized by this finite automaton.
###### Lemma 5.2
Injectivity is decidable for p$\nu\text{-}\mathcal{CA}$ with structural period
$p=1$.
###### Proof 10
Let $H$ be a p$\nu\text{-}\mathcal{CA}$ with period $p=1$ and let $D$ its
reduced product graph. We prove that $H$ is injective if and only if
$L(D)\subseteq 0^{*}$.
If $L(D)\not\subseteq 0^{*}$, there exists a word $w\in L(D)$ such that
$w_{i}=1$, for some $i$. By definition of $D$, this means that there are at
least two distinct configurations which have the same image by $H$. Hence $H$
is not injective.
If $H$ is not injective, then there are two distinct configurations $x$ and
$y$ such that $H(x)=H(y)$. Let $i\in\mathbb{Z}$ be such that $x_{i}\neq y_{i}$
and set $m=\max(|i|,k+1)$. For any $j\in\mathbb{Z}$, define
$u^{j}=x_{[j-r,j+r)}$ et $v^{j}=y_{[j-r,j+r)}$. Then, the path on $D$
$(((u^{-m},-k),(v^{-m},-k)),\dots,((u^{-k},-k),(v^{-k},-k)),\dots,((u^{0},0),(v^{0},0)),\dots,\\\
((u^{k+1},k+1),(v^{k+1},k+1)),\dots,((u^{m},k+1),(v^{m},k+1)))$
starts from an initial state, ends at a final state, and contains an edge
labelled with $1$. Hence, $L(D)\not\subseteq 0^{*}$.
###### Theorem 5.3
Surjectivity and injectivity are decidable for p$\nu\text{-}\mathcal{CA}$.
###### Proof 11
Let $H$ be a p$\nu\text{-}\mathcal{CA}$ of radius $r$. If $p=1$, by Lemma 5.1
(resp., Lemma 5.2) we can decide surjectivity (resp., injectivity) of $H$.
Otherwise, without loss of generality, assume $p=r$. By Proposition 3.2, $H$
is topologically conjugated to a p$\nu\text{-}\mathcal{CA}$ $H^{\prime}$ with
structural period 1. By Lemma 5.1 (resp., Lemma 5.2) and since $H$ is
surjective (resp., injective) iff $H^{\prime}$ is as well, we can decide
surjectivity (resp., injectivity) of $H$.
### 5.1 Injectivity and surjectivity: structural implications
We now study how informations (about surjectivity or injectivity) on the
global rule $H$ of a p$\nu\text{-}\mathcal{CA}$ with structural period 1
relate to properties of the composing local rules.
###### Proposition 5.4
Let $F$ and $G$ be two CA of rules $f$ and $g$, respectively. For any
p$\nu\text{-}\mathcal{CA}$ $H$ with period $p=1$ having $f$ and $g$ as right
and left default rules, it holds that
1. 1.
$H$ surjective $\Rightarrow F$ surjective and $G$ surjective
2. 2.
$H$ injective $\Rightarrow F$ surjective and $G$ surjective
3. 3.
$H$ injective on finite configurations $\Rightarrow F$ surjective and $G$
surjective
###### Proof 12
Let $k$ be the largest natural such that $h_{k}\neq f$ or $h_{-k}\neq g$.
Without loss of generality, assume that $F$ is not surjective.
1. 1.
There exists a block $u$ which has no pre-image by $f$. Let $y$ be any
configuration belonging to $[u]_{k+1}$. By definition of $H$, there is no
configuration $x\in A^{\mathbb{Z}}$ with $H(x)=y$.
2. 2.
By a theorem in [12], $f$ admits a diamond, i.e., there exist $u,v,w\in A^{+}$
with $u\neq v$ of same length such that $f(wuw)=f(wvw)$. Build
$x\in[wuw]_{k+1}$ and $y\in[wvw]_{k+1}$ such that $x_{i}=y_{i}$ for all $i$
different from the cylinder positions. By definition of $H$, $H(x)=H(y)$.
3. 3.
the proof is similar to item 2.
For d$\nu\text{-}\mathcal{CA}$ a stronger result holds.
###### Proposition 5.5
Let $F$ be a CA of local rule $f$. For any d$\nu\text{-}\mathcal{CA}$ $H$ with
default rule $f$, it holds that
1. 1.
$H$ injective $\Rightarrow F$ injective
2. 2.
$H$ injective $\Rightarrow H$ surjective
###### Proof 13
Let $k$ be the largest natural such that $h_{k}\neq f$ or $h_{-k}\neq f$. Fix
a configuration $y\in A^{\mathbb{Z}}$ and for any $u\in A^{2k+1}$ let
$y^{u}\in[u]_{-k}$ be the configuration such that $y^{u}_{i}=y_{i}$ for all
$i\in\mathbb{Z}$, $|i|>k$. Define $Y=\\{y^{u}:u\in A^{2k+1}\\}$ and
$X=F^{-1}(Y)$.
1. 1.
If $H$ is injective then $|X|=|H(X)|$ and by Proposition 5.4 $F$ is
surjective. So $X\geq|A|^{2k+1}$. By definition of $H$, it holds that
$H(X)\subseteq Y$. Hence,
$|A|^{2k+1}\leq|X|=|H(X)|\leq|Y|=|A|^{2k+1}$
which gives $|X|=|A|^{2k+1}$ Thus, $F$ is injective.
2. 2.
if $H$ is injective we also have $H(X)=Y$. Thus $y\in Y$ has a pre-image by
$H$.
## 6 Dynamics
In order to study equicontinuity and almost equicontinuity, we introduce an
intermediate class between d$\nu\text{-}\mathcal{CA}$ and
r$\nu\text{-}\mathcal{CA}$.
###### Definition 6.1 ($n$-compatible r$\nu\text{-}\mathcal{CA}$)
A r$\nu\text{-}\mathcal{CA}$ $H$ is _$n$ -compatible with a local rule_ $f$ if
for any $k\in\mathbb{N}$, there exist two integers $k_{1}>k$ and $k_{2}<-k$
such that $\forall i\in[k_{1},k_{1}+n)\cup[k_{2},k_{2}+n),\;h_{i}=f$.
In other words, a $\nu\text{-}\mathcal{CA}$ is $n$-compatible with $f$ if,
arbitrarily far from the center of the lattice, there are intervals of length
$n$ in which the local rule $f$ is applied.
The notion of blocking word and the related results cannot be directly
restated in the context of $\nu\text{-}\mathcal{CA}$ because some words are
blocking just thanks to the uniformity of CA. To overcome this problem we
introduce the following notion.
###### Definition 6.2 (Strongly blocking word)
A word $u\in A^{l}$ is said to be _strongly $s$-blocking_ ($0<s\leq l$) for a
CA $F$ of local rule $f$ if there exists an offset $d\in[0,l-s]$ such that for
any $\nu\text{-}\mathcal{CA}$ $H$ with $\forall i\in[0,l)$, $h_{i}=f$ it holds
that
$\forall x,y\in[u]_{0},\forall n\geq 0,\quad
H^{n}(x)_{[d,d+s)}=H^{n}(y)_{[d,d+s)}\enspace.$
Roughly speaking, a word is strongly blocking if it is blocking whatever be
the perturbations involving the rules in its neighborhood. The following
extends Proposition $5.12$ in [16] to strongly $r$-blocking words.
###### Theorem 6.1
Let $F$ be a CA of local rule $f$ and radius $r$. The following statements are
equivalent:
1. (1)
$F$ is equicontinuous;
2. (2)
there exists $k>0$ such that any word $u\in A^{k}$ is strongly $r$-blocking
for $F$;
3. (3)
any d$\nu\text{-}\mathcal{CA}$ $H$ of default rule $f$ is ultimately periodic.
###### Proof 14
(1) $\Rightarrow$ (2). Suppose that $F$ is equicontinuous. By [15, Th. 4],
there exist $p>0$ and $q\in\mathbb{N}$ such that $F^{q+p}=F^{q}$. As a
consequence, we have that
$\forall u\in A^{*},|u|>2(q+p)r\Rightarrow
f^{p+q}(u)=f^{q}(u)_{[pr,|u|-(2q+p)r)}\enspace.$
Let $H$ be a $\nu\text{-}\mathcal{CA}$ such that $h_{j}=f$ for each
$j\in[0,(2p+2q+1)r)$. For any $x\in A^{\mathbb{Z}}$ and $i\in\mathbb{N}$,
consider the following words:
$\displaystyle s^{(i)}=$ $\displaystyle H^{i}(x)_{[0,qr)}$ $\displaystyle
t^{(i)}=$ $\displaystyle H^{i}(x)_{[qr,(q+p)r)}$ $\displaystyle u^{(i)}=$
$\displaystyle H^{i}(x)_{[(q+p)r,(q+p+1)r)}$ $\displaystyle v^{(i)}=$
$\displaystyle H^{i}(x)_{[(q+p+1)r,(q+2p+1)r)}$ $\displaystyle w^{(i)}=$
$\displaystyle H^{i}(x)_{[(q+2p+1)r,(2q+2p+1)r)}\enspace.$
For all $i\in[0,q+p]$, $u^{(i)}$ is completely determined by
$s^{(0)}t^{(0)}u^{(0)}v^{(0)}w^{(0)}=x_{[0,(2q+2p+1)r)}$ (see Figure 1).
Moreover, for any natural $i$, we have
$\displaystyle u^{(i+q+p)}$
$\displaystyle=f^{q+p}(s^{(i)}t^{(i)}u^{(i)}v^{(i)}w^{(i)})$
$\displaystyle=f^{q}(s^{(i)}t^{(i)}u^{(i)}v^{(i)}w^{(i)})_{[pr,(p+1)r)}$
$\displaystyle=(t^{(i+q)}u^{(i+q)}v^{(i+q)})_{[pr,(p+1)r)}$
$\displaystyle=u^{(i+q)}\enspace.$
$u^{(i)}$$t^{(i)}$$s^{(i)}$$v^{(i)}$$w^{(i)}$$u^{(i+q)}$$t^{(i+q)}$$v^{(i+q)}$$u^{(i+q)}$$qr$$pr$$r$$pr$$qr$$q$iterations$p$iterations
Figure 1: A strongly blocking word.
Summarizing, for all $i\in\mathbb{N}$, $u^{(i)}$ is determined by the word
$x_{[0,(2q+2p+1)r)}$ which is then strongly $r$-blocking. Since $x$ had been
chosen arbitrarily, $(2)$ is true.
(2) $\Rightarrow$ (3). Since any word of length $k$ is strongly $r$-blocking,
any word $u\in A^{2k+1}$ of length $2k+1$ is ($2r+1$)-blocking, i.e., roughly
speaking, $u$ blocks the column $(H^{t}(z)_{[d,d+2r]})_{t\in\mathbb{N}}$ which
appears under itself inside the dynamical evolution of $H$ from any
configuration $z\in[u]_{0}$.
For any $u\in A^{2k+1}$, consider the configuration
$y={}^{\infty}u^{\infty}\in[u]_{-k}$ (bi-infinite concatenation of $u$). There
exist $q_{u}$ and $p_{u}$ such that $F^{p_{u}+q_{u}}(y)=F^{q_{u}}(y)$. Set
$q=\max\\{q_{u}:u\in A^{2k+1}\\}\text{ and }p=lcm\\{p_{u}:u\in A^{2k+1}\\}$
For any word $u\in A^{2k+1}$, the column blocked by $u$ admits $q$ and $p$ as
pre-period and period, respectively.
Let $H$ be a d$\nu\text{-}\mathcal{CA}$ of default rule $f$ and let $n$ be
such that $\forall i,|i|>n,h_{i}=f$. Choose $x\in A^{\mathbb{Z}}$ and
$i\in\mathbb{Z}$ such that $|i|>n+k$. So $h_{i-k}=\dots=h_{i+k}=f$ and
$x_{[i-k,i+k]}$ is a strongly blocking word, then
$H^{p+q}(x)_{i}=H^{q}(x)_{i}$. On the other hand, the sequence
$(H^{j}(x)_{[-n-k,n+k]})_{j\in\mathbb{N}}$ is completely determined by
$u=x_{[-m,m]}$ where $m=n+2k+\max\\{r_{i}:i\in\mathbb{Z}\\}$ (see Figure 2).
Figure 2: Dynamics of a d$\nu\text{-}\mathcal{CA}$ in presence of strongly
blocking words.
Moreover, there exist $\alpha_{u}>0$ and $\beta_{u}\geq q$ such that
$H^{\beta_{u}}(x)_{[-n-k,n+k]}=H^{\beta_{u}+p\alpha_{u}}(x)_{[-n-k,n+k]}$
leading to $H^{\beta_{u}}(x)=H^{\beta_{u}+p\alpha_{u}}(x)$. Set now
$q^{\prime}=\max\\{\beta_{u}:u\in A^{2m+1}\\}\text{ and
}p^{\prime}=lcm\\{p\alpha_{u}:u\in A^{2m+1}\\}.$
Hence, $\forall x\in
A^{\mathbb{Z}},H^{q^{\prime}+p^{\prime}}(x)=H^{q^{\prime}}(x)$ and $H$ is
ultimately periodic.
$(3)\Rightarrow(1)$ Since $H$ is an ultimately periodic
d$\nu\text{-}\mathcal{CA}$ of default rule $f$, the CA $F$ is ultimately
periodic too. By [15, Th. 4], $F$ is equicontinuous.
###### Theorem 6.2
Let $F$ be a CA with local rule $f$ admitting a strongly $r$-blocking word
$u$. Let $H$ be a r$\nu\text{-}\mathcal{CA}$ of radius $r$. If $H$ is
$|u|$-compatible with $f$ then $H$ is almost equicontinuous.
###### Proof 15
Let $p$ and $n$ be the offset and the length of $u$, respectively. For any
$k\in\mathbb{N}$, consider the set $T_{u,k}$ of configurations $x\in
A^{\mathbb{Z}}$ having the following property $\mathcal{P}$: there exist $l>k$
and $m<-k$ such that $x_{[l,l+n)}=x_{[m,m+n)}=u$ and $\forall
i\in[l,l+n)\cup[m,m+n)\;h_{i}=f$. Remark that $T_{u,k}$ is open, being a union
of cylinders. Clearly, each $T_{u,k}$ is dense, thus the set
$T_{u}=\bigcap_{k\geq n}T_{u,k}$ is residual. We claim that any configuration
in $T_{u}$ is an equicontinuity point. Indeed, choose arbitrarily $x\in
T_{u}$. Set $\epsilon=2^{-k}$, where $k\in\mathbb{N}$ is such that $x\in
T_{u,k}$. Then, there exist $k_{1}>k$ and $k_{2}<-k-n$ satisfying
$\mathcal{P}$ (see Figure 3).
$u$$k_{2}$$k_{2}+p$$k_{2}+p+r$$k_{2}+|u|$$-k$$k$$k_{1}$$f$$u$$k_{1}+p$$k_{1}+p+r$$k_{1}+|u|$$f$
Figure 3: An equicontinuity point (see Theorem 6.2).
Fix $\delta=\min\\{2^{-(k_{1}+n)},2^{-k_{2}}\\}$ and let $y\in A^{\mathbb{Z}}$
be such that $d(x,y)<\delta$. Then
$y_{[k_{2},k_{1}+|u|)}=x_{[k_{2},k_{1}+|u|)}$. Since $u$ is $r$-blocking,
$\forall t\in\mathbb{N}$, $H^{t}(x)$ and $H^{t}(y)$ are equal inside the
intervals $[k_{1}+p,k_{1}+p+r]$ and $[k_{2}+p,k_{2}+p+r]$, then
$d(H^{t}(x),H^{t}(y))<\epsilon$.
In a similar manner one can prove the following.
###### Theorem 6.3
Let $F$ be an equicontinuous CA of local rule $f$. Let $k\in\mathbb{N}$ be as
in Proposition 6.1. Any r$\nu\text{-}\mathcal{CA}$ $k$-compatible with $f$ is
equicontinuous.
### 6.1 Perturbing almost equicontinuous CA
In the sequel, we show how the loss of uniformity may lead to a dramatic
change in the dynamical behavior of the automata network.
###### Example 7 (An almost equicontinuous CA)
Let $A=\\{0,1,2\\}$ and define a local rule $f:A^{3}\to A$ as follows:
$\forall x,y\in A$,
$\begin{array}[]{c}f(x,0,y)=\begin{cases}1&\text{if $x=1$ or $y=1$}\\\
0&\text{otherwise}\end{cases}\\\ f(x,1,y)=\begin{cases}2&\text{if $x=2$ or
$y=2$}\\\ 1&\text{otherwise}\end{cases}\\\ f(x,2,y)=\begin{cases}0&\text{if
$x=1$ or $y=1$}\\\ 2&\text{otherwise}\enspace.\end{cases}\end{array}$
We show that the CA defined in Example 7 is almost equicontinuous.
###### Proof 16
Just remark that the number of 0s inside the word $20^{i}2$ is non-decreasing.
Thus $202$ is a $1$-blocking word (see Figure 4).
ht 22002200220022000220022000220022000010-21110-20-20-2
Figure 4: Evolution of words $20^{i}2$ according to $F^{(9)}$.
The following example defines a $\nu\text{-}\mathcal{CA}$ which is sensitive
to the initial conditions although its default rule give rise to an almost
equicontinuous CA.
###### Example 8 (A sensitive $\nu\text{-}\mathcal{CA}$ with an almost
equicontinuous default rule)
Consider the d$\nu\text{-}\mathcal{CA}$ $H^{(8)}:A^{\mathbb{Z}}\to
A^{\mathbb{Z}}$ defined as follows
$\forall x\in A^{\mathbb{Z}},\forall i\in\mathbb{Z},\quad
H^{(8)}(x)_{i}=\begin{cases}1&\text{if $i=0$}\\\
f(x_{i-1},x_{i},x_{i+1})&\text{otherwise\enspace,}\end{cases}$
where $f$ and $A$ are as in Example 7.
(a)
(b)
(c) (d)
Figure 5: Space-time diagrams for $F^{(9)}$ and $H^{(8)}$.
Remark that positive and negative cells do not interact each other under the
action of $H^{(8)}$. Therefore, in order to study the behavior of $H^{(8)}$,
it is sufficient to consider the action of $H^{(8)}$ on $A^{\mathbb{N}}$. In
the sequel, we will simply note by $H$ the map $H^{(8)}$.
In order to prove that $H^{(8)}$ is sensitive, we need some technical Lemmata.
###### Lemma 6.4
For any $u\in A^{*}$, consider the sequence $(u^{(n)})_{n\in\mathbb{N}}$
defined as:
$\begin{cases}u^{(n+1)}=f(1u^{(n)}0)&\forall n\in\mathbb{N}\\\
u^{(0)}=u2&\end{cases}$
Then,
$\exists m,\,\forall n\geq m,\quad u^{(n)}=1^{|u|+1}$
###### Proof 17
We proceed in 4 steps.
1. 1.
First of all, we are going to show that there exists $n_{0}\in\mathbb{N}$ such
that $u^{(n_{0})}$ does not contain any $1$. In particular, we prove that
there exists $n_{0}\in\mathbb{N}$ such that $\forall n\leq n_{0}$ the integer
$i^{(n)}=\min\left\\{i\leq|u|:u^{(n)}_{i}=2\text{ and
}u^{(n)}_{[i+1,|u|]}\in\\{0,2\\}^{*}\right\\}$
is well defined with the property $P(n)=\left(\forall k\in[0,n),i^{(k+1)}\leq
i^{(k)}\right)$ and $i^{(n_{0})}=0$. By definition, $i^{(0)}$ is well defined
and the property $P(0)$ is true. Suppose now that, for some $n\in\mathbb{N}$,
$i^{(n)}$ is well defined and the property $P(n)$ is true. We deal with the
following cases.
1. (a)
If $i^{(n)}=0$, then we set $n_{0}=n$ and we are done.
2. (b)
If $i^{(n)}\neq 0$ does not contain any $1$, then we can write
$u^{(n)}=0^{i^{(n)}}2w$ with $w\in\\{0,2\\}^{*}$ and we have
$u^{(n+1)}=10^{i^{(n)}-1}2w$. So, $i^{(n+1)}=i^{(n)}$ is well defined with
$P(n+1)$ true and, having the element $n+1$ as a starting point, we fall in
the next case.
3. (c)
If $i^{(n)}\neq 0$ contains at least one $1$, let $h\in[0,i^{(n)})$ be the
greatest position in which it appears and set $s=i^{(n)}-h-1$. We can write
$u^{(n)}=v^{(n)}10^{s}2w$, for some $v^{(n)}\in A^{h}$ and
$w\in\\{0,2\\}^{*}$. Then, for each $j\in[1,s]$, we obtain
$u^{(n+j)}=v^{(n+j)}10^{s-j}2w$, for some $v^{(n+j)}\in A^{h+j}$. So, for each
$j\in[1,s]$, $i^{(n+j)}=i^{(n)}$ is well defined with $P(n+j)$ true.
Furthermore, it holds that $u^{(n+s+1)}=v^{(n+s+1)}20w$, for some
$v^{(n+s+1)}\in A^{h+s+1}$, and $i^{(n+s+1)}=i^{(n+s+1)}-1$ is well defined
with $P(n+s+1)$ true, and we can reconsider the three cases having the element
$n+s+1$ as starting point.
By considering iteratively the three cases, we are sure to reach a natural
$n_{0}$ such that $i^{(n_{0})}=0$ since whenever we fall in the third case the
value of $i^{(n)}$ decreases.
2. 2.
Proceeding by induction, we now show that
$\forall n\geq n_{0},\exists k\in\mathbb{N},\exists
v\in\\{0,2\\}^{*},\quad\text{s.t.}\quad u^{(n)}=1^{k}v.$
Clearly, this is true for $n=n_{0}$ with $k=0$. Assume now that the statement
is true for some $n\geq n_{0}$ and consider the following cases.
* (a)
If $v=\epsilon$, then $u^{(n+1)}=u^{(n)}=1^{k}v$
* (b)
If $v_{0}=0$, then $u^{(n+1)}=1^{k+1}v_{[1,|v|-1]}$
* (c)
If $v_{0}=2$ and $k\neq 0$, then $u^{(n+1)}=1^{k-1}20v_{[1,|v|-1]}$
* (d)
If $v_{0}=2$ and $k=0$, then $u^{(n+1)}=0v_{[1,|v|-1]}$
In all the cases, the statement is true for $n+1$.
As a consequence, we also have that the number $|u^{(n)}|_{2}$ of $2$ inside
$u^{(n)}$ is a (non strictly) decreasing sequence:
$\forall n\geq n_{0},\quad|u^{(n+1)}|_{2}\leq|u^{(n)}|_{2}$
Indeed, $u^{(n)}$ does not contain the block $121$, which, transforming itself
into $202$, is the unique one able to increase the number of $2$.
3. 3.
We now prove that there exists $n_{1}\geq n_{0}$, such that $u^{(n_{1})}$ no
longer contains any $2$, and then $u^{(n_{1})}=1\cdots 10\cdots 0$. This is
assured by showing that
$\forall n\geq n_{0},|u^{(n)}|_{2}>0\Rightarrow\exists
s\in\mathbb{N},|u^{(n+s)}|_{2}<|u^{(n)}|_{2}$
Let $n\geq n_{0}$ such that $|u^{(n)}|_{2}>0$. Since $u^{(n)}=1^{k}v$ for some
$k\in\mathbb{N},v\in\\{0,2\\}^{*}$, we can write $u^{(n)}=1^{k}0^{h}2w$ for
some $h\in\mathbb{N},w\in\\{0,2\\}^{*}$. Thus, we have $u^{(n+h)}=1^{k+h}2w$
and $u^{(n+h+i)}=1^{k+h-i}20^{i}w$ for each $i\in[1,h+k]$. So,
$u^{(n+2h+k)}=20^{h+k}w$ and, setting $s=2h+k+1$, we obtain
$u^{(n+s)}=0^{s-h}w$, assuring that $|u^{(n+s)}|_{2}<|u^{(n)}|_{2}$.
4. 4.
Since $u^{(n_{1})}=1^{k}0^{h}$ for some $h,k\in\mathbb{N}$, it is easy to
observe that $u^{(n_{1}+i)}=1^{k+i}0^{h-i}$, $i\in[1,h]$. In particular,
setting $m=n_{1}+h$, we obtain $u^{(m)}=1^{|u|+1}$ and $\forall n\geq
m,u^{(n)}=u^{(m)}$. This concludes the proof.
###### Lemma 6.5
Consider the rewriting system on $A^{*}\times\\{0,1\\}$ defined by the
following rules:
1. 1.
$(u0,0)\xrightarrow{1}(u,0)$
2. 2.
$(u1,0)\xrightarrow{2}(u,1)$
3. 3.
$(u2,0)\xrightarrow{3}(f(1u20),0)$
4. 4.
$(u0,1)\xrightarrow{4}(f(1u0),1)$
5. 5.
$(u1,1)\xrightarrow{5}(u,1)$
6. 6.
$(u2,1)\xrightarrow{6}(f(1u2),0)$
7. 7.
$(\epsilon,x)\xrightarrow{7}(\epsilon,1)$
Starting from any $(u,x)\in A^{*}\times\\{0,1\\}$, after a certain number $m$
of rule applications, the system ultimately falls into $(\epsilon,1)$.
###### Proof 18
This system is non ambiguous and then, for any $(u,x)\in
A^{*}\times\\{0,1\\}$, it (well) defines the sequence
$(u^{(n)},x^{(n)})_{n\in\mathbb{N}}$ such that
$\begin{cases}(u^{(n)},x^{(n)})\rightarrow(u^{(n+1)},x^{(n+1)})&\forall
n\in\mathbb{N}\\\ (u^{(0)},x^{(0)})=(u,x)&\end{cases}$
where $\rightarrow$ is the unique possible application of a system rule.
Consider the sequence
$(l^{(n)})_{n\in\mathbb{N}}=(|u^{(n)}|)_{n\in\mathbb{N}}$. By definition, it
is a (non strictly) decreasing sequence and then it converges to some
$l\in\mathbb{N}$, or, equivalently, there exists $m\in\mathbb{N}$ such that
$\forall n\geq m$, $l^{(n)}=l$. We show that $l=0$ and this also prove the
thesis. For a sake of argument, suppose that $l>0$. Thus, there exists
$k\in\mathbb{N}$ such that $\forall n\geq
k,(u^{(n)},x^{(n)})\xrightarrow{3}(u^{(n+1)},x^{(n+1)})$ since, except rule 7,
rule 3 is the only one leaving $l^{(n)}$ unchanged. Furthermore, the sequence
$(u^{(n+k)})_{n\in\mathbb{N}}$ verifies the hypothesis of Lemma 6.4 and so it
is ultimately equal to $1^{l}$, that is contrary to the fact that $\forall
n\geq k,u^{(n)}_{l}=2$, since rule 3 is always applied
###### Lemma 6.6
Let $\mathcal{F}=\\{01,12,20,22\\}$. For any $x\in A^{\mathbb{N}}$ and any
$i\in\mathbb{N}$, if no element of $\mathcal{F}$ appears inside
$x_{[i,\infty)}$, then no element of $\mathcal{F}$ appears inside
$H^{(8)}(x)_{[i+1,\infty)}$.
###### Proof 19
The $f$–pre-images of words in $\mathcal{F}$ are :
* 1.
$f^{-1}(01)$ = {0001, 1201, 2001}
* 2.
$f^{-1}(12)$ = {0012, 0112, 1012, 1020, 1022, 1112, 2012}
* 3.
$f^{-1}(20)$ = {0120, 0121, 0122, 0200, 0202, 0221, 1120, 1121, 1122, 2120,
2121, 2122, 2200, 2202, 2221}
* 4.
$f^{-1}(22)$ = {0220, 0222, 2112, 2220, 2222}
So, if there exists $w\in\mathcal{F}$ appearing in
$H^{(8)}(x)_{[i+1,\infty)}$, necessarily a word $u\in\mathcal{F}$ is inside
$x_{[i,\infty)}$.
###### Lemma 6.7
For any $u\in A^{*}$, there exists $n_{0}\in\mathbb{N}$ s.t. $\forall
n>n_{0},H^{n}(u0^{\infty})_{1}=1$.
###### Proof 20
Consider the sequences $(u^{(n)},x^{(n)})_{n\in\mathbb{N}}$ and
$(l^{(n)})_{n\in\mathbb{N}}$ from Lemma 6.5 in which
$(u^{(0)},x^{(0)})=(u_{[1,|u|-1]},0)$. Define the sequences
$(k^{(n)})_{n\in\mathbb{N}}$ and $(y^{(n)})_{n\in\mathbb{N}}$ as follows:
$\begin{cases}k^{(n+1)}=k^{(n)}+1&\forall
n\in\mathbb{N}\;\text{s.t.}\;(u^{(n)},x^{(n)})\xrightarrow{a}(u^{(n+1)},x^{(n+1)})\quad
a=3,4,6,7\\\ k^{(n+1)}=k^{(n)}&\forall
n\in\mathbb{N}\;\text{s.t.}\;(u^{(n)},x^{(n)})\xrightarrow{a}(u^{(n+1)},x^{(n+1)})\quad
a=1,2,5\\\ k^{(0)}=0&\end{cases},$
and, $\forall n\in\mathbb{N}$, $y^{n}=H^{k^{(n)}}(u0^{\infty})$, respectively.
First of all, we are going to prove that the property
$L(n)=\left(y^{n}_{[1,l^{(n)}+1]}=u^{(n)}x^{(n)}\right),$
linking the dynamics of $H$ with the one induced by the rewriting system, and
the property
$M(n)=\left(\forall w\in\mathcal{F},\quad\text{$w$ does not appears
inside}\;y^{n}_{[l^{(n)}+1,\infty)}\right),$
are valid for all $n\in\mathbb{N}$. We proceed by induction.
The properties $L(n)$ and $M(n)$ statement are clearly true for $n=0$. Suppose
now that they are valid for some $n\in\mathbb{N}$ and let the rewriting system
evolve on $(u^{(n)},x^{(n)})$ according to the rule $a$, for some
$a=1,\ldots,7$.
If $a\in\\{1,2,5\\}$, then $k^{(n+1)}=k^{(n)}$ and $l^{(n+1)}=l^{(n)}-1$.
Hence, $L(n+1)$ is true. Moreover, when the restriction passes from
$[l^{(n)}+1,\infty)$ to $[l^{(n+1)}+1,\infty)$, the additional word inside the
configuration $y^{n+1}=y^{n}$ is either $00$, or $10$, or $11$, depending on
the value of $a$. So, $M(n+1)$ is also valid.
If $a=7$, then $k^{(n+1)}=k^{(n)}+1$ and $l^{(n+1)}=l^{(n)}=0$. Since $L(n)$
and $M(n)$ are true, $y^{n}_{[0,2]}\in\\{100,102,110,111\\}$, and then
$x^{(n+1)}=1=y^{n+1}_{1}$. So, $L(n+1)$ is valid. Moreover, no element
$\mathcal{F}$ appears inside $y^{n}_{[1,\infty)}$ and neither inside
$y^{n}_{[0,\infty)}$, and, hence, by Lemma 6.6, neither inside
$y^{n+1}_{[1,\infty)}$. Thus, $M(n+1)$ is true.
If $a\in\\{3,4,6\\}$, then $k^{(n+1)}=k^{(n)}+1$. By the fact that $L(n)$ is
true, we have $u^{(n+1)}=f(y^{n}_{[0,l^{(n+1)}+1]})$, or, equivalently,
$u^{(n+1)}=y^{n+1}_{[1,l^{(n+1)}]}$. Since $M(n)$ is true, by Lemma 6.6,
$y^{n+1}_{[l^{(n)}+2,\infty)}$ does not contain any element of $\mathcal{F}$.
It remains to prove that $x^{(n+1)}=y^{n+1}_{l^{(n+1)}+1}$, and there is no
word of $\mathcal{F}$ inside $y^{n+1}_{[l^{(n+1)}+1,l^{(n)}+2]}$.
If $a=3$, $l^{(n+1)}=l^{(n)}$ and we have $y^{n+1}_{l^{(n+1)}+1}=f(20a)$ where
necessarily $a\neq 1$ since $M(n)$ is valid and $01\in\mathcal{F}$. Hence,
$y^{n+1}_{l^{(n+1)}+1}=0=x^{(n+1)}$ and $L(n+1)$ is true. For a sake of
argument, assume that the word
$w=y^{n+1}_{[l^{(n+1)}+1,l^{(n)}+2]}\in\mathcal{F}$. Necessarily, we obtain
$w=01$ and so, by definition of $f$, $y^{n}_{[l^{(n)}+2,l^{(n)}+3]}=01$, that
is a contradiction. Hence, $M(n+1)$ is valid.
If $a=4$ (resp., $a=6$), $l^{(n+1)}=l^{(n)}-1$ and we have
$y^{n+1}_{l^{(n+1)}+1}=f(a01)$ (resp., $f(a21)$). So,
$y^{n+1}_{l^{(n+1)}+1}=0=x^{(n+1)}$ and $L(n+1)$ is true. Since $M(n)$ is
true, $y^{n}_{[l^{(n)},l^{(n)}+3]}=01bc$ (resp., $21bc$) with
$bc\in\\{00,02,10,11\\}$. By definition of $f$, it follows that
$y^{n+1}_{[l^{(n+1)},l^{(n+1)}+2]}=111$ and then $M(n+1)$ is valid.
Summarizing, we have proved that $L(n)$ and $M(n)$ are valid for all
$n\in\mathbb{N}$ and, in particular,
$\forall n\in\mathbb{N},\quad
H^{k^{(n)}}(u0^{\infty})_{[1,l^{(n)}+1]}=u^{(n)}x^{(n)}$
Now, let $m$ be the integer from Lemma 6.5. Recall that $\forall n\geq m$,
$l^{(n)}=0$ and $(u^{(n)},x^{(n)})=(\epsilon,1)$. Thus, setting
$n_{0}=k^{(m)}$, we obtain
$\forall n\geq n_{0},\quad H^{n}(u0^{\infty})_{1}=1.$
###### Lemma 6.8
For any $u\in A^{*}$ and any $n\in\mathbb{N}$, there exists $m>n$ such that
$H^{m}(u2^{\infty})_{1}=2$.
###### Proof 21
For any $u\in A^{*}$ and any $n\in\mathbb{N}$, define the configuration
$z^{n}=H^{n}(u2^{\infty})$ and the integers
$a^{(n)}=\max\\{i\in\mathbb{N}:{z_{i}^{n}}=1\\}$ and
$b^{(n)}=\min\\{i\in\mathbb{N}:{z_{i}^{n}}=2\wedge\forall j>i,{z_{j}^{n}}\neq
1\\}$. Remark that $a^{(n)}$ and $b^{(n)}$ are well defined and $\forall
n\in\mathbb{N}$, $a^{(n)}<b^{(n)}$. We want to prove that $\forall
n\in\mathbb{N},(b^{(n)}>1\Rightarrow\exists k>n,b^{(k)}<b^{(n)})$. Let
$n\in\mathbb{N}$ be such that $b^{(n)}>1$. Since
$z^{n}_{[a^{(n)},b^{(n)}]}=10^{b^{(n)}-a^{(n)}-1}2$ and $\forall i>b^{(n)}$,
${z_{i}^{n}}\neq 1$, by definition of $f$, it follows that
$a^{(n+i)}=a^{(n)}+i$, $b^{(n+i)}=b^{(n)}$, for $i\in[0,b^{(n)}-a^{(n)}-1]$.
Hence, $b^{(n+b^{(n)}-a^{(n)})}=b^{(n)}-1<b^{(n)}$.
We conclude stating that $H^{(8)}$ is sensitive.
###### Proof 22
It is a direct consequence of Lemmata 6.7 and 6.8.
The following example shows that default rules individually defining almost
equicontinuous CA can also constitute $\nu\text{-}\mathcal{CA}$ that have a
completely different behavior from the one in Example 8.
###### Example 9 (An equicontinuous $\nu\text{-}\mathcal{CA}$ made by almost
equicontinuous CA)
Let $A=\\{0,1,2\\}$ and define the local rule $f:A^{3}\to A$ as: $\forall
x,y,z\in A$, $f(x,y,z)=2$ if $x=2$ or $y=2$ or $z=2$, $z$ otherwise. The CA
$F$ of local rule $f$ is almost equicontinuous since $2$ is a blocking word.
The restriction of $F$ to $\\{0,1\\}^{\mathbb{Z}}$ gives the shift map which
is sensitive. Thus $F$ is not equicontinuous. Define now the following
d$\nu\text{-}\mathcal{CA}$ $H^{(9)}$:
$\forall x\in A^{\mathbb{Z}},\forall i\in\mathbb{Z},\quad
H^{(9)}(x)_{i}=\begin{cases}2&\text{if $i=0$}\\\
f(x_{i-1},x_{i},x_{i+1})&\text{otherwise}\enspace.\end{cases}$
We now prove that $H^{(9)}$ is equicontinuous.
###### Proof 23
Let $n\in\mathbb{N},x,y\in A^{\mathbb{Z}}$ be such that
$x_{[-2n,2n]}=y_{[-2n,2n]}$. Since $H$ is of radius $1$, $\forall k\leq
n,H^{k}(x)_{[-n,n]}=H^{k}(y)_{[-n,n]}$ and $\forall k>n$,
$H^{k}(x)_{[-n,n]}=2^{2n+1}=H^{k}(y)_{[-n,n]}$. So, $H$ is equicontinuous.
## 7 Conclusions
This paper have introduced $\nu\text{-}\mathcal{CA}$, an extension of CA model
obtained by relaxing the uniformity property (i.e., the fact that the same
local rule is applied to all sites of the CA lattice). The study of how this
change affects the dynamics of the systems has just started. We proved several
results concerning basic set properties like injectivity and surjectivity.
Moreover, we studied how informations about the $\nu\text{-}\mathcal{CA}$ can
determine properties on the underlying CA or vice-versa.
The study of $\nu\text{-}\mathcal{CA}$ can be continued along several
different directions. Of course, it would be interesting to progress in the
analysis of the dynamical behavior. In particular we believe it would be
worthwhile to study how information moves along the space-time diagrams and
how the density of changes affects the entropy of the system.
It is well-known that CA cannot be used a random generator and, in general,
they are poor (but fast) random generators. Can $\nu\text{-}\mathcal{CA}$ be a
better tool in this context?
## 8 Acknowledgements
This work has been supported by the French ANR Blanc Projet “EMC”, by the
“Coopération scientifique international Région Provence–Alpes–Côte d’Azur”
Project “Automates Cellulaires, dynamique symbolique et décidabilité”, and by
the PRIN/MIUR project “Mathematical aspects and forthcoming applications of
automata and formal languages”.
## References
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* [3] J. Cervelle, A. Dennunzio, and E. Formenti. Chaotic behavior of cellular automata. In B. Meyers, editor, Mathematical basis of cellular automata, volume 1 of Encyclopedia of Complexity and System Science, pages 978–989. Springer Verlag, 2009.
* [4] P. Chaudhuri, D. Chowdhury, S. Nandi, and S. Chattopadhyay. Additive Cellular Automata Theory and Applications, volume 1. IEEE Press, 1997.
* [5] A. Dennunzio, P. Di Lena, E. Formenti, and L. Margara. On the directional dynamics of additive cellular automata. Theoretical Computer Science, 410:4823–4833, 2009.
* [6] A. Dennunzio and E. Formenti. Decidable properties of 2d cellular automata. In Twelfth Conference on Developments in Language Theory (DLT 2008), volume 5257 of Lecture Notes in Computer Science, pages 264–275. Springer-Verlag, 2008.
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* [12] G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Mathematical System Theory, 3:320–375, 1969.
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* [15] P. Kůrka. Languages, equicontinuity and attractors in cellular automata. Ergodic Theory & Dynamical Systems, 17:417–433, 1997.
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* [17] P. Kůrka. Topological dynamics of one-dimensional cellular automata. In B. Meyers, editor, Mathematical basis of cellular automata, Encyclopedia of Complexity and System Science. Springer Verlag, 2008.
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* [19] T. Nakano, T. Morie, M. Nagata, and A. Iwata. A cellular automaton-type image extraction algorithm and its implementation using a fpga. In IEEE Asia-Pacific Conference on Circuits and Systems, volume 2, pages 197– 200. IEEE Press, 2002.
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|
arxiv-papers
| 2011-07-26T14:47:15 |
2024-09-04T02:49:20.947628
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alberto Dennunzio, Enrico Formenti, Julien Provillard",
"submitter": "Alberto Dennunzio",
"url": "https://arxiv.org/abs/1107.5228"
}
|
1107.5266
|
# Identifying Overlapping and Hierarchical Thematic Structures
in Networks of Scholarly Papers:
A Comparison of Three Approaches
Frank Havemann1,∗ Jochen Gläser2 Michael Heinz1 Alexander Struck1
1 Institut für Bibliotheks- und Informationswissenschaft, Humboldt-Universität
zu Berlin, Berlin, Germany
2 Zentrum Technik und Gesellschaft, Technische Universität Berlin, Berlin,
Germany
$\ast$ E-mail: Frank (dot) Havemann (at) ibi.hu-berlin.de
## Abstract
We implemented three recently proposed approaches to the identification of
overlapping and hierarchical substructures in graphs and applied the
corresponding algorithms to a network of 492 information-science papers
coupled via their cited sources. The thematic substructures obtained and
overlaps produced by the three hierarchical cluster algorithms were compared
to a content-based categorisation, which we based on the interpretation of
titles and keywords. We defined sets of papers dealing with three topics
located on different levels of aggregation: h-index, webometrics, and
bibliometrics. We identified these topics with branches in the dendrograms
produced by the three cluster algorithms and compared the overlapping topics
they detected with one another and with the three predefined paper sets. We
discuss the advantages and drawbacks of applying the three approaches to paper
networks in research fields.
## 1 Introduction
The delineation of scientific fields is a pertinent problem of science studies
in general and bibliometrics in particular (cf. e.g. van Raan, 2004 [1, p.
39]). Bibliometric research has shown that clusters in networks of papers do
not have natural’ boundaries (cf. Zitt et al., 2005 [2]). This is why fields
must be delineated by applying thresholds for parameters, which are chosen
arbitrarily in terms of ‘good structures’ for the purposes of the analysis at
hand (cf. e.g. references [3, 4]).
However, the problem of delineation might be a consequence of the overlap of
thematic structures. The overlap of themes in publications is well known to
science studies. Sullivan et al. (1977) [5, p. 235] observed that in the
literature of the field of weak interaction half of the references were
articles outside the specialty. Amsterdamska and Leydesdorff (1989) [6, p.
461] provide an example of an article that targeted two different specialties
at once. If disjoint clusters of co-cited sources (Marshakova 1973 [7], Small
1973 [8]) are projected forward to their citing papers, the clusters of citing
papers inevitably overlap—a phenomenon that has never been explored by
bibliometrics. Taken together, these observations suggest that the sciences
consist of numerous fields of different sizes that partially or totally
overlap, i.e. feature hierarchies as well as mutually overlapping ‘neighbours’
with fuzzy boundaries.
If thematic structures have boundaries that are hidden by their overlaps,
delineation is not impossible in principle but rather depends on tools that
enable the identification of overlapping fields and topics.
So far, only one such tool, namely co-citation analysis, has been applied to
the delineation task. However, it assumes disjoint source clusters and locates
thematic overlaps only in citing papers. This unrealistic assumption makes it
unsuitable to detect overlapping topics. An attempt to obtain overlapping
thematic structures by singular value decomposition (SVD) of paper-source
matrices failed because SVD produces—at least if arbitrary thresholds are
avoided—as many thematic substructures as there are papers [9], which again is
an unrealistic assumption.
The aim of this paper is to introduce and preliminarily assess three
algorithms for the identification of overlapping thematic structures in
networks of papers. We derived these algorithms from three recently proposed
approaches to the detection of overlapping and hierarchical substructures in
networks—which in network analysis are called communities. For a concise
description of the current state of finding communities in networks see the
introduction of reference [10]. Our selection and specification of the general
approaches is based on the assumption that the thematic substructures both
overlap and build hierarchies.
We further had to take into account the information utilised by the different
approaches. Thematic structures can be determined top-down using global
information or bottom-up using either global and local or only local
information. This corresponds to different ways in which scientific
perspectives are used in the construction of thematic structures. Since the
production of contributions to scientific knowledge is based on the
interpretation of that knowledge by individual producers [11], thematic
structures in paper sets are always constructed from the individual
perspectives of the authors. A bottom-up approach using only local information
enables the reconstruction of thematic structures from the perspective of
those contributing knowledge to these themes. The use of global information in
the top-down or bottom-up construction of thematic structures, e.g. by
spectral and modularity-based methods [12, p. 41, p. 27], is akin to including
the perspective of ‘outsiders’, i.e. of authors/papers not contributing to the
specific topic. Such a ‘democratic’ procedure can be justified as well but is
likely to lead to different results (for an attempt to justify the global
perspective see Klavans and Boyack, 2011 [4]).
These considerations made us select three approaches that enable the
identification of overlapping and hierarchical structures in networks on the
basis of local information. A first approach starts from hard clusters
obtained by any clustering method and fractionally assigns the nodes at the
borders between clusters to these clusters (cf. e.g. Wang et al., 2009 [13]).
Another approach is based on a hard clustering of links between nodes into
disjoint modules, which makes nodes members of all modules (or communities)
that their links belong to (cf. e.g. Ahn et al., 2010 [14]). The third
approach constructs natural communities of all nodes, which can overlap with
each other, by applying a greedy algorithm that maximises local fitness (cf.
e.g. Lancichinetti et al., 2009 [15]).
We introduce our implementations of the three approaches and discuss their
basic features using a small benchmark graph (the karate club, Zachary, 1977
[16]) as an example. The comparative analysis applies the algorithms to a
network of 492 bibliographically coupled papers published 2008 in six
information-science journals. The use of information-science papers enabled
the construction of paper sets of selected topics by manually assigning papers
to the topics h-index, webometrics, and bibliometrics on the basis of titles,
abstracts, and keywords. The clustering solutions and the overlap of modules
were then assessed by comparing them to the paper sets. On the basis of this
comparison we discuss advantages and disadvantages of the three algorithms.
## 2 Communities in Networks
In network analysis, communities are understood as cohesive subgroups of nodes
separated from the rest of the graph. Thus, communities can only be found in
networks if there are groups of densely interconnected nodes that are only
loosely connected to each other. Most community definitions are based on these
two aspects, i.e. cohesion and separation [12, pp. 83–87]. In order to apply
algorithms for the detection of communities, cohesion and separation must be
defined [17]. Owing to the continuous nature of the two properties,
communities cannot be detected unequivocally. Instead, structures of varying
‘communityness’ can be identified [18].
In the case of thematic structures in networks of papers, the communities to
be detected do not only overlap each other but are also hierarchically
ordered. For hierarchies of communities, both cohesion and separation can be
measured directly in the dendrogram. The simplest measure of separation of a
community is the similarity level $s_{u}$ at which its branch in the
dendrogram unites with another branch. The simplest measure of cohesion of a
community is the similarity level $s_{d}$ at which its branch in the
dendrogram decays into two branches but here low level means high cohesion.
Thus, good communities are those with high level $s_{u}$ and low level
$s_{d}$. This corresponds nicely to the usual selection of long branches as
important ones. They have large differences $s_{u}-s_{d}$, i.e. are stable
over relatively large similarity intervals. Using this difference, we can
order branches with respect to their quality as communities, i.e. combined
cohesion and separation.
In our experiments, the stability is negatively correlated with community
size. Many small branches are very stable and many larger branches are very
unstable. In order to find ‘interesting’ communities, we plot branch length
$s_{u}-s_{d}$ over community size and identify communities that are unusually
stable for their size, i.e. are represented by branches far from the axes of
the plot.
An alternative approach to community delineation associates cohesion with high
internal and separation with low external degrees of community members. The
internal degree $\kappa_{\mathrm{in}}(C,V_{i})$ of a node $V_{i}$ is defined
as the sum of weights of edges linking this node with nodes in community $C$,
its external degree $\kappa_{\mathrm{out}}=\kappa-\kappa_{\mathrm{in}}$, where
$\kappa$ is the node’s total degree. Radicchi et al. (2004) [19] define a
community in the strong sense as a set of nodes all of which have higher
internal than external degree. For a community in the weak sense they only
demand that the sum of internal degrees exceeds the sum of external degrees.
These sums are usually referred to as the internal and external degrees of
community $C$:
$k_{\mathrm{in,out}}(C)=\sum_{V_{i}\in C}\kappa_{\mathrm{in,out}}(C,V_{i}).$
(1)
These internal and external degrees of a community can be used to define its
fitness (see the approaches ‘natural communities’ and ‘fuzzification’ for
applications). By combining cohesion and separation, the fitness measure
evaluates the quality of a community in a similar way as $s_{u}-s_{d}$ does
based on a community’s branch in a dendrogram.
When applied to overlapping communities, the measures used in the delineation
of weak communities must take the nature of overlaps into account. Following
Steve Gregory (2011) [20], we distinguish between crisp and fuzzy overlapping
communities. If a network has crisp overlapping communities, nodes either
belong or don’t belong to a community. Communities are fuzzy if individuals’
grades of membership vary. This type of structure is appropriate for the
relationship between papers and topics because most papers cover several
topics in varying intensities, which led us to the application of fuzzy set
theory.
Fuzzy set theory operates with membership grades that are real numbers between
zero and one but does not assume that a node’s grades of membership in
different sets sum up to unity. A node could also be a full member in more
than one community.
To determine whether a fuzzy community $C$ is a community in the weak sense we
have to redefine its internal and external degree $k_{\mathrm{in,out}}(C)$ by
weighting the degrees with node membership grades. With $\mu_{i}(C)=1$ if
$V_{i}\in C$ and $\mu_{i}(C)=0$ otherwise, we can rewrite the definitions
given above for crisp communities as
$k_{\mathrm{in}}(C)=\sum_{i,j=1}^{n}\mu_{i}(C)a_{ij}\mu_{j}(C)$ (2)
and
$k_{\mathrm{out}}(C)=\sum_{i,j=1}^{n}\mu_{i}(C)a_{ij}[1-\mu_{j}(C)],$ (3)
where $a_{ij}$ is the weight of edge $(i,j)$ and $n$ the graph size. These
formulae can also be used for a fuzzy community $C$ if $\mu_{i}(C)$ is
identified with node’s $V_{i}$ membership grade in $C$. Then $1-\mu_{i}(C)$ is
its membership grade in $C$’s fuzzy complement. Fuzzy set $C$ is a community
in the weak sense if $k_{\mathrm{in}}(C)>k_{\mathrm{out}}(C)$.
## 3 Three Approaches
We introduce the three approaches and explain their basic mechanisms with a
simple example, namely the social network of 34 members of a karate club
analysed by Zachary (1977) [16].111using the unweighted graph:
http://networkx.lanl.gov/examples/graph/karate_club.html Members of the karate
club were asked about friendship ties. The network turned out to have two
central actors who, after the split of the original club, founded separate new
clubs. Authors who implemented algorithms based on the three approaches
applied them to the network described by Zachary.
### 3.1 Natural Communities
Figure 1: Natural communities. Karate club graph with overlapping communities
of two nodes (red and violet).
A natural community of a node can be constructed by any ‘greedy’ algorithm
which evaluates the inclusion of neighbouring nodes into the community using
an appropriate metric or fitness function. If a community with a neighbour
node is fitter than without it, the neighbour will be included. The essence of
this local approach is that independently constructed natural communities of
nodes can overlap. Figure 1 shows two overlapping communities of karate club
members. On the left-hand side, the red node’s community has all yellow and
green nodes as its members. On the right-hand side, the violet node’s
community has all blue and green nodes as members. Thus, we have five (green)
nodes in the overlap of both natural communities.
The idea to identify overlapping communities as sub-graphs which are locally
optimal with respect to some given metric was first published by Baumes et al.
(2005) [21]. It can be implemented in several ways. In the same year, Baumes
et al. [22] tested a combination of two greedy algorithms which both use the
same metric. The _initialisation_ produces disjoint seed clusters, the metric
of which is then improved by an iterative procedure leading to overlapping
communities.
Lancichinetti et al. (2009) [15] combined the concept of locally optimal sub-
graphs with the idea of variable resolution to enable their algorithm to
reveal hierarchical community structures. They introduced a resolution
parameter into their fitness function. Higher resolution results in smaller,
lower in larger natural communities. The fitness function includes only local
information. It is defined as the ratio of the sum of internal degrees
$k_{\mathrm{in}}(C)$ to the sum of all degrees
$k(C)=k_{\mathrm{in}}(C)+k_{\mathrm{out}}(C)$ of nodes in a community $C$. The
denominator is taken to the power of $\alpha$, the resolution parameter:
$f(C,\alpha)=\frac{k_{\mathrm{in}}(C)}{k(C)^{\alpha}}.$ (4)
Figure 1 displays a cover of the karate-club network obtained by Lancichinetti
et al. with a stochastic version of their algorithm for the resolution
interval $0.76<\alpha<0.84$. Their LFM (local fitness maximisation) algorithm
has to be repeated for all resolution levels of interest.
The construction of a scientific paper’s natural community in a similarity
network of papers can be interpreted as the construction of its thematic
environment from its own ‘scientific perspective’. This idea is attractive
from a conceptual point of view because it mimics the way in which scientists
apply their individual perspectives when constructing their fields. This is
why locality is a realistic assumption for topic extraction in paper networks.
At the same time, the strictly local approach enables the local exploration of
networks which are too big for global analysis like the Web or the complete
citation network of scientific papers. A node’s natural community is a local
structure that can be constructed without knowing the whole graph. The idea to
find local community structures without knowing the whole graph by using a
greedy local cluster algorithm goes back to Clauset (2005) [23]. His procedure
can also be used to construct overlapping graph modules [24]. In contrast to
the resolution-depending fitness function of Lancichinetti et al. (2009) [15]
Clauset evaluated modules with a function that does not depend on resolution.
### 3.2 Link Clustering
If links instead of nodes are clustered, nodes with more than one link can be
fractional members of clusters, as figure 2 shows for the karate club. For
example, vertex 1 (violet point) has four edges belonging to one and twelve
edges belonging to another hard cluster of links. Thus, it has membership
grades 4/16 and 12/16, respectively, in the two clusters.
Figure 2: Hard clusters of links. Karate club graph with overlapping node
communities induced by three hard link clusters.
For clustering links we need a measure of link similarity. Ahn et al. (2010
[14, eq. 2, p. 5]) chose the Jaccard index of neighbourhoods of nodes attached
to two links (a node itself is included into its neighbourhood).
In a different approach to link clustering, Evans and Lambiotte (2009) [25]
used the line graph of an undirected graph. To get a graph’s line graph first
a bipartite graph of the graph’s nodes and edges is constructed by putting an
edge node on each edge. The bipartite graph can then be projected onto the
line graph, a graph where nodes and edges have interchanged their roles.
Recently Ball et al. (2011) [26] successfully tested an algorithm which finds
overlapping node communities with a generative stochastic model of hard link
clusters. Kim and Jeong (2011) [27] applied the fast _Infomap_ [28] algorithm
to link clustering.
The clustering of citation links instead of papers is of high interest to
bibliometrics because a citation is probably the conceptually most homogenous
bibliometric unit. Since many references are referred to only once in a paper,
it can be assumed that these links between the citing and the cited
publication can be assigned to one theme. Even though there are many cases in
which a paper cites a source for several different reasons, a citation link
can be assumed to have a higher thematic homogeneity than a publication. Based
on this assumption of homogeneity, citation links can be hard-clustered, which
leads to overlapping clusters of papers. The membership grade of a paper to a
module corresponds to the part of outgoing citation links of this paper within
this link cluster.
We applied the hierachical link clustering (HLC) method suggested by Ahn et
al. (2010) [14] to cluster citation links in the bipartite network of papers
and their cited sources. Ghosh et al. (2011) [29] have generalised HLC to
tripartite graphs.
We did not consider the line-graph approach because it is not local (due to
its use of modularity).
### 3.3 Fuzzification of Hard Clusters
Figure 3: Fuzzification of hard clusters. Karate club graph with overlapping
communities from two hard clusters.
The approach assumes that hard-cluster algorithms validly identify disjoint
community cores which just need to be ‘softened’ at the borders. If this is
the case, modifying a hard cluster by evaluating the inclusion of its nodes
and neighbouring nodes with regard to some metric or fitness is a plausible
method for constructing overlapping communities. The fitness balance of a node
with respect to a cluster can then be used to decide about its membership and
to calculate its membership grade. Thus, we construct fuzzy overlapping
communities.
Figure 3 shows the karate-club result Wang et al. (2009) [13] obtained with
their implementation of the fuzzification approach, which they applied to two
hard clusters.
However, the simplest approach to making hard clusters overlapping and fuzzy
is to redefine the membership grades of all nodes with links crossing borders.
If some of a node’s links end within cluster $C$ and some outside $C$ then its
membership in $C$ can be defined as the ratio
$\kappa_{in}(C,V_{i})/\kappa(V_{i})$. We use this definition to calculate
fractional grades after we constructed overlapping communities by fitness
improvement. Such an approach can be criticised as being inconsistent because
fractional memberships are calculated using non-fractional (zero or full)
memberships of nodes as obtained by fitness evaluation.
## 4 Data
### 4.1 The Paper Network
We apply the three algorithms to a network of papers in the 2008 volume of six
information-science journals with a high proportion of bibliometrics papers
(for details of data see reference [30], papers downloaded from Web of
Science).
We start from the affiliation matrix $M$ of the bipartite network of papers
and their cited sources. Here we neglect that a few cited sources are also
citing papers in the 2008 volume but this minor simplification can be avoided
in future analyses. Link clustering is done with $M$ itself, the other two
algorithms analyse a bibliographic-coupling network constructed from $M$ as
follows. In the network of papers, two nodes (papers) are linked
(bibliographically coupled) if they both have at least one cited source in
common. To account for different lengths of reference lists we normalise the
paper vectors of $M$ to an Euclidean length of one. With this normalisation,
the element $a_{ij}$ of matrix $A=MM^{T}$ equals Salton’s cosine index of
bibliographic coupling between paper $i$ and $j$.
The symmetric adjacency matrix $A$ describes a weighted undirected network of
bibliographically coupled papers. The main component of the network of 533
information-science papers 2008 (528 articles and five letters) contains 492
papers. Three small components and 34 isolated papers are of no interest for
our cluster experiments.
Figure 4: Information science 2008. Three topics and their overlaps in a
network of 492 bibliographically coupled papers. Topics assigned manually to
papers by inspection of their keywords, titles and abstracts. The nodes’
colours correspond to four sets: (i) green to $h$-index, (ii) blue to
bibliometrics without $h$-index and without webometrics, (iii) red to
webometrics without bibliometrics, and (iv) violet to the overlap of
webometrics and bibliometrics. Transparent nodes are papers dealing with other
information-science topics, mainly with information retrieval and information
behaviour.
### 4.2 Three Topics
For the evaluation of the three algorithms, we used keywords, titles, and
abstracts of papers to identify those that belong to three topics, namely
$h$-index, bibliometrics, and webometrics. The $h$-index is an indicator for
the evaluation of a researcher’s performance, which has been proposed by the
physicist J. E. Hirsch in 2005. Since then, the use of the $h$-index for
evaluating individual researchers, proposals for $h$-index derivatives and for
$h$-indices of journals or other aggregates of papers have been discussed in
the literature. 46 of the 492 papers cite the 2005 paper by Hirsch, which is
the most cited source in our sample. The $h$-index is clearly an invention in
the field of bibliometrics. About 200 other papers are also addressing
bibliometric themes. For the purposes of this evaluation, we excluded analyses
of patents from bibliometrics. In a smaller webometrics set, internet
activities of (mainly academic) institutions and individuals are analysed.
We first assigned papers to the three topics on the basis of their keywords
and subsequently checked the classification by inspecting titles and
abstracts. This led to 42 papers assigned to the $h$-index and its
derivatives, further 182 bibliometric papers not mentioning the $h$-index in
title or abstract, 24 webometric papers, and eight papers in the overlap
between webometrics and bibliometrics. In figure 4 we display the graph of the
sample of 492 bibliographically coupled papers using the force-directed
placement algorithm by Fruchterman and Reingold (as implemented in the
R-package _igraph_).222cf. http://www.r-project.org
## 5 Fuzzy Natural Communities
### 5.1 MONC Algorithm
MONC [30] uses ideas from Lancichinetti et al. (2009) [15] but replaces their
numerical approach by a faster and more precise parameter-free analytical
solution.
Lancichinetti et al. proposed an algorithm which rests on a greedy expansion
of natural communities of nodes by local fitness maximisation (LFM algorithm).
Communities of different nodes can overlap each other. The size of a natural
community of a node depends on resolution $\alpha$. LFM has to be repeated for
each relevant resolution level to reveal the hierarchical structure of the
network. Our parameter-free MONC algorithm exactly calculates resolution
levels at which communities change by including a node that improves their
fitness. To save further computing time, MONC merges overlapping natural
communities when they become identical during the iteration process [30].
Similar to Lee et al. (2010) [31]—who tested a variant of LFM—we found that
using cliques as seeds gives better results than starting from single nodes.
While Lee et al. use maximal cliques (i.e. cliques which are not sub-graphs of
other cliques), we optimise clique size by excluding nodes that are only
weakly integrated [30, p. 6].
From MONC results, we construct _fuzzy natural communities_ of nodes i.e.
fuzzy sets in which each graph node has a definite membership grade. Each
fuzzy natural community represents its seed node’s perspective on the whole
network. Since the emphasis on local perspectives lets MONC construct many
natural communities that are very similar, the fuzzy natural communities are
hard-clustered hierarchically using the fuzzy-set Jaccard index as a
similarity measure for, e.g., single-linkage clustering. Branches in
dendrograms derived from MONC results do not represent disjoint sets of nodes
but overlapping fuzzy communities.
### 5.2 MONC Post-Processing
Greedy algorithms which locally maximise a resolution depending fitness (or
density) function can reveal hierarchies of overlapping modules. Lancichinetti
et al. (2009) [15, pp. 7–9] have successfully tested their LFM algorithm on a
simple benchmark graph with two hierarchical levels.
MONC (like LFM) needs some postprocessing to reveal a graph’s hierarchy. We
successfully tested the following procedure for detecting a graph’s hierarchy
from MONC results. We define the grade a node is a member in a community from
the resolution level at which it becomes a member (cf. next subsection). With
this definition communities become fuzzy sets over the universe of all nodes
called _fuzzy natural communities_. Two communities are similar if their fuzzy
intersection is large. As a relative measure, we use the fuzzy Jaccard index
to define the similarity of two natural communities. Then communities can be
clustered by any hard-cluster algorithm to reveal the graph’s hierarchy.
Here we should add a comment. We construct a node’s perspective on the whole
graph i.e. its natural community as a fuzzy set over the universe of all
nodes. We hierarchically cluster the fuzzy sets which is equivalent to node
clustering based on a variant of the concept of structural equivalence [12, p.
86]. Nodes are structurally equivalent if their neighbourhoods are equal, they
are structurally similar if their neighbourhoods are similar in some sense. We
operationalise structural similarity of two nodes as the fuzzy Jaccard index
of their fuzzy natural communities representing their perspectives on the
whole graph. Despite the equivalence of our method to the concept of
structural similarity of nodes we insist on the definitions given above: we do
not cluster nodes but their fuzzy natural communities.
We have also to comment on our earlier claim MONC would reveal the graph’s
hierarchy automatically, i.e. without postprocessing [30]. MONC’s merging of
overlapping natural communities can be visualised in a dendrogram. Two
communities merge at a resolution level where they become identical sets of
nodes. In the case of Zachary’s karate club we discussed graph covers on
different resolution levels by going through this dendrogram starting from its
root. This inspection of a dendrogram of merging communities seemed to confirm
our expectation that MONC’s community merging is determined by the graph’s
actual hierarchy [30, pp. 11–13]. Tests with the information-science network
convinced us that this assumption is not true. If two communities merge at a
low level of inverse resolution $\gamma=1/\alpha$, they are very similar and
are therefore located close to each other in the graph’s hierarchy. But
merging at low $\gamma$ is not a necessary condition for communities to be
similar. There are many very similar communities which merge at high $\gamma$
because of very small differences in membership up to this level. This
phenomenon leads to many near-duplicates when we determine modules at one
selected resolution level (as we have already found when testing MONC on non-
hierarchical benchmark graphs [30, p. 16]).
### 5.3 Grades of Membership
MONC’s greedy expansion of seeds can be discussed in terms of ‘hosts inviting
guests’ to their communities. Each node $i$ of the (connected) graph is
‘invited’ to each community $j$ at some level of inverse resolution
$\gamma_{ij}$. To construct fuzzy communities with various grades of node
membership we propose to define the membership grade of node $i$ in the
community of node $j$ as
$\mu_{ij}=\exp(-\gamma_{ij}^{2}).$ (5)
Using the decreasing exponential function of squared $\gamma_{ij}$ (as in the
density function of the normal distribution) ensures that (1) the host is full
member in its own community ($\mu_{jj}=1$), (2) ‘late guests’ get lower
grades, and (3) the ‘first guests’ get membership grades near one (the
function starts from one, its derivation from zero).
We assume that the dendrogram of fuzzy natural communities reflects the
graph’s hierarchical structure. For each branch we define a community as the
fuzzy union of all fuzzy sets of the branch’s nodes. This means that all host
nodes of the branch are full members of the branch community. This definition
ensures the hierarchical order of branches: if two branches unite then their
communities are fuzzy subsets of their fuzzy union. Thus, each branch of the
dendrogram of fuzzy natural communities, i.e. each vertical line, represents a
fuzzy community.
Figure 5: Stability over size of all MONC branch communities. Stable
communities corresponding to our three topics in information-science papers
2008 are marked: bibliometrics (blue), webometrics (red), $h$-index (green).
Figure 6: MONC communities of the three topics in information-science papers
2008: (a) $h$-index, (b) bibliometrics, (c) webometrics. Saturation of points
correlates with membership grade. Colours of circles denote manually
determined topics (cf. fig. 4).
Figure 7: Scree plots of MONC communities of the three topics in information-
science papers 2008: (a) $h$-index, (b) bibliometrics, (c) webometrics. Red
lines mark possible thresholds.
### 5.4 MONC Communities of Topics
For each branch community we plot its stability i.e. its branch’s length
$s_{u}-s_{d}$ over community size, which is estimated by the number of full
members (figure 5, cf. also above, p. 2). Three of the outliers correspond to
our predefined topics, and will be evaluated in comparison with results of the
other algorithms (see below, section Comparison of Algorithms). The two stable
communities with about 400 full members unite bibliometrics, webometrics,
information retrieval and some other smaller topics but do not include a set
of less central graph nodes.
Figure 6 shows the relationship between MONC communities corresponding to the
three topics and the manually determined paper sets of topics. All grades
below a threshold are set to zero. We derive the thresholds from scree plots
of membership grades (figure 7). These plots show that at some critical
membership grade the node sets of each branch inflate to nearly the whole
graph. We argue that this inflation marks the border of a community of a
branch. We set the grade’s threshold on a value that cuts the scree at the
last steepest gradient before inflation ($\mu_{\mathrm{thr}}=.229,\,.355,\,.1$
for $h$-index, bibliometrics, and webometrics, respectively).
## 6 Hierarchical Clustering
of Citation Links
### 6.1 HLC Algorithm
on Bipartite Graphs
We consider the bipartite network of papers and cited sources. Citation links
between the two types of nodes can be hard-clustered, which leads to induced
overlapping communities of papers (and also to communities of sources which,
however, are not analysed here). The membership grade of a paper to a
thematic community equals the fraction of its citation links belonging to the
corresponding link cluster.
Links can be seen as similar if the neighbourhoods of their nodes overlap to a
high degree. Thus, the Jaccard index of these neighbourhoods can be used as a
similarity measure (cf. Ahn et al., 2010 [14, eq. 2, p. 5]). We discuss the
definition of similarity between links in a bipartite graph in terms of papers
and cited sources. The neighbourhood of a paper $p_{i}$ is the set of its
references $R_{i}$, the neighbourhood of a cited source $s_{i}$ is the set of
papers $C_{i}$ citing it. The neighbourhood $N_{i}$ of citation link $i$ is
then the union of these disjoint333We neglect that a few cited sources are
also citing papers, cf. above. sets: $N_{i}=R_{i}\cup C_{i}$. The size of the
intersect of two link neighbourhoods is given by
$|N_{i}\cap N_{j}|=|C_{i}\cap C_{j}|+|R_{i}\cap R_{j}|$ (6)
and the size of their union by
$|N_{i}\cup N_{j}|=|C_{i}\cup C_{j}|+|R_{i}\cup R_{j}|.$ (7)
The distance metrics used for clustering is then
$d_{ij}=1-\frac{|C_{i}\cap C_{j}|+|R_{i}\cap R_{j}|}{|C_{i}\cup
C_{j}|+|R_{i}\cup R_{j}|}.$ (8)
Ahn et al. calculate similarities only for link pairs which have a node in
common because they “expect” disconnected link pairs to be less similar then
pairs connected over a node [14, p. 5]. Since counterexamples disproving this
assumption can be constructed, we decided to calculate similarities for all
pairs of nodes. Such a procedure uses more information but is also more time-
consuming.
Figure 8: Stability over size of all 5004 HLC branch communities. Stable
communities corresponding to our three topics in information-science papers
2008 are marked: bibliometrics as blue, webometrics as red, and $h$-index as
green point, respectively.
Figure 9: HLC communities of the three topics in information-science papers
2008: (a) $h$-index, (b) bibliometrics, (c) webometrics. Saturation of points
correlates with membership grade. Colours of circles denote manually
determined topics (cf. fig. 4).
Hard clustering of links can be done with any hierarchical clustering method.
We tested four standard methods. The dendrograms of Ward and average
clustering of citation links seem to reflect the graph’s hierarchy more
adequately than those of single-linkage and complete-linkage clustering. The
latter two methods impose too low or too high restrictions, respectively, on
finding clusters.
### 6.2 HLC Communities of Topics
For all pairs of citation links from the 492 citing papers to all sources we
determine link similarities. Pairs of citation links to sources cited only
once have zero distance within their reference list (cf. equation 8). They are
clustered at zero-distance level with one another. Next, these zero-distance
clusters are joined with links to the least cited source in their reference
list. This allows us to restrict clustering to all $m=5005$ citation links to
sources which are cited more then once.
We applied the average-clustering method to this set. The corresponding
dendrogram does not give a clear picture of the graph’s hierarchy unless we
re-parametrise the distance axis. We choose $d\to d^{\log_{2}m}$ to de-skew
distances $d$. Using these rescaled data, we plot branch length over community
size to find relatively stable and large communities (figure 8). We measure
community size by the sum of fractional membership grades of papers attached
to the clustered citation links. Like in the MONC case, we find our three
topics as exceptional points in the plot.
Figure 9 shows the graph of 492 papers coloured proportional to their
membership grade in the three topics, respectively.
## 7 Fuzzification of
Hard Clusters
### 7.1 Fuzzification Algorithm
Figure 10: Stability over size of branch communities. Stable Ward clusters
corresponding to our three topics in information-science papers 2008 are
marked: bibliometrics as blue, webometrics as red, and $h$-index as green
point, respectively.
Figure 11: Fuzzy communities of the three topics in information-science
papers 2008: (a) $h$-index, (b) bibliometrics, (c) webometrics. Saturation of
points correlates with membership grade. Colours of circles denote manually
determined topics (cf. fig. 4).
The fuzzification approach assumes that hard cluster algorithms validly
identify disjoint community cores which just need to be ‘softened’ at the
borders. We implemented an algorithm that evaluates border nodes of each hard
cluster with regard to their connectiveness with it. Border nodes have edges
crossing the cluster’s border and can be located inside or outside the
cluster.
The algorithm uses an evaluation metric that is based on the fitness function
defined by Lancichinetti et al. (2009) [15, 13] (see above, equation 4, page
4). For each border node of a cluster we calculate the clusters’s fitness with
and without this node. The fitness balance of a node with respect to a cluster
determines its membership. Negative balance means exclusion from the cluster.
We evaluate all border nodes of a cluster without changing it during the
evaluation (in contrast to the greedy LFM algorithm which updates the
community after deciding about a node’s membership). The fitness-inherent
resolution parameter controls the extent of the overlap, where lower values
cause a wider area to be considered for the inclusion into the former hard
cluster. While MONC uses resolution levels to calculate membership grades, the
fitness-inherent parameter is arbitrary here. Thus, we didn’t apply it in our
comparison and set $\alpha=1$. In a second step the crisp overlapping
communities are made fuzzy. The fractional membership grade of a node could be
defined using its fitness balance as input but this did not lead to fuzzy
communities that match the three predefined topics. Hence, we used a
definition that ignores the value of (positive) fitness balances: The
membership grade $\mu_{i}(C)$ of vertex $V_{i}$ in community $C$ is
$\mu_{i}(C)=\kappa_{in}(C,V_{i})/\kappa(V_{i}),$ (9)
where $\kappa_{in}(C,V_{i})$ is the sum of weights of edges between vertex
$V_{i}$ and vertices in $C$ and $\kappa(V_{i})$ the sum of all its edge
weights.
### 7.2 Fuzzy Communities of Topics
We applied standard Ward and average clustering on the network of $n=492$
bibliographically coupled information-science papers. Complete and single
linkage failed to provide acceptable results as can be already deduced from
the dendrograms. Average clustering also results in a dendrogram which is not
easy to interpret. The Ward dendrogram shows a very stable and clear $h$-index
cluster which is united with the rest of the graph in the last merging step
(cf. figure 10). Fitness-based optimisation with resolution $\alpha=1$
enlarges this cluster extremely and lowers precision without gain in recall
with respect to the set of manually selected $h$-index papers. Thus, fitness
maximisation is not a successful strategy for this topic that has been well
matched (by e.g. Ward) clustering already and is highly connected to its
network environment.
If we omit fitness maximisation and only calculate fractional membership
grades according to equation 9 the result is not better. Many external border
nodes become partial members of the fuzzy $h$-index community.
On the other hand, the hard bibliometrics cluster is much smaller than
expected and needs fitness maximisation or at least fractional membership
grades to match the topic. Figure 11 shows how the fitness-optimised and
fuzzyfied Ward clusters of the three topics fit the topics as paper sets.
## 8 Comparison of Algorithms
### 8.1 Comparison of the
Identified Communities
To compare the results we calculate fuzzy Salton’s cosine of manually defined
topics with fuzzy communities identified by the three algorithms considered.
In addition, table 1 gives values of fuzzy $k_{\mathrm{in}}$ and
$k_{\mathrm{out}}$, the geometric mean of which equals the cosine. Table 2
shows how fuzzy communities constructed by the algorithms overlap each other.
In table 3 we list the fuzzy internal and external degrees,
$k_{\mathrm{in}}(C)$ and $k_{\mathrm{out}}$ (cf. equations 2 and 3, p. 2),
together with their ratio $k_{\mathrm{in}}(C)/k_{\mathrm{out}}(C)$ for each
fuzzy topic community constructed by the three algorithms. All fuzzy
communities are communities in the weak sense. The ratio can be interpreted as
a measure of ‘communityness’.
The assumption that hard clusters can be improved by fitness-based
optimisation and fuzzification could not be validated with Ward clusters as
input. While the optimised and fuzzified bibliometrics cluster gained slightly
better similarity results than the other two algorithms, the clearly
identified $h$-index hard-cluster did not improve because both optimisation
and fuzzification included too many nodes which were related but were not
assigned to the topic. The fitness-inherent resolution parameter could improve
similarity values but would have to be choosen differently for different
clusters—a procedure which cannot be applied when target topics are not known
in advance. The fact, that fuzzification results in an $h$-index community
with best ratio $k_{\mathrm{in}}(C)/k_{\mathrm{out}}(C)$ should be interpreted
with care. It only means, that the algorithm finds a big cluster which is
relatively separated from the rest of the graph. It (partly) includes many
papers which do not refer to the $h$-index.
Table 1: Topic matches by algorithms
topic | MONC | HLC | fuzzy
---|---|---|---
$h$-index | .71 | .93 | .59
precision | .56 | .91 | .35
recall | .89 | .95 | 1.00
bibliometrics | .79 | .82 | .83
precision | .72 | .83 | .87
recall | .86 | .81 | .80
webometrics | .58 | .60 | .46
precision | .53 | .85 | .45
recall | .64 | .43 | .47
bib-web overlap | .46 | .29 | .30
precision | .34 | .24 | .14
recall | .64 | .36 | .65
Fuzzy cosine indices, precision, and recall of paper sets and fuzzy
communities (and of bibliometrics-webometrics overlap) found by the three
algorithms
Table 2: Community matching between algorithms
| MONC | HLC | fuzzy
---|---|---|---
topic | HLC | fuzzy | MONC
$h$-index | .73 | .60 | .62
bibliometrics | .76 | .84 | .78
webometrics | .63 | .46 | .55
bib-web overlap | .51 | .41 | .43
Fuzzy cosine indices of fuzzy communities (and of bibliometrics-webometrics
overlap) found by the three algorithms
Table 3: Fuzzy $k_{\mathrm{in}}/k_{\mathrm{out}}$ of communities
$C$ | variable | MONC | HLC | fuzzy
---|---|---|---|---
$h$-index | $k_{\mathrm{in}}/k_{\mathrm{out}}$ | 5.97 | 7.41 | 9.70
| $k_{\mathrm{in}}$ | 244.65 | 245.66 | 352.21
| $k_{\mathrm{out}}$ | 40.98 | 33.17 | 36.31
biblio- | $k_{\mathrm{in}}/k_{\mathrm{out}}$ | 3.41 | 19.03 | 15.37
metrics | $k_{\mathrm{in}}$ | 314.23 | 466.97 | 456.97
| $k_{\mathrm{out}}$ | 92.03 | 24.54 | 29.74
webo- | $k_{\mathrm{in}}/k_{\mathrm{out}}$ | 1.43 | 1.21 | 1.32
metrics | $k_{\mathrm{in}}$ | 21.04 | 10.74 | 45.01
| $k_{\mathrm{out}}$ | 14.71 | 8.85 | 34.19
The ratio $k_{\mathrm{in}}(C)/k_{\mathrm{out}}(C)$, $k_{\mathrm{in}}(C)$, and
$k_{\mathrm{out}}(C)$ of fuzzy communities found by the three algorithms
Hierarchical clustering of citation links gave better results than MONC. Link
clustering classifies $h$-index as a bibliometric topic whereas MONC only
includes some $h$-index papers into bibliometrics. Fuzzy cosines of HLC
communities and manually selected topics are always better than the
corresponding MONC values (s. table 1).
### 8.2 Assumptions Used
All three algorithms implemented by us are based on the assumptions that a
graph’s communities (1) are best determined locally, (2) overlap each other,
(3) are best described by fractional membership grades, and (4) form a
hierarchy. We used these four assumptions as criteria in our selection of
approaches to community detection. Nonetheless, with respect to all four
criteria there are differences between the selected algorithms. Another
criterion was that results should not—at least not strongly—depend on
arbitrary parameters. Furthermore, each of the algorithms is based on specific
assumptions, which we already mentioned in the respective sections of this
paper.
The fuzzification procedure based on hard clusters whose fitness is improved
assumes that the hierarchical cluster algorithm delivers essentially valid but
improvable hard clusters. Our results do not confirm the improvement using
standard fitness measures. For the membership grades we have not found a
local, consistent, and realistic definition. With respect to its input data,
this procedure—like MONC but unlike HLC—assumes that a network of scholarly
papers weighted with paper similarity (based on references and/or text) can be
used to identify hierarchical thematic structures.
Hierarchical link clustering: Paper networks are projections of bipartite
graphs and thus do not use the full information content of the raw data.
Hierarchical link clustering (HLC) rests on a broader information basis when
applied to links in bipartite networks of papers and their cited sources or in
tripartite networks of papers, cited sources, and terms used in papers and
sources. HLC only assumes that a source is cited for only one reason or for
only very few similar reasons in one paper. In the case of terms, the
assumption is that authors use one term in one paper with only one meaning.
These assumptions are both not only very plausible but could even be tested in
case studies. A further advantage of link clustering is that it allows to
combine citation and textual information in tripartite graphs—a very ‘natural’
solution of this longstanding problem (cf. e.g. the introduction of reference
[32] and sources cited there).
For MONC there are no further assumptions beyond the four mentioned above and
the one about paper-similarity networks. However, we found that MONC needs
some post-processing to reveal the hierarchy of a graph. Thus, it is assumed
that hierarchical clustering of the nodes’ perspectives results in a realistic
hierarchy of topics.
### 8.3 Methodological Aspects
Our implementations of the three approaches to overlapping communities all
involve a hard clustering procedure. The fuzzification algorithm uses hard
clusters of nodes as input, i.e. clustering has to be done as pre-processing.
Hard clustering of fuzzy natural communities is part of MONC’s post-
processing. In the case of HLC, the agorithm itself is a hard clustering
procedure. For HLC we only need to calculate link similarities as some kind of
pre-processing.
We have presented results obtained with only one standard hard-cluster
algorithm per approach but tested also other ones. Fuzzification and link
clustering also worked with Louvain algorithm [33] but we abandoned this fast
modularity-driven method due to its use of global information (and the poor
hierarchical structure obtained). Fuzzification could perform better with
average-linkage clustering but its dendrogram showed only very small stable
communities. In the case of average link clustering (HLC) we succeeded in
finding relatively stable communities of some size after re-parametrising the
dendrogram’s similarity axis.
When it comes to defining grades of a node’s memberships in different
communities, link clustering implies a very plausible and consistent
definition. MONC membership grades could also be defined alternatively to the
ansatz used here (equation 5). We see this methodological ambiguity as a
disadvantage (arbitrary parameters are only a special case of such an
ambiguity). In our fuzzification experiments we calculated fractional
membership grades using non-fractional (zero or full) membership of nodes as
input (equation 9). This inconsistent definition could possibly be avoided by
an iterative algorithm that in each step uses the fractional grades as input
to calculate new ones. We did not yet test such an iteration procedure and
hence do not know whether it would converge or not. An alternative would be to
use the fitness balances of a node as input for a membership definition. Our
attempts to define grades this way led to communities with only very few full
members, which could be a desired feature for topic extraction that cannot be
achieved by HLC membership grades.
MONC membership grades fit into the framework of fuzzy set theory because a
node’s grades in general do not sum up to unity. Link clustering leads to node
grades which are normalised. Thus, an HLC grade is more adequately interpreted
as a probability.
## 9 Discussion
We implemented three local approaches to the identification of overlapping and
hierarchically ordered communities in networks as algorithms and tested their
ability to extract manually defined thematic substructures from a network of
information-science papers and their cited sources.
Hierarchical clustering of citation links proved to be the most satisfactory
approach—with regard to the test results, to its methodological simplicity, to
its ability to work with the broadest information basis (the bipartite graph
of papers and sources), and to its potential for a simple inclusion of text
information in addition to citation data—an issue on top of our agenda.
Clustering citation links does not need to be restricted to a small period of
time but can also be applied to a longer time period. This might make it
possible to solve the problem of tracing the development of topics over time.
The only limitation HLC encounters is the limited coverage of publication
databases, i.e. the existence of citation links to publications that are not
included in the database.
MONC was found to be useful for overcoming the longstanding problem of field
delineation by greedily expanding the paper set downloaded from a citation
database [30, p. 19]. Instead of delineating research fields by journal sets,
they can be identified with a large enough natural community—obtained with low
enough resolution—of an appropriate seed node.
Hierarchical clustering of citation links can be applied to this problem too.
We only have to cluster the environment of a seed set and then to omit all
branches in the dendrogram which are not sub-branches of the most stable
community. The next iteration starts from this reduced paper set. As in the
MONC case, we would alternate between expansion (inclusion of neighbours) and
partial reduction of the sample (exclusion of neighbours which do not improve
community fitness or—in the HLC case—stability).
While the fuzzification algorithm only sometimes creates good clusters in
terms of target topics, iterating fitness-based optimisation may lead to more
consistent clusters by removing loosely connected nodes. If the iteration is
done node by node this leads to a version of the LFM algorithm [15] applied to
hard clusters instead of single nodes or cliques, an approach already proposed
by Baumes et al. (2005) [22].
## Acknowledgements
This work is part of a project in which we develop methods for measuring the
diversity of research. The project is funded by the German Ministry for
Education and Research (BMBF). We would like to thank all developers of
R.444http://www.r-project.org
## Author Contributions
Conceived and designed the experiments: all authors. Performed the
experiments: AS (fuzzification), MH (link clustering), FH (MONC). Analysed the
data: AS (fuzzification), MH (link clustering, comparison), FH (MONC). Wrote
the paper: FH, JG. Discussed the text: all authors.
## References
* 1. Van Raan A (2004) Measuring science. In: Moed HF, Glänzel W, Schmoch U, editors, Handbook of quantitative science and technology research: The use of publication and patent statistics in studies of S&T systems, Dordrecht etc.: Kluwer, chapter 11. pp. 19–50.
* 2. Zitt M, Ramanana-Rahary S, Bassecoulard E (2005) Relativity of citation performance and excellence measures: From cross-field to cross-scale effects of field-normalisation. Scientometrics 63: 373–401.
* 3. Janssens F, Glänzel W, De Moor B (2008) A hybrid mapping of information science. Scientometrics 75: 607–631.
* 4. Klavans R, Boyack K (2011) Using global mapping to create more accurate document-level maps of research fields. Journal of the American Society for Information Science and Technology 62: 1–18.
* 5. Sullivan D, White D, Barboni E (1977) Co-citation analyses of science: An evaluation. Social Studies of Science 7: 223–240.
* 6. Amsterdamska O, Leydesdorff L (1989) Citations: Indicators of significance? Scientometrics 15: 449–471.
* 7. Marshakova IV (1973) Sistema svyazey mezhdu dokumentami, postroyennaya na osnove ssylok (po ukazatelyu “Science Citation Index“). Nauchno-Tekhnicheskaya Informatsiya Seriya 2 – Informatsionnye Protsessy i Sistemy 6: 3–8.
* 8. Small H (1973) Co-citation in the Scientific Literature: A New Measure of the Relationship Between Two Documents. Journal of the American Society for Information Science 24: 265–269.
* 9. Mitesser O, Heinz M, Havemann F, Gläser J (2008) Measuring Diversity of Research by Extracting Latent Themes from Bipartite Networks of Papers and References. In: Kretschmer H, Havemann F, editors, Proceedings of WIS 2008: Fourth International Conference on Webometrics, Informetrics and Scientometrics & Ninth COLLNET Meeting. Humboldt-Universität zu Berlin, Berlin: Gesellschaft für Wissenschaftsforschung. ISBN: 978-3-934682-45-0, http://www.collnet.de/Berlin-2008/MitesserWIS2008mdr.pdf.
* 10. Lancichinetti A, Radicchi F, Ramasco JJ, Fortunato S (2011) Finding statistically significant communities in networks. PLoS ONE 6: e18961.
* 11. Gläser J (2006) Wissenschaftliche Produktionsgemeinschaften: Die soziale Ordnung der Forschung. Campus Verlag.
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* 13. Wang X, Jiao L, Wu J (2009) Adjusting from disjoint to overlapping community detection of complex networks. Physica A: Statistical Mechanics and its Applications 388: 5045–5056.
* 14. Ahn Y, Bagrow J, Lehmann S (2010) Link communities reveal multiscale complexity in networks. Nature 466: 761–764.
* 15. Lancichinetti A, Fortunato S, Kertesz J (2009) Detecting the overlapping and hierarchical community structure in complex networks. New Journal of Physics 11: 033015.
* 16. Zachary W (1977) An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33: 452–473.
* 17. Tibély G (2011) Criterions for locally dense subgraphs. Arxiv preprint arXiv:11033397 .
* 18. Friggeri A, Chelius G, Fleury E (2011) Ego-munities, exploring socially cohesive person-based communities. Arxiv preprint arXiv:11022623 .
* 19. Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. Proceedings of the National Academy of Sciences of the United States of America 101: 2658–2663.
* 20. Gregory S (2011) Fuzzy overlapping communities in networks. Journal of Statistical Mechanics: Theory and Experiment 2011: P02017.
* 21. Baumes J, Goldberg M, Krishnamoorthy M, Magdon-Ismail M, Preston N (2005) Finding communities by clustering a graph into overlapping subgraphs. In: International Conference on Applied Computing (IADIS 2005).
* 22. Baumes J, Goldberg M, Magdon-Ismail M (2005) Efficient identification of overlapping communities. Intelligence and Security Informatics 3495: 27–36.
* 23. Clauset A (2005) Finding local community structure in networks. Physical Review E 72: 26132.
* 24. Lee C, Reid F, McDaid A, Hurley N (2011) Seeding for pervasively overlapping communities. Arxiv preprint arXiv:11045578 .
* 25. Evans T, Lambiotte R (2009) Line graphs, link partitions, and overlapping communities. Physical Review E 80: 16105.
* 26. Ball B, Karrer B, Newman M (2011) An efficient and principled method for detecting communities in networks. ArXiv preprint arXiv11043590 .
* 27. Kim Y, Jeong H (2011) The map equation for link community. Arxiv preprint arXiv:11050257 .
* 28. Rosvall M, Bergstrom C (2008) Maps of random walks on complex networks reveal community structure. Proceedings of the National Academy of Sciences 105: 1118.
* 29. Ghosh S, Kane P, Ganguly N (2011) Identifying overlapping communities in folksonomies or tripartite hypergraphs, WWW 2011, Poster, March 28–April 1, 2011, Hyderabad, India : 39–40.
* 30. Havemann F, Heinz M, Struck A, Gläser J (2011) Identification of Overlapping Communities by Locally Calculating Community-Changing Resolution Levels. Journal of Statistical Mechanics: Theory and Experiment 2011: P01023.
* 31. Lee C, Reid F, McDaid A, Hurley N (2010) Detecting highly overlapping community structure by greedy clique expansion. In: Proceedings of the 4th SNA-KDD Workshop. ArXiv: http://arxiv.org/abs/1002.1827.
* 32. Janssens F, Zhang L, Moor B, Glänzel W (2009) Hybrid clustering for validation and improvement of subject-classification schemes. Information Processing & Management 45: 683–702.
* 33. Blondel V, Guillaume J, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment 2008: P10008.
|
arxiv-papers
| 2011-07-26T17:00:06 |
2024-09-04T02:49:20.959262
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Frank Havemann, Jochen Gl\\\"aser, Michael Heinz, and Alexander Struck",
"submitter": "Frank Havemann",
"url": "https://arxiv.org/abs/1107.5266"
}
|
1107.5305
|
# Universal constants and
equations of turbulent motion
by Helmut Z. Baumert Institute for Applied Marine and Limnic Studies (IAMARIS
e.V.), Hamburg, Germany
###### Abstract
In the spirit of Prandtl’s [1926] conjecture, for turbulence at
$Re\rightarrow\infty$ we present an analogy with the kinetic theory of gases,
with dipoles made of quasi-rigid and “dressed” vortex tubes as frictionless,
incompressible but deformable quasi-particles. Their movements are governed by
Helmholtz’ elementary vortex rules applied locally. A contact interaction or
“collision” leads either to random scatter of a trajectory or to the formation
of two likewise rotating, fundamentally unstable whirls forming a dissipative
patch slowly rotating around its center of mass which is almost at rest. This
approach predicts von Karman’s constant as $\kappa=1/\sqrt{2\pi}\simeq 0.399$
and the spatio-temporal dynamics of energy-containing time and length scales
controlling turbulent mixing [[]]baumert09 in agreement with observations. A
link to turbulence spectra was missing so far. In the present paper it is
shown that the above image of random vortex-dipole movements is compatible
with Kolmogorov’s turbulence spectra if dissipative patches, beginning as two
$likewise$ rotating eddies, evolve locally into a space-filling bearing in the
sense of [Herrmann (1990)], i.e. into an “Apollonian gear” consisting of
incompressible and flexibly deformable vortex tubes which are frictionless,
excepting the dissipative scale of size zero.
For steady and locally homogeneous conditions our approach predicts the pre-
factor in the three-dimensional Eulerian wavenumber spectrum,
$E(k)=\alpha_{1}\,\varepsilon^{2/3}\,k^{-5/3}$, as
$\alpha_{1}=\frac{1}{3}(4\,\pi)^{2/3}\simeq 1.802$, and in the Lagrangian
frequency spectrum, $E(\omega)=\beta_{1}\,\varepsilon\,\omega^{-2}$, as
$\beta_{1}=2$. The unique values for $\alpha_{1},\beta_{1}$ and $\kappa$ are
situated well within the broad scatter range of observational, experimental
and approximative results. Our derivations rest on geometry, methods from
many-particle physics, and on elementary conservation laws.
Baumert Universal constants and equations H. Z. Baumert, IAMARIS, Bei den
Mühren 69A,
D-20457 Hamburg, Germany (baumert@iamaris.org) Submitted as v2 to arXiv.org
## 1 Introduction
In the present paper we show that fluid turbulence can be understood in an
idealised sense as a statistical many-body ensemble – a tangle of vortex tubes
taken as discrete particles. We follow an early conjecture by Ludwig [Prandtl
(1926)] who discussed an analogy between molecular diffusion and turbulence.
He related his mixing length111According to [Hinze (1953)] also G. I. Taylor
made early use of the notion mixing length. or Mischungsweg with the mean-free
path of kinetic gas theory and considered his fluid elements or fluid lumps
(Flüssigkeitsballen) of locally nearly same size as relatives of gas
molecules. He further assumed his mixing length to scale with the “diameter”
of his fluid elements.
Although purely heuristic, his concept became popular in the years before WW2.
The question how to compute the details could not be answered without
reference to measurements. After WW2 the gas-kinetic analogy found thus strong
criticism from the continuous-image side [[, see Introduction in]]batchelor53.
This was plausible because the gas analogy represents a discrete concept that,
if useful, would label a Copernicanian turn or a paradigm shift in our grown
view of turbulence as an exclusive matter of continuum mechanics.
In this sitution Werner [Albring (1981)], in the footsteps of Prandtl,
explicitely challenged the continuous paradigm of [Reynolds (1895)], [Keller
and Friedmann (1924)], [Taylor (1935)], [Batchelor (1953)] and their many
followers, when he posed the fundamental question:
> Can the Navier-Stokes equation be used to calculate turbulent flows?
His doubts were based on the feeling that the continuous image of RANS does
possibly override features of most elementary vortex interactions at small
scales.
In analogy to Albring one might ask: Can we deduce a rose’s blossom from the
periodic system of chemical elements? Intuitively we answer with no. However,
this answer is justified because generally “more is different” [Anderson
(1972)].
Looking in this sense at Albring’s above question we may add that the
prediction horizon of the Navier-Stokes equation (NSE) is strongly limited by
a series of higher-order non-equilibrium phase transitions when a growing
Reynolds number goes through a number of critical values. In these super-
critical regions the sensitivity against initial conditions becomes relevant
and leads to the famous butterfly effect [[]]lorenz63. This is closely related
with irreversibility. At least in the limit of vanishing viscosity the
initial-value problem for NSE looks like a reversible one. But we know that
turbulence at $Re=\infty$ is an irreversible process.
In principle this problem was already known by Poincare and others as an
aspect of the three-body problem of celestial mechanics, but in a fluid-
mechanical context it has been demonstrated only after WW2 by [Lorenz (1960),
Lorenz (1963)]. Without going into details of the onset of turbulence and non-
equilibrium phase transitions in hydrodynamics and their reflections in
various branches of pure mathematics, together with Prandtl and Albring we
hypothesize that NSE is not sufficient to understand the secrets of turbulence
or to even predict turbulent flows, in particular not in the limit
$Re\rightarrow\infty$. Some readers might even go a step further and argue
that higher-order elements of the Friedman-Keller expansions of NSE, e.g. the
second and third-order turbulence closures discussed in [Voropayeva (2007)]
and the literature quoted therein, as well as the various “corrections” of
those closures populating turbulence theory, are modern analogues of the
epicycles of geocentric times. But this would exceed the limits of the present
report. In this respect we better refer to the agreeable picture of today’s
status of turbulence science drawn by [Davidson (2004)] in the preface of his
book where he mentions even “religious wars…between the different camps” of
turbulence theory.
There are not many physical phenomena resisting theoreticians so long like
turbulence. In the past it was mainly supra-conductivity which took about 30
years [[]]feynman63b. Today similarly obstinate problems are dark matter, dark
energy, and super-symmetry which have also now an age of about up to 40 years.
But turbulence waits still much longer for redemption and remained
particularly as the very last unsolved enigma of classical physics.
Below we develop an asymptotically invariant alternative to RANS and higher-
order closures, which is not primarily based on NSE but does not violate NSE
either.
Although the theory of many-particle physics offers a huge reservoir of
potentially helpful methods and tools, a closer inspection reveals that the
number of directly applicable tools is less impressive. Neither Liouville
theorem, ergodic hypothesis or Hamilton formalism nor other concepts for
thermodynamic equilibrium are applicable in their classical forms. Turbulence
is essentially an open-system, thermodymically non-equilibrium phenomenon. In
best cases we have a steady state or Fließgleichgewicht in the sense of
[Bertalanffy (1953)], [Glansdorff and Prigogine (1971)] and [Haken (1978),
Haken (1983)]. However, the concepts of Ising about particle dressing, quasi-
particles and renormalization – in the broader sense of [Dresden (1993)] and
[McComb (2004)] – appeared as essential guidelines in the slow evolution of
our thoughts.
Surely, turbulence can successfully be attacked from more than one
side222Theories generally compromise portrait and design aspects and in some
case mathematically different images of turbulence will show up eventually as
fully equivalent in their physical predictions, like Heisenberg’s matrix
mechanics and Schrödinger’s wave mechanics.. Whereas the continuous image has
led to a number of important results, for instance RANS and the broad spectrum
of different heuristic closure schemes discussed e.g. by [Wilcox (2006)],
without additional phenomenological input it gave neither unique values of the
universal constants of turbulent motion like von-Karman’s and Kolmogorov’s
spectral constants nor a closed image of turbulent flows. This has led to the
following view shared by many theoreticians [[]p. 173]landaulifshitz_eng87:
> …and $\kappa$ (is) a numerical constant, the von Karman constant, whose
> value cannot be calculated theoretically and must be determined
> experimentally. It is found to be $\kappa=0.4$.
In this respect one can actually be more optimistic. Frictionless turbulence
is governed in a weak sense by the Euler equation, a special case of NSE. The
Euler equation together with the conservation of mass represent pure “inert
geometry”. We can thus expect that the universal constants of turbulent motion
are ruled by certain geometrical constants, e.g. by irrational numbers like
$\sqrt{2}$ and/or transcendental numbers like $\pi$. Indeed, this is possible.
In a precursor study based on the particle picture used also below, von
Karman’s constant could be derived as $\kappa=1\left/\sqrt{2\pi}\right.\simeq
0.399$ [Baumert (2009)]. This interesting result led to the conjecture that
the particle concept can give also insight into spectral aspects of
turbulence.
Below we make explicit use of theoretico-physical thinking, which differs from
other scientific activities like computer modelling and mathemtical physics,
from applied mathematics, and from the measuring and observational
disciplines.
Computer modelling and measurements have in common that they deal with really
existing, finite systems and data, in particular with real and integer
numbers, with variables of finite size.
Theoretical physics, however, deals with “things” which never did, which do
not and never will exist: e.g. with point masses, homogeneous continua,
idealized vortex tubes, linear waves, frictionless fluids, plane and
impenetrable walls, infinitely small or large variables. A further building
block of theoretical physics is an often unspoken principle which we apply
also here: it “dictates that, all things being equal, one goes for the
simplest possibility–a rule that has worked remarkably well.” [[]p. 85, see
also Chandrasekhar, 1979]zee86. Sometimes this principle is called Occam’s
razor.
Mathematics also deals with similar non-existing objects, but their relations
to the real world are outside its responsibility. It is the responsibility of
theoretical physics to relate idealized thought systems with the real world.
The following report concentrates solely on the latter aspects.
With one exception, the following text contains no “fancy shmancy
mathematics”. It is the relation between equations (7), (8) and (9) that would
need some time to be derived from scratch. Furtunately, this has already been
done by other authors many years ago and went into the textbooks on
stochastic-dynamic systems [[,
e.g.]]kraichnan68,haken78,haken1983,stratonovich92,stratonovich94 to which we
refer the theoretically interested reader. We further suggest to begin first
with [Baumert (2009)] because it is actually the basis of the present report.
Everywhere the pronomen “we” is used in this text, it means the two of us, the
dear reader and the author.
## 2 Particles
If we take Prandtl’s discrete particle concept seriously but not literally,
then a number of questions need to be posed – and answered. A first question
to be addressed is the following.
#### Where do the particles come from?
With respect to their substance our particles are indistinguishable from their
fluid or gaseous environment. It is only their state of motion which makes the
difference to the sourrounding fluid so that we better should talk about
quasi-particles, but keep for brevity the term particles in the following.
They are characterized by their geometry, their specific kinetic energy, by
their vectors of linear and angular momentum.
Our particles are generated as representatives of turbulent fluctuations such
that their energy has to be drawn from the mean flow field via so-called shear
production333Convective turbulence and internal-wave breaking will be treated
in another study. to keep the total kinetic energy of the flow in balance.
I.e. if there is a shear in the mean flow and non-vanishing turbulent
viscosity, then the mean flow looses kinetic energy which reappears in form of
TKE444In the following the word turbulent kinetic energy or TKE is often used.
In the given context it means actually a kinetic energy density, i.e. energy
per unit of mass. Therefore we measure TKE most comfortably in the units m2
s-2., i.e. as energy of vortex motions. Hereby an exact process description of
vortex generation is not needed as long as we know how much energy is lost and
how much vorticity is generated so that we can place vortices of corresponding
properties into the flow. In a way we may talk about an emergence of the
particles.
#### What do the particles look like?
Due to Helmholtz’ principle of conservation of circulation, a circulation-free
fluid volume should exhibit zero circulation over the course of time. I.e.
particle generation cannot add any circulation to the volume. This implies
that turbulence production can take place only in form of the generation of
couples of counter-rotating vortices with zero total circulation. Consequently
we specify Prandtl’s fluid elements as dipoles where the simplest form is
locally fully symmetric, which is to be understood in a statistical sense.
This assumption is central for our theory but deserves a comment. It is well-
known that in laminar and turbulent flows wall friction generates velocity
gradients or current shear, which is identical with circulaton induction into
the mean flow. This process is governed by RANS, more specifically by its
turbulent friction term containing correlators like $\langle
u^{\prime}\,w^{\prime}\rangle$, i.e. by the Reynolds tensor. This tensor is
calculated further below on the basis of a particle theory. If a particle
would have own circulation on the local level, it would add extra circulation
to the flow, which would violate RANS. Our particles can therefore have no own
circulation.
Thus we have the challenging theory situation that a mean flow exhibits
turbulent friction and circulation while the turbulent vortices responsible
for fluctuations and fricition have no circulation and are essentially
frictionless, excepting the singular dissipation scale of size zero, see
further below.
With respect to the specific form of vortices in our dipoles we refer to the
classical notion of vortex tubes wherein vorticity is confined to the interior
of a tube of smaller or larger cross section [[, see further below and]who
quote papers by Kuo & Corrsin, 1972, and Brown & Roshko, 1974, showing tubes
as dominating characteristic structures]pullinsaffman98. A nice and more
recent study of vortex tubes has been presented by [Wilczek (2011)] who also
published on the internet a number of valuable animations of vortex ensembles
in motion555http://pauli.uni-muenster.de/tp/menu/forschen/ ag-
friedrich/mitarbeiter/wilczek-michael.html. We note that, as always, the
centerlines of our vortex tubes form either closed loops or they are attached
to boundaries.
Ordinary linear momentum of a dipole is known from classical theory in local
approximation and can be imagined in analogy to the motions of smoke rings. It
is conserved in our many-particle image if each particle, generated in a
locally homogeneous volume element, exhibits no preferred direction of motion.
The linear momentums of two dipoles with opposite directions compensate each
other. Assuming a locally “sufficiently high” and even number of dipoles thus
guarantees that the particle-induced linear momentum and circulation are zero
and total momentum of the flow exclusively governed by the mean-flow momentum
balance including the Reynolds stress terms.
#### How do the particles move?
The motion of a single isolated dipole [[, a “naked” particle in the
terminology of many-particle physics, see e.g.]]dresden93,mccomb2004 can be
given by classical rules [[, see]]lamb32, albring81,saffman92,
baumert05b,baumert09. However, if the selected dipole is embedded in a dense
tangle and thus surrounded by a cloud of similar dipoles, its properties are
“screened” by the cloud and thus modified: it appears to be “dressed” –
without violation of governing conservation laws. The solution is discussed in
the next section.
## 3 Vortices, energy, and scales
### Traditional vortex models
The hydrodynamic literature presents a larger number of elementary analytic
models for single isolated vortices under different conditions. Whereas the
so-called potential vortex is less realistic, the Oseen vortex, the Rankine
and the Taylor vortex, the Burgers, the Lundgren, and the Long, the Sullivan
and the spherical Hill vortex are more realistic models of isolated vortices
far from others and from boundaries [[, for overviews
see]]lamb32,lugt79,albring81,saffman92,pullinsaffman98,davidson04.
Excepting the potential vortex, all other vortex models are principally
realistic and show first a core with radially increasing tangential velocity,
then a saddle, and then a tail wherein the tangential velocity decreases
radially down to zero. However, in our context these models are not applicable
because they hold for conditions of isolation only. They cannot be transferred
to conditions of a dense vortex tangle where the distance between vortex
dipoles is small and the surrounding cloud of similar vortices screens the
effects from the rest. Therefore a new approach is needed, without violating
governing conservation laws, neither on a global nor on the scale of the
locally homogeneous and isotropic fluid volume.
I.e. an ideally “dressed” vortex dipole moving frictionless within a tangle of
similar objects should necessarily be characterized by a finite effective
radius, $r$, within which all the kinetic energy, $\cal K$, and vorticity of
the vortex is concentrated so that we may talk of an “energy-containing
radius”. For simplicity, in a statistical sense $r$ and $\omega$ are taken
identical for the two vortices forming the internally symmetric dipole.
We summarize as follows:
* •
The effective scales $\omega$ and $r$ are governed by local conservation of
energy and angular momentum.
* •
The radius $r$ defines a boundary within which kinetic energy and vorticity of
a vortex are confined.
* •
The effective tangential velocity of a vortex is $u=\omega\,r$ and equals the
propagation velocity of the dipole. The resulting kinetic energy of the dipole
is thus ${\cal K}=2\times u^{2}/2=2\times
r^{2}\,\omega^{2}/2=(r\,\omega)^{2}$.
* •
The vortices are incompressible but deformable quasi-solid bodies which move
frictionless in the vortex tangle. I.e. vorticity is uniformly distributed
within the cross-section of a vortex tube and has the value $2\times\omega$.
* •
At solid boundaries vortices perform frictionless roll motions. Dissipation
happens exclusively $within$ the fluid at scale zero. (In reality the
dissipation interacts with boundaries through heat and sound generation.
These assumptions describe our image of vortices in a turbulent vortex tangle
without violating the conservation laws of kinetic energy, momentum, angular
momentum and circulation. Compared with the above-mentioned vortex models our
approach may be interpreted as a renormalization procedure leading to a finite
kinetic energy and a finite spatial extension.
While on a first glance this procedure looks somewhat arbitrary, exactly this
image is well established since long in physical oceanography, which is a
world of highest Reynolds numbers. In physical oceanography, meteorology and
physical limnology the a.m. characteristic radius, $r$, is traditionally
related with the “energy-containing” scale, which is a direct observable under
conditions of stable stratification and related with the so-called overturning
and Thorpe scales. Central aspects are discussed in detail in the following
section.
### Vortex tubes and rigid-body rotation
In stable stratification turbulent vortices are directly recognizable in
vertical profiles of density, $\rho$, and other scalar variables, such as
temperature and salinity, as “overturning” of density profiles. Within certain
energetic limits they move heavier over lighter fluid and thus produce a
direct imprint of the vortices in instantaneous depth profiles of density,
$\rho(z)$. This imprint has a unique character in stratified flows because the
buoyancy force implies that there should be a stably stratified “reference”
state, $\bar{\rho}$, a state in the absence of turbulent overturning.
This imprint appears typically as a Z-pattern, see Fig. 1, which obviously
stands for a solid-body rotation wherein the tangential (azimutal) velocity
grows linearly with the radius coordinate up to a maximum at its outer radius
and then (outside the body) drops rapidly down to zero. We interprete a
Z-pattern as a realisation of a vortex embedded in a dense vortex ensemble.
i.e. as a so-called rigid vortex as discussed by [Lugt (1979)] and further
below666 The idea of a rigid vortex is compatible with fluid mechanics because
Helmholtz’ laws also apply here. But the connection with continuum mechanics
is not trivial because the velocity distribution in a rigid vortex exhibits a
singularity at the outer radius where it drops from a finite value down to
zero. Further, the isolated rigid vortex is not stable..
To our knowledge, the use of overturning deviations from the stable state to
derive vortex radii was first introduced and explored by Thorpe (1977) in an
analysis of temperature profiles from Loch Ness, Scotland. Thorpe showed how
the reference state can be reconstructed from measurements and how overturns
and overturning scales can be detected. His approach is a solid basis to
measure the radii (or diameters) of vortices in stably stratified fluids
directly.
Thorpe (1977) described the general case of a monotonous reference density
distribution and the overturning fluid body as frozen for the time of the
overturn. This is justified as long as the length achieved through molecular
diffusion during the time of rotation, $l_{d}=\sqrt{2\,\nu_{m}\,t_{ot}}$, is
small compared with the energy-containing radius $r$ of the overturn777Here
$\nu_{m}$ is molecular viscosity and $t_{ot}$ the duration of an overturn at
the energy-containing scale $r\propto L$.. In water this condition is always
satisfied as $l_{d}$ remains here in the range of centimeters and less.
Later authors like Imberger and Boashash (1986) considered a subset of the
monotonous case, the linear reference-density distribution as depicted in our
Fig. 1. Itsweire et al. (1986) in their Fig. 1 were the first to discribe the
overturning motion explicitely as the effect of a rigid-vortex motion (cf.
also Pullin and Saffman, 1998).
Figure 1: Overturning and overturning scales in the sense of (Imberger and
Boashash, 1986, Fig. 6 there). The cartoon depicts a $180\deg$ overturning of
a 20 m thick layer that brings relatively heavy water up and lighter water
down. Note the characteristic Z-pattern of the vertical displacement $\zeta$.
The original stable density profile is the blue line, the unstable density in
the overturn is red and dashed. By courtesy of Dr. Hartmut Peters, Earth and
Space Research, Seattle, USA.
### Thorpe’s method in oceanography
Background. A short digression on the background of Thorpe’s method might be
in place. Geophysical fluid dynamics is a domain of pioneering turbulence
studies at high Reynolds numbers because atmosphere and oceans offer the
necessary conditions for free. The first massive demonstration of measured
Kolmogorov spectra was carried out by Grant et al. (1959) at a Reynolds number
of about $Re\approx 10^{8}$ in a 100 m deep tidal channel888Today’s DNS of
turbulent flows do not exceed $Re\approx 10^{5}$ and even the European
superpipe CiCLOPE will in the best case not significantly exceed $Re\approx
10^{6}$ Rüedi et al. (2009).. They demonstrated the validity of the 5/3 law
over an intervall of 3.5 orders of magnitude (about 12 octaves). Together with
the logarithmic law of the wall and the decay of TKE in homogeneous isotropic
turbulence according to $t^{-1}$ (or $(x/U)^{-1}$ in the wind tunnel; cf.
Batchelor, 1953, Chapter VII) the 5/3 law belongs until now to the most
prominent universal features of high-$Re$ turbulence.
Many observational techniques and dynamical concepts of geophysical fluid
dynamics were first developed in meteorology and later adopted in physical
oceanography and limnology. That this was not the case with turbulent
overturning is likely related to the fact that the troposphere, the most
accessible part of the atmosphere, does not show consistent mean
stratification. In contrast, oceans and lakes are almost everywhere stably
stratified outside of boundary layers. It further makes sense that turbulent
overturning was first explored in a lake where temperature, $T$, is the sole
stratifying agent. In contrast, ocean stratification depends on temperature as
well as salinity, $S$, and density $\rho=\rho(p,T,S)$ cannot be measured with
the same resolution as $T$ alone. We note in passing that real fluids are
compressible such that analyzes of overturning scales have to be based on
potential density and potential temperature.
The imprint of turbulence in vertical density profiles allows defining and
extracting overturning scales such as the Thorpe scale, $L_{Th}$ Thorpe
(1977). The cartoon of Fig. 1 illustrates the overturning of a $L_{ot}=20$ m
thick layer of the water column by a single vortex in solid body rotation of
diameter $L_{ot}$. According to our above discussion of vortex dipoles as
quasi-particles we conclude that $L_{ot}=2\,r$.
The graph depicts the moment after a 180$\deg$ rotation that brings heavy
water up by vertical displacements with a range of $\zeta=0\,...+L_{ot}$ and
moves lighter water down by $\zeta=0\,...-L_{ot}$. The Thorpe scale is defined
as the r.m.s. of $\zeta$. The linear original density profile of the cartoon
implies
$L_{Th}=L_{ot}/\sqrt{3}\;=\frac{2}{\sqrt{3}}\,r\;=\;1.15\;r\;.$ (1)
We define $\zeta$ by the path from its original depth to a displaced depth.
That is, $\zeta$ carries the same sign as the vertical turbulent velocity
$w^{\prime}$ that has caused it.
Figure 2: A big overturn in the Pacific Equatorial Undercurrent at
$0\deg,140\deg$W during the Tropic Heat II cruise (Peters et al., 1995,
adapted from ): (a) potential temperature ($\Theta$, red) and potential
density ($\sigma_{\Theta}$, black), Thorpe-sorted $\sigma_{\Theta}$ (blue),
and Thorpe scale $L_{Th}$ (shaded); (b) turbulent vertical displacement
$\zeta$. By courtesy of Dr. Hartmut Peters.
Fig. 1 may seem simplistic. It can easily be made more realistic by adding
that, in the course of the overturning, and owing to the unstable
stratification within overturns, flow instability will occur and generate
turbulence and a range of smaller scales than $L_{ot}$, inside the big
overturn.
The ocean is full of overturns that look just like this scenario. Fig. 2
depicts a big overturn in the Pacific Equatorial Undercurrent, EUC, on the
equator at $140\deg$W from Peters et al. (1995). Note the Z-shape of the big
overturn and its sharp upper and lower edges.
In oceanography and limnology, Thorpe’s concept of vortices overturning parts
of the water column are applied to observations as in Fig. 2. Measured
potential density ($\sigma_{\Theta}$; by convention 1000 kg m-3 is subtracted)
or potential temperature ($\Theta$) data points are “Thorpe-sorted” into
monotonically rising or falling sequences corresponding to stable density
stratification. The sorted profiles are taken as a proxy for the reference, or
“mean” profile that gave rise to the observed turbulence and overturning. The
vertical distance over which data points have to be moved to make the profile
monotonic is $-\zeta$. I.e. Thorpe-sorting “undoes” the overturning.
Individual overturns are defined by $\sum_{i}\,\zeta_{i}=0$ Dillon (1982).
The shaded bars in Fig. 2(a) show $L_{Th}$ averaged over individual overturns.
This definition is highly robust. A squared buoyancy frequency $N^{2}$
computed from the sorted $\sigma_{\Theta}$ or $\Theta$ is non-negative999 Here
we may restrict our considerations to incompressible fluids where
$N^{2}=-g/\langle\rho\rangle d\langle\rho\rangle/dz$ with $g$ as gravitational
acceleration, $g=9,81$ m s-2., $N^{2}\geq 0$.
Turbulent length scales related to overturning scales are unaffected by the
presence of internal gravity waves (IGWs). IGWs are ubiquitous in stratified
geophysical flows and dominate velocity and scalar spectra at vertical scales
of the order of O(100 m) and smaller (e.g. Peters et al., 1995). The TKE
accounts only for a tiny fraction of the integral of these “red” spectra.
Owing to this property, length scales commonly used in laboratory experiments,
such as the Ellison (1957) scale,
$L_{E}=\overline{\rho^{\prime}}/\langle\partial\langle\rho\rangle/\partial
z\rangle$, are unsuitable for a characterization of turbulent length scales in
geophysical flows because density fluctuations
$\rho^{\prime}=\rho-\langle\rho\rangle$ inevitably are dominated by IGW
signals so that $L_{E}$ can no longer be interpreted as a characteristic
length scale of turbulent fluctuations. Here, angular brackets denote ensemble
averages $\overline{\rho^{\prime}}=\langle\rho^{\prime 2}\rangle^{1/2}$ is an
r.m.s. value.
Thorpe scale and Taylor scaling. The great power of Thorpe’s [1977] concept
can be demonstrated by considering the so-called Taylor scaling, with $q$ as
an r.m.s. measure of turbulent velocity fluctuations. Dimensional analysis
strictly gives for $l$ as an energy-related length scale the following:
$l\propto\,{q^{3}}/{\varepsilon}\;.$ (2)
Here $l$ is obviously related with TKE because of $\bar{\cal K}\propto q^{2}$,
and $\varepsilon$ is the TKE dissipation rate101010We use the convention that
the symbol ${\cal K}$ denotes the energy of one selected dipole whereas
$\bar{\cal K}$ is the local average of the same variable for a whole ensemble.
If the Thorpe scale $is$ indeed a measure of the size of the energy-containing
vortices and thus related with the energy-containing scale, i.e.
$L_{Th}\propto l$, then we can write (2) as follows,
$\varepsilon\times L_{Th}\propto\,{q^{3}}\;,$ (3)
or
$q\;\propto\,(\varepsilon\times L_{Th})^{1/3}\;.$ (4)
Instead of $q$ one can alternatively use $\bar{\cal K}^{1/2}$ with turbulent
kinetic or mean vortex kinetic energy, $\bar{\cal K}$. Fig. 3 shows
measurements in the ocean demonstrating that relation (4) is true, in a
statistical sense. The Thorpe scale characterizes indeed the size of the
energy-containing eddies.
8.2cm
Figure 3: Turbulent velocity $q_{\epsilon}\\!=\\!(\epsilon\,l)^{1/3}$ derived
from measured $\epsilon$ via Taylor scaling, (4), versus measured turbulent
velocity fluctuation $q$. Adapted from Peters et al. (1995) where
$l\\!=\\!1.6\,L_{th}$ and where $q$ is the spectral velocity variance at
vertical wavenumbers $\geq\,1/l$. Shown are data for individual overturns at
least 1 m thick from depths 60–350 m in the Equatorial Undercurrent at
$0\deg,\,140\deg$W. The gray line indicates the median ratio of $q_{\epsilon}$
over $q$. By courtesy of Dr. Hartmut Peters.
However, this statement needs a comment. Thorpe-sorting as a technique to
measure the radius or diameter of energy-containing turbulent vortices rests
on the existence, and hence works only, in stratified flows. Under geophysical
conditions this automatically implies that a certain coexistence of IGWs and
turbulence is inescapable. The $q$ in (2) must only reflect TKE and must not
be contaminated by the much larger IGW energy.
While the separation of waves and turbulence is beyond the scope of this
note111111The reader is referred to Peters et al. (1995) and D’Asaro and Lien
(2000)., a workaround is to consider spectra and to study only the ultra-
violett or short-wavelengths part where IGW existence is excluded through
their dispersion relation. Peters et al. (1995) did exactly this and extracted
$q$ from oceanic observations as the velocity variance at scales of $l$ and
smaller. On this basis, Taylor scaling clearly emerges from their observations
(Fig. 3).
Thorpe’s [1977] concept of turbulent vortices and their imprint through
vertical overturning on density profiles allows extracting energy-scale
variables even in conditions where velocity and scalar spectra are heavily
dominated by internal gravity waves. It allows relating energy-scale- to
dissipation scale variables through the turbulent length scale that carries
Thorpe’s name. These directly observable energy-containing length scales
stand for the radius or diameter of our renormalized vortices discussed
further above.
### Dressing a rigid-vortex tube
As we learned from the considerations and examples above, turbulence may be
imagined as vortex tubes resembling a dense “local cloud” of dipoles in
chaotic motion, the cloud having zero angular and linear momentum. So far the
rigid-vortex tubes have been considered as somewhat arbitrary idealizations of
real-world vortices.
Translated into the language of home cooking, a snapshot of rigid-vortex
turbulence may be imagined as a dense, entangled heap of hot spaghetti
arrabiata. The spaghetti rotate around their inner centerlines and move
frictionless within the (inviscid) sauce arrabiata. Therefore the only
interactions between individual spaghetti occurs when they touch each other
randomly.
It arises the question whether the rigid-vortex tube and a corresponding
dipole is a stable solution of the Euler equations. We leave it to the
interested reader to derive from these equations that an isolated rigid vortex
indeed solves these equations, but, as long as it exists, generates the
following pressure head as a consequence of inertial (centrifugal) forces:
$p=p_{0}+\frac{\rho}{2}\times\omega^{2}r^{2}.$ (5)
Here $p_{0}$ is the background pressure of a laminar reference flow, e.g. in
the ocean the depth-depending hydrostatic pressure. If the pressure outside
the vortex would simply be $p_{0}$ then, due to the action of the outwards-
directed pressure head of the vortex motion given by the second term in (5),
the vortex would loose stability.
The stability of quasi-rigid vortices as observed in real-world turbulence can
thus be guaranteed only by the help of a compensating force of equal strength.
Here a concept of many-particle physics comes into play: dressing. As far as
we will embed our initially isolated rigid vortex into a locally homogeneous
and isotropic large ensemble (“cloud”) of similar vortices (more precisely:
dipols made of rigid vortices), the members of the cloud generate more or less
exactly the counter pressure needed to compensate (5) and keep the vortex
“stable enough”.
According to (5) and with $u=\omega\,r$ the pressure deviation can be written
as follows,
$\delta p=p-p_{0}=\frac{\rho}{2}\times u^{2}=\rho\times\cal K\,,$ (6)
where $\cal K$ is the kinetic energy density in a vortex. Obviously the
measurement of turbulent pressure fluctuations can help estimating turbulent
kinetic energy. This corresponds to an early classical but approximate result
of continuum theory (eq. (8.3.21) on p. 182 in Batchelor, 1953; Hinze, 1953,
p. 242, last eq.). We will not explore the potential of (6) further and leave
it to the interested reader.
## 4 Turbulence and kinetic gas theory
### Similarities
About 150 years ago, James Clerk Maxwell presented his kinetic theory of
gases. Even if the details of molecular interaction forces in vacuum would
have been known that time, it would have not been helpful. An integration of
Newtons law of motion in terms of ordinary differential equations for each one
of the billion particles was practically excluded. But the creative use of
symmetries and conservation laws radically simplified the situation and led
eventually to a closed description of the most important macroscopic
properties of gases. Main elements are the following:
* (i)
The particles in a gas are perfectly elastic points with non-zero mass.
* (ii)
They are in permanent random motion which is to be described in terms of
statistical moments and sometimes called “molecular chaos”. The scatter
motions exhibit no preferred dirctions.
* (iii)
The collision results depend only on the local angular orientation of the
collision partners.
* (iv)
Between collisions they move uniformly and independently, without preferred
direction.
* (v)
Due to chemical neutrality, collisions lead only to the scatter of
trajectories.
The corresponding “turbulent relatives” of the above are:
1. (i’)
The particles are locally symmetric vortex-dipole tubes with finite cross-
sectional area, with vorticity and kinetic energy confined in the tubes, and
with zero circulation.
2. (ii’)
Their random translatory motions prefer no directions and may be termed ‘dipol
chaos’ Marmanis (1998).
3. (iii’)
The result of collisions depends only on the local angular orientation of the
colliding dipole elements.
4. (iv’)
Between collisions the particles move along complex trajectories which may be
curved.
5. (v’)
For symmetry reasons, 50 % of all collisions occur between two counter-
rotating dipole elements leading to dipole recombinations (or reconnections,
like those reported in a turbulent superfluid by Paoletti et al., 2010) and at
the end to quasi-elastic, random scatter motions resembling turbulent
diffusion and mixing, see left branch in Figure 14.
We underline that these rules shall apply only locally in space and time. Of
course, the prediction of the global pathway of a vortex dipole is impossible.
### Differences
Major differences between ideal gases and turbulence are the following:
* (a)
The number of chemically neutral gas particles in a countainer is constant in
time. They form a closed thermodynamic system. There is no particle
annihilation.
* (b)
The particles of turbulence are excited energy states, their number decreases
via collisions and subsequent energy dissipation. They need to be replaced by
shear generation of new particles if their number shall be kept more or less
in balance.
* (c)
In a gas, trajectories of point masses are simple lines in space, trajectories
of vortex-dipole tubes produce curved areas.
While gases evolve towards static thermodynamic equilibria, turbulence evolves
either towards $dynamic$ equilibria or steady states121212This neither implies
homogeneity, equifinality nor uniqueness of steady states., which may have
periodic character, or turbulence dies off.
Figure 4: Two possible collision results of two dipoles: the left branch is
“diffusive” because it leads to a recombination of the dipole elements and
scatter of trajectories which is known as turbulent diffusion. The right
branch is “dissipative” because it leads to an unstable vortex configuration
which decays “somehow” into heat141414The details of this decay process are
discussed later below.. For symmetry reasons both branches have identical
probabilities of 0.5. Note that the circular form of the vortex cross sections
presented here are chosen for reasons of clarity. They hold as an statistical
average picture only. Real vortices have elliptic or even strongly deformatted
cross sections.
To specify this important difference we have to supplement our above property
list for “turbulence particles” as follows:
1. (d)
For the same symmetry reasons like in (v’) of section 4 above, the remaining
50 % of all dipole collisions occur between two likewise rotating dipole
elements. Each case generates a fundamentally unstable vortex couple forming a
slowly rotating dissipative patch with its center of mass more or less at
rest. This patch decays through turbulent dissipation in the special form of a
“devil’s gear” or Kolmogorov spectrum. For details see further below and the
right branch in Fig. 14.
## 5 Equations of turbulent motion
### Brownian and turbulent motions
The turbulence properties (iii’) and (v’) of section 4 above establish an
analogy of turbulent dipole movements with the Brownian motion of particles
suspended in a fluid at rest (in the sense of Einstein, 1905). In the present
turbulent case it is not the kinetic theory of heat which governs the motions,
it rather is Helmholtz’ theory of dipol motions.
The (local) temporal path increments $\delta\vec{Y}_{j}$ of a vortex dipole
$j$ may be found by integration of the following Langevin equation over the
time increment $\delta t$,
$\frac{d{\vec{Y}_{j}}}{dt}=\vec{V}_{j}\;\;,\;j=1\dots{\cal N},$ (7)
where $\vec{V}_{j}$ is the random center-of-mass velocity of the selected
dipole $j$.
To specify the stochastic process $\vec{V}_{j}=\vec{V}_{j}(t)$ as simple as
possible, we choose a zero-mean, white-noise Gaussian process. Its strength is
controlled by the locally averaged dipole properties $\bar{\cal K}$ and
$\bar{\omega}$. More elaborate random processes like the Ornstein-Uhlenbeck or
the Kraichnan model Kraichnan (1968) and others are beyond the focus of the
present paper.
We now need to make a bigger jump over the broad river of stochastic dynamic
systems theory where the probability density function for the solution of a
dynamic system151515A dynamic system is here understood as a set of possibly
non-linear ordinary differential equations driven at their right-hand sides by
stochastic processes. may easily be taken from applied textbooks like those of
Haken (1978, 1983) in form of solutions of the Fokker-Planck equation (FPE)
corresponding to the extremely simple stochastic-dynamic system (7). In the
present case the FPE reads as follows,
$\frac{\partial{\cal N}}{\partial
t}-\frac{\partial}{\partial\vec{x}}\left(\nu\;\frac{\partial{\cal
N}}{\partial\vec{x}}\right)=0\,,$ (8)
where $\cal N$ is the probability density mentioned above, an equivalent of
the particle number density itself. $\nu$ characterizes the strength or
intensity of the noise $V_{j}(t)$ in (7). As already mentioned it is governed
by the control variables ($\bar{\cal K}$, $\bar{\omega}$) and appears later as
coefficient of eddy diffusivity of momentum, i.e. as eddy viscosity.
In the transition from (7) to (8) it has implicitely provided that all
gradients exhibit sufficiently smooth and slow behavior. Those local quasi-
equilibrium conditions are typically assumed in non-equilibrium thermodynamics
and many-particle physics. This clearly excludes shocks and steep fronts from
our considerations.
A further comment concerns an assumption used implicitely above. It is an
analogue of the so-called Ising assumption known from the theory of magnetism:
we treat triple or higher interactions between vortex filaments as
negligible161616This is not fully trivial. Under specific conditions of Bose-
Einstein condensates stable configurations consisting of one vortex and two
anti-vortices have been observed in the laboratory, either in linear setups or
equilateral triangles Seman et al. (2009). The latter is most symmetric and
called a $tripole$. Here we assume that in our very dense “vortex gas”
tripole-tripole interactions are controlled by dipole-dipole interactions of
their subsets. Note that tripoles might violate the principle of zero
circulation on the level of elementary interactions needed to keep the mean-
flow momentum balance in correct balance..
Figure 5: Local cross section through the first developmental stage of an
unstable pair of likewise rotating vortices resulting from a dipole-dipole
collision (right branch in Fig. 14). The green circles represent the primary
energy-containing vortices with identical radii $r$. Note that the green
circles do not touch each other! They are separated by the red circles who
symbolize secondary vortices in the beginning phase of a whole vortex cascade.
The dipole cloud surrounding the above structure not only generates the
necessary pressure head to keep individual vortices stable but it also acts as
a source of perturbations initiating roll-up instabilities and thus tertiary
and higher-order vortices and eventually a fully developed dissipative patch,
see text. The broken blue line and the arrows symbolize the slow rotation of
the whole patch around the common center of mass.
### Generation and annihilation of particles
Property (d) in Section 4 above allows to supplement the right-hand side of
equation (8) with sink and source terms, leading to a reaction-diffusion type
equation:
$\frac{\partial{\cal N}}{\partial
t}-\frac{\partial}{\partial\vec{x}}\left(\nu\;\frac{\partial{\cal
N}}{\partial\vec{x}}\right)=\Pi-\beta\;{\cal N}^{2}\;.$ (9)
Here $\Pi$ is the rate of quasi-particle generation. It is to be expressed in
terms of kinetic energy per unit time and is thus necessarily proportional to
the energy loss of the mean flow.
The second term corresponds to the energy-dissipation rate of TKE,
$\varepsilon$, i.e. TKE conversion into heat and/or sound, with $\beta$ being
a constant (for details see Baumert, 2009). $\varepsilon$ scales with the rate
of particle annihilation resulting from collisions. As we learn from chemical
kinetics (see Haken, 1978, 1983, for details), the annihilation term is
quadratic in the particle number, $\cal N$, because $two$ particles need to
collide to generate $one$ unstable couple which is then converted into heat
and/or sound. This is discussed in the next Section.
## 6 Dissipative patches
### Formation of vortex spectra
Our Fig. 14 and the concept behind it may seem simplistic. It demonstrates the
only two possible results of a dipole collision. While the left half of the
Figure shows the recombination of counter-rotating vortices from counter-
rotating vortices, the right half shows the the formation of a couple of
likewise rotating vortices from counter-rotating vortices. If isolated or
naked, the likewise rotating couple revolves around a common center of mass
and remais thus nearly at rest. This couple is known to be unstable, to form
stationary dissipative patches Sommerfeld (1948)171717It somehow resembles the
“dissipative elements” discussed by the group around Norbert Peters (see
Schäfer et al., 2010)..
This picture with the dissipative patch almost at rest implies that
dissipation is a spatially patchy phenomenon called intermittency and studied
extensively by various authors (for an overview see e.g. Frisch, 1995). We do
not go into details because intermittency is outside the focus of this study.
We discuss instead the way how the unstable configurations at the right half
of Fig. 14 and in Fig. 5 could be transformed into a Kolmogorov-Richardson
spectrum. This problem has attracted early attention by Taylor (1937) and
Kolmogorov (1941). The latter found on dimensional grounds that for a steady
energy flux from large to small scales the kinetic energy spectrum as function
of wavenumber may be presented as follows181818Actually, the original
arguments in Kolmogorov’s derivation were more subtle, but the use of
arguments in the sense of Rayleigh’s method of dimensional analysis or,
stricter, Buckingham’s $\pi$ theorem, is “strict enought” for the present
discussion.:
$d{\mathcal{K}}=\alpha_{1}\,\varepsilon^{\alpha_{2}}\,k^{-\alpha_{3}}\,dk\;,$
(10)
where here $k=2\pi/\lambda$ is the wave number and $\lambda$ the wavelength.
$\varepsilon$ is the dissipation rate of TKE, $\cal K$. Based on strict
dimensional arguments, Kolmogorov (1941) proved that $\alpha_{2}=2/3$ and
$\alpha_{3}=5/3$, in excellent agreement with the famous observations by Grant
et al. (1959) in a tidal inlet with $Re\approx 10^{8}$ and a depth of about
100 m. A theoretically sound value of $\alpha_{1}$ was open until today and
given now below.
### Devil’s gear
Our view of the Kolmogorov-Richardson cascade has been changed through a study
by Herrmann (1990) who has shown that Kolmogorov’s value for $\alpha_{3}$
corresponds numerically to the data of a space-filling bearing (see also
Herrmann et al., 1990). The latter is the densest non-overlapping (Apollonian)
circle packing in the plane, with side condition that the circles are
pointwise in contact but able to rotate freely, without friction or slipping.
One may call it a “devil’s gear” Pöppe (2004).
The contact condition for two different “wheels” with indizes 1 and 2 of the
gear reads
$u=\omega_{1}\;r_{1}=\omega_{2}\;r_{2}\;,$ (11)
where $u$ is necessarily constant throughout the gear and governed by the
energy of the decaying vortex pair as $u=\sqrt{2\,\cal K}$. It follows that
$\omega_{2}=\omega_{1}\;\frac{r_{1}}{r_{2}}\;,$ (12)
so that for very small $r_{2}$ the frequency $\omega_{2}$ may become
acustically relevant.
If the above gear is frictionless then the next question arises $where$ –
within this picture – energy could be dissipated. Clearly, in a fluid with
vanishing but non-zero viscosity, dissipation happens at scales where velocity
gradients are high enough, here: at a scale of measure zero. Our dissipative
patch (Fig. 5 shows the first stage of its formation) is thus “almost
frictionless” and therefore a Hamiltonian system, excepting scales of size
zero.
The formation of a fully developed spectrum of “wheels” from Fig. 5 deserves
certain perturbations “from the sides”, a condition which is guaranteed by the
random reconnection/recombination and scatter processes sketched in the left
half of Fig. 14 and also by the incomplete mutual pressure compensation of the
vortices in our vortex ensemble.
Without speculating too much we may expect that in a quasi-steady state
patches like in Fig. 5 are formed via roll-up instabilities at the “surface”
of the respective larger vortices. They steadily evolve into a fully developed
gear. Its energy, dissipated at the smallest radii, will decrease the energy
content of the primary (initial) vortex pair unless it is fed by mean flow.
The outer limits of such a patch are sketched in Fig. 6 for the begin of the
cascade process. The most important message which we gain from this figure is
that the $longest$ or energy-containing wavelength of the dissipative patch
equals $\lambda_{0}=2\,r$. The wavelength in a dipole is $4\,r$ and it forms
no patch or spectrum. This difference is essential. Further below we use
$\lambda_{0}$ as a lower integration limit for the spectral energy
distribution. It is important to underline that $\lambda_{0}$ labels the upper
wavelength limit in a dissipative patch. This limit is actually not influenced
by the formation details of the spectrum.
We finally notice that the transformation of the unstable configuration of two
likewise rotating vortices deserves time to set the greater masses of the
smaller scales into motion. This inertia effect might play a role in highly
dynamic scenarios.
Figure 6: Outer limits of a dissipative patch ($c.f.$ Fig. 5). The maximum
wavelength is obviously equal to $\lambda_{0}=2\,r$.
## 7 Universal constants
In a precursor study Baumert (2009) it has been shown that the geometric-
mechanical concept given in the present paper allows the derivation of von-
Karman’s constant as $\kappa=1/\sqrt{2\,\pi}$. It fueled hopes that other
universal constants of turbulent motion might also be derived from that
apparatus.
### Kolmogorov constant $\alpha_{1}$
in the wavenumber spectrum
The TKE can be calculated by integrating (10) over the dissipatve patch in the
sense sketched in Figures 5 and 6 yielding
${\cal
K}=\alpha_{1}\,\varepsilon^{2/3}\,\int_{k_{0}}^{\infty}k^{-5/3}\;dk=\alpha_{1}\,\frac{3}{2}\,\left(\frac{\varepsilon}{k_{0}}\right)^{2/3},$
(13)
where $k_{0}=2\,\pi/\lambda_{0}$ characterizes the lower end of the
$turbulence$ spectrum in the wavenumber space. We loosely assign the
wavenumber range $k=0\dots k_{0}$ to the mean flow which may basically be
resolved in numerical models.
The dissipation rate $\varepsilon$ in (13) can be expressed as follows,
$\varepsilon={\cal K}/\tau\,,$ (14)
with $\tau$ being the lifetime of a dissipative patch. Inserting (14) in (13)
and rearranging gives the following:
$\alpha_{1}=\frac{2}{3}\,\left(2\,\pi\right)^{2/3}{\cal
K}^{1/3}\left(\frac{\tau}{2\,r}\right)^{2/3}\,.$ (15)
Here we took from Fig. 6 that the energy-containing initial or primary
wavelength of a dissipative patch is given by $\lambda_{0}=2\,r$.
Now we make a local quasi-equilibrium assumption for conditions of extremely
dense vortex packing: Our marching dipoles can occupy only those places which
are simultaneously “emptied” from dissipative patches by decay. This means
that the life time of a dissipative patch, $\tau={\cal K}/\varepsilon$, should
equal the time of “free flight” of a dipole over a distance $2\,r$ (cf.
Baumert, 2009):
$\tau={\cal K}/\varepsilon=2\,r/u\;.$ (16)
Here we used the scalar dipole velocity $u$,
$u=\omega\,r=\sqrt{2\,{\cal K}}.$ (17)
After some algebra, the dimensionless pre-factor of the three-dimensional
wavenumber spectrum reads as follows:
$\alpha_{1}\;=\;\frac{1}{3}\,(4\,\pi)^{2/3}\;=\;1.802\,.$ (18)
The corresponding value of an ideal one-dimensional spectrum is one third of
the above, i.e. 0.60.
### Kolmogorov constant $\beta_{1}$
in the frequency spectrum
The generalized form of the Lagrangian frequency spectrum of fluid turbulence
is the following,
$d{\mathcal{K}}=\beta_{1}\,\varepsilon^{\beta_{2}}\,\omega^{-\beta_{3}}\,d\omega\;,$
(19)
where $\omega$ is the angular frequency. Tennekes and Lumley (1972) derived
$\beta_{2}=1$ and $\beta_{3}=2$ in a similar fashion like Kolmogorov (1941)
derived $\alpha_{2}$ and $\alpha_{3}$ (see also McComb, 2004). The integration
of (19) from $\omega=\omega_{0}$ to $\omega=\infty$ gives after rearrangement
and using (14) the following,
$\beta_{1}\;=\;\omega_{0}\;\tau\,.$ (20)
Now we take from the right part of (16) $\tau=2\,r/u$, from the left part of
(17) $\omega_{0}=u/r$ and insert both in (20) to get finally
$\beta_{1}\;=\;\omega_{0}\;\tau\,=\;\frac{u}{r}\,\frac{2\,r}{u}\;=\;2\,.$ (21)
### Pre-factor in the velocity
autocorrelation function
The spatial autocorrelation function $B(\rho)$ of fluctuating velocities is a
special second-order case of a structure function191919For details see §34 in
Landau and Lifshitz (1987).. With the abbreviation
$\rho=\left|\vec{\rho}\right|$, it is defined as follows,
$B(\rho)=\langle u_{1}(\vec{x})\times u_{2}(\vec{x}+\vec{\rho})\rangle.$ (22)
Here $u_{1},u_{2}$ are the (scalar) velocity components of the flow velocity
$\vec{u}$ in the direction $\vec{\rho}$ connecting the points $\vec{x}$ and
$\vec{x}+\vec{\rho}$ where the velocities $u_{1}$ and $u_{2}$ are taken
respectively:
$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\vec{u}(\vec{x})\cdot{\vec{\rho}}/{\rho}\;,$ (23) $\displaystyle
u_{2}$ $\displaystyle=$
$\displaystyle\vec{u}(\vec{x}+\vec{\rho})\cdot{\vec{\rho}}/{\rho}\;.$ (24)
The central dot denotes the scalar product (or dot product). Notice that,
rather than a density, $\rho$ is here a spatial distance.
In their §34 on p. 145 Landau and Lifshitz (1987) have shown that, based on
early results by Kolmogorov, $B(r)$ in (22) may be written as follows,
$B(r)=C\times(\varepsilon\,\rho)^{2/3},$ (25)
where $C$ is a dimensionless numerical constant which is related with
$\alpha_{1}$ from the universal wavenumber spectrum (10) as follows,
$\displaystyle C$ $\displaystyle=$
$\displaystyle\alpha_{1}\,\frac{27}{55}\,\Gamma(1/3)$ (26) $\displaystyle=$
$\displaystyle\frac{1}{3}\,(4\,\pi)^{2/3}\,\frac{27}{55}\,\Gamma(1/3)$ (27)
$\displaystyle\approx$ $\displaystyle 2.37\;.$ (28)
Here $\Gamma(z)$ is the Euler gamma function.
In contrast to our derivations of $\kappa$, $\alpha_{1}$ and $\beta_{1}$ the
value (28) for $C$ should be taken with care because the derivation of its
relation with $\alpha_{1}$ by Landau and Lifshitz uses an approximation and is
thus valid for small values of the distance variable $\rho$ only, i.e. for
$\ll\lambda_{0}=2\,\bar{r}$.
## 8 Discussion
### Comparison with observations
The rounded numerical values $\kappa=0.4$, $\alpha_{1}=1.8$,
$\alpha_{1}/3=0.6$, and $\beta_{1}=2$ for von Karman’s and Kolmogorov’s
universal constants predicted by our theory are situated well within the error
bars of many high-$Re$ number observations, NSE and RG based analytical
approximations, laboratory and DNS experiments. Based on observations,
Tennekes and Lumley (1972) gave the values $\alpha_{1}=1.62$ and
$\beta_{1}=2.02$, but with greater uncertainty.
Later works have been analysed in an important study by Sreenivasan (1995) who
possibly gave the most comprehensive literature review of experimental and
observational values for the number $\alpha_{1}/3$ until now. Later Yeung and
Zhou (1997) reported a value of $\alpha_{1}=1.62$ based on high-resolution DNS
studies with up to 5123 grid points.
Figure 7: Experimental and observational results for $\alpha_{1}/3$ measured,
collected from the literature, and analysed by Sreenivasan (1995). The solid
green line follows our somewhat arbitrary approximation
$0.6\times{\sqrt{Re_{\lambda}}}/{\left(\sqrt{Re_{*}}+\sqrt{Re_{\lambda}}\right)}$
wherein $\alpha_{1}/3=0.6$ is the theoretically derived asymptotic value. Here
we took $Re_{*}=10$. A similar presentation has been chosen by Sreenivasan
(1995) for $\alpha_{1}/3$ in his Fig. 3, and by Yeung et al. (2006) for
$\pi\beta_{1}$ in their Fig. 2.
The results of their later efforts suggest, again with DNS but based on a grid
of 20483 points, the value $\beta_{1}=2.1$ (Yeung et al., 2006, their Fig. 3).
In a recent study by Donzis and Sreenivasan (2010) a DNS grid of $4096^{3}$
has led to $\alpha_{1}\approx 1.58$.
Based on their oceanic measurements (with much higher Reynolds numbers
compared with DNS) Lien and D’Asaro (2002) found that $\beta_{1}=1.75\dots
2.04$. They state that
> …since the present uncertainty is comparable to that between high quality
> estimates of the Eulerian one-dimensional longitudinal Kolmogorov constant
> measured by many dozen investigators over the last 50 years, large
> improvements in the accuracy of the estimate of $\beta_{1}$ seem unlikely.
Beginning with an initiating work by Forster et al. (1977), systematic
analytical approximations using RG methods and related techniques for NSE
became further sources of estimates for the universal constants. E.g. Yakhot
and Orszag (1986b, a) reported $\alpha_{1}\approx 1.62$ whereas McComb and
Watt (1992) derived $\alpha_{1}=1.60\pm 0.01$ and Park and Deem (2003)
obtained $\alpha_{1}=1.68$. We note that these approximations are technically
extremely complex and neither unique nor part of an integrated descriptive
concept for turbulence.
Turbulence plays a crucial role in almost all fields of engineering, including
medical applications, and in geophysical fluid dynamics up to climate-change
studies. It plays an essential part in our everyday life. When we leave our
house or ride our bike, when we jump into our pool – we always are literally
embedded in a turbulent fluid. Compared with most other fields of modern
physics the present scatter in the values of the universal constants is
uniquely high and therefore actually not longer acceptable. However, our
theoretical results containing irrational and transcendental numbers suggest
that the universal constants of turbulence can principally not be measured in
real-world fluids. Or would measuring $\pi$ be a reasonable task?
A possible solution of this dilemma is the asymptotic analysis of measurements
at higher and higher Reynolds numbers so that an extrapolation to
$Re\rightarrow\infty$ becomes feasible with some certainty. The data analyses
by Sreenivasan (1995) and the report by Yeung et al. (2006) could serve as a
methodical model. Fig. 7 shows a re-plot of Sreenivasan’s data with $Re$ on
the abscissa and measured $\alpha_{1}/3$ on the ordinate. The data exhibit a
visible tendency to grow with increasing $Re$ to a saturation value which is
statistically indistinguishable from our 0.6.
### Universality and fundamentality;
turbulence, physics, and geometry
Besides problems and tasks posed in engineering and geophysical fluid dynamics
where prediction accuracy and extrapolability matter most, there is a question
posed by natural philosophy: how are universal constants of turbulent motion,
fundamental physics constants like the electron’s elementary charge (see e.g.
Fritzsch, 2009), and mathematical constants like $\pi$ related with each
other?
First, the measurement accuracy of “turbulent numbers” is extremely poor.
Second, “turbulent constants” characterize universal properties of a specific
(turbulent) form of dynamic and self-similar motions rather than more “static”
properties like the elementary charge. They are thus closer to the vacuum
speed of light or the Hubble constant describing the (accelerated) expansion
motion of the universe.
Third, like mathematical constants, “turbulent numbers” are dimensionless
whereas fundamental physics motion constants like the Hubble or the vacuum
speed of light are given in kilometers per second. The latter are absolute
values. “Our” constants characterize self-similar motions202020 If we exclude
from our consideration physical cosmology and arbitrarily chosen ratios
between masses of the various atoms and molecules then physics has actually
only one dimensionless exception: Sommerfeld’s fine-structure constant, with a
value of about $137$. Wolfgang Pauli has long been preoccupied with the
question of why, and Richard Feynman (1985) even speculated about a relation
with $\pi$. Here we have shown that $\pi$ is at least related with constants
of turbulent motion..
This provocates the question whether geometry is part of physics or vice
versa, as discussed for instance by Palais (1981). One might argue that the
discovery of the constant angle sum in triangles was historically the first
discovery of a physical conservation law by man, made and explicitly
formulated much earlier than the conservation laws of volume, mass etc.212121
These thoughts are actually outside the scope of the present report. Some
patience is needed as a “theory of everything” is not in sight Laughlin
(2005), and just-answered questions typically give birth to new conundrums.
### The completed image of turbulent flows
For the sake of clarity we concentrate on the most simple non-trivial
situation, a spatially one-dimensional channel flow with velocity component
$U$ in horizontal ($x$) direction, with vertical variation along $z$, and with
horizontal, $\tilde{u}$, and vertical velocity fluctuations, $\tilde{w}$. An
example of this situation has been presented earlier (see eq. (4.20) and
(4.21) in Baumert, 2005a, where stratification is already considered through
the squared buoyancy frequency, $N^{2}$). The Reynolds decomposition of our
flow field then reads as follows:
$\displaystyle U(z,t)$ $\displaystyle=$ $\displaystyle\langle
U\rangle+\tilde{u}(z,t)\,,$ (29) $\displaystyle W(z,t)$ $\displaystyle=$
$\displaystyle\langle W\rangle+\tilde{w}(z,t)\,.$ (30)
Mean flow: RANS. One may now insert (29, 30) into the corresponding two-
dimensional Euler equation to find together with mass conservation the
following,
$\frac{\partial\langle U\rangle}{\partial
t}+\frac{\partial\langle\tilde{u}\,\tilde{w}\rangle}{\partial
z}=-\frac{\partial\langle p\rangle}{\partial x}\,,$ (31)
where $\langle p\rangle$ is the pressure. Equation (31) is usually called a
Reynolds-averaged Navier-Stokes equation (RANS). Here we take the case of
vanishing molecular viscosity.
Turbulent mixing: downgradient flux. Equation (31) is not yet closed because
the correlator describing a diffusive flux,
$\langle\tilde{u}\,\tilde{w}\rangle$, needs to be specified, a task which
consumed substantial efforts over the last 60 years. The following quasi-
linear flux-gradient relation is theoretically well established in many
branches of many-particle physics and reads in our case as follows:
$-\,\langle\tilde{u}\,\tilde{w}\rangle\;=\;\nu\,\frac{\partial}{\partial
z}\,\langle U\rangle\,.$ (32)
Kolmogorov-Prandtl relation. However, also (32) does not yet close the problem
because now the so-called turbulent viscosity, $\nu$, needs to be specified.
In our picture the latter is again presented in a many-particle format in
analogy to Einstein’s theory of Brownian motion as follows (for details see
Baumert, 2009):
$\nu=L^{2}/\tau={\cal K}/(\pi\,\Omega)\,.$ (33)
Here $L=r/\kappa$ and $\tau=1/\Omega$ are locally averaged222222For reasons of
transparency the overbars are omitted. space and time scales which are
expressed within the framework of our mechanistic dipol-chaos model in terms
of $\cal K$ and $\omega=2\,\pi\,\Omega$.
Equations of turbulent motion, neutral stratification. We skip here the
technical details of derivations given in Baumert (2009) and quote only the
result that the variables $\cal K$ and $\Omega$ are governed by a specific
system of nonlinear partial differential equations as follows:
$\displaystyle\frac{\partial\cal K}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\frac{\partial\cal K}{\partial z}\right)$ $\displaystyle=$
$\displaystyle\nu\left[\left(\frac{\partial\langle U\rangle}{\partial
z}\right)^{2}-\Omega^{2}\right],$ (34)
$\displaystyle\frac{\partial\Omega}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\,\frac{\partial\Omega}{\partial z}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left[\frac{1}{2}\left(\frac{\partial\langle
U\rangle}{\partial z}\right)^{2}-\Omega^{2}\right]$ (35)
These equations resemble at least the structure of the so-called $\cal
K$-$\Omega$ closure model used by many authors (loc. cit. Wilcox, 2006). They
use differing empirically gained sets of prefactors of the terms and cannot
identify the physical nature of $\Omega$. Our derivation of (34, 35) should
not be confused with a new “scheme” or closure of this sort because – as we
have shown – our equations are founded solidly on the most simple principles
of hydrodynamics and many-particle physics, without use of phenomenological
data and, for the first time, giving even the universal constants of turbulent
motion. On the other hand, the experience of Wilcox (2006) and colleagues
shows that practical experiments and application needs guide the creative
engineer very close to the physically correct solution, of course, without
guiding further to the universal constants.
Turbulence spectra. With the dissipation rate $\varepsilon$,
$\varepsilon={\cal K}\,\Omega/\pi\,,$ (36)
we may express the spectra in the admissible turbulent ranges
$k=k_{0}\dots\infty$ and $\omega=\omega_{0}\dots\infty$ as follows:
$\displaystyle\frac{d{\cal K}}{dk}$ $\displaystyle=$
$\displaystyle\alpha_{1}\,\varepsilon^{2/3}\,k^{-5/3}\,\;,$ (37)
$\displaystyle\frac{d{\mathcal{K}}}{d\omega}$ $\displaystyle=$
$\displaystyle\beta_{1}\,\varepsilon\;\omega^{-2}\,\;,$ (38)
where turbulence begins at
$\displaystyle k_{0}$ $\displaystyle=$
$\displaystyle\pi/\kappa\,L\;=\;\pi^{2}\,\Omega\sqrt{2/{\cal K}}\;,$ (39)
$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\Omega/\kappa^{2}\;=\;2\pi\,\Omega\,,$ (40)
respectively. Longer scales or slower motions are possibly turbulent, but not
in the sense of a fully developed spectrum. In the case of slow variations of
the driving forces the motions outside the validity range belong to the mean
flow. Notice that the lower limits of the turbulent spectral ranges are
$dynamic$ quantities which are controlled by the dynamic variables $\cal K$
and $\Omega$.
Universal constants. The values of our theoretically derived universal
constants of turbulent motion are
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,(4\,\pi)^{2/3}\;=\;1.802\,,$ (41)
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle 2\,,$ (42)
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
1/\sqrt{2\,\pi}=0.399\,.$ (43)
Solid boundaries. TKE cannot penetrate solid boundaries like walls, which is
sometimes called an ‘adiabatic boundary condition’, in analogy to heat
conduction. Assuming the wall at $z=z_{*}$ this means that the following
condition has to be satisfied:
$\left.\frac{\partial{\cal K}}{\partial z}\right|_{z=z_{*}}=0\,.$ (44)
This is already sufficient to solve (34, 35) at solid walls and produce the
logarithmic law of the wall. A second condition for $\Omega$ is not needed due
to the a.m. nonlinearities of the equations (for details see Baumert, 2005a).
Stable stratification. Stable stratification may be described by the following
modification of (34):
$\frac{\partial\cal K}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\frac{\partial\cal K}{\partial
z}\right)=\nu\left[\left(\frac{\partial\langle U\rangle}{\partial
z}\right)^{2}-2N^{2}-\Omega^{2}\right].$ (45)
This description is valid as long as $\Omega\geq N$. In the case of free decay
and linear stratification, $\Omega$ approaches $N$ “from above”. Due to the
dispersion relation for internal waves, slow disordered turbulent motions at
$\Omega\approx N$ and slower can only exist as waves so that turbulence is
converted into internal waves when $\Omega\rightarrow N$. The stratification
aspects are discussed by Baumert and Peters (2004, 2005) in greater detail and
in relation to observations and measurements.
Limitations. A trivial limitation for (45) is $N^{2}\geq 0$. For unstable or
convective situations these equations are possibly not applicable, but we did
not yet test this case.
Equations (29 – 44) form a closed, complete description of mean and turbulent
motions, including their wavenumber and frequency spectra with their universal
spectral constants, applicable under neutrally stratified conditions. For
stably stratified flows (34) needs to be replaced with (45), provided that
there are $no$ external sources of internal-wave energy. This means that, due
to the ubiquituous presence of internal waves in most geophysical flows, the
latter are $not$ covered by (45). Some modifications are necessary and a
certain knowledge of the sources of internal-wave energy is needed. The
necessary modifications are not part of this report.
The difference between (34, 35) or (45, 35) on the one hand, and traditional
phenomenologically based closure schemes discussed e.g. by Wilcox (2006) on
the other, is mathematically minimal but physically relevant. Those schemes
use for instance quantities like $\varepsilon$ or $\tau$ as primary variables
in balance equations although conservation laws for those quantities do not
exist. The differences are becoming generally relevant and visible at solid
boundaries and, in the stratified case, also within the turbulent fluid
volume.
An important limitation has already been mentioned above. Further, variations
in the mean flow field should be relatively slow and spatial gradients not too
strong so that we may talk about time-dependent but quasi-steady behavior with
sufficiently homogeneous state variables on local scales and sufficiently
developed Kolmogorov spectra. This excludes shock-waves from our
considerations. But alos this limitation has not been tested yet.
The free decay of turbulence might be such a case. Our theory predicts the
decay of TKE to scale with $t^{-1}$ [or with $(x/U)^{-1}$ in the wind tunnel].
This is also supported through a group-theoretical analysis by Oberlack
(2002). Whereas wind-tunnel data Batchelor (1953) and selected free-decay
measurements Dickey and Mellor (1980) agree surprisingly well with our theory,
in some experiments if rapid free decay the exponent is somewhat greater than
unity. This situation is challenging and needs a further study.
We may summarize as follows:
* •
The present theory is based on conservation laws and geometry only, without
use of empirical data.
* •
Besides spectrally integrated parameters like turbulent viscosity, dissipation
rate etc., it gives full turbulent spectra, even under non-stationary
conditions.
* •
The theory predicts fundamental constants of turbulent motion.
* •
The approach is limited to slow and smooth mean flows under neutral or stable
stratification.
* •
Internal gravity waves are taken into account only so far as they are
intrinsically coupled with local current shear.
Our methodology differs from past concepts. The latter use specific series
expansions of the Navier-Stokes equation into a system of partial differential
equations for higher and higher moments in the perturbations, with the zeroth
hierarchy element being the Reynolds equation (RANS). The $k$-$\varepsilon$,
the Mellor-Yamada, $k$-$\tau$, the $k$-$\omega$ and the many other turbulence-
closure schemes populating the literature are examples (e.g. Baumert et al.,
2005; Wilcox, 2006).
The present picture includes a perturbation step, too, but it stops at the
zeroth level, at RANS. Here the turbulent fluxes still remain unknown. We then
took into account that each fluid element experiences an effective total force
field consisting additively of the external mean field $\bar{\mathcal{H}}$
acting in the total fluid volume, and of the field $\tilde{\mathcal{H}}$
controlled by nearest neighboring vortex dipoles (“Ising assumption”),
${\mathcal{H}}=\bar{\mathcal{H}}+\tilde{\mathcal{H}}.$ (46)
Our results show that $\tilde{\mathcal{H}}$ is locally best described by a
simple white-noise stochastic force. This approach has successfully been used
about 100 years ago by Langevin, Smoluchowsky and Einstein in theoretical
analyses of Brownian motion. This assumption directly delivers Fokker-Planck
equations for the expectation-value distribution functions of particle number
and of particle properties $\cal K$ and $\Omega$.
The theory applies exclusively to locally homogeneous, locally isotropic and
weakly unsteady turbulent flows. To extreme non-stationarities and/or sharp
spatial gradients like in shockwaves it is possibly not applicable. As a rule,
temporal changes of the mean flow should take place on time scales
sufficiently greater than $1/\Omega$ because otherwise the Kolmogorov spectra
(37, 38) are not yet well enough established.
Coda. The theory completed here will hopefully not form a Procrustean bed in
Saffman’s (1977) sense (loc. cit. Davidson, 2004, p. 107) who even feared that
“in searching for a theory of turbulence, perhaps we are looking for a chimera
…”. This view has just recently been enforced by Hunt (2011) as follows: “But
there are good reasons why the answer to the big question that Landau and
Batchelor raised about whether there is a general theory of turbulence is
probably ‘no’. ”
As the above smacks of defeatism, we pass the word to Sir Winston Churchill
who demanded in a critical situation “We shall never surrender!” He surely
would encourage a new generation of men and women to follow their own fresh
ideas towards a general theory of turbulence at asymptotically high Reynolds
numbers.
###### Acknowledgements.
This report profited substantially from the scientific cooperation between the
author and Hartmut Peters at Earth and Space Research in Seattle, WA, who even
contributed Figures 1 to 3, and major paragraphs of Chapter 3 of the present
report. The cooperation took place within the framework of the Department of
the Navy Grant N62909-10-1-7050 issued by the Office of Naval Research Global.
The United States Government has a royalty-free license throughout the world
in all copyrightable material contained herein. The author is thankful to
Eckhard Kleine who directed the attention to the remarkable works of Hans J.
Herrmann and helped as a robust sparring partner in turbulent debates. Special
thanks are due to Stephen A. Thorpe who critically commented the precursor
version of this report. Thanks are further due to David I. Benjamin, Peter
Braun, Eric D’Asaro, Bruno Eckhardt, Philippe Fraunier, Harald Fritzsch, Boris
Galperin, Rupert Klein, Ren-Chieh Lien, Jim Riley, Gisbert Stoyan, John
Simpson, Jürgen Sündermann, Oleg F. Vasiliev, Michael Wilczek, and Sergej S.
Zilitinkevich. The author thanks particularly the conference organizers of
Turbulent Mixing and Beyond 2011 at the Abdus Salam International Centre of
Theoretical Physics, Trieste/Italy, where the above text has been presented
for the first time. Special thanks are due to Snezhana Abarzhi and Joseph J.
Niemela who managed to create a fair, open and creative scientific atmosphere.
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|
arxiv-papers
| 2011-07-25T23:02:59 |
2024-09-04T02:49:20.971239
|
{
"license": "Public Domain",
"authors": "Helmut Z. Baumert",
"submitter": "Helmut Z. Baumert",
"url": "https://arxiv.org/abs/1107.5305"
}
|
1107.5474
|
# Selecting Attributes for Sport Forecasting using Formal Concept Analysis
Gonzalo A. Aranda-Corral1, Joaquín Borrego-Díaz2 Juan Galán-Páez2
1Department of Information Technology, Universidad de Huelva, Spain
gonzalo.aranda@dti.uhu.es
2Department of Computer Science and Artificial Intelligence, Universidad de
Sevilla, Spain
jborrego@us.es, juangalan@us.es
###### Abstract
In order to address complex systems, apply pattern recongnition on their
evolution could play an key role to understand their dynamics. Global patterns
are required to detect emergent concepts and trends, some of them with
qualitative nature. Formal Concept Analysis (FCA) is a theory whose goal is to
discover and to extract Knowledge from qualitative data. It provides tools for
reasoning with implication basis (and association rules). Implications and
association rules are usefull to reasoning on previously selected attributes,
providing a formal foundation for logical reasoning. In this paper we analyse
how to apply FCA reasoning to increase confidence in sports betting, by means
of detecting temporal regularities from data. It is applied to build a
Knowledge Based system for confidence reasoning.
## Introduction
Formal Concept Analysis (FCA) Ganter & Wille (1999) is a mathematical theory
for data analysis using formal contexts and concept lattices as key tools.
Domains can be formally modelled according to the extent and the intent of
each formal concept. In FCA, the basic data structure is a formal context
(with a qualitative nature) which represents a set of objects and their
properties and it is useful both to detect and to describe regularities and
structures of concepts. It also provides a sound formalism for reasoning with
such structures, mainly Stem Basis and association rules. Therefore, it is
interesting to consider its application for reasoning with temporal
qualitative data in order to discover temporal trends Aranda-Corral et al.
(2011).
In this paper, FCA application scope is the challenge of sports betting,
specifically, the forecasting of soccer league’s results. Forecasting sport
results is a fast growing research area, because of its economic impact in
betting markets as well as for its potential application to problems with
similar behaviour (markets) Inst. Engineering and Technology (2010).
Considering sports betting as a complex system, soccer leagues represent a
challenging system with a huge amount of knowledge, available through WWW, and
its behaviour is weekly exhaustive analysed by journalists, betting companies
and supporters. Roughly speaking, three dimensions have been considered for
analysing/synthesizing prediction systems: 1)Those which analyse information
on teams (endogenous) versus those which analyse results (exogenous); 2)Those
which exploit quantitative data versus those which exploit qualitative
knowledge, and finally, 3)Statistic-based ones versus other methods. Usually,
one can work with hybrid models, and rarely with pure qualitative and
exogenous reasoning systems appear in literature, although their use is
considered for experiments (for example, frugal methods Goldstein & Gigerenzer
(2009) and based on the recognition heuristic Goldstein & Gigerenzer (2002))
or as part of hybrid systems (see e.g. Min et al. (2008)). There are two
reasons that may justify this point.
On the one hand, transformation from a large quantitative dataset to a
qualitative problem is faced with the selection of an acceptable threshold and
the discovery of better relations (see e.g. Imberman et al. (1999)). On the
other hand, a qualitative dataset must be accomplished with some amount of
information based on confidence, trust or probability of these data sets.
Figure 1: FCA based model for prediction of qualitative features of Complex
Systems
The aim of this paper is to describe all researching work made for selecting
and computing attribute sets related to soccer results, into a specific
framework: FCA, and starting from soccer match results, with no previous
analysis of any other specific attributes. This task is previous to build an
Expert System for advising sport betting which could detects some kind of
regularities on data. Concept lattices, which are computed from attribute
values, represent a mathematical structure of relationships among the concepts
which are involved in selected sport events to study. Since this method is
bet-oriented, its performance is evaluated within a confidence-based reasoning
system. This sistem increases number of hits in soccer matches forecasting,
discovering temporal trends by means of data mining and association rules
reasoning. The analysis of attributes has been used in Aranda-Corral et al.
(2011) to describe a confidence-based (and contextual) reasoning system for
forecasting sports betting. In this paper we analyse the attribute selection
problem as a problem of selection of features that shape the behaviour of the
complex system that represents professional soccer leagues. Theoretical
framework, on which this model is based on, will be presented at Aranda-Corral
et al. 2011b . Due to a really huge amount of information, attribute selection
advised by experts is mandatory. In fact, the system can be considered as a
reasoning model based on bounded rationality and recognizition heuristics. and
focused on features which were considered as important by human experts.
Therefore, the system aims to forecast results, but it is designed based on
bounded rationality models, instead of statistic models (although in the
future hybrid models will be considered.
The system is a first prototype from a more general system, which are building
to analyse qualitative features of Complex Systems (see Fig. 1), using FCA.
The idea is to isolate qualitate attributes from (past) local interactions
among components of complex system and to apply FCA tools in order to predict
properties system’s behavior in a near future.
## Background: Formal Concept Analysis
According to R. Wille, FCA Ganter & Wille (1999) mathematizes the
philosophical understanding of a concept as a unit of thoughts composed of two
parts: the extent and the intent. The extent covers all objects belonging to
this concept, while the intent comprises of all common attributes valid for
all the objects under consideration. It also allows the computation of concept
hierarchies from data tables. In this section, we succinctly present basic FCA
elements (the fundamental reference is Ganter & Wille (1999)).
A formal context $M=(O,A,I)$ consists of two sets, $O$ (objects) and $A$
(attributes) and a relation $I\subseteq O\times A$. Finite contexts can be
represented by a 1-0-table (identifying $I$ with a Boolean function on
$O\times A$). See Fig. 2 for an example of formal context about live beings.
Figure 2: Formal context, associated concept lattice and Stem Basis
The FCA main goal is the computation of the concept lattice associated to the
context. Given $X\subseteq O$ and $Y\subseteq A$ it defines
$X^{\prime}:=\\{a\in A\ |\ oIa\mbox{ for all }o\in X\\}$
$Y^{\prime}:=\\{o\in O\ |\ oIa\mbox{ for all }a\in Y\\}$
A (formal) concept is a pair $(X,Y)$ such that $X^{\prime}=Y$ and
$Y^{\prime}=X$. For example, concepts from living beings formal context (Fig.
2, left) is depicted in Fig. 2, right.
Using this Fig. 2, each node is a concept, and its intension (or extension)
can be formed by the set of attributes (or objects) included along the path to
the top (or bottom). E.g. The node tagged with the attribute Legs represents
to the concept $(\\{Legs,Mobility,NeedWater\\},\\{Cat,Frog\\})$.
In this paper it works with logical relations on attributes which are valid in
the context. Logical expressions in FCA are implications between attributes.
An implication is a pair of sets of attributes, written as $Y_{1}\to Y_{2}$,
which is true with respect to $M=(O,A,I)$ according to the following
definition. A subset $T\subseteq A$ respects $Y_{1}\to Y_{2}$ if
$Y_{1}\not\subseteq T$ or $Y_{2}\subseteq T$. It says that $Y_{1}\to Y_{2}$
holds in $M$ ($M\models Y_{1}\to Y_{2}$) if for all $o\in O$, the set
$\\{o\\}^{\prime}$ respects $Y_{1}\to Y_{2}$. In that case, it is said that
$Y_{1}\to Y_{2}$ is an implication of $M$.
###### Definition. 1
Let ${\mathcal{L}}$ be a set of implications and $L$ be an implication.
1. 1.
$L$ follows from ${\mathcal{L}}$ (${\mathcal{L}}\models L$) if each subset of
$A$ respecting ${\mathcal{L}}$ also respects $L$.
2. 2.
${\mathcal{L}}$ is complete if every implication of the context follows from
${\mathcal{L}}$.
3. 3.
${\mathcal{L}}$ is non-redundant if for each $L\in{\mathcal{L}}$,
${\mathcal{L}}\setminus\\{L\\}\not\models L$.
4. 4.
${\mathcal{L}}$ is a (implication) basis for $M$ if ${\mathcal{L}}$ is
complete and non-redundant.
It can obtain a basis from the pseudo-intents Guigues & Duquenne (1986) called
Stem Basis (SB):
${\mathcal{L}}=\\{Y\to Y^{\prime\prime}\ :\ Y\mbox{ is a pseudointent}\\}$
A SB for the formal context on live beings is provided in Fig. 2 (right). It
is important to remark that SB is only an example of a basis for a formal
context. In this paper any specific property of the SB can be used, and it can
be replaced by any implication basis.
It is possible to extend $\models$ in relation to any propositional formula
with propositional variables in $A$, by considering each object
$o\in\mathbb{M}$ as a valuation $v_{o}$ on $\mathbb{A}$ defining
$v_{o}(A)=1\Longleftrightarrow(o,A)\in\mathbb{I}$
Thus $M\models F$ if and only if for any $o\in O$ it holds that $v_{o}\models
F$.
The Armstrong rules Armstrong (1974) provides a formal basis for implicational
reasoning:
$\displaystyle\frac{}{X\rightarrow X}\ \ \ \frac{X\rightarrow Y}{X\cup
Z\rightarrow Y},\ \ \ \frac{X\rightarrow Y,\ Y\cup Z\rightarrow W}{X\cup
Z\rightarrow W}$
A set of implications is closed if and only if the set is closed by these
rules Armstrong (1974). By defining $\vdash_{A}$ as the proof relation by
Armstrong rules, it holds that the implicational bases are
$\vdash_{A}$-complete:
###### Theorem 2
Let $\mathcal{L}$ be an implicational basis for $M$, and $L$ an implication.
Then $M\models L$ if and only if ${\mathcal{L}}\vdash_{A}L$
In order to work with formal contexts, stem basis and association rules, the
Conexp111http://sourceforge.net/projects/conexp/ software has been selected.
It is used as a library to build the module which provides the implications
(and association rules) to the reasoning module of our system. The reasoning
module is a production system based on which was designed for Aranda-Corral &
Borrego-Díaz (2010). Initially it works with SB, and entailment is based on
the following result:
###### Theorem 3
Let $\mathcal{L}$ be a basis for $M$ and $\\{A_{1},\dots,A_{n}\\}\cup
Y\subseteq A$. The following conditions are equivalent:
1. 1.
$\mathcal{S}\cup\\{A_{1},\dots A_{n}\\}\vdash_{p}Y$ ($\vdash_{p}$ is the
entailment with the production system).
2. 2.
$S\vdash_{A}A_{1},\dots A_{n}\rightarrow Y$
3. 3.
$M\models\\{A_{1},\dots A_{n}\\}\rightarrow Y$.
### Association rules for a a formal context
We can consider a Stem Basis as an adequate production system in order to
reason. However, Stem Basis is designed for entailing true implications only,
without any exceptions into the object set nor implications with a low number
of counterexamples in the context.
Another more important question arises when it works on predictions. In this
case we are interested in obtaining methods for selecting a result among all
obtained results (even if they are mutually incoherent), and theorem 3 does
not provide such a method. Therefore, it is better to consider association
rules (with confidence) instead of true implications and the initial
production system must be revised for working with confidence.
Researching on logical reasoning methods for association rules is a relatively
recent promising research line Balcázar (2010). In FCA, association rules are
implications between sets of attributes. Confidence and support are defined as
usual. Recall that the support of $X$, $supp(X)$ of a set of attributes X is
defined as the proportion of objects which satisfy every attribute of $X$, and
the confidence of a association rule is $conf(X\to Y)=supp(X\cup
Y)/{supp}(X)$. Confidence can be interpreted as an estimate of the probability
$P(Y|X)$, the probability of an object satisfying every attribute of $Y$ under
the condition that it also satisfies every one of $X$. Conexp software
provides association rules (and their confidence) for formal contexts.
Figure 3: Context based reasoning system
## Reasoning under contextual selection. Logical Foundations
The model (described in Aranda-Corral et al. 2011b ) is composed of events
(objects) which have a number of properties (attributes). They consitute a
universal formal context $\mathbb{M}$ (which we call monster context following
the tradition in Model Theory). Thus $\mathbb{M}$ can be considered as the
global memory from which subcontexts are extracted. Once the specific context
is considered, it is also possible to consider background knowledge $\Delta$
(in form of propositional logic formulas) which would be combined with the
knowledge extracted from formal context (Stem basis or association rules).
###### Definition. 4
Let $\mathbb{M}=(\mathbb{O},\mathbb{A},\mathbb{I})$ be the monster context,
and let $O$ be a set of objects.
1. 1.
A context on $O$ is a context $M=(O_{1},A,I)$ where $O\subseteq
O_{1}\subseteq\mathbb{O}$
2. 2.
A contextual selection on $O$ and $M$ is a map
$s:O\to{\mathcal{P}}(O_{1})\times{\mathcal{P}}(A)$
3. 3.
A contextual KB for an object $o\in O$ w.r.t. a selection $s$ with confidence
$\gamma$ is a subset of association rules with confidence greater or equal to
$\gamma$ of the formal context associated to $s(o)=(s_{1}(o),s_{2}(o))$, that
is, to the context $M(s(o)):=\displaystyle(s_{1}(o),s_{2}(o),I_{\restriction
s_{1}(o)\times s_{2}(o)})$ (note that when condifence is 1 the contextual KB
is a implicational basis).
Contextual KBs is useful for entailing attributes on an object. The reasoning
model on $\mathbb{M}$ is argumentative, where the argument is based on KBs
extracted from subcontexts Aranda-Corral et al. 2011b :
###### Definition. 5
Let $L$ be an implication and $\Delta$ a background knowledge. It is said that
$L$ is a possible consequence of $\mathbb{M}$ under $\Delta$,
$\mathbb{M}\models^{\Delta}_{\exists}L$, if there exists $M$ a nonempty
subcontext of $\mathbb{M}$ such that $M\models\Delta\cup\\{L\\}$.
Note that by theorem 3, when $\Delta$ is a set of implications, it holds that
$\models_{\exists}$ is equivalent to $\vdash_{\exists}$ which is defined by:
$\mathbb{M}\vdash_{\exists}L$ if there exists $M\models\Delta$ a subcontext of
$\mathbb{M}$ such that $S\vdash_{p}L$ (where $S$ is a stem basis for $M$).
### The role of attribute selection for formal contexts
Attributes are essentials in the contextual selection to build good formal
contexts. Association rules are extracted from the contexts and those are used
by the production system. By means of these association rules and some initial
facts based on the match we want to forecast the production system infers the
confidence (probability) for each one of the three possible results of a
match, home team wins, draw or away team wins. Thus attributes constitute one
of the most important and sensitive parts of the system. They are sensitive
because on how they represent the behavior of the teams will depend the
accuracy of the inferred results.
Figure 4: Concept Lattice for the match Málaga-Sevilla (week 31, season
2009-10)
## Confidence-based reasoning system
The reasoning system works on facts of the type $(a,c)$, where $a$ is an
attribute and $c$ is the estimated probability of the trueness of $a$, which
we also call confidence (by similarity with the same term for association
rules). See Aranda-Corral et al. (2011) for a more detailed description of the
reasoning system.
The system has a module for a confidence-based reasoning system (Fig. 3). Its
entries for a match $Team_{1}$ \- $Team_{2}$ are: the contextual Knowledge
basis for a threshold given as rule set and attribute values for the current
match (except 1,X,2) as facts, all of them with a confidence (whose value
depends on the reasoning mode, see below). The production system is executed
and the output is a triple $<(1,c_{1}),(X,c_{x}),(2,c_{2})>$ of attribute,
confidence for this match. The attribute with greater confidence is selected
as the prediction. Production system execution is standard, with several modes
for confidence computing of results based in uncertain reasoning in Expert
Systems Giarratano & Riley (2005). Any attribute/fact $a$ is initialized with
confidence
$conf(a):=\displaystyle\frac{|\\{o\ :\ oIa\\}|+1}{|O|+1}$
## Attributes and formal contexts for soccer league
For both selecting data and building contexts, some assumptions on forecasting
in soccer league matches have been considered. Reconsiderations of such
decisions can be easily computed in the system. First, we consider that the
regularity of team’s behaviour only depends on the contextual selection that
has been considered. This contextual selection is obtained by taking matches
from the last $X$ weeks backwards, starting from the week just before the one
we want to forecast.Second, since FCA methods are used to discover regularity
features, thus it does not consider forecasting exceptions (unexpected
results). Therefore, the model can be considered as a starting point for
betting expert who would adjust attributes, in order to more personalised
criteria.
These attributes have to be computed and used to entail the forecasting. This
analysis is assisted by Conexp. ConExp software is used to compute and analyze
the concept laticces associated to the temporal contexts. In order to select
most interesting attributes for the system, starting from an initial
configuration, user can compute the associated concept lattice and check it.
In this way, attributes goodness (and thresholds) can be evaluated to
reconsider current attribute selection. For example, in Fig. 4, the concept
lattice associated to contextual selection for Málaga-Sevilla match is shown.
This contextual selection is obtained from a given attribute selection and
last 38 weeks matches before. In this concept lattice, the attribute
$ID\\_1\\_T\\_16$ is defined by: ’the budget of $team_{2}$ is greater than
$\gamma_{1}$ times the budget of $team_{1}$’, where $\gamma_{1}$ is the
threshold the expert must estimate. In the concept lattice we can observe that
the biggest concept containing the attributes $team_{2}\\_wins$ and
$ID\\_1\\_T\\_16$ covers the about the 10% of the objects owned by the first
attribute, therefore it is suggested to use the second attribute for reasoning
with association rules to get a prediction.
The system computes the value of an amount of attributes on objects.
Experimentally a boolean combination of attributes is possible. Once the
temporal context has been computed, the system can build contextual selections
by selecting the match and the attribute set. The selection of attributes was
made by considering four kinds of factors: those related with the
classification, the history of teams’ matches in the recent past, results of
direct matches and other non related results, as for example the difference
between team budgets. Seventeen relevant attributes were selected.The
attribute set has three special attributes, $Team_{1}$ wins (1), $Team_{2}$
wins (2) and draws (X).
With respect to data source, they are automatically extracted from RSSSF
Archive222http://www.rsssf.com. Objects are matches and attributes are a list
of features, including temporal stamp (week, year). Data was collected for the
past four years. Actually the size of the context is about 300 objects and 18
attributes (although several of them are parametrized, see section bellow).
Thus, $|I|$ is about 5,100 pairs.
## Attribute selection
We have chosen a small set of attributes with many possibilities through a few
customizable parameters. When these parameters are having set up with proper
values, the set of attributes will represent team’s logical behavior.
Recall that formal concept analysis works with qualitative attributes and all
teams information which we work with are quantitative data. Thus it is
necessary to convert quantitative attributes into qualitative ones. This task
is left to users by choosing a proper threshold to each attribute.
Before choosing the set of base attributes , we have carried out a analysis on
information about soccer results . The aim have been to discover which factors
are more influential in teams behavior and which ones are less influential.
First of all, we have collected any interesting factor found, and after
analyzing each one, individually, we have chosen most suitable ones. Examined
factors can be classified in four different categories (see Table 1): those
related to season’s classification, those related to previous team’s results,
those related to historical direct matches and any other factors. It is worth
to note that to increase possibilities of the attribute set, and considering
the Boolean nature of formal context attributes, we have added the option to
create new ones by means of logical combinations of these attributes.
According to considered factors, the system computes a base set of 18
attributes, which are customizable by some parameters. This will let us to
obtain a diverse set of attributes. In Table 2 attributes are specified. Four
parameters are used:
* •
Threshold: Parameter to be used to translate quantitative attribute values
into qualitative ones.
* •
Team: Recall that in the formal context considered, objects are matches but
attributes belongs to team properties. This parameter will set the team from
object (match) on which attribute will be considered. It has two possible
values: {HOME, AWAY}. Thus, usually, we will have twice each attribute at
context, once for home team and once for away.
* •
Number of Matches: sets the number of past matches to be considered when some
attributes are computed, e.g. the ones associated to previous team results.
* •
Kind of matches: sets past matches type to be taken into account to compute
some attributes, considering home/away team’s condition at matches. Three
possible values: {MATCHES AS HOME TEAM, MATCHES AS AWAY TEAM, ALL MATCHES}
With these parameters, and the possibility to compound attributes, it is
possible to build a detailed attributes set. Note that experiments show that
simplest and most logical attributes give a good team behavior representation.
Although we consider that a versatile attributes set, as above described, was
necessary because of a huge number of factors can determine the result of a
soccer match. Task of customizing the attribute set is left to users, and it
is the most important one in forecasting process. Thus, a basic soccer
knowledge should be required. The goodness of customization will determine
system results.
Factors | Correlation Degree | Used?
---|---|---
Associated to the classification in the league | |
Team in the first classification level | medium/high | yes
Team in the last classification level | medium/high | yes
Difference between team’s classifications | medium/high | yes
Team was in a different league last year | medium | no
Team socred a important number of goals (in the last matches) | medium/low | no
Associated to previous results of the team | |
Number of consecutive won matches. | high | yes
Number of consecutive lost matches. | high | yes
Number of consecutive draws. | medium | yes
Number of non consecutive won matches in previous weeks. | high | yes
Number of non consecutive lost matches in previous weeks. | high | yes
Number of non consecutive draws in previous weeks. | medium/high | yes
Points collected in previous weeks. | medium/high | yes
Factors related with directed matches (incluidas previous years) | |
Number of wins in previous directed matchs | medium/high | yes
number of losts in previous directed matchs | medium/high | yes
number of draws in previous directed matchs | medium/high | yes
Other Factors | |
Number of red cards collected by the team’s players. | low | no
Wheather the day and the city where the match took place | medium | no (hard to parametrize)
Motivation because of the fans support when playing as home team. | high | no (hard to parametrize, subjective)
Team hires a new coach. | high | no (only useful when new coach hired)
Some players of the team are selected for their National Team. | medium/Low | no (relevant for some nationalities)
Difference between team’s budgets. | high | yes
One or more important team’s players are injured. | medium | no (hard to automatically collect the data)
Cups collected in the lasts years. | low | no (only for a few of teams)
Table 1: Factors considered for selecting/building attributes
## Computing problems
The way of competition causes to take into account some special situations for
computing attributes values. In this section we describe the main problems
emerged and how they were fixed. Roughly speaking, these main problems
concerns to initial matches in season.
### Beginning of a new season: week 0
This problem is not hard, but as many others unavoidable, and a solution
becomes essential. It happens when computing an attribute value related to
league standings to forecast first week of a season. As any previous week has
been played yet, there is not way to build a standing table.
When teams in current season remain in the same league as last, a trivial
solution is to take into account positions and matches in last weeks of
previous season. If the team played in a higher division than last season, it
will be at the first position in the standing. Otherwise, if the team played
in a lower division, it will be considered at last position.
### Missing matches in attribute computation
Other problem, closely related to previous one, is when not enough previous
matches are available to compute an attribute. Solution pass through taking
lasts matches of last season as if they were in a continuous temporal line.
This is not so simple, because of some teams were not playing at same division
last season. Indeed, when playing in a lower or higher division, difficulty of
division changes and matches cannot be compared into the same way. Therefore,
we need to handle the situation of a team playing in a different division from
current season division.
Other troubled situation where there are not enough matches for attribute
computation is to compute results for directed matches between two teams
because of there is only a few of such matches in the data source.
For these two related situations we offer two solutions. First is to compute
attribute with a null value, but in this way we are giving a fake information
to the system. We are setting that attribute is not true but, in fact, we have
not information enought to determine it, so a better approach is required.
Chosen solution is based on adjusting attribute’s threshold. The value of this
threshold is decreased proportionally to relation between number of required
matches and number of available matches. Threshold $\gamma$ is revised by
$\gamma_{new}=\gamma_{old}\cdot\frac{\mbox{number of match results
available}}{\mbox{number of match results needed}}$
When number of required matches is too high and number of available matches is
low, it looks like we are giving fake information to system again, but our
experience shows that collateral effects of this approach are worthless
compared to compute attributes with a null value.
## Attribute selection vs expert system behaviour
In general terms, current base attribute set behavior forecast the most
possible results of a match is quite good, in regular conditions. Even so,
some experiments, in order to study attribute’s behavior, have been developed.
### Strict attributes
An attribute is strict when only a few objects can satisfy it, because of its
threshold is too high. By working with sets of strict attributes, we can
assure that they estimate the teams behavior better than other sets. Thus,
with strict attributes, we will have very reliable estimates, but just only
for very few matches, and non for most of others. In the other hand, using
less strict attributes, system will produce less reliable estimations but for
a big scope. So it is essential to find a balance between these two opposite
situations: reliability of attribute set against number of matches without
information. A good solution could be to build and use different attribute
sets, ones more strict and others less. Thus, less strict attribute sets will
be used when strict ones fail doing an estimation.
Attribute | Configurable parameters
---|---
1) Number of non consecutive won matches in previous weeks $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
2) Number of non consecutive lost matches in previous weeks $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
3) Number of non consecutive draws in previous weeks $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
4) Points collected in previous matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
5) Position in the classification based on previous matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
6) Number of positions over the opponent in the classification based on previous matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
7) Number of positions under the opponent in the classification based on previous matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
8) Number of wins in previous directed matchs (included previos leagues) $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
9) Number of losts in previous directed matchs (included previos leagues) $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Number of Matchs$>$ $<$Matchs$>$
10) Number of drawns in previous directed matchs (included previos leagues) $>$ threshold | $<$threshold$>$ $<$Number of Matchs$>$ $<$Matchs$>$
11) Position in the classification $>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
12) Number of positions over the opponent in the classification$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
13) Number of positions under the opponent in the classification$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
14) Number of consecutive won matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
15) Number of consecutive lost matches$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
16) Number of consecutive draws$>$ threshold | $<$threshold$>$ $<$Team$>$ $<$Matchs$>$
17) Team’s budget Y times bigger than opponent’s budget (Y $>$ threshold) | $<$threshold$>$ $<$Team$>$
18) Team’s budget Y times smaller than opponent’s budget (Y $>$ threshold) | $<$threshold$>$ $<$Team$>$
Table 2: Attributes and parameters
## Trends towards the victory of the home team
It is a fact that, in soccer, it is more probable a victory from home team
than away team. To deal with this, we offer two different approaches. First,
modelling the teams behavior and second computing confidence values. For
modeling teams behavior (attribute set customization) it is a good practice to
use attributes with low exhaustive thresholds for home team and more exigent
threshold for attributes related to away team. Therefore, it will be easier
for home team to satisfy an attribute than away team. It is possible to
imitate this trend based on this approach.
Around 50% of played matches finish with victory of home team. This means that
the attribute value, corresponding to matches result, will be ’home team
victory’ around 50% of objects from formal context. As consequence of the
former, many rules from the inferred association rules will contain the
attribute ’result = home team victory’ within their conclusions. Thus when
forecasting a match the system will infer, in most of cases, ’home team
victory’ as consequence of overestimation confidence value for this result. It
is possible to avoid this effect easily, just applying a decreasing
(reduction) factor over confidence for ’home team victory’. It is estimated by
means of experiments.
Figure 5: KB fragment from Fig. 4
## Results
Following the process described above, an experiment was run for the Spanish
premier soccer league from 2009-10. Attributes were selected according the
experience of an expert, and contextual KB is computed (in Fig. 5 a KB
fragment for Málaga-Sevilla match is shown). From this selection
$\vdash_{\exists}$ is computed for each match in each week.
Figure 6: Correct predictions on the last 17 weeks of the season 2010-11
2009-10 season: Experiments with the system show forecasts of about 58.16% by
a contextual selection based on the previous 38 matches of each team. Such a
percentage of hits for a qualitative reasoning system may be considered as an
acceptable result comparable with expectable results of experts Goldstein &
Gigerenzer (2009); Andersson et al. (2003). Experiments with other contextual
selections shows an increase in the number of hits by about 7% in the second
half of the season. The reason is that data from the first half provides more
recent information on teams and past matches.
2010-11 season: According to the idea commented above, we have evaluated the
system in the second half of 2010-11 soccer season. A way to evaluate how good
is this forecasting sistem is comparing number of successes in our pool with
the most popular betting selections. This popular selections are collected
from the most voted results for each match, published at state agency web that
controls soccer pools. In Fig. 6 both results are compared. Our hits are in
blue and popular ones in green and last seventeen weeks from 2010-11 season
are represented. Note that Spanish soccer pools are over 15 matches.
## Conclusions and Future Work
Figure 7: Comparative of correct predictions on the whole season 2010-11.
Percentages
The challenge to detect emergent concepts for reasoning about complex systems
represents an exciting researching field. Concepts with qualitative nature are
extracted from data only considering partial features of complex system
dynamics, a partial understanding of system. In this paper, FCA is applied to
this aim with a specific application. The selection attribute problem based on
FCA-based reasoning system for sport forecasting is analysed. In fact, the
reasoning system is a computational logic model for bounded rationality. The
model is concerned with association rule reasoning and it does not use -in its
current form- more sophisticated probability tools (as for example Min et al.
(2008)). As is stated in Goldstein & Gigerenzer (1996), the theory of
probabilistic mental models assumes that inferences about unknown states of
the world are based on probability cues Brunswik (1955). It can say that
confidence of association rules extracted from subcontexts play the role of
probability cues.
Any statistical approaches have been taken into account, because of it was not
the aim of this paper. Although a comparative study of our system against C4.5
classifier has been done. For this, two different attribute selections have
been considered and used for both, C4.5 classifier and our system. The
experiment is to forecast all matches (380) in season 2010-11. In order to
stimate each match result, considering $N$ (weeks) as timestamp, previous
matches are used to build contextual selection (or trainining set in C4.5)
from weeks $N-1$ to $N-19$ (190 objects). Fig. 7 shows the percentage of
correct predictions for our system and C4.5 classifier, using both attribute
selections. Other cols are also shown: ’user’s most voted results’, local team
always win and two random generated. These ’random generated’ cols were built
assuming different weigths per result. It means, $<1:55\%,X:23\%,2:22\%>$ and
$<1:65\%,X:18\%,2:17\%>$ were used, where $1,X,2$ are the probabilities for
forecasting a match with the result: local team wins, drawn and away team
wins, respectively.
It is worth to note that, while classifier achieves highest performances
(58,68%) when number of matches increase from 190 to 380, our system reaches
this highest performance (59,74%) using only 190 instances. This conclusion is
based on our system use some fast and frugal Goldstein & Gigerenzer (1996)
methods, and these are designed to achieve aceptable results using as less as
possible resources.
The relationship of our proposal with Recognition Heuristics Goldstein &
Gigerenzer (2002) (roughly speaking, if one of the possibilities is recognized
and the other is not, then infer that the recognized object has the higher
value with respect to the criterion) is not clear. We may assert that our
model recognises trends in contexts. Trends (represented as association rules)
can be considered as a kind of recognizing method, though. The system is based
on bounded rationality models instead of statistic models, although in future
hybrid models will be considered.
In the short term, we carry on extending our system in order to be able to
combine the results of two or more attribute sets with different exigency
level. Therefore the system will return only one result and more reliable. In
the long term, we aim to extend the model in orfer to obtain a general system
to detect emergent concepts in Complex Systems
After some real betting experiments during current season (2010-2011) with one
customized attribute set, we have observed another intriguing fact. If we take
a look to number of successful predictions per week, we are able to
distinguish some groups of consecutive weeks in which number of correct
predictions is under or over the average. Recall that these predictions are
the logical inferred results by one customized attribute set. This suggests
that it could be possible to find another attribute set, with a different
parameters customization, which it will accomplish the correct predictions of
first attribute set. It means that when first attribute set produce bad
forecasting, second should produce good ones, and vice versa. The reason of
this is that each match there is not only one possible logical result. It
means, when one of firsts teams of current ranking plays against one of lasts
team, attending to ranking criteria, the logical result of this match would be
that first one wins. But if we attend to others, like first team lost last
week and second team won last 5 weeks, this results would be different. Future
works pass through for finding these complementary attribute sets and
detecting when their behaviors change during season in order to select the
proper attribute set to forecast each week.
Finally, we are also analyzing how to finde a weight for matches which allows
the system to work with matches from different divisions, simultaneously. Note
that a winning match at first division will have a higher weight than a
winning at second. This will be really useful at the beginning of season
because of we need to compute attributes related to previous matches results
and teams which are involved played at different divisions last season.
## Acknowledgements
Supported by TIN2009-09492 project of Spanish Ministry of Science and
Innovation, and Excellence project TIC-6064 of Junta de Andalucía cofinanced
with FEDER founds.
## References
* Inst. Engineering and Technology (2010) Why Spain will win…, Engineering & Technology 5 June - 18 June 2010.
* Alonso et al. (2008) J. A. Alonso-Jiménez, G. A. Aranda-Corral, J. Borrego-Díaz, and M. M. Fernández-Lebrón, M. J. Hidalgo-Doblado, Extending Attribute Exploration by Means of Boolean Derivatives, Proc. 6th Int. Conf. Concept Lattices and Their Applications (CLA2008), pp. 121-132 (2008).
* Andersson et al. (2003) P. Andersson, M. Ekman, J. Edman, Forecasting the fast and frugal way: A study of performance and information-processing strategies of experts and non-experts when predicting the World Cup 2002 in soccer, Working Paper Series in Business Administration 2003:9, Stockholm School of Economics.
* Aranda-Corral & Borrego-Díaz (2010) G. A. Aranda-Corral, J. Borrego-Díaz, Reconciling Knowledge in social tagging web services. Proc. 5th Int. Conf. Hybrid AI Systems (HAIS 2010), LNAI, vol. 6077. Springer-Verlag, Berlin, 383-390 (2010).
* Aranda-Corral et al. (2011) G. A. Aranda-Corral, J. Borrego-Díaz, J. Galán-Páez, Confidence-Based Reasoning with Local Temporal Formal Contexts. to appear in IWANN 2011, LNCS (2011).
* (6) G. A. Aranda-Corral, J. Borrego-Díaz, J. Galán-Páez, Bounded Rationality for Data Reasoning based on Formal Concept Analysis. To appear in DEXA Workshop DALI (2011).
* Armstrong (1974) W. Armstrong, Dependency structures of data base relationships. Proc. of IFIP Congress, Geneva, 580-583 (1974).
* Balcázar (2010) J.L. Balcázar, Redundancy, Deduction Schemes, and Minimum-Size Bases for Association Rules, Logical Methods in Computer Science 6(2):1-23 (2010).
* Brunswik (1955) E. Brunswik, Representative design and probabilistic theory in a functional psychology. Psychological Review, (62):193-217 (1955).
* Ganter & Wille (1999) B. Ganter and R. Wille. Formal Concept Analysis - Mathematical Foundations. Springer, 1999.
* Giarratano & Riley (2005) J. C. Giarratano, G.D. Riley, Expert Systems: Principles and Programming. Brooks/Cole Publishing Co ( 2005).
* Goldstein & Gigerenzer (1996) D. G. Goldstein, G. Gigerenzer, Reasoning The Fast and Frugal Way: Models of Bounded Rationality, Psychological Review 103(4): 650-669 (1996).
* Goldstein & Gigerenzer (2002) D. G. Goldstein, G. Gigerenzer, Models of ecological rationality: the recognition heuristic, Psychological review, 109(1): 75-90 (2002).
* Goldstein & Gigerenzer (2009) D.G. Goldstein, G. Gigerenzer, Fast and frugal forecasting. International Journal of Forecasting, 25, 760-772 (2009).
* Guigues & Duquenne (1986) Guigues, J.-L., Duquenne, V.: Familles minimales d’ implications informatives resultant d’un tableau de donnees binaires. Math. Sci. Humaines 95, 5–18 (1986).
* Imberman et al. (1999) S. P. Imberman, B. Domanski, R. A. Orchard: Using Booleanized Data To Discover Better Relationships Between Metrics. Int. CMG Conference 1999: 530-539
* Min et al. (2008) B. Min, J. Kim, C. Choe, H. Eom, R. I. McKay, A compound framework for sports results prediction: A football case study. Know.-Based Syst. 21(7):551-562. 2008
|
arxiv-papers
| 2011-07-27T13:52:20 |
2024-09-04T02:49:20.988849
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gonzalo A. Aranda-Corral, Joaqu\\'in Borrego-D\\'iaz and Juan\n Gal\\'an-P\\'aez",
"submitter": "Gonzalo A. Aranda-Corral",
"url": "https://arxiv.org/abs/1107.5474"
}
|
1107.5525
|
# The generating function of amplitudes with $N$ twisted and $M$ untwisted
states
Igor Pesando1
1Dipartimento di Fisica Teorica, Università di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
ipesando@to.infn.it
###### Abstract:
We show that the generating function of all amplitudes with $N$ twisted and
$M$ untwisted states, i.e. the Reggeon vertex for magnetized branes on
$\mathbb{R}^{2}$ can be computed once the correlator of $N$ non excited
twisted states and the corresponding Green function are known and we give an
explicit expression as a functional of the these objects.
D-branes, Conformal Field Theory
††preprint: DFTT-20-2011
## 1 Introduction and conclusions
In the late 80s a lot of work was done in computing the generating functions
of all amplitudes for the bosonic string and superstring. Many methods were
(further) developed such as the sewing method ([1]), the group theoretic
method ([2]) and conserved charges method ([3]). Following the main idea of
([4],[5]) in this paper we would like to compute the generating function for
$N$ generic excited twisted states and $M$ generic untwisted states on
$\mathbb{R}^{2}$ for the open string in presence of magnetic fields in the
upper half plane using the path integral approach. Much work has been already
done in computing non excited twisted states correlation functions, especially
on $T^{2}$ (see for example [6], [7], [8] and [9]) but not so much on the
computation of correlators involving excited twisted fields ([15], [16] see
for earlier work) which remain quite mysterious.
In this paper we want to show that there is a quite simple way of labeling
excited twisted states which is deeply connected with the operator-state map
and that few ingredients are actually needed for computing all correlators
involving excited twisted state and arbitrary untwisted ones on
$\mathbb{R}^{2}$. To obtain any correlator is only necessary the knowledge of
the full (i.e. classical and quantum) $N$ non excited twist correlator on the
disk111 The twist fields in this and the following correlators are actually
$\sigma_{\epsilon,\kappa=0}(x,\bar{x})$, see the in the main text.
$C(x_{1},\dots
x_{N})=\langle\sigma_{\epsilon_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N}}(x_{N},\bar{x}_{N})\rangle_{disk,~{}full}~{}~{}~{}~{}x_{t}\in\mathbb{R}$
(1)
and the boundary Green function in presence of such operators
$G^{ij}_{bou}(x;y;\\{x_{t}\\}_{t=1\dots
N})=G^{ji}_{bou}(y;x;\\{x_{t}\\}_{t=1\dots
N})=G^{ij}(x,\bar{x};y,\bar{y};\\{x_{t}\\}_{t=1\dots N})$ (2)
which can be derived from
$G^{ij}(z,\bar{z};w,\bar{w};\\{x_{t}\\}_{t=1\dots N})=\frac{\langle
X^{i}(z,\bar{z})X^{j}(w,\bar{w})\sigma_{\epsilon_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N}}(x_{N},\bar{x}_{N})\rangle_{disk}}{\langle\sigma_{\epsilon_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N}}(x_{N},\bar{x}_{N})\rangle_{disk}}.$
(3)
by setting $z=x,w=y\in\mathbb{R}$.
The main result of the paper is the generating function for the above
mentioned amplitudes given in eq.s (32) and (33). These two expressions have
exactly the same contain but the latter is written in a more usual way, i.e.
using auxiliary expansion variables while the former has an expression like
those used in the previous literature ([1]). Let us now explain the building
blocks of this last version of the main formula (32).
* •
To any (excited) twisted operator inserted at $x_{t}$ ($t=1\dots N$) in the
amplitude we associate an auxiliary Hilbert space ${\cal H}_{t}$. On ${\cal
H}_{t}$ act the quantum fields $X^{i}_{(t)}(z,\bar{z})$222 In the following
quantum fields have attached the label of the Hilbert space they act on, e.g.
$X^{i}_{(t)}(z,\bar{z})$ while classical fields in path integral have no
label, i.e. $X^{i}(z,\bar{z})$. ($i=1,2$ or $i=z,\bar{z}$)
$\displaystyle Z_{(t)}(z,\bar{z})$
$\displaystyle=X^{z}_{(t)}(z,\bar{z})=\frac{X^{1}_{(t)}+iX^{2}_{(t)}}{\sqrt{2}}=\frac{1}{2}\left(Z_{L(t)}(z)+Z_{R(t)}(\bar{z})\right),$
$\displaystyle\bar{Z}_{(t)}(z,\bar{z})$
$\displaystyle=X^{\bar{z}}_{(t)}(z,\bar{z})=\frac{X^{1}_{(t)}-iX^{2}_{(t)}}{\sqrt{2}}=\frac{1}{2}\left(\bar{Z}_{(t)L}(z)+\bar{Z}_{(t)R}(\bar{z})\right)$
(4)
which have expansions
$\displaystyle Z_{(t)L}(z)$ $\displaystyle=$ $\displaystyle
z_{(t)0}+i\sqrt{2\alpha^{\prime}}e^{-i\gamma_{t}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(t)n+1-\epsilon_{t}}}{\,{n+1-\epsilon_{t}}}z^{-(n+1-\epsilon_{t})}-\frac{\alpha_{(t)n+\epsilon_{t}}^{\dagger}}{\,{n+\epsilon_{t}}}z^{+(n+\epsilon_{t})}\right]$
$\displaystyle Z_{(t)R}(\bar{z})$ $\displaystyle=$ $\displaystyle
z_{(t)0}+i\sqrt{2\alpha^{\prime}}e^{+i\gamma_{t}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(t)n+1-\epsilon_{t}}}{\,{n+1-\epsilon_{t}}}{\bar{z}}^{-(n+1-\epsilon_{t})}-\frac{\alpha_{(t)n+\epsilon_{t}}^{\dagger}}{\,{n+\epsilon_{t}}}{\bar{z}}^{+(n+\epsilon_{t})}\right]$
(5)
and
$\displaystyle\bar{Z}_{(t)L}(z)$ $\displaystyle=$
$\displaystyle\bar{z}_{(t)0}+i\sqrt{2\alpha^{\prime}}e^{+i\gamma_{t}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n+1-\epsilon_{t}}^{\dagger}}{\,{n+1-\epsilon_{t}}}z^{+(n+1-\epsilon_{t})}+\frac{\alpha_{(t)n+\epsilon_{t}}}{\,{n+\epsilon_{t}}}z^{-(n+\epsilon_{t})}\right]$
$\displaystyle\bar{Z}_{(t)R}(\bar{z})$ $\displaystyle=$
$\displaystyle\bar{z}_{(t)0}+i\sqrt{2\alpha^{\prime}}e^{-i\gamma_{t}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n+1-\epsilon_{t}}^{\dagger}}{\,{n+1-\epsilon_{t}}}{\bar{z}}^{+(n+1-\epsilon_{t})}+\frac{\alpha_{(t)n+\epsilon_{t}}}{\,{n+\epsilon_{t}}}{\bar{z}}^{-(n+\epsilon_{t})}\right]$
(6)
The previous fields satisfy the boundary conditions333 These can also be
written as $e^{i\gamma_{t}}\partial
Z_{(t)L}(x)=\frac{1}{\cos\gamma_{t}}\partial Z_{(t)}(x,x)$ when $x>0$ and
$e^{i\gamma_{t-1}}\partial Z_{(t)L}(y)=\frac{1}{\cos\gamma_{t-1}}\partial
Z_{(t)}(y,{\bar{y}})$ when $y<0$. These expressions are those used to connect
the open string operators when naturally expressed as function of
$X(x,{\bar{x}})$ to their expressions as functional of $X_{L}(x)$.
$\displaystyle e^{i\gamma_{t}}\partial Z_{(t)L}|_{x}$ $\displaystyle=$
$\displaystyle
e^{-i\gamma_{t}}\bar{\partial}Z_{(t)R}|_{x}~{}~{}x\in\mathbb{R}^{+}$ (7)
$\displaystyle e^{+i\gamma_{t-1}}\partial Z_{(t)L}|_{y}$ $\displaystyle=$
$\displaystyle
e^{-i\gamma_{t-1}}\bar{\partial}Z_{(t)R}|_{y}~{}~{}y=|y|e^{i\pi}\in\mathbb{R}^{-}$
(8)
where we have defined the phases ($-\frac{\pi}{2}<\gamma_{t}<\frac{\pi}{2}$)
$\displaystyle e^{i\gamma_{t}}=\frac{1+iB_{t}}{\sqrt{1+B_{t}^{2}}}$
$\displaystyle\rightarrow$ $\displaystyle
B_{t}=\tan\gamma_{t}=2\pi\alpha^{\prime}~{}q_{(0)}F_{12(0)}$ $\displaystyle
e^{i\gamma_{t-1}}=\frac{1+iB_{t-1}}{\sqrt{1+B_{t-1}^{2}}}$
$\displaystyle\rightarrow$ $\displaystyle
B_{t-1}=\tan\gamma_{t-1}=2\pi\alpha^{\prime}~{}q_{(\pi)}F_{12(\pi)}$ (9)
where $B_{t-1}=2\pi\alpha^{\prime}~{}q_{(\pi)}F_{12(\pi)}$ and
$B_{t}=2\pi\alpha^{\prime}~{}q_{(0)}F_{12(0)}$ are the adimensional magnetic
fields which are on the $x<0$ ($\sigma=\pi$) and $x>0$ ($\sigma=0$)
boundaries. In the field expansion the shift $\epsilon_{t}$ is given by
$\epsilon_{t}=\left\\{\begin{array}[]{l
c}\frac{1}{\pi}\left(\gamma_{t}-\gamma_{t-1}\right)&\gamma_{t}>\gamma_{t-1}\\\
1+\frac{1}{\pi}\left(\gamma_{t}-\gamma_{t-1}\right)&\gamma_{t}<\gamma_{t-1}\end{array}\right.~{}~{}~{}~{}0\leq\epsilon_{t}<1$
(10)
The previous operators act on the ${\cal H}_{t}$ twisted ground state defined
by
$\bar{\alpha}_{(t)n+1-\epsilon_{t}}|T_{t}\rangle=\alpha_{(t)m+\epsilon_{t}}|T_{t}\rangle=x^{2}_{(t)0}|T_{t}\rangle=0$
(11)
and have the non vanishing commutation relations 444Since the annihilator and
creator operators have flat indexes this holds independently of our choice of
the taking the metric diagonal; in particular from definition of the complex
fields we have $(dX^{1})^{2}+(dX^{2})^{2}=2dZd\bar{Z}$, i.e. $G_{z\bar{z}}=1$.
$\displaystyle[\bar{\alpha}_{(t)n+1-\epsilon_{t}},\bar{\alpha}_{(t)m+1-\epsilon_{t}}^{\dagger}]$
$\displaystyle=$ $\displaystyle(n+1-\epsilon_{t})\delta_{n,m}~{}~{}n,m\geq 0$
$\displaystyle{}[\alpha_{(t)n+\epsilon_{t}},\alpha_{(t)m+\epsilon_{t}}^{\dagger}]$
$\displaystyle=$ $\displaystyle(n+\epsilon_{t})\delta_{n,m}~{}~{}n,m\geq 0$
$\displaystyle{}[z_{(t)0},\bar{z}_{(t)0}]$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$ (12)
Notice that the choice of the definition of the zero modes vacuum is somewhat
arbitrary since they do not change the energy, our choice is dictated by our
gauge choice for the background magnetic field $A=B~{}x^{1}~{}dx^{2}$ which
implies the translational invariance $X^{2}\rightarrow X^{2}+\epsilon$ and by
the observation that is almost the proper choice in toroidal
compactifications. The existence of the zero modes imply that the vacuum is
degenerate since all the states
$|T_{t},\kappa_{t}\rangle=e^{i\kappa_{t}x^{1}_{(t)0}}|T_{t}\rangle$ have
exactly the same energy of the vacuum and therefore there exists a one
parameter family of twist fields ([12])
$\sigma_{\epsilon_{t},\kappa_{t}}(x,{\bar{x}})$.
Given the previous vacuum definition we have the following twisted Green
functions
$\displaystyle G_{T(t)}^{zz}(z,\bar{z};w,\bar{w})$
$\displaystyle=[Z^{(+)}(z,\bar{z}),Z^{(-)}(w,\bar{w})]|_{an.cont}=\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$
$\displaystyle G_{T(t)}^{\bar{z}\bar{z}}(z,\bar{z};w,\bar{w})$
$\displaystyle=[\bar{Z}^{(+)}(z,\bar{z}),\bar{Z}^{(-)}(w,\bar{w})]|_{an.cont}=-\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$
$\displaystyle G_{T(t)}^{z\bar{z}}(z,\bar{z};w,\bar{w})$
$\displaystyle=[Z^{(+)}(z,\bar{z}),\bar{Z}^{(-)}(w,\bar{w})]|_{an.cont}$
$\displaystyle=\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$
$\displaystyle~{}~{}-\frac{\alpha^{\prime}}{2}\left[\,g_{\epsilon_{t}}\left(\frac{w}{z}\right)+\,g_{\epsilon_{t}}\left(\frac{\bar{w}}{\bar{z}}\right)+e^{-2i\gamma_{t}}\,g_{\epsilon_{t}}\left(\frac{\bar{w}}{z}\right)+e^{2i\gamma_{t}}\,g_{\epsilon_{t}}\left(\frac{w}{\bar{z}}\right)\right]$
$\displaystyle G_{T(t)}^{\bar{z}z}(z,\bar{z};w,\bar{w})$
$\displaystyle=[\bar{Z}^{(+)}(z,\bar{z}),Z^{(-)}(w,\bar{w})]|_{an.cont}$
$\displaystyle=-\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$
$\displaystyle~{}~{}-\frac{\alpha^{\prime}}{2}\left[\,g_{1-\epsilon_{t}}\left(\frac{w}{z}\right)+\,g_{1-\epsilon_{t}}\left(\frac{\bar{w}}{\bar{z}}\right)+e^{2i\gamma_{t}}\,g_{1-\epsilon_{t}}\left(\frac{\bar{w}}{z}\right)+e^{-2i\gamma_{t}}\,g_{1-\epsilon_{t}}\left(\frac{w}{\bar{z}}\right)\right]$
(13)
which can be obtained by analytically continuing their operatorial expression
from $|z|>|w|$ to the whole upper plane in such a way to preserve the symmetry
$G^{ij}(z,\bar{z};w,\bar{w})=G^{ij}(w,\bar{w};z,\bar{z})$. In the previous
expressions we have defined $\,g_{\nu}(z)$ as the analytic continuation of
$\,g_{\nu,s}(z)=-\sum_{n-\nu>0}\frac{1}{n-\nu}z^{n-\nu}~{}~{}~{}|z|<1,~{}~{}~{}-\pi+2\pi
s<\phi=arg(z)\leq\pi+2\pi s.$ (14)
in the properly chosen sheet $s$. Notice that the symmetry of the Green
function $G^{ij}(z,\bar{z};w,\bar{w})=G^{ji}(w,\bar{w};z,\bar{z})$ is not
obvious in the zero modes sector, i.e. for the constant terms but it holds due
to the $\,g$ transformation property ([12])
$\displaystyle\,g_{\nu,s}(z)=\,C_{\nu,s}(\phi)+\,g_{1-\nu,-s}\left(\frac{1}{z}\right),~{}~{}~{}~{}\,C_{\nu,s}(\phi)=\left\\{\begin{array}[]{cc}\frac{\pi
e^{-i\pi\nu}}{sin\pi\nu}e^{-i2\pi\nu s}&2\pi s<\phi<\pi+2\pi s\\\ \frac{\pi
e^{+i\pi\nu}}{sin\pi\nu}e^{-i2\pi\nu s}&-\pi+2\pi s<\phi<2\pi
s\end{array}\right.$ (17) (18)
This fact implies that we cannot really completely separate the zero modes and
non zero modes also for the twisted sector as it already happens for the
untwisted one.
For $x>0>y$ and $|y/x|<1$ the previous Green functions become on the
boundary555 When $|y/x|>1$ we must be more careful since we want to evaluate
the $\,g$ on a cut; for example when $0<x,y$ the expression which is valid for
all ranges is
$G_{T(t)~{}bou}^{z\bar{z}}(x;y)=\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}-\alpha^{\prime}~{}\cos\gamma_{t}[e^{-i\gamma_{t}}\,g_{\epsilon_{t}}\left(\frac{y}{x}e^{-i0}\right)+e^{i\gamma_{t}}\,g_{\epsilon_{t}}\left(\frac{y}{x}e^{+i0}\right)].$
In any case we can always use the symmetry property for the Green functions to
reduce the computation in the range where we can apply the given expressions.
$\displaystyle G_{T(t)~{}bou}^{zz}(x;y)$
$\displaystyle=\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$ $\displaystyle
G_{T(t)~{}bou}^{\bar{z}\bar{z}}(x;y)$
$\displaystyle=-\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}$ $\displaystyle
G_{T(t)~{}bou}^{z\bar{z}}(x;y)$
$\displaystyle=\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}-2\alpha^{\prime}~{}\cos\gamma_{t}~{}\cos\gamma_{t-1}~{}e^{i\gamma_{t}-i\gamma_{t-1}}~{}\,g_{\epsilon_{t}}\left(\frac{y}{x}\right)$
$\displaystyle G_{T(t)~{}bou}^{\bar{z}z}(x;y)$
$\displaystyle=-\frac{\pi\alpha^{\prime}}{B_{t}-B_{t-1}}-2\alpha^{\prime}~{}\cos\gamma_{t}~{}\cos\gamma_{t-1}~{}e^{-i\gamma_{t}+i\gamma_{t-1}}~{}\,g_{1-\epsilon_{t}}\left(\frac{y}{x}\right)$
(19)
The other cases can be obtained with the substitution rule
$x>0~{}~{}\cos\gamma_{t}~{}e^{i\gamma_{t}}\leftrightarrow
x<0~{}~{}\cos\gamma_{t-1}~{}e^{-i\gamma_{t-1}}$ and the same for $y$ in the
$G^{z\bar{z}}$ propagator. For the $G^{\bar{z}z}$ propagator one takes the
complex conjugate of the previous substitution rule.
* •
In a similar way to any untwisted operator we insert in the amplitude we
associate an auxiliary Hilbert space ${\cal H}_{a,t_{a}}$. This Hilbert space
as well as the position where the untwisted vertex is inserted $x_{a,t_{a}}$
are better labeled by both a counting label $a=1\dots M$ and a further label
$t_{a}\in\\{1,\dots N\\}$ which specify which is the magnetic field felt by
the untwisted state (dipole string). This could seem irrelevant but it is
important in defining the regularized Green functions (30) and in computing
the non commutative phases. In the following we will use a lighter notation as
$x_{a,t_{a}}\rightarrow x_{a}$ when there is not possibility of confusion.
On the auxiliary Hilbert space ${\cal H}_{a,t_{a}}$ act the quantum fields
$Z_{(a,t_{a})}(z,\bar{z})=\frac{1}{\sqrt{2}}\left(X^{1}_{(a,t_{a})}(z,\bar{z})+iX^{2}_{(a,t_{a})}(z,\bar{z})\right)=\frac{1}{2}(Z_{(a,t_{a})L}(z)+Z_{(a,t_{a})R}(\bar{z}))$
(20)
which have expansions
$\displaystyle Z_{(a,t_{a})L}$ $\displaystyle=$ $\displaystyle
e^{-i\gamma_{t_{a}}}\Bigg{(}z_{(a,t_{a})0}-2\alpha^{\prime}{\bar{p}}_{(a,t_{a})}~{}i\ln(z)+i\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}+\frac{\bar{\alpha}_{(a,t_{a})n}}{\,{n}}z^{-n}-\frac{\alpha_{(a,t_{a})n}^{\dagger}}{\,{n}}z^{n}\Bigg{)}$
$\displaystyle Z_{(a,t_{a})R}$ $\displaystyle=$ $\displaystyle
e^{+i\gamma_{t_{a}}}\left(z_{(a,t_{a})0}-2\alpha^{\prime}{\bar{p}}_{(a,t_{a})}~{}i\ln(\bar{z})+i\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}+\frac{\bar{\alpha}_{(a,t_{a})n}}{\,{n}}\bar{z}^{-n}-\frac{\alpha_{(a,t_{a})n}^{\dagger}}{\,{n}}\bar{z}^{n}\right)$
and
$\displaystyle\bar{Z}_{(a,t_{a})L}$ $\displaystyle=$ $\displaystyle
e^{+i\gamma_{t_{a}}}\left(\bar{z}_{(a,t_{a})0}-2\alpha^{\prime}{p}_{(a,t_{a})}~{}i\ln(z)+i\sqrt{2\alpha^{\prime}}\sum_{n\neq
0}-\frac{\bar{\alpha}_{(a,t_{a})n}^{\dagger}}{\,{n}}z^{n}+\frac{\alpha_{(a,t_{a})n}}{\,{n}}z^{-n}\right)$
$\displaystyle\bar{Z}_{(a,t_{a})R}$ $\displaystyle=$ $\displaystyle
e^{-i\gamma_{t_{a}}}\left(\bar{z}_{(a,t_{a})0}-2\alpha^{\prime}{p}_{(a,t_{a})}~{}i\ln(\bar{z})+i\sqrt{2\alpha^{\prime}}\sum_{n\neq
0}-\frac{\bar{\alpha}_{(a,t_{a})n}^{\dagger}}{\,{n}}\bar{z}^{n}+\frac{\alpha_{(a,t_{a})n}}{\,{n}}\bar{z}^{-n}\right)$
The previous quantum fields satisfy the boundary conditions
$\displaystyle e^{+i\gamma_{t_{a}}}\partial Z_{(a,t_{a})}|_{x}$
$\displaystyle=$ $\displaystyle
e^{-i\gamma_{t_{a}}}\bar{\partial}Z_{(a,t_{a})}|_{x}~{}~{}x\in\mathbb{R}^{+}$
$\displaystyle e^{+i\gamma_{t_{a}}}\partial Z_{(a,t_{a})}|_{y}$
$\displaystyle=$ $\displaystyle
e^{-i\gamma_{t_{a}}}\bar{\partial}Z_{(a,t_{a})}|_{y}~{}~{}y=|y|e^{i\pi}\in\mathbb{R}^{-}$
where we have defined the angle $\gamma_{t_{a}}$, in a similar way for the
twisted scalar (dicharged string), as
$e^{i\gamma_{t_{a}}}=\frac{1+iB_{t_{a}}}{\sqrt{1+B_{t_{a}}}}\Rightarrow
B_{t_{a}}=\tan\gamma_{t_{a}},~{}~{}~{}~{}-\frac{\pi}{2}<\gamma_{t_{a}}<\frac{\pi}{2}$
(23)
The creation and destruction operators act on the dipole ground state defined
by
$\bar{\alpha}_{(a,t_{a})n}|0_{(a,t_{a})}\rangle=\alpha_{(a,t_{a})n}|0_{(a,t_{a})}\rangle={\bar{p}}_{(a,t_{a})}|0_{(a,t_{a})}\rangle={p}_{(a,t_{a})}|0_{(a,t_{a})}\rangle=0$
(24)
and have non trivial commutation relations
$\displaystyle[z_{(a,t_{a})0},\bar{z}_{(a,t_{a})0}]$ $\displaystyle=$
$\displaystyle 2\pi\alpha^{\prime}B_{t_{a}}$
$\displaystyle{}[z_{(a,t_{a})0},{p}_{(a,t_{a})}]$ $\displaystyle=$
$\displaystyle i$
$\displaystyle{}[\alpha_{(a,t_{a})n},{\alpha}^{\dagger}_{(a,t_{a})m}]$
$\displaystyle=$ $\displaystyle n\delta_{m,n}$
$\displaystyle{}[\bar{\alpha}_{(a,t_{a})n},\bar{\alpha}^{\dagger}_{(a,t_{a})m}]$
$\displaystyle=$ $\displaystyle n\delta_{m,n}$ (25)
The normal ordering is the usual one but it worth noticing that in the zero
modes sector is defined as
$:e^{i(\bar{k}Z_{(a)zm}+k\bar{Z}_{(a)zm})(x,{\bar{x}})}:=\left\\{\begin{array}[]{c
r}e^{i\cos\gamma_{t}(\bar{k}z_{(a)0}+k\bar{z}_{(a)0})}~{}e^{2\alpha^{\prime}\ln(|x|)\cos\gamma_{t}(\bar{k}{\bar{p}}_{(a)0}+k{p}_{(a)0})}&x>0\\\
e^{i\cos\gamma_{t}(\bar{k}\hat{z}_{(a)0}+k\hat{\bar{z}}_{(a)0})}~{}e^{2\alpha^{\prime}\ln(|x|)\cos\gamma_{t}(\bar{k}{\bar{p}}_{(a)0}+k{p}_{(a)0})}&x<0\end{array}\right..$
(26)
with $\hat{z}_{(a)0}=z_{(a)0}-i2\pi\alpha^{\prime}\tan\gamma_{t}~{}{\bar{p}}$
which have the property that their commutation relations are the opposite of
the $z_{(a)0}$ ones. Finally the untwisted Green functions in a magnetic
background $B_{t_{a}}$ are given by ($0<arg(z-\bar{w})<\pi$)
$\displaystyle G_{U(t_{a})}^{zz}(z,\bar{z},w,\bar{w})$
$\displaystyle=G_{U(t_{a})}^{\bar{z}\bar{z}}(z,\bar{z},w,\bar{w})=0$
$\displaystyle G_{U(t_{a})}^{z\bar{z}}(z,\bar{z},w,\bar{w})$
$\displaystyle=[Z^{(+)}(z,\bar{z}),\bar{Z}^{(-)}(w,\bar{w})]|_{an.cont}$
$\displaystyle=+\frac{1}{2}\pi\alpha^{\prime}\sin(2\gamma_{t_{a}})-\alpha^{\prime}\left[\ln|z-w|+\cos(2\gamma_{t_{a}})\ln|z-\bar{w}|+\sin(2\gamma_{t_{a}})arg(z-\bar{w})\right]$
$\displaystyle G_{U(t_{a})}^{\bar{z}z}(z,\bar{z},w,\bar{w})$
$\displaystyle=[\bar{Z}^{(+)}(z,\bar{z}),Z^{(-)}(w,\bar{w})]|_{an.cont}$
$\displaystyle=-\frac{1}{2}\pi\alpha^{\prime}\sin(2\gamma_{t_{a}})-\alpha^{\prime}\left[\ln|z-w|+\cos(2\gamma_{t_{a}})\ln|z-\bar{w}|-\sin(2\gamma_{t_{a}})arg(z-\bar{w})\right]$
(27)
The constant terms can be obtained by rewriting
$z_{(a)0}=z_{(a)00}+i\pi\alpha^{\prime}\tan\gamma_{t}~{}{\bar{p}}$ (28)
so that $[z_{(a)00},\bar{z}_{(a)00}]=0$ an considering the additional term
proportional to ${\bar{p}}$ coming from this rewriting as belonging to
$Z^{(+)}(z,\bar{z})$. Notice however once again that the constant terms are
needed to ensure the symmetry
$G^{ij}(z,\bar{z};w,\bar{w})=G^{ji}(w,\bar{w};z,\bar{z})$.
The previous Green functions become on the boundary $z=x,w=y\in\mathbb{R}$ 666
When using these Green functions in eq. (3) in absence of twist fields we
recover the results from the operatorial formalism.
$\displaystyle G_{U(t_{a}),~{}bou}^{z\bar{z}}(x;y)$
$\displaystyle=\frac{1}{2}\pi\alpha^{\prime}\sin(2\gamma_{t_{a}})-2\alpha^{\prime}\left[\cos^{2}(\gamma_{t_{a}})\ln|x-y|+\frac{1}{2}\sin(2\gamma_{t_{a}})arg(x-\bar{y})\right]$
$\displaystyle G_{U(t_{a}),~{}bou}^{\bar{z}z}(x;y)$
$\displaystyle=-\frac{1}{2}\pi\alpha^{\prime}\sin(2\gamma_{t_{a}})-2\alpha^{\prime}\left[\cos^{2}(\gamma_{t_{a}})\ln|x-\bar{y}|-\frac{1}{2}\sin(2\gamma_{t_{a}})arg(x-\bar{y})\right]$
(29)
From these expressions we can read the open string metric ${\cal
G}^{z\bar{z}}_{(t)}=\cos^{2}(\gamma_{t})$ and the non commutativity parameter
$\Theta^{z\bar{z}}_{(t)}=\frac{1}{2}\sin(2\gamma_{t})$, we can also read the
$\mathbb{R}^{2}$ vielbein ${\cal V}^{\underline{z}\bar{z}}_{(t)}={\cal
V}^{\underline{\bar{z}}z}_{(t)}=\frac{1}{\cos(\gamma_{t})}$ where
$\underline{z},\underline{\bar{z}}$ are the flat indexes which are also
implicit in the creation and destruction operators.
* •
We define the boundary Green function regularized by the untwisted Green
function for a background $B_{t_{a}}$ as
$\displaystyle G_{bou,~{}reg~{}U(t_{a})}^{ij}(x;y;\\{x_{v}\\})=$
$\displaystyle
G^{ij}_{bou}(x;y;\\{x_{v}\\})-G_{U(t_{a}),~{}bou}^{ij}(x;y)~{}~{}~{}~{}x,y\in\mathbb{R}$
(30)
where $G_{U(t_{a}),~{}bou}^{ij}(x,y)$ are defined in eq.s (• ‣ 1). The choice
of the background $B_{t}$ in the regularization would seem arbitrary but it is
not since these regularized Green functions (and their derivatives) enter only
where an untwisted dipole state is emitted and this is on a well defined
interval of the boundary.
We also define the analogous twisted boundary Green function regularized by
the twisted Green function at the twist insertion point $t$ as777The
symmetrization is because we a symmetric function in $x\leftrightarrow y$,
i.e. independent on the way we take the limit $x>y$ or $y>x$.
$\displaystyle G^{ij}_{bou,~{}reg~{}T(t)}(x,y;\\{x_{v}\\})$
$\displaystyle=G^{ij}_{bou}(x,y;\\{x_{v}\\})-G^{ij}_{T(t)~{}bou}(x,y;\\{x_{0}=x_{t},x_{\infty}=\infty\\})$
(31)
where $G^{ij}_{t~{}bou}$ are given in eq.s (• ‣ 1) with the substitution
$\frac{y}{x}\rightarrow\frac{y-x_{t}}{x-x_{t}}$.
Given the previous building blocks the main formula is given by ($x_{t}\neq
x_{a}~{}~{}~{}\forall t,a$)
$\displaystyle\langle V_{N+M}(\\{x_{t}\\}_{t=1,\dots
N};\\{x_{a,t_{a}}\\}_{a=1,\dots M})|=C(x_{1},\dots x_{N})$
$\displaystyle\prod_{a=1}^{M}\langle
0_{(a)a},z_{(a)00}=\bar{z}_{(a)00}=0|\prod_{t=1}^{N}\langle
T_{(t)},x^{1}_{(t)}=0|\delta(i\sum_{a}(\alpha_{(a)0}-\bar{\alpha}_{(a)0})+i\sum_{t}(z_{(t)0}-\bar{z}_{(t)0}))$
$\displaystyle\prod_{a}\exp\Big{\\{}-\frac{1}{4\alpha^{\prime}}\alpha_{(a)0}^{2}~{}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})-\frac{1}{4\alpha^{\prime}}\bar{\alpha}_{(a)0}^{2}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}^{2}~{}G^{zz}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a}\exp\Big{\\{}}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\alpha_{(a)n}~{}\bar{\alpha}_{(a)m}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=y=x_{a}}$
$\displaystyle\prod_{t}\exp\Big{\\{}\frac{1}{2}\left(\frac{\tan\gamma_{t}-\tan\gamma_{t-1}}{2\pi\alpha^{\prime}}{x_{(t)0~{}i=2}}{}\right)^{2}~{}G^{22}_{bou,~{}reg~{}T(t)}(x;y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt-\frac{1}{2\alpha^{\prime}}\sum_{n,m=1}^{\infty}\frac{\bar{\alpha}_{(t)n}}{n-\epsilon_{t}}\frac{\alpha_{(t)m}}{m-1+\epsilon_{t}}~{}{\cal
V}_{(t)~{}\underline{z}\bar{z}}{\cal V}_{(u)~{}\underline{\bar{z}}z}$
$\displaystyle\hskip
70.0001pt~{}\frac{\partial^{n-1}_{x}}{(n-1)!}~{}\frac{\partial^{m-1}_{y}}{(m-1)!}\Big{[}(x-x_{t})^{1-\epsilon_{t}}(y-x_{t})^{\epsilon_{t}}~{}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou,~{}reg~{}T(t)}(x,y;\\{x_{v}\\})\Big{]}\Big{\\}}\Big{|}_{x=y=x_{t}}$
$\displaystyle\prod_{a<b}\exp\Big{\\{}-\frac{1}{2\alpha^{\prime}}\alpha_{(a)0}~{}\alpha_{(b)0}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{b})~{}\underline{\bar{z}}z}~{}G^{zz}_{bou}(x;y;\\{x_{v}\\})-\frac{1}{2\alpha^{\prime}}\bar{\alpha}_{(a)0}~{}\bar{\alpha}_{(b)0}~{}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}{\cal
V}_{(t_{b})~{}\underline{z}\bar{z}}~{}G^{\bar{z}\bar{z}}_{bou}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\alpha_{(a)n}~{}\bar{\alpha}_{(b)m}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{b})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\bar{\alpha}_{(a)n}~{}\alpha_{(b)m}~{}{\cal
V}_{(t_{b})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{\bar{z}z}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=x_{a},y=x_{b}}$
$\displaystyle\prod_{t<u}\exp\Big{\\{}\left(\frac{\tan\gamma_{t}-\tan\gamma_{t-1}}{2\pi\alpha^{\prime}}{x_{(t)0~{}i=2}}{}\right)\left(\frac{\tan\gamma_{u}-\tan\gamma_{u-1}}{2\pi\alpha^{\prime}}{x_{(u)0~{}i=2}}{}\right)~{}G^{22}_{bou}(x;y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt-\frac{1}{2\alpha^{\prime}}\sum_{n,m=1}^{\infty}\frac{\bar{\alpha}_{(t)n}}{n-\epsilon_{t}}\frac{\alpha_{(u)m}}{m-1+\epsilon_{u}}~{}{\cal
V}_{(t)~{}\underline{z}\bar{z}}{\cal V}_{(u)~{}\underline{\bar{z}}z}$
$\displaystyle\hskip
70.0001pt~{}\frac{\partial^{n-1}_{x}}{(n-1)!}~{}\frac{\partial^{m-1}_{y}}{(m-1)!}\left[(x-x_{t})^{\epsilon_{t}}(y-x_{u})^{1-\epsilon_{u}}\partial_{x}\partial_{y}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]$
$\displaystyle\hskip
30.00005pt-\frac{1}{2\alpha^{\prime}}\sum_{n,m=1}^{\infty}\frac{\alpha_{(t)n}}{n-1+\epsilon_{t}}\frac{\bar{\alpha}_{(u)m}}{m-\epsilon_{u}}~{}{\cal
V}_{(t)~{}\underline{\bar{z}}z}{\cal V}_{(u)~{}\underline{z}\bar{z}}$
$\displaystyle\hskip
70.0001pt~{}\frac{\partial^{n-1}_{x}}{(n-1)!}~{}\frac{\partial^{m-1}_{y}}{(m-1)!}\left[(x-x_{t})^{1-\epsilon_{t}}(y-x_{u})^{\epsilon_{u}}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou}(x,y;\\{x_{v}\\})\right]\Big{\\}}\Big{|}_{x=x_{t},y=x_{v}}$
$\displaystyle\prod_{t,a}\exp\Big{\\{}-\frac{1}{2\alpha^{\prime}}\left(\frac{\tan\gamma_{t}-\tan\gamma_{t-1}}{\pi}\frac{x_{(t)0~{}i=2}}{\sqrt{2\alpha^{\prime}}}\right)\sum_{m=0}^{\infty}~{}\alpha_{(a)m}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}~{}\frac{\partial^{m}_{y}}{m!}G^{2z}_{bou}(x,y;\\{x_{v}\\})$
$\displaystyle-\frac{1}{2\alpha^{\prime}}\left(\frac{\tan\gamma_{t}-\tan\gamma_{t-1}}{\pi}\frac{x_{(t)0~{}i=2}}{\sqrt{2\alpha^{\prime}}}\right)\sum_{m=0}^{\infty}~{}\bar{\alpha}_{(a)m}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{m}_{y}}{m!}G^{2\bar{z}}_{bou}(x,y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt-\frac{1}{2\alpha^{\prime}}\sum_{n=1,m=0}^{\infty}\frac{\bar{\alpha}_{(t)n}}{n-\epsilon_{t}}~{}\alpha_{(a)m}~{}{\cal
V}_{(t)~{}\underline{z}\bar{z}}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}~{}\frac{\partial^{n-1}_{x}}{(n-1)!}~{}\frac{\partial^{m}_{y}}{m!}\left[(x-x_{t})^{\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]$
$\displaystyle\hskip
30.00005pt-\frac{1}{2\alpha^{\prime}}\sum_{n=1,m=0}^{\infty}\frac{\alpha_{(t)n}}{n-1+\epsilon_{t}}~{}\bar{\alpha}_{(a)m}~{}{\cal
V}_{(t)~{}\underline{\bar{z}}z}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n-1}_{x}}{(n-1)!}~{}\frac{\partial^{m}_{y}}{m!}\left[(x-x_{t})^{1-\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]\Big{\\}}\Big{|}_{x=x_{t},y=x_{a}}$
(32)
where the operator indexes are raised an lowered using the flat metric while
Green function indexes are raised an lowered using $\mathbb{R}^{2}$ metric.
The previous expression can also be written without using the auxiliary
operators as a more conventional generating function. In order to do so we
introduce the auxiliary parameters $d_{(t)n}$, $\bar{d}_{(t)n}$ and $c_{(a)n}$
and $\bar{c}_{(a)n}$ which roughly correspond to $\alpha_{(t)n+1-\epsilon}$,
$\bar{\alpha}_{(t)n-\epsilon}$ and $\alpha_{(a)n}$, $\bar{\alpha}_{(a)n}$ (see
eq.s (4) and (3) for a precise mapping) of the previous expression. Then we
can write the generating function as
$\displaystyle{\cal V}_{N+M}(\\{d_{(t)}\\}_{t=1,\dots
N};\\{c_{(a)}\\}_{a=1,\dots M};\\{x_{t}\\}_{t=1,\dots
N};\\{x_{a,t_{a}}\\}_{a=1,\dots M})=$ $\displaystyle=$
$\displaystyle\delta\left(Re\left(\sum_{t}d_{(t)0}+\sum
c_{(a)0})\right)\right)~{}C(x_{1},\dots x_{N})$
$\displaystyle\prod_{a}\exp\Big{\\{}\frac{1}{2}c_{(a)0}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})+\frac{1}{2}\bar{c}_{(a)0}^{2}~{}G^{zz}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a}\exp\Big{\\{}}\sum_{n,m=0}^{\infty}c_{(a)n}~{}\bar{c}_{(a)m}~{}\partial^{n}_{x}\partial^{m}_{y}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=y=x_{a}}$
$\displaystyle\prod_{t}\exp\Big{\\{}\frac{1}{2}d_{(t)0}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}T(t)}(x;y;\\{x_{v}\\})+\frac{1}{2}\bar{d}_{(t)0}^{2}~{}G^{zz}_{bou,~{}reg~{}T(t)}(x;y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt+d_{(t)0}~{}\bar{d}_{(t)0}~{}G^{z\bar{z}}_{bou,~{}reg~{}T(t)}(x;y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt+\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}~{}d_{(t)m}~{}\partial^{n-1}_{x}\partial^{m-1}_{y}\Big{[}(x-x_{t})^{1-\epsilon_{t}}(y-x_{t})^{\epsilon_{t}}~{}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou,~{}reg~{}T(t)}(x,y;\\{x_{v}\\})\Big{]}\Big{\\}}\Big{|}_{x=y=x_{t}}$
$\displaystyle\prod_{a<b}\exp\Big{\\{}\bar{c}_{(a)n}~{}\bar{c}_{(b)m}~{}G^{zz}_{bou}(x;y;\\{x_{v}\\})+c_{(a)n}~{}c_{(b)m}~{}G^{\bar{z}\bar{z}}_{bou}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}+\sum_{n,m=0}^{\infty}\bar{c}_{(a)n}~{}c_{(b)m}~{}\partial^{n}_{x}~{}\partial^{m}_{y}~{}G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}+\sum_{n,m=0}^{\infty}c_{(a)n}~{}\bar{c}_{(b)m}~{}\partial^{n}_{x}~{}\partial^{m}_{y}~{}G^{\bar{z}z}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=x_{a},y=x_{b}}$
$\displaystyle\prod_{t<u}\exp\Big{\\{}d_{(t)0}~{}d_{(u)0}~{}G^{\bar{z}\bar{z}}_{bou}(x_{t},x_{u};\\{x_{v}\\})+\bar{d}_{(t)0}~{}\bar{d}_{(u)0}~{}G^{zz}_{bou}(x_{t},x_{u};\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt+d_{(t)0}~{}\bar{d}_{(u)0}~{}G^{\bar{z}z}_{bou}(x_{t},x_{u};\\{x_{v}\\})+\bar{d}_{(t)0}~{}d_{(u)0}~{}G^{z\bar{z}}_{bou}(x_{t},x_{u};\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt+\sum_{n,m=1}^{\infty}d_{(t)n}~{}\bar{d}_{(u)m}~{}\partial^{n-1}_{x}\partial^{m-1}_{y}\left[(x-x_{t})^{\epsilon_{t}}(y-x_{u})^{1-\epsilon_{u}}\partial_{x}\partial_{y}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]$
$\displaystyle\hskip
30.00005pt+\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}~{}d_{(u)m}~{}\partial^{n-1}_{x}\partial^{m-1}_{y}~{}\left[(x-x_{t})^{1-\epsilon_{t}}(y-x_{u})^{\epsilon_{u}}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou}(x,y;\\{x_{v}\\})\right]\Big{\\}}\Big{|}_{x=x_{t},y=x_{v}}$
$\displaystyle\prod_{t,a}\exp\Big{\\{}\sum_{m=0}^{\infty}d_{(t)0}~{}\bar{c}_{(a)m}~{}\partial^{m}_{y}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})+\bar{d}_{(t)0}~{}c_{(a)m}~{}\partial^{m}_{y}G^{z\bar{z}}_{bou}(x,y;\\{x_{v}\\})$
$\displaystyle\hskip
30.00005pt+\sum_{n=1,m=0}^{\infty}d_{(t)n}~{}\bar{c}_{(a)m}~{}\partial^{n-1}_{x}\partial^{m}_{y}\left[(x-x_{t})^{\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]$
$\displaystyle\hskip
30.00005pt+\sum_{n=1,m=0}^{\infty}\bar{d}_{(t)n}~{}c_{(a)m}~{}\partial^{n-1}_{x}\partial^{m}_{y}\left[(x-x_{t})^{1-\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{bou}(x,y;\\{x_{v}\\})\right]\Big{\\}}\Big{|}_{x=x_{t},y=x_{a}}$
(33)
Notice that all the previous expressions are meaningful because of the
behavior of the Green functions
$\displaystyle G^{zz}_{bou}(x;y;\\{x_{v}\\})$ $\displaystyle=const$
$\displaystyle G^{\bar{z}\bar{z}}_{bou}(x;y;\\{x_{v}\\})$
$\displaystyle=const$ $\displaystyle
G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})=G^{\bar{z}z}_{bou}(y;x;\\{x_{v}\\})$
$\displaystyle{\sim}_{x\rightarrow
x_{t}}const+(x-x_{t})^{\epsilon_{t}}[g^{z\bar{z}}_{0}(y;\\{x_{v}\\})+O(x-x_{t})]$
$\displaystyle
G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})=G^{\bar{z}z}_{bou}(y;x;\\{x_{v}\\})$
$\displaystyle\sim_{y\rightarrow
x_{u}}const+(y-x_{u})^{1-\epsilon_{u}}[g^{z\bar{z}}_{0}(x;\\{x_{v}\\})+O(y-x_{u})]$
$\displaystyle
G^{\bar{z}z}_{bou}(x;y;\\{x_{v}\\})=G^{z\bar{z}}_{bou}(y;x;\\{x_{v}\\})$
$\displaystyle\sim_{x\rightarrow
y;x,y\in(x_{t},x_{t+1})}const-2\alpha^{\prime}\cos^{2}\gamma_{t}log|x-y|+O(x-y)$
(34)
where $g^{z\bar{z}}_{0}$ and $g^{z\bar{z}}_{0}$ are some functions of the
given variables and the last line is strictly speaking true when $x\rightarrow
y$ but not at the same time when $x\rightarrow x_{t}$ and $y\rightarrow
x_{t}$. It is anyhow true that
$(x-x_{t})^{1-\epsilon_{t}}(y-x_{t})^{\epsilon_{t}}~{}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou,~{}reg~{}T(t)}(x,y;\\{x_{v}\\})$
is well defined for $x=y=x_{t}$ as discussed in the appendix B.
The rest of the paper is organized in the following way: in the next section
we make some examples of the use of the previous formulae and we clarify the
operator to state mapping we use in the twisted sector. In section 3 we derive
the previous formulae for the case with non excited twisted matter and finally
in section 4 we consider excited twisted matter.
## 2 Examples
We want now apply the main formulae stated in the previous section to some
examples while elucidating the nature of excited twisted states.
We start from the simplest example and then move to some more complex ones
while in appendix A we check the $N=2$ not excited states and $M$ tachyons
amplitude against the result found in ([12]).
### 2.1 Example 1: $N$ not excited twisted states
From the operator to auxiliary state map
$\sigma_{\epsilon_{t},\kappa_{t}}(x_{t},\bar{x}_{t})\leftrightarrow|T_{t},\kappa_{t}\rangle=\lim_{x\rightarrow
0}\sigma_{\epsilon_{t},\kappa_{t}}(x,\bar{x})|0_{SL}\rangle$ (35)
we deduce that
$\displaystyle\langle\sigma_{\epsilon_{1},\kappa_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\rangle$
$\displaystyle=\langle V_{N+0}(\\{x_{t}\\}_{t=1,\dots
N}|\prod_{t}|T_{t},\kappa_{t}\rangle$
$\displaystyle=\delta(\sum_{t}\kappa_{t})e^{-\frac{1}{2}\sum_{t}\kappa_{t}^{2}~{}G^{22}_{bou,~{}reg~{}T(t)}(x_{t};x_{t})}e^{-\frac{1}{2}\sum_{u,t}\kappa_{t}\kappa_{u}~{}G^{22}_{bou}(x_{t};x_{u})}~{}C(x_{1},\dots
x_{N})$ (36)
where the phases proportional to $\kappa$ probably vanish as they do in the
$N=2$ case but this must be checked with an explicit computation of the Green
function which can, in principle, be extracted from ([6]) after T-dualizing.
The same computation can be performed using the more conventional generating
function as
$\displaystyle\langle\sigma_{\epsilon_{1},\kappa_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\rangle={\cal
V}_{N+M}\prod_{t}e^{i\kappa_{t}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(t)0}^{2}}}\Big{|}_{c=0;d=0}.$ (37)
### 2.2 Example 2: $N$ not excited twisted and $2$ untwisted states
Similarly to what done in the previous example from the maps
$\displaystyle\sigma_{\epsilon_{t}}(x_{t},\bar{x}_{t})\leftrightarrow|T_{t}\rangle=\lim_{x\rightarrow
0}\sigma_{\epsilon_{t}}(x,\bar{x})|0_{SL}\rangle$ $\displaystyle\partial
X^{z}(y_{a},\bar{y}_{a})\leftrightarrow-i\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{a}}~{}\alpha^{\dagger}_{(a)1}|0_{(a)}\rangle=\lim_{y\rightarrow
0^{+}}\partial Z_{(a)}(y,\bar{y})|0_{SL}\rangle$ (38)
where we have made the choice $x_{t_{a}}<y_{a}<x_{t_{a}}+1$ (which fixes the
magnetic field felt by the untwisted state) , we deduce
$\displaystyle\langle\sigma_{\epsilon_{1},\kappa_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\partial_{y_{1}}Z(y_{1},\bar{y}_{1})\partial_{y_{2}}{\bar{Z}}(y_{2},\bar{y}_{2})\rangle$
$\displaystyle=\langle V_{N+2}(\\{x_{t}\\}_{t=1,\dots
N},\\{x_{a}\\}_{a=1.2}|\otimes_{t}|T_{t}\rangle~{}\otimes(-i\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{1}})\alpha_{(1)1}^{\dagger}|0_{(1)}\rangle~{}\otimes(-i\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{2}}~{})\bar{\alpha}_{(2)1}^{\dagger}|0_{(2)}\rangle$
$\displaystyle=\delta(\sum_{t}\kappa_{t})e^{-\frac{1}{2}\sum_{t}\kappa_{t}^{2}~{}G^{22}_{bou,~{}reg~{}T(t)}(x_{t};x_{t})}e^{-\frac{1}{2}\sum_{u,t}\kappa_{t}\kappa_{u}~{}G^{22}_{bou}(x_{t};x_{u})}~{}C(x_{1},\dots
x_{N})~{}\partial_{y_{1}}\partial_{y_{2}}G^{z\bar{z}}_{bou}(y_{1};y_{2};\\{x_{t}\\})$
(39)
where ${\bar{y}}$ is a function of $y$ as in eq. (46). The previous result is
an immediate consequence of the the definition of $G$ given in (3) but it can
also be interpreted as a “proof” of eq. (3) since the Green function entering
in the previous formula is the Green function obtained from the path integral.
### 2.3 Example 3: $N-1$ not excited twisted, $1$ excited twisted and $2$
untwisted states
We can now discuss the excited twisted states. The easiest way to denote an
excited twisted state is by writing from which untwisted state it can be
obtained by OPE, for example
$\left(\partial_{x}Z~{}\partial_{x}^{2}Z~{}\sigma_{\epsilon,\kappa}\right)(x,{\bar{x}})$
can be obtained by taking the finite part of the OPE
$(\partial_{y}Z~{}\partial_{y}^{2}Z)(y,{\bar{y}})\sigma_{\epsilon,\kappa}(x,{\bar{x}})$
as $y\rightarrow x$. This limit can be taken in a clearer and easier way when
we realize $(\partial_{y}Z~{}\partial_{y}^{2}Z)(y,{\bar{y}})$ as a normal
ordered operator in a twisted auxiliary Hilbert space where
$\sigma_{\epsilon,\kappa}(x,{\bar{x}})\leftrightarrow|T,\kappa\rangle$. In
particular we can define the finite part of the limit as
$\lim_{y\rightarrow
0^{+}}:\left[y^{1-\epsilon}\partial_{y}Z(y,y)~{}\partial_{y}[y^{1-\epsilon}\partial_{y}Z(y,y)]\right]:|T,\kappa\rangle=\left(-i\sqrt{2\alpha^{\prime}}\,\cos\gamma\right)^{2}\alpha^{\dagger}_{\epsilon}\alpha^{\dagger}_{1+\epsilon}|T,\kappa\rangle$
(40)
Similarly we can consider the map
$\displaystyle\left(\partial\bar{Z}\sigma_{\epsilon_{t},\kappa_{t}}\right)(x_{t},\bar{x}_{t})\leftrightarrow-i\sqrt{2\alpha^{\prime}}\,\cos\gamma_{t}\bar{\alpha}^{\dagger}_{1-\epsilon}|T_{t},\kappa_{t}\rangle$
(41)
and compute the correlator
$\displaystyle\langle\left(\partial_{x}{\bar{Z}}\sigma_{\epsilon_{1},\kappa_{1}}\right)(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\partial_{y_{1}}Z(y_{1},\bar{y}_{1})\rangle$
$\displaystyle=\langle V_{N+1}(\\{x_{t}\\}_{t=1,\dots
N},\\{x_{a}\\}_{a=1.2}|(-i\sqrt{2\alpha^{\prime}}\cos\gamma_{1}~{})\bar{\alpha}_{(1)\epsilon_{1}}^{\dagger}|T_{1},\kappa_{1}\rangle\otimes_{t>1}|T_{t},\kappa_{t}\rangle~{}\otimes(-i\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{1}})\alpha_{(1)1}^{\dagger}|0_{(1)}\rangle$
(42)
to be
$\displaystyle\delta(\sum_{t}\kappa_{t})e^{-\frac{1}{2}\sum_{t}\kappa_{t}^{2}~{}G^{22}_{bou,~{}reg~{}T(t)}(x_{t};x_{t})}e^{-\frac{1}{2}\sum_{u,t}\kappa_{t}\kappa_{u}~{}G^{22}_{bou}(x_{t};x_{u})}~{}C(x_{1},\dots
x_{N})~{}$
$\displaystyle~{}~{}\partial_{y_{1}}\left[(x-x_{1})^{\epsilon_{1}}\partial_{x}G^{\bar{z}z}_{bou}(x,y_{1};\\{x_{v}\\})\right]|_{x=x_{1}}.$
(43)
The same computation can be performed using the generating function as
$\displaystyle\langle\left(\partial_{x}{\bar{Z}}\sigma_{\epsilon_{1},\kappa_{1}}\right)(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\partial_{y_{1}}Z(y_{1},\bar{y}_{1})\rangle={\cal
V}_{N+M}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(1)1}}e^{i\kappa_{1}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(1)0}^{2}}}\prod_{t>1}e^{i\kappa_{t}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(t)0}^{2}}}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
c_{(1)1}}\Big{|}_{c=0;d=0}$ (44)
## 3 Derivation for untwisted matter
The starting point is very similar to ([4],[5]) where it was recognized that
the generator for all closed (super)string amplitudes is a quadratic path
integrals. The idea in the previous papers is that the appropriate boundary
condition for R and/or NS sector can be obtained simply by inserting linear
sources with the desired boundary conditions. Because of this assumption the
quantum fluctuations are the same for all the amplitudes: from the purely NS
to the mixed ones. It was later realized that this prescription misses a
proper treatment of the quantum fluctuations ([10]) and that when this part is
considered the amplitudes factorize correctly ([11]).
Here we consider open strings and we realize the proper twisted boundary
conditions by quadratic boundary terms which are nothing else but the coupling
of the string to the magnetic field background. Therefore a non excited twist
field is realized by a discontinuity in the magnetic field888 And in
particular the transition from an eigenstate $|\alpha\rangle$ in magnetic
field $B_{t}$ to an eigenstate $|\beta\rangle$ in magnetic $B_{t+1}$ at
worldsheet time $\tau_{t}$ can be computed as in usual quantum mechanics as
$\langle\beta|\alpha\rangle=\int{\cal D}X~{}\langle
X(\sigma),\tau_{t}^{-}|\alpha\rangle\left(\langle
X(\sigma),\tau_{t}^{+}|\beta\rangle\right)^{*}$. .
The $N$ non excited twist field amplitudes with Euclidean worldsheet metric is
then computed by the quadratic path integral999 We define
$d^{2}z=2dz~{}d\bar{z}$ and $z=e^{\tau_{E}+i\sigma}\in H$. 101010This is the
path integral corresponding to all twist fields with zero “momentum” and
therefore it is proportional to a $\delta(0)$ which arises from $X^{2}$ zero
mode. The general case is treated in the next section.
$\displaystyle C(x_{1},\dots x_{N})\delta(0)$ $\displaystyle={\cal N}\int{\cal
D}X~{}e^{-\frac{1}{2\pi\alpha^{\prime}}\left[\int_{H}d^{2}z~{}\frac{1}{2}G_{ij}\partial_{z}X^{i}\partial_{\bar{z}}X^{j}-i\int_{\partial
H}dx~{}B(x)X^{1}(x,{\bar{x}})\partial_{x}X^{2}(x,{\bar{x}})\right]}$ (45)
where $dx\partial_{x}X^{2}(x,{\bar{x}})$ must be interpreted as the pullback
on the boundary of $dX^{2}$ in such a way that $\bar{x}$ depends on $x$ as
${\bar{x}}=\left\\{\begin{array}[]{ll}x&x=|x|>0\\\
xe^{-i2\pi}&x=|x|e^{i\pi}<0\end{array}\right.,$ (46)
$H$ is the superior half plane and the adimensional magnetic field $B(x)$ is
given by
$B(x)=2\pi\alpha^{\prime}~{}qF_{12}(x)=\sum_{t=1}^{N}B_{t}~{}\theta(x-x_{t})~{}\theta(x_{t+1}-x)$
(47)
so that the dicharged string at $x=x_{t}$ feels a magnetic field $B_{t-1}$ on
the left, i.e. $\sigma=\pi$ and $B_{t}$ on the right, i.e. $\sigma=0$. Here we
have set $B_{-1}=B_{N+1}$ and $x_{-1}=-\infty$, $x_{N+1}=+\infty$.
In the previous equation (45) we have chosen the gauge
$A_{1}=0,~{}~{}~{}~{}A_{2}=Bx^{1}$ (48)
in order to make clear that we have only one zero mode which is associated
with the shift $X^{2}\rightarrow X^{2}+\epsilon$ and therefore there is only
one conserved momentum as it is the case in the Landau levels problem on
$\mathbb{R}^{2}$.
Now we can add the untwisted states vertexes. This can be done by considering
the generating function of all untwisted vertexes at $x=x_{a}\in\mathbb{R}$
as111111 This formulation involving the full field on the boundary is only
right for NN boundary condition. For DD boundary condition one should use a
slightly different one ([13]).
$\displaystyle{\cal S}(c_{(a)})$
$\displaystyle=\exp\\{\sum_{n=0}^{\infty}c_{(a)\,n\,i}\partial^{n}_{x}|_{x=x_{a}}X^{i}(x,{\bar{x}})\\}=\exp\\{\int_{\partial
H}dx~{}J_{i}(x;x_{a})~{}X^{i}(x,{\bar{x}})\\}$ $\displaystyle J_{i}(x;x_{a})$
$\displaystyle=\sum_{n=0}^{\infty}c_{(a)ni}\partial^{n}_{x_{a}}\delta(x-x_{a})$
(49)
where $c$ are arbitrary complex numbers (or functions as we use in the next
section). To understand how the previous generating vertex work let us take
the example from ([12]). Consider the vertex which describes the fluctuations
of the gauge vector around the dipole string background we can derive it from
a generating functional for the dipole string as 121212 Notice that here we
are talking about the abstract path integral representation of the vertex and
not of the operatorial representation. The operatorial representation of the
vertex can be realized in an auxiliary space with both twisted and untwisted
boundary conditions. The untwisted auxiliary Hilbert space representation is
the usual operatorial representation while the twisted one is the one derived
in ([12]). Just because of these different realizations this auxiliary Hilbert
space must not be confused with ${\cal H}_{(a,t_{a})}$ introduced before which
is a way of representing the $c_{(a)ni}$, see (3). 131313 It is worth
noticing how vertexes for dipole strings have the same functional form
independently on the magnetic backgrounds $B_{t_{a}}$ nevertheless they differ
because different conditions for physical states, in the previous example we
have $k_{\mu}\eta^{\mu\nu}k_{\nu}+k_{i}{\cal
G}^{ij}(B_{t_{a}})k_{j}=\epsilon_{\mu}\eta^{\mu\nu}k_{\nu}+\epsilon_{i}{\cal
G}^{ij}(B_{t_{a}})k_{j}=0$ where $\mu,\nu\neq 1,2$.
$\displaystyle
V(x_{a};\epsilon,k)=\epsilon_{i}\partial_{x}X^{i}(x,{\bar{x}})e^{ik_{j}X^{j}(x,{\bar{x}})}\Big{|}_{x=x_{a}}=S(c_{(a)},x_{a})~{}\epsilon_{i}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
c_{(a)1i}}e^{ik_{j}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
c_{(a)0j}}}\Big{|}_{c_{(a)}=0}$ (50)
As a matter of facts the previous vertex gives an indefinite result when
inserted in the path integral even when $B=0$. We must therefore regularize it
and consider
$\displaystyle{\cal S}_{reg}(c_{(a)})$ $\displaystyle={\cal
N}_{(a)}(x_{a})\exp\Big{\\{}\sum_{n=0}^{\infty}c_{(a)\,n\,i}\partial^{n}_{x}|_{x=x_{a}}\langle
X^{i}(x,{\bar{x}})\rangle\Big{\\}}$ $\displaystyle={\cal
N}_{(a)}(x_{a})\exp\Big{\\{}\int_{\partial
H}dx~{}J_{i,reg}(x;x_{a})~{}X^{i}(x,{\bar{x}})\Big{\\}}$ (51)
where the regularized curred is given by
$\displaystyle J_{i,reg}(x;x_{a})$
$\displaystyle=\sum_{n=0}^{\infty}c_{(a)ni}\partial^{n}_{x_{a}}\delta_{reg}(x-x_{a}),$
(52)
the averaged field by
$\langle X^{i}(x,x)\rangle=\int_{\partial H}dy~{}\delta_{reg}(x-y)X(y,y)$ (53)
and the normalization factor is
$\displaystyle{\cal N}_{(a)}(x_{a})=\exp\Big{\\{}$
$\displaystyle-\frac{1}{2}\sum_{n,m=0}^{\infty}c_{(a)ni}~{}c_{(a)mj}$
$\displaystyle\int_{\partial H}dx\int_{\partial
H}dy~{}\partial^{n}_{x}\delta_{reg}(x-x_{a})~{}\partial^{m}_{y}\delta_{reg}(y-x_{a})~{}G_{U(t_{a}),~{}bou}^{ij}(x;y)\Big{\\}}\Big{|}_{x=x_{a},y=x_{a}}$
(54)
with $G_{U(t_{a}),~{}bou}^{ij}(x;y)$ the boundary Green functions in the
dipole case with magnetic field $B_{t_{a}}$ given in eq.s (• ‣ 1). There are
two reasons why we have introduced the previous definitions. The first is that
it works in reproducing the amplitudes for $N=2$ as discussed in appendix A.
The second is connected to the way the regularization terms is suggested from
the operatorial formalism. In operatorial formalism the simplest approach is
to consider a point splitting, i.e.
$[\exp\left(c_{(a)i}X^{i}_{(a)}(x_{a},x_{a})\right)]_{p.s.}=\exp\left(c_{(a)i}[X^{i(-)}_{(a)}(x_{a}e^{-\eta},x_{a}e^{-\eta})+X^{i(+)}_{(a)}(x_{a},x_{a})]\right)$
which implies a regularization factor ${\cal
N}_{(a)}(x_{a})=\exp\left(-\frac{1}{2}c_{(a)i}c_{(a)j}G^{ij}_{bou}(x;y)\right)\Big{|}_{x=x_{a},y=x_{a}e^{-\eta}}$.
When we smooth the fields the previous regularization factor becomes
$\displaystyle{\cal N}_{(a)}(x_{a})=$
$\displaystyle\exp\left(-\frac{1}{2}c_{(a)i}c_{(a)j}\int_{x>y}dx~{}dy~{}2~{}\delta_{reg}(x-x_{a})\delta_{reg}(y-x_{a})G^{ij}_{bou}(x;y)\right)$
$\displaystyle=$ $\displaystyle\exp\left(-c_{(a)i}c_{(a)j}\int
dx~{}dy~{}\delta_{reg}(x-x_{a})\delta_{reg}(y-x_{a})\frac{G^{ij}_{bou}(x;y)~{}\theta(x-y)+G^{ji}_{bou}(y;x)~{}\theta(y-x)}{2}\right)$
$\displaystyle=$ $\displaystyle\exp\left(-\frac{1}{2}c_{(a)i}c_{(a)j}\int
dx~{}dy~{}\delta_{reg}(x-x_{a})\delta_{reg}(y-x_{a})G^{ij}_{bou}(x;y)\right)$
(55)
where the factor $2$ in the first line is due to the fact we are using one of
the two $\delta$ just one half because of the constraint $x>y$ as it can be
directly verified by using a step function regularization of the delta. In the
last step we have used the property $G^{ji}_{bou}(y;x)=G^{ij}_{bou}(x;y)$.
In conclusion the path integral we want to compute in order to get the
generating function for all the $M$ untwisted correlators in presence of $N$
twists is
$\displaystyle Z(\\{x_{t}\\};\\{x_{a}\\})$ $\displaystyle={\cal N}\int{\cal
D}X~{}e^{-\frac{1}{2\pi\alpha^{\prime}}\left[\int_{H}d^{2}z~{}\frac{1}{2}G_{ij}\partial_{z}X^{i}\partial_{\bar{z}}X^{j}+i\int_{\partial
H}dx~{}B(x)X^{1}(x,{\bar{x}})\partial_{x}X^{2}(x,{\bar{x}})\right]}$
$\displaystyle~{}~{}~{}~{}\times\prod_{a=1}^{M}{\cal
N}_{(a)}(x_{a})e^{\int_{\partial H}dx~{}J_{i,reg}(x;x_{a})X^{i}(x,{\bar{x}})}$
(56)
This path integral can then be performed to get
$\displaystyle Z(\\{x_{t}\\};\\{x_{a}\\})$ $\displaystyle=C(x_{1},\dots
x_{N})~{}\delta(i\sum_{a}c_{(a)0\,i=2})$
$\displaystyle\prod_{a}\exp\Big{\\{}\frac{1}{2}\int_{\partial
H}dx\int_{\partial H}dyJ_{i,reg}(x;x_{a})~{}J_{j,reg}(y;x_{a})$
$\displaystyle\hskip
50.00008pt~{}[G^{ij}_{bou}(x;y;\\{x_{v}\\})-G_{U(t_{a})~{},bou}^{ij}(x;y)]\Big{\\}}$
$\displaystyle\prod_{a,b;a\neq b}\exp\Big{\\{}\frac{1}{2}\int_{\partial
H}dx\int_{\partial H}dyJ_{i,reg}(x;x_{a})~{}J_{j,reg}(y;x_{b})$
$\displaystyle\hskip 50.00008pt~{}G^{ij}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$ (57)
where $G^{ij}_{bou}(x;y;\\{x_{v}\\})$ is the Green function of the quadratic
operator which turns out to be the one defined in (2). Notice that the
quadratic operator is not even hermitian exactly as it happens for Landau
levels in the plain quantum mechanics. The previous result can be also
rewritten after the regularization has been removed as
$\displaystyle Z(\\{x_{t}\\};\\{x_{a}\\})=$ $\displaystyle C(x_{1},\dots
x_{N})~{}\delta(i\sum_{a}c_{(a)0\,i=2})$
$\displaystyle\prod_{a}\exp\Big{\\{}\frac{1}{2}\sum_{n,m=0}^{\infty}c_{(a)ni}~{}c_{(a)mj}~{}\partial^{n}_{x}|_{x=x_{a}}~{}\partial^{m}_{y}|_{y=x_{a}}~{}G^{ij}_{bou~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})\Big{\\}}$
$\displaystyle\prod_{a,b;a\neq
b}\exp\Big{\\{}\frac{1}{2}\sum_{n,m=0}^{\infty}c_{(a)ni}~{}c_{(b)mj}~{}\partial^{n}_{x}|_{x=x_{a}}~{}\partial^{m}_{y}|_{y=x_{b}}~{}G^{ij}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$
(58)
where we have defined the boundary Green function regularized by the untwisted
Green function for a background $B_{t_{a}}$
$\displaystyle G^{z\bar{z}}_{bou~{}reg~{}U(t_{a})}(x,y;\\{x_{v}\\})$
$\displaystyle=\left[G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})-G^{z\bar{z}}_{U(t_{a}),~{}bou}(x;y)\right]$
(59)
The previous expression can be simplified a little using the symmetry
$G_{bou}^{ij}(x;y)=G_{bou}^{ji}(y;x)$ and by rewriting it in the complex basis
as
$\displaystyle Z(\\{x_{t}\\};\\{x_{a}\\})=$ $\displaystyle C(x_{1},\dots
x_{N})~{}\delta\left(\sum_{a}(c_{(a)0}-\bar{c}_{(a)0})\right)$
$\displaystyle\prod_{a}\exp\Big{\\{}\frac{1}{2}c_{(a)0}^{2}~{}G^{\bar{z}\bar{z}}_{bou~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})+\frac{1}{2}\bar{c}_{(a)0}^{2}~{}G^{zz}_{bou~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a}\exp\Big{\\{}}\sum_{n,m=0}^{\infty}\bar{c}_{(a)n}~{}c_{(a)m}~{}\partial^{n}_{x}|_{x=x_{a}}~{}\partial^{m}_{y}|_{y=x_{a}}~{}G^{z\bar{z}}_{bou~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})\Big{\\}}$
$\displaystyle\prod_{a<b}\exp\Big{\\{}\bar{c}_{(a)n}~{}\bar{c}_{(b)m}~{}G^{zz}_{bou}(x_{a};x_{b};\\{x_{v}\\})+c_{(a)n}~{}c_{(b)m}~{}G^{\bar{z}\bar{z}}_{bou}(x_{a};x_{b};\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}+\sum_{n,m=0}^{\infty}\bar{c}_{(a)n}~{}c_{(b)m}~{}\partial^{n}_{x}|_{x=x_{a}}~{}\partial^{m}_{y}|_{y=x_{b}}~{}G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}+\sum_{n,m=0}^{\infty}c_{(a)n}~{}\bar{c}_{(b)m}~{}\partial^{n}_{x}|_{x=x_{a}}~{}\partial^{m}_{y}|_{y=x_{b}}~{}G^{\bar{z}z}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}$
(60)
with $\bar{c}_{(a)n}=c_{(a)nz}$ and $c_{(a)n}=c_{(a)n\bar{z}}$.
We can now give a different formulation of the previous result if we realize
the algebra
$[c_{(a)ni},\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
c_{(b)mj}}]=\delta_{i}^{j}~{}\delta_{n,m}~{}\delta_{a,b}$ (61)
on the untwisted scalar (dipole string) auxiliary Hilbert spaces ${\cal
H}_{a,t_{a}}$ with backgrounds $B_{t_{a}}$ introduced before as
$\displaystyle 1\rightarrow$ $\displaystyle\langle
z_{(a)00}=\bar{z}_{(a)00}=0|\langle 0_{(a)}|$
$\displaystyle\bar{c}_{(a)n\,}\rightarrow\frac{i}{\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{a}}}\frac{\alpha_{(a)n}}{n!},$
$\displaystyle~{}~{}c_{(a)n\,}\rightarrow\frac{i}{\sqrt{2\alpha^{\prime}}\cos\gamma_{t_{a}}}\frac{\bar{\alpha}_{(a)n}}{n!}~{}~{}~{}n\geq
0$
$\displaystyle\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial\bar{c}_{(a)m}}\rightarrow-i\sqrt{2\alpha^{\prime}}\,(m-1)!\,\cos\gamma_{t_{a}}\,\alpha_{(a)m}^{\dagger},$
$\displaystyle~{}~{}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
c_{(a)m}}\rightarrow-i\sqrt{2\alpha^{\prime}}\,(m-1)!\,\cos\gamma_{t_{a}}\,\bar{\alpha}_{(a)m}^{\dagger}~{}~{}m>0$
(62)
where the “strange” choice of the normalization is due to the last expressions
which arise from the desire of identifying
$-i\sqrt{2\alpha^{\prime}}\,(m-1)!\,\cos\gamma_{t_{a}}\,\alpha_{(a)m}^{\dagger}\sim\partial^{m}X^{(-)z}_{(a)}(x,{\bar{x}})|_{x=0}.$
(63)
Using these auxiliary Hilbert spaces we can now rewrite the previous
expression for the $M$ untwisted correlators (3) as
$\displaystyle Z(\\{x_{t}\\};\\{x_{a}\\})$ $\displaystyle=C(x_{1},\dots
x_{N})~{}\delta\left(i\sum_{a}(\alpha_{(a)0}-\bar{\alpha}_{(a)0})\right)~{}\prod_{a=1}^{M}\langle
z_{(a)00}=\bar{z}_{(a)00}=0|\langle 0_{(a)}|$
$\displaystyle\prod_{a}\exp\Big{\\{}-\frac{1}{4\alpha^{\prime}}\alpha_{(a)0}^{2}~{}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a}\exp\Big{\\{}}-\frac{1}{4\alpha^{\prime}}\bar{\alpha}_{(a)0}^{2}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}^{2}~{}G^{zz}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a}\exp\Big{\\{}}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\alpha_{(a)n}~{}\bar{\alpha}_{(a)m}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=y=x_{a}}$
$\displaystyle\prod_{a<b}\exp\Big{\\{}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\alpha_{(a)n}~{}\bar{\alpha}_{(b)m}~{}{\cal
V}_{(t_{a})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{b})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{z\bar{z}}_{bou}(x;y;\\{x_{v}\\})$
$\displaystyle\phantom{\prod_{a<b}\exp\Big{\\{}}-\frac{1}{2\alpha^{\prime}}\sum_{n,m=0}^{\infty}\bar{\alpha}_{(a)n}~{}\alpha_{(b)m}~{}{\cal
V}_{(t_{b})~{}\underline{\bar{z}}z}{\cal
V}_{(t_{a})~{}\underline{z}\bar{z}}~{}\frac{\partial^{n}_{x}}{n!}~{}\frac{\partial^{m}_{y}}{m!}~{}G^{\bar{z}z}_{bou}(x;y;\\{x_{v}\\})\Big{\\}}\Big{|}_{x=x_{a},y=x_{b}}$
(64)
where ${\cal V}$ are the $\mathbb{R}^{2}$ vielbein which connect the $\alpha$
flat index with the Green function $G$ curved index.
## 4 Derivation for twisted matter
The strategy we are going to follow is to consider the amplitude derived in
previous section with $N+M$ untwisted states at the positions
$\\{x_{a}\\}_{a=1\dots M},\\{x_{f}\\}_{f=1\dots N}$ and unexcited twists at
positions $\\{x_{t}\\}_{t=1\dots N}$. Then we choose $N$ of untwisted states
at the positions $\\{x_{f}\\}_{f=1\dots N}$ for which we take the limit
$x_{f}\rightarrow x_{t}$. In order to get the desired amplitude with $M$
untwisted and $N$ excited twisted states we must choose in a proper way the
$c_{(f)ni}$. This amounts not only to choose $c_{(f)ni}$ in (3) as a function
of $x_{f}$ as in eq. (4) but to introduce a further normalization ${\cal
R}(x_{f})$ as in eq. (74) in such a way that we can “undo” the OPE and get a
result which is a generating function for the twisted states
$\exp\Big{\\{}d^{2}_{(t)0}x_{0}^{i=2(aux\,t)}-i\sqrt{2\alpha^{\prime}}\,\cos\gamma_{t}\,\sum_{n=1}^{\infty}\left[d_{(t)n}(n-1)!\bar{\alpha}^{\dagger(aux\,t)}_{n-\epsilon_{t}}+\bar{d}_{(t_{f})n}(n-1)!\alpha^{\dagger(aux\,t)}_{n-1+\epsilon_{t}}\right]\Big{\\}}|T_{(aux\,t)}\rangle$
(65)
when expressed in a chart where $x_{t}=0$. The “strange” normalization is
chosen because it is the easiest map from operators to states, f.x. the
twisted excited state which can be obtained by subtracting the divergences of
the limit $y\rightarrow x_{t}^{+}$ of
$[\partial_{y}^{3}Z(y,y)]^{2}\sigma_{\epsilon_{t}}(x_{t},x_{t})$ gives the
state
$\lim_{y\rightarrow
0^{+}}\left[\partial_{y}^{2}[y^{1-\epsilon_{t}}\partial_{y}Z^{(aux\,t)}(y,y)]\right]^{2}|T_{(aux\,t)}\rangle=\left(-i\sqrt{2\alpha^{\prime}}~{}\cos\gamma_{t}~{}2!~{}\alpha^{\dagger(aux\,t)}_{2+\epsilon_{t}}\right)^{2}|T_{(aux\,t)}\rangle$
(66)
thus making contact between eq. (65) and eq. (4).
Let us start studying the OPE ${\cal S}(c,x_{f})\sigma_{t}(0,0)$. This can be
studied in an auxiliary Hilbert space ${\cal H}_{aux\,t}$ (not to be confused
the the Hilbert space ${\cal H}_{t}$ which we introduced in the first section
and which is associated with coefficients $d_{(t)}$) where
$\sigma_{\epsilon_{t}}(0,0)$ is represented by the twisted vacuum
$|T_{(aux\,t)}\rangle$ and the generating function ${\cal S}(c,x_{f})$ as
$\displaystyle{\cal S}_{(aux\,t)}(c_{(f)},x_{f})=$ $\displaystyle
e^{-\frac{1}{2}c_{(f)0}^{2}~{}G^{zz}_{bou,~{}reg~{}U(t_{a})}(x_{f};x_{f};\\{x=0,x=\infty\\})-\frac{1}{2}\bar{c}_{(f)0}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x_{f};x_{f};\\{x=0,x=\infty\\})}$
$\displaystyle
e^{-\frac{1}{2}\sum_{m,n=0}^{\infty}c_{(f)n}~{}\bar{c}_{(f)m}~{}\partial^{n}_{x}|_{x=x_{f}}~{}\partial^{m}_{y}|_{y=x_{f}}~{}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x=0,x=\infty\\})}$
$\displaystyle:e^{\sum_{n=0}^{\infty}\bar{c}_{(f)n}~{}\partial^{n}_{x}|_{x=x_{f}}Z_{(aux\,t)}(x,\bar{x})+c_{(f)n}~{}\partial^{n}_{x}|_{x=x_{f}}\bar{Z}_{(aux\,t)}(x,\bar{x})}:$
(67)
In the previous equation the normal ordering is performed with respect to the
operators entering the expansion of the quantum fields
$Z_{(aux\,t)}(z,\bar{z})$ and $\bar{Z}_{(aux\,t)}(z,\bar{z})$ which act on
${\cal H}_{aux\,t}$. Then the OPE can be computed as
$\displaystyle{\cal S}(c,x_{f})\sigma_{\epsilon_{t}}(0,0)\leftrightarrow$
$\displaystyle
e^{-\frac{1}{2}c_{(f)0}^{2}~{}G^{zz}_{bou,~{}reg~{}U(t_{a})}(x_{f};x_{f};\\{x=0,x=\infty\\})-\frac{1}{2}\bar{c}_{(f)0}^{2}~{}G^{\bar{z}\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x_{f};x_{f};\\{x=0,x=\infty\\})}$
$\displaystyle
e^{-\frac{1}{2}\sum_{m,n=0}^{\infty}\partial^{n}_{x}|_{x=x_{f}}~{}\partial^{m}_{y}|_{y=x_{f}}\left[~{}c_{(f)n}~{}\bar{c}_{(f)m}~{}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x=0,x=\infty\\})\right]}$
$\displaystyle
e^{\sum_{n=0}^{\infty}\partial^{n}_{x}|_{x=x_{f}}\left[\bar{c}_{(f)n}~{}Z_{(aux\,t)}^{(-)}(x,x)\right]+\partial^{n}_{x}|_{x=x_{f}}\left[c_{(f)n}~{}\bar{Z}_{(aux\,t)}^{(-)}(x,x)\right]}|T_{(aux\,t)}\rangle$
(68)
which is similar to a rewriting of eq. (65) as
$\displaystyle\lim_{x_{f}\rightarrow 0^{+}}$ $\displaystyle
e^{\bar{d}_{(t)0}~{}\left[Z_{(aux~{}t)}^{(-)}(x_{f},x_{f})\right]+d_{(t)0}~{}\left[\bar{Z}_{(aux~{}t)}^{(-)}(x_{f},x_{f})\right]}$
$\displaystyle
e^{\sum_{n=1}^{\infty}\bar{d}_{(t)n}~{}\partial^{n-1}_{x}|_{x=x_{f}}\left[x^{1-\epsilon_{t}}\partial_{x}Z_{(aux~{})}^{(-)}(x,x)\right]+d_{(t)n}~{}\partial^{n-1}_{x}|_{x=x_{f}}\left[x^{\epsilon_{t}}\partial_{x}\bar{Z}_{(aux~{})}^{(-)}(x,x)\right]}|T_{(aux\,t)}\rangle$
(69)
where in the second line we have written $\partial Z$ since we want to get rid
of zero modes and in the limit it is necessary to write $x_{f}\rightarrow
0^{+}$ since the the behavior of $\partial Z$ changes by an overall
normalization when $x<0$.
Comparison between the two previous expressions suggests to consider then the
operator acting on the Hilbert space ${\cal H}_{(aux~{}t)}$
$\displaystyle{\cal T}_{(aux~{}t)}(d_{(t)},x_{f})$ $\displaystyle={\cal
N}(d_{(t)},x_{f},x_{t})~{}e^{\bar{d}_{(t)0}~{}\left[Z_{(aux~{}t,reg)}(x_{f},x_{f})\right]+d_{(t)0}~{}\left[\bar{Z}_{(aux~{}t,reg)}(x_{f},x_{f})\right]}$
$\displaystyle
e^{\sum_{n=1}^{\infty}\bar{d}_{(t)n}~{}\partial^{n-1}_{x}\left[x^{1-\epsilon_{t}}\partial_{x}Z_{(aux~{}t,reg)}(x,x)\right]+d_{(t)n}~{}\partial^{n-1}_{x}\left[x^{\epsilon_{t}}\partial_{x}\bar{Z}_{(aux~{}t,reg)}(x,x)\right]}\Big{|}_{x=x_{f}^{+}}$
(70)
where $Z_{(aux~{}t,reg)}(x_{f},x_{f})$ is point split regularized of
$Z_{(aux~{}t)}(x_{f},x_{f})$ defined as
$Z_{(aux~{}t,reg)}(x_{f},x_{f})=Z_{(aux~{}t)}^{(-)}(x_{f}e^{-\eta},x_{f}e^{-\eta})+Z_{(aux~{}t)}^{(+)}(x_{f},x_{f}),$
no normal ordering is performed and the normalization factor is given by
$\displaystyle{\cal N}^{-1}(d_{(t)},x_{f},x_{t})$
$\displaystyle=\Big{\\{}e^{\frac{1}{2}\bar{d}_{(t)0}^{2}G^{zz}_{T(t)~{}bou}(x;y)+\frac{1}{2}d_{(t)0}^{2}G^{\bar{z}\bar{z}}_{T(t)~{}bou}(x;y)+\bar{d}_{(t)0}d_{(t)0}\frac{G^{z\bar{z}}_{T(t)~{}bou}(x;y)+G^{\bar{z}z}_{T(t)~{}bou}(x;y)}{2}}$
$\displaystyle
e^{\frac{1}{2}\sum_{n=1}^{\infty}\bar{d}_{(t)0}d_{(t)n}\big{[}\partial^{n-1}_{x}(x^{\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{T(t)~{}bou}(x;y))+\partial^{n-1}_{y}(y^{\epsilon_{t}}\partial_{y}G^{z\bar{z}}_{T(t)~{}bou}(x;y))\big{]}}$
$\displaystyle
e^{\frac{1}{2}\sum_{n=1}^{\infty}d_{(t)0}\bar{d}_{(t)n}\big{[}\partial^{n-1}_{x}(x^{1-\epsilon_{t}}\partial_{x}G^{z\bar{z}}_{T(t)~{}bou}(x;y))+\partial^{n-1}_{y}(y^{1-\epsilon_{t}}\partial_{y}G^{\bar{z}z}_{T(t)~{}bou}(y;x))\big{]}}$
$\displaystyle
e^{\frac{1}{2}\sum_{n,l=1}^{\infty}d_{(t)l}\bar{d}_{(t)n}\big{[}\partial^{l-1}_{y}\partial^{n-1}_{x}(y^{\epsilon_{t}}x^{1-\epsilon_{t}}\partial_{x}\partial_{y}G^{z\bar{z}}_{T(t)~{}bou}(x;y))+\partial^{l-1}_{x}\partial^{n-1}_{y}(x^{\epsilon_{t}}y^{1-\epsilon_{t}}\partial_{x}\partial_{y}G^{\bar{z}z}_{T(t)~{}bou}(x;y))\big{]}}$
$\displaystyle\Big{\\}}\Big{|}_{x=x_{f};y=x_{f}e^{-\eta}}$ (71)
where
$G^{ij}_{T(t)~{}bou}(y;x)=G^{ji}_{T(t)~{}bou}(x;y)=G^{ij}_{bou}(y;x;\\{x_{1}=0,x_{2}=\infty\\})$
are the (analytic continuation of the) boundary Green functions defined in
eq.s (• ‣ 1).
The reason why we have written the previous expression in a non normal ordered
way is to understand the expression of the regularization factor of the
corresponding “classical” vertex (4) which we want to insert in the path
integral.
The previous operator can also be written in a way to make its connection with
the idea of undoing the OPE clearer as
$\displaystyle{\cal T}_{(aux~{}t)}(d_{(t)},x_{f})=$ $\displaystyle{\cal
R}(d_{(t)},x_{f})~{}{\cal S}_{(aux~{}t)}(c_{(f)}(d_{(t)},x_{f}),x_{f})$
$\displaystyle=$ $\displaystyle{\cal
R}(d_{(t)},x_{f})e^{\sum_{n=0}^{\infty}\bar{d}_{(t)n}~{}\partial^{n}_{x}|_{x=x_{f}}\left[x^{-\epsilon_{t}}Z_{(aux\,t)}(x,x)\right]+d_{(f)n}~{}\partial^{n}_{x}|_{x=x_{f}}\left[x^{-(1-\epsilon_{t})}\bar{Z}_{(aux\,t)}(x,x)\right]}$
(72)
where
$\displaystyle c_{(f)0}(d_{(t)},x_{f})=d_{(t)0}~{}~{}$
$\displaystyle~{}~{}\bar{c}_{(f)0}(d_{(t)},x_{f})=\bar{d}_{(t)0}$
$\displaystyle c_{(f)n}(d_{(t)},x_{f})=$
$\displaystyle\sum_{k=n}^{\infty}\binom{k-1}{n-1}d_{(t)k}\partial^{k-n}x_{f}^{-(1-\epsilon_{t})}$
$\displaystyle\bar{c}_{(f)n}(d_{(t)},x_{f})=$
$\displaystyle\sum_{k=n}^{\infty}\binom{k-1}{n-1}\bar{d}_{(t)k}\partial^{k-n}x_{f}^{\epsilon_{t}}$
(73)
and the normalization factor is
$\displaystyle{\cal R}(d_{(t)},x_{f})=$ $\displaystyle
e^{-\sum_{m,n=0}^{\infty}d_{(t)n}~{}\bar{d}_{(t)m}~{}\partial^{n-1}_{x}~{}\partial^{m-1}_{y}\left[x^{1-\epsilon_{t}}y^{\epsilon_{t}}~{}\partial_{x}\partial_{y}G^{z\bar{z}}_{bou,~{}reg~{}U(t_{a})}(x;y;\\{x=0,x=\infty\\})\right]}\Big{|}_{x=y=x_{f}}$
(74)
in order to undo the OPE and get the desired result as in eq. (4).
We shall now translate the previous operator (4) into an abstract operator we
can insert in the path integral at an arbitrary point $x_{t}$, therefore we
move it from $x_{t}=0$ to a generic $x_{t}$ and we consider a generating
vertex as
$\displaystyle{\cal T}(d_{(t)},x_{t})=$ $\displaystyle\lim_{x\rightarrow
x_{t}^{+}}{\cal N}_{\cal T}(d_{(t)},x,x_{t})~{}e^{\bar{d}_{(t)0}~{}\langle
Z(x,{\bar{x}})\rangle+d_{(t)0}~{}\langle\bar{Z}(x,{\bar{x}})\rangle}$
$\displaystyle
e^{\sum_{n=1}^{\infty}\bar{d}_{(t)n}~{}\partial^{n-1}_{x}\langle(x-x_{t})^{1-\epsilon_{t}}\partial_{x}Z(x,{\bar{x}})\rangle+d_{(t)n}~{}\partial^{n-1}_{x}\langle(x-x_{t})^{\epsilon_{t}}\partial_{x}\bar{Z}(x,{\bar{x}})\rangle}$
(75)
where $x\rightarrow x_{t}$ has to be understood as taking the limit after the
path integral has been computed. We have defined the averaged fields such as
$\langle(x-x_{t})^{1-\epsilon_{t}}\partial_{x}Z(x,{\bar{x}})\rangle=\int_{\partial
H}dy~{}\delta_{reg}(x-y)(y-x_{t})^{1-\epsilon_{t}}\partial_{y}Z(y,{\bar{y}})$
(76)
because we want a well defined regulated expression after performing the path
integral and introduced the normalization factor
$\displaystyle{\cal N}_{\cal T}(d_{(t)},x,x_{t})=$ $\displaystyle
e^{-\frac{1}{2}\bar{d}_{(t)0}^{2}\langle\langle
G^{zz}_{N=2~{}bou}(x;x)\rangle\rangle-\frac{1}{2}d_{(t)0}^{2}\langle\langle
G^{\bar{z}\bar{z}}_{N=2~{}bou}(x;x)\rangle\rangle-\bar{d}_{(t)0}d_{(t)0}\langle\langle
G^{z\bar{z}}_{N=2~{}bou}(x;x)\rangle\rangle}$ $\displaystyle
e^{-\frac{1}{2}\sum_{n=1}^{\infty}\bar{d}_{(t)0}d_{(t)n}\partial^{n-1}_{x}\langle\langle(x-x_{t})^{\epsilon_{t}}\partial_{x}G^{\bar{z}z}_{N=2~{}bou}(x;y)\rangle\rangle}$
$\displaystyle
e^{-\frac{1}{2}\sum_{n=1}^{\infty}d_{(t)0}\bar{d}_{(t)n}\partial^{n-1}_{x}\langle\langle(x-x_{t})^{1-\epsilon_{t}}\partial_{x}G^{z\bar{z}}_{N=2~{}bou}(x;y)\rangle\rangle}$
$\displaystyle
e^{-\frac{1}{2}\sum_{n,l=1}^{\infty}d_{(t)l}\bar{d}_{(t)n}\partial^{l-1}_{y}\partial^{n-1}_{x}\langle\langle(y-x_{t})^{\epsilon_{t}}(x-x_{t})^{1-\epsilon_{t}}\partial_{x}\partial_{y}G^{z\bar{z}}_{N=2~{}bou}(x;y)\rangle\rangle}\Big{|}_{y=x}$
(77)
where the doubly regularized Green functions are defined such as
$\displaystyle\langle\langle G^{zz}_{N=2~{}bou}(x;x)\rangle\rangle$
$\displaystyle=\int dy_{1}\int
dy_{2}~{}\delta_{reg}(x-y_{1})~{}\delta_{reg}(x-y_{2})$ $\displaystyle\hskip
70.0001ptG^{zz}_{bou}(y_{1};y_{2};\\{x_{1}=x_{t},x_{2}=\infty\\})$
$\displaystyle\langle\langle(y-x_{t})^{\epsilon_{t}}(x-x_{t})^{1-\epsilon_{t}}\partial_{x}\partial_{y}G^{z\bar{z}}_{N=2~{}bou}(x;y)\rangle\rangle$
$\displaystyle=\int dy_{1}\int
dy_{2}~{}\delta_{reg}(x-y_{1})~{}\delta_{reg}(y-y_{2})$
$\displaystyle(y_{2}-x_{t})^{\epsilon_{t}}~{}(y_{1}-x_{t})^{1-\epsilon_{t}}\partial_{1}\partial_{2}G^{z\bar{z}}_{bou}(y_{1};y_{2};\\{x_{1}=x_{t},x_{2}=\infty\\})$
(78)
Since the previous expression is in nuce the same as for the untwisted matter,
we can immediately deduce from (3) the result of inserting and integrating
over the $X$ to be eq. (33).
In analogy with what done for the untwisted states we can realize the algebra
$\left[d_{(t)n},\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(u)m}}\right]=\left[\bar{d}_{(t)n},\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial\bar{d}_{(u)m}}\right]=\delta_{m,n}`\delta_{u,t}$
(79)
with operators acting on the twisted scalar (dicharged string) auxiliary
Hilbert spaces ${\cal H}_{t}$ as
$\displaystyle 1\rightarrow$ $\displaystyle\langle
T_{\epsilon_{t}},x^{1}_{(t)0}=0|$
$\displaystyle\bar{d}_{(t)n\,}\rightarrow\frac{i}{\sqrt{2\alpha^{\prime}}\cos\gamma_{t}}\frac{\alpha_{(t)n-1+\epsilon_{t}}}{(n-1)!~{}(n-1+\epsilon_{t})},$
$\displaystyle~{}~{}d_{(t)n\,}\rightarrow\frac{i}{\sqrt{2\alpha^{\prime}}\cos\gamma_{t}}\frac{\bar{\alpha}_{(t)n-\epsilon_{t}}}{(n-1)!~{}(n-\epsilon_{t})}~{}~{}~{}n>0$
$\displaystyle\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial\bar{d}_{(t)m}}\rightarrow-i\sqrt{2\alpha^{\prime}}\,(m-1)!\,\cos\gamma_{t}\,\alpha_{(t)m-1+\epsilon_{t}}^{\dagger},$
$\displaystyle~{}~{}\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
d_{(t)m}}\rightarrow-i\sqrt{2\alpha^{\prime}}\,(m-1)!\,\cos\gamma_{t}\,\bar{\alpha}_{(t)m-\epsilon_{t}}^{\dagger}~{}~{}m>0$
(80)
which gives eq. (32) when substituted into eq. (33).
Acknowledgments We would like to thank P. Di Vecchia and F. Pezzella for
discussions. The author thanks the Nordita for hospitality during different
stages of this work.
## Appendix A Check of the $N=2$ amplitudes
We would now check that the operatorial amplitudes with $N=2$ and the path
integral approach give the same result, phases included. Let us consider the
tachyonic amplitude
$\displaystyle\langle\sigma_{-\epsilon,\lambda}(x_{\infty},x_{\infty})~{}\sigma_{\epsilon,\kappa}(x_{0},x_{0})~{}V_{T}(x_{1};k_{(1)})\dots~{}V_{T}(x_{M};k_{(M)})\rangle$
(81)
with $x_{t=1}=x_{0}$, $x_{t=2}=x_{\infty}$ and $\gamma_{0}$ and $\gamma_{1}$
arbitrary but $\gamma_{2}=\gamma_{0}$ so that
$\pi\epsilon_{1}=-\pi\epsilon_{2}$. Using the results from ([12]) we can
compute it in the limit $x_{0}\rightarrow 0$ and
$x_{\infty}\rightarrow\infty$, for $x_{1}>\dots x_{M}>0$ and when multiplied
by the appropriate power of $x_{\infty}$ as
$\displaystyle\langle
T_{\epsilon},-\lambda|e^{-\frac{1}{2}R^{2}(\epsilon)~{}\Delta(k_{(1)})}x_{1}^{-\Delta(k_{(1)})}e^{i(\bar{k}_{(1)}z_{0}+k_{(1)}\bar{z}_{0})}e^{i\cos\gamma_{1}[\bar{k}_{(1)}Z_{nzm}(x_{1},x_{1})+k_{(1)}\bar{Z}_{nzm}(x_{1},x_{1})]}\dots|T_{\epsilon},\kappa\rangle$
$\displaystyle=$ $\displaystyle\delta(\kappa+\kappa+\sum_{a}k_{(a)2})$
$\displaystyle\prod_{a}e^{\frac{1}{2}\pi\alpha^{\prime}\frac{1}{\tan\gamma_{1}-\tan\gamma_{0}}(k_{(a)}^{2}-\bar{k}_{(a)}^{2})}$
$\displaystyle\prod_{a}\Big{[}e^{-\frac{1}{2}R^{2}(\epsilon)~{}\Delta(k_{(a)})}x_{a}^{-\Delta(k_{(a)})}\Big{]}$
$\displaystyle\prod_{a}e^{-\pi\alpha^{\prime}\frac{1}{\tan\gamma_{1}-\tan\gamma_{0}}\frac{\kappa}{\sqrt{2}}(k_{(a)}-\bar{k}_{(a)})}$
$\displaystyle\prod_{a<b}\Big{[}e^{\pi\alpha^{\prime}\frac{1}{\tan\gamma_{1}-\tan\gamma_{0}}(k_{(a)}-\bar{k}_{(a)})(k_{(b)}+\bar{k}_{(b)})}e^{\alpha^{\prime}\cos^{2}\gamma_{1}(k_{(a)}\bar{k}_{(b)}\,g_{1-\epsilon}\left(\frac{x_{b}}{x_{a}}\right)+\bar{k}_{(a)}k_{(b)}\,g_{\epsilon}\left(\frac{x_{b}}{x_{a}}\right))}\Big{]}$
(82)
where we have used the commutation relations (• ‣ 1),
$R^{2}(\epsilon)=\lim_{u\rightarrow
1^{-}}[\,g_{\epsilon}(u)+\,g_{1-\epsilon}(u)-2\log(1-u)]=-\left(\psi(\epsilon)+\psi(1-\epsilon)-2\psi(1)\right)$
and $~{}\Delta(k_{(a)})=2\alpha^{\prime}\cos^{2}\gamma~{}k_{(a)}\bar{k}_{(a)}$
is the conformal dimension of the tachyonic vertex.
We can now compare with the general expression (33) with the identifications
$c_{(a)0}\rightarrow i~{}k_{(a)}$, $\bar{c}_{(a)0}\rightarrow
i~{}\bar{k}_{(a)}$, $d_{(1)0}\rightarrow i~{}\frac{\kappa}{\sqrt{2}}$,
$\bar{d}_{(1)0}\rightarrow i~{}\frac{\kappa}{\sqrt{2}}$, $d_{(2)0}\rightarrow
i~{}\frac{\lambda}{\sqrt{2}}$ and $\bar{d}_{(2)0}\rightarrow
i~{}\frac{\lambda}{\sqrt{2}}$. We can also compare with the expression (32)
upon the product with the state…
in order to understand where the different terms come from the path integral
point of view. In matching these terms is important to be careful in rewriting
all the Green functions $G(x;y)$ in such a way that $x>y$ by using the
symmetry (2) since this is the natural way they appear from the operatorial
formalism. We recognize that
* •
the factor
$\exp\\{\frac{1}{2}\pi\alpha^{\prime}\frac{1}{\tan\gamma_{1}-\tan\gamma_{0}}(k_{(a)}^{2}-\bar{k}_{(a)}^{2})\\}$
come from the $G^{zz}_{bou~{}reg~{}U(t_{a})}$ and
$G^{\bar{z}\bar{z}}_{bou~{}reg~{}U(t_{a})}$ terms.
* •
The factors
$\prod_{a}\Big{[}e^{-\frac{1}{2}R^{2}(\epsilon)~{}\Delta(k_{(a)})}x_{a}^{-\Delta(k_{(a)})}\Big{]}$
come from the $G^{z\bar{z}}_{bou~{}reg~{}U(t_{a})}$ terms. In particular the
result follows from the following steps
$\displaystyle G^{z\bar{z}}_{bou~{}reg~{}U(t_{a})}(x_{a}^{+},x_{a})=$
$\displaystyle\frac{\pi\alpha^{\prime}}{\tan\gamma_{1}-\tan\gamma_{0}}-\pi\alpha^{\prime}\sin\gamma_{1}\cos\gamma_{1}+2\alpha^{\prime}\cos^{2}\gamma_{1}\ln|x_{a}|$
$\displaystyle-2\alpha^{\prime}\cos^{2}\gamma_{1}\left(\,g_{\epsilon}\left(\frac{x_{a}}{x_{a}^{+}}\right)-\log\left(1-\frac{x_{a}}{x_{a}^{+}}\right)\right)$
$\displaystyle=\frac{\pi\alpha^{\prime}}{\tan(\gamma_{1}-\gamma_{0})}\cos^{2}\gamma_{1}+2\alpha^{\prime}\cos^{2}\gamma_{1}\ln|x_{a}|$
$\displaystyle~{}~{}-2\alpha^{\prime}\cos^{2}\gamma_{1}\left(\psi(1-\epsilon)-\psi(1)\right)$
$\displaystyle=2\alpha^{\prime}\cos^{2}\gamma_{1}\ln|x_{a}|-\alpha^{\prime}\cos^{2}\gamma_{1}\left(\psi(\epsilon)+\psi(1-\epsilon)-2\psi(1)\right)$
(83)
where we have used the expression for the Green function given in eq. (• ‣ 1)
since we have chosen $x=x_{a}^{+}$ and in the last line we have used the
digamma property $\psi(1-\epsilon)=\psi(\epsilon)+\pi\cot(\pi\epsilon)$. It is
also worth stressing that the result is independent on setting $x_{a}^{+}$ in
the first argument since the function $G^{z\bar{z}}_{bou~{}reg~{}U(t_{a})}$ is
continued analytically at $x=y=x_{a}$ in such a way that
$G^{ij}(x;y)=G^{ji}(y;x)$ so that would we have chosen the first argument to
be $x_{a}^{-}$ we would have got the same result computing
$G^{\bar{z}z}_{bou~{}reg~{}U(t_{a})}(x_{a},x_{a}^{-})$.
* •
The terms
$e^{-\pi\alpha^{\prime}\frac{1}{\tan\gamma_{1}-\tan\gamma_{0}}\frac{\kappa}{\sqrt{2}}(k_{(a)}-\bar{k}_{(a)})}$
arise from the $\prod_{t,a}$ terms. While rewriting the Green functions
$G(x;y)$ in such a way that $x>y$ we see that the terms with $t=2$ (at
$x_{\infty}$) cancel while those from $t=1$ (at $x_{0}$) do not and reproduce
the operatorial result.
* •
The terms $\prod_{a<b}$ come trivially from the corresponding ones in eq.
(33).
* •
The terms $\prod_{t}$ in (33) give a trivial result in a non trivial way. It
is immediate to find that
$G^{zz}_{bou,~{}reg~{}(t)}=G^{\bar{z}\bar{z}}_{bou,~{}reg~{}(t)}=0$. On the
other side we get for $0<\frac{y-x_{0}}{x-x_{0}}<1$ and
$0<\omega=\frac{y-x_{0}}{x-x_{0}}\frac{x-x_{\infty}}{y-x_{\infty}}<1$
$\displaystyle G^{z\bar{z}}_{bou,~{}reg~{}(t)}(x;y;\\{x_{0},x_{\infty}\\})$
$\displaystyle=-2\alpha^{\prime}\cos^{2}\gamma\left[\,g_{\epsilon}(\omega)-\,g_{\epsilon}(\frac{y-x_{0}}{x-x_{0}})\right]$
(84)
Now we can write $x=x_{0}+\alpha$, $y=x_{0}+\alpha(1-\delta)$ with $\alpha>0$
and $0<\delta<1$ and expand in $\delta$ to get
$\displaystyle G^{z\bar{z}}_{bou,~{}reg~{}(t)}(x;y;\\{x_{0},x_{\infty}\\})$
$\displaystyle=-2\alpha^{\prime}\cos^{2}\gamma\left[-\log\left(1+\frac{\alpha}{x_{\infty}-x_{0}}\right)+\frac{(\epsilon-1)\alpha}{x_{\infty}-x_{0}+\alpha}\delta+O(\delta^{2})\right]$
(85)
which vanishes when $\alpha=\delta=0$. In a similar way can be treated the
case when $x,y\rightarrow x_{\infty}$.
* •
The terms $\prod_{t<u}$ in (33) give also a trivial result in a not completely
trivial fashion. We have to evaluate the Green functions for
$\omega=\frac{y-x_{0}}{x-x_{0}}\frac{x-x_{\infty}}{y-x_{\infty}}$ when
$x=x_{0}$ and $y=x_{\infty}$ so $\omega=0$ and we are left with only the
constant terms,as we expect from the general asymptotic (1) hence
$\prod_{t<u}=\exp\\{\frac{\pi\alpha^{\prime}}{\tan\gamma_{1}-\tan\gamma_{0}}\
[-d_{(1)0}d_{(2)0}+\bar{d}_{(1)0}\bar{d}_{(2)0}-d_{(1)0}\bar{d}_{(2)0}+\bar{d}_{(1)0}d_{(2)0}]\\}$
which vanishes when evaluated with the previously stated substitutions for
which $d_{(t)0}=\bar{d}_{(t)0}$.
## Appendix B Behavior of the Green function when $x,y\rightarrow x_{t}$
In this section we follow and adapt the computation done in ([6]). We start
considering the derivative of Green function of the left moving part defined
as
$\partial_{z}\partial_{w}G^{z\bar{z}}_{LL}(z;w;\\{x_{t}\\}_{t=1\dots
N})=\frac{\langle\partial_{z}Z_{L}(z)\partial_{w}\bar{Z}_{L}(w)\sigma_{\epsilon_{1},\kappa_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\rangle_{disk}}{\langle\sigma_{\epsilon_{1},\kappa_{1}}(x_{1},\bar{x}_{1})\dots\sigma_{\epsilon_{N},\kappa_{N}}(x_{N},\bar{x}_{N})\rangle_{disk}}$
(86)
which has asymptotics
$\displaystyle-\frac{1}{2\alpha^{\prime}}\partial_{z}\partial_{w}G^{z\bar{z}}_{LL}(z;w;\\{x_{t}\\}_{t=1\dots
N})$ $\displaystyle\sim_{z\rightarrow w}\frac{1}{(z-w)^{2}}+O(1)$
$\displaystyle\sim_{z\rightarrow x_{t}}(z-x_{t})^{\epsilon_{t}-1}$
$\displaystyle\sim_{w\rightarrow x_{t}}(w-x_{t})^{-\epsilon_{t}}$ (87)
then we can write
$\displaystyle-\frac{1}{2\alpha^{\prime}}\partial_{z}\partial_{w}G^{z\bar{z}}_{LL}(z;w;\\{x_{t}\\}_{t=1\dots
N})=$
$\displaystyle\prod_{u}\left(\frac{w-x_{u}}{z-x_{u}}\right)^{1-\epsilon_{u}}\Big{[}\frac{1}{(z-w)^{2}}\sum_{u<v}a_{uv}\frac{(z-x_{u})(z-x_{v})}{(w-x_{u})(w-x_{v})}$
$\displaystyle+\sum_{u_{1}<u_{2}<u_{3}<u_{4}}\frac{b_{u_{1}u_{2}u_{3}u_{4}}}{(w-x_{u_{1}})(w-x_{u_{2}})(w-x_{u_{3}})(w-x_{u_{4}})}\Big{]}$
(88)
Now we can study the behavior $x,y\rightarrow x_{1}$ by setting
$z=x_{1}+\alpha,~{}~{}~{}~{}w=x_{1}+\alpha(1-\delta)$ (89)
and letting $\alpha,\delta\rightarrow 0^{+}$. A simple computation gives
$\displaystyle-\frac{1}{2\alpha^{\prime}}\partial_{z}\partial_{w}G^{z\bar{z}}_{LL}(z;w;\\{x_{t}\\}_{t=1\dots
N})\sim$
$\displaystyle\frac{\sum_{u<v}a_{uv}}{\delta^{2}}+\frac{1}{\alpha}\frac{1}{1-\delta}\sum_{1<u_{2}<u_{3}<u_{4}}\frac{b_{1u_{2}u_{3}u_{4}}}{(x_{1}-x_{u_{2}})(x_{1}-x_{u_{3}})(x_{1}-x_{u_{4}})}+O(1)$
(90)
from which we deduce the constraint
$\sum_{u<v}a_{uv}=1.$ (91)
Then the regularized Green function which is obtained by subtracting the
corresponding Green function with only two twist, one of which in $x_{1}$ and
the other at $\infty$ is of the form
$\displaystyle\partial_{z}\partial_{w}G^{z\bar{z}}_{LL,~{}reg}(z;w;\\{x_{t}\\}_{t=1\dots
N})\sim$ $\displaystyle\frac{B_{1}(x_{u})}{\alpha}\frac{1}{1-\delta}+O(1)$
(92)
hence the terms in $\prod_{t}$ are well defined since
$\displaystyle(z-x_{1})^{1-\epsilon_{1}}~{}(w-x_{1})^{\epsilon_{1}}~{}\partial_{z}\partial_{w}G^{z\bar{z}}_{LL,~{}reg}(z;w;\\{x_{t}\\}_{t=1\dots
N})\sim$
$\displaystyle\alpha~{}(1-\delta)^{\epsilon_{1}}\left[\frac{B_{1}(x_{u})}{\alpha}\frac{1}{1-\delta}+O(1)\right].$
(93)
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|
arxiv-papers
| 2011-07-27T16:31:15 |
2024-09-04T02:49:20.999939
|
{
"license": "Public Domain",
"authors": "Igor Pesando",
"submitter": "Pesando Igor",
"url": "https://arxiv.org/abs/1107.5525"
}
|
1107.5784
|
# Persistence of Topological Order and Formation of Quantum Well States in
Topological Insulators Bi2(Se,Te)3 under Ambient Conditions
Chaoyu Chen1, Shaolong He1, Hongming Weng1, Wentao Zhang1, Lin Zhao1, Haiyun
Liu1, Xiaowen Jia1, Daixiang Mou1, Shanyu Liu1, Junfeng He1, Yingying Peng1,
Ya Feng1, Zhuojin Xie1, Guodong Liu1, Xiaoli Dong1, Jun Zhang1, Xiaoyang
Wang2, Qinjun Peng2, Zhimin Wang2, Shenjin Zhang2, Feng Yang2, Chuangtian
Chen2, Zuyan Xu2, Xi Dai1, Zhong Fang1, X. J. Zhou1,∗
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
2Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
(July 28, 2011)
The topological insulators represent a unique state of matter where the bulk
is insulating with an energy gap while the surface is metallic with a Dirac
cone protected by the time reversal symmetryTin3D ; ZhangQiPT ; HasanKaneRev ;
ZhangQiRMP ; JMoorePer . These characteristics provide a venue to explore
novel quantum phenomena in fundamental physicsXLQiNP ; RDLiAxions ; XLQiMomo ;
LFuMajorana ; RYuSience and show potential applications in spintronics and
quantum computingRevnextgeneration ; ZhangQiPT ; JMoorePer . One critical
issue directly related with the applications as well as the fundamental
studies is how the topological surface state will behave under ambient
conditions (1 atmosphere air and room temperature). In this paper, we report
high resolution angle-resolved photoemission measurements on the surface state
of the prototypical topological insulators, Bi2Se3, Bi2Te3 and
Bi2(Se0.4Te2.6), upon exposing to ambient conditions. We find that the
topological order persists even when the surface is exposed to air at room
temperature. However, the surface state is strongly modified after such an
exposure. Particularly, we have observed the formation of two-dimensional
quantum well states near the surface of the topological insulators after the
exposure which depends sensitively on the original composition, x, in
Bi2(Se3-xTex). These rich information are crucial in utilizing the surface
state and in probing its physical properties under ambient conditions.
The angle-resolved photoemission spectroscopy (ARPES) is a powerful
experimental tool to directly identify and characterize topological
insulatorsHasanARPES . A number of three-dimensional topological insulators
have been theoretically predicted and experimentally identified by
ARPESBiSbARPES ; BiSeTheo ; Bi2Se3ARPES ; Bi2Te3ARPES ; TlBiSe2Theo ;
TlBiSe2ARPES ; some of their peculiar properties have been revealed by
scanning tunneling microscopy (STM)TZhangSTM ; YazdaniSTM ; KaptulnikSTM ;
PCheng ; Hanaguri . The application of the topological surface states depends
on the surface engineering that can be manipulated by incorporation of non-
magneticLAWraypn ; mnmi ; ccvb ; crs or magneticYLChenFe ; ZHasanPertur ;
LAWraypn ; mnmi impurities or gas adsorptionsZHasanPertur ; LPlucinski ;
LAWraypn ; HMBenia . While the ARPES and STM measurements usually involve the
fresh surface obtained by cleaving samples in situ under ultra-high vacuum,
for the transport measurements which are widely used to investigate the
intrinsic quantum behaviors of the topological surface stateDXQu ; JGAnalytis
; ZRen ; JChen , and particularly the ultimate utilizations of the topological
insulators, the surface may be exposed to ambient conditions (1 atmosphere air
and room temperature) or some gas protection environment. It is therefore
crucial to investigate whether the topological order can survive under the
ambient conditions, and furthermore, whether and how the surface state may be
modified after such exposures.
We start by first looking at the electronic structure of the prototypical
topological insulators Bi2(Se,Te)3 under ultra-high vacuum. The Fermi surface
and the band structure of the Bi2(Se3-xTex) topological insulators depend
sensitively on the composition, x, as shown in Fig. 1. The single crystal
samples here were all cleaved in situ and measured at 30 K in an ultra-high
vacuum (UHV) chamber with a base pressure better than 5 $\times$ 10-11 Torr.
For Bi2Se3, a clear Dirac cone appears near -0.36 eV (Figs. 1d and 1e); the
corresponding Fermi surface (Fig. 1a) is nearly circular but with a clear
hexagon-shape in the measured dataHexBi2Se3 . It is apparently of n-type
because the Fermi level intersects with the bulk conduction band. On the other
hand, the Dirac cone of the Bi2Te3 sample lies near -0.08 eV (Figs. 1h and
1i), much closer to the Fermi level than that reported before (-0.34 eV in
Bi2Te3ARPES ). The corresponding Fermi surface (Fig. 1c) becomes rather small,
accompanied by the appearance of six petal-like bulk Fermi surface sheets.
These results indicate that our Bi2Te3 sample is of p-type because the Fermi
level intersects the bulk valence band along the
$\overline{\Gamma}$-$\overline{M}$ direction. This is also consistent with the
positive Hall coefficient measured on the same Bi2Te3 sampleCZhang . This
difference of the Fermi surface topology and the location of the Dirac cone
from othersBi2Te3ARPES may be attributed to the different carrier
concentration in Bi2Te3 due to different sample preparation conditions. In our
Bi2(Se3-xTex) samples, we have seen a crossover from n-type Bi2Se3 to p-type
Bi2Te3. In order to eliminate the interference of the bulk bands on the
surface state near the Fermi level, we fine tuned the composition x in
Bi2(Se3-xTex) and found that, for x=2.6, nearly no spectral weight can be
discerned from the bulk conduction band, as seen from both the Fermi surface
(Fig. 1b) and the band structure (Figs. 1f and 1g). A slight substitution of
Te by Se in Bi2(Se0.4Te2.6) causes a dramatic drop of the Dirac point to -0.31
eV (Figs. 1f and 1g) and an obvious hexagon-shaped Fermi surface (Fig. 1b). It
is interesting to note that the hexagon-shape of Bi2(Se0.4Te2.6) (Fig. 1b) is
rather pronounced, although its Fermi surface size is smaller than that of
Bi2Se3 (Fig. 1a). The hexagonally-shaped Fermi surface observed in the
topological surface states reflects the hybridization of surface electronic
states with the bulk states and can be theoretically explained by considering
the higher order terms in the k $\cdot$ p HamiltonianHexWarp .
In order to directly examine how the topological surface state behaves under
ambient conditions in the topological insulators, we carried out our ARPES
measurements in different ways. (1). We first cleaved the sample _in situ_ and
performed ARPES measurement in the ultra-high vacuum (UHV) chamber. The sample
was then pulled out to another chamber filled with 1 atmosphere N2 gas,
exposed for about 5 minutes, before transferring back to the UHV chamber to do
ARPES measurements; (2). We cleaved and measured the sample in the UHV
chamber, and then pulled the sample out to air for 5 minutes before
transferring back to the UHV chamber for the ARPES measurements; (3). We
cleaved the sample in air and then transferred it to the UHV chamber to do the
ARPES measurement. Our measurements show that the above procedures of exposure
to air or N2 produce similar and reproducible results for a given sample.
The surface exposure of the topological insulators to air or N2 gives rise to
a dramatic alteration of the surface state, as shown in Figs. 2, 3 and 4, for
Bi2Se3, Bi2(Se0.4Te2.6), and Bi2Te3, respectively, when compared with those
for the fresh surface (Fig. 1). The first obvious change is the shifting of
the Dirac cone position relative to the Fermi level. For Bi2Se3,
Bi2(Se0.4Te2.6) and Bi2Te3, it shifts from the original -0.36 eV (Figs. 1d and
1e), -0.31 eV (Figs. 1f and 1g), -0.08 eV (Figs. 1h and 1i) for the fresh
surface to -0.48 eV (Fig. 2b), -0.40 eV (Figs. 3a and 3b), and -0.28 eV [Figs.
4(c-f)] at 30 K for the exposed surface, respectively. In all these cases, the
shift of the Dirac cone to a larger binding energy indicates an additional
doping of electrons into the surface state. The exposure also gives rise to a
dramatic change of the surface Fermi surface. For Bi2Se3, in addition to a
slight Fermi surface size increase, an obvious change occurs in the Fermi
surface shape that the hexagon-shape becomes much more pronounced in the
exposed surface (Fig. 2d) than that in the fresh sample (Fig. 1a). For
Bi2(Se0.4Te2.6), one clearly observes the much-enhanced warping effect in the
exposed surface (Fig. 3c) when compared with the nearly standard hexagon in
the fresh surface (Fig. 1b). The most dramatic change occurs for Bi2Te3 where
not only the Fermi surface size increases significantly, but also the warping
effect in the exposed surface (Fig. 4i) becomes much stronger. Overall, the
exposure causes the lowering of the Dirac cone position, an increase of the
surface Fermi surface size, and an obvious enhancement of the Fermi surface
warping effect in the Bi2(Se3-xTex) system.
The topological order in the Bi2(Se3-xTex) topological insulators is robust
even when the surface is exposed to ambient conditions, in spite of all the
alterations mentioned above. One clearly observes the persistence of the Dirac
cone in the exposed surface as in Bi2Se3 (Figs. 2b and 2c), in Bi2(Se0.4Te2.6)
(Figs. 3a and 3b), and Bi2Te3 (Figs. 4(c-h)). Particularly, this is the case
for the surface exposed to air and measured at room temperature (Fig. 2c for
Bi2Se3, and Figs. 4g and 4h for Bi2Te3). On the other hand, after the
exposure, although the signal of the surface state gets weaker for Bi2Se3
(Fig. 2), it remains rather strong for Bi2(Se0.4Te2.6) (Fig. 3) and Bi2Te3
(Fig. 4). This is in a stark contrast to the conventional trivial surface
state where minor surface contamination will cause the extinction of the
surface stateHuefner . The robustness of the topological order to Coulomb,
magnetic and disorder perturbations has been reported beforeZHasanPertur ;
LPlucinski . Our present observations directly demonstrate the robustness of
the topological order against absorption and thermal process under ambient
conditions due to the protection of the time-reversal symmetryHasanKaneRev ;
ZhangQiRMP .
The surface exposure to air or N2 in the Bi2(Se3-xTex) topological insulators
produces two-dimensional electronic states near the surface. In Bi2Se3, the
exposure gives rise to additional parabolic bands, as schematically marked by
the dashed line in Figs. 2b and 2c. Correspondingly, this leads to additional
Fermi surface sheet(s) inside the regular topological surface state (Figs. 2d
and 2e). In Bi2(Se0.4Te2.6), this effect gets more pronounced and the newly
emerged bulk conduction band splits into several discrete bands, as marked by
the dashed lines in Fig. 3b. While the band quantization effect occurs in the
bulk conduction band in Bi2(Se0.4Te2.6), it shows up in the valence band in
the exposed Bi2Te3 surface, as shown in Figs. 4(c-h), where one can see a
couple of discrete “M”-shaped bands. The quantized bands are obvious at low
temperature and get slightly smeared out when the temperature rises to room
temperature (Figs. 4g and 4h).
The formation of the split bands in the exposed surface of the topological
insulators is reminiscent of the quantum well states observed in the quantum
confined systemsSiAg and in some topological insulatorstdeg ; mnmi ; crs ;
ccvb ; HMBenia ; KingReshba . There are a couple of possibilities that the
quantum well states may be formed. One usual way is due to band bending
effect. As mentioned before, the surface exposure to air or N2 causes an
electron transfer to the surface of the topological insulators. The
accumulation of these additional electrons near the surface would lead to a
downward bending of the bulk bands near the surface region, as schematically
shown in Fig. 3d, resulting in a “V”-shaped potential well where the bulk
conduction band of electrons can be confined. This picture, as proposed
beforetdeg ; ccvb , seems to be able to explain the two-dimensional quantum
well states in the conduction bands in Bi2(Se0.4Te2.6) (Figs. 3a and 3b).
However, it becomes questionable to explain the quantum well states observed
in the bulk valence band of Bi2Te3 (Figs. 4c-j). In this case, the downward
band bending no longer acts as a quantum well potential for the valence band
top because the charge carriers are hole-like. The band-bending is therefore
not a general picture that can explain the formation of the two-dimensional
quantum well states in Bi2(Se0.4Te2.6) (Figs. 3a and 3b) and Bi2Te3 (Fig. 4)
topological insulators on the same footing. An alternative scenario is the
expansion of van der Waals spacings in between the quintuple layers (QLs)
caused by the intercalation of gasesSVEremeev . The observation of multiple
split bands with different spacings would ask for multiple van der Waals gaps
with different expansions. Whether and how these can be realized in the
exposed surface remains to be investigated. We note that our observations of
multiple split bands are similar to those seen in the ultra-thin films of
Bi2Se3YZhang and Bi2Te3YYLi . From our first principle band structure
calculations on Bi2Te3 with different number of quintuple layers, we also find
that a detached slab with a thickness of 7 quintuple layers can give a rather
consistent description (Fig. 4l) of our observed results in terms of the
quantitative spacings between the 3 resolved bands (VB0, VB1, and VB2 bands as
marked in Figs. 4c and 4l). In addition, the distance between the conduction
band bottom (CB0 band in Figs. 4i and 4l) and the first valence sub-band
bottom (VB0 band in Figs. 4i and 4l) is rather consistent between the measured
and calculated results. These seem to suggest that a “confined surface slab”
with nearly 7 quintuple layers may be formed after the exposure that acts more
or less independently from the bulk. More work needs to be done to further
investigate whether such a confined surface slab can be thermodynamically
stable. Overall, the formation of the two-dimensional quantum well states is a
general phenomenon for the exposed surface of the Bi2(Se3-xTex) topological
insulators; the effect depends sensitively on the composition x of the samples
which may facilitate manipulation of these quantum well states.
The present work has significant implications on the fundamental study and
ultimate applications of the topological insulators. Many experimental
measurements, such as some transport measurements, involve samples exposed to
ambient conditions. The practical applications may involve sample surface
either exposed to ambient condition, or be in contact with other magnetic or
superconducting materials. On the one hand, the robustness of the topological
order under ambient conditions sends a good signal for these experimental
characterization and practical utilizations. The formation of the quantum well
states may give rise to new phenomena to be studied and utilized. The
sensitivity of the surface state to the Bi2(Se3-xTex) composition provides a
handle to manipulate these quantum states. On the other hand, the strong
modification of the electronic structure and the formation of additional
quantum well states in the exposed surface have to be considered seriously in
interpreting experimental data and in surface engineering. It is critical to
realize before hand that the surface under study or to be utilized may exhibit
totally different behaviors as those from the fresh surface cleaved in ultra-
high vacuum. In addition to the alteration of electronic states upon exposure,
the transport properties of the topological surface state may be further
complicated by the formation of quantum well states.
We thank Prof. X. H. Chen for providing us samples at the initial stage of the
project, and Prof. Liling Sun and Prof. Zhong-xian Zhao for their help in the
characterization of the samples. This work is supported by the NSFC (91021006)
and the MOST of China (973 program No: 2011CB921703).
METHODS
Crystal growth methods Single crystals of $Bi_{2}(Se_{3-x}Te_{x})$ (x=0, 2.6
and 3) were grown by the self-flux method. Bismuth, selenium and tellurium
powders were weighed according to the stoichiometric $Bi_{2}(Se_{3-x}Te_{x})$
(x=0, 2.6 and 3) composition. After mixing thoroughly, the powder was placed
in alumina crucibles and sealed in a quartz tube under vacuum. The materials
were heated to 1000 ∘C, held for 12 hours to obtain a high degree of mixing,
and then slowly cooled down to 500 ∘C over 100 hours before cooling to room
temperature. Single crystals of several millimeters in size were obtained. The
crystal structure of the resulting crystals was examined by use of a rotating
anode x-ray diffractometer with Cu _K α_ radiation ($\lambda$ = 1.5418 Å). The
chemical composition of the crystals was analyzed by the energy dispersive
X-ray spectroscopy (EDAX) and the induction-coupled plasma atomic emission
spectroscopy (ICP-AES). The resistivity of the crystals was measured by the
standard four-probe method.
Laser-ARPES methods. The angle-resolved photoemission measurements were
carried out on our vacuum ultra-violet (VUV) laser-based angle-resolved
photoemission systemLiuARPES . The photon energy of the laser is 6.994 eV with
a bandwidth of 0.26 meV. The energy resolution of the electron energy analyzer
(Scienta R4000) is set at 1 meV, giving rise to an overall energy resolution
of $\sim$1 meV which is significantly improved from 10$\sim$15 meV from
regular synchrotron radiation systemsBi2Se3ARPES ; Bi2Te3ARPES . The angular
resolution is $\sim$0.3∘, corresponding to a momentum resolution $\sim$0.004
$\AA$-1 at the photon energy of 6.994 eV, more than twice improved from 0.009
$\AA$-1 at a regular photon energy of 21.2 eV for the same angular resolution.
Our superior instrumental resolution of laser ARPES has made the measured
features of topological insulators in this work much sharper. The Fermi level
is referenced by measuring on a clean polycrystalline gold that is
electrically connected to the sample. The samples were all measured in vacuum
with a base pressure better than 5$\times$10-11 Torr.
∗Corresponding author: XJZhou@aphy.iphy.ac.cn
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Figure 1: Fermi surface and band structure of Bi2(Se3-xTex) (x=0, 2.6, 3)
topological insulators cleaved in situ and measured at 30 K in ultra-high
vacuum. (a-c) show Fermi surface of Bi2Se3, Bi2(Se0.4Te2.6) and Bi2Te3,
respectively. The Fermi surface here, and in other figures below, are original
data without involving artificial symmetrization. The band structures along
two high symmetry lines $\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ are
shown in (d,e) for Bi2Se3, in (f,g) for Bi2(Se0.4Te2.6) and (h,i) for Bi2Te3.
Figure 2: Fermi surface and band structure of Bi2Se3 cleaved in air and
measured in the ultra-high vacuum (UHV) chamber. (a). Band structure of the
fresh Bi2Se3 cleaved and measured in the UHV chamber at 30 K along
$\bar{\Gamma}-\bar{M}$ direction. (b). Band structure of Bi2Se3 cleaved in air
and measured in UHV at 30 K along $\bar{\Gamma}-\bar{M}$ direction. (c). Band
structure of Bi2Se3 cleaved in air and measured in UHV at 300 K along
$\bar{\Gamma}-\bar{M}$ direction. (d,e). Fermi surface of Bi2Se3 cleaved in
air and measured in UHV at 30K and 300 K, respectively. Black dashed lines in
(b) and (c) mark the parabolic bands above the Dirac point from the two-
dimensional electron gas. Figure 3: Emergence of quantum well states in
Bi2Te2.6Se0.4 after exposing to N2. (a,b). Band structure measured at 30 K
along $\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$, respectively. Black
dashed lines in (b) mark the quantum well states formed in the bulk conduction
band (BCB) above the Dirac point. (c). The corresponding Fermi surface. It
shows three-fold symmetry where three corners of M points are strong while the
other three are weak. This is also in agreement with the asymmetric band
structure in Fig. 3b. (d). Schematic band structure showing the possible
formation of the quantum well states near the sample surface in the bulk
conduction band. The blue dotted lines between the bulk valence band (BVB) and
bulk conduction band (BCB) represent the topological surface states while the
blue solid lines represent quantum well states. Figure 4: Persistence of
topological surface state and formation of quantum well states in Bi2Te3 after
exposure to N2 or air. The sample was first cleaved and measured in UHV at 30
K. (a,b) show the corresponding band structure along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. The sample was
then pulled out from the UHV chamber and exposed to N2 at 1 atmosphere for 5
minutes before transferring back into UHV chamber for the ARPES measurement.
(c,d) show the band structure of the N2-exposed sample along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. The black dashed
lines in (c) illustrate the quantum well states formed in the bulk valence
band below the Dirac point. The sample was then pulled out again and exposed
to air for 5 minutes before putting back in vacuum for ARPES measurement.
(e,f) show the band structure of the air-exposed sample at 30 K along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. (g,h) show the
measurements at 300 K and (i,j) show their corresponding second-derivative
images in order to highlight the bands. (k) Fermi surface of N2-exposed
sample. (l). First principle calculation of the band structure of Bi2Te3 slab
with seven quintuple layers.
|
arxiv-papers
| 2011-07-28T18:18:42 |
2024-09-04T02:49:21.015729
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chaoyu Chen, Shaolong He, Hongming Weng, Wentao Zhang, Lin Zhao,\n Haiyun Liu, Xiaowen Jia, Daixiang Mou, Shanyu Liu, Junfeng He, Yingying Peng,\n Ya Feng, Zhuojin Xie, Guodong Liu, Xiaoli Dong, Jun Zhang, Xiaoyang Wang,\n Qinjun Peng, Zhimin Wang, Shenjin Zhang, Feng Yang, Chuangtian Chen, Zuyan\n Xu, Xi Dai, Zhong Fang and X. J. Zhou",
"submitter": "Xingjiang Zhou",
"url": "https://arxiv.org/abs/1107.5784"
}
|
1107.5840
|
# Higher symmetries of the Laplacian via quantization
J.-P. Michel University of Luxembourg, Campus Kirchberg, Mathematics Research
Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Grand Duchy
of Luxembourg jean-philippe.michel@uni.lu
###### Abstract.
We develop a new approach, based on quantization methods, to study higher
symmetries of invariant differential operators. We focus here on conformally
invariant powers of the Laplacian over a conformally flat manifold, and
recover results of Eastwood, Leistner, Gover and Šilhan.
In particular, conformally equivariant quantization establishes a clear
correspondence between hamiltonian symmetries of the null geodesic flow and
the algebra of higher symmetries of the conformal Laplacian. Resorting to
symplectic reduction, this leads to a quantization of the minimal nilpotent
coadjoint orbit of the conformal group and allows to identify the latter
algebra of symmetries in terms of the Joseph ideal. By the way, we obtain a
tangential star-product for a family of coadjoint orbits of the conformal
group.
###### Key words and phrases:
Symmetry algebra, Laplacian, Quantization, Conformal geometry, Minimal
nilpotent orbit, Symplectic reduction.
###### 2010 Mathematics Subject Classification:
58J10, 53A30, 70S10, 17B08, 53D20, 53D55
I thank the Luxembourgian NRF for support via the AFR grant PDR-09-063.
## 1\. Introduction
The higher symmetries of a differential operator $P$ are the differential
operators $D_{1}$ satisfying $PD_{1}=D_{2}P$ for some differential operator
$D_{2}$. They clearly form an algebra and preserve the kernel of $P$. The
first order ones define a Lie algebra ${\mathfrak{g}}$ generalizing the
invariance Lie algebra of $P$. The determination of the space of higher
symmetries of $P$, together with its algebra and ${\mathfrak{g}}$-module
structure, is of interest from at least two point of view: the integrability
of the equation $P\phi=0$, with $\phi$ in the source space of $P$, and the
representation theory of the ${\mathfrak{g}}$-module $\ker P$.
The first example which deserves to be investigated is certainly the conformal
(or Yamabe) Laplacian, that we consider on a conformally flat manifold
$(M,\mathrm{g})$ of arbitrary signature $(p,q)$ and dimension $n=p+q\geq 3$.
It writes as
$\Delta=\nabla_{i}\,\mathrm{g}^{ij}\nabla_{j}+\frac{n-2}{4(n-1)}R$, where
$\nabla$ is the Levi-Civita connection and $R$ the scalar curvature, and it
sends $\lambda$\- to $\mu$-densities, with $\lambda=\frac{n-2}{2n}$,
$\mu=\frac{n+2}{2n}$. On one hand, $\Delta\phi=0$ is the most basic wave
equation, describing e.g. a free massless quantum particle on
$(M,\mathrm{g})$. Its integration has been achieved in various contexts
resorting to symmetries, and we highlight here only two seminal works. On
$\mathbb{R}^{3}$, Boyer, Kalnins and Miller have classified all the second
order symmetries of the Laplacian [10], which allows them to get all the
possible coordinates systems separating the equation $\Delta\phi=0$. For
curved but Ricci-flat manifolds, Carter has built up a way to get symmetries,
by quantization of Killing tensors [13], and has successfully used it to
integrate on Kerr space-time the wave equation $(\Delta+m^{2})\phi=0$, with a
mass term $m\in\mathbb{R}$. On the other hand, the first order symmetries of
the conformal Laplacian are given by
$\Delta(X+\lambda\mathrm{Div}X)=(X+\mu\mathrm{Div}X)\Delta,$
where $\mathrm{Div}$ is the divergence operator and
$X\in{\mathfrak{g}}\simeq\mathrm{o}(p+1,q+1)$ is a conformal Killing vector.
This provides a representation of ${\mathfrak{g}}$ on $\ker\Delta$ which
integrates, if $M=\mathbb{S}^{p}\times\mathbb{S}^{q}$ and $p+q$ even, to a
unitary irreducible representation of the Lie group $G=\mathrm{O}(p+1,q+1)$
[5]. This is the intensively studied minimal representation of $G$, see e.g.
[27, 28]. It is linked to the minimal nilpotent coadjoint orbit
$\mathcal{O}_{00}$ of $G$ through the Joseph ideal [25], which appears as the
kernel of the representation of $\mathfrak{U}({\mathfrak{g}})$, the universal
enveloping algebra of ${\mathfrak{g}}$, on $\ker\Delta$. Nevertheless, this
representation cannot be obtained via the Kirillov’s orbit method [26] since
$\mathcal{O}_{00}$ admits no invariant polarization [43]. In the complex
setting, the alternative deformation program has been applied successfully by
Arnal, Benamor and Cahen, who have proved existence and uniqueness of a graded
${\mathfrak{g}}$-equivariant star product on regular functions on
$\mathcal{O}_{00}^{\mathbb{C}}$ [1]. From this star product, Astashkevich and
Brylinski have built a unitary irreducible representation of $G^{\mathbb{C}}$
[2], but the way to recover the real representation of $G$ on $\ker\Delta$ is
unclear.
Resorting to conformal ambient space, Eastwood has been able to determine the
space of higher symmetries of the conformal Laplacian [19]. The latter
corresponds as a ${\mathfrak{g}}$-module to the space of conformal Killing
tensors, which are the Hamiltonian symmetries of the null geodesic flow.
Besides, as an algebra, it identifies to a quotient
$\mathfrak{U}({\mathfrak{g}})/J$ of the universal enveloping algebra.
Surprisingly, this ideal $J$ coincides with the previously mentioned Joseph
ideal [22]. The correspondence obtained by Eastwood might then be interpreted
as a quantization of the minimal coadjoint orbit of $G$.
Our aim is precisely to prove that the results of Eastwood stem from the now
well-developed theory of equivariant quantization of cotangent bundles [17, 9,
33, 12], and that after a basic symplectic reduction, this quantization
provides a supplement to the orbit method in the case of the minimal nilpotent
coadjoint orbit of $G$. In particular, it induces on that orbit the star
product of Arnal, Benamor and Cahen. The present approach can be generalized
to number of cases, indeed, equivariant quantization is available for any
$|1|$-graded parabolic geometry and for differential operators acting on any
irreducible natural bundles [12]. To illustrate its efficiency, we deal with
the determination of higher symmetries of the conformal powers of the
Laplacian, denoted $\Delta^{\ell}$. Thus, we recover recent results of Gover
and Šilhan [40] obtained via tractor calculus and highly non-trivial
computations. Let us now detail the content of this article.
In Section $2$, we introduce our main tools, namely the classification of
conformally invariant operators on symbols [21, 4, 35], the conformally
equivariant quantization [17] and the induced star product on symbols [16].
In Section $3$ lies our first main result. We characterize the space
$\mathcal{A}^{\lambda,\ell}$ of higher symmetries of $\Delta^{\ell}$ and the
space $\mathcal{K}^{\ell}$ of $\ell$-generalized conformal Killing tensors
[36] as kernels of conformally invariant operators. Then, we prove that
conformally equivariant quantization maps one operator to the other, and then
one space to the other. This correspondence can be made explicit, since
formulae have been derived for this quantization [18, 32], even in the curved
case [37, 39].
In section $4$, we first introduce algebras $\mathcal{K}$ and
$\mathcal{A}^{\lambda}$ generated by ${\mathfrak{g}}$, the Lie algebra of
conformal vector fields. The spaces of symmetries $\mathcal{K}^{\ell}$ and
$\mathcal{A}^{\lambda,\ell}$ are obtained as quotients of the latter. Then, we
describe all the coadjoint orbits of $G$ in the image of the moment map
$\mu:T^{*}\mathbb{R}^{p+1,q+1}\rightarrow{\mathfrak{g}}^{*}$ as symplectic
reductions of the source manifold. Their algebras of regular functions are
determined, and two of them identify to $\mathcal{K}$ and $\mathcal{K}^{1}$.
As a consequence, we get an explicit description of the symmetry algebras
$\mathcal{A}^{\lambda,\ell}$. In particular, the algebra $\mathcal{K}^{1}$ is
associated to the minimal nilpotent coadjoint orbit $\mathcal{O}_{00}$ and the
corresponding symmetry algebra $\mathcal{A}^{\lambda,1}$ of the Laplacian is
isomorphic to the universal enveloping algebra of ${\mathfrak{g}}$ factored by
the Joseph ideal. Finally, we build a star product on each coadjoint orbit in
the image of $\mu$. In particular, the conformally equivariant quantization
induces the star-product of Arnal, Benamor and Cahen [1] on $\mathcal{O}_{00}$
and furnishes a representation of it on $\ker\Delta$, which integrates into
the minimal representation of $G$ if $M=\mathbb{S}^{p}\times\mathbb{S}^{q}$
and $p+q\geq 4$ is even.
## 2\. Conformal geometry of differential operators and of their symbols
We introduce in this section the basic notions that we need and the two key
facts leading to our main theorem in the next section, namely: the existence
and uniqueness of the conformally equivariant quantization [17] and the
classification of the conformally invariant operators on the space of symbols
[35].
### 2.1. Actions of vector fields
Let $M$ be a smooth manifold. We start with the definitions of the algebra
$\mathcal{D}(M)$ of differential operators on $M$ and of its algebra of
symbols ${\mathcal{S}}(M)$. The first one is the union of the spaces
$\mathcal{D}_{k}(M)$ of differential operators of order $k$, defined as the
spaces of operators $A$ on ${\mathcal{C}}^{\infty}(M)$ satisfying
$[\cdots[A,f_{0}],\cdots],f_{k}]=0$ for all functions
$f_{0},\ldots,f_{k}\in{\mathcal{C}}^{\infty}(M)$. The second one,
${\mathcal{S}}(M)$, is its canonically associated graded algebra, defined by
${\mathcal{S}}(M)=\bigoplus_{k=0}^{\infty}\mathcal{D}_{k}(M)/\mathcal{D}_{k-1}(M)$.
It identifies to the algebra of functions on $T^{*}M$ which are polynomial in
the fibers, the grading corresponding to the polynomial degree. The canonical
projection
$\sigma_{k}:\mathcal{D}_{k}(M)\rightarrow\mathcal{D}_{k}(M)/\mathcal{D}_{k-1}(M)$
is called the principal symbol map.
We are interested in the action of the Lie algebra $\mathrm{Vect}(M)$ of
vector fields on these both algebras. The diffeomorphisms of $M$ lift
canonically into automorphisms of ${\mathrm{GL}}(M)$, the principal bundle of
linear frames over $M$. Consequently, they act canonically on sections of
every associated bundles to ${\mathrm{GL}}(M)$. The corresponding
infinitesimal actions of $\mathrm{Vect}(M)$ are given by Lie derivatives. In
particular, we get a $\mathrm{Vect}(M)$-module structure on the spaces of
symbols ${\mathcal{S}}(M)$ and of $\lambda$-densities
${\mathcal{F}}^{\lambda}:=\Gamma(|\Lambda^{n}T^{*}M|^{\otimes\lambda})$, with
$\lambda\in\mathbb{R}$. Via a global section $|\mathrm{vol}|$, the latter
identifies to the module $({\mathcal{C}}^{\infty}(M),\ell^{\lambda})$, endowed
with the $\mathrm{Vect}(M)$-action
$\ell_{X}^{\lambda}=X+\lambda\mathrm{Div}(X)$, the operator $\mathrm{Div}$
being the divergence with respect to $|\mathrm{vol}|$. Therefore, the
$\mathrm{Vect}(M)$-module $\mathcal{D^{\lambda,\mu}}$ of differential
operators from $\lambda$\- to $\mu$-densities identifies to
$(\mathcal{D}(M),\mathcal{L}^{\lambda,\mu})$, with
$\mathcal{L}_{X}^{\lambda,\mu}A=\ell_{X}^{\mu}A-A\ell_{X}^{\lambda},$ for all
$X\in\mathrm{Vect}(M)$ and $A\in\mathcal{D}(M)$. This adjoint action preserves
the filtration of $\mathcal{D}(M)$, hence the algebra of symbols inherits of a
$\mathrm{Vect}(M)$-action compatible with the grading which coincides with the
one by Lie derivative on
${\mathcal{S}}(M)\otimes_{{\mathcal{C}^{\infty}}(M)}{\mathcal{F}}^{\delta}$,
for $\delta=\mu-\lambda$. We denote by ${\mathcal{S}}^{\delta}$ the obtained
module of symbols and by ${\mathcal{S}}^{\delta}_{k}$ the submodule of
homogeneous symbols of degree $k$.
### 2.2. Conformal Lie algebra
A conformal structure on a smooth manifold $M$ is given by the equivalence
class $[\mathrm{g}]$ of a pseudo-Riemannian metric $\mathrm{g}$, where two
metrics $h$ and $\mathrm{g}$ are considered equivalent if $h=F\mathrm{g}$ for
some positive function $F\in{\mathcal{C}^{\infty}}(M)$. The signature $(p,q)$
of the metric $\mathrm{g}$ is an invariant of the conformal structure, and to
each signature corresponds a canonical flat model $(\mathbb{R}^{p,q},[\eta])$,
with $\eta=\mathbb{I}_{p}\otimes-\mathbb{I}_{q}$. If there exists an atlas
$(U_{i},\phi_{i})$ on $(M,[\mathrm{g}])$ such that the pull-back by every
chart $\phi_{i}$ of the canonical flat conformal structure coincides with the
restriction of $[\mathrm{g}]$ to $U_{i}$, the conformal manifold
$(M,[\mathrm{g}])$ is said to be conformally flat. The vector fields that
preserve a conformal class $[\mathrm{g}]$ are called conformal Killing vector
fields, they satisfy $L_{X}\mathrm{g}=F\mathrm{g}$, with $L_{X}\mathrm{g}$ the
Lie derivative of $\mathrm{g}$ along $X$ and $F$ a function. They form a sheaf
of Lie algebras, and if $(M,[\mathrm{g}])$ is conformally flat of dimension
$p+q\geq 3$, it is locally isomorphic to ${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$
the conformal Lie algebra of $(\mathbb{R}^{p,q},[\eta])$.
An important example of conformally flat manifold is
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, viewed as a homogeneous space of
$G=\mathrm{O}(p+1,q+1)$. Starting from the isometric action of $G$ on the
pseudo-Euclidean space $\mathbb{R}^{p+1,q+1}$, we get an action of $G$ on the
space of isotropic half-lines, which identifies naturally to the manifold
$\mathbb{S}^{p}\times\mathbb{S}^{q}$. Via this construction, the flat metric
on $\mathbb{R}^{p+1,q+1}$ induces a conformally flat structure on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, preserved by the $G$-action.
### 2.3. Conformal invariants
Over a conformally flat manifold $(M,[\mathrm{g}])$, a conformal invariant in
a $\mathrm{Vect}(M)$-module is an invariant element under the action of
conformal Killing vector fields. Using local conformal coordinates $(x^{i})$,
which are such that $\mathrm{g}_{ij}=F\eta_{ij}$ for a positive function $F$,
the classification of conformal invariants over $(M,[\mathrm{g}])$ amounts to
the one of ${\mathfrak{g}}$-invariants over the flat space. For modules of
tensors, this can be performed thanks to the Weyl’s theory of invariants [42]
and for modules of differential operators this can be done resorting to the
results on morphisms of Verma modules obtained in [6, 7]. However, all results
presented here can be derived from the more elementary Weyl’s theory of
invariants.
First, we provide the well-known classification of the conformal invariants of
the $\mathrm{Vect}(M)$-modules of differential operators
$\mathcal{D^{\lambda,\mu}}$ and of symbols ${\mathcal{S}}^{\delta}$. We write
them in terms of the local conformal coordinates $(x^{i},p_{i})$ on $T^{*}M$,
and of their corresponding derivatives $(\partial_{i},\partial_{p_{i}})$.
###### Proposition 2.1.
On a conformally flat manifold $(M,[\mathrm{g}])$, the conformal invariants of
$({\mathcal{S}}^{\delta})_{\delta\in\mathbb{R}}$ and
$(\mathcal{D^{\lambda,\mu}})_{\lambda,\mu\in\mathbb{R}}$ are given locally, up
to a multiplicative constant, by
* •
$\mathrm{R}^{\ell}\in{\mathcal{S}}^{\frac{2\ell}{n}}$ for $\ell\in\mathbb{N}$,
* •
$\Delta^{\ell}\in\mathcal{D^{\lambda,\mu}}$ for $\ell\in\mathbb{N}$ and
$\lambda=\frac{n-2\ell}{2n}$, $\mu=\frac{n+2\ell}{2n}$,
where $\mathrm{R}=\eta^{ij}p_{i}p_{j}$, and
$\Delta=\eta^{ij}\partial_{i}\partial_{j}$ is the conformal Laplacian.
We refer to [24] and references inside for global expressions of the conformal
powers of the Laplacian. Since the principal symbol map is
$\mathrm{Vect}(\mathbb{R}^{n})$-equivariant, conformally invariant
differential operators give rise to conformally invariant symbols, but the
fact that they are in correspondence is remarkable.
Second, we present the classification of the conformally invariant
differential operators on the space of symbols, as it appears in [35]. It
relies on the harmonic decomposition of the ${\mathfrak{g}}$-module of
symbols. Namely, we have
${\mathcal{S}}^{\delta}=\bigoplus_{k,s\in\mathbb{N},\,2s\leq
k}{\mathcal{S}}^{\delta}_{k,s}$, where ${\mathcal{S}}^{\delta}_{k,s}$ is the
submodule of homogeneous symbols of degree $k$ of the form $P=R^{s}Q$ with $Q$
a traceless symbol, i.e. $TQ=0$ for
$T=\eta_{ij}\partial_{p_{i}}\partial_{p_{j}}$.
###### Theorem 2.2.
[35] Let $k\geq 2s$ and $k^{\prime}\geq 2s^{\prime}$ be integers, and
$\delta,\delta^{\prime}\in\mathbb{R}$. The space of conformal invariant
differential operators:
${\mathcal{S}}^{\delta}_{k,s}\rightarrow{\mathcal{S}}^{\delta^{\prime}}_{k^{\prime},s^{\prime}}$,
is either trivial or of dimension $1$. In the latter case
$j=\frac{n}{2}(\delta^{\prime}-\delta)$ is an integer and the space is
generated by
* •
$R^{s^{\prime}}D^{d}T^{s}$, if $s^{\prime}-s=j$, $k-k^{\prime}=d-2j$ and
$\delta=1+\frac{2(k-s)-d-1}{n}$,
* •
$R^{s^{\prime}}G_{0}^{g}T^{s}$, if $g+s^{\prime}-s=j$,
$k-k^{\prime}=s-s^{\prime}-j$ and $\delta=\frac{2s+1-g}{n}$,
* •
$R^{s^{\prime}}\mathcal{L}_{\ell}T^{s}$, if $\ell+s^{\prime}-s=j$,
$k-k^{\prime}=2(\ell-j)$ and $\delta=\frac{1}{2}+\frac{k-\ell}{n}$,
where, locally, $D=\partial_{i}\partial_{p_{i}}$ is the divergence operator,
$G_{0}$ is the composition of the gradient operator
$G=\eta^{ij}p_{i}\partial_{j}$ with the projection on traceless symbols and
$\mathcal{L}_{\ell}=\Delta^{\ell}+a_{1}GD\Delta^{\ell-1}+\cdots+a_{\ell}G^{\ell}D^{\ell}$
for given real coefficients $a_{1},\ldots,a_{\ell}$.
Global expression for divergence and gradient operators can be find in [15],
and we refer to [45] for $\mathcal{L}_{1}$.
### 2.4. Conformally equivariant quantization
Let $\lambda,\mu\in\mathbb{R}$ and $\delta=\mu-\lambda$. We call quantization
the linear isomorphisms
${\mathcal{Q}}:{\mathcal{S}}^{\delta}\rightarrow\mathcal{D^{\lambda,\mu}}$
which are right inverses of the principal symbol map on homogeneous symbols.
If the both $\mathrm{Vect}(M)$-modules of symbols and of differential
operators are indeed isomorphic as vector spaces by the very definition of
symbols, they are not as $\mathrm{Vect}(M)$-modules [29, 30]. A natural
question is whether they are isomorphic as modules over given Lie subalgebras
of $\mathrm{Vect}(M)$, and whether this extra structure allows to single out
one quantization. This has been studied in various contexts under the name
equivariant quantization, and one of the fundamental result is the following.
###### Theorem 2.3.
[17] On a conformally flat manifold, there exists a unique conformally
equivariant quantization for generic values of $\delta=\mu-\lambda$, i.e. a
unique ${\mathfrak{g}}$-module morphism
$\mathcal{Q}^{\lambda,\mu}:{\mathcal{S}}^{\delta}\rightarrow\mathcal{D^{\lambda,\mu}}$
which is a right inverse of the principal symbol map on homogeneous symbols.
The exceptional values of $\delta$ leading to a non-unique or a non-existing
conformally equivariant quantization have been classified in [39, 35]. They
can be obtained via Theorem 2.2 and the following result.
###### Proposition 2.4.
[35] The conformally equivariant quantization exists and is unique on
${\mathcal{S}}^{\delta}_{k,s}$ if and only if there is no conformally
invariant differential operators from ${\mathcal{S}}^{\delta}_{k,s}$ to
${\mathcal{S}}^{\delta}$.
We will need a slightly stronger result than theorem 2.3, straightforwardly
deduced from [17].
###### Proposition 2.5.
Let $F$ be a ${\mathfrak{g}}$-submodule of ${\mathcal{S}}^{\delta}$. For a
generic shift $\delta$, e.g. $\delta=0$, there exists a unique morphism of
${\mathfrak{g}}$-modules from $F$ to $\mathcal{D^{\lambda,\mu}}$, which is a
right inverse of the principal symbol map on homogeneous symbols.
Explicit formulae are available for the conformally equivariant quantization.
As an example, we recall the one obtained by Radoux [37] on the space
${\mathcal{S}}^{\delta}_{*,0}=\bigoplus_{k\in\mathbb{N}}{\mathcal{S}}^{\delta}_{k,0}$
of traceless symbols. It relies on the divergence operator
$D=\partial_{i}\partial_{p_{i}}$ and the normal ordering
${\mathcal{N}}:P^{i_{1}\cdots i_{k}}(x)p_{i_{1}}\cdots p_{i_{k}}\mapsto
P^{i_{1}\cdots i_{k}}(x)\partial_{i_{1}}\cdots\partial_{i_{k}}$.
###### Proposition 2.6.
[37] Let $\delta\notin\\{1+\frac{2k-1-m}{n}|\,m=1,\ldots,k\\}$. On the space
${\mathcal{S}}^{\delta}_{k,0}$ of traceless symbols of degree $k$, the
conformally equivariant quantization is given by
(2.1)
$\mathcal{Q}^{\lambda,\mu}={\mathcal{N}}\circ\left(\sum_{m=0}^{k}c^{k}_{m}D^{m}\right),$
with $c^{k}_{0}=1$ and
$c^{k}_{m}=\frac{k-m+n\lambda}{m(2k-m-1+n(1-\delta))}\,c^{k}_{m-1},$ for
$m=1,\ldots,k$.
In the general case, including symbols with non-vanishing trace, fully
explicit formulae are known only for symbols up to the order $3$ in momenta
variables $p$, they are available in conformal coordinates [31] as well as in
covariant terms [18, 32]. Finally, let us mention that Šilhan has obtained the
expression of the conformally equivariant quantization in the curved case on
all symbols, resorting to tractor calculus [39].
### 2.5. Conformally equivariant graded star product
Let us start with standard definitions. The algebra of symbols
${\mathcal{S}}^{0}$ is commutative and graded, moreover, as a subalgebra of
${\mathcal{C}^{\infty}}(T^{*}M)$, it carries a Poisson bracket denoted by
$\\{\cdot,\cdot\\}$. A graded (or homogeneous) star product on
${\mathcal{S}}^{0}$ is an associative $\mathbb{C}[[\hbar]]$-linear product
$\star$ on ${\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]$, with $\hbar$ a
formal parameter. For $P,Q\in{\mathcal{S}}^{0}$, it is of the form $P\star
Q=\sum_{m\in\mathbb{N}}\left(\mathsf{i}\hbar\right)^{m}B_{m}(P,Q)$ and
satisfies:
1. (1)
$B_{0}(P,Q)=PQ$,
2. (2)
$B_{1}(P,Q)-B_{1}(Q,P)=\\{P,Q\\}$,
3. (3)
for all integers $k,l,m$,
$B_{m}:{\mathcal{S}}^{0}_{k}\otimes{\mathcal{S}}^{0}_{l}\rightarrow{\mathcal{S}}^{0}_{k+l-m}$
is a bidifferential operator.
A frequently required extra property is the symmetry (or parity) of the star
product, namely $B_{m}(P,Q)=(-1)^{m}B_{m}(Q,P)$ for all integers $m$, or
equivalently $\overline{P\star Q}=\overline{Q}\star\overline{P}$, where
$\overline{\cdot}$ is the complex conjugation.
One source of star products on ${\mathcal{S}}^{0}$ is quantizations. Let us
introduce two maps, the linear map
$\Im:{\mathcal{S}}^{0}\rightarrow{\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]$
defined by $(\mathsf{i}\hbar)^{k}\mathrm{Id}$ on ${\mathcal{S}}^{0}_{k}$ and
the $\mathbb{C}[[\hbar]]$-linear extension of some quantization
${\mathcal{Q}}\otimes\mathrm{Id}:{\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]\rightarrow\mathcal{D}^{\lambda,\lambda}\otimes\mathbb{C}[[\hbar]]$.
Clearly, the composition
${\mathcal{Q}}_{\hbar}=({\mathcal{Q}}\otimes\mathrm{Id})\circ\Im$ gives rise
to a graded star product on ${\mathcal{S}}^{0}$ as the pull back by
${\mathcal{Q}}_{\hbar}$ of the composition of differential operators,
(2.2) $P\star
Q={\mathcal{Q}}_{\hbar}^{-1}({\mathcal{Q}}_{\hbar}(P)\circ{\mathcal{Q}}_{\hbar}(Q)).$
Moreover, for $\lambda=\frac{1}{2}$, this star product is symmetric if and
only if the quantization satisfies
${\mathcal{Q}}_{\hbar}(\overline{P})={\mathcal{Q}}_{\hbar}(P)^{*}$ for all
$P\in{\mathcal{S}}^{0}$. Here, the superscript ∗ denotes the adjoint operation
with respect to the Hermitian product on complex compactly supported half-
densities, given by $(\phi,\psi)=\int_{M}\overline{\phi}\psi$.
The action of $X\in\mathrm{Vect}(M)$ on
${\mathcal{S}}^{0}\subset{\mathcal{C}^{\infty}}(T^{*}M)$ is given by the
Hamiltonian vector field $\\{\mu_{X},\cdot\\}$ where $\mu_{X}=X^{i}p_{i}$.
Thanks to the star product on ${\mathcal{S}}^{0}$, we can define a new action
of $\mathrm{Vect}(M)$ on ${\mathcal{S}}^{0}$ via the star bracket, i.e.
$X\in\mathrm{Vect}(M)$ acts on $P\in{\mathcal{S}}^{0}$ by
$[\mu_{X},P]_{\star}=\mu_{X}\star P-P\star\mu_{X}$. The star product is said
conformally equivariant (or strongly ${\mathfrak{g}}$-invariant) if both
induced ${\mathfrak{g}}$-actions coincide, namely
$[\mu_{X},P]_{\star}=\mathsf{i}\hbar\\{\mu_{X},P\\}$ for all
$X\in{\mathfrak{g}}$. As one can expect, conformally equivariant quantizations
gives rise to ${\mathfrak{g}}$-equivariant star products.
###### Proposition 2.7.
[17, 16] Let $(M,[\mathrm{g}])$ be a conformally flat manifold. The star
product $\star_{\lambda}$ induced by the conformally equivariant quantization
${\mathcal{Q}}^{\lambda,\lambda}$ via equation (2.2) is a graded
${\mathfrak{g}}$-equivariant star product on ${\mathcal{S}}^{0}$. It is
symmetric if and only if $\lambda=\frac{1}{2}$.
It is easy to prove that all graded ${\mathfrak{g}}$-equivariant star products
on ${\mathcal{S}}^{0}$ arise in that way.
## 3\. Classification of the higher symmetries of the conformal powers of the
Laplacian
The aim of this section is to show how conformally equivariant quantization
sheds new light on the determination of higher symmetries of conformal
Laplacian, initiated by Eastwood [19] and pursued in [20] and [40] for
conformal powers of the Laplacian, in the conformally flat case. In all this
section we work over a conformally flat manifold $(M,[\mathrm{g}])$ of
dimension $n\geq 3$ and $\Delta^{\ell}$ denotes the conformal
$\ell^{\text{th}}$ power of the Laplacian, pertaining to
$\mathcal{D^{\lambda,\mu}}$ for values of the weights henceforth fixed to
$\lambda=\frac{n-2\ell}{2n}$, $\mu=\frac{n+2\ell}{2n}$.
### 3.1. Definition of higher symmetries of $\Delta^{\ell}$
There is a number of different notions of symmetries for a differential
operator $P$ on $M$. The most basic one is given by vector fields
$X\in\mathrm{Vect}(M)$ preserving the considered operator: $[P,X]=0$. We can
broaden this notion by considering rather differential operators
$D\in\mathcal{D}(M)$ commuting to $P$. Such symmetries preserve obviously the
eigenspaces of $P$. Here we are interested in the more general notion of
higher symmetries, which act on the kernel of $P$. We give their definitions
for $P=\Delta^{\ell}$.
###### Definition 3.1.
Let $\lambda=\frac{n-2\ell}{2n}$, $\mu=\frac{n+2\ell}{2n}$ and let
$(\Delta^{\ell})=\\{A\Delta^{\ell}|\,A\in\mathcal{D}^{\mu,\lambda}\\}$ be the
left ideal generated by the conformal $\ell^{\text{th}}$ power of the
Laplacian on the conformally flat manifod $(M,[\mathrm{g}])$. A higher
symmetry of $\Delta^{\ell}$ is a class of differential operators
$[D_{1}]\in\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})$, such that
$\Delta^{\ell}D_{1}=D_{2}\Delta^{\ell}$, for some
$D_{2}\in\mathcal{D}^{\mu,\mu}$.
This definition does not depend on the chosen representative since
$\Delta^{\ell}(A\Delta^{\ell})=(\Delta^{\ell}A)\Delta^{\ell}$ for any
differential operator $A$. Resorting to conformal coordinates, higher
symmetries prove to be locally the same on flat and conformally flat
manifolds, but global existence can nevertheless be problematic in this more
general setting. We do not address this issue and work only locally. The
higher symmetries of $\Delta^{\ell}$ form then an algebra
$\mathcal{A}^{\lambda,\ell}$, which is characterized as the kernel of the
conformally invariant map
$\displaystyle\mathrm{QHS}:\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})$
$\displaystyle\rightarrow$
$\displaystyle\mathcal{D}^{\lambda,\mu}/(\Delta^{\ell})$ (3.3)
$\displaystyle\left[D\right]$ $\displaystyle\mapsto$
$\displaystyle[\Delta^{\ell}D]$
where $\mathrm{QHS}$ stands for Quantum Higher Symmetries. Let us mention that
the left ideal generated by $\Delta^{\ell}$ can be defined in the modules
$\mathcal{D}^{\lambda,\lambda^{\prime}}$ for any $\lambda^{\prime}$, and in
particular for $\lambda^{\prime}=\mu$, as the subspace
$(\Delta^{\ell})=\\{A\Delta^{\ell}|\,A\in\mathcal{D}^{\mu,\lambda^{\prime}}\\}$.
###### Example 3.2.
The higher symmetries of $\Delta^{\ell}$ given by first order differential
operators are the constants, acting by multiplication as zero order
differential operators, and the Lie derivatives $\ell_{X}^{\lambda}$ for
$X\in{\mathfrak{g}}$. In accordance with Proposition 2.1, we have indeed
$\Delta^{\ell}\ell_{X}^{\lambda}=\ell_{X}^{\mu}\Delta^{\ell}$.
### 3.2. Symmetries of the null geodesic flow and generalizations
Using the properties of the principal symbol map $\sigma$, the equality
$[D,\Delta]=A\Delta$ defining higher symmetries of the Laplacian implies
$\\{\sigma(D),R\\}=\sigma(A)R$, where $\\{\cdot,\cdot\\}$ denotes the
canonical Poisson bracket on $T^{*}M$. Consequently, the principal symbol of
higher symmetries of the Laplacian is a constant of motion for the null
geodesic flow. The latter are known to be given by conformal Killing tensors,
which are symmetric tensors and identify to symbols of weight $0$. In the
following, round bracket denotes symmetrization of indices, $(\partial_{i})$
the partial derivatives associated to local conformal coordinates $(x^{i})$
and $L$ is an arbitrary tensor.
###### Definition 3.3.
A conformal Killing $k$-tensor $K$ is defined equivalently as
* •
A symmetric traceless tensor of order $k$ s.t.
$\partial_{(i_{0}}K_{i_{1}\cdots
i_{k})}=\mathrm{g}_{(i_{0}i_{1}}L_{i_{2}\cdots i_{k})}$,
* •
A traceless symbol of degree $k$ satisfying $\\{R,K\\}=fR$ for some function
$f$,
* •
A traceless symbol of degree $k$ in the kernel of $G_{0}$.
As for higher symmetries, we are not concerned by global existence questions
and work locally. For $k=1$, we recover the notion of conformal Killing
vectors which identifies to the Lie algebra ${\mathfrak{g}}$. The conformal
Killing tensors of higher orders correspond to transformations of the phase
space $T^{*}M$ not preserving the configuration manifold $M$. Resorting to
Theorem 2.2, the operator
$G_{0}:{\mathcal{S}}^{0}_{k,0}\rightarrow{\mathcal{S}}^{\frac{2}{n}}_{k+1,0}$
is conformally invariant for every $k$, and this is the only one on
${\mathcal{S}}^{0}_{k,0}$. Hence, the space of conformal Killing tensors of a
given order is an indecomposable ${\mathfrak{g}}$-module. Since it is finite
dimensional, the semi-simplicity of ${\mathfrak{g}}$ implies that it is an
irreducible ${\mathfrak{g}}$-module. We can generalize this picture to tensors
(or symbols) with trace, using the conformal invariance of $G_{0}^{2s+1}T^{s}$
on ${\mathcal{S}}^{0}_{k,s}$. The following definition is due to Nikitin and
Prilipko [36].
###### Definition 3.4.
A $s$-generalized conformal Killing $k$-tensor $K$ is defined equivalently as
* •
A symmetric traceless tensor of order $(k-2s)$ s.t.
$\partial_{(i_{0}}\cdots\partial_{i_{2s}}K_{i_{2s+1}\cdots
i_{k})}=\mathrm{g}_{(i_{0}i_{1}}L_{i_{2}\cdots i_{k})}$,
* •
A symbol $R^{s}K\in{\mathcal{S}}^{0}_{k,s}$ which is in the kernel of
$G_{0}^{2s+1}T^{s}$.
The equivalence of the two assertions relies on the equality
$T^{s}(R^{s}K)=cK$, where $c$ is a constant. Again, the space of
$s$-generalized conformal Killing tensors of order $k$ is an irreducible
${\mathfrak{g}}$-module. We denote this subspace of ${\mathcal{S}}^{0}_{k,s}$
by $\mathcal{K}_{k,s}$ and we introduce the spaces of classical higher
symmetries $\mathcal{K}^{\ell}=\bigoplus_{s=0}^{\ell-1}\mathcal{K}_{*,s}$,
with $\mathcal{K}_{*,s}=\bigoplus_{k\geq 2s}\mathcal{K}_{k,s}$, as well as
their limit $\mathcal{K}=\bigoplus_{s\in\mathbb{N}}\mathcal{K}_{*,s}$.
### 3.3. From classical to quantum symmetries
Gover and Šilhan have proved that the two types of symmetries defined in the
two previous paragraphs are in bijection [40]. We propose a completely
different and far less technical proof, showing that the latter correspondence
is given by equivariant quantization. We assume that $\ell\in\mathbb{N}^{*}$,
$\lambda=\frac{n-2\ell}{2n}$ and $\mu=\frac{n+2\ell}{2n}$.
###### Theorem 3.5.
The conformally equivariant quantization descends to an isomorphism of
${\mathfrak{g}}$-modules
${\mathcal{Q}}^{\lambda,\lambda}:\mathcal{K}^{\ell}\rightarrow\mathcal{A}^{\lambda,\ell}$,
identifying higher symmetries of $\Delta^{\ell}$ with $s$-generalized
conformal Killing tensors for $s<\ell$. Moreover, every local section $P$ of
$\mathcal{K}$ satisfies
$\Delta^{\ell}{\mathcal{Q}}^{\lambda,\lambda}(P)={\mathcal{Q}}^{\mu,\mu}(P)\Delta^{\ell}$.
###### Proof.
The conformal invariance of $\Delta^{\ell}\in\mathcal{D^{\lambda,\mu}}$ and
the uniqueness of the conformally equivariant quantization on the
${\mathfrak{g}}$-module $(R^{\ell})$ lead to the following factorization:
${\mathcal{Q}}^{\lambda,\lambda^{\prime}}(PR^{\ell})={\mathcal{Q}}^{\mu,\lambda^{\prime}}(P)\Delta^{\ell}$,
for any symbol $P$ and any generic $\lambda^{\prime}\in\mathbb{R}$.
Consequently, we get an isomorphism of ${\mathfrak{g}}$-modules on the
quotient spaces
${\mathcal{Q}}^{\lambda,\lambda^{\prime}}:\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{\lambda^{\prime}-\lambda}_{*,s}\rightarrow\mathcal{D}^{\lambda,\lambda^{\prime}}/(\Delta^{\ell})$,
where
${\mathcal{S}}^{\delta}_{*,s}=\bigoplus_{k\in\mathbb{N}}{\mathcal{S}}^{\delta}_{k,s}$.
We keep notation ${\mathcal{Q}}^{\lambda,\lambda^{\prime}}$ in the latter case
for simplicity. According to Theorem 2.2 and Proposition 2.4,
${\mathcal{Q}}^{\lambda,\lambda}$ always exists but not
$\mathcal{Q}^{\lambda,\mu}$, which exists if $n$ is odd or
$3\ell\leq\frac{n}{2}+1$.
The idea of the proof is to use the conformally equivariant quantization to
identify the kernel $\mathcal{A}^{\lambda,\ell}$ of the operator
$\mathrm{QHS}$, see (3.3), to the one of an operator $\mathrm{CHS}$ on
symbols, its name standing for Classical Higher Symmetries.
In the easier case where $n$ is odd or $3\ell\leq\frac{n}{2}+1$,
$\mathcal{Q}^{\lambda,\mu}$ exists and the latter operator is determined by
the following commutative diagram of ${\mathfrak{g}}$-modules
(3.4)
$\textstyle{\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}}$$\textstyle{\mathcal{D}^{\lambda,\mu}/(\Delta^{\ell})}$$\textstyle{\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{0}_{*,s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\scriptstyle{\mathrm{CHS}}$$\textstyle{\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{\frac{2\ell}{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\mu}}$
Consequently, $\mathrm{CHS}$ is a conformally invariant operator and resorting
to the classification given in Theorem 2.2, it is necessarily equal on
${\mathcal{S}}^{0}_{*,s}$ to $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, up to a
multiplicative constant. This constant cannot be zero since $\mathrm{QHS}$
does not vanish on the image of ${\mathcal{S}}^{0}_{*,s}$. Hence, by
Definition 3.4, the kernel of $\mathrm{QHS}$ is isomorphic to the space
$\mathcal{K}^{\ell}$ of $s$-generalized conformal Killing tensors for $0\leq
s<\ell$.
In the other case, the conformally invariant operator $\mathrm{CHS}$ is
defined via the following commutative diagram of ${\mathfrak{g}}$-modules
(3.5)
$\textstyle{\mathcal{D}^{\lambda,\lambda}_{k}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}}$$\textstyle{\mathcal{D}^{\lambda,\mu}_{k^{\prime}}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{k^{\prime}}}$$\textstyle{{\mathcal{S}}^{0}_{k,s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\scriptstyle{\mathrm{CHS}}$$\textstyle{{\mathcal{S}}^{\frac{2\ell}{n}}_{k^{\prime}},}$
where $\sigma_{k^{\prime}}$ is the principal symbol map and
$k^{\prime}\in\mathbb{N}$ is taken as small as possible, so that
$\mathrm{CHS}$ does not vanish. According to Theorem 2.2 it is proportional to
$R^{\ell-s-1}G_{0}^{2s+1}T^{s}$ or to
$R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$, and the second
case can occur only if $n/2+(k-s)\leq\ell$. Contrary to
$\mathcal{Q}^{\lambda,\mu}$, the principal symbol map is not an isomorphism,
hence, a priori, we just get that the space
${\mathcal{Q}}^{\lambda,\lambda}(\ker\mathrm{CHS})$ contains the kernel of
$\mathrm{QHS}$. If $\mathrm{CHS}$ is again proportional to
$R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, we obtain the converse inclusion by
irreducibility of the kernel of $\mathrm{CHS}$. Else, $\mathrm{CHS}$ is
proportional to $R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$ and
we have $k\leq\ell+s-n/2$. According to proposition 2.4,
$\mathcal{Q}^{\lambda,\mu}$ is then well-defined on
${\mathcal{S}}_{k^{\prime\prime}}^{\frac{2\ell}{n}}$ for $k^{\prime\prime}\leq
k^{\prime}$ and we are back to the situation in (3.4). Thus, the kernels of
$\mathrm{CHS}$ and $\mathrm{QHS}$ are always in correspondence via
${\mathcal{Q}}^{\lambda,\lambda}$. We deduce that
${\mathrm{gr}}\mathcal{A}^{\lambda,\ell}\simeq\ker\mathrm{CHS}$ is a
subalgebra of ${\mathcal{S}}^{0}/(R^{\ell})$, with first degree elements given
by ${\mathfrak{g}}$. If its intersection with ${\mathcal{S}}^{0}_{k,s}$ is the
kernel of $R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$, then it
is infinite dimensional and so will be its intersection with
${\mathcal{S}}^{0}_{m}$ for $m\geq k$. However, for $m$ big enough, we have
proved that the kernel of $\mathrm{CHS}$ in ${\mathcal{S}}^{0}_{m,s}$ is the
one of $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, which is finite dimensional. In
conclusion, $\mathrm{CHS}$ is proportional to $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$
on any ${\mathcal{S}}^{0}_{k,s}$ and we get the desired correspondence between
$\mathcal{K}^{\ell}$ and $\mathcal{A}^{\lambda,\ell}$ in all cases.
Now, we can define on $\mathcal{D}^{\lambda,\lambda}$ a new conformally
invariant operator
$\mathrm{QHS}_{0}:D\mapsto\Delta^{\ell}D-{\mathcal{Q}}^{\mu,\mu}\circ({\mathcal{Q}}^{\lambda,\lambda})^{-1}(D)\Delta^{\ell}$.
If $\mathcal{Q}^{\lambda,\mu}$ exists, the commutative diagram
$\textstyle{\mathcal{D}^{\lambda,\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}_{0}}$$\textstyle{\mathcal{D}^{\lambda,\mu}}$$\textstyle{{\mathcal{S}}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{{\mathcal{S}}^{\frac{2\ell}{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\mu}}$
leads to a non-vanishing conformally invariant operator on
${\mathcal{S}}^{0}_{*,s}$ for every $s$. According to Theorem 2.2, it is
proportional to the same operator $\mathrm{CHS}$ as before if $s<\ell-1$ and
this is the null operator else. We conclude that, in particular,
$\Delta^{\ell}{\mathcal{Q}}^{\lambda,\lambda}(P)={\mathcal{Q}}^{\mu,\mu}(P)\Delta^{\ell}$
for any $P\in\mathcal{K}$. The proof in the remaining case is analogous. ∎
###### Remark 3.6.
For $\lambda=\frac{n-2}{2n}$, classical and quantum symmetries for the
equations of motion of a free massless particle correspond to each other via
${\mathcal{Q}}^{\lambda,\lambda}$: $\\{R,K\\}\propto
R\Longleftrightarrow[\Delta,{\mathcal{Q}}^{\lambda,\lambda}(K)]=A\Delta$.
###### Remark 3.7.
We can obtain explicit expressions for the higher symmetries of
$\Delta^{\ell}$ via the formulae for the conformally equivariant quantization
given in (2.1) for $\ell=1$ or in [39] for the general case. The obtained
differential operators admit analogs in the curved case, which are not
necessarily higher symmetries anymore. E.g. all the conformal Killing
$2$-tensors do not give rise to higher symmetries of the Laplacian in general
[34].
## 4\. Algebras of symmetries: geometric realizations and deformations
The aim of this section is to provide geometric interpretations to the
algebras of classical and quantum symmetries, as well as identifying the star
product induced by the composition of quantum symmetries. In all this section
we work over a conformally flat manifold $(M,[\mathrm{g}])$ of dimension
$n\geq 3$.
### 4.1. Algebras of symmetries are generated by ${\mathfrak{g}}$
Let us give a brief reminder on universal enveloping algebra
$\mathfrak{U}({\mathfrak{g}})$ and symmetric algebra
$\mathrm{S}({\mathfrak{g}})$ of an arbitrary Lie algebra ${\mathfrak{g}}$.
From the tensor algebra of ${\mathfrak{g}}$, they inherit respectively a
filtration $\\{\mathfrak{U}_{k}({\mathfrak{g}})\\}_{k}$ and a grading
$\mathrm{S}({\mathfrak{g}})=\bigoplus_{k}\mathrm{S}_{k}({\mathfrak{g}})$ such
that
${\mathrm{gr}}\,\mathfrak{U}({\mathfrak{g}})\simeq\mathrm{S}({\mathfrak{g}})$.
Consequently, the canonical projections
$\mathfrak{U}_{k}({\mathfrak{g}})\rightarrow\mathfrak{U}_{k}({\mathfrak{g}})/\mathfrak{U}_{k-1}({\mathfrak{g}})$
define a principal symbol map, whose right inverses are called quantizations
of $\mathrm{S}({\mathfrak{g}})$. The symmetrization map
$\mathrm{Sym}:\mathrm{S}({\mathfrak{g}})\rightarrow\mathfrak{U}({\mathfrak{g}})$
defined by
$\mathrm{Sym}:X_{i_{1}}\cdots
X_{i_{k}}\mapsto\frac{1}{k!}\sum_{\tau\in\mathfrak{S}_{k}}X_{\tau(i_{1})}\cdots
X_{\tau(i_{k})}$
is known to define a ${\mathfrak{g}}$-equivariant quantization of
$\mathrm{S}({\mathfrak{g}})$ for the canonical extensions of the adjoint
action of ${\mathfrak{g}}$ to $\mathrm{S}({\mathfrak{g}})$ and
$\mathfrak{U}({\mathfrak{g}})$. Any other ${\mathfrak{g}}$-equivariant
quantization is then of the form $\Phi=\mathrm{Sym}\circ\phi$, with
$\phi=\mathrm{Id}+N$ and $N$ a ${\mathfrak{g}}$-equivariant map on
$\mathrm{S}({\mathfrak{g}})$ lowering the degree. Analogously to the case of
symbols, a ${\mathfrak{g}}$-equivariant graded star product $\star_{\Phi}$ can
be obtained on $\mathrm{S}({\mathfrak{g}})$, as the pull-back of the product
on $\mathfrak{U}({\mathfrak{g}})\otimes\mathbb{C}[[\hbar]]$ by the map
$\Phi_{\hbar}=(\Phi\otimes\mathrm{Id})\circ\Im$, where
$\Im:\mathrm{S}({\mathfrak{g}})\rightarrow\mathrm{S}({\mathfrak{g}})\otimes\mathbb{C}[[\hbar]]$
is the linear map defined by $(\mathsf{i}\hbar)^{k}\mathrm{Id}$ on
$\mathrm{S}_{k}({\mathfrak{g}})$. Denoting by $\tau$ and $\gamma$ the anti-
automorphisms of $\mathfrak{U}({\mathfrak{g}})$ and
$\mathrm{S}({\mathfrak{g}})$ defined by $-\mathrm{Id}$ on ${\mathfrak{g}}$,
the symmetry of the star product on $\mathrm{S}({\mathfrak{g}})$ is equivalent
to $\Phi_{\hbar}(\bar{\cdot})=\tau\circ\Phi_{\hbar}(\cdot)$, or simply
$\Phi\circ\gamma=\tau\circ\Phi$.
We return to the case ${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$. Via the defining
universal properties of the both algebras $\mathrm{S}({\mathfrak{g}})$ and
$\mathfrak{U}({\mathfrak{g}})$, the pull-back defined by the moment map
$\mu^{*}:{\mathfrak{g}}\rightarrow{\mathcal{S}}^{0}_{1}$ and the Lie
derivative
$\ell^{\lambda}:{\mathfrak{g}}\rightarrow\mathcal{D}^{\lambda,\lambda}_{1}$
extend to algebra morphisms
$\mu^{*}:\mathrm{S}({\mathfrak{g}})\rightarrow{\mathcal{S}}^{0}$ and
$\ell^{\lambda}:\mathfrak{U}({\mathfrak{g}})\rightarrow\mathcal{D}^{\lambda,\lambda}$.
###### Theorem 4.1.
Let $\lambda\in\mathbb{R}$. The space $\mathcal{K}$ of classical symmetries is
equal to the algebra
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\simeq\mathrm{S}({\mathfrak{g}})/I$, with
$I$ a graded ideal of $\mathrm{S}({\mathfrak{g}})$. Its image by the
conformally equivariant quantization, namely
$\mathcal{A}^{\lambda}:={\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K})$, is an
algebra satisfying
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda}$,
with $J^{\lambda}$ a filtered ideal such that
${\mathrm{gr}}\,J^{\lambda}\simeq I$. The star product $\star_{\lambda}$
induced by ${\mathcal{Q}}^{\lambda,\lambda}$ on ${\mathcal{S}}^{0}$ restricts
then to a ${\mathfrak{g}}$-equivariant graded star product on $\mathcal{K}$.
Moreover, the conformally equivariant quantization of $\mathcal{K}$ lifts to a
${\mathfrak{g}}$-equivariant quantization $\Phi^{\lambda}$ of
$\mathrm{S}({\mathfrak{g}})$, such that the following diagram commutes
(4.6)
$\textstyle{\mathrm{S}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}}$$\scriptstyle{\Phi^{\lambda}}$$\textstyle{\mathfrak{U}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{\lambda}}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}}$
###### Proof.
We start with proving $\mathcal{K}=\mu^{*}(\mathrm{S}({\mathfrak{g}}))$. Since
they are ${\mathfrak{g}}$-modules, this amounts to proving that
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}=\mathcal{K}_{k,s}$
for all $k,s$. Let $\ell\in\mathbb{N}^{*}$. The space
$\mathcal{A}^{\lambda,\ell}$ is a subalgebra of
$\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})$ containing
$\ell^{\lambda}({\mathfrak{g}})$. As the principal symbol map commutes with
the product, we get that $\mathcal{K}^{\ell}$ is a subalgebra of
${\mathcal{S}}^{0}/(R^{\ell})$ containing $\mu^{*}({\mathfrak{g}})$. Hence,
$\mathcal{K}$ is a subalgebra of ${\mathcal{S}}^{0}$, which contains the
algebra $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ generated by the conformal
Killing vector fields, in particular
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}\subset\mathcal{K}_{k,s}$
for all $k,s$. Using the elements generated by constant Killing vector fields,
we get that $\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}$
is non-empty and since $\mathcal{K}_{k,s}$ is irreducible, we are done.
Now, we prove that
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$. By semi-
simplicity of ${\mathfrak{g}}$, its finite dimensional representations are
completely reducibles. In particular, for any $k$,
$\mathcal{K}\cap{\mathcal{S}}^{0}_{k}$ can be viewed as a submodule of
$\mathrm{S}_{k}({\mathfrak{g}})$, leading to the decomposition
$\mathrm{S}({\mathfrak{g}})\simeq\mathcal{K}\oplus I$ of the symmetric
algebra. In other words, $\mu^{*}$ admits a ${\mathfrak{g}}$-equivariant
section. Using the embedding of $\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$
into $\mathcal{D}^{\lambda,\lambda}$, we get then the following diagram of
${\mathfrak{g}}$-modules
$\textstyle{\mathrm{S}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Sym}}$$\textstyle{\mathfrak{U}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{\lambda}}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{\lambda,\lambda}}$
Each arrow in the latter diagram is ${\mathfrak{g}}$-equivariant and preserves
the principal symbol. Hence, uniqueness of ${\mathcal{Q}}^{\lambda,\lambda}$
on the ${\mathfrak{g}}$-module $\mathcal{K}$ implies that the diagram is
commutative, proving
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$.
Since $\mu^{*}$ respects the grading, its kernel $I$ is a graded ideal, and
since $\ell^{\lambda}$ preserves the filtration, its kernel $J^{\lambda}$ is
filtered. Using the commutativity of the following diagram,
$\textstyle{\mathfrak{U}_{k}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}^{\lambda}\cap\mathcal{D}^{\lambda,\lambda}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{S}_{k}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}_{k},}$
where the vertical arrows denote principal symbol maps, we get that
${\mathrm{gr}}\,J^{\lambda}=I$.
We have proved $\mathrm{S}({\mathfrak{g}})\simeq\mathcal{K}\oplus I$, and
along the same line we get
$\mathfrak{U}({\mathfrak{g}})\simeq\mathcal{A}^{\lambda}\oplus J^{\lambda}$.
Using again the semi-simplicity of ${\mathfrak{g}}$, the isomorphism
$J^{\lambda}_{k}/J^{\lambda}_{k-1}\simeq I_{k}$ leads to
$J_{k}^{\lambda}\simeq I_{k}\oplus J^{\lambda}_{k-1}$. Thus, it exists an
isomorphism of ${\mathfrak{g}}$-modules between $I$ and $J^{\lambda}$, inverse
to the symbol map. Together with the previous decomposition of
$\mathrm{S}({\mathfrak{g}})$ and $\mathfrak{U}({\mathfrak{g}})$, this ensures
the existence of the quantization $\Phi^{\lambda}$ and the commutativity of
the diagram (4.6). ∎
The proof shows that
$\mathcal{K}_{k,s}=\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}$.
Thus, on conformally flat manifolds, the $s$-generalized conformal Killing
$k$-tensors are algebraically generated from the conformal Killing vectors.
This widely generalizes a result in [38], stating the same fact for second
order conformal Killing tensors.
###### Corollary 4.2.
Let $\ell\in\mathbb{N}^{*}$ and $\lambda=\frac{n-2\ell}{2n}$. The following
diagram of ${\mathfrak{g}}$-modules is commutative
(4.7)
$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}^{\ell}=\mathcal{K}/(R^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}/(\Delta^{\ell})=\mathcal{A}^{\lambda,\ell}}$
The algebra of $s$-generalized conformal Killing tensors, for $s<\ell$,
satisfies $\mathcal{K}^{\ell}\simeq\mathrm{S}({\mathfrak{g}})/I^{\ell}$, with
$I^{\ell}$ the ideal generated by $I$ and $(\mu^{*})^{-1}(R^{\ell})$, and the
algebra of higher symmetries of $\Delta^{\ell}$ satisfies
$\mathcal{A}^{\lambda,\ell}\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda,\ell}$
with $J^{\lambda,\ell}$ the ideal generated by $J^{\lambda}$ and
$(\ell^{\lambda})^{-1}(\Delta^{\ell})$. The star product $\star_{\lambda}$
projects accordingly onto $\mathcal{K}^{\ell}$, defining there a graded
${\mathfrak{g}}$-equivariant star product.
###### Proof.
By definition $\mathcal{K}^{\ell}=\mathcal{K}/(R^{\ell})$ and by Theorem 3.5
we have
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{\ell})=\mathcal{A}^{\lambda,\ell}$.
Adding that $R^{\ell}\in\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ and
${\mathcal{Q}}^{\lambda,\lambda}((R^{\ell}))=(\Delta^{\ell})$, we get the
announced commutative diagram. The remaining results trivially follow. ∎
### 4.2. A family of coadjoint orbits of $\mathrm{O}(p+1,q+1)$
We restrict in this paragraph to the case where $M$ is the homogeneous space
$\mathbb{S}^{p}\times\mathbb{S}^{q}$ of the conformal group
$G=\mathrm{O}(p+1,q+1)$. The latter acts linearly and in a Hamiltonian way on
$T^{*}\mathbb{R}^{p+1,q+1}$, and forms a Howe dual pair with
$\mathrm{SL}(2,\mathbb{R})$, its centralizer in the symplectic linear group
$\mathrm{Sp}(2n+2,\mathbb{R})$. More precisely, they form a symplectic dual
pair in the sense of [3]. Denoting their moment maps by
$\mu:T^{*}\mathbb{R}^{p+1,q+1}\longrightarrow{\mathfrak{g}}^{*}\qquad\text{and}\qquad
J:T^{*}\mathbb{R}^{p+1,q+1}\longrightarrow\mathrm{sl}(2,\mathbb{R})^{*},$
the symplectic reduction of $T^{*}\mathbb{R}^{p+1,q+1}$ with respect to
$\mathrm{SL}(2,\mathbb{R})$ for different regular values of the moment map $J$
gives rise to (finite coverings of) all the coadjoint orbits in the image of
$\mu$. We rather describe them as symplectic reductions at $0$ with respect to
Lie subgroups of $\mathrm{SL}(2,\mathbb{R})$ generated by the flow of
Hamiltonian functions in
$J^{*}\big{(}\mathrm{S}(\mathrm{sl}(2,\mathbb{R}))\big{)}$, i.e. polynomial
functions in $x^{2}=\eta_{AB}x^{A}x^{B}$, $xp=x^{A}p_{A}$ and
$p^{2}=\eta^{AB}p_{A}p_{B}$, where $(x^{A},p_{A})$ are cartesian coordinates
on $T^{*}\mathbb{R}^{p+1,q+1}$. Important such functions are given by the
Casimir elements of ${\mathfrak{g}}$ and $\mathrm{sl}(2,\mathbb{R})$ in
${\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$. They are equal to
$C=(xp)^{2}-x^{2}p^{2}$ and $C/4$ respectively, if we define the Killing form
by the map $(X,Y)\mapsto\frac{1}{2}\mathrm{Tr}(\rho(X)\rho(Y))$ with $\rho$
their standard representation.
We denote by $\left\langle f_{1},\ldots,f_{k}\right\rangle$ the Lie group
generated by the flow of Hamiltonian functions
$f_{1},\ldots,f_{k}\in{\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$ and
by $T^{*}\mathbb{R}^{p+1,q+1}//\left\langle f_{1},\ldots,f_{k}\right\rangle$
the corresponding symplectic quotient at $0$. If those functions are linearly
closed under the Poisson bracket, the latter space is then the quotient of the
zero locus of $f_{1},\ldots,f_{k}$ by their Hamiltonian flows. By the Marsden-
Weinstein theorem, this quotient space is a symplectic manifold if $0$ is a
regular value of the involved Hamiltonian functions. E.g., we have
(4.8) $T^{*}\left(\mathbb{R}^{p+1,q+1}\setminus\\{0\\}\right)//\left\langle
xp,x^{2}\right\rangle\simeq T^{*}M.$
Notice that $T^{*}M$ splits in three stable submanifolds under the Hamiltonian
$G$-action, according to the sign of the norm of the covectors, with
straightforward notations: $T^{*}M=T^{*}_{+}M\sqcup T^{*}_{0}M\sqcup
T^{*}_{-}M$.
###### Theorem 4.3.
Let $p,q\geq 1$, $n\geq 3$ and $P(\alpha,\beta)$ be the space of planes in
$\mathbb{R}^{p+1,q+1}$ of signature $(\alpha,\beta)$. The coadjoint orbits of
$G$ in the image of $\mu$ are classified along four families:
1. (1)
the one parameter family of semi-simple orbits $\mathcal{O}_{a^{+}}$ and
$\mathcal{O}_{a^{-}}$ for $a\in\mathbb{R}^{*}_{+}$ such that
$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}//\left\langle
xp,C-a\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{a^{+}}\sqcup\mathcal{O}_{a^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq\quad}$$\textstyle{P(2,0)\sqcup
P(0,2),}$
2. (2)
the one parameter family of semi-simple orbits $\mathcal{O}_{a}$ for
$a\in\mathbb{R}^{*}_{-}$ such that
$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}//\left\langle
xp,C-a\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq\quad}$$\textstyle{P(1,1),}$
3. (3)
the two nilpotent orbits $\mathcal{O}_{0^{+}}$ and $\mathcal{O}_{0^{-}}$ such
that
$\textstyle{T^{*}_{+}M\sqcup
T^{*}_{-}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{0^{+}}\sqcup\mathcal{O}_{0^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}\quad}$$\textstyle{P(1,0)\sqcup
P(0,1),}$
4. (4)
The minimal nilpotent orbit $\mathcal{O}_{00}$ such that
$\textstyle{(T^{*}M\setminus M)//\left\langle
R\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}}$$\textstyle{P(0,0).}$
All the arrows denote $G$-equivariant coverings, whose fibers are indicated as
superscript. The first ones are symplectomorphisms.
###### Proof.
Through the $G$-module isomorphisms
$\Lambda^{2}\mathbb{R}^{p+1,q+1}\simeq{\mathfrak{g}}\simeq{\mathfrak{g}}^{*}$,
coadjoint orbits are identified to $G$-orbits in the space of bivectors,
endowed with the natural $G$-action. The moment map $\mu$ is then given by
$T^{*}\mathbb{R}^{p+1,q+1}\ni(u,v)\mapsto u\wedge v$, and valued in the space
of simple bivectors $\mathrm{Bv}=\\{u\wedge
v|\,u,v\in\mathbb{R}^{p+1,q+1}\\}$. Our key tool is the $G$-equivariant
projection of $\mathrm{Bv}$ on the Grassmannian $\mathrm{Gr}(2,n+2)$ of planes
in $\mathbb{R}^{p+1,q+1}$. This is encompass in the following sequence of
$G$-spaces:
(4.9)
$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathrm{SL}(2,\mathbb{R})}$$\textstyle{\mathrm{Bv}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}\quad\qquad}$$\textstyle{\mathrm{Gr}(2,n+2)\cup\\{0\\},}$
where the superscripts denote the fibers of the coverings over
$\mathrm{Gr}(2,n+2)$. The moment map preserves the Poisson structure, hence a
$G$-stable submanifold of $T^{*}\mathbb{R}^{p+1,q+1}$ projects onto coadjoint
orbits of $G$, which themselves project onto $G$-orbits of
$\mathrm{Gr}(2,n+2)\cup\\{0\\}$. Thanks to the Witt Theorem, the latter are
known to be $\\{0\\}$ and the $6$ spaces $P(\alpha,\beta)$ of planes of given
signature $(\alpha,\beta)$ for the induced metric. E.g., the $G$-stable
embedding of $T^{*}_{+}M\sqcup T^{*}_{-}M$ into $T^{*}\mathbb{R}^{p+1,q+1}$
projects onto coadjoint orbits, which project onto the orbits $P(1,0)\sqcup
P(0,1)$ in $\mathrm{Gr}(2,n+2)$. The fibers of these two projections are
easily computed. In the three other cases, the zero locus of the given
Hamiltonian functions have the announced images in $\mathrm{Gr}(2,n+2)$. Their
Hamiltonian flows act only in the fibers of $\mu$, so the map $\mu$ descends
to the symplectic quotients. The fibers are proved to reduce to
$\mathbb{Z}_{2}$, resorting to the explicit expressions of the one parameter
groups generated by $xp$, $C$ and $R$. Namely, they respectively are
$(u,v)\mapsto(tu,t^{-1}v)$, $(u,v)\mapsto(u+(tv^{2})v,v-(tu^{2})u)$ and
$(u,v)\mapsto(u+tv,v)$ for $t\in\mathbb{R}^{*}$. We end with the four
sequences $(1)$, $(2)$, $(3)$, and $(4)$. There, a unique coadjoint orbit lies
over each orbit in $\mathrm{Gr}(2,n+2)$, since the action of the group $G$ is
transitive in the fibers of each arrow. For a proof of the minimality of
$\mathcal{O}_{00}$ we refer to [44]. ∎
The two last points in the latter theorem combine, according to Cordani [14],
to provide a conformal regularization by $T^{*}M$ of the cone
$\mathcal{O}_{0^{+}}\cup\mathcal{O}_{00}\cup\mathcal{O}_{0^{-}}$, with
singularity in $\mathcal{O}_{00}$.
###### Remark 4.4.
The used symplectic reductions of $T^{*}\mathbb{R}^{p+1,q+1}$ correspond
clearly to symplectic reduction with respect to $\mathrm{SL}(2,\mathbb{R})$
at, respectively, the points $(0,\sqrt{a},\pm\sqrt{a})$,
$(0,-\sqrt{|a|},-\sqrt{|a|})$, $(0,0,\pm a)$ and $(0,0,0)$. Hence, we obtain a
bijection between the coadjoints orbits of $\mathrm{SL}(2,\mathbb{R})$ and the
ones in the image of $\mu$, in accordance with [3].
Now, we determine the algebra of regular functions on each coadjoint orbit of
$G$ in the image of $\mu$. We have
${\mathfrak{g}}\simeq\Lambda^{2}\mathbb{R}^{p+1,q+1}$, that we represent by
the Young diagram ${\Yboxdim{5pt}\yng(1,1)}$. Accordingly, elementary
representation theory of the orthogonal Lie algebra leads to
(4.10)
${\mathfrak{g}}\odot{\mathfrak{g}}={\mbox{\tiny$\yng(2,2)\oplus\yng(1,1,1,1)$}}\quad\text{and}\quad{\mbox{\tiny$\yng(2,2)={\yng(2,2)}_{\,0}\oplus{\yng(2)}_{\,0}$}}\oplus\mathbb{R}.$
In the second decomposition, the index $0$ denotes the trace free part, and
the three components correspond to $\mathcal{K}_{2,0}$, $\mathcal{K}_{2,1}$
and the one-dimensional space generated by the Casimir element in
$\mathrm{S}_{2}({\mathfrak{g}})$, still denoted by $C$. The extra term in the
decomposition of ${\mathfrak{g}}\odot{\mathfrak{g}}$ is generated by exterior
products in $\Lambda\mathbb{R}^{p+1,q+1}$ of elements of
${\mathfrak{g}}\simeq\Lambda^{2}\mathbb{R}^{p+1,q+1}$.
###### Lemma 4.5.
The kernel of the pull-back
$\mu^{*}:\mathrm{S}({\mathfrak{g}})\rightarrow{\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$
by the moment map of ${\mathfrak{g}}$ is the ideal generated by
${\Yboxdim{5pt}\yng(1,1,1,1)}$.
###### Proof.
Since elements of ${\mathfrak{g}}$ are skew-symmetric $2$-tensors $V^{AB}$ on
$\mathbb{R}^{p+1,q+1}$, the map $\mu^{*}$ is explicitly given by $V^{AB\cdots
CD}\mapsto x_{A}\cdots x_{C}V^{AB\cdots CD}p_{B}\cdots p_{D}$, and vanishes
then on tensors $V^{AB\cdots CD}$ which are skew-symmetric in any $3$ indices.
Hence, the module ${\Yboxdim{5pt}\yng(1,1,1,1)}$ is in the kernel of $\mu^{*}$
and $\mu^{*}(\mathrm{S}_{k}({\mathfrak{g}}))$ is contained in the module
$\mathrm{S}_{k}({\mathfrak{g}})/\left(\,{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$,
described by the Young diagram with $2$ lines and $k$ columns. But none of the
irreducible components of such a Young diagram is in the kernel of $\mu^{*}$,
as all the traces, $x^{2},xp,p^{2}$, can occur in
${\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$. In conclusion, the
algebra $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ is isomorphic to
$\mathrm{S}({\mathfrak{g}})/\left(\,{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$. ∎
###### Proposition 4.6.
Let $a\in\mathbb{R}$ and
$\mathcal{I}_{a}=\left([C-a]\mathbb{R}\oplus{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$.
The algebras of regular functions are given by
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$ on $\mathcal{O}_{a^{(\pm)}}$, and
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{00}$ on $\mathcal{O}_{00}$, with
$\mathcal{I}_{00}=\left({\Yboxdim{5pt}{\yng(2)}_{\,0}}\right)+\mathcal{I}_{0}$.
Moreover, the algebras of regular functions on $\mathcal{O}_{0^{\pm}}$ and
$\mathcal{O}_{00}$ are isomorphic to $\mathcal{K}$ and $\mathcal{K}^{1}$
respectively.
###### Proof.
According to (4.8), we get that
$T_{\pm}^{*}(\mathbb{R}^{p+1,q+1}\setminus\\{0\\})//\left\langle
xp,C\right\rangle$ is a $\mathbb{Z}_{2}$-covering of $T^{*}_{\pm}M$. Together
with theorem 4.3, this proves that each coadjoint orbit of $G$ in the image of
$\mu$ is finitely covered by a symplectic reduction of
$T^{*}\mathbb{R}^{p+1,q+1}$. Therefore, its algebra of regular functions is
isomorphic to the corresponding reduction of
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))$. The reduction with respect to $xp$
modifies only the fibers of $\mu$ and the Casimir $C$ Poisson commutes with
all elements in $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$, so that reduction with
respect to $\left\langle xp,C-a\right\rangle$ amounts to modding out by
$(C-a)$. This gives the result for the orbits $\mathcal{O}_{a^{(\pm)}}$ for
every $a\in\mathbb{R}$. By Theorem 4.1, the algebra $\mathcal{K}$ is generated
by ${\mathfrak{g}}$. The isomorphism
$\mathcal{K}\simeq\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{0}$ follows then
from the local diffeomorphism between $T^{*}_{\pm}M$ and
$\mathcal{O}_{0^{\pm}}$.
Resorting to Theorem 4.3, the coadjoint orbit $\mathcal{O}_{00}$ is locally
diffeomorphic to the symplectic quotient $(T^{*}M\setminus M)//\left\langle
R\right\rangle$, i.e. to the quotient of the submanifold $T^{*}_{0}M\setminus
M$ of null covectors by the Hamiltonian flow of $R$. Thus, the algebra of
regular functions on $\mathcal{O}_{00}$ arises as a reduction of
$\mathcal{K}$. Since $\\{R,\mathcal{K}\\}\subset(R)$, this reduced algebra is
$\mathcal{K}/(R)$, which is isomorphic to $\mathcal{K}_{1}$. As
$R\in\mathcal{K}_{2,1}$ is the pull-back of an element in
${\Yboxdim{5pt}{\yng(2)}_{\,0}}$, we finally obtain $\mathcal{I}_{00}$. ∎
### 4.3. The Joseph ideal
We return now to a general conformally flat manifold $(M,[\mathrm{g}])$, and
use notation of Paragraph 4.1, in particular
$\mathcal{K}\simeq\mathrm{S}({\mathfrak{g}})/I$ and
$\mathcal{K}^{1}\simeq\mathrm{S}({\mathfrak{g}})/I^{1}$. The algebraic
description of these symmetry algebras obtained in Proposition 4.6 are of
local nature and thus carry on over $(M,[\mathrm{g}])$. Hence, the ideals
$I,I^{1}$ identify to $\mathcal{I}_{0},\mathcal{I}_{00}$. We compute now the
corresponding ideals $J^{\lambda}$ and $J^{\lambda,1}$ in
$\mathfrak{U}({\mathfrak{g}})$, which define the algebras
$\mathcal{A}^{\lambda}$ and $\mathcal{A}^{\lambda,1}$. Recall that we define
the Killing form by $\frac{1}{2}\mathrm{Tr}(XY)$, for every
$X,Y\in{\mathfrak{g}}$. The corresponding Casimir operator ${\mathcal{C}}$ in
$\mathfrak{U}({\mathfrak{g}})$ is given by the symmetrization of the Casimir
element $C$ in $\mathrm{S}({\mathfrak{g}})$.
###### Proposition 4.7.
For every $\lambda\in\mathbb{R}$, the ideals $J^{\lambda}$ are equal to
$\big{(}\mathrm{Sym}({\Yboxdim{5pt}\yng(1,1,1,1)})\oplus[{\mathcal{C}}-\rho(\lambda)]\mathbb{R}\big{)}$,
where ${\mathcal{C}}$ is the Casimir operator of ${\mathfrak{g}}$ and
$\rho(\lambda)=n^{2}\lambda(1-\lambda)$ its eigenvalue on $\lambda$-densities.
For $\lambda=\frac{n-2}{n}$,
$J^{\lambda,1}=\left(\mathrm{Sym}({\Yboxdim{5pt}\yng(2)}_{\,0})\right)+J^{\lambda}$
is the Joseph ideal.
###### Proof.
Since the graded ideal associated to $J^{\lambda}$ is $I$, we deduce that
$J^{\lambda}$ is also quadratic and resorting to Theorem 4.1, we have
$J^{\lambda}_{2}=\Phi^{\lambda}(I_{2})$ with
$\Phi^{\lambda}=\mathrm{Sym}\circ\phi^{\lambda}$. The map $\phi^{\lambda}$
being ${\mathfrak{g}}$-equivariant, the space $\phi^{\lambda}(I_{2})$ is a
${\mathfrak{g}}$-submodule of
$\mathfrak{U}_{2}({\mathfrak{g}})\simeq\mathbb{R}\oplus{\mathfrak{g}}\oplus\mathrm{S}_{2}({\mathfrak{g}})$.
Hence, $J_{2}^{\lambda}$ is generated by
$\mathrm{Sym}\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$ and the Casimir
operator ${\mathcal{C}}$ of $\mathfrak{U}({\mathfrak{g}})$, modified by some
real number. Since this element projects onto $0$ on
$\mathcal{D}^{\lambda,\lambda}$, this real number is necessarily the
eigenvalue of $\ell^{\lambda}({\mathcal{C}})$ on $\lambda$-densities. The
latter has been computed in [17], where the opposite Killing form is used. The
formula giving $J^{\lambda,1}$ is trivially deduced from Corollary 4.2. Thanks
to Theorem 3.5, we have the isomorphism of algebras
${\mathrm{gr}}\,(\mathfrak{U}({\mathfrak{g}})/J^{\lambda,1})\simeq\mathrm{S}({\mathfrak{g}})/I^{1}$.
As $I_{1}$ is prime, we deduce that $J^{\lambda,1}$ is completely prime.
Besides, their common characteristic variety is the closure of the minimal
nilpotent coadjoint orbit of $G$. These two properties characterize the Joseph
ideal [25]. ∎
The identification of the Joseph ideal in the context of the higher symmetries
of the Laplacian was already obtained in different manners [22, 41], but never
from its original definition like here. From the latter proposition, the
ideals $J^{\lambda,\ell}$ associated to symmetries of $\Delta^{\ell}$ are
straightforwardly deduced as they are generated by $J^{\lambda}$ and
$(\ell^{\lambda})^{-1}(\Delta^{\ell})$. The latter corresponds to the Young
diagram … 0 of length $2\ell$. The determination of those ideals has been
already performed in the context of higher symmetries of $\Delta^{\ell}$ in
[19, 20, 40], but in different terms. Let us make clear the link between the
two approaches. We denote by $\left\langle\cdot,\cdot\right\rangle$ the chosen
Killing form and $C$ the associated Casimir element in
$\mathrm{S}({\mathfrak{g}})$. In the previous works, the projections of
$X\odot Y\in{\mathfrak{g}}\odot{\mathfrak{g}}$ on each irreducible component
are used. Following
${\mathfrak{g}}\odot{\mathfrak{g}}={\Yboxdim{5pt}\yng(2,2)}_{\,0}\oplus{\Yboxdim{5pt}\yng(2)}_{\,0}\oplus\mathbb{R}\oplus{\Yboxdim{5pt}\yng(1,1,1,1)}$,
we have $X\odot Y=X\boxtimes Y+X\bullet Y+\frac{\left\langle
X,Y\right\rangle}{2\dim{\mathfrak{g}}}C+X\wedge Y$. Then, the ideal
$J^{\lambda}$ is clearly generated by $\mathrm{Sym}\big{(}\frac{\left\langle
X,Y\right\rangle}{2\dim{\mathfrak{g}}}(C-\rho(\lambda))+X\wedge Y\big{)}$ for
$X,Y\in{\mathfrak{g}}$ or equivalently by
$\mathrm{Sym}\big{(}X\odot Y-X\boxtimes Y-X\bullet
Y+\frac{\rho(\lambda)}{2\dim{\mathfrak{g}}}\left\langle
X,Y\right\rangle\big{)},$
which is the obtained expression in [19, 20, 40], modulo the extra generator
associated to $R^{\ell}$.
### 4.4. Quantization of a family of coadjoint orbits of $G$
We have described the algebras of regular functions on the coadjoint orbits of
$G$ in the image of $\mu$ as quotients
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}$ for various ideals $\mathcal{I}$. We
perform now their deformation quantization, which should arise from
quantization-like maps
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}\rightarrow\mathfrak{U}({\mathfrak{g}})/\mathcal{J}$,
for ideals $\mathcal{J}$ to be defined. This means to find deformations of the
Poisson algebra $\mathrm{S}({\mathfrak{g}})$ which restrict to coadjoint
orbits, and this is in general not trivial. Thus, for ${\mathfrak{g}}$ a semi-
simple Lie algebra, Cahen, Gutt and Rawnsley have proved the non-existence of
a star product on ${\mathcal{C}^{\infty}}({\mathfrak{g}}^{*})$ tangent to each
coadjoint orbit [11]. Here, for the graded algebras $\mathcal{K}$ and
$\mathcal{K}^{1}$, such a quantization map is provided by the conformally
equivariant quantization, we have proved that it induces a graded
${\mathfrak{g}}$-equivariant star-product on them and determines the
corresponding ideals $\mathcal{J}$. As the remaining algebras
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$ for $a\neq 0$ are only filtered,
this direct approach fails and we resort to the following Lemma.
###### Lemma 4.8.
Let $a\in\mathbb{R}$. There exists a ${\mathfrak{g}}$-equivariant linear map
$\phi_{a}=\mathrm{Id}+N_{a}$ on $\mathrm{S}({\mathfrak{g}})$, such that
$N_{a}$ lowers the degree and $\phi_{a}(\mathcal{I}_{a})=\mathcal{I}_{0}$,
hence $\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}\simeq\mathcal{K}$.
###### Proof.
We know that $\mathrm{S}({\mathfrak{g}})\simeq I\oplus\mathcal{K}$ and
$I=(C)+\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$. Resorting to the semi-
simplicity of ${\mathfrak{g}}$ and the filtration of
$\mathcal{I}_{a}=(C-a)+\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$, we get that
$\mathrm{S}({\mathfrak{g}})\simeq\mathcal{I}_{a}+\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$
and $(C-a)$ admits a ${\mathfrak{g}}$-stable complement in $\mathcal{I}_{a}$.
The map $\phi_{a}$ defined by by $\frac{C}{C-a}\mathrm{Id}$ on $(C-a)$ and the
identity on a ${\mathfrak{g}}$-stable complementary space satisfies the
required properties. ∎
###### Theorem 4.9.
There exists a family of ${\mathfrak{g}}$-equivariant quantizations
$(\Phi_{a}^{\lambda})_{a,\lambda\in\mathbb{R}}$ of
$\mathrm{S}({\mathfrak{g}})$ such that: (i) it lifts
$({\mathcal{Q}}^{\lambda,\lambda})_{\lambda\in\mathbb{R}}$ to
$\mathrm{S}({\mathfrak{g}})$ for $a=0$, (ii) it induces a family of symmetric
${\mathfrak{g}}$-invariant star products on the coadjoint orbits
$\mathcal{O}_{a^{(\pm)}}$ for $a\in\mathbb{R}$, (iii) if $a=0$ and
$\lambda=\frac{n-2}{2n}$, it induces the unique graded
${\mathfrak{g}}$-equivariant star-product on $\mathcal{O}_{00}$, introduced by
Arnal-Benhamor-Cahen [1] and Astashkevich-Brylinski [2].
###### Proof.
The Theorem 4.1 ensures the existence of a ${\mathfrak{g}}$-equivariant
quantization $\Phi_{\lambda}$ of $\mathrm{S}({\mathfrak{g}})$ lifting
${\mathcal{Q}}^{\lambda,\lambda}$ for every $\lambda\in\mathbb{R}$. The lift
property is equivalent to $\Phi^{\lambda}(I)=J^{\lambda}$. We define then the
family of ${\mathfrak{g}}$-equivariant quantizations
$\Phi^{\lambda}_{a}=\Phi^{\lambda}\circ\phi_{a}$, where $\phi_{a}$ is
introduced in Lemma 4.8. It can be chosen such that $\phi_{0}=\mathrm{Id}$, so
(i) is trivially satisfied. The Lemma 4.8 ensures that
$\Phi^{\lambda}_{a}(\mathcal{I}_{a})$ is an ideal and a
${\mathfrak{g}}$-module, hence the ${\mathfrak{g}}$-invariant star product
$\star_{\Phi_{a}^{\lambda}}$ on $\mathrm{S}({\mathfrak{g}})$, induced by
$\Phi_{a}^{\lambda}$, descends on the quotient
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$. We recall that
$\star_{\Phi_{a}^{\lambda}}$ is symmetric if $\Phi_{a}^{\lambda}$ satisfies
$\tau\circ\Phi_{a}^{\lambda}=\Phi_{a}^{\lambda}\circ\gamma$. Redefining
$\Phi_{a}^{\lambda}$ by
$\frac{1}{2}(\Phi_{a}^{\lambda}+\tau\circ\Phi_{a}^{\lambda}\circ\gamma)$ this
is trivially the case, and the quantization $\Phi_{0}^{\lambda}$ is still a
lift of ${\mathcal{Q}}^{\lambda,\lambda}$ by uniqueness of the latter. This
proves (ii). The last point follows then from Corollary 4.2, Proposition 4.6
and the uniqueness result in [1, 2]. ∎
###### Remark 4.10.
For two distinct coadjoint orbits, the star products obtained above do not
coincide in general. This is reminiscent to the work of Fioresi and Lledo
[23], dealing with star products tangential to semi-simple coadjoint orbits of
semi-simple Lie groups.
Quantization of a coadjoint orbit of a Lie group is more usually understood as
building an irreducible unitary representation of this group from that orbit.
Since the Hamiltonian $G$-action on
$T^{*}(\mathbb{S}^{p}\times\mathbb{S}^{q})$ preserves its vertical
polarization, the Kirillov’s orbit method [26] applies to the coadjoint orbits
$\mathcal{O}_{0^{\pm}}$. It leads to the unitary representation of $G$ onto
the pre-Hilbert space of compactly supported half-densities on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, endowed with its canonical Hermitian
product $(\phi,\psi)=\int_{M}\bar{\phi}\psi$. The corresponding infinitesimal
action of the Lie algebra ${\mathfrak{g}}$ by Lie derivative extends to a
deformed representation of the whole Poisson algebra $\mathcal{K}$ of regular
functions on $\mathcal{O}_{0^{\pm}}$, via the conformally equivariant
quantization ${\mathcal{Q}}^{\frac{1}{2},\frac{1}{2}}$.
In contradistinction, the minimal coadjoint orbit cannot be quantize via the
orbit method as it admits no $G$-invariant polarization. This key fact is
proved in [43] and stems from the following result of [8]:
${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$ can be identified to a Lie subalgebra of
polynomial vector fields on $\mathbb{R}^{m}$ iff $m\geq p+q$. The following
proposition shows how conformally equivariant quantization provides a
supplement to the orbit method for the minimal nilpotent orbit.
###### Proposition 4.11.
Let $\lambda=\frac{n-2}{2n}$. The algebra $\mathcal{K}^{1}$ of conformal
Killing tensors on $\mathbb{S}^{p}\times\mathbb{S}^{q}$ identifies to the
algebra of regular functions on the minimal coadjoint orbit
$\mathcal{O}_{00}$. Its conformally equivariant quantization
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{1})\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda,1}$
is the algebra of higher symmetries of the conformal Laplacian on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, which acts faithfully on its kernel.
Moreover, $J^{\lambda,1}$ is the Joseph ideal.
###### Proof.
Since the Joseph ideal is a maximal primitive ideal, the action of
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{1})\simeq\mathcal{A}^{\lambda,1}$
on the kernel of $\Delta$ is trivial or faithful, hence it is faithful. ∎
In particular, the infinitesimal action of ${\mathfrak{g}}$ on harmonic
functions is given by the Lie derivative of $\frac{n-2}{2n}$-densities and the
kernel of the induced representation of $\mathfrak{U}({\mathfrak{g}})$ is the
Joseph ideal. This result dates back to [5] and was so far the only link
between the minimal coadjoint orbit of $G$ and its minimal representation,
which integrates the one of ${\mathfrak{g}}$ on $\ker\Delta$.
### Acknowledgements
It is a pleasure to acknowledge Christian Duval and Valentin Ovsienko for
their constant interest in this work, and Josef Šilhan for invaluable
discussions.
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|
arxiv-papers
| 2011-07-28T22:20:03 |
2024-09-04T02:49:21.026067
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jean-Philippe Michel",
"submitter": "Jean-Philippe Michel",
"url": "https://arxiv.org/abs/1107.5840"
}
|
1107.5919
|
# The Productive Ligurian Pool
###### Abstract
In contrast with the behavior of the eddies in the open-ocean, the sub-
mesoscale eddies generated in the constricted Ligurian Basin (NW
Mediterranean), are unproductive but their combined effect, arranged in a rim-
like fashion , contributes to the containment of a Productive Ligurian Pool
(PLP). Data derived from MODIS satellite sensor showed persistent higher
chlorophyll concentrations in the centre of the basin, concurrent with high
EKE values in its surroundings, derived from AVISO altimetry merged products.
This suggested that this ’productive pool’ is maintained by the intense
(sub)mesoscale eddy activity in the rim. Numerical realistic experiments,
using a Regional Ocean Model System, forced by MERCATOR and by a high-
resolution COSMO-l7 atmospheric model, also showed that most of the sub-
mesoscale eddies, during 2009 and 2010, are concentrated in the rim
surrounding the basin, contributing to the formation of a basin-scale cyclonic
gyre. We hypothesized that the interaction between eddies in the rim might
contribute to import of nutrients into the pool in two ways: (i) by advection
of nutrients from the nearby coastal regions into the pool; (ii) by
concentrating eddy upwelled nutrients inside the pool; or by a combination
thereof.
Gindraft=false CASELLA ET AL Productive Ligurian Pool E. Casella, P. Tepisch
CIMA Research Foundation, Via Armando Magliotto, 2 - 17100 Savona,
Italy(elisa.casella@cimafoundation.org) X. Couvelard, CCM-Center for
Mathematical Sciences, University of Madeira, Campus da Penteada, 9000-390,
Funchal, Madeira, Portugal (xavierc@uma.pt) R. M.A. Caldeira, CIIMAR-
Interdisciplinary Centre of Marine and Environmental Research, Rua dos Bragas,
289, 4050 - 123 Porto, Portugal (rcaldeira@ciimar.up.pt)
11affiliationtext: Corresponding author
## 1 Introduction
The northwest part of the Mediterranean Sea, the Ligurian Sea, is known as an
oligotrophic region (e.g. Nezlin et al., 2004; D’Ortenzio and Ribera d’Alcalà,
2009), and therefore any supply of nutrients is extremely valuable to sustain
the whole trophic chain. The biological production of the Ligurian Sea is
dominated by a marked seasonal cycle (Arnone, 1994) and a significant inter-
annual variability (La Violette, 1994; Marty et al., 2002). Nevertheless, the
system heterogeneity is closely associated with local and regional
hydrodynamical factors, especially those responsible for a high variability of
the mixed layer depth and for the transport processes (Nezlin et al., 2004;
Casella et al., 2011). The regional hydrodynamics is dominated by a cyclonic
circulation system feed by the East Corsica Current (ECC) and the West Corsica
Current (WCC) which is known as the Liguro-Provençal-Catalan (LPC) current
(Figure 1). The LPC-current flows along the costal slopes of Italy, France and
Spain, and is affected by instability processes which generate (sub)mesoscale
eddies, capable of inducing relatively intense shelf-edge flows, producing
significant dynamical heterogeneity (Millot, 1991).The presence of mesoscale
eddies in the Western Mediterranean Sea is well documented (Santoleri et al.,
1983; Marullo et al., 1985; Gasparini et al., 1999; Robinson and Leslie, 2001;
Echevin et al., 2003; Casella et al., 2011). Nevertheless, the peculiar
feature of the Northwest Mediterranean Sea emphasized herein is that, in
contrast to many other marginal seas rich in mesoscale eddy activity, the
nearshore zones are poor in surface chlorophyll (Nezlin et al., 2004). The
LPC-current and the presence of mesoscale eddy activity, is reflected in the
high values of EKE (Figure 2). In this discussion, we focus on role of a rim
of eddies along the Ligurian Pronvençal basin, for the maintenance of a
cyclonic gyre, and for the containment of the Productive Ligurian Pool (PLP,
hereafter).
## 2 Data and methods
### 2.1 Satellite products
Eddy Kinetic Energy (EKE) was computed from altimetry data. Surface velocities
were computed from weekly merged products of absolute dynamic topography
(ADT), at $1/8^{\circ}$ resolution on a Mercator projection, distributed by
AVISO (www.aviso.oceanobs.com). ADT maps are obtained by merging measurements
from all available altimeter missions (Ducet et al., 2000). Combining data
from different missions significantly improves the estimation of mesoscale
signals (Pascual et al., 2006).We used the processed series which considers up
to 4 satellites at a given time, thus it has the best possible regional
sampling. Furthermore, the period of our study (2009-2010) has one of the best
altimetric coverage with a four-satellite constellation (Jason 1, Jason 2,
Envisat and Cryostat). ADT is obtained adding along track Sea Level Anomaly to
the Mean dynamic Topography (Rio and Hernandez, 2004).
Chlorophyl data was derived from the MODIS-Moderate Resolution Imaging
Spectroradiometer flying onboard of the Aqua platform. MODIS data was
processed by the Ocean Biology Processing Group (OBPG) at Goddard Space Flight
Center. Merged Level-3 chlorophyll products are being created routinely for
daily, 8-day, monthly, seasonal and annual time periods, processed at a 9km
spatial resolution. Although seasonal and monthly averages were analyzed, only
the annual average product is shown, in order to emphasize the persistence of
the PLP event during 2009 and 2010.
### 2.2 The Ligurian-Provençal Regional Ocean Modeling System
The Regional Ocean Modeling System (ROMS) (Shchepetkin and McWilliams, 2003;
Shchepetkin and McWilliams, 2005) was configured for the Northwestern part of
the Mediterranean Sea. ROMS solves the primitive equations based on the
Boussinesq approximation. In the Ligurian model solutions for 2009 and 2010,
the model domain extends from $37.8^{\circ}$ N to $44.5^{\circ}$ N and from
$2^{\circ}$ E to $16.5^{\circ}$ E. The bottom topography is derived from a 30
arc-second resolution database GEBCO08. In the model version adopted here,
there are two open lateral boundaries: at $37.8^{\circ}$ N and $2^{\circ}$ E.
We have chosen a horizontal resolution of $1/32^{\circ}$. At this resolution,
the Rossby radius of deformation (of the order of $5-12km$ in the whole
Mediterranean and for different seasons, see Grilli and Pinardi (1998) is
resolved and consequently the model configuration is adequate to simulate
mesoscale structures. The model grid has 35 vertical levels with vertical
refinement near the surface, to obtain a satisfactory representation of the
surface layer and the euphotic zone. At the open lateral boundaries, the model
is forced with temperature, salinity and velocity fields obtained from the
MERCATOR product PSY2V3 (www.mercator-ocean.fr). MERCATOR, has an horizontal
spatial resolution of $1/12^{\circ}$, with daily outputs.
At the sea surface, the regional ocean circulation model was forced with the
monthly mean climatologies of heat and freshwater fluxes, derived from the
Comprehensive Ocean-Atmosphere Data Set, COADS(da Silva A.M. et al., 1994).
For the atmospheric momentum, wind-stress was extracted from the Limited Area
Model Italy (COSMO-I7). COSMO-I7 (Montani et al., 2003), is a non-hydrostatic
and fully compressible numerical weather prediction model, which is a regional
version of the Lokal Model (Schattler and Doms, 2000), regularly used for
operational and research applications. The COSMO-I7, 3 hourly solutions, have
an horizontal resolution of $1/16^{\circ}$. Validation of the COSMO-I7 wind
fields has been widely performed (Steppeler et al., 2003). As shown in Casella
et al. (2011), forcing the Ligurian ocean circulation model with high-
resolution winds, substantially contributed to the formation of mesoscale and
sub-mesoscale eddies, which is an important characteristic of the regional
dynamics (Gasparini et al., 1999; Robinson and Leslie, 2001; Echevin et al.,
2003; Nezlin et al., 2004).
### 2.3 ROMS data comparison
Non-spite the scarce availability of public data, we have compared the two
years (2009, 2010) of ROMS numerical simulations with current meter data,
deployed in the Corsica Channel. Data has been collected by the ‘Istituto di
Scienze Marine Consiglio Nazionale delle Ricerche’ (ISMAR-CNR, La Spezia). We
considered data from two moorings, located at $9.685^{\circ}$ E, Longitude and
$43.033^{\circ}$ N, Latitude, measuring at two different depths (70m, 125m).
From March 2010, the current meters have been replaced by an Acoustic Doppler
Current Profiler (ADCP), which measured currents from 13m to about 380m,
processed in 20m bins. During the study period, we compared the simulated
current in the Corsica Channel with current meters and ADCP data. The model
equivalent mean current, are in good agreement with the measured data, for the
same approximate location. The current between 70-125m has the same general
direction (north), and comparable magnitudes (0.4-0.2 $ms^{-1}$ in winter;
$0.1-0.2ms^{-1}$ during summer), as well as the same seasonal variability.
ROMS also reproduced the general SST and altimetry, monthly and seasonal
trends, measured with satellite sensors.
### 2.4 Eddy detection algorithm
In order to detect and track eddies from our numerical solution, the ‘Find
Okubo-Weiss Eddies in Aviso SSH Product tool contained in the MGET tool for
ArcGis’ (Roberts et al., 2010) was adapted to be used with the ROMS output. As
defined by Henson and Thomas (2008), an eddy consists of a region of high
vorticity (the core), surrounded by a circulation cell of high strain (the
ring). Such regions can be detected using the Okubo-Weiss parameter (Okubo,
1970; Weiss, 1991). The eddy detection algorithm used the Okubo-Weiss
parameter (Q) computed for ROMS with Romstools (Penven et al., 2008), and
filter the candidate eddy cores using various techniques designed to remove
small, ephemeral eddies and other unwanted features. These techniques include
removing eddies that are smaller than a minimum size and tracking eddies
through the time-series, and removing eddies that are too short in duration.
The final analysis classifies eddies as either anticyclonic or cyclonic based
on the curvature of the sea surface height (anticyclonic is concave down,
cyclonic is concave up). After a sensitivity analysis for the study region, a
threshold value of Q = ($-1.5e^{-9}s^{-2}$) was chosen, and only eddies with a
minimum duration of 3 days, and a minimum area of 8 pixels (about 5km radius
of the vorticity core), were considered.
## 3 The Productive Ligurian Pool
The analysis of yearly mean ocean color data, confirmed the persistent
existence of productive zone in centre of the Ligurian Sea, during 2009 and
2010. Figure 2 shows the spatial distribution of the yearly averaged
chlorophyll concentration in 2010, (2009 has a similar pattern, figure 2-c).
The analysis of the concurrent altimetry data for 2010 also suggests that the
region surrounding the LPB has the most intense EKE values (Figure 2-b). The
yearly averaged chlorophyll concentration maintains the imprint of the spatial
distribution during the spring bloom. Figure 2 shows a dynamic region at the
rim dominated by the ECC, the WCC and the LPC-current, with a low chlorophyll
concentration, whereas the central area (Productive Ligurian Pool), shows a
high chlorophyll concentration. In fact, the ROMS simulations confirm the
presence of the well known cyclonic gyre surrounding the LPB, as shown by
streams lines for the 2009 and 2010 model solutions (Figure 2). The model
clearly shows a cyclonic circulation which surrounds the productive area, with
intense eddy activity in the rim. To note is the inter-annual variability, the
main patch of chlorophyll is not located in the same location inside the
cyclonic gyre, thus suggesting the heterogeneity previously discussed in the
literature.
Eddies are located along the path of the ECC, the WCC and the LPC currents, as
it is shown in figure 2. A highly dynamic rim of eddies can permit the
transport of nutrients from the eddy-induced vertical mixing and/or advected
from a nearby coastal regions to be contained in the central, and less
turbulent zone (hence the designative name of pool). This transport is
expected to have a strongest enriching effect when MLD in the central zone is
shallower and nutrient can remain in the euphotic layer (i.e. during spring).
Thus it is not surprising that productivity is twice as high during the spring
bloom, when compared to the yearly averaged values, nevertheless the general
pattern is maintained. During 2009, 810 eddies were detected in the realistic
ROMS solutions of the LPB, 316 (40%) were cyclonic and 485 (60%) anticyclonic;
whereas during the 2010 simulation, 935 eddies were counted, 357 (38%)
cyclonic and 578 (62%) anticyclonic. The strongest wind-stress peaks were also
calculated for the 2010 solution, suggesting that the wind plays a role on the
generation of (sub)mesoscale features, which in turn seem to condition the
overall productivity of the region. In fact, the 2010 spring bloom showed
higher chlorophyll concentration values, compared to 2009.
Constricted basins and marginal seas in contrast with the open-ocean can not
assume steady-state conditions, hence the recent developments on strait-
marginal seas systems theories. Pratt and Spall (2008), used numerical models
to test a theory, which explains the exchanged occurring between a schematic
buoyancy-forced marginal sea and the open ocean. The incoming surface layer
formed a baroclinic unstable boundary current, that circles the marginal sea
in a cyclonic manner and feeding heat to the interior, by way of mesoscale
eddies. As in the Ligurian-Provençal Basin, eddies play a fundamental role in
the maintenance and in the exchange processes with the marginal sea.
Consistent with the overall heat and volume balances for the marginal sea,
Pratt and Spall (2008) found a continuous family of hydraulically controlled
states including critical flows, mediated by eddies, in a strait between the
marginal sea and the open-ocean. Nevertheless, the behavior of eddies in a
constricted basin and/or marginal sea is not yet fully understood, further
investigation of the LPB could continue to provide new insights. Kida et al.
(2009) showed that the eddies accompanying baroclinic instability in the Faroe
Bank Channel can set up a double-gyre circulation in the upper ocean, an eddy-
driven topographic beta plume. In regions where baroclinic instability is
growing, the momentum flux from the overflow into the upper ocean can act as a
drag on the overflow causing the overflow to descend the slope at a steeper
angle than what would arise from bottom friction alone. In contrast, the upper
layer of the Mediterranean overflow is likely to be dominated by an
entrainment-driven topographic beta plume. The difference arises because
entrainment occurs at a much shallower location for the Mediterranean case and
the background potential vorticity gradient of the upper ocean is expected to
be much larger (Kida et al., 2009).
###### Acknowledgements.
The authors are grateful for travel funds provided by CIMA Foundation, which
enabled researchers to materialized this collaboration, and to CCM for hosting
the researchers in Madeira. Numerical model solutions were calculated at
CIIMAR HPC unit, constructed using funds the FCT-Portuguese National Science
Foundation pluriannual, and from RAIA ($0313\\_RAIA\\_1\\_E$) and RAIA.co
projects, co-funded by INTERREG IV and by FEDER (‘Fundo Europeu de
Desenvolvimento Regional, 2007 2013’), through the POCTEP regional initiative.
The altimeter products were produced by SSALTO/DUACS and distributed by AVISO
with support from CNES. MODIS-Aqua data was extracted using the Giovanni
online data system, developed and maintained by the NASA GES DISC.
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Figure 1: Bathymetry (meters) and general circulation of the Ligurian-
Provençal Basin. TC - Thyrenian Current; ECC - East Corsica Current; WCC -
West Corsica Current; CC - Corsica Channel; LPB - Ligurian Provençal Basin;
LPCC - Ligurian Provençal Catalan Current; GL - Golf of Lion. The circular
patterns represent the rim-eddies and the small arrows the nutrient
entrainment transport process.
Figure 2: Top Panel: (a) Yearly averaged chlorophyll ($mgm^{-3}$), derived
from MODIS-Aqua; and (b) yearly averaged EKE ($m^{2}s^{-2}$) derived from
AVISO altimetry, for 2010. Bottom panel: Yearly averaged ROMS streamlines and
eddy detected (black spots),representative of the averaged circulation for the
Ligurian-Provençal Basin for (c) 2009 and for (d) 2010. Mode solutions are
overlaid onto yearly averaged chlorophyll data
|
arxiv-papers
| 2011-07-29T10:16:31 |
2024-09-04T02:49:21.043793
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Casella, P. Tepisch, X. Couvelard, R. M. A. Caldeira",
"submitter": "Rui Caldeira",
"url": "https://arxiv.org/abs/1107.5919"
}
|
1107.5924
|
# Reachability in Biochemical Dynamical Systems by Quantitative Discrete
Approximation
L. Brim Faculty of Informatics
Masaryk University
Botanická 68a, Brno, Czech Republic J. Fabriková Faculty of Informatics
Masaryk University
Botanická 68a, Brno, Czech Republic S. Dražan Faculty of Informatics
Masaryk University
Botanická 68a, Brno, Czech Republic and D. Šafránek The work has been
supported by the Grant Agency of Czech Republic grant No. 201/09/1389 and by
the Czech ministry of education intent No. MSM0021622419.Faculty of
Informatics
Masaryk University
Botanická 68a, Brno, Czech Republic safranek@fi.muni.cz
###### Abstract
In this paper a novel computational technique for finite discrete
approximation of continuous dynamical systems suitable for a significant class
of biochemical dynamical systems is introduced. The method is parameterized in
order to affect the imposed level of approximation provided that with
increasing parameter value the approximation converges to the original
continuous system. By employing this approximation technique, we present
algorithms solving the reachability problem for biochemical dynamical systems.
The presented method and algorithms are evaluated on several exemplary
biological models and on a real case study. This is a full version of the
paper published in the proceedings of CompMod 2011.
## 1 Introduction
Under the modern holistic paradigm provided by _systems biology_ [6], genome-
scale knowledge of individual components is combined with knowledge of
interactions underlying the physiology of living organisms. The central goal
of systems biology is to integrate all available biological data and to
reconstruct _executable models_ [20] which allow to investigate complicated
behaviour emerging from the underlying biochemistry. An important dimension is
the quantitative aspect of the data and processes being modeled.
With respect to [19], we consider biological models to be captured by the
notion of a _biochemical dynamical system_ consisting of variables describing
a certain quantity of the respective species in time (e.g., number of
molecules or molar concentration). Variable values evolve in time with respect
to rules modeling the effect of reactions. The space of all possible
configurations of variable values is referred as the _state space_.
There exist several modeling approaches that differ in abstraction employed
for modeling of time, variable values, and molecular interaction effects. The
most commonly used approach concerns systems of ordinary differential
equations (ODE) [29] where both time and model variables are interpreted as
continuous quantities. Effects of interactions are modeled in terms of
continuous deterministic updates of variables. Variable values represent molar
concentrations of the species. In general, the ODE approach relies on many
physical and chemical assumptions simplifying thermodynamic conditions under
which particular biochemical phenomena can be modeled correctly [26]. It is
important to note that even simple interactions such as second order reactions
lead to non-linear ODEs. However, under certain assumptions, biological
systems make specific subclasses of general non-linear dynamical systems. Such
a specialization motivated development of specific analysis techniques [19, 8,
5, 25].
Nevertheless, dimensionality and complexity of biological models preclude
satisfactory application of analysis methods implying that to explore the
model dynamics the only practicable method is numerical simulation. Since
numerical simulation generates an approximate solution (a trajectory) starting
from a single initial point in the continuous state space, the scope of such
exploration is limited to the particular trajectory only. This is sufficient
for “local” analysis provided that initial conditions are precisely known.
However, studied systems are typically under-determined in terms of uncertain
quantitative parameters and initial conditions. Therefore generalization of
the exploration scope is necessary to reveal and understand the complicated
emergent behaviour. An important example of a problem which cannot be
effectively solved by local methods is _global temporal property_ – the
problem to decide whether a given dynamical phenomenon, e.g., oscillation or
variables correlation, is globally present/absent for all considered initial
conditions [17, 10].
In this paper we limit ourselves to a subclass of dynamical phenomena
representing _reachability_ of a given portion of the state space. Example of
a global temporal property problem that belongs to this subclass is to
identify minimal or maximal concentration of species reachable from a
particular set of initial conditions.
In general, the reachability problem is undecidable due to unboundedness and
uncountability of the state space. However, since concentrations of species
cannot expand infinitely, state spaces of biological systems dynamics can be
considered bounded in most cases. Analysis can be therefore considered
indirectly on suitable finite discrete approximations of continuous state
spaces [25, 4].
For a significant class of biochemical dynamical systems determined by multi-
affine vector fields (i.e., affine in each variable), there has been developed
an over-approximative abstraction technique based on partitioning the
continuous state space by a finite set of _rectangles_. Rectangles determine
states of a _rectangular transition system_ representing the finite discrete
(over)approximation of the continuous state space [15], as shown in Figure 1a.
The rectangular abstraction has been employed in [25] for reachability
analysis and further elaborated by model checking methods in [7]. The results
show that the extent of spurious behaviour introduced by the abstraction is
typically very high thus limiting satisfactory application of the method. The
problem is based mainly on the fact that a transition between any two
individual rectangles over-approximates the vector field on the border between
the rectangles (a so-called _facet_ , see Figure 1b) provided that the
information regarding which trajectories starting in an entry facet evolve
through a particular exit facet is abstracted out. This causes the rectangular
transition system to generate many rectangle sequences in which there is no
corresponding trajectory of the original continuous system embedded. Moreover,
the extent of such spurious behaviour is not directly eliminated by increasing
the partition density.
When analysing approximate models as in systems biology, the need for precise
results critically required in systems verification can be relaxed provided
that a suitable approximation of the solution can be even more efficient to
obtain useful results. Henceforth, in the field of complex systems exhaustive
techniques are often combined with approximative methods thus making a certain
shift in the way of applying formal methods [30, 12, 11].
### 1.1 Our Contribution
We present a new technique for discrete approximation of biochemical systems
with dynamics given by a system of ODEs with multi-affine right hand side. Our
discrete approximation is not an exact abstraction wrt the original continuous
system, but rather an approximation that approaches exact reachability with
decreasing approximation granularity. While still assuming the rectangular
partition at the background, we employ a _measure_ that enables local
quantification of the amount of trajectories evolving on a rectangle in a
particular facet-to-facet direction. To this end, every rectangle is augmented
with a local memory representing the information at which part (_entry set_)
of the entry facet it has been entered. On each entry set, we identify _focal
subsets_ from which all trajectories lead to the same exit facet. In Figure
1c, there are two different states of a _quantitative discrete approximation
automaton_ (QDAA) depicted. Both states share the same rectangle
$[1,1.5]\times[1,1.5]$ and they differ in entry sets (marked yellow). The
upper state with entry set $\\{1.5\\}\times[1,1.5]$ has only one focal set -
all trajectories from its entry set exit the state through the facet
$[1,1.5]\times\\{1\\}$. The second state with entry set
$[1,1.5]\times\\{1.5\\}$ has two focal subsets made by the green and the red
part of the entry facet, respectively.
Transitions from a state with given entry set have weights assigned to
themselves. Consider a transition from a state $A$ to a state $B$. The
transition exists if there is a part $P$ of the entry set of $A$ such that the
trajectories of ODE solutions go from $P$ to $B$. Weight of a transition from
$A$ to $B$ corresponds to the ($n-1$-dimensional) volume of $P$ divided by the
volume of the entry set of $A$. In this manner, the measure reflects amounts
of trajectories proceeding in a particular direction. Rectangle regions
related by weighted transitions make the QDAA which is a discrete-time Markov
chain. (See Theorem 3.2 and its proof.)
From a computational viewpoint, the continuous volumes are finitely
approximated by discretization on a uniform grid. Local numerical simulations
are employed to identify the entry regions and focal subsets. The density of
facet discretization grid is considered as the _method parameter_. Because of
combining numerical simulation with rectangular abstraction, the resulting
QDAA makes neither an over- nor an under-approximation of the original
continuous system. Since for every sequence of states the approximate volume
measure converges to the continuous volume with increasing discretization
parameter, the parameter indirectly affects the correspondence between the
original continuous behaviour and its approximation. This makes the method
sufficient for approximating reachability in complex biochemical dynamical
systems.
In general, the following main contributions are brought by this paper.
1. 1.
A novel computational technique for finite discrete approximation of multi-
affine dynamical systems by means of QDAA.
2. 2.
Showing that QDAA converges to the original continuous system behaviour. (See
Theorem 3.3 and its proof.)
3. 3.
A reachability algorithm for QDAA.
4. 4.
Evaluation on elementary models and an _E. Coli_ case study.
Since the most common application of the considered systems class is the
domain of biochemical dynamical systems modeled directly by rules of mass
action kinetics [24], evaluation of the method and algorithms is realized on
biological models fitting this framework.
Figure 1: (a) Vector field of a linear system partitioned by thresholds, (b)
the principle of rectangular abstraction, (c) and quantification of the extent
of over-approximation in terms of transition weights. The dashed line inside
the rectangle demonstrates the approximate border separating trajectories
exiting through different facets.
### 1.2 Related Work
Discrete approximation methods are commonly used in continuous and hybrid
systems analysis (see [3] for an overview regarding reachability) to handle
the uncountability of the state space. Direct methods work on the original
system and rely on a successor operation iteratively computing the reachable
set whereas indirect methods abstract from the continuous model by a finite
structure for which the analysis is simpler. Our method belongs to the latter
class, since it uses numerical simulations and creates the abstraction
automaton. Considering a fixed set of initial conditions, there is a certain
overhead with generating states of the automaton in comparison with simple
numerical simulations. However, the advantage of constructing the automaton is
obtaining a global view of the dynamics. Moreover, in addition to rectangular
abstraction, the automaton is augmented with weighted transitions which
represent quantitative information describing volumes of subsets of initial
conditions belonging to attraction basins of different parts of the phase
space.
An indirect method based on rectangular abstraction automaton making the
finite quotient of the continuous state space has been employed, e.g., in [25,
2, 4]. In general, these methods rely on results [15, 21] and are applicable
to (piece-wise) affine or (piece-wise) multi-affine systems. Although not
addressed formally in this paper, our technique can be considered as a
refinement of [25]. However, we focus on obtaining satisfactory approximate
results eliminating the extent of spurious behaviour coming from
conservativeness of rectangular abstraction. Our technique can be employed for
the recognition of spurious behaviour of the rectangular abstraction
transition system.
The technique presented in [28] employes timed automata for the finite
quotient of a continuous system as an alternative to piece-wise linear
approximations. Another indirect technique adapted to multi-affine biological
models is [16]. The approach also employes rectangular abstraction, but
results in less conservative reachable sets by means of polyhedral operations.
In [3, 9] there are techniques proposed for rectangular refinement that go
towards reduction of over-conservativeness. These techniques work fine for
linear systems while leaving the non-linear systems as a challenge.
Direct methods are mostly based on hybridization realized by partitioning the
system state space into domains where the local continuous behaviour is
linearized [13]. This method, in an improved form, has been applied to non-
linear biochemical dynamical systems [18]. In general, direct methods give
good results for low-dimensional systems and small initial sets. In comparison
with indirect approaches, they are computationally harder. From this
viewpoint, our approach lies between both extremes.
## 2 Preliminaries
### 2.1 Basic definitions and facts
Let $\mathbb{N}$ denote the set of positive integers, $\mathbb{N}_{0}$ the set
$\mathbb{N}\cup\\{0\\}$, and $\mathbb{R}^{+}_{0}$ the set of nonnegative real
numbers. For $n\in\mathbb{N}$, denote $\mathbb{R}^{n}$ the standard
$n$-dimensional Euclidean space with standard topology and Euclidean norm
$\left|\cdot\right|:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}_{0}$. For an
arbitrary function $f$ we use the common notation $\mathit{dom}(f)$ for the
domain of $f$.
For every $i\in\\{1,\dotsc,n\\}$ assume $a_{i},b_{i}\in\mathbb{R}$ such that
$a_{i}\leq b_{i}$. Denote $I=\prod_{i=1}^{n}[a_{i},b_{i}]$ an _$n$
-dimensional closed interval in_ $\mathbb{R}^{n}$ and $\mathit{vol}(I)$ the
_$n$ -dimensional volume of $I$_ defined as
$\mathit{vol}(I)=\prod_{i=1}^{n}(b_{i}-a_{i})$. Further denote
$\mathit{Inter}(I)$ the _interior of $I$_, defined as the cartesian product of
open intervals $\prod_{i=1}^{n}(a_{i},b_{i})$.
For any $X\subseteq\mathbb{R}^{n}$ denote $\lambda^{*}_{n}(X)$ the _Lebesgue
outer measure_ (on $\mathbb{R}^{n}$) of the set $X$. Basically
$\lambda^{*}_{n}(X)$ is the minimal nonnegative real number such that whenever
$X$ can be covered by a sequence of closed intervals in $\mathbb{R}^{n}$ the
sum of volumes of these intervals is greater then or equal to
$\lambda^{*}_{n}(X)$. (For precise definitions see [31].) Note that
$\lambda^{*}_{n}(X)<\infty$ for every bounded set $X$ and
$\lambda^{*}_{n}(I)=\mathit{vol}(I)$ for every $n$-dimensional interval $I$.
Let $n\geq 2,i\leq n,c\in\mathbb{R}$. We use $\mathbb{R}^{n-1}_{i}(c)$ to
denote the hyper-plane $\mathbb{R}^{n-1}_{i}(c)=\\{\langle
x_{1},\dotsc,x_{n}\rangle\in\mathbb{R}^{n}\mid x_{i}=c\\}$. Denote
$\hat{\pi}_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n-1}$ the projection
omitting the $i$th variable, $\hat{\pi}_{i}(\langle
x_{1},\dotsc,x_{n}\rangle)=\langle x_{1},\dotsc,x_{i-1},x_{i+1},\dotsc
x_{n}\rangle$. Let $X\subseteq\mathbb{R}^{n-1}_{i}(c)$. We extend the notion
of the $(n-1)$-dimensional Lebesgue outer measure to such sets $X$ and denote
$\lambda^{*}_{n-1}(X)$ the $(n-1)$-dimensional Lebesgue outer measure of
$\hat{\pi}_{i}(X)$.
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a continuous function (an
autonomous vector field). We say that
$\dot{x}=f(x)$ (1)
is an _autonomous ODE system_. An important property of autonomous systems is
the fact that if $y(t)$ is a solution of (1) on an open interval $(a,b)$, then
$y(t+t_{0})$ is also a solution (defined on interval $(a-t_{0},b-t_{0})$).
A function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ _satisfies the
Lipschitz condition locally_ on $\mathbb{R}^{n}$, if for every
$x\in\mathbb{R}^{n}$ there exists an open set $U\subseteq\mathbb{R}^{n}$,
$x\in U$ and a constant $L\in\mathbb{R}$ such that for every two points
$x_{1},x_{2}\in U$ the inequality $\left|f(x_{1})-f(x_{2})\right|\leq
L\cdot\left|x_{1}-x_{2}\right|$ holds.
###### Theorem 2.1 (Trajectories of solutions of an autonomous system)
Let (1) be an autonomous ODE system, where $f$ is defined on $\mathbb{R}^{n}$
and let $f$ satisfy the Lipschitz condition locally on $\mathbb{R}^{n}$. Let
$x$ be an inextendible solution of system (1). Then $dom(x)$ is an open
interval, and for every point $\alpha\in\mathbb{R}^{n}$ there exists exactly
one trajectory of an inextendible solution $x(t)$ of system (1) coming through
$\alpha$.
###### Theorem 2.2 (Continuous dependency on initial conditions)
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be continuous on an open set
$E\subseteq\mathbb{R}^{n}$ with the property that for every $y_{0}\in E$, the
initial value problem $\dot{x}=f(x),x(0)=y_{0}$ has a unique solution
$y(t)=\eta(t,y_{0})$ ($\eta$ is a function of variables $t,y_{0}$ ). Let
$w_{\bot},w_{\top}\in\mathbb{R}$ such that $(w_{\bot},w_{\top})$ is the
maximal interval of existence of $y(t)=\eta(t,y_{0})$.
Then the bounds $w_{\bot},w_{\top}$ are (lower, resp. upper semicontinuous)
functions of $y_{0}$ in $E$ and $\eta(t,y_{0})$ is continuous on the set
$\\{\langle t,y_{0}\rangle\mid y_{0}\in
E,w_{\bot}(y_{0})<t<w_{\top}(y_{0})\\}\subseteq\mathbb{R}^{n+1}$.
We restrict ourselves to multi-affine autonomous systems. That is, systems of
the form (1), such that the vector field $f$ is a _multi-affine_ function,
defined as a polynomial of variables $x_{1},\dotsc,x_{n}\in\mathbb{R}^{n}$ of
degree at most one in every variable. The assumptions of Theorems 2.1 and 2.2
(from [22]) are satisfied for systems of this class, therefore the properties
stated in the above theorems can be used for reasoning about autonomous
systems with multi-affine vector fields.
### 2.2 Biochemical dynamical system
According to [19], by a biochemical dynamical system we understand a
collection of $n$ biochemical species interacting in biochemical reactions.
Species concentrations are represented by variables $x_{1},\dotsc,x_{n}$
attaining values from $\mathbb{R}_{0}^{+}$. If the stoichiometric coefficients
in reactions do not exceed one and the reaction dynamics respects the law of
mass action kinetics [24], the dynamical system can be described by a multi-
affine autonomous system in the form (1).
In a biochemical dynamical system we are typically interested in a bounded
part ($n$-dimensional interval) of the phase space in $\mathbb{R}^{n}$.
Further, we consider the phase space partitioned by a (non-uniform)
rectangular grid. In particular, for each variable there is defined a finite
set of _thresholds_ , making the system _partition_. Thresholds determine
$(n-1)$-dimensional hyper-planes in $\mathbb{R}^{n}$ and can be freely
specified according to particular questions that should be addressed by the
model analysis, e.g., specification of unsafe or attracting sets. Cells laid
out by $2n$ adjacent threshold hyper-planes (cells are again intervals in
$\mathbb{R}^{n}$) are called _hyper-rectangles_ , for short we refer to them
as _rectangles_.
###### Definition 2.1
Define a _biochemical dynamical system_ (_biochemical system_ for short) as a
tuple $\mathcal{B}=\langle n,f,\mathcal{T},\mathcal{I}_{C}\rangle$, where
* •
$n\in\mathbb{N}$ is the _dimension_ of $\mathcal{B}$,
* •
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the multi-affine vector field
of $\mathcal{B}$,
* •
$\mathcal{T}=\langle T_{1},\dotsc,T_{n}\rangle$ is the _partition_ of
$\mathcal{B}$ where each $T_{i}$ is a finite subset of $\mathbb{R}^{+}_{0}$,
and define the _set of rectangles_ given by $\mathcal{T}$ as
$\mathit{Rect}(\mathcal{T})=\\{\prod_{j=1}^{n}I_{j}\mid\forall j\exists a,b\in
T_{j}:I_{j}=[a,b],\forall c\in T_{j}:c\leq a\vee c\geq b\\},$
* •
$\mathcal{I}_{C}\subseteq\mathit{Rect}(\mathcal{T})$ is the set of _initial
conditions_ (_initial set_) of $\mathcal{B}$.
###### Definition 2.2
Let $\mathcal{B}=\langle n,f,\mathcal{T},\mathcal{I}_{C}\rangle$ be a
biochemical system and let $H\in Rect(\mathcal{T})$ be a rectangle such that
$H=I_{1}\times\ldots\times I_{n}$, where $I_{i}=[a_{i},b_{i}]$. For every
$i\in\\{1,\ldots,n\\}$ define the _lower_ (resp. _upper_) _facet_ of $H$ wrt
the $i$th variable:
$\begin{array}[]{l}\mathit{Facet}^{\bot}_{i}(H)=\\{\langle
x_{1},\dotsc,x_{n}\rangle\in H\mid x_{i}=a_{i}\\},\\\
\mathit{Facet}^{\top}_{i}(H)=\\{\langle x_{1},\dotsc,x_{n}\rangle\in H\mid
x_{i}=b_{i}\\}.\end{array}$
Denote $Facets_{i}(H)$ the _set of $i$th dimension facets of H_,
$Facets_{i}(H)=Facet^{\bot}_{i}(H)\cup Facet^{\top}_{i}(H)$, and $Facets(H)$
the _set of (all) facets of $H$_, $Facets(H)=\bigcup_{i=1}^{n}Facets_{i}(H)$.
###### Definition 2.3
Let $H$, $H^{\prime}\in\mathit{Rect}(\mathcal{T})$. We say that $H$ is a
_neighbour of_ $H^{\prime}$, denoted $H\bowtie H^{\prime}$, if there exists
$F\in\mathrm{Facets}(H)$ such that $H\cap H^{\prime}=F$.
## 3 Quantitative Discrete Approximation
$\begin{array}[]{c}A\stackrel{{\scriptstyle 0.5}}{{\rightarrow}}B\\\
B\stackrel{{\scriptstyle 0.8}}{{\rightarrow}}A\\\ \end{array}$
$\begin{array}[]{c}\frac{d[A]}{dt}=-0.5\cdot[A]+0.8\cdot[B]\\\\[2.84526pt]
\frac{d[B]}{dt}=0.5\cdot[A]-0.8\cdot[B]\\\\[11.38109pt] \text{thresholds on
}[A]:\\{0,2.5,5\\}\\\ \text{thresholds on }[B]:\\{0,2.5,5\\}\end{array}$
Figure 2: Example of a biochemical system with two species and two reactions.
Dynamics given by a system of two ODEs and the system of thresholds are in the
left part of the figure. Vector field is visualized in the middle, and its
Rectangular Abstraction Transition System on the right.
Given a biochemical system $\mathcal{B}=\langle
n,f,\mathcal{T},\mathcal{I}\rangle$, we aim to define a finite automaton
reflecting the behaviour of $\mathcal{B}$, and for each state, to assign every
transition a weight quantifying probability of proceeding to a particular
successor.
A state is defined as a pair $\langle H,E\rangle$ – a rectangle $H$, and a
subset $E$ of a particular facet of $H$. The set $E$ represents a so-called
_entry set_ , a region through which trajectories of the system (1) enter the
interior of $H$. Intuitively, we can say that $E$ encodes the history of
previous evolution of the system from initial set $\mathcal{I}_{C}$ to $H$.
Entry sets are either subsets of $(n-1)$-dimensional facets of $H$ or (in case
of initial states) the whole $n$-dimensional rectangle $H$.
Since entry sets can be arbitrary sets in Euclidean space, we approximate them
by a finite discrete structure. Each facet is provided with a uniform grid on
which we approximate any subset of the facet by the set of rectangular
fragments, so-called _tiles_ (Figure 3). The grid is $n$-dimensional or
$(n-1)$ dimensional depending on the dimension of approximated entry sets.
When following the trajectories of solutions of differential equations of the
models dynamics in time, entry sets are identified by trajectories of
solutions passing through them on their way from preceding rectangles. In
following definitions we treat this intuitive perception of entry sets
formally.
Let $\kappa\in\mathbb{N}$, let $\mathcal{B}=\langle
n,f,\mathcal{T},\mathcal{I}_{C}\rangle$ be a biochemical system,
$H\in\mathit{Rect}(\mathcal{T})$, and $F\in\mathit{Facets}(H)$ for all
definitions and theorems from this section.
###### Definition 3.1
Let $H$ be of the form $H=\prod_{j=1}^{n}I_{j}$, where $\forall
j:I_{j}=[a_{j},b_{j}]$. Let $B\in\\{H\\}\cup\mathit{Facets}(H)$. Set either
$n^{\prime}=n$, if $B=H$, or $n^{\prime}=(n-1)$, if
$B\in\mathit{Facets}_{i}(H)$ for some $1\leq i\leq n$ (in this case $\exists
c\in\\{a_{i},b_{i}\\}:B\subset\mathbb{R}^{n-1}_{i}(c)$).
Define the _set of $\kappa$-tiles_ of $B$ as
$\mathit{Tiles}^{\kappa}_{n^{\prime}}(B)=\\{A\subseteq B\mid
A=\prod_{j=1}^{n}A_{j}\\}$, where $A_{i}=\\{c\\}$, if
$B\in\mathit{Facets}_{i}(H)$, and otherwise ($j\neq i$ or $B=H$) $A_{j}$ is a
closed interval in $\mathbb{R}^{+}_{0}$ of the form
$[a_{j}+\frac{k_{j}}{\kappa}(b_{j}-a_{j}),a_{j}+\frac{k_{j}+1}{\kappa}(b_{j}-a_{j})]$,
where for all $j\in\\{1,\dotsc,n\\},j\neq i$ the nonnegative integer
$k_{j}\in\mathbb{N}_{0}$ satisfies $k_{j}<\kappa$.
The following definition introduces the notion of general entry sets.
###### Definition 3.2
Define the _set of entry points into a rectangle $H$ through facet $F$_, as
the set $\mathit{Entry}(F,H)=$
$\bigl{\\{}y_{0}\in F\mid\exists\mathrm{\ a\ trajectory\ }y(t)\mathrm{\ of\ a\
solution\ of~{}(\ref{eq:autsystem})}\mathrm{\ such\ that\ }y(0)=y_{0}\mathrm{\
and\ }\exists\epsilon>0:y(t)\in H\mathrm{\ for\ }\forall
t\in(0,\epsilon)\bigr{\\}}.$
Next we define the approximation of entry sets on a grid of $\kappa$-tiles.
Additionally, we define the respective (discrete) volume measure of a set (see
Figure 3 c),d)).
###### Definition 3.3
Let $X\subset H$. Let $n^{\prime}=n-1$, if there exists
$i\in\\{1,\dotsc,n\\},F\in Facets_{i}(H)$ such that $X\subseteq F$, and let
$n^{\prime}=n$, otherwise. Let $M=F$, if $X\subseteq F$, and let $M=H$, if
there is no such facet $F$. Define
* •
the _set of $\kappa$-tiles_ approximating the set $X$ as
$\mathit{Tiles}^{\kappa}_{n^{\prime}}(X)=\Biggl{\\{}A\in\mathit{Tiles}^{\kappa}_{n^{\prime}}(M)\mid\frac{\lambda^{*}_{n^{\prime}}(A\cap
X)}{\lambda^{*}_{n^{\prime}}(A)}\geq\frac{1}{2}\Biggr{\\}},$
* •
the _rectangular $\kappa$-grid measure_ of the set $X$ as
$\lambda_{n^{\prime}}^{\kappa}(X)=\sum_{A\in\mathit{Tiles}^{\kappa}_{n^{\prime}}(X)}\mathit{vol}(A).$
a) b) c) d)
Figure 3: a) Let $H=[0,2.5]\times[0,2.5]$ be a rectangle. The blue areas
depict elements of $\mathit{Tiles}^{3}_{2}(H)$.
b) Let $F=\mathit{Facet}^{\top}_{1}(H)=\\{2.5\\}\times[0,2.5]$. The red line
segments are elements of $\mathit{Tiles}^{3}_{1}(F)$.
The set $\mathit{EntrySets}_{3}(H)$ has
$2+4\cdot(1+\binom{3}{2}+\binom{3}{1})=30$ elements: $\emptyset,H,$ and $7$
for every facet of $H$ (the facet itself, $3$ segments and $3$ unions of pairs
of segments of the facet).
c) Let $X$ be a subset of $H$ (the shaded polygon). Let $\kappa=5$.
d) The set of $\kappa-$tiles approximating $X$ is the set of five blue
intervals (each satisfying that at least half of its area is in $X$). The
cardinality of $\mathit{Tiles}^{\kappa}_{2}(X)$ is $5$. Thus
$\lambda^{\kappa}_{2}(X)=5\cdot(0.5\cdot 0.5)=1.25.$
The following definition declares the set of all discretized entry sets for a
given rectangle.
###### Definition 3.4
For $H$, define _set of (approximate) entry sets_
$\mathit{EntrySets}_{\kappa}(H)=$
$\Bigl{\\{}E\subseteq H\mid E=\emptyset\vee E=H\vee\exists
F\in\mathit{Facets}(H),\mathcal{E}\subseteq\mathit{Tiles}_{\kappa}\bigl{(}\mathit{Entry}(F,H)\bigr{)}:E=\bigcup\mathcal{E}\Bigr{\\}}.$
For an example of a set of (approximate) entry sets of a rectangle see Figure
3 a),b). Note that set of approximate entry sets is always finite. Further
note that also the empty set and the entire rectangle are considered as entry
sets. These represent singular cases needed in the subsequent construction of
the automaton. In particular, states with the empty entry set approximate
fixed point behaviour not leaving the rectangle (steady state memory) whereas
the rectangle-form entry set is employed for initial rectangles.
###### Definition 3.5
Let
$E\in\mathit{EntrySets}_{\kappa}(H),H^{\prime}\in\mathit{Rect}(\mathcal{T}),F^{\prime}\in\mathit{Facets}(H)$
such that $H^{\prime}\bowtie H,F^{\prime}=H\cap H^{\prime}$.
Define the _focal subset of $E$ on $H$ targeting $F^{\prime}$_, denoted
$\mathit{Focal}(H,E,F^{\prime})$, as the set of all $y_{0}\in E$ such that
there exist $\epsilon,\epsilon^{\prime},c>0$ and a trajectory of a solution
$y(t)$ of system (1) with inital conditions $y(0)=y_{0}$ satisfying $y(t)\in
H$ for $t\in(0,c),y(t)\in\mathit{Inter}(H)$ for $t\in(0,\epsilon)$, $y(c)\in
F^{\prime}$, and $y(t)\in\mathit{Inter}(H^{\prime})$ for
$t\in(c,c+\epsilon^{\prime})$. Let $\mathit{ExitSet}(H,E,F^{\prime})$ denote
the set of all such (targeted) points $y(c)\in F^{\prime}$.
Define _focal subset of $E$ on $H$ not leaving $H$_,
$\mathit{Focal}(H,E,\emptyset)$, as the set of all points $y_{0}\in E$ such
that there exists a trajectory of a solution $y(t)$ of system (1) with initial
conditions $y(0)=y_{0}$ satisfying $y(t)\in H$ for all $t>0$.
Next we define the successor function for any pair $\langle H,E\rangle$ and
subsequently the quantitative discrete approximation automaton.
###### Definition 3.6
Let $E\in\mathit{EntrySets}_{\kappa}(H)$. Define the _successors of_ $\langle
H,E\rangle$ as the set of pairs $\langle H^{\prime},E^{\prime}\rangle$ with
$H^{\prime}\in\mathit{Rect}(\mathcal{T}),E^{\prime}\in\mathit{EntrySets}_{\kappa}(H^{\prime})$
such that
$\mathit{Succs}(\langle H,E\rangle)=\bigl{\\{}\langle
H^{\prime},E^{\prime}\rangle\mid H^{\prime},E^{\prime}\mathrm{\ satisfy\ one\
of\ conditions\ 1.-3.\ below}\bigr{\\}}$
1. 1.
$H^{\prime}\bowtie H$, $E\neq\emptyset$. Denote $F^{\prime}$ the facet of $H$
satisfying $F^{\prime}=H\cap H^{\prime}$. Let $n^{\prime}=n$, if $E=H$, and
$n^{\prime}=(n-1)$, otherwise. Moreover,
$E^{\prime}=\bigcup\mathit{Tiles}_{\kappa}\bigl{(}\mathit{ExitSet}(H,E,F^{\prime})\bigr{)}$
and
$\lambda^{\kappa}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,\emptyset)\bigr{)}>0$.
2. 2.
$H^{\prime}=H$, $E\neq\emptyset$, and $E^{\prime}=\emptyset$. Further, it
holds that either $E\subseteq F$ and
$\lambda^{\kappa}_{n-1}\bigl{(}\mathit{Focal}(H,E,\emptyset)\bigr{)}>0$, or
$E=H$ and
$\lambda^{\kappa}_{n}\bigl{(}\mathit{Focal}(H,E,\emptyset)\bigr{)}>0$.
3. 3.
$H^{\prime}=H$ and $E^{\prime}=E=\emptyset$.
###### Definition 3.7 (The Quantitative Discrete Approximation Automaton)
Let $\kappa,\mathcal{B}$ be as above. The _quantitative abstraction automaton_
$QDAA_{\kappa}(\mathcal{B})$ of a biochemical system $\mathcal{B}$ with
parameter $\kappa$ is a tuple $QDAA_{\kappa}(\mathcal{B})=\langle
S,\mathcal{I}_{C},\delta,p\rangle$, where
* •
the _set of states_ $S=\\{\langle H,E\rangle\mid
H\in\mathit{Rect}(\mathcal{T}),E\in\mathit{EntrySets}_{\kappa}(H)\\},$
* •
the _set of initial conditions_ $I_{C}=\left\\{\langle H,H\rangle\mid
H\in\mathcal{I}_{C}\right\\}$,
* •
the _transition function_ $\delta:S\rightarrow 2^{S}$ is defined as
$\delta(\langle H,E\rangle)=\mathit{Succs}(\langle H,E\rangle)$,
* •
the _weight function_ $p:S\times S\rightarrow[0,1]$ is defined by the
following expression, where $S=\langle H,E\rangle,S^{\prime}=\langle
H^{\prime},E^{\prime}\rangle$. Suppose $n^{\prime}=n$, in case $E=H,$ and
$n^{\prime}=n-1$, otherwise.
$p(S,S^{\prime})=\begin{cases}1,&\mathrm{if\
}H=H^{\prime},\,E=E^{\prime}=\emptyset,\\\
\vspace{3mm}\dfrac{\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,\emptyset)\bigr{)}}{\sum_{A\in\mathit{Facets}(H)\cup\\{\emptyset\\}}\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,A)\bigr{)}},&\mathrm{if\
}H=H^{\prime},\,E\neq\emptyset,E^{\prime}=\emptyset,\\\
\dfrac{\lambda^{*}_{n^{\prime}}(\mathit{Focal}(H,E,F^{\prime}))}{\sum_{A\in\mathit{Facets}(H)\cup\\{\emptyset\\}}\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,A)\bigr{)}},&\mathrm{if\
}H\bowtie H^{\prime},E^{\prime}\subseteq F^{\prime}=H\cap H^{\prime},\\\
0,&\mathrm{otherwise.}\end{cases}$
###### Example 3.1
Assume the biochemical system from Figure 2. See Figure 4 a) for an example of
focal subsets described below. Let $R=[0,2.5]\times[2.5,5]$ be a rectangle and
let
$F_{0}=\mathit{Facet}^{\top}_{2}(R),F_{1}=\mathit{Facet}^{\top}_{1}(R),F_{2}=\mathit{Facet}^{\bot}_{2}(R),F_{3}=\mathit{Facet}^{\bot}_{1}(R).$
For the state $\langle R,F_{0}\rangle$ the focal set of $F_{1}$ equals
$F_{0}$, whereas $Focal(F_{0})=Focal(F_{2})=Focal(F_{3})=\emptyset$.
Let $H=[0,2.5]\times[0,2.5]$ and $F=\mathit{Facet}^{\top}_{1}(R).$ For the
state $\langle H,H\rangle$ the set $\mathit{Focal}(F)$ is the blue area inside
$H$ and $\mathit{Focal}(\emptyset)$ is the yellow area. All the solutions of
the biochemical systems dynamics with initial conditions in
$\mathit{Focal}(\emptyset)$ approach the yellow line of fixed points and stay
in $H$ forever. All the solutions starting in the blue area leave $H$ in
finite time through $F$.
In the right part of Figure 4 is the set of reachable states of the
quantitative discrete approximation automaton (QDAA) obtained from the
biochemical system described in Figure 2 with initial conditions
$\mathcal{I}_{C}=\\{[0,2.5]\times[0,2.5]\\}$.
Let $H,R$ be the same as above. Let $S=[2.5,5]\times[0,2.5]$ and let
$\mathcal{I}_{C}=\\{H\\}$. The QDAA successor states of $\langle H,H\rangle$
are $\langle H,\emptyset\rangle$ (a selfloop state) and $\langle S,E\rangle$
(where $E$ denotes the $\kappa$-tiles approximation of the red segment in
$\mathit{Facet}^{\bot}_{1}(S)$). For $\kappa\rightarrow\infty$ the weights of
these two transitions approach the area ratios of yellow and blue regions of
$H$ respectively. The only successor of $\langle H,\emptyset\rangle$ is (by
definition) itself. The state $\langle S,E\rangle$ has one successor $\langle
S,\emptyset\rangle$, since all the trajectories beginning in $E$ approach the
line of fixed points and stay inside $S$ forever.
Therefore the set of concentrations reachable from initial rectangle $H$ is
$[0,5]\times[0,2.5]$. See the rectangular abstraction transition system from
Figure 2 where the set reachable from $H$ is
$[0,5]\times[0,2.5]\cup[2.5,5]\times[2.5,5],$ although there exists no
trajectory of a solution of the biochemical systems dynamics that starts in
$H$ and reaches a point inside $[2.5,5]\times[2.5,5].$
On the other hand, if $\kappa$ is too small, some behaviours of the system are
not reflected in QDAA, because the set of $\kappa$-tiles corresponding to the
entry set may be empty. With finer partition into $\kappa$-tiles smaller entry
sets can be captured and approximation of the biochemical system by a QDAA is
more realistic.
a) b)
Figure 4: a) Focal sets examples, b) QDAA example.
In the Theorem 3.1 we ensure correctness of using the Lebesque measure in
Definition 3.7. We ensure that there is no non-zero volume entry set such that
all trajectories from this set lead to a facet without entering the interior
of a neighbouring rectangle.
Following notations and lemmas provide the background that is used in the
proof of Theorem 3.1.
###### Notation 3.1
Let us denote $B_{\epsilon}(x)$ the _open sphere_ with centre
$x\in\mathbb{R}^{n}$ and radius $\epsilon>0$
($B_{\epsilon}(x)=\\{y\in\mathbb{R}^{n}|\left|x-y\right|<\epsilon\\}$).
The following theorems from mathematical analysis and differential topology
(see for example [23], [22]) will be used in the proofs.
###### Lemma 3.1 (Peano)
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be continuous on open set
$E\subseteq\mathbb{R}^{n}$ such that $f$ possesses continuous first order
partial derivatives. Then for every $y_{0}\in E$, the initial value problem
$\dot{x}=f(x),x(0)=y_{0}$ has a unique solution $y(t)=\eta(t,y_{0})$ and the
unique solution has continuous first order derivative wrt the variable $t$ (is
of class $C^{1}$) on its open domain of definition.
###### Lemma 3.2
The following statements about zero measure sets hold. (A set has Lebesgue
outer measure zero iff it has Lebesgue measure zero.)
1. 1.
If $S\subset\mathbb{R}^{n},\lambda_{n}^{*}(S)=0$ and
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a smooth map (with continuous
first order derivatives), then $\lambda_{n}^{*}(f(S))=0$.
2. 2.
Being of zero measure is a diffeomorphism-invariant property of subsets of
$\mathbb{R}^{n}$.
###### Lemma 3.3 (A version of Fubini theorem)
Let $U$ be a compact subset of $\mathbb{R}^{n}$. Denote by
$\mathbb{R}_{t}^{n}$ the subset
$\\{t\\}\times\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$, and denote
$U_{t}=U\cap\mathbb{R}_{t}^{n}$. If every set $U_{t}$ satisfies
$\lambda_{n-1}^{*}(U_{t})=0$, then $\lambda_{n}^{*}(U)=0$.
###### Lemma 3.4
Let $g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a multiaffine function. Let
$F\in\mathit{Facets}_{i}(H)$. The following statements hold.
1. 1.
If there exists a set $U\subseteq H$ such that $\lambda_{n}^{*}(U)>0$ and
$\forall x\in U:g(x)=0$, then $g\equiv 0$ on $H$ (and on $\mathbb{R}^{n}$).
2. 2.
If there exists a set $V\subseteq F$ such that $\lambda_{n-1}^{*}(V)>0$ and
$\forall x\in V:g(x)=0$, then $g\equiv 0$ on $F$ (and on the hypherplane
$\mathbb{R}_{i}^{n}(c)$ containing $F$).
###### Proof.
The proof of statement _1._ will be done by mathematical induction wrt $n$.
For $n=1$ is $H$ a line segment, and the function $g$ is a linear function of
one variable. Either there is one point (a set of measure zero in
$\mathbb{R}^{1}$) where the function attains zero value, or the function is
zero on the whole line.
For $n=2$, $H$ is a rectangle in plane. The multi-affine function of two
variables is either zero on whole plane, or on two intersecting lines, or a
hyperbolic curve or empty set. Therefore either the function is zero on the
whole plane, or on a subset of plane of measure zero. Only the second case is
compatible with $g$ being nonzero on a set of non-zero Lebesgue outer measure.
Assume that the statement holds for $1,2,\dotsc,n-1$ and let us prove it for
$n$. By Theorem 3.3.
_Proof of 2._ is analogous, we can identify $\mathbb{R}_{i}^{n}(c)$ with
$\mathbb{R}^{n-1}$ and assume that $g$ is a multiaffine function of $n-1$
variables (with $x_{i}=c$ constant). ∎∎
###### Definition 3.8
Let $U\subset H$, let $I$ be an (bounded or unbounded) interval in
$\mathbb{R}$. Define the _$I$ -trajectories unoin_ of $U$ as the set
$\Xi_{I}(U)\bigl{\\{}x(s)|s\in I,x(t)\mathrm{\ is\ a\ solution\ of\
system~{}(\ref{eq:autsystem})\ defined\ on\ }I,x(0)\in U\bigr{\\}}.$
###### Lemma 3.5
Let $U\subset H$ be a $(n-1)$-dimensional closed disc in $H$ (i.e.
$\exists i\in\\{1,\dotsc,n\\}\exists c\in\mathbb{R}\exists\delta>0\exists x\in
H:U=\\{y\in\mathbb{R}_{i}^{n}(c)|\left|x-y\right|\leq\delta\\},$
we do not assume there is a facet $F$ of $H$ with $U\subseteq F$).
Let $\lambda_{n-1}^{*}(U)>0$ and let either $\forall y\in U:f(y)\cdot e_{i}>0$
or $\forall y\in U:f(y)\cdot e_{i}<0$.
Then either $\exists U^{\prime}\subseteq U$ with
$\lambda_{n-1}^{*}(U^{\prime})>0$ such that all the solutions of system (1)
stay forever in $H$, or there exists $F^{\prime}\in\mathit{Facets}(H)$ such
that $\lambda_{n-1}^{*}(F^{\prime}\cap\Xi_{[0,+\infty)}(U))>0$.
Moreover, in the second case, $\lambda_{n-1}^{*}(F^{\prime}\cap W)>0,$ where
$W$ is the set of points of the boundary of $H$ where the trajectories leave
the rectangle $H$ for the first time (a subset of the connected component of
$\Xi_{[0,+\infty)}(U)\cap H$ containing $U$).
###### Proof.
Consider the case, when there is no $U^{\prime}\subseteq U$ with
$\lambda_{n-1}^{*}(U^{\prime})>0$ such that all the solutions of system (1)
stay forever in $H$.
Since all compactly supported vector fields are complete, we can restrict
ourselves on the multi-affine vector field $f$ defined on a compact
neighbourhood of $H$ and assume that flow of this (complete) vector field
defines a one-parametric group of diffeomorphisms
$Fl_{t}^{f}:(t,y_{0})\rightarrow y(t),$ where $y(t)$ is the respective
solution of system 1.
We consider a diffeomorphism $\varphi_{U}$ between $\Xi_{[0,+\infty)}(U)\cap
H$ and a special subset $V$ of $U\times\mathbb{R}$. We prove that for $U$
satisfying the assumptions of Lemma 3.5 the set $V$ is of nonzero measure.
Therefore its image by diffeomorphism $\varphi$ must be of nonzero measure
(due to Theorem 3.2). If on the other hand the set
$\Xi_{(-\infty,0]}(F^{\prime}\cap W)\cap H$ was of measure zero, its image in
$\varphi_{F^{\prime}\cap W}$ must be of zero measure also, but since
$\Xi_{[0,+\infty)}(U)\cap H$ is the union of finitely many such sets
$\Xi_{(-\infty,0]}(F^{\prime}\cap W)\cap H$, there has to exist $F^{\prime}$
such that the set $\Xi_{(-\infty,0]}(F^{\prime}\cap W)\cap H$ has nonzero
measure. ∎∎
###### Lemma 3.6
Let $E\in\mathit{EntrySets}_{\kappa}(F,H),E\neq\emptyset$.
Then either $\exists U^{\prime}\subseteq E$ with
$\lambda_{n-1}^{*}(U^{\prime})>0$ such that all the solutions of system (1)
stay forever in $H$, or there exists $F^{\prime}\in\mathit{Facets}(H)$ such
that
$\lambda_{n-1}^{*}(F^{\prime}\cap\Xi_{[0,+\infty)}(E\cap\mathit{Entry}(F,H)))>0.$
Moreover, in the second case, $\lambda_{n-1}^{*}(F^{\prime}\cap W)>0,$ where
$W$ is the set of points of the boundary of $H$ where the trajectories leave
the rectangle $H$ for the first time (a subset of the connected component of
$\Xi_{[0,+\infty)}(E\cap\mathit{Entry}(F,H))\cap H$ containing the set
$E\cap\mathit{Entry}(F,H)$).
###### Proof.
Consider again the case, when there is no $U^{\prime}\subseteq U$ with
$\lambda_{n-1}^{*}(U^{\prime})>0$ such that all the solutions of system (1)
stay forever in $H$.
The situation is easier if there exists a point $x\in E$ with
$f(x)\cdot\nu_{H}(F)<0$. The multi-affine function $f$ is continuous,
therefore there exists am open sphere containing $x$ such that
$f(y)\cdot\nu_{H}(F)<0$ for all points $y$ from this sphere. As a subset of
the intersection of this sphere and $F$ there exists a closed disc like $U$
from the assumption of Lemma 3.5 and the statement of this lemma holds.
Otherwise the whole set $E\cap\mathit{Entry}(F,H)$ is a subset of $F$ with
nonzero $(n-1)$-dimensional outer Lebesgue measure on which $f\equiv 0$. By
Lemma 3.4 $f\equiv 0$ on the whole $(n-1)$-dimensional hypher-plane containing
$F$.
The idea of proof for this configuration is to use the definitorial property
of $\mathit{Entry}(F,H)$, that insures existence of $\epsilon_{z}$ and a point
$y\in\mathit{Inter}(H)$ reachable by a trajectory from a point $z$ in
$E\cap\mathit{Entry}(F,H)$ in time $\epsilon_{z}$ with $f(z)\cdot\nu_{H}(F)<0$
(that is implied by the properties of multi-affine function $f$). By
continuous dependency on the initial conditions we find a disc around $z$ that
cosists of points reachable from $E\cap\mathit{Entry}(F,H)$ (continuous
dependence) and satisfies assumptions of Lemma 3.5. Then we use Lemma 3.5 and
this lemma is proved. ∎∎
###### Theorem 3.1
Let $E\in\mathit{EntrySets}_{\kappa}(H),E\neq\emptyset$. Further, let
$n^{\prime}=n$, if $E=H$, and $n^{\prime}=(n-1)$, otherwise. Then
$\sum_{A\in\mathit{Facets}(H)\cup\\{\emptyset\\}}\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,A)\bigr{)}>0,$
(2)
###### Proof.
The proof will be divided into two parts.
First, for $E\subseteq F,E\neq\emptyset.$ Second, for $E=H$.
_Proof of 1._ The statement of theorem is obtained by using Lemma 3.6.
_Proof of 2._ Let $E=H$. In the case when all the trajectories with initial
points in $H$ stay forever in $H$ the inequality
$\lambda^{*}_{n}\bigl{(}\mathit{Focal}(H,E,\emptyset)\bigr{)}>0$ holds.
In the case when there exists a point $x_{0}\in H$ and a trajectory of a
solution $x(t)$ of system (1) with $x(0)=x_{0},\exists c,\epsilon>0:\forall
t\in[0,c],x(t)\in H,\forall t\in(c,c+\epsilon)x(t)\notin H.$ Let us denote $y$
the point $x(c+\frac{\epsilon}{2})\notin H$. From Theorem 2.2 (continuous
dependency on initial conditions) there exists an $n$-dimensional sphere $B$
with centre $x_{0}$ such that trajectories of all solutions of (1) with
initial point in the sphere $B$ leave the rectangle $H$ and continue into a
neighbourhood of point $y$. The intersection $H\cap B$ surely contains a disc
satisfying assumptions of Lemma 3.5 and therefore there exists
$F^{\prime}\in\mathit{Facets}(H)$ such that the intersection of
$\Xi_{[0,+\infty)(B)}$ and $F^{\prime}$ is of nonzero measure, i.e. it has a
subset of nonzero $(n-1)$-dimensional measure in the interior of the facet
$F^{\prime}$. Then there exists an open sphere $B^{\prime}$ contained in $B$
such that all trajectories of solutions with initial points in $B^{\prime}$
leave the rectangle $H$ through the interior of facet $F^{\prime}$, and
$\lambda^{*}_{n}\bigl{(}\mathit{Focal}(H,E,F^{\prime})\bigr{)}\geq\lambda_{n}^{*}(B^{\prime})>0$.
∎∎
###### Theorem 3.2
The quantitative abstraction automaton $\mathrm{QDAA}_{\kappa}(\mathcal{B})$
of a biochemical system $\mathcal{B}$ is a discrete time Markov chain.
###### Proof.
The number of states is finite, bounded by
$\left|\mathit{Rect}(\mathcal{T})\right|\Big{(}2+2n\cdot\big{(}2^{\kappa^{n-1}}-1\big{)}\Big{)}$.
Sum of probabilities of transitions from one state $\langle H,E\rangle$ equals
$1$ for $E=\emptyset$ and
$\frac{\sum_{A\in\mathit{Facets}(H)\cup\\{\emptyset\\}}\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,A)\bigr{)}}{\sum_{A\in\mathit{Facets}(H)\cup\\{\emptyset\\}}\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,A)\bigr{)}}$.
The later sum equals $1$, whenever its denominator is nonzero (it is the case
because of Theorem 3.1). The probabilities of transitions from a given state
are independent of previous states of the automaton. ∎∎
Finally, we provide a theorem suggesting that for sufficiently large values of
parameter $\kappa$, the rectangular $\kappa$-grid measure of a bounded set $X$
contained in the phase space of biochemical system approaches its Lebesque
outer measure (Theorem 3.3). Following lemmas and definition will be used in
the proof of this theorem.
Let $\mathcal{B}=\langle n,f,\mathit{T},\mathit{I}_{C}\rangle$ be a
biochemical system, $H\in\mathit{Rect}(\mathcal{T}),F\in\mathit{Facets}(H),$
and $\kappa\in\mathbb{N}$ throughout this section.
###### Definition 3.9
Let $X\subseteq\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})].$ Define
$\lambda_{n}^{\kappa}(X)$ as the sum
$\sum_{H\in\mathit{Rect}(\mathcal{T})}\lambda_{n}^{\kappa}(X\cap H).$
Analogously define $\lambda_{n-1}^{\kappa}(X)$ as the sum
$\sum_{F\in\bigcup_{H\in\mathit{Rect}(\mathcal{T})}\mathit{Facets}(H)}\lambda_{n-1}^{\kappa}(X\cap
F).$
###### Lemma 3.7
Let $m\in\\{n-1,n\\}$, let $J$ be an $m$-dimensional interval in
$n$-dimensional space, $J\subseteq\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})].$
Then
$\lim_{\kappa\rightarrow\infty}\lambda_{m}^{\kappa}(J)=\lambda_{m}^{*}(J)$
###### Proof.
For given $\kappa$ there are at most $2n\cdot\kappa^{n-1}$ distinct tiles $R$
satisfying $R\nsubseteq J$ and $R\cap J\neq\emptyset$.
The difference between $\lambda_{m}^{\kappa}(J)$ and
$\lambda_{m}^{*}(J)=\mathit{vol}(J)$ is bounded by
$\dfrac{1}{2}\sum
vol(R_{i})=\dfrac{1}{2}2n\cdot\kappa^{n-1}\mathit{vol}(R_{1})=$
$=n\kappa^{n-1}\dfrac{V}{\kappa^{n}}=\dfrac{nV}{\kappa}$
The expression approaches zero as $\kappa\rightarrow\infty$. ∎∎
###### Lemma 3.8
Let $M$ be a positive integer number ($M<\infty$). Let
$U=\bigcup_{i=1}^{M}I_{i}$ be a union of $M$ $n$-dimensional bounded
rectangles, subsets of $\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})]$. Let this
the intersection of every two intervals be of zero Lebesgue outer measure.
Then
$\lim_{\kappa\rightarrow\infty}\lambda_{n}^{\kappa}(U)=\lambda_{n}^{*}(U)$
###### Proof.
There are maximally $M\cdot 2n\kappa^{n-1}$ distinct tiles $R$ satisfying
$R\nsubseteq U$ and $R\cap U\neq\emptyset$.
The difference between $\lambda_{m}^{\kappa}(U)$ and
$\lambda_{m}^{*}(U)=\mathit{vol}(U)$ is bounded by
$M\dfrac{1}{2}\sum\mathit{vol}(R_{i})=M\dfrac{1}{2}2n\cdot\kappa^{n-1}\mathit{vol}(R_{1})=$
$=Mn\kappa^{n-1}\dfrac{V}{\kappa^{n}}=M\dfrac{nV}{\kappa}$
The expression approaches zero as $\kappa\rightarrow\infty$. ∎∎
###### Lemma 3.9
Let $X$ be a (bounded) subset of $\mathbb{R}^{n}$ and let $J_{1},J_{2},\ldots$
be a (countable) sequence of intervals such that
$X\subseteq\bigcup_{j=1}^{\infty}J_{j}$. Then there exists a sequence
$I_{1},I_{2},\ldots$ of intervals with $\lambda_{n}^{*}(I_{i}\cap I_{j})=0$
for each pair $i\neq j$ such that
$\bigcup_{i=1}^{\infty}I_{i}=\bigcup_{j=1}^{\infty}J_{j}$.
###### Proof.
The sequence of pairs of intervals from the first sequence is also countable.
Replace every two rectangles that overlap by finitely many new non-overlapping
rectangles. ∎∎
###### Lemma 3.10
Let $X\subseteq\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})]$ be a bounded subset
of $\mathbb{R}^{n}$ with lebesgue outer measure
$\lambda^{*}_{n}(X)=r\in\mathbb{R}^{+}_{0}$.
Let $\epsilon>0$. Let $J_{1},J_{2},\ldots$ be a cover of $X$ by intervals such
that $\sum_{i=1}^{\infty}\mathit{vol}(J_{i})-r<\epsilon$.
Let $\delta>0$ be a positive real number. Then there exists a number
$k_{\delta}$ such that
$\sum_{i=1}^{k_{\delta}}vol(J_{i})\in(r-\delta,r+\delta)$.
(That means $\sum_{i=k_{\delta}}^{\infty}vol(J_{i})<\delta$.)
###### Proof.
The sequence of partiall sums of the sequence of volumes
$(vol(J_{i}))_{i=1}^{\infty}$ converges to $r$. That implies the the existence
of such $k_{\delta}$. ∎∎
###### Corollary 3.1
Let $X$ and $\delta$ be the same as in the previous lemma. Let
$I_{1},I_{2},\ldots$ be a cover of $X$ by countably many intervals with
$\lambda_{n}^{*}(I_{i}\cap I_{j})=0$ for each pair $i\neq j$. Let $k_{\delta}$
be as in the previous lemma. Then there exist two subsets of $X$ denoted
$X^{\prime}$ and $X^{\prime\prime}$ such that $X=X^{\prime}\cup
X^{\prime\prime}$, $\lambda^{*}_{n}(X^{\prime}\cap X^{\prime\prime})=0$,
$X^{\prime\prime}\subseteq\bigcup_{i=1}^{k_{\delta}}I_{i}$ and
$X^{\prime}\subseteq\bigcup_{i=k_{\delta}}^{\infty}I_{i}$.
###### Proof.
The existence of $X^{\prime}$ and $X^{\prime\prime}$ is obvious (can be
defined as $X^{\prime\prime}=\bigcup_{i=1}^{k_{\delta}}I_{i}\cap X$ and
$X^{\prime}=\bigcup_{i=k_{\delta}}^{\infty}I_{i}\cap X$ respectively), there
remains the proof of the equality $\lambda^{*}_{n}(X^{\prime}\cap
X^{\prime\prime})=0$.
The set $X^{\prime}\cap X^{\prime\prime}$ contains only countably many
intersections of intervals $I_{1},I_{2},\ldots$, and these intersection are of
measure zero, therefore their union has also measure zero,
$\lambda_{n}^{*}(X^{\prime}\cap X^{\prime\prime})=0$. ∎∎
###### Lemma 3.11
Let $X$ be a subset of $\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})]$ with
Lebesgue outer measure $\lambda_{n}^{*}(X)=r$. Then for every $\kappa$ the
following statement is true:
$\lambda_{n}^{\kappa}(X)\leq 2r$
###### Proof.
For any tile $J$ to be counted into the $\kappa$-grid measure of a set, the
inequalities $\lambda_{n}^{*}(X\cap
J)\geq\frac{1}{2}\mathit{vol}(J)\Leftrightarrow 2\lambda_{n}^{*}(X\cap
J)\geq\mathit{vol}(J)$ must hold. Therefore the set $X$ with Lebesgue outer
measure $r$ can saturate at most a set of tiles of the overall volume $2r$. ∎∎
###### Theorem 3.3
Let $X\subseteq H\in\mathit{Rect}(\mathcal{T})$ (or more generally
$X\subseteq\prod_{i=1}^{n}[\min(T_{i}),\max(T_{i})]$, where $\langle
T_{1},\dotsc,T_{n}\rangle=\mathcal{T}$ is the partition of $\mathcal{B}$).
Then
$\lim_{K\rightarrow\infty}\lambda^{\kappa}_{n}(X)=\lambda^{*}_{n}(X).$ (3)
###### Proof.
The proof uses the definition of outer Lebesgue measure and our aim is to
prove that for every $\epsilon>0$ there exists such $\kappa$ that the
difference $|\lambda^{\kappa}_{n}(X)-\lambda^{*}_{n}(X)|<\epsilon$.
Let $\epsilon>0$ be a positive real number, denote $r=\lambda^{*}_{n}(X)$.
For $X$ in the type of interval the proof is easy (Lemma 3.7).
For general $X$ consider sufficiently accurate (with less than
$\frac{\epsilon}{2}$ difference of sum of volumes and $r$) cover of $X$ with
countable collection of intervals whose interiors do not intersect (Lemma
3.9).
There exist two subsets of $X$ denoted $X^{\prime}$ and $X^{\prime\prime}$
such that $X=X^{\prime}\cup X^{\prime\prime},\lambda^{*}_{n}(X^{\prime}\cap
X^{\prime\prime})=0$, such that the collection of intervals can be divided
into a finite part $I_{1},\ldots,I_{k}$ and the remainder $I_{k+1},\ldots$
such that the sum of volumes of the remainder is sufficiently small (less than
$\frac{\epsilon}{8}$) (Corollary 3.1).
For a bounded set $Y$ with $\lambda^{*}_{n}(Y)=s$ the inequality
$\lambda^{\kappa}_{n}(Y)\leq 2s$ holds for every $\kappa$ (Lemma 3.11).
For the finite set of $k$ intervals we can find such $\kappa$ that
$|\sum_{i=1}^{k}\lambda^{\kappa}_{n}(I_{i})-\sum_{i=1}^{k}\lambda^{*}_{n}(I_{n})|\leq\frac{\epsilon}{4}$
(applying Lemma 3.7 finitely many times, taking the maximal of obtained values
of $\kappa$) and the overall difference of $\lambda^{\kappa}_{n}(X)$ and
$\lambda^{*}_{n}(X)$ is less than $\epsilon$. ∎∎
Note that the result applies also to the case with $X\subseteq
F\in\mathit{Facets}(H)$ and $\lambda^{\kappa}_{n-1},\lambda^{*}_{n-1}$.
## 4 Algorithm
This section introduces procedures for obtaining the reachable state space of
the quantitative discrete approximation automaton. Algorithm 1 is a procedure
of computing the set of reachable states. Algorithm 2 describes the
computation of transitions from one state (i.e. successors) together with
their weights using numerical simulations.
The procedure of computing reachable state space (Algorithm 1) is based on
breadth first search. States corresponding to initial conditions of the
biological system are enqueued first and a list of states already visited is
maintained. The computation is always finite, because there are only finitely
many possible states of the automaton and each of them can be at most once
added and after the computation of its successors removed from the queue.
Algorithm 1 Computing the set of reachable states
0: $\mathcal{B}=(n,f,\mathcal{T},\mathcal{I}_{C})$, $\kappa\in\mathbb{N}$
0: $\mathrm{Reachable}=\mathrm{set\ of\ all\ reachable\ states\ of\ the\
automaton\ }QDAA_{\kappa}(\mathcal{B})$
1: Reachable $\leftarrow\emptyset$
2: for all $H\in\mathcal{I}_{C}$ do
3: s $\leftarrow\langle H,H\rangle$
4: Reachable $\leftarrow$ Reachable $\cup\\{s\\}$
5: Queue.pushBack($s$)
6: while Queue $\neq\emptyset$ do
7: $s\leftarrow$ Queue.firstElement
8: $A\leftarrow$ getSuccessors($s$)
9: for all $a\in A$ do
10: if $a\notin$ Reachable then
11: Reachable $\leftarrow$ Reachable $\cup\\{a\\}$
12: Queue.pushBack($a$)
13: return Reachable
Computation of the successors (Algorithm 2) of one state requires determining
the rectangles and the entry sets of the successors and weights of the
transitions. This can be done approximately using numerical simulations. We
sample the entry set of the state and perform numerical simulations with the
sampled points as initial conditions and the dynamics of the given biological
system as the vector field. For each simulated trajectory we watch whether it
leaves the rectangle before given maximal time interval elapses. If this is
the case then the location of the exit point through which the trajectory
leaves the rectangle is of interest.
Entry sets of the successor states are also determined within Algorithm 2. If
the successor is a selfloop state the entry set is empty. For a neighbouring
rectangle successor with one common facet the entry set is computed using the
exit points locations and more numerical simulations. From the set of exit
points in a facet we can estimate the set of $\kappa$-tiles of the facet that
surely have nonempty intersection with the exit set. It remains to decide in
which of the $\kappa$-tiles the intersection of the tile with the exit set
takes at least one half of the volume of the tile.
To this end we use numerical simulations and the fact that for an autonomous
system of ODEs $\dot{x}=f(x)$ with a solution $x(t)$ the function $x(-t)$ is a
solution of autonomous system $\dot{x}=-f(x)$. For determining whether to
include a $\kappa$-tile in the entry set of a successor state, we sample the
tile and perform numerical simulations of the trajectories of system
$\dot{x}=-f(x)$. If more than one half of the simulated trajectories go
through the rectangle and the entry set of the original state, then the
$\kappa$-tile is included in the entry set of successor state, otherwise the
$\kappa$-tile is not included.
Weights of the transitions correspond to portions of the set of performed
simulations that leave the rectangle to the respective neighbouring
rectangles. Weight of the transition from the state to the so-called selfloop
state with the same rectangle is determined as the portion of trajectories
that do not leave the rectangle in given maximal time interval.
Algorithm 2 Procedure getSuccessors
0: $\mathcal{B}=(n,f,\mathcal{T},\mathcal{I})$, $\kappa,M\in\mathbb{N}$,
$H\in\mathit{Rect}(\mathcal{T})$, $E\in\mathit{EntrySets}(H)$
0: $\mathrm{Successors}=\mathit{Succs}_{\kappa}(\langle H,E\rangle)$
1: if $E=\emptyset$ then
2: Successors $\leftarrow\\{\langle H,\emptyset\rangle\\}$
3: return Successors
4: $A\leftarrow$ set of $M$ random points in $E$
5: ExitPoints $\leftarrow\emptyset$
6: StaysInside $\leftarrow 0$
7: for all $x_{0}\in A$ do
8: simulate trajectory from $x_{0}$ until it leaves $H$ through a point
$x_{1}$ or given time elapses
9: if $x_{1}$ exists then
10: ExitPoints $\leftarrow$ ExitPoints $\cup\\{x_{1}\\}$
11: else
12: StaysInside $\leftarrow$ StaysInside $+1$
13: for all F$\in\mathit{Facets}(H)$, F$=H\cap H^{\prime}$ do
14: if ExitPoints $\cap$ F $\neq\emptyset$ then
15: EntryTiles $\leftarrow\\{Z\in\mathrm{Tiles}_{\kappa}(F)\mid Z\cap$
ExitPoints $\neq\emptyset\\}$
16: for all Z $\in$ EntryTiles do
17: $B\leftarrow$ set of $M$ random points in Z
18: RealPointsCount $\leftarrow 0$
19: for all $y_{0}\in B$ do
20: simulate trajectory from $y_{0}$ until it leaves $H$ through a point
$y_{1}$ or given time elapses
21: if $y_{1}\in E$ then
22: RealPointsCount $\leftarrow$ RealPointsCount $+1$
23: if RealPointsCount $<\frac{M}{2}$ then
24: EntryTiles $\leftarrow$ EntryTiles $\setminus\\{$Z$\\}$
25: if EntryTiles $\neq\emptyset$ then
26: Successors $\leftarrow$ Successors $\cup\langle
H^{\prime},EntryTiles\rangle$
27: Weight[$\langle H,E\rangle$][$\langle H,EntryTiles\rangle$]
$\leftarrow\dfrac{|\mathrm{ExitPoints}\cap\mathrm{F}|}{|A|}$
28: return Successors
Performing the backward simulations (lines 16–24 of Algorithm 2) can be
switched off. The resulting transition system differs from the QDAA in the
entry sets, that can be larger. Difference of the outputs can be seen on
Figure 5. The algorithm with backward simulations computes the QDAA and for
$(\kappa\rightarrow\infty)$ approaches the real behaviour of the solutions of
dynamics ODE system. On the other hand the algorithm without backward
simulations overapproximates the entry sets, therefore the transitions are
included even if the entry set of a state is smaller than half of one
$\kappa$-tile. Both options still lead to automatons with reachable states
whose rectangles are included in the set of reachable rectangles of the
rectangular abstraction with the same initial rectangles.
The worst case complexity of the algorithms follows. There are at most $k^{n}$
rectangles in the phase space of the biochemical system, where $k$ is the
maximal number of thresholds on one variable. The maximal number of states of
QDAA of the form $\langle H,E\rangle$ for a fixed rectangle $H$ is
$2n\cdot\big{(}2^{\kappa^{n-1}}-1\big{)}$, where $n$ is the dimension of the
biochemical system. For the average numbers of visited different states of
QDAA with the same rectangle encountered while analysing our evaluation models
see the line labeled $\varrho$ in Table 1. Complexity of the computation of
successors of a given state depends on the dimension of the system, the
$\kappa$ parameter and on the number of simulations $M$ used per one tile. In
the worst case when all the tiles are examined (either as a part of entry set
or potential exit set) there are $2n\cdot\kappa^{n-1}\cdot M$ simulations.
Visualization of the state space of QDAA involves highlighting the borders of
the rectangles $H$ such that there is at least one state $\langle H,E\rangle$
visited during the computation. The intensity of the fill colour of a
rectangle $H$ is calculated proportional to the sum of weights of all possible
paths from initial set $\mathcal{I}_{C}$ to the first appearance of states
with $H$ as the rectangle. The weight of a finite path is obtained as the
product of weights of the subsequent transitions in the path. The sum is
always between zero and one.
### 4.1 Entering and leaving conditions
The computation of successor states proceeds in two steps.
First, the probabilities of potential successors are computed for all
$\kappa$-tiles of $E$ and summed up to get the probabilities of successors for
the whole set $E$. For each $\kappa$-tile of $E$ several numerical simulations
of a solution of system (1), with initial point $x(0)$ placed randomly in the
tile, are performed. If the computed trajectory satisfies entering and leaving
conditions (Definition 4.2) and leaves the box $H$ or the maximal time
interval elapses, the number of trajectories leaving $H$ through the
particular facet (resp. the number of trajectories assumed to stay forever in
$H$ and leading to the transition $\langle H,E\rangle\rightarrow\langle
H,\emptyset\rangle$) is increased.
Second step of the algorithm takes into account the rectangles
$H^{\prime}\bowtie H$ with nonzero probability of transition $\langle
H,E\rangle\rightarrow\langle H^{\prime},\mathrm{yet\ unknown\
}E^{\prime}\rangle$ computed in the first step and determines the entry sets
$E^{\prime}$ of this successors. Denote $F=H^{\prime}\cap H$. For every
$\kappa$-tile of $F$ the algorithm decides if the tile is a subset of
$E^{\prime}$.
Replacing $\mathit{Entry}(F,H)$ with more easily computed set
$\mathit{Entry^{\prime}}(F,H)$ (defined in Definition 4.2) of points
satisfying the entering condition in the definition of QDAA does not lead to a
nonzero difference in the values of transitions probabilities due to Theorem
4.1.
###### Definition 4.1
Let $F\in\mathit{Facets}_{i}(H)$ be a facet of the form
$F=[a_{1},b_{1}]\times\ldots\\{c\\}\ldots\times[a_{n},b_{n}].$ Define the
_normal vector_ $\nu_{H}(F)$ to the facet $F$ with respect to the rectangle
$H$ as the vector $-e_{i}$ for $c=a_{i}$ and $e_{i}$ for $c=b_{i}$, where
$e_{i}$ denotes the $i$th vector of the standard basis of $\mathbb{R}^{n}$
(i.e. the vector orthogonal to $F$ and pointing outside from $H$).
###### Definition 4.2
Consider rectangle $H,F,F^{\prime}\in\mathit{Facets}(H)$, a mutiaffine vector
field $f$, a solution $x(t)$ of the system (1) and $r>0$ satisfying $x(0)\in
F,x(r)\in F^{\prime},\forall t\in(0,r):x(t)\in H.$ We say that $x(t)$
satisfies the _entering (resp. leaving) condition_ with respect to $H$ and
$f$, if $f(x(0))\cdot\nu_{H}(F)<0)$ (resp.
$f(x(r))\cdot\nu_{H}(F^{\prime})>0$).
Define $\mathit{Entry}^{\prime}(F,H)=\\{x\in F|\nu_{H}(F)\cdot f(x)<0\\}$.
Our next step is to show (in Theorem 4.1) that the focal set of
$\mathit{Entry^{\prime}}(F,H)\setminus\mathit{Entry}(F,H)$ is of measure zero,
thus the set difference is insignificant for our volume based notions.
Now we will extend our definition of $\mathit{Focal}(H,E,F^{\prime})$ set of
points from which the trajectories leave the box through a facet $F^{\prime}$
(Definition 3.6) to $\mathit{Focal}(H,E,X)$ for an arbitrary subset $X$ of
facet $F^{\prime}$.
###### Definition 4.3
Let $E\in\mathit{EntrySets}_{\kappa}(H),E\neq\emptyset,F^{\prime}\in
Facets(H)$. Let $X\subseteq F^{\prime}$. Define $\mathit{Focal}(H,E,X)$ the
set of all points $y_{0}\in E$ such that there exist
$\epsilon,c,\epsilon^{\prime}>0$ and a solution $y(t)$ of the system (1)
satisfying $y(t)\in H$ for $t\in(0,c),y(t)\in\mathit{Inter}(H)$ for
$t\in(0,\epsilon)$, $y(c)\in X$, and $y(t)\in\mathit{Inter}(H^{\prime})$ for
$t\in(c,c+\epsilon^{\prime})$.
###### Theorem 4.1
Let
$E\in\mathit{EntrySets}_{\kappa}(H),E\neq\emptyset,F^{\prime}\in\mathit{Facets}(H)$.
Let $V_{+}=\\{x\in\mathit{ExitSet}(H,E,F^{\prime})|\eta_{F^{\prime}}\cdot
f(x)>0\\}$ and let $V_{0}=\mathit{ExitSet}(H,E,F^{\prime})\setminus V_{+}$.
Then $\lambda^{*}_{n^{\prime}}\bigl{(}\mathit{Focal}(H,E,V_{0})\bigr{)}=0,$
where $n^{\prime}$ denotes $n$ for $E=H$, and $(n-1)$ otherwise.
###### Proof.
First, we have to observe that trajectories from such initial points have to
leave the rectangle through a set of measure zero (in fact it is a zero set of
a multi-affine polynomial).
Then the proof is simillar to proof of Lemma 3.5. ∎∎
###### Remark 4.1
Theorem 4.1 implies that replacing $\mathit{Entry}(F,H)$ with
$\mathit{Entry^{\prime}}(F,H)$ in the definition of QDAA (recall that these
can be only entry sets of the successor states, not of the initial states
$\langle H,H\rangle$), does not lead to a nonzero difference in the values of
transitions probabilites.
###### Remark 4.2
Deciding if a point $x$ is an element of $\mathit{Entry}^{\prime}(F,H)$ is
straightforward (compared to checking the $\epsilon$-condition of Definition
3.2), and $\mathit{Entry}^{\prime}(F,H)\subseteq\mathit{Entry}(F,H)$.
###### Remark 4.3
Moreover, there is the following symmetry property. Let
$F,F^{\prime}\in\mathit{Facets}(H),x_{0}\in F,x_{1}\in F^{\prime}$. There is a
solution $y(t)$ of system (1) satisfying $y(0)=x_{0},\exists
t_{1}:y(t_{1})=x_{1},f(y(0))\cdot\nu_{H}(F)<0,f(y(t_{1}))\cdot\nu_{H}(F^{\prime})>0$
and $y(t)\in H$ for $t\in(0,t_{1})$, if and only if there is a solution $x(t)$
of system $\dot{x}(t)=-f(x)$ satisfying $x(0)=x_{1},\exists
t_{1}:x(t_{1})=x_{0},-f(x(0))\cdot\nu_{H}(F^{\prime})<0,-f(x(t_{1}))\cdot\nu_{H}(F)>0$
and $x(t)\in H$ for $t\in(0,t_{1})$.
## 5 Evaluation and Case Study
In this section the state spaces of several biological models (of dimensions
two, four and seven) are explored. Using our prototype implementation of the
algorithms from Section 4 implemented in C++, we evaluate our approach on two
exemplary biochemical systems. Additionally, we provide a case study held on a
biochemical pathway studied in _E. coli_ and compare the reachability results
of the case study and one of the smaller models with results obtained using
the rectangular abstraction approach.
Before we proceed with the models, let us introduce several terms useful for
the evaluation. For a biochemical system $\mathcal{B}=\langle
n,f,\mathcal{T},\mathcal{I}_{C}\rangle$ we denote
$\mathcal{R}(\mathcal{I}_{C})\subseteq\mathit{Rect}(\mathcal{T})$ the _set of
all rectangles reachable from initial set $\mathcal{I}_{C}$_. For each
$H\in\mathit{Rect}(\mathcal{T})$ we denote $\mathit{mem}(H)$ the subset of
$\mathcal{R}(\mathcal{I}_{C})$ consisting of all states reachable from the
initial set with $H$ as rectangle, the so-called _memory of the rectangle_
$H$, $\mathit{mem}(H)=\\{\langle R,E\rangle\in\mathcal{R}(\mathcal{I}_{C})\mid
R=H\\}$. Further we denote $\varrho$ the average number of memory states
(cardinality of $\mathit{mem}(H)$ averaged over all
$H\in\mathcal{R}(\mathcal{I}_{C})$). The number of QDAA states representing
the memory of a rectangle is in the worst case equal to the number of all its
possible entry sets. However, the actual values of $\varrho$ in our examples
are much smaller (see Table 1).
Let us focus on the effect of parameter $\kappa$ on cardinality of
$\mathcal{R}(\mathcal{I}_{C})$ and on $\varrho$. Expected behaviour of the
approximation is the following. Every facet is divided into $\kappa^{n-1}$
tiles. A tile is included in the entry set $E$ of some reachable state
$\langle H,E\rangle$ if the focal subset $\mathit{Focal}(H,E,A)$ fills at
least half of the volume of the tile. For higher values of $\kappa$, the set
$\mathit{Tiles}_{n^{\prime}}^{\kappa}(\mathit{Focal}(H,E,A))$ better
approximates the set $\mathit{Focal}(H,E,A)$ because of the higher
$\kappa$-grid resolution. Thus with increasing $\kappa$, the quantitative
information denoting the probability of reaching states in
$\mathcal{R}(\mathcal{I}_{C})$ can be computed more precisely. We demonstrate
that on models examined below.
### 5.1 Oscillatory model
First, we consider a $2$-dimensional model which is a variant of Lotka-
Volterra model with oscillatory behaviour. The oscillatory model has the form
of the following multi-affine system:
$\begin{array}[]{l}\frac{dX}{dt}=5\cdot X-1\cdot X\cdot Y\\\\[5.69054pt]
\frac{dY}{dt}=0.4\cdot X\cdot Y-5.4\cdot Y\end{array}$
We consider the following partition and initial conditions for this model:
$\begin{array}[]{l}T_{X}=\\{i|i\in\langle
0,30\rangle\subseteq\mathbb{N}_{0}\\}\\\\[5.69054pt] T_{Y}=\\{i|i\in\langle
0,12\rangle\subseteq\mathbb{N}_{0}\\}\end{array}$
$\mathcal{I}_{C}:X\in\langle 20,21\rangle,Y\in\langle 5,6\rangle$
Results achieved on our implementation are presented in Table 1 and visualized
in Figure 5. Black rectangles denote the initial set.
### 5.2 Enzyme kinetics
Similarly, we examined a $4$-dimensional model of basic enzyme kinetics based
on the following set of reactions:
$\begin{array}[]{rl}S+E&\stackrel{{\scriptstyle
k_{1}}}{{\rightarrow}}ES\\\\[5.69054pt] ES&\stackrel{{\scriptstyle
k_{2}}}{{\rightarrow}}S+E\\\\[5.69054pt] ES&\stackrel{{\scriptstyle
k_{3}}}{{\rightarrow}}P+E\end{array}$
The corresponding multi-affine ODE model considered in the paper is the
following:
$\begin{array}[]{l}\frac{dS}{dt}=-0.01\cdot S\cdot E+1\cdot ES\\\\[5.69054pt]
\frac{dE}{dt}=1\cdot ES-0.01\cdot S\cdot E+1\cdot ES\\\\[5.69054pt]
\frac{dES}{dt}=-1\cdot ES+0.01\cdot E\cdot S-1\cdot ES\\\\[5.69054pt]
\frac{dP}{dt}=1\cdot ES\end{array}$
We consider the following partition and initial conditions for this model:
$\begin{array}[]{l}T_{S}=\\{0.01,5,10,15,25,50,60,85,95,100\\}\\\\[5.69054pt]
T_{E}=\\{0.01,5,10,15,25,50,60,85,95,100\\}\\\\[5.69054pt]
T_{ES}=\\{0.01,5,10,15,25,50,60,85,95,100\\}\\\\[5.69054pt]
T_{P}=\\{0.01,5,10,15,25,50,60,85,95,100\\}\end{array}$
$\mathcal{I}_{C}:S\in\langle 25,50\rangle,E\in\langle
95,100\rangle,ES\in\langle 0.01,5\rangle,P\in\langle 0.01,10\rangle$
Projection of the approximated phase space to the enzyme/substrate plane is
shown in Figure 6. For both the oscillatory model and the enzyme kinetics
model full version of Algorithm 2 (with backward simulations) was used.
$\kappa=4$ $\kappa=16$ $\kappa=60$
Figure 5: Reachability in oscillatory model and comparison with numerical simulation, first two figures were obtained using the full version of Algorithm 2, the third one with lines 16–24 omitted. For comparison: using the rectangular abstraction transition system on this biochemical model, the whole phase space $[0,30]\times[0,12]$ is reachable from the same inital conditions. | Oscillatory | Enzyme
---|---|---
$\kappa$ | 4 | 8 | 16 | 32 | 64 | 128 | 4 | 5 | 6 | 7
$|\mathcal{R}(\mathcal{I}_{C})|$ | $52$ | $46$ | $40$ | $39$ | $37$ | $35$ | $76$ | $104$ | $123$ | $166$
$\varrho$ | $1.63$ | $2.2$ | $3.78$ | $2.9$ | $4.57$ | $6$ | $4.36$ | $10.76$ | $16.8$ | $53.6$
Table 1: Results for the two models and several different settings of the
discretization parameter $\kappa$.
$\kappa=4$ $\kappa=6$ Numerical simulation
Figure 6: Enzyme kinetics model – projection of the reachable set to the
enzyme/substrate plane and comparison with numerical simulation.
### 5.3 Case Study on E.Coli Ammonium Assimilation Model
We consider a model specifying the ammonium transport from the external
environment into cells of _E. Coli_ [27]. The model describes the ammonium
transport process that takes effect at very low external ammonium
concentrations. In such conditions, the transport process complements the
deficient ammonium diffusion. The process is driven by a membrane-located
ammonium transport protein $AmtB$ that binds external ammonium cations
$NH_{4}ex$ and uses their electrical potential to conduct $NH_{3}$ into the
cytoplasm. In Figure 7, biochemical reactions of this model and the scheme of
the transport channel are shown (left and middle).
The level of pH and external ammonium concentration are considered constant.
The system of differential equations:
$\begin{array}[]{c@{$\,=\,$}l}\frac{d[AmtB]}{dt}&-k_{1}[AmtB][NH_{4}ex]+k_{2}[AmtB:NH_{4}]+k_{4}[AmtB:NH_{3}]\\\\[2.84526pt]
\frac{d[AmtB:NH_{3}]}{dt}&k_{3}[AmtB:NH_{4}]-k_{4}[AmtB:NH_{3}]\\\\[2.84526pt]
\frac{d[AmtB:NH_{4}]}{dt}&k_{1}[AmtB][NH_{4}ex]-k_{2}[AmtB:NH_{4}]-k_{3}[AmtB:NH_{4}]\\\\[2.84526pt]
\frac{d[NH_{3}in]}{dt}&k_{4}[AmtB:NH_{3}]-k_{6}[NH_{3}in][H_{in}]+k_{7}[NH_{4}in]+k_{9}[NH_{3}ex]\\\\[2.84526pt]
\frac{d[NH_{4}in]}{dt}&k_{6}[NH_{3}in][H_{in}]-k_{5}[NH_{4}in]-k_{7}[NH_{4}in]\end{array}$
Constant species: $NH_{3}ex,NH_{4}ex,H_{in},H_{ex}.$
Initial conditions and threshold numbers:
$\begin{array}[]{ccl}T_{NH_{3}ex}&=&\\{0,28\cdot 10^{-9},29\cdot
10^{-9},1\cdot 10^{-5}\\}\\\ T_{NH_{4}ex}&=&\\{0,49\cdot 10^{-7},5\cdot
10^{-6},1\cdot 10^{-5}\\}\\\ T_{AmtB}&=&\\{0,1\cdot 10^{-12},1\cdot
10^{-10},5\cdot 10^{-6},9.9\cdot 10^{-6},1\cdot 10^{-5}\\}\\\
T_{AmtB:NH_{3}}&=&\\{0,1\cdot 10^{-7},1\cdot 10^{-5}\\}\\\
T_{AmtB:NH_{4}}&=&\\{0,1\cdot 10^{-7},1\cdot 10^{-5}\\}\\\
T_{NH_{3}in}&=&\\{0,1\cdot 10^{-8},1\cdot 10^{-7},1\cdot 10^{-6},11\cdot
10^{-7},1\cdot 10^{-5},1\cdot 10^{-4},1\cdot 10^{-3}\\}\\\
T_{NH_{4}in}&=&\\{0,1\cdot 10^{-8},1\cdot 10^{-7},2\cdot 10^{-6},2.1\cdot
10^{-6},1\cdot 10^{-6},1\cdot 10^{-5},1\cdot 10^{-4},1\cdot 10^{-3}\\}\\\
\end{array}$
$\begin{array}[]{cl}\mathcal{I}_{C}:&NH_{3}ex\in\langle 28\cdot
10^{-9},29\cdot 10^{-9}\rangle,\\\ &NH_{4}ex\in\langle 49\cdot 10^{-7},5\cdot
10^{-6}\rangle,\\\ &AmtB\in\langle 0,1\cdot 10^{-5}\rangle,\\\
&AmtB:NH_{3}\in\langle 0,1\cdot 10^{-5}\rangle,\\\ &AmtB:NH_{4}\in\langle
0,1\cdot 10^{-5}\rangle,\\\ &NH_{3}in\in\langle 1\cdot 10^{-6},11\cdot
10^{-7}\rangle,\\\ &NH_{4}in\in\langle 2\cdot 10^{-6},21\cdot
10^{-7}\rangle.\\\ \end{array}$
$\begin{array}[]{cl}AmtB+NH_{4}ex\stackrel{{\scriptstyle
k_{1}}}{{\leftarrow}}\stackrel{{\scriptstyle
k_{2}}}{{\rightarrow}}AmtB:NH_{4}&k_{1}=5\cdot 10^{8},k_{2}=5\cdot 10^{3}\\\
AmtB:NH_{4}\stackrel{{\scriptstyle
k_{3}}}{{\rightarrow}}AmtB:NH_{3}+H_{ex}&k_{3}=50\\\
AmtB:NH_{3}\stackrel{{\scriptstyle
k_{4}}}{{\rightarrow}}AmtB+NH_{3}in&k_{4}=50\\\
NH_{4}in\stackrel{{\scriptstyle k_{5}}}{{\rightarrow}}&k_{5}=80\\\
NH_{3}in+H_{in}\stackrel{{\scriptstyle
k_{6}}}{{\leftarrow}}\stackrel{{\scriptstyle
k_{7}}}{{\rightarrow}}NH_{4}in&k_{6}=1\cdot 10^{15},k_{7}=5.62\cdot 10^{5}\\\
NH_{3}in\stackrel{{\scriptstyle k_{8}}}{{\leftarrow}}\stackrel{{\scriptstyle
k_{9}}}{{\rightarrow}}NH_{3}ex&k_{8}=k_{9}=1.4\cdot 10^{4}\end{array}$
Figure 7: Ammonium transport model (left). Simulations of the ammonium
assimilation model from $20$ randomly sampled points in $\mathcal{I}_{C}$
projected on the concentration of $NH_{4}in$, blue lines represent bounds on
this concentration found by the QDAA - two subsequent thresholds
$10^{-6},10^{-5}$ (right).
The upper bounds on concentrations of $NH_{3}in$ and $NH_{4}in$ considering
the biological system with given initial conditions were estimated as
$1.1\cdot 10^{-6}$ ($NH_{3}in$ does not exceed the initial concentration) and
$5.4\cdot 10^{-4}$ by the rectangular abstraction (overapproximation).
Reachable intervals using Algorithm 2 without the backward simulations were
$[10^{-8},1.1\cdot 10^{-6}]$ for $NH_{3}in$ ($NH_{3}in$ does not exceed the
initial concentration), and $[10^{-6},10^{-5}]$ for $NH_{4}in$. This results
are in agreement with simulated data and in the case of the concentration of
$NH_{4}in$ the QDAA results are by one order closer to numerical simulations
than the rectangular abstraction results as can be seen in the right part of
Figure 7.
## 6 Conclusion
We have presented a new theoretical method for finite discrete approximation
of autonomous continuous systems equipped with a measure that indirectly
quantifies correspondence of the approximated behaviour with the original
continuous behaviour. We have provided a computational technique which we
implemented in a prototype software. We have examined the implementation on
small dimensional models which showed satisfactory results for computing
reachability.
The method can be either used as a parameterized simulation technique or
employed with rectangular abstraction to quantify the extent of spurious
counterexamples. Thus the method can improve the current possibilities of
analysis based on model checking techniques. We leave for future work
integration of this method into the software for model checking of biochemical
dynamical systems [14].
At the theoretical side, we leave for future work precise clarification of our
method wrt the rectangular abstraction. From the computational viewpoint, we
aim to develop a parallel reachability algorithm that would make the method
scalable and applicable to systems of larger dimensions.
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|
arxiv-papers
| 2011-07-29T10:51:15 |
2024-09-04T02:49:21.051185
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lubos Brim and Jana Fabrikova and Sven Drazan and David Safranek",
"submitter": "David \\v{S}afr\\'anek",
"url": "https://arxiv.org/abs/1107.5924"
}
|
1107.5950
|
# A unified Generating function of the $q$-Genocchi polynomials with Their
Interpolation Functions
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Açıkgöz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr , Hassan Jolany
School of Mathematics, Statistics and Computer Science, University of Tehran,
Iran hassan.jolany@khayam.ut.ac.ir and Jong Jin Seo Department of Applied
Mathematics, Pukyong National University, Busan, 608-737,Korea
seo2011@pknu.ac.kr
(Date: April 18, 2011)
###### Abstract.
The purpose of this paper is to construct of the unification $q$-extension
Genocchi polynomials. We give some interesting relations of this type of
polynomials. Finally, we derive the $q$-extensions of Hurwitz-zeta type
functions from the Mellin transformation of this generating function which
interpolates the unification of $q$-extension of Genocchi polynomials.
###### Key words and phrases:
Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials
###### 1991 Mathematics Subject Classification:
05A10, 11B65, 28B99, 11B68, 11B73.
## 1\. Introduction, Definitions and Notations
Recently, many mathematician have studied to unification Bernoulli, Genocchi,
Euler and Bernstein polynomials (see [20,21]). Ozden [20] introduced $p$-adic
distribution of the unification of the Bernoulli, Euler and Genocchi
polynomials and derived some properties of this type of unification
polynomials.
In [20], Ozden constructed the following generating function:
(1.1)
$\sum_{n=0}^{\infty}y_{n,\beta}\left(x;k,a,b\right)\frac{t^{n}}{n!}=\frac{2^{1-k}t^{k}e^{xt}}{\beta^{b}e^{t}-a^{b}},\text{
}\left|t+b\ln\left(\frac{\beta}{a}\right)\right|<2\pi$
where $k\in\mathbb{N}=\left\\{1,2,3,...\right\\},$ $a,b\in\mathbb{R}$ and
$\beta\in\mathbb{C}.$ The polynomials $y_{n,\beta}\left(x;k,a,b\right)$ are
the unification of the Bernoulli, Euler and Genocchi polynomials.
Ozden showed, $\beta=b=1,$ $k=0$ and $a=-1$ into (1.1), we have
$y_{n,1}\left(x;0,-1,1\right)=E_{n}\left(x\right),$
where $E_{n}\left(x\right)$ denotes classical Euler polynomials, which are
defined by the following generating function:
(1.2)
$\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi,\text{ }$
In [5,20], classical Genocchi polynomials defined as follows:
(1.3)
$\frac{2te^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi,\text{ }$
From (1.2) and (1.3), we easily see,
(1.4) $G_{n}\left(x\right)=nE_{n-1}\left(x\right),$
For a fixed real number $\left|q\right|<1,$ we use the notation of $q$-number
as
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{, \ \ (see [1-4,6-24]),}$
Thus, we note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$.
In [1,7,8,10], $q$-extension of Genocchi polynomials are defined as follows:
$G_{n+1,q}\left(x\right)=\left(n+1\right)\left[2\right]_{q}\sum_{l=0}^{\infty}\left(-1\right)^{l}q^{l}\left[x+l\right]_{q}^{n}.$
In [2], $\left(h,q\right)$-extension of Genocchi polynomials are defined as
follows:
$G_{n+1,q}^{\left(h\right)}\left(x\right)=\left(n+1\right)\left[2\right]_{q}\sum_{l=0}^{\infty}\left(-1\right)^{l}q^{\left(h-1\right)l}\left[x+l\right]_{q}^{n}.$
In this paper, we shall construct unification of $q$-extension of the Genocchi
polynomials, however we shall give some interesting relationships. Moreover,
we shall derive the $q$-extension of Hurwitz-zeta type functions from the
Mellin transformation of this generating function which interpolates.
## 2\. Novel Generating Functions of $q$-extension of Genocchi polynomials
###### Definition 1.
Let $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ and
$k\in\mathbb{N}=\left\\{1,2,3,...\right\\},$Then the unification of
$q$-extension of Genocchi polynomials defined as follows:
(2.1)
$\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$
and
(2.2)
$\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left[m+x\right]_{q}t}.$
where into (2.1) substituting $x=0,$
$S_{n,\beta,q}\left(0\left|k,a,b\right.\right)=S_{n,\beta,q}\left(k,a,b\right)$
are called unification of $q$-extension of Genocchi numbers.
As well as, from (2.1) and (2.2) Ozden’s constructed the following generating
function, namely, we obtain (1.1),
$\lim_{q\rightarrow
1}\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=\frac{2^{1-k}t^{k}e^{xt}}{\beta^{b}e^{t}-a^{b}}\text{.}$
By (2.2), we see readily,
(2.3)
$\displaystyle\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left[m+x\right]_{q}t}$
$\displaystyle=$
$\displaystyle\frac{e^{\left[x\right]_{q}t}}{q^{kx}}\left(-\left[2\right]_{q}^{1-k}\left(q^{x}t\right)^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left(q^{x}t\right)\left[m\right]_{q}}\right)$
$\displaystyle=$ $\displaystyle
q^{-kx}\left(\sum_{n=0}^{\infty}\left[x\right]_{q}^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}q^{nx}S_{n,\beta,q}\left(k,a,b\right)\frac{t^{n}}{n!}\right)$
From (2.3) by using Cauchy product we get
$\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}=q^{-kx}\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\binom{n}{l}q^{lx}S_{l,\beta,q}\left(k,a,b\right)\left[x\right]_{q}^{n-l}\right)\frac{t^{n}}{n!}$
By comparing coefficients of $\frac{t^{n}}{n!}$ in the both sides of the above
equation, we arrive at the following theorem:
###### Theorem 1.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\sum_{l=0}^{n}\binom{n}{l}q^{lx}S_{l,\beta,q}\left(k,a,b\right)\left[x\right]_{q}^{n-l}$
As well as, we obtain corollary 1:
###### Corollary 1.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,
(2.4)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\left(S_{\beta,q}\left(k,a,b\right)+\left[x\right]_{q}\right)^{n}$
with usual the convention about replacing
$\left(S_{\beta,q}\left(x\left|k,a,b\right.\right)\right)^{n}$by
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$
By applying the definition of generating function of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right),$ we have
$\displaystyle\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left(\sum_{n=0}^{\infty}\left[m+x\right]_{q}^{n}\frac{t^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(-\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left[m+x\right]_{q}^{n}\right)\frac{t^{n+k}}{n!}$
So we derive the Theorem 2 which we state hear:
###### Theorem 2.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is positive integer.
We obtain,
(2.5)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=-\frac{n!\left[2\right]_{q}^{1-k}}{\left(n-k\right)!}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left[m+x\right]_{q}^{n-k}$
With regard to (2.5), we see after some calculations
(2.6) $\displaystyle S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$
$\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\sum_{m=0}^{\infty}\left(\frac{\beta^{b}}{a^{b}}\right)^{m}q^{lm}$
$\displaystyle=$
$\displaystyle\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\frac{1}{\beta^{b}q^{l}-a^{b}}$
$\displaystyle=$
$\displaystyle\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n}{k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\frac{1}{\beta^{b}q^{l}-a^{b}}$
From (2.6) and well known identity
$\left[\binom{n}{k}\binom{n-k}{l}=\binom{n}{k+l}\binom{k+l}{k}\right],$ we
obtain the following theorem:
###### Theorem 3.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain
(2.7)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{\left(l-k\right)x}\frac{1}{\beta^{b}q^{l-k}-a^{b}}$
We put $x\rightarrow 1-x,\beta\rightarrow\beta^{-1},q\rightarrow q^{-1}$ and
$a\rightarrow a^{-1}$ into (2.7), namely,
$\displaystyle S_{n,\beta^{-1},q^{-1}}\left(1-x\left|k,a^{-1},b\right.\right)$
$\displaystyle=$
$\displaystyle\frac{k!\left[2\right]_{q^{-1}}^{1-k}}{a^{-b}}\left(\frac{1}{1-q^{-1}}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{-\left(l-k\right)\left(1-x\right)}\frac{1}{\beta^{-b}q^{-\left(l-k\right)}-a^{-b}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k}q^{k-1}q^{n-k}k!\left[2\right]_{q}^{1-k}a^{b}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{k-l}q^{\left(l-k\right)x}\frac{\beta^{b}q^{l-k}a^{b}}{a^{b}-\beta^{b}q^{l-k}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}\left(\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{x\left(l-k\right)}\frac{1}{\beta^{b}q^{l-k}-a^{b}}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$
So, we obtain symmetric properties of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ as follows:
###### Theorem 4.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain
$S_{n,\beta^{-1},q^{-1}}\left(1-x\left|k,a^{-1},b\right.\right)=\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$
By (2.2), we see
(2.8)
$\displaystyle\frac{\beta}{a}\tciFourier_{\beta,q}\left(t,1\left|k,a,b\right.\right)-\tciFourier_{\beta,q}\left(t,0\left|k,a,b\right.\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{\beta}{a}\right)S_{n,\beta,q}\left(1\left|k,a,b\right.\right)-S_{n,\beta,q}\left(k,a,b\right)\right)\frac{t^{n}}{n!}=-\frac{\left[2\right]_{q}^{1-k}}{a^{b}}t^{k}$
From (2.8), we obtain the following theorem:
###### Theorem 5.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain
(2.9)
$S_{n,\beta,q}\left(k,a,b\right)-\left(\frac{\beta}{a}\right)S_{n,\beta,q}\left(1\left|k,a,b\right.\right)=\left\\{\QATOP{0,\text{
\ \ }n\neq k}{\frac{\left[2\right]_{q}^{1-k}}{a^{b}}k!,n=k}\right.$
From (2.4) and (2.9), we obtain corollary as follows:
###### Corollary 2.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$, which is $k$ positive integer.
We get
$S_{n,\beta,q}\left(k,a,b\right)-\frac{\beta}{aq^{k}}\left(qS_{\beta,q}\left(k,a,b\right)+1\right)^{n}=\left\\{\QATOP{0,n\neq
k}{\frac{\left[2\right]_{q}^{1-k}}{a^{b}}k!,n=k}\right.$
with the usual convention about replacing
$\left(S_{\beta,q}\left(k,a,b\right)\right)^{n}$ by
$S_{n,\beta,q}\left(k,a,b\right).$
From $\left(9\right),$ now, we shall obtain distribution relation for
unification $q$-extension of Genocchi polynomials, after some calculations,
namely,
$\displaystyle S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ $\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\left(\frac{\beta}{a}\right)^{bm}\left[m+x\right]_{q}^{n-k}$
$\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{b\left(l+md\right)}\left[l+md+x\right]_{q}^{n-k}$
$\displaystyle=$
$\displaystyle\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}\left(-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\left(\frac{\beta^{d}}{a^{d}}\right)^{bm}\left[m+\frac{x+l}{d}\right]_{q^{d}}^{n-k}\right)$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}^{1-k}}{\left[2\right]_{q^{d}}^{1-k}}\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}S_{n,\beta^{d},q^{d}}\left(\frac{x+l}{d}\left|k,a^{d},b\right.\right)$
Therefore, we obtain the following theorem:
###### Theorem 6.
$\left(distribution\text{ \ }formula\text{ }for\text{
}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\right)$For $a,b\in\mathbb{R}$ ,
$\beta\in\mathbb{C}$ which $k$ is $\mathop{\mathrm{p}ositive}$ integer. We
obtain,
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\frac{\left[2\right]_{q}^{1-k}}{\left[2\right]_{q^{d}}^{1-k}}\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}S_{n,\beta^{d},q^{d}}\left(\frac{x+l}{d}\left|k,a^{d},b\right.\right)$
## 3\. Interpolation function of the polynomials
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$
In this section, we give interpolation function of the generating functions of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ however, this function is
meromorphic function. This function interpolates
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ at negative integers.
For $s\in\mathbb{C}$ , by applying the Mellin transformation to (2.2), we
obtain
$\displaystyle\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k+1}}{\Gamma\left(s\right)}\mathop{\displaystyle\oint}t^{s-k-1}\tciFourier_{\beta,q}\left(-t,x\left|k,a,b\right.\right)dt$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-t\left[m+x\right]_{q}t}$
So, we have
$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)=\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\frac{\beta^{bm}a^{-bm-b}}{\left[m+x\right]_{q}^{s}}$
We define $q-$extension Hurwitz-zeta type function as follows theorem:
###### Theorem 7.
For $a,b\in\mathbb{R}$ , $\beta,s\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,
(3.1)
$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)=\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\frac{\beta^{bm}a^{-bm-b}}{\left[m+x\right]_{q}^{s}}$
for all $s\in\mathbb{C}.$ We note that
$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)$ is analytic function in the
whole complex $s$-plane.
###### Theorem 8.
For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,
$\boldsymbol{\Im}_{\beta,q}\left(-n;x,a,b\right)=-\frac{\left(n-k\right)!}{n!}S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$
###### Proof.
Let $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ and $k\in\mathbb{N}$ with
$k\in\mathbb{N}=\left\\{1,2,3,...\right\\}$. $\Gamma\left(s\right)$, has
simple poles at $z=-n=0,-1,-2,-3,\cdots.$ The residue of
$\Gamma\left(s\right)$ is
$\mathop{\mathrm{R}e}s\left(\Gamma\left(s\right),-n\right)=\frac{\left(-1\right)^{n}}{n!}.$
We put $s\rightarrow-n$ into (3.1) and using the above relations, the desired
result can be obtained.
## References
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* [2] Araci, S., Seo, J.J., Erdal, D., Different Approach On The $\left(h,q\right)$ Genocchi Numbers and Polynomials Associated with $q$-Bernstein Polynomials, (Submitted)
* [3] Araci, S., Seo, J.J., Erdal, D., New Construction weighted $\left(h,q\right)$-Genocchi numbers and Polynomials Related to Zeta Type Functions, Discrete Dynamics in Nature and Society(in press).
* [4] Araci, S., Erdal D., and Seo, J-J., A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ Associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis(in press).
* [5] Jolany, H., and Sahrifi, H., Some Results for the Apostol-Genocchi polynomials of Higher Order, Accepted in Bulletin of the Malaysian Mathematical Sciences Society, arXiv:1104.1501v1 [math. NT].
* [6] Ryoo. C-S., Kim, T., and Lee, B., $q$-Bernoulli numbers and polynomials revisited, Accepted in Advances in Difference Equations.
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|
arxiv-papers
| 2011-07-29T12:31:00 |
2024-09-04T02:49:21.063755
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet A\\c{c}ikg\\\"oz, Hassan Jolany, and Jong Jin Seo",
"submitter": "Hassan Jolany",
"url": "https://arxiv.org/abs/1107.5950"
}
|
1108.0007
|
# A Invertible Dimension Reduction of Curves on a Manifold
Sheng Yi
North Carolina State University
Hamid Krim
North Carolina State University
Larry K. Norris
North Carolina State University
###### Abstract
In this paper, we propose a novel lower dimensional representation of a shape
sequence. The proposed dimension reduction is invertible and computationally
more efficient in comparison to other related works. Theoretically, the
differential geometry tools such as moving frame and parallel transportation
are successfully adapted into the dimension reduction problem of high
dimensional curves. Intuitively, instead of searching for a global flat
subspace for curve embedding, we deployed a sequence of local flat subspaces
adaptive to the geometry of both of the curve and the manifold it lies on. In
practice, the experimental results of the dimension reduction and
reconstruction algorithms well illustrate the advantages of the proposed
theoretical innovation.
## 1 Introduction
For many pattern recognition problems, the feature space is nonlinear and of
high dimensionality and nonlinearity, for example, face recognition, image
classification and shape recognition. Many dimension reduction techniques [1]
are therefore naturally introduced to ease the modeling, computation and
visualization of the feature space. Dimension reduction has been widely used
with different vision techniques such as contour tracking [3], face
recognition [4] and object recognition [2].
One of the key points of dimension reduction is how to represent the geometry
of the original space in a lower dimensional space [9, 10]. Most of the
previous dimension reduction research [1] focuses on embedding the points in
the original space into a lower dimensional space. Beyond the point-wise
dimension reduction, the dimension reduction for a sequence of points or an
evolving curve in the original feature space is more important in applications
where features of interest are time varying such as, face expression [4],
events indexing in video sequence as a shape process [11] and human activity
as a shape sequence [3]. To the best of our knowledge, only a few works [3, 2]
address the problem of how to represent a curve in the original feature space
in a lower dimensional space.
The dimension reduction of high dimensional curves may be categorized into two
classes. One is to reduce the whole curve space to a lower dimension as in [3,
12, 13]. The other one is to reduce a particular the curve to a corresponding
lower dimensional curve as in [2]. The drawback of the first one is the
Euclidean assumption by using an approximation with a spline function.
Moreover, as in [3] by directly applying PCA to the spline coefficients, the
global dimension reduction does not consider the intrinsic dimensionality of a
particular high dimensional curve. A curve sequence like walking, for example,
is much simpler than a dancing sequence. Thus the dynamics of walking might be
represented in a much lower dimensional space in comparison to other complex
activities. The second category of dimension reduction as in [2], considered
the geometry of a particular curve in the original space, which, due to the
chosen implementation of the Whitney embedding theorem, is computationally
heavy on account of the iterative subspace search. At the same time, there
exists no invertible mapping from the embedding space to the original one,
thus limiting its extension to a generative process.
Motivated by the importance of a lower dimensional representation of a
manifold-valued curve and the limitations of the current techniques, in this
paper a novel dimension reduction of a shape manifold valued curve is
proposed. In contrast with previous curve dimension reduction techniques [3,
2], the proposed method has the following advantages:
1. 1.
the dimension reduction is adaptive to the Riemannian geometry of a shape
space.
2. 2.
the computation is linear in the length of the curve times the dimension of
the shape space.
3. 3.
with a proper dimensionality, there exists a reconstruction as a subspace
parallel transport on a manifold.
To introduce the idea of the proposed dimension reduction framework for the
manifold valued curves, this paper follows the moving frame formulation in [6,
7]. According to [7, 6], every curve on a manifold may be represented by a
differential equation,
$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t)$ (1)
where, $V(X_{t})$ is a vector field on a manifold, and $Z_{t}$ is a curve in a
Euclidean space. Under such representation, to introduce a lower dimensional
representation of $X_{t}$, a method is proposed to find the optimal vector
field $V(X_{t})$ such that:
1. 1.
The sequence of $V(X_{t})$ is path dependent and uniquely determined by
$X_{t}$ and initial condition $V(X_{0})$ by a efficient computation.
2. 2.
The resulting $dZ_{t}$ from $X_{t}$ and $V(X_{t})$ have a distribution that
lies in a lower dimensional subspace.
The first requirement on $V(X_{t})$ means that $X_{t}$ can be represented by
only the initial condition $V(X_{0}),X_{0}$ and the embedded curve $Z_{t}$.
According to [6] such property yields the 1-1 correspondence between $X_{t}$
and $Z_{t}$. The second requirement is more intuitive that among all the
$V(X_{t})$ satisfied the first requirement, the optimal one is selected such
that the variance of tangents is best represented in a subspace.
The first requirement is satisfied by select $V(X_{t})$ that is a horizontal
vector field [7] defined by the Levi-Civita connection as in Section 2.2 with
a $L^{2}$ metric induced to the shape manifold $M$ from the ambient space.
According to [6, 7], under a defined connection, for $X_{t}\in M$ and initial
condition $V(X_{0})$, there exists a unique selection of moving frame
$V(X_{t})$. Thus given $V(X_{0})$ the resulting curve in Euclidean space have
one-one correspondence to $X_{t}$. The reason of using the Levi-Civita
connection have two folds. The first one is that it is a non-flat connection.
In contrast to the other category: flat connection, it is path dependent. Such
property is important for dimension reduction because it provides more
adaptivity to the geometry of the path on the manifold. The second reason is
that in comparison with the general nonflat connections defined by a PDE on
manifold, such particular Levi-Civita connection in Section 2.2 is consistent
with the metric of the shape manifold in [5] that this paper is based on, and
is computationally more efficient for a known ambient space and normal space
of the manifold, as is the case of our shape manifold.
The second requirement is achieved by the subspace representation resulting
from maximizing the variance of the corresponding curve development $dZ(t)$
with different choices of $V(X_{0})$. In theory, without knowing the initial
condition $V(X_{0})$ the Levi-Civita connection only determines the vector
field $V(X_{t})$ up to a group action within the fiber of the Frame bundle
(the total space of frames of the tangent space of a manifold) [7]. Among
different choices for $V(X_{t})$ along the fiber in the frame bundle, a
optimal frame is selected such that the previously discussed second
requirement is satisfied to make the resulted $Z{t}$ better representable in a
lower dimensional space. A
$\tilde{V}(X_{0})=\\{x_{0},e_{1},e_{2},\cdots,e_{dim(M)}\\}$ is selected by
carried out PCA of all the parallel transported tangents along $X_{t}$.let
$\\{e_{1},e_{2},\cdots,e_{L}\\}$ are the eigenvectors corresponding to the $L$
largest Eigenvalues in the PCA calculation. Then A subspace spanned by the
subset $\\{e_{1},e_{2},\cdots,e_{L}\\}$ is constructed to better represent the
variance of all the tangents along $X_{t}$ in a lower dimensional space.
To summarize, a curve in the shape manifold $M$ is represented as
$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{s})d\tilde{W}_{i}(s)$
(2)
where $\tilde{V}_{i}(X_{t})$ can be represented by $L$ curves in $R^{3}$ and
$\tilde{Z}_{i}(t)$ is represented by a curve in $R^{L}$. Our proposed
dimensional reduction achieves the following points,
1. 1.
a lower dimensional representation of curves on a manifold.
2. 2.
The moving frame representation can be generated to a stochastic
representation of random process on a manifold. Thus the propose method can be
well adapted to a generative modeling in a lower dimensional space.
3. 3.
the computation is linear in the dimension of $X_{t}$.
4. 4.
there exists a reconstruction from the lower dimensional representation to the
original curve on the shape manifold.
The rest of the paper is organized as following. Section 2 provide necessary
background knowledge of shape manifolds and the geometric calculation on it.
In Section 3 and 4, the dimension reduction framework is introduced, which
includes the curve embedding and the curve reconstruction.
## 2 Preliminary
In this section, we first introduce in Subsection 2.1 the shape manifold that
the dimension reduction is based on. Then in Subsection 2.2 a fundamental
moving frame representation of curves on manifold is briefly introduced.
### 2.1 Shape Manifold
According to [5], a planar shape is a simple and closed curve in
$\mathbb{R}^{2}$,
$\alpha(s):I\rightarrow\mathbb{R}^{2},$ (3)
where an arc-length parameterization is adopted. A shape is represented by a
_direction index function_ $\theta(t)$. With such a parameterization,
$\theta(s)$ may be associated to the shape by
$\frac{\partial\alpha}{\partial s}=e^{j\theta(s)}.$ (4)
The ambient space of the manifold of $\theta$ is an affine space based on
$\mathbb{L}^{2}$. Thus
$\theta\in A(\mathbb{L}^{2}).$ (5)
The restriction of a shape is that it must be a closed and simple curve, and
invariant over rigid Euclidean transformations. The shape manifold $M$ is
defined by a level function $\phi$ as
$\phi(\theta)=\left(\int_{0}^{2\pi}\theta
ds,\int_{0}^{2\pi}\cos(\theta)ds,\int_{0}^{2\pi}\sin(\theta)ds\right).$ (6)
$M=\phi^{-1}(\pi,0,0).$ (7)
One of the most important properties of $M$ is that the tangent space $TM$ is
well defined. Such a property not only simplifies the analysis, but also makes
the incremental computation possible,
$T_{\theta}M=\\{f\in\mathbb{L}^{2}|f~{}\bot~{}span\\{1,\cos(\theta),\sin(\theta)\\}\\}.$
(8)
In addition, an iterative projection is proposed in [5] to project the point
in ambient space back to the shape manifold $M$. The idea is that each time
$\theta$ is updated as $\theta+d\theta$, where $d\theta$ is orthogonal to the
level set $\phi^{-1}(\phi(\theta))$. The $d\theta$ is calculated as
$d\phi^{-1}((\pi,0,0)-\phi(\theta))$. For the detailed form of the Jacobian of
$d\phi$, one could refer to [5].
The problem of this manifold for our stochastic modeling is the infinite
dimension. The mapping of random process in [6] from a manifold to a flat
space is only defined on a finite dimensional manifold. Therefore, a Fourier
approximation of the shape manifold we discussed above is developed, such that
the dimension is reduced to a finite number.
### 2.2 Moving Frame Geometry
Let $X_{t}$ be a curve on a manifold $M$, then following the moving frame
representation [8], the tangent $dX_{t}$ of manifold valued curve $X_{t}$ may
be written as:
$dX_{t}=\sum_{i}V_{i}(X_{t})dZ_{t},$ (9)
where $\\{V_{i}(X_{t})\\}_{i=1,2,\cdots,dim(M)}$ is a frame of the tangent
space at $X_{t}$, which is denoted as $T_{X_{t}}M$. Consequently $dZ_{t}$ may
be understood as a linear coefficient of $dX_{t}$ under the representation of
$V(X_{t})$. Such moving frame representation is widely used in geometry
studies of curves. For example in the Frenet theorem [8] the geometry of curve
is uniquely identified by the curvature of $V(X_{t})$ up to a rigid Euclidean
transformation. Recently in computer graphics, a rotation minimizing moving
frame [17] are developed to avoid the singularity of original moving frame
used in Frenet theorem.
In this paper, the moving frame representation is utilized for the similar
purpose of representing a curve on a manifold. In contrast to the previous
designs of moving frame, in this paper we elaborate to propose a adaptive
moving frame to represent a high dimensional curve in a lower dimensional
space. Vector field $V(X(t))$ are developed as a sequence of parallel frames
along $X_{t}$. The parallelism is defined under a Levi-Civita connection. The
advantages of such innovation are briefly introduced in the following:
1. 1.
Once $V(X_{t})$ are parallel according to the curve development theory in [6],
$X_{t}$ can be represented uniquely as $(X_{0},V_{0},Z_{t})$. Such property
yields the invertibility of the proposed dimension reduction of $X_{t}$. In
other words, by the representative curve $Z_{t}$ in a lower dimensional space,
$X_{t}$ can be reconstructed when the initial conditions $(X_{0},V_{0})$ are
available.
2. 2.
Under Levi-Civita connection, the parallel frames are calculated according to
metric of the manifold $M$. Such property implies that the shape of $Z_{t}$
will reflect well the shape of $X_{t}$ on $M$. For example, if $X_{t}$ is a
geodesic curve, then in the proposed method $Z_{t}$ will be a straight line in
Euclidean space
3. 3.
The particular shape manifold [5] we adopted in this paper have a known
tangent and normal space in the ambient space which greatly simplified the
calculation of the moving frames under Levi-Civita connection.
The Levi-Civita connection may be viewed as the Christoffel Symbol $\Gamma$
that defines the directional derivative in $M$.
$D_{\partial x_{i}}\partial x_{j}=\sum_{k}\Gamma_{i,j}^{k}\partial x_{k}$ (10)
where $x_{i}$ is the coordinate function of an arbitrary point $m$ in $M$. The
tangents of $M$ can be written as linear combination of $\\{\partial
x_{i}\\}_{i=1,2,\cdots,dim(M)}$. For example at $X_{0}$,
$V_{i}(X_{0})=\sum_{j}a_{j}^{i}\partial x_{j}$ and
$dX_{0}=\sum_{j}b_{j}\partial x_{j}$. Thus according to Equation
$(\ref{eq.direc_direv})$, we can analytically calculate the derivative of
$V_{i}(X_{t})$ along $X_{t}$ in direction $dX_{t}$, which is denoted as
$D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))$. $V(X(t))$ is parallel
if $\forall i$,
$D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))=0.$ (11)
However in practical problems, it is usually impossible to chart the manifold.
In other words, the coordinate functions $x$ of $M$ are usually unknown. In
this paper, the shape manifold [5] we introduced in Section 2.1 have some nice
properties that allows us to calculate $D_{\frac{\partial X_{t}}{\partial
t}}(V_{i}(X_{t}))$ in terms of a projection of Euclidean calculus in the
ambient space onto the tangent space of the manifold $M$.
$D_{\frac{\partial X_{t}}{\partial
t}}(V_{i}(X_{t}))=\it{Proj}_{T_{X_{t}}M}\left(\frac{\partial\bar{V}_{i}(X_{t})}{\partial
t}\right),$ (12)
where $\bar{V}_{i}$ is the vector representing $V_{i}$ in the ambient space of
$M$. Since the ambient space is an affine space based on $\mathbb{L}^{2}$,
thus $\bar{V}_{i}$ is calculable as a real vector in $\mathbb{L}^{2}$. The
$\it{Proj}_{T_{X_{t}}M}$ is a mapping of vectors in $\mathbb{L}^{2}$ onto the
tangent space of $M$.
## 3 Dimension Reduction
The dynamics of some common human activities such as walking and running, are
relatively simpler in comparison to a complex activity like dancing. Thus it
is nature to speculate that the simple activity could be represented in a
lower dimensional subspace on the shape manifold. To learn such sub-manifold,
a dimension reduction is proposed for a curve on a shape manifold. As
described in the previous section, first the horizontal vector field
$V(X_{t})$ is defined by a Levi-Civita connection. Then Principle Component
Analysis (PCA) is carried out to compute the optimal vector field
$\tilde{V}(X_{t})=\\{x_{t},e_{1},e_{2},\cdots,e_{L}\\}$ such that all the
parallel transported tangents along $X_{t}$ could be represented in a lower
dimensional space.
The shape sequence on a manifold $M$ can be represented by a differential
equation as,
$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t),$ (13)
where, $V(X_{t})$ is selected to be the horizontal vector field on a manifold
and $Z_{t}$ is the curve development in a Euclidean space. Such a formulation
have the property according to [7, 6] that the resulting $Z_{t}$ is 1-1
corresponded to $X_{t}$ with the initial condition $V(X_{0}),X_{0}$.
The idea of the proposed dimension reduction is to reduce the dimension for a
manifold valued curve $X_{t}$ by learning a sequence of subspaces
$\tilde{V}(X_{t})$ on the shape manifold $M$, such that the resulting $dZ_{t}$
have a distribution that concentrate in a lower dimensional subspace.
A Levi-Civita connection is adapted on the shape manifold according to the
$L^{2}$ metric induce from the ambient space. According to the definition of
covariant derivative under Levi-Civita connection as in Section 2.2, the
horizontal vector field $V(X_{t})$ is the solution of the following
differential equation.
$\forall i,D_{\frac{\partial X_{t}}{\partial t}}(V_{i}(X_{t}))=0,$ (14)
with initial condition $(X_{0},V(X_{0}))$.
The subspace moving frames $\tilde{V}(X_{t})$ for dimension reduction is
constructed by select the optimal initial condition
$V(X_{0})=\tilde{V}(X_{0})$ . First parallel transport all the tangents
$\frac{\partial X_{t}}{t}$ to the tangent space at $X_{0}$. With the moving
frame formulation in Equation $(\ref{eq.mov_frame})$, the parallel
transportation results is a set $\tau$
$\tau=\\{\sum_{i=1}^{dim(M)}V_{i}(X_{t})dZ_{i}(t)\\}_{t=1,2,\cdots,N}$ (15)
According to [7], the set $\tau$ is invariant to the choice of $V(X_{0})$.
Thus in the first step an arbitrary initial condition is assigned.
Then Let the $\\{e_{1},e_{2},e_{3},\cdots,e_{L}\\}$ be the eigenvectors that
corresponding to the $L$ largest eigenvalues of vectors in $\tau$. The optimal
initial condition is constructed as
$\tilde{V}_{i}(X_{0})=e_{i}.$ (16)
The solution $\tilde{V}(X_{t})$ of Equation $(\ref{eq.pde})$ with initial
condition $(X_{0},\tilde{V}(X_{0}))$ is the moving frames we used to represent
the $X_{t}$ by the corresponding $\tilde{Z}_{t}$ in a lower dimensional space
$R^{L}$
In the following, we provide the solution to the differential equation
$(\ref{eq.pde})$. According to Equation $(\ref{eq.proj_deri})$ and normal
space expression of the shape manifold $M$ in Section 2.1, the above equation
can be solved as the following. Let $\\{B_{1}(t),B_{2}(t),B{3}(t)\\}$ be the
orthogonalization of $\\{1,cos(X_{t}),sin(X_{t})\\}$, which is the basis of
normal space of the shape manifold $M$. $\forall i$,
$\frac{\partial V_{i}(X_{t})}{\partial
t}=a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}).$ (17)
In the numerical calculation of the Euclidean derivatives
$\left(\frac{V_{i}(X_{t})}{\partial t}\right)$ are implemented as follows. For
a small enough $h>0$,
$\left(\frac{\partial V_{i}(X_{t})}{\partial
t}\right)=\frac{V_{i}(X_{t+h})-V_{i}(X_{t})}{h}.$ (18)
Substitute the Equation $(\ref{eq.direv_app})$ into the covariant derivative
Equation $(\ref{eq.cond1})$, $V_{i}(X_{t+h})$ could be written as,
$V_{i}(X_{t+h})=V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}).$
(19)
Since $V_{i}(X_{t+h})\in T_{X_{t+h}}M$, the parameter $a_{i}$ should satisfy
that,
$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{1}(t_{0}+h))>=0,$
$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{2}(t_{0}+h)>=0,$
and
$<V_{i}(X_{t})+a_{1}B_{1}(t_{0})+a_{2}B_{2}(t_{0})+a_{3}B_{3}(t_{0}),B_{3}(t_{0}+h)>=0.$
From the above three equations, we have,
$\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\ a_{3}\end{array}\right)=-A^{-1}V,$
(20)
where $A=$
$\left(\begin{array}[]{ccc}B_{1}(X_{t})B_{1}(X_{t+h})&B_{2}(X_{t})B_{1}(X_{t+h})&B_{3}(X_{t})B_{1}(X_{t+h})\\\
B_{1}(X_{t})B_{2}(X_{t+h})&B_{2}(X_{t})B_{2}(X_{t+h})&B_{3}(X_{t})B_{2}(X_{t+h})\\\
B_{1}(X_{t})B_{3}(X_{t+h})&B_{2}(X_{t})B_{3}(X_{t+h})&B_{3}(X_{t})B_{3}(X_{t+h})\\\
\end{array}\right).$
So the parameter $a_{i}$ could be solved in a linear fashion. Thus the shape
sequence can be represented as follows,
$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{t})d\tilde{Z}_{i}(t),$
(21)
where $d\tilde{Z}_{i}(s)=<\frac{\partial X_{t}}{\partial
t},\tilde{V}_{i}(X_{t})>$ is a curve development in $R^{L}$.
Figure 2 illustrates the $R^{3}$ embedding of the shape sequence in Figure 1.
The curvature of the embedded curve in $R^{3}$ in Figure 2 represent well the
change of original shape sequence along time.
Figure 1: The original shape sequence for activity: Running
Figure 2: The embedding of a shape sequence in Figure 1 into $R^{3}$
In the following, the additional different activities from database [16] is
considered, such as Jumping, Siding, Skipping and Waving. The dimension
reduction result in Figures 3,5,4,6, also well visualized the evolution on the
shape manifold $M$.
Figure 3: (a) The original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from the embedding curve in $R^{3}$.
Figure 4: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
the reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.
Figure 5: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.
Figure 6: (a) A original shape sequence; (b) The embedding into $R^{3}$; (c)
The reconstruction of the shape sequence from its corresponding embedding
curve in $R^{3}$.
## 4 Reconstruction from the Lower Dimensional Representation
As introduced in the previous sections, one of the claimed advantages of the
proposed dimension reduction is that it is possible to generate the original
shape sequence from the dimension reduction results. Such an invertible
property makes our lower dimensional shape process embedding a compelling way
to learn the generative model of the shape sequence.
Let $\tilde{Z}(t)$ be the embedded shape process in $R^{L}$. A approximation
of the original sequence on the shape manifold $M$ may be generated by the
following differential equation:
$X_{t}=X_{0}+\int_{0}^{t}\sum_{i=1}^{L}\tilde{V}_{i}(X_{0})d\tilde{Z}_{i}(t)$
(22)
where the horizontal vector field $\tilde{V}_{i}(X_{t}))$ is generated
uniquely by the differential equation 14 with the initial frame
$X_{0},\tilde{V}_{i}(X_{0})$.
Numerically Equation $(\ref{eq.gen1})$ is implemented as a difference
function,
$X_{k}=f(X_{k-1})+\sum_{i=1}^{L}\tilde{V}_{i}(X_{0})d\tilde{Z}_{i}(k-1))$ (23)
where $f$ is the iterative projection from the ambient space to the shape
manifold as in [5], which is also briefly introduced in Section 2.1.
Figure 7 illustrates the reconstruction result of the shape sequence of
activity: running. The reconstruction is from the embedding curve
$\tilde{Z}(t)\in R^{3}$ and the initial condition $X_{0}$,
$\tilde{V}_{i}(X_{0})$ to the shape manifold $M$.
Figure 7: The reconstruction result for original sequence in Figure 1 with
embedding result in Figure 2
The Figure 7 demonstrates that the reconstruction approximates well the
original shape sequence. More reconstruction results of the shape sequence of
other human activities are shown in Figures 3,5,4,6.
## 5 Conclusion
Dimension reduction of points on a high dimensional manifold is well studied
in the past decades. Only a few works, however, address the problem of
dimension reduction of curves on a manifold. In this paper a novel dimension
reduction technology is proposed for shape dynamics with the following
advantages that, as far as we know, no previous work has ever achieved:
1. 1.
The subspace is learned nonlinearly according to the geometry of the
underlying manifold.
2. 2.
The computation is linear in the size of data.
3. 3.
There exists an efficient reconstruction from the lower dimensional
representation to the original curve on a shape manifold.
4. 4.
The proposed dimension reduction technique provides both analytical and
practical foundations for generative modeling of shape dynamics.
This work also first apply the differential geometry tools such as moving
frame representation and Levi-Civita connection to dimension reduction of
curves on a manifold. It provides a new perspective on the dimension reduction
with a moving frame formulation to naturally characterize the nonlinear
features of a manifold valued dynamics in a linear subspace.
## References
* [1] I. K. Fodor. “A survey of dimension reduction techniques”, Technical Report UCRL-ID-148494, Lawrence Livermore National Laboratory, 2002.
* [2] Aouada, D., Krim, H. , “Squigraphs for Fine and Compact Modeling of 3-D Shapes”, Image Processing, IEEE Transactions on , vol.19, no.2, pp.306-321, Feb. 2010.
* [3] A.M. Baumberg, “Learning deformable models for tracking human motion”, Ph.D. dissertation, University of Leeds, 1995.
* [4] Ya Chang, Changbo Hu, Matthew Turk, “Manifold of Facial Expression,” amfg, pp.28, IEEE International Workshop on Analysis and Modeling of Faces and Gestures, 2003
* [5] E Klassen, A Srivastava, W Mio, SH Joshi, “Analysis of planar shapes using geodesic paths on shape spaces”,IEEE Trans. Pattern Analysis and Machine Intelligence, 2004.
* [6] Elton P. Hsu,“ stochastic analysis on manifold, Graduate studies in mathematics”, vol 38.
* [7] Shoshichi Kobayashi and katsumi Nomizu, “Foundations of differential geometry”, vol1, John Wiley& Sons, 1996\.
* [8] Richard L. Bishop, Richard J. Crittenden, “Geometry of Manifold, Academic Press”, 1964.
* [9] Joshua B. Tenenbaum, Vin de Silva and John C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction”, Science 22, pp. 2319-2323, Dec, 2000
* [10] Sam T. Roweis, Lawrence K. Saul, “Nonlinear Dimensionality Reduction by Locally Linear Embedding”, Science, vol. 22, pp. 2323-2326, Dec, 2000.
* [11] Somboon Hongeng, Ram Nevatia, Francois Bremond, “Video-based event recognition: activity representation and probabilistic recognition methods”, Computer Vision and Image Understanding, Volume 96, Issue 2, Special Issue on Event Detection in Video, November 2004, pp. 129-162.
* [12] Blake A., Curwen R., and Zisserman A., “A framework for spatio-temporal control in the tracking of visual contours”, International Journal of computer Msion, 1993.
* [13] Curwen R. and Blake A., “Dynamic contours: Realtime active splines”, In Blake A. and Yuille A., editors, Active Vision, chapter 3, pages 39-57. MIT Press, 1992.
* [14] Kass M., Witkin A., and Terzopoulos D. “Snakes: Active contour models”, In First International Conference on Computer Vision, pp. 259-268, 1987.
* [15] Terzopoulos D. and Szeliski R., “Tracking with kalman snakes”, In Blake A. and Yuille A., editors, Active Vision, chapter 1, pp. 3-20. MIT Press, 1992.
* [16] Moshe Blank, Lena Gorelick, Eli Shechtman, Michal Irani Ronen Basri, “Action as Space-Time Shapes”, IEEE ICCV, 2005.
* [17] W. Wang and B. Joe, Robust computation of the rotation minimizing frame for sweep surface modeling. Computer Aided Design 29 5 (1997), pp. 379 391
|
arxiv-papers
| 2011-07-29T20:06:59 |
2024-09-04T02:49:21.077777
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sheng Yi, Hamid Krim, Larry K. Norris",
"submitter": "Sheng Yi",
"url": "https://arxiv.org/abs/1108.0007"
}
|
1108.0122
|
On Negative Order KdV Equations
Zhijun Qiao111E-mail address: qiao@utpa.edu
Department of Mathematics, The University of Texas-Pan American, 1201 W
University Drive, Edinburg, TX 78539, USA
Engui Fan222 E-mail address: faneg@fudan.edu.cn
School of Mathematical Sciences, Institute of Mathematics and Key Laboratory
of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, P.R.
China
Abstract.
In this paper, we study negative order KdV (NKdV) equations and give their
Hamiltonian structures, Lax pairs, infinitely many conservation laws, and
explicit multi-soliton and multi-kink wave solutions thorough bilinear
Bäcklund transformations. The NKdV equations studied in the paper are
differential and can be derived from the first member in the negative order
KdV hierarchy. The NKdV equations are not only gauge-equivalent to the
Camassa-Holm equation through some hodograph transformations, but also closely
related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-
Hamiltonian structures and a Darboux transformation of the NKdV equations are
constructed with the aid of trace identity and their Lax pairs, respectively.
The 1- and 2- kink wave and soliton solutions are given in an explicit formula
through the Darboux transformation. The 1-kink wave solution is expressed in
the form of $tanh$ while the 1-bell soliton is in the form of $sech$, and both
forms are very standard. The collisions of 2-kink-wave and 2-bell-soliton
solutions, are analyzed in details, and this singular interaction is a big
difference from the regular KdV equation. Multi-dimensional binary Bell
polynomials are employed to find bilinear formulation and Bäcklund
transformations, which produce $N$-soliton solutions. A direct and unifying
scheme is proposed for explicitly building up quasi-periodic wave solutions of
the NKdV equations. Furthermore, the relations between quasi-periodic wave
solutions and soliton solutions are clearly described. Finally, we show the
quasi-periodic wave solution convergent to the soliton solution under some
limit conditions.
Keywords: Negative order KdV equations, bilinear Bäcklund transformation,
Darboux transformation, kink wave solution, soliton solution, quasi-periodic
solution.
Table of Contents
1\. Introduction
2\. Hamiltonian structures of the NKdV hierarchy
2.1. The NKdV hierarchy
2.2. Bi-Hamiltonian structures
3\. Relations to other important equations
3.1. Kupershmidt deformation
3.2. NKdV hierarchy with self-consistent sources
3.3. Reduction of NKdV equation
3.4. Ermakov-Pinney systems
4\. Darboux transformation of NKdV equations
4.1. Darboux transformation
4.2. Reduction of Darboux transformation
5\. Applications of Darboux transformation
5.1. Kink waves and interaction
5.2. Bell solitons and interaction
6\. Bilinearization of NKdV equations
6.1. Binary Bell polynomials
6.2. Bilinear formulation
6.3. $N$-soliton solutions
7\. Bilinear Bäcklund transformation of NKdV equations
7.1. Bilinear Bäcklund transformation
7.2. Inverse scattering formulation
8\. Darboux covariant Lax pair of NKdV equations
9\. Conservation laws of NKdV equations
10\. Quasi-periodic solutions of NKdV equations
10.1. Theta functions
10.2. Bilinear formulas
10.3. One-periodic wave solutions and long wave limit
10.4. Two-periodic wave solutions and long wave limit
10.5. Multi-periodic wave solutions
1\. Introduction
The Korteweg-de Vries (KdV) equation
$u_{t}+6uu_{x}+u_{xxx}=0$
was proposed by Korteweg and de Vries in fluid dynamics [39], starting from
the observation and subsequent experiments by Russell [77]. There are many
excellent sources for the highly interesting background and historical
development of the KdV equation, which brings it to the forefront of modern
mathematical physics. In 1967, Gardner, Greener, Kruskal and Miura found the
inverse scattering transformation method to solve the Cauchy problem of the
KdV equation with sufficiently decaying initial data [19]. Soon thereafter,
Lax explained the magical isospectral property of the time dependent family of
Schrodinger operators which is now called the Lax pair, and introduced the KdV
hierarchy through a recursive procedure [44]. In the same year a sequence of
infinitely many polynomial conservation laws were obtained with the help of
Miura’s transformation [54, 55].
There are tools to view the KdV equation as a completely integrable system by
Gardner, and Zakharov and Faddeev [20, 82]. The bilinear derivative method was
developed by Hirota to find $N$-soliton solutions of the KdV equation [29].
The KdV hierarchy was constructed by Lax [43] through a recursive approach,
and further studied by Gel’fand and Dikii [21]. On the base of the inverse
spectral theory and algebro-geometric methods, the inverse scattering method
was extended to periodic initial data by by Novikov, Dubrovin, Lax, Its,
Matveev et al [12, 37, 50, 58]. For more recent reviews on the KdV equation
one may refer to literature [2, 4, 5, 6, 9, 26, 47, 60, 78, 79].
All the work done in the above mentioned publications dealt with the positive
order KdV hierarchy, which includes the KdV equation as a special member.
However, there was only little work on the NKdV hierarchy. Verosky [81]
studied symmetries and negative powers of recursion operator and gave the
following negative order KdV equation (called the NKdV equation thereafter)
$None$ $\begin{array}[]{l}v_{t}=w_{x},\\\\[8.0pt]
w_{xxx}+4vw_{x}+2v_{x}w=0.\end{array}$
and Lou [48] presented additional symmetries based on the invertible recursion
operator of the KdV system and particularly provided the following NKdV
equation (called the NKdV-1 equation thereafter)
$None$ $v_{t}=2uu_{x},\ \
u_{xx}+vu=0,\Longleftrightarrow\left(\frac{u_{xx}}{u}\right)_{t}+2uu_{x}=0,$
which can be reduced from the NKdV equation (1.1) under the following
transformation
$None$ $w=u^{2},\ \ v=-\frac{u_{xx}}{u}.$
Moreover, the second part of NKdV-1 equation (1.2) is a linear Schrödinger
equation or Hill equation
$u_{xx}+vu=0.$
Fuchssteiner [18] pointed out the gauge-equivalent relation between the NKdV
equation (1.1) and the Camassa-Holm (CH) equation [7]
$m_{t}+m_{x}u+2mu_{x}=0,\ \ m=u-u_{xx}$
through some hodograph transformation, and later on Hone proposed the
associate CH equation, which is actually equivalent to the NKdV equation
(1.1), and gave soliton solutions through the KdV system [33]. Zhou [86]
generalized the Kupershmidt deformation and proposed a kind of the mixed KdV
hierarchy, which contains the NKdV equation (1.1) as a special case.
Very recently, Qiao and Li [62] gave a unifying formulation of the Lax
representations for both negative and positive order KdV hierarchies, and
furthermore studied all possible traveling wave solutions, including soliton,
kink wave, and periodic wave solutions, of the integrable NKdV-1 equation
(1.2) with the following Lax pair
$None$ $\begin{array}[]{l}L\psi\equiv\psi_{xx}+v\psi=\lambda\psi,\\\ \\\
\psi_{t}=\frac{1}{2}u^{2}\lambda^{-1}\psi_{x}-\frac{1}{2}uu_{x}\lambda^{-1}\psi.\end{array}$
The most interesting [62] is: the NKdV-1 equation has both soliton and kink
solutions, which is the first integrable example, within our knowledge, having
such a property in soliton theory.
Studying negative order integrable hierarchies plays an important role in the
theory of peaked soliton (peakon) and cusp soliton (cuspon). For instance, the
well-known CH peakon equation is actually produced through its negative order
hierarchy while its positive order hierarchy includes the remarkable Harry-Dym
type equation [63]. The Degasperis-Procesi (DP) peakon equation [10] can also
be generated through its negative order hierarchy [65]. Both the CH equation
and the DP equation are typical integrable peakon and cupson systems with
nonlinear quadratic terms [7, 11, 49, 63, 84]. Recently, some nonlinear cubic
integrable equations have also been found to have peakon and cupson solutions
[34, 59, 64, 66].
In this paper, we study the NKdV hierarchy, in particular, focus on the NKdV
equation (1.1) and the NKdV-1 equation (1.2). Actually, as per [48, 67], the
NKdV equation (1.1) can embrace other possible differential-integro forms
according to the kernel of operator
$K=\frac{1}{4}\partial_{x}^{3}+\frac{1}{2}(v\partial_{x}+\partial_{x}v)$. Here
we just list the NKdV-1 equation (1.2) as it is differential and also
equivalent to a nonlinear quartic integrable system:
$uu_{xxt}-u_{xx}u_{t}-2u^{3}u_{x}=0.$
The purpose of this paper is to investigate integrable properties, $N$-soliton
and $N$-kink solutions of the NKdV equation (1.1) and NKdV-1 equation (1.2).
In section 2, the trace identity technique is employed to construct the bi-
Hamiltonian structures of the NKdV hierarchy. In section 3, we show that the
NKdV equation (1.1) is related to the Kupershmidt deformation and the Ermakov-
Pinney systems, and is also able to reduced to the NKdV-1 equation (1.2) under
a transformation. In section 4, a Darboux transformation of the NKdV equation
(1.1) is provided with the help of its Lax pairs. In section 5, as a direct
application of the Darboux transformation, the kink-wave and bell soliton
solutions are explicitly given, and the collision of two soliton solutions is
analyzed in detail through two-solitons. In section 6, an extra auxiliary
variable is introduced to bilinearize the NKdV equation (1.1) through binary
Bell polynomials. In section 7, the bilinear Bäcklund transformations are
obtained and Lax pairs are also recovered. In section 8, we will give a kind
of Darboux covariant Lax pair, and in section 9, infinitely many conservation
laws of the NKdV equation (1.1) are presented through its Lax equation and a
generalized Miura transformation. All conserved densities and fluxes are
recursively given in an explicit formula. In sections 10, a direct and
unifying scheme is proposed for building up quasi-periodic wave solutions of
the NKdV equation (1.1) in an explicit formula. Furthermore, the relations
between quasi-periodic wave solutions and soliton solutions are clearly
described. Finally, we show the quasi-periodic wave solution convergent to the
soliton solution under some limit conditions.
2\. Hamiltonian structures of the NKdV hierarchy
To find the Hamiltonian structures of the NKdV hierarchy, let us re-derive the
NKdV hierarchy in matrix form.
2.1. The NKdV hierarchy
Consider the Schrödinger-KdV spectral problem
$None$ $\displaystyle\psi_{xx}+v\psi=\lambda\psi,$
where $\lambda$ is an eigenvalue, $\psi$ is the eigenfunction corresponding to
the eigenvalue $\lambda$, and $v$ is a potential function.
Let $\varphi_{1}=\psi,\ \ \varphi_{2}=\psi_{x}$, then the spectral problem
(2.1) becomes
$None$
$\displaystyle\varphi_{x}=U\varphi=\left(\begin{matrix}0&1\cr\lambda-v&0\end{matrix}\right)\varphi,$
where $\varphi=(\varphi_{1},\varphi_{2})^{T}$ is a two-dimensional vector of
eigenfunctions.
The Gateaux derivative of spectral operator $U$ in direction $\xi$ at point
$v$ is
$None$
$U^{\prime}[\xi]=\frac{d}{d\varepsilon}U(v+\varepsilon\xi)|_{\varepsilon=0}=\left(\begin{matrix}0&0\cr-\xi&0\end{matrix}\right),$
which is injective and linear with respect to the variable $\xi$.
The Lenard recursive sequence $\\{G_{m}\\}$ of the spectral problem (2.1) is
defined by
$None$ $\displaystyle G_{-1}\in KerK=\\{G|KG=0\\},\ \ G_{0}\in
KerJ=\\{G|JG=0\\}$ $\displaystyle KG_{m-1}=JG_{m},\ \ m=0,-1,-2\cdots,$
which directly produces the NKdV hierarchy:
$None$ $v_{t}=KG_{m-1}=JG_{m},\ \ m=-1,-2\cdots$
where
$None$ $\displaystyle
K=\frac{1}{4}\partial_{x}^{3}+\frac{1}{2}(v\partial_{x}+\partial_{x}v),\ \ \
J=\partial_{x},$
and $K$ is exactly a recursion operator of the well-known KdV hierarchy
$v_{t}=K^{n}v_{x},\ \ n=0,1,2,\cdots.$
The first equation ($m=0$) in the NKdV hierarchy (2.5) is trivial equation
$v_{t}=JG_{0}=0,\ \ JG_{0}=KG_{-1}=0.$
The second equation ($m=-1$) in the NKdV hierarchy (2.5) takes
$v_{t}=G_{-1,x},\ \ \ KG_{-1}=0,$
which is exactly the NKdV equation (1.1) by replacing $G_{-1}=w$.
In a similar way to the paper [62], we construct zero curvature representation
for NKdV hierarchy.
Proposition 1. Let $U$ be the spectral matrix defined in (2.2), then for an
arbitrarily smooth function $G\in C^{\infty}(\mathbb{R})$, the following
operator equation
$None$ $\displaystyle V_{x}-[U,V]=U^{\prime}[KG]-\lambda U^{\prime}[JG]$
admits a matrix solution
$V=V(G)=\left(\begin{matrix}\displaystyle{-\frac{1}{4}G_{x}}&\displaystyle{\frac{1}{2}G}\cr\cr\displaystyle{-\frac{1}{4}G_{xx}-\frac{1}{2}vG+\frac{1}{2}\lambda
G}&\displaystyle{\frac{1}{4}G_{x}}\end{matrix}\right)\lambda^{-1},$
which is a linear function with respect to $G$, and Gateaux derivative is
defined by (2.3).
Theorem 1. Suppose that $\\{G_{j},\ \ j=-1,-2,\cdots\\}$ is the first Lenard
sequence defined by (2.4), and $V_{j}=V(G_{j})$ is a corresponding solution to
the operator equation (2.7) for $G=G_{j}$. With $V_{j}$ being its
coefficients, a $m$th matrix polynomial in $\lambda$ is constructed as follows
$W_{m}=\sum_{j=1}^{m}V_{j}\lambda^{-m+j}.$
Then we conclude that the NKdV hierarchy (2.5) admits zero curvature
representation
$U_{t}-W_{m,x}+[U,W_{m}]=0,$
which is equivalent to
$None$
$\displaystyle\varphi_{x}=U\varphi=\left(\begin{matrix}0&1\cr\displaystyle{\lambda-v}&0\end{matrix}\right)\varphi,$
$\displaystyle\varphi_{t}=W_{m}\varphi=\sum_{j=1}^{m}\left(\begin{matrix}-{\frac{1}{4}G_{j,x}}&\frac{1}{2}G_{j}\cr\cr{-\frac{1}{4}G_{j,xx}-\frac{1}{2}vG_{j}+\frac{1}{2}\lambda
G_{j}}&{\frac{1}{4}G_{j,x}}\end{matrix}\right)\lambda^{-m+j-1}\varphi.$
This theorem actually provides an unified formula of the Lax pairs for the
whole NKdV hierarchy (2.5).
According to theorem 1, the NKdV equation (1.1) admits Lax pair with parameter
$\lambda$
$\begin{array}[]{l}L\psi\equiv\psi_{xx}+v\psi=\lambda\psi,\\\ \\\
\psi_{t}=\frac{1}{2}w\lambda^{-1}\psi_{x}-\frac{1}{4}w_{x}\lambda^{-1}\psi,\end{array}$
or equivalently,
$None$
$\begin{array}[]{l}L\psi=(\partial_{x}^{2}+v)\psi=\lambda\psi,\\\\[8.0pt]
M\psi=(4\partial_{x}^{2}\partial_{t}+4v\partial_{t}+2w\partial_{x}+3w_{x})\psi=0.\end{array}$
The NKdV equation (1.1) also possesses Lax pair without parameter
$None$ $\begin{array}[]{l}L\psi=(\partial_{x}^{2}+v)\psi=0,\\\\[8.0pt]
M\psi=(4\partial_{x}^{2}\partial_{t}+4v\partial_{t}+2w\partial_{x}+3w_{x})\psi=0.\end{array}$
Especially, taking the constraint $v=-{u_{xx}}/{u}$ and $w=u^{2}\in KerK$, we
then further get the NKdV equation (1.2) and its Lax parir (1.4).
2.2. Hamiltonian structures
Proposition 2. [78] For the spectral problem (2.2), assume that $V$ is a
solution to the following stationary zero curvature equation with the given
homogeneous rank
$None$ $V_{x}=[U,V].$
Then there exists a constant $\beta$, such that
$None$ $\frac{\delta}{\delta v}\left\langle V,\frac{\partial
U}{\partial\lambda}\right\rangle=\left(\lambda^{-\beta}\frac{\partial}{\partial\lambda}\lambda^{\beta}\right)\left\langle
V,\frac{\partial U}{\partial v}\right\rangle,$
holds, where $\langle\cdot,\cdot\rangle$ stands for the trace of the product
of two matrices.
Let $\\{G_{m},\ \ m=-1,-2\cdots\\}$ be the negative order Lenard sequence
recursively given through (2.4) and
$None$ $G_{\lambda}=\sum_{m=-\infty}^{-1}G_{m}\lambda^{-m},$
be a series with respect to $\lambda$. Assume that
$V_{\lambda}=V(G_{\lambda})$ is the matrix solution for the operator equation
(2.9) corresponding to $G=G_{\lambda}$. So, $V_{\lambda}$ can be written as
$V_{\lambda}=\sum_{m=-\infty}^{-1}V_{m}\lambda^{-m}.$
Then, we have the following proposition.
Proposition 3. $V_{\lambda}$ satisfies the following Lax form
$V_{\lambda,x}=[U,V_{\lambda}].$
Proof. By (2.4), we have
$\displaystyle(K-\lambda
J)G_{\lambda}=\sum_{m=-\infty}^{-1}KG_{m}\lambda^{-m}-\sum_{m=-\infty}^{-1}JG_{m}\lambda^{-m+1}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=KG_{-1}\lambda^{-1}+\sum_{m=-\infty}^{-1}(KG_{m-1}-JG_{m})\lambda^{-m}=0.$
Therefore, Proposition 1 implies
$V_{\lambda,x}-[U,V_{\lambda}]=U^{\prime}[KG_{\lambda}]-\lambda
U^{\prime}[JG_{\lambda}]=U^{\prime}[KG_{\lambda}-\lambda JG_{\lambda}]=0.$
$\square$
Next, we discuss the Hamiltonian structures of the hierarchy (2.5). It is
crucial to find infinitely many conserved densities.
Theorem 2.
1. (1)
The hierarchy (2.5) possesses the bi-Hamiltonian structures
$None$ $v_{t}=K\frac{\delta H_{m-1}}{\delta v}=J\frac{\delta H_{m}}{\delta
v},\ \ m=-1,-2\cdots,$
where the Hamiltonian functions $H_{m}$ are implicitly given through the
following formulas
$None$ $\ H_{-1}=G_{-1}\in KerK,\ \ H_{m}=\frac{G_{m}}{m},\ \ \
m=-1,-2\cdots.$
2. (2)
The hierarchy (2.5) is integrable in the Liouville sense.
3. (3)
The Hamiltonian functions $\\{H_{m}\\}$ are conserved densities of the whole
hierarchy (2.5) and therefore they are in involution in pairs.
Proof. A direction calculation leads to
$\left\langle V_{\lambda},\frac{\partial
U}{\partial\lambda}\right\rangle=\frac{1}{2}G_{\lambda},\ \ \ \left\langle
V_{\lambda},\frac{\partial U}{\partial
v}\right\rangle=-\frac{1}{2}G_{\lambda}.$
By using the trace identity (2.12) and the expansion (2.13), we obtain
$None$ $\displaystyle\frac{\delta}{\delta
v}\left(\sum_{m=-\infty}^{-1}G_{m}\lambda^{-m}\right)=\sum_{m=-\infty}^{-1}(m-1-\beta)G_{m-1}\lambda^{-m}+(-1-\beta)G_{-1},$
$\displaystyle\ m=-1,-2\cdots.$
If taking $G_{-1}\not=0$, form (2.16) we find $\beta=-1$ and
$None$ $\frac{\delta H_{m}}{\delta v}=G_{m-1},\ \ \ m=-1,-2\cdots,$
where $H_{m}$ are given by (2.15). Substituting (2.17) into (2.5) yields the
bi-Hamiltonian structures (2.14).
Next, we consider infinitely many conserved densities to guarantee
integrability of the hierarchy (2.16). Since $J$ and $K$ are skew-symmetric
operators, we infer that
$\mathcal{L}^{*}J=(J^{-1}K)^{*}J=-K^{*}=K=J\mathcal{L},$
which implies
$\begin{array}[]{l}\displaystyle{\\{H_{n},H_{m}\\}=\left(\frac{\delta
H_{n}}{\delta v},J\frac{\delta H_{m}}{\delta
v}\right)=(\mathcal{L}^{n}G_{-1},J\mathcal{L}^{m}G_{-1})=(\mathcal{L}^{n}G_{-1},\mathcal{L}^{*}J\mathcal{L}^{m-1}G_{-1})}\\\
\displaystyle{=(\mathcal{L}^{n+1}G_{-1},J\mathcal{L}^{m-1}G_{0})=\\{H_{n+1},H_{m-1}\\}},\
\ m,n\leq-1.\end{array}$
Repeating the above argument gives
$None$ $\\{H_{n},H_{m}\\}=\\{H_{m},H_{n}\\}=\\{H_{m+n},H_{-1}\\}.$
On the other hand, we find
$None$
$\\{H_{m},H_{n}\\}=(\mathcal{L}^{m}G_{-1},J\mathcal{L}^{n}G_{-1})=(J^{*}\mathcal{L}^{m}G_{-1},\mathcal{L}^{n}G_{-1})=-\\{H_{n},H_{m}\\}.$
Then combining (2.18) with (2.19) leads to
$\\{H_{m},H_{n}\\}=0,$
which implies that $\\{H_{m}\\}$ are in involution, and therefore the
hierarchy (2.14) are integrable in Liouville sense.
Especially, under the constraint (1.3), we obtain bi-Hamilton structures of
the NKdV equation (1.2)
$v_{t}=K\frac{\delta H_{-1}}{\delta v}=J\frac{\delta H_{0}}{\delta u},$
where two Hamiltonian functions are given by
$H_{0}=\frac{1}{3}u^{3},\ \ \ \ H_{-1}=-u^{2},$
which can also be written in a conserved density form in the sense of
equivalence class
$H_{0}\sim-\frac{1}{3}\int u^{3}dx,\ \ \ \ H_{-1}\sim-\int u^{2}dx.$
3\. Relations to other important equations
3.1. Kupershmidt deformation
Recently a class of new integrable systems, known as the Kupershmidt
deformation of soliton equations, have attracted much attention. This topic
starts from Kupershmidt, Karasu-Kalkani’ work [28, 38, 40].
For the Lenard operator pair (2.6), we define Lenard gradients recursively by
$KG_{j}=JG_{j+1},\ KG_{-1}=JG_{0}=0,\ \ j=0,\pm 1,\pm 2,\cdots,$
then KdV hierarchy is
$None$ $v_{t}=KG_{m-1}=JG_{m},\ \ m=0,\pm 1,\pm 2,\cdots$
which contains both the NKdV hierarchy and the positive order KdV hierarchy.
The first equation ($m=0$) in the KdV hierarchy (3.1) is trivial system
$None$ $v_{t}=JG_{0}=0,\ \ KG_{-1}=JG_{0}=0,$
which can be regarded as is a “sharp threshold” equation of the NKdV hierarchy
and positive order KdV hierarchy.
A Kupershmidt nonholonomic deformation of the hierarchy (3.1) takes
$None$ $\begin{array}[]{l}v_{t}=JG_{m}+Jw,\ \ m=0,\pm 1,\pm
2,\cdots,\\\\[8.0pt] Kw=0,\end{array}$
where two operators $K$ and $J$ are given by (1.4). Then the first flow
($m=0$) of the hierarchy (3.3) is exactly the NKdV equation (1.1)
$\begin{array}[]{l}v_{t}=w_{x},\\\\[8.0pt]
w_{xxx}+4vw_{x}+2v_{x}w=0,\end{array}$
which may be regarded as a Kupershmidt nonholonomic deformation of the
threshold equation (3.2)
3.2. NKdV hierarchy with self-consistent sources
Soliton equations with self-consistent sources have important physical
applications, for example, the KdV equation with self-consistent source
describes the interaction of long and short capillary-gravity waves [45, 51,
52, 53].
For the $N$ distinct $\lambda_{j}$ of the spectral problem (2.1), the
functional gradient of $\lambda_{j}$ with respect to $v$ is
$\frac{\delta\lambda_{j}}{\delta v}=\psi_{j}^{2}.$
Here we define the whole KdV hierarchy with self-consistent sources as follows
$None$ $\begin{array}[]{l}\displaystyle{v_{t}=JG_{m}+\alpha
J\frac{\delta\lambda}{\delta v}=JG_{m}+\alpha
J\sum_{j=1}^{N}\psi_{j}^{2},}\\\\[8.0pt]
\psi_{j,xx}+(v+\lambda_{j})\psi_{j}=0,\\\\[8.0pt] m=0,\pm 1,\pm 2,\cdots;\
j=1,\cdots,N.\end{array}$
Taking $m=1$ in the hierarchy (3.4) leads to the KdV equation with self-
consistent sources
$\begin{array}[]{l}\displaystyle{v_{t}=\frac{1}{4}(v_{xxx}+6vv_{x})+\alpha\partial_{x}\sum_{j=1}^{N}\psi_{j}^{2},}\\\\[8.0pt]
\psi_{j,xx}+(v+\lambda_{j})\psi_{j}=0,\ j=1,\cdots,N,\end{array}$
while choosing $m=-1$ in the hierarchy (3.4) gives the NKdV equation with
self-consistent sources
$\begin{array}[]{l}\displaystyle{v_{t}=w_{x}+\alpha\partial_{x}\sum_{j=1}^{N}\psi_{j}^{2},}\\\\[8.0pt]
w_{xxx}+4vw_{x}+2v_{x}w=0,\\\\[8.0pt] \psi_{j,xx}+(v+\lambda_{j})\psi_{j}=0,\
j=1,\cdots,N.\end{array}$
Obviously, taking $N=1,\ m=0,\ \alpha=1,\ v\rightarrow v+\lambda_{1}$ in the
hierarchy (3.4), then we get the NKdV equation (1.2)
$\begin{array}[]{l}v_{t}=(\psi_{1}^{2})_{x},\ \
\psi_{1,xx}+v\psi_{1}=0,\end{array}$
which may be regarded as the threshold equation (3.2) with self-consistent
sources.
3.3. Reduction of the NKdV equation (1.1)
Theorem 3. $(u,v)$ is a solution of NKdV-1 equation (1.2) if and only if
$(w,v)$ with $w=u^{2}$ is a solution of NKdV equation (1.1) under the
transformation
$None$ $u_{xx}+vu=0,$
which is actually a linear Schrödinger equation or Hill equation.
Proof. Let
$None$ $w=u^{2},$
then by (1.1), we have
$v_{t}=w_{x}=2uu_{x},$
which is the first equation of (1.2). By (3.6), the second equation of (1.1)
leads to
$3u_{x}(u_{xx}+vu)+u(u_{xx}+vu)_{x}=0,$
or equivalently,
$None$ $[u^{3}(u_{xx}+vu)]_{x}=0,$
Apparently, according to (3.7), if $(u,v)$ is a solution of the NKdV-1
equation (1.2), then $(w,v)$ is a solution of the NKdV equation (1.1) where
$w=u^{2}$. Reversely, if $(w,v)$ is a solution of the NKdV equation (1.1),
then $(u,v)$ is also a solution of the NKdV-1 equation (1.2) under the
transformation (3.5).
For a given function $\phi$, let us define the following Baker-Akhiezer
function
$None$ $u=\exp\left(\int_{0}^{x}\phi dx\right),$
then (3.2) yields the following Riccati equation
$None$ $\phi_{x}+\phi^{2}+v=0.$
So, we have
Theorem 4. $(u,v)$ is a solution of the NKdV-1 equation (1.2) if and only if
$(w,v)$ is a solution of the NKdV equation (1.1) as $\phi$ is a solution of
the Riccati equation (3.9) while $u$ is the Baker-Akhiezer function (3.8) and
$w=u^{2}$.
3.4. Ermakov-Pinney equation
The Ermakov-Ray-Reid systems
$\displaystyle\psi_{xx}+\omega^{2}(x)\psi=\frac{1}{\psi^{2}\phi}F(\frac{\phi}{\psi}),$
$\displaystyle\phi_{xx}+\omega^{2}(x)\phi=\frac{1}{\psi\phi^{2}}G(\frac{\psi}{\phi}),$
were originally introduced by Ermakov [13, 71]. Due to their nice mathematical
properties of Ermakov systems admitting a novel integral of motion together
with a concomitant nonlinear superposition principle and extensively physical
applications, there has been numerous an extensive literature devoting to the
analysis of the Ermakov systems [3, 61, 72, 73, 74, 75]. The most simple case
is equation
$\psi_{xx}+\omega^{2}(x)\psi=\frac{c}{\psi^{3}},$
which is called the Ermakov-Pinney equation. The Ermakov-Pinney equation is a
quite famous example of a nonlinear ordinary differential equation. Such an
equation (and generalizations thereof) have been shown to be relevant to a
number of physical contexts including quantum cosmology, quantum field theory,
nonlinear elasticity and nonlinear optics [17, 76, 80]. A recent account of
some of its properties along with applications in cosmological settings can be
found in Ref. [70].
Proposition 4. Suppose that $(w,v)$ is a solution of the NKdV equation (1.1).
Let
$w=p_{t}=\psi^{2},\ v=p_{x},$
then $\psi$ satisfies a Ermakov-Pinney equation
$None$ $\psi_{xx}+v\psi=\frac{\mu}{\psi^{3}},$
where $\mu$ is an integration constant.
Especially, if $(u,v)$ is the solution of the NKdV-1 equation (1.2), let
$None$ $u=\phi\exp\left(i\int\mu\phi^{-2}dx\right),$
then $\phi$ satisfies the Ermakov-Pinney equation
$None$ $\phi_{xx}+v\phi=\frac{\mu}{\phi^{3}}.$
Proof. Substituting transformation $w=\psi^{2}$ into the second equation of
the NKdV equation (1.1) yields
$\begin{array}[]{l}w_{xxx}+4vw_{x}+2v_{x}w=2\psi(\psi_{xx}+v\psi)_{x}+6\psi_{x}(\psi_{xx}+v\psi)\\\\[8.0pt]
=\displaystyle{\frac{2}{\psi^{2}}}[(\psi_{xx}+v\psi)\psi^{3}]_{x}=0,\end{array}$
which leads to (3.9).
Substituting transformation (3.10) into the second equation of the NKdV
equation (1.2) yields
$\displaystyle
u_{x}+vu=(\phi_{xx}+v\phi-\frac{\mu}{\phi^{3}})\exp\left(i\int\mu\phi^{-2}dx\right)=0,$
which implies (3.11).
Proposition 5. For given function $v$, let $\psi_{1},\psi_{2}$ are two
solutions of linear Schrödinger equation
$None$ $u_{xx}+vu=0,$
then equation
$None$ $w_{xxx}+4w_{x}v+2v_{x}w=0$
admits a general solution
$None$ $w=a\psi_{1}^{2}+2b\psi_{1}\psi_{2}+c\psi_{2}^{2},$
where
$ac-b^{2}=\frac{\mu}{2W},\ \ W=\psi_{1}\psi_{2,x}-\psi_{1,x}\psi_{2}.$
Proof. Let $w=\psi^{2}$, by proposition 4, then $\psi$ satisfies a Ermakov-
Pinney equation (3.9). It is easy to check that if $\psi_{1}$ and $\psi_{2}$
are two solutions of equation (3.14), then
$\psi=\sqrt{a\psi_{1}^{2}+2b\psi_{1}\psi_{2}+c\psi_{2}^{2}}$
is a solution of equation (3.9). So (3.14) is a general solution of the
equation (3.13).
4\. Darboux transformation of NKdV equations
In this section, we shall construct a Darboux transformation for general NKdV
equation (1.1), and then reduce it to the NKdV-1 equation (1.2).
4.1. Darboux transformation
A Darboux transformation is actually a special gauge transformation
$None$ $\tilde{\psi}=T\psi$
of solutions of the Lax pair (2.9), here $T$ is a differential operator (For
the Lax pair (2.10), the Darboux transformation with $\lambda=0$ can be
obtained). It requires that $\tilde{\psi}$ also satisfies the same Lax pair
(2.9) with some $\tilde{L}$ and $\tilde{M}$, i. e.
$None$ $\begin{array}[]{l}\tilde{L}\tilde{\psi}=\lambda\tilde{\psi},\ \ \ \
\tilde{L}=TLT^{-1},\\\\[8.0pt] \tilde{M}\tilde{\psi}=0,\ \ \ \
\tilde{M}=TMT^{-1}\end{array}$
Apparently, we have
$[\tilde{L},\tilde{M}]=T[L,M]T^{-1},$
which implies that $\tilde{L}$ and $\tilde{M}$ are required to have the same
forms as $L$ and $M$, respectively, in order to make system (2.9) invariant
under the gauge transformation (3.4). At the same time the old potentials $u$
and $v$ in $L$, $M$ will be mapped into new potentials $\tilde{u}$ and
$\tilde{v}$ in $\tilde{L}$, $\tilde{M}$. This process can be done continually
and usually it may yield a series of multi-soliton solutions.
Let us now set up a Darboux transformation for the system (2.9). Let
$\psi_{0}=\psi_{0}(x,t)$ be a basic solution of Lax pair (2.9) for
$\lambda_{0}$, and use it to define the following gauge transformation
$None$ $\tilde{\psi}=T\psi,$
where
$None$ $T=\partial_{x}-\sigma,\ \ \sigma=\partial_{x}\ln\psi_{0}.$
From (2.9) and (4.4), one can see that $\sigma$ satisfies
$None$ $\sigma_{x}+\sigma^{2}+v-\lambda=0$ $None$
$4\sigma_{xxt}+12\sigma_{x}\sigma_{t}+4v\sigma_{t}+2w\sigma_{x}+6\sigma\sigma_{xt}+3w_{xx}=0.$
Proposition 6. The operator $\tilde{L}$ determined by (4.2) has the same form
as $L$, that is,
$\tilde{L}=\partial_{x}^{2}+\tilde{v},$
where the transformation between $v$ and $\tilde{v}$ is given by
$None$ $\tilde{v}=v+2\sigma_{x}.$
The transformation: $(\psi,v)\rightarrow(\tilde{\psi},\tilde{v})$ is called a
Darboux transformation of the first spectral problem of Lax pair (2.9).
Proof. According to (4.2), we just prove
$\tilde{L}T=TL,$
that is,
$(\partial_{x}^{2}+\tilde{v})(\partial_{x}-\sigma)=(\partial_{x}-\sigma)(\partial_{x}^{2}+v),$
which is true through (4.5) and (4.7).
Proposition 7. Under the transformation (4.3), the operator $\tilde{M}$
determined by (4.2) has the same form as $M$, that is,
$None$
$\tilde{M}=4\partial_{x}^{2}\partial_{t}+4\tilde{v}\partial_{t}-2\tilde{w}\partial_{x}-3\tilde{w}_{x},$
where the transformations between $w$, $v$ and $\tilde{w}$, $\tilde{v}$ are
given by
$None$ $\tilde{w}=w+2\sigma_{t},\ \ \ \tilde{v}=v+2\sigma_{x}.$
The transformation: $(\psi,w,v)\rightarrow(\tilde{\psi},\tilde{w},\tilde{v})$
is Darboux transformation of the second spectral problem of Lax pair (2.9).
Proof. To see that $\tilde{M}$ has the form (4.8) same as $M$, we just prove
$None$ $\tilde{M}T=TM,$
where
$None$
$\tilde{M}=4\partial_{x}^{2}\partial_{t}+f\partial_{t}+g\partial_{x}+h,$
with three functions $f,g$, and $h$ to be determined. Substituting
$\tilde{M},\ M,\ L$ into (4.10) and comparing the coefficients of all distinct
operators lead to:
coefficient of operator $\partial_{x}\partial_{t}$
$f=4v+8\sigma_{x}=4\tilde{v},$
which holds by using (4.9).
coefficient of operator $\partial_{x}^{2}$
$g=2w+4\sigma_{t}=2\tilde{w},$
which implies from (4.9).
coefficient of operator $\partial_{x}$
$\begin{array}[]{l}h=8\sigma_{xt}+5w_{x}-2\sigma
w+g\sigma=6\sigma_{xt}+3w_{x}+2(\sigma_{x}+\sigma^{2}+v)_{t}\\\\[4.0pt]
=6\sigma_{xt}+3w_{x}=3\tilde{w}_{x},\end{array}$
here we have used equation (4.5) and (4.9).
coefficient of operator $\partial_{t}$
$-4\sigma_{xx}-f\sigma=4v_{x}-4v\sigma,$
that is,
$\sigma_{xx}+2\sigma\sigma_{x}+v_{x}=0.$
which holds by using (4.5).
coefficient of non-operator:
$4\sigma_{xxt}+f\sigma_{t}+g\sigma_{x}+\sigma h+3w_{xx}-3\sigma w_{x}=0,$
that is,
$4\sigma_{xxt}+12\sigma_{x}\sigma_{t}+4v\sigma_{t}+2w\sigma_{x}+6\sigma\sigma_{xt}+3w_{xx}=0,$
which is the equation (4.6). We complete the proof. $\square$
Propositions 4 and 5 tell us that the transformations (4.3) and (4.9) send the
Lax pair (2.9) to another Lax pair (4.2) in the same type. Therefore, both of
the Lax pairs lead to the same NKdV equation (1.1). So, we call the
transformation $(\psi,w,v)\rightarrow(\tilde{\psi},\tilde{w},\tilde{v})$ a
Darboux transformation of the NKdV equation (1.1). In summary, we arrive at
the following theorem.
Theorem 5. A solution $w,\ v$ of the NKdV equation (1.1) is mapped into its
new solution $\tilde{w},\ \tilde{v}$ under the Darboux transformations (4.3)
and (4.9).
4.2. Reduction of Darboux transformations
To get Darboux transformations for the NKdV-1 equation (1.2), we consider two
reductions of the Darboux transformations (4.3) and (4.9).
Corollary 1. Let $\lambda=k^{2}>0$, then under the constraints $w=u^{2},\
v=-u_{xx}/u$, the Darboux transformations (4.3) and (4.9) are reduced to a
Darboux transformation of the NKdV-1 equation (1.2)
$(\psi,v,u)\rightarrow(\tilde{\psi},\tilde{v},\tilde{u})$, where
$None$ $\begin{array}[]{l}\tilde{\psi}=T\psi,\ \ \tilde{v}=v+2\sigma_{x},\ \
\tilde{u}=k^{-1}(u_{x}-\sigma u)=k^{-1}Tu.\end{array}$
Proof. For $\lambda>0$, suppose that $(v,u)$ is a solution of the NKdV-1
equation and $\psi$ is an eigenfunction of the Lax pair (1.4), then we have
$\lambda^{-1}(u\psi_{x}-u_{x}\psi)=\partial_{x}^{-1}(u\psi).$
Therefore, the Lax pair (1.4) can be rewritten as
$None$ $\displaystyle\psi_{xx}+v\psi=\lambda\psi,$
$\displaystyle\psi_{t}=\frac{1}{2}u\lambda^{-1}(u\psi_{x}-u_{x}\psi)=\frac{1}{2}u\partial_{x}^{-1}(u\psi)=N(u,\lambda)\psi,$
where $N=N(u,\lambda)=\frac{1}{2}u\partial_{x}^{-1}u$.
According to Proposition 6, the spectral problem of Lax pair (4.12) is
covariant under the transformation (4.13), that is,
$\tilde{\psi}_{xx}+\tilde{v}\tilde{\psi}=\lambda\tilde{\psi}.$
So, we only need to prove
$None$ $\tilde{\psi}_{t}=N(\tilde{u},\lambda)\tilde{\psi}$
Substituting (4.13) into the left hand side of (4.14) yields
$None$
$\displaystyle\tilde{\psi}_{t}=(\psi_{t})_{x}-(\sigma\psi)_{t}=(N\psi)_{x}-\sigma
N\psi-(\psi_{0}^{-1}N\psi_{0})_{x}\psi,$
$\displaystyle=\frac{1}{2}[(u_{x}-\sigma
u)\partial_{x}^{-1}(u\psi)-\psi_{0}^{-1}\psi(u_{x}-\sigma
u)\partial_{x}^{-1}(u\psi_{0})]$
$\displaystyle=\frac{1}{2}k\tilde{u}[\partial_{x}^{-1}(u\psi)+k^{-2}(u_{x}-\sigma
u)\psi].$
In the same way, substituting (4.13) into the right hand side of (4.14)
produces
$None$ $\displaystyle
N(\tilde{u},\lambda)\tilde{\psi}=\frac{1}{2}\tilde{u}\partial_{x}^{-1}[k^{-1}(u_{x}-\sigma
u)(\psi_{x}-\sigma\psi)]$
$\displaystyle=\frac{1}{2}k^{-1}\tilde{u}[u_{x}\psi-\partial_{x}^{-1}(u_{xx}\psi)-\sigma
u\psi+\partial_{x}^{-1}(\psi_{0}^{-1}\psi_{0,xx}u\psi)]$
$\displaystyle=\frac{1}{2}k^{-1}\tilde{u}[k^{2}\partial_{x}^{-1}(u\psi)+(u_{x}-\sigma
u)\psi].$
Combining (4.15) with (4.16) implies that (4.14) holds. $\square$
In a similar way, we also have the following result.
Corollary 2. Let $\lambda=0$, then under the constraints $w=u^{2},\
v=-u_{xx}/u$, the Darboux transformations (4.3) and (4.9) are reduced to
another Darboux transformation of the NKdV-1 equation (1.2)
$(\psi,v,u)\rightarrow(\tilde{\psi},\tilde{v},\tilde{u})$, where
$None$ $\displaystyle\tilde{v}=v+2\sigma_{x},\ \
\tilde{\psi}=\psi-\psi_{0}^{-1}\sigma\partial_{x}^{-1}(\psi_{0}\psi),$
$\displaystyle\tilde{u}=\left\\{\begin{matrix}\displaystyle{\psi_{0}^{-1}\sigma},&u=0,\cr\cr\displaystyle{u-\psi_{0}^{-1}\sigma\partial_{x}^{-1}(\psi_{0}u)},&u\not=0.\end{matrix}\right.$
with $\sigma=\partial_{x}\ln(1+\partial_{x}^{-1}\psi_{0}^{2})$.
5\. Applications of the Darboux transformation
In this section, we shall apply the Darboux transformations (4.3) and (4.9) to
obtain kink-type and bell-type of explicit solutions for the NKdV equation
(1.1).
5.1. The kink-wave solutions
For the case of $\lambda=k^{2}>0$, we substitute $v=0,w=1$ into the Lax pair
(2.9) and choose the following basic solution
$None$ $\psi=e^{\xi}+e^{-\xi}=2\cosh\xi,\ \ \xi=kx-\frac{1}{2k}t+\gamma,$
where $\gamma$ and $k$ are two arbitrary constants.
Taking $\lambda=k_{1}^{2}$, then (4.4) and (5.1) lead to
$\sigma_{1}=\partial_{x}\ln\psi=k_{1}\tanh\xi_{1},\ \
\xi_{1}=k_{1}x-\frac{1}{2k_{1}}t+\gamma_{1}.$
The Darboux transformation (4.9) gives bell-type solution for the NKdV
equation (1.1)
$None$ $\begin{array}[]{l}\tilde{v}^{I}=2\sigma_{1,x}=2k_{1}^{2}{\rm
sech}^{2}\xi_{1},\\\\[8.0pt]
\tilde{w}^{I}=1-2\sigma_{1,t}=\tanh^{2}\xi_{1}.\end{array}$
By using Darboux trasformation (4.13), we get a kink-type wave solution for
the NKdV equation (1.2)
$None$ $\tilde{u}^{I}=k_{1}^{-1}(u_{x}-\sigma u)=-\tanh\xi_{1},\ \
\xi_{1}=k_{1}x-\frac{1}{2k_{1}}t+\gamma_{1}.$
Remark 1. There is much difference between traveling waves of the NKdV
equation (1.2) and of the classical KdV equation. For the NKdV equation (1.2),
its one-wave solution is a negative-moving (i.e. from right to left) kink-wave
with velocity $-{1}/{2k_{1}^{2}}$, amplitude $\pm 1$ and width ${1}/{k_{1}}.$
Its amplitude is independent of velocity, and width is directly proportional
to the velocity. For the KdV equation
$None$ $u_{t}+6uu_{x}+u_{xxx}=0,$
one-soliton solution is
$None$ $u=\frac{k^{2}}{2}{\rm sech}^{2}\frac{k(x-k^{2}t)}{2},$
which is a bell-type positive-moving wave with velocity $k^{2}$, amplitude
$k^{2}/2$ and width $1/k$, respectively. Its amplitude is directly
proportional to velocity, and width is inversely proportional to the velocity.
Let us now construct two-kink solutions to see the interaction of two kink
solutions. According to (4.4),
$None$ $\tilde{\psi}=T\psi=(\partial_{x}-\sigma_{1})(e^{\xi}+e^{-\xi})$
is also an eigenfunction of Lax pair (2.9). Taking $\lambda=k_{2}^{2}$, we
have
$None$
$\sigma_{2}=-k_{1}\tanh\xi_{1}+\frac{k_{1}^{2}-k_{2}^{2}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}}.$
Repeating the Darboux transformation (4.9) one more time, we get two soliton
solution for the the NKdV equation (1.1)
$\tilde{v}^{II}=\tilde{v}^{I}+2\sigma_{2,x}=\frac{(k_{1}^{2}-k_{2}^{2})(k_{2}^{2}{\rm
sech}^{2}\xi_{2}-k_{1}^{2}{\rm
sech}^{2}\xi_{1})}{(k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2})^{2}},$
$\tilde{w}^{II}=\tilde{w}^{I}-2\sigma_{2,t}=\left(\frac{k_{1}\tanh\xi_{2}-k_{2}\tanh\xi_{1}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}}\right)^{2}.$
Therefore, we obtain a two-kink wave solution of the NKdV equation (1.2)
$None$
$\tilde{\tilde{u}}=\frac{k_{2}\tanh\xi_{1}-k_{1}\tanh\xi_{2}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}}.$
(a) (b)
Figure 1. The two-kink wave solution $u(x,t)$ with parameters: $k_{1}=1,$
$k_{2}=0.6$. (a) Perspective view of the wave. (b) Overhead view of the wave,
with contour plot shown. The bright lines are crests and the dark lines are
troughs.
(a) (b)
(c) (d) (e)
Figure 2. Interaction between singular soliton ${\rm csch}\xi_{1}$ and smooth
soliton ${\rm sech}\xi_{2}$ with parameters: (a) $t=-3$, (b) $t=-0.05$, (c)
$t=0$, (d) $t=0.05$, (e) $t=3$.
Let us use the two-kink wave solution (5.8) to analyze interaction of the two
one-soliton solutions. Without loss of generality, we suppose $k_{1}>k_{2}>0$,
then we have
$\xi_{2}=\frac{k_{2}}{k_{1}}\left[\xi_{1}-\frac{k_{1}}{2}(\frac{1}{k_{2}^{2}}-\frac{1}{k_{1}^{2}})t\right].$
Therefore, on the fixed line $\xi_{1}=$constant, we get
$\tanh\xi_{2}\sim-1,\ \ t\rightarrow+\infty,$
and it follows (5.8) that
$None$
$\tilde{\tilde{u}}\sim\frac{k_{2}\tanh\xi_{1}+k_{1}}{k_{1}\tanh\xi_{1}+k_{2}}=\coth\left(\xi_{1}-\frac{1}{2}\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right),\
t\rightarrow+\infty.$
In a similar way, one can get
$\tanh\xi_{2}\sim 1\ \ {\rm as}\ t\rightarrow-\infty,$
which are main parts compared with terms $1$ and $e^{2\xi_{1}}$, and it
follows (3.19) that
$None$
$\tilde{\tilde{u}}\sim\frac{k_{2}e^{2\xi_{1}}-k_{1}}{k_{1}e^{2\xi_{1}}-k_{2}}=\coth\left(\xi_{1}+\frac{1}{2}\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right),\
t\rightarrow-\infty.$
In a similar way, on the line $\xi_{2}=$constant, we will arrive at
$None$
$\tilde{\tilde{u}}\sim\tanh\left(\xi_{2}+\frac{1}{2}\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right),\
\ {\rm as}\ \ t\rightarrow+\infty,$ $None$
$\tilde{\tilde{u}}\sim\tanh\left(\xi_{2}-\frac{1}{2}\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right),\
\ {\rm as}\ \ t\rightarrow-\infty.$
Remark 2. From expressions (5.9)-(5.12), we see that the two-kink wave
solution (5.8) is a singular solution, which is able to be decomposed into a
singular kink-type solution and a smooth kink wave solutions. The expressions
(5.10) and (5.12) show that the wave $\tanh\xi_{2}$ is on the left of the wave
$\coth\xi_{1}$ before their interaction, while the expressions (5.9) and
(5.11) show that the wave $\coth\xi_{1}$ is on the left of the wave
$\tanh\xi_{2}$ after their interaction. The shapes of the two kink waves
$\coth\xi_{1}$ and $\tanh\xi_{2}$ don’t change except their phases. Their
phases of the two waves $\coth\xi_{1}$ and $\tanh\xi_{2}$ are
$\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}>0$ and
$-\ln\frac{k_{1}-k_{2}}{k_{1}+k_{2}}<0$, respectively as the wave is
negatively going along the $x-$axis. Very interesting case is particular at
$t=0$: collision of such two kink waves forms a smooth bell-type soliton and
its singularity disappears (See Figure 2).
After their interaction It can be seen that the the two kink waves resume
their original shapes. At the right moment of interaction, the two kink waves
are fused into a smooth bell-type soliton. The two-kink wave interactions
possess the regular elastic-collision features and pass through each other,
and their shapes keep unchanged with a phase shift after the interaction.
Here, we also demonstrate a fact that the large-amplitude kink wave with
faster velocity overtakes the small-amplitude one, after collision, the
smaller one is left behind.
5.2. The bell-type soliton solutions
(i) For the case of $\lambda=0$ (i.e. without parameter $\lambda$), we
substitute $v=-k^{2},\ w=0$ into the Lax pair (2.10), and choose the following
basic solution as
$\psi=e^{\xi}+e^{-\xi},\ \ \xi=kx+\frac{1}{2k}t,$
where $k$ is an arbitrary constant.
Taking $k=k_{1}$, (4.4) gives
$None$ $\sigma=\sigma_{1}=\partial_{x}\ln\psi=k_{1}\tanh\xi_{1},\ \
\xi_{1}=k_{1}x+\frac{1}{2k_{1}}t.$
Using the Darboux transformation (4.9), we have one-soliton solution for the
NKdV equation (1.1)
$None$ $\displaystyle\tilde{v}=v+2\sigma_{1,x}=2k_{1}^{2}{\rm
sech}^{2}\xi_{1}-k_{1}^{2},$ $\displaystyle\tilde{w}=-2\sigma_{1,t}={\rm
sech}^{2}\xi_{1}.$
So, we get a one-soliton solution for the NKdV-1 equation (1.2) by using
Darboux transformation (4.17)
$None$ $\tilde{u}={\rm sech}\xi_{1},\ \ \xi_{1}=k_{1}x+\frac{1}{2k_{1}}t.$
Remark 3. For the negative order KdV equation (1.2), its one-soiton solution
(5.15) is a smooth bell-type negative-moving wave, whose velocity, amplitude
and width are ${1}/{2k_{1}^{2}}$, $\pm 1$ and ${1}/{k_{1}}$, respectively. Its
amplitude is independent of velocity, and width is directly proportional to
the velocity.
(ii) For the case of $\lambda=-k^{2}$, we take a seed solution of
$v=-2k^{2},w=1$ in the Lax pair (2.9), and choose the following basic solution
as
$\psi=e^{\xi}+e^{-\xi},\ \ \xi=kx-\frac{1}{2k}t+\gamma,$
where $k$ is an arbitrary constant.
Taking $k=k_{1}$ sends (3.7) to
$None$ $\sigma=\sigma_{1}=\partial_{x}\ln\psi=k_{1}\tanh\xi_{1},\ \
\xi_{1}=k_{1}x-\frac{1}{2k_{1}}t+\gamma_{1}.$
Using the Darboux transformation (3.12), we then get one-soliton solution
$None$ $\displaystyle\tilde{v}^{I}=v+2\sigma_{1,x}=-2k_{1}^{2}{\rm
tanh}^{2}\xi_{1}+\gamma_{1},$
$\displaystyle\tilde{w}^{I}=1-2\sigma_{1,t}=1+{\rm sech}^{2}\xi_{1},$
which cannot satisfies the constraint (3.3), so $\sqrt{\tilde{w}^{I}}$ is not
soliton for the NKdV equation (1.2).
Remark 4. For the NKdV equation (1.1), its one-soiton solution (5.14) is a
smooth bell-type positive-moving wave, whose velocity, amplitude and width are
${1}/{2k_{1}^{2}}$, $\pm 1$ and ${1}/{k_{1}}$, respectively. Its amplitude is
independent of velocity, and width is directly proportional to the velocity.
Let’s construct a two-soliton solution of the NKdV equation (1.1). According
to the gauge transformation (4.4),
$\tilde{\psi}=T\psi=(\partial_{x}-\sigma_{1})(e^{\xi}+e^{-\xi})$
is also an eigenfunction of Lax (2.9). We have
$\sigma_{2}=-k_{1}\tanh\xi_{1}+\frac{k_{1}^{2}-k_{2}^{2}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}}.$
Repeating the Darboux transformation (4.9) one more time, we obtain
$\tilde{v}^{II}=\tilde{v}^{I}+2\sigma_{2,x}=\frac{(k_{1}^{2}-k_{2}^{2})(k_{2}^{2}{\rm
sech}^{2}\xi_{2}-k_{1}^{2}{\rm
sech}^{2}\xi_{1})}{(k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2})^{2}},$
$\tilde{w}^{II}=\tilde{w}^{I}-2\sigma_{2,t}=\left(\frac{k_{1}\tanh\xi_{2}-k_{2}\tanh\xi_{1}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}}\right)^{2},$
which is the same for NKdV-1 equation (1.2). So we get two-soliton solution
with (5.8)
$\tilde{\tilde{u}}=\pm\frac{k_{1}\tanh\xi_{2}-k_{2}\tanh\xi_{1}}{k_{1}\tanh\xi_{1}-k_{2}\tanh\xi_{2}},$
but here $\xi_{j}=k_{j}x-\frac{1}{2k_{j}}t,\ j=1,2$.
6\. Bilinearization of the NKdV equation
The bilinear derivative method, developed by Hirota [29], has become a
powerful approach to construct exact solutions of nonlinear equations. Once a
nonlinear equation is written in a bilinear form by using some transformation,
then multi-solitary wave solutions or quasi-periodic wave solutions can
usually be obtained [30, 31, 35, 36, 46, 85, 14]. However, unfortunately, this
method is not as direct as many people might wish because the original
equation is reduced to two or more bilinear equations under new variables
called Hirota’s variables. Since no a general rule to select Hirota’s
variables, there is no rule to choose some essential formulas (such as
exchange formulas), either. Especially the construction of bilinear Bäcklund
transformation relies on a particular skill and appropriate exchange formulas.
On the other hand, in recent years Lambert and his co-workers have found a
kind of the generalized Bell polynomials playing important role in seeking the
characterization of bilinearized equations. Based on the Bell polynomials,
they presented an alternative procedure to obtain parameter families of a
bilinear Bäcklund transformation and Lax pairs for soliton equations in a
quick and short way [27, 41, 42].
6.1. Multi-dimensional binary Bell polynomials
The main tool we use here is a class of generalized multi-dimensional binary
Bell polynomials [27]-[42].
Definition 1. Let $n_{k}\geq 0,\ k=1,\cdots,\ell$ denote arbitrary integers,
$f=f(x_{1},\cdots,x_{\ell})$ be a $C^{\infty}$ multi-variable function, then
$None$
$Y_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(f)\equiv\exp({-f})\partial_{x_{1}}^{n_{1}}\cdots\partial_{x_{\ell}}^{n_{\ell}}\exp(f)$
is a polynomial in the partial derivatives of $f$ with respect to
$x_{1},\cdots,x_{\ell}$, which we call a multi-dimensional Bell polynomial (a
generalized Bell polynomial or $Y$-polynomial).
For the two dimensional case, let $f=f(x,t)$, then the associated Bell
polynomials through (6.1) can produce the following representatives:
$\displaystyle{Y}_{x}(f)=f_{x},\ {Y}_{2x}(f)=f_{2x}+f_{x}^{2},\ \ \
{Y}_{3x}(f)=f_{3x}+3f_{x}f_{2x}+f_{x}^{3},$
$\displaystyle{Y}_{x,t}(f)=f_{x,t}+f_{x}f_{t},\ \
{Y}_{2x,t}(f)=f_{2x,t}+f_{2x}f_{t}+2f_{x,t}f_{x}+f_{x}^{2}f_{t},\cdots.$
Definition 2. Based on the use of above Bell polynomials (6.21), the multi-
dimensional binary Bell polynomials ( $\mathcal{Y}$-polynomials) are defined
by
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g,h)=Y_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(f)\mid_{f_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}=\left\\{\begin{matrix}g_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}},&r_{1}+\cdots+r_{\ell}\
\ {\rm is\ \ odd},\cr\cr
h_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}},&r_{1}+\cdots+r_{\ell}\ \ {\rm is\ \
even},\end{matrix}\right.}$
which is a multi-variable polynomial with respect to all partial derivatives
$g_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}$ ($r_{1}+\cdots+r_{\ell}\ \ {\rm is\
odd}$) and $h_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}$ ($r_{1}+\cdots+r_{\ell}\ \
{\rm is\ even}$), $r_{k}=0,\cdots,n_{k},\ k=0,\cdots,\ell$.
The binary Bell polynomials also inherit partial structures of the Bell
polynomials. The first few lower order binary Bell Polynomials are
$None$ $\displaystyle\mathcal{Y}_{x}(g)=g_{x},\
\mathcal{Y}_{2x}(g,h)=h_{2x}+g_{x}^{2},\ \
\mathcal{Y}_{x,t}(g,h)=h_{xt}+g_{x}g_{t}.$
$\displaystyle\mathcal{Y}_{3x}(g,h)=g_{3x}+3g_{x}h_{2x}+g_{x}^{3},\cdots.$
Proposition 8. The link between binary Bell polynomials
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g,h)$ and the standard
Hirota bilinear expression $D_{x_{1}}^{n_{1}}\cdots
D_{x_{\ell}}^{n_{\ell}}F\cdot G$ can be given by an identity
$None$ $\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g=\ln F/G,h=\ln
FG)=(FG)^{-1}D_{x_{1}}^{n_{1}}\cdots D_{x_{\ell}}^{n_{\ell}}F\cdot G,$
where${n_{1}}+n_{2}+\cdots+n_{\ell}\geq 1$, and operators
$D_{x_{1}},\cdots,D_{x_{\ell}}$ are classical Hirota’s bilinear operators
defined by
$D_{x_{1}}^{n_{1}}\cdots D_{x_{\ell}}^{n_{\ell}}F\cdot
G=(\partial_{x_{1}}-\partial_{x_{1}^{\prime}})^{n_{1}}\cdots(\partial_{x_{\ell}}-\partial_{x_{\ell}^{\prime}})^{n_{\ell}}F(x_{1},\cdots,x_{\ell})G(x_{1}^{\prime},\cdots,x_{\ell}^{\prime})|_{x_{1}^{\prime}=x_{1},\cdots,x_{\ell}^{\prime}=x_{\ell}}.$
In the special case of $F=G$, the formula (6.4) becomes
$None$ $\displaystyle F^{-2}D_{x_{1}}^{n_{1}}\cdots
D_{x_{\ell}}^{n_{\ell}}G\cdot
G=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(0,q=2\ln G)$
$\displaystyle=\left\\{\begin{matrix}0,&n_{1}+\cdots+n_{\ell}\ \ {\rm is\ \
odd},\cr\cr P_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(q),&n_{1}+\cdots+n_{\ell}\
\ {\rm is\ \ even}.\end{matrix}\right.$
The first few $P$-polynomial are
$None$ $\displaystyle P_{2x}(q)=q_{2x},\ P_{x,t}(q)=q_{xt},\
P_{4x}(q)=q_{4x}+3q_{2x}^{2},$ $\displaystyle
P_{6x}(q)=q_{6x}+15q_{2x}q_{4x}+15q_{2x}^{3},\cdots.$
The formulas (6.4) and (6.5) will prove particularly useful in connecting
nonlinear equations to their corresponding bilinear forms. This means that if
a nonlinear equation is expressedby a linear combination of $P$-polynomials,
then the nonlinear equation can be transformed into a linear equation.
Proposition 9. The binary Bell polynomials
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,w)$ can be separated into
$P$-polynomials and $Y$-polynomials
$None$ $\displaystyle(FG)^{-1}D_{x_{1}}^{n_{1}}\cdots
D_{x_{\ell}}^{n_{\ell}}F\cdot
G=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g,h)|_{g=\ln F/G,h=\ln FG}$
$\displaystyle=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g,g+q,)|_{g=\ln
F/G,q=2\ln G}$
$\displaystyle=\sum_{n_{1}+\cdots+n_{\ell}=even}\sum_{r_{1}=0}^{n_{1}}\cdots\sum_{r_{\ell}=0}^{n_{\ell}}\prod_{i=1}^{\ell}\left(\begin{matrix}n_{i}\cr
r_{i}\end{matrix}\right)P_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}(q)Y_{(n_{1}-r_{1})x_{1},\cdots,(n_{\ell}-r_{\ell})x_{\ell}}(v).$
The key property of the multi-dimensional Bell polynomials
$None$
$Y_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g)|_{g=\ln\psi}={\psi_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}}/{\psi},$
implies that the binary Bell polynomials
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(g,h)$ can still be
linearized by means of the Hopf-Cole transformation $g=\ln\psi$, that is,
$\psi=F/G$. The formulas (6.6) and (6.7) will then provide the shortest way to
the associated Lax system of nonlinear equations.
6.2. Bilinearization
Theorem 6. Under the transformation
$v=v_{0}+2(\ln G)_{2x},\ \ w=w_{0}+2(\ln G)_{xt},$
the NKdV equation (1.1) can be bilinearized into
$None$ $\displaystyle(D_{x}^{4}+12v_{0}D_{x}^{2}-D_{x}D_{y})G\cdot G=0,$
$\displaystyle(2D_{t}D_{x}^{3}+6w_{0}D_{x}^{2}+D_{t}D_{y})G\cdot G=0.$
where $y$ ia an auxiliary variable, and $u_{0}$, $v_{0}$ are two constant
solutions of the NKdV equation (1.1).
Proof. The invariance of the NKdV equation (1.1) under the scale
transformation
$x\rightarrow\lambda x,\ \ t\rightarrow\lambda^{\alpha}t,\ \
v\rightarrow\lambda^{-2}v,\ \ w\rightarrow\lambda^{-\alpha-1}w$
shows that the dimensions of the fields $v$ and $w$ are $-2$ and
$-(\alpha+1)$, respectively. So we may introduce a dimensionless potential
field $q$ by setting
$None$ $v=v_{0}+q_{2x},\ \ \ w=w_{0}-q_{xt}.$
Substituting the transformation (6.9) into the equation (1.1), we can write
the resulting equation in the following form
$\displaystyle
q_{4x,t}+4q_{2x}q_{2x,t}+2q_{3x}q_{xt}+4v_{0}q_{2x,t}+2w_{0}q_{3x}=0,$
which is regrouped as follows
$None$
$\displaystyle\frac{2}{3}q_{4x,t}+2(q_{2x}q_{2x,t}+q_{xt}q_{3x})+\frac{1}{3}q_{4x,t}+2q_{2x}q_{2x,t}+4v_{0}q_{2x,t}+2w_{0}q_{3x}=0,$
where we will see that Such an expression is necessary to get a bilinear form
of the equation (1.1). Further integrating the equation (6.10) with respect to
$x$ yields
$None$ $\displaystyle
E(q)\equiv\frac{2}{3}(q_{3x,t}+3q_{2x}q_{xt}+3w_{0}q_{2x})+\frac{1}{3}\partial_{x}^{-1}\partial_{t}(q_{4x}+3q_{2x}^{2}+12v_{0}q_{2x})=0.$
In order to write the equation (6.11) in a local bilinear form, let us first
get rid of the integral operator $\partial_{x}^{-1}$. To do so, we introduce
an auxiliary variable $y$ and impose a subsidiary constraint condition
$None$ $q_{4x}+3q_{2x}^{2}+12v_{0}q_{2x}-q_{xy}=0.$
Then, the equation (6.10) becomes
$None$ $\displaystyle 2(q_{3x,t}+3q_{2x}q_{xt}+3w_{0}q_{2x})+q_{yt}=0.$
According to the formula (6.5), the equations (6.12) and (6.13) are then cast
into a pair of equations in the form of $P$-polynomials
$\displaystyle P_{4x}(q)+12v_{0}P_{2x}(q)-P_{xy}(q)=0,$ $\displaystyle
2P_{3x,t}(q)+6w_{0}P_{2x}(q)+P_{yt}(q)+3\gamma=0.$
Finally, by the property (6.4), making the following variable
$q=2\ln G\ \ \Longleftrightarrow\ \ v=v_{0}+2(\ln G)_{2x},\ \ w=w_{0}+2(\ln
G)_{xt},$
change above system to the following bilinear forms of the NKdV equation (1.1)
as follows
$None$ $\displaystyle(D_{x}^{4}+12v_{0}D_{x}^{2}-D_{x}D_{y})G\cdot G=0,$
$\displaystyle(2D_{t}D_{x}^{3}+6w_{0}D_{x}^{2}+D_{t}D_{y})G\cdot G=0,$
which is also simultaneously bilinear system in $y$. This system is easily
solved with multi-soliton solutions by using the Hirota’s bilinear method.
$\square$
Finally, we show that the NKdV-1 equation (1.1) can be directly bilinearized
through a transformation, not Bell polynomials. Making dependent variable
transformation
$None$ $v=v_{0}+2(\ln F)_{xx},\ \ u=G/F,$
we can change the equation (1.2) into
$\displaystyle 2(F_{xt}-F_{x}F_{t})=G^{2},$ $\displaystyle
F_{xx}G-2F_{x}G_{x}+G_{xx}F+v_{0}FG=0,$
which is equivalent to the bilinear form
$None$ $\displaystyle D_{x}D_{t}F\cdot F=G^{2},\ \ (D_{x}^{2}+v_{0})F\cdot
G=0.$
It is obvious that the bilinear form of the NKdV-1 (6.16) is more simple than
the bilinear form of NKdV (6.15).
6.3. N-soliton solutions
As usual as the normal perturbation method, let us expand $G$ in the power
series of a small parameter $\varepsilon$ as follows
$G=1+\varepsilon g^{(1)}+\varepsilon^{2}g^{(2)}+\varepsilon^{3}g^{(3)}+\cdots$
Substituting the above equation into (6.7) and sorting each order of
$\varepsilon$, we have
$None$ $\displaystyle\varepsilon:\ \
(D_{x}^{4}+12v_{0}D_{x}^{2}-D_{x}D_{y})g_{1}\cdot 1=0,$ $\displaystyle\ \ \ \
(2D_{t}D_{x}^{3}+6w_{0}D_{x}^{2}+D_{t}D_{y})g^{(1)}\cdot 1=0,$ $None$
$\displaystyle\varepsilon^{2}:\ \
(D_{x}^{4}+12v_{0}D_{x}^{2}-D_{x}D_{y})(2g^{(2)}\cdot 1+g^{(1)}\cdot
g^{(1)})=0,$ $\displaystyle\ \ \ \
(2D_{t}D_{x}^{3}+6w_{0}D_{x}^{2}+D_{t}D_{y})(2g^{(2)}\cdot 1+g^{(1)}\cdot
g^{(1)})=0,$ $None$ $\displaystyle\varepsilon^{3}:\ \
(D_{x}^{4}+12v_{0}D_{x}^{2}-D_{x}D_{y})(g^{(3)}\cdot 1+g^{(1)}\cdot
g^{(2)})=0,$ $\displaystyle\ \ \ \
(2D_{t}D_{x}^{3}+6w_{0}D_{x}^{2}+D_{t}D_{y})(g^{(3)}\cdot 1+g^{(1)}\cdot
g^{(2)})=0,$ $\displaystyle\cdots\cdots$
By employing formulae mentioned above, the system (6.16) is equivalent to the
following linear system
$\displaystyle g^{(1)}_{xxxx}+12v_{0}g^{(1)}_{xx}-g^{(1)}_{xy}=0,$
$\displaystyle 2g^{(1)}_{xxxt}+6w_{0}g^{(1)}_{xx}+g^{(1)}_{yt}=0,$
which has solution
$None$ $g^{(1)}=e^{\xi},\ \ \
\xi=kx-\frac{2kw_{0}}{k^{2}+4v_{0}}t+(k^{3}+12v_{0}k)y+\sigma,$
where $k$ and $\sigma$ are two arbitrary parameters.
Substituting (6.12) into (6.10) and (6.11) and choosing
$g^{(2)}=g^{(3)}=\cdots=0$, then the $G$’s expansion is truncated with a
finite sum as
$G=1+e^{\xi},$
which gives regular one-soliton solution of the NKdV equation (1.1)
$None$ $\displaystyle
v=v_{0}+2\partial_{x}^{2}\ln(1+e^{\xi})=v_{0}+\frac{k^{2}}{2}{\rm
sech}^{2}\xi/2,$ $\displaystyle
w=w_{0}+2\partial_{t}\partial_{x}\ln(1+e^{\xi})=w_{0}+\frac{k^{2}w_{0}}{k^{2}+4v_{0}}{\rm
sech}^{2}\xi/2,$ $\displaystyle\xi=kx-\frac{2kw_{0}}{k^{2}+4v_{0}}t+\gamma,$
where $\gamma=(k^{3}+12v_{0}k)y+\sigma$, and $k$, $v_{0}$, $w_{0}$ are
constants.
Let $w_{0}=1,\ v_{0}=0$, then the solution (6.20) reads as a kink-type
solution of the NKdV-I equation (1.2)
$u=\pm\tanh\xi/2,\ \ \ \xi=kx-\frac{2}{k}t+\gamma.$
In a similar way, taking
$g^{(1)}=e^{\xi_{1}}+e^{\xi_{2}},\ \
\xi_{j}=k_{j}x-\frac{2k_{j}w_{0}}{k_{j}^{2}+4v_{0}}t+\gamma_{j},\ \ j=1,2,$
we get a two-soliton wave solution
$None$ $\displaystyle
v=v_{0}+2\partial_{x}^{2}\ln(1+e^{\xi_{1}}+e^{\xi_{2}}+e^{\xi_{1}+\xi_{2}+A_{12}})$
$\displaystyle
w=w_{0}-2\partial_{t}\partial_{x}\ln(1+e^{\xi_{1}}+e^{\xi_{2}}+e^{\xi_{1}+\xi_{2}+A_{12}}),$
$\displaystyle A_{12}=\ln\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2}.$
In general, we can get a N-soliton solution of the NKdV equation (1.1)
$\displaystyle
v=v_{0}+2\partial_{x}^{2}\ln\left(\sum_{\mu_{j}=0,1}\exp(\sum_{j=1}^{N}\mu_{j}\xi_{j}+\sum_{1\leq
j\leq N}^{N}\mu_{j}\mu_{l}A_{jl}\right),$ $\displaystyle
w=w_{0}-\partial_{t}\partial_{x}\ln\left(\sum_{\mu_{j}=0,1}\exp(\sum_{j=1}^{N}\mu_{j}\xi_{j}+\sum_{1\leq
j\leq N}^{N}\mu_{j}\mu_{l}A_{jl}\right),$ $\displaystyle
A_{jl}=\ln\left(\frac{k_{j}-k_{l}}{k_{j}+k_{l}}\right)^{2}.$
where the notation $\sum_{\mu_{j}=0,1}$ represents all possible combinations
$\mu_{j}=0,1$, and
$\xi_{j}=k_{j}x-\frac{2k_{j}w_{0}}{k_{j}^{2}+4v_{0}}t+\gamma_{j},\ \
j=1,2,\cdots,N.$
In the following, we discuss the soliton solutions for NKdV-1 equation by
using bilinear equation (6.16). Let us expand $F$ and $G$ in the power series
of a small parameter $\varepsilon$ as follows
$F=1+f^{(2)}\varepsilon^{2}+f^{(4)}\varepsilon^{4}+f^{(6)}\varepsilon^{6}+\cdots$
$G=g^{(1)}\varepsilon+g^{(3)}\varepsilon^{3}+g^{(5)}\varepsilon^{5}+\cdots$
Substituting the above equation into (6.16) and arranging each order of
$\varepsilon$, we have
$None$ $\displaystyle g^{(1)}_{xx}+v_{0}g^{(1)}=0,$ $\displaystyle
g^{(3)}_{xx}+v_{0}g^{(3)}=-(D_{x}^{2}+v_{0})f^{(2)}\cdot g^{(1)},$
$\displaystyle g^{(5)}_{xx}+v_{0}g^{(5)}=-(D_{x}^{2}+v_{0})(f^{(2)}\cdot
g^{(3)}+f^{(4)}\cdot g^{(1)}),$ $\displaystyle\cdots\cdots$ $None$
$\displaystyle 2f^{(2)}_{xt}=(g^{(1)})^{2},$ $\displaystyle
2f^{(4)}_{xt}=2g^{(1)}g^{(3)}-D_{x}D_{t}f^{(2)}\cdot f^{(2)},$ $\displaystyle
2f^{(6)}_{xt}=2g^{(1)}g^{(5)}+2(g^{(3)})^{2}-2D_{x}D_{t}f^{(3)}\cdot f^{(3)},$
$\displaystyle\cdots\cdots$
Let $v_{0}=-k^{2}$, it follows from the first equation of (6.23) and (6.24)
that
$None$ $g^{(1)}=e^{\xi},\ \ f^{(2)}=\frac{1}{4}e^{2\xi},\ \
\xi=kx+\frac{1}{2k}t+\gamma.$
Substituting (6.25) into the second equation of (6.23) leads to
$g^{(3)}_{xx}-k^{2}g^{(3)}=0,$
from which we may take $g^{(3)}=0$, further choose $g^{(5)}=\cdots=0,\
f^{(4)}=\cdots=0.$ So $F$ and $G$ are truncated with a finite sum as
$F=1+\frac{1}{4}e^{2\xi},\ \ G=e^{\xi}.$
Finally, the formula (6.14) gives one-soliton solution of the NKdV-1 equation
(1.2)
$v=2k^{2}{\rm sech}^{2}\xi-k^{2},\ \ \ u={\rm sech}\xi.$
7\. Bilinear Bäcklund transformation
In this section, we search for the bilinear Bäcklund transformation and Lax
pair of the NKdV equation (1.1).
7.1. Bilinear Bäcklund transformation
Theorem 7. Suppose that $F$ is a solution of the bilinear equation (6.8), and
if $G$ satisfying
$None$ $\displaystyle(D_{x}^{2}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}D_{x}^{2}+2w_{0}D_{x}+(4v_{0}+3\lambda)D_{t}]F\cdot G=0,$
then G is another solution of the equation (6.8).
Proof. Let
$q=2\ln G,\ \ \ \tilde{q}=2\ln F$
be two different solutions of the equation (6.10). Introducing two new
variables
$h=(\tilde{q}+q)/2=\ln(FG),\ \ g=(\tilde{q}-q)/2=\ln(F/G),$
makes the function $E$ invariant under the two fields $\tilde{q}$ and $q$:
$None$ $\displaystyle E(\tilde{q})-E(q)=E(h+g)-E(h-g)$
$\displaystyle=8v_{0}g_{xt}+4w_{0}g_{2x}+2g_{3x,t}+4h_{2x}g_{x,t}+4h_{x,t}g_{2x}+4\partial_{x}^{-1}(h_{2x}g_{2x,t}+h_{2x,t}g_{2x})$
$\displaystyle=2\partial_{x}(\mathcal{Y}_{2x,t}(g,h)+4v_{0}\mathcal{Y}_{t}(g)-2w_{0}\mathcal{Y}_{x}(g))+R(g,h)=0,$
where
$R(g,h)=-2\partial_{x}[(h_{2x}+g_{x}^{2})g_{t}]+4h_{2x}g_{xt}-4h_{2x,t}g_{x}+4\partial_{x}^{-1}(h_{2x}g_{2x,t}+h_{2x,t}g_{2x}).$
This two-field invariant condition can be regarded as a natural ansatz for a
bilinear Bäcklund transformation and may produce some required transformations
under additional appropriate constraints.
In order to decouple the two-field condition (7.2),let us impose a constraint
so as to express $R(g,h)$ in the form of $x$-derivative of
$\mathcal{Y}$-polynomials. The simple choice of the constraint may be
$None$ $\mathcal{Y}_{2x}(g,h)=h_{2x}+g_{x}^{2}=\lambda,$
which directly leads to
$None$ $\displaystyle R(g,h)=2\lambda$ $\displaystyle
g_{xt}+4h_{2x}g_{xt}-4h_{2x,t}g_{x}-4g_{x}^{2}g_{xt}=6\lambda g_{xt},$
where $h_{2x,t}=-2g_{x}g_{xt}$ and $h_{2x}=\lambda-g_{x}^{2}$ are used.
Using the relations (7.2)-(7.4), we derived a coupled system of
$\mathcal{Y}$-polynomials
$None$ $\displaystyle\mathcal{Y}_{2x}(g,h)-\lambda=0,$
$\displaystyle\mathcal{Y}_{2x,t}(g,h)+(4v_{0}+3\lambda)\mathcal{Y}_{t}(g)+2w_{0}\mathcal{Y}_{x}(g)=0,$
where we prefer the second equation to be expressed in the form of conserved
quantity without integration with respect to $x$. This is very useful to
construct conservation laws. Apparently, the identity (6.2) directly sends the
system (7.5) to the following bilinear Bäcklund transformation
$None$ $\displaystyle(D_{x}^{2}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}D_{x}^{2}+2w_{0}D_{x}+(4v_{0}+3\lambda)D_{t}]F\cdot G=0,$
where we have integrated the second equation in the system (7.5) with respect
to $x$, and $w_{0}$ is the corresponding integration constant. $\square$
7.2. Inverse scattering formulation
Theorem 8. The NKdV equation (1.1) admits a Lax pair
$None$ $\displaystyle\psi_{2x}+v\psi=\lambda\psi,$ $\displaystyle
4\psi_{2x,t}+4v\psi_{t}-2w\psi_{x}-3w_{x}\psi=0.$
Proof. By the transformation $v=\ln\psi$, it follows from the formulas (6.5)
and (6.6) that
$\displaystyle\mathcal{Y}_{t}(g)=\psi_{t}/\psi,\ \
\mathcal{Y}_{x}(g)=\psi_{x}/\psi,\ \
\mathcal{Y}_{2x}(g,h)=q_{2x}+\psi_{2x}/\psi,$
$\displaystyle\mathcal{Y}_{2x,t}(g,h)=2q_{xt}\psi_{x}/\psi+q_{2x}\psi_{t}/\psi+\psi_{2x,t}/\psi,\
\ $
which make the system (7.5) linearized into a Lax pair with parameter
$\lambda$
$None$ $\displaystyle L\psi\equiv(\partial_{x}^{2}+q_{2x})\psi=\lambda\psi,$
$None$ $\displaystyle
M\psi\equiv[\partial_{t}\partial_{x}^{2}+(4v_{0}+q_{2x})\partial_{t}+2(q_{xt}+w_{0})\partial_{x}+3\lambda\partial_{t}]\psi,$
or equivalently,
$\displaystyle\psi_{2x}+v\psi=\lambda\psi,$ $\displaystyle
4\psi_{2x,t}+4v\psi_{t}-2w\psi_{x}-3w_{x}\psi=0,$
where the equation (7.8) is used to get the second equation. One can easily
verify from equations (7.8) and (7.9) that
$[L,M]=q_{4x,t}+4(v_{0}+q_{2x})q_{2x,t}+2q_{3x}(q_{xt}+w_{0})=0$
exactly gives the NKdV equation (1.1) through replacing $v_{0}+q_{2x}$ and
$w_{0}+q_{xt}$ by $v$ and $w$, respectively. $\square$
8\. Darboux covariant Lax pair
In this section, we will give a kind of Darboux covariant Lax pair, whose form
is invariant under the gauge transformation (4.3).
Theorem 9. The NKdV equation (1.1) possesses the following Darboux covariant
Lax pair
$\displaystyle L\psi=\lambda\psi,$ $\displaystyle{M}_{{\rm cov}}\psi=0,\ \
{M}_{{\rm cov}}=M+3\partial_{x}L,$
under the gauge transformation $\tilde{\psi}=T\psi$. This is actually
equivalent to the Lax pair (2.9).
Proof. In section 4, we have shown that the gauge transformation (4.1) maps
the operator $L(q)$ onto a similar operator
$\tilde{{L}}(\tilde{q})=TL(q)T^{-1},$
which satisfies the following covariance condition
$\tilde{{L}}(\tilde{q})=L(q+\Delta q),\ \tilde{q}=q+\Delta q,\ {\rm with}\ \
\Delta q=2\ln\phi.$
Next, we want to find a third order operator ${M}_{{\rm cov}}(q)$ with
appropriate coefficients, such that $M_{{\rm cov}}(q)$ is mapped by gauge
transformation (8.1) onto a similar operator $\tilde{M}_{{\rm{\rm
cov}}}(\tilde{q})$, which satisfies the covariance condition
$\tilde{M}_{{\rm cov}}(\tilde{q})={M}_{{\rm cov}}(q+\Delta q),\
\tilde{q}=q+\Delta q.$
Suppose that $\phi$ is a solution of the following Lax pair
$None$ $\displaystyle L\psi=\lambda\psi,$ $\displaystyle{M}_{{\rm
cov}}\psi=0,\ \ {M}_{{\rm
cov}}=4\partial_{t}\partial_{x}^{2}+b_{1}\partial_{x}+b_{2}\partial_{t}+b_{3},$
where $b_{1},\ b_{2}$ and $b_{3}$ are functions to be determined. Then, the
transformation $T$ is required to map the operator $M_{{\rm cov}}$ to the
similar one
$None$ $TM_{{\rm cov}}T^{-1}=\tilde{M}_{{\rm cov}},\ \ \tilde{L}_{2,{\rm
cov}}=4\partial_{t}\partial_{x}^{2}+\tilde{b}_{1}\partial_{x}+\tilde{b}_{2}\partial_{t}+\tilde{b}_{3},$
where $\tilde{b}_{1},\tilde{b}_{2}$ and $\tilde{b}_{3}$ satisfy the covariant
condition
$None$ $\tilde{b}_{j}={b}_{j}(q)+\Delta b_{j}={b}_{j}(q+\Delta q),\ \
j=1,2,3.$
It follows from (8.2) and (5.3) that
$None$ $\displaystyle\Delta b_{1}=\tilde{b}_{1}-b_{1}=4\sigma_{t},\ \ \Delta
b_{2}=\tilde{b}_{2}-b_{2}=8\sigma_{x},$ $None$ $\displaystyle\Delta
b_{3}=\tilde{b}_{3}-b_{3}=\sigma\Delta b_{1}+8\sigma_{xt}+b_{1,x},$
and $\sigma$ satisfy
$None$
$4\sigma_{2x,t}+\tilde{b}_{1}\sigma_{x}+\tilde{b}_{2}\sigma_{t}+\sigma\Delta
b_{3}+b_{3,x}=0.$
According to the relation (8.4), it remains to determine $b_{1},\ b_{2}$ and
$b_{3}$ in the form of polynomial expressions in terms of $q$’s derivatives
$b_{j}=F_{j}(q,q_{x},q_{y},q_{xy},q_{2x},q_{2y},q_{2x,y},\cdots),\ \ j=1,2,3$
such that
$None$ $\Delta F_{j}=F_{j}(q+\Delta q,q_{x}+\Delta q_{x},q_{t}+\Delta
q_{t},\cdots)-F_{j}(q,q_{x},q_{t},\cdots)=\Delta b_{j},$
with $\Delta q_{kx,lt}=2(\ln\phi)_{kx,lt},\ k,l=1,2,\cdots$, and $\Delta
b_{j}$ being given through the relations (8.5)-(8.7).
Expanding the left hand of the equation (8.8), we obtain
$\Delta b_{1}=\Delta F_{1}=F_{1,q}\Delta q+F_{1,q_{x}}\Delta
q_{x}+F_{1,q_{y}}\Delta q_{y}+F_{1,q_{xy}}\Delta
q_{xt}+\cdots=4\sigma_{t}=2\Delta q_{xt},$
which implies that we can determine $b_{1}$ up to a arbitrary constant
$c_{1}$, namely,
$None$ $b_{1}=F_{1}(q_{xt})=2q_{xt}+c_{1},\ c_{1}\ {\rm is\ an\ arbitrary\
constant}$
Proceeding in the same way deduce the function $b_{2}$ as follows
$None$ $b_{2}=F_{2}(q_{2x})=4q_{2x}+c_{2},$
where $c_{2}$ is an arbitrary constant.
We see from the relation (8.6) that $\Delta b_{3}$ contains the term
$b_{1,x}=q_{2x,t}$, which should be eliminated such that $\Delta b_{3}$ admits
the form (8.8). By the Lax pair (8.2), we have the following relation
$None$ $q_{2x,t}=-\sigma_{xt}-2\sigma\sigma_{t}.$
Substituting (8.9) and (8.11) into (8.6) yields
$\Delta b_{3}=4\sigma\sigma_{t}+8\sigma_{xt}+2q_{2x,t}=6\sigma_{xt}=3\Delta
q_{2x,t}.$
If choosing
$None$ $b_{3}=F_{3}(q_{2x,t})=3q_{2x,t}+c_{3},$
the third condition
$\Delta F_{3}=F_{3,q}\Delta q+F_{3,q_{x}}\Delta q_{x}+F_{3,q_{t}}\Delta
q_{t}\cdots=\Delta b_{3}$
holds, where $c_{3}$ is an arbitrary constant.
Letting $c_{1}=-2v_{0},\ c_{2}=0,\ c_{3}=w_{0}$ in (8.9), (8.10) and (8.12),
then it follows from (8.2) that we have the following Darboux covariant
evolution equation
$M_{{\rm cov}}\psi=0,\ \ M_{{\rm
cov}}=4\partial_{t}\partial_{x}^{2}+2q_{xt}\partial_{x}+4q_{2x}\partial_{t}+3q_{2x,t},$
which coincides with the equation (8.7). Moreover, the relation between two
operators $L_{2,{\rm cov}}$ and $L_{2}$ are related through
$M_{{\rm cov}}=M+3\partial_{x}L.$
The compatibility condition of the Darboux covariant Lax pair (8.2) exactly
gives the NKdV equation(1.1) in Lax representation
$\displaystyle[M_{{\rm
cov}},L]=q_{4x,t}+4(v_{0}+q_{2x})q_{2x,t}+q_{3x}(q_{xt}+w_{0})$
$\displaystyle=v_{xxx}+4vw_{x}+2v_{x}w=0.$
$\square$
In the above repeated procedure, we are able to obtain higher order operators,
which are also Darboux covariant with respect to $T$, to produce higher order
members of the negative order KdV hierarchy.
9\. Conservation laws of NKdV equations
In this section, we will present infinitely many conservation laws in a local
form for the NKdV equation (1.1) based on a generalized Miura transformation.
Theorem 10. The NKdV equation (1.1) possesses the following infinitely many
conservation laws
$None$ $F_{n,t}+G_{n,x}=0,\ n=1,2,\cdots.$
where the conversed densities $F_{n}^{\prime}s$ are recursively given by
recursion formulas explicitly
$None$ $\displaystyle F_{0}=v_{xx}-v^{2},\ \ F_{1}=-v_{xxx}+2vv_{x},$
$\displaystyle
F_{n}=I_{n,xx}-\sum_{k=0}^{n}I_{k}I_{n-k}+\sum_{k=0}^{n-2}I_{k}I_{n-2-k,x},\ \
n=2,3,\cdots.$
and the fluxes $G_{n}^{\prime}s$ are
$None$ $\displaystyle G_{0}=2wI_{0}=2wv,\ \ G_{1}=2wI_{1}=-2wv_{x},$
$\displaystyle G_{n}=2wI_{n},\ \ n=2,3,\cdots.$
Proof. For the simplicity, let us select $v_{0}=w_{0}=0$ in the transformation
(6.9). We introduce a new potential function
$None$ $q_{2x}=\eta+\varepsilon\eta_{x}+\varepsilon^{2}\eta^{2},$
where $\varepsilon$ is a constant parameter. Substituting (9.4) into the Lax
equation (7.10) leads to
$\displaystyle
0=[L,M]=(1+\varepsilon\partial_{x}+2\varepsilon^{2}\eta)[-4(\eta+\varepsilon^{2}\eta^{2})\eta_{t}-2(q_{x}-\varepsilon\eta)_{t}\eta_{x}+\eta_{2x,t}],$
which implies that $v=q_{2x},\ w=q_{xt}$ given by (9.4) are a solution of the
NKdV equation (1.1) if $\eta$ satisfies the following equation
$None$
$-4(\eta+\varepsilon^{2}\eta^{2})\eta_{t}-2(q_{x}-\varepsilon\eta)_{t}\eta_{x}+\eta_{2x,t}-4\eta_{t}=0.$
On the other hand, it follows from (9.5) that
$[(q_{x}-\varepsilon\eta)_{t}]_{x}=-(\eta+\varepsilon^{2}\eta^{2})_{t}.$
Therefore, the equation (9.5) can be rewritten as
$(\eta_{2x}-\eta^{2})_{t}+[2\eta(\varepsilon^{2}\eta-q_{x})_{t}]_{x}=0,$
or a divergent-type form
$None$ $(\eta_{2x}+2\varepsilon^{2}\eta\eta_{x}-\eta^{2})_{t}+(2\eta w)_{x}=0$
by replacing $q_{xt}=w$.
Inserting the expansion
$None$ $\eta=\sum_{n=0}^{\infty}I_{n}(q,q_{x},q_{t}\cdots)\varepsilon^{n},$
into the equation (9.4) and comparing the coefficients for power of
$\varepsilon$, we obtain the recursion relations to calculate $I_{n}$ in an
explicit form
$None$ $\displaystyle I_{0}=q_{2x}=v,\ \ I_{1}=-I_{0,x}=-v_{x},$
$\displaystyle I_{n}=-I_{n-1,x}-\sum_{k=0}^{n-2}I_{k}I_{n-2-k},\ \
n=2,3,\cdots.$
Substituting (9.7) into (9.6) and simplifying terms in the power of
$\varepsilon$ provide us infinitely many conservation laws
$F_{n,t}+G_{n,x}=0,\ n=1,2,\cdots$
where the conversed densities $F_{n}^{\prime}s$ and the fluxes
$G_{n}^{\prime}s$ are by (9.2) and (9.3), respectively. $\square$
Here, we already give recursion formulas (9.7) and (9.8) to show how to
generate conservation laws (9.6) based on the first few explicitly provided.
Apparently, the first equation in conservation laws (9.6)
$v_{xxt}-2vv_{t}+2wv_{x}+2w_{x}v=0$
is exactly the NKdV equation (1.1)
$\displaystyle v_{t}+w_{x}=0,$ $\displaystyle w_{xxx}+4vw_{x}+2wv_{x}=0.$
which is reduced to the NKdV equation (1.2) under the constraints
$v=-u_{xx}/u$ and $w=u^{2}.$
In conclusion, the NKdV equation (1.1) is completely integrable and admits
bilinear Bäcklund transformation, Lax pair and infinitely many local
conservation laws.
10\. Quasi-periodic solutions of the NKdV equation
In this section, we study quasi-periodic wave solutions of the NKdV equation
(1.1) by using bilinear Bäcklund transformation (7.1) and bilinear formulas
derived in section 9.
In fact, a quasi-periodic solution, also called algebro-geometric solutions or
finite gap solutions, was originally studied in the late 1970s by Novikov,
Dubrovin, McKean, Lax, Its, and Matveev et al [12, 37, 50, 58], based on the
inverse spectral theory and algebro-geometric method. In recent years, this
theory has been extended to a large class of nonlinear integrable equations
including sine-Gordon equation, Camassa-Holm equation, Thirring model
equation, Kadomtsev-Petviashvili equation, Ablowitz-Ladik lattice, and Toda
lattice [8, 22, 23, 24, 26, 25, 32, 68, 63, 69, 83, 88, 87]. The algebro-
geometric theory, however, needs Lax pairs and is also involved in complicated
analysis procedure on the Riemann surfaces. It is rather difficult to directly
determine the characteristic parameters of waves, such as frequencies and
phase shifts for a function with given wave-numbers and amplitudes. On the
other hand, the bilinear derivative method developed by Hirota is a powerful
approach for constructing exact solution of nonlinear equations in an explicit
form. If a nonlinear equation is able to be written in a bilinear form by a
dependent variable transformation, then multi-solitary wave solutions are
usually obtained for the equation [30, 31, 35, 36, 85]. Based on the Hirota
forms, Nakamura proposed a convenient way to find a kind of explicit quasi-
periodic solutions of nonlinear equations [57], where the periodic wave
solutions of the KdV equation and the Boussinesq equation were obtained. Such
a method indeed displays some advantages over algebro-geometric methods. For
example, it does not need any Lax pair and Riemann surface for the given
nonlinear equation, and is also able to find the explicit construction of
multi-periodic wave solutions. The method relies on the existence of the
Hirota’s bilinear form as well as arbitrary parameters appearing in Riemann
matrix [14, 15].
10.1. Multi-dimensional Riemann theta functions
Let us first begin with some preliminary work about multi-dimensional Riemann
theta functions and their quasi-periodicity. The multi-dimensional Riemann
theta function is defined by
$None$
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{s}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{{Z}}^{N}}\exp\\{2\pi
i\langle\boldsymbol{\zeta}+\boldsymbol{\varepsilon},\boldsymbol{n}+\boldsymbol{s}\rangle-\pi\langle\boldsymbol{\tau}(\boldsymbol{n}+\boldsymbol{s}),\boldsymbol{n}+\boldsymbol{s}\rangle\\},$
where $\boldsymbol{n}=(n_{1},\cdots,n_{N})^{T}\in\mathbb{Z}^{N}$ is an integer
value vector, and
$\boldsymbol{s}=(s_{1},\cdots,s_{N})^{T},\boldsymbol{\varepsilon}=(\varepsilon_{1},\cdots,\varepsilon_{N})^{T}\in\mathbb{{C}}^{N}$
is a complex parameter vector.
$\boldsymbol{\zeta}=(\zeta_{1},\cdots,\zeta_{N})^{T},\
\zeta_{j}=\alpha_{j}x+\beta_{j}t+\delta_{j}$, $\
\alpha_{j},\beta_{j},\delta_{j}\in\Lambda_{0}$, $j=1,2,\cdots,N$ are complex
phase variables, where $x,t$ are ordinary real variables and $\theta$ is a
Grassmann variable. The inner product of two vectors
$\boldsymbol{f}=(f_{1},\cdots,f_{N})^{T}$ and
$\boldsymbol{g}=(g_{1},\cdots,g_{N})^{T}$ is defined by
$\langle\boldsymbol{f},\boldsymbol{g}\rangle=f_{1}g_{1}+f_{2}g_{2}+\cdots+f_{N}g_{N}.$
The matrix $\boldsymbol{\tau}=(\tau_{ij})$ is a positive definite and real-
valued symmetric $N\times N$ matrix. The entries $\tau_{ij}$ of the periodic
matrix $\boldsymbol{\tau}$ can be considered as free parameters of the theta
function (10.1).
In this paper, we choose $\tau$ to be purely imaginary matrix to make the
theta function (10.1) real-valued. In definition (10.1) for the case of
$\boldsymbol{s}=\boldsymbol{\varepsilon}=\boldsymbol{0}$, we denote
$\vartheta(\boldsymbol{\zeta},{\boldsymbol{\tau}})=\vartheta(\boldsymbol{\zeta},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})$
for simplicity. Therefore, we have
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\vartheta(\boldsymbol{\zeta}+\boldsymbol{\varepsilon},\boldsymbol{\tau})$.
Remark 4. The above periodic matrix $\boldsymbol{\tau}$ is different from the
one in the algebro-geometric approach discussed in [58]-[15], where it is
usually constructed on a compact Riemann surface $\Gamma$ with genus
$N\in\mathbb{N}$. One may see that the entries in the matrix
$\boldsymbol{\tau}$ are not free and difficult to be explicitly given.
$\square$
Definition 3. A function $g(\boldsymbol{x},t)$ on
$\mathbb{C}^{N}\times\mathbb{C}$ is said to be quasi-periodic in $t$ with
fundamental periods $T_{1},\cdots,T_{k}\in\mathbb{C}$ if $T_{1},\cdots,T_{k}$
are linearly dependent over $\mathbb{Z}$ and there exists a function
$G(\boldsymbol{x},\boldsymbol{y})\in\mathbb{C}^{N}\times\mathbb{C}^{k}$ such
that
$G(\boldsymbol{x},y_{1},\cdots,y_{j}+T_{j},\cdots,y_{k})=G(\boldsymbol{x},y_{1},\cdots,y_{j},\cdots,y_{k}),\
\ {\rm for\ all}\ y_{j}\in\mathbb{C},\ j=1,\cdots,k.$
$G(\boldsymbol{x},t,\cdots,t,\cdots,t)=g(\boldsymbol{x},t).$
In particular, $g(\boldsymbol{x},t)$ becomes periodic with $T$ if and only if
$T_{j}=m_{j}T$. $\square$
Let’s first see periodicity of the theta function
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$.
Proposition 10. [56] Let $\boldsymbol{e_{j}}$ be the $j-$th column of $N\times
N$ identity matrix $I_{N}$; ${\tau_{j}}$ be the $j-$th column of
$\boldsymbol{\tau}$, and $\tau_{jj}$ the $(j,j)$-entry of $\boldsymbol{\tau}$.
Then the theta function $\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$ has
the periodic properties
$\displaystyle\vartheta(\boldsymbol{\zeta}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})=\exp(-2\pi
i\zeta_{j}+\pi\tau_{jj})\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau}).$
The theta function $\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$ which
satisfies the condition (5.4) is called a multiplicative function. We regard
the vectors $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau_{j}},\ \ j=1,\cdots,N\\}$ as periods of the theta
function $\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$ with multipliers
$1$ and $\exp({-2\pi i\zeta_{j}+\pi\tau_{jj}})$, respectively. Here, only the
first $N$ vectors are actually periods of the theta function
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$, but the last $N$ vectors
are the periods of the functions
$\partial^{2}_{\zeta_{k},\zeta_{l}}\ln\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})$
and
$\partial_{\zeta_{k}}\ln[\vartheta(\boldsymbol{\zeta}+\boldsymbol{e},\boldsymbol{\tau})/\vartheta(\boldsymbol{\zeta}+\boldsymbol{h},\boldsymbol{\tau})],\
k,l=1,\cdots,N$.
Proposition 11. Let $\boldsymbol{e_{j}}$ and $\boldsymbol{\tau_{j}}$ be
defined as above proposition 2. The meromorphic functions
$f(\boldsymbol{\zeta})$ are as follow
$\displaystyle(i)\ \ \ \ \
f(\boldsymbol{\zeta})=\partial_{\zeta_{k}\zeta_{l}}^{2}\ln\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau}),\
\ \boldsymbol{\zeta}\in C^{N},\ \ \ k,l=1,\cdots,N,$ $\displaystyle(ii)\ \ \ \
f(\boldsymbol{\zeta})=\partial_{\zeta_{k}}\ln\frac{\vartheta(\boldsymbol{\zeta}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\zeta}+\boldsymbol{h},\boldsymbol{\tau})},\
\ \boldsymbol{\zeta},\ \boldsymbol{e},\ \boldsymbol{h}\in C^{N},\ \
j=1,\cdots,N.$
then in all two cases (i) and (ii), it holds that
$\displaystyle
f(\boldsymbol{\zeta}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}})=f(\boldsymbol{\zeta}),\
\ \ \boldsymbol{\zeta}\in C^{N},\ \ \ j=1,\cdots,N,$
which implies that $f(\boldsymbol{\zeta})$ is a quasi-periodic function.
10.2. Bilinear formulae of theta functions
To construct a kind of explicitly quasi-periodic solutions of the NKdV
equation (1.1), we propose some important bilinear formulas of multi-
dimensional Riemann theta functions, whose derivations are similar to the case
of super bilinear equations [16], so we just list them without proofs.
Theorem 11. Suppose that
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are two Riemann theta functions, in which
$\boldsymbol{\varepsilon}=(\varepsilon_{1},\dots,\varepsilon_{N})$,
$\boldsymbol{\varepsilon^{\prime}}=(\varepsilon_{1}^{\prime},\dots,\varepsilon_{N}^{\prime})$,
and $\boldsymbol{\zeta}=(\zeta_{1},\cdots,\zeta_{N})$,
$\zeta_{j}=\alpha_{j}x+\omega_{j}t+\delta_{j},\ \ j=1,2,\cdots,N$. Then
operators $D_{x},D_{t}$ and $S$ exhibit the following perfect properties when
they act on a pair of theta functions
$None$ $\displaystyle
D_{x}\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
$\displaystyle=\sum_{\boldsymbol{\mu}}\partial_{x}\vartheta(2\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon},-\boldsymbol{\mu}/2|2\boldsymbol{\tau})|_{\boldsymbol{\zeta}=\boldsymbol{0}}\vartheta(2\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
where $\boldsymbol{\mu}=(\mu_{1},\cdots,\mu_{N})$, and the notation
$\sum_{\boldsymbol{\mu}}$ represents $2^{N}$ different transformations
corresponding to all possible combinations $\mu_{1}=0,1;\cdots;\mu_{N}=0,1$.
In general, for a polynomial operator $H(D_{x},D_{t})$ with respect to $D_{x}$
and $D_{t}$, we have the following useful formula
$None$ $\displaystyle
H(D_{x},D_{t})\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{\mu}}C(\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\varepsilon},\boldsymbol{\mu})\vartheta(2\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
in which, explicitly
$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}H(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$
where we denote $\boldsymbol{\mathcal{M}}=(4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle,\ 4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\omega}\rangle).$
Remark 6. The formulae (10.3) and (10.4) show that if the following equations
are satisfied
$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=0,$
for all possible combinations $\mu_{1}=0,1;\mu_{2}=0,1;\cdots;\mu_{N}=0,1$, in
other word, all such combinations are solutions of equation (10.5), then
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are $N$-periodic wave solutions of the bilinear equation
$H(D_{x},D_{t})\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\zeta},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=0.$
We call the formula (10.5) constraint equations, whose number is $2^{N}$. This
formula actually provides us an unified approach to construct multi-periodic
wave solutions for supersymmetric equations. Once a supersymmetric equation is
written bilinear forms, then its multi-periodic wave solutions can be directly
obtained by solving system (10.5).
Theorem 12. Let
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
and $H(D_{x},D_{t})$ be given in Theorem 10, and make a choice such that
$\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\ j=1,\cdots,N$. Then
(i) If $H(D_{x},D_{t})$ is an symmetric operator, i. e.
$H(-D_{x},-D_{t})=H(D_{x},D_{t}),$
then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an odd
number, namely
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2.$
(ii) If $H(D_{x},D_{t})$ is a skew-symmetric operator, i.e.
$H(-D_{x},-D_{t})=-H(D_{x},D_{t}),$
then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an even
number, namely
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$
Proposition 12. Let $\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\
j=1,\cdots,N$. Assume $H(D_{x},D_{t})$ is a linear combination of even and odd
functions
$H(D_{x},D_{t})=H_{1}(D_{x},D_{t})+H_{2}(D_{x},D_{t}),$
where $H_{1}$ is even and $H_{2}$ is odd. In addition,
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
corresponding (10.8) is given by
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})+C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),$
where
$C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}H_{1}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right],$
$C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}H_{2}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$
Then
$\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2,$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$
The theorem 2 and corollary 1 are very useful to deal with coupled super-
Hirota’s bilinear equations, which will be seen in the following section 10.
By introducing differential operators
$\displaystyle\nabla=(\partial_{\zeta_{1}},\partial_{\zeta_{2}},\cdots,\partial_{\zeta_{N}}),$
$\displaystyle\partial_{x}=\alpha_{1}\partial_{\zeta_{1}}+\alpha_{2}\partial_{\zeta_{2}}+\cdots+\alpha_{N}\partial_{\zeta_{N}}=\boldsymbol{\alpha}\cdot\nabla,$
$\displaystyle\partial_{t}=\beta_{1}\partial_{\zeta_{1}}+\beta_{2}\partial_{\zeta_{2}}+\cdots+\beta_{N}\partial_{\zeta_{N}}=\boldsymbol{\beta}\cdot\nabla,$
then we have
$\displaystyle\partial_{x}^{k}\partial_{t}^{l}\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau})=(\boldsymbol{\alpha}\cdot\nabla)^{k}(\boldsymbol{\beta}\cdot\nabla)^{l}\vartheta(\boldsymbol{\zeta},\boldsymbol{\tau}),\
\ k,l=0,1,\cdots.$
10.3. One-periodic waves and asymptotic analysis
Let us first construct one-periodic wave solutions of the NKdV equation (1.1)
by using bilinear Bácklund transformation (7.6). As a simple case of the theta
function (10.1) with $N=1,s=0$, we choose $F$ and $G$ as follows
$None$ $\displaystyle
F=\vartheta(\zeta,0,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in\zeta-\pi
n^{2}\tau}),$ $\displaystyle
G=\vartheta(\zeta,1/2,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi
in(\zeta+1/2)-\pi n^{2}\tau})$ $\displaystyle\ \ \
=\sum_{n\in\mathbb{Z}}(-1)^{n}\exp({2\pi in\zeta-\pi n^{2}\tau}),$
where $\zeta=\alpha x+\beta t+\delta$ is the phase variable, and $\tau>0$ is a
positive parameter.
By Theorem 6, the operator $H_{1}=D_{x}^{2}-\lambda$ in bilinear equation
(7.6) is symmetric, and its corresponding constraint equation in the formula
(10.5) automatically vanishes for $\mu=1$. Meanwhile,
$H_{2}=D_{t}D_{x}^{2}-2w_{0}D_{x}+(4v_{0}+3\lambda)D_{t}$ are skew-symmetric,
and its corresponding constraint equation automatically vanishes for $\mu=0$.
Therefore, the Riemann theta function (10.6) is a solution of the bilinear
equation (7.6), provided the following equations
$None$ $\displaystyle\sum_{n\in\mathbb{Z}}\\{[4\pi
i(n-\mu/2)]^{2}\alpha^{2}-\lambda\\}\exp(-2\pi\tau(n-\mu/2)^{2}+\pi
i(n-\mu/2))|_{\mu=0}=0,$ $\displaystyle\sum_{n\in\mathbb{Z}}\\{[4\pi
i(n-\mu/2)]^{3}\alpha^{2}\beta+8\pi i(n-\mu/2)\alpha w_{0}+4\pi
i(n-\mu/2)(4v_{0}+3\lambda)\beta\\}$
$\displaystyle\times\exp(-2\pi\tau(n-\mu/2)^{2}+\pi i(n-\mu/2))|_{\mu=1}=0.$
hold.
Let
$\displaystyle\rho=e^{-\pi\tau/2},$
$\displaystyle\vartheta_{1}(\zeta,\rho)=\vartheta(2\mathbf{\zeta},1/4,-1/2|2\tau)=\sum_{n\in\mathbb{Z}}\rho^{(2n-1)^{2}}\exp[4i\pi(n-1/2)(\zeta+1/4)],$
$\displaystyle\vartheta_{2}(\zeta,\rho)=\vartheta(2\zeta,1/4,0|2\tau)=\sum_{n\in\mathbb{Z}}\rho^{4n^{2}}\exp[4i\pi
n(\zeta+1/4)],$
then, the equation (10.7) can be written as a linear system about $\beta$ and
$\lambda$
$None$
$\displaystyle\vartheta_{2}^{\prime\prime}\alpha^{2}-\vartheta_{2}\lambda=0,$
$\displaystyle\vartheta_{1}^{\prime\prime\prime}\alpha^{2}\beta+2\vartheta_{1}^{\prime}\alpha
w_{0}+(4v_{0}+3\lambda)\vartheta_{1}^{\prime}\beta=0,$
where the derivative value of $\vartheta_{j}(\zeta,\rho)$ at $\zeta=0$ is
denoted by simple notations
$\vartheta_{j}^{\prime}=\vartheta_{j}^{\prime}(0,\rho)=\frac{d\vartheta_{j}(\zeta,\rho)}{d\zeta}|_{\zeta=0},\
\ j=1,2.$
It is not hard to see that the system (10.8) admits the following solution for
the NKdV equation (1.1)
$None$
$\displaystyle\lambda=\frac{\vartheta_{2}^{\prime\prime}\alpha^{2}}{\vartheta_{2}},\
\ \
\beta=\frac{-2\vartheta_{1}^{\prime}\vartheta_{2}w_{0}}{\vartheta_{1}^{\prime\prime\prime}\vartheta_{2}\alpha^{2}+4\vartheta_{1}^{\prime}\vartheta_{2}v_{0}+3\vartheta_{1}^{\prime}\vartheta_{2}^{\prime\prime}\alpha^{2}}.$
So, we obtain the following one-periodic wave solution
$None$ $V=v_{0}+2\partial_{x}^{2}\ln\vartheta(\zeta,0,0|\tau),\ \
W=w_{0}+2\partial_{x}\partial_{t}\ln\vartheta(\zeta,0,0|\tau),$
where $\zeta=\alpha x+\beta t+\delta$ and parameter $\beta$ is given by
(10.9), while other parameters $\alpha,\tau,v_{0},w_{0}$ are arbitrary. Among
the four parameters, the two ones $\alpha$ and $\tau$ completely dominate a
one-periodic wave.
In summary, one-periodic wave (10.10) is one-dimensional and has two
fundamental periods $1$ and $i\tau$ in phase variable $\zeta$ (see Figure 3).
(a) (b)
(c) (d)
Figure 3. One-periodic wave for the NKdV equation (1.1) with parameters:
$\alpha=0.6,$ $\tau=2,v_{0}=0.5,w_{0}=1$. (a) and (b) show that every one-
periodic wave is periodic in both $x$ and $y$ directions. (c) Perspective view
of the wave. (d) Overhead view of the wave, with contour plot shown. The
bright hexagons are crests and the dark hexagons are troughs.
In the following theorem, we will see that the one-periodic wave solution
(10.10) can be broken into soliton solution (6.20) under a long time limit and
their relation can be established as follows.
Theorem 13. In the one-periodic wave solution (10.6), the parameter $\beta$ is
given by (10.9), other parameters are chosen as
$None$ $\alpha=\frac{k}{2\pi i},\ \ \ \delta=\frac{\gamma+\pi\tau}{2\pi i},$
where $k_{1}$ and $\gamma$ are the same as those in (6.20). Then under a small
amplitude limit, one-periodic wave solution (10.10) can be broken into the
single soliton solutions (6.20), that is,
$None$ $V\longrightarrow v,\ \ W\longrightarrow w,\ \ {\rm as}\ \
\rho\rightarrow 0.$
In particular, in the case of $v_{0}=0,\ w_{0}=1$, the one-periodic solution
(10.5) tends to the kink-type soliton solution (5.2), that is,
$None$ $V\longrightarrow\tilde{v}^{I},\ \ W\longrightarrow\tilde{w}^{I},\ \
{\rm as}\ \ \rho\rightarrow 0.$
Proof. Here we use the system (10.8) to analyze asymptotic properties of the
one-periodic solution (10.10). Let us expand the coefficients of the system
(10.8) as follows
$None$
$\displaystyle\vartheta_{1}^{\prime}=-4\pi\rho+12\pi\rho^{9}+\cdots,\quad\vartheta_{1}^{\prime\prime\prime}=16\pi^{3}\rho+432\pi^{3}\rho^{9}+\cdots,$
$\displaystyle\vartheta_{2}=1+2\rho^{4}+\cdots,\quad\vartheta_{2}^{\prime\prime}=32\pi^{2}\rho^{4}+\cdots,$
Suppose that the solution of the system (10.8) has the following form
$None$
$\displaystyle\lambda=\lambda_{0}+\lambda_{1}\rho+\lambda_{2}\rho^{2}+\cdots=\lambda_{0}+o(\rho),$
$\displaystyle\beta=\beta_{0}+\beta_{1}\rho+\beta_{2}\rho^{2}+\cdots=\beta_{0}+o(\rho).$
Substituting the expansions (10.14) and (10.15) into the system (10.8) and
letting $\rho\rightarrow 0$, we immediately obtain the following relation
$None$ $\displaystyle\lambda_{0}=0,\ \ \beta_{0}=\frac{-\alpha
w_{0}}{-2\pi^{2}\alpha^{2}+2v_{0}}.$
Combining (10.11) and (10.16) leads to
$\displaystyle\lambda\longrightarrow 0,$ $\displaystyle 2\pi
i\beta\longrightarrow 2\pi i\beta_{0}=\frac{-2\pi i\alpha
w_{0}}{-2\pi^{2}\alpha^{2}+2v_{0}}=\frac{-2kw_{0}}{k^{2}+4v_{0}},\ \ {\rm as}\
\ \rho\rightarrow 0,$
or equivalently rewritten as
$None$ $\displaystyle\hat{\zeta}=2\pi i\zeta-\pi\tau=kx+2\pi i\beta t+\gamma$
$\displaystyle\quad\longrightarrow
kx-\frac{2kw_{0}}{k^{2}+4v_{0}}t+\gamma=\xi,\ \ {\rm as}\ \ \rho\rightarrow
0.$
It remains to verify that the one-periodic wave (10.11) has the same form as
the one-soliton solution (6.20) under the limit $\rho\rightarrow 0$. Let us
expand the function $F$ in the following form
$F=1+\rho^{2}(e^{2\pi i\zeta}+e^{-2\pi i\zeta})+\rho^{8}(e^{4\pi
i\zeta}+e^{-4\pi i\zeta})+\cdots.$
It follows from (10.11) and (10.17)
$None$ $\displaystyle
F=1+e^{\hat{\zeta}}+\rho^{4}(e^{-\hat{\zeta}}+e^{2\hat{\zeta}})+\rho^{12}(e^{-2\hat{\zeta}}+e^{3\hat{\zeta}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\zeta}}\longrightarrow
1+e^{\xi},\ \ {\rm as}\ \ \rho\rightarrow 0.$
So, combining (10.11) and (10.18) yields
$\displaystyle v\longrightarrow v_{0}+2\partial_{xx}\ln(1+e^{\xi}),$
$\displaystyle w\longrightarrow
w_{0}+2\partial_{t}\partial_{x}\ln(1+e^{\xi}),\ \ {\rm as}\ \ \rho\rightarrow
0.$
Thus, we conclude that the one-periodic solution (10.10) may go to a bell-type
soliton solutions (6.20) as the amplitude $\rho\rightarrow 0$. $\square$
10.4. Two-periodic waves and asymptotic properties
Let us now consider two-periodic wave solutions to the NKdV equation (1.1).
For the case of $N=2,\ \boldsymbol{s}=\boldsymbol{0},\
\boldsymbol{\varepsilon}=\boldsymbol{1/2}=(1/2,1/2)$ in the Riemann theta
function (10.1), we choose $F$ and $G$ as follows
$None$ $\displaystyle
F=\vartheta(\boldsymbol{\zeta},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\zeta},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
$\displaystyle
G=\vartheta(\boldsymbol{\zeta},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\zeta}+\boldsymbol{1/2},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
$\displaystyle\ \
=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}(-1)^{n_{1}+n_{2}}\exp\\{2\pi
i\langle\boldsymbol{\zeta},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
where $\boldsymbol{n}=(n_{1},n_{2})\in Z^{2},\ \
\boldsymbol{\zeta}=(\zeta_{1},\zeta_{2})\in\mathcal{C}^{2},\ \
\zeta_{i}=\alpha_{j}x+\beta_{j}t+\delta_{j},\ \ j=1,2$, and
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2}),\
\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathcal{C}^{2}$. The matrix
$\boldsymbol{\tau}$ is a positive definite and real-valued symmetric ${2\times
2}$ matrix that is,
$\boldsymbol{\tau}=(\tau_{ij})_{2\times 2},\ \ \tau_{12}=\tau_{21},\ \
\tau_{11}>0,\ \ \tau_{22}>0,\ \ \tau_{11}\tau_{22}-\tau_{12}^{2}>0.$
According to Theorem 5, constraint equations associated with
$H_{1}=D_{x}^{2}-\lambda$ and
$H_{2}=D_{t}D_{x}^{2}-2w_{0}D_{x}+(4v_{0}+3\lambda)D_{t}$ automatically vanish
for $(\mu_{1},\mu_{2})=(0,1),(1,0)$ and for $(\mu_{1},\mu_{2})=(0,0),(1,1)$,
respectively. Hence, making the theta functions (10.19) satisfy the bilinear
equation (7.6) gives the following constraint equations
$None$
$\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[-16\pi^{2}\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle^{2}-\lambda\right]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle$
$\displaystyle+\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,\
\ {\rm for}\ \ (\mu_{1},\mu_{2})=(0,0),\ (1,1)=0,$
$\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[-64\pi^{3}i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle^{2}\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\beta}\rangle+8\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle
w_{0}+4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\beta}\rangle(4v_{0}+3\lambda)\right]$
$\displaystyle\times\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle+\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,$
$\displaystyle{\rm for}\ \ (\mu_{1},\mu_{2})=(0,1),\ (1,0).$
Let
$\displaystyle\rho_{kl}=e^{-\pi\tau_{kl}/2},k,l=1,2,\boldsymbol{\rho}=(\rho_{11},\rho_{12},\rho_{22})$
$\displaystyle\vartheta_{j}(\boldsymbol{\zeta},\boldsymbol{\rho})=\vartheta(2\zeta,\boldsymbol{1/4},-\boldsymbol{s}_{j}/2|2\tau)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ =\sum_{n_{1},n_{2}\in Z}\exp\\{4\pi
i\langle\boldsymbol{\zeta+1/4},\boldsymbol{n}-\boldsymbol{s_{j}}/2\rangle\\}\prod_{k,l=1}^{2}\rho_{kl}^{(2n_{k}-s_{j,k})(2n_{j}-s_{j,l})},$
$\displaystyle\boldsymbol{s_{j}}=(s_{j,1},s_{j,2}),\quad j=1,2,\ \
\boldsymbol{s}_{1}=(0,1),\ \ \boldsymbol{s_{2}}=(1,0),\ \
\boldsymbol{s_{3}}=(0,0),\ \ \boldsymbol{s_{4}}=(1,1)$
then the system (10.20) can be rewritten as a linear system
$None$
$\displaystyle(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}-\lambda\vartheta_{j}=0,\
j=3,4,$ $None$
$\displaystyle(\boldsymbol{\beta}\cdot\nabla)(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}+2w_{0}(\boldsymbol{\alpha}\cdot\nabla)\vartheta_{j}+(4v_{0}+3\lambda)(\boldsymbol{\beta}\cdot\nabla)\vartheta_{j}=0,\
j=1,2,$
where $\vartheta_{j}$ represent the derivative values of functions
$\vartheta_{j}(\boldsymbol{\zeta},\boldsymbol{\rho})$ at
$\zeta_{1}=\zeta_{2}=0$.
The system (10.22) admits a unique solution
$None$
$\displaystyle\left(\begin{matrix}\beta_{1}\cr\beta_{2}\end{matrix}\right)=\left[\frac{\partial(f,g)}{\partial(\zeta_{1},\zeta_{2})}\right]^{-1}\left(\begin{matrix}2w_{0}(\boldsymbol{\alpha}\cdot\nabla)\vartheta_{1}\cr
2w_{0}(\boldsymbol{\alpha}\cdot\nabla)\vartheta_{2}\end{matrix}\right)$
where $\frac{\partial(f,g)}{\partial(\zeta_{1},\zeta_{2})}$ is the Wronskinan
matrix given by
$\displaystyle\frac{\partial(f,g)}{\partial(\zeta_{1},\zeta_{2})}=\left(\begin{matrix}\partial_{\zeta_{1}}f&\partial_{\zeta_{2}}f\cr\partial_{\zeta_{1}}g&\partial_{\zeta_{2}}g\end{matrix}\right),$
$\displaystyle
f=[(\boldsymbol{\alpha}\cdot\nabla)^{2}+4v_{0}+3\lambda]\vartheta_{1},\ \
g=[(\boldsymbol{\alpha}\cdot\nabla)^{2}+4v_{0}+3\lambda]\vartheta_{2}.$
With the help of the above $(\beta_{1},\beta_{2})$, we are able to get a two-
periodic wave solution to the NKdV equation (1.1)
$None$
$V=v_{0}+\partial_{x}^{2}\ln\vartheta(\boldsymbol{\zeta},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau}),\
\
W=w_{0}+\partial_{x}\partial_{t}\vartheta(\boldsymbol{\zeta},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau}),$
where $\alpha_{1},$ $\alpha_{2},\tau_{12},\delta_{1}$ and $\delta_{2}$ are
arbitrary parameters, while other parameters $\beta_{1},\beta_{2}$ and
$\tau_{11}$, $\tau_{22}$ are given by (10.23) and (10.21), respectively.
In summary, the two-periodic wave (10.24) is a direct generalization of two
one-periodic waves. Its surface pattern is two-dimensional with two phase
variables $\zeta_{1}$ and $\zeta_{2}$. The two-periodic wave (10.24) has $4$
fundamental periods $\\{e_{1},e_{2}\\}$ and $\\{i\tau_{1},i\tau_{2}\\}$ in
$(\zeta_{1},\zeta_{2})$, and is spatially periodic in two directions
$\zeta_{1},\zeta_{2}$. Its real part is not periodic in $\theta_{1}$
direction, while its imaginary part and modulus are all periodic in both $x$
and $t$ directions.
(a) (b)
(c) (d)
Figure 4. Two-periodic wave for the NKdV equation (1.1). (a) and (b) show that
every one-periodic wave is periodic in both $x$\- and $y$-directions. (c)
Perspective view of the wave. (d) Overhead view of the wave, with contour plot
shown. The bright hexagons are crests and the dark hexagons are troughs.
Finally, we study the asymptotic properties of the two-periodic solution
(10.24). In a similar way to Theorem 5, we figure out the relation between the
two-periodic solution (10.24) and the two-soliton solution (6.21) as follows.
Theorem 14. Assume that $(\beta_{1},\beta_{2})$ is a solution of the system
(10.22), and in the two-periodic wave solution (10.24), parameters
$\alpha_{j},\delta_{j},\tau_{12}$ are chosen as
$None$ $\displaystyle\alpha_{j}=\frac{k_{j}}{2\pi i},\ \
\delta_{j}=\frac{\gamma_{j}+\pi\tau_{jj}}{2\pi i},\ \
\tau_{12}=-\frac{A_{12}}{2\pi},\ \ j=1,2,$
where $k_{j},\gamma_{j},j=1,2$ and $A_{12}$ are those given in (6.21). Then,
we have the following asymptotic relations
$None$ $\displaystyle\lambda\longrightarrow 0,\ \ \
\zeta_{j}\longrightarrow\frac{\eta_{j}+\pi\tau_{jj}}{2\pi i},\ \ j=1,2,$
$\displaystyle F\longrightarrow
1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},\ \ {\rm as}\ \
\rho_{11},\rho_{22}\rightarrow 0.$
So, the two-periodic wave solution (10.24) just tends to the two-soliton
solution (6.21) under a limit condition
$V\longrightarrow v,\ \ W\longrightarrow w,\ \ {\rm as}\ \
\rho_{11},\rho_{22}\rightarrow 0.$
Proof. Using (10.20), we may expand the function $F$ in the following explicit
form
$\displaystyle F=1+(e^{2\pi i\zeta_{1}}+e^{-2\pi
i\zeta_{1}})e^{-\pi\tau_{11}}+(e^{2\pi i\zeta_{2}}+e^{-2\pi
i\zeta_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ +(e^{2\pi
i(\zeta_{1}+\zeta_{2})}+e^{-2\pi
i(\zeta_{1}+\zeta_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$
Furthermore, adopting (10.25) and making a transformation we infer that
$\displaystyle
F=1+e^{\hat{\zeta}_{1}}+e^{\hat{\zeta}_{2}}+e^{\hat{\zeta}_{1}+\hat{\zeta}_{2}-2\pi\tau_{12}}+\rho_{11}^{4}e^{-\hat{\zeta}_{1}}+\rho_{22}^{4}e^{-\hat{\zeta}_{2}}+\rho_{11}^{4}\rho_{22}^{4}e^{-\hat{\zeta}_{1}-\hat{\zeta}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1+e^{\hat{\zeta}_{1}}+e^{\hat{\zeta}_{2}}+e^{\hat{\zeta}_{1}+\hat{\zeta}_{2}+A_{12}},\
\ {\rm as}\ \ \rho_{11},\rho_{22}\rightarrow 0,$
where $\hat{\zeta}_{j}=\alpha_{j}x+\hat{\beta}_{j}t+\delta_{j},\ \ j=1,2,$ and
$\hat{\beta}_{j}=2\pi i\beta_{j},j=1,2$.
Now, we need to prove
$None$
$\displaystyle\hat{\beta}_{j}\longrightarrow\frac{-2k_{j}w_{0}}{k_{j}^{2}+4v_{0}},\
\ \hat{\zeta}_{j}\longrightarrow\xi_{j},\ \ j=1,2,\ \ \ {\rm as}\ \
\rho_{11},\rho_{22}\rightarrow 0.$
As in the case of $N=1$, the solution of the system (10.23) has the following
form
$None$
$\displaystyle\beta_{1}=\beta_{1,0}+\beta_{1,1}\rho_{11}+\beta_{2,2}\rho_{22}+o(\rho_{11},\rho_{22}),$
$\displaystyle\beta_{2}=\beta_{2,0}+\beta_{2,1}\rho_{11}+\beta_{2,2}\rho_{22}+o(\rho_{11},\rho_{22}),$
$\displaystyle\lambda=\lambda_{0}+\lambda_{1}\rho_{11}+\lambda_{2}\rho_{22}+o(\rho_{11},\rho_{22}).$
Expanding functions $\vartheta_{j},\ j=1,2,3,4$ in equations (10.21) and
(10.22) with substitution of assumption (10.28), and letting
$\rho_{11},\rho_{22}\longrightarrow 0$ , we will obtain
$None$ $\displaystyle\lambda_{0}=0,$ $\displaystyle 16\pi
i(-\pi^{2}\alpha_{1}^{2}+v_{0})\beta_{1,0}-8\pi iw_{0}\alpha_{1}=0,$
$\displaystyle 16\pi i(-\pi^{2}\alpha_{2}^{2}+v_{0})\beta_{2,0}-8\pi
iw_{0}\alpha_{2}=0.$
Using (10.28) and (10.29), we conclude that
$\displaystyle\lambda=o(\rho_{11},\rho_{22})\longrightarrow 0,$
$\displaystyle\beta_{j}=\frac{-2k_{j}w_{0}}{k_{j}^{2}+4v_{0}}+o(\rho_{11},\rho_{22})\longrightarrow\frac{-2k_{j}w_{0}}{k_{j}^{2}+4v_{0}},\
\ \ {\rm as}\ \ \rho_{11},\rho_{22}\rightarrow 0,$
and therefore we have (10.26). So, the two-periodic wave solution (10.24)
tends to the two-soliton solution (6.21) as $\rho_{11},\rho_{22}\rightarrow
0$. $\square$
10.5. Multi-periodic wave solutions
The system (10.5) indicates that constructing multi-periodic wave solutions
depends on the solvability of the system (10.5). Obviously, the number of
constraint equations of the type (10.5) is $2^{N-1}+1$. On the other hand, we
have $\frac{1}{2}N(N+1)+3N+3$ parameters
$\tau_{ii},\tau_{ij},\alpha_{i},\omega_{i},\lambda,u_{0},v_{0}$. Among them,
$2N$ parameters $\tau_{ii},\alpha_{i}$ may be the given parameters relate to
the amplitudes and wave numbers of $N$-periodic waves. Therefore, the number
of the unknown parameters is $\frac{1}{2}N(N+1)+N+3$ while $\frac{1}{2}N(N+1)$
parameters $\tau_{ij}$, implicitly appearing in the series form, can not to be
solved explicitly in general. So, the number of the explicit unknown
parameters is only $N+3$, and the number of equations is larger than the
unknown parameters in the case of $N>4$. This fact means that if equation
(10.5) is satisfied, then we have at least $N$-periodic wave solutions ($N\leq
4$). In this paper, we only consider one- and two-periodic wave solutions of
the NKdV equation (1.1). There are still certain computation difficulties in
the calculation for the case of $N>2$, which will be studied in the future.
Acknowledgment
This work is supported by the U. S. Army Research Office under contract/grant
No. W911NF- 08-1-0511 and by the Texas Norman Hackerman Advanced Research
Program under Grant No. 003599- 0001-2009, the National Science Foundation of
China (No. 10971031) and Shanghai Shuguang Tracking Project (No. 08GG01).
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|
arxiv-papers
| 2011-07-31T01:46:37 |
2024-09-04T02:49:21.087920
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhijun Qiao and Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1108.0122"
}
|
1108.0134
|
# Ricci Flow Equation on $(\alpha,\beta)$-Metrics
A. Tayebi, E. Peyghan and B. Najafi
###### Abstract
In this paper, we study the class of Finsler metrics, namely
$(\alpha,\beta)$-metrics, which satisfies the un-normal or normal Ricci flow
equation.
Keywords: Finsler metric, Einstein metric, Ricci flow equation.111 2010
Mathematics subject Classification: 53B40, 53C60.
## 1 Introduction
In 1982, R. S. Hamilton for a Riemannian metric $g_{ij}$ introduce the
following geometric evolution equation
$\frac{d}{dt}(g_{ij})=-2Ric_{ij},\ \ g(t=0)=g_{0},$
where $Ric_{ij}$ is the Ricci curvature tensor and is known as the un-
normalised Ricci flow in Riemannian geometry [7]. Hamilton showed that there
is a unique solution to this equation for an arbitrary smooth metric on a
closed manifold over a sufficiently short time. He also showed that Ricci flow
preserves positivity of the Ricci curvature tensor in three dimensions and the
curvature operator in all dimensions [6]. The Ricci flow theory related
geometric analysis and various applications became one of the most intensively
developing branch of modern mathematics [5, 7, 8, 12, 17]. The most important
achievement of this theory was the proof of W. Thurston’s geometrization
conjecture by G. Perelman [14, 15, 16]. The main results on Ricci flow
evolution were proved originally for (pseudo) Riemannian and Kähler
geometries. Thus the Ricci flow theory became a very powerful method in
understanding the geometry and topology of Riemannian and Kählerian manifolds.
On the other hand, Finsler geometry is a natural extension of Riemannian
geometry without quadratic restriction [19]. But it is not simple to define
Ricci flows of mutually compatible fundamental geometric structures on Finsler
manifolds. The problem of constructing the Finsler-Ricci flow theory contains
a number of new conceptual and fundamental issues on compatibility of
geometrical and physical objects and their optimal configurations. The same
equation can be used in the Finsler setting, since both the fundamental tensor
$g_{ij}$ and Ricci tensor $Ric_{ij}$ have been generalized to that broader
framework, albeit gaining a $y$ dependence in the process [2][18]. However,
there are some reasons why we shall refrain from doing so: (i) not every
symmetric covariant 2-tensor $g_{ij}(x,y)$ arises from a Finsler metric
$F(x,y)$; (ii) there is more than one geometrical context in which $g_{ij}$
makes sense. Thus, Bao called this equation as an un-normalised Ricci flow for
Finsler geometry. Using the elegance work of Akbar-Zadeh in [1], Bao proposed
the following normalised Ricci flow equation for Finsler metrics
$\frac{d}{dt}\log F=-R+\frac{1}{Vol(SM)}\int_{SM}R\ dV,\ \ \ F(t=0)=F_{0},$
(1)
where the underlying manifold $M$ is compact [2].
In a series of papers, Vacaru studied Ricci flow evolutions of geometries and
physical models of gravity with symmetric and nonsymmetric metrics and
geometric mechanics, when the field equations are subjected to nonholonomic
constraints and the evolution solutions, mutually transform as Riemann and
Finsler geometries [20, 21, 22, 23, 24, 25, 26].
It is remarkable that, Chern had asked whether every smooth manifold admits a
Ricci-constant Finsler metric? The weaker case of this question is that
whether every smooth manifold admits a Einstein Finsler metric? His question
has already been settled in the affirmative for dimension 2 because, by a
construction of Thurston s, every Riemannian metric on a two-dimensional
manifold admits a complete Riemannian metric of constant Gaussian curvature.
## 2 Preliminaries
Let $M$ be an $n$-dimensional $C^{\infty}$ manifold. Denote by $T_{x}M$ the
tangent space at $x\in M$, and by $TM=\cup_{x\in M}T_{x}M$ the tangent bundle
of $M$.
A Finsler metric on $M$ is a function $F:TM\rightarrow[0,\infty)$ which has
the following properties:
(i) $F$ is $C^{\infty}$ on $TM_{0}:=TM\setminus\\{0\\}$;
(ii) $F$ is positively 1-homogeneous on the fibers of tangent bundle $TM$,
(iii) for each $y\in T_{x}M_{0}$, the following form $g_{y}$ on $T_{x}M$ is
positive definite,
$g_{y}(u,v):={1\over 2}\left[F^{2}(y+su+tv)\right]|_{s,t=0},\ \ u,v\in
T_{x}M.$
For a Finsler metric $F=F(x,y)$ on a manifold $M$, the spray ${\bf
G}=y^{i}\frac{\partial}{\partial x^{i}}-2G^{i}\frac{\partial}{\partial y^{i}}$
is a vector field on $TM$, where $G^{i}=G^{i}(x,y)$ are defined by
$G^{i}={g^{il}\over
4}\Big{\\{}[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}\Big{\\}},$
Let $x\in M$ and $F_{x}:=F|_{T_{x}M}$. To measure the non-Euclidean feature of
$F_{x}$, define ${\bf C}_{y}:T_{x}M\otimes T_{x}M\otimes
T_{x}M\rightarrow\mathbb{R}$ by
${\bf C}_{y}(u,v,w):={1\over 2}\frac{d}{dt}\left[{\bf
g}_{y+tw}(u,v)\right]|_{t=0},\ \ u,v,w\in T_{x}M,$
The family ${\bf C}:=\\{{\bf C}_{y}\\}_{y\in TM_{0}}$ is called the Cartan
torsion. It is well known that ${\bf{C}}=0$ if and only if $F$ is Riemannian.
For $y\in T_{x}M_{0}$, define mean Cartan torsion ${\bf I}_{y}$ by ${\bf
I}_{y}(u):=I_{i}(y)u^{i}$, where $I_{i}:=g^{jk}C_{ijk}$,
$C_{ijk}=\frac{1}{2}\frac{{\partial}g_{ij}}{{\partial}y^{k}}$ and
$u=u^{i}\frac{\partial}{\partial x^{i}}|_{x}$. By Deicke’s Theorem, $F$ is
Riemannian if and only if ${\bf I}_{y}=0$.
Regarding the Cartan tensors of these metrics, M. Matsumoto introduced the
notion of C-reducibility and proved that any Randers metric $F=\alpha+\beta$
and Kropina metric $F=\alpha^{2}/\beta$ are C-reducible, where
$\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric and
$\beta=b_{i}(x)y^{i}$ is a 1-form on $M$. Matsumoto-Hōjō proved that the
converse is true [10]. Furthermore, by considering Kropina and Randers
metrics, Matsumoto introduced the notion of $(\alpha,\beta)$-metrics [9]. An
$(\alpha,\beta)$-metric is a Finsler metric on $M$ defined by
$F:=\alpha\phi(s)$, where $s=\beta/\alpha$, $\phi=\phi(s)$ is a $C^{\infty}$
function on the $(-b_{0},b_{0})$ with certain regularity, $\alpha$ is a
Riemannian metric and $\beta$ is a 1-form on $M$.
In [11], Matsumoto-Shibata introduced the notion of semi-C-reducibility by
considering the form of Cartan torsion of a non-Riemannian
$(\alpha,\beta)$-metric on a manifold $M$ with dimension $n\geq 3$. A Finsler
metric is called semi-C-reducible if its Cartan tensor is given by
$C_{ijk}={\frac{p}{1+n}}\\{h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\\}+\frac{q}{C^{2}}I_{i}I_{j}I_{k},$
where $p=p(x,y)$ and $q=q(x,y)$ are scalar function on $TM$,
$h_{ij}:=g_{ij}-F^{-2}y_{i}y_{j}$ is the angular metric and
$C^{2}=I^{i}I_{i}$. If $q=0$, then $F$ is just C-reducible metric.
## 3 Ricci Flow Equation
In 1982, R. S. Hamilton introduce the following geometric evolution equation
$\frac{d}{dt}(g_{ij})=-2Ric_{ij},\ \ g(t=0)=g_{0}$
which is known as the un-normalised Ricci flow in Riemannian geometry [7]. The
same equation can be used in the Finsler setting, since both the fundamental
tensor $g_{ij}$ and Ricci tensor $Ric_{ij}$ have been generalized to that
broader framework, albeit gaining a $y$ dependence in the process. However,
there are some reasons why we shall refrain from doing so: (i) Not every
symmetric covariant 2-tensor $g_{ij}(x,y)$ arises from a Finsler metric
$F(x,y)$; (ii) There is more than one geometrical context in which $g_{ij}$
makes sense. Thus, Bao called this equation as an un-normalised Ricci flow for
Finsler geometry.
Professor Chern had asked, on several occasions, whether every smooth manifold
admits a Ricci-constant Finsler metric. It is hoped that the Ricci flow in
Finsler geometry eventually proves to be viable for addressing Chern s
question. How to formulate and generalize these constructions for non-
Riemannian manifolds and physical theories is a challenging topic in
mathematics and physics. Bao studied Ricci flow equation in Finsler spaces
[2]. In the following a scalar Ricci flow equation is introduced according to
the Bao’s paper.
A deformation of Finsler metrics means a 1-parameter family of metrics
$g_{ij}(x,y,t)$, such that $t\in[-\epsilon,\epsilon]$ and $\epsilon>0$ is
sufficiently small. For such a metric $\omega=u_{i}dx^{i}$, the volume element
as well as the connections attached to it depend on $t$. The same equation can
be used in the Finsler setting. We can also use another Ricci flow equation
instead of this tensor evolution equation [2]. By contracting
$\frac{d}{dt}g_{ij}=-2Ric_{ij}$ with $y^{i}$ and $y^{j}$ gives, via Euler s
theorem, we get
$\frac{\partial F^{2}}{\partial t}=-2F^{2}R,$
where $R=\frac{1}{F^{2}}Ric$. That is,
$d\log F=-R,\ \ F(t=0)=F_{0}.$
This scalar equation directly addresses the evolution of the Finsler metric
$F$, and makes geometrical sense on both the manifold of nonzero tangent
vectors $TM_{0}$ and the manifold of rays. It is therefore suitable as an un-
normalized Ricci flow for Finsler geometry.
## 4 Un-Normal Ricci Flow Equation on $(\alpha,\beta)$-Metrics
Here, we study $(\alpha,\beta)$-metrics satisfying un-normal Ricci flow
equation and prove the following.
###### Theorem 4.1.
Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$. Suppose that
$F=\Phi(\frac{\beta}{\alpha})\alpha$ be an $(\alpha,\beta)$-metric on $M$.
Then every deformation $F_{t}$ of the metric $F$ satisfying un-normal Ricci
flow equation is an Einstein metric.
To prove the Theorem 4.1, we need the following.
###### Lemma 4.2.
Let $F_{t}$ be a deformation of an $(\alpha,\beta)$-metric $F$ on manifold $M$
of dimension $n\geq 3$. Then the variation of Cartan tensor is given by
following
$\displaystyle
C^{\prime}_{ijk}I^{i}I^{j}I^{k}=-\frac{2R(1+nq)}{1+n}||I||^{4}-\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}-3||I||^{2}I^{m}R_{,m}$
(2)
where $||I||^{2}=I_{m}I^{m}$.
###### Proof.
First assume that $F_{t}$ be a deformation of a Finsler metric on a two-
dimensional manifold $M$ satisfies Ricci flow equation, i.e.
$\frac{d}{dt}g_{ij}:=g^{\prime}_{ij}=-2Ric_{ij},\ \ \ d\log
F:=\frac{F^{\prime}}{F}=-R,$ (3)
where $R=\frac{1}{F^{2}}Ric$. By definition of Ricci tensor, we have
$\displaystyle Ric_{ij}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ \frac{1}{2}[RF^{2}]_{y^{i}y^{j}}$ (4)
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
Rg_{ij}+\frac{1}{2}F^{2}R_{,i,j}+R_{,i}y_{j}+R_{,i}y_{i}$
where $R_{,i}=\frac{\partial R}{\partial y^{i}}$ and
$R_{,i,j}=\frac{\partial^{2}R}{\partial y^{i}\partial y^{j}}$. Taking a
vertical derivative of (4) and using $y_{i,j}=g_{ij}$ and $FF_{k}=y_{k}$
yields
$\displaystyle Ric_{ij,k}=2RC_{ijk}+\frac{1}{2}F^{2}R_{,i,j,k}\\!\\!\\!\\!$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
\\{g_{jk}R_{,i}+g_{ij}R_{,k}+g_{ki}R_{,j}\\}$ (5) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\ \\{R_{,j,k}y_{i}+R_{,i,j}y_{k}+R_{,k,i}y_{j}\\}.$
Contracting (5) with $I^{i}I^{j}I^{k}$ and using $y_{i}I^{i}=y^{i}I_{i}=0$
implies that
$\displaystyle
Ric_{ij,k}I^{i}I^{j}I^{k}=2RC_{ijk}I^{i}I^{j}I^{k}+\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}+3||I||^{2}I^{m}R_{,m}.$
(6)
The Cartan tensor of an $(\alpha,\beta)$-metric on a $n$-dimensional manifold
$M$ is given by
$C_{ijk}={\frac{p}{1+n}}\\{h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\\}+\frac{q}{||I||^{2}}I_{i}I_{j}I_{k},$
(7)
where $p=p(x,y)$ and $q=q(x,y)$ are scalar function on $TM$ with $p+q=1$.
Multiplying (7) with $I^{i}I^{j}I^{k}$ yields
$C_{ijk}I^{i}I^{j}I^{k}=({\frac{p}{1+n}}+q)||I||^{4}=\frac{1+nq}{1+n}||I||^{4}.$
(8)
Then by (6) and (8), we get
$\displaystyle
Ric_{ij,k}I^{i}I^{j}I^{k}=\frac{2R(1+nq)}{1+n}||I||^{4}+\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}+3||I||^{2}I^{m}R_{,m}.$
(9)
On the other hand, since $F_{t}$ satisfies Ricci flow equation then
$\displaystyle C^{\prime}_{ijk}=\frac{1}{2}\frac{\partial
g^{\prime}_{ij}}{\partial
y^{k}}=\frac{1}{2}\frac{\partial(-2Ric_{ij})}{\partial y^{k}}=-Ric_{ij,k}.$
(10)
By (9) and (10) we get (2). ∎
###### Lemma 4.3.
Let $F_{t}$ be a deformation of an $(\alpha,\beta)$-metric $F$ on a
$n$-dimensional manifold $M$. Then $C^{\prime}_{ijk}I^{i}I^{j}I^{k}$ is a
factor of $||I||^{2}$.
###### Proof.
Since $g^{ij}g_{jk}=\delta^{i}_{k}$, then we have
$0=(g^{ij}g_{jk})^{\prime}=g^{\prime ij}g_{jk}+g^{ij}g^{\prime}_{jk}=g^{\prime
ij}g_{jk}-2g^{ij}Ric_{jk},$
or equivalently $g^{\prime ij}g_{jk}=2g^{ij}Ric_{jk}$ which contracting it
with $g^{lk}$ implies that
$g^{\prime il}=2Ric^{il}.$ (11)
Then we have
$\displaystyle I^{\prime}_{i}=(g^{jk}C_{ijk})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{jk})^{\prime}C_{ijk}+g^{jk}C^{\prime}_{ijk}$ (12) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{jk}C_{ijk}-g^{jk}Ric_{jk,i}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
2Ric^{jk}g_{jk,i}-(g^{jk}Ric_{jk})_{,i}+g^{jk}_{\ \ ,i}Ric_{jk}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\ -(g^{jk}Ric_{jk})_{,i}=-\rho_{i}$
where $\rho:=g^{jk}Ric_{jk}$ and $\rho_{i}=\frac{\partial\rho}{\partial
y^{i}}$. Thus
$\displaystyle I^{\prime i}=(g^{ij}I_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{ij})^{\prime}I_{j}+g^{ij}I^{\prime}_{j}$ (13) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{ij}I_{j}-g^{ij}\rho_{j}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{ij}I_{j}-\rho^{i}.$
The variation of $y_{i}:=FF_{y^{i}}$ with respect to $t$ is given by
$y^{\prime}_{i}=-2Ric_{im}y^{m}.$
Therefore, we can compute the variation of angular metric $h_{ij}$ as follows
$\displaystyle h^{\prime}_{ij}=(g_{ij}-F^{-2}y_{i}y_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-2Ric_{ij}-2F^{-2}Ry_{i}y_{j}+2F^{-2}(Ric_{im}+Ric_{jm})y^{m}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-2Ric_{ij}+2R(h_{ij}-g_{ij})+2(Ric_{im}\ell_{j}+Ric_{jm}\ell_{i})\ell^{m},$
where $\ell_{i}:=F^{-1}y_{i}$. Thus
$h^{\prime}_{ij}=2Rh_{ij}-2Rg_{ij}-2Ric_{ij}+2(Ric_{im}\ell_{j}+Ric_{jm}\ell_{i})\ell^{m}.$
(14)
Now, we consider the variation of Cartan tensor
$\displaystyle C^{\prime}_{ijk}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\
\big{\\{}\frac{p}{1+n}[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]+\frac{q}{||I||^{2}}I_{i}I_{j}I_{k}\big{\\}}^{\prime}$
(15) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
\frac{p^{\prime}}{1+n}[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]+\frac{q^{\prime}}{||I||^{2}}I_{i}I_{j}I_{k}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
\frac{p}{1+n}\Big{[}h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\Big{]}^{\prime}+q\Big{[}\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}\Big{]}^{\prime}$
We have
$[\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}]^{\prime}=\frac{-(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{||I||^{2}}C_{ijk}-\frac{1}{||I||^{2}}(\rho_{i}I_{j}I_{k}+\rho_{j}I_{i}I_{k}+\rho_{k}I_{i}I_{j})$
(16)
Multiplying (16) with $I^{i}I^{j}I^{k}$ implies that
$\displaystyle[\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}]^{\prime}I^{i}I^{j}I^{k}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\ -\big{[}\frac{(nq+1)(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{(n+1)||I||^{2}}+3\rho_{m}I^{m}\big{]}||I||^{2}$
(17) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
\big{[}\frac{2(nq+1)(\rho^{m}I_{m}-Ric^{pq}I_{p}I_{q})}{(n+1)||I||^{2}}-3\rho_{m}I^{m}\big{]}||I||^{2}$
On the other hand
$\displaystyle[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-(\rho_{i}h_{jk}+\rho_{j}h_{ik}+\rho_{k}h_{ij})-2R[I_{i}g_{jk}+I_{j}g_{ik}+I_{k}g_{ij}]$
(18) $\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
2[I_{i}Ric_{jk}+I_{j}Ric_{ik}+I_{k}Ric_{ij}]$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\ 2R[I_{i}h_{jk}+I_{j}h_{ik}+I_{k}h_{ij}]$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
2[I_{i}\Lambda_{jk}+I_{j}\Lambda_{ik}+I_{k}\Lambda_{ij}],$
where $\Lambda_{jk}:=(Ric_{jr}\ell_{k}+Ric_{kr}\ell_{j})\ell^{r}$. Multiplying
(18) with $I^{i}I^{j}I^{k}$ implies that
$\displaystyle[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]^{\prime}I^{i}I^{j}I^{k}=-3(\rho_{m}I^{m}+2Ric_{pq}I^{p}I^{q})||I||^{2}.$
(19)
On the other hand, since $p^{\prime}+q^{\prime}=0$ then we get
$\displaystyle\Big{[}\frac{p^{\prime}}{1+n}(h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j})+\frac{q^{\prime}}{||I||^{2}}I_{i}I_{j}I_{k}\Big{]}I^{i}I^{j}I^{k}=\frac{nq^{\prime}}{1+n}||I||^{4}$
(20)
Putting (17), (19) and (20) in (15) implies that
$C^{\prime}_{ijk}I^{i}I^{j}I^{k}$ is a factor of $||I||^{2}$. More precisely,
we have the following
$\displaystyle
C^{\prime}_{ijk}I_{i}I_{j}I_{k}=\Big{[}\frac{nq^{\prime}}{n+1}||I||^{2}\\!\\!\\!\\!$
$\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
q\Big{(}\frac{2(nq+1)(\rho^{m}I_{m}-Ric^{pq}I_{p}I_{q})}{(n+1)||I||^{2}}-3\rho_{m}I^{m}\Big{)}$
(21) $\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
\frac{3p}{n+1}(\rho_{m}I^{m}+2Ric_{pq}I^{p}I^{q})\Big{]}||I||^{2}.$
This completes the proof. ∎
Proof of Theorem 4.1: By Lemmas 4.2 and 4.3, it follows that
$R_{,i,j,k}I^{i}I^{j}I^{k}$ is a factor of $||I||^{2}$. Thus
$R_{,i,j,k}I^{i}I^{j}I^{k}=A_{ij}I_{k}+B_{i}g_{jk}.$
It is remarkable that, since $R_{,i,j,k}$ is symmetric with respect to indexes
i, j and k, then the order of indexes in this relation doesn t matter. Now,
multiplying $R_{,i,j,k}$ with $y^{k}$ or $y^{j}$ implies that $R_{,i}=0$. It
means that $R=R(x)$ and then $F_{t}$ is an Einstein metric.
## 5 Normal Ricci Flow Equation on $(\alpha,\beta)$-Metrics
If M is a compact manifold, then $S(M)$ is compact and we can normalize the
Ricci flow equation by requiring that the flow keeps the volume of $SM$
constant. Recalling the Hilbert form $\omega:=F_{y^{i}}dx^{i}$, that volume is
$Vol_{SM}:=\int_{SM}\frac{(-1)}{(n-1)!}^{\frac{n(n-1)}{2}}\omega\wedge(d\omega)^{n-1}:=\int_{SM}dV_{SM}.$
During the evolution, $F$, $\omega$ and consequently the volume form $dV_{SM}$
and the volume $Vol_{SM}$, all depend on $t$. On the other hand, the domain of
integration $SM$, being the quotient space of $TM_{0}$ under the equivalence
relation $z\sim y$ , $z=\lambda y$ for some $\lambda>0$, is totally
independent of any Finsler metric, and hence does not depend on t. We have
$\frac{d}{dt}(dV_{SM})=\big{[}g_{ij}\frac{d}{dt}g_{ij}-n\frac{d}{dt}\log
F\big{]}dV_{SM}$
A normalized Ricci flow for Finsler metrics is proposed by Bao as follows
$\frac{d}{dt}\log F=-R+\frac{1}{Vol(SM)}\int_{SM}R\ dV,\ \ \ F(t=0)=F_{0},$
(22)
where the underlying manifold $M$ is compact. Now, we let $Vol(SM)=1$. Then
all of Ricci-constant metrics are exactly the fixed points of the above flow.
Let
$Ric_{ij}=\frac{1}{2}(F^{2}R)_{.y^{i}.y^{j}}$
and differentiating (22) with respect to $y^{i}$ and $y^{j}$ the following
normal Ricci flow tensor evaluation equation is concluded
$\frac{d}{dt}g_{ij}=-2Ric_{ij}+\frac{2}{Vol(SM)}\int_{SM}R\ dVg_{ij},\ \ \
g(t=0)=g_{0},$ (23)
Starting with any familiar metric on M as the initial data $F_{0}$, we may
deform it using the proposed normalized Ricci flow, in the hope of arriving at
a Ricci constant metric.
###### Theorem 5.1.
Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$. Suppose that
$F=\Phi(\frac{\beta}{\alpha})\alpha$ be an $(\alpha,\beta)$-metric on $M$.
Then every deformation $F_{t}$ of the metric $F$ satisfying normal Ricci flow
equation is an Einstein metric.
###### Proof.
Now, we consider Finsler surfaces that satisfies the normal Ricci flow
equation. Then
$\frac{dg_{ij}}{dt}=-2Ric_{ij}+2\int_{SM}R\ dVg_{ij},\ \ d\log
F:=\frac{F^{\prime}}{F}=-R+\int_{SM}R\ dV.$ (24)
By the same argument used in the un-normal Ricci flow case, we can calculate
the variation of mean Cartan tensor as follows
$\displaystyle I^{\prime}_{i}=(g^{jk}C_{ijk})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{jk})^{\prime}C_{ijk}+g^{jk}C^{\prime}_{ijk}$ (25) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2[Ric^{jk}-\int_{SM}R\
dVg^{jk}]C_{ijk}+g^{jk}[Ric_{jk,i}+2\int_{SM}R\ dVC_{ijk}]$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{jk}g_{jk,i}-(g^{jk}Ric_{jk})_{,i}+g^{jk}_{\
\ ,i}Ric_{jk}$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-(g^{jk}Ric_{jk})_{,i}$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-\rho_{i}$
Then we have
$\displaystyle I^{\prime i}=(g^{ij}I_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{ij})^{\prime}I_{j}+g^{ij}I^{\prime}_{j}$ (26) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2[Ric^{ij}-\int_{SM}R\
dVg^{ij}]I_{j}-g^{ij}\rho_{j}$
By the same way that we used in un-normal Ricci flow, it follows that
$\displaystyle C^{\prime}_{ijk}=\frac{-(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{||I||^{2}}C_{ijk}-\frac{1}{||I||^{2}}(\rho_{i}I_{j}I_{k}+\rho_{j}I_{i}I_{k}+\rho_{k}I_{i}I_{j})$
(27)
Contracting it with $I^{i}I^{j}I^{k}$ yields
$C^{\prime}_{ijk}I^{i}I^{j}I^{k}=(\Omega||I||^{2}-3\rho_{m}I^{m})||I||^{2},$
(28)
where
$\Omega:=-\frac{I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m}}{||I||^{2}}=\frac{2\rho^{m}I_{m}-2Ric^{ml}I_{l}}{||I||^{2}}+2\int_{SM}R\
dV.$
By Lemma 4.2, we deduce that $R_{,i,j,k}I^{i}I^{j}I^{k}$ is a factor of
$||I||^{2}$. By the same argument, it result that every deformation $F_{t}$ of
the metric $F$ satisfying normal Ricci flow equation is an Einstein metric. ∎
## References
* [1] H. Akbar-Zadeh, Generalized Einstein manifolds, J. Geom. Phys. 17(1995), 342-380.
* [2] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics XX, 2007.
* [3] D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry. In A Sampler of Riemann Finsler Geometry, Mathematical Sciences Research Institute Publications, vol. 50, Cambridge University Press, 2004, 197-259.
* [4] L. Berwald, On Cartan and Finsler Geometries, III, Two Dimensional Finsler Spaces with Rectilinear Extremal, Ann. Math. 42 No. 2 (1941) 84 122.
* [5] H. -D. Cao and X. -P. Zhu, Hamilton–Perelman’s proof of the Poincar’e conjecture and the geometrization conjecture, Asian J. Math. 10 (2006) 165–495; arXiv: math. DG/ 0612069.
* [6] R. S. Hamilton, Four Manifolds with Positive Curvature Operator, J. Diff. Geom. 24, 153-179, 1986.
* [7] R. S. Hamilton, Three–manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255-306.
* [8] B. Kleiner and J. Lott, Notes on Perelman’s papers, arXiv: math.DG/ 0605667.
* [9] M. Matsumoto, Theory of Finsler spaces with $(\alpha,\beta)$-metric, Rep. Math. Phys. 31(1992), 43-84.
* [10] M. Matsumoto and S. Hōjō, A conclusive theorem for C-reducible Finsler spaces, Tensor. N. S. 32(1978), 225-230.
* [11] M. Matsumoto and C. Shibata, On semi-C-reducibility, T-tensor and S4-1ikeness of Finsler spaces, J. Math. Kyoto Univ. 19(1979), 301-314.
* [12] J. W. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, arXiv: math.DG/ 0607607.
* [13] B. Najafi and A. Tayebi, On a family of Einstein Randers metric, International Journal of Geometric Methods in Modern Physics, (2011), DOI No: 10.1142/S021988781100552X.
* [14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/ 0211159.
* [15] G. Perelman, Ricci flow with surgery on three–manifolds, arXiv: math. DG/ 03109.
* [16] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three–manifolds, arXiv: math.DG/ 0307245.
* [17] N. SadeghZadeh and A. Razavi, Ricci Flow equation on C-reducible metrics, International Journal of Geometric Methods in Modern Physics, (2011), DOI No: 10.1142/S0219887811005385
* [18] A. Tayebi and E. Peyghan, On Ricci tensors of Randers metrics, J. Geom. Phys. 60(2010), 1665-1670.
* [19] A. Tayebi and E. Peyghan, On a class of Riemannian metrics arising from Finsler structures, C. R. Acad. Sci. Paris, Ser. I 349 (2011), 319-322.
* [20] S. Vacaru, Nonholonomic ricci flows: II. evolution equations and dynamics, J. Math. Phys. 49 (2008), 043504.
* [21] S. Vacaru, Nonholonomic Ricci flows, exact solutions in gravity, and Symmetric and Nonsymmetric Metrics, Int. J. Theor. Phys. 48 (2009) 579-606
* [22] S. Vacaru, The Entropy of Lagrange-Finsler spaces and Ricci flows, Rep. Math. Phys. 63 (2009) 95-110
* [23] S. Vacaru, Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows, J. Math. Phys. 50 (2009) 073503
* [24] S. Vacaru, Fractional Nonholonomic Ricci Flows, arXiv:1004.0625.
* [25] S. Vacaru, Nonholonomic Ricci flows and parametric deformations of the solitonic pp–waves and Schwarzschild solutions, Electronic Journal of Theoretical Physics 6, N21 (2009), 63-93.
* [26] S. Vacaru, Nonholonomic Ricci flows: Exact solutions and gravity, Electronic Journal of Theoretical Physics 6, N20 (2009) 27-58.
Akbar Tayebi
Faculty of Science, Department of Mathematics
Qom University
Qom. Iran
Email: akbar.tayebi@gmail
Esmail Peyghan
Faculty of Science, Department of Mathematics
Arak University
Arak. Iran
Email: epeyghan@gmail.com
Behzad Najafi
Faculty of Science, Department of Mathematics
Shahed University
Tehran. Iran
Email: najafi@shahed.ac.ir
|
arxiv-papers
| 2011-07-31T04:58:50 |
2024-09-04T02:49:21.102715
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Tayebi, E. Peyghan and B. Najafi",
"submitter": "Esmaeil Peyghan",
"url": "https://arxiv.org/abs/1108.0134"
}
|
1108.0147
|
# Brane world regularization of point particle classical self-energy
Román Linares1 lirr@xanum.uam.mx Hugo A. Morales-Técotl1 hugo@xanum.uam.mx
Omar Pedraza2 omarp@uaeh.edu.mx Luis O. Pimentel1 lopr@xanum.uam.mx 1
Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa,
San Rafael Atlixco 186, C.P. 09340, México D.F., México, 2 Área Académica de
Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo,
Carretera Pachuca-Tulancingo Km. 4.5, C P. 42184, Pachuca, México.
###### Abstract
Physical effects in brane worlds models emerge by the incorporation of field
modes coming from extra dimensions with the usual four dimensional ones. Such
effects can be tested with well established experiments to set bounds on the
parameters of the brane models. In this work we extend a previous result which
gave finite electromagnetic potentials and self energies for a source looking
pointlike to an observer sitting in a 4D Minkowski subspace of a single brane
of a Randall-Sundrum spacetime including compact dimensions, and along which
the source stretches uniformly. We show that a scalar particle produces a
nonsingular static potential, possess a finite self-energy and that
technically its analysis is very similar to the electrostatic case. As for the
latter, we use the deviations from the Coulomb potential to set bounds on the
anti de Sitter radius of the brane model on the basis of two experiments,
namely, one of the Cavendish type and other being the scattering of electrons
by Helium atoms. We found these are less stringent than others previously
obtained using the Lamb shift in Hydrogen.
###### pacs:
11.25.Wx, 11.10Kk, 11.25.Mj,04.25.-g, 03.50.-z
## I Introduction
Despite the extraordinarily rich accuracy with which the predictions of
electrodynamics have been experimentally tested over the years (see for
instance Goldhaber and Nieto (2010, 1971) and references therein), efforts to
place limits on deviations from its standard formulation continue nowadays.
The nature of the experiments cover a big range of possibilities which include
among others: a) Testing the power in the inverse-square law of Coulomb, b)
Seeking a nonzero value for the rest mass of the photon and c) Considering
more degrees of freedom, allowing mass for the photon while preserving
explicit gauge invariance. It is worth to mention that all these experiments
have probed length scales increasing dramatically over time.
Now, historically, once the Maxwell theory of electromagnetic fields was
established, one of the main concerns in physics was the construction of a
consistent description of electrodynamics and charged particles. The first
serious proposals in this direction were developed by Lorentz Lorentz (1909)
and Abraham Abraham (1903). These proposals and subsequent attempts based on
classical electrodynamics, special relativity and the Lorentz force law led to
the theory known as Classical Electron Theory (CET). Whereas in the Maxwell
theory charges are considered as punctual which produce infinite Lorentz self-
force and infinite electromagnetic self-energies associated with the
singularities of the Liénard-Wiechert potential, in the CET charges are
considered as extended objects that experience a volume-averaged Lorentz
force. Parallel to the development of the CET, quantum mechanics was developed
giving origin to one of the most spectacular theory we have in physics:
Quantum Electrodynamics (QED). This theory is awesome due to the impressive
range of electromagnetic phenomena it covers with spectacular precision. After
the experimental achievements of QED, CET dropped from the list of contenders
for a fundamental theory of electrodynamics interacting with matter. However
despite the great success of QED there are still some features of the theory
that could be waiting for a better explanation, for instance, its property of
renormalizability. It turns out that QED is defined by a perturbative series
that is renormalizable in each order, but it is most likely to be merely
asymptotic in character rather than convergent Dyson (1952), in such a way
that the precision results are obtained only when computations are made to
some order in the expansion series, but without any a priori prescription to
stop the series at some order.
It is thus tempting to investigate theories that avoid singularities. These
are not expected to solve the problem but at least they may contribute to a
better understanding of the singularity issue.
In this context, recently, in a previous work, some of us found that a source
lying on the single brane of a Randall-Sundrum spacetime including compact
dimensions, and which looks pointlike to an observer sitting in usual 3D
space, produces a static potential which is non singular at 3D point position.
Moreover it matches Coulomb’s potential outside a small neighborhood Morales-
Técotl et al. (2007). The presence of the compact dimensions in this setup
serve to localize the gauge field on the brane Dubovsky et al. (2000a, b); Oda
(2000). The aim of this paper is to investigate further some consequences of
the above property to set bounds to the AdS curvature radius $\epsilon$ using
the experimental results from the Cavendish experiment for electromagnetism
and the scattering of electrons by Helium atoms. For the sake of clarity, the
simpler case of a scalar particle is first considered. Remarkably, the
nonsingular character of the potential holds together with the finiteness of
the selfenergy. Indeed, technically, the study of the potentials for both
scalar and electromagnetic is very similar.
Our interest in this work is twofold, on one side it is interesting to explore
how the old problem of divergences acquires a different character in light of
the brane world models, at least classically, and, on the other hand, it is
also interesting from the perspective of the brane-world scenarios Randall and
Sundrum (1999a, b); Antoniadis et al. (1998); Arkani-Hamed et al. (1998);
Antoniadis (1990), which have recently been matter of a copious research,
mainly in high energy physics (see e.g. Allanach et al. (2004); Csaki (2004),
and references therein) and cosmology (see e.g. Maartens (2004); Elizalde
(2006); Maartens and Koyama (2010), and references therein). More recently,
the possibility to obtain information from models with extra dimensions
studying low energy physical phenomena has also been addressed. In particular
we mention the ones that have been performed in the RSII-$p$ setup, such as
the electric charge conservation Dubovsky et al. (2000b), the Casimir effect
between parallel plates Linares et al. (2010); Frank et al. (2008) and the
Hydrogen Lamb shift Morales-Técotl et al. (2007).
The paper is organized as follows. In section II we briefly describe the
RSII-$p$ setup, section III is dedicated to obtain the static potential for a
scalar field whereas in section IV we do the same for the electric case. In
section V.1 we set bounds to the AdS radius $\epsilon$ comparing our
electrostatic results with the experimental values obtained in Cavendish like
experiments of the Coulomb force. Section V.2 is dedicated to the same purpose
but this time we use the experimental results of the scattering process of
electrons by Helium atoms. Finally section VI is devoted to a brief
discussion.
## II Randall-Sundrum II-$p$ scenarios
The Randall-Sundrum II-$p$ scenarios consist of a ($3+p$)-brane with $p$
compact dimensions and positive tension $\sigma$, embedded in a ($5+p$)
spacetime whose metrics are two patches of anti-de Sitter (AdS5+p) having
curvature radius $\epsilon$. The interest in these models comes from its
property of localizing on the brane: scalar, gauge and gravity fields due to
the gravity produced by the brane itself. This property is valid whenever
there are $p$ extra compact dimensions Dubovsky et al. (2000b); Oda (2000). In
the limiting case $p=0$, the model only localizes scalar and gravity fields.
With this setup and appropriate fine-tuning between the brane tension $\sigma$
and the bulk cosmological constant $\Lambda$, which are related to $\epsilon$
as follows
$\sigma=\frac{2(3+p)}{8\pi\epsilon
G_{5+p}},\quad\Lambda=-\frac{(3+p)(4+p)}{16\pi\epsilon^{2}G_{5+p}}=-\frac{(4+p)\sigma}{4\epsilon},$
(1)
there exists a solution to (5+$p$)D Einstein equations with metric
$ds_{5+p}^{\,2}=e^{-2|y|/\epsilon}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}-\sum_{i=1}^{p}R^{2}_{i}d\theta_{i}^{2}\right]-dy^{2}.$
(2)
Here $\eta_{\mu\nu}$ is the 4D Minkowski tensor, $\theta_{i}\in[0,2\pi]$ are
$p$ compact coordinates, $R_{i}$ are the sizes of compact dimensions,
$G_{5+p}$ is the ($5+p$)D Newton constant. Throughout the paper we will use
the following notation for the 5+$p$ coordinates
$X^{M}\equiv(x^{\mu},\theta_{i},y)$, where $\mu=0,1,2,3$, and $i=1,\dots p$.
In this work we consider two different $(5+p)D$ field theories on RSII-$p$: a
massless scalar field and electrodynamics. They will be subjected to a hybrid
of the two well known consistent compactifications, namely Kaluza-Klein (KK)
Kaluza (1921); Klein (1926a) and warped Randall and Sundrum (1999a). These two
differ among them on whether the compactified manifold is factorizable or not.
The corresponding effective field theories in 4D Minkowski space-time will be
given.
In regard to the KK compactification, it is well known toroidal dimensional
reductions lead to consistent lower dimensional theories which nonetheless can
be questioned in that they do not come with a mechanism to fix the moduli, or
equivalently, the radii of the $p$D torus $T^{p}$ Duff et al. (1986); Green et
al. (1987). Historically, a way out in such cases, has been to conform with
the corresponding phenomenology at low enough energies and set a bound for the
radii (e.g. the use of the classical value of the electron charge required a
radius of the order Planck length in the original KK setting Klein (1926a,
b)). We will adhere to this approach by considering a low energy approximation
so that we truncate the massive KK modes of the compact dimensions but keep
those corresponding to the noncompact dimension just meaning that we assume
the energy scale of the former is much smaller than that of the latter. This
is explicitly performed in the Green’s function in III.1 for the scalar field
and in IV for the gauge field. As for the consistency of the Randall-Sundrum
compactification it has been discussed in Rubakov (2001) (and references
therein).
For completeness we only mention other mechanisms adopted in the literature to
perform a generalized KK compactification. One of them is the so called
Scherk-Schwarz compactification Scherk and Schwarz (1979a, b) or flux
compactifications Grana (2006); Blumenhagen et al. (2007); Douglas and Kachru
(2007). In this mechanism the symmetries of the compactification manifold
and/or the fields are used to produce an effective potential for stabilizing
the size of the extra dimensions. There also exists a quantum proposal by
Candelas and Weinberg Candelas and Weinberg (1984) where the effective
potential for the moduli fields is produced by the Casimir energy of matter
fields or gravity. It remains open to study these possibilities for our
present setup.
A remark regarding the stability of the scenario described by the metric (2)
is here in order. Concerning the world volume of the (3+$p$)-brane,
$M_{4}\otimes T^{p}$, it is clear that the space is stable since it is flat.
On the other hand the stability of the space-time (2), without the $T^{p}$
structure, was studied long ago in Randall and Sundrum (1999a); Garriga and
Tanaka (2000) for static perturbations of the metric and in Sasaki et al.
(2000) for general space-time dependent sources. The stability of other warped
compactifications has also been addressed, for instance in Goldberger and Wise
(1999); Lesgourgues and Sorbo (2004); Maity et al. (2006); Das et al. (2008)
it was discussed the moduli stabilization of the RSI model whereas in Flachi
et al. (2003) it was discussed for more general metrics.
Before ending this section is worth to mention that this setup has been
considered in different low energy physics effects such as the electric charge
conservation Dubovsky et al. (2000b), the Casimir effect between two conductor
hyperplates Linares et al. (2008a, b); Frank et al. (2008); Linares et al.
(2010) and the Liennard-Wiechert potentials and Hydrogen Lamb shift Morales-
Técotl et al. (2007) among others.
## III Static potential for a scalar field
In this section we compute the potential produced by a static source which is
seen as punctual by an observer living on the usual 3D subspace of the
(3+$p$)-brane. It stretches however along the $p$ compact dimensions thus
forming a $p$-dimensional torus (1).
Figure 1: Schematic view of the charge source for $p$=2. The source is
effectively pointlike from the perspective of an observer sitting in the usual
3d space.
### III.1 The Green’s function
Let us consider a massless scalar field $\Phi$ described by the action in
(5+$p$)D
$S=\int\,d\,{}^{4}x\,\prod_{i=1}^{p}\,R_{i}d\theta_{i}\,dy\,\sqrt{|g|}\,\left(\frac{1}{2}g^{MN}\partial_{M}\Phi\,\partial_{N}\Phi+\Phi
J_{scalar}\right).$ (3)
The equation of motion for the scalar field is
$\frac{1}{\sqrt{|g|}}\partial_{M}\left(\sqrt{|g|}g^{MN}\partial_{N}\Phi\right)=J_{scalar}$
(4)
where the source is given by
$J_{scalar}=\lambda^{(p)}\delta^{3}\left(\vec{x}-\vec{x}_{0}\right)\delta\left(y-y_{0}\right).$
(5)
Here $\lambda^{(p)}$ is a constant whose dimensions are [charge]/[length]p,
explicitly: $\lambda^{(p)}=\frac{\lambda}{(2\pi)^{p}R_{1}\cdots R_{p}}$, with
$\lambda$ the total charge.
In the background (2), the equation of motion (4) becomes
$e^{2|y|/\epsilon}\left[\Box\Phi-\sum_{i=1}^{p}\frac{1}{R_{i}^{2}}\,\partial_{\theta_{i}}^{\,2}\Phi\right]-\frac{1}{\sqrt{|g|}}\,\partial_{y}\left[\sqrt{|g|}\partial_{y}\Phi\right]=J_{scalar},$
(6)
where $\Box$ stands for the flat 4D D’Alambertian. The corresponding Green’s
equation is
$e^{2|y|/\epsilon}\left[\Box
G-\sum_{i=1}^{p}\frac{1}{R_{i}^{2}}\,\partial_{\theta_{i}}^{\,2}G\right]-\frac{1}{\sqrt{|g|}}\,\partial_{y}\left[\sqrt{|g|}\partial_{y}G\right]=\frac{\delta(y-y^{\prime})\delta^{p}(R_{i}\theta_{i}-R_{i}\theta^{\prime}_{i})\delta^{4}(x-x^{\prime})}{\sqrt{|g|}},$
(7)
where $G$ is the $(5+p)$D Green’s function. This can be expressed in terms of
the eigenfunctions of the differential operators for the different
coordinates. Assuming $\Psi(X^{M})\equiv
e^{ik_{\mu}x^{\mu}}\prod_{i=1}^{p}\Theta_{i}(\theta_{i})\psi(y)$, where the
modes $\Theta_{n}$ and $\psi_{m}$ account for the $\theta$ and $y$ dependence
respectively. These have been discussed previously (see for instance Dubovsky
et al. (2000a); Linares et al. (2010)) and here we only give a summary. The
differential equations governing the $p$ compact modes depending on
$\theta_{i}$ are
$(\partial_{\theta_{i}}^{2}+m_{\theta_{i}}^{2}R_{i}^{2})\Theta_{i}(\theta_{i})=0,\hskip
28.45274pti=1,\cdots,p,$ (8)
whereas for the noncompact modes depending on $y$ one gets
$(\partial_{y}^{2}-\frac{(4+p)}{\epsilon}\mbox{sgn}(y)\partial_{y})+m^{2}e^{2|y|/\epsilon})\psi(y)=0.$
(9)
The $(p+1)$ constants of separation, $m_{\theta_{i}},m$, fulfill the following
dispersion relation
$k^{2}=\sum_{i=1}^{p}m_{\theta_{i}}^{2}+m^{2}\equiv m_{p}^{2}+m^{2}.$ (10)
To account for the compactness of the $p$ dimensions Eq. (8) is solved under
the periodic boundary conditions
$\Theta_{n_{i}}(\theta_{i})=\Theta_{n_{i}}(\theta_{i}+2\pi),$ (11)
and the solutions turn out to be
$\Theta_{n_{i}}(\theta_{i})=\frac{1}{\sqrt{2\pi
R_{i}}}e^{in_{i}\theta_{i}},\,\,\,\mbox{where}\,\,\,n_{i}=m_{\theta_{i}}R_{i}\in\mathbb{Z}.$
(12)
To match the modes across the brane along the non-compact dimension, equation
(9) is solved with the following boundary conditions
$\psi(y=0^{+})=\psi(y=0^{-})\qquad{\rm
and}\qquad\partial_{y}\psi(y=0^{+})=\partial_{y}\psi(y=0^{-}).$ (13)
In this case the solutions include a massless zero mode localized on the brane
$\psi_{0}(y)=\sqrt{\frac{2+p}{2\epsilon}}$ (14)
which satisfies the normalization condition
$2\int_{0}^{\infty}dye^{-(2+p)|y|/\epsilon}\psi_{0}^{2}(y)=1$, as well as
massive modes given by
$\psi_{m}(y)=e^{\gamma
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}\left[a_{m}J_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right)+b_{m}N_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right)\right],$
(15)
where $J_{\gamma}$ and $N_{\gamma}$ are the Bessel and Neumann functions,
respectively. In this expression
$\gamma\equiv\frac{4+p}{2},$ (16)
and the coefficients $a_{m}$ y $b_{m}$ are
$a_{m}=-\frac{A_{m}}{\sqrt{1+A_{m}^{2}}},\quad
b_{m}=\frac{1}{\sqrt{1+A_{m}^{2}}},\quad
A_{m}=\frac{N_{\gamma-1}\left(m\epsilon\right)}{J_{\gamma-1}\left(m\epsilon\right)}.$
(17)
Notice that in this case, the localization of the massive modes on the brane
is better for increasing $p$ , since the modes are modulated exponentially by
a factor of $e^{-p|y|/(2\epsilon)}$. The normalization condition for the
massive modes is
$\int_{-\infty}^{\infty}dye^{-(p+2)|y|/\epsilon}\psi_{m}(y)\psi_{m^{\prime}}(y)=\delta(m-m^{\prime})$.
With the eigenfunctions at hand it is straightforward to use them to write
down the Green’s function. It takes the form
$G(x,\theta_{i},y;x^{\prime},\theta^{\prime}_{i},y^{\prime})=\prod_{i=1}^{p}\sum_{\\{n\\}}\frac{e^{in_{i}\theta_{i}}e^{-in_{i}\theta_{i}^{\prime}}}{2\pi
R_{i}}\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}\left[\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{k^{2}-m_{p}^{2}}+\int_{0}^{\infty}dm\frac{\psi_{m}(y)\psi_{m}(y^{\prime})}{k^{2}-m^{2}-m_{p}^{2}}\right],$
(18)
where $\\{n\\}$ denotes
$\\{n_{1},n_{2},\dots,n_{p}\,|\,n_{1}\in\mathbb{Z},\dots,n_{p}\in\mathbb{Z}\\}$.
At this point it is convenient to introduce an approximation that will allow
us to obtain analytic expressions of the potential. We have massive modes from
both compact dimensions (12) and the noncompact one (15). Since we are
interested in the low energy regime we assume $m_{\theta_{i}}\ll
m\ll\epsilon^{-1}$. Hence we will set $n_{1}=\dots=n_{p}=0$, and, as for the
noncompact modes (15) we use
$\psi_{m}(y)\approx-e^{\gamma
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}J_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right).$
(19)
In such low energy regime and upon integrating over the $p$ compact extra
dimensions, we end up with an effective 5D Green’s function
$G_{5D}(x,y;x^{\prime},y^{\prime})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}\left[\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{k^{2}}+\int_{0}^{\infty}dm\frac{\psi_{m}(y)\psi_{m}(y^{\prime})}{k^{2}-m^{2}}\right],$
(20)
where the massless mode is given by (14) and the massive modes by (19).
Although we have only taken the zero modes of the compact extra dimensions,
notice that their imprints remain in the 5D Green’s function through (16) and
(19).
### III.2 Static potential
Now we are in position to compute the static potential. In this case the
useful Green’s function is
$\displaystyle G(\vec{x},y;\vec{x^{\prime}},y^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt^{\prime}G(\vec{x},t=0,y;\vec{x^{\prime}},t^{\prime},y^{\prime})$
(21) $\displaystyle=$
$\displaystyle\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{4\pi
r}+\int_{0}^{\infty}dm\psi_{m}(y)\psi_{m}(y^{\prime})\frac{e^{-mr}}{4\pi r}$
where $r=|\vec{x}-\vec{x}^{\prime}|$. As usual the potential is obtained upon
integrating the Green’s function times the source, Eq. (5), and we are
interested in its form at the brane, i.e. $y=0$, namely,
$\displaystyle\varphi(r,y=0)$ $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}dy^{\prime}G(\vec{x},y=0;\vec{x^{\prime}},y^{\prime})\,J_{scalar}(\vec{x^{\prime}},y^{\prime};\vec{x_{0}},y_{0})$
(22) $\displaystyle=$
$\displaystyle\frac{\lambda^{(p)}\psi_{0}(0)\psi_{0}(y_{0})}{4\pi
r}+\lambda^{(p)}\int_{0}^{\infty}dm\psi_{m}(0)\psi_{m}(y_{0})\frac{e^{-mr}}{4\pi
r},$
where now $r=|\vec{x}-\vec{x}_{0}|$, and (19) takes the asymptotic value
$\psi_{m}(0)\approx\frac{1}{\Gamma(\gamma-1)}\sqrt{\frac{m\epsilon}{2}}\left(\frac{m\epsilon}{2}\right)^{\gamma-2}.$
(23)
Finally, the potential becomes
$\varphi(r)=\frac{\lambda^{(p)}}{4\pi
r}\left(\frac{2+p}{2\epsilon}\right)-\lambda^{(p)}\int_{0}^{\infty}dm\frac{1}{\Gamma(\gamma-1)}\left(\frac{m\epsilon}{2}\right)^{\gamma-1}e^{\gamma
y_{0}/\epsilon}J_{\gamma}\left({m}{\epsilon}e^{y_{0}/\epsilon}\right)\frac{e^{-mr}}{4\pi
r}.$ (24)
Next we further assume the source to be located at the brane, i.e. $y_{0}=0$.
The explicit form of (24) now depends on whether the number of extra compact
dimensions, $p$, is odd or even, and so we discuss each case separately.
### III.3 Odd number of extra compact dimensions
In the case that $p$ takes odd values, $\gamma$ takes semi-integer values and
is useful to use the relation
$m^{l+1/2}J_{l+1/2}\left(m\epsilon\right)=(-1)^{l}\sqrt{\frac{2}{\pi}}\epsilon^{l+1/2}\left(\frac{d}{\epsilon
d\epsilon}\right)^{l}\frac{\sin(m\epsilon)}{\epsilon},$ (25)
in the integrand of (24). Upon evaluation of the integral we get
$\varphi(r)=\frac{\lambda^{(p)}}{4\pi
r}\frac{2+p}{2\epsilon}-\frac{(-1)^{\gamma-\frac{1}{2}}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-3/2}\sqrt{\pi}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-\frac{1}{2}}\left(\frac{\pi}{2\epsilon}-\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right).$
(26)
Let us notice the first term of this expression is divergent at $r=0$. However
such a term cancels out with the first term within parenthesis for every odd
$p$. This leads to the form of the effective potential
$\varphi(r)=\frac{(-1)^{\gamma-\frac{1}{2}}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-3/2}\sqrt{\pi}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-\frac{1}{2}}\left(\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right).$
(27)
As an example, let us work out the case in which we have only one compact
extra dimension, ie $p=1\,\Rightarrow\,\gamma=5/2$. From (27) we obtain
$\varphi(r)=\frac{2q_{s}^{(1)}}{3\pi\epsilon}\left[3\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}+\frac{5}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)}+2\frac{\frac{r^{2}}{\epsilon^{2}}}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{2}}\right],$
(28)
where $q_{s}^{(1)}=\frac{3\lambda^{(1)}}{8\pi\epsilon}$. The finite value of
the potential at the 3D point position of the source takes the value
$\lim_{r\rightarrow 0}\varphi(r)=\frac{16q_{s}^{(1)}}{3\pi\epsilon},$ (29)
evidently regularized by the existence of $\epsilon$ and $R$. Using (28) we
can compute the effective self-energy of the point charge, as determined by a
3D observer
$E_{self}^{(p=1)}:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\nabla\varphi\right)^{2}=\frac{85\left(q_{s}^{(1)}\right)^{2}}{9\epsilon}.$
(30)
### III.4 Even number of extra compact dimensions
In the case that $p$ takes even values $\gamma$ is integer and we can use the
relation
$m^{l}J_{l}(m\epsilon)=(-1)^{l}\epsilon^{l}\left(\frac{d}{\epsilon
d\epsilon}\right)^{l-1}\left(-\frac{J_{1}(m\epsilon)m}{\epsilon}\right),$ (31)
in (24), to obtain 111It is worth mentioning here an alternative approach to
get the same results for both odd an even $p$. It amounts to using the
completeness relation of the noncompact modes. In such a case the coefficient
in front of $1/r$ is proportional to the square of a Dirac delta and by using
either dimensional regularization or distribution operations the coefficient
vanishes thus obtaining the same result, ie that the potential is finite at
the 3D position of the source Morales-Técotl et al. (2007).
$\varphi(r)=\frac{(-1)^{\gamma+1}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-1}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-1}\left(\frac{r}{\epsilon^{2}\sqrt{r^{2}+\epsilon^{2}}}\right).$
(32)
As an example, let us consider the lowest even value for $p$:
$p=2\,\Rightarrow\gamma=3$. From (32) we obtain
$\varphi(r)=\frac{q_{s}^{(2)}}{8\epsilon}\left[\frac{8}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{4}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}+\frac{3}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{5}{2}}}\right],$
(33)
where $q_{s}^{(2)}=\frac{\lambda^{(2)}}{2\pi\epsilon}$. In this case the
finite value of the potential at the 3D position of the source is
$\lim_{r\rightarrow 0}\varphi(r)=\frac{15q_{s}^{(2)}}{8\epsilon},$ (34)
whereas the source self-energy is given by
$E_{self}^{(p=2)}=\frac{51975\pi^{2}\left(q_{s}^{(2)}\right)^{2}}{65536\epsilon}.$
(35)
## IV Electrostatic potential
The procedure to compute this potential is similar to the one we used in the
scalar case. In the spirit of avoiding repetition, we describe briefly the
computation giving special emphasis to the aspects that are different with
respect to the scalar case. A previous discussion of the photon Green’s
function analysis in the RSII-$p$ scenario can be found in Dubovsky et al.
(2000b). We begin by considering the (5+$p$)D action
$S=\int\,d\,{}^{4}x\,\prod_{i=1}^{p}\,R_{i}d\theta_{i}\,dy\,\sqrt{|g|}\,\left(\frac{1}{4}g^{MN}g^{PQ}F_{MP}F_{NQ}+A_{M}J^{N}_{gauge}\right),$
(36)
leading to the equation of motion
$\displaystyle\frac{1}{\sqrt{|g|}}\partial_{M}\bigg{(}\sqrt{|g|}\,\,g^{MP}g^{NQ}F_{PQ}\bigg{)}=-j^{N}_{gauge}.$
(37)
We consider a static source along the brane, uniformly distributed along the
$p$ extra compact dimensions, namely
$\displaystyle\sqrt{|g|}j_{gauge}^{N}=\rho^{(p)}\delta^{N}_{0}\delta^{3}(\vec{x}-\vec{x}_{0})\delta(y-y_{0}),$
(38)
where $\rho^{(p)}$ is the charge density. Now we write down the equation of
motion for the gauge field in the background (2). In order to do this, it is
convenient to fix the gauge $A^{y}=0$ and $A^{\theta_{i}}=0$, which is
consistent with the value $J^{\theta_{i}}=0$ for the components of the current
density in the directions of the compact extra dimensions. Thus Eq. (37)
becomes
${\cal
O}\hat{A}^{\sigma}-e^{-p|y|/\epsilon}\partial^{\sigma}\partial_{\mu}\hat{A}^{\mu}=-R^{-p}e^{p|y|/\epsilon}\sqrt{g}\,\,j^{\sigma},$
(39)
where we assume equal size compact dimensions, $R_{i}=R,i=1,\dots,p$, and the
differential operator $\cal O$ is defined as
${\cal
O}:=e^{-(p+2)|y|/\epsilon}\left(-\partial^{2}_{y}+\frac{p+2}{\epsilon}\,sgn(y)\partial_{y}+e^{2|y|/\epsilon}\Box\right),$
(40)
and $\hat{A}^{\nu}=\eta^{\nu\mu}A_{\mu}$. Inspection of equation (39) reveals
the term $\partial_{\mu}\hat{A}^{\mu}$ is pure gauge on the brane, so we drop
it from now on Dubovsky et al. (2000b).
To solve (39) let us notice that the differential operator (40) is invariant
under the change $y\to-y$, so the solutions will inherit such symmetry. This
is important since we are looking for the potential on the brane. We shall
adopt again the Green’s function method. As in the scalar case, the necessary
tools are the eigenfunctions and eigenvalues of the differential equation.
The eigenfunctions and eigenvalues for the $p$ compact modes are the same as
those for the scalar field, Eq. (12). As for the noncompact modes depending
upon $y$ and subject to the boundary conditions (13) they fulfill again a
Bessel equation and have the following form
$\phi_{0}=\sqrt{\frac{p}{2\epsilon}},\quad\quad\phi_{m}(y)=e^{\nu
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}\left[a_{m}J_{\nu}\left({m\epsilon}\,e^{y/\epsilon}\right)+b_{m}N_{\nu}\left({m\epsilon}\,e^{y/\epsilon}\right)\right],$
(41)
where
$\nu\equiv\frac{p+2}{2},$ (42)
and the constants $a_{m}$, $b_{m}$ are defined as in (17), with $\gamma$
replaced by $\nu$. The modes are normalized in the form
$\int_{-\infty}^{\infty}dy\,e^{-p|y|/\epsilon}\phi_{0}^{2}=1$ and
$\int_{-\infty}^{\infty}dye^{-p|y|/\epsilon}\phi_{m}(m\epsilon)\phi_{m^{\prime}}(m^{\prime}\epsilon)=\delta(m-m^{\prime})\,$.
Formally the Green’s function, its low energy approximation and the static
potential on the brane are obtained from (18), (20) and (24), replacing the
scalar modes $\psi$ by the gauge modes $\phi$ as well as the factor $\gamma$
by $\nu$. The electrostatic Green’s function on the brane takes the form
$G_{gauge}(\vec{x}-\vec{x}^{\prime},y=0,y^{\prime})=\frac{p}{2\epsilon}\frac{1}{4\pi
r}-\frac{1}{4\pi r}\frac{e^{\nu
y^{\prime}/\epsilon}}{\Gamma(\nu-1)}\left(\frac{\epsilon}{2}\right)^{\nu-1}\int_{0}^{\infty}dm\,m^{\nu-1}J_{\nu}\left(my^{\prime}\right)e^{-mr}\,.$
(43)
Since we are interested in the potential for a source located on the brane, we
have to evaluate the above expression in the limit $y^{\prime}\rightarrow 0$.
As in the scalar case this limit is different depending on whether $p$ is
either even or odd. They are given explicitly below.
#### IV.0.1 $p$ odd
In this case the potential gets the form
$A^{0}(r)=\frac{\sigma^{(5+p)}}{4\pi
R^{p}r}\sqrt{\frac{2}{\pi}}\frac{(-1)^{\nu}\epsilon^{2\nu-1}}{\Gamma(\nu-1)(2)^{\nu-1}}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\nu-\frac{1}{2}}\left[\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right]\,,$
(44)
where $r=|\vec{x}-\vec{x}_{0}|$. As an example notice that for one extra
compact dimension $p=1$ one gets
$A^{0}(r)=\frac{2e}{\epsilon\pi}\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}\right),\quad
e=\frac{e^{(6)}}{2R\epsilon^{2}}\,,$ (45)
which reduces to the Coulomb potential for $r\gg\epsilon$ and is finite at the
3D source position
$\lim_{r\rightarrow 0}A^{0}(r)=\frac{4e}{\pi\epsilon}.$ (46)
The self-energy in this case is
$E_{self}^{(p=1)}:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\nabla
A^{0}\right)^{2}=\frac{5e^{2}}{32\pi^{3}\epsilon}.$ (47)
#### IV.0.2 $p$ even
Now $\nu$ is an integer and
$A^{0}(r)=\frac{(-1)^{\nu}\sigma^{(5+p)}\epsilon^{2\nu-1}}{2^{\nu-1}\Gamma(\nu-1)R^{p}}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\nu-1}\left(\frac{r}{\epsilon^{2}\sqrt{r^{2}+\epsilon^{2}}}\right)\,.$
(48)
Notice that for $p=2$,
$A^{0}(r)=\frac{e}{\epsilon}\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}\right),\quad
e=\frac{e^{(7)}}{R^{2}\epsilon}\,,$ (49)
which becomes the Coulomb potential for $r\gg\epsilon$ and its finite at the
3D source position:
$\lim_{r\rightarrow 0}A^{0}(r)=\frac{3e}{2\epsilon}.$ (50)
The source self-energy is now
$E_{self}^{(p=2)}=\frac{315e^{2}}{16384\pi\epsilon}.$ (51)
The static potentials for $p=1$, $p=2$ and Coulomb’s are compared in Fig. (2).
Remarkably as we have mentioned, the electrostatic potentials corrected by the
extra dimensions are finite at the 3D position of the charge.
It is interesting and natural to explore possible consequences of the modified
electrostatic potentials we just obtained using known experiments like the
Cavendish and scattering ones. We do so in the following section.
Figure 2: Electrostatic potential of the point particle for the standard 4D
Coulomb case and $p=1,2$.
## V Phenomenology of the electrostatic potential
### V.1 Cavendish experiment
From the different results obtained to verify the accuracy of the
electrostatic force, we have chosen the ones obtained by Plimpton and Lawton
Plimpton and Lawton (1936) and more recent modifications (see Tu et al.
(2005); Tu and Luo (2004) for a review of the different experiments). The
reason is that this belongs to a series of experiments in which the main idea
was to test the accuracy of Coulomb’s force between charged particles using
similar techniques as the one used by Cavendish to test the gravitational
force (see for instance Goldhaber and Nieto (2010) for a recent review on the
different perspectives and experiments performed to test different aspects of
electrodynamics). In the modern version of Cavendish experiment we have a
modified electromagnetic potential for a charge $Q$
$V=V_{C}+\delta V,$
where $\delta V$ is the modification to the Coulomb potential. The idea behind
the concentric charged spheres experiments is that only for the Coulomb
potential the interior of a charged sphere is field free and therefore the
potential there is a constant. Then the potential difference between a charged
outer sphere and the uncharged inner sphere is zero only if the potential is
that of Coulomb. Any deviation from this would imply a nonvanishing potential
difference between the spheres that can be measured.
The potential of a sphere with a charge $Q$ and radius $c$ at a distance $r$
from the center is
$U(Q,r,c)=\frac{Q}{2cr}\left[f(r+c)-f(|r-c|)\right],$
where the function $f$ is given by
$f(r)=\int_{0}^{r}{ds\;sV(s,Q=1)},$
It is easy to verify that for $V=\alpha/r$, $U(Q,r<c,c)=const.$, that is the
potential is constant in the interior.
In the simplest version of the Cavendish experiment one has an outer sphere of
radius $b$, charged to a certain voltage, and then measures the relative
voltage difference to the uncharged inner sphere of radius $a<b$,
$\gamma_{ab}=\left|\frac{{\mathcal{V}}_{b}-{\mathcal{V}}_{a}}{{\mathcal{V}}_{b}}\right|\;\;=\;\;\left|\frac{U(Q,b,b)-U(Q,a,b)}{U(Q,b,b)}\right|.$
(52)
Plimpton and Lawton found that $|\gamma_{ab}|\leq 3\times 10^{-10}$ with
$a=0.696$ m, $b=0.762$m. Here we will calculate $|\gamma_{ab}|$ for the two
potentials corresponding to $p=1$ and $p=2$, namely
$\displaystyle v_{1}$ $\displaystyle=$
$\displaystyle\frac{2Q}{\epsilon\pi}\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan(\frac{r}{\epsilon})}{\frac{r}{\epsilon}}\right),$
$\displaystyle v_{2}$ $\displaystyle=$
$\displaystyle\frac{Q}{\epsilon}\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2(1+\frac{r^{2}}{\epsilon^{2}})^{\frac{3}{2}}}\right).$
(53)
The results, to first order in $\epsilon$ are
$|\gamma_{1ab}|=\frac{\epsilon}{\pi
b},\;\;|\gamma_{2ab}|=\frac{\epsilon}{4b}.$ (54)
Taking into account the experimental bound of Plimpton and Lawton this means
that $\epsilon\leq 7.18\times 10^{-10}m$ or $\epsilon\leq 9.14\times
10^{-10}m$ for $p=1$ and $p=2$, respectively.
The more recent version of the Cavendish experiment employs four concentric
spheres of radii a, b, c, d in increasing order. The Outer sphere has a charge
$Q$ and the next one -$Q$. Then the potential at radius r is given by
$U(Q,r,c,d)=\frac{Q}{2dr}\left[f(r+d)-f(|r-d|)\right]-\frac{Q}{2cr}\left[f(r+c)-f(|r-c|)\right],$
(55)
The experiment sets a bound for the ratio of the potential differences between
the two uncharged spheres and the two outer spheres
$\gamma_{abcd}=\left|\frac{{\mathcal{V}}_{b}-{\mathcal{V}}_{a}}{{\mathcal{V}}_{c}-{\mathcal{V}}_{d}}\right|=\left|\frac{U(Q,b,c,d)-U(Q,a,c,d)}{U(Q,c,c,d)-Q(Q,d,c,d)}\right|.$
(56)
Williams et al. Williams et al. (1971) found that $|\gamma_{abcd}|\leq 2\times
10^{-16}$ with $a=0.60$ m, $b=0.94$m, $c=0.947$m and $d=1.27$m. Here we will
calculate $|\gamma_{abcd}|$ for the two potentials corresponding to $p=1$ and
$p=2$, and using the experimental limits to constrain $\epsilon$. A
straightforward calculation gives, to leading order in $\epsilon$
$\gamma_{1abcd}=\frac{cd\left(\frac{4(c-d)(c+d)\left(-2a^{2}+c^{2}+d^{2}\right)}{\left(a^{2}-c^{2}\right)^{2}\left(a^{2}-d^{2}\right)^{2}}+\frac{4\left(2b^{2}(c-d)(c+d)-c^{4}+d^{4}\right)}{\left(b^{2}-c^{2}\right)^{2}\left(b^{2}-d^{2}\right)^{2}}\right)}{3\pi(c-d)}\epsilon^{3}+O\left(\epsilon^{4}\right),$
(57)
$\gamma_{2abcd}=\frac{cd\left(\frac{\frac{\frac{1}{(a+c)^{3}}+\frac{1}{(a-c)^{3}}}{c}+\frac{\frac{1}{(d-a)^{3}}-\frac{1}{(a+d)^{3}}}{d}}{a}+\frac{\frac{\frac{1}{(c-b)^{3}}-\frac{1}{(b+c)^{3}}}{c}+\frac{\frac{1}{(b+d)^{3}}+\frac{1}{(b-d)^{3}}}{d}}{b}\right)}{16(c-d)}\epsilon^{4}+O\left(\epsilon^{5}\right).$
(58)
Taking into account the experimental value obtained by Williams et al.
($|\gamma_{abcd}|\leq 2\times 10^{-16}$ ) the corresponding bounds for
$\epsilon$ are $\epsilon\leq 2.80\times 10^{-7}m$ or $\epsilon\leq 4.02\times
10^{-6}m$ for $p=1$ and $p=2$, respectively. In this case the two sphere
experiment gives a tighter constraint on $\epsilon$. The reason for this may
be the peculiarities of the modification of the Coulomb potentials that in our
case contains positive powers of $r$.
### V.2 Scattering by Helium atoms
We shall study the collision of a particle of charge $ze$ and mass $m$ with an
atom of atomic number $Z$. Notice that an exact formulation of this problem
requires the use of a many-body Hamiltonian which describes all the particles
of the system, however we shall make the assumption that the complicated
interaction of the incident particle with the constituents of the atom can be
accounted for by an effective electrostatic potential $V(r)$ in which the
incident particle travels.
It is physically reasonable that the electrostatic potential in which the
incident particle travels is well approximated by
$V(\vec{r})=ze\left[Zev_{1,2}(\vec{r})+e\int\rho(\vec{r}^{\prime})v_{1,2}\left(|\vec{r}-\vec{r}^{\prime}|\right)d^{3}\vec{r}^{\prime}\right],$
(59)
were $\vec{r}$ is the position vector of the incident particle and
$v_{1,2}(\vec{r})$ are given by (45) and (49). The first term is due to the
field of the nucleus and the second term is the potential of the atomic
electrons, described in terms of an effective electron density $\rho$. It is
worth mentioning that in this description we are neglecting all effects of
symmetry and spin. For neutral atoms, the density satisfies
$\int\rho(\vec{r})d^{3}\vec{r}=Z.$ (60)
When the incident particle carries sufficiently high energy, the scattering
amplitudes can be easily evaluated by the Born approximation
$f(\theta)=-\frac{m}{2\pi\hbar^{2}}\int
e^{i\vec{q}\cdot\vec{r}}V(\vec{r})d^{3}\vec{r},$ (61)
where $\vec{q}=\vec{k}_{0}-\vec{k}$, and $\vec{k}_{0}$ and $\vec{k}$ are the
initial and final momentum, respectively. Since the scattering is elastic,
$|\vec{k}|=|\vec{k}_{0}|=k$. Thus introducing Eq. (59) in (61) and making the
following change of variable $\vec{R}=\vec{r}-\vec{r}^{\prime}$ we have
$\displaystyle f_{1,2}(\theta)$ $\displaystyle=$
$\displaystyle-\frac{me^{2}}{2\pi\hbar^{2}}\left[zZ\int
e^{i\vec{q}\cdot\vec{r}}v_{1,2}(\vec{r})d^{3}\vec{r}-z\int
e^{i\vec{q}\cdot\vec{R}}v_{1,2}\left(\vec{R}\right)d^{3}\vec{R}\int\rho(\vec{r}^{\prime})e^{i\vec{q}\cdot\vec{r}^{\prime}}d^{3}\vec{r}^{\prime}\right],$
(62) $\displaystyle=$
$\displaystyle-\frac{me^{2}z}{2\pi\hbar^{2}}\left[Z-F(\vec{q})\right]\int
e^{i\vec{q}\cdot\vec{r}}v_{1,2}(\vec{r})d^{3}\vec{r},$
$F(\vec{q})$ is called the form factor of the atom. We defined $F(\vec{q})$ as
$F(\vec{q})=\int\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^{3}\vec{r}.$ (63)
When the potential is spherically symmetric, the angular integration can be
performed to give
$f_{1,2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\frac{\sin(qr)}{qr}v_{1,2}(r)r^{2}dr,$
(64)
with $q=|\vec{q}|=2k\sin(\theta/2)$ and $r=|\vec{r}|$. The evaluation of this
integral depends on the form that $v_{1,2}(r)$ takes. We first calculate
$f_{1}(\theta)$
$f_{1}(\theta)=-\frac{4me^{2}z}{\pi\hbar^{2}}\frac{1}{\epsilon
q}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\sin(qr)\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}\right)rdr,$
(65)
using the following relations (see Gradshteyn and Ryzhik (1980))
$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{1+\frac{r^{2}}{\epsilon^{2}}}rdr$
$\displaystyle=$ $\displaystyle\frac{\pi}{2}\epsilon^{2}e^{-q\epsilon},$ (66)
$\displaystyle\int_{0}^{\infty}\sin(qr)\frac{\arctan\left(\frac{r}{\epsilon}\right)}{r}rdr$
$\displaystyle=$ $\displaystyle\frac{\pi}{2}\frac{e^{-q\epsilon}}{q},$ (67)
$f_{1}(\theta)$ can be written as
$f_{1}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\left[\frac{1}{q^{2}}+\frac{\epsilon}{q}\right]e^{-q\epsilon}.$
(68)
Considering the form of $v_{2}(r)$, $f_{2}(\theta)$ can be expressed as
$f_{2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\frac{1}{\epsilon
q}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\sin(qr)\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}\right)rdr.$
(69)
Now let us consider the integrals
$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}rdr$
$\displaystyle=$ $\displaystyle\epsilon^{2}K_{1}(q\epsilon),$ (70)
$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}rdr$
$\displaystyle=$ $\displaystyle\epsilon^{3}qK_{0}\left(q\epsilon\right),$ (71)
where $K_{0}(x)$ and $K_{1}(x)$ are Bessel functions of zeroth and first
order, respectively. Thus $f_{2}(\theta)$ takes the form
$f_{2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\left[\frac{\epsilon}{q}K_{1}(q\epsilon)+\frac{\epsilon^{2}}{2}K_{0}(q\epsilon)\right].$
(72)
For Helium we can calculate the electron density as
$\rho(r)=Z\left(\frac{b^{3}}{\pi a_{0}^{3}}\right)e^{\frac{-2br}{a_{0}}},$
(73)
with $b$ being the effective charge and having the value 1.69 for Helium while
$a_{0}$ is the Bohr radius. The form factor becomes
$F(q)=\frac{Z}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}.$ (74)
The differential scattering cross section for elastic processes thus become
$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)^{(p=1)}$ $\displaystyle=$
$\displaystyle\left(\frac{2zZ}{a_{0}q^{2}}\right)^{2}\left[1-\frac{1}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}\right]^{2}\left[1+q\epsilon\right]^{2}e^{-2q\epsilon},$
(75) $\displaystyle\left(\frac{d\sigma}{d\Omega}\right)^{(p=2)}$
$\displaystyle=$
$\displaystyle\left(\frac{2zZ}{a_{0}q^{2}}\right)^{2}\left[1-\frac{1}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}\right]^{2}\left[q\epsilon
K_{1}(q\epsilon)+\frac{q^{2}\epsilon^{2}}{2}K_{0}(q\epsilon)\right]^{2}.$ (76)
For incident electrons, we set $z=-1$ and $Z=2$ for the Helium atom. To
complete the analysis we compare the theoretical results with the
corresponding experimental ones. This comparison is made explicit in Figures
(3) and (4). For both $p=1,2$ a best agreement is attained when $\epsilon\sim
10^{-10}m$.
Figure 3: Comparison of experimental differential cross section Jansen et al.
(1976) with that corresponding to one compact dimension, Eq. (75). Figure 4:
Comparison of experimental differential cross section Jansen et al. (1976)
with that corresponding to two compact dimensions, Eq. (76).
## VI Discussion
The ever increasing accuracy with which electrodynamics has been tested
naturally lends itself to consider it as a probe to set bounds for possible
deviations coming from the existence of extra dimensions. Amongst different
models the so called Randall-Sundrum ones including a single 3-brane and $p$
extra compact dimensions (RSII-$p$) have provided simple scenarios that yield
effects well under control. Take for example the Casimir force Linares et al.
(2010); Frank et al. (2008): In a nutshell the field modes corresponding to
the extra dimensions add up to modify the usual Casimir force expression and
the deviations are assumed to be bounded by the uncertainties in the
experimental data. This in turn sets bounds for the parameters of the brane
model.
In this work we have explored the static potential produced by a scalar and a
charged sources, respectively, in RSII-$p$. These sources are effectively
pointlike from the perspective of an observer sitting in the usual 3D space.
However they stretch uniformly along the $p$ compact dimensions thus having
the structure of a $T^{p}$ torus. Remarkably the effective potentials turn out
to be non-singular at the position in 3D space. At first one may think this is
related to the fact the sources stretch along the extra dimensions, similarly
as in models of charged spherical shells Rohrlich (2007). This is not the case
as a more careful look reveals: the potential produced by either a charged
ring or a torus is not finite at the source itself Andrews (2006); Kondratev
et al. (2008); Bannikova et al. (2010). The RSII-p scenario thus allows to
regularize the 3D potentials and selfenergies. Indeed the combined limit
having AdS radius and compact size going to zero yields the usual standard
divergent result.
We have determined the potentials in the low energy regime in terms of light
modes; this entails approximating the continuous modes given in terms of
Bessel functions by their small argument form whereas for the compact modes we
keep the zero mode only. Within this approximation a delicate balance occurs
between part of the massive sector contribution to the potential and the zero
mode. Since the zero mode is responsible for the usual singular $1/r$ term,
the potential characteristic of massless fields, the balance just described
regularizes such a divergence. Moreover the remaining effective potential
becomes the usual $1/r$ within a few times $\epsilon$ away from $r=0$ and
provides finite selfenergies as determined from the usual 3D formulae.
To probe the effective potentials we proposed to consider two types of
experiments. First we adopted the long known Cavendish experiment with two and
four conducting spheres that is used to test the form of the Coulomb force. To
be consistent with know experimental results for the case of two spheres a
value of $\epsilon\sim 10^{-10}m$ is required. The four spheres setting
however turns out to produce a milder bound $\epsilon\sim 10^{-7}m$, probably
due to the positive powers of the correcting terms of the effective potentials
when developing around $1/r$. The second possibility we studied to test our
effective potentials was to consider electrons scattered off by Helium. A
comparison of the differential cross section modified by the RSII-p scenario
with the curve fitting experimental data indicates consistency with a value of
$\epsilon\sim 10^{-11}m$. In a previous work Morales-Técotl et al. (2007) we
used the Lamb shift to set a bound of $\epsilon\sim 10^{-14}m$ for $p=1$, and
$\epsilon\sim 10^{-13}m$, for $p=2$, which clearly are stronger than the ones
obtained in the present work.
The fact that for both the scalar and electromagnetic case the potentials
become well behaved leads naturally to the question of whether the same
results holds for the gravitational case. This is work under study and will be
reported elsewhere. Indeed, historically, finiteness of the potentials have
led in the past to the idea that gravity regulates the self-energy of the
charged point particle Arnowitt et al. (1962) as well as nonlinear field
equations to achieve the finiteness of the electric field Born and Infeld
(1934).
In the low energy regime we have focused on in this work there are some other
possible directions which can be pursued. These include a reanalysis of the
radiation reaction problem in both electromagnetic an gravitational cases as
well as the complete understanding of the regularization of the potentials and
in particular its relation to the topology of the sources together with their
dimensionality.
###### Acknowledgements.
HAMT acknowledges partial support from grant CONACyT-NSF Strong backreaction
effects in quantum cosmology.
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|
arxiv-papers
| 2011-07-31T06:39:44 |
2024-09-04T02:49:21.110153
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rom\\'an Linares, Hugo A. Morales-T\\'ecotl, Omar Pedraza, Luis O.\n Pimentel",
"submitter": "Omar Pedraza",
"url": "https://arxiv.org/abs/1108.0147"
}
|
1108.0156
|
# Room-temperature generation of giant pure spin currents
using Co2FeSi spin injectors
Takashi Kimura kimura@ifrc.kyushu-u.ac.jp Advanced Electronics Research
Division, INAMORI Frontier Research Center, Kyushu University, 744 Motooka,
Fukuoka, 819-0395, Japan CREST, Japan Science and Technology Agency,
Sanbancho, Tokyo 102-0075, Japan Naoki Hashimoto Department of Electronics,
Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan
Shinya Yamada Department of Electronics, Kyushu University, 744 Motooka,
Fukuoka 819-0395, Japan Masanobu Miyao Department of Electronics, Kyushu
University, 744 Motooka, Fukuoka 819-0395, Japan CREST, Japan Science and
Technology Agency, Sanbancho, Tokyo 102-0075, Japan Kohei Hamaya
hamaya@ed.kyushu-u.ac.jp Department of Electronics, Kyushu University, 744
Motooka, Fukuoka 819-0395, Japan PRESTO, Japan Science and Technology Agency,
Sanbancho, Tokyo 102-0075, Japan
Generation, manipulation, and detection of a pure spin current, i.e., the flow
of spin angular momentum without a charge currentJedema01-1 ; Jedema01-2 ;
Chappert ; Zutic ; Saitoh ; Kimura02 , are prospective approaches for
realizing next-generation spintronic devices with ultra low electric power
consumptions. Conventional ferromagnetic electrodes such as Co and NiFe have
so far been utilized as a spin injector for generating the pure spin currents
in nonmagnetic channelsJedema01-1 ; Jedema01-2 ; Kimura02 ; Kimura03 ;
Kimura04 ; Yang ; Kimura05 ; Tinkham ; Johnson ; Mihajlovic ; idzuchi .
However, the generation efficiency of the pure spin currents is extremely low
at room temperature, giving rise to a serious obstacle for device
applications. Here, we demonstrate the generation of giant pure spin currents
at room temperature in lateral spin valve devices with a highly ordered
Heusler-compound Co2FeSi spin injector. The generation efficiency of the pure
spin currents for the Co2FeSi spin injectors reaches approximately one hundred
times as large as that for NiFe ones , indicating that Heusler-compound spin
injectors enable us to materialize a high-performance lateral spin device. The
present study is a technological jump in spintronics and indicates the great
potential of ferromagnetic Heusler compounds with half metallicity for
generating pure spin currents.
Electrical spin injection from a ferromagnet (F) into a nonmagnet (N) can
generate a spin current, i.e., the flow of spin angular momentum, even in a
nonmagnetic channelJohnson1 . In general, the spin current is induced by
diffusing non-equilibrium spin accumulations in the vicinity of the F/N
interface under the spin injection. However, since the difference in the
density of states between majority and minority spins, i.e., spin polarization
$P$, is not so large for a conventional F such as Co or NiFe (Py), the induced
spin current in the N mainly returns back to the F (Fig. 1a). This gives rise
to an extremely low injection efficiency of the spin current in the NKimura02
; van Son ; Schmidt . If we utilize a perfectly spin-polarized F, so called a
half-metallic ferromagnet (HMF)Groot , as a spin injector, fully spin-
polarized electrons can be injected into the N and the backflow of the spin
currents can be completely suppressed, resulting in a dramatical improvement
of the injection efficiency of the spin currents in the N (Fig. 1b). Also, one
can extract a charge current by using nonlocal electrical spin injection in a
mesoscopic lateral geometry, and can transfer only a spin current without the
charge current, i.e., a pure spin current (Fig. 1c), in the nonmagnetic
channel. In this scheme, using HMF spin injectors is a key for generating a
giant pure spin current in the N (Fig. 1d).
As materials with half metallicity, we focus on Co-based Heusler compounds
which enable huge tunnel magnetoresistance (TMR) and giant magnetoresistance
(GMR) effects in vertical stacking device structuresInomata ; Felser ;
Sakuraba1 . Despite these high performances, none of the lateral spin
transports using the Heusler-compound electrodes have been reported yet. Thus,
the combination of the high-performance Co-based Heusler compounds with
laterally configured device structures is a prospective challenge for highly
efficient generation of the pure spin currents. In this work, we show that a
Co-based Heusler compound, Co2FeSi (CFS), enables the highly efficient
injection of the spin currents.
Our device structure is a lateral spin valve (LSV) consisting of the CFS spin
injector and detector bridged by a Cu strip (Fig. 2a), where the CFS thin film
with highly ordered $L2_{1}$ structures has been epitaxially grown on
Si(111)Yamada . Details of the growth of the CFS thin films and the
fabrication processes of the LSVs are given in the Methods sections. As shown
in Fig. 2b, a pure spin current generated by the nonlocal spin injection from
CFS1 can be detected by CFS2 after the propagation of 600-nm distance in the
Cu strip. Figure 2c shows a nonlocal magnetoresistance of the CFS/Cu LSV
measured at room temperature (RT), together with that of a Py/Cu LSV. Here,
the size of the CFS/Cu junction is three times as large as that of the Py/Cu
junction. Note that a giant spin signal ($\Delta R_{\rm S}$) of 2.3 m$\Omega$
is seen for the CFS/Cu LSV (Fig. 2c), which is approximately ten times as
large as that for the Py/Cu LSV. Since the spin injection efficiency is
inversely proportional to the size of the F/N junctions Kimura02 , the giant
$\Delta R_{\rm S}$ demonstrated in the CFS/Cu LSV with larger sizes in the
junctions implies a great possibility of the present CFS/Cu LSV. We also show
the local spin valve signal of 4.5 m$\Omega$ at RT for the same CFS/Cu LSV
(Fig. 2d). The value of 4.5 m$\Omega$ is almost twice of the non-local $\Delta
R_{\rm S}$, in reasonable agreement with the previous reports Jedema01-2 ;
Kimura02 . This means that one dimensional spin diffusion model well describes
the spin transport in the present CFS/Cu LSV. Figure 2e shows a dependence of
$\Delta R_{\rm S}$ on the bias current density ($J_{\rm inj}$) at the
injecting junction for the CFS/Cu LSV, which is almost same as that for the
Py/Cu LSV. The reduction of the $\Delta R_{\rm S}$ is less than 20 % even
under a high bias current density ($\sim$ 1011 A/m2), indicating much superior
property compared to the LSV consisting of the high resistive tunnel
junctions, where the $\Delta R_{\rm S}$ drastically decreases even at low bias
current density ($\sim 10^{8}$ A/m2)Tinkham . The temperature dependence of
$\Delta R_{\rm S}$ for the CFS/Cu LSV is also almost same as that for the
Py/Cu LSV, where the $\Delta R_{\rm S}$ takes a maximum value around 20 K,
below which the $\Delta R_{\rm S}$ decreases with decreasing temperature (Fig.
2f). This behavior can be explained by an enhancement in the spin-flip
scattering at the Cu surface for the CFS/Cu LSV below 20 K, as discussed in
Ref Kimura05 . Surprisingly, the $\Delta R_{\rm S}$ for the CFS/Cu exceeds 10
m$\Omega$ below 70 K (inset of Fig. 2f). From these results, we recognize that
the present CFS/Cu LSV can be treated as conventional ohmic LSVs and can
generate a giant pure spin current with a much less electric power than
previously reported LSVsJedema01-1 ; Jedema01-2 ; Kimura02 ; Kimura03 ;
Kimura04 ; Yang ; Kimura05 ; Tinkham ; Johnson ; Mihajlovic ; idzuchi .
To quantitatively evaluate the device performance of the present LSVs from the
nonlocal spin signals, we measured $\Delta R_{\rm S}$ of CFS/Cu LSV devices
with various distances ($d$), together with Py/Cu LSV devices as references,
where $d$ is the centre-centre distance between spin injector and detector.
Here, we introduce a characteristic value in the LSV devices by extending the
resistance change area product, commonly utilized to characterize the device
performances in the vertical spin devicesNakatani2 ; Sakuraba2 . The
resistance change area product for the nonlocal spin signal, i.e., $\Delta
R_{\rm S}A$, is defined as $\Delta R_{\rm S}(S_{\rm inj}S_{\rm det}/S_{\rm
N})$, where $S_{\rm inj}$, $S_{\rm det}$, and $S_{\rm N}$ are the junction
sizes in the spin injector and detector, and the cross section of the
nonmagnetic strip. This $\Delta R_{\rm S}A$ allows us to equivalently compare
the generation efficiency of the pure spin current between our CFS/Cu and the
other conventional F/N LSV devices. The plot of $\Delta R_{\rm S}A$ versus $d$
at RT for CFS/Cu LSVs and Py/Cu LSVs is shown in Fig. 3, together with that at
80 K in the inset. The $\Delta R_{\rm S}A$ is increased with decreasing $d$
for both series of the CFS/Cu and Py/Cu LSV devices. By solving one
dimensional spin diffusion equationTakahashi ; Kimura02 (see Supplementary
information), $\Delta R_{\rm S}A$ can be expressed as
$\Delta R_{\rm S}A\approx\frac{\left(\frac{P_{F}}{(1-P_{F}^{2})}\rho_{\rm
F}\lambda_{\rm F}+\frac{P_{I}}{(1-P_{I}^{2})}{\rm RA}_{\rm
F/N}\right)^{2}}{\rho_{\rm N}\lambda_{\rm N}\sinh\left({d}/{\lambda_{\rm
N}}\right)},$ (1)
where $P_{\rm F}$ and $P_{\rm I}$ are, respectively, the bulk and interface
spin polarizations for F, $\lambda_{\rm F}$ and $\lambda_{\rm N}$ are the spin
diffusion lengths for F and N, and, $\rho_{\rm F}$ and $\rho_{\rm N}$ are the
resistivities for F and N, respectively. As shown in Fig. 3, the plots of
$\Delta R_{\rm S}A$ versus $d$ for both series of the CFS/Cu and Py/Cu LSV
devices are well reproduced by the fitting curves with $\lambda_{\rm Cu}=$ 500
nm and $\lambda_{\rm Cu}=$ 1300 nm at RT and 80 KJedema01-1 ; Jedema01-2 ;
Kimura02 ; Kimura03 , respectively. For the Py/Cu LSVs, assuming $\lambda_{\rm
Py,RT}=$ 3 nm and $\lambda_{\rm Py,80K}=$ 5 nm, we obtained a reasonable
$P_{\rm Py}$ of $0.3$ and 0.35 at RT and 80 K, respectivelyJedema01-1 ;
Jedema01-2 ; Kimura02 . Thus, the above equation is a reliable for expressing
the generation efficiency of the pure spin current among various LSVs. We then
roughly estimate the spin polarization for CFS ($P_{\rm CFS}$). Since it is
impossible to determine the spin polarization and the spin diffusion length
independently from the present results, we assume that $\lambda_{\rm CFS}$ is
the same order of that for CFSA (see Supplementary information). If we use
$\lambda_{\rm CFS,RT}=$ $2\sim 4$ nm and $\lambda_{\rm CFS,80K}=$ $3\sim 6$
nm, $P_{\rm CFS}$ can be estimated to be 0.56 $\pm$ 0.10 at RT and 0.67 $\pm$
0.11 at 80 K, similar to the $P$ estimated from the analysis of the current
perpendicular GMR effectsSakuraba2 ; Nakatani2 . Although the present CFS
epitaxial layers have highly ordered structures with a high magnetic moment
above 5 $\mu_{\rm B}$/f.uYamada , the value is still smaller than 6 $\mu_{\rm
B}$/f.u. in the perfectly ordered CFSInomata ; Felser . Since further
enhancement in $P_{\rm CFS}$ will be achieved by improving the crystal growth
technique, a scaling characteristic with $P=1$ will be obtainable ultimately
(see dashed line). It should be noted that the resistivity for the
ferromagnetic electrode ($\rho_{\rm F}$) and the interface resistance area
product (RAF/N) are also important factors in Eq. (1). Therefore, the
relatively large $\rho_{\rm CFS}$ and RACFS/Cu compared to those in the Py/Cu
LSV (see Method) are also advantages for obtaining large $\Delta R_{\rm S}A$.
Our data for the CFS/Cu LSVs is approximately one hundred times as large as
that for the Py/Cu LSV, indicating a significant improvement of the generation
efficiency of the pure spin current using ohmic junctions. The present result
is a markedly technological advance in spintronics using pure spin currents,
generated by Heusler-compound spin injectors.
Figure 1: Concept of efficient generation of a pure spin current. a,b,
Schematic diagrams of the electrical spin injection from a conventional
ferromagnet (F) or a half-metallic (HM) F into a nonmagnet (N). For the
conventional F, most of the original spin currents go back to the F (a
backflow of the spin current), giving rise to a significant reduction in the
injected spin current. For the HMF, the original spin current is fully
injected into the N without the backflow. c, Generation of a pure spin current
by using nonlocal spin injection. The electron charges are extracted toward
left hand side while the spin currents diffuse into both side symmetrically.
d, Spatial distributions of the spin-dependent electro-chemical potentials,
($\mu_{\uparrow}$, $\mu_{\downarrow}$), in the N. Although the charge current
$\propto\partial(\mu_{\uparrow}+\mu_{\downarrow})/\partial x$ is zero in the
right hand side, a finite spin current
$\propto\partial(\mu_{\uparrow}-\mu_{\downarrow})/\partial x$ is generated
over the spin diffusion length. Thus, the pure spin current can be generated
in the right hand side of the N. Figure 2: Giant nonlocal spin valve effect.
a, A scanning electron microscope image of the fabricated Co2FeSi(CFS)/Cu
lateral spin valve. b, Schematic of a nonlocal spin valve measurement. Spin-
polarized electrons are injected from contact 2, and electron charges are
extracted from contact 1. A nonlocal voltage is measured between contact 3 and
contact 4. c, A room-temperature nonlocal spin-valve signal for the CFS/Cu
LSV, together with that for the Py/Cu LSV. The signal varies according to the
relative magnetization orientation of two wire-shaped CFS electrodes, as shown
in the inset illustrations. d, A room-temperature local spin-valve signal for
the CFS/Cu LSV. The inset shows the current-voltage probe configuration, i.e.
the current is injected from contact 2 and extracted from contact 3, and the
voltage is measured between contact 5 and contact 6. The low and high
resistance states correspond to the parallel and anti-parallel magnetization
alignments, respectively. The expected magnetization configurations agree with
those observed in the nonlocal spin valve signal. e, Nonlocal spin signal
$\Delta R_{\rm S}$ as a function of $J_{\rm inj}$ at the injecting junction,
normalized by $\Delta R_{\rm S}$ at a small bias current density of $J_{0}\sim
10^{9}$ A/m2, for the CFS/Cu LSV (red solid squares) and the Py/Cu LSV (blue
open circles). f, Temperature dependence of $\Delta R_{\rm S}$ for the CFS/Cu
LSV (red solid squares) and the Py/Cu LSV (blue open circles), normalized by
$\Delta R_{\rm S}$ at 20 K. The inset shows a nonlocal spin-valve effect of
the CFS/Cu LSV at $T=70$ K. The scale bars in c, d, and the inset of f are 1,
2, and 5 m$\Omega$, respectively. Figure 3: Scaling plot for lateral spin
valve devices with metallic junctions. The resistance change area product for
the nonlocal spin signal ($\Delta R_{\rm S}A$) as a function of $d$ for CFS/Cu
LSVs (filled squares), together with that for Py/Cu LSVs (open circles). The
main panel and inset show the data at RT and 80 K, respectively. The solid
curves are fitting results with $\lambda_{\rm Cu}=500$ nm at RT (red line) and
$\lambda_{\rm Cu}=1300$ nm at 80 K (blue line), and the dashed curves are
theoretical upper limits using $P=$ 1, calculated from the equation, $\Delta
R_{\rm S}A=P^{2}(S_{\rm Inj}S_{\rm Det}/S_{\rm N})\rho_{\rm Cu}\lambda_{\rm
Cu}e^{-d/\lambda_{\rm Cu}}$.
## Method
As a spin injector- and detector-material, 25-nm-thick Co2FeSi (CFS) films
were grown on Si(111) templates by low temperature molecular beam
epitaxyYamada . Prior to the growth, surface cleaning of substrates was
performed with an aqueous HF solution (HF : H2O = 1 : 40), and then, they were
heat-treated at 450 ∘C for 20 min in an MBE chamber with a base pressure of 2
$\times$ 10-9 Torr. After the reduction in the substrate temperature down to
100 ∘C, we co-evaporated Co, Fe, and Si with stoichiometric chemical
compositions by using Knudsen cells. During the growth, two-dimensional
epitaxial growth was confirmed by observing reflection high energy electron
diffraction patterns. The formed epitaxial CFS films were characterized by
means of cross-sectional transmission electron microscopy (TEM), nanobeam
electron diffraction (ED), and 57Fe conversion electron Mössbauer
spectroscopy. From these detailed characterizations, we have observed highly
ordered $L2_{1}$ structures in the CFS layersYamada . Next, we patterned
submicronsized resist mask structures on the CFS films using a conventional
electron-beam lithography. Then an Ar ion milling technique is employed to
form the wire-shaped CFS spin injector and detector with 300 nm in width. One
CFS wire is connected to two square pads to facilitate domain wall nucleation,
while the other has pointed-end edges. Using the two different wire shapes, we
can control the magnetization configuration by adjusting external magnetic
fields ($H$), where $H$ is applied along the CFS wires. Finally, top Cu
strips, 200 nm in width and 100 nm in thikness, bridging the CFS wires and
bonding pads were patterned by a conventional lift-off technique. Prior to the
Cu deposition, the surfaces of the CFS wires were well cleaned by the Ar ion
milling with a low accelerating voltage, resulting in low resistive ohmic
interfaces with $R_{\rm CFS/Cu}$ $\approx$ 1 ${\rm f}\Omega{\rm m}^{2}$.
Nonlocal and local spin valve measurements were carried out by a conventional
current-bias lock-in technique ($\sim 200$ Hz). The resistivities for the
prepared CFS and Cu wires are 90.5 $\mu\Omega$cm and 2.5 $\mu\Omega$ cm at RT
and 54.6 $\mu\Omega$cm and 1.2 $\mu\Omega$cm at 80 K, respectively.
## Acknowledgements
This work was supported by Fundamental Research Grants from CREST-JST and
PRESTO-JST.
## Author Information
T. K., N. H. and K. H. carried out the device fabrications and the transport
measurements. S. Y., M. M. and K. H. prepared the epitaxial Co2FeSi films. T.
K. and K. H. performed the data analysis. T. K., M. M. and K. H. planned the
present project and wrote the paper.
## References
* (1) Jedema, J, F., Filip, T. A. & van Wees, J. B. Electrical spin injection and accumulation at room temperature in an all-metal mesoscopic spin valve. Nature 410, 345-348 (2001).
* (2) Jedema, J. F., Nijboer, S. M., Filip, T. A. & van Wees, J. B. Spin injection and spin accumulation in all-metal mesoscopic spin valves. Phys. Rev. B 67, 085319 (2003).
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* (7) Kimura, T., Otani, Y. & Hamrle, J. Enhancement of spin accumulation in a nonmagnetic layer by reducing junction size. Phys. Rev. B 73, 132405 (2006).
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## Supplementary information
### Introducing the resistance area product for nonlocal spin signals
In the lateral spin valve consisting of two ferromagnetic wire bridged by a
nonmagnetic strip as shown in Figs. 2a and 2b, the nonlocal spin signal
$\Delta R_{\rm S}$ based on a one-dimensional spin diffusion model is
givenVFmodel ; Takahashi_B by
$\Delta R_{\rm S}=e^{-\frac{d}{\lambda_{\rm N}}}\frac{R_{\rm SN}\left(P_{\rm
I}\frac{R_{\rm SI_{inj}}}{R_{\rm SN}}+P_{\rm F}\frac{R_{\rm SF_{ini}}}{R_{\rm
SN}}\right)\left(P_{\rm I}\frac{R_{\rm SI_{det}}}{R_{\rm SN}}+P_{\rm
F}\frac{R_{\rm SF_{det}}}{R_{\rm SN}}\right)}{\left(1+2\frac{R_{\rm
SI_{inj}}}{R_{\rm SN}}+2\frac{R_{\rm SF_{ini}}}{R_{\rm
SN}}\right)\left(1+2\frac{R_{\rm SI_{det}}}{R_{\rm SN}}+2\frac{R_{\rm
SF_{det}}}{R_{\rm SN}}\right)-e^{-\frac{2d}{\lambda_{\rm N}}}.}$ (2)
Here $P_{\rm F}$ and $P_{\rm I}$ are the bulk and interface spin polarizations
of the ferromagnetic electrode, respectively, and $R_{\rm SF_{\rm inj}}$,
$R_{\rm SF_{\rm det}}$ and $R_{\rm SN}$ are the spin resistances for the
ferromagnetic injector, detector and the nonmagnetic strip, respectively.
Also, $R_{\rm SI_{\rm inj}}$ and $R_{\rm SI_{\rm det}}$ are the interface spin
resistances for the injecting and detecting junctions. $d$ and $\lambda_{\rm
N}$ are the separation distance between the injector and detector and the spin
diffusion length for the nonmagnetic strip. The spin resistance is defined as
$2\rho\lambda/((1-P^{2})S)$, where $\rho$, $\lambda$ and $S$ are the
resistivity, the spin diffusion length and the effective cross section for the
spin current, respectively. The interface spin resistance $R_{\rm SI}$ is
defined by $2{\rm RA}/((1-P_{\rm I}^{2})S)$, where RA is the resistance area
product.
In the nonmagnetic strip with a long spin diffusion length over a few hundred
nanometer, $S$ is given by the cross section of the strip. On the other hand,
in the ferromagnets with a short spin diffusion length less than 10 nm, $S$ is
given by the size of the junction in contact with the nonmagnetic strip
because the spin current abruptly decays in the vicinity of the F/N
interfaceKimura03 . Moreover, for the ohmic junction, the interface resistance
is typically a few hundred mili ohm, which is also much smaller than $R_{\rm
SN}$. When $R_{\rm SN}\gg R_{\rm SF},R_{\rm SI}$, the above equation can be
simplified as
$\Delta R_{\rm S}\approx\frac{(P_{\rm F}R_{\rm SF_{inj}}+P_{\rm I}R_{\rm
SI_{inj}})(P_{\rm F}R_{\rm SF_{det}}+P_{\rm I}R_{\rm SI_{det}})}{2R_{\rm
SN}\sinh\left({d}/{\lambda_{\rm N}}\right)}.$ (3)
It should be noted that although $R_{\rm SF}$ increases with $P$ and diverges
at $P=1$, the condition of $R_{\rm SF}$ $\ll$ $R_{\rm SN}$ is still valid for
$P<0.9$ for the CFS film.
By introducing the junction sizes $(S_{\rm inj}$, $S_{\rm det}$ and $S_{\rm
N})$, this equation can be revised as
$\Delta R_{\rm S}\approx\frac{S_{\rm N}}{S_{\rm inj}S_{\rm
det}}\frac{\left(\frac{P_{F}}{(1-P_{F}^{2})}\rho_{\rm F}\lambda_{\rm
F}+\frac{P_{I}}{(1-P_{I}^{2})}{\rm RA}_{\rm F/N}\right)^{2}}{\rho_{\rm
N}\lambda_{\rm N}\sinh\left({d}/{\lambda_{\rm N}}\right)}.$ (4)
By defining $\Delta R_{\rm S}A$ as $\Delta R_{\rm S}(S_{\rm inj}S_{\rm
det}/{S_{\rm N}})$, we obtain
$\Delta R_{\rm S}A\approx\frac{\left(\frac{P_{F}}{(1-P_{F}^{2})}\rho_{\rm
F}\lambda_{\rm F}+\frac{P_{I}}{(1-P_{I}^{2})}{\rm RA}_{\rm
F/N}\right)^{2}}{\rho_{\rm N}\lambda_{\rm N}\sinh\left({d}/{\lambda_{\rm
N}}\right)}.$ (5)
In Eq. (4), the influence of the junction sizes on the spin signal can be
normalized. Thus, $\Delta R_{\rm S}A$ allows us to fairly evaluate the
generation efficiency of the pure spin current for various combinations of a
ferromagnetic metal and a nonmagnetic one.
### Estimation of interface resistance
The interface resistance was estimated by measuring the 4-terminal resistances
with local spin valve configuration. In this configuration, the total
resistance consists of the resistance of the Cu wire and the two interface
resistances. Since the resistance of the Cu wire can be estimated from the
resistivity for Cu, we can roughly calculate the interface resistance by
subtracting the resistance for the Cu wire from that in the local spin valve
configuration. By using the relation that the difference in the resistance
$\Delta R$ is given by ${\rm RA}_{\rm F/N}(S_{\rm inj}+S_{\rm det})$, we can
obtain ${\rm RA}_{\rm F/N}$. For the CFS/Cu LSVs, $\Delta R$ was $\sim$ 30
m$\Omega$, indicating ${\rm RA_{CFS/Cu}}$ $\sim$ 1 f$\Omega$m2.
### Estimation of spin polarization
By fitting the experimental data on the distance dependences of the $\Delta
R_{\rm S}A$ using Eq. (4), we can estimate the spin polarization of the spin
injector in the LSV systems. For the Py/Cu LSVs, $R_{\rm Py/Cu}$ is less than
0.1 ${\rm f}\Omega{\rm m}^{2}$, much smaller than $\rho_{\rm Py}\lambda_{\rm
Py}$ (0.75 ${\rm f}\Omega{\rm m}^{2}$). Thus, we can neglect the second term
in the numerator of Eq. (3), then obtain $P_{\rm Py}\sim$ 0.3 at RT and 0.35
at 80 K, respectively with assuming $\lambda_{\rm Py,RT}=$ 3 nm and
$\lambda_{\rm Py,80K}=5$ nmKimura01 .
For the CFS/Cu LSVs, as described in the previous section, $R_{\rm CFS/Cu}$
can be approximately estimated as $\sim$1 ${\rm f}\Omega{\rm m}^{2}$. Because
of the following reasons, we assumed that the spin diffusion length for CFS
($\lambda_{\rm CFS}$) is the same order of that for CFSA ($\lambda_{\rm
CFSA}$), which was reported as 2.2 nm at RT and 3 nm at 14 K in recent study
of the vertical magnetoresitance deviceNakatani2_B ; Taniguchi . The spin-
diffusion length is proportional to the magnitude of the spin-orbit
interactions. Since the atomic number of Al is close to Si, we expect that the
magnitude of spin-orbit interaction in CFS should be almost same order of
CFSA. From these considerations, we expect that $\lambda_{\rm CFS}$ is the
same level or shorter than $\lambda_{\rm CFSA}$. Since the use of the longer
$\lambda_{\rm CFS}$ prevents an overestimation of $P_{\rm CFS}$, we use
$\lambda_{\rm CFS,RT}=2\sim 4$ nm and $\lambda_{\rm CFS,80K}=3$ $\sim$ 6 nm,
where the minimum value corresponds to $\lambda_{\rm CFSA}$. We then obtained
$P_{\rm CFS}=0.50\sim 0.71$ at RT and $P_{\rm CFS}=0.66\sim 0.81$ at 80 K with
assuming $P_{I}=0.5\sim 0.7$, which is typical interface spin polarization
between the Co-based alloy and Cu CPP1 ; CPP2 .
### Comparison of the device performance between LSVs with metallic and
resistive interface resistances
In the present paper, we discuss the LSVs only with metallic junctions, since
Eq. (1) is valid only for the condition of $\rho_{\rm F}\lambda_{\rm F}$,
${\rm RA_{F/N}}\ll\rho_{\rm N}\lambda_{\rm N}$. When the above condition is
not satisfied, for example, the LSVs with the resistive interface (${\rm
RA_{F/N}}\gg\rho_{\rm N}\lambda_{\rm N}$) Wang ; Fukuma ; Hoffmann , the
device performance cannot be evaluated by Eq (1). In order to fairly compare
the device performance of different type LSVs, one should focus on the
injection efficiency of the pure spin current. For the metallic junctions, the
injection efficiency $\eta_{I_{S}}$ of the pure spin current, which is defined
by the ratio of the spin current $I_{S}$ injected into the ferromagnetic
contact to the excited charge current $I_{C}$, can be calculated as
$\eta_{I_{S}}\equiv\frac{I_{S}}{I_{C}}\approx\frac{1}{2}\frac{S_{\rm
N}}{S_{\rm Inj}}\frac{\left(\frac{P_{F}}{(1-P_{F}^{2})}\rho_{\rm
F}\lambda_{\rm F}+\frac{P_{I}}{(1-P_{I}^{2})}{\rm RA}_{\rm
F/N}\right)}{\rho_{\rm N}\lambda_{\rm N}\sinh\left({d}/{\lambda_{\rm
N}}\right)}.$ (6)
For example, the injection efficiency of the present CFS/Cu LSV is estimated
to be $\sim 0.5$ with $d=100$ nm.
On the other hand, the efficiency for the LSV with the resistive interface,
where ${\rm RA_{F/N}}\gg\rho_{\rm N}\lambda_{\rm N}$, is given by
$\eta_{I_{S}}\approx\frac{1}{4}P_{\rm I}\frac{S_{\rm Det}}{S_{\rm
N}}\frac{\rho_{\rm N}\lambda_{\rm N}}{{\rm RA_{\rm
F/N}}}\exp^{-\frac{d}{\lambda_{\rm N}}}.$ (7)
In this case, the efficiency decreases with increasing the interface
resistance area product $\rm RA_{\rm F/N}$. For example, for the Py/Ag LSVs
with moderate interface resistances, where the large spin signals comparable
to the present CFS/Cu LSV have been reported,Fukuma ; Hoffmann the interface
resistance area product $\rm RA_{\rm F/N}$ is 100 f$\Omega$m2. Thus, the
injection efficiency is estimated to be $\sim$ 0.005 with $d=$ 100 nm.
## References
* (1) Valet, T. & Fert, A. Theory of the perpendicular magnetoresistance in magnetic multilayers. Phys. Rev. B 48, 7099-7113 (1993).
* (2) Takahashi, S. & Maekawa, S. Spin injection and detection in magnetic nanostructures. Phys. Rev. B 67, 052409 (2003).
* (3) Kimura, T. & Otani, Y. Spin transport in lateral ferromagnetic/nonmagnetic hybrid structure. J. Phys. Cond. Mat. 19, 165216 (2007).
* (4) Vouille, C., Barth$\rm{\acute{e}}$l$\rm{\acute{e}}$my, A., Elokan, M. F., Fert A., Schroeder, P. A., Hsu, S. Y., Reilly, A. & Loloee, R. Microscopic mechanisms of giant magnetoresistance. Phys. Rev. B 60, 6710-6722 (1999).
* (5) Baxter, J. R., Pettifor, G. D. & Tsymbal., Y. E. Interface proximity effects in current-perpendicular-to-plane magnetoresistance. Phys. Rev. B 71, 024415 (2005).
* (6) Nakatani, M. T., Furubayashi, T., Kasai, S., Sukegawa, H., Takahashi, Y. K., Mitani. S. & Hono K. Bulk and interfacial scatterings in current-perpendicular-to-plane giant magnetoresistance with Co2Fe(Al0.5Si0.5) Heusler alloy layers and Ag spacer. Appl. Phys. Lett. 96, 212501 (2010).
* (7) Taniguchi T., Imamura H., Nakatani T., & Hono K., Effect of the number of layers on determination of spin asymmetries in current-perpendicular-to-plane giant magnetoresistance, Appl. Phys. Lett. 98, 042503 (2011).
* (8) Wang X. J., Zou H., Ocola L. E., and Ji Y. High spin injection polarization at an elevated dc bias in tunnel-junction-based lateral spin valves. Appl. Phys. Lett. 95, 022519 (2009).
* (9) Fukuma Y., Wang L., Idzuchi H. & Otani Y. Enhanced spin accumulation obtained by inserting low-resistance MgO interface in metallic lateral spin valves. Appl. Phys. Lett. 97, 012507 (2010), Y. Fukuma, L. Wang, H. Idzuchi, S. Takahashi, S. Maekawa, and Y. Otani. Giant enhancement of spin accumulation and long-distance spin precession in metallic lateral spin valves. Nature Mater. 10, 527 (2011)
* (10) Mihajlovic G., Schreiber D. K. , Liu Y., Pearson J. E., Bader S. D. , Petford-Long A. K., & Hoffmann A. Enhanced spin signals due to native oxide formation in Ni80Fe20/Ag lateral spin valves. Appl. Phys. Lett. 97, 112502 (2010).
|
arxiv-papers
| 2011-07-31T08:09:53 |
2024-09-04T02:49:21.119308
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. Kimura, N. Hashimoto, S. Yamada, M. Miyao, and K. Hamaya",
"submitter": "Takashi Kimura",
"url": "https://arxiv.org/abs/1108.0156"
}
|
1108.0174
|
# Lectures and notes: Mirzakhani’s volume recursion and approach for the
Witten-Kontsevich theorem on moduli tautological intersection numbers
Scott A. Wolpert111Partially supported by National Science Foundation grant
DMS - 1005852.
Figure 1: The boundary pants configurations for the length identity.
The following materials were presented in a short course at the 2011 Park City
Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann
Surfaces. Brad Safnuk assisted in the preparation for and running of the
course. It is my pleasure to thank Brad for his assistance.
## 1 Introduction.
The papers.
Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered
Riemann surfaces. Invent. Math., 167(1):179-222, 2007.
Weil-Petersson volumes and intersection theory on the moduli space of curves.
J. Amer. Math. Soc., 20(1):1-23, 2007.
The goals of the papers.
1\. Derive an explicit recursion for WP moduli space volume polynomials.
2\. Apply symplectic reduction to show that the polynomial coefficients are
intersection numbers.
3\. Show that the volume recursion satisfies the Virasoro relations - Witten’s
conjecture.
The trio of Maryam Mirzakhani papers [Mir07a, Mir07b, Mir08] are distinguished
for involving a large number of highly developed considerations. The first
work requires a detailed description of Teichmüller space, the action of the
mapping class group, formulas for Weil-Petersson (WP) symplectic geometry,
classification of simple geodesic arcs on a pair of pants, the length infinite
sum identity and exact calculations of integrals. The second work involves a
description of the moduli space tautological classes $\kappa_{1}$ and $\psi$,
as characteristic classes for $S^{1}$ principal bundles in hyperbolic
geometry, the moment map and exact symplectic reduction, as well as
combinatorial calculations. The third work uses the $PL$ structure of
Thurston’s space of measured geodesic laminations $\mathcal{MGL}$, the train-
track symplectic form and Masur’s result that the mapping class group acts
ergodically on $\mathcal{MGL}$. A fine feature of the works is that they
showcase important aspects of the geometry, topology and deformation theory of
Riemann/hyperbolic surfaces. Mirzakhani’s recursion for volume is applied in
all three works and in a current preprint. A discussion placing Mirzakhani’s
work into the context of approaches to Witten-Kontsevich theory is given in
the first section of the Mulase-Safnuk paper [MS08]. A brief exposition of
Kontsevich’s original solution [Kon92] of Witten’s conjecture, including the
basic geometry of tautological classes on moduli space and the ribbon graph
expansion of matrix integrals is given in the Bourbaki Seminar of Looijenga
[Loo93]. A brief exposition of Mirzakhani’s volume recursion, solution of
Witten-Kontsevich and applications of WP volume limits are given in [Do11]. An
overall exposition of Mirzakhani’s prime simple geodesic theorem [Mir08] is
given in [Wlp10, Chaps. 9, 10]. Mirzakhani’s work is just one part of a
subject with high activity and many active researchers; Google Scholar shows
200 citations to the three Mirzakhani works, and much more generally over 1100
citations to Witten’s original papers [Wit91, Wit92] on two-dimensional
gravity and gauge theories.
First paper brief. A symplectic fibered product decomposition for covers of
the moduli space of bordered Riemann/hyperbolic surfaces is combined with the
$d\ell\wedge d\tau$ formula for the symplectic form and a universal identity
for sums of geodesic-lengths to derive an explicit recursion for computing
volume. The volume of the moduli space of genus $g$, $n$ boundaries surfaces
is shown to be a polynomial with positive coefficients in surface boundary
lengths of total degree $6g-6+2n$.
Second paper brief. Twist Hamiltonian flows are combined with the boundary
length moment map to apply symplectic reduction for the family of moduli
spaces of bordered hyperbolic surfaces. The consequence is that the
coefficients of the volume polynomials are moduli space characteristic
numbers. A geometric construction shows that the characteristic numbers are
tautological intersection numbers. The volume recursion is shown to satisfy
Virasoro algebra constraints.
The following theorems are the immediate take away results of the papers.
###### Theorem.
The WP volume polynomials. The volume polynomials are determined recursively
from the volume polynomials of smaller total degree, [Mir07a, Formula (5.1) &
Theorem 8.1]. The volume $V_{g,n}(L_{1},\dots,L_{n})$ of the moduli space of
genus $g$, $n$ boundaries, hyperbolic surfaces with boundary lengths
$L=(L_{1},\dots,L_{n})$ is a polynomial
$V_{g,n}(L)\,=\,\sum_{\stackrel{{\scriptstyle\alpha}}{{|\alpha|\leq
3g-3+n}}}C_{\alpha}\,L^{2\alpha},$
for multi index $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and where
$C_{\alpha}>0$ lies in $\pi^{6g-6+2n-2|\alpha|}\mathbb{Q}$, [Mir07a, Theorems
1.1 & 6.1]. The coefficients are intersection numbers given as
$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$
where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form,
$\alpha!=\prod_{j=1}^{n}\alpha_{j}!$, and $\delta_{**}$ is the Kronecker
indicator delta, [Mir07b, Theorem 4.4].
###### Theorem.
[Mir07a, Theorems 6.3 & 6.4]. Recursive relations for the volume polynomial
leading coefficients. For a multi index $\alpha$, define
$(\alpha_{1},\dots,\alpha_{n})_{g}=C_{\alpha}\times
2^{-\delta_{1g}\delta_{1n}}\times\prod_{i=1}^{n}\alpha_{i}!\times
2^{|\alpha|},$
then for $n>0$ and $\sum_{i}\alpha_{i}=3g-3+n$,
$\mbox{the dilaton
equation}\qquad\quad(1,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,(2g-2+n)(\alpha_{1},\dots,\alpha_{n})_{g}$
and for $n>0$ and $\sum_{i}\alpha_{i}=3g-2+n$,
$\mbox{the string
equation}\qquad(0,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,\sum_{\alpha_{i}\neq
0}(\alpha_{1},\dots,\alpha_{i}-1,\dots,\alpha_{n})_{g}.$
For the intersection number generating function
$\mathbf{F}(\lambda,t_{0},t_{1},\dots)\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}\sum_{\\{d_{j}\\}}\,\langle\prod_{j=1}^{\infty}\tau_{d_{j}}\rangle_{g}\,\prod_{r\geq
0}t_{r}^{n_{r}}/n_{r}!\,,$
with $n_{r}=\\#\\{j\mid d_{j}=r\\}$, and
$\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}},$
for $\sum_{j=1}^{n}d_{j}=3g-3+n$ and the product $\langle\tau_{*}\rangle$
otherwise zero, then the exponential $e^{\mathbf{F}}$ satisfies Virasoro
algebra constraints, [Mir07b, Theorem 6.1].
## 2 The organizational outline and reading guide.
The following outline combines [Mir07a, Mir07b] with the exposition of [Wlp10,
Chapter 9]. The lectures are presented in the next section.
* •
Teichmüller spaces, moduli spaces, mapping class groups and the symplectic
geometry.
* –
The Teichmüller space $\mathcal{T}_{g,n}$ and moduli space
$\mathcal{M}_{g,n}$; the Teichmüller space
$\mathcal{T}_{g}(L_{1},\dots,L_{n})$ and moduli space
$\mathcal{M}_{g}(L_{1},\dots,L_{n})$ of prescribed length geodesic bordered
hyperbolic surfaces; the augmented Teichmüller space and Deligne-Mumford type
compactifications; Dehn twists and the mapping class group
$\operatorname{MCG}$ action.
* –
The WP symplectic geometry, [Wlp10].
* *
The symplectic form $\omega=2\omega_{\tiny{\mbox{WP\ K\"{a}hler}}}$ and
normalizations.
* *
Geodesic-length functions $\ell_{\alpha}$, Fenchel-Nielsen infinitesimal twist
deformations $t_{\alpha}$ and the duality formula $\omega(\
,t_{\alpha})=d\ell_{\alpha}$.
* *
Fenchel-Nielsen (FN) twist-length coordinates $(\tau_{j},\ell_{j})$ for
Teichmüller space and the formula $\omega=\sum_{j}d\ell_{j}\wedge d\tau_{j}$.
* –
The intermediate moduli space $\mathcal{M}^{\gamma}_{g,n}$ of pairs
$(R,\gamma)$ \- a surface and a weighted multicurve
$\gamma=\sum_{j}c_{j}\gamma_{j}$.
* *
The covering tower
$\mathcal{T}_{g,n}\longrightarrow\mathcal{M}^{\gamma}_{g,n}\longrightarrow\mathcal{M}_{g,n}.$
* *
The stabilizer subgroup $\operatorname{Stab}(\gamma)\subset\operatorname{MCG}$
for a weighted multicurve. The $\operatorname{MCG}$ deck cosets for the
covering tower.
* *
Symplectic structures for $\mathcal{M}^{\gamma}_{g,n}$ and
$\mathcal{M}_{g}(L)$.
* –
The $\mathcal{T}_{g,n}$ and $\mathcal{M}^{\gamma}_{g,n}$ level sets of the
total length $\ell=\sum_{j}c_{j}\ell_{\gamma_{j}}$.
###### Lemma.
[Mir07a, Lemma 7.2]. Preparation for volume recursion and symplectic
reduction. A finite cover of $\mathcal{M}^{\gamma}_{g,n}$ is a fibered product
of symplectic planes and lower dimensional moduli spaces.
* •
The McShane-Mirzakhani length identity.
* –
The set $\mathcal{B}$ of homotopy classes rel boundary of simple arcs with
endpoints on the boundary. Classification of geodesic arcs normal to a
boundary: simple geodesics normal to a boundary at each endpoint
$\Longleftrightarrow$ disjoint pairs of boundary intervals
$\Longleftrightarrow$ wire frames for pants.
* –
Birman-Series: simple geodesics have measure zero.
* –
The rational exponential function $H$; the hyperbolic trigonometric functions
$\mathcal{D}$ and $\mathcal{R}$; relations.
* –
The length identity.
###### Theorem.
[Mir07a, Theorem 1.3 & 4.2] and [TWZ06, Thrm. 1.8]. For a hyperbolic surface
$R$ with boundaries $\beta_{j}$ with lengths $L_{j}$,
$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$
where the first sum is over all unordered pairs of simple closed geodesics
with $\beta_{1},\alpha_{1},\alpha_{2}$ bounding an embedded pair of pants, and
the double sum is over simple closed geodesics with
$\beta_{1},\beta_{j},\alpha$ bounding an embedded pair of pants.
* –
[Mir07a, Section 8] \- Recognizing and understanding the identity as a smooth
analog to a $\operatorname{MCG}$ fundamental domain. Reducing to the action of
smaller $\operatorname{MCG}$ groups.
* •
The Mirzakhani volume recursion.
* –
A covolume formula - writing a moduli integral as a length level set integral
- the role of the intermediate moduli space $\mathcal{M}_{g,n}^{\gamma}$ \-
[Mir07a, Theorem 7.1] and [Wlp10, Theorem 9.5].
* –
The application for the McShane-Mirzakhani identity.
* *
The integrals $\mathcal{A}_{g,n}^{connected}$,
$\mathcal{A}_{g,n}^{disconnected}$ and $\mathcal{B}_{g,n}$ of lower-
dimensional moduli volumes.
* *
The corresponding connected and disconnected boundary pants configurations.
* *
Combining the length identity and covolume formula for the main result; see
[Wlp10, pg. 91 bottom, pg. 92].
###### Theorem.
[Mir07a, Section 5 and Theorem 8.1]. For $(g,n)\neq(1,1),(0,3),$ the volume
$V_{g,n}(L)$ satisfies
$\frac{\partial\ }{\partial
L_{1}}L_{1}V_{g,n}(L)\,=\,\mathcal{A}_{g,n}^{connected}(L)\,+\mathcal{A}_{g,n}^{disconnected}(L)\,+\mathcal{B}_{g,n}(L).$
* *
The integrals $\mathcal{A}^{*}_{*}(L)$ and $\mathcal{B}_{*}(L)$ are
polynomials in boundary lengths with coefficients sums of special values of
Riemann zeta; the coefficients are positive rational multiples of powers of
$\pi$.
* •
Symplectic reduction and the Duistermaat-Heckman theorem.
* –
The WP kappa equation $\omega=2\pi^{2}\kappa_{1}$ on Deligne-Mumford [Wlp90].
* –
The geometry and topology of the Teichmüller space
$\widehat{\mathcal{T}}_{g}(L)=\widehat{\mathcal{T}}_{g}(L_{1},\dots,L_{n})$ of
hyperbolic surfaces with geodesic boundaries with points.
* –
A symplectic structure for $\widehat{\mathcal{T}}_{g}(L)$ by summing on almost
tight pants. The boundary length moment map
$R\in\widehat{\mathcal{T}}_{g}(L)\mapsto L^{2}/2\in\mathbb{R}_{+}^{n}$.
Twisting boundary points as Hamiltonian flows.
* –
Symplectic reduction, [FO06]. $\mathcal{T}_{g}(L)$ as the reduced space
$\widehat{\mathcal{T}}_{g}(L)/(S^{1})^{n}$. $\widehat{\mathcal{T}}_{g}(L)$ as
a principal $(S^{1})^{n}$ bundle over $\mathcal{T}_{g,n}(L)$ and the small $L$
equivalence $\mathcal{T}_{g}(L)\approx\mathcal{T}_{g,n}(0)$,
$\textstyle{(S^{1})^{n}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{T}}_{g}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{g}(L)\approx\mathcal{T}_{g,n}(0)\,.}$
* –
Points on circles, cotangent $\mathbb{C}$-lines at punctures, circle bundles
and homotopic structure groups. The tautological cotangent line class $\psi$.
* –
$\operatorname{MCG}$ equivariant maps and quotients. Deligne-Mumford type
compactifications and finite covers. The elliptic stack $2$.
* –
The Duistermaat-Heckman normal form, [Mir07b, Theorem 3.2], [MS08, Section
2.5] and [Wlp10, pg. 95],
$2\omega_{\overline{\widehat{\mathcal{M}}}_{g}(L)/(S^{1})^{n}}\,\equiv\,2\omega_{\overline{\mathcal{M}}_{g,n}(0)}\
+\ \sum_{j}\frac{L_{j}^{2}}{2}\psi_{j}\ .$
* –
The consequence of symplectic reduction, [Mir07b, Theorem 4.4].
###### Theorem.
The volume polynomial $V_{g,n}(L)$ coefficients are
$\overline{\mathcal{M}}_{g,n}$ intersection numbers given as
$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$
where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form and $\delta_{**}$ is the Kronecker
indicator.
* •
The pattern of intersection numbers.
* –
The general intersection number symbol
$\langle\kappa_{1}^{d_{0}}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}}\kappa_{1}^{d_{0}}$
and volume polynomial expansions.
* –
Examples: $V_{1,1}(L)=\frac{\pi^{2}}{6}\,+\,\frac{L^{2}}{24}$,
$V_{0,4}(L_{1},L_{2},L_{3},L_{4})=(4\pi^{2}+L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2})/2$
and
$V_{2,1}(L)=\frac{1}{2211840}(L^{2}+4\pi^{2})(L^{2}+12\pi^{2})(5L^{4}+384\pi^{2}L^{2}+6960\pi^{4})$.
* –
The partition function $\mathbf{F}=\sum_{g}\langle
e^{\sum_{j}t_{j}\tau_{j}}\rangle_{g}$ and Virasoro constraint differential
operators $\mathbf{L}_{k}$ [Mir07b]. The partition function
$\mathbf{G}=\sum_{g}\langle e^{s\kappa_{1}+\sum_{j}t_{j}\tau_{j}}\rangle_{g}$
and Virasoro constraint differential operators $\mathbf{V}_{k}$ [MS08, Theorem
1.1].
* –
The Do remove a boundary relation
$\frac{\partial V_{g,n+1}}{\partial L_{n+1}}(L,2\pi i)\,=\,2\pi
i(2g-2+n)V_{g,n}(L),\qquad\cite[cite]{[\@@bibref{}{Dothe,DoNo}{}{}]}.$
* –
The Manin-Zograf volumes generating function [MZ00]. The punctures asymptotic
- for positive constants $c,\,a_{g}$, genus fixed and large $n$, then
$V_{g,n}\,=\,c^{n}n!\,n^{(5g-7)/2}(a_{g}\,+\,O(1/n)),\qquad\cite[cite]{[\@@bibref{}{MZ}{}{}]}.$
* –
The Schumacher-Trapani genera asymptotic - for $n$ fixed, there are positive
constants, then
$c_{1}^{\,g}(2g)!\,<\,V_{g,n}\,<\,c_{2}^{\,g}(2g)!,\qquad\cite[cite]{[\@@bibref{}{Gru,
SchTr}{}{}]}.$
* –
The Zograf conjecture
$V_{g,n}\,=\,(4\pi^{2})^{2g+n-3}(2g+n-3)!\frac{1}{\sqrt{g\pi}}\big{(}1\,+\,\frac{c_{n}}{g}\,+\,O(1/g^{2})\big{)},$
for fixed $n$ and $g$ tending to infinity, [Zog08]. Expected values of
geometric invariants [Mir10].
## 3 The lectures.
The following is an exposition of Mirzakhani’s proof of the Witten-Kontsevich
theorem, including the immediate background material on Teichmüller theory,
moduli space theory and on symplectic reduction. The lectures are presented as
from a graduate text - the exposition follows the development of concepts, and
does not consider the historical development of the material. The goals are
general treatment of the material and overall understanding for the reader. In
places, the approaches of several authors are combined for a simpler treatment
of the material. Only immediate references to the literature are included. The
reader should consult the literature for the historical development, for
complete references and for consequences of the material.
1. 1.
The background and overview.
2. 2.
The McShane-Mirzakhani identity.
3. 3.
The covolume formula and recursion.
4. 4.
Symplectic reduction, principal $S^{1}$ bundles and the normal form.
5. 5.
The pattern of intersection numbers and Witten-Kontsevich.
Lecture 1: The background and overview.
General background.
By Uniformization, for a surface of negative Euler characteristic, a conformal
structure is equivalent to a complete hyperbolic structure. We consider
Riemann surfaces $R$ of finite topological type with hyperbolic metrics,
possibly with punctures and geodesic boundaries, if boundaries are non empty.
Fix a topological reference surface $F$, and consider a marking, an
orientation preserving homeomorphism $f:F\rightarrow R$ up to homotopy. If
boundaries are non empty, homotopy is rel boundary setwise. Write
$\mathcal{T}$ for the Teichmüller space of $R$ \- the space of equivalence
classes of pairs $\\{(f,R)\\}$, where pairs are equivalent if there is a
homotopy mapping triangle with a conformal map (a hyperbolic isometry) between
Riemann surfaces.
Figure 2: A genus $3$ surface with $2$ geodesic boundaries.
We consider the following Teichmüller spaces $\mathcal{T}$: $\mathcal{T}_{g}$
\- for compact genus $g$ surfaces; $\mathcal{T}_{g}(L)$ \- for genus $g$
surfaces with labeled geodesic boundaries of prescribed lengths
$L=(L_{1},\dots,L_{n})$; $\mathcal{T}_{g,n}$ \- for genus $g$ surfaces with
$n$ labeled punctures. In the case of $\mathcal{T}_{g}(L)$, homotopies of
surfaces are rel boundaries setwise. $\mathcal{T}_{g}$ and $\mathcal{T}_{g,n}$
are complex manifolds, while $\mathcal{T}_{g}(L)$ is a real analytic manifold.
A non trivial, non puncture peripheral, free homotopy class $\alpha$ on $F$
has a unique geodesic representative for $f(\alpha)$ on the surface $R$ \- the
geodesic length $\ell_{\alpha}(R)$ provides a natural function on Teichmüller
space. Collections of geodesic-length functions provide local coordinates and
global immersions to Euclidean space for $\mathcal{T}$. The differential of
geodesic-length for a simple curve is nowhere vanishing. At each point of
$\mathcal{T}$, the differentials of geodesic-lengths of simple curves are
dense in the cotangent bundle.
A surface can be cut open on a simple closed geodesic - the boundaries are
isometric circles. Since a neighborhood of a simple geodesic has an $S^{1}$
symmetry, the boundaries can be reassembled with a relative rotation to form a
new hyperbolic structure. The deformation is the Fenchel-Nielsen (FN) twist.
Figure 3: A positive Fenchel-Nielsen twist deformation.
The infinitesimal deformation for unit speed hyperbolic displacement of
initial adjacent points, is the Fenchel-Nielsen infinitesimal twist vector
field $t_{\alpha}$ on $\mathcal{T}$. (A positive twist corresponds to
displacing to the right when crossing the geodesic.)
Geodesic boundaries of hyperbolic surfaces of common length can be assembled
to form new surfaces. Given boundary reference points, the relative rotation
is measured in terms of arc length. A hyperbolic pair of pants is a genus zero
surface with three geodesic boundaries. For pants, boundary reference points
are provided by considering the unique orthogonal connecting geodesics between
boundaries. At a gluing, the common boundary length $\ell$ and relative
rotation, the FN twist parameter $\tau$, are unrestricted parameters ($\tau$
is defined by continuation from an initial configuration). The length $\ell$
varies in $\mathbb{R}_{>0}$ and twist $\tau$ varies in $\mathbb{R}$. Each
finite topological type hyperbolic surface can be assembled from pairs of
pants.
###### Theorem 1.
Fenchel-Nielsen coordinates. Fixing the topological type of a pants
decomposition and an initial configuration, the FN parameters
$\prod_{j=1}^{3g-3+n}(\ell_{j},\tau_{j})$ define a real analytic equivalence
of $\mathcal{T}$ to $\prod_{j=1}^{3g-3+n}\mathbb{R}_{>0}\times\mathbb{R}$.
The Weil-Petersson (WP) metric is Kähler. The symplectic geometry begins with
the symplectic form $\omega=2\omega_{\tiny{\mbox{WP\ K\"{a}hler}}}$ and the
basic twist-length duality
$\omega(\ ,t_{\alpha})=d\ell_{\alpha}.$
It follows from the Lie derivative equation $L_{X}\omega(\ ,\ )=d\omega(X,\ ,\
)+d(\omega(X,\ ))$ that the form $\omega$ is invariant under all twist flows.
It follows that geodesic-length functions are Hamiltonian potentials for $FN$
infinitesimal twists. Symmetry reasoning shows that $\ell$ and $\tau$ provide
action-angle coordinates for $\omega$.
###### Theorem 2.
(W), [Wlp10]. The $d\ell\wedge d\tau$ formula. The WP symplectic form is
$\omega\,=\,\sum_{j=1}^{3g-3+n}\,d\ell_{j}\wedge d\tau_{j}.$
Frontier spaces are adjoined to $\mathcal{T}$ corresponding to allowing
$\ell_{j}=0$ with the FN angle $\theta_{j}=2\pi\tau_{j}/\ell_{j}$ then
undefined (in polar coordinates, angle is undefined at the origin). The
vanishing length describes pairs of pants with corresponding boundaries
represented by punctures - the equation $\ell_{\alpha}=0$ describes hyperbolic
structures with $\alpha$ represented by pairs of punctures. For a subset of
indices $J\subset\\{1,\dots,3g-3+n\\}$, the $J$-null stratum is
$\mathcal{S}(J)\,=\,\\{R\mbox{ degenerate}\mid\ell_{j}(R)=0\mbox{ iff }j\in
J\\}$. Each null stratum is a product of lower dimensional Teichmüller spaces.
A basis of neighborhoods in $\mathcal{T}\cup\mathcal{S}(J)$ is defined in
terms of the parameters $(\ell_{k},\theta_{k},\ell_{j})_{k\notin J,\,j\in J}$.
The augmented Teichmüller space is the stratified space
$\overline{\mathcal{T}}=\mathcal{T}\cup_{\tiny{pants\ decompositions\
}\mathcal{P}}\cup_{J\subset\mathcal{P}}\,\mathcal{S}(J).$
The space $\overline{\mathcal{T}}$ is also described as the Chabauty topology
closure of the discrete faithful type-preserving representations of
$\pi_{1}(F)$ into $PSL(2;\mathbb{R})$, modulo $PSL(2;\mathbb{R})$ conjugation.
The augmentation construction is valid for
$\mathcal{T}_{g},\mathcal{T}_{g}(L)$ and $\mathcal{T}_{g,n}$.
$\overline{\mathcal{T}}$ is a Baily-Borel type partial compactification.
$\overline{\mathcal{T}}$ is never locally compact. The $d\ell\wedge d\tau$
formula provides for the extension of the symplectic structure to the
augmented Teichmüller space $\overline{\mathcal{T}}$. Each strata is
symplectic.
The mapping class group ($\operatorname{MCG}$) $Homeo^{+}(F)/Homeo_{0}(F)$
acts on markings by precomposition and thus acts on $\mathcal{T}$. For
$Homeo^{+}(F)$ we consider type-preserving (boundary point, boundary curve),
boundary label preserving, orientation preserving homeomorphisms.
$Homeo_{0}(F)$ is the normal subgroup of elements homotopic to the identity
rel boundary setwise. A Dehn twist is a homeomorphism that is the identity on
the complement of a tubular neighborhood of a simple closed curve, non trivial
in homotopy, and rotates one boundary of the tubular neighborhood relative to
the other. Dehn twist classes generate $\operatorname{MCG}$.
$\operatorname{MCG}$ acts properly discontinuously on $\mathcal{T}$ and by
biholomorphisms for $\mathcal{T}_{g}$ and $\mathcal{T}_{g,n}$. Except for a
finite number of topological types the action is effective. Finite
$\operatorname{MCG}$ subgroups act with fixed points. $\operatorname{MCG}$
acts on the stratified space $\overline{\mathcal{T}}$. Bers observed that
there are constants $b_{g,n}$, depending on topological type, such that a
genus $g$, $n$ punctured hyperbolic surface has a pants decomposition with
seam lengths at most $b_{g,n}$. It follows that the domain $\\{\ell_{j}\leq
b_{g,n},\,0<\tau_{j}\leq\ell_{j}\\}$ in FN coordinates is a rough fundamental
set - each $\operatorname{MCG}$ orbit intersects the domain a bounded positive
number of times. $\overline{\mathcal{T}}/\operatorname{MCG}$ is a compact real
analytic orbifold; $\overline{\mathcal{T}}_{g}/\operatorname{MCG}$ and
$\overline{\mathcal{T}}_{g,n}/\operatorname{MCG}$ are topologically the
Deligne-Mumford stable curve compactifications of $\mathcal{M}_{g}$ and
$\mathcal{M}_{g,n}$. The Bers fundamental set observation combines with the
$d\ell\wedge d\tau$ formula to provide that the WP volume of $\mathcal{M}$ is
finite.
The Bers fiber space $\mathcal{C}$ (specifically $\mathcal{C}_{g}$ and
$\mathcal{C}_{g,n}$) is the complex disc holomorphic bundle over $\mathcal{T}$
with fiber over $\\{(f,R)\\}$ the universal cover $\widetilde{R}$. A point on
a fiber can be considered as a puncture and determines a curve from basepoint
for the fundamental group - so $\mathcal{C}_{g}\approx\mathcal{T}_{g,1}$ and
$\mathcal{C}_{g,n}\approx\mathcal{T}_{g,n+1}$. An extension
$\operatorname{MCG}_{\mathcal{C}}$ of $\operatorname{MCG}(F)$ by the
fundamental group $\pi_{1}(F)$ acts properly discontinuously and
holomorphically on $\mathcal{C}$. The group $\operatorname{MCG}_{\mathcal{C}}$
is isomorphic to $\operatorname{MCG}_{g,n+1}$. For the epimorphism from
$\operatorname{MCG}_{\mathcal{C}}$ to $\operatorname{MCG}(F)$, the first group
acts equivariantly on the fibration of $\mathcal{C}$ over $\mathcal{T}$. The
resulting map
$\pi:\mathcal{C}/\operatorname{MCG}\rightarrow\mathcal{T}/\operatorname{MCG}$
describes an orbifold bundle, the universal curve, with orbifold fibers - the
fibers are Riemann surfaces modulo their full automorphism group. Manifold
finite local covers and the quotient can be described by starting with a
surface with locally maximal symmetries and introducing a local trivialization
of the bundle by canonical (extremal, harmonic) maps of surfaces.
The augmentation construction applies to the Bers fiber space to give
$\overline{\mathcal{C}}$. $\operatorname{MCG}_{\mathcal{C}}$ acts on the
stratified space $\overline{\mathcal{C}}$.
Figure 4: A puncture section of the universal curve
$\overline{\mathcal{C}}/\operatorname{MCG}_{\mathcal{C}}$ over
$\overline{\mathcal{T}}/\operatorname{MCG}(F)$.
The augmentation quotient $\overline{\mathcal{C}}/\operatorname{MCG}$ is an
orbifold and almost an orbifold bundle over $\overline{\mathcal{M}}$ \- at a
node (a pair of punctures) of a Riemann surface, the fiber becomes vertical -
the local model of the fibration is the germ at the origin of the projection
$\\{(z,w)\\}\rightarrow\\{t=zw\\}$, the family of complex hyperbolas. The
turning of the fibers of the almost orbifold bundle
$\overline{\mathcal{C}}/\operatorname{MCG}\rightarrow\overline{\mathcal{M}}$
is measured by line bundles on $\overline{\mathcal{M}}$. The family of tangent
$\mathbb{C}$-lines to the fibers $(Ker\,d\pi)$ is the tangent bundle along a
smooth Riemann surface and the relative dualizing sheaf along a noded Riemann
surface. The hyperbolic metrics of the individual fibers provide a line bundle
metric for $(Ker\,d\pi)$ on $\overline{\mathcal{C}}$, that although not smooth
is sufficiently regular for calculation of the Chern form $\mathbf{c}_{1}$.
The kappa forms/cohomology classes
$\kappa_{k}=\int_{\pi^{-1}(\\{R\\})}\mathbf{c}_{1}^{k+1}$ given by integration
over fibers are basic to moduli geometry. The geometry and algebra of the
kappa classes is studied in the Carel Faber lectures. Explicit calculation of
the Chern form and integration provides the following.
Figure 5: Tangents along a fiber of the universal curve
$\overline{\mathcal{C}}/\operatorname{MCG}_{\mathcal{C}}$.
###### Theorem 3.
(W), [Wlp90]. For the hyperbolic metrics on fibers,
$2\pi^{2}\kappa_{1}=\omega$ pointwise on $\mathcal{M}$ and in cohomology on
$\overline{\mathcal{M}}$.
A conformal structure has a unique extension to fill in a puncture. A labeled
puncture defines a section $s$ of
$\overline{\mathcal{C}}_{g,n}/\operatorname{MCG}\rightarrow\overline{\mathcal{M}}_{g,n}$.
A section satisfies $\pi\circ s=id$, differentiating gives $d\pi\circ
ds=d\,id$. At a node, $d\pi$ vanishes in the node opening direction - for
$\pi(z,w)=t$ then $d\pi=wdz\,+\,zdw$ vanishes at the origin. Sections of
$\overline{\mathcal{C}}/\operatorname{MCG}$ over $\overline{\mathcal{M}}$ are
consequently disjoint from nodes. Along a puncture section
$s:\overline{\mathcal{M}}\rightarrow\overline{\mathcal{C}}/\operatorname{MCG}$,
we consider the family of tangent lines $(Ker\,d\pi)|_{s}$ or the dual family
$(Ker\,d\pi)^{*}|_{s}$. In the Carel Faber lectures, the Chern class is
denoted as $K$. -The pullback to $\overline{\mathcal{M}}$ by a puncture
section $s$ of the dual family $(Ker\,d\pi)^{*}$ is the moduli geometry
canonical psi class $\psi$ .-
In these lectures, to emphasize concepts and the underlying geometry, we will
at times informally interchange a line bundle and its Chern class, informally
refer to the moduli space as a manifold, the universal curve as a fiber
bundle, and at times refer to the open moduli space when actually the
augmentation quotient is required. Our goal is to discuss the central matters.
In spite of the informal approach, an experienced reader will find that the
treatment is complete.
Basic references for the above material are [Bus92] and [Wlp10].
Volume results overview.
Mirzakhani shows that the WP volume
$V_{g,n}(L)\,=\,V(\overline{\mathcal{M}}_{g}(L))$ is a polynomial in $L$, with
coefficients given by the intersection numbers of powers of $\kappa_{1}$ and
powers of $\psi$. She further shows that her recursion for determining the
volume polynomials satisfies the defining relations for the Witten-Kontsevich
conjecture. The following theorems are the immediate results of the two
papers.
###### Theorem 4.
The WP volume polynomials. The volume polynomials are determined recursively
from the volume polynomials of smaller total degree, [Mir07a, Formula (5.1) &
Theorem 8.1]. The volume $V_{g,n}(L_{1},\dots,L_{n})$ of the moduli space of
genus $g$, $n$ boundaries, hyperbolic surfaces with boundary lengths
$L=(L_{1},\dots,L_{n})$ is a polynomial
$V_{g,n}(L)\,=\,\sum_{\stackrel{{\scriptstyle\alpha}}{{|\alpha|\leq
3g-3+n}}}C_{\alpha}\,L^{2\alpha},$
for multi index $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and where
$C_{\alpha}>0$ lies in $\pi^{6g-6+2n-2|\alpha|}\mathbb{Q}$, [Mir07a, Theorems
1.1 & 6.1]. The coefficients are intersection numbers given as
$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$
where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form,
$\alpha!=\prod_{j=1}^{n}\alpha_{j}!$, and $\delta_{**}$ is the Kronecker
indicator delta, [Mir07b, Theorem 4.4].
###### Theorem 5.
[Mir07a, Theorems 6.3 & 6.4]. Recursive relations for the volume polynomial
leading coefficients. For a multi index $\alpha$, define
$(\alpha_{1},\dots,\alpha_{n})_{g}=C_{\alpha}\times
2^{-\delta_{1g}\delta_{1n}}\times\prod_{i=1}^{n}\alpha_{i}!\times
2^{|\alpha|},$
then for $n>0$ and $\sum_{i}\alpha_{i}=3g-3+n$,
$\mbox{the dilaton
equation}\qquad\quad(1,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,(2g-2+n)(\alpha_{1},\dots,\alpha_{n})_{g}$
and for $n>0$ and $\sum_{i}\alpha_{i}=3g-2+n$,
$\mbox{the string
equation}\qquad(0,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,\sum_{\alpha_{i}\neq
0}(\alpha_{1},\dots,\alpha_{i}-1,\dots,\alpha_{n})_{g}.$
For the intersection number generating function
$\mathbf{F}(\lambda,t_{0},t_{1},\dots)\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}\sum_{\\{d_{j}\\}}\,\langle\prod_{j=1}^{\infty}\tau_{d_{j}}\rangle_{g}\,\prod_{r\geq
0}t_{r}^{n_{r}}/n_{r}!\,,$
with $n_{r}=\\#\\{j\mid d_{j}=r\\}$, and
$\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}},$
for $\sum_{j=1}^{n}d_{j}=3g-3+n$ and the product $\langle\tau_{*}\rangle$
otherwise zero, then the exponential $e^{\mathbf{F}}$ satisfies Virasoro
algebra constraints, [Mir07b, Theorem 6.1].
A fine structure for volumes is suggested by the Zograf conjecture
$V_{g,n}\,=\,(4\pi^{2})^{2g+n-3}(2g+n-3)!\frac{1}{\sqrt{g\pi}}\big{(}1\,+\,\frac{c_{n}}{g}\,+\,O(1/g^{2})\big{)},$
for fixed $n$ and $g$ tending to infinity.
As an application of the method for recursion of volumes and intersection
numbers, Do derives a remove a boundary relation for the volume polynomials
$\frac{\partial V_{g,n+1}}{\partial L_{n+1}}(L,2\pi i)\,=\,2\pi
i(2g-2+n)V_{g,n}(L),\qquad\cite[cite]{[\@@bibref{}{Dothe}{}{}]}.$
The relation gives the compact case volume $V_{g}$.
Statement of the volume recursion [Mir07a, Sec. 5]. The WP volume
$V_{g}(L_{1},\dots,L_{n})$ of the moduli space
$\mathcal{T}_{g}(L_{1},\dots,L_{n})/\operatorname{MCG}$ is a symmetric
function of boundary lengths as follows.
* •
For $L_{1},L_{2},L_{3}\geq 0$, formally set
$V_{0,3}(L_{1},L_{2},L_{3})=1$
and
$V_{1,1}(L_{1})=\frac{\pi^{2}}{12}+\frac{L_{1}^{2}}{48}.$
* •
For $L=(L_{1},\dots,L_{n})$, let $\widehat{L}=(L_{2},\dots,L_{n})$ and for
$(g,n)\neq(1,1)$ or $(0,3)$, the volume satisfies
$\frac{\partial\ }{\partial
L_{1}}L_{1}V_{g}(L)=\mathcal{A}_{g}^{con}(L)+\mathcal{A}_{g}^{dcon}(L)+\mathcal{B}_{g}(L)$
where
$\mathcal{A}_{g}^{*}(L)=\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\widehat{\mathcal{A}}_{g}^{*}(x,y,L)\,xy\,dxdy$
and
$\mathcal{B}_{g}(L)=\int_{0}^{\infty}\widehat{\mathcal{B}}_{g}(x,L)\,x\,dx.$
The quantities
$\widehat{\mathcal{A}}_{g}^{con},\,\widehat{\mathcal{A}}_{g}^{dcon}$ are
defined in terms of the function
$H(x,y)=\frac{1}{1+e^{\frac{x+y}{2}}}+\frac{1}{1+e^{\frac{x-y}{2}}}$
and moduli volumes for subsurfaces
$\widehat{\mathcal{A}}_{g}^{con}(x,y,L)=H(x+y,L_{1})V_{g-1}(x,y,\widehat{L})$
and surface decomposition sum
$\quad\quad\quad\widehat{\mathcal{A}}_{g}^{dcon}(x,y,L)=\sum\limits_{\stackrel{{\scriptstyle
g_{1}+g_{2}=g}}{{I_{1}\amalg
I_{2}=\\{2,\dots,n\\}}}}H(x+y,L_{1})V_{g_{1}}(x,L_{I_{1}})V_{g_{2}}(y,L_{I_{2}}),$
where in the second sum only decompositions for pairs of hyperbolic structures
are considered and the unordered sets $I_{1},I_{2}$ provide a partition. The
third quantity $\widehat{\mathcal{B}}_{g}$ is defined by the sum
$\quad\quad\quad\quad\frac{1}{2}\sum_{j=1}^{n}\big{(}H(x,L_{1}+L_{j})+H(x,L_{1}-L_{j})\big{)}\,V_{g}(x,L_{2},\dots,\widehat{L_{j}},\dots,L_{n}),$
where $L_{j}$ is omitted from the argument list of $V_{g}$.
The basic point: the volume $V_{g}(L_{1},\dots,L_{n})$ is an appropriate
integral of volumes for surfaces formed with one fewer pairs of pants.
Lecture 2: The McShane-Mirzakhani identity.
In 1991, Greg McShane discovered a universal identity for a sum of lengths of
simple geodesics for a once punctured torus, [McS98]. A generalization of the
identity serves as the analog of a partition of unity for the action of the
mapping class group. The identity enables reduction of the action to the
actions of smaller mapping class groups. Consideration of the identity begins
with a surface with geodesic boundaries and a study of arcs from the boundary
to itself.
Introduce $\mathcal{B}$, the set of non trivial free homotopy classes of
simple curves from the boundary to the boundary, homotopy rel the boundary. We
illustrate the approach by considering simple curves with endpoints on a
common boundary $\beta$; the analysis is similar for simple curves connecting
distinct boundaries. Each homotopy class contains a unique shortest geodesic,
orthogonal to $\beta$ at end points - refer to these geodesics as ortho
boundary geodesics. If the surface is doubled across its boundary, then the
ortho boundary geodesics double to simple closed geodesics.
The set $\mathcal{B}$ is in bijection to the set of topological pants embedded
in the surface with $\beta$ as one boundary - refer to these pants as
$\beta$-cuff pants. First note that the endpoints of an ortho boundary
geodesic $\gamma,[\gamma]\in\mathcal{B}$ are distinct. A small
neighborhood/thickening of $\gamma\cup\beta$ is the corresponding topological
pair of pants. Geometrically, the curve $\gamma$ separates $\beta$ into proper
sub arcs; the union of each sub arc with $\gamma$ is a simple curve, that
defines a free homotopy class containing a unique geodesic. The corresponding
geometric pair of pants $\mathcal{P}$ has boundaries $\beta$ and the two
determined geodesics. We will see below that a geometric pair of pants
contains a unique ortho boundary geodesic. The unions of ortho boundary
geodesics and $\beta$ are the spines, the wire frames, for the embedded
geometric $\beta$-cuff pants.
We now describe how the behavior of geodesics emanating orthogonally from
$\beta$ defines a Cantor subset of $\beta$. The Cantor set will have measure
zero and the length identity is simply the sum of lengths of the complementary
intervals. The following description follows the analysis by Tan-Wong-Zhang,
[TWZ06].
Consider the maximal continuations of geodesics emanating orthogonally from
$\beta$ \- refer to these geodesics as ortho emanating geodesics. In addition
to the ortho boundary geodesics, there are three types of ortho emanating
geodesics: non simple, simple infinite length and simple crossing the boundary
obliquely at a second endpoint. We will see that the types are detected by
considering initial segments in a pair of pants.
Consider the geometric pants $\mathcal{P}$, obtained from an ortho boundary
geodesic $\gamma$ (see Figure 6). The boundaries are $\beta$ and the two
defined geodesics $\alpha$ and $\lambda$ (in the special case $(g,n)=(1,1)$
then $\alpha=\lambda$). A spiral is an infinite simple geodesic ray that
accumulates to a simple closed geodesic. Two ortho emanating geodesics are
spirals with accumulation set $\alpha$ and two are spirals with accumulation
set $\lambda$. The two spirals accumulating to a boundary wind in opposite
directions around the boundary. The $\mathcal{P}$ main gaps are the two
disjoint subarcs of $\beta$ that each contain in their interior an endpoint of
$\gamma$ and have spiral initial points as endpoints. The main gaps will be
the components of the Cantor set complement corresponding to the pants
$\mathcal{P}$. From the geometry of pants, geodesics ortho emanating from the
main gaps are either the spiral endpoints, $\gamma$, non simple with self
intersection in $\mathcal{P}$ or simple crossing $\beta$ obliquely at a second
endpoint. The complement in $\beta$ of the main gaps are a pair of open
intervals. From the geometry of pants, for a given open interval all ortho
emanating geodesics exit the pants by crossing one of the boundaries $\alpha$
or $\lambda$. For a given open interval, the initial segments in $\mathcal{P}$
are simple and these geodesics are classified by their subsequent behavior
elsewhere on the surface, by their behavior on some other pair of pants. The
ortho boundary geodesic connecting $\beta$ to $\alpha$ is contained in one of
the open intervals, and the ortho boundary geodesic from $\beta$ to $\lambda$
is contained in the other. A pair of pants has an equatorial reflection,
stabilizing each boundary. The equatorial reflection acts naturally on the
decomposition of $\beta$, interchanging or stabilizing elements.
Figure 6: A pair of pants with equators, main gaps, two spirals and an
orthoboundary geodesic $\gamma$.
In the above, associated to a main gap are the ortho emanating geodesics that
self intersect in the pants, and the simple geodesics that obliquely cross
$\beta$ a second time - refer to the second type geodesics as boundary
oblique. The next observation is that the associations can be reversed, the
associations define bijections between main gaps and geodesics with particular
behaviors on the surface.
A boundary oblique geodesic and $\beta$ form a crooked wire frame that
determines a pair of pants, similar to an ortho boundary geodesic determining
a pair of pants. Boundary oblique geodesics come in continuous families with
each family limiting to an ortho boundary geodesic. A family and its limit
determine the same pair of pants. The initial points (the $\beta$ orthogonal
points) of family elements lie in a common main gap interval - this
observation reverses the association of segments of main gaps to boundary
oblique geodesics.
Next we describe reversing the association of segments of main gaps to non
simple ortho emanating geodesics. The first self intersection of such a
geodesic is contained in a unique embedded pair of pants. To see this,
consider the lasso subarc beginning at $\beta$ and ending where the geodesic
passes through its first self intersection point a second time. The boundary
of a small neighborhood/thickening of the lasso is the union of a simple
closed curve and an element of $\mathcal{B}$. A geometric argument shows that
the self intersection point is contained in the pants determined by the
element of $\mathcal{B}$ and the lasso initial point lies in the main gap for
the pants. This observation reverses the association of segments of main gaps
to non simple ortho emanating geodesics.
Figure 7: A small neighborhood of a lasso.
We recall that non simple with interior intersection is an open condition on
the space of geodesics and an open condition on the space of ortho emanating
geodesics. By considering the double of the surface, simple with all boundary
intersections orthogonal is a closed condition on the space of geodesics. The
set simple with orthogonal single boundary intersection is a Cantor set. The
classification of ortho emanating geodesics is complete.
###### Theorem 6.
(Tan-Wong-Zhang [TWZ06], and Mirzakhani [Mir07a], all following McShane
[McS98].) There is a Cantor set partition of boundary points by the behavior
of ortho emanating geodesics:
$\beta\,=\,\\{\mbox{simple with orthogonal single boundary intersection}\\}\\\
\,\cup\,\\{\mbox{ortho boundary geodesics}\\}\,\cup\,\\{\mbox{simple boundary
oblique geodesics}\\}\,\cup\,\\{\mbox{non simple}\\}.$
An $\epsilon$-neighborhood of the simple complete geodesics, orthogonal to the
boundary at intersections, is a countable union of thin corridors. In the
universal cover the corridors are described by reduced bi infinite words in
the fundamental group. By analyzing the number and width of corridors, Birman-
Series show that the set is very thin.
###### Theorem 7.
(Birman-Series, [BS85].) Simple geodesics have measure zero. The set
$\mathcal{S}$ of simple complete geodesics, orthogonal to the boundary at
intersections, has Hausdorff dimension $1$. The intersection of $\mathcal{S}$
and the boundary has Hausdorff dimension and measure $0$.
The basic summand for the length identity is a rational exponential function.
Define the function $H$ on $\mathbb{R}^{2}$ by
$H(x,y)=\frac{1}{1+e^{\frac{x+y}{2}}}+\frac{1}{1+e^{\frac{x-y}{2}}}$ (1)
and the corresponding functions $\mathcal{D},\mathcal{R}$ on $\mathbb{R}^{3}$
by
$\begin{split}\mathcal{D}(x,y,z)\,&=\,2\log\Bigg{(}\frac{e^{\frac{x}{2}}\,+\,e^{\frac{y+z}{2}}}{e^{\frac{-x}{2}}\,+\,e^{\frac{y+z}{2}}}\Bigg{)}\quad\mbox{and}\\\
\mathcal{R}(x,y,z)\,&=\,x\,-\,\log\Bigg{(}\frac{\cosh\frac{y}{2}\,+\cosh\frac{x+z}{2}}{\cosh\frac{y}{2}\,+\cosh\frac{x-z}{2}}\Bigg{)}.\end{split}$
(2)
The functions $\mathcal{D}$ and $\mathcal{R}$ are related to $H$ as follows,
$\begin{split}\frac{\partial\ }{\partial
x}\mathcal{D}(x,y,z)&=H(y+z,x),\quad\mathcal{D}(0,0,0)=0\quad\mbox{and}\\\
2\,\frac{\partial\ }{\partial
x}\mathcal{R}(x,y,z)&=H(z,x+y)+H(z,x-y),\quad\mathcal{R}(0,0,0)=0.\end{split}$
(3)
###### Theorem 8.
[Mir07a, Theorem 1.3 & 4.2] and [TWZ06, Thrm. 1.8]. The Mirzakhani-McShane
identity. For a hyperbolic surface $R$ with boundaries $\beta_{j}$ with
lengths $L_{j}$,
$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$
where the first sum is over all unordered pairs of simple closed geodesics
with $\beta_{1},\alpha_{1},\alpha_{2}$ bounding an embedded pair of pants, and
the double sum is over simple closed geodesics with
$\beta_{1},\beta_{j},\alpha$ bounding an embedded pair of pants.
###### Proof.
By the above Theorems, $\ell_{\beta}$ equals the sum over embedded
$\beta$-cuff pants of main gap lengths and the counterpart lengths for double
boundary cuff pants. To find the main gap lengths, begin with a formula for
the lengths of the complementary intervals. In a pair of pants, the ortho
boundary geodesic $\delta_{\alpha}$ from $\beta$ to $\alpha$ bisects a
complementary interval (see Figure 6). Let $\beta_{\alpha}$ be one of the
resulting half intervals. The segment $\beta_{\alpha}$ has the geodesic
$\delta_{\beta}$ emanating at one end and a spiral $\sigma$ to $\alpha$
emanating at the other end. In the universal cover, consider contiguous lifts
$\widetilde{\delta_{\alpha}}$, $\widetilde{\beta_{\alpha}}$ and
$\widetilde{\sigma}$. The three lifts and a half infinite ray lift
$\widetilde{\alpha}$ of $\alpha$, combine to form a quadrilateral
$\widetilde{\alpha},\widetilde{\delta_{\alpha}},\widetilde{\beta_{\alpha}},\widetilde{\sigma}$
with angles $\pi/2,\pi/2,\pi/2$ and $0$ between $\widetilde{\sigma}$ and
$\widetilde{\alpha}$. By hyperbolic trigonometry of quadrilaterals [Bus92], it
follows that
$\tanh\ell_{\beta_{\alpha}}\,=\,\operatorname{sech}\delta_{\alpha}\,=\,\frac{\sinh(\ell_{\beta}/2)\,\sinh(\ell_{\alpha}/2)}{\cosh(\ell_{\lambda}/2)\,+\,\cosh(\ell_{\beta}/2)\,\cosh(\ell_{\alpha}/2)}\,.$
The complementary interval length is
$2\ell_{\beta_{\alpha}}=\ell_{\beta}-\mathcal{R}(\ell_{\beta},\ell_{\lambda},\ell_{\alpha})$.
The formula for main gap lengths now follows from the general relation
$\mathcal{R}(x,y,z)\,+\,\mathcal{R}(x,z,y)=x\,+\,\mathcal{D}(x,y,z)$. For
double boundary cuff pants, the main gap lengths are added to the
complementary interval length. The result is
$\mathcal{R}(\ell_{\beta},\ell_{\alpha},\ell_{\lambda})$. ∎
Lecture 3: The covolume formula and recursion.
The main step is application of the length identity to reduce the action of
the mapping class group to an action of smaller mapping class groups, and
consequently express the volume as an integral over a length level set. The
result is an integral of products of lower dimensional volume functions - the
recursion.
The approach is illustrated by computing the genus one, one boundary, volume.
The length identity is
$L\,=\,\sum_{\alpha\
\tiny{simple}}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha}).$
Introduce $\operatorname{Stab}(\alpha)\subset\operatorname{MCG}$, the
stabilizer for $\operatorname{MCG}$ acting on free homotopy classes. A torus
is an elliptic curve with universal cover $\mathbb{C}$ with involution
$z\rightarrow-z$ stabilizing the deck transformation lattice. The involution
acts on tori and tori with one puncture or boundary. The involution reverses
orientation for the free homotopy class of each simple closed geodesic and the
stabilizer $\operatorname{Stab}(\alpha)$ is the semi direct product of the
Dehn twists by the involution $\mathbb{Z}/2\mathbb{Z}$ subgroup. The
involution acts trivially on Teichmüller space. (The torus is one of the
exceptional cases where the $\operatorname{MCG}$ action on $\mathcal{T}$ is
not effective. We will also discuss the torus case below, where a multiplicity
is involved.) A Dehn twist acts on the Teichmüller space in FN coordinates by
$(\ell,\tau)\rightarrow(\ell,\tau+\ell)$. The sector $\\{0\leq\tau<\ell\\}$ is
a fundamental domain for the $\operatorname{Stab}(\alpha)$ action. A mapping
class $h\in\operatorname{MCG}$ acts on a geodesic-length function by
$\ell_{\alpha}\circ h^{-1}=\ell_{h(\alpha)}$.
Write the length identity as
$L\,=\,\sum_{\alpha}\,\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,=\,\sum_{h\in\operatorname{MCG}/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{h(\alpha)},\ell_{h(\alpha)}),$
use the $\operatorname{MCG}$ action on geodesic-length functions, to find
$LV(L)\,=\,\int_{\mathcal{T}(L)/\operatorname{MCG}}\sum_{\operatorname{MCG}/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{\alpha}\circ
h^{-1},\ell_{\alpha}\circ h^{-1})\,\omega,$
change variables on $\mathcal{T}$ by $p=h(q)$ to find
$\sum_{h\in\operatorname{MCG}/\operatorname{Stab}(\alpha)}\int_{h(\mathcal{T}(L)/\operatorname{MCG})}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,d\tau
d\ell\,=\,\int_{\mathcal{T}(L)/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,d\tau
d\ell,$
and use the $\operatorname{Stab}(\alpha)$ fundamental domain, to obtain the
integral
$\int_{0}^{\infty}\int_{0}^{\ell}\,\mathcal{D}(L,\ell,\ell)\,d\tau d\ell.$
The integral in $\tau$ gives a factor of $\ell$.
The derivatives $\partial\mathcal{D}(x,y,z)/\partial x$ and
$\partial\mathcal{R}(x,y,z)/\partial x$ are simpler than the original
functions $\mathcal{D}$ and $\mathcal{R}$ \- apply this observation and
differentiate in $L$ to obtain a formula for the derivative of $LV(L)$,
$\frac{\partial}{\partial
L}LV(L)\,=\,\int_{0}^{\infty}\frac{1}{1+e^{\ell+\frac{L}{2}}}\,+\,\frac{1}{1+e^{\ell-\frac{L}{2}}}\,\ell
d\ell\,=\,\frac{\pi^{2}}{6}\,+\,\frac{L^{2}}{8}.$
The formula $V(L)=\frac{\pi^{2}}{6}+\frac{L^{2}}{24}$ results.
We prepare for the general case. In algebraic geometry intersection theory,
the elliptic involution gives rise to multiplying elliptic intersection counts
by a factor of $1/2$. The factor corresponds to the generic fiber of the
universal elliptic curve being the quotient of the elliptic curve by its
involution. Along this line, the general volume recursion is simplified if
$V_{1,1}(L)$ is formally defined to be $1/2$ of the given value $V(L)$. In
mapping class group theory, the elliptic involution appears as the half Dehn
twist for simple closed curves bounding a torus. In particular, consider the
fundamental group $\pi_{1}(R)$ of a surface, with the standard presentation
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots
a_{g}b_{g}a_{g}^{-1}b_{g}^{-1}c_{1}\cdots c_{n}=1$, with $c_{j}$ a loop about
the $j^{th}$ boundary. The half Dehn twist about the curve
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}$ is the automorphism of $\pi_{1}(R)$ given by:
$a_{1}\rightarrow b_{1}a_{1}^{-1}b_{1}^{-1},b_{1}\rightarrow
b_{1}^{-1};a_{j}\rightarrow a_{1}^{-1}a_{j}a_{1},b_{j}\rightarrow
a_{1}^{-1}b_{j}a_{1},\,\mbox{for }j=2\dots g,\mbox{ and }c_{j}\rightarrow
a_{1}^{-1}c_{j}a_{1},\,\mbox{for }j=1\dots n$. The square of a half Dehn twist
is a Dehn twist and a half Dehn twist acts on the associated FN parameters by
$(\ell,\tau)\rightarrow(\ell,\tau+\ell/2)$.
We now set up for the covolume formula. Let $R$ be a hyperbolic surface with
geodesic boundaries $\beta_{1},\,\dots,\beta_{n}$. Consider a weighted
multicurve
$\gamma\,=\,\sum_{j=1}^{m}a_{j}\gamma_{j},$
where $a_{j}$ are real weights and $\gamma_{j}$ are distinct, disjoint, simple
closed geodesics. Define
$\operatorname{Stab}(\gamma)\subset\operatorname{MCG}$ to be the mapping
classes stabilizing the collection of unlabeled, weighted geodesics - elements
of $\operatorname{Stab}(\gamma)$ may permute components of the multicurve with
equal weights. Write $\operatorname{Stab}(\gamma_{j})$ for the stabilizer of
an individual geodesic and $\operatorname{Stab}_{0}(\gamma_{j})$ for the
subgroup of elements preserving orientation.
Write $R(\gamma)$ for the surface cut open along the $\gamma$ \- each
$\gamma_{j}$ gives rise to two new boundaries - $R(\gamma)$ may be
disconnected. Write
$\mathcal{T}(R(\gamma);\mathbf{x}),\,\mathbf{x}=(x_{1},\dots,x_{m})$ for the
(product) Teichmüller space of the cut open surface with the pair of
boundaries for $\gamma_{j}$ having length $x_{j}$. Denote by
$\operatorname{MCG}(R(\gamma))$ the product of mapping class groups of the
components of $R(\gamma)$ and by
$\mathcal{T}(R(\gamma);\mathbf{x})/\operatorname{MCG}(R(\gamma))$ the
corresponding product of moduli spaces. For the product of symplectic forms on
$\mathcal{T}(R(\gamma);\mathbf{x})$ corresponding to the components of
$R(\gamma)$, the volume $V(R(\gamma);\mathbf{x})$ is the product of volumes of
the component moduli spaces, where again the pair of boundaries for
$\gamma_{j}$ have common length $x_{j}$. Considerations also involve the
finite symmetry group
$\operatorname{Sym}(\gamma)\,=\,\operatorname{Stab}(\gamma)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})$
of mapping classes that possibly permute and reverse orientation of the
$\gamma$ elements.
Summing the translations of a function over a group gives a group action
invariant function. Begin with a function $f$, suitably small at infinity, and
introduce the $\operatorname{MCG}$ sum
$f_{\gamma}(R)=\sum_{\operatorname{MCG}/\operatorname{Stab}(\gamma)}f\big{(}\sum_{j=1}^{m}a_{j}\ell_{h(\gamma_{j})}(R)\big{)}.$
(4)
The next theorem expresses the moduli space integral
$\int_{\mathcal{M}(R)}\,f_{\gamma}\,dV$
as a weighted integral of lower dimensional moduli space volumes.
###### Theorem 9.
[Mir07a, Thrm. 7.1] The covolume formula. For a weighted
$\gamma=\sum_{j=1}^{m}a_{j}\gamma_{j}$ and the $\operatorname{MCG}$ sum of a
function $f$, small at infinity, then
$\int_{\mathcal{T}(R)/\operatorname{MCG}}f_{\gamma}\,dV=(|\mbox{Sym}(\gamma)|)^{-1}\int_{\mathbb{R}_{>0}^{m}}f(|\mathbf{x}|)V(R(\gamma);\mathbf{x})\,\mathbf{x}\cdot
d\mathbf{x}$
where $|\mathbf{x}|=\sum_{j}a_{j}x_{j}$ and $\mathbf{x}\cdot
d\mathbf{x}=x_{1}\cdots x_{m}dx_{1}\cdots dx_{m}$.
###### Proof.
Corresponding to the components $R^{\prime}$ of the cut open surface
$R(\gamma)$, consider the short exact sequence for mapping class groups,
$1\longrightarrow\prod_{j}\mbox{Dehn}(\gamma_{j})\longrightarrow\bigcap_{j}\operatorname{Stab}_{0}(\gamma_{j})\longrightarrow\prod_{R(\gamma)\
components}\operatorname{MCG}(R^{\prime})\longrightarrow 1,$
and the associated fibration of Teichmüller spaces from Fenchel-Nielsen
coordinates,
$\textstyle{\prod_{R(\gamma)\ components}\mathcal{T}(R^{\prime})\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\gamma_{j}}\mathbb{R}_{>0}\times\mathbb{R}\,.}$
(The short exact sequence places half Dehn twists in the mapping class groups
of the tori with single boundaries.) The $d\ell\wedge d\tau$ formula provides
that the fibration is a fibration of symplectic manifolds.
To establish the formula, first write for coset sums
$\sum\limits_{\operatorname{MCG}/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,=\,\sum\limits_{\operatorname{MCG}/\operatorname{Stab}(\gamma)}\
\sum\limits_{\operatorname{Stab}(\gamma)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,=\,|\operatorname{Sym}(\gamma)|\,f_{\gamma},$
using that $f_{\gamma}$ is $\operatorname{Sym}(\gamma)$ invariant for the
second equality. Substitute the resulting formula for $f_{\gamma}$ into the
integral, and unfold the sum (express the
$\operatorname{MCG}/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})$ translation
sum as a sum of translates of a $\operatorname{MCG}$ fundamental domain) to
obtain the equality
$\int_{\mathcal{T}(R)/\operatorname{MCG}}f_{\gamma}\,dV=(|\operatorname{Sym}(\gamma)|)^{-1}\int_{\mathcal{T}(R)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,dV.$
Substitute the fibration
$\textstyle{\prod_{R(\gamma)\
components}\mathcal{T}(R^{\prime})/\operatorname{MCG}(R^{\prime})\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(R)/\bigcap_{j}\operatorname{Stab}_{0}(\gamma_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\gamma_{j}}(\mathbb{R}_{>0}\times\mathbb{R})/\mbox{Dehn}_{*}(\gamma_{j})\,,}$
where $\mbox{Dehn}_{*}(\gamma_{j})$ is generated by a half twist if the curve
bounds a torus with a single boundary and otherwise is generated by a simple
twist. Substitute the factorization of the volume element
$dV=\prod_{R(\gamma)\
components}dV(R^{\prime})\times\prod_{\gamma_{j}}d\ell_{j}\wedge d\tau_{j}.$
The function $f$ depends only on the values $\mathbf{x}$. For the values
$\mathbf{x}$ fixed, perform the
$\prod\mathcal{T}(R^{\prime})/\operatorname{MCG}(R^{\prime})$ integration to
obtain the product volume $V(R(\gamma);\mathbf{x})$. Finally
$\mbox{Dehn}_{*}(\gamma_{j})$ acts only on the variable $\tau_{j}$ with
fundamental domain $0<\tau_{j}<\ell_{j}/2$ if $\gamma_{j}$ bounds a torus with
a single boundary or otherwise with fundamental domain $0<\tau_{j}<\ell_{j}$.
For a torus with a single boundary, the action is accounted for by using the
volume value that is $1/2$ the original $V(L)$. The right hand side of the
formula is now established. ∎
We are now ready to apply the covolume formula to the length identity. The
application follows the genus one example. Again - it is essential to use the
$\operatorname{MCG}$ geodesic-length function action $\ell_{\alpha}\circ
h^{-1}=\ell_{h(\alpha)}$, to consider a sum over topological configurations as
a sum of $\operatorname{MCG}$ translates of a function. Consider the action on
configurations. The mapping class group naturally acts on $\mathcal{B}$, the
set of non trivial free homotopy classes of simple curves with endpoints on
the boundary, rel the boundary. Recall the correspondences: elements of
$\mathcal{B}$ $\Longleftrightarrow$ wire frames $\Longleftrightarrow$ boundary
pants $\mathcal{P}$. Viewing Figure 1, the $\operatorname{MCG}$ orbits on
$\mathcal{B}$ are of three types, describing location of the boundary pants
$\mathcal{P}$.
* •
Orbits for simple curves from $\beta_{1}$ to $\beta_{1}$.
* –
A single orbit for $R-\mathcal{P}$ connected, with
$\operatorname{Sym}(\beta_{1},\alpha_{1},\alpha_{2})=\mathbb{Z}/2\mathbb{Z}$.
* –
A collection of orbits for $R-\mathcal{P}$ disconnected. The orbits are
classified by the joint partitions of genus $g=g_{1}+g_{2}$ and of labeled
boundaries $\\{\beta_{2},\dots,\beta_{n}\\}$, with each resulting component
with negative Euler characteristic. In general
$\operatorname{Sym}(\beta_{1},\alpha_{1},\alpha_{2})=1$, except in the special
case of one boundary and $g_{1}=g_{2}$.
* •
A collection of orbits, one for each choice of a second boundary. In
particular, an orbit for simple curves from $\beta_{1}$ to $\beta_{j},\,j\neq
1$. A resulting surface $R-\mathcal{P}$ is connected, with
$\operatorname{Sym}(\beta_{1},\beta_{j},\alpha)=1$.
Consider Theorem 8, and integrate each side of the identity
$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$
over the moduli space of $R$ relative to the volume $dV$. Form the $L_{1}$
partial derivative of each side to simplify the quantities $\mathcal{D}$ and
$\mathcal{R}$. Apply formulas (3) for the right hand side. Express the right
hand side as individual sums for given orbit types. Apply Theorem 9 for each
orbit type to find integrals in terms of lower dimensional moduli volumes as
follows.
* •
For the sum over simple curves from $\beta_{1}$ to $\beta_{1}$ with
$R-\mathcal{P}$ connected, the summand function is $\mathcal{D}$ with
$\frac{\partial\mathcal{D}}{\partial
L_{1}}\,=\,H(\ell_{\alpha_{1}}+\ell_{\alpha_{2}},L_{1}),$
a function of length of a multicurve, and the resulting integral is
$\int_{0}^{\infty}\int_{0}^{\infty}H(x+y,L_{1})V_{g-1}(x,y,\widehat{L})\,xy\,dxdy,$
for $\widehat{L}=(L_{2},\dots,L_{n})$.
* •
For the sum over simple curves from $\beta_{1}$ to $\beta_{1}$ with
$R-\mathcal{P}$ disconnected, the summand function is $\mathcal{D}$ with
$\frac{\partial\mathcal{D}}{\partial
L_{1}}\,=\,H(\ell_{\alpha_{1}}+\ell_{\alpha_{2}},L_{1}),$
a function of length of a multicurve, and the resulting integral is
$\int_{0}^{\infty}\int_{0}^{\infty}\sum_{\stackrel{{\scriptstyle
g_{1}+g_{2}=g}}{{I_{1}\amalg
I_{2}=\\{2,\dots,n\\}}}}H(x+y,L_{1})V_{g_{1}}(x,L_{I_{1}})V_{g_{2}}(y,L_{I_{2}})\,xy\,dxdy.$
* •
For the sum over simple curves from $\beta_{1}$ to $\beta_{j},\,j\neq 1$, the
summand function is $\mathcal{R}$ with
$\frac{\partial\mathcal{R}}{\partial
L_{1}}\,=\,\frac{1}{2}H(\ell_{\alpha},L_{1}+L_{j})\,+\,\frac{1}{2}H(\ell_{\alpha},L_{1}-L_{j}),$
a sum of functions of weighted length of a multicurve, and the resulting
integral is
$\int_{0}^{\infty}\frac{1}{2}\sum_{j=2}^{n}\big{(}H(x,L_{1}+L_{j})+H(x,L_{1}-L_{j})\big{)}V_{g}(x,L_{2},\dots,\widehat{L_{j}},\dots,L_{n})\,x\,dx,$
where $L_{j}$ is omitted from the argument list of $V_{g}$.
Compare to the end of Lecture 1 \- the volume recursion is established. The
volume function $V_{g,n}(L)$ is recursively determined.
What type of function is $V_{g,n}(L)$? The recursion involves two elementary
integrals, see formula (1) above for the definition of $H$,
$\int_{0}^{\infty}x^{2j+1}H(x,t)\,dx\qquad\mbox{and}\qquad\int_{0}^{\infty}\int_{0}^{\infty}x^{2j+1}y^{2k+1}H(x+y,t)\,dxdy.$
By direct calculation, see [Mir07a, formula (6.2) and Lemma 6.2], each
integral is a polynomial in $t^{2}$ with each coefficient a product of
factorials and the Riemann zeta function at a non negative even integer, each
coefficient is a positive rational multiple of an appropriate power of $\pi$.
The first polynomial has degree $j+1$ in $t^{2}$, while the second has degree
$i+j+2$. The first part of Theorem 4 is now established. The second part is
the subject of the next lecture.
Lecture 4: Symplectic reduction, principal $S^{1}$ bundles and the normal
form.
The goal of the lecture is to establish the following theorem. The formula
combines with Theorem 3, the WP kappa equation $\omega=2\pi^{2}\kappa_{1}$, to
provide that the coefficients of the volume polynomial $V_{g}(L)$ are
tautological intersection numbers. The result completes the proof of Theorem
4.
###### Theorem 10.
For
$d=\frac{1}{2}\dim_{\mathbb{R}}\mathcal{T}_{g}(L)=\dim_{\mathbb{C}}\mathcal{T}_{g,n}$,
then
$V_{g}(L)\ =\
\frac{1}{d!}\int_{\mathcal{T}_{g}(L)/\operatorname{MCG}}\omega_{\mathcal{T}_{g}(L)}^{d}\
=\
\frac{1}{d!}\int_{\mathcal{T}_{g,n}/\operatorname{MCG}}\big{(}\omega\,+\,\sum_{j=1}^{n}\frac{L_{j}^{2}}{2}\psi_{j}\big{)}^{d}.$
The proof is essentially by establishing a cohomology equivalence between
symplectic spaces - combining symplectic reduction, the Duistermaat-Heckman
theorem and explicit geometry to obtain the formula.
The considerations of the lecture are presented for the appropriate
Teichmüller spaces $\mathcal{T}$ and the open moduli spaces
$\mathcal{T}/\operatorname{MCG}$. The constructions are compatible with the
augmentation construction. The results are valid for the appropriate
Teichmüller spaces and compactified moduli spaces. The compactified moduli
spaces are orbifolds. The following results are for cohomology statements over
$\mathbb{Q}$; cohomology arguments over $\mathbb{Q}$ for manifolds are in
general also valid for orbifolds. Alternatively, the orbifold matter can be
bypassed by applying the general result that compactified moduli spaces have
manifold finite covers, [BP00, Loo94].
The Teichmüller spaces. We consider the trio.
* •
$\mathcal{T}_{g}(L)$ \- the space of marked genus $g$ hyperbolic surfaces,
with geodesic boundaries $\beta_{1},\dots,\beta_{n}$ of prescribed lengths
$L_{1},\dots,,L_{n}$. A hyperbolic surface can be doubled across its geodesic
boundary to obtain a compact hyperbolic surface of higher genus. Accordingly,
$\mathcal{T}_{g}(L)$ can be considered as a locus in $\mathcal{T}_{2g+n-1}$.
The symplectic form $\omega$ of the image Teichmüller space restricts to the
locus and defines a symplectic form on the locus. A pants decomposition for a
surface with boundary, can be doubled to give a pants decomposition for a
doubled surface. The doubled pants decomposition is characterized by
containing the geodesics $\beta_{1},\dots,\beta_{n}$ and being symmetric.
Fenchel-Nielsen coordinates and the $d\ell\wedge d\tau$ formula are applied
for doubled decompositions to obtain a description of the locus
$\mathcal{T}_{g}(L)\subset\mathcal{T}_{2g+n-1}$, and to define a symplectic
from $\omega_{\mathcal{T}_{g}(L)}$. The symplectic form is given as
$\sum_{j}d\ell_{j}\wedge d\tau_{j}$ (without boundary parameters) for any
pants decomposition of a surface with boundaries. $\operatorname{MCG}$
invariance is immediate.
* •
$\widehat{\mathcal{T}}_{g,n}$ \- the space of marked genus $g$ hyperbolic
surfaces, with pointed geodesic boundaries $\beta_{1},\dots,\beta_{n}$ \-
boundary lengths are allowed to vary and a variable point is given on each
boundary. The $\mathbb{R}$ dimension of $\widehat{\mathcal{T}}_{g,n}$ is $2n$
greater than the $\mathbb{R}$ dimension of $\mathcal{T}_{g}(L)$. Symplectic
reduction requires a symplectic form on $\widehat{\mathcal{T}}_{g,n}$, that is
equivalent to $\omega_{\mathcal{T}_{g}(L)}$ on $L$ level sets and is invariant
under rotating points on boundaries. A form is given by describing
$\widehat{\mathcal{T}}_{g,n}$ as a higher dimensional Teichmüller space.
Figure 8: A genus $2$ surface with three pointed boundaries.
To this purpose, introduce almost tight pants - pairs of pants with two
labeled boundaries being punctures (length zero) and a third boundary of
prescribed length. An almost tight pants will be glued to each surface
boundary $\beta_{j}$. The pants equatorial reflection defines symmetric points
on the pants boundary; the puncture labeling uniquely determines an equatorial
boundary point. A standard model for a surface with pointed geodesic
boundaries is given by gluing on almost tight pants - at each boundary glue on
an almost tight pants with matching boundary length and the equatorial point
aligned with the point on the boundary. The construction does not involve
choices, so is natural with respect to marking homeomorphisms and the
$\operatorname{MCG}$ action.
Figure 9: Aligning boundary and equatorial points to glue on almost tight
pants.
For a surface $R$ with labeled, pointed boundaries, write $\widehat{R}$ for
the standard model surface with glued on almost tight pants. The punctures of
$\widehat{R}$ are labeled in pairs. By the general hyperbolic collar result,
small length geodesics are necessarily disjoint [Bus92, Wlp10]. For the
lengths $L_{1},\dots,L_{n}$ suitably small, the labeled geodesics
$\beta_{1},\dots,\beta_{n}$ are uniquely determined on the surface
$\widehat{R}$ by having small length and bounding labeled punctures. The
pointed boundary, marked surface $R$ is equivalent to the marked surface
$\widehat{R}$ modulo Dehn twists about the $\beta_{j}$ (Dehn twists, since the
boundary points are given on a circle). In particular, for $c$ suitably small,
the equivalence is between the open subset $\\{L<c\\}$ of
$\widehat{\mathcal{T}}_{g,n}$ and the open subset $\\{L<c\\}$ in
$\mathcal{T}_{g,2n}/\prod_{j}\mbox{Dehn}(\beta_{j})$.
Definition and equivalence of $S^{1}$ principal bundles are next.
Considerations begin with the short exact sequence from Theorem 9,
$1\longrightarrow\mbox{Dehn}(\beta)=\prod_{j}\mbox{Dehn}(\beta_{j})\longrightarrow\mbox{Stab}(\beta)=\bigcap_{j}\operatorname{Stab}(\beta_{j})\longrightarrow\operatorname{MCG}(R)\longrightarrow
1,$
(now $\operatorname{Stab}(\beta_{j})=\operatorname{Stab}_{0}(\beta_{j})$,
since an orientation preserving pants homeomorphism preserves boundary
orientation). The geodesics $\beta_{1},\dots,\beta_{n}$ define subsets of the
Riemann surface bundles (the universal curves) over
$\widehat{\mathcal{T}}_{g,n}$ and over
$\mathcal{T}_{g,2n}/\mbox{Dehn}(\beta)$. The subsets define oriented circle
bundles, provided automorphisms of the Riemann surfaces act at most as
rotations on the individual geodesics. The small lengths $L_{1},\dots,L_{n}$
and labeled boundaries provide the condition. The geodesics define circle
bundles over Teichmüller bases. We see below that rotation along geodesics
defines an $S^{1}$ principal structure. Next, from the above short exact
sequence and the definition of marking - the equivalence between geodesics
$\beta\subset R$ and $\beta\subset\widehat{R}$, and the projections of circle
bundles to bases, commute with the actions of
$\operatorname{MCG}(R)\approx\operatorname{Stab}(\beta)/\mbox{Dehn}(\beta)$.
The geodesics define equivalent orbifold $S^{1}$ principal bundles over the
quotients
$\\{L<c\\}\subset\widehat{\mathcal{T}}_{g,n}/\operatorname{MCG}(R)\times(S^{1})^{n}$
(the $\operatorname{MCG}(R)$ and $(S^{1})^{n}$ actions on
$\widehat{\mathcal{T}}_{g,n}$ commute) and
$\\{L<c\\}\subset\mathcal{T}_{g,2n}/\operatorname{Stab}(\beta)$.
The symplectic form of $\mathcal{T}_{g,2n}$ defines a symplectic form
$\omega_{\widehat{\mathcal{T}}_{g,n}}$ on the open subset $\\{L<c\\}$.
Fenchel-Nielsen coordinates and the $d\ell\wedge d\tau$ formula are applied.
The form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given as
$\sum_{k}d\ell_{k}\wedge d\tau_{k}$ for any pants decomposition of
$\widehat{R}$ containing the multicurve $\beta$. Importantly, the form
$\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given for surfaces $R$ with pointed
boundaries $\beta_{j}$, by an extended interpretation of the $d\ell\wedge
d\tau$ formula, with a term for each boundary, now with the interpretation
that $\tau(\beta_{j})$ parameterizes the location of the specified point. See
Figure 3, the parameter $\tau(\beta_{j})$ increasing corresponds to the point
moving on the boundary with the surface interior on the right.
$\operatorname{MCG}(R)$ invariance of the symplectic form is immediate.
Restriction of the form to $L$ level sets and invariance under rotating
boundary points are discussed below.
* •
$\mathcal{T}_{g,n}$ \- the space of marked genus $g$ hyperbolic surfaces with
$n$ punctures. $\mathcal{T}_{g,n}$ has the $\operatorname{MCG}$ invariant
symplectic form $\omega$. $\mathcal{T}_{g,n}$ coincides with the Teichmüller
space $\mathcal{T}_{g}(0)$, where surface boundary lengths are zero.
We will relate the three symplectic manifolds.
Symplectic reduction for $\widehat{\mathcal{T}}_{g,n}$. We consider the
Hamiltonian geometry of FN twists, geodesic-lengths and especially the moment
map
$\widehat{\mathcal{T}}_{g,n}\ \stackrel{{\scriptstyle\mu}}{{\longrightarrow}}\
\widehat{L}=(L^{2}_{1}/2,\dots,L^{2}_{n}/2)\in\mathbb{R}_{\geq 0}^{n}.$
Write $t_{j}$ for the unit speed infinitesimal rotation of the point on the
boundary $\beta_{j}$; $t_{j}$ is a vector field on
$\widehat{\mathcal{T}}_{g,n}$. In terms of the standard model surfaces,
$t_{j}$ is the FN infinitesimal twist vector field for $\beta_{j}$, and
$t_{j}$ is the infinitesimal rotation of the $j^{th}$ almost tight pants. By
twist-length duality, we have $\omega_{\widehat{\mathcal{T}}_{g,n}}(-t_{j},\
)=dL_{j}$ and the scaled $-L_{j}t_{j}$ is unit infinitesimal rotation (unit
time flow is a full rotation). The function $\frac{1}{2}L_{j}^{2}$ is the
corresponding Hamiltonian potential, since
$\omega_{\widehat{\mathcal{T}}_{g,n}}(-L_{j}t_{j},\ )=d(\frac{1}{2}L_{j}^{2})$
(the momentum $\frac{1}{2}L^{2}$ determines the twist sign/orientation). The
vector fields $-L_{j}t_{j}$ are the infinitesimal generators for the
$(S^{1})^{n}$ action on $\widehat{\mathcal{T}}_{g,n}$, given by rotating the
boundary points. The symplectic form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is
twist invariant and we are ready for symplectic reduction, ready to consider
the quotient $\widehat{\mathcal{T}}_{g,n}/(S^{1})^{n}$.
Figure 10: Positive rotations for an $(S^{1})^{3}$ action.
A level set of the moment map
$\mu:\widehat{\mathcal{T}}_{g,n}\longrightarrow\mathbb{R}^{n}$ is a locus of
prescribed $\beta$ length hyperbolic surfaces. The group $(S^{1})^{n}$ acts on
level sets by rotating almost tight pants. The quotient of a level set by the
group is naturally $\mathcal{T}_{g}(L)$ \- the level set prescribes the
boundary lengths and the group action removes the location information for the
points.
###### Proposition 11.
Symplectic reduction. For
$\widehat{\mathcal{T}}_{g,n}/(S^{1})^{n}\approx\mathcal{T}_{g}(L)$, then
$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(\widehat{L})}/(S^{1})^{n}\
\approx\ \omega_{\mathcal{T}_{g}(L)}.$
###### Proof.
The form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given by the $d\ell\wedge
d\tau$ formula for any pants decomposition of a standard model surface
containing the multicurve $\beta$. The differentials $dL_{j}$ vanish on $\mu$
level sets and the formula reduces to the sum for a pants decomposition of a
surface with boundary, a sum without boundary parameters - the
$\omega_{\mathcal{T}_{g}(L)}$ formula. ∎
$S^{1}$ principal bundles. We review basics about characteristic classes.
###### Definition 12.
Let $\pi:P\longrightarrow M$ be a smooth circle bundle over a smooth compact
manifold $M$. The bundle is $S^{1}$ principal provided,
1. 1.
$S^{1}$ acts freely on $P$,
2. 2.
$\pi(p_{1})=\pi(p_{2})$ if and only if there exists $s\in S^{1}$, such that
$p_{1}\cdot s=p_{2}$.
A connection for an $S^{1}$ principal bundle is a smooth distribution
$\mathcal{H}\subset\mathbf{T}P$ of tangent subspaces such that,
1. 1.
$\mathbf{T}_{p}P=\mathcal{H}_{p}\oplus\ker\pi_{*}\big{|}_{p}$, for each $p\in
P$,
2. 2.
$s^{*}\mathcal{H}_{p}=\mathcal{H}_{p\cdot s}$.
A connection is uniquely given as $\mathcal{H}=\ker A$, for a $1$-form $A$ on
$P$, provided $A$ is $S^{1}$ invariant and $A(\dot{s})=1$. An $S^{1}$
invariant inner product $\langle\ ,\ \rangle$ provides an example of an
invariant $1$-form by $A(v)=\langle
v,\dot{s}\rangle/\langle\dot{s},\dot{s}\rangle$. The curvature $2$-form on $P$
for a connection is $\Phi(v,w)=dA(\operatorname{hor}v,\operatorname{hor}w)$,
for $\operatorname{hor}$ the horizontal projection of $\mathbf{T}P$ to
$\mathcal{H}$.
###### Theorem 13.
[MS74]. There exists a unique closed $2$-form $\Omega$ on $M$, such that
$\Phi=\pi^{*}\Omega$. The cohomology class of $\Omega$ is independent of the
choice of $S^{1}$ principal connection for $P$ and the first Chern class is
$c_{1}(P)\,=\,[\Omega]\in H^{2}(M,\mathbb{Z})$.
As above, the variable point on the boundary $\beta_{j}$ of the surface $R$
defines an $S^{1}$ principal bundle $\widehat{\beta}_{j}$ over
$\mathcal{T}_{g}(L)$; $S^{1}$ acts by moving the point with the surface
interior on the left. A choice of connection for the bundle gives a first
Chern class $c_{1}(\widehat{\beta}_{j})$.
Applying the Duistermaat-Heckman theorem. We extend the definition of
$\widehat{\mathcal{T}}_{g,n}$ to include $L=0$; geodesic boundaries of $R$ can
be replaced with punctures. Hyperbolic structures converge for boundary
lengths tending to zero; in particular collar regions converge to cusp
regions. The extension of $\widehat{\mathcal{T}}_{g,n}$ is given by
parameterizing boundary points by points on a collar/cusp region boundary.
We recall basics about collars and cusps. For a geodesic $\alpha$ of length
$\ell_{\alpha}$, the standard collar in the upper half plane $\mathbb{H}$ is
$\mathcal{C}(\ell_{\alpha})=\\{d(z,i\mathbb{R}_{+})\leq w(\alpha)\\}$, for the
half width $w(\alpha)$ given by $\sinh w(\alpha)\sinh\ell_{\alpha}/2=1$. The
quotient cylinder $\\{d(z,i\mathbb{R}_{+})\leq w(\alpha)\\}/\langle z\mapsto
e^{\ell_{\alpha}}z\rangle$ embeds into $R$ to give a collar neighborhood of
the geodesic. For a cusp, the standard cusp in $\mathbb{H}$ is
$\mathcal{C}_{\infty}=\\{\Im z\geq 1/2\\}$. The quotient cylinder $\\{\Im
z\geq 1/2\\}/\langle z\mapsto z+1\rangle$ embeds into $R$ to give a cusp
region. The boundary of a collar, for $\ell_{\alpha}$ bounded, and boundary of
a cusp region have length approximately 2. Collars and cusp regions are
foliated by geodesics normal to the boundary. For geodesic-lengths tending to
zero, half collar neighborhoods Gromov-Hausdorff converge to a cusp region
(convergence is uniform on bounded distance neighborhoods of the boundary);
boundaries and geodesics normal to the boundary converge.
Figure 11: Projecting along geodesics to a collar and a cusp region boundary.
The geodesics normal to the boundary of a collar provide a projection from the
core geodesic to each collar boundary. The projection is used to note that
prescribing a point on a geodesic boundary of $R$ is equivalent to prescribing
a point on the boundary of the half collar neighborhood of the geodesic. Since
collars and their boundaries converge to a cusp region and its boundary, for
core geodesic length tending to zero, we have a description for the extension
of the definition of $\widehat{\mathcal{T}}_{g,n}$ to include surfaces with
collections of lengths $L_{j}$ zero.
The standard cusp region is uniformized by the variable $w=e^{2\pi iz}$. A
point on the cusp region boundary $\Im z=1/2$ corresponds to a point on
$|w|=e^{-\pi}$ and given the factor $e^{-\pi}$, a point on the circle
corresponds to a tangent vector at the origin. The variable $w$ is unique
modulo multiplication by a unimodular number; the identification of the circle
with tangent vectors at the origin is canonical. For an $S^{1}$ infinitesimal
generator $-L_{j}t_{j}$, displacement is to the left when crossing the
geodesic, (compare to the Figure 3 positive twist, right displacement) the
reference point moves with the surface interior on its left, the tangent
vector at the origin rotates clockwise, and a dual cotangent vector rotates
counter clockwise (the positive direction for a $\mathbb{C}$-line). Combining
equivalences, the $S^{1}$ principal bundle of a point on a cusp region
boundary is equivalent to a non zero vector in the cotangent line for the
puncture.
Proof of Theorem 10. The $\operatorname{MCG}(R)$ and $(S^{1})^{n}$ actions on
$\widehat{\mathcal{T}}_{g,n}$ commute; consider the quotient
$\widehat{\mathcal{T}}_{g,n}/\operatorname{MCG}(R)\times(S^{1})^{n}$. By the
Duistermaat-Heckman theorem, [CdS01, Chapter 30, Theorem 30.8], for small
values of $\widehat{L}$, including $0$, the reduced level sets
$\mu^{-1}(\widehat{L})/(S^{1})^{n}$ are mutually diffeomorphic. Furthermore by
Duistermaat-Heckman, the $L$ level set reduced symplectic form
$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(\widehat{L})}/(S^{1})^{n}$
is cohomologous to the sum of the $0$ level set reduced form
$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(0)}/(S^{1})^{n}$ and
the contributions $(L_{j}^{2}/2)\,c_{1}(\widehat{\beta}_{j})$, for
$c_{1}(\widehat{\beta}_{j})$ the first Chern class for the $S^{1}$ principal
bundle of a point varying on the $j^{th}$ cusp region boundary. Combining with
Proposition 11, gives the desired cohomology equivalence,
$\omega_{\mathcal{T}_{g}(L)}\ \equiv\ \omega_{\mathcal{T}_{g}(0)}\ +\
\sum_{j=1}^{n}\frac{L_{j}^{2}}{2}c_{1}(\widehat{\beta}_{j}).$
By the description of collars and cotangent lines at punctures, the circle
bundle $\widehat{\beta}_{j}$ is topologically equivalent to the psi line
bundle $\psi_{j}$ (see Lecture 1) with equality of first Chern classes. The
proof is finished.
Lecture 5: The pattern of intersection numbers and Witten-Kontsevich.
We begin with the discussion of Harris-Morrison [HM98, pgs. 71-75]. For a
finite sequence of non negative integers $\\{\alpha_{j}\\}$, define the top
$\psi$-intersection number by
$\langle\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}.$
For a non trivial pairing, the genus $g$, number of punctures $n$, and
exponents $\alpha_{j}$ are related by $3g-3+n=\sum_{j=1}^{n}\alpha_{j}$,
otherwise the pairing is defined as zero. More generally using exponents to
denote powers (repetitions) of the variables $\tau$, define
$\langle\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\tau_{m}^{d_{m}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,d}}\prod_{j=0}^{m}\,\prod_{k=1}^{d_{j}}\,\psi_{(j,k)}^{\,j}.$
($\tau_{j}^{d_{j}}$ denotes that for $d_{j}$ punctures, the associated $\psi$
is raised to the $j^{th}$ power; the subscripts $(j,k)$ are distinct puncture
labels.) For the second pairing, the formal count of punctures is
$d=\sum_{j=0}^{m}d_{j}$, and the formal degree of the product is the count of
$\psi$ factors $\sum_{j=0}^{m}jd_{j}$. For a non trivial pairing, the genus,
number of punctures and degree are related by $3g-3+d=\sum_{j=0}^{m}jd_{j}$,
otherwise the pairing is zero. The psi classes are known to be positive -
integrals of products over subvarieties are positive; the non trivial pairings
are positive, matching Mirzakhani’s positivity of volume polynomial
coefficients, see Theorem 4.
Witten considered a partition function (probability of states), for two-
dimensional gravity. For an infinite vector
$\mathbf{t}=(t_{0},t_{1},\dots,t_{n},\dots)$, and $\gamma$ the formal sum
$\gamma=\sum_{j=0}^{\infty}t_{j}\tau_{j}$, Witten introduced a genus $g$
generating function for $\tau$ products,
$F_{g}(\mathbf{t})=\sum_{n=0}^{\infty}\frac{\langle\gamma^{n}\rangle_{g}}{n!},$
in which the numerator is defined by monomial expansion, resulting in the
formal power series
$F_{g}(\mathbf{t})=\sum_{\\{d_{j}\\}}\
\langle\prod_{j=0}^{\infty}\tau_{j}^{d_{j}}\rangle_{g}\,\prod_{j=0}^{\infty}\frac{t_{j}^{d_{j}}}{d_{j}!},$
where the sum is over all sequences of non negative integers $\\{d_{j}\\}$
with only finitely many non zero terms. –By Theorem 4, the intersection
numbers
$\langle\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\tau_{m}^{d_{m}}\rangle_{g}$ are
the coefficients of the leading terms of the volume polynomials $V_{g,n}(L)$.–
The quantum gravity partition function is
$\mathbf{F}(\lambda,\mathbf{t})\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}F_{g}(\mathbf{t}).$
Based on a realization of the function in terms of matrix integrals, Witten
conjectured that the partition function should satisfy two forms of the
Korteweg-deVries (KdV) equations. Kontsevich gave a proof of the conjecture
using a cell decomposition of the moduli spaces $\mathcal{M}_{g,n}$, [Kon92].
Cells are enumerated by ribbon graphs/fat graphs. Kontsevich encoded the
intersection numbers in an enumeration of trivalent ribbon graphs. He then
used Feynman diagram techniques and a matrix Airy integral to establish
Witten’s conjectures.
Two basic relations for the intersection numbers are: for $n>0$ and
$\sum_{i}\alpha_{i}=3g-2+n>0$, the
$\mbox{string
equation}\qquad\langle\tau_{0}\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,\sum_{\alpha_{i}\neq
0}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{i}-1}\cdots\tau_{\alpha_{n}}\rangle_{g},$
and for $n\geq 0$ and $\sum_{i}\alpha_{i}=3g-3+n\geq 0$, the
$\mbox{dilaton
equation}\qquad\quad\langle\tau_{1}\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,(2g-2+n)\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}.$
The first equation is for adding a new puncture without an associated factor
of $\psi$ in the product, while the second equation is for adding a new
puncture with a single associated factor of $\psi$.
Similar to setting $V_{0,3}(L)=1$, the intersection symbol for the thrice
punctured sphere is normalized to $\langle\tau_{0}^{3}\rangle_{0}=1$. The
general genus $0$ formula is
$\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{0}\,=\,\bigg{(}\frac{n-3}{\alpha_{1}!\cdots\alpha_{n}!}\bigg{)},$
with the right hand side a multinomial coefficient for $n-3$. The genus $0$
string equation is simply Pascal’s multinomial neighbor relation.
For genus $1$, Theorem 10 and the WP kappa equation, Theorem 3, give
$V(L)=\int_{\mathcal{M}_{1,1}}2\pi^{2}\kappa_{1}+\frac{L^{2}}{2}\psi$. The
formula combines with the Lecture 3 calculation
$V_{1,1}(L)=\frac{\pi^{2}}{12}+\frac{L^{2}}{48}$ (now including the elliptic
involution $\frac{1}{2}$ factor) to provide the evaluations,
$\frac{1}{2}\int_{\mathcal{M}_{1,1}}\kappa_{1}\,=\,\langle\tau_{1}\rangle_{1}\,=\,\frac{1}{24}.$
General genus $1$ evaluations are found from the single evaluation by applying
the string and dilaton equations. A consequence of the Witten conjecture is
that all $\langle\tau\rangle$ intersections can be calculated from the initial
values $\langle\tau_{0}^{3}\rangle_{0}=1$ and
$\langle\tau_{1}\rangle_{1}=\frac{1}{24}$, using the Virasoro equations
$L_{n}(e^{\mathbf{F}})=0$ for the partition function described below.
In the volume recursion, leading coefficients are obtained from leading
coefficients - the recursion specializes to leading coefficients, see [Mir07b,
Lemma 5.3]. We now sketch the application of the specialized recursion to
relations for the partition function and a solution of Witten’s conjecture.
Relations come from the Virasoro Lie algebra. The Witt subalgebra is generated
by the differential operators $\mathcal{L}_{n}=-z^{n+1}\partial/\partial
z,\,n\geq-1$, with commutators
$[\mathcal{L}_{n},\mathcal{L}_{m}]=(n-m)\mathcal{L}_{n+m}$. The string and
dilaton equations can be written as linear homogeneous differential equations
for the exponential $e^{\mathbf{F}}$ of the partition function. The
differential operator for the string equation is
$L_{-1}\,=\,-\frac{\partial}{\partial
t_{0}}\,+\,\frac{\lambda^{-2}}{2}\,t_{0}^{2}\,+\,\sum_{j=0}^{\infty}t_{j+1}\frac{\partial}{\partial
t_{j}},$
and the differential operator for the dilaton equation is
$L_{0}\,=\,-\frac{3}{2}\frac{\partial\quad}{\partial
t_{1}}\,+\,\sum_{j=0}^{\infty}\frac{2j+1}{2}t_{j}\frac{\partial}{\partial
t_{j}}\,+\,\frac{1}{16}.$
With simple conditions, there is a unique way to extend operator definitions
to obtain a representation of the $\\{\mathcal{L}_{n}\\}$ subalgebra. The
general operator is
$L_{n}\,=\,-\frac{(2n+3)!!}{2^{n+1}}\frac{\partial\quad}{\partial
t_{n+1}}\,+\,\sum_{j=0}^{\infty}\frac{(2j+2n+1)!!}{(2j-1)!!\,2^{n+1}}\,t_{j}\,\frac{\partial\quad}{\partial
t_{j+n}}\\\
+\,\frac{\lambda^{2}}{2}\,\sum_{j=0}^{n-1}\frac{(2j+1)!!(2n-2j-1)!!}{2^{n+1}}\,\frac{\partial^{2}\qquad\quad}{\partial
t_{j}\partial t_{n-j-1}},$
with commutator $[L_{n},L_{m}]\,=\,(n-m)L_{n+m}$.
###### Theorem 14.
[Mir07b, Theorem 6.1]. The Witten-Kontsevich conjecture: Virasoro constraints.
For $n\geq-1$, then $L_{n}(e^{\mathbf{F}})\,=\,0.$
###### Proof.
For an exponents multi index $\mathbf{k}=(k_{1},\dots,k_{n})$, the volume
recursion formula becomes the coefficient relation
$(2k_{1}+1)V_{g,n}(L)[\mathbf{k}]\,=\,\mathcal{A}_{g,n}^{con}(L)[\mathbf{k}]\,+\,\mathcal{A}_{g,n}^{dcon}(L)[\mathbf{k}]\,+\,\mathcal{B}_{g,n}(L)[\mathbf{k}].$
The leading coefficient relation takes the following explicit form (following
the labeling of boundaries, the punctures are labeled $1,\dots,n$)
$(2k_{1}+1)!!\langle\tau_{k_{1}}\cdots\tau_{k_{n}}\rangle\\\
=\frac{1}{2}\,\sum_{i+j=k_{1}-2}(2i+1)!!(2j+1)!!\sum_{I\subset\\{2,\dots,n\\}}\langle\tau_{i}\tau_{\mathbf{k}_{I}}\rangle\,\langle\tau_{j}\tau_{\mathbf{k}_{I^{c}}}\rangle\\\
+\,\frac{1}{2}\sum_{i+j=k_{1}-2}(2i+1)!!(2j+1)!!\langle\tau_{i}\tau_{j}\tau_{k_{2}}\cdots\tau_{k_{n}}\rangle\\\
+\,\sum_{j=2}^{n}\frac{(2k_{1}+2k_{j}-1)!!}{(2k_{j}-1)!!}\langle\tau_{k_{2}}\cdots\tau_{k_{1}+k_{j}-1}\cdots\tau_{k_{n}}\rangle.$
Rearranging the explicit relation provides that
$L_{k_{1}-1}(e^{\mathbf{F}})=0$. ∎
Mulase and Safnuk consider a generating function for the intersections of
combinations of the $\kappa_{1}$ and $\psi$ classes [MS08]
$\mathbf{G}(s,t_{0},t_{1},\dots)\,=\,\sum_{g}\,\langle e^{s\kappa_{1}+\sum
t_{j}\tau_{j}}\rangle_{g}\,=\,\sum_{g}\sum_{m,\\{d_{j}\\}}\langle\kappa_{1}^{m}\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\rangle_{g}\frac{s^{m}}{m!}\prod_{j=0}^{\infty}\frac{t_{j}^{d_{j}}}{d_{j}!},$
where again products, other than $3g-3+n$-products, are defined as zero.
Mulase and Safnuk use the volume recursion and rearrangement of terms to prove
the following.
###### Theorem 15.
[MS08, Thrm. 1.1] Virasoro constraints. For each $k\geq-1$, define
$\mathcal{V}_{k}=-\frac{1}{2}\sum_{i=0}^{\infty}(2(i+k)+3)!!\frac{(-2s)^{i}}{(2i+1)!}\frac{\partial\qquad}{\partial
t_{i+k+1}}+\frac{1}{2}\sum_{j=0}^{\infty}\frac{(2(j+k)+1)!!}{(2j-1)!!}t_{j}\frac{\partial\
}{\partial t_{j+k}}\\\ +\frac{1}{4}\sum_{\stackrel{{\scriptstyle
d_{1}+d_{2}=k-1}}{{d_{1},d_{2}\geq
0}}}(2d_{1}+1)!!(2d_{2}+1)!!\frac{\partial^{2}\quad}{\partial
t_{d_{1}}\partial
t_{d_{2}}}+\frac{\delta_{k,-1}t_{0}^{2}}{4}+\frac{\delta_{k,0}}{48},$
for the double factorial and Kronecker delta function $\delta_{*,*}$. Then
* •
the operators $\mathcal{V}_{k}$ satisfy the Virasoro commutator relations
$[\mathcal{V}_{n},\mathcal{V}_{m}]=(n-m)\mathcal{V}_{n+m}$;
* •
the generating function $\mathbf{G}$ satisfies
$\mathcal{V}_{k}(e^{\mathbf{G}})=0$ for $k\geq-1$.
The initial conditions and second system of equations uniquely determine the
generating function.
In a direct display that the intersection numbers for $\kappa_{1}$ and $\psi$
classes are equivalent intersection numbers for $\psi$ classes, Mulase and
Safnuk show that
$\mathbf{G}(s,t_{0},t_{1},t_{2},t_{3}\dots)\,=\,\mathbf{F}(t_{0},t_{1},t_{2}+\gamma_{2},t_{3}+\gamma_{3},\dots),$
where $\gamma_{j}=-(-s)^{j-1}/(2j+1)j!$ [MS08, Thrm. 1.2]. An explicit proof
of the relation also comes from a formula of Faber, expressing kappa classes
in terms of psi classes on moduli spaces for a greater number of punctures.
In his thesis [Do08], Norman Do presents a
$\mbox{generalized string equation}\qquad V_{g,n+1}(L,2\pi
i)\,=\,\sum_{k=1}^{n}\int L_{k}V_{g,n}(L)\,dL_{k},$
and
$\mbox{generalized dilaton equation}\qquad\frac{\partial V_{g,n+1}}{\partial
L_{n+1}}(L,2\pi i)\,=\,2\pi i(2g-2+n)V_{g,n}(L),$
where on the left hand side, the value $2\pi i$ is substituted for the
$(n+1)^{st}$ boundary length and $L=(L_{1},\dots,L_{n})$. By Theorem 4, the
second equation, for appropriate non negative multi indices
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$ and integers $m$, is
equivalent to the relations
$\int_{\overline{\mathcal{M}}_{g,n+1}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}\psi_{n+1}(\kappa_{1}-\psi_{n+1})^{m}=(2g-2+n)\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}\kappa_{1}^{m}.$
A proof of the generalized equations is based on the pullback relations for
psi and kappa classes, and general considerations for images of classes. In
particular for
$\pi:\overline{\mathcal{M}}_{g,n+1}\longrightarrow\overline{\mathcal{M}}_{g,n}$,
the morphism of forgetting the last puncture, then the classes
$\widetilde{\kappa}_{m},\ \widetilde{\psi}_{k}$ on
$\overline{\mathcal{M}}_{g,n+1}$ and $\kappa_{m},\ \psi_{k}$ on
$\overline{\mathcal{M}}_{g,n}$, satisfy
$\widetilde{\kappa}_{m}=\pi^{*}\kappa_{m}+\psi_{n+1}^{m}$ and
$\widetilde{\psi}_{k}=\pi^{*}\psi_{k},+D_{k}$ (for $D_{k}$ the divisor of the
$k^{th}$ puncture on the universal curve over $\overline{\mathcal{M}}_{g,n}$),
[HM98].
A second proof of the equations is based on exact formulas for the operations
in the volume recursion. For the generalized dilaton equation, consider the
following four operators acting on the ring
$\mathbb{C}[x^{2},y^{2},L_{1}^{2},\dots,L_{n}^{2},\dots]$,
$2\frac{\partial\ }{\partial
L_{1}}L_{1}[\cdot],\qquad\frac{\partial\quad}{\partial
L_{n+1}}[\cdot]\bigg{|}_{L_{n+1}=2\pi
i},\qquad\int_{0}^{\infty}\int_{0}^{\infty}xyH(x+y,L_{1})[\cdot]dxdy,\\\
\mbox{and}\quad\int_{0}^{\infty}x\big{(}H(x,L_{1}+L_{k})+H(x,L_{1}-L_{k})\big{)}[\cdot]dx.\qquad$
Formulas for the operators are developed. For the proof, the generalized
dilaton equation is written using the second operator, the volume recursion is
applied for the left hand side, operator formulas are applied, and terms are
gathered to give the right hand side. The generalized dilaton equation gives
WP volumes for the compact case, including the following examples
$\displaystyle V_{2,0}=\frac{43\pi^{6}}{2160},$ $\displaystyle V_{3,0}$
$\displaystyle=\frac{176557\pi^{12}}{1209600},$ $\displaystyle
V_{4,0}=\frac{1959225867017\pi^{18}}{493807104000}\qquad\mbox{and}\ $
$\displaystyle V_{5,0}$
$\displaystyle=\frac{84374265930915479\pi^{24}}{355541114880000}.$
We close the lecture by noting that there is extensive research on Witten’s
conjecture for the moduli space. The Kontsevich and Mirzakhani approaches are
analytic in nature. Okounkov and Pandharipande [OP09] transformed the question
to counting Hurwitz numbers, topological types of branched covers of the
sphere, and used a combinatorial approach to count factorizations of
permutations into transpositions. Their combinatorial approach gives
Kontsevich’s formula. There is an even greater body of research on
intersection numbers and relations on the moduli space [HM98].
## References
* [BP00] M. Boggi and M. Pikaart. Galois covers of moduli of curves. Compositio Math., 120(2):171–191, 2000.
* [BS85] J. Birman and C. Series. Simple curves have Hausdorff dimension one. Topology, 24(1):217–225, 1985.
* [Bus92] Peter Buser. Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1992.
* [CdS01] Ana Cannas da Silva. Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001.
* [DN09] Norman Do and Paul Norbury. Weil-Petersson volumes and cone surfaces. Geom. Dedicata, 141:93–107, 2009.
* [Do08] Norman Do. Intersection theory on moduli spaces of curves via hyperbolic geometry. PhD Thesis, The University of Melbourne, 2008.
* [Do11] Norman Do. Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes. Arxiv:math/1103.4674, 2011.
* [FO06] José Figueroa-O’Farrill. PG minicourse: BRST cohomology, Lecture 2: Symplectic reduction. Edinburgh Mathematical Physics Group, http://empg.maths.ed.ac.uk/Activities/BRST/Lect2.pdf, 2006.
* [Gru01] Samuel Grushevsky. An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces. Math. Ann., 321(1):1–13, 2001.
* [Har77] W. J. Harvey, editor. Discrete groups and automorphic functions. Academic Press [Harcourt Brace Jovanovich Publishers], London, 1977.
* [HM98] Joe Harris and Ian Morrison. Moduli of curves, volume 187 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998.
* [IT92] Y. Imayoshi and M. Taniguchi. An introduction to Teichmüller spaces. Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors.
* [Kon92] Maxim Kontsevich. Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys., 147(1):1–23, 1992.
* [Loo93] Eduard Looijenga. Intersection theory on Deligne-Mumford compactifications (after Witten and Kontsevich). Astérisque, (216):Exp. No. 768, 4, 187–212, 1993. Séminaire Bourbaki, Vol. 1992/93.
* [Loo94] Eduard Looijenga. Smooth Deligne-Mumford compactifications by means of Prym level structures. J. Algebraic Geom., 3(2):283–293, 1994.
* [McS98] Greg McShane. Simple geodesics and a series constant over Teichmuller space. Invent. Math., 132(3):607–632, 1998.
* [Mir07a] Maryam Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math., 167(1):179–222, 2007.
* [Mir07b] Maryam Mirzakhani. Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc., 20(1):1–23 (electronic), 2007.
* [Mir08] Maryam Mirzakhani. Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. of Math. (2), 168(1):97–125, 2008.
* [Mir10] Maryam Mirzakhani. Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus. Arxiv:math/1012.2167, 2010.
* [MS74] John W. Milnor and James D. Stasheff. Characteristic classes. Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.
* [MS08] Motohico Mulase and Brad Safnuk. Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy. Indian J. Math., 50(1):189–218, 2008.
* [MZ00] Yuri I. Manin and Peter Zograf. Invertible cohomological field theories and Weil-Petersson volumes. Ann. Inst. Fourier (Grenoble), 50(2):519–535, 2000.
* [OP09] A. Okounkov and R. Pandharipande. Gromov-Witten theory, Hurwitz numbers, and matrix models. In Algebraic geometry—Seattle 2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 325–414. Amer. Math. Soc., Providence, RI, 2009.
* [ST01] Georg Schumacher and Stefano Trapani. Estimates of Weil-Petersson volumes via effective divisors. Comm. Math. Phys., 222(1):1–7, 2001.
* [TWZ06] Ser Peow Tan, Yan Loi Wong, and Ying Zhang. Generalizations of McShane’s identity to hyperbolic cone-surfaces. J. Differential Geom., 72(1):73–112, 2006.
* [Wit91] Edward Witten. Two-dimensional gravity and intersection theory on moduli space. In Surveys in differential geometry (Cambridge, MA, 1990), pages 243–310. Lehigh Univ., Bethlehem, PA, 1991.
* [Wit92] Edward Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9(4):303–368, 1992.
* [Wlp82] Scott A. Wolpert. The Fenchel-Nielsen deformation. Ann. of Math. (2), 115(3):501–528, 1982.
* [Wlp83] Scott A. Wolpert. On the symplectic geometry of deformations of a hyperbolic surface. Ann. of Math. (2), 117(2):207–234, 1983.
* [Wlp90] Scott A. Wolpert. The hyperbolic metric and the geometry of the universal curve. J. Differential Geom., 31(2):417–472, 1990.
* [Wlp10] Scott A. Wolpert. Families of Riemann surfaces and Weil-Petersson Geometry, volume 113 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2010.
* [Zog08] Peter Zograf. On the large genus asymptotics of Weil-Petersson volumes. Arxiv:math/0812.0544, 2008.
|
arxiv-papers
| 2011-07-31T13:06:27 |
2024-09-04T02:49:21.128399
|
{
"license": "Public Domain",
"authors": "Scott A. Wolpert",
"submitter": "Scott Wolpert",
"url": "https://arxiv.org/abs/1108.0174"
}
|
1108.0179
|
# Effect of $\Sigma$-beam Asymmetry Data on Fits to Single Pion
Photoproduction off Neutron
I.I. Strakovsky111Electronic address: igor@gwu.edu, R.A. Arndt222Deceased,
W.J. Briscoe, M.W. Paris, R.L. Workman The George Washington University,
Washington, DC 20052, USA
###### Abstract
We investigate the influence of new GRAAL $\Sigma$-beam asymmetry measurements
on the neutron in multipole fits to the single-pion photoproduction database.
Results are compared to those found with the addition of a double-polarization
quantity associated with the sum rule.
###### pacs:
13.60.Le, 25.20.Lj, 13.88.+e, 11.80.Et
## I Introduction
Only with good data on both proton and neutron targets one can hope to
disentangle the isoscalar and isovector EM couplings of the various baryon
resonances, as well as the isospin properties of the non-resonant background
amplitudes. In particular, the simple quark model predicts several resonances
that couple much stronger to the neutron than to the proton. The lack of
$\gamma n\to\pi^{-}p$ and $\gamma n\to\pi^{0}n$ data does not allow us to be
as confident about the determination of neutron couplings relative to those of
the proton.
Some of the N∗ baryons, N(1675)5/2-, for instance, have stronger
electromagnetic couplings to the neutron than to the proton but parameters are
very uncertain. PDG PDG estimate for the A1/2 and A3/2 decay amplitudes of
the N(1720)3/2+ state are consistent with zero, while the recent SAID
determination Dugger gives small but non-vanishing values. The reason for the
disagreement between the PDG estimate for the A1/2 decay amplitude and the
recent SAID determination Dugger is also unclear. Other unresolved issues
relate to the N(1700)$3/2^{-}$ and second P11, N(1710)1/2+, that are not seen
in the recent GW $\pi$N partial-wave analysis (PWA) Ar06 , contrary to other
PWAs used by the Particle Data Group PDG .
New, high quality data on $\gamma n\to\pi^{-}p$ and $\gamma n\to\pi^{0}n$ are
needed to shed light on these issues, and the tagged-photon hall at GRAAL
offered a state-of-the-art facility to obtain such data. Here we report on an
analysis included novel $\Sigma$-measurements, covering incident photon
energies from threshold ($E_{\gamma}=707$ MeV) up to $E_{\gamma}=1500$ MeV.
The present measurement of $\Sigma$s for $\vec{\gamma}n\to\pi^{-}p$ Ma10 and
for $\vec{\gamma}n\to\pi^{0}n$ Sa09 is part of an extensive program at the
GRAAL to provide data of unrivaled quality on charged and neutral meson
photoproduction on the neutron, which includes polarized beam observable in
addition to the cross sections.
## II New GRAAL Measurements for $\Sigma$ on the Neutron
Figure 1: $\Sigma$-beam asymmetry for $\vec{\gamma}n\to\pi^{0}n$. Data from
GRAAL Collaboration Sa09 . Solid lines correspond to the SAID-MA09 solution
(GRAAL data included in the database) Ma10 . Dash-dotted (dashed) lines show
the SAID-SP09 Dugger (MAID2007 maid ) (GRAAL data excluded in the database).
To gauge the influence of new GRAAL data and their compatibility with previous
measurements, the GRAAL $\Sigma$s have been included in a number of fits using
the full SAID database for $\gamma N\to\pi N$ up to $E_{\gamma}=2.7$ GeV said
. The impact of new data on the SAID PWA can be understood from the comparison
of the new SAID fit MA09 Ma10 , which involves new GRAAL data, with the
previous SAID fit SP09 Dugger and MAID2007 results maid .
216 $\Sigma$s for $\pi^{0}n$ final state at Eγ=703– 1475 MeV and
$\theta$=53–164∘ with 99 $\Sigma$s for $\pi^{-}p$ final state at Eγ=753–1439
MeV and $\theta$=33–163∘ GRALL data have been added to the GW SAID database
said . We have to notice that this GRAAL $\pi^{0}n$ contribution doubled the
World database for this reaction. Our best fit MA09 Ma10 for $\pi^{0}n$ and
$\pi^{-}p$, reduced initial $\chi^{2}$/dp=223 and 89 (SP09 Dugger ) to 3.1 and
4.9, respectively. It shows, in particular, that previous $\pi^{-}p$
measurements provided a better constraint vs. $\pi^{0}n$ case.
In Figs. 1 and 2, we show the excitation functions for several production
angles. The number of the distributions shown is enough to illustrate the
quality of new GRAAL data, the main features of the $\gamma n\to\pi N$
dynamics at the measured energy range, and the impact of the present data on
PWAs. The most noticeable effect of the present data on the new MA09 is due to
very good measurements of the medium-angle (65–140∘) $\Sigma$s for W in the
range above 1650 MeV. Earlier, this angular region either had been measured
with worse accuracy or could only be reached by extrapolation.
Figure 2: $\Sigma$-beam asymmetry for $\vec{\gamma}n\to\pi^{-}p$. Data from
GRAAL Collaboration Ma10 . The notation of the PWA solutions is the same as in
Fig. 1
The difference between our MA09 and SP09 results for the neutron target is
visible specifically for $S_{11}$nE (Fig. 3). It is observed above
E${}_{\gamma}\sim$400 MeV while modified MAID2007 shown a significant changes
vs. MAID2007 Sa09 above 1 GeV (see Fig. 7 at Ref. Sa09 ).
Figure 3: The multipole amplitude $S_{11}$nE (${}_{n}E_{0+}^{1/2}$). (a) Re
and (b) Im parts show. The vertical arrows indicate $W_{R}$ (Breit-Wigner
mass) and the horizontal bars show the full and partial width $\Gamma$ and
$\Gamma_{\pi N}$ associated with the SAID solution SP06 for $\pi N$ Ar06 .
The difference between previous pion photoproduction and new GRAAL
measurements may result in significant changes in the neutron couplings.
## III Helicity-Dependent Photoabsorption Cross Sections on the Neutron
The amplitudes obtained in our analyses can be used to evaluate the single-
pion production component of several sum rules, in particular GDH, Baldin, and
forward spin polarizability Ar02 . In Table 1, we summarized our results for
the neutron target.
The running integrals are shown in Fig. 4. The evaluation of sum rules (GDH,
Baldin, and forward spin polarizability) for the neutron target and for a
single pion contribution exhibits convergence by 1 GeV. Agreement with Mainz
is good. Clearly, calculations above 450 MeV have to take into account
contributions beyond single-pion photoproduction.
Figure 4: Running (a) GDH, (b) Baldin, and (c) forward spin polarizability $\gamma_{0}$ integrals for the neutron target. The solid (dash-dotted) lines represent the SAID-MA09 (SAID-SP09) solution. Dashed lines show the MAID2007 predictions. Table 1: Comparison of the recent SAID-MA09, SAID-SP09, and MAID2007 calculations for the GDH, Baldin and the forward spin polarizability from threshold up to 2.5 GeV in W (for MAID up to 2 GeV) and displayed as MA09/SP09/MAID2007. Reaction | GDH | Baldin | $\gamma_{0}$
---|---|---|---
| ($\mu b$) | ($10^{-4}fm^{3}$) | ($10^{-4}fm^{4}$)
$\gamma n\to\pi^{-}p$ | 21/ 21/ 20 | 8.4/8.4/8.7 | 1.4/ 1.5/ 1.4
$\gamma n\to\pi^{0}n$ | $-$159/$-$157/$-$151 | 4.8/4.8/4.6 | $-$1.5/$-$1.5/$-$1.5
###### Acknowledgements.
This work was supported in part by the U. S. Department of Energy under Grant
DE-FG02-99ER41110.
## References
* (1) K. Nakamura, et al. (Particle Data Group), _J. Phys. G_ 37, 075021 (2010).
* (2) M. Dugger, et al. (CLAS Collaboration), _Phys. Rev. C_ 79, 065206 (2009).
* (3) R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, _Phys. Rev. C_ 74, 0605082 (2006).
* (4) G. Mandaglio, et al. (GRAAL Collaboration), _Phys. Rev. C_ 82, 045209 (2010).
* (5) R. Di Salvo, et al. (GRAAL Collaboration), _Eur. Phys. J. A_ 42, 151 (2009).
* (6) D. Drechsel, S.S. Kamalov, and L. Tiator, _Eur. Phys. J. A_ 34, 69 (2007).
* (7) The SAID website contains data and fits for this and a number of other medium-energy reactions: http://gwdac.phys.gwu.edu.
* (8) R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, _Phys. Rev. C_ 66, 055213 (2002).
|
arxiv-papers
| 2011-07-31T14:57:21 |
2024-09-04T02:49:21.141940
|
{
"license": "Public Domain",
"authors": "I.I. Strakovsky, R.A. Arndt, W.J. Briscoe, M.W. Paris, R.L. Workman\n (GWU)",
"submitter": "Igor Strakovsky",
"url": "https://arxiv.org/abs/1108.0179"
}
|
1108.0185
|
Orthogonalizing Penalized Regression
Shifeng Xiong1, Bin Dai2, and Peter Z. G. Qian222Corresponding author: Peter
Z. G. Qian. Email: peterq@stat.wisc.edu
1 Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing 100190
2 Department of Statistics
University of Wisconsin-Madison, Madison, WI 53706
Abstract
Since the penalized likelihood function of the smoothly clipped absolute
deviation (SCAD) penalty is highly non-linear and has many local optima,
finding a local solution to achieve the so-called oracle property is an open
problem. We propose an iterative algorithm, called the OEM algorithm, to fill
this gap. The development of the algorithm draws direct impetus from a
missing-data problem arising in design of experiments with an orthogonal
complete matrix. In each iteration, the algorithm imputes the missing data
based on the current estimates of the parameters and updates a closed-form
solution associated with the complete data. By introducing a procedure called
active orthogonization, we make the algorithm broadly applicable to problems
with arbitrary regression matrices. In addition to the SCAD penalty, the
proposed algorithm works for other penalties like the MCP, lasso and
nonnegative garrote. Convergence and convergence rate of the algorithm are
examined. The algorithm has several unique theoretical properties. For the
SCAD and MCP penalties, an OEM sequence can achieve the oracle property after
sufficient iterations. For various penalties, an OEM sequence converges to a
point having grouping coherence for fully aliased regression matrices. For
computing the ordinary least squares estimator with a singular regression
matrix, an OEM sequence converges to the Moore-Penrose generalized inverse-
based least squares estimator.
KEY WORDS: Design of experiments; MCP; Missing data; Optimization; Oracle
property; Orthogonal design; SCAD; The EM algorithm; The Lasso.
## 1 INTRODUCTION
Fan and Li (2001) proposed the smoothly clipped absolute deviation (SCAD)
penalty to achieve simultaneous estimation and variable selection. Consider a
linear model
$\mathbf{Y}=\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}+\mbox{\boldmath$\upvarepsilon$\unboldmath},$
(1)
where $\boldsymbol{X}=(x_{ij})$ is the $n\times p$ regression matrix,
$\mathbf{Y}\in{\mathbb{R}}^{n}$ is the response vector,
$\mbox{\boldmath$\upbeta$\unboldmath}=(\beta_{1},\ldots,\beta_{p})^{\prime}$
is the vector of regression coefficients and the distribution of the vector of
random error
$\mbox{\boldmath$\upvarepsilon$\unboldmath}=(\varepsilon_{1},\ldots,\varepsilon_{n})^{\prime}$
is $N(\mathbf{0}_{n},\ \sigma^{2}\boldsymbol{I}_{n})$ with $\mathbf{0}_{n}$
being the $n$th zero vector and $\boldsymbol{I}_{n}$ being the $n\times n$
identity matrix. Throughout, let $\|\cdot\|$ denote the Euclidean norm. A
_regularized least squares estimator_ of $\upbeta$ with this penalty is given
by solving
$\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}}\left[\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\right],$
(2)
where for $\theta>0$,
$P_{\lambda}^{\prime}(\theta)=\lambda
I(\theta\leqslant\lambda)+(a\lambda-\theta)_{+}I(\theta>\lambda)/(a-1),$ (3)
$a>2$, $\lambda>0$ is the tuning parameter and $I$ is the indicator function.
In order to apply the penalty $P_{\lambda}$ equally on all the variables,
$\boldsymbol{X}$ can be standardized so that
$\sum_{i=1}^{n}x_{ij}^{2}=1,\;\mbox{for}\;j=1,\ldots,p.$ (4)
Both theory and computation of the estimator in (2) have been actively
studied. On the theoretical side, Fan and Li (2001) introduced an important
concept, called the _oracle_ property. An estimator of $\upbeta$ having this
property can not only select the correct submodel asymptotically, but also
estimate the nonzero coefficients as efficiently as if the correct submodel
were known in advance. On the computational side, existing algorithms for
solving this optimization problem include local quadratic approximation (Fan
and Li 2001; Hunter and Li 2005), local linear approximation (Zou and Li
2008), the coordinate descent algorithm (Tseng 2001; Tseng and Yun 2009;
Breheny and Huang 2010; Mazumder, Friedman, and Hastie 2010) and the
minimization by iterative soft thresholding (MIST) algorithm (Schifano,
Strawderman, and Wells 2010), among others.
Departing from the existing work, we study the SCAD penalty from a new
perspective, targeting on the _interface_ between theory and computing. Fan
and Li (2001) proved that there exists a local solution to (2) with the oracle
property. From the optimization viewpoint, (2) can have many local minima (Huo
and Chen 2010) and it is very challenging to find one of them to achieve the
oracle property. To the best of our knowledge, no theoretical results are
available to show that any existing algorithm can provide such a local
minimum. We propose an iterative algorithm, called orthogonalizing EM (OEM),
to fill this gap. We will show in Section 4 that the OEM solution to (2) can
indeed achieve the oracle property under regularity conditions. OEM draws its
direct impetus from a missing data problem with a complete orthogonal design
arising in design of experiments. Throughout, a matrix is orthogonal if its
columns are orthogonal. In each iteration, the algorithm imputes the missing
data based on the current estimate of $\upbeta$ and updates a closed-form
solution to (2) associated with the complete data. Much beyond this orthogonal
design formulation, the OEM algorithm applies to general data structures by
_actively orthogonalizing_ arbitrary regression matrices.
Though the inspiration of the OEM algorithm stems from the SCAD penalty, it,
not surprisingly, works for the general penalized regression problem:
$\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}\in\Theta}\left[\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)\right],$
(5)
where ${{\mbox{\boldmath$\upbeta$\unboldmath}}\in\Theta}$, $\Theta$ is a
subset of ${\mathbb{R}}^{p}$ and $\lambda$ is the vector of tuning parameters.
Besides the SCAD penalty, choices for
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ include the ridge regression
(Hoerl and Kennard 1970), the nonnegative garrote (Breiman 1995), the lasso
(Tibshirani 1996) and the MCP (Zhang 2010). Algorithms for solving the problem
in (5) include those developed in Fu (1998), Grandvalet (1998), Osborne,
Presnell, and Turlach (2000), the LARS algorithm introduced in Efron, Hastie,
Johnstone, and Tibshirani (2004) and the coordinate descent algorithm (Tseng
2001; Friedman, Hastie, Hofling and Tibshirani 2007; Wu and Lange 2008; Tseng
and Yun 2009), and are available in R packages like lars (Hastie and Efron
2011), glmnet (Friedman, Hastie, and Tibshirani 2011) and scout (Witten and
Tibshirani 2011).
In addition to achieving the oracle property for the SCAD and MCP penalties,
the OEM algorithm has several other unique theoretical features. 1. _Having
grouping coherence_ : An estimator
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}$ of $\upbeta$ in (1) is said to
have grouping coherence if it has the same coefficient for full aliased
columns in $\boldsymbol{X}$ (Zou and Hastie 2005). For the lasso, SCAD and
MCP, an OEM sequence converges to a point having grouping coherence. 2.
_Convergence in singular case_ : When $\boldsymbol{X}$ in (1) is singular, the
ordinary least squares estimator given by (5) without any penalty is not
unique. For this singular case, an OEM solution, or essentially the Healy-
Westmacott estimator (Healy and Westmacott 1956), converges to the Moore-
Penrose generalized inverse-based least squares estimator.
The remainder of the article will unfold as follows. Section 2 derives the OEM
algorithm for a missing data problem with a complete orthogonal design.
Section 3 significantly broadens the applicability of the algorithm by
introducing an idea for actively expanding any regression matrix to an
orthogonal matrix. Section 4 establishes the oracle property of the OEM
solution for the SCAD and MCP. Section 5 provides convergence properties of
the OEM algorithm. Section 6 shows that for a regression matrix with full
aliased columns, an OEM sequence for the lasso, SCAD or MCP converges to a
solution with grouping coherency and illustrates how to use the OEM algorithm
to compute the ordinary least squares estimator for a singular regression
matrix. Section 7 provides some discussion.
## 2 THE OEM ALGORITHM AS A PENALIZED HEALY-WESTMACOTT PROCEDURE
Orthogonal designs are widely used in science and engineering. Such designs
have been intensively studied in different branches of statistics including
design of experiments, information theory (MacWilliams and Sloane 1977), liner
models, sampling survey and computer experiments. Popular classes of
orthogonal designs include orthogonal arrays (Hedayat, Sloane, and Stufken
1999), orthogonal main-effect plans (Addelman 1962; Wu and Hamada 2009) and
orthogonal Latin hypercube designs (Ye 1998; Steinberg and Lin 2006; Bingham,
Sitter, and Tang 2009; Lin, Mukerjee, and Tang 2009; Pang, Liu, and Lin, 2009;
Sun, Liu, and Lin 2009; Lin, Bingham, Sitter, and Tang 2010) from the computer
experiments literature (Santner, Williams, and Notz 2003; Fang, Li, and
Sudjianto 2005). In this section, we motivate the OEM algorithm by using a
missing data problem with an orthogonal complete design. Suppose that the
matrix $\boldsymbol{X}$ in (1) for this problem is a submatrix of an $m\times
p$ _complete_ orthogonal matrix
$\boldsymbol{X}_{c}=(\boldsymbol{X}^{\prime}\
\boldsymbol{\Delta}^{\prime})^{\prime},$ (6)
where $\boldsymbol{\Delta}$ is the $(m-n)\times p$ _missing_ matrix. Let
$\mathbf{Y}_{c}=(\mathbf{Y}^{\prime},\
\mathbf{Y}_{\mathrm{miss}}^{\prime})^{\prime}$ (7)
define the vector of complete observations with $\mathbf{Y}_{\mathrm{miss}}$
corresponding to $\boldsymbol{\Delta}$. If $\mathbf{Y}_{\mathrm{miss}}$ were
observable, then the ordinary least square estimator of $\upbeta$ based on the
complete data ($\boldsymbol{X}_{c}$, $\mathbf{Y}_{c}$) has a closed form as
$\boldsymbol{X}_{c}$ is orthogonal. In light of this fact, Healy and
Westmacott (1956) proposed an iterative procedure to compute the ordinary
least squares estimator $\mbox{\boldmath$\upbeta$\unboldmath}_{OLS}$ of
$\upbeta$. In each iteration, their procedure imputes the values of
$\mathbf{Y}_{\mathrm{miss}}$ and updates the closed-form ordinary least
squares estimator associated with the complete data. The OEM algorithm follows
the same idea but solves (2) with the SCAD penalty. If
$\mathbf{Y}_{\mathrm{miss}}$ were observable, then $\boldsymbol{X}$ in (2) and
$\mathbf{Y}$ can be replaced by $\boldsymbol{X}_{c}$ and $\mathbf{Y}_{c}$,
yielding a closed-form solution to (2). Much beyond this orthogonal design
formulation, we will significantly broaden the applicability of the algorithm
in Section 3 by introducing an idea, called active orthogonalization, to
actively expand any regression matrix into an orthogonal matrix.
Define
$\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}.$ (8)
Let $(d_{1},\ldots,d_{p})$ denote the diagonal elements of
$\boldsymbol{X}_{c}^{\prime}\boldsymbol{X}_{c}$. The OEM algorithm for solving
the optimization problem in (2) proceeds as follows. Let
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$ be an initial estimate of
$\upbeta$. For $k=0,1,\ldots$, impute $\mathbf{Y}_{\mathrm{miss}}$ as
$\mathbf{Y}_{I}=\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$,
let $\mathbf{Y}_{c}=(\mathbf{Y}^{\prime},\ \mathbf{Y}_{I}^{\prime})^{\prime}$,
and solve
$\displaystyle\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\arg\\!\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}}\left[\|\mathbf{Y}_{c}-\boldsymbol{X}_{c}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\right]$
(9)
until $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ converges. Letting
$\mathbf{u}=(u_{1},\ldots,u_{p})^{\prime}=\boldsymbol{X}^{\prime}\mathbf{Y}+\boldsymbol{A}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)},$
(10)
(9) becomes
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\arg\\!\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}}\left[\sum_{j=1}^{p}(d_{j}\beta_{j}^{2}-2u_{j}\beta_{j})+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\right],$
(11)
which is _separable_ in the dimensions of $\upbeta$. If $\boldsymbol{X}$ in
(1) is standardized as in (4) with $d_{j}\geqslant 1$ for all $j$, (11) has a
closed-form
$\beta_{j}^{(k+1)}=\left\\{\begin{array}[]{ll}{\mathrm{sign}}(u_{j})\big{(}|u_{j}|-\lambda\big{)}_{+}/d_{j},&\text{when}\
|u_{j}|\leqslant(d_{j}+1)\lambda,\\\
{\mathrm{sign}}(u_{j})\big{[}(a-1)|u_{j}|-a\lambda\big{]}/\big{[}(a-1)d_{j}-1\big{]},&\text{when}\
(d_{j}+1)\lambda<|u_{j}|\leqslant a\lambda d_{j},\\\ u_{j}/d_{j},&\text{when}\
|u_{j}|>a\lambda d_{j}.\end{array}\right.$ (12)
As pointed out in Dempster, Laird, and Rubin (1977) that the Healy-Westmacott
procedure is essentially an EM algorithm, OEM is an EM algorithm as well. The
complete data $\mathbf{Y}_{c}=(\mathbf{Y}^{\prime},\
\mathbf{Y}_{\mathrm{miss}}^{\prime})^{\prime}$ in (7) follow a regression
model
$\mathbf{Y}_{c}=\boldsymbol{X}_{c}\mbox{\boldmath$\upbeta$\unboldmath}+\mbox{\boldmath$\upvarepsilon$\unboldmath}_{c}$,
where $\mbox{\boldmath$\upvarepsilon$\unboldmath}_{c}$ is from
$N(\mathbf{0}_{m},\ \boldsymbol{I}_{m})$. Let
$\mbox{\boldmath$\upbeta$\unboldmath}_{\mathrm{SCAD}}$ be a solution to (2),
where
$\mbox{\boldmath$\upbeta$\unboldmath}_{\mathrm{SCAD}}=\arg\\!\max_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}}L(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mathbf{Y})$
and the penalized likelihood function
$L(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mathbf{Y})$ is
$(2\pi)^{-n/2}\exp\left(-\frac{1}{2}\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}\right)\exp\left[-\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\right].$
Given $\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$, the E-step of the OEM
algorithm for the SCAD is
$\displaystyle
E\big{[}\log\\{L(\mbox{\boldmath$\upbeta$\unboldmath}|\mathbf{Y}_{c})\\}\mid\mathbf{Y},\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\big{]}$
$\displaystyle=$
$\displaystyle-C\big{\\{}\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+E\big{(}\|\mathbf{Y}_{\mathrm{miss}}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\big{)}+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\big{\\}}$
$\displaystyle=$
$\displaystyle-C\big{\\{}n+\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+\|\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|)\big{\\}}$
for some constant $C>0$. Define
$Q_{\mathrm{SCAD}}(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})=\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+\|\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+2\sum_{j=1}^{p}P_{\lambda}(|\beta_{j}|).$
(13)
The M-step of the OEM algorithm for the SCAD is
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\arg\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}}Q_{\mathrm{SCAD}}(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}),$
which is equivalent to (11).
For the general penalized regression problem in (5), the M-step of the OEM
algorithm becomes
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\arg\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}}\in\Theta}Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}),$
(14)
where $Q$ replaces $Q_{\mathrm{SCAD}}$ in (13) for the corresponding penalty
function. If $\Theta$ and $P$ in (5) are _decomposable_ as
$\Theta=\prod_{j=1}^{p}\Theta_{j}$ and
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)=\sum_{j=1}^{p}P_{j}(\beta_{j};\lambda)$,
similarly to (11), (14) reduces to $p$ one-dimensional problems
$\beta_{j}^{(k+1)}=\arg\\!\\!\min_{\beta_{j}\in\Theta_{j}}\big{[}d_{j}\beta_{j}^{2}-2u_{j}\beta_{j}+P_{j}(\beta_{j};\lambda)\big{]},\
\text{for}\ j=1,\ldots,p,$ (15)
with $\mathbf{u}=(u_{1},\ldots,u_{p})^{\prime}$ defined in the same way as in
(10). This shortcut applies to the following penalties:
1\. The lasso (Tibshirani 1996), where $\Theta_{j}={\mathbb{R}}$,
$P_{j}(\beta_{j};\lambda)=2\lambda|\beta_{j}|,$ (16)
and (15) becomes
$\beta_{j}^{(k+1)}={\mathrm{sign}}(u_{j})\left(\frac{|u_{j}|-\lambda}{d_{j}}\right)_{+}.$
(17)
Here, for $a\in\mathbb{R}$, $(a)_{+}$ denotes $\max\\{a,0\\}$.
2\. The nonnegative garrote (Breiman 1995), where
$\Theta_{j}=\\{x:x\hat{\beta}_{j}\geqslant 0\\}$,
$P_{j}(\beta_{j};\lambda)=2\lambda\beta_{j}/\hat{\beta}_{j}$,
$\hat{\beta}_{j}$ is the ordinary least squares estimator of $\beta_{j}$, and
(15) becomes
$\beta_{j}^{(k+1)}=\left(\frac{u_{j}\hat{\beta}_{j}-\lambda}{d_{j}\hat{\beta}_{j}^{2}}\right)_{+}\hat{\beta}_{j}.$
3\. The elastic-net (Zou and Hastie 2005), where $\Theta_{j}={\mathbb{R}}$,
$P_{j}(\beta_{j};\lambda)=2\lambda_{1}|\beta_{j}|+\lambda_{2}\beta_{j}^{2}.$
(18)
and (15) becomes
$\beta_{j}^{(k+1)}={\mathrm{sign}}(u_{j})\left(\frac{|u_{j}|-\lambda_{1}}{d_{j}+\lambda_{2}}\right)_{+}.$
(19)
5\. The MCP (Zhang 2010), where $\Theta_{j}={\mathbb{R}}$,
$P_{j}(\beta_{j};\lambda)=2P_{\lambda}(|\beta_{j}|)$, and
$P_{\lambda}^{\prime}(\theta)=(\lambda-\theta/a)I(\theta\leqslant a\lambda)$
(20)
with $a>1$ and $\theta>0$. If $\boldsymbol{X}$ in (1) is standardized as in
(4) with $d_{j}\geqslant 1$ for all $j$, (15) becomes
$\beta_{j}^{(k+1)}=\left\\{\begin{array}[]{ll}{\mathrm{sign}}(u_{j})a\big{(}|u_{j}|-\lambda\big{)}_{+}/(ad_{j}-1),&\text{when}\
|u_{j}|\leqslant a\lambda d_{j},\\\ u_{j}/d_{j},&\text{when}\ |u_{j}|>a\lambda
d_{j}\big{.}\end{array}\right.$ (21)
6\. The “Berhu” penalty (Owen 2006), where $\Theta_{j}={\mathbb{R}}$,
$P_{j}(\beta_{j};\lambda)=2\lambda\big{\\{}|\beta_{j}|I(|\beta_{j}|<\delta)+(\beta_{j}^{2}+\delta^{2})I(|\beta_{j}|\geqslant\delta)/(2\delta)\big{\\}}$
for some $\delta>0$, and (15) becomes
$\beta_{j}^{(k+1)}=\left\\{\begin{array}[]{ll}{\mathrm{sign}}(u_{j})\big{(}|u_{j}|-\lambda\big{)}_{+}/d_{j},&\text{when}\
|u_{j}|<\lambda+d_{j}\delta,\\\
u_{j}\delta/(\lambda+d_{j}\delta),&\text{when}\
|u_{j}|\geqslant\lambda+d_{j}\delta.\end{array}\right.$
Obviously, if the penalty on $\upbeta$ disappears, the OEM algorithm reduces
to the Healy-Westmacott procedure. Quite interestingly, Theorem 6 in Section 5
shows that, for the same $\boldsymbol{X}$ and $\mathbf{Y}$ in (1), the OEM
algorithm for the elastic-net and lasso numerically converges faster than the
Healy-Westmacott procedure.
###### Example 1.
For the model in (1), let the complete matrix $\boldsymbol{X}_{c}$ be a
fractional factorial design from Xu (2009) with 4096 runs in 30 factors.
Clearly, $\boldsymbol{X}_{c}$ is an orthogonal design. Let $\boldsymbol{X}$ in
(1) be the submatrix of $\boldsymbol{X}_{c}$ consisting of the first 3000 rows
and let $\mathbf{Y}$ be generated with $\sigma=1$ and
$\beta_{j}=(-1)^{j}\exp\big{[}-2(j-1)/20\big{]}\ \text{for}\ j=1,\ldots,p.$
(22)
Here, $p=30$ and $n=3000$. Assume the response values corresponding to the
last 1096 rows of $\boldsymbol{X}_{c}$ are missing. We used the OEM algorithm
to solve the optimization problem in (2) with an initial value
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}=\mathbf{0}$ and a criterion to
stop when relative changes in all coefficients are less than $10^{-6}$. For
$\lambda=1$ and $a=3.7$ in (3), Figure 1 plots values of the objective
function in (2) of the OEM sequence against iteration numbers, where the
algorithm converges at iteration 13.
[0.5]
Figure 1. Values of the objective function of an OEM sequence for the SCAD
against iterations for Example 1.
## 3 THE GENERAL FORMULATION WITH ACTIVE ORTHOGONALIZATION
The OEM algorithm in Section 2 was derived for a missing-data problem where
$\boldsymbol{X}$ in (1) is imbedded in a _pre-specified_ orthogonal matrix. We
drop this assumption in this section and further develop the algorithm for
general data structures by introducing a procedure to _actively_ expand an
arbitrary matrix to an orthogonal matrix. The general idea of augmenting extra
data has been used for EM problems before. For example, for a covariance
estimation problem in Rubin and Szatrowski (1982), extra data are added
elaborately to make the maximum likelihood estimator of the expanded patterned
covariance matrices have an explicit form. To facilitate the use of the OEM
algorithm in Section 2, the contribution here is to develop a scheme to
_orthogonalize_ the matrix $\boldsymbol{X}$ with an arbitrary structure.
Take $\boldsymbol{S}$ to be a $p\times p$ diagonal matrix with non-zero
diagonal elements $s_{1},\ldots,s_{p}$. Define
$\boldsymbol{Z}=\boldsymbol{X}\boldsymbol{S}^{-1}.$ (23)
The eigenvalue decomposition of $\boldsymbol{Z}^{\prime}\boldsymbol{Z}$
(Wilkinson 1965) is
$\boldsymbol{V}^{\prime}\boldsymbol{\Gamma}\boldsymbol{V},$
where $\boldsymbol{V}$ is an orthogonal matrix and $\boldsymbol{\Gamma}$ is a
diagonal matrix whose diagonal elements,
$\gamma_{1}\geqslant\cdots\geqslant\gamma_{p}$, are the nonnegative
eigenvalues of $\boldsymbol{Z}^{\prime}\boldsymbol{Z}$. Let
$t=\\#\\{j:\ \gamma_{j}=\gamma_{1},\ j=1,\ldots,p\\}$ (24)
denote the number of the $\gamma_{j}$ equal to $\gamma_{1}$. For example, if
$\gamma_{1}=\gamma_{2}$ and $\gamma_{1}>\gamma_{j}$ for $j=3,\ldots,p$, then
$t=2$. Define
$\boldsymbol{B}={\mathrm{diag}}(\gamma_{1}-\gamma_{t+1},\ldots,\gamma_{1}-\gamma_{p})$
(25)
and
$\boldsymbol{\Delta}=\boldsymbol{B}^{1/2}\boldsymbol{V}_{1}\boldsymbol{S},$
(26)
where $\boldsymbol{V}_{1}$ is the submatrix of $\boldsymbol{V}$ consisting of
the last $p-t$ rows. Let $\boldsymbol{X}_{c}$ be the augmented matrix of
$\boldsymbol{\Delta}$ and $\boldsymbol{X}$.
###### Lemma 1.
The matrix $\boldsymbol{X}_{c}$ constructed above is orthogonal.
###### Proof.
Note that
$\boldsymbol{X}_{c}^{\prime}\boldsymbol{X}_{c}=\boldsymbol{X}^{\prime}\boldsymbol{X}+\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta},$
which, by plugging (25) and (26), becomes
$\boldsymbol{S}[\boldsymbol{Z}^{\prime}\boldsymbol{Z}+\boldsymbol{V}^{\prime}(\gamma_{1}\boldsymbol{I}_{p}-\boldsymbol{\Gamma})\boldsymbol{V}]\boldsymbol{S}=\gamma_{1}\boldsymbol{S}^{2}.$
(27)
Now, because
$\gamma_{1}\boldsymbol{I}_{p}-\boldsymbol{\Gamma}=\left(\begin{array}[]{cc}\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{B}\end{array}\right),$
(27) is orthogonal, which completes the proof. ∎
The underlying geometry of the active orthogolization in Lemma 1 can be
described as follows. For a vector $\mathbf{x}\in{\mathbb{R}}^{m}$, let
$P_{\omega}\mathbf{x}$ denote its projection onto a subspace $\omega$ of
${\mathbb{R}}^{m}$. This lemma implies that for the column vectors of
$\boldsymbol{X}$ in (1),
$\mathbf{x}_{1},\ldots,\mathbf{x}_{p}\in{\mathbb{R}}^{n}$, there exists a set
of mutually orthogonal vectors
$\mathbf{x}_{c1},\ldots,\mathbf{x}_{cp}\in{\mathbb{R}}^{n+p-t}$, essentially
the column vectors of $\boldsymbol{X}_{c}$ in (6), satisfy the condition that
$P_{{\mathbb{R}}^{n}}\mathbf{x}_{ci}=\mathbf{x}_{i}$, for $j=1,\ldots,p$.
Proposition 1 makes this precise.
###### Proposition 1.
Let $\omega$ be an $n$-dimensional subspace of ${\mathbb{R}}^{m}$ with
$n\leqslant m$. If $p\leqslant m-n+1$, then for any $p$ vectors
$\mathbf{x}_{1},\ldots,\mathbf{x}_{p}\in\omega$, there exist $p$ vectors
$\mathbf{x}_{c1},\ldots,\mathbf{x}_{cp}\in{\mathbb{R}}^{m}$ such that
$P_{\omega}\mathbf{x}_{ci}=\mathbf{x}_{i}$ for $j=1,\ldots,p$ and
$\mathbf{x}_{ci}^{\prime}\mathbf{x}_{cj}=0$ for $i\neq j$.
Figure 2 expands two vectors $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ in
$\mathbb{R}^{2}$ to two orthogonal vectors $\mathbf{x}_{c1}$ and
$\mathbf{x}_{c2}$ in $\mathbb{R}^{3}$.
[0.8]
Figure 2. Expand two two-dimensional vectors $\mathbf{x}_{1}$ and
$\mathbf{x}_{2}$ to two three-dimensional vectors $\mathbf{x}_{c1}$ and
$\mathbf{x}_{c2}$ with $\mathbf{x}_{c1}^{\prime}\mathbf{x}_{c2}=0$.
Now, if $\boldsymbol{X}_{c}$ from Lemma 1 is treated as the complete matrix
defined in (6), the OEM algorithm in Section 2 follows through immediately.
When using the OEM algorithm to solve (5), in (10) instead of computing
$\boldsymbol{\Delta}$ in (26), one may compute
$\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$ and the
diagonal entries $d_{1},\ldots,d_{p}$ of
$\boldsymbol{X}_{c}^{\prime}\boldsymbol{X}_{c}$. Note that
$\boldsymbol{A}=\gamma_{1}\boldsymbol{S}^{2}-\boldsymbol{X}^{\prime}\boldsymbol{X}$
(28)
and
$d_{j}=\gamma_{1}s_{j}^{2}\ \text{for}\ j=1,\ldots,p,$ (29)
where $\boldsymbol{S}$ and $\boldsymbol{Z}$ are defined in (23) and
$\gamma_{1}$ is the largest eigenvalue of
$\boldsymbol{Z}^{\prime}\boldsymbol{Z}=\boldsymbol{S}^{-1}\boldsymbol{X}^{\prime}\boldsymbol{X}\boldsymbol{S}^{-1}$.
One way to compute $\gamma_{1}$ is to use the power method (Wilkinson 1965)
described below. Given a nonzero initial vector
$\mathbf{a}^{(0)}\in{\mathbb{R}}^{p}$, let
$\gamma_{1}^{(0)}=\|\mathbf{a}^{(0)}\|$. For $k=0,1,...$, compute
$\mathbf{a}^{(k+1)}=\boldsymbol{X}^{\prime}\boldsymbol{X}\mathbf{a}^{(k)}/\gamma_{1}^{(k)}$
and $\gamma_{1}^{(k+1)}=\|\mathbf{a}^{(k+1)}\|$ until convergence. If
$\mathbf{a}^{(0)}$ is not an eigenvector of any $\gamma_{j}$ that does not
equal $\gamma_{1}$, then $\gamma_{1}^{(k)}$ converges to $\gamma_{1}$. For $t$
defined in (24), the convergence rate is linear (Watkins 2002) specified by
$\lim_{k\rightarrow\infty}\frac{\|\gamma_{1}^{(k+1)}-\gamma_{1}\|}{\|\gamma_{1}^{(k)}-\gamma_{1}\|}=\frac{\gamma_{t+1}}{\gamma_{1}}.$
An easy way to make
$\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$ in (28)
positive definite is to replace $\boldsymbol{B}$ in (25) by
$\boldsymbol{B}={\mathrm{diag}}(d-\gamma_{t+1},\ldots,d-\gamma_{p})$
with $d\geqslant\gamma_{1}$, which changes (28) and (29) to
$\boldsymbol{A}=d\boldsymbol{S}^{2}-\boldsymbol{X}^{\prime}\boldsymbol{X}$
(30)
and
$d_{j}=ds_{j}^{2},\ \text{for}\ j=1,\ldots,p,$ (31)
respectively. If $d>\gamma_{1}$, then
$\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$ is positive
definite.
###### Remark 1.
The matrix $\boldsymbol{S}$ in (23) can be chosen flexibly. One possibility is
to use $\boldsymbol{S}=\boldsymbol{I}_{p}$ so that
$\boldsymbol{X}^{\prime}\boldsymbol{X}+\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}=d\boldsymbol{I}_{p}$
(32)
with $d\geqslant\gamma_{1}$, and $\boldsymbol{X}_{c}/\sqrt{d}$ is standardized
as in (4).
###### Example 2.
Suppose that $\boldsymbol{X}$ in (1) is orthogonal. Take
$\boldsymbol{S}={\mathrm{diag}}\left[(\sum_{i=1}^{n}x_{i1}^{2})^{1/2},\ldots,(\sum_{i=1}^{n}x_{ip}^{2})^{1/2}\right].$
(33)
Since $t=p$, $\boldsymbol{\Delta}$ in (26) is empty. This result indicates the
active orthogonalization procedure will not overshoot: if $\boldsymbol{X}$ is
orthogonal already, it adds no row.
###### Example 3.
Let
$\boldsymbol{X}=\left(\begin{array}[]{rrrrr}0&&0&&3/2\\\ -4/3&&-2/3&&1/6\\\
2/3&&4/3&&1/6\\\ -2/3&&2/3&&-7/6\end{array}\right).$
If $\boldsymbol{S}=\boldsymbol{I}_{3}$, (26) gives
$\boldsymbol{\Delta}=(-2/\sqrt{3},\ 2/\sqrt{3},\ 1/\sqrt{3}).$
###### Example 4.
Consider a two-level design in three factors, $A$, $B$ and $C$:
$\left(\begin{array}[]{rrr}-1&\ -1&\ -1\\\ -1&\ 1&\ 1\\\ 1&\ -1&\ 1\\\ 1&\ 1&\
-1\end{array}\right).$
The regression matrix including all main effects and two-way interactions is
$\boldsymbol{X}=\left(\begin{array}[]{rrrrrr}-1&\ -1&\ -1&\ 1&\ 1&\ 1\\\ -1&\
1&\ 1&-1&-1&1\\\ 1&\ -1&\ 1&\ -1&\ 1&\ -1\\\ 1&\ 1&\ -1&\ 1&\ -1&\
-1\end{array}\right),$
where the last three columns for the interactions are fully aliased with the
first three columns for the main effects. For
$\boldsymbol{S}=\boldsymbol{I}_{3}$, (26) gives
$\boldsymbol{\Delta}=\left(\begin{array}[]{rrrrrr}0&\ -2&\ 0&\ 0&\ -2&\ 0\\\
0&\ 0&\ -2&\ -2&\ 0&\ 0\\\ -2&\ 0&\ 0&\ 0&\ 0&\ -2\end{array}\right).$
The elements in $\boldsymbol{\Delta}$ are chosen flexibly, not restricted to
$\pm 1$.
###### Example 5.
Consider a $1000\times 10$ random matrix $\boldsymbol{X}=(x_{ij})$ with
entries independently drawn from the uniform distribution on $[0,1)$. Using
$\boldsymbol{S}$ in (33), (26) gives
$\boldsymbol{\Delta}=\left(\begin{array}[]{rrrrrrrrrr}-7.99&16.06&-6.39&-18.26&12.91&-8.67&7.56&34.08&-17.04&-11.81\\\
26.83&-12.09&7.91&1.02&-22.75&-6.90&-19.98&26.10&-0.86&0.88\\\
-4.01&1.48&9.51&-21.99&19.46&-10.27&-25.12&-3.39&7.29&27.90\\\
21.77&10.72&-0.61&-6.46&28.00&1.28&-6.86&-7.04&11.13&-30.64\\\
-15.78&5.60&-15.26&-7.67&-9.76&23.93&-14.71&12.25&29.45&-7.89\\\
16.34&10.61&-41.82&11.82&6.49&-7.38&-6.14&-1.82&-1.86&13.09\\\
-8.15&24.97&12.11&24.35&3.66&-2.59&-27.84&-3.45&-9.40&-13.72\\\
-5.35&-21.70&-4.16&7.42&13.98&29.84&-10.26&7.60&-25.13&7.78\\\
-19.62&-22.43&-2.61&22.58&11.80&-22.08&1.25&15.87&14.94&0.31\\\
\end{array}\right).$
Only nine rows need to be added to make this large $\boldsymbol{X}$ matrix
orthogonal.
## 4 ACHIEVING THE ORACLE PROPERTY WITH NONCONVEX PENALTIES
Fan and Li (2001) introduced an important concept called the oracle property
and showed that there exists one local minimum of (2) with this property.
However, because the optimization problem in (2) has an exponential number of
local optima (Huo and Ni 2007; Huo and Chen 2010), no theoretical results in
the literature claim that an existing algorithm can provide such a local
minimum. In this section, we prove that an OEM sequence for the SCAD and MCP
can indeed achieve this property. The theoretical results in this and the
following sections work for the OEM algorithm in both Sections 2 and 3.
First, we describe the oracle concept. A penalized least squares estimator of
$\upbeta$ in (1) has this property if it can not only select the correct
submodel asymptotically, but also estimate the nonzero coefficients
$\mbox{\boldmath$\upbeta$\unboldmath}_{1}$ in (34) as efficiently as if the
correct submodel were known in advance. Suppose that the number of nonzero
coefficients of $\upbeta$ in (1) is $p_{1}$ (with $p_{1}\leqslant p$) and
partition $\upbeta$ as
$\mbox{\boldmath$\upbeta$\unboldmath}=(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{\prime},\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{\prime})^{\prime},$
(34)
where $\mbox{\boldmath$\upbeta$\unboldmath}_{2}=\mathbf{0}$ and no component
of $\mbox{\boldmath$\upbeta$\unboldmath}_{1}$ is zero. Divide the regression
matrix $\boldsymbol{X}$ in (1) to $(\boldsymbol{X}_{1}\ \boldsymbol{X}_{2})$
with $\boldsymbol{X}_{1}$ corresponding to
$\mbox{\boldmath$\upbeta$\unboldmath}_{1}$. If all the variables that
influence the response in (1) are known, an _oracle_ estimator can be given as
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}=(\hat{\mbox{\boldmath$\upbeta$\unboldmath}}_{1}^{\prime},\hat{\mbox{\boldmath$\upbeta$\unboldmath}}_{2}^{\prime})^{\prime}$
with $\hat{\mbox{\boldmath$\upbeta$\unboldmath}}_{2}=\mathbf{0}$, where
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}\sim
N(\mathbf{0},\
\sigma^{2}(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}).$
We need several assumptions.
###### Assumption 1.
As $n\rightarrow\infty$,
$\frac{\boldsymbol{X}^{\prime}\boldsymbol{X}}{n}\rightarrow\boldsymbol{\Sigma}=\left(\begin{array}[]{cc}\boldsymbol{\Sigma}_{1}&\boldsymbol{\Sigma}_{12}\\\
\boldsymbol{\Sigma}_{21}&\boldsymbol{\Sigma}_{2}\end{array}\right),$
where $\boldsymbol{\Sigma}$ is positive definite and $\boldsymbol{\Sigma}_{1}$
is $p_{1}\times p_{1}$. Furthermore, $\boldsymbol{X}/\sqrt{n}$ is standardized
such that each entry on the diagonal of
$\boldsymbol{X}^{\prime}\boldsymbol{X}/n$ is 1, and
$\boldsymbol{X}^{\prime}\boldsymbol{X}/n+\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}=d\boldsymbol{I}_{p}$
with $d\geqslant\gamma_{1}$, where $d=O(1)$ and $\gamma_{1}$ is the largest
eigenvalue of $\boldsymbol{X}^{\prime}\boldsymbol{X}/n$.
###### Assumption 2.
The tuning parameter $\lambda=\lambda_{n}$ in (3) satisfies the condition
that, as $n\rightarrow\infty$, $\lambda_{n}/n\rightarrow 0$ and
$\lambda_{n}/\sqrt{n}\rightarrow\infty$.
Let $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)},\ k=0,1,\ldots,\\}$ be the
OEM sequence from (12) for the SCAD with a fixed $a>2$ in (3). We need an
assumption on $k=k_{n}$. Let $\eta$ be the largest eigenvalue of
$\boldsymbol{I}_{p_{1}}-\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}/(nd)$.
Under Assumption 1, $\eta$ tends to a limit lying between 0 and 1 as
$n\rightarrow\infty$.
###### Assumption 3.
As $n\rightarrow\infty$, $\eta^{k_{n}}\lambda_{n}/\sqrt{n}\rightarrow 0$ and
$k_{n}^{2}\exp(-c(\lambda_{n}/\sqrt{n})^{2})\rightarrow 0$ for any $c>0$.
One choice for $k_{n}$ to satisfy Assumption 3 is
$k_{n}=\left(\frac{\lambda_{n}}{\sqrt{n}}\right)^{\nu}\ \text{for some}\
\nu>0.$
Under Assumption 2, $k_{n}\rightarrow\infty$ as $n\rightarrow\infty$.
Under the above assumptions, Theorem 1 shows that
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}=(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)^{\prime}}_{1},\mbox{\boldmath$\upbeta$\unboldmath}^{(k)^{\prime}}_{2})^{\prime}$
can achieve the oracle property.
###### Theorem 1.
Suppose that Assumption 1-3 hold. If
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}=(\boldsymbol{X}^{\prime}\boldsymbol{X})^{-1}\boldsymbol{X}^{\prime}\mathbf{Y}$,
then as $n\rightarrow\infty$,
(i)
${\mathrm{P}}(\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)}=\mathbf{0})\rightarrow
1$;
(ii)
$\sqrt{n}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})\rightarrow
N(\mathbf{0},\sigma^{2}\boldsymbol{\Sigma_{1}^{-1}})$ in distribution.
The proof of Theorem 1 is deferred to the Appendix.
###### Remark 2.
From (64) in the proof of Theorem 1, for $k=1,2,\ldots$,
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ is _consistent_ in variable
selection. That is, ${\mathrm{P}}(\beta^{(k)}_{j}\neq 0\ \text{for}\
j=1,\ldots,p_{1})\rightarrow 1$ and
${\mathrm{P}}(\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)}=\mathbf{0})\rightarrow
1$ as $n\rightarrow\infty$.
###### Remark 3.
The proof of Theorem 1 uses the convergence rates of ${\mathrm{P}}(A_{k}),\
{\mathrm{P}}(B_{k})$ and ${\mathrm{P}}(C_{k}^{h})$. If an OEM sequence
satisfies the condition that $\beta_{j}^{(k+1)}=0$ when $|u_{j}|<\lambda$ and
$\beta_{j}^{(k+1)}=u_{j}/d$ when $|u_{j}|>c\lambda$ for some $c=O(1)$, then
${\mathrm{P}}(A_{k+1})={\mathrm{P}}(|u_{j}|<\lambda)$ and
${\mathrm{P}}(B_{k+1})={\mathrm{P}}(|u_{j}|>c\lambda)$. Since an OEM sequence
for the MCP satisfies the above condition, an argument very similar to the
proof in the Appendix shows that the convergence rates of
${\mathrm{P}}(A_{k}),\ {\mathrm{P}}(B_{k})$ and ${\mathrm{P}}(C_{k}^{h})$ for
the MCP are the same as those with the SCAD. Thus, under Assumption 1-3,
Theorem 1 holds for the MCP with a fixed $a>1$ in (20).
Huo and Chen (2010) showed that, for the SCAD penalty, solving the global
minimum of (5) leads to an NP-hard problem. Theorem 1 indicates that as far as
the oracle property is concerned, the local solution given by OEM will
suffice.
## 5 CONVERGENCE OF THE OEM ALGORITHM
In this section, we derive theoretical results on convergence properties of
the OEM algorithm and compare the convergence rates of OEM for the ordinary
least squares estimator and the elastic-net and lasso. The general penalty in
(5) is considered here. Our derivations employs the main tool in Wu (1983) in
conjunction with special properties of the penalties mentioned in Section 2.
We make several assumptions for $\Theta$ and
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ in (5).
###### Assumption 4.
The parameter space $\Theta$ is a closed convex subset of ${\mathbb{R}}^{p}$.
###### Assumption 5.
For a fixed $\lambda$, the penalty
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)\rightarrow+\infty$ as
$\|\mbox{\boldmath$\upbeta$\unboldmath}\|\rightarrow+\infty$.
###### Assumption 6.
For a fixed $\lambda$, the penalty
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ is continuous with respect
to $\mbox{\boldmath$\upbeta$\unboldmath}\in\Theta$.
All penalties discussed in Section 2 satisfy these assumptions. The
assumptions cover the case in which the iterative sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ defined in (14) may fall on
the boundary of $\Theta$ (Nettleton 1999), like the nonnegative garrote
(Breiman 1995) and the nonnegative lasso (Efron et al. 2004). The bridge
penalty (Frank and Friedman 1993) in (37) also satisfies the above
assumptions.
For the model in (1), denote the objective function in (5) by
$\displaystyle
l(\mbox{\boldmath$\upbeta$\unboldmath})=\|\mathbf{Y}-\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}\|^{2}+P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda).$
(35)
For some penalizes like the bridge, it may be numerically infeasible to
perform the M-step in (14). For this situation, following the generalized EM
algorithm in Dempster, Laird, and Rubin (1977), we define a _generalized OEM_
algorithm to be an iterative scheme
$\displaystyle\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\rightarrow\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}\in\mathcal{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}),$
(36)
where
$\mbox{\boldmath$\upbeta$\unboldmath}\rightarrow\mathcal{M}(\mbox{\boldmath$\upbeta$\unboldmath})\subset\Theta$
is a point-to-set map such that
$Q(\mbox{\boldmath$\upphi$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath})\leqslant
Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}),\quad\text{for
all}\
\mbox{\boldmath$\upphi$\unboldmath}\in\mathcal{M}(\mbox{\boldmath$\upbeta$\unboldmath}).$
Here, $Q$ is given by replacing the SCAD with
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ in (13). The OEM sequence
defined by (14) is a special case of (36). For example, the generalized OEM
algorithm can be used for the bridge penalty, where $\Theta_{j}={\mathbb{R}}$
and
$P_{j}(\beta_{j};\lambda)=\lambda|\beta_{j}|^{a}$ (37)
for some $a\in(0,1)$ in (5). Since the solution to (15) with the bridge
penalty has no closed form, one may use one-dimensional search to compute
$\beta_{j}^{(k+1)}$ that satisfies (36). By Assumption 1,
$\\{\mbox{\boldmath$\upbeta$\unboldmath}\in\Theta:l(\mbox{\boldmath$\upbeta$\unboldmath})\leqslant
l(\mbox{\boldmath$\upbeta$\unboldmath}^{(0)})\\}$ is compact for any
$l(\mbox{\boldmath$\upbeta$\unboldmath}^{(0)})>-\infty$. By Assumption 6,
$\mathcal{M}$ is a closed point-to-set map (Zangwill 1969; Wu 1983).
The objective functions of the EM algorithms in the literature like those
discussed in Wu (1983), Green (1990) and McLachlan and Krishnan (2008) are
typically continuously differentiable. This condition does not hold for the
objective function in (5) with the lasso and other penalties. A more general
definition of stationary points is needed here. We call
$\mbox{\boldmath$\upbeta$\unboldmath}\in\Theta$ a stationary point of $l$ if
$\liminf_{t\rightarrow
0_{+}}\frac{l\big{(}(1-t)\mbox{\boldmath$\upbeta$\unboldmath}+t\mbox{\boldmath$\upphi$\unboldmath}\big{)}-l(\mbox{\boldmath$\upbeta$\unboldmath})}{t}\geqslant
0\quad\text{for all}\ \mbox{\boldmath$\upphi$\unboldmath}\in\Theta.$
Let $S$ denote the set of stationary points of $l$. Analogous to Theorem 1 in
Wu (1983) on the global convergence of the EM algorithm, we have the following
result.
###### Theorem 2.
Let $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ be a generalized OEM
sequence generated by (36). Suppose that
$l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)})<l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})\quad\text{for
all }\ \mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\in\Theta\setminus S.$ (38)
Then all limit points of $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$
are elements of $S$ and $l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$
converges monotonically to $l^{*}=l(\mbox{\boldmath$\upbeta$\unboldmath}^{*})$
for some $\mbox{\boldmath$\upbeta$\unboldmath}^{*}\in S$.
###### Theorem 3.
If $\mbox{\boldmath$\upbeta$\unboldmath}^{*}$ is a local minimum of
$Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{*})$,
then $\mbox{\boldmath$\upbeta$\unboldmath}^{*}\in S$.
This theorem follows from the fact that
$l(\mbox{\boldmath$\upbeta$\unboldmath})-Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{*})$
is differentiable and
$\frac{\partial\big{[}l(\mbox{\boldmath$\upbeta$\unboldmath})-Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{*})\big{]}}{\partial\mbox{\boldmath$\upbeta$\unboldmath}}\Big{|}_{\mbox{\boldmath$\upbeta$\unboldmath}=\mbox{\boldmath$\upbeta$\unboldmath}^{*}}=0.$
###### Remark 4.
By Theorem 3, if $\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\notin S$, then
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ cannot be a local minimum of
$Q(\mbox{\boldmath$\upbeta$\unboldmath}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$.
Thus, there exists at least one point
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}\in\mathcal{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$
such that
$Q(\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})<Q(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$
and therefore satisfies the condition in (38). As a special case, an OEM
sequence generated by (14) satisfies (38) in Theorem 2.
Next, we consider the convergence of a generalized OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ in (36). By Theorem 3, such
results will automatically hold for an OEM sequence as well. If the penalty
function $P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ is convex and
$l(\mbox{\boldmath$\upbeta$\unboldmath})$ has a unique minimum, Theorem 4
shows that $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ converges to the
global minimum.
###### Theorem 4.
Let $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ be defined in Theorem
2. Suppose that $l(\mbox{\boldmath$\upbeta$\unboldmath})$ in (35) is a convex
function on $\Theta$ with a unique minimum
$\mbox{\boldmath$\upbeta$\unboldmath}^{*}$ and that (38) holds for
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$. Then
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\rightarrow\mbox{\boldmath$\upbeta$\unboldmath}^{*}$
as $k\rightarrow\infty$.
###### Proof.
We only need to show that $S=\\{\mbox{\boldmath$\upbeta$\unboldmath}^{*}\\}$.
For $\mbox{\boldmath$\upphi$\unboldmath}\in\Theta$ with
$\mbox{\boldmath$\upphi$\unboldmath}\neq\mbox{\boldmath$\upbeta$\unboldmath}^{*}$
and $t>0$, we have
$\frac{l\big{(}(1-t)\mbox{\boldmath$\upphi$\unboldmath}+t\mbox{\boldmath$\upbeta$\unboldmath}^{*}\big{)}-l(\mbox{\boldmath$\upbeta$\unboldmath}^{*})}{t}\leqslant\frac{tl(\mbox{\boldmath$\upbeta$\unboldmath}^{*})+(1-t)l(\mbox{\boldmath$\upphi$\unboldmath})-l(\mbox{\boldmath$\upphi$\unboldmath})}{t}=l(\mbox{\boldmath$\upbeta$\unboldmath}^{*})-l(\mbox{\boldmath$\upphi$\unboldmath})<0.$
This implies $\mbox{\boldmath$\upphi$\unboldmath}\notin S$. ∎
Theorem 5 discusses the convergence of an OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ for more general penalties.
For $a\in{\mathbb{R}}$, define $S(a)=\\{\mbox{\boldmath$\upphi$\unboldmath}\in
S:\ l(\mbox{\boldmath$\upphi$\unboldmath})=a\\}$. From Theorem 2, all limit
points of an OEM sequence are in $S(l^{*})$, where $l^{*}$ is the limit of
$l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$ in Theorem 2. Theorem 5 states
that the limit point is unique under certain conditions.
###### Theorem 5.
Let $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ be a generalized OEM
sequence generated by (36) with
$\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}>0$. If (38) holds, then all
limit points of $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ are in a
connected and compact subset of $S(l^{*})$. In particular, if the set
$S(l^{*})$ is discrete in that its only connected components are singletons,
then $\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ converges to some
$\mbox{\boldmath$\upbeta$\unboldmath}^{*}$ in $S(l^{*})$ as
$k\rightarrow\infty$.
###### Proof.
Note that
$Q(\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})=l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)})+\|\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}-\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\|^{2}\leqslant
Q(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\mid\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})=l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})$.
By Theorem 2,
$\|\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}-\boldsymbol{\Delta}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\|^{2}\leqslant
l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-l(\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)})\rightarrow
0$ as $k\rightarrow\infty$. Thus,
$\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}-\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\|\rightarrow
0$. This theorem now follows immediately from Theorem 5 of Wu (1983).∎
Since the bridge, SCAD and MCP penalties all satisfy the condition that
$S(l^{*})$ is discrete, an OEM sequence for any of them converges to the
stationary points of $l$. Theorem 5 is obtained under the condition
$\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}>0$. Since the error
$\upvarepsilon$ in (1) has a continuous distribution, it is easy to show that
Theorem 5 holds with probability one if
$\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$ is singular when $d$ defined
in (30) and (31) equals $\gamma_{1}$.
We now derive the convergence rate of the OEM sequence in (14). Following
Dempster, Laird, and Rubin (1977), write
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}),$
where the map
$\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath})=(M_{1}(\mbox{\boldmath$\upbeta$\unboldmath}),\ldots,M_{p}(\mbox{\boldmath$\upbeta$\unboldmath}))^{\prime}$
is defined by (14). We capture the convergence rate of the OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ through $\mathbf{M}$.
Assume that (32) holds for $d\geqslant\gamma_{1}$, where $\gamma_{1}$ is the
largest eigenvalue of $\boldsymbol{X}^{\prime}\boldsymbol{X}$. For the active
orthogolization in (30) and (31), taking $\boldsymbol{S}=\boldsymbol{I}_{p}$
satisfies this assumption; see Remark 1.
Let $\mbox{\boldmath$\upbeta$\unboldmath}^{*}$ be the limit of the OEM
sequence $\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$. As in Meng
(1994), we call
$R=\limsup_{k\rightarrow\infty}\frac{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}=\limsup_{k\rightarrow\infty}\frac{\|\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}_{k})-\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{*})\|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}$
(39)
the global rate of convergence for the OEM sequence. If there is no penalty in
(5), i.e., computing the ordinary least squares estimator, the global rate of
convergence $R$ in (39) becomes the largest eigenvalue of
$\boldsymbol{J}(\mbox{\boldmath$\upbeta$\unboldmath}^{*})$, denoted by
$R_{0}$, where $\boldsymbol{J}(\mbox{\boldmath$\upphi$\unboldmath})$ is the
$p\times p$ Jacobian matrix for
$\mathbf{M}(\mbox{\boldmath$\upphi$\unboldmath})$ having $(i,j)$th entry
$\partial
M_{i}(\mbox{\boldmath$\upphi$\unboldmath})/\partial\mbox{\boldmath$\upphi$\unboldmath}_{j}$.
If (32) holds, then
$\boldsymbol{J}(\mbox{\boldmath$\upbeta$\unboldmath}^{*})=\boldsymbol{A}/d$
with $\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$. Thus,
$R_{0}=\frac{d-\gamma_{p}}{d}.$ (40)
For (5), the penalty function
$P(\mbox{\boldmath$\upbeta$\unboldmath};\lambda)$ typically is not
sufficiently smooth and $R$ in (39) does not have an analytic form. Theorem 6
gives an upper bound of $R_{\mathrm{net}}$, the value of $R$ for the elastic-
net penalty in (18) with $\lambda_{1},\lambda_{2}\geqslant 0$.
###### Theorem 6.
For $\boldsymbol{\Delta}$ from (6), if (32) holds, then
$R_{\mathrm{NET}}\leqslant R_{0}$.
###### Proof.
Let $\mathbf{x}_{j}$ denote the $j$th column of $\boldsymbol{X}$ and
$\mathbf{a}_{j}$ denote the $j$th column of
$\boldsymbol{A}=\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}$,
respectively. For an OEM sequence for the elastic-net, by (19),
$M_{j}(\mbox{\boldmath$\upbeta$\unboldmath})=f(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}),\
\text{for}\ j=1,\ldots,p,$
where
$f(u)={\mathrm{sign}}(u)\left(\frac{|u|-\lambda_{1}}{d+\lambda_{2}}\right)_{+}.$
For $j=1,\ldots,p$, observe that
$\displaystyle\frac{|M_{j}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-M_{j}(\mbox{\boldmath$\upbeta$\unboldmath}^{*})|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}$
$\displaystyle=$
$\displaystyle\frac{|f(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-f(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{*})|}{|(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{*})|}$
$\displaystyle\cdot\frac{|(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-(\mathbf{x}_{j}^{\prime}\mathbf{Y}+\mathbf{a}_{j}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{*})|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{d}\cdot\frac{|\mathbf{a}_{j}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*})|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}.$
Thus,
$\frac{\|M(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-M(\mbox{\boldmath$\upbeta$\unboldmath}^{*})\|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}\leqslant\frac{1}{d}\cdot\frac{\|A(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*})\|}{\|\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}^{*}\|}\leqslant\frac{d-\gamma_{p}}{d}.$
This completes the proof.∎
###### Remark 5.
Theorem 6 indicates that, for the same $\boldsymbol{X}$ and $\mathbf{Y}$ in
(1), the OEM solution for the elastic-net converges faster than its
counterpart for the ordinary least squares. Since the lasso is a special case
of the elastic-net with $\lambda_{2}=0$ in (18), this theorem holds for the
lasso as well.
###### Remark 6.
From (40) and Theorem 6, the convergence rate of the OEM algorithm depends on
the ratio of the smallest eigenvalue, $\gamma_{p}$, and the largest
eigenvalue, $\gamma_{1}$, of $\boldsymbol{X}^{\prime}\boldsymbol{X}$. This
rate is the fastest when $\gamma_{1}=\gamma_{p}$, i.e., if $\boldsymbol{X}$ is
orthogonal and standardized. This result suggests that OEM converges faster if
$\boldsymbol{X}$ is orthogonal or nearly orthogonal like from a supersaturated
design or a nearly orthogonal Latin hypercube design (Owen 1994; Tang 1998).
This result is in agreement with the recent finding in the design of
experiments community that the use of orthogonal or nearly orthogonal designs
can significantly improve the accuracy of penalized variable selection methods
(Phoa, Pan, and Xu 2009; Deng, Lin, and Qian 2010; Zhu 2011).
[0.6]
Figure 3. (Left) the average values of $R_{0}$ in (40) against increasing $n$
for Example 6; (right) the average iteration numbers against increasing $n$
for Example 6, where the dashed and solid lines denote the ordinary least
squares estimator and the lasso, respectively.
###### Example 6.
We generate $\boldsymbol{X}$ from $p$ dimensional Gaussian distribution
$N(\mathbf{0},\boldsymbol{V})$ with $n$ independent observations, where the
$(i,j)$th entry of $\boldsymbol{V}$ is 1 for $i=j$ and $\rho$ for $i\neq j$.
Values of $\mathbf{Y}$ and $\upbeta$ are generated by (1) and (22). The same
setup was used in Friedman, Hastie, and Tibshirani (2009). For $p=10,\
\rho=0.1,\ \lambda=0.5$ and increasing $n$, the left panel of Figure 3 depicts
the average values of $R_{0}$ in (40) against increasing $n$ and the right
panel of the figure depicts the average iteration numbers against increasing
$n$, with the dashed and solid lines corresponding to the ordinary least
squares estimator and the lasso, respectively. Quite strikingly, this figure
indicates that OEM requires _fewer_ iterations as $n$ becomes larger, which
makes OEM particulary attractive for situations with massive data (SAMSI
2012). It is important to note that here the OEM sequence for the lasso
requires fewer iteration than its counterpart for the ordinary least squares,
empirically validating Theorem 6.
## 6 POSSESSING GROUPING COHERENCE
In this section, we consider the convergence of the OEM algorithm when the
regression matrix $\boldsymbol{X}$ in (1) is singular due to fully aliased
columns or other conditions. Let $\boldsymbol{X}$ be standardized as in (4)
with columns $\mathbf{x}_{1},\ldots,\mathbf{x}_{p}$. If $\mathbf{x}_{i}$ and
$\mathbf{x}_{j}$ are fully aliased, i.e., $|\mathbf{x}_{i}|=|\mathbf{x}_{2}|$,
then the objective function in (5) for the lasso is not strictly convex and
has many minima (Zou and Hastie 2005). Data with fully aliasing structures
commonly appear in observational studies and various classes of experimental
designs like supersaturated designs (Wu 1993; Lin 1993; Tang and Wu 1997; Li
and Lin 2002; Xu and Wu 2005) and factorial designs (Dey and Mukerjee 1999;
Mukerjee and Wu 2006).
Zou and Hastie (2005) states that if some columns of $\boldsymbol{X}$ are
identical, it is desirable to have grouping coherence by assigning the same
coefficient to them. Definition 1 makes this precise.
###### Definition 1.
An estimator
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}=(\hat{\beta}_{1},\ldots,\hat{\beta}_{p})^{\prime}$
of $\upbeta$ in (1) has _grouping coherence_ if
$\mathbf{x}_{i}=\mathbf{x}_{j}$ implies $\hat{\beta}_{i}=\hat{\beta}_{j}$ and
$\mathbf{x}_{i}=-\mathbf{x}_{j}$ implies $\hat{\beta}_{i}=-\hat{\beta}_{j}$.
Let $\mathbf{0}_{p}$ denote the zero vector in ${\mathbb{R}}^{p}$. Let
$\mathbf{e}_{ij}^{+}$ be the vector obtained by replacing the $i$th and $j$th
entries of $\mathbf{0}_{p}$ with $1$. Let $\mathbf{e}_{ij}^{-}$ be the vector
obtained by replacing the $i$th and $j$th entries of $\mathbf{0}_{p}$ with $1$
and $-1$, respectively. Let ${\mathcal{E}}$ denote the set of all
$\mathbf{e}_{ij}^{+}$ and $\mathbf{e}_{ij}^{-}$. By Definition 1, an estimator
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}$ has grouping coherence if and
only if for any $\mbox{\boldmath$\upalpha$\unboldmath}\in{\mathcal{E}}$ with
$\boldsymbol{X}\mbox{\boldmath$\upalpha$\unboldmath}=\mathbf{0}$,
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\hat{\mbox{\boldmath$\upbeta$\unboldmath}}=0$.
###### Lemma 2.
Suppose that (32) holds. For the OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ of the lasso, SCAD or MCP,
if $\boldsymbol{X}\mbox{\boldmath$\upalpha$\unboldmath}=\mathbf{0}$ and
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}=0$
for $\mbox{\boldmath$\upalpha$\unboldmath}\in{\mathcal{E}}$, then
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=0$.
###### Proof.
For $\mathbf{u}$ defined in (10), we have that
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mathbf{u}=\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{X}^{\prime}\mathbf{Y}+\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(d\boldsymbol{I}_{p}-\boldsymbol{X}^{\prime}\boldsymbol{X})\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}=0$
for any $\mbox{\boldmath$\upalpha$\unboldmath}\in{\mathcal{E}}$ with
$\boldsymbol{X}\mbox{\boldmath$\upalpha$\unboldmath}=\mathbf{0}$ and
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}=0$.
Then by (17), (12) and (21), an OEM sequence of the lasso, SCAD or MCP
satisfies the condition that if
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mathbf{u}=0$, then
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=0$
for $\mbox{\boldmath$\upalpha$\unboldmath}\in{\mathcal{E}}$. This completes
the proof.∎
###### Remark 7.
Lemma 2 implies that, for $k=1,2,\ldots$,
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ has grouping coherence if
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$ has grouping coherence. Thus, if
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ converges, then its limit
has grouping coherence. By Theorem 5, if $d>\lambda_{1}$ in (32), then an OEM
sequence for the SCAD or MCP converges to a point with grouping coherence.
When $\boldsymbol{X}$ in (1) has fully aliased columns, the objective function
in (5) for the lasso has many minima and hence the condition in Theorem 4 does
not hold. Theorem 7 shows that, even with full aliasing, an OEM sequence (17)
for the lasso converges to a point with grouping coherence.
###### Theorem 7.
Suppose that (32) holds. If $\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$ has
grouping coherence, then as $k\rightarrow\infty$, the OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ of the lasso converges to a
limit that has grouping coherence.
###### Proof.
Partition the matrix $\boldsymbol{X}$ in (1) as $(\boldsymbol{X}_{1}\
\boldsymbol{X}_{2})$, where no two columns of $\boldsymbol{X}_{2}$ are fully
aliased and any column of $\boldsymbol{X}_{1}$ is fully aliased with at least
one column of $\boldsymbol{X}_{2}$. Let $J$ denote the number of columns in
$\boldsymbol{X}_{1}$. Partition $\upbeta$ as
$(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{\prime},\
\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{\prime})^{\prime}$ and
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ as
$(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)^{\prime}},\
\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)^{\prime}})^{\prime}$,
corresponding to $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$, respectively.
For $j=1,\ldots,p$, let
$\omega(j)=\\#\\{i=1,\ldots,p:|\mathbf{x}_{i}|=|\mathbf{x}_{j}|\\}.$
By Lemma 2, for $j=1,\ldots,J$, $\beta_{j}^{(k)}=\beta_{j^{\prime}}^{(k)}$ if
$\mathbf{x}_{j}=\mathbf{x}_{j^{\prime}}$ and
$\beta_{j}^{(k)}=-\beta_{j^{\prime}}^{(k)}$ otherwise, where
$j^{\prime}\in\\{J+1,\ldots,p\\}$. It follows that
$\\{\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)}\\}$ can be viewed as an OEM
sequence for solving
$\min_{{\scriptsize\mbox{\boldmath$\uptheta$\unboldmath}}}\|\mathbf{Y}-\tilde{\boldsymbol{X}}\mbox{\boldmath$\uptheta$\unboldmath}\|^{2}+2\sum_{j=1}^{p-J}|\theta_{j}|,$
(41)
where
$\mbox{\boldmath$\uptheta$\unboldmath}=(\theta_{1},\ldots,\theta_{p-J})^{\prime}$,
and the columns of $\tilde{\boldsymbol{X}}$ are
$\omega(J+1)\mathbf{x}_{J+1},\ldots,\omega(p)\mathbf{x}_{p}$. Because the
objective function in (41) is strictly convex, by Theorem 4,
$\\{\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)}\\}$ converges to a limit
with grouping coherence. This completes the proof.∎
###### Example 7.
Consider $\boldsymbol{X}$ in Example 4. Let
$\mathbf{Y}=(2,1,-4,1.5)^{\prime}$. Using an initial point
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}=0$, the OEM sequence of the lasso
with $\lambda=1$ converges to
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}=(-0.5625,\ 0.4375,\ -0.6875,\
0.6875,\ -0.4375,\ 0.5625)^{\prime},$
which has grouping coherence. For the same data and the same initial point,
the coordinate descent sequence converges to $2(-0.5625,\ 0.4375,\ -0.6875,\
0,\ 0,\ 0)^{\prime}$, which does not have grouping coherence.
Theorem 7 shows that, if the initial point has grouping coherence, then every
limit point of the OEM sequence for the lasso inherits this property. It is
now tempting to ask whether such a result holds for the OEM sequence with
$\lambda=0$ in (16), i.e., the Healy and Westmacott procedure. Since full
aliasing is just one possible culprit for making the matrix $\boldsymbol{X}$
in (1) lack full column rank and hence $\boldsymbol{X}^{\prime}\boldsymbol{X}$
become singular, Theorem 8 provides an answer to this question for the general
singular situation.
Let $r$ denote the rank of $\boldsymbol{X}$. When $r<p$, the singular value
decomposition (Wilkinson 1965) of $\boldsymbol{X}$ is
$\boldsymbol{X}=\boldsymbol{U}^{\prime}\left(\begin{array}[]{cc}\boldsymbol{\Gamma}_{0}^{1/2}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right)\boldsymbol{V},$
where $\boldsymbol{U}$ is an $n\times n$ matirx, $\boldsymbol{V}$ is a
$p\times p$ orthogonal matrices and $\boldsymbol{\Gamma}_{0}$ is a diagonal
matrix whose diagonal elements are the positive eigenvalues,
$\gamma_{1}\geqslant\cdots\geqslant\gamma_{r}$, of
$\boldsymbol{X}^{\prime}\boldsymbol{X}$. Define
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}=(\boldsymbol{X}^{\prime}\boldsymbol{X})^{+}\boldsymbol{X}^{\prime}\mathbf{Y},$
(42)
where + denotes the Moore-Penrose generalized inverse (Ben-Israel and Greville
2003).
###### Theorem 8.
Suppose that
$\boldsymbol{X}^{\prime}\boldsymbol{X}+\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}=\gamma_{1}\boldsymbol{I}_{p}$.
If $\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$ lies in the linear space
spanned by the first $r$ columns of $\boldsymbol{V}^{\prime}$, then as
$k\rightarrow\infty$, for the OEM sequence
$\\{\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\\}$ of the ordinary least
squares,
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\rightarrow\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}$.
###### Proof.
Denote
$\boldsymbol{D}=\boldsymbol{I}_{p}-\gamma_{1}^{-1}\boldsymbol{X}^{\prime}\boldsymbol{X}$.
Note that
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\gamma_{1}^{-1}\boldsymbol{X}^{\prime}\mathbf{Y}+\boldsymbol{D}\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$.
By induction,
$\displaystyle\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}$ $\displaystyle=$
$\displaystyle\gamma_{1}^{-1}(\boldsymbol{I}_{p}+\boldsymbol{D}+\cdots+\boldsymbol{D}^{k-1})\boldsymbol{X}^{\prime}\mathbf{Y}+\boldsymbol{D}^{k}\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$
$\displaystyle=$
$\displaystyle\gamma_{1}^{-1}\boldsymbol{V}^{\prime}\left\\{\boldsymbol{I}_{p}+\left(\begin{array}[]{cc}\boldsymbol{I}_{r}-\gamma_{1}^{-1}\boldsymbol{\Gamma}_{0}&\boldsymbol{0}\\\
\boldsymbol{0}&-\boldsymbol{I}_{p-r}\end{array}\right)+\cdots+\left(\begin{array}[]{cc}(\boldsymbol{I}_{r}-\gamma_{1}^{-1}\boldsymbol{\Gamma}_{0})^{k-1}&\boldsymbol{0}\\\
\boldsymbol{0}&(-1)^{k-1}\boldsymbol{I}_{p-r}\end{array}\right)\right\\}\boldsymbol{V}$
$\displaystyle\
\cdot\boldsymbol{V}^{\prime}\left(\begin{array}[]{cc}\boldsymbol{\Gamma}_{0}^{1/2}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right)\boldsymbol{U}\mathbf{Y}+\boldsymbol{D}^{k}\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}$
$\displaystyle=$
$\displaystyle\gamma_{1}^{-1}\boldsymbol{V}^{\prime}\left(\begin{array}[]{cc}\big{\\{}\boldsymbol{I}_{r}+(\boldsymbol{I}_{r}-\gamma_{1}^{-1}\boldsymbol{\Gamma}_{0})+\cdots+(\boldsymbol{I}_{r}-\gamma_{1}^{-1}\boldsymbol{\Gamma}_{0})^{k-1}\big{\\}}\boldsymbol{\Gamma}_{0}^{1/2}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right)\boldsymbol{U}\mathbf{Y}+\boldsymbol{D}^{k}\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}.$
As $k\rightarrow\infty$, we have that
$\boldsymbol{D}^{k}\rightarrow\boldsymbol{V}^{\prime}\left(\begin{array}[]{cc}\boldsymbol{0}&\\\
&\boldsymbol{I}_{p-r}\end{array}\right)\boldsymbol{V}$
and $\boldsymbol{D}^{k}\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}\rightarrow
0$, which implies that
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}\rightarrow\boldsymbol{V}^{\prime}\left(\begin{array}[]{cc}\boldsymbol{\Gamma}_{0}^{-1/2}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right)\boldsymbol{U}\mathbf{Y}=\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}.$
This completes the proof.∎
The condition
$\boldsymbol{X}^{\prime}\boldsymbol{X}+\boldsymbol{\Delta}^{\prime}\boldsymbol{\Delta}=\gamma_{1}\boldsymbol{I}_{p}$
holds if $\boldsymbol{S}=\boldsymbol{I}_{p}$ in (26).
###### Remark 8.
Computing the Moore-Penrose generalized inverse
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}$ in (42) is a difficult
problem. Theorem 8 says that the OEM algorithm provides an efficient solution
to this problem. When $r<p$, the limiting matrix
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}$ given by an OEM sequence has
the following properties. First, it has the minimal Euclidean norm among the
least squares estimators
$(\boldsymbol{X}^{\prime}\boldsymbol{X})^{-}\boldsymbol{X}^{\prime}\mathbf{Y}$
(Ben-Israel and Greville 2003). Second, its model error has a simple form,
$E\big{[}(\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}-\mbox{\boldmath$\upbeta$\unboldmath})^{\prime}(\boldsymbol{X}^{\prime}\boldsymbol{X})(\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}-\mbox{\boldmath$\upbeta$\unboldmath})\big{]}=r\sigma^{2}$.
Third, $\boldsymbol{X}\mbox{\boldmath$\upalpha$\unboldmath}=\mathbf{0}$
implies
$\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}=0$
for any vector $\upalpha$, which immediately implies that
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}$ has grouping coherence.
###### Example 8.
Use the same data and the same initial point in Example 7. The OEM sequence of
the ordinary least squares converges to
$\hat{\mbox{\boldmath$\upbeta$\unboldmath}}^{*}=(-0.6875,\ 0.5625,\ -0.8125,\
0.8125,\ -0.5625,\ 0.6875)^{\prime},$
which has grouping coherence.
## 7 DISCUSSION
For the regularized least squares method with the SCAD penalty, finding a
local solution to achieve the oracle property is a well-known open problem. We
have proposed an algorithm, called the OEM algorithm, to fill this gap. For
the SCAD and MCP penalties, this algorithm can provide a local solution with
the oracle property. The discovery of the algorithm is quite accidental,
drawing direct impetus from a missing-data problem arising in design of
experiments. The introduction of the active orthogonization procedure in
Section 3 makes the algorithm applicable to very general data structures from
observational studies and experiments. Recent years have witnessed an
explosive interest in both the theoretical and computational aspects of
penalized methods. Our introduction of an algorithm that not only has
desirable numerical convergence properties but also possesses an important
theoretical property suggests a new interface between these two aspects.
Subsequent work in this direction is expected. The active orthogonization idea
is general and may have potential applications beyond the scope of the OEM
algorithm, such as other EM algorithms (Meng and Rubin 1991; Meng and Rubin
1993; Meng 2007), data augmentation (Tanner and Wong 1987), Markov chain Monte
Carlo (Liu 2001), smoothing splines (Wahba and Luo 1997; Luo 1998), mesh-free
methods in numerical analysis (Fasshauer 2007) and parallel computing (Kumar,
Grama, Gupta, and Karypis 2003). This procedure may also has intriguing
mathematical connection with complementary theory in design of experiments
(Tang and Wu 1996; Chen and Hedayat 1996; Xu and Wu 2005).
The result on the oracle property in Section 4 uses the assumption that the
sample size $n$ goes to infinity. This result is appealing for practical
situations with massive data (SAMSI 2012), such as the data deluge in
astronomy, the Internet and marketing (the Economist 2010), large-scale
industrial experiments (Xu 2009) and modern simulations in engineering (NAE
2008), to just name a few. For applications like micro-array and image
analysis, one might be interested in extending the result to the small $n$ and
large $p$ case, like in Fan and Peng (2004). Such an extension, however, poses
significant challenges. Even for a fixed $p$, the penalized likelihood
function for the SCAD can have a large number of local minima (Huo and Chen
2010). When $p$ goes to infinity, that number can be prohibitively large,
which makes it very difficult to sort out a local minima with the oracle
property.
In addition to achieving the oracle property for nonconvex penalties, an OEM
sequence has other unique theoretical properties, including convergence to a
point having grouping coherence for the lasso, SCAD or MCP and convergence to
the Moore-Penrose generalized inverse-based least squares estimator for
singular regression matrices. These theoretical results together with the
active orthogonization scheme form the main contribution of the article.
A computer package for distributing the OEM algorithm to the general audience
is under development and will be released. We now remark on the acceleration
issue and directions for future work. The algorithm can be speeded up by using
various methods from the EM literature (McLachlan and Krishnan 2008). For
example, following the idea in Varadhan and Roland (2008), one can replace the
OEM iteration in (14) by
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}-2\gamma\mathbf{r}+\gamma^{2}\mathbf{v},$
where
$\mathbf{r}=\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-\mbox{\boldmath$\upbeta$\unboldmath}^{(k)},\
\mathbf{v}=\mathbf{M}(\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}))-\mathbf{M}(\mbox{\boldmath$\upbeta$\unboldmath}^{(k)})-r$
and $\gamma=-\|\mathbf{r}\|/\|\mathbf{v}\|$. This scheme is found to lead to
significant reduction of the running time in several examples. For problems
with very large $p$, one may consider a hybrid algorithm to combine the OEM
and coordinate descent ideas. It partitions $\upbeta$ in (1) into $G$ groups
and in each iteration, it minimizes the objective function $l$ in (35) by
using the OEM algorithm with respect to one group while holding the other
groups fixed. Here are some details. Group $\upbeta$ as
$\mbox{\boldmath$\upbeta$\unboldmath}=(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{\prime},\ldots,\mbox{\boldmath$\upbeta$\unboldmath}_{G}^{\prime})^{\prime}$.
For $k=0,1,\ldots$, solve
$\mbox{\boldmath$\upbeta$\unboldmath}_{g}^{(k+1)}=\arg\min_{{\scriptsize\mbox{\boldmath$\upbeta$\unboldmath}_{g}}}l(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k+1)},\ldots,\mbox{\boldmath$\upbeta$\unboldmath}_{g-1}^{(k+1)},\mbox{\boldmath$\upbeta$\unboldmath}_{g},\mbox{\boldmath$\upbeta$\unboldmath}_{g+1}^{(k)},\ldots,\mbox{\boldmath$\upbeta$\unboldmath}_{G}^{(k)})\
\text{for}\ g=1,\ldots,G$ (46)
by OEM until convergence. Note that (46) has a much lower dimension than the
iteration in (14). For $G=1$, the hybrid algorithm reduces to the OEM
algorithm and for $G=p$, it becomes the coordinate descent algorithm.
Theoretical properties of this hybrid algorithm will be studied and reported
elsewhere.
Extension of the OEM algorithm can be made by imposing special structures on
regression matrices, such as grouped variables (Yuan and Lin 2006; Zhou and
Zhu 2008; Huang, Ma, Xie, and Zhang 2009; Zhao, Rocha, and Yu 2009; Wang,
Chen, and Li 2009; Xiong 2010), mixtures (Khalili and Chen 2007) and heredity
constraints (Yuan, Joseph, and Lin 2007; Yuan, Joseph, and Zou 2009; Choi, Li,
and Zhu 2010), among many other possibilities.
APPENDIX: PROOF OF THEOREM 1
###### Proof.
We first give some definitions and notation. Let $\Phi$ be the cumulative
distribution function of the standard normal random variable. For $a>2$ and
$\lambda$ in (3) and $d\geqslant\gamma_{1}$ in Assumption 1, define
$s(u;\lambda)=\left\\{\begin{array}[]{ll}{\mathrm{sign}}(u)\big{(}|u|-\lambda\big{)}_{+}/d,&\text{when}\
|u|\leqslant(d+1)\lambda,\\\
{\mathrm{sign}}(u)\big{\\{}(a-1)|u|-a\lambda\big{\\}}/\big{\\{}(a-1)d-1\big{\\}},&\text{when}\
(d+1)\lambda<|u|\leqslant ad\lambda,\\\ u/d,&\text{when}\
|u|>ad\lambda,\end{array}\right.$
Let
$\mathbf{s}(\mathbf{u};\lambda)=\big{[}s(u_{1};\lambda),\ldots,s(u_{p};\lambda)\big{]}.^{\prime}$
The OEM sequence from (12) satisfies the condition that
$\mbox{\boldmath$\upbeta$\unboldmath}^{(k+1)}=\mathbf{s}(\mathbf{u}^{(k)};\lambda_{n}/n)$,
where
$\displaystyle\mathbf{u}^{(k)}=(\mathbf{u}^{(k)^{\prime}}_{1},\mathbf{u}^{(k)^{\prime}}_{2})^{\prime}=\frac{\boldsymbol{X}^{\prime}\mathbf{Y}}{n}+\left(d\boldsymbol{I}_{d}-\frac{\boldsymbol{X}^{\prime}\mathbf{Y}}{n}\right)\mbox{\boldmath$\upbeta$\unboldmath}^{(k)}.$
(47)
For $k=1,2,\ldots$, define two sequences of events
$A_{k}=\\{\mbox{\boldmath$\upbeta$\unboldmath}_{2}^{(k)}=\mathbf{0}\\}$ and
$B_{k}=\\{\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}=\mathbf{u}_{1}^{(k-1)}/d\\}$.
For $h>0$ and $k=0,1,\ldots$, let
$C_{k}^{h}=\\{\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}\|\leqslant
h\lambda_{n}/n\\}$. The flow of the proof is to first show that
${\mathrm{P}}(A_{k}),\ {\mathrm{P}}(B_{k})$ and ${\mathrm{P}}(C_{k}^{h})$ all
tend to one at exponential rates as $n$ goes to infinity, thus establishing
Theorem 1 (i), then show that ${\mathrm{P}}(\cap_{i=1}^{k}A_{i})$ and
${\mathrm{P}}(\cap_{i=1}^{k+1}B_{i})$ tend to one and finally establish
Theorem 1(ii) by noting that the asymptotic normality of
$\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}$ follows when
$\cap_{i=1}^{k}A_{i}$ and $\cap_{i=1}^{k+1}B_{i}$ both occur.
_Step 1_. Let
$\boldsymbol{G}=\boldsymbol{X}(\boldsymbol{X}^{\prime}\boldsymbol{X})^{-1}$
with columns $\mathbf{g}_{1},\ldots,\mathbf{g}_{p}$. Let $\boldsymbol{G}_{1}$
denote its submatrix with the first $p_{1}$ columns. Let $\tau\geqslant 0$
denote the largest eigenvalue of
$\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{2}\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{1}/n^{2}$.
Define
$h_{A}=\left\\{\begin{array}[]{ll}1/(2\tau),&\text{when}\ \tau>0,\\\
+\infty,&\text{when}\ \tau=0.\end{array}\right.$ (48)
Let
$(\mathbf{v}_{1},\ldots,\mathbf{v}_{p_{1}})=d\boldsymbol{I}_{p_{1}}-n^{-1}\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}$
and $b=\max\\{\|\mathbf{v}_{1}\|,\ldots,\|\mathbf{v}_{p_{1}}\|\\}$. Define
$h_{B}=\frac{ad}{b},$ (49)
with $a$ and $d$ given in (3) and Assumption 1, respectively. For $h>0$,
define
$h_{C}=\frac{h}{2\eta},$ (50)
where $\eta$, used in Assumption 3, is the largest eigenvalue of
$\boldsymbol{I}_{p_{1}}-\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}/(nd)$.
For $C_{0}^{h}$, since
$\mbox{\boldmath$\upbeta$\unboldmath}^{(0)}=\mbox{\boldmath$\upbeta$\unboldmath}+\boldsymbol{G}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}$,
we have that
$\displaystyle{\mathrm{P}}(C_{0}^{h})$ $\displaystyle=$
$\displaystyle{\mathrm{P}}(\|\boldsymbol{G}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\|\leqslant
h\lambda_{n}/n)$ (51) $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}|\mathbf{g}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\leqslant
h\lambda_{n}/(n\sqrt{p_{1}})\ \text{for}\ j=1,\ldots.p_{1}\big{)}$
$\displaystyle\geqslant$ $\displaystyle
1-\sum_{j=1}^{p_{1}}\Big{[}1-{\mathrm{P}}\big{(}|\mathbf{g}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\leqslant
h\lambda_{n}/(n\sqrt{p_{1}})\big{)}\Big{]}$ $\displaystyle=$ $\displaystyle
1-2\sum_{j=1}^{p_{1}}\left[1-\Phi\left(\frac{h\lambda_{n}}{n\sqrt{p_{1}}\sigma\|\mathbf{g}_{j}\|}\right)\right].$
For $A_{1}$, note that
$\displaystyle{\mathrm{P}}(A_{1})$ $\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}|\beta_{j}^{(0)}|\leqslant\lambda_{n}/(nd)\
\text{for}\ j=p_{1}+1,\ldots,p\big{)}$ (52) $\displaystyle\geqslant$
$\displaystyle
1-\sum_{j=p_{1}+1}^{p}\Big{[}1-{\mathrm{P}}\big{(}|\mathbf{g}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\leqslant\lambda_{n}/(nd)\big{)}\Big{]}$
$\displaystyle=$ $\displaystyle
1-2\sum_{j=p_{1}+1}^{p}\left[1-\Phi\left(\frac{\lambda_{n}}{nd\sigma\|\mathbf{g}_{j}\|}\right)\right].$
For $B_{1}$, note that
$\displaystyle{\mathrm{P}}(B_{1})$ $\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}|\beta_{j}^{(0)}|\geqslant a\lambda_{n}/n\
\text{for}\ j=1,\ldots,p_{1}\big{)}$ (53) $\displaystyle\geqslant$
$\displaystyle
1-\sum_{j=1}^{p_{1}}\Big{[}1-{\mathrm{P}}\big{(}|\beta_{j}+\mathbf{g}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\geqslant
a\lambda_{n}/n\big{)}\Big{]}$ $\displaystyle=$ $\displaystyle
1-\sum_{j=1}^{p_{1}}\left[\Phi\left(\frac{n\beta_{j}+a\lambda_{n}}{n\sigma\|\mathbf{g}_{j}\|}\right)-\Phi\left(\frac{n\beta_{j}-a\lambda_{n}}{n\sigma\|\mathbf{g}_{j}\|}\right)\right].$
For any $h>0$, by (51),
$\displaystyle{\mathrm{P}}(C_{1}^{h})$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}(C_{1}^{h}\cap B_{1})\ =\ {\mathrm{P}}(C_{0}^{h}\cap
B_{1})$ (54) $\displaystyle\geqslant$ $\displaystyle
1-2\sum_{j=1}^{p_{1}}\left[1-\Phi\left(\frac{h\lambda_{n}}{n\sqrt{p_{1}}\sigma\|\mathbf{g}_{j}\|}\right)\right]$
$\displaystyle-\sum_{j=1}^{p_{1}}\left[\Phi\left(\frac{n\beta_{j}+a\lambda_{n}}{n\sigma\|\mathbf{g}_{j}\|}\right)-\Phi\left(\frac{n\beta_{j}-a\lambda_{n}}{n\sigma\|\mathbf{g}_{j}\|}\right)\right].$
Next, consider $A_{k},\ B_{k}$ and $C_{k}^{h}$, for $k=2,3,\ldots$. If
$A_{k-1}$ occurs, then by (47),
$\displaystyle\mathbf{u}^{(k-1)}$ $\displaystyle=$
$\displaystyle\frac{\boldsymbol{X}^{\prime}\boldsymbol{X}\mbox{\boldmath$\upbeta$\unboldmath}}{n}+\frac{\boldsymbol{X}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}}{n}+\left(\begin{array}[]{cc}d\boldsymbol{I}_{p_{1}}-\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}/n&-\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{2}/n\\\
-\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{1}/n&d\boldsymbol{I}_{p-p_{1}}-\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{2}/n\end{array}\right)\left(\begin{array}[]{c}\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}\\\
\mathbf{0}\end{array}\right).$
Thus,
$\displaystyle\mathbf{u}_{1}^{(k-1)}=d\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}+\frac{\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}}{n}[\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}]+\frac{\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}}{n},$
(56)
$\displaystyle\mathbf{u}_{2}^{(k-1)}=\frac{\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{1}}{n}[\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}]+\frac{\boldsymbol{X}_{2}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}}{n}.$
(57)
By (57),
$\displaystyle{\mathrm{P}}(A_{k})$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}(A_{k}\cap A_{k-1}\cap C_{k-1}^{h_{A}})$ (58)
$\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}\\{|u_{j}^{(k-1)}|\leqslant\lambda_{n}/n\
\text{for}\ j=p_{1}+1,\ldots,p\\}\cap A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{\|\mathbf{u}_{2}^{(k-1)}\|\leqslant\lambda_{n}/n\\}\cap
A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$ $\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}\\{\|n^{-1}\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{1}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)})+n^{-1}\boldsymbol{X}_{2}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\|\leqslant\lambda_{n}/n\\}\cap
A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{\|n^{-1}\boldsymbol{X}_{2}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\|\leqslant\lambda_{n}/n-\tau\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}\|\\}\cap
A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$ $\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}\\{\|n^{-1}\boldsymbol{X}_{2}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\|\leqslant\lambda_{n}/(2n)\|\\}\cap
A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{|\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\leqslant\lambda_{n}/(2\sqrt{p-p_{1}})\
\text{for}\ j=p_{1}+1,\ldots,p\\}\cap A_{k-1}\cap C_{k-1}^{h_{A}}\big{)}$
$\displaystyle\geqslant$ $\displaystyle
1-2\sum_{j=p_{1}+1}^{p}\left[1-\Phi\left(\frac{\lambda_{n}}{2\sqrt{p-p_{1}}\sigma\|\mathbf{x}_{j}\|}\right)\right]$
$\displaystyle-\big{[}1-{\mathrm{P}}(A_{k-1})\big{]}-\big{[}1-{\mathrm{P}}(C_{k-1}^{h_{A}})\big{]},$
where $h_{A}$ is defined in (48).
By (56),
$\displaystyle{\mathrm{P}}(B_{k})$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}(B_{k}\cap A_{k-1}\cap C_{k-1}^{h_{B}})$ (59)
$\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}\\{|d\beta_{j}+\mathbf{u}_{j}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)})+n^{-1}\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\geqslant
ad\lambda_{n}/n\ \text{for}\ j=1,\ldots,p_{1}\\}\cap A_{k-1}\cap
C_{k-1}^{h_{B}}\big{)}$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{|d\beta_{j}+n^{-1}\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\geqslant|\mathbf{u}_{j}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)})|+ad\lambda_{n}/n\
\text{for}\ j=1,\ldots,p_{1}\\}\cap A_{k-1}\cap C_{k-1}^{h_{B}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{|d\beta_{j}+n^{-1}\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\geqslant
b\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}\|+ad\lambda_{n}/n\
\text{for}\ j=1,\ldots,p_{1}\\}\cap A_{k-1}\cap C_{k-1}^{h_{B}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{|d\beta_{j}+n^{-1}\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|\geqslant
2ad\lambda_{n}/n\ \text{for}\ j=1,\ldots,p_{1}\\}\cap A_{k-1}\cap
C_{k-1}^{h_{B}}\big{)}$ $\displaystyle\geqslant$ $\displaystyle
1-\sum_{j=1}^{p_{1}}\left[\Phi\left(\frac{nd\beta_{j}+2ad\lambda_{n}}{\sigma\|\mathbf{x}_{j}\|}\right)-\Phi\left(\frac{nd\beta_{j}-2ad\lambda_{n}}{\sigma\|\mathbf{x}_{j}\|}\right)\right]-\big{[}1-{\mathrm{P}}(A_{k-1})\big{]}$
$\displaystyle-\big{[}1-{\mathrm{P}}(C_{k-1}^{h_{B}})\big{]},$
where $h_{B}$ is defined in (49).
For any $h>0$, we have that
$\displaystyle{\mathrm{P}}(C_{k}^{h})$ $\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}(C_{k}^{h}\cap B_{k}\cap A_{k-1}\cap
C_{k-1}^{h_{C}})$ (60) $\displaystyle=$
$\displaystyle{\mathrm{P}}\big{(}\\{\|(\boldsymbol{I}_{p_{1}}-\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}/(nd))(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})+\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}/(nd)\|\leqslant
h\lambda_{n}/n\\}\cap B_{k}\cap A_{k-1}\cap C_{k-1}^{h_{C}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{\eta\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}\|+\|\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}/(nd)\|\leqslant
h\lambda_{n}/n\\}\cap B_{k}\cap A_{k-1}\cap C_{k-1}^{h_{C}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{\|\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}/(nd)\|\leqslant
h\lambda_{n}/(2n)\\}\cap B_{k}\cap A_{k-1}\cap C_{k-1}^{h_{C}}\big{)}$
$\displaystyle\geqslant$
$\displaystyle{\mathrm{P}}\big{(}\\{|\mathbf{x}_{j}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}/(nd)|\leqslant
h\lambda_{n}/(2n\sqrt{p_{1}})\ \text{for}\ j=1,\ldots,p_{1}\\}\cap B_{k}\cap
A_{k-1}\cap C_{k-1}^{h_{C}}\big{)}$ $\displaystyle\geqslant$ $\displaystyle
1-2\sum_{j=1}^{p_{1}}\left[1-\Phi\left(\frac{h\lambda_{n}}{2n\sqrt{p_{1}}\sigma\|\mathbf{x}_{j}\|}\right)\right]-\big{[}1-{\mathrm{P}}(A_{k-1})\big{]}-\big{[}1-{\mathrm{P}}(B_{k})\big{]}$
$\displaystyle-\big{[}1-{\mathrm{P}}(C_{k-1}^{h_{C}})\big{]},$
where $h_{C}$ is defined in (50).
Since $1-\Phi(x)=o\big{(}\exp(-x^{2}/2)\big{)}$ as $x\rightarrow+\infty$,
(52), (53) and (54) imply that
$\displaystyle
1-P(A_{1})=o\big{(}\exp[-c_{1}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$
$\displaystyle
1-P(B_{1})=o\big{(}\exp(-c_{2}n)\big{)}=o\big{(}\exp[-c_{2}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$
$\displaystyle
1-P(C_{1}^{h})=o\big{(}\exp[-c_{3}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$
where $c_{1},\ c_{2}$ and $c_{3}$ are positive constants. By induction, it now
follows from (58), (59) and (60) that
$\displaystyle 1-P(A_{k})$ $\displaystyle=$
$\displaystyle\big{[}1-P(A_{k-1})\big{]}+\big{[}1-{\mathrm{P}}(C_{k-1}^{h_{A}})\big{]}+o\big{(}\exp[-c_{4}(\lambda_{n}/\sqrt{n})^{2}]\big{)}$
(61) $\displaystyle=$ $\displaystyle
o\big{(}k\exp[-c_{4}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$
where $c_{4}$ and $c_{5}$ are positive constants. Similarly,
$\displaystyle
1-P(B_{k})=o\big{(}k\exp[-c_{6}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$ (62)
$\displaystyle
1-P(C_{k}^{h})=o\big{(}k\exp[-c_{7}(\lambda_{n}/\sqrt{n})^{2}]\big{)},$ (63)
where $c_{6}$ and $c_{7}$ are positive constants. By (61), a sufficient
condition for ${\mathrm{P}}(A_{k})\rightarrow 1$ is
$k\exp\big{[}-c(\lambda_{n}/\sqrt{n})^{2}\big{]}\rightarrow 0$ (64)
for any $c>0$, which is covered by Assumption 3.
_Step 2_. Consider the asymptotic normality of
$\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}$. When the events $A_{k-1},\
A_{k},\ B_{k}$ and $B_{k+1}$ all occur, by (56),
$\displaystyle\frac{\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}}{n}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})=\frac{\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}}{n}+d(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k+1)}),$
(65)
$\displaystyle\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k+1)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}\|=\left\|\left(\boldsymbol{I}_{p_{1}}-\frac{\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}}{nd}\right)(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)})\right\|\leqslant\eta\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k-1)}\|.$
(66)
When the events $C_{1}^{1/2},\ C_{2}^{1/2},\ A^{(k)}=\bigcap_{i=1}^{k}A_{i}$
and $B^{(k+1)}=\bigcap_{i=1}^{k+1}B_{i}$ all occur, by (66),
$\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k+1)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}\|\leqslant\eta^{k-1}\|\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(2)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(1)}\|\leqslant\eta^{k-1}\lambda_{n}/n.$
(67)
For any $\mbox{\boldmath$\upalpha$\unboldmath}\in{\mathbb{R}}^{p}$ with
$\mbox{\boldmath$\upalpha$\unboldmath}\neq\mathbf{0}$, by (65) and (67),
$\displaystyle|\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})-\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}|$
$\displaystyle=$ $\displaystyle
dn^{3/2}|\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k+1)})|$
$\displaystyle\leqslant$ $\displaystyle
n^{-1/2}\lambda_{n}\eta^{k-1}d\|\mbox{\boldmath$\upalpha$\unboldmath}\|\|n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\|.$
For any $x\in{\mathbb{R}}$, note that
$\displaystyle\left|{\mathrm{P}}\big{(}\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})\leqslant
x\big{)}-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
(68) $\displaystyle\leqslant$
$\displaystyle\left|{\mathrm{P}}\big{(}\\{\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\mbox{\boldmath$\upbeta$\unboldmath}_{1}^{(k)}-\mbox{\boldmath$\upbeta$\unboldmath}_{1})\leqslant
x\\}\cap C_{1}^{1/2}\cap C_{2}^{1/2}\cap A^{(k)}\cap
B^{(k+1)}\big{)}-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
$\displaystyle+\big{[}1-{\mathrm{P}}(C_{1}^{1/2})\big{]}+\big{[}1-{\mathrm{P}}(C_{1}^{1/2})\big{]}+\big{[}1-{\mathrm{P}}(A^{(k)})\big{]}+\big{[}1-{\mathrm{P}}(B^{(k+1)})\big{]}$
$\displaystyle\leqslant$
$\displaystyle\left|{\mathrm{P}}\big{(}\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\leqslant
x-n^{-1/2}\lambda_{n}\eta^{k-1}d\|\mbox{\boldmath$\upalpha$\unboldmath}\|\|n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\|\big{)}-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
$\displaystyle+\left|{\mathrm{P}}\big{(}\sqrt{n}\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\boldsymbol{X}_{1}^{\prime}\mbox{\boldmath$\upvarepsilon$\unboldmath}\leqslant
x+n^{-1/2}\lambda_{n}\eta^{k-1}d\|\mbox{\boldmath$\upalpha$\unboldmath}\|\|n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\|\big{)}-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
$\displaystyle+3\big{[}1-{\mathrm{P}}(C_{1}^{1/2})\big{]}+3\big{[}1-{\mathrm{P}}(C_{1}^{1/2})\big{]}+3\big{[}1-{\mathrm{P}}(A^{(k)})\big{]}+3\big{[}1-{\mathrm{P}}(B^{(k+1)})\big{]}$
$\displaystyle=$
$\displaystyle\left|\Phi\left(\frac{x-n^{-1/2}\lambda_{n}\eta^{k-1}d\|\mbox{\boldmath$\upalpha$\unboldmath}\|\|n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\|}{\sigma[\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\mbox{\boldmath$\upalpha$\unboldmath}]^{1/2}}\right)-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
$\displaystyle+\left|\Phi\left(\frac{x+n^{-1/2}\lambda_{n}\eta^{k-1}d\|\mbox{\boldmath$\upalpha$\unboldmath}\|\|n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\|}{\sigma[\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}n(\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1})^{-1}\mbox{\boldmath$\upalpha$\unboldmath}]^{1/2}}\right)-\Phi\left(\frac{x}{\sigma(\mbox{\boldmath$\upalpha$\unboldmath}^{\prime}\boldsymbol{\Sigma}_{1}^{-1}\mbox{\boldmath$\upalpha$\unboldmath})^{1/2}}\right)\right|$
$\displaystyle+6\big{[}1-{\mathrm{P}}(C_{1}^{1/2})\big{]}+3\big{[}1-{\mathrm{P}}(A^{(k)})\big{]}+3\big{[}1-{\mathrm{P}}(B^{(k+1)})\big{]}.$
Now, under Assumption 3, by (61), (62) and (63), (68) converges to zero as
$n\rightarrow\infty$. This completes the proof.∎
ACKNOWLEDGEMENTS
Xiong is supported by grant 10801130 of the National Natural Science
Foundation of China. Dai is partially support by Grace Wahba through NIH grant
R01 EY009946, ONR grant N00014-09-1-0655 and NSF grant DMS-0906818. Qian is
partially supported by NSF grant CMMI 0969616, an IBM Faculty Award and an NSF
Faculty Early Career Development Award DMS 1055214. The authors thank Xiao-Li
Meng and Grace Wahba for comments and suggestions.
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|
arxiv-papers
| 2011-07-31T15:50:02 |
2024-09-04T02:49:21.148349
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shifeng Xiong, Bin Dai, and Peter Z. G. Qian",
"submitter": "Shifeng Xiong Doc",
"url": "https://arxiv.org/abs/1108.0185"
}
|
1108.0203
|
# Expanding Universe: slowdown or speedup?
Yu. L. Bolotin ybolotin@gmail.com O. A. Lemets oleg.lemets@gmail.com D. A.
Yerokhin denyerokhin@gmail.com A.I.Akhiezer Institute for Theoretical
Physics, National Science Center ”Kharkov Institute of Physics and
Technology”, Akademicheskaya Str. 1, 61108 Kharkov, Ukraine
###### Abstract
The purpose of the review - to provide affordable to a wider audience a
description of kinematics of the cosmological expansion, and dynamic
interpretation of this process. The focus will be the accelerated expansion of
the Universe. Will be considered by virtually all major opportunity to explain
the accelerated expansion of the Universe, including an introduction to the
energetic budget of the universe of so-called dark energy, a modification of
Einstein’s equations, and a new approach based on the holographic principle.
expanding Universe, deceleration parameter, cosmological acceleration, dark
energy
###### pacs:
98.80.-k
## I Introduction
There are two fundamental reasons making the question put in the title so
principal. Firstly, in common sense, the correct answer is necessary (though
insufficient) to predict the ultimate fate of the Universe. To be more
specific, one must know current values of the kinematic parameters (velocity,
acceleration etc.) in order to fix initial conditions necessary for solution
of the differential equations describing dynamics of the Universe. Secondly,
description of either decelerated or accelerated expansion of the Universe
requires absolutely distinct cosmological models. Conventional substances
(non-relativistic matter and radiation) from one hand and general relativity
from the other explain the decelerated expansion. In order to obtain the
accelerated expansion one must either change the composition of the Universe
rather radically, or make even more responsible conclusion: the fundamental
physical theories lying in the basis of our understanding of the World, are
essentially wrong. Common experience tells us that it is very unlikely to
answer the posed question in near future. Already $80$ years passed after the
discovery of the Universe expansion, and we still cannot establish the source
responsible for the initial velocity distribution. ”The Big Bang” is nothing
but euphemism used to hide limitation of our knowledge. What does actually
expand? This question is still sharply controversial: ”How can vacuum expand?”
RootofAllEvil . Even more miraculous, how can vacuum expand with acceleration
or deceleration? “…how is it possible for space, which is utterly empty, to
expand? How can nothing expand? The answer is: space does not expand.
Cosmologists sometimes talk about expanding space, but they should know
better.” Rees_Weinberg .
No generation of homo sapiens can vanquish the temptation to believe that it
is chosen to acquire complete and final understanding of Nature and Universe.
In particular, the first half of the last century, with discovery of Universe
expansion by Hubble and due to now fundamental physical theories, relativity
and quantum mechanics first of all, was marked by new cosmological paradigm —
the Big Bang model. The main basis of the model is the celebrated Hubble law,
discovered in twenties of the last century and supported by all types of up-
to-date cosmological observations. The Big Bang model was able to give
satisfactory explanation for thermal evolution of cooling Universe, predicted
existence of relict radiation, correctly described relative abundance of light
elements and many other features of the Universe.
Towards the end of the last century it was commonly hoped that the Big Bang
model supplemented by the Inflation Theory represented an adequate (at least
as a first approximation) model of Universe. However this hope was never to
come true. The cosmological paradigm had to change for the reason of
observations performed with ever increasing precision. Right after the Hubble
discovery the cosmologists tempted to measure the deceleration of the
expansion, attributed to gravitation. They were so firmly confident to
discover exactly above stated effect, that the corresponding observable was
named the deceleration parameter. However, in the year $1998$ two independent
collaborations Riess ; Perlmutter42 , studying distant supernovae, presented
convincing evidence for the fact that the Universe expansion is accelerated.
It turned out that the brightness decreases in average much faster than it was
commonly believed according to the Big Bang model. Such additional dimming
means that the distance, attributed to given redshift, was somewhat
underestimated. And it in turn means that the universe expansion accelerates:
in the past the Universe expanded slower than nowadays. The discovery of
cosmological acceleration is probably one of the most important observations
not only in modern cosmology but in physics in general. Accelerated expansion
of Universe the most directly demonstrates the fact that our fundamental
theories are either incomplete, or, even worse, misleading Trodden ; Smolin.
Physical origin of the cosmological acceleration still remains greatest
miracle. As we already mentioned above, if the Universe is solely composed of
matter and radiation, it must decelerate the expansion. If the expansion in
fact accelerates, we have to choices, each of which forces us to revise our
basic physical concepts.
1. 1.
Up to $75\%$ of energy density in the Universe exists in form of unknown
substance (commonly called the dark energy) with high negative pressure,
providing the accelerated expansion.
2. 2.
General Relativity theory must be revised on cosmological scales.
We should remark that, besides the two above cited radical possibilities to
put the theory in agreement with the observations, there still exist an
obvious conservative way to get rid of the problem: it implies more adequate
utilization of available theoretical possibilities. D.Wiltshire Wiltshire
expressed this idea in the following way: I will take the viewpoint that
rather than adding further epicycles to the gravitational action 111generally
speaking, ”epicycles” here stand not only for additional terms in the Hilbert-
Einstein action, but also for such new substances as dark energy and dark
matter the cosmological observations which we currently interpret in terms of
dark energy are inviting us to think more deeply about the foundations of
general relativity. The above mentioned term ”more adequate utilization of the
available possibilities” need an explanation. Let us give an example. In the
context of cosmological models based on homogeneous and isotropic Universe, in
order to explain the observed acceleration one have to involve a new form of
matter with negative pressure — the dark energy. According to an alternative
interpretation Wiltshire ; Buchert ; Ishibashi ; Backreaction , the
accelerated expansion of the Universe follows from deviations from the
homogeneity. It is assumed that mass density in the Universe is considerably
inhomogeneous on the scales smaller than the Hubble radius. In order to
transit to effective description of homogeneous and isotropic Universe, one
should average and/or smooth out the inhomogeneities up to some properly
chosen scale of the averaging. In such an averaged Universe one can define the
”effective cosmological parameters”. After that it turns out that so obtained
equations of motion in general differ from the equations with the same
parameters in the models based on the cosmological principle of uniform and
isotropic Universe. If the difference looks like an effective dark energy
contribution, then one can try to explain the accelerated expansion of
Universe in frames of General Relativity without any dark energy.
The present review considers all the three above listed possibilities. It aims
to present a comprehensive description for both the kinematics of the
cosmological expansion and the dynamical description of the process. We
concentrate our attention on the acceleration of the cosmological expansion.
It should be emphasized that the equivalence principle directly links the
acceleration with the gravity nature and space-time geometry. Therefore its
value is extremely important for verification of different cosmological
models. It is the quantity which to great extent determines the ultimate fate
of the Universe.
## II Cosmography — the kinematics of expanding Universe
This section is devoted to a particular way of Universe description, called
the ”cosmography”Weinberg_GC , entirely based on the cosmological principle.
The latter states that the Universe is homogeneous and isotropic on the scales
larger than hundreds megaparsecs, and it allows, among the diversity of models
for description of the Universe, to select a specific class of uniform and
isotropic ones. The most general space-time metrics consistent with the
cosmological principle is the Friedmann-Robertson-Walker (FRW) one:
$ds^{2}=dt^{2}-a^{2}(t)\left\\{\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right\\}.$ (1)
Here $a(t)$ is the scale factor and $r$ is coordinate of a point, which does
not take part in any motion except the global expansion of the Universe. In
some sense the main problem of cosmology is to determine the dependence
$a(t).$
The cosmological principle helps to construct the metrics of the Universe and
to make the first steps in interpretation of cosmological observations. Recall
that kinematics means the part of mechanics that describes motion of bodies
irrelative to forces causing it. In that sense the cosmography is nothing but
kinematics of cosmological expansion. In order to construct the principal
cosmological quantity — the time dependence of the scale factor $a(t)$ — one
have to have the equations of motion (the Einstein equations of general
relativity) and an assumption on material composition of the Universe,
allowing to construct the energy-momentum tensor. The efficiency of
cosmography lies in the fact that it allows to verify arbitrary cosmological
models obeying the cosmological principle. Modifications of general relativity
or introduction of new components (dark matter or dark energy) will certainly
change the dependence $a(t),$ but will not affect the relations between the
kinematic characteristics. Velocity of the Universe expansion, determined by
the Hubble parameter $H(t)\equiv{\dot{a}(t)}/{a(t)},$ depends on time. The
time dependence of the latter is measured by the deceleration parameter
$q(t).$ Let us define it by Taylor expansion of the time dependence of the
scale factor $a(t)$ in vicinity of the current moment of time $t_{0}$:
$a(t)=a\left({{t}_{0}}\right)+\dot{a}\left({{t}_{0}}\right)\left[t-{{t}_{0}}\right]+\frac{1}{2}\ddot{a}\left({{t}_{0}}\right){{\left[t-{{t}_{0}}\right]}^{2}}+\cdots$
(2)
Let us rewrite the relation in the following form
$\frac{a(t)}{a\left(t_{0}\right)}=1+H_{0}\left[t-t_{0}\right]-\frac{q_{0}}{2}H_{0}^{2}{{\left[t-{{t}_{0}}\right]}^{2}}+\cdots,$
(3)
where the deceleration parameter reads
$q(t)\equiv-\frac{\ddot{a}(t)a(t)}{\dot{a}^{2}(t)}=-\frac{\ddot{a}(t)}{a(t)}\frac{1}{H^{2}(t)}.$
(4)
As will be shown below, accelerated growth of the scale factor occurs at
$q<0,$ while accelerated growth of the expansion velocity $\dot{H}>0$
corresponds to $q<-1.$ When the sign of the deceleration parameter was first
defined, it seemed evident that gravitation — the only force governing the
dynamics of Universe — dumps its expansion. A natural desire to deal with a
positive parameter predetermined the choice of the sign. Afterwards it turned
out that the choice made is inconsistent with the observed dynamics of
expansion and it is rather an example of historical curiosity.
For more detailed kinematical description of cosmological expansion it is
useful to consider the extended parameter set Visser_Cosmography ; Visser_Jerk
$\displaystyle H(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a}\frac{da}{dt};$ $\displaystyle q(t)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{a}\frac{{{d}^{2}}a}{d{{t}^{2}}}{{\left[\frac{1}{a}\frac{da}{dt}\right]}^{-2}};$
$\displaystyle j(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a}\frac{{{d}^{3}}a}{d{{t}^{3}}}{{\left[\frac{1}{a}\frac{da}{dt}\right]}^{-3}};$
$\displaystyle s(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a}\frac{{{d}^{4}}a}{d{{t}^{4}}}{{\left[\frac{1}{a}\frac{da}{dt}\right]}^{-4}};$
$\displaystyle l(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a}\frac{{{d}^{5}}a}{d{{t}^{5}}}{{\left[\frac{1}{a}\frac{da}{dt}\right]}^{-5}}.$
Note that the latter four parameters are dimensionless. Derivatives of lower
parameters can be expressed through higher ones. So, for example,
$\frac{dq}{d\ln(1+z)}=j-q(2q+1).$
Let us Taylor expand the time dependence of the scale factor using the above
introduced parameters
$\begin{matrix}a(t)={{a}_{0}}\left[\right.1+{{H}_{0}}\left(t-{{t}_{0}}\right)-\frac{1}{2}{{q}_{0}}H_{0}^{2}{{\left(t-{{t}_{0}}\right)}^{2}}+\\\
\frac{1}{3!}{{j}_{0}}H_{0}^{3}{{\left(t-{{t}_{0}}\right)}^{3}}+\frac{1}{4!}{{s}_{0}}H_{0}^{4}{{\left(t-{{t}_{0}}\right)}^{4}}+\frac{1}{5!}{{l}_{0}}H_{0}^{5}{{\left(t-{{t}_{0}}\right)}^{5}}\
+{\mathrm{O}}\left({{\left(t-{{t}_{0}}\right)}^{6}}\right)\left.\right].\\\
\end{matrix}$ (5)
In terms of the same parameters the Taylor series for redshift reads
$1+z={{\left[\begin{matrix}&1+{{H}_{0}}(t-{{t}_{0}})-\frac{1}{2}{{q}_{0}}H_{0}^{2}{{(t-{{t}_{0}})}^{2}}+\frac{1}{3!}{{j}_{0}}H_{0}^{3}{{\left(t-{{t}_{0}}\right)}^{3}}+\frac{1}{4!}{{s}_{0}}H_{0}^{4}{{\left(t-{{t}_{0}}\right)}^{4}}\\\
&+\frac{1}{5!}{{l}_{0}}H_{0}^{5}{{\left(t-{{t}_{0}}\right)}^{5}}\
+{\mathrm{O}}\left({{\left(t-{{t}_{0}}\right)}^{6}}\right)\\\
\end{matrix}\right]}^{-1}};$ (6)
$z={{H}_{0}}({{t}_{0}}-t)+\left(1+\frac{{{q}_{0}}}{2}\right)H_{0}^{2}{{(t-{{t}_{0}})}^{2}}+\cdots.$
(7)
Let us cite few useful relations for the deceleration parameter
$\displaystyle q(t)$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\left(\frac{1}{H}\right)-1;$ $\displaystyle q(z)$
$\displaystyle=$ $\displaystyle\frac{1+z}{H}\frac{dH}{dz}-1;$ $\displaystyle
q(z)$ $\displaystyle=$ $\displaystyle\frac{d\ln H}{dz}(1+z)-1;$ $\displaystyle
q(a)$ $\displaystyle=$
$\displaystyle-\left(1+\frac{\frac{dH}{dt}}{{{H}^{2}}}\right)=-\left(1+\frac{a\frac{dH}{da}}{H}\right);$
$\displaystyle q$ $\displaystyle=$ $\displaystyle-\frac{d\ln(aH)}{d\ln a}.$
For the case of single component liquid with density $\rho$ one has
$q(a)=-1-\frac{a\frac{d\rho}{da}}{2\rho}.$ (8)
The derivatives $\frac{dH}{dz},$ $\frac{{{d}^{2}}H}{d{{z}^{2}}},$
$\frac{d^{3}H}{dz^{3}}$ and $\frac{d^{4}H}{dz^{4}}$ can be expressed through
the parameters $q$ and $j$ solely
$\displaystyle\frac{dH}{dz}$ $\displaystyle=$ $\displaystyle\frac{1+q}{1+z}H;$
$\displaystyle\frac{{{d}^{2}}H}{d{{z}^{2}}}$ $\displaystyle=$
$\displaystyle\frac{j-q^{2}}{\left(1+z\right)^{2}}H;$
$\displaystyle\frac{d^{3}H}{dz^{3}}$ $\displaystyle=$
$\displaystyle\frac{H}{(1+z)^{3}}\left(3q^{2}+3q^{3}-4qj-3j-s\right);$
$\displaystyle\frac{d^{4}H}{dz^{4}}$ $\displaystyle=$
$\displaystyle\frac{H}{(1+z)^{4}}\left(-12q^{2}-24q^{3}-15q^{4}+32qj+25q^{2}j+7qs+12j-4j^{2}+8s+l\right).$
Let us present for convenience the useful relations for transformations from
higher time derivatives to derivatives with respect to red shift:
$\displaystyle\frac{d^{2}}{dt^{2}}=(1+z)H\left[H+(1+z)\frac{dH}{dz}\right]\frac{d}{dz}+(1+z)^{2}H^{2}\frac{d^{2}}{dz^{2}};$
(9) $\displaystyle\frac{d^{3}}{dt^{3}}$ $\displaystyle=$
$\displaystyle-(1+z)H\left\\{H^{2}+(1+z)^{2}\left(\frac{dH}{dz}\right)^{2}+(1+z)H\left[4\frac{dH}{dz}+(1+z)\frac{d^{2}H}{dz^{2}}\right]\right\\}\frac{d}{dz}-3(1+z)^{2}H^{2}$
(10) $\displaystyle\times$
$\displaystyle\left[H+(1+z)\frac{dH}{dz}\right]\frac{d^{2}}{dz^{2}}-(1+z)^{3}H^{3}\frac{d^{3}}{dz^{3}};$
$\displaystyle\frac{d^{4}}{dt^{4}}$ $\displaystyle=$
$\displaystyle(1+z)H\left[H^{2}+11(1+z)H^{2}\frac{dH}{dz}+11(1+z)H\frac{dH}{dz}+(1+z)^{3}\left(\frac{dH}{dz}\right)^{3}+7(1+z)^{2}H\frac{d^{2}H}{dz^{2}}\right.$
(11) $\displaystyle+$
$\displaystyle\left.4(1+z)^{3}H\frac{dH}{dz}\frac{d^{2}H}{d^{2}z}+(1+z)^{3}H^{2}\frac{d^{3}H}{d^{3}z}\right]\frac{d}{dz}+(1+z)^{2}H^{2}\left[7H^{2}+22H\frac{dH}{dz}+7(1+z)^{2}\left(\frac{dH}{dz}\right)^{2}\right.$
$\displaystyle+$
$\displaystyle\left.4H\frac{d^{2}H}{dz^{2}}\right]\frac{d^{2}}{dz^{2}}+6(1+z)^{3}H^{3}\left[H+(1+z)\frac{dH}{dz}\right]\frac{d^{3}}{dz^{3}}+(1+z)^{4}H^{4}\frac{d^{4}}{dz^{4}}+(1+z)^{4}H^{4}\frac{d^{4}}{dz^{4}};$
Derivatives of the Hubble parameter squared with respect to the red shift
$\frac{{{d^{(i)}}{H^{2}}}}{{d{z^{(i)}}}},\;i=1,2,3,4$ expressed through the
cosmographic parameters take the form
$\frac{d(H^{2})}{dz}=\frac{2H^{2}}{1+z}(1+q)$
$\frac{d^{2}(H^{2})}{dz^{2}}=\frac{2H^{2}}{(1+z)^{2}}(1+2q+j);$
$\frac{d^{3}(H^{2})}{dz^{3}}=\frac{2H^{2}}{(1+z)^{3}}(-qj-s);$
$\frac{d^{4}(H^{2})}{dz^{4}}=\frac{2H^{2}}{(1+z)^{4}}(4qj+3qs+3q^{2}j-j^{2}+4s+l).$
Time derivatives of the Hubble parameter can be expressed through above
defined cosmographic parameters $H,q,j,s,l$ as
$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle-{{H}^{2}}(1+q);$ (12)
$\displaystyle\ddot{H}$ $\displaystyle=$
$\displaystyle{{H}^{3}}\left(j+3q+2\right);$ $\displaystyle\dddot{H}$
$\displaystyle=$ $\displaystyle{{H}^{4}}\left[s-4j-3q(q+4)-6\right];$
$\displaystyle\dddot{H}$ $\displaystyle=$
$\displaystyle{{H}^{5}}\left[l-5s+10\left(q+2\right)j+30(q+2)q+24\right].$
From the relations (12) one can easily see that the accelerated growth of the
acceleration velocity $\dot{H}>0$ corresponds to the case $q<-1.$
The Hubble parameter, as can be seen from the relation (II), is linked to the
deceleration parameter with the integral relation
$H={{H}_{0}}\exp\left[\int_{0}^{z}{\left[q({z}^{\prime})+1\right]d\ln(1+{z}^{\prime})}\right].$
It immediately follows that in order to calculate the principal characteristic
of the expanding Universe $H(z)$ one needs the information on dynamics of
cosmological expansion, coded in the quantity $q(z).$
## III Dynamics of cosmological expansion briefly
Dynamics of Universe in frames of General Relativity is described by the
Einstein equations
${{R}_{\mu\nu}}-\frac{1}{2}R{{g}_{\mu\nu}}=8\pi G{{T}_{\mu\nu}}.$
The energy-momentum tensor ${{T}_{\mu\nu}}$ describes the space distribution
of mass(energy), while the curvature tensor components ${{R}_{\mu\nu}}$ and
its trace $R$ are expressed through the metric tensor ${{g}_{\mu\nu}}$ and its
derivatives of first and second order. The Einstein equations generally are
non-linear and hard to analyze. The problem is simplified when considering
mass distribution with special symmetry properties provided by the metric. For
homogeneous and isotropic Universe described by the FRW-metric, the Einstein
equations reduce to the system of Friedmann equations:
$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{k}{a^{2}},$
(13) $\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+3p\right).$ (14)
Here $\rho$ and $p$ are total energy density and pressure respectively for all
the components present in the Universe at the considered time moment.
The system (13,14) is sufficient for complete description of the Universe
dynamics. The conservation equation
$\dot{\rho}+3H(\rho+p)=0,$ (15)
follows from the Lorentz-invariance of the energy-momentum tensor
$T_{\nu,\mu}^{\mu}=0$ and, as can be easily seen, it is nothing but the first
principle of thermodynamics for ideal liquid with constant entropy $dE+pdV=0.$
We remark, that this equation can be derived from the Friedman equations
(13,14).
Let us now consider relative density for $i$th component of the Universe:
${{\Omega}_{i}}\equiv\frac{{{\rho}_{i}}}{{{\rho}_{c}}},\,\quad{{\rho}_{c}}\equiv\frac{3{{H}^{2}}}{8\pi
G},\quad\Omega\equiv\sum\limits_{i}{{{\Omega}_{i}}.}$ (16)
Setting the relative curvature density by definition
${{\Omega}_{k}}\equiv-\frac{k}{{{a}^{2}}{{H}^{2}}},$ the first Friedmann
equation can be presented in the form
$\sum\limits_{i}{{{\Omega}_{i}}=1}.$
In order to solve the Friedmann equations one has to define the matter
composition of the Universe and construct the state equation for each
component. In the simplest linear form the state equation reads
$p_{i}=w_{i}\rho_{i}.$ (17)
Solving the Friedmann equations for the case $w=const,$ $k=0$ one obtains
$a(t)\propto{{\left(t/{{t}_{0}}\right)}^{\frac{2}{3(1+w)}}},\
\rho\propto{{a}^{-3(1+w)}}.$
(The scale factor is normalized by the condition $a\left({{t}_{0}}\right)=1.$)
The above cited solutions exist only provided $w\neq-1,$ the otherwise case
will be considered separately. For the Universe dominated by radiation
(relativistic gas of photons and neutrinos) one has $w=1/3,$ while for the
matter dominated case $w=0.$ As a result of such state equations, for the
matter dominated case one gets
$a(t)\propto{{\left(t/{{t}_{0}}\right)}^{2/3}},\ \rho\propto{{a}^{-3}}.$
which is easily interpreted as the particle number conservation law. For the
radiation-dominated case the solution reads
$\quad
a(t)\propto{{\left(t/{{t}_{0}}\right)}^{1/2}},\quad\rho\propto{{a}^{-4}}.$
The latter result is a consequence of the fact that the energy density of
radiation decreases as ${{a}^{-3}}$ due to growth of occupied volume (as
Universe expands) and additionally as ${{a}^{-1}}$ due to the redshift. Note
that from the equation (15) it follows that $\rho=const$ for $w=-1.$ In the
latter case the hubble parameter is constant leading to exponential growth of
the scale factor
$a(t)\propto{{e}^{Ht}}.$
Therefore in the case of traditional cosmological components in form of matter
and radiation with $w=0$ and $w=1/3$ respectively, the Universe expansion can
only decelerate, i.e. $\ddot{a}<0.$
Using the definition of deceleration parameter, one can see that for flat
Universe filled by single component with state equation $p=w\rho$
$q=\frac{1}{2}(1+3w).$
In generic case $(k=(0,\pm 1),\,\quad\rho=\sum\limits_{i}{{{\rho}_{i}}},\quad
p=\sum\limits_{i}{{{\rho}_{i}}}{{w}_{i}})$ one gets
$q=\frac{\Omega}{2}+\frac{3}{2}\sum\limits_{i}{{{w}_{i}}}{{\Omega}_{i}}.$ (18)
Using (16), the latter relation can be rewritten in the form
$q=\frac{1}{2}\left(1+\frac{k}{{{a}^{2}}{{H}^{2}}}\right)\left(1+3\frac{p}{\rho}\right).$
Since the deceleration parameter $q$ is a slowly varying quantity (e.g.
$q=1/2$ for matter-dominated case and $q=-1$ in the Universe dominated by dark
energy in form of cosmological constant), then the useful information is
contained in its time average value, which is very interesting to obtain
without integration of the equations of motions for the scale factor. Let us
see how it is possible LimaAge . Let us define the time average $\bar{q}$ on
the interval $\left[0,t_{0}\right]$ by the expression
$\bar{q}\left(t_{0}\right)=\frac{1}{t_{0}}\int_{0}^{t_{0}}{q(t)dt}.$
Substituting the definition of the deceleration parameter
$q(t)=-\frac{\ddot{a}a}{{{{\dot{a}}}^{2}}}=\frac{d}{dt}\left(\frac{1}{H}\right)-1,$
it is easy to see that
$\bar{q}(t_{0})=-1+\frac{1}{{{t}_{0}}{{H}_{0}}}$ (19)
or, equivalently,
$t_{0}=\frac{H_{0}^{-1}}{1+\bar{q}}.$ (20)
As was expected, the present age of universe scales as $H_{0}^{-1},$ but the
proportionality coefficient depends only on the average value of the
deceleration parameter. It is worth to mention that this purely kinematic
result does not dependent on curvature of space in the Universe, neither on
number of component filling it, nor on the particular type of the gravity
theory. Let us somewhat reformulate the obtained results for the average
deceleration parameter. For single-component flat Universe the Friedman
equations can be presented in the following form
$\displaystyle 8\pi G\rho$ $\displaystyle=$ $\displaystyle
3\frac{\dot{a}^{2}}{a^{2}};$ (21) $\displaystyle 8\pi Gp$ $\displaystyle=$
$\displaystyle-2\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}.$ (22)
For the single component Universe with the state equation $p=w\rho,\quad
w=const$ the equation for the scale factor
$a\ddot{a}+\left(\frac{1+3w}{2}\right){{\dot{a}}^{2}}=0,$
has an exact solution
$a(t)={{a}_{0}}{{\left[\frac{3}{2}\left(1+w\right){{H}_{0}}t\right]}^{\frac{2}{3(1+w)}}}.$
(23)
Now it immediately follows that
$q=\frac{1+3w}{2}=const,\quad t_{0}=\frac{2H_{0}^{-1}}{3(1+w)}.$ (24)
In particular, one has for the cases of radiation $\left(w=1/3\right),~{}q=1$
and non-relativistic matted $\left(w=0\right),~{}q=1/2.$ From the other hand,
accounting that in the considered case $q=\bar{q},$ the latter relation can be
rewritten as $t_{0}=H_{0}^{-1}/\left(1+\bar{q}\right),$ which coincides with
the relation (20). The resulting expression (24) can be presented in the
following form
$T=\frac{H^{-1}}{1+\bar{q}},$ (25)
where $T,H,\bar{q}$ are age of Universe, Hubble parameter and average
deceleration parameter respectively. Since $\bar{q}$ is of order of unity,
then it immediately follows from (23) that on any stage of Universe evolution
the Hubble time $H_{0}^{-1}$ represents the characteristic time scale.
Further model-independent dynamical restrictions on the Universe kinematics
can be derived from the so-called energy conditions HawkingEllis ;
CarrollSpacetime ; VisserBarcelro ; SantosAlcaniz ; Ming-JianZhang . Those
conditions based on rather general physical principles, impose restrictions on
the energy-momentum tensor components $T_{\mu\nu}.$ Specifying a particular
model for the medium222specific model does not specify the state equation!
those conditions can be transformed into inequalities, limiting possible
values of pressure and density in the medium. In the Friedmann model the
medium is an ideal fluid, where
$T_{\mu\nu}=\left(\rho+p\right){{u}_{\mu}}{{u}_{\nu}}-p{{g}_{\mu\nu}},$ (26)
where $u_{\mu}$ is the fluid four-velocity, with total density $\rho$ and
pressure $p$ given, respectively, by
$\rho=\frac{3}{8\pi
G}\left(\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}+\frac{k}{{{a}^{2}}}\right),~{}~{}p=-\frac{1}{8\pi
G}\left(2\frac{{\ddot{a}}}{a}+\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}+\frac{k}{{{a}^{2}}}\right).$
(27)
The most common energy conditions (see, e.g., HawkingEllis ; CarrollSpacetime
; VisserBarcelro ; SantosAlcaniz ) reduce to
$\displaystyle NEC$ $\displaystyle\Rightarrow$ $\displaystyle\rho+p\geq 0;$
$\displaystyle WEC$ $\displaystyle\Rightarrow$ $\displaystyle\rho\geq 0\quad
and\quad\rho+p\geq 0;$ $\displaystyle SEC$ $\displaystyle\Rightarrow$
$\displaystyle\rho+3p\geq 0,and\quad\rho+p\geq 0;$ $\displaystyle DEC$
$\displaystyle\Rightarrow$ $\displaystyle\rho\geq 0\quad and\quad-\rho\leq
p\leq\rho,$
where NEC, WEC, SEC and DEC correspond, respectively, to the null, weak,
strong and dominant energy conditions. Because these conditions do not require
a specific equation of state of the matter in the Universe, they provide very
simple and model-independent bounds on the behavior of the energy density and
pressure. Therefore, the energy conditions are among of many approaches to
understand the evolution of Universe. From Eqs. (27), one easily obtains that
these energy conditions can be translated into the following set of dynamical
constraints relating the scale factor
$\displaystyle NEC$ $\displaystyle\Rightarrow$
$\displaystyle-\frac{{\ddot{a}}}{a}+\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}+\frac{k}{{{a}^{2}}}\geq
0;$ $\displaystyle WEC$ $\displaystyle\Rightarrow$
$\displaystyle\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}+\frac{k}{{{a}^{2}}}\geq 0;$
(28) $\displaystyle SEC$ $\displaystyle\Rightarrow$
$\displaystyle\frac{{\ddot{a}}}{a}\leq 0;$ $\displaystyle DEC$
$\displaystyle\Rightarrow$
$\displaystyle\frac{{\ddot{a}}}{a}+2\left[\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}+\frac{k}{{{a}^{2}}}\right]\geq
0.$
For flat Universe the conditions (28) can be transformed to restrictions on
the deceleration parameter $q$
$\displaystyle NEC$ $\displaystyle\Rightarrow$ $\displaystyle q\geq-1;$
$\displaystyle SEC$ $\displaystyle\Rightarrow$ $\displaystyle q\geq 0;$ (29)
$\displaystyle DEC$ $\displaystyle\Rightarrow$ $\displaystyle q\leq 2,$
while WEC is always satisfied for arbitrary real $a(t).$
The conditions (29), considered separately, leave a principal possibility for
both decelerated $\left(q>0\right)$ and accelerated $\left(q<0\right)$
expansion of Universe. The sense of the restrictions for NEC in (29) is quite
clear. As it follows from the second Friedmann equation, the condition of
accelerated expansion of Universe reduce to the inequality $\rho+3p\leq 0,$
i.e. accelerated expansion of Universe is possible only in presence of
components with high negative pressure $p<-1/3\rho.$ The energetic condition
$SEC$ excludes existence of such components. As a result $q\geq 0$ in that
case. At the same time the conditions $NEC$ and $DEC$ are compatible with the
condition $p<-1/3\rho$ and therefore regimes with $q<0$ are allowed here. In
conclusion of the section let us pay attention to an interesting feature of
the expanding Universe dynamics. According to Hubble law, the galaxies
situated on the Hubble sphere, recede from us with light speed. Velocity of
the Hubble sphere itself equals to time derivative of the Hubble radius
${{R}_{H}}=c/H,$
$\frac{d}{dt}\left({{R}_{H}}\right)=c\frac{d}{dt}\left(\frac{1}{H}\right)=-\frac{c}{{{H}^{2}}}\left(\frac{{\ddot{a}}}{a}-\frac{{{{\dot{a}}}^{2}}}{{{a}^{2}}}\right)=c(1+q).$
(30)
In the Universe with decelerated expansion $(q>0)$ the Hubble sphere has
superluminal velocity $c(q+1)$ and it outruns those galaxies. Therefore the
galaxies initially situated outside the Hubble sphere, gradually enter inside
it. Any observer in arbitrary point of the Universe will see that the number
of galaxies constantly increases. In the case of accelerated expanding
Universe $(q<0)$ the Hubble sphere has subluminal $c(1-|q|)$ and it lags
behind the galaxies. Therefore now the galaxies initially contained in the
Hubble sphere will gradually leave it and become unaccessible for observation.
Should we consider those unobservable galaxies as physically real? The
distinction between physics and metaphysics is the possibility of experimental
verification for the physical theories. Physics have no deal with unobservable
objects. However, the science constantly expands its boundaries, thus
including to consideration more and more abstract concepts, once been
metaphysical before: atoms, electromagnetic waves, black holes… The list can
be continued further.
We are most likely inhabitants of accelerated expanding Universe. Likewise in
the decelerated expanding one, there are galaxies so distant from us that no
signal from them can be detected by terrestrial observer. However, if the
cosmic expansion accelerates, then we recede from those galaxies with
superluminal velocity. Therefore, if their light have not reach us till now
then it will never come. Such galaxies are unaccessible for observation not
only temporarily, they are forever unobservable. Such ”never observable
galaxies” descent from the same ”Big Bang” like our Milky Way. Should we
attribute them to physical objects or to metaphysics? Those who consider the
science fiction as a realization of most unbounded fantasy, are absolutely
wrong. Compared to modern cosmology, the science fiction is dull and lacks
imagination.
Accelerated expansion of the Universe was first introduced in cosmological
models with creation of the inflation theory. It was developed to fix multiple
defeats of the Big Bang model. It turns out that in order to resolve most of
them it is sufficient to have exponentially fast accelerated expansion of
Universe at the very beginning of its evolution, during just about
$10^{-35}s.$ The simplest way to obtain expansion of such type is to consider
dynamics of Universe with scalar field. The inflation theory is formulated in
many ways, starting from the models based on quantum gravity and high-
temperature phase transitions theory with supercooling and exponential
expansion in the false vacuum state. To illustrate the main ideas of the
theory let us consider flat, homogeneous and isotropic Universe filled by the
scalar field $\varphi$ with the potential $V(\varphi),$ independent of the
coordinates. The first Friedmann equation (13) in the considered case takes
the following form
$H^{2}=\frac{8\pi}{3M_{Pl}^{2}}\left(\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)\right).$
The conservation equation (15), written in terms of the scalar field, takes
the form of Klein-Gordon equation, and in the case of non-stationary
background one obtains:
$\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=0.$ (31)
In the fast expanding Universe the scalar field rolls down very slowly, like a
ball in viscous medium, while the effective viscosity turns out to be
proportional to the expansion velocity. In the slow-roll regime
$H\dot{\varphi}\gg\ddot{\varphi},\quad
V\left(\varphi\right)\gg\dot{\varphi}^{2}.$
In the same limit the equations of motion take the form
$\displaystyle 3H\dot{\varphi}+V^{\prime}(\varphi)=0;$ $\displaystyle
H^{2}=\frac{{8\pi}}{{3M_{Pl}^{2}}}V(\varphi).$
To be more specific, let us consider the simplest model of the scalar field
with mass $m$ and potential energy $V(\varphi)=\frac{{m^{2}}}{2}\varphi^{2}.$
Right after the start of inflation $\ddot{\varphi}\ll
3H\dot{\varphi};~{}\dot{\varphi}^{2}\ll m^{2}\varphi^{2},$ therefore
$\displaystyle 3\frac{{\dot{a}}}{a}\dot{\varphi}+m^{2}\varphi=0;$
$\displaystyle
H=\frac{{\dot{a}}}{a}=\frac{{2m\varphi}}{{M_{Pl}}}\sqrt{\frac{\pi}{3}}.$
Due to fast growth of the scale factor and slow variation of the field because
of strong friction, one has
$a\propto e^{Ht},\quad H=\frac{2m\varphi}{M_{Pl}}\sqrt{\frac{\pi}{3}}.$
For better clarity let us obtain the state equation for the scalar field in
the slow-roll regime. For homogeneous scalar field in the potential
$V\left(\varphi\right)$ in the local Lorentz frame the non-zero components of
the energy momentum tensor are
$T_{00}=\frac{1}{2}\dot{\varphi}^{2}+V\left(\varphi\right)=\rho_{\varphi};\;T_{ij}=\left({\frac{1}{2}\dot{\varphi}^{2}-V\left(\varphi\right)}\right)\delta_{ij}=p_{\varphi}\delta_{ij}.$
In the slow-roll regime $\dot{\varphi}^{2}\ll V(\varphi)$ and, consequently,
$p_{\varphi}\approx-\rho_{\varphi}.$ Therefore the energy-momentum tensor in
the slow-roll regime approximately coincides with the vacuum one with
$p=-\rho.$ Using (18) and taking into account the fact that the inflation
scenery with exponential growth of the scale factor leads to the flat space
case, and therefore $\Omega=1,$ one obtains that $q=-1$ during the period of
the inflation expansion.
Therefore the scalar field provides rather natural way to obtain the
accelerated expansion of Universe at least on early stages of its evolution.
As the field intensity decrease in the slow-roll regime, viscosity falls down
and the Universe comes out the inflation regime with exponential growth of the
scale factor.
We remark that the scalar field can provide decelerated expansion of Universe
as well as the accelerated one. Near the minimum of the inflation potential
the inflation conditions definitely violate and the Universe leaves the
inflation regimes. The scalar field start to oscillate near the minimum.
Assuming that period of the oscillations is much less than the cosmological
time scale, one can neglect the expansion term in the equation (31), and it
easy to come up with effective state equation near the minimum of inflation
potential. Let us rewrite the scalar field equation(31) in the form
$\frac{d}{{dt}}\left({\varphi\dot{\varphi}}\right)-\dot{\varphi}^{2}+\varphi
V^{\prime}_{\varphi}=0.$
After averaging over the oscillation period the first term turns to zero and
then
$\left\langle{\dot{\varphi}^{2}}\right\rangle\simeq\left\langle{\varphi
V^{\prime}_{\varphi}}\right\rangle.$
The effective (averaged) state equation now reads
$w\equiv\frac{p}{\rho}\simeq\frac{{\left\langle{\varphi
V^{\prime}_{\varphi}}\right\rangle-\left\langle{2V}\right\rangle}}{{\left\langle{\varphi
V^{\prime}_{\varphi}}\right\rangle+\left\langle{2V}\right\rangle}}.$
For the case of quadratic potential $V\propto\varphi^{2}$ one obtains $w\simeq
0,$ which corresponds to the state equation of non-relativistic matter.
We note also that the scalar field models are very wide-spread in modern
cosmology, allowing to describe not only accelerated expansion, but much more
complicated dynamics of Universe. Besides, most of the scalar field models are
well motivated in the particle physics and concurring theories.
## IV Evidence for the accelerated expansion of Universe
The Hubble law tells nothing about the magnitude, sign or very possibility of
non-uniform expansion of Universe. The approximation where it works is
insensible to acceleration. Investigation of the non-linear effects requires
high red-shift data. If the observation detect deviations from the linear law,
then the magnitude and sign of the deviation allow to judge about the sign of
the cosmological acceleration. If the detected deviation lies towards larger
distances at fixed redshift then the acceleration is positive. The distance is
estimated by brightness of the source in assumption that the considered kind
of sources represent the standard candles — an ensemble of object with
practically equal luminosity. Therefore the observed brightness of the objects
depends only on the distance to observer. The supernova outbursts of type Ia
(exploding white dwarfs) present an example of such kind of objects. As the
white dwarfs have very low mass dispersion, their luminosity is practically
the same. An additional advantage is the huge power of
$\left(\sim{{10}^{36}}W\right),$ released in explosion. Thanks to that, they
can be detected on distances compatible to size of the observable Universe.
Given the emitted light intensity $L,$ or internal luminosity of the object,
and having measured the light intensity $F,$ which have reached us, or
observed flow, one can calculate the distance to the object. The quantity
defined this way is called the luminosity distance ${d}_{L}$
$d_{L}^{2}=\frac{L}{4\pi F}.$ (32)
In order to determine the acceleration of the Universe expansion one needs to
express the luminosity distance in terms of the redshift for the registered
radiation. Let $E$ be internal (absolute) luminosity of some source. A
terrestrial observer registers the photon flow $F.$ Increasing of the photon
wavelength, and therefore decrease of its energy, in the expanding Universe
during its pass from the source to the observer results in effective
(apparent) luminosity of the source $L=E/a(t).$ The conservation law for the
energy emitted on time interval $dt$ and absorbed on the interval
$d{{t}_{0}},$ implies
$F4\pi{{r}^{2}}d{{t}_{0}}=Edt=La(t)dt,$ (33)
where $r$ is the comoving distance between the source and the observer at time
moment $t_{0},$ which coincides with the physical distance in normalization of
the form $a\left(t_{0}\right)=1.$ Since the comoving distance between source
and observer does not change, the conformal time interval $d\eta=dt/a$ between
two light pulses is the same at the point of emission and at the point
observation
$\frac{dt}{a(t)}=\frac{dt_{0}}{a_{0}}.$
Therefore from (33) follows that
$F=\frac{La^{2}(t)}{4\pi r^{2}}.$ (34)
Comparing this expression with the definition of the luminosity distance (32),
one finds
$d_{L}=\frac{r}{a(t)}=(1+z)r.$ (35)
The obtained result is physically clear. In the expanding Universe the
registered flow decreases by factor $(1+z)^{2}:$ first, due to increase of the
photon wave length $(1+z)$ times and, second, due to increase of the time
interval for arrival of the fixed energy portion also $(1+z)$ times.
Let us now determine the comoving distance to the currently observed light
source as the function of its redshift. The equation of motion for photon is
$ds^{2}=0.$ Let us consider a radial trajectory with the observer at the
origin of the coordinates. In the case of spatially flat metrics one has
$ds^{2}=a^{2}(t)\left(d\eta^{2}-dr^{2}\right)=0.$ (36)
Accounting that
$d\eta=\frac{d\eta}{dt}\frac{dt}{da}\frac{da}{dz}dz=-\frac{dz}{H(z)},$ (37)
one finds
$r(z)=\int_{0}^{z}{\frac{dz^{\prime}}{H(z^{\prime})}}.$ (38)
Therefore for spatially flat Universe
$d_{L}=(1+z)\int_{0}^{z}{\frac{d{z}^{\prime}}{H({z}^{\prime})}}.$ (39)
In general case,
$d_{L}(z)=c(1+z){{\left(1-{{\Omega}_{0}}\right)}^{-1/2}}H_{0}^{-1}S\left[{{\left(1-{{\Omega}_{0}}\right)}^{1/2}}{{H}_{0}}\int_{0}^{z}{\frac{d{z}^{\prime}}{H({z}^{\prime})}}\right],$
(40)
where
$S(x)=\left\\{\begin{array}[]{c}\sin x,\quad{{\Omega}_{0}}>1;\\\
x,\quad\quad{{\Omega}_{0}}=1;\\\ \sinh x,\ {{\Omega}_{0}}<1.\\\
\end{array}\right.$ (41)
The quantities $H_{0},\,\Omega_{0}\equiv\frac{\rho_{0}}{\rho_{cr}},\
\left(\rho_{cr}\equiv\frac{3H_{0}^{2}}{8\pi G}\right)$ refer to the present
moment of time. For the multi-component flat case
$d_{L}=\frac{1+z}{{{H}_{0}}}\int_{0}^{z}{\frac{d{z}^{\prime}}{\sqrt{\sum\limits_{i}{\Omega_{0i}{{\left(1+{z}^{\prime}\right)}^{3\left(1+{{w}_{i}}\right)}}}}}}.$
(42)
The relation (40). can be rewritten in terms of the deceleration parameter.
For the flat case it is
$d_{L}(z)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}=(1+z)H_{0}^{-1}\int_{0}^{z}{du}\exp\left[-\int_{0}^{u}\left[1+q(v)d\ln(1+v)\right]\right].$
(43)
Let us use normal definition for the absolute value $\mu$ of the distance to a
standard candle (say, SNe1a):
$\mu(z)\equiv\left[{{m}_{B}}(z)-{{M}_{B}}\right]=5\log\left({{d}_{L}}/Mpc\right)+25.$
(44)
Using (43), one can link that quantity to the acceleration history $q(z)$ via
$\mu(z)=25+5\log\left[\frac{1+z}{H_{0}Mpc}\right]\int_{0}^{z}du\exp\left(-\int_{0}^{u}{\left[1+q(u)\right]d\ln
u}\right).$ (45)
Here $M_{B}$ and $m_{B}$ are absolute and apparent stellar magnitude of the
source respectively. The latter expression represents a fundamental relation,
linking the deceleration parameter history with the $SNe1a$ measurements.
We remark that the relation (43) is based solely on the FRW-metrics. It means
that the acceleration-deceleration dilemma can be resolved without any
assumption about applicability of general relativity. However, supernova
observations cannot help to determine current deceleration parameter
immediately. In order to interpret the data, we must know $H(z)$ or $q(z),$
which in turns requires to specify the dynamical equations and material
composition of the Universe.
It is useful to cite the expression for the luminosity distance with accuracy
up to $z^{2}$ terms:
$d_{L}=\frac{z}{H_{0}}\left[1+\left(\frac{1-q_{0}}{2}\right)z+{\mathrm{O}}(z^{2})\right],$
(46)
where in the flat case $q_{0}=\frac{1}{2}\sum\limits_{i}\Omega_{i}(1+3w_{i}).$
It follows from (46) that for small $z$ the luminosity distance is
proportional to the redshift, with the proportionality coefficient equal to
inverse value of the Hubble constant. For more distant cosmological objects
the higher order corrections for the luminosity distance depend on the current
value of the deceleration parameter $q_{0},$ ore, in other words, on quantity
and type of components, filling the Universe.
The expression for luminosity distance in the next order correction over the
redshift reads
$\displaystyle d_{L}(z)$ $\displaystyle=$
$\displaystyle\frac{cz}{H_{0}}[1+\frac{1}{2}\left(1-q_{0}\right)z-\frac{1}{6}\left(1-q_{0}-3q_{0}^{2}+j_{0}\right)z^{2}+$
$\displaystyle+$
$\displaystyle\frac{1}{24}\left(2-2q_{0}-15q_{0}^{2}-15q_{0}^{3}+5j_{0}+10j_{0}q_{0}+s_{0}\right)z^{3}+{\mathrm{O}}(z^{4})].$
Let us briefly discuss the methodology for verification of the cosmological
models using the type $SNe1a$ supernova outbursts. Let us consider more
closely Perlmutter42 ; Riess two supernovae 1992P at low redshift $z=0.026$
with $m=16.08$ and 1997ap at high redshift $z=0.83$ with $m=24.32.$ As
indicated earlier, $d_{L}(z)\simeq z/H_{0}$ for $z\ll 1.$ Using (44) we find
$M=-19.09.$ Then the luminosity distance of 1997ap is obtained by substituting
$m=24.32$ and $M=-19.09$ for (44)
$H_{0}d_{L}\simeq 1.16\quad for\quad z=0.83.$ (48)
From the other hand, for the case of Universe filled by non-relativistic
matter, using (42), one finds
${{H}_{0}}{{d}_{L}}\simeq 0.95.$
The latter result apparently contradicts the observations (48).
Therefore, registration of outbursts for any kind of light sources with the
same internal luminosity, i.e. standard candles, allows to determine the
velocity of Universe expansion in different moments of its evolution.
Comparing the obtained results with the predictions of different theoretical
models, one can select the most adequate ones. Despite of the conceptual
simplicity, the problem have been continuously facing many obstacles. Let us
list just a few of them. The supernova outbursts are rare and random. To
collect sufficient statistics one has to cover rather large part of the sky.
The outbursts takes place for a limited time, therefore it is important to
detect a supernova as soon as possible in order to follow the dynamics of its
brightness variation. And of course the main problem is still connected to
disputable applicability of the type Ia supernova for a standard candle. In
the early ninetieth the United States created two groups for detection and
analysis of the type $SNeIa$ supernova outbursts: SuperNova Cosmology Project
and High-Z SuperNova Search. It were their results Perlmutter42 ; Riess which
in the years 1998 and 1999 gave base for conclusion on accelerated expansion
of Universe, so dramatically affected not only modern cosmology, but all
physics as a whole. During the passed decade the results Perlmutter42 ; Riess
were multiple times reproduced with ever improving statistics. The main
conclusion was always reconfirmed: comparable recently (at $z\sim 0.5)$ our
Universe passed through the transition from decelerated expansion to
accelerated one. Let us present in more details the analysis carried out by
Riess RiessIazl1 , where he used the so-called golden set of $SNe1a.$ The set
contained $157$ well studied $SNe1a$ supernova with redshifts ranged in
$0.1<z<1.76.$ The analysis was based on the relation (43) which is exact
expression for the luminosity distance in a geometrically flat Universe. In
the case of linear two-parameter decomposition
$q(z)=q_{0}+q_{1}z,$ (49)
the integral in (43) can be exactly evaluated and the expression for the
luminosity distance takes the form CunhaLima
$d_{L}(z)=\frac{1+z}{H_{0}}e^{q_{1}}q_{1}^{q_{0}-q_{1}}\left[\gamma\left(q_{1}-q_{0},(1+z)q_{1}\right)-\gamma\left(q_{1}-q_{0},q_{1}\right)\right],$
(50)
where ${{q}_{0}},{{q}_{1}}$ are values of $q(z),\frac{dq(z)}{dz}$ at $z=0,$
$\gamma$ is the reduced gamma-function. Using (50) one can obtain information
about $q_{0},q_{1}$ and, therefore, about global behavior of $q(z).$ The
dynamical ”phase transition” occurs at $q\left({{z}_{t}}\right)=0$ or
equivalently at $z_{t}=-q_{0}/q_{1}.$
The second widely used parameterization is
$q(z)=q_{0}+q_{1}\frac{z}{1+z}.$ (51)
Its advantage is in the fact that it is well-behaved at large $z,$ while the
linear approximation suffers of divergences. In such parameterization one has
$d_{L}(z)=\frac{1}{(1+z)H_{0}}e^{q_{1}}q_{1}^{-(q_{0}+q_{1})}\left[\gamma\left(q_{1}+q_{0},q_{1}\right)-\gamma\left(q_{1}+q_{0},q_{1}/(1+z)\right)\right].$
(52)
Now the parameter $q_{1}$ determines the correction to $q_{0}$ in the distant
past: $q(z)=q_{0}+q_{1}$ at $z\gg 0.$ The likelihood for the parameters
$q_{0}$ and $q_{1}$ can be determined from $\chi^{2}$ statistic,
$\chi^{2}\left(H_{0},q_{0},q_{1}\right)=\sum\limits_{i}{\frac{{{{\left({{\mu_{p,i}}\left({{z_{i}};{H_{0}},{q_{0}},{q_{1}}}\right)-{\mu_{0,i}}}\right)}^{2}}}}{{\sigma_{{\mu_{0,i}}}^{2}+\sigma_{v}^{2}}}},$
(53)
where $\sigma_{\mu_{0,i}}$ is the uncertainty in the individual distance
modules, $\sigma_{v}$ is the dispersion in SNe redshifts due to peculiar
velocities. The obtained results give evidence in favor of the Universe with
recent acceleration $\left(q_{0}<0\right)$ and previous deceleration
$(q_{1}>0)$ with $99.2\%$ and $99.8\%$ likelihood. In the case of linear
decomposition of the parameter $q$ the transition from decelerated expansion
in the past to current accelerated expansion took place at the redshift
$z_{t}=0.46\pm 0.13.$ Unfortunately one should not take this result too
serious: the linear approximation always leads to the transition, provided the
two parameters have opposite signs.
The consequent and statistically more reliable analysis qualitatively
supported the above cited results. Of course, the quantitative results depend
on the used set of supernova, but the main result remains the same: we live in
accelerated expanding Universe, which passed from decelerated expansion to
accelerated one in the recent past.
Nowadays the method based on the supernova outbursts observations is
undoubtedly leading. But it turned out to have competitors. A promising and
completely independent alternative (and no way a surrogate) is the observation
of angular diameter distances $D_{A}(z)$ for a given set of distant objects.
The combination of the Sunyaev-Zeldovich effect with measurements of surface
brightness in X-ray range provides the method for determination of angular
diameter distances for galactic clusters Bonamente . The Sunyaev-Zeldovich
stands for small perturbation of CMB spectrum produced by inverse Compton
scattering of relict photons passed through clouds of hot electrons. Observing
the temperature decrement of the CMB spectrum towards galaxy clusters together
with the X-rays observations, it is possible to break the degeneracy between
concentration and temperature thereby obtaining $D_{A}(z).$ Therefore, such
distances are fully independent of the one given by the luminosity distance.
For the case of spatially flat Universe described by FRW-metric, the angular
diameter distance is
$d_{A}(z)=\frac{1}{(1+z)H_{0}}\int_{0}^{z}\frac{du}{H\left(u\right)}=\frac{1}{(1+z)H_{0}}\int_{0}^{z}\exp\left[-\int_{0}^{u}\left[(1+q({u}^{\prime})\right]d\ln(1+{u}^{\prime}\right]du,$
(54)
CunhaLimaHolanda considered the 38 measurements of angular diameter distances
from galaxy clusters as obtained through SZE/X-ray method by Bonamente and
coworkers Bonamente Lima et al used a maximum likelihood determined by a
$\chi^{2}$ statistics
$\chi^{2}\left(z,p\right)=\sum\limits_{i}\frac{\left({{D}_{A}}\left({{z}_{i}},p\right)-{{D}_{{{A}_{0,i}}}}\right)}{\sigma_{{{D}_{A0i}}}^{2}+\sigma_{stat}^{2}},$
(55)
where $D_{A_{0,i}}$ is the observational angular diameter distance,
$\sigma_{D_{A0i}}$is the uncertainty in the individual distance,
$\sigma_{stat}$ is the contribution of the statistical errors added in
quadrature and the complete set of parameters is given by
$p\equiv\left(H_{0},q_{0},q_{1}\right).$ For the sake consistency, the Hubble
parameter $H_{0}$ has been fixed by its best it value
$H_{0}^{*}=80km/\sec\cdot Mpc^{-1}.$ In the case of linear parameterization
the results are the following: best its to the free parameters are
$q_{0}=-1.35,~{}q_{1}=4.2,~{}z_{t}=0.32.$ Such results favor a Universe with
recent acceleration $\left(q_{0}<0\right)$ and a previous decelerating stage
$\left(dq/dz>0\right).$ In the case of the parameterization
$q(z)={{q}_{0}}+{{q}_{1}}\frac{z}{1+z},$
one gets $q_{0}=-1.43,~{}q_{1}=6.18,~{}z_{t}=0.3.$ In both cases the results
well agree with the ones obtained using the $SNe1a$ data.
It was recently shown that the so-called luminous red galaxies (LRG’s) provide
another possibility for direct measurements of the expansion velocity
JimenezLoeb ; ZhangMa ; MaZhangAstr . Idea of the method is to reconstruct the
Hubble parameter from the time derivative of the redshift
$H(z)=-\frac{1}{1+z}\frac{dz}{dt}.$ (56)
The derivative can be found in measurements of the ”age difference ” between
two passively evolving galaxies at different but close redshifts. The method
was realized for $0.1<z<1.75.$ We stress that the range includes the
interesting for us transition region, where $z\sim 0.5.$ The results of
analysis for available data CarvalhoAlkaniz_Cosmography agree with the ones
based on the supernova and angular diameter. In nearest future a set of some
$2000$ passively evolving galaxies is going to be measured in the range
$0<z<1.5.$ Those observations will give about $1000$ values of $H(z)$ with
$15\%$ accuracy, provided the age of galaxies is determined with $10\%$
precision.
Concluding the present section we present a novel and promising direction to
research the history of cosmological expansion of Universe. We remind that
astrophysicists initially used cepheids — the variable stars with intensity
proportional to period of brightness variation — as the standard candles. A
typical example of cepheids is the Polar Star, the brightest and closest to
Earth variable star with period of $3.97$ days. The cepheids are perfect
standard candles for galactic distances. They allowed to determine size of our
Galaxy and distance to our closest neighbor — the Andromeda galaxy. Research
of the Universe dynamics involves principally larger scales, and thus it
requires to use essentially more powerful standard candles. Recall, that the
cosmological principle, which postulates homogeneity and isotropy of the
Universe, and which all our above used equations of Universe dynamics are
based on, is valid on scales of $100\,Mpc$ and larger. Using considerably more
powerful radiation sources as the standard candles, namely type Ia supernovae,
already allowed us to advance remarkably deeper to the history of Universe.
However the possibilities of the new standard candles are also limited. Up to
present time the type Ia supernovae were observed only up to the redshift
$z<2,$ though more reliable reconstruction of the cosmological expansion
history requires even higher redshifts and therefore more powerful standard
candles. It turns out that objects with required properties are already
available! It is the so-called gamma-ray bursts(GRB) — giant energy emissions
of explosive type with duration from tree to hundred second, observed in the
hardest part of the electromagnetic spectrum. The typical energy, emitted by
the gamma-ray burst is $\sim 10^{54}\,erg$ which is one order of magnitude
higher than at a supernova outburst. It is already compatible to the Sun rest
mass! The events producing the gamma-ray bursts are so powerful that they can
be observed even by naked eye, though they occur on distances of billions
light years from the Earth! The energy yields in form of a collimated flow,
called a jet. Occurrence of the jets means that we can see only small number
of all bursts that happen in Universe. Distribution of GRB durations has
pronounced bimodal character.
Origin of short GRB is probably linked to coalescence of two neutron stars, or
a neutron star and a black hole. The events with longer duration are
presumably due to creation of a black hole in collapse of massive stellar
nucleus (¿25 Solar mass) with high angular momentum — it is the so-called
collapsar model. The possibility to use the GRB as the standard candles is
based on occurrence of the so-called ”Amati relation”, which relates the peak
frequency of the burst with its total energy. This relation is a direct analog
of the period-luminosity relation for cepheids. Strong dispersion (see Fig.1)
yet limits applications of the GRB as the standard candles, but the
possibility to advance to the region of considerably higher redshifts makes
this direction potentially very attractive.
Figure 1: 15 GRB with measured redshifts and collimation angle(compared to Ia
supernova data in SCM)
.
## V Scale factor dynamics in SCM
Now let us consider evolution of the deceleration parameter in Standard
Cosmological model (SCM). Recall that the Big Bang model included only two
substances – matter and radiation – which could give only decelerated
expansion of the Universe. According to SCM, the Universe is presently
dominated by the dark energy — a component with negative pressure. It is the
component responsible for the observed accelerated expansion of Universe. Let
us determine the redshift and time moment of the transition to the accelerated
expansion, i.e. find the inflection point on the curve (see Fig. 2),
describing the time dependence of the scale factor
Figure 2: Time dependence of the scale factor in SCM.
The second Friedmann equation in the SCM (14) can be transformed to the
following
$\frac{{\ddot{a}}}{a}=\frac{1}{2}H_{0}^{2}\left[2{{\Omega}_{\Lambda
0}}-{{\Omega}_{m0}}{{(1+z)}^{3}}\right],$ (57)
where $\Omega_{\Lambda 0}$ and ${{\Omega}_{m0}}$ are present values of the
relative density for dark energy in form of cosmological constant and for
matter respectively. Now it follows for the redshift at the point of
transition from decelerated expansion to accelerated one
$z^{*}={{\left(\frac{2{{\Omega}_{\Lambda
0}}}{{{\Omega}_{m0}}}\right)}^{1/3}}-1.$ (58)
With the SCM parameters $\Omega_{\Lambda 0}\simeq 0.73,\ {{\Omega}_{m0}}\simeq
0.27$ one gets $z^{*}\simeq 0.745.$ Note that the result (58) can be obtained
using the fact that for a multi-component Universe with state equations
$p_{i}=w_{i}\rho_{i}$ respectively, the deceleration parameter (18) equals
(recall that $\Omega=\sum\limits_{i}\Omega_{i}=1$ in the flat case )
$q=\frac{1}{2}-\frac{3}{2}\frac{{{\Omega}_{\Lambda
0}}}{{{\left(1+z\right)}^{3}}{{\Omega}_{m0}}+{{\Omega}_{\Lambda 0}}}.$ (59)
The condition $q=0$ allows to reproduce the relation (58). Let us discuss its
asymptotes. For early Universe $(z\to\infty),$ filled by components with
positive pressure , $q\left(z\to\infty\right)=\frac{1}{2},$ i.e. as it was
expected the expansion is decelerated, while in the distant future with
domination of the cosmological constant the accelerated expansion is observed:
$q\left(z\to-1\right)=-1.$ The latter result is a trivial consequence of the
exponential expansion $a\propto{{e}^{Ht}}$ in the case of Universe dominated
by dark energy in the form of cosmological constant.
We remark that the dependence $q(t)$ can be immediately obtained from the
definition $q=-\ddot{a}/a{{H}^{2}},$ using the SCM solutions for the scale
factor:
$\displaystyle a(t)$ $\displaystyle=$
$\displaystyle{{A}^{1/3}}{{\sinh}^{2/3}}\left(t/{{t}_{\Lambda}}\right);$
$\displaystyle A$ $\displaystyle\equiv$
$\displaystyle\frac{{{\Omega}_{m0}}}{{{\Omega}_{\Lambda
0}}},\quad{{t}_{\Lambda}}\equiv\frac{2}{3}H_{0}^{-1}\Omega_{\Lambda
0}^{-1/2}.$ (60)
It results in the following
$q(t)=\frac{1}{2}\left[1-3\tanh^{2}\left(\frac{t}{t_{\Lambda}}\right)\right],$
(61)
Figure 3: Time dependence of the deceleration parameter $q(t)$ in SCM.
The dependence $q(t)$ is presented on Fig.3. Note that the time asymptotes at
$t\to 0$ and $t\to\infty$ correspond to the above obtained ones at
$z\to\infty$ and $z\to-1.$
Let us now determine the time of the transition to the accelerated expansion.
Inserting the relation (V), one obtains
$t\left(a\right)=\frac{2}{3}\Omega_{\Lambda
0}^{-1/2}H_{0}^{-1}{arsh}\left[{{\left(\frac{{{\Omega}_{\Lambda
0}}}{{{\Omega}_{m0}}}\right)}^{1/2}}{a}^{3/2}\right].$ (62)
Transforming from the redshift to the scale factor
$a^{*}={{(1+z^{*})}^{-1}}={{\left(\frac{{{\Omega}_{m0}}}{2{{\Omega}_{\Lambda
0}}}\right)}^{1/3}}$ , one gets
$t^{*}\equiv(a^{*})=\frac{2}{3}\Omega_{\Lambda
0}^{-1/2}H_{0}^{-1}\sinh^{-1}\left(1/2\right)\simeq 5.25\,Gyr.$ (63)
Because of physical importance of the obtained result we present another one,
probably the simplest, interpretation of it. If the quantity $aH=\dot{a}$
growth, then $\ddot{a}>0,$ which corresponds to accelerated expansion of
Universe. According to the first Friedmann equation,
$\frac{aH}{{{H}_{0}}}=\sqrt{\frac{{{a}^{3}}{{\Omega}_{\Lambda
0}}+{{\Omega}_{m0}}}{a}}\simeq\sqrt{\frac{0.73{{a}^{3}}+0.27}{a}}.$
It is easy to show that the function under radical starts to grow at
$a^{*}\simeq 0.573,$ which corresponds to $z^{*}=0.745.$ It is interesting to
note that the transition to accelerated expansion of Universe $(z\simeq 0.75)$
occurred remarkably earlier than the dark energy started to dominate $(z\simeq
0.4).$
In the SCM the dark energy exist in form of cosmological constant with the
state equation ${{p}_{\Lambda}}=-{{\rho}_{\Lambda}},$ i.e. the state parameter
equals ${{w}_{\Lambda}}=-1.$ A natural question rises: what is the limiting
value of that parameter, which still provides accelerated expansion of
Universe in the present time? As we have seen above, the condition for
accelerated expansion is the following
$\sum\limits_{i}{\left({{\rho}_{i}}+3{{p}_{i}}\right)}<0.$ In SCM it is
transformed to the following
$w_{{}_{DE}}<-\frac{1}{3}\Omega_{DE}^{-1};\quad w_{{}_{DE}}<-0.46.$
Of course, a substance with such state equation is not the cosmological
constant, and it can be realized, for instance, with the help of scaler fields
(see next section).
Now let us estimate absolute magnitude of the cosmological acceleration using
the SCM parameters. Taking time derivative of the Hubble law, one gets
$\dot{V}=\left(\dot{H}+{{H}^{2}}\right)R.$ (64)
The time derivative of the Hubble parameter is
$\dot{H}=\frac{\ddot{a}a-{{{\dot{a}}}^{2}}}{{{a}^{2}}}=\frac{{\ddot{a}}}{a}-{{H}^{2}}.$
Therefore
$\dot{H}+{{H}^{2}}=\frac{{\ddot{a}}}{a}=-\frac{4\pi
G}{3}\left({{\rho}_{m}}-2{{\rho}_{\Lambda}}\right)=\frac{8\pi
G}{3}\left({{\rho}_{\Lambda}}-\frac{1}{2}{{\rho}_{m}}\right)=H^{2}\left({{\Omega}_{\Lambda}}-\frac{1}{2}{{\Omega}_{m}}\right).$
And finally we get the analog of the Hubble law for the acceleration $\dot{V}$
$\dot{V}=\tilde{H}R;\quad\tilde{H}=H^{2}\left({{\Omega}_{\Lambda}}-\frac{1}{2}{{\Omega}_{m}}\right).$
(65)
In the present time $({{\Omega}_{m}}={{\Omega}_{m0}},\
{{\Omega}_{\Lambda}}={{\Omega}_{\Lambda 0}}),$ and on the distance, for
example, $R=1$ Mpc one gets
$\dot{V}\simeq{{10}^{-11}}cm\>{\sec}^{-2}.$ (66)
A possible way to observe such effect is based on the fact that the redshift
for any cosmological object slowly changes due to the acceleration (or
deceleration) of the Universe expansion. Let us estimate the magnitude of that
effect. By definition
$z=\frac{a\left({{t}_{0}}\right)}{a(t)}-1,$
where $t$ is the time of emission and ${{t}_{0}}$ is the detection time, and
it follows that
$\frac{dz}{d{{t}_{0}}}=\frac{\dot{a}({{t}_{0}})a(t)-a({{t}_{0}})\dot{a}(t)\frac{dt}{d{{t}_{0}}}}{{{a}^{2}}(t)}.$
(67)
Taking into account that
$\frac{dt}{d{{t}_{0}}}=\frac{a(t)}{a({{t}_{0}})}=\frac{1}{(1+z)},$ one gets
$\dot{z}=\frac{\dot{a}({{t}_{0}})}{a(t)}-\frac{a({{t}_{0}})\dot{a}(t)}{{{a}^{2}}(t)}\frac{1}{1+z}.$
(68)
Thus, the rate of redshift variation for the light emitted at time $t$ and
registered at the present time $t_{0},$ is determined by the relation
$\dot{z}\equiv\frac{dz}{d{{t}_{0}}}={{H}_{0}}(1+z)-H(t).$
In the SCM
$H={{H}_{0}}{{\left({{\Omega}_{m0}}(1+z\right)}^{3}}+{\Omega}_{\Lambda
0})^{1/2}.$
Therefore for variation of the redshift on the time interval $\Delta t$ one
obtains
$\Delta z=\dot{z}\Delta
t={{H}_{0}}\left[1+z-\left(\Omega_{m0}(1+z\right)^{3}+\Omega_{\Lambda
0})^{1/2}\right].$ (69)
Note that for the two limiting cases ${{\Omega}_{\Lambda
0}}=1,\,{{\Omega}_{m0}}=0$ (accelerated expansion) and ${{\Omega}_{\Lambda
0}}=0,\,{{\Omega}_{m0}}=1$ (decelerated expansion), as was expected $\Delta z$
has opposite sign. For the SCM parameters the redshift variation $\Delta z$
and the velocity increment $\Delta V$ for a source with the redshift $z=4$ and
duration of observation $\Delta{{t}_{0}}=10$ equal respectively
$\Delta z\approx{{10}^{-9}},~{}~{}\Delta V=c\frac{\Delta z}{1+z}\approx
6\,cm/\sec.$
The result discourages by its smallness. However, taking into account fast
progress in precision of observable cosmology, we should not despair. Let us
give an example. Today the list of exoplanets – planets out the Solar system –
counts more than 300 items. The most successful method to detect the
exoplanets is to measure radial velocity of the stars. A star accompanied by
an exoplanet experiences velocity oscillations ”to us and back from us”, which
can be measured observing the Doppler shift of the star spectrum. For the
first look it is impossible: affected by the Earth, velocity of the sun
annually oscillates with amplitude of centimeters per second. Even effect of
Jupiter produces variations of only meters per second, while thermal
broadening of the spectral lines for the star corresponds to dispersion of
velocity of order of thousands kilometers per second. It means that even in
the case of Jupiter one have to measure shift of the spectral lines on a
thousandth of its width. It seems incredible, but the task was successfully
worked out.
## VI Dynamical forms of dark energy and evolution of Universe
The cosmological constant represents one of many possible realizations of the
hypothetic substance called the dark energy, introduced for explanation of the
accelerated expansion of Universe. As we have seen above, such substance have
to have the parameter $w$ in the state equation $p=w\rho,$ which satisfies the
condition $w<-1/3$ (in absence of other components). Unfortunately the nature
of dark energy is absolutely unknown, which produces huge number of hypotheses
and candidates for the role of fundamental contributor to the energy budget of
the Universe. We told many times about impressive progress of observational
cosmology in the last decade. However we still cannot answer the question
about time evolution of the state parameter $w.$ If it changes with time, we
must seek an alternative to the cosmological constant. For quite a short time
a plenty of such alternative (with respect to $w=-1$) possibilities were
investigated. The scalar fields, formed the post-inflation Universe, are
considered to be one of the principal candidates for the role of dark energy.
The most popularity acquired the version of the scalar field $\varphi$ with
appropriately chosen potential $V\left(\varphi\right).$ In such models, unlike
the cosmological constant one, the scalar field is truly dynamical variable,
and dark energy density depends on time. The models differ by the choice of
the scalar field Lagrangian. Let us start from probably the simplest dark
energy model of such type, named the quintessence. We define quintessence as
the scalar field $\varphi$ in the potential $V\left(\varphi\right),$ minimally
coupled to gravity, i.e. experiencing effect of the space-time curvature.
Besides that we chose canonical form for the kinetic energy term. The action
for such field takes the form
$S=\int{{{d}^{4}}x\sqrt{-g}L=\int{{{d}^{4}}x\sqrt{-g}\left[\frac{1}{2}{{g}^{\mu\nu}}\frac{\partial\varphi}{\partial{{x}_{\mu}}}\frac{\partial\varphi}{\partial{{x}_{\nu}}}-V\left(\varphi\right)\right]}},$
(70)
where $g\equiv\det{{g}_{\mu\nu}}.$ Equation of motion for the scalar field is
obtained by variation of the action with respect to the field,
$\frac{1}{\sqrt{-g}}{{\partial}_{\mu}}\left(\sqrt{-g}{{g}^{\mu\nu}}\frac{\partial\varphi}{\partial{{x}_{\nu}}}\right)=-\frac{dV}{d\varphi}.$
(71)
In the case of flat Friedmannian Universe, namely in FRW-metric (1) one obtain
for homogeneous field $\varphi(t)$ the following
$\ddot{\varphi}+3H\dot{\varphi}+{V}^{\prime}(\varphi)=0,$ (72)
where ${V}^{\prime}(\varphi)\equiv\frac{dV}{d\varphi}.$ This relation is
sometimes called the Klein-Gordon equation.
The energy-momentum tensor for the scalar field can be found by variation of
(70) with respect to metrics ${{g}^{\mu\nu}},$ resulting in the following
${{T}_{\mu\nu}}=\frac{2}{\sqrt{-g}}\frac{\delta
S}{\delta{{g}^{\mu\nu}}}=\frac{\partial\varphi}{\partial{{x}_{\mu}}}\frac{\partial\varphi}{\partial{{x}_{\nu}}}-{{g}_{\mu\nu}}L.$
(73)
In the case of homogeneous field$\varphi(t)$ in locally Lorentzian reference
frame, where metrics${{g}_{\mu\nu}}$ can be replaced by the Minkowski one, we
obtain the density and pressure of the scalar field in the following form
${{\rho}_{\varphi}}={{T}_{00}}=\frac{1}{2}{{\dot{\varphi}}^{2}}+V\left(\varphi\right);\quad{{p}_{\varphi}}={{T}_{ii}}=\frac{1}{2}{{\dot{\varphi}}^{2}}-V\left(\varphi\right).$
(74)
The Friedmann equations for the flat Universe, filled by the scalar field,
take the form
${{H}^{2}}=\frac{8\pi
G}{3}\left[\frac{1}{2}{{{\dot{\varphi}}}^{2}}+V(\varphi)\right];$ (75a)
$\dot{H}=-4\pi G\dot{\varphi}^{2}.$ (75b)
The system (75a – 75b) should be complemented by the Klein-Gordon equation
(72) for the scalar field. Note, that the latter can be obtained from the
conservation equation for the scalar field
${{\dot{\rho}}_{\varphi}}+3H\left({{\rho}_{\varphi}}+{{p}_{\varphi}}\right)=0.$
(76)
by substitution of the expressions (74) for energy density and pressure.
Using (74), one get the state equation for the scalar field
${{w}_{\varphi}}=\frac{{{p}_{\varphi}}}{{{\rho}_{\varphi}}}=\frac{{{{\dot{\varphi}}}^{2}}-2V}{{{{\dot{\varphi}}}^{2}}+2V}.$
(77)
As one can see, the parameter of the state equation for the scalar field
${{w}_{\varphi}}$ varies in the range
$-1\leq{{w}_{\varphi}}\leq 1.$ (78)
The state equation for the scalar field is conveniently transformed into the
following
$w(x)=\frac{x-1}{x+1},\quad
x\equiv\frac{\frac{1}{2}{{{\dot{\varphi}}}^{2}}}{V(\varphi)}.$ (79)
The function $w(x)$ increases monotonously from the minimum value
${{w}_{\min}}=-1$ at $x=0$ up to the maximum asymptotic value ${{w}_{\max}}=1$
at $x\to\infty,$ corresponding to $V=0.$ In the slow-roll limit, $x\ll 1$
$({{\dot{\varphi}}^{2}}\ll V(\varphi))$ the scalar field behaves as the
cosmological constant, ${{w}_{\varphi}}=-1.$ It is easy to see that in such
case ${{\rho}_{\varphi}}=const.$ The opposite limit $x\gg 1$
$(\dot{\varphi}\gg V\left(\varphi\right))$ correspond to the hard matter
${{w}_{\varphi}}=1.$ In that case the scalar field energy density evolves as
${{\rho}_{\varphi}}\propto{{a}^{-6}}.$ The intermediate case $x\sim 1,\,p\sim
0$ corresponds to non-relativistic matter.
Transforming (76) into the integral form
$\rho={{\rho}_{0}}\exp\left[-3\int{\left(1+{{w}_{\varphi}}\right)\frac{da}{a}}\right].$
(80)
one finds that in the general case the scalar field energy density behaves as
$\rho_{\phi}\propto a^{-m},\quad 0<m<6.$ (81)
The value ${{w}_{\varphi}}=-1/3$ represents boundary between the regimes of
decelerated and accelerated Universe expansions. Thus, the case of accelerated
expansion takes place for $0\leq m<2.$ A natural question arises: what are the
potentials for scalar fields that can provide the accelerated expansion of
Universe? The question can be posed another way: what potentials can produce
the quintessence applicable for the role of dark energy? Consider simplified
version of the problem. Let us find the scalar field potential resulting in
power law for the scale factor growth:
$a(t)\propto{{t}^{p}}.$ (82)
For accelerated expansion the condition $p>1$ must be satisfied. Recall that
$p=2/3$ for the case of non-relativistic matter dominated Universe, and
$p=1/2$ for radiation-dominated one, so in both cases the expansion will be
decelerated. Making use of the Friedmann equations, one can express the
potential $V(\varphi)$ and $\dot{\varphi}$ in terms of $H$ and $\dot{H}.$ It
allows to write down the system of equations, which describe the parametric
dependence $V$ on $\varphi:$
$V=\frac{3{{H}^{2}}}{8\pi G}\left(1+\frac{{\dot{H}}}{3{{H}^{2}}}\right);$
(83a) $\varphi={{\int{dt\left(-\frac{{\dot{H}}}{4\pi G}\right)}}^{1/2}}.$
(83b)
Excluding the time variable with the help of the relation
$\frac{\varphi}{{{m}_{Pl}}}=\sqrt{\frac{p}{4\pi}}\ln t,$ for the case of the
power law (82) one finds
$V\left(\varphi\right)={{V}_{0}}\exp\left(-\sqrt{\frac{16\pi}{p}}\frac{\varphi}{{{m}_{Pl}}}\right).$
(84)
The obtained result implies that the scalar field in the potential (84) under
condition $p>1$ can be treated as the dark energy, i.e. it provides
accelerated expansion of Universe. As we can see above, in the quintessence
model the required dynamical behavior is achieved by the choice of the scalar
field potential. Another dark energy model, realized by the scalar field with
modified kinetic term, is called the $k$-essence k-Inflation ; k-Inflation2 .
Let us define the quantity
$X\equiv\frac{1}{2}{{g}^{\mu\nu}}\frac{\partial\varphi}{\partial{{x}_{\mu}}}\frac{\partial\varphi}{\partial{{x}_{\nu}}}$
and consider the action for the scalar field of the form
$S=\int{{{d}^{4}}x\sqrt{-g}}L\left(\varphi,X\right).$ (85)
where $L$ is generally speaking an arbitrary function of the variables
$\varphi$ and $X.$ The traditional scalar field action corresponds to the case
$L\left(\varphi,X\right)=X-V(\varphi).$ (86)
We restrict our consideration to the following subset of Lagrangians:
$L\left(\varphi,X\right)=K(X)-V(\varphi),$ (87)
where $K(X)$ is positive defined function of the kinetic energy $X.$ For
description of homogeneous and uniform Universe we should choose
$X=\frac{1}{2}{{\dot{\varphi}}^{2}}.$ Using the standard definition (74), one
finds
${{p}_{\varphi}}=L\left(\varphi,X\right)=K(X)-V(\varphi);$ (88a)
${{\rho}_{\varphi}}=2X\frac{\partial K(X)}{\partial X}-K(X)+V(\varphi).$ (88b)
Accordingly, the state equation for $k$-essence takes the form
${{w}_{\varphi}}=\frac{K(X)-V(\varphi)}{2X\frac{\partial K(X)}{\partial
X}-K(X)+V(\varphi)}.$ (89)
We demonstrate the main features of $k$-essence in example of the simplified
model putter , where the Lagrangian has the form $L=F(X).$ Such model is
called purely kinetic $k$-essence. In that case one has
${{\rho}_{\varphi}}=2X{{F}_{X}}-F;\quad{{F}_{X}}\equiv\frac{\partial
F}{\partial X};$ (90a) $p=F;$ (90b) ${{w}_{\varphi}}=\frac{F}{2X{{F}_{X}}-F}.$
(90c)
The equations of motion for the field can be obtained either from the Euler-
Lagrange equation for the action (85), or by substitution of density and
pressure expressions (90a and 90b) respectively into the conservation equation
for the $k$-essence. It results in the following
${{F}_{X}}\ddot{\varphi}+{{F}_{XX}}{{\dot{\varphi}}^{2}}\ddot{\varphi}+3H{{F}_{X}}\dot{\varphi}=0,$
(91)
or in terms of the kinetic energy $X$
$\left({{F}_{X}}+2{{F}_{XX}}X\right)\dot{X}+6H{{F}_{X}}X=0.$ (92)
The latter equation can be exactly solved
$XF_{X}^{2}=k{{a}^{-6}}.$ (93)
where constant $k>0.$
The solution (93) $X(a)$ has one important property: all principal
characteristics of $k$-essence
(${{\rho}_{\varphi}},{{p}_{\varphi}},{{w}_{\varphi}};$ see (90a \- 90c)) as
functions of the scale factor are completely determined by the function $F(X)$
and do not depend on evolution of other energy densities. All the dependence
of $k$-essence on other components appears only by means of $a(t).$ But this
dependence is trivially $a(t)\propto{{\rho}_{tot}},$ and it is determined by
the dominant component in the energy density. Solutions of such type are
called the tracker solutions and its occurrence allows to approach the
solution of the coincidence problem. It can be shown Armendariz-Picon , that
such property is present not only in purely kinetic $k$-essence, but also its
general case.
Extensive set of available cosmological observations shows that the parameter
$w$ in the state equation for dark matter lies in narrow vicinity of $w=-1.$
We considered above the region $-1\leq w<-1/3.$ The lower bound of the region
$w=-1$ corresponds to the cosmological constant, and all remaining interval
can be realized by scalar fields with canonic Lagrangian. Recall that the
upper bound $w=-1/3$ appears because of requirement to provide the observed
accelerated expansion of Universe. But can we go beyond the bounds of that
interval?
It is rather difficult question for the component of energy we know so little
about. General relativity usually impose certain restrictions on possible
values of the energy-momentum tensor components, which are called the energy
conditions(see section III). One of the simplest restrictions of such type is
the condition $\rho+p\geq 0.$ Its physical motivation is to prevent the
instability of vacuum. Applied to dynamics of Universe this condition requires
that density of any allowed energy component should not grow as Universe
expands. The cosmological constant with ${{\dot{\rho}}_{\Lambda}}=0$
represents the limiting. Taking into account our ignorance about the nature of
dark energy, it is reasonable to pose a question: may such a mysterious
substance differ so much from already known ”good” energy sources and violate
the condition $\rho+p\geq 0$? Accounting the requirement of positive defined
energy density, valid even for dark one (it is necessary to make the Universe
flat), and negative pressure (in order to explain the accelerated Universe
expansion), the above mentioned violation will result in $w<-1.$ Some time ago
such component, called the fantom energy, attracted attention of
physicistsCaldwell12 . Action for the fantom field $\varphi,$ minimally
coupled to gravity, differs only by sign of the kinetic term from the canonic
action for the scalar field. In that case the energy density and pressure of
the fantom field are determined by the expressions
${{\rho}_{\varphi}}={{T}_{00}}=-\frac{1}{2}{{\dot{\varphi}}^{2}}+V\left(\varphi\right);\quad{{p}_{\varphi}}={{T}_{ii}}=-\frac{1}{2}{{\dot{\varphi}}^{2}}-V\left(\varphi\right),$
(94)
and state equation takes the form
${{w}_{\varphi}}=\frac{{{p}_{\varphi}}}{{{\rho}_{\varphi}}}=\frac{{{{\dot{\varphi}}}^{2}}+2V(\varphi)}{{{{\dot{\varphi}}}^{2}}-2V(\varphi)}.$
(95)
If ${{\dot{\varphi}}^{2}}<2V(\varphi),$ then ${{w}_{\varphi}}<-1.$ As an
example let us consider the case of Universe, containing only non-relativistic
matter $(w=0)$ and phantom field $({{w}_{\varphi}}<-1).$ Densities of the two
components evolve independently: ${{\rho}_{m}}\propto{{a}^{-3}}$ and
${{\rho}_{\varphi}}\propto{{a}^{-3\left(1+{{w}_{\varphi}}\right)}}.$ If matter
domination ends at the time moment ${{t}_{m}},$ then the solution for the
scale factor at $t>{{t}_{m}}$ reads
$a(t)=a({{t}_{m}}){{\left[-{{w}_{\varphi}}+(1+{{w}_{\varphi}})\left(\frac{t}{{{t}_{m}}}\right)\right]}^{\frac{2}{3(1+{{w}_{\varphi}})}}}.$
(96)
It immediately follows that for ${{w}_{\varphi}}<-1$ at the time moment
${{t}_{BR}}=\frac{{{w}_{\varphi}}}{(1+{{w}_{\varphi}})}{{t}_{m}}$ the scale
factor and whole set of the cosmological characteristics (such as scalar
curvature and energy density of the fantom field) for the Universe diverge.
This catastrophe was called the Big Rip, which is precede by a specific regime
which is called super-acceleration. Let us explain the origin of the super-
acceleration regime on a simple example. Consider differential equation
$\frac{dx}{dt}=A{{x}^{2}}.$ (97)
In the case $A>0$ the equation (93) realizes the non-linear positive feedback.
Fast growth of the function $x(t)$ leads to the “Big Rip” (divergence of the
function to infinity) on finite time period. Indeed, the general solution of
the equation reads
$x(t)=-\frac{1}{A(t+B)}.$ (98)
where $B$ is the integration constant. At $t=-B$ the Big Rip occurs.
It is easy to see that the model (97) represents a particular version of the
Friedmann equation for ${{w}_{\varphi}}<-1.$ Since
${{\rho}_{\varphi}}\propto{{a}^{-3(1+{{w}_{\varphi}})}},$ then the first
Friedmann equation can be presented in the form
$\dot{a}=A{{a}^{-\frac{3}{2}\left(1+{{w}_{\varphi}}\right)+1}}.$ (99)
For instance, with ${{w}_{\varphi}}=-\frac{5}{3}$ the equation (99) exactly
coincides with (98).
## VII $f(R)$-Gravity, Braneworld Cosmology and MOND
Although SCM can explain the present accelerated expansion of Universe and
well agrees with current observational data, theoretical motivation for this
model can be regarded as quite poor. Consequently, there have been several
attempts to propose dynamical alternatives to dark energy. We considered them
in the previous section. Unfortunately, none of these attempts are problem-
free. An alternative and more radical approach is based on assumption that
there is no dark energy, and the acceleration is generated because of gravity
weakened on very large scales due to modification of general relativity. In
frames of such broad approach three main directions can be selected: $f(R)$
gravity, braneworld cosmology and modified Newtonian dynamics (MOND). Let us
briefly consider those alternatives to SCM from the point of view under our
interest: slowdown or speedup?
### VII.1 $f(R)-$ gravity
Theory of $f(R)$ gravity is created by direct generalization of the Einstein-
Hilbert action with the replacement $R\to f(R).$ The transformed action reads
$S=\frac{1}{16\pi G}\int{{{d}^{4}}x\sqrt{-g}f(R)}.$ (100)
For the generalization we take the function $f(R)$ depending only on Ricci
scalar $R,$ not including other invariants, such as
${{R}_{\mu\nu}}{{R}^{\mu\nu}}.$ Motivation for such choice is the following:
the action with $f(R)$ is sufficiently generic in order to reflect main
features of the gravity, and with all that it is simple enough to avoid
technical difficulties in calculations. We remark that the function $f(R)$
must satisfy the stability conditions
${f}^{\prime}(R)>0,\quad{f}^{\prime\prime}(R)>0,$ (101)
were the primes denote differentiation with respect to the scalar Ricci
curvature $R.$ Complete action for $f(R)$ gravity thus reads
$S=\frac{1}{2\kappa}\int{{{d}^{4}}x\sqrt{-g}f(R)+{{S}_{m}}\left({{g}_{\mu\nu}},\psi\right)\quad}.$
(102)
Here $\psi$ is common symbol for all material fields. Variation with respect
to metrics after some manipulations gives
${f}^{\prime}(R){{R}_{\mu\nu}}-\frac{1}{2}f(R){{g}_{\mu\nu}}-\left({{\nabla}_{\mu}}{{\nabla}_{\nu}}-{{g}_{\mu\nu}}\square\right){f}^{\prime}(R)=8\pi
G{{T}_{\mu\nu}},$ (103)
where
${{T}_{\mu\nu}}=\frac{-2}{\sqrt{-g}}\frac{\delta{{S}_{m}}}{\delta{{g}^{\mu\nu}}},$
(104)
and ${{\nabla}_{\mu}}$ is the covariant derivative, associated with the Levi-
Civita connection of the metric and
$\square\,\equiv{{\nabla}_{\mu}}{{\nabla}^{\mu}}.$
Leaving aside the complications connecting with variation procedure, let us
concentrate our attention on the field equations (103). They represent partial
differential equation of fourth order in metrics, as the Ricci scalar $R$
already contains second order derivatives of the latter. For the action linear
on $R,$ the fourth-order derivatives (the two latter terms on the left hand
side of (103)) turn to zero and theory reduces to the standard general
relativity.
Note that the trace of (103) gives
${f}^{\prime}(R)R-2f(R)+3\square{f}^{\prime}(R)=8\pi GT,$ (105)
where $T={{g}^{\mu\nu}}{{T}_{\mu\nu}}$ links $R$ with $T$ differential
relation, unlike algebraic one in general relativity, where $R=-8\pi GT.$ This
a direct hint on the fact that the field equation of the $f(R)$-theory allow
much wider set of solutions compared to general relativity. To illustrate such
statement we remark that the Jebsen-Birkhoff’s theorem, stating that the
Schwarzschild solution represents the unique spherically symmetric vacuum
solution, does not hold in the $f(R)$-theory. Leaving aside the details, we
note, that now $T=0$ does not require that $R=0$ or even is constant.
The equation (105) appears very useful for investigation of different aspects
of $f(R)$-gravity, especially for stability of solutions and weak field limit.
In particular it is conveniently used for analysis of the so-called maximally
symmetric solutions. It is the solutions with $R=const.$ For $R=const$ and
${{T}_{\mu\nu}}=0$ the equation (105) reduces to
${f}^{\prime}(R)R-2f(R)=0.$ (106)
For a given function $f(R)$ this equation is algebraic for $R.$ If $R=0$ is a
root of the equation, then (103) reduces in that case to ${{R}_{\mu\nu}}=0$
and maximally symmetric solution represents the Minkovski space-time. If
otherwise the root of equation (106) equals $R=const=C,$ then (103) reduces to
${{R}_{\mu\nu}}=\frac{C}{4}{{g}_{\mu\nu}}$ and maximally symmetric solution
corresponds to the space of de Sitter or anti-de Sitter (the cosmological
constant in general relativity) depending on sign of $C.$
Now let us consider how dynamics of Universe are immediately described in
$f(R)$ cosmology. Inserting FRW-metric into (103) and using the stress-energy
tensor in the form
${{T}_{\mu\nu}}=\left(\rho+p\right){{u}_{\mu}}{{u}_{\nu}}-p{{g}_{\mu\nu}},$
(107)
one obtains
${{H}^{2}}=\frac{8\pi
G}{3{f}^{\prime}}\left[\rho+\frac{1}{2}\left(R{f}^{\prime}-f\right)-3H\dot{R}{f}^{\prime\prime}\right];$
(108) $2\dot{H}+3{{H}^{2}}=-\frac{8\pi
G}{{{f}^{\prime}}}\left[p+{{{\dot{R}}}^{2}}{f}^{\prime\prime\prime}+2H\dot{R}{f}^{\prime\prime}+\ddot{R}{f}^{\prime\prime}+\frac{1}{2}\left(f-R{f}^{\prime}\right)\right].$
(109)
As was mentioned above, the main motivation for $f(R)$-gravity is in the fact
that it leads to accelerated expansion of Universe without any dark energy.
The easiest way to see it is to introduce the effective energy density and
effective pressure
${{\rho}_{eff}}=\frac{R{f}^{\prime}-f}{2{f}^{\prime}}-\frac{3H\dot{R}{f}^{\prime\prime}}{{{f}^{\prime}}};$
(110)
${{p}_{eff}}=\frac{{{{\dot{R}}}^{2}}{f}^{\prime\prime\prime}+2H\dot{R}{f}^{\prime\prime}+\ddot{R}{f}^{\prime\prime}+\frac{1}{2}\left(f-R{f}^{\prime}\right)}{{{f}^{\prime}}}.$
(111)
In the spatially flat Universe the density ${{\rho}_{eff}}$ must be non-
negative, as it follows (108) in the limit $\rho\to 0.$ Then the equations
(108), (109) take the form of standard Friedmann equations
${{H}^{2}}=\frac{8\pi G}{3}{{\rho}_{eff}};$ (112)
$\frac{{\ddot{a}}}{a}=-\frac{4\pi
G}{3}\left({{\rho}_{eff}}+3{{p}_{eff}}\right).$ (113)
The effective parameter ${{w}_{eff}}$ in the state equation in that case
equals
${{w}_{eff}}\equiv\frac{{{p}_{eff}}}{{{\rho}_{eff}}}=\frac{{{{\dot{R}}}^{2}}{f}^{\prime\prime\prime}+2H\dot{R}{f}^{\prime\prime}+\ddot{R}{f}^{\prime\prime}+\frac{1}{2}\left(f-R{f}^{\prime}\right)}{\frac{R{f}^{\prime}-f}{2}-3H\dot{R}{f}^{\prime\prime}}.$
(114)
The denominator in (114) is definitely positive, so sign of ${{w}_{eff}}$ is
determined by the numerator. In general case of metric $f(R)$-model in order
to reproduce (mimic) the equation of state for de Sitter (cosmological
constant) with ${{w}_{eff}}=-1,$ the following condition must be satisfied
$\frac{{{f}^{\prime\prime\prime}}}{{{f}^{\prime\prime}}}=\frac{\dot{R}H-\ddot{R}}{{{{\dot{R}}}^{2}}}.$
(115)
Let us consider two examples (regardless its realizability). The firs is the
function of the form $f(R)\propto{{R}^{n}}.$ We can easily calculate
${{w}_{eff}}$ as function of $n,$ if we assume the power law for time
dependence of the scale factor
$a(t)={{a}_{0}}{{\left(t/{{t}_{0}}\right)}^{\alpha}}$ (arbitrary dependence
$a(t)$ results in time dependence for ${{w}_{eff}}$). The result for $n\neq 1$
is
${{w}_{eff}}=-\frac{6{{n}^{2}}-7n-1}{6{{n}^{2}}-9n+3},$ (116)
and $\alpha$ in terms of $n$ reads
$\alpha=\frac{-2{{n}^{2}}+3n-1}{n-2}.$ (117)
Appropriate choice of $n$ leads to desired value for ${{w}_{eff}}.$ For
instance, $n=2$ leads to
${{w}_{eff}}=-1,\quad\alpha=\infty.$ (118)
This result is expected from consideration of quadratic corrections to
Einstein-Hilbert action, which were used in the inflation scenery by
Starobinski starobin12 .
The second example is
$f(R)=R-\frac{{{\mu}^{2\left(n+1\right)}}}{{{R}^{n}}}.$ (119)
In that case, assuming again the power law for time dependence of the scale
factor, one gets
${{w}_{eff}}=-1+\frac{2(n+2)}{3(2n+1)(n+1)}.$ (120)
The condition for accelerated expansion $w_{eff}<-1/3$ for the two above
considered models is transformed into the form
$n^{2}\mp n-1>0,$
where the signs $\mp$ respond to the first and the second examples
respectively. In particular, in the second case, for $n=1$ we find that
${{w}_{eff}}=-2/3.$ We remark that in such case the positive values of $n$
imply the presence of terms inversely proportional to $R,$ unlike the above
considered case.
### VII.2 Braneworld Cosmology
All models explaining the accelerated expansion of Universe share one common
property: they involve one or another way to weaken the gravity. It is the
negative pressure in SCM and in the scalar field models, transformation of the
universal gravity law in the modified gravitation model or voids in the models
with inhomogeneities. An original approach to suppress the gravity is realized
in the Braneworld scenario Randall . According to latter we live on a three-
dimensional brane, which is embedded in the volume of higher dimension (in the
simplest case it is four-dimensional). All material fields are restricted to
the brane, while the gravitation penetrates both the brane and whole embedding
volume (see Lecture notes by R.Maartens Maartens ). Higher dimension of the
space where the gravity acts, results in its suppression
($F\propto{{R}^{-(D-1)}},$ where $D$ is the dimension of the space).
According such scenario the equation of motion for the scalar field on the
brane reads
$\ddot{\varphi}+3H\dot{\varphi}+{V}^{\prime}(\varphi)=0.$ (121)
where
${{H}^{2}}=\frac{8\pi}{3{{M_{Pl}}^{2}}}\rho\left(1+\frac{\rho}{2\sigma}\right)+\frac{{{\Lambda}_{4}}}{3}+\frac{\varepsilon}{{{a}^{4}}};$
(122a) $\rho=\frac{1}{2}{{{\dot{\varphi}}}^{2}}+V\left(\varphi\right).$ (122b)
Here $\varepsilon$ is the integration constant, which transforms the volume
gravity to the brane restricted one. The brane tension $\sigma$ provides
connection between the for-dimensional Planck mass $M_{Pl}$ and five-
dimensional one $M_{Pl}^{(5)}:$
$M_{Pl}=\sqrt{\frac{3}{4\pi}}\left(\frac{{{M_{Pl}^{(5)}}^{3}}}{\sqrt{\sigma}}\right).$
(123)
The tension $\sigma$ also links the four-dimensional cosmological constant
${{\Lambda}_{4}}$ on the brane with the five-dimensional volume one
${{\Lambda}_{b}}$ by:
${{\Lambda}_{4}}=\frac{4\pi}{{{M_{Pl}^{(5)}}^{3}}}\left({{\Lambda}_{b}}+\frac{4\pi}{3{{M_{Pl}^{(5)}}^{3}}}{{\sigma}^{2}}\right).$
(124)
Remark, that the equations (122a \- 122b) contain an additional term
$\propto{{\rho}^{2}}/\sigma,$ whose presence is connected with the conditions
on the brane-volume boundary. The latter term is responsible for dynamical
enhancement of the decay experienced by the scalar field while it rolls down
the potential. In such a case inflation regime can be realized even for those
potentials which are too steep to satisfy the slow-roll inflation conditions
in the standard approach. Indeed, the slow-roll parameters in the braneworld
models in the limit $V/\sigma\gg 1$ read:
$\varepsilon\simeq
4{{\varepsilon}_{FRW}}{{\left(\frac{V}{\sigma}\right)}^{-1}};\quad\eta\simeq
2{{\eta}_{FRW}}{{\left(\frac{V}{\sigma}\right)}^{-1}}.$ (125)
It follows now that the slow-roll inflation condition
$\left(\varepsilon,\eta\ll 1\right)$ is easier satisfied than in the FRW
cosmology under the condition $V/\sigma\gg 1.$ The inflation thus can take
place for very steep quintessence potentials, such as
$V\propto{{e}^{-\lambda\varphi}},\ V\propto{{\varphi}^{-\alpha}}$ and others.
This in turn let us hope that inflation and quintessence can be generated by
the same scalar field.
A radically different approach to provide the accelerated expansion of
Universe was proposed in papers Deffayet , Deffayet2 , sahni21 .
The sharp distinction of the DDG model from the RS one is in the fact that
both the volume cosmological constant and brane tension are set to zero, while
a curvature term is introduced into the brane action. The theory is based on
the following action
$S={{M_{Pl}^{(5)}}^{3}}\int_{bulk}{R+{{M_{Pl}}^{2}}\int_{brane}{R+}}\int_{brane}{{{L}_{matter}}}.$
(126)
The physical sense of the term $\int_{brane}{R}$ is unclear. Probably, it
accounts for quantum effects caused by material fields, which lead to the same
term in the Einstein action, as was first noted by A.D. Sakharov in its
induced gravity theorySakharov .
The Hubble parameter in the DDG braneworld takes the form
$H=\sqrt{\frac{8\pi
G{{\rho}_{m}}}{3}+\frac{1}{l_{c}^{2}}}+\frac{1}{{{l}_{c}}},$ (127)
where ${{l}_{c}}={{M_{Pl}}^{2}}/{{M_{Pl}^{(5)}}^{3}}$ is a new length scale,
defined by the 4D Planck mass $M_{Pl}$ and 5D one $M_{Pl}^{(5)}.$ An important
property of such a model is that the accelerated expansion of Universe does
require presence of dark energy. Instead, since gravity becomes five
dimensional on length scales
$R>{{l}_{c}}=2{{H}^{-1}}{{\left(1-{{\Omega}_{m}}\right)}^{-1}},$ one finds
that the expansion of the Universe is modified during late times instead of
early times as in RS model.
More general class of braneworld models, which includes both RS and DDG models
as particular cases, is described by the action
$S={{M_{Pl}^{(5)}}^{3}}\int_{bulk}{(R-2{{\Lambda}_{b}})+\int_{brane}{({{M_{Pl}}^{2}}R-2\sigma)+}}\int_{brane}{{{L}_{matter}}}.$
(128)
For $\sigma={{\Lambda}_{b}}=0$ the action (128) transforms into the DDG model
action (126), and for $m=0$ it reduces to RS-model.
As was shown in sahni21 , the action (128) describes the Universe, where the
accelerated expansion era occurs on later evolution stages with the Hubble
parameter
$\frac{{{H}^{2}}(z)}{H_{0}^{2}}={{\Omega}_{m}}{{(1+z)}^{3}}+{{\Omega}_{\sigma}}+2{{\Omega}_{l}}\mp
2\sqrt{{{\Omega}_{l}}}\sqrt{{{\Omega}_{m}}{{(1+z)}^{3}}+{{\Omega}_{\sigma}}+{{\Omega}_{l}}+{{\Omega}_{{{\Lambda}_{b}}}}},$
(129)
where
${{\Omega}_{l}}=\frac{1}{l_{c}^{2}H_{0}^{2}},\
{{\Omega}_{m}}=\frac{{{\rho}_{0m}}}{3{{M_{Pl}}^{2}}H_{0}^{2}},\
{{\Omega}_{\sigma}}=\frac{\sigma}{3{{M_{Pl}}^{2}}H_{0}^{2}},\
{{\Omega}_{{{\Lambda}_{b}}}}=-\frac{{{\Lambda}_{b}}}{6H_{0}^{2}}.$ (130)
Signs $\mp$ correspond to two possible ways to embed a brane in to the volume.
As it was in DDG model, ${{l}_{c}}\sim H_{0}^{-1}$ if $M_{Pl}^{(5)}\sim
100MeV.$ On shorter length scales $r\ll{{l}_{c}}$ and on early times we
recover the general relativity, while on large length scales $r\gg{{l}_{c}}$
and long time periods the brane effects start to be important. Indeed, setting
$M_{Pl}^{(5)}=0\ ({{\Omega}_{l}}=0)$ the equation (129) reduces to the
$\Lambda CDM$ model:
$\frac{{{H}^{2}}(z)}{H_{0}^{2}}={{\Omega}_{m}}{{(1+z)}^{3}}+{{\Omega}_{\sigma}}.$
(131)
while for the case $\sigma={{\Lambda}_{b}}=0$ the relation (129) reproduces
the DDG model. An important feature of the action (128) is the fact that it
generates the effective state equation ${{w}_{eff}}\leq-1.$ It can be easily
seen sahni21 from the expression for current value of the effective parameter
in the state equation
${{w}_{0}}=\frac{2{{q}_{0}}-1}{3\left(1-{{\Omega}_{m}}\right)}=-1\pm\frac{{{\Omega}_{m}}}{1-{{\Omega}_{m}}}\sqrt{\frac{{{\Omega}_{l}}}{{{\Omega}_{m}}+{{\Omega}_{\sigma}}+{{\Omega}_{l}}+{{\Omega}_{{{\Lambda}_{b}}}}}}.$
(132)
Taking the lower sign, one gets ${{w}_{0}}<-1.$
It is worth noting that in the latter model the accelerated Universe expansion
phase is the transient phenomenon, which comes to end when the Universe
returns to the matter dominated phase.
### VII.3 MOND
The so-called modified Newtonian dynamics (MOND) is sometimes considered as an
alternative to the DM (dark matter) concept. MOND represents such modification
of Newton physics, which allows to explain the flat rotational curves for
galaxies without attributing to any assumptions on DM. MOND assumes that the
second Newton law $F=ma$ must be modified for sufficiently low accelerations
$(a\ll{{a}_{0}})$ so that
$\vec{F}=m\vec{a}\mu(a/{{a}_{0}}).$ (133)
where $\mu(x)=x$ if $x\ll 1,$ and $\mu(x)=1$ if $x\gg 1.$ It is easy to see
that such law leads to the modification of the traditional formula for the
gravitational acceleration $\vec{F}=m{{\vec{g}}_{N}}\
({{g}_{N}}=GM/{{r}^{2}}).$ Relation between the ”correct” acceleration and
traditional Newtonian one reads
$a=\sqrt{{{a}_{0}}{{g}_{N}}}.$ (134)
For a rotating point mass one gets $a={{v}^{2}}/r$ (this purely kinematic
relation is independent on choice of the dynamic model). But here $a$ stands
already for ”correct” acceleration. It follows that
${{v}^{4}}=GM{{a}_{0}}.$ (135)
i.e. for sufficiently small accelerations the rotational curves for isolated
body of mass $M$ do not depend on radial distance $r,$ where the velocity is
measured. In other words, the considered theory predicts not only flat
rotational curves but also the fact that individual halo, associated with a
galaxy, has infinite extension. This prediction may cause a serious problem
for MOND, as recent observations of galaxy-galaxy lensing support the result
that the maximum halo extension is about 0.5 Mpc. The value ${{a}_{0}},$
needed to explain the observations
${{a}_{0}}\sim{{10}^{-8}}\,cm/{{s}^{2}}.$ (136)
is of the same order as $c{{H}_{0}}.$ It supports the hypothesis that MOND
“can reflect the influence of cosmology on local particle dynamics”.
Although the results of MOND agree very well with the observations of
individual galaxies, it is unclear whether they will be as well successful for
description of cluster structure, where strong gravitational lensing points
out considerably denser mass concentration in the cluster center than was
predicted by MOND. Another difficulty connected with MOND, is the fact that it
is quite problematic to incorporate it in more general relativistic gravity
theory. Presently it is not clear what predictions are given by the MOND-like
theories for complicated gravity effects like the gravitational lensing.
## VIII Dynamics of Universe with interaction in the dark sector
SCM considers dark matter and dark energy as independent components of energy
budget of Universe. The dark energy postulated in form of cosmological
constant, introduced as early as by Albert Einstein in order to make possible
the creation of stationary Universe model. The assumed absence of interaction
between the two components means that the energy densities for each component
obey independent conservation equations
${{\dot{\rho}}_{i}}+3H\left({{\rho}_{i}}+{{p}_{i}}\right)=0.$
Coupling between the two components leads to modified evolution of Universe.
In particular, the energy density for non-relativistic component will not
evolve according the law ${{a}^{-3}},$ and the dark energy density (in form of
the cosmological constant) will not remain constant any more. From one hand,
such modification of the theory opens for us new possibilities for solution of
principal problems in cosmology. So, for example, solution of the coincidence
problem333the problem is that during all the Universe history the two
densities decay by different laws, so it is necessary to impose very strict
limitations on their values in early Universe in order to make them be of
comparable order nowadays reduces to appropriate choice of interaction
parameters, which can provide satisfaction of the condition
$\frac{{{\Omega}_{de}}}{{{\Omega}_{dm}}}={\mathrm{O}}(1),$
in the present time consistently with the condition $\ddot{a}>0.$ From the
other hand, introduction of the interaction will modify the relations between
the observable parameters. In particular, the modification will affect the
fundamental relation between the photometric distance and the redshift, which
the evidence of accelerated expansion of Universe is mainly based on. It
imposes strict limitations both on the form of interaction and its parameters.
Taking into account that dark matter and dark energy represent dominant
components in the Universe, then from point of view of field theory it is
naturally to consider interaction between them InteractionQ1 ; InteractionQ2 ;
InteractionQ3 ; InteractionQ4 ; InteractionQ5 ; InteractionQ6 ; InteractionQ7
; 0801.1565 ; ZKGuo . Appropriate interaction between the dark matter and dark
energy can facilitate solution of the coincidence problem InteractionQ10 ;
InteractionQ8 ; InteractionQ9 . Non-minimal coupling in the dark sector can
considerably affect the history of cosmological expansion of Universe and
evolution of density fluctuations, thus modifying the rate of the cosmological
structure growth. The possibility of dark matter and dark energy to interact
with each other is widely discussed in recent literature InteractionQ1 ;
InteractionQ2 ; InteractionQ3 ; InteractionQ4 ; InteractionQ5 ; InteractionQ6
; InteractionQ7 ; InteractionQ10 ; InteractionQ8 ; InteractionQ9 ; 0801.1565 ;
ZKGuo . Different and independent data of many observations, such as Wilkinson
Microwave Anisotropy Probe, SNe Ia and BAO, were specially analyzed in order
to investigate the limitations on the intensity and form of the interaction in
the dark sector. Some researchers also suggest BaldiClusters , that dynamical
equilibrium of collapsing structures, such as clusters, will essentially
depend on form and sign of the interaction between dark matter and dark
energy.
### VIII.1 Model of Universe with time dependent cosmological constant
The simplest example of the model with interacting DM and DE is the
cosmological model with decaying vacuum. Actually the $\Lambda$(t)CDM
cosmology represents one of the cases where parameter $w$ for dark energy
equals to $-1.$
This model is based on the assumption that the dark energy is nothing that
physical vacuum, and energy density of the latter is calculated on the curved
space background with subtraction of renormalized energy density of physical
vacuum in the flat space 0711.2686 . The resulting effective energy density of
physical vacuum depends on space-time curvature and decays from high initial
values in early Universe (at strong curvature) to almost zero magnitude in the
present time.
Due to the Bianchi identity, the decay of vacuum must be accompanied by
creation or mass increase of dark energy particles, which is common property
of the decaying vacuum models, or in more general case, for the models with
interacting dark matter and dark energy.
Now let us give the formulation of the $\Lambda$(t)CDM. In the present section
we will use the system of units where the reduced Planck mass equals to unity:
${{M}_{Pl}}={{\left(8\pi G\right)}^{-1/2}}=1.$
For the case of flat Universe described by FRW-metric, the first Friedmann
(13) and the conservation equation can be presented in the form
$\rho_{tot}=3H^{2},~{}~{}\dot{\rho}_{tot}+3H(\rho_{tot}+p_{tot})=0,$
where $\rho_{tot}=\rho_{m}+\rho_{\Lambda},$ and taking into account that
$p_{m}=0$ and $p_{\Lambda}=-\rho_{\Lambda},$ the conservation equation takes
the form:
$\dot{\rho}_{m}+3H\rho_{m}=-\dot{\rho}_{\Lambda}.$ (137)
The right hand side of the latter contains an additional term which plays role
of a source — it is the decaying cosmological constant. The problem with such
approach is in fact that we end up with the same number of equations, but
acquire additional unknown function $\Lambda(t).$ The above described approach
to define the physical vacuum density, though intuitively clear, faces certain
principal difficulties BertolamiLdec . That is why a phenomenological approach
prevails in the literature. Below we present a simple but exactly solvable
model. Consider the case when $\Lambda$ depends on time as Carneiro :
$\Lambda=\sigma H.$ (138)
It is interesting to note that with the choice $\sigma\approx m_{\pi}^{3}$
(${{m}_{\pi}}$ is the energy scale of the QCD vacuum condensation) the
relation (138) gives the value of $\Lambda$ which is very close to the
observed one.
For the considered components the first Friedmann equation takes the form
${{\rho}_{\gamma}}+\Lambda=3{{H}^{2}}.$ (139)
The equations (137) and (139) together with the conservation equation
${{p}_{\gamma}}=w{{\rho}_{\gamma}}\equiv\left(\gamma-1\right){{\rho}_{\gamma}},$
and the decay law for the cosmological constant (137) completely determine the
scale factor time evolution. Combining those equation, one obtains the
evolution equation in the following form
$2\dot{H}+3\gamma{{H}^{2}}-\sigma\gamma H=0.$
Under condition $H>0$ the solution for the latter equation reads
$a(t)=C{{\left[\exp\left(\sigma\gamma
t/2\right)-1\right]}^{\frac{2}{3\gamma}}},$
where $C$ is one of the two integration constants (recall that the equation
for the scale factor is of second order). The second integration constant is
determined from the condition $a(t=0)=0.$ Using such solution, one can find
the densities of matter (radiation) and time-dependent cosmological constant
${{\rho}_{\gamma}}=\frac{{{\sigma}^{2}}}{3}{{\left(\frac{C}{a}\right)}^{\frac{3}{2}\gamma}}\left[1+{{\left(\frac{C}{a}\right)}^{\frac{3}{2}\gamma}}\right];$
(140)
$\Lambda=\frac{{{\sigma}^{2}}}{3}\left[1+{{\left(\frac{C}{a}\right)}^{\frac{3}{2}\gamma}}\right].$
(141)
Let us now analyze the history of Universe expansion in the scenario under
consideration. In the radiation-dominated epoch $\gamma=w+1=4/3$ and,
therefore,
$a(t)=C{{\left[{{e}^{\frac{2}{3}\sigma t}}-1\right]}^{\frac{1}{2}}}.$
For small values of time $\left(\sigma t\ll 1\right)$ and we recover the well-
known dependence $a\propto{{t}^{1/2}},$
$a(t)=C\sqrt{\frac{2}{3}}\sigma{{t}^{\frac{1}{2}}}.$
The densities ${{\rho}_{\gamma}}$(140) and $\Lambda$ (141) in the radiation
dominated epoch transform into
${{\rho}_{\gamma}}\to{{\rho}_{r}}=\frac{{{\sigma}^{2}}{{C}^{4}}}{3}\frac{1}{{{a}^{4}}}+\frac{{{\sigma}^{2}}{{C}^{2}}}{3}\frac{1}{{{a}^{2}}};$
$\Lambda=\frac{{{\sigma}^{2}}}{3}+\frac{{{\sigma}^{2}}{{C}^{2}}}{3}\frac{1}{{{a}^{2}}}.$
In the linit $a\to 0\ \left(t\to 0\right)$ one has
${{\rho}_{r}}=\frac{{{\sigma}^{2}}{{C}^{4}}}{3}\frac{1}{{{a}^{4}}}=\frac{3}{4{{t}^{2}}};$
$\Lambda=\frac{{{\sigma}^{2}}{{C}^{2}}}{3}\frac{1}{{{a}^{2}}}=\frac{\sigma}{2t}.$
Now consider the matter dominated era. In that case $\gamma=w+1=1$ and
$a(t)=C{{\left[{{e}^{\frac{1}{2}\sigma t}}-1\right]}^{\frac{2}{3}}}.$
For $\sigma t\ll 1$ we reproduce the standard law for matter evolution
$a(t)=C{{\left(\frac{\sigma}{2}\right)}^{\frac{2}{3}}}{{t}^{\frac{2}{3}}}.$
For the densities ${{\rho}_{\gamma}}$ and $\Lambda$ in the matter dominated
era one finds
${{\rho}_{\gamma}}\to{{\rho}_{m}}=\frac{{{\sigma}^{2}}{{C}^{3}}}{3}\frac{1}{{{a}^{3}}}+\frac{{{\sigma}^{2}}{{C}^{3/2}}}{3}\frac{1}{{{a}^{3/2}}};$
$\Lambda=\frac{{{\sigma}^{2}}}{3}+\frac{{{\sigma}^{2}}{{C}^{3/2}}}{3}\frac{1}{{{a}^{3/2}}}.$
Note that in the limit of long times $(\sigma t\gg 1)$ the scale factor growth
exponentially as
$a(t)=C{{e}^{\frac{\sigma}{3}t}}.$
In the same limit the matter density tends to zero, and the time-dependent
function $\Lambda(t)$ transforms into the ”real” cosmological constant.
In the matter dominated epoch the Friedmann equation (139) can be presented in
the form
$H(z)={{H}_{0}}\left[1-{{\Omega}_{m0}}+{{\Omega}_{m0}}{{(1+z)}^{3/2}}\right].$
This expression can be used for calculation of the deceleration parameter
$\left(q=\frac{1+z}{H}\frac{dH}{dz}-1\right),$
$q(z)=\frac{\frac{3}{2}{{\Omega}_{m0}}{{(1+z)}^{3/2}}}{1-{{\Omega}_{m0}}+{{\Omega}_{m0}}{{(1+z)}^{3/2}}}-1.$
(142)
One thus finds for the current value of the deceleration parameter
$q(z=0)=\frac{3}{2}{{\Omega}_{m0}}-1.$
Therefore the accelerated expansion of Universe occurs under the following
condition
${{\Omega}_{m0}}<\frac{2}{3},$
This condition is in fact satisfied for the observed value
${{\Omega}_{m0}}\approx 0.23.$ From (142) it follows that the transition from
the decelerated expansion to the accelerated one took place at
$z^{*}={{\left[2\frac{1-{{\Omega}_{m0}}}{{{\Omega}_{m0}}}\right]}^{2/3}}-1.$
This value $\left(z^{*}\approx 1.2\right)$ exceeds (being however of the same
order ${\mathrm{O}}\,(1)$) the corresponding value for SCM $\left(z^{*}\approx
0.75\right),$ which is the result of the matter production in the vacuum decay
process.
Therefore the model based on the time-dependent cosmological constant, where
vacuum density linearly depends on Hubble parameter, appears to be quite
competitive. It sufficiently accurately reproduces the ”canonic” results,
relative both to the radiation dominated and matter dominated eras. The
present Universe expansion is accelerated according to the model under
consideration. Verification of the model with the latest observational data
obtained for SN1a leads to the results (e.g., $0.27\leq{{\Omega}_{m0}}\leq
0.37$), that agree very well with the currently accepted estimates for the
accelerated Universe expansion parameters.
### VIII.2 Interacting dark matter and dark energy
One of the most interesting properties of dark matter is its possible
interaction with dark energy. Although the most realistic models (SCM in
particular) postulate that the dark matter and dark energy are uncoupled,
there is no serious evidence to consider such assumption as a principle. In
the present time many active researches are provided to investigate the
possible consequences of such interaction InteractionQ1 ; InteractionQ2 ;
InteractionQ3 ; InteractionQ4 ; InteractionQ5 ; InteractionQ6 ; InteractionQ7
; InteractionQ10 ; InteractionQ8 ; InteractionQ9 ; 0801.1565 ; ZKGuo ;
BaldiClusters ; Chimento ; Lima . As we have mentioned above, the interaction
of dark components can at least soften some sharp cosmological problems, such
as the coincidence problem for example. The dark energy density is
approximately three times as high as the dark matter one. Such coincidence
could be explained if dark matter was somehow sensitive to dark energy.
Remark that the possibility of interaction between the dark energy in form of
scalar field and dark matter lies in the basis of warm inflation model. Unlike
the cold inflation scenario, the former does not assume that the scalar field
in the inflation period is isolated (uncoupled) from other fields. That is why
instead of overcooled Universe in the inflation period, in the Universe model
under discussion a certain quantity of radiation is always conserved, which is
sufficiently remarkable to manifest in post-inflation dynamics.
Interaction of the scalar field quantum, responsible for the inflation start –
the inflaton – with other fields imply that its master equation contains the
terms describing the process of energy dissipation from inflaton system to
other particles. Barera and Fang Berera initially assumed that consistent
description of the inflaton field with energy dissipation requires the master
equation in form of the Langeven equation, where the fluctuation-dissipation
relation is present, which uniquely links the field fluctuations and
dissipated energy. Such equation lies in the the basis of description of the
warm inflation process.
Interaction between the components in the Universe must be introduced in such
a way that preserves the covariance of the energy-momentum tensor
$T^{\,\,\mu\nu}_{(tot)\,\,;\nu}=0,$ therefore
$T^{\,\,\mu\nu}_{{}_{DM}\,\,;\nu}=-T^{\,\,\mu\nu}_{{}_{DE}\,\,;\nu}\neq 0,$
where $u_{\nu}$ is the 4-velocity. The conservation equations in that case
take the form:
$u_{\nu}T^{\,\,\mu\nu}_{{}_{DM}\,;\mu}=-u_{\nu}T^{\,\,\mu\nu}_{{}_{DE}\,;\mu}=-Q.$
(143)
For FRW-metric the equations (143) transform to:
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (144)
$\displaystyle\dot{\rho}_{{}_{DE}}+3H\rho_{{}_{DE}}(1+w_{{}_{DE}})=-Q,$ (145)
where $\rho_{m}$ and $\rho_{{}_{DE}}$ are densities of dark matter and dark
energy respectively, $w_{{}_{DE}}$ is the parameter of state equation for for
dark energy, $H\equiv\dot{a}/a$ is the Hubble parameter.
$Q~{}\left\\{\begin{array}[]{l}>0\\\
<0\end{array}\right.~{}~{}\Rightarrow~{}~{}\mbox{energy
transits}~{}\left\\{\begin{array}[]{l}\mbox{DE $\to$ DM};\\\ \mbox{DM $\to$
DE}\end{array}\right.$
If $Q<0,$ then dark matter continuously decays into dark energy, if $Q>0$ then
vice versa. Note that the equations (144) and (145) obey the conservation
equation:
$\dot{\rho}_{tot}+3H\rho_{tot}(\rho_{tot}+p_{tot})=0,$ (146)
where $\rho_{tot}=\rho_{{}_{DE}}+\rho_{m}$ is the total energy density.
The interaction between the dark matter and dark energy is effectively
equivalent to modification of the state equation for the interacting
components. Indeed, the equations (144) can be transformed to the standard
form of the conservation law written for uncoupled components:
$\begin{array}[]{l}{\dot{\rho}_{{}_{DE}}+3H\rho_{{}_{DE}}(1+w_{{}_{DE,eff}})=0}\\\
{\dot{\rho}_{{}_{DM}}+3H\rho_{{}_{DM,eff}}=0},\end{array},$
where
$w_{{}_{DE,eff}}=w_{{}_{DE}}-\frac{Q}{3H\rho_{{}_{DE}}};\quad
w_{{}_{DM,eff}}=\frac{Q}{3H\rho_{{}_{DM}}},\quad$ (147)
play role of effective state equations for dark energy and dark matter
respectively.
As we do not know the nature of both the dark energy and dark matter, we
cannot derive the coupling $Q$ from the first principles secondref . However
it is clear from dimension analysis that this quantity must depend on the
energy density of one of the dark component, or a combination of both
components with dimension of energy density, times the quantity of inverse
time dimension. For the latter it is natural to take the Hubble parameter $H.$
Different forms of $Q$ the most often used in the literature, are the
following:
$Q=3H\gamma\rho_{m},~{}~{}Q=3H\gamma\rho_{{}_{DE}},~{}~{}Q=3H\gamma(\rho_{m}+\rho_{{}_{DE}}).$
(148)
For example we consider the simplest case for such model
$\begin{array}[]{l}\dot{\rho}_{m}+3H\rho_{m}=\delta H\rho_{m},\\\
\dot{\rho}_{{}_{DE}}+3H\left({\rho_{{}_{DE}}+p_{{}_{DE}}}\right)=-\delta
H\rho_{m},\end{array}$
where $\delta$ is the dimensionless coupling constant.
Integration of the latter equations yields
$\begin{array}[]{l}\rho_{{}_{DE}}=\rho_{{}_{DE0}}a^{-\left[{3(1+w_{{}_{DE}})+\delta}\right]};\hfill\\\
\rho_{m}=\frac{{-\delta\rho_{{}_{DE0}}}}{{3w_{{}_{DE}}+\delta}}a^{-\left[{3(1+w_{{}_{DE}})+\delta}\right]}+\left({\rho_{m0}+\frac{{\delta\rho_{{}_{DE0}}}}{{3w_{{}_{DE}}+\delta}}}\right)a^{-3}.\end{array}$
(149)
Inserting the obtained densities into the first Friedmann equation (13) and
transforming from scale factor to redshift, one gets for $H^{2}(z)$ the
following
$H^{2}=\frac{(1+z)^{3}H_{0}^{2}}{3(3w_{{}_{DE}}+\delta)}\left[3w_{{}_{DE}}\Omega_{DE}(1+z)^{(3w_{{}_{DE}}+\delta)}+\Omega_{m}(3w_{{}_{DE}}+\delta)\right].$
(150)
Note that in the considered case the deceleration parameter $q(z)$ can be
easily determined by the following
$q(z)=\frac{1+z}{2H^{2}}\frac{dH^{2}}{dz}-1,$
and direct substitution of (150) results in explicit dependence of
deceleration parameter on the redshift:
$q(z)=-1+\frac{3}{2}\frac{w_{{}_{DE}}\Omega_{DE}(3(1+w_{{}_{DE}})+\delta)(1+z)^{(3w_{{}_{DE}}+\delta)}+\Omega_{m}+\delta/w_{{}_{DE}}}{3w_{{}_{DE}}\Omega_{DE}(1+z)^{(3w_{{}_{DE}}+\delta)}+\Omega_{m}+\delta/w_{{}_{DE}}}.$
(151)
With $w_{{}_{DE}}=-1$ and $\delta=0,$ this expression coincides with the
deceleration parameter value obtained in SCM. Note that in the considered
model appropriate choice of the coupling parameter value $\delta$ can
essentially soften or even get rid of the compatibility problem for the
densities of dark matter and dark energy. Indeed, using the densities (149),
one finds the following relation $R=\rho_{m}/\rho_{{}_{DE}}:$
$R=-\frac{\delta}{{3w_{{}_{DE}}+\delta}}+\left(R_{0}+\frac{\delta}{3w_{{}_{DE}}+\delta}\right)a^{3w_{{}_{DE}}+\delta},$
where $R_{0}=\rho_{m0}/\rho_{{}_{DE0}}$ is the ratio of the densities in the
present time. In SCM it is known to be $R\sim a^{-3},$ which differs from the
considered model on the $\delta$ in exponent.
Let us now consider the inverse problem, and instead of interaction we
postulate the ratio
$\frac{\rho_{m}}{\rho_{{}_{DE}}}=f(a),$ (152)
where $f(a)$ is arbitrary differentiable function of scale factor. Thus one
obtains:
$\displaystyle\rho_{m}=\rho_{{}_{DE}}f(a),$ (153)
$\displaystyle\rho_{{}_{DE}}=\frac{\rho_{m}}{f(a)},$ (154)
$\displaystyle\dot{\rho}_{m}=\dot{\rho}_{{}_{DE}}f+\rho_{{}_{DE}}f^{\prime}\dot{a},$
(155)
$\displaystyle\dot{\rho}_{{}_{DE}}=\frac{\dot{\rho}_{m}}{f}-\frac{\rho_{m}\dot{a}f^{\prime}}{f^{2}}.$
(156)
Inserting the expressions (155) and (153) into the equation (144) we obtain:
$\dot{\rho}_{{}_{DE}}f+\rho_{{}_{DE}}f^{\prime}\dot{a}+3H\rho_{{}_{DE}}f=Q,$
(157)
where prime denotes the differentiation with respect to the scale factor.
Using here the expression for $\dot{\rho}_{{}_{DE}},$ obtained from (145), one
gets:
$Q=\frac{H\rho_{{}_{DE}}f}{1+f}\left(\frac{f^{\prime}a}{f}-3w_{{}_{DE}}\right).$
(158)
The first Friedmann equation takes on the form:
$3H^{2}=\rho_{{}_{DE}}+\rho_{m}=\rho_{cr},$ (159)
where $\rho_{cr}$ is the critical density. Accordingly one can write down that
$\displaystyle\Omega_{{}_{DE}}=\frac{\rho_{{}_{DE}}}{\rho_{cr}}=\frac{1}{1+f};$
(160) $\displaystyle\Omega_{m}=\frac{\rho_{m}}{\rho_{cr}}=\frac{f}{1+f}.$
(161)
Now, substituting (161) into (158), we finally obtain the expression for $Q$:
$Q=H\rho_{{}_{DE}}\Omega_{m}\left(\frac{f^{\prime}a}{f}-3w_{{}_{DE}}\right)=H\rho_{m}\Omega_{{}_{DE}}\left(\frac{f^{\prime}a}{f}-3w_{{}_{DE}}\right).$
(162)
It is worth noting that in the case when $f(a)\propto a^{\xi}$ we obtain the
expression
$Q=H\rho_{m}\Omega_{{}_{DE}}\left(\xi-3w_{{}_{DE}}\right)=H\rho_{{}_{DE}}\Omega_{m}\left(\xi-3w_{{}_{DE}}\right).$
(163)
From the equation (146) it follows that:
$\frac{d\ln\rho_{tot}}{d\ln a}=-3(1+w_{eff}),$ (164)
where
$w_{eff}=\frac{p_{tot}}{\rho_{tot}}=\frac{\rho_{{}_{DE}}w_{{}_{DE}}}{\rho_{{}_{DE}}+\rho_{m}}=\frac{w_{{}_{DE}}}{1+f}=\Omega_{{}_{DE}}w_{{}_{DE}},$
(165)
and therefore
$\frac{d\ln\rho_{tot}}{d\ln a}=-3(1+\Omega_{{}_{DE}}w_{{}_{DE}}).$ (166)
From the latter equation we obtain:
$\rho_{tot}=Ca^{-3}\exp\left(-\int 3\Omega_{{}_{DE}}w_{{}_{DE}}\,d\ln
a\right)$ (167)
where $C$ is the integration constant determined from the requirement that
$\rho_{tot}(a=1)=\rho_{tot,0}=3H_{0}^{2}/(8\pi G).$ Using the expression for
$\rho_{tot},$ one can easily obtain from the Friedmann equations the relation
for $E\equiv H/H_{0},$ which is used to verify cosmological models and
establish restrictions on cosmological parameters. Expression for
$\rho_{{}_{DE}}=\Omega_{{}_{DE}}\rho_{tot}$ and
$\rho_{m}=\Omega_{m}\rho_{tot}$ can also easily obtained from the equations
(160) and (167).
At last it is worth noting that one can obtain from the equation (152),
$f_{0}=f(a=1)=\frac{\rho_{m0}}{\rho_{{}_{DE0}}}=\frac{\Omega_{m0}}{1-\Omega_{m0}}=\frac{1}{\Omega_{{}_{DE0}}}-1.$
(168)
Let it be now
$f(a)=f_{0}\,a^{\xi},$ (169)
where $\xi$ is a constant and $f_{0}$ is defined above. Substituting the
equation (169) into the equation (167) and requiring that
$\rho_{tot}(a=1)=\rho_{tot,0},$ we can determine the integration constant and
find that
$\rho_{tot}=\rho_{tot,0}\,a^{-3}\left[\,\Omega_{m0}+\left(1-\Omega_{m0}\right)a^{\xi}\,\right]^{-3w_{eff}/\,\xi}.$
(170)
Inserting this expression into the Friedmann equation, one finds:
$\begin{gathered}E^{2}=\frac{H^{2}}{H_{0}^{2}}=a^{-3}\left[\,\Omega_{m0}+\left(1-\Omega_{m0}\right)a^{\xi}\,\right]^{-3w_{{}_{X}}/\,\xi}\\\
=(1+z)^{3}\left[\,\Omega_{m0}+\left(1-\Omega_{m0}\right)(1+z)^{-\xi}\,\right]^{-3w_{{}_{X}}/\,\xi}.\end{gathered}$
(171)
Using the above obtained formulae, it is easy to find the following expression
for the deceleration parameter in the considered model
$q=1+\frac{3}{2}\left(w_{{}_{DE}}\Omega_{{}_{DE}}+w_{m}\Omega_{m}\right),$
(172)
where relative densities $\Omega_{{}_{DE}},~{}\Omega_{{}_{DM}}$ has the
following form
$\Omega_{{}_{DE}}=\frac{\Omega_{{}_{DE0}}}{\Omega_{{}_{DE0}}+(1-\Omega_{{}_{DE0}})a^{\xi}},~{}~{}~{}\Omega_{{}_{DM}}=\frac{(1-\Omega_{{}_{DE0}})a^{\xi}}{\Omega_{{}_{DE0}}+(1-\Omega_{{}_{DE0}})a^{\xi}},$
(173)
and we finally obtain
$q=\frac{1}{2}+\frac{3}{2}\frac{w_{{}_{DE}}\Omega_{{}_{DE0}}(1+z)^{\xi}}{1-\Omega_{{}_{DE0}}+\Omega_{{}_{DE0}}(1+z)^{\xi}},$
(174)
As the dark energy density is not constant in the considered model, unlike the
SCM, it facilitates solution of the coincidence problem under condition
$\xi<3.$ In the model under consideration the accelerated expansion phase
starts at the redshift $z^{*},$ which is defined by the relation $q(z^{*})=0,$
and it equals to
$z^{*}=\left(\frac{1-\Omega_{{}_{DE0}}}{{(1+3w_{{}_{DE}}})\Omega_{{}_{DE0}}}\right)^{\frac{1}{\xi}}-1.$
(175)
This relation naturally generalizes the expression (58), obtained for SCM. The
difference of the value (175) from the corresponding one for SCM depends on
magnitude of the distinction of the parameters $\xi$ and $w_{{}_{DE}}$ from 3
and -1 respectively. In the case of zero difference the quantities (58) and
(175) coincide. Note that at $z\to\infty$ Universe expanded with deceleration
$q\to\frac{1}{2},$ the case $z\to-1$ responds to
$q\to\frac{1}{2}-\frac{2}{3}w_{{}_{DE}}.$ Therefore, as it was expected,
dynamics of the considered model is asymptotically (at $z\to\infty$ and
$z\to-1$) identical to the case of two-component Universe, filled by non-
relativistic matter and dark energy with the state equation
$p=w_{{}_{DE}}\rho.$
### VIII.3 Cosmological models with the new type of interaction
In the present subsection we consider one more type of interaction $Q,$
HaoWei_Q(q) which change its sign, i.e. direction of energy transfer, when
the expansion changes from decelerated regime to accelerated one and vice
versa.
Some recently appeared papers r14 make attempt to determine both the very
possibility of interaction in the dark sector and specify its form and sign,
basing solely on analysis o observational data. The analysis splits all
available set of redshift data $z$ into some parts, within which the function
$\delta(z)=Q/(3H)$ is considered to be constant. This analysis allowed to
establish that the function $\delta(z)$ the most likely takes zero values
$(\delta=0)$ in the interval of redshifts $0.45\lesssim z\lesssim 0.9.$ It
turns out that this remarkable result produces new problems. Indeed, as we
have already mentioned, the researchers mostly consider the interaction of the
form (148):
$Q=3H\gamma\rho_{m},~{}Q=3H\gamma\rho_{{}_{DE}},~{}Q=3H\gamma(\rho_{m}+\rho_{{}_{DE}})$
and for the model under consideration it is always either positive or negative
defined, and therefore it cannot change its sign. Sign changes are possible
only in the case when $\gamma=f(t),$ which can change the sign of $Q$ or
$Q=3H(\alpha\rho_{m}+\beta\rho_{{}_{DE}}),$ where $\alpha$ and $\beta$ have
opposite signs.
As authors of r14 mention, solution of such problem requires to introduce a
new type of interaction, which can change its sign during the evolution of
Universe.
An interaction $Q$ of such type was proposed in the paper HaoWei_Q(q) , where
its cosmological consequences were considered. As it was noted in the above
cited paper, the redshift interval, where the function $\delta(z)$ must change
its sign, includes the transition point when the Universe expansion stopped to
decelerate and started the acceleration (see (58)). Therefore the simplest
type of the interaction which can explain the above mentioned property is the
case when the source $Q$ is proportional to the deceleration parameter $q:$
$Q=q(\alpha\dot{\rho}+3\beta H\rho)\,$ (176)
where $\alpha$ and $\beta$ are dimensionless constants, and the sign of $Q$
will change with the transition of universe from the decelerated expansion
stage $(q>0)$ to the accelerated one $(q<0).$ The authors also consider the
case
$\displaystyle Q=q(\alpha\dot{\rho}_{m}+3\beta H\rho_{m}),$ (177)
$\displaystyle Q=q(\alpha\dot{\rho}_{tot}+3\beta H\rho_{tot}),$ (178)
$\displaystyle Q=q(\alpha\dot{\rho}_{{}_{DE}}+3\beta H\rho_{{}_{DE}}).$ (179)
The paper HaoWei_Q(q)1 considers a model of Universe with decaying
cosmological constant
$\dot{\rho}_{\Lambda}=-Q\,.$
The Friedmann and Raychaudhuri equation take thus the form
$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{3}\rho_{tot}=\frac{\kappa^{2}}{3}(\rho_{\Lambda}+\rho_{m})\,,$
(180) $\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}}{2}(\rho_{tot}+p_{tot})=-\frac{\kappa^{2}}{2}\rho_{m},$
(181)
where $\kappa^{2}\equiv 8\pi G.$ Following the paper HaoWei_Q(q)1 , in the
succeeding subsections we consider cosmological models with interaction of the
type (177)-(179).
#### VIII.3.1 Case $Q=q(\alpha\dot{\rho}_{m}+3\beta H\rho_{m})$
For the beginning we consider the case when the interaction takes the form
(177) and insert such expression into the conservation equation (144),
resulting in the following
$\dot{\rho}_{m}=\frac{\beta q-1}{1-\alpha q}\cdot 3H\rho_{m}\,.$ (182)
Substituting the obtained expression into the equation (177), we finally get
$Q=\frac{\beta-\alpha}{1-\alpha q}\cdot 3qH\rho_{m}.$ (183)
From the equation (181) one obtains
$\rho_{m}=-\frac{2}{\kappa^{2}}\dot{H}.$ (184)
Inserting it into the equation (182) one finds that
$\ddot{H}=\frac{\beta q-1}{1-\alpha q}\cdot 3H\dot{H}\,,$ (185)
Thus we obtained the second order differential equation for the function
$H(t).$ Now transform from the time derivative to differentiation with respect
to the scale factor (denoted by the prime ′), then the equation (185) takes on
the form
$aH^{\prime\prime}+\frac{a}{H}H^{\prime\,2}+H^{\prime}=\frac{\beta
q-1}{1-\alpha q}\cdot 3H^{\prime}\,.$ (186)
This expression represents a second order differential expression for the
function $H(a).$ Note that the deceleration parameter
$q=-1-\frac{\dot{H}}{H^{2}}=-1-\frac{a}{H}H^{\prime}\,,$
is also function of $H$ and $H^{\prime}$ and, except the case $\alpha\not=0,$
the equation has no exact solution and it represents a transcendental
differential equation of second order. That is why we consider solely the case
$\alpha=0.$ Thus the interaction (177) takes the form
$Q=3\beta qH\rho_{m}.$
With $\alpha=0$ the solution (186) can be presented in the form
$H(a)=C_{12}\left[\,3C_{11}(1+\beta)-(2+3\beta)\,a^{-3(1+\beta)}\,\right]^{1/(2+3\beta)},$
(187)
where $C_{11}$ and $C_{12}$ are the integration constants determined below. We
find the relative density of dark matter as the following
$\Omega_{m}\equiv\frac{\kappa^{2}\rho_{m}}{3H^{2}}=-\frac{2\dot{H}}{3H^{2}}=-\frac{2aH^{\prime}}{3H}\,.$
(188)
Inserting the equation (187) into (188), one gets
$\Omega_{m}=\frac{2\left(1+\beta\right)}{2+3\beta-3C_{11}\left(1+\beta\right)\,a^{3\left(1+\beta\right)}}\,.$
(189)
With the requirements $\Omega_{m}(a=1)=\Omega_{m0}$ and $H(a=1)=H_{0},$ the
integration constants take the form
$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{m0}(2+3\beta)-2(1+\beta)}{3\Omega_{m0}(1+\beta)},$
(190) $\displaystyle C_{12}$ $\displaystyle=$ $\displaystyle
H_{0}\left[3C_{11}(1+\beta)-(2+3\beta)\right]^{-1/(2+3\beta)}.$ (191)
Substitution of the expressions (190) and (191) into the equation (187)
finally gives the result
$E\equiv\frac{H}{H_{0}}=\left\\{1-\frac{2+3\beta}{2(1+\beta)}\,\Omega_{m0}\left[1-(1+z)^{3(1+\beta)}\right]\right\\}^{1/(2+3\beta)}.$
(192)
The model contains two free parameters $\Omega_{m0}$ and $\beta.$ We note that
if $\beta=0,$ then the equation (192) reduces to
$E(z)=\left[\Omega_{m0}(1+z)^{3}+\left(1-\Omega_{m0}\right)\right]^{1/2},$
which is equivalent to $\Lambda$CDM model. Using the relation
$q(z)=-\frac{(1+z)}{E(z)}\frac{d}{dz}\left(\frac{1}{E(z)}\right)-1,$
one finds the dependence of deceleration parameter on the redshift in the
considered model
$q(z)=-1+\frac{3}{2}\Omega_{m0}\frac{(1+z)^{3(1+\beta)}}{E^{(2+3\beta)}}.$
(193)
The effective parameter of the state equation is known to equal
$w_{\rm eff}\equiv\frac{p_{tot}}{\rho_{tot}}=\frac{(2q-1)}{3}.$
Figure 4: $\Omega_{m},$ $\Omega_{\Lambda},$ $q$ and $w_{\rm eff}$ as function
of the redshift $z$ at $\Omega_{m0}=0.2738$ and $\beta=-0.010$ in the case
$Q=3\beta qH\rho_{m}.$
The figure 4 presents the plots for dependence of some cosmological parameters
on the redshift $z.$ The free parameters $\Omega_{m0}$ and $\beta$ were chosen
to provide the best agreement with observations. One can find that in the
considered model the transition from the decelerated expansion $(q>0)$ to the
accelerated one $(q<0)$ took place at $z_{t}=0.7489,$ the parameter $\beta$ is
negative and therefore dark matter decays into dark energy when $z>z_{t},$ and
vice versa at $z<z_{t}.$ The Universe is interaction-free in dark sector at
$z_{t}.$
#### VIII.3.2 Case $Q=q(\alpha\dot{\rho}_{tot}+3\beta H\rho_{tot})$
Now we consider the case (178), and proceeding completely analogous the above
considered case one obtains
$Q=\frac{3qH^{3}}{\kappa^{2}}\left(2\alpha\frac{\dot{H}}{H^{2}}+3\beta\right).$
(194)
Inserting the equations (184) and (194) into (144), and transforming to
differentiation with respect to the scale factor, one finds
$aH^{\prime\prime}+\frac{a}{H}H^{\prime\,2}+\left(4+3\alpha
q\right)H^{\prime}+\frac{9\beta qH}{2a}=0\,.$ (195)
As in the previous case, we obtained again the differential equation of second
order for the function $H(a).$ Exact solution exists only in the case
$\alpha=0:$
$H(a)=C_{22}\cdot
a^{-3(2-3\beta+r_{1})/8}\cdot\left(a^{3r_{1}/2}+C_{21}\right)^{1/2},$ (196)
where $C_{21},$ $C_{22}$ are integration constants and
$r_{1}\equiv\sqrt{4+\beta\left(4+9\beta\right)}.$ Inserting (196) into (188),
we get
$\Omega_{m}=\frac{1}{4}\left[2-3\beta+\left(\frac{2C_{21}}{a^{3r_{1}/2}+C_{21}}-1\right)r_{1}\right].$
(197)
The integration constants are determined as usual from the condition
$\Omega_{m}(a=1)=\Omega_{m0},~{}H(a=1)=H_{0}:$
$C_{21}=-1+\frac{2\,r_{1}}{2-3\beta-4\Omega_{m0}+r_{1}},~{}~{}C_{22}=H_{0}\left(1+C_{21}\right)^{-1/2}.$
(198)
We finally get
$E\equiv\frac{H}{H_{0}}=(1+z)^{3(2-3\beta+r_{1})/8}\cdot\left[\frac{(1+z)^{-3r_{1}/2}+C_{21}}{1+C_{21}}\right]^{1/2}.$
(199)
In the considered model there are also two free parameters $\Omega_{m0}$ and
$\beta.$ Using the condition $0\leq\Omega_{m}\leq 1$ with $a\to 0,$ from the
equation (197) one gets $\beta\geq 0.$ The best agreement of the model under
consideration with observational data occurs with $\Omega_{m0}=0.2701$ and
$\beta=0.0.$ It means that the considered interaction model gives worse
agreement with observations than $\Lambda CDM.$ More detailed discussion an
interested reader finds in the paper HaoWei_Q(q)1 by the author of the
considered model. The transition form the decelerated expansion phase $(q>0)$
to the accelerated one $(q<0)$ occurs at $z_{t}=0.7549.$
Figure 5: The same as on Fig. 4, but in the case of interaction of the form
$Q=3\beta qH\rho_{tot}$ under condition $\beta\geq 0.$
#### VIII.3.3 The case of $Q=q(\alpha\dot{\rho}_{\Lambda}+3\beta
H\rho_{\Lambda})$
For the conclusion we consider the case (179). Following the same procedure as
in the two preceding case, one obtains
$Q=\frac{3\beta qH\rho_{\Lambda}}{1+\alpha q}\,.$ (200)
With the equation (184) one has
$\rho_{\Lambda}=\frac{3}{\kappa^{2}}H^{2}-\rho_{m}=\frac{1}{\kappa^{2}}\left(3H^{2}+2\dot{H}\right).$
(201)
Therefore the equation for the Hubble parameter in terms of the scale factor
takes the form:
$aH^{\prime\prime}+\frac{a}{H}H^{\prime\,2}+\left(4+\frac{3\beta q}{1+\alpha
q}\right)H^{\prime}+\frac{9\beta qH}{2a(1+\alpha q)}=0.$ (202)
Exact solution can be obtained in the case, $Q=3\beta qH\rho_{\Lambda},$
namely
$H(a)=C_{32}\cdot
a^{-3(2-5\beta+r_{2})/[4(2-3\beta)]}\cdot\left(a^{3r_{2}/2}+C_{31}\right)^{1/(2-3\beta)},$
(203)
where $C_{31},$ $C_{32}$ are the integration constants, and
$r_{2}\equiv\sqrt{\left(2-\beta\right)^{2}}=\left|\,2-\beta\,\right|\,.$
Inserting (203) into (188), we get
$\Omega_{m}=\frac{1}{2\left(2-3\beta\right)}\left[2-5\beta+\left(\frac{2C_{31}}{a^{3r_{2}/2}+C_{31}}-1\right)r_{2}\right]\,.$
(204)
Assuming that $\Omega_{m}(a=1)=\Omega_{m0}$ and $H(a=1)=H_{0},$ we can write
$C_{31}=-1+\frac{2\,r_{2}}{2-5\beta+r_{2}+2\Omega_{m0}\left(3\beta-2\right)},~{}~{}C_{32}=H_{0}\left(1+C_{31}\right)^{1/(3\beta-2)},$
(205)
and finally get
$E\equiv\frac{H}{H_{0}}=(1+z)^{3\left(2-5\beta+r_{2}\right)/\left[4\left(2-3\beta\right)\right]}\cdot\left[\frac{(1+z)^{-3r_{2}/2}+C_{31}}{1+C_{31}}\right]^{1/\left(2-3\beta\right)}.$
(206)
As before, the model has two free parameters $\Omega_{m0},\beta.$ The maximum
plausibility method for the free parameters of the considered model gives the
result HaoWei_Q(q)1 $\Omega_{m0}=0.2717,~{}\beta=0.0136.$ Unlike the two
preceding models, the observational data analyzed in HaoWei_Q(q)1 , give
evidence in favor of $\beta>0.$ More detailed discussion the interested reader
can find in the paper HaoWei_Q(q)1 of the author of the model.
Figure 6: The same as Fig. 4, but for the case $Q=3\beta qH\rho_{\Lambda}.$
The plot 6 displays the dependence for deceleration parameter and effective
parameter of the state equation $w_{\rm eff}\equiv
p_{tot}/\rho_{tot}=(2q-1)/3$ as functions of the redshift $z$ with the
parameters obtained by the maximum plausibility method. It is easy to show
that the transition from the decelerated expansion phase $(q>0)$ to the
accelerated one $(q<0)$ takes place at $z_{t}=0.7398.$ As the parameter
$\beta$ obtained from observations satisfies $\beta>0,$ then dark energy
decays into dark matter ($Q>0$) for $z>z_{t},$ and vice versa $(Q<0)$ for
$z<z_{t}.$
### VIII.4 Statefinders for the interacting dark energy
In this part of the review, we discuss recently introduced “statefinder
parameters” sahni , that include the third derivative of the cosmic scale
factor, are useful tools to characterize interacting quitessence models pavon
.
Accelerated expansion of Universe is normally described by the deceleration
parameter $q=-\ddot{a}/(aH^{2}).$ As was mentioned above, current value of the
Universe deceleration parameter $q$ is negative, however its amplitude is
difficult to determine from the observations. Since the accelerated dynamics
of Universe is predicted by many models, they can be used to obtain an
additional information on $q.$
In particular, an example of such models is given by the cosmological models
where the dominant components – dark matter and dark energy – interact with
each other. The models, where the dominating components do not evolve
independently and interact with each other instead, are of special interest
because, as was mentioned above, they can facilitate or even completely solve
the “coincidence problem”.
Let us consider the Universe filled by two components of non-relativistic
matter (indexed by $m$) with negligibly small pressure $p_{m}\ll\rho_{m}$ and
dark energy (indexed by $x$) with the state equation $p_{x}=w\rho_{x},$ where
$w<0.$ The dark energy decays into dark matter according to
$\displaystyle\dot{\rho}_{m}$ $\displaystyle+$ $\displaystyle 3H\rho_{m}=Q\,,$
$\displaystyle\dot{\rho}_{x}$ $\displaystyle+$ $\displaystyle
3H(1+w)\rho_{x}=-Q\,,$ (207)
where $Q\geq 0$ measures the strength of the interaction. For further
convenience we will write it as $Q=-3\Pi H,$ where $\Pi$ is a new variable
with dimension of pressure.
The Einstein equations for spatially flat Universe have the form
$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\,\pi G}{3}\rho\
,$ (208) $\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle-\frac{8\,\pi
G}{2}(\rho+p_{x})\,,$ (209)
where $\rho=\rho_{m}+\rho_{x}$ is the total energy density. The quantity
$\dot{H}$ is connected by the deceleration parameter $q$ by the relation
$q=-1-(\dot{H}/H^{2})=(1+3w\Omega_{x})/2,$ where $\Omega_{x}\equiv 8\,\pi
G\rho_{x}/(3H^{2}).$ Evidently the deceleration parameter does not depend on
interaction between the components. Nevertheless, differentiation of $\dot{H}$
yields
$\frac{\ddot{H}}{H^{3}}=\frac{9}{2}\left(1+\frac{p_{x}}{\rho}\right)+\frac{9}{2}\left[w\left(1+w\right)\frac{\rho_{x}}{\rho}-w\frac{\Pi}{\rho}-\frac{\dot{w}}{3H}\frac{\rho_{x}}{\rho}\right]\
.$ (210)
In contrast with $H$ and $\dot{H},$ the second order derivative $\ddot{H}$ in
fact depends on the interaction of the components. Therefore, in order to
distinguish models with different types of interaction, or between interacting
and free models, we need the cosmology dynamical description involving the
parameters explicitly dependent on $\ddot{H}.$ In the papers alam and sahni
Sahni and Alam introduced the pair of parameters (the so-called
”statefinders”), which seem to be promising candidates for investigation of
distinction between the models with interacting components.
$r=\frac{\dddot{a}}{aH^{3}},\qquad
s=\frac{r-1}{3(q-\textstyle{1\over{2}})}\,.$ (211)
In the considered context of interacting components they take on the form
$r=1+\frac{9}{2}\frac{w}{1+\kappa}\left[1+w-\frac{\Pi}{\rho_{x}}-\frac{\dot{w}}{3wH}\right]\
,$ (212)
where $\kappa\equiv\rho_{m}/\rho_{x},$ and
$s=1+w-\frac{\Pi}{\rho_{x}}-\frac{\dot{w}}{3wH}\,.$ (213)
For the interaction-free models with $\Pi=0,$ the statefinders reduce to the
expressions that were studied in sahni and alam . Let us show how the
statefinders can be useful for analysis of cosmological models, where the
dominant components interact with each other. Note that the third order
derivative of the scale factor is necessary for description of any variations
in the general state equation for the cosmic medium Visser_Jerk ; Dabrowski .
It is obvious from the general relation alam
$r-1=\frac{9}{2}\frac{\rho+P}{P}\frac{\dot{P}}{\dot{\rho}}\ ,$ (214)
where $P$ is the total density of cosmic medium which in our case leads to the
relation $P\approx p_{x}.$ Since
$\left(\frac{d}{dt}\right)\left(\frac{P}{\rho}\right)=\frac{\dot{\rho}}{\rho}\left[\frac{\dot{P}}{\dot{\rho}}-\frac{P}{\rho}\right]\
,$ (215)
it is evident that the interaction term in the relation
$\dot{P}\approx\dot{p}_{x}=\dot{w}\rho_{x}+w\dot{\rho}_{x}$ in accordance with
(207) will additionally affect the time dependence of the common state
parameter $P/\rho.$
Models with interacting components suggest a dynamical approach to the
coincidence problem. The principal quantity in that context is the density
ratio $\kappa,$ which was defined above in expression (212). Strict solution
of the coincidence problem requires that the density ratio should be of order
of unity on late stages of Universe evolution. At least it has to vary
sufficiently slowly for time interval of order $H^{-1}.$ The ratio $\kappa$ is
governed by the equation (207)
$\dot{\kappa}=-3H\left[\left(\frac{\rho_{x}+\rho_{m}}{\rho_{m}\,\rho_{x}}\right)\Pi-w\right]\kappa\,.$
(216)
Now let us consider the statefinders for different solutions of that equation.
### VIII.5 Scaling solutions
In the paper scaling it was shown by the authors that the so-called scaling
solutions, i.e. the solutions of the form $\rho_{m}/\rho_{x}\propto a^{-\xi},$
where $\xi$ is constant parameter, lying in the range $[0,3],$ can be
calculated in the case when when dark energy decays into dark matter (see
eq.(207)). The solutions of such type make interest because they facilitate
solution of the coincidence problem dalal . Indeed, the model with $\xi=3$
reduces to $\Lambda$CDM one for $w=-1$ and $\Pi=0.$ For $\xi=0$ the Universe
dynamics becomes stable with $\kappa=const,$ and the coincidence problem does
not appear ZPC . Thus for the parameter values $\xi<3$ the coincidence problem
weakens. In this scheme with $w=const,$ one can easily see that the
interaction which generates the scaling solutions, can be calculated as the
following
$\frac{\Pi}{\rho_{x}}=\left(w+\frac{\xi}{3}\right)\,\frac{\kappa_{0}(1+z)^{\xi}}{1+\kappa_{0}(1+z)^{\xi}}\
,$ (217)
where $\kappa_{0}$ denotes the current ratio of the energy densities and
$z=(a_{0}/a)-1$ is the redshift. Substituting this expression into the
equations (212) and (213), one finds the statefinder parameters:
$r=1+\frac{9}{2}\frac{w}{1+\kappa_{0}(1+z)^{\xi}}\left[1+w-\left(w+\frac{\xi}{3}\right)\frac{\kappa_{0}(1+z)^{\xi}}{1+\kappa_{0}(1+z)^{\xi}}\right]\,,$
(218)
and
$s=1+w-\left(w+\frac{\xi}{3}\right)\frac{\kappa_{0}(1+z)^{\xi}}{1+\kappa_{0}(1+z)^{\xi}}.$
(219)
Figure 7 presents the dependence $r(s)$ for different values of $\xi.$ The
less is the value $\xi,$ the lower is the corresponding curve on $s$–$r$
diagram and the less actual is the coincidence problem. Remark that those
curves are qualitatively identical to those drawn for the models with
interaction-free components (see Fig.8 sahni ). Compare also that $\Lambda$CDM
model ($\Pi=0,$ $w=-1$) corresponds to the point $s=0,$ $r=1$ (not shown on
the figure).
Figure 7: The curves $r(s)$ are drawn for the redshift range $[0,6]$ (from
left to right) with $w=-0.95$ and $\kappa_{0}=3/7$ for 3 different values of
the $\xi$ parameter: (a) $2.5$; (b) $1.5$; (c) $0.5.$
Figure 8: The Statefinder pair ($r,s$) is shown for different forms of dark
energy. In quintessence (Q) models ($w=$constant $\neq-1$) the value of $s$
remains fixed at $s=1+w$ while the value of $r$ asymptotically declines to
$r(t\gg t_{0})\simeq 1+\frac{9w}{2}(1+w).$ Two models of quintessence
corresponding to $w_{Q}=-0.25,-0.5$ are shown. Quintessence (K) models are
presented by a scalar field (quintessence) rolling down the potential
$V(\phi)\propto\phi^{-\alpha}$ with $\alpha=2,4.$ $\Lambda$CDM ($r=1,s=0$) and
SCDM in the absence of $\l$ ($r=1,s=1$) are the fixed points of the system.
The hatched region is disallowed in quintessence models and in the
quintessence model which we consider. The filled circles show the current
values of the statefinder pair ($r,s$) for the Q and K models with
$\Omega_{m0}=0.3.$
The present values $r_{0}$ and $s_{0}$ of the statefinder parameters are
important both from the observational point of view and for the purpose of
distinction between different models. For the scaling models one has
$r_{0}=1+\frac{9}{2}\frac{w}{1+\kappa_{0}}s_{0}\,,\qquad{\rm and}\qquad
s_{0}=1+w-\left(w+\frac{\xi}{3}\right)\frac{\kappa_{0}}{1+\kappa_{0}}\,.$
(220)
Having in mind that
$q_{0}=\frac{1}{2}\frac{1+\kappa_{0}+3w}{1+\kappa_{0}}\ ,$ (221)
and introducing
$q_{0\Lambda}\equiv
q_{0}\left(w=-1\right)=-\frac{1}{2}\frac{2-\kappa_{0}}{1+\kappa_{0}}\Leftrightarrow\frac{3}{2}\frac{\kappa_{0}}{1+\kappa_{0}}=1+q_{0\Lambda}\
,$ (222)
we can classify different models by its dependence $s_{0}(q_{0}),$ which is
$s_{0}=\frac{2}{3}\left[q_{0}-q_{0\Lambda}+\left(\frac{\xi}{3}-1\right)\left(1+q_{0\Lambda}\right)\right]\
.$ (223)
The first part of the bracket in the righthand side describes deviation from
$w=-1,$ and the second responds for deviation from $\Lambda$CDM value $\xi=3.$
For models with $w=-1,$ which have the same deceleration parameter
$q_{0}=q_{0\Lambda},$ we obtain
$s_{0}=\frac{2}{3}\left(\frac{\xi}{3}-1\right)\left(1+q_{0\Lambda}\right).$ Of
course, $\xi=3$ corresponds to $\Lambda$CDM model with $s_{0}=0.$ If the
conditions $\kappa_{0}=3/7$ and $\xi=1$ hold, then
$s_{0}\left(\xi=1\right)=-0.2,$ while for $\xi=0$ one has
$s_{0}\left(\xi=0\right)=-0.3.$
Analogous considerations are valid also for other values of $w.$ Therefore the
parameter $s_{0}$ enable us to distinguish between different scaling models,
which have the same deceleration parameter.
### VIII.6 Concluding remarks
The statefinder parameters introduced in sahni and alam , as was assumed,
became useful tools for verification of models with interacting components,
which in turn solve or weaken the coincidence problem — a stone of obstacle
for many models with late acceleration. While the deceleration parameter does
not depend on interaction between the dark energy and dark matter, the state
finders $(r,s)$ contain the dependence explicitly.
## IX Holographic dynamics: entropic acceleration
In the previous section we presented the description of Universe dynamics
based on general relativity with some generalizations. The present section is
intended to show that there exists a principally independent approach, which
enables us both to reproduce all achievements of the traditional description
and resolve a number of problems the latter faced to.
The traditional point of view assumed that space filling fields constitute the
dominant part of degrees of freedom in our World. However it became clear soon
that such estimate much hardened the development of the quantum gravity
theory: for the latter to make sense one had to cutoff on small distances all
the integrals appeared in the theory. Consequently our World was described on
a three-dimensional discrete lattice with cell size of order of Planck length.
Recently some physicists came up with even more radical point of view:
complete description of Nature required only two-dimensional lattice, situated
on space edge of our World, instead of the three-dimensional one. Such
approach is based on so-called ”holographic principle” Hooft ; Susskind ;
Witten ; Maldacena_Gravity ; Susskind_Hologram ; Gubser_Klebanov_Polyakov ;
TASI_AdS_CFT . The term comes from the optical holography, which represent
nothing but two-dimensional recording of three-dimensional objects. The
holographic principle is composed of the two main statements:
1. 1.
all information contained in some region of space can be ”recorded”
(presented) on boundary of that region;
2. 2.
the theory contains at most one degree of freedom per Planck area on
boundaries of the considered space region
$N\leq\frac{A{{c}^{3}}}{G\hbar}.$ (224)
Therefore central place in the holographic principle is occupied by the
assumption that all information about the Universe can be coded on some two-
dimensional surface — the holographic screen. Such approach leads to possibly
new interpretation of cosmological acceleration and to completely novel
concept of gravity. The gravitation is now defined as entropic force generated
by variation of information connected to positions of material bodies.
Let us consider a small piece of holographic screen and approaching particle
of mass $m.$ According to the holographic principle, the particle affects the
information amount (and thus on entropy), which is stored on the screen. It is
naturally to assume that the entropy increment near the screen is linear on
the displacement $\Delta x,$
$\Delta S=2\pi{{k}_{B}}\frac{mc}{\hbar}\Delta x.$ (225)
The factor $2\pi$ is introduced for convenience reasons, to be clear below. To
make clear why that quantity is proportional to mass, let us imagine that the
particle splits on two or more particles of lower mass. Each of them then
carries its own entropy increment with displacement on $\Delta x.$ As both
entropy and mass are additive function, then they are proportional to each
other. According to the first principle of thermodynamics, the entropic force,
caused by information variation, reads
$F\Delta x=T\Delta S.$ (226)
If the entropy gradient is given, or calculated using the relation (225), and
the screen temperature is fixed, then one can find the entropy force. As is
known, an observer, moving with acceleration $a,$ effectively feels the so-
called Unruh temperature Unruh
${{k}_{B}}T=\frac{1}{2\pi}\frac{\hbar}{c}a.$ (227)
Let us assume that the total energy of the system equals $E.$ A simple
assumption follows, that it is uniformly distributed on all $N$ bits of
information on the holographic screen. The temperature is then defined as
average energy per bit
$E=\frac{1}{2}N{{k}_{B}}T.$ (228)
The formulae (224)-(228) enable us to formulate the holographic dynamics, and
in particular dynamics of Universe, without any notion of gravity.
The above presented ideology is essentially based on successful description of
black holes physics Bekenstein ; Bekenstein1 ; Bekenstein2 ; Bekenstein3 . On
the first face, there is nothing common between the extremely diluted Universe
with density $\rho\sim{{10}^{-29}}g/c{{m}^{3}}$ and ”typical” black hole of
stellar mass with critical density
$\rho\sim{{10}^{14}}g/c{{m}^{3}}\,(M=10{{M}_{\odot}}).$ But the situation
sharply changes if we transit to more massive black holes. As we have seen
above, the black hole radius scales ${{r}_{g}}\propto M,$ therefore for the
black hole critical density one has $\rho\propto{{M}^{-2}}.$ Super-massive
black holes situated in the galactic nuclei reach masses of order
${{10}^{10}}{{M}_{\odot}},$ and their gravitational radius (${{r}_{g}}\sim
3\times{{10}^{15}}cm$) five times exceeds the dimensions of Solar system. The
critical density in the latter case equals $\rho\sim{{10}^{-4}}g/c{{m}^{3}},$
which is one order of magnitude less than the room air density
($1.3\times{{10}^{-3}}g/c{{m}^{3}}$). Let us consider, how close are the
parameters of our observed Universe to those of black hole. We estimate mass
of the observed Universe taking the Hubble radius for its radial dimension
${{H}^{-1}}$ The mass contained inside the Hubble sphere thus reads
${{M}_{univ}}=\frac{4\pi}{3}R_{H}^{3}\rho.$ (229)
In the case of flat space 444as we have repeatedly mentioned above, deviations
from the flatness are very small, the density $\rho$ of the observable part of
Universe enclosed by Hubble sphere can be reasonably replaced by the critical
one (16) $\rho={3{{H}^{2}}}/{(8\pi G)},$ and therefore the formula (229) takes
the form
${{M}_{univ}}=\frac{{{R}_{H}}}{2G}.$
Inserting the obtained mass value into the expression for gravitational radius
of the Universe, ${{r}_{g}}=2G{{M}_{univ}},$ one finds
$r_{g}=R_{H}.$ (230)
The obtained result for many reasons is nothing but a rough estimate, however
it obviously favors the above cited black hole argumentation in application to
the observable Universe. Remark that in the case of the Sun the ratio of the
physical radius ${{R}_{\odot}}=695\,500km$ to the gravitational one
${{r}_{g\odot}}=3km$ counts more than five orders of magnitude.
Now let us identify the holographic screen with the Hubble sphere of radius
$R={{H}^{-1}}$ (which is valid for flat Universe) and we try, making use of
the holographic principle, to reproduce the Friedmann equations, using neither
Einstein equations nor Newtonian mechanics. The holographic screen spans the
area $A=4\pi{{R}^{2}}$ and carries the (maximum) information
$N=4\pi{{R}^{2}}/L_{Pl}^{2}$ bits. The change of information quantity $dN$ for
the time period $dt,$ caused by the Universe expansion $R\to R+dR,$ equals
$dN=\frac{dA}{L_{Pl}^{2}}=\frac{8\pi R}{L_{Pl}^{2}}dR.$
Here we use the system of units $c={{k}_{B}}=1.$ Variation of Hubble radius
leads to change in Hawking temperature $\left(T=\frac{\hbar}{2\pi R}\right)$
$dT=-\frac{\hbar}{2\pi{{R}^{2}}}dR.$
From the equipartition law it follows that
$dE=\frac{1}{2}NdT+\frac{1}{2}TdN=\frac{\hbar}{L_{Pl}^{2}}dR=\frac{dR}{G},$
(231)
$\left(L_{Pl}^{2}=\frac{\hbar G}{{{c}^{3}}}\right).$ The quantity $dR$ can be
presented in the form
$dR=-H\dot{H}{{R}^{3}}dt.$ (232)
From the other hand, the energy flow through the Hubble sphere can be
calculated if the energy-momentum tensor is given for the substance filling
the Universe. Considering the latter substance as an ideal liquid, (see
chapter 2) and using the relation
${{T}_{\mu\nu}}=(\rho+p){{u}_{\mu}}{{u}_{\nu}}+p{{g}_{\mu\nu}},$ we obtain
$dE=A\left(\rho+p\right)dt.$ (233)
Equating (231) to (233) accounting (234), one finds
$\dot{H}=-4\pi G\left(\rho+p\right).$
It is well known that the system of equations
$\dot{H}=-4\pi G\left(\rho+p\right),~{}\dot{\rho}+3H\left(\rho+p\right)=0.$
(234)
is equivalent to the system (13)-(14). Thus the declared aim is achieved. We
remark that already in the year 1995 a more general problem was solved
Jacobson : the Einstein equations were obtained from the thermodynamic
considerations. This important result followed from the fact that entropy was
proportional to horizon area and from the assumption that the relation (227)
holds for any accelerated moving observer situated inside her own casual
horizon, and $T$ stands for Unruh temperature.
Derivation of the Friedmann equations from the holographic principle is of
course an important result, but by itself it represents nothing but
reproduction of something already well-known. A natural question is whether
one can develop a novel approach to description of Universe dynamic basing on
the holographic principle? If yes, than is it possible to overcome the
unsolvable difficulties of the traditional approach in frames of the new one?
Let us start from the logic scheme which the holographic dynamics of Universe
is based on.
The standard procedure to develop any field theory involves the formulation of
equations of motion obtained from some effective action corresponding to the
energy scale characteristic to the phenomenon to be described by the theory.
General relativity theory is not an exception from the rule: equations of
motion in general relativity are obtained by variation of effective action
with respect to dynamical variables present in it. The standard approach to
evaluate the integrals that arise in variations of the action is the
integration by parts, which produces the terms corresponding to boundaries of
the region under consideration. According to such approach a quite reasonable
assumption is made that on boundary of the considered region, far from the
field sources, all the dynamical variables, be it either fields, or
gravitational potentials or connectivity components (as in general
relativity), turn to zero. In holographic physics instead the surface is the
place where the main action takes goes on. The natural conclusion follows that
the contribution of surface terms now should be necessarily accounted.
Let us show that consideration of the boundary term in the Einstein-Hilbert
action is equivalent to inclusion of non-zero energy-momentum tensor in the
standard Einstein equations. The action for gravitational field takes the form
†††We use the signature $(+,-,-,-),$ and definition
$R^{\rho}_{~{}\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma},$
${R}_{\nu\mu}=R_{\nu\beta\mu}^{~{}~{}~{}~{}~{}\beta},$
$R=g^{\mu\nu}R_{\mu\nu}.$
$S_{EH}\,=\,-\frac{c^{3}}{16\pi G}\int d^{4}x\,\sqrt{-g}\,R\,.$ (235)
Variation of that action with respect to metrics $g_{\mu\nu}$ in a compact
region $\Omega,$ one obtains
$\displaystyle\delta\int_{\Omega}d^{4}x\sqrt{-g}\,R\,=$ (236)
$\displaystyle\int_{\Omega}d^{4}x\sqrt{-g}\Biggl{[}\left(g^{\mu\nu}\nabla^{2}\delta
g_{\mu\nu}\,-\nabla^{\mu}\nabla^{\nu}\delta
g_{\mu\nu}\right)-\left(R^{\mu\nu}-\frac{1}{2}\,g^{\mu\nu}R\right)\delta
g_{\mu\nu}\Biggr{]}\,,$
where $\nabla^{2}\,=\,g^{\alpha\beta}\nabla_{\alpha}\,\nabla_{\beta}.$ Full
derivative in (235) can be presented as the contribution of the region
boundary $\partial\,\Omega,$ where those contribution do not cancel. In order
to obtain the Einstein equations from the least action principle without
boundary terms, the Einstein-Hilbert action should be complemented by a
functional intended to compensate the contribution of full derivative in
(235). We denote it as $S_{Boundary}[g]..$ Then the full action takes on the
form
$S\,=\,-\frac{1}{16\pi
G}\int_{\Omega}d^{4}x\sqrt{-g}\,R+\,S_{Boundary}[g]\,+\,S_{Source}\,,$ (237)
where we included possible sources $S_{Source}$ of gravitational field,
coupled with the material fields. The functional $S_{Boundary}[g],$ thus takes
the form
$\delta S_{Boundary}[g]\,=\,-\,\frac{1}{16\pi
G}\int_{\Omega}d^{4}x\sqrt{-g}\Bigl{(}g^{\mu\nu}\nabla^{2}\delta
g_{\mu\nu}\,-\,\nabla^{\mu}\nabla^{\nu}\delta g_{\mu\nu}\Bigr{)}\,,$ (238)
In the context of holographic physics where the boundaries play crucial role,
the functional $S_{Boundary}[g]$ can be treated as action for holographic dark
energy. The Einstein equations including the contributions of material fields
and boundary terms have the following form
$\displaystyle R^{\mu\nu}-\frac{1}{2}\,g^{\mu\nu}R$ $\displaystyle=$
$\displaystyle 8\pi
G\,\left(T^{\mu\nu}_{Source}+T^{\mu\nu}_{Boundary}\right)\,,$ $\displaystyle
T^{\mu\nu}_{Source}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{-g}}\,\frac{\delta S_{Source}}{\delta
g_{\mu\nu}},~{}T^{\mu\nu}_{Boundary}=\frac{2}{\sqrt{-g}}\,\frac{\delta
S_{Boundary}}{\delta g_{\mu\nu}}\,.$ (239)
Remark that in the flat space case, when $g_{\mu\nu}=0,$ the boundary action
$S_{Boundary}[g]=0,$ and Einstein equations take the standard form. We also
note that such modification of Einstein equations preserves the general
structure of the equations, which, as we will see below, guarantees the
conservation of the Friedmann equation structure.
Inclusion of the additional terms in the Einstein equations requires the so-
called ”holographic correction” of the Friedmann equations. The easiest way to
perform the correction is to insert the entropic force into the second
Friedmann equation EntropicCosmology . The entropic term structure can be
guessed from the dimensional considerations
$\frac{{\ddot{a}}}{a}=-\frac{4\pi
G}{3}\left(\rho+3p\right)+\frac{{{a}_{e}}}{{{L}_{b}}}.$ (240)
Here ${{a}_{e}}$=${{F}_{e}}/m$ is the acceleration caused by the entropic
force and ${{L}_{b}}$ is the scale length, determined by the holographic
screen position. If one takes the Hubble sphere as the holographic screen,
then the scale length coincides with the Hubble radius:
${{L}_{b}}={{R}_{H}}={{H}^{-1}}.$ The entropic acceleration magnitude can be
expressed through the holographic screen temperature ${{T}_{b}}$
${{a}_{e}}=\frac{{{F}_{e}}}{m}={{T}_{b}}\frac{\Delta S}{\Delta
x}\frac{1}{m}=2\pi{{T}_{b}}.$ (241)
Does the considered scheme agree with the main observational data at least? We
have shown above in frames of SCM that absolute value of the cosmological
acceleration of the Hubble sphere equals $\dot{V}\simeq
4\times{{10}^{-10}}m{{\sec}^{-2}}.$ Let us estimate that magnitude in the
holographic approach.
It is naturally to assign the Unruh temperature to the holographic screen
${{T}_{b}}={{T}_{U}}.$ Let us link it with the Hawking radiation temperature
${{T}_{H}},$ which is equal to
${{T}_{H}}=\frac{\hbar}{8\pi{{k}_{B}}GM}=\frac{\hbar g}{2\pi{{k}_{B}}},$ (242)
where $g$ is the free fall acceleration on the Hubble sphere. Comparing the
latter expression with Unruh temperature
${{T}_{U}}=\frac{\hbar a}{2\pi{{k}_{B}}},$ (243)
one gets
${{T}_{H}}={{T}_{U}}(a=g).$ (244)
Thus the Unruh temperature coincides with that for the Hawking radiation, but
depends on the reference frame acceleration instead of surface gravitation
(free fall acceleration). According to the equivalence principle, the free
fall acceleration on the Hubble surface is equivalent to acceleration of a
reference frame and we can relate it to each other. Treating the Hubble sphere
as an analogue of events horizon for a black hole, we can present the formula
(242) in the following form
${{T}_{H}}=\frac{\hbar}{{{k}_{B}}}\frac{H}{2\pi}\sim 3\times{{10}^{-30}}\,K.$
(245)
Comparing (245) and (243), one finds that the Hubble sphere acceleration
${{a}_{H}}$ equals
${{a}_{H}}=H\simeq{{10}^{-9}}m\,{{\sec}^{-2}}.$ (246)
The latter result is in convincing qualitative agreement with the one obtained
in SCM.
Certainly, quality of a novel model is not defined by its ability to reproduce
the results established by its predecessors — it is nothing but minimum task.
Its main purpose and virtue is to solve the problems untouchable by former
models. The holographic dynamics is very promising from that point of view as
it gives hope to solve both main challenges of the SCM — they are cosmological
constant magnitude and coincidence problem.
Let us start from the former. The cosmological constant problem stands for the
huge disagreement 666about $120$ orders of magnitude !!! between the observed
density of dark energy in form of the cosmological constant and its ”expected”
value. The expectations are based on rather natural assumptions on the cutoff
parameter for the integral, representing the zero fluctuations of vacuum. The
holographic principle enables us to replace the ”natural assumptions” by more
thorough quantitative estimates.
### IX.1 Universe with holographic dark energy
In any effective quantum field theory defined in space region with typical
length scale $L$ and using the ultraviolet cutoff $\Lambda,$ the system
entropy takes the form $S\propto\Lambda^{3}L^{3}.$ For instance, fermions,
placed in nodes of space lattice with characteristic size $L$ and period
$\Lambda^{-1},$ occupy one of the number $2^{{(L\Lambda)}^{3}}$ states.
Therefore entropy of such system is $S\propto\Lambda^{3}L^{3}.$ According to
the holographic principle, this quantity must obey the inequality
CohenKaplanNelson
$L^{3}\Lambda^{3}\leq S_{BH}\equiv\frac{1}{4}\frac{A_{BH}}{l_{Pl}^{2}}=\pi
L^{2}M_{Pl}^{2},$ (247)
where $S_{BH}$ is the black hole entropy and $A_{BH}$ stands for its event
horizon area, which in the simplest case coincides with the surface of sphere
with radius $L.$ This reasoning shows that magnitude of the infrared cutoff
cannot be chosen independently of the ultraviolet one.
Thus we formulate the important result CohenKaplanNelson : in frames of
holographic dynamics the infrared cutoff magnitude is strictly linked to the
ultraviolet one. In other words, physical properties on small UV-scales
depends on physical parameters on large IR-scales. In particular, if the
inequality (247) holds, then one gets
$L\sim{{\Lambda}^{-3}}M_{Pl}^{2}.$ (248)
Effective field theories with UV-cutoff (248) obviously include many states
with the gravitational radius exceeding the region where the theory was
initially defined. In other words, for arbitrary cutoff parameter one can find
sufficiently large volume where entropy in an effective field theory exceeds
the Bekenstein limit. In order check it let us remark that usually an
effective quantum field theory must be able to describe the system under
temperature $T\leq\Lambda.$ While $T\gg 1/L,$ the system has thermal energy
$M\sim L^{3}T^{4}$ and entropy $M\sim L^{3}T^{3}.$ The condition (247) is
satisfied at $T\leq\left(M_{Pl}^{2}/L\right)^{1/3},$ which responds to
gravitational radius $r_{g}\sim L(LM_{Pl})\gg L.$ In order to get rid of that
difficulty an even more strict limitation CohenKaplanNelson is imposed on the
infrared cutoff $L\sim\Lambda^{-1},$ which excludes all the states localized
within limits of their gravitational radii. Taking into account all above
mentioned and using the expression (249)
$\rho_{vac}\approx\frac{\Lambda^{4}}{16\pi^{2}},$ (249)
the condition (247) can be presented in the form
$L^{3}\rho_{\Lambda}\leq LM_{Pl}^{2}\equiv 2M_{BH},$ (250)
where $M_{BH}$ is the black hole mass with gravitational radius $L.$ Thus
total energy contained in the region of dimensions $L$ cannot exceed the mass
of a black hole of roughly the same size. This result agrees surprisingly well
with the expression (230) provided the IR-cutoff scale is identified with the
Hubble radius $H^{-1},$ besides that such choice is well motivated in
cosmology context. The quantity $\rho_{\Lambda}$ is commonly called the
holographic dark energy.
In the cosmological aspect under interest the link between the small and large
scales can be obtained from rather natural condition, that total energy
contained within a volume of dimension $L$ cannot exceed mass of a black hole
of the same size:
$L^{3}\rho_{\Lambda}\leq M_{BH}\sim LM_{Pl}^{2}.$ (251)
Here $\rho_{\Lambda}$ is the energy density. Would this inequality be
violated, the Universe was composed of black holes solely. Applying this
relation to the Universe as whole it is naturally to identify the IR-scale
with the Hubble radius $H^{-1}.$ Then for the upper bound of the energy
density one finds
$\rho_{\Lambda}\sim L^{-2}M_{Pl}^{2}\sim H^{2}M_{Pl}^{2}.$ (252)
The quantity $\rho_{\Lambda}$ is usually called the holographic dark energy.
We will below denote its density as $\rho_{DE}.$ Accounting that
$M_{Pl}\simeq 1.2\times{{10}^{19}}GeV;~{}~{}~{}H_{0}\simeq 1.6\times
10^{-42}GeV,$
one finds
$\rho_{DE}\sim 10^{-46}\,GeV^{4}.$ (253)
This expression will agrees with the observed value of the dark energy density
$\rho_{DE}\simeq 3\times 10^{-47}GeV^{4}.$ Therefore the holographic dynamics
is free from the cosmological constant problem.
Though the obtained dark energy density corresponds to the correct one, the
choice of the IR-cutoff scale equal to the Hubble radius a problem arises
connected with the state equation problem: the holographic dark energy cannot
provide the accelerated expansion in that case Hsu . It can be easily seen
from the following simple arguments. Consider the Universe composed of
holographic dark energy with density given by the relation (252) and of normal
matter component. In that case $\rho={{\rho}_{\Lambda}}+{{\rho}_{M}}.$ From
the first Friedmann equation it follows that $\rho\propto{{H}^{2}}.$ If
${{\rho}_{\Lambda}}\propto{{H}^{2}},$ then dynamical behavior of holographic
dark energy coincides with that of normal matter, thus the accelerated
expansion is impossible. If the Universe was initially dominated by matter,
then the dark energy defined by (252) is in fact a tracker solution, as it
reproduces dynamics of the dominating component. The presence of tracker
solutions in frames of the holographic models gives a hope to solve the
coincidence problem, but it contradicts to the observed dynamics of Universe.
In order to produce the accelerated expansion of Universe in frames of
holographic dark energy model we will try to use the IR-cutoff spatial scale
different from the Hubble radius. First which comes in mind is to replace the
Hubble radius by the particle horizon $R_{p}$:
$R_{p}=a\int_{0}^{t}\frac{dt}{a}=a\int_{0}^{a}\frac{da}{Ha^{2}}.$ (254)
Unfortunately such substitution cannot give the desired result. It can be made
clear again using the initially matter-dominated Universe model. Due to the
causality principle the gravity influence cannot extend to the regions
situated on distances exceeding the causality horizon size. Therefore the
vacuum energy density present in the Friedmann equations cannot be an
arbitrary function of $L.$ In the matter dominated era the causality
(particle) horizon scales as $a^{3/2},$ therefore $\rho_{{}_{DE}}(L)\sim
a^{-3}.$ As $\rho(t)\sim a(t)^{-3(1+w)},$ then it immediately follows that for
the case of holographic energy with IR-cutoff on the particle horizon one gets
$w=0,$ which again clearly contradicts the observations.
Let us convince ourself now that the above discovered problem preserves also
in the case of Universe dominated by holographic dark energy. Present the
density of the latter in the following form Hsu
$\rho_{DE}=3c^{2}M_{Pl}^{2}L^{-2}.$ (255)
The coefficient $3{{c}^{2}}\,(c>0)$ is introduced for convenience, and
$M_{Pl}$ further stands for reduced Planck mass: $M_{Pl}^{-2}=8\pi G.$
Replacing $L\to{{R}_{h}}$ in (255), one obtains the first Friedmann equation
in the form
$R_{p}H=c,$ (256)
and it immediately follows that
$\frac{1}{Ha^{2}}=c\frac{d}{da}\left(\frac{1}{Ha}\right).$ (257)
One now easily finds that
$H^{-1}=\alpha a^{1+\frac{1}{c}},$ (258)
where $\alpha$ is constant. As $R_{h}=cH^{-1},$ and
$\rho_{\Lambda}=3{{c}^{2}}M_{Pl}^{2}R_{h}^{-2},$ then
$\rho_{DE}=3\alpha^{2}M_{Pl}^{2}a^{-2\left(1+\frac{1}{c}\right)}.$ (259)
How can we find the parameter $w$ in the state equation if the dependence
$\rho=f(a)$ is given? One possible approach is the following. Let us use the
conservation equation
$\dot{\rho}+3H\rho(1+w)=0.$
In the case $\rho=f(a)$ it takes the form
${f}^{\prime}(a)\dot{a}+3Hf(a)(1+w)=0,$
and it follows that
$w=-\frac{1}{3}\frac{{f}^{\prime}(a)}{f(a)}a-1=-\frac{1}{3}\frac{d\ln
f(a)}{da}a-1.$ (260)
It is easy to see that for well-known cases of matter $\rho\propto{{a}^{-3}},$
radiation $\rho\propto{{a}^{-4}}$ and cosmological constant $\rho=const$ one
obtains the correct values $w=0,1/3,-1.$ From the expression (260) it follows
that
$w=-\frac{1}{3}+\frac{2}{3c}>-\frac{1}{3}.$ (261)
Thus one can see that the above described component does not deserves the name
of “dark energy” in proper sense, as it cannot serve its main purpose — to
provide the accelerated expansion of universe. The difficulties originate from
the fact that, as it follows from (257), the derivative obeys
$\frac{d}{da}({{H}^{-1}}/a)>0.$ In order to obtain the accelerated expansion
of Universe, one should ”slow down” the growth of the IR-cutoff scale. It
turns out Holographic_Dark_Energy , that it can be done if one replaces the
particle horizon by the events one. Recall (see Chapter 2), that the size of
spatial region such that a signal emitted at time $t$ from any point of it
will reach the immobile observer at infinitely far future, equals
$R_{e}=a(t)\int_{t}^{\infty}{\frac{d{t}^{\prime}}{a\left(t^{\prime}\right)}}.$
(262)
Assuming again that the dark energy dominates and finding solution of the
first Friedmann equation in the form
$\int_{a}^{\infty}{\frac{da}{H{{a}^{2}}}}=\frac{c}{Ha},$ (263)
one finds
$\rho_{DE}=3{{c}^{2}}M_{Pl}^{2}R_{e}^{-2}=3{{\alpha}^{2}}M_{Pl}^{2}{{a}^{-2\left(1-\frac{1}{c}\right)}},$
(264)
or $w=-\frac{1}{3}-\frac{2}{3c}<-\frac{1}{3}.$ We obtained a component which
behaves as the dark energy, i.e. it provides the accelerated expansion of
Universe. If $c=1$ then it behaves as the cosmological constant. If $c<1,$
then $w<-1.$ Such value of $w$ responds to phantom model in the traditional
approach.
In the above considered cases we investigated properties of holographic dark
energy in two limiting cases: the matter dominated and dark energy dominated
one. Let us now consider the common case situation, i.e. the Universe dynamics
at arbitrary relation between the both components densities Q-GHuang . For
simplicity we restrict to the case of flat Universe and set the IR-cutoff
scale to the event horizon ${{R}_{e}}$ (35). Introducing the relative density
of the holographic dark energy ${{\Omega}_{DE}}={{\rho}_{DE}}/{{\rho}_{cr}}$
$({{\rho}_{cr}}=3M_{Pl}^{2}{{H}^{2}}),$ we represent (255) for $L={{R}_{e}}$
in the form
$HR_{e}=\frac{c}{\sqrt{\Omega_{DE}}}.$ (265)
Of course, with $\Omega_{DE}=1$ and replacement $R_{e}\to R_{h}$ the equation
(265) transits into (256). Taking derivative with respect to time from both
sides of (262), one gets
$\dot{R}_{e}=H{{R}_{e}}-1=\frac{c}{\sqrt{\Omega_{D}E}}-1$ (266)
From the definition (255) it follows that
$\frac{d{{\rho}_{{DE}}}}{dt}=-6{{c}^{2}}M_{Pl}^{2}R_{e}^{-3}{{\dot{R}}_{e}}=-2H\left(1-\frac{\sqrt{{{\Omega}_{{DE}}}}}{c}\right){{\rho}_{{DE}}}.$
(267)
Due to the energy conservation law
$\frac{d}{da}\left({{a}^{3}}{{\rho}_{{DE}}}\right)=-3{{a}^{2}}{{p}_{{DE}}}.$
(268)
and it follows that
${{p}_{{DE}}}=-\frac{1}{3}\frac{d{{\rho}_{{DE}}}}{d\ln a}-{{\rho}_{DE}}.$
(269)
Thus the state equation reads
${{w}_{{DE}}}=\frac{{{p}_{{DE}}}}{{{\rho}_{{DE}}}}=-\frac{1}{3}\frac{d\ln{{\rho}_{{DE}}}}{d\ln
a}-1=-\frac{1}{3}\left(1+\frac{2}{c}\sqrt{{{\Omega}_{{DE}}}}\right).$ (270)
We used the fact that $d\ln a=Hdt.$ This result can be obtained without
calculation of the pressure ${{p}_{\Lambda}}$ by the relation (260). The
obtained expression for $w_{{}_{DE}}$ is a consequence of the holographic dark
energy definition in form of (255), therefore it is independent on other
energy components. It follows from the obtained result that
$w_{{}_{DE}}\simeq-1/3$ in the case of domination of other energy components
and $w_{{}_{DE}}=-\frac{1}{3}\left(1+\frac{2}{c}\right)$ in the dark energy
dominated case. The latter result coincides with the expression (260),
obtained above for the Universe filled by solely holographic dark energy.
For the first impression the declared task is completed. The holographic dark
energy with density (255) from the one hand provides correspondence between
the observed density and the theoretical estimate, and from the other it leads
to the state equation which is able to generate the accelerated expansion of
Universe. However the holographic dark energy with IR-cutoff on the event
horizon still leaves unsolved problems connected with the causality principle:
according to the definition of the event horizon the holographic dark energy
dynamics depends on future evolution of the scale factor. Such dependence is
hard to agree with the causality principle.
In order to seek the way out from that dead end let us address once more the
cosmological constant problem, which is the huge gap between the theoretical
estimates and observed values of the dark energy density. The simplest type of
the dark energy — the cosmological constant — is attribute to vacuum average
of quantized fields and can be measured in gravitational experiments.
Therefore the cosmological constant problem is the quantum gravity one.
Although the complete quantum gravity theory is absent, a combination of
quantum mechanics with general relativity can shed light on the question under
discussion.
From the first days of quantum mechanics the concept of measurements, either
real or gedanken, played fundamental role for understanding of physical
reality. General relativity states that classical physical laws can be
verified with arbitrary unlimited precision. The above revealed connection
between the macroscopic (infrared) and microscopic scales dictates the
necessity of deeper analysis of the measurement process. The uncertainty
relations combined with general relativity generates a fundamental space-time
scale — the planck length ${{L}_{Pl}}\sim{{10}^{-33}}\,cm.$
The existence of a fundamental length scale critically influence the
measurement process PCBGFL . Assume that the fundamental length scale is
$L_{f}.$ As the space-time reference frame must make sense, it must be linked
to physical bodies. Therefore the fundamental length postulate is equivalent
to limitations for possibilities to realize the precise reference frames. In
terms of light signal experiments this means, for instance, that the time
interval required for the light signal to propagate from A to b and back,
measured by clock in A, is subject to uncontrolled fluctuations. The latter
should be treated as evidence for the fluctuations of metric, i.e. of the
gravitational field. Thus the fundamental length postulate is equivalent to
gravity field fluctuations.
The existence of quantum fluctuations in the metric Sasakura ; Maziashvili_1 ;
Maziashvili_2 ; 0707.4049 directly leads to the following conclusion, related
to the problem of distance measurements in the Minkowski space: the distance
$t$ 777recall that we use the system of units where the light speed equals
$c=\hbar=1,$ so that ${{L}_{Pl}}={{t}_{Pl}}=M_{Pl}^{-1}$ cannot be measured
with precision exceeding Karolyhazy the following
$\delta t=\beta t_{Pl}^{2/3}{{t}^{1/3}},$ (271)
where $\beta$ is a factor of order of unity. Following CohenKaplanNelson we
can consider the result (271) as the relation between the UV and IR scales in
frames of effective quantum field theory, which correctly describes the
entropy features of black holes. Indeed, presenting the relation (252) in
terms of length and replacing $\Lambda\to\delta t,$ we now recover (271) in
holographic context.
The relation (271) together with quantum mechanical energy-time uncertainty
relation enables us to estimate the quantum fluctuations energy density in the
Minkowski space-time. According to (247) we can consider the region of volume
$t^{3}$ as composed of cells $\delta{{t}^{3}}\sim t_{Pl}^{2}t.$ Therefore such
a cell represents the minimally detectable unit of space-time for the scale
$t.$ If the age of the chosen region equals $t,$ than it follows from the
energy-time uncertainty relations that its existence cannot be realized with
energy less than $\sim{{t}^{-1}}.$ Thus we come to conclusion: if lifetime
(age) of some spatial region of linear size $t$ equals $t,$ then there exists
a minimal cell $\delta{{t}^{3}},$ with energy that cannot be less than
$E_{\delta t^{3}}\sim t^{-1}.$ (272)
From (271) and (272) it immediately follows that due to the energy-time
uncertainty principle the energy density of quantum fluctuating metric in the
Minkowski space equals Sasakura ; Maziashvili_2 ; 0707.4049
${{\rho}_{q}}\sim\frac{{{E}_{\delta{{t}^{3}}}}}{\delta{{t}^{3}}}\sim\frac{1}{t_{Pl}^{2}{{t}^{2}}}.$
(273)
A fact of principal importance is that the dynamical behavior of the metric
fluctuations density (273) coincides with the above defined holographic dark
energy (252), (255), though the expression were derived based on completely
different physical principles. The holographic dark energy density was
obtained on the basis of entropy restrictions — the holographic principle,
while the metric fluctuations energy density of the Minkowski space is caused
only by their quantum nature, namely with the uncertainty principle.
The relation (273) allows to introduce an alternative model for holographic
dark energy 0707.4049 , which uses the age of Universe $T$ for IR-cutoff
scale. In such a model
$\rho_{q}=\frac{3{{n}^{2}}M_{Pl}^{2}}{{{T}^{2}}},$ (274)
where $n$ is a free parameter of model, and the number coefficient $3$ is
introduced for convenience. So defined energy density (274) with $T\sim
H_{0}^{-1},$ where ${{H}_{0}}$ is the current value of the Hubble parameter,
leads to the observed value of the dark energy density with the coefficient
$n$ value of order of unity. Thus in SCM, where ${{H}_{0}}\simeq
72km\,{{\sec}^{-1}}Mp{{c}^{-1}},\ {{\Omega}_{DE}}\simeq 0.73,\ T\simeq
13.7\,Gyr,$ one finds that $n\simeq 1.15.$
Now let us address the crucial question, whether the holographic energy
density in form (274) result in accelerated expansion of Universe. For
simplicity consider the Universe free of other energy components. In such case
the first Friedmann equation reads
$H^{2}=\frac{1}{3M_{Pl}^{2}}{{\rho}_{q}}.$ (275)
The Universe age $T$ in (274) is linked with scale factor by the relation
$T=\int_{0}^{a}{\frac{d{a}^{\prime}}{H{a}^{\prime}}}.$ (276)
Solution of the equation (275) with energy density (274) reads
$a={{\left[n\left({{H}_{0}}t+\alpha\right)\right]}^{n}}.$ (277)
The integration constant can be determined from the conditions $a_{0}=1.$
Evaluating the second order derivative from the scale factor, it can be shown
that the accelerated expansion of Universe takes place under the condition
$n>1.$ We note that the above obtained value $n\simeq 1.15,$ corresponding to
the observed value of the dark energy density, satisfies the above mentioned
condition, which can be obtained from the conservation equation for dark
energy, easily transformed to the following form
${{w}_{q}}=-1-\frac{{\dot{\rho}}}{3H\rho}.$ (278)
Using (274), (274) and (275) we present the expression (278) in the form
${{w}_{q}}=-1+\frac{2}{3n}.$ (279)
As was repeatedly mentioned above, the accelerated expansion of Universe
requires $w<-1/3,$ which is equivalent to the above obtained condition $n>1.$
Like in the previous model of the holographic dark energy, we transit to more
general case: the Universe where dark energy coexists with matter with density
${{\rho}_{m}}.$ Such Universe is described by the Friedmann equation
${{H}^{2}}=\frac{1}{3M_{Pl}^{2}}\left({{\rho}_{q}}+{{\rho}_{m}}\right).$ (280)
Transforming to relative densities
${{\Omega}_{m}}=\frac{{{\rho}_{m}}}{3{{H}^{2}}M_{pl}^{2}}$ and
${{\Omega}_{q}}=\frac{{{\rho}_{q}}}{3{{H}^{2}}M_{Pl}^{2}}=\frac{{{n}^{2}}}{{{T}^{2}}{{H}^{2}}},$
we present the Friedmann equation (280) in the form
$\frac{d\Omega_{q}}{d\ln
a}=(3-\frac{2}{n}\sqrt{\Omega_{q}})(1-\Omega_{q})\Omega_{q}.$ (281)
The equation (281) can be solved exactly to give
$\displaystyle\frac{1}{n}\ln a+c_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{3n-2}\ln(1-\sqrt{\Omega_{q}})-\frac{1}{3n+2}\ln(1+\sqrt{\Omega_{q}})$
(282)
$\displaystyle+\frac{1}{3n}\ln\Omega_{q}+\frac{8}{3n(9n^{2}-4)}\ln(\frac{3n}{2}-\sqrt{\Omega_{q}}).$
The integration constant can be determined from the condition
${{\Omega}_{q}}\simeq 0.73$ at $a=1.$ Let us analyze the dynamics of relative
dark energy density in two limiting cases: the matter-dominated and dark
energy-dominated ones. In the former case it follows from (281) that
${\Omega_{q}}\approx{c_{1}}{a^{3}}.$ (283)
Fast growth of relative contribution of the dark energy, irrespective of $n$
value, results in the dark energy dominated era, when
${{\Omega}_{q}}\approx 1-{{c}_{2}}{{a}^{-(3n-2)/n}}.$ (284)
The state equation for the dark energy can be obtained from the relation
(278).
${{w}_{q}}=-1+\frac{2}{3n}\sqrt{{{\Omega}_{q}}}.$ (285)
In the early Universe during matter-dominated era one gets ${{\Omega}_{q}}\to
0$ and ${{w}_{q}}\to-1,$ i.e. during that period the holographic dark energy
in the considered model behaves like cosmological constant. In later epoch of
dark energy domination, when ${{\Omega}_{q}}\to 1,$ the state equation (285)
naturally transforms to the above obtained relation (279). Remark that fate of
the Universe filled by matter and holographic dark energy with density (274)
is constant accelerated expansion with power law time dependence (277) of the
scale factor. Thus the holographic model for dark energy with IR-cutoff scale
set to the Universe age, allows the following:
1. 1.
to obtain the observed value of the dark energy density;
2. 2.
provide the accelerated expansion regime on later stages of the Universe
evolution;
3. 3.
resolve contradictions with the causality principle.
However the reader should not hurry with the ultimate conclusions. The first
successes of holographic principle application from one hand awoke hopes to
create on that basis an adequate description of the Universe dynamics, free of
a number of problems which the traditional approach suffers from. From the
other hand, it was those successes which provoke we would say unreasonable
optimism. We believe that papers entitled somewhat like “Solution of dark
energy problem” 1004.1285 represent manifestation of specific “holographic
extremism”. We would like to remind the saying of W.Churchill: “Success means
motion from failure to failure, not loosing the enthusiasm”. The holographic
dynamics is one of the youngest directions of theoretical physics. On that
path the physicists experienced yet too few failures in order to pretend for
complete success.
## X Transient acceleration
Unlike the fundamental theories, the physical models only reflect our current
understanding of a process or a phenomenon, to describe which they were
created. The model efficiency significantly depends on its flexibility, i.e.
its ability to be modernized using the new-coming information. That is why
evolution of any actively living model includes multiple generalizations,
directed both to resolve the conceptual problems and to describe ever growing
set of observational data. In the case of SCM those generalizations can be
divided on two types. The first includes replacements of cosmological constant
by more complicated dynamical forms of dark energy, taking into account the
possibility of interaction of the latter with dark matter. Generalizations of
the second type are of more radical nature and pretend one consequent change
of cosmological paradigm. The ultimate (explicit or hidden) aim of them is to
reject the dark components due to modifications of Einstein equations, and
consequently the Newtonian laws two. The generalizations of both types can be
well viewed on example of the phenomenon called the “transient acceleration”.
As we have seen above in frames of SCM, the dependence of the deceleration
parameter $q$ on the redshift $z$ has the characteristic monotonous trend to
the limiting value $q(z)=-1$ for $z\to-1.$ It physically means that after the
beginning of dark energy-dominated era (at $z\sim 1$), the Universe in SCM is
condemned to eternal acceleration.
Below we consider some cosmological models with dynamical forms of dark
energy, which lead to transient acceleration, and we also discuss the
information on current Universe expansion rate extracted from observational
data.
### X.1 Theoretical background
J. Barrow Barrow was one of the first to consider principal possibility of
the transient acceleration. He showed that many well established scenarios,
consistent with the current accelerated expansion of Universe, do not exclude
the possibility of recurrence to non-relativistic matter-dominated era, and
thus to the decelerated expansion regime. Therefore the transition to the
accelerated expansion does not yet imply the eternal accelerated expansion.
In order to show that we, following the Barrow work Barrow , consider a
homogeneous and isotropic flat Universe, filled by non-relativistic matter and
scalar field with the potential energy $V(\varphi)$ and state equation
$p=w\rho.$ We take the scalar field potential in the form
$V(\varphi)=V_{p}(\varphi)e^{-\lambda\varphi}.$ (286)
In some versions of low-energy limits of string theory the potential
$V_{p}(\varphi)$ represents a polynomial. The exponential potential with a
shallow minimum was first suggested by Albrecht and Skordis AS . This minimum
on the exponential potential background was created due to the polynomial
factor $V_{p}(\varphi)$ of the simplest form
$V_{p}(\varphi)=(\varphi-\varphi_{0})^{2}+A.$ (287)
In the considered case the potential takes the following form
$V(\varphi)=e^{-\lambda\varphi}\left(A+(\varphi-\varphi_{0})^{2}\right).$
(288)
In order to relate the above considered potential to that of string theory the
constant parameters $A$ and $\varphi_{0}$ should be of order of unity in
Planck units. In such quintessence models the accelerated expansion of
Universe appears naturally on late stages of evolution without any fitting of
initial parameters, thus this model is free of the precise tuning problem.
The accelerated expansion starts when the field rolls down the local potential
minimum at $\phi=\phi_{0}+(1\pm\sqrt{1-\lambda^{2}A})/\lambda,$ which is
formed by the quadratic factor in (288), where $1\geq\lambda^{2}A.$ While the
field stays in the false vacuum state its kinetic energy is negligibly small
$(\phi\approx const),$ and consequent domination $\rho_{\phi}$ is caused by
almost constant value of the potential energy, which runs the era of the
accelerated expansion of Universe which never ends.
It was discovered in the paper Barrow that it is neither solely possible nor
most probable scenario.
The transient vacuum domination appears in the two following cases. When
$A\lambda^{2}<1,$ the field $\varphi$ reaches the local minimum with kinetic
energy sufficient to overcome the potential barrier and continues roll down
the exponential part of the potential to the region where
$\varphi\gg\varphi_{0}.$ The kinetic energy is defined by the scaling regime
and thus by the parameters of potential rather than by the initial conditions.
The transition acceleration appears also in the case when the condition
$A\lambda^{2}>1$ holds. Since $A$ grows proportional to $\lambda^{-2},$ the
potential looses its local minimum and flattens near the inflection point. It
is sufficient to cause the temporal acceleration of the Universe expansion,
however the field never stops rolling down the potential and once the Universe
will be again dominated by matter with the dependence $a(t)\propto t^{2/3}.$
Therefore the Universe quits the regime of eternal accelerated expansion and
returns to the decelerated expansion. Besides the the well motivated family
Albrecht-Skordis potentials the possibility of transient acceleration regime
appears to be more probable than the eternal accelerated expansion.
### X.2 Different models with transient acceleration
In order to show explicitly that the transient acceleration represents a
natural feature of different cosmological models, we briefly consider some of
them below.
Barrow considered in his work Barrow a model of Universe where dark energy is
present in form of the scalar field with potential (288). We consider some
other examples of cosmological models that provide alteration of the
accelerated expansion phase by the decelerated one.
#### X.2.1 Scalar field, multidimensional cosmology and transient
acceleration
In the paper Russo it was shown that the transition acceleration phase can be
realized in the exponentail potential as well, and besides that a
$d$-dimensional cosmological model was also considered in the work. The action
of the latter reads
$S=\int
d^{d}x\sqrt{-g}\bigg{(}\frac{R}{2\kappa^{2}}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-
V_{0}\exp(-\lambda\phi)\bigg{)},$ (289)
where $\kappa_{N}^{2}=8\pi G_{N}=1/2$ and $V_{0}>0,$ $\lambda>0$ (the case
$\lambda<0$ is connected to the case $\lambda>0$ by the replacement
$\phi\to-\phi$). Following the paper Russo we consider below the FRW-metric
for flat Universe $(k=0)$
$ds^{2}=dt^{2}-a^{2}(t)dx^{i}dx^{i}\ ,\ \ \ \ i=1,...,d-1.$ (290)
In that case the action takes the form
$S=\int
d^{d}x\bigg{(}(d-1)(d-2)a^{d-3}\dot{a}^{2}+a^{d-1}\big{(}\frac{1}{2}\dot{\phi}^{2}-V(\phi)\big{)}\bigg{)},$
(291)
and in variables $(u,v)$
$\phi=\frac{1}{\kappa}\sqrt{d-2\over d-1}(v-u)\ ,~{}~{}a^{d-1}=e^{v+u},$ (292)
one obtains
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{d-1}xdt\
e^{u+v}\bigg{(}\frac{2(d-2)}{\kappa^{2}(d-1)}\dot{u}\dot{v}-V_{0}e^{-2\alpha(v-u)}\bigg{)},$
$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\kappa}\sqrt{\frac{d-2}{d-1}}\lambda.$ (293)
Let us transform to new time variable $\tau$
$\frac{d\tau}{dt}=\kappa\sqrt{\frac{(d-1)V_{0}}{2(d-2)}}\ e^{\alpha(u-v)}.$
(294)
and therefore
$S=\frac{1}{\kappa}\sqrt{\frac{2(d-2)V_{0}}{d-1}}\int d^{d-1}xd\tau\
e^{u+v}e^{\alpha(u-v)}\big{(}u^{\prime}v^{\prime}-1\big{)}.$
Using (293), (294) and (292) it can be shown Russo , that the general solution
for the case $\alpha<1$ takes the form:
$\displaystyle d\mathfrak{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{2(d-2)}{\kappa^{2}(d-1)V_{0}}\
e^{\frac{4\alpha^{2}\tau}{w}}\frac{(1+me^{-2w\tau})^{\frac{2\alpha}{(1-\alpha)}}}{(1-me^{-2w\tau})^{\frac{2\alpha}{(1+\alpha)}}}d\tau^{2}-e^{\frac{4\tau}{sw}}(1+me^{-2w\tau})^{\frac{2}{s(1-\alpha)}}(1-me^{-2w\tau})^{\frac{2}{s(1+\alpha)}}dx^{i}dx^{i};$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\sqrt{\frac{d-2}{d-1}}\bigg{(}\frac{2\alpha\tau}{w}-\frac{1}{1+\alpha}\log(1-me^{-2w\tau})+\frac{1}{1-\alpha}\log(1+me^{-2w\tau})\bigg{)},$
(295)
where $s=d-1,$ and $m$ is the integration constant.
The solution (295) has the asymptotes:
$\displaystyle a\sim t^{\frac{1}{d-1}}\
,~{}~{}~{}\phi=-\frac{1}{\kappa}\sqrt{\frac{d-2}{d-1}}\log t\ ,\ \ \mbox{\rm
for}\ \ t\cong 0\ ,$ $\displaystyle a\sim
t^{\frac{4\kappa^{2}}{(d-2)\lambda^{2}}}\ ,\ \ \ \ \phi=\frac{2}{\lambda}\log
t\ ,\ \ \ \ \ \mbox{\rm for}\ \ t\gg 1\ .$ (296)
On early stages of evolution the state equation is extremely rigid $p=\rho,$
and on later stages it is easy to show that
$p=\omega\rho,\ \ \ \ \omega=\frac{d-2}{2\kappa^{2}(d-1)}\lambda^{2}-1\ .$
In accordance with the with the equation (296), the accelerated expansion will
continue forever under condition $\lambda<\frac{2\kappa}{\sqrt{d-2}}.$ Let us
show that under condition
$\frac{2\kappa}{\sqrt{d-2}}<\lambda<2\kappa\sqrt{\frac{d-1}{d-2}}$ the
solution (295) with $m>0$ has a transient acceleration stage. First let us
find $\frac{da}{dt}=\frac{da}{d\tau}\frac{d\tau}{dt}.$ Using (295) it is easy
to show that $\dot{a}$ is proportional to positively defined quantities $m$
and $\tau.$ Setting $|m|=1,$ which can be easily done by shift along the
$\tau$ axis, one finds $\ddot{a}$ in the following form:
$\ddot{a}=-({\rm
positive})\bigg{(}\big{(}(d-1)\alpha^{2}-1\big{)}Z^{2}-2(d-2)\,{\rm
sign}(m)\alpha Z+d-1-\alpha^{2}\bigg{)},$ (297)
where $Z\equiv\cosh(2w\tau).$ If $(d-1)\alpha^{2}<1,$ which corresponds to the
case $\lambda<\frac{2\kappa}{\sqrt{d-2}},$ then we obtain the eternal
accelerated expansion for arbitrary values of $m,$ because only the first term
in (297)dominates on the later stages of evolution. Remark that such solution
represents an attractor. If $(d-1)\alpha^{2}>1$ holds for later times then the
solution always corresponds to to decelerated expansion of Universe. For
$(d-1)\alpha^{2}>1$ and $m<0$ the right hand side of the equation (297) is
negatively defined, which corresponds to decelerated expansion of Universe
during whole evolution time. At last, in the case $(d-1)\alpha^{2}>1$ and
$m>0,$ the solution always provides the transient acceleration phase, which is
mentioned intown . Indeed, the equation (297) has two roots
$Z_{\pm}={(d-2)\alpha\pm\sqrt{d-1}(1-\alpha^{2})\over(d-1)\alpha^{2}-1},$
(298)
defined in the interval $\tau_{-}(\alpha)<\tau<\tau_{+}(\alpha),$
corresponding to the accelerated expansion.
The root $\tau_{\pm}$ is real (and positive), because on the interval
$\alpha\in(1/\sqrt{d-1},1)$ we have $Z_{\pm}>1.$ In the limit $\alpha\to 1$
the two roots coincide and duration of the accelerated expansion phase tend to
zero. In the opposite limit $\alpha\to 1/\sqrt{d-1}$ one obtains
$Z_{-}=d/(2\sqrt{d-1})$ and $Z_{+}\to\infty,$ which corresponds to infinite
period of accelerated expansion. For higher dimensions of space one obtains
$Z_{\pm}=1/\alpha\pm(1-\alpha^{2})/(\alpha^{2}\sqrt{d})+O(1/d),$ and it means
that the transient acceleration duration is shorter in the Universe with
larger number of spatial dimensions.
#### X.2.2 Transient acceleration in models with multiple scalar fields
The existence of an event horizon in the case of a de Sitter phase is an
obstacle to the implementation of string theory because the S-matrix
formulation is no longer possible, hence eternal acceleration, leading to a de
Sitter space in the asymptotic future is problematic HKS . The transient
acceleration models allow to avoid that contradiction, because they produce
the accelerated expansion today and in recent past, without any contradiction
in future. In the paper Polarski the dynamics of homogeneous and isotropic
Universe was considered, where the dark energy is realized by two (generally
speaking, turn by turn) scalar fields $\Phi$ and $\psi.$ Let us write down the
equations of motion for that system
$\displaystyle\dot{\rho_{b}}$ $\displaystyle=$
$\displaystyle-3H\gamma_{b}\rho_{b}$ $\displaystyle\ddot{\varphi}$
$\displaystyle=$ $\displaystyle-3H\dot{\varphi}-\partial_{\varphi}V$ (299)
$\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle-3H\dot{\psi}-\partial_{\psi}V,$
and also the first Friedmann equation:
$H^{2}=\frac{8\pi G}{3}(\rho_{b}+\rho_{Q})-\frac{k}{a^{2}}~{}.$
Here the dot denoted the derivative with respect to time $t,$ the subscript
$b$ marks the background components such as dark matter ($m$) or radiation
($r$), and $Q$ stands for dark energy in form of the two scalar fields. It is
convenient also to define
$p_{b,{}_{Q}}=(\gamma_{b,{}_{Q}}-1)\rho_{b,{}_{Q}}~{},$
then $\gamma_{m}=1$ and $\gamma_{r}=\frac{4}{3},$ where, for any component
$i,$ we have introduced for convenience the quantity $\gamma_{i}\equiv
1+w_{i}.$ The energy density and pressure for the quintessence fields in the
potential $V(\varphi,\psi)$ have the form:
$\rho_{Q}=\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{2}\dot{\psi}^{2}+V(\varphi,\psi)~{}~{}p_{Q}=\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{2}\dot{\psi}^{2}-V(\varphi,\psi).$
As always, the possibility of the transient acceleration crucially depends on
the potential form. Below we consider several possibilities to obtain the
transient acceleration. The paper Polarski considers a number of examples,
including the case when the two fields interact through the coupling potential
$V(\varphi,\psi),$ as well as the case when one of the fields is free. Here we
consider only the models with potentials depending on both fields. Direct
generalization of the potential from the paper (288) for the case of minimal
coupling between the two scalar fields enables us to obtain the transient
acceleration.
Consider a potential of the form
$V(\varphi,\psi)=M^{4}e^{-\lambda\varphi}(P_{0}+f(\psi)(\varphi-\varphi_{c})^{2}+g(\psi)).$
(300)
The above introduced additional scalar field $\psi$ will control absence or
presence o the potential minimum for $\varphi.$ The main idea is that, from
one hand, a potential minimum initially exists for the scalar $\varphi,$ which
provides accelerated expansion of Universe, and from the other hand evolution
of the field $\psi$ results in the extinction of the minimum and the Universe
returns to the decelerated expansion regime. The model of AS is recovered
with $f\equiv 1$ and $g\equiv 0.$ For the potential (300) the position of the
minimum is determined by the expression
$\varphi_{\pm}=\varphi_{c}+\frac{1}{\lambda}\left(1\pm\sqrt{1-\lambda^{2}\frac{P_{0}+g(\psi)}{f(\psi)}}\right).$
(301)
The function $g$ ($g>0$) is responsible for the scalar field mass and it
usually takes the form $g\propto\psi^{2},$ but as it does not affect the
system dynamics, we set for simplicity $g\equiv 0.$
The minimum (301) disappears under the condition
$f(\psi)<\lambda^{2}P_{0}\equiv f(\psi_{c}).$ (302)
Note that the potential (300) can be presented in the form
$V(\varphi,\psi)=M^{4}/\lambda^{2}e^{-\lambda\varphi}(f(\psi_{c})+f(\psi)(\lambda\varphi-\lambda\varphi_{c})^{2}).$
We assume that $f$ is positive, continuous and monotonous function for
$\psi>0$ and/or $\psi<0.$ From the condition (340) it is easy to see that if
$f$ decreases (increases), then for $\psi<\psi_{i}$ ($\psi_{i}-$ is the
initial value of the scalar field), ($\psi<\psi_{c}$) the accelerated
expansion takes place, because the minimum (301) exists. One is free to use
different functions $f$ to obtain the transient acceleration, for instance we
present the case $p,~{}\alpha\geq
0,~{}f=1+\alpha\psi^{p},~{}f=\tanh(\alpha\psi^{p}),$ or for arbitrary $p$ and
$\alpha,~{}f=\exp(\alpha\psi^{p}),~{}f=\cosh(\alpha\psi^{p}).$ During the
evolution of Universe the scalar field $\varphi$ rolls down the potential and
dominates in the exponential part so that $M_{Pl}^{2}m^{2}_{\varphi}\sim
M_{Pl}^{2}m^{2}_{\psi}\sim V\sim\dot{\varphi}^{2}\sim M_{Pl}^{2}H^{2}$ with
$\Omega_{Q}=4/\lambda^{2},$ $w_{Q}=1/3$ during the radiation dominated era,
and with $\Omega_{Q}=3/\lambda^{2},$ $w_{Q}=0$ for the matter-dominated one,
while $\varphi$ approach the minimum. When the field $\varphi$ is in the
minimum (301) and oscillates near its minimum value, the universe expands with
acceleration with $V\gg\dot{\varphi}^{2}$ and therefore $w_{Q}\simeq-1$ until
$\psi$ obeys $\psi\leq\psi_{c},$ $(\psi\geq\psi_{c}),$ if the function $f$
grows (decays). In that moment the minimum (301) disappears and the field
$\varphi$ starts free roll acquiring large values (speeding up) and the the
Universe expansion becomes decelerated one $q>0.$
When $\psi$ is initially larger (smaller) than $\psi_{c}$, provided $f$ is
increasing (decreasing) $\psi$ passes through $\psi_{c}$ because
$\partial_{\psi}V=M^{4}e^{-\lambda\phi}(\phi-\phi_{c})^{2}\frac{df}{d\psi}$ is
positive (negative), acceleration occurs which is always transient; if on the
contrary $\psi_{i}\leq\psi_{c}$ ($\psi_{i}\geq\psi_{c}$) quintessence
domination is not possible. The critical value $\psi_{c}$ controls the
presence or not of the minimum for $\phi$.
For a particular example let us consider the simplest case
$f(\psi)=\psi^{2}~{},$ (303)
then the minimum for $\varphi$ disappears under condition
$-\psi_{c}\leq\psi\leq\psi_{c},$ where $\psi_{c}\equiv\lambda\sqrt{P_{0}}.$ In
analogy with the model (288), one can obtain $\lambda\gtrsim 9,$ which agrees
with the limitations imposed by the cosmological observations, provided the
initial value $\varphi_{i}$ is fixed and the critical value $\varphi_{c}$ must
be finely tuned in order to make the quintessence dominate in the present
time.
The diagram 9 shows the state equation parameter $w_{Q,0}\simeq-0.491$ and
$w_{eff,0}=-0.874.$ When the field $\varphi$ gets into the minimum, then
$w_{Q}\simeq-1.$ The deceleration parameter $q$ is also shown on the plot, and
it points out on the fact that in the present time the Universe already passed
the transient acceleration stage and currently it expands with deceleration
$q_{0}\simeq 0.013.$ The accelerated expansion ($q<0$) starts at $z\simeq
0.658$ and stops at $z\simeq 0.0035,$ when the Universe age equals
$t_{end}/t_{0}\simeq 0.996,$ while in the present time one has
$H_{0}t_{0}\simeq 0.912.$
Figure 9: The diagram plots the dependence of the state equation parameter
$w_{{}_{Q}}$ (solid line) and deceleration parameter $q$ (dashed line) as
functions of the redshift $z$ for the potential (300) with the fixed parameter
values $\lambda=10,$ $\phi_{c}=23.8$ and $P_{0}$ taking the following values
values (upward from below) $0.160,0.162,0.163,0.164,0.166,0.168,0.170$
$(\phi_{i}=0$ and $\psi_{i}=5).$ In the models with
$p_{0}=0.164,0.166,0.168,0.170$ the accelerated expansion is replaced by the
decelerated one in the modern epoch.
#### X.2.3 Decaying dark energy as a scalar field
As was already mentioned above, the cosmological models where the dark energy
decays have many attractive features, one of which being the presence of
transient acceleration — alteration of the accelerated Universe expansion
period by the decelerated one, which takes place when the dark energy density
becomes sufficiently low. Dynamics of such Universe clearly differs from its
evolution predicted by SCM. The considered model with the decaying dark energy
represents a pre-image of the commonly accepted inflation model, where the
field, generated the inflationary expansion of Universe, experiences the
decay.
Consider a scalar field model with the potential that takes both positive and
negative values 0302302 :
$V(\varphi)={V_{0}}\cos\frac{\varphi}{f},\,\,\,f=\frac{{\sqrt{{V_{0}}}}}{m}.$
(304)
The assumption that the scalar field potential can take negative values is
very curious, but one should remember that except the motivation to explain
the observed transient acceleration this model lacks any other observational
support. Nevertheless such effective potentials often appear in the super-
gravity and M-theory. It is be shown below how to obtain the transient
acceleration in frames of that model. Remark also that evolution of Universe
in cosmological models with negative potentials sharply differs from that of
SCM. So, for example, in the case when the Friedmann equation is dominated by
potential energy (304), taking negative values
${\left({\frac{{\dot{a}}}{a}}\right)^{2}}=\frac{1}{3}\left({\rho+\frac{1}{2}{{\dot{\varphi}}^{2}}+V(\varphi)}\right),$
(305)
even the spatially flat Universe can collapse, which is principally impossible
in SCM, but it can happen only on later stages of Universe evolution, far
before the transient acceleration ends. Up to the moment when the Universe
starts collapsing, many stages of transient acceleration manage to occur (see
Fig.10), which become more often when approaching the collapse moment.
Figure 10: Dependence of the deceleration parameter on the scale factor for
the Universe filled by matter and dark energy in form of scalar field in the
potential $V(\varphi)={V_{0}}\cos\left({\varphi m/\sqrt{{V_{0}}}}\right),$
with $m=0.74;\,{V_{0}}=150,\varphi(0)=0.23,\varphi^{\prime}(0)=0.$ The present
time corresponds to $a=1.$ The potential parameters are chosen so that the
deceleration parameter value appears to be the same as in SCM,
$q(1)\approx-0.6.$
#### X.2.4 Transient Acceleration In the Universe with the Interacting
Components
Consider the spatially flat Universe composed from three components cosmolog :
dark energy, dark matter and baryons. The first friedmann equation then takes
the form:
$3H^{2}=\rho_{{}_{DE}}+\rho_{m}+\rho_{b},$ (306)
where as usual $\rho_{{}_{DE}}$ is the dark energy density, $\rho_{m}$ – dark
matter energy density, $\rho_{b}$ – the density of baryons,
$H=\frac{\dot{a}}{a}$ – Hubble parameter. The state equation for the dark
energy takes the form $p_{{}_{DE}}=w\rho_{{}_{DE}}.$
The conservation equations in that case take the form:
$\begin{gathered}{\dot{\rho}_{{}_{DE}}+3H(1+w)\rho_{{}_{DE}}=-Q;}\hfill\\\
{\dot{\rho}_{m}+3H\rho_{m}=Q,}\end{gathered}$ (307)
where $Q$ characterizes the interaction. The conservation equation for the
baryons reads:
$\dot{\rho}_{b}+3H\rho_{b}=0\,\,\,\Rightarrow\,\,\,\rho_{b}=\rho_{b0}\left(\frac{a_{0}}{a}\right)^{3}.$
(308)
Total density equals $\rho=\rho_{m}+\rho_{b}+\rho_{{}_{DE}}.$ Without lack of
generality, we assume that the dark matter energy density satisfies
$\rho_{m}=\tilde{\rho}_{m0}\left(\frac{a_{0}}{a}\right)^{3}f\left(a\right),$
(309)
where $\tilde{\rho}_{m0}$ and $a_{0}$ are constants and $f(a)$ is arbitrary
time dependent function. From (307) and (309) one obtains
$Q=\rho_{m}\frac{\dot{f}}{f}=\tilde{\rho}_{m0}\left(\frac{a_{0}}{a}\right)^{3}\dot{f}.$
(310)
Let $f(a)$ takes the form
$f(a)=1+g(a).$ (311)
Interaction free case corresponds to $f(a)=1,$ thus the function $g(a)$
responds for the interaction. Taking into account that
$\dot{f}=\dot{g}=\frac{dg}{da}\dot{a},$ (312)
one obtains
$Q=\tilde{\rho}_{m0}\frac{dg}{da}\dot{a}\left(\frac{a_{0}}{a}\right)^{3}.$
(313)
For $\rho_{m}$ it means that
$\rho_{m}=\tilde{\rho}_{m0}\left(1+g\right)\left(\frac{a_{0}}{a}\right)^{3},$
(314)
where $\rho_{m0}=\rho_{m}(a_{0})$ in presence of interaction, and analogously
$\tilde{\rho}_{m0}$ in uncoupled case. The two initial values of dark matter
density are linked by the following relation
$\rho_{m0}=\tilde{\rho}_{m0}\left(1+g_{0}\right),$ (315)
where $g_{0}\equiv g(a_{0}).$ As can be seen from (310), in the case when $Q$
is positive the dark energy decays into dark matter $\frac{dg}{da}>0$ and vice
versa $\frac{dg}{da}<0$ otherwise. It follows from the equations (307) and
(313)
$\dot{\rho}_{{}_{DE}}+3H\left(1+w\right)\rho_{{}_{DE}}=-\tilde{\rho}_{m0}\frac{dg}{da}\dot{a}\left(\frac{a_{0}}{a}\right)^{3}.$
(316)
Assuming that $w=const$ the solution of equation (316) takes the form
$\begin{gathered}\rho_{{}_{DE}}=\left(\rho_{{}_{DE0}}+\tilde{\rho}_{m0}g_{0}\right)\left(\frac{a_{0}}{a}\right)^{3\left(1+w\right)}-\\\
-\tilde{\rho}_{m0}\left(\frac{a_{0}}{a}\right)^{3}g+3w\tilde{\rho}_{m0}a_{0}^{3}a^{-3\left(1+w\right)}\int_{a_{0}}^{a}daga^{3w-1}.\end{gathered}$
(317)
Rewrite the second Friedmann equation in the terms of $g(a)$
$\begin{array}[]{l}{\frac{\ddot{a}}{a}=-\frac{1}{6}\left\\{\tilde{\rho}_{m0}\left(1+g\right)\left(\frac{a_{0}}{a}\right)^{3}+\rho_{b0}\left(\frac{a_{0}}{a}\right)^{3}+\left(1+3w\right)\right.\times}\\\
{\times\left[\left(\rho_{{}_{DE0}}+\tilde{\rho}_{m0}g_{0}\right)\left(\frac{a_{0}}{a}\right)^{3\left(1+w\right)}\right.\left.\left.-\tilde{\rho}_{m0}\left(\frac{a_{0}}{a}\right)^{3}g+3w\tilde{\rho}_{m0}a_{0}^{3}a^{-3\left(1+w\right)}\int_{a_{0}}^{a}daga^{3w-1}\right]\right\\}.}\end{array}$
(318)
In order to solve the equation (318) one have to explicitly define the
function $g(a).$ For the lack of knowledge about nature of the dark energy, as
well as the dark matter, the function $g(a)$ cannot be derived from the first
principles. Therefore for the model under consideration we introduce the
interaction such that the resulting dynamics in the model corresponds to the
observed data.
Consider interaction with the function $g(a)$ in the form
$g\left(a\right)=a^{n}\exp\left(-a^{2}/\sigma^{2}\right),$ where $n$ is
natural number, and $\sigma$ is positive real number. The transient
acceleration implies that the dark energy density starts to diminish, i.e.
experiences decay $\frac{dg}{da}>0.$ This condition requires that $n$ and
$\sigma$ satisfy the inequality $n\sigma^{2}>2.$ The dependencies of relative
densities on the scale factor for $n=7$ and $\sigma=1.5$ are shown on Fig. 11.
Figure 11: The dependencies of relative densities on the scale factor for
$n=7$ and $\sigma=1.5$ (on the left). On the right: the dependencies of
deceleration parameter on the scale factor in the model with interacting dark
energy and dark matter $q(a)$(solid line) with $n=7$ and $\sigma=1.5$ in
comparison to SCM (dashed line).
The considered model which allows to produce the transient acceleration with
certain choice of the interaction parameters, however for large values of
scale factor as well for small ones it is indistinguishable from SCM.
#### X.2.5 Decaying cosmological constant and transient acceleration phase
As a simple example of the transient acceleration we consider a model cosmolog
with decaying cosmological constant:
$\dot{\rho}_{dm}+3\frac{\dot{a}}{a}\rho_{dm}=-\dot{\rho}_{\Lambda}\;,$ (319)
where $\rho_{dm}$ and $\rho_{\Lambda}$ are energy densities of dark matter and
cosmological constant $\Lambda$ respectively. On early stages of Universe
expansion, when $\rho_{\Lambda}$ is sufficiently small, such decay has almost
no influence on the cosmic evolution. On later stages, the more the dark
energy contribution grows, the more its decay affects the standard dependence
for the dark matter energy density $\rho_{dm}\propto a^{-3}$ on the scale
factor $a.$ Let us assume that such deviation can be described by a scale
factor function $\epsilon(a),$ then
$\rho_{dm}=\rho_{dm,0}a^{-3+\epsilon(a)}\;,$ (320)
where in the present epoch $a_{0}=1.$ Other material fields (radiation and
baryons) evolve independently and conserve. Therefore the dark energy density
takes the form:
$\rho_{\Lambda}=\rho_{dm,0}\int_{a}^{1}{\epsilon(\tilde{a})+\tilde{a}\epsilon^{\prime}\ln(\tilde{a})\over\tilde{a}^{4-\epsilon(a)}}d\tilde{a}+{\rm{X}},$
(321)
where prime denotes the derivative with respect to the scale factor, and
${\rm{X}}$ – is the integration constant. Neglecting the contribution of
radiation, the first Friedmann equation takes the form
${{H}}=H_{0}\left[\Omega_{b,0}{a}^{-3}+\Omega_{dm,0}\varphi(a)+{\Omega}_{{\rm{X,0}}}\right]^{1/2}.$
(322)
The function $\varphi(a)$ reads
$\varphi(a)=a^{-3+\epsilon(a)}+\int_{a}^{1}{\epsilon(\tilde{a})+\tilde{a}\epsilon^{\prime}\ln(\tilde{a})\over\tilde{a}^{4-\epsilon(a)}}d\tilde{a},$
(323)
where ${\Omega}_{{\rm{X,0}}}$ denotes relative contribution of the constant
${\rm{X}}$ into the total relative density. In order to proceed further, some
assumptions on the explicit form of the function $\epsilon(a)$ should be done.
In the present review we follow the original paper cosmolog on $\Lambda$-dark
matter interaction and consider the simplest case
$\displaystyle\epsilon(a)$ $\displaystyle=$
$\displaystyle\epsilon_{0}a^{\xi};\quad$ (324) $\displaystyle=$
$\displaystyle\epsilon_{0}(1+z)^{-\xi},$
where $\epsilon_{0}$ and $\xi$ can take both positive and negative values.
From the above cited expression (321) it follows that
$\rho_{\Lambda}=\rho_{m0}\epsilon_{0}\int_{a}^{1}{[1+\ln(\tilde{a}^{\xi})]\over\tilde{a}^{4-\xi-\epsilon_{0}\tilde{a}^{\xi}}}d\tilde{a}+{\rm{X}}.$
(325)
Remark that the case $\epsilon_{0}=0$ corresponds to SCM, i.e.
${\rm{X}}\equiv{\rho}_{\Lambda 0}.$
Using the above cited formulae it is easy to obtain the dependence of the
relative densities $\Omega_{b}(a),$ $\Omega_{dm}(a)$ and
$\Omega_{\Lambda}(a)$:
$\Omega_{b}(a)=\frac{a^{-3}}{{\rm{A}}+a^{-3}+{\rm{B^{-1}}}\varphi(a)};$ (326a)
$\Omega_{dm}(a)=\frac{a^{-3+\epsilon(a)}}{{\rm{D}}+{\rm{B}}a^{-3}+\varphi(a)};$
(326b)
$\Omega_{{\rm{\Lambda}}}(a)=\frac{{\rm{D}}+\varphi(a)-a^{-3+\epsilon(a)}}{{\rm{D}}+{\rm{B}}a^{-3}+\varphi(a)};$
(326c)
where ${\rm{A}}={\Omega_{{\rm{X}},0}}/{\Omega_{b,0}},$
${\rm{B}}={\Omega_{b,0}}/{\Omega_{dm,0}}$ and
${\rm{D}}={\Omega_{{\rm{X}},0}}/{\Omega_{dm,0}}.$
In such simple model, appropriate choice of the parameters $\epsilon_{0}$ and
$\xi$ enables to obtain practically any kind of Universe dynamics. In the
context of the present review it is especially interesting to consider the
case when $\epsilon_{0}>0$ and $\xi$ takes large positive values ($\xi\gtrsim
0.8$). The figure 12 shows the dependence of the scale factor for the case
$\xi=1.0$ and $\epsilon_{0}=0.1.$ Note that with such parameters in the
present time $a\sim 1$ the Universe expands with acceleration, but unlike the
case of SCM, the dark energy domination will not last forever and at $a\gg 1$
the Universe enters into new era of non-relativistic matter domination with
$a\to\infty.$ Such type of dynamical behavior is impossible for most models
with $\Lambda(t)$ or those with interacting quintessence, widely discussed in
literature, but it is an attribute of the so-called thawing thaw and hybrid
cqg potentials, which follow from string or M-theory fischler (see also ed
)888Authors of the paper fischler present arguments for the fact that forever
expanding Universe, which is common feature of most quintessence models
(including the standard $\Lambda$CDM-model), contradicts the predictions of
string/M-theories, because those models contain cosmological horizon, which
make impossible to apply the usual procedure of S-matrix formalism, describing
the interaction of the particles..
Figure 12: The deceleration parameter as function of $\log(a)$ for some
selected values of $\epsilon_{0}$ and $\xi.$
In order to give clearer idea of transient acceleration phenomenon we find the
explicit form of the deceleration parameter $q=-a\ddot{a}/\dot{a}^{2},$ in the
considered model
$q(a)=\frac{3}{2}\frac{\Omega_{b,0}a^{-3}+\Omega_{dm,0}a^{\epsilon(a)-3}}{\Omega_{b,0}{a}^{-3}+\Omega_{dm,0}\varphi(a)+{\Omega}_{\rm{X},0}}-1.$
(327)
Its dependence on $\log(a)$ for selected values of $\xi$ and $\epsilon_{0}$ is
shown on Fig.12. Note that large values of the parameter $\xi$ corresponds to
the Universe dominated by matter in the past ($q(a)\rightarrow 1/2$ for $a\gg
1$), afterwards (at $a_{acc}<1$) a long-during era of accelerated expansion
takes place, which, unlike SCM, does not last forever, ending at some
$a_{dec}>1.$
### X.3 Transient acceleration: holographic limitations
The novel concept of the holographic Universe considered in the previous
section turns out to impose strict limitations to Universe dynamics. In order
to see that we consider the matter entropy contained within the observed
horizon in the case when the matter is composed of non-relativistic substance.
Each particle of the substance carries one unit of information, and therefore
$S_{m}=K_{B}N,$ where $N=(4\pi/3)r^{3}_{A}n,$ and $n$ is the particle
concentration, for which $n=n_{0}a^{-3}.$ Thus one obtains for entropy the
following
$S_{m}=k_{B}\,\frac{4\pi}{3}\,\tilde{r}_{A}^{3}\,n_{0}\,a^{-3}\propto
a^{3/2}\,.$ (328)
and it follows that $S^{\prime\prime}_{m}\propto\frac{3}{4\,\sqrt{a}}>0.$ It
is interesting to note that as the Universe expands the particle number
contained inside the horizon increases too, which provides the entropy growth
and therefore $S^{\prime\prime}_{m}\,+\,S^{\prime\prime}_{A}>0.$ It is easy to
see that the Universe tends to maximum entropy provided its horizon decreases
with time. In the simplest case the horizon equals to Hubble radius and the
above condition is satisfied if the Universe expands with acceleration. In
such case the deceleration parameter is negative ($q<0$), thus
$\frac{d}{dt}(R_{H})=c(1+q)<c,$
and Hubble sphere has velocity which is less then light speed by the quantity
$cq,$ and thus it is left behind those galaxies. Therefore the particles
initially placed inside the Hubble sphere, gradually leave it.
From the latter fact that a conclusion follows that in order to obey the
second law of thermodynamics, the universe must be dominated by dark energy
with state equation parameter satisfying the condition $-1\leq w\leq-2/3.$
Another alternative is to modify the gravitation theory in order to obtain the
accelerated dynamics without the dark energy.
Likewise, since every isolated system is expected to evolve in a such a way
that it tends to thermodynamic equilibrium the curvature of the ${\cal A}(a)$
function should never spontaneously change from negative to positive values.
In order to specify the set of possible cosmological values, or equivalently
to impose certain limitations on the values taken by the state equation
parameter $w,$ it is necessary to use the laws which are not immediately
connected to cosmology. In the context of holographic cosmology it is
reasonable to use the generalized law of the black hole thermodynamics — the
generalized square law (GSL). According to the latter the total entropy of the
system $S$ must obey the condition $S^{\prime\prime}<0.$ As will be shown
below, (see 1012.0474v1 ) application of this criterion on later stages of
Universe evolution helps considerable specify the range of values allowable
for the state equation parameter $w.$
For further convenience we express the horizon area $\cal A$ and its
derivatives through the deceleration parameter $q\equiv-\ddot{a}/(a\,H^{2}),$
which can be presented in the above mentioned form:
$q=-\left(1\,+\,\frac{a\,H^{\prime}}{H}\right)\,.$ (329)
For the case of spatially flat Universe ${\cal A}=4\pi H^{-2},$ therefore
${\cal A}^{\prime}=8\pi(1+q)/H^{2}$ and
${\cal A}^{\prime\prime}=2{\cal
A}\,\left[(1\,+\,q)\,(1\,+\,2q)\,+\,\frac{q^{\prime}}{a}\right]\,.$ (330)
It follows that for the case $q\geq-1/2$ the second order derivative of the
horizon area, ${\cal A}^{\prime\prime},$ cannot change from positive to
negative values while $q^{\prime}<0.$ It excludes the dependence for $q$ shown
on Fig.13. Such dependence of the deceleration parameter appears in the
cosmological models where the current accelerated expansion is transient, and
after a short dark matter dominated period the dark matter starts dominating
again, which leads to the decelerated Universe expansion era. jcap-julio2010 ;
bose-majumdar .
Figure 13: The deceleration parameter dependence in some models with transient
acceleration.
### X.4 Transient acceleration in models with holographic dark energy
Current literature usually considers the models where the required dynamics of
Universe is provided by one or another, and always only one, type of dark
energy. As was multiply mentioned above, in order to explain the observed
dynamics of Universe, the action for gravitational field is commonly
complemented, besides the conventional matter fields (both matter and baryon),
by either the cosmological constant, which plays role of physical vacuum in
SCM, or more complicated dynamical objects — scalar fields, $K$-essence and so
on. In the context of holographic cosmology, the latter term is usually
neglected, restricting to contribution of the boundary terms. Nevertheless
such restriction has no theoretical motivation.
In the present subsection we consider the cosmological model which contains
both volume and surface termsJCAPphase ; CosmicAccel . The role of former is
played by homogeneous scalar field in exponential potential, which interacts
with dark matter. The boundary term responds to holographic dark energy in
form of (281). Such model is shown to produce the transient acceleration phase
— the evolution stage when the accelerated expansion of Universe changes to
the decelerated one, and afterwards the Universe comes to the stage of eternal
accelerated expansion, thus removing the limitations imposed in the previous
subsection.
To describe the dynamical properties of the Universe it is convenient to to
transform to dimensionless variables as the following:
$\displaystyle x=\frac{\dot{\varphi}}{\sqrt{6}M_{Pl}H},\quad
y=\frac{1}{M_{Pl}H}\sqrt{\frac{V(\varphi)}{3}},\quad
z=\frac{1}{M_{Pl}H}\sqrt{\frac{\rho_{m}}{3}},\quad
u=\frac{1}{M_{Pl}H}\sqrt{\frac{\rho_{{}_{q}}}{3}}.$ (331)
The evolution of scalar field is described by the Klein-Gordon equation, which
in the case of interaction between the scalar field and matter takes the
following form:
$\ddot{\varphi}+3H\dot{\varphi}+\frac{dV}{d\varphi}=-\frac{Q}{\dot{\varphi}}.$
(332)
In the present section we consider the case when the interaction parameter $Q$
is a linear combination of energy density for scalar field and dark energy
$Q=3H(\alpha\rho_{\varphi}+\beta\rho_{m}),$ (333)
where $\alpha,$ $\beta$ are constant parameter. For given model, regardless
the explicit form of the scalar field potential $V(\varphi),$ the system of
dynamical equations takes the following form
$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3x}{2}g(x,z,u)-3x+\sqrt{\frac{3}{2}}\lambda
y^{2}-\gamma,\hfill$ $\displaystyle y^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3y}{2}g(x,z,u)-\sqrt{\frac{3}{2}}\lambda xy,\hfill$ (334)
$\displaystyle z^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3z}{2}g(x,z,u)-\frac{3}{2}z+\gamma\frac{x}{z},\hfill$
$\displaystyle u^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3u}{2}g(x,z,u)-\frac{u^{2}}{n},\hfill$
where
$g(x,z,u)=2x^{2}+z^{2}+\frac{2}{3n}u^{3},~{}\lambda\equiv-\frac{1}{V}\frac{dV}{d\varphi}M_{Pl}.$
(335)
and
$\displaystyle Q$ $\displaystyle=$ $\displaystyle
9H^{3}M_{Pl}^{2}\left[\alpha(x^{2}+y^{2})+\beta z^{3}\right];$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{\alpha(x^{2}+y^{2})+\beta z^{3}}{x}.$ (336)
As was mentioned above, here we consider the simplest case of exponential
potential
$V=V_{0}\exp\left(\sqrt{\frac{2}{3}}\frac{\mu\varphi}{M_{Pl}}\right),$ (337)
where $\mu$ is constant. Taking into account the expression (336), the system
of equations (334) reads
$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3x}{2}\left[g(x,z,u)-\frac{\alpha(x^{2}+y^{2})+\beta
z^{2}}{x^{2}}\right]-3x-\mu y^{2},\hfill$ $\displaystyle y^{\prime}$
$\displaystyle=$ $\displaystyle\frac{3y}{2}g(x,z,u)+\mu xy,\hfill$ (338)
$\displaystyle z^{\prime}$ $\displaystyle=$
$\displaystyle\frac{3z}{2}\left[g(x,z,u)+\frac{\alpha(x^{2}+y^{2})+\beta
z^{2}}{z^{2}}\right]-\frac{3}{2}z,\hfill$ $\displaystyle u^{\prime}$
$\displaystyle=$ $\displaystyle\frac{3u}{2}g(x,z,u)-\frac{u^{2}}{n}.\hfill$
In the considered model the deceleration parameter takes the form
$q=-1+\frac{3}{2}\left[2x^{2}+z^{2}+\frac{2}{3n}u^{3}\right].$ (339)
Note that the cosmological parameters have no explicit dependence on the
interaction form, and only determine the evolution dynamical variables. This
fact essentially complicates analysis of such system. Let us make few remarks
on the system of equations (338). First consider the case $y=0,$ which
corresponds to free scalar field. It easy to obtain some restricting relations
on the interaction parameters, which follow from the requirement for energy
density to be real
$2\sqrt{\frac{\beta}{\alpha}}>1+\alpha+\beta,$ (340)
which in turn requires $0>\beta>\alpha,|\alpha|+|\beta|<1.$ This unstable
critical point corresponds to Universe filled by dark matter. The case of
multiple such points is also possible but irrelevant. To conclude we note that
any of such critical points is easily shown to exist also in the interval
$x_{0}<0.$ For the case $z\neq 0$ with the restrictions imposed by (340), one
obtains
$x_{c}=\left[\left(a+\sqrt{{\beta}/{\alpha}}\right)^{1/2}+a\right]z_{c},$
(341)
where $a=\left(2\sqrt{{\beta}/{\alpha}}-(1+\alpha+\beta)\right)^{1/4}.$
#### X.4.1 Case $Q=0$
In the present section we consider in more details the interaction-free case
with scalar field and dark matter. The critical points of the system (338) in
the case $(\alpha=\beta=0)$ are given in the table 1. The phase space
generated by the system of equations (338) contains six physically relevant
critical points, the last of which is an attractor. The first critical point
$(1,0,0,0)$ is unstable and corresponds to Universe dominated by scalar field
with extremely rigid state equation $(w_{\varphi}=1),$ the second critical
point is also unstable and responds to the evolution period when the scalar
field is dynamically equivalent to cosmological constant.
The next third point $(0,0,1,0)$ is of no interest because it corresponds to
Universe composed solely of dark matter which is also unstable. The fourth
critical point $(0,0,0,1)$ corresponds to Universe solely composed of
holographic dark energy in form of (281), and was considered above in details.
The most physical interest is presented by the last sixth critical point which
is an attractor. It corresponds to the Universe filled by scalar field and
holographic dark energy. This critical point is completely determined by the
scalar field potential parameter $\mu$ and magnitude of $n:$
$\begin{array}[]{cccccl}x_{*}&=&\frac{2}{3n\mu}u_{*},&y_{*}&=&\sqrt{1-\left(1+\frac{4}{9n^{2}\mu^{2}}\right)u_{*}^{2}},\\\
z_{*}&=&0,&u_{*}&=&\frac{3}{2n\mu^{2}}\left(-1+\sqrt{1+\frac{4n^{2}\mu^{4}}{9}}\right).\end{array}$
(342)
The obtained property $x_{*}\propto u_{*}$ is typical for the so-called
tracking solutions Tracking_Solutions . Remark also there is also so-called
background interaction between the scalar field and dark energy, caused by the
fact that the scalar field dynamics is affected by the holographic dark energy
which has negative pressure and influences on the Universe expansion rate by
means of Hubble parameter, which enters into the Klein-Gordon equation for the
scalar field.
Table 1: Critical points for the autonomous system of equations (338) $(x_{c},y_{c},z_{c},u_{c})$ | Stability | $q$ | $w_{\varphi}$ | $w_{tot}$
---|---|---|---|---
coordinates | character | | |
$\;(1,0,0,0)$ | unstable | $2$ | $1$ | $1$
$\;(0,1,0,0)$ | unstable | $-1$ | $-1$ | $-1$
$\;(0,0,1,0)$ | unstable | $\frac{1}{2}$ | $\nexists$ | 0
$\;(0,0,0,1)$ | stable | $-1+\frac{1}{n}$ | $\nexists$ | $-1+\frac{2}{3n}$
$(-\frac{3}{2\mu},\frac{3}{2\mu},\sqrt{1-\frac{3}{2\mu^{2}}},0)$ | unstable | $\frac{1}{2}$ | $\nexists$ | 0
$(x_{*},y_{*},0,u_{*})$ | attractor | $q_{*}<0$ | $w_{\varphi*}$ | $w_{tot*}$
In the attractor point the dark matter density turns to zero. To fit that
model with the observations one has to strictly define the initial conditions
such that the accelerated expansion of Universe starts before then the
considered evolution phase begins.
#### X.4.2 Review of the case $Q=3H\alpha\rho_{\varphi}$
In the above case, the phenomenon of transient acceleration that occurs in
such a Universe does not match the observations. In order to fix that problem
we consider a model where scalar field interacts with dark matter. In the
present section we consider the case with the interaction parameter of the
form (333) with $\beta=0.$ The Figure 14 shows the dependencies
$\Omega_{q},\Omega_{m}$ and $\Omega_{\varphi}$ for the case when
$\alpha=0.005,$ $\mu=-5$ and $n=3.$
Figure 14: Behavior of $\Omega_{\varphi}$ (dot line), $\Omega_{q}$ (dash line)
and $\Omega_{m}$ (solid line) as a function of $N=\ln a$ for
$n=3,\;\alpha=0.005$ and $\mu=-5$ (left side). Evolution of deceleration
parameter for this model (right side).
From the explicit form of the equations and interaction character it is easy
to see that neither type nor position of the above obtained critical points
change when the interaction disappears. The latter affects only behavior of
the dynamical variables, which corresponds to different trajectories in the
phase space between the critical points. This corresponds to the fact that the
interaction parameters enter only in the Hubble parameter derivatives of
second order and higher with respect to time.
With such values of interaction parameters the transient acceleration begins
almost in the current era. Like in the conventional cosmological models with
dark energy in form of scalar field, the latter starts to dominate and causes
the accelerated universe expansion phase. Along with the Universe expansion
the contribution of $\Omega_{q}$ increases, which leads to the fact that the
background (space) starts increasing faster than the field and becomes
asymptotically free. Such field is known to have the so-called super-strict
state equation and forces the Universe to decrease its expansion rate. Soon
however, when the contribution $\Omega_{q}$ increases enough that the scalar
field cannot any more prevent the expansion, Universe starts to speed up
again.
### X.5 Observational evidence
Starobinsky starobinsky with co-authors, based on independent observational
data, including the brightness curves for SNe Ia, cosmic microwave background
temperature anisotropy and baryon acoustic oscillations (BAO), were able to
show (see Fig.15), that the acceleration of Universe expansion reached its
maximum value and now decreases. In terms of the deceleration parameter it
means that the latter reached its minimum value and started to increase. Thus
the main result of the analysis is the following: SCM is not unique though the
simplest explanation of the observational data, and the accelerated expansion
of Universe presently dominated by dark energy is just a transient phenomenon.
The term ”transient acceleration” is increasingly often used with respect to
the Universe dynamics.
Remark, that the paper starobinsky also showed that with the use of CPL
parametrization
$w(z)=w_{0}+\frac{w_{a}\,z}{1+z}.$ (343)
for the state equation parameter, it is impossible avoid contradictions in
attempt to combine the data obtained from observations of near supernovae of
type SN1a with those of the relict radiation anisotropy. One of the way out of
that contradiction is to reject the above mentioned parametrization and invent
a new one. Starobinsky group suggested a novel parametrization which can
combine those data sets:
$w(z)=-\frac{1+\tanh\left[(z-z_{t})\Delta\right]}{2}.$ (344)
Within such approximation $w=-1$ on early times of Universe evolution and
increases up to its maximum value $w\sim 0$ for small values of $z.$
Figure 15 shows the dependence of the deceleration parameter $q$ reconstructed
by the parametrization (344).
Figure 15: The deceleration parameter dependence $q(z)$ reconstructed from
independent observational data, including the brightness curves for SNe Ia,
cosmic microwave background temperature anisotropy and baryon acoustic
oscillations (BAO), SN and parametrization (344). The red solid line shows the
best fit on the confidence level $1\sigma$ CL starobinsky .
In the year 2010 in frames of Supernova Cosmology Project (SCP), the most
recent data set on supernovae outbursts was published amanullah . It contains
557 events and is largest up to date. Besides that the data set concerning
supernovae with moderate red shift $(z<0.3)$ was remarkably extended. Today
there are several works LiWuYu ; LiWuYu1 , making analysis of the above
mentioned data in order to test the transient acceleration hypothesis. All the
authors share common opinion that the ultimate answer will be given only by
remade more accurate measurements. Moreover, obtain a convincing result one
will probably have to remake all the data treatment procedure. Thus it was
shown in LiWuYu ; LiWuYu1 that there is a contradiction between the data
obtained from (SNe Ia+BAO) for small redshifts and those from (CMB) for high
ones. The contradiction is in the fact that the analysis of two data sets
separately one obtains opposite results. For example, using only (SNe Ia+BAO)
data, one obtains evidence in support of the fact that the Universe expansion
rate reached the maximum value at $(z\sim 0.3)$ and presently starts to
decrease. At the same time, if the data are complemented by the CMB data set,
the results of analysis drastically change and no deviations from $\Lambda$CDM
can be revealed.
Figure 16: On the left: the deceleration parameter dependence $q(z)$
reconstructed from results of analysis of Union2+BAO data sets with confidence
level $2\sigma.$ The grey region and the one between the two dashed lines
corresponds to presence and absence of systematic errors in the SNe Ia
observations. On the right: the confident regions for $68.3\%$ and $95\%$
levels are shown for $w_{0}$ and $w_{1}$ in the parametrization CPL with
$w=w_{0}+w_{1}z/(1+z).$ Dashed, solid, thin solid lines corresponds to the
results of Union2S, Union2S+BAO and Union2S+BAO+CMB respectively. The point
$w_{0}=-1,$ $w_{1}=0$ corresponds to spatially flat $\Lambda$CDM model(SCM).
Therefore the result of reconstruction the evolutionary dependence of dark
energy and answer to the question whether our Universe will expand with
deceleration or accelerated expansion will continue forever (like in SCM),
strongly depends on data obtained from SNe Ia observation, its quality, the
particular method of reconstruction for the cosmological parameters such as
$q(z),w(z)$ and $\Omega_{DE}$), as well as on explicit form of the state
equation parametrization. For detailed answer to that question we have to wait
more precise observational data and seek less model-dependent ways of their
analysis.
## Acknowledgements
We are grateful V.A. Cherkaskiy for qualitative translation of this review.
Without his extraordinary help this review could not be appear. Work is
supported in part by the Joint DFFD-RFBR Grant # F40.2/040.
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|
arxiv-papers
| 2011-07-31T20:50:30 |
2024-09-04T02:49:21.165296
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu.L. Bolotin, O.A. Lemets and D.A. Yerokhin",
"submitter": "Danylo Yerokhin",
"url": "https://arxiv.org/abs/1108.0203"
}
|
1108.0239
|
# Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time
Linear Switched Systems
Xiongping Dai Department of Mathematics, Nanjing University, Nanjing 210093,
People’s Republic of China xpdai@nju.edu.cn Yu Huang Department of
Mathematics, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, People’s
Republic of China stshyu@mail.sysu.edu.cn Mingqing Xiao Department of
Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, USA
mxiao@math.siu.edu
###### Abstract
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$ have a
common, but not necessarily strict, Lyapunov matrix (i.e. there exists a
symmetric positive-definite matrix $P$ such that $P-S_{k}^{T}PS_{k}\geq 0$ for
$k=1,2$). Based on a splitting theorem of the state space $\mathbb{R}^{d}$
(Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several
stability criteria for the discrete-time linear switched dynamics
$x_{n}=S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}(x_{0}),\quad
x_{0}\in\mathbb{R}^{d}\textrm{ and }n\geq 1$
governed by the switching signal $\sigma\colon\mathbb{N}\rightarrow\\{1,2\\}$.
More specifically, let $\rho(A)$ stand for the spectral radius of a matrix
$A\in\mathbb{R}^{d\times d}$, then the outline of results obtained in this
paper are: (1) For the case $d=2$, $\boldsymbol{S}$ is absolutely stable
(i.e., $\|S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}\|\to 0$ driven by all
switching signals $\sigma$) if and only if $\rho(S_{1}),\rho(S_{2})$ and
$\rho(S_{1}S_{2})$ all are less than $1$; (2) For the case $d=3$,
$\boldsymbol{S}$ is absolutely stable if and only if $\rho(A)<1\;\forall
A\in\\{S_{1},S_{2}\\}^{\ell}$ for $\ell=1,2,3,4,5,6$, and $8$. This further
implies that for any
$\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$ with the
generalized spectral radius $\rho(\boldsymbol{S})=1$ where $d=2$ or $3$, if
$\boldsymbol{S}$ has a common, but not strict in general, Lyapunov matrix,
then $\boldsymbol{S}$ possesses the spectral finiteness property.
###### keywords:
Linear switched/inclusion dynamics, non-strict Lyapunov matrix, asymptotic
stability , finiteness property
###### MSC:
[2010] 93D20, 37N35
††journal: xxxlabel1label1footnotetext: Project was supported partly by
National Natural Science Foundation of China (Grant Nos. 11071112 and
11071263), the NSF of Guangdong Province and in part by NSF 0605181 and
1021203 of the United States.
## 1 Introduction
### 1.1 Motivations
Let $\mathbb{R}^{d\times d}$ be the standard topological space of all
$d$-by-$d$ real matrices where $2\leq d<+\infty$, and for any
$A\in\mathbb{R}^{d\times d}$, by $\rho(A)$ we denote the spectral radius of
$A$. In addition, we identify $A$ with its induced operator $A(\cdot)\colon
x\mapsto Ax$ for $x\in\mathbb{R}^{d}$. Let
$\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$ be a
finite set with $2\leq K<+\infty$. We consider the stability and stabilization
of the linear inclusion/control dynamics
$x_{n}\in\left\\{S_{1},\dotsc,S_{K}\right\\}(x_{n-1}),\quad
x_{0}\in\mathbb{R}^{d}\textrm{ and }n\geq 1.$ (1.1)
As in [12, 10], we denote by $\varSigma_{\\!K}^{+}$ the set of all admissible
control signals $\sigma\colon\mathbb{N}\rightarrow\\{1,\dotsc,K\\}$, equipped
with the standard product topology. Here and in the sequel
$\mathbb{N}=\\{1,2,\dotsc\\}$ and for any $\sigma\in\varSigma_{\\!K}^{+}$ we
will simply write $\sigma(n)=\sigma_{\\!n}$ for all $n\geq 1$.
For any input $(x_{0},\sigma)$, where $x_{0}\in\mathbb{R}^{d}$ is an initial
state and $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ a
control (switching) signal, there is a unique output $\langle
x_{n}(x_{0},\sigma)\rangle_{n=1}^{+\infty}$, called an orbit of the system
(1.1), which corresponds to the unique solution of the discrete-time linear
switched dynamics
$x_{n}=S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}(x_{0}),\quad
x_{0}\in\mathbb{R}^{d}\textrm{ and }n\geq 1$ (1.2)
driven/governed by the switching signal $\sigma$. Then as usual,
$\boldsymbol{S}$ is called (asymptotically) _stable_ driven by $\sigma$ if
$\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}(x_{0})\|=0\;\forall x_{0}\in\mathbb{R}^{d};\quad\textrm{or
equivalently},\quad\|S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}\|\to 0\textrm{
as }n\to+\infty.$
$\boldsymbol{S}$ is said to be _absolutely stable_ if it is stable driven by
all switching signals $\sigma\in\varSigma_{\\!K}^{+}$; see, e.g., [16]. We
note that the stability of $\boldsymbol{S}$ is independent of the norm
$\|\cdot\|$ used here.
It is a well-known fact that if each member $S_{k}$ of $\boldsymbol{S}$ shares
a common Lyapunov matrix; i.e., there exists a symmetric positive-definite
matrix $Q\in\mathbb{R}^{d\times d}$ such that
$Q-S_{k}^{T}QS_{k}>0\quad(1\leq k\leq K),$
then $\boldsymbol{S}$ is absolutely stable. Here T stands for the transpose
operator of matrices or vectors. An essentially weak condition is that each
member $S_{k}$ of $\boldsymbol{S}$ shares a common, “but not necessarily
strict,” Lyapunov matrix; that is, there exists a symmetric positive-definite
matrix $P$ such that
$\displaystyle P-S_{k}^{T}PS_{k}\geq 0,\quad 1\leq k\leq K.$ (1.3a) Here
“$A\geq 0$” means $x^{T}Ax\geq 0\,\forall x\in\mathbb{R}^{d}$. Associated to
the weak Lyapunov matrix $P$ as in (1.3a), we define the vector norm on
$\mathbb{R}^{d}$ as $\displaystyle\|x\|_{P}=\sqrt{x^{T}Px}\quad\forall
x\in\mathbb{R}^{d}.$ (1.3b)
(We also write its induced operator/matrix norm on $\mathbb{R}^{d\times d}$ as
$\|\cdot\|_{P}$.) Then, $\|S_{k}\|_{P}\leq 1$ for all $1\leq k\leq K$.
Condition (1.3a) is both practically important and academically challenging,
for example, [20, 1, 18, 2, 25] for the continuous-time case and [16] for
discrete case. Indeed, it is desirable in many practical issues and is closely
related to periodic solutions and limit cycles, see, e.g., [5, 6] and [22,
Proposition 18]; in addition, if $S_{k},1\leq k\leq K$, are paracontractive
(i.e., $x^{T}S_{k}^{T}S_{k}x\leq x^{T}x$ for all $x\in\mathbb{R}^{d}$, and
“$=$” holds if and only if $S_{k}(x)=x$, see, e.g., [24]), then condition
(1.3a) holds.
In this paper, we will study the stability of $\boldsymbol{S}$ that satisfies
condition (1.3a). Even under condition (1.3a), the stability of every
subsystems $S_{k}$ does not implies the absolute stability of
$\boldsymbol{S}$, as shown by Example 6.6 constructed in Section 6. So, our
stability criteria — Theorems A, B, C, and D — established in this paper, are
nontrivial.
### 1.2 Stability driven by nonchaotic switching signals
Under condition (1.3a), in [3] for the continuous-time case, Balde and Jouan
provided a large class of switching signals for which a large class of
switched systems are stable, by considering nonchaotic inputs and the geometry
of $\omega$-limit sets of the matrix sequences $\langle
S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}\rangle_{n=1}^{+\infty}$.
Recall from [3, Definition 1] that a switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ is said to be
_nonchaotic_ , if to any sequence $\langle n_{i}\rangle_{i\geq
1}\nearrow+\infty$ and any $m\geq 1$ there corresponds some integer $\delta$
with $2\leq\delta\leq m+1$ such that $\forall\ell_{0}\geq 1$,
$\exists\ell\geq\ell_{0}$ so that $\sigma$ is constant restricted to some
subinterval of $[n_{\ell},n_{\ell}+m]$ of length greater than or equal to
$\delta$. A switching signal $\sigma\in\varSigma_{\\!K}^{+}$ is said to be
_generic_ [16] (or _regular_ in [3]) if each alphabet in $\\{1,\dotsc,K\\}$
appears infinitely many times in the sequence
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$.
Then our first stability criterion obtained in this paper can be stated as
follows:
###### Theorem A.
Let $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$
satisfy condition (1.3a) with $\rho(S_{k})<1$ for all $1\leq k\leq K$. Then
$\|S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}\|\to 0\quad\textrm{as
}n\to+\infty$
for any nonchaotic switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$.
We note that in Theorem A, if $\sigma$ is additionally generic (regular), then
this statement is a direct consequence of [3, Theorem 3]. However, without the
genericity of $\sigma$, here we need to explore an essential property of a
nonchaotic switching signal; see Lemma 2.1 below. In the case of $d=2$ and
$K=2$, an ergodic version of Theorem A will be stated in Corollary 5.3 in
Section 5.
As is shown by Example 6.6 mentioned before, under the assumption of Theorem
A, one cannot expect the stability of $\boldsymbol{S}$ driven by an arbitrary
switching signal.
### 1.3 A splitting theorem driven by recurrent signals
Next, we consider another type of switching signal — recurrent switching
signal, which does not need to be nonchaotic and balanced and which seems more
general from the viewpoint of ergodic theory. In fact, all recurrent switching
signals form a set of total measure $1$.
Corresponding to a switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$, for the
system $\boldsymbol{S}$ we define two important subspaces of the state space
$\mathbb{R}^{d}$:
$\displaystyle
E^{s}(\sigma)=\left\\{x_{0}\in\mathbb{R}^{d}\colon\|S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}(x_{0})\|_{P}\to 0\textrm{ as }n\to+\infty\right\\}$ and
$\displaystyle
E^{c}(\sigma)=\left\\{x_{0}\in\mathbb{R}^{d}\colon\exists\,\langle
n_{i}\rangle_{i=1}^{+\infty}\nearrow+\infty\textrm{ such that
}\lim_{i\to+\infty}S_{\sigma_{\\!n_{i}}}\cdots
S_{\sigma_{\\!1}}(x_{0})=x_{0}\right\\};$
called, respectively, the _stable_ and _central manifolds_ of $\boldsymbol{S}$
driven by $\sigma$. Here $E^{s}(\sigma)$ and $E^{c}(\sigma)$ are indeed
independent of the norm $\|\cdot\|_{P}$.
A switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ is called
_recurrent_ under the classical one-sided Markov shift transformation,
$\theta\colon\sigma(\cdot)\mapsto\sigma(\cdot+1)$, of $\varSigma_{\\!K}^{+}$,
if for any $\ell\geq 1$ there exists some $m$ sufficiently large such that
$(\sigma_{\\!1},\dotsc,\sigma_{\\!\ell})=(\sigma_{\\!1+m},\dotsc,\sigma_{\\!\ell+m}).$
We have then, for $\boldsymbol{S}$, the following important splitting theorem
of the state space $\mathbb{R}^{d}$ based on a recurrent switching signal:
###### Splitting Theorem ([13]).
Let $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$
satisfy condition (1.3a). Then, for any recurrent switching signal
$\sigma\in\varSigma_{\\!K}^{+}$ it holds
$\mathbb{R}^{d}=E^{s}(\sigma)\oplus E^{c}(\sigma)\quad\textrm{and}\quad
S_{\sigma_{\\!1}}(E^{s/c}(\sigma))=E^{s/c}(\sigma(\cdot+1)).$
This theorem is a special case of a more general result [13, Theorem B′′]. So
in this case, if the central manifold $E^{c}(\sigma)=\\{0\\}$ then
$\boldsymbol{S}$ is stable driven by the recurrent switching signal $\sigma$.
This splitting is in fact unique under the Lyapunov norm $\|\cdot\|_{P}$.
### 1.4 Almost sure stability
Under condition (1.3a), let
$\mathbb{K}_{\|\cdot\|_{P}}(S_{k})=\\{x\in\mathbb{R}^{d}\colon\|S_{k}(x)\|_{P}=\|x\|_{P}\\}$
for $1\leq k\leq K$. We note that if $\|S_{k}\|_{P}<1$ then
$\mathbb{K}_{\|\cdot\|_{P}}(S_{k})=\\{0\\}$.
Next, using the above splitting theorem, we can obtain the following almost
sure stability criterion:
###### Theorem B.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$ satisfy
(1.3a) and
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\}$,
where $d=2$ or $3$. Then, if $\mathbb{P}$ is a non-atomic ergodic probability
measure of the one-sided Markov shift
transformation$\theta\colon\varSigma_{2}^{+}\rightarrow\varSigma_{2}^{+}$
defined by $\sigma(\cdot)\mapsto\sigma(\cdot+1)$, there holds
$\|S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}\|_{P}\to 0\quad\textrm{as
}n\to+\infty$
for $\mathbb{P}$-a.e. $\sigma\in\varSigma_{2}^{+}$.
We consider a simple example. Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}$ with
$S_{1}=\mathrm{diag}(\frac{1}{2},\frac{1}{2})$ and $S_{2}=\mathrm{diag}(1,1)$.
Then, $\mathbb{K}_{\|\cdot\|_{2}}(S_{1})=\\{0\\}$ and
$\mathbb{K}_{\|\cdot\|_{2}}(S_{2})=\mathbb{R}^{2}$, where $\|\cdot\|_{2}$
stands for the usual Euclidean norm. So,
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\}$.
Clearly, $\boldsymbol{S}$ is not absolutely stable. This shows that under the
situation of Theorem B, it is necessary to consider the almost sure stability.
### 1.5 Absolute stability and finiteness property
For absolute stability, we can obtain the following two criteria Theorems C
and D, which show the stability is decidable in the cases of $d=2,3$ under
condition (1.3a).
###### Theorem C.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{2\times 2}$ satisfy
condition (1.3a). Then, $\boldsymbol{S}$ is absolutely stable if and only if
$\rho(A)<1$ for all $A\in\\{S_{1},S_{2}\\}^{\ell}$ for $\ell=1,2$.
###### Theorem D.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{3\times 3}$ satisfy
condition (1.3a). Then, $\boldsymbol{S}$ is absolutely stable if and only if
$\rho(A)<1$ for all $A\in\\{S_{1},S_{2}\\}^{\ell}$ for $\ell=1,2,3,4,5,6$, and
$8$.
On the other hand, the accurate computation of the _generalized spectral
radius_ of $\boldsymbol{S}$, introduced by Daubechies and Lagarias in [15] as
$\rho(\boldsymbol{S})=\lim_{n\to+\infty}\max_{\sigma\in\varSigma_{\\!K}^{+}}\sqrt[n]{\rho(S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}})}\quad\left(~{}=\sup_{n\geq
1}\max_{\sigma\in\varSigma_{\\!K}^{+}}\sqrt[n]{\rho(S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}})}\right),$
is very important for many subjects. If one can find a finite-length word
$(w_{1},\dotsc,w_{n})\in\\{1,\dotsc,K\\}^{n}$ for some $n\geq 1$, which
realizes $\rho(\boldsymbol{S})$, i.e.,
$\rho(\boldsymbol{S})=\sqrt[n]{\rho(S_{w_{n}}\dotsm S_{w_{1}})},$
then $\boldsymbol{S}$ is said to have the _spectral finiteness property_. A
brief survey for some recent progresses regarding the finiteness property can
be found in [14, $\S$1.2].
Under condition (1.3a), we have $\rho(\boldsymbol{S})\leq 1$. If
$\rho(\boldsymbol{S})<1$ then $\boldsymbol{S}$ is absolutely stable; see,
e.g., [16]. If $\rho(\boldsymbol{S})=1$ then $\|\cdot\|_{P}$ is just an
extremal norm for $\boldsymbol{S}$ (see [4, 28, 9] for more details). In [16],
Gurvits proved that if $\boldsymbol{S}$ has a _polytope_ 111A norm $\|\cdot\|$
on $\mathbb{R}^{d}$ is called a (real) _polytope norm_ , if the unit sphere
$\mathbb{S}_{\|\cdot\|}=\left\\{x\in\mathbb{R}^{d}\colon\|x\|=1\right\\}$ is a
polytope in $\mathbb{R}^{d}$; see, e.g., [16]. extremal norm on
$\mathbb{R}^{d}$, then it has the spectral finiteness property. However, the
Lyapunov norm $\|\cdot\|_{P}$ defined as in (1.3b) does not need to be a
polytope norm, for example, $P=I_{d}$ the identity matrix which is associated
with the usual Euclidean norm $\|\cdot\|_{2}$ on $\mathbb{R}^{d}$.
As a consequence of the statements of Theorems C and D, we can easily obtain
the following spectral finiteness result.
###### Corollary.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$ satisfy
condition (1.3a) with $\rho(\boldsymbol{S})=1$. Then the following two
statements hold.
1. (1)
For the case $d=2$, there follows
$1=\max\left\\{\rho(S_{1}),\rho(S_{2}),\sqrt{\rho(S_{1}S_{2})}\right\\}$.
2. (2)
In the case $d=3$, there holds $1=\max\left\\{\sqrt[n]{\rho(S_{w_{n}}\dotsm
S_{w_{1}})}\,|\,w\in\\{1,2\\}^{n},n=1,2,3,4,5,6,8\right\\}$.
###### Proof.
Let $d=2$. Assume
$\max\left\\{\rho(S_{1}),\rho(S_{2}),\sqrt{\rho(S_{1}S_{2})}\right\\}<1$. Then
Theorem C implies that $\boldsymbol{S}$ is absolutely stable and so
$\rho(\boldsymbol{S})<1$, a contradiction. Similarly, we can prove the
statement in the case $d=3$. ∎
It should be pointed out that if $\rho(\boldsymbol{S})<1$, then
$\rho(\boldsymbol{S})$ does not need to be attained by these maximum values
defined as in the above corollary.
### 1.6 Outline
The paper is organized as follows. We shall prove Theorem A in Section 2. In
fact, we will prove a more general result (Theorem 2.3) than Theorem A there.
Since the above Splitting Theorem is very important for the proofs of Theorems
B, C, and D, we will give some notes on it in Section 3. Then, Theorem B will
be proved in Section 4. Section 5 will be devoted to proving Theorems C and D.
We will construct some examples in Section 6 to illustrate applications of our
Theorems stated here. Finally, we will end this paper with some concluding
remarks in Section 7.
## 2 Switched systems driven by nonchaotic switching signals
This section is devoted to proving Theorem A stated in Section 1.2 under the
guise of a more general result.
For any integer $2\leq K<+\infty$, we recall that a switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ is called
_nonchaotic_ , if to any sequence $\langle n_{i}\rangle_{i\geq
1}\nearrow+\infty$ and any $m\geq 1$ there corresponds some $\delta$ with
$2\leq\sigma\leq m+1$ such that for all $\ell_{0}\geq 1$, there exists an
$\ell\geq\ell_{0}$ so that $\sigma$ is constant restricted to some subinterval
of $[n_{\ell},n_{\ell}+m]$ of length greater than or equal to $\delta$.
Clearly, a constant switching signal $\sigma$ with $\sigma(n)\equiv k$ is
nonchaotic.
Then from definition, we can obtain the following lemma, which discovers the
essential property of a nonchaotic switching signal.
###### Lemma 2.1.
Let $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ be a
nonchaotic switching signal. Then, there exists some alphabet
$k\in\\{1,\dotsc,K\\}$ such that for any $\ell\geq 1$ and any
$\ell^{\prime}\geq 1$, there exists an $n_{\ell}\geq\ell^{\prime}$ so that
$\sigma_{n_{\ell}+1}=\dotsm=\sigma_{n_{\ell}+\ell}=k$.
###### Proof.
First, we can choose a sequence $\langle n_{i}\rangle_{i\geq
1}\nearrow+\infty$ and some $k\in\\{1,\dotsc,K\\}$, which are such that
$n_{i+1}-n_{i}\nearrow+\infty$ and $\sigma_{n_{i}}=k$ for all $i\geq 1$. Now
from the definition of nonchaotic property with $m=1$, it follows that we can
choose a subsequence of $\langle n_{i}\rangle_{i\geq 1}$, still write, without
loss of generality, as $\langle n_{i}\rangle_{i\geq 1}$, such that
$\sigma_{n_{i}}=\sigma_{n_{i}+1}=k$ for all $i\geq 1$. Repeating this
procedure for $\langle n_{i}+1\rangle_{i\geq 1}$ proves the statement. ∎
Lemma 2.1 shows that the $\omega$-limit set of a nonchaotic switching signal
contains at least one constant switching signal, under the sense of the
classical Markov shift transformation.
The following fact is a simple consequence of the classical Gel’fand spectral
formula, which will be refined in Section 5 for the Lyapunov norm
$\|\cdot\|_{P}$.
###### Lemma 2.2.
For any $A\in\mathbb{R}^{d\times d}$ and any matrix norm $\|\cdot\|$ on
$\mathbb{R}^{d\times d}$, if $\rho(A)<1$ then there is an integer $N\geq 1$
such that $\|A^{N}\|<1$.
For $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$, it
is said to be _product bounded_ , if there is a universal constant $\beta\geq
1$ such that
$\|S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}\|\leq\beta\quad\forall\sigma\in\varSigma_{\\!K}^{+}\textrm{
and }n\geq 1.$
This property does not depend upon the norm $\|\cdot\|$ used here.
If $\boldsymbol{S}$ is product bounded, then one always can choose a vector
norm $\|\cdot\|$ on $\mathbb{R}^{d}$ such that its induced operator norm
$\|\cdot\|$ on $\mathbb{R}^{d\times d}$ is such that $\|S_{k}\|\leq 1$ for all
$1\leq k\leq K$. Then the norm $\|\cdot\|$ on $\mathbb{R}^{d}$ acts as a
Lyapunov function for $\boldsymbol{S}$. However, there does not need to exist
a common, not strict in general, “quadratic” Lyapunov function/matrix $P$ as
in (1.3a). So, the following theorem is more general than Theorem A stated in
Section 1.2.
###### Theorem 2.3.
Let $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$ be
product bounded. If $\rho(S_{k})<1$ for all $1\leq k\leq K$, then
$\boldsymbol{S}$ is stable driven by any nonchaotic switching signals
$\sigma\in\varSigma_{\\!K}^{+}$.
###### Proof.
Without loss of generality, let $\|\cdot\|$ be a matrix norm on
$\mathbb{R}^{d\times d}$ such that $\|S_{k}\|\leq 1$ for all $1\leq k\leq K$.
Let $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ be an
arbitrary nonchaotic switching signal. Let $k$ be given by Lemma 2.1. Since
$\rho(S_{k})<1$, by Lemma 2.2 we have some $m\geq 1$ such that
$\|S_{k}^{m}\|<1$.
Thus, for an arbitrary $\varepsilon>0$ there is an $\ell\geq 1$ such that
$\|S_{k}^{m\ell}\|<\varepsilon$. From Lemma 2.1, it follows that as
$n\to+\infty$,
$\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}\|\leq\|S_{\sigma_{\\!n_{m\ell}+m\ell}}\dotsm
S_{\sigma_{\\!n_{m\ell}+1}}\|<\varepsilon.$
So, $\|S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}\|\to 0$ as $n\to+\infty$,
since $\varepsilon>0$ is arbitrary. This completes the proof of Theorem 2.3. ∎
Under condition (1.3a), the statement of Theorem 2.3 will be strengthened by
Corollary 5.3 in Section 5.
## 3 $\omega$-limit sets for product bounded systems
In this section, we will introduce $\omega$-limit sets and give some notes on
our splitting theorem stated in Section 1.3 that is very important for our
arguments in the next sections.
### 3.1 $\omega$-limit sets of a trajectory
We now consider the linear inclusion (1.1) generated by
$\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$ where
$2\leq K<+\infty$, as in Section 1. The classical one-sided Markov shift
transformation
$\theta\colon\varSigma_{\\!K}^{+}\rightarrow\varSigma_{\\!K}^{+}$
is defined as
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\mapsto\theta(\sigma)=(\sigma_{\\!n+1})_{n=1}^{+\infty}\qquad\forall\sigma\in\varSigma_{\\!K}^{+}.$
###### Definition 3.1 ([23, 24, 3]).
Let $x_{0}\in\mathbb{R}^{d}$ be an initial state and
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ a switching
signal. The set of all limit points of the sequence $\langle
S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}(x_{0})\rangle_{n=1}^{+\infty}$ in
$\mathbb{R}^{d}$ is called the $\omega$-limit set of $\boldsymbol{S}$ at the
input $(x_{0},\sigma)$. We denote it by $\omega(x_{0},\sigma)$ here.
It is easy to see that for any switching signal $\sigma$, the corresponding
switched system is asymptotically stable if and only if
$\omega(x_{0},\sigma)=\\{0\\}\,\forall x_{0}\in\mathbb{R}^{d}$. Thus we need
to consider the structure of $\omega(x_{0},\sigma)$ in order to study the
stability of the switched dynamics induced by $\boldsymbol{S}$.
###### Lemma 3.2.
Assume $\boldsymbol{S}$ is product bounded; that is, there is a matrix norm
$\|\cdot\|$ on $\mathbb{R}^{d\times d}$ such that $\|S_{k}\|\leq 1$ for all
$1\leq k\leq K$. Then, for any initial data $x_{0}\in\mathbb{R}^{d}$ and any
switching signal $\sigma$, the following two statements hold.
1. (1)
The $\omega$-limit set $\omega(x_{0},\sigma)$ is a compact subset contained in
a sphere $\\{x\in\mathbb{R}^{d};\ \|x\|=r\\}$, for some $r\geq 0$.
2. (2)
The trajectory $\langle x_{n}(x_{0},\sigma)\rangle_{n=1}^{+\infty}$ in
$\mathbb{R}^{d}$ tends to $0$ as $n\rightarrow\infty$ if and only if there
exists a subsequence of it which tends to $0$.
###### Proof.
Since the sequence $\langle\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|\rangle_{n=1}^{+\infty}$ is nonincreasing in
$\mathbb{R}$ for any $\sigma\in\varSigma_{\\!K}^{+}$, it is convergent as
$n\rightarrow+\infty$. Denoted by $r$ its limit, we have the statement (1).
The statement (2) follows immediately from the statement (1). This proves
Lemma 3.2. ∎
In the case (2) of this lemma, we call the orbit $\langle
x_{n}(x_{0},\sigma)\rangle_{n=1}^{+\infty}$ with initial value $x_{0}$ is
asymptotically stable.
We note here that Lemma 3.2 is actually proved in [24, 3] for the continuous-
time case, but [3] is under the condition that each member of $\boldsymbol{S}$
shares a common, not strict in general, quadratic Lyapunov function and [24]
under an additional assumption of “paracontraction” except the Lyapunov
function. In Section 3.3, we will consider the $\omega$-limit set of a matrix
trajectory $\langle S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}\rangle_{n=1}^{+\infty}$. In addition, in the continuous-time
case, $\omega(x_{0},\sigma)$ is a connected set. This is an important property
needed in [24, 3].
For a given switching signal, to consider the stability of the corresponding
switched system, we need to classify which kind of initial values in
$\mathbb{R}^{d}$ makes the corresponding orbits asymptotically stable. It is
difficult to have such classification for a general switching signal. In the
following, for the recurrent switching signal, we have a classification
result.
### 3.2 Decomposition for general extremal norm
In this subsection, we will introduce a preliminary splitting theorem of the
state space $\mathbb{R}^{d}$ which plays the key in our classification.
First, we recall from [21, 27] that for a topological dynamical system
$T\colon\Omega\rightarrow\Omega$ on a separable metrizable space $\Omega$, a
point $w\in\Omega$ is called “recurrent”, provided that one can find a
positive integer sequence $n_{i}\nearrow+\infty$ such that $T^{n_{i}}(w)\to w$
as $i\to+\infty$. And $w\in\Omega$ is said to be “weakly Birkhoff recurrent”
[29] (also see [10]), provided that for any $\varepsilon>0$, there exists an
integer $N_{\varepsilon}>1$ such that
$\sum_{i=0}^{jN_{\varepsilon}-1}I_{\mathbf{B}(w,\varepsilon)}(T^{i}(w))\geq
j\qquad\forall j\in\mathbb{N},$
where $I_{\mathbf{B}(w,\varepsilon)}\colon\Omega\to\\{0,1\\}$ is the
characteristic function of the open ball $\mathbf{B}(w,\varepsilon)$ of radius
$\varepsilon$ centered at $w$ in $\Omega$. We denote by $R(T)$ and $W(T)$,
respectively, the set of all recurrent points and weakly Birkhoff recurrent
points of $T$. It is easy to see that $R(T)$ and $W(T)$ both are invariant
under $T$ and $W(T)\subset R(T)$.
In the qualitative theory of ordinary differential equation, this type of
recurrent point is also called a “Poisson stable” motion, for instance, in
[21].
For the one-sided Markov shift $(\varSigma_{\\!K}^{+},\theta)$, it is easily
checked that every periodically switched signal is recurrent. And
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in R(\theta)$ means that there exists
a subsequence $n_{i}\nearrow+\infty$ such that
$\theta^{n_{i}}(\sigma)\to\sigma$ as $i\to+\infty$. This implies that
$S_{\sigma_{n_{i}+n}}\cdots S_{\sigma_{n_{i}+1}}\to S_{\sigma_{n}}\cdots
S_{\sigma_{1}}\quad\textrm{as }i\to+\infty$
for any $n\geq 1$. We should note that for any two finite-length words
$w\not=w^{\prime}$, the switching signal $\sigma=(w^{\prime},w,w,w,\dotsc)$ is
not recurrent.
For any function $A\colon\Omega\rightarrow\mathbb{R}^{d\times d}$, the cocycle
$A_{T}\colon\mathbb{N}\times\Omega\rightarrow\mathbb{R}^{d\times d}$ driven by
$T$ is defined as
$A_{T}(n,w)=A(T^{n-1}w)\cdots A(w)$
for any $n\geq 1$ and all $w\in\Omega$. Now, our basic decomposition theorem
can be stated as follows:
###### Theorem 3.3 ([13, Theorem B′]).
Let $T\colon\Omega\rightarrow\Omega$ be a continuous transformation of a
separable metrizable space $\Omega$. Let
$A\colon\Omega\rightarrow\mathbb{R}^{d\times d}$ be a continuous family of
matrices with the property that there exists a norm $\|\cdot\|$ such that
$\|A_{T}(n,w)\|\leq 1\quad\forall n\geq 1\textrm{ and }w\in\Omega.$
Then for any recurrent point $w$ of $T$, there corresponds a splitting of
$\mathbb{R}^{d}$ into subspaces
$\mathbb{R}^{d}=E^{s}(w)\oplus E^{c}(w),$
such that
$\displaystyle\lim_{n\to+\infty}\|A_{T}(n,w)(x)\|=0$ $\displaystyle\forall
x\in E^{s}(w)$ and $\displaystyle\|A_{T}(n,w)(x)\|=\|x\|\;\forall n\geq 1$
$\displaystyle\forall x\in E^{c}(w).$
Here $\|\cdot\|$ does not need to be a Lyapunov norm $\|\cdot\|_{P}$ as in
(1.3b) and further the central manifold $E^{c}(\sigma)$ is not necessarily
unique and invariant. Although $\|A_{T}(n,w)|E^{s}(w)\|$ converges to $0$, yet
$\|A_{T}(n,w)|E^{s}(w)\|$ does not need to converge exponentially fast, as is
shown by [13, Example 4.6].
However, under the assumptions of Theorem 3.3, if $w$ is a weakly Birkhoff
recurrent point of $T$, we have the following alternative results:
###### Theorem 3.4.
Let $T\colon\Omega\rightarrow\Omega$ be a continuous transformation of a
separable metrizable space $\Omega$. Let
$A\colon\Omega\rightarrow\mathbb{R}^{d\times d}$ be a continuous family of
matrices with the property that there exists a norm $\|\cdot\|$ such that
$\|A_{T}(n,w)\|\leq 1$ for all $n\geq 1$ and $w\in\Omega$. If $w\in\Omega$ is
a weakly Birkhoff recurrent point of $T$, Then either
$\displaystyle\|A_{T}(n,w)\|\xrightarrow[]{\textrm{exponentially fast}}0$
$\displaystyle\textrm{ as }n\to+\infty,$ or
$\displaystyle\|A_{T}(n,T^{i}(w))\|=1\;\forall i\geq 0$
$\displaystyle\textrm{for }n\geq 1.$
###### Proof.
If there exist $i\geq 0$ and $n\geq 1$ such that $\|A_{T}(n,T^{i}(w))\|<1$
then from $T^{i}(w)\in W(T)$ and [10, Theorem 2.4], it follows that
$\|A_{T}(m,T^{i}(w))\|\xrightarrow[]{\textrm{exponentially
fast}}0\qquad\textrm{ as }m\to+\infty.$
This completes the proof of Theorem 3.4. ∎
### 3.3 Decomposition under a weak Lyapunov matrix
For a recurrent switching signal $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$ of
$\boldsymbol{S}$, to consider its stability, it is essential to compute the
stable manifold $E^{s}(\sigma)$. From the proof of Theorem 3.3 presented in
[13], we know that $E^{s}(\sigma)$ is the kernel of an idempotent matrix that
is a limit point of $S_{\sigma_{n_{i}}}\cdots S_{\sigma_{\\!1}}$ with
$\theta^{n_{i}}(\sigma)\to\sigma$ as $i\to+\infty$.
However, in applications, it is not easy to identify which subsequence
$\langle n_{i}\rangle_{i\geq 1}$ with this property. In this subsection,
instead of the product boundedness, we assume the more strong condition (1.3a)
with induced norm $\|\cdot\|_{P}$ on $\mathbb{R}^{d}$.
In this case, we can calculate the stable manifold $E^{s}(\sigma)$ for any
switching signal $\sigma$ (not necessarily recurrent) of $\boldsymbol{S}$. To
do this end, we first consider the geometry of the limit sets
$\omega(x_{0},\sigma)$ of $\boldsymbol{S}$ driven by $\sigma$. For the similar
results in continuous-time switched linear systems, see [3].
For any switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$, on the other
hand, we will consider the sequence $\langle S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}\rangle_{n=1}^{+\infty}$ of matrices and let $\omega(\sigma)$
denote the set of all limit points of this sequence in $\mathbb{R}^{d\times
d}$.
###### Definition 3.5 ([28, 3]).
The set $\omega(\sigma)$ is called the $\omega$-limit set of $\boldsymbol{S}$
driven by $\sigma$, for any $\sigma\in\varSigma_{\\!K}^{+}$.
From condition (1.3a), it follows immediately that $\omega(\sigma)$ is non-
empty and compact. But it may not be a semigroup in the sense of matrix
multiplication when $\sigma$ is not a recurrent switching signal. We note that
if $\sigma\in R(\theta)$ then from the proof of [13, Theorem 4.2],
$\omega(\sigma)$ contains a nonempty compact semigroup and so there is an
idempotent element in $\omega(\sigma)$.
Parallel to Lemma 3.2, we can obtain the following result.
###### Lemma 3.6.
Under condition (1.3a), there follows the following statements.
1. (a)
For any switching signal $\sigma\in\varSigma_{\\!K}^{+}$ of $\boldsymbol{S}$,
it holds that
$\omega(\sigma)\subset\\{M\in\mathbb{R}^{d\times d}\colon\|M\|_{P}=r\\},$
for some constant $0\leq r\leq 1$; if $\sigma$ is further recurrent, then
either $r=0$ or $1$.
2. (b)
For any input $(x_{0},\sigma)\in\mathbb{R}^{d}\times\varSigma_{\\!K}^{+}$ for
$\boldsymbol{S}$, we have
$\omega(x_{0},\sigma)=\\{M(x_{0})\,|\,M\in\omega(\sigma)\\}=\omega(\sigma)(x_{0}).$
3. (c)
For any two elements $M$ and $N$ in $\omega(\sigma)$, it holds that
$M^{T}PM=N^{T}PN.$
We note that the continuous-time cases of the statements (b) and (c) of Lemma
3.6 have been proved in [3, $\S$3] using the polar decomposition of matrices.
We here present a simple treatment for the sake of self-closeness.
###### Proof.
We first note that from (1.3a) and (1.3b), it follows immediately that
$\|S_{k}\|_{P}\leq 1$ for all indices $1\leq k\leq K$.
For the statement (b), we let
$(x_{0},\sigma)\in\mathbb{R}^{d}\times\varSigma_{\\!K}^{+}$ be arbitrary. If
$M\in\omega(\sigma)$, it is clear that $M(x_{0})\in\omega(x_{0},\sigma)$.
Conversely, let $y\in\omega(x_{0},\sigma)$ be arbitrary. By the definition of
$\omega(x_{0},\sigma)$ there exists an increasing sequence $\\{n_{i}\\}$ such
that
$y=\lim_{i\to\infty}S_{\sigma_{n_{i}}}\cdots S_{\sigma_{\\!1}}(x_{0}).$
The product boundedness condition implies that the sequence $\langle
S_{\sigma_{n_{i}}}\cdots S_{\sigma_{\\!1}}\rangle_{i=1}^{+\infty}$ has a
convergent subsequence, whose limit element is denoted by $M$. Thus
$y=M(x_{0})$.
For the statement (c) of Lemma 3.6, let $M,\ N\in\omega(\sigma)$ be arbitrary.
As $\|S_{k}\|_{P}\leq 1$ for all $1\leq k\leq K$, from Lemma 3.2 we have
$\|M(x)\|_{P}=\|N(x)\|_{P}\quad\forall x\in\mathbb{R}^{d}.$
That is
$x^{T}(M^{T}PM-N^{T}PN)x=0\quad\forall x\in\mathbb{R}^{d}.$
It follows, from the symmetry of the matrix $M^{T}PM-N^{T}PN$, that
$M^{T}PM=N^{T}PN.$
This proves the statement (c) of Lemma 3.6.
Finally, the statement (a) of Lemma 3.6 comes from the statement (c) and
Theorem 3.3. In fact, let $M,\ N\in\omega(\sigma)$ be arbitrary. Then there
are vectors $x,y\in\mathbb{R}^{d}$ such that
$\|x\|_{P}=\|y\|_{P}=1,\quad\|M\|_{P}=\|M(x)\|_{P},\quad\textrm{and}\quad\|N\|_{P}=\|N(y)\|_{P}.$
So, from (c) it follows that
$\|M\|_{P}=\sqrt{x^{T}M^{T}PMx}=\sqrt{x^{T}N^{T}PNx}\leq\|N\|_{P}=\sqrt{y^{T}N^{T}PNy}=\sqrt{y^{T}M^{T}PMy}\leq\|M\|_{P}.$
This together with Theorem 3.3 proves the statement (a) of Lemma 3.6.
Thus the proof of Lemma 3.6 is completed. ∎
Let $M\in\omega(\sigma)$. Then $\sqrt{M^{T}PM}$ is a nonnegative-definite
matrix which does not depend on the choice of the matrix $M\in\omega(\sigma)$
by the statement (c) of Lemma 3.6 and is uniquely decided by the switching
signal $\sigma$. So, we write
$Q_{\sigma}=\sqrt{M^{T}PM}\quad\forall M\in\omega(\sigma).$ (3.1)
The continuous-time case of the following statement (1) of Proposition 3.7 has
already been proved by Balde and Jouan [3, Theorem 1] using the polar
decomposition of matrices.
###### Proposition 3.7.
Under condition (1.3a), for any switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$ of $\boldsymbol{S}$, there hold the
following two statements.
1. (1)
The switching signal $\sigma$ is asymptotically stable for $\boldsymbol{S}$;
that is,
$\lim_{n\to\infty}S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})=0\quad\forall x_{0}\in\mathbb{R}^{d},$
if and only if $Q_{\sigma}=0$;
2. (2)
If $Q_{\sigma}\neq 0$, then
$\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}=\|Q_{\sigma}(x_{0})\|_{2}\quad\forall
x_{0}\in\mathbb{R}^{d}.$
So, the stable manifold of $\boldsymbol{S}$ at $\sigma$ is such that
$E^{s}(\sigma)=$ kernel of $Q_{\sigma}$; that is
$\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}=0\quad\forall x_{0}\in E^{s}(\sigma).$
Here $\|\cdot\|_{2}$ denotes the Euclidean vector norm on $\mathbb{R}^{d}$.
###### Proof.
The statement (1) holds trivially from the statement (a) of Lemma 3.6 or from
the statement (2) to be proved soon. We next will prove the statement (2). For
that, let $Q_{\sigma}\neq 0$. For an arbitrary $x_{0}\in\mathbb{R}^{d}$, by
the definition of $Q_{\sigma}$ as in (3.1) there exists a subsequence $\langle
n_{i}\rangle_{i\geq 1}$ and some $M\in\omega(\sigma)$ such that
$\lim_{i\to+\infty}\|S_{\sigma_{n_{i}}}\cdots
S_{\sigma_{\\!1}}(x_{0})\|_{P}=\|M(x_{0})\|_{P}=\sqrt{x_{0}^{T}Q_{\sigma}^{2}x_{0}}=\sqrt{x_{0}^{T}Q_{\sigma}^{T}Q_{\sigma}x_{0}}=\|Q_{\sigma}(x_{0})\|_{2}.$
Therefore, by (1.3a) we have
$\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}=\|Q_{\sigma}(x_{0})\|_{2}.$
This thus completes the proof of Proposition 3.7. ∎
We note here that if $Q_{\sigma}$ is idempotent, then from Proposition 3.7 we
have $E^{c}(\sigma)=\mathrm{Im}(Q_{\sigma})$ and
$\mathbb{R}^{d}=E^{s}(\sigma)\oplus E^{c}(\sigma)$. Because in general there
lacks the recurrence of $\sigma$, one cannot define a central manifold
$E^{c}(\sigma)$ satisfying $\mathbb{R}^{d}=E^{s}(\sigma)\oplus E^{c}(\sigma)$
as done in Theorem 3.3. However, we will establish another type of splitting
theorem in the case $d=2$ for $\boldsymbol{S}$ driven by a general switching
signal, not necessarily recurrent.
For that, we first introduce several notations for the sake of our
convenience. For any given $A\in\mathbb{R}^{d\times d}$ and any vector norm
$\|\cdot\|$ on $\mathbb{R}^{d}$, write
$\|A\|_{\mathrm{co}}=\min\left\\{\|A(x)\|\colon x\in\mathbb{R}^{d}\textrm{
with }\|x\|=1\right\\},$ (3.2)
called the _co-norm_ (also _minimum norm_ in some literature) of $A$ under
$\|\cdot\|$.
###### Definition 3.8.
Under condition (1.3a), for any switching signal
$\sigma\in\varSigma_{\\!K}^{+}$ the numbers
$r_{E}(\sigma):=\|M\|_{P}\quad\textrm{and}\quad
r_{I}(\sigma):=\|M\|_{P,\mathrm{co}},$
for $M\in\omega(\sigma)$, are called the $\omega$-exterior and
$\omega$-interior radii of $\boldsymbol{S}$ driven by $\sigma$, respectively.
According to the statement (c) of Lemma 3.6, $r_{E}(\sigma)$ and
$r_{I}(\sigma)$ both are well defined independent of the choice of $M$.
Motivated by $\mathbb{E}^{c}(\sigma)$ in [10, $\S$5.2.2] and by
$\mathcal{V}_{i}$ in [3, Lemma 1], for any given $A\in\mathbb{R}^{d\times d}$
and any vector norm $\|\cdot\|$ on $\mathbb{R}^{d}$, let
$\displaystyle\mathbb{K}_{\|\cdot\|}(A)=\left\\{x\in\mathbb{R}^{d}\colon\|A(x)\|=\|A\|\cdot\|x\|\right\\}$
(3.3a) and
$\displaystyle\mathbb{K}_{\|\cdot\|_{\mathrm{co}}}(A)=\left\\{x\in\mathbb{R}^{d}\colon\|A(x)\|=\|A\|_{\mathrm{co}}\cdot\|x\|\right\\}.$
(3.3b)
Clearly, if $\ker(A)\not=\\{0\\}$, then $\|A\|_{\mathrm{co}}=0$ and so
$\mathbb{K}_{\|\cdot\|_{\mathrm{co}}}(A)=\ker(A)$ in this case.
For a general norm $\|\cdot\|$ on $\mathbb{R}^{d}$,
$\mathbb{K}_{\|\cdot\|_{\mathrm{co}}}(A)$ and $\mathbb{K}_{\|\cdot\|}(A)$ are
not necessarily linear subspaces. However, for a Lyapunov norm, we can obtain
the following.
###### Lemma 3.9.
Under the Lyapunov norm $\|\cdot\|_{P}$ as in (1.3b), there
$\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(A)$ and
$\mathbb{K}_{\|\cdot\|_{P}}(A)$ both are linear subspaces of $\mathbb{R}^{d}$
for any $A\in\mathbb{R}^{d\times d}$.
###### Proof.
Let $A\in\mathbb{R}^{d\times d}$ be arbitrarily given. By definitions, we have
$\begin{split}x\in\mathbb{K}_{\|\cdot\|_{P}}(A)&\Leftrightarrow
x^{T}\|A\|_{P}Px-x^{T}A^{T}PAx=0\\\ &\Leftrightarrow
x^{T}(\|A\|_{P}P-A^{T}PA)x=0\\\ &\Leftrightarrow\|G(x)\|_{2}=0\\\
&\Leftrightarrow x\in\ker(G).\end{split}$
Here $G^{2}=\|A\|_{P}P-A^{T}PA\geq 0$ is symmetric. Since $\ker(G)$, the
kernel of $x\mapsto Gx$, is a linear subspace of $\mathbb{R}^{d}$,
$\mathbb{K}_{\|\cdot\|_{P}}$ is also a linear subspace of $\mathbb{R}^{d}$.
On the other hand, for any $x\in\mathbb{R}^{d}$ we have
$\|A(x)\|_{P}\geq\|A\|_{P,\mathrm{co}}\cdot\|x\|_{P}$. So,
$x^{T}(A^{T}PA-\|A\|_{P,\mathrm{co}}P)x\geq 0\quad\forall x\in\mathbb{R}^{d}.$
Let $H^{2}=A^{T}PA-\|A\|_{P,\mathrm{co}}P$, which is symmetric and
nonnegative-definite. Then it holds that
$\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(A)=\ker(H)$, a linear subspace.
Thus, the proof of Lemma 3.9 is completed. ∎
Now, the improved splitting theorem can be stated as follows:
###### Theorem 3.10.
Let $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{2\times 2}$
satisfy condition (1.3a). Then, for any switching signal
$\sigma\in\varSigma_{\\!K}^{+}$, not necessarily recurrent, there exists a
splitting of $\mathbb{R}^{2}$ into subspaces
$\mathbb{R}^{2}=\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)\oplus\mathbb{K}_{\|\cdot\|_{P}}(\sigma)$
such that
$\displaystyle\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}=r_{I}\|x_{0}\|_{P}$ $\displaystyle\forall
x_{0}\in\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma),$
$\displaystyle\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}=r_{E}\|x_{0}\|_{P}$ $\displaystyle\forall
x_{0}\in\mathbb{K}_{\|\cdot\|_{P}}(\sigma),$ and $\displaystyle
r_{I}\|x_{0}\|_{P}<\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\|_{P}<r_{E}\|x_{0}\|_{P}$ $\displaystyle\forall
x_{0}\in\mathbb{R}^{2}-\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)\cup\mathbb{K}_{\|\cdot\|_{P}}(\sigma).$
###### Proof.
Let $r_{I}<r_{E}$ and $M\in\omega(\sigma)$. Define
$\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)=\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(M)$
and $\mathbb{K}_{\|\cdot\|_{P}}(\sigma)=\mathbb{K}_{\|\cdot\|_{P}}(M)$. From
the statement (2) of Proposition 3.7, it follows that
$\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)$ and
$\mathbb{K}_{\|\cdot\|_{P}}(\sigma)$ both are independent of the choice of
$M$. So,
$\mathbb{R}^{2}=\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)\oplus\mathbb{K}_{\|\cdot\|_{P}}(\sigma)$
from Lemma 3.9. We note that if $r_{I}=r_{E}$, then
$\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)=\mathbb{K}_{\|\cdot\|_{P}}(\sigma)=\mathbb{R}^{2}$.
This completes the proof of Theorem 3.10. ∎
In the case where $\sigma$ is recurrent, one can easily see that
$E^{s}(\sigma)=\mathbb{K}_{\|\cdot\|_{P,\mathrm{co}}}(\sigma)\quad\textrm{and}\quad
E^{c}(\sigma)=\mathbb{K}_{\|\cdot\|_{P}}(\sigma).$
## 4 Asymptotical stability under a weak Lyapunov matrix
In this section, we will discuss the stability of switched linear system with
a common, but not necessarily strict, quadratic Lyapunov function. In this
case, a criteria for stability is derived without computing the limit matrix
$Q_{\sigma}$ as in (3.1). We still assume $\boldsymbol{S}$ is composed of
finitely many subsystems. That is, $\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}$
with $2\leq K<+\infty$.
### 4.1 Stability of generic recurrent switching signals
Now for $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$, if
$\mathrm{Card}\\{n\,|\,\sigma_{\\!n}=k\\}=\infty$ for all $1\leq k\leq K$ then
$\sigma$ is called “generic.” Recall that a switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{\\!K}^{+}$ is said to be
_stable_ for $\boldsymbol{S}$ if
$\|S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}\|\rightarrow 0\quad\textrm{as
}n\to+\infty.$
(Note that the stability is independent of the chosen norm $\|\cdot\|$.) As is
known, a switching system which is asymptotically stable for all periodically
switching signals does not need to be asymptotically stable for all switching
signals in general [8, 7, 19, 17]. However we can obtain the following result.
###### Lemma 4.1.
If all recurrent switching signals are stable for $\boldsymbol{S}$, then it is
asymptotically stable driven by all switching signals in
$\varSigma_{\\!K}^{+}$.
###### Proof.
Since the set $R(\theta)$ of all recurrent switching signals has full measure
$1$ for all ergodic measures with respect to $(\varSigma_{\\!K}^{+},\theta)$,
the result follows from [11, Lemma 2.3]. ∎
By Lemma 4.1, to obtain the asymptotic stability of $\boldsymbol{S}$, it
suffices to prove that it is only asymptotically stable driven by all
recurrent switching signals.
In addition, we need the following lemma.
###### Lemma 4.2.
Under condition (1.3a), if $\|S_{k}\|_{P}=1$ and
$\mathbb{K}_{\|\cdot\|_{P}}(S_{k})$ is $S_{k}$-invariant, then
$\rho(S_{k})=1$.
Here $\mathbb{K}_{\|\cdot\|_{P}}(S_{k})$ is defined as in (3.3).
###### Proof.
The statement comes obviously from Lemma 3.9. ∎
In the following, for simplicity, we just consider a switched system which is
composed of two subsystems. That is, $K=2$.
###### Lemma 4.3.
Under condition (1.3a) with $K=2$ (i.e., $\boldsymbol{S}=\\{S_{1},S_{2}\\}$),
if $\|S_{1}\|_{P}=\|S_{2}\|_{P}=1$ and
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\},$
(4.1)
and at least one of them is invariant (i.e.,
$S_{1}(\mathbb{K}_{\|\cdot\|_{P}}(S_{1}))=\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$
or
$S_{2}(\mathbb{K}_{\|\cdot\|_{P}}(S_{2}))=\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$),
then every generic switching signal is stable for $\boldsymbol{S}$.
###### Proof.
Assume that $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ is $S_{1}$-invariant.
(Otherwise, if $\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ is $S_{2}$-invariant, the
proof is the same.) Let $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$ be a generic
switching signal; that is, in $(\sigma_{\\!n})_{n=1}^{+\infty}$, both $1$ and
$2$ appear infinitely many times. Then there exists a subsequence
$\\{\sigma_{n_{i}}\\}$ such that
$\sigma_{n_{i}}=1\quad\textrm{and}\quad\sigma_{n_{i}+1}=2\quad\forall i\geq
1.$
For a given initial value $x_{0}\in\mathbb{R}^{d}$, consider the subsequence
$\\{S_{\sigma_{n_{i}-1}}\dotsm S_{\sigma_{\\!1}}(x_{0})\\}_{i=1}^{+\infty}$.
By the assumption (1.3a), it has a convergent subsequence in $\mathbb{R}^{d}$.
Without loss of generality, we assume that
$S_{\sigma_{n_{i}-1}}\cdots S_{\sigma_{\\!1}}(x_{0})\rightarrow
y\in\mathbb{R}^{d}\quad\textrm{as }i\rightarrow+\infty.$
Thus
$S_{\sigma_{n_{i}}}S_{\sigma_{n_{i}-1}}\dotsm S_{\sigma_{\\!1}}(x_{0})\to
S_{1}(y),$ $S_{\sigma_{n_{i}+1}}S_{\sigma_{n_{i}}}S_{\sigma_{n_{i}-1}}\dotsm
S_{\sigma_{\\!1}}(x_{0})\to S_{2}S_{1}(y),$
as $i\to+\infty$. By the statement (1) of Lemma 3.2, we have
$\|S_{2}S_{1}(y)\|_{P}=\|S_{1}(y)\|_{P}=\|y\|_{P}.$
Thus $y\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ and
$S_{1}(y)\in\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$. From the $S_{1}$-invariance of
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ it follows that
$S_{1}(y)\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2}).$
So $S_{1}(y)=0$ and so is $y$. From the statement (2) of Lemma 3.2, we have
$S_{\sigma_{\\!n}}\cdots S_{\sigma_{\\!1}}(x_{0})\to 0\quad\textrm{as
}n\to+\infty.$
That is, $\sigma$ is a stable switching signal for $\boldsymbol{S}$. This
proves Lemma 4.3. ∎
Both $\mathbb{E}^{c}(\sigma)$ in [10, $\S$5.2.2] and $\mathcal{V}_{i}$ in [3,
Lemma 1] are invariant. Unfortunately, here our subspace
$\mathbb{K}_{\|\cdot\|_{P}}(S_{k})$ does not need to be $S_{k}$-invariant in
general. See Example 6.2 in Section 6. If this is the case, we still have,
however, the following criterion.
###### Theorem 4.4.
Under conditions (1.3a) and (4.1) with
$\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$, the following
two statements hold.
1. (1)
If $d=2$, then all generic recurrent switching signals
$\sigma\in\varSigma_{2}^{+}$, which satisfy
$\sigma\not=(\widehat{1,2},\widehat{1,2},\dotsc),$
are stable for $\boldsymbol{S}$;
2. (2)
if $d=3$, then all generic recurrent switching signals
$\sigma\in\varSigma_{2}^{+}$ such that
$\sigma\not=(w,w,w,\dotsc),\qquad\textrm{where
}w\in\\{(1,2),(2,1),(1,2,2),(2,1,1)\\},$
are stable for $\boldsymbol{S}$.
###### Proof.
First, if $\|S_{1}\|_{P}<1$ or $\|S_{2}\|_{P}<1$, then every generic switching
signal is stable for $\boldsymbol{S}$ and hence the statements (1) and (2)
trivially hold. So, we next assume $\|S_{1}\|_{P}=\|S_{2}\|_{P}=1$. This
implies that $\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{k})\geq 1$ for $k=1,2$.
For the statement (1) of Theorem 4.4, from (4.1) it follows that
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{k})=1$ for $k=1,2$. Let
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$ be a given generic recurrent
switching signal such that
$\sigma(\cdot+n)\not=(\widehat{1,2},\widehat{1,2},\dotsc,\widehat{1,2},\dotsc)\quad\forall
n\geq 1.$ (4.2)
From Theorem 3.3, there corresponds a splitting of $\mathbb{R}^{2}$ into
subspaces
$\mathbb{R}^{2}=E^{s}(\sigma)\oplus E^{c}(\sigma),$
such that
$\displaystyle\lim_{n\to+\infty}\|S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}(x_{0})\|_{P}=0$ $\displaystyle\forall x_{0}\in
E^{s}(\sigma)$ and $\displaystyle\|S_{\sigma_{\\!n}}\cdots
S_{\sigma_{\\!1}}(x_{0})\|_{P}=\|x_{0}\|_{P}\;\forall n\geq 1$
$\displaystyle\forall x\in E^{c}(\sigma).$
To prove that $\sigma$ is a stable switching signal for $\boldsymbol{S}$, we
need to prove that $E^{c}(\sigma)=\\{0\\}$. By the genericity of $\sigma$ and
(4.2), $\sigma$ must contains the word $(1,1,2)$ or $(2,2,1)$. Without loss of
generality, we assume that
$(\sigma_{\\!1},\sigma_{\\!2},\sigma_{\\!3})=(1,1,2).$
Thus we have
$\|S_{2}S_{1}S_{1}(x_{0})\|_{P}=\|S_{1}S_{1}(x_{0})\|_{P}=\|S_{1}(x_{0})\|_{P}=\|x_{0}\|_{P}\quad\forall
x_{0}\in E^{c}(\sigma)$
These imply that
$\\{x_{0},\ S_{1}(x_{0})\\}\subset\mathbb{K}_{\|\cdot\|_{P}}(S_{1}),\quad
S_{1}S_{1}(x_{0})\in\mathbb{K}_{\|\cdot\|_{P}}(S_{2}).$
Suppose that $x_{0}\neq 0$. It follows from
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=1$ that there exists a real number
$\lambda$ with $|\lambda|=1$ such that
$S_{1}(x_{0})=\lambda x_{0}.$
This means that $x_{0}$ is an eigenvector of $S_{1}$ with eigenvalue
$\lambda$. So
$S_{1}S_{1}(x_{0})=\lambda^{2}x_{0}\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1}).$
Therefore
$S_{1}S_{1}(x_{0})\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\}$.
Thus we have $S_{1}S_{1}(x_{0})=0$, which implies $x_{0}=0$, a contradiction.
Next, for proving the statement (2) of Theorem 4.4 that $d=3$, by (4.1), we
have that one of
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1}),\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ has
dimension $1$ and the other has dimension at least $1$ and at most $2$.
If both $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ and
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ have dimension $1$, then by the same
argument as in the statement (1), all generic recurrent switching signals
satisfying (4.2) are stable for $\boldsymbol{S}$.
Next, we assume that, for example,
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=1\quad\textrm{and}\quad\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=2.$
We claim that for any generic recurrent switching signal
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{2}^{+}$, if
$\sigma(\cdot+n)\not\in\left\\{(\widehat{1,2},\widehat{1,2},\dotsc,\widehat{1,2},\dotsc),(\widehat{1,2,2},\widehat{1,2,2},\dotsc,\widehat{1,2,2},\dotsc)\right\\}\quad\forall
n\geq 1.$ (4.3)
then $\sigma$ is stable for $\boldsymbol{S}$. There is no loss of generality
in assuming $\sigma_{\\!1}=1$; otherwise replacing $\sigma$ by
$\sigma(\cdot+n)$ for some $n\geq 1$. Then,
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=E^{c}(\sigma)\quad\textrm{if
}E^{c}(\sigma)\not=\\{0\\},$
where $E^{c}(\sigma)$ is given by Theorem 3.3.
Whenever the word $11$ appears in the sequence
$(\sigma_{\\!n})_{n=1}^{+\infty}$, $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ is
$S_{1}$-invariant. Then, Lemma 4.3 follows that $\sigma$ is stable for
$\boldsymbol{S}$. Next, we assume $11$ does not appear in
$(\sigma_{\\!n})_{n=1}^{+\infty}$. If $121$ appears in
$(\sigma_{\\!n})_{n=1}^{+\infty}$ then
$\widehat{12}\widehat{12}\widehat{12}\dotsm$ must appear too, a contradiction.
So, $121$ cannot appear in $(\sigma_{\\!n})_{n=1}^{+\infty}$. Then $122$ must
appear. If $1221$ appears in $(\sigma_{\\!n})_{n=1}^{+\infty}$ then
$\widehat{122}\widehat{122}\widehat{122}\dotsm$ must appear too, a
contradiction. Thus, the word $1222$ must appear in
$(\sigma_{\\!n})_{n=1}^{+\infty}$.
When $\sigma$ contains the word $(2,2,2,1)$, assume that, for example,
$(\sigma_{\\!{n+1}},\sigma_{\\!{n+2}},\sigma_{\\!{n+3}},\sigma_{\\!{n+4}})=(2,2,2,1).$
Then we have
$\|S_{1}S_{2}S_{2}S_{2}(x_{0})\|_{P}=\|S_{2}S_{2}S_{2}(x_{0})\|_{P}=\|S_{2}S_{2}(x_{0})\|_{P}=\|S_{2}(x_{0})\|_{P}=\|x_{0}\|_{P}\quad\forall
x_{0}\in E^{c}(\sigma(\cdot+n)),$
which show that for all $x_{0}\in E^{c}(\sigma(\cdot+n))$,
$\\{x_{0},S_{2}(x_{0}),S_{2}S_{2}(x_{0})\\}\subset\mathbb{K}_{\|\cdot\|_{P}}(S_{2}),\quad
S_{2}S_{2}S_{2}(x_{0})\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1}).$
If $x_{0}$ and $S_{2}(x_{0})$ are linear dependent, that is,
$S_{2}(x_{0})=\lambda x_{0},$
for some $\lambda$ with $|\lambda|=1$, then
$S_{2}S_{2}S_{2}(x_{0})=\lambda^{3}x_{0}\in\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$.
So
$S_{2}S_{2}S_{2}(x_{0})\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\},$
which implies that $x_{0}=0$. On the other hand ,if $x_{0}$ and $S_{2}(x_{0})$
are linear independent, then
$S_{2}S_{2}(x_{0})=\lambda x_{0}+\alpha S_{2}(x_{0}),$
for some $\lambda$ and $\alpha$, since
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=2$. Thus $S_{2}S_{2}S_{2}(x_{0})$ is a
linear combination of $S_{2}(x_{0})$ and $S_{2}S_{2}(x_{0})$. So it is also in
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$. Therefore
$S_{2}S_{2}S_{2}(x_{0})\in\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\cap\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=\\{0\\},$
which shows $x_{0}=0$. Thus $E^{c}(\sigma(\cdot+n))=\\{0\\}$ and then
$E^{c}(\sigma)=\\{0\\}$.
Similarly, when $\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=2$ and
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=1$, we can prove that all generic
recurrent switching signals, but the following four periodic switching signals
$(1,1,1,\dotsc),\ (2,2,2,\dotsc),\ (\widehat{2,1},\widehat{2,1},\dotsc),\
(\widehat{2,1,1},\widehat{2,1,1},\dotsc),$
are stable for $\boldsymbol{S}$.
This completes the proof of Theorem 4.4. ∎
We have the following remarks on Theorem 4.4.
###### Remark 1.
Similarly, we can consider a switched linear system composed of two subsystems
on $\mathbb{R}^{d}$ with $d\geq 4$. In this case, under the assumptions (1.3a)
and (4.1), if either $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ or
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ has dimension $1$, then all generic
recurrent switching signals but finitely many periodic signals are stable for
$\boldsymbol{S}$.
###### Remark 2.
Under the assumptions on Theorem 4.4, in order to obtain the stability for all
recurrent switching signals, we just need to check finitely many periodic
signals to see whether they are stable for $\boldsymbol{S}$.
###### Remark 3.
Theorem 4.4 suggests a easy computable sufficient condition of asymptotically
stable for switched linear systems which are composed of two subsystems. In
fact, Remark 2 provides a direct way to check the stability of all recurrent
signals, which implies the asymptotically stable of the systems by Lemma 4.1.
We can also discuss the stability of switched linear systems composed of
finite many subsystems similarly. But it is troublesome to formulate the
corresponding assumptions. Here we will give an example to illustrate such
conditions in Section 6.
### 4.2 Almost sure stability
Let $(\varSigma_{\\!K}^{+},\mathscr{B})$ be the Borel $\sigma$-field of the
space $\varSigma_{\\!K}^{+}$ and then the one-sided Markov shift map
$\theta\colon\sigma(\cdot)\mapsto\sigma(\cdot+1)$ is measurable. A Borel
probability measure $\mathbb{P}$ on $\varSigma_{\\!K}^{+}$ is said to be
_$\theta$ -invariant_, if $\mathbb{P}=\mathbb{P}\circ\theta^{-1}$, i.e.
$\mathbb{P}(B)=\mathbb{P}(\theta^{-1}(B))$ for all $B\in\mathscr{B}$. A
$\theta$-invariant probability measure $\mathbb{P}$ is called _$\theta$
-ergodic_, provided that for $B\in\mathscr{B}$,
$\mathbb{P}\left((B\setminus\theta^{-1}(B))\cup(\theta^{-1}(B)\setminus
B)\right)=0$ implies $\mathbb{P}(B)=1$ or $0$.
An ergodic measure $\mathbb{P}$ is called _non-atomic_ , if every singleton
set $\\{\sigma\\}$ has $\mathbb{P}$-measure $0$.
Using Theorem 4.4, we can easily prove Theorem B stated in Section 1.4.
###### Proof of Theorem B.
Let $\mathbb{P}$ be an arbitrary non-atomic $\theta$-ergodic measure on
$\varSigma_{2}^{+}$. Then from the Poincaré recurrence theorem (see, e.g.,
[27, Theorem 1.4]), it follows that $\mathbb{P}$-a.e.
$\sigma\in\varSigma_{2}^{+}$ are recurrent. In addition, sine $\mathbb{P}$ is
non-atomic, we obtain that $\mathbb{P}$-a.e. $\sigma\in\varSigma_{2}^{+}$ are
non-periodic and generic. This completes the proof of Theorem B from Theorem
4.4. ∎
We note that in the proof of Theorem B presented above, the deduction of the
genericity of $\sigma$ needs the assumption $K=2$.
## 5 Absolute stability of a pair of matrices with a weak Lyapunov matrix
We now deal with the case
$\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d}$, where $S_{1}$
and $S_{2}$ both are stable and share a common, but not necessarily strict,
quadratic Lyapunov function. For any $A\in\mathbb{R}^{d\times d}$, we denote
by $\rho(A)$ the spectral radius of $A$.
Our first absolute stability result Theorem C is restated as follows:
###### Theorem 5.1.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{2\times 2}$ satisfy
condition (1.3a). Then, $\boldsymbol{S}$ is absolutely stable (i.e.,
$\|S_{\sigma_{\\!n}}\dotsc S_{\sigma_{\\!1}}\|\to 0$ as $n\to+\infty$, for all
switching signals $\sigma\in\varSigma_{2}^{+}$) if and only if there holds
that $\rho(S_{1})<1,\rho(S_{2})<1$, and $\rho(S_{1}S_{2})<1$.
###### Proof.
We only need to prove the sufficiency. Let $\rho(S_{1})<1,\rho(S_{2})<1$, and
$\rho(S_{1}S_{2})<1$. Let
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{2}^{+}$ be an arbitrary
recurrent switching signal. Clearly, if $\sigma$ is not generic, then it is
stable for $\boldsymbol{S}$. So we assume $\sigma$ is generic and recurrent.
Then, from Theorem 3.3 there exists a splitting of $\mathbb{R}^{2}$ into
subspaces:
$\mathbb{R}^{2}=E^{s}(\sigma)\oplus E^{c}(\sigma).$
If $\dim E^{c}(\sigma)=0$, then $\sigma$ is stable for $\boldsymbol{S}$; and
if $\dim E^{c}(\sigma)=2$ then either $\rho(S_{1})=1$ or $\rho(S_{2})=1$, a
contradiction. We now assume $\dim E^{c}(\sigma)=1$.
Then, $\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=1$ and
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=1$. It might be assumed, without loss
of generality, that $\sigma_{\\!1}=1$ and then we have
$\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=E^{c}(\sigma)$. From this, we see
$\sigma_{2}=2,\;\sigma_{3}=1,\;\dotsc,\;\sigma_{2n}=2,\;\sigma_{2n+1}=1,\;\dotsc.$
This contradicts $\rho(S_{1}S_{2})=\rho(S_{2}S_{1})<1$.
Therefore, $E^{c}(\sigma)=\\{0\\}$ and $\boldsymbol{S}$ is absolutely stable
from Lemma 4.1. ∎
So, Theorem C is proved.
Next, we need a simple fact for considering higher dimensional cases.
###### Lemma 5.2 ([26, Corollary]).
Let $A\in\mathbb{R}^{d\times d}$ be a stable matrix (i.e., $\rho(A)<1$) such
that
$D-A^{T}DA\geq 0$
for some symmetric, positive-definite matrix $D$. Then
$D-(A^{d})^{T}DA^{d}>0$.
This lemma refines Lemma 3.2. From it, we can obtain a simple result which
improves the statement of Theorem A in the case of $d=2$ and $K=2$.
###### Corollary 5.3.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{2\times 2}$ satisfy
condition (1.3a). If $\rho(S_{1})<1$ and $\rho(S_{2})<1$, then for any
$\theta$-ergodic probability measure $\mathbb{P}$ on $\varSigma_{2}^{+}$,
$\boldsymbol{S}$ is stable driven by $\mathbb{P}$-a.e.
$\sigma\in\varSigma_{2}^{+}$ as long as $\mathbb{P}$ satisfies
$\mathbb{P}(\\{(12,12,12,\dotsc),(21,21,21,\dotsc)\\})=0$.
###### Proof.
Since $\mathbb{P}$ is ergodic and
$\mathbb{P}(\\{(12,12,12,\dotsc),(21,21,21,\dotsc)\\})=0$, we have
$\mathbb{P}(\\{\sigma\in\varSigma_{2}^{+}\,|\,\sigma(\cdot+n)=(12,12,12,\dotsc)\textrm{
or }(21,21,21,\dotsc)\textrm{ for some }n\geq 1\\})=0.$
Now, let $\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{2}^{+}$ be
arbitrary. Then, $\sigma$ can consist of the following $2$-length words:
$11,\;22,\;12,\;21.$
If $11$ (or $22$) appears infinitely many times in
$(\sigma_{\\!n})_{n=1}^{+\infty}$, then from Lemma 5.2 it follows that
$\boldsymbol{S}$ is stable driven by $\sigma$. Next, assume $11$ and $22$ both
only appear finitely many times in $(\sigma_{\\!n})_{n=1}^{+\infty}$ and let
$a=12$ and $b=21$. Then, one can find some $N\geq 1$ such that
$\sigma(\cdot+N)=(a,a,a,\dotsc).$
Note here that if $ab$ appears $m$ times in $(\sigma_{\\!n})_{n=1}^{+\infty}$
then $22$ must appear $m$ times; if $ba$ appears $m$ times in
$(\sigma_{\\!n})_{n=1}^{+\infty}$ then $11$ must appear $m$ times. So,
$\boldsymbol{S}$ is stable driven by $\mathbb{P}$-a.e.
$\sigma\in\varSigma_{2}^{+}$.
This completes the proof of Corollary 5.3. ∎
The condition $\mathbb{P}(\\{(12,12,12,\dotsc),(21,21,21,\dotsc)\\})=0$ means
that $\mathbb{P}$ is not distributed on the periodic orbit of the one-sided
Markov shift $(\varSigma_{\\!K}^{+},\theta)$:
$\\{(12,12,\dotsc),\;(21,21,\dotsc)\\}.$
This corollary shows that $\boldsymbol{S}$ is “completely” almost sure stable
up to only one ergodic measure supported on a periodic orbit generated by the
word $12$.
In addition, Theorem C can be directly deduced from Corollary 5.3 and Lemma
4.1.
For the sake of our convenience, we now restate our second absolute stability
result Theorem D as follows:
###### Theorem 5.4.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{3\times 3}$ satisfy
condition (1.3a). Then, $\boldsymbol{S}$ is absolutely stable if and only if
there holds the following conditions:
$\rho(S_{1})<1,\quad\rho(S_{2})<1,$ $None$ $\rho(S_{1}S_{2})<1,$ $None$
$\rho(S_{w_{1}}S_{w_{2}}S_{w_{3}})<1\quad\forall(w_{1},w_{2},w_{3})\in\\{1,2\\}^{3},$
$None$ $\rho(S_{w_{1}}\dotsm
S_{w_{4}})<1\quad\forall(w_{1},\dotsc,w_{4})\in\\{1,2\\}^{4},$ $None$
$\rho(S_{w_{1}}\dotsm
S_{w_{5}})<1\quad\forall(w_{1},\dotsc,w_{5})\in\\{1,2\\}^{5},$ $None$
$\rho(S_{w_{1}}\dotsm
S_{w_{6}})<1\quad\forall(w_{1},\dotsc,w_{6})\in\\{1,2\\}^{6},$ $None$
$\rho(S_{w_{1}}\dotsm
S_{w_{8}})<1\quad\forall(w_{1},\dotsc,w_{8})\in\\{1,2\\}^{8}.$ $None$
We note here that it is somewhat surprising that we do not need to consider
the words of length $7$.
###### Proof.
We need to consider only the sufficiency. Let conditions (C1) – (C8) all hold.
According to Lemma 4.1, we let
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}\in\varSigma_{2}^{+}$ be an arbitrary
recurrent switching signal. There is no loss of generality in assuming
$\sigma_{\\!1}=1$.
It is easily seen that $0\leq\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})\leq 2$ and
$0\leq\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})\leq 2$ by condition (C1). Then
from Theorem 3.3 with $\|\cdot\|=\|\cdot\|_{P}$, there exists a splitting of
$\mathbb{R}^{3}$ into subspaces:
$\mathbb{R}^{3}=E^{s}(\sigma)\oplus E^{c}(\sigma)\quad\textrm{such that }\dim
E^{c}(\sigma)\leq\dim\mathcal{K}_{k,\|\cdot\|_{P}}\textrm{ for }k=1,2.$
There is only one of the following three cases occurs.
* •
$\dim E^{c}(\sigma)=2$;
* •
$\dim E^{c}(\sigma)=1$;
* •
$\dim E^{c}(\sigma)=0$.
Clearly, if $\sigma$ is not generic, then it is stable for $\boldsymbol{S}$.
So we let $\sigma$ be generic in what follows. We also note that
$E^{c}(\sigma)\subseteq\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$.
Case (a): Let $\dim E^{c}(\sigma)=2$. Then
$\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=\dim\mathbb{K}_{\|\cdot\|_{P}}(S_{2})=2$
and further we have $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})=E^{c}(\sigma)$. If
$\sigma_{2}=1$ then it follows that $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ is
$S_{1}$-invariant and so $\rho(S_{1})=1$ by Lemma 4.2, a contradiction. Thus,
$\sigma_{2}=2$. If $\sigma_{3}=2$ it follows that
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ is $S_{2}$-invariant and so
$\rho(S_{2})=1$ by Lemma 4.2, also a contradiction. So, $\sigma_{3}=1$.
Repeating this, we can see $\sigma=(1,2,1,2,1,2,\dotsc)$, a contradiction to
condition (C2). Thus, the case (a) cannot occur.
Case (b): Let $\dim E^{c}(\sigma)=1$. (This is the most complex case needed to
discussion.) We first claim that $\sigma$ does not contain any one of the
following two words:
$(1,1,1),\;(2,2,2).$
In fact, without loss of generality, we let
$(\sigma_{n+1},\sigma_{n+2},\sigma_{n+3})=(2,2,2)$. Choose a vector $x\in
E^{c}(\sigma)$ with $\|x\|_{P}=1$. Then, $v:=S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}(x)\in\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ with $\|v\|_{P}=1$.
Moreover, $S_{2}(v)$ and $S_{2}(S_{2}(v))$ both belong to
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ such that with
$\|S_{2}(v)\|_{P}=\|S_{2}(S_{2}(v))\|_{P}=1$. Since $S_{2}(v)\not=\pm v$
(otherwise $\rho(S_{2})=1$), we see $\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ is
$S_{2}$-invariant. So, $\rho(S_{2})=1$ by Lemma 4.2, a contradiction to
condition (C1).
Secondly, we claim that if $\sigma$ contains the word of the form
$(1,1,w_{1},\dotsc,w_{m},1,1)$ then
$\rho(S_{w_{m}}\dotsc,S_{w_{1}}S_{1}S_{1})=1;$
and if $\sigma$ contains the word of the form $(2,2,w_{1},\dotsc,w_{m},2,2)$
then
$\rho(S_{w_{m}}\dotsc,S_{w_{1}}S_{2}S_{2})=1.$
In fact, without loss of generality, we assume that
$\sigma=(1,\sigma_{2},\dotsc,\sigma_{\\!n},2,2,w_{1},\dotsc,w_{m},2,2,\dotsc).$
Then, take arbitrarily a vector $x\in E^{c}(\sigma)$ with $\|x\|_{P}=1$ and
write $v_{n}:=S_{\sigma_{\\!n}}\dotsm S_{\sigma_{\\!1}}(x)$. So, $v_{n}$ and
$S_{2}(v_{n})$ both belong to $\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ such that
$\|v_{n}\|_{p}=\|S_{2}(v_{n})\|_{P}=1$. On the other hand,
$v^{\prime}:=S_{w_{m}}\dotsm S_{w_{1}}S_{2}S_{2}(v_{n})$ and
$S_{2}(v^{\prime})$ both belong to $\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ with
$\|v^{\prime}\|_{P}=\|S_{2}(v^{\prime})\|_{P}=1$. If $v_{n}\not=\pm
v^{\prime}$ then $\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ is $S_{2}$-invariant and
so $\rho(S_{2})=1$ by Lemma 4.2, a contradiction to condition (C1). Thus, we
have $v_{n}=\pm v^{\prime}$ and then
$\rho(S_{w_{m}}\dotsc,S_{w_{1}}S_{2}S_{2})=1$.
Thirdly, we show the case (b), i.e., $\dim E^{c}(\sigma)=1$, does not occur
too. In fact, from the above claims, it follows that
$\sigma=(\sigma_{\\!n})_{n=1}^{+\infty}$ only possesses the following forms:
$1\to\begin{cases}12\to\dotsm~{}\textrm{(case (A))}\\\
2\to\begin{cases}1\to\dotsm~{}\textrm{(case (B))}\\\
21\to\dotsm~{}\textrm{(case (C))}\end{cases}\end{cases}$ (5.1)
Here and in the sequel, “$a\to b$” means that $b$ follows $a$; i.e.,
$\sigma_{\\!n}=a$ and $\sigma_{\\!{n+1}}=b$ for some $n$. For example, in the
above figure, “$1\to 2\to 21$” means $\sigma_{\\!1}=1,\sigma_{\\!2}=2$ and
$(\sigma_{\\!3},\sigma_{\\!4})=(2,1)$. In addition, in the following three
figures, the symbol “$\times$” means “This case does not happen.” For the case
(A) in the above figure (5.1), we have the following:
$112\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C3)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C5)})\\\
2~{}(\times\textrm{ by (C2)}\textrm{ and Lemma}~{}\ref{lemV.2})\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C8)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by (C2) and Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C4)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by (C2) and Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}$
Thus,
$(\sigma_{\\!1},\sigma_{\\!2},\sigma_{\\!3})\not=(1,1,2)$
and then
$(\sigma_{\\!1},\sigma_{\\!2})\not=(1,1).$ (5.2)
For the case (C) in the figure (5.1), we have
$1221\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C3)})\\\ 2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C5)})\\\
2~{}(\times\textrm{ by (C6)})\end{cases}\\\ 2~{}(\times\textrm{ by
(C6)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by (C4)})\end{cases}\\\
2\to\begin{cases}1\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{
by (C3)})\\\ 2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C5)})\\\ 2~{}(\times\textrm{ Lemma~{}\ref{lemV.2}})\end{cases}\\\
2~{}(\times\textrm{ by (C8)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C6)})\end{cases}\\\ 2\to\begin{cases}1~{}(\times\textrm{ by
Lemma~{}\ref{lemV.2}})\\\ 2~{}(\times\textrm{ by
(C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}$
Thus
$(\sigma_{\\!1},\sigma_{\\!2},\sigma_{\\!3},\sigma_{\\!4})\not=(1,2,2,1)$
and then
$(\sigma_{\\!1},\sigma_{\\!2},\sigma_{\\!3})\not=(1,2,2).$ (5.3)
Finally, for the case (B) in the figure (5.1),
$121\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C3)})\\\ 2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C5)})\\\
2~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C8)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C4)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\\\
2\to\begin{cases}1\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{
by (C3)})\\\ 2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C5)})\\\ 2~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C8)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\end{cases}\\\
21\to\begin{cases}1~{}(\times\textrm{ by (C4)})\\\
2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by (C6)})\\\
2\to\begin{cases}1~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\\\
2~{}(\times\textrm{ by (C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
Lemma~{}\ref{lemV.2}})\end{cases}\\\
21\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C3)})\\\ 21\to\begin{cases}1~{}(\times\textrm{ by (C5)})\\\
2~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\end{cases}\end{cases}\\\
2~{}(\times\textrm{ by (C4)})\end{cases}\\\
2\to\begin{cases}1\to\begin{cases}12\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{
by (C3)})\\\ 2\to\begin{cases}1\to\begin{cases}1~{}(\times\textrm{ by
(C5)})\\\ 2~{}(\times\textrm{ by Lemma~{}\ref{lemV.2}})\end{cases}\\\
2~{}(\times\textrm{ by (C8)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C6)})\end{cases}\\\ 2\to\begin{cases}1~{}(\times\textrm{ by
Lemma~{}\ref{lemV.2}})\\\ 2~{}(\times\textrm{ by
(C5)})\end{cases}\end{cases}\\\ 2~{}(\times\textrm{ by
(C3)})\end{cases}\end{cases}\end{cases}\end{cases}$
Thus, $(\sigma_{\\!1},\sigma_{\\!2},\sigma_{\\!3})\not=(1,2,1)$. Further, from
(5.3) it follows $(\sigma_{\\!1},\sigma_{\\!2})\not=(1,2)$. This together with
(5.2) implies that $(\sigma_{\\!1},\sigma_{\\!2})\not\in\\{(1,1),(1,2)\\}$, a
contradiction.
So, $\dim E^{c}(\sigma)\not=1$ and hence case (b) does not occur.
Therefore, $\dim E^{c}(\sigma)=0$. This implies that $\sigma$ is stable for
$\boldsymbol{S}$. Therefore $\boldsymbol{S}$ is absolutely stable from Lemma
4.1.
This completes the proof of Theorem 5.4. ∎
## 6 Examples
We in this section shall give several examples to illustrate applications of
our results. In what follows, let $\|\cdot\|_{2}$ be the usual Euclidean norm
on $\mathbb{R}^{d}$; that is, $P=I_{d}$ in (1.3b).
First, a very simple example is the following.
###### Example 6.1.
Let $\boldsymbol{S}=\\{S_{1},\ S_{2}\\}$ with
$S_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&\alpha\end{array}\right),\qquad
S_{2}=\left(\begin{array}[]{cc}\beta&0\\\ 0&1\end{array}\right),$
where $|\alpha|<1,\ \ |\beta|<1$. It is easy to see that
$\|S_{1}\|_{2}=\|S_{2}\|_{2}=1,$
and that
$\mathbb{K}_{\|\cdot\|_{2}}(S_{1})=\\{(x_{1},0)^{T}\in\mathbb{R}^{2}\mid
x_{1}\in\mathbb{R}\\},\
\mathbb{K}_{2,\|\cdot\|_{2}}(S_{2})=\\{(0,x_{2})^{T}\in\mathbb{R}^{2}\mid
x_{2}\in\mathbb{R}\\}$. So, we can obtain that
$\mathbb{K}_{\|\cdot\|_{2}}(S_{1})\bigcap\mathbb{K}_{\|\cdot\|_{2}}(S_{2})=\\{0\\}$
and $\mathbb{K}_{\|\cdot\|_{2}}(S_{k})$ is $S_{k}$-invariant. Thus the
switched linear system $\boldsymbol{S}$ is asymptotically stable for all
switching signals in which each $k$ in $\\{1,2\\}$ is stable by Lemma 4.3.
Also, from Theorem 4.4, it follows that all recurrent signals but the fixed
signals $(1,1,1,\dotsc)$ and $(2,2,2,\dotsc)$ are stable for $\boldsymbol{S}$.
We note here that the periodic switching signal $(1,2,1,2,\dotsc)$ is stable
for $\boldsymbol{S}$.
A more interesting example is the following
###### Example 6.2.
Let $\boldsymbol{S}=\\{S_{1},\ S_{2}\\}$ with
$S_{1}=\alpha\left(\begin{array}[]{cc}1&0\\\ 1&1\end{array}\right),\quad
S_{2}=\beta\left(\begin{array}[]{cc}1&\frac{3}{2}\\\ 0&1\end{array}\right),$
where
$\alpha=\sqrt{\frac{3-\sqrt{5}}{2}},\quad\beta=\frac{1}{2}.$
Then, $\|S_{1}\|_{2}=\|S_{2}\|_{2}=1$. A direct computation shows that
$\mathbb{K}_{\|\cdot\|_{2}}(S_{1})=\left\\{(x_{1},x_{2})^{T}\in\mathbb{R}^{2}\mid
x_{1}=\frac{\sqrt{5}+1}{2}x_{2}\right\\}$
$\mathbb{K}_{\|\cdot\|_{2}}(S_{2})=\left\\{(x_{1},x_{2})^{T}\in\mathbb{R}^{2}\mid
x_{2}=2x_{1}\right\\}.$
Thus
$\mathbb{K}_{\|\cdot\|_{2}}(S_{1})\bigcap\mathbb{K}_{\|\cdot\|_{2}}(S_{2})=\\{0\\}$.
But they are not invariant. Thus $\boldsymbol{S}$ is asymptotically stable for
all generic recurrent switching signals but the periodic signal
$(1,2,1,2,\dotsc)$ by Theorem 4.4. Note that the two subsystems themselves are
asymptotically stable.
Next, we give an example which is the discretization of the switched linear
continuous system borrowed from [3].
###### Example 6.3.
Let $\boldsymbol{S}=\\{S_{1},\ S_{2},S_{3}\\}$ with
$S_{1}=\left(\begin{array}[]{ccc}\alpha&0&0\\\ 0&0&-1\\\
0&1&0\end{array}\right),\qquad S_{2}=\left(\begin{array}[]{ccc}\alpha&0&0\\\
0&\alpha&0\\\ 0&0&1\end{array}\right),\qquad
S_{3}=\left(\begin{array}[]{ccc}\alpha&0&0\\\ 0&1&0\\\
0&0&\alpha\end{array}\right),$
where $|\alpha|<1$. It is easy to see that
$\|S_{1}\|_{2}=\|S_{2}\|_{2}=\|S_{3}\|_{2}=1$ and
$\displaystyle\mathbb{K}_{\|\cdot\|_{2}}(S_{1})$
$\displaystyle=\left\\{(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}\mid
x_{1}=0\right\\},$ $\displaystyle\mathbb{K}_{\|\cdot\|_{2}}(S_{2})$
$\displaystyle=\left\\{(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}\mid
x_{1}=x_{2}=0\right\\},$ $\displaystyle\mathbb{K}_{\|\cdot\|_{2}}(S_{3})$
$\displaystyle=\left\\{(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}\mid
x_{1}=x_{3}=0\right\\}.$
Since
$\mathbb{K}_{\|\cdot\|_{2}}(S_{2})\bigcap\mathbb{K}_{\|\cdot\|_{2}}(S_{3})=\\{0\\}$
and they are invariant respect to $S_{2}$ and $S_{3}$, respectively, we have
that any generic switching signal in which either the word $(2,3)$ or the
$(3,2)$ appears infinitely many times are stable by Lemma 4.3. For the any
other generic switching signals $\sigma=(\sigma_{\\!1},\sigma_{\\!2},\dotsc)$,
that is, in which both the word $(2,3)$ and the $(3,2)$ appear at most finite
many times, the matrix $Q_{\sigma}$ defined in (3.1) is
$Q_{\sigma}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&\alpha^{k_{0}}&0\\\
0&0&\alpha^{j_{0}}\end{array}\right).$
for some nonnegative integers $k_{0}$ and $j_{0}$ which depend on the times of
appearance of $(2,3)$ and $(3,2)$ in $\sigma$. Thus by Proposition 3.7, we
have
$\displaystyle\lim_{n\to\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}x\|_{2}=0,$ $\displaystyle\forall x\in\\{(x_{1},0,0)^{T}\mid
x_{1}\in\mathbb{R}\\}=\ker(Q_{\sigma}),$
$\displaystyle\lim_{n\to\infty}\|S_{\sigma_{\\!n}}\dotsm
S_{\sigma_{\\!1}}x\|_{2}=\|Q_{\sigma}(x)\|_{2},$ $\displaystyle\forall
x\in\\{(0,x_{2},x_{3})^{T}\mid x_{2},\
x_{3}\in\mathbb{R}\\}=\mathrm{Im}(Q_{\sigma}),$
for such kind of generic switching signals.
The following Example 6.4 is associated to Theorem C.
###### Example 6.4.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}$ with
$S_{1}=\frac{1}{2}\left(\begin{matrix}1&0\\\
\frac{3}{2}&-1\end{matrix}\right),\quad
S_{2}=\sqrt{\frac{3-\sqrt{5}}{2}}\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right).$
Then, using $\sqrt{\rho(A^{T}A)}=\|A\|_{2}$ we have
$\rho(S_{1})=\frac{1}{2}<1,\quad\|S_{1}\|_{2}=1\quad\textrm{and}\quad\rho(S_{2})=\sqrt{\frac{3-\sqrt{5}}{2}}<1,\quad\|S_{2}\|_{2}=1.$
In addition,
$\rho(S_{1}S_{2})=\sqrt{\frac{3-\sqrt{5}}{2}}=\rho(S_{2})<1.$
Therefore, $\boldsymbol{S}$ is absolutely stable by Theorem C.
The interesting [22, Proposition 18] implies that if
$\mathcal{S}=\\{A_{1},\dotsc,A_{m}\\}\subset\mathbb{R}^{d\times d}$ is
symmetric (i.e. $A^{T}\in\mathcal{S}$ whenever $A\in\mathcal{S}$), then
$\mathcal{S}$ has the spectral finiteness property; in fact, it holds that
$\rho(\mathcal{S})=\sqrt{\rho(A^{T}A)}$ for some $A\in\mathcal{S}$. This
naturally motivates us to extend an arbitrary $\boldsymbol{S}$ into a
symmetric set $\mathcal{S}=\boldsymbol{S}\cup\boldsymbol{S}^{T}$. Let us see a
simple example.
###### Example 6.5.
Let
$\boldsymbol{S}=\left\\{A=\sqrt{\frac{3-\sqrt{5}}{2}}\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)\right\\}$. Then, $\boldsymbol{S}$ satisfies (1.3a) with
$\|A\|_{2}=1$ such that $\rho(\boldsymbol{S})=\sqrt{\frac{3-\sqrt{5}}{2}}<1$.
But for $\mathcal{S}=\\{A,A^{T}\\}$,
$\rho(\mathcal{S})=\sqrt{\rho(A^{T}A)}=1\not=\rho(\boldsymbol{S})$.
This example shows that the extension $\mathcal{S}$ does not work for the
original system $\boldsymbol{S}$ needed to be considered here.
Finally, the following Example 6.6 is simple. Yet it is very interesting to
the stability analysis of switched systems.
###### Example 6.6.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{2\times 2}$ with
$S_{1}=\sqrt{\frac{3-\sqrt{5}}{2}}\left(\begin{matrix}1&0\\\
1&1\end{matrix}\right),\quad
S_{2}=\sqrt{\frac{3-\sqrt{5}}{2}}\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right).$
Then, using $\sqrt{\rho(A^{T}A)}=\|A\|_{2}$ we have
$\rho(S_{1})=\rho(S_{2})=\sqrt{\frac{3-\sqrt{5}}{2}}<1,\quad\|S_{1}\|_{2}=\|S_{2}\|_{2}=1,\quad\textrm{and}\quad\rho(S_{1}S_{2})=1.$
So, $\boldsymbol{S}$ is not absolutely stable. Yet from Corollary 5.3,
$\boldsymbol{S}$ is stable driven by $\mathbb{P}$-a.e.
$\sigma\in\varSigma_{2}^{+}$, for any $\theta$-ergodic probability measure
$\mathbb{P}$ on $\varSigma_{2}^{+}$, as long as $\mathbb{P}$ is not the
ergodic measure distributed on the periodic orbit
$\\{(12,12,12,\dotsc),(21,21,21,\dotsc)\\}.$
## 7 Concluding remarks
In this paper, we have considered the asymptotic stability of a discrete-time
linear switched system, which is induced by a set
$\boldsymbol{S}=\\{S_{1},\dotsc,S_{K}\\}\subset\mathbb{R}^{d\times d}$ such
that each $S_{k}$ shares a common, but not necessarily strict, Lyapunov matrix
$P$ as in (1.3a).
We have shown that if every subsystem $S_{k}$ is stable then $\boldsymbol{S}$
is stable driven by a nonchaotic switching signal. Particularly, in the cases
$K=2$ and $d=2,3$, we have proven that $\boldsymbol{S}$ has the spectral
finiteness property and so the stability is decidable.
Recall that $\boldsymbol{S}$ is called _periodically switched stable_ , if
$\rho(S_{w_{n}}\dotsm S_{w_{1}})<1$ for all finite-length words
$(w_{1},\dotsc,w_{n})\in\\{1,\dotsc,K\\}^{n}$ for $n\geq 1$; see, e.g., [16,
12, 10].
Finally, we end this paper with a problem for further study.
###### Conjecture.
Let $\boldsymbol{S}=\\{S_{1},S_{2}\\}\subset\mathbb{R}^{d\times d},d\geq 4$,
be an arbitrary pair such that condition (1.3a). If $\boldsymbol{S}$ is
periodically switched stable, then it is absolutely stable. Equivalently, if
$\rho(\boldsymbol{S})=1$ there exists at least one word
$(w_{1},\dotsc,w_{n})\in\\{1,2\\}^{n}$ for some $n\geq 1$ such that
$\sqrt[n]{\rho(S_{w_{n}}\dotsm S_{w_{1}})}=1$.
Since there exist uncountable many pairs $(\alpha,\gamma)\in(0,1)\times(0,1)$,
for which
$\boldsymbol{S}_{\alpha,\gamma}=\left\\{S_{1}=\alpha\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right),\quad S_{2}=\gamma\left(\begin{matrix}1&0\\\
1&1\end{matrix}\right)\right\\}$
is periodically switched stable such that $\|S_{1}\|=\|S_{2}\|=1$ under some
extremal norm $\|\cdot\|$ on $\mathbb{R}^{2}$; but
$\boldsymbol{S}_{\alpha,\gamma}$ is not absolutely stable with
$\rho(\boldsymbol{S}_{\alpha,\gamma})=1$. See, for example, [8, 7, 19, 17].
So, condition (1.3a) is very important for our Theorems B, C and D and for the
above conjecture. In fact, the essential good of $\|\cdot\|_{P}$ is to
guarantee that $\mathbb{K}_{\|\cdot\|_{P}}(S_{1})$ and
$\mathbb{K}_{\|\cdot\|_{P}}(S_{2})$ are linear subspaces of $\mathbb{R}^{d}$
in our arguments.
## References
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* [2] A. Bacciotti and R. Lionel, Liapunov Functions and Stability in Control Theory, 2nd ed., Comm. Control Engrg. Ser., Springer-Verlag, Berlin 2005.
* [3] M. Balde and P. Jouan, Geometry of the limit sets of linear switched systems, SIAM J. Control Optim., 49 (2011), 1048–1063.
* [4] N. Barabanov, Lyapunov indicators of discrete inclusions I–III, Autom. Remote Control, 49 (1988), 152–157, 283–287, 558–565.
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* [7] V. D. Blondel, J. Theys and A. A. Vladimirov, An elementary counterexample to the finiteness conjecture, SIAM J. Matrix Anal. Appl., 24 (2003), 963–970.
* [8] T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77–111.
* [9] X. Dai, Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices, J. Math. Anal. Appl., 379 (2011), 827–833.
* [10] X. Dai, Weakly recurrent switching signals, almost sure and partial stability of linear switched systems, J. Differential Equations, 250 (2011), 3584-3629.
* [11] X. Dai, Y. Huang and M. Xiao, Realization of joint spectral radius via ergodic theory, Electron. Res. Announc. Math. Sci., 18 (2011), 22-30.
* [12] X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities, Automatica, 47 (2011), 1512–1519.
* [13] X. Dai, Y. Huang and M. Xiao, Pointwise stabilization of discrete-time matrix-valued stationary Markov chains, Preprint 2011, arXiv:1107.0132v1 [math.PR].
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* [18] D. J. Hartfiel, Nonhomogeneous Matrix Products, World Scientific, New Jersey London, 2002.
* [19] V. S. Kozyakin, Structure of extremal trajectories of discrete linear systems and the finiteness conjecture, Autom. Remote Control, 68 (2007), 174–209.
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* [24] U. Serres, J.-C. Vivalda and P. Riedinger, On the convergence of linear switched systems, IEEE Trans. Automat. Control, 56 (2011), 320–332.
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* [26] P. P. Vaidyanathan and V. Liu, An improved sufficient condition for absence of limit cycles in digital filters, IEEE Trans. Circuits Systems, VOL. CAS-34 (1987), 319–322.
* [27] P. Walters, An Introduction to Ergodic Theory, GTM 79, Springer-Verlag, New York, 1982.
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|
arxiv-papers
| 2011-08-01T05:35:06 |
2024-09-04T02:49:21.195407
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiongping Dai, Yu Huang and Mingqing Xiao",
"submitter": "Xiongping Dai",
"url": "https://arxiv.org/abs/1108.0239"
}
|
1108.0246
|
Estimations of changes of the Sun’s mass and
the gravitation constant from the modern observations
of planets and spacecraft
E. V. Pitjeva1∗, N. P. Pitjev2∗∗
1Institute of Applied Astronomy of Russian Academy of Sciences, St.Petersburg
2St.Petersburg State University, St.Petersburg
More than 635 000 positional observations of planets and spacecraft of
different types, mostly radiotechnical ones (1961-2010), have been used for
estimating possible changes of the gravitation constant, the solar mass, and
semi-major axes of planets, as well as the value of the astronomical unit,
related to them. The analysis of the observations has been performed on the
basis of the EPM2010 ephemerides of IAA RAS in post-newtonian approximation.
The EPM ephemerides are computed by numerical integration of the equations of
motion of the nine major planets, the Sun, the Moon, asteroids and trans-
neptunian objects.
The estimation of change for the geliocentric gravitation constant
$GM_{\odot}$ has been obtained
$(\dot{GM_{\odot}})/GM_{\odot}=(-5.0\pm 4.1)\cdot 10^{-14}\quad\hbox{in
year}\quad\ (3\sigma$).
The positive century changes of semi-major axes $\dot{a}_{i}/a_{i}$ have been
determined simultaneously for the planets Mercury, Venus, Mars, Jupiter,
Saturn provided with high-accuracy sets of the observations, as expected if
the geliocentric gravitation constant is decreasing in century wise. Perhaps,
loss of the mass of the Sun $M_{\odot}$ the produces change of $GM_{\odot}$
due to the solar radiation and the solar wind compensated partially by the
matter dropping on the Sun.
It was been found from the obtained $GM_{\odot}$ change and taking into
account the maximal limits of the possible $M_{\odot}$ change that the
gravitation constant $\dot{G}/G$ falls within the interval
$-4.2\cdot 10^{-14}<\dot{G}/G<+7.5\cdot 10^{-14}$ in year
with the 95$\%$ probability.
The astronomical unit (au) is only connected with the geliocentric gravitation
constant by its definition. The decrease of $GM_{\odot}$ obtained in this
paper should correspond to the century decrease of au. However, it has been
shown that the present accuracy level of observations does not permit to
evaluate the au change. The attained possibility of fining the $GM_{\odot}$
change from high-accuracy observations points that fixing the connection
between $GM_{\odot}$ and au at the certain time moment is desirable, as it is
inconvenient highly to have the changing value of the astronomical unit.
PACS: 96.60.Bn, 06.20Jr, 95.10.Ce, 95.10.Eg.
∗E-mail: evp@ipa.nw.ru
∗∗E-mail: ai@astro.spbu.ru
INTRODUCTION
The whole array of high-precision observations of the planets and the
development of modern planetary ephemeris form prerequisites for the study of
very subtle effects, in particular, a change of the geliocentric gravitational
constant $GM_{\odot}$ with time. The question of variability and the possible
rate of the change of gravitational constant $G$ is regularly raised and
considered in some cosmological theories (Uzan, 2003; 2009). The Sun’s mass
$M_{\odot}$, can not be absolutely constant too. On the one hand, it decreases
due to continuous thermonuclear reactions and production of the radiant
energy, with the matter carried away by the solar wind. On the other hand,
there is a regular drop of interplanetary substances on the Sun, including
dust, meteoroids, asteroids and comets.
In the history of the Sun there have possibly been the periods of positive and
negative changes of the solar mass. In the initial period, shortly after the
formation of the protosun and the beginning of nuclear reactions in it, the
mass of the central body probably increased due to the continuing compression
to the center of the initial cloud. During the formation of the planetary
system, until the interplanetary space was cleared, the mass of the matter
falling on the Sun was higher than the mass of the Sun reduced due to the
light and corpuscular radiation. Now traces of the original, protosolar cloud
are only likely to remain on the periphery of the Solar system beyond Neptune,
the issue of the quantifying changes of the mass of the Sun still remains open
because of the uncertainty of the overall balance including the mass loss due
to radiation and the matter carried away by the solar wind, and the mass
increase due to the matter falling on the Sun, in particular, comets whose
collision with the photosphere of the Sun was repeatedly recorded by the SOHO
space observatory (http://ares.nrl.navy.mil/sungrazer/). Any estimation is
difficult because of the complexity of reliable estimation of falling matter
mass as well as the time-varied intensity and the angular distribution of the
solar wind in space. In this paper, we attempt to obtain experimental
estimates of the change of the solar mass, namely, the geliocentric
gravitational constant $GM_{\odot}$ from the analysis of observational data of
motion of planets and spacecraft.
The change of the value of the astronomical unit (au) is related to the change
of the geliocentric gravitational constant. The astronomical unit is close in
its magnitude to the average distance from the Earth to the Sun, but, by its
definition it is only related to the heliocentric gravitational constant
$GM_{\odot}$. From the obtained estimate $\dot{GM_{\odot}}$, we can estimate
the possible change in time of the au value. This estimate can be compared
with the direct change value of au obtained from observations.
EXPECTED EFFECTS OF THE CHANGE OF THE SUN’S MASS
Estimations of the mass change of the Sun and its rate repeatedly are cited in
the papers related to the exploration of solar physics, the solar wind and
radiation (e.g. Sunyaev, 1986; Livingston, 2000). The Sun’s luminosity
$L_{\odot}$ somewhat varies during the eleven year cycle and the $\sim$ 27-day
rotation around its axis; however, the fluctuations $L_{\odot}$ do not exceed
$0.1\div 0.2\%$ (Frohlich, Lean, 1998; 2004). If we take the average total
solar luminosity to be $L_{\odot}=3.846\cdot 10^{33}$ erg/s and the mass of
the Sun $M_{\odot}=1.9891\cdot 10^{33}$ g (Brun et al, 1998), then the
decrease of the mass of the Sun due to radiation as a fraction of the solar
mass is equal to $\dot{M}_{\odot}=-6.789\cdot 10^{-14}M_{\odot}$ per year.
The mass carried away with the solar wind was also repeatedly evaluated. The
basic composition of the solar wind is as follows: approximately 95% —
protons, 4% — nuclei of the helium atoms, and less than 1% — nuclei of atoms
of other elements (C, N, O, Ne, Mg, Ca, Si, Fe) (Brandt, 1973; Hundhauzen,
1976). The total number of particles flying away every second, is approximate
by estimated as $1.3\cdot 10^{36}$ (Kallenrode, 2004). The flow of the solar
wind affects the activity of the Sun, coronal mass ejections. Typically, the
average loss per year through the solar wind is estimated as $2\cdot
10^{-14}M_{\odot}$ (Hundhauzen, 1976; Hundhausen, 1997; Meyer-Vernet, 2007),
that is, less than a third of the mass loss due to radiation. There are
estimates $(2\div 3)\cdot 10^{-14}M_{\odot}$ per year (Sunyaev, 1986; Carroll,
Ostlie, 1996; Livingston, 2000), where a value of $3\cdot 10^{-14}M_{\odot}$
can be considered the upper limit of the mass carried away by the solar wind.
The cumulative effect of the relative annual decrease of the mass of the Sun
due to radiation and the solar wind can be restricted by the inequality
$-9.8\cdot 10^{-14}<\dot{M}_{\odot}/M_{\odot}<-8.8\cdot 10^{-14}.$ (1)
The reverse process occurs due to the fall of the dust, meteor, asteroid and
comet substance on the Sun. The dusty environment can not make a significant
contribution to the mass of the material fallen. According to the current
data, the density of the interplanetary dust decreases with distance from the
Sun, so that at the distance greater than 3 au there is not dust practically,
with the 2/3ds of the interplanetary dust concentrating in particles of
$10^{-5}\div 10^{-3}$ g, and the size of dust particles being mostly $1\div
10\ {\mu}m$ (Mann et al, 2010). The total mass of the dust matter is estimated
approximately as $10^{19}\div 10^{20}$ g (Sunyaev, 1986). Even with the
assumption that all the mass reaches the Sun in several thousands years, the
rate of the dust fall-out particles will be less than $10^{-16}M_{\odot}$ per
year. However, the dust particles smaller than 2 microns are swept by the
solar pressure, while the ones greater than 2 microns move towards the Sun.
The most part of the approaching dust sublimates within 0.1 au ($\sim
20R_{\odot}$) and can not reach the surface of the Sun. It is also necessary
to note, that a substantial part of the dust is carried away by the solar wind
to the periphery of the Solar system (Mann et al., 2010). Therefore, the
possible rate of the dust component falling out on the Sun is much smaller
than $(10^{-16}\div 10^{-17})\cdot M_{\odot}$ per year.
The larger particles, meteoroids and asteroids may fall on the Sun. Studies
show that there is a constant migration of asteroids with an opportunity to
complete the orbit evolution by the collision with the Sun (Farinella and et
al, 1994; Gladman et al, 1997). The total number of small bodies is very
large; the number of bodies larger than 1 km is about 1 million. A significant
part of asteroids move within a region close to the orbit planes of major
planets of the Solar system, mostly situated in the belt between the orbits of
Mars and Jupiter. The current estimates of the total mass of the asteroid belt
give $(13\pm 2)\cdot 10^{-10}M_{\odot}$ (Pitjeva, 2010b), i.e. less than
$10^{-3}$ mass of the Earth. For the ring to be able to exist for tens or
hundreds millions of years, the fraction of the outgoing annually material
should be significantly smaller than $10^{-7}\div 10^{-8}$ of the total mass
of the asteroid belt, in case the main asteroid belt is not replenished from
the outside. With the outgoing material falling on the Sun not regularly, we
find that the upper limit of possible mass of the Sun drop-down material from
the main belt is less than $(10^{-16}\div 10^{-17})\cdot M_{\odot}$ per year.
Thus, we obtain a significantly lower value (two to three orders of magnitude)
than the decrease of the solar mass by radiation and solar wind, so in the
solar neighbourhood and in the field of the main asteroid belt there is no
sufficient interplanetary matter migrating to the Sun to be compared with the
decrease of the solar mass due to radiation.
The mass of the matter that can come from distant regions of the solar system,
mainly in the form of comets (Bailey et al 1992), is more uncertain. There are
trans-Neptunian areas — the Kuiper Belt, a cloud of the Hills, the Oort cloud.
Currently, a large number of comets is detected in the immediate vicinity of
the Sun (sungrazing comets) using the LASCO coronagraph (http://lasco-
www.nrl.navy.mil/) installed at the SOHO solar space observatory (Marsden,
1989; 2005). Comets close to and often passing near the Sun are not long-
living. They can disintegrate into fragments, or completely ”fall apart”. An
example is a large family of Kreutz comets (Sekanina, Chodas, 2007). Some of
them enter directly to the dense layers of the Sun. On the pictures of the
SOHO observatory (http://sungrazer.nrl.navy.mil/index.php) the death of small
fragments of comets in the solar photosphere is regularly recorded (kamikaze
comet). The falling rocky bodies are more difficult to register, as during
their approach to the Sun the glowing gas tail like that of icy objects and
comets, does not form and the rocky bodies remain invisible. The overall
contribution of the visible and invisible objects may be significant, although
a reliable estimate of the total mass of substance reaching the Sun is
extremely difficult. Nevertheless, the upper limits to the total mass can be
specified. Statistics of comets discovered by the SOHO observatory gives about
500 comets for $30\div 35$ months, on the average $170\div 200$ comets per
year. Many of them completely evaporate during their passage through the lower
layers of the solar corona. There are registered repeatedly instances when the
comets reached the photosphere. We assume that all the detected comets
“vanished” and their mass increased the mass of the Sun, an overestimated
value will result. The relatively small comets with the size from tens to
hundreds meters are usually recorded, but for the upper estimate, we assume
that the diameters of their nuclei ($d_{com}$) are several kilometers (as for
the comets, which approach the spacecraft). If we take the average value of
$d_{com}=5$ km, the density of 3 g/cm3 and double the result due to the
missing and invisible falling objects, the annual upper bound is
$\dot{M}_{\odot,com}/M_{\odot}<+3.2\cdot 10^{-14}.$ (2)
This estimate is comparable to values obtained for the mass loss of the Sun
due to radiation and the solar wind, although it seems to be overestimated, as
the large falling objects have not been actually registered. This value can be
considered the upper limit of the possible increase of the solar mass due to
the material falling in the form of comets, meteors, asteroids and dust.
Now one can point to the two sides of the common interval, probably
overestimated, which should contain the value of $\dot{M}_{\odot}/M_{\odot}$.
To obtain the lower limit, let us take the maximum loss estimate due to the
solar wind, and at the same time, the zero drop of the material on the Sun. To
find the upper limit, we use the maximum estimate (2) for the material falling
on the Sun and the assumption that there is no mass loss due to the solar
wind. Then, we obtain
$-9.8\cdot 10^{-14}<\dot{M}_{\odot}/M_{\odot}<-3.6\cdot 10^{-14}\quad\hbox{
}.$ (3)
There are the restrictions to be kept in mind when trying to estimate the
experimental changes of the Sun’s mass.
THE INFLUENCE OF THE CHANGE OF THE SOLAR MASS ON ORBITAL ELEMENTS OF PLANETS
The change of the $M_{\odot}$ solar mass must lead to the appearance of
variations of elements of the planet orbits, but a small and monotonic change
affects only some certain elements. The effect is sought from extremely the
small expected change (of the order or less than $10^{-13}M_{\odot}$ per
year), so it is sufficiently to consider the influence within the framework of
the two-body problem (the Sun and the planet), as it is usually done, because
the correction due to the influence of other bodies in the weak effect
associated with the change of the central mass, is still several orders of
magnitude smaller. The two-body problem with the variable mass has a long
standing history and goes back to papers at the turn of the XIX and XX
centuries by Gulden (1884), Meshchersky (1893), Stromgren (1903), Plummer
(1906), etc. (see the detailed overview by Polyakhova, 1989). Variants of the
isotropic mass variation in the two-body problem without the appearance of
reactive forces, when no a particle is passed any pulse, were considered by
MacMillan, Jeans, Armellini, Duboshin, Levi-Civita. A similar problem arises
considering a possible change in time of the gravitational constant $G(t)$
under the Dirac’s hypothesis (Dirac, 1938) leads to the change of the mutual
attraction forces and accelerations between bodies, and the analogous
equations of the two-body problem. The analysis of equations for the central
field of a variable mass body within the framework of General Relativity is
given in the paper by Krasinsky and Brumberg (Krasinsky, Brumberg, 2004).
If we denote $\mu(t)$ the product of $G(M_{\odot}+m)$, where $m$ – the planet
mass, the vector equation of the relative motion for a body with mass $m$ is
written
${\bf\ddot{r}}=-{\mu(t)\over{r^{3}}}\bf r.$ (4)
In general, we can assume that the gravitational constant $G$, incoming in
$\mu(t)$, may depend on time. Since the central field remains while $\mu(t)$
(4) changes, then the area integral remains too, which is obtained immediately
if the left and right side of (4) to vector multiply on $\bf r$:
${\bf r}\times{\bf\ddot{r}}=0\qquad\hbox{or}\qquad{d\over{dt}}({\bf
r}\times\bf\dot{r})=0,$
then
${\bf r}\times\bf\dot{r}=\bf c.$ (5)
The flat motion follows from the existence of the vector area integral (5).
Energy for the case of the time-dependent value of $\mu(t)$ is changing and is
not more the integral of motion. Taking into account the monotony and
smallness of the change of $\mu(t)$, it was shown (Jeans, 1924), that the
invariant holds
$\mu(t)*a(t)=const,$ (6)
where $a$ – is the orbital semi-major planet axis. Sometimes this relation is
called the adiabatic invariant of Poincare-Jeans, so as the first conclusion
(6) was made by Poincare (Poincaré, 1911). The assumption, that $\mu(t)$
varies rather slowly, is essential for derivation of this result (Gelfgat,
1965). In our case, the expected value for Sun
$\mid\dot{\mu}_{\odot}(t)/\mu_{\odot}(t)\mid\sim 10^{-13}$ per year is much
orders of magnitude smaller than it is required by the Gelfgat’s constraints.
Since the plane of motion is preserved, then the elements ($i,\Omega$),
determining the position of the orbital plane, do not change. It should be
considered the dependence of semi-axis $a=a(t)$, the eccentricity $e=e(t)$ and
the argument of pericentre $g=g(t)$ of the osculating elliptical orbits from
the characteristics of the $\mu(t)$ change for instantaneous values of
coordinates, velocities, and a mass. From (6) we find a relationship between
the changes of $\mu(t)$ and $a(t)$:
${\dot{\mu}\over\mu}=-{\dot{a}\over a}.$ (7)
The relationship obtained from the integral of areas for the osculating orbit
is
$\mu(t)*a(t)*(1-e^{2})=c^{2},\qquad\hbox{where}\qquad c=\mid\bf c\mid,$ (8)
therefore, using (6), the $e$ eccentricity of the osculate orbit remains
constant under the given conditions $e=const$ (Jeans, 1925). Investigation of
the change of the $g$ pericentre position for the small and monotonic $\mu(t)$
change was made in the paper (Kevorkian, Cole, 1996), where it was shown that
under accepted conditions of the smallness and monotony of ${\dot{\mu}}$, the
$g$ value does not have a secular trend, and there may be only small
oscillations with the small amplitude of order $({\dot{\mu}}/\mu)^{2}$.
The change in time of $a=a(t)$, as $\mu(t)$, leads to the change of the period
of the body $m$ that is leaving from the position for the case $\mu=const$,
and the increase of deviation will depend quadratically on the time interval.
The possible $\mu_{\odot}(t)=GM_{\odot}$ change in the Solar system should be
appeared in a systematic, progressive, although very small deviation of the
body position on the orbit (that is their longitude) and the change of the
$a_{i}$ semi-major axes proportionally to the $\mu_{\odot}(t)$ change with
opposite sign (7). The fact that the value $\mu(t)=G(M_{\odot}+m)$ includes
the $m$ mass of a planet does not change the situation, since taking into
account the mass $m$ of a planet leads to the correction by several orders
less.
Thus, when the area integral remains and the attractive force from the main
body decrease/increase monotone, the second body is moving along the
trajectory gradually receding/approaching from/to the central body. The
relative increase of the distance is equal to the relative decreasing of mass
of the central body and vice versa. An orbit is transforming gradually
remaining identical to itself, and is of a spiral form. The $GM_{\odot}(t)$
change does not lead to secular trends of the eccentricity and the longitude
of perihelion. The semi-axis is century changing $a=a(t)$. Thus it is
necessary to look for the effect caused by possible change in time of the
heliocentric gravitational constant in the corresponding secular variation of
the semi-axes of the planet orbits.
OBSERVATIONAL MATERIAL, REDUCTION OF OBSERVATIONS
More than 635,000 positional observations of planets and spacecraft of various
types (Table 1), mainly radiotechnical (1961-2010) have been used to construct
high-precision ephemerides of planets and to determine the change of the
heliocentric gravitational constant. The very accurate observations are
required to find the very small effects, and it is most important and
desirable to have observational data for planets, close to the Sun and having
shorter periods, in the first place, the data for Mercury and Venus.
Radiotechnical measurements which began in 1961 and are continuing with rising
numbers since, first, yielded two new types of measurements in astrometry: the
distance and the relative speed, and secondly, the accuracy of the
measurements became several orders of magnitude greater than the accuracy of
the optical observations.
Table 1. The observations used
Observation type | Time interval | Observation number
---|---|---
Optical | 1913–2009 | 57768
Radiotechnical | 1961–2010 | 577763
Total | | 635531
For this reason, the ephemerides of the inner planets provided by
radiotechnical observations (mostly, data of time delays) are based fully on
these data. At present, the radiolocation of planet surfaces is not carried
out, but trajectory data of various spacecraft that orbiting around planets or
passing near them are received regularly. Accuracy of observations of ranging
has improved from 6 km to several meters for today’s data of the spacecraft.
It is necessary to say that the ephemerides of the outer planets so far,
mainly, are based on optical measurements since 1913, when at the Naval
Observatory of USA the improved micrometer was introduced, and observations
become more accurate ($0\hbox to0.0pt{.\hss}^{\prime\prime}5$).
Table 2. The distribution of optical observation and rms residuals in mas,
1913–2009
Planet | Observation number | $\sigma$
---|---|---
Jupiter | 13038 | 190
Saturn | 16246 | 150
Uranus | 11672 | 188
Neptune | 11342 | 177
Pluto | 5470 | 141
Total | 57768 |
However, until now a complete rotation period of Neptune and Pluto is not
provided by the observations. In addition to optical observations of these
planets, for the construction of ephemerides and estimation of their
parameters the absolute observation satellites of the outer planets are used,
as these observation are more precisely, and practically free from the phase
effect hard taking into account, which is in observations of the planets
themselves. Modern optical data are the CCD observations, their accuracy
reaches $0\hbox to0.0pt{.\hss}^{\prime\prime}05$. The number of used optical
and radio observations, their planet distribution, as well as mean square
error of residuals of observations are given in Table. 1 - 3.
Table 3. Distribution, time interval, number and rms residuals for radiometric
observations
Planet | Data type | Time interval | Observation number | $\sigma$
---|---|---|---|---
Mercury | $\tau$ [m] | 1964–1997 | 937 | 575
| $\tau$ [m] | 1973–2009 | 5 | 31.0
| $\alpha,\delta$ [mas] | 2008–2009 | 6 | 2.1
Venus | $\tau$ [m] | 1961–1995 | 2324 | 584
| Magellan $dr$ [mm/s] | 1992–1994 | 195 | 0.007
| MGN,VEX VLBI $\alpha,\delta$ [mas] | 1990–2007 | 22 | 3.0
| VEX $\tau$ [m] | 2006–2009 | 28163 | 3.6
| Cassini $\tau$ [m] | 1998–1999 | 2 | 2.4
| Cassini $\alpha,\delta$ [mas] | 1998–1999 | 4 | 105
Mars | $\tau$ [m] | 1965–1995 | 54851 | 738
| Viking $\tau$ [m] | 1976–1982 | 1258 | 8.8
| Viking $d\tau$ [mm/s] | 1976–1978 | 14978 | 0.89
| Pathfinder $\tau$ [m] | 1997 | 90 | 2.8
| Pathfinder $d\tau$ [mm/s] | 1997 | 7569 | 0.09
| MGS $\tau$ [m] | 1998–2006 | 165562 | 1.4
| Odyssey $\tau$ [m] | 2002–2008 | 293707 | 1.2
| MRO $\tau$ [m] | 2006–2008 | 7775 | 1.6
| spacecraft VLBI $\alpha,\delta$ [mas] | 1984–2010 | 136 | 0.6
Jupiter | $\tau$ [m] | 1973–2000 | 7 | 13.8
| spacecraft $\alpha,\delta$ [mas] | 1973–2000 | 16 | 5.0
| spacecraft VLBI $\alpha,\delta$ [mas] | 1996–1997 | 24 | 9.5
Saturn | $\tau$ [m] | 1979–2006 | 34 | 3.5
| $\alpha,\delta$ [mas] | 1979–2006 | 92 | 0.4
Uranus | Voyager-2 $\tau$ [m] | 1986 | 1 | 7.4
| Voyager-2 $\alpha,\delta$ [mas] | 1986 | 2 | 11.0
Neptune | Voyager-2 $\tau$ [m] | 1989 | 1 | 22.9
| Voyager-2 $\alpha,\delta$ [mas] | 1989 | 2 | 3.7
Total | | | 577763 |
The most accurate and long series of observations are available for Mars for
which spacecraft and landers were launched repeatedly. Radiotechnical
observations relating to Venus are much smaller, there are spacecraft Magellan
and Venus Express. A situation for observations of Mercury is much worse. The
Messenger spacecraft (NASA) on the orbit around it have just appeared and we
have not these data. There were only one-time conjunctions of Mariner-10
(1974–1975) and Messenger (2008–2009) spacecraft and ranging for the Mercury
surface (1964–1997). Situation should be changed after the new data from the
Messenger spacecraft and from the future spacecraft BepiColombo (ESA, launch
2014) will be available. There are a number of radiotechnical observations for
Jupiter and Saturn: the data for several spacecraft for Jupiter, and data of
the Cassini spacecraft for Saturn. For Uranus and Neptune there are one 3-D
points ($\alpha,\delta,R$), resulting from the conjunction of Voyager-2 with
these planets. The data were taken from the JPL database
(http:/ssd.jpl.nasa.gov/iau-comm4/), created by Dr. Standish, and now
supported by Dr. Folkner and the VEX data sent due to kindness of Dr. Fienga,
as well as supplemented by rows of American and Russian radar observation of
planets 1961–1995, taken from various sources. Russian radar observations of
planets, along with their references are stored on the site of IAA RAS
http://www.ipa.nw.ru/PAGE/DEPFUND/LEA/ENG/englea.htm. The brief description of
astrometric radio observations can be found in Table. 2 by Pitjeva (2005).
EPM2010 PLANETARY EPHEMERIDES, DETERMINED PARAMETERS
This work is based on the EPM2010 planetary ephemerides of IAA RAS. Numerical
ephemerides of motion of the planets and the Moon (EPM — Ephemerides of
Planets and the Moon) began to create in the seventieth years of last century
under the leadership of G.A. Krasinsky for support of Russian space flights,
and has successfully developed since then. The version of the EPM2004
ephemerides has used to release the Russian “Astronomical Yearbook” described
in (Pitjeva, 2005), the version of the EPM2008 ephemerides in the paper
(Pitjeva, 2010a). The EPM2010 ephemerides were constructed using more than 635
thousands of observations (1913-2010) of different types. EPM2010 differ from
the previous versions by the improved dynamic model of motion of Solar system
bodies, adding the perturbation from the ring of Trans-Neptunian objects
(TNO), the new value of the Mercury mass, defined due to the three encounters
of the Messenger spacecraft with Mercury, improvement of reductions of
observations with the addition of the relativistic delay effect from Jupiter
and Saturn, and the expanded database of observations, including
radiotechnical (2008 - 2010) and CCD (2009) measurements.
Ephemerides were constructed by the simultaneous numerical integration of
equations of motion of all the major planets, the Sun, the Moon, the largest
301 asteroids, 21 TNO, the lunar libration, taking into account the
perturbations from the oblateness of the Sun and the asteroid belt, lying in
the ecliptic plane and consisting of the remaining smaller asteroids, as well
as the ring of the TNO rest at the mean distance of 43 au. The equations of
motion of the bodies were taken in the post-Newtonian approximation in the
Schwarzschild field. Integration in the barycentric system of coordinates for
the epoch J2000.0 performed by the Everhard method over 400 years (1800-2200)
by the lunar and planetary integrator of the software package ERA-7
(Krasinsky, Vasilyev, 1997). The accuracy of numerical integration was
verified by comparing the results of the forward and backward integrations
over the century of the time interval. The errors were at least order of the
magnitude smaller than the accuracy of observations. Thus, the accuracy of the
ephemeris is determined mainly the accuracy of observations and their
reductions.
In the basic version of the improved EPM2010 planetary ephemeris about 260
parameters are determined: elements of the orbits of the planets and the 18
satellites of the outer planets; value of the astronomical unit; three angles
of the orientation with respect to the ICRF frame; 13 parameters of the
rotation of Mars and the coordinates for the three martian landers; masses of
10 asteroids, the mean density for the three taxonomic classes of asteroids
(C, S, M), the mass and radius of the asteroid belt, the mass of the TNO ring,
the ratio of the mass of the Earth and the Moon; quadrupole moment of the Sun
($J_{2}$) and 23 parameters for the solar corona of different conjunctions of
the planets with the Sun; eight coefficients of the topography of Mercury and
the corrections to the level surface of Venus and Mars, constant shifts for
the three series of planetary radar observations and for 7 spacecraft; 5
coefficients for the additional effect of the phase of the outer planets.
The accuracy of EPM ephemerides was tested by comparison with the observations
(all residuals do not superior to their a priori errors), as well as
comparison with the DE421 (JPL) independent ephemerides, those are in a good
agreement (Pitjeva, 2010a).
After constructing the EPM2010 ephemerides to all the observations the some
other parameters can be to estimate: the changes of the $GM_{\odot}$
heliocentric gravitational constant, the $G$ gravitational constant, semi-axes
of the planet orbits, and the astronomical unit.
OBTAINED ESTIMATES OF THE $\quad\bf M_{\odot},G,a_{i}\quad$ CHANGES
The main problem of this case consists in the smallness of the effects that
need to be revealed. It was impossible to do this before an appearance in
recent years a quite large number of high-precision observations, including
data from spacecraft. Accuracy of determination of the parameters increased
significantly also due to extension of the time interval for which there are
high-precision sets of planet observations.
The parameters $\dot{G}$ and $\dot{GM_{\odot}}$ were fitted by the least
squares method simultaneously with all basic parameters of ephemerides, but
each separately, i.e. they are considered in different solution versions. If
$\dot{GM_{\odot}}$ is found then it is taken into account that the
accelerations between the Sun and other bodies change, and the mutual
attractions between other pairs of bodies remain. This differs from the
situation when we find the change of $G$ and when the forces between all
bodies vary accordingly. It should be noted that for the version of the
$\dot{G}$ definition from the planet motions, the main contribution makes by
the Sun, since the equations of the planet motions include products of the
masses on the gravitational constant, among them the term for the Sun
($GM_{\odot}$) is the main one of several orders of magnitude more than the
others. Therefore, separating the change of $G$ from the change $M_{\odot}$
with the dominant term of the $GM_{\odot}$ is impossible. In this regard, it
is more correctly (and reliably) to determine from planet motions the change
of $GM_{\odot}$ instead of $\dot{G}$ or $\dot{M}_{\odot}$ separately.
Table 4. The secular change values of the semi-major axes for the 6 planets
provided with the high-accuracy observations
Planet | $\dot{a}/a$ (century-1) | Correlation coefficients
---|---|---
| | between $\dot{a}$ and $a$
Mercury | (3.30 $\pm$ 5.95)$\cdot 10^{-12}$ | 56.5%
Venus | (3.74 $\pm$ 2.90)$\cdot 10^{-12}$ | 95.8%
Earth | (1.35 $\pm$ 0.32)$\cdot 10^{-14}$ | 0.6%
Mars | (2.35 $\pm$ 0.54)$\cdot 10^{-14}$ | 0.4%
Jupiter | (3.63$\pm$ 2.24)$\cdot 10^{-9}$ | 20.2%
Saturn | (9.44$\pm$ 1.38)$\cdot 10^{-10}$ | 35.9%
Table. 4 shows the values obtained for the relative change of semi-major axes
of the planet orbits. The most accurate results connect with availability of
observations obtained with using radiotechnical equipment, in particular using
the spacecraft observations and the duration of larger time intervals.
Accordingly, the most reliable relative values of $\dot{a}/a$ have been
received for them. These are the results for all the inner planets from
Mercury to Mars. For Jupiter and Saturn the accuracy of ${\dot{a}/a}$ is less.
Values for the planets, not provided by radiotechnical data are unreliable. It
is important that all the values obtained for the planets from Mercury to
Saturn show the positive values of the $\dot{a}/a$ ratio, i.e., indicate the
decrease in time of the $GM_{\odot}$ heliocentric gravitational constant
$(\ref{f-7})$.
The change of the $GM_{\odot}$ heliocentric gravitational constant has been
determined from fitting all observations:
$(\dot{GM_{\odot}})/GM_{\odot}=(-5.04\pm 4.14)\cdot 10^{-14}\quad\hbox{per
year}\quad(3\sigma).$ (9)
This was made similarly to find $\dot{G}/G$:
$\dot{G}/G=(-4.96\pm 4.14)\cdot 10^{-14}\quad\hbox{per year}\quad(3\sigma).$
(10)
The closeness of the results (9) and (10) is not surprising, since while
finding the $\dot{G}/G$, when the forces between all pairs of bodies change,
the effect of the central body is dominant and again the effect of
$GM_{\odot}$ is found practically, instead of $G$. From the result (9)
obtained for $GM_{\odot}$, it is possible to estimate the $\dot{G}$ value
using the relation
$\dot{\mu}_{\odot}/\mu_{\odot}=\dot{G}/G+\dot{M}_{\odot}/M_{\odot}.$ (11)
This relation is valid with the 95$\%$ ($2\sigma$) probability:
$-7.8\cdot 10^{-14}<\dot{G}/G+\dot{M}_{\odot}/M_{\odot}<-2.3\cdot
10^{-14}\quad\hbox{year}^{-1}.$ (12)
Hence, using the limits (3) found for the value $\dot{M}_{\odot}/M_{\odot}$,
we obtain the $\dot{G}/G$ value with the 95$\%$ probability is within the
interval
$-4.2\cdot 10^{-14}<\dot{G}/G<+7.5\cdot 10^{-14}\quad\hbox{per year}.$ (13)
Note, the $\dot{G}/G$ estimate, obtained in 2004 from to the lunar laser
ranging (Williams et al, 2004), which in any case is not complicated by the
possible change of the solar mass, gives the following values of the limits
for the gravitational constant change: $\dot{G}/G=(4\pm 13)\cdot 10^{-13}$ per
year. The estimate of $(\dot{GM_{\odot}})/GM_{\odot}$ obtained by us (9) has
the opposite sign and its value is an order of magnitude smaller.
The obtained change of $GM_{\odot}$, most probably, is related to the change
of the Sun’s mass $M_{\odot}$, rather than to the $G$ change. Thus, we have
$\dot{M_{\odot}}/M_{\odot}=(-5.0\pm 4.1)\cdot 10^{-14}\quad\hbox{per
year}\quad(3\sigma).$ (14)
Note that this value hits the limitation interval (3) for
$\dot{M_{\odot}}/M_{\odot}.$ The obtained change of $GM_{\odot}$ (9),
possibly, reflects the balance between the mass loss due the radiation and the
solar wind and the falling material contained in comets, rocky debris and
asteroids, which do not produce the visible glowing gas tail.
THE POSSIBLE CHANGE OF THE ASTRONOMICAL UNIT
The change of the astronomical unit is connected with the change of the
heliocentric gravitational constant. The astronomical unit, although is close
in magnitude to the average distance of the Earth from the Sun, but by its
definition (resolution MAS 1976 - IAU, 1976) is connected with the
heliocentric gravitational constant:
$GM_{\odot}[m^{3}s^{-2}]=k^{2}\cdot\hbox{au}^{3}[m^{3}]/86400^{2}[s^{2}],$
(15)
where k = 0.01720209895 is Gaussian gravitational constant. Currently, au is
determined from ranging data with very high real accuracy, allowing to deduce
the value of the heliocentric gravitational constant from the of value
${\hbox{au}~{}=}\ (149597870700\,{\pm~{}3}$) m, using the relation (15):
${GM_{\odot}~{}=}\,(1327124404\,{\pm~{}1})[km^{3}s^{-2}]$ (e.g., Pitjeva and
Standish, 2009), which coincides with the value $GM_{\odot}$, proposed by
W.Folkner and obtained by the same method. These values of au and $GM_{\odot}$
for the TDB time scale were approved at the XXVII IAU General Assembly (2009)
as the best current values of astronomical constants
$(http://maia.usno.navy.mil/NSFA2/NSFA\\_cbe.html)$.
In the paper by Krasinsky and Brumberg (Krasinsky, Brumberg, 2004) from raging
data 1961 – 2003, using a numerical theory of planetary motion, about
coinciding with the EPM2004 (Pitjeva, 2005), the authors obtained the secular
increase of the astronomical unit $\dot{au}$ = 15 m per century, which should
correspond to the increase of the heliocentric gravitational constant
$\dot{GM_{\odot}}/GM_{\odot}\simeq 3\cdot 10^{-12}\quad\hbox{per year}.$ (16)
The positive change of au should correspond to the decrease of semi-major axes
of planet orbits, and not vice versa, as sometimes this is claimed, and
alternative theories of gravity (Miura T. et al, 2009; Nyambuya G.G., 2010)
are even constructed on this incorrect basis. Such the large positive change
(16) of the does not correspond to estimations of physical processes in the
Solar system (the solar radiation and wind, the matter falling on the Sun),
and also to the estimate (9) obtained in this study:
$\dot{GM_{\odot}}/GM_{\odot}=-5.04\cdot 10^{-14}$ per year. However, authors
considered themselves that the increase of au and the heliocentric
gravitational constant (16) are rather parameters of agreement than the real
change of the physical parameters.
Analysis of the obtained results based on the observations described in this
paper, and the EPM2010 ephemerides, shows that the present level of
observational accuracy does not permit to evaluate the au change. In the paper
by Krasinsky and Brumberg the au change was determined simultaneously with all
other parameters, specifically, with the orbital elements of planets and the
value of the au astronomical unit itself. However, at present it is impossible
to determine simultaneously two parameters: the value of the astronomical
unit, and its change. In this case, the correlation between $au$ and its
change $\dot{au}$ reaches 98.1 $\%$, and leads to incorrect values of both of
these parameters, in particular, gives $\dot{au}$ the order of 15 m per
century.
Without the simultaneous determination of $au$ and $\dot{au}$, i.e. if only
the change of the astronomical unit is estimated, together with other
parameters, the $\dot{au}$ value is about 1 m per century, and does not exceed
its formal uncertainty, thus it is determined:
$\dot{au}$ = (1.2 $\pm$ 3.2) m/cy (3 $\sigma$).
Furthermore, including or excluding the $\dot{au}$ value from the number of
the solution parameters does not change the observation residuals, the mean
error of the unit weight is also not changed ($\Delta\sigma\simeq 0.2\%$), so
there is no reason to assume that $\dot{au}$ is the necessary parameter of
agreement, and include it in the number parameters to be estimated.
The modern accuracy has approached to the level when it is possible to
estimate the change of the heliocentric gravitational constant $GM_{\odot}$,
therefore it is desirable to specify the definition of the astronomical unit,
for example, by fixing the connection between $GM_{\odot}$ and au at the
certain time moment, as it is inconvenient highly to have the changing value
of the astronomical unit.
CONCLUSION
Modern radiotechnical observations of the planets and spacecraft having the
meter accuracy (relative error of $10^{-12}\div 10^{-11}$) make it possible to
obtain estimates of very small effects in the Solar system. The significant
progress is related to several factors: increase in accuracy of reduction
procedures for observations and in dynamical models of planet motion, as well
as improvement of the quality of observational data, increasing their accuracy
and the time interval in which these observations are obtained.
The results obtained indicate on the decrease of the heliocentric gravitation
constant per year at the level
$\dot{GM_{\odot}}/GM_{\odot}=(-5.0\pm 4.1)\cdot 10^{-14}\quad(3\sigma).$
For the gravitation constant it is found that value of $\dot{G}/G$ falls in
the interval
$-4.2\cdot 10^{-14}<\dot{G}/G<+7.5\cdot 10^{-14}$
with the 95$\%$ probability.
The obtained change of $GM_{\odot}$ seems to be due to the change of the solar
mass $M_{\odot}$, rather than the $G$ change and reflects the balance between
the loss of the solar mass due to by the radiation and the solar wind and
matter falling on the Sun. It is possible to make the cautious conclusion that
at present in the Solar system there is still the significant effect of matter
falling on the Sun, that compensate partly the effect of reducing the solar
mass due to the radiation and the solar wind.
In the future, the connection between $GM_{\odot}$ and au (15) should be fixed
at the certain time moment, as it is inconvenient highly to have the changing
value of the astronomical unit; moreover, it should be to define the changes
$GM_{\odot},\,M_{\odot},\,G$ rather than the change of the astronomical unit.
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|
arxiv-papers
| 2011-08-01T06:38:12 |
2024-09-04T02:49:21.207058
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E.V. Pitjeva, N.P. Pitjev",
"submitter": "Nicolai Pitjev",
"url": "https://arxiv.org/abs/1108.0246"
}
|
1108.0483
|
#
Thermodynamics of the Lee-Wick partners: An alternative approach
Kaushik Bhattacharya$, Suratna Das†
$Department of Physics, Indian Institute of Technology, Kanpur,
Kanpur 208016, India
† Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba, Mumbai 400005, India111Major part of the work has
been carried out while at Theoretical Physics Division, Physical Research
Laboratory, Navrangpura, Ahmedabad 380009, India email: $kaushikb@iitk.ac.in,
†suratna@tifr.res.in
###### Abstract
It was pointed out some time ago that there can be two variations in which the
divergences of a quantum field theory can be tamed using the ideas presented
by Lee and Wick. In one variation the Lee-Wick partners of the normal fields
live in an indefinite metric Hilbert space but have positive energy and in the
other variation the Lee-Wick partners can live in a normal Hilbert space but
carry negative energy. Quantum mechanically the two variations mainly differ
in the way the fields are quantized. In this article the second variation of
Lee and Wick’s idea is discussed. Using statistical mechanical methods the
energy density, pressure and entropy density of the negative energy Lee-Wick
fields have been calculated. The results exactly match with the thermodynamic
results of the conventional, positive energy Lee-Wick fields. The result sheds
some light on the second variation of Lee-Wick’s idea. The result seems to say
that the thermodynamics of the theories do not care about the way they are
quantized.
## 1 Introduction
The Lee-Wick field theories [1, 2] originated in an attempt to address the
problem related with the infinities in quantum field theories. Recently some
authors have tried to implement Lee-Wick’s idea in a higher derivative version
of a quantum field theory [3, 4]. All these theories assumed the existence of
some partners of the Standard model particles. The main ideas of the Lee-Wick
Standard Model in Ref. [3] have been extended in Ref. [5] where the authors
use two Lee-Wick partners for each standard model field: one with negative and
the other with positive norm. Later, this idea has been used in Ref. [6] to
improve gauge coupling unification without introducing additional fields in
the higher-derivative theory. The Higgs sector of the Lee-Wick Standard Model
has also been constrained in Ref. [7].
There has been at least one attempt [8] to use the concepts of these Lee-Wick
constructions in cosmology where the authors were able to show the bouncing
nature of the universe whose energy is dominated by the energies of a scalar
field and its Lee-Wick partner. In Ref. [9] the authors tried to formulate a
possible thermodynamic theory of particles which includes the Lee-Wick
partners using a method of statistical field theory previously formulated by
Dashen, Ma and Bernstein in Ref. [10].
The present article mainly focusses on a variant of the original Lee-Wick
idea,222Some work in this direction was started much before by Pauli in Ref.
[11] which was concerned about the taming of the divergences in a quantum
field theory. In 1984 Boulware and Gross [12] tried to show that the original
proposal of Lee and Wick was related to a complex implementation of the Pauli-
Villars regularization scheme [13]. The complexity of the idea arises from the
fact that the Pauli-Villars regulator fields in the Lee-Wick theories are not
just ad hoc regulator fields, they also have dynamics. To implement the Pauli-
Villars idea Lee and Wick introduced massive partner fields for all the normal
fields in the theory. The scheme becomes involved when one tries to quantize
the partner fields. It turns out that the partner fields can be quantized in
two ways. In one way the norm of the states of the partner fields on the
underlying Hilbert space remains definite and in the other case the norm of
the states of the partner fields on the Hilbert space becomes indefinite. In
the former case the energy of the Lee-Wick partner fields turns out to be
negative and in the later case the energy of the Lee-Wick partner fields
remain positive.
Both the options, as stated above, have their merits and demerits. In the
first option, where the partner field states have positive definite norms but
negative energy, the theory remains quantum mechanically understandable but
does not have a proper ground state. There are run-away solutions. In the
other option, where the Lee-Wick partner field states live on an indefinite
metric Hilbert space but carry positive energy, there are zero-norm states
which can grow indefinitely. This option also gives rise to run-away
solutions. Historically Lee and Wick preferred to work with the theory defined
on an indefinite metric. The difficulties of the run-away solutions were
addressed by applying future boundary conditions which again made the theory
non-causal.
The present article deals with the option which Lee and Wick discarded, a
quantum field theory of Lee-Wick fields whose states do live on a definite
metric Hilbert space but has negative energy. The motivation for such an
unconventional work comes from a very interesting result related to the
thermodynamics of the standard Lee-Wick theory as given in Ref. [9]. It turns
out that the thermodynamics of the indefinite metric, positive energy Lee-Wick
partners is exactly the same as definite metric but negative energy Lee-Wick
fields. This similarity of the thermodynamics of the two different scenarios
gives us a glimpse of the path which Lee and Wick did not take historically.
There are a plethora of problems related with the option which is presented in
this article, the most important of them being that the theory is
energetically unstable. Presently we do not give all the pathological
properties of the alternative Lee-Wick prescription, nor do we know the cures
of all the formal (pathological) diseases of the theory. The formal aspects of
the negative energy Lee-Wick sector remains mostly open for further
investigation in the near future. Inspite of all the conceptual difficulties
related to run-away solutions the result presented is too strong to be taken
as a coincidence. Interestingly, the energy instability of the model plays an
important role in the thermodynamics of the unusual fields and indirectly
affects the results presented in this article which match surprisingly with
the thermodynamics of the positive energy Lee-Wick fields. Readers who are
purely interested in the formal aspects of the alternative Lee-Wick
prescription can go through Ref. [12] for a more lucid and formal development
of the basic ideas.
The present article is presented in the following manner. A brief introduction
on the unusual regulator fields and their properties is presented in the next
section. Section 3 discusses the technique to find out the thermal
distribution function of the regulator fields. In section 4 the energy
density, pressure and entropy density of a gas comprising of elementary
particles and their unusual field partners are calculated using the thermal
distribution functions. The last section 5 summarizes the important results
obtained in this article.
## 2 Canonical quantization of the positive energy and negative energy Lee-
Wick fields
In the paper written by Grinstein, O’Connell and Wise, [3] on Lee-Wick
standard model, the authors proposed a higher derivative field theory as the
underlying theory of nature. The quadratic kinetic terms of the normal field
theories, both bosonic and fermionic, are regained by introducing new degrees
of freedom. It turns out that the Lagrangians of the new fields have wrong
signs. In their work Grinstein, O’Connell and Wise [3] did not give any
prescription for the canonical quantization of the new fields. In a later work
by Fornal, Grinstein and Wise [9] the authors derived the thermodynamics of
the unusual new degrees of freedom using methods of statistical field theory.
In this section we first canonically quantize these new partner fields. This
exercise will immediately show that these fields do carry positive energy. In
the next step we give the prescription for obtaining negative energy fields in
the Lee-Wick paradigm.
### 2.1 Indefinite metric positive energy case
We do the quantization for the scalar field with the understanding that the
other bosons in the theory do follow the same quantization rules. If the
Lagrangian of the Lee-Wick partner of an usual scalar field be represented as
$\xi$, then according to [3], the non-interactive part of its Lagrangian is
given as
$\displaystyle{\mathcal{L}_{\xi}}=-\frac{1}{2}\partial_{\mu}\xi\partial^{\mu}\xi+\frac{1}{2}m_{\xi}^{2}\xi^{2}\,,$
(1)
where $m_{\xi}$ is the mass of the $\xi$ field. The field $\xi$ can be
expanded, in the Fourier space, in the same fashion as a standard scalar
field:
$\displaystyle\xi(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}2\epsilon({\bf
p})}}\left[a({\bf p})e^{-ip\cdot x}+\bar{a}({\bf p})e^{ip\cdot x}\right]\,,$
(2)
where $\epsilon({\bf p})=\sqrt{{\bf p}^{2}+m_{\xi}^{2}}$ is the dispersion
relation of the Lee-Wick partner excitation. The partner field can be
quantized by the following condition:
$\displaystyle[\xi(t,{\bf x})\,,\,\pi_{\xi}(t,{\bf y})]=i\delta^{3}({\bf
x-y})\,,$ (3)
where $\pi_{\xi}\equiv\delta L_{\xi}/\delta\dot{\xi}=-\dot{\xi}$. The
quantization condition for the partner fields yields the following unusual
commutation relation
$\displaystyle\left[a({\bf p})\,,\,\bar{a}({\bf q})\right]=-\delta^{3}({\bf
p}-{\bf q})\,,$ (4)
while the other commutators involving $a({\bf p})$ and $\bar{a}({\bf q})$ are
all zero. The equation above predicts that the field excitations of the
partner fields will have indefinite norm in the Hilbert space. The unusual
commutation relation between the creation and annihilation operators of the
$\xi$ field excitations is related to the negative sign of the canonically
conjugate momentum corresponding to the $\xi$ field.
If eigenstates of the number operator,
$\displaystyle N({\bf p})=-\bar{a}({\bf p})a({\bf p})\,,$ (5)
are defined by the following way
$\displaystyle N({\bf p})|n({\bf p})\rangle=n({\bf p})|n({\bf p})\rangle\,,$
(6)
where $n({\bf p})$ is a positive integer interpreted as the number of
particles with momentum ${\bf p}$ then the Hamiltonian of the field
configuration is
$\displaystyle H=\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,,$ (7)
where we have dropped the zero-point contribution in the Hamiltonian. The
important point to note is the negative sign in the definition of the the
number operator. In this case if we stick to the conventional definition of
the number operator (the same operator without the negative sign) then it can
be shown that it will have negative eigenvalues. The quantity $-\bar{a}({\bf
p})a({\bf p})$ has a positive spectrum. This unconventional behavior of the
number operator originates from the unconventional commutation relation of the
creation and annihilation operators of the field excitations as given in Eq.
(4). The negative sign of the original Lagrangian of the $\xi$ field is
balanced by the negative sign of the number operator and consequently the
Hamiltonian of the field turns out to be positive.
For the fermionic case one can take the Lagrangian of the Lee-Wick partner
field to be
$\displaystyle{\mathcal{L}}=-{\psi}^{\dagger}\gamma^{0}(i{\hbox
to0.0pt{/\hss}\partial}-m_{\psi})\psi\,,$ (8)
where $m_{\psi}$ is the mass of $\psi$ field excitations333The interested
reader can consult Refs. [14, 15] related to fermions leaving in an indefinite
metric. The negative sign of the Lagrangian in Eq. (8) gives the unusual sign
in conjugate momentum. The fermionic field can be expanded in the Fourier
basis as
$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}2\epsilon({\bf
p})}}\sum_{s=1,2}\left[a_{s}({\bf p})u_{s}({\bf p})e^{-ip\cdot
x}+\bar{b}_{s}({\bf p})v_{s}({\bf p})e^{ip\cdot x}\right]\,,$ (9)
where $a_{s}({\bf p})$ and $b_{s}({\bf p})$ are the annihilation operators of
the fermionic and anti-fermionic excitations of the $\psi$ field. Quantizing
the fermion field $\psi$ in the conventional sense
$\displaystyle\left\\{\psi(t,{\bf x}),\pi_{\psi}(t,{\bf
y})\right\\}=\delta^{3}({\bf x-y})\,,$ (10)
one gets
$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=-\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf k}\right)\,.$
(11)
In the fermionic case one can define the number operator for the fermions and
anti-fermions exactly in the same way as given in Eq. (6) and the form of the
Hamiltonian will be similar to that in Eq. (7). In this case also the Lee-Wick
excitations will have positive energy but indefinite norm. An excellent
exposition of the relationship between the unusual commutation relations of
the creation and annihilation operators (of the bosonic/fermionic fields) and
the indefinite metric it induces in the Hilbert space is presented in the
first two appendices of Ref. [2].
In Ref. [9] the authors tried to envisage the Lee-Wick partners as
intermediate resonances. According to them the normal scalar particles scatter
with each other as $\phi({\bf p}_{1})+\phi({\bf p}_{2})\to\phi({\bf
p^{\prime}}_{1})+\phi({\bf p^{\prime}}_{2})$ through two Lee-Wick resonances.
The scattering in reality happens like $\phi({\bf p}_{1})+\phi({\bf
p}_{2})\to\xi(\tilde{{\bf p}}_{1})+\xi(\tilde{{\bf p}}_{2})$ and then
$\xi(\tilde{{\bf p}}_{1})+\xi(\tilde{{\bf p}}_{2})\to\phi({\bf
p^{\prime}}_{1})+\phi({\bf p^{\prime}}_{2})$ where $\xi(\tilde{{\bf p}}_{1})$
and $\xi(\tilde{{\bf p}}_{2})$ stands for the Lee-Wick fields. In this
formalism the Lee-Wick partners are unstable resonances, with a negative decay
width, and their existence is ephemeral. Writing the $S$-matrix as
$S=1-i\mathcal{T}$ one can write the $\mathcal{T}$ matrix amplitude for
$\phi({\bf p}_{1})+\phi({\bf p}_{2})\to\xi(\tilde{{\bf
p}}_{1})+\xi(\tilde{{\bf p}}_{2})$ as
$\displaystyle\langle\tilde{{\bf p}_{1}},\tilde{{\bf
p}_{2}}|\mathcal{T}(E)|{\bf p}_{1},{\bf
p}_{2}\rangle=2\pi\delta(E_{1}+E_{2}-\epsilon_{1}-\epsilon_{2})\delta^{3}({\bf
p}_{1}+{\bf p}_{2}-\tilde{{\bf p}_{1}}-\tilde{{\bf p}_{2}})\mathcal{M}(E)\,,$
(12)
where the energy $E=E_{1}+E_{2}$ and momentum ${\bf p}={\bf p}_{1}+{\bf
p}_{2}$ are such that $(p_{1}+p_{2})^{2}=m^{2}_{\xi}$ which sets the threshold
value for the creation of the Lee-Wick resonances. Here $\epsilon_{i}$ stands
for the energy of the Lee-Wick fields and $m_{\xi}$ is the mass of the Lee-
Wick excitations. The important ingredient in Ref. [9] lies in the
prescription used for writing $\mathcal{M}(E)$:
$\displaystyle\mathcal{M}(E)=-\frac{1}{2}\frac{g^{2}}{E^{2}-{\bf
p}^{2}-m^{2}_{\xi}+im_{\xi}\Gamma}\,,$ (13)
where $g$ specifies the coupling of the normal fields with the Lee-Wick
fields. The interesting part of the above prescription lies in the overall
minus sign of $\mathcal{M}(E)$ and the negative sign in front of the decay
width $\Gamma$ in the denominator. The authors of Ref. [9] then utilize a
conventional relation between the scattering matrix elements and the grand
partition function of a thermodynamic system to predict the various
thermodynamic parameters. Thermodynamics of ephemeral resonance fields with
negative decay widths is highly non-trivial but in principle one can find it
out as done in Ref. [9].
### 2.2 When the states have positive definite norm but negative energy
In the present case we will try to find out the thermodynamics of the negative
energy Lee-Wick fields which live in a normal Hilbert space. To do this one
can start from the same Lagrangian for the real scalars as used in Ref. [3]
and as given in Eq. (1). The field expansion of the $\xi$ field can be kept
exactly the same as in Eq. (2). To have a definite metric starting with a
negative Lagrangian one has to quantize the fields with the wrong sign as
$\displaystyle[\xi(t,{\bf x})\,,\,\pi_{\xi}(t,{\bf y})]=-i\delta^{3}({\bf
x-y})\,.$ (14)
This quantization condition yields the usual commutation relation between the
the creation and annihilation operators of the $\xi$ field excitations. The
$\xi$ field excitations will now lie in a normal Hilbert space (i.e. the
states having definite norm). The price one has to pay for a cross over from
the indefinite norm landscape to the positive norm landscape is the ground
state. If we stick to the definition of the number operator as defined in Eq.
(5) then $N({\bf p})$ will have negative eigenvalues. Consequently the
Hamiltonian of the field configuration can still be written as
$\displaystyle H=\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,,$
but unlike the previous case the Hamiltonian will be having negative
eigenvalues. Although this Hamiltonian will have zero as an eigenvalue, but
unlike the indefinite norm case, this eigenvalue is not the least but the
maximum of the eigenspectrum. Consequently the definite norm Lee-Wick field
excitations do not have a proper ground state. If we interpret Eq. (5), for
the indefinite metric case, as the sum of positive energy excitations then Eq.
(5) has to be interpreted as a sum of positive energy de-excitations of the
definite metric field.
From a very similar analysis it can be shown that if one changes the
quantization condition for the fermions from that given in Eq. (10) to
$\displaystyle\left\\{\psi(t,{\bf x}),\pi_{\psi}(t,{\bf
y})\right\\}=-i\delta^{3}({\bf x-y})\,,$ (15)
and defines the number operator for the particles as $N({\bf p})=\bar{a}({\bf
p})a({\bf p})$, which has positive eigenvalues, one gets a Hamiltonian as
$\displaystyle H=-\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,.$ (16)
The negative sign in the Hamiltonian arises because the Lagrangian of the
fermionic field as given in Eq. (8) carries a negative sign. Similarly the
anti-fermions also carry negative energy. In this article we are following the
convention of Lee and Wick as given in the first appendix in Ref. [2]. In
their convention the fermionic algebra can be set by a set of two
anticommutation relations (one for the positive definite norm, the other for
the indefinite norm) for the creation and annihilation operators of the
fermions and correspondingly there will be two expressions of the number
operator. According to Lee and Wick a fermionic system can be quantized by the
following set of rules:
$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=\pm\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf
k}\right)\,,\,\,\,N({\bf p})=\pm\bar{a}_{s}({\bf p})\,a_{s}({\bf p})\,.$ (17)
When dealing with the indefinite metric theory we chose the minus sign in the
anticommutator and the minus sign in the number operator. On the other hand
when we are dealing with fermion fields whose states live in a normal Hilbert
space we choose the positive sign of the anticommutator and the positive
signed number operator.
## 3 Thermal distribution functions of the regulator fields
If some one assumes the existence of such exotic negative energy Lee-Wick
fields, may be during the earliest phases of the universe, then one can
calculate the thermodynamics of the negative energy Lee-Wick fields. The
calculation of the thermodynamics of the negative energy fields starts with
the prediction of a thermal distribution function of such fields. In this
section the thermal distribution function of both the bosonic and the
fermionic Lee-Wick partners are calculated.
For the bosonic excitations of the negative energy Lee-Wick fields we know
that the number operator and Hamiltonian has the form as given in Eq. (5) and
Eq. (7). In the present case the Hamiltonian of the fields as given in Eq. (7)
has negative eigenvalues because the fields are assumed to be quantized with
the wrong sign as in Eq. (14). Due to the presence of on-shell excitations the
thermal vacuum becomes $|\Omega\rangle\equiv|n({\bf p_{1}}),n({\bf
p_{2}}),\cdots\rangle$ where $n({\bf p_{1}})$ is the number of excitations
carrying momentum ${\bf p_{1}}$. The action of the number operator on such a
vacuum is
$\displaystyle N({\bf p})|\Omega\rangle=n({\bf p})|\Omega\rangle\,.$ (18)
As the value of $n({\bf p})$ actually turns out to be negative for the kind of
Lee-Wick theory we are considering so in reality $|n({\bf p})|$ gives the
number of particles with momentum ${\bf p}$ and energy $\epsilon({\bf p})$
which are missing from the thermal vacuum. In this analysis we will consider a
non-interacting real scalar field for which the chemical potential $\mu=0$. In
the present case with a Hamiltonian of the form as given in Eq. (7), where the
number operator has negative eigenvalues, the single particle partition
function will be,
$\displaystyle z_{\rm LW}^{B}={\rm Tr}e^{-\beta H}=\sum_{|n({\bf
p})|=0}^{\infty}e^{\beta\left|n({\bf p})\right|\epsilon(\bf p)},$ (19)
where $\beta=\frac{1}{T}$. From the last equation it is seen that the series
representing the single particle partition function for the Lee-Wick partner
of a Standard model boson does not converge for $\beta>0$. Consequently we
regularize the last expression by cutting off the summation for a finite value
of $n({\bf p})$ as:
$\displaystyle z_{\rm LW}^{B}=\sum_{|n({\bf p})|=0}^{M-1}e^{\beta\left|n({\bf
p})\right|\epsilon({\bf p})}=\frac{1-e^{\beta\epsilon({\bf
p})M}}{1-e^{\beta\varepsilon({\bf p})}}\,,$ (20)
where $M$ is a dimensionless cut-off which can be made indefinitely big at the
end of the calculation.
Next we calculate the thermal distribution function of the field excitations
from the expression of the single cell partition function of the field
excitations as given in Eq. (20). In conventional statistical mechanics we can
find the single cell distribution function via
$\displaystyle f({\bf p})=\frac{1}{\beta}\left(\frac{\partial\ln
z}{\partial\mu}\right)_{V,\beta}\,,$ (21)
where $\mu$ is an auxiliary chemical potential whose exact nature is not
important for our purpose. In presence of an auxiliary chemical potential the
single particle partition function can be written as
$\displaystyle z_{\rm LW}^{B}=\sum_{|n({\bf p})|=0}^{M-1}e^{\beta\left|n({\bf
p})\right|\\{\epsilon({\bf p})-\mu\\}}=\frac{1-e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}M}}{1-e^{\beta\left\\{\epsilon({\bf p})-\mu\right\\}}}\,.$
(22)
Applying conventional methods, the distribution function can also be written
as
$\displaystyle f_{\rm B}({\bf p})=\frac{1}{\beta}\left(\frac{\partial\ln
z^{B}_{\rm LW}}{\partial\mu}\right)_{V,\beta}\,,$ (23)
which comes out to be,
$\displaystyle f_{\rm B}({\bf p})=-\frac{e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}}{1-e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}}+M\frac{e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}M}}{1-e^{\beta\left\\{\epsilon({\bf p})-\mu\right\\}M}}\,.$
(24)
Now setting the auxiliary chemical potential to be zero we get the
distribution function of the fields as:
$\displaystyle f_{\rm B}({\bf p})=-\frac{1}{e^{-\beta\epsilon({\bf
p})}-1}+\frac{M}{e^{-\beta\epsilon({\bf p})M}-1}\,.$ (25)
Figure 1: The plot of the distribution function as given in Eq. (25). The
topmost curve is for a normal Bose-Einstein distribution and the lower two
curves correspond for $f_{\rm B}({\bf p})$ for the ultraviolet cutoff $M=100$
and $200$. The inverse temperature in all the cases is $.01{\rm GeV}^{-1}$ and
the energy $\epsilon({\bf p})$ is in GeV.
This is the distribution function of the fields whose Lagrangian is as given
in Eq. (1). These fields are quantized via Eq. (14). These are not the fields
which appear in the Standard model of particle physics. The distribution
function as plotted in Fig. 1 shows that the average excitation per energy
level is negative definite. Obviously these systems describe a physical theory
which is non-trivial and the negative sign of the distribution is only
meaningful when compared with the positive definite distribution of the normal
Standard model bosons. In general these kind of distributions will produce
negative energy density and pressure but once these energy density and
pressure is added with the positive energy density and pressure of the
Standard model bosons we get a net positive energy density and pressure. The
important point to notice about the distributions is that there is no pile up
of quantas near $\epsilon({\bf p})=0$ as is in the case of the Bose-Einstein
distribution. The reason being that the spectra of the Lee-Wick excitations
with negative energy, as given in Eq. (25), has two infinite spikes as the
energy tends to zero and they cancel each other near the origin.
If we take the Lagrangian of the fermionic fields as given in Eq. (8) and
quantize them via Eq. (15) then the anticommutators of the creation and the
annihilation operators, which define the fermionic excitations of the $\psi$
field discussed in the last section, is given as
$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf k}\right)\,,$
(26)
where $s$, $s^{\prime}$ may be some internal quantum numbers. We can proceed
in a similar way as done before and calculate the thermal distribution of
these excitations. If we use the conventional number operator $N({\bf
p})\equiv\bar{a_{s}}({\bf p})a_{s}({\bf p})$, which has positive eigenvalues
as 0 and 1, then the Hamiltonian of a single oscillator is
$\displaystyle H({\bf p})=-\frac{1}{2}\epsilon({\bf p})\left[\bar{a}_{s}({\bf
p}){a}_{s}({\bf p})-{a}_{s}({\bf p})\bar{a}_{s}({\bf p})\right]=-\epsilon({\bf
p})\left[N({\bf p})-\frac{1}{2}\,\delta^{3}({\bf 0})\right]\,,$ (27)
where $\frac{1}{2}\,\epsilon({\bf p})\,\delta^{3}({\bf 0})$ is the zero point
energy. The negative sign of the Hamiltonian is due to the negative sign of
the Lagrangian of the $\psi$ field. A similar analysis can be done for the
anti-fermions also. If we use this oscillator Hamiltonian to calculate the
single particle partition function there will be no problem related to the
convergence of the series. The single particle partition function for the
anticommuting fields turns out to be
$\displaystyle z_{\rm LW}^{F}=\sum_{n({\bf p})=0}^{1}e^{\beta n({\bf
p})\\{\epsilon({\bf p})-\mu\\}}=1+e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}\,,$ (28)
where $\mu$ is an auxiliary chemical potential. Now applying the formula in
Eq. (23) and setting $\mu=0$ at the end we get the distribution function for
the definite metric Lee-Wick partners of the Standard model fermions as:
$\displaystyle f_{\rm F}({\bf p})=-\frac{1}{e^{-\beta\epsilon({\bf p})}+1}\,.$
(29)
Figure 2: The plot of the distribution function as given in Eq. (29). The
topmost curve is for a normal Fermi-Dirac distribution at $\beta=.01{\rm
GeV}^{-1}$ and the lower two curves correspond for $f_{\rm F}({\bf p})$ for
$\beta$ values $.01{\rm GeV}^{-1}$ and $.1{\rm GeV}^{-1}$ and the energy
$\epsilon{({\bf p})}$ is in GeV.
Unlike the previous case, in the present scenario the distribution function
has no dependence on the dimensionless regulator $M$.
The negative sign of the distribution function signifies that the present
field configurations arises due to a de-excitation or loss of positive energy
particles. The vacuum defined is not stable and there exists much less
energetic states than the vacuum itself. These kind of fields are unstable.
The maximum energy of the field configurations is zero.
## 4 Energy density, pressure and entropy density from the distribution
function.
### 4.1 The bosonic case
To calculate the relevant thermodynamic quantities for the bosonic field from
a statistical mechanical point of view we will employ Eq. (25). The energy
density can be calculated using the following known equation
$\displaystyle\rho=\frac{g}{\left(2\pi\right)^{3}}\int\epsilon({\bf p})f_{\rm
B}(\epsilon)d^{3}p=\frac{g}{2\pi^{2}}\int_{0}^{\infty}\epsilon({\bf p})f_{\rm
B}(\epsilon)|{\bf p}|^{2}d{|\bf p}|\,,$ (30)
where $g$ stands for any intrinsic degree of freedom of the particle. For a
relativistic excitation $\epsilon^{2}={\bf p}^{2}+m^{2}$ where $m$ is the mass
of the bosonic excitations. Changing the integration variable from $|{\bf p}|$
to $\epsilon$ one gets
$\displaystyle\rho=-\frac{g}{2\pi^{2}}\int_{0}^{\infty}\left(\epsilon^{3}-\frac{m^{2}}{2}\epsilon\right)\frac{d\epsilon}{e^{-\beta\epsilon}-1}+\frac{Mg}{2\pi^{2}}\int_{0}^{\infty}\left(\epsilon^{3}-\frac{m^{2}}{2}\epsilon\right)\frac{d\varepsilon}{e^{-\beta\epsilon
M}-1}\,.$ (31)
In the above integral it is assumed that $|{\bf p}|\gg m$ and to have a closed
integral the lower limit of the integral is assumed to be zero. In the extreme
relativistic limit the system temperature $T\gg m$. Both of the integrals can
only be done when $\beta<0$, and in that case the result of the last integral
is
$\displaystyle\rho=-\frac{g}{2\pi^{2}}\left(\frac{\pi^{4}T^{4}}{15}-\frac{m^{2}\pi^{2}T^{2}}{12}\right)+\frac{g}{2\pi^{2}}\left(\frac{\pi^{4}T^{4}}{15M^{3}}-\frac{m^{2}\pi^{2}T^{2}}{12M}\right)\,.$
(32)
Analytically continuing the above result for $\beta>0$ and taking $M\to\infty$
we see that for normal temperatures the energy density for extreme
relativistic excitations of the bosonic fields is of the following form
$\displaystyle\rho=-g\left(\frac{\pi^{2}T^{4}}{30}-\frac{m^{2}T^{2}}{24}\right)\,.$
(33)
As expected, the energy density turns out to be negative for the excitations
in this case. The pressure of the bosonic field excitations can be found out
from
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{g}{\left(2\pi\right)^{3}}\int\frac{|{\bf
p}|^{2}}{3\epsilon}f_{\rm
B}(\epsilon)d^{3}p=\frac{g}{2\pi^{2}}\int_{0}^{\infty}\frac{|{\bf
p}|^{4}}{3\epsilon}f_{\rm B}(\varepsilon)d|{\bf p}|\,.$ (34)
Following similar steps as in the case of the energy density, it is seen that
the pressure of extremely relativistic excitations of the bosonic fields turns
out to be
$\displaystyle
p=-g\left(\frac{\pi^{2}T^{4}}{90}-\frac{m^{2}T^{2}}{24}\right)\,.$ (35)
The entropy density of the bosonic field is simply given by
$\displaystyle
s=\frac{\rho+p}{T}=-g\left(\frac{2\pi^{2}T^{3}}{45}-\frac{m^{2}T}{12}\right)\,.$
(36)
These values of the energy density, pressure and entropy density exactly match
the corresponding values calculated for the conventional, positive energy Lee-
Wick partners. In [9] the authors were trying to formulate thermodynamics for
a higher-derivative theory. The higher derivative theory was converted into
standard theory (theory up to a second derivative) with the introduction of
Lee-Wick partners whose states have indefinite norms. The authors in the
previous work did not quantize the system explicitly but were working with the
form of the propagators of the Lee-Wick partners.
If we assume that in the early universe for each bosonic degrees of freedom in
the Standard model there exist a corresponding Lee-Wick bosonic degree of
freedom whose field configuration has negative energy then the net energy
density, pressure and entropy density of the early universe turns out to be
$\displaystyle\rho_{B}=\rho_{\rm
SM}+\rho=\frac{gm^{2}T^{2}}{24}\,,\,\,\,\,p_{B}=p_{\rm
SM}+p=\frac{gm^{2}T^{2}}{24}\,,\,\,\,\,s_{B}=s_{\rm
SM}+s=\frac{gm^{2}T}{12}\,,$ (37)
which are all positive as expected. Here the energy density, pressure and
entropy density for the Standard model bosonic particles are $\rho_{\rm
SM}=g\frac{\pi^{2}T^{4}}{30}$, $p_{\rm SM}=g\frac{\pi^{2}T^{4}}{90}$ and
$s_{\rm SM}=g\frac{2\pi^{2}T^{3}}{45}$ respectively [16].
### 4.2 The fermionic case
In this subsection we apply Eq. (29) to find the energy density, pressure and
entropy density of the fermionic excitations. In this case the distribution
function do not have any dependence on the regulator $M$. For relativistic
excitations the integrals which give the energy density and pressure for the
fermionic case are exactly similar with the bosonic case except that now we
have to use the distribution for the fermions. The integrals can be easily
done, granted $\beta<0$, but the results can be analytically continued for
positive temperatures. The results in this case are listed below. The energy
density, pressure and entropy density of the Lee-Wick partners are as follows:
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{4}}{240}-\frac{m^{2}T^{2}}{48}\right)\,,$
(38) $\displaystyle p$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{4}}{720}-\frac{m^{2}T^{2}}{48}\right)\,,$
(39) $\displaystyle s$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{3}}{180}-\frac{m^{2}}{24}\right)\,.$
(40)
The energy density and pressure quoted above are equivalent to the energy
density and pressure for the positive energy Lee-Wick partners as calculated
in [9] for the special case of $g=2$. If we assume that to each unusual
fermionic degree of freedom there corresponds one standard fermionic degree
from the Standard model, then the total fermionic contribution is
$\displaystyle\rho_{F}=\rho_{\rm
SM}+\rho=\frac{gm^{2}T^{2}}{48}\,,\,\,\,p_{F}=p_{\rm
SM}+p=\frac{gm^{2}T^{2}}{48}\,,\,\,\,s_{F}=s_{\rm SM}+s=\frac{gm^{2}T}{24}\,$
(41)
which are all positive. Here the energy density, pressure density and entropy
density for the Standard model fermionic particles are $\rho_{\rm
SM}=g\frac{7\pi^{2}T^{4}}{240}$, $p_{\rm SM}=g\frac{7\pi^{2}T^{4}}{720}$ and
$s_{\rm SM}=g\frac{7\pi^{2}T^{3}}{180}$ respectively [16].
It is worth pointing out here that there is some confusion regarding higher
derivative theories of fermions. The confusion is regarding the number of Lee-
Wick partners (one left-handed and the other right-handed) of the chiral
fermions. The authors of Ref. [9] claim that there will be two positive energy
Lee-Wick partners of a chiral fermion which are interrelated. Where as in Ref.
[17] the author claims that the two positive energy Lee-Wick partners of the
chiral fermion may not be interdependent. In that case the Lee-Wick degrees of
freedom exceeds the one of its Standard model partner yielding negative
energy, pressure and entropy density. This issue is yet to be resolved.
## 5 Discussion and conclusion
Initially it was pointed out that Lee-Wick’s idea of implementing the Pauli-
Villars regularization scheme can be implemented in two ways. This idea was
presented by Boulware and Gross [12] way back in 1984\. In one way the
regulator fields live in a indefinite metric space but carry positive energy
and in the other way the regulator fields live in a definite metric space but
carry negative energy. Lee and Wick took the first option and tried to redress
the issue of indefinite norm in such a way that unitarity is preserved in the
theory. The second option remained uncultivated. In this article we explored
the second option with limited means. No cures for the energy instability of
these kind of theories are known to the present authors. The results presented
in the article show the dubious nature of the energy instability of the
fields, but instead of making the theory meaningless the same instabilities
conspire to produce a result which matches with the thermodynamics of the Lee-
Wick partners living in the indefinite metric space.
In this article we have studied a system of bosonic and fermionic fields,
whose Lagrangians have the wrong sign and, which are quantized with the wrong
sign of the commutators and the anticommutators. These fields are Lee-Wick
partners who live in a normal Hilbert space but have negative energy
excitations. The negative energy of the field configuration is not due to any
particular form of the potential but solely an outcome of the negative sign of
the Lagrangian and the modified quantization process. The vacuum of the theory
is not the state with the lowest energy, it is rather the state with the
maximal energy making the field configuration unstable. The bosonic and
fermionic degrees of freedom do still follow commutation and anticommutation
relations and specifically the fermionic fields still follow the Pauli
exclusion principle. In this article the emphasize had been on the calculation
of energy density, pressure and entropy density of the unusual field
configurations.
To calculate the above mentioned thermodynamic quantities one requires to have
a statistical mechanics of the field excitations. One encounters the
difficulty of a diverging sum when calculating the single particle partition
function of the bosonic fields. Keeping to conventional ways, where the
temperature of the system is positive definite, the partition function can
only be summed when one uses an ultraviolet cutoff. The distribution function
calculated from the partition function turns out to be negative definite,
which is a nontrivial result. The negative nature of the distribution function
implies that there must be an average loss of particles in any energy level.
The energy density, pressure calculated from the distribution functions of the
unusual fields discussed in this article match exactly with the results
calculated by Fornal, Grinstein and Wise in [9]. The derivation of the new
distribution functions and the connection between the regulator field
thermodynamics as presented in this article and the thermodynamics of the Lee-
Wick partners, as presented in Ref. [9], is one of the main motivations for
this work. The main emphasize of the present article has not been to
recalculate the results obtained in Ref. [9] as the theory presented in this
article is not the same as that of Ref. [9]. The two theories are quantized
differently. The similarity of the thermodynamic results of the two variants
of the Lee-Wick model implies that the different kind of instabilities
plaguing the theories are related in presence of a thermal bath. From the main
analysis of this article it can be inferred that the conventional Lee-Wick
prescription is equivalent to its variant in the thermodynamic sector. The
analysis of the thermodynamics of the positive definite metric Lee-Wick
partners gives a clear idea about the origin of the negative energy density
and pressure of these field configurations. The theory presented in the
article is amenable to the standard techniques of finite temperature field
theory. One can utilize the results of finite-temperature quantum field theory
to calculate various thermal effects, like the thermal mass of the Standard
model particles in presence of the thermalized Lee-Wick partners, in the early
universe. The thermal distribution functions of the Lee-Wick partners
calculated in the present article can be used to write the propagators of the
unusual fields in the real-time formalism [18].
The fact that the thermodynamics of the indefinite metric, positive energy
Lee-Wick fields and the thermodynamics of the definite metric but negative
energy Lee-Wick partners turns out to be the same remains an interesting
result which invites further work in these fields in the future.
Acknowledgement: We want to thank Subhendra Mohanty for pointing out some
subtle ideas involved in this article.
## References
* [1] T. D. Lee and G. C. Wick, Nucl. Phys. B 9, 209 (1969).
* [2] T. D. Lee and G. C. Wick, Phys. Rev. D 2, 1033 (1970).
* [3] B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev. D 77, 025012 (2008) [arXiv:0704.1845 [hep-ph]].
* [4] C. D. Carone and R. F. Lebed, Phys. Lett. B 668, 221 (2008) [arXiv:0806.4555 [hep-ph]].
* [5] C. D. Carone and R. F. Lebed, JHEP 0901, 043 (2009) [arXiv:0811.4150 [hep-ph]].
* [6] C. D. Carone, Phys. Lett. B 677, 306 (2009) [arXiv:0904.2359 [hep-ph]].
* [7] C. D. Carone and R. Primulando, Phys. Rev. D 80, 055020 (2009) [arXiv:0908.0342 [hep-ph]].
* [8] Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D 80, 023511 (2009) [arXiv:0810.4677 [hep-th]].
* [9] B. Fornal, B. Grinstein and M. B. Wise, Phys. Lett. B 674, 330 (2009) [arXiv:0902.1585 [hep-th]].
* [10] R. Dashen, S. K. Ma and H. J. Bernstein, Phys. Rev. 187, 345 (1969).
* [11] W. Pauli, Rev. Mod. Phys. 15, 175 (1943).
* [12] D. G. Boulware and D. J. Gross, Nucl. Phys. B 233, 1 (1984).
* [13] W. Pauli and F. Villars, Rev. Mod. Phys. 21 (1949) 434.
* [14] G. Sudarshan, Phys. Rev. 123 (1961) 2183.
* [15] M. E. Arons, M. Y. Han and G. Sudarshan, Phys. Rev. 137 (1965) B1085.
* [16] Kolb & Turner, The Early Universe (Westview Press, 1994) Chapter: 3
* [17] M. B. Wise, Int. J. Mod. Phys. A 25, 587 (2010) [arXiv:0908.3872 [hep-ph]].
* [18] J. F. Nieves, Phys. Rev. D 42, 4123 (1990) [Erratum-ibid. D 49, 3067 (1994)] [Phys. Rev. D 49, 3067 (1994)].
|
arxiv-papers
| 2011-08-02T05:21:30 |
2024-09-04T02:49:21.220581
|
{
"license": "Public Domain",
"authors": "Kaushik Bhattacharya, Suratna Das",
"submitter": "Kaushik Bhattacharya",
"url": "https://arxiv.org/abs/1108.0483"
}
|
1108.0836
|
# A new type of reflected backward doubly stochastic differential equations
††thanks: The work of Auguste Aman is support by TWAS Research Grants to
individuals (No. 09-100 RG/MATHS/AF/AC-I- UNESCO FR: 3240230311) and it been
dedicated to all the dead of the post-election crisis in his country.
The work of Yong Ren is supported by the National Natural Science Foundation
of China (No 10901003), the Key Project of Chinese Ministry of Education (No
211077) and the Anhui Provincial Natural Science Foundation (No 10040606Q30).
Auguste Aman1 and Yong Ren 2
1\. U.F.R Mathématiques et Informatique, Université de Cocody,
582 Abidjan 22, Côte d’Ivoire
2\. Department of Mathematics, Anhui Normal University,
Wuhu 241000, China augusteaman5@yahoo.fr, corresponding authorrenyong@126.com
and brightry@hotmail.com
###### Abstract
In this paper, we introduce a new kind of "variant" reflected backward doubly
stochastic differential equations (VRBDSDEs in short), where the drift is the
nonlinear function of the barrier process. In the one stochastic case, this
type of equations have been already studied by Ma and Wang [25]. They called
it as "variant" reflected BSDEs (VRBSDEs in short) based on the general
version of the Skorohod problem recently studied by Bank and El Karoui [6].
Among others, Ma and Wang [25] showed that VRBSDEs is a novel tool for some
problems in finance and optimal stopping problems where no existing methods
can be easily applicable. Since more of those models have their stochastic
counterpart, it is very useful to transpose the work of Ma and Wang [25] to
doubly stochastic version. In doing so, we firstly establish the stochastic
variant Skorohod problem based on the stochastic representation theorem, which
extends the work of Bank and El Karoui [6]. We prove the existence and
uniqueness of the solution for VRBDSDEs by means of the contraction mapping
theorem. By the way, we show the comparison theorem and stability result for
the solutions of VRBDSDEs.
AMS Subject Classification: 60H15; 60H20
Keywords: Reflected backward doubly stochastic differential equation,
stochastic Skorohod problem, stochastic representation theorem.
## 1 Introduction
The theory of backward stochastic differential equations (BSDEs in short) was
developed by Pardoux and Peng [31]. Precisely, given a data $(\xi,f)$
consisting of a progressively measurable process $f$, so-called the generator,
and a square integrable random variable $\xi$, they proved the existence and
uniqueness of an adapted process $(Y,Z)$ solution to the following BSDEs:
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})\,{\rm d}s-\int_{t}^{T}Z_{s}\,{\rm
d}B_{s},\quad 0\leq t\leq T.$
These equations have attracted great interest due to their connections with
mathematical finance [16, 17], stochastic control and stochastic games [20,
21, 22]. Furthermore, it was shown in various papers that BSDEs give the
probabilistic representation for the solution (at least in the viscosity
sense) of a large class of systems of semi-linear parabolic partial
differential equations (PDEs in short) [29, 30, 32, 34].
Further, other settings of BSDEs have been proposed. Especially, El-Karoui et
al. [15] have introduced the notion of reflected BSDEs (RBSDEs in short),
which is a BSDE but the solution is forced to stay above a lower barrier. In
details, a solution of such equations is a triple of processes $(Y,Z,K)$
satisfying that
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})\,{\rm
d}s+K_{T}-K_{t}-\int_{t}^{T}Z_{s}\,{\rm d}B_{s},\ Y_{t}\geq S_{t},$ (1.1)
where $S$, so-called the barrier, is a given stochastic process. The role of
the continuous increasing process $K$ is to push the state process upward with
the minimal energy, in order to keep it above $S$; in this sense, it satisfies
$\int_{0}^{T}(Y_{t}-S_{t})\,{\rm d}K_{t}=0.$ RBSDEs have been proven to be
powerful tools in mathematical finance [13, 19], the mixed game problems [12,
23], providing a probabilistic formula for the viscosity solution of an
obstacle problem for a class of parabolic PDEs ([14, 15, 37]) and so on. On
other interesting results on RBSDEs driven by a Brownian motion with different
barrier conditions, one can see Hamadène [18], Lepeltier and Xu [24] and Peng
and Xu [35].
Very recently, Ma and Wang [25] introduced the so-called Variant Reflected
Backward Stochastic Differential Equations (VRBSDEs in short) associated with
the notion of variant Skorohod problem studied by Bank and El Karoui [6], that
is
$Y_{t}=E\left\\{X_{T}+\int_{t}^{T}f(s,Y_{s},A_{s})ds|\mathcal{F}_{t}\right\\},0\leq
t\leq T,$ (1.2)
where $X=\\{X_{t}\\}_{t\geq 0}$ is an optional process of class (D) and the
solution $(Y,A)$ satisfies that
1. (i)
$Y_{t}\leq X_{t},0\leq t\leq T,Y_{T}=X_{T};$
2. (ii)
$A=\\{A_{t}\\}$ is an adapted, increasing process such that $A_{0-}=-\infty$,
and the following flat-off condition holds
$E\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0.$ (1.3)
In addition, if the filtration $\mathcal{F}$ is generated by a Brownian motion
$B$, then (1.2) has the following extension form
$dY_{t}=-f(t,Y_{t},Z_{t},A_{t})dt+Z_{t}dB_{t},\ Y_{t}\leq X_{t},0\leq t\leq
T,\ Y_{T}=X_{T}.$ (1.4)
Unlike the role of $K$ in (1.1) as stated previously, the process $A$ here
could be regarded as a density of a reflecting force, which acts through the
drift $f$ in a nonlinear manner. From the above statements, we can see that
there is great difference between the frameworks of RBSDEs and VRBSDEs. Also,
even the fundamental well-posedness property of the VRBSDE cannot be obtained
by means of the usual ways used in BSDE and RBSDE. This brand new kind of
BSDEs has some important applications in finance and optimal stopping problems
([25]).
In [33], Pardoux and Peng proposed another class of BSDEs, named backward
doubly stochastic differential equations (BDSDEs in short) with the form:
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})dB_{s}-\int_{t}^{T}Z_{t}dW_{t},0\leq
t\leq T,$ (1.5)
where the integral with respect to $\\{B_{t}\\}$ is a backward Itô integral
and the integral with respect to $\\{W_{t}\\}$ is a standard forward Itô
integral. Those two types of integrals are particular cases of the It
-Skorohod integral, see Nualart and Pardoux [28]. Following it, some well-
known works have been done in the probabilistic representation of certain
quasi-linear stochastic partial differential equations by means of BDSDEs from
different aspects, one can see Bally and Matoussi [5], Boufoussi et al. [7,
8], Buckdahn and Ma [9, 11, 10], Matoussi and Scheutzow [26], Zhang and Zhao
[38] and the references therein. Based on the reflected framework of El-Karoui
et al. [15], Bahlali et al. [4], Aman [2] and Ren [36] respectively proved the
existence and uniqueness of the solution for a class of reflected BDSDEs
(RBDSDEs in short) driven by Brownian motions and Lévy processes. Especially,
very recently, Matoussi and Stoica [27] proved the existence and uniqueness
result for the obstacle problem of quasi-linear parabolic stochastic PDEs by
means of the RBDSDEs.
Motivated by the aforementioned works, in this paper, we study a class of
variant reflected backward doubly stochastic differential equations (VRBDSDEs
in short). In doing so, we firstly establish the stochastic variant Skorohod
problem based on the stochastic representation theorem, which extends the work
of Bank and El Karoui [6]. We prove the existence and uniqueness of the
solution for VRBDSDEs by means of the contraction mapping theorem. In
addition, we show the comparison theorem and the stability result for the
solutions of VRBDSDEs.
Let us describe our plan. First, the formulation of the problems is proposed
in Section 2. The main results are presented in Section 3.
## 2 Formulation of the problems
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $T>0$ be
fixed throughout this paper. Let $\\{W_{t},\,0\leq t\leq T\\}$ and
$\\{B_{t},\,0\leq t\leq T\\}$ be two mutually independent standard Brownian
motion processes, with values respectively in $\mathbb{R}^{d}$ and in
$\mathbb{R}^{\ell}$, define on $(\Omega,\mathcal{F},\mathbb{P})$. Let
$\mathcal{N}$ denote the class of $\mathbb{P}$-null sets of $\mathcal{F}$. For
each $t\in[0,T]$, let us define
$\mathcal{F}_{t}=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{t,T}^{B},$
where for any process $\\{\eta_{t};0\leq t\leq
T\\},\;\mathcal{F}_{s,t}^{\eta}=\sigma\\{\eta_{r}-\eta_{s};s\leq r\leq
t\\}\vee\mathcal{N}$ and, $\mathcal{F}_{t}^{\eta}=\mathcal{F}_{0,t}^{\eta}$.
Knowing that $\\{\mathcal{F}_{t}^{W},t\in[0,T]\\}$ is an increasing filtration
and $\\{\mathcal{F}_{t,T}^{B},t\in[0,T]\\}$ is a decreasing filtration, the
collection $\\{\mathcal{F}_{t},t\in[0,T]\\}$ is neither increasing nor
decreasing so it does not constitute a filtration.
Let us describe the following spaces will frequently used in the sequel.
$\bullet$ For any $n\in\mathbb{N}$, $\mathcal{M}^{2}(0,T,\mathbb{R}^{n})$
denotes the set of (class of $d\mathbb{P}\otimes dt$ a.e.) $n$-dimensional
jointly measurable random processes $\\{\varphi_{t};0\leq t\leq T\\}$ such
that
$\|\varphi\|_{\mathcal{M}^{2}}^{2}=\mathbb{E}\left(\int_{0}^{T}\mid\varphi_{t}\mid^{2}dt\right)<+\infty.$
$\bullet$ $\mathcal{S}^{\infty}([0,T],\mathbb{R})$ denotes the set of one
dimensional continuous $\mathcal{F}_{t}$-measurable bounded random processes.
$\bullet$ $\mathbb{L}^{\infty}(\mathbb{R})$ denotes the space of all
$\mathcal{F}_{T}$-measurable bounded random variables.
$\bullet$ $\mathcal{M}_{0,T}$ denotes the space of all stopping times taking
values in $[0,T]$.
$\bullet$ The process $X$ is said to belong to Class (D) on $[0,T]$ if the
family of random variables $\\{X_{\tau},\,\tau\in\mathcal{M}_{0,T}\\}$ is
uniformly integrable.
Next, let us give the standing assumptions relative to VRBDSDE.
(A1) The boundary processes $X=\\{X_{t},0\leq t\leq T\\}$ is assumed to be an
optional process of class (D) and is lower-semi-continuous in expectation,
(A2) The coefficients
$f:[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ and
$g:[0,T]\times\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy the
following assumptions:
1. (i)
for fixed $(\omega,t,y)\in\Omega\times[0,T]\times\mathbb{R}$, the function
$f(t,\omega,y,\cdot)$ is continuous and strictly decreasing from $+\infty$ to
$-\infty$;
2. (ii)
for fixed $(y,l)\in\mathbb{R}^{2}$, the processes $f(\cdot,\cdot,y,l)$ and
$g(\cdot,\cdot,y)$ are jointly measurable with
$\displaystyle\mathbb{E}\int_{0}^{T}[|f(t,\omega,y,l)|+|g(t,\omega,y)|^{2}]dt<+\infty;$
3. (iii)
there exists a constant $L>0$, such that for all fixed $t,\omega,l,$ it holds
that
$\displaystyle|f(t,\omega,y,l)-f(t,\omega,y^{\prime},l)|$ $\displaystyle\leq$
$\displaystyle L|y-y^{\prime}|,$
$\displaystyle|g(t,\omega,y)-g(t,\omega,y^{\prime})|$ $\displaystyle\leq$
$\displaystyle L|y-y^{\prime}|,\;\;\forall\,y,\,y^{\prime}\in\mathbb{R};$
4. (iv)
there exist two constants $k>0$ and $K>0$, such that for all fixed
$t,\omega,y,$ it holds that
$\displaystyle
k|l-l^{\prime}|\leq|f(t,\omega,y,l)-f(t,\omega,y,l^{\prime})|\leq
K|l-l^{\prime}|,\;\;\forall\,l,\,l^{\prime}\in\mathbb{R}.$
Given $\xi\in\mathbb{L}^{2}(\mathbb{R})$ and the boundary process $X$, we
consider the following VRBDSDE.
(i)
$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},A_{s})ds+\int_{t}^{T}g(s,Y_{s})dB_{s}-\int_{t}^{T}Z_{s}dW_{s},\;0\leq
t\leq T;$ (2.1)
(ii)
$Y_{t}\leq X_{t},\;0\leq t\leq T,\;\;Y_{T}=X_{T}=\xi;$
(iii)
the process $(A_{t})_{t}$ is $\mathcal{F}_{t}$-measurable, increasing, càdlàg
(right continuous with left limits), and $A_{0^{-}}=-\infty$, such that
$\displaystyle{\mathbb{E}\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0}$.
The study of this new type of BDSDEs is based on the extension of Stochastic
Representation Theorem initiated by Bank and El Karoui [6]. To do this, let us
consider the following filtration $(\mathcal{G}_{t})_{t\geq 0}$ defined by
$\displaystyle\mathcal{G}_{t}=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{T}^{B}.$
###### Theorem 2.1.
Assume (A2)–(i), (ii). Then, every optional process $X$ of class (D) which is
lower semi-continuous in expectation admits a representation of the form
$\displaystyle X_{S}=\mathbb{E}\left\\{X_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq
u}L_{v}\right)du+\int_{S}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{S}\right\\}$
(2.2)
for any stopping times $S\in\mathcal{M}_{0,T}$, where $L$ is an optional
process taking values in $\mathbb{R}\cup\\{-\infty,+\infty\\}$, and it can be
characterized as follows
$(i)$
$\,f\left(u,\sup_{S\leq v\leq u}L_{v}\right)\in L^{1}(\mathbb{P}\otimes
dt),\,g\left(u\right)\in L^{2}(\mathbb{P}\otimes dt)$ for any stopping times
$S$,
$(ii)$
$\,L_{S}=\mbox{ess}\inf_{\tau>S}l_{S,\tau}$, where the "$\mbox{ess}\inf$" is
taken over all stopping times $S\in\mathcal{M}_{0,T}$ such that $S<T$, a.s.;
and $l_{S,\tau}$ is the unique $\mathcal{F}_{S}$-measurable random variable
satisfying that
$\displaystyle\mathbb{E}\\{X_{S}-X_{\tau}|\mathcal{F}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int_{S}^{\tau}g\left(u\right)dB_{u}|\mathcal{F}_{S}\right\\},$
$(iii)$
if $\;V(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{X_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du+\int_{t}^{\tau}g(u)dB_{u}|\mathcal{F}_{t}\right\\},\,t\in[0,T]$,
is the value functions of a family of optimal stopping problems indexed by
$l\in\mathbb{R}$, then
$\displaystyle L_{t}=\sup\\{l:V(t,l)=X_{t}\\},\;\;\;t\in[0,T].$
###### Proof.
Let $X$ be a optional process $X$ of class (D) which is lower semi-continuous
in expectation and $g$ be a function given above. Setting
$\displaystyle\widetilde{X}_{t}=X_{t}+\int^{t}_{0}g(u)dB_{u},$
according to assumption $({\bf A1})$ and $({\bf A2})$, it is clear that
$\widetilde{X}$ is a optional process of class (D) and is lower semi-
continuous. Therefore, it follows from Theorem 3 in [6] that there exists an
optional process $L$ taking values in $\mathbb{R}\cup\\{-\infty,+\infty\\}$
such that for any stopping times $S\in\mathcal{M}_{0,T}$,
$\displaystyle\widetilde{X}_{S}=\mathbb{E}\left\\{\widetilde{X}_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq u}L_{v}\right)du|\mathcal{G}_{S}\right\\}.$ (2.3)
Moreover, $L$ is characterized as follows:
* •
$\,f\left(u,\sup_{S\leq v\leq u}L_{v}\right)\in L^{1}(\mathbb{P}\otimes dt)$
for any stopping times $S$, which satisfy $(i)$.
* •
$\,L_{S}=\mbox{ess}\inf_{\tau>S}l_{S,\tau}$, where the "$\mbox{ess}\inf$" is
taken over all stopping times $S\in\mathcal{M}_{0,T}$ such that $S<T$, a.s.;
and $l_{S,\tau}$ is the unique $\mathcal{G}_{S}$-measurable random variable
satisfying that
$\displaystyle\mathbb{E}\\{\widetilde{X}_{S}-\widetilde{X}_{\tau}|\mathcal{G}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du|\mathcal{G}_{S}\right\\}.$
(2.4)
* •
If $\widetilde{V}(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{\widetilde{X}_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du|\mathcal{G}_{t}\right\\},\,t\in[0,T]$,
is the value functions of a family of optimal stopping problems indexed by
$l\in\mathbb{R}$, then
$\displaystyle
L_{t}=\sup\\{l:\widetilde{V}(t,l)=\widetilde{X}_{t}\\},\;\;\;t\in[0,T].$
Since $\widetilde{X}_{S}$ is $\mathcal{F}_{S}$-measurable and
$\mathcal{F}_{S}\subset\mathcal{G}_{S}$, and according to the definition of
$\widetilde{X}$, it follows from equalities (2.3) and (2.4) that
$\displaystyle X_{S}=\mathbb{E}\left\\{X_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq u}L_{v}\right)du+\int^{T}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\}$
and
$\displaystyle\mathbb{E}\\{X_{S}-X_{\tau}|\mathcal{F}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\},$
(2.5)
respectively. To prove $(ii)$, it remains to show that $l_{S,\tau}$ is a
$\mathcal{F}_{S}$-measurable random variable, which is clear by (2.4). To end
the proof let us show $(iii)$. In fact, equalities (2.4) and (2.5) provide
$\displaystyle\mathbb{E}\left\\{X_{\tau}+\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{G}_{S}\right\\}$
$=\mathbb{E}\left\\{X_{\tau}+\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\}.$
Hence, denoting
$\displaystyle V(t,l)=\widetilde{V}(t,l)-\int^{\tau}_{t}g(u)dB_{u},$
we have
$\displaystyle V(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{X_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{t}\right\\}$
and
$\displaystyle L_{t}=\sup\\{l:\,V(t,l)=X_{t}\\},\;\;\;t\in[0,T],$
which prove $(iii)$. ∎
A direct consequence of the previous stochastic representation theorem is the
following stochastic variant Skorohod problem.
###### Theorem 2.2.
Assume (A2)–(i), (ii). Then, for every optional process $X$ of class (D) which
is lower semi-continuous in expectation, there exists a unique pair of
$\mathcal{F}_{t}$-measurable processes $(Y,A)$, where $Y$ is continuous and
$A$ is increasing such that
$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{t}\right\\},\,\,t\in[0,T].$
Furthermore, the process $A$ can be expressed as $A_{t}=\sup_{0\leq s\leq
t^{+}}L_{s}$, where $L$ is the process in Theorem 2.1.
Before give the proof of the above theorem, let us give a remark.
###### Remark 2.3.
The previous theorem can be enounced as follows: there exists a unique pair of
$\mathcal{F}_{t}$-measurable processes $(Y,Z,A)$, where $Y$ is continuous and
$A$ is increasing such that
$\displaystyle
Y_{t}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},\,\,t\in[0,T].$
###### Proof.
Let us define $A_{t}=\sup_{0\leq s\leq t^{+}}L_{s}$, where $L$ is the process
appears in (2.2) and the $\mathcal{G}_{t}$-square integrable martingale
$\displaystyle
M_{t}=\mathbb{E}\left\\{X_{T}+\int_{0}^{T}f\left(u,A_{u}\right)du+\int_{0}^{T}g\left(u\right)dB_{u}|\mathcal{G}_{t}\right\\},\;\;0\leq
t\leq T.$
An obvious extension of the Itô martingale representation theorem yields the
existence of a $\mathcal{G}_{t}$-progressively measurable process
$\\{Z_{t}\\}$ with values in $\mathbb{R}^{d}$ such that
$\displaystyle\mathbb{E}\left(\int^{T}_{0}|Z_{s}|^{2}ds\right)<+\infty,$
$\displaystyle M_{t}=M_{0}+\int_{0}^{t}Z_{s}dW_{s},\;\;\;0\leq t\leq T.$
Hence,
$\displaystyle M_{T}=M_{t}+\int_{t}^{T}Z_{s}dW_{s},\;\;\;0\leq t\leq T.$
Replacing $M_{T}$ and $M_{t}$, by their defining formulas and subtracting
$\int_{0}^{t}f\left(u,A_{u}\right)du+\int_{0}^{t}g\left(u\right)dB_{u}$ from
both sides of the equality yields that
$\displaystyle
Y_{t}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},$
where
$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{G}_{t}\right\\}.$
(2.6)
It remains to show that $\\{Y_{t}\\}$ and $\\{Z_{t}\\}$ are
$\mathcal{F}_{t}$-measurable. For $Y_{t}$, this is obvious since for each $t$,
$\displaystyle
Y_{t}=\mathbb{E}\left\\{\Theta|\mathcal{F}_{t}\vee\mathcal{F}^{B}_{t}\right\\}.$
where $\Theta$ is $\mathcal{F}^{W}_{T}\vee\mathcal{F}^{B}_{t,T}$-measurable.
Hence $\mathcal{F}^{B}_{t}$ is independent of
$\mathcal{F}_{t}\vee\sigma(\Theta)$, and
$\displaystyle Y_{t}=\mathbb{E}\left\\{\Theta|\mathcal{F}_{t}\right\\}.$
Now
$\displaystyle\int^{T}_{t}Z_{u}dW_{u}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-Y_{t},$
and the right side is
$\mathcal{F}^{W}_{T}\vee\mathcal{F}^{B}_{t,T}$-measurable. Hence, from the Itô
martingale representation theorem $\\{Z_{s},t<s<T\\}$ is
$\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{t,T}$-adapted. Consequently, $Z_{s}$
is $\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{t,T}$-measurable, for any $t<s$ so
it is $\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{s,T}$ measurable. Therefore,
the equality (2) becomes
$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{t}\right\\},$
which shows the desired result. ∎
## 3 Main results
The main objective of this section is to prove the existence and uniqueness
result to the new type of reflected BDSDEs. As mentioned in [25], we use the
well-known contraction mapping theorem, to provide the existence and
uniqueness of the solution. Next, like as in [25], we derive the comparison
theorem and a stability result of such equations.
### 3.1 Existence and uniqueness
Let us make the following extra assumptions on the boundary process $X$ and
the coefficients $f$ and $g$.
(A3) There exists a constant $\Gamma>0$, such that
1. (i)
for any $\mu\in\mathcal{M}_{0,T}$, it holds that
$\displaystyle\mbox{ess}\sup_{\overset{\tau>\mu}{\tau\in\mathcal{M}_{0,T}}}\left\\{\left|\frac{\mathbb{E}\left\\{X_{\tau}-X_{\mu}|\mathcal{F}_{\mu}\right\\}}{\mathbb{E}\left\\{\tau-\mu|\mathcal{F}_{\mu}\right\\}}\right|+\left|\frac{\left[\left(\mathbb{E}\int_{\mu}^{\tau}|g(u,0)|^{2}du|\mathcal{F}_{\mu}\right)\right]^{1/2}}{\mathbb{E}\left\\{\tau-\mu|\mathcal{F}_{\mu}\right\\}}\right|\right\\}\leq\Gamma,\;\;a.s.;$
2. (ii)
$|f(t,0,0)|\leq\Gamma,\;\;\;t\in[0,T]$.
Let us consider the following mapping $\Phi$ on
$\mathcal{S}^{\infty}([0,T],\mathbb{R})$: for a given process
$y\in\mathcal{S}^{2}([0,T],\mathbb{R})$, we define
$\Phi(y)_{t}=Y_{t},\,t\in[0,T]$, where $(Y,Z,A)$ is the unique solution of the
variant Skorohod problem:
$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f\left(u,y_{u},A_{u}\right)du+\int_{t}^{T}g\left(u,y_{u}\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},\,\,t\in[0,T],$
(3.1) $\displaystyle\mathbb{E}\int^{T}_{0}|Y_{t}-X_{t}|dA_{t}=0.$
It follows from Theorem 2.1 and Theorem 2.2 that the reflecting process $A$ is
exactly determined by $y$ in this sense: $A_{t}=\sup_{0\leq s\leq t^{+}}L_{s}$
and $L$ satisfies the stochastic representation:
$\displaystyle X_{t}=\xi+\int_{t}^{T}f\left(u,y_{u},\sup_{t\leq v\leq
u}L_{v}\right)du+\int_{t}^{T}g\left(u,y_{u}\right)dB_{u}-\int_{t}^{T}\overline{Z}dW_{u},\;\;t\in[0,T].$
Our goal is to prove that the mapping $\Phi$ is a contraction from
$\mathcal{S}^{2}([0,T],\mathbb{R})$ to itself. However, it should be noted
that the contraction can only show the existence and uniqueness of $Y$; the
uniqueness of $A$ must be established separately.
We now derive some priori estimates that will be useful in the sequel. To
begin with, let us consider the stochastic representation
$\displaystyle X_{t}=\xi+\int_{t}^{T}f\left(u,0,\sup_{t\leq v\leq
u}L^{0}_{v}\right)du+\int_{t}^{T}g\left(u,0\right)dB_{u}-\int^{T}_{t}Z^{0}_{u}dW_{u}.$
Let us denote $A^{0}_{t}=\sup_{0\leq s\leq t^{+}}L_{s}^{0}$. Then we have the
following result.
###### Lemma 3.1.
Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Then, it holds that
$\displaystyle\|A^{0}\|_{\infty}\leq\frac{3\sqrt{3}\Gamma}{k},$ (3.2)
where $k$ and $\Gamma$ are the constants appearing in the previous
assumptions.
###### Proof.
For fixed $s\in[0,T]$ and any stopping times $\tau>s$, let us denote by
$l^{0}_{s,\tau}$ the $\mathcal{F}_{s}$-measurable random variable such that
$\displaystyle
X_{s}-X_{\tau}=\int_{s}^{\tau}f\left(u,0,l^{0}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,0\right)dB_{u}-\int^{\tau}_{s}Z^{0}_{u}dW_{u}.$
Then, it follows from Theorem 2.1 that
$L^{0}_{s}=\mbox{ess}\inf_{\tau>s}l_{s,\tau}^{0}$ and $A_{t}^{0}=\sup_{0\leq
s\leq t^{+}}L^{0}_{s}$. On the other hand, we have
$\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})-\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)-\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)$
$\displaystyle=\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,0,l^{0}_{s,\tau}\right)-f\left(u,0,0\right)]du|\mathcal{F}_{s}\right).$
(3.3)
On the set $\\{\omega,\,l_{s,\tau}^{0}(\omega)<0\\}$, since $f(t,0,\cdot)$ is
decreasing and $l^{0}_{s,\tau}$ is $\mathcal{F}_{s}$-measurable, we have
$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,0,l^{0}_{s,\tau}\right)-f\left(u,0,0\right)]du|\mathcal{F}_{s}\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}k|l^{0}_{s,\tau}|du|\mathcal{F}_{s}\right)$
$\displaystyle\geq$ $\displaystyle
k|l^{0}_{s,\tau}|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right).$
According to (3.3), we get
$\displaystyle\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})-\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)-\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)\geq
k|l^{0}_{s,\tau}|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right).$
In other words, on $\\{l_{s,\tau}^{0}<0\\}$, we have
$\displaystyle|l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{k}\left\\{\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}-\frac{\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}-\frac{\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right\\}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(3.4)
We can show similarly that on the set $\\{l_{s,\tau}^{0}>0\\}$ the following
relation holds
$\displaystyle l^{0}_{s,\tau}$ $\displaystyle\leq$
$\displaystyle\frac{1}{k}\left\\{-\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}+\frac{\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}+\frac{\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right\\}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(3.5)
Putting (3.4) and (3.5) together, we have
$\displaystyle|l^{0}_{s,\tau}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{3}{k^{2}}\Bigg{\\{}\left|\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|^{2}+\left|\frac{\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,0,0\right)|du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|^{2}$
(3.6)
$\displaystyle+\frac{\mathbb{E}\left(\left|\int_{s}^{\tau}g\left(u,0\right)dB_{u}\right|^{2}|\mathcal{F}_{s}\right)}{\left|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)\right|^{2}}\Bigg{\\}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
Using conditional expectation version of isometry property, we get
$\displaystyle\mathbb{E}\left(\left|\int_{s}^{\tau}g\left(u,0\right)dB_{u}\right|^{2}|\mathcal{F}_{s}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}|g\left(u,0\right)|^{2}du|\mathcal{F}_{s}\right)$
which together with (3.6) leads to
$\displaystyle|l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{k}\Bigg{\\{}\left|\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|+\frac{\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,0,0\right)|du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}$
(3.7)
$\displaystyle+\frac{\left[\mathbb{E}\left(\int_{s}^{\tau}|g\left(u,0\right)|^{2}du|\mathcal{F}_{s}\right)\right]^{1/2}}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\Bigg{\\}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
Since
$\displaystyle|A^{0}_{t}|=\left|\sup_{0\leq s\leq
t^{+}}L^{0}_{s}\right|\leq\sup_{0\leq s\leq t^{+}}|L^{0}_{s}|=\sup_{0\leq
s\leq t^{+}}\left\\{\mbox{ess}\inf_{\tau>s}|l_{s,\tau}|\right\\},$
we derive from (3.6) and $({\bf A3})$ that
$\displaystyle|A^{0}_{t}|\leq\sup_{0\leq s\leq
t^{+}}\left\\{\mbox{ess}\inf_{\tau>s}|l_{s,\tau}|\right\\}\leq\frac{3\,\sqrt{3}\,\Gamma}{k}$
and ends the proof. ∎
###### Lemma 3.2.
Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Then, for any
$t\in[0,T]$, it holds almost surely that
$\displaystyle|A_{t}-A^{\prime}_{t}|\leq\frac{\sqrt{2}\,L}{k}(1+\sqrt{T})\|y-y^{\prime}\|_{\infty}.$
###### Proof.
Again, we fix $s\in[0,T]$ and let $\tau\in\mathcal{M}(0,T)$ be such that
$\tau>s$ a.s. Let us consider, according to Theorem 2.1,
$l_{s,\tau},\,l^{\prime}_{s,\tau}$ two $\mathcal{F}_{s}$-measurable random
variables such that
$\displaystyle\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})=\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,y_{u},l_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,y_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,y^{\prime}_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}.$
(3.8)
Let us denote
$D^{\tau}_{s}=\left\\{\omega/l_{s,\tau}(\omega)>l^{\prime}_{s,\tau}(\omega)\right\\}$,
thus $D^{\tau}_{s}\in\mathcal{F}_{s}$, for any stopping times $\tau>s$. Since
${\bf 1}_{D^{\tau}_{s}}$ is $\mathcal{F}_{s}$-measurable, it follows from
(3.8) that
$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right)\right]^{2}$
$\displaystyle=\left[\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)]{\bf
1}_{D^{\tau}_{s}}du+\int_{s}^{\tau}(g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)){\bf
1}_{D^{\tau}_{s}}dB_{u}|\mathcal{F}_{s}\right)\right]^{2}.$ (3.9)
By assumption $({\bf A2})$-$(iv)$, we have
$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right)\right]^{2}\geq
k^{2}|l_{s,\tau}-l^{\prime}_{s,\tau}|^{2}[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}{\bf
1}_{D^{\tau}_{s}}]^{2}.$ (3.10)
Next, assumption (A2)-(iii) together with conditional expectation version of
isometry property lead to
$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)]{\bf
1}_{D^{\tau}_{s}}du+\int_{s}^{\tau}(g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)){\bf
1}_{D^{\tau}_{s}}dB_{u}|\mathcal{F}_{s}\right)\right]^{2}$
$\displaystyle\leq$ $\displaystyle
2\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du\right)\right]^{2}$ (3.11)
$\displaystyle+2\left[\mathbb{E}\int_{s}^{\tau}|g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)|^{2}{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right]$ $\displaystyle\leq$ $\displaystyle
2L^{2}\|y-y^{\prime}\|^{2}_{\infty}[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}{\bf
1}_{D^{\tau}_{s}}]^{2}+2L^{2}\|y-y^{\prime}\|^{2}_{\infty}\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right){\bf
1}_{D^{\tau}_{s}}.$
Combining (3.10) and (3.11) with (3.9), we obtain
$\displaystyle
k|l_{s,\tau}-l^{\prime}_{s,\tau}|\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}\leq\sqrt{2}L\|y-y^{\prime}\|_{\infty}\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}+\sqrt{2}L\|y-y^{\prime}\|_{\infty}[\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)]^{1/2},$
on $D^{\tau}_{s}$. Thus,
$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{2}\,L}{k}(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2})\|y-y^{\prime}\|_{\infty}$
on $D^{\tau}_{s}$, since $\tau>s$. Similarly, we can show that the inequality
holds on the complement of $D^{\tau}_{s}$ as well. Therefore, we have
$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{2}\,L}{k}(1+[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}]^{-1/2})\|y-y^{\prime}\|_{\infty}.$
(3.12)
Next, since
$L_{s}=\mbox{ess}\inf_{\tau>s}l_{s,\tau},\;L^{\prime}_{s}=\mbox{ess}\inf_{\tau>s}l^{\prime}_{s,\tau},\;A_{t}=\sup_{0\leq
s\leq t}L_{s}$ and $A^{\prime}_{t}=\sup_{0\leq s\leq t}L^{\prime}_{s}$, we
conclude from (3.12) that
$\displaystyle|A_{t}-A^{\prime}_{t}|=\left|\sup_{0\leq s\leq
t}L_{s}-\sup_{0\leq s\leq t}L^{\prime}_{s}\right|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\left|\mbox{ess}\inf_{\tau>s}l_{s,\tau}-\mbox{ess}\inf_{\tau>s}l^{\prime}_{s,\tau}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}|l_{s,\tau}-l^{\prime}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\frac{\sqrt{2}L}{k}\left[1+\left(\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}\right)^{-1/2}\right]\|y-y^{\prime}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{2}L}{k}(1+\sqrt{T})\|y-y^{\prime}\|_{\infty}.$
∎
We are now ready to prove the main result of this section, the existence and
uniqueness of the solution to the VRBDSDE.
###### Theorem 3.3.
Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Assume further that
${2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}<1},$
then the VRBDSDE (2.1) admits a unique solution $(Y,A)$.
###### Proof.
First, let us show that the mapping $\Phi$ defined by (3.4) is from
$\mathcal{S}^{\infty}$ to itself. To do this, we note that by using assumption
$({\bf A1})$ and Lemmas 3.1 and 3.2, we derive
$\displaystyle|Y_{t}|^{2}=|\Phi(y)_{t}|^{2}\leq
3\mathbb{E}\left\\{|\xi|+\left(\int^{T}_{t}|f(s,y_{s},A_{s})|ds\right)^{2}+\left(\int_{t}^{T}g(s,y_{s})dB_{s}\right)^{2}|\mathcal{F}_{t}\right\\}.$
(3.13)
We have
$\displaystyle\mathbb{E}\left\\{\left(\int^{T}_{t}f(s,y_{s},A_{s})ds\right)^{2}|\mathcal{F}_{t}\right\\}$
$\displaystyle\leq$ $\displaystyle
4T^{2}\left(K^{2}\|A-A^{0}\|^{2}_{\infty}+L^{2}\|y\|^{2}_{\infty}+K^{2}\|A^{0}\|^{2}_{\infty}+\Gamma^{2}\right)$
(3.14) $\displaystyle\leq$ $\displaystyle
4T^{2}L^{2}\left(1+2\frac{K^{2}}{k^{2}}\left(1+\sqrt{T}\right)\|y\|^{2}_{\infty}\right)+4T^{2}\left(1+27\frac{K^{2}}{k^{2}}\right)\Gamma^{2}$
and
$\displaystyle\mathbb{E}\left\\{\left(\int^{T}_{t}g(s,y_{s})|ds\right)^{2}|\mathcal{F}_{t}\right\\}$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left\\{\left(\int_{0}^{T}|g(s,0)|^{2}ds\right)|\mathcal{F}_{t}\right\\}+2L^{2}T\|y\|_{\infty}^{2}.$
(3.15)
It follows from (3.13), (3.14) and (3.15) that
$\displaystyle|Y_{t}|$ $\displaystyle\leq$
$\displaystyle\|\xi\|+\sqrt{2}\left[\mathbb{E}\left(\int_{0}^{T}|g(s,0)|^{2}ds|\mathcal{F}_{t}\right)\right]^{1/2}+L\left[2T\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+\sqrt{2T}\right]\|y\|_{\infty}$
$\displaystyle+2T\left(1+3\sqrt{3}\frac{K}{k}\right)\Gamma.$
As it is known by assumption that $\xi$ belongs to $\mathbb{L}^{\infty}$, we
deduce from $({\bf A3})$-$(i)$ that $Y=\Phi(y)$ belongs to $S^{\infty}$. Now,
let us prove that $\Phi$ is a contraction. For $y,\,y^{\prime}\in S^{\infty}$,
we denote $Y=\Phi(y)$ and $Y^{\prime}=\Phi(y^{\prime})$. Then, for
$t\in[0,T]$, we have
$\displaystyle|\Phi(y)-\Phi(y^{\prime})|^{2}$ $\displaystyle=$
$\displaystyle\left|\mathbb{E}\left\\{\int_{t}^{T}[f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})]ds+\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}|\mathcal{F}_{t}\right\\}\right|^{2}$
(3.16) $\displaystyle\leq$ $\displaystyle
2\left|\mathbb{E}\left\\{\int_{t}^{T}|f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})|ds|\mathcal{F}_{t}\right\\}\right|^{2}$
$\displaystyle+2\mathbb{E}\left\\{\left|\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}\right|^{2}|\mathcal{F}_{t}\right\\}.$
Applying assumption on $f$ and Lemma 3.2, we derive that
$\displaystyle\left|\mathbb{E}\left(\int_{t}^{T}|f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})|ds|\mathcal{F}_{t}\right)\right|^{2}$
$\displaystyle\leq$ $\displaystyle
2T^{2}\left(L^{2}\|y^{\prime}-y\|^{2}_{\infty}+K^{2}\|A-A^{\prime}\|_{\infty}^{2}\right)$
$\displaystyle\leq$ $\displaystyle
2T^{2}\left[L^{2}+K^{2}\frac{2L^{2}}{k^{2}}\left(1+\sqrt{T}\right)^{2}\right]\|y^{\prime}-y\|_{\infty}.$
Moreover, it follows from conditional expectation version of isometry property
that
$\displaystyle\mathbb{E}\left\\{\left|\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}\right|^{2}|\mathcal{F}_{t}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\left(\int_{t}^{T}|g(s,y_{s})-g(s,y^{\prime}_{s})|^{2}ds\right)|\mathcal{F}_{t}\right\\}$
(3.18) $\displaystyle\leq$ $\displaystyle
L^{2}T\|y-y^{\prime}\|^{2}_{\infty}.$
Finally, putting (LABEL:eq9) and (3.18) into (3.16), we obtain
$\displaystyle|\Phi(y)-\Phi(y^{\prime})|\leq\left[2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}\right]\|y-y^{\prime}\|_{\infty}.$
Since we assume that
$2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}<1$,
it is not difficult to see that $\Phi$ is a contraction.
Let us denote by $Y\in\mathcal{S}^{\infty}$ the unique fixed point and by $A$
the associating reflecting process defined by $A_{t}=\sup_{0\leq v\leq
t^{+}}L_{v}$, where $L$ satisfies the representation
$\displaystyle
X_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},\sup_{t\leq v\leq
s}L_{v}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{u}|\mathcal{F}_{t}\right\\}.$
(3.19)
Let us now prove that $(Y,A)$ is the solution to the VRBDSDE (2.1). For this
instance, it follows from (3.19), the definition of $A$, and the monotonicity
of $f$ on the third variable that for all $t\in[0,T]$,
$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},A_{s}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},\sup_{t\leq
v\leq
s}L_{v}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{s}|\mathcal{F}_{s}\right\\}=X_{t}.$
To end the proof of existence, it remains to show that the flat-off conditions
holds. The properties of optional projection and definition of $A$ and $L$
lead to
$\displaystyle\mathbb{E}\int_{0}^{T}(Y_{t}-X_{t})dA_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\left\\{\int_{t}^{T}\left[f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)-f\left(s,Y_{s},\sup_{t\leq v\leq
s}L_{v}\right)\right]ds\right\\}dA_{t}.$
Next, using the Fubini theorem and the fact that Lebesgue measure does not
charge the discontinuities of the path $u\mapsto\sup_{t\leq v\leq u}L_{v}$,
which are only countably many, we have
$\displaystyle\mathbb{E}\int_{0}^{T}(Y_{t}-X_{t})dA_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\left\\{\int_{0}^{s}\left[f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)-f\left(s,Y_{s},\sup_{t\leq v\leq
s^{+}}L_{v}\right)\right]dA_{t}\right\\}ds,$
which provide by the same argument used in [25] that
$\displaystyle\mathbb{E}\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0.$
For the uniqueness, let us suppose that there is another solution
$(Y^{\prime},A^{\prime})$ to the VRBDSDE such that $Y^{\prime}_{t}\leq
X_{t},\;t\in[0,T]$, and
$\displaystyle
Y^{\prime}_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y^{\prime}_{s},A^{\prime}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{\prime}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},\;\;\;\mathbb{E}\int_{0}^{T}|Y^{\prime}_{t}-X_{t}|dA_{t}=0.$
Since both $Y$ and $Y^{\prime}$ are the unique fixed points of the mapping
$\Phi$, it follows that $Y=Y^{\prime}$. Let us consider the stochastic variant
Skorohod problem
$\displaystyle\widetilde{Y}_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f^{Y}\left(s,\widetilde{A}_{s}\right)ds+\int_{t}^{T}g^{Y}\left(s\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle\widetilde{Y}_{t}\leq X_{t},\;\;\;\;\;Y_{T}=X_{T}=\xi,$ (3.20)
$\displaystyle\mathbb{E}\int_{0}^{T}|\widetilde{Y}_{t}-X_{t}|d\widetilde{A}_{t}=0,$
where $f^{Y}\left(s,l\right)=f\left(s,Y_{s},l\right)$ and
$g^{Y}\left(s\right)=g\left(s,Y_{s}\right)$. Thanks to Theorem 2.2, there
exists a unique pair of process $(\widetilde{Y},\widetilde{A})$ that solves
the stochastic variant Skorohod problem. Moreover, since $(Y,A)$ and
$(Y^{\prime},A^{\prime})$ are the solutions to the variant BDSDE (3.20), it
follows that $Y_{t}=\widetilde{Y}_{t}$ and
$A_{t}=\widetilde{A}_{t}=A^{\prime}_{t},\;t\in[0,T]$, a.s., which proves the
uniqueness, whence the theorem. ∎
###### Corollary 3.4.
Suppose that $(Y,A)$ is a solution to VRBDSDE with generator $f$ and $g$ and
upper boundary $X$. Then $A_{0^{-}}=-\infty$ and $Y_{0}=X_{0}$.
###### Proof.
Since the existence and uniqueness proof depends heavily on the well-posedness
result of the extended stochastic representation theorem, we must require that
$A_{0^{-}}=-\infty$. On the other hand, since $Y$ is a fixed point of the
mapping $\Phi$ defined by (3.4), it not difficult to see that $Y_{0}$ and
$X_{0}$ satisfy the following equalities:
$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},\sup_{0\leq
v\leq s}L_{v}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\},$
$\displaystyle Y_{0}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},A_{s}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\}.$
Hence, by the same argument that the paths of the increasing process
$u\mapsto\sup_{t\leq v\leq u}L_{v}$ has only countably many discontinuities,
which are negligible under the Lebesgue measure, we prove that $Y_{0}=X_{0}$.
∎
### 3.2 Comparison theorems
This section is devoted to study the comparison theorem of the VRBDSDE, one of
the very important tools in the theory of BSDEs. Let us remark that our method
follows closely to one appeared in [25], which is quite different from all the
existing arguments in the BSDE literature.
To state, let us consider the following two VRBDSDEs for $i=1,2$,
$\displaystyle
Y^{i}_{t}=\mathbb{E}\left\\{\xi^{i}+\int_{t}^{T}f^{i}\left(s,Y^{i}_{s},A^{i}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle Y^{i}_{t}\leq X^{i}_{t},\;\;\;\;\;Y^{i}_{T}=X^{i}_{T}=\xi^{i},$
(3.21) $\displaystyle\mathbb{E}\int_{0}^{T}|Y^{i}_{t}-X^{i}_{t}|dA^{i}_{t}=0.$
In the sequel, we call $(f^{i},g,X^{i}),\;i=1,2$ as the "parameters" of the
VRBDSDE (3.21), $i=1,2$, respectively. We also define the two following
stopping times
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\inf\left\\{t\in[0,T),\;A_{t}^{2}>A^{1}_{t}+\varepsilon\right\\}\wedge
T;$ $\displaystyle\tau$ $\displaystyle=$
$\displaystyle\inf\left\\{t\in[\mu,T),\;A_{t}^{1}>A^{2}_{t}-\frac{\varepsilon}{2}\right\\}\wedge
T.$ (3.22)
We recall the following result appear in [25].
###### Lemma 3.5.
The stopping times $\mu$ and $\tau$ defined by (3.22) have the standing
properties:
(i)
$\;\mu$ and $\tau$ are points of increase for $A^{2}$ and $A^{1}$,
respectively. In other word, for any $\delta>0$, it holds that
$A^{2}_{\mu^{-}}<A^{2}_{\mu+\delta}$ and
$A^{1}_{\tau^{-}}<A^{1}_{\tau+\delta}$.
(ii)
$\;\mathbb{P}(\mu<\tau)=1$, and $A^{1}_{t}\leq
A^{2}_{t}-\frac{\varepsilon}{2}$, for all $t\in[\mu,\tau],\,\mathbb{P}$-a.s.,
(iiii)
it holds that $Y^{2}_{\mu}=X^{2}_{\mu}$ and
$Y^{1}_{\tau}=X^{1}_{\tau},\;\mathbb{P}$-a.s.
Before give the comparison theorem, in order to simplify the notations, let us
give the following. For $(Y^{i},A^{i}),\;i=1,2$ be the solution to two
VRBDSDEs with boundaries $X^{1}$ and $X^{2}$ respectively, we denote
$\Delta\Theta=\Theta^{1}-\Theta^{2},\;\Theta=X,Y,A$, and $\xi$. Furthermore,
recall
$\displaystyle\mathcal{G}_{t}=\mathcal{F}^{W}_{t}\vee\mathcal{F}^{B}_{T},$
we define two martingales
$\displaystyle
M^{i}_{t}=\mathbb{E}\left\\{\int_{0}^{T}f^{i}\left(s,Y^{i}_{s},A^{i}_{s}\right)ds+\int_{0}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{G}_{t}\right\\},\;t\in[0,T],\,i=1,2.$
###### Theorem 3.6.
Assume that the parameters of the VRBDSDEs (3.21) $(f^{i},g,X^{i}),\,i=1,2$,
satisfy $({\bf A1})$ and $({\bf A2})$. Assume further that
(i)
$\;f^{1}(t,y,l)\geq f^{2}(t,y,l),\;d\mathbb{P}\otimes dt$ a.s.,
(ii)
$\;X^{1}_{t}\leq X^{2}_{t},\;\;0\leq t\leq T$, a.s.,
(iii)
$\;\Delta X_{s}\leq\mathbb{E}\\{{\rm e}^{(L+\frac{1}{2}L^{2})(t-s)}\Delta
X_{t}|\mathcal{G}_{s}\\}$ a.s. for all $s$ and $t$ such that $s<t$.
Then, we have $A^{1}_{t}\geq A^{2}_{t},\;t\in[0,T],\ \mathbb{P}$-a.s.
###### Remark 3.7.
As it is explained in [25], the assumption $(iii)$ in above theorem
significate that the process ${\rm e}^{Ls}\Delta X_{s}$, is a submartingale
and it does not add restrictive on the regularity of the boundary processes
$X^{1}$ and $X^{2}$, which are only required to be the optional processes
satisfying $({\bf A3})$.
###### Proof of Theorem 3.6.
According to (3.21) and the previous notations, we can write, on the set
$\\{\mu<T\\}$
$\displaystyle\Delta Y_{\mu}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\Delta
Y_{\tau}+\int_{\mu}^{\tau}\left[f^{1}\left(s,Y^{1}_{s},A^{1}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right)\right]ds\right.$
(3.23)
$\displaystyle+\left.\int_{\mu}^{\tau}\left[g\left(s,Y^{1}_{s}\right)-g\left(s,Y^{2}_{s}\right)\right]dB_{s}+(\Delta
M_{\tau}-\Delta M_{\mu})|\mathcal{F}_{\mu}\right\\},$
where $\Delta M=M^{1}-M^{2}$, and
$\displaystyle\nabla_{y}f^{1}_{s}$ $\displaystyle=$
$\displaystyle\frac{f^{1}\left(s,Y^{1}_{s},A^{1}_{s}\right)-f^{1}\left(s,Y^{2}_{s},A^{1}_{s}\right)}{Y^{1}_{s}-Y^{2}_{s}}{\bf
1}_{\\{Y^{1}_{s}\neq Y^{2}_{s}\\}},$ $\displaystyle\nabla_{y}g_{s}$
$\displaystyle=$
$\displaystyle\frac{g\left(s,Y^{1}_{s}\right)-g\left(s,Y^{2}_{s}\right)}{Y^{1}_{s}-Y^{2}_{s}}{\bf
1}_{\\{Y^{1}_{s}\neq Y^{2}_{s}\\}},$ $\displaystyle\Delta_{l}f^{1}_{s}$
$\displaystyle=$ $\displaystyle
f^{1}\left(s,Y^{2}_{s},A^{1}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right),$
$\displaystyle\Delta_{2}f_{s}$ $\displaystyle=$ $\displaystyle
f^{1}\left(s,Y^{2}_{s},A^{2}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right).$
It is clear that $({\bf A2})$ implies that $\nabla_{y}f^{1}$ and $\nabla_{y}g$
are bounded progressively measurable processes, and by the definition of
$\mu,\,\tau$ and the monotonicity of $f$ on it variable $l$, we have
$\Delta_{l}f^{1}>0$ on the interval $[\mu,\tau]$. Hence, $\Delta Y$ is a
unique solution of the following linear BDSDE
$\displaystyle\Delta Y_{\mu}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\Delta
Y_{\tau}+\int_{\mu}^{\tau}\nabla_{y}f^{1}_{s}\Delta
Y_{s}ds+\int_{\mu}^{\tau}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds\right.$
$\displaystyle+\left.\int_{\mu}^{\tau}\nabla_{y}g_{s}\Delta
Y_{s}dB_{s}+(\Delta M_{\tau}-\Delta M_{\mu})|\mathcal{F}_{\mu}\right\\}$
Setting
$\Gamma_{t}=\exp\left(\int_{0}^{t}\nabla_{y}f^{1}_{s}ds+\int_{0}^{t}\nabla_{y}g_{s}dB_{s}-\frac{1}{2}\int_{0}^{t}|\nabla_{y}g_{s}|^{2}ds\right)$,
as it done in [3], one can derive
$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}=\mathbb{E}\left\\{\int_{\mu}^{\tau}\Gamma_{s}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds-\int_{\mu}^{\tau}\Gamma_{s}d(\Delta
M_{s})|\mathcal{F}_{\mu}\right\\}.$
Therefore, since $f^{1}\geq f^{2},\;\Delta_{2}f\geq 0,\,d\mathbb{P}\otimes
dt$-a.s., and consequently, since $M^{i},\;i=1,2$, is a martingale, we get
$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}=\mathbb{E}\left\\{\int_{\mu}^{\tau}\Gamma_{s}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds|\mathcal{F}_{\mu}\right\\}>0.$
(3.24)
On the other hand, by the flat-off condition and Lemma 3.5-$(iii)$, one can
check that $Y^{1}_{\mu}-Y^{2}_{\mu}\leq X^{1}_{\mu}-X^{2}_{\mu}$ and
$Y^{1}_{\tau}-Y^{2}_{\tau}\leq X^{1}_{\tau}-X^{2}_{\tau}$,
$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}\leq\mathbb{E}\left\\{\Gamma_{\mu}\Delta
X_{\mu}-\Gamma_{\tau}\Delta X_{\tau}|\mathcal{F}_{\mu}\right\\}.$ (3.25)
It is now clear that if the right hand side of (3.25) is non-positive, then
(3.25) contradicts to (3.24), and therefore one must have
$\mathbb{P}(\mu<T)=0$. In other words, $A^{2}_{t}\leq A^{1}_{t}+\varepsilon$,
for all $t\in[0,T],\,\mathbb{P}$-a.s. Since $\varepsilon$ is taken arbitrary,
entails that
$\displaystyle A^{2}_{t}\leq
A^{1}_{t},\;\;\;\;t\in[0,T],\;\;\mathbb{P}\mbox{-a.s.}$
Now it remain to show that the right hand side of (3.25) is non-positive. To
do this, let us note that since by assumption $(ii)$ we have $\Delta
X_{\tau}\leq 0$, it follows from (3.25) and assmption $(iii)$ that
$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}$ $\displaystyle\leq$
$\displaystyle\Gamma_{\mu}\mathbb{E}\left\\{\Delta X_{\mu}-{\rm
e}^{\int_{\mu}^{\tau}\nabla_{y}f^{1}_{s}ds+\int_{\mu}^{\tau}\nabla_{y}g_{s}dB_{s}-\frac{1}{2}\int_{\mu}^{\tau}|\nabla_{y}g_{s}|^{2}ds}\Delta
X_{\tau}|\mathcal{F}_{\mu}\right\\}$ $\displaystyle\leq$
$\displaystyle\Gamma_{\mu}\mathbb{E}\left\\{\Delta X_{\mu}-{\rm
e}^{(L+\frac{1}{2}L^{2})(\tau-\mu)}\Delta X_{\tau}|\mathcal{F}_{\mu}\right\\}$
$\displaystyle\leq$ $\displaystyle 0.$
∎
As it is emphasized in [25], Theorem 3.6 only gives the comparison between the
two reflecting processes $A^{1}$ and $A^{2}$. This is still one step away from
comparison between $Y^{1}$ and $Y^{2}$, which is much desirable for obvious
reason. But, the latter is not true in general, due do the "opposite"
monotonicity on $f^{i}$’s on the variable $l$. We nevertheless have the
following corollary of Theorem 3.6.
###### Corollary 3.8.
Assume all the assumptions of Theorem 3.6 hold and further $f^{1}=f^{2}$. Then
$Y^{1}_{t}\leq Y^{2}_{t}$, for all $t\in[0,T],\,\mathbb{P}$-a.s.
###### Proof.
Let us denote $f=f^{1}=f^{2}$ and define two random functions
$\widetilde{f}^{i}(t,\omega,y)=f(t,\omega,y,A^{i}_{t}(\omega))$, for
$(t,\omega,y)\in[0,T]\times\Omega\times\mathbb{R},i=1,2$. Then $Y^{1}$ and
$Y^{2}$ can be seen as the solution of BDSDEs
$\displaystyle
Y^{i}_{t}=\mathbb{E}\left\\{\xi^{i}+\int_{t}^{T}\widetilde{f}^{i}\left(s,Y^{i}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},\;t\in[0,T],\;\;i=1,2.$
It follows from the fact $A^{1}\geq A^{2}$ that
$\widetilde{f}^{1}(t,\omega,y)=f(t,\omega,y,A^{1}_{t}(\omega))\leq
f(t,\omega,y,A^{2}_{t}(\omega))=\widetilde{f}^{2}(t,\omega,y)$. Therefore,
since $\xi^{1}=X^{1}_{T}\leq X^{2}_{T}=\xi^{2}$, and according to the
comparison theorem of BDSDEs, we have $Y^{1}_{t}\leq Y^{2}_{t}$, for all
$t\in[0,T],\ \mathbb{P}$-a.s. ∎
### 3.3 Stability results
In this section, we study another useful aspect of the well-posedness of the
VRBDSDE, which it is called the continuous dependence of the solution on the
boundary process whence the terminal process as well. For this instance, let
us introduce, for any optional process $X$ and any stopping time $\mu$ and
$\tau$ satisfy that $\mu<\tau$,
$\displaystyle
m_{\mu,\tau}(X)=\frac{\mathbb{E}\\{X_{\tau}-X_{\mu}|\mathcal{F}_{\mu}\\}}{\mathbb{E}\\{\tau-\mu|\mathcal{F}_{\mu}\\}}.$
Let us note that the random variable $m_{\mu,\tau}(X)$ measures the path
regularity of the "nonmartingale" part of the boundary process $X$. In the
sequel, we will show that this will be a major measurement for the "closeness"
of the boundary processes, as far as the continuous dependence is concerned.
Let us consider $\\{X^{n}\\}_{n=1}^{\infty}$, a sequence of optional processes
satisfying that $({\bf A3})$. We suppose that $\\{X^{n}\\}_{n=1}^{\infty}$
converges to $X^{0}$ in $\mathcal{S}^{\infty}$, and that $X^{0}$ satisfies
$({\bf A3})$ as well. Let $(Y^{n},A^{n})$ be the unique solution to the
VRBDSDE’s with parameters $(f,g,X^{n})$, for
$n=0,1,2,\cdot\cdot\cdot\cdot\cdot$. Roughly speaking, for
$n=0,1,2,\cdot\cdot\cdot\cdot$, we have
$\displaystyle
X^{n}_{t}=\mathbb{E}\left\\{\xi^{n}+\int_{t}^{T}f\left(s,Y^{n}_{s},\sup_{t\leq
v\leq
s}L^{n}_{v}\right)ds+\int_{t}^{T}g\left(s,Y^{n}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle A^{n}_{s}=\sup_{0\leq v\leq s^{+}}L^{n}_{v},$ $\displaystyle
Y^{n}_{t}=\mathbb{E}\left\\{\xi^{n}+\int_{t}^{T}f\left(s,Y^{n}_{s},A^{n}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{n}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\}.$
Next, let us give the following lemma that provides the control of
$|A^{n}_{t}-A_{t}^{0}|$, which is needed in the sequel.
###### Lemma 3.9.
Assume $({\bf A2})$ and $({\bf A3})$ hold. Then, for all $t\in[0,T]$, it holds
that
$\displaystyle|A^{n}_{t}-A_{t}^{0}|\leq\frac{\sqrt{3}}{k}\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}L}{k}(1+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}.$
###### Proof.
The proof follows the similar step as the proof of Lemma 3.2. Let us consider,
$l^{n}_{s,\tau},\;n=0,1,\cdot\cdot\cdot$ the $\mathcal{F}_{s}$-measurable
random variable such that
$\displaystyle\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,Y^{n}_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}.$
(3.26)
Therefore, for $n=1,\cdot\cdot\cdot$, we have
$\displaystyle\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\int_{s}^{\tau}[f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)]du\right.$
$\displaystyle\left.+\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}.$
On the set
$D^{\tau}_{s}=\left\\{\omega/l^{n}_{s,\tau}(\omega)>l^{0}_{s,\tau}(\omega)\right\\}\in\mathcal{F}_{s}$,
we get
$\displaystyle{\bf
1}_{D^{\tau}_{s}}\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$ $\displaystyle{\bf
1}_{D^{\tau}_{s}}\mathbb{E}\left\\{\int_{s}^{\tau}[f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)+f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)]du\right.$
$\displaystyle\left.+\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}.$
From $({\bf A2})$, it clear that on
$D^{\tau}_{s},\,f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)\geq
k|l^{n}_{s,\tau}-l^{0}_{s,\tau}|$ and hence
$\displaystyle
k^{2}[|l^{n}_{s,\tau}-l^{0}_{s,\tau}|\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}]^{2}{\bf
1}_{D^{\tau}_{s}}$ $\displaystyle\leq$ $\displaystyle
3|\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})|^{2}{\bf
1}_{D^{\tau}_{s}}$
$\displaystyle+3\left|\mathbb{E}\left\\{\int_{s}^{\tau}L|Y^{n}_{u}-Y^{0}_{u}|du\mathcal{F}_{s}\right\\}\right|^{2}{\bf
1}_{D^{\tau}_{s}}$
$\displaystyle\left.+3\left|\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}\right|^{2}{\bf
1}_{D^{\tau}_{s}}.$
Next, assumption $({\bf A2})$-$(iii)$ together with conditional expectation
version of isometry
property lead to
$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{3}}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}\,L}{k}\left(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2}\right)\|y-y^{\prime}\|_{\infty}$
on $D^{\tau}_{s}$. Similarly, we can show that the inequality holds on the
complement of $D^{\tau}_{s}$ as well. Therefore, we have
$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{3}}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}\,L}{k}\left(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2}\right)\|y-y^{\prime}\|_{\infty}$
Finally, according to the definition of $A^{n},\,n=0,1,\cdot\cdot\cdot$, we
conclude that for $n=1,2,\cdot\cdot\cdot$,
$\displaystyle|A_{t}-A^{\prime}_{t}|$ $\displaystyle=$
$\displaystyle\left|\sup_{0\leq s\leq t}L^{n}_{s}-\sup_{0\leq s\leq
t}L^{0}_{s}\right|$ $\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\left|\mbox{ess}\inf_{\tau>s}l^{n}_{s,\tau}-\mbox{ess}\inf_{\tau>s}l^{0}_{s,\tau}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}|l^{n}_{s,\tau}-l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\frac{\sqrt{3}}{k}\left[\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+L\left(1+\left(\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}\right)^{-1/2}\right)\right]\|Y^{n}-Y^{0}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\frac{\sqrt{3}}{k}\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}L}{k}(1+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}.$
∎
Now, we are ready to derive the main result of this subsection.
###### Theorem 3.10.
Assume $({\bf A2})$ and $({\bf A3})$ hold. Further, assume that
${\sqrt{6}TL\left(1+\sqrt{3}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{3T}<1}.$
Then, it holds that
$\displaystyle\|Y^{n}-Y^{0}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{1-\left[\sqrt{6}TL\left(1+\sqrt{3}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{3T}\right]}\times$
$\displaystyle\times\left\\{\left\|\xi^{n}-\xi^{0}\right\|_{\infty}+\frac{\sqrt{6}TK}{k}\left\|\sup_{\mu\in[0,T]}ess\sup_{\tau>\mu}\frac{1}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|\right\|_{\infty}\right\\}.$
###### Proof.
Using the similar arguments as Theorem 3.3, we obtain this estimation
$\displaystyle|Y^{n}_{t}-Y^{0}_{t}|\leq\sqrt{3}\|\xi^{n}-\xi^{0}\|_{\infty}+\sqrt{3}L(\sqrt{2}T+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}+\sqrt{6}TK\|A^{n}-A^{0}\|_{\infty},$
which, together with Lemma 3.9, proves the desired result. ∎
###### Remark 3.11.
Let us emphasize that, since all the model study in section 6 of [25] have
their stochastic counterpart, we can with no more difficulty establish
respectively the stochastic version of recursive intertemporal utility
minimization, optimal stopping problems. It suffice to follows the similar
step as in [25] with some additional argument due to the presence of the
backward stochastic integral with respect the Brownian motion $B$.
Acknowledgement The work was partially done while the second author visited
Shandong University. He would like to thank Prof. Shige Peng for providing a
stimulating working environment.
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|
arxiv-papers
| 2011-08-03T12:58:17 |
2024-09-04T02:49:21.235635
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Auguste Aman and Yong Ren",
"submitter": "Auguste Aman",
"url": "https://arxiv.org/abs/1108.0836"
}
|
1108.0842
|
Светлой памяти Владимира Александровича Жегалина (1947–2002) — учителя физики,
который умел заинтересовать своих учеников, и тем самым определил мою
дальнейшую жизнь.
Специальная теория относительности
Андрей Грозин
###### Abstract
В этой статье кратко и (я надеюсь) доступно излагаются основные положения
специальной теории относительности. Я старался показать, что теория
относительности не только не противоречит здравому смыслу, но, наоборот,
логически следует из него. Подробно обсуждается геометрия пространства–времени
Минковского. Геометрический подход (с минимумом формул) позволяет сделать
выводы теории относительности наглядными и интуитивно очевидными. Я
использовал радиолокационный подход Бонди, который позволяет прийти к выводам
теории кратчайшим путём.
Использовать скорость света $c$ в теории относительности — это всё равно что
изучать евклидову геометрию, измеряя координату $x$ в сантиметрах, а $y$ в
дюймах. Во всех формулах будет назойливо присутствовать “фундаментальная
константа” $c=2.54\;$см/дюйм; естественность и очевидность результатов
полностью исчезнут.
## 1 Пространство–время
Пространство–время — это множество всех событий. Событие характеризуется тем,
что произошло где-то и когда-то. Оно однозначно определяется четырьмя числами
— моментом времени и тремя пространственными координатами. То есть
пространство–время четырёхмерно. Все события, произошедшие с частицей,
образуют её мировую линию. В каждый момент времени частица где-то находилась,
а это уже событие. Мировая линия представляет собой одномерное подмножество
пространства–времени.
Рисовать четырёхмерные картинки довольно неудобно, поэтому мы будем в основном
рассматривать движение частиц вдоль прямой. Тогда пространство–время двумерно,
и его легче себе представить. Например, на рис. 1 показано движение поездов
метро.
$t$$x$Станция 1Станция 2 Figure 1: Движение поезда описывается двумерной
областью между мировыми линиями его головы и хвоста.
Каждый наблюдатель имеет при себе часы, и может измерять время событий на
своей мировой линии. Мы примем постулат, что существуют правильно идущие часы.
Показания двух правильных часов одного наблюдателя могут отличаться лишь
выбором начала отсчёта (естественно, предполагается, что тиканье разных часов
приведено к единой единице измерения времени). Вопрос о том, как сделать
правильные часы, относится не к физике пространства–времени, а к физике
населяющей его материи. Он кратко обсуждается в приложении A (необязательном
для понимания основного текста). Наблюдатель не может непосредственно измерить
время события вне своей мировой линии. Этот процесс неизбежно включает в себя
передачу сигналов, и будет подробно рассмотрен в § 2.
$t_{1}$$t_{2}$12 Figure 2: Обмен сигналами.
Допустим, наблюдатель 1 посылает сигнал наблюдателю 2 в момент $t_{1}$, а тот
немедленно посылает сигнал обратно (рис. 2). Логически, можно представить себе
три возможности:
* •
Возможна ситуация, когда $t_{2}<t_{1}$, то есть ответный сигнал приходит
раньше отправления первого сигнала. К счастью, эта возможность не
соответствует нашему реальному миру; иначе про причинность можно было бы
забыть.
* •
Разность $t_{2}-t_{1}$ положительна, но может быть сделана сколь угодно малой
путём выбора подходящих сигналов и технических усовершенствований. В этом
случае можно ввести абсолютное время, одинаковое для всех наблюдателей. Именно
так все и думали до создания теории относительности.
* •
Разность $t_{2}-t_{1}$ не может быть сделана меньше некоторого положительного
значения. То есть существует наибыстрейший сигнал. Как показывают
эксперименты, именно так устроен реальный мир.
Таким наибыстрейшим сигналом является свет (а также гравитационные волны). Это
связано с тем, что фотоны (и гравитоны) — безмассовые частицы (§ 5). Однако,
конкретная природа наибыстрейших сигналов не является важной для теории
относительности. Если когда-нибудь у фотона обнаружат очень маленькую
ненулевую массу, то это будет серьёзным потрясением для теории
электромагнетизма, но никак не затронет основ теории относительности. Важен
только сам факт, что никакой сигнал не может быть быстрее чего-то. Мы будем
для определённости называть наибыстрейшие сигналы световыми.
$O$$A$ Figure 3: Взрыв бомбы.
Пусть событие $O$ будет взрывом бомбы. Тогда наблюдатель сначала увидит
вспышку (событие $A$), а потом уже на него посыплются осколки (рис. 3).
Мировые линии световых сигналов, испущенных в точке $O$, образуют световой
полуконус будущего этого события. Событие $O$ может влиять на события, лежащие
внутри (и на границе) этого полуконуса, при помощи света и других сигналов
(например, осколков). Эта область пространства–времени называется _будущим_
события $O$.
Аналогично, мировые линии световых сигналов, приходящих в точку $O$, образуют
световой полуконус прошлого этого события. События внутри (и на границе) этого
полуконуса могут влиять на событие $O$. Эта область называется _прошлым_
события $O$. Область пространства–времени вне светового конуса называется
_удалённым_ события $O$ (рис. 4). События из этой области не могут влиять на
$O$, и $O$ не может влиять на них.
будущеепрошлоеудалённое$O$ Figure 4: Будущее, прошлое и удалённое события $O$.
Световой конус в трёхмерном пространстве–времени (две пространственных
координаты) показан на рис. 5. Будущее находится внутри светового полуконуса
будущего; прошлое — внутри светового полуконуса прошлого; удалённое — вне
светового конуса. В четырёхмерном пространстве–времени световой конус — это
трёхмерная поверхность, нарисовать которую труднее.
Figure 5: Световой конус.
## 2 Преобразования Лоренца
Частица, на которую не действуют никакие силы, движется по инерции. Принцип
относительности Галилея гласит, что _все инерциальные наблюдатели
равноправны_. Если один инерциальный наблюдатель поставил какой-то эксперимент
и получил некоторый результат, а другой инерциальный наблюдатель поставил
такой же эксперимент, то он получит такой же результат.
12$t_{1}$$t_{2}$$\varphi$$O$ Figure 6: Два инерциальных наблюдателя.
Допустим, два инерциальных наблюдателя пролетели мимо друг друга (событие $O$,
рис. 6). Их часы установлены в 0 в точке $O$. Наблюдатель 1 послал световой
сигнал в момент времени $t_{1}$ (по своим часам); наблюдатель 2 его принял в
момент $t_{2}$ (по своим часам). Тогда
$t_{2}=t_{1}\,e^{\varphi}\,,$ (2.1)
где величина $\varphi$ называется углом между мировыми линиями наблюдателей 1
и 2.
123$t_{1}$$t_{2}$$t_{3}$$\varphi_{12}$$\varphi_{23}$$\varphi_{13}$ Figure 7:
Три инерциальных наблюдателя в одной плоскости.
Почему эту величину естественно назвать углом? Рассмотрим трёх инерциальных
наблюдателей, движущихся в одной плоскости. Пусть их мировые линии
пересекаются в одной точке ($O$, рис. 7). Тогда
$t_{2}=t_{1}\,e^{\varphi_{12}}\,,\qquad
t_{3}=t_{1}\,e^{\varphi_{13}}=t_{2}\,e^{\varphi_{23}}\,,$
откуда следует
$\varphi_{13}=\varphi_{12}+\varphi_{23}\,,$ (2.2)
как в обычной евклидовой геометрии. Подчеркнём ещё раз, что простой закон
сложения углов (2.2) верен только для мировых линий в одной плоскости (как и в
евклидовой геометрии).
Пусть теперь наблюдатель 1 посылает световой сигнал в момент $t_{1}$ (по своим
часам); наблюдатель 2 получает его в момент $t$ (событие $A$), и сразу
посылает ответный сигнал; наблюдатель 1 получает его в момент $t_{2}$ (рис.
8). Как мы знаем, $t=t_{1}\,e^{\varphi}$; кроме того, $t_{2}=t\,e^{\varphi}$,
ведь наблюдатели 1 и 2 равноправны. Значит,
$t_{1}=t\,e^{-\varphi}\,,\qquad t_{2}=t\,e^{\varphi}\,.$ (2.3)
12$t_{1}$$t_{2}$$t$$A$$O$$\varphi$ Figure 8: Обмен сигналами между двумя
инерциальными наблюдателями.
Событие $A$ произошло вне мировой линии наблюдателя 1, поэтому он не может
непосредственно измерить время этого события по своим часам. Наиболее
естественно определить время события $A$ с точки зрения наблюдателя 1 как
середину интервала $[t_{1},t_{2}]$, так как световой сигнал распространяется в
обе стороны одинаково быстро. По той же причине, наиболее естественно
определить координату события $A$ с точки зрения наблюдателя 1 как половину
этого промежутка времени. Скорость наибыстрейшего сигнала мы принимаем за 1,
что фиксирует естественную единицу измерения расстояния; сигнал
пропутешествовал от наблюдателя 1 до события $A$ и обратно. Таким образом, по
определению, временная и пространственная координаты события $A$ с точки
зрения наблюдателя 1 есть
$x^{0}=\frac{t_{1}+t_{2}}{2}\,,\qquad x^{1}=\frac{t_{2}-t_{1}}{2}\,.$ (2.4)
Подставляя (2.3), получим
$x^{0}=t\cosh\varphi\,,\qquad x^{1}=t\sinh\varphi\,.$ (2.5)
Мы пришли к очень важному результату: величина
$x^{2}\equiv(x^{0})^{2}-(x^{1})^{2}=t^{2}$ (2.6)
не зависит от того, как движется наблюдатель 1 (т. е. от $\varphi$).
Компоненты $x^{0}$, $x^{1}$ вектора $x$ (из точки $O$ в точку $A$) различны
для разных наблюдателей; инвариантную величину $x^{2}$ естественно назвать
квадратом длины вектора $x$, т. е. квадратом расстояния от $O$ до $A$. Это
расстояние есть $t$, то есть интервал времени между событиями $O$ и $A$ по
часам наблюдателя 2 (оба события лежат на его мировой линии). Формула (2.6)
для квадрата длины вектора отличается от привычной евклидовой тем, что вместо
знака $+$ между двумя членами стоит знак $-$. Поэтому геометрия
пространства–времени называется псевдоевклидовой (или геометрией Минковского).
Из формулы (2.5) видно, что привычная ньютоновская скорость наблюдателя 2 по
отношению к наблюдателю 1 есть
$\frac{x^{1}}{x^{0}}=\tanh\varphi$
(она всегда $<1$, и стремится к 1 при $\varphi\to\infty$). Эта величина,
однако, неудобна; удобнее использовать угол $\varphi$ между мировыми линиями,
обладающий естественным свойством аддитивности (2.2).
12$t_{1}$$t_{2}$$t_{1}^{\prime}$$t_{2}^{\prime}$$\varphi$$A$$O$ Figure 9:
Преобразование Лоренца.
Как связаны координаты $x^{0}$, $x^{1}$ события $A$ с точки зрения наблюдателя
1 и координаты $x^{0\prime}$, $x^{1\prime}$ того же события с точки зрения
наблюдателя 2, мировая линия которого образует угол $\varphi$ с мировой линией
наблюдателя 1? Из рис. 9 мы видим
$x^{0\prime}=\frac{t_{1}^{\prime}+t_{2}^{\prime}}{2}\,,\qquad
x^{1\prime}=\frac{t_{2}^{\prime}-t_{1}^{\prime}}{2}\,,$
где
$t_{1}^{\prime}=t_{1}\,e^{\varphi}\,,\qquad
t_{2}^{\prime}=t_{2}\,e^{-\varphi}\,.$ (2.7)
Подставляя
$t_{1}=x^{0}-x^{1}\,,\qquad t_{2}=x^{0}+x^{1}\,,$
мы окончательно получаем
$x^{0\prime}=x^{0}\cosh\varphi-x^{1}\sinh\varphi\,,\qquad
x^{1\prime}=-x^{0}\sinh\varphi+x^{1}\cosh\varphi.$ (2.8)
Это преобразование Лоренца. Легко проверить, что квадрат длины вектора $x$
инвариантен: $(x^{0\prime})^{2}-(x^{1\prime})^{2}=(x^{0})^{2}-(x^{1})^{2}$.
Преобразование Лоренца проще выглядит в координатах светового фронта
(приложение B).
## 3 Геометрия Минковского
Рассмотрим вектор $x$ из точки $O$ в точку $A$. Возможны три случая (рис. 4):
* •
$x^{2}>0$ — времениподобный вектор, может быть направлен в будущее или в
прошлое (событие $A$ лежит в будущем или прошлом события $O$). Произведя
подходящее преобразование Лоренца, можно добиться того, чтобы единственной
ненулевой компонентой была $x^{0}$ (для этого достаточно провести мировую
линию наблюдателя через $O$ и $A$).
* •
$x^{2}=0$ — светоподобный вектор, направлен вдоль светового конуса в будущее
или прошлое (событие $A$ лежит на световом конусе события $O$). Два
светоподобных вектора невозможно сравнивать (который длиннее, а который
короче), за исключением случая, когда они коллинеарны.
* •
$x^{2}<0$ — пространственноподобный вектор (событие $A$ лежит в удалённом
события $O$). Произведя подходящее преобразование Лоренца, можно добиться
того, чтобы единственной ненулевой компонентой была $x^{1}$ (т. е. чтобы
события $O$ и $A$ были одновременны).
Мы знаем, что такое квадрат вектора $x^{2}$ (2.6). А как насчёт скалярного
произведения двух векторов $x\cdot y$? Его можно определить как
$x\cdot y=\frac{(x+y)^{2}-x^{2}-y^{2}}{2}=x^{0}y^{0}-x^{1}y^{1}\,.$
В четырёхмерном пространстве–времени $x\cdot
y=x^{0}y^{0}-x^{1}y^{1}-x^{2}y^{2}-x^{3}y^{3}$. Поэтому наряду с
контравариантными компонентами $x^{\mu}$ вектора $x$ ($\mu=0$, 1, 2, 3) вводят
ковариантные компоненты $x_{\mu}$:
$x_{0}=x^{0}\,,\quad x_{1}=-x^{1}\,,\quad x_{2}=-x^{2}\,,\quad
x_{3}=-x^{3}\,.$
Тогда скалярное произведение имеет простой вид
$x\cdot y=x^{\mu}y_{\mu}=x_{\mu}y^{\mu}\,,$ (3.1)
где по повторяющемуся индексу (один раз сверху и один раз снизу) всегда
подразумевается суммирование от 0 до 3.
Векторы $x$ и $y$ называются ортогональными, если $x\cdot y=0$. Два
времениподобных вектора не могут быть ортогональны друг другу: их скалярное
произведение всегда $>0$, если оба направлены в будущее (или в прошлое), и
$<0$, если один направлен в будущее, а другой в прошлое. Светоподобный вектор
ортогонален сам себе, а также всем коллинеарным с ним светоподобным векторам.
Он не может быть ортогонален времениподобному вектору или светоподобному
вектору, не коллинеарному с ним. Времениподобный вектор $x$ ортогонален
пространственноподобному вектору $y$, если их направления симметричны друг
другу относительно светоподобной прямой (рис. 10): если $y^{0}=x^{1}$,
$y^{1}=x^{0}$, то $x\cdot y=0$; ортогональность не нарушится, если $y$
умножить на скаляр. Два пространственноподобных вектора могут быть
ортогональны друг другу, если имеется не менее двух пространственных
координат: если натянутая на них плоскость пространственноподобна, то её
геометрия евклидова, и эти два вектора могут быть ортогональны.
$x^{0}$$x^{1}$$x$$y$$\varphi$$\varphi$ Figure 10: Ортогональные векторы.
Времениподобный вектор $x$ длины $t$ ($x^{2}=t^{2}$), направленный под углом
$\varphi$ к оси времени, имеет компоненты (2.5)
$x^{\mu}=t(\cosh\varphi,\sinh\varphi)$ (рис. 11). То есть его проекция на
направление $e^{\mu}=(1,0)$ есть $t\cosh\vartheta$. Она всегда $\geq t$;
равенство достигается при $\varphi=0$.
$x^{0}$$x^{1}$$x$$\varphi$$t\cosh\varphi$$t\sinh\varphi$ Figure 11: Проекции
времениподобного вектора $x$ на оси координат.
Скалярное произведение двух времениподобных векторов $x$ и $y$ (будем считать,
что оба направлены в будущее) равно длине вектора $x$ (то есть
$\sqrt{x^{2}}$), умноженной на проекцию $y$ на направление $x$ (она равна
$\sqrt{y^{2}}\cosh\varphi$):
$x\cdot y=\sqrt{x^{2}}\sqrt{y^{2}}\cosh\varphi\,,$ (3.2)
где $\varphi$ — угол между направлениями $x$ и $y$. Скалярное произведение
всегда $\geq\sqrt{x^{2}}\sqrt{y^{2}}$; равенство достигается, когда они
коллинеарны ($\varphi=0$).
“Окружность” — это геометрическое место точек $A$, удалённых от центра $O$ на
расстояние $t$. Она определяется уравнением $x^{2}=t^{2}$, и представляет
собой гиперболу (рис. 12a). Она имеет две ветви, в будущем и в прошлом,
представляющие собой пространственноподобные кривые; асимптотами являются
образующие светового конуса. Верхнюю ветвь можно параметрически задать как
$x^{\mu}=t(\cosh\varphi,\sinh\varphi)\,.$ (3.3)
a$O$$A$$x$$t$$\varphi$b$O$$A$$x$$r$$\varphi$ Figure 12: “Окружности”: (a)
$x^{2}=t^{2}$; (b) $x^{2}=-r^{2}$.
Если же “окружность” имеет пространственноподобный радиус ($x^{2}=-r^{2}$,
рис. 12b), то ветви — времениподобные кривые, расположенные в области
удалённого. Правая ветвь параметрически задаётся как
$x^{\mu}=r(\sinh\varphi,\cosh\varphi)\,.$ (3.4)
Времениподобный вектор (3.3) ортогонален пространственноподобному вектору
(3.4) с тем же $\varphi$ (рис. 10).
$d\varphi$$\varphi$$\varphi+d\varphi$ Figure 13: Дуги, соответствующие малых
углам $d\varphi$.
Точки на “окружности” (3.4), соответствующие углу 0 и малому углу $d\varphi$,
разделены вектором $x^{\mu}=(r\,d\varphi,0)$ длины $r\,d\varphi$ (Рис. 13).
Длина дуги между углами $\varphi$ и $\varphi+d\varphi$ равна $r\,d\varphi$ при
любом $\varphi$, потому что поворотом можно превратить угол $\varphi$ в 0, а
$\varphi+d\varphi$ в $d\varphi$. Это легко увидеть и по-другому: точки,
соответствующие углам $\varphi$ и $\varphi+d\varphi$, отстоят друг от друга на
$dx^{\mu}=(\cosh\varphi,\sinh\varphi)\,r\,d\varphi\,,$
и разделены времениподобным интервалом
$dx^{2}=r^{2}d\varphi^{2}\,.$ (3.5)
То есть длина дуги с углом $d\varphi$ есть $r\,d\varphi$ (как и в евклидовой
геометрии).
В трёхмерном пространстве–времени (две пространственных координаты) “сфера”
$x^{2}=t^{2}$ представляет собой двухполостный гиперболоид (рис. 14a). Он
состоит из двух пространственноподобных поверхностей, одна в будущем, а другая
в прошлом. “Сфера” $x^{2}=-r^{2}$ — однополостный гиперболоид (рис. 14b),
времениподобная поверхность, лежащая в области удалённого. В четырёхмерном
пространстве–времени их нарисовать сложнее.
ab Figure 14: “Сферы”: (a) $x^{2}=t^{2}$; (b) $x^{2}=-r^{2}$.
Рассмотрим сумму двух времениподобных векторов $x=OB$ и $y=BA$, направленных в
будущее (рис. 15). На рисунке изображены дуги “окружностей”: расстояния от $O$
до $B$ и до $C$ одинаковы, как и расстояния от $A$ до $B$ и до $D$. Очевидно,
что расстояние от $O$ до $A$ больше, чем сумма расстояний от $O$ до $B$ и от
$B$ до $A$:
$\sqrt{(x+y)^{2}}\geq\sqrt{x^{2}}+\sqrt{y^{2}}\,.$ (3.6)
Равенство достигается, когда $x$ и $y$ коллинеарны; если векторы $x$ и $y$
светоподобны, то $\sqrt{x^{2}}+\sqrt{y^{2}}=0$. Иными словами, проекция $OE$
вектора $OB$ на направление $OA$ длиннее самого вектора $OB$ (рис. 11); точно
так же, проекция $EA$ вектора $BA$ на направление $OA$ длиннее самого вектора
$BA$. Это неравенство легко получить и по-другому: достаточно возвести его в
квадрат и использовать (3.2).
$O$$A$$C$$E$$D$$B$ Figure 15: Неравенство треугольника.
Любую времениподобную мировую линию, соединяющую события $O$ и $A$, можно
сколь угодно точно приблизить ломаной. Из неравенства треугольника (3.6)
следует, что прямая $OA$ имеет наибольшую длину среди всех этих мировых линий.
То есть время между $O$ и $A$ по часам инерциального наблюдателя больше, чем
по часам любого другого наблюдателя, двигавшегося с ускорением111Это иногда
называют “парадоксом близнецов”. По-моему, в том, что две точки можно
соединить кривыми разной длины, нет ничего парадоксального..
Если у нас есть ортонормированный базис $e_{0}$, $e_{1}$, т. е. такой, что
$e_{0}^{2}=1\,,\qquad e_{1}^{2}=-1\,,\qquad e_{0}\cdot e_{1}=0\,,$ (3.7)
то любой вектор $x$ можно по нему разложить:
$x=x^{0}e_{0}+x^{1}e_{1}\,.$ (3.8)
Компоненты вектора получаются проецированием на векторы базиса:
$x_{0}=x\cdot e_{0}\,,\qquad x_{1}=x\cdot e_{1}$ (3.9)
(как мы уже обсуждали, $x_{0}=x^{0}$, $x_{1}=-x^{1}$).
В системе отсчёта инерциального наблюдателя 1, вектор $e_{0}$ направлен вдоль
мировой линии этого наблюдателя, т. е. вдоль его оси времени:
$e_{0}^{\mu}=(1,0)$. Иными словами, это вектор скорости наблюдателя 1 (§ 4).
Вектор $e_{1}$ ему ортогонален: $e_{1}^{\mu}=(0,1)$. Пусть имеется ещё один
инерциальный наблюдатель, мировая линия которого образует угол $\varphi$ с
мировой линией первого наблюдателя (рис. 16). Орт оси времени второго
наблюдателя $e_{0}^{\prime\mu}=(\cosh\varphi,\sinh\varphi)$ направлен из
начала координат в точку на единичной окружности под углом $\varphi$. Орт оси
$x$ ортогонален ему: $e_{1}^{\prime\mu}=(\sinh\varphi,\cosh\varphi)$, и его
конец лежит на окружности $x^{2}=-1$. То есть орты второго наблюдателя
получаются из ортов первого поворотом на угол $\varphi$:
$e_{0}^{\prime}=e_{0}\cosh\varphi+e_{1}\sinh\varphi\,,\qquad
e_{1}^{\prime}=e_{0}\sinh\varphi+e_{1}\cosh\varphi\,.$ (3.10)
Компоненты вектора $x$ (3.8) в повёрнутой системе координат легко найти
проецированием:
$x_{0}^{\prime}=x\cdot
e_{0}^{\prime}=x_{0}\cosh\varphi+x_{1}\sinh\varphi\,,\qquad
x_{1}^{\prime}=x\cdot e_{1}^{\prime}=x_{0}\sinh\varphi+x_{1}\cosh\varphi\,.$
(3.11)
Это ни что иное как преобразование Лоренца (2.8).
$O$$A$$x$$x^{0}$$x^{1}$$e_{0}$$e_{1}$$x^{0\prime}$$x^{1\prime}$$e_{0}^{\prime}$$e_{1}^{\prime}$$\varphi$$\varphi$
Figure 16: Преобразование Лоренца.
## 4 Скорость и ускорение
$x^{\mu}(t)$$x^{\mu}(t+dt)$$dx^{\mu}$$v^{\mu}$ Figure 17: Скорость.
Мировую линию частицы можно задать параметрически как $x^{\mu}(t)$. Её
скорость — это вектор
$v^{\mu}=\frac{dx^{\mu}}{dt}\,,$ (4.1)
где вектор $dx^{\mu}=x^{\mu}(t+dt)-x^{\mu}(t)$, $dx^{2}=dt^{2}$ (рис. 17).
Поэтому
$v^{2}=1\,,$ (4.2)
то есть скорость $v^{\mu}(t)$ — это единичный касательный вектор к мировой
линии в точке $x^{\mu}(t)$, направленный в будущее. Мировая линия частицы,
движущейся по инерции, прямая:
$x^{\mu}=x_{0}^{\mu}+v^{\mu}t\,.$ (4.3)
Ускорение — это
$a^{\mu}=\frac{dv^{\mu}}{dt}\,.$ (4.4)
Поскольку $(v+dv)^{2}=v^{2}+2v\cdot dv=v^{2}=1$,
$v\cdot a=0\,.$ (4.5)
То есть ускорение — пространственноподобный вектор, ортогональный скорости.
$x,a$$v$$\varphi$$\varphi$ Figure 18: Равноускоренное движение.
Пусть мировой линией частицы будет правая ветвь гиперболы
$x^{2}=-r^{2}$ (4.6)
(рис. 18). Параметрически она задаётся как
$x^{\mu}=r(\sinh\varphi,\cosh\varphi)$. Длина дуги (т. е. время) для точки с
углом $\varphi$ есть $t=r\varphi$ (3.5) (если выбрать начало отсчёта времени в
точке $\varphi=0$). Поскольку $(x+dx)^{2}=x^{2}+2x\cdot dx=x^{2}=-r^{2}$,
$x\cdot v=0$, то есть скорость — времениподобный вектор, ортогональный $x$.
При изменении $t$ от $-\infty$ до $+\infty$ конец вектора $x$ пробегает
мировую линию
$x^{\mu}(t)=r\left(\sinh\frac{t}{r},\cosh\frac{t}{r}\right)$ (4.7)
— правую ветвь гиперболы. При этом конец вектора $v(t)$ пробегает верхнюю
ветвь гиперболы $v^{2}=1$, и в каждый момент $t$ остаётся ортогональным
$x(t)$, то есть характеризуется тем же углом $\varphi$ (рис. 18). Конечно, тот
же результат можно получить, продифференцировав (4.7):
$v^{\mu}(t)=\left(\cosh\frac{t}{r},\sinh\frac{t}{r}\right)\,.$ (4.8)
Ускорение $a^{\mu}(t)$ ортогонально $v^{\mu}(t)$, и потому направлено вдоль
$x^{\mu}(t)$. То же можно получить дифференцированием (4.8):
$a^{\mu}(t)=\frac{1}{r}\left(\sinh\frac{t}{r},\cosh\frac{t}{r}\right)=\frac{x^{\mu}(t)}{r^{2}}\,.$
(4.9)
Поэтому $a^{2}=-1/r^{2}$ постоянно; такое движение естественно назвать
равноускоренным.
Рассмотрим теперь пример с двумя пространственными координатами. Протон в
коллайдере равномерно движется по окружности радиуса $r$ с периодом
$T_{0}=2\pi/\omega_{0}$ по часам покоящегося наблюдателя. Его мировая линия —
спираль (рис. 19a)
$x^{1}=r\cos(\omega_{0}x^{0})\,,\quad x^{2}=r\sin(\omega_{0}x^{0})\,.$ (4.10)
$x^{1}$$x^{2}$$x^{0}$ab Figure 19: Мировая линия частицы, равномерно
движущейся по окружности.
Если мысленно развернуть цилиндр, на который намотана спиральная мировая
линия, то станет ясно, что за период протон проходит $T_{0}$ вдоль оси $x^{0}$
и $2\pi r$ в пространственном направлении. Поэтому период движения по часам
протона равен
$T=\sqrt{T_{0}^{2}-(2\pi r)^{2}}=\frac{T_{0}}{\cosh\varphi}\,,$ (4.11)
где
$\cosh\varphi=\frac{1}{\sqrt{1-(r\omega_{0})^{2}}}\,,$ (4.12)
то есть протон движется под углом $\varphi$ к оси $x^{0}$. То же можно
получить дифференцированием (4.10):
$dt^{2}=(dx^{0})^{2}-(dx^{1})^{2}-(dx^{2})^{2}=(1-r^{2}\omega_{0}^{2})(dx^{0})^{2}\,,$
(4.13)
Мировую линию можно задать через собственное время $t$:
$x^{\mu}=(t\cosh\varphi,r\cos(\omega t),r\sin(\omega t))\,,$ (4.14)
где $\omega=2\pi/T=\omega_{0}\cosh\varphi$ (энергия протона $E=m\cosh\varphi$,
см. § 5). Вектор скорости описывает конус, расположенный внутри светового
конуса (Рис. 19b):
$v^{\mu}=(\cosh\varphi,-r\omega\sin(\omega t),r\omega\cos(\omega t))$ (4.15)
(что получается дифференцированием (4.14)). Его конец пробегает окружность
радиуса $\sinh\vartheta=\omega r$, то есть длины $2\pi\omega r$, за период
собственного времени $T=2\pi/\omega$. Поэтому длина вектора ускорения
(пространственноподобная) равна $r\omega^{2}$, и
$a^{\mu}=-r\omega^{2}(0,\cos(\omega t),\sin(\omega t))$ (4.16)
(что получается дифференцированием (4.15)).
## 5 Импульс и волновой вектор
Каждая частица характеризуется своей массой $m$. Вектор
$p^{\mu}=mv^{\mu}$ (5.1)
называется импульсом. Иными словами, импульс — это касательный вектор к
мировой линии длины $m$, направленный в будущее:
$p^{2}=m^{2}\,.$ (5.2)
Все возможные векторы импульса частицы образуют часть “сферы” (5.2), лежащую в
будущем (рис. 14a), она называется массовой поверхностью.
Безмассовые частицы имеют светоподобный импульс: $p^{2}=0$. Например, фотон
имеет $m=0$. Мировые линии безмассовых частиц светоподобны. Любой интервал
времени вдоль такой мировой линии равен 0; поэтому для них невозможно
определить скорость. Импульс, однако, хорошо определён, и является касательным
вектором к мировой линии. Массовая поверхность частицы с $m=0$ — это световой
полуконус будущего (рис. 5).
В системе отсчёта некоторого инерциального наблюдателя $p^{\mu}=(E,\vec{p})$,
где временная компонента $E$ называется энергией, а пространственные
компоненты образуют трёхмерный импульс $\vec{p}$. Они связаны как $p^{2}\equiv
E^{2}-\vec{p}\,^{2}=m^{2}$. То есть энергия частицы, имеющей импульс
$p^{\mu}$, с точки зрения наблюдателя со скоростью $v^{\mu}$ есть $E=p\cdot
v$.
Рассмотрим распад частицы с массой $m$ на две частицы с массами $m_{1}$ и
$m_{2}$. Их импульсы связаны законом сохранения
$p=p_{1}+p_{2}\,.$ (5.3)
По неравенству треугольника (3.6) $m>m_{1}+m_{2}$. Какова энергия частицы 1 в
системе покоя распадающейся частицы? Перепишем закон сохранения (5.3) в виде
$p-p_{1}=p_{2}$ и возведём в квадрат: $(p-p_{1})^{2}=m^{2}+m_{1}^{2}-2p\cdot
p_{1}=m_{2}^{2}$. Но $p\cdot p_{1}=mE_{1}$, поэтому
$E_{1}=\frac{m^{2}+m_{1}^{2}-m_{2}^{2}}{2m}\,.$ (5.4)
Точно так же
$E_{2}=\frac{m^{2}+m_{2}^{2}-m_{1}^{2}}{2m}\,;$ (5.5)
легко проверить сохранение энергии $E_{1}+E_{2}=m$. Трёхмерные импульсы
продуктов распада в системе покоя $p$ противоположны:
$\vec{p}_{1}+\vec{p}_{2}=0$; по модулю они равны
$\vec{p}\,^{2}=E_{1}^{2}-m_{1}^{2}=E_{2}^{2}-m_{2}^{2}=\frac{(m+m_{1}+m_{2})(m-m_{1}-m_{2})(m+m_{1}-m_{2})(m-m_{1}+m_{2})}{4m^{2}}\,.$
(5.6)
Например, в распаде $\pi^{0}\to\gamma\gamma$ массы фотонов $m_{1}=m_{2}=0$, и
$E_{1}=E_{2}=m/2$.
Ещё один пример — эффект Комптона. Фотон с энергией $E$ рассеивается на угол
$\vartheta$ на покоящемся электроне. Найдём энергию $E^{\prime}$ рассеянного
фотона. Пусть импульсы начальных электрона и фотона будут $p$ и $k$, а
конечных $p^{\prime}$ и $k^{\prime}$:
$p+k=p^{\prime}+k^{\prime}\,.$ (5.7)
Перепишем закон сохранения импульса в виде $p+k-k^{\prime}=p^{\prime}$ и
возведём его в квадрат: $(p+k-k^{\prime})^{2}=m^{2}+2p\cdot k-2p\cdot
k^{\prime}-2k\cdot k^{\prime}=m^{2}$. В системе покоя начального электрона
$p=(m,\vec{0})$, $k=E(1,\vec{n}\,)$,
$k^{\prime}=E^{\prime}(1,\vec{n}\,^{\prime})$, где $\vec{n}$,
$\vec{n}\,^{\prime}$ — единичные трёхмерные векторы в направлении движения
начального и конечного фотонов. Поэтому $p\cdot k=mE$, $p\cdot
k^{\prime}=mE^{\prime}$, $k\cdot k^{\prime}=EE^{\prime}(1-\cos\vartheta)$, и
мы получаем
$\frac{1}{E^{\prime}}-\frac{1}{E}=\frac{1-\cos\vartheta}{m}\,.$ (5.8)
И ещё один пример. Пион с энергией $E_{\pi}$ налетает на покоящийся нуклон.
При каких $E_{\pi}$ может происходить реакция $N\pi\to\Lambda K$
($m_{\Lambda}+m_{K}>m_{N}+m_{\pi}$)? Закон сохранения импульса
$p_{N}+p_{\pi}=p_{\Lambda}+p_{K}\,.$ (5.9)
Квадрат массы конечного состояния
$(p_{\Lambda}+p_{K})^{2}\geq(m_{\Lambda}+m_{K})^{2}$. Перепишем его через
импульсы начальных частиц:
$(p_{N}+p_{\pi})^{2}=m_{N}^{2}+m_{\pi}^{2}+2m_{N}E_{\pi}\geq(m_{\Lambda}+m_{K})^{2}$,
откуда
$E_{\pi}\geq\frac{(m_{\Lambda}+m_{K})^{2}-m_{N}^{2}-m_{\pi}^{2}}{2m_{N}}\,.$
(5.10)
На пороге рождающиеся $\Lambda$ и $K$ имеют одинаковые скорости, т. е.
$p_{\Lambda}$ и $p_{K}$ коллинеарны.
Теперь мы рассмотрим волны. Плоская волна
$e^{-ik\cdot x}$ (5.11)
характеризуется волновым вектором $k$. Если $k^{2}>0$, то можно перейти в
систему покоя, где $k=(k^{0},\vec{0})$. В ней волна (5.11) представляет собой
колебания $e^{-ik^{0}x^{0}}$, синхронные во всём пространстве (рис. 20a). В
произвольной системе отсчёта, поверхности постоянной фазы (например,
максимумы, минимумы и нули действительной части волны (5.11)) представляют
собой пространственноподобные плоскости, ортогональные волновому вектору $k$
(рис. 20b). Если $k^{2}=0$ (как для электромагнитных волн), то системы покоя
не существует. В этом случае плоскости постоянной фазы светоподобны, и
светоподобный вектор $k$ лежит на такой плоскости (рис. 20c).
a$k$b$k$$\varphi$$\varphi$c$k$ Figure 20: Плоские волны: волновой вектор $k$ и
плоскости постоянной фазы.
Наблюдатель со скоростью $v$ имеет мировую линию $x=vt$. Он воспримет волну
(5.11) как колебания $e^{-i\omega t}$ с частотой $\omega=k\cdot v$. Частоты
волны с точки зрения разных наблюдателей различны (эффект Допплера).
В квантовой теории нет традиционного для классической физики различия между
частицами и полями — квантовые поля объединяют эти понятия. Элементарное
возбуждение (квант поля), соответствующее плоской волне (5.11) с волновым
вектором $k$, есть частица с импульсом
$p=k\,.$ (5.12)
Это соотношение де Бройля в естественных единицах $\hbar=1$. Так что два
вектора, которые мы обсуждали в этом параграфе, есть на самом деле одно и то
же.
## Appendix A Часы
Если наблюдатель несёт с собой частицу массы $m$, то её волновая функция
зависит от времени как $e^{-imt}$. Однако фаза волновой функции не может быть
измерена, и поэтому не может использоваться в качестве часов.
Допустим, наблюдатель взял с собой мешок свежеприготовленных $B_{s}^{0}$
мезонов. Они (как и $\bar{B}_{s}^{0}$ мезоны) не имеют определённой массы, а
представляют собой суперпозицию $B_{sH}^{0}$ и $B_{sL}^{0}$ мезонов:
$|B_{s}^{0}{>}=\frac{|B_{sH}^{0}{>}+|B_{sL}^{0}{>}}{\sqrt{2}}\,,\qquad|\bar{B}_{s}^{0}{>}=\frac{|B_{sH}^{0}{>}-|B_{sL}^{0}{>}}{\sqrt{2}}\,.$
(A.1)
$B_{sH}^{0}$ и $B_{sL}^{0}$ мезоны имеют определённые массы $m_{sH}$ и
$m_{sL}$. Поэтому по прошествии времени $t$ состояние частицы, которая была
$B_{s}^{0}$ мезоном при $t=0$, будет
$\begin{split}|B_{s}^{0}(t){>}&{}=\frac{e^{-im_{sH}t}|B_{sH}^{0}{>}+e^{-im_{sL}t}|B_{sL}^{0}{>}}{\sqrt{2}}\\\
&{}=e^{-imt}\left[\cos\frac{\Delta m\,t}{2}\,|B_{s}^{0}{>}-i\sin\frac{\Delta
m\,t}{2}\,|\bar{B}_{s}^{0}{>}\right]\,,\end{split}$ (A.2)
где $m=(m_{sH}+m_{sL})/2$, $\Delta m=m_{sH}-m_{sL}$. То есть эта частица будет
$B_{s}^{0}$ мезоном с вероятностью $\cos^{2}(\Delta m\,t/2)$ и
$\bar{B}_{s}^{0}$ мезоном с вероятностью $\sin^{2}(\Delta m\,t/2)$.
Вероятности распадов, характерных для $B_{s}^{0}$ и $\bar{B}_{s}^{0}$, будут
соответственно осциллировать. Подсчитав число этих осцилляций, можно измерить
промежуток времени между двумя событиями.
Реальные атомные часы подобным образом используют разность масс двух
стационарных состояний атома.
## Appendix B Координаты светового фронта
Иногда удобно вместо компонент $x^{0}$, $x^{1}$ вектора $x$ в
ортонормированном базисе использовать
$x_{+}=x^{0}+x^{1}\,,\qquad x_{-}=x^{0}-x^{1}\,.$ (B.1)
Для события $A$ на рис. 8, $x_{+}=t_{2}=te^{\varphi}$,
$x_{-}=t_{1}=te^{-\varphi}$. Квадрат вектора $x$
$x^{2}=x_{+}x_{-}$ (B.2)
равен $t^{2}$. Для времениподобного вектора $x$, направленного в будущее,
$x_{+}>0$ и $x_{-}>0$; аналогично, область прошлого определяется неравенствами
$x_{+}<0$, $x_{-}<0$, а в области удалённого эти компоненты имеют
противоположные знаки (рис. 4).
Координаты $x^{\prime}_{+}$, $x^{\prime}_{-}$ события $A$ относительно
наблюдателя 2 (рис. 9) выражаются через его координаты $x_{+}$, $x_{-}$
относительно наблюдателя 1 формулой (2.7):
$x^{\prime}_{+}=x_{+}e^{-\varphi}\,,\qquad x^{\prime}_{-}=x_{-}e^{\varphi}\,.$
(B.3)
При этом $x^{2}$ (B.2) остаётся инвариантным. Эта форма записи преобразования
Лоренца эквивалентна (2.8), но выглядит проще.
“Окружность” $x^{2}\equiv x_{+}x_{-}=t^{2}$ представляет собой гиперболу (рис.
12a); её верхнюю ветвь можно параметрически задать как
$x_{+}=te^{\varphi}\,,\qquad x_{-}=te^{-\varphi}\,.$
Аналогично, правую ветвь “окружности” $x^{2}\equiv x_{+}x_{-}=-r^{2}$ (рис.
12b) параметрически можно задать как
$x_{+}=re^{\varphi}\,,\qquad x_{-}=-re^{-\varphi}\,.$
$e_{+}$$e_{-}$$e_{0}$$e_{1}$$x$$\frac{1}{2}x_{+}e_{-}$$\frac{1}{2}x_{-}e_{+}$
Figure 21: Координаты светового фронта.
Скалярное произведение (3.1) в координатах светового фронта имеет вид
$x\cdot y=\frac{1}{2}\left(x_{+}y_{-}+x_{-}y_{+}\right)\,.$ (B.4)
Вместо ортонормированного базиса $e_{0}$, $e_{1}$ введём базис из двух
светоподобных векторов
$\begin{split}&e_{+}=e_{0}-e_{1}\,,\qquad e_{-}=e_{0}+e_{1}\,;\\\
&e_{0}=\frac{1}{2}\left(e_{+}+e_{-}\right)\,,\qquad
e_{1}=\frac{1}{2}\left(e_{-}-e_{+}\right)\,.\end{split}$ (B.5)
Они удовлетворяют свойствам
$e_{+}^{2}=e_{-}^{2}=0\,,\qquad e_{+}\cdot e_{-}=2\,.$ (B.6)
Любой вектор $x$ можно разложить по этому базису (рис. 21):
$x=\frac{1}{2}\left(x_{-}e_{+}+x_{+}e_{-}\right)\,;$ (B.7)
компоненты выделяются проецированием:
$x_{+}=x\cdot e_{+}\,,\qquad x_{-}=x\cdot e_{-}\,.$ (B.8)
$e_{+}$$e^{\prime}_{+}$$e_{-}$$e^{\prime}_{-}$$e_{0}$$e^{\prime}_{0}$$e_{1}$$e^{\prime}_{1}$$x$
Figure 22: Преобразование Лоренца в координатах светового фронта.
Если мировая линия второго инерциального наблюдателя образует угол $\varphi$ с
мировой линией первого, то
$e_{-}^{\prime}=e_{-}e^{\varphi}\,,\qquad e_{+}^{\prime}=e_{+}e^{-\varphi}\,,$
(B.9)
(рис. 22), и компоненты вектора $x$ во второй системе отсчёта даются формулой
(B.3).
|
arxiv-papers
| 2011-08-03T13:30:17 |
2024-09-04T02:49:21.246154
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrey Grozin",
"submitter": "Andrey Grozin",
"url": "https://arxiv.org/abs/1108.0842"
}
|
1108.0924
|
arxiv-papers
| 2011-08-03T12:59:25 |
2024-09-04T02:49:21.287189
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Israfil I. Guseinov, Nimet Zaim and Nursen Seckin Goorgun",
"submitter": "Israfil Guseinov",
"url": "https://arxiv.org/abs/1108.0924"
}
|
|
1108.1020
|
# Optical Variability and Colour Behaviour of 3C 345
Jianghua Wu1, Xu Zhou1, Jun Ma1 and Zhaoji Jiang1
1Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences,
20A Datun Road, Chaoyang District, Beijing 100012, China E-mail:
jhwu@nao.cas.cn
###### Abstract
The colour behaviour of blazars is a subject of much debate. One argument is
that the BL Lac objects show bluer-when-brighter chromatism while the flat-
spectrum radio quasars (FSRQs) display redder-when-brighter trend. Base on a
3.5-year three-colour monitoring programme, we studied the optical variability
and colour behaviour of one FSRQ, 3C 345. There is at least one outburst in
this period. The overall variation amplitude is 2.640 mags in the $i$ band.
Intra-night variability was observed on two nights. The bluer-when-brighter
and redder-when-brighter chromatisms were simultaneously observed in this
object when using different pairs of passbands to compute the colours. The
bluer-when-brighter chromatism is a shared property with the BL Lacs, while
the redder-when-brighter trend is likely due to two less variable emission
features, the Mg ii line and the blue bump, at short wavelengths. With
numerical simulations, we show that some other strong but less variable
emission lines in the spectrum of FSRQs may also significantly alter their
colour behaviour. Then the colour behaviour of an FSRQ is linked not only to
the emission process in the relativistic jet, but also to the redshift, the
passbands used for computing the colour and the strengths of the less variable
emission features relative to the strength of the non-thermal continuum.
###### keywords:
galaxies: active — quasars: individual (3C 345).
## 1 Introduction
Blazars are the most variable subclass of active galactic nuclei (AGNs). The
most prominent property of blazars is the rapid and strong variability in
their continuum emission. The continuum emission and its variability is
believed to originate from the relativistic jet pointed basically to our line
of sight. In the frame work of leptonic model, the low energy continuum
emission of blazars has a synchrotron origin, while the high energy continuum
emission is from inverse Compton upscattering of the low energy emission by
the relativistic electrons in the jet. The low energy seed photons may either
be produced in the jet (the synchrotron self-Compton or SSC process) or come
from the accretion disc, broad line region (BLR), or dusty torus (for a recent
review, see Böttcher, 2007). Blazars can be subdivided into flat-spectrum
radio quasars (FSRQs) and BL Lacs, depending on whether or not there are
strong emission lines in their spectra.
3C 345 is the first established variable quasar (Burbidge, 1965) and is
classified as FSRQs later. Its redshift is 0.5928 (Marziani et al., 1996) and
its position is 16:42:58.8, 39:48:37 (J2000.0). The high declination enables
short monitoring gap ($\sim$ 80 days) each year. Ever since its discovery, it
has been monitored intensively and a wealth of data have been collected. Its
optical flux shows rapid variations occurring in a few days or weeks
superimposed upon a slowly varying component (McGimsey et al., 1975). Several
large-amplitude outbursts ($\geq 1.5$ mags) were observed (e.g. Schramm et
al., 1993; Webb et al., 1994). Based on a collection of the historical data
and on their own observations, Belokon & Babadzhanyants (1999) found that 3C
345 varied by more than 3.0 mags in the $B$ band from 1965 to 1995. On short
time-scales, Babadzhanyants et al. (1985) reported a brightening of 0.48 $B$
mags in just half an hour on JD 245 4230. Kidger & de Diego (1990) even
claimed a 0.47 $B$ mags brightness drop in 13 minutes. Despite these rapid
variations, Mihov et al. (2008) did not find significant intra-night optical
variability (INOV) in this object. The optical polarization of 3C 345 is
highly variable between 5% and 35% and is strongly correlated with brightness
and wavelength (Smith et al., 1986).
In the radio domain, several large amplitude outbursts were observed in this
object (see the website of Radio Astronomical Observatory, University of
Michigan, http://www.astro.lsa.umich.edu/obs/radiotel/umrao.php). The
outbursts occurred at high frequency and propagated gradually to lower
frequencies with gradually decreasing amplitudes (Aller et al., 1985; Bregman
et al., 1986; Webb et al., 1994). The high-frequency ($\geq 14$ GHz) radio
flares occurred almost simultaneously with the optical ones, while the lower-
frequency (4.8 and 8 GHz) flares lagged the optical-infrared flares by roughly
1 years (Bregman et al., 1986; Webb et al., 1994; Lobanov & Roland, 2005).
Superluminal components were identified in the jet of this object (Perley,
Fomalont & Johnston, 1982; Kollgaard, Wardle & Roberts, 1989).
The period was searched in the variability of 3C 345 since late 1960’s. The
claimed periods range from 80 days (Kinman et al., 1968) to 11.4 years (Webb
et al., 1988). However, some more recent results didn’t show any period (e.g.,
Kidger & Beckman, 1986; Kidger, 1989; Schramm et al., 1993). A few models were
proposed to explain the quasi-periodic variability of 3C 345, such as the
lighthouse model by Schramm et al. (1993) and the binary black hole models by
Caproni & Abraham (2004) and by Lobanov & Roland (2005).
The colour behaviour of blazars is a subject of much debate. Some authors
found a bluer-when-brighter (BWB) chromatism (e.g., Vagnetti et al., 2003; Wu
et al., 2005, 2007), some others claimed the opposite, namely, a redder-when-
brighter (RWB) trend (e.g., Ramírez et al., 2004), or no clear tendency (e.g.,
Böttcher et al., 2007, 2009). The same object may show different trends in
different variation modes (e.g., Poon et al., 2009) or on different time-
scales (e.g., Ghisellini et al., 1997; Raiteri et al., 2003). Several authors
argued that BL Lacs display BWB chromatism while FSRQs show RWB trend (Fan &
Lin, 2000; Gu et al., 2006; Hu et al., 2006; Rani et al., 2010).
We have monitored 3C 345 since 2006. More than 700 data points were collected
on 53 nights. The long- and short-term variability of this object was studied
and the colour behaviour was investigated. Here we present the results.
Figure 1: Finding chart of 3C 345, taken in the $i$ band on JD 245 4954. The
blazar and three comparison stars are labelled. North is to the up and east,
to the left.
## 2 Observations and Data Reductions
The monitoring was performed with a 60/90 cm Schmidt telescope at Xinglong
station, National Astronomical Observatories of China. The telescope is
equipped with a $4096\times 4096$ E2V CCD and 15 intermediate-band filters,
which are used to do the Beijing-Arizona-Taiwan-Connecticut (BATC) survey
(Zhou, 2005). The CCD has a pixel size of 12 $\micron$ and a spatial
resolution of 1.3 $\arcsec{\rm pixel}^{-1}$. When used for blazar monitoring,
only the central $512\times 512$ pixels are read out as a frame in order to
reduce the readout time and to increase the sampling rate. Each such frame has
a field of view of about $11^{\prime}\times 11^{\prime}$. The monitoring of 3C
345 was made in five intermediate bands, $c$, $e$, $i$, $m$ and $o$ ($e$, $i$
and $m$ bands in 2006 and $c$, $i$ and $o$ bands from 2007 to 2009). Their
central wavelengths and bandwidths are listed in Table 1. The exposure times
are generally 540, 300, 180, 300 and 480 s in the $c$, $e$, $i$, $m$ and $o$
bands, respectively, and may vary slightly depending on the weather and moon
phase. An example frame in the $i$ band is shown in Fig. 1. 3C 345 and three
reference stars are labelled. The reference stars are adopted from Smith et
al. (1984). Their $c$, $e$, $i$, $m$ and $o$ magnitudes were obtained with
observations on several photometric nights and are presented in Table 2. The
data presented here cover the period from 2006 February 16 (JD 245 3783) to
2009 June 1 (JD 245 4984). More than 700 data points were collected on 53
nights.
Table 1: Central Wavelengths and Bandwidths of 5 BATC Filters Filter | Central Wavelength | Bandwidth
---|---|---
| (Å) | (Å)
$c$ | 4206 | 289
$e$ | 4885 | 372
$i$ | 6685 | 514
$m$ | 8013 | 287
$o$ | 9173 | 248
The data reduction procedures include bias subtraction, flat-fielding,
extraction of instrumental aperture magnitude, and flux calibration. The radii
of the aperture and the sky annuli were adopted as 3, 7 and 10 pixels,
respectively. The brightness of 3C 345 was calibrated relative to the average
brightness of stars B and D. Star E acts as a check star. Its differential
magnitude was also calculated relative to the average brightness of stars B
and D, and was used to verify the accuracy of our observations.
Table 2: BATC $c$, $e$, $i$, $m$ and $o$ magnitudes of reference stars Passband | B | D | E
---|---|---|---
$c$ | 14.933 | 16.286 | 16.951
$e$ | 14.782 | 15.885 | 16.183
$i$ | 14.118 | 15.069 | 14.728
$m$ | 13.974 | 14.886 | 14.332
$o$ | 13.852 | 14.706 | 14.115
Table 3: Statics on INOVs JD | Band | Duration | $C$ | Amplitude
---|---|---|---|---
| | (hour) | | (mag)
245 3783 | $e$ | 1.62 | 2.750 | 0.104
245 3783 | $i$ | 1.62 | 8.286 | 0.153
245 3783 | $m$ | 1.62 | 5.545 | 0.154
245 3786 | $i$ | 2.24 | 7.889 | 0.437
245 3786 | $m$ | 2.25 | 5.048 | 0.252
## 3 Light Curves and Variation Amplitudes
Figure 2: Intra-night light curves of 3C 345 in the $e$, $i$ and $m$ bands on
JD 245 3783 (large panels) and of the check stars (small panels).
Figure 3: Intra-night light curves of 3C 345 in the $i$ and $m$ bands on JD
245 3786 (large panels) and of the check stars (small panels).
Among the 53 nights, 3C 345 shows INOVs on only 2 nights in 2006. The light
curves on these 2 nights are displayed in Figs. 2 and 3. The large panels show
the light curves of 3C 345, while the small panels give those of the check
star. Following Jang & Miller (1997) and Romero, Cellone & Combi (1999), a
variability parameter is defined as $C=\sigma_{\mathrm{T}}/\sigma$, where
$\sigma_{\mathrm{T}}$ is the standard deviation of the magnitudes of the
target blazar and $\sigma$ is that of the check star. The latter can be taken
as the typical measurement error on a certain night. When $C\geqslant 2.576$,
the object can be claimed to be variable at the 99% confidence level. The
variability parameters were calculated for the five intra-night light curves,
and the results are listed in Table 3. All five $C$ values are greater than
2.576, thus confirming the INOVs on these two nights. The variation
amplitudes, as defined by Heidt & Wagner (1996), are also given in Table 3.
The amplitude in the $e$ band is much less than those in the $i$ and $m$ bands
on the same night.
Figure 4: Nightly-mean night curves of 3C 345 in five bands. For clarity, the
$c$-, $e$-, $m$\- and $o$-band magnitudes are shifted by 1.0, 1.0, $-$1.0 and
$-$1.0 magnitudes, respectively.
3C 345 did not show INOV on the remaining 51 nights. Part of the reason may be
the relatively short monitoring period on each single night in 2007–2009. Then
its nightly average magnitudes were calculated and used for the following
analyses. The nightly-averaged light curves are plotted in Fig. 4. Data in 5
different passbands are denoted with different symbols and are connected with
different lines. One can see that there is at least one outburst from 2006 (JD
245 3800) to the first half of 2007 (JD 245 4250). Our recorded peak is at JD
245 3909, with $i=15.011\pm 0.011$, which corresponds to $R\sim 15.008$ with a
simple interpolation between the fluxes in the $e$ and $i$ bands. However, we
have very few observations during this period. So we cannot identify the exact
time(s) and amplitude(s) (and/or number) of the outburst(s). From around JD
245 4200 to JD 245 4600, the object was monitored more or less constantly and
declined gradually in brightness. After that, we again have few data and the
object showed a tendency to recover till the end of our monitoring. The
overall amplitude in the $i$ band (used in the whole monitoring period) is
2.640 mags. In 2006, where the source was monitored in the $e$, $i$ and $m$
bands, the variation amplitudes in the three bands are respectively 1.656,
1.829 and 1.731 mags. In 2007–2009 (or from JD 245 4192 to JD 245 4591), where
the source was monitored in the $c$, $i$ and $o$ bands, the amplitudes are
respectively 1.652, 1.964 and 1.947 mags. Then two conclusions can be drawn.
Firstly, the amplitude in the $e$ or $c$ band is the smallest among those in
the three bands ($e$, $i$ and $m$, or $c$, $i$ and $o$). This is similar to
the case of the INOVs in this object mentioned above. Secondly, the $i$ band
amplitude is greater than the corresponding $m$ or $o$ band amplitude.
Figure 5: Colour-magnitude diagrams of 3C 345 in 2006 (left panels) and in
2007–2009 (right panels). The upper four panels show RWB chromatism while the
bottom two panels show BWB chromatism.
## 4 Colour Behaviour
The long-term colour behaviours of 3C 345 were studied by using the nightly-
average magnitudes. For the 2006 data, the colour indices of $e-i$, $e-m$ and
$i-m$ were calculated and plotted against $e$, $e$ and $i$, respectively, in
the left panels of Fig. 5. For the 2007–2009 data, the colour indices of
$c-i$, $c-o$ and $i-o$ were calculated and plotted against $c$, $c$ and $i$,
respectively, in the right panels of Fig. 5. The $e-i$, $e-m$, $c-i$ and $c-o$
colours got redder when the source became brighter, whereas the $i-m$ and
$i-o$ colours got bluer when the source became brighter. The correlation
coefficients are respectively $-$0.958, $-$0.557, $-$0.794, $-$0.702, 0.707
and 0.307, and the chance probabilities are respectively $6.83\times 10^{-4}$,
$1.94\times 10^{-1}$, $2.16\times 10^{-8}$, $1.06\times 10^{-5}$, $4.98\times
10^{-2}$ and $7.72\times 10^{-2}$. There are much fewer data points in the
left panels than in the right ones, so the correlations in the left ones are
weaker than those in the right ones, but the overall trends are the same.
For the two nights (JDs 245 3783 and 245 3786) showing INOVs, the colour-
magnitude relation was also explored. Because there is no $e$-band observation
on the latter night, we only studied the $i-m$ colour versus $i$ magnitude
correlation. When calculating the $i-m$ colour, the $m$-band light curve on
the two nights were linearly interpolated or extrapolated so as to get the $m$
magnitudes at exactly the same time when the $i$-band observations were made.
The result is displayed in Fig. 6. The object tends to be BWB. The correlation
coefficient is 0.809 and the chance probability is $8.37\times 10^{-5}$,
indicating a strong BWB chromatism.
Figure 6: Colour-magnitude diagram of 3C 345 on JDs 245 3783 and 245 3786, the
two nights showing INOVs. A strong BWB chromatism is visible.
Therefore, both RWB and BWB chromatisms were observed in the variability of 3C
345. This is quite different from the previous results on the spectral or
colour behaviour of blazars, in which usually only one kind of colour
behaviour is reported for one object. Or at most, an object, except for
displaying either RWB or BWB chromatism in its middle- and short-term
variability, may be achromatic for its long-term variability (e.g., Ghisellini
et al., 1997; Villata et al., 2002; Raiteri et al., 2003) or during a certain
episode of time (Wu et al., 2005; Poon, Fan & Fu, 2009). One exception is that
Raiteri et al. (2003) have recognized both RWB and BWB trends in S5 0716+714,
but at different times. Our study, however, revealed that the RWB and BWB
chromatisms appeared in 3C 345 at the same time and/or in both the long-term
variability and INOVs. This is the first report of such a phenomenon in the
study of the colour behaviour of blazars.
## 5 Discussions
Figure 7: Composite quasar spectra and Transmission curves of three filter
systems.
Several authors have studied the colour or spectral behaviour of 3C 345. They
all found a RWB or steeper-when-brighter behaviour in the optical domain in
this object (e.g., Kidger & Takalo, 1990; Schramm et al., 1993; Zhang et al.,
2000; Mihov et al., 2008). However, there may be a cutoff magnitude ($R\approx
15.5$) in the colour-magnitude correlation, as noted by Mihov et al. (2008).
When the object is brighter than that magnitude, its colour is much less
dependent on the magnitude. This cutoff magnitude may also exist in the
colour-magnitude diagrams in Mihov et al. (2008) and Zhang et al. (2000), but
is not so obvious as in Schramm et al. (1993).
It has been suggested that a few components in the UV to blue band of the
spectrum of 3C 345 may be responsible for the RWB or steeper-when-brighter
behaviour (Kidger & Takalo, 1990; Schramm et al., 1993; Mihov et al., 2008).
These components include the Mg ii emission line at 2798 Å and the blue bump
(BB) from 1000 to 4000 Å in the rest frame of this object. The BB itself
consists of a forest of the Fe ii emission lines, the Balmer continuum re-
emission and the 10 000–40 000 K black body emission. Unlike the underlying
non-thermal continuum (hereafter NTC), which is believed to come from the jet,
these features come either from the accretion disc or from the BLR. Their
variation properties should resemble those of the same emission features of
normal quasars, i.e., they are variable but have smaller amplitudes and longer
time-scales than the NTC variability of blazars (e.g. Ulrich, Maraschi & Urry,
2008; Benítez et al., 2010). When 3C 345 is in low state, these less variable
emission features (hereafter LVEFs) dominate the fluxes and dilute the
variability at short wavelengths. So the variation amplitudes at short
wavelengths are smaller than those at long wavelengths, as manifested by
Bregman et al. (1986) and this work (§3), leading naturally to a RWB behaviour
when the measurements at short wavelengths ($U$, $B$, $V$, $c$ or $e$) are
included in the colour calculation (Dai et al., 2011). When 3C 345 is in high
state (brighter than the cutoff magnitude of $R=15.5$), the underlying NTC
become strong and may match the LVEFs in strength. Then the object will show a
stable colour, as illustrated in Schramm et al. (1993).
On the other hand, when the colour or spectral index is computed by using two
measurements at long wavelengths where the LVEFs are not included in, we’ll
get a BWB or flatter-when-brighter behaviour for 3C 345, as for its BWB
behaviour in the $i-m$ and $i-o$ colours in this work and its flatter-when-
brighter trend in the infrared regime in Bregman et al. (1986).
Similar colour behaviour has been observed for at least two more FSRQs. In 3C
454.3, the colour is RWB in faint state, reaches a ‘saturation’ at $R\approx
14$, and turns into BWB in bright state (Villata et al., 2006; Raiteri et al.,
2008; Sasada et al., 2010a). These behaviours were also explained in terms of
a thermal component, probably from the accretion disc, in the emission of 3C
454.3 (Raiteri et al., 2007, 2008; Sasada et al., 2010a). For PKS 1510-089,
which has a pronounced BB in its spectrum (Singh et al., 1997; Kataoka et al.,
2008), it exhibits a RWB trend except for its prominent flare, which can be
explained by the strong contribution of thermal emission from the accretion
disc (Sasada et al., 2010b).
Therefore, the Mg ii+BB emission can significantly change the colour behaviour
of FSRQs. Except for the Mg ii line and the BB, there are some other strong
emission features in the spectrum of FSRQs, such as the Ly$\alpha\,\lambda
1216$, C iv $\lambda 1549$, C iii]$\,\lambda 1909$, H$\beta\,\lambda 4861$+[O
iii] $\lambda\lambda 4959,5007$+Fe ii and H$\alpha\,\lambda 6563$ lines. These
features emanate from the BLR rather than from the jet, and are thus expected
to be less variable than the underlying NTC. They may also significantly
change the colour behaviour of FSRQs when one or more of them is included in
the passbands that are used in the computation of the colour index, especially
when the observations are made with an intermediate- or narrow-band filter
system.
In order to assess how the emission lines change the measured fluxes in the
passbands they reside, we then made a number of simulations by convolving the
FSRQ spectrum with the transmission curves of several filter systems. The
spectrum was shifted from 0 to 3 with a step of 0.01. Each spectrum was
convolved with the transmission curves, so as to see the redshift effects on
the measured fluxes.
We used a composite quasar spectrum to mimic the FSRQ spectrum in order to
have a high signal-to-noise ratio and a large enough wavelength coverage. The
FSRQ spectrum resembles the normal quasar spectrum in the content of the
emission lines. However, their continua may differ much. The quasar spectrum
usually has a strong BB, while the FSRQ spectrum has no or only weak BB due to
the prominence of the strongly beamed jet emission (Jolley et al., 2009). We
kept the emission features of the composite quasar spectrum but partly removed
the BB from the continuum. The modified composite quasar spectrum was then
used in the convolutions.
The composite quasar spectrum was adopted as a combination of the HST UV
composite from 650 to 1150 Å (Zheng et al., 1997), the 2dF quasar composite
from 1150 to 6930 Å (Croom et al., 2002) and a near-infrared composite from
6930 to 10600 Å (Glikman et al., 2006). We assigned a spectral index of 1.52
(Fossati et al., 1998) to the continuum from 1000 to 10600 Å and a spectral
index of 2.6 for the continuum from 650 to 1000 Å, so as to add a weak BB into
the continuum. The spectral index $\alpha$ is defined as
$f_{\nu}\sim\nu^{-\alpha}$. This modified composite spectrum is plotted in the
left panel of Fig. 7. For a comparison, the composite of the Large Bright
Quasar Survey (LBQS, Francis et al., 1991) is also plotted, demonstrating the
strong BB peaked at around 1300 Å. The right panel of Fig. 7 displays the
transmission curves of the BATC (15 intermediate bands), SDSS (five broad
bands, $u,g,r,i$ and $z$) and Johnson-Cousins (J-C, five broad band, $U,B,V,R$
and $I$) filter systems.
Figure 8: Observed fluxes of a FSRQ in the 15 intermediate BATC bands. The
fluxes is plotted with colour in logarithm scale. The open circles denote the
redshifts at which the emission lines move to the weighted centers of the
passbands. The dashed lines are used to guide the ’movements’ of the emission
lines with increasing redshift.
The convolutions were made at first between the composite spectrum and the
BATC transmission curves. The resulting fluxes in the 15 intermediate bands
were plotted in logarithm scale as colours in Fig. 8. The open circles label
the redshifts at which the emission lines move to the weighted centers of the
passbands. One can see from that figure that the strong emission lines move
gradually from the short- to long-wavelength BATC passband with increasing
redshift, as shown by the dashed lines. This figure demonstrates clearly that
the strong emission lines can significantly enhance the measured fluxes in the
passbands they are included in and that the enhancement changes with redshift.
At the redshift close to 0.6, the fluxes of 3C 345 at the $c$, $d$ and $e$
bands are enhanced and dominated by the Mg ii line and the Fe ii lines on its
both sides, especially during its faint state. On the other hand, the fluxes
in the $i$, $m$ and $o$ bands are not enhanced by any strong emission lines
and are dominated by the emission from the jet. So the $e-i$, $e-m$, $c-i$ and
$c-o$ colours become RWB, while the $i-m$ and $i-o$ colours tend BWB, as
mentioned at the beginning of this section.
Figure 9: Same as Fig. 8 but for the SDSS and J-C filter systems.
Similar convolutions were made between the composite spectrum and the SDSS and
J-C transmission curves. The results are plotted in Fig. 9. Due to the much
broader passbands, which lower the flux dominance of the emission lines over
the underlying NTC, and the partly overlapping of the adjacent passbands (see
Fig. 7, right), the ’movements’ of the emission lines on the two maps are not
so manifest as in Fig. 8, but the overall trends can still be seen. At some
redshifts, more than one emission lines can be included in a same passband.
For example, both the Ly$\alpha$ and C iv lines are included in the SDSS $g$
band and in the J-C $U$ band at a redshift of about 2.40 and 1.45,
respectively, leading to a considerably enhanced fluxes at those two
passbands. At the redshift close to 0.6, the $B$-band flux of 3C 345 is
enhanced by the flux of Mg ii line, so its $B-V$ and $B-R$ colours show RWB
trends when it is in low state. In the case of 3C 454.3, which is at a
redshift of 0.859, its $V$-band flux is enhanced by the fluxes of the Mg ii
line. So its $V-R$ and $V-K_{\mathrm{s}}$ colours show RWB trends when it is
in low state.
The simulations clearly illustrate that the strong emssion lines can change
the colour behaviours of FSRQs. Previous explanations for the RWB of FSRQs,
either by the Mg ii+BB or by a thermal component, have in fact included the
contributions from the strong emission lines of the Ly$\alpha$, C iv and C
iii] lines. On the other hand, our simulations focus mainly on the emission
lines, whose equivalent widths are usually less than 80 Å in the rest frame
(Peterson, 1997). For the much broader ’emission’ feature, the BB, which can
span from less than 1000 to about 4000 Å, a wavelength range much broader than
the FWHM ($<1500$ Å) of a broad-band filter, the simulations cannot
demonstrate whether it can change the colour behaviours. However, in the real
cases, the broad (the BB) and narrow (the emission lines) LVEFs usually act
simultaneously, especially in the $B$ and $V$ bands at low redshifts. This
will lead to a RWB trend for the $B-R$, $V-R$, $V-J$ and $V-K_{\mathrm{s}}$
colours when the object is in faint state, as in the cases of 3C 345, 3C 454.3
and PKS 1510-089.
The simulations are not conclusive due to the different strengths of the LVEFs
in different objects. In some objects, the LVEFs are intrinsically weak. Then
the colour behaviour is expected to be BWB. This is in analogy to the case of
BL Lac objects. Also, because of the broad passbands of the SDSS and J-C
filters, there is a good chance for both the blue ($u$, $g$, $U$ and $B$) and
the red passbands ($r$, $i$, $z$, $V$, $R$ and $I$) to include a strong
emission feature. Then whether the object is RWB or BWB will depend on the
interplay between the enhancements of the fluxes in the blue and red
passbands. The most reliable way to study the spectral behaviour of FSQRs is
to fit the continuum of their spectra. This will eliminate the impact of the
LVEFs to the observed fluxes.
Most recently, Gu & Ai (2011) assembled a sample of 29 FSRQs in the SDSS
Stripe 82 region. By fitting a power-law to the SDSS $ugriz$ photometric data,
they found only one FSRQ showing RWB trend. This is quite different from
previous results (Fan & Lin, 2000; Gu et al., 2006; Hu et al., 2006; Rani et
al., 2010). The fitting of five photometric data points is somewhat similar to
the fitting of the continuum, because the fluxes of the LVEFs cannot dominate
in all five passbands, as can be seen in Fig. 9. So their results on the
colour behaviour of FSRQs are more reasonable than the previous results.
## 6 Conclusions
We carried out a three-colour monitoring programme on the FSRQ, 3C 345 from
2006 February to 2009 June. There is at least one outburst during this period.
The overall variation amplitude is 2.640 mags in the $i$ band. INOVs was
observed on two nights. The BWB and RWB chromatisms were simultaneously
detected in this object when using different pairs of passbands to compute the
colours. The BWB chromatism is a shared property with the BL Lacs, while the
RWB trend is likely due to two LVEFs, the Mg ii line and the blue bump, at
short wavelengths.
We made numerical simulations by convolving a composite quasar spectrum with
three filter systems. The results indicate that some other strong but less
variable emission lines, such as the Ly$\alpha$, C iv, C iii], H$\beta$+[O
iii] and H$\alpha$ lines, in the spectrum of FSRQs may also significantly
enhance the measured fluxes of the passbands they are included in, and hence
change the colour behaviour of the object.
The emission of BL Lac objects is dominated by the beamed NTC from the
relativistic jet. However, the emission from FSRQs should be a combination of
the NTC from the jet, the thermal emission from the accretion disc and the
line emission from the BLR. The two latter components are collectively name as
LVEFs in this paper. The beamed NTC from the jet tends to show BWB chromatism,
no matter in BL Lacs or FSRQs. The LVEFs in FSRQs can significantly change
their colour behaviours, especially when they are in faint state.
The factors that govern the colour behaviour of FSRQs can be summarized as
follows.
1. 1.
The relative strengths of the LVEFs and the NTC. The relative strengths of the
two components can be different for different objects, and are related to the
brightness states of the objects. Strong LVEFs but weak NTC will result in RWB
trend, while weak LVEFs and strong NTC will lead to in BWB chromatism.
2. 2.
The redshift. The redshift decides the wavelengths of the observed LVEFs.
3. 3.
The passbands used for colour calculation. Different passbands will include
different LVEFs.
## 7 Acknowledgements
The authors thank the anonymous referee for insightful suggestions and
comments. We thank M. Gu for useful discussions. This work has been supported
by Chinese National Natural Science Foundation grants 10873016, and 11073032,
and by the National Basic Research Programme of China (973 Programme) No.
2007CB815403.
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|
arxiv-papers
| 2011-08-04T08:34:24 |
2024-09-04T02:49:21.292182
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianghua Wu, Zhou Xu, Jun Ma, and Zhaoji Jiang",
"submitter": "Jianghua Wu",
"url": "https://arxiv.org/abs/1108.1020"
}
|
1108.1097
|
# Thulium and ytterbium-doped titania thin films deposited by MOCVD
S bastien Forissier sebastien.forissier@grenoble-inp.org Herv Roussel Carmen
Jimenez Odette Chaix Laboratoire des Mat riaux et du G nie Physique, CNRS,
Grenoble Institute of Technology, MINATEC, 3 parvis Louis N el, 38016
Grenoble, France Antonio Pereira Amina Bensalah-Ledoux Laboratoire de
Physico-Chimie des Mat riaux Luminescents UMR 5620 CNRS / UCBL Domaine
Scientifique de la Doua, Universit Claude Bernard Lyon 1 B timent Alfred
Kastler 10 rue Ada Byron 69622 Villeurbanne cedex, France Jean-Luc
Deschanvres Laboratoire des Mat riaux et du G nie Physique, CNRS, Grenoble
Institute of Technology, MINATEC, 3 parvis Louis N el, 38016 Grenoble, France
Bernard Moine Laboratoire de Physico-Chimie des Mat riaux Luminescents UMR
5620 CNRS / UCBL Domaine Scientifique de la Doua, Universit Claude Bernard
Lyon 1 B timent Alfred Kastler 10 rue Ada Byron 69622 Villeurbanne cedex,
France
(September 2, 2024)
###### Abstract
In this study we synthesized thin films of titanium oxide doped with thulium
and/or ytterbium to modify the incident spectrum on the solar cells. This
could be achieved either by photoluminescence up-converting devices, or down-
converting devices. As down-converter thin films our work deals with thulium
and ytterbium-doped titanium dioxide. Thulium and ytterbium will act as
sensitizer and emitter, respectively. The rare-earth doped thin films are
deposited by aerosol-assisted MOCVD using organo-metallic precursors such as
titanium dioxide acetylacetonate, thulium and ytterbium
tetramethylheptanedionate solved in different solvents. These films have been
deposited on silicon substrates under different deposition conditions
(temperature and dopant concentrations for example). Adherent films have been
obtained for deposition temperatures ranging from 300°C to 600°C. The
deposition rate varies from 0.1 to 1 $\upmu$m/h. The anatase phase is obtained
at substrate temperature above 400°C. Further annealing is required to exhibit
luminescence and eliminate organic remnants of the precursors. The
physicochemical and luminescent properties of the deposited films were
analyzed versus the different deposition parameters and annealing conditions.
We showed that absorbed light in the near-UV blue range is re-emitted by the
ytterbium at 980 nm and by a thulium band around 800 nm.
CVD, thulium, ytterbium, down-conversion, thin film, titanium oxide,
photovoltaic
## I Introduction
Titanium dioxide has received much interest because of its various
applications: photocatalysis, pigment, Transparent Conducting Oxide for
photovoltaic applications Richards2006 . In this field we focused on doping
titanium dioxide with rare-earths (thulium and ytterbium in this work) to
study the down-conversion between those ions from blue-visible and near-
ultraviolet light to near-infrared light for solar spectrum engineering. The
modification of the solar spectrum would allow us to achieve better yields for
silicon solar cells Badescu2007 ; Trupke2002 . Thin films were grown using
aerosol assisted Metal Oxide Chemical Vapor Deposition (MOCVD) method.
## II Experimental
Thulium and ytterbium-doped titanium oxide thin films were deposited by mean
of aerosol assisted Metal Oxide Chemical Vapor Deposition method
Deschanvres1998 ; Deschanvres1989 . Liquid source solution, were composed of
titanium (IV) oxide bis(acetylacetonate), TiO2(acac)2, thulium (III)
tris(2,2,6,6-tetramethyl-3,5-heptanedionate) and ytterbium (III)(acac)
dissolved in high purity butanol (99%). All these precursors were purchased
from STREM Chemicals, butanol was purchased from Alfa Aesar. Precursors were
selected for non-toxicity, good stability at room temperature, easy handling,
high volatility and low cost Ryabova1968 . The films were deposited on (100)
silicon substrate. The source solution is delivered to the piezoelectric
transducer Viguie1975 through a constant level burette to ensure a constant
pulverization during the whole deposition. The aerosol was produced by means
of a flat piezoelectric transducer excited at 800 kHz which generated an
ultrasonic beam in a solution containing the reactant of the material to be
deposited. This ultrasonic spraying system guarantees a narrow dispersion of
the droplet size (4-10 $\upmu$m). Droplets are carried with two air fluxes,
dried and purified, to the heated sample holder, the lower air flux (12.7 l/m)
assure the main propulsion when the upper air flux (10.1 l/m) lengthen the
vapour stay in the vicinity of the sample holder. The overall air flow is
parallel to the substrate’s surface. For the deposition, the substrate was
fixed by clips on the sample holder heated by an electrical resistance with a
built-in thermocouple.
We made different series of samples with varying cation concentration in
solution. Both Tm-doped, Yb-doped and co-doped samples were successfully
synthesized. The compositions of the doped films were measured by electron
probe microanalysis (EPMA) and computed by help of special software dedicated
to the thin film analysis, called Stratagem and edited by the SAMx society
Pouchou1984 . The properties of the films reported in this paper are described
in section 3.2 and summarized in table 1. The X-ray diffraction profile was
obtained with a Bruker D8 Advance using Cu K$\alpha_{1}$ radiation in
$\theta/2\theta$ configuration. Absorption FT-IR spectroscopy was used to
study the structural evolution of the films versus the deposition conditions.
Spectra were obtained between 250 and 4000 cm${}^{-}1$ with 4 cm${}^{-}1$
resolution with a Bio-Rad Infrared Fourier Transform spectrometer FTS165 and
after performing Si substrate subtraction. Fluorescence and decay time
measurements were taken using a laboratory-built apparatus. It is composed of
a tunable laser NT342-10-AW from Ekspla, a monochromator TRIAX 190 from Jobin
Yvon, a photomultiplier C4877 from Hamamatsu and a multichannel scaler SR430
from Stanford Research. Excitation spectra were taken using a F900
spectrofluorimeter Edinburgh with a high spectral resolution. A Xenon Arc lamp
(450 W) is used for the excitation, the detector is a photomultiplier
Hamamatsu R2658P cooled by Peltier effect.
## III Results and discussion
### III.1 Structural properties
Figure 1: Crystallization versus deposition temperature. Figure 2:
Crystallization effect of annealing.
As shown by XRD diffraction the samples deposited at temperature lower than
400°C are amorphous (Figure 1). Above 400°C as deposition temperature, the
films crystallize in the anatase phase of titanium oxide. As shown on Figure 2
the crystalline quality of the films is improved by annealing, first at 500°C
for 1 h and then at 800°C for 1 h. With increasing annealing temperature the
anatase peaks are becoming higher and thinner. After annealing at 800°C
undoped samples crystallize in the rutile phase, whereas the presence of rare-
earth dopants prevents the phase transition as reported in Graf2007 .
### III.2 Composition
Figure 3: Doping versus temperature.
Due to the reactor’s geometry, the thicknesses of the sample are not
homogeneous on the whole surface but it shows a good uniformity on half of it
as seen with interferential colors depending on their thickness. However
despite this geometry the doping level is quite uniform on the whole surface.
We synthesized different samples from the same solution at different
temperature and the electron microprobe measurement showed that the optimal
doping efficiency is obtained at 400°C (Figure 3). In this condition the rare-
earth precursor reactivity is lower than the titanium oxide precursor
reactivity.
### III.3 Luminescence properties
Figure 4: Excitation spectrum of a Tm-doped sample (red curve) and excitation
spectrum of a Yb-doped sample (black curve). Spectra were recorded in separate
experiments. Figure 5: Emission spectrum of a Tm, Yb co-doped sample.
The emission scan (Figure 5) were taken exciting at 330 nm in the TiO2 matrix
(3.2 eV gap) near the${}^{1}D_{2}$ level of thulium. On a Tm, Yb co-doped
sample we see both the Yb ${}^{2}F_{5/2}-^{2}F_{7/2}$ transition at 980 nm and
the Tm ${}^{3}F_{4}-^{3}H_{6}$ transition around 800 nm. On mono-doped samples
(not figured) we only see the respective ion transitions. It appears that the
absorbed energy is transferred through the matrix to the rare-earth dopant
ions leading to a down-conversion mechanism with thulium and ytterbium. The Yb
luminescence is interesting for solar cells because it’s right before the band
gap of the silicon; the Tm luminescence is still in the wavelength range of
good absorption for silicon solar cells.
The excitations scans (Figure 4) were taken looking at the ytterbium
${}^{2}F_{5/2}-^{2}F_{7/2}$ transition and then the thulium
${}^{3}F_{4}-^{3}H_{6}$ transition. Both transitions are activated from the
near-UV region between 300 and 350 nm. This corroborates the idea of the
energy path we have between the matrix and the rare-earth ions.
The lifetime of the ${}^{3}F_{4}-^{3}H_{6}$ transition of Tm was measured on
Tm-doped only and Tm, Yb co-doped samples. For Tm-doped only samples we see
that the lifetime of the transition decreases with the increasing percentage
of Tm (see Table 1). When we compare co-doped and mono-doped samples we see
that at a constant level of Tm when we increase the percentage of Yb the
lifetime is decreasing (Figure 6), leading us to conclude of energy transfer
between Tm and Yb. The transfer rate was computed with the relation
$1-\tau_{x}/\tau$ where $\tau_{x}$ is the lifetime of a co-doped sample and
$\tau$ the lifetime of a Tm-doped sample with the same concentration, with
samples A and E this gives a 10% transfer rate.
Figure 6: Decay time of Tm,Yb-doped samples. Sample | Tm (%) | Yb (%) | Lifetime ($\upmu$)
---|---|---|---
A | 0.86 | 0 | 50
B | 2.08 | 0 | 48
C | 0.41 | 0.69 | 56
D | 0.37 | 0.85 | 46
E | 0.83 | 0.79 | 45
F | 1.21 | 3.63 | 43
Table 1: Cationic doping and lifetimes of the thulium
## IV Conclusion
In summary, we succeeded in doping titanium dioxide with thulium and ytterbium
by Metal Oxide Chemical Vapour and to grow them partially crystallized. After
proper annealing the luminescence of the samples was studied both with
excitation and emission scans. The luminescence study showed energy transfer
between the titanium oxide matrix and the rare-earth ions and two interesting
emission for silicon solar cells. Lifetime measurements showed the energy
exchange between thulium and ytterbium with an estimated transfer rate of 10%.
## V Acknowledgements
This work has been supported by French Research National Agency (ANR) through
Habitat intelligent et solaire photovolta que program (project MULTIPHOT
n°ANR-09-HABISOL-009), the CARNOT Energie du future and the cluster Energies
Rh ne-Alpes.
## References
* [1] BS Richards. Luminescent layers for enhanced silicon solar cell performance: Down-conversion. Solar energy materials and solar cells, 90(9):1189–1207, may 2006\.
* [2] Viorel Badescu, Alexis De Vos, Alina Mihaela Badescu, and Aleksandra Szymanska. Improved model for solar cells with down-conversion and down-shifting of high-energy photons. Journal of physics D-applied physics, 40(2):341–352, JAN 21 2007\.
* [3] T Trupke, MA Green, and P Wurfel. Improving solar cell efficiencies by up-conversion of sub-band-gap light. J. Appl. Phys., 92(7):4117–4122, OCT 1 2002.
* [4] J. L. Deschanvres, W. Meffre, J. C. Joubert, J. P. Senateur, F. Robaut, J. E. Broquin, and R. Rimet. Rare earth-doped alumina thin films deposited by liquid source cvd processes. J. Alloys Compd., 275-277:742 – 745, 1998.
* [5] J-L Deschanvres, F Cellier, G Delabouglise, M Labeau, M Langlet, and J-C Joubert. Thin-film of ceramic oxides by modified CVD. Journal de physique, 50(C-5, 5):695–705, May 1989.
* [6] L.A. Ryabova and Y. S. Savitskaya. The preparation of thin films of some oxides by the pyrolysis method. Thin Solid Films, 2:141, 1968.
* [7] J.C. Viguié and J Spitz. Chemical vapor deposition at low temperatures. J. Electrochem. Soc., 122(4):585, 1975.
* [8] J. L. Pouchou and F. Pichoir. La Recherche Aérospatiale, 5:349–367, 1984.
* [9] Corinna Graf, Renate Ohser-Wiedemann, and Guenter Kreisel. Preparation and characterization of doped metal supported tio2-layers. Journal of photochemistry and photobiology A-chemistry, 188(2-3):226–234, MAY 20 2007.
|
arxiv-papers
| 2011-08-04T14:09:12 |
2024-09-04T02:49:21.298634
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S\\'ebastien Forissier, Herv\\'e Roussel, Carmen Jimenez, Odette Chaix,\n Antonio Pereira, Amina Bensalah-Ledoux, Jean-Luc Deschanvres, and Bernard\n Moine",
"submitter": "S\\'ebastien Forissier",
"url": "https://arxiv.org/abs/1108.1097"
}
|
1108.1141
|
# A Formalism for Scattering of Complex Composite Structures. 2 Distributed
Reference Points
Carsten Svaneborg1,2∗ and Jan Skov Pedersen2 1Center for Fundamental Living
Technology, Department of Physics, Chemistry and Pharmacy, University of
Southern Denmark, Campusvej 55, DK-5320 Odense, Denmark 2Department of
Chemistry and Interdisciplinary Nanoscience Center (iNANO), University of
Aarhus, Langelandsgade 140, DK-8000 Århus, Denmark.
###### Abstract
Recently we developed a formalism for the scattering from linear and acyclic
branched structures build of mutually non-interacting sub-units.[C. Svaneborg
and J. S. Pedersen, J. Chem. Phys. 136, 104105 (2012)] We assumed each sub-
unit has reference points associated with it. These are well defined positions
where sub-units can be linked together. In the present paper, we generalize
the formalism to the case where each reference point can represent a
distribution of potential link positions. We also present a generalized
diagrammatic representation of the formalism. Scattering expressions required
to model rods, polymers, loops, flat circular disks, rigid spheres and
cylinders are derived. and we use them to illustrate the formalism by deriving
the generic scattering expression for micelles and bottle brush structures and
show how the scattering is affected by different choices of potential link
positions.
## I Introduction
Light scattering, small-angle neutron or X-ray scattering (LS, SANS and SAXS,
respectively) techniques are ideal for obtaining detailed information about
self-assembled molecular and colloidal structures.Guinier and Fournet (1955);
Higgins and Benoit (1994); Lindner and Zemb (2002) However, these techniques
provide reciprocal space intensity spectra. Typically such spectra can not be
interpreted directly, instead extensive modeling is required to infer
structural information. Such an analysis necessitates the availability of a
large tool-box of model expressions characterizing the scattering spectra
expected from many different well defined structures. Fitting such expressions
to experimental scattering spectra allows the experimentalist to infer and
accurately quantify which structures are most likely to be present in a given
sample.
While scattering theory and statistical mechanics provides a general framework
for how to derive such models, there are no general analytical methods for
deriving the expected scattering spectra for complex self-assembled
structures. Often unchecked approximations need to be introduced to obtain
analytical results. Alternatively computer simulations can be employed to make
virtual scattering experiments from ensembles of well defined structures,
which can then be compared to the experimental scattering spectra. Our aim
here is to present a formalism for deriving the scattering expressions
characterizing a large class of structures.
We assume the structures are build out of well defined components which we
call sub-units. We make no assumptions as to the internal structure a sub-
units, nor on the number of sub-unit types that can be present in a given
structure. Sub-units have well defined reference points by which they can be
joined to other sub-units. The formalism in its present form requires that all
such joints are completely flexible. Finally the formalism requires that the
structures formed by joining sub-units does not contain loops. For structures
that meet these requirements, the formalism allows exact scattering
expressions characterizing the corresponding scattering spectrum to be derived
with great ease. The central idea is to express the scattering of the whole
structure in terms of scattering expressions characterizing the sub-units
instead of the scattering from the individual scattering sites comprising the
structure. This idea was previously used by Benoit and HadziioannouBenoit and
Hadziioannou (1988) to calculate the scattering from various block-copolymer
structures, and by ReadRead (1998) who applied it to calculate the scattering
from H-polymers and stochastic branched polymer structures. Teixeira et al.
used this idea to calculate the scattering from structures composed of
polymer/rod polycondensatesTeixeira et al. (2000, 2007).
In a previous paperSvaneborg and Pedersen (2012), we derived and presented a
versatile formalism for predicting the scattering from linear and branched
structures composed of arbitrary functional sub-units. We argued that the
formalism is complete in the following sense: Three functions describe the
scattering from a sub-unit, and we derive three analogous scattering
expressions that describe the scattering from a structure. Hence, we can 1)
build bottom-up hierarchical structures by building structures by joining sub-
units to well defined sites in a structure and 2) build top-down hierarchical
structures by substituting sub-units by more complex sub-structures composed
of sub-units. Furthermore the formalism is generic, in the sense that
scattering contributions from structural connectivity and the internal sub-
unit structures are decoupled. This allows generic structural scattering
expressions to be derived and composed to describe the scattering from complex
hierarchical structures independently of which sub-units we build the
structure. We also developed a diagrammatic interpretation of the formalism
that allows us to map structural transformations onto algebraic
transformations of the corresponding scattering expressions. We illustrated
this with the transformation rules producing the scattering expression for an
$n$’th generation dendrimer, by successive replacement of the outermost leafs
by star shaped sub-structures. In this previous paperSvaneborg and Pedersen
(2012), we focused on structural complexity and illustrated the formalism by
deriving the structural scattering expressions of linear chain structures,
stars, pom-poms, bottle-brushes (i.e. chains of stars) and dendritic
structures (i.e. stars build of stars).
In the present paper our focus is to present the expressions characterizing a
large variety of possible sub-units such as rigid rods, flexible and semi-
flexible polymers, loops of flexible polymers, and excluded volume polymers.
We also present general expressions for geometric sub-units together with the
expressions for special cases such as disks, spherical shells, solid spheres
and cylinders. While the form factors for most of the sub-units are well
known, the form factor amplitudes and phase factors depend crucially on how we
choose to link the sub-units together. Different choices of linking (for
instance center-to-center or surface-to-surface) cause additional geometric
factors to appear in the form factor amplitudes and phase factors. To
illustrate the formalism, we use it to predict the scattering for chains of
identical sub-units or alternating sub-units. Furthermore, we address the
situation where a single sub-unit can have a distribution of reference points,
and hence form factor amplitudes and phase factors need to be averaged over
linking probability distributions. Together, paper I and the present paper
allow the scattering expressions for complex heterogeneous structures of a
variety of sub-units to be derived with great ease.
The paper is structured as follows: In Sect. II we present the formalism and
generalize it for structures where reference points can be distributed. We
illustrate the formalism by deriving the generic scattering for a block
copolymer micelle. In Sect. III we present the general scattering expressions
characterizing an arbitrary linear sub-unit with internal conformational
degrees of freedom, and in Sect. IV we give examples of the scattering from
chains and bottle-brush structures. In Sect. V we present the general
scattering expressions characterizing an arbitrary geometrical sub-unit
without internal conformations, and in Sect. VI we give examples of the
scattering from block copolymer micelles with different core sub-units and
tethering geometries as well as the scattering from end-linked cylinders. We
present our conclusions in Sect. VII. In two Appendices, we derive the
scattering terms for polymers, rods, and closed polymer loops, and for
spheres, disks and cylinders taking different tethering geometries into
account.
## II Theory
The present theory pertains to the small-angle scattering for structures build
out of sub-units and how to efficiently derive the scattering spectra
characterizing such structures. The formalism is identical for light, X-ray or
neutron scattering experiments within the Rayleigh-Debye-Ganz approximation.
We define an excess scattering length for each sub-unit. This parameter
captures the experimental details of the interactions between the incident
radiation and the scatterers inside the sub-unit, and also the scattering
properties of the solvent in which we assume the structures are dissolved.
Each sub-unit comprises a specific number of scattering sites. We equip each
sub-unit with an arbitrary number of reference points, these are positions on
the sub-unit where we can join two or more sub-units together. Later we will
generalize each reference point to represent a distribution of such positions.
If the sub-unit is a polymer molecule, then a natural choice could be to have
the two ends as reference points, if we are interested in deriving the
scattering from end-linked polymer structures. Assume that the $I$’th sub-unit
is composed of point-like scatterers, where the $j$’th scatterer in the sub-
unit is located at a position ${\bf r}_{Ij}$ and has excess scattering length
$b_{Ij}$. Let ${\bf R}_{I\alpha}$ denote the position of the $\alpha$’th
reference point associated with the $I$’th sub-unit. Once two or more sub-
units are connected at the same reference point, we refer to it as a vertex in
the resulting structure, e.g. if sub-units $I$ and $J$ are joined at reference
point $\alpha$ then ${\bf R}_{I\alpha}={\bf R}_{J\alpha}$ denotes the same
position in space and a vertex in the structure. Here and in the following
capital letters refers to sub-units, lower case letters refers to scatterers
inside a sub-unit, and Greek letters refers to vertices and reference points.
Scattering experiments measures the distribution of pair-distances between
scatterers in a structure. For a given structure $S$ we can define three types
of pair-distance distributions. The form factor $F_{S}(q)$ is the excess
scattering length weighted Fourier transformed and conformationally averaged
pair-distance distribution between all scatterers in the structure; this is
what is measured in a scattering experiment. We can also define two auxiliary
pair-distance distributions. The form factor amplitude $A_{S\alpha}(q)$ which
is the scattering length weighted Fourier transform of the pair-distance
distribution between all scatterers in the structure and a specified vertex
$\alpha$. Finally, the phase factor $\Psi_{S\alpha\omega}$ is the Fourier
transform of the pair-distance distribution between two vertices $\alpha$ and
$\omega$ in the structure.
We can define the form factor, form factor amplitudes and phase factors of a
structure $S$ in terms of the scattering sites and reference points as
$F_{S}(q)=\left(\beta_{S}\right)^{-2}\left\langle\sum_{j,k}b_{Sj}b_{Sk}e^{i{\bf
q}\cdot({\bf r}_{Sj}-{\bf r}_{Sk})}\right\rangle_{S},$ (II.1)
$A_{S\alpha}(q)=\left(\beta_{S}\right)^{-1}\left\langle\sum_{j}b_{Sj}e^{i{\bf
q}\cdot({\bf r}_{Sj}-{\bf R}_{S\alpha})}\right\rangle_{S},$ (II.2)
and
$\Psi_{S\alpha\omega}(q)=\left\langle e^{i{\bf q}\cdot({\bf R}_{S\alpha}-{\bf
R}_{S\omega})}\right\rangle_{S}.$ (II.3)
The $\langle\text{$\cdots$}\rangle_{S}$ averages are over internal
conformations and orientations. The total scattering length of the whole
structure is $\beta_{S}=\sum_{j}b_{Sj}$. Due to the orientational average, all
the functions only depend on the magnitude of the momentum transfer $q$, which
is given by the angle between the incident and scattered beam and the wave
length of the radiation. We also have
$\Psi_{S\alpha\omega}=\Psi_{S\omega\alpha}$ due to the orientational average.
Here and in the rest of the paper, the form factor, form factor amplitudes,
and phase factors are normalized to unity in the limit $q\rightarrow 0$.
The derivation of scattering expressions for complex structures can be vastly
simplified by describing the structure not in terms of fundamental scattering
sites, but instead in terms of logical structural sub-units of the structure.
Each sub-unit corresponds to a well defined group of the scattering sites, and
is characterized by its own form factor, form factor amplitudes, and phase
factors defined by eqs. II.1-II.3. For instance, to derive the scattering
expression for a block-copolymer micelle, we can group all the scattering
sites of the core into one sub-unit, and let each polymer in the corona be
described by a sub-unit.
The fundamental result of the present formalism, is to express the form
factor, form factor amplitudes and phase factors of a whole structure in terms
of the same functions characterizing the sub-units. An exact and generic
expression can only be derived in the case where the internal conformations of
all sub-units are uncorrelated, since in this case the structural average
factorizes into single-sub unit averages. This allows generic scattering
expressions to be derived for a large class of complex structures. The
assumption of uncorrelated sub-units corresponds to assuming that sub-units
are mutually non-interacting, that joints are completely flexible, and that
the structure does not contain loops. Subject to these assumptions, we can
succinctly express the form factor, form factor amplitudes and phase factors
of a structure $S$ as
$F_{S}(q)=\beta_{S}^{-2}\left[\sum_{I}\beta_{I}^{2}F_{I}(q)+\sum_{\begin{subarray}{c}I\neq
J\\\ \alpha\in I\>\mbox{near}\>\omega\in
J\end{subarray}}\beta_{I}\beta_{J}A_{I\alpha}(q)A_{J\omega}(q)\prod_{\begin{subarray}{c}(K,\tau,\eta)\\\
\in\mbox{P}(\alpha,\omega)\end{subarray}}\Psi_{K\tau\eta}(q)\right],$ (II.4)
$A_{S\alpha}(q)=\beta_{s}^{-1}\left[\sum_{\begin{subarray}{c}I\\\ \omega\in
I\,\mbox{near}\,\alpha\end{subarray}}\beta_{I}A_{I\omega}(q)\prod_{\begin{subarray}{c}(K,\tau,\eta)\\\
\in\mbox{P}(\alpha,\omega)\end{subarray}}\Psi_{K\tau\eta}(q)\right],$ (II.5)
and
$\Psi_{S\alpha\omega}(q)=\prod_{\begin{subarray}{c}(K,\tau,\eta)\\\
\in\mbox{P}(\alpha,\omega)\end{subarray}}\Psi_{K\tau\eta}(q).$ (II.6)
Here $F_{I}$ denotes the form factor of the $I$’th sub-unit, $A_{I\alpha}$
denotes the form factor amplitude of the $I$’th sub-unit relative to the
reference point $\alpha$, $\Psi_{I\tau\eta}$ denotes the phase factor of the
$I$’th sub-unit between reference points $\tau$ and $\eta$,
and$\beta_{I}=\sum_{j}b_{Ij}$ the total excess scattering length of the $I$’th
sub-unit. These terms are defined as eqs. II.1-II.3 with $S$ replaced by $I$.
In the form factor we have a double sum over distinct sub-unit pairs, and in
the form factor amplitude a single sum over sub-units. In the form factor sum,
the restriction $\alpha\in I$ near $\omega\in J$ means that we for have to
identify the reference point $\alpha$ on $I$ nearest to $J$ in terms of the
structural connectivity, and similarly the reference point $\omega$ on $J$
nearest to $I$. Having done this, we can identify the path $P(\alpha,\omega)$
of sub-units ($K$) and reference point pairs ($\tau$,$\eta$) that has to be
traversed to walk between the $I$’th and $J$’th sub-unit to from the reference
point $\alpha$ to the reference point $\omega$ on the structure. Since we are
assuming acyclic branched structures this path definition is always unique.
Details of the derivation of this expression is given in ref. Svaneborg and
Pedersen (2012). In the example section below, we will show how to derive the
scattering for a few concrete structures using eqs. II.4-II.6. First we will
generalize the formalism to the case of random linking positions, and present
a diagrammatic illustration of the physical interpretation of the generalized
formalism.
Above we have assumed that sub-units are always linked at reference points
which correspond to specific sites on the sub-units. In the following, we
refer to this as regular reference points. For many structures there is an
element of randomness to where sub-units are joined together. For an example
of a bottle-brush structure, we can for instance imagine a main rod with
polymer sub-units linked at random positions along the rod. While the
structure has an element of randomness, the connectivity remains well-defined.
Note, that this situation differs from random linking, where the structure
will have a random connectivity. Here and below we only address the first
situation. Random linking can be described by cascade theoryGordon (1962);
Burchard (1983) or Markov chain modelsTeixeira et al. (2000, 2007).
The formalism can easily be generalized to the case where some or all
reference points refers to distributions of potential link positions on a sub-
unit. Above, we have assumed that a reference point $\alpha$ on a sub-unit
refers to a unique fixed position ${\bf R}_{I\alpha}$ on the sub-unit, and
that sub-unit pairs $I$ and $J$ are joined at the $\alpha$ vertex when ${\bf
R}_{I\alpha}={\bf R}_{J\alpha}$. Below we consider the situation where a
reference points can be picked from a given distribution. In this case we
refer to the reference point as an distributed reference point. Assuming we
are given a set of potential positions for the $\alpha$th reference point on
sub-unit $I$ where the $m$th possible position is ${\bf R}_{I\alpha m}$ and is
associated with a probability $Q_{I\alpha m}$ and similar ${\bf R}_{J\alpha
n}$ and $Q_{J\alpha n}$ for the $n$th possible position of the $\alpha$th
reference point on sub-unit $J$. Then the probability of joining two sub-units
$I$ and $J$ at the $\alpha$ vertex at specific positions ${\bf R}_{I\alpha
m}={\bf R}_{J\alpha n}$ is given by the product $Q_{I\alpha m}Q_{J\alpha n}$.
This is tantamount to assuming that the two linking positions on the two
joined sub-units are statistically uncorrelated. Note that one or both of
these distributions can still refer to a single potential position, hence
regular reference points remains a special case of the generalized formalism.
To derive eqs. II.4-II.6, we had to assume that the internal conformations of
all sub-units were uncorrelated. This allowed structural averages to be
factorized into single-sub unit averages. When we have assigned a probability
distribution to some or all reference points we have to calculate $\langle
A_{S\alpha}(q)\rangle_{{\bf Q}}$, $\langle\Psi_{S\alpha\omega}(q)\rangle_{{\bf
Q}}$ where $\langle\cdots\rangle_{{\bf Q}}$ denotes the additional averages
over link position distributions. Since we have assumed that linking positions
on different sub-units are statistically independent, the structural and
linking position averages again factorize into single sub-unit averages over
internal conformations as well as linking degrees of freedom. Sub-unit form
factors are independent of reference points, and hence unaffected by this
average. For a sub-unit with a finite set of potential random linking
positions, we can define the reference point averaged form factor amplitudes
and phase factors as
$A_{I\langle\alpha\rangle}(q)=\left\langle A_{I\alpha}(q;{\bf
R}_{I\alpha})\right\rangle_{Q_{I\alpha}}\equiv\sum_{n}Q_{I\alpha
n}A_{I\alpha}(q,{\bf R}_{I\alpha n}),$ (II.7)
$\Psi_{I\langle\alpha\rangle\langle\omega\rangle}(q)=\left\langle\Psi_{I\alpha\omega}(q;{\bf
R}_{I\alpha},{\bf
R}_{I\omega})\right\rangle_{Q_{I\alpha}Q_{I\omega}}\equiv\sum_{n,m}Q_{I\alpha
n}Q_{I\omega m}\Psi_{I\alpha}(q;{\bf R}_{I\alpha n},{\bf R}_{I\omega m}).$
(II.8)
Figure II.1: Illustration of an example structure where five sub-units have
been joined at four vertices $\alpha$, $\gamma$, $\delta$, $\epsilon$. Each
sub-unit is illustrated by an ellipse where we associate the interior with the
internal conformation of the scattering sites and the circumference with the
reference points on the sub-unit. Sub-units are joined to each other by
reference points, and they are illustrated as a single dot (e.g. $\epsilon$,
$\delta$, $\gamma$) for a regular reference point or a thick line (e.g.
$\langle\alpha\rangle$, $\langle\delta\rangle$) for distributed reference
points. Note that we use the same Greek letter to label vertices in the
structure, and regular or distributed reference points on different sub-units
at the same vertex.
Here we write explicitly the reference points, that are to be averaged over in
the form factor amplitudes and phase factors. Here and below we use
$\langle\alpha\rangle$ to denote the case where the $\alpha$th reference point
label on the $I$th sub-unit has been averaged. We continue to denote by
$\alpha$ a regular reference point, where no average is to be performed. The
formalism (eqs. II.4-II.6) remains valid when some or all form factor
amplitudes and phase factors include distributed reference points.
Fig. II.1 schematically illustrates an example structure for which the
formalism can provide the corresponding scattering expression. It shows how
sub-units can be joined either at regular reference points or at distributed
reference points. Note how a reference point is associated with a sub-unit,
hence $\delta$ refers to a regular reference point on sub-unit $L$, that is
joined to any of the $\delta$ reference points on the $I$th sub-unit denoted
by the $\langle\delta\rangle$ average reference point. In general, a vertex
have an arbitrary functionality, and a sub-unit can have an arbitrary number
of reference points at which it can join with other sub-units. Hence the
structures described by formalism is not limited to two-functional graph-like
structures, but to any hyper-graph structures that does not contain loops.
In the special case, where we can make a one-to-one identification between
reference points and scattering sites such that $n=i$, ${\bf R}_{I\alpha
n}={\bf r}_{Ii}$, and $Q_{I\alpha n}=b_{Ii}/\beta_{I}$. Then II.7, II.2, II.8,
and II.3 are identical to the form factor eq. II.1. Hence we conclude that
$A_{I\langle\alpha\rangle}(q)=\Psi_{I\langle\alpha\rangle\langle\omega\rangle}(q)=F_{I}(q)$
in this case. For a polymer chain, for instance, this means that the form
factor amplitude relative to a random position on the polymer, and the phase
factor relative to two random positions on the polymer are identical to the
polymer form factor. This is not a surprise since the site-to-site, site-to-
reference point and reference-to-reference point pair-distance distributions
all are identical in this case. When deriving form factors for a given
structure, we often assume that all sites in a sub-unit has equal excess
scattering length, hence all the known expressions for form factors of
structures can be uses as reference point averaged form factor amplitudes and
phase factors to describe structures with distributed link positions.
Figure II.2: Definition of diagrams representing the sub-unit form factor,
form factor amplitude, and phase factor terms expressed for the different
possibilities of reference point averages for the $I$th sub-unit.
Fig. II.2 introduces a diagrammatic interpretation of the form factor, form
factor amplitude and phase factor of a sub-unit. As in fig. II.1 we associate
each sub-unit with an ellipse, where reference points are associated with the
circumference while the scattering sites are associated with the interior. The
form factor is the Fourier transform of the pair-distance distribution between
all scattering sites in a sub-unit, and we illustrate this by a straight line
inside the sub-unit ellipse. Form factor amplitudes and phase factors depend
on reference points. Regular reference points are shown as dots, while
distributed reference points are illustrated as a thick line on the
circumference. The form factor amplitude is Fourier transform of the pair-
distance distribution between a specified reference point and all scattering
sites inside the sub-unit, and this is illustrated as a line from dot to the
inside of the sub-unit. In the case, of a reference point average, we
illustrate the reference point not as a dot but by a thick line illustrating
all the possible reference points, and the form factor amplitude as a line
from anywhere along the thick line to the inside of the sub-unit. The phase
factor is the Fourier transform of the pair-distance distribution between two
specified reference points, and this is illustrated as a line traversing the
sub-unit connecting two reference points or reference point averages.
Using eq. II.1, we can calculate the scattering form factor for a given
structure composed of sub-units joined by regular or distributed reference
points. The first term is just a sum over the form factors of all the sub-
units weighted by their excess scattering lengths. The second term is more
complicated, and it describes the scattering interference contributed by
different sub-unit pairs. For each distinct pair of sub-units $I$ and $J$ in
the double sum, we identify which vertex $\alpha$ at sub-unit $I$ that is
nearest sub-unit $J$ and which vertex $\omega$ at sub-unit $J$ that is nearest
to sub-unit $I$. Here “near” means in terms of the shortest path originating
at a reference point $\alpha$ (or $\langle\alpha\rangle$) on $I$ and
terminating on a reference point $\omega$ (or $\langle\omega\rangle$) on $J$.
We denote the path connecting $\alpha$ and $\omega$ through the structure
$P(\alpha,\omega)$. For the product, we have to identify each sub-unit $K$ on
this path and also identify the reference points $\tau$ and $\eta$ (or
$\langle\tau\rangle$, $\langle\eta\rangle$) across which the sub-unit is
traversed. For some structures the path can traverse a sub-unit by the same
reference point. In the case of a well defined reference point
$\Psi_{K\alpha\alpha}(q)\equiv 1$ and we can neglect the contribution, however
in the case of an distributed reference point the corresponding term
$\Psi_{K\langle\alpha\rangle\langle\alpha\rangle}(q)$ will contribute to the
product. The path construction is always unique and well defined for
structures that does not contain loops.
The form factor expression (eq. II.4) has a quite simple physical
interpretation, despite the complex notation required to describe branched
reference point distributed structures. The structural form factor is the
pair-correlation function between all scattering sites in the structure. It
can be obtained by propagating position information between all scattering
sites in the structure. When both sites belong to the same sub-unit it is
given by the sub-unit form factors and is described by the first term in eq.
II.4. The distance information between sites on different sub-units is
obtained by propagating position information along paths through the
structure. To propagate site-to-site position information between sites in
sub-unit $I$ and sites in sub-unit $J$, we first have to propagate the
position information between the sites in sub-unit $I$ to the vertex $\alpha$
at $I$. This is done by the form factor amplitude $\beta_{I}A_{I\alpha}$ or
$\beta_{I}A_{I\langle\alpha\rangle}$. The position information is then
propagated step-by-step along the path of intervening sub-units towards the
vertex $\omega$ at sub-unit $J$. Each time a sub-unit is traversed it
contributes a phase factor $\Psi_{K\tau\eta}$,
$\Psi_{K\langle\tau\rangle\eta}$,$\Psi_{K\tau\text{$\langle$}\eta\rangle}$, or
$\Psi_{K\langle\tau\rangle\langle\eta\rangle}$ to account for the
conformationally averaged distance between the two (potentially distributed)
reference points. Finally the position information is propagated between the
vertex $\omega$ and all the sites inside the $J$ sub-unit. This is done by the
final form factor amplitude $\beta_{J}A_{J\omega}$ or
$\beta_{J}A_{J\langle\omega\rangle}$. Only the amplitudes has an excess
scattering length prefactor, since they represent the amplitudes of scattered
waves from all the scatterers inside the sub-units relative to the $\alpha$
and $\omega$ vertices while the product of phase factors represent excess
phase contributed by the path between the vertices. The product of all these
propagators describe the scattering length weighted site-to-site scattering
interference contribution from the $I$’th and $J$’th sub-units. By summing
over all such pair contributions all the possible site-to-site pair-distances
in the structure are accounted for.
Figure II.3: Diagrammatic representation of some of the terms contributing to
the form factor of the structure shown in fig. II.1 using the definition of
diagrams shown in fig. II.2.
Fig. II.3 diagrammatically illustrates some of the terms that contributes to
the form factor (eq. II.4) using the diagrammatic definitions in fig. II.2. We
can generate all the diagrams by picking a pair of sites inside the same or
two different sub-units and drawing a line between them using reference points
to step between sub-units and to traverse across sub-units (diagram 1). A line
between two sites within the same sub-unit contributes the form factor of that
sub-unit. Sub-units that are joined directly to each other will contribute the
product of two form factor amplitudes and excess scattering lengths of the two
sub-units (diagram 2, 3, 5). For sub-units that are not directly joined to
each other, the form factor amplitude product is further multiplied by the
phase factors of all the sub-units on the intervening path (diagram 4, 6). For
all the reference point labels of the form factors and phase factors, we
either have a regular reference point, or a distributed reference point which
depend on the details of the given structure. In general, for a structure of
$N$ sub-units, there will be $N$ form factor contributions and $N(N-1)/2$
different scattering interference contributions that has to be determined. The
longest possible path is $N-2$ sub-units which occurs in the case of a linear
chain of sub-units.
Similar diagrammatic interpretations apply to the structural form factor
amplitude and phase factors (eqs. II.5 and II.6). For the form factor
amplitude we have to propagate position information between a specified vertex
and all sites in the structure. Diagrammatically this can be done by picking a
site in any sub-unit and drawing a line between the site and the specified
vertex using reference points to step between sub-units and to traverse across
sub-units. Summing over all the $N$ diagrams will produce the form factor
amplitude of the structure. For the phase factor we have to propagate position
information between two specified vertices. Diagrammatically this is done by
drawing a line between the two vertices using reference points to step between
sub-units and to traverse across sub-units. The resulting structural phase
factor is the product of the phase-factors of all the intervening sub-units.
With the diagrammatic interpretation of the formalism, it becomes quite easy
to draw a structure, and write down the corresponding scattering expressions.
The price of this simplicity is that we had to assume that sub-units are
mutually non-interacting, that the joints between sub-units are completely
flexible, and that the structures does not contain loops. However, no
assumptions were made about the internal structure of the sub-units. The
formalism is complete in the sense that the three structural scattering
expressions allows a whole structure to be used as a sub-unit to build more
complex structures. This we utilized in Paper 1 but will not use
here.Svaneborg and Pedersen (2012) The formalism is also generic in the sense
that scattering contributions due to structural connectivity and the internal
structure of the sub-units have been completely decoupled. This allows us to
write down generic scattering expressions for structures without knowing what
sub-units they are made of. This information can be specified at a later point
when concrete expressions for the sub-unit form factor, form factor
amplitudes, and phase factors are inserted. Below we give some generic
examples, and then dedicate the rest of the paper to derive and present
scattering expressions characterizing specific sub-unit structures.
### II.1 Example structures
Figure II.4: Diagrammatic representation of the form factor of a block
copolymer micelle with a core sub-unit “C” and a number of identical sub-units
“T” tethered at random points on the surface. All the scattering contributions
are shown using the diagrammatics in fig. II.2. They correspond to
$\beta_{T}^{2}F_{T}$ (solid line),
$\beta_{T}^{2}A_{T\alpha}^{2}\Psi_{C\langle\alpha\rangle\langle\alpha\rangle}$
(short dashed line), $\beta_{T}\beta_{C}A_{T\alpha}A_{C\langle\alpha\rangle}$
(long dashed line), and $\beta_{C}^{2}F_{C}$ terms (dot dashed line) in eq.
II.9. Figure II.5: Diagrammatic representation of the form factor amplitude
and phase factor of a block copolymer micelle. The scattering contributions to
$A_{mic,\omega}$ are shown by lines starting at an $\omega$ reference point on
a tethered sub-unit and using the diagrammatics in fig. II.2. They correspond
to the $\beta_{T}A_{T\omega}$ (solid line),
$\beta_{C}A_{C\langle\alpha\rangle}\Psi_{T\omega\alpha}$ (dash dotted line),
and
$\beta_{T}A_{T\alpha}\Psi{}_{C\langle\alpha\rangle\langle\alpha\rangle}\Psi_{T\alpha\omega}$
terms (short dashed line) in eq. II.10. The phase factor (eq. II.11) between
the tips of two tethered sub-units is given by
$\Psi_{T\omega\alpha}^{2}\Psi_{C\langle\alpha\rangle\langle\alpha\rangle}$
(long dashed line) .
We can regard a micelle as $N$ identical two-functional sub-units tethered by
one end to a random site on the surface of geometric structure representing
the core. Such a structure is shown in fig. II.4. Note that the core surface
still is referred to by a single reference point label $\alpha$. Similarly, we
can regard a bottle-brush polymer as $N$ identical two-functional sub-units
tethered by one end to a random point on a main structure such as a polymer
chain. The connectivity of the two structures is identical, and hence they are
characterized by the same generic scattering expression:
$F_{mic}(q)=\left(N\beta_{T}+\beta_{C}\right)^{-2}\left(\beta_{C}^{2}F_{C}+N\beta_{T}^{2}F_{T}\right.$
$\left.+2N\beta_{C}\beta_{T}A_{C\langle\alpha\rangle}A_{T\alpha}+N(N-1)\beta_{T}^{2}A_{T\alpha}^{2}\psi_{C\langle\alpha\rangle\langle\alpha\rangle}\right)$
(II.9)
Here the tethered sub-units are denoted by $T$ and the end attached to the
core surface is denoted “$\alpha$” while “$\omega$” denotes the free end. The
core sub-unit is denoted by $C$ and the average over random surface points is
denoted by $\langle\alpha\rangle$. The form factor consists of terms
representing all the possible pair distributions between sub-units in the
structure. These are shown in fig. II.4. Each term has a prefactor which for
form factor terms is the number of corresponding sub-units in the structure.
The form factor amplitude product terms represent pair distributions between
different sub-units and they are counted twice. Hence there is both an
$A_{C\langle\alpha\rangle}A_{T\alpha}$ contribution and an identical
$A_{T\alpha}A_{C\langle\alpha\rangle}$ contribution for each of the $N$
tethered sub-units. For the pair distribution between two tethered sub-units,
we note that we have $N$ tethered sub-units to pick the starting point from,
and $N-1$ tethered sub-units to pick end ending point. This also counts each
pair twice. The prefactor of the form factor ensures it is normalized to unity
in the limit of small $q$ values.
We have chosen to express the form factor amplitude relative to the tip of a
tethered sub-unit. The form factor amplitude represents the pair distribution
between the reference point at the tip of a tethered sub-unit and the sites in
the same sub-unit, the sites in the core, and in the sites in the other
tethered sub-units. These are shown in fig. II.5 and when taking into account
the multiplicity of the sub-units the normalized form factor becomes
$A_{mic,\omega}(q)=\left(N\beta_{T}+\beta_{C}\right)^{-1}\left(\beta_{C}\Psi_{T,\omega\alpha}A_{C,\langle\alpha\rangle}\right.$
$\left.+\beta_{T}A_{T\omega}+\beta_{T}(N-1)\Psi_{T\omega\alpha}\psi_{C\langle\alpha\rangle\langle\alpha\rangle}A_{T\alpha}\right).$
(II.10)
The contribution to the tip-to-tip phase factor is also shown in fig. II.5 and
is given by
$\Psi_{mic,\omega\omega}(q)=\psi_{T\alpha\omega}^{2}\Psi_{C\langle\alpha\rangle\langle\alpha\rangle}.$
(II.11)
Since exactly the same diagrams are required to describe a bottle-brush
structure where side-structures are randomly tethered along some main chain
structure (corresponding to the core of the micelle), the generic scattering
expressions for these two structures are identical. They will first differ
when we make choices of which sub-unit structures are involved and insert the
corresponding form factor, form factor amplitudes, and phase factors in the
expressions.
Figure II.6: Diagrammatic representation of chain of identical two-functional
sub-units with regular reference points (top) or linked at two random
positions (bottom).
PreviouslySvaneborg and Pedersen (2012), we derived the form factor of a chain
of identical two functional sub-units using eq. II.4 as
$F_{chain}(q)=\frac{F}{N}+2\frac{\Psi_{\alpha\omega}^{N}-N\Psi_{\alpha\omega}+N-1}{N^{2}(\Psi_{\text{$\alpha\omega$}}-1)^{2}}A_{\alpha}A_{\omega}.$
(II.12)
Here we have discarded the superfluous sub-unit index, and also omitted the
$q$ dependence on the right hand side for sake of brevity. A diagrammatic
representation is shown in fig. II.6. $\alpha$ and $\omega$ denotes the two
ends of the sub-unit. The sub-units can be asymmetric with regard to
exchanging the two ends, for instance if the sub-unit is a di-block copolymer.
The sub-units are joined as $\omega-\alpha$, leaving one free $\alpha$ end and
one free $\omega$ end of the structure. If we assume that each of the
reference points are picked from two distributions, then we obtain the
scattering expression for the corresponding randomly joined chain by replacing
the regular reference points by distributed reference points as
$F_{chain}(q)=\frac{F}{N}+2\frac{\Psi_{\langle\alpha\rangle\langle\omega\rangle}^{N}-N\Psi_{\langle\alpha\rangle\langle\omega\rangle}+N-1}{N^{2}(\Psi_{\langle\alpha\rangle\langle\omega\rangle}-1)^{2}}A_{\langle\alpha\rangle}A_{\langle\omega\rangle}.$
(II.13)
Fig. II.6 shows a diagrammatical representation of such a structure, where a
random reference point $\omega$ is joined with a random reference point
$\alpha$ on the next sub-unit. Again this leaves a structure with two ends
characterized by $\langle\alpha\rangle$ and $\langle\omega\rangle$. If the
sub-units are block-copolymers and the linking can be anywhere along the
copolymer, then the corresponding structure is one where each polymer is
randomly cross-linked with the previous and next polymers in the chain.
Alternative, if the $\alpha$ link is anywhere in the $A$ block, and the
$\omega$ link anywhere in the $B$ block, then the result will be a chain where
each di-block copolymer (except for the ends) has one link on the each of the
two blocks. These different choices correspond to different expressions for
$\Psi_{\langle\alpha\rangle\langle\omega\rangle}$, $A_{\langle\alpha\rangle}$,
and $A_{\langle\omega\rangle}$.
Note that the formalism is generic. We have made absolutely no assumptions as
to the internal structure of the sub-units in the expressions above. These
scattering expressions we have presented are completely generic and only
encode the structural connectivity. The formalism is also complete, in the
sense that a whole structure described by the formalism can be used as a sub-
unit to build more complex structures within the formalism . For example, we
can regard the three functions $F_{mic}(q)$, $A_{mic,\omega}(q)$, and
$\Psi_{mic,\omega\omega}(q)$ given by eqs. II.9-II.11 as defining a micelle
sub-unit. We could then obtain the scattering expression for a chain of
micelles, by inserting the micelle sub-unit expressions into the form factor
of a chain eq. II.12. This illustrates the versatility of the formalism.
## III Sub-units with internal conformations
Each sub-unit is characterized by a three types of pair-distance distribution
functions, the site-to-site, site-to-reference, and the reference-to-reference
point pair-distribution functions, denoted $P_{ss}(\sigma,\rho;r)$,
$P_{s\alpha}(\sigma;r)$, and $P_{\alpha\omega}(r)$, respectively. Here
$\sigma$ and $\rho$ are running labels denoting scattering sites such as e.g.
an index of a point scatterer or a contour-length, surface or volume element,
respectively, while the $\alpha$ and $\omega$ labels denotes fixed reference
points. The corresponding positions are denoted ${\bf r}_{\sigma}$, ${\bf
r}_{\rho}$, ${\bf R}_{\alpha}$, and ${\bf R}_{\omega}$, respectively. In a
rigid structure, the pair-distance $r=|{\bf r}_{\sigma}-{\bf r}_{\rho}|$ is
constant and the pair-distance distributions reduce to delta functions. In a
flexible structure with internal conformational degrees of freedom such as a
polymer, the distance between two sites will in general be given by a
distribution. Let the excess scattering length density of a scattering site
$\sigma$ be denoted $b(\sigma)$, and $\beta=\int\mbox{d}\sigma b(\sigma)$
denotes the total excess scattering length of the sub-unit. The 3D
isotropically averaged Fourier transform is $\mathcal{F}(P)=\int\mbox{d}r4\pi
r^{2}\frac{\sin(qr)}{qr}P$. Hence the sub-unit scattering expressions are
$F(q)=\beta^{-2}\int\mbox{d}\sigma\mbox{d}\rho
b(\sigma)b(\rho)\int\mbox{d}r4\pi
r^{2}\frac{\sin(qr)}{qr}P_{ss}(\sigma,\rho;r),$ (III.1)
$A_{\alpha}(q)=\beta^{-1}\int\mbox{d}\sigma b(\sigma)\int\mbox{d}r4\pi
r^{2}\frac{\sin(qr)}{qr}P_{s\alpha}(\sigma;r),$ (III.2)
$\Psi_{\text{$\alpha\omega$}}(q)=\int\mbox{d}r4\pi
r^{2}\frac{\sin(qr)}{qr}P_{\alpha\omega}(r).$ (III.3)
In the case of a linear sub-unit with translational invariance along the
contour, the pair-distance distributions functions only depend on the relative
contour distance. Let $L$ denote the total contour length, such that
$\sigma,\rho\in[0;L]$ denote a pair of sites along the sub-unit separated by a
contour length distance $l=\text{|$\sigma$-$\rho$|}$ and a spatial separation
$r$. The two ends are located at $\alpha=0$ and $\omega=L$, respectively. Then
$P_{ss}(\sigma,\rho;r)=P(|\sigma-\rho|;r)$,
$P_{s\alpha}(\sigma;r)=P(|\sigma-\alpha|;r)$, and
$P_{\alpha\omega}(r)=P(|\alpha-\omega|;r)$ where $P(l;r)$ denotes the pair-
distance distribution between two sites separated by a contour length $l$ and
a direct distance $r$. In Appendix IX, we use these expressions to derive the
form factor, form factor amplitudes, and phase factors of polymers, rods, and
closed polymeric loops.
## IV Scattering examples
?figurename? IV.1: Illustration of chains and tethered structures. a) end-
linked polymers, b) contour-linked polymers, c) contour-linked loops, d) rods
contour-linked to a polymer, and e) loops contour-linked to a loop.
?figurename? IV.2: Normalized form factors from a chain of $N=100$ identical
sub-units for end-linked polymers (red solid line), contour-linked polymers
(green dashed), contour-linked polymer loops (blue dotted), end-linked rods
(magenta short dashed), contour-linked rods (brown medium dashed). All sub-
units has the same radius of gyration $R_{g}$. ?figurename? IV.3: Normalised
form factor for bottle brush structures where the main chain structure is
either a rod (green) or a polymer loop (blue). The main chain structure has
$N=20$ rods (dotted), polymers (solid), or polymer loops (dashed) tethered at
random positions. The main chain and tethered sub-unit radii of gyration,
$R_{g,C}$ and $R_{g}$, are fixed at $R_{g,C}/R_{g}=10$ for all the structures.
The excess scattering length of the core and tethered sub-units are the same
$\beta_{C}=\beta_{T}$.
Fig. IV.1 a-c illustrates some of the possible structures obtained by linking
sub-units into chains. When identical polymers are end-linked the result is a
long linear polymer. A very different structure is obtained, when polymers are
allows to link anywhere along their contour. The result resembles a bottle-
brush structure, where each sub-unit in the chain has two pendant chains of a
random length. Note that just as in the end-linked case, all the internal sub-
units in the contour-linked chain have exactly two links. This is very
different from a truly randomly linked structure, which would form a gel-like
network. The scattering from a gel-like network can be described by the Random
Phase Approximation (RPA)Benoit and Benmouna (1984), and the diagrammatic
representation of the RPA form factor corresponds to a weighed sum over
contour-linked chain diagrams of varying number of sub-units.
The scattering from these structures are obtained from eq. II.13 by inserting
the corresponding sub-unit form factor, form factor amplitude and phase
factors. We have derived these terms for a flexible polymer chain, a rod, and
a closed polymer loop in appendix IX. Fig. IV.2 shows the scattering from end-
linked and contour-linked polymers and rods, as well as contour-linked loops.
At small $q$ values we observe the Guinier $F(q)\approx
1+\frac{(qR_{g})^{2}}{3}$ regime, where $R_{g}$ is the radius of gyration of
the entire structure, at intermediate $q$ values we see the power law
characteristic of the fractal dimension of the structure, while at large $q$
values we see the scattering from the sub-units. The end-linked chain shows
the expected Debye scattering behavior corresponding to a random walk with an
asymptotic behavior $2(qR_{g}){}^{-2}$ at large $q$ values. The contour-linked
polymers have a smaller radius of gyration since the chain structure is more
at intermediate length scales, however at small length scales we again see the
same sub-unit scattering as for the end-linked polymers. The chain of polymer
loops is more compact than the chain of end-linked polymers, which is why
their scattering is larger at large $q$ values, however at large $q$ values we
observe the asymptotic $(qR_{g})^{-2}$ behaviour expected for a polymer loop.
The end-linked chain of rods is observed to have the largest radius of
gyration of all structures. At intermediate length scales the end-linked rod
chain has a random-walk like structure, while at small length scales shows a
cross-over to the $\pi(Lq)^{-1}$ asymptotic behavior characterizing the rod-
like sub-units. Chains of contour-linked polymers, polymer loops, and rods
have the same radius of gyration, since we have fixed the size of the sub-
units to produce the same radius of gyration.
Fig. IV.1d-e shows the bottle-brush structures that are obtained by tethering
rods to a long main chain polymer and by tethering polymer loops to a main
chain polymer loop. The scattering from these structures is obtained from eq.
II.9 by inserting the corresponding form factor, form factor amplitudes and
phase factors of the main chain and tethered sub-units. The scattering form
factors are shown in fig. IV.3. Again we have fixed the radius of gyration of
the main chain and of the tethered sub-units to the same values independent of
their structure and for this reason all the bottle-brush structures has the
same radius of gyration. At intermediate length scales we see a small regime
with power law behavior corresponding to the fractal dimension of the main
chain $q^{-2}$ for the random-walk like polymer loop and $q^{-1}$ for the
straight rod, while at small length scales we observe the power law
corresponding to the fractal dimensions of the tethered sub-units. Again we
observe that the polymer loop sub-unit scattering is a factor one half lower
than that of the linear polymer sub-unit.
## V Geometric sub-units
We assume that a sub-unit is a rigid geometric body without internal degrees
of freedom. In this case, it is more convenient to express the sub-unit
scattering expressions as the orientational average of the phase integral over
all the scattering sites as
$F_{rigid}(q)=\left\langle{\cal F}_{\beta}({\bf q},{\bf O}){\cal
F}_{\beta}(-{\bf q},{\bf O})\right\rangle_{o},\quad A_{rigid}(q,{\bf
O})=\left\langle{\cal F}_{\beta}({\bf q},{\bf O})\right\rangle_{o},$ (V.1)
here $\left\langle\cdots\right\rangle_{o}$ denotes an orientational average.
While the form factor is independent of the choice of origin ${\bf O}$, it is
useful when expressing form factor amplitudes, since we have
$A_{rigid,\alpha}(q)=A_{rigid}(q,{\bf R}_{\alpha})$ for a particular regular
reference point ${\bf R}_{\alpha}$. The phase integral is defined as
${\cal F}_{\beta}({\bf q},{\bf O})=\left(\int\mbox{d}{\bf r}\beta({\bf
r})\right)^{-1}\int\mbox{d}{\bf r}\beta({\bf
r})\exp\left(i\mathbf{q}\cdot(\mathbf{r}-\mathbf{O})\right),$ (V.2)
which is the Fourier transform of the excess scattering length density
distribution $\beta({\bf r})$ of the sub-unit relative to the origin ${\bf
O}$. We normalize the phase integral such that ${\cal F}_{\beta}({\bf
q}=0,{\bf O})=1$. The major challenge when calculating the scattering from
geometric objects is to calculate the phase integral analytically and then
perform the orientational averages.
Since we are not only interested in regular reference points, but also
reference points that are averaged over distributions of potential reference
point sites, we will focus on the situation where these site distributions are
also characterized by a geometric objects. For instance, we could be
interested in the form factor amplitude of a sphere relative to a random site
on its surface, or the phase factor between two random sites on the surface of
a sphere. By generalizing the averages eqs. (II.7 and II.8) into integrals
over reference point distributions, and recognizing that these averages can be
recast into the form of phase factor integrals, we can express the reference
point distribution averaged form factor amplitude and phase factors as
$A_{rigid\langle\alpha\rangle}(q)=\left\langle\int\mbox{d}{\bf
r}^{\prime}Q_{\alpha}({\bf r}^{\prime}){\cal F}_{\beta}({\bf q};{\bf
r}^{\prime})\right\rangle_{o}=\left\langle{\cal F}_{\beta}({\bf q},{\bf
O}){\cal F}_{Q_{\alpha}}(-{\bf q},{\bf O})\right\rangle_{o},$ (V.3)
$\Psi_{rigid\langle\alpha\rangle\omega}(q)=\left\langle\int\mbox{d}{\bf
r}Q_{\alpha}({\bf r})e^{i{\bf q}\cdot({\bf r}-{\bf
R}_{\omega})}\right\rangle_{o}=\left\langle{\cal F}_{Q_{\alpha}}({\bf q},{\bf
R}_{\omega})\right\rangle_{o},$ (V.4)
and
$\Psi_{rigid\langle\alpha\rangle\langle\omega\rangle}(q)=\left\langle\int\mbox{d}{\bf
r}\mbox{d}{\bf r}^{\prime}Q_{\alpha}({\bf r})Q_{\omega}({\bf
r}^{\prime})e^{i{\bf q}\cdot({\bf r}-{\bf
r}^{\prime})}\right\rangle_{o}=\left\langle{\cal F}_{Q_{\alpha}}({\bf q},{\bf
O}){\cal F}_{Q_{\omega}}(-{\bf q},{\bf O})\right\rangle_{o}.$ (V.5)
Here ${\cal F}_{Q_{\alpha}}({\bf q},{\bf O})$ denotes the Fourier transform
(eq. V.2) of the reference point site probability distribution
$Q_{\alpha}({\bf r})$. The form factor amplitude $A_{\langle\alpha\rangle}(q)$
and double averaged phase factor
$\Psi_{\langle\alpha\rangle\langle\omega\rangle}(q)$ are independent of the
choice of origin ${\bf O}$ by construction. Again we recognize that if the
normalized excess scattering length distribution and the reference point site
probability distribution are proportional $\beta({\bf r})\propto
Q_{\alpha}({\bf r})$, then the form factor, averaged form factor amplitude,
and double averaged phase factor reduce to the same function. Finally, the
phase factor between two regular reference points ${\bf R}_{\alpha}$ and ${\bf
R}_{\omega}$ is given by
$\Psi_{rigid,\alpha\omega}(q)=\frac{\sin(q|\mathbf{R}_{\alpha}-\mathbf{R}_{\omega}|)}{q|\mathbf{R}_{\alpha}-\mathbf{R}_{\omega}|}.$
(V.6)
For a large number of geometric objects the scattering form factor is known,
see e.g. Pedersen (2002). In an appendix, we derive the scattering expressions
characterizing spheres, flat disks, spheres, and cylinders with special
emphasis on how phase factors and form factor amplitudes change depending on
different choices of reference point distributions. This is relevant in many
applications e.g. for structures such as block-copolymer micelles and polymer
brushes end-grafted to a surface or an interface.Pedersen and Gerstenberg
(1996); Pedersen (2000) Below we give some examples.
## VI Scattering examples
?figurename? VI.1: Illustration of tethering geometries: a) disk surface, b)
disk rim, c) cylinder equator, d) end, e) side, and f) surface tethering.
?figurename? VI.2: Normalized form factors for spherical core sub-unit with
$N=100$ polymers tethered on the surface (black solid line), at the equator
(red dotted line), and for a disk-like core sub-unit with polymers tethered on
the surface (green short dashed line) and on the rim (blue long dashed line).
The sub-units has the same radii of gyration with $R_{g,C}/R_{g}=10$, and the
excess scattering lengths $\beta_{C}=100\beta_{T}$ . ?figurename? VI.3:
Normalized form factors of cylinders with $N=100$ polymer sub-units for
equator (black solid line), end (red dotted line), side (green short dashed
line), and surface (blue long dashed line) tethering geometries. The sub-units
has fixed radii of gyration with $R_{g,C}/R_{g}=10$, the length $L$ and radius
$R$ of the cylinder are equal, and the excess scattering lengths are
$\beta_{C}=100\beta_{T}$. ?figurename? VI.4: Normalised form factors for
chains of $N=100$ cylinders as function of their aspect ratio for $R/L=0.001$
(red dotted line), $0.01$ (green short dashed), $0.1$ (blue long dashed). Also
shown are the scattering from a polymer (black dot-dashed) and a rod (magenta
solid line). The horizontal axis of each form factor has been scaled with the
radius of gyration of each of the structures.
Fig. VI.1 shows some of the possible tethering geometries for disk-like and
cylindrical micelles. For a disk we can either have the sub-units tethered to
anywhere on the surface, or only at the rim of the surface. For a cylinder we
could for instance tether polymers to the equator, the two ends, the side, or
the entire surface of the cylinder. The corresponding scattering expressions
are obtained from eq. II.9 by inserting the form factor, form factor
amplitudes, and phase factors corresponding to the chosen sub-units and
tethering geometry. In Appendix X, we have derived the required expressions to
characterize these tethering geometries.
Fig. VI.2 shows the form factors for disk-like and spherical micelles. At
small $q$ values we observe the Guinier regime which characterizes the radius
of gyration of the whole structure, while at large $q$ values we observe the
scattering due to the tethered chain sub-unit. In an intermediate regime, the
scattering is both due to the micellar core geometry and the tethered sub-
units. Even though the number of chains is the same, significant differences
are observed in the scattering for the different tethering geometries, but
coincidentally the sphere with equatorial tethering and disk with rim
tethering produce very similar scattering patterns. Fig. VI.3 shows the form
factors for cylindrical micelles for different choices of tethering geometry.
Again the tethering geometry is observed to introduce significant differences
in the spectra.
The scattering expression for a chain of thick end-linked cylinders are
obtained from eq. II.12 by inserting the form factor, form factor amplitude
relative to the reference point where the axis crosses the end, and phase
factor between the two ends. In fig. VI.4, the scattering from this thick
random walk is compared to that of a thin polymer and a rod. At large
distances in the Guinier regime we see the crossover from a point like
structure at the very largest scales to a random walk like structure with
scaling behavior $(qR_{g})^{-2}$. At intermediate length scales the cylinder
structure is probed and shows a scaling behavior like $(qL)^{-1}$ comparable
to the rod. At length scales at and below the radius of the rod, we see strong
oscillations due to cross section of the cylinder, and the envelope of the
scattering curve shows the $q^{-4}S^{2}$ behavior of Porod scattering from the
surface, where $S$ denotes the total surface area of the cylinders.
## VII Conclusions
In a previous paperSvaneborg and Pedersen (2012), we presented a formalism for
predicting the scattering from a linear and branched structures composed of
mutually non-interacting sub-units. A sub-unit can have an arbitrary number of
reference points. Sub-units are connected into structure by joining their
reference points to the reference points of other sub-units. In the present
paper, we have briefly presented the formalism, and generalized it to the case
where reference points can be characterized by a distribution of potential
link positions on a sub-unit. For instance, one reference point of a polymer
could be a random site along the contour, or a reference point of a sphere
could be a random point on the surface. To generalize the formalism, we had to
assume that reference point distributions on different sub-units are mutually
statistically uncorrelated.
We used the generalized formalism to derive the generic scattering expression
for a micelle / bottle-brush structure with a core sub-unit and a number of
identical sub-units tethered to random positions on the core / main chain.
Since the connectivity of a micelle and a bottle-brush is the same, the
generic scattering expressions are also identical. We presented the general
scattering expressions for the form factor, form factor amplitudes and phase
factors of a structure with internal conformations. We illustrated the
scattering the expression using end-linked and contour-linked chains of
polymers, rods, and polymeric loops and bottle-brush structures of rods and
polymer loops with tethered polymers, rods, or polymer loops. All these
structures are special cases of the generic scattering expression, which are
obtained when the form factor, form factor amplitudes, and phase factors of
the corresponding sub-units are inserted into the generic structural
scattering expression. We derived these terms in an appendix.
We also presented the general expressions for the form factor, form factor
amplitudes, and phase factors for rigid sub-units without internal
conformations. We derived the scattering terms for spheres, disks, and
cyllinders for a variety of different reference point distributions in an
appendix. While the form factors for all these sub-units are known, the form
factor amplitudes and phase factors are not necessarily known, and these are
required by the formalism. This allowed us to predict the scattering from
micelles with different core structures and geometries of tethering the corona
sub-units.
Taken together, the formalism presented in ref. Svaneborg and Pedersen (2012),
the present generalization to distributed reference points, and the sub-unit
scattering expressions derived in the appendix enables the scattering from a
large class of regular or random-linked, homogeneous or heterogeneous, linear
and branched structures to be derived with great ease. With this, we hope to
have provided a valuable tool for analyzing scattering data in the future.
## VIII Acknowledgments
C.S. and J.S.P. gratefully acknowledges discussions with C.L.P. Oliveira.
Funding for this work is provided in part by the Danish National Research
Foundation through the Center for Fundamental Living Technology (FLinT). The
research leading to these results has received funding from the European
Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement
n° 249032, MatchIT - Matrix for Chemical IT.
## IX Appendix: Sub-units with internal conformations
In the following we derive the scattering expressions for rigid rods, flexible
polymers, and closed polymer loops using eqs. III.1-III.3.
### IX.1 Rigid rods
The most basic example is a randomly orientated infinitely thin rigid rod of
length $L$. The rigid rod is special as the contour length $l$ and direct
distance $r$ between a pair sites are degenerate parameters, hence the
$\delta(r-l)$ factor in the rod pair-distance distribution:
$P(l;r)=\delta(r-l)\Theta(L-r)/(4\pi l^{2})$, where $\Theta(x)$ denotes the
Heaviside step function. Performing the contour length integrations
(III.1)-(III.3), the rod scattering triplet becomes
$F_{rod}(q,L)=\frac{2\mbox{Si}(x)}{x}-\frac{4}{x^{2}}\sin^{2}\left(\frac{x}{2}\right),$
(IX.1)
$A_{rod}(q,L)=\frac{\mbox{Si}(x)}{x}\quad\Psi_{rod}(q,L)=\frac{\sin(x)}{x},$
(IX.2)
where $x=qL$ and $\mbox{Si}(x)=\int_{0}^{x}\mbox{d}y\sin(y)/y$ is the Sin
integralAbramowitz and Stegun (1964). The expression for the rod form factor
was previous derived by NeugebauerNeugebauer (1953) and Teixera et al Teixeira
et al. (2007). In the case where the reference points are distributed along
the rod contour with a constant probability (denoted contour-linking and shown
with sub-script $\langle c\rangle$ and $\langle c\rangle\langle c\rangle$),
the rod phase factor and form factor amplitudes are given by
$\Psi_{rod,\langle c\rangle\langle c\rangle}(q)=A_{rod,\langle
c\rangle}(q)=F_{rod}(q,L).$
### IX.2 Flexible polymers
Flexible polymers can be modeled as random walks with a effective step length
or Kuhn length $b$. The pair-distance distributions between two sites
separated by a contour length $l$ are given by the Gaussian distribution
$P_{rw}(l;r)=\left(\frac{3}{2\pi
bl}\right)^{\frac{3}{2}}\exp\left(-\frac{3}{2}\frac{r^{2}}{bl}\right)$
Inserting this distribution into eqs. (III.1)-(III.3) yields the scattering
triplet characterizing a flexible polymer
$F_{pol}(x)=\frac{2[e^{-x}-1+x]}{x^{2}},\quad
A_{pol}(x)=\frac{1-e^{-x}}{x},\quad\mbox{and}\quad\Psi_{pol}(x)=e^{-x}$ (IX.3)
here $x=(qR_{g,rw})^{2}$, and the radius of gyration is given by
$R_{g,rw}^{2}=bL/6$. The result for the form factor amplitude has previously
been given by Hammouda Hammouda (1992) and the form factor was calculated by
Debye Debye (1947). These expressions can also be obtained as a self-
consistency requirement of this formalism requiring that the scattering form
factor, form factor amplitude and phase factor from a di-block copolymer with
two identical blocks of length $L/2$ match the scattering expressions for one
block of length $L$.Svaneborg and Pedersen (2012) Expressions for the
scattering from poly disperse flexible polymer characterized by a Schultz-Zimm
distribution are given in Ref. Benoit and Hadziioannou (1988). In the case
where the reference points are distributed randomly along the polymer contour
(denoted $\langle c\rangle$ and $\langle c\rangle\langle c\rangle$), the
polymer phase factor and form factor amplitude are given by
$\Psi_{pol,\langle c\rangle\langle c\rangle}(x)=A_{pol,\langle
c\rangle}(x)=F_{pol}(x).$
### IX.3 Polymer loops
While the formalism does not apply for structures that contains loops, no
assumptions are made as to the internal structure of the sub-units, which can
contain loops. The simplest loop is formed by linking the two ends of a
flexible polymer chain. We can model a polymer loop as two random walks with
contour lengths $l$ and $L-l$, respectively, starting at ${\bf R}_{\alpha}$
and ending at the link $\mathbf{R}_{\omega}$. In this case, the pair-distance
distribution is given by
$P_{loop}(l;|\mathbf{R}_{\omega}-\mathbf{R}_{\alpha}|)\propto
P_{rw}(l;|\mathbf{R}_{\omega}-\mathbf{R}_{\alpha}|)P_{rw}(L-l;|\mathbf{R}_{\omega}-\mathbf{R}_{\alpha}|),$
and the corresponding phase factor becomes
$\Psi_{loop,\alpha\omega}(q;l)=\exp\left(-\frac{bl(L-l)q^{2}}{6L}\right).$
Since the link at $l$ can be anywhere along the loop $[0;L]$, we need to
average over the link position to get the loop scattering:
$\Psi_{loop,\langle\alpha\rangle\langle\omega\rangle}(q)=\int_{0}^{L}\frac{\mbox{d}l}{L}\Psi_{loop}(q;l)=\frac{\exp\left(-2y^{2}\right)D[y]}{y}.$
Here $y=q\sqrt{bL}/\sqrt{24}$ and $D[y]=\exp(-y^{2})\int_{0}^{y}\exp(t^{2})dt$
is the Dawson integral, which is related to the imaginary part of the complex
error functionAbramowitz and Stegun (1964). By construction, the form factor
amplitude, form factor and average phase factor are all identical when the
reference point(s) is averaged over all sites in the structure, since eqs.
(III.1)-(III.3) become identical
$A_{loop,\langle\alpha\rangle}(q)=F_{loop}(q)=\Psi_{loop,\langle\alpha\rangle\langle\omega\rangle}(q)$.
The form factor of a flexible polymer loop was previous derived by Zimm and
Stockmayer.Zimm and Stockmayer (1949)
## X Appendix: geometric sub-units
Below we will derive the scattering from a solid sphere, a flat disk, and a
cylinder for a variety of reference point distributions using eqs. V.1-V.5.
### X.1 Solid sphere
For a solid homogeneous sphere with excess scattering length density $\beta$,
we can characterize any scattering site by its spherical coordinate
$\sigma=(r,\phi,\theta)$. Then
$\mathbf{r}(\sigma)=(r\cos\phi\sin\theta,r\sin\phi\sin,r\cos\theta)$. We can
choose a coordinate system such that the sphere center is located at the
origin ${\bf O}=0$. Due to the spherical symmetry the scattering vector
$\mathbf{q}$ is pointing towards the pole ($\theta=0)$, then
$\mathbf{q}\cdot\mathbf{r}_{\sigma}=qr\cos\theta$. The phase integral becomes
${\cal F}_{sphere}({\bf q},0;R)=\left(\frac{4\pi
R^{3}}{3}\right)^{-1}\int_{-\pi}^{\pi}\mbox{d}\phi\int_{-1}^{1}\mbox{d}\cos\theta\int_{0}^{R}\mbox{d}rr^{2}e^{iqr\cos\theta}$
$=\frac{3\left(\sin(qR)-qR\cos(qR)\right)}{(qR)^{3}},$ (X.1)
Due to the spherical symmetry, we do not need to perform an additional
orientational average $\langle\cdots\rangle_{o}$ when using eqs. V.1-V.6.
Hence, the form factor, the form factor amplitude and phase factor of a solid
sphere with $\mathbf{R}_{\alpha}=\mathbf{R}_{\omega}=0$ fixed at the center
(denoted subscript “c”) are given by
$F_{sphere}(q;R)=A_{sphere,c}^{2}(q;R),$ (X.2)
$A_{sphere,c}(q;R)=\frac{3\left(\sin(qR)-qR\cos(qR)\right)}{(qR)^{3}},$
$\Psi_{sphere,cc}(q)=1.$ (X.3)
The scattering from a solid sphere was derived by Reyleigh in 1911Rayleigh
(1911). Having the reference points at the center is the simplest choice,
however, for the derivation of e.g. the scattering from spherical micelles it
is more relevant let the surface of the sphere be a reference point. In this
case the corresponding normalized reference point distributions are
$Q_{\alpha}({\bf R})=Q_{\omega}({\bf R})=\delta(|{\bf R}|-R)/(4\pi R^{2})$
representing a spherical shell. We can calculate ${\cal F}_{shell}({\bf
q},0;R)$ by integration of eq. V.2. However, since the shell corresponds to
the upper limit of the radial integral in eq. X.1, we can trivially obtain its
Fourier transform by differentiation of ${\cal F}_{sphere}$ as
${\cal F}_{shell}({\bf q},0;R)=\left(4\pi
R^{2}\right)^{-1}\frac{\mbox{d}}{\mbox{d}R}\left[\frac{4\pi R^{3}}{3}{\cal
F}_{sphere}({\bf q},0,R)\right]=\frac{\sin qR}{qR}.$ (X.4)
Here we have introduced an area and volume prefactor to account for the
normalizations of the two Fourier transforms, such that ${\cal
F}_{shell}\rightarrow 1$ when $q\rightarrow 0$. We can obtain the form factor
amplitude and phase factors relative to the surface reference point (denoted
by subscript “$\langle s\rangle$”) combining eqs. X.1, X.4, V.3, and V.5 as
$A_{sphere,\langle
s\rangle}(q,R)=\frac{3\left(\sin(qR)-qR\cos(qR)\right)}{(qR)^{3}}\times\frac{\sin(qR)}{qR},$
and
$\Psi_{sphere,\langle s\rangle\langle
s\rangle}(q,R)=\left(\frac{\sin(qR)}{qR}\right)^{2}.$ (X.5)
The spherical phase factor was previously derived by Pedersen and
Gerstenberg.Pedersen and Gerstenberg (1996)
### X.2 Flat circular disk
Due to rotational symmetry, we can choose a geometry where the disk is in the
$xy$ plane, and $\mathbf{q}$ in the $xz$ plane. Expressing the scattering site
in polar coordinates $\sigma=(r,\phi)$, such that
$\mathbf{r}=(r\cos\phi,r\sin\phi,0)$, and
$\mathbf{q}=(q\sin\theta,0,q\cos\theta)$ then
$\mathbf{q}\cdot\mathbf{r}=qr\cos\phi\sin\theta$.
${\cal F}_{disk}({\bf q},0;R)=\frac{2J_{1}(qR\sin\theta)}{qR\sin\theta}$ (X.6)
Expressing the integrals in cylindrical coordinates immediately provides the
form factor, form factor amplitude and phase factor for
$\mathbf{R}_{\alpha}=\mathbf{R}_{\omega}=0$ fixed at the center as
$F_{disk}(q)=\left\langle\left(\frac{2J_{1}(qR\sin\theta)}{qR\sin\theta}\right)^{2}\right\rangle_{o}=\frac{2}{q^{2}R^{2}}\left[1-\frac{J_{1}(2qR)}{qR}\right],$
$A_{disk,c}(q)=\left\langle\frac{2J_{1}(qR\sin\theta)}{qR\sin\theta}\right\rangle_{o},\quad\Psi_{disk,cc}(q)=1$
(X.7)
Here $J_{n}(x)$ denotes the $n$’th Bessel function of the first kindAbramowitz
and Stegun (1964), and
$\langle\cdots\rangle_{o}=\frac{1}{2}\int_{-1}^{1}\mbox{d}\cos(\theta)\cdots$
denotes the remaining orientational average, which needs to be performed
numerically. The form factor of a disk first derived by Kratky and PorodKratky
and Porod (1949). We could also choose to put the reference point anywhere on
the circular rim of the disk. In this case, we can again obtain the Fourier
transform of the points on a circular rim ${\cal F}_{rim}$ by differentiation
of ${\cal F}_{disk}$ as
${\cal F}_{rim}({\bf q},0,R)=\left(2\pi
R\right)^{-1}\frac{\mbox{d}}{\mbox{d}R}\left[\pi R^{2}{\cal F}_{disk}({\bf
q},0,R)\right]$
$=J_{0}(qR\sin\theta)$ (X.8)
The corresponding form factor amplitude of the disk relative to any site on
the rim and the phase factor between two sites on the rim (denoted by
subscript $\langle r\rangle$)
$A_{disk,\langle
r\rangle}(q)=\left\langle\frac{2J_{1}(qR\sin\theta)}{qR\sin\theta}\times
J_{0}(qR\sin\theta)\right\rangle_{o},$ (X.9)
$\Psi_{disk,\langle r\rangle\langle r\rangle}(q)=\left\langle
J_{0}^{2}(qR\sin\theta)\right\rangle_{o}.$ (X.10)
If we instead chooses to put the two reference points at any point on the
surface of the disk (again denoting an average surface reference point by
$\langle s\rangle$), the result is again given by the disk form factor as we
have seen previously:
$A_{disk,\langle s\rangle}(q)=\Psi_{disk,\langle s\rangle\langle
s\rangle}(q)=F_{disk}(q).$ (X.11)
### X.3 Solid cylinder
We can choose a cylinder with its center at the origin and the axis along $z$,
the natural choice is to describe it with polar coordinates
$\sigma=(r,\phi,z)$ such that $\mathbf{r}(\sigma)=(r\cos\phi,r\sin\phi,z)$ and
we define $\mathbf{q}=(q\sin\theta,0,q\cos\theta)$ then
$\mathbf{q}\cdot\mathbf{r}(\sigma)=qr\cos\phi\sin\theta+qz\cos\theta$. The
phase integral becomes
${\cal F}_{cyl}({\bf q},{\bf
O};R,L)=\frac{4J_{1}(qR\sin\theta)\sin(\frac{qL}{2}\cos\theta)}{LRq^{2}\sin\theta\cos\theta}\times
e^{-i\mathbf{q}\cdot\mathbf{O}}.$ (X.12)
We have chosen to explicitly write the origin ${\bf O}$, since this will be
required to calculate the form factor amplitude. With regular reference points
at the ends of the cylinder axis $\mathbf{R}_{\alpha}=(0,0,-L/2)$ and
$\mathbf{R}_{\omega}=(0,0,+L/2)$ (denoted by subscript “$a$”), the form
factor, form factor amplitude and phase factor can be derived as
$F_{cyl}(q;R,L)=\left\langle\left(\frac{4J_{1}(qR\sin\theta)\sin(\frac{qL}{2}\cos\theta)}{LRq^{2}\sin\theta\cos\theta}\right)^{2}\right\rangle_{o},$
(X.13)
$A_{cyl,a}(q;R,L)=\left\langle{\cal F}_{cyl}({\bf q},{\bf
R}_{\alpha};R,L)\right\rangle_{o}=\left\langle{\cal F}_{cyl}({\bf q},{\bf
R}_{\omega};R,L)\right\rangle_{o}$ (X.14)
$=\left\langle\frac{4J_{1}(qR\sin\theta)\sin(\frac{qL}{2}\cos\theta)}{LRq^{2}\sin\theta\cos\theta}\cos\left(\frac{qL}{2}\cos\theta\right)\right\rangle_{o},$
(X.15)
$\Psi_{cyl,aa}(q;R,L)=\frac{\sin qL}{qL}.$ (X.16)
The form factor of a solid cylinder was previously derived by FournetFournet
(1951). We can also derive the Fourier transform of the end and side of the
cylinder by differentiation as
${\cal F}_{cyl,end}({\bf q},0;R,L)=\left(\pi
R^{2}\right)^{-1}\frac{\mbox{d}}{\mbox{d}L}\left[\pi LR^{2}{\cal F}_{cyl}({\bf
q},0;R,L)\right]$
$=\frac{2J_{1}(qR\sin\Theta)\cos(\frac{Lq}{2}\cos\Theta)}{qR\sin\Theta}$
(X.17)
${\cal F}_{cyl,side}({\bf q},0;R,L)=\left(2\pi
RL\right)^{-1}\frac{\mbox{d}}{\mbox{d}R}\left[\pi LR^{2}{\cal F}_{cyl}({\bf
q},0;R,L)\right]$
$=\frac{2J_{0}(qR\sin\Theta)\sin(\frac{Lq}{2}\cos\Theta)}{Lq\cos\Theta}$
(X.18)
By combining eqs. X.17 and X.18 weighting the terms by their relative areas
and normalizing, we obtain the Fourier transform of the surface of a cylinder
asPedersen (2000)
${\cal F}_{cyl,sur\\!face}({\bf q},0;R,L)=(R+L)^{-1}\left(R{\cal
F}_{cyl,end}+L{\cal F}_{cyl,side}\right)$ (X.19)
With these Fourier transforms we can write down the form factor amplitudes and
phase factors of the cylinder relative to a reference point distributed on the
ends (denoted by subscript $\langle e\rangle)$, on the hull (denoted by
$\langle h\rangle$) or anywhere on the surface (denoted by $\langle s\rangle$)
as
$A_{cyl,\langle
e\rangle}(q;R,L)=\left\langle\frac{8J_{1}^{2}(qR\sin\theta)\sin(\frac{qL}{2}\cos\theta)\cos(\frac{Lq}{2}\cos\theta)}{LR^{2}q^{3}\sin^{2}\theta\cos\theta}\right\rangle_{o},$
$A_{cyl,\langle
h\rangle}(q;R,L)=\left\langle\frac{8J_{0}(qR\sin\theta)J_{1}(qR\sin\theta)\sin^{2}(\frac{qL}{2}\cos\theta)}{RL^{2}q^{3}\sin\theta\cos^{2}\theta}\right\rangle_{o},$
$A_{cyl,\langle
s\rangle}(q;R,L)=\left\langle\frac{4J_{1}(qR\sin\theta)\sin(\frac{qL}{2}\cos\theta)}{LRq^{2}\sin\theta\cos\theta}\right.$
$\left.\times\frac{2}{q(R+L)}\left(\frac{J_{1}(qR\sin\Theta)\cos(\frac{Lq}{2}\cos\Theta)}{\sin\Theta}+\frac{J_{0}(qR\sin\Theta)\sin(\frac{Lq}{2}\cos\Theta)}{\cos\Theta}\right)\right\rangle_{o},$
$\Psi_{cyl,\langle e\rangle\langle
e\rangle}=\left\langle\left(\frac{2J_{1}(qR\sin\Theta)\cos(\frac{Lq}{2}\cos\Theta)}{qR\sin\Theta}\right)^{2}\right\rangle_{o},$
$\Psi_{cyl,\langle h\rangle\langle
h\rangle}=\left\langle\left(\frac{2J_{0}(qR\sin\Theta)\sin(\frac{Lq}{2}\cos\Theta)}{Lq\cos\Theta}\right)^{2}\right\rangle_{o},$
and
$\Psi_{cyl,\langle s\rangle\langle
s\rangle}=\left\langle\frac{4}{q^{2}(R+L)^{2}}\left(\frac{J_{1}(qR\sin\Theta)\cos(\frac{Lq}{2}\cos\Theta)}{\sin\Theta}\right.\right.$
$\left.\left.+\frac{J_{0}(qR\sin\Theta)\sin(\frac{Lq}{2}\cos\Theta)}{\cos\Theta}\right)^{2}\right\rangle_{o}.$
## ?refname?
* Guinier and Fournet (1955) A. Guinier and G. Fournet, _Small angle scattering of X-rays_ (Wiley, New York, 1955).
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|
arxiv-papers
| 2011-08-04T17:08:12 |
2024-09-04T02:49:21.306181
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carsten Svaneborg and Jan Skov Pedersen",
"submitter": "Carsten Svaneborg",
"url": "https://arxiv.org/abs/1108.1141"
}
|
1108.1257
|
# Multi-channel Hybrid Access Femtocells:
A Stochastic Geometric Analysis
Yi Zhong and Wenyi Zhang, _Senior Member, IEEE_ The authors are with
Department of Electronic Engineering and Information Science, University of
Science and Technology of China, Hefei 230027, China (email:
geners@mail.ustc.edu.cn, wenyizha@ustc.edu.cn). The research has been
supported by the National Basic Research Program of China (973 Program)
through grant 2012CB316004, National Natural Science Foundation of China
through grant 61071095, Research Fund for the Doctoral Program of Higher
Education of China through grant 20103402120023, and by MIIT of China through
grants 2010ZX03003-002 and 2011ZX03001-006-01.
###### Abstract
For two-tier networks consisting of macrocells and femtocells, the channel
access mechanism can be configured to be open access, closed access, or hybrid
access. Hybrid access arises as a compromise between open and closed access
mechanisms, in which a fraction of available spectrum resource is shared to
nonsubscribers while the remaining reserved for subscribers. This paper
focuses on a hybrid access mechanism for multi-channel femtocells which employ
orthogonal spectrum access schemes. Considering a randomized channel
assignment strategy, we analyze the performance in the downlink. Using
stochastic geometry as technical tools, we model the distribution of
femtocells as Poisson point process or Neyman-Scott cluster process and derive
the distributions of signal-to-interference-plus-noise ratios, and mean
achievable rates, of both nonsubscribers and subscribers. The established
expressions are amenable to numerical evaluation, and shed key insights into
the performance tradeoff between subscribers and nonsubscribers. The
analytical results are corroborated by numerical simulations.
###### Index Terms:
Channel management, femtocell, hybrid access, Neyman-Scott cluster process,
spatial Poisson process, two-scale approximation, two-tier network
## I Introduction
In current cellular network services, about 50% of phone calls and 70% of data
services take place indoors [1]. For such indoor use cases, network coverage
is a critical issue. One way to improve the indoor performance is to deploy
the so-called femtocell access points (FAPs) besides macrocell base stations
(MBSs). Femtocells are small cellular base stations, typically designed for
use in home or small business [2][3]. The use of femtocells not only benefits
the users, but also the operators. As the distance between transmitter and
receiver is reduced, users will enjoy high quality links and power savings.
Furthermore, the reduced transmission range also creates more spatial reuse
and reduces electromagnetic interference.
Among the many challenges faced by femtocells, and more generally, two-tier
networks, is the issue of interference; see Figure 1. The two-tier
interference problem differs from that in traditional single-tier networks in
several important aspects: First, due to the limitations of access mechanism,
a user equipment (UE) may not be able to connect to the access point which
offers the best service. Second, since femtocells connect to operator’s core
network via subscribers’ private ISP, coordination between macrocells and
femtocells and among femtocells is limited. Finally, compared to planned
macrocell deployments, femtocells are usually deployed in an ad hoc manner,
and the randomly placed femtocells make it difficult to manage the
interference. In two-tier networks, interference can be categorized into two
types: (a) cross-tier, referring to the interference from one tier to the
other; (b) co-tier, referring to the interference within a tier.
Figure 1: Downlink two-tier network model for hybrid access femtocell.
In this paper, we consider two-tier networks based on multicarrier techniques,
for example those deploying LTE or WiMAX standards, which use orthogonal
frequency-division multiple access (OFDMA) techniques. In multicarrier
systems, the available spectrum is divided into orthogonal subcarriers, which
are then grouped into multiple subchannels, assigned to different users. Due
to the flexibility in channel assignment, the interference may be alleviated.
The access mechanism of femtocells (see, e.g., [4]) is a key factor that
affects the performance of two-tier networks, and generally can be classified
as follows, where we call the UEs registered to a femtocell as subscribers,
and those not registered to any femtocell as nonsubscribers.
* •
Closed access: An FAP only allows its subscribers to connect.
* •
Open access: An FAP allows all its covered UEs, no matter registered or not,
to connect.
* •
Hybrid access: An FAP allows its covered nonsubscribers to connect via a
subset of its available subchannels, and reserves the remaining subchannels
for its subscribers.
Hybrid access [5] is an intermediate approach, in which a fraction of resource
is allocated to nonsubscribers. By doing so, nonsubscribers near an FAP may
handover into the femtocell to avoid high interference; meanwhile, with
certain amount of resource reserved for subscribers, the performance of
subscribers may be well assured even in the presence of nonsubscribers.
In hybrid access, a central issue is how to allocate the resource between
subscribers and nonsubscribers. Previous studies [6] [7] indicate that hybrid
access improves the network performance at the cost of reduced performance for
subscribers, therefore suggesting a tradeoff between the performance of
nonsubscribers and subscribers. In this paper, we consider a hybrid access
mechanism that uses a randomized channel assignment strategy, and analyze the
performance in the downlink of both macrocells and femtocells. We employ
stochastic geometry to characterize the spatial distributions of users as well
as access points; see, e.g., [8] and references therein for its recent
applications in wireless networks. In order to make the work integral, we will
carry out the analysis in two different cases. As a general assumption, we
first assume that the FAPs are distributed as a Poisson point process (PPP).
Then, we switch to the case when the FAPs are distributed as a Neyman-Scott
cluster process. The cluster process is likely to be more realistic because
the FAPs are typically deployed in populous locations, like commercial or
residential area. Accordingly, we derive the key performance indicators
including mean achievable rates and distributions of the signal-to-
interference-plus-noise ratios (SINRs) of both nonsubscribers and subscribers.
In our study, we establish general integral expressions for the performance
indicators, and closed form expressions under specific model parameters. With
the obtained results, we reveal how the performance of subscribers and
nonsubscribers trades off each other.
The introduction of stochastic geometry in the analysis of wireless network is
not our original, and an overview of related works is as follows. In [9], the
authors proposed to study key performance indicators for cellular networks,
such as coverage probabilities and mean achievable rates. In [10], the
considered scheme divides the spectrum resource into two orthogonal parts
which are assigned to macrocells and femtocells, respectively, with femtocells
being closed access. In [11], the authors considered two-tier femtocell
networks using time-hopped CDMA, examining the uplink outage probability and
the interference avoidance capability. In [12], the success probabilities
under Rayleigh fading for both macrocells and femtocells are derived in uplink
and downlink respectively. In [13] and [14], stochastic geometry tools are
applied in the coexistence analysis of cognitive radio networks. In [15] and
[16], the authors studied the performance of various femtocell access
mechanisms, under substantially different system models from ours. More
explicitly, the work in [15], which does not make use of stochastic geometry,
focused on the uplink with only one MBS and one FAP in the model. Though the
work in [16] also applies the Laplace transform of interference in the
derivation, the substantially difference of our work lies in that we take the
load (measured by the number of UEs in a cell) into consideration, utilizing
the size distribution of Voronoi cells to derive the distribution of the load.
Moreover, we model the mechanism for sharing sub-channels in multi-channel
systems, and propose to use a two-scale approximation which substantially
simplifies the analysis. All the works mentioned above are based on the PPP
assumption. The works applying the clustered model can be found in [17] which
derived the success probability for transmission in clustered ad-hoc networks
and in [18] which discussed the property of interference with clustered
interferers.
The main contribution of our work is detailed as follows. The existing works
mostly ignore the network load which is a key factor that affects the
distribution of interfering access points (APs). For example, when the load is
uniformly distributed in the plane, the APs with larger coverage may
experience more load, thus leading to more interference to the network.
Moreover, the optimal proportion of shared resources of a femtocell also
depends on the distribution of network load. In our work, we focus on the
performance analysis in the context of multi-channel systems, in which case
not all sub-channels are occupied and not all APs cause interference to a
given subchannel. We evaluate the two-tier interference when the FAPs are
distributed as the PPP and the Neyman-Scott cluster respectively. In addition,
we propose two-scale approximation to simplify the analysis and verify the
effectiveness of the approximation by simulation.
The remaining part of the paper is organized as follows. Section II describes
the two-tier network model, the channel assignment strategy, and the hybrid
access mechanism. Based on a two-scale approximation for the spatial
distributions of FAPs, Section III analyzes the statistical behavior of UEs,
deriving the distributions of the number of UEs connecting to either an FAP or
an MBS, as well as the probabilities of a subchannel being used by either an
FAP or an MBS. Built upon those statistics, Section IV and V establish
expressions for the distributions of SINRs, and mean achievable rates in the
cases when the FAPs are distributed as PPP and Neyman-Scott cluster
respectively. Section VII illustrates the aforementioned analysis by numerical
results, which are also corroborated by simulations. Finally, Section VIII
concludes the paper.
## II Network Model
### II-A Hybrid Access Femtocells
In the two-tier network, we consider two types of access points, MBSs and
FAPs. The MBSs constitute the macrocell tier, and they induce a Voronoi
tessellation of the plane (see Figure 2). When a UE attempts to access the
macrocell network, it chooses to connect to the MBS in the Voronoi cell in
which the UE is situated. An FAP provides network access to UEs in its
vicinity, and we assume that all FAPs have a covering radius of $R_{f}$.
Within the covered circular area of each FAP are two types of UEs, called
subscribers and inside nonsubscribers. Inside nonsubscribers are those UEs who
gather around an FAP without subscribing to its service; for example,
transient customers in a shop or a restaurant. Besides those two types of UEs,
we also consider a third type of UEs, outside nonsubscribers, who are
uniformly scattered over the whole plane, corresponding to those regular
macrocell network users.
(a) FAPs are distributed as PPP.
(b) FAPs are distributed as Neyman-Scott cluster.
Figure 2: The Voronoi macrocell topology, in which each Voronoi cell is the
coverage area of a macrocell and each small circle represents a femtocell.
The available spectrum is evenly divided into $M$ subchannels, which are to be
shared by both macrocell tier and femtocell tier. Each FAP is configured to
allocate a fixed number, $M_{s}$, of subchannels for its covered inside
nonsubscribers. These $M_{s}$ subchannels are called shared subchannels, and
the remaining $M_{r}=M-M_{s}$ subchannels are called reserved subchannels as
they are reserved for the subscribers. In the considered hybrid access
mechanism, each FAP selects its shared subchannels randomly, and independently
of other FAPs. We assume that each UE, whether subscriber or nonsubscriber,
needs one subchannel for transmitting. When a UE accesses an MBS or an FAP,
the serving subchannel is selected randomly (see Figure 3).
Figure 3: Spectrum allocation in each hybrid access femtocell. All the $M_{s}$
shared subchannels are randomly selected by each FAP.
The hybrid access mechanism operates as follows.
* •
A subscriber accesses to one of the $M_{r}$ reserved subchannels of its
corresponding FAP. When there are more than $M_{r}$ subscribers in an FAP,
they are served by time-sharing with equal time proportion.
* •
An inside nonsubscriber accesses to one of the $M_{s}$ shared subchannels of
its covering FAP. When there are more than $M_{s}$ inside nonsubscribers in an
FAP, they are served by time-sharing with equal time proportion.
* •
An outside nonsubscriber accesses the MBS located in the Voronoi cell in which
the outside nonsubscriber is situated. When there are more than $M$ outside
nonsubscribers in the Voronoi cell, they are served by time-sharing with equal
proportion.
### II-B Mathematical Model and Two-scale Approximation
To formulate the aforementioned hybrid access scenario mathematically, we
model the spatial distributions of the nodes using spatial point processes as
follows. The MBSs constitute a homogeneous Poisson point process (PPP)
$\Phi_{m}$ of intensity $\lambda_{m}$ on the plane. The distribution of FAPs
will be divided into two cases for discussion:
* •
Case 1: the FAPs constitute another homogeneous PPP $\Phi_{f}$ of intensity
$\lambda_{f}$.
* •
Case 2: the FAPs are distributed as a Neyman-Scott cluster process $\Phi_{f}$
[19]. The center of the clusters are assumed to be distributed according to a
stationary PPP $\Phi_{p}$ of intensity $\lambda_{p}$, which is called the
parent process. For each cluster center $x\in\Phi_{p}$, the FAPs are
distributed according to an independent PPP $\Phi^{x}$ of intensity
$\lambda_{c}$ in the circular covered area of radius $R_{c}$ around the center
$x$. The complete distribution of all FAPs is given as
$\Phi_{f}=\bigcup_{x\in\Phi_{p}}\Phi^{x}.$ (1)
In this case, the number of FAPs in a typical cluster is a Poisson random with
parameter $\pi R_{c}^{2}\lambda_{c}$ and the intensity of all FAPs is
$\lambda_{f}=\pi R_{c}^{2}\lambda_{c}\lambda_{p}$.
In the circular covered area of radius $R_{f}$ of each FAP, the subscribers
are distributed according to a homogeneous PPP of intensity $\lambda_{s}$, and
the inside nonsubscribers are distributed according to another homogeneous PPP
of intensity $\lambda_{\mathrm{in}}$. Outside nonsubscribers constitute on the
whole plane a homogeneous PPP of intensity $\lambda_{\mathrm{out}}$. All the
PPPs are mutually independent.
In this paper, we focus on the downlink performance. The transmit power is set
to a constant value $P_{m}$ for an MBS, and $P_{f}$ for an FAP. For the sake
of convenience, we adopt a standard path loss propagation model with path loss
exponent $\alpha>2$. Regarding fading, we assume that the link between the
serving access point (either an MBS or an FAP) and the served UE experiences
Rayleigh fading with parameter $\mu$. The received signal power of a UE at a
distance $r$ from its serving access point therefore is $P_{m}hr^{-\alpha}$
(MBS) or $P_{f}hr^{-\alpha}$ (FAP) where $h\sim\mathrm{Exp}(\mu)$. The fading
of interference links may follow an arbitrary probability distribution, and is
denoted by $g$. Furthermore, considering the typical scenario of indoor
femtocell deployment, we introduce a wall isolation at the boundary of each
FAP coverage area, which incurs a wall penetration loss factor $W<1$. For all
receivers, the noise power is $\sigma^{2}$.
The different point processes corresponding to different entities in the
network interact in a complicated way, thus making a rigorous statistical
analysis extremely difficult. For example, an inside nonsubscriber may be
covered by more than one FAPs, thus leading to the delicate issue of FAP
selection, and furthermore rendering the subchannel usage distributions among
FAPs and MBSs intrinsically correlated. To overcome the technical difficulties
due to spatial interactions, in the subsequent analysis we propose a two-scale
approximation for the network model, motivated by the fact that the covered
area of an FAP is significantly smaller than that of an MBS. The two-scale
approximation consists of two views, the macro-scale view and the micro-scale
view. The macro-scale view concerns an observer outside the coverage area of
an FAP, and in that view the whole coverage area of the FAP shrinks to a
single point, marked by the numbers of subscribers and inside nonsubscribers
therein. The micro-scale view concerns an observer inside the coverage area of
an FAP, and in that view the coverage area is still circular with radius
$R_{f}$ in which the subscribers and inside nonsubscribers are spatially
distributed. By such a two-scale approximation, an inside nonsubscriber can
only be covered by a unique FAP, and the coverage area of an FAP can only be
within a unique Voronoi cell of an MBS. Meanwhile, at the cell edge the
outside nonsubscribers are clearly divided by the boundary and the subscribers
and inside nonsubscribers are all attached to the corresponding FAPs which are
also clearly divided. These consequences substantially simplify the
performance analysis. In Section VII, we validate the two-scale approximation
method through comparing analytical results and simulation results, for
network parameters of practical interest.
## III Statistics of UEs and Subchannels
In this section, we characterize the distributions of UEs connecting to
different types of access points, and the distributions of used subchannels in
MBSs and FAPs. The analysis is based on a snapshot of the network model, and
the obtained results will then be applied for characterizing the distributions
of SINRs and achievable rates in Section IV.
### III-A Distributions of UEs
Let $U_{s}$ be the number of subscribers accessing a given FAP and from our
model we have $U_{s}\sim\mathrm{Poisson}(\lambda_{s}\pi R_{f}^{2})$.
Similarly, let $U_{\mathrm{in}}$ be the number of inside nonsubscribers
accessing a given FAP, and we have
$U_{\mathrm{in}}\sim\mathrm{Poisson}(\lambda_{\mathrm{in}}\pi R_{f}^{2})$.
The number of outside nonsubscribers who access a given MBS, denoted by
$U_{\mathrm{out}}$, is characterized as follows. We note that the macrocell
coverage area is a Voronoi cell, and denote by $S$ the area of the Voronoi
cell. There is no known closed form expression of the probability density
function (pdf) of $S$, whereas a simple approximation [20] has proven
sufficiently accurate for practical purposes. Considering scaling, the
approximate pdf of the size of a macrocell coverage area is given by
$f(S)=\frac{343}{15}\sqrt{\frac{7}{2\pi}}(S\lambda_{m})^{\frac{5}{2}}\exp(-\frac{7}{2}S\lambda_{m})\lambda_{m}.$
(2)
Conditioning upon $S$, the number of outside nonsubscribers is a Poisson
random variable with mean $\lambda_{\mathrm{out}}S$. The probability
generating function of the unconditioned $U_{\mathrm{out}}$ is thus given by
$G(z)=\int_{0}^{\infty}\exp\Big{(}\lambda_{\mathrm{out}}(z-1)S\Big{)}f(S)dS.$
(3)
Plugging in the approximate pdf of $S$ and simplifying the integral, we get
$G(z)=\frac{343}{8}\sqrt{\frac{7}{2}}\Big{(}\frac{7}{2}-\frac{\lambda_{\mathrm{out}}}{\lambda_{m}}(z-1)\Big{)}^{-\frac{7}{2}}.$
(4)
The distribution of $U_{\mathrm{out}}$ is therefore given by the derivatives
of $G(z)$,
$\mathbb{P}\\{U_{\mathrm{out}}=i\\}=\frac{G^{(i)}(0)}{i!},\quad i=0,1,\ldots.$
(5)
### III-B Distributions of Subchannel Usage
Since the subchannels are uniformly and independently selected by each FAP, it
suffices to analyze an arbitrary one of them. Let us examine the probability
that a given subchannel is used by an MBS or an FAP. First, we evaluate the
average number of subchannels used by an MBS or an FAP, and then we normalize
the average number by the total number of subchannels, $M$.
The probability that a subchannel is used by an FAP is
$\displaystyle
P_{\mathrm{busy},f}=\frac{1}{M}\Big{(}\sum_{i=0}^{\infty}\min\\{i,M_{r}\\}\mathbb{P}\\{U_{s}=i\\}$
$\displaystyle+\sum_{j=0}^{\infty}\min\\{j,M_{s}\\}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\Big{)}.$
(6)
For a Poisson random variable $N\sim\mathrm{Poisson}(\lambda)$, its cumulative
distribution function (cdf) is
$\sum_{i=0}^{n}\mathbb{P}\\{N=i\\}=\sum_{i=0}^{n}\frac{\lambda^{i}}{i!}e^{-\lambda}=\frac{\Gamma(n+1,\lambda)}{n!},$
(7)
where $\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\mathrm{d}t$ is the
incomplete gamma function. Using (7) to simplify $P_{\mathrm{busy},f}$, we get
$\displaystyle
P_{\mathrm{busy},f}=1+\frac{M_{r}}{M}\frac{1}{M_{r}!}\Big{(}\lambda_{s}\pi
R_{f}^{2}\Gamma(M_{r},\lambda_{s}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{r}+1,\lambda_{s}\pi R_{f}^{2})\Big{)}$
$\displaystyle+\frac{M_{s}}{M}\frac{1}{M_{s}!}\Big{(}\lambda_{\mathrm{in}}\pi
R_{f}^{2}\Gamma(M_{s},\lambda_{\mathrm{in}}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{s}+1,\lambda_{\mathrm{in}}\pi R_{f}^{2})\Big{)}.$ (8)
The probability that a subchannel is used by an MBS is
$\displaystyle
P_{\mathrm{busy},m}=\frac{1}{M}\sum_{i=0}^{\infty}\min\\{i,M\\}\mathbb{P}\\{U_{\mathrm{out}}=i\\},$
(9)
where $\mathbb{P}\\{U_{\mathrm{out}}=i\\}$ is given by (5).
The spatial point process of FAPs that use a given subchannel is the
independent thinning of the original process of FAPs $\Phi_{f}$ by the
probability $P_{\mathrm{busy},f}$, denoted by $\Phi_{f}^{\prime}$. The term
“independent thinning” means that $\Phi_{f}^{\prime}$ can be viewed as
obtained from $\Phi_{f}$ by independently removing points with probability
$1-P_{\mathrm{busy},f}$. When the FAPs are distributed as a homogeneous PPP,
the resulting point process is still a homogeneous PPP with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. For the case when the
FAPs are distributed as Neyman-Scott cluster process, the resulting point
process is a Neyman-Scott cluster process with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. Moreover, the intensity
of the parent process is still $\lambda_{p}$ and the intensity of the FAPs in
each cluster is reduced to
$\lambda_{c}^{\prime}=\lambda_{c}P_{\mathrm{busy},f}$. As for the MBSs, the
correlations between the sizes of neighboring cells may lead to the dependence
in the thinning of the original PPP of MBSs. However, in order to facilitate
the analysis, we assume that the independent thinning assumption still holds.
Therefore, the spatial process of MBSs that use a given subchannel is the
independent thinning of the original PPP of MBSs $\Phi_{m}$ by the probability
$P_{\mathrm{busy},m}$, denoted by $\Phi_{m}^{\prime}$ with intensity
$\lambda_{m}^{\prime}=\lambda_{m}P_{\mathrm{busy},m}$. Meanwhile, the
computation of $P_{\mathrm{busy},m}$ and the SINR distribution can also be
decoupled. The simulation results validate that our independent thinning
assumption is rather reliable. These two independently thinned point processes
will prove useful in the subsequent analysis.
## IV Performance With Poisson Distribution Of FAPs
In this section, we derive the distributions of the SINRs and the mean
achievable rates of UEs served by MBS and FAP respectively when the FAPs are
distributed as the PPP. For each type of UEs, we begin with general settings,
and then simplify the general results under specific parameters to gain
insights. The mean achievable rates are the averaged instantaneous achievable
rates over both channel fading and spatial distributions of UEs and access
points.
### IV-A Macrocell UEs
#### IV-A1 General Case
For an active UE served by an MBS, it must be occupying one subchannel of the
MBS. The following theorem gives the cdf of SINR and the mean achievable rate
of each active macrocell UE in general case.
###### Theorem 1
The cdf of the SINR of a macrocell UE, denoted by
$Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is given by
$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(T,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
Tv^{\frac{\alpha}{2}}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$ (10)
The mean achievable rate of a macrocell UE is given by
$\displaystyle\tau_{m}=\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(e^{t}-1,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
v^{\frac{\alpha}{2}}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}t\mathrm{d}v.$
(11)
In (10) and (11), $\beta(T,\alpha)$ is given by
$\displaystyle\beta(T,\alpha)=\frac{2(\mu
T)^{\frac{2}{\alpha}}}{\alpha}\mathbb{E}_{g}\Big{(}g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},\mu
Tg)$
$\displaystyle-(1+\frac{\lambda_{f}^{\prime}(WP_{f})^{\frac{2}{\alpha}}}{\lambda_{m}^{\prime}P_{m}^{\frac{2}{\alpha}}})\Gamma(-\frac{2}{\alpha})\Big{)}\Big{)}.$
(12)
The proof of Theorem 1 is in Appendix A.
In Theorem 1, $Z_{m}(T)$ in (10) gives the probability that the SINR is below
a given target level $T$, and $\tau_{m}$ in (11) gives the mean achievable
rate of a macrocell UE. The integrals in (10) and (11) can be evaluated by
numerical methods, and furthermore, they can be simplified to concise forms in
the special case when the interference experiencing Rayleigh fading and the
path loss exponent being $\alpha=4$ with no noise, $\sigma^{2}=0$.
#### IV-A2 Special Case When Interference Experiences Rayleigh Fading
Here we consider the case where the interference experiences Rayleigh
distribution with mean $\mu$, i.e. $g\sim\mathrm{Exp}(\mu)$. In this case, the
results are as follows.
###### Corollary 1
When the interference follows Rayleigh fading, the cdf of the SINR is
$\displaystyle\\!\\!\\!Z_{m}(T)$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$ $\displaystyle
1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(T,\alpha)$ (13)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}WT}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$
The mean achievable rate is
$\displaystyle\\!\\!\\!\tau_{m}$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)$ (14)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}W(e^{t}-1)}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}v\mathrm{d}t,$
where
$\varphi(T,\alpha)=T^{2/\alpha}\int_{T^{-2/\alpha}}^{\infty}\frac{1}{1+u^{\alpha/2}}\mathrm{d}u.$
(15)
The results in Corollary 1 are further simplified in the following special
cases.
Specifically, when $\alpha=4$ we obtain
$\displaystyle Z_{m}(T)$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!1-\frac{1}{1+\sqrt{T}\Big{(}\arctan\sqrt{T}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}^{\prime}}\sqrt{\frac{WP_{f}}{P_{m}}}\Big{)}P_{\mathrm{busy},m}}.$
The mean achievable rate of a macrocell UE is simplified into
$\tau_{m}=\int_{0}^{\frac{\pi}{2}}\frac{2}{\tan
y+(\frac{\pi}{2}-y)P_{\mathrm{busy},m}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}}\mathrm{d}y.$
(17)
From a practical perspective, it is desirable to shape $Z_{m}(T)$ to make it
small for small values of $T$. From (LABEL:equ:SINR_Simple), we see that there
are two approaches to shape $Z_{m}(T)$. First, $Z_{m}(T)$ decreases as
$P_{\mathrm{busy},m}$, the probability that a subchannel is used by an MBS,
decreases. This may be interpreted as an effect of frequency reuse. Second,
$Z_{m}(T)$ decreases as the whole term,
$\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}$,
decreases, which corresponds to a number of network parameters, representing
the effect due to the deployment of the femtocell tier.
### IV-B Femtocell UEs
#### IV-B1 General Case
A UE served by an FAP occupies one subchannel of the FAP. The following
theorem gives the cdf of the SINR and the mean achievable rate of each active
femtocell UE in general case.
###### Theorem 2
The cdf of the SINR of a femtocell UE in general case is given by
$\displaystyle
Z_{f}(T)=1-\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\exp\Big{(}-\rho(\alpha)T^{2/\alpha}v$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{f}}\Big{)}\mathrm{d}v.$
(18)
The mean achievable rate of a femtocell UE is given by
$\displaystyle\tau_{f}=\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\exp\Big{(}-\rho(\alpha)(e^{t}-1)^{2/\alpha}v$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{f}}\Big{)}\mathrm{d}t\mathrm{d}v.$ (19)
in (18) and (2), $\rho(\alpha)$ is given by
$\displaystyle\rho(\alpha)=-\frac{2\pi\mu^{2/\alpha}}{\alpha}\Gamma\Big{(}-\frac{2}{\alpha}\Big{)}\Big{(}\lambda_{m}^{\prime}\Big{(}\frac{WP_{m}}{P_{f}}\Big{)}^{2/\alpha}$
$\displaystyle+\lambda_{f}^{\prime}W^{4/\alpha}\Big{)}\mathbb{E}_{g}(g^{2/\alpha}).$
(20)
The proof of Theorem 2 is in Appendix B.
#### IV-B2 Special Case When Interference Experiences Rayleigh Fading
For Rayleigh fading, the results are essentially the same as that in the
general fading case. The only additional simplification is the evaluation of
$\rho(\alpha)$. We just give the result in the very special case when
$\sigma^{2}=0$ and $\alpha=4$.
When the interference experiences Rayleigh fading, we have
$\mathbb{E}_{g}(g^{\frac{1}{2}})=\mu\int_{0}^{\infty}g^{\frac{1}{2}}e^{-\mu
g}\mathrm{d}g=\frac{1}{2}\sqrt{\frac{\pi}{\mu}}$, and consequently,
$\rho(4)=\frac{\pi^{2}}{2}\Big{(}\lambda_{m}^{\prime}\sqrt{\frac{WP_{m}}{P_{f}}}+\lambda_{f}^{\prime}W\Big{)}.$
(21)
The SINR distribution is then given by
$Z_{f}(T)=1-\frac{1-e^{-\rho(4)\sqrt{T}R_{f}^{2}}}{\rho(4)\sqrt{T}R_{f}^{2}}.$
(22)
Let $y=\rho(4)R_{f}^{2}\sqrt{e^{t}-1}$, the mean achievable rate is simplified
into
$\tau_{f}=2\int_{0}^{\infty}\frac{1-e^{-y}}{y^{2}+\rho^{2}(4)R_{f}^{4}}\mathrm{d}y.$
(23)
These expressions are convenient for numerical evaluation.
## V Performance With Clustered FAPs
In this section, we derive the distributions of SINRs and the mean achievable
rates in the case when the FAPs are distributed as the Neyman-Scott cluster
process. To facilitate the analysis, we only focus on the case when the
interference links experience exponential fading.
### V-A Macrocell UEs
The following theorem gives the cdf of SINR and the mean achievable rate of
each active macrocell UE.
###### Theorem 3
In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
macrocell UE, denoted by $Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by
$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{Tv^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(T,\alpha)\bigg{)}\mathrm{d}v.$ (24)
The mean achievable rate of a macrocell UE is
$\displaystyle\tau_{m}=\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{(e^{t}-1)v^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu(e^{t}-1)v^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)\bigg{)}\mathrm{d}v\mathrm{d}t,$
(25)
where $\varphi(T,\alpha)$ is given by (15). Let $C(o,R_{c})$ be the circle
centered at the origin with radius $R_{c}$ and $\eta(s,x)$ is given as
$\eta(s,x)=\exp\Big{(}\int_{C(o,R_{c})}\frac{-\lambda_{c}^{\prime}}{1+\frac{1}{s}|x+y|^{\alpha}}\mathrm{d}y\Big{)}.$
(26)
The proof of Theorem 3 is in Appendix C.
### V-B Femtocell UEs
The following theorem gives the cdf of SINR and the mean achievable rate of
each active femtocell UE.
###### Theorem 4
In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
femtocell UE, denoted by $Z_{f}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by
$\displaystyle Z_{f}(T)=1-\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}$ $\displaystyle-\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{WTP_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta(Tr^{\alpha}W^{2},x)\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta(Tr^{\alpha}W^{2},y-z)\mathrm{d}y\bigg{)}r\mathrm{d}r.$
(27)
The mean achievable rate of a femtocell UE is given by
$\displaystyle\tau_{f}=\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{\infty}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu(e^{t}-1)r^{\alpha}\sigma^{2}}{P_{f}}-$
$\displaystyle\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{W(e^{t}-1)P_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\big{(}(e^{t}-1)r^{\alpha}W^{2},x\big{)}\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta\Big{(}(e^{t}-1)r^{\alpha}W^{2},y-z\Big{)}\mathrm{d}y\bigg{)}r\mathrm{d}r\mathrm{d}t,$
(28)
where $z=(r,0)$ and $\eta(s,x)$ is given by (26).
The proof of Theorem 4 is in Appendix D.
## VI Mean Achievable Rates of Nonsubscribers and Subscribers
There are two types of nonsubscribers, outside nonsubscribers who access MBSs
and inside nonsubscribers who access FAPs. When the number of outside
nonsubscribers in a macrocell is no greater than the total number of
subchannels (i.e., $U_{\mathrm{out}}\leq M$), each nonsubscriber UE
exclusively occupies a subchannel, and its mean achievable rate is $\tau_{m}$.
However, when $U_{\mathrm{out}}>M$, those $U_{\mathrm{out}}$ UEs share the $M$
subchannels with mean achievable rate $\frac{M}{U_{\mathrm{out}}}\tau_{m}$.
Since the evaluation is conditioned upon the existence of at least one UE, the
mean achievable rate of an outside nonsubscriber UE is given by
$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{out}}=\frac{\sum_{i=1}^{M}\mathbb{P}\\{U_{\mathrm{out}}=i\\}+\sum_{i=M+1}^{\infty}\mathbb{P}\\{U_{\mathrm{out}}=i\\}\frac{M}{i}}{1-\mathbb{P}\\{U_{\mathrm{out}}=0\\}}\tau_{m}.$
(29)
Similarly, the mean achievable rate of an inside nonsubscriber is given by
$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{in}}=\frac{\sum_{j=1}^{M_{s}}\mathbb{P}\\{U_{\mathrm{in}}=j\\}+\sum_{j=M_{s}+1}^{\infty}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\frac{M_{s}}{j}}{1-\mathbb{P}\\{U_{\mathrm{in}}=0\\}}\tau_{f}.$
(30)
When averaged over both inside and outside nonsubscribers, the overall mean
achievable rate of nonsubscriber is obtained as
$\\!\\!\\!\\!\tau_{n}=\frac{\lambda_{out}\tau_{\mathrm{out}}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}\tau_{\mathrm{in}}}{\lambda_{out}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}}.$ (31)
Regarding subscribers, note that they are exclusively served by FAPs. Similar
to the analysis of nonsubscribers, the mean achievable rate of a subscriber is
given by
$\tau_{s}=\frac{\sum_{i=1}^{M_{r}}\mathbb{P}\\{U_{s}=i\\}+\sum_{i=M_{r}+1}^{\infty}\mathbb{P}\\{U_{s}=i\\}\frac{M_{r}}{i}}{1-\mathbb{P}\\{U_{s}=0\\}}\tau_{f}.$
(32)
## VII Numerical Results
The numerical results are obtained according to both the analytical results we
have derived and Monte Carlo simulation. The default configurations of system
model are as follows (also see Table I). The total number of subchannels is
$M=20$ and the coverage radius of each femtocell is $R_{f}=10$m. The transmit
power of FAP is $P_{f}=13$dBm, and that of MBS is $P_{m}=39$dBm. We set the
path loss exponent as $\alpha=4$, with all links experiencing Rayleigh fading
of normalized $\mu=1$. The wall penetration loss is set as $W=-6$dB. We focus
on the interference-limited regime, and for simplicity we ignore the noise
power (i.e., $\sigma^{2}=0$). The intensity of MBSs is set as
$\lambda_{m}=0.00001$ and of FAPs $\lambda_{f}=0.0001$. In the clustered case,
the intensity of parent process is set as $\lambda_{p}=0.00001$ and of FAPs in
each cluster $\lambda_{c}=0.00127$. By setting the radius of each cluster as
$R_{c}=50$m, we get the intensity of FAPs as $\lambda_{f}=0.0001$. So the
average coverage area of an MBS is roughly equal to a circle with a radius of
$180$m, and on average there are ten FAPs within the coverage area of an MBS.
Unless otherwise specified, the subscribers and inside nonsubscribers are
distributed within an FAP coverage are with intensities
$\lambda_{s}=\lambda_{\mathrm{in}}=0.015$. The intensity of outside
nonsubscribers is set as $\lambda_{\mathrm{out}}=0.0001$.
TABLE I: SYSTEM PARAMETERS Symbol | Description | Value
---|---|---
$\Phi_{m},\Phi_{f}$ | Point processes defining the MBSs and FAPs | N/A
$\lambda_{m}$ | Density of MBSs | 0.00001 MBS/m2
$\lambda_{f}$ | Density of FAPs | 0.0001 FAP/m2
$\lambda_{p}$ | Density of parent process for the clustered FAPs | 0.00001 center/m2
$\lambda_{c}$ | Density of FAPs in each cluster | 0.00127 FAPs/m2
$R_{c}$ | Radius of each cluster | $50$m
$R_{f}$ | Radius of femtocell | $10$m
$\lambda_{\mathrm{out}}$ | Densities of outside nonsubscribers | 0.0001 user/m2
$\lambda_{s},\lambda_{\mathrm{in}}$ | Densities of subscribers and inside nonsubscribers | 0.015 user/m2
$P_{m},P_{f}$ | Transmit power at MBS and FAP | $39$dBm,$13$dBm
$M$ | Number of subchannels at each access point | $20$
$M_{s},M_{r}$ | Number of subchannels shared and reserved by each femtocell | Not fixed
$\alpha$ | Path loss exponent | $4$
$\mu$ | Rayleigh fading parameter | $1$ (normalized)
$W$ | Wall penetration loss | $-6$dB
$\sigma^{2}$ | Noise power | $0$ (interference-limited regime)
$P_{\mathrm{busy,m}},P_{\mathrm{busy,f}}$ | Probabilities that a given subchannel is used by an MBS and an FAP | Not fixed
$\Phi_{m}^{\prime},\Phi_{f}^{\prime}$ | Point processes defining MBSs and FAPs that interfere a given subchannel | N/A
$\lambda_{m}^{\prime},\lambda_{f}^{\prime}$ | Densities of MBSs and FAPs that interfere a given subchannel | Not fixed
$U_{\mathrm{out}},U_{\mathrm{in}}$ | Numbers of nonsubscribers that access a given MBS and a given FAP | Not fixed
$U_{s}$ | Numbers of subscribers that access a given FAP | Not fixed
$\tau_{f},\tau_{m}$ | Mean achievable rates of femtocell UEs and macrocell UEs | Not fixed
$\tau_{s},\tau_{n}$ | Mean achievable rates of subscribers and nonsubscribers | Not fixed
Figure 4 displays the SINR distributions of macrocell UEs and femtocell UEs,
when the number of shared subchannels is set as $M_{s}=10$. We plot in dotted
curves the analytical results, and we also plot the empirical cdfs obtained
from Monte Carlo simulation. The curves reveal that the simulation results
match the analytical results well, thus corroborating the accuracy of our
theoretical analysis. From the SINR distributions, we observe that while the
macrocell UEs experience a fair amount of interference, due to the shrinking
cell size, the interference for femtocell UEs is substantially alleviated. We
also observe that the performance of femtocell UEs is worse in the clustered
case than in the Poisson case when the intensity of the FAPs is set as the
same value; however, the performance of the macrocell UEs is just the reverse.
This result reveals that the gathering of FAPs leads to more interference to
femtocell UEs and at the same time reduces the chance that a macrocell UE
being interfered by the nearby FAPs.
Figure 4: Cdfs of SINRs for macrocell UEs and femtocell UEs, when $M_{s}=10$.
Figure 5 displays the mean achievable rates of macrocell UEs and femtocell UEs
as the outside nonsubscribers intensity $\lambda_{\mathrm{out}}$ increases. We
observe that the mean achievable rates of macrocell and femtocell UEs drop
initially and then tend to be stable. To interpret this behavior, we note that
as the intensity of outside nonsubscribers begins to increase, more
subchannels become occupied by MBSs, incurring more macrocell tier
interference; however, when the intensity of outside nonsubscribers is
sufficiently large, almost all the subchannels are persistently occupied by
MBSs with the UEs served by time-sharing, and then the interference saturates
thus leading to stable performance for UEs. In the case when the FAPs are
clustered, we observe that the mean achievable rate of femtocell UEs only
mildly decreases with the increasing of $\lambda_{\mathrm{out}}$, suggesting
that the performance of femtocell UEs is mainly limited by the interference
from the nearby FAPs and has little correlation with the intensity of
interfering MBSs.
Figure 5: Mean achievable rates of macrocell UEs and femtocell UEs as
functions of the intensity of outside nonsubscribers $\lambda_{\mathrm{out}}$.
Figure 6 displays the mean achievable rates of nonsubscribers and subscribers
as functions of the number of shared subchannels, in which the rate of
nonsubscribers is averaged over both outside and inside nonsubscribers. When
few subchannels are to be shared by each FAP (i.e., small $M_{s}$), the mean
achievable rate of nonsubscribers is small while subscribers enjoy a good
spectral efficiency. On the contrary, when most of the subchannels are shared
to nonsubscribers, the mean achievable rate of subscribers deteriorates
seriously. Nevertheless, we observe from the figure that there exists a stable
compromise at which the rates of both subscribers and nonsubscribers do not
drop much from their maxima. For the default configuration in our numerical
study, the stable compromise in the PPP case as well as in the clustered case
occurs when the value of $M_{s}$ lies in the range of $[7,12]$, corresponding
to a reasonably wide tuning range for system designer when provisioning the
resource.
Figure 6: Performance of nonsubscribers and subscribers as a function of the
number of shared subchannals $M_{s}$.
Figure 7 displays the mean achievable rates of nonsubscribers and subscribers
with different proportions of inside nonsubscribers and subscribers in
femtocells. For a fair comparison, we fix the sum intensity of the two types
of femtocell UEs, as $\lambda_{s}+\lambda_{\mathrm{in}}=0.03$. Figure 7(a)
gives the performance in the case when most of the femtocell UEs are
nonsubscribers, while Figure 7(b) corresponds to the opponent case. From the
curves, we observe that in order to achieve a good performance for both types
of UEs, the number of shared subchannels $M_{s}$ should be adjusted based on
the intensity of inside nonsubscribers. Moreover, we also find that the tuning
range of $M_{s}$ in the PPP case is almost the same as that in the clustered
case; thus illustrating that it is the load of the network rather than the
spatial distribution of FAPs that contributes the major impact on the choice
of $M_{s}$.
(a) $\lambda_{s}=0.003$ and $\lambda_{\mathrm{in}}=0.027$.
(b) $\lambda_{s}=0.027$ and $\lambda_{\mathrm{in}}=0.003$.
Figure 7: Performance of nonsubscribers and subscribers with changing
proportions of inside nonsubscribers and subscribers.
## VIII Conclusions
In this paper, we explored the application of stochastic geometry in the
analysis of hybrid access mechanisms for multi-channel two-tier networks,
focusing on the evaluation of the tradeoff between nonsubscribers and
subscribers. We characterized several key statistics of UEs and subchannels,
and established SINR distribution and mean achievable rate for each type of
UEs.
Our analysis revealed the interaction among the various parameters in the
network model, and thus shed useful insights into the choice of network
parameters and the provisioning of resource in system design. From our
numerical study, we observe that although there is an apparent conflict
between the interests of nonsubscribers and subscribers, there usually exists
a reasonably wide tuning range over which nonsubscribers and subscribers
attain a stable compromise at which the rates of both subscribers and
nonsubscribers do not drop much from their maxima. We also found that although
the spatial distributions of FAPs are different, the tuning range is almost
the same when the intensities of different types of UEs are fixed.
## Appendix A
Assume that the typical marcocell UE is located at the origin $o$, and let $r$
be the distance between it and its serving MBS. Since an outside nonsubscriber
always chooses its nearest MBS to access, the cdf of $r$ is obtained by
$\displaystyle\mathbb{P}\\{r\leq R\\}$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\\{\mathrm{no\ MBS\ closer\ than\ }R\\}$ (33) $\displaystyle=$
$\displaystyle 1-e^{-\lambda_{m}\pi R^{2}}.$
Then, the pdf of $r$ is $f(r)=e^{-\lambda_{m}\pi r^{2}}2\pi\lambda_{m}r$.
Assuming that the considered UE is at distance $r$ from its serving MBS,
$g_{\cdot}$ is the fading of an interference link, and $R_{\cdot}$ is the
distance between the UE and an interfering access point, the SINR experienced
by the UE is $\mathrm{SINR}=\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}$,
where
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
is the interference from the macrocell tier (excluding the serving MBS itself
which is denoted by $b_{0}$) and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$ is the
interference from the femtocell tier with the wall penetration loss taken into
account.
Thus, the cdf of the SINR is given by
$\displaystyle Z_{m}(T)=1-\mathbb{P}\\{\mathrm{SINR}>T\\}$ (34)
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}>T\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}h>\frac{Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{E}\Big{\\{}e^{-\frac{\mu
Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})}\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{m}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{m}}\Big{)}\mathrm{d}r.$
Now, we evaluate the Laplace transform for the interference conditioning on
the fact that the typical UE is served by the nearest MBS $b_{0}$ which is at
the distance $r$. First, we derive the Laplace transform for the interference
of MBSs, denoted as $I_{m}$.
$\displaystyle\mathcal{L}_{I_{m}}(s)$ $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{\\{}\exp(-s\\!\\!\\!\\!\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!P_{m}g_{i}R_{i}^{-\alpha})\Big{\\}}$
(35) $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!E_{g_{i}}\\{\exp(-sg_{i}R_{i}^{-\alpha}P_{m})\\}\Big{)}$
$\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!\mathcal{L}_{g}(sR_{i}^{-\alpha}P_{m})\Big{)},$
The probability generating functional of PPP $\Phi$ in the region $D$ with
intensity $\lambda$, denoted by
$G_{p}(v)=\mathbb{E}\\{\prod_{x\in\Phi}v(x)\\}$, is given by [19] as follows
$G_{p}(v)=\exp\Big{(}-\lambda\int_{D}(1-v(x))\mathrm{d}x\Big{)}.$ (36)
Let $C(o,r)$ be the circle centered at the origin $o$ with radius $r$. As
there is no MBS in the circle $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. This implies that the interfering MBSs
are distributed as PPP on the space $R^{2}$ exclusive of the region $C(o,r)$.
Let $D=R^{2}\setminus C(o,r)$ and $v(x)=\mathcal{L}_{g}(s|x|^{-\alpha}P_{m})$,
by applying the probability generating functional of PPP we get
$\displaystyle\mathcal{L}_{I_{m}}(s)=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\int_{r}^{\infty}(1-\mathcal{L}_{g}(sx^{-\alpha}P_{m}))x\mathrm{d}x\Big{)}$
$\displaystyle=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\underbrace{\int_{0}^{\infty}\\!\\!\int_{r}^{\infty}\\!\\!(1-e^{-sx^{-\alpha}P_{m}g})f(g)x\mathrm{d}x\mathrm{d}g}_{(A)}\Big{)}.$
(37)
where the last equation follows from the exchange of the integral order.
Let $y=sP_{m}gx^{-\alpha}$ and integrate by parts, from the properties of
Gamma function we obtain
$\displaystyle(A)=-\frac{r^{2}}{2}+\frac{1}{\alpha}(sP_{m})^{\frac{2}{\alpha}}\mathbb{E}_{g}\Big{\\{}$
$\displaystyle
g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},sP_{m}gr^{-\alpha})-\Gamma(-\frac{2}{\alpha})\Big{)}\Big{\\}}.$
(38)
The evaluation of the Laplace transform for $I_{f}$ is almost the same except
that the interfering FAPs are distributed on the whole space $R^{2}$. Similar
to (37), we get
$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\exp\Big{(}-2\pi\lambda_{f}^{\prime}\int_{0}^{\infty}\\!\\!(1-\mathcal{L}_{g}(sx^{-\alpha}WP_{f}))x\mathrm{d}x\Big{)}$
(39) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}(sWP_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}\Big{\\{}g^{\frac{2}{\alpha}}\Big{\\}}.$
Substituting $\mathcal{L}_{I_{m}}(s)$ and $\mathcal{L}_{I_{f}}(s)$ into
$Z_{m}(T)$ hence leads to (10).
Now we evaluate the mean achievable rate. Since for a positive random variable
$X$, $\mathbb{E}\\{X\\}=\int_{t>0}\mathbb{P}\\{X>t\\}\mathrm{d}t$, we have
$\displaystyle\tau_{m}$ $\displaystyle=$
$\displaystyle\mathbb{E}\\{\ln(1+\mathrm{SINR})\\}$ (40) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\ln(1+\mathrm{SINR})>t\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\mathrm{SINR}>e^{t}-1\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}(1-Z_{m}(e^{t}-1))\mathrm{d}t.$
Plugging $Z_{m}(T)$ into (40), we arrive at (11) and thus establish Theorem 1.
## Appendix B
Assume that the typical UE is located at the origin $o$. Let $r$ be the
distance between a femtocell UE and its serving FAP. Because the femtocell UEs
are uniformly distributed in the circular coverage area of radius $R_{f}$ of
each FAP, the pdf of $r$ is given by $f(r)={2r}/{R_{f}^{2}}$.
Denote by $I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$
the interference strengths from MBSs and FAPs respectively. Similar to the
derivation of (34), we have
$Z_{f}(T)=1-\int_{0}^{R_{f}}\frac{2r}{R_{f}^{2}}e^{-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{f}}\Big{)}\mathrm{d}r.$ (41)
Since $\Phi_{m}^{\prime}$ is a homogeneous PPP with intensity
$\lambda_{m}^{\prime}$, we obtain the Laplace transform for $I_{m}$ similar as
the derivation of (39)
$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}(sWP_{m})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(42)
The FAPs are distributed as a homogeneous PPP; however, the serving FAP is not
included when calculating the interference. By the Slivnyak-Mecke Theorem, the
reduced Palm distribution of the Poisson p.p. is equal to its original
distribution. Thus, the Laplace transform for $I_{f}$ can still be obtained
similar as the derivation of (39)
$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{f}}\big{\\}}=\exp\Big{(}-\pi\lambda_{f}^{\prime}(sW^{2}P_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(43)
In the above, it is noteworthy that the interference from an FAP penetrates
two walls thus the loss becoming $W^{2}$ instead of $W$. Substituting the
Laplace transform for $I_{m}$ and $I_{f}$ into (41) with $v=r^{2}$, we get the
SINR distribution, and similar to (40), we get the mean achievable rate.
## Appendix C
The derivation is exactly the same as Appendix A till the equation (34). The
essential distinction of the derivation lies in that the Laplace transform for
the interference is different from that in the PPP case. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
and $I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$. Since
there is no MBS in the disk $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. Referring to the previous derivation in
the PPP case, we obtain the Laplace transform for $I_{m}$ as follows
$\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}r^{2}\varphi\Big{(}\frac{sP_{m}}{\mu
r^{\alpha}},\alpha\Big{)}\Big{)}$ (44)
The FAPs are distributed as a Neyman-Scott cluster process and the generating
functional $G(v)=\mathbb{E}(\prod_{x\in\Phi}v(x))$ is given by [19, Page 157]
$\displaystyle G(v)=\exp\Big{(}-\lambda_{p}\int_{R^{2}}\Big{(}1-$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x+y))\mathrm{d}y\Big{)}\Big{)}\mathrm{d}x\Big{)}$
(45)
Similar to the derivation of (35), we get the Laplace transform for $I_{f}$
$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\mathcal{L}_{g}(sR_{j}^{-\alpha}WP_{f})\Big{\\}}$
(46) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\frac{\mu}{\mu+sR_{j}^{-\alpha}WP_{f}}\Big{\\}}$
Let $v(x)=\frac{\mu}{\mu+sP_{f}W|x|^{-\alpha}}$ and plugging into the
generating functional of Neyman-Scott cluster process (45), we get the Laplace
transform for $I_{f}$. Having derived the Laplace transform for $I_{m}$ and
$I_{f}$, similar to the derivations in Appendix A, we obtain the results in
Theorem 3.
## Appendix D
Different from the above proofs, we assume that the serving FAP rather than
the typical UE is located at the origin. The typical UE is distributed in the
circle centered at the origin with radius $R_{f}$. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$.
First, we evaluate the Laplace transform for the interference conditioned on
the fact that the typical UE is located at distance $r$ from the serving FAP
located at the origin. Without loss of generality, we assume that the typical
UE is located at $z=(r,0)$. Since $\Phi_{m}^{\prime}$ is a homogeneous PPP
with intensity $\lambda_{m}^{\prime}$, we obtain the Laplace transform for
$I_{m}$ similar as the derivation of (39)
$\\!\\!\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}\Big{(}\frac{sWP_{m}}{\mu}\Big{)}^{\frac{2}{\alpha}}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})\Big{)}$
(47)
The FAPs are distributed as a Neyman-Scott cluster process. Since the serving
FAP, located at the origin, is not included when calculating the interference,
we should consider the reduced Palm distribution of the cluster process when
evaluating the Laplace transform for $I_{f}$. Let
$G_{o}^{!}(v)=\mathbb{E}_{o}^{!}(\prod_{x\in\Phi}v(x))$ denotes the generating
functional of the reduced Palm distribution of the cluster process. The
notation $\mathbb{E}_{o}^{!}(\cdot)$ denotes the conditional expectation for
the point process given that there is a point of the process at the origin but
without including the point. The conditional generating functional
$G_{o}^{!}(v)$ is given by the Lemma 1 in [17] as follows
$\displaystyle G_{o}^{!}(v)=\frac{1}{\pi R_{c}^{2}}G(v)\int_{C(o,R_{c})}$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x-y))\mathrm{d}x\Big{)}\mathrm{d}y$
(48)
where $G(v)$ is the generating functional of Neyman-Scott cluster process and
is given by (45).
Referring to (46), let $v(x)=\frac{\mu}{\mu+sP_{f}W|x-z|^{-\alpha}}$. Plugging
$v(x)$ into the conditional generating functional of Neyman-Scott cluster
process (48), we get the Laplace transform for $I_{f}$. Having derived the
Laplace transform for $I_{m}$ and $I_{f}$, similar to the derivation in
Appendix B, we obtain the results in Theorem 4.
## Acknowledgement
The authors wish to thank the anonymous reviewers for their valuable comments
on this work.
## References
* [1] G. Mansfield, “Femtocells in the US Market-Business Drivers and Consumer Propositions,” _Femtocells Europe_ , pp. 1927–1948, 2008.
* [2] V. Chandrasekhar, J. Andrews, and A. Gatherer, “Femtocell networks: a survey,” _IEEE Communications Magazine_ , vol. 46, no. 9, pp. 59–67, 2008\.
* [3] H. Claussen, L. Ho, and L. Samuel, “An overview of the femtocell concept,” _Bell Labs Technical Journal_ , vol. 13, no. 1, pp. 221–245, 2008.
* [4] Nortel, Vodafone, “Open and closed access for home NodeBs,” _3GPP document reference R4-071231_ , 2007.
* [5] Vodafone, “Open and semi-open access support for UTRA Home NB,” _3GPP-TSG RAN, Prague, Czech Republic, Tech. Rep. R2-085280_ , 2008.
* [6] G. De La Roche, A. Valcarce, D. López-Pérez, and J. Zhang, “Access control mechanisms for femtocells,” _IEEE Communications Magazine_ , vol. 48, no. 1, pp. 33–39, 2010.
* [7] A. Valcarce, D. López-Pérez, G. De La Roche, and J. Zhang, “Limited access to OFDMA femtocells,” in _Proc. IEEE 20th Int. Symp. Personal, Indoor and Mobile Radio Commun. (PIMRC)_ , 2009.
* [8] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 27, no. 7, pp. 1029–1046, 2009.
* [9] J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” _IEEE Transactions on Communications_ , no. 99, pp. 1–13, 2010.
* [10] V. Chandrasekhar and J. Andrews, “Spectrum allocation in tiered cellular networks,” _IEEE Transactions on Communications_ , vol. 57, no. 10, pp. 3059–3068, 2009.
* [11] ——, “Uplink capacity and interference avoidance for two-tier femtocell networks,” _IEEE Transactions on Wireless Communications_ , vol. 8, no. 7, pp. 3498–3509, 2009.
* [12] W. Cheung, T. Quek, and M. Kountouris, “Throughput optimization, spectrum allocation, and access control in two-tier femtocell networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 30, no. 3, pp. 561–574, 2012\.
* [13] C. Yin, C. Chen, T. Liu, and S. Cui, “Generalized results of transmission capacities for overlaid wireless networks,” _2009 IEEE International Symposium on Information Theory (ISIT)_ , pp. 1774–1778, 2009.
* [14] R. Vaze, “Transmission Capacity of Spectrum Sharing Ad Hoc Networks with Multiple Antennas,” _IEEE Transactions on Wireless Communications_ , vol. 10, no. 7, pp. 2334–2340, 2011.
* [15] P. Xia, V. Chandrasekhar, and J. Andrews, “Open vs. closed access femtocells in the uplink,” _IEEE Transactions on Wireless Communications_ , vol. 9, no. 12, pp. 3798–3809, 2010.
* [16] H. Jo, P. Xia, and J. Andrews, “Open, closed, and shared access femtocells in the downlink,” _EURASIP Journal on Wireless Communications and Networking_ , no. 1, pp. 1–16, 2012.
* [17] R. Ganti and M. Haenggi, “Interference and outage in clustered wireless ad hoc networks,” _IEEE Transactions on Information Theory_ , vol. 55, no. 9, pp. 4067–4086, 2009.
* [18] K. Gulati, B. Evans, J. Andrews, and K. Tinsley, “Statistics of co-channel interference in a field of poisson and poisson-poisson clustered interferers,” _IEEE Transactions on Signal Processing_ , vol. 58, no. 12, pp. 6207–6222, 2010.
* [19] D. Stoyan, W. Kendall, J. Mecke, and D. Kendall, _Stochastic geometry and its applications_. Wiley New York, 1995\.
* [20] J. Ferenc and Z. Néda, “On the size distribution of Poisson Voronoi cells,” _Physica A: Statistical Mechanics and its Applications_ , vol. 385, no. 2, pp. 518–526, 2007.
|
arxiv-papers
| 2011-08-05T05:55:55 |
2024-09-04T02:49:21.316045
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi Zhong, Wenyi Zhang",
"submitter": "Yi Zhong",
"url": "https://arxiv.org/abs/1108.1257"
}
|
1108.1262
|
¡!DOCTYPE HTML PUBLIC ”-//W3C//DTD HTML 4.01 Transitional//EN”
”http://www.w3.org/TR/html4/loose.dtd”¿ ¡html¿ ¡head¿ ¡meta http-
equiv=”Content-Type” content=”text/html; charset=iso-8859-15”¿ ¡title¿1st
International Workshop on Complex Systems in Sports¡/title¿ ¡/head¿
¡body¿ ¡h1¿1st International Workshop on Complex Systems in Sports -
Proceedings ¡/h1¿
¡p¿¡a href=’http://cs-sports.ugr.es/’ title=’Workshop site’¿The first CS-
Sports workshop will take place in August 12th, 2011¡/a¿, in conjunction with
¡a href=’http://ecal2011.org’ title=’Whole conference site’¿the European
Conference on Artificial Life¡/a¿. This is the first edition, and we had 5
submissions, all of them accepted after revision by two reviewers and the
organizing committee; these are the full-length version of those
abstracts¡/a¿.
¡h2¿List of accepted papers¡/h2¿
LIST:1108.0779
LIST:1108.0782
LIST:1107.5474
LIST:1108.1065
LIST:1108.0261
¡/body¿ ¡/html¿
|
arxiv-papers
| 2011-08-05T07:30:45 |
2024-09-04T02:49:21.322229
|
{
"license": "Public Domain",
"authors": "Juan Juli\\'an Merelo Guerv\\'os, Carlos Cotta, Antonio M. Mora",
"submitter": "Juan Juli\\'an Merelo-Guerv\\'os Pr.",
"url": "https://arxiv.org/abs/1108.1262"
}
|
1108.1323
|
# Sub-barrier capture with quantum diffusion approach
V.V.Sargsyan1, R.A.Kuzyakin1, G.G.Adamian1, N.V.Antonenko1, W.Scheid2 and
H.Q.Zhang3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
With the quantum diffusion approach the behavior of capture cross sections and
mean-square angular momenta of captured systems are revealed in the reactions
with deformed and spherical nuclei at sub-barrier energies. With decreasing
bombarding energy under the barrier the external turning point of the nucleus-
nucleus potential leaves the region of short-range nuclear interaction and
action of friction. Because of this change of the regime of interaction, an
unexpected enhancement of the capture cross section is found at bombarding
energies far below the Coulomb barrier. This effect is shown its worth in the
dependence of mean-square angular momentum on the bombarding energy. From the
comparison of calculated capture cross sections and experimental capture or
fusion cross sections the importance of quasifission near the entrance channel
is demonstrated for the actinide-based reactions and reactions with medium-
heavy nuclei at extreme sub-barrier energies.
###### pacs:
25.70.Ji, 24.10.Eq, 03.65.-w
Key words: astrophysical $S$-factor; dissipative dynamics; sub-barrier capture
## I Introduction
The measurement of excitation functions down to the extreme sub-barrier energy
region is important for studying the long range behavior of nucleus-nucleus
interaction as well as the coupling of relative motion with other degrees of
freedom BeckermanNi58Ni64 ; ZhangOth ; ScarlassaraNi58Zr94 ; ZhangOU ; Og ;
ZuhuaFTh ; Nadkarni ; Mor ; trotta ; Ji1 ; Tr1 ; DasguptaS32Pb208 ; NishioOU ;
Ji22 ; Ji2 ; NishioSiU ; Vino ; Dg ; Ca2 ; NishioSU ; StefaniniS36Ca48 ;
HindeSTh ; Shri ; ItkisSU ; akn ; Nishionew ; Siga ; MontagnoliS36Ni64 . The
experimental data obtained are of interest for solving astrophysical problems
related to nuclear synthesis. Indications for an enhancement of the
$S$-factor, $S=E_{\rm c.m.}\sigma\exp(2\pi\eta)$ Zvezda ; Zvezda2 , where
$\eta(E_{\rm c.m.})=Z_{1}Z_{2}e^{2}\sqrt{\mu/(2\hbar^{2}E_{\rm c.m.})}$ is the
Sommerfeld parameter, at energies $E_{\rm c.m.}$ below the Coulomb barrier
have been found in Refs. Ji1 ; Ji2 ; Dg . Its origin is still under
discussion.
From the comparison of capture cross sections and fusion cross sections one
can show a significant role of the quasifission channel in the reactions with
various medium-light and heavy nuclei at sub-barrier energies. The competition
between the complete fusion and quasifission can strongly reduce the value of
the fusion cross section and, respectively, the value of the evaporation
residue cross section Volkov ; nasha ; Avaz . This effect is especially
crucial in the production of superheavy nuclei. It worth remembering that
first evidences of hindrance for compound nucleus formation in the reactions
with massive nuclei at low energies near the Coulomb barrier were observed at
GSI already long time ago GSI .
To clarify the behavior of capture and fusion cross sections at sub-barrier
energies, a further development of the theoretical methods is required Avaz ;
Gomes ; our ; Den2 . The conventional coupled-channel approach with realistic
set of parameters is not able to describe the fusion cross sections either
below or above the Coulomb barrier Dg . The use of a quite shallow nucleus-
nucleus potential Es with an adjusted repulsive core considerably improves
the agreement between the calculated and experimental data. Besides the
coupling with collective excitations, the dissipation, which is simulated by
an imaginary potential in Ref. Es or by damping in each channel in Ref. Hag1
, seems to be important. The quantum diffusion approach based on the quantum
master-equation for the reduced density matrix has been suggested in Ref.
EPJSub for the describing capture process. The collisions of nuclei are
treated in terms of a single collective variable: the relative distance
between the colliding nuclei. Our approach takes into consideration the
fluctuation and dissipation effects in collisions of heavy ions which model
the coupling with various channels. In the present paper the capture model
EPJSub is applied.
Figure 1: The calculated (lines) capture cross section (upper part), average
angular momenta of captured system (middle part) versus $E_{\rm c.m.}$, and
partial capture cross sections (lower part) versus $J$ at $E_{\rm c.m.}$=65
(dash-dot-dotted line), 69.5 (dash-dotted line), 73 (dotted line), 83 (dashed
line), and 100 (solid line) MeV for the 16O+208Pb reaction with the spherical
nuclei. The experimental cross sections marked by closed squares, circles,
rhombus, stars are from Refs. Dg ; Mor ; Tr1 ; Og , respectively. The
experimental values of $\langle J^{2}\rangle$ (solid squares) are taken from
Ref. Vand . The value of the Coulomb barrier $V_{b}$ is indicated by arrow.
Figure 2: The calculated (solid lines) capture cross section versus $E_{\rm
c.m.}$ for the reactions 48Ca+208Pb (upper part) and 34S+238U (lower part).
The experimental cross sections are taken from Refs. Ca1 (closed squares and
triangle), Ca2 (open squares), and Nishionew (open circles). The value of
the Coulomb barrier $V_{b}$ is indicated by arrow. The static quadrupole
deformation parameters are: $\beta_{1}$(48Ca)=$\beta_{2}$(208Pb)=0,
$\beta_{1}$(34S) =0.125, and $\beta_{2}$(238U)=0.286 Ram .
Figure 3: The same as in Fig. 2, but for the reactions for the reactions 16O
+ 232Th and 4He + 238U. The experimental data in the upper part are taken from
Refs. BackOTh (open triangles), ZhangOth (closed triangles), MuakamiOTh
(open squares), KailasOTh (closed squares), ZuhuaFTh (open stars) and
Nadkarni (closed stars). The fission cross sections from Refs. trotta and
ViolaOU are shown in the lower part by open circles and solid squares,
respectively. The value of the Coulomb barrier $V_{b}$ for the spherical
nuclei is indicated by arrow. The dashed curve represents the calculation with
the Wong formula Wong . The static quadrupole deformation parameters are:
$\beta_{2}$(232Th)=0.261 $\beta_{2}$(238U)=0.286 Ram and $\beta_{1}$(
4He)=$\beta_{1}$(16O)=0.
Figure 4: The same as in Fig. 2, but for the reactions 16O + 238U and 36S +
238U. The experimental cross sections are taken from Refs. NishioOU (open
triangles), TokeOU (closed triangles), ZuhuaFTh (open squares), ZhangOU
(closed squares), ViolaOU (open stars), ItkisSU (closed stars), and NishioSU
(rhombuses). The dashed curve represents the calculation with the Wong formula
Wong . The static quadrupole deformation parameters are:
$\beta_{2}$(238U)=0.286 and $\beta_{1}$(16O)=$\beta_{1}$(36S)=0.
Figure 5: The calculated mean-square angular momenta versus $E_{\rm c.m.}$
for the reactions 16O + 232Th,238U are compared with experimental data
ZuhuaFTh . The dashed curve represents the calculation by the Wong-type
formula Wong .
## II Comparison with experimental data and predictions
One can see in Figs. 1–4 that with decreasing $E_{\rm c.m.}$ up to about 3 -
18 MeV below the Coulomb barrier the regime of interaction is changed because
at the external turning point the colliding nuclei do not reach the region of
nuclear interaction where the friction plays a role. As result, at smaller
$E_{\rm c.m.}$ the capture cross sections $\sigma_{cap}$ fall with a smaller
rate. Therefore, an effect of the change of fall rate of sub-barrier capture
cross section should be in the data if we assume that the friction starts to
act only when the colliding nuclei approach the barrier. Note that at sub-
barrier energies the experimental data have still large uncertainties to make
a firm experimental conclusion about this effect. The effect seems to be more
pronounced in collisions of spherical nuclei, where the regime of interaction
is changed at $E_{\rm c.m.}$ up to about 3.5 - 5 MeV below the Coulomb barrier
EPJSub .
The calculated mean-square angular momenta $\langle J^{2}\rangle$ of captured
systems versus $E_{\rm c.m.}$ are presented in Figs. 1 and 5 for the reactions
16O+208Pb and 16O + 232Th,238U. At energies below the barrier $\langle
J^{2}\rangle$ has a minimum. This minimum depends on the deformations of
nuclei and on the factor $Z_{1}\times Z_{2}$. For the reactions 16O +
232Th,238U, these minima are about 7 – 8 MeV below the corresponding Coulomb
barriers, respectively. The experimental data Vand indicate the presence of
the minimum as well. On the left-hand side of this minimum the dependence of
$\langle J^{2}\rangle$ on $E_{\rm c.m.}$ is rather weak. Note that the found
behavior of $\langle J^{2}\rangle$, which is related to the change of the
regime of interaction between the colliding nuclei, would affect the angular
anisotropy of the products of fission-like fragments following capture.
Indeed, the values of $\langle J^{2}\rangle$ are extracted from data on
angular distribution of fission-like fragments akn .
The agreement between the experimental $\langle J^{2}\rangle$ and those
calculated with Wong-type formula is rather bad. At energies below the barrier
the $\langle J^{2}\rangle$ has no a minimum (see Fig. 5). However, for the
considered reactions the saturation values of $\langle J^{2}\rangle$ are close
to those obtained in our formalism.
In Fig. 6 the calculated astrophysical $S$–factor versus $E_{\rm c.m.}$ is
shown for the 16O+238U reaction. The $S$-factor has a maximum which is seen in
experiments Ji1 ; Ji2 ; Es . After this maximum $S$-factor slightly decreases
with decreasing $E_{\rm c.m.}$ and then starts to increase. This effect seems
to be more pronounced in collisions of spherical nuclei. The same behavior has
been revealed in Refs. LANG by extracting the $S$-factor from the
experimental data.
In Fig. 6, the so-called logarithmic derivative, $L(E_{\rm
c.m.})\break=d(\ln(E_{\rm c.m.}\sigma_{cap}))/dE_{\rm c.m.},$ and the barrier
distribution$d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm c.m.}^{2}$ are presented
for the 16O+238U reaction. The logarithmic derivative strongly increases below
the barrier and then has a maximum at $E_{\rm c.m.}\approx
V_{b}^{orient}$(sphe-re-pole)-3 MeV (at $E_{\rm c.m.}\approx V_{b}$-3 MeV for
the case of spherical nuclei). The maximum of $L$ corresponds to the minimum
of the $S$-factor. The barrier distributions calculated with an energy
increment 0.2 MeV have only one maximum at $E_{\rm c.m.}\approx
V_{b}^{orient}$(sphere-sphere)$=V_{b}$ as in the experiment DH . With an
increasing increment the barrier distribution is shifted to lower energies.
Assuming a spherical target nucleus in the calculations, we obtain a more
narrow barrier distribution.
Figure 6: The calculated values of the astrophysical $S$-factor with
$\eta_{0}=\eta(E_{\rm c.m.}=V_{b})$ (middle part), the logarithmic derivative
$L$ (upper part) and the fusion barrier distribution $d^{2}(E_{\rm
c.m.}\sigma_{cap})/dE_{\rm c.m.}^{2}$ (lower part) for the 16O+238U reaction.
The value of $L$ calculated with the assumption of
$\beta_{1}$(16O)=$\beta_{2}$(238U)=0 is shown by a dashed line. The solid and
dotted lines show the values of $d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm
c.m.}^{2}$ calculated with the increments 0.2 and 1.2 MeV, respectively. The
closed squares are the experimental data of Ref. DH .
Figure 7: The same as in Fig. 4, but for the 48Ca + 232Th,238U reactions. The
excitation energies $E^{*}_{CN}$ of the corresponding nuclei are indicated.
The experimental data are taken from Refs. Itkis1 (marked by squares) and
Shen (marked by circles). The static quadrupole deformation parameters are:
$\beta_{2}$(238U)=0.286, $\beta_{2}$(232Th)=0.261, and $\beta_{1}$(48Ca)=0.
## III Capture cross sections in reactions with large fraction of
quasifission
In the case of large values of $Z_{1}\times Z_{2}$ the quasifission process
competes with complete fusion at energies near barrier and can lead to a large
hindrance for fusion, thus ruling the probability for producing superheavy
elements in the actinide-based reactions nasha ; trota . Since the sum of the
fusion cross section $\sigma_{fus}$, and the quasifission cross section
$\sigma_{qf}$ gives the capture cross section,
$\sigma_{cap}=\sigma_{fus}+\sigma_{qf},$
and $\sigma_{fus}\ll\sigma_{qf}$ for actinide-based reactions 48Ca + 232Th,
238U,244Pu,246,248Cm and 50Ti + 244Pu nasha , we have
$\sigma_{cap}\approx\sigma_{qf}.$
In a wide mass-range near the entrance channel, the quasifission events
overlap with the products of deep-in-elastic collisions and can not be firmly
distinguished. Because of this the mass region near the entrance channel is
taken out in the experimental analyses in Refs. Itkis1 ; Itkis2 . Thus, by
comparing the calculated and experimental capture cross sections one can study
the importance of quasifission near the entrance channel for the actinide-
based reactions leading to superheavy nuclei.
The capture cross sections for the quasifission reactions Itkis1 ; Itkis2 ;
Shen are shown in Figs. 7 and 8. One can observe a large deviations of the
experimental data of Refs. Itkis1 ; Itkis2 from the the calculated results.
The possible reason is an underestimation of the quasifission yields measured
in these reactions. Thus, the quasifission yields near the entrance channel
are important. Note that there are the experimental uncertainties in the
bombarding energies.
One can see in Fig. 9 that the experimental and the theoretical cross sections
become closer with increasing bombarding energy. This means that with
increasing bombarding energy the quasifission yields near the entrance channel
mass-region decrease with respect to the quasifission yields in other mass-
regions. As seen in Fig. 9, the quasifission yields near the entrance channel
mass-region increase with $Z_{1}\times Z_{2}$.
Figure 8: The same as in Fig. 7, but for the indicated 48Ca,50Ti + 244Pu
reactions. The experimental data are from Refs. Itkis2 (squares) and Itkis1
(circles). The static quadrupole deformation parameters are:
$\beta_{2}$(244Pu)=0.293, and $\beta_{1}$(48Ca)=$\beta_{1}$(50Ti)=0.
Figure 9: The ratio of theoretical and experimental capture cross sections
versus the excitation energy $E_{\rm c.m.}$ of the compound nucleus for the
reactions 48Ca+238U (closed stars), 48Ca+244Pu (closed triangles), 48Ca+246Cm
(closed squares), 48Ca+248Cm (closed circles), and 50Ti+244Pu (closed
rhombuses).
Figure 10: The calculated capture cross sections versus $E_{\rm c.m.}$ for
the indicated reactions. The experimental fusion cross sections marked by
closed and open squares are taken from Refs. StefaniniS36Ca48 ;
MontagnoliS36Ni64 , respectively. The values of the Coulomb barrier are
indicated by arrows. The static quadrupole deformation parameters are:
$\beta_{1}$(36S)=$\beta_{2}$(48Ca)=0 and $\beta_{2}$(64Ni)=0.087 Moel1 .
Figure 11: The calculated capture cross sections versus $E_{\rm c.m.}$ for
the indicated reaction. The experimental fusion cross sections marked by
closed squares and circles are taken from Refs. BeckermanNi58Ni64 ; Ji22 ,
respectively. Here, $\beta_{1,2}$(64Ni)=0.087.
## IV Origin of fusion hindrance in reactions with medium-mass nuclei at sub-
barrier energies
In Figs. 10 and 11 the calculated capture cross section are presented for the
reactions 36S + 48Ca,64Ni and 64Ni + 64Ni. The values of $V_{b}$ are adjusted
to the experimental data for the fusion cross sections shown as well. For the
systems mentioned above, the difference between the sub-barrier capture and
fusion cross sections becomes larger with decreasing bombarding energy $E_{\rm
c.m.}$. The same effect one can see for the 16O + 208Pb reaction. Assuming
that the estimated capture and the measured fusion cross sections are correct,
the small fusion cross section at energies well below the Coulomb barrier may
indicate that other reaction channel is open and the system evolves by other
reaction mechanism after the capture. The observed hindrance factor may be
understood in term of quasifission whose cross section should be added to the
one of fusion to obtain a meaningful comparison with the calculated capture
cross section. 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 quantal tunneling through the Coulomb barrier in a binary event with mass
and charge close to the entrance channel. In this sense the quasifission is
the general phenomenon which takes place in the reactions with the massive
Volkov ; nasha ; Avaz ; GSI , medium-mass and, probably, light nuclei. For the
medium-mass and light nuclei, this reaction mode has to be studied in the
future experiments: from the mass (charge) distribution measurements one can
show the distinct components due to quasifission. The low-energy experimental
data would probably provide straight information since the high-energy data
may be shaded by competing reaction processes such as quasifission and deep-
inelastic collisions. Note that the binary decay events were already observed
experimentally in Henning for the 58Ni + 124Sn reaction at energies below the
Coulomb barrier but assumed to be related to deep-inelastic scattering.
## V Summary
The quantum diffusion approach is applied to study the capture process in the
reactions with deformed and spherical nuclei at sub-barrier energies. Due to a
change of the regime of interaction (the turning-off of the nuclear forces and
friction) at sub-barrier energies, the curve related to the capture cross
section as a function of bombarding energy has smaller slope. This change is
also reflected in the functions $\langle J^{2}\rangle$, $L(E_{\rm c.m.})$, and
$S(E_{\rm c.m.})$. The mean-square angular momentum of captured system versus
$E_{\rm c.m.}$ has a minimum and then saturates at sub-barrier energies. This
behavior of $\langle J^{2}\rangle$ would increase the expected anisotropy of
the angular distribution of the products of fission and quasifission following
capture. The astrophysical factor has a maximum and a minimum at energies
below the barrier. The maximum of $L$-factor corresponds to the minimum of the
$S$-factor. The importance of quasifission near the entrance channel is shown
for the actinide-based reactions at near barrier energies and reactions with
medium-mass nuclei at extreme sub-barrier energies. One can suggest the
experiments to check these predictions.
We thank H. Jia and D. Lacroix for fruitful discussions and suggestions. We
are grateful to K. Nishio for providing us his experimental data. This work
was supported by DFG, NSFC, and RFBR. The IN2P3-JINR and Polish-JINR
Cooperation programs are gratefully acknowledged.
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|
arxiv-papers
| 2011-08-05T13:25:41 |
2024-09-04T02:49:21.330307
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V.Sargsyan, R.A.Kuzyakin, G.G.Adamian, N.V.Antonenko, W.Scheid and\n H.Q.Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1108.1323"
}
|
1108.1379
|
# Cold collision shift cancelation and inelastic scattering in a Yb optical
lattice clock
A. D. Ludlow, N. D. Lemke, J. A. Sherman, and C. W. Oates National Institute
of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA
ludlow@boulder.nist.gov G. Quéméner, J. von Stecher, and A. M. Rey JILA,
National Institute of Standards and Technology and University of Colorado,
Boulder, CO 80309-0440
###### Abstract
Recently, $p$-wave cold collisions were shown to dominate the density-
dependent shift of the clock transition frequency in a 171Yb optical lattice
clock. Here we demonstrate that by operating such a system at the proper
excitation fraction, the cold collision shift is canceled below the $5\times
10^{-18}$ fractional frequency level. We report inelastic two-body loss rates
for ${}^{3}\\!P_{0}\,$-${}^{3}\\!P_{0}$ and
${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$ scattering. We also measure interaction
shifts in an unpolarized atomic sample. Collision measurements for this
spin-1/2 171Yb system are relevant for high performance optical clocks as well
as strongly-interacting systems for quantum information and quantum simulation
applications.
###### pacs:
34.50.Cx; 42.62.Eh; 32.70.Jz; 42.62.Fi
††preprint: APS/123-QED
Large ensembles of ultracold atoms offer atomic clocks a measurement of the
atomic state with high signal-to-noise ratio. For clocks utilizing optical
transitions, this has the potential to yield time and frequency measurements
with new levels of precision and speed (e.g. Jiang et al. (2011); Takamoto et
al. (2011)). However, large ensembles of cold atoms can lead to high number
density and thus significant interatomic interactions. These interactions can
perturb the clock transition frequency, compromising the accuracy of the
atomic standard. For example, in cesium fountain primary standards, cold
collision shifts can become significant Gibble and Chu (1993); Leo et al.
(2001), influencing clock operation (e.g. Dos Santos et al. (2002); Heavner et
al. (2005); Szymaniec et al. (2007)).
Optical lattice clocks, which probe the ultra-narrow
${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$ transition in two-valence-electron atoms
held in an optical potential, are also susceptible to cold collisions. Non-
negligible collision effects have been observed in lattice clocks using
nuclear-spin-polarized, fermionic samples of 87Sr Ludlow et al. (2008);
Campbell et al. (2009) and 171Yb Lemke et al. (2009). Measurement and control
of these collisions therefore play a key role in the continued development of
these standards. At the same time, the control of these interactions are an
integral part of proposals for quantum information Hayes et al. (2007);
Gorshkov et al. (2009) and quantum simulation of solid-state-analog
Hamiltonians Gorshkov et al. (2010); Cazalilla et al. (2009); Foss-Feig et al.
(2010). For 87Sr, the primary interaction giving the cold collision shift was
identified as an $s$-wave interaction between atoms in non-identical
superpositions of the clock states Campbell et al. (2009); Rey et al. (2009);
Gibble (2009). It was recently observed that strong interactions can isolate
and suppress the shift Rey et al. (2009); Swallows et al. (2011).
For 171Yb, it was shown that a $p$-wave interaction between atoms in the
${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ electronic states was the dominant
mechanism responsible for the cold collision shift Lemke et al. . Unlike the
$s$-wave case, such an interaction is less sensitive to small particle
distinguishability between the ultracold fermions. Consequently the observed
shift exhibits roughly a linear dependence on the ${}^{3}\\!P_{0}$ excitation
fraction (Fig. 1(a)). Moreover, in the weakly interacting regime of the 1-D
lattice, the cold collision shift crossed zero at a mean excitation fraction
close to 0.5. In this work, we exploit this zero-crossing to demonstrate
cancelation of the 171Yb cold collision shift in a 1-D optical lattice below
the $5\times 10^{-18}$ fractional frequency level. The cancelation is enabled
by the fact that near 50$\%$ excitation, population of the two electronic
states of one atom induce equal collision shifts on the opposite states of
another atom, leaving zero net shift for the clock transition Footnote1 . The
collision shift cancelation can thus be likened to the Stark shift cancelation
which optical lattice clocks exploit by operating at the “magic” wavelength
Katori et al. (2003); Ye et al. (2008). Our uncertainty in this cancelation
reaches below the smallest total uncertainty levels reported to date for any
type of atomic clock, making it a powerful and pragmatic technique for
mitigating the collisional shift in a lattice clock.
The 171Yb ($I=1/2$) optical lattice clock is described in detail elsewhere
Lemke et al. ; Lemke et al. (2009). After dual laser-cooling stages, atoms are
loaded into a 1-D optical lattice using a laser at
$\lambda_{\text{magic}}\simeq 759$ nm. Atomic population, initially split
between the two ground magnetic substates, is optically pumped to a single
spin state using $\sigma$-polarized light resonant with the ${}^{1}\\!S_{0}$
($I=1/2$) - ${}^{3}P_{1}$ ($I=3/2$) transition in the presence of a bias
magnetic field (5 G). We estimate the average atomic density can reach
$\rho_{0}\approx 3\times 10^{11}$ atoms/cm3, with an atomic temperature of
$\sim 10$ $\mu$K. The atoms are then spectroscopically probed on the
${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$ transition using Ramsey spectroscopy. The
Ramsey pulse times used here are 5-10 ms, while the dark time is 80-150 ms. A
cavity-stabilized probe laser is actively stabilized to the center Ramsey
fringe using an acousto-optic modulator. The cold collision shift is the
measured frequency difference between interleaved atomic samples of high and
low density.
Figure 1: (color online) (a) Cold collision shift as a function of excitation
fraction, for a polarized sample in a 1-D optical lattice oriented vertically
(filled triangles) or horizontally (filled circles). Open squares are with an
unpolarized atomic sample. (b) Cold collision shift cancelation: the
measurement uncertainty for this cancelation is shown, given by the total
deviation (similar to two-sample Allan deviation Greenhall et al. (1999)). (c)
Measurement of the residual cold collision shift when operating at $51\%$
excitation and an atomic density of $\rho_{1}$. The red dashed line indicates
the weighted mean of the seven measurements, +2.5 mHz, while the solid lines
gives the one-$\sigma$ error bars of $\pm 2.4$ mHz.
Measurement of the collision shift in a 1-D lattice is shown as filled points
in Fig. 1(a), as a function of excitation fraction (during the Ramsey dark
time). The theoretical calculation of the shift using a $p$-wave model is also
shown (solid line). In the model, the dominant interaction, $V_{eg}$, is
between ${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ atoms, and a weaker interaction
between two excited state atoms, $V_{ee}=0.1\times V_{eg}$, is also included
Lemke et al. . The shift has a zero value near 50$\%$ excitation, and this
zero-crossing is the focus of our attention here. To determine the precise
excitation corresponding to zero shift, we made real-time measurements of the
excitation fraction by turning off the second Ramsey pulse and measuring the
ground and excited atomic populations after the dark time. These measurements
were interspersed between cycles of usual Ramsey spectroscopy. We then
slightly adjusted the probe laser power to keep the measured excitation
fraction constant during each shift measurement.
The precision of the zero-crossing measurement is given by the clock
instability during interleaved measurements of high and low atomic density.
Our measurement stability benefits from recent improvements to the cavity-
stabilized laser used to probe the clock transition Jiang et al. (2011). Fig.
1(b) shows the precision of one such measurement, where we observed a
collision shift of -1.4 (3) mHz after 14500 s of averaging. Fig. 1(c) shows
the result of seven similar measurements, taken sequentially over the course
of several weeks and with different measurement durations. For all but one
measurement, the mean excitation fraction was controlled to $51\pm 0.3\pm 1.3$
$\%$, where the first uncertainty is the fluctuation in the measured
excitation fraction and the second is the systematic uncertainty in the
absolute excitation value. (For data point number two, the excitation fraction
was set to $1\%$ higher; we thus applied a $-5$ mHz correction to the measured
collision shift, as determined from the slope of the curve in Fig. 1(a).) The
weighted mean of the seven measurements (red lines) is 2.5 (2.4) mHz (reduced
$\chi^{2}=1.04$). This corresponds to a fractional shift of 4.8 (4.6) $\times
10^{-18}$ of the transition frequency, and demonstrates the smallest
measurement of a collision shift in a lattice clock.
In order to routinely implement this collision shift cancelation, we consider
the robustness of this technique to relevant experimental conditions. For an
operational density of $\rho_{1}\simeq 3\times 10^{10}$ atoms/cm3, a 1$\%$
change in the excitation fraction leads to a change in collision shift of
$\sim 1\times 10^{-17}$. As described above, it is straightforward to keep the
excitation fixed at or below the $1\%$ level. However, a complication arises
from time-dependent excitation due to trap loss. Particularly, we have
observed inelastic two-body losses involving both
${}^{3}\\!P_{0}$-${}^{3}\\!P_{0}$ and ${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$
interactions. With both ${}^{1}\\!S_{0}$ ($g$) and ${}^{3}\\!P_{0}$ ($e$)
populations present, the number density rate equations are:
$\displaystyle\dot{n}_{g}(t)=-\Gamma_{g}n_{g}(t)-\beta_{eg}n_{g}(t)n_{e}(t)$
$\displaystyle\dot{n}_{e}(t)=-\beta_{ee}n_{e}(t)^{2}-\Gamma_{e}n_{e}(t)-\beta_{eg}n_{g}(t)n_{e}(t)$
For a single population ($n_{g}$ or $n_{e}$) $\beta_{eg}$ loss can be ignored.
Fig. 2(a) shows ${}^{1}\\!S_{0}$ trap loss (black triangles, for a pure sample
of ${}^{1}\\!S_{0}$ atoms), with a one body loss fit yielding a trap lifetime
of $1/\Gamma_{g}=$ 480 (20) ms. Also shown is ${}^{3}\\!P_{0}$ trap loss (blue
circles, for a pure sample of ${}^{3}\\!P_{0}$ atoms). This population
experiences notably stronger decay at high densities, and good fit requires
both one-body ($\Gamma_{e}$) and two-body ($\beta_{ee}$) losses. After
integrating the losses over the spatial extent of a single lattice site, as
well as across the distribution of occupied sites, we find $1/\Gamma_{e}=$ 520
(28) ms and $\beta_{ee}=5\times 10^{-11}$ cm3/s, somewhat larger than the
${}^{3}\\!P_{0}$-${}^{3}\\!P_{0}$ decay measured in 88Sr Traverso et al.
(2009); Lisdat et al. (2009). Because the absolute atomic density is difficult
to calibrate, the uncertainty in $\beta_{ee}$ is estimated as 60$\%$.
Figure 2: (color online) (a) ${}^{1}\\!S_{0}$ trap loss (black triangles) with
an exponential fit (dashed curve). ${}^{3}\\!P_{0}$ trap loss (blue circles)
with a one- and two-body loss fit (solid curve). The inset shows
${}^{1}\\!S_{0}$ loss for a pure ${}^{1}\\!S_{0}$ sample (black triangles) and
a mixed sample of ${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ atoms (blue circles).
The former is fit with a simple exponential, the latter with
${}^{3}\\!P_{0}\,$-${}^{3}\\!P_{0}$ and ${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$
two body losses. (b) Two body loss rates as a function of temperature,
calculated with $p_{\text{ls}}=1$ (dashed curves) and with $p_{\text{ls}}=0.8$
and $\delta=0.51\,\pi$ (solid curves). Black is for a nuclear-spin-polarized
sample of ${}^{3}\\!P_{0}$ atoms ($\beta_{ee}$), red an unpolarized sample of
${}^{3}\\!P_{0}$ atoms ($[\beta_{ee}+\tilde{\beta}_{ee}]/2$). Circles give
experimental measurements, black for $\beta_{ee}$ and blue for $\beta_{eg}$.
Inelastic two-body loss for ${}^{3}\\!P_{0}$ atoms can be considered with a
single atom-atom channel, time-independent quantum formalism similar to
Idziaszek and Julienne (2010); Idziaszek et al. (2010). Lacking an accurate
Yb2 potential, the short range physics is described here by a boundary
condition at an interatomic separation $R_{0}=20$ a0 (a0 is the Bohr radius),
represented by two parameters: (i) an accumulated phase shift,
$0\leq\delta\leq\pi$, due to the unknown short-range potential from $R=0$ to
$R=R_{0}$, and (ii) a loss probability, $0\leq p_{\text{ls}}\leq 1$ at
$R=R_{0}$. The long range physics is accurately described by a van der Waals
interaction, with $C_{6}=3886$ a.u. given by Dzuba and Derevianko (2010). The
Schrödinger equation is numerically solved from $R_{0}$ to $R\to\infty$, and
the cross section is computed as a function of the collision energy. The
thermalized rate coefficients are found by averaging the cross sections with a
Maxwell–Boltzmann distribution of collision energies for a given temperature.
With full loss probability ($p_{\text{ls}}=1$), $\beta_{ee}$ is shown in Fig.
2(b) (black dashed curve) as a function of temperature. Here, the rates are
universal, independent of $\delta$ Idziaszek and Julienne (2010); Idziaszek et
al. (2010). The interaction involves two identical fermions, and thus quantum
statistics dictates that the interactions must be odd partial waves, notably
$p$-wave at these ultracold temperatures. As two excited atoms approach each
other, those which successfully tunnel through the $p$-wave barrier are
assumed to inelastically scatter with unit probability, a reasonable
assumption due to the large number of exit collision channels available to a
pair of atoms with high internal energy. For an unpolarized sample of
${}^{3}\\!P_{0}$ atoms (here with equal $m_{I}=\pm 1/2$ populations),
interactions between distinguishable atoms also include an $s$-wave term, and
the loss rate for distinguishable ${}^{3}\\!P_{0}$ atoms is labeled
$\tilde{\beta}_{ee}$. The total loss rate for the unpolarized sample, given by
the average of $\beta_{ee}$ and $\tilde{\beta}_{ee}$, is shown in Fig. 2(b) as
a red dashed curve ($p_{\text{ls}}=1$). Experimentally, we observed identical
loss rates within $10\%$ for a polarized and an unpolarized sample at 10
$\mu$K (Fig. 2(b), black circle). Since full loss predicts unequal loss rates
for polarized and unpolarized samples at 10 $\mu$K, ${}^{3}\\!P_{0}$ atoms may
not be lost with full unit probability at short range. Instead, using short
range parameters of $p_{\text{ls}}=0.8$ and $\delta=0.51\,\pi$, the loss rates
are shown as solid curves in Fig. 2(b) for polarized (black) and unpolarized
(red) cases. Here, we found better agreement with the experimental data,
suggesting a deviation from the universal regime.
For a mixed population of ${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ atoms, we
observed additional loss through $\beta_{eg}$. Fig. 2(a) inset highlights this
by comparing ${}^{1}\\!S_{0}$ trap loss for two different atomic samples: with
(blue circles) and without (black triangles) the presence of ${}^{3}\\!P_{0}$
atoms. The additional loss is particularly notable at short times where both
${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ populations are large and is consistent
with a ${}^{1}\\!S_{0}\,$-${}^{3}\\!P_{0}$ two-body loss rate at the level of
$\beta_{eg}=3\times 10^{-11}$ cm3/s. The ${}^{3}\\!P_{0}$ population is
excited coherently from the ${}^{1}\\!S_{0}$ state, but due to excitation
inhomogeneity, the lossy collisions are between partially distinguishable
atoms Lemke et al. . The magnitude of the inelastic loss is noteworthy because
for molecular states correlating to a ${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$
atom pair, only the ${}^{1}\Sigma_{g}^{+}$ ground state (correlating to the
${}^{1}\\!S_{0}$-${}^{1}\\!S_{0}$ state) lies at lower energy, and long-range
coupling to this state is spin-forbidden.
In general, all of these loss processes lead to a time-dependent excitation
fraction during the Ramsey dark time. This, in turn, affects the balance
between ${}^{1}\\!S_{0}$ and ${}^{3}\\!P_{0}$ collisionally induced energy
shifts. For simplicity, we can easily operate at a lower atom number density,
reducing not only the collisionally-induced shifts, but also the two-body
inelastic losses. At the same time, the number of quantum absorbers remains
well in excess of 1000, in order to accommodate a high signal-to-noise ratio.
At a density of $\rho_{1}$, the excitation fraction over a dark time of
$T=150$ ms changes by only several percent. We therefore take the time-
averaged value to indicate the excitation fraction where the shift is
canceled.
Any degree of imperfect polarization of the nuclear spin state introduces a
host of other possible atomic interactions. In $p$-wave, the $V_{eg}^{-}$
interaction, which shifts the singlet state, becomes allowed Lemke et al. .
Furthermore, distinguishability between different nuclear spin states allows
$s$-wave interactions $U_{gg}$, $U_{ee}$, and $U_{eg}^{+}$. In 171Yb, both the
$s$\- and $p$-wave $|gg\rangle$ interaction terms are small Kitagawa et al.
(2008). However, the remaining interactions can alter the observed collision
shift from the spin-polarized case. For this reason, high polarization purity
is important for optical lattice clocks. Here, we benefit from the simple
structure ($I=1/2$) of 171Yb and can readily optically pump to a single spin
ground state with $99\%$ purity. The polarization purity is directly measured
by observing the absence of a clock excitation spectrum for the unpopulated
spin state.
To quantify the effect of imperfect polarization, we measure the collision
shift as a function of excitation for unpolarized atoms (open squares in Fig.
1(a), equal mixture of both spin states). Since, during spectroscopy, a weak
bias field (1 G) lifts the degeneracy of the $\pi$-transitions from the two
spin states, here the excitation fraction is defined as that for the
particular spin state being resonantly excited, not accounting for the other
‘spectator’ spin state. In general, the measured shifts are smaller than for
the spin-polarized case, implying that competing interactions have the
opposite sign as those in the polarized case. Notably, the measured zero-
crossing in the shift occurs at a lower excitation fraction (around $40\%$),
leaving a net positive shift at $51\%$, where the shift is zero for the spin
polarized case. Based upon these measurements, we determine a $1\%$
polarization impurity will not affect the shift zero-crossing at or above the
level measured in Fig. 1(c).
Figure 3: (color online) Cold collision shift as a function of excitation for
a polarized sample in a 2-D lattice, for different Ramsey times. $\rho_{2}$ is
a typical 2-D lattice density, where $25\%$ of the atoms are doubly occupied
in a lattice site with density of $4\times 10^{12}$ atoms/cm3. Circles
correspond to experimental measurements, and solid curves are theoretical
calculations Lemke et al. . In both cases, blue is a Ramsey time of $T=10$ ms,
red is 40 ms, black is 80 ms, green is 160 ms, and orange is 210 ms.
We have shown that the cold collision shift can be canceled at the $5\times
10^{-18}$ level in a 1-D optical lattice clock of 171Yb. Other implementations
may be suitable for reducing the clock shift at or below this level. In a 2-D
optical lattice clock, strong interactions can lead to a decay of the
collision shift Lemke et al. ; Gibble (2010). In particular, longer Ramsey
dark times lead to smaller shifts and can reduce the shift dependence on
excitation fraction (Fig. 3), making it attractive for shift reduction. Care
must be taken since, as we have observed both experimentally and
theoretically, the interactions can also reduce Ramsey fringe contrast.
Alternatively, the 2-D lattice system also exhibits a zero crossing in the
shift versus excitation. However, strong interactions can move the zero-
crossing excitation with a nonlinear dependence on interaction strength (Fig.
3), and thus the weak interactions of the 1-D lattice may be easier to
control.
The 3-D optical lattice continues to be an interesting choice Katori et al.
(2003), where it is straightforward to achieve high atom number with an
average of $\ll 1$ atom per lattice site. The small fraction of lattice sites
with double occupancy will exhibit very strong interactions. Doubly-occupied
sites could also be eliminated using photoassociation losses Akatsuka et al.
(2010) or directly via the two-body losses observed here. Furthermore, the 3-D
optical lattice system may exhibit kinematic suppression of the atomic
interactions which yield the shifts Gunter et al. (2005). Indeed, these rich
atomic systems will undoubtedly continue to offer many interesting phenomena
in 1-, 2-, and 3-D confinement.
The authors gratefully acknowledge assistance from Y. Jiang, and financial
support from NIST, NSF-PFC, AFOSR, and ARO with funding from the DARPA-OLE.
## References
* Jiang et al. (2011) Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. S. Ma, and C. W. Oates, Nature Photonics 5, 158 (2011).
* Takamoto et al. (2011) M. Takamoto, T. Takano, and H. Katori, Nature Photonics 5, 288 (2011).
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|
arxiv-papers
| 2011-08-05T18:30:50 |
2024-09-04T02:49:21.337893
|
{
"license": "Public Domain",
"authors": "A. D. Ludlow, N. D. Lemke, J. A. Sherman, C. W. Oates, G. Quemener, J.\n von Stecher, and A. M. Rey",
"submitter": "Andrew Ludlow",
"url": "https://arxiv.org/abs/1108.1379"
}
|
1108.1406
|
LMU-ASC 36/11
ROM2F/2011/10
# Introducing the Slotheon: a slow Galileon scalar field in curved space-time
Cristiano Germani cristiano.germani@lmu.de Arnold Sommerfeld Center, Ludwig-
Maximilians-University, Theresienstr. 37, 80333 Muenchen, Germany Luca
Martucci luca.martucci@roma2.infn.it I.N.F.N. Sezione di Roma “Tor Vergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”, Via della Ricerca
Scienti ca, 00133 Roma, Italy Parvin Moyassari
parvin.moyassari@physik.lmu.de Excellence Cluster Universe, Boltzmannstr. 2,
85748 Garching, Germany Arnold Sommerfeld Center, Ludwig-Maximilians-
University, Theresienstr. 37, 80333 Muenchen, Germany
###### Abstract
In this paper, we define covariant Galilean transformations in curved
spacetime and find all scalar field theories invariant under this symmetry.
The Slotheon is a Galilean invariant scalar field with a modified propagator
such that, whenever gravity is turned on and energy conditions are not
violated, it moves “slower” than in the canonical set-up. This property is
achieved by a non-minimal derivative coupling of the Slotheon to the Einstein
tensor. We prove that spherically symmetric black holes cannot have Slotheonic
hairs. We then notice that in small derivative regimes the theory has an
asymptotic local shift symmetry whenever the non-canonical coupling dominates
over the canonical one.
## I Introduction
Undoubtedly, the search for theories with special symmetries is a key issue in
theoretical physics. Indeed, usually, such theories have the advantage of
being quantum mechanically under control.
One of the simplest possible symmetries is the shift invariance of a scalar
field $\pi$, i.e. the symmetry under the shift
$\displaystyle\pi\rightarrow\pi+c\ ,$ (1)
where $c$ is a constant. However, such a symmetry is not very interesting as
any theory involving only derivatives of $\pi$ would be invariant under (1).
The question is then whether such symmetry may be generalized to a more
complicated shift
$\displaystyle\pi\rightarrow\pi+f(x)\ ,$ (2)
where $f(x)$ is some specific function of space-time coordinates, depending on
some constants parametrizing the independent symmetries encoded in (2). The
class of Lagrangians invariant under (2) will be more and more constrained,
depending on the degree of arbitrariness of $f(x)$. The extreme case is the
one in which $f(x)$ is a completely arbitrary function and then (2) can be
regarded as a gauge symmetry.
The next to trivial shift symmetry is what is commonly called Galileon shift
galilean . This symmetry, formulated solely in flat (Minkowski) space, is an
on-shell symmetry. In other words, the equation of motion are invariant under
the Galileon shift
$\displaystyle\pi\rightarrow\pi+c+c_{\mu}x^{\mu}\ ,$ (3)
with $c$ and $c_{\mu}$ respectively a constant and a constant vector, whereas
the action shifts by a total derivative which gives a non-vanishing boundary
contribution. Mainly inspired by the decoupling limit of the Dvali-Gabadadze-
Porrati (DGP) model dgp , the Authors of galilean showed that in flat space,
there exist only four forms of scalar field Lagrangians with second order
field equations and invariant under the Galileon symmetry. These theories,
turned out to admit a non-renormalization theorem. In other words, it is
proven that the mass parameters in the Galileon terms do not get renormalized
non ; non2 111Note however that any other operator can be generated by loops,
see for example nr ; referee ..
In a subsequent analysis, the Authors of covariant showed that healthy
covariantization of the flat space Galileon invariant theories, would
generically break the flat space Galileon invariance. This is mainly due to
the fact that the constant form $c_{\mu}$, is not shear free, i.e.
$\nabla_{(\alpha}c_{\beta)}\neq 0$, where we defined
$v_{(\alpha\beta)}=\frac{1}{2}\left(v_{\alpha\beta}+v_{\beta\alpha}\right)$.
Indeed, by inserting the transformation (3) in the equation of motion for the
scalar field one always get terms proportional to the shear of
$c_{\alpha}$222Note that since we like to obtain a scalar equation, only shear
and not vorticity (the antisymmetric part) of the covariant derivative of
$c_{\alpha}$ enters in the shifted equations..
The question is then whether any symmetric scalar field theory under the shift
(2) can be constructed in a fixed curved space-time. Moreover, we will always
implicitly consider only theories with equations of motion (for both gravity
and scalar field) that are up to second order, although we will not explicitly
state it anymore.
In a manifold with covariantly constant Killing vectors, a Galileon symmetry
similar to (3), may indeed be realized. In this case however, as we shall
prove it, only the following Lagrangians can be constructed in contrast to the
flat space case. They are 333Note that in addition there are also “tadpole”
terms such as ${\cal L}_{\rm
tp}=M_{tp}^{3}\pi\left(1+\frac{R}{\mu_{1}^{2}}+\frac{GB}{\mu_{2}^{4}}\right)\
,$ where $M_{tp}$ and $\mu_{1,2}$ are mass scales and $GB$ is the Gauss-Bonnet
term. We will not discuss them here as we focus on source free equation of
motion for $\pi$. Nevertheless, even considering them, they will not be
invariant under the approximate gauged shift symmetry (2).
$\displaystyle{\cal L}_{2}={\cal L}_{2}^{\text{m}}+{\cal
L}_{2}^{\text{nm}}\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi+\frac{1}{2M_{2}^{2}}G^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi\
,$ (4) $\displaystyle{\cal L}_{3}={\cal L}_{3}^{\text{m}}+{\cal
L}_{3}^{\text{nm}}\equiv\pm\frac{1}{2M_{3}^{3}}(\partial\pi)^{2}\square\pi\pm\frac{1}{2M_{5}^{5}}{}^{**}R^{\alpha\beta\mu\nu}\partial_{\alpha}\pi\partial_{\mu}\pi\nabla_{\beta}\nabla_{\nu}\pi\
.$ (5)
where $M_{i}$ are mass scales for the operators of dimension $i+4$.
$G_{\mu\nu}$ and ${}^{**}R^{\alpha\beta\mu\nu}$ are respectively the Einstein
and double dual Riemann tensors 444The double dual Riemann tensor is defined
as
${}^{**}R^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\equiv-\frac{1}{4}\mathcal{E}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}~{}\mathcal{E}^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}R_{\mu_{3}\mu_{4}\nu_{3}\nu_{4}}\
,$ where
$\mathcal{E}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}=-\frac{1}{\sqrt{-g}}\delta_{1}^{[\mu_{1}}\delta_{2}^{\mu_{2}}\delta_{3}^{\mu_{3}}\delta_{4}^{\mu_{4}]}\
.$ . Note that, at this level, only the quadratic canonical Lagrangian may
have a definite sign in order to avoid ghost propagation around Minkowski.
Instead, the sign of ${\cal L}_{2}^{\rm nm}$ is chosen in such a way to avoid
ghosts when the weak energy condition $G^{tt}\geq 0$ is satisfied 555We would
like to stress that this condition is not enough to guarantee the absence of
ghost propagation whenever the metric is dynamical..
Next, one can remove the requirement of the existence of covariantly conserved
Killing vector and couple the theory (4) to a dynamical metric, by adding the
standard Einstein-Hilbert term and possibly also a potential for $\pi$. In
this case, we will show that for parity invariant Lagrangians
($\pi\rightarrow-\pi$), in the small derivatives regime of the scalar, an
approximate infinitesimal shift symmetry (2) emerges for the theory ${\cal
L}_{2}^{\text{nm}}$, if and only if an appropriate shift of the metric is also
considered. The theory ${\cal L}_{2}^{\text{nm}}$, in this regime, is the base
for the Gravitationally-Enhanced-Friction (GEF) models of inflation yuki ; new
; uv . Therefore, thanks to the additional symmetry, the GEF models are
endowed, during inflation (small scalar field derivatives) with a protection
against quantum corrections to the effective Lagrangian up to the Planck
scales, if the potential terms only softly break the gauged shift symmetry
(2). In fact, although other terms may be generated by loops that are
invariant under the symmetry (2), they will be generically suppressed by
either slow roll or higher powers of Planck mass. This is mainly due to the
fact that in this theory, the gravity strong coupling is still at the Planck
scale, as we shall show.
By adding the standard Einstein-Hilbert term to ${\cal L}_{2}$, and possibly a
non trivial potential for $\pi$, one gets a simple though rich gravitational
theory, with some nice peculiarities. In particular, in regimes in which the
analogue of the weak energy condition is valid, the field $\pi$ moves ‘slower’
than in the cousin canonical theory. For this reason, we dub $\pi$ as the
Slotheon and the Slotheonic nature of this theory is, in fact, at the origin
of the efficiency of the GEF models.
Furthermore, we show that the Slotheonic theory has only spherically symmetric
black hole solutions with no scalar hairs and we find indications that this
property should hold for any black hole solutions. This important result
combined with previous analysis in homogeneous and isotropic space-times yuki
, is a step forward to prove the stability of this theory.
## II Shift and accidental symmetries
In this section we would like to find all the possible scalar field
Lagrangians (collectively denoted as ${\cal L}$), that do not contain purely
potential terms, according to some symmetry principle.
First of all, being $\pi$ a real scalar, it is natural to impose on ${\cal L}$
the shift symmetry (1). In other words, we assume that ${\cal L}$ depends only
on derivatives of $\pi$. Furthermore, in order to avoid problems with ghosts,
due to the so called Ostrogradski instability ghost we also impose that the
equations of motion contain at most second order derivatives.
These requirements still leaves a large number of possible ${\cal L}$. In
order to further restricts the form of the action we require that, for certain
background metrics, the shift symmetry (1) enhances to accidental point-
dependent shift-symmetries of the form (2).
The prototype example of such an effect is provided by the Galileon symmetry
for flat space-time galilean , in which (1) enhances to (3). We would like to
preserve the Galileon symmetry for the case of flat space-time and generalized
it to more general, though still restricted, curved space-times.
As we shall discuss, the on-shell flat space-time symmetry (3) can be
naturally generalized in background metrics with a certain number of
covariantly constant vectors $\xi_{a}=\xi_{a}^{\mu}\partial_{\mu}$. Requiring
that the equations of motion derived from ${\cal L}$ preserve such accidental
symmetries will lead us to consider only the set ${\cal L}_{2,3}$.
### II.1 Galileon symmetry in curved space-time
In flat spacetime, the transformations (3) can be written in a covariant way
by introducing the translation Killing vectors of Minkowski, i.e.
$\xi_{a}^{\mu}=\delta_{a}^{\mu}$ in Cartesian coordinates. These Killing
vectors are very special as they have the properties of being convariantly
constant, i.e. $\nabla_{\mu}\xi_{a}=0$.
Our definition of Galilean transformation in curved spacetime will just be the
straightforward generalization of the flat case. Let us introduce the one
forms $\xi^{a}=\xi^{a}_{\mu}\mbox{{\rm d}}x^{\mu}$ dual to the Killing
vectors: $\xi^{a}_{\mu}=g_{\mu\nu}\xi^{\nu}_{a}$. Hence, we will require that
our theories are invariant under
$\displaystyle\pi_{\mu}\rightarrow\pi_{\mu}+c_{a}\,\xi^{a}_{\mu}\ ,$ (6)
where $c_{a}$ are constants, latin indices are contracted with Euclidean
metric and we have used the notation
$\displaystyle\pi_{\mu_{1}\ldots\mu_{k}}\equiv\nabla_{\mu_{1}}\ldots\nabla_{\mu_{k}}\pi\
.$ (7)
Consistency of (6) requires the one forms $\xi^{a}$ to be closed:
$\displaystyle\mbox{{\rm d}}\xi^{a}=0\ .$ (8)
In other words, $\xi_{a}$ must be covariantly constant and we will loosely say
that the Killing vectors must be integrable 666Note, that one could find
different symmetries. For example, if one relaxes the constant shift
invariance (1) in curved space-time, one can find other theories invariant
under specific shifts $\pi\rightarrow\pi+c(x)$, where $c(x)$ is a function of
curvatures. A typical example is given in burrage ; goon1 . We thank Claudia
de Rham for pointing this out..
Space-times admitting integrable Killing vectors are of particular type
Stefani . A Killing vector $\xi^{\mu}$ can be covariantly constant only if
$\xi$ satisfies the algebraic condition
$\displaystyle R^{\mu}{}_{\nu\rho\sigma}\xi^{\nu}=0\ ,$ (9)
which can be obtained from the consistency condition
$[\nabla_{\rho},\nabla_{\sigma}]\xi^{\mu}=0$. In other words, the holonomy
group of space-time must be reduced to a subgroup of SO$(1,3)$. Explicitly, if
the vector is non-null, the space-time metric is of the form
$\displaystyle\mbox{{\rm d}}s^{2}=g_{ij}(x^{k})\mbox{{\rm d}}x^{i}\mbox{{\rm
d}}x^{j}+\kappa\,\mbox{{\rm d}}y^{2}\,,~{}~{}~{}i,j,k=1,2,3\ ,$ (10)
where $\kappa=+1,-1$ for spacelike or timelike $\xi^{\mu}$, respectively, or
for a null $\xi^{\mu}$
$\displaystyle\mbox{{\rm d}}s^{2}=g_{ij}(x^{k})\mbox{{\rm d}}x^{i}\mbox{{\rm
d}}x^{j}+\mbox{{\rm d}}z\mbox{{\rm d}}y\,,~{}~{}~{}i,j,k=1,2,3\ ,$ (11)
where $z$ is any coordinate in the $i$’s directions.
Given a set of integrable Killing vectors $\xi_{a}$ we can easily integrate
(6) into a curved Galileon transformation
$\displaystyle\pi(x)\rightarrow\pi(x)+c+c_{a}\int^{x}_{\gamma,x_{0}}\xi^{a},$
(12)
where we have chosen a certain reference point $x_{0}$ and a curve $\gamma$
connecting $x$ with $x_{0}$. Thanks to (8), this quantity is well defined.
Indeed, it does not change under continuous deformation of the curve $\gamma$.
Furthermore, the change of the reference point $x_{0}$ can be reabsorbed into
a shift of $c$.
The transformation (12) represents our proposal of curved Galileon symmetry.
Let us revisit the Minkowski case in this covariant language. In that case,
the integrable Killing vectors are the four generators of the translations.
Fixing $x^{\mu}$ to be the Minkowskian coordinates, the associated one-forms
take the form
$\displaystyle\xi^{a}\Big{|}_{\rm Mink_{4}}=\delta^{a}_{\mu}\mbox{{\rm
d}}x^{\mu}\ .$ (13)
By choosing $x_{0}$ as the origin $x^{\mu}=0$ it is immediate to see that (12)
reproduces (3), with $c_{\mu}\equiv c_{a}\delta^{a}_{\mu}$.
### II.2 Galileon invariant theories in curved space-time
We would like now to find a Galilean invariant theory in curved spacetime
where the metric is non-dynamical. Later on we will drop this last
requirement. The Lagrangian defining the theory we look for is either 1) the
covariantized version of the Galileon Lagrangian in Minkowski or 2) made of
terms which vanish once restricted to flat space.
We start with theories that are not trivial once restricted to flat space,
i.e. the case 1). In flat space, Galilean invariant theories were classified
by galilean . To keep the equation of motion second order in curved spacetime,
the Authors in covariant showed that the original flat space self couplings
of the scalar field derivatives must be supplemented by non-minimal couplings
to curvatures.
We can now check directly what theories among the one classified in covariant
are invariant under the Galilean symmetries in curved spacetime (6). The key
point to bare in mind is that the shift of the scalar field derivative is
covariantly constant and, the same shift, once contracted to curvatures
vanishes as in (9). Therefore, if the scalar field equation of motion contains
terms proportional to one derivative of $\pi$ without contraction to
curvatures, then, the theory is not Galilean invariant in curved spacetime. An
example of those terms is $R(\partial\pi)^{2}$.
Inspecting the set of four Lagrangians found in covariant , the only Galilean
invariant theories in curved spacetime which are not trivial in the flat limit
are ${\cal L}_{2}^{\rm m},{\cal L}_{3}^{\rm m}$.
We now consider theories that are solely non-minimally coupled, i.e. they
vanish in the flat limit. These theories, and in fact all possible scalar-
tensor theories with second order equation of motion, are classified by
Hordenski in hord . We can then easily check that the only theories invariant
under the Galilean symmetry in curved spacetime are ${\cal L}_{2}^{\rm
nm},{\cal L}_{3}^{\rm nm}$. This is again due to the fact that terms
proportional to one derivative of $\pi$ without contraction to curvatures are
not Galilean invariant in curved spacetime.
In conclusion, the only invariant theory under Galilean symmetry in curved
spacetime (6), parameterized by the masses $M_{i}$, is
$\displaystyle S=-\frac{1}{2}\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left({\cal
L}_{2}+{\cal L}_{3}\right)\ .$ (14)
It is interesting to note that the theory (14), as in the flat case non ,
follows a non-renormalization theorem for the mass parameters $M_{i}$,
whenever gravity is non dynamical. This is due to the fact that in each cubic
vertex, the algebraic structure of derivatives is not different from the one
of the flat space, thanks to Bianchi identities. In other words, one can
easily check that cubic interactions only produce effective higher-derivatives
operators and therefore cannot renormalize $M_{i}$. Specifically, one sees,
following non for the flat case, that vertexes of type
$\displaystyle\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\partial_{\alpha}\pi_{\rm
ext}\partial_{\mu}\pi_{\rm int}\nabla_{\beta\nu}\pi_{\rm
int}=-\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\nabla_{\alpha\nu}\pi_{\rm
ext}\partial_{\mu}\pi_{\rm int}\partial_{\beta}\pi_{\rm int}+{\rm boundaries}\
,$ (15)
where $\pi_{\rm ext}$ and $\pi_{\rm int}$ are respectively the external and
the internal legs of a diagram involving loops, cannot renormalize $M_{3}$ as
they are equivalent to higher-derivatives powers in $\pi_{\rm ext}$.
Obviously, as the quadratic Lagrangian has no any interactions in the case of
non-dynamical metric, $M_{2}$ is not renormalized as well.
This conclusion would change in the case in which gravity is dynamical.
Nevertheless, in this case, the parameters $M_{i}$ would only have runnings
suppressed by $M_{\rm P}$, as we shall discuss later on.
## III Gauging the shift invariance in curved space-time
As we showed before, only few manifolds may support a Galileon symmetry. We
may however ask whether the Galileon invariance introduced earlier can be
recovered, even in some approximate sense, in space-times with no integrable
Killing vectors.
An obvious requirement is that, if such an approximate symmetry exists, it
should be also realized point-wise. Locally indeed we can always define
Riemann coordinates $(x_{R}^{\mu})$ around any point $P$ such that, for any
constant form $c_{\mu}$ in this coordinates
$\displaystyle\nabla_{\mu}c_{\nu}={\cal O}(x_{R})\ .$ (16)
In this case we can ask whether there exists any theory invariant under the
local Galileon symmetry
$\displaystyle\pi\rightarrow\pi+c+c_{\mu}x_{R}^{\mu}\ ,$ (17)
up to order ${\cal O}(x_{R})$. Note that, although the Christoffel symbols
vanish up to order ${\cal O}(x_{R})$, curvatures do not vanish. Let us then
consider the scalar field equations of motion of theories (4) and (5)
$\displaystyle E_{2}:$ $\displaystyle=$
$\displaystyle\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}_{2}}\right)\pi_{\mu\nu}\
,$ (18) $\displaystyle E_{3}:$ $\displaystyle=$
$\displaystyle\pm\left(\pm\frac{M_{3}^{3}}{M_{5}^{5}}{}^{**}R^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}+g^{\mu_{1}\nu_{1}}g^{\mu_{2}\nu_{2}}-g^{\mu_{1}\nu_{2}}g^{\mu_{2}\nu_{1}}\right)\left(\pi_{\mu_{1}\nu_{1}}\pi_{\mu_{2}\nu_{2}}+R^{\alpha}{}_{\mu_{1}\nu_{1}\mu_{2}}\pi_{\alpha}\pi_{\nu_{2}}\right)\
.$
We can easily see that only $E_{2}$ is invariant under the approximate shift
(17) up to distance $x_{R}\sim\ell$ where $\ell$ is the local curvature radius
of the spacetime.
At this level however, the approximate symmetry (17) cannot be extended far
away from the point $P$. It is then clear that gravity should participate to
the Galileon shift in order to extend this symmetry at distances such that the
Christoffel symbols cannot be neglected. We will then only focus on the theory
${\cal L}_{2}$. This theory can be singled out by requiring the action to be
invariant under the following additional discrete symmetry
$\displaystyle\pi\rightarrow-\pi\ ,$ (19)
which we will loosely call $\pi$-parity.
Let us then study the following action:
$\displaystyle S(g,\pi)=\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left[\frac{M_{\rm
P}^{2}}{2}R(g)+{\cal L}_{2}\right]\ .$ (20)
Now we can make the metric $g_{\mu\nu}$ participate actively, enlarging the
possibility of identifying the relevant (approximate) symmetries of the form
(2).
Consider the following small derivative expansion regime
$\displaystyle\varepsilon\sim\frac{(\partial\pi)^{2}}{M_{2}^{2}M^{2}_{\rm
P}}\ll 1\ ,$ (21)
and note that fab
$\displaystyle\int d^{4}x\sqrt{-g}G^{\alpha\beta}\pi_{\alpha}\pi_{\beta}$
$\displaystyle=$ $\displaystyle M_{2}^{2}M^{2}_{\rm P}\int
d^{4}x\frac{\delta{\sqrt{-g}R}}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}\Big{|}_{\delta
g_{\alpha\beta}=-\frac{\partial_{\alpha}\pi\partial_{\beta}\pi}{M_{2}^{2}M_{\rm
P}^{2}}}+{\rm boundaries}\ .$ (22)
We find that (20) can be found as an expansion of ${\cal O}(\varepsilon^{2})$
of the following action 777Our quadratic action (23) agrees with kurt and
disagrees with claudia . This can be seen by noticing that the purely
derivative quadratic terms in $\pi$ of kurt (Eq. (31) of the cited paper) is
nothing else than the Ricci scalar coupled to the kinetic term of $\pi$ plus
boundary terms. The disagreement is due to a missing factor upon passing from
the correct expansion Eq.(77) to the Lagrangian Eq.(79) of claudia .
$\displaystyle\hat{S}(h,\pi)=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\,\sqrt{-h}\left[M_{\rm
P}^{2}R(h)-h^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi\right]\ ,$ (23)
where
$\displaystyle h_{\mu\nu}\equiv
g_{\mu\nu}-\frac{\partial_{\mu}\pi\partial_{\nu}\pi}{M_{2}^{2}M_{\rm P}^{2}}\
,$ (24)
and $h^{\mu\nu}$ is the inverse of $h_{\mu\nu}$ which is also known as Finsler
metric. Explicitly, we have
$\displaystyle\hat{S}(h,\pi)=S(g,\pi)+{\cal O}\left(\varepsilon^{2}\right)\ .$
(25)
Notice that, in the regime (21) and to leading order in the perturbative
$\varepsilon$-expansion, the canonical action $\hat{S}(h,\pi)$ can be regarded
as the Einstein frame action of the theory (20). Clearly, this is not true if
(21) is violated, as the two theories are substantially different.
The metric $h_{\mu\nu}$ is exactly invariant if we consider the combined
transformation
$\displaystyle\pi\rightarrow\pi+f(x)\ ,\quad g_{\mu\nu}\rightarrow
g_{\mu\nu}+2\frac{\partial_{(\mu}f\partial_{\nu)}\pi}{M_{2}^{2}M_{\rm P}^{2}}\
,$ (26)
where we can assume that
$\displaystyle\frac{\partial f}{M_{2}M_{\rm P}}\sim{\cal
O}(\sqrt{\varepsilon})\ .$ (27)
In this way, the transformed $\pi$ continues to satisfy the small derivative
condition (21). This simple observation has an immediate consequence in a
regime in which the Lagrangian $G^{\mu\nu}\pi_{\mu}\pi_{\nu}/M_{2}^{2}$
dominates over the canonical kinetic term in (20), i.e.
$\displaystyle\frac{{\cal L}_{2}^{\rm nm}}{{\cal L}_{2}^{\rm m}}\gg 1\ .$ (28)
We call this the high friction regime for reasons which will become clear in
the following discussions.
If the system is in high friction regime the theory (20) can be recast as a
first order expansion of the Einstein-Hilbert action for the metric $h$
$\displaystyle S_{\rm EH}(h)=\frac{M_{\rm P}^{2}}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-h}\,R(h)\ .$ (29)
It is now easy to see that, in the small derivative high-friction regime
defined by (21) and (28), the Slotheon action $S(g,\pi)$ is invariant under
the transformation (26) up to terms of order ${\cal
O}\left(\varepsilon^{2}\right)$. Hence, we conclude that in this regime the
action (20) has an approximate symmetry (26) which ‘gauges’ the constant shift
symmetry $\pi\rightarrow\pi+c$ by mixing $\pi$ and metric degrees of freedom.
As it is clear from the above discussion, this gauge symmetry simply removes
the physical degrees of freedom encoded in $\pi$, which recombines with $g$
into the physical Einstein metric $h$, at least to first order in
$\varepsilon$.
It is interesting to compare this symmetry with the curved Galileon symmetry
discussed in section II. Consider a certain metric $g$ with a certain set of
Killing vectors $\xi_{a}$ and take $f(x)$ to have the form given by (12), i.e.
$\displaystyle f(x)=c+c_{a}\int^{x}_{x_{0},\gamma}\xi^{a}\ .$ (30)
By requiring (21), we must impose $c_{a}\xi^{a}_{\mu}/(M_{2}M_{\rm
P})\sim\sqrt{\varepsilon}$. Having fixed the metric, the equation of motion
for $\pi$ is clearly invariant under (30). However now, differently from the
curved Galileon symmetry in section II, the symmetry (26) acts also on the
metric $g_{\mu\nu}$, under which the equations should be approximately
invariant by construction, if (21) and (28) are satisfied. This effect can be
understood by observing that the transformation for $g_{\mu\nu}$ can be
regarded as an infinitesimal $\pi$-dependent ‘diffeomorphism’
$\displaystyle g_{\mu\nu}\rightarrow g_{\mu\nu}+\nabla_{(\mu}w_{\nu)}\,,\qquad
w_{\mu}=2\frac{c_{a}\xi^{a}_{\mu}\pi}{M_{2}^{2}M_{\rm P}^{2}}\ .$ (31)
We would like to end this section by commenting on ${\cal L}_{3}$. As
discussed in fab ,
$\displaystyle\int
d^{4}x\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\pi_{\alpha}\pi_{\mu}\pi_{\beta\nu}$
$\displaystyle=$ $\displaystyle-\frac{1}{4}M_{2}^{2}M_{\rm P}^{2}\int
d^{4}x\frac{\delta{\sqrt{-g}\pi GB}}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}\Big{|}_{\delta
g_{\alpha\beta}=-\frac{\partial_{\alpha}\pi\partial_{\beta}\pi}{M_{2}^{2}M_{\rm
P}^{2}}}+{\rm boundaries}\ ,$ (32)
where $GB$ is the Gauss-Bonnet combination. Therefore, ${\cal L}_{3}$ might be
also rewritten in terms of a Finsler metric in the high friction regime.
However, the presence of the tadpole term $\pi GB$, would not be invariant
under the symmetry (26).
We then focus in the rest of the paper only on the following $\pi$-parity
invariant Lagrangian:
$\displaystyle S_{\rm sloth}=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}R-\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\partial_{\alpha}\pi\partial_{\beta}\pi\right]\
,$ (33)
where we replaced $M_{2}\rightarrow M$ for notational simplicity.
### III.1 On the strong coupling in high friction regime
Because of the non-trivial coupling of gravity with the Slotheon, the
identification of the tree-level strong coupling scale of the theory (33) is
in general strongly background dependent. In other words, in order to
calculate the perturbative cut-off scale of the theory (33) as an expansion of
fields in specific backgrounds, one should take care upon identifying the
correct propagating degrees of freedom which are generically a combination of
the Slotheon, the graviton and the background quantities.
In order to identify the physical perturbative degrees of freedom one should
rewrite the theory as an expansion around gaussian fixed points. Although the
obvious Minkowski cut-off of the theory (33) is $\Lambda_{\text{cut-
off}}=(M^{2}M_{\rm P})^{1/3}$, it has been shown in yuki ; new that for a
slow rolling Slotheon in a homogeneous and isotropic background, (i.e. when
$\varepsilon\ll 1$) the strong coupling scale of the theory is enhanced to
$\Lambda_{\text{cut-off}}=M_{\rm P}+{\cal O}(\varepsilon)$. This result can be
readily generalized by using the previous arguments. Indeed, let us consider
an expansion around the $\varepsilon\ll 1$ solution, which we loosely call
small derivative regime.
If there exists a non-trivial background for the scalar field, one can always
reparameterize time in order to reabsorb the perturbative scalar degree of
freedom into the metric. Explicitly, let us consider the expansion of the
Slotheon around a background solution $\pi_{0}$. At first order (higher orders
are easily generalizable) we have
$\displaystyle\pi=\pi_{0}(t,\vec{x})+\delta\pi(t,\vec{x})\ ,$ (34)
where $\delta\pi$ is the perturbation.
We can now consider the first order coordinate transformation $t\rightarrow
t+\delta t$ to obtain, at first order
$\displaystyle\pi=\pi_{0}+\dot{\pi}_{0}\delta t+\delta\pi\ .$ (35)
Therefore, by choosing the gauge $\delta t=-\delta\pi/\dot{\pi}_{0}$ we obtain
the desired result of reabsorbing the scalar degree of freedom into the
metric. This gauge is called unitary gauge in cosmology and widely used to
calculate (quantum) correlation functions maldacena .
Using the unitary gauge, during the small derivative regime of the background
scalar field, the theory (33) is well approximated by (29) plus the canonical
kinetic term, i.e.
$\displaystyle S=\frac{1}{2}\int d^{4}x\sqrt{-h}\left[M_{\rm
P}^{2}R(h)-(\partial\pi)^{2}\right]\ .$ (36)
Therefore, the true (gaussian) degrees of freedom become a self interacting
$h_{\mu\nu}$ with cut-off scale $M_{\rm P}$ and the free scalar $\pi$.
We thus proved that the strong coupling scale of the Slotheonic theory in a
background in which $\varepsilon\ll 1$ is
$\displaystyle\Lambda_{\text{cut-off}}=M_{\rm P}+{\cal O}(\varepsilon)\ ,$
(37)
which matches direct computations in homogeneous and isotropic backgrounds
yuki ; new . The presence of a possible (renormalizable) potential term for
$\pi$ obviously does not alter this result.
In this sense then, the background $\varepsilon\ll 1$ is always in weak
coupling if $M$ and curvatures are below the Planck scale.
## IV The Slotheon: a “slow” scalar field
Let us now investigate the properties of the theory (33). In particular, we
focus on the dynamics which governs the temporal evolution of the scalar field
$\pi$.
Let us take the ADM decomposition ADM ; wald where the metric can be written
as
$\displaystyle\mbox{{\rm d}}s^{2}=-N^{2}\mbox{{\rm
d}}t^{2}+\gamma_{ij}(\mbox{{\rm d}}x^{i}-N^{i}\mbox{{\rm d}}t)(\mbox{{\rm
d}}x^{j}-N^{j}\mbox{{\rm d}}t)\ .$ (38)
In this parameterization of the metric the action related to ${\cal L}_{2}$
looks like
$\displaystyle S=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\,N\sqrt{\gamma}\left[\left(\frac{1}{N^{2}}+\frac{G^{tt}}{M^{2}}\right)\dot{\pi}^{2}+2\left(\frac{N^{i}}{N}+\frac{G^{ti}}{M^{2}}\right)\dot{\pi}\partial_{i}\pi-(\gamma^{ij}-\frac{G^{ij}}{M^{2}})\partial_{i}\pi\partial_{j}\pi\right]\
.$ (39)
The momentum conjugate to $\pi$ is therefore defined as
$\displaystyle\Pi=\frac{\delta
S}{\delta\dot{\pi}}=N\sqrt{\gamma}\left[\left(\frac{1}{N^{2}}+\frac{G^{tt}}{M^{2}}\right)\dot{\pi}+\left(\frac{N^{i}}{N}+\frac{G^{ti}}{M^{2}}\right)\partial_{i}\pi\right]\
.$ (40)
The contribution of the Hamiltonian density coming from the scalar field
kinetic term is then
$\displaystyle{\cal
K}_{\pi}=\frac{1}{2}\sqrt{\gamma}\,\alpha^{2}\frac{\dot{\pi}^{2}}{N^{2}}\ ,$
(41)
where we defined
$\displaystyle\alpha^{2}\equiv 1+N^{2}\frac{G^{tt}}{M^{2}}\ .$ (42)
We would now like to focus on the regimes in which $G^{tt}\geq 0$. This
condition can be regarded as the analogous of standard weak energy condition
in our non-canonical theory and immediately implies that
$\displaystyle\alpha^{2}\geq 1\ .$ (43)
Considering the same background geometry, we would like now to compare the
kinetic energies of a canonical scalar field and of the Slotheon. In order to
do that we must fix the time lapse to be the same for the two theories. The
simplest choice is to use the synchronous gauge $N=1$. In this case, for a
given kinetic energy per unit volume (${\cal K}_{\pi}$) we have
$\displaystyle\dot{\pi}^{2}\sim\frac{{\cal K}_{\pi}}{\alpha^{2}}\leq{\cal
K}_{\pi}\ .$ (44)
It is then clear that the time derivative of the Slotheon is smaller than the
corresponding one (with the same energy density) of a canonical scalar field
(where $\alpha=1$). In this sense the Slotheon is slower than a canonical
scalar field.
The same conclusion can be readily drawn also by adding a positive definite
potential. Although we imposed the $\pi$-parity invariance to select the
theory all our results in this section and in the following sections are also
valid for potentials breaking this invariance.
The Slotheonic theory is then
$\displaystyle\tilde{S}=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}R-\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}}\right)\pi_{\mu}\pi_{\nu}-2V(\pi)\right]\
,$ (45)
with $V(\pi)\geq 0$ and it is easy to see that this modification does not
modify the above arguments. Notice also that the slowing of the Slotheon is
due solely to gravitational interaction. This is profoundly different from
self-interacting theories which have similar properties only in specific
backgrounds (see for instance selfinteraction ).
A typical example of a Slotheonic theory in action can be seen on de Sitter or
almost-de Sitter (inflationary) backgrounds, where $G^{tt}=3\Lambda^{2}$ and
$\Lambda$ is roughly constant. In this case the scalar field kinetic energy of
a canonical scalar field is modified as
$\displaystyle\dot{\pi}^{2}\rightarrow(1+3\frac{\Lambda^{2}}{M^{2}})\dot{\pi}^{2}\
.$ (46)
Redefining the effective time of the scalar field as
$\displaystyle\mbox{{\rm d}}t_{\rm Slotheon}=\frac{\mbox{{\rm
d}}t}{\sqrt{1+3\frac{\Lambda^{2}}{M^{2}}}}\ ,$ (47)
we find
$\displaystyle\mbox{{\rm d}}t_{\rm Slotheon}\leq\mbox{{\rm d}}t\ .$ (48)
Therefore, one may interpret the proper clock of the Slotheon to be slower
than the clock of an observer tight to the Universe expansion. From a
different point of view, the slowness of the Slotheon in the previous example
was obtained by increasing the friction term acting on the scalar field. This
mechanism has been dubbed the Gravitational-Enhanced-Friction mechanism in
yuki for inflationary scenarios and it is the base of New Higgs Inflation new
and UV-Protected inflation uv ; yuki .
## V No-(Slotheonic) Hair Theorem
In this section we will prove that the only spherically symmetric black hole
solution of the Slotheonic theory (45) is the vacuum solution, i.e. the
Schwarzschild solution. This result is an important step to prove the
stability of the Slotheonic theory in general curved space-time (which we
postpone for future work). In fact, it is widely believed that ghost-like or
unstable scalar theories may support scalar hairs outside a black hole horizon
bekenstein95 .
Since we assumed that the potential does not violate energy conditions, we can
restrict our proof to the massless case. In fact, since a mass implies a
faster decay of the scalar field than the massless case, proving the
impossibility of massless scalar hairs will be enough. We will then restrict
our attention to the theory
$\displaystyle S=\frac{1}{2}\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}\,R-\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}}\right)\pi_{\mu}\pi_{\nu}\right]\
.$ (49)
In order to prove that the only spherically symmetric solution is trivial for
the Slotheon we will closely follow bekenstein with the help of the gravity
and scalar field equations obtained by varying the action (49) with respect to
$\pi$ and metric. The equations are respectively (see also shush )
$\displaystyle(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}})\pi_{\mu\nu}=0,$ (50)
$\displaystyle G_{\mu\nu}=M_{\rm P}^{-2}T_{\mu\nu}\ .$
Where
$\displaystyle
T_{\mu\nu}=\pi_{\mu}\pi_{\nu}-\frac{1}{2}g_{\mu\nu}(\partial\pi)^{2}+\frac{\Theta_{\mu\nu}}{M^{2}}\
,$ (51)
and
$\displaystyle\Theta_{\mu\nu}=\frac{1}{2}\pi_{\mu}\pi_{\nu}R-2\pi_{\alpha}\pi_{(\mu}R^{\alpha}_{\nu)}+\frac{1}{2}\pi_{\alpha}\pi^{\alpha}G_{\mu\nu}-\pi^{\alpha}\pi^{\beta}R_{\mu\alpha\nu\beta}-\pi_{\alpha\mu}\pi^{\alpha}_{\nu}+\pi_{\mu\nu}\pi_{\alpha}^{~{}\alpha}+\frac{1}{2}g_{\mu\nu}[\pi_{\alpha\beta}\pi^{\alpha\beta}-(\pi_{\alpha}^{~{}\alpha})^{2}+2\pi_{\alpha}\pi_{\beta}R^{\alpha\beta}]\
.$
### V.1 Spherically symmetric case
Let us start by imposing spherical symmetry. In this case the metric will be
$\displaystyle\mbox{{\rm d}}s^{2}=-A(r)^{2}\mbox{{\rm
d}}t^{2}+B(r)^{2}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm d}}\Omega^{2}\ ,$ (52)
where $d\Omega^{2}=\mbox{{\rm d}}\theta^{2}+\sin\theta^{2}\mbox{{\rm
d}}\phi^{2}$.
The equation of motion for the scalar field reads
$\displaystyle\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\nabla_{\beta}\pi=0\
.$ (53)
Multiplying it by the scalar field $\pi$ and integrating in the closed region
$S$ of Fig.1, delimited by an horizon at $r_{H}$ and two time slices
$\Sigma_{\pm}$, we get
$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\pi\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\nabla_{\beta}\pi=0\
.$ (54)
Figure 1: Integration region.
Integrating by parts (54) we obtain
$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\pi\nabla_{\beta}\pi=\int_{H}\mbox{{\rm
d}}^{3}x\sqrt{-g}n_{\alpha}\left(g^{r\alpha}-\frac{G^{r\alpha}}{M^{2}}\right)\pi\pi^{\prime}\
,$ (55)
where the sum of the boundary integrals over $\Sigma_{\pm}$ vanish because of
staticity and the integral at infinity vanishes because the assumption of
asymptotic flatness. In (55) $H$ is the horizon surface, $n^{\alpha}$ is the
normal to the horizon and ${}^{\prime}=d/dr$. By definition an horizon is a
light-like surface, i.e. $n_{\alpha}n^{\alpha}=0$ and for a static metric
$n_{t}=0$ on the horizon. By using the Cauchy inequality
$\displaystyle 0\leq(n_{i}A^{i})^{2}\leq n_{i}n^{i}A_{j}A^{j}=0\ ,$ (56)
where the last equality is valid if and only if $A_{j}A^{j}<\infty$, we find
that the left hand side of (55) vanishes. Taking
$\displaystyle A^{j}=\left(g^{rj}-\frac{G^{rj}}{M^{2}}\right)\pi\pi^{\prime}\
,$ (57)
we see indeed that $A_{j}A^{j}$ cannot diverge for a smooth space-time and
non-divergent scalar field. We are then left with the integral equation
$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\pi\nabla_{\beta}\pi=\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{rr}-\frac{G^{rr}}{M^{2}}\right)\pi^{\prime 2}=0\ .$
(58)
We are now interested in finding the form of $G^{rr}$. The gravity equations
are (we fix here $M_{\rm P}$=1)
$\displaystyle G_{\alpha\beta}=T_{\alpha\beta}\ ,$ (59)
where $T_{\alpha\beta}$ is given in Eq. (51). With the metric (52) we find
$\displaystyle
G_{rr}=\frac{\frac{1}{2}-\frac{1}{M^{2}r^{2}}}{1+\frac{3}{2}\frac{\pi^{\prime
2}}{B^{2}M^{2}}}\pi^{\prime 2}\ .$ (60)
Plugging the previous result into the integral (58) we get
$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\frac{B^{2}+\frac{\pi^{\prime
2}}{M^{2}}(1+\frac{1}{r^{2}})}{B^{4}(1+\frac{3}{2}\frac{\pi^{\prime
2}}{B^{2}M^{2}})}\pi^{\prime 2}=0\ .$ (61)
Since the integrand is positive definite in (61), the only solution is
$\pi^{\prime}=0$, i.e. the only solution for a spherically symmetric black
hole is with no Slotheonic hairs. The black hole solution is then a solution
of the Einstein equation in vacuum that has as the unique solution the
Schwarzschild metric
$\displaystyle\mbox{{\rm d}}s^{2}=-(1-\frac{2m}{r})\mbox{{\rm
d}}t^{2}+(1-\frac{2m}{r})^{-1}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm
d}}\Omega^{2}\ ,$ (62)
where $m$ is the black hole mass.
### V.2 No-hair theorem: a re-interpretation and a conjecture
We can now re-interpret the no-hair theorem proved previously in the theory
(33). If we consider the canonical theory (36) we can obviously use the
standard no-hair theorem. In that case there is no non-trivial solution for
the scalar field $\pi$ and the only spherically symmetric solution is the
Schwarzschild solution
$\displaystyle\mbox{{\rm d}}s^{2}=-(1-\frac{2m}{r})\mbox{{\rm
d}}t^{2}+(1-\frac{2m}{r})^{-1}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm
d}}\Omega^{2}\ .$ (63)
Of course, the theories (33) and (36) are equivalent only up to fist order in
the small derivative perturbative regime (21). Hence, the no-hair theorem for
the canonical theory (36) can only be used to easily conclude that black hole
solutions in (33) cannot have perturbative Slotheonic hairs. This provides a
non-trivial confirmation of our direct proof of the no-hair theorem in Sec.V
for spherically symmetric black holes and it automatically extends to non-
spherically symmetric black holes, within the perturbative regime. This
encourages us to conjecture that there are no black hole solutions with non-
perturbative Slotheonic hairs. We leave the investigation of this important
conjecture for future work.
## VI Asymptotic local shift symmetry and inflation
Standard inflationary models enjoy an asymptotic shift symmetry of the scalar
creminelli
$\displaystyle\pi\rightarrow\pi+c\ ,$ (64)
due to the fact that, under such a shift, the Inflaton potential only shifts
at next to leading order in the slow roll expansion. This shift however, does
not protect the theory under new derivative couplings and it is expected to be
anyway broken by Quantum Gravity effects. In order to avoid these potential
problems one may then try to “gauge” the symmetry (64) to
$\displaystyle\pi\rightarrow\pi+f(x)\ .$ (65)
In a spatially flat Friedamn-Robertson-Walker (FRW) geometry
$\displaystyle\mbox{{\rm d}}s^{2}=-\mbox{{\rm d}}t^{2}+a(t)^{2}\mbox{{\rm
d}}\vec{x}\cdot\mbox{{\rm d}}\vec{x}\ ,$ (66)
the field and gravity evolution equations are new
$\displaystyle H^{2}=\frac{1}{3M_{\rm
P}^{2}}\left[\frac{\dot{\pi}^{2}}{2}(1+9\frac{H^{2}}{M^{2}})+V\right],$ (67)
$\displaystyle\partial_{t}\left[a^{3}\dot{\pi}(1+3\frac{H^{2}}{M^{2}})\right]=-a^{3}V^{\prime}\
,$ (68)
where $H=\frac{\dot{a}}{a}$ and $(\dot{})=\mbox{{\rm d}}/\mbox{{\rm d}}t$.
In GEF of yuki ; new ; uv , the Inflaton (a Slotheon) is non-minimally coupled
to gravity as in (33) so that slow roll may be naturally obtained. With this
coupling, even very steep potentials for the scalar field, $V(\pi)$, would
produce a successful inflationary scenario, thanks to a huge gravitational
friction acting on the Inflaton during inflation. Specifically, one can then
always choose the mass $M$ small enough such that, during inflation,
$H^{2}/M^{2}\gg 1$. Note, as explained before, that no strong coupling happens
here thanks to the canonical normalization of the field $\pi$ new ; yuki .
This regime is called the high friction regime yuki . In this regime, for any
given potential $V$, a quasi-de Sitter solution always exists for $M$ small
enough. This is the basis for the New Higgs Inflation new and the UV-
protected Inflation uv . A quasi-de Sitter background implies that the slow
roll parameters are small, i.e.
$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}\ll 1\ ,\
\delta\equiv\Big{|}\frac{\ddot{\pi}}{H\dot{\pi}}\Big{|}\ll 1\ .$ (69)
We will firstly focus on the case in which the Inflaton potential has small
curvatures (chaotic type inflation), at least during inflation. We then ask
that the “canonical” slow roll conditions are satisfied
$\displaystyle\epsilon_{\rm can}\equiv\frac{V^{\prime 2}}{2V^{2}}M_{\rm
P}^{2}\ll 1\ ,\ \eta_{\rm can}\equiv\frac{V^{\prime\prime}}{V}M_{\rm P}^{2}\ll
1\ ,$ (70)
and assume a monomial potential for the Inflaton so that the above conditions
generically require $\pi\gg M_{\rm P}$.
In high friction limit ($H\gg M$), during slow roll, one finds that yuki
$\displaystyle\epsilon\simeq\frac{3}{2}\frac{\dot{\pi}^{2}}{M^{2}M_{\rm
P}^{2}}\simeq\epsilon_{\rm can}\frac{M^{2}}{3H^{2}}\ll 1\ ,\
V=V_{0}\left[1+{\cal O}(\sqrt{\epsilon})\frac{\delta\pi}{M_{\rm P}}\right]\ ,$
(71)
where $\delta\pi$ is a shift on the background value for $\pi$ and $\epsilon$
is the true slow roll parameter defined in (69). Thus, it is exactly in this
regime that the symmetry (26) is realized for the kinetic and gravitational
parts of the action as, in this regime, $\varepsilon\sim\epsilon\ll 1$. The
potential term also would break the symmetry (26) only at higher order in slow
roll. This can be easily seen from the action. There, the potential term would
shift as
$\displaystyle V\sqrt{-g}\rightarrow V\sqrt{-g}\left(1+{\cal
O}(\sqrt{\epsilon})\frac{f}{M_{\rm P}}\right)\ ,$ (72)
thanks to (71). In other words, the local shift symmetry (26) is only softly
broken by the potential if (70) are satisfied.
Let us now suppose that the potential generating inflation violates the
conditions (70). For monomial potential this would mean sub-Planckian field
values. The GEF mechanism would nevertheless work in order to fulfill (69) for
$M$ small enough. This can be easily seen from the first equation in (71),
which is always valid in high friction limit yuki . In this case the symmetry
(26) would in general be badly broken by the potential, unless the potential
does not introduce any self interactions. In other words, the symmetry (26)
may still be softly broken by a mass term, i.e. in the case in which
$V=V_{0}\pm\frac{1}{2}m^{2}\pi^{2}$, for any field value during inflation.
Because no self-interactions are introduced in the potential, one would indeed
expect that quantum corrections to the propagator would still be suppressed by
slow roll, i.e. they would still obey the asymptotic symmetry (26) during
inflation 888For a similar discussion see also galinf .. Thanks to that, the
UV-protected inflation of uv ; yuki , has an extra quantum protection in the
high friction limit: the local shift symmetry (26).
We then found that during inflation and in high friction regime, the
Slotheonic Lagrangian (45) enjoys an asymptotic gauge symmetry (26) protecting
chaotic type inflationary set-up and inflationary set-up with mass potentials
from quantum corrections to both the potential and the kinetic terms. Note
that extra-derivative couplings that could be added and are invariant under
the approximate symmetry (26), can only come from further expanding the action
(23). Therefore, extra-derivative couplings may only modify the equation to
higher order in slow roll, in this sense they are completely negligible and
the inflationary trajectory is stable. One may still wonder about couplings of
the Slotheon to matter fields. These couplings would generically produce a
(Coleman-Weinberg) logarithmically corrected potential for the effective
canonically normalized field $\psi=\frac{H}{M}\pi$. The strength of these
corrections depends upon the particular couplings chosen and may or may not be
important for the Inflationary evolution. We leave this important discussion
for a future work.
Finally, let us comment on possible renormalization of the mass parameter
suppressing the non-minimal coupling ${\cal L}_{2}^{\rm nm}$. Since, during
the high friction regime, the graviton is still canonically normalized with
the Planck scale (see Sec. III.1), we expect that the running of $M$ is
suppressed by the Planck scale and therefore negligible for an inflationary
trajectory where the total energy is far below $M_{\rm P}$. The study of the
exact running of $M$ is left for future work.
## Acknowledgements
We wish to thank Alex Vikman for careful reading and important comments on the
first draft of the paper. CG wishes to thank Paolo Creminelli and Guido
D’Amico, for important comments. CG and PM would also like to thank Yuki
Watanabe for useful discussions. PM wishes to thanks Claudia de Rham for
comments on the first draft. The Authors thanks Alex Kehagias for earlier
participation to the project and important discussions. Finally, the Authors
wish to thank the anonymous referee for pointing out an extra Galilean
invariant term. CG and PM are supported by Alexander Von Humboldt Foundation.
LM is partially supported by the ERC Advanced Grant n.226455 “Superfields”, by
the Italian MIUR-PRIN contract 20075ATT78 and by the NATO grant
PST.CLG.978785.
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|
arxiv-papers
| 2011-08-05T20:16:41 |
2024-09-04T02:49:21.343702
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cristiano Germani, Luca Martucci, Parvin Moyassari",
"submitter": "Luca Martucci",
"url": "https://arxiv.org/abs/1108.1406"
}
|
1108.1428
|
# Fusion symmetric spaces and subfactors
Hans Wenzl Department of Mathematics
University of California
San Diego, California hwenzl@ucsd.edu
###### Abstract.
We construct analogs of the embedding of orthogonal and symplectic groups into
unitary groups in the context of fusion categories. At least some of the
resulting module categories also appear in boundary conformal field theory. We
determine when these categories are unitarizable, and explicitly calculate the
index and principal graph of the resulting subfactors.
This paper is a sequel of our previous paper [W4], where we introduced a
$q$-deformation of Brauer’s centralizer algebra for orthogonal and symplectic
groups; this algebra had already appeared more or less before in [Mo], see
also discussion in [W4]. It is motivated by finding a deformation of
orthogonal or symplectic subgroups of a unitary group which is compatible with
the standard quantum deformation of the big group. This has been done before
on the level of coideal subalgebras of Hopf algebras by Letzter. However, our
categorical approach also allows us to extend this to the level of fusion
tensor categories, where we find finite analogs of symmetric spaces related to
the already mentioned groups. Moreover, we can establish $C^{*}$ structures,
necessary for the construction of subfactors, in this categorical setting;
this is not so obvious to see in the setting of co-ideal algebras.
It is well-known how one can use a subgroup $H$ of a (for simplicity here)
finite group $G$ to construct a module category of the representation category
${\rm Rep\ }G$ of $G$. This module category also appears in the context of
subfactors of II1 von Neumann factors as follows: Let $R$ be the hyperfinite
II1 factor, and let $\mathcal{N}=R^{G}\subset\mathcal{M}=R^{H}$ be the fixed
points under outer actions of $G$ and $H$. Then the category of
$\mathcal{N}-\mathcal{N}$ bimodules is equivalent to ${\rm Rep\ }G$, and the
module category is given via the $\mathcal{M}-\mathcal{N}$ bimodules of the
inclusion $\mathcal{N}\subset\mathcal{M}$; its simple objects are labeled by
the irreducible representations of $H$. In particular, an important invariant
called the principal graph of the subfactor is determined by the restriction
rules for representations from $G$ to $H$. Important examples of subfactors
were constructed from fusion categories whose Grothendieck semirings are
quotients of the ones of semisimple Lie groups. So a natural question to ask
is whether one can perform a similar construction in this context. More
precisely, can we find restriction rules for type $A$ fusion categories which
describe a subfactor as before, and which will approach in the classical limit
the usual restriction rules from $U(N)$ to $O(N)$.
We answer this question in the positive in this paper via a fairly elementary
construction. We show that certain semisimple quotients of the $q$-Brauer
algebras have a $C^{*}$ structure and contain $C^{*}$-quotients of Hecke
algebras of type $A$. The subfactor is then obtained as the closure of
inductive limits of such algebras. Due to its close connection to Lie groups,
we can give very explicit general formulas for its index and its first
principal graph. Observe that the Lie algebra $\mathfrak{s}l_{N}$ decomposes
as an $\mathfrak{s}o_{N}$ module into the direct sum
$\mathfrak{s}o_{N}\oplus\mathfrak{p}$, where $\mathfrak{p}$ is a simple
$\mathfrak{s}o_{N}$-module. Then the index can be expressed explicitly in
terms of the weights of $\mathfrak{p}$, see Theorem 3.4. As before in the
group case, it can be interpreted as the quotient of the dimension of the
given fusion category by the sum of the squares of $q$-dimensions of
representations of orthogonal or symplectic subgroups whose labels are in the
alcove of a certain affine reflection group; however in our case, there is no
corresponding tensor category for the denominator, and the $q$-dimensions
differ from the ones of the corresponding quantum groups. Also, the
restriction rules for the corresponding bimodules of this subfactor, the first
principal graph, can be derived from the classical restriction rules via an
action of the already mentioned affine reflection group, similarly as it was
done before for tensor product rules for fusion categories. However, in our
case, the affine reflection comes from the highest short root of the
corresponding Lie algebra in the non-simply laced case; it is also different
from the one for fusion categories in the even dimensional orthogonal case.
In the case corresponding to $O(2)$, we obtain the Goodman-de la Harpe-Jones
subfactors for Dynkin diagrams $D_{n}$. We similarly also obtain a series of
subfactors for other even-dimensional orthogonal groups. Perhaps a little
surprisingly, there does not seem to be an analogous construction which would
correspond to the group $SO(N)$ with $N$ even. Our examples for the odd-
dimensional orthogonal group and for symplectic groups seem to be closely
related to examples constructed by Feng Xu [X] and Antony Wassermann [Wa] by
completely different methods. Several explicit cases are discussed in detail
at the end of the paper, as well as their connections to other approaches,
coming from boundary conformal field theory, subfactors and tensor categories.
The first chapter mostly contains basic material from subfactor theory which
will be needed later. In the second chapter we review and expand material on
the $q$-Brauer algebra as defined in [W4], see also [Mo]. In particular, we
define $C^{*}$-structures for certain quotients and use that to construct
subfactors. The third chapter is mainly concerned with the finer structure of
these subfactors, such as explicit closed formulas for the index and
calculation of the first principal graph. The same techniques would also
extend to other examples, such as the ones in [X].
$Acknowledgments:$ It is a pleasure to thank Antony Wassermann, David Jordan,
Viktor Ostrik and Feng Xu for useful references, and Fred Goodman also for
technical advice.
## 1\. II1 factors
### 1.1. Periodic commuting squares
We will construct subfactors using the set-up of periodic commuting squares as
in [W1]. More precisely, we assume that we have increasing sequences of finite
dimensional $C^{*}$ algebras $A_{1}\subset A_{2}\subset\ ...$ and
$B_{1}\subset B_{2}\subset\ ...$ such that $A_{n}\subset B_{n}$ for all
$n\in{\mathbb{N}}$. Let $\Lambda_{n}$ resp $\tilde{\Lambda}_{n}$ be labeling
sets for the simple components of $B_{n}$ and $A_{n}$ respectively. Let
$G_{n}$ be the inclusion matrix for $A_{n}\subset B_{n}$. If we write a
minimal idempotent $p_{\mu}\in A_{n,\mu}$ as a sum of minimal mutually
commuting idempotents of $B_{n}$, then the entry $g_{\lambda\mu}$ of $G_{n}$
denotes the number of those idempotents which are in $B_{n,\lambda}$. We say
that our sequences of algebras are periodic with period $d$, if there exists
an $n_{0}\in{\mathbb{N}}$ such that for any $n>n_{0}$ we have bijections $j$
between $\Lambda_{n}$ and $\Lambda_{n+d}$ as well as between
$\tilde{\Lambda}_{n}$ and $\tilde{\Lambda}_{n+d}$ which do not change the
inclusion matrices for $A_{n}\subset B_{n}$ as well as for $A_{n}\subset
A_{n+1}$ and $B_{n}\subset B_{n+1}$. This means, in particular, that
$g_{j(\lambda)j(\mu)}=g_{\lambda\mu}$ for all $\lambda\in\Lambda_{n}$,
$\mu\in\tilde{\Lambda}_{n}$, $n>n_{0}$.
The trace functional defines inner products on the algebras $A_{n}$ and
$B_{n}$ by $(b_{1},b_{2})=tr(b_{1}^{*}b_{2})$. Let $e_{A_{n+1}}$ and
$e_{B_{n}}$ be the orthogonal projections onto the subspaces $A_{n+1}$ and
$B_{n}$ of $B_{n+1}$. Then the commuting square condition says that
$e_{A_{n+1}}e_{B_{n}}=e_{A_{n}}=e_{B_{n}}e_{A_{n+1}}$ for all
$n\in{\mathbb{N}}$. Finally, we also note that the trace $tr$ is uniquely
determined on $A_{n}$ and $B_{n}$ by its weight vectors ${\bf a}_{n}$ and
${\bf b}_{n}$ which are defined as follows: Let $p_{\mu}$ be a minimal
idempotent in the simple component of $A_{n}$ labeled by $\mu$. Then we define
$a_{n,\mu}=tr(p_{\mu})$, and ${\bf a}_{n}=(a_{n,\mu})_{\mu}$, where $\mu$ runs
through a labeling set of the simple components of $A_{n}$. The weight vector
${\bf b}_{n}$ for $B_{n}$ is defined similarly. The following proposition
follows from [W1], Theorem 1.5 (where the matrix $G=(g_{\lambda\mu})$ defined
here would correspond to the matrix $G^{t}$ in [W1]):
###### Proposition 1.1.
Under the given conditions, we get a subfactor $\mathcal{N}\subset\mathcal{M}$
whose index $[\mathcal{M}:\mathcal{N}]$ is equal to $\|{\bf
a}_{n}\|^{2}/\|{\bf b}_{n}\|^{2}$ for any sufficiently large $n$. Moreover, we
have $\sum g_{\lambda\mu}a_{n,\lambda}=[\mathcal{M}:\mathcal{N}]b_{n,\mu}$.
### 1.2. Special periodic algebras
In general, it can be quite hard to determine finer invariants of the
subfactors, the so-called higher relative commutants (or centralizers) from
the generating sequence of algebras. However, under certain circumstances,
this can become quite easy. We describe such a set-up. It is a moderate
abstraction of an approach which has already been used before by a number of
authors. The reader familiar with tensor categories and module categories
should think of the algebras $A_{n}={\rm End}_{\mathcal{C}}(X^{\otimes n})$
and the algebras $B_{n}={\rm End}_{\mathcal{D}}(Y\otimes X^{\otimes n})$ for
$X$ an object in a $C^{*}$ tensor category $\mathcal{C}$, and $Y$ an object in
a module category ${\mathcal{D}}$ over $\mathcal{C}$. In the following, we
will make the following assumptions beyond the ones in the previous
subsection:
1. 1.
The algebras $A_{n}$ will be monoidal $C^{*}$-algebras. This means we have
canonical embeddings of $C^{*}$ algebras $A_{m}\otimes A_{n}\to A_{n+m}$ with
multiplicativity of the trace, i.e. $tr(a_{1}\otimes
a_{2})=tr(a_{1})tr(a_{2})$.
2. 2.
We have canonical embeddings $B_{m}\otimes A_{n}\to B_{n+m}$, again with
multiplicativity of the trace.
3. 3.
We have the commuting square condition for the sequences of algebras
$A_{n}\subset B_{n}$ and $1\otimes A_{n-1}\subset A_{n}$.
4. 4.
There exists $d\in{\mathbb{N}}$ and a projection $p\in A_{d}$ such that
$(1_{m}\otimes p)A_{m+d}(1\otimes p)\cong A_{m}$ and $(1_{m}\otimes
p)B_{m+d}(1\otimes p)\cong B_{m}$ for all $m\in{\mathbb{N}}$.
Examples for this set-up will be given at the end of this section and in
Section 2. Moreover, any finite depth subfactor
$\mathcal{N}\subset\mathcal{M}$ (see e.g. [GHJ], [EK] for definitions)
produces algebras for such a set-up as follows: Let $\mathcal{M}^{\otimes
n}=\mathcal{M}\otimes_{\mathcal{N}}\mathcal{M}\otimes_{\mathcal{N}}\ ...\
\otimes\mathcal{M}$ ($n$ factors). Obviously, $\mathcal{M}^{\otimes n}$ is an
$\mathcal{N}-\mathcal{N}$ as well as an $\mathcal{M}-\mathcal{N}$ bimodule.
One can check that for $A_{n}={\rm
End}_{\mathcal{N}-\mathcal{N}}\mathcal{M}^{\otimes n}\subset B_{n}={\rm
End}_{\mathcal{M}-\mathcal{N}}\mathcal{M}^{\otimes n+1}$ the axioms above are
satisfied; here the embedding is defined by letting the elements of $A_{n}$
act on the second to $(n+1)-st$ factor of $\mathcal{M}^{\otimes n+1}$. It is
also possible to define these algebras in connection of relative commutants in
the Jones tower of relative commutants (see e.g. [Bi] for details). Recall
that for factors $\mathcal{N}\subset\mathcal{M}$ the relative commutant (or
centralizer) $\mathcal{N}^{\prime}\cap\mathcal{M}$ is defined to be the set
$\\{b\in\mathcal{M},ab=ba$ for all $a\in\mathcal{N}\\}$.
###### Lemma 1.2.
The subfactor $\mathcal{N}\subset\mathcal{M}$ generated from the sequences of
algebras $1_{m}\otimes A_{n}\subset B_{n+m}$ has relative commutant $B_{m}$.
The same statement also holds with $B_{n+m}$ and $B_{m}$ in the last sentence
replaced by $A_{n+m}$ and $A_{m}$.
$Proof.$ This is essentially the proof used for Theorem 3.7 in [W1]. Observe
that by induction on $r$ and assumption 4 above, we also have $(1_{m}\otimes
p^{\otimes r})X_{m+rd}(1_{m}\otimes p^{\otimes r})\cong X_{m}$ for $X=A,B$. It
follows from Theorem 1.6 of [W1] that the dimension of the relative commutant
$\mathcal{N}^{\prime}\cap\mathcal{M}$ is at most equal to the dimension of
$B_{m}$. The claim follows from the fact that $B_{m}\otimes 1_{n}$ commutes
with $1_{m}\otimes A_{n}$ for all $n$.
### 1.3. Bimodules and principal graphs
We calculate the first principal graph for subfactors constructed in our set-
up, using fairly elementary methods from [W1] as well as the bimodule
approach. The latter was first used in the subfactor context by Ocneanu, see
e.g. [EK]. For the connection between bimodules and principal graphs, see [Bi]
and for more details compatible with our notation, see also [EW]. While most
of this section has already appeared before implicitly or explicitly, the
presentation in our set-up might be useful also in other contexts.
Pick $k$ large enough so that $m=kd>n_{0}$. Hence the inclusion matrices for
$A_{rd}\subset B_{rd}$ coincide for all $r\geq k$ using the bijection of
simple components as described in Section 1.1. Let $\Lambda_{m}$ and
$\tilde{\Lambda}_{m}$ be labelling sets for the simple components of $B_{m}$
and $A_{m}$ respectively. Let $\mathcal{N}$ and $\mathcal{M}$ be the factors
generated by the increasing sequences of algebras $A_{n}$ and $B_{n}$
respectively, see Prop. 1.1 or Lemma 1.2, with the $m$ there equal 0. Both of
these factors have a subfactor $\tilde{\mathcal{N}}$ generated by the
subalgebras $1_{m}\otimes A_{n}\subset A_{n+m}\subset B_{n+m}$. We now define
for each $\lambda\in\tilde{\Lambda}_{m}$ an $\mathcal{N}-\tilde{\mathcal{N}}$
bimdule $N_{\lambda}$ as follows: It is the Hilbert space completion of
$\mathcal{N}p_{\lambda}$ with respect to the inner product induced by $tr$,
where $p_{\lambda}$ is a minimal idempotent in $A_{m,\lambda}$, the simple
component of $A_{m}$ labeled by $\lambda$, with obvious left and right actions
by $\mathcal{N}$ and $\tilde{N}$. To ease notation, we shall often refer to it
as an $\mathcal{N}-\mathcal{N}$ bimodule, using the isomorphism between
$\tilde{\mathcal{N}}$ and $\mathcal{N}$ given by the trace preserving maps
$a\in A_{n}\mapsto 1_{m}\otimes a\in A_{n+m}$.
Similarly, we define $\mathcal{M}-\tilde{\mathcal{N}}$ bimodules $M_{\mu}$ for
any $\mu\in\Lambda_{m}$ which are Hilbert space completions of
$\mathcal{M}p_{\mu}$, where $p_{\mu}$ is a minimal idempotent in the simple
component $B_{m,\mu}$ of $B_{m}$. Finally, we define the inclusion numbers
$b_{\mu}^{\lambda}$ for elements $\lambda\in\tilde{\Lambda}$ and
$\mu\in\Lambda_{m}$ as usual (see Section 1.1).
###### Lemma 1.3.
The bimodules $N_{\lambda}$ and $M_{\mu}$ are irreducible
$\mathcal{N}-\tilde{\mathcal{N}}$ resp $\mathcal{M}-\tilde{\mathcal{N}}$
bimodules. We have the decomposition
$M_{\mu}\cong\bigoplus_{\lambda}b^{\lambda}_{\mu}N_{\lambda}$ as
$\mathcal{N}-\tilde{\mathcal{N}}$ modules.
$Proof.$ This is well-known (see e.g. [EW] for more details). It follows from
Lemma 1.2 that the endomorphism ring of the $\mathcal{M}-\tilde{\mathcal{N}}$
bimodule $\mathcal{M}$ is given by $B_{m}$. Hence the
$\mathcal{M}-\tilde{\mathcal{N}}$ bimodules $M_{\mu}$ are simple, as $p_{\mu}$
was chosen to be a minimal idempotent in $B_{m}$. One shows similarly that
also the $N_{\lambda}$’s are simple $\mathcal{N}-\tilde{\mathcal{N}}$
bimodules.
Observe that $\dim_{\mathcal{N}}N_{\lambda}=tr(p_{\lambda})$ and
$\dim_{\mathcal{M}}M_{\mu}=tr(p_{\mu})$ (see e.g. [J]). Now if $p_{\lambda}$
is a minimal idempotent in $A_{m}$, it follows from the definitions that ${\rm
Ind}_{\mathcal{N}}^{\mathcal{M}}N_{\lambda}:=\mathcal{M}p_{\lambda}$ is
isomorphic as an $\mathcal{M}-\tilde{\mathcal{N}}$ bimodule to the direct sum
$\oplus b^{\lambda}_{\mu}M_{\mu}$. By Frobenius reciprocity, see e.g. [EK],
[Bi] it follows that the module $N_{\lambda}$ appears with multiplicity
$b^{\lambda}_{\mu}$ in $\mathcal{M}_{\mu}$, viewed as an
$\mathcal{N}-\tilde{\mathcal{N}}$ bimodule. Hence the
$\mathcal{N}-\tilde{\mathcal{N}}$ bimodule $M_{\mu}$ has a submodule which is
isomorphic to $\bigoplus_{\lambda}b^{\lambda}_{\mu}N_{\lambda}$. But as
$M_{\mu}$ has $\mathcal{N}$-dimension $[\mathcal{M}:\mathcal{N}]tr(p_{\mu})$,
it coincides with this submodule, by Proposition 1.1.
###### Theorem 1.4.
Let $\mathcal{N}\subset\mathcal{M}$ be the subfactor generated by sequences of
algebras $A_{n}\subset B_{n}$ satisfying the conditions in Section 1.2. Then
its first principal graph is given by the inclusion graph for $A_{kd}\subset
B_{kd}$ for sufficiently large $k$.
$Proof.$ It is well-known that the first principal graph is given by the
induction-restriction graph of $\mathcal{M}-\mathcal{N}$ and
$\mathcal{N}-\mathcal{N}$ bimodules appearing in the tensor products
$\mathcal{M}^{\otimes n}$, $n\in{\mathbb{N}}$, where $\mathcal{M}^{\otimes
n}=\mathcal{M}\otimes_{\mathcal{N}}\mathcal{M}\otimes\ ...\
\otimes_{\mathcal{N}}\mathcal{M}$ ($n$ factors), see [EK], [Bi]. Obviously,
this graph does not change if we replace all $X-\mathcal{N}$ bimodules $H$ in
this setting by $X-q\mathcal{N}q$ bimodules $Hq$, for
$X=\mathcal{M},\mathcal{N}$ and $q$ a nonzero projection in $\mathcal{N}$. The
claim can now be shown for $q=p^{\otimes k}$ where $k$ is chosen large enough
so that $kd>n_{0}$, using Lemma 1.3.
Recall that many examples come from module tensor categories, where
$A_{n}={\rm End}_{\mathbb{C}}(X^{\otimes n})$ and $B_{n}={\rm
End}(Y\otimes(X^{\otimes n})$ for an object $X$ in a tensor category
${\mathbb{C}}$ and an object $Y$ in the module category ${\mathcal{D}}$ over
${\mathbb{C}}$. In this setting, the weight vectors of our trace are given by
$a_{n,\lambda}=\tilde{d}_{\lambda}/x^{n}$ and $b_{n,\mu}=d_{\mu}/yx^{n}$ for
positive quantities $d_{\mu},\tilde{d}_{\lambda},x$ and $y$. Then we have
###### Corollary 1.5.
Assuming the conditions for the trace weights as just given, we have
subfactors $\mathcal{N}\subset\mathcal{M}_{\mu}$ with index
$[\mathcal{M}_{\mu}:\mathcal{N}]=d_{\mu}^{2}[\mathcal{M}:\mathcal{N}]$, with
$\mathcal{N}\subset\mathcal{M}$ as in Theorem 1.4.
###### Remark 1.6.
There is also a second important invariant for
$\mathcal{N}\subset\mathcal{M}$, the dual principal graph. It can be
analogously defined as an induction-restriction graph between irreducible
$\mathcal{M}-\mathcal{M}$ and $\mathcal{M}-\mathcal{N}$ bimodules appearing in
the tensor powers $\mathcal{M}^{\otimes n}$. Its calculation is more difficult
than the first principal graph. This is quite similar to the corresponding
problem for subfactors coming from conformal inclusions and related
constructions, see e.g [X1], [BEK], [EW]. We plan to study this problem in a
future publication via suitable adaptions of techniques in those papers.
### 1.4. The $GHJ$-construction
We give a well-known and well-studied example for our current set-up, which
was first constructed in [GHJ]. Let $G$ be a matrix with nonnegative integer
entries and norm less than 2. It is well-known that such matrices are
classified by Coxeter graphs of type $ADE$. We assume that the columns of $G$
are indexed by the even vertices, and the rows by the odd vertices. We define
$C^{*}$-algebras $B_{n}$ by $B_{0}={\mathbb{C}}^{v_{e}}$, and $B_{1}=\oplus
M_{d_{j}}$, where $v_{e}$ is the number of even vertices, and the summands of
$B_{1}$ are labelled by the odd vertices $j$, whose dimension $d_{j}$ is equal
to the number of even vertices to which $j$ is connected. The embedding
$B_{o}\subset B_{1}$ is given by the inclusion matrix $G$. Then we define
recursively $B_{n+1}$ via Jones’ basic construction [J] for $B_{n-1}\subset
B_{n}$. Here the trace on $B_{n}$ is the unique normalized trace whose values
on minimal idempotents are given by the Perron-Frobenius vector of $G^{t}G$ or
$GG^{t}$, depending on whether $n$ is even or odd, and the vector is
normalized such that $tr(1)=1$. Then the algebra $B_{n+1}$ is generated by
$B_{n}$, acting on itself via left multiplication, and the orthogonal
projection $e_{n}$ onto the subspace $B_{n-1}$ of $B_{n}$, with respect to the
inner product coming from the trace. The algebra $A_{n}$ is defined to be the
subalgebra generated by the identity 1 and the projections $e_{i}$, $1\leq
i<1$. It is well-known that these algebras satisfy the commuting square
condition, that they are periodic with periodicity 2, and that the Jones
projections $e_{i}$ satisfy the conditions of the projection $p$ in Section
1.2. This has already been shown in [GHJ].
## 2\. $q$-Brauer algebras
### 2.1. Definitions
Fix $N\in{\mathbb{Z}}$ and let $[N]=(q^{N}-q^{-N})/(q-q^{-1})$, where $q$ is
considered to be a complex number. We denote by $H_{n}(q^{2})$ the Hecke
algebra of type $A_{n-1}$. It is given by generators $g_{1},g_{2},\ ...\
g_{n-1}$ which satisfy the usual braid relations and the quadratic relation
$g_{i}^{2}=(q^{2}-1)g_{i}+q^{2}$. The $q$-Brauer algebra $Br_{n}(N)$ is the
complex algebra defined via generators $g_{1},g_{2},\ ...\ g_{n-1}$ and $e$
and relations
1. (H)
The elements $g_{1},g_{2},\ ...\ g_{n-1}$ satisfy the relations of the Hecke
algebra $H_{n}(q^{2})$.
2. (E1)
$e^{2}=[N]e$,
3. (E2)
$eg_{i}=g_{i}e$ for $i>2$, $eg_{1}=q^{2}e$, $eg_{2}e=q^{N+1}e$ and
$eg_{2}^{-1}e=q^{-1-N}e$.
4. (E3)
$g_{2}g_{3}g_{1}^{-1}g_{2}^{-1}e_{(2)}=e_{(2)}=e_{(2)}g_{2}g_{3}g_{1}^{-1}g_{2}^{-1}$,
where $e_{(2)}=e(g_{2}g_{3}g_{1}^{-1}g_{2}^{-1})e$.
It is easy to see that this algebra coincides with the algebra defined in [W4]
after substituting $q$ there by $q^{2}$, and $e$ there by $q^{1-N}e$ (with the
$q$ of this paper); this is also compatible with the different definition of
$[N]$ in [W4]. We have chosen this parametrization as it will make it easier
to define a $*$ structure on it. More precisely, if $|q|=1$, there exists a
complex conjugate antiautomorphism $b\mapsto b^{*}$ on $Br_{n}(N)$ defined by
(2.1) $e^{*}=e,\quad g_{i}^{*}=g_{i}^{-1},\ 1\leq i<n.$
It is easy to check at the relations that this operation is well-defined.
### 2.2. Molev representation
We give a representation of our algebra $Br_{n}(N)$ in ${\rm End}(V^{\otimes
n})$, where $V={\mathbb{C}}^{N}$. For this we use the matrices used by Molev
in [Mo] for the definition of his $q$-deformation of Brauers’ centralizer
algebra. His defining relations are slightly different from ours; but Molev
has informed the author that our algebra satisfies the relations of his
algebra. It turns out that also his matrices satisfy the relations of our
algebras, which we will outline here. Let $R$ be the well-known solution of
the quantum Yang-Baxter equation for type $A$. For simplicity we will use this
notation for what is often denoted as $\check{R}$. If $E_{ij}$ are the matrix
units for $n\times n$ matrices, we define the following elements in ${\rm
End}(V^{\otimes 2})$:
$R\ =\ \sum_{i}qE_{ii}\otimes E_{ii}\ +\ \sum_{i\neq j}E_{ij}\otimes E_{ji}\
+\ \sum_{i<j}(q-q^{-1})E_{ii}\otimes E_{jj},$
and
$Q=\sum_{i,j}q^{N+1-2i}\ E_{ij}\otimes E_{ij}.$
Moreover, if $A\in{\rm End}(V^{\otimes 2})$, we define the operator
$A_{i}\in{\rm End}(V^{\otimes n})$ by
$A_{i}\ =\ 1_{i-1}\otimes A\otimes 1_{n-1-i},$
where $1_{k}$ is the identity on $V^{\otimes k}$. Then we have the following
proposition, all of whose essential parts were already proved in [Mo].
However, the relations for our algebras are slightly different, so we give
some of the adjustments of the work in [Mo] to our context below.
###### Proposition 2.1.
The map $g_{i}\mapsto qR_{n-i}$ and $e\mapsto Q_{n-1}$ defines a
representation $\Phi$ of $Br_{n}(N)$. It specializes to the usual
representation of Brauer’s centralizer algebra in ${\rm End}(V^{\otimes n})$
for $q=1$.
$Proof.$ Most of the relations are already known or are easy to check. E.g. it
is well-known that the matrices $qR_{i}$ satisfy the relations of the Hecke
algebra $H_{n}(q^{2})$. Relation $(E1)$ is checked easily, and also the
relations in $(E2)$ are fairly straightforward to check. It suffices to check
$(E3)$ for $n=4$. For this observe that by [Mo], (4.16), we have
$Q_{3}R_{2}R_{3}R_{1}R_{2}Q_{3}=Q_{1}Q_{3}+q^{N+1}(q-q^{-1})Q_{3}(R_{1}+q^{-1}1),$
in our notation. Using the relation $R_{i}=R_{i}^{-1}+(q-q^{-1})1$ for the
second and third factor of the left hand side, one derives from this
$Q_{3}R_{2}^{-1}R_{3}^{-1}R_{1}R_{2}Q_{3}=Q_{1}Q_{3}.$
To check relation $(E3)$, observe that
$R_{1}R_{2}(v_{i}\otimes v_{i}\otimes v_{j}\otimes v_{j})\ =\
R_{3}R_{2}(v_{i}\otimes v_{i}\otimes v_{j}\otimes v_{j}),$
where $(v_{i})$ is the standard basis for ${\mathbb{C}}^{N}=V$. One derives
from this that $R_{2}^{-1}R_{1}^{-1}R_{3}R_{2}Q_{1}Q_{3}=Q_{1}Q_{3}$.
Moreover, the same calculations above also work with $R_{i}$ replaced by
$R_{i}^{-1}$ and $Q_{j}$ replaced by its transpose $Q_{j}^{T}$. Hence one can
show as before that
$R_{2}R_{3}R_{1}^{-1}R_{2}^{-1}Q_{1}^{T}Q_{3}^{T}=Q_{1}^{T}Q_{3}^{T}$.
Transposing this, using $R_{i}^{T}=R_{i}$ shows the last part of the claim.
### 2.3. Quotients
We can now rephrase the main results of [W4] in our notation as follows:
###### Theorem 2.2.
(a) There exists a well-defined functional $tr$ on $Br_{n}(N)$ defined
inductively by $tr(g_{1})=q^{N+1}/[N]$, $tr(e)=1/[N]$ and
$tr(bg_{n})=tr(b)tr(g_{n})$ for all $b\in Br_{n}(N)$.
(b) Let $\overline{Br}_{n}(N)=Br_{n}(N)/I_{n}$, where $I_{n}$ is the
annihilator ideal of $tr$. Then $\overline{Br}_{n}(N)$ is semisimple and the
inclusion $\overline{Br}_{n}(N)\subset\overline{Br}_{n+1}(N)$ is well-defined
for all $n$.
It is possible to explicitly describe the structure of the quotients
$\overline{Br}_{n}=\overline{Br}_{n}(N)$. To do so, we need the following
definitions for the labeling sets of simple representations. More
conceptually, the labeling sets $\Lambda(N,\ell)$ consist of all such diagrams
$\lambda$ for which the quantities $d_{\mu}(q)\neq 0$ for any subdiagrams
$\mu\subset\lambda$ including $\lambda$ itself, where $q^{2}$ is a primitive
$\ell$-th root of unity and the $d_{\mu}$’s are defined in Section 2.5.
###### Definition 2.3.
Fix integers $N$ and $\ell$ satisfying $1<|N|<\ell$.
(i) The set $\tilde{\Lambda}(N,\ell)$ consists of all Young diagrams with
$\leq N$ rows such that the first and $N$-th row differ by at most $\ell-N$
boxes for $N>0$. If $N<0$, the Young diagrams have at most $|N|$ columns,
where the first and $|N|$-th column differ by at most $\ell-|N|$ boxes.
(ii) The set $\Lambda(N,\ell)$ consists of all Young diagrams $\lambda$ with
$\lambda_{i}$ boxes in the $i$-th row and $\lambda_{j}^{\prime}$ boxes in the
$j$-th column which satisfy
(a) $\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq N$ and
$\lambda_{1}\leq(\ell-N)/2$ if $N>0$ and $\ell-N$ even,
(b) $\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq N$ and
$\lambda_{1}+\lambda_{2}\leq\ell-N$ if $N>0$ and $\ell-N$ odd,
(c) $\lambda_{1}\leq|N|/2$ and
$\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq\ell-|N|$ if $N<0$ is even,
(d) $\lambda_{1}+\lambda_{2}\leq|N|$ and
$\lambda^{\prime}_{1}+\lambda^{\prime}_{2}\leq\ell-|N|$ if $N<0$ is odd.
Diagrams which miss one of these inequalities only by the quantity one are
called boundary diagrams of $\Lambda(N,\ell)$; e.g. in case (a) if
$\lambda_{1}^{\prime}+\lambda_{2}^{\prime}=N+1$.
###### Theorem 2.4.
([W4], Section 5) Let $q^{2}$ be a primitive $\ell$-th root of unity, and let
$N$ be an integer satisfying $1<|N|<\ell$. Then the simple components of
$\overline{Br}_{n}=\overline{Br}_{n}(N)$ are labeled by the Young diagrams in
$\Lambda(N,\ell)$ with $n,n-2,n-4,...$ boxes. If $V_{n,\lambda}$ is a simple
$\overline{Br}_{n}$ module for such a diagram $\lambda$, it decomposes as a
$Br_{n-1}$ module as
(2.2) $V_{n,\lambda}\ \cong\ \bigoplus_{\mu}V_{n-1,\mu},$
where $\mu$ runs through diagrams in $\Lambda(N,\ell)$ obtained by removing
or, if $|\lambda|<n$, also by adding a box to $\lambda$.
### 2.4. Path idempotents and matrix units
We will give some details about the proof of Theorem 2.4 which will also be
needed for further results. Observe that the restriction rule 2.2 implies that
a minimal idempotent $p_{\mu}$ in $\overline{Br}_{n-1,\mu}$ can be written as
a sum of minimal idempotents with exactly one in $\overline{Br}_{n,\lambda}$
for each diagram $\lambda$ in $\Lambda(N,\ell)$ which can be obtained by
adding or subtracting a box from $\mu$. This inductively determines a system
of minimal idempotents and matrix units of $\overline{Br}_{n}(q^{N},q)$
labeled by paths resp. pairs of paths in $\Lambda(N,\ell)$ of length $n$. Such
a path is defined to be a sequence of Young diagrams
$(\lambda^{(i)})_{i=0}^{n}$ where $\lambda^{(0)}$ is the empty Young diagram,
and $\lambda^{(i+1)}$ is obtained from $\lambda^{(i)}$ by adding or removing a
box. It follows from the restriction rule above that the dimension of
$V_{n,\lambda}$ is equal to the number of paths of length $n$ with
$\lambda^{(n)}=\lambda$, and that we can label a complete system of matrix
units for the simple component $\overline{Br}_{n,\lambda}$ by pairs of such
paths. We then have the following lemma:
###### Lemma 2.5.
For each pair of paths $t_{1},t_{2}$ in $\Lambda(N,\infty)$ with the same
endpoint we can define the matrix unit $E_{t_{1},t_{2}}$ as a linear
combination of products of generators over algebraic functions (rational for
path idempotents) in $q$ with poles only at roots of unity. More precisely,
the formula for $E_{t_{1},t_{2}}$ is well-defined for $q^{2}$ a primitive
$\ell$-th root of unity if both $t_{1}$ and $t_{2}$ are paths in
$\Lambda(N,\ell)$.
$Proof.$ This was proved in [W4], Section 5. As the result is not explicitly
stated as such, we give some details here. One observes that the two-sided
ideal generated by the element $\bar{e}\in\overline{Br}_{n+1}$ is isomorphic
to Jones’ basic construction for the algebras
$\overline{Br}_{n-1}\subset\overline{Br}_{n}$ (or, strictly speaking, by
certain conjugated subalgebras which are denoted by $i_{1}(\overline{Br}_{n})$
and $i_{2}(\overline{Br}_{n-1})$, see Section 5.2 in [W4]). One can then
define path idempotents and matrix units inductively as it was done in [RW],
Theorem 1.4 using the formulas for the weights of traces, which will also be
reviewed in Section 2.5; this is closely related to what is also known in
subfactor theory as the Ocneanu-Sunder path model [Su]. The complement of this
ideal is a quotient of the Hecke algebra $\bar{H}_{n+1}$ for which matrix
units already were more or less defined in [W1], p. 366.
###### Lemma 2.6.
Let $p_{[1^{N}]}$ be the minimal idempotent in $H_{N}$ corresponding to its
one-dimensional sign representation. Then we have $\bar{p}_{[1^{N}]}^{\otimes
2}\overline{Br}_{m+2N}\bar{p}_{[1^{N}]}^{\otimes 2}\cong\overline{Br}_{m}$ for
all $m>0$.
$Proof.$ Observe that if $p\in\overline{Br}_{2N,\emptyset}(N)$, the simple
component labeled by the empty Young diagram $\emptyset$, then it follows from
the restriction rule 2.2 (see also the equivalent version below Theorem 2.4)
by induction on $m$ that $p\overline{Br}_{m+2N}p\cong\overline{Br}_{m}$ for
all $m\geq 0$. Hence it suffices to show that $p_{[1^{N}]}^{\otimes 2}$ is
such an idempotent.
If $q=1$ and $N>0$, $\Phi(Br_{n}(N))$ coincides with the commutant of the
action of the orthogonal group $O(N)$ on $V^{\otimes n}$, which is semisimple.
Moreover, the trace $tr$ is just a multiple of the pull-back of the natural
trace on ${\rm End}(V^{\otimes n})$, so
$\Phi(Br_{n}(N))\cong\overline{Br}_{n}(N)$ at $q=1$. As $\Phi(p_{[1^{N}]})$
projects onto the one-dimensional determinant representation in $V^{\otimes
N}$, the claim follows easily in that case, using Brauer duality, i.e. the
fact that $\Phi(Br_{n}(N))$ is equal to the commutant of $O(N)$ on $V^{\otimes
n}$ for all $n$.
We will now use the fact that we can also define $Br_{n}(N)$ over the field of
rational functions ${\mathbb{C}}(q)$, see [W4]. It follows from Lemma 2.5 that
we can also define the path idempotents for $\overline{Br}_{n}(N)$ over that
field for paths of length $n$ in $\Lambda(N,\infty)$. As the rank of an
idempotent is an integer, the claim follows as well for $q$ a variable, and
for $q\in{\mathbb{C}}$ not a root of unity. But as
$p_{t}\bar{p}_{[1^{N}]}^{\otimes 2}p_{t}=0$ for any path $t$ of length $2N$ in
$\Lambda(N,\ell)$ which ends in $\lambda\neq\emptyset$, we also get the rank 0
for $\bar{p}_{[1^{N}]}^{\otimes 2}$ at $q^{2}$ a primitive $\ell$-th root of
unity in $\overline{Br}_{2N,\lambda}(N)$. This finishes the proof for $N>0$.
The proof for the symplectic case $N<0$ even goes the same way.
### 2.5. Weights of the trace
Using the character formulas of orthogonal groups, one can calculate the
weights of $tr$ for the algebras $Br_{n}(N)$, i.e. its values at minimal
idempotents of $Br_{n}(N)$. We will need the following quantities for a given
Young diagram $\lambda$
(2.3) $d(i,j)\ =\begin{cases}\lambda_{i}+\lambda_{j}-i-j&\text{if $i\leq
j$,}\\\ -\lambda_{i}^{\prime}-\lambda_{j}^{\prime}+i+j-2&\text{if
$i>j$.}\end{cases}$
Moreover, we define $h(i,j)$ to be the length of the hook in the Young diagram
$\lambda$ whose corner is the box in the $i$-th row and $j$-th column. We can
now restate [W4], Theorem 4.6 in the notations of this paper as follows:
###### Theorem 2.7.
The weights of the Markov trace $tr$ for the Hecke algebra
$\bar{H}_{n}(q^{2})$ are given by
$\tilde{\omega}_{\lambda}=\tilde{d}_{\lambda}/[N]^{n}$, where $|\lambda|=n$,
and for $\overline{Br}_{n}(N)$ they are given by
$\omega_{\lambda,n}=d_{\lambda}/[N]^{n}$, where
$\displaystyle\tilde{d}_{\lambda}\ =\
\prod_{(i,j)\in\lambda}\frac{[N+j-i]}{[h(i,j)]},\quad d_{\lambda}\ =\
\prod_{(i,j)\in\lambda}\frac{[N+d(i,j)]}{[h(i,j)]},$
where $\lambda$ runs through all the Young diagrams in
$\tilde{\Lambda}(N,\ell)$ with $n$ boxes for $\bar{H}_{n}(q^{2})$, and through
all Young diagrams in $\Lambda(N,\ell)$ with $n,n-2,n-4,...$ boxes. for
$\overline{Br}_{n}$.
###### Lemma 2.8.
The weights $\omega_{\lambda,n}$ are positive for all
$\lambda\in\Lambda(N,\ell)$ if and only if $q^{2}=e^{\pm 2\pi i/\ell}$ with
$\ell>N$ and
(a) $N>0$ and $\ell-N$ even or
(b) $N<0$ odd.
$Proof.$ The weights can be rewritten for our choice of $q$ as
$\omega_{\lambda,n}\ =\ \frac{\sin^{n}(\pi/\ell)}{\sin^{n}(N\pi/\ell)^{n}}\
\prod_{(i,j)\in\lambda}\frac{\sin(N+d(i,j))\pi/\ell}{\sin(h(i,j)\pi/\ell)}.$
As $h(i,j)\leq h(1,1)=\lambda_{1}+\lambda_{1}^{\prime}-1<\ell$ for all boxes
$(i,j)$ of $\lambda$, it follows that all factors in the formula above are
positive for $N>0$ (negative for $N<0$) except possibly the ones in the
numerator under the product. If $N>0$ and $\ell-N$ odd, one checks that for
the diagram $[\ell-N+1)/2]$ we have $\omega_{\lambda,|\lambda|}<0$. By the
same argument, one shows that $\omega_{\lambda,|\lambda|}<0$ for
$\lambda=[(|N|+1)/2]$ and $N<0$. In the other two cases, one checks that
$0<|d(i,j)|<\ell$ for all boxes $(i,j)$ of a diagram
$\lambda\in\Lambda(N,\ell)$.
### 2.6. $C^{*}$-quotients
###### Proposition 2.9.
If the weights $\omega_{\lambda,n}$ are positive for all
$\lambda\in\Lambda(N,\ell)$, the star operation defined by $e^{*}=e$ and by
$g_{i}^{*}=g_{i}^{-1}$ makes the quotients $\overline{Br}_{n}$ into $C^{*}$
algebras.
$Proof.$ The proof goes by induction on $n$, with the claims for $n=1$ and
$n=2$ easy to check. By [W4], the two-sided ideal $I_{n+1}$ generated by $e$
in $\overline{Br}_{n+1}$ is isomorphic to Jones’ basic construction for
$\overline{Br}_{n-1}\subset\overline{Br}_{n}$, see also the remarks before
Lemma 2.6. In particular, this ideal is spanned by elements of the form
$b_{1}eb_{2}$, with $b_{1},b_{2}\in i_{1}(\overline{Br}_{n})$, where
$i_{1}(a)=\Delta_{n+1}a\Delta_{n+1}^{-1}$, with $\Delta=(g_{1}g_{2}\
...g_{n-1})(g_{1}\ ...\ g_{n-2})\ ...\ g_{1}$. By induction assumption and
properties of Jones’ basic construction, this ideal has a $C^{*}$ structure
given by $(b_{1}eb_{2})^{*}=b_{2}^{*}eb_{1}^{*}$. This coincides with the $*$
operation defined before algebraically. It was shown in [W4] that
$\overline{Br}_{n+1}\cong I_{n+1}\oplus\bar{H}_{n+1}$, where $\bar{H}_{n+1}$
is a semisimple quotient of the Hecke algebra $H_{n+1}$ whose simple
components are labeled by the Young diagrams $\lambda\in\Lambda(N,\ell)$ with
$n+1$ boxes. All these simple representations satisfy the $(k,\ell)$ condition
in [W1]. It follows from that paper that the map $g_{i}^{*}=g_{i}^{-1}$
induces a $C^{*}$ structure for any such representation. This finishes the
proof.
###### Theorem 2.10.
For each choice of $N$ and $\ell$ with $q^{2}=e^{\pm 2\pi i/\ell}$, and for
each nonnegative integer $m$ we obtain a subfactor
$\mathcal{N}\subset\mathcal{M}$ with
$\mathcal{N}^{\prime}\cap\mathcal{M}=\overline{Br}_{m}$ and with index
$[\mathcal{M}:\mathcal{N}]\ =\ [N]^{m}\
\frac{\sum_{\mu\in\tilde{\Lambda}(N,\ell)}\tilde{d}_{\mu}^{2}}{\sum_{\lambda\in\Lambda(N,\ell)}d_{\lambda}^{2}},$
with notations as in Section 2.5. Moreover, its first principal graph is given
by the inclusion graph for $\bar{H}_{2Nk}\subset\overline{Br}_{2nk+m}$ for any
sufficiently large $k$.
$Proof.$ Let us first check conditions 1-4 in Section 1.2 with
$A_{n}=\bar{H}_{n}$ and $B_{n}=\overline{Br}_{n}(N)$ for $q=e^{\pi i/\ell}$
and $1<|N|<\ell$. Condition 1 is well-known and was checked in e.g.[W1].
Similarly, Cond. 2 follows from the results in [W4], using the map $b\otimes
g_{i}\in\overline{Br}_{m}\otimes\bar{H}_{n}\mapsto bg_{m+i}$. Condition 3
means that the conditional expectation from $\overline{Br}_{n+1}$ to
$\overline{Br}_{n}$ maps $\bar{H}_{n+1}$ onto $\bar{H}_{n}$. But as any
element of $\bar{H}_{n+1}$ can be written as a linear combination of elements
of the form $ag_{n}b$, with $a,b\in\bar{H}_{n}$, we have for any
$c\in\overline{Br}_{n}$ that
$tr(ag_{n}bc)=tr(g_{n})tr(abc)=tr(E_{\bar{H}_{n}}(ag_{n}b)c).$
Hence the commuting square condition is satisfied for any four algebras of the
type above. Finally, Condition 4 follows for $d=2N$ and the projection
$p=p_{[1^{N}]}^{\otimes 2r}$ from Lemma 2.6.
The periodicity condition for $\bar{H}_{n}$ was shown in [W1] by proving that
$\bar{p}_{[1^{N}]}\bar{H}_{m+N}\bar{p}_{[1^{N}]}\cong\bar{H}_{m}$. This
induces an injective map
$\tilde{\Lambda}(N,\ell)_{m}\to\tilde{\Lambda}(N,\ell)_{m+N}$ by adding a
column of $N$ boxes to the given Young diagram which has to become surjective
for sufficiently large $m$ by definition of $\tilde{\Lambda}(N,\ell)$. The
$2N$ periodicity for the algebras $\overline{Br}_{n}(N)$ follows similarly
using Lemma 2.6, or see [W4].
## 3\. $S$-matrix
We will need certain well-known identities, which can be found in [Kc], except
for one case, which is a variation of the other ones. Because of this, we
review the material in more detail. This might also be useful to the non-
expert reader, as the identities needed here can be derived by completely
elementary methods.
### 3.1. Lattices
Let $M\subset L\subset{\mathbb{R}}^{k}$ be two lattices of full rank. This
means that they are isomorphic to ${\mathbb{Z}}^{k}$ as abelian groups, and
each of them spans ${\mathbb{R}}^{k}$ over ${\mathbb{R}}$. Moreover, we assume
that we have an inner product on ${\mathbb{R}}^{k}$ such that $({\bf x},{\bf
y})\in{\mathbb{Z}}$ for all ${\bf x},{\bf y}\in M$. We define the dual lattice
$M^{*}$ to be the set of all ${\bf y}\in{\mathbb{R}}^{k}$ such that $({\bf
x},{\bf y})\in{\mathbb{Z}}$ for all ${\bf x}\in M$; the dual lattice $L^{*}$
is defined similarly. Obviously $M\subset L$ implies $L^{*}\subset M^{*}$.
Finally, we also assume that $A=L/M$ is a finite abelian group. Then each
$\gamma\in M^{*}$ defines a character of $A$ via the map $e^{\gamma}:{\bf
x}\in L\mapsto e^{2\pi i(\gamma,{\bf x})}$. In particular, one can identify
the group dual of $A$ with $M^{*}/L^{*}$. Define the matrix
$\SS=\frac{1}{|L:M|^{1/2}}(e^{\gamma}({\bf x}))$, where $\gamma$ and ${\bf x}$
are representatives for the cosets $M^{*}/L^{*}$ and $L/M$. Then $\SS$ is the
character matrix of $A$ up to a multiple and one easily concludes that it is
unitary. More precisely, we can view it as a unitary operator between Hilbert
spaces $V$ and $V^{*}$ with orthonormal bases labeled by the elements of $L/M$
and $M^{*}/L^{*}$ respectively.
### 3.2. Weights of traces
We will primarily be interested in lattices related to root, coroot and weight
lattices of orthogonal and symplectic groups. We define the lattices
(3.1) $Q=\\{{\bf x}\in{\mathbb{Z}}^{k},\ 2|\sum x_{i}\\}\quad{\rm and}\quad
P={\mathbb{Z}}^{k}\cup(\varepsilon+{\mathbb{Z}}^{k}),$
where $\varepsilon$ is the element in ${\mathbb{R}}^{k}$ with all its
coordinates equal to $1/2$. Observe that $P^{*}=Q$ with respect to the usual
scalar product of ${\mathbb{R}}^{k}$. Moreover, one can identify coroot and
weight lattices of $\mathfrak{s}o_{2k}$ or $\mathfrak{s}o_{2k+1}$ with $Q$ and
$P$ respectively. In particular, we define for any $\gamma\in P$ the
functional $e^{\gamma}:{\mathbb{R}}^{k}\to{\mathbb{C}}$ by $e^{\gamma}({\bf
x})=e^{2\pi i(\gamma,{\bf x})}$. The Weyl group of type $B_{k}$ acts as usual
via permutations and sign changes on the coordinates. Let
$a_{W}=\sum_{w}\varepsilon(w)w$, where $\varepsilon(w)$ is the sign of the
element $w$. Then the characters $\chi_{\lambda}$ for $\mathfrak{s}o_{2k+1}$
resp for $\mathfrak{s}p_{2k}$ are given by
$\chi_{\lambda}=a_{w}(e^{\lambda+\rho})/a_{w}(e^{\rho})$, where
$\rho=(k+1/2-i)$ for $\mathfrak{s}o_{2k+1}$ and $\rho=(k+1-i)$ for
$\mathfrak{s}p_{2k}$, and $W$ is the Weyl group of type $B_{k}$.
We will also need the somewhat less familiar character formulas for the full
orthogonal group $O(N)$: Recall that the irreducible representations of $O(N)$
are labeled by Young diagrams $\lambda$ with at most $N$ boxes in the first
two columns. $O(N)$-modules labeled by Young diagrams
$\lambda\neq\lambda^{\dagger}$ restrict to isomorphic $SO(N)$-modules if and
only if $\lambda_{1}^{\prime}=N-(\lambda^{\dagger})_{1}^{\prime}$ and
$\lambda_{i}^{\prime}=(\lambda^{\dagger})_{i}^{\prime}$ for $i>1$. Hence if
$g=exp({\bf x})$ is an element in $SO(N)$, it suffices to consider the
quantities $\chi_{\lambda}(g)=\chi_{\lambda}({\bf x})$ for $\lambda$ with at
most $k$ rows for $N=2k$ or $N=2k+1$. We can now express the weights of
Theorem 2.7 in terms of these characters; in fact the formulas in Theorem 2.7
were derived from these characters, see [Ko] and [W4].
###### Lemma 3.1.
Let $d_{\lambda},\ \tilde{d}_{\lambda}$ be as in Theorem 2.7 for $q=e^{\pi
i/\ell}$. Moreover, we define for $|N|=2k$ or $N=2k+1$ the vector
${\rho\check{}}\in{\mathbb{R}}^{k}$ by ${\rho\check{}}=((|N|+1)/2-i)_{i}$. By
the discussion above, it suffices to evaluate
$\chi^{O(N)}({\rho\check{}}/\ell)$ for Young diagrams $\lambda$ with
$\lambda_{1}^{\prime}\leq N/2$, which will be assumed in the following.
(a) If $N=2k+1>0$, then
$d_{\lambda}=\chi_{\lambda}^{O(N)}({\rho\check{}}/\ell)=\chi_{\lambda}^{SO(N)}({\rho\check{}}/\ell)$.
(b) If $N=2k>0$ and $\lambda_{1}^{\prime}\leq k$, then
$d_{\lambda}=m(\lambda)\det(\cos(l_{j}{\rho\check{}}_{i})/\det(\cos(k-j){\rho\check{}}_{i})$,
where $l_{j}=(\lambda+\rho)_{j}=\lambda_{j}+k-j$ and where $m(\lambda)=2$ or
$1$, depending on whether $\lambda$ has exactly $k$ rows or not.
(c) If $N=-2k$, then
$d_{\lambda}=(-1)^{|\lambda|}\chi_{\lambda^{t}}({\rho\check{}}/\ell)$ for the
symplectic character labelled by the transposed diagram $\lambda^{T}$.
(d) We have $\tilde{d}_{\lambda}=\chi_{\lambda}^{SU(N)}(\rho/\ell)$ for $N>0$
and
$\tilde{d}_{\lambda}=(-1)^{|\lambda|}\chi_{\lambda^{T}}^{SU(N)}(\rho/\ell)$
for $N<0$, where $\rho=((|N|+1)/2-i)\in{\mathbb{R}}^{|N|}$.
$Proof.$ Observe that ${\rho\check{}}$ is the element $\rho$ of the Cartan
subalgebra of $\mathfrak{s}l_{N}$, viewed as an element of the Cartan
subalgebra of the Lie subalgebra $\mathfrak{s}o_{N}$ or $\mathfrak{s}p_{N}$,
depending on the case. The proof now goes as e.g the proof of Theorem 4.6 in
[W4], which is essentially the one of [Ko]. The fact that these arguments also
work for the special quotients $\overline{Br}_{n}$ follows from the proof of
[W4], Theorem 5.5.
###### Remark 3.2.
Let $\Delta_{+}$ be the set of positive roots of a semisimple Lie algebra and
$|\Delta_{+}|$ be its cardinality. As usual, we can express the Weyl
denominator in $\chi_{\lambda}({\rho\check{}}/\ell)$ in product form as
(3.2) $\Delta({\rho\check{}}/\ell)\ =\
\prod_{\alpha>0}(e^{(\alpha,{\rho\check{}})\pi
i/\ell}-e^{-(\alpha,{\rho\check{}})\pi i/\ell})\ =\ (-i)^{|\Delta_{+}|}\
\prod_{\alpha>0}2\sin((\alpha,{\rho\check{}})\pi/\ell).$
### 3.3. Usual $S$-matrices
As usual, we pick as dominant chamber $C_{+}$ the regions given by
$x_{1}>x_{2}>\ ...\ x_{k}>0$ for Lie types $B_{k}$ and $C_{k}$. We also choose
the fundamental domains $D$ with respect to the translation actions of
$M,M^{*},L,L^{*}$ such that it has 0 in its center; here the lattices $M$ and
$L$ will be certain multiples of the lattices $P$, $Q$ or ${\mathbb{Z}}^{k}$
to be specified later. Let $\bar{P}_{+}$ be the intersection of $M^{*}$ with
the fundamental alcove $D\cap C_{+}$.
Observe that we also obtain a representation of the Weyl group $W$ on the
vector spaces $V$ and $V^{*}$. Then it is easy to check that $a_{W}(V^{*})$
has an orthonormal basis $|W|^{-1/2}a_{w}(e^{\gamma})$, with
$\gamma\in\bar{P}_{+}$, and we can define a similar basis $a_{W}({\bf x})$ for
$a_{W}(V)$. Let $S$ be the matrix which describes the action of
$\SS_{|a_{W}(V)}$ with respect to that basis. Then it is not hard to check
(and we will do a slightly more complicated case below) that its coefficents
are given by
(3.3) $s_{\gamma,{\bf x}}=\frac{1}{|L:M|^{1/2}}\sum_{w}\varepsilon(w)e^{2\pi
i(w.\gamma,{\bf x})}.$
If $L$ is the weight lattice of a simple Lie algebra, the entry
$s_{\gamma,{\bf x}}$ is the numerator of Weyl’s character formula for the
dominant weight $\lambda=\gamma-\rho$, up to the factor $|L:M|^{-1/2}$. As the
columns of the unitary matrix $S$ have norm one, it follows that
(3.4) $\sum_{\lambda}\chi_{\lambda}^{2}({\bf x})=\frac{|L:M]}{\Delta^{2}({\bf
x})},$
where $\Delta$ is the Weyl denominator, and the summation goes over the
dominant weights $\lambda$ such that $\lambda+\rho\in\bar{P}_{+}$. We are now
in the position to prove some cases of the following proposition:
###### Proposition 3.3.
Let $\Lambda(N,\ell)_{ev}$ be the subset of $\Lambda(N,\ell)$ consisting of
Young diagrams with an even number of boxes. Then we have
$\sum_{\lambda\in\Lambda(N,\ell)_{ev}}d_{\lambda}^{2}\ =\
\frac{\ell^{k}}{b(N)}\
\prod_{\alpha>0}\frac{1}{4\sin^{2}(\alpha,{\rho\check{}})\pi/\ell},$
where ${\rho\check{}}=((|N|+1)/2-i)$ and $\alpha>0$ runs through the positive
roots of $\mathfrak{s}o_{N}$ for $N>0$ and of $\mathfrak{s}p_{|N|}$ for $N<0$
even, and $b(N)=2$ for $N=2k>0$, and $b(N)=1$ otherwise.
$Proof.$ Let us consider the case $N=2k+1>0$, with $P$ and $Q$ as in 3.1. Let
$L=\ell^{-1}{\mathbb{Z}}^{k}$ and let $M_{1}=Q$ and $M_{2}={\mathbb{Z}}^{k}$.
Then we have $M_{1}^{*}=P$, and $M_{2}^{*}={\mathbb{Z}}^{k}$. Now observe that
$M_{1}^{*}$ is the weight lattice of $\mathfrak{s}o_{N}$, and the elements
$\gamma\in\bar{P}_{+}$ are in 1-1 correspondence with the dominant weights
$\lambda$ of $\mathfrak{s}o_{N}$ satisfying $\lambda_{1}\leq(\ell-N)/2$, via
the correspondence $\gamma=\lambda+\rho$. Moreover, $|L:M_{1}|=2\ell^{k}$.
Hence it follows from Eq 3.4 that
$\sum\chi_{\lambda}^{2}({\rho\check{}})=2\ell^{k}/\Delta^{2}({\rho\check{}})$.
Playing the same game for the lattice $M_{2}$, we now only get the sum over
the characters $\chi_{\lambda}^{2}$ for which $\lambda+\rho$ is in
${\mathbb{Z}}^{k}$, which is only half as large as before. Hence also the sum
over the characters $\chi_{\lambda}^{2}$ for which
$\lambda\in{\mathbb{Z}}^{k}$ has to have the same value. This sum coincides
with the right hand side of the statement for $N>0$ odd, by the restriction
rules for $O(N)$ to $SO(N)$ (see Lemma 3.1 and its preceding discussion).
The symplectic case $N=-2k<0$ goes similarly. Here we define $M\subset
L=\ell^{-1}P$, and with $L^{*}=\ell Q\subset M^{*}={\mathbb{Z}}^{k}$. Then it
follows that $\sum d_{\lambda}^{2}=2\ell^{k}/\Delta^{2}({\rho\check{}}/\ell)$,
where the summation goes over all diagrams $\lambda$ such that
$\lambda^{T}\in\Lambda(N,\ell)$. Playing the same game for $M=P$ and
$M^{*}=Q$, we get $\sum d_{\lambda}^{2}=\ell^{k}/\Delta({\rho\check{}}/\ell)$,
where now the summation goes over all even, or over all odd diagrams in
$\Lambda(N,\ell)$, depending on whether the sum of coordinates of
$\rho=(k+1-i)$ is odd or even. In each case, we obtain that
$\sum_{ev}d_{\lambda}^{2}=\ell^{k}/\Delta({\rho\check{}}/\ell)$. We have
proved the proposition except for the case $N=2k>0$, for which we need a
little more preparation.
### 3.4. Another $S$-matrix
We now consider a slight generalization of the above. Observe that we can
define a second sign function $\tilde{\varepsilon}$ for $W=W(B_{k})$ which
coincides with the usual sign function on its normal subgroup $W(D_{k})$,
while we have $\tilde{\varepsilon}(w)=-\varepsilon(w)$ for $w\not\in
W(D_{k})$. It is easy to see that also in this case we have
$\tilde{\varepsilon}(vw)=\tilde{\varepsilon}(v)\tilde{\varepsilon}(w)$ for all
$v,w\in W$. We define $\tilde{a}_{W}=\sum\tilde{\varepsilon}(w)w$, and also
denote the corresponding operators on the various (quotient) lattices and on
the vector spaces $V$ and $V^{*}$ by the same symbol. One observes that now we
get an orthonormal basis for $\tilde{a}_{W}(V^{*})$ of the form ${\bf
b}_{\gamma}=|Stab(\gamma)|^{-1/2}|W|^{-1/2}\tilde{a}_{W}(e^{\gamma})$, labeled
by the elements of $\bar{P}_{+}$ which now consist of the $\gamma\in D$ such
that $\gamma_{1}>\gamma_{2}>\ ...\ \gamma_{k}\geq 0$. Observe that
$|Stab(\gamma)|$ is equal to 1 or 2, depending on whether $\gamma_{k}>0$ or
$\gamma_{k}=0$. One similarly defines a basis for $\tilde{a}_{W}(V)$. Let
${\bf x}$ be such that $Stab({\bf x})=1$, i.e. $x_{k}>0$, and let $b_{\bf
x}=|W|^{-1/2}\tilde{a}_{W}({\bf x})$. Then, writing $M^{*}/\ell L^{*}$ as a
collection of $W$ orbits, we obtain
$\displaystyle\SS{\bf b}_{\bf x}\ $ $\displaystyle\
=|W|^{-1/2}\sum_{\lambda\in\bar{P}_{+}}\sum_{v,w\in
W}\frac{1}{|Stab_{W}(\gamma)|}\tilde{\varepsilon}(w)\tilde{s}_{v.\gamma,w.{\bf
x}}v.\gamma$ $\displaystyle\ =\
\sum_{\lambda\in\bar{P}_{+}}\sum_{v}(\sum_{w}\tilde{\varepsilon}(w)\tilde{s}_{w.\gamma,{\bf
x}}\frac{1}{|Stab_{W}(\lambda)|})\tilde{\varepsilon}(v)v.\gamma.\,$
where we replaced $\tilde{\varepsilon}(w)$ by
$\tilde{\varepsilon}(v)\tilde{\varepsilon}(w^{-1}v)$,
$\tilde{s}_{v.\lambda,w.{\bf x}}$ by $\tilde{s}_{w^{-1}v.\gamma,{\bf x}}$ and
finally also substituted $w^{-1}v$ by $w$. We see from this that the
coefficient of $v.\gamma$ is equal to 0 if $\gamma$ has a nontrivial
stabilizer except in the case when $\gamma_{k}=0$. Hence it follows that $\SS$
maps $a_{W}(V)$ into $a_{W}(V^{*})$. Taking bases
$(\tilde{a}_{W}(\gamma))_{\gamma\in P_{+}}$ and $(\tilde{a}_{W}({\bf x}))$, we
see that $\SS_{|a_{W}(V)}$ can be described by the matrix $S=(s_{\gamma,{\bf
x}})$ whose coefficients are given for ${\bf x}$ with trivial stabilizer by
(3.5) $s_{\gamma,{\bf
x}}=|Stab(\gamma)|^{-1/2}|L:M|^{-1/2}\sum_{w}\varepsilon(w)e^{2\pi
i(w.\gamma,{\bf x})}.$
### 3.5. Squares of characters
Using the discussion before and the formulas of Lemma 3.1 it is not hard to
see that for $N$ even and $\lambda_{1}^{\prime}\leq N/2$ we can write
$\chi_{\lambda}^{O(N)}=m(\lambda)\tilde{a}_{W}(e^{\lambda+\rho})/\tilde{a}_{W}(e^{\rho}),$
where $m(\lambda)=2$ or $1$ depending on whether $\lambda$ has exactly $k$
rows or not. In particular, applying this to the trivial representation, we
obtain $2\Delta(\rho)=\tilde{a}_{W}(e^{\rho})$.
Let $P$ and $Q$ be as in 3.1, and set $L=\ell^{-1}P$ and $M={\mathbb{Z}}^{k}$.
Then $L^{*}=\ell Q\subset M^{*}={\mathbb{Z}}^{k}$, and it is easy to see that
all of these lattices are $W=W(B_{k})$-invariant. Moreover, let
${\rho\check{}}/\ell=(k+1/2-i)/\ell\in\ell^{-1}P=M^{*}$. Then it follows for
$N=2k$ and $\ell$ even that
$\sum_{\lambda\in\Lambda(N,\ell)}\chi^{2}_{\lambda}({\rho\check{}}\ell)\ =\
\frac{1}{\Delta^{2}({\rho\check{}}\ell)}\sum_{\lambda_{k+1}=0,\lambda_{1}\leq(\ell-N)/2}(\tilde{a}_{W}(e^{\lambda+\rho})({\rho\check{}})\ell)^{2}\
=\
\frac{|L:M]}{2\Delta^{2}({\rho\check{}})}\sum_{\lambda}s_{\lambda,{\rho\check{}}/\ell}^{2}.$
Now observe that the matrix $S$ is unitary and that $[L:M]=2\ell^{k}$.
Moreover, by e.g. Prop. 1.1 and Theorem 2.10 the square sum over odd diagrams
must be equal to the square sum over even diagrams. Hence we obtain for $N>0$
even, and $\ell$ even that
(3.6) $\sum_{\lambda\in\Lambda(N,\ell)_{ev}}d_{\lambda}^{2}\ =\
\frac{\ell^{k}}{2\Delta^{2}({\rho\check{}})},$
where $\Lambda(N,\ell)_{ev}$ denotes the set of diagrams in $\Lambda(N,\ell)$
with an even number of boxes. This finishes the last case of the proof of
Proposition 3.3
### 3.6. Calculation of index
As usual, identify the Cartan algebra of $\mathfrak{s}l_{N}$ with the diagonal
$N\times N$ matrices with zero trace. The embedding of the Cartan algebras of
an orthogonal or symplectic subalgebra is given via diagonal matrices for
which the $N+1-i$-th entry is the negative of the $i$-th entry, for $1\leq
i\leq N/2$. Hence, if $\epsilon_{i}$ is the $\mathfrak{s}l_{N}$ weight given
by the projection onto the $i$-th diagonal entry, we have
$(\epsilon_{N+1-i})_{|\mathfrak{s}o_{N}}=(-\epsilon_{i})_{|\mathfrak{s}o_{N}}$,
with a similar identity also holding for symplectic subalgebras. Using our
description of coroot and weight lattices of orthogonal and symplectic Lie
algebras as sublattices of ${\mathbb{R}}^{k}$, and defining $\phi_{i}$ to be
the projection onto the $i$-coordinate, we see that
$(\epsilon_{N+1-i})_{|\mathfrak{s}o_{N}}=-\phi_{i}=(-\epsilon_{i})_{|\mathfrak{s}o_{N}}$.
This allows us to describe the decomposition of $\mathfrak{s}l_{N}$ as an
$\mathfrak{s}o_{N}$ resp. $\mathfrak{s}p_{N}$ module as follows: We have
(3.7) $\mathfrak{s}l_{N}=\mathfrak{s}o_{N}\oplus\mathfrak{p},\quad{\rm
resp.}\quad\mathfrak{s}l_{N}=\mathfrak{s}p_{N}\oplus\mathfrak{p}$
where $\mathfrak{p}$ is the nontrivial irreducible submodule in the
symmetrization of the vector representation of $\mathfrak{s}o_{N}$, resp.
$\mathfrak{p}$ is the nontrivial irreducible submodule in the
antisymmetrization of the vector representation of $\mathfrak{s}p_{N}$. The
nonzero weights $\omega>0$ of $\mathfrak{p}$ coming from positive roots of
$\mathfrak{s}l_{N}$ and the multiplicity $n(\mathfrak{p})$ of the weight 0 in
$\mathfrak{p}$ are given by
(a) $2\phi_{i},\phi_{i}$ and $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with
$n(\mathfrak{p})=k$ for $\mathfrak{s}o_{N}$ with $N=2k+1$ odd,
(b) $2\phi_{i}$ and $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with
$n(\mathfrak{p})=k-1$ for $\mathfrak{s}o_{N}$ with $N=2k$ even,
(c) $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with $n(\mathfrak{p})=k-1$ for
$\mathfrak{s}p_{|N|}$ with $N=-2k<0$ even.
###### Theorem 3.4.
The index of the subfactor $\mathcal{N}\subset\mathcal{M}$ obtained from the
inclusions of algebras $\bar{H}_{n}(q)\subset\overline{Br}_{n}(q^{N},q)$ is
given by
$[\mathcal{M}:\mathcal{N}]=b(N)\ell^{n(\mathfrak{p})}\prod_{\omega>0}\frac{1}{4\sin^{2}(\omega,{\rho\check{}})\pi/\ell},$
where the product goes over the weights $\omega>0$ of $\mathfrak{p}$ coming
from positive roots of $\mathfrak{s}l_{N}$, as listed above, $n(\mathfrak{p})$
is the multiplicity of the zero weight in $\mathfrak{p}$, and $b(N)$ and
${\rho\check{}}$ are as in Prop. 3.3.
###### Corollary 3.5.
If $q=e^{\pi i/\ell}\to 1$, the index $[\mathcal{M}:\mathcal{N}]$ goes to
$\infty$ with asymptotics $\ell^{\dim\mathfrak{p}}$.
$Proof.$ We use Theorem 2.10, where the denominator has been calculated in
Proposition 3.3. The numerator follows from a standard argument for
$S$-matrices for Lie type $A$, see [Kc], versions of which have also been used
in this section. For an elementary calculation, see [E].
###### Remark 3.6.
It is straightforward to adapt our index formula to subfactors related to
other fixed points $H=G^{\alpha}$ of an order two automorphism $\alpha$ of a
compact Lie group $G$, up to some integer (or perhaps rational) constant
$b(H,G)$. Again, $\mathfrak{p}$ would be the $-1$ eigenspace of the induced
action of $\alpha$ on the Lie algebra $\mathfrak{g}$, and the same $S$-matrix
techniques applied in this section would go through. E.g. our formulas for
$N=3$ and $\ell$ odd coincide with the ones at the end of [X] for even level
of $SU(3)$, up to a factor 3. This is to be expected as in our case only those
diagrams appear in the principal graph (see next section) which also label
representations of the projective group $PSU(3)$.
### 3.7. Restriction rules and principal graph
It follows from Theorem 1.4 that the principal graph of
$\mathcal{N}\subset\mathcal{M}$ is given by the inclusion matrix for
$\bar{H}_{2k}\subset\overline{Br}_{2k}$ for $k$ sufficiently large. This still
leaves the question how to explicitly calculate these graphs. Observe that in
the classical case $q=1$ these would be given by the restriction rules from
the unitary group $U(N)$ to $O(N)$, for $N>0$. Formulas for these restriction
coefficients have been well-known, see e.g. [Wy] (see Theorems 7.8F and 7.9C),
Littlewood’s formula (see e.g. [KT], Section 1.5, and the whole paper for
additional results). Another approach closely related to the setting of fusion
categories can also be found in [W5].
Let $b^{\lambda}_{\mu}(N)$ be the multiplicity of the simple $O(N)$-module
$V_{\mu}$ in the $U(N)$ module $F_{\lambda}$, for $N>0$, where $\lambda$,
$\mu$ are Young diagrams. It is well-known that for fixed Young diagrams
$\lambda$ and $\mu$, the number $b_{\mu}^{\lambda}(N)$ will become a constant
$b_{\mu}^{\lambda}$ for $N$ large enough. Fix now also $\ell>|N|$. We define
similar coefficients in our setting as follows: Recall that the simple
components of $\bar{H}_{n}$ are labeled by the diagrams in
$\tilde{\Lambda}(N,\ell)_{n}$ and the ones of $\overline{Br}_{n}$ by the
diagrams in $\Lambda(N,\ell)$. We then define for
$\lambda\in\tilde{\Lambda}(N,\ell)$ and $\mu\in\Lambda(N,\ell)$ the number
$b^{\lambda}_{\mu}(N,\ell)$ to be the multiplicity of a simple
$\bar{H}_{n,\lambda}$ module in a simple $\overline{Br}_{n,\mu}$ module.
In the following lemma the symbol $\chi_{\mu}$ will also be used for the
$O(N)$ character corresponding to the simple representation labeled by the
Young diagram $\mu$. Moreover, we also denote by $\overline{Br}_{\infty}$ the
inductive limit of the finite dimensional algebras $\overline{Br}_{n}$ under
their standard inclusions, for fixed $N$ and $\ell$.
###### Lemma 3.7.
(a) Each $g\in O(N)$ for which $\chi_{\mu}(g)=0$ for all boundary diagrams
$\mu$ of $\Lambda(N,\ell)$ defines a trace on $\overline{Br}_{\infty}$
determined by $tr(p_{\mu})=\chi_{\mu}(g)/\chi_{[1]}(g)^{n}$, where $p_{\mu}$
is a minimal projection of $\overline{Br}_{n,\mu}$.
(b) For given $\lambda\in\tilde{\Lambda}(N,\ell)_{n}$ the coefficients
$b^{\lambda}_{\mu}(N,\ell)$ are uniquely determined by the equations
$\chi^{U(N)}_{\lambda}(g)=\sum_{\mu}b^{\lambda}_{\mu}(N,\ell)\chi_{\mu}(g)$
for all $g$ as in (a), where the summation goes over all diagrams $\mu$ in
$\Lambda(N,\ell)$ with $n,n-2,\ ...$ boxes.
$Proof.$ The formula in statement (a) determines a trace on
$\overline{Br}_{n}$ for each $n$. To show that these formulas are compatible
with the standard embeddings we observe that a minimal idempotent
$p_{\mu}\in\overline{Br}_{n,\mu}$ is the sum of minimal idempotents
$e_{\lambda}\in\overline{Br}_{n+1,\lambda}$ where $\lambda$ runs through all
diagrams in $\Lambda(N,\ell)$ obtained by adding or removing a box to/from
$\lambda$, see Eq 2.2 and the remarks below that theorem. Evaluating the
traces of these idempotents and multiplying everything by
$\chi_{\lambda}(g)^{n+1}$, equality of the traces is equivalent to
$\chi_{\mu}(g)\chi_{[1]}(g)=\sum_{\lambda}\chi_{\lambda}(g).$
By the usual tensor product rule for orthogonal groups, the left hand side
would be equal to the sum of characters corresponding to $all$ diagrams
$\lambda$ which differ from $\mu$ by only one box. It is easy to check that
this differs from the sum above only by boundary diagrams, for which the
characters at $g$ is equal to 0. This shows (a).
For (b), we first show that
$tr(p_{\lambda})=\chi_{\lambda}^{U(N)}(g)/\chi^{U(N)}_{[1]}(g)^{n}$ for
$p_{\lambda}\in\bar{H}_{n,\lambda}$ a minimal idempotent and $tr$ a trace as
in (a). As the weight vector for $\overline{Br}_{n+2N}$ is a multiple of the
one of $\overline{Br}_{n}$, for $n$ large enough, the same must also hold for
the weight vectors of $\bar{H}_{n}$ and $\bar{H}_{n+2N}$, by periodicity of
the inclusions. Hence these weight vectors must be eigenvectors of the
inclusion matrix for $\bar{H}_{n}\subset\bar{H}_{n+2N}$. As this inclusion
matrix is just a block of the $2N$-th power of the fusion matrix of the vector
representation for the corresponding type $A$ fusion category, its entries
must be given by $U(N)$ characters of a suitable group element. To identify
these elements, it suffices to observe that the antisymmetrizations of the
vector representation, labeled by the Young diagrams $\lambda=[1^{j}]$, $1\leq
j\leq N$, remain irreducible as $O(N)$ modules. This means the corresponding
Hecke algebra idempotent remains a minimal idempotent also in
$\overline{Br}_{j}$. Hence $tr(p_{\lambda})=\chi_{\lambda}^{U(N)}(g)$ for
$\lambda=[1^{j}]$ and $1\leq j\leq N$. But as the antisymmetrizations generate
the representation ring of $U(N)$, and also of the corresponding fusion ring,
the claim follows for general $\lambda$. For more details, see e.g. [GW]
Recall that the coefficient $b^{\lambda}_{\mu}(N,\ell)$ can be defined as the
rank of $p_{\lambda}$ in an irreducible $\overline{Br}_{n,\mu}$
representation. So obviously the formula in the statement holds for any $g$ as
in (a). Examples for such $g$ come from $exp({\bf x})$ with ${\bf x}\in
M^{*}=\ell^{-1}Q$ for which the character is given by the expression
$\chi_{\lambda}({\bf x})$ as in Section 2.5. As the columns of the orthogonal
$S$-matrix are linearly independent, this would identify $SO(N)$
representations. If $N$ is odd, the two $O(N)$ representations which reduce to
the same $SO(N)$ representation are labeled by Young diagrams with opposite
parities. Hence only one of them can occur in the decomposition of a given
$U(N)$ representation. A similar argument also works in the symplectic case.
For $N$ even, we can have two diagrams $\lambda$ and $\lambda^{\dagger}$ with
the same $SO(N)$ character, where one of them, say $\lambda$ has less than $k$
rows. They can be distinguished by elements $g\in O(N)\backslash{}SO(N)$ for
which $\chi_{\lambda^{\dagger}}(g)=-\chi_{\lambda}(g)$. It is well-known that
such elements $g$ must have eigenvalues $\pm 1$, and $\chi_{\lambda}(g)$ is
given by the character formula for $Sp(2k-2)$ in the remaining $2k-2$
eigenvalues (see [Wy]). It follows from the invertibility of the $S$-matrix
for $Sp(2k-2)$ at level $\ell/2-k$ (see [Kc]) that we can identify those
diagrams $\lambda$ by evaluating $\chi_{\lambda}^{Sp(2k-2)}({\bf x}/\ell)$ for
${\bf x}\in{\mathbb{Z}}^{k-1}$ with $\ell/2>x_{1}>x_{2}>\ ...\ >x_{k-1}>0$,
and that those elements satisfy the boundary condition
$\chi^{Sp(2k-2)}_{\lambda}({\bf x})=0$ for any boundary diagram $\lambda$.
The lemma above is illustrated in the following section for a number of
explicit examples. We can also give a closed formula for the restriction
coefficients, using a well-known quotient map for fusion rings (even though in
our case, the quotient ring does not correspond to a tensor category as far as
we know). In the context of fusion rings, this is known as the Kac-Walton
formula; for type $A$ see also e.g. [GW]. In our case, we need to use a
slightly different affine reflection group $\mathcal{W}$. In the orthogonal
case $N=2k$ and $N=2k+1$ it is given by the semidirect product of
$\ell{\mathbb{Z}}^{k}$ with the Weyl group of type $B_{k}$. In the symplectic
case, it is given by the semidirect product of $\ell Q$ with the Weyl group of
type $B_{k}$. As usual, we define the dot action of $\mathcal{W}$ on
${\mathbb{R}}^{k}$ by $w.{\bf x}=w({\bf x}+\rho)-\rho$, where $\rho$ is half
the sum of the positive roots of the corresponding Lie algebra, with the roots
embedded into ${\mathbb{R}}^{k}$ as described above, and $\varepsilon$ is the
usual sign function for reflection groups. This can be extended to an action
on the labeling set of $O(N)$ representations by identifying a Young diagrams
with $\leq k$ rows with the corresponding vector in ${\mathbb{Z}}^{k}$, and by
using the restriction rule from $O(N)$ to $SO(N)$ in the other cases. See also
e.g. Lemma 1.7 in [W5] for more details.
###### Theorem 3.8.
With notations as above, the restriction multiplicity
$b^{\lambda}_{\mu}(N,\ell)$ for $N=2k+1>0$ and $N=-2k$ is given by
$b^{\lambda}_{\mu}(N,\ell)=\sum_{w\in\mathcal{W}}\varepsilon(w)b^{\lambda}_{w.\mu}(N).$
If $N=2k>0$, we have to replace $\varepsilon$ by $\tilde{\varepsilon}$ (see
Section 3.4) in the formula above.
$Proof.$ Looking at the character formulas, we see that an action of an
element $w$ of the finite reflection group on $\lambda$ just changes the
character by the sign of $w$. Moreover, by definition of the elements ${\bf
x}$ we have that $\chi_{\lambda}({\bf x})=\chi_{\lambda+\mu}({\bf x})$ for any
$\mu\in M$. It follows that $\chi_{w.\lambda}({\bf
x})=\varepsilon(w)\chi_{\lambda}({\bf x})$ for all ${\bf x}\in M^{*}$ and
$w\in\mathcal{W}$. Hence summing over the $\mathcal{W}$-orbits, we obtain for
any ${\bf x}\in M^{*}$, $\lambda\in\tilde{\Lambda}(N,\ell)$ and
$\mu\in\Lambda(N,\ell)$ that
$\chi^{U(N)}_{\lambda}({\bf
x})=\sum_{\gamma}b^{\lambda}_{\gamma}(N)\chi_{\gamma}=\sum_{\mu}(\sum_{w}b^{\lambda}_{w.\mu}(N))\chi_{\mu}.$
The claim now follows from this and Lemma 3.7.
## 4\. Examples and other approaches
### 4.1. The case $N=2$:
This corresponds to the Goodman-de la Harpe-Jones subfactors for type
$D_{\ell/2+1}$, where $\ell>2$ has to be even. It follows from our theorem
that the even vertices of the principal graph are labeled by the Young
diagrams $\lambda$ with an even number $n$ of boxes, at most two rows and with
$\lambda_{1}-\lambda_{2}\leq\ell-2$; there are $(\ell-2)/2$ such diagrams.
Their dimensions are given by $\tilde{d}_{k}=[2k+1]$, $0\leq k<(\ell-2)/2$.
Moreover, one checks that $\Lambda(2,\ell)$ consists of Young diagrams $[j]$
with $0\leq j\leq\ell/2-1$ and of $[1^{2}]$, one column with 2 boxes, with
dimensions $d_{[j]}=2\cos j\pi/\ell$ for $j>0$ and dimension equal to 1 for
the remaining cases (i.e. for $\emptyset$ and for $[1^{2}]$). The restriction
rule (i.e. principal graph) follows from writing the dimensions as
$\tilde{d}_{k}=2\cos\tilde{k}\pi/\ell\ +\ 2\cos(\tilde{k}-2)\pi/\ell+\ ...\
+1,$
where $\tilde{k}=$ min $\\{k,\ell/2-k\\}$. Indeed, this determines the graph
completely except for whether to pick the diagram $\emptyset$ or $[1^{2}]$ for
the object with dimension 1. It follows from the restriction rule $O(2)\subset
U(2)$ that we take $\emptyset$ for $j$ even, and $[1^{2}]$ for $j$ odd. To
calculate the index one can check by elementary means that $\sum_{\lambda\
even}d_{\lambda}^{2}=\ell/2$. Moreover, it is well-known that the sum
$\sum_{\lambda\ even}\tilde{d}_{\lambda}^{2}$ over even partitions for
$\mathfrak{s}l_{2}$ is equal to $\ell/4\sin^{2}\pi/\ell$. Hence we obtain as
index $[\mathcal{M}:\mathcal{N}]=1/2\sin^{2}\pi/\ell$.
### 4.2. The case $N=3$
It is also fairly elementary to work out this case in detail. Recall that by
Weyl’s dimension formula we have
$\tilde{d}_{\lambda}=\frac{[\lambda_{1}-\lambda_{2}+1][\lambda_{2}-\lambda_{3}+1][\lambda_{1}-\lambda_{3}+2]}{[1]^{2}[2]}.$
Now observe that the product of two $q$-numbers is given by the tensor product
rules for $\mathfrak{s}l_{2}$, i.e. we have for $n\geq m$ that
$[n][m]=[n+m-1]+[n+m-3]+...+[n-m+1]$. As an example, we have
$\tilde{d}_{[4]}\ =\ \frac{[6][5]}{[2]}\ =\ \frac{[10]+[8]+[6]+[4]+[2]}{[2]}\
=\ [9]+[5]+[1],$
i.e. the fourth antisymmetrization of the vector representation of $U(3)$
decomposes as a direct sum of the one-, five- and nine-dimensional
representation of $SO(3)$. One similarly can show the well-known result that
the adjoint representation of $SU(3)$, labeled by the Young diagram $[2,1]$
decomposes into the direct sum of the three- and the five-dimensional
representation of $SO(3)$, i.e. $\mathfrak{p}$ is the five dimensional
representation of $SO(3)$. Hence we get from Theorem 3.4 that the index is
equal to
$[\mathcal{M}:\mathcal{N}]\ =\ \frac{\ell}{4^{2}\sin^{2}(2\pi/\ell)\
\sin^{2}(\pi/\ell)}.$
We note that here as well as in the other examples, the dimensions (i.e.
entries of the Perron-Frobenius vectors) are given by $|\tilde{d}_{\lambda}|$
for even vertices, and by $\sqrt{[\mathcal{M}:\mathcal{N}]}|d_{\mu}|$ for odd
vertices, with $\tilde{d}_{\lambda}$ and $d_{\mu}$ as in Lemma 3.1. To
consider explicit examples, the first nontrivial case for $N=3$ occurs for
$\ell=7$. We leave it to the reader to check that in this case the first
principal graph is given by the Dynkin graph $D_{8}$. A more interesting graph
is obtained for $\ell=9$, see Fig 4.1. Here we have the three invertible
objects of the $SU(3)_{6}$ fusion category, including the trivial object
(often denoted as $*$) on the left; they generate a group isomorphic to
${\mathbb{Z}}/3$. The vertices with the double edge are labeled by the object
corresponding to the 5-dimensional representation of $SO(3)$ and the diagram
$[4,2]$ for $SU(3)_{6}$. This is the only fixed point under the
${\mathbb{Z}}/3$ action given by the invertible objects (or, in physics
language, the currents). It would be interesting to see whether one can carry
out an orbifold construction in this context related to the one in [EK1].
Figure 4.1. SO(3) for $\ell=9$
### 4.3. The case $N=4$
This is the first nontrivial and, apparently, new case corresponding to even-
dimensional orthogonal groups. As we shall see, somewhat surprisingly, the
corresponding construction for $SO(4)$ does not seem to work. We do the case
with $\ell=8$ in explicit detail. It is not hard to check that we already get
the periodic inclusion matrix for $n=12$. As we consider an analog of the
restriction to $O(4)$ for which the determinant can be $\pm 1$, we should,
strictly speaking, consider a fusion category for $SU(4)\times\\{\pm 1\\}$. We
shall actually use the Young diagram notation for representations of $U(4)$.
For $n=12$ we have the invertible objects labeled by $[3^{4}]$, $[4^{3}]$,
$[5^{2}1^{2}]$ and $[62^{3}]$ (i.e. e.g. the last diagram has six boxes in the
first and two boxes in the second, third and fourth row). They generate a
subgroup isomorphic to ${\mathbb{Z}}/4$. It follows from the $O(4)$
restriction rules that $[3^{4}]$ and $[5^{2}1^{2}]$ contain the determinant
representation, and $[4^{3}]$ and $[62^{3}]$ contain the trivial
representation as one-dimensional $O(4)$ subrepresentations. This allows us to
calculate the restrictions for representations of each ${\mathbb{Z}}/4$ orbit
simultaneously. As usually for at least one element of each orbit the ordinary
restriction rules still hold, it makes the general calculations easier. The
principal graph can be seen in Fig. 4.2. As in the $N=3$ example, the one-
dimensional currents, including the trivial object $*$ appear as the left- and
right-most vertices in the graph. The lowest vertex corresponds to the
$O(4)$-object $[2]$ which is connected to the objects in the
${\mathbb{Z}}/4$-orbit $\\{[2,1^{2}],[3,1],[4,3,1],[3,3,2]\\}$. We also note
that we get the same graph for the $Sp(4)$ case $N=-4$ for $\ell=8$. However,
for other roots of unities, already the indices of the subfactors differ which
are given by
$O(4):\ \frac{2\ell}{4\sin^{2}(3\pi/\ell)\ 4\sin^{2}(2\pi/\ell)\
16\sin^{4}(\pi/\ell)}\hskip 30.00005ptSp(4):\
\frac{\ell}{4\sin^{2}(2\pi/\ell)\ 4\sin^{2}(\pi/\ell)}.$
It was originally thought that we should also be able to get fusion category
analogs for the restriction from $SU(N)$ to $SO(N)$ for $N$ even. It is easy
to check that this is not possible for $O(2)$. Some initial checks also seem
to suggest a similar phenomenon for higher ranks. E.g. using the same element
${\rho\check{}}$ in the $SO(N)$ character formula would give dimension
functions which are not invariant under the $D_{N}$ diagram automorphism.
Figure 4.2. O(4) and Sp(4) for $\ell=8$
### 4.4. Related results
We discuss several results related to our findings. Our original motivation
was to construct subfactors related to twisted loop groups. It was shown in R.
Verill’s PhD thesis [V] that it is not possible to construct a fusion tensor
product for representations of twisted loop groups. However, it seemed
reasonable to expect that representations of twisted loop groups could become
a module category over representations of their untwisted counterparts. Many
results, in particular about the combinatorics of such categories can be found
in the context of boundary conformal field theory in papers by Gaberdiel,
Gannon, Fuchs, Schweigert, diFrancesco, Petkova, Zuber and others (see e.g.
[GG], [FS], [PZ] and the papers cited therein).
In the mathematics literature, one can find closely related results in the
papers [X] and [Wa]. Here the authors construct module categories via a
completely different approach in the context of loop groups. E.g. the formulas
at the end of [X] for the special case $N=3$ differ only by a factor 3 (which
can be explained, see Remark 3.6) from our formulas for $N=3$ for even level
(together with Corollary 1.5), modulo misprints; similar formulas for the
symplectic case as well as restriction coefficients also appear at the end of
[Wa]. We can not get results corresponding to the odd level cases in [X]. The
combinatorics there suggests that this would require considering an embedding
of $Sp(N-1)$ into $SU(N)$ under which the vector representation would not
remain irreducible. In contrast, we can also construct module categories for
$\ell-N$ odd, which would correspond to odd level; however, these categories
are not unitarizable (which follows from Lemma 2.8) and they have different
fusion rules. However, we do get fairly general formulas for the index and
principal graphs of this type of subfactors in the unitary case, which was one
of the problems posed in [X]. These formulas were known to this author as well
as to Antony Wassermann at least back in 2008 when they had discussions about
their respective works in Oberwolfach and at the Schrödinger Institute.
We close this section by mentioning that while our results for $N>0$ odd and
$N<0$ even are in many ways parallel to results obtained via other approaches
in connection with twisted loop groups, there does not seem to be an obvious
analog for our results for $N>0$ even. E.g. the combinatorial results in [GG]
for that case seem to be different to ours.
### 4.5. Conclusions and further explorations
We have constructed module categories of fusion categories of type $A$ via
deformations of centralizer algebras of certain subgroups of unitary groups.
We have also classified when they are unitarizable, and we have constructed
the corresponding subfactors. These deformations are compatible with the
Drinfeld-Jimbo deformation of the unitary group but not with the Drinfeld-
Jimbo deformation of the subgroup. Most of the deformation was already done in
[W4] via elementary methods. In principle, at least, it should be possible to
use this elementary approach also for other inclusions. However, this might
become increasingly tedious.
As we have seen already in Section 2.2, it should be possible to get a
somewhat more conceptual approach using different deformations of the
subgroup, see [N], [Mo], [L1], [L2], [IK] and references therein. In
particular in the work of Letzter, such deformations via co-ideal algebras
have been defined for a large class of embeddings of a semisimple Lie algebra
into another one. At this point, it does not seem obvious how to define
$C^{*}$-structures in this setting, and additional complications arise as
these coideal algebras are not expected to be semisimple at roots of unity.
Nevertheless, the results in this and other papers such as [X], [Wa] would
seem to suggest that similar constructions might be possible also in a more
general setting. It would also be interesting to see whether this could lead
to a general solution of Problem 5.5 in [O].
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|
arxiv-papers
| 2011-08-06T00:06:33 |
2024-09-04T02:49:21.353006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hans Wenzl",
"submitter": "Hans Wenzl",
"url": "https://arxiv.org/abs/1108.1428"
}
|
1108.1601
|
CDF Collaboration222With visitors from aIstituto Nazionale di Fisica Nucleare,
Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy, bUniversity of CA
Irvine, Irvine, CA 92697, USA, cUniversity of CA Santa Barbara, Santa Barbara,
CA 93106, USA, dUniversity of CA Santa Cruz, Santa Cruz, CA 95064, USA,
eCERN,CH-1211 Geneva, Switzerland, fCornell University, Ithaca, NY 14853, USA,
gUniversity of Cyprus, Nicosia CY-1678, Cyprus, hOffice of Science, U.S.
Department of Energy, Washington, DC 20585, USA, iUniversity College Dublin,
Dublin 4, Ireland, jUniversity of Fukui, Fukui City, Fukui Prefecture, Japan
910-0017, kUniversidad Iberoamericana, Mexico D.F., Mexico, lIowa State
University, Ames, IA 50011, USA, mUniversity of Iowa, Iowa City, IA 52242,
USA, nKinki University, Higashi-Osaka City, Japan 577-8502, oKansas State
University, Manhattan, KS 66506, USA, pUniversity of Manchester, Manchester
M13 9PL, United Kingdom, qQueen Mary, University of London, London, E1 4NS,
United Kingdom, rUniversity of Melbourne, Victoria 3010, Australia, sMuons,
Inc., Batavia, IL 60510, USA, tNagasaki Institute of Applied Science,
Nagasaki, Japan, uNational Research Nuclear University, Moscow, Russia,
vUniversity of Notre Dame, Notre Dame, IN 46556, USA, wUniversidad de Oviedo,
E-33007 Oviedo, Spain, xTexas Tech University, Lubbock, TX 79609, USA,
yUniversidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile, zYarmouk
University, Irbid 211-63, Jordan, hhOn leave from J. Stefan Institute,
Ljubljana, Slovenia,
# Measurement of the top-quark mass in the lepton+jets channel using a matrix
element technique with the CDF II detector
T. Aaltonen Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki,
Finland B. Álvarez Gonzálezw Instituto de Fisica de Cantabria, CSIC-
University of Cantabria, 39005 Santander, Spain S. Amerio Istituto Nazionale
di Fisica Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131
Padova, Italy D. Amidei University of Michigan, Ann Arbor, Michigan 48109,
USA A. Anastassov Northwestern University, Evanston, Illinois 60208, USA A.
Annovi Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica
Nucleare, I-00044 Frascati, Italy J. Antos Comenius University, 842 48
Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice,
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Batavia, Illinois 60510, USA A. Apresyan Purdue University, West Lafayette,
Indiana 47907, USA T. Arisawa Waseda University, Tokyo 169, Japan A.
Artikov Joint Institute for Nuclear Research, RU-141980 Dubna, Russia J.
Asaadi Texas A&M University, College Station, Texas 77843, USA W. Ashmanskas
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA B.
Auerbach Yale University, New Haven, Connecticut 06520, USA A. Aurisano
Texas A&M University, College Station, Texas 77843, USA F. Azfar University
of Oxford, Oxford OX1 3RH, United Kingdom W. Badgett Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A. Barbaro-Galtieri
Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA V.E. Barnes Purdue University, West Lafayette, Indiana 47907, USA
B.A. Barnett The Johns Hopkins University, Baltimore, Maryland 21218, USA P.
Barriadd Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa,
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Bartos Comenius University, 842 48 Bratislava, Slovakia; Institute of
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Padova, Italy I. Bizjakhh University College London, London WC1E 6BT, United
Kingdom K.R. Bland Baylor University, Waco, Texas 76798, USA B. Blumenfeld
The Johns Hopkins University, Baltimore, Maryland 21218, USA A. Bocci Duke
University, Durham, North Carolina 27708, USA A. Bodek University of
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West Lafayette, Indiana 47907, USA J. Boudreau University of Pittsburgh,
Pittsburgh, Pennsylvania 15260, USA A. Boveia Enrico Fermi Institute,
University of Chicago, Chicago, Illinois 60637, USA L. Brigliadoriaa Istituto
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Bologna, Italy A. Brisuda Comenius University, 842 48 Bratislava, Slovakia;
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Michigan State University, East Lansing, Michigan 48824, USA E. Brucken
Division of High Energy Physics, Department of Physics, University of Helsinki
and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland M.
Bucciantoniocc Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of
Pisa, ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa,
Italy J. Budagov Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia H.S. Budd University of Rochester, Rochester, New York 14627, USA S.
Budd University of Illinois, Urbana, Illinois 61801, USA K. Burkett Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA G. Busettobb
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of Padova, I-35131 Padova, Italy P. Bussey Glasgow University, Glasgow G12
8QQ, United Kingdom A. Buzatu Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 C. Calancha Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain S. Camarda Institut de Fisica d’Altes
Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra
(Barcelona), Spain M. Campanelli University College London, London WC1E 6BT,
United Kingdom M. Campbell University of Michigan, Ann Arbor, Michigan
48109, USA F. Canelli11 Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA B. Carls University of Illinois, Urbana, Illinois 61801,
USA D. Carlsmith University of Wisconsin, Madison, Wisconsin 53706, USA R.
Carosi Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa,
ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy S.
Carrillok University of Florida, Gainesville, Florida 32611, USA S. Carron
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA B. Casal
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain M. Casarsa Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA A. Castroaa Istituto Nazionale di Fisica Nucleare
Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy P. Catastini
Harvard University, Cambridge, Massachusetts 02138, USA D. Cauz Istituto
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Udine, I-33100 Udine, Italy V. Cavaliere University of Illinois, Urbana,
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Berkeley, California 94720, USA L. Cerritoq University College London, London
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Normale Superiore, I-56127 Pisa, Italy G. Chlachidze Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA F. Chlebana Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA K. Cho Center
for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University,
Suwon 440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon 305-806, Korea; Chonnam National University, Gwangju 500-757, Korea;
Chonbuk National University, Jeonju 561-756, Korea D. Chokheli Joint
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Wisconsin, Madison, Wisconsin 53706, USA Y.S. Chung University of Rochester,
Rochester, New York 14627, USA C.I. Ciobanu LPNHE, Universite Pierre et
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and eeScuola Normale Superiore, I-56127 Pisa, Italy A. Clark University of
Geneva, CH-1211 Geneva 4, Switzerland C. Clarke Wayne State University,
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Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131 Padova,
Italy M.E. Convery Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA J. Conway University of California, Davis, Davis, California
95616, USA M.Corbo LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS,
UMR7585, Paris, F-75252 France M. Cordelli Laboratori Nazionali di Frascati,
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University of California, Davis, Davis, California 95616, USA D.J. Cox
University of California, Davis, Davis, California 95616, USA F. Cresciolicc
Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity
of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy C. Cuenca
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Barbaro University of Rochester, Rochester, New York 14627, USA S. De Cecco
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Università di Roma, I-00185 Roma, Italy G. De Lorenzo Institut de Fisica
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ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
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ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy J.R. Dittmann Baylor University, Waco, Texas 76798, USA
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ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy P.
Dong Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA M.
Dorigo Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste,
ggUniversity of Udine, I-33100 Udine, Italy T. Dorigo Istituto Nazionale di
Fisica Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131
Padova, Italy K. Ebina Waseda University, Tokyo 169, Japan A. Elagin Texas
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University, College Station, Texas 77843, USA H.C. Fang Ernest Orlando
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA S.
Farrington University of Oxford, Oxford OX1 3RH, United Kingdom M. Feindt
Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology,
D-76131 Karlsruhe, Germany J.P. Fernandez Centro de Investigaciones
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Ferrazzaee Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa,
ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy R.
Field University of Florida, Gainesville, Florida 32611, USA G. Flanagans
Purdue University, West Lafayette, Indiana 47907, USA R. Forrest University
of California, Davis, Davis, California 95616, USA M.J. Frank Baylor
University, Waco, Texas 76798, USA M. Franklin Harvard University,
Cambridge, Massachusetts 02138, USA J.C. Freeman Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA Y. Funakoshi Waseda University,
Tokyo 169, Japan I. Furic University of Florida, Gainesville, Florida 32611,
USA M. Gallinaro The Rockefeller University, New York, New York 10065, USA
J. Galyardt Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
J.E. Garcia University of Geneva, CH-1211 Geneva 4, Switzerland A.F.
Garfinkel Purdue University, West Lafayette, Indiana 47907, USA P. Garosidd
Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity
of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy H. Gerberich
University of Illinois, Urbana, Illinois 61801, USA E. Gerchtein Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA S. Giaguff
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, ffSapienza
Università di Roma, I-00185 Roma, Italy V. Giakoumopoulou University of
Athens, 157 71 Athens, Greece P. Giannetti Istituto Nazionale di Fisica
Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola
Normale Superiore, I-56127 Pisa, Italy K. Gibson University of Pittsburgh,
Pittsburgh, Pennsylvania 15260, USA C.M. Ginsburg Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA N. Giokaris University of Athens,
157 71 Athens, Greece P. Giromini Laboratori Nazionali di Frascati, Istituto
Nazionale di Fisica Nucleare, I-00044 Frascati, Italy M. Giunta Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy G. Giurgiu The Johns
Hopkins University, Baltimore, Maryland 21218, USA V. Glagolev Joint
Institute for Nuclear Research, RU-141980 Dubna, Russia D. Glenzinski Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA M. Gold
University of New Mexico, Albuquerque, New Mexico 87131, USA D. Goldin Texas
A&M University, College Station, Texas 77843, USA N. Goldschmidt University
of Florida, Gainesville, Florida 32611, USA A. Golossanov Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA G. Gomez Instituto de
Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain G.
Gomez-Ceballos Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA M. Goncharov Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA O. González Centro de Investigaciones
Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain I. Gorelov
University of New Mexico, Albuquerque, New Mexico 87131, USA A.T. Goshaw
Duke University, Durham, North Carolina 27708, USA K. Goulianos The
Rockefeller University, New York, New York 10065, USA S. Grinstein Institut
de Fisica d’Altes Energies, ICREA, Universitat Autonoma de Barcelona, E-08193,
Bellaterra (Barcelona), Spain C. Grosso-Pilcher Enrico Fermi Institute,
University of Chicago, Chicago, Illinois 60637, USA R.C. Group55 Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA J. Guimaraes da
Costa Harvard University, Cambridge, Massachusetts 02138, USA Z. Gunay-
Unalan Michigan State University, East Lansing, Michigan 48824, USA C. Haber
Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA S.R. Hahn Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA E. Halkiadakis Rutgers University, Piscataway, New
Jersey 08855, USA A. Hamaguchi Osaka City University, Osaka 588, Japan J.Y.
Han University of Rochester, Rochester, New York 14627, USA F. Happacher
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy K. Hara University of Tsukuba, Tsukuba, Ibaraki 305,
Japan D. Hare Rutgers University, Piscataway, New Jersey 08855, USA M. Hare
Tufts University, Medford, Massachusetts 02155, USA R.F. Harr Wayne State
University, Detroit, Michigan 48201, USA K. Hatakeyama Baylor University,
Waco, Texas 76798, USA C. Hays University of Oxford, Oxford OX1 3RH, United
Kingdom M. Heck Institut für Experimentelle Kernphysik, Karlsruhe Institute
of Technology, D-76131 Karlsruhe, Germany J. Heinrich University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA M. Herndon University of
Wisconsin, Madison, Wisconsin 53706, USA S. Hewamanage Baylor University,
Waco, Texas 76798, USA D. Hidas Rutgers University, Piscataway, New Jersey
08855, USA A. Hocker Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA W. Hopkinsf Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA D. Horn Institut für Experimentelle Kernphysik,
Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany S. Hou
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
R.E. Hughes The Ohio State University, Columbus, Ohio 43210, USA M. Hurwitz
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
U. Husemann Yale University, New Haven, Connecticut 06520, USA N. Hussain
Institute of Particle Physics: McGill University, Montréal, Québec, Canada H3A
2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6;
University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF,
Vancouver, British Columbia, Canada V6T 2A3 M. Hussein Michigan State
University, East Lansing, Michigan 48824, USA J. Huston Michigan State
University, East Lansing, Michigan 48824, USA G. Introzzi Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy M. Ioriff Istituto Nazionale
di Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di Roma, I-00185
Roma, Italy A. Ivanovo University of California, Davis, Davis, California
95616, USA E. James Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA D. Jang Carnegie Mellon University, Pittsburgh, Pennsylvania
15213, USA B. Jayatilaka Duke University, Durham, North Carolina 27708, USA
E.J. Jeon Center for High Energy Physics: Kyungpook National University,
Daegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea;
Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of Science and
Technology Information, Daejeon 305-806, Korea; Chonnam National University,
Gwangju 500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea
M.K. Jha Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity of
Bologna, I-40127 Bologna, Italy S. Jindariani Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA W. Johnson University of
California, Davis, Davis, California 95616, USA M. Jones Purdue University,
West Lafayette, Indiana 47907, USA K.K. Joo Center for High Energy Physics:
Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea S.Y. Jun Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213, USA T.R. Junk Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA T. Kamon Texas A&M University,
College Station, Texas 77843, USA P.E. Karchin Wayne State University,
Detroit, Michigan 48201, USA A. Kasmi Baylor University, Waco, Texas 76798,
USA Y. Katon Osaka City University, Osaka 588, Japan W. Ketchum Enrico
Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA J. Keung
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA V.
Khotilovich Texas A&M University, College Station, Texas 77843, USA B.
Kilminster Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA D.H. Kim Center for High Energy Physics: Kyungpook National University,
Daegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea;
Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of Science and
Technology Information, Daejeon 305-806, Korea; Chonnam National University,
Gwangju 500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea
H.S. Kim Center for High Energy Physics: Kyungpook National University, Daegu
702-701, Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea H.W. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea J.E. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea M.J. Kim
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy S.B. Kim Center for High Energy Physics: Kyungpook
National University, Daegu 702-701, Korea; Seoul National University, Seoul
151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute
of Science and Technology Information, Daejeon 305-806, Korea; Chonnam
National University, Gwangju 500-757, Korea; Chonbuk National University,
Jeonju 561-756, Korea S.H. Kim University of Tsukuba, Tsukuba, Ibaraki 305,
Japan Y.K. Kim Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637, USA N. Kimura Waseda University, Tokyo 169, Japan M. Kirby
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA S.
Klimenko University of Florida, Gainesville, Florida 32611, USA K.
Kondo††footnotemark: Waseda University, Tokyo 169, Japan D.J. Kong Center
for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University,
Suwon 440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon 305-806, Korea; Chonnam National University, Gwangju 500-757, Korea;
Chonbuk National University, Jeonju 561-756, Korea J. Konigsberg University
of Florida, Gainesville, Florida 32611, USA A.V. Kotwal Duke University,
Durham, North Carolina 27708, USA M. Kreps Institut für Experimentelle
Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany J.
Kroll University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA D.
Krop Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637,
USA N. Krumnackl Baylor University, Waco, Texas 76798, USA M. Kruse Duke
University, Durham, North Carolina 27708, USA V. Krutelyovc Texas A&M
University, College Station, Texas 77843, USA T. Kuhr Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany M. Kurata University of Tsukuba, Tsukuba, Ibaraki 305,
Japan S. Kwang Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637, USA A.T. Laasanen Purdue University, West Lafayette, Indiana
47907, USA S. Lami Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity
of Pisa, ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa,
Italy S. Lammel Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA M. Lancaster University College London, London WC1E 6BT, United
Kingdom R.L. Lander University of California, Davis, Davis, California
95616, USA K. Lannonv The Ohio State University, Columbus, Ohio 43210, USA
A. Lath Rutgers University, Piscataway, New Jersey 08855, USA G. Latinocc
Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity
of Siena and eeScuola Normale Superiore, I-56127 Pisa, Italy T. LeCompte
Argonne National Laboratory, Argonne, Illinois 60439, USA E. Lee Texas A&M
University, College Station, Texas 77843, USA H.S. Lee Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637, USA J.S. Lee
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea S.W. Leex
Texas A&M University, College Station, Texas 77843, USA S. Leocc Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy S. Leone Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy J.D. Lewis Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA A. Limosanir
Duke University, Durham, North Carolina 27708, USA C.-J. Lin Ernest Orlando
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA J.
Linacre University of Oxford, Oxford OX1 3RH, United Kingdom M. Lindgren
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA E.
Lipeles University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA A.
Lister University of Geneva, CH-1211 Geneva 4, Switzerland D.O. Litvintsev
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA C. Liu
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA Q. Liu Purdue
University, West Lafayette, Indiana 47907, USA T. Liu Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA S. Lockwitz Yale
University, New Haven, Connecticut 06520, USA A. Loginov Yale University,
New Haven, Connecticut 06520, USA D. Lucchesibb Istituto Nazionale di Fisica
Nucleare, Sezione di Padova-Trento, bbUniversity of Padova, I-35131 Padova,
Italy J. Lueck Institut für Experimentelle Kernphysik, Karlsruhe Institute
of Technology, D-76131 Karlsruhe, Germany P. Lujan Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA P. Lukens
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA G. Lungu
The Rockefeller University, New York, New York 10065, USA J. Lys Ernest
Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
R. Lysak Comenius University, 842 48 Bratislava, Slovakia; Institute of
Experimental Physics, 040 01 Kosice, Slovakia R. Madrak Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Maeshima Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA K. Makhoul
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA S.
Malik The Rockefeller University, New York, New York 10065, USA G. Mancaa
University of Liverpool, Liverpool L69 7ZE, United Kingdom A. Manousakis-
Katsikakis University of Athens, 157 71 Athens, Greece F. Margaroli Purdue
University, West Lafayette, Indiana 47907, USA C. Marino Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany M. Martínez Institut de Fisica d’Altes Energies, ICREA,
Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain R.
Martínez-Ballarín Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain P. Mastrandrea Istituto Nazionale di
Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di Roma, I-00185
Roma, Italy M.E. Mattson Wayne State University, Detroit, Michigan 48201,
USA P. Mazzanti Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity
of Bologna, I-40127 Bologna, Italy K.S. McFarland University of Rochester,
Rochester, New York 14627, USA P. McIntyre Texas A&M University, College
Station, Texas 77843, USA R. McNultyi University of Liverpool, Liverpool L69
7ZE, United Kingdom A. Mehta University of Liverpool, Liverpool L69 7ZE,
United Kingdom P. Mehtala Division of High Energy Physics, Department of
Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014,
Helsinki, Finland A. Menzione Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy C. Mesropian The Rockefeller University, New York, New
York 10065, USA T. Miao Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA D. Mietlicki University of Michigan, Ann Arbor, Michigan
48109, USA A. Mitra Institute of Physics, Academia Sinica, Taipei, Taiwan
11529, Republic of China H. Miyake University of Tsukuba, Tsukuba, Ibaraki
305, Japan S. Moed Harvard University, Cambridge, Massachusetts 02138, USA
N. Moggi Istituto Nazionale di Fisica Nucleare Bologna, aaUniversity of
Bologna, I-40127 Bologna, Italy M.N. Mondragonk Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA C.S. Moon Center for High Energy
Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea R. Moore Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA M.J. Morello Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Morlock Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany P. Movilla Fernandez Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A. Mukherjee Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA Th. Muller Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany P. Murat Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA M. Mussiniaa Istituto Nazionale di Fisica Nucleare
Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy J. Nachtmanm Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA Y. Nagai
University of Tsukuba, Tsukuba, Ibaraki 305, Japan J. Naganoma Waseda
University, Tokyo 169, Japan I. Nakano Okayama University, Okayama 700-8530,
Japan A. Napier Tufts University, Medford, Massachusetts 02155, USA J. Nett
Texas A&M University, College Station, Texas 77843, USA C. Neu University of
Virginia, Charlottesville, Virginia 22906, USA M.S. Neubauer University of
Illinois, Urbana, Illinois 61801, USA J. Nielsend Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA L. Nodulman
Argonne National Laboratory, Argonne, Illinois 60439, USA O. Norniella
University of Illinois, Urbana, Illinois 61801, USA E. Nurse University
College London, London WC1E 6BT, United Kingdom L. Oakes University of
Oxford, Oxford OX1 3RH, United Kingdom S.H. Oh Duke University, Durham,
North Carolina 27708, USA Y.D. Oh Center for High Energy Physics: Kyungpook
National University, Daegu 702-701, Korea; Seoul National University, Seoul
151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute
of Science and Technology Information, Daejeon 305-806, Korea; Chonnam
National University, Gwangju 500-757, Korea; Chonbuk National University,
Jeonju 561-756, Korea I. Oksuzian University of Virginia, Charlottesville,
Virginia 22906, USA T. Okusawa Osaka City University, Osaka 588, Japan R.
Orava Division of High Energy Physics, Department of Physics, University of
Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland L.
Ortolan Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de
Barcelona, E-08193, Bellaterra (Barcelona), Spain S. Pagan Grisobb Istituto
Nazionale di Fisica Nucleare, Sezione di Padova-Trento, bbUniversity of
Padova, I-35131 Padova, Italy C. Pagliarone Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine, I-33100 Udine,
Italy E. Palenciae Instituto de Fisica de Cantabria, CSIC-University of
Cantabria, 39005 Santander, Spain V. Papadimitriou Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A.A. Paramonov Argonne
National Laboratory, Argonne, Illinois 60439, USA J. Patrick Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA G. Paulettagg Istituto
Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of
Udine, I-33100 Udine, Italy M. Paulini Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213, USA C. Paus Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA D.E. Pellett University of
California, Davis, Davis, California 95616, USA A. Penzo Istituto Nazionale
di Fisica Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine,
I-33100 Udine, Italy T.J. Phillips Duke University, Durham, North Carolina
27708, USA G. Piacentino Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy E. Pianori University of Pennsylvania, Philadelphia,
Pennsylvania 19104, USA J. Pilot The Ohio State University, Columbus, Ohio
43210, USA K. Pitts University of Illinois, Urbana, Illinois 61801, USA C.
Plager University of California, Los Angeles, Los Angeles, California 90024,
USA L. Pondrom University of Wisconsin, Madison, Wisconsin 53706, USA K.
Potamianos Purdue University, West Lafayette, Indiana 47907, USA O.
Poukhov††footnotemark: Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia F. Prokoshiny Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia A. Pronko Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA F. Ptohosg Laboratori Nazionali di Frascati, Istituto Nazionale di
Fisica Nucleare, I-00044 Frascati, Italy E. Pueschel Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA G. Punzicc Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy J. Pursley University of
Wisconsin, Madison, Wisconsin 53706, USA A. Rahaman University of
Pittsburgh, Pittsburgh, Pennsylvania 15260, USA V. Ramakrishnan University
of Wisconsin, Madison, Wisconsin 53706, USA N. Ranjan Purdue University,
West Lafayette, Indiana 47907, USA I. Redondo Centro de Investigaciones
Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain P. Renton
University of Oxford, Oxford OX1 3RH, United Kingdom M. Rescigno Istituto
Nazionale di Fisica Nucleare, Sezione di Roma 1, ffSapienza Università di
Roma, I-00185 Roma, Italy T. Riddick University College London, London WC1E
6BT, United Kingdom F. Rimondiaa Istituto Nazionale di Fisica Nucleare
Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy L. Ristori44 Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA A. Robson
Glasgow University, Glasgow G12 8QQ, United Kingdom T. Rodrigo Instituto de
Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain T.
Rodriguez University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
E. Rogers University of Illinois, Urbana, Illinois 61801, USA S. Rollih
Tufts University, Medford, Massachusetts 02155, USA R. Roser Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA M. Rossi Istituto
Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of
Udine, I-33100 Udine, Italy F. Rubbo Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA F. Ruffinidd Istituto Nazionale di Fisica
Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola
Normale Superiore, I-56127 Pisa, Italy A. Ruiz Instituto de Fisica de
Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain J. Russ
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA V. Rusu
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA A.
Safonov Texas A&M University, College Station, Texas 77843, USA W.K.
Sakumoto University of Rochester, Rochester, New York 14627, USA Y. Sakurai
Waseda University, Tokyo 169, Japan L. Santigg Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ggUniversity of Udine, I-33100 Udine,
Italy L. Sartori Istituto Nazionale di Fisica Nucleare Pisa, ccUniversity of
Pisa, ddUniversity of Siena and eeScuola Normale Superiore, I-56127 Pisa,
Italy K. Sato University of Tsukuba, Tsukuba, Ibaraki 305, Japan V.
Savelievu LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris,
F-75252 France A. Savoy-Navarro LPNHE, Universite Pierre et Marie
Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France P. Schlabach Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A. Schmidt Institut für
Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany E.E. Schmidt Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA M.P. Schmidt††footnotemark: Yale University,
New Haven, Connecticut 06520, USA M. Schmitt Northwestern University,
Evanston, Illinois 60208, USA T. Schwarz University of California, Davis,
Davis, California 95616, USA L. Scodellaro Instituto de Fisica de Cantabria,
CSIC-University of Cantabria, 39005 Santander, Spain A. Scribanodd Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy F. Scuri Istituto
Nazionale di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena
and eeScuola Normale Superiore, I-56127 Pisa, Italy A. Sedov Purdue
University, West Lafayette, Indiana 47907, USA S. Seidel University of New
Mexico, Albuquerque, New Mexico 87131, USA Y. Seiya Osaka City University,
Osaka 588, Japan A. Semenov Joint Institute for Nuclear Research, RU-141980
Dubna, Russia F. Sforzacc Istituto Nazionale di Fisica Nucleare Pisa,
ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale Superiore,
I-56127 Pisa, Italy A. Sfyrla University of Illinois, Urbana, Illinois
61801, USA S.Z. Shalhout University of California, Davis, Davis, California
95616, USA T. Shears University of Liverpool, Liverpool L69 7ZE, United
Kingdom P.F. Shepard University of Pittsburgh, Pittsburgh, Pennsylvania
15260, USA M. Shimojimat University of Tsukuba, Tsukuba, Ibaraki 305, Japan
S. Shiraishi Enrico Fermi Institute, University of Chicago, Chicago, Illinois
60637, USA M. Shochet Enrico Fermi Institute, University of Chicago,
Chicago, Illinois 60637, USA I. Shreyber Institution for Theoretical and
Experimental Physics, ITEP, Moscow 117259, Russia A. Simonenko Joint
Institute for Nuclear Research, RU-141980 Dubna, Russia P. Sinervo Institute
of Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8;
Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University
of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British
Columbia, Canada V6T 2A3 A. Sissakian††footnotemark: Joint Institute for
Nuclear Research, RU-141980 Dubna, Russia K. Sliwa Tufts University,
Medford, Massachusetts 02155, USA J.R. Smith University of California,
Davis, Davis, California 95616, USA F.D. Snider Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A. Soha Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA S. Somalwar Rutgers University,
Piscataway, New Jersey 08855, USA V. Sorin Institut de Fisica d’Altes
Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra
(Barcelona), Spain P. Squillacioti Istituto Nazionale di Fisica Nucleare
Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola Normale
Superiore, I-56127 Pisa, Italy M. Stancari Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA M. Stanitzki Yale University, New
Haven, Connecticut 06520, USA R. St. Denis Glasgow University, Glasgow G12
8QQ, United Kingdom B. Stelzer Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 O. Stelzer-Chilton Institute of Particle Physics: McGill University,
Montréal, Québec, Canada H3A 2T8; Simon Fraser University, Burnaby, British
Columbia, Canada V5A 1S6; University of Toronto, Toronto, Ontario, Canada M5S
1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3 D. Stentz
Northwestern University, Evanston, Illinois 60208, USA J. Strologas
University of New Mexico, Albuquerque, New Mexico 87131, USA G.L. Strycker
University of Michigan, Ann Arbor, Michigan 48109, USA Y. Sudo University of
Tsukuba, Tsukuba, Ibaraki 305, Japan A. Sukhanov University of Florida,
Gainesville, Florida 32611, USA I. Suslov Joint Institute for Nuclear
Research, RU-141980 Dubna, Russia K. Takemasa University of Tsukuba,
Tsukuba, Ibaraki 305, Japan Y. Takeuchi University of Tsukuba, Tsukuba,
Ibaraki 305, Japan J. Tang Enrico Fermi Institute, University of Chicago,
Chicago, Illinois 60637, USA M. Tecchio University of Michigan, Ann Arbor,
Michigan 48109, USA P.K. Teng Institute of Physics, Academia Sinica, Taipei,
Taiwan 11529, Republic of China J. Thomf Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA J. Thome Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA G.A. Thompson University of
Illinois, Urbana, Illinois 61801, USA E. Thomson University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA P. Ttito-Guzmán Centro de
Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid,
Spain S. Tkaczyk Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA D. Toback Texas A&M University, College Station, Texas 77843, USA
S. Tokar Comenius University, 842 48 Bratislava, Slovakia; Institute of
Experimental Physics, 040 01 Kosice, Slovakia K. Tollefson Michigan State
University, East Lansing, Michigan 48824, USA T. Tomura University of
Tsukuba, Tsukuba, Ibaraki 305, Japan D. Tonelli Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA S. Torre Laboratori Nazionali di
Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy D.
Torretta Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
P. Totaro Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento,
bbUniversity of Padova, I-35131 Padova, Italy M. Trovatoee Istituto Nazionale
di Fisica Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and
eeScuola Normale Superiore, I-56127 Pisa, Italy Y. Tu University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA F. Ukegawa University of
Tsukuba, Tsukuba, Ibaraki 305, Japan S. Uozumi Center for High Energy
Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon 305-806,
Korea; Chonnam National University, Gwangju 500-757, Korea; Chonbuk National
University, Jeonju 561-756, Korea A. Varganov University of Michigan, Ann
Arbor, Michigan 48109, USA F. Vázquezk University of Florida, Gainesville,
Florida 32611, USA G. Velev Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA C. Vellidis University of Athens, 157 71 Athens, Greece
M. Vidal Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain I. Vila Instituto de Fisica de
Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain R. Vilar
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain J. Vizán Instituto de Fisica de Cantabria, CSIC-University
of Cantabria, 39005 Santander, Spain M. Vogel University of New Mexico,
Albuquerque, New Mexico 87131, USA G. Volpicc Istituto Nazionale di Fisica
Nucleare Pisa, ccUniversity of Pisa, ddUniversity of Siena and eeScuola
Normale Superiore, I-56127 Pisa, Italy P. Wagner University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA R.L. Wagner Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA T. Wakisaka Osaka City University,
Osaka 588, Japan R. Wallny University of California, Los Angeles, Los
Angeles, California 90024, USA S.M. Wang Institute of Physics, Academia
Sinica, Taipei, Taiwan 11529, Republic of China A. Warburton Institute of
Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8; Simon
Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University of
Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British
Columbia, Canada V6T 2A3 D. Waters University College London, London WC1E
6BT, United Kingdom M. Weinberger Texas A&M University, College Station,
Texas 77843, USA W.C. Wester III Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA B. Whitehouse Tufts University, Medford,
Massachusetts 02155, USA D. Whitesonb University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA A.B. Wicklund Argonne National
Laboratory, Argonne, Illinois 60439, USA E. Wicklund Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA S. Wilbur Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637, USA F. Wick
Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology,
D-76131 Karlsruhe, Germany H.H. Williams University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA J.S. Wilson The Ohio State University,
Columbus, Ohio 43210, USA P. Wilson Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA B.L. Winer The Ohio State University, Columbus,
Ohio 43210, USA P. Wittichg Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA S. Wolbers Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA H. Wolfe The Ohio State University, Columbus,
Ohio 43210, USA T. Wright University of Michigan, Ann Arbor, Michigan 48109,
USA X. Wu University of Geneva, CH-1211 Geneva 4, Switzerland Z. Wu Baylor
University, Waco, Texas 76798, USA K. Yamamoto Osaka City University, Osaka
588, Japan J. Yamaoka Duke University, Durham, North Carolina 27708, USA T.
Yang Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
U.K. Yangp Enrico Fermi Institute, University of Chicago, Chicago, Illinois
60637, USA Y.C. Yang Center for High Energy Physics: Kyungpook National
University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of
Science and Technology Information, Daejeon 305-806, Korea; Chonnam National
University, Gwangju 500-757, Korea; Chonbuk National University, Jeonju
561-756, Korea W.-M. Yao Ernest Orlando Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA G.P. Yeh Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Yim Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Yoh Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA K. Yorita Waseda
University, Tokyo 169, Japan T. Yoshidaj Osaka City University, Osaka 588,
Japan G.B. Yu Duke University, Durham, North Carolina 27708, USA I. Yu
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju
500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea S.S. Yu
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA J.C. Yun
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA A.
Zanetti Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste,
ggUniversity of Udine, I-33100 Udine, Italy Y. Zeng Duke University, Durham,
North Carolina 27708, USA S. Zucchelliaa Istituto Nazionale di Fisica
Nucleare Bologna, aaUniversity of Bologna, I-40127 Bologna, Italy
###### Abstract
A measurement of the top-quark mass is presented using Tevatron data from
proton-antiproton collisions at center-of-mass energy $\sqrt{s}=1.96$ TeV
collected with the CDF II detector. Events are selected from a sample of
candidates for production of $t\bar{t}$ pairs that decay into the lepton+jets
channel. The top-quark mass is measured with an unbinned maximum likelihood
method where the event probability density functions are calculated using
signal and background matrix elements, as well as a set of parameterized jet-
to-parton transfer functions. The likelihood function is maximized with
respect to the top-quark mass, the signal fraction in the sample, and a
correction to the jet energy scale (JES) calibration of the calorimeter jets.
The simultaneous measurement of the JES correction ($\Delta_{\mathrm{JES}}$)
amounts to an additional in situ jet energy calibration based on the known
mass of the hadronically decaying $W$ boson. Using the data sample of 578
lepton+jets candidate events, corresponding to 3.2 fb-1 of integrated
luminosity, the top-quark mass is measured to be $m_{t}=\rm 172.4\pm
1.4\>(stat+\Delta_{\mathrm{JES}})\pm 1.3\>(syst)~{}GeV/{\it c}^{2}$.
###### pacs:
14.65.Ha,13.85.-t,12.15.Ff
The top-quark mass, $m_{t}$, is an intrinsic parameter of the standard model
(SM) of particle physics and is of particular importance due to its strikingly
large value. As a result, the top quark has a large effect on radiative
corrections to electroweak processes and has a Yukawa coupling to the Higgs
field of $\mathcal{O}(1)$, which may provide insight into the mechanism of
electroweak symmetry breaking Bhattacharyya (2011).
The Higgs boson mass, $m_{H}$, is not predicted by the SM, but constraints on
its value can be derived from the calculation of radiative corrections to the
$W$ boson mass, $m_{W}$, and from the values of other precision electroweak
variables LEP and Tevatron Electroweak Working Groups, and SLD electroweak
heavy flavour groups (2010). These corrections depend primarily on $\ln m_{H}$
and $m_{t}^{2}$, and thus precision measurements of $m_{W}$ and $m_{t}$
provide important constraints on $m_{H}$.
The dominant top-quark production process is pair production via the strong
interaction. At Fermilab’s Tevatron this process is initiated by $p\bar{p}$
collisions at center-of-mass energy $\sqrt{s}=\rm 1.96~{}TeV$. Because of its
large mass, the top quark decays rapidly with lifetime $\tau_{t}\sim 10^{-25}$
s Abazov et al. (2011) — fast enough that it has essentially no time to
interact and may be considered as a free quark. This allows a direct
measurement of its mass from the daughter particles from its decay, and as a
result $m_{t}$ has the lowest relative uncertainty of all of the quark masses
Nakamura et al. (2010).
In the SM top quarks decay via the weak interaction, predominantly to $W$
bosons and $b$ quarks as $t\bar{t}\to W^{+}b\>W^{-}\bar{b}$. $W$ bosons decay
into lower-mass fermion-antifermion pairs: a charged lepton and a neutrino
($W^{+}\to\bar{\ell}\nu_{\ell}$ or $W^{-}\to\ell\bar{\nu}_{\ell}$), “leptonic
decay”; or an up-type quark and a down-type quark ($W^{+}\to
q\bar{q}^{\prime}$ or $W^{-}\to\bar{q}q^{\prime}$), “hadronic decay”. The
result presented here uses the lepton+jets decay channel (with
$q\bar{q}^{\prime}b\ell\bar{\nu_{\ell}}\bar{b}$ or
$\bar{\ell}\nu_{\ell}b\bar{q}q^{\prime}\bar{b}$ in the final state), where one
of the two $W$ bosons decays leptonically into an electron or a muon, and the
other decays hadronically. All the quarks in the final state evolve into jets
of hadrons. Events with tau leptons are not selected directly, but may
contribute a few percent of the total sample via leptonic cascade decays or
fake jets. The most recent $m_{t}$ measurements obtained at the Tevatron using
the lepton+jets topology are reported in Ref. Tevatron Electroweak Working
Group (2011), while the results of an earlier version of the present analysis
using 955 pb-1 of integrated luminosity are reported in Ref. Abulencia et al.
(2007). The distinctive feature of this analysis is the use of matrix element
calculations to describe the dominant background contribution. The result
presented here uses a more than three times larger data sample than the
earlier version, and employs a more detailed likelihood function.
The leptons and jets resulting from the top-antitop quark pair ($t\bar{t}$)
decay are detected in the CDF II general-purpose particle detector that is
described in detail elsewhere Blair et al. (1996). Azimuthally and forward-
backward symmetric about the beam-line, the detector contains a high precision
particle tracking system immersed in a 1.4 T magnetic field and surrounded by
calorimetry, with muon detectors on the outside. A right-handed spherical co-
ordinate system is employed, with the polar angle $\theta$ measured from the
proton beam direction, the azimuthal angle $\phi$ in the plane perpendicular
to the beam-line, and the distance $r$ from the center of the detector.
Transverse energy and momentum are defined as $E_{T}\equiv E\sin\theta$ and
$p_{T}\equiv p\sin\theta$, where $E$ and $p$ denote energy and momentum.
Pseudorapidity is defined as $\eta\equiv-\ln\tan\left(\theta/2\right)$.
Electron | $E_{T}$ | $>\rm 20~{}GeV$ | $|\eta|<1.1$
---|---|---|---
or Muon | $p_{T}$ | $>\rm 20~{}GeV/{\it c}$ | $|\eta|<1.0$
$\not\\!\\!E_{T}$ | $\not\\!\\!E_{T}$ | $>\rm 20~{}GeV$ | $|\eta|<3.6$
Jets | $E_{T}$ | $>\rm 20~{}GeV$ | $|\eta|<2.0$
Four jets; at least one from a $b$ quark
Table 1: Event selection criteria.
This measurement makes use of CDF II data collected between February 2002 and
August 2008, representing approximately 3.2 fb-1 of integrated luminosity. The
event selection criteria (Table 1) are tuned to select the lepton+jets final-
state particles, requiring that each event must have exactly one high-$E_{T}$
electron or high-$p_{T}$ muon, exactly four high-$E_{T}$ jets, and a
significant amount of missing $E_{T}$, $\not\\!\\!E_{T}$ Abulencia et al.
(2006), characteristic of the undetected neutrino. Jets are reconstructed
using a cone algorithm Abe et al. (1992), with the cone radius $\Delta
R\equiv\sqrt{\left(\Delta\eta\right)^{2}+\left(\Delta\phi\right)^{2}}=0.4$. At
least one of the four jets must be identified as originating from a $b$ quark
via the SECVTX algorithm Acosta et al. (2005a), which detects displaced
secondary vertices characteristic of the decay of long-lived $b$ hadrons. A
total of 578 events are selected, of which 76% are expected to be $t\bar{t}$
events (Table 2). Of the 24% of events expected to be background, it is
predicted that 69% arise from the production of a $W$ boson in conjunction
with 4 jets ($W$+jets), 19% come from multi-jet QCD production (non-$W$),
while the remaining 12% are from sources such as diboson and single-top-quark
production. These fractions are estimated using theoretical cross-sections,
Monte Carlo (MC) simulated events, and data. The $t\bar{t}$ events are
generated using the Lund Monte Carlo program PYTHIA Sjostrand et al. (2006),
with a top-quark mass of $\rm 175~{}GeV/{\it c}^{2}$ and a $t\bar{t}$
production cross section of $6.7\pm 0.8$ pb Kidonakis and Vogt (2003). The
$W+$jets and $Z+$jets events are generated using the ALPGEN generator Mangano
et al. (2003) while the single-top-quark events are generated using the
MADEVENT package Alwall et al. (2007), in both cases also using PYTHIA to
perform the parton showering and hadronization. Diboson events are also
generated using PYTHIA. In addition, data are used for non-$W$ events Acosta
et al. (2005b).
sample | # of events | % of total | % of bkg
---|---|---|---
$t\bar{t}$ signal | 425.0 $\pm$ 58.9 | 76.0% | -
$W+$jets | 92.6 $\pm$ 15.9 | 16.6% | 69.0%
non-$W$ | 25.0 $\pm$ 12.5 | 4.5% | 18.7%
single top quark | 6.6 $\pm$ 0.4 | 1.2% | 4.9%
diboson | 6.0 $\pm$ 0.6 | 1.1% | 4.5%
$Z+$jets | 3.9 $\pm$ 0.5 | 0.7% | 2.9%
total | 559.2 $\pm$ 67.0 | 100% | -
Observed | 578 | |
Table 2: Number of expected signal and background events, corresponding to the
total integrated luminosity of 3.2 fb-1. The percentages are used when
generating Monte Carlo simulated experiments.
This analysis employs an unbinned maximum likelihood method Abulencia et al.
(2007); Abazov et al. (2004, 2005). The $m_{t}$-dependent probability density
function (p.d.f.) is calculated for each event in the data sample:
$\mathcal{P}(k)=\nu_{\mathrm{sig}}P_{s}(k)+(1-\nu_{\mathrm{sig}})P_{b}(k)\mathrm{,}$
(1)
where $k\equiv(E_{i},\vec{p}_{i})$ represents the measured kinematic
quantities of the event, $P_{s}$ and $P_{b}$ are respectively the normalized
p.d.f.s for signal and background events, and $\nu_{\mathrm{sig}}$ is the
signal fraction parameter (constrained $0\leq\nu_{\mathrm{sig}}\leq 1$).
Signal events are defined as events consistent with $q\bar{q}\rightarrow
t\bar{t}$ production and $t\bar{t}$ decay into the lepton+jets channel, as
described by the leading-order (LO) matrix element evaluated by Mahlon and
Parke Mahlon and Parke (1996). Background events are assumed to be described
by a matrix element for $W+$jets production, which is calculated using a sum
of 1286 $W+$4-partons amplitudes for 592 subprocesses encoded in the VECBOS MC
event generator Berends et al. (1991). This approximation does mean that there
are some events that, in principle, are not described by either $P_{s}$ or
$P_{b}$, including non-$W$, single top, diboson, $Z+$jets, and
$W+bb+$2-partons events, as well as $W+$jets events from $W+$0-, 1-, 2- and
3-partons processes. However, studies with MC simulated events show that the
ratio $P_{b}/P_{s}$ calculated for all of these event types is similar to that
for $W+$4-partons events, and that, in practice, such events mostly contribute
to the likelihood function via the $P_{b}$ term and do not add any more bias
than the $W+$4-partons events or than the poorly reconstructed $t\bar{t}$
events themselves Linacre (2010). Any residual bias in the measured top-quark
mass is removed at the end, as described later in the paper.
The signal and background p.d.f.s, $P_{s}$ and $P_{b}$, are constructed in
analogous fashions, starting with the appropriately normalized parton-level
differential cross-section Nakamura et al. (2010), $d\hat{\sigma}_{s}$ or
$d\hat{\sigma}_{b}$, which is then convolved with parton distribution
functions (PDFs) and a jet-to-parton transfer function $W(k,\varkappa)$.
$P_{s}$ is thus given by
$\displaystyle
P_{s}(k;m_{t},\Delta_{\mathrm{JES}})=\frac{1}{n_{jp}}\sum_{\mathrm{jet\>perm.}}^{n_{jp}}\frac{1}{\hat{\sigma}_{s}(m_{t})}\frac{1}{A_{s}(m_{t},\Delta_{\mathrm{JES}})}\times$
(2) $\displaystyle\int
d\hat{\sigma}_{s}(\varkappa;m_{t})\;dx_{\mathrm{Bj}}^{1}dx_{\mathrm{Bj}}^{2}\;W(k,\varkappa;\Delta_{\mathrm{JES}})\;f(x_{\mathrm{Bj}}^{1})(x_{\mathrm{Bj}}^{2})\mathrm{,}$
where $\varkappa\equiv(\varepsilon_{i},\vec{\pi}_{i})$ represents the actual
event parton-level kinematic quantities corresponding to the measured
quantities $k$, and parameter $\Delta_{\mathrm{JES}}$ is defined in the next
paragraph. The PDFs $f(x_{\mathrm{Bj}})$ define the probability density for a
colliding parton to carry a longitudinal momentum fraction $x_{\mathrm{Bj}}$
and are given by CTEQ5L Lai et al. (2000). $A_{s}$ is the mean acceptance
function for signal events, a normalization term that is the consequence of
the constriction of the phase-space of the integral by the event selection
cuts and by the detector acceptance. The average over the jet permutations,
$n_{jp}$, is due to ambiguity in assigning final state jets to partons. The
fact that the two light quarks in the final state are indistinguishable allows
the reduction from the original 24 permutations to 12 in the expression for
$P_{s}$, and the $b$-tagging information allows a further reduction to 6
assignments for events with one identified $b$-jet and 2 for events with both
$b$-jets identified. In the similar expression for $P_{b}$, all 24
permutations are averaged.
The jet-to-parton transfer function $W(k,\varkappa)$ is a p.d.f. describing
the probability density for an event with out-going partons and charged lepton
with $\varkappa$ to be measured as reconstructed $k$. The charged lepton is
assumed to be well-measured, allowing the use of a Dirac $\delta$-function to
represent the mapping between its parton-level momentum, $\vec{\pi}_{\ell}$,
and its reconstructed momentum, $\vec{p}_{\ell}$. For the four jets, the
function is obtained by parameterizing the jet-to-parton mapping observed in
fully simulated PYTHIA $t\bar{t}$ events. These events contain all of the
information about the original partons as well as the measured jets. The
simulation includes physical effects, such as radiation and hadronization, as
well as the effects of measurement resolution and of the jet reconstruction
algorithm. The parameterization is made in two parts that are assumed to be
independent: the energy transfer function $W_{E}$, describing the jet energies
$E$, and the angular transfer function $W_{A}$, describing the mapping for the
jet angles. The jet-to-parton transfer function is thus given by
$\displaystyle W(k,\varkappa;\Delta_{\mathrm{JES}})$
$\displaystyle=\delta^{3}(\vec{p}_{\ell}-\vec{\pi}_{\ell})\>W_{A}$ (3)
$\displaystyle\times\prod_{i=1}^{4}{\left(\frac{1}{E_{i}p_{i}}W_{E}^{i}(E_{i},\varepsilon_{i};\Delta_{\mathrm{JES}})\right)}\mathrm{.}$
The reconstructed jet energies, $E_{i}$, used in the function $W_{E}$ are not
just the raw calorimeter energy deposits, but are first calibrated so that
they represent the combined energies released in the calorimeter by the many
particles constituting each jet. This is achieved using the CDF jet energy
scale (JES) calibration Bhatti et al. (2006), which is subject to a
significant systematic uncertainty. The uncertainties of individual jet energy
measurements, $\sigma(E_{i})$, are therefore correlated, and their fractional
JES uncertainty, $\sigma(E_{i})/E_{i}$, is typically $\sim 3\%$. If this were
included as a systematic uncertainty on the measured $m_{t}$ it would reduce
the measurement precision drastically; in fact, each 1% of fractional JES
uncertainty would add about $\rm 1~{}GeV/{\it c}^{2}$ uncertainty to the
measured $m_{t}$ Affolder et al. (2001). However, such a treatment
overestimates the uncertainty because the energies of the two daughter jets of
the hadronically decaying $W$ boson can be constrained based on the known $W$
boson mass. Applying this constraint to all events in the data sample while
allowing the jet energies to be shifted results in the in situ measurement of
the JES correction, $\Delta_{\mathrm{JES}}$, defined as the number of
$\sigma(E_{i})$ values by which the energy of each jet is shifted in the
likelihood fit. This effectively re-calibrates the measured jet energies based
on the known $W$ boson mass and replaces a large component of the JES
systematic uncertainty with a much smaller statistical uncertainty on the
$\Delta_{\mathrm{JES}}$. The $\Delta_{\mathrm{JES}}$ dependence of the jet
energies is included in the parameterization of the function $W_{E}$. This
parameterization is made in eight bins in pseudorapidity $|\eta|$, separately
for light and $b$-jets, using a sum of two Gaussians as a function of the
difference between the parton energies and the corrected jet energies as
measured in a sample of PYTHIA $t\bar{t}$ events that pass the same selection
criteria as the data.
In an earlier version of this analysis Abulencia et al. (2007), the jet-to-
parton transfer functions for all jet angles were approximated by Dirac
$\delta$-functions. The introduction of the function $W_{A}$ was motivated by
a discrepancy noticed in simulated $t\bar{t}$ events in the 2-jet effective
invariant mass of the hadronically decaying $W$ boson, $m_{W}$. Even when the
true simulated parton-level jet energies are used, instead of the
corresponding reconstructed detector-level values, the use of the measured jet
angles rather than their parton-level values causes a significant shift of the
reconstructed $m_{W}$ from its nominal value, as illustrated in Fig. 1.
Figure 1: The reconstructed 2-jet invariant mass of the hadronically decaying
$W$ boson, $m_{W}$, for measured jet angles (solid line) and for parton-level
angles (dotted line), obtained after assuming the primary parton energy as jet
energy. For ease of comparison, the parton-level distribution is normalized so
that the maxima of the two distributions are the same.
There is also a negative skewness in the distribution for measured angles, and
since parton-level jet energies are used, the observed effects are due to the
differences between the measured angles and the parton-level angles alone. The
peak of the $m_{W}$ distribution, when fit by a Breit-Wigner distribution,
corresponds to a $W$ boson pole mass of $\rm 79.5~{}GeV/{\it c}^{2}$, a
$\rm-0.9~{}GeV/{\it c}^{2}$ shift from its parton-level value of $\rm
80.4~{}GeV/{\it c}^{2}$. This is found to be a result of a correlation between
the measured jet directions: the measured angle, $\alpha_{12}$, between the
two jets is, on average, reduced so that the two jets appear closer together
than their parent partons, which can be seen in Fig. 2. Since the apparent $W$
boson mass is utilized to measure $\Delta_{\mathrm{JES}}$ and thus calibrate
the measured jet energies, a jet-to-parton transfer function describing the
change in the angle $\alpha_{12}$ is important in making an accurate
measurement of $\Delta_{\mathrm{JES}}$ and thus the top-quark mass. The
function $W_{A}$ also describes a much smaller correlation effect seen in the
angle $\alpha_{Wb}$ between the hadronic-side $b$-jet and the hadronically
decaying $W$ boson. The function $W_{A}$ is thus parameterized using two
different functions, $W^{12}_{A}$ and $W^{Wb}_{A}$, describing the mappings
for the angles $\alpha_{12}$ and $\alpha_{Wb}$. The remaining angles describe
resolution effects rather than the correlations and, due to computational
constraints, are assumed to be well measured with their contributions to
$W_{A}$ approximated by Dirac delta functions.
The functions $W^{12}_{A}$ and $W^{Wb}_{A}$ are both fit using a sum of a
skew-Cauchy distribution and two Gaussians, describing the change in the
cosine of the relevant angle, $\Delta\cos{(\alpha_{12})}$ and
$\Delta\cos{(\alpha_{Wb})}$, from partons to measured jets. Since the
correlation effects are stronger in jets that are closer together, the
functions are parameterized in bins of $\cos(\alpha_{12})$ and
$\cos(\alpha_{Wb})$, respectively; one example for each function is shown in
Fig. 2.
Figure 2: Examples of parameterization of the functions $W^{12}_{A}$ and
$W^{Wb}_{A}$ in the bins where $0.2<\cos{(\alpha_{12})}<0.4$ and
$0.2<\cos{(\alpha_{Wb})}<0.4$. The histograms show MC simulation events and
the curves represent the parameterization.
The $m_{W}$ distribution after convolution with the function $W_{A}$ is shown
in Fig. 3. The skewness is removed and the mean value agrees well with the
parton-level distribution.
Figure 3: The $m_{W}$ distribution for measured angles from Fig. 1 is plotted
(solid line) after convolution with the function $W_{A}$. For ease of
comparison, the parton-level distribution (dotted line) is normalized so that
the maxima of the two distributions are the same.
The 20 integration variables (3 for each final-state particle and the
$x_{\mathrm{Bj}}$ for each initial state parton, assuming zero transverse
momentum for the $t\bar{t}$ pair) in the expression for the signal and
background p.d.f.s (Eq. 2) are reduced to 16 by integrating over the
4-momentum conservation Dirac $\delta$-function inherent in the expression for
$d\hat{\sigma}_{s}$. The charged lepton 3-momentum integration and all but two
of the jet angular integrations are made trivial by the Dirac
$\delta$-functions in the function $W(k,\varkappa)$, leaving 7 integration
variables. In $P_{s}$, this is further reduced to 5 variables via a change of
variables to the squared masses of the top quarks and by using the narrow-
width approximation for the Breit-Wigner distributions of both top-quark
decays in the $t\bar{t}$ matrix element. The integral is then evaluated using
the VEGAS Lepage (1978) adaptive Monte Carlo integration algorithm Abulencia
et al. (2007), which uses importance sampling, which means that the sample
points are concentrated in the regions that make the largest contribution to
the integral.
The treatment of $P_{b}$ is unchanged since the previous version of this
analysis Abulencia et al. (2007), except for the updated energy transfer
function $W_{E}$. The integrand in the expression for $P_{b}$ is much more
computationally intensive than for $P_{s}$ and a simplified Monte Carlo method
of integration is employed, giving reasonable convergence with an execution
time comparable to that of $P_{s}$. The simplifications used in this
computation of $P_{b}$ include setting the function $W_{A}$ to a Dirac
$\delta$-function for all angles, using a narrow width approximation for the
$W$ boson decay, and neglecting the $\Delta_{\mathrm{JES}}$ dependence of the
function $W_{E}$. Therefore, the value of $P_{b}$ for each event does not
depend on the likelihood parameters $m_{t}$ and $\Delta_{\mathrm{JES}}$, while
$P_{s}$ is a two-dimensional function of those parameters Abulencia et al.
(2007). In this approximation, the product of the background p.d.f.
normalization terms (corresponding to the variables $\hat{\sigma}_{b}\cdot
A_{b}$ in Eq. 2) is set to a constant, whose value is chosen to optimize the
statistical sensitivity of the method, effectively providing an appropriate
relative normalization with respect to $P_{s}$.
The log-likelihood function is given as a sum over the 578 events in the
sample:
$\displaystyle\ln\,$
$\displaystyle\mathcal{L}(k;m_{t},\Delta_{\mathrm{JES}},\nu_{\mathrm{sig}})=$
(4)
$\displaystyle\sum^{578}_{i=1}\ln\left[\nu_{\mathrm{sig}}{P_{s}(k_{i};m_{t},\Delta_{\mathrm{JES}})}+(1-\nu_{\mathrm{sig}})P_{b}(k_{i})\right]\mathrm{.}$
It is calculated on a two-dimensional $31\times 17$ grid in $m_{t}$ and
$\Delta_{\mathrm{JES}}$, spanning $145\leq m_{t}\leq\rm 205~{}GeV/{\it c}^{2}$
and $-4.8\leq\Delta_{\mathrm{JES}}\leq 4.8$, with a spacing between grid
points of $\rm 2~{}GeV/{\it c}^{2}$ in $m_{t}$ and 0.6 in
$\Delta_{\mathrm{JES}}$. To optimize computational time, the bin size is
chosen to be as large as possible without appreciably affecting the fit
result. The third likelihood parameter, the signal fraction parameter
$\nu_{\mathrm{sig}}$, is allowed to vary continuously (within the constraint
$0\leq\nu_{\mathrm{sig}}\leq 1$), and the likelihood function is maximized
with respect to $\nu_{\mathrm{sig}}$ at each point on the grid using the
MINUIT program James (2000). The resulting surface described on the grid is
the profile log-likelihood, maximized for $\nu_{\mathrm{sig}}$. The top-quark
mass, $m_{t}$, and the jet energy scale correction, $\Delta_{\mathrm{JES}}$,
are measured by making a two-dimensional parabolic fit to the surface,
consistent with the expectation for the likelihood function to be Gaussian
near its maximum. The maximum of the parabola gives the measured $m_{t}$ and
$\Delta_{\mathrm{JES}}$, while the measured $\nu_{\mathrm{sig}}$ is taken from
its value at the grid point of maximum likelihood. The estimated one-$\sigma$
statistical uncertainty of the measurement is represented by the ellipse
corresponding to a change in log-likelihood $\Delta\ln\mathcal{L}=0.5$ from
the maximum of the fitted parabola. The values of $m_{t}$ and
$\Delta_{\mathrm{JES}}$ are anti-correlated (Fig. 4). No correlation is
observed between $\nu_{\mathrm{sig}}$ and $m_{t}$ or $\Delta_{\mathrm{JES}}$.
The accuracy of the measured $m_{t}$ and $\Delta_{\mathrm{JES}}$, and their
uncertainties, are checked using ensembles of MC simulated experiments, using
the MC samples previously mentioned with the addition of 22 $t\bar{t}$ samples
generated with values of $m_{t}$ between $161$ and $\rm 185~{}GeV/{\it
c}^{2}$. The numbers of $t\bar{t}$ events and those of the various backgrounds
are Poisson fluctuated around the values shown in Table 2. Studies of the
relationships between the known input simulation parameters and their
corresponding measurements show no evidence of bias when a clean sample of MC
simulated $t\bar{t}$ events is used, containing only lepton+jets events with
correct jet-parton matching. However, the presence of signal events with jets
which are poorly or incorrectly matched to partons and events which do not
match the decay hypothesis biases the likelihood fit result and increases the
pull width. The presence of background events also biases the fit, due to the
backgrounds that are not well described by $P_{b}$ and the approximations in
$P_{b}$. The bias is removed using a set of functions obtained from a fit to
the MC simulation and parameterized in terms of the measured
$\Delta_{\mathrm{JES}}$ and $\nu_{\mathrm{sig}}$ Linacre (2010). This amounts
to adding $\rm 1.1~{}GeV/{\it c}^{2}$ to the $m_{t}$ value produced by the
likelihood fit and multiplying the uncertainty by 1.26 so that the pull width
is consistent with unity. The systematic uncertainty due to this measurement
calibration is small, as shown in Table 3.
Despite the reduction from the in situ $\Delta_{\rm JES}$ calibration, the
remaining uncertainty from JES obtained by varying the parameters in JES
Bhatti et al. (2006) is among the largest systematic uncertainties of the
measurement (Table 3). Other significant systematic uncertainties are mainly a
result of assumptions made in the simulation of the events that are used in
the tuning and calibration of the measurement method. In most cases, they are
evaluated by varying different aspects of the MC simulation, such as signal MC
generator (PYTHIA versus HERWIG Corcella et al. (2002)), color reconnection
model tune (Apro versus ACRpro Skands and Wicke (2007); Wicke and Skands
(2008); Skands (2009)), and parameters of initial and final state radiation
(ISR and FSR). A detailed description of the systematic effects has been
published elsewhere Aaltonen et al. (2009). The systematic uncertainties for
each effect are added in quadrature, resulting in a total estimated systematic
uncertainty of $\rm 1.3~{}GeV/{\it c}^{2}$ (Table 3).
Systematic | (GeV/c2)
---|---
MC Generator | 0.70
Residual JES | 0.65
Color Reconnection | 0.56
$b$-jet energy | 0.39
Background | 0.45
ISR and FSR | 0.23
Multiple Hadron Interactions | 0.22
PDFs | 0.13
Lepton Energy | 0.12
Measurement Calibration | 0.12
Total | 1.31
Table 3: Contributions to the total expected systematic uncertainty.
The measurement is made using the data sample of 578 events, yielding
$\displaystyle m_{t}$ $\displaystyle=172.4\rm\pm
1.4\>(stat\mathrm{+}\Delta_{\mathrm{JES}})\pm 1.3\>(sys)~{}GeV/{\it c}^{2}$
$\displaystyle m_{t}$ $\displaystyle=172.4\rm\pm 1.9\>(total)~{}GeV/{\it
c}^{2}\mathrm{,}$ (5)
with $\rm\Delta_{\mathrm{JES}}=0.3\pm 0.3\>(stat)$. The central value and the
contour ellipses corresponding to the one-, two- and three-$\sigma$
statistical confidence intervals of the measurement are illustrated in Fig. 4.
The overall statistical uncertainty on the measured top-quark mass is labeled
“stat+$\Delta_{\mathrm{JES}}$” because it includes the uncertainty on $m_{t}$
due to the statistical uncertainty on the measured $\Delta_{\mathrm{JES}}$,
i.e. the uncertainty is given by half of the full width of the one-$\sigma$
contour of Fig. 4.
In conclusion, a precise measurement of the top-quark mass has been presented
using CDF lepton+jets candidate events corresponding to an integrated
luminosity of 3.2 fb-1. Using an improved matrix element method with an in
situ jet energy calibration, the top quark mass is measured to be $m_{t}=\rm
172.4\pm 1.9~{}GeV/{\it c}^{2}$.
Figure 4: The measurement result and the contour ellipses of the parabolic fit
corresponding to the one-, two- and three-$\sigma$ confidence intervals for
the statistical uncertainty on $m_{t}$ and $\Delta_{\mathrm{JES}}$.
###### Acknowledgements.
We thank the Fermilab staff and the technical staffs of the participating
institutions for their vital contributions. This work was supported by the
U.S. Department of Energy and National Science Foundation; the Italian
Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture,
Sports, Science and Technology of Japan; the Natural Sciences and Engineering
Research Council of Canada; the National Science Council of the Republic of
China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the
Bundesministerium für Bildung und Forschung, Germany; the Korean World Class
University Program, the National Research Foundation of Korea; the Science and
Technology Facilities Council and the Royal Society, UK; the Institut National
de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation
for Basic Research; the Ministerio de Ciencia e Innovación, and Programa
Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; the Academy of Finland;
and the Australian Research Council (ARC).
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|
arxiv-papers
| 2011-08-08T04:30:42 |
2024-09-04T02:49:21.366443
|
{
"license": "Public Domain",
"authors": "CDF Collaboration: T. Aaltonen, B. \\'Alvarez Gonz\\'alezv, S. Amerio,\n D. Amidei, A. Anastassov, A. Annovi, J. Antos, G. Apollinari, J.A. Appel, A.\n Apresyan, T. Arisawa, A. Artikov, J. Asaadi, W. Ashmanskas, B. Auerbach, A.\n Aurisano, F. Azfar, W. Badgett, A. Barbaro-Galtieri, V.E. Barnes, B.A.\n Barnett, P. Barriaee, P. Bartos, M. Baucecc, G. Bauer, F. Bedeschi, D.\n Beecher, S. Behari, G. Bellettinidd, J. Bellinger, D. Benjamin, A. Beretvas,\n A. Bhatti, M. Binkley, D. Bisellocc, I. Bizjakii, K.R. Bland, C. Blocker, B.\n Blumenfeld, A. Bocci, A. Bodek, D. Bortoletto, J. Boudreau, A. Boveia, B.\n Braua, L. Brigliadoribb, A. Brisuda, C. Bromberg, E. Brucken, M.\n Bucciantoniodd, J. Budagov, H.S. Budd, S. Budd, K. Burkett, G. Busettocc, P.\n Bussey, A. Buzatu, S. Cabrerax, C. Calancha, S. Camarda, M. Campanelli, M.\n Campbell, F. Canelli, A. Canepa, B. Carls, D. Carlsmith, R. Carosi, S.\n Carrillok, S. Carron, B. Casal, M. Casarsa, A. Castrobb, P. Catastini, D.\n Cauz, V. Cavaliereee, M. Cavalli-Sforza, A. Cerrif, L. Cerritoq, Y.C. Chen,\n M. Chertok, G. Chiarelli, G. Chlachidze, F. Chlebana, K. Cho, D. Chokheli,\n J.P. Chou, W.H. Chung, Y.S. Chung, C.I. Ciobanu, M.A. Ciocciee, A. Clark, D.\n Clark, G. Compostellacc, M.E. Convery, J. Conway, M.Corbo, M. Cordelli, C.A.\n Cox, D.J. Cox, F. Cresciolidd, C. Cuenca Almenar, J. Cuevasv, R. Culbertson,\n D. Dagenhart, N. d'Ascenzot, M. Datta, P. de Barbaro, S. De Cecco, G. De\n Lorenzo, M. Dell'Orsodd, C. Deluca, L. Demortier, J. Dengc, M. Deninno, F.\n Devoto, M. d'Erricocc, A. Di Cantodd, B. Di Ruzza, J.R. Dittmann, M.\n D'Onofrio, S. Donatidd, P. Dong, T. Dorigo, K. Ebina, A. Elagin, A. Eppig, R.\n Erbacher, D. Errede, S. Errede, N. Ershaidataa, R. Eusebi, H.C. Fang, S.\n Farrington, M. Feindt, J.P. Fernandez, C. Ferrazzaff, R. Field, G. Flanaganr,\n R. Forrest, M.J. Frank, M. Franklin, J.C. Freeman, I. Furic, M. Gallinaro, J.\n Galyardt, J.E. Garcia, A.F. Garfinkel, P. Garosiee, H. Gerberich, E.\n Gerchtein, S. Giagugg, V. Giakoumopoulou, P. Giannetti, K. Gibson, C.M.\n Ginsburg, N. Giokaris, P. Giromini, M. Giunta, G. Giurgiu, V. Glagolev, D.\n Glenzinski, M. Gold, D. Goldin, N. Goldschmidt, A. Golossanov, G. Gomez, G.\n Gomez-Ceballos, M. Goncharov, O. Gonz\\'alez, I. Gorelov, A.T. Goshaw, K.\n Goulianos, A. Gresele, S. Grinstein, C. Grosso-Pilcher, R.C. Group, J.\n Guimaraes da Costa, Z. Gunay-Unalan, C. Haber, S.R. Hahn, E. Halkiadakis, A.\n Hamaguchi, J.Y. Han, F. Happacher, K. Hara, D. Hare, M. Hare, R.F. Harr, K.\n Hatakeyama, C. Hays, M. Heck, J. Heinrich, M. Herndon, S. Hewamanage, D.\n Hidas, A. Hocker, W. Hopkinsg, D. Horn, S. Hou, R.E. Hughes, M. Hurwitz, U.\n Husemann, N. Hussain, M. Hussein, J. Huston, G. Introzzi, M. Iorigg, A.\n Ivanovo, E. James, D. Jang, B. Jayatilaka, E.J. Jeon, M.K. Jha, S.\n Jindariani, W. Johnson, M. Jones, K.K. Joo, S.Y. Jun, T.R. Junk, T. Kamon,\n P.E. Karchin, Y. Katon, W. Ketchum, J. Keung, V. Khotilovich, B. Kilminster,\n D.H. Kim, H.S. Kim, H.W. Kim, J.E. Kim, M.J. Kim, S.B. Kim, S.H. Kim, Y.K.\n Kim, N. Kimura, S. Klimenko, K. Kondo, D.J. Kong, J. Konigsberg, A. Korytov,\n A.V. Kotwal, M. Kreps, J. Kroll, D. Krop, N. Krumnackl, M. Kruse, V.\n Krutelyovd, T. Kuhr, M. Kurata, S. Kwang, A.T. Laasanen, S. Lami, S. Lammel,\n M. Lancaster, R.L. Lander, K. Lannonu, A. Lath, G. Latinoee, I. Lazzizzera,\n T. LeCompte, E. Lee, H.S. Lee, J.S. Lee, S.W. Leew, S. Leodd, S. Leone, J.D.\n Lewis, C.-J. Lin, J. Linacre, M. Lindgren, E. Lipeles, A. Lister, D.O.\n Litvintsev, C. Liu, Q. Liu, T. Liu, S. Lockwitz, N.S. Lockyer, A. Loginov, D.\n Lucchesicc, J. Lueck, P. Lujan, P. Lukens, G. Lungu, J. Lys, R. Lysak, R.\n Madrak, K. Maeshima, K. Makhoul, P. Maksimovic, S. Malik, G. Mancab, A.\n Manousakis-Katsikakis, F. Margaroli, C. Marino, M. Mart\\'inez, R.\n Mart\\'inez-Ballar\\'in, P. Mastrandrea, M. Mathis, M.E. Mattson, P. Mazzanti,\n K.S. McFarland, P. McIntyre, R. McNultyi, A. Mehta, P. Mehtala, A. Menzione,\n C. Mesropian, T. Miao, D. Mietlicki, A. Mitra, H. Miyake, S. Moed, N. Moggi,\n M.N. Mondragonk, C.S. Moon, R. Moore, M.J. Morello, J. Morlock, P. Movilla\n Fernandez, A. Mukherjee, Th. Muller, P. Murat, M. Mussinibb, J. Nachtmanm, Y.\n Nagai, J. Naganoma, I. Nakano, A. Napier, J. Nett, C. Neuz, M.S. Neubauer, J.\n Nielsene, L. Nodulman, O. Norniella, E. Nurse, L. Oakes, S.H. Oh, Y.D. Oh, I.\n Oksuzian, T. Okusawa, R. Orava, L. Ortolan, S. Pagan Grisocc, C. Pagliarone,\n E. Palenciaf, V. Papadimitriou, A.A. Paramonov, J. Patrick, G. Paulettahh, M.\n Paulini, C. Paus, D.E. Pellett, A. Penzo, T.J. Phillips, G. Piacentino, E.\n Pianori, J. Pilot, K. Pitts, C. Plager, L. Pondrom, K. Potamianos, O.\n Poukhov, F. Prokoshiny, A. Pronko, F. Ptohosh, E. Pueschel, G. Punzidd, J.\n Pursley, A. Rahaman, V. Ramakrishnan, N. Ranjan, I. Redondo, P. Renton, M.\n Rescigno, F. Rimondibb, L. Ristori, A. Robson, T. Rodrigo, T. Rodriguez, E.\n Rogers, S. Rolli, R. Roser, M. Rossi, F. Ruffiniee, A. Ruiz, J. Russ, V.\n Rusu, A. Safonov, W.K. Sakumoto, L. Santihh, L. Sartori, K. Sato, V.\n Savelievt, A. Savoy-Navarro, P. Schlabach, A. Schmidt, E.E. Schmidt, M.P.\n Schmidt, M. Schmitt, T. Schwarz, L. Scodellaro, A. Scribanoee, F. Scuri, A.\n Sedov, S. Seidel, Y. Seiya, A. Semenov, F. Sforzadd, A. Sfyrla, S.Z.\n Shalhout, T. Shears, P.F. Shepard, M. Shimojimas, S. Shiraishi, M. Shochet,\n I. Shreyber, A. Simonenko, P. Sinervo, A. Sissakian, K. Sliwa, J.R. Smith,\n F.D. Snider, A. Soha, S. Somalwar, V. Sorin, P. Squillacioti, M. Stanitzki,\n R. St. Denis, B. Stelzer, O. Stelzer-Chilton, D. Stentz, J. Strologas, G.L.\n Strycker, Y. Sudo, A. Sukhanov, I. Suslov, K. Takemasa, Y. Takeuchi, J. Tang,\n M. Tecchio, P.K. Teng, J. Thomg, J. Thome, G.A. Thompson, E. Thomson, P.\n Ttito-Guzm\\'an, S. Tkaczyk, D. Toback, S. Tokar, K. Tollefson, T. Tomura, D.\n Tonelli, S. Torre, D. Torretta, P. Totarohh, M. Trovatoff, Y. Tu, N.\n Turiniee, F. Ukegawa, S. Uozumi, A. Varganov, E. Vatagaff, F. V\\'azquez, G.\n Velev, C. Vellidis, M. Vidal, I. Vila, R. Vilar, M. Vogel, G. Volpidd, P.\n Wagner, R.L. Wagner, T. Wakisaka, R. Wallny, S.M. Wang, A. Warburton, D.\n Waters, M. Weinberger, W.C. Wester III, B. Whitehouse, D. Whitesonc, A.B.\n Wicklund, E. Wicklund, S. Wilbur, F. Wick, H.H. Williams, J.S. Wilson, P.\n Wilson, B.L. Winer, P. Wittichg, S. Wolbers, H. Wolfe, T. Wright, X. Wu, Z.\n Wu, K. Yamamoto, J. Yamaoka, T. Yang, U.K. Yangp, Y.C. Yang, W.-M. Yao, G.P.\n Yeh, K. Yim, J. Yoh, K. Yorita, T. Yoshidaj, G.B. Yu, I. Yu, S.S. Yu, J.C.\n Yun, A. Zanetti, Y. Zeng, S. Zucchelli",
"submitter": "Eva Halkiadakis",
"url": "https://arxiv.org/abs/1108.1601"
}
|
1108.1668
|
# INTEGRAL observations of the $\gamma$-ray binary 1FGL J1018.6-5856
Jian Li11affiliation: Laboratory for Particle Astrophysics, Institute of High
Energy Physics, Beijing 100049, China. , Diego F. Torres22affiliation:
Institució Catalana de Recerca i Estudis Avançats (ICREA). 33affiliation:
Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta,
08193 Barcelona, Spain , Yupeng Chen11affiliation: Laboratory for Particle
Astrophysics, Institute of High Energy Physics, Beijing 100049, China. ,
Diego Götz44affiliation: AIM (UMR 7158 CEA/DSM-CNRS-Université Paris Diderot)
Irfu/Service d’Astrophysique, Saclay, 91191 Gif-sur-Yvette Cedex, France ,
Nanda Rea33affiliation: Institut de Ciències de l’Espai (IEEC-CSIC), Campus
UAB, Torre C5, 2a planta, 08193 Barcelona, Spain , Shu Zhang11affiliation:
Laboratory for Particle Astrophysics, Institute of High Energy Physics,
Beijing 100049, China. , G. Andrea Caliandro33affiliation: Institut de
Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193
Barcelona, Spain , Jianmin Wang11affiliation: Laboratory for Particle
Astrophysics, Institute of High Energy Physics, Beijing 100049, China.
55affiliation: Theoretical Physics Center for Science Facilities (TPCSF), CAS,
China
###### Abstract
The Fermi-LAT collaboration has recently reported that one of their detected
sources, namely, 1FGL J1018.6-5856, is a new gamma-ray binary similar to LS
5039. This has prompted efforts to study its multi-frequency behavior. In this
report, we present the results from 5.78-Ms INTEGRAL IBIS/ISGRI observations
on the source 1FGL J1018.6-5856.
By combining all the available INTEGRAL data, a detection is made at a
significance level of 5.4$\sigma$ in the 18–40 keV band, with an average
intensity of 0.074 counts s-1. However, we find that, there is non-statistical
noise in the image that effectively reduces the significance to about
4$\sigma$ and a significant part of the signal appears to be located in a
0.2-wide phase region, at phases 0.4–0.6 (where even the corrected
significance amounts to 90% of the total signal found). Given the scarcity of
counts, a variability is hinted at about $3\sigma$ at the hard X-rays, with an
anti-correlation with the Fermi-LAT periodicity. Should this behavior be true,
it would be similar to that found in LS 5039, and prompt observations with TeV
telescopes at phases anti-correlated with the GeV maximum.
X-rays: binaries, X-rays: individual (1FGL J1018.6-5856)
## 1 Introduction
The Fermi Large Area telescope (Fermi-LAT) is a pair-production detector with
large effective area ($\sim$ 8000 cm2 on axis for $E>1$ GeV) and field of view
($\sim$ 2.4 sr at 1 GeV), sensitive to gamma rays in the energy range from 20
MeV to $>$ 300 GeV (Atwood et al. 2009). Precisely because of its capability
of covering a wide region of the sky, its normal mode of operation is
surveying, which facilitates serendipitous discoveries and simultaneous
observations of many sources. The Fermi-LAT collaboration has recently
released (Corbet et al. 2011) the results for continuing data analysis of the
gamma-ray emission from 1FGL J1018.6-5856, for which earlier information was
also reported in the Fermi-LAT source catalog (Abdo et al., 2010). The
considered data, obtained between MJD 54682 and 55627 in the energy range 100
MeV to 200 GeV show the presence of periodic modulation with a period of 16.58
$\pm$ 0.04 days, and an epoch of maximum gamma-ray flux at MJD 55403.3 $\pm$
0.4 (Corbet et al. 2011). A coincident X-ray flux was found using Swift-XRT
observations, which features also a high degree of variability, with the
0.3–10 keV count rates ranging from approximately 0.01 to 0.05 counts/s, as
well as a star of magnitude B2 which in turn coincides with the Swift-XRT
detection. Based on all of the former, Corbet et al. (2011) reported that 1FGL
J1018.6-5856 is a new gamma-ray binary like, for instance, LS 5039 (see Abdo
et al. 2009). In this Letter we present the results of the analysis of 5.78-Ms
INTEGRAL IBIS/ISGRI data on the source 1FGL J1018.6-5856.
## 2 Observations and data analysis
INTEGRAL (Winkler et al. 2003) is optimized to work between 15 keV–10 MeV. Its
main instruments are the IBIS (15 keV–10 MeV; Ubertini et al. 2003) and the
Spectrometer on board INTEGRAL (SPI, 20 keV–8 MeV; Vedrenne et al. 2003). At
the lower energies (15 keV–1 MeV), the cadmium telluride array ISGRI (Lebrun
et al. 2003) of IBIS has a better continuum sensitivity than SPI below $\sim
300$ keV. The INTEGRAL observations were carried out per pointing, called as
individual Science Windows (SCWs), with a typical time duration of about 2000
s each. The data reduction was performed by using the standard Online Science
Analysis (OSA), version 9.0. The results were obtained by running the pipeline
from the raw to the image level. In this analysis, only IBIS/ISGRI public data
are taken into account. The available INTEGRAL observations when 1FGL
J1018.6-5856 had offset angle less than 14o comprise about 2014 SCWs, adding
up to a total exposure time of 5.78 Ms. Our dataset covers rev. 30–867, from
2003-01-11 to 2009-11-20 (MJD 52650-55155). This large amount of data allow
for an in-depth investigation on the hard X-ray emission from 1FGL
J1018.6-5856.
## 3 Results
Figure 1: Mosaic image of the 1FGL J1018.6-5856 sky region, derived by
combining all ISGRI data in the 18–40 keV band. The strongest source is GRO
J1008-57 whereas the faintest one, visible only in the image is 1FGL
J1018.6-5856. The significance level is given by the color scale, with the
contours start at $3\sigma$, and following steps of $1\sigma$. The position of
1FGL J1018.6-5856 (magenta) as well as its updated center following the 2FGL
Fermi-LAT Catalog (blue)222 See the website:
http://fermi.gsfc.nasa.gov/ssc/data/access/lat/2yr_catalog/ for more
information on 2FGL Fermi-LAT Catalog.are shown, while the cross represent the
counterparts identified using XRT (from Corbet et al. 2011).
An INTEGRAL detection of 1FGL J1018.6-5856 is derived from combining all the
ISGRI data, with a significance level of 5.4$\sigma$ and an average intensity
of 0.074 counts s-1 in the 18–40 keV band. 1FGL J1018.6-5856 is also
consistently detected in the 18–60 keV; its map significance is reduced only
to a fluctuation at higher energies. We only obtain 1.63$\sigma$ in the 40–100
keV. The significance and flux measurement of 1FGL J1018.6-5856 are obtained
from the pixel comprising the most prominent position determined by Swift-XRT
(Corbet et al. 2011), which is within Fermi-LAT 1FGL and 2FGL error circle.
The pixel size of INTEGRAL is about 4.93 arcmin $\times$ 4.93 arcmin, whereas
the uncertainty in position of Fermi, i.e., the 95% confidence radius is
$\sim$1.76 arcmin and is thus fully included in the pixel used for our
significance estimation. However, we would like to notice that the source
significance should be taken with care. The pixel significance distribution of
each 18-40 keV map has a Gaussian shape with a width parameter, instead of 1
as expected from a purely statistical dominated image, of about 1.1, which may
due to an unknown systematic error. Typically, systematic noise is produced
from being close to the very bright sources and or very crowded regions. As
more maps from different observations are combined, the width parameter
increases and finally reaches up to 1.4 for all data considered. Thus, we find
that there is a non-statistical noise in the image that effectively reduces
the significance to 5.4/1.4, or about 4$\sigma$. Given the spatial coincidence
of the source with 1FGL J1018.6-5856 we can claim a detection at 99.9937 %, or
alternatively, that the probability of a chance coincidence is of the order
$\sim$ $6.3\times 10^{-5}$.
If we assume (although there is no a prior reason to, and we caveat that this
is just done for estimation to an order of magnitude) that the spectral shape
of 1FGL J1018.6-5856 is same with that of the Crab nebula in the corresponding
energy range (a power-law with $\Gamma\sim$2.14), a flux of 1FGL J1018.6-5856
is derived as $2.25\times 10^{-11}$ erg cm-2 s-1, or about 0.35 mCrab in flux
units. If we instead assume that the spectrum of 1FGL J1018.6-5856 is similar
to that of LS 5039 at inferior conjunction passage as given by Hoffmann et al.
(2009), we would obtain 1FGL J1018.6-5856 flux of about $0.75\times 10^{-11}$
erg cm-2 s-1, or a factor of 0.23 of the LS 5039 flux. The neighboring source
GRO J1008-57, 1.36∘ away from 1FGL J1018.6-5856 is detected with a
significance of $54.6\sigma$ and with an intensity of 4.2 mCrab. It likely
constitutes a problem for the analysis of 1FGL J1018.6-5856 with non-imaging
detectors.
Table 1: Exposure times and obtained significance for the whole INTEGRAL observation (all phase bins) and separated in orbital bins. phase | flux | significance | exposure | width of combined sig.
---|---|---|---|---
| (counts s-1) | (in 18–40 keV) | (Ms) | map distribution
0.0–1.0 | 0.074$\pm$0.014 | 5.4 | 5.78 | 1.39
0.0–0.2 | -0.0073$\pm$0.032 | – | 1.16 | 1.36
0.2–0.4 | 0.094$\pm$0.030 | 3.12 | 1.13 | 1.32
0.4–0.6 | 0.154$\pm$0.031 | 4.89 | 1.13 | 1.37
0.6–0.8 | 0.050$\pm$0.032 | 1.54 | 1.05 | 1.30
0.8–1.0 | 0.061$\pm$0.028 | 2.15 | 1.31 | 1.36
Since 1FGL J1018.6-5856 is much weaker than other neighboring sources in its
crowded sky region, it is impossible for OSA 9.0 to derive a lightcurve
directly out of the standard reduction procedure. However, one can combine the
images from different science windows based on orbital phase-bins and, via
inferring the flux from each of the combined image, produce an orbital
lightcurve manually. Thus, to see how 1FGL J1018.6-5856 varies along its 16.58
days orbit, we divide the INTEGRAL data into phase bins. The reference time at
phase zero is set to the peak flux that observed by Fermi-LAT at $T_{max}={\rm
MJD}\;55403.3\pm n\times 16.58$ days, as reported at GeV energies in Corbet et
al. (2011). We show the INTEGRAL results on 1FGL J1018.6-5856 for each phase
bin in Table 1, where the flux, the corresponding exposure and the
significance are provided. Again, given that the pixel distribution has a
Gaussian width of about 1.34 in each of the images (see Table 1), we
conservatively consider that the significance should be lowered by this
factor. Hinted from Table 1 is a trend of having an anti-correlation between
the hard X-ray emission and the Fermi-LAT periodicity. Actually, we find that
a significant portion of the signal comes only from a 0.2-wide phase region,
at phases 0.4–0.6 (where even the corrected significance amounts to about 90%
of the signal found in total). However, the scarcity of counts makes it
difficult to have a definitive proof of the variability: a constant fit to the
count rate yields a reduced $\chi^{2}$ of 14.09/4, suggesting that the
significance of variability is at 99.64%, or only 2.7$\sigma$ level.
Though it is impossible to derive a lightcurve from standard reduction
procedure, we could read pixels corresponding to 1FGL J1018.6-5856’ position
and derived a light curve manually. To search for a periodic signal in the
light curve data, we used the Lomb–Scargle periodogram method. Power spectrum
is generated for the light curve using the PERIOD subroutine (Press & Rybicki
1989). No significant signal is seen at an orbit period of 16.58 days beyond
90% white–noise confidence level. This is consistent with having only an
overall weak detection of the source in our imaging analysis and the orbit
modulation is not apparent in lightcurve.
## 4 Concluding remarks
We have carried out an analysis of all INTEGRAL data available to 1FGL
J1018.6-5856, a new gamma-ray binary with a period of $\sim$ 16.5 days
discovered blindly by means of a power spectrum analysis of Fermi-LAT
detection (100 MeV–200 GeV). The total effective exposure extracted on the
source from the archive amounts to 5.78 Ms, and leads to a credible detection
of the source at hard X-rays (18–40 keV). The count rate is however very low,
preventing from extracting strong conclusions about orbital variability,
although it is hinted with an anti-correlation with the gamma-ray emission
detected by Fermi-LAT in 100 MeV – 200 GeV. In fact, 1FGL J1018.6-5856 seems
to significantly show up only during a small part of its orbital evolution,
far from the gamma-ray maximum at 100 MeV–200 GeV. This would represent a more
marked distinction with respect to what was obtained in the case of LS I
+61∘303 which presents a displacement of the maximum between hard X-rays (by
INTEGRAL) and gamma-rays (by Fermi-LAT and MAGIC), without being completely
anti-correlated (Zhang et al. 2010). Instead this is in line with the results
for LS 5039 (see, e.g., Hoffman et al. 2009) where the hard X-ray emission as
measured with INTEGRAL is correlated with the TeV emission measured with HESS
(Aharonian et al. 2006), and thus it is fully anti-correlated with the GeV
emission as measured by Fermi-LAT (Abdo et al. 2009). The case of 1FGL
J1018.6-5856 appears as another incarnation of this behavior, albeit with the
conservative caveat of yet a low significance for an strong claim —the maximum
count rate of 1FGL J1018.6-5856 is about five times lower than that of LS
5039—, emphasizing a possible physical similarity of the two sources. TeV
observations at what appears to be the maximum of the hard X-ray lightcurve
are thus encouraged.
We acknowledge support from the grants AYA2009-07391 and SGR2009-811, as well
as the Formosa Program TW2010005. NR is supported by a Ramon y Cajal
Fellowship. This work was also subsidized by the National Natural Science
Foundation of China via NSFC-10325313,10521001,10733010,11073021 and 10821061,
the CAS key Project KJCX2-YW-T03, and 973 program 2009CB824800
## References
* Abdo A. et al. (2009) Abdo A. et al., 2009, ApJ, 706, L56
* Abdo A. et al. (2008) Abdo A. et al., 2010, ApJS, 188, 405
* Aharonian, F. A. , et al. (2006) Aharonian, F. A. , et al. 2006, A&A, 460, 743
* Atwood, W. B., et al. (2009) Atwood, W. B., et al. 2009, ApJ, 697, 1071
* Corbet R. et al. (2011) Corbet R. et al. 2011, ATel, 3221
* Hoffmann A. D. et al. (2009) Hoffmann A. D. et al. 2009, A&A, 494, L37
* Ubertini P. et al. (2003) Ubertini P. et al., 2003, A&A, 411, L131
* Press, W. H., & Rybicki, G. B. (1989) Press, W. H., & Rybicki, G. B. 1989, ApJ, 338, 277
* Vedrenne G. et al. (2003) Vedrenne G. et al., 2003, A&A, 411, L63
* Winkler C. et al. (2003) Winkler C. et al., 2003, A&A, 411, L1
* Zhang, S. et al. (2010) Zhang, S., Torres, D. F., Li, J., Chen, Y. P., Rea, N., & Wang, J. M. 2010, MNRAS, 408, 642
|
arxiv-papers
| 2011-08-08T10:16:00 |
2024-09-04T02:49:21.406520
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Li, Diego F. Torres, Yupeng Chen, Diego G\\\"otz, Nanda Rea, Shu\n Zhang, G. Andrea Caliandro, Jianmin Wang",
"submitter": "Jian Li",
"url": "https://arxiv.org/abs/1108.1668"
}
|
1108.1722
|
# A Standard Law for the Equatorward Drift of the Sunspot Zones
D. H. Hathaway1 1 NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
email: david.hathaway@nasa.gov
###### Abstract
The latitudinal location of the sunspot zones in each hemisphere is determined
by calculating the centroid position of sunspot areas for each solar rotation
from May 1874 to June 2011. When these centroid positions are plotted and
analyzed as functions of time from each sunspot cycle maximum there appears to
be systematic differences in the positions and equatorward drift rates as a
function of sunspot cycle amplitude. If, instead, these centroid positions are
plotted and analyzed as functions of time from each sunspot cycle minimum then
most of the differences in the positions and equatorward drift rates
disappear. The differences that remain disappear entirely if curve fitting is
used to determine the starting times (which vary by as much as 8 months from
the times of minima). The sunspot zone latitudes and equatorward drift
measured relative to this starting time follow a standard path for all cycles
with no dependence upon cycle strength or hemispheric dominance. Although
Cycle 23 was peculiar in its length and the strength of the polar fields it
produced, it too shows no significant variation from this standard. This
standard law, and the lack of variation with sunspot cycle characteristics, is
consistent with Dynamo Wave mechanisms but not consistent with current Flux
Transport Dynamo models for the equatorward drift of the sunspot zones.
###### keywords:
Solar Cycle, Observations; Sunspots, Statistics; Sunspots, Velocity
## 1 Introduction
The equatorward drift of the sunspot zones is now a well known characteristic
of the sunspot cycle. While Carrington58 noted the disappearance of low
latitude spots followed by the appearance of spots confined to mid-latitudes
during the transition from Cycle 9 to Cycle 10, and Sporer80 calculated and
plotted the equatorward drift of sunspot zones over Cycles 10 and 11, the very
existence of the sunspot zones was still in question decades later [Maunder
(1903)]. The publication of the “Butterfly Diagram” by Maunder04 laid this
controversy to rest and revealed a key aspect of the sunspot cycle – sunspots
appear in zones on either side of the equator that drift toward the equator as
each sunspot cycle progresses.
Cycle-to-cycle variations in the sunspot latitudes have been noted previously.
Becker54 and Waldmeier55 both noted that, at maximum, the sunspot zones are at
higher latitudes in the larger sunspot cycles. More recently, Hathaway_etal03
found an anti-correlation between the equatorward drift rate and cycle period
and suggested that this was evidence in support of flux transport dynamos
[Nandy & Choudhuri (2002)]. However, Hathaway10 noted that all these results
are largely due to the fact that larger sunspot cycles reach maximum sooner
than smaller sunspot cycles and that the drift rate is faster in the earlier
part of both small and large cycles. Nonetheless, Hathaway10 did find that the
sunspot zones appeared at slightly higher latitudes (with slightly higher
drift rates) in the larger sunspot cycles when comparisons were made relative
to the time of sunspot cycle minimum.
The equatorward drift of the sunspot zones is a key characteristic of the
sunspot cycle. It must be reproduced in viable models for the Sun’s magnetic
dynamo and can be used to discriminate between the various models.
In the Babcock61 and Leighton69 dynamo models the latitudinal positions of the
sunspot zones are determined by the latitudes where the differential rotation
and initial magnetic fields produce magnetic fields strong enough to make
sunspots. This “critical” latitude moves equatorward from the position of
strongest latitudinal shear as the cycle progresses. The initial strength of
the magnetic field in these models is determined by the polar field strength
at cycle minimum so we might expect a delayed start for cycles starting with
weak polar fields and the equatorward propagation might depend on both the
differential rotation profile (which doesn’t vary substantially) and the
initial polar fields (which do vary substantially).
In a number of dynamo models (both kinematic and magnetohydrodynamic) the
equatorward drift of the sunspot zones is produced by a “Dynamo Wave” (cf.
Yoshimura75) which propagates along iso-rotation surfaces at a rate and
direction given by the product of the shear in the differential rotation and
the kinetic helicity in the fluid motions. In these models the equatorward
propagation is a function of the differential rotation profile and the profile
of kinetic helicity - both of which don’t vary substantially.
In flux transport dynamo models (cf. NandyChoudhuri02) the equatorward drift
is produced primarily by the equatorward return flow of a proposed deep
meridional circulation. In these models, variations in the meridional flow
speed (which does vary substantially with cycle amplitude and duration in
these models) should be observed as variations in the equatorward drift rate
of the sunspot zones.
Here we reexamine the latitudes of the sunspot zones and find that cycle-to-
cycle and hemispheric variations vanish when time is measured relative to a
cycle starting time derived from fitting the monthly sunspot numbers in each
cycle to a functional form for the cycle shape.
## 2 The Sunspot Zones
Sunspot group positions and areas have been measured daily since May 1874. The
Royal Observatory Greenwich carried out the earlier observations using a small
network of solar observatories from May 1874 to December 1976. The United
States Air Force and National Oceanic and Atmospheric Administration continued
to acquire similar observations from a somewhat larger network starting in
January 1977. We calculate the average daily sunspot area over each Carrington
rotation for 50 equal area latitude bins (equi-spaced in $\sin\lambda$ where
$\lambda$ is the latitude). The sunspot zones are clearly evident in the
resulting Butterfly Diagram - Figure 1.
Figure 1.: Sunspot areas as functions of sin(latitude) and time for each
Carrington rotation from 1880 to 2010. These data include four small cycles
(Cycles 12, 13, 14, and 16), four average cycles (Cycles 15,17, 18, and 20),
and four large cycles (Cycles 19, 21, 22, and 23).
We divide the data into separate sunspot cycles by attributing low-latitude
groups to the earlier cycle and high-latitude groups to the later cycle when
the cycles overlap at minima. We further divide the data by hemisphere and
then calculate the latitude, $\bar{\lambda}$, of the centroid of sunspot area
for each hemisphere for each rotation of each sunspot cycle using
$\bar{\lambda}=\sum{A(\lambda_{i})\lambda_{i}}/\sum{A(\lambda_{i})}$ (1)
where $A(\lambda_{i})$ is the average daily sunspot area in the latitude bin
centered on latitude $\lambda_{i}$ and the sums are over the 25 latitude bins
for a given hemisphere and Carrington rotation. These centroid positions then
provide the sunspot zone latitudes and drift rates for each hemisphere as a
function of time through each cycle.
## 3 The Sunspot Zone Equatorward Drift
Previous work on cycle-to-cycle variations in the positions and drift rates of
the sunspot zones [Becker (1954), Waldmeier (1955), Hathaway et al. (2003)]
made those measurements relative to the sunspot cycle maxima. The centroid
position data are plotted as functions of time from cycle maxima in Figure 2.
The data encompass 12 sunspot cycles which, fortuitously, include four cycles
much smaller than average (Cycles 12, 13, 14, and 16 with smoothed sunspot
cycle maxima below 90), four cycles much larger than average (Cycles 18, 19,
21, and 22 with smoothed sunspot cycle maxima above 150), and four cycles
close to the average (Cycles 15, 17, 20, and 23).
Figure 2 illustrates why the earlier studies concluded that larger cycles tend
to have sunspot zones at higher latitudes. The centroid positions for the
large cycles (in red) are clearly at higher latitudes than those for the
medium cycles which, in turn, are at higher latitudes than those for the small
cycles. While this conclusion is technically correct, it is somewhat
misleading since large cycles reach their maxima sooner than small cycles (the
“Waldmeier Effect” Waldmeier35 and Hathaway10) and the sunspot zones are
always at higher latitude earlier in each cycle.
Figure 2.: The centroid (area weighted) positions of the sunspot zones in each
hemisphere for each solar rotation are plotted as functions of time from each
sunspot cycle maximum. The individual data points are shown in the upper
panel. The size of the symbol varies with the average daily sunspot area for
each solar rotation and hemisphere. The color of the symbol varies with the
amplitude of the sunspot cycle associated with the data. The average centroid
positions of the sunspot zones for small (blue) medium (green) and large (red)
cycles plotted at 6-month intervals in time from sunspot cycle maximum are
shown in the lower panel.
In Figure 3 the centroid positions are plotted as functions of time from
sunspot cycle minima. Since large cycles reach maximum earlier than small
cycles, the data points for the large cycles are shifted to the left (closer
to minimum) relative to the medium and small cycles. The size of the shift is
different for each cycle depending on the dates of minimum and maximum.
Comparing Figures 2 and 3 shows that: 1) the latitudinal scatter is smaller in
Figure 3 than in Figure 2 and; 2) the differences in the centroid positions
for the different cycle amplitudes are diminished in Figure 3. This suggests
that there is a more general, cycle amplitude independent, law for the
latitudes (and consequently latitudinal drift rates) of the sunspot zones. A
slight additional shift in the adopted times for sunspot cycle minima (with
earlier times for small cycles) would appear to further diminish any cycle
amplitude differences.
Figure 3.: The centroid positions of the sunspot zones in each hemisphere for
each solar rotation are plotted as functions of time from sunspot cycle
minimum with the same method as in Figure 2.
Determinations of the dates of sunspot cycle minima are not well defined. Many
investigators simply take the date of minimum in some smoothed sunspot cycle
index (e. g. sunspot number, sunspot area, 10.7 cm radio flux). Unfortunately,
this can give dates that are clearly not representative of the actual cycle
minima. This problem led earlier investigators to define the date of minimum
as some (undefined) average of the dates of: 1) minimum in the monthly sunspot
number; 2) minimum in the smoothed monthly sunspot number; 3) maximum in the
number of spotless days per month; 4) predominance of new cycle sunspot groups
over old cycle sunspot groups [Waldmeier (1961), McKinnon (1987), Harvey &
White (1999)]. Even neglecting the fact the the nature of the average is not
defined, it is clear from the published dates for previous cycle minima that
the first criterium is never used (probably due to the wide scatter it gives)
and that even the simple average of the remaining criteria doesn’t give the
published dates [Hathaway (2010)].
An alternative to using this definition for the dates of sunspot cycle minima
is to use curve fitting to either the initial rise of activity or to the
complete sunspot cycle. Curve fitting is less sensitive to the noise
associated minimum cycle behavior (e.g. discretized data and missing data from
the unseen hemisphere). Hathaway_etal94 described earlier attempts at fitting
solar cycle activity levels (monthly sunspot numbers) to parameterized
functions and arrived at a function of just two parameters (cycle starting
time $t_{0}$ and cycle amplitude $R_{max}$) as the most useful function for
characterizing and predicting solar cycle behavior. This function:
$F(t;t_{0},R_{max})=R_{max}\ 2({t-t_{0}\over b})^{3}/\left[exp({t-t_{0}\over
b})^{2}-0.71\right]$ (2)
is a skewed Gaussian with an initial rise that follows a cubic in time from
the starting time (measured in months). The width parameter, $b$, is a
function of cycle amplitude $R_{max}$ that mimics the “Waldmeier Effect.” This
function is
$b(R_{max})=27.12+25.15(100/R_{max})^{1/4}$ (3)
Fitting $F(t;t_{0},R_{max})$ to the monthly averages of the daily
International Sunspot Numbers using the Levenberg-Marquardt method [Press et
al. (1986)] gives the amplitudes and starting times given by Hathaway_etal94
and reproduced in Table 1 with the addition of results for Cycle 23.
On average the small cycles have starting times about 7 months earlier than
minimum while medium cycles and large cycles have starting times about equal
to minimum. However, since minimum is determined by the behavior of both the
old and the new cycles, there are substantial differences between the dates of
minima and the starting times even among the medium and large cycles. For
example, Cycles 21 and 22 were both large but the minimum was 3 months earlier
than the starting time in Cycle 21 and 4 months later in Cycle 22. This is
illustrated in Figure 4.
Figure 4.: Monthly sunspot numbers for the Cycle 20/21 minimum (top) and Cycle
21/22 minimum (bottom). The curves fit to each cycle are shown with the
colored lines with the sum of both contributions indicated by the dashed black
line. Dates of minima and starting times are indicated to illustrate the
differences.
Measuring the time through each cycle relative to these starting times (rather
than minimum or maximum) removes the scatter and cycle amplitude dependence on
the centroid positions as shown in Figure 5.
Figure 5.: The centroid positions of the sunspot zones in each hemisphere for each solar rotation are plotted as functions of time from sunspot cycle start as determined by fitting a parameterized function to each cycle. The individual data points are shown in the upper panel. The average centroid positions of the sunspot zones for small (blue) medium (green) and large (red) cycles plotted at 6-month intervals in time from sunspot cycle start are shown in the lower panel. The average centroid positions for all of the data are shown with $2\sigma$ error bars. All three curves fall within the $2\sigma$ errors, criss-crossing each other along a common, standard trajectory given by the exponential fit in Equation 4 (dashed line) Table 1.: Sunspot cycle number, amplitude, minimum date, starting date, difference (starting date - minimum date in months), and dominant hemisphere. Cycle | Amplitude | Min | $t_{0}$ | $\Delta$ | Hemisphere
---|---|---|---|---|---
12 | 75 (small) | 1878/12 | 1878/05 | -7 | South
13 | 88 (small) | 1890/01 | 1889/05 | -8 | South
14 | 64 (small) | 1901/12 | 1901/08 | -4 | Balanced
15 | 105 (medium) | 1913/06 | 1913/03 | -3 | North
16 | 78 (small) | 1923/09 | 1923/02 | -7 | North
17 | 119 (medium) | 1933/10 | 1933/11 | +1 | Balanced
18 | 151 (large) | 1944/02 | 1944/03 | +1 | South
19 | 201 (large) | 1954/04 | 1954/04 | 0 | North
20 | 111 (medium) | 1964/10 | 1964/11 | +1 | North
21 | 164 (large) | 1976/03 | 1976/06 | +3 | Balanced
22 | 158 (large) | 1986/07 | 1986/03 | -4 | Balanced
23 | 121 (medium) | 1996/08 | 1996/08 | 0 | South
The lack of any substantial cycle amplitude dependence on the centroid
positions when time is measured relative to the curve fitted cycle starting
time suggests that the equatorward drift of the sunspot zones follows a
standard path or law. This path is well represented by an exponential function
with
$\bar{\lambda}(t)=28^{\circ}\exp\left[-(t-t_{0})/90\right]$ (4)
where time, $t$, is measured in months. This exponential fit and the data for
the small, medium, and large cycles are plotted as functions of time from the
cycle starting time in the lower panel of Figure 5.
Hemispheric differences in solar activity were first noted by Sporer89 not
long after the discovery of the sunspot cycle itself. Much has been made of
these differences and their possible connection to a variety of sunspot cycle
phenomena. NortonGallagher10 recently revisited these connections and found
little evidence for any of them. Nonetheless we are compelled to examine
possible differences in the sunspot zone locations and equatorward drift
relative to the hemispheric asymmetries. We keep the same starting time for
each hemisphere of each cycle as determined from the curve fitting of the
sunspot numbers but separate the data by the strength of the activity in the
hemisphere. Using the data shown in NortonGallagher10 for the sunspot area
maximum and total sunspot area for each hemisphere in each cycle we identify
cycles in which the northern hemisphere dominates as Cycles 15, 16, 19, and
20, cycles in which the southern hemisphere dominates as Cycles 12, 13, 18,
and 23 with Cycles 14, 17, 21, and 22 having fairly balanced hemispheric
activity. (The relevant sunspot cycle characteristics are listed in Table 1.)
This gives 8 stronger hemispheres, 8 weaker hemispheres, and 8 balanced
hemispheres. The latitude positions of the sunspot zones for the stronger
hemispheres, weaker hemispheres,and balanced hemispheres are shown separately
in Figure 6. We find no significant differences in the sunspot zone latitude
positions associated with hemispheric asymmetry in spite of the fact that for
the unbalanced cycles the same starting time is used for both the strong and
the weak hemisphere.
Figure 6.: The average centroid positions of the sunspot zones for weaker
hemispheres (blue) balanced hemispheres (green) and stronger hemispheres (red)
plotted at 6-month intervals in time from sunspot cycle start. The average
centroid positions for all of the data are shown with $2\sigma$ error bars.
Here too, all three curves fall within the $2\sigma$ errors, criss-crossing
each other along a common, standard trajectory given by the exponential fit in
Equation 4 (dashed line).
## 4 Cycle 23
Cycle 23 had a long, low, extended minimum prior to the start of Cycle 24.
This delayed start of Cycle 24 left behind the lowest smoothed sunspot number
minimum and the largest number of spotless days in nearly a century. The polar
fields during this minimum were the weakest seen in the four cycle record and
the flux of galactic cosmic rays was the highest seen in the six cycle record.
One explanation for both the weak polar fields and the long cycle has been
suggested by flux transport dynamos [Nandy, Muñoz-Jaramillo, & Martens
(2011)]. This model can produce both these characteristics if the meridional
flow was faster than average during the first half of Cycle 23 and slower than
average during the second half. The meridional flow measured by the motions of
magnetic elements in the near surface layers (HathawayRightmire10 and
HathawayRightmire11) exhibited the opposite behavior - slow meridional flow at
the beginning of Cycle 23 and fast meridional flow at the end. Although the
speed of the near surface meridional flow was used to estimate the speed of
the proposed deep meridional return flow in their flux transport dynamo
models, Nandy_etal11 suggest that the variations seen near the surface are
unrelated to variations at the base of the convection zone. However, with
their model the latitudinal drift of the sunspot zones during Cycle 23 should
provide a direct measure of the deep meridional flow and its variations.
Figure 7.: The average centroid positions for Cycle 23 are shown with the
solid line and the exponential fit is shown with the dashed line. The average
centroid positions for all of the data are shown with $2\sigma$ error bars.
Cycle 23 data falls within the $2\sigma$ errors for the full dataset and
follows the standard trajectory. The trajectory (fast then slow) suggested by
Nandy et al. (2011) is shown by the red line. The trajectory (slow then fast)
derived from the observed meridional flow variations (Hathaway & Rightmire
2010, 2011) is shown by the green line.
Figure 7 shows the latitudinal positions of the sunspot zones for Cycle 23
along with those for the full 12 cycle dataset (with $2\sigma$ error bars).
The latitudinal drift of the sunspot zones during Cycle 23 follows within the
$2\sigma$ error range for the average of the last 12 cycles and follows the
standard exponential given by Equation 4. A drift rate that was 30% higher
than average at the start and 30% lower than average at the end of Cycle 23
(the red line in Figure 7) as proposed by Nandy_etal11 is inconsistent with
the data. A drift rate governed by the observed meridional flow variations in
the near surface layers (Hathaway & Rightmire 2010, 2011 - the green line in
Figure 7) is also inconsistent with the data for Cycle 23. This indicates that
the meridional flow is not connected to the latitudinal drift of the sunspot
zones.
## 5 Conclusions
We find that if time is measured relative to a cycle starting time determined
by fitting the monthly sunspot numbers to a parametric curve, then the
latitude positions of the sunspot zones follow a standard path with time. We
find no significant variations from this path associated with sunspot cycle
amplitude or hemispheric asymmetry.
This standard behavior suggests that the equatorward drift of the sunspot
zones is not produced by the Sun’s meridional flow - which is observed (and
theorized) to vary substantially from cycle-to-cycle. This regularity thus
questions the viability of flux transport dynamos as models of the Sun’s
activity cycle. The lack of the variations in drift rate during Cycle 23 in
spite of observed and theorized variations in the meridional flow also argues
against these models.
The earlier kinematic dynamo models of Babcock61 and Leighton69 may be
consistent with the regularity of the sunspot zone drift due to their
dependence on the fairly constant differential rotation profile. However, it
is unclear how the variability of the initial polar fields might influence the
latitudinal drift in these models.
It is clear, however, that this regularity is consistent with dynamo models in
which a Dynamo Wave produces the equatorward drift of the sunspot zones. The
speed of a Dynamo Wave depends on the product of the differential rotation
shear and the kinetic helicity - both of which are not observed or expected to
vary substantially.
### Acknowledgements
The author would like to thank NASA for its support of this research through a
grant from the Heliophysics Causes and Consequences of the Minimum of Solar
Cycle 23/24 Program to NASA Marshall Space Flight Center. He is also indebted
to Lisa Rightmire, Ron Moore, and an anonymous referee whose comments and
suggestions improved both the figures and the manuscript. Most importantly, he
would like to thank the American taxpayers for supporting scientific research
in general and this research in particular.
## References
* Babcock (1961) Babcock, H. W.: 1961 Astrophys. J. 133 572\.
* Becker (1954) Becker, U.: 1954 Z. Astrophys. 34 129\.
* Carrington (1858) Carrington, R. C.: 1858 Mon. Not. Roy. Astron. Soc. 19 1\.
* Harvey & White (1999) Harvey, K. L. & White, O. R.: 1999 J. Geophys. Res. 104 (A9) 19,759.
* Hathaway (2010) Hathaway, D. H.: 2010 Living Rev. Solar Phys. 7 1\.
* Hathaway et al. (2003) Hathaway, D. H., Nandy, D., Wilson, R. M., & Reichmann, R. J.: 2003 Astrophys. J. 589 665\.
* Hathaway & Rightmire (2010) Hathaway, D. H. & Rightmire, L.: 2010 Science 327 1350\.
* Hathaway & Rightmire (2011) Hathaway, D. H. & Rightmire, L.: 2011 Astrophys. J. 729 80\.
* Hathaway, Wilson, & Reichmann (1994) Hathaway, D. H., Wilson, R. M., & Reichmann, R. J.: 1994 Solar Phys. 151 177\.
* Leighton (1969) Leighton, R. B.: 1969 Astrophys. J. 156 1\.
* Maunder (1903) Maunder, E. W.: 1903 Observatory 26 329\.
* Maunder (1904) Maunder, E. W.: 1904 Mon. Not. Roy. Astron. Soc. 64 747\.
* McKinnon (1987) McKinnon, J. A.: 1987 Sunspot Numbers 1610-1986 (based on The Sunspot-Activity in the Years 1610-1960, by Prof. M. Waldmeier, Copyright 1961 Swiss Federal Observatory, Zurich, Switzerland) UAG Reports UAG-95, National Geophysical Data Center, NOAA, Boulder.
* Nandy & Choudhuri (2002) Nandy, D. & Choudhuri, A. R.: 2002 Science 296 1671\.
* Nandy, Muñoz-Jaramillo, & Martens (2011) Nandy, D. & Muñoz-Jaramillo, A., & Martens, P. C. H.: 2011 Nature 471 80\.
* Norton & Gallagher (2010) Norton, A. A., & , Gallagher, J. C.: 2010 Solar Phys. 261 193\.
* Press et al. (1986) Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T.: 1986 Numerical Recipes Cambridge University Press, Cambridge, 181pp.
* Spörer (1880) Spörer, G.: 1880 Publicationen des Astrophysikalischen zu Potsdam 2 No. 5 1.
* Spörer (1889) Spörer, G.: 1889 Bulletin Astronomique, Serie I 6 60\.
* Waldmeier (1935) Waldmeier, M.: 1935 Astron. Mitt. Zurich 14 (133) 105.
* Waldmeier (1955) Waldmeier, M.: 1955 Ergebnisse und Probleme der Sonnenforschung Geest & Portig, Leipzig, 2nd edn.
* Waldmeier (1961) Waldmeier, M.: 1961 The Sunspot-Activity in the Years 1610-1960 Schulthess Co., Swiss Federal Observatory, Zurich.
* Yoshimura (1975) Yoshimura, H.: 1975 Astrophys. J. 201 740\.
|
arxiv-papers
| 2011-08-08T14:39:44 |
2024-09-04T02:49:21.413646
|
{
"license": "Public Domain",
"authors": "D. H. Hathaway",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1108.1722"
}
|
1108.1764
|
# A bound on Universal Extra Dimension Models
from up to $2\,\text{fb}^{-1}$ of LHC Data at 7 TeV
Kenji Nishiwaki, Kin-ya Oda,
Naoya Okuda, and Ryoutaro Watanabe
∗Department of Physics, Kobe University, Kobe 657-8501, Japan
${}^{\dagger}{}^{\ddagger}{}^{\S}$Department of Physics, Osaka University,
Osaka 560-0043, Japan
E-mail: nishiwaki@stu.kobe-u.ac.jp E-mail: odakin@phys.sci.osaka-u.ac.jp
E-mail: okuda@het.phys.sci.osaka-u.ac.jp E-mail:
ryoutaro@het.phys.sci.osaka-u.ac.jp
###### Abstract
The recent up to $\sim 2\,\text{fb}^{-1}$ of data from the ATLAS and CMS
experiments at the CERN Large Hadron Collider at 7 TeV put an upper bound on
the production cross section of a Higgs-like particle. We translate the
results of the $H\to WW\to l\nu l\nu$ and $H\to\gamma\gamma$ as well as the
combined analysis by the ATLAS and CMS into an allowed region for the Kaluza-
Klein (KK) mass $M_{\text{KK}}$ and the Higgs mass $M_{H}$ for all the known
Universal Extra Dimension (UED) models in five and six dimensions. Our bound
is insensitive to the detailed KK mass splitting and mixing and hence
complementary to all other known signatures.
KOBE-TH-11-07
OU-HET-722/2011
## 1 Introduction
The ATLAS and CMS experiments at the CERN Large Hadron Collider (LHC) have
presented their latest results for the ${\color[rgb]{0,0,0}\lesssim
2}\,\text{fb}^{-1}$ of data at the center of mass energy 7 TeV at the XXV
International Symposium on Lepton Photon Interactions at High Energies (Lepton
Photon 11), Mumbai, India, 22–27 August 2011. One of the most remarkable among
them is the bound on the Higgs mass in the Standard Model (SM). A combined
analysis of the ATLAS experiment excludes the existence of the SM Higgs in
mass ranges $146\,\text{GeV}<M_{H}<232\,\text{GeV}$,
$256\,\text{GeV}<M_{H}<282\,\text{GeV}$, and
$296\,\text{GeV}<M_{H}<466\,\text{GeV}$ within the 95% Confidence Level (CL)
based on 1.0–$2.3\,\text{fb}^{-1}$ data [1] and that of the CMS experiment
excludes $145\,\text{GeV}<M_{H}<216\,\text{GeV}$,
$226\,\text{GeV}<M_{H}<288\,\text{GeV}$, and
$310\,\text{GeV}<M_{H}<400\,\text{GeV}$ within the 95% CL based on
1.1–$1.7\,\text{fb}^{-1}$ data [2]. Further the production cross section of a
Higgs-like particle, a particle that decays the same way as the SM Higgs, is
severely constrained by these data in the still-allowed regions, namely light
$115\,\text{GeV}<M_{H}<145\,\text{GeV}$, middle
$288\,\text{GeV}<M_{H}<296\,\text{GeV}$, and heavy $M_{H}>466\,\text{GeV}$
windows.
In this Letter, we translate the above constraint on the production cross
section into that on the Kaluza-Klein (KK) scale of various 5-Dimensional (5D)
and 6D Universal Extra Dimension (UED) models, namely the minimal UED (mUED)
[3] and the Dirichlet Higgs (DH) [4, 5] models in 5D and the 6D UED models on
$T^{2}/Z_{2}$ [3], $T^{2}/Z_{4}$ [6, 7], $T^{2}/(Z_{2}\times Z^{\prime}_{2})$
[8], ${RP}^{2}$ [9], $S^{2}/Z_{2}$ [10], Projective Sphere (PS) [11], and
$S^{2}$ [12].111 In [11] the terminology “real projective plane” is employed
for a sphere with its antipodal points being identified. In order to
distinguish [11] from [9], we call the former the Projective Sphere (PS). We
note that the PS and $S^{2}$ UED models have no orbifold fixed point and hence
no localized interaction on it. Concretely, we bound the UED parameter space
of $M_{\text{KK}}$ (first-level KK mass) and $M_{H}$ (zero mode Higgs mass)
based on the leading ATLAS and CMS constraints on the total cross section and
on that of each channel [1, 2].
One of the biggest advantages of the UED models is the existence of a natural
Dark Matter (DM) candidate, the Lightest KK Particle (LKP) [13]. The 6D UED
models have further advantages of the requirement of the number of generations
to be (zero modulo) three [14] and the assurance of the proton stability [15].
There exist several bounds on the 5D mUED model, within which the brane-
localized interactions are assumed to be vanishing at the 5D Ultra-Violet (UV)
cutoff scale ${\color[rgb]{0,0,0}\Lambda_{5D}}$. The latest analysis on DM
relic abundance including the effects from second KK resonances gives the
preferred KK scale at around $M_{\text{KK}}\sim 1.3\,\text{TeV}$ [16]. It is
noted that the first KK charged Higgs becomes the LKP when $M_{H}\gtrsim
240$–300 GeV, depending on the KK scale [17]. The electroweak precision data
suggests that the KK scale should be $M_{\text{KK}}\gtrsim 800\,\text{GeV}$
($300\,\text{GeV}\lesssim M_{\text{KK}}\lesssim 400\,\text{GeV}$) at the 95%
CL for $M_{H}=115$ (700) GeV [18, 19, 20]. The observed branching ratio of
$B_{d}\rightarrow X_{s}\gamma$ confines the KK scale as
$M_{\text{KK}}>600\,\text{GeV}$ [21] at the $95\%\,\text{CL}$. Recent study
puts a constraint ${M_{\text{KK}}>{\color[rgb]{0,0,0}600}\,\text{GeV}}$ for
$10<\Lambda_{{\color[rgb]{0,0,0}5D}}/M_{\text{KK}}<40$ at the
$95\%\,\text{CL}$ {} [22], from the ATLAS SUSY search result in
multijet$+E_{\text{T}}^{\text{miss}}$ with $1\,\text{fb}^{-1}$ data [23].222
Inclusion of the decay channel into KK Higgs, if allowed, might significantly
affect the result. We thank K. Tobioka on this point. We see that current LHC
bound from jets plus missing $E_{T}$ is not severe even for the most
constraint mUED. This is because we have typically smaller mass splitting
between the LKP and other new particles than the one between the lightest
supersymmetric particle and other sparticle in the minimal supersymmetric
standard model.
We note that all of these bounds are strongly dependent on the mass splitting
and mixing within the first KK level and therefore on the boundary mass
structure which is derived from the above-mentioned assumption that all of
them are zero at the 5D UV cutoff scale. The bound on the KK scale put in this
Letter is complementary to them in the sense that this is depending only on
the Higgs mass. That is, our bound is insensitive to the boundary masses if
they are smaller than the KK scale, as is necessary to have a higher
dimensional picture at all.
## 2 Procedure to obtain the bound
The ATLAS and CMS groups have shown the results for the combined analyses for
the ratio $\sigma^{95\%}_{pp\to H}/\sigma^{\text{SM}}_{pp\to H}$ as a function
of the Higgs mass $M_{H}$, where $\sigma^{95\%}_{pp\to H}$ is an upper bound
on the production cross section of a particle that decays the same as the SM
Higgs, at the 95% CL [1, 2]. In our case the constrained production cross
section is that of the UED Higgs. In UED models, a process can be affected by
KK-loops when it is loop-induced in the SM. In particular, the dominant Higgs
production channel via the gluon fusion process can be greatly enhanced, see
[12] and references therein. For middle and heavy Higgs mass regions, the
constraint is mainly from the $H\to WW$ and $ZZ$ channels, which are dominated
by the tree-level SM processes and therefore the result of the combined
analysis can be applied directly.
For the light Higgs mass region, the severest bound on $\sigma_{pp\to
H}^{\text{95\%}}/\sigma_{pp\to H}^{\text{SM}}$ comes from $H\to WW\to l\nu
l\nu$ or $H\to\gamma\gamma$. The latter is loop-induced in the SM and can be
affected by the KK-loops. Further, the loop-induced decay into gluons is not
negligible in this region in computation of the total decay width. Therefore,
we cannot trust the combined analysis which assumes that the branching ratios
are not changed from the SM. In the light mass range, we apply the CMS bounds
on $\sigma_{pp\to H\to\gamma\gamma}^{\text{95\%}}/\sigma_{pp\to
H\to\gamma\gamma}^{\text{SM}}$ and $\sigma_{pp\to H\to
WW}^{\text{95\%}}/\sigma_{pp\to H\to WW}^{\text{SM}}$ [2].
For the Higgs production, we focus on the the gluon fusion process via the
(KK) top quark loops, which is the dominant Higgs production channel in the SM
and the UED models [12, 24, 25, 26]. The parton level cross section of each
model $\hat{\sigma}^{\text{model}}_{gg\to H}$ is
$\displaystyle\hat{\sigma}^{\text{model}}_{gg\to H}(\hat{s})$
$\displaystyle={\pi^{2}\over 8M_{H}}\,\Gamma_{H\to
gg}^{{\text{model}}}(M_{H})\,{\delta(\hat{s}-M_{H}^{2})},$ (1)
where $\hat{s}$ is the parton level center-of-mass-energy-squared and
$\Gamma_{H\to gg}^{{\text{model}}}$ is the partial decay width into a pair of
gluons in each model:
$\displaystyle\Gamma^{\text{model}}_{H\to gg}(M_{H})$
$\displaystyle={\color[rgb]{0,0,0}K}{\alpha_{s}^{2}\over
8\pi^{3}}{M_{H}^{3}\over
v_{\text{EW}}^{2}}\,\left|J_{t}^{\text{model}}\\!\left({\color[rgb]{0,0,0}M_{H}^{2}}\right)\right|^{2}{\color[rgb]{0,0,0},}$
(2)
where $\alpha_{s}$ is the QCD coupling strength and $v_{\text{EW}}$ is the
Higgs vacuum expectation value $\simeq 246\,\text{GeV}$ and $K$ is the
$K$-factor to take into account the higher order QCD corrections, whose NNLO
value is $\simeq 1.9$ at the relevant energies, see e.g. Ref. [27]. When we
consider a ratio such as $\sigma_{pp\to H}^{\text{95\%}}/\sigma_{pp\to
H}^{\text{SM}}$ from the gluon fusion process, the overall K-factor does not
influence the result. However it contributes to the decay branching ratios of
the light Higgs boson non-negligibly.
For each model, the loop function $J_{t}^{\text{model}}$ describes the
contributions of all the zero and KK modes for the top quark in the triangle
loops:
$\displaystyle J_{t}^{\text{SM}}(\hat{s})$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right),$ (3) $\displaystyle
J_{t}^{\text{mUED}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left({m_{t(n)}^{2}\over\hat{s}}\right)\right\\},$ (4)
$\displaystyle J_{t}^{\text{DH}}(\hat{s})$
$\displaystyle=\sqrt{2}\varepsilon_{1}\left[\left|I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}+\left|2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}\tilde{I}\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}\right]^{1/2},$
(5) $\displaystyle{J_{t}^{T^{2}/Z_{2}}(\hat{s})=J_{t}^{{RP}^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{{m+n\geq
1\atop\text{or\ }m=-n\geq 1}}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$
(6) $\displaystyle{J_{t}^{T^{2}/Z_{4}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
1,n\geq 0}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t({m,n})}^{2}\over\hat{s}\right)\right\\},$
(7) $\displaystyle{J_{t}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
0,n\geq 0,\atop(m,n)\not=(0,0)}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$ (8)
$\displaystyle J_{t}^{S^{2}/Z_{2}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}n^{\color[rgb]{0,0,0}S^{2}/Z_{2}}(j)\,I\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(9) $\displaystyle J_{t}^{\text{PS}}(\hat{s})={J_{t}^{S^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}(2j+1)\,I\\!\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(10)
where $I$ and $\tilde{I}$ are given by
$\displaystyle I(\lambda)$
$\displaystyle={\color[rgb]{0,0,0}-2\lambda+\lambda(1-4\lambda)}\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right],$ (11)
$\displaystyle\tilde{I}(\lambda)$
$\displaystyle={\color[rgb]{0,0,0}(+\lambda)}\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right],$ (12)
the $n^{\text{model}}(j)$ counts the number of degeneracy:
$\displaystyle n^{S^{2}/Z_{2}}(j)$ $\displaystyle=\begin{cases}j+1,\\\
j,\end{cases}$ $\displaystyle n_{\text{even}}^{\text{PS}}(j)$
$\displaystyle=\begin{cases}2j+1,\\\ 0,\end{cases}$ $\displaystyle
n_{\text{odd}}^{\text{PS}}(j)$ $\displaystyle=\begin{cases}0,&\quad\text{for
}j=\text{{\color[rgb]{0,0,0}even}},\\\ 2j+1,&\quad\text{for
}j=\text{{\color[rgb]{0,0,0}odd}},\end{cases}$ (13)
and we write the KK top and $W$ masses ($X=t,W$)
$\displaystyle m_{{\color[rgb]{0,0,0}X}(n)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{n^{2}}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+n^{2}{M_{\text{KK}}^{2}}},$
(14) $\displaystyle m_{{\color[rgb]{0,0,0}X}(m,n)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{m^{2}+n^{2}}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\left(m^{2}+n^{2}\right){M_{\text{KK}}^{2}}}},$
(15) $\displaystyle m_{{\color[rgb]{0,0,0}X}(j)}$
$\displaystyle\equiv\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{j(j+1)}{R^{2}}}}=\sqrt{m_{{\color[rgb]{0,0,0}X}}^{2}+{\frac{j(j+1){M_{\text{KK}}^{2}}}{2}}},$
(16)
with ${M_{\text{KK}}}$ being the first KK mass: $M_{\text{KK}}=1/R$ for the
$S^{1}/Z_{2}$ (mUED), an interval (DH), and $T^{2}$-based compactifications
(namely $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$, $T^{2}/Z_{4}$ and
$RP^{2}$) and being ${M_{\text{KK}}}=\sqrt{2}/R$ for the $S^{2}$-based ones
(namely $S^{2}/Z_{2}$, PS and $S^{2}$). The range of the KK summation reflects
the structure of each extra dimensional background.333 The origin of the
factor 2 in front of each KK summation is the fact that there are both left
and right handed (namely, vector-like) KK modes for each chiral quark zero
mode corresponding to a SM quark.
The factor $\sqrt{2}\varepsilon_{1}$ in Eq. (5) is equal to $2\sqrt{2}/\pi\sim
0.90$. Readers who want more explanations on the above expressions should
consult Ref. [12].
As is mentioned above, we compute the decay rate into a photon pair, following
[26]. The result is
$\begin{split}&\Gamma^{\text{model}}_{H\to\gamma\gamma}(M_{H})={G_{F}\over
8\sqrt{2}\pi}M_{H}^{3}\cdot{\alpha^{2}\over\pi^{2}}\left|J_{W}^{\text{model}}\\!\left(M_{H}^{2}\right)+{4\over
3}J_{t}^{\text{model}}\\!\left(M_{H}^{2}\right)\right|^{2},\end{split}$ (17)
where
$\displaystyle J_{W}^{\text{SM}}(M_{H}^{2})$ $\displaystyle=L\\!\left({1\over
2},3,3,6,0;{M_{W}^{2}\over M_{H}^{2}},{M_{W}^{2}\over M_{H}^{2}}\right),$ (18)
$\displaystyle J_{W}^{\text{mUED}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{n=1}^{\infty}L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(n)}^{2}\over M_{H}^{2}}\right),$
(19) $\displaystyle J_{W}^{T^{2}/Z_{4}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{m\geq 1,n\geq
0}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (20) $\displaystyle
J_{W}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{m\geq 0,n\geq
0\atop(m,n)\neq(0,0)}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (21) $\displaystyle
J_{W}^{T^{2}/Z_{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{{m+n\geq 1\atop\text{or\
}m=-n\geq 1}}L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (22) $\displaystyle
J_{W}^{{RP}^{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{(m,n)}^{A}L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(m,n)}^{2}\over
M_{H}^{2}}\right)+\sum_{(m,n)}^{B}L\\!\left(0,1,0,2,0;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(m,n)}^{2}\over M_{H}^{2}}\right),$ (23) $\displaystyle
J_{W}^{S^{2}/Z_{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}n^{S^{2}/Z_{2}}(j)\,L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right),$ (24) $\displaystyle
J_{W}^{S^{2}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}(2j+1)\,L\\!\left({1\over 2},5,4,10,1;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right),$ (25) $\displaystyle
J_{W}^{\text{PS}}(M_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}\Bigg{[}n_{\text{even}}^{\text{PS}}(j)\,L\\!\left({1\over
2},4,4,8,1;{M_{W}^{2}\over M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right)$
$\displaystyle\phantom{=J_{W}^{\text{SM}}(M_{H}^{2})+\sum_{j\geq
1}\Bigg{[}}+n_{\text{odd}}^{\text{PS}}(j)\,L\\!\left(0,1,0,2,0;{M_{W}^{2}\over
M_{H}^{2}},{M_{W(j)}^{2}\over M_{H}^{2}}\right)\Bigg{]},$ (26)
with
$\displaystyle L(a,b,c,d,e;\lambda_{1},\lambda_{2})$
$\displaystyle=a+b\lambda_{1}-\left[\lambda_{1}\\!\left(c-d\lambda_{2}\right)-e\lambda_{2}\right]\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda_{2}}+1-i\epsilon\right].$ (27)
The $A$-summation for $RP^{2}$ are over the region that satisfies both $m\geq
1$ and $n\geq 1$ as well as over the ranges $(m,n)=(0,2),(0,4),(0,6),\dots$
and $(m,n)=(2,0),(4,0),(6,0),\dots$. Similarly, the $B$-summation are over
$m\geq 1$ and $n\geq 1$ as well as over $(m,n)=(0,1),(0,3),(0,5),\dots$ and
$(m,n)=(1,0),(3,0),(5,0),\dots$. The Dirichlet Higgs model only allows the
heavy mass region in which $H\to\gamma\gamma$ is irrelevant and hence we do
not compute the process for it.
In six dimensional UED models, KK summation in Eqs. (6)–(10) and (20)–(26)
must be terminated by a UV cutoff, for which we take the maximum and minimum
possible values consistent with the Naive Dimensional Analysis (NDA), shown in
Table 1. Let us briefly explain this treatment hereafter. For more details,
see Ref. [12]. Since the electroweak symmetry is broken by the Higgs mechanism
in the SM and UED models (except for the DH model), the gluon fusion process
is described by a dimension-six operator at lowest in 4D point of view after
KK expansion. This means that the calculation is UV logarithmic-divergent
(convergent) in six (five) dimensions.444 Of course 5D UED is non-
renormalizable and we have to introduce a cutoff scale in theory. However,
this does not appear in our analysis because we can calculate the gluon fusion
process with no UV divergence in 5D UEDs. Therefore we need to put an upper
limit of the summation over KK indices in 6D. We do it by adopting the NDA. In
both the $T^{2}$ and $S^{2}$-based geometries, the most stringent bound turns
out to be the one from the perturbativity of the $U(1)_{Y}$ gauge interaction,
which results in the following allowed regions of KK indices [12]:
$\displaystyle m^{2}+n^{2}$ $\displaystyle\lesssim{\color[rgb]{0,0,0}30},$
$\displaystyle\text{for $T^{2}$-case}\
\left(m_{(m,n)}^{2}=\frac{m^{2}+n^{2}}{R^{2}}\right),$ (28) $\displaystyle j$
$\displaystyle\lesssim{\color[rgb]{0,0,0}9},$ $\displaystyle\text{for
$S^{2}$-case}\ \left(m_{(j,m)}^{2}=\frac{j(j+1)}{R^{2}}\right),$ (29)
where the index $m$ for the $S^{2}$-case discriminates the degenerate states
of each $j$-th level. The cutoff scale of 6D UED theory $\Lambda_{6D}$ must be
lower than that in Eq. (28) or (29). In Table 1, we list the values that we
take.
Based on the knowledge sketched above, we can evaluate the total cross section
of the Higgs production of the UED models $\sigma_{pp\to H}^{\text{model}}$
and the ratio to that of the SM $\sigma_{pp\to
H}^{\text{model}}/\sigma^{\text{SM}}_{pp\to H}$ to be compared to the
experimental result.
## 3 Results
| $T^{2}$-based | $S^{2}$-based
---|---|---
| max | min | max | min
KK index | $m^{2}+n^{2}<28$ | $m^{2}+n^{2}\leq 10$ | $j(j+1)\leq 90$ | $j(j+1)\leq 30$
UV cutoff | $\Lambda_{6D}\sim 5M_{\text{KK}}$ | $\Lambda_{6D}\sim 3M_{\text{KK}}$ | $\Lambda_{6D}\sim 7M_{\text{KK}}$ | $\Lambda_{6D}\sim 4M_{\text{KK}}$
Table 1: Our choices of maximum and minimum upper bounds for KK indices and
for the corresponding UV cutoff scale.
First, we show our results for the light region:
$115\,{\text{GeV}}<M_{H}<145\,\text{GeV}$. We apply the CMS bounds on
$H\to\gamma\gamma$ and $H\to WW$ channels that are dominant in this range. In
Fig. 1, we list the contour plots for the excluded region in the
$M_{\text{KK}}$-$M_{H}$ plane for various UED models in 5 and 6 dimensions.
Plots for the maximum and minimum choices of the UV cutoff scale are presented
for the 6D UED models. In general, UED models enhance Higgs production via
gluon fusion and reduce the Higgs decay into a pair of photons. Therefore,
$\sigma^{\text{UED}}_{pp\to H\to\gamma\gamma}$ receives nontrivial
contributions from such effects. Typically, the enhancement of Higgs
production cross section overcomes the suppression of the di-photon branching
ratio in the $H\to\gamma\gamma$ excluded range (with orange and red colors),
whereas the region for smaller $M_{\text{KK}}$ is not excluded because of the
suppression of the di-phton branching ratio. (For example, in the case of
$S^{2}$ UED, the di-photon cross section is suppressed for
$M_{\text{KK}}\lesssim 400\,\text{GeV}$.) We find that all the suppressed
region is already excluded by $WW$ channel. It is natural that the lower the
cutoff scale becomes, the more the allowed parameter region is enlarged since
smaller numbers of KK tops contribute to the process. In 6D, we have more
light KK top quarks running in the loop, and get stronger constraints than in
5D. The $\operatorname{BR}(H\to WW)$ is also affected by the enhancement of
the total Higgs decay rate due to the increase of $H\to gg$.
Second, we move on to the middle region:
$288\,\text{GeV}<M_{H}<296\,\text{GeV}$. The Standard Model is still allowed
in this range whereas we find that all the UED models below $M_{KK}=1.4$ TeV
is excluded.
Finally, let us discuss the heavy region: $M_{H}>446\,\text{GeV}$. We choose
severer bound on $\sigma^{\text{95\%}}_{pp\to H}/\sigma^{\text{SM}}_{pp\to H}$
between ATLAS and CMS data for each $M_{H}$. That is, we use ATLAS and CMS
bounds for $M_{H}<500\,\text{GeV}$ and $M_{H}\geq 500\,\text{GeV}$,
respectively. In Fig. 2 we plot our results. We note that in all the allowed
region, we get $M_{H}<2M_{\text{KK}}$ and hence the Higgs does not decay into
a pair of KK particles.
Now let us comment on the DH model. In this model, the bound is put only on
$M_{\text{KK}}$ ($=M_{H}$) and the theoretical value of the ratio
$\sigma_{pp\to H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}$ decreases when one
increases $M_{\text{KK}}$, while the experimental upper bound
$\sigma_{95\%}/\sigma_{\text{SM}}$ is an increasing function of $M_{H}$ in the
high-mass region. The cross-over occurs at $M_{\text{KK}}=480\,\text{GeV}$
which gives $\sigma_{pp\to H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}\simeq
1.2$ and we conclude that the allowed parameter region of $M_{\text{KK}}$ is:
$\displaystyle M_{\text{KK}}>480\,\text{GeV}\quad(\text{95\% CL in Dirichlet
Higgs model}).$ (30)
In the $M_{\text{KK}}$ region of
$110\,{\text{GeV}}<M_{\text{KK}}<149\,\text{GeV}$ and
$206\,{\text{GeV}}<M_{\text{KK}}<300\,\text{GeV}$, the value of $\sigma_{pp\to
H}^{\text{DH}}/\sigma^{\text{SM}}_{pp\to H}$ grows significantly and thereby
these regions are rejected by the CMS result at the 95% CL. Noting that the
indirect electroweak constraint gives
$430\,\text{GeV}<M_{\text{KK}}<500\,\text{GeV}$ at the 90% CL [4], the allowed
region of $M_{\text{KK}}$ roughly lies between $480\,\text{GeV}\lesssim
M_{\text{KK}}\lesssim 500\,\text{GeV}$.
## 4 Summary and Discussions
In this Letter we have constrained the UED models in 5D and 6D by use of the
latest ATLAS and CMS bounds on the Higgs production cross section. The bound
on 6D UED is severer than that on 5D UED because 6D KK mass spectrum is denser
than that in 5D and therefore the KK top modes contribute to the gluon fusion
process larger. The KK (Higgs) mass of the Dirichlet Higgs model is pinned
down at around 500 GeV.
In the light mass range $115\,\text{GeV}<M_{H}<145\,\text{GeV}$, one of the
dominant constrains on the Higgs production cross section is coming from the
$H\to\gamma\gamma$ decay, which is reduced when KK scale is not large, due to
the interference between the SM gauge boson and KK top loops, with some
corrections from SM top and KK gauge bosons. We have taken into account the
CMS constraints from $H\to\gamma\gamma$ and $H\to WW\to l\nu l\nu$ instead of
the combined analysis. We find in this light region that the Higgs mass above
140 GeV is already excluded in the 5D and 6D $T^{2}/Z_{4}$ UED models, while
the mass above 130 GeV is ruled out in other 6D UED models, both within a
reasonably small KK scale $<1.4\,\text{TeV}$.
Our analyses on the 6D UED models cannot evade ambiguities from the NDA, but
the plots in Figs. 1 and 2 imply that the dependence on the cutoff is rather
mild $\lesssim 10\%$. For a low cutoff scale, there can also be contributions
from higher dimensional operators that must be taken into account. We have
ignored these possible contributions in our analysis.
In both 5D and 6D cases, the bound is insensitive to the detailed boundary
mass structure. In this sense this constraint is complementary to other ones
such as the relic abundance of the LKP and the $M_{T2}$ analysis of the decay
of the colored KK into the LKP.
When the KK scale is not much heavier than the weak scale $\simeq
246\,\text{GeV}$, the UED models tend to prefer much heavier Higgs mass than
in the SM in order to cancel the KK top loops in the $T$-parameter. (This
contribution has the same origin as the gluon fusion process discussed in this
Letter.) In this regard it would be important to put an experimental bound for
the Higgs mass beyond 600 GeV.
There is the triviality bound if the Higgs is heavy and the UED scale is light
which are being studied by the authors, along with the vacuum stability bound
for the case of light Higgs, and will be presented in a separate publication.
### Acknowledgment
We are grateful to Abdelhak Djouadi, Koichi Hamaguchi and Kohsaku Tobioka for
helpful communications. We also thank the Yukawa Institute for Theoretical
Physics for hosting the workshop “Field Theory and String Theory,”
YITP-W-11-05, during which this work was initiated. K.O. is partially
supported by Scientific Grant by Ministry of Education and Science (Japan),
Nos. 23104009, 23740192 and 20244028.
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Figure 1: Excluded regions in $M_{\text{KK}}$-$M_{H}$ plane by the CMS
constraints on $\sigma^{95\%}_{pp\to H\to WW}/\sigma^{\text{SM}}_{pp\to H\to
WW}$ and $\sigma^{95\%}_{pp\to H\to\gamma\gamma}/\sigma^{\text{SM}}_{pp\to
H\to\gamma\gamma}$ for the light Higgs. The bound from $WW$ channel (cyan and
blue respectively for minimum and maximum UV cutoffs, former of which is
superimposed on the latter) is superimposed on that from $\gamma\gamma$
channel (orange and red, the same as above) leaving its outline. The UV cutoff
scales of our choice are summarized in Table 1.
Figure 2: Constraints from the combined analysis from each ATLAS and CMS
experiment in the heavy mass region, drawn the same as Fig. 1 with maximum UV
cutoff (blue) being superimposed by the minimum UV cutoff (cyan).
|
arxiv-papers
| 2011-08-08T17:58:23 |
2024-09-04T02:49:21.419212
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenji Nishiwaki, Kin-ya Oda, Naoya Okuda and Ryoutaro Watanabe",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1108.1764"
}
|
1108.1765
|
# Heavy Higgs at Tevatron and LHC
in Universal Extra Dimension Models
Kenji Nishiwaki, Kin-ya Oda,
Naoya Okuda, and Ryoutaro Watanabe
∗Department of Physics, Kobe University, Kobe 657-8501, Japan
∗Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019,
India
${}^{\dagger}{}^{\ddagger}{}^{\S}$Department of Physics, Osaka University,
Osaka 560-0043, Japan
E-mail: nishiwaki@hri.res.in E-mail: odakin@phys.sci.osaka-u.ac.jp E-mail:
okuda@het.phys.sci.osaka-u.ac.jp E-mail: ryoutaro@het.phys.sci.osaka-u.ac.jp
###### Abstract
Universal Extra Dimension (UED) models tend to favor a distinctively heavier
Higgs mass than in the Standard Model (SM) and its supersymmetric extensions
when the Kaluza-Klein (KK) scale is not much higher than the electroweak one,
which we call the weak scale UED, in order to cancel the KK top contributions
to the $T$-parameter. Such a heavy Higgs, whose production through the gluon
fusion process is enhanced by the KK top loops, is fairly model independent
prediction of the weak scale UED models regardless of the brane-localized mass
structure at the ultraviolet cutoff scale. We study its cleanest possible
signature, the Higgs decay into a $Z$ boson pair and subsequently into four
electrons and/or muons, in which all the four-momenta of the final states can
be measured and both the $Z$ boson masses can be checked. We have studied the
Higgs mass 500 GeV (and also 700 GeV with $\sqrt{s}=14\,\text{TeV}$) and have
found that we can observe significant resonance with the integrated luminosity
$10\,\text{fb}^{-1}$ for six dimensional UED models.
KOBE-TH-11-05
OU-HET-721/2011
## 1 Introduction
The Universal Extra Dimension (UED) scenario [1], in which all the Standard
Model (SM) fields propagate in bulk of compactified extra dimension(s), is an
attractive possibility whose simplest five dimensional realization on orbifold
$S^{1}/Z_{2}$, the minimal UED model (mUED), may account for the existence of
the dark matter as the Lightest Kaluza-Klein Particle (LKP) [2] and can give a
loose gauge coupling unification at around 30 TeV [3]. See also Refs. [4, 5]
for review on mUED. For the mUED, the latest analysis including the effects
from second KK resonances gives the preferred Kaluza-Klein (KK) scale at
around $M_{\text{KK}}\sim 1.3\,\text{TeV}$ [6]. As is mentioned in [2, 6],
this result strongly depends on the brane-localized mass structure, which is
assumed to be vanishing at the UV cutoff scale [7] in mUED.
One of the most important signature to establish the model would be the direct
search of KK resonances at the CERN Large Hadron Collider (LHC). See Refs.
[8]–[14] for mUED and Refs. [15]–[21] for 6D UED models. We note that some of
them also pertain to the International Linear Collider (ILC), see also Refs.
[22]–[27]. The LHC already puts a lower bound on the KK scale for mUED as
$M_{\text{KK}}\gtrsim 500\,\text{GeV}$ at the 95% CL from $M_{T2}$ analysis of
cascade decay of first KK particles into the LKP [14, 28]. It is noted that
bounds on mUED [29] and $T^{2}/Z_{4}$ UED [30] from $b\to s\gamma$ processes
claim $M_{\text{KK}}\gtrsim 600\,\text{GeV}$ and 650 GeV, respectively. Again
all the above bounds strongly depend on the KK mass splitting and flavor
mixing patterns and hence on the boundary mass structure.111 A 95% CL bound on
the KK scale $M_{\text{KK}}>961\,\text{GeV}$ is put on a “UED” model, assuming
existence of large additional extra dimensions compactified with radius of
order $\text{eV}^{-1}$, in which SM fields cannot propagate, so that the LKP
decays into KK-gravitons [31]. In this paper, we do not assume such additional
large extra dimensions. In particular, we cannot see a decay product unless
there are enough mass splitting among the first KK modes so that it becomes
sufficiently energetic. In this paper, we present a complementary signal that
is insensitive to such detailed boundary structure.
In the SM, the electroweak data constrain the Higgs mass to be $M_{H}\lesssim
170\,\text{GeV}$ at the 95% CL, see e.g. [32, 33]. On the contrary, mUED
prefer heavier Higgs when the KK scale is not much higher than the electroweak
scale $v_{\text{EW}}\simeq 246\,\text{GeV}$, namely, the KK scale should be
$M_{\text{KK}}\gtrsim 800\,\text{GeV}$ ($300\,\text{GeV}\lesssim
M_{\text{KK}}\lesssim 400\,\text{GeV}$) at the 95% CL for $M_{H}=115$ (700)
GeV [34, 35, 36]. We note that this is fairly model independent feature of a
general UED model since KK top modes always contribute positively to the
$T$-parameter and such an effect requires a heavy Higgs in order to cancel
these KK top contributions by the ordinary negative $\log M_{H}$ dependence.
In this paper, we call such a natural UED model without big mass splitting
among electroweak, Higgs and KK scales: $M_{H}\sim
M_{\text{KK}}\sim\mathcal{O}(10^{2})$ GeV, the weak scale UED model.
Concretely, we will pick up the cases: $M_{H}=330,500$ and $700$ GeV. To
summarize, existence of a heavy Higgs is a model independent prediction of the
weak scale UED models in contrast to the relic abundance of the dark matter
and the cascade decay signature of the KK particles that are dependent on the
detailed boundary mass structure at the Ultra-Violet (UV) cutoff scale of the
higher-dimensional gauge theory.
In five and six dimensional UED models, the Higgs production cross section via
the gluon fusion process is enhanced by the KK top loops [37, 38, 39]. In this
paper, we analyze its cleanest possible signature, the Higgs decay into a $Z$
boson pair and subsequently into four electrons and/or muons, in which all the
four momenta of the final states can be measured and both the $Z$ boson masses
can be checked.
In Section 2, we review the relevant part of all the known 5D and 6D UED
models to the Higgs production process via the gluon fusion through the KK top
loops. Considered models are the 5D mUED model on $S^{1}/Z_{2}$ [1], the
Dirichlet Higgs (DH) model on an interval [40, 41], the 6D $T^{2}$-based
models on $T^{2}/Z_{2}$ [1], $T^{2}/(Z_{2}\times Z_{2}^{\prime})$ [42],
$T^{2}/Z_{4}$ [43, 44], $RP^{2}$ [45], and the 6D $S^{2}$-based models on
Projective Sphere (PS) [46], $S^{2}$ (see Section 2.3.2), $S^{2}/Z_{2}$
[47].222 In [46] the terminology “real projective plane” is employed for the
compactified space, the sphere with its antipodal points being identified. In
order to distinguish [46] from [45], we call the former the Projective Sphere.
In Section 3, we present the concrete computation of the process. The cross
sections for the DH, $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$,
$RP^{2}$, and $S^{2}$ UED models are newly obtained. We also review the
estimation of the UV cutoff scale in the $T^{2}$-based geometry [48] and
extend it to the $S^{2}$-based one. In Section 4, we show our numerical
results. The last section is for summary and discussions. In Appendix A, we
present the relevant Feynman rules for our computation in the DH model. In
Appendix B we explain our estimation of the UV cutoff scale for 6D UED models,
based on the Renormalization Group Equation (RGE) analysis in the
renormalizable KK picture. In Appendix C, we review the way to take into
account the width in the amplitude and justify our approximation.
## 2 Review on known 5D and 6D UED models
In this section, we give a brief review on various UED models. Readers who are
not interested in the details of these models may skip this section. In the
first part of this section, we briefly review the 5D minimal UED model on
$S^{1}/Z_{2}$ [1] and Dirichlet Higgs model on an interval [40, 41]. The
remaining of the section is devoted to an overview of various types of 6D UED
models.
### 2.1 5D UED models
#### 2.1.1 Minimal UED model on $S^{1}/Z_{2}$
First we review the 5D UED model [1]. The matter contents of the model are the
same as those of the SM, but they are living in the bulk of flat five-
dimensional space, compactified on the orbifold $S^{1}/Z_{2}$. The action $S$
is written down as
$\displaystyle S$ $\displaystyle=\int d^{4}x\int_{-\pi R}^{\pi
R}dy\left[\mathcal{L}_{\text{bulk}}+\delta(y)\mathcal{L}_{0}+\delta(y-\pi
R)\mathcal{L}_{\pi R}\right]{.}$ (1)
Usually when one say mUED model, it is implied that all the boundary masses
are zero at the UV cutoff scale and are generated through radiative
corrections [7]. Hereafter, when we call mUED model, we do not assume any
boundary mass structure and concentrate on the signal that is independent of
it. In particular, we do not include the constraints from the direct KK search
[14, 28] and from the relic abundance of the LKP [6] that are dependent on the
KK mass splitting pattern.
The $Z_{2}$ twist conditions on the bulk SM fields are put as
$\displaystyle\mathcal{B}_{\mu}(x,-y)$ $\displaystyle=\mathcal{B}_{\mu}(x,y),$
$\displaystyle\mathcal{B}_{5}(x,-y)$ $\displaystyle=-\mathcal{B}_{5}(x,y),$
$\displaystyle\mathcal{W}_{\mu}(x,-y)$ $\displaystyle=\mathcal{W}_{\mu}(x,y),$
$\displaystyle\mathcal{W}_{5}(x,-y)$ $\displaystyle=-\mathcal{W}_{5}(x,y),$
$\displaystyle\mathcal{G}_{\mu}(x,-y)$ $\displaystyle=\mathcal{G}_{\mu}(x,y),$
$\displaystyle\mathcal{G}_{5}(x,-y)$ $\displaystyle=-\mathcal{G}_{5}(x,y),$
(2) $\displaystyle L(x,-y)$ $\displaystyle=\gamma^{5}L(x,y),$ $\displaystyle
E(x,-y)$ $\displaystyle=-\gamma^{5}E(x,y),$ $\displaystyle Q(x,-y)$
$\displaystyle=\gamma^{5}Q(x,y),$ $\displaystyle U(x,-y)$
$\displaystyle=-\gamma^{5}U(x,y),$ $\displaystyle D(x,-y)$
$\displaystyle=-\gamma^{5}D(x,y),$ (3)
and
$\displaystyle\Phi(x,-y)$ $\displaystyle=\Phi(x,y),$ (4)
where $x$ and $y$ ($=x^{5}$) denote four and extra dimensional coordinates,
respectively.333 We follow the metric and spinor conventions of [49]. We can
see that the wanted zero modes remain after the twist (2)–(4). There are fixed
points of the $Z_{2}$ orbifolding at $y=0,\pi R$. If the boundary Lagrangians
at $y=0,\pi R$ are equal at the UV cutoff scale, there remains an additional
accidental symmetry under the reflection ${\pi R\over 2}-y\to{\pi R\over
2}+y$, called the KK parity, which ensures the stability of the LKP and makes
it a dark matter candidate.
The gauge and Yukawa interactions for the KK-top quarks, which we need for
later calculation, are:
$\displaystyle\mathcal{L}_{\text{KK top}}$
$\displaystyle=-ig_{4s}\sum_{n=1}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(n)}\gamma^{\mu}\mathcal{G}_{\mu}^{(0)}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(n)}$ $\displaystyle\quad-{m_{t}\over
v_{\text{EW}}}H^{(0)}\sum_{n=1}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(n)}\begin{bmatrix}\sin{2\alpha^{(n)}}&-\gamma^{5}\cos{2\alpha^{(n)}}\\\
\gamma^{5}\cos{2\alpha^{(n)}}&\sin{2\alpha^{(n)}}\end{bmatrix}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(n)},$ (5)
where ${g_{4s}=g_{s}/\sqrt{2\pi R}}$ is a dimensionless 4D $SU(3)_{C}$
coupling constant and $v_{\text{EW}}\simeq 246\,\text{GeV}$ is the 4D Higgs
vacuum expectation value which appear after the KK expansion;
$\mathcal{G}^{(0)}(H^{(0)})$ shows zero-mode gluon (zero-mode physical Higgs);
$t_{1}^{(n)}$ and $t_{2}^{(n)}$ are mass eigenstates of $n$-th KK top quarks
and each mixing angle $\alpha^{(n)}$ is determined to be
$\cos{2\alpha^{(n)}}=m_{(n)}/\sqrt{m_{t}^{2}+m^{2}_{(n)}}$,
$\sin{2\alpha^{(n)}}=m_{t}/\sqrt{m_{t}^{2}+m^{2}_{(n)}}$, with $m_{(n)}:=n/R$.
Each KK state is twofold degenerate and n-th KK top mass is
$\displaystyle m_{t,(n)}=\sqrt{m_{t}^{2}+m^{2}_{(n)}}.$ (6)
KK tops give dominant contribution to the gluon fusion process due to their
large Yukawa coupling to the Higgs. We note that $\gamma^{5}$ is put in Eq.
(5) merely to arrange the sign of both the KK masses positive.
#### 2.1.2 Dirichlet Higgs (DH) model
Dirichlet Higgs model is defined on an interval: $0\leq y\leq\pi R$. The
action $S$ is as follows:
$\displaystyle S$ $\displaystyle=\int d^{4}x\int_{0}^{\pi
R}dy\left[\mathcal{L}_{\text{bulk}}+\delta(y)\mathcal{L}_{0}+\delta(y-\pi
R)\mathcal{L}_{\pi R}\right],$ (7)
where $R$ is a radius of the extra spacial direction. The structure of the
bulk Lagrangian, covariant derivatives and field strength of gauge bosons are
the same as that of the mUED model. There is no difference between the matter
contents of this model and those of the mUED model. As in the mUED model, we
neglect the possible boundary interactions in this paper. The zero-mode sector
of the UED on an interval becomes the same as that of the mUED on the orbifold
$S^{1}/Z_{2}$ when we choose the boundary conditions for the SM degrees of
freedom
$\Psi^{N}=\mathcal{G}_{\mu},\mathcal{W}_{\mu},\mathcal{B}_{\mu};L_{L},Q_{L};E_{R},U_{R},D_{R}$
to be Neumann (at $y=0$ and $\pi R$):
$\displaystyle\partial_{5}\Psi^{N}(x,0)=\partial_{5}\Psi^{N}(x,\pi R)=0$ (8)
and for other non-SM modes
$\Psi^{D}=\mathcal{B}_{5},\mathcal{W}_{5};L_{R},Q_{R};E_{L},U_{L},D_{L}$ to be
Dirichlet:
$\displaystyle\Psi^{D}(x,0)=\Psi^{D}(x,\pi R)=0.$ (9)
We note that mode functions with Dirichlet and Neumann boundary conditions are
not orthogonal to each other, unlike the orbifolding on $S^{1}/Z_{2}$.444 In
other words the KK mass-squared operator $\partial_{5}^{2}$ is not hermitian
in this setup, though the kinetic term is still positive definite. Kinetic
terms turn out to be diagonal even though the expansion is not orthonormal. We
can explicitly check that the non-orthogonality does not lead to extra mixing
for spinors even after the EWSB because non-orthogonal terms drop out due to
the 4D-chirality.
If we had put the Neumann condition on the Higgs $\Phi$, we would get exactly
the same zero-mode sector as in the mUED model on $S^{1}/Z_{2}$. In the
Drichlet Higgs model on interval, the EWSB is caused by a non-zero Dirichlet
boundary condition on the $SU(2)_{W}$-doublet Higgs field [40, 41]. We assume
that the KK-parity is respected by the boundary conditions on the Higgs field
too. The advantage of the Dirichlet EWSB is that we do not need to assume the
negative mass-squared in the bulk Lagrangian nor the quartic coupling which is
a higher dimensional operator in 5D. Throughout this paper, we consider the
minimal case: $\mathcal{V}(\Phi)=0$. We list the necessary Feynman rules in
Appendix A.
### 2.2 6D UED models based on $T^{2}$
We consider a gauge theory on six-dimensional spacetime $M^{4}\times T^{2}$,
which is a direct product of the four-dimensional Minkowski spacetime $M^{4}$
and two-torus $T^{2}$: $0\leq y\leq 2\pi R_{y},0\leq z\leq 2\pi R_{z}$. We
assume that the two radii of $T^{2}$ have the same value $R=R_{y}=R_{z}$ for
simplicity.555 In the $T^{2}/Z_{4}$ orbifold case, the condition $R_{y}=R_{z}$
is imposed by the consistency with the $Z_{4}$ discrete symmetry. See also
[50] for a realization of CP violation from the complex structure of
$T^{2}/Z_{4}$, which appears in 4D effective interactions after KK
decomposition.
When we use 6D Weyl spinor for 6D UED model construction, there is a
constraint on the choice of 6D chiralities. The origin of this constraint is
the cancellation of 6D gravitational and SU(2)L global anomalies that cannot
be removed by use of the Green-Schwarz mechanism. This constraint requires the
number of matter generation to be (multiple of) three [51]. A suitable choice
of the 6D chirality for a single matter generation is as follows:
$\displaystyle({Q}_{+},{U}_{-},{D}_{-};{L}_{+},{E}_{-},{N}_{-}),$ (10)
where the $\pm$ suffixes represent 6D chirality of each field. Number of
d.o.f. of 6D Weyl fermion is 4, the same as that of a 4D Dirac fermion.
Therefore we can construct 6D UED models on $T^{2}$ following the orbifolding
method of the 5D UED model. We have several options for the orbifolding to
realize the SM chiral fermions in the zero mode sector of (10). Let us review
them in turn. The range for KK summation is listed in Table 2.
#### 2.2.1 Orbifold ${T^{2}/Z_{2}}$, $T^{2}/(Z_{2}\times Z^{\prime}_{2})$,
and $T^{2}/Z_{4}$
type of orbifolding | identification | fixed points $(y_{i},z_{i})$
---|---|---
$T^{2}/Z_{2}$ | $(y,z)\sim(-y,-z)$ | $(0,0),\ (\pi R,0),\ (0,\pi R),\ (\pi R,\pi R)$
$T^{2}/(Z_{2}\times Z^{\prime}_{2})$ | $(y,z)\sim(-y,z)$ and $(y,z)\sim(y,-z)$ | $(0,0),\ (\pi R,0),\ (0,\pi R),\ (\pi R,\pi R)$
$T^{2}/Z_{4}$ | $(y,z)\sim(-z,y)$ | $(0,0),\ (\pi R,\pi R)$
Table 1: Fixed points which stem from each identification.
We consider ${T^{2}/Z_{2}}$ [1], $T^{2}/(Z_{2}\times Z^{\prime}_{2})$ [42],
and $T^{2}/Z_{4}$ [43, 44] orbifolds. Let us write down the action
$\displaystyle S$ $\displaystyle=\int d^{4}x\int_{-\pi R}^{\pi R}dy\int_{-\pi
R}^{\pi R}dz\left[\mathcal{L}_{\text{bulk}}(x,y,z)+\sum_{\vec{y}\ \in\
\vec{y}_{i}}\delta(\vec{y}_{i})\mathcal{L}_{\vec{y}_{i}}(x)\right],$ (11)
where $\vec{y}_{i}=(y_{i},z_{i})$ are orbifold fixed points. We note that
terms localized at the fixed points are induced at quantum level even if we
assume that they are vanishing at tree level [7, 52, 53]. For the orbifold we
consider in this paper, the projections are:
$\displaystyle(y,z)$ $\displaystyle\sim(-y,-z)$ $\displaystyle\text{for
$T^{2}/Z_{2}$},$ (12) $\displaystyle(y,z)\sim(-y,z)\ $
$\displaystyle\text{and}\ (y,z)\sim(y,-z)$ $\displaystyle\text{for
$T^{2}/(Z_{2}\times Z^{\prime}_{2})$},$ (13) $\displaystyle(y,z)$
$\displaystyle\sim(-z,y)$ $\displaystyle\text{for $T^{2}/Z_{4}$}.$ (14)
See also Table. 1. In each case, we can choose a suitable boundary condition
of 6D Weyl fermions, whose exact forms are not discussed in this paper, to
generate 4D Weyl fermions at the zero modes.
The bulk Lagrangian, covariant derivatives and field strengths of gauge bosons
are essentially the same as that of the 5D mUED model except for the structure
of spinors. For 6D Weyl fermions, the kinetic and Yukawa terms are
$\displaystyle\mathcal{L}_{\text{kinetic}}$
$\displaystyle=-\overline{Q_{+}}\Gamma^{M}\mathcal{D}_{M}Q_{+}-\overline{U_{-}}\Gamma^{M}\mathcal{D}_{M}U_{-}-\overline{D_{-}}\Gamma^{M}\mathcal{D}_{M}D_{-}$
$\displaystyle\quad-\overline{L_{+}}\Gamma^{M}\mathcal{D}_{M}L_{+}-\overline{E_{-}}\Gamma^{M}\mathcal{D}_{M}E_{-}-\overline{N_{-}}\Gamma^{M}\mathcal{D}_{M}N_{-},$
(15) $\displaystyle\mathcal{L}_{\text{Yukawa}}$
$\displaystyle=-\lambda_{U}\overline{U_{-}}\left(Q_{+}\cdot\Phi\right)-\lambda_{D}\left(\overline{Q_{+}}\Phi\right)D_{-}-\lambda_{E}\left(\overline{L_{+}}\Phi\right)E_{-}+\text{h.c.},$
(16)
where contraction of $SU(2)$ indices are understood.666 We leave the neutrino
sector untouched since it is irrelevant for the Higgs signal considered in
this paper. Resultant interactions relevant for our discussion are
$\displaystyle\mathcal{L}_{\text{KK top}}$
$\displaystyle=-ig_{4s}\sum_{(m,n)}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(m,n)}\gamma^{\mu}\mathcal{G}_{\mu}^{(0)}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(m,n)}$
$\displaystyle\quad-\frac{m_{t}}{v_{\text{EW}}}H^{(0)}\sum_{(m,n)}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(m,n)}\begin{bmatrix}\sin{2\alpha^{(m,n)}}&-\gamma^{5}\cos{2\alpha^{(m,n)}}\\\
\gamma^{5}\cos{2\alpha^{(m,n)}}&\sin{2\alpha^{(m,n)}}\end{bmatrix}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(m,n)},$ (17)
where ${g_{4s}=g_{s}/(2\pi R)}$ is the dimensionless 4D $SU(3)_{C}$ coupling
constant and $v_{\text{EW}}\simeq 246\,\text{GeV}$ is the 4D Higgs vev,
$\mathcal{G}^{(0)}(H^{(0)})$ shows zero-mode gluon (zero-mode physical Higgs),
and $t_{1}^{(m,n)},t_{2}^{(m,n)}$ are mass eigenstates of $(m,n)$-th KK top
quarks. Again we only consider the KK top quark loops since contributions from
other flavors are suppressed by the small Yukawa coupling. Each mixing angle
$\alpha^{(m,n)}$ is determined to be
$\cos{2\alpha^{(m,n)}}=m_{(m,n)}/\sqrt{m_{t}^{2}+m^{2}_{(m,n)}}$,
$\sin{2\alpha^{(m,n)}}=m_{t}/\sqrt{m_{t}^{2}+m^{2}_{(m,n)}}$. Each KK state is
twofold degenerate and $(m,n)$-th KK top mass is
$\displaystyle m_{t,(m,n)}=\sqrt{m_{t}^{2}+m^{2}_{(m,n)}},$ (18)
with
$\displaystyle m_{(m,n)}:=\frac{\sqrt{m^{2}+n^{2}}}{R}.$ (19)
It should be mentioned that the difference from the mUED case appears only in
the form of KK mass and the number of d.o.f. in each KK level when we consider
the gluon fusion process. We adopt m(n) as the y(z)-directional KK index,
whose parameter region is determined by the way of the orbifolding. This
information has a great influence on the enhancement of the Higgs production
through the gluon fusion.
type of orbifolding | range of $(m,n)$
---|---
$T^{2}/Z_{2}$ | $m+n\geq 1,$ or $m=-n\geq 1$
$T^{2}/(Z_{2}\times Z^{\prime}_{2})$ | $0\leq m<\infty,\ 0\leq n<\infty;\ (m,n)\not=(0,0)$
$T^{2}/Z_{4}$ | $1\leq m<\infty,\ 0\leq n<\infty$
Table 2: The range of the parameter $(m,n)$ except the zero mode case
$(m,n)=(0,0)$ in each case of the orbifolding.
#### 2.2.2 Real Projective Plane $(RP^{2})$
We can construct a UED model on a non-orientable geometry: Real Projective
Plane ($RP^{2}$) [45]. $RP^{2}$ is defined by two types of identifications: a
$\pi$-rotation $r$ and a glide $g$:
$\displaystyle r:(y,z)\sim(-y,-z),\quad g:(y,z)\sim(y+\pi R,-z+\pi R).$ (20)
The system is invariant under each manipulation in Eq. (20). Note that the
shifts $y\sim y+2\pi R$ and $z\sim z+2\pi R$ can be obtained as different
combinations of $r$ and $g$, respectively. Note also that no fixed point
exists globally in this background geometry.
Under $r$ and $g$, Weyl fermions transforms as
$\displaystyle r:\Psi_{{\pm}}(x;-y,-z)$
$\displaystyle=p_{r}\Gamma_{r}\Psi_{{\pm}}(x;y,z),\quad\Gamma_{r}={i}\Gamma^{5}\Gamma^{6}\Gamma^{7},$
(21) $\displaystyle g:\tilde{\Psi}_{{\pm}}(x;y+\pi R,-z+\pi R)$
$\displaystyle=p_{g}\Gamma_{g}\Psi_{{\mp}}(x;y,z),\quad\Gamma_{g}={\Gamma^{6}\Gamma^{7}},$
(22)
where $\Gamma^{7}$ is the 6D chirality operator and $p_{r},p_{g}$
($Z_{2}$-parities) can take the value $\pm 1$. The $\tilde{\Psi}_{{\pm}}$ is
what we call the “mirror” fermion. Eq. (21) has the same form with that of the
$T^{2}/Z_{2}$ orbifold condition for 6D fermion. An essential point of this
model is that the condition (22) does not generate a 4D Weyl fermion in the
zero mode sector. In other words, the 6D chirality of both sides of Eq. (22)
are different from each other. This means that we have to introduce new
fermions $\tilde{\Psi}_{{\pm}}$ which have opposite 6D chirality and the same
SM quantum number compared to each corresponding field $\Psi_{{\mp}}$.
Concretely, “mirror” fermions:
$\displaystyle{\mathcal{Q}}_{-},{\mathcal{U}}_{+},{\mathcal{D}}_{+};{\mathcal{L}}_{-},{\mathcal{E}}_{+},\mathcal{N}_{+},$
(23)
are identified with $\\{{Q}_{+},{U}_{-},{D}_{-};{L}_{+},{E}_{-},{N}_{-}\\}$,
respectively. The choice of 6D chiralities in Eq. (23) obeys the condition for
realizing the 6D anomaly cancellation which we have argued before.
The bulk Lagrangian is the same as that of the $T^{2}$-based models using
orbifold except for the existence of the mirror fermions:
$\displaystyle\mathcal{L}_{\text{kinetic}}$
$\displaystyle=\frac{1}{2}\Big{[}-\overline{Q_{+}}\Gamma^{M}\mathcal{D}_{M}Q_{+}-\overline{U_{-}}\Gamma^{M}\mathcal{D}_{M}U_{-}-\overline{D_{-}}\Gamma^{M}\mathcal{D}_{M}D_{-}$
$\displaystyle\phantom{=\frac{1}{2}\Big{[}\
}-\overline{L_{+}}\Gamma^{M}\mathcal{D}_{M}L_{+}-\overline{E_{-}}\Gamma^{M}\mathcal{D}_{M}E_{-}-\overline{N_{-}}\Gamma^{M}\mathcal{D}_{M}N_{-}$
$\displaystyle\phantom{=\frac{1}{2}\Big{[}\
}-\overline{\mathcal{Q}_{-}}\Gamma^{M}\mathcal{D}_{M}\mathcal{Q}_{-}-\overline{\mathcal{U}_{+}}\Gamma^{M}\mathcal{D}_{M}\mathcal{U}_{+}-\overline{\mathcal{D}_{+}}\Gamma^{M}\mathcal{D}_{M}\mathcal{D}_{+}$
$\displaystyle\phantom{=\frac{1}{2}\Big{[}\
}-\overline{\mathcal{L}_{-}}\Gamma^{M}\mathcal{D}_{M}\mathcal{L}_{-}-\overline{\mathcal{E}_{+}}\Gamma^{M}\mathcal{D}_{M}\mathcal{E}_{+}-\overline{\mathcal{N}_{+}}\Gamma^{M}\mathcal{D}_{M}\mathcal{N}_{+}\Big{]},$
(24) $\displaystyle\mathcal{L}_{\text{Yukawa}}$ $\displaystyle={1\over
2}\Big{[}-\lambda_{U}\overline{U_{-}}\left(Q_{+}\cdot\Phi\right)-\lambda_{D}\left(\overline{Q_{+}}\Phi\right)D_{-}-\lambda_{E}\left(\overline{L_{+}}\Phi\right)E_{-}$
$\displaystyle\phantom{={1\over 2}\big{[}\
}-\lambda_{U}\overline{\mathcal{U}_{+}}\left(\mathcal{Q}_{-}\cdot\Phi\right)-\lambda_{D}\left(\overline{\mathcal{Q}_{-}}\Phi\right)\mathcal{D}_{+}-\lambda_{E}\left(\overline{\mathcal{L}_{-}}\Phi\right)\mathcal{E}_{+}+\text{h.c.}\Big{]},$
(25)
where we introduce the $``1/2"$ factors for later convenience. The neutrino
sector is again left untouched as it is irrelevant for our discussion. By use
of Eq. (22), we can erase all the mirror fermions and obtain the ordinary form
of Lagrangian same as $T^{2}$ cases. The form of $\pi$-rotation $r$ given in
Eq. (21) is the same as that of $Z_{2}$ orbifolding in Eq. (12). Therefore,
the interactions of $RP^{2}$ model, needed to calculate the gluon fusion
process, take the same form as that of $T^{2}/Z_{2}$ one given in Eq. (17).
### 2.3 6D UED models based on $S^{2}$
Let us review UED models based on the $S^{2}$ compactification. We span the
extra dimension by the zenith and azimuthal angles $\theta$ and $\phi$,
respectively. The two-sphere $S^{2}$ has a positive curvature and to stabilize
the radius $R$, we introduce an extra $U(1)_{X}$ gauge field which has a
monopole-like classical configuration [54]
$\displaystyle[\mathcal{X}^{c}_{\phi}(x^{\mu},\theta,\phi)]^{{N}\atop{S}}$
$\displaystyle={n\over 2g_{X}}(\cos{\theta}\mp 1),$ $\displaystyle(\text{other
components})$ $\displaystyle=0,$ (26)
where the superscript $c$ denotes the classical configuration, $g_{X}$ is the
6D $U(1)_{X}$ gauge coupling, the integer $n$ is the (negative) monopole
charge, and the superscripts $N$ and $S$ indicate that the field is given in
north (involving the $\theta=0$ point) and south (involving the $\theta=\pi$
point) charts, respectively. The $U(1)_{X}$ transition function from the north
to the south chart is given by
$[\mathcal{X}_{M}(x^{\mu},\theta,\phi)]^{S}=[\mathcal{X}_{M}(x^{\mu},\theta,\phi)]^{N}+\frac{1}{g_{X}}\partial_{M}\alpha(x^{\mu},\theta,\phi)$
(27)
with $\alpha(x^{\mu},\theta,\phi)=n\phi$. Because of the monopole-like
configuration, the radius of $S^{2}$ is stabilized spontaneously at
$R^{2}=\left({n\over 2g_{X}M_{\ast}^{2}}\right)^{2},$ (28)
where $M_{\ast}$ is the 6D Planck scale.
We mention that any 6D field ${\Xi}$ on $S^{2}$ is KK expanded by use of the
spin-weighted spherical harmonics ${}_{s}Y_{jm}(\theta,\phi)$ as follows:
${\Xi}(x,\theta,\phi)^{N\atop
S}=\sum_{j=|s|}^{\infty}\sum_{m=-j}^{j}{\xi}^{(j,m)}(x)f_{{\Xi}}^{(j,m)}(\theta,\phi)^{N\atop
S},\quad f_{{\Xi}}^{(j,m)}(\theta,\phi)^{N\atop
S}:={{}_{s}Y_{jm}(\theta,\phi)e^{\pm is\phi}\over R},$ (29)
where $\xi^{(j,m)}$ is the $(j,m)$-th expanded 4D field, $f^{(j,m)}_{\Xi}$ is
the corresponding mode function and $s$ is the spin weight of the field
${\Xi}$. The spin-weighted spherical harmonics ${}_{s}Y_{jm}(\theta,\phi)$
matches the orthonormal condition as
$\displaystyle\int_{0}^{2\pi}d\phi\int_{-1}^{1}d\cos{\theta}\
\overline{{}_{s}Y_{jm}(\theta,\phi)}\,{}_{s}Y_{j^{\prime}m^{\prime}}(\theta,\phi)=\delta_{jj^{\prime}}\delta_{mm^{\prime}}.$
(30)
A spin weight of fermion is closely related to its $U(1)_{X}$ charge. When we
assign $U(1)_{X}$ charges of 6D Weyl fermions $\Psi_{\pm}$ as
$q_{\Psi_{\pm}}$, the corresponding spin weights of 4D Weyl fermions
$\\{\psi_{+{L\atop R}},\psi_{-{L\atop R}}\\}$ are given as follows in our
convention:
$s_{+{L\atop R}}={nq_{\Psi_{+}}\pm 1\over 2},\quad s_{-{L\atop
R}}={nq_{\Psi_{-}}\mp 1\over 2}.$ (31)
Note that if a 6D Weyl fermion takes a spin weight $s=0$, a $j=0$ mode appears
as a 4D Weyl fermion with vanishing KK mass. This means that we can get chiral
SM fermions without orbifolding in the case of $S^{2}$. When we take the
values:
$\displaystyle(s_{+R},s_{+L},s_{-R},s_{-L})=(0,1,1,0),$ (32)
we can create the same situation as in the $T^{2}$-based models discussed
before. The spin weight of the 4D-vector component of a 6D gauge boson is
always $s=0$ and then there is a zero mode that can be identified as the SM
gauge boson. On the other hand, extra dimensional components of the 6D gauge
boson are expanded by the $|s|=1$ spin-weighted spherical harmonics and has no
zero-mode.
In our configuration, any $(j,m)$-th KK mode has the KK mass:
$m_{(j,m)}=\frac{\sqrt{j(j+1)}}{R}.$ (33)
An important point is that the form of the above KK mass is independent of the
index of $m$. This means that there are $2j+1$ degenerate modes for each $j$.
Note that the lightest KK mode has the mass $\sqrt{2}/R$.
As discussed above, the 4D-vector component of a 6D gauge boson has a zero
mode. This is the case for the extra $U(1)_{X}$ gauge boson too.
Phenomenologically the existence of an extra $U(1)$ interaction, under which
SM fields are charged, is problematic [46]. In the following, let us see how
to get rid of this massless $U(1)_{X}$ vector.
#### 2.3.1 Projective Sphere (PS)
We can construct a UED model compactified on the Projective Sphere $(PS)$, a
sphere $S^{2}$ with its antipodal points being identified by
$(\theta,\phi)\sim(\pi-\theta,\phi+\pi)$ [46]. In the UED model based on PS,
the 6D action takes a different form from that of the 6D orbifold UED models.
One of the remarkable points of this model is that there is no fixed point on
the background geometry PS. As in the $RP^{2}$ model, we introduce “mirror” 6D
Weyl fermions:
$\displaystyle{\mathcal{Q}}_{-},{\mathcal{U}}_{+},{\mathcal{D}}_{+};{\mathcal{L}}_{-},{\mathcal{E}}_{+},\mathcal{N}_{+}$
(34)
which have opposite 6D chirality and opposite SM and $U(1)_{X}$ charges,
compared to the fields
$\\{{Q}_{+},{U}_{-},{D}_{-};{L}_{+},{E}_{-},{N}_{-}\\}$. Because of the
existence of mirror fermions the kinetic term takes the same form as in the
$RP^{2}$ model (24) and the Yukawa interaction is modified to
$\displaystyle\mathcal{L}_{\text{Yukawa}}$ $\displaystyle={1\over
2}\Big{[}-\lambda_{U}\overline{U_{-}}\left(Q_{+}\cdot\Phi\right)-\lambda_{D}\left(\overline{Q_{+}}\Phi\right)D_{-}-\lambda_{E}\left(\overline{L_{+}}\Phi\right)E_{-}$
$\displaystyle\phantom{={1\over
2}\Big{[}}-\lambda_{U}^{*}\overline{\mathcal{U}_{+}}\left(\mathcal{Q}_{-}\cdot\Phi\right)-\lambda_{D}^{*}\left(\overline{\mathcal{Q}_{-}}\Phi\right)\mathcal{D}_{+}-\lambda_{E}^{*}\left(\overline{\mathcal{L}_{-}}\Phi\right)\mathcal{E}_{+}+\text{h.c.}\Big{]}.$
(35)
Like the $RP^{2}$ case which we have discussed before, we introduce the
$``1/2"$ factors for a later convenience. The covariant derivatives in this
model are given as
$\displaystyle{\mathcal{D}_{M}}$
$\displaystyle=\partial_{M}+ig_{s}\mathcal{G}_{M}^{a}T^{a}_{s}+ig\mathcal{W}_{M}^{a}T^{a}+ig_{Y}\mathcal{B}_{M}Y$
$\displaystyle\hskip 227.62204pt(\text{for $\Phi$}),$ (36)
$\displaystyle{\mathcal{D}_{M}}$
$\displaystyle=\partial_{M}+ig_{s}\mathcal{G}_{M}^{a}T^{a}_{s}+ig\mathcal{W}_{M}^{a}T^{a}+ig_{Y}\mathcal{B}_{M}Y+ig_{X}q_{\Psi}(\mathcal{X}_{M}^{c}+\mathcal{X}_{M})+\Omega_{M}$
$\displaystyle\hskip 227.62204pt(\text{for
${Q}_{+},{U}_{-},{D}_{-};{L}_{+},{E}_{-},{N}_{-}$}),$ (37)
$\displaystyle{\mathcal{D}_{M}}$
$\displaystyle=\partial_{M}+ig_{s}\mathcal{G}_{M}^{a}[-T^{a}_{s}]^{\text{T}}+ig\mathcal{W}_{M}^{a}[-T^{a}]^{\text{T}}+ig_{Y}\mathcal{B}_{M}[-Y]+ig_{X}q_{\Psi}(\mathcal{X}_{M}^{c}+\mathcal{X}_{M})+\Omega_{M}$
$\displaystyle\hskip 227.62204pt(\text{for
${\mathcal{Q}}_{-},{\mathcal{U}}_{+},{\mathcal{D}}_{+};{\mathcal{L}}_{-},{\mathcal{E}}_{+},\mathcal{N}_{+}$}),$
(38)
where $\Omega_{M}$ is the spin connection. The covariant derivative of Higgs
is the same as that in the $S^{2}/Z_{2}$ case, but there is a difference
between those of fermions and these “mirror” fermions. We discuss these points
shortly below.
As we mentioned before, projective sphere is a non-orientable manifold and has
no fixed point. Let us consider the 6D $P$ and $CP$ transformations. Under the
antipodal projection,
$\left\\{\begin{array}[]{lcc}\mathcal{X}_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&\mathcal{X}_{\mu}^{C}(x,\theta,\phi)^{S\atop N},\\\
\mathcal{X}_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&-\mathcal{X}_{\theta}^{C}(x,\theta,\phi)^{S\atop N},\\\
\\{\mathcal{X}_{\phi}^{c},\mathcal{X}_{\phi}\\}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&\\{(\mathcal{X}_{\phi}^{c})^{C},\mathcal{X}_{\phi}^{C}\\}(x,\theta,\phi)^{S\atop
N},\end{array}\right.$ (39)
where the superscript $C$ denotes the 6D C transformation. Recall that the
superscript $c$ denotes the classical configuration. These conditions leave
the monopole-like configuration invariant under the antipodal identification
and projects out the unwanted $U(1)_{X}$ 4D-vector zero mode. In contrast,
identification of a SM gauge boson $\mathcal{A}^{(i)}_{M}$ should be done by
another condition since we want the corresponding 4D-vector zero mode, where
$i$ shows the type of gauge group. We adopt the 6D $P$ transformation and
those identifications are written as
$\left\\{\begin{array}[]{lcc}\mathcal{A}^{(i)}_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&\mathcal{A}^{(i)}_{\mu}(x,\theta,\phi)^{S\atop N},\\\
\mathcal{A}^{(i)}_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&-\mathcal{A}^{(i)}_{\theta}(x,\theta,\phi)^{S\atop N},\\\
\mathcal{A}^{(i)}_{\phi}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&\mathcal{A}^{(i)}_{\phi}(x,\theta,\phi)^{S\atop N},\end{array}\right.$
(40)
where it is evident that the zero mode of ${\mathcal{A}_{\mu}^{(i)}}$
survives. We also identify Higgs with the 6D $P$ transformation to obtain its
zero mode:
$\Phi(x,\pi-\theta,\phi+\pi)^{N\atop S}=\Phi(x,\theta,\phi)^{S\atop N}.$ (41)
Finally, we discuss the identification of 6D Weyl fermions. Since 6D Weyl
fermions have $U(1)_{X}$ charge and interact with the $U(1)_{X}$ gauge boson,
they should be identified by the 6D $CP$ transformation. The specific form of
the 6D $CP$ transformation, for example in the case of $U_{-}$, is as follows:
$\mathcal{U}_{+}(x,\pi-\theta,\phi+\pi)^{N\atop
S}=P{U}_{-}^{C}(x,\theta,\phi)^{S\atop N},$ (42)
where the matter field $U_{-}$ is identified to the mirror $\mathcal{U}_{+}$.
We decide the forms of covariant derivatives (37) and (38) on the criterion of
invariance of the action under the 6D $CP$ transformation. Using the
identification conditions (39)–(42), we can see that the mirror fermions drop
out of the action after the identifications and eventually we obtain the usual
type of UED model action. This can be interpreted that all the modes of mirror
fermions
$\\{{\mathcal{Q}}_{-},{\mathcal{U}}_{+},{\mathcal{D}}_{+};{\mathcal{L}}_{-},{\mathcal{E}}_{+},\mathcal{N}_{+}\\}$
are erased and no mode of
$\\{{Q}_{+},{U}_{-},{D}_{-};{L}_{+},{E}_{-},{N}_{-}\\}$ is projected out. The
interaction terms which we need for calculation in this model is the same as
those in Eq. (17). Only difference is the number of degenerate top KK modes in
each $j$-level.
#### 2.3.2 $S^{2}$ UED with a Stueckelberg Field ($S^{2}$)
As a solution to the massless $U(1)_{X}$ problem, we can simply give a
Stueckelberg mass [55, 56], see also [57, 58] for reviews, to the $U(1)_{X}$
field. We can make the unwanted $U(1)_{X}$ 4D-vector zero mode to be massive
while preserving the classical monopole structure in Eq. (26). This way, we
can formulate a UED model on $S^{2}$ with no field identification. Let us call
this simple model the $S^{2}$ UED model. In the $S^{2}$ UED model, the matter
contents, bulk Lagrangian, definition of field strengths and covariant
derivatives and the configuration of the classical $U(1)_{X}$ field are the
same as the PS model after removing the mirror fermions, except for the
Stueckelberg field part. In contrast to the $S^{2}/Z_{2}$ orbifold below,
there are no fixed point nor a localized Lagrangian anywhere on $S^{2}$, as in
the case of PS UED. The d.o.f. of KK fermions has no difference between the
$S^{2}$ UED and PS one (after the antipodal projection). There is no need for
an additional computation; all we have to do is to borrow the PS result as a
whole when we are only interested in the gluon fusion process.
#### 2.3.3 $S^{2}/Z_{2}$ orbifold
Although the above $S^{2}$ UED model with a Stueckelberg $U(1)_{X}$ mass is
already phenomenologically viable, we may further perform a $Z_{2}$
orbifolding on it [47].777 This extra $Z_{2}$ cannot project out the
$U(1)_{X}$ gauge field. In [47], the $U(1)_{X}$ is assumed to be broken by an
anomaly. Since we need a classical configuration of the $U(1)_{X}$, it would
be theoretically preferable to break it by a tiny Stueckelberg mass. On this
orbifold, the point ${(\theta,\phi)}$ is identified with $(\pi-\theta,-\phi)$.
The 6D action $S$ is as follows:
$\displaystyle S$ $\displaystyle=\int
d^{4}x\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\sqrt{-g}\left[\mathcal{L}_{\text{bulk}}(x,y,z)+\delta\left(\theta-{\pi\over
2}\right)\delta\left(\phi\right)\mathcal{L}_{(\pi/2,0)}(x)+\delta\left(\theta-{\pi\over
2}\right)\delta\left(\phi-\pi\right)\mathcal{L}_{(\pi/2,\pi)}(x)\right],$ (43)
where $\sqrt{-g}=R^{2}\sin{\theta}$. This system has two fixed points of the
$Z_{2}$ symmetry at $(\theta,\phi)=(\frac{\pi}{2},0),(\frac{\pi}{2},\pi)$ and
we describe the localized terms with
$\mathcal{L}_{(\pi/2,0)},\mathcal{L}_{(\pi/2,\pi)}$, respectively. Like the
$T^{2}$-case, we do not discuss those parts in this paper.
We can easily construct mode functions of $S^{2}/Z_{2}$
$f_{s,t}^{(j,m)}(\theta,\phi)$ with spin weight $s$ in both north and south
charts following the general prescription [59] as follows:
$f_{s,t}^{(j,m)}(\theta,\phi)^{N\atop
S}=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)+(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm
is\phi}&\text{for}\ t=+1\\\
\displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)-(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm
is\phi}&\text{for}\ t=-1\end{array}\right.,$ (44)
where $t=\pm 1$ is the $Z_{2}$ parity. These mode functions have the property
that $f_{s,t=\pm 1}^{(j,m)}(\pi-\theta,-\phi)^{N\atop S}=\pm f_{s,t=\pm
1}^{(j,m)}(\theta,\phi)^{S\atop N}$. To realize the $Z_{2}$ symmetry, we
identify a field at $(\theta,\phi)$ in north chart with the same field at
$(\pi-\theta,-\phi)$ in south chart.
The range of the summation over $m$ shrinks from $[-j,j]$ to $[0,j]$ after the
$Z_{2}$ identification. Under the transformation of
$(\theta,\phi)\rightarrow(\theta,\phi+\pi)$, mode functions behave as
$f_{s=0,t=+1}^{(j,m)}(\theta,\phi+\pi)^{N\atop
S}=(-1)^{m}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S},\quad f_{s=\pm
1,t=-1}^{(j,m)}(\theta,\phi+\pi)^{N\atop S}=-(-1)^{m}f_{s=\pm
1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S}.$ (45)
After some fields redefinition, we can find that each KK field has a KK parity
$(-1)^{m}$, which is a remnant of the KK angular momentum conservation.
We focus on the $m=0$ modes of each $j$ level. When we see the concrete forms
of mode functions in $m=0$, which are
$\displaystyle f_{s=0,t=+1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{0}Y_{j0}(\theta,\phi),$ (46)
$\displaystyle f_{s=+1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{1}Y_{j0}(\theta,\phi)e^{\pm
i\phi},$ (47) $\displaystyle f_{s=-1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{-1}Y_{j0}(\theta,\phi)e^{\mp
i\phi},$ (48)
where we find that $m=0$ modes appear only in the case of even $j$. Then
degeneracy of KK masses is
$\begin{array}[]{cl}j+1&\text{for}\quad j:\text{even},\\\ j&\text{for}\quad
j:\text{odd},\end{array}$ (49)
since $m$ runs from $0$ to $j$. Again, this mode counting is the only
important point when computing the enhancement of the Higgs production via
gluon fusion process. We do not discuss the form of interactions which we need
for calculating the gluon fusion process because there is essentially no
difference from the $T^{2}$ case (17).
## 3 Higgs production and decay into four leptons in UED models
Figure 1: Feynman diagram which describes the dominant contribution to the
gluon fusion Higgs production process and the subsequent decay to 2Z.
In the SM, the cross section for the Leading Order (LO) one-loop Higgs
production via the gluon fusion process and its subsequent decay into a $Z$
boson pair: $gg\to H\to ZZ$ is given by [60]. The LO parton-level cross
section shown in Fig. 1 is:
$\displaystyle\hat{\sigma}^{\text{SM}}_{gg\to H\to ZZ}$
$\displaystyle={\alpha_{{4s}}^{2}\over 256\pi^{3}}\left(m_{Z}\over
v_{\text{EW}}\right)^{4}\left[1+{\left(\hat{s}-2m_{Z}^{2}\right)^{2}\over
8m_{Z}^{4}}\right]{\hat{s}\over\left(\hat{s}-M_{H}^{2}\right)^{2}+\Delta^{2}}\sqrt{1-{4m_{Z}^{2}\over\hat{s}}}\,\left|{I\\!\left({\hat{s}}\right)}\right|^{2},$
(50)
where $m_{W},m_{Z},m_{t}$, and $M_{H}$ are respectively the $W$, $Z$, top
quark, and Higgs boson masses, $\alpha_{4s}={g_{4s}^{2}/4\pi}$ is the $4D$ QCD
gauge coupling, $\hat{s}$ is the center-of-mass-energy-squared of the
scattering partons, we employ the normalization for the Higgs vev:
$v_{\text{EW}}^{2}=1/\sqrt{2}G_{F}\simeq\left(246\,\text{GeV}\right)^{2}$, and
the loop functions are defined as
$\displaystyle I(\lambda)$
$\displaystyle={-}2\lambda+\lambda({1-4\lambda})\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right],$ (51)
$\displaystyle\tilde{I}(\lambda)$ $\displaystyle=\lambda\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right].$ (52)
Explicit result of the integral is
$\displaystyle\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right]$
$\displaystyle=\begin{cases}\displaystyle-2\left[\arcsin{1\over\sqrt{4\lambda}}\right]^{2}&\text{(for
$\lambda\geq{1\over 4}$)},\\\ \displaystyle{1\over
2}\left[\ln{1+\sqrt{1-4\lambda}\over
1-\sqrt{1-4\lambda}}-i\pi\right]^{2}&\text{(for $\lambda<{1\over
4}$)}.\end{cases}$ (53)
We have also defined $\tilde{I}$ for later use for the Dirichlet Higgs model.
In Eq. (50), we have taken into account the total decay width of the Higgs in
its propagator:
$\displaystyle\Delta=M_{H}\Gamma_{H}.$ (54)
In the current analysis, we take into account the Higgs decay into $W$, $Z$
and top quark pairs, which are dominant when we consider the heavy SM Higgs
boson: $M_{H}\geq 2m_{W}$. Explicit form is shown in Eq. (124) in appendix.
Note that we take into account only the top quark loop in the SM cross section
(50), given by the diagram shown in Fig. 1, since the Yukawa coupling to
others are negligible compared to the top one. We have also ignored the
contributions from the sub-leading box diagrams [61]. See Appendix C for
further discussion on how to take into account the width.
### 3.1 Gluon fusion process in UED models
In this paper, we consider the KK-top loop contributions to the gluon fusion
process in several UED models, namely, 5D UED model on $S^{1}/Z_{2}$ (mUED)
[1], Dirichlet Higgs (DH) [40], 6D UED model on
$T^{2}/Z_{2}~{}\cite[cite]{[\@@bibref{}{Appelquist:2000nn}{}{}]},\
T^{2}/Z_{4}~{}\cite[cite]{[\@@bibref{}{Dobrescu:2004zi,Burdman:2005sr}{}{}]},\
T^{2}/(Z_{2}\times
Z^{\prime}_{2})~{}\cite[cite]{[\@@bibref{}{Mohapatra:2002ug}{}{}]}$, Real
Projective Plane (${RP}^{2}$) [45], $S^{2}/Z_{2}$ [47], Projective Sphere (PS)
[46] and $S^{2}$ with a Stueckelberg Field ($S^{2}$). We have given a brief
review on these models in the previous section. The contribution from KK-top
loops to the gluon fusion process is analogous to that of the top-loop in SM
and the difference resides only in the loop function. Relevant Feynman diagram
is shown in Figs. 2 and 3. The effective vertex, which is represented by the
lined blob in the diagram, includes the contributions to the gluon fusion from
the zero mode top quark and the KK top quarks. Since the zero mode sector of
UED model regenerates the SM configuration, the result of the former
contribution is the same as that of the SM in Eq. (50). The forms of the
latter contribution will be shown soon later. We note that the letter “$H$” in
Figs. 2 and 3 shows the zero mode physical Higgs boson in all the cases except
for the DH model where $H$ stands for the first KK Higgs.
Figure 2: A schematic description of the dominant contribution to the gluon
fusion Higgs production process and the subsequent decay to 2Z. The lined blob
indicates the effective vertex. Figure 3: The effective vertex which describes
the Higgs production from the gluon fusion.
For each model, we get the following result, where $J_{\text{model}}$
indicates the corresponding loop function:
$\displaystyle\hat{\sigma}^{\text{model}}_{gg\to H\to ZZ}$
$\displaystyle={\alpha_{{4s}}^{2}\over 256\pi^{3}}\left(m_{Z}\over
v_{\text{EW}}\right)^{4}\,{1+\left(\hat{s}-2m_{Z}^{2}\right)^{2}\over
8m_{Z}^{4}}\,{\hat{s}\over\left(\hat{s}-M_{H}^{2}\right)^{2}+\left({M_{H}\Gamma_{H}}\right)^{2}}\sqrt{1-{4m_{Z}^{2}\over\hat{s}}}\,{K}\left|J_{\text{model}}\left(\hat{s}\right)\right|^{2},$
(55)
with
$\displaystyle J_{\text{mUED}}(\hat{s})$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left({m_{t(n)}^{2}\over\hat{s}}\right),$ (56)
$\displaystyle J_{\text{DH}}(\hat{s})$
$\displaystyle=\sqrt{2}\varepsilon_{1}\sqrt{\left|I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}+\left|2\sum_{n=1}^{\infty}\left({m_{t}\over
m_{t(n)}}\right)^{2}\tilde{I}\\!\left(m_{t(n)}^{2}\over\hat{s}\right)\right|^{2}},$
(57) $\displaystyle{J_{T^{2}/Z_{2}}(\hat{s})=J_{{RP}^{2}}(\hat{s})}$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{{m+n\geq
1\atop\text{or\ }m=-n\geq 1}}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t(m,n)}^{2}\over\hat{s}\right),$ (58)
$\displaystyle{J_{T^{2}/Z_{4}}(\hat{s})}$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq 1,n\geq
0}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t({m,n})}^{2}\over\hat{s}\right),$ (59)
$\displaystyle{J_{T^{2}/Z_{2}\times Z^{\prime}_{2}}(\hat{s})}$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq 0,n\geq
0,\atop(m,n)\not=(0,0)}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\left(m_{t(m,n)}^{2}\over\hat{s}\right),$ (60)
$\displaystyle J_{S^{2}/Z_{2}}(\hat{s})$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j=1}^{j_{\text{max}}}\left({m_{t}\over
m_{t(j)}}\right)^{2}n(j)\,I\left(m_{t(j)}^{2}\over\hat{s}\right),$ (61)
$\displaystyle J_{\text{PS}}(\hat{s})={J_{S^{2}}(\hat{s})}$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j=1}^{j_{\text{max}}}\left({m_{t}\over
m_{t(j)}}\right)^{2}(2j+1)\,I\\!\left(m_{t(j)}^{2}\over\hat{s}\right),$ (62)
where the KK top masses are given by
$\displaystyle m_{t(n)}$
$\displaystyle:=\sqrt{m_{t}^{2}+{\frac{n^{2}}{R^{2}}}}=\sqrt{m_{t}^{2}+{{n^{2}}{M_{\text{KK}}^{2}}}},$
(63) $\displaystyle m_{t(m,n)}$
$\displaystyle:=\sqrt{m_{t}^{2}+{\frac{m^{2}+n^{2}}{R^{2}}}}=\sqrt{m_{t}^{2}+{{\left(m^{2}+n^{2}\right)M_{\text{KK}}^{2}}{}}},$
(64) $\displaystyle m_{t(j)}$
$\displaystyle:=\sqrt{m_{t}^{2}+{\frac{j(j+1)}{R^{2}}}}=\sqrt{m_{t}^{2}+{\frac{j(j+1)M_{\text{KK}}^{2}}{2}}}.$
(65)
Here the $M_{\text{KK}}$ is the first KK mass, which is written as
$\displaystyle M_{\text{KK}}={1\over R}$ (66)
for the compactifications based on $S^{1}/Z_{2}$, interval, and $T^{2}$
(namely, the mUED, DH, $T^{2}/Z_{2}$, $RP^{2}$, $T^{2}/Z_{4}$, and
$T^{2}/(Z_{2}\times Z_{2}^{\prime})$ models) and is written as
$\displaystyle M_{\text{KK}}={\sqrt{2}\over R}$ (67)
for the $S^{2}$-based ones (namely, the $S^{2}/Z_{2}$, PS and $S^{2}$ models).
The gluon fusion process for mUED in $S^{1}/Z_{2}$ is first shown in Ref. [37]
and for $S^{2}/Z_{2}$ in Ref. [38]. Also it has been calculated for
$T^{2}/Z_{4}$ and PS in Ref. [39]. The results for DH, $T^{2}/Z_{2}$,
$RP^{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$ and $S^{2}$ are newly presented
in this paper. The factor $\sqrt{2}\varepsilon_{1}$ in Eq. (57) is equal to
$2\sqrt{2}/\pi\sim 0.9$. The origin of this suppression factor is non-
orthonormality of mode functions on an interval. In the case of $S^{2}$-based
compactification, there are some degenerated states, the number of which is
described with $n(j)$ on $S^{2}/Z_{2}$ and $(2j+1)$ on PS or $S^{2}$ for each
KK-index $j$. The specific form of $n(j)$ for the orbifold $S^{2}/Z_{2}$ in
Eq. (61) is as follows:
$\displaystyle n(j)=\begin{cases}j+1&\text{for}\ j:\text{even},\\\
j&\text{for}\ j:\text{odd}.\end{cases}$ (68)
Several comments are in order.
* •
The origin of the factor 2 in front of each KK summation is the fact there are
both left and right handed (namely, vector-like) KK modes for each chiral
quark zero mode.
* •
All the KK contributions are positive and hence always enhance the Higgs
production rate via the gluon fusion process, except for the DH model in which
the zero mode Higgs contribution is absent.
* •
Each value of Yukawa couplings of KK quarks to the Higgs is the same as that
of the coupling between the corresponding zero mode fermion and the Higgs. We
only consider triangle loop diagrams of the SM top quark and its KK excited
modes because their Yukawa coupling to the Higgs is dominant compared to that
of other fermions.
* •
In each KK summation infinite numbers of KK modes contribute to the process in
principle. In 6D, these summations are divergent and a suitable scheme of
regularization is required. $j_{\text{max}}$ in Eqs. (61) and (62) shows an
upper bound of the summation over the index $j$. Further discussion will be
shown in the following subsection.
* •
$K$ in Eq. (55) is the so-called K-factor, a phenomenological approximation in
order to naively take into account higher order QCD corrections. One may take
$K\sim 2$ for Tevatron and $K\sim 1.5\text{--}1.6$ for LHC, respectively [32].
In the limit where the KK-loop is viewed as a contribution to the effective
Higgs-gluon-gluon coupling, the QCD corrections to Higgs production are very
similar between the SM and the new physics contributions. The reason is that
the Higgs-gluon-gluon coupling always has the same structure, and only its
coefficient changes. This is discussed in detail e.g. in Ref. [62] in the
context of SUSY (but it works the same way in UED as in SUSY). Therefore we
have included a K factor also for the new physics terms as in Eq. (55).
* •
When the compactification radius is too large, namely, when the first KK $W$
is lighter than half the Higgs mass, the Higgs can decay into a pair of KK
particles and its decay width $\Gamma_{H}$ becomes broader. In this situation
it becomes harder to find the evidence of the Higgs boson and hence we
restrict ourselves within the region where the Higgs mass is smaller than
twice the first KK $W$ mass so that such a decay mode does not open up.
* •
Although the one-loop gluon fusion process is the dominant production channel
of the Higgs, its contribution to the Higgs total decay width is smaller at
least by three orders of magnitude comparing to the decay into a $W$ pair in
the case of the SM with $M_{H}\geq 300$ GeV [32]. Even after enhancement of
$\mathcal{O}(10)$ from KK-top contributions, decay into gluon pair is still
negligible.
* •
In Eq. (55), the Higgs decay width is taken into account by the naive Breit-
Wignar formula in the denominator. When the Higgs mass is large, say
$M_{H}=700\,\text{GeV}$, the Higgs decay width is as large as 180 GeV. In some
literature, the expression $M_{H}\Gamma_{H}$ in the the Breit-Wignar formula
is replaced by $\hat{s}\Gamma_{H}/M_{H}$. In Appendix C we discuss reliability
of our treatment.
### 3.2 UV cutoff scale in six dimensions
In 6D UED models, since the gluon fusion process is UV divergent, we must
consider upper limit of the summations of KK number in such models.888 In 5D
UED model, we can execute this mode summation with no divergence. First let
us briefly review how the Naive Dimensional Analysis (NDA) is applied to the
higher dimensional theory. Following the concept of NDA, a loop expansion
parameter $\epsilon$ in D-dimensional SU(N) gauge theory at a scale $\mu$ is
obtained as
$\displaystyle\epsilon{(\mu)}$
$\displaystyle=\frac{1}{2}\frac{2\pi^{D/2}}{(2\pi)^{D}\Gamma(D/2)}N_{g}\,g_{Di}^{2}(\mu)\,\Lambda^{D-4},$
(69)
where $N_{g}$ is a group index, $g_{Di}$ is a dimensionful gauge coupling in
$D$-dimensions and $\Lambda$ is a UV cutoff scale. The index $i$ is introduced
to express the type of gauge interaction and the remaining part originates
from $D$-dimensional momentum loop integral. The cutoff scale $\Lambda$ is the
scale where the perturbation breaks down $\epsilon(\Lambda)\sim 1$.
Precisely speaking, the dimensionful higher dimensional gauge coupling does
not “run.” Let us explain what is meant by the running coupling in (69),
basically following Ref. [48]. When we consider a 6D theory $(D=6)$ with two
compact spacial dimensions, an effective 4D gauge coupling $g_{4i}$ emerges
after KK decomposition: $g_{4i}={g_{6i}}/{\sqrt{V_{2}}}$, where $V_{2}$ is the
volume of two extra dimensions. Concretely, $V_{2}=(2\pi R)^{2}$ and $4\pi
R^{2}$ for $T^{2}$ and $S^{2}$, respectively. In this paper, we employ a
bottom-up approach for the running gauge coupling. At energies below the first
KK scale, theory is purely four dimensional (after integrating out all the
massive modes of order KK scale) and the gauge coupling runs logarithmically.
Let us then increase the energy scale. Every time we cross a KK mass scale,
there open up the corresponding KK modes to run in the loops in the gauge
boson two-point function. In all the scales, the theory is renormalizable and
the running of the gauge coupling is logarithmic. However, due to the increase
of the number of particles in the loops, the running of the gauge coupling
becomes _effectively_ power-law at the energy scales much above the first KK
scale. This way, we get the effective power-law running of the gauge coupling,
within purely renormalizable approach. In this paper, we neglect possible
threshold corrections at the UV cutoff scale, since we are interested in what
is the highest possible $\Lambda$ that can be consistent with the low energy
theory.
In the above stated strategy, we get the following running of the 4D effective
gauge coupling strength $\alpha_{4i}(\mu)$
$\displaystyle{\alpha_{4i}^{{-1}}(\mu)\ {\simeq}\
\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln{\mu\over
m_{Z}}+2C\,{\textsf{b}_{i}^{\text{6D}}\over 2\pi}\ln{\mu\over
M_{\text{KK}}}-C\,\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\left[\left(\frac{\mu}{M_{\text{KK}}}\right)^{2}-1\right],}$
(70)
where $C=\pi/2$ and 1 for $T^{2}$ and $S^{2}$, respectively. In Eq. (70), we
have approximated that all the masses are degenerate in each KK level.
Depending on models, some fraction of the KK modes are projected out, but we
assume that all of them contribute to the running, in order to give the most
conservative upper bound on the UV cutoff scale. More detailed explanation is
given in Appendix B.
| $T^{2}$-based | $S^{2}$-based
---|---|---
| max | min | max | min
KK index | $m^{2}+n^{2}\leq 28$ | $m^{2}+n^{2}\leq 10$ | $j(j+1)\leq 90$ | $j(j+1)\leq 30$
UV cutoff | $\Lambda_{6D}\sim 5M_{\text{KK}}$ | $\Lambda_{6D}\sim 3M_{\text{KK}}$ | $\Lambda_{6D}\sim 7M_{\text{KK}}$ | $\Lambda_{6D}\sim 4M_{\text{KK}}$
Table 3: Our choices of maximum and minimum upper bounds for KK indices and
for the corresponding UV cutoff scale.
While the described procedure gives a reasonable estimate of the UV cutoff
scale, one has to be aware that this is not much more than an order-of-
magnitude estimate. We will plot our results for maximum and minimum values of
the UV cutoff scale that are theoretically reasonable. In Table 3, we list our
choice of bounds at which the KK mode summation is truncated to regularize the
process.
### 3.3 Convolution of parton distribution
Let us briefly review the standard prescription to estimate the event number
of the Higgs production in $pp\to ZZ\to 4\ell$ (four leptons) via the gluon
fusion process as a function of the invariant mass of $ZZ$, given the parton
level cross section, where $4\ell$ denotes two pairs consisting of either
$e^{-}e^{+}$ or $\mu^{-}\mu^{+}$, since a tau pair is less visible at the LHC.
That is, the final state is possibly $e^{-}e^{+}e^{-}e^{+}$,
$\mu^{-}\mu^{+}\mu^{-}\mu^{+}$, or $e^{-}e^{+}\mu^{-}\mu^{+}$. Because of the
large mass difference between Z boson and two electrons or two muons, the
subsequent processes: $Z\to e^{+}e^{-}$ and $Z\to\mu^{+}\mu^{-}$ are well
treated with on-shell approximation.
By using parton distribution function of gluon $f_{g}(x,\hat{s})$, we give the
formula of the total cross section of $pp\to ZZ$:
$\sigma_{pp\to
ZZ}^{\text{model}}(s)=\int_{0}^{1}d\tau\hat{\sigma}^{\text{model}}_{gg\to H\to
ZZ}(\tau s)L(\tau,s),$ (71)
where we define
$L(\tau,s):=\int_{-\ln{1\over\sqrt{\tau}}}^{\ln{1\over\sqrt{\tau}}}dyf_{g}(\sqrt{\tau}e^{y},\tau
s)f_{g}(\sqrt{\tau}e^{-y},\tau s).$ (72)
The concrete form of $\hat{\sigma}_{pp\to ZZ}^{\text{model}}$ has been shown
in Eqs. (55)–(62). The invariant mass of $ZZ$ is represented as
$M^{2}_{ZZ}=\hat{s}=\tau s$. Then, the differential cross section of $pp\to
ZZ\to 4\ell$ as a function of $M_{ZZ}$ is written as
${d\sigma_{pp\to ZZ\to 4\ell}^{\text{model}}(M_{ZZ})\over
dM_{ZZ}}={2M_{ZZ}\over s}\times\hat{\sigma}^{\text{model}}_{gg\to H\to
ZZ}(M^{2}_{ZZ})\times L\left({M^{2}_{ZZ}\over s},s\right)\times
4\operatorname{Br}(Z\to 2\ell)^{2},$ (73)
where $\operatorname{Br}(Z\to 2\ell):=\operatorname{Br}(Z\to
e^{+}e^{-})=\operatorname{Br}(Z\to\mu^{+}\mu^{-})=0.034$ is the branching
ratio.
## 4 Numerical results
Geometry | Allowed region of KK indices
---|---
$T^{2}/Z_{2}$ or $RP^{2}$ | $(m,n)=(1,0),\,(2,0),\,(3,0),\,(4,0),\,(5,0),$
| $(m,n)=$ $(0,1),\,(1,1),\,(2,1),\,(3,1),\,(4,1),\,(5,1),$
| $(m,n)=$ $(-1,2),\,(0,2),\,(1,2),\,(2,2),\,(3,2),\,(4,2),$
| $(m,n)=$ $(-2,3),\,(-1,3),\,(0,3),\,(1,3),\,(2,3),\,(3,3),\,(4,3),$
| $(m,n)=$ $(-3,4),\,(-2,4),\,(-1,4),\,(0,4),\,(1,4),\,(2,4),\,(3,4),$
| $(m,n)=$ $(-1,5),\,(0,5),\,(1,5),$
| $(m,n)=$ $(1,-1),\,(2,-1),\,(3,-1),\,(4,-1),\,(5,-1),$
| $(m,n)=$ $(2,-2),\,(3,-2),\,(4,-2),$
| $(m,n)=$ $(3,-3),\,(4,-3),$
$T^{2}/(Z_{2}\times Z^{\prime}_{2})$ | $(m,n)=(1,0),\,(2,0),\,(3,0),\,(4,0),\,(5,0),$
| $(m,n)=$ $(0,1),\,(1,1),\,(2,1),\,(3,1),\,(4,1),\,(5,1),$
| $(m,n)=$ $(0,2),\,(1,2),\,(2,2),\,(3,2),\,(4,2),$
| $(m,n)=$ $(0,3),\,(1,3),\,(2,3),\,(3,3),\,(4,3),$
| $(m,n)=$ $(0,4),\,(1,4),\,(2,4),\,(3,4),$
| $(m,n)=$ $(0,5),\,(1,5),$
$T^{2}/Z_{4}$ | $(m,n)=(1,0),\,(2,0),\,(3,0),\,(4,0),\,(5,0),$
| $(m,n)=$ $(1,1),\,(2,1),\,(3,1),\,(4,1),\,(5,1),$
| $(m,n)=$ $(1,2),\,(2,2),\,(3,2),\,(4,2),$
| $(m,n)=$ $(1,3),\,(2,3),\,(3,3),\,(4,3),$
| $(m,n)=$ $(1,4),\,(2,4),\,(3,4),$
| $(m,n)=$ $(1,5),$
$S^{2}/Z_{2}$ | $j=1\sim{9}$ ($n(j)=j+1$ for $j$: even or $n(j)=j$ for $j$: odd)
PS or $S^{2}$ | $j=1\sim{9}$ ($2j+1$ degenerated states)
Table 4: The region of KK summation for the maximum UV cutoff.
We use the top quark mass $m_{t}=172$ GeV and the LO QCD running coupling. We
apply the CTEQ5 LO PDF of gluon [63]. In our analysis, we consider only KK-
scales large enough not to open up the Higgs decay channel into a pair of KK-
top, $Z$, and $W$ particles. When the decay channel into two KK particles
opens up, the Higgs resonance tends to become too broad and hard to be seen at
Tevatron and LHC. We list in Table 4 the KK modes that satisfy the maximum
cutoff criterion given in Table 3.
### 4.1 Tevatron
Figure 4: The $H\to ZZ\to 4\ell$ event number per each 25 GeV bin of $M_{ZZ}$
for $M_{H}=330$ GeV expected at Tevatron with an integrated luminosity
$8\,\text{fb}^{-1}$ at $\sqrt{s}=1.96$ TeV. The grey, black, and magenta lines
represent the expected event number with $M_{\text{KK}}=200,\,400$, and
${800}$ GeV, respectively. For 6D UED models, we consider dependency on the UV
cutoff, whose range is from minimum (lower side of band) to maximum (upper
sider of band) given in Table 3.
We evaluate the $H\to ZZ\to 4\ell$ event number per each 25 GeV bin of
$M_{ZZ}$, expected at Tevatron with an integrated luminosity
$8\,\text{fb}^{-1}$ at $\sqrt{s}=1.96$ TeV in the above mentioned UED models.
We show the results for the Higgs mass $M_{H}=330$ GeV. We consider the KK
scales $M_{\text{KK}}=200,\,400$, and ${800}$ GeV, except for the Dirichlet
Higgs (DH) model which does not have a zero-mode Higgs. For the DH model, we
take first-KK Higgs as the Higgs field, that is, $M_{\text{KK}}=M_{H}=330$
GeV.999 Note that the coupling of the Dirichlet Higgs to the SM modes is
decreased by a factor 0.9 compared to the SM Higgs. The results are shown in
Fig. 4. We see that the event number is enhanced in all the UED models from
the SM one. In particular, the 6D Projective Sphere model can give a large
enhancement in the Higgs production by a factor as large as hundred compared
to the SM when the KK-scale is low at 200 GeV.101010 Even if the one-loop
$H\to gg$ decay rate is enhanced by the factor 100 from the SM one, it is
still subdominant compared with the tree-level $H\to WW$ [32] and we neglect
its contribution to the total decay width of the Higgs. Two $ZZ\to 4\ell$
events are observed in a 300–350 GeV bin at CDF with the integrated luminosity
$4.8\,\text{fb}^{-1}$ [64] and two such events are observed in 325–375 GeV
bins at D0 with $6.4\,\text{fb}^{-1}$ [65]. Recently two more events are
reported around 330 GeV [66]. However, the above large cross section to
explain the Tevatron data is found to be inconsistent with the LHC data [67,
68].
Figure 5: The $H\to ZZ\to 4\ell$ event number per each 25 GeV bin of $M_{ZZ}$
for $M_{H}=500$ GeV, expected at LHC with an integrated luminosity
${10}\,\text{fb}^{-1}$ at $\sqrt{s}=7$ TeV. The black, magenta, and cyan lines
represent the expected event number with $M_{\text{KK}}=400,\,800$, and $1200$
GeV, respectively. Dependence on the 6D UV cutoff scale is shown the same as
in Fig. 4.
Figure 6: The $H\to ZZ\to 4\ell$ event number per each 25 GeV bin of $M_{ZZ}$
for $M_{H}=700$ GeV, expected at LHC with an integrated luminosity
${10}\,\text{fb}^{-1}$ at $\sqrt{s}=14$ TeV. The black, magenta, and cyan
lines represent the expected event number with $M_{\text{KK}}=400,\,800$, and
$1200$ GeV, respectively. Dependence on the 6D UV cutoff scale is shown the
same as in Fig. 4.
### 4.2 LHC
We plot the event number of $H\to ZZ\to 4\ell$ for the Higgs mass
$M_{H}=500\,\text{GeV}$ at the LHC with an integrated luminosity
${10}\,\text{fb}^{-1}$ at $\sqrt{s}=7$ TeV in Fig. 5. When
$\sqrt{s}=7\,\text{TeV}$, we have checked that we cannot see sizable number of
events for all the UED models with $M_{H}=700\,\text{GeV}$ even for an
integrated luminosity $10\,\text{fb}^{-1}$, expected by the end of 2012 after
which the LHC is planned to be (shut-down for a year and then) upgraded to
13–14 TeV. Therefore, we show corresponding results for
$M_{H}=700\,\text{GeV}$ at $\sqrt{s}=14\,\text{TeV}$ with an integrated
luminosity ${10}\,\text{fb}^{-1}$ in Fig. 6. We have chosen several KK scales:
400, 800, 1200 GeV. We show our plots in logarithmic scale so that one can
easily see the results for different luminosities by simply shifting them
upward/downward.
In Fig. 5 with an integrated luminosity ${10}\,\text{fb}^{-1}$, we can see in
total a few events in 5D UED models (mUED and DH) and
$\mathcal{O}(1)$–$\mathcal{O}(10)$ events per each 25 GeV bin for 6D UED
models. Note that even if we can only see at best in total a few events for 5D
UED by the end of 2012, this $ZZ\to 4\ell$ channel is virtually background
free at 500 GeV and the result would be still significant.
In Fig. 6 for $M_{H}=700\,\text{GeV}$, we have plotted the results for the
upgraded energy $\sqrt{s}=14\,\text{TeV}$. We see that even with the
integrated luminosity $10\,\text{fb}^{-1}$, 6D UED models can have a few
events per each 25 GeV bin if the KK scale is relatively low
$M_{\text{KK}}=400\,\text{GeV}$. The 6D $T^{2}/Z_{2}$, $RP^{2}$, $S^{2}$ and
PS models can have in total few events for higher KK mass
$M_{\text{KK}}={800}\,\text{GeV}$. When the integrated luminosity adds up to
$100\,\text{fb}^{-1}$, we can see a few events per each bin for 5D mUED and DH
models, and even for the SM (though the SM itself cannot satisfy the
electroweak constraints on the $S,T$ parameters, contrary to the UED models).
We see that the Dirichlet Higgs model has slightly smaller cross section than
the Standard Model. This is because the KK scale is fixed to be large, 700
GeV, and hence the enhancement of the KK top loop is small, while the Yukawa
coupling of the (first KK) Higgs to the top quark is decreased by the factor
$2\sqrt{2}/\pi\simeq 0.9$.
Let us emphasize that the enhancement of Higgs production $gg\to H$ in UED
does not depend on the details of the model such as the mass structure at the
orbifold fixed points. Parameter dependence is only on the Higgs mass and the
KK scale. In this sense, this Higgs channel signal is complementary to the
direct search of the KK modes decaying into LKP [14, 28], which is nice
because of the directness but is dependent on the details of KK mass splitting
from the boundary terms.
## 5 Summary and discussions
We have presented a review on the known 5D and 6D UED models focusing on the
relevant part to the gluon fusion process. We have explained our computation
of the gluon fusion process including the KK top-quark loops, which is new for
the $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$, $RP^{2}$, and $S^{2}$
UED models. For 6D UED models, we have shown an NDA analysis of the highest
possible UV cutoff scale in the $S^{2}$-based compactification, extending the
analysis of Ref. [48, 3] on the $T^{2}$ compactification.
For Higgs mass $M_{H}=500\,\text{GeV}$, we can see a few (virtually background
free) $H\to ZZ\to 4\ell$ events in 5D UED models with $10\,\text{fb}^{-1}$ of
integrated luminosity. The 6D UED models can further exhibit the shape of the
resonance if the KK scale is relatively low. When Higgs mass is as large as
$M_{H}=700\,\text{GeV}$, we found no parameter region that can be seen within
the integrated luminosity of $\mathcal{O}(10)\,\text{fb}^{-1}$ at
$\sqrt{s}=7\,\text{TeV}$. We have also studied the event rate for the upgraded
energy $\sqrt{s}=14\,\text{TeV}$ when $M_{H}=700\,\text{GeV}$. We see that the
5D and 6D UED models typically require $100\,\text{fb}^{-1}$ and
$10\,\text{fb}^{-1}$ of data, respectively, in order to establish the
existence of the resonance. As is reported in Ref. [39], the 6D UED model on
the Projective Sphere (or $S^{2}$) shows the greatest enhancement of the Higgs
production via gluon fusion process, among all the known UED models.
Let us again emphasize that the presented Higgs signal of UED needs only the
Higgs mass and KK scale as input parameters, is independent of the detailed KK
mass splitting, and hence is unaffected by the boundary/fixed-point mass
structures. Therefore, this Higgs signal of UED is complementary to the direct
KK resonance production and to the dark matter signal. As the enhancement of
the Higgs production via the gluon fusion process can be so large, recent data
from the LHC [67, 68] can already significantly exclude the parameter space of
the UED models. This analysis is presented in a separate publication [69]. We
will also show a combined bound from the triviality and the electroweak
precision constraints in addition to that of Ref. [69]. (The triviality bound
would lower the maximum allowed UV cutoff scale when the Higgs mass is heavy.)
In this paper, we have studied the cleanest possible signature $H\to ZZ\to
4\ell$. It is expected that a combined analysis including other decay channels
such as $H\to WW$ and $H\to ZZ\to\ell\ell\nu\nu$ will provide a large gain
over all individual analyses [68]. Such a combined analysis for the UED models
will be presented elsewhere.
### Acknowledgment
We are grateful to Shoji Asai for valuable comments and thank Abdelhak
Djouadi, Kaoru Hagiwara, Kazunori Hanagaki, Shigeki Matsumoto, Kazuki Sakurai,
and Kohsaku Tobioka for helpful communications. We appreciate Referee’s
thorough reading and constructive comments. The work of K.O. is partially
supported by Scientific Grant by Ministry of Education and Science (Japan),
Nos. 20244028, 23104009 and 23740192.
## Appendix
## Appendix A Feynman rules for Dirichlet Higgs model
In this appendix, we describe the Feynman rules that are necessary for our
computation. The mass terms of KK fermions are
$\displaystyle-\sum_{n=1}^{\infty}\begin{bmatrix}\overline{Q_{t}}&\overline{U_{t}}\end{bmatrix}^{(n)}\begin{bmatrix}\frac{n}{R}&m_{t}\\\
m_{t}&-\frac{n}{R}\end{bmatrix}\begin{bmatrix}{Q_{t}}\\\
{U_{t}}\end{bmatrix}^{(n)},$ (74)
where $Q_{t}$ is an upper component of the quark doublet in third generation
and $U_{t}$ is the top quark singlet. Transforming each KK states by the
following unitary transformation including chiral rotation:
$\displaystyle\begin{bmatrix}t_{1}\\\ t_{2}\end{bmatrix}^{(n)}$
$\displaystyle=\begin{bmatrix}\gamma^{5}\\\
&1\end{bmatrix}\begin{bmatrix}\cos\alpha^{(n)}&-\sin\alpha^{(n)}\\\
\sin\alpha^{(n)}&\cos\alpha^{(n)}\end{bmatrix}\begin{bmatrix}Q_{t}\\\
U_{t}\end{bmatrix}^{(n)},$ (75)
we can obtain the ordinary diagonalized Dirac mass terms, where $t_{1}^{(n)}$
and $t_{2}^{(n)}$ are mass eigenstates of n-th KK top quarks and each mixing
angle $\alpha^{(n)}$ is determined to be
$\cos{2\alpha^{(n)}}=(n/R)/\sqrt{m_{t}^{2}+n^{2}/R^{2}}$,
$\sin{2\alpha^{(n)}}=m_{t}/\sqrt{m_{t}^{2}+n^{2}/R^{2}}$. Each KK state is
twofold degenerate and n-th KK top mass is
$m_{t,(n)}=\sqrt{m_{t}^{2}+n^{2}/R^{2}}$. The corresponding interaction terms
are
$\displaystyle\mathcal{L}_{\text{KK top}}$
$\displaystyle=-ig_{{4s}}\sum_{n=1}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(n)}\gamma^{\mu}\mathcal{G}_{\mu}^{(0)}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(n)}$ $\displaystyle\quad-{m_{t}\over
v_{\text{EW}}}\sum_{n,m=1}^{\infty}\begin{bmatrix}\overline{t_{1}}&\overline{t_{2}}\end{bmatrix}^{(n)}H^{(m)}\left(\sqrt{2}\varepsilon_{m}+\frac{1}{\sqrt{2}}(\varepsilon_{2n+m}-\varepsilon_{2n-m})\gamma^{5}\right)\begin{bmatrix}\sin{2\alpha^{(n)}}&-\gamma^{5}\cos{2\alpha^{(n)}}\\\
\gamma^{5}\cos{2\alpha^{(n)}}&\sin{2\alpha^{(n)}}\end{bmatrix}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}^{(n)},$ (76)
where $g_{{4s}}$ is a dimensionless 4D $SU(3)_{C}$ coupling constant and
$v_{\text{EW}}$ is the 4D Higgs vacuum expectation value which appears after
KK expansion. $\mathcal{G}^{(0)}_{\mu}$ is massless gluon and $H^{(m)}$ is
$m$-th KK Higgs bosons. The concrete shape of the factor of
$\sqrt{2}\varepsilon_{n}$ is $2\sqrt{2}/n\pi$, whose origin is the non-
orthonormality of mode functions in Dirichlet Higgs model. In Dirichlet Higgs
model there is no zero mode Higgs because of choosing Dirichlet boundary
condition in Higgs field. The first KK Higgs boson behaves like a heavy SM
Higgs except that its interaction with the SM fields are multiplied by
$\sqrt{2}\varepsilon_{1}$. The explicit form of Feynman rules is:
$\displaystyle\quad={i\over-p^{2}-\left(m/R\right)^{2}+i\epsilon},$ (77)
$\displaystyle\quad={i\over i\not{p}+\sqrt{m_{t}^{2}+n^{2}/R^{2}}+i\epsilon},$
(78)
$\displaystyle\quad={g_{4s}\gamma_{\mu}\left[\frac{\lambda^{a}}{2}\right]},$
(79)
$\displaystyle\quad={-i\frac{m_{t}}{v_{\text{EW}}}\left(\sqrt{2}\varepsilon_{m}+\frac{1}{\sqrt{2}}(\varepsilon_{2n+m}-\varepsilon_{2n-m})\gamma^{5}\right)\sin{2\alpha^{(n)}}},$
(80)
where $a$ is a gluon (adjoint) color index and $\lambda^{a}$ are the Gell-Mann
matrices. It is noted that we can find an interaction which is proportional to
$\gamma^{5}$ matrix at the KK Higgs Yukawa couplings to KK top quarks, which
generates another type of contribution to the Higgs production through the
gluon fusion process.111111 This situation is similar to the famous fact that
$\pi^{0}\rightarrow 2\gamma$ decay is enhanced through chiral anomaly [70].
This type of contribution do not exist in ordinary 5D or 6D UED models.
## Appendix B UV cutoff based on RGE analysis in KK picture
In this section, we show the details of the renormalization group analysis,
based on the strategy stated in Section 3.2 that follows Ref. [48]. We note
that we do not need any regularization of the infinite KK sum, employed in
Ref. [48], in our bottom-up approach. A higher-dimensional gauge theory is
equivalent to the corresponding 4D theory with infinite tower of KK modes. It
suggest that all we have to do for deriving running effect of the 4D effective
gauge coupling at leading order is to count the d.o.f. of fields whose masses
are lower compared to a reference energy $\mu$. This important information is
encoded into the coefficient $\textsf{b}_{i}$. The formula for the running
coupling of 4D $SU(N)$ or $U(1)$ gauge theory is well-known as follows:
$\displaystyle\frac{d}{d\ln{\mu}}\alpha_{4i}^{-1}$
$\displaystyle=-\frac{1}{2\pi}\textsf{b}_{i},$ (81)
$\displaystyle\textsf{b}_{SU(N)}$
$\displaystyle=\left(-\frac{11}{3}{\sum_{\text{4D
vectors}}}C_{2}(\text{Adjoint})+\frac{2}{3}\sum_{\text{4D
Weyl}\atop\text{fermions}}C\left(r\right)+\frac{1}{3}\sum_{\text{4D
Higgs}}C\left(r\right)+\frac{1}{6}\sum_{\text{4D
adjoint}\atop\text{scalars}}C_{2}(\text{Adjoint})\right),$ (82)
$\displaystyle\textsf{b}_{U(1)}$
$\displaystyle=\left(\frac{2}{3}\sum_{\text{4D
Weyl}\atop\text{fermions}}Y_{f}^{2}+\frac{1}{3}\sum_{\text{4D
Higgs}}Y_{H}^{2}\right),$ (83)
where $\alpha_{4i}$ is a 4D gauge coupling strength, whose group is discerned
by $i$; $C\left(r\right)$ is defined by
$\operatorname{tr}\left[T^{a}_{r}T^{b}_{r}\right]=C\left(r\right)\delta^{ab}$
for each representation of $SU(N)$ group and the specific value is $1/2$ for
the fundamental representation; $C_{2}(\text{Adjoint})$ is the quadratic
Casimir operator for the adjoint representation, whose value is $N$;
$Y_{f}\,(Y_{H})$ shows $U(1)$ charge of 4D Weyl fermion (Higgs). The
contribution from KK vector bosons is included in the first term in Eq. (82).
We note that a 6D gauge field has two extra dimensional components and that a
4D adjoint scalar is left as a physical mode after the KK decomposition. (When
counting the number of adjoint scalar degrees of freedom in the running (81),
it is two rather than one in our treatment.)
We can obtain the solution of Eq. (81) by the integration of the both sides
over the region $[m_{Z},\mu]$. The essential point is that the coefficient
$\textsf{b}_{i}$ becomes altered when the reference energy $\mu$ cross a
threshold and the number of the effectively massless d.o.f. of fields under
the $\mu$ changes. Neglecting the KK mass splitting from the electroweak
symmetry breaking, the threshold correction from $l$-th KK particles arrises
simultaneously when $\mu$ exceeds the value of $l$-th KK mass. This
approximation simplifies the calculation to estimate the effect from $l$-th KK
particles all at once. The values of $\textsf{b}_{i}$ in the zero modes
($\textsf{b}_{i}^{\text{SM}}$) and $l$-th KK modes
($\textsf{b}_{i}^{\text{6D}}$) from the bulk SM matter contents are summarized
in Table 5.121212 Note that we do not employ the GUT normalization for the
$U(1)_{Y}$ coupling and the beta function.
gauge group | SM contribution ($\textsf{b}^{\text{SM}}_{i}$) | KK contribution ($\textsf{b}^{\text{6D}}_{i}$)
---|---|---
$SU(3)_{C}$ | $\displaystyle-7$ | $-2$
$SU(2)_{W}$ | $\displaystyle-{19}/{6}$ | $\displaystyle{3}/{2}$
$U(1)_{Y}$ | $\displaystyle{41/6}$ | $\displaystyle{27}/{2}$
Table 5: RGE coefficients in Eq. (70).
This way, the shape of $\alpha_{4i}(\mu)$ is determined as
$\displaystyle\alpha_{4i}^{{-1}}(\mu)=\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln{\mu\over
m_{Z}}-\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\sum_{l}\ln\frac{\mu}{m_{(l)}},$
(84)
where $m_{(l)}$ shows $l$-th KK mass and the upper bound of the summation is
$\displaystyle m_{(l)}\leq\mu.$ (85)
In the case of $S^{1}$ in five dimensions, where the spectrum of KK masses is
equally-spaced, the above calculation is executed with no difficulty. In 6D
cases, by contrast, the KK mass spectrum is not equally-spaced and we use the
following approximation.
$\square$ $T^{2}$-case:
In the $T^{2}$-case, the form of $(m,n)$-th KK mass is
$m_{(m,n)}=\sqrt{m^{2}+n^{2}}/R$ and eventually that of the first (lightest)
KK mass is $M_{\text{KK}}=1/R$. The exact form of the summation is as follows:
$\displaystyle\sum_{(m,n)}\ln{\mu\over\sqrt{m^{2}+n^{2}}M_{\text{KK}}}$
$\displaystyle\text{for}\quad 1\leq m^{2}+n^{2}\leq\left(\mu\over
M_{\text{KK}}\right)^{2}.$ (86)
We approximate the summation over $m$ and $n$ by integral over $r$ and
$\theta$ in two-dimensional polar coordinates:
$\displaystyle\sum_{(m,n)}\left(\ln{\mu\over
M_{\text{KK}}}-\frac{1}{2}\ln\left(m^{2}+n^{2}\right)\right)$
$\displaystyle\simeq\int_{1}^{\mu/M_{\text{KK}}}2\pi dr\cdot
r\left(\ln{\mu\over M_{\text{KK}}}-\ln{r}\right)$
$\displaystyle=\frac{\pi}{2}\left[\left(\frac{\mu}{M_{\text{KK}}}\right)^{2}-1{-2\ln{\mu\over
M_{\text{KK}}}}\right],$ (87)
which results in
$\displaystyle\alpha_{4i}^{{-1}}(\mu)\ {\simeq}\
\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln{\mu\over
m_{Z}}-\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\cdot\frac{\pi}{2}\left[\left(\frac{\mu}{M_{\text{KK}}}\right)^{2}-1{{}-2\ln{\mu\over
M_{\text{KK}}}}\right].$ (88)
$\square$ $S^{2}$-case:
In the $S^{2}$-case, the form of $(j,m)$-th KK mass is
$m_{(j,m)}=\sqrt{j(j+1)}/R$ and that of the first (lightest) KK mass is
$M_{\text{KK}}=\sqrt{2}/R$. There are $2j+1$ numbers of degenerated states in
each $j$-level. The exact form of the summation is as follows:
$\displaystyle\sum_{j=1}^{j_{\text{max}}}(2j+1)\ln\left(\mu\over\sqrt{\frac{j(j+1)}{2}}M_{\text{KK}}\right)$
$\displaystyle\text{for}\quad 2\leq j(j+1)\leq 2\left(\mu\over
M_{\text{KK}}\right)^{2}.$ (89)
When we use the approximation: $j(j+1)\simeq j^{2}$, $j_{\text{max}}$ and
$j_{\text{min}}$ are determined as
$\displaystyle j_{\text{max}}\simeq\sqrt{2}\frac{\mu}{m_{\text{KK}}},\quad
j_{\text{min}}\simeq\sqrt{2}.$ (90)
Let us approximate the summation over $j$ by an integral over $j$:
$\displaystyle\sum_{j=1}^{l_{\text{max}}}(2j+1)\ln\left(\mu\over\sqrt{\frac{j(j+1)}{2}}M_{\text{KK}}\right)$
$\displaystyle\simeq\int_{j_{\text{min}}}^{j_{\text{max}}}{dj\,}(2j)\ln\left(j_{\text{max}}\over
j\right)$ $\displaystyle=\left(\mu\over
M_{\text{KK}}\right)^{2}{-1}{-2\ln{\mu\over M_{\text{KK}}}}.$ (91)
Thereby we obtain the final form:
$\displaystyle\alpha_{4i}^{{-1}}(\mu)\ {\simeq}\
\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln\left(\mu\over
m_{Z}\right)-\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\cdot
1\left[\left(\mu\over M_{\text{KK}}\right)^{2}-1{{}-2\ln{\mu\over
M_{\text{KK}}}}\right].$ (92)
Combining Eq. (88) and Eq. (92), we get the final form (70). Neglecting the
logarithmic terms in Eq. (70), we obtain
$\displaystyle\alpha_{4i}^{-1}(\Lambda)\sim\alpha_{4i}^{-1}({m_{Z}})-\frac{C\textsf{b}^{\text{6D}}_{i}}{{2}\pi}\frac{\Lambda^{2}}{M_{\text{KK}}^{2}}.$
(93)
We note that the coefficient of the quadratic term for $T^{2}$ coincide with
that in Ref. [48] obtained from a different regularization scheme. Putting Eq.
(93) into the condition $\epsilon(\Lambda)\sim 1$, we get
$\displaystyle\Lambda^{2}\sim{{4\pi M_{\text{KK}}^{2}\over
C\left(N_{g}+2\textsf{b}_{i}^{\text{6D}}\right)\alpha_{4i}(m_{Z})},}$ (94)
where we have used $V_{2}=8\pi C/M_{\text{KK}}^{2}$. Concretely, we get
$\displaystyle\Lambda\lesssim\begin{cases}5.3\,M_{\text{KK}}&\text{for
$T^{2}$,}\\\ 6.6\,M_{\text{KK}}&\text{for $S^{2}$,}\end{cases}$ (95)
: from the $U(1)_{Y}$ cutoff.
In addition to this analysis, we also make consideration for the Landau poles
of the gauge interactions. If the value of the energy where a Landau pole
emerges is smaller than that of the cutoff which we have discussed before, we
should treat the position of the Landau pole:
$\alpha_{4i}^{-1}(\Lambda_{\text{Landau}})=0$, which is easily obtained with
leading order approximation as
$\displaystyle\Lambda_{\text{Landau}}^{2}\sim\frac{{2}\pi
M_{\text{KK}}^{2}}{C\textsf{b}^{\text{6D}}_{i}\alpha_{4i}({m_{Z}})},$ (96)
as a cutoff scale instead. The concrete forms of each value are show in Table
6.
| Type of geometry
---|---
types | $T^{2}$-based | $S^{2}$-based
$SU(3)_{C}$ cutoff | no cutoff | no cutoff
$SU(2)_{W}$ cutoff | $6.9\,M_{\text{KK}}$ | $8.6\,M_{\text{KK}}$
$U(1)_{Y}$ cutoff | $5.3\,M_{\text{KK}}$ | $6.6\,M_{\text{KK}}$
$SU(3)_{C}$ Landau pole | no cutoff | no cutoff
$SU(2)_{W}$ Landau pole | $8.9\,M_{\text{KK}}$ | $11\,M_{\text{KK}}$
$U(1)_{Y}$ Landau pole | $5.4\,M_{\text{KK}}$ | $6.8\,M_{\text{KK}}$
Table 6: The values of the cutoff scales and the positions of the Landau poles
in $T^{2}$ and $S^{2}$ cases.
In the analysis above, we have taken values of $N_{g}$ as $3$, $2$ and $1$ in
each case of $SU(3)_{C}$, $SU(2)_{W}$ and $U(1)_{Y}$, respectively, and have
employed the values
$\left\\{\begin{array}[]{rcl}\alpha_{U(1)_{Y}}^{{-1}}(m_{Z})&=&97.9,\\\
\alpha_{SU(2)_{W}}^{{-1}}(m_{Z})&=&29.4,\\\
\alpha_{SU(3)_{C}}^{{-1}}(m_{Z})&=&8.44,\end{array}\right.$ (97)
at $m_{Z}=91.1\,\text{GeV}$. We do not consider a TeV-scale gauge coupling
unification condition as a UV cutoff in this paper.
In the both $T^{2}$ and $S^{2}$ cases, the most stringent bounds come from the
$U(1)_{Y}$ cutoff scales, which restrict the effective range of the
perturbation the most severely. It is natural that the scale emerging the
$U(1)_{Y}$ Landau pole is near the upper limit of the perturbativity but a
little bit higher.
## Appendix C Breit-Wignar formula
We review how the Breit-Wigner formula emerges from the resummation of the
one-particle irreducible (1PI) Higgs two-point function, in order to be
careful of possible systematic errors when the width becomes broad. In this
paper, we assume that the Higgs mass is larger than twice the $W$ KK mass so
that it is sufficient to limit ourselves to the SM case when discussing the
Higgs total width. It is straightforward to extend the result for the UED when
one wants to take the KK loops into account.
In the SM, the Higgs production cross section via the gluon fusion process is
obtained as
$\displaystyle\hat{\sigma}_{gg\to H}$ $\displaystyle={\pi^{2}\over
8M_{H}}\,\Gamma_{H\to gg}(M_{H})\,\delta(\hat{s}-M_{H}^{2}),$ (98)
where
$\displaystyle\Gamma_{H\to gg}(M_{H})$ $\displaystyle={\alpha_{s}^{2}\over
8\pi^{3}}{M_{H}^{3}\over v_{\text{EW}}^{2}}\,\left|I\\!\left(m_{t}^{2}\over
M_{H}^{2}\right)\right|^{2}.$ (99)
Then we get
$\displaystyle\hat{\sigma}_{gg\to H\to ZZ}$
$\displaystyle\simeq\hat{\sigma}_{gg\to H}\,\operatorname{Br}_{H\to
ZZ}(M_{H})$ $\displaystyle={\pi^{2}\over 8M_{H}}\Gamma_{H\to
gg}(M_{H})\operatorname{Br}_{H\to ZZ}(M_{H})\,\delta(\hat{s}-M_{H}^{2}),$
(100)
where
$\displaystyle\operatorname{Br}_{H\to ZZ}(M_{H})$ $\displaystyle={\Gamma_{H\to
ZZ}(M_{H})\over\Gamma_{H}(M_{H})}={{M_{H}^{3}\over 32\pi
v_{\text{EW}}^{2}}\left[1-{4m_{Z}^{2}\over M_{H}^{2}}+{12m_{Z}^{4}\over
M_{H}^{4}}\right]\sqrt{1-{4m_{Z}^{2}\over M_{H}^{2}}}\over\Gamma_{H}(M_{H})}.$
(101)
The expression (98) is obtained in the limit of the vanishing decay width
$\Gamma_{H}\to 0$. We may introduce a narrow width in Eq. (100) by the Breit-
Wigner type replacement
$\displaystyle\delta(\hat{s}-M_{H}^{2})$
$\displaystyle\to{1\over\pi}{\Delta\over\left(\hat{s}-M_{H}^{2}\right)^{2}+\Delta^{2}}$
(102)
to get
$\displaystyle\hat{\sigma}_{gg\to H\to ZZ}$ $\displaystyle\simeq{\pi\over
8M_{H}}{\Gamma_{H\to gg}(M_{H})\,\Gamma_{H\to
ZZ}(M_{H})\over\Gamma_{H}(M_{H})}{\Delta\over\left(\hat{s}-M_{H}^{2}\right)^{2}+\Delta^{2}}.$
(103)
When we want to reproduce the delta function, $\Delta$ in Eq. (102) cannot
depend on $\hat{s}$, otherwise we cannot get the correct normalization: $\int
d\hat{s}\,\delta(\hat{s}-M_{H}^{2})=1$. One should then perform the
replacement of the delta function (102) as
$\displaystyle\Delta\to M_{H}\Gamma_{H}(M_{H})$ (104)
in Eqs. (98) and (100). In literature, see e.g. [32], $\Delta$ in Eq. (103) is
sometimes replaced as
$\displaystyle\Delta$ $\displaystyle\to{\hat{s}\over M_{H}}\Gamma_{H}(M_{H}).$
(105)
Instead of the truncation (100), we have already obtained the full $gg\to H\to
ZZ$ cross section (50) by the naive Breit-Wignar type replacement $\Delta\to
m_{H}\Gamma_{H}$ in and only in the denominator of the Higgs propagator
$\displaystyle{i\over Q^{2}-M_{H}^{2}+i\Delta}.$ (106)
Hereafter, let us see that this treatment gives sufficiently good
approximation to the full result (126).
### C.1 Resummed propagator
First let us review how the resummed propagator is obtained in the SM. We
write the bare Higgs mass and field in terms of the renormalized ones and the
counter terms
$\displaystyle M_{B}^{2}$ $\displaystyle=M_{H}^{2}+\delta M_{H}^{2},$
$\displaystyle H_{B}$ $\displaystyle=\sqrt{Z_{H}}H,$ $\displaystyle Z_{H}$
$\displaystyle=1+\delta Z_{H}.$ (107)
The resummed bare propagator reads
$\displaystyle D_{B}$ $\displaystyle={i\over Q^{2}-M_{B}^{2}+\Pi_{H}(Q^{2})},$
(108)
where $M_{B}$ and $\Pi_{H}(Q^{2})$ are the bare mass and the 1PI two-point
function, respectively, both of which contain ultraviolet (UV) divergences,
and $Q^{2}:=-q^{2}$ for a Higgs four-momentum $q$. The renormalized propagator
then becomes
$\displaystyle D_{R}$ $\displaystyle={D_{B}\over Z_{H}}={i\over
Q^{2}-M_{H}^{2}+\hat{\Sigma}_{H}(Q^{2})},$ (109)
where
$\displaystyle\hat{\Sigma}_{{H}}(Q^{2})$
$\displaystyle:=\Sigma_{H}(Q^{2})-Z_{H}\,\delta M_{H}^{2}+\delta
Z_{H}\left(Q^{2}-M_{H}^{2}\right)$ (110)
is the renormalized (finite) 1PI two-point function, with
$\Sigma_{H}(Q^{2}):=Z_{H}\,\Pi_{H}(Q^{2})$ being the 1PI two-point function
that is given in terms of the renormalized fields but still contains UV
divergences. Note that it is sufficient to consider the case $Q^{2}>0$ for our
purpose since we have only $s$-channel Higgs propagator, though $Q^{2}$ can be
negative when the Higgs is virtual, e.g. when exchanged in $t$-channel. The
Higgs two point function in the SM is given by [71]
$\displaystyle\Sigma_{H}(Q^{2})$ $\displaystyle=-{1\over
16\pi^{2}v^{2}}\Bigg{\\{}{6m_{t}^{2}}\left[2A_{0}(m_{t}^{2})+(4m_{t}^{2}-Q^{2})B_{0}(Q^{2},m_{t}^{2})\right]$
$\displaystyle\phantom{=-{1\over
16\pi^{2}v^{2}}\Bigg{\\{}}-2\left[\left(6m_{W}^{4}-2Q^{2}m_{W}^{2}+{M_{H}^{4}\over
2}\right)B_{0}(Q^{2},m_{W}^{2})+\left(3m_{W}^{2}+{M_{H}^{2}\over
2}\right)A_{0}(m_{W}^{2})-6m_{W}^{4}\right]$ $\displaystyle\phantom{=-{1\over
16\pi^{2}v^{2}}\Bigg{\\{}}-\left[\left(6m_{Z}^{4}-2Q^{2}m_{Z}^{2}+{M_{H}^{4}\over
2}\right)B_{0}(Q^{2},m_{Z}^{2})+\left(3m_{Z}^{2}+{M_{H}^{2}\over
2}\right)A_{0}(m_{Z}^{2})-6m_{Z}^{4}\right]$ $\displaystyle\phantom{=-{1\over
16\pi^{2}v^{2}}\Bigg{\\{}}-{3\over
2}\left[{3M_{H}^{4}}B_{0}(Q^{2},M_{H}^{2})+{M_{H}^{2}}A_{0}(M_{H}^{2})\right]\Bigg{\\}},$
(111)
where the loop functions are
$\displaystyle A_{0}(m^{2})$
$\displaystyle=m^{2}\left(\Delta-\ln{m^{2}\over\mu^{2}}+1\right),$ (112)
with $\Delta:={2\over\epsilon}-\gamma+\ln 4\pi$ for $D=4-\epsilon$, and
$\displaystyle B_{0}(Q^{2},m^{2})$
$\displaystyle=\Delta-\int_{0}^{1}dx\ln{Q^{2}x(x-1)+m^{2}-i\varepsilon\over\mu^{2}}$
$\displaystyle=\Delta-\ln{m^{2}\over\mu^{2}}+\mathcal{I}\\!\left(4m^{2}\over
Q^{2}\right),$ (113)
where
$\displaystyle\mathcal{I}(\kappa)$
$\displaystyle:=\int_{0}^{1}dx{x\left(2x-1\right)\over\left(x-{1\over
2}\right)^{2}-{1-\kappa\over 4}-i\varepsilon}=\int_{0}^{1}dx{2x^{2}-x\over
x^{2}-x+{\kappa\over 4}-i\varepsilon}.$ (114)
For concreteness we write down
$\displaystyle\mathcal{I}(\kappa)$
$\displaystyle=\begin{cases}2\left(1-\sqrt{\kappa-1}\arctan\left(1\over\sqrt{\kappa-1}\right)\right)&\text{($\kappa\geq
1$ or $\operatorname{Im}\kappa\neq 0$),}\\\
2-\sqrt{1-\kappa}\,\ln\left(1+\sqrt{1-\kappa}\over
1-\sqrt{1-\kappa}\right)+i\pi\sqrt{1-\kappa}&(0<\kappa\leq 1),\end{cases}$
(115) $\displaystyle\mathcal{I}^{\prime}(\kappa)$
$\displaystyle=\begin{cases}{1\over\kappa}-{\arctan\\!\left(1\over\sqrt{\kappa-1}\right)\over\sqrt{\kappa-1}}&\text{($\kappa>1$
or $\operatorname{Im}\kappa\neq 0$),}\\\ {1\over\kappa}+{1\over
2\sqrt{1-\kappa}}\ln\left(1+\sqrt{1-\kappa}\over
1-\sqrt{1-\kappa}\right)-{i\pi\over
2\sqrt{1-\kappa}}&(0<\kappa<1).\end{cases}$ (116)
Note that when $\operatorname{Im}\kappa=0$,
$\displaystyle\operatorname{Im}\mathcal{I}(\kappa)$
$\displaystyle=\pi\sqrt{1-\kappa}\,\theta(1-\kappa),$
$\displaystyle\operatorname{Im}\mathcal{I}^{\prime}(\kappa)$
$\displaystyle=-{\pi\over 2\sqrt{1-\kappa}}\,\theta(1-\kappa).$ (117)
In particular,
$\displaystyle\mathcal{I}(4)$ $\displaystyle={6-\sqrt{3}\pi\over 3}\simeq
0.18,$ $\displaystyle\mathcal{I}^{\prime}(4)$
$\displaystyle=-{2\sqrt{3}\pi-9\over 36}\simeq-0.052,$ (118)
and for small and large $\kappa$,
$\displaystyle\mathcal{I}(\kappa)$ $\displaystyle=2+\ln{\kappa\over
4}+i\pi+O\\!\left(\kappa\ln\kappa\right)$ $\displaystyle(\kappa\ll 1),$ (119)
$\displaystyle\mathcal{I}(\kappa)$ $\displaystyle={2\over
3\kappa}+O\\!\left(\kappa^{-2}\right)$ $\displaystyle(\kappa\gg 1).$ (120)
### C.2 On-shell scheme renormalization
In the on-shell scheme, we put the following renormalization conditions:
$\displaystyle\left.\operatorname{Re}\hat{\Sigma}_{H}(Q^{2})\right|_{Q^{2}=M_{H}^{2}}$
$\displaystyle=0,$ $\displaystyle\left.\operatorname{Re}{\partial\over\partial
Q^{2}}\hat{\Sigma}_{H}(Q^{2})\right|_{Q^{2}=M_{H}^{2}}$ $\displaystyle=0,$
(121)
which gives
$\displaystyle\hat{\Sigma}_{H}(Q^{2})$
$\displaystyle=\Sigma_{H}(Q^{2})-\operatorname{Re}\Sigma_{H}(M_{H}^{2})-\operatorname{Re}\Sigma_{H}^{\prime}(M_{H}^{2})\left(Q^{2}-M_{H}^{2}\right)$
$\displaystyle=-{1\over
16\pi^{2}v^{2}}\Bigg{(}6m_{t}^{2}\Bigg{\\{}\left(4m_{t}^{2}-Q^{2}\right)\left[\mathcal{I}\\!\left(4m_{t}^{2}\over
Q^{2}\right)-\operatorname{Re}\mathcal{I}\\!\left(4m_{t}^{2}\over
M_{H}^{2}\right)\right]$ $\displaystyle\phantom{=-{1\over
16\pi^{2}v^{2}}\Bigg{(}6m_{t}^{2}\Bigg{\\{}}+\left(4m_{t}^{2}-M_{H}^{2}\right){4m_{t}^{2}\over
M_{H}^{4}}\operatorname{Re}\mathcal{I}^{\prime}\\!\left(4m_{t}^{2}\over
M_{H}^{2}\right)\left(Q^{2}-M_{H}^{2}\right)\Bigg{\\}}$
$\displaystyle\phantom{=-{1\over 16\pi
v^{2}}\Bigg{(}}-2\Bigg{\\{}\left(6m_{W}^{4}-2Q^{2}m_{W}^{2}+{M_{H}^{4}\over
2}\right)\left[\mathcal{I}\\!\left(4m_{W}^{2}\over
Q^{2}\right)-\operatorname{Re}\mathcal{I}\\!\left(4m_{W}^{2}\over
M_{H}^{2}\right)\right]$ $\displaystyle\phantom{=-{1\over 16\pi
v^{2}}\Bigg{(}-2\Bigg{\\{}}+\left(6m_{W}^{4}-2M_{H}^{2}m_{W}^{2}+{M_{H}^{4}\over
2}\right){4m_{W}^{2}\over
M_{H}^{4}}\operatorname{Re}\mathcal{I}^{\prime}\\!\left(4m_{W}^{2}\over
M_{H}^{2}\right)\left(Q^{2}-M_{H}^{2}\right)\Bigg{\\}}$
$\displaystyle\phantom{=-{1\over 16\pi
v^{2}}\Bigg{(}}-\Bigg{\\{}\left(6m_{Z}^{4}-2Q^{2}m_{Z}^{2}+{M_{H}^{4}\over
2}\right)\left[\mathcal{I}\\!\left(4m_{Z}^{2}\over
Q^{2}\right)-\operatorname{Re}\mathcal{I}\\!\left(4m_{Z}^{2}\over
M_{H}^{2}\right)\right]$ $\displaystyle\phantom{=-{1\over 16\pi
v^{2}}\Bigg{(}-\Bigg{\\{}}+\left(6m_{Z}^{4}-2M_{H}^{2}m_{Z}^{2}+{M_{H}^{4}\over
2}\right){4m_{Z}^{2}\over
M_{H}^{4}}\operatorname{Re}\mathcal{I}^{\prime}\\!\left(4m_{Z}^{2}\over
M_{H}^{2}\right)\left(Q^{2}-M_{H}^{2}\right)\Bigg{\\}}$
$\displaystyle\phantom{=-{1\over 16\pi v^{2}}\Bigg{(}}-{9M_{H}^{4}\over
2}\Bigg{\\{}\left[\mathcal{I}\\!\left(4M_{H}^{2}\over
Q^{2}\right)-\mathcal{I}\\!\left(4\right)\right]+{4\over
M_{H}^{2}}\mathcal{I}^{\prime}\\!\left(4\right)\left(Q^{2}-M_{H}^{2}\right)\Bigg{\\}}\Bigg{)},$
(122)
and hence
$\displaystyle\operatorname{Im}\hat{\Sigma}_{H}(Q^{2})$
$\displaystyle=-{1\over
16\pi^{2}v^{2}}\Bigg{\\{}-6m_{t}^{2}Q^{2}\pi\left[1-{4m_{t}^{2}\over
Q^{2}}\right]^{3/2}\,\theta\\!\left(1-{4m_{t}^{2}\over Q^{2}}\right)$
$\displaystyle\phantom{=-{1\over 16\pi
v^{2}}\Bigg{(}}-M_{H}^{4}\left({12m_{W}^{4}\over
M_{H}^{4}}-{4Q^{2}m_{W}^{2}\over
M_{H}^{4}}+1\right)\pi\sqrt{1-{4m_{W}^{2}\over
Q^{2}}}\,\theta\\!\left(1-{4m_{W}^{2}\over Q^{2}}\right)$
$\displaystyle\phantom{=-{1\over 16\pi v^{2}}\Bigg{(}}-{M_{H}^{4}\over
2}\left({12m_{Z}^{4}\over M_{H}^{4}}-{4Q^{2}m_{Z}^{2}\over
M_{H}^{4}}+1\right)\pi\sqrt{1-{4m_{Z}^{2}\over
Q^{2}}}\,\theta\\!\left(1-{4m_{Z}^{2}\over Q^{2}}\right)$
$\displaystyle\phantom{=-{1\over 16\pi v^{2}}\Bigg{(}}-{9M_{H}^{4}\over
2}\pi\sqrt{1-{4M_{H}^{2}\over Q^{2}}}\,\theta\\!\left(1-{4M_{H}^{2}\over
Q^{2}}\right)\Bigg{\\}}.$ (123)
We can compare this result with the tree-level Higgs decay width
$\displaystyle\Gamma_{H}^{\text{tree}}(M_{H})$ $\displaystyle={M_{H}^{3}\over
16\pi v_{\text{EW}}^{2}}\left[1-{4m_{W}^{2}\over M_{H}^{2}}+{12m_{W}^{4}\over
M_{H}^{4}}\right]\sqrt{1-{4m_{W}^{2}\over
M_{H}^{2}}}\,\theta\\!\left(M_{H}-2m_{W}\right)$
$\displaystyle\quad+{M_{H}^{3}\over 32\pi
v_{\text{EW}}^{2}}\left[1-{4m_{Z}^{2}\over M_{H}^{2}}+{12m_{Z}^{4}\over
M_{H}^{4}}\right]\sqrt{1-{4m_{Z}^{2}\over
M_{H}^{2}}}\,\theta\\!\left(M_{H}-2m_{Z}\right)$
$\displaystyle\quad+{3M_{H}m_{t}^{2}\over 8\pi
v_{\text{EW}}^{2}}\left[1-{4m_{t}^{2}\over
M_{H}^{2}}\right]^{3/2}\theta\\!\left(M_{H}-2m_{t}\right).$ (124)
We see from Eq. (123) that the leading term for the $Q\sim M_{H}\gg 2m_{t}$
limit is not proportional to $Q^{2}$ for the $W$ and $Z$ contributions and
therefore the replacement (105), namely
$\displaystyle D_{R}\to{i\over Q^{2}-M_{H}^{2}+i{Q^{2}\over M_{H}}\Gamma_{H}}$
(125)
does not give a good fit.
When computing the full cross section for the process $gg\to H\to ZZ$, we may
employ the resummed propagator (109). Neglecting the contributions from box
diagrams, we get the cross section
$\displaystyle\hat{\sigma}_{gg\to H\to ZZ}$
$\displaystyle={\alpha_{s}^{2}\over 256\pi^{2}}\left(m_{Z}\over
v_{\text{EW}}\right)^{4}\left[1+{\left(\hat{s}-2m_{Z}^{2}\right)^{2}\over
8m_{Z}^{4}}\right]\sqrt{1-{4m_{Z}^{2}\over\hat{s}}}\,\left|I\\!\left(m_{t}^{2}\over\hat{s}\right)\right|^{2}$
$\displaystyle\quad\times{1\over\pi}{\hat{s}\over\left(\hat{s}-m_{H}^{2}+\operatorname{Re}\hat{\Sigma}_{H}\\!\left(\hat{s}\right)\right)^{2}+\left(\operatorname{Im}\hat{\Sigma}_{H}(\hat{s})\right)^{2}}.$
(126)
In Fig. 7, we show the results for various replacement. We see that any
replacement suffices when Higgs mass is not very large,
$M_{H}=300\,\text{GeV}$ (left). (This is the case for the Higgs mass above the
top threshold $m_{H}\gtrsim 2m_{t}$ too.) For the large Higgs mass (right),
the decay width becomes larger and we see that our approximation (50) with
Breit-Wignar type replacement in the denominator $\Delta\to M_{H}\Gamma_{H}$
gives a good fit to the cross section with full two-point function (126).
Figure 7: Parton level cross section for the process $gg\to H\to ZZ$ with
$m_{H}=300\,\text{GeV}$ (left) and $700\,\text{GeV}$ (right). Inclusion of
full two point function (126) (blue solid line), the full one (50) with
$\Delta\to M_{H}\Gamma_{H}$ (red dashed), and the truncated one (103) with
$\Delta\to\hat{s}\Gamma_{H}/M_{H}$ (yellow dot-dashed) are shown. In the right
figure, the full one (50) with $\Delta\to\hat{s}\Gamma_{H}/M_{H}$ (thin dot-
dashed) and the truncated one (103) with $\Delta\to M_{H}\Gamma_{H}$ (thin
dotted) are also drawn.
Finally, just for comparison with Eq. (103), let us present the resummed cross
section (126) in a rewritten form
$\displaystyle\hat{\sigma}_{gg\to H\to ZZ}$ $\displaystyle={\pi\over
8M_{H}}\,{\alpha_{s}^{2}M_{H}^{3}\over
8\pi^{3}v_{\text{EW}}^{2}}\left|{\hat{s}\over
M_{H}^{2}}I\\!\left(m_{t}^{2}\over\hat{s}\right)\right|^{2}$
$\displaystyle\quad\times{{\hat{s}\over
M_{H}}\Gamma_{H}(M_{H})\over\left(\hat{s}-M_{H}^{2}+\operatorname{Re}\Pi_{H}(\hat{s})\right)^{2}+\left(\operatorname{Im}\Pi_{H}(\hat{s})\right)^{2}}$
$\displaystyle\quad\times{{M_{H}^{3}\over 32\pi
v_{\text{EW}}^{2}}\left[1-{4m_{Z}^{2}\over\hat{s}}+{12m_{Z}^{4}\over\hat{s}^{2}}\right]\sqrt{1-{4m_{Z}^{2}\over\hat{s}}}\over\Gamma_{H}(M_{H})}.$
(127)
### C.3 Pole scheme renormalization
Figure 8: Ratio of the pole mass $m_{H}$ to the on-shell one $M_{H}$ as a
function of the on-shell mass (left); same for the ratio of the pole-scheme
width $\Gamma_{H}$ to the tree-level one $\Gamma_{H}^{\text{tree}}$ (right).
Let us review the pole scheme renormalization [72]. Instead of the on-shell
condition (121), the pole scheme renormalization condition fixes the pole of
the propagator (108) at $\overline{Q^{2}}$ which we parametrize by two real
constants $m_{H}$ and $\Gamma_{H}$ as:
$\overline{Q^{2}}=m_{H}^{2}-im_{H}\Gamma_{H}$, namely,
$\displaystyle\overline{Q^{2}}-M_{B}^{2}+\Pi_{H}(\overline{Q^{2}})$
$\displaystyle=0.$ (128)
As the pole position of the bare propagator (108) is the same as that of the
renormalized one (109), we see that the on-shell renormalized two-point
function satisfies
$\displaystyle\overline{Q^{2}}-M_{H}^{2}+\hat{\Sigma}_{H}(\overline{Q^{2}})=0,$
(129)
that is,
$\displaystyle M_{H}^{2}$
$\displaystyle=m_{H}^{2}+\operatorname{Re}\hat{\Sigma}_{H}(\overline{Q^{2}}),$
$\displaystyle m_{H}\Gamma_{H}$
$\displaystyle=\operatorname{Im}\hat{\Sigma}_{H}(\overline{Q^{2}}).$ (130)
We see that the real and imaginary parts determine the pole scheme mass
$m_{H}$ and decay rate $\Gamma_{H}$ as functions of the on-shell scheme mass
$M_{H}$. Note that the renormalized two-point function $\hat{\Sigma}_{H}$ has
implicit dependence on the on-shell mass $M_{H}$.
In Fig 8, we plot the ratio of the pole to on-shell mass $m_{H}/M_{H}$ (left)
and the pole-scheme width $\Gamma_{H}$ to the tree-level width (124) (right)
as functions of the on-shell mass $M_{H}$, computed within the SM.
It is known that the on-shell scheme mass is gauge dependent at the NNLO, see
references in [72]. In contrast, the pole position of the amplitude is a gauge
independent physical notion. Therefore, in principle we should utilize the
pole scheme mass and width. However, the on-shell mass and the pole mass are
identical at consistent 1-loop order (which is the highest order considered
here), and they start differing only at 2-loop order and above. We see from
Fig. 8 that the both schemes agree within 1% accuracy in our approximated
treatment neglecting non-resonant box contributions, which are the same order
as $O(\alpha)$-corrections to the resonant ones.131313The difference in Fig. 8
stems from the fact that a complex value is inserted for $\overline{Q^{2}}$ in
Eq. (130), i.e. $\overline{Q^{2}}=m_{H}^{2}-im_{H}\Gamma_{H}$. However,
$\Gamma_{H}$ is formally a higher-order term, since it corresponds to the
imaginary part of the self-energy, which first occurs at one-loop order.
Therefore, when inserted into the one-loop self-energies in Eq. (130), this
leads to a contribution that is formally at the NNLO level, which causes the
small numerical difference between the two schemes in Fig. 8. We safely
utilize the on-shell mass and the tree-level width (124) even though the Higgs
decay width becomes as large as 180 GeV when $M_{H}=700\,\text{GeV}$.
Note that our treatment to include the decay width in the numerical
calculation (50) with (54) corresponds to the (gauge invariant) complex-mass
scheme, see e.g. Refs. [73, 74], where the real Higgs mass is replaced by a
complex value everywhere in the amplitude, which in our case leads to a
replacement of the Higgs mass in its propagator only.
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|
arxiv-papers
| 2011-08-08T17:59:32 |
2024-09-04T02:49:21.425845
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenji Nishiwaki, Kin-ya Oda, Naoya Okuda and Ryoutaro Watanabe",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1108.1765"
}
|
1108.1981
|
# Weak lensing measurement of galaxy clusters in the CFHTLS-Wide survey
HuanYuan Shan$\dagger$1$\dagger$1affiliationmark: 22affiliation: Laboratoire
d’Astrophysique de Marseille, CNRS-Université de Provence, 38 rue Frédéric
Joliot-Curie, 13 388 Marseille Cedex 13, France 33affiliation: Centre de
Physique des Particules de Marseille, CNRS/IN2P3-Luminy and Université de la
Méditerranée, Case 907, F-13288 Marseille Cedex 9, France , Jean-Paul
Kneib22affiliation: Laboratoire d’Astrophysique de Marseille, CNRS-Université
de Provence, 38 rue Frédéric Joliot-Curie, 13 388 Marseille Cedex 13, France ,
Charling Tao11affiliation: Department of Physics and Tsinghua Center for
Astrophysics, Tsinghua University, Beijing, 100084, China 33affiliation:
Centre de Physique des Particules de Marseille, CNRS/IN2P3-Luminy and
Université de la Méditerranée, Case 907, F-13288 Marseille Cedex 9, France ,
Zuhui Fan44affiliation: Department of Astronomy, Peking University, Beijing,
100871, China , Mathilde Jauzac22affiliation: Laboratoire d’Astrophysique de
Marseille, CNRS-Université de Provence, 38 rue Frédéric Joliot-Curie, 13 388
Marseille Cedex 13, France , Marceau Limousin22affiliation: Laboratoire
d’Astrophysique de Marseille, CNRS-Université de Provence, 38 rue Frédéric
Joliot-Curie, 13 388 Marseille Cedex 13, France 5 5affiliationmark: , Richard
Massey66affiliation: Institute for Astronomy, Royal Observatory, Blackford
Hill, Edinburgh EH9 3HJ, UK , Jason Rhodes77affiliation: California Institute
of Technology, MC 350-17, 1200 East California Boulevard, Pasadena, CA 91125,
USA 88affiliation: Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91109, USA , Karun Thanjavur99affiliation: Canada
France Hawaii Telescope, 65-1238 Mamalahoa Hwy, Kamuela, HI 96743, USA
1010affiliation: Department of Physics & Astronomy, University of Victoria,
Victoria, BC, V8P 1A1, Canada 1111affiliation: National Research Council of
Canada, Herzberg Institute of Astrophysics, 5071 West Saanich Road, Victoria,
BC, V9E 2E7, Canada and Henry J. McCracken1212affiliation: Institude
d’Astrophysique de Paris, UMR 7095, 98 bis Boulevard Arago, 75014 Paris,
France
###### Abstract
We present the first weak gravitational lensing analysis of the completed
Canada-France-Hawaii Telescope Legacy Survey (CFHTLS). We study the $64~{}{\rm
deg}^{2}$ W1 field, the largest of the CFHTLS-Wide survey fields, and present
the largest contiguous weak lensing convergence “mass map” yet made.
2.66 million galaxy shapes are measured, using a Kaiser, Squires and
Broadhurst (KSB) pipeline verified against high-resolution Hubble Space
Telescope imaging that covers part of the CFHTLS. Our $i^{\prime}$-band
measurements are also consistent with an analysis of independent
$r^{\prime}$-band imaging. The reconstructed lensing convergence map contains
$301$ peaks with signal-to-noise ratio $\nu>3.5$, consistent with predictions
of a $\Lambda$CDM model. Of these peaks, $126$ lie within $3\arcmin.0$ of a
brightest central galaxy identified from multicolor optical imaging in an
independent, red sequence survey. We also identify seven counterparts for
massive clusters previously seen in X-ray emission within $6~{}\rm deg^{2}$
XMM-LSS survey.
With photometric redshift estimates for the source galaxies, we use a
tomographic lensing method to fit the redshift and mass of each convergence
peak. Matching these to the optical observations, we confirm $85$
groups/clusters with $\chi^{2}_{\mathrm{reduced}}<3.0$, at a mean redshift
$\langle z_{c}\rangle=0.36$ and velocity dispersion
$\langle\sigma_{c}\rangle=658.8~{}\rm km~{}s^{-1}$. Future surveys, such as
DES, LSST, KDUST and EUCLID, will be able to apply these techniques to map
clusters in much larger volumes and thus tightly constrain cosmological
models.
###### Subject headings:
cosmology: observations, - galaxies: clusters: general, -gravitational
lensing: weak, -X-rays: galaxies: clusters
55affiliationtext: Dark Cosmology Centre, Niels Bohr Institute, University of
Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen,
Denmark$\dagger$$\dagger$affiliationtext: Email address: shanhuany@gmail.com
## 1\. Introduction
Clusters of galaxies are the largest gravitationally bound structures in the
universe. The number and mass of the biggest clusters are highly sensitive to
cosmological parameters including the mass density $\Omega_{m}$, the
normalization of the mass power spectrum $\sigma_{8}$ (e.g., Press & Schechter
1974; Frenk et al. 1990; Eke et al. 1996; Sheth & Tormen 1999), and the
dynamics of dark energy (e.g., Bartelmann et al. 2006; Francis et al. 2009;
Grossi & Springel 2009). Understanding the properties of clusters is vital to
test theories of structure formation and to map the distribution of cosmic
matter on scales of $\sim$1–10 Mpc.
Theoretical predictions of structure formation deal directly with the total
mass of clusters; measurements are restricted to indirect proxies that can be
observed. Contaminating the translation between theory and observation are
large uncertainties in the interpretation of galaxy richness, X-ray
luminosity/temperature and the Sunyaev-Zeldovich decrement (e.g. Bode et al.
2007; Leauthaud et al. 2010). Weak gravitational lensing, the coherent
distortion of galaxies behind a cluster, can potentially provide direct
measurements of the total mass regardless of its baryon content, dynamical
state, and star formation history.
By measuring the shear (coherent elongation) of many background galaxies, we
can reconstruct the two-dimensional (2D) weak lensing convergence map, which
is proportional to density projected along each line of sight. Peaks in the
convergence map with high signal-to-noise ratio $\nu$ generally correspond to
massive clusters (Hamana et al. 2004; Haiman et al. 2004). Since the three-
dimensional (3D) shear signal should increase behind those clusters in a
predictable way that depends upon only the lens-source geometry, we can also
use photometric redshift estimates of the background galaxies (from multi-band
imaging) to measure the redshift and mass of each foreground cluster (Wittman
et al. 2001, 2003; Hennawi & Spergel 2005; Gavazzi & Soucail 2007).
Systematic weak lensing cluster searches have only recently become
practicable. Miyazaki et al. (2002) used Subaru/Suprime-Cam in excellent
seeing conditions to find an excess of $4.9\pm 2.3$ convergence peaks with
$\nu>5$ in an area of $2.1~{}\rm deg^{2}$. Dahle et al. (2003) and Schirmer et
al. (2003) each identified several shear-selected clusters with redshifts
$z\sim 0.5$ determined from two-color photometry. Hetterscheidt et al. (2005)
reported the detection of five cluster candidates over a set of $50$
disconnected Very Large Telescope/FORS images covering an effective area of
$0.64~{}\rm deg^{2}$, while Wittman et al. (2006) found eight detections in
the first $8.6~{}\rm deg^{2}$ of the Blanco Deep Lens Survey. Gavazzi &
Soucail (2007) presented a weak lensing analysis of initial Canada-France-
Hawaii Telescope Legacy Survey (CFHTLS) Deep data covering $4~{}\rm deg^{2}$.
They demonstrated that the image quality at CFHT is easily sufficient for
cluster finding. Miyazaki et al. (2007) presented the first large sample of
weak lensing-selected clusters in the Subaru weak lensing survey, with $100$
significant convergence peaks in a $16.7~{}\rm deg^{2}$ effective survey area.
Hamana et al. (2009) reported results from a multi-object spectroscopic
campaign to target $36$ of these cluster candidates, of which $28$ were
confirmed (and $6$ were projections along a line of sight of multiple, small
groups).
The main astrophysical systematic effect afflicting weak lensing cluster
surveys is the projection of large-scale structure along the line of sight.
Random noise is also added due to the finite density of resolved source
galaxies and the scatter of their intrinsic shapes. Numerical studies (White
et al. 2002; Hamana et al. 2004) show that these contaminants significantly
reduce the purity of cluster detection. To improve our analysis, we shall
combine our weak lensing results with multi-wavelength imaging. Simultaneous
detection of a weak lensing signature plus an overdensity of galaxies with a
single red sequence provides an unambiguous cluster identification.
Furthermore, 3D lensing tomography using photometric redshifts from the multi-
wavelength data can remove the other potential hurdles of: lensing signal
dilution by cluster member galaxies, and identifying the redshift of weak
lensing peaks when no corresponding galaxy overdensity is apparent.
Here we present a weak gravitational lensing analysis of the $64~{}\rm
deg^{2}$ CFHTLS-Wide W1 field, which is sufficiently large to contain several
hundred galaxy clusters. Compared to the analysis of the CFHTLS-Deep survey by
Gavazzi & Soucail (2007), our shallower CFHTLS-Wide imaging (and lower source
galaxy density) will favor the detection of higher mass, nearby clusters. The
huge increase in survey area over any previous survey is expected to yield
many more systems overall. In this paper, we shall primarily study the
properties of the detected clusters, rather than the cosmology in which they
are embedded. For this purpose, we adopt a default cosmological model with
$\Omega_{m}=0.27$, $\Omega_{\Lambda}=0.73$, $\sigma_{8}=0.809$,
$H_{0}=100~{}h~{}\mathrm{km}~{}{\mathrm{s}}^{-1}~{}\mathrm{Mpc}^{-1}$, and
$h=0.71$.
This paper is organized in the following way. In Section 2, we describe the
CFHT and Hubble Space Telescope (HST) data used. In Section 3, we present the
measurement of galaxy shapes in the CFHT imaging, and their calibration
against measurements of the same galaxies in the HST imaging. In Section 4, we
reconstruct the 2D lensing convergence “mass map” signal, and extract a
catalog of local maxima that represent cluster candidates. In Section 5, we
search for optical counterparts of these candidates, dramatically cleaning the
catalog. In Section 6, we investigate the full 3D lensing signal around each
cluster, further cleaning the catalog when the lensing signal behind spurious
peaks does not increase as expected with redshift - but obtaining an
independent estimate of the cluster redshift when it does. We finally explore
global scaling relations between cluster mass observables, then conclude in
Section 7.
## 2\. Data
### 2.1. CFHTLS-Wide T0006 imaging
The CFHT Legacy Survey is a joint Canadian-French program to make efficient
use of the CFHT wide field imager MegaPrime, simultaneously addressing several
fundamental questions in astronomy. Each MegaPrime/MegaCam image consists of
an array of $9\times 4$ e2v CCDs with a pixel scale of $0\arcsec.187$ and a
total field of view of $\sim 1~{}\mathrm{deg}^{2}$. The survey used most of
the telescope dark and gray time from 2003 to 2008. We analyze CFHTLS-Wide
imaging from the Terapix T0006 processing run, which is the first to include
the complete survey and was publicly released on 2010 November 15 (Goranova et
al. 2009). These data cover $\sim 171~{}\rm deg^{2}$ in four fields (W1, W2,
W3, and W4) of which the $72$-pointing, $\sim 64~{}\rm deg^{2}$ W1 field is
the largest, and in five passbands
($u^{\prime},g^{\prime},r^{\prime},i^{\prime}$,and $z^{\prime}$) down to
$i\sim 24.5$ and $r\sim 25.0$.
Fu et al. (2008) showed that the $i^{\prime}$-band exposures, taken in sub-
arcsecond seeing conditions, provide the best image quality and resolve the
galaxy population with highest median redshift. Resolving the shapes of more
distant galaxies is vital for weak lensing analysis, since the strength of the
shear signal is proportional to the ratio of the Lens–Source and
Observer–Source distances. We therefore choose to analyze the
$i^{\prime}$-band images (mean seeing $0".73$) in the contiguous W1 field. We
also analyze the independent $r^{\prime}$-band imaging to check the
calibration of our shear measurements. We also use photometric redshift
estimates for source galaxies obtained from the multicolor imaging (Ilbert et
al. 2006; Coupon et al. 2009; Arnouts et al. 2010).
Early releases of smaller regions of the CFHTLS have also been used to measure
the weak lensing cosmic shear signal (Semboloni et al. 2006; Hoekstra et al.
2006; Fu et al. 2008). As the survey size has increased, the statistical
errors have shrunk, and difficulty measuring shapes at a precision better than
the statistical error has so far prevented publication of a cosmic shear
analysis of the complete survey. However, weak lensing cluster searches are
restricted by construction to regions of the survey where the signal is
strongest, and the circular symmetry of our analysis removes the negative
impact of additive shear measurement errors (cf. Mandelbaum et al. 2005).
### 2.2. HST COSMOS Imaging
The HST COSMOS survey (Scoville et al. 2007) is the largest contiguous optical
imaging survey ever conducted from space. High resolution ($0\arcsec.12$)
imaging in the $I_{F814W}$ band was obtained during 2003–2005 across an area
of 1.64 $\rm deg^{2}$ that also corresponds to the CFHTLS D2 deep field. Any
galaxies resolved by CFHT are very easily resolved by HST, which therefore
provides highly accurate shape measurements almost without the need for point-
spread function (PSF) correction. We shall calibrate our CFHTLS shape
measurements against those from COSMOS by Leauthaud et al. (2010).
Note that measurements of the shapes of individual galaxies from ground-based
and space-based observations need not necessarily match exactly, even without
shape measurement errors, because the different noise properties of the data
sets may make them most sensitive to different isophotes, which can be twisted
relative to each other. The slightly different passbands may also emphasize
different regions of a galaxy’s morphology. However, across a large population
of galaxies, these differences should average out, and a comparison of
successful shear measurements between the two data sets should agree.
## 3\. Galaxy Shape Measurement
Table 1SExtractor Configuration Parameters Parameter | Value
---|---
DETECT_MINAREA | 3
DETECT_THRESH | 1.0
DEBLEND_NTHRESH | 32
DEBLEND_MINCONT | 0.002
CLEAN_PARAM | 1.0
BACK_SIZE | 512
BACK_FILTERSIZE | 9
BACKPHOTO_TYPE | local
BACKPHOTO_THICK | 30
### 3.1. Object Detection and Masking
Figure 1.— “W1+2+3” pointing from the CFHTLS-Wide W1 field in $i^{\prime}$
band, showing masked regions. This pointing is representative of those with
fairly poor image quality: the seeing of $0\arcsec.78$ is worse in only 24 of
72 (1 in 3) pointings. In our automated algorithm for masking diffraction
spikes around bright stars, the basic shape of the star mask is predefined,
and its size is scaled with the observed major axis of each star.
We conduct shape measurement in both CFHTLS $i^{\prime}$ and $r^{\prime}$
bands. We detect astronomical sources in the images using SExtractor (Bertin &
Arnouts 1996). Our choice of the main SExtractor parameters is listed in Table
LABEL:tab:tab1, and the data are filtered prior to detection by a $3$ pixel
Gaussian kernel.
Figure 2.— Star selection (red points) in the planes of magnitude vs. flux
radius (top) and magnitude vs. peak surface brightness (bottom). We find the
latter more robust. The red points denote objects selected as stars for PSF
modeling. The blue objects are spurious detection.
Near saturated stars, many spurious objects are found due to detector effects
and optical ghosting. It would also be difficult to measure the shapes of real
stars or galaxies in these regions, because of the steep background gradients.
We have developed an automatic pipeline to define polygonal-shaped masks
around saturated stars, and all objects inside the masks are removed from our
catalog. The masks in all images are then visually inspected; our automated
pipeline fails in a few cases (mainly very saturated stars for which the
centroid of the star measured by SExtractor was widely offset from the
diffraction spikes) and those stellar masks are corrected by hand. An example
of the masks for one CFHT pointing is shown in Figure 1. This pointing has
slightly worse than average image quality, so we shall use it throughout this
paper as a conservative representation of our analysis. After applying all of
our masks across the entire survey, the final effective sky coverage drops
from $64~{}\rm deg^{2}$ to $\sim 51.3~{}\rm deg^{2}$ and $55.0~{}\rm deg^{2}$
for $i^{\prime}$ and $r^{\prime}$ bands, respectively.
We shall employ the popular KSB method for galaxy shear measurement (Kaiser et
al. 1995; Luppino & Kaiser 1997; Hoekstra et al. 1998). In this method, the
observed galaxy shape is modeled as a convolution of the (sheared) galaxy with
the PSF, which is modeled as an isotropic, circular profile convolved with a
small anisotropy.
### 3.2. PSF Modelling
Figure 3.— Spatial variation of measured stellar ellipticities in the
representative CFHTLS-Wide W1+2+3 field, before (top) and after (bottom) PSF
anisotropy correction. The longest tick marks represent ellipticities of $\sim
11\%$. The mean absolute ellipticity after correction is $0.62\%$.
To measure the shapes of galaxies, it is first necessary to correct them for
convolution with the PSF imposed by the telescope optics and Earth’s
atmosphere. The changing size and shape of the PSF across the field of view
and between exposures can be traced from stars, which are intrinsically point
sources. We identify stars from their constrained locus within the size-
magnitude plane (Figure 2(a)). Heymans et al. (2006) suggest using the full
width at half maximum (FWHM). However, we find that FWHM is not robustly
measured by SExtractor, so we instead use the $\mu_{\rm max}$-magnitude plane
(Bardeau et al. 2005, 2007; Leauthaud et al. 2007), where $\mu_{\rm max}$ is
the peak surface brightness (Figure 2(b)). The red points in Figure 2 indicate
the selected stars; our chosen locus reflects a careful balance between
obtaining sufficient stars to model the small-scale variations that we observe
in the PSF pattern, and introducing spurious noise by including faint stars.
The blue points are spurious detections of noise, cosmic rays, etc. (cf.
Leauthaud et al. 2007).
Figure 4.— Projection of the stellar ellipticities in the $(e_{1},e_{2})$
plane before (black) and after (red) PSF anisotropy correction. The post-
correction residuals are consistent with featureless white noise.
We then measure the Gaussian-weighted shape moments of the stars, and
construct their ellipticity. In addition to cuts in $\mu_{\mathrm{max}}$ and
magnitude, we also exclude noisy outliers with signal-to-noise $\nu<100$ or
absolute ellipticity $e^{*}$ more than $2\sigma$ away from the mean local
value, and we iteratively remove objects very different from neighboring
stars. In 15 pointings with the worst image quality, including W1+2+3, the PSF
becomes larger than $r_{g}\sim 0\arcsec.5$ in the corners of the field of
view, so we finally add these regions to the survey mask (and exclude galaxies
in them from our weak lensing analysis).
Having obtained our clean sample of stars, we construct a spatially varying
model of the PSF across the field of view. In most pointings, we fit the $\sim
30$ stars in each of the $36$ individual CCDs composing the MegaCam focal
plane, using a polynomial of second order in $x$ and $y$. For stacked data
with large dithers, we use a higher order polynomial. Figures 3 and 4 show the
stellar ellipticity before and after correction for the W1+2+3 pointing, using
a weight function of default size $r_{g}$ to measure the PSF shape moments.
The residual stellar ellipticity after correction is a consistent random
scatter around zero, of width $\sigma_{e_{i}}\sim 0.01$.
Figure 5.— Ellipticity of the PSF changes from the core to the wings. This
shows the mean PSF ellipticity in the $i^{\prime}$-band of the CFHTLS-Wide W1
pointing W1+2+3 as a function of the size of the Gaussian weight function with
which it is measured, before (black) and after (red) PSF anisotropy
correction. The error bars show the rms scatter throughout that pointing.
The ellipticity of the PSF changes from the core to the wings. We measure the
PSF shape using differently-sized weight functions and, when correcting each
galaxy, use the same size weight function to measure both the PSF and galaxy
shapes. Figure 5 shows the variation of mean stellar ellipticity as a function
of the weight function size $r_{g}$, before and after PSF anisotropy
correction.
### 3.3. Galaxy Shape Measurement
Figure 6.— Galaxy magnitude and redshift distributions. Top: number counts of
galaxies per magnitude bin immediately after star-galaxy separation (solid
histograms) and after all the lensing cuts (dotted histograms) for imaging in
$i^{\prime}$ (black lines) and $r^{\prime}$ (red lines) bands. Bottom: the
redshift distribution of galaxies used from the $i^{\prime}$ (black lines) and
$r^{\prime}$ (red lines) bands.
Galaxies are selected as those objects with half light radius
$1.1r^{\mathrm{PSF}}_{h}<r_{h}<4$ pixels, where $r^{\mathrm{PSF}}_{h}$ is the
size of the largest star, signal-to-noise $\nu>10$, magnitude
$21.5<i^{\prime}<24.5$ and SExtractor flag $\rm{\tt FLAGS=0}$. To also exclude
blended or close pairs that could bias ellipticity measurements, we also cut
objects with corrected ellipticity $|e_{\rm cor}|>1$ and pairs of galaxies
within $3\arcsec$. After survey masking and catalog cuts, the galaxy number
density is $n_{g}\sim 11.5~{}{\rm arcmin}^{-2}$ in an area of
$A_{i^{\prime}}\sim 51.3~{}\rm deg^{2}$ of $i^{\prime}$-band imaging; and
$n_{g}\sim 7.9~{}{\rm arcmin}^{-2}$ in $A_{r^{\prime}}\sim 55.0~{}\rm deg^{2}$
of $r^{\prime}$-band data. Note that both are lower than the galaxy density
$n_{g}\sim 38~{}{\rm arcmin}^{-2}$ obtained in CFHTLS-Deep imaging by Gavazzi
& Soucail (2007). This will restrict our detections to generally more massive
clusters. Figure 6 shows the magnitude distribution of the galaxies, and the
redshift distribution of the $72~{}\%$ ($76~{}\%$) of galaxies selected in the
$i^{\prime}$ ($r^{\prime}$) bands that also have photometric redshift
estimates by Arnouts et al. (2010).
We then measure the shapes of all the selected galaxies. Our implementation of
KSB is based on the KSBf90111http://www.roe.ac.uk/ heymans/KSBf90/Home.html
pipeline (Heymans et al. 2006). This has been generically tested on simulated
images containing a known shear signal as part of the Shear Testing Programme
(STEP; Heymans et al. 2006; Massey et al. 2007) and the Gravitational lensing
Accuracy Testing (GREAT08; Bridle et al. 2010) challenge. In all cases, the
method was found to have small and repeatable systematic errors.
If the PSF anisotropy is small, the shear $\gamma$ can be recovered to first-
order from the observed ellipticity $e^{\rm obs}$ of each galaxies via
$\gamma=P_{\gamma}^{-1}\left(e^{\rm obs}-\frac{P^{\rm sm}}{P^{\rm
sm*}}e^{*}\right),$ (1)
where asterisks indicate quantities that should be measured from the PSF model
interpolated to the position of the galaxy, $P^{\rm sm}$ is the smear
polarizability, and $P_{\gamma}$ is the correction to the shear polarizability
that includes smearing with the isotropic component of the PSF. The
ellipticities are constructed from a combination of each object’s weighted
quadrupole moments, and the other quantities involve higher order shape
moments. All definitions are taken from Luppino & Kaiser (1997). Note that we
approximate the matrix $P_{\gamma}$ by a scalar equal to half its trace. Since
measurements of $\mathrm{Tr}~{}P_{\gamma}$ from individual galaxies are noisy,
we follow Fu et al. (2008) and fit it as a function of galaxy size and
magnitude, which are more robustly observable galaxy properties.
Following Hoekstra et al. (2000), we weight the shear contribution from each
galaxy as
$w=\frac{1}{\sigma_{e,i}^{2}}=\frac{P_{\gamma}^{2}}{\sigma_{0}^{2}P_{\gamma}^{2}+\sigma_{e,i}^{2}},$
(2)
where $\sigma_{e,i}$ is the error in an individual ellipticity measurement
obtained via the formula in Appendix A of Hoekstra et al. (2000), and
$\sigma_{0}\sim 0.278$ is the dispersion in galaxies’ intrinsic ellipticities.
### 3.4. Calibration of Multiplicative Shear Measurement Biases
Figure 7.— Choices for the way shear polarizability $P_{\gamma}$ can be fitted
to a galaxy population in the CFHT KSBf90 pipeline, to reduce noise and bias
in individual measurements. Linearly spaced contours compare our shear
measurements of galaxies in a subset of the CFHTLS-Deep imaging, stacked to
the depth of the CFHTLS-Wide survey, against measurements from the Hubble
Space Telescope. Dashed lines show the best-fit relation
$\gamma_{1}^{\mathrm{CFHT}}=(m-1)\gamma_{1}^{\mathrm{HST}}+c$. The four panels
illustrate various fitting schemes. Top-left: raw (noisy) $P_{\gamma}$
measurements from each galaxy, without fitting. Top-right: fitted as a
polynomial in galaxy size $P_{\gamma}(r_{h})$. Bottom-left: fitting function
$P_{\gamma}(r_{h},\mathrm{mag})$ from Fu et al. (2008). Bottom-right: best-fit
rational function $P_{\gamma}(r_{h},\mathrm{mag})$, as described in the text.
Figure 8.— Robustness of the calibration of our shear measurement as a
function of image quality. Linearly-spaced contours compare our shear
measurements of galaxies in subsets of CFHTLS-Deep imaging with varying mean
seeing (black solid: $0\arcsec.90$, red dashed: $0\arcsec.69$, blue dotted:
$0\arcsec.57$) to measurements from the Hubble Space Telescope. Shears are
consistently underestimated by our pipeline, but the calibration is remarkably
robust.
We exploit the opportunity that the CFHTLS-Deep D2 field includes the HST
COSMOS survey field, and verify the calibration of our shear measurement
pipeline for ground-based data against an independent analysis of the much
higher resolution space-based data (Leauthaud et al. 2007, 2010). We stack
subsets of the CFHTLS-Deep D2 imaging to the same depth as the CFHTLS-Wide
survey and analyze it using the same pipeline applied to the CFHTLS-Wide W1
field. Since any galaxy seen by CFHT is very well resolved by HST, and imaged
to very high signal-to-noise ratio by the COSMOS survey, the space-based shear
measurements require only negligible PSF correction and suffer from only
negligible shot noise. A consistent shear measurement between ground and space
for this subset of galaxies would therefore indicate a robust shear
measurement across the CFHTLS.
Multiplicative shear measurement biases $m$ are the most problematic for
circularly-symmetric cluster measurements. Multiplicative biases cannot be
internally diagnosed within a shear catalog, so our comparison against
external data is most useful for checking that $m$ is sufficiently small that
it corresponds to a bias smaller than our statistical errors. Within the KSB
framework, difficulties in shear calibration mainly rest in measurement of the
shear polarizability $P_{\gamma}$, so we first investigate different
possibilities for fitting $P_{\gamma}$ across a galaxy population. Figure 7
compares shear measurements from a subset of the CFHTLS-Deep imaging with mean
seeing $0\arcsec.69$ (similar to the mean seeing in our survey) stacked to the
depth of the CFHTLS-Wide imaging against shear measurements obtained from HST.
The dashed lines show the best-fit linear relations
$\gamma_{1}^{\mathrm{CFHT}}=(1+m)\gamma_{1}^{\mathrm{HST}}+c$, which are
obtained using a total least-squares fitting method (e.g. Kasliwal et al.
2008) that accounts for the noise present in both shear catalogs. The top-left
panel shows the CFHT shear measurements with $P_{\gamma}$ naïvely obtained
from each raw, noisy galaxy without any fitting. This results in a large bias
on shear measurements and a large amount of extra noise. The top-right panel
shows the shear measurements if $P_{\gamma}$ is fitted as a function of galaxy
size, $P_{\gamma}(r_{h})=a_{0}+a_{1}r_{h}+a_{2}r_{h}^{2}$. The bottom-left
panel shows shears if
$P_{\gamma}(r_{h},\mathrm{mag})=a_{0}+a_{1}r_{h}+a_{2}r_{h}^{2}+a_{3}m_{\nu}$
(Fu et al. 2008). The bottom-right panel shows the matched shear with
$P_{\gamma}(r_{h},\mathrm{mag})$ the best-fit rational function
$P_{\gamma}=\frac{a_{0}+a_{1}m_{\nu}+a_{2}m_{\nu}^{2}+a_{3}r_{h}}{1+a_{4}m_{\nu}+a_{5}m_{\nu}^{2}+a_{6}r_{h}+a_{7}r_{h}^{2}}~{}.$
(3)
In this example, the coefficients are $a_{0}=25.07$, $a_{1}=-2.19$,
$a_{2}=0.045$, $a_{3}=0.53$, $a_{4}=0.58$, $a_{5}=-0.022$, $a_{6}=-0.85$, and
$a_{7}=0.14$. The more sophisticated fits produce a shear catalog that is a
marginally better match to the reliable HST measurements, and this is even
more true if we redo the analysis using the full CFHTLS-Deep depth, in which
galaxies are fainter and smaller. We henceforth choose to adopt the rational
function fit to $P_{\gamma}$ for all subsequent analyses, obtaining new best-
fit coefficients for each pointing.
To quantify the performance of our shear measurement pipeline as a function of
image quality, we stack subsets of the CFHTLS-Deep D2 imaging with low-,
medium-, and high-seeing to the same depth as the CFHTLS-Wide survey, and
analyze each separately (Figure 8). We find that our CFHTLS pipeline
consistently underestimates shear, but that the calibration is remarkably
robust to seeing conditions. We can therefore simply recalibrate our pipeline
for all images by multiplying all measured shears by $1/(1-0.21)$.
### 3.5. Assessment of Residual Additive Shear Systematics
Figure 9.— Mean shear measurements from galaxies in $i^{\prime}$-band
observations of the entire CFHTLS-Wide W1 field. In the absence of additive
systematics, these should be consistent with zero. In practice, they always
remain within the dashed lines than indicate an order of magnitude lower than
the $1\%$–$10\%$ shear signal around clusters. Upper and lower panels show
components $\gamma_{1}$ and $\gamma_{2}$, respectively. Left, middle, and
right panels show trends as a function of galaxy magnitude, size, and
detection signal-to-noise.
Additive shear measurement systematics $c$ generally cancel out in circularly-
symmetric cluster measurements (Mandelbaum et al. 2006). However, to double-
check for significant additive systematics, we first measure the mean shears
$\langle\gamma\rangle$ across all $72$ pointings of the CFHTLS-Wide W1 field.
Figure 9 demonstrates that the mean shear is consistent with zero as expected,
for galaxies of all sizes, magnitudes, and signal-to-noise ratios.
Figure 10.— The cross-correlation between shear measurements and stellar
ellipticities, as a function of the separation between galaxies and stars,
averaged throughout the CFHTLS-Wide W1 field. If all residual influence of the
observational PSF has been successfully removed from the galaxy shape
measurements, the red (lower) points should be consistent with zero.
We also look for residual systematics left in the weak lensing cosmic shear
signal due to imperfect PSF correction. Figure 10 shows the correlation
$\xi_{\mathrm{sys}}$ between the corrected shapes of galaxies and the
uncorrected shapes of stars. Following Bacon et al. (2003) and Massey et al.
(2005), we normalize the star-galaxy ellipticity correlation by the
uncorrected star-star ellipticity correlation to assess its impact on shear
measurements
$\xi_{\rm
sys}(\theta)=\frac{<e^{*}(\mathbf{x})\,\gamma(\mathbf{x}+\mathbf{\theta})>^{2}}{<e^{*}(\mathbf{x})\,e^{*}(\mathbf{x}+\mathbf{\theta})>},$
(4)
where $e^{*}$ is the ellipticity of the stars before PSF correction and
$\gamma$ is the shear estimate from galaxies. We find that our PSF correction
is well within requirements for our analysis because on cluster scales $1$–$5$
$\rm arcmin$, the amplitude of $\xi_{\rm sys}$ is at least one order of
magnitude smaller than the cosmic shear signal
$\xi_{\pm}=\xi_{\rm tt}(\theta)\pm\xi_{\rm
xx}(\theta)=\frac{1}{2\pi}\int_{0}^{\infty}\ell\,P_{\kappa}(\ell)\,{\rm
J}_{0,4}(\ell\theta)\,\mathrm{d}\ell~{},$ (5)
where $\xi_{\rm tt}(\theta)$ ($\xi_{\rm xx}(\theta)$) are the correlation
functions between components of shear rotated tangentially (at $45^{\circ}$)
to the line between pairs of galaxies separated by an angle $\theta$ and $\rm
J_{0}$, $\rm J_{4}$ are Bessel functions of the first kind.
Figure 11.— Gravitational lensing is achromatic, so measurements of galaxy
shapes from imaging in different colors should on average be consistent. This
shows a comparison of shear measurements obtained from CFHTLS-Wide
$r^{\prime}$-band and $i^{\prime}$-band imaging of the whole W1 field. The
dashed line shows the best-fit linear relation
$\gamma_{1}^{i}=(m-1)\gamma_{1}^{r}+c$.
Figure 12.— Distribution of foreground mass in the W1+2+3 pointing,
reconstructed from shear measurements via the KS93 method. Left: the physical
$E$-mode convergence signal. Right: the $B$-mode systematics signal, created
by rotating the shears by $45^{\circ}$ then remaking the map. Contours are
drawn at detection significances of $3\sigma$, $4\sigma$, and $5\sigma$, with
dashed lines for negative values. Figure 13.— Reconstructed “dark matter
mass” convergence map for the entire $64$ $\rm deg^{2}$ CFHTLS-Wide W1 field.
This has been smoothed by a Gaussian filter of width $\theta_{G}=1\arcmin$.
Black contours are drawn at detection signal-to-noise ratios $\nu=3.0$, $3.5$,
$4.0$, and red contours continue this sequence from $\nu\geq 4.5$.
Figure 14.— Reconstructed “dark matter mass” convergence map for the entire
$64$ $\rm deg^{2}$ CFHTLS-Wide W1 field showing the same data as Figure 13,
but smoothed with a Gaussian filter of width $\theta_{G}=6\arcmin$ (left) and
multi-scale wavelet filtering (right) to highlight the large-scale features.
Figure 15.— Numbers of local maxima (solid line) and minima (dashed line) in
our $E$-mode (black) and $B$-mode (red) convergence map of the CFHTLS-Wide W1
field, with smoothing scales $\theta_{G}=1\arcmin$ (top-panel) and
$\theta_{G}=2\arcmin$ (bottom-panel). Local maxima can still have a slightly
negative peak height if they occur along the same line of sight as a negative
noise fluctuation (or a huge void), and local minima can similarly have a
slightly positive peak height. To eliminate spurious noise peaks, we shall
mainly consider maxima or minima with $|\nu|>3.5$. In this regime, there is an
excess of local maxima over local minima, and an excess of $E$-mode peaks over
$B$-mode peaks (see Table 2).
Figure 16.— Cumulative density of local maxima $N(>\nu)$ (solid line) and
corresponding density of local minima $N(<-\nu)$ (dashed line) in the CFHTLS-
Wide W1 $E$-mode (black) and $B$-mode (red) convergence map with smoothing
scale $\theta_{G}=1\arcmin$ (top-panel) and $\theta_{G}=2\arcmin$ (bottom-
panel). Error bars are simply $1\sigma$ uncertainties assuming Poisson shot
noise. The dot-dashed line shows the prediction from Gaussian random field
theory (van Waerbeke 2000). Circles with additional error bars and the dotted
curves show an analytical prediction in a $\Lambda$CDM universe (Fan et al.
2010), with and without the influence of random noise. The excess of positive
maxima over negative minima demonstrates the non-Gaussianity of the
convergence field. Figure 17.— Reconstructed lensing convergence signal-to
noise map for the W1+2+3 pointing, plus overlays showing optically and X-ray
selected cluster counterparts. The smoothing scale of the background map is
$\theta_{G}=1\arcmin$. Symbols indicate the positions of
$+$: lensing peaks detected with $\nu>3.5$ in the $\theta_{G}=1\arcmin$ map,
$\square$: lensing peaks detected with $\nu>3.5$ in the $\theta_{G}=2\arcmin$
map,
$\triangle$: optically-detected clusters in the K2 catalog, and
$\times$: X-ray selected clusters found in XMM-LSS (Adami et al. 2011).
Figure 18.— Distribution of the offsets between matched pairs of weak lensing
peaks and K2-detected clusters. Left: offsets for all peaks in maps with
smoothing scales of $0\arcmin.5$ (solid black), $1\arcmin$ (dotted red), and
$2\arcmin$ (dashed blue). Right: offsets for peaks with $\nu>3$ (solid black
histograms), $\nu>3.5$ (dotted red histograms) and $\nu>4$ (dashed blue
histograms), all with the smoothing scale $\theta=1~{}\rm arcmin$. Figure
19.— Cluster redshift distribution for the matched clusters of weak lensing
and K2 (black solid histogram) and the total K2 detected clusters with K2
detection significance $(r-i)>3$ (red dashed histogram).
Because gravitational lensing is achromatic while systematics are typically
not, we can also assess the robustness of our measurements by comparing shears
measured from independent imaging acquired in multiple bands. The first
attempt at comparing multicolor shear measurements was made by Kaiser et al.
(2000) using the CFHT12K camera. The $I$ and $V$ bands showed significantly
different signals that were inconsistent with the change in redshift
distribution between the two filters. After a great deal of algorithmic
progress, Semboloni et al. (2006) obtained consistent shear measurements from
$i^{\prime}$-band and $r^{\prime}$-band CFHTLS-Deep data. Gavazzi & Soucail
(2007) extracted consistent shear from the $g^{\prime}$, $r^{\prime}$,
$z^{\prime}$ and $i^{\prime}$ bands of CFHTLS-Deep. Gavazzi et al. (2009) also
measured consistent values of the PSF-corrected ellipticities of central Coma
cluster galaxies in MegaCam-$u^{*}$ and CFH12k-$I$ bands. Our analysis
pipeline measures shears in the independent CFHTLS $r^{\prime}$-band and
$i^{\prime}$-band imaging that are consistent within the $\sim 0.1$ rms noise
(Figure 11). To maximize the total number of galaxies in our shear catalog, we
therefore combine $r^{\prime}$-band and $i^{\prime}$-band measurements. Only
the unique $r^{\prime}$-band galaxies are added to the $i^{\prime}$-band
catalog. The combined catalog includes $\sim 2.66$ million galaxies, with
$n_{g}\sim 14.5~{}{\rm arcmin}^{-2}$ for lensing measurements.
We shall continue testing for systematics at each stage of our analysis, by
checking that the shear signal behaves as expected, and is consistent with
external data sets. An important example of this is the (nonphysical) $B$-mode
signal, which we shall compute wherever we reconstruct the (physical)
$E$-mode. Pure gravitational fields produce zero $B$-mode for isolated
clusters and only tiny $B$-modes through coupling between multiple systems
along adjacent lines of sight (Schneider et al. 2002). The $B$-mode signal
corresponds to the imaginary component of $P_{\kappa}(\ell)$; it can be
conveniently measured by rotating all galaxy shears through $45^{\circ}$ then
remeasuring the $E$-mode signal (Crittenden et al. 2002).
## 4\. Mass reconstructions
### 4.1. Kaiser-Squires Inversion and Masking
The shear field $\gamma_{i}(\mathbf{\theta})$ is sparsely and noisily sampled
by measurements of the shapes of galaxies at positions $\mathbf{\theta}$. The
smooth, underlying shear field $\gamma_{i}(\mathbf{\theta})$ can be written in
terms of the lensing potential $\phi(\mathbf{\theta})$ as
$\gamma_{1}=\frac{1}{2}(\partial_{1}^{2}-\partial_{2}^{2})\phi,$ (6)
$\gamma_{2}=\partial_{1}\partial_{2}\phi,$ (7)
where the partial derivatives $\partial_{i}$ are with respect to $\theta_{i}$.
The convergence field $\kappa(\mathbf{\theta})$, which is proportional to the
mass projected along a line of sight, can also be expressed in terms of the
lensing potential as
$\kappa=\frac{1}{2}(\partial_{1}^{2}+\partial_{2}^{2})\phi.$ (8)
We shall reconstruct the convergence field from our shear measurements via the
Kaiser & Squires (1993) (KS93) method. This is obtained by inverting Equations
(6) and (7) in Fourier space:
$\hat{\gamma}_{i}=\hat{P}_{i}\hat{\kappa}$ (9)
for $i=1,2$, where the hat symbol denotes Fourier transforms, we define
$k^{2}=k_{1}^{2}+k_{2}^{2}$ and
$\hat{P}_{1}(k)=\frac{k_{1}^{2}-k_{2}^{2}}{k^{2}},$ (10)
$\hat{P}_{2}(k)=\frac{2k_{1}k_{2}}{k^{2}}.$ (11)
This inversion is non-local, so we deal with masked regions of the shear field
by masking out the same area in the convergence field, plus a $1\arcmin.5$
border. We shall ignore any signal within these regions, and set the
convergence to zero in relevant figures.
For the finite density of source galaxies resolved by CFHT, the scatter of
their intrinsic ellipticities means that a raw, unsmoothed convergence map
$\kappa(\mathbf{\theta})$ will be very noisy. Following Miyazaki et al.
(2002), we smooth the convergence map by convolving it (while still in Fourier
space) with a Gaussian window function,
$W_{G}(\theta)=\frac{1}{\pi\theta_{G}^{2}}{\rm
exp}\left(-\frac{\theta^{2}}{\theta_{G}^{2}}\right),$ (12)
As shown by van Waerbeke (2000), if different galaxies’ intrinsic
ellipticities are uncorrelated, the statistical properties of the resulting
noise field can be described by Gaussian random field theory (Bardeen et al.
1986; Bond & Efstathiou 1987) on scales where the discreteness effect of
source galaxies can be ignored. The Gaussian field is uniquely specified by
the variance of the noise, which is in turn controlled by the number of
galaxies within a smoothing aperture (Kaiser & Squires 1993; Van Waerbeke
2000)
$\sigma_{\rm
noise}^{2}=\frac{\sigma_{e}^{2}}{2}\frac{1}{2\pi\theta_{G}^{2}n_{g}},$ (13)
where $\sigma_{e}$ is the rms amplitude of the intrinsic ellipticity
distribution and $n_{g}$ is the density of source galaxies. We define the
signal-to-noise ratio for weak lensing detections by
$\nu\equiv\frac{\kappa}{\sigma_{\rm noise}}.$ (14)
To define the noise level in theoretical calculations of $\nu$, we adopt a
constant effective density of galaxies equal to the mean within our survey.
For observational calculations of $\nu$, we use the mean galaxy density in
each pointing — but do not consider the non-uniformity of the density within
each field due to masks or galaxy clustering.
It turns out that a simple Gaussian filter of width $\theta_{G}\approx
1\arcmin$ is close to the optimal linear filter for cluster detection, and
this choice has been extensively studied in simulations (White et al. 2002;
Hamana et al. 2004; Tang & Fan 2005). Because of our relatively low source
galaxy density, the galaxies’ random intrinsic shapes will produce spurious
noise peaks, degrading the completeness and purity of our cluster detection.
To reduce contamination, we repeat our mass reconstruction using two smoothing
scales $\theta_{G}=1\arcmin$ and $\theta_{G}=2\arcmin$. The map with greater
smoothing will be less noisy; to help remove spurious peaks from the higher
resolution map, we consider only those peaks detected above a signal-to-noise
threshold in both maps.
Figure 12 shows the reconstructed convergence field corresponding to
foreground mass in the W1+2+3 pointing. The left panel shows the $E$-mode
reconstruction with KS93 method after smoothing by a $1\rm arcmin$ Gaussian
kernel. This contains several high signal-to-noise ratio peaks, while the
associated $B$-mode systematics measurement in the right panel is
statistically consistent with zero, with fewer peaks. As weak lensing produces
only curl-free or $E$-mode distortions, a detection (significant above
statistical noise) of curl or $B$-mode signal would have indicated
contamination from residual systematics, e.g. imperfect PSF correction.
### 4.2. Large-scale Lensing Mass Map
A reconstructed “dark matter mass” convergence map for the entire $64~{}{\rm
deg}^{2}$ CFHTLS-Wide W1 field is presented in Figure 13. We detect $301$
peaks with $\nu>3.5$ in maps with both smoothing scales $\theta_{G}=1\arcmin$
and $\theta_{G}=2\arcmin$. The same information is reproduced in Figure 14,
with $\theta_{G}=6\arcmin$ and after multi-scale entropy restoration filtering
(MRLens; Starck et al. 2006), to better display large-scale features. The
MRLens filtering effectively suppresses noise peaks, but results in non-
Gaussian noise that complicates the peak selection (Jiao et al. 2011), so we
shall not use it further.
Table 2The Number of Local Maxima and Minima in the Convergence Map of the CFHTLS-Wide W1 Field, as a Function of Smoothing Scale. Smoothing | $E$-mode | $E$-mode | $B$-mode | $B$-mode
---|---|---|---|---
Scale $\theta_{G}$ | $\nu>3.5$ | $\nu<-3.5$ | $\nu>3.5$ | $\nu<-3.5$
$0\arcmin.5$ | $1512$ | $1270$ | $1244$ | $1033$
$1\arcmin.0$ | $543$ | $445$ | $361$ | $282$
$2\arcmin.0$ | $281$ | $233$ | $148$ | $126$
To assess the reliability of this map, we shall first investigate the
statistical properties of local maxima and minima. Figure 15 shows the
distribution of peak heights, as a function of detection signal-to-noise. The
bimodal distribution in both the $B$-mode and $E$-mode signals is dominated by
positive and negative noise fluctuations, but an asymmetric excess in the
$E$-mode signal is apparent at both $\nu>3.5$ and, at lower significance,
$\nu<-3.5$. The amplitude, slope and non-Gaussianity of this excess are all
powerful discriminators between values of parameters in cosmological models
(Pires et al. 2009). Positive peaks correspond mainly to dark matter halos
around galaxy clusters. Local minima could correspond to voids (Jain & van
Waerbeke 2000; Miyazaki et al. 2002), but the large size of voids is ill-
matched to our $\theta_{G}=1\arcmin$ filter width, and their density contrast
can never be greater than unity, so this aspect of our data is likely just
noise.
Figure 16 recasts the peak distribution into a cumulative density of positive
maxima or negative minima. As expected, we find a non-Gaussian mass
distribution with more highly significant positive maxima (corresponding to
mass overdensities) than highly significant negative minima (see Table 2).
Analytic predictions of peak counts are also overlaid. Following van Waerbeke
(2000), dot-dashed lines show the expected density of pure noise peaks, and
dotted lines show the expected number of true dark matter halos. Predictions
from Fan et al. (2010), which also take into account the effect of noise on
the heights of true peaks and the clustering of noise peaks near dark matter
halos, are shown as circles with error bars. In these theoretical
calculations, we model the population of background galaxies as having an
intrinsic ellipticity dispersion $\sigma_{e}=0.278$, density $n_{g}=14.5~{}\rm
arcmin^{-2}$ and the redshift distribution from Fu et al. (2008) in a
$\Lambda$CDM universe. At $\nu>4.5$, it appears that theory may begin to
predict more peaks than are observed. However, these are very small number
statistics, and our observations are consistent with analytical predictions
within Poisson noise. At very low $\nu$, the number of peaks is washed out and
actually decreases when noise is superimposed, because it is impossible to
extract very low-$\nu$ peaks. However, if larger and deeper surveys still find
fewer low-$\nu$ or high-$\nu$ peaks than expected, and systematic effects such
as the consequences of masked regions are more fully understood, it may
indicate a cosmology with, e.g., lower $\Omega_{m}$ and $\sigma_{8}$ than the
values used for our predictions.
Our main conclusions about the distribution of convergence peaks are as
follows.
* •
Peak counts detected in CFHTLS-Wide are consistent with predictions from a
$\Lambda$CDM cosmological model, once noise effects are properly included (van
Waerbeke 2000; Fan et al. 2010).
* •
The convergence field is non-Gaussian. The excess of local maxima with
$\nu>3.5$ compared to local minima with $\nu<-3.5$ is also consistent with
models (Miyazaki et al. 2002; Gavazzi & Soucail 2007).
* •
Noise peaks dominate the expected peak counts due to cosmological weak lensing
below $|\nu|\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3$.
We expect that weak lensing peak counts will become reliably employed to
constrain cosmological parameters in future lensing analyses. Although it is
difficult to remove the contributions of noise from intrinsic galaxy shapes
and the projection of large scale structures, these effects can be
analytically predicted. Pushing these predictions into the low signal-to-noise
regime might also allow constraints to use a much higher number density of
less massive peaks, tightening predictions by reducing Poisson noise.
## 5\. Optical/X-ray counterparts
We shall now compare our weak lensing peak detections with catalogs of
overdensities detected via optical or X-ray emission. We adopt the K2 optical
cluster catalog constructed using photometric redshifts from the same CFHTLS-
Wide W1 imaging (Thanjavur et al. 2009), and the XMM-LSS X-ray selected
cluster catalog (Adami et al. 2011), which partially overlaps our survey
field.
Table 3The Detection Purity of Various Weak Lensing Cluster Surveys Described in the Literature. | Weak Lensing Cluster Sample | Comparison Sample | Purity
---|---|---|---
Hamana et al. (2004) | $n_{\mathrm{gal}}=30$ arcmin-2, $\nu>4$ | Halos (simulations) | $60\%$
Dietrich et al. (2007) | $n_{\mathrm{gal}}=18$ arcmin-2, | Halos (simulations) | $75\%$
Miyazaki et al. (2007) | $\nu>3.69$ | X-ray clusters (XMM-LSS) | $80\%$
Gavazzi & Soucail (2007) | $n_{\mathrm{gal}}\sim 35$ arcmin-2, $\nu>3.5$ | Photometric clusters | $\sim 65\%$
Schirmer et al. (2007) | $\nu>4$ | BCG | $\sim 45\%$
Geller et al. (2010) | $\nu>3.5$ | Spectroscopic clusters (SHELS) | $\sim 33\%$
We define the purity $f_{p}$ of our blind weak lensing cluster search as the
fraction of peaks above a given detection threshold $\nu_{\rm th}$ that are
associated with an optically detected cluster
$f_{p}=\frac{N_{\rm matched}(\nu>\nu_{\rm th})}{N(\nu>\nu_{\rm th})}~{}.$ (15)
The expected purity depends upon the survey depth (galaxy density),
systematics, and the extent of multiband or spectroscopic follow-up. Using
numerical simulations with $n_{\mathrm{gal}}=30$ arcmin-2, Hamana et al.
(2004) predict a purity of more than $60\%$ for convergence peaks with
$\nu>4$. The Bonn Lensing, Optical, and X-ray selected galaxy clusters (BLOX)
simulations by Dietrich et al. (2007) with $n_{\mathrm{gal}}=18$ arcmin-2 also
predict that $75\%$ of matches between convergence map peaks and massive halos
are within $2\arcmin.15$. In practice, Miyazaki et al. (2007) achieve $80\%$
purity for the $\sim 100$ peaks in Subaru convergence maps with $\nu>3.69$.
Gavazzi & Soucail (2007) obtain $\sim 65\%$ purity from $14$ peaks in CFHTLS-
Deep with $\nu>3.5$. Schirmer et al. (2007) obtain $\sim 45\%$ purity for the
$158$ possible mass concentrations identified in the Garching-Bonn Deep Survey
(GaBoDS) at $\nu>4$, consistent with an earlier evaluation of a subsample of
the survey (Maturi et al. 2007). Geller et al. (2010) find only $\sim 33\%$
purity by combining the Deep Lens Survey with the Smithsonian Hectospec
Lensing Survey (SHELS) (see Table 3).
Figure 17 shows a subset of our cluster search in the representative W1+2+3
pointing, which also overlaps with the XMM-LSS survey (Pacaud et al. 2007;
Adami et al. 2011). The observed weak lensing peak positions may not coincide
exactly with cluster centers defined from optical emission, because of a
combination of noise, substructure, and physical processes associated with
cluster mergers (Fan et al. 2010; Hamana et al. 2004). We therefore search for
matched pair candidates in K2 within a $3\arcmin.0$ radius of peaks that
appear in both the $\theta_{G}=1\arcmin$ and $\theta_{G}=2\arcmin$ lensing
map. This search radius is chosen to be larger than the smoothing scale, but
smaller than the angular virial radius of a massive cluster at $0.1<z<0.9$
(Hamana et al. 2004). If more than one pair exists within $3\arcmin.0$, we
adopt the closest match as the primary candidate.
Table 4Cluster Search Purity as a Function of Peak Height Threshold $\nu_{\rm th}$. $\nu_{\rm th}$ | $N(\nu>\nu_{\rm th})$ | $N_{\rm matched}(\nu>\nu_{\rm th},\rm K2)$ | $f_{p}$
---|---|---|---
3.5 | 301 | 126 | 42%
4.0 | 125 | 67 | 54%
4.5 | 51 | 30 | 59%
We obtain $126$ matches between weak lensing peaks with $\nu>3.5$ and optical
K2 clusters: corresponding to a purity of $42\%$. This is lower than in
CFHTLS-Deep (Gavazzi & Soucail 2007) because of the much lower source galaxy
density. The purity is listed as a function of detection threshold $\nu_{\rm
th}$ in Table 4, and the complete catalog of matched pairs is presented in
Table LABEL:tab:matchpeaks. Figure 18 shows the separations between the
matched weak lensing peaks and K2 centers. The left panel shows the offsets
for peaks with $\nu>0$ in maps with various smoothing scales; the separation
between peaks typically matches the smoothing scale. The right panel shows the
offsets for only the reliably-detected peaks with high $\nu$; the finite
number of clusters is too low to draw solid conclusions about the typical
separation. Figure 19 shows the redshift distribution of matched clusters,
relative to the overall K2 sample. Our lensing selection preferentially
detects clusters at $0.2<z<0.4$, and becomes inefficient above $z>0.5$.
We also compare our $\nu>3.5$ lensing peaks with X-ray observations from the
XMM-LSS survey. Adami et al. (2011) present $66$ spectroscopically confirmed
clusters $(0.05<z<1.5)$ within the $6~{}\rm deg^{2}$ XMM-LSS survey. This
partially overlaps with the CFHTLS-W1 field ($53$ X-ray clusters are within
W1). In this overlap region, we find $31$ lensing peaks, $7$ of which are
within $5\arcmin.0$ of X-ray clusters. Note that we increase the distance
threshold for matches because of additional noise in the X-ray centers and the
common phenomenon of separation between the gravitational field and X-ray gas
— Shan et al. (2010) found that $45\%$ of a sample of $38$ clusters had X-ray
offsets $>10\arcsec$. Indeed, in our new sample, we also find that the offsets
between weak lensing and X-ray centers are always comparable to or much larger
than the offset between weak lensing and optical centers.
Miyazaki et al. (2007) also perform a weak lensing analysis of
$0.5~{}\mathrm{deg}^{2}$ of the XMM-LSS survey. They find $15$ lensing peaks
with optical counterparts, of which $10$ match X-ray selected clusters (Adami
et al. 2011). Many of the X-ray clusters are simply at too high a redshift to
be detected by the CFHTLS lensing data. However, three of these clusters are
detected in both our analysis (c77, c92 and c93 in Table LABEL:tab:matchxray)
and that of Miyazaki et al. (2007).
## 6\. Tomographic analysis of lensing peaks
Since multi-band photometric redshift estimates are available for $76\%$ of
the source galaxies, we shall now perform a 3D, tomographic shear analysis
around all $\nu>3.5$ cluster candidates with an optical or X-ray counterpart.
This process yields an estimate of the cluster redshift (and mass)
independently of its visible emission. It also further cleans the cluster
catalog of spurious peaks created by either noise or the projection of
multiple small systems along a single line of sight.
We fit the 3D shear signal around each cluster candidate to both a singular
isothermal sphere (SIS) and a Navarro et al. (1996; NFW) model, with the
cluster lens as an additional free parameter. We assume that the center of
each cluster candidate is at the position of the brightest central galaxy
(BCG). This may not be optimal (Johnston et al. 2007), but it is much more
precisely known than the peak of the lensing signal (Fan et al. 2010). For
each source galaxy, we adopt the best-fit redshift from the cleaned Arnouts et
al. (2010) catalog. Following Gavazzi & Soucail (2007), we measure the mean
tangential shear in a $1\arcmin$–$5\arcmin$ annulus from the cluster center,
excluding the core to minimize dilution of the signal from any cluster member
galaxies with incorrect photometric redshifts.
For an SIS model, the component of shear tangential to the cluster center is
$\gamma_{t}(\theta,z_{s};z_{l})=\frac{D_{ls}}{D_{s}}\frac{4\pi\sigma^{2}_{v}}{c^{2}}\frac{1\arcsec}{2\theta}~{},$
(16)
where $\theta$ is the angular distance from the center. We simultaneously fit
the unknown lens redshift and characteristic cluster velocity dispersion
$\sigma_{v}$ by minimizing
$\chi_{\rm
SIS}^{2}(z_{l},\sigma_{v})=\sum_{i}\frac{\big{(}\gamma_{t,i}-\gamma_{{\rm
SIS},i}(z_{l},\sigma_{\rm tomo})\big{)}^{2}}{\sigma_{e,i}^{2}}~{},$ (17)
where $w_{i}=w(z_{l},z_{s,i})$ and $\sigma_{e,i}^{2}$ is given by Equation.
(2).
The full NFW model has two free parameters, but we assume the Bullock et al.
(2001) relation between concentration $c$ and virial mass $M_{\rm vir}$ seen
in numerical simulations
$c_{\rm NFW}(M_{\rm vir},z)=\frac{c_{*}}{1+z}\left(\frac{M_{\rm
vir}}{10^{14}h^{-1}M_{\odot}}\right)^{-0.13},$ (18)
where $c_{*}=8$. As for the SIS model, there is then only one free parameter,
and we fit the shear field for lens redshift and halo mass by minimizing
$\chi_{\rm NFW}^{2}(z_{l},M_{\rm
vir})=\sum_{i}\frac{\big{(}\gamma_{t,i}-\gamma_{\rm NFW}(z_{l},M_{\rm
vir})\big{)}^{2}}{\sigma_{e,i}^{2}}.$ (19)
To aid comparison with other work, for each cluster we calculate both the
virial mass $M_{\rm vir}$ and the mass $M_{\rm 200}$ enclosed within a radius
$r_{\rm 200}$ in which the mean density of the halo is $200$ times the
critical density at the redshift of the cluster.
Figure 20.— Results of a tomographic weak lensing analysis around one example
peak (c3 in our catalog), which has an optical counterpart at redshift
$z=0.28$. The solid and dashed lines are the best-fit SIS and NFW models. Top:
projection in the redshift direction showing the characteristic increase with
redshift of a real signal. The best-fit lens redshift is
$0.26^{+0.11}_{-0.10}$ for an SIS model and $0.26^{+0.12}_{-0.12}$ for an NFW
model. Middle: radial profile projected onto the plane of the sky. Black
circles show the $E$-mode tangential shear signal. Red circles (triangles)
show positive (negative) values of the $B$-mode systematic signal, which
oscillates about zero. Bottom: joint redshift-mass constraints from the best
fit NFW model. Contours show 68%, 95% and 99% confidence limits. Figure 21.—
Composite $3\times 3$ arcmin2 CFHTLS $g^{\prime}$, $r^{\prime}$, $i^{\prime}$
color images for four clusters detected in both weak lensing and the K2
optical catalog. Contours show signal-to-noise in reconstructed convergence,
starting at $\nu=3.0$ and increasing in steps of $0.5$ and the images are
centered on the convergence peak. Top-left: cluster c3 with $\nu=5.395$. Top-
right: cluster c49 with $\nu=4.728$. Bottom-left: cluster c21 with
$\nu=4.142$. Bottom-right: cluster c103 with $\nu=3.725$. Candidate c21 is
included here as an example of a 2D lensing peak that is probably spurious: it
does not appear obviously associated with an overdensity of galaxies in the
optical imaging, and is not well-fit by a 3D shear signal.
Figure 20 illustrates our 3D tomographic results on one cluster
(identification c3). The mean tangential shear is consistent with zero for
$z_{s}\leq 0.26$. The subsequent increase with redshift is clear and allows
for an unambiguous identification of the lens redshift. Error bars are derived
from the scatter in observed ellipticities (intrinsic+measurement error), as
determined by Equation (1). The observed tangential shear profile is
consistent with either an SIS or NFW model. Due to our low density of source
galaxies and our exclusion of galaxies within the central $1\arcmin$, we have
insufficient signal-to-noise for individual clusters to distinguish between
the two models, which differ most noticeably near the core. The amplitude of
the systematic $B$-mode signal, computed by rotating all shear estimates by
$45^{\circ}$, is always at least one order of magnitude smaller than the
tangential shear, and it oscillates about zero.
Figure 21 shows the signal-to-noise contours of the convergence signal
reconstructed around four clusters from the 3D shear signal.
Figure 22.— Comparison between cluster redshifts derived directly from the
photometric redshifts of cluster member galaxies $z_{\rm opt}$ and from weak
lensing tomography $z_{\rm tomo}$. This figure includes all $85$ $\nu>3.5$
weak lensing peaks with optical counterparts, whose 3D shear signal is
consistent at $\chi^{2}_{\mathrm{red}}<3$ with the expected increase as a
function of redshift.
Our tomographic analysis confirms the identification of $85$ clusters with
$\chi^{2}_{\mathrm{red}}=\chi^{2}/{\rm dof}<3$. The mean redshift and velocity
dispersion of these clusters are $\langle z_{c}\rangle=0.36$ and
$\langle\sigma_{v}\rangle=658.8\rm km~{}s^{-1}$. Their full properties are
listed in Table LABEL:tab:matchpeaks. Reassuringly, we find that the inferred
lens redshifts are effectively identical for either SIS or NFW profile fits,
and are consistent (although noisy) with the (independent) photometric
redshifts of the cluster member galaxies. Figure 22 compares the redshift
estimates from tomographic gravitational lensing and optical spectroscopy of
member galaxies for these $85$ clusters.
At the faint end of our source galaxy sample, photo-$z$ estimates will be
unreliable because of noise in the photometry and degeneracies in the broad-
band colors of galaxies with different spectral energy distributions at
different redshifts. This will show up as a “double peak” in the posterior
probability of the redshift distribution. Using only the best-fit peak might
be randomly picking whichever of these peaks is higher because of noise. This
often biases lensing analyses because the expected lensing signal may be much
higher at one redshift than the other. To check for such an effect, we redo
the tomographic analysis without any photo-$z$s that have a double peak
(Arnouts et al. 2010). For cluster c3 in our catalog, we get very similar fit
results: $z_{\rm SIS}=0.26^{+0.13}_{-0.12}$ and
$\sigma_{v}=724.0^{+128.3}_{-131.5}$ for the SIS model, and $z_{\rm
NFW}=0.26^{+0.09}_{-0.11}$ and $m_{\rm NFW}=4.14^{+0.81}_{-0.76}$ for the NFW
model. This suggests that the fit is not very sensitive to that population of
galaxies with “double peaks” in photometric redshifts.
In cases where the data are poorly fit ($\chi^{2}_{\mathrm{red}}>3$) by a 3D
lensing signal, the inferred velocity dispersion or virial mass are typically
very small (often consistent with zero). These systems are probably spurious
peaks due to noise or projection effects. In addition, two peaks (c71 and
c104) have an unphysically high velocity dispersion $\sigma_{\rm
tomo}>1300~{}\rm km~{}s^{-1}$. After careful examination of the image data, we
found that they lie near two strongly saturated stars whose flux extends
beyond the masked regions, possibly degrading galaxy shape measurements.
One important goal of cluster lensing is to measure the total mass of systems,
which can also be estimated from the velocity dispersion of its member
galaxies or from its X-ray emission, under various assumptions about the state
of the intra-cluster medium and hydrostatic equilibrium (e.g., Bahcall et al.
1995; Carlberg et al. 1996; Carlstrom et al. 2002).
Figure 23.— Scaling relation between X-ray and weak lensing tomographic
measurements of the velocity dispersion of clusters. Black filled circles show
our data for four lensing-selected clusters. Red filled squares show the
lensing-selected clusters of Hamana et al. (2009). Blue open circles and green
open squares denote the X-ray selected clusters from Cypriano et al. (2004)
and Hoekstra (2007) respectively.
Figure 23 shows the relation between one mass proxy obtained from X-ray
observations and another obtained from weak lensing, for the $4$ of $7$
clusters in our sample with an X-ray counterpart and an SIS velocity
dispersion parameter well constrained by lensing. These are overlaid on the
results of another lensing-selected cluster sample (Hamana et al. 2009) and
two X-ray selected cluster samples (Cypriano et al. 2004; Hoekstra 2007). Even
combining these samples, the scatter is large. However, the consistency of the
scatter between the four samples suggests that neither detection method
produces a strong selection bias. Note that, since the density (and shear)
profile of real cluster is not necessarily a single power law, the best-fit
SIS model may depend on details of the fitting method, and the range over
which data are fit. Our results therefore are somewhat method-dependent. A
corollary of this issue is that it might also be possible to minimize scatter
in the $\sigma_{\rm xray}$-$\sigma_{\rm tomo}$ relation by optimizing the
tomographic lens fitting method.
## 7\. Conclusions
We have presented a tomographic weak gravitational lensing analysis of the
completed $64~{}\rm deg^{2}$ CFHTLS-Wide W1 field, demonstrating some of the
power of lensing to probe mass in galaxy clusters. We measured the shapes of
distant galaxies using the KSB shape measurement method, which we verified
against shape measurements from high-resolution HST imaging of an overlapping
sky area. We also obtained consistent shape measurements using two independent
imaging bands. The level of residual shape measurement systematics is an order
of magnitude lower than the $1\%$–$10\%$ shear signal expected in galaxy
clusters, so this analysis is acceptable for cluster studies.
We have reconstructed the largest contiguous “dark matter map” convergence
field to date, using two different smoothing scales to help remove spurious
noise peaks. From this map, we performed the largest lensing-selected blind
cluster search to date, finding $301$ local maxima in the lensing map with
detection significance $\nu>3.5$. Once sources of noise are properly modeled
from the intrinsic shapes of galaxies and substructure, this is consistent
with predictions from a $\Lambda$CDM cosmology. Note that our theoretical
calculations (Fan et al. 2010) do not consider projection effects of
structures along the line of sight.
To identify counterparts of the weak lensing peaks, we match our cluster
candidates to the K2 galaxy cluster catalog, created using photometric
redshift estimates across the CFHTLS-Wide W1 field (Thanjavur et al. 2009). Of
the $301$ peaks with $\nu>3.5$, $126$ have a corresponding optically-detected
BCG within $3\arcmin.0$. In the (much smaller) survey area that overlaps the
XMM-LSS survey, we also find matches for seven lensing peaks with X-ray
selected clusters. Thus, many of the candidate peaks are indeed likely just
noise.
Tomographic weak lensing techniques dramatically improve standard 2D
algorithms. In a full 3D lensing analysis of the $\nu>3.5$ peaks, we further
distinguish real clusters from noise fluctuations, and confirm (at
$\chi^{2}_{\rm red}<3$) the identification of $85$ clusters. Importantly, we
obtain independent measurements of the cluster lens redshifts, which are
consistent with the redshifts of their previously-identified optical
counterparts. For each cluster, we fit NFW and SIS radial profiles to the
lensing data to measure the mass or velocity dispersion $\sigma_{\rm tomo}$.
The clusters’ mean redshift and velocity dispersion is $\langle
z_{c}\rangle=0.36$ and $\langle\sigma_{c}\rangle=658.8~{}\rm km~{}s^{-1}$.
Weak lensing measurements of the total mass in the four of our clusters with
X-ray counterparts are also in reasonable agreement with mass estimates
obtained from X-ray emission. Future surveys, such as DES, LSST, KDUST and
EUCLID, will be able to apply these techniques to map clusters in much larger
volumes. Such large catalogs will also be able to tightly constrain
cosmological models (Takada & Bridle 2007; Dietrich & Hartlap 2010).
## Acknowledgments
The authors thank Bernard Fort, Liping Fu, Bo Qin, Catherine Heymans, and
Ludovic Van Waerbeke for useful discussions. H.Y.S. acknowledges support from
Sino French laboratories FCPPL and Origins and CPPM hospitality during stays
in France. H.Y.S. acknowledges the support from NSFC of China under grants
11103011 and China Postdoctoral Science Foundation. J.P.K. acknowledges
supports from CNRS as well as PNCG and CNES. Z.H.F. acknowledges the support
from NSFC of China under grants 10773001, 11033005, and 973 program
2007CB815401. M.L. acknowledges the Centre National de la Recherche
Scientifique (CNRS) for its support. The Dark Cosmology Centre is funded by
the Danish National Research Foundation.
This work is based on observations obtained with MegaPrime/MegaCam, a joint
project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT)
which is operated by the National Research Council (NRC) of Canada, the
Institut National des Science de l’Univers of the Centre National de la
Recherche Scientifique (CNRS) of France, and the University of Hawaii. This
work is based in part on data products produced at TERAPIX and the Canadian
Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy
Survey, a collaborative project of NRC and CNRS.
This work also uses observations obtained with the Hubble Space Telescope. The
HST COSMOS Treasury program was supported through NASA grant HST-GO-09822. We
thank Tony Roman, Denise Taylor, and David Soderblom for their assistance in
planning and scheduling the extensive COSMOS observations. We thank the NASA
IPAC/IRSA staff (Anastasia Laity, Anastasia Alexov, Bruce Berriman, and John
Good) for providing online archive and server capabilities for the COSMOS data
sets. It is also our pleasure to gratefully acknowledge the contributions of
the entire COSMOS collaboration, consisting of more than 70 scientists. More
information on the COSMOS survey is available at
http://www.astro.caltech.edu/~cosmos.
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Table 5Catalog of the Matching Groups/Clusters between $\nu>3.5$ Convergence Peaks and K2-detected Groups/Clusters in CFHTLS-Wide W1 Fields. K2 redshift denotes the median redshift of bright ($i\leq 20$) cluster members ID | $z_{\rm K2}$ | $\alpha_{\rm K2}$ | $\delta_{\rm K2}$ | $\rm sig_{r-i}$ | $\alpha$ | $\delta$ | d | $z_{\rm SIS}$ | $\sigma_{v}$ | $\chi^{2}_{\rm SIS}$ | $z_{\rm NFW}$ | $m_{\rm NFW}$ | $m_{200}$ | $\chi^{2}_{\rm NFW}$ | $\nu$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | | | arcsec | | $\rm km/s$ | | | $10^{14}M_{\odot}/h$ | $10^{14}M_{\odot}/h$ | |
c1-w1 | 0.18 | 30.990 | -4.223 | 3.19 | 30.976 | -4.231 | 59.2 | $0.28^{+0.17}_{-0.10}$ | $829.7^{+225.1}_{-308.9}$ | 0.42 | $0.29^{+0.16}_{-0.14}$ | $2.88^{+6.22}_{-2.15}$ | $2.50^{+5.39}_{-1.86}$ | 0.41 | 4.458
c2-w1 | 0.48 | 30.625 | -3.966 | 4.09 | 30.619 | -3.970 | 29.6 | $0.26^{+0.08}_{-0.10}$ | $945.5^{+173.8}_{-212.5}$ | 1.52 | $0.26^{+0.07}_{-0.08}$ | $11.0^{+17.8}_{-7.20}$ | $9.32^{+15.1}_{-6.10}$ | 1.50 | 4.113
c3-w1 | 0.28 | 30.421 | -5.030 | 7.45 | 30.420 | -5.030 | 1.62 | $0.26^{+0.11}_{-0.10}$ | $701.6^{+134.2}_{-132.8}$ | 0.85 | $0.26^{+0.12}_{-0.12}$ | $3.51^{+0.63}_{-0.66}$ | $3.02^{+0.54}_{-0.57}$ | 0.88 | 5.395
c4-w1 | 0.72 | 30.296 | -6.089 | 9.85 | 30.260 | -6.088 | 128.5 | $0.69^{+0.21}_{-0.22}$ | $\sim 0$ | 4.33 | $0.69^{+0.23}_{-0.20}$ | $\sim 0$ | $\sim 0$ | 4.25 | 3.732
c5-w1 | 0.38 | 30.860 | -5.925 | 4.72 | 30.867 | -5.945 | 74.3 | $0.41^{+0.10}_{-0.09}$ | $503.1^{+102.5}_{-104.1}$ | 1.37 | $0.40^{+0.10}_{-0.11}$ | $0.78^{+0.16}_{-0.17}$ | $0.70^{+0.14}_{-0.15}$ | 1.37 | 3.510
c6-w1 | 0.27 | 30.840 | -7.331 | 3.68 | 30.836 | -7.341 | 38.2 | $0.20^{+0.09}_{-0.12}$ | $559.8^{+131.8}_{-139.6}$ | 0.68 | $0.22^{+0.11}_{-0.10}$ | $0.79^{+0.18}_{-0.17}$ | $0.69^{+0.16}_{-0.15}$ | 0.51 | 4.184
c7-w1 | 0.33 | 31.077 | -7.213 | 3.20 | 31.080 | -7.216 | 15.5 | $0.39^{+0.30}_{-0.23}$ | $646.1^{+183.9}_{-222.3}$ | 1.05 | $0.40^{+0.25}_{-0.35}$ | $1.38^{+2.09}_{-1.18}$ | $1.23^{+1.86}_{-1.05}$ | 1.03 | 3.677
c8-w1 | -99.0 | 30.779 | -7.142 | 4.28 | 30.785 | -7.141 | 20.1 | $0.76^{+0.15}_{-0.19}$ | $162.3^{+137.0}_{-125.4}$ | 0.78 | $0.80^{+0.16}_{-0.17}$ | $0.11^{+0.08}_{-0.09}$ | $0.10^{+0.07}_{-0.08}$ | 0.73 | 4.586
c9-w1 | 0.81 | 30.884 | -6.736 | 5.78 | 30.882 | -6.749 | 46.8 | $0.77^{+0.25}_{-0.26}$ | $892.1^{+274.1}_{-291.7}$ | 1.25 | $0.78^{+0.24}_{-0.26}$ | $4.07^{+0.98}_{-1.04}$ | $3.82^{+0.92}_{-0.97}$ | 1.21 | 3.722
c10-w1 | 0.36 | 30.628 | -6.524 | 5.11 | 30.649 | -6.536 | 85.3 | $0.41^{+0.13}_{-0.12}$ | $\sim 0$ | 9.34 | $0.41^{+0.12}_{-0.11}$ | $\sim 0$ | $\sim 0$ | 8.30 | 3.897
c11-w1 | 0.31 | 30.364 | -8.283 | 4.20 | 30.369 | -8.275 | 35.2 | $0.35^{+0.16}_{-0.13}$ | $\sim 0$ | 5.22 | $0.35^{+0.15}_{-0.17}$ | $\sim 0$ | $\sim 0$ | 4.28 | 3.710
c12-w1 | 0.14 | 30.325 | -7.651 | 4.43 | 30.346 | -7.659 | 81.9 | $0.22^{+0.12}_{-0.11}$ | $396.6^{+128.6}_{-102.1}$ | 1.96 | $0.42^{+0.15}_{-0.14}$ | $0.28^{+0.07}_{-0.06}$ | $0.25^{+0.06}_{-0.05}$ | 1.59 | 4.728
c13-w1 | 0.23 | 30.708 | -9.337 | 4.83 | 30.704 | -9.356 | 69.7 | $0.21^{+0.16}_{-0.10}$ | $473.8^{+221.3}_{-239.5}$ | 2.15 | $0.21^{+0.13}_{-0.11}$ | $0.16^{+0.08}_{-0.09}$ | $0.14^{+0.07}_{-0.08}$ | 2.01 | 4.084
c14-w1 | 0.40 | 30.534 | -8.437 | 7.52 | 30.537 | -8.417 | 71.4 | $0.43^{+0.09}_{-0.08}$ | $691.1^{+102.1}_{-97.8}$ | 0.14 | $0.43^{+0.09}_{-0.11}$ | $1.50^{+0.19}_{-0.17}$ | $1.34^{+0.17}_{-0.15}$ | 0.14 | 4.045
c15-w1 | 0.32 | 30.870 | -9.817 | 18.02 | 30.881 | -9.838 | 86.2 | $0.36^{+0.14}_{-0.13}$ | $483.9^{+120.9}_{-112.7}$ | 0.68 | $0.36^{+0.13}_{-0.15}$ | $0.70^{+0.16}_{-0.18}$ | $0.62^{+0.14}_{-0.16}$ | 0.59 | 4.835
c16-w1 | 0.66 | 30.452 | -10.853 | 5.81 | 30.463 | -10.840 | 59.6 | $0.62^{+0.08}_{-0.09}$ | $747.7^{+114.3}_{-117.2}$ | 1.15 | $0.62^{+0.10}_{-0.08}$ | $2.24^{+0.33}_{-0.35}$ | $2.06^{+0.30}_{-0.32}$ | 1.07 | 3.511
c17-w1 | 0.32 | 32.021 | -4.583 | 3.85 | 32.045 | -4.542 | 171.4 | $0.35^{+0.10}_{-0.09}$ | $931.9^{+218.7}_{-239.2}$ | 1.12 | $0.35^{+0.11}_{-0.14}$ | $5.52^{+5.32}_{-3.20}$ | $4.81^{+4.63}_{-2.79}$ | 0.93 | 3.628
c18-w1 | 0.25 | 31.797 | -4.002 | 3.04 | 31.772 | -3.963 | 165.9 | $0.51^{+0.45}_{-0.17}$ | $874.1^{+175.9}_{-340.5}$ | 0.69 | $0.50^{+0.21}_{-0.14}$ | $5.09^{+6.41}_{-3.42}$ | $4.56^{+5.75}_{-3.07}$ | 0.61 | 5.011
c19-w1 | 0.29 | 31.793 | -3.894 | 4.37 | 31.792 | -3.854 | 142.4 | $0.39^{+0.19}_{-0.18}$ | $\sim 0$ | 6.94 | $0.38^{+0.20}_{-0.17}$ | $\sim 0$ | $\sim 0$ | 5.82 | 4.540
c20-w1 | 0.19 | 31.283 | -4.972 | 3.17 | 31.308 | -4.993 | 118.5 | $0.12^{+0.12}_{-0.13}$ | $\sim 0$ | 5.12 | $0.11^{+0.11}_{-0.12}$ | $\sim 0$ | $\sim 0$ | 4.64 | 3.899
c21-w1 | 0.42 | 31.558 | -6.167 | 4.34 | 31.538 | -6.167 | 70.2 | $0.32^{+0.14}_{-0.16}$ | $\sim 0$ | 3.73 | $0.31^{+0.13}_{-0.11}$ | $\sim 0$ | $\sim 0$ | 4.16 | 4.142
c22-w1 | 0.41 | 31.272 | -5.923 | 4.63 | 31.254 | -5.943 | 95.4 | $0.51^{+0.26}_{-0.25}$ | $273.0^{+164.5}_{-157.3}$ | 1.51 | $0.49^{+0.15}_{-0.18}$ | $0.24^{+0.16}_{-0.15}$ | $0.22^{+0.15}_{-0.14}$ | 1.22 | 3.860
c23-w1 | 0.35 | 31.667 | -7.384 | 4.89 | 31.665 | -7.400 | 57.0 | $0.26^{+0.21}_{-0.22}$ | $\sim 0$ | 3.14 | $0.26^{+0.19}_{-0.20}$ | $\sim 0$ | $\sim 0$ | 3.30 | 4.211
c24-w1 | 0.40 | 31.358 | -7.588 | 10.03 | 31.353 | -7.589 | 20.6 | $0.88^{+0.55}_{-0.80}$ | $659.3^{+141.3}_{-181.0}$ | 0.82 | $0.88^{+0.11}_{-0.12}$ | $1.16^{+0.70}_{-0.45}$ | $1.11^{+0.67}_{-0.43}$ | 0.71 | 5.311
c25-w1 | 0.42 | 31.927 | -7.502 | 3.33 | 31.953 | -7.489 | 101.7 | $0.32^{+0.31}_{-0.28}$ | $\sim 0$ | 3.48 | $0.32^{+0.29}_{-0.27}$ | $\sim 0$ | $\sim 0$ | 3.48 | 3.860
c26-w1 | 0.24 | 31.320 | -7.680 | 3.00 | 31.326 | -7.664 | 63.5 | $0.22^{+0.13}_{-0.12}$ | $417.4^{+138.7}_{-131.5}$ | 0.86 | $0.22^{+0.12}_{-0.12}$ | $0.49^{+0.17}_{-0.16}$ | $0.43^{+0.15}_{-0.14}$ | 0.66 | 4.430
c27-w1 | 0.26 | 31.595 | -8.807 | 4.22 | 31.617 | -8.806 | 75.8 | $0.17^{+0.09}_{-0.14}$ | $646.2^{+103.0}_{-135.1}$ | 1.07 | $0.19^{+0.11}_{-0.16}$ | $1.95^{+1.54}_{-1.27}$ | $1.67^{+1.32}_{-1.09}$ | 1.12 | 4.423
c28-w1 | 0.08 | 31.769 | -8.495 | 3.43 | 31.724 | -8.512 | 171.6 | $0.06^{+0.10}_{-0.11}$ | $426.0^{+153.1}_{-166.8}$ | 2.41 | $0.08^{+0.12}_{-0.10}$ | $0.65^{+0.21}_{-0.18}$ | $0.55^{+0.18}_{-0.15}$ | 1.54 | 3.651
c29-w1 | 0.27 | 31.230 | -10.400 | 14.90 | 31.230 | -10.407 | 24.4 | $0.21^{+0.16}_{-0.18}$ | $\sim 0$ | 5.94 | $0.22^{+0.13}_{-0.15}$ | $\sim 0$ | $\sim 0$ | 4.60 | 4.104
c30-w1 | 0.19 | 32.773 | -4.129 | 3.19 | 32.758 | -4.151 | 96.1 | $0.37^{+0.13}_{-0.15}$ | $455.5^{+132.6}_{-138.9}$ | 0.34 | $0.40^{+0.14}_{-0.15}$ | $0.48^{+0.11}_{-0.13}$ | $0.43^{+0.10}_{-0.12}$ | 0.43 | 4.674
c31-w1 | 0.34 | 32.634 | -4.123 | 6.94 | 32.653 | -4.132 | 76.5 | $0.28^{+0.12}_{-0.11}$ | $1134.9^{+243.1}_{-237.0}$ | 1.38 | $0.28^{+0.12}_{-0.12}$ | $12.3^{+2.41}_{-2.35}$ | $10.5^{+2.05}_{-2.00}$ | 1.17 | 4.899
c32-w1 | 0.28 | 32.981 | -4.688 | 3.95 | 32.957 | -4.652 | 157.7 | $0.22^{+0.17}_{-0.14}$ | $\sim 0$ | 3.59 | $0.23^{+0.16}_{-0.17}$ | $\sim 0$ | $\sim 0$ | 3.15 | 3.737
c33-w1 | 0.23 | 32.435 | -5.730 | 3.36 | 32.456 | -5.726 | 77.9 | $0.21^{+0.07}_{-0.10}$ | $599.6^{+97.8}_{-117.5}$ | 1.04 | $0.22^{+0.11}_{-0.14}$ | $1.09^{+0.12}_{-0.13}$ | $0.94^{+0.10}_{-0.11}$ | 1.11 | 3.576
c34-w1 | 0.48 | 32.669 | -7.455 | 5.61 | 32.675 | -7.449 | 31.3 | $0.36^{+0.14}_{-0.11}$ | $703.7^{+150.7}_{-193.7}$ | 0.73 | $0.39^{+0.25}_{-0.17}$ | $1.71^{+2.27}_{-1.28}$ | $1.52^{+2.02}_{-1.14}$ | 0.68 | 4.474
c35-w1 | 0.45 | 32.986 | -7.123 | 3.48 | 32.941 | -7.132 | 163.9 | $0.41^{+0.24}_{-0.23}$ | $1071.5^{+262.2}_{-239.6}$ | 1.89 | $0.42^{+0.17}_{-0.18}$ | $6.89^{+1.07}_{-1.13}$ | $6.10^{+0.95}_{-1.00}$ | 1.81 | 3.911
c36-w1 | 0.40 | 32.865 | -6.694 | 3.31 | 32.839 | -6.726 | 145.2 | $0.36^{+0.15}_{-0.16}$ | $580.5^{+121.5}_{-124.3}$ | 1.51 | $0.36^{+0.16}_{-0.17}$ | $1.27^{+0.23}_{-0.25}$ | $1.13^{+0.20}_{-0.22}$ | 1.28 | 4.430
c37-w1 | 0.62 | 32.599 | -6.543 | 6.99 | 32.575 | -6.541 | 85.3 | $0.51^{+0.29}_{-0.27}$ | $\sim 0$ | 4.12 | $0.51^{+0.28}_{-0.27}$ | $\sim 0$ | $\sim 0$ | 4.54 | 3.504
c38-w1 | 0.33 | 32.837 | -8.396 | 8.91 | 32.824 | -8.403 | 55.2 | $0.39^{+0.10}_{-0.10}$ | $\sim 0$ | 3.28 | $0.41^{+0.11}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 3.17 | 3.774
c39-w1 | 0.34 | 32.135 | -7.730 | 3.54 | 32.167 | -7.750 | 133.5 | $0.37^{+0.17}_{-0.19}$ | $682.9^{+225.9}_{-241.1}$ | 1.34 | $0.36^{+0.16}_{-0.18}$ | $2.50^{+2.01}_{-1.67}$ | $2.19^{+1.76}_{-1.46}$ | 1.13 | 3.632
c40-w1 | 0.46 | 33.012 | -7.676 | 9.58 | 33.020 | -7.659 | 69.4 | $0.76^{+0.13}_{-0.13}$ | $1101.3^{+261.1}_{-263.8}$ | 1.21 | $0.83^{+0.13}_{-0.15}$ | $4.49^{+1.18}_{-1.26}$ | $4.23^{+1.11}_{-1.19}$ | 1.05 | 4.486
c41-w1 | 0.51 | 32.658 | -7.471 | 6.09 | 32.655 | -7.445 | 93.5 | $0.55^{+0.21}_{-0.18}$ | $\sim 0$ | 4.11 | $0.55^{+0.17}_{-0.16}$ | $\sim 0$ | $\sim 0$ | 3.65 | 4.157
c42-w1 | 0.71 | 32.313 | -9.244 | 15.30 | 32.324 | -9.237 | 44.9 | $0.78^{+2.12}_{-0.32}$ | $530.9^{+138.9}_{-208.4}$ | 1.68 | $0.77^{+0.13}_{-0.12}$ | $1.10^{+0.60}_{-0.40}$ | $1.04^{+0.57}_{-0.38}$ | 1.31 | 3.792
c43-w1 | 0.65 | 32.795 | -9.139 | 6.56 | 32.815 | -9.157 | 95.8 | $0.61^{+0.20}_{-0.12}$ | $678.1^{+112.3}_{-143.9}$ | 0.83 | $0.66^{+0.11}_{-0.09}$ | $1.80^{+1.15}_{-0.95}$ | $1.67^{+1.07}_{-0.88}$ | 0.66 | 3.815
c44-w1 | 0.29 | 32.553 | -8.562 | 11.58 | 32.575 | -8.560 | 78.4 | $0.25^{+0.12}_{-0.11}$ | $971.1^{+342.1}_{-382.5}$ | 0.74 | $0.26^{+0.09}_{-0.08}$ | $10.9^{+5.61}_{-6.85}$ | $6.85^{+9.30}_{-4.78}$ | 0.71 | 4.367
c45-w1 | 0.30 | 32.961 | -9.994 | 4.98 | 32.971 | -9.978 | 68.7 | $0.35^{+0.15}_{-0.16}$ | $\sim 0$ | 3.51 | $0.34^{+0.19}_{-0.17}$ | $\sim 0$ | $\sim 0$ | 3.38 | 4.109
c46-w1 | 0.62 | 32.847 | -9.875 | 3.04 | 32.883 | -9.877 | 129.3 | $0.67^{+0.17}_{-0.16}$ | $521.8^{+133.9}_{-129.1}$ | 1.67 | $0.67^{+0.16}_{-0.18}$ | $0.66^{+0.12}_{-0.14}$ | $0.62^{+0.11}_{-0.13}$ | 1.56 | 4.238
c47-w1 | 0.28 | 33.607 | -6.461 | 9.74 | 33.596 | -6.454 | 47.5 | $0.24^{+0.11}_{-0.10}$ | $698.8^{+198.2}_{-211.7}$ | 1.41 | $0.25^{+0.10}_{-0.10}$ | $2.35^{+1.12}_{-1.09}$ | $2.04^{+0.97}_{-0.95}$ | 1.28 | 3.562
c48-w1 | 0.65 | 34.017 | -6.137 | 11.48 | 34.001 | -6.095 | 161.7 | $0.64^{+0.29}_{-0.14}$ | $152.8^{+162.4}_{-139.6}$ | 0.65 | $0.64^{+0.19}_{-0.15}$ | $0.07^{+0.11}_{-0.10}$ | $0.06^{+0.10}_{-0.09}$ | 0.63 | 3.930
c49-w1 | 0.30 | 33.780 | -5.981 | 7.22 | 33.779 | -5.978 | 11.9 | $0.25^{+0.13}_{-0.13}$ | $547.5^{+120.8}_{-117.9}$ | 2.59 | $0.27^{+0.14}_{-0.11}$ | $0.78^{+0.18}_{-0.16}$ | $0.69^{+0.16}_{-0.14}$ | 1.83 | 4.728
c50-w1 | 0.19 | 33.941 | -5.930 | 3.46 | 33.956 | -5.976 | 174.8 | $0.37^{+0.16}_{-0.17}$ | $667.4^{+156.3}_{-198.7}$ | 1.28 | $0.37^{+0.12}_{-0.14}$ | $2.49^{+2.22}_{-1.55}$ | $2.20^{+1.96}_{-1.37}$ | 1.18 | 3.579
c51-w1 | 0.35 | 33.293 | -5.625 | 4.20 | 33.295 | -5.613 | 40.8 | $0.25^{+0.07}_{-0.10}$ | $629.1^{+98.4}_{-118.7}$ | 1.96 | $0.25^{+0.11}_{-0.13}$ | $1.86^{+2.20}_{-1.44}$ | $1.61^{+1.90}_{-1.25}$ | 1.70 | 4.205
c52-w1 | 0.69 | 33.528 | -7.145 | 5.13 | 33.509 | -7.148 | 67.8 | $0.31^{+0.29}_{-0.30}$ | $\sim 0$ | 7.76 | $0.33^{+0.21}_{-0.22}$ | $\sim 0$ | $\sim 0$ | 6.81 | 3.739
c53-w1 | 0.28 | 33.402 | -6.953 | 3.15 | 33.384 | -6.944 | 70.6 | $0.23^{+0.12}_{-0.11}$ | $694.7^{+133.5}_{-129.8}$ | 0.86 | $0.22^{+0.11}_{-0.11}$ | $1.65^{+0.29}_{-0.21}$ | $1.44^{+0.25}_{-0.18}$ | 0.61 | 4.166
c54-w1 | 0.31 | 33.728 | -6.539 | 4.65 | 33.748 | -6.495 | 171.4 | $0.38^{+0.14}_{-0.15}$ | $193.1^{+109.6}_{-101.3}$ | 1.77 | $0.37^{+0.16}_{-0.14}$ | $0.19^{+0.10}_{-0.11}$ | $0.17^{+0.09}_{-0.10}$ | 1.73 | 3.590
c55-w1 | 0.32 | 33.113 | -9.926 | 9.79 | 33.130 | -9.950 | 106.0 | $0.83^{+2.07}_{-0.30}$ | $606.1^{+163.2}_{-231.8}$ | 1.63 | $0.84^{+0.14}_{-0.13}$ | $0.97^{+0.33}_{-0.29}$ | $0.92^{+0.31}_{-0.28}$ | 1.58 | 3.809
c56-w1 | 0.34 | 33.720 | -9.936 | 3.35 | 33.737 | -9.908 | 118.3 | $0.50^{+0.21}_{-0.19}$ | $597.4^{+147.1}_{-138.2}$ | 0.93 | $0.49^{+0.19}_{-0.19}$ | $1.09^{+0.19}_{-0.20}$ | $0.99^{+0.17}_{-0.18}$ | 0.94 | 3.872
c57-w1 | 0.39 | 33.427 | -9.769 | 4.34 | 33.426 | -9.770 | 4.689 | $0.42^{+0.15}_{-0.14}$ | $514.7^{+128.7}_{-123.5}$ | 0.16 | $0.43^{+0.13}_{-0.15}$ | $0.74^{+0.16}_{-0.18}$ | $0.66^{+0.14}_{-0.16}$ | 0.16 | 3.570
c58-w1 | 0.51 | 34.244 | -4.554 | 6.43 | 34.233 | -4.585 | 118.0 | $0.56^{+0.25}_{-0.16}$ | $449.3^{+211.9}_{-163.0}$ | 1.34 | $0.55^{+0.21}_{-0.18}$ | $0.45^{+0.20}_{-0.14}$ | $0.42^{+0.18}_{-0.13}$ | 1.21 | 6.448
c59-w1 | 0.75 | 34.071 | -4.157 | 9.06 | 34.030 | -4.131 | 174.9 | $0.82^{+0.22}_{-0.23}$ | $\sim 0$ | 3.87 | $0.85^{+0.21}_{-0.21}$ | $\sim 0$ | $\sim 0$ | 3.70 | 3.564
c60-w1 | 0.62 | 34.400 | -3.946 | 13.75 | 34.436 | -3.978 | 176.1 | $0.65^{+0.19}_{-0.17}$ | $830.3^{+196.8}_{-225.4}$ | 1.32 | $0.64^{+0.16}_{-0.17}$ | $4.63^{+1.15}_{-1.17}$ | $4.24^{+1.05}_{-1.07}$ | 1.23 | 4.363
c61-w1 | 0.37 | 34.327 | -3.772 | 3.13 | 34.311 | -3.749 | 100.0 | $0.27^{+0.12}_{-0.14}$ | $\sim 0$ | 3.56 | $0.29^{+0.13}_{-0.12}$ | $\sim 0$ | $\sim 0$ | 3.21 | 3.791
c62-w1 | 0.36 | 34.657 | -5.570 | 4.17 | 34.684 | -5.572 | 98.4 | $0.57^{+0.13}_{-0.14}$ | $909.7^{+61.5}_{-60.1}$ | 1.55 | $0.55^{+0.15}_{-0.13}$ | $5.31^{+0.32}_{-0.30}$ | $4.82^{+0.29}_{-0.27}$ | 1.47 | 4.194
c63-w1 | 0.79 | 34.511 | -6.743 | 15.30 | 34.531 | -6.488 | 89.8 | $0.70^{+0.31}_{-0.28}$ | $\sim 0$ | 4.33 | $0.71^{+0.26}_{-0.25}$ | $\sim 0$ | $\sim 0$ | 4.09 | 3.931
c64-w1 | 0.33 | 34.655 | -5.674 | 9.59 | 34.653 | -5.626 | 170.7 | $0.42^{+0.14}_{-0.11}$ | $\sim 0$ | 4.18 | $0.41^{+0.13}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 3.87 | 4.186
c65-w1 | 0.35 | 34.092 | -7.369 | 5.17 | 34.070 | -7.380 | 87.7 | $0.27^{+0.12}_{-0.12}$ | $\sim 0$ | 4.28 | $0.26^{+0.10}_{-0.09}$ | $\sim 0$ | $\sim 0$ | 4.09 | 5.230
c66-w1 | 0.33 | 34.483 | -6.755 | 3.35 | 34.449 | -6.742 | 129.5 | $0.31^{+0.10}_{-0.19}$ | $\sim 0$ | 3.26 | $0.31^{+0.11}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 3.13 | 4.692
c67-w1 | 0.22 | 34.350 | -6.687 | 3.84 | 34.390 | -6.717 | 180.0 | $0.37^{+0.26}_{-0.19}$ | $794.7^{+235.7}_{-340.1}$ | 1.72 | $0.39^{+0.06}_{-0.05}$ | $3.63^{+0.90}_{-1.71}$ | $3.20^{+0.79}_{-1.51}$ | 1.51 | 4.515
c68-w1 | 0.33 | 34.219 | -8.229 | 7.14 | 34.219 | -8.223 | 21.6 | $0.38^{+0.16}_{-0.17}$ | $\sim 0$ | 3.09 | $0.37^{+0.15}_{-0.16}$ | $\sim 0$ | $\sim 0$ | 3.06 | 3.967
c69-w1 | 0.71 | 34.948 | -8.120 | 13.80 | 34.965 | -8.091 | 117.5 | $0.75^{+0.21}_{-0.18}$ | $514.0^{+123.4}_{-109.9}$ | 1.27 | $0.74^{+0.19}_{-0.18}$ | $0.63^{+0.15}_{-0.14}$ | $0.59^{+0.14}_{-0.13}$ | 1.22 | 3.737
c70-w1 | 0.70 | 34.837 | -7.585 | 13.38 | 34.829 | -7.570 | 59.7 | $0.76^{+0.07}_{-0.08}$ | $545.4^{+94.2}_{-98.4}$ | 0.62 | $0.76^{+0.06}_{-0.08}$ | $0.90^{+0.13}_{-0.17}$ | $0.85^{+0.12}_{-0.16}$ | 0.58 | 4.497
c71-w1 | 0.28 | 34.415 | -9.099 | 6.19 | 34.419 | -9.104 | 21.4 | $0.43^{+0.05}_{-0.06}$ | $1342.7^{+242.7}_{-249.2}$ | 0.95 | $0.44^{+0.06}_{-0.06}$ | $33.1^{+5.21}_{-5.32}$ | $28.7^{+4.51}_{-4.61}$ | 0.85 | 5.520
c72-w1 | 0.31 | 34.476 | -9.849 | 4.14 | 34.439 | -9.837 | 138.6 | $0.31^{+0.13}_{-0.14}$ | $1079.6^{+251.6}_{-266.8}$ | 1.24 | $0.27^{+0.16}_{-0.14}$ | $13.2^{+2.86}_{-2.73}$ | $11.3^{+2.44}_{-2.33}$ | 1.23 | 3.705
c73-w1 | 0.32 | 34.692 | -9.463 | 3.50 | 34.729 | -9.454 | 132.6 | $0.22^{+0.11}_{-0.15}$ | $\sim 0$ | 3.61 | $0.22^{+0.13}_{-0.15}$ | $\sim 0$ | $\sim 0$ | 3.58 | 3.800
c74-w1 | 0.41 | 34.280 | -9.354 | 15.23 | 34.290 | -9.367 | 60.1 | $0.47^{+0.17}_{-0.18}$ | $\sim 0$ | 3.33 | $0.49^{+0.16}_{-0.17}$ | $\sim 0$ | $\sim 0$ | 3.30 | 3.990
c75-w1 | 0.21 | 34.029 | -10.419 | 3.37 | 34.055 | -10.421 | 92.8 | $0.53^{+0.97}_{-0.17}$ | $506.9^{+140.8}_{-207.8}$ | 1.45 | $0.57^{+0.92}_{-0.18}$ | $0.57^{+0.82}_{-0.45}$ | $0.53^{+0.76}_{-0.42}$ | 1.42 | 4.841
c76-w1 | -99.0 | 35.401 | -4.420 | 3.72 | 35.374 | -4.391 | 140.7 | $0.50^{+0.23}_{-0.25}$ | $\sim 0$ | 6.14 | $0.42^{+0.23}_{-0.21}$ | $\sim 0$ | $\sim 0$ | 5.80 | 4.355
c77-w1 | 0.43 | 35.441 | -3.772 | 22.39 | 35.456 | -3.768 | 57.2 | $0.34^{+0.11}_{-0.12}$ | $766.1^{+182.5}_{-229.9}$ | 0.86 | $0.35^{+0.15}_{-0.15}$ | $2.04^{+2.78}_{-1.54}$ | $1.79^{+2.44}_{-1.35}$ | 0.82 | 4.209
c78-w1 | 0.44 | 35.595 | -4.890 | 3.26 | 35.601 | -4.892 | 23.0 | $0.31^{+0.15}_{-0.13}$ | $\sim 0$ | 4.45 | $0.31^{+0.14}_{-0.15}$ | $\sim 0$ | $\sim 0$ | 4.17 | 3.788
c79-w1 | 0.04 | 35.216 | -4.782 | 3.08 | 35.229 | -4.743 | 150.3 | $0.06^{+0.12}_{-0.11}$ | $573.8^{+113.9}_{-108.7}$ | 1.16 | $0.07^{+0.11}_{-0.10}$ | $2.78^{+0.33}_{-0.31}$ | $2.30^{+0.27}_{-0.26}$ | 1.09 | 4.204
c80-w1 | 0.59 | 35.436 | -6.358 | 6.80 | 35.436 | -6.354 | 14.1 | $0.45^{+0.19}_{-0.17}$ | $\sim 0$ | 6.63 | $0.43^{+0.20}_{-0.19}$ | $\sim 0$ | $\sim 0$ | 5.90 | 4.710
c81-w1 | 0.22 | 35.590 | -5.682 | 3.09 | 35.569 | -5.723 | 164.9 | $0.59^{+0.11}_{-0.12}$ | $1266.7^{+108.9}_{-117.3}$ | 1.35 | $0.67^{+0.13}_{-0.11}$ | $10.9^{+1.70}_{-1.50}$ | $9.97^{+1.56}_{-1.37}$ | 1.29 | 4.870
c82-w1 | 0.37 | 35.650 | -7.270 | 3.34 | 35.680 | -7.281 | 112.4 | $0.32^{+0.13}_{-0.11}$ | $628.4^{+106.1}_{-104.6}$ | 1.22 | $0.33^{+0.12}_{-0.10}$ | $1.80^{+0.22}_{-0.25}$ | $1.59^{+0.19}_{-0.22}$ | 1.21 | 3.923
c83-w1 | 0.32 | 35.830 | -7.997 | 5.21 | 35.826 | -7.992 | 21.4 | $0.08^{+0.07}_{-0.09}$ | $954.6^{120.0}_{132.6}$ | 1.68 | $0.10^{+0.04}_{-0.05}$ | $11.1^{+1.26}_{-1.30}$ | $9.10^{+1.03}_{-1.07}$ | 1.67 | 4.287
c84-w1 | 0.02 | 35.924 | -7.989 | 3.40 | 35.931 | -7.994 | 31.5 | $0.16^{+0.04}_{-0.05}$ | $789.9^{+64.3}_{-69.1}$ | 0.15 | $0.12^{+0.02}_{-0.01}$ | $7.50^{+0.82}_{-0.77}$ | $6.21^{+0.68}_{-0.64}$ | 0.19 | 4.127
c85-w1 | 0.35 | 35.486 | -8.950 | 7.92 | 35.486 | -8.937 | 48.4 | $0.25^{+0.15}_{-0.18}$ | $\sim 0$ | 3.98 | $0.24^{+0.17}_{-0.19}$ | $\sim 0$ | $\sim 0$ | 3.91 | 4.372
c86-w1 | 0.27 | 35.868 | -8.865 | 5.82 | 35.883 | -8.864 | 55.4 | $0.34^{+0.16}_{-0.14}$ | $523.9^{+122.8}_{-118.9}$ | 0.65 | $0.36^{+0.15}_{-0.16}$ | $0.79^{+0.12}_{-0.13}$ | $0.70^{+0.11}_{-0.12}$ | 0.41 | 4.779
c87-w1 | 0.32 | 35.821 | -9.347 | 6.25 | 35.805 | -9.344 | 61.1 | $0.38^{+0.19}_{-0.22}$ | $\sim 0$ | 3.52 | $0.39^{+0.18}_{-0.20}$ | $\sim 0$ | $\sim 0$ | 2.94 | 3.690
c88-w1 | 0.24 | 35.941 | -10.924 | 5.53 | 35.935 | -10.888 | 129.8 | $0.26^{+0.08}_{-0.09}$ | $538.0^{+112.3}_{-107.2}$ | 0.27 | $0.25^{+0.10}_{-0.08}$ | $0.90^{+0.15}_{-0.16}$ | $0.78^{+0.13}_{-0.14}$ | 0.22 | 3.543
c89-w1 | 0.34 | 35.827 | -10.399 | 3.79 | 35.828 | -10.406 | 26.4 | $0.43^{+0.17}_{-0.12}$ | $\sim 0$ | 4.77 | $0.42^{+0.16}_{-0.13}$ | $\sim 0$ | $\sim 0$ | 3.70 | 5.260
c90-w1 | 0.19 | 35.520 | -10.346 | 4.55 | 35.488 | -10.353 | 118.1 | $0.08^{+0.11}_{-0.10}$ | $\sim 0$ | 2.96 | $0.10^{+0.12}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 3.11 | 3.640
c91-w1 | 0.20 | 35.284 | -10.330 | 4.14 | 35.317 | -10.305 | 149.6 | $0.28^{+0.13}_{-0.14}$ | $\sim 0$ | 4.19 | $0.26^{+0.14}_{-0.12}$ | $\sim 0$ | $\sim 0$ | 3.72 | 4.422
c92-w1 | 0.65 | 36.372 | -4.258 | 4.79 | 36.372 | -4.248 | 36.9 | $0.65^{+0.26}_{-0.26}$ | $\sim 0$ | 5.69 | $0.66^{+0.22}_{-0.20}$ | $\sim 0$ | $\sim 0$ | 4.63 | 4.548
c93-w1 | 0.30 | 36.121 | -4.165 | 3.36 | 36.097 | -4.167 | 87.5 | $0.26^{+0.13}_{-0.12}$ | $359.6^{+131.2}_{-121.7}$ | 0.25 | $0.25^{+0.11}_{-0.11}$ | $0.77^{+0.17}_{-0.18}$ | $0.67^{+0.15}_{-0.16}$ | 0.20 | 4.063
c94-w1 | 0.55 | 36.108 | -5.088 | 3.13 | 36.077 | -5.102 | 122.2 | $0.20^{+0.17}_{-0.15}$ | $344.8^{+97.5}_{-89.1}$ | 0.78 | $0.20^{+0.18}_{-0.16}$ | $0.23^{+0.06}_{-0.04}$ | $0.20^{+0.05}_{-0.04}$ | 0.81 | 3.521
c95-w1 | 0.35 | 36.617 | -4.998 | 10.12 | 36.636 | -4.990 | 72.8 | $0.22^{+0.14}_{-0.15}$ | $\sim 0$ | 4.55 | $0.21^{+0.13}_{-0.14}$ | $\sim 0$ | $\sim 0$ | 4.15 | 3.948
c96-w1 | 0.50 | 36.121 | -4.821 | 8.92 | 36.116 | -4.852 | 114.5 | $0.47^{+0.12}_{-0.14}$ | $701.3^{+131.3}_{-129.0}$ | 0.61 | $0.47^{+0.13}_{-0.12}$ | $2.10^{+0.31}_{-0.27}$ | $1.89^{+0.28}_{-0.24}$ | 0.57 | 4.211
c97-w1 | 0.29 | 36.455 | -5.896 | 10.48 | 36.465 | -5.892 | 38.6 | $0.35^{+0.11}_{-0.12}$ | $430.4^{+116.3}_{-120.2}$ | 0.79 | $0.35^{+0.13}_{-0.12}$ | $0.53^{+0.12}_{-0.11}$ | $0.47^{+0.11}_{-0.10}$ | 0.76 | 4.505
c98-w1 | 0.32 | 36.631 | -5.695 | 3.81 | 36.635 | -5.692 | 17.3 | $0.23^{+0.10}_{-0.11}$ | $\sim 0$ | 4.40 | $0.24^{+0.12}_{-0.15}$ | $\sim 0$ | $\sim 0$ | 4.36 | 4.147
c99-w1 | 0.51 | 36.890 | -7.462 | 3.34 | 36.903 | -7.483 | 88.4 | $0.48^{+0.13}_{-0.19}$ | $1055.6^{+263.4}_{-305.1}$ | 1.73 | $0.49^{+0.14}_{-0.18}$ | $6.03^{+1.27}_{-1.32}$ | $5.41^{+1.14}_{-1.18}$ | 1.51 | 3.546
c100-w1 | 0.31 | 35.996 | -8.595 | 18.74 | 36.008 | -8.599 | 45.3 | $0.36^{+0.16}_{-0.14}$ | $905.3^{+182.9}_{-193.5}$ | 0.87 | $0.35^{+0.15}_{-0.16}$ | $4.53^{+0.85}_{-0.91}$ | $3.92^{+0.74}_{-0.79}$ | 0.73 | 4.631
c101-w1 | 0.34 | 36.405 | -9.775 | 5.22 | 36.415 | -9.758 | 72.7 | $0.30^{+0.09}_{-0.07}$ | $670.8^{+103.7}_{-92.8}$ | 1.36 | $0.31^{+0.07}_{-0.06}$ | $1.26^{+0.19}_{-0.22}$ | $1.11^{+0.17}_{-0.19}$ | 1.23 | 3.856
c102-w1 | 0.33 | 36.402 | -11.158 | 6.97 | 36.398 | -11.166 | 30.2 | $0.24^{+0.11}_{-0.19}$ | $\sim 0$ | 2.95 | $0.27^{+0.12}_{-0.11}$ | $\sim 0$ | $\sim 0$ | 3.08 | 4.075
c103-w1 | 0.27 | 36.785 | -11.058 | 4.54 | 36.791 | -11.053 | 27.7 | $0.29^{+0.12}_{-0.15}$ | $301.5^{+152.2}_{-158.7}$ | 0.59 | $0.29^{+0.14}_{-0.13}$ | $0.15^{+0.09}_{-0.11}$ | $0.13^{+0.08}_{-0.10}$ | 0.50 | 3.725
c104-w1 | 0.29 | 36.473 | -10.992 | 3.88 | 36.476 | -10.988 | 15.2 | $0.23^{+0.05}_{-0.05}$ | $1567.2^{+188.5}_{-185.9}$ | 0.96 | $0.24^{+0.06}_{-0.05}$ | $46.6^{+3.32}_{-2.45}$ | $38.9^{+2.77}_{-2.04}$ | 0.90 | 3.838
c105-w1 | 0.29 | 37.199 | -5.589 | 6.37 | 37.200 | -5.618 | 103.7 | $0.32^{+0.07}_{-0.08}$ | $729.8^{+93.6}_{-89.8}$ | 0.19 | $0.33^{+0.08}_{-0.08}$ | $3.43^{+0.31}_{-0.22}$ | $2.97^{+0.27}_{-0.19}$ | 0.18 | 3.713
c106-w1 | 0.32 | 37.662 | -4.991 | 10.71 | 37.670 | -4.988 | 36.7 | $0.79^{+3.20}_{-0.26}$ | $700.2^{+231.9}_{-322.5}$ | 1.53 | $0.75^{+1.23}_{-0.22}$ | $1.77^{+2.77}_{-1.56}$ | $1.66^{+2.60}_{-1.47}$ | 1.60 | 4.449
c107-w1 | 0.34 | 37.720 | -4.855 | 7.63 | 37.724 | -4.863 | 30.6 | $0.27^{+0.08}_{-0.06}$ | $324.6^{+92.8}_{-83.7}$ | 0.69 | $0.26^{+0.08}_{-0.07}$ | $0.26^{+0.10}_{-0.11}$ | $0.23^{+0.09}_{-0.10}$ | 0.62 | 4.784
c108-w1 | 0.30 | 37.358 | -4.816 | 3.08 | 37.359 | -4.835 | 71.2 | $0.33^{+0.13}_{-0.09}$ | $348.6^{+113.3}_{-104.2}$ | 0.56 | $0.33^{+0.12}_{-0.10}$ | $0.24^{+0.07}_{-0.09}$ | $0.21^{+0.06}_{-0.08}$ | 0.53 | 3.655
c109-w1 | -99.0 | 37.000 | -6.397 | 3.17 | 37.022 | -6.429 | 117.5 | $0.46^{+0.16}_{-0.09}$ | $741.4^{+152.6}_{-194.8}$ | 1.72 | $0.43^{+0.16}_{-0.11}$ | $3.17^{+3.32}_{-2.07}$ | $2.82^{+2.95}_{-1.84}$ | 1.25 | 3.946
c110-w1 | 0.49 | 37.812 | -5.572 | 3.10 | 37.784 | -5.587 | 116.9 | $0.52^{+0.17}_{-0.15}$ | $303.5^{+141.2}_{-133.1}$ | 0.89 | $0.52^{+0.15}_{-0.13}$ | $0.16^{+0.10}_{-0.07}$ | $0.15^{+0.09}_{-0.06}$ | 0.83 | 4.335
c111-w1 | 0.28 | 37.767 | -7.269 | 10.60 | 37.776 | -7.272 | 33.9 | $0.37^{+0.12}_{-0.10}$ | $\sim 0$ | 3.17 | $0.36^{+0.11}_{-0.11}$ | $\sim 0$ | $\sim 0$ | 3.18 | 3.708
c112-w1 | 0.36 | 37.541 | -7.536 | 3.04 | 37.540 | -7.513 | 81.0 | $0.42^{+0.16}_{-0.18}$ | $817.2^{+164.2}_{-177.0}$ | 0.87 | $0.43^{+0.17}_{-0.17}$ | $3.76^{+0.72}_{-0.74}$ | $3.34^{+0.64}_{-0.66}$ | 0.81 | 4.646
c113-w1 | 0.19 | 37.354 | -7.494 | 5.18 | 37.340 | -7.486 | 58.8 | $0.39^{+0.09}_{-0.10}$ | $\sim 0$ | 4.83 | $0.38^{+0.10}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 4.41 | 3.725
c114-w1 | 0.37 | 37.088 | -9.226 | 3.35 | 37.096 | -9.242 | 62.4 | $0.35^{+0.11}_{-0.13}$ | $672.0^{+114.3}_{-119.5}$ | 1.21 | $0.35^{+0.12}_{-0.13}$ | $2.71^{+0.56}_{-0.51}$ | $2.39^{+0.49}_{-0.45}$ | 1.10 | 3.616
c115-w1 | 0.27 | 37.299 | -8.901 | 3.48 | 37.294 | -8.892 | 35.3 | $0.31^{+0.12}_{-0.14}$ | $947.2^{+155.7}_{-168.3}$ | 2.47 | $0.32^{+0.13}_{-0.14}$ | $5.55^{+0.72}_{-0.75}$ | $4.77^{+0.62}_{-0.64}$ | 2.44 | 4.734
c116-w1 | 0.28 | 37.355 | -8.841 | 5.10 | 37.347 | -8.835 | 34.1 | $0.26^{+0.16}_{-0.15}$ | $174.9^{+194.5}_{-189.3}$ | 2.81 | $0.25^{+0.16}_{-0.17}$ | $0.08^{+0.13}_{-0.14}$ | $0.07^{+0.12}_{-0.13}$ | 2.67 | 4.513
c117-w1 | 0.64 | 37.828 | -11.147 | 5.49 | 37.802 | -11.140 | 93.2 | $0.77^{+0.27}_{-0.25}$ | $\sim 0$ | 5.64 | $0.74^{+0.24}_{-0.25}$ | $\sim 0$ | $\sim 0$ | 5.46 | 3.684
c118-w1 | 0.27 | 37.922 | -4.883 | 12.07 | 37.936 | -4.880 | 54.2 | $0.29^{+0.13}_{-0.12}$ | $311.2^{+103.4}_{-97.4}$ | 0.56 | $0.30^{+0.11}_{-0.12}$ | $0.14^{+0.05}_{-0.06}$ | $0.12^{+0.04}_{-0.05}$ | 0.53 | 3.555
c119-w1 | 0.49 | 38.525 | -4.728 | 3.14 | 38.483 | -4.737 | 153.6 | $0.46^{+0.14}_{-0.11}$ | $422.1^{+126.1}_{-135.6}$ | 0.48 | $0.46^{+0.13}_{-0.14}$ | $0.24^{+0.11}_{-0.08}$ | $0.22^{+0.10}_{-0.07}$ | 0.43 | 3.567
c120-w1 | 0.31 | 38.048 | -6.491 | 5.70 | 38.022 | -6.501 | 102.0 | $0.12^{+0.11}_{-0.12}$ | $\sim 0$ | 3.77 | $0.16^{+0.12}_{-0.11}$ | $\sim 0$ | $\sim 0$ | 3.71 | 3.850
c121-w1 | 0.33 | 37.925 | -7.790 | 3.65 | 37.938 | -7.828 | 145.0 | $0.37^{+0.12}_{-0.11}$ | $725.9^{+152.9}_{-143.8}$ | 0.16 | $0.37^{+0.14}_{-0.12}$ | $2.12^{+0.39}_{-0.43}$ | $1.86^{+0.34}_{-0.38}$ | 0.16 | 3.853
c122-w1 | 0.43 | 37.998 | -7.624 | 3.44 | 37.986 | -7.658 | 130.0 | $0.33^{+0.15}_{-0.17}$ | $\sim 0$ | 3.83 | $0.32^{+0.16}_{-0.15}$ | $\sim 0$ | $\sim 0$ | 3.74 | 3.725
c123-w1 | 0.37 | 38.688 | -8.801 | 3.62 | 38.737 | -8.797 | 172.1 | $0.41^{+0.13}_{-0.13}$ | $547.5^{+117.8}_{-102.7}$ | 1.19 | $0.40^{+0.12}_{-0.13}$ | $1.81^{+0.29}_{-0.31}$ | $1.60^{+0.26}_{-0.27}$ | 1.13 | 3.563
c124-w1 | 0.23 | 38.669 | -9.835 | 3.70 | 38.630 | -9.843 | 141.1 | $0.26^{+0.15}_{-0.15}$ | $756.0^{+143.0}_{-139.6}$ | 0.57 | $0.26^{+0.15}_{-0.14}$ | $7.14^{+1.32}_{-1.27}$ | $6.09^{+1.13}_{-1.08}$ | 0.56 | 4.467
c125-w1 | 0.63 | 37.902 | -9.562 | 4.82 | 37.907 | -9.595 | 117.5 | $0.59^{+0.29}_{-0.18}$ | $814.3^{+313.2}_{-187.3}$ | 1.53 | $0.57^{+0.33}_{-0.15}$ | $3.00^{+3.14}_{-1.90}$ | $2.74^{+2.86}_{-1.73}$ | 1.47 | 4.915
c126-w1 | 0.26 | 38.639 | -10.472 | 5.31 | 38.647 | -10.458 | 57.3 | $0.44^{+0.10}_{-0.10}$ | $536.5^{+102.1}_{-108.7}$ | 1.62 | $0.40^{+0.11}_{-0.10}$ | $0.96^{+0.15}_{-0.13}$ | $0.86^{+0.13}_{-0.12}$ | 1.16 | 4.764
Table 6Catalog of the Matching Groups/Clusters between $\nu>3.5$ Convergence Peaks and X-Ray/K2-detected Groups/Clusters in CFHTLS-Wide W1 Fields. K2 redshift denotes the median redshift of bright ($i\leq 20$) cluster members. X-ray redshift denotes the photometric redshifts of X-ray clusters (Adami et al. 2011). The parameters $d_{\rm xray}$ and $d_{\rm k2}$ are the offset between X-ray/optical and weak lensing center, respectively. ID | $\alpha_{\rm xray}$ | $\delta_{\rm xray}$ | $\alpha_{\rm K2}$ | $\delta_{\rm K2}$ | $\alpha$ | $\delta$ | $d_{\rm xray}$ | $d_{\rm k2}$ | $\nu$ | $z_{\rm K2}$ | $z_{\rm xray}$ | $\sigma_{v-xray}$ | $z_{\rm SIS}$ | $\sigma_{v}$ | $\chi^{2}_{\rm SIS}$ | $z_{\rm NFW}$ | $m_{\rm NFW}$ | $m_{200}$ | $\chi^{2}_{\rm NFW}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | | | $\rm arcsec$ | $\rm arcsec$ | | | | $\rm km/s$ | | $\rm km/s$ | | | $10^{14}M_{\odot}/h$ | $10^{14}M_{\odot}/h$ |
J022145.2-034617 | 35.438 | -3.772 | 35.441 | -3.772 | 35.456 | -3.768 | 66.5 | 57.2 | 4.209 | 0.43 | 0.429$\pm$0.001 | 977$\pm$157 | $0.34^{+0.11}_{-0.12}$ | $766.1^{+182.5}_{-229.9}$ | 0.86 | $0.35^{+0.15}_{-0.15}$ | $2.04^{+2.78}_{-1.54}$ | $1.79^{+2.44}_{-1.35}$ | 0.82
J022402.0-050525 | 36.008 | -5.090 | 36.108 | -5.088 | 36.077 | -5.101 | 251.1 | 122.2 | 3.521 | 0.55 | 0.324$\pm$0.001 | 364$\pm$69 | $0.20^{+0.17}_{-0.15}$ | $344.8^{+97.5}_{-89.1}$ | 0.78 | $0.20^{+0.18}_{-0.16}$ | $0.23^{+0.06}_{-0.04}$ | $0.20^{+0.05}_{-0.04}$ | 0.81
J022433.8-041405 | 36.141 | -4.234 | 36.121 | -4.165 | 36.097 | -4.161 | 289.1 | 87.5 | 4.063 | 0.30 | 0.262$\pm$0.001 | 483$\pm$100 | $0.26^{+0.13}_{-0.12}$ | $359.6^{+131.2}_{-121.7}$ | 0.25 | $0.25^{+0.11}_{-0.11}$ | $0.77^{+0.17}_{-0.18}$ | $0.67^{+0.15}_{-0.16}$ | 0.20
J022530.6-041420 | 36.377 | -4.239 | 36.372 | -4.258 | 36.372 | -4.248 | 36.3 | 36.9 | 4.548 | 0.65 | 0.14$\pm$0.002 | 899$\pm$218 | $0.65^{+0.26}_{-0.26}$ | $\sim 0$ | 5.69 | $0.66^{+0.22}_{-0.20}$ | $\sim 0$ | $\sim 0$ | 4.63
J021837.0-054028 | 34.654 | -5.675 | 34.655 | -5.674 | 34.653 | -5.626 | 175.7 | 170.7 | 4.186 | 0.33 | 0.275$\pm$0.0 | - | $0.31^{+0.14}_{-0.11}$ | $\sim 0$ | 4.18 | $0.32^{+0.13}_{-0.10}$ | $\sim 0$ | $\sim 0$ | 3.87
J021842.8-053254 | 34.678 | -5.548 | 34.657 | -5.570 | 34.684 | -5.572 | 91.1 | 98.4 | 4.194 | 0.36 | 0.38$\pm$0.001 | 847$\pm$279 | $0.57^{+0.13}_{-0.14}$ | $909.7^{+61.5}_{-60.1}$ | 1.55 | $0.55^{+0.15}_{-0.13}$ | $5.31^{+0.32}_{-0.30}$ | $4.82^{+0.29}_{-0.27}$ | 1.47
J022632.4-050003 | 36.638 | -5.007 | 36.678 | -4.950 | 36.643 | -4.959 | 62.6 | 72.8 | 3.156 | 0.22 | 0.494$\pm$0.0 | - | $0.23^{+0.19}_{-0.17}$ | $\sim 0$ | 3.80 | $0.25^{+0.16}_{-0.21}$ | $\sim 0$ | $\sim 0$ | 3.81
|
arxiv-papers
| 2011-08-09T16:56:08 |
2024-09-04T02:49:21.451588
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "HuanYuan Shan, Jean-Paul Kneib, Charling Tao, Zuhui Fan, Mathilde\n Jauzac, Marceau Limousin, Richard Massey, Jason Rhodes, Karun Thanjavur and\n Henry J. McCracken",
"submitter": "HuanYuan Shan",
"url": "https://arxiv.org/abs/1108.1981"
}
|
1108.2170
|
# Discontinuous Galerkin Method for the Air Pollution Model
Lite Zhao zhaolite@gmail.com Xijian Wang wangxj1980426@gmail.com Qinzhi Hou
q.hou@tue.nl School of Applied Physics and Materials, Wuyi University,
People’s Republic of China School of Mathematics and Computing Science, Wuyi
University, People’s Republic of China Department of Mathematics and Computer
Science, Eindhoven University of Technology, The Netherlands
###### Abstract
In this paper we present the discontinuous Galerkin method to solve the
problem of the two-dimensional air pollution model. The resulting system of
ordinary differential equations is called the semidiscrete formulation. We
show the existence and uniqueness of the ODE system and provide the error
estimates for the numerical error.
###### keywords:
air pollution model, discontinuous Galerkin method, error estimate
††journal: Computers and Mathematics with Applications
## 1 Introduction
Air pollution is the introduction of chemicals, particulate matter, or
biological materials that cause harm or discomfort to humans or other living
organisms, or cause damage to the natural environment or built environment,
into the atmosphere. The basic technology for analyzing air pollution is
through the mathematical models and numerical methods for predicting the
transport of air pollutants in the lower atmosphere[1, 2, 3, 4, 5]. Different
air pollution models have been developed in the last decades by the National
Environmental Research Institute (http://www.dmu.dk/en/air/models/). In the
present paper we consider the following Danish Eulerian model [2, 4, 5]
$\displaystyle\frac{{\partial u}}{{\partial t}}+\frac{\partial}{{\partial
x}}(cu)+\frac{\partial}{{\partial y}}(eu)-\frac{\partial}{{\partial
x}}({k_{x}}\frac{{\partial u}}{{\partial x}})-\frac{\partial}{{\partial
y}}({k_{y}}\frac{{\partial u}}{{\partial y}})=f(u),$ (1a) $\displaystyle
f(u)=-({k_{1}}+{k_{2}})u+E+Q(u),$ (1b) $\displaystyle
u(x,\;y,\;0)={u_{0}}(x,\;y),\;\;(x,\;y)\in\Omega,$ (1c) $\displaystyle
u(x,\;y,\;t)\left|{{}_{\partial\Omega}}\right.=0,\;\;t\in[0,\;T].$ (1d)
The different quantities involved in the mathematical model have the following
meaning:
* 1.
the concentration is denoted by $u$;
* 2.
$c$ and $e$ are wind velocities;
* 3.
$k_{x}$ and $k_{y}$ are diffusion coefficients;
* 4.
the emission source is described by $E$;
* 5.
$k_{1}$ and $k_{2}$ are constant deposition coefficients;
* 6.
the chemical reaction is denoted by $Q$.
Meanwhile, we give the following assumptions:
* 1.
$u\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega),\;\;u_{t},u_{tt}\in L^{2}(\Omega)$;
* 2.
$Q(u)$ satisfy the Lipschitz condition;
* 3.
$0<{k_{*}}\leqslant\min\\{\left|{{k_{x}}}\right|,\;\left|{{k_{y}}}\right|\\}\leqslant\max\\{\left|{{k_{x}}}\right|,\;\left|{{k_{y}}}\right|\\}\leqslant{k^{*}},\;0<{c_{*}}\leqslant\min\\{\left|c\right|,\;\left|e\right|\\}\leqslant\;\;\;\max\\{\left|c\right|,\;\left|e\right|\\}\leqslant{c^{*}},\;{k_{*}},\;{k^{*}},\;{c_{*}},\;{c^{*}}$are
constants.
A general description of the Danish Eulerian Model and its numerical treatment
is given in [5, 6, 7]. Research on the finite difference method and finite
volume element method for this air pollution model already has good results
[8, 9, 10, 11]. In this article, we use the discontinuous Galerkin method (DG
method) to analyse and solve the air pollution model.
DG methods in mathematics form a class of numerical methods for solving
partial differential equations. They have recently gained popularity due to
many of their attractive properties, refer to [12, 13, 14, 15, 16, 17, 18, 19,
20]. First of all, the flexibility of the methods allows for general non-
conforming meshes with variable degree of approximation. This makes the
implementation of h-p adaptivity for DG easier than that for conventional
approaches. Moreover, the DG methods are locally mass conservative at the
element level. In addition, they have less numerical diffusion than most
conventional algorithms, thus are likely to offer more accurate solution for
at least advection-dominated transport problems. They handle rough coefficient
problems and capture the discontinuity in the solution very well by the nature
of discontinuous function space. Furthermore, the DG methods are easier to
implement than most traditional finite element methods. The trial and test
spaces are easier to construct than conforming methods because they are local.
The paper is organized as follows: In Section 2, the variational formulation
of the DG method is stated. And we show the existence and uniqueness of the
resulting ordinary differential equations system. Finally we provide the error
estimates for the numerical error in Section 3.
## 2 Semidiscrete formulation
In this section, we approximate the solution $u(t)$ by a function $U_{h}(t)$
that belongs to the finite-dimensional space
$\mathcal{D}_{k}(\varepsilon_{h})$ for all $t\geq 0$. The solution $U_{h}$ is
referred to as the semidiscrete solution. In what follows, we assume that
$s>\frac{3}{2}$. We introduce a bilinear form $J_{0}^{\sigma_{0},\beta_{0}}$:
$H^{s}({\varepsilon_{h}})\times H^{s}({\varepsilon_{h}})\rightarrow\mathbb{R}$
that penalize the jump of the function values:
$J_{0}^{\sigma_{0},\beta_{0}}(w,v)=\sum\limits_{e\in\Gamma_{h}\cup\partial{\Omega}}\frac{\sigma^{0}_{e}}{\left|{e}\right|^{\beta_{0}}}\int_{e}{[w][v]}$
The parameter $\sigma_{e}^{0}$ is called penalty parameter. It is nonnegative
real number. The power $\beta_{0}$ is positive number. $\left|{e}\right|$
simply means the length of $e$. We now define the DG bilinear form
$a_{\epsilon}:H^{s}({\varepsilon_{h}})\times
H^{s}({\varepsilon_{h}})\rightarrow\mathbb{R}$
$\displaystyle a_{\epsilon}(w,v)=$
$\displaystyle\sum\limits_{E\in\varepsilon_{h}}\int_{E}(k_{x}\frac{{\partial
w}}{{\partial x}}\frac{\partial v}{\partial x}+k_{y}\frac{{\partial
w}}{{\partial y}}\frac{\partial v}{\partial
y})-\sum\limits_{e\in\Gamma_{h}}\int_{e}(\\{k_{x}\frac{{\partial w}}{{\partial
x}}\overrightarrow{n_{1}}\\}+\\{k_{y}\frac{{\partial w}}{{\partial
y}}\overrightarrow{n_{2}}\\})[v]$
$\displaystyle-\epsilon\sum\limits_{e\in\Gamma_{h}}\int_{e}{(\\{k_{x}\frac{{\partial
v}}{{\partial x}}\overrightarrow{n_{1}}\\}+\\{k_{y}\frac{{\partial
v}}{{\partial
y}}\overrightarrow{n_{2}}\\})[w]}+J_{0}^{\sigma_{0},\beta_{0}}(w,v).$
The bilinear form $a_{\epsilon}$ contains another parameter $\epsilon$ that
may take the value -1,0, or 1. $a_{\epsilon}$ is symmetric if $\epsilon=-1$
and it is nonsymmetric otherwise.
This bilinear form yields the following energy seminorm:
${\left\|v\right\|_{\varepsilon}}={(\sum\limits_{E\in{\varepsilon_{h}}}{\left\|{{D^{1/2}}\nabla
v}\right\|_{{L^{2}}(E)}^{2}}+\sum\limits_{e\in\Gamma_{h}}\frac{\sigma^{0}_{e}}{\left|{e}\right|^{\beta_{0}}}\left\|[v]\right\|_{{L^{2}}(e)}^{2})^{1/2}}$
Second, the convection term $\frac{\partial}{{\partial
x}}(cu)+\frac{\partial}{{\partial y}}(eu)$ is approximated by an upwind
discretization. Let us denote the upwind value of a function $w$ by $w^{up}$.
We recall that $\left({\begin{array}[]{*{20}{c}}{\overrightarrow{{n_{1}}}}\\\
{\overrightarrow{{n_{2}}}}\end{array}}\right)$ is a unit normal vector
pointing from $E_{e}^{1}$ to $E_{e}^{2}$:
${w^{up}}=\left\\{\begin{gathered}w\left|{{}_{E_{e}^{1}}}\right.,\;\;if\;\;c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}}\geq
0\;\;\hfill\\\
w\left|{{}_{E_{e}^{2}}}\right.,\;\;if\;\;c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}}\leq
0\;\;\hfill\\\ \end{gathered}\right.\;\;\;\;\;\;\;\forall e=\partial
E^{1}_{e}\cap\partial E^{2}_{e}.$
Let
$b(c,e;w,v)=-\sum\limits_{E\in\varepsilon_{h}}{\int_{E}{(cw\frac{\partial
v}{\partial x}+ew\frac{\partial v}{\partial
y})}}+\sum\limits_{e\in\Gamma_{h}}{\int_{e}{(c\overrightarrow{n_{1}}w^{up}[v]+e\overrightarrow{n_{2}}w^{up}[v])}}$
The general semidiscrete DG variational formulation of problem (1a)-(1d) is as
follows: Find $U_{h}\in L^{2}(0,T;\mathcal{D}_{k}(\varepsilon_{h}))$, such
that
$\displaystyle\forall t>0,\forall
v\in\mathcal{D}_{k}(\varepsilon_{h}),(\frac{\partial U_{h}}{\partial
t},v)_{\Omega}+a_{\epsilon}(U_{h}(t),v)+b(c,e;U_{h}(t),v)=L(U_{h}(t),v),$ (2a)
$\displaystyle\forall
v\in\mathcal{D}_{k}(\varepsilon_{h}),(U_{h}(0),v)_{\Omega}=(u_{0},v)_{\Omega},$
(2b)
where the form $L$ is
$L(w;v)=\int_{\Omega}f(w)v.$
The next lemma establishes the consistency between the model problem and the
variational formulation.
###### Lemma 2.1.
Assume that the weak solution $u$ of problem (1a)-(1d) belongs to
$H^{1}(0,T;H^{2}(\varepsilon_{h}))$, then $u$ satisfies the variational
problem (2a)-(2b).
###### Proof.
Let $v$ be a test function in $\mathcal{D}_{k}(\varepsilon_{h})$. We multiply
by $v|_{E}$ and integrate by parts on one element $E\in\varepsilon_{h}$, and
use Green’s theorem:
$\displaystyle{(\frac{{\partial u}}{{\partial
t}},\;v)_{E}}-\int_{E}{(-k_{x}\frac{{\partial u}}{{\partial x}}\frac{\partial
v}{\partial x}-k_{y}\frac{{\partial u}}{{\partial y}}\frac{\partial
v}{\partial y}+cu\frac{\partial v}{\partial x}+eu\frac{\partial v}{\partial
y})}+$ $\displaystyle\int_{\partial{E}}{(-k_{x}\frac{{\partial u}}{{\partial
x}}\overrightarrow{n_{1}}v-k_{y}\frac{{\partial u}}{{\partial
y}}\overrightarrow{n_{2}}v+cu\overrightarrow{n_{1}}v+eu\overrightarrow{n_{2}}v)}=\int_{E}{f(u)v}$
Summing over all elements and using the regularity of the exact solution, we
obtain
$\displaystyle{(\frac{{\partial u}}{{\partial
t}},\;v)_{\Omega}}-\sum\limits_{E\in\varepsilon_{h}}{\int_{E}{(-k_{x}\frac{{\partial
u}}{{\partial x}}\frac{\partial v}{\partial x}-k_{y}\frac{{\partial
u}}{{\partial y}}\frac{\partial v}{\partial y}+cu\frac{\partial v}{\partial
x}+eu\frac{\partial v}{\partial y})}}+$
$\displaystyle\sum\limits_{e\in\Gamma_{h}}{\int_{e}{(-\\{k_{x}\frac{{\partial
u}}{{\partial x}}\overrightarrow{n_{1}}\\}[v]-\\{k_{y}\frac{{\partial
u}}{{\partial
y}}\overrightarrow{n_{2}}\\}[v]+cu\overrightarrow{n_{1}}[v]+eu\overrightarrow{n_{2}}[v])}}+$
$\displaystyle\epsilon\sum\limits_{e\in\Gamma_{h}}\int_{e}{(-\\{k_{x}\frac{{\partial
v}}{{\partial x}}\overrightarrow{n_{1}}\\}[u]-\\{k_{y}\frac{{\partial
v}}{{\partial
y}}\overrightarrow{n_{2}}\\}[u])}+\sum\limits_{e\in\Gamma_{h}}\frac{\sigma^{0}_{e}}{\left|{e}\right|^{\beta_{0}}}\int_{e}{([u][v])}=\int_{\Omega}{f(u)v}.$
Since $u_{up}=u$, we clearly have our result. ∎
### 2.1 Existence and uniqueness of the solution
Because of the lack of continuity constraints between mesh elements for the
test functions, the basic functions of $\mathcal{D}_{k}(\varepsilon_{h}))$
have a support contained in one element. We write
$\mathcal{D}_{k}(\varepsilon_{h})=span\\{\phi_{i}^{E}:\;1\leqslant
i\leqslant{N_{loc}},\;E\in{\varepsilon_{h}}\\}$
with
$\phi_{i}^{E}(x)=\left\\{\begin{gathered}\widetilde{{\phi_{i}}}\circ{F_{E}}(x),\;\;\;x\in
E,\hfill\\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\notin E.\hfill\\\
\end{gathered}\right.$
In 2D, we have
$\widehat{\phi}(\widehat{x},\widehat{y})=\widehat{x}^{I}\widehat{y}^{I},I+J=i,0\leq
i\leq k$. This yields the local dimension
$N_{loc}=\frac{(k+1)(k+2)}{2}.$
using the global basis functions, we can expand the semidiscrete solution
$\forall
t\in(0,\;T),\;\forall(x,\;y)\in\Omega,\;{U_{h}}(t,\;x,\;y)=\sum\limits_{E\in{\varepsilon_{h}}}{\sum\limits_{i=1}^{{N_{loc}}}{\xi_{i}^{E}(t)\phi_{i}^{E}(x,\;y)}}.$
(3)
The degrees of freedom $\xi^{E}$’s are functions of time. Let $N_{el}$ denote
the number of elements in the mesh. We can rename the basis functions and the
degrees of freedom such that
$\displaystyle\\{\phi_{i}^{E}:1\leq i\leq
N_{loc},E\in\varepsilon_{h}\\}=\\{\widetilde{\phi}_{j}:1\leq j\leq
N_{loc}N_{el}\\},$ $\displaystyle\\{\xi_{i}^{E}:1\leq i\leq
N_{loc},E\in\varepsilon_{h}\\}=\\{\widetilde{\xi}_{j}:1\leq j\leq
N_{loc}N_{el}\\}.$
Plugging (3) into the variational problem (2a)-(2b) yields a linear system of
ordinary differential equations as follows:
$\displaystyle
M\frac{d\widetilde{\xi}}{dt}(t)+(A+B)\widetilde{\xi}=G(\widetilde{\xi}),$
$\displaystyle M\widetilde{\xi}(0)=\widetilde{U}_{0}.$
The matrices $M,A$ are called the mass and stiffness matrices, and they are
defined by
$\forall 1\leqslant
i,\;j\leqslant{N_{loc}}{N_{el}},\;\;{M_{ij}}={(\widetilde{{\phi_{j}}},\;\widetilde{{\phi_{i}}})_{\Omega}},\;\;{A_{ij}}={a_{\epsilon}}(\widetilde{{\phi_{j}}},\;\widetilde{{\phi_{i}}}).$
The matrix $B$ results from the convective term, and the vector
$G(\widetilde{\xi})$ depends on the vector solution
$\displaystyle\forall 1\leqslant
i,\;j\leqslant{N_{loc}}{N_{el}},\;\;\;{(B)_{ij}}=b(c;\;e;\;{\widetilde{\phi}_{j}},\;{\widetilde{\phi}_{i}}),$
$\displaystyle\forall 1\leqslant
i\leqslant{N_{loc}}{N_{el}},\;\;\;{(G)_{i}}=L(\widetilde{\xi};\;{\widetilde{\phi}_{i}}).$
Since the matrix $M$ is invertible and the vector function
$G(\widetilde{\xi})$ is Lipschitz with respect to $\widetilde{\xi}$, there
exists a unique solution to the variational problem (2a)-(2b).
## 3 Error estimates
In this section, we first present the Gronwall’s inequalities [21], which are
important tools for analyzing time-dependent problems.
###### Lemma 3.2 (Continuous Gronwall inquality).
Let $f,g,h$ be piecewise continuous nonnegative functions defined on (a, b).
Assume that $g$ is nondereasing. Assume that there is a positive constant $C$
independent of $t$ such that
$\forall t\in(a,b),\;f(t)+h(t)\leq g(t)+C\int_{a}^{t}f(s)ds.$
Then,
$\forall t\in(a,b),\;f(t)+h(t)\leq e^{C(t-a)}g(t).$
Now we state a priori error estimates for the semidiscrete scheme [22].
###### Theorem 3.3.
Assume that the solution $u$ to problem (1a)-(1d) belongs to
$H^{1}(0,T;H^{2}(\varepsilon_{h}))$ and that $u_{0}$ belongs to
$H^{s}(\varepsilon_{h})$ for $s>3/2$. Assume that $\beta_{0}\geq 1$. In the
case of SIPG and IIPG, assume that $\sigma_{e}^{0}$ is sufficiently large for
all $e$. Then, there is a constant $C$ independent of $h$ such that
$\displaystyle{\left\|{u-{U_{h}}}\right\|_{{L^{\infty}}({L^{2}}(\Omega))}}+{\left(\int_{0}^{T}{\left\|{u(t)-{U_{h}}(t)}\right\|_{\varepsilon}^{2}dt}\right)^{1/2}}$
$\displaystyle\leqslant
C{h^{\min(k+1,\;s)-1}}({\left\|u\right\|_{{H^{1}}(0,\;T;\;{H^{s}}({\varepsilon_{h}}))}}+{\left\|{{u_{0}}}\right\|_{{H^{s}}({\varepsilon_{h}})}}).$
###### Proof.
We omit some details which is similar to the proof of Theorem 3.4.([13]). We
write $u-U_{h}=\rho-\chi$ with $\rho=u-\widetilde{u}$ and
$\chi=U_{h}-\widetilde{u}$. The function
$\widetilde{u}\in\mathcal{D}_{k}(\varepsilon_{h}))$ is an approximation of $u$
that satisfies good error bounds. The error equation is satisfied for all $v$
in $\mathcal{D}_{k}(\varepsilon_{h})$:
$\displaystyle(\frac{\partial\chi}{\partial t},v)_{\Omega}$
$\displaystyle+a_{\epsilon}(\chi,v)+b(c,e;\chi,v)=(\frac{\partial\rho}{\partial
t},v)_{\Omega}+a_{\epsilon}(\rho,v)$
$\displaystyle+b(c,e;\rho,v)+(f(U_{h})-f(u),v)_{\Omega}.$
Now, by choosing $v=\chi$ and using the coercivity of $a_{\epsilon}$, we
obtain
$\displaystyle\frac{1}{2}\frac{d}{{dt}}\left\|{\chi}\right\|_{{L^{2}}(\Omega)}^{2}+\kappa\left\|{\chi}\right\|_{\varepsilon}^{2}+b(c;e;\chi,\chi)\leqslant(\frac{\partial\rho}{\partial
t},\chi)_{\Omega}$
$\displaystyle+a_{\epsilon}(\rho,\chi)+b(c,e;\rho,\chi)+(f(U_{h})-f(u),\chi)_{\Omega}.$
We use Green’s formula and the fact that
$\nabla\cdot\left({\begin{array}[]{*{20}{c}}c\\\ e\end{array}}\right)=0$:
$\displaystyle\sum\limits_{E\in{\varepsilon_{h}}}{\int_{E}{\left({\begin{array}[]{*{20}{c}}c\\\
e\end{array}}\right)\chi\cdot\nabla\chi}}$
$\displaystyle=\frac{1}{2}\sum\limits_{E\in{\varepsilon_{h}}}{\int_{E}{\left({\begin{array}[]{*{20}{c}}c\\\
e\end{array}}\right)\cdot\nabla\chi^{2}}}$
$\displaystyle=\frac{1}{2}\sum\limits_{E\in{\varepsilon_{h}}}\int_{\partial
E}{\left({\begin{array}[]{*{20}{c}}c\\\
e\end{array}}\right)\cdot{\left({\begin{array}[]{*{20}{c}}\overrightarrow{n_{1}}\\\
\overrightarrow{n_{2}}\end{array}}\right)_{E}\chi^{2}}}$
$\displaystyle=\frac{1}{2}\sum\limits_{e\in{\Gamma_{h}}}\int_{e}(c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}})[\chi^{2}].$
Thus we obtain
$\displaystyle b(c,e;\chi,\chi)=$
$\displaystyle-\sum\limits_{E\in{\varepsilon_{h}}}{\int_{E}{\left({\begin{array}[]{*{20}{c}}c\\\
e\end{array}}\right)\chi\cdot\nabla\chi}}+\sum\limits_{e\in{\Gamma_{h}}}\int_{e}(c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}})\chi^{up}[\chi]$
$\displaystyle=\sum\limits_{e\in{\Gamma_{h}}}\int_{e}(c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}})(\chi^{up}[\chi]-\frac{1}{2}[\chi^{2}])$
$\displaystyle=\sum\limits_{e\in{\Gamma_{h}}}\int_{e}(c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}})(\chi^{up}[\chi]-\\{\chi\\}[\chi])$
$\displaystyle=\frac{1}{2}\sum\limits_{e\in{\Gamma_{h}}}\int_{e}|c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}}|[\chi]^{2}\geq
0.$
We now bound each term in $b(c,e;\rho,\chi)$. Using Cauchy-Schwarz’s and
Young’s inequalities, we have
$\sum\limits_{E\in{\varepsilon_{h}}}{\int_{E}{\left({\begin{array}[]{*{20}{c}}c\\\
e\end{array}}\right)\rho\cdot\nabla\chi}}\leq
C\sum\limits_{E\in{\varepsilon_{h}}}{\left\|\rho\right\|_{L^{2}(E)}}{\left\|{\nabla\chi}\right\|_{L^{2}(E)}}\leq\frac{\kappa}{8}{\left\|{\chi}\right\|_{\varepsilon}^{2}}+C{\left\|{\rho}\right\|_{L^{2}(\Omega)}^{2}}$
and
$\displaystyle\sum\limits_{e\in{\Gamma_{h}}}\int_{e}(c\overrightarrow{n_{1}}+e\overrightarrow{n_{2}})\chi^{up}[\chi]$
$\displaystyle\leq{\sum\limits_{e\in\Gamma_{h}}{\left\|{{{\left|{c\overrightarrow{{n_{1}}}+e\overrightarrow{{n_{2}}}}\right|}^{\frac{1}{2}}}[\chi]}\right\|}_{0,\;e}}{\left\|{{{\left|{c\overrightarrow{{n_{1}}}+e\overrightarrow{{n_{2}}}}\right|}^{\frac{1}{2}}}\rho_{*}}\right\|}_{0,\;e}$
$\displaystyle\leq\frac{1}{4}{\sum\limits_{e\in\Gamma_{h}}{\left\|{{{\left|{c\overrightarrow{{n_{1}}}+e\overrightarrow{{n_{2}}}}\right|}^{\frac{1}{2}}}[\chi]}\right\|}_{0,\;e}^{2}}+C\sum\limits_{e\in\Gamma_{h}}{\left\|\rho^{up}\right\|}_{L^{2}(e)}^{2}.$
Finally, we bound the nonlinear source term, using the Lipschitz property:
$\int_{\Omega}(f(U_{h})-f(u))\chi\leq
C{\left\|({U_{h}-u})\right\|}_{L^{2}(\Omega)}{\left\|\chi\right\|}_{L^{2}(\Omega)}\leq
C{\left\|\chi\right\|}_{L^{2}(\Omega)}^{2}+C{\left\|\rho\right\|}^{2}_{L^{2}(\Omega)}.$
The other terms are identical to the ones in the proof of Theorem 2.13 and 3.4
([13]). Then the main result is obtained by combining all bounds and using
Gronwall’s inequality of Lemma 3.2. ∎
We can choose any of the time discretizations such as backward Euler and
forward Euler and some that are of high order such as Crank-Nicolson and
Runge-Kutta methods. The analysis of the resulting fully discrete schemes can
be done in a common way.
## Acknowledgement
The second author wishes to thank the financial support from the Erasmus
Mundus Scholarship of the European Union during his visiting study in Europe.
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* Baumann and Oden [1999] C. Baumann, J. Oden, A discontinuous hp finite element method for convection–diffusion problems, Computer Methods in Applied Mechanics and Engineering 175 (1999) 311–341.
* Oden et al. [1998] J. T. Oden, I. Babuška, C. E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys. 146 (1998) 491–519.
* Rivière [2000] B. Rivière, Discontinuous galerkin methods for solving the miscible displacement problem in porous media, Ph. D. Thesis, The University of Texas at Austin (2000).
* Rivière et al. [1999] B. Rivière, M. Wheeler, V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous galerkin methods for elliptic problems. part i, Computational Geosciences 3 (1999) 337–360.
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* Wheeler and Darlow [1980] M. Wheeler, B. Darlow, Interior penalty galerkin procedures for miscible displacement problems in porous media, in: Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979), pp. 485–506.
* Heywood and Rannacher [1990] J. Heywood, R. Rannacher, Finite-element approximation of the nonstationary navier-stokes problem part iv: Error analysis for second-order time discretization, SIAM Journal on Numerical Analysis (1990) 353–384.
* Rivière and Wheeler [2002] B. Rivière, M. Wheeler, Non conforming methods for transport with nonlinear reaction, in: Fluid flow and transport in porous media, mathematical and numerical treatment: proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment, June 17-21, 2001, Mount Holyoke College, South Hadley, Massachusetts, volume 295, Amer Mathematical Society, p. 421.
|
arxiv-papers
| 2011-08-10T13:08:09 |
2024-09-04T02:49:21.465997
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lite Zhao, Xijian Wang, Qinzhi Hou",
"submitter": "Xijian Wang Mr",
"url": "https://arxiv.org/abs/1108.2170"
}
|
1108.2248
|
# A simple and objective method for reproducible resting state network (RSN)
detection in fMRI
Gautam V. Pendse 1, David Borsook 1,2, and Lino Becerra 1,2
1 P.A.I.N Group, Imaging and Analysis Group (IMAG), McLean Hospital, Harvard
Medical School
2 A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital
To whom correspondence should be addressed. e-mail: gpendse@mclean.harvard.edu
(June 15, 2011)
###### Contents
1. 1 Introduction
2. 2 Methods
1. 2.1 ICA background
1. 2.1.1 Estimating the model order $q$
2. 2.1.2 Identifiability of the noisy ICA model
3. 2.1.3 Why is there a run-to-run variability in estimated ICs?
2. 2.2 ICA algorithms, single subject ICA and group ICA
1. 2.2.1 Single subject ICA
2. 2.2.2 Group ICA
3. 2.3 The original RAICAR algorithm
4. 2.4 The RAICAR-N enhancement
5. 2.5 How many subjects should be used per group ICA run in RAICAR-N?
6. 2.6 How to display the estimated non-Gaussian spatial structure in ICA maps?
3. 3 Experiments and Results
1. 3.1 Human rsfMRI data
2. 3.2 Preprocessing
3. 3.3 RAICAR-N analysis with 1 ICA run per subject
4. 3.4 RAICAR-N on random sets of 5 subjects - 50 group ICA runs
5. 3.5 RAICAR-N on random sets of 5 subjects - 100 group ICA runs
4. 4 Group comparison of ICA results
1. 4.1 Approaches based on a single group ICA run or no ICA run
1. 4.1.1 Advantages of single group ICA based approaches
2. 4.2 Approaches based on multiple single subject or group ICA runs
1. 4.2.1 Advantages of multiple ICA run approaches
5. 5 Discussion
1. 5.1 Results for publicly available rsfMRI data
2. 5.2 Single subject ICA vs Group ICA
3. 5.3 How should subjects be grouped for group ICA?
4. 5.4 RAICAR-N for group comparisons of reproducible ICs
1. 5.4.1 Discussion of single group ICA based approaches
2. 5.4.2 Discussion of multiple ICA run approaches
6. 6 Conclusions
7. 7 Figure Legends
## Abstract
Spatial Independent Component Analysis (ICA) decomposes the time by space
functional MRI (fMRI) matrix into a set of 1-D basis time courses and their
associated 3-D spatial maps that are optimized for mutual independence. When
applied to resting state fMRI (rsfMRI), ICA produces several spatial
independent components (ICs) that seem to have biological relevance - the so-
called resting state networks (RSNs). The ICA problem is well posed when the
true data generating process follows a linear mixture of ICs model in terms of
the identifiability of the mixing matrix. However, the contrast function used
for promoting mutual independence in ICA is dependent on the finite amount of
observed data and is potentially non-convex with multiple local minima. Hence,
each run of ICA could produce potentially different IC estimates even for the
same data. One technique to deal with this run-to-run variability of ICA was
proposed by Yang et al. (2008) in their algorithm RAICAR which allows for the
selection of only those ICs that have a high run-to-run reproducibility. We
propose an enhancement to the original RAICAR algorithm that enables us to
assign reproducibility $p$-values to each IC and allows for an objective
assessment of both within subject and across subjects reproducibility. We call
the resulting algorithm RAICAR-N (N stands for null hypothesis test), and we
have applied it to publicly available human rsfMRI data
(http://www.nitrc.org). Our reproducibility analyses indicated that many of
the published RSNs in rsfMRI literature are highly reproducible. However, we
found several other RSNs that are highly reproducible but not frequently
listed in the literature.
## Notation
* •
Scalars variables and functions will be denoted in a non-bold font (e.g.,
$\sigma^{2},L,p$ or $\Psi,f$). Vectors will be denoted in a bold font using
lower case letters (e.g.,
$\boldsymbol{y},\boldsymbol{\mu},\boldsymbol{\eta}$). Matrices will be denoted
in bold font using upper case letters (e.g.,
$\boldsymbol{A},\boldsymbol{\Sigma},\boldsymbol{W}$). The transpose of a
matrix $\boldsymbol{A}$ will be denoted by $\boldsymbol{A^{T}}$ and its
inverse will be denoted by $\boldsymbol{A^{-1}}$. $\boldsymbol{I_{p}}$ will
denote the $p\times p$ identity matrix and $\mathbf{0}$ will denote a vector
or matrix of all zeros whose size should be clear from context. ${N\choose L}$
is the number of ways of choosing $L$ objects from $N$ objects when order does
not matter.
* •
The $j$th component of vector $\boldsymbol{t_{i}}$ will be denoted by $t_{ij}$
whereas the $j$th component of vector $\boldsymbol{t}$ will be denoted by
$t_{j}$. The element $(i,j)$ of matrix $\boldsymbol{G}$ will be denoted by
$G(i,j)$ or $G_{ij}$. Estimates of variables will be denoted by putting a hat
on top of the variable symbol. For example, an estimate of $\boldsymbol{s}$
will be denoted by $\boldsymbol{\hat{s}}$.
* •
If $\boldsymbol{x}$ is a random vector with a multivariate Normal distribution
with mean $\boldsymbol{\mu}$ and covariance $\boldsymbol{\Sigma}$ then we will
denote this distribution by
$\mathcal{N}\left(\boldsymbol{x}\mid\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
The joint density of vector $\boldsymbol{s}$ will be denoted by
$\boldsymbol{p}_{\boldsymbol{s}}(\boldsymbol{s})$ whereas the marginal density
of $s_{i}$ will be denoted as $p_{s_{i}}(s_{i})$.
$\mathbf{E}\left[f(\boldsymbol{s},\boldsymbol{\eta})\right]$ denotes the
expectation of $f(\boldsymbol{s},\boldsymbol{\eta})$ with respect to both
random variables $\boldsymbol{s}$ and $\boldsymbol{\eta}$.
## 1 Introduction
Independent component analysis (ICA) (Jutten and Herault, 1991; Comon, 1994;
Bell and Sejnowski, 1995; Attias, 1999) models the observed data as a linear
combination of a set of statistically independent and unobservable sources.
(McKeown et al., 1998) first proposed the application of ICA to the analysis
of functional magnetic resonance imaging (fMRI) data. Subsequently, ICA has
been applied to fMRI both as an exploratory tool for the purpose of
identifying task related components (McKeown et al., 1998) as well as a signal
clean up tool for the purpose of removing artifacts from the fMRI data (Tohka
et al., 2008). Recently, it has been shown that ICA applied to resting state
fMRI (rsfMRI) in healthy subjects reveals a set of biologically meaningful
spatial maps of independent components (ICs) that are consistent across
subjects - the so called resting state networks (RSNs) (Beckmann et al.,
2005). Hence, there is a considerable interest in applying ICA to rsfMRI data
in order to define the set of RSNs that characterize a particular group of
human subjects, a disease, or a pharmacological effect.
Several variants of the linear ICA model have been applied to fMRI data
including square ICA (with equal number of sources and sensors) (McKeown et
al., 1998), non-square ICA (with more sensors than sources) (Calhoun et al.,
2001), and non-square ICA with additive Gaussian noise (noisy ICA) (Beckmann
and Smith, 2004). All of these models are well known in the ICA literature
(Jutten and Herault, 1991; Cardoso, 1998; Comon, 1994; Attias, 1999). Since
the other ICA models are specializations of the noisy ICA model, we will
assume a noisy ICA model henceforth.
Remarkably, the ICA estimation problem is well posed in terms of the
identifiability of the mixing matrix given several non-Gaussian and at most 1
Gaussian source in the overall linear mixture (Rao, 1969; Comon, 1994; Theis,
2004; Davies, 2004). In the presence of more than 1 Gaussian source, such as
in noisy ICA, the mixing matrix corresponding to the non-Gaussian part of the
linear mixture is identifiable (upto permutation and scaling). In addition,
the source distributions are uniquely identifiable (upto permutation and
scaling) given a noisy ICA model with a particular Gaussian co-variance
structure, for example, the isotropic diagonal co-variance. For details, see
section 2.1.2.
While these uniqueness results are reassuring, a number of practical
difficulties prevent the reliable estimation of ICs on real data. These
difficulties include (1) true data not describable by an ICA model, (2) ICA
contrast function approximations, (3) multiple local minima in the ICA
contrast function, (4) confounding Gaussian noise and (5) model order
overestimation. See section 2.1.3 for more details. A consequence of these
difficulties is that multiple ICA runs on the same data or different subsets
of the data produce different estimates of the IC realizations.
One technique to account for this run-to-run variability in ICA was proposed
by (Himberg et al., 2004) in their algorithm ICASSO. Using repeated runs of
ICA with bootstrapped data using various initial conditions, ICASSO clusters
ICs across ICA runs using agglomerative hierarchical clustering and also helps
in visualizing the estimated ICs. The logic is that reliable ICs will show up
in almost all ICA runs and thus will form a tight cluster well separated from
the rest. (Esposito et al., 2005) proposed a technique similar to ICASSO
called self-organizing group ICA (sogICA) which allows for clustering of ICs
via hierarchical clustering in across subject ICA runs. When applied to
multiple ICA runs across subjects, ICASSO does not restrict the IC clusters to
contain only 1 IC from each subject per ICA run. In contrast, sogICA allows
the user to select the minimum number of subjects for a ”group representative”
IC cluster containing distinct subjects. By labelling each ICA run as a
different ”subject” sogICA can also be applied to analyze multiple ICA runs
across subjects.
Similar in spirit to ICASSO and sogICA, Yang et al. (2008) proposed an
intuitive approach called RAICAR (Ranking and Averaging Independent Component
Analysis by Reproducibility) for reproducibility analysis of estimated ICs.
The basic idea in RAICAR is to select only those ICs as ”interesting” or
”stable” which show a high run-to-run ”reproducibility”. RAICAR uses simple
and automated spatial cross-correlation matrix based IC alignment which has
been shown to be more accurate compared to ICASSO (Yang et al., 2008). RAICAR
is applicable to both within subject as well as across subjects
reproducibility analysis.
A few limitations of ICASSO, sogICA and RAICAR are worth noting:
* •
ICASSO requires the user to select the number of IC clusters and is
inapplicable without modification for across subjects analysis of ICA runs
since the IC clusters are not restricted to contain only 1 IC per ICA run.
* •
sogICA requires the user to select the minimum number of subjects for a ”group
representative” cluster and also a cutoff on within cluster distances.
* •
RAICAR uses an arbitrary threshold on the reproducibility indices selected ”by
eye” or set at an arbitrary value, such as $50\%$ of the maximum
reproducibility value.
We propose a simple extension to RAICAR that avoids subjective user decisions
and allows for an automatic reproducibility cutoff. The reproducibility
indices calculated in RAICAR differ in magnitude significantly depending on
whether the input to RAICAR:
* •
(a) is generated using multiple ICA runs on the same data
* •
(b) comes from multiple ICA runs on varying data sets (e.g. between and across
subject runs)
Figure 1: Figure illustrates the variation in normalized reproducibility from
RAICAR depending on whether the input to RAICAR is (a) Multiple ICA runs on
single subject data or (b) Multiple ICA runs across subjects. Notice that the
normalized reproducibility is much lower for across subjects analysis compared
to within subject analysis.
See Figure 1 for an illustration of this effect. Obviously, the
reproducibility indices are much lower in case (b) since we account for both
within subject and between subjects variability in estimating ICs. Case (b) is
also of great interest from a practical point of view since we are often
interested in making statements about a group of subjects. Hence, it is clear
that a cutoff on RAICAR reproducibility values for the purposes of selecting
the ”highly reproducible” components should be data dependent. In this work,
1. 1.
We propose a modification of the original RAICAR algorithm by introducing an
explicit ”null” model of no reproducibility.
2. 2.
We use this ”null” model to automatically generate $p$-values for each IC via
simulation. This allows for an objective cutoff specification for extracting
reproducible ICs (e.g. reproducible at $p<0.05$) within and across subjects.
We call the resulting algorithm RAICAR-N (N stands for ”null” hypothesis
test).
3. 3.
We validate RAICAR-N by applying it to publicly available human rsfMRI data.
## 2 Methods
The organization of this article revolves around the following sequence of
questions which ultimately lead to the development of RAICAR-N:
1. 1.
Why is a reproducibility assessment necessary in ICA analysis? In order to
answer this question, we cover the fundamentals of ICA including
identifiability issues in sections 2.1 and 2.2.
2. 2.
How does the original RAICAR algorithm assess reproducibility? The answer to
this question in section 2.3 will set up the stage for RAICAR-N.
3. 3.
How does RAICAR-N permit calculation of reproducibility $p$-values? In section
2.4, we describe the RAICAR-N ”null” model and a simulation based approach for
assigning $p$-values to ICs.
4. 4.
How to promote diversity in group ICA runs given a limited number of subjects
when using RAICAR-N and how to display the non-Gaussian spatial structure in
estimated ICs? These issues are covered in section 2.5 and 2.6.
5. 5.
How can RAICAR-N be extended for between group comparison of ICs and how does
it compare to other approaches in the literature? This question is addressed
in section 5.4.
### 2.1 ICA background
In this section, we provide a brief introduction to ICA along with a
discussion of associated issues related to model order selection,
identifiability and run-to-run variability. The noisy ICA model assumes that
observed data $\boldsymbol{y}$ is generated as a linear combination of
unobservable independent sources confounded with Gaussian noise:
$\boldsymbol{y}=\boldsymbol{\mu}+\boldsymbol{A}\,\boldsymbol{s}+\boldsymbol{\eta}$
(2.1)
In this model,
$\displaystyle\boldsymbol{y}$ $\displaystyle=p\times 1\mbox{ observed signal
vector}$ (2.2) $\displaystyle\boldsymbol{\mu}$ $\displaystyle=p\times 1\mbox{
mean vector}$ $\displaystyle\boldsymbol{A}$ $\displaystyle=p\times q\mbox{
mixing matrix with $p>q$ (more sensors than sources) and rank $q$ }$
$\displaystyle\boldsymbol{\eta}$ $\displaystyle=p\times 1\mbox{ Gaussian noise
vector with density
}\mathcal{N}\left(\boldsymbol{\eta}\mid\boldsymbol{\boldsymbol{0}},\boldsymbol{\boldsymbol{\Sigma}}\right)$
$\displaystyle\boldsymbol{s}$ $\displaystyle=q\times 1\mbox{ vector of
independent random variables (the ICs) }$ $\displaystyle\mbox{ $\,\,\,\,$ with
}\mathbf{E}(\boldsymbol{s}\boldsymbol{s}^{T})=\boldsymbol{D}\mbox{
(diagonal)}\mbox{ and }\mathbf{E}(\boldsymbol{s})=\boldsymbol{0}$
$\displaystyle\mbox{ $\,\,\,\,$ and with }\boldsymbol{s}\mbox{ and
}\boldsymbol{\eta}\mbox{ independent }$
If the marginal density of the $i$th source $s_{i}$ is $p_{s_{i}}(s_{i})$ then
the joint source density $\boldsymbol{p}_{\boldsymbol{s}}(\boldsymbol{s})$
factorizes as $\prod_{i=1}^{q}p_{s_{i}}(s_{i})$ because of the independence
assumption but is otherwise assumed to be unknown. Also, since the elements of
$\boldsymbol{s}$ are independent their co-variance matrix $\boldsymbol{D}$ is
diagonal. The set of variables
$\mathcal{F}=\left\\{\boldsymbol{\mu},\boldsymbol{A},\boldsymbol{D},\boldsymbol{\Sigma}\right\\}$
represents the unknown parameters in the noisy ICA model. Before discussing
the identifiability of model 2.1, we briefly discuss the choice of model order
or the assumed number of ICs $q$.
#### 2.1.1 Estimating the model order $q$
Rigorous estimation of the model order $q$ in noisy ICA is difficult as the IC
densities $p_{s_{i}}(s_{i})$ are unknown. This means that
$\boldsymbol{p}\left(\boldsymbol{y}\mid q,\mathcal{F}\right)$, the marginal
density of the observed data given the model order and the ICA parameters
cannot be derived in closed form (by integrating out the ICs) without making
additional assumptions on the form of IC densities. Consequently, standard
model selection criteria such as Bayes information criterion (BIC) (Kass and
Raftery, 1993) cannot be easily applied to the noisy ICA model to estimate
$q$. One solution is to use a factorial mixture of Gaussians (MOG) joint
source density model as in (Attias, 1999), and use the analytical expression
for $\boldsymbol{p}\left(\boldsymbol{y}\mid q,\mathcal{F}\right)$ in
conjunction with BIC. This solution is quite general in terms of allowing for
an arbitrary Gaussian noise co-variance $\boldsymbol{\Sigma}$, but maximizing
$\boldsymbol{p}\left(\boldsymbol{y}\mid q,\mathcal{F}\right)$ with respect to
$\mathcal{F}$ becomes computationally intractable using an expectation
maximization (EM) algorithm for $q>13$ ICs (Attias, 1999). Another rigorous
non-parametric approach for estimating $q$ that is applicable to the noisy ICA
model with isotropic diagonal Gaussian noise co-variance i.e., with
$\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$ is the random matrix theory
based sequential hypothesis testing approach of Kritchman and Nadler (2009).
To the best of our knowledge, these are the only 2 rigorous approaches for
estimating $q$ in the noisy ICA model.
Approximate approaches for estimating $q$ commonly used in fMRI literature
(e.g., (Beckmann and Smith, 2004)) consist of first relaxing the isotropic
diagonal noisy ICA model (with
$\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$) into a probabilistic PCA
(PPCA) model of (Tipping, 1999) where the source densities are assumed to be
Gaussian i.e., where
$\boldsymbol{p}_{\boldsymbol{s}}(\boldsymbol{s})=\mathcal{N}\left(\boldsymbol{\boldsymbol{s}}\mid\boldsymbol{\boldsymbol{0}},\boldsymbol{\boldsymbol{I}_{q}}\right)$.
When using the PPCA model, it becomes possible to integrate out the Gaussian
sources to get an expression for $\boldsymbol{p}\left(\boldsymbol{y}\mid
q,\mathcal{F}\right)$ that can be analytically maximized (Tipping, 1999).
Subsequently, methods such as BIC can be applied to estimate $q$. Alternative
approaches for estimating $q$ in the PPCA model consist of the Bayesian model
selection of Minka (2000), or in data-rich situations such as fMRI, even the
standard technique of cross-validation (Hastie et al., 2009).
From a biological point of view, it has been argued (Cole et al., 2010) that
the number of extracted ICs simply reflect the various equally valid views of
the human functional neurobiology - smaller number of ICs represent a coarse
view while a larger number of ICs represent a more fine grained view. However,
it is worth noting that from a statistical point of view, over-specification
of $q$ will lead to over-fitting of the ICA model which might render the
estimated ICs less generalizable across subjects. On the other hand, under-
specification of $q$ will result in incomplete IC separation. Both of these
scenarios are undesirable.
#### 2.1.2 Identifiability of the noisy ICA model
To what extent is the noisy linear ICA model identifiable? Consider a
potentially different decomposition of the noisy ICA model 2.1:
$\boldsymbol{y}=\boldsymbol{\mu_{1}}+\boldsymbol{A_{1}}\,\boldsymbol{s_{1}}+\boldsymbol{\eta_{1}}$
(2.3)
where
$\displaystyle\boldsymbol{y}$ $\displaystyle=p\times 1\mbox{ observed signal
vector}$ (2.4) $\displaystyle\boldsymbol{\mu_{1}}$ $\displaystyle=p\times
1\mbox{ mean vector}$ $\displaystyle\boldsymbol{A_{1}}$ $\displaystyle=p\times
q\mbox{ mixing matrix with $p>q$ (more sensors than sources) and rank $q$ }$
$\displaystyle\boldsymbol{\eta_{1}}$ $\displaystyle=p\times 1\mbox{ Gaussian
noise vector with density
}\mathcal{N}\left(\boldsymbol{\eta_{1}}\mid\boldsymbol{\boldsymbol{0}},\boldsymbol{\boldsymbol{\Sigma_{1}}}\right)$
$\displaystyle\boldsymbol{s_{1}}$ $\displaystyle=q\times 1\mbox{ vector of
independent random variables (the ICs) }$ $\displaystyle\mbox{ $\,\,\,\,$ with
}\mathbf{E}(\boldsymbol{s}\boldsymbol{s}^{T})=\boldsymbol{D_{1}}\mbox{
(diagonal)}\mbox{ and }\mathbf{E}(\boldsymbol{s})=\boldsymbol{0}$
$\displaystyle\mbox{ $\,\,\,\,$ and with }\boldsymbol{s_{1}}\mbox{ and
}\boldsymbol{\eta_{1}}\mbox{ independent }$
What can be said about the equivalence between the parameterizations in 2.1
and 2.3?
##### Identifiability of $\boldsymbol{\mu}$:
Equating the expectations of the right hand size of 2.3 and 2.1 and noting
that
$\boldsymbol{s},\boldsymbol{\eta},\boldsymbol{s_{1}},\boldsymbol{\eta_{1}}$
have mean $\boldsymbol{0}$ we get:
$\boldsymbol{\mu_{1}}=\boldsymbol{\mu}$ (2.5)
Thus the mean vector $\boldsymbol{\mu}$ is exactly identifiable.
##### Identifiability of $\boldsymbol{A}$:
A fundamental decomposition result states that the noisy ICA problem is well-
posed in terms of the identifiability of the mixing matrix $\boldsymbol{A}$
upto permutation and scaling provided that the components of $\boldsymbol{s}$
are independent and non-Gaussian (Rao, 1969; Comon, 1994; Theis, 2004; Davies,
2004). If $\boldsymbol{\Lambda}$ is a diagonal scaling matrix and
$\boldsymbol{P}$ is a permutation matrix then the identifiability result can
be stated as:
$\boldsymbol{A_{1}}=\boldsymbol{A}\,\boldsymbol{\Lambda}\,\boldsymbol{P}$
(2.6)
where 2.3 is another decomposition of $\boldsymbol{y}$ with
$\boldsymbol{s_{1}}$ containing independent and non-Gaussian components. In
other words, the mixing matrix $\boldsymbol{A}$ is identifiable upto
permutation and scaling.
##### Identifiability of $\boldsymbol{D}$ and $\boldsymbol{\Sigma}$:
Equating the second moments of the right hand side of 2.3 and 2.1 and noting
the equality of means 2.5 and the independence of
$\boldsymbol{s},\boldsymbol{\eta}$ and
$\boldsymbol{s_{1}},\boldsymbol{\eta_{1}}$ we get:
$\boldsymbol{E}\left[(\boldsymbol{y}-\boldsymbol{\mu})(\boldsymbol{y}-\boldsymbol{\mu})^{T}\right]=\boldsymbol{A}\boldsymbol{D}\boldsymbol{A^{T}}+\boldsymbol{\Sigma}=\boldsymbol{A_{1}}\boldsymbol{D_{1}}\boldsymbol{A_{1}^{T}}+\boldsymbol{\Sigma_{1}}$
(2.7)
Let $\boldsymbol{W}$ be a $q\times p$ matrix and $\boldsymbol{\tilde{Q}}$ be a
$p\times(p-q)$ orthogonal matrix such that:
$\displaystyle\boldsymbol{W}$
$\displaystyle=(\boldsymbol{A^{T}}\boldsymbol{A})^{-1}\boldsymbol{A^{T}}$
(2.8) $\displaystyle\boldsymbol{\tilde{Q}^{T}}\boldsymbol{A}$
$\displaystyle=\boldsymbol{0}$
$\displaystyle\boldsymbol{\tilde{Q}^{T}}\boldsymbol{\tilde{Q}}$
$\displaystyle=\boldsymbol{I_{p-q}}$
From 2.8 and 2.7 we get:
$\displaystyle\boldsymbol{D}+\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}}$
$\displaystyle=\boldsymbol{\Lambda}\,\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}\boldsymbol{\Lambda^{T}}+\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}}$
(2.9)
$\displaystyle\boldsymbol{\tilde{Q}^{T}}\boldsymbol{\Sigma}\boldsymbol{\tilde{Q}}$
$\displaystyle=\boldsymbol{\tilde{Q}^{T}}\boldsymbol{\Sigma_{1}}\boldsymbol{\tilde{Q}}$
Case 1: $\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$ and
$\boldsymbol{\Sigma_{1}}=\sigma_{1}^{2}\boldsymbol{I_{p}}$
The second equation in 2.9 along with the orthogonality of
$\boldsymbol{\tilde{Q}}$ gives $\sigma^{2}=\sigma_{1}^{2}$ and thus
$\boldsymbol{\Sigma}=\boldsymbol{\Sigma_{1}}$. If we fix the scaling of
$\boldsymbol{A_{1}}$ by selecting
$\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$ then from the first equation in
2.9 we get:
$\displaystyle\boldsymbol{D}$
$\displaystyle=\boldsymbol{\Lambda}\,\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}\boldsymbol{\Lambda^{T}}$
(2.10)
$\displaystyle=\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}\boldsymbol{\Lambda^{2}}$
($\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}$ is diagonal)
$\displaystyle=\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}$
In other words, the noise co-variance
$\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$ is uniquely determined and
for a fixed scaling $\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$, the source
variances $\boldsymbol{D}$ are also uniquely determined upto permutation.
Case 2: $\boldsymbol{\Sigma}$ and $\boldsymbol{\Sigma_{1}}$ arbitrary positive
definite matrices
Suppose $\boldsymbol{X}$ is a square matrix and let
$\mbox{diag}(\boldsymbol{X})$ be the diagonal matrix obtained by setting the
non-diagonal elements of $\boldsymbol{X}$ to $0$ and similarly let
$\mbox{offdiag}(\boldsymbol{X})$ be the matrix obtained by setting the
diagonal elements of $\boldsymbol{X}$ to 0. The noise-covariance is partially
identifiable by the following conditions:
$\displaystyle\boldsymbol{\tilde{Q}^{T}}\boldsymbol{\Sigma}\boldsymbol{\tilde{Q}}$
$\displaystyle=\boldsymbol{\tilde{Q}^{T}}\boldsymbol{\Sigma_{1}}\boldsymbol{\tilde{Q}}$
(2.11)
$\displaystyle\mbox{offdiag}\left(\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}}\right)$
$\displaystyle=\mbox{offdiag}\left(\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}}\right)$
For a fixed scaling $\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$, the sources
variances $\boldsymbol{D},\boldsymbol{D_{1}}$ are constrained by:
$\boldsymbol{D}+\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}})=\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}+\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}})\\\
$ (2.12)
In general, the source variances $\boldsymbol{D}$ cannot be uniquely
determined as noted in (Davies, 2004).
##### Identifiability of the distribution of $\boldsymbol{s}$:
Is the distribution of the non-Gaussian components of $\boldsymbol{s}$
identifiable? From 2.1 and 2.3:
$\boldsymbol{\mu}+\boldsymbol{A}\,\boldsymbol{s}+\boldsymbol{\eta}=\boldsymbol{\mu_{1}}+\boldsymbol{A_{1}}\,\boldsymbol{s_{1}}+\boldsymbol{\eta_{1}}$
(2.13)
Substituting 2.5 and 2.6 in 2.13 we get:
$\boldsymbol{A}\,\boldsymbol{s}+\boldsymbol{\eta}=\boldsymbol{A}\,\boldsymbol{\Lambda}\,\boldsymbol{P}\,\boldsymbol{s_{1}}+\boldsymbol{\eta_{1}}$
(2.14)
Premultiplying both sides by $\boldsymbol{W}$ from 2.8 we get:
$\boldsymbol{s}+\boldsymbol{W}\boldsymbol{\eta}=\boldsymbol{\Lambda}\,\boldsymbol{P}\,\boldsymbol{s_{1}}+\boldsymbol{W}\boldsymbol{\eta_{1}}$
(2.15)
Let
$\Psi_{\boldsymbol{s}},\Psi_{\boldsymbol{\eta}},\Psi_{\boldsymbol{s_{1}}},\Psi_{\boldsymbol{\eta_{1}}}$
be the characteristic functions of
$\boldsymbol{s},\boldsymbol{\eta},\boldsymbol{s_{1}}$ and
$\boldsymbol{\eta_{1}}$ respectively. Then
$\displaystyle\Psi_{\boldsymbol{s}}(\boldsymbol{t})$
$\displaystyle=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\boldsymbol{s}\right)\right]$
$\displaystyle\Psi_{\boldsymbol{\eta}}(\boldsymbol{t})$
$\displaystyle=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\boldsymbol{\eta}\right)\right]$
(2.16) $\displaystyle\Psi_{\boldsymbol{s_{1}}}(\boldsymbol{t})$
$\displaystyle=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\boldsymbol{s_{1}}\right)\right]$
$\displaystyle\Psi_{\boldsymbol{\eta_{1}}}(\boldsymbol{t})$
$\displaystyle=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\boldsymbol{\eta_{1}}\right)\right]$
where $i=\sqrt{-1}$ and $\boldsymbol{t}$ is a vector of real numbers of length
equal to that of the corresponding random vectors in 2.16. Using 2.15, we can
write:
$\displaystyle\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{s}+\boldsymbol{W}\boldsymbol{\eta}\right\\}\right)\right]=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{\Lambda}\,\boldsymbol{P}\,\boldsymbol{s_{1}}+\boldsymbol{W}\boldsymbol{\eta_{1}}\right\\}\right)\right]\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$ (2.17)
Noting the independence of $\boldsymbol{s},\boldsymbol{\eta}$ and
$\boldsymbol{s_{1}},\boldsymbol{\eta_{1}}$:
$\displaystyle\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{s}\right\\}\right)\right]\,\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{W}\boldsymbol{\eta}\right\\}\right)\right]$
$\displaystyle=\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{\Lambda}\,\boldsymbol{P}\,\boldsymbol{s_{1}}\right\\}\right)\right]\,\mathbf{E}\left[\mbox{exp}\left(i\boldsymbol{t^{T}}\left\\{\boldsymbol{W}\boldsymbol{\eta_{1}}\right\\}\right)\right]$
(2.18)
$\displaystyle\Rightarrow\Psi_{\boldsymbol{s}}\left(\boldsymbol{t}\right)\,\,\Psi_{\boldsymbol{\eta}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}}\left(\boldsymbol{P^{T}}\boldsymbol{\Lambda^{T}}\boldsymbol{t}\right)\,\,\Psi_{\boldsymbol{\eta_{1}}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$
Now $\boldsymbol{\eta}$ and $\boldsymbol{\eta_{1}}$ are multivariate Gaussian
random vectors both with mean $\boldsymbol{0}$ and co-variance matrix
$\boldsymbol{\Sigma}$ and $\boldsymbol{\Sigma_{1}}$ respectively. Hence, their
characteristic functions are given by (Feller, 1966; Wlodzimierz, 1995):
$\displaystyle\Psi_{\boldsymbol{\eta}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)=\mbox{
exp
}\left(-\frac{1}{2}\boldsymbol{t^{T}}\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}}\boldsymbol{t}\right)\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$ (2.19)
$\displaystyle\Psi_{\boldsymbol{\eta_{1}}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)=\mbox{
exp
}\left(-\frac{1}{2}\boldsymbol{t^{T}}\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}}\boldsymbol{t}\right)\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$
###### Claim 2.1.
A sufficient condition for identifiability upto permutation and scaling of the
non-Gaussian distributions in $\boldsymbol{s}$ given two different
parameterizations in 2.1 and 2.3 is:
$\displaystyle\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}})=\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}})$
(2.20)
###### Proof.
From 2.20 and 2.11, we get:
$\displaystyle\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}}=\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}}$
(2.21)
Thus from 2.19,
$\displaystyle\Psi_{\boldsymbol{\eta}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)=\Psi_{\boldsymbol{\eta_{1}}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$ (2.22)
From 2.19,
$\Psi_{\boldsymbol{\eta}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)$ and
$\Psi_{\boldsymbol{\eta_{1}}}\left(\boldsymbol{W^{T}}\boldsymbol{t}\right)$
are not equal to 0 for any finite $\boldsymbol{t}$, therefore, from 2.22 and
2.18 we get:
$\displaystyle\Psi_{\boldsymbol{s}}\left(\boldsymbol{t}\right)\,\,$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}}\left(\boldsymbol{P^{T}}\boldsymbol{\Lambda^{T}}\boldsymbol{t}\right)\mbox{
for all }\boldsymbol{t}\in\mathbf{R^{q}}$ (2.23)
Note that $\boldsymbol{\Lambda}$ is a diagonal scaling matrix with entries
$\lambda_{1},\lambda_{2},\ldots,\lambda_{q}$ on the diagonal and
$\boldsymbol{P}$ is a permutation matrix. Thus,
$\displaystyle\boldsymbol{P}^{T}\boldsymbol{\Lambda^{T}}\boldsymbol{t}=\begin{pmatrix}\lambda_{i_{1}}t_{i_{1}}\\\
\lambda_{i_{2}}t_{i_{2}}\\\ \vdots\\\ \lambda_{i_{q}}t_{i_{q}}\end{pmatrix}$
(2.24)
where $i_{1},i_{2},\ldots,i_{q}$ is some permutation of integers
$1,2,\ldots,q$. Suppose $\Psi_{\boldsymbol{s}(j)}$ is the characteristic
function of the $j$th component of $\boldsymbol{s}$ and
$\Psi_{\boldsymbol{s_{1}}(j)}$ is the characteristic function of the $j$th
component of $\boldsymbol{s_{1}}$. Since the components of $\boldsymbol{s}$
and $\boldsymbol{s_{1}}$ are independent by assumption, the joint
characteristic functions $\Psi(\boldsymbol{s})$ and $\Psi(\boldsymbol{s_{1}})$
factorize:
$\displaystyle\Psi_{\boldsymbol{s}}\left(\boldsymbol{t}\right)$
$\displaystyle=\Psi_{\boldsymbol{s}(1)}(t_{1})\,\Psi_{\boldsymbol{s}(2)}(t_{2})\ldots\Psi_{\boldsymbol{s}(j)}(t_{j})\ldots\Psi_{\boldsymbol{s}(q)}(t_{q})$
(2.25)
$\displaystyle\Psi_{\boldsymbol{s_{1}}}\left(\boldsymbol{P^{T}}\boldsymbol{\Lambda^{T}}\boldsymbol{t}\right)$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}(1)}(\lambda_{i_{1}}t_{i_{1}})\Psi_{\boldsymbol{s_{1}}(2)}(\lambda_{i_{2}}t_{i_{2}})\ldots\Psi_{\boldsymbol{s_{1}}(j)}(\lambda_{i_{j}}t_{i_{j}})\ldots\Psi_{\boldsymbol{s_{1}}(q)}(\lambda_{i_{q}}t_{i_{q}})$
From 2.25 and 2.23
$\displaystyle\Psi_{\boldsymbol{s}(1)}(t_{1})\ldots\Psi_{\boldsymbol{s}(j)}(t_{j})\ldots\Psi_{\boldsymbol{s}(q)}(t_{q})$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}(1)}(\lambda_{i_{1}}t_{i_{1}})\ldots\Psi_{\boldsymbol{s_{1}}(j)}(\lambda_{i_{j}}t_{i_{j}})\ldots\Psi_{\boldsymbol{s_{1}}(q)}(\lambda_{i_{q}}t_{i_{q}})$
(2.26)
All characteristic functions satisfy (Feller, 1966; Wlodzimierz, 1995):
$\displaystyle\Psi_{\boldsymbol{s}(k)}(0)$ $\displaystyle=1$ (2.27)
$\displaystyle\Psi_{\boldsymbol{s_{1}}(k)}(0)$ $\displaystyle=1\mbox{ for all
$k$ }$
Since $i_{1},i_{2},\ldots,i_{q}$ is simply a permutation of integers
$1,2,\ldots,q$, there exists a $j$ such that $i_{j}=1$. Then set
$t_{2}=0,t_{3}=0,\ldots,t_{q}=0$ in 2.26. Then 2.27 and 2.26 imply:
$\displaystyle\Psi_{\boldsymbol{s}(1)}(t_{1})$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}(j)}(\lambda_{i_{j}}t_{i_{j}})=\Psi_{\boldsymbol{s_{1}}(j)}(\lambda_{1}t_{1})\mbox{
for all }t_{1}\in\mathbf{R}$ (2.28)
Select the scaling matrix as $\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$ and
thus $\boldsymbol{\Lambda}$ is a diagonal matrix with elements $\pm 1$ on the
diagonal. Thus $\lambda_{1}=\pm 1$ and 2.28 can be re-written as:
$\displaystyle\Psi_{\boldsymbol{s}(1)}(t_{1})=\Psi_{\boldsymbol{s_{1}}(j)}(\pm
t_{1})\mbox{ for all }t_{1}\in\mathbf{R}$ (2.29)
Therefore,
$\displaystyle\Psi_{\boldsymbol{s}(1)}(t_{1})$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}(j)}(t_{1})\mbox{ for all
}t_{1}\in\mathbf{R}$ (2.30) or $\displaystyle\Psi_{\boldsymbol{s}(1)}(t_{1})$
$\displaystyle=\Psi_{\boldsymbol{s_{1}}(j)}(-t_{1})=\Psi_{-\boldsymbol{s_{1}}(j)}(t_{1})\mbox{
for all }t_{1}\in\mathbf{R}$
Hence the characteristic function of the $1$st component of $\boldsymbol{s}$
is identical to the characteristic function of the (possibly sign-flipped)
$j$th component of $\boldsymbol{s_{1}}$. Since characteristic functions
uniquely characterize a probability distribution (Feller, 1966), the
distribution of $\boldsymbol{s}(1)$ and $\pm\boldsymbol{s_{1}}(j)$ is
identical. Next, by setting $t_{1}=0,t_{3}=0,\ldots,t_{q}=0$, we can find a
distribution from $\boldsymbol{s_{1}}$ that matches the $2$nd component
$\boldsymbol{s}(2)$ of $\boldsymbol{s}$. Proceeding in a similar fashion, it
is clear that the distribution of each component of $\boldsymbol{s}$ is
uniquely identifiable upto sign flips for the choice
$\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$. For a general
$\boldsymbol{\Lambda}$, the source distributions are uniquely identifiable
upto permutation and (possibly negative) scaling, as claimed. ∎
While the source distributions might not be uniquely identifiable for
arbitrary co-variance matrices $\boldsymbol{\Sigma}$, they are indeed uniquely
identifiable upto permutation and scaling for the noisy ICA model with
isotropic Gaussian noise co-variance. For more general conditions that
guarantee uniqueness of source distributions, please see Eriksson and Koivunen
(2004, 2006).
###### Corollary 2.2.
If $\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$ and
$\boldsymbol{\Sigma_{1}}=\sigma_{1}^{2}\boldsymbol{I_{p}}$, then the source
distributions are uniquely identifiable upto sign flips for
$\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$.
###### Proof.
Suppose $\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$ and
$\boldsymbol{\Sigma_{1}}=\sigma_{1}^{2}\boldsymbol{I_{p}}$. Then from 2.9
$\boldsymbol{\Sigma}=\boldsymbol{\Sigma_{1}}$ and thus
$\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}})=\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}})$.
The corollary then follows from Claim 2.1. ∎
###### Corollary 2.3.
If $\boldsymbol{D}=\boldsymbol{D_{1}}=\boldsymbol{I_{q}}$, then the source
distributions are uniquely identifiable up to sign flips for
$\boldsymbol{\Lambda^{2}}=\boldsymbol{I_{q}}$.
###### Proof.
If $\boldsymbol{D}=\boldsymbol{D_{1}}=\boldsymbol{I_{q}}$, then noting that
$\boldsymbol{P}\boldsymbol{P}^{T}=\boldsymbol{I_{q}}$, we get
$\boldsymbol{D}=\boldsymbol{P}\boldsymbol{D_{1}}\boldsymbol{P^{T}}$. Hence
from 2.12, we get
$\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma}\boldsymbol{W^{T}})=\mbox{diag}(\boldsymbol{W}\boldsymbol{\Sigma_{1}}\boldsymbol{W^{T}})$.
The corollary then follows from Claim 2.1. ∎
#### 2.1.3 Why is there a run-to-run variability in estimated ICs?
From the discussion in section 2.1.2, it is clear that for a noisy ICA model
with isotropic diagonal additive Gaussian noise co-variance:
1. 1.
The noisy ICA parameters
$\mathcal{F}=\left\\{\boldsymbol{\mu},\boldsymbol{A},\boldsymbol{D},\boldsymbol{\Sigma}\right\\}$
are uniquely identifiable up to permutation and scaling.
2. 2.
The source distributions in $\boldsymbol{s}$ are uniquely identifiable upto
permutation and scaling.
While the above theoretical properties of ICA are reassuring, there are a
number of practical difficulties that prevent the reliable estimation of ICs
on real data:
1. 1.
Validity of the ICA model:
The assumption that the observed real data is generated by an ICA model is
only that - an ”assumption”. If this assumption is not valid, then the
uniqueness results do not hold anymore.
2. 2.
Mutual information approximations:
From an information theoretic point of view, the ICA problem is solved by
minimizing a contrast function which is an approximation to the mutual
information (Hyvarinen, 1998) between the ICs that depends on the finite
amount of observed data. Such an approximation is necessary, since we do not
have access to the marginal source densities $p_{s_{i}}$. Different
approximations to mutual information will lead to different objective
functions and hence different solutions. This is one of the reasons why
different ICA algorithms often produce different IC estimates even for the
same data.
3. 3.
Non-convexity of ICA objective functions:
The ICA contrast function is potentially non-convex and hence has multiple
local minima. Since global minimization is a challenging problem by itself,
most ICA algorithms will only converge to local minima of the ICA contrast
function. The run-to-run variability of IC estimates will also depend on the
number of local minima in a particular ICA contrast function.
4. 4.
IC estimate corruption by Gaussian noise:
For noisy ICA, the IC realizations cannot be recovered exactly even if the
true mixing matrix $\boldsymbol{A}$ and mean vector $\boldsymbol{\mu}$ are
known in 2.1. Commonly used estimators for recovering realization of ICs
include the least squares (Beckmann and Smith, 2004) as well as the minimum
mean square error (MMSE) (Davies, 2004). Consider the least squares estimate
$\boldsymbol{\hat{s}}$ of a realization of $\boldsymbol{s}$ based on
$\boldsymbol{y}$:
$\displaystyle\boldsymbol{\hat{s}}=(\boldsymbol{A^{T}}\boldsymbol{A})^{-1}\boldsymbol{A^{T}}(\boldsymbol{y}-\boldsymbol{\mu})=\boldsymbol{s}+(\boldsymbol{A^{T}}\boldsymbol{A})^{-1}\boldsymbol{A^{T}}\boldsymbol{\eta}$
(2.31)
This means that even for known parameters, IC realization estimates
$\boldsymbol{\hat{s}}$ will be corrupted by correlated Gaussian noise. Hence
using different subsets of the data under the true model will also lead to
variability in estimated ICs.
5. 5.
Over-fitting of the ICA model:
Over specification of the model order leads to the problem of over-fitting in
ICA. As we describe below, this can lead to (1) the phenomenon of IC
”splitting” and (2) an increase in the variance of the IC estimates.
1\. IC ”splitting”
Suppose that the true model order or the number of non-Gaussian sources in an
ICA decomposition of $\boldsymbol{y}$ such as 2.1 is $q$. Then a fundamental
result in (Rao, 1969, Theorem 1) states that for any other ICA decomposition
of $\boldsymbol{y}$, the number of non-Gaussian sources remains the same while
the number of Gaussian sources can change. In other words, $\boldsymbol{y}$
cannot have two different ICA decompositions containing different number of
non-Gaussian sources.
In view of this fact, how can a model order $q$ ICA decomposition containing
$q$ non-Gaussian sources be ”split” into a $(q+1)$ ICA decomposition
containing $(q+1)$ non-Gaussian sources when performing ICA estimation using
an assumed model order of $(q+1)$? As we describe below, the order $(q+1)$ ICA
decomposition is only an approximation to the order $q$ ICA decomposition.
Let $\boldsymbol{a_{i}}$ be the $i$th column of $\boldsymbol{A}$ in 2.1. In
the presence of noise, it might be possible to approximate:
$\boldsymbol{a_{i}}s_{i}\approx\boldsymbol{a_{i}^{1}}s_{i}^{1}+\boldsymbol{a_{i}^{2}}s_{i}^{2}$
(2.32)
Here:
* •
$\boldsymbol{a_{i}}s_{i}$ is the contribution of the $i$th non-Gaussian source
$s_{i}$ to the ICA model 2.1.
* •
$s_{i}^{1}$ and $s_{i}^{2}$ are independent non-Gaussian random variables that
are also independent with respect to all non-Gaussian sources $s_{j},j\neq i$
in 2.1.
* •
$\boldsymbol{a_{i}^{1}}$ and $\boldsymbol{a_{i}^{2}}$ are the basis time
courses corresponding to $s_{i}^{1}$ and $s_{i}^{2}$ respectively.
* •
The time courses $\boldsymbol{a_{i}^{1}}$ and $\boldsymbol{a_{i}^{2}}$ look
similar to each other.
Note that if $\boldsymbol{a_{i}^{1}}=\boldsymbol{a_{i}^{2}}$, then 2.32 can be
made into an equality by choosing $s_{i}=s_{i}^{1}+s_{i}^{2}$. By replacing
$\boldsymbol{a_{i}}s_{i}$ in 2.1 using 2.32, we arrive at an approximate model
order $(q+1)$ decomposition of $\boldsymbol{y}$. In this decomposition, the
component $s_{i}$ from a model order $q$ decomposition appears to be ”split”
into two sub-components: $s_{i}^{1}$ and $s_{i}^{2}$.
2\. Inflated variance of IC estimates
Overestimation of model order will lead to over-fitting of the mixing matrix
$\boldsymbol{A}$. In other words, $\boldsymbol{A}$ could have several columns
that are highly correlated with each other. This could happen as a result of
IC ”splitting” as discussed above. Now, for a given realization
$\boldsymbol{s}$, the variance of $\boldsymbol{\hat{s}}$ is given by
$\mbox{Var}(\boldsymbol{\hat{s}})=\sigma^{2}(\boldsymbol{A^{T}}\boldsymbol{A})^{-1}$
(for isotropic Gaussian co-variance). An increase in number of columns of
$\boldsymbol{A}$ and the fact that many of them are highly correlated implies
that the variability of IC estimates $\mbox{Var}(\boldsymbol{\hat{s}})$ is
inflated.
In other words, running ICA multiple times on the same data or variations
thereof with random initialization could produce different ICs.
### 2.2 ICA algorithms, single subject ICA and group ICA
In this section, we give a brief summary of how the ICA parameters are
estimated in practice and also summarize the two most common modes of ICA
application to fMRI data - single subject ICA (section 2.2.1) and temporal
concatenation based group ICA (section 2.2.2).
Given several independent observations $\boldsymbol{y}$ as per the noisy ICA
model 2.1, most ICA algorithms estimate the ICA parameters
$\mathcal{F}=\left\\{\boldsymbol{\mu},\boldsymbol{A},\boldsymbol{D},\boldsymbol{\Sigma}\right\\}$
and the realizations of $\boldsymbol{s}$ in 2 steps. We only consider the case
with $\boldsymbol{\Sigma}=\sigma^{2}\boldsymbol{I_{p}}$, since as shown in
section 2.1.2, the mixing matrix $\boldsymbol{A}$ and source distributions of
$\boldsymbol{s}$ are identifiable upto permutation and scaling for this case.
1. 1.
First, the diagonal source co-variance is arbitrarily set as
$\boldsymbol{D}=\boldsymbol{I_{q}}$. The mean vector $\boldsymbol{\mu}$ is
estimated as $\mathbf{E}\left(\boldsymbol{y}\right)$. Then, using PCA or PPCA
(Tipping, 1999), the mixing matrix $\boldsymbol{A}$ is estimated, upto an
orthogonal rotation matrix $\boldsymbol{O}$, to be in a signal subspace which
is spanned by the principal eigenvectors corresponding to the largest
eigenvalues of the data co-variance matrix
$\boldsymbol{E}\left[(\boldsymbol{y}-\boldsymbol{\mu})(\boldsymbol{y}-\boldsymbol{\mu})^{T}\right]$.
The noise variance $\sigma^{2}$ is estimated in this step as well.
2. 2.
Next, an estimator $\boldsymbol{\hat{s}}$ for the source realizations is
defined using techniques such as least squares or MMSE. The only unknown
involved in these estimates is the orthogonal rotation matrix
$\boldsymbol{O}$.
3. 3.
Finally, the non-Gaussianity of the empirical density of components of
$\boldsymbol{\hat{s}}$ is optimized with respect to $\boldsymbol{O}$ using
algorithms such as fixed point ICA Hyvarinen (1998, 1999).
For more details on noisy ICA estimation, please see (Beckmann and Smith,
2004) and for more details on ICA algorithms, please see (Hyvarinen et al.,
2001).
#### 2.2.1 Single subject ICA
How is ICA applied to single subject fMRI data? Suppose we are given a single
subject fMRI scan which we rearrange as a $p\times n$ 2D matrix
$\boldsymbol{Y}$ in which column $i$ is the $p\times 1$ observed time-course
$\boldsymbol{y_{i}}$ in the brain at voxel $i$. Observed time-courses
$\boldsymbol{y_{1}},\boldsymbol{y_{2}},\ldots,\boldsymbol{y_{n}}$ are
considered to be $n$ independent realizations of $\boldsymbol{y}$ as per the
linear ICA model 2.1. Suppose
$\boldsymbol{\hat{S}}=[\boldsymbol{\hat{s}_{1}},\boldsymbol{\hat{s}_{2}},\ldots,\boldsymbol{\hat{s}_{n}}]$
is the $q\times n$ matrix containing the estimated source realizations at the
$n$ voxels. The $j$th row of $\boldsymbol{\hat{S}}$ is the $j$th IC. In other
words, we decompose the time by space fMRI 2D matrix into a set of basis time-
courses and a set of $q$ 3D IC maps using ICA.
#### 2.2.2 Group ICA
How is ICA applied to data from a group of subjects in fMRI? Suppose we
collect fMRI images from $m$ subjects. First, we register all subjects to a
common space using a registration algorithm (e.g., affine registration). Next,
we rearrange each of the fMRI scans into $m$ 2D matrices
$\boldsymbol{Y_{1}}\ldots\boldsymbol{Y_{m}}$, each of size $p\times n$. Column
$j$ in $\boldsymbol{Y_{i}}$ is the demeaned time-course observed at voxel
location $j$ for subject $i$. The matrices
$\boldsymbol{Y_{1}}\ldots\boldsymbol{Y_{m}}$ are temporally concatenated to
get a $pm\times n$ matrix $\boldsymbol{Z}$ as follows:
$\boldsymbol{Z}=\begin{pmatrix}\boldsymbol{Y_{1}}\\\ \vdots\\\
\boldsymbol{Y_{i}}\\\ \vdots\\\ \boldsymbol{Y_{m}}\end{pmatrix}$ (2.33)
Column $i$ of $\boldsymbol{Z}$ is the $pm\times 1$ vector $\boldsymbol{z_{i}}$
which is assumed to follow a linear ICA model 2.1.
$\boldsymbol{z_{1}},\boldsymbol{z_{2}},\ldots,\boldsymbol{z_{n}}$ are
considered to be independent realizations of the model 2.1. Suppose
$\boldsymbol{\hat{S}_{G}}=[\boldsymbol{\hat{s}_{1}},\boldsymbol{\hat{s}_{2}},\ldots,\boldsymbol{\hat{s}_{n}}]$
is a $q\times n$ matrix containing the estimated source realizations at the
$n$ voxels. The $j$th row of $\boldsymbol{\hat{S}_{G}}$ is the $j$th group IC.
In group ICA, the joined time-series across subjects is modeled using noisy
linear ICA. In practice, $\boldsymbol{Y_{i}}$ is the PCA reduced data set for
subject $i$. The PCA reduction is either done separately for each subject
using subject specific data co-variance (Calhoun et al., 2001) or an average
data co-variance across subjects (Beckmann et al., 2005). The average co-
variance approach requires each subject to have the same number of time points
in fMRI scans.
### 2.3 The original RAICAR algorithm
In this section, we give a brief introduction to the RAICAR algorithm of (Yang
et al., 2008). Suppose we are given a data set which we decompose into $n_{C}$
ICs using ICA (e.g., single subject or group ICA). Our goal is to assess which
ICs consistently show up in multiple ICA runs i.e., the reproducibility of
each of these $n_{C}$ ICs. To that extent, we run the ICA algorithm $K$ times.
Suppose $\boldsymbol{x}_{j}^{(m)}$ is the $n\times 1$ vector (e.g. spatial ICA
map re-arranged into a vector) of the $j$th IC from $m$th ICA run. Suppose
$\boldsymbol{G}_{lm}$ is a $n_{C}\times n_{C}$ absolute spatial cross-
correlation coefficient matrix between the ICs from runs $l$ and $m$:
$\boldsymbol{G}_{lm}(i,j)=|\mbox{corrcoef}(\boldsymbol{x}_{i}^{(l)},\boldsymbol{x}_{j}^{(m)})|$
(2.34)
where $|.|$ denotes absolute value. $\boldsymbol{G}_{lm}(i,j)$ is the absolute
spatial cross-correlation coefficient between IC $i$ from run $l$ and IC $j$
from run $m$. The matrices $\boldsymbol{G}_{lm}$ are then arranged as elements
of a $K\times K$ block-matrix $\boldsymbol{G}$ such that the $l$th row and
$m$th column of $\boldsymbol{G}$ is $\boldsymbol{G}(l,m)=\boldsymbol{G}_{lm}$
(see Figure 2). This block matrix $\boldsymbol{G}$ is the starting point for a
RAICAR across-run component matching process.
Since ICs within a particular run cannot be matched to each other, the
$n_{C}\times n_{C}$ matrices $\boldsymbol{G}(l,l),l=1\ldots K$ along the
block-diagonal of $\boldsymbol{G}$ are set to $\boldsymbol{0}$ as shown in
Figure 2 with a gray color. The following steps are involved in a RAICAR
analysis:
1. 1.
Find the maximal element of $\boldsymbol{G}$. Suppose this maximum occurs in
matrix $\boldsymbol{G}_{lm}$ at position $(i,j)$. Hence component $i$ from run
$l$ matches component $j$ from run $m$. Let us label this matched component by
$MC_{1}$ (the first matched component).
2. 2.
Next, we attempt to find from each run $s$ ($s\neq l$ and $s\neq m$) a
component that matches with component $MC_{1}$. Suppose element $(a_{s},j)$ is
the maximal element in the $j$th column of $\boldsymbol{G}_{sm}$. Then
component $a_{s}$ is the best matching component from run $s$ with the $j$th
component from run $m$.
Similarly, suppose element $(i,b_{s})$ is the maximal element in the $i$th row
of $\boldsymbol{G}_{ls}$. Then component $b_{s}$ is the best matching
component from run $s$ with component $i$ from run $l$. As noted in (Yang et
al., 2008), in most cases $a_{s}=b_{s}$. However, it is possible that
$a_{s}\neq b_{s}$. Hence the component number $e_{s}$ matching $MC_{1}$ from
run $s$ is defined as follows:
$e_{s}=\begin{cases}a_{s}&\text{if
$\boldsymbol{G}_{sm}(a_{s},j)\geq\boldsymbol{G}_{ls}(i,b_{s})$},\\\
b_{s}&\text{if
$\boldsymbol{G}_{sm}(a_{s},j)<\boldsymbol{G}_{ls}(i,b_{s})$}.\end{cases}$
(2.35)
We would also like to remove component $e_{s}$ of run $s$ from further
consideration during the matching process. To that extent, we zero out the
$e_{s}$th row from $\boldsymbol{G}_{sr},r=1\ldots K$ and the $e_{s}$th column
from $\boldsymbol{G}_{rs},r=1\ldots K$.
3. 3.
Once a matching component $e_{s}$ has been found for all runs $s\neq l,m$, we
also zero out the $i$th row from $\boldsymbol{G}_{lr},r=1\ldots K$ and the ith
column from $\boldsymbol{G}_{rl},r=1\ldots K$. Similarly, we zero out the
$j$th column from $\boldsymbol{G}_{rm},r=1\ldots K$ and the $j$th row from
$\boldsymbol{G}_{mr},r=1\ldots K$. This eliminates component $i$ from run $l$
and component $j$ from run $m$ from further consideration during the matching
process.
4. 4.
Steps 1-3 complete the matching process for one IC component across runs.
These steps are repeated until $n_{C}$ components are matched across the $K$
runs. We label the matched component $s$ as $MC_{s}$ which contains a set of
$K$ matching ICs one from each of the $K$ ICA runs.
Suppose matched component $s$, $MC_{s}$ consists of the matched ICs
$\boldsymbol{x}_{i_{1}}^{(1)},\boldsymbol{x}_{i_{2}}^{(2)},\ldots,\boldsymbol{x}_{i_{K}}^{(K)}$.
Form the $K\times K$ cross-correlation matrix $H_{MC_{s}}$ between the matched
components in $MC_{s}$. The $(a,b)$th element of this matrix is simply:
$H_{MC_{s}}(a,b)=|\mbox{corrcoef}\left(\boldsymbol{x}_{i_{a}}^{(a)},\boldsymbol{x}_{i_{b}}^{(b)}\right)|$
(2.36)
The normalized reproducibility of $MC_{s}$ is then defined as:
$\mbox{Reproducibility}(MC_{s})=\left(\frac{2}{(K-1)K}\right)\sum_{a=1}^{K}\sum_{b=a+1}^{K}H_{MC_{s}}(a,b)$
(2.37)
The double sum in 2.37 is simply the sum of the upper triangular part of
$H_{MC_{s}}$ excluding the diagonal. The normalizing factor $\frac{(K-1)K}{2}$
is simply the maximum possible value of this sum. Hence the normalized
reproducibility satisfies: $\mbox{Reproducibility}(MC_{s})\leq 1$.
Note that our definition of normalized reproducibility is slightly different
from that in Yang et al. (2008). Whereas Yang et al. (2008) averages the
thresholded absolute correlation coefficients, we simply average the un-
thresholded absolute correlation coefficients to compute reproducibility
thereby avoiding the selection of a threshold on the absolute correlation
coefficients.
Figure 2: Pictorial depiction of the original RAICAR algorithm (Yang et al.,
2008). The ICA algorithm is run $K$ times with each run producing $n_{C}$ ICs.
$\boldsymbol{G}$ is a $K\times K$ block matrix with elements
$\boldsymbol{G}(l,m)=\boldsymbol{G}_{lm}$ where $\boldsymbol{G}_{lm}$ is the
$n_{C}\times n_{C}$ absolute spatial cross-correlation matrix between ICs from
runs $l$ and $m$. The numbered green circles indicate the sequence of steps in
applying RAICAR to a given data set. Our definition of normalized
reproducibility in box 7 averages un-thresholded correlation coefficients
thereby avoiding the selection of a correlation coefficient threshold prior to
averaging.
### 2.4 The RAICAR-N enhancement
In this section, we describe how to compute reproducibility $p$-values for
each matched component in RAICAR. Note that the RAICAR ”component matching”
process can be used to assess the reproducibility of any spatial component
maps - not necessarily ICA maps. For instance, RAICAR can be used to assess
the reproducibility of a set of PCA maps across subjects.
In order to generate reproducibility $p$-values for the matched component
maps:
1. 1.
We need to determine the distribution of normalized reproducibility that we
get from the RAICAR ”component matching” process when the input to RAICAR
represents a set of ”non-reproducible component maps” across the $K$ runs.
2. 2.
In addition, we would also like to preserve the overall structure seen in the
observed sets of spatial component maps across the $K$ runs when generating
sets of ”non-reproducible component maps” across the $K$ runs.
Hence for IC reproducibility assessment, we propose to use the original set of
ICs across the $K$ runs to generate the ”non-reproducible component maps”
across the $K$ runs.
Suppose $K$ ICA runs are submitted to RAICAR which gives us a $n_{C}\times 1$
vector of observed normalized reproducibility values
$\mbox{Reproducibility}(MC_{i}),i=1\ldots n_{C}$ \- one for each IC. We
propose to attach $p$-values for measuring the reproducibility of each IC in a
data-driven fashion as follows:
1. 1.
First, we label the $Kn_{C}$ ICs across the $K$ runs using unique integers. In
run 1, the ICs are labelled using integers $1,\ldots,n_{C}$. In run 2, the ICs
are labelled using integers $(n_{C}+1),\ldots,2n_{C}$ and so on. In run $K$,
the ICs are labelled using integers $(K-1)n_{C}+1,\ldots,Kn_{C}$.
2. 2.
Our ”null” hypothesis is:
$\displaystyle\mathbf{H_{0}}:\,\,\,$ None of the ICs are reproducible (2.38)
Hence, we can randomly label component $i$ from run $l$ as component $d$ from
run $s$
To do this, we randomly permute the integers $1,2,\ldots,Kn_{C}$ to get the
permuted integers $p(1),p(2),\ldots,p(Kn_{C})$. Obviously $p(i)\neq p(j)\mbox{
if }i\neq j$.
3. 3.
The $K$ sets ”non-reproducible component runs under $\mathbf{H_{0}}$” are
constructed by assigning components with labels:
* •
$p(1),\ldots,p(n_{C})$ to run 1 under $\mathbf{H_{0}}$.
* •
$p(n_{C}+1),\ldots,p(2n_{C})$ to run 2 under $\mathbf{H_{0}}$
* •
$p\left((K-1)n_{C}+1\right),\ldots,p(Kn_{C})$ to run $K$ under
$\mathbf{H_{0}}$
4. 4.
After $K$ runs have been generated under $\mathbf{H_{0}}$, we subject these to
a RAICAR analysis. This gives us $n_{C}$ values of normalized reproducibility,
one for each matched component under $\mathbf{H_{0}}$.
5. 5.
Steps 1-4 are repeated $R$ times to build up a pooled $Rn_{C}\times 1$ vector
of normalized reproducibility $\mbox{{Reproducibility}}_{Null}$ under
$\mathbf{H_{0}}$.
6. 6.
Finally, we assign a $p$-value for reproducibility to each matched IC across
the $K$ runs. The observed reproducibility for $i$th matched IC is
$\mbox{Reproducibility}(MC_{i})$ and its $p$-value is:
$\mbox{Reproducibility}_{pval}(MC_{i})=\frac{\left\\{\mbox{no. of
}\mbox{{Reproducibility}}_{Null}\geq\mbox{Reproducibility}(MC_{i})\right\\}+1}{Rn_{C}+1}$
(2.39)
7. 7.
Only those components with $\mbox{Reproducibility}_{pval}(MC_{i})<p_{crit}$
are considered to be significantly reproducible. We can use a fixed and
objective value for $p_{crit}$ such as $0.05$. Note that this fixed cutoff is
independent of the amount of variability in the input to RAICAR-N. Please see
Figure 3 for a pictorial depiction of this process.
Figure 3: Pictorial depiction of the process for generating a ”null”
distribution in RAICAR-N. Our ”null” hypothesis is: ”$\mathbf{H_{0}}$: None of
the ICs are reproducible. Hence, we can randomly label IC $i$ from run $l$ as
IC $d$ from run $s$”. Therefore we randomly split the $Kn_{C}$ ICs across $K$
runs into $K$ parts and run the RAICAR algorithm on each set of randomly split
ICs. This gives us a set of ”null” reproducibility values which can be used to
compute $p$-values for the observed reproducibility of ICs in the original
RAICAR run. The green circles indicate the sequence of steps for generating
the ”null” distribution after the steps in Figure 2.
### 2.5 How many subjects should be used per group ICA run in RAICAR-N?
The input to RAICAR-N can either be single subject ICA runs or group ICA runs
across a set of subjects. Note that the individual subject ICA runs are
spatially unconstrained whereas a group ICA spatially constrains the group ICs
across a set of subjects. Hence the number of ICs that can be declared as
significantly reproducible at the group level are usually more than those that
can be declared significantly reproducible at the single subject level. Hence
the following question is relevant:
Figure 4: Flowchart for a group ICA based RAICAR-N analysis. The $N$ single
subject data sets are first pre-processed and subsequently bootstrapped to
create $K$ groups, each group containing $L$ distinct subjects. Each group of
$L$ subjects is submitted to a temporal concatenation group ICA analysis. The
resulting IC maps (either raw ICs or ICs scaled by noise standard deviation)
are subjected to a RAICAR analysis. The cross-realization cross correlation
matrix (CRCM) is randomly permuted multiple times:
$\boldsymbol{G}\rightarrow\boldsymbol{G}(\boldsymbol{g},\boldsymbol{g})$ where
$\boldsymbol{g}$ is a random permutation of integers from $1,\ldots,Kn_{C}$.
The permuted CRCMs are subjected to a RAICAR analysis to generate a
realization of reproducibility values under the ”null” hypothesis. The
computed ”null” distribution of reproducibility values is used to assign $p$
values to the observed reproducibility of the original RAICAR run. Finally,
reproducible ICs are averaged using a random effects analysis and the
resulting $t$-statistic images are subjected to Gammaneg, Student $t$ and
Gammapos mixture modeling.
Suppose we have a group of $N$ subjects. We randomly select $L$ subjects and
form a single group of subjects. We repeat this process $K$ times to get $K$
groups of $L$ subjects each of which is subjected to a group ICA analysis.
Given the number of subjects $N$, how should we choose $L$ and $K$?
First, we discuss the choice of $L$. If $L=N$ then each of the $K$ groups will
contain the same $N$ subjects and hence there will be no diversity in the $K$
groups. We would like to control the amount of diversity in the $K$ groups of
$L$ subjects. Consider any 2 subjects $X$ and $Y$. The probability $P_{XY}(L)$
that both $X$ and $Y$ appear in a set of $L$ randomly chosen subjects from $N$
subjects is given by:
$P_{XY}(L)=\frac{{N-2\choose L-2}}{{N\choose L}}$ (2.40)
The expected number of times that $X$ and $Y$ appear together in sets of $L$
subjects out of $K$ independently drawn sets is:
$E_{XY}(L)=K\,P_{XY}(L)$ (2.41)
Ideally, we would like $E_{XY}(L)$ to be only a small fraction of $K$. Hence
we impose the restriction:
$E_{XY}(L)=K\,P_{XY}(L)\leq\alpha_{max}\,K$ (2.42)
where $\alpha_{max}$ is a user defined constant such as $\alpha_{max}=0.05$.
This implies that the chosen value of $L$ must satisfy:
$P_{XY}(L)\leq\alpha_{max}$ (2.43)
In practice, we choose the largest value of $L$ that satisfies this
inequality. As shown in Figure 5, if $N=23$ and $\alpha_{max}=0.05$ then the
largest value of $L$ that satisfies 2.43 is $L=5$.
Figure 5: Figure shows a plot of $P_{XY}(L)$ vs $L$ for $N=23$ in blue. The
red line shows the $\alpha_{max}=0.05$ cutoff. The largest value of $L$ for
which $P_{XY}(L)\leq 0.05$ is $L=5$.
The number of group ICA runs $K$ should be as large as possible. From our
experiments on real fMRI data we can roughly say that values of $K>50$ give
equivalent results.
### 2.6 How to display the estimated non-Gaussian spatial structure in ICA
maps?
The ICs have been optimized for non-Gaussianity. However, there can be many
types of non-Gaussian distributions. It has been empirically found that the
non-Gaussian distributions of ICs found in fMRI data have the following
structure:
1. 1.
A central Gaussian looking part and
2. 2.
A tail that extends out on either end of the Gaussian
It has been suggested in (Beckmann and Smith, 2004) that a Gaussian/Gamma
mixture model can be fitted to this distribution and the Gamma components can
be thought of as representatives of the non-Gaussian structure. We follow a
similar approach:
1. 1.
The output of a RAICAR-N analysis is a set of spatial ICA maps (either
$z$-transformed maps or raw maps) concatenated into a 4-D volume.
2. 2.
We do a voxelwise transformation to Normality using the voxelwise empirical
cumulative distribution function as described in (van Albada and Robinson,
2007).
3. 3.
Next, we submit the resulting 4-D volume to a voxelwise group analysis using
ordinary least squares. The design matrix for group analysis depends on the
question being considered. In our case, the design matrix was simply a single
group average design.
4. 4.
The resulting $t$-statistic maps are subjected to Student $t$, Gammapos and
Gammaneg mixture modeling. The logic is that if the original ICA maps are pure
Gaussian (i.e., have no interesting non-Gaussian structure) then the result of
a group average analysis will be a pure Student $t$ map which will be captured
by a single Student $t$ (i.e., the Gammapos and Gammaneg will be driven to $0$
class fractions). Hence the ”null” hypothesis will be correctly accounted for.
5. 5.
If the Gamma distributions have $>0.5$ posterior probability at some voxels
then those voxels are displayed in color to indicate the presence of
significant non-Gaussian structure over and above the background Student $t$
distribution.
Examples of Student $t$, Gammapos and Gammaneg mixture model fits are shown in
Figure 6.
Figure 6: Examples of displaying non-Gaussian spatial structure using a
Student $t$, Gammapos and Gammaneg mixture model. Notice how the Gammaneg
density is driven to near $0$ class fraction in the absence of significant
negative non-Gaussian structure. Figure 7: $p$-value cutoffs for within and
across single subject analysis using RAICAR-N. This figure illustrates the
intuitive fact that within subject ICA runs are much more reproducible
compared to across subject ICA runs.
## 3 Experiments and Results
### 3.1 Human rsfMRI data
rsfMRI data titled: `Baltimore (Pekar, J.J./Mostofsky, S.H.; n = 23 [8M/15F];
ages: 20-40;`
` TR = 2.5; # slices = 47; # timepoints = 123)`, a part of the 1000 functional
connectomes project, was downloaded from the Neuroimaging Informatics Tools
and Resources Clearinghouse (NITRC): http://www.nitrc.org/projects/fcon_1000/.
### 3.2 Preprocessing
Data was analyzed using tools from the FMRIB software library (FSL:
http://www.fmrib.ox.ac.uk/fsl/). Preprocessing steps included motion
correction, brain extraction, spatial smoothing with an isotropic Gaussian
kernel of 5mm FWHM and 100s high-pass temporal filtering. Spatial ICA was
performed using a noisy ICA model as implemented in FSL MELODIC (Beckmann and
Smith, 2004) in either single subject or multi-subject temporal concatenation
mode also called group ICA. Please see section 2.2 for a brief summary of
single subject ICA and group ICA. In each case, we fixed the model order of
ICA at $q=40$ to be consistent with the model order range typically extracted
in rsfMRI and fMRI (Smith et al., 2009; Esposito et al., 2005). For temporal
concatenation based group ICA, single subject data was first affinely
registered to the MNI 152 brain and subsequently resampled to 4x4x4 resolution
(MNI 4x4x4) to decrease computational load.
### 3.3 RAICAR-N analysis with 1 ICA run per subject
Spatial ICA was run once for each of the $N=23$ subjects in their native
space. The resulting set of ICA components across subjects were transformed to
MNI 4x4x4 space and were submitted to a RAICAR-N analysis.111In all RAICAR-N
analyses reported in this article, we used the $z$-transformed IC maps - which
are basically the raw IC maps divided by a voxelwise estimate of noise
standard deviation (named as melodic_IC.nii.gz in MELODIC). It is also
possible to use the raw IC maps as inputs to RAICAR-N. ICA components were
sorted according to their reproducibility and $p$-values were computed for
each ICA component. Please see Figure 8.
Figure 8: Single subject rsfMRI ICA runs across 23 subjects were combined
using a RAICAR-N analysis. Figure (a) shows the observed values of normalized
reproducibility (bottom) as well as the ”null” distribution of normalized
reproducibility across $R=100$ simulations (top). Figure (b) shows the
$p$-values for each IC along with the $0.05$ and $0.1$ cutoff lines.
We compared the reproducible RSNs from the single subject RAICAR-N analysis to
the group RSN maps reported in literature (Beckmann et al., 2005). Please see
Figure 9.
Figure 9: The top 8 ”reproducible” ICs from a RAICAR-N analysis on single
subject ICA runs compared with standard RSN maps reported in literature
(Beckmann et al., 2005). We are able to declare 4 ”standard” RSNs as
significantly reproducible at a $p$-value $<0.05$. There are 2 other
”standard” RSNs that achieve a reproducibility $p$-value between $0.05$ and
$0.06$ as well as 2 ”non-standard” RSNs that achieve $p$-values of $0.0125$
and $0.05699$ respectively. We also could not find 2 of the published RSNs in
(Beckmann et al., 2005) as reproducible in single subject ICA runs.
To summarize, when single subject ICA runs are combined across subjects:
* •
We are able to declare 4 ”standard” RSNs as significantly reproducible at a
$p$-value $<0.05$.
* •
There are 2 other ”standard” RSNs that achieve a reproducibility $p$-value
between 0.05 and 0.06.
* •
There are 2 other ”non-standard” RSNs that are of interest: one achieves a
$p$-value of 0.0125 and the other achieves a $p$-value of 0.05699.
### 3.4 RAICAR-N on random sets of 5 subjects - 50 group ICA runs
To promote diversity across the group ICA runs, as discussed in section 2.5,
$L=5$ subjects were drawn at random from the group of $N=23$ subjects and
submitted to a temporal concatenation based group ICA. This process was
repeated $K=50$ times and the resulting set of 50 group ICA maps were
submitted to a RAICAR-N analysis. ICA components were sorted according to
their reproducibility and $p$-values were computed for each ICA component.
Please see Figure 10.
Figure 10: $L=5$ subjects were randomly drawn from the set of $N=23$ subjects
and submitted to a temporal concatenation based group ICA. This process was
repeated $K=50$ times and the resulting ICA maps were submitted to a RAICAR-N
analysis. Figure (a) shows the observed values of normalized reproducibility
(bottom) as well as the ”null” distribution of normalized reproducibility
across $R=100$ simulations (top). Figure (b) shows the $p$-values for each IC
along with the $0.05$ and $0.1$ cutoff lines.
We compared the reproducible RSNs from the single subject RAICAR-N analysis to
the RSN maps reported in literature (Beckmann et al., 2005). Please see Figure
11.
Figure 11: The top 15 ”reproducible” ICs from $K=50$ runs of $L=5$ subject
group ICA RAICAR-N analysis compared with standard RSN maps reported in
literature (Beckmann et al., 2005). We are able to declare 8 ”standard” RSNs
as significantly reproducible at a $p$-value of $<0.05$. There are 6 other
”non-standard” RSNs that can be declared as significantly reproducible at a
$p$-value of $<0.05$ and 1 other ”non-standard” RSN that achieves a $p$-value
of $0.05299$.
In summary, when 50 random 5 subject group ICA runs (from a population of 23
subjects) are combined using RAICAR-N:
* •
We are able to declare 8 ”standard” RSNs as significantly reproducible at a
$p$-value $<0.05$.
* •
There are 6 other ”non-standard” RSNs that can be declared as significantly
reproducible at a $p$-value $<0.05$.
* •
There is 1 other ”non-standard” RSN that achieves a $p$-value of 0.05299.
### 3.5 RAICAR-N on random sets of 5 subjects - 100 group ICA runs
To promote diversity across the group ICA runs, as discussed in section 2.5,
$L=5$ subjects were drawn at random from the group of $N=23$ subjects and
submitted to a temporal concatenation based group ICA. This process was
repeated $K=100$ times and the resulting set of 100 group ICA maps were
submitted to a RAICAR-N analysis. ICA components were sorted according to
their reproducibility and $p$-values were computed for each ICA component.
Please see Figure 12.
Figure 12: $L=5$ subjects were randomly drawn from the set of $N=23$ subjects
and submitted to a temporal concatenation based group ICA. This process was
repeated $K=100$ times and the resulting ICA maps were submitted to a RAICAR-N
analysis. Figure (a) shows the observed values of normalized reproducibility
(bottom) as well as the ”null” distribution of normalized reproducibility
across $R=100$ simulations (top). Figure (b) shows the $p$-values for each IC
along with the $0.05$ and $0.1$ cutoff lines.
We compared the reproducible RSNs from the single subject RAICAR-N analysis to
the RSN maps reported in literature (Beckmann et al., 2005). Please see Figure
13.
Figure 13: The top 15 ”reproducible” ICs from $K=100$ runs of $L=5$ subject
group ICA RAICAR-N analysis compared with standard RSN maps reported in
literature (Beckmann et al., 2005). We are able to declare 8 ”standard” RSNs
as significantly reproducible at a $p$-value of $<0.05$. There are 6 other
”non-standard” RSNs that can be declared as significantly reproducible at a
$p$-value of $<0.05$ and 1 other ”non-standard” RSN that achieves a $p$-value
of $0.05824$.
In summary, when 100 random 5 subject group ICA runs (from a population of 23
subjects) are combined using RAICAR-N:
* •
We are able to declare 8 ”standard” RSNs as significantly reproducible at a
$p$-value $<0.05$.
* •
There are 6 other ”non-standard” RSNs that can be declared as significantly
reproducible at a $p$-value $<0.05$.
* •
There is 1 other ”non-standard” RSN that achieves a $p$-value of 0.05824.
## 4 Group comparison of ICA results
In this section, we summarize the main approaches for group analysis of ICA
results which can be broadly classified into two categories: (1) Approaches
based on a single ICA run or no ICA run and (2) Approaches based on multiple
ICA runs. To make things concrete, suppose we have two groups of subjects $A$
and $B$.
### 4.1 Approaches based on a single group ICA run or no ICA run
The main idea in these approaches is to use the results of a group ICA using
all subjects to derive subject specific spatial maps for group comparison. A
typical sequence of steps is as follows:
1. 1.
The first step involves extraction of a set of template IC maps or a set of
template mixing matrix time courses. This can be accomplished using two
techniques:
1. (a)
Group ICA based template IC maps or time courses:
A temporal concatenation based group ICA is run using data from all subjects
in group $A$ and $B$. This usually involves two PCA data reductions. The first
reduction is based on a subject wise PCA decomposition (Calhoun et al., 2001)
or an average PCA decomposition (Beckmann et al., 2005) as discussed in
section 2.2.2. The next reduction is based on PCA reduced temporally
concatenated data. Subsequently, the group ICs and the dual PCA reduced mixing
matrix time courses are estimated using an ICA algorithm.
2. (b)
User supplied set of template IC maps:
The user supplies a set of spatial maps, perhaps corresponding to an ICA
decomposition on an independent data set.
2. 2.
The next step either uses template IC maps or time courses.
1. (a)
Template time course based approach:
First, the mixing matrix is PCA back projected and partitioned into subject
specific sub matrices. Next, subject specific spatial maps corresponding to
the group ICs are estimated via least-squares and a second PCA back projection
is used to estimate the corresponding subject specific time courses. This is
the approach proposed in (Calhoun et al., 2001), which we will refer to as the
group ICA back projection approach.
2. (b)
Template IC based approach:
First, spatial multiple regression using the template ICs as regressors is
used against the original data of each subject to derive subject specific time
courses corresponding to each template IC. Next, a second multiple regression
using the subject specific time courses is used against the original data of
each subject to derive subject specific spatial maps corresponding to each
template IC. This approach called ”dual-regression” has been proposed by
(Beckmann et al., 2009). A similar approach called fixed average spatial ICA
(FAS ICA) had also been proposed earlier in (Calhoun et al., 2004). Both dual-
regression and FAS ICA involve the first spatial regression stage, but dual-
regression also includes a second temporal regression stage.
3. 3.
Once subject specific spatial maps and time courses corresponding to group ICs
have been determined, they are entered into a random effects analysis for
group comparison.
#### 4.1.1 Advantages of single group ICA based approaches
1. 1.
Much reduced computational load compared to multiple ICA based approaches.
2. 2.
Ability to take advantage of constrained spatial IC estimation across all
subjects via group ICA.
Please see section 5 for discussion.
### 4.2 Approaches based on multiple single subject or group ICA runs
In these approaches results of multiple ICA runs in groups $A$ and $B$ are
used for a between group analysis. A typical sequence of steps is as follows:
1. 1.
The first step involves:
* •
running a separate single subject ICA for all subjects from groups $A$ and $B$
(possibly with multiple runs per subject) or
* •
running a set of group ICA runs across various sets of subjects separately,
with each set containing subjects either from group $A$ or group $B$
2. 2.
The next step is to establish a correspondence between the ICs within and
across groups. There are two main techniques of establishing this
correspondence:
1. (a)
Template based methods:
In these approaches, the user defines a template or a spatial map containing
the network of interest. Examples of templates include a spatial map of the
default mode network (DMN) derived from a separate ICA analysis, a spatial map
from a separate PCA analysis, or even a binary mask defining the regions of
interest. The template is then used to select from each run of ICA (single
subject or group ICA) in each group ($A$ and $B$), an IC that best matches the
template using a predefined metric such as spatial correlation coefficient or
goodness of fit (GOF) (Greicius et al., 2004).
2. (b)
Template free methods:
These approaches do not need a pre-defined template from the user, but instead
attempt to match or cluster all ICs simultaneously within and across groups.
Examples of such approaches include self organizing group ICA (sogICA,
(Esposito et al., 2005)) and RAICAR (Yang et al., 2008). Each matched
component or IC cluster includes one IC from each ICA run (single subject or
group ICA) in each group ($A$ and $B$).
3. 3.
Finally, the selected ICs in template based methods or ICs from a selected IC
cluster/matched component in template free methods are then entered into a
random effects group analysis (with repeated measures for multiple single
subject ICA runs) for between group comparison.
#### 4.2.1 Advantages of multiple ICA run approaches
1. 1.
They account for both algorithmic and data set variability of ICA.
2. 2.
Group comparisons happen on true ICs i.e., optimal solutions for the ICA
problem.
Please see section 5 for discussion.
## 5 Discussion
As discussed in section 2.1.2, in the noisy linear ICA model with isotropic
diagonal Gaussian noise co-variance, for a given true model order, the mixing
matrix and the source distributions are identifiable upto permutation and
scaling. However, as pointed out in section 2.1.3, various factors prevent the
convergence of ICA algorithms to unique IC estimates. These factors include
ICA model not being the true data generating model, approximations to mutual
information used in ICA algorithms, multiple local minima in ICA contrast
functions, confounding Gaussian noise as well as variability due to model
order over-estimation. A practical implication of these factors is that ICA
algorithms converge to different IC estimates depending on how they are
initialized and on the specific data used as input to ICA. Hence, there is a
need for a rigorous assessment of reproducibility or generalizability of IC
estimates. A set of reproducible ICs can then be used as ICA based
characteristics of a particular group of subjects.
We proposed an extension to the original RAICAR algorithm for reproducibility
assessment of ICs within or across subjects. The modified algorithm called
RAICAR-N builds up a ”null” distribution of normalized reproducibility values
under a random assignment of observed ICs across the $K$ runs. This ”null”
distribution is used to compute reproducibility $p$-values for each observed
matched component from RAICAR. An objective cutoff such as $p<0.05$ can be
used to detect ”significantly reproducible” components. This avoids subjective
user decisions such as selection of the number of clusters in ICASSO or the
reproducibility cutoff in RAICAR or a cutoff on intra cluster distance in
sogICA.
### 5.1 Results for publicly available rsfMRI data
We applied RAICAR-N to publicly available $N=23$ subject rsfMRI data from
http://www.nitrc.org/. We analyzed the data in 2 different ways:
1. 1.
$n_{C}=40$ ICs were extracted for each of the $N=23$ subjects. The $K=23$
single subject ICA runs were subjected to a RAICAR-N analysis (after
registration to standard space).
In single subject ICA based RAICAR-N analysis (see Figures 8 \- 9), we are
able to declare 6 out of the 8 ICs reported in (Beckmann et al., 2005) (which
used group ICA) as ”reproducible” (4 ICs have $p$-values $<0.05$ and 2 ICs
have $p$-values $<0.06$). This is consistent with the 5 reproducible RSNs
reported in (DeLuca et al., 2005) using single subject ICA analysis.
2. 2.
$L=5$ subjects were randomly drawn from $N=23$ subjects to create one group of
subjects which was subjected to a group ICA analysis in which $n_{C}=40$
components were extracted. This process was repeated $K=50$ or $100$ times and
the resulting group ICA runs were subjected to a RAICAR-N analysis.
In group ICA based RAICAR-N analysis (see Figures 10 \- 13), we are able to
declare all 8 components reported in (Beckmann et al., 2005) as ”reproducible”
(at $p<0.05$). Some of the ICs detected as ”reproducible” in the group ICA
based RAICAR-N on human rsfMRI data are not shown in (Beckmann et al., 2005)
but do appear in the more recent paper (Smith et al., 2009). RAICAR-N results
for $K=50$ are almost identical to those for $K=100$ suggesting that $K=50$
runs of group ICA are sufficient for a RAICAR-N reproducibility analysis.
### 5.2 Single subject ICA vs Group ICA
Based on our results, it appears that single subject ICA maps are less
reproducible compared to group ICA maps as illustrated in Figures 8 and 10. A
single subject ICA based analysis is more resistant to subject specific
artifacts. On the other hand, a group ICA based analysis makes the strong
assumption that ICs are spatially identical across subjects. If this
assumption is true, group ICA takes advantage of temporal concatenation to
constrain the ICs spatially across subjects thereby reducing their variance.
Hence, when there are no gross artifacts in individual rsfMRI data sets, group
ICA is expected to be more sensitive for reproducible IC detection. As seen in
Figures 9 and 11, our results agree with this proposition. All ICs declared as
”reproducible” in the single subject based RAICAR-N analysis continue to
remain ”reproducible” in the group ICA based RAICAR-N analysis.
### 5.3 How should subjects be grouped for group ICA?
This raises the question of how the subjects should be grouped together for
individual group ICA runs in preparation for RAICAR-N. If all $N$ subjects are
used in all group ICA analyses then there is no diversity in the individual
group ICA runs. In this case, a RAICAR-N analysis will capture algorithmic
variability due to non-convexity of ICA objective function but not dataset
variability. Hence, our conclusions might not be generalizable to a different
set of $N$ subjects.
Another option is to randomly select $L$ subjects out of $N$ for each group
ICA run and submit the resulting $K$ group ICA runs to RAICAR-N. In this case,
we will account for both algorithmic and data set variability via a RAICAR-N
analysis. In other words, we will be able to determine those ICs that are
”reproducible” across different sets of $L$ subjects and across multiple ICA
runs. A key question is: How should we choose $L$ and $K$? In section 2.5, we
proposed a simple method to determine the number of subjects $L$ to be used in
a single group ICA run out of the $N$ subjects - the key idea is to form
groups with enough ”diversity”. Multiple such group ICA runs can then be
submitted to a RAICAR-N analysis for reproducibility assessment. Clearly, the
larger the value of $N$, the larger the value of $L$. Hence, increasing the
number of subjects $N$ in a study will allow us to make conclusions that are
generalizable to a larger set of $L$ subjects. Also, conclusions generalizable
to $L_{1}$ subjects are expected to hold for $L_{2}>L_{1}$ subjects but not
vice versa.
### 5.4 RAICAR-N for group comparisons of reproducible ICs
In the present work, our focus was on enabling the selection of reproducible
ICs for a given single group of subjects. However, RAICAR-N can be extended
for between group analysis of reproducible components as well. Before we
describe how to do so, it is useful to discuss other approaches for group
analysis of RSNs described in section 5.4. Suppose we have two groups of
subjects $A$ and $B$.
#### 5.4.1 Discussion of single group ICA based approaches
1. 1.
Subject specific maps corresponding to group ICA maps derived using ICA back
projection or dual regression are not true ICs, i.e., they are not solutions
to an ICA problem.
2. 2.
These approaches do not account for either the algorithmic or the data set
variability of an ICA decomposition. The single group ICA decomposition will
contain both reproducible and non-reproducible ICs, but there is no systematic
way to differentiate between the two.
3. 3.
Both dual regression and ICA back projection using data derived IC templates
are circular analyses. First, group ICA using all data is used to derive
template IC maps or template time courses. Next least-squares based ICA back
projection or dual regression using a subset of the same data is used to
derive subject specific maps and time courses corresponding to each IC. Thus
model $1$ (group ICA) on data $\mathcal{D}$ is used to learn an assumption
$\mathcal{A}$ (template IC maps or template time courses) that is then used to
fit model $2$ (dual regression or ICA back projection) on a subset of the same
data $\mathcal{D}$. This is circular analysis (Kriegeskorte et al., 2009; Vul
and Kanwisher, 2010).
It is easy to avoid circular analysis in a dual regression approach via cross-
validation. For example, one can split the groups $A$ and $B$ into two random
parts, a ”training” set and a ”test” set. First, the ”training” set can be
used to derive template IC maps using group ICA. Next, the ”training” set
based template IC maps can be used as spatial regressors for dual regression
on the ”test” set. Alternatively, the template ICs for dual regression can
also come from a separate ICA decomposition on a independent data set
unrelated to groups $A$ and $B$ such as human rsfMRI data. This train/test
approach cleanly avoids the circular analysis problem. It is not clear how to
use cross-validation for an ICA back projection approach since template time
courses cannot be assumed to remain the same across ICA decompositions.
4. 4.
Subject specific structured noise is quite variable in terms of its spatial
structure. Hence, a group ICA analysis cannot easily model or account for
subject specific structured noise via group level ICs. Consequently, subject
specific spatial maps in ICA back projection or dual regression will have a
noise component that is purely driven by the amount of structured noise in
individual subjects. On the other hand, a single subject ICA based analysis
can accurately model subject specific structured noise via single subject ICs.
#### 5.4.2 Discussion of multiple ICA run approaches
1. 1.
(Zuo et al., 2010) report that using different sets of template ICs in
template based methods using spatial correlation such as (Harrison et al.,
2008) can result in the selection of different ICs in individual ICA runs.
This is not surprising since IC correspondence derived from template based
methods does depend on the particular template used. This is similar to a seed
based correlation analysis being dependent on the particular seed ROI used. It
is worth noting that template free approaches such as sogICA and RAICAR do not
rely on any template.
2. 2.
(Cole et al., 2010) state that individual runs across subjects (or groups of
subjects) can be quite variable in terms of the spatial structure of the
estimated ICs. For example, (Cole et al., 2010) point out that an IC might be
apparently split into two sub-components in some subjects but not others. The
real problem is that the same model order could lead to over-fitting in some
subjects (or groups of subjects) but not in others. Hence, the observed
differences in a group comparison might be biased by the unknown difference in
the amount of over-fitting across groups $A$ and $B$.
As described in 2.1.3, over-fitting can lead to the phenomenon of component
”splitting” in ICA. This is not limited to single subject ICA but can also
occur in group ICA. For instance, (Zuo et al., 2010) report the ”default mode”
network as split into three sub networks using group ICA and note that
component ”splitting” can also reflect functional segregation or hierarchy
within a particular IC and is not necessarily a consequence of model order
overestimation in every case.
Over-fitting can be correctly accounted for by a reproducibility analysis.
This is because we expect the real and stable non-Gaussian sources to be
reproducible across multiple ICA runs (algorithmic variability) and across
different subjects or groups of subjects (data set variability).
If we want the results of a between group ICA analysis to be generalizable to
an independent group of subjects then we must account for both the algorithmic
and data variability of ICA. We propose to modify RAICAR-N for enabling
between group comparisons of ”reproducible” ICs as follows:
1. 1.
Enter multiple within and across subject (or within and across sets of
subjects) ICA runs for groups $A$ and $B$ into a RAICAR analysis. Perform the
RAICAR component matching process across groups $A$ and $B$.
2. 2.
Use RAICAR-N to compute reproducibility $p$-values separately for group $A$
and $B$ for each matched component across groups $A$ and $B$.
3. 3.
Only ICs that are separately reproducible in both groups $A$ and $B$ and that
are maximally similar to each other are used for between group comparisons.
To summarize, a RAICAR-N analysis:
* ✓
can be applied for ”reproducible” component detection either within or across
subjects in any component based analysis - not necessarily ICA. For instance,
a set of PCA maps across subjects can be submitted to a RAICAR-N analysis.
* ✓
is simple to implement and accounts for both algorithmic and data set
variability of an ICA decomposition.
* ✓
avoids any user decisions except the final $p$-value cutoff which can be
objectively pre-set at standard values such as $0.05$.
* ✓
can be extended to enable comparisons of reproducible ICs between groups $A$
and $B$.
## 6 Conclusions
Multiple group ICA runs using groups of subjects with enough ”diversity” can
be used to account for the run-to-run variability in ICA algorithms both due
to the non-convex ICA objective function as well as across subjects data
variability. These group ICA runs can be subjected to a RAICAR-N
”reproducibility” analysis. RAICAR-N enables the objective detection of
”reproducible components” in any component based analysis of fMRI data such as
ICA and can also be used for a between group comparison of ”reproducible” ICs.
## Acknowledgements
We gratefully acknowledge financial support from the Pain and Analgesia
Imaging and Neuroscience (P.A.I.N) group, McLean Hospital, Harvard Medical
School, Belmont MA, USA under the grants K24NS064050 (DB) and R01NS065051
(DB). We would also like to thank Dr. Christian Beckmann for making the IC
image files from his 2005 paper (Beckmann et al., 2005) available to us.
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## 7 Figure Legends
Figure 1: Figure illustrates the variation in normalized reproducibility from
RAICAR depending on whether the input to RAICAR is (a) Multiple ICA runs on
single subject data or (b) Multiple ICA runs across subjects. Notice that the
normalized reproducibility is much lower for across subjects analysis compared
to within subject analysis. Figure 2: Pictorial depiction of the original
RAICAR algorithm Yang et al. [2008]. The ICA algorithm is run $K$ times with
each run producing $n_{C}$ ICs. $\boldsymbol{G}$ is a $K\times K$ block matrix
with elements $\boldsymbol{G}(l,m)=\boldsymbol{G}_{lm}$ where
$\boldsymbol{G}_{lm}$ is the $n_{C}\times n_{C}$ absolute spatial cross-
correlation matrix between ICs from runs $l$ and $m$. The numbered green
circles indicate the sequence of steps in applying RAICAR to a given data set.
Our definition of normalized reproducibility in box 7 averages un-thresholded
correlation coefficients thereby avoiding the selection of a correlation
coefficient threshold prior to averaging. Figure 3: Pictorial depiction of the
process for generating a ”null” distribution in RAICAR-N. Our ”null”
hypothesis is: ”$\mathbf{H_{0}}$: None of the ICs are reproducible. Hence, we
can randomly label IC $i$ from run $l$ as IC $d$ from run $s$”. Therefore we
randomly split the $Kn_{C}$ ICs across $K$ runs into $K$ parts and run the
RAICAR algorithm on each set of randomly split ICs. This gives us a set of
”null” reproducibility values which can be used to compute $p$-values for the
observed reproducibility of ICs in the original RAICAR run. The green circles
indicate the sequence of steps for generating the ”null” distribution after
the steps in Figure 2. Figure 4: Flowchart for a group ICA based RAICAR-N
analysis. The $N$ single subject data sets are first pre-processed and
subsequently bootstrapped to create $K$ groups, each group containing $L$
distinct subjects. Each group of $L$ subjects is submitted to a temporal
concatenation group ICA analysis. The resulting IC maps (either raw ICs or ICs
scaled by noise standard deviation) are subjected to a RAICAR analysis. The
cross-realization cross correlation matrix (CRCM) is randomly permuted
multiple times:
$\boldsymbol{G}\rightarrow\boldsymbol{G}(\boldsymbol{g},\boldsymbol{g})$ where
$\boldsymbol{g}$ is a random permutation of integers from $1,\ldots,Kn_{C}$.
The permuted CRCMs are subjected to a RAICAR analysis to generate a
realization of reproducibility values under the ”null” hypothesis. The
computed ”null” distribution of reproducibility values is used to assign $p$
values to the observed reproducibility of the original RAICAR run. Finally,
reproducible ICs are averaged using a random effects analysis and the
resulting $t$-statistic images are subjected to Gammaneg, Student $t$ and
Gammapos mixture modeling. Figure 5: Figure shows a plot of $P_{XY}(L)$ vs $L$
for $N=23$ in blue. The red line shows the $\alpha_{max}=0.05$ cutoff. The
largest value of $L$ for which $P_{XY}(L)\leq 0.05$ is $L=5$. Figure 6:
Examples of displaying non-Gaussian spatial structure using a Student $t$,
Gammapos and Gammaneg mixture model. Notice how the Gammaneg density is driven
to near $0$ class fraction in the absence of significant negative non-Gaussian
structure. Figure 7: $p$-value cutoffs for within and across single subject
analysis using RAICAR-N. This figure illustrates the intuitive fact that
within subject ICA runs are much more reproducible compared to across subject
ICA runs. Figure 8: Single subject rsfMRI ICA runs across 23 subjects were
combined using a RAICAR-N analysis. Figure (a) shows the observed values of
normalized reproducibility (bottom) as well as the ”null” distribution of
normalized reproducibility across $R=100$ simulations (top). Figure (b) shows
the $p$-values for each IC along with the $0.05$ and $0.1$ cutoff lines.
Figure 9: The top 8 ”reproducible” ICs from a RAICAR-N analysis on single
subject ICA runs compared with standard RSN maps reported in literature
Beckmann et al. [2005]. We are able to declare 4 ”standard” RSNs as
significantly reproducible at a $p$-value $<0.05$. There are 2 other
”standard” RSNs that achieve a reproducibility $p$-value between $0.05$ and
$0.06$ as well as 2 ”non-standard” RSNs that achieve $p$-values of $0.0125$
and $0.05699$ respectively. We also could not find 2 of the published RSNs in
Beckmann et al. [2005] as reproducible in single subject ICA runs. Figure 10:
$L=5$ subjects were randomly drawn from the set of $N=23$ subjects and
submitted to a temporal concatenation based group ICA. This process was
repeated $K=50$ times and the resulting ICA maps were submitted to a RAICAR-N
analysis. Figure (a) shows the observed values of normalized reproducibility
(bottom) as well as the ”null” distribution of normalized reproducibility
across $R=100$ simulations (top). Figure (b) shows the $p$-values for each IC
along with the $0.05$ and $0.1$ cutoff lines. Figure 11: The top 15
”reproducible” ICs from $K=50$ runs of $L=5$ subject group ICA RAICAR-N
analysis compared with standard RSN maps reported in literature Beckmann et
al. [2005]. We are able to declare 8 ”standard” RSNs as significantly
reproducible at a $p$-value of $<0.05$. There are 6 other ”non-standard” RSNs
that can be declared as significantly reproducible at a $p$-value of $<0.05$
and 1 other ”non-standard” RSN that achieves a $p$-value of $0.05299$. Figure
12: $L=5$ subjects were randomly drawn from the set of $N=23$ subjects and
submitted to a temporal concatenation based group ICA. This process was
repeated $K=100$ times and the resulting ICA maps were submitted to a RAICAR-N
analysis. Figure (a) shows the observed values of normalized reproducibility
(bottom) as well as the ”null” distribution of normalized reproducibility
across $R=100$ simulations (top). Figure (b) shows the $p$-values for each IC
along with the $0.05$ and $0.1$ cutoff lines. Figure 13: The top 15
”reproducible” ICs from $K=100$ runs of $L=5$ subject group ICA RAICAR-N
analysis compared with standard RSN maps reported in literature Beckmann et
al. [2005]. We are able to declare 8 ”standard” RSNs as significantly
reproducible at a $p$-value of $<0.05$. There are 6 other ”non-standard” RSNs
that can be declared as significantly reproducible at a $p$-value of $<0.05$
and 1 other ”non-standard” RSN that achieves a $p$-value of $0.05824$.
|
arxiv-papers
| 2011-08-10T19:02:29 |
2024-09-04T02:49:21.472967
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gautam V. Pendse, David Borsook, Lino Becerra",
"submitter": "Gautam Pendse",
"url": "https://arxiv.org/abs/1108.2248"
}
|
1108.2312
|
# Effects of the vortices and impurities on the nuclear spin relaxation rate
in iron-based superconductors
Hong-Min Jiang Department of Physics, Hangzhou Normal University, Hangzhou
310036, China National Laboratory of Solid State of Microstructure and
Department of Physics, Nanjing University, Nanjing 210093, China Jia Guo
National Laboratory of Solid State of Microstructure and Department of
Physics, Nanjing University, Nanjing 210093, China Jian-Xin Li National
Laboratory of Solid State of Microstructure and Department of Physics, Nanjing
University, Nanjing 210093, China
###### Abstract
The effects of magnetic vortices and nonmagnetic impurities on the low energy
quasiparticle excitations and the spin-lattice relaxation rate are examined in
the iron-based superconductors for the $s_{\pm}$-, $s$\- and $d$-wave pairing
symmetries, respectively. The main effect of the vortices is to enhance the
quasiparticle excitations and the spin-lattice relaxation rate for all
symmetries, and leads to a $T^{3}$ dependence of the relaxation rate followed
by a nearly $T$-linearity at lower temperatures. This enhancement can only be
seen for the $s_{\pm}$\- and $d$-wave symmetries in the presence of
nonmagnetic impurities. These results suggest that the $s_{\pm}$-wave and
$d$-wave pairing states behave similarly in response to the magnetic field and
nonmagnetic impurities, therefore it may be impossible to distinguish them on
the basis of the measurements of spin-lattice relaxation rates when a magnetic
field and/or impurity scatterings are present.
###### pacs:
74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj
## I introduction
Recently much attention has been payed to the newly discovered iron arsenide
superconductors, kami1 ; xhchen1 ; zaren1 ; gfchen1 ; wang1 which display
superconducting transition temperature as high as more than 50K, appear to
share a number of general features with high-$T_{c}$ cuprates, including the
layered structure and proximity to a magnetically ordered state. kami1 ; cruz1
; jdong These observations suggest that the conventional phonon-mediated
pairing mechanism appears to be unlikely and the magnetic correlations may be
relevant for superconductivity. So far, unconventional superconductivity with
pairing symmetry $s_{\pm}$ mediated by the interband spin fluctuations has
been proposed by a number of theories for this layered iron superconductors.
Mazin ; kuroki ; Yao ; fczhang Although, such a popular proposal can explain
some experimental findings, the situation is complicated by the power-law
temperature dependence of the spin-lattice relaxation rate $T_{1}^{-1}\sim
T^{n}$ below $T_{c}$ with the doping dependent $n$ varying from 6 to 1.5.
hslee ; grafe1 ; Hammerath1 ; Nakai1 ; Matano1 ; Kawasaki ; Matano2 ; swzhang
; Yashima ; Fukazawa1 ; Fukazawa2 Because a simple theoretical analysis shows
that the relaxation rate $T_{1}^{-1}$ should exhibit an exponential
temperature dependence for a fully gaped superconductors, while power law
relation $T_{1}^{-1}\sim T^{3}$ holds in the presence of line nodes in the SC
gap. Moreover, several experiments have reported the evidence of a residual
density-of-state (DOS) at zero energy in the SC state, where the temperature
dependent relaxation rate deviates from the $T_{1}^{-1}\sim T^{3}$ relation
and exhibits $T_{1}^{-1}\sim T$ behavior at very low temperature. Nakai1 ;
Michioka1 ; Hammerath1 The disparity between the theoretical proposal and the
experimental fact presents a puzzle that should be resolved for the
determination of the SC pairing symmetry in these high-$T_{c}$
superconductors.
The relaxation rate has been studied in the presence of the impurity-enhanced
quasiparticle scattering at zero field in the band representation, and it
shows the power law temperature dependence $T_{1}^{-1}\sim T^{3}$ for the
$s_{\pm}=\Delta_{0}\cos(k_{x})\cos(k_{y})$-wave SC pairing. Parker ; YSenga ;
YBang Although the enhancement of the quasiparticle scattering has been well
included in the previous studies, the relationship between the SC gap and the
impurities addressed in the band representation were debated controversially.
Onari1 ; Onari2 More importantly, actual spin-lattice relaxation rate
measured in NMR experiments are conducted under a magnetic field of several
Tesla, where the relaxation rate shows the temperature dependence
$T_{1}^{-1}\sim T^{3}$ and even $T_{1}^{-1}\sim T$. Nakai1 ; Fukazawa1 ; lma ;
Michioka1 At the same time, H. J. Grafe et al. has found that the
superconducting vortices contribute to the spin-lattice relaxation rate when
the magnetic field is perpendicular to the conducting plane but not for the
parallel direction. grafe1 ; Kitagawa1 In fact, it is unclear up to now
whether the different value of $n$ in the temperature dependence of
$T_{1}^{-1}$ is caused by a change of the pairing symmetry with doping, a
disorder scattering effect in an $s_{\pm}$ gap superconductor, or other
unknown mechanism. In view of these theoretical and experimental facts, it is
necessary to compare and contrast different pairing symmetries by studying the
effects of impurities and magnetic field on the spin-lattice relaxation rate
in the analysis of the standardized procedure extracting the gap symmetry.
The purpose of this study is to present such a contrastive study on three
different superconducting (SC) pairing symmetries with $s_{\pm}$-, $s$\- and
$d$-wave, respectively. For this purpose, we calculate the DOS and the spin-
lattice relaxation rate in the Fe-based superconductors in the presence of
magnetic field and nonmagnetic impurities by self-consistently solving the
Bogoliubov-de Gennes equations based on the simple two-orbital model. It is
shown that the characteristic low energy quasiparticle excitations depend on
the gap functions in the presence of magnetic vortices and nonmagnetic
impurities. The magnetic vortices contribute significantly to the spin-lattice
relaxation rate and lead to the relation $T_{1}^{-1}\sim T^{3}$ followed by a
nearly linear dependence at lower temperatures for all symmetries. While in
the presence of nonmagnetic impurities, this enhancement of $T_{1}^{-1}$ can
only be seen for the $s_{\pm}$\- and $d$-wave symmetries. Thus, in the
presence of magnetic field and nonmagnetic impurities, the $s_{\pm}$-wave
pairing behaves much like what the $d$-wave does.
The remainder of the paper is organized as follows. In Sec. II, we introduce
the model Hamiltonian and carry out analytical calculations. In Sec. III, we
present numerical calculations and discuss the results. In Sec. IV, we make a
conclusion.
## II THEORY AND METHOD
We start with an effective two-orbital model that takes only the iron $d_{xz}$
and $d_{yz}$ orbitals into account. ragh1 By assuming an effective attraction
that causes the superconducting pairing and including the possible
interactions between the two orbitals’ electrons, one can construct an
effective model to study the vortex and impurity physics of the iron-based
superconductors in the superconducting state:
$\displaystyle H=H_{0}+H_{pair}.$ (1)
The first term is a tight-binding model
$\displaystyle H_{0}=$
$\displaystyle-\sum_{ij,\alpha\beta,\sigma}e^{i\varphi_{ij}}t_{ij,\alpha\beta}c^{{\dagger}}_{i,\alpha,\sigma}c_{j,\beta,\sigma}$
(2)
$\displaystyle-\mu\sum_{i,\alpha,\sigma}c^{{\dagger}}_{i,\alpha,\sigma}c_{i,\alpha,\sigma}+\sum_{i,\alpha,\sigma}U_{i}c^{{\dagger}}_{i,\alpha,\sigma}c_{i,\alpha,\sigma},$
which describes the electron effective hoppings between sites $i$ and $j$ of
the Fe ions on the square lattice, including the intra-
($t_{ij,\alpha\alpha}$) and inter-orbital
($t_{ij,\alpha,\beta},\alpha\neq\beta$) hoppings with the subscripts $\alpha$,
$\beta$ ($\alpha,(\beta)=1,2$ for $xz$ and $yz$ orbital, respectively)
denoting the orbitals and $\sigma$ the spin. $c^{{\dagger}}_{i,\alpha\sigma}$
creates an $\alpha$ orbital electron with spin $\sigma$ at the site $i$
($i\equiv(i_{x},i_{y})$), and $\mu$ is the chemical potential. The magnetic
field is introduced through the Peierls phase factor $e^{i\varphi_{ij}}$ with
$\varphi_{ij}=\frac{\pi}{\Phi_{0}}\int^{r_{i}}_{r_{j}}\mathbf{A(r)}\cdot
d\mathbf{r}$ in the vortex state, where $A=(-Hy,0,0)$ stands for the vector
potential in the Landau gauge and $\Phi_{0}=hc/2e$ is the superconducting flux
quantum. In the case of the SC state with impurities, we randomly select the
half number of total sites, at where the random disorder potentials $U_{i}$
uniformly distributed over [$-U,U$] are set in. Then $U$ is a parameter to
characterize the strength of the disorder. The hopping integrals are chosen as
to capture the essence of the density function theory (DFT) results. gxu1
Taking the hopping integral between the $d_{yz}$ orbitals $|t_{1}|=1$ as the
energy unit, we have,
$\displaystyle t_{i,i\pm\hat{x},xz,xz}=$ $\displaystyle
t_{i,i\pm\hat{y},yz,yz}=t_{1}=-1.0$ $\displaystyle t_{i,i\pm\hat{y},xz,xz}=$
$\displaystyle t_{i,i\pm\hat{x},yz,yz}=t_{2}=1.25$ $\displaystyle
t_{i,i\pm\hat{x}\pm\hat{y},xz,xz}=$ $\displaystyle
t_{i,i\pm\hat{x}\pm\hat{y},yz,yz}=t_{3}=-0.9$ $\displaystyle
t_{i,i+\hat{x}-\hat{y},xz,yz}=$ $\displaystyle
t_{i,i+\hat{x}-\hat{y},yz,xz}=t_{i,i-\hat{x}+\hat{y},xz,yz}$ $\displaystyle=$
$\displaystyle t_{i,i+\hat{x}-\hat{y},yz,xz}=t_{4}=-0.85$ $\displaystyle
t_{i,i+\hat{x}+\hat{y},xz,yz}=$ $\displaystyle
t_{i,i+\hat{x}+\hat{y},yz,xz}=t_{i,i-\hat{x}-\hat{y},xz,yz}$ $\displaystyle=$
$\displaystyle t_{i,i-\hat{x}-\hat{y},yz,xz}=-t_{4}.$ (3)
Here, $\hat{x}$ and $\hat{y}$ denote the unit vector along the $x$ and $y$
direction, respectively.
The second term accounts for the superconducting pairing. Considering that a
main purpose here is to address the effects of the magnetic vortices and
nonmagnetic impurities on the spin-lattice relaxation rate in the iron-based
superconductors, we take a phenomenological form for the intra-orbital pairing
interaction,
$\displaystyle H_{pair}=$
$\displaystyle\sum_{ij,\alpha}V_{ij}(\Delta_{ij,\alpha\alpha}c^{{\dagger}}_{i,\alpha\uparrow}c^{{\dagger}}_{j,\alpha\downarrow}+h.c.)$
(4)
with $V_{ij}$ as the strengths of effective attractions.
Thus, we obtain the Bogoliubov-de Gennes equations for this model Hamiltonian
hmjiang
$\displaystyle\sum_{j,\alpha<\beta}\left(\begin{array}[]{cccc}H_{ij,\alpha\alpha,\sigma}&\Delta_{ij,\alpha\alpha}&H_{ij,\alpha\beta,\sigma}&0\\\
\Delta^{\ast}_{ij,\alpha\alpha}&-H^{\ast}_{ij,\alpha\alpha,\bar{\sigma}}&0&-H^{\ast}_{ij,\alpha\beta,\bar{\sigma}}\\\
H_{ij,\alpha\beta,\sigma}&0&H_{ij,\beta\beta,\sigma}&\Delta_{ij,\beta\beta}\\\
0&-H^{\ast}_{ij,\alpha\beta,\bar{\sigma}}&\Delta^{\ast}_{ij,\beta\beta}&-H^{\ast}_{ij,\beta\beta,\bar{\sigma}}\end{array}\right)$
(9) $\displaystyle\times\left(\begin{array}[]{cccc}u^{n}_{j,\alpha,\sigma}\\\
v^{n}_{j,\alpha,\bar{\sigma}}\\\ u^{n}_{j,\beta,\sigma}\\\
v^{n}_{j,\beta,\bar{\sigma}}\end{array}\right)=E_{n}\left(\begin{array}[]{cccc}u^{n}_{i,\alpha,\sigma}\\\
v^{n}_{i,\alpha,\bar{\sigma}}\\\ u^{n}_{i,\beta,\sigma}\\\
v^{n}_{i,\beta,\bar{\sigma}}\end{array}\right),$ (18)
where,
$\displaystyle H_{ij,\alpha\alpha,\sigma}=$
$\displaystyle-e^{i\varphi_{ij}}t_{ij,\alpha\alpha}-\mu$ $\displaystyle
H_{ij,\alpha\beta(\beta\neq\alpha),\sigma}=$
$\displaystyle-e^{i\varphi_{ij}}t_{ij,\alpha\beta(\beta\neq\alpha)}.$ (19)
$u^{n}_{j,\alpha,\sigma}$ ($u^{n}_{j,\beta,\bar{\sigma}}$),
$v^{n}_{j,\alpha,\sigma}$ ($v^{n}_{j,\beta,\bar{\sigma}}$) are the Bogoliubov
quasiparticle amplitudes on the $j$-th site with corresponding eigenvalues
$E_{n}$.
The pairing amplitude and electron densities are obtained through the
following self-consistent equations,
$\displaystyle\Delta_{ij,\alpha\alpha}=$
$\displaystyle\frac{V_{ij}}{4}\sum_{n}(u^{n}_{i,\alpha,\sigma}v^{n\ast}_{j,\alpha,\bar{\sigma}}+v^{n\ast}_{i,\alpha,\bar{\sigma}}u^{n}_{j,\alpha,\sigma})\times$
$\displaystyle\tanh(\frac{E_{n}}{2k_{B}T})$ $\displaystyle
n_{i,\alpha,\uparrow}=$
$\displaystyle\sum_{n}|u^{n}_{i,\alpha,\uparrow}|^{2}f(E_{n})$ $\displaystyle
n_{i,\alpha,\downarrow}=$
$\displaystyle\sum_{n}|v^{n}_{i,\alpha,\downarrow}|^{2}[1-f(E_{n})].$ (20)
The site-averaged DOS $N(E)$ is calculated by
$\displaystyle N(E)=$
$\displaystyle-\frac{1}{N}\sum_{i}\sum_{n,\alpha}[|u_{i,\alpha,\uparrow}^{n}|^{2}f^{{}^{\prime}}(E_{n}-E)$
(21)
$\displaystyle+|v_{i,\alpha,\downarrow}^{n}|^{2}f^{{}^{\prime}}(E_{n}+E)],$
where, $f^{{}^{\prime}}(E)$ is the derivative of the Fermi-Dirac distribution
function with respect to energy. The nuclear spin-lattice relaxation rate is
given by Takigawa
$\displaystyle R(r_{i},r_{i^{\prime}})=$
$\displaystyle\textmd{Im}\chi_{+,-}(r_{i},r_{i^{\prime}},i\Omega_{n}\rightarrow\Omega+i\eta)/(\Omega/T)|_{\Omega\rightarrow
0}$ $\displaystyle=$
$\displaystyle-\sum_{n,n^{\prime}}[\mathcal{U}^{n}_{i}\mathcal{U}^{n\ast}_{i^{\prime}}\mathcal{V}^{n^{\prime}}_{i}\mathcal{V}^{n^{\prime}\ast}_{i^{\prime}}-\mathcal{V}^{n}_{i}\mathcal{U}^{n\ast}_{i^{\prime}}\mathcal{U}^{n^{\prime}}_{i}\mathcal{V}^{n^{\prime}\ast}_{i^{\prime}}]$
(22) $\displaystyle\times\pi Tf^{\prime}(E_{n})\delta(E_{n}-E_{n^{\prime}}).$
Here, $\mathcal{U}^{n}_{i}=u^{n}_{i,\alpha}+u^{n}_{i,\beta}$ and
$\mathcal{V}^{n}_{i}=v^{n}_{i,\alpha}+v^{n}_{i,\beta}$. We choose
$\textbf{r}_{i}=\textbf{r}_{i^{\prime}}$ because the nuclear spin-lattice
relaxation at a local site is dominant. Then the site-dependent relaxation
time is given by $T_{1}(r)=1/R(r,r)$ and the bulk relaxation time
$T_{1}=(1/N)\sum_{r}T_{1}(r)$.
In numerical calculations, $V_{ij}$ is chosen to give a short coherence length
of a few lattice spacing in the SC state being consistent with experiments.
takeshita Under the conditions $V_{ij}\sim 2.0$, $\mu=1.2$ at temperature
$T=1\times 10^{-5}$, the filling factor
$n=\sum_{i,\alpha,\sigma}(n_{\alpha,\sigma})/(N_{x}N_{y})=1.9$ and the
coherent peak of the SC order parameter in the DOS is at $\Delta_{max}\sim
0.25$. Thus, we estimate the coherence length $\xi_{0}\sim
E_{F}a/|\Delta_{max}|\sim 5a$, ydzhu with $a$ being the Fe-Fe distance on the
square lattice. Due to this short coherence length, presumably the system will
be a type-II superconductor. To study the vortex states, we employ the
magnetic unit cell with size $N_{x}\times N_{y}=48\times 24$ that accommodates
two magnetic vortices, unless otherwise specified. In view of these
parameters, we estimate the upper critical field $B_{c2}\sim 100T$. Therefore,
the model calculation is particularly suitable for the iron-based type-II
superconductors such as CaFe1-xCoxAsF, Eu0.7Na0.3Fe2As2 and FeTe1-xSx, where
the typical coherence length $\xi_{0}$ deduced from the experiments is of a
few lattice spacing and the upper critical field achieves as high as dozens of
Tesla. takeshita
## III results and discussion
Since no final consensus on the SC pairing symmetry has yet been achieved, we
choose three possible singlet pairing symmetries, i.e., the most popular sign-
reversed $s_{\pm}$-wave, the on-site $s$-wave and the $d$-wave symmetries with
their respective gap functions
$\Delta_{s_{\pm}}=\Delta_{0}\cos(k_{x})\cos(k_{y})$, $\Delta_{s}=\Delta_{0}$
and $\Delta_{d}=\Delta_{0}[\cos(k_{x})-\cos(k_{y})]$ to carry out the
contrasting study. At the end of this section, we will also touch on another
possibility of the $s_{\pm}$-wave with angular variation along the electron
Fermi pockets, which will be referred to as $s_{h}$-wave. In order to obtain
comparable values of critical temperature $T_{c}$ in the self-consistent
calculations, we set respectively $V_{ij}=1.6$ for $s_{\pm}$-wave,
$V_{ij}=2.0$ for $s$-wave, and $V_{ij}=1.8$ for $d$-wave pairing symmetries.
To begin with, we briefly summarize the site-averaged DOS spectra $N(E)$ in
the uniform SC state at $T=1\times 10^{-5}$ as shown by the solid lines in
Fig. 1. For the $s_{\pm}$\- and $s$-wave symmetries, no node exists in the gap
along the Fermi surfaces. Correspondingly, the full gap structures can be seen
in the DOS as shown in Figs. 1(a) and 1(b). In the $d$-wave symmetry, the gap
function has line nodes at $k_{x}=k_{y}$, which cross the hole pocket but do
not intersect the electron pocket. As a result, the DOS consists of a small
V-shaped gap structure at very low energy and a U-shaped gap structure at
higher energy as shown in Fig. 1(c), which exhibits a difference from that in
high-$T_{c}$ cuprates.
Next, we show the DOS in the magnetic vortex state and in the presence of
nonmagnetic impurities as presented by the dotted and dashed lines in Fig. 1,
respectively. In the vortex state, the application of a magnetic field will
induce the quasiparticle flow around the vortex core, such that the nodal line
will appear in the otherwise fully gaped SC state due to an additional Doppler
shift in the quasiparticle energy, giving rise to the V-shaped gap structures
for the $s_{\pm}$\- and $s$-wave symmetries, as shown in Figs. 1(a) and 1(b).
In contrast, in the presence of impurities, the DOS shows the V-shaped
structure for the $s_{\pm}$-wave symmetry but the U-shaped for the $s$-wave
symmetry [Figs. 1(a) and 1(b)]. This is due to the fact that the impurity
potential has intra- and inter-band components for the multiband materials.
The intra-band components scatter fermions that have the same sign for the
$s_{\pm}$-wave SC order parameter and therefore do not affect the
superconductivity. Whereas the inter-band components scatter fermions with
opposite SC order parameters, thus have the pairing breaking effect. As a
result, they yield an obvious decrease in $T_{c}$ and simultaneously introduce
the V-shaped feature in DOS. In the case of the $s$-wave symmetry, no sign
change in the SC order parameters occurs on both the electron and hole
pockets, so that no obvious pairing breaking effect is induced by the
impurities and the U-shaped DOS is untouched.
For the $d$-wave symmetry, the small V-shaped superconducting gap is filled by
the low energy quasiparticles induced by either vortices or impurities,
resulting in a pseudogap-like U-shaped feature with finite density at Fermi
energy, as indicated by the dotted and dashed lines in Fig. 1(c). The line
nodes existing on the hole-pocket in the $d$-wave symmetry make it vulnerable
to the impurities, resulting in the disappearance of the small V-shaped
structure. However, the full gap opening on the electron-pocket is robust
against impurities, so that the pseudogap-like U-shaped feature is obtained.
In the vortex state, since the quasiparticles induced by the vortex come
preferably from the line nodes, the pseudogap-like U-shaped structure remains.
The results in Fig. 1 indicate that the characteristic low energy
quasiparticle excitations depend on the gap functions in the presence of
magnetic vortices and nonmagnetic impurities.
Now, we turn to the discussion of the temperature $(T)$ dependence of the
nuclear relaxation rate. For the uniform SC state, both the $s_{\pm}$\- and
$s$-wave symmetries produce a power law relation $T_{1}^{-1}\sim T^{5}$ below
$T_{c}$ and it evolves into an exponential dependence at very low temperature,
as shown by the solid lines in Figs. 2 and 3, Vorontsov1 which are the
consequence of the full-gap DOS in Fig. 1. In the vortex state, due to the
similar V-shaped DOS for both the $s_{\pm}$\- and $s$-wave symmetries,
$T_{1}^{-1}$ changes it’s $T$ dependence to $T^{3}$ below $T_{c}$ and becomes
nearly proportional to $T$ for the $s_{\pm}$-wave symmetry while $T^{1.5}$ for
the $s$-wave symmetry at low temperature, as denoted by the dotted lines in
Figs. 2(a) and 3(a), respectively. [The dotted lines in Figs. 2(a)-5(a) show
the results for magnetic unit cell with size $48\times 24$, while the dash-
dotted lines in these figures the results for magnetic unit cell with size
$40\times 20$.]
We note that both the $T^{3}$ dependence and the low-$T$ feature with nearly
linear slop are reminiscent of the experimental observations, Hammerath1 ;
Nakai1 where the $T$-linear dependence has been regarded as the evidence for
a residual density of states at zero energy in SC state. As mentioned above,
although the $s_{\pm}$-wave pairing is basically the full gap, the application
of a magnetic field will cause the quasiparticle flow around the vortex core,
such that the nodal line will appear in the SC state due to an additional
Doppler shift quasiparticle energy, giving rise to the V-shaped DOS and
$T^{3}$ dependent relaxation rate. In view of this, we may expect that the
slop of temperature dependent $T_{1}^{-1}$ should be insensitive to the
strength of the magnetic field, which has already been observed in experiment.
lma This is evident by the comparison between the dotted- and dash-dotted-
curves shown in Fig. 2(a), where the magnetic field for the dash-dotted-curve
is about $1.5$ times as large as that for the dotted-curve.
A striking difference between the $s_{\pm}$-wave and the $s$-wave states in
the $T$-dependence of $T_{1}^{-1}$ appears when the effect of the impurity
scattering is considered, as shown in Figs. 2(b) and 3(b). For the
$s_{\pm}$-wave state, $T_{1}^{-1}$ deviates gradually from $T^{5}$ to a
overall $T^{3}$ behavior at weak disorder such as $U=1$, then to a $T$ linear
dependence at low temperatures as the disorder strength $U$ is increased to
about $U=1.5$, which depicts the sensitivity of the SC order to the impurities
and is in accordance with the DOS results. On the other hand, $T_{1}^{-1}$
changes little upon the introduction of the disorder for the $s$-wave
symmetry. We also notice that $T_{c}$ is reduced substantially for the
$s_{\pm}$-wave symmetry at the moderate and even the weak disorder strength,
as can be seen from the lower-$T$ shift of the inflexion point on the curves
in Fig. 2(b). This again reflects the fact that the sign-reversed
$s_{\pm}$-wave pairing is fragile against the nonmagnetic impurities, which
has also been predicted theoretically by adopting a more sophisticated orbital
model and conformed experimentally in the specific heat and resistivity
measurements. Onari1 ; Mu1 ; Guo1
In Fig. 4, we present the results for the $d$-wave symmetry. Unlike the
formers, the overall $T$-dependence of $T_{1}^{-1}$ in the uniform SC state
roughly follows $T^{3}$ relation due to the existing of nodal line in the gap
structure. When a magnetic field is applied, $T_{1}^{-1}$ exhibits a
$T$-linear dependence at low temperatures, though a $T^{3}$ behavior following
$T_{c}$ still remains. We notice that this trend is rather robust against the
magnitude of the magnetic field, as the results for two cases are nearly the
same though their magnitude differs about $1.5$ times, as shown in Fig. 4(a).
Thus, for the three different pairing states, $T_{1}^{-1}$ exhibits nearly
similar $T$-dependence in the magnetic vortex state, due to the presence of
quasiparticles induced by the vortex. In the presence of impurities,
$T_{1}^{-1}$ for both the $d$-wave and the $s_{\pm}$-wave symmetries exhibit a
consecutive change from the $T^{3}$ to $T$-linear dependence when temperature
is decreased, which contrasts with that for the $s$-wave state.
We notice that the actual multiple Fermi surface sheets in iron-based
superconductors are not exactly reproduced by the two-orbital model, so the
general structure of the gap in the $s_{\pm}$-wave channel may involve the
angular variation along the electron Fermi pockets and have the form
$\Delta_{h}=\Delta_{0}\\{\cos(k_{x})\cos(k_{y})+h[\cos(k_{x})+\cos(k_{y})]\\}$,
where the factor $h$ measures the strength of the angular dependent variations
along the electron Fermi pockets. Chubukov1 ; SMaiti1 ; SMaiti2 ; Basov1 ;
AFKemper1 In the case of $h>1$, there will be accidental nodes along the
electron Fermi pockets, SMaiti3 which will be focused here. Such accidental
nodes are reflected in the V-shaped DOS in Fig. 1(d) and nearly $T^{4}$
dependence of $T_{1}^{-1}$ for the typical results with $h=1.5$ in the uniform
SC state [Fig. 5]. This result differs from the $T^{3}$ dependence for the
symmetry imposed nodal behavior such as in the case of the $d$-wave state.
Although the $T$-dependence of $T_{1}^{-1}$ in the vortex state is much like
that of the $s_{\pm}$-wave case, it is less influenced by the impurities [see
Fig. 5].
## IV conclusion
In conclusion, we have investigated the effect of the magnetic field and
nonmagnetic impurities on the DOS and the spin-lattice relaxation rate
$T_{1}^{-1}$ in the iron-based superconductors. It is shown that the
characteristic site-averaged DOS depends on the gap functions in the presence
of magnetic vortices and nonmagnetic impurities. The magnetic vortices have a
significant contribution to the spin-lattice relaxation rate and lead to the
relation $T_{1}^{-1}\sim T^{3}$ followed by a nearly $T$-linear dependence at
low temperatures for all three symmetries ($s_{\pm}$-, $s$\- and $d$-wave)
considered here, though in the clean uniform state a $T^{3}$ dependence for
the $d$-wave symmetry differentiates from the others with a $T^{5}$
dependence. In the presence of nonmagnetic impurities, this enhancement of
$T_{1}^{-1}$ can only be seen for the $s_{\pm}$\- and $d$-wave symmetries,
whereas it is almost unaffected for the $s$-wave symmetry. Our results suggest
that it is impossible to distinguish the $s_{\pm}$\- and $d$-wave symmetries
on the basis of the measurements of spin-lattice relaxation rates when a
magnetic field and/or impurity scatterings are present.
## V acknowledgement
This work was supported by the National Natural Science Foundation of China
(Grant No. 10904062 and No. 91021001), Hangzhou Normal University (HSKQ0043,
HNUEYT), and the Ministry of Science and Technology of China (973 project
Grants Nos. 2011CB922101, 2011CB605902).
Figure 1: Site-averaged DOS for the $s_{\pm}$-wave in (a), the $s$-wave in
(b), the $d$-wave in (c), and the $s_{h}$-wave in (d) (see text). The DOS in
the uniform SC state is plotted with solid lines. The results in the magnetic
vortex state and those in the presence of nonmagnetic impurities are shown
with the dotted and dashed lines, respectively. Figure 2: $T$-dependence of
$T_{1}^{-1}$ shown in the double logarithmic chart for $s_{\pm}$-wave symmetry
in the vortex state (a), and in the presence of nonmagnetic impurities (b).
The dotted and dash-dotted curves denote different strength of the magnetic
field (see text) and the impurity scattering. The results for the uniform SC
state are also plotted with solid line in each figure. Figure 3:
$T$-dependence of $T_{1}^{-1}$ shown in the double logarithmic chart for
$s$-wave symmetry in the vortex state (a), and in the presence of nonmagnetic
impurities (b). The dotted and dash-dotted curves denote different strength of
the magnetic field (see text) and the impurity scattering. The results for the
uniform SC state are also plotted with solid line in each figure. Figure 4:
$T$-dependence of $T_{1}^{-1}$ shown in the double logarithmic chart for
$d$-wave symmetry in the vortex state (a), and in the presence of nonmagnetic
impurities (b). The dotted and dash-dotted curves denote different strength of
the magnetic field (see text) and the impurity scattering. The results for the
uniform SC state are also plotted with solid line in each figure. Figure 5:
$T$-dependence of $T_{1}^{-1}$ shown in the double logarithmic chart for
$s_{h}$-wave symmetry (see text) in the vortex state (a), and in the presence
of nonmagnetic impurities (b). The dotted and dash-dotted curves denote
different strength of the magnetic field (see text) and the impurity
scattering. The results for the uniform SC state are also plotted with solid
line in each figure.
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|
arxiv-papers
| 2011-08-11T00:41:00 |
2024-09-04T02:49:21.482579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-Min Jiang, Jia Guo, and Jian-Xin Li",
"submitter": "Hong-Min Jiang",
"url": "https://arxiv.org/abs/1108.2312"
}
|
1108.2537
|
# Enhancements to cavity quantum electrodynamics system
A. D. Cimmarusti , J. A. Crawford , D. G. Norris and L. A. Orozco Joint
Quantum Institute, Department of Physics, University of Maryland and National
Institute of Standards and Technology, College Park, MD 20742-4111, U.S.A.
###### Abstract
We show the planned upgrade of a cavity QED experimental apparatus. The system
consists of an optical cavity and an ensemble of ultracold 85Rb atoms coupled
to its mode. We propose enhancements to both. First, we document the building
process for a new cavity, with a planned finesse of $\sim$20000\. We address
problems of maintaining mirror integrity during mounting and improving
vibration isolation. Second, we propose improvements to the cold atom source
in order to achieve better optical pumping and control over the flux of atoms.
We consider a 2-D optical molasses for atomic beam deflection, and show
computer simulation results for evaluating the design. We also examine the
possibility of all-optical atomic beam focusing, but find that it requires
unreasonable experimental parameters.
###### pacs:
42.15.Eq, 42.50.Pq, 32.60.+i
## I Introduction
The simplest realization of a cavity quantum electrodynamic system consists of
a single material fermion coupled to a boson field. Our work in the optical
regime couples Rb atoms and a finite number of modes of an optical cavity
berman94 . These systems have numerous applications in quantum information
science turchette95b ; cirac97 ; gheri98 ; wilk07b , and also enable the study
of quantum optics effects difficult to observe in free space norris10 ;
norris09a ; terraciano09 ; bishop08 ; guerlin07 ; harochebook ; hennrich05 .
Our current cavity QED experimental setup incorporates two independent vacuum
chamber components: a spherical hexagon and a cube (See Fig. 1). The former
houses an unbalanced Magneto-Optical Trap (MOT), from which we obtain a Low-
Velocity Intense Source (LVIS) of 85Rb atoms lu96 . The MOT uses a pair of
coils to generate a quadrupole magnetic field, with two retro-reflected laser
beams in the horizontal plane, and a third along the axis of the strong
magnetic field gradient (vertical direction). At the bottom of the chamber,
the vertical beam strikes a gold mirror and quarter-wave plate that have a
1.5-mm diameter hole, yielding a region of unbalanced laser intensity that
pushes out a beam of cold atoms. Directly below the hexagon lies a cubic
chamber which contains the optical cavity. Atoms propagate 3-4 cm through the
mirror to the cavity and are optically pumped to a desired $m$ state en route.
A fraction of the atoms pass between the cavity mirrors (2.2 mm spacing) and
couple to the TEM00 mode ($1/e$ field radius 56 $\mu$m) for $\sim$ 5 $\mu$s
before striking the bottom of the vacuum chamber.
Figure 1: (color online) Current vacuum chamber (not drawn to scale).
Unbalanced MOT (red beams) by hole in mirror creates LVIS.
The current arrangement has room for improvement in several fronts: Light from
the vertical MOT beam propagates collinearly with the atoms, complicating the
desired optical pumping distribution and driving the atoms weakly while they
are coupled to the cavity mode. We are unable to access the regime of
collective strong coupling in this system, due to the small number of atoms
that couple fully to the cavity mode. This is primarily due to the
distribution of transverse velocities of the atoms coming from the LVIS, which
results in a broadening spatial profile and lower atom density as the beam
propagates. To increase the single atom coupling we decrease the mode volume
by a factor of two and keep the decay rate of the cavity increasing the
finesse also by a factor of two. However, the reduction of the volume
complicates the collective strong coupling.
The structure of the paper is as follows: Section 2 talks about the progress
in the construction of the new optical cavity. In section 3, we discuss
deflection of an LVIS by a 2-D optical molasses (OM). We explore the
possibility of focusing an atomic beam in section 4, and we end with
conclusions in section 5.
## II Building the optical cavity
At the heart of every optical cavity QED apparatus lies a high finesse (low
loss) resonator. As the optical transmission coefficients for the mirrors used
in the resonator may be as low as a few parts per million, reducing the
absorptive losses to this level is a great experimental challenge. Our present
cavity (decay rate $\kappa/2\pi=2.8\times 10^{6}$ s-1 and finesse of 11,000)
suffers from higher than expected losses, reducing its finesse by about a
factor of two from the intended value ( $\sim$25,000, as calculated using the
reflectivity of the mirrors). We believe the losses are due to adhesive
residue inadvertently deposited on the mirror surfaces during construction, as
well as the aging of the high-reflectivity coatings over the nearly two
decades since deposition. We are constructing a new cavity with small and
well-characterized losses, as well as slightly stronger coupling to the atoms
and enhanced vibration isolation.
Two spherical mirrors with Research Electro-Optics (REO) high-reflectivity
coatings form the cavity. For precise alignment, these circular mirrors sit in
a square groove machined into a single piece of MACOR, a low-outgassing
nasa_outgas , machinable ceramic that can withstand high temperatures. The
MACOR is adhered atop two separate piezoelectric transducers (PZTs), bridging
the 2 mm gap between them. With the MACOR affixed to the PZTs and the PZTs
affixed to a stainless steel base, we cut the MACOR piece in two, thereby
allowing for independent translation of each half of the cavity. It is
desireable to cut the MACOR with minimal force, so as to avoid putting stress
on the PZTs or comprimising the alignment of the groove in the MACOR. This
procedure provides good alignment of the MACOR grooves between the two halves,
so that simply resting the mirrors in the grooves before gluing keeps the
reflective surfaces parallel. To maintain high vacuum ($\sim 10^{-8}$ Torr or
lower) with the cavity mount inserted in the chamber, we use only adhesives
that NASA rates as low-outgassing based on percent total mass loss at low
pressure nasa_outgas such as Loctite Hysol 1C. In place of solder, we use
EPO-TEK H20E, a silver-filled electrically conductive low-outgassing epoxy, to
bond the PZTs to wires and the stainless steel base. Figure 2 shows the
various steps followed.
Figure 2: (color online) Photographs of the cavity building process. (A) Non-
magnetic steel base, (B) PZTs on base, (C) MACOR with slit in the middle and
(D) high reflectivity mirrors.
The choice of curvature and coating for the two mirrors depends on the desired
cavity properties. We aim to construct a cavity with coupling in the
intermediate regime (i.e. cavity decay rate $\kappa$ of the same order as
single-atom coupling rate $g$ and spontaneous emission rate $\gamma$), and
with a finesse of about 20,000. After nearly two decades of aging, the REO
mirrors require direct measurement of their transmission properties in order
to determine the expected finesse.
We use a 5 mW, 780 nm laser beam and a mounted optical power meter to measure
the intensity transmission coefficient $T$ directly. A 780 nm filter reduces
background light into the detector to below 0.01 nW. After measuring the power
of the unimpeded beam, we move a mirror into the path of the laser such that
it is incident on the coated side, and the mirror reflects the majority of the
power back through the optical isolator before the laser. Finally, we use an
aperture after the mirror to carefully block as much of the scatter as
possible (without clipping the transmitted beam itself), and record the
intensity of transmitted light. Taking the ratio of the transmitted optical
power to the incident optical power (over several independent trials), we
determine the current transmissions of the high-reflectivity mirrors. The
labeled $T$ values for the mirrors did not always agree with our measurements
as Table 1 shows.
# | $R$ | Date | Target $T$ | Range $T$
---|---|---|---|---
6 | 45 cm | 11/98 | 300 ppm | 235 - 297 ppm
3 | 45 cm | 11/98 | 15 ppm | 7.57 - 8.06 ppm
Table 1: Mirror parameters: # number of mirrors, $R$ radius of curvature,
Target transmission and Range of trasmission.
Choosing mirrors with transmissions of $T_{1}$ = 8 ppm and $T_{2}$= 297 ppm
(so that over 97% of photons in the cavity will exit through one mirror,
making detection simpler), we calculate finesse assuming that the absorption
losses are very small with Eq. (1)
$F=\frac{\pi}{1-\sqrt{(1-T_{1})(1-T_{2})}}.$ (1)
These directly measured transmissions give an expected finesse of
$\sim$20,000.
We use a method of impedance mismatching in which the stainless steel cavity
mount sits atop a stack of materials with very different resonant frequencies,
such that vibrations do not easily propagate through the entire stack, to
improve the mechanical decoupling of the cavity. The materials include lead,
copper, and Sorbothane (a shock absorbing synthetic viscoelastic urethane
polymer). In order to find the optimal configuration of the materials, we
place the cavity base atop a test stack and use a PZT attached to the base as
a microphone, connecting its output to a spectrum analyzer (Stanford Research
Systems SR770 FFT Network Analyzer). We then strike the tabletop with a
hammer, delivering controlled “delta function” impulses, several times over a
span of ten seconds. We collect PZT voltage and average the spectrum over a
range of frequencies from 0 to 100 kHz, and we compare dozens of different
combinations of damping materials.
We obtain a drastic improvement to the effectiveness of the damping stack by
altering the geometry of the Sorbothane layer, based on advice from the
manufacturers of Sorbothane sorbothane . Instead of a solid sheet of
Sorbothane, we cut the material into twelve roughly 0.25-inch squares, and
space the squares out over the area of the damping stack. As shown in Fig. 3,
the small pieces of Sorbothane make this geometry significantly more effective
at damping, as they have more room to deform sideways and dissipate
vibrational energy.
Figure 3: (color online) Power spectra after impulse excitation. Damping
layers: Lead-Sorbothane-Lead.
## III Atomic beam deflection
We propose a new atomic beam design to address the problem of the vertical MOT
beam that propagates together with the atoms, inspired by wang10 ; nellessen89
; witte92 ; ashkin70 , in which a 2-D optical molasses deflects the atomic
beam from its initial trajectory and into the cavity mode (see Fig. 4). The
benefits of deflection are threefold: It rids us of the unwanted MOT light, it
allows for better control of the atom number, and it provides a way to lower
the average speed of the atoms traversing the cavity mode. The latter
translates into longer transit times for the atoms in the cavity mode.
A pair of retro-reflected laser beams, intersecting at right angles in the
$x$-$y$ plane, create a 2-D optical molasses (see Fig. 4). The molasses acts
as a viscous damping force of form $\vec{F}=-\beta\vec{v}_{T}$, opposing the
motion of the atoms in the $x$-$y$ plane, and rapidly damping the velocity in
these directions, $\vec{v}_{T}$, to a mean of zero. Aiming the initial atom
beam from the LVIS at some angle with respect to the $x$, $y$, and $z$ axes,
the molasses damps all velocity components except those along $z$, giving an
effective deflection into the $z$ direction while also reducing the mean
longitudinal velocity of the beam.
Figure 4: (color online) LVIS of atoms exit (a) hole angled to the vertical
axis of our optical cavity, scatter light from a (b) 2-D optical molasses.
Implementing this geometry requires a major modification of our apparatus. We
simulate the beam deflection process to find the optimal experimental
parameters before changing the system. The code for this simulation is in C++
with the aid of Root, an open source object-oriented data analysis framework
developed by CERN and used extensively in particle physics. The code currently
relies on histogramming, graphing, geometry and physics libraries of Root and
can be obtained from the website of the author under the terms of the GNU
General Public License andres_sim_beam
The first stage of the simulation involves the modeling of an atomic beam in
our current LVIS setup. The main goal of the simulation is to give a
reasonable estimate for the ratio of atoms that couple to the cavity mode to
the total number of atoms that exit the unbalanced MOT. The code does not
simulate a MOT and assumes no atom-atom interactions.
We create the beam by randomly assigning $(x,y)$ starting coordinates to each
atom from a uniform distribution across the area of the hole in the LVIS
mirror (see Fig. 1), defining the vertical starting position as $z=0$. In a
similar way, we randomly specify $(v_{x},v_{y})$ initial transverse
velocities, constrained by the geometric collimation mechanism of the LVIS
mirror aperture to a maximum value as in Eq. (2):
$v_{T,max}=\frac{D_{h}}{d_{mh}}v_{0},$ (2)
where $D_{h}$ is the diameter of the hole, $d_{mh}$ is the distance from the
center of the MOT to the center of the hole and $v_{0}$ is the mean
longitudinal speed of the atom beam from the unbalanced MOT. The initial
vertical velocity components $v_{z}$ come from a Gaussian distribution with
mean $v_{0}=14$ m/s and FWHM 2.7 m/s, in accordance with lu96 .
Figure 5 shows 1000 atomic trajectories using parameters from our current LVIS
and cavity setup. Each atom propagates until it reaches within one waist of
the cavity mode center or misses and reaches the chamber wall. The white spot
at the top center of the figure corresponds to the position of the cavity
mode. The simulations show that 7 - 9 % of atoms coming from the LVIS couple
to the cavity mode, where the range corresponds to the uncertainty in the
actual distance between LVIS aperture and cavity mode (between 3 and 4 cm.)
For the planned smaller optical cavity, the range drops to 4 - 6 %, down by
approximately a factor of two as expected from the mode volume.
Figure 5: (color online) One thousand atomic trajectories a beam exiting a
1.5 mm hole. The optical cavity mirror spacing is 2.2 mm and sits at
$z_{cav}=4$ cm. Colors correspond to different atoms.
We next simulate the atomic beam deflection proposal. We follow the treatment
of metcalf99 for the calculation of the scattering rates in steady-state for
a two-level atom in the presence of a laser. We give each of the four beams a
transverse Gaussian profile as in Eq. (3):
$s_{0,i}^{(g)}=s_{0}\left(\frac{w_{0}}{w(z_{i})}\right)^{2}\exp{\left[-2\left(\frac{r_{i}^{2}}{w^{2}(z_{i})}\right)\right]},$
(3)
where $z_{i}$ and $r_{i}$ are the longitudinal and transverse coordinates with
respect to the $k$-vector of the $i$th beam, $w_{0}$ is the waist of each
beam, $s_{0}$ is the on-resonance saturation parameter at the center of each
waist (i.e., at $z_{i}$=$r_{i}$= 0), and $w(z_{i})$ is the beam spot size as a
function of longitudinal position along each beam. $s_{0,i}^{(g)}$ is the
Gaussian beam correction to the on-resonance saturation parameter. We use it
to calculate the saturation parameter per beam for each atom as Eq. (4):
$s_{i}=\frac{s_{0,i}^{(g)}}{1+\left(2\left(\delta_{i}-\vec{k_{i}}\cdot\vec{v}\right)/\gamma\right)^{2}},$
(4)
where $\delta=\omega_{\ell}-\omega_{a}$ is the laser detuning with respect to
the atomic transition frequency, $\vec{k}$ is the laser wavevector, $\vec{v}$
is the velocity of the atom, $\gamma$ is the transition linewidth. We make the
approximation that the molasses beams do not interfere, which can be
accomplished experimentally by using orthogonal polarizations and spatially
separating the $x$ and $y$ beam pairs. In the simplified model we add the
contribution from each beam independently and find the total saturation
parameter $s_{T}=\sum_{i=1}^{4}s_{i}$. We then calculate the total scattering
rate $\gamma_{p}$ as in Eq. (5):
$\gamma_{p}=\frac{\gamma}{2}\frac{s_{T}}{1+s_{T}},$ (5)
When an atom enters the molasses region (defined within the $1/e^{2}$
intensity region of the beams), it immediately begins scattering at a rate
$\gamma_{p}$. A scattering event involves a recoil velocity of magnitude
$\hbar k/m$ in the direction of the absorbed beam, and a recoil of the same
magnitude but random direction for the reemission. We determine which beam the
atom absorbs from by comparing a random number to the ratio
$\gamma_{p,i}/\gamma_{p}$, i.e. the relative weight of each beam compared to
the total scattering rate. The beam with the largest weight will be most
likely to give the atom a momentum kick. We then select two more random
numbers to determine the angle of reemission in three dimensions. Updating the
velocity of the atom with the result of the two recoils, we allow the atom to
propagate freely until the next scattering event. To determine this time step,
we pick random numbers from the exponential waiting-time distribution (Eq. 6),
$W(t)=\gamma_{p}e^{-\gamma_{p}t},$ (6)
which governs the probability of the next event occuring in time $t$ for a
Poisson random process. At each time step we record the position and velocity
of the atom. The algorithm repeats until the atom exits the molasses.
We use very weakly focused Gaussian beams with parameters $w_{0}=w(z_{i})=7.5$
mm, for simulating the proposed atomic beam setup, such that the beam
intensity has negligible $z_{i}$ dependence. We choose a slightly smaller
aperture size in the LVIS mirror, 1 mm, to achieve a smaller initial beam, and
we set the angles of the initial LVIS beam trajectory as $\theta=30^{\circ}$
(polar) and $\phi=45^{\circ}$ (azimuthal) with respect to the cavity
coordinate system.
Figure 6 shows the effect of laser detuning on the beam profile. We graph the
average polar angle of the atomic beam velocity at a fixed position just after
it exits the molasses region.
Figure 6: Average polar angle of atoms after 2-D OM as a function of laser
detuning for beams with $w_{0}=w=7.5$ mm and $s_{0}=3$.
Using the same parameters as before, but choosing the optimal detuning of
$\delta/\gamma=-0.5$, Fig. 7 shows atomic trajectories under the action of the
2-D optical molasses. For this simulation, we placed the cavity at $z_{cav}=1$
cm and obtained 11 % of atoms coupling to the mode.
Figure 7: (color online) Atomic beam trajectories deflected by 2-D OM with
$\delta/\gamma=-0.5$ and $s_{0}=3$.
Figure 8 shows the $x$ component of the velocity for the atoms. We calculate
the RMS transverse speed of the collimated atomic beam and find 15.3 cm/s,
yielding a Doppler temperature of 120 $\mu$K. This value agrees with a simple
calculation. For a true 1-D system with two beams, we write the heating rate
as $4\gamma_{p}E_{r}$, where $E_{r}=\hbar\omega_{r}$ is the recoil energy
metcalf99 . One factor of two comes from the presence of two beams and the
second because each absorption and re-emission cycle involves two recoils in
that dimension. In three dimensions, the recoil energy from isotropic re-
emission is divided equally among all directions, so this second factor of two
becomes $(1+N/3)$, where $N$ is the number of dimensions with a pair of
molasses beams. When equated to the cooling rate along any one dimension, this
gives Doppler temperature $T=\hbar(4/12,5/12,6/12)\gamma/k_{b}$ for
$N=(1,2,3)$ pairs of molasses beams. In our case $N=2$ and the result is in
agreement with the simulation.
Figure 8: (color online) Atomic beam $v_{x}$ damped by 2-D OM with
$\delta/\gamma=-0.5$ and $s_{0}=3$.
## IV Laser lens: Focusing
The 2-D OM induces a deflection in the atomic beam (see Fig. 7), and
collimates the beam down to the Doppler temperature in the transverse
directions, but does not focus the atoms. This requires a position dependent
force. Eq. 3 shows the possibility of introducing spatial dependence through
the factor $w(z_{i})$. By focusing the laser beams tightly at distance $p$
away from the central axis as shown in Fig. 9, we can create an intensity
gradient over the interaction region and induce atomic beam focusing balykin88
.
It is possible to achieve focusing by using a 2-D MOT as a magneto-optical
lens berthoud98 ; labeyrie99 . This gives a position dependent force from the
magnetic field gradient that shifts the atomic resonances, allowing spatial
compression. However, its setup carries undesirable consequences. The
proximity of the 2-D MOT coils to the optical cavity, make accurate control of
the magnetic fields in the cavity very difficult.
Figure 9: Simplified schematic of a 1-D laser lens.
Figure 9 shows that when $\vec{v}=0$, $F_{1}-F_{2}=-\hbar
k\left(\gamma_{p,2}-\gamma_{p,1}\right)\neq 0$ away from the axis. We estimate
how tightly focused the beams have to be to achieve a usable intensity
imbalance. Fig. 10 shows our results under the following assumptions. We take
a fixed point for the calculation, at $z=1$ mm. We choose this value because
it represents a typical position of an atom in the beam as it exits the 2-D
OM. The beam spot size at the center of the laser lens region is 5 mm and an
intensity per beam at the center of $s_{0}=2$. We set $\vec{v}=0$, to
characterize the position dependent part of the interaction.
Figure 10: Scattering rate ratio for an atom at $z=1$ mm ($\vec{v}=0$) in a
laser lens with $s_{0}(0,0)=3$ and $w(0)=5$ mm.
These results indicate that a successful laser lens requires very tight
focusing ($\sim$0.5 $\mu$m). We separate the deflection and the focusing, and
elaborate a more sophisticated model for a two-dimensional laser that includes
more of the atomic structure through a six level atom model correction
bouyer94 (See Fig. 11).
Figure 11: (color online) 6-level atomic structure. Driven by $\sigma^{+}$
and $\sigma^{-}$ polarized light. Decays and excitations take place according
the Clebsch-Gordan coefficients shown metcalf99 .
The algorithm for the laser lens builds upon the one for the 2-D OM. The
6-level atom correction weighs the scattering rates per beam and spontaneous
emission using the Clebsch-Gordan coefficients. We use two different
quantization axes, $x$ and $y$, depending from which beam the atom most
recently absorbed. This approach makes the code simpler.
Results from the simulation are complicated. They show focusing for very small
waists. The refinements in the model (i.e. 6-level correction) do not seem to
loosen such tight requirements. We will continue to explore this, and evaluate
its practical benefits.
## V Conclusions
We have demonstrated enhancements to increase the quality of our cavity and
allow us to couple more atoms to its modes.
In cavity construction, we expect that careful experimental measurement of the
transmissions of our mirrors have allowed us to make significantly more
accurate predictions of finesse. Additionally, our progress in reducing
mechanical vibrations via impedance mismatching of damping materials will help
to ensure stable operation of the cavity.
We show encouraging simulation results for the implementation of 2-D optical
molasses for the deflection and collimation of a Low Velocity Intense Source
of atoms. Parameter explorations pointed at an optimal molasses laser beam
detuning at $\delta/\gamma=-0.5$. By placing the cavity at a moderately short
distance from the molasses, we showed that we can get up to 11 % of atoms into
the new cavity mode, in contrast with our simulation predictions for our
current system of about 6 %.
Despite initial evidence that the laser lens technique can achieve atomic
focusing, we consider it highly difficult to implement in our experiment, due
primarily to the very small laser beam waists required for its effectiveness
($<1$ $\mu$m).
## Acknowledgments
This work was supported by the National Science Foundation (NSF). We thank J.
R. Ramos for his participation in early stages of the optical cavity building
process. We are grateful to S. L. Rolston, H. J. Carmichael and M. Scholten
for their stimulating discussions and guidance. Thanks to the Root developers
and users for their invaluable advice.
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|
arxiv-papers
| 2011-08-11T23:33:35 |
2024-09-04T02:49:21.492269
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. D. Cimmarusti, J. A. Crawford, D. G. Norris and L. A. Orozco",
"submitter": "Andres Cimmarusti",
"url": "https://arxiv.org/abs/1108.2537"
}
|
1108.2544
|
# A Simple Model of Direct Gauge Mediation
Sibo Zheng and Yao Yu
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
In the context of direct gauge mediation Wess-Zumino models are very
attractive in supersymmetry model building. Besides the spontaneous
supersymmetry and $R$-symmetry breaking, the problems of small gaugino mass as
well as $\mu$ and B$\mu$ terms should be solved so as to achieve a viable
model. In this letter, we propose a simple model as an existence proof, in
which all these subjects are realized simultaneously, with no need of fine
tuning. This completion also implies that much of parameter space for direct
gauge medition can be directly explored at LHC.
08/ 2011
Supersymmetry (SUSY) is an appealing candidate for explaining the mass
hierarchy and providing the unification of gauge couplings. Experimental
searches at colliders such as LHC give arise to strigent constraints on
supersymmetric physical parameters at low energy where SUSY must be broken.
Some difficulties can be avoided by adjusting the mechanism of SUSY breaking
or allowing a few fine tunings.
On the realm of SUSY model building, gauge mediated SUSY breaking , is one of
the most well studied scenarios for a few reasons. At first, the problem of
flavor changing neutral currents can be naturally solved with supersymmetric
particles $\sim 1$TeV in this paradigm. Second, this makes the supersymmetry
testable at a few current colliders such as LHC. Finally, microscopic models
which trigger SUSY breaking can be constructed in a wide SUSY theories.
In particular, chiral theories, or concretely Wess-Zumino models [1] are very
attractive and powerful. One reason is that they often serve as the effective
theories of (strongly coupled ) microscopic theories, for example the well
known massive SQCD theory with a dual description [2]. Furthermore, they are
calculable and under control, thus might be applied to viable SUSY
phenomenology. Very recently, it is generally argued that Wess-Zumino models
in which SUSY is spontaneously broken are actually type of O’ Raifeartaigh
models [3],
$\displaystyle{}W=fX+(\lambda_{ij}X+m_{ij})\varphi_{i}\tilde{\varphi}_{j}+\cdots$
(1)
where neglected terms denote the cubic terms.
What is of more interest is to directly apply these O’ Raifeartaigh models to
models buildings in the context of gauge mediation, i.e, direct gauge
mediation (DGM) [4, 5, 6]. In contrast with the minimal gauge mediation (see
review [7] and references therein), the is no need to introduce additional
messenger sector in DGM. At first sight, it is observed that the gaugino mass
of order $\mathcal{O}(F)$ ( $\sqrt{F}$ refers to the supersymmetry breaking
scale) often vanishes in direct gauge mediated O’ Raifeartaigh models, Now it
is understood [3] that this phenomena is tied to the global vacuum structure
composed of pseudomoduli space $X$, whether $R$-symmetry is spontaneous broken
or not. In light of this new finding, various O’ Raifeartaigh models where the
gaugino mass problem can be resolved are proposed [8, 9, 10, 11, 12, 13, 14,
15, 16], some of which even have microscopic completions [10, 16].
In this letter, we discuss another important subject left in DGM, that is the
generation of viable $\mu$ and B$\mu$ terms as in ordinary gauge mediation
[17, 18, 19, 20, 21, 22, 23] 111In [22], the authors discuss the strongly
coupled generation of $\mu$ term in context of direct gauge mediation.. This
subject have been less addressed in comparison with the gaugino mass problem.
We restrict us to most generalized O’ Raifeartaigh models that respect
renormalization and $R$-symmetry, and then address the gaugino mass and $\mu$
problem simultaneously.
First, what kind of O’ Raifeartaigh models in (1) can solve the gaugino mass
problem ? We can take a few limits in (1) for illustration. If $\lambda$ is
diagonal (via bi-unitary transformation ) and $m=0$, this actually reduces to
the minimal gauge mediation, in which det$\mathcal{M}=X^{N}$det$\lambda$. As
well known there is no small gaugino mass problem in this context . However,
it is nerve considered as starting point of direct gauge mediation, as
spontaneous SUSY breaking can not be realized in this setup. If $m$ is
diagonal then det$\mathcal{M}=$det$m$ 222The field space composed of the
pseudomoduli $X$ is stable globally in this type of O’ Raifeartaigh models,
which implies that the determinant $\mathcal{M}$ is a constant [3]., which
results in spontaneously broken SUSY and vanishing gaugino mass at order of
$\mathcal{O}(F)$.
Therefore, in order to render the O’ Raifeartaigh model to generate the one-
loop gaugino masses, or equivalently guarantee the determinant to depend on
$X$, there must be at least one non-zero diagonal element in $\lambda$. So the
superpotential can be constructed as the mixing of those of minimal setup and
tree-level mass terms of messengers.
Now, we consider a concrete model in light of above observations, whose
superpotential is given by,
$\displaystyle{}W_{1}$ $\displaystyle=$ $\displaystyle
fX+\lambda_{1}X\left(S\tilde{S}+T\tilde{T}\right)+m_{1}S\tilde{T}$ (2)
$\displaystyle+$
$\displaystyle\lambda_{2}X\left(\varphi_{1}\tilde{\varphi}_{1}+\varphi_{3}\tilde{\varphi}_{3}\right)+m_{2}\varphi_{1}\tilde{\varphi}_{3}+\lambda_{3}X\left(\varphi_{2}\tilde{\varphi}_{2}+\varphi_{4}\tilde{\varphi}_{4}\right)+m_{3}\varphi_{2}\tilde{\varphi}_{4}$
This is the minimal setup as we will find. We assume all the masses and
couplings in (2) are real without loss of generality. The couplings in (2) can
be realized via imposing global symmetries $[SU(2)\times~{}SU(2)]^{2}$ as
follows,
$\displaystyle{}\Phi=\left(\begin{array}[]{c}\varphi_{1}\\\
\varphi_{2}\end{array}\right),~{}~{}~{}~{}\tilde{\Phi}=\left(\begin{array}[]{c}\tilde{\varphi}_{1}\\\
\tilde{\varphi}_{2}\end{array}\right),~{}~{}~{}~{}\Sigma=\left(\begin{array}[]{c}\varphi_{3}\\\
\varphi_{4}\end{array}\right),~{}~{}~{}~{}\tilde{\Sigma}=\left(\begin{array}[]{c}\tilde{\varphi}_{3}\\\
\tilde{\varphi}_{4}\end{array}\right).$ (11)
Also the global symmetry assignment results in the degeneracies
$\lambda_{2}=\lambda_{3}$ and $m_{2}=m_{3}$. Thus, eq(2) can be rewritten as,
$\displaystyle W_{1}$ $\displaystyle=$ $\displaystyle
fX+\lambda_{1}X\left(S\tilde{S}+T\tilde{T}\right)+m_{1}S\tilde{T}$ (12)
$\displaystyle+$
$\displaystyle\lambda_{2}X\left(\Phi\tilde{\Phi}+\Sigma\tilde{\Sigma}\right)+m_{2}\Phi\tilde{\Sigma}$
We assume $S,\tilde{S}$ and $T,\tilde{T}$ as standard model singlets.
Additional coupling associated with the Higgs fields can be introduced when
the $SU(2)$ global symmetries are directly gauged as the standard model
electroweak groups,
$\displaystyle{}W_{2}=\lambda_{\mu}\tilde{S}\tilde{\Phi}H_{\mu}+\lambda_{d}S\Phi
H_{d}$ (13)
In particular, either $S,\tilde{S}$ fields or $T,\tilde{T}$ can couple to the
Higgs doublets, but they can not be allowed to appear in (13) at same time as
a result of $R$-symmetry. 333 The reason is due to the absence of quardratic
mass terms for $S^{2}$ and $T^{2}$ in (2). We undersatand this fact as a
consequence of $R(S)\neq 1$ and $R(T)\neq 1$. Similar understanding can also
be applied to singelt fields $\tilde{S}$ and $\tilde{T}$. Similarly, either
$\Phi,\tilde{\Phi}$ or $\Sigma,\tilde{\Sigma}$ can be coupled to Higgs fields.
We choose the set in (13) for example. The superpotential of O’ Raifeartaigh
models we consider is $W=W_{1}+W_{2}$, which respects gauge symmetries of
standard model and $R$-symmetry involved. Once all the subjects involved in
SUSY model buildings are realized in such kind of models, one can add triplet
fields of QCD gauge group in (2) so as to complete the model.
According to (2) and (13), the vacuum is represented by,
$\displaystyle{}S=\tilde{S}=0,~{}~{}~{}~{}\left(\begin{array}[]{c}\Phi\\\
\Sigma\end{array}\right)=0,~{}~{}~{}~{}~{}~{}\left(\begin{array}[]{c}\tilde{\Phi}\\\
\tilde{\Sigma}\end{array}\right)=0,~{}~{}~{}X~{}~{}arbitrary$ (18)
with potential $V=f^{2}$. To achieve this vacuum, the property that there are
diagonal $\lambda$ and non-zero mass terms in (2) is crucial in above
analysis. At this SUSY breaking vacuum (18) the gauge symmetries of standard
model is unbroken. Even without studying the details of pseudomoduli space
$X$, one expects that there is no gaugino mass problem in this model. Since in
the region $X\rightarrow 0$, some freedoms in messengers become tachyonic.
This means that the vacuum (13) is not stable globally. From (2), the
eigenvalues of messenger fermion mass squared $\mathcal{M}^{2}_{F}$ are given
by,
$\displaystyle{}m^{2}_{1/2,i^{\pm}}=m^{2}\left(\frac{1}{2}+x_{i}^{2}\pm\sqrt{\frac{1}{4}+x_{i}^{2}}\right)$
(19)
for a given basis $i$. In (19) we have defined the dimensionless coefficients
$x_{i}=\lambda_{i}X/m_{i}$. Similarly, it is straightforward to evaluate the
messenger boson mass squared $\mathcal{M}^{2}_{B}$. From these eigenvalues we
verify that some fermions are massless while some bosons tachyonic at small
$X<\sqrt{f}$. So the physical parameter space is given by
$\displaystyle{}\sqrt{f}<<X<\min{(m_{i})}$ (20)
Actually, a ratio of order $\mathcal{O}(10)$ for the first constraint in (20)
is sufficient to guarantee the positive masses of messenger fields. In this
note, we will take the small $F$ limit in order to simplify the analysis of
Coleman-Weinberg potential in the next paragraph.
Let us examine the $R$-symmetry breaking in our model. In (2) one finds that
there must be $R$-charge assignments other than 0 or 2 in (2). Following the
argument in [24], which states that $R$-symmetry can not be broken except
there are fields with $R$-charge other than 0 and 2, one can see that the
$R$-symmetry breaking or equivalently negative mass squared $m_{X}^{2}$ is not
difficult to be realized. According to discussions in the previous paragraph,
it is sufficient to study the region of moderate $X$ value, we will focus on
this region with small $F$-term. Under limit (20) the one-loop Coleman-
Weinberg potential $V_{CW}(X)$ for the pseudomoduli at moderate $X$ is
approximately given by,
$\displaystyle{}V_{CW}(X)=\frac{5f^{2}}{32\pi^{2}}\sum_{i=1}^{i=3}\lambda_{i}^{2}V_{2}(x_{i}),~{}~{}~{}~{}x_{i}=\lambda_{i}X/m_{i}$
(21)
where
$\displaystyle{}V_{2}(x_{i})=-\frac{2}{1+4x_{i}^{2}}+4\log~{}x_{i}+\frac{2x_{i}^{2}+1}{\left(4x_{i}^{2}+1\right)^{\frac{3}{2}}}\log\frac{2x^{2}_{i}+1+\sqrt{4x_{i}^{2}+1}}{2x^{2}_{i}+1-\sqrt{4x_{i}^{2}+1}}$
(22)
The Coleman-Weinberg potential is plotted in fig. 1; one finds that $V_{CW}$
is minimized at $x_{1}=x_{2}=x_{3}\simeq 0.25$ or $X_{0}\sim 0.1$ m.
Figure 1: $V_{CW}$ varies as function of $x_{i}$ in unit of $f^{2}$. For
illustration, take the particular values
$\lambda_{1}=\lambda_{2}=\lambda_{3}=3$ and $m_{1}=m_{2}=m_{3}=$ m.
Figure 2: Unified masses $m_{i}=$m and $X_{0}=0.1$ m in $(a)$. The parameter
space composed of $\lambda_{2}$ and $\lambda_{3}$ is shown in the region
$0.1\leq\lambda_{1}\leq 3$. Non-degenerate masses among $m_{2}$ and $m_{1}$ in
$(b)$. We set the messenger scale $X_{0}=0.1m_{1}$, while $\lambda_{1}$ varies
in the region $0.1-3$.
Firstly, we set all masses $m_{i}$ are unified in order to simplify the
analysis. As shown in fig. 2$(a)$, if one wants to obtain $X_{0}=0.1m_{1}$,
$\lambda_{2}$ and $\lambda_{3}$ should be chosen around the window $0.1-2.5$,
when $\lambda_{1}$ varies from 0.1 to 3 444Large Yukawa couplings often give
rise to the problem of Landau pole in the context of direct gauge mediation
[25, 26]. If $X_{0}\leq 0.01$ m, we find that there are no parameter space
allowed. Relax the condition $m_{i}=$ m and allow deviation of $m_{2}$ from
$m_{1}$, we show in fig. 2.$(b)$ the parameter space when
$\lambda_{2}=\lambda_{3}$, $X_{0}=0.1m_{1}$ and $\lambda_{1}$ varies from 0.1
to 3.
Now we proceed to discuss the soft terms induced by superpotential (13), which
can be read from the one-loop effective Kahler potential $K_{eff}$ after
integrating out the messenger fields involved [27],
$\displaystyle{}K_{eff}=-\frac{1}{32\pi^{2}}Tr\left(\mathcal{M}^{{\dagger}}\mathcal{M}\log\frac{\mathcal{M}^{{\dagger}}\mathcal{M}}{\Lambda^{2}}\right)$
(23)
From (13), in the case $m_{3}=m_{2}$ and $\lambda_{3}=\lambda_{2}$ matrix
$\mathcal{M}^{{\dagger}}\mathcal{M}$ is reduced to $4\times 4$ and given by,
$\displaystyle{}\mathcal{M}^{{\dagger}}\mathcal{M}=\left(\begin{array}[]{cccc}\lambda_{2}^{2}\mid
X\mid^{2}+\lambda_{\mu}^{2}\mid
H_{\mu}\mid^{2}&\lambda_{2}\lambda_{\mu}X^{*}H_{\mu}+\lambda_{1}\lambda_{\mu}XH_{\mu}^{*}&0&\lambda_{2}m_{2}X^{*}\\\
\lambda_{2}\lambda_{d}XH^{*}_{d}+\lambda_{1}\lambda_{\mu}X^{*}H_{\mu}&\lambda_{1}^{2}\mid
X\mid^{2}+\lambda_{d}^{2}\mid
H_{d}\mid^{2}+m_{1}^{2}&\lambda_{1}m_{1}X&\lambda_{d}m_{2}H_{d}^{*}\\\
0&\lambda_{1}m_{1}X^{*}&\lambda_{1}^{2}\mid X\mid^{2}&0\\\
\lambda_{2}m_{2}X&\lambda_{d}m_{2}H_{d}&0&\lambda^{2}_{2}\mid
X\mid^{2}\end{array}\right)$ (28)
under basis
$\left(\Phi,\tilde{S},\tilde{T},\Sigma\right)\mathcal{M}\left(\tilde{\Phi},S,T,\tilde{\Sigma}\right)^{T}$.
Relevant mass terms can be read from (23) as,
$\displaystyle{}\mu$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\bar{\theta}^{2}}\mathcal{Z}_{\mu
d}\mid_{\theta=\bar{\theta}=0},$ $\displaystyle B\mu$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{2}}\frac{\partial}{\partial\theta^{2}}\mathcal{Z}_{\mu
d}\mid_{\theta=\bar{\theta}=0}$ $\displaystyle m^{2}_{H_{\mu}}$
$\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{2}}\frac{\partial}{\partial\theta^{2}}\log\mathcal{Z}_{\mu}\mid_{\theta=\bar{\theta}=0}$
(29) $\displaystyle m^{2}_{H_{d}}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{2}}\frac{\partial}{\partial\theta^{2}}\log\mathcal{Z}_{d}\mid_{\theta=\bar{\theta}=0}$
where $Z_{\mu d}$ and $Z_{\mu,d}$ are given by,
$\displaystyle{}Z_{\mu d}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial(H_{\mu}H_{d})}K_{eff}\mid_{H_{\mu}=H_{d}=0},$
$\displaystyle Z_{\mu,d}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial(H^{{\dagger}}_{\mu,d}H_{\mu,d})}K_{eff}\mid_{H_{\mu,d}=H^{{\dagger}}_{\mu,d}=0}$
(30)
Since these soft terms are generated through one hidden sector in our
framework, our model belongs to what is known as one-scale gauge medaition. As
discussed in [28], one roughly expects a relation as
$\displaystyle\mid B\mu\mid\sim m^{2}_{H_{\mu,d}}>>\mu^{2}$ (31)
which plagues these one-scale models and indicates the failure of EWSB.
However, more precise estimates needs to be done so as to verify this relation
given a specific model, and it is not impossible to avoid this relation in
some circumstances.
Here we point out some possibilities. One choice is that $m^{2}_{H_{\mu}}$ is
negative, with its absolute value smaller than positive $m^{2}_{H_{d}}$ but
larger than $\mu^{2}$. Another choice is that one allows a large
$m^{2}_{H_{d}}$ and small $m^{2}_{H_{\mu}}$, with a small hierarchy
$m^{2}_{H_{d}}>>B\mu$ so that it can balance the influence coming from the
small hierachy $B\mu>>\mu^{2}$ [31]. We refer [30] to the reads for more
discussions about this issue. As we will see the model we discuss here is a
new example in the first choice.
Since the matrix (28) is quite complicated so that the effective Kahler
potential can not be generally evaluated, we take the limit $m_{2}=m_{3}$ and
$\lambda_{2}=\lambda_{3}$ to simplify the simulation. Note that these choices
correspond to a favored parameter space, as seen in fig 2.$(b)$.
The leading contributions to $m^{2}_{H_{\mu,d}}$ are composed of two parts.
One arises from the ordinary gauge mediation. The other comes from the
superpotential (13). The later contribution induced at one-loop, generally
dominates over the former. By using the conditions of electroweak symmetry
breaking,
$\displaystyle{}(c.1)$ $\displaystyle:$
$\displaystyle~{}~{}(B\mu)^{2}>(\mid\mu\mid^{2}+m^{2}_{H_{\mu}})(\mid\mu\mid^{2}+m^{2}_{H_{d}})$
$\displaystyle(c.2)$ $\displaystyle:$
$\displaystyle~{}~{}2B\mu<2\mid\mu\mid^{2}+m_{H_{\mu}}^{2}+m_{H_{d}}^{2}$ (32)
For $\lambda_{2}=\lambda_{3}=1$ and fixed scale $X=0.1m_{2}$ , it turns out
that the allowed parameter space is given by 555Since couplings
$\lambda_{\mu}$ and $\lambda_{d}$ are overall coefficients in $\mu$ and $B\mu$
terms, we have taken $\lambda_{\mu}=\lambda_{d}=1$ for simplicity. Also note
that large deviation from $\lambda_{2}\sim\lambda_{3}\sim 1$ is not consistent
with the choice $X\sim 0.1$m, as shown in fig. 2. ,
$\displaystyle{}\lambda_{1}\sim 0.26,~{}~{}~{}~{}~{}m_{1}/m_{2}\sim 0.12$ (33)
which results in the following spectra in our model,
$\displaystyle{}m^{2}_{H_{\mu}}:\mu^{2}:B\mu:m^{2}_{H_{d}}\sim 1:2:500:10^{3}$
(34)
after we put values of (33) into (28) and (A Simple Model of Direct Gauge
Mediation). The spectra (34) suggests that our model is an example of large
$m_{H_{d}}$ and small $m_{H_{\mu}}$ mentioned above.
What about the RG effects on the spectra given by (28) when one runs from
$X_{0}$ to the electroweak scale ? Since there are no multiple messenger
threshold corrections in our model, the RG effects are quite simple. According
to the RG equations of MSSM given in [29], one observes that the $m_{H_{\mu}}$
receive its quantum corrections more substantially than $\mu$, $m_{H_{d}}$ and
$B_{\mu}$. If we restrict us to low-scale gauge mediation with $X_{0}\sim
10^{3}-10^{7}$TeV, the correction can be estimated through linear
approximation. For the spectra given by (34) , $\delta
m^{2}_{H_{\mu}}\sim-0.1\times m^{2}_{H_{d}}/16\pi^{2}\sim-\mu^{2}$. This
negative contribution implies that the first condition in (A Simple Model of
Direct Gauge Mediation) can be still satisifed, while the second condition
does not substantially modified . We refer the readers to the recent work [30]
on this subject through effective field theory analysis.
What about the other choices such as $m_{2}=m_{3}<<m_{1}$ and
$\lambda_{2}=\lambda_{3}<<\lambda_{1}$, or $m_{2}=m_{3}<<m_{1}$ and
$\lambda_{2}=\lambda_{3}>>\lambda_{1}$, or $m_{2}=m_{3}>>m_{1}$ and
$\lambda_{2}=\lambda_{3}<<\lambda_{1}$ ? We find that it is often impossible
to both satisfy the electroweak symmetry breaking conditions $(c.1)$ and
$(c.2)$ in these cases. What is worse is that the parameter space to generate
the one-loop gaugino masses is substantially suppressed under these limits, as
shown in fig. 2$(b)$.
In summary, we propose a simple Wess-Zumino model, which can serve as viable
SUSY model of direct gauge mediation. In this scenario, all messengers involve
in supersymmetry breaking. The $R$-symmetry is also spontaneously broken as a
result of the specific choices of $R$-charges. Phenomenologically, We find the
gaugino mass is induced at one-loop, with the same order of the scalar masses.
Also, there is no $\mu$ problem associated with soft masses in Higgs sector,
which can be naturally solved in our model, with no need of fine tunings among
Yukawa couplings in the SUSY breaking hidden sector. Since most of
supersymmetric particles $\sim 1$TeV, this makes part of direct gauge mediated
SUSY models testable at LHC.
As far as we know in the literature, our model is the first example that
address all these important issues. We do not discuss the possible searches of
relevant signals at the LHC. The main goal in this paper is to provide an
existence proof in SUSY model building in the context of direct gauge
mediation.
$\bf{Acknowledgement}$
We thank Jia-Hui Huang for discussions and Jin Min Yang for reading the
manuscript. This work is supported in part by the Fundamental Research Funds
for the Central Universities with project number CDJRC10300002.
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|
arxiv-papers
| 2011-08-12T01:02:40 |
2024-09-04T02:49:21.497672
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng and Yao Yu",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1108.2544"
}
|
1108.2585
|
# Malthusian assumptions, Boserupian response in models of the transitions to
agriculture 111To appear in: “Society, Nature and History: The Legacy of Ester
Boserup”, Springer, Vienna
Carsten Lemmen
(Institut für Küstenforschung, Helmholtz-Zentrum Geesthacht, Max-Planck Straße
1, 21501 Geesthacht, Germany (carsten.lemmen@hzg.de) )
### Abstract.
In the many transitions from foraging to agropastoralism it is debated whether
the primary drivers are innovations in technology or increases of population.
The driver discussion traditionally separates Malthusian (technology driven)
from Boserupian (population driven) theories. I present a numerical model of
the transitions to agriculture and discuss this model in the light of the
population versus technology debate and in Boserup’s analytical framework in
development theory. Although my model is based on
ecological—Neomalthusian—principles, the coevolutionary positive feedback
relationship between technology and population results in a seemingly
Boserupian response: innovation is greatest when population pressure is
highest. This outcome is not only visible in the theory-driven reduced model,
but is also present in a corresponding “real world” simulator which was tested
against archaeological data, demonstrating the relevance and validity of the
coevolutionary model. The lesson to be learned is that not all that acts
Boserupian needs Boserup at its core.
## 1 Transitions to agriculture
The relationship between humans and their environment underwent a radical
change during the last 10,000 years: from mobile and small groups of foraging
people to sedentary extensive cultivators and on to high-density intensive
agriculture modern society; these transitions fundamentally turned the
formerly predominantly passive human user of the environment into an active
component of the Earth system. The most striking global impact is only visible
and measurable during the last 150 years (Crutzen and Stoermer, 2000; Crutzen,
2002); much earlier, however, the use of forest resources for metal smelting
from early Roman times and the medieval extensive agricultural system had
already changed the landscape (Barker, 2011; Kaplan et al., 2009); global
climate effects of these early extensive cultivation and harvesting practices
are yet under debate (Ruddiman, 2003; Lemmen, 2010; Kaplan et al., 2011;
Stocker et al., 2011).
Transitions to agriculture occurred in almost every region of the world,
earliest in China and the Near East over 9000 years ago (Kuijt and Goring-
morris, 2002; Londo et al., 2006), and latest in Australia and Oceania with
the arrival of Polynesian and European immigrants few hundred years ago
(Diamond and Bellwood, 2003). While each local transition can be considered
revolutionary, the many diverse mechanisms, environments, and cultural
contexts of each agricultural transition make it difficult to speak of the one
’Neolithic revolution’, as the transition to farming and herding was termed by
V. G. Childe almost a century ago (Childe, 1925). The transitions from
foraging to farming were not only one big step, but may have consisted of
intermediary stages: Bogaard (2005) looks at the transition in terms of the
land use system: she sees first inadvertent cultivation then horticulture then
simple and then advanced agriculture, while Boserup (1965) discriminates these
stages by the management practice ranging from forest, bush and short fallow
to annual and multi cropping.
In contemporary hunting-gathering societies much less time has to be devoted
to procuring food from hunting and gathering opposed to agriculture and
herding (e.g., Sahlins, 1972); less labor is required for long fallow systems
compared to intensive multi-cropping agriculture(Boserup, 1965). So why farm?
Different responses from archaeology (Barker, 2011), demography (Turchin and
Nefedov, 2009), historical economy (Weisdorf, 2005), and ecosystem modeling
(Wirtz and Lemmen, 2003) call upon processes such as social reorganization,
the value of leisure, changing resources, or coevolutionary thresholds.
The probably simplest relationship was proposed by Malthus (1798, p.11),
namely that more production sustains larger population. With larger
population, more production is possible, thereby constituting a positive
feedback loop, which ideally results in ever greater (geometric) growth and
productivity. That this is not the case in a world with finite resources was
expressed by Malthus (1798, p. 4) by stating that “Population, when unchecked,
increases at a geometrical ratio. Subsistence increases only in an
arithmetical ratio. A slight acquaintance with numbers will show the immensity
of the first power in comparison with the second”. Malthus identified the need
for positive and preventive checks to balance population increase with the
limited capacity of resources.
How does an increase in productivity come about? First and foremost, the input
of more labor increases productivity (Malthus, 1798, p. 11), subject to the
constraints of finite resources and diminishing returns. Where Malthus,
however, focused on extensive productivity increase, the intensification
component of productivity increase was highlighted by Boserup (1965).
Investments in a more intensive production system would, however, require
large additional labor, and the benefits of such investments were often small.
To stimulate an investment in more intensive agriculture, Boserup requires
population pressure.
Both Malthus (1798, 1826) and Boserup (1965, 1981) concentrate on the role of
labor (and later division of labor and social/family organization) and
innovations which increase area productivity (like storage or tools, requiring
relatively more labor for harvesting, building, and tool processing). Both
authors neglect the role of labor-independent innovation, or innovations which
increase both area and labor productivity; these are innovations in the
resources themselves, such as cultivation of higher-yielding grains or
imported high yield varieties, or their management such as water rights; this
distinction may not be unambiguous for all innovations, it is used here
conceptually. Labor-independent innovation can be stimulated by diversity and
density of a population, both of which are positively related to population
size. Already Darwin (1859, p. 156) wrote “The more diversified [..], by so
much will they be better enabled to seize on many and widely diversified
places in the polity of nature”. Translated into the realm of innovativity,
Darwin’s “seizing of places”, or niche occupation, would be the realization of
technical and scientific opportunities. As for density as a stimulus of
innovation, it is aggregation which constitutes a motor of technological and
cultural change (Smith, 1776; Boyd and Richerson, 1995)222This does not,
however, give the reason for a particular choice of one innovation over
another (Sober, 1992)..
## 2 Models of population, production, and innovation
In 1996, E. Boserup reflected on the problems arising from the differences in
terminology and methodology when comparing different models of development
theories (Boserup, 1996). She suggested a common framework to facilitate
interdisciplinary cooperation based on six structures: Environment ($E$),
Population ($P$), technology, occupational structure, family structure and
culture. In this framework, she then interpreted the major works of Adam
Smith, Thomas Malthus, Max Weber, Karl Marx, David Ricardo, and Neomalthusian
thinking, as well as her own view on different stages of the developmental
process.
For many of the theories and models discussed by Boserup in this framework,
the partitioning in six structures can be simplified by (a) aggregating
technology and occupational structure into a single entity technology ($T$),
and by (b) aggregating culture and family structure into a single entity
culture ($C$). Aggregating technology and occupational structure means that I
assume here that changes in technology are equivalent to changes in
organization and that the location of technological change is the occupational
sector. By aggregating family structure and culture I assume that values and
social conventions penetrate from the society into the family and are governed
by similar dynamics. The reduced framework then consists of the four
compartments population, environment, technology, and culture (PETC, Figure
1).
In this PETC framework, the one referring to Malthus (1798) involves only
population and environment. Population growth exerts pressure on the
environment, and failure to provide adequate resources from the environment
acts as a positive check on population through higher mortality (Figure 1a).
Technology does not play a role in this simplest Malthusian model333Malthus
considered the increase of carrying capacity by autonomously occurring
inventions (Lee, 1986), however, this was not discussed by Boserup (1996) in
her model intercomparison.. Culture in the form of preventive checks—such as
birth control—acts on population only in later versions of his theory
(Malthus, 1826). At its core remains “the dependent role he assigns to
population growth” (Marquette, 1997). D. Ricardo (1821) proposed that the
incentive to intensify and develop technologies comes from a stimulus in
population pressure. The demand for more land ($E$), however, leads to
declining marginal benefits of and a negative feedback on innovation ($T$) due
to high costs of renting the land (Figure 1b). In Ricardo’s work, population
is independent, and technology and environment are the dependent variables.
Population is also the driving factor in Boserup’s (1965; 1981) works. Of the
six transitions considered by Boserup (1996), five can be accommodated within
my PETC framework as a succession of population, environment, technology, and
culture: foraging to crop production, village development, Eastern hemisphere
pastoralism, urbanization, and industrialization (Figure 1c)444The sixth
transition—Western European fertility decline—follows a different path as a
succession of technology, environment, culture, and last population; it is not
considered here.. In all these transitions, population growth leads to
pressure felt from the limited environmental resources, which in turn
stimulates technological and organizational change, and later results in
cultural changes evident in cults, social hierarchies, women’s status, and
status symbols. Within this group of five transitions, her model of village
development, in addition, has a direct population–technology link, and allows
for a feedback of the land resources on occupational structure (dotted lines
in Figure 1c). Furthermore, her model of the foraging to farming transition
includes a feedback from culture to organizational structure (not shown).
Figure 1: Four compartment framework for the interrelationship between
population, environment, technology, and culture. Four economic theories are
contrasted: the essays on the principles of population by T. Malthus (1798,
1826) (panel a, dotted line indicates the revised essay including culture
change); D. Ricardo (1821)’s principles of economy (panel b); E. Boserup
(1965, 1981)’s theories for five transitions explained in Boserup (1996)
(panel c, 1981 refinements shown as dotted lines); and the ecological model
proposed in this chapter. The framework is a simplification of the six
compartment framework originally proposed by Boserup (1996).
## 3 A combined model and ‘real’ world application
I suggest here a different model of population development taking the foraging
to farming transition as an example (Figure 1d). This model is a reduced form
of the Global Land Use and technological Evolution Simulator (GLUES, described
below), which has been operationally applied to a number of problems in
archaeology and climate research (Kaplan et al., 2011; Lemmen, 2010; Lemmen
and Wirtz, 2012; Lemmen et al., 2011). The reduced model shares the functional
characteristics of the full model, but it is not spatially explicit and the
biogeographic and climate background is regarded as constant (see Appendix for
equations).
In terms of the PETC framework, the dynamics between population, environment,
technology and culture is the following (Figure 1d, cmp. Boserup 1996, p.
509).
1. 1.
P$\rightarrow$ T$\rightarrow$ P Population growth stimulates innovation by
aggregation and diversity. Innovations in, e.g., health care increase
population;
2. 2.
P$\rightarrow$ E $\rightarrow$ P Population growth uses ever more land for
hunting and exerts pressure on the game stock, higher population densities
damage the environment, and food shortage leads to reduced fertility
(preventive check) or higher mortality (positive check). The rising capacity
of the environment supports higher population;
3. 3.
T$\rightarrow$ E More intensive foraging or farming strategies damage the
environment, while efficiency gains lead to higher capacity of the
environment;
4. 4.
T$\rightarrow$ C Adoption of novel technologies induces changes in social
structure where specialists and leaders or cults emerge;
5. 5.
C$\rightarrow$ P Family and social structure change reproduction rates.
Richerson and Boyd (1998) claim that basically all models which are rooted in
ecology are Neomalthusian in essence, i.e., they can be characterized by a
P$\rightarrow$ T $\rightarrow$ E loop in Boserup’s (1996) framework. This loop
can be detected in my model, as well; in fact, historically it developed from
ecosystem models of tree stands or algal communities(Wirtz and Eckhardt,
1996). Unlike many other models, however, GLUES is based on coevolutionary
dynamics of technologies and population, and as such has no a priori
information on whether there is a (Malthusian) “invention-pull view of
population history” (Lee, 1986, p. 98), or whether population is the
(Boserupian) driver of development555See also Simon (1993) for a detailed
discussion.. Applications of GLUES show that there is an emergent emancipation
of population development from the environment with increasing population and
innovation (Lemmen and Wirtz, 2010, 2012; Lemmen et al., 2011).
GLUES mathematically resolves the dynamics of population density and three
population-averaged characteristic sociocultural traits: technology $T_{A}$,
share of agropastoral activities $C$, and economic diversity $T_{B}$. These
are defined for preindustrial societies as follows:
1. 1.
Technology $T_{A}$ is a trait which describes the efficiency of food
procurement—related to both foraging and farming—and improvements in health
care. In particular, technology as a model describes the availability of
tools, weapons, and transport or storage facilities. It aggregates over
various relevant characteristics of early societies and also represents social
aspects related to work organization and knowledge management. It quantifies
improved efficiency of subsistence, which is often connected to social and
technological modifications that run in parallel. An example is the technical
and societal skill of writing as a means for cultural storage and
administration, with the latter acting as a organizational lubricant for food
procurement and its optimal allocation in space and among social groups.
$T_{A}$ is labour dependent.
2. 2.
A second model variable $C$ represents the share of farming and herding
activities, encompassing both animal husbandry and plant cultivation. It
describes the allocation of energy, time, or manpower to agropastoralism with
respect to the total food sector.
3. 3.
Economic diversity $T_{B}$ resolves the number of different agropastoral
economies available to a regional population. This trait is in the full model
closely tied to regional vegetation resources and climate constraints; in this
reduced model, it denotes a labour-independent technology. A larger economic
diversity offering different niches for agricultural or pastoral practices
enhances the reliability of subsistence and the efficacy in exploiting
heterogeneous landscapes.
The temporal change of each of these characteristic traits follows the
direction of increased benefit for success (i.e. growth) of its associated
population (Appendix equation 2); this concept had been derived for genetic
traits in the works of Fisher (1930), and was recently more stringently
formulated by Metz and colleagues (Metz et al., 1992; Kisdi and Geritz, 2010)
as adaptive dynamics (AD). In AD, the population averaged value of a trait
changes at a rate which is proportional to the gradient of the fitness
function evaluated at the mean trait value. The AD approach was extended to
functional traits of ecological communities (Wirtz and Eckhardt, 1996; Merico
et al., 2009), and was first applied to cultural traits of human communities
by Wirtz and Lemmen (2003).
The adaptive coevolution of the food production system $\\{T_{A},T_{B},C\\}$
and population $P$ (Appendix equations 1–4), which is at the heart of this
model’s implementation, had also been found empirically by Boserup (1981, p.
15): “The close relationship which exists today between population density and
food production system is the result of two long-existing processes of
adaptation. On the one hand, population density has adapted to the natural
conditions for food production []; on the other hand, food supply systems have
adapted to changes in population density.”
## 4 Innovation in the transition to agriculture
Figure 2: Trajectories of population $P$, environment $E$, and technologies
$T_{A},T_{B}$ (panel a) and phase diagram of innovation rate versus population
pressure (panel b) from a simulation with a simplified version of the Global
Land Use and technological Evolution Simulator. The trajectories describe the
temporal evolution of population density, capacity denoted as environment, a
labour dependent technology $T_{A}$, and a labour-independent technology
$T_{B}$. Numbers identify the different stages of development in the both
diagrams. In the phase diagram b), the innovation rate, derived as the
cumulative change in $T_{A}+T_{B}$, is shown in relation to population
pressure, calculated as $1-E+P$.
The outcome of the coevolutionary model simulation with the reduced GLUES is
shown in Figure 2. I divided both the trajectories (temporal evolution of
state variables, panel a) and the the phase space (panel b) into six stages,
which I discuss below.
1. 1.
Growth phase: Starting from a Malthusian perspective, and looking only at
population and environment (quantified here as the ecosystem capacity, i.e.
the ratio of birth over mortality terms in the growth rate equation 3),
population grows towards its capacity with diminishing returns as $P$
approaches $E$; this first phase spans only a short period of time but covers
a large area in phase space;
2. 2.
Persistent innovation in technology $T_{A}$ and associated investments in tool
making and administration allow sustained slow growth of population $P$ and
alleviates the built-up population pressure; in contrast to the growth phase,
the phase space coverage is very small while the temporal extent of this phase
is large;
3. 3.
Transition phase: rapid innovation in a labour-independent technology $T_{B}$
(e.g. domestication successes) leads to
4. 4.
Pressure relief, but induces also a change in culture (not shown);
5. 5.
Equilibration: Innovation slows but has led to a wider gap between $P$ and $E$
because of the investments made in manufacturing and organization during the
transition: accordingly, population pressure increases more slowly and up to a
lower value than in the growth phase (1.).
6. 6.
Persistent innovation: corresponds to phase (2.) and is again characterized by
persistent innovation in technology $T_{A}$ and a slow population pressure
relief.
What can be learned about the relationship between population pressure and
innovativity from Figure 2? (i) Innovation is greatest at high population
pressure. (ii) In this model there is always innovation, at no time is
technology change negative. (iii) The relationship between innovativity and
population pressure changes profoundly during the foraging-farming transition;
three different regimes can be identified: (i) a positive relationship where
acceleration of innovation corresponds to population pressure increases
(phases 1., 2., 6.), (ii) a negative relationship with pressure relief during
accelerating innovation (phase 3., 4.), and (iii) a negative relationship with
deceleration of innovativity at increasing pressure (phase 5.).
A superficial analysis would find that population pressure is the motor of
innovation in this example: population increase seemingly precedes the
stepwise technological change (Figure 2a). Only a detailed look at the phase
space (Figure 2b)—especially at the transition phases 2. and 3.—shows that
innovativity decelerates at very high population pressure and that the largest
innovation occurs slightly below the highest population pressure. In fact, the
driver in the transition depicted here is not population, but
technology666There would be no evolution of $T$ without $P$ due to the
coevolutionary definition of the system. The dynamics of $T$, however, leads
the dynamics of $P$ at the foraging farming transition.. Only the different
coevolutionary time scales of population growth (fast) and innovation (slow)
yield the seemingly Boserupian, i.e., population driven, response.
The same mathematical model—plus spatial and biogeographic aspects—has been
used to successfully simulate the many transitions to agriculture in Neolithic
Europe (Lemmen et al., 2011), with good agreement with the radiocarbon record.
Also there, the transitions appear Boserupian with critical innovations
occurring at high population pressure. If the numerical analysis had not been
available (and proved that this is in fact technology driven), such as it is
in the discretely sampled data from observations of technological change, one
would have to have come to the erroneous conclusion that this type of
innovation was population driven.
## 5 Conclusion
I presented a reduced version of the Global Land Use and technological
Evolution Simulator, a numerical model which is capable of realistically
simulating regional foraging-farming transitions worldwide. The simulated—and
possibly also observed—transitions are seemingly Boserupian, i.e., population
driven: innovation is greatest when population pressure is high. Analytical
examination of the model, however, shows that technological change is the
driver, and that in the context of a simplified version of Boserup’s (1996)
framework in development theory the model should be classified as
Neomalthusian. I thus demonstrated that Boserupian appearance may be based on
Malthusian assumptions; I caution not to infer too quickly a Boserupian
mechanism for an observed real world system when its dynamics appears to be
population pressure driven.
## Appendix: the reduced GLUES model
A coevolutionary system of population $P$ and characteristic traits
$X\in\\{T_{A},T_{B},C\\}$ is defined by the evolution equations
$\displaystyle\frac{\mathrm{d}P}{\mathrm{d}t}$ $\displaystyle=$ $\displaystyle
P\cdot r$ (1) $\displaystyle\frac{\mathrm{d}X}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle\delta_{X}\cdot\frac{\partial{}r}{\partial{}X},$ (2)
where $r$ denotes the specific growth rate of population $P$, and the
$\delta_{X}$ are variability measures for each $X$. Growth rate $r$ is defined
as
$r=\mu\cdot(1-\omega
T_{A})\cdot(1-\gamma\sqrt{T_{A}}P)\cdot\mathrm{SI}-\rho\cdot T^{-1}_{A}\cdot
P,$ (3)
with coefficients $\mu,\rho,\omega,\gamma$. In this formulation, the positive
term including food production SI is modulated by labour loss for
administration $(-\omega T_{A})$ and by overexploitation of the environment
$(-\gamma\sqrt{T_{A}}P)$. Food production depends on the cultural system $C$
and available technologies as follows:
$SI=(1-C)\cdot\sqrt{(}T_{A})+C\cdot T_{A}\cdot T_{B},$ (4)
where the left summand denotes foraging activities and the right summand
agropastoral practice.
To produce the results for Figure 2, I assumed the following parameter values:
$\mu=\rho=0.004$, $\omega=0.04$, $\gamma=0.12$, $\delta_{T_{A}}=0.025$,
$\delta_{T_{B}}=0.9$; a variable $\delta_{C}=C\cdot(1-C)$; and initial values
for $P_{0}=0.01$, $T_{A,0}=1.0$, $T_{B,0}=0.8$, and $C_{0}=0.04$.
### Acknowledgments.
This study was partly funded by the German National Science Foundation (DFG
priority project 1266 Interdynamik) and by the PACES program of the Helmholtz
Gemeinschaft. The paper received great stimulus from discussions during the
Ester Boserup Conference 2010—A Centennial Tribute: Long-term trajectories in
population, gender relations, land use, and the environment, November 15–17,
2010 in Vienna, Austria. I received helpful comments from two anonymous
reviewers. GLUES is free and open source software and can be obtained from
http://glues.sourceforge.net/.
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* Wirtz and Lemmen (2003) Wirtz, K. W., and C. Lemmen (2003), A global dynamic model for the neolithic transition, Climatic Change, (1998), 333–367.
|
arxiv-papers
| 2011-08-12T08:21:06 |
2024-09-04T02:49:21.503478
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carsten Lemmen",
"submitter": "Carsten Lemmen",
"url": "https://arxiv.org/abs/1108.2585"
}
|
1108.2598
|
# Traces on symmetrically normed operator ideals
F. Sukochev School of Mathematics and Statistics, University of New South
Wales, Sydney, 2052, Australia. f.sukochev@unsw.edu.au and D. Zanin School
of Mathematics and Statistics, University of New South Wales, Sydney, 2052,
Australia. d.zanin@unsw.edu.au
###### Abstract.
For every symmetrically normed ideal $\mathcal{E}$ of compact operators, we
give a criterion for the existence of a continuous singular trace on
$\mathcal{E}$. We also give a criterion for the existence of a continuous
singular trace on $\mathcal{E}$ which respects Hardy-Littlewood majorization.
We prove that the class of all continuous singular traces on $\mathcal{E}$ is
strictly wider than the class of continuous singular traces which respect
Hardy-Littlewood majorization. We establish a canonical bijection between the
set of all traces on $\mathcal{E}$ and the set of all symmetric functionals on
the corresponding sequence ideal. Similar results are also proved in the
setting of semifinite von Neumann algebras.
###### Key words and phrases:
Symmetric functionals, singular traces
###### 2000 Mathematics Subject Classification:
47L20, 47B10, 46L52
## 1\. Introduction
In his groundbreaking paper [6], J. Dixmier proved the existence of positive
singular traces (that is, linear positive unitarily invariant functionals
which vanish on all finite dimensional operators) on the algebra $B(H)$ of all
bounded linear operators acting on infinite-dimensional separable Hilbert
space $H.$ Namely, if $\psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a concave
increasing function such that
(1) $\lim_{t\to\infty}\frac{\psi(2t)}{\psi(t)}=1,$
then there is a singular trace $\tau_{\omega},$ defined for every positive
compact operator $A\in B(H)$ by setting
(2) $\tau_{\omega}(A)=\omega(\frac{1}{\psi(n)}\sum_{k=1}^{n}s_{k}(A)).$
Here, $\\{s_{k}(A)\\}_{k\in\mathbb{N}}$ is the sequence of singular values of
the compact operator $A\in B(H)$ taken in the descending order and $\omega$ is
an arbitrary dilation invariant generalised limit on the algebra $l_{\infty}$
of all bounded sequences. This trace is finite on $0\leq A\in B(H)$ if and
only if $A$ belongs to the Marcinkiewicz ideal (see e.g. [14],[15],[27])
$\mathcal{M}_{\psi}:=\\{A\in B(H):\
\sup_{n\in\mathbb{N}}\frac{1}{\psi(n)}\sum_{k=1}^{n}s_{k}(A)<\infty\\}.$
In [18], Dixmier’s result was extended to an arbitrary Marcinkiewicz ideal
$\mathcal{M}_{\psi}$ with the following condition on $\psi$
(3) $\liminf_{t\to\infty}\frac{\psi(2t)}{\psi(t)}=1.$
All the traces defined above by formula (2) vanish on the ideal
$\mathcal{L}_{1}$ consisting of all compact operators $A\in B(H)$ such that
$\sum_{k=1}^{\infty}s_{k}(A)<\infty$.
An ideal $\mathcal{E}$ of algebra $B(H)$ is said to be symmetrically normed if
$\\{s_{k}(B)\\}_{k\in\mathbb{N}}\leq\\{s_{k}(A)\\}_{k\in\mathbb{N}}$ and
$A\in\mathcal{E}$ implies that $\|B\|_{\mathcal{E}}\leq\|A\|_{\mathcal{E}}$
(see [14], [15], [29]111 We have to caution the reader that in Theorem 1.16 of
[29] the assertion $(b)$ does not hold for the norm of an arbitrary
symmetrically normed ideal $\mathcal{E}$ (see e.g. corresponding
counterexamples in [19, p. 83])., [28], [20]). Since the ideal
$\mathcal{M}_{\psi}$ is just a special example of symmetrically normed
operator ideal, the following question (suggested in [18], [16], [17], [7])
arises naturally.
###### Question 1.
Which symmetrically normed operator ideals admit a nontrivial singular
trace222 In this paper, we exclusively deal with positive traces?
In analyzing Dixmier’s proof of the linearity of $\tau_{\omega}$ given by (1),
it was observed in [18] (see also [3]) that $\tau_{\omega}$ possesses the
following fundamental property, namely if $0\leq A,B\in\mathcal{M}_{\psi}$ are
such that
(4) $\sum_{k=1}^{n}s_{k}(B)\leq\sum_{k=1}^{n}s_{k}(A),\quad\forall
n\in\mathbb{N},$
then $\tau_{\omega}(B)\leq\tau_{\omega}(A).$ Such a class of traces was termed
“fully symmetric”in [20], [30] (see also earlier papers [8],[25], where the
term “symmetric”was used). It is natural to consider such traces only on fully
symmetrically normed operator ideals $\mathcal{E}$ (that is, on symmetrically
normed operator ideals $\mathcal{E}$ satisfying the condition: if $A,B$
satisfy (4) and $A\in\mathcal{E},$ then $B\in\mathcal{E}$ and
$\|B\|_{\mathcal{E}}\leq\|A\|_{\mathcal{E}}$). In fact, it was established in
[8] that every Marcinkiewicz ideal $\mathcal{M}_{\psi}$ with $\psi$ satisfying
the condition (3) possesses fully symmetric traces. Furthermore, in the recent
paper [18], the following unexpected result was established. If $\psi$
satisfies the condition (3), then every fully symmetric trace on
$\mathcal{M}_{\psi}$ is a Dixmier trace $\tau_{\omega}$ for some $\omega.$
The following question ( also suggested in [18], [7], [16], [17]) arises
naturally.
###### Question 2.
Which fully symmetrically normed operator ideals admit a nontrivial singular
trace which is fully symmetric?
In papers [16],[17] the following two problems (closely related to Question 1
and Question 2) were also suggested.
###### Question 3.
Which fully symmetrically normed operator ideals admit a trace which is not
fully symmetric?
Let us fix an orthonormal basis $\\{e_{n}\\}_{n\in\mathbb{N}}$ in $H.$ An
operator $A\in B(H)$ is called diagonal if $(Ae_{n},e_{m})=0$ for every $n\neq
m.$
###### Question 4.
Let the mapping $\varphi:\mathcal{E}\to\mathbb{C}$ be unitarily invariant.
Suppose that $\varphi$ is linear on the subset of all diagonal operators from
$\mathcal{E}.$ Does it imply that $\varphi$ is a trace on $\mathcal{E}?$
In some very special cases (for principal ideals contained in
$\mathcal{L}_{1},$ which are, strictly speaking, not symmetrically normed
ideals), Question 3 was answered in the affirmative333We are grateful to the
referee for this remark. in [33]. In [20], question 3 was answered in the
affirmative for the special case of Marcinkiewicz ideals under the assumption
(1). It should be pointed out that the method used in [20] cannot be extended
to an arbitrary Marcinkiewicz ideal $\mathcal{M}_{\psi}$ and, furthermore,
cannot be extended to a general symmetrically normed operator ideal. Question
4 was answered in [20] in full generality using deep results from [11, 10]
(see also [9]).
The following theorem is the main result of this paper. It yields answers to
Questions 1–3. In the course of the proof of Theorem 5, we also present a new
(and very simple) proof answering Question 4. Prior to stating Theorem 5, we
make a few preliminary observations, for which we are grateful to the referee.
Any trace $\varphi:\mathcal{E}\to\mathbb{C}$ obeys the condition
$\frac{1}{m}\varphi(A^{\oplus m})=\varphi(A),\quad A\in\mathcal{E},m\geq 1.$
Here, the direct sum $A^{\oplus m}$ is formed with respect to some arbitrary
Hilbert space isomorphism $H^{\oplus m}\simeq H.$ Thus, traces are closely
related to the following convex (see Lemma 11 below) functional on
$\mathcal{E}.$
$\pi:A\to\lim_{m\to\infty}\frac{1}{m}\|A^{\oplus m}\|_{\mathcal{E}},\quad
A\in\mathcal{E}.$
The non-triviality of the functional $\pi:\mathcal{E}\to\mathbb{R}$ is an
obvious necessary condition for the existence of a trace.
###### Theorem 5.
Let $\mathcal{E}$ be a symmetrically normed operator ideal. Consider the
following conditions.
1. (1)
There exist nontrivial singular traces on $\mathcal{E}.$
2. (2)
There exist nontrivial singular traces on $\mathcal{E},$ which are fully
symmetric.
3. (3)
There exist nontrivial singular traces on $\mathcal{E},$ which are not fully
symmetric.
4. (4)
$\mathcal{E}\neq\mathcal{L}_{1}$ and there exist an operator $A\in\mathcal{E}$
such that
(5) $\lim_{m\to\infty}\frac{1}{m}\|A^{\oplus m}\|_{\mathcal{E}}>0.$
1. (i)
The conditions $(1)$ and $(4)$ are equivalent for every symmetrically normed
operator ideal $\mathcal{E}.$
2. (ii)
The conditions $(1),$ $(2)$ and $(4)$ are equivalent for every fully
symmetrically normed operator ideal $\mathcal{E}.$
3. (iii)
The conditions $(1)-(4)$ are equivalent for every fully symmetrically normed
operator ideal $\mathcal{E}$ equipped with a Fatou norm.
Recall that the norm on a symmetrically normed operator ideal $\mathcal{E}$ is
called a Fatou norm if the unit ball of $\mathcal{E}$ is closed with respect
to strong (or, equivalently, weak) operator convergence. Observe that
classical ideals (such as Schatten-von Neumann ideals $\mathcal{L}_{p},$
Marcinkiewicz, Orlicz and Lorentz ideals [14], [15], [29]) have a Fatou norm.
In fact, in some standard references on the subject (e.g. Simon’s book [29]),
the requirement that symmetrically normed operator ideal has a Fatou norm
appears to be a part of the definition. Similarly, in the book [24], devoted
to the study of symmetric444termed there “rearrangement invariant”. function
spaces (which are a commutative counterpart of symmetrically normed operator
ideals), an assumption that the norm is a Fatou norm is incorporated into the
definition [24, p. 118].
The proof of Theorem 5 is given in Section 7. In fact, in this paper we will
prove a more general result for symmetric spaces associated with semifinite
von Neumann algebras. The precise statements are given in Section 4 (see
Theorems 23, 28, 29), Section 5 (see Theorems 33, 35, 36) and Section 6 (see
Theorems 47, 48). The appendix contains the proof of important technical
results for which we were unable to find a suitable reference. We also present
a new and short proof of the Figiel-Kalton theorem from [13].
Finally, we say a few words about our proof and its relation to the previous
results in the literature. Our strategy is based on the approach from recent
papers [30] and [21], where condition (5) was connected to the geometry of
$\mathcal{E}$ (see also [2]). The condition (5) is easy to verify in concrete
situations. For example, the following corollary of Theorem 5 strengthens the
main result of [20] and complements earlier results of J. Varga [32].
###### Corollary 6.
Every Marcinkiewicz ideal $\mathcal{M}_{\psi}$ with $\psi$ satisfying the
condition (3) admits a trace which is not fully symmetric.
Indeed, it is proved in [1, Proposition 2.3] that the condition $(4)$ of
Theorem 5 is equivalent to the condition (3) for the Marcinkiewicz ideal
$\mathcal{M}_{\psi}.$ Some examples of symmetrically normed operator ideals,
which are not Marcinkiewicz ideals, possessing symmetric traces were presented
in [7]. These results are also an immediate corollary of Theorem 5.
For completeness, we note that the assertion $(ii)$ in Theorem 5 holds for a
wider class of relatively fully symmetrically normed operator ideals. The
latter class is defined as follows: if $A,B\in\mathcal{E}$ are such that (4)
holds, then $\|B\|_{\mathcal{E}}\leq\|A\|_{\mathcal{E}}.$ It coincides with
the class of all symmetrically normed subspaces of a fully symmetric operator
ideal (see [19])
## 2\. Definitions and preliminaries
The theory of singular traces on symmetric operator ideals rests on some
classical analysis which we now review for completeness.
As usual, $L_{\infty}(0,\infty)$ is the set of all bounded Lebesgue measurable
functions on the semi-axis equipped with the uniform norm. Given a function
$x\in L_{\infty}(0,\infty),$ one defines its decreasing rearrangement
$t\to\mu(t,x)$ by the formula (see e.g. [22])
$\mu(t,x)=\inf\\{s\geq 0:\ m(\\{x>s\\})\leq t\\}.$
Let $H$ be a Hilbert space and let $B(H)$ be the algebra of all bounded
operators on $H$ equipped with the uniform norm.
Let $\mathcal{M}\subset B(H)$ be a semi-finite von Neumann algebra equipped
with a fixed faithful and normal semi-finite trace $\tau.$ $\mathcal{M}$ is
said to be atomic (see [31, Definition 5.9]) if every nonzero projection in
$\mathcal{M}$ contains a nonzero minimal projection. $\mathcal{M}$ is said to
be atomless if there is no minimal projections in $\mathcal{M}.$
For every $A\in\mathcal{M},$ the generalised singular value function
$t\to\mu(t,A)$ is defined by the formula (see e.g. [12])
$\mu(t,A)=\inf\\{\|Ap\|:\ \tau(1-p)\leq t\\}.$
If, in particular, $\mathcal{M}=B(H),$ then $\mu(A)$ is a step function and,
therefore, can be identified with the sequence of singular numbers of the
operators $A$ (the singular values are the eigenvalues of the operator
$|A|=(A^{*}A)^{1/2}$ arranged with multiplicity in decreasing order).
Equivalently, $\mu(A)$ can be defined in terms of the distribution function
$d_{A}$ of $A.$ That is, setting
$d_{A}(s)=\tau(E^{|A|}(s,\infty)),\quad s\geq 0,$
we obtain
$\mu(t,A)=\inf\\{s\geq 0:\ d_{A}(s)\leq t\\},\quad t>0.$
Here, $E^{|A|}$ denotes the spectral measure of the operator $|A|.$
Using the Jordan decomposition, every operator $A\in B(H)$ can be uniquely
written as
$A=(\Re A)_{+}-(\Re A)_{-}+i(\Im A)_{+}-i(\Im A)_{-}.$
Here, $\Re A:=1/2(A+A^{*})$ (respectively, $\Im A:=1/2i(A-A^{*})$) for any
operator $A\in B(H)$ and $B_{+}=BE^{B}(0,\infty)$ ( respectively,
$B_{−}=-BE^{B}(-\infty,0)$) for any self-adjoint operator $B\in B(H).$ Recall
that $\Re A,\Im A\in\mathcal{M}$ for every $A\in\mathcal{M}$ and
$B_{+},B_{-}\in\mathcal{M}$ for every self-adjoint $B\in\mathcal{M}.$
Further, we need to recall the important notion of Hardy–Littlewood
majorization. Let $A,B\in(L_{1}+L_{\infty})(\mathcal{M}).$ The operator $B$ is
said to be majorized by $A$ and written $B\prec\prec A$ if and only if
$\int_{0}^{t}\mu(s,B)ds\leq\int_{0}^{t}\mu(s,A)ds,\quad t\geq 0.$
We have (see [12])
$A+B\prec\prec\mu(A)+\mu(B)\prec\prec 2\sigma_{1/2}\mu(A+B)$
for every positive operators $A,B\in(L_{1}+L_{\infty})(\mathcal{M}).$
If $s>0,$ the dilation operator $\sigma_{s}$ is defined by setting
$(\sigma_{s}(x))(t)=x(\frac{t}{s}),\quad t>0$
in the case of the semi-axis. In the case of the interval $(0,1),$ the
operator $\sigma_{s}$ is defined by
$(\sigma_{s}x)(t)=\begin{cases}x(t/s),&t\leq\min\\{1,s\\}\\\ 0,&s<t\leq
1.\end{cases}$
Similarly, in the sequence case, we define an operator $\sigma_{n}$ by setting
$\sigma_{n}(a_{1},a_{2},\cdots)=(\underbrace{a_{1},\cdots,a_{1}}_{\mbox{$n$
times}},\underbrace{a_{2},\cdots,a_{2}}_{\mbox{$n$ times}},\cdots)$
and an operator $\sigma_{1/2}$ by setting
$\sigma_{1/2}:(a_{1},a_{2},a_{3},a_{4},\cdots)\to(\frac{a_{1}+a_{2}}{2},\frac{a_{3}+a_{4}}{2},\cdots).$
###### Definition 7.
The Banach space $E(\mathcal{M},\tau)\subset(L_{1}+L_{\infty})(\mathcal{M})$
is said to be a symmetric operator space if the following conditions hold.
1. (1)
Given $A\in E(\mathcal{M},\tau)$ and $B\in(L_{1}+L_{\infty})(\mathcal{M})$
with $\mu(B)=\mu(A),$ we have $B\in E(\mathcal{M},\tau)$ and
$\|B\|_{E}=\|A\|_{E}.$
2. (2)
Given $0\leq A\in E(\mathcal{M},\tau)$ and $0\leq
B\in(L_{1}+L_{\infty})(\mathcal{M})$ with $B\leq A,$ we have $B\in
E(\mathcal{M},\tau)$ and $\|B\|_{E}\leq\|A\|_{E}.$
The space $E(\mathcal{M},\tau)$ is called fully symmetric if for every $A\in
E(\mathcal{M},\tau)$ and every $B\in(L_{1}+L_{\infty})(\mathcal{M})$ with
$B\prec\prec A,$ we have $B\in E(\mathcal{M},\tau)$ and
$\|B\|_{E}\leq\|A\|_{E}.$
The norm on a symmetric space $E(\mathcal{M},\tau)$ is a Fatou norm if the
unit ball of $E(\mathcal{M},\tau)$ is closed with respect to strong (or,
equivalently, weak) operator convergence. Every symmetric space equipped with
a Fatou norm is necessarily fully symmetric.
A linear functional $\varphi:E(\mathcal{M},\tau)\to\mathbb{C}$ is said to be
symmetric if $\varphi(B)=\varphi(A)$ for every positive $A,B\in
E(\mathcal{M},\tau)$ such that $\mu(B)=\mu(A).$ A linear functional
$\varphi:E(\mathcal{M},\tau)\to\mathbb{C}$ is said to be fully symmetric if
$\varphi(B)\leq\varphi(A)$ for every positive $A,B\in E(\mathcal{M},\tau)$
such that $B\prec\prec A.$ Every fully symmetric functional is symmetric and
bounded. The converse fails [20].
A functional $\varphi:E(\mathcal{M},\tau)\to\mathbb{C}$ is called singular if
$\varphi=0$ on $(L_{1}\cap L_{\infty})(\mathcal{M}).$ If
$E(\mathcal{M},\tau)\not\subset L_{1}(\mathcal{M}),$ then every symmetric
functional is singular.
If $E=E(0,\infty)$ and if $\varphi:E\to\mathbb{R}$ is a symmetric functional,
then $s\varphi(x)=\varphi(\sigma_{s}x)$ for every $x\in E.$ If $E=E(0,1)$ and
if $\varphi:E\to\mathbb{R}$ is a singular symmetric functional, then
$s\varphi(x)=\varphi(\sigma_{s}x)$ for every $x=\mu(x)\in E.$
Let $E$ be a fully symmetric Banach space either on the interval $(0,1)$ or on
the semi-axis. We need the notion of an expectation operator (see [2]).
Let $\mathcal{A}=\\{A_{k}\\}$ be a (finite or infinite) sequence of disjoint
sets of finite measure and denote by $\mathfrak{A}$ the collection of all such
sequences. Denote by $A_{\infty}$ the complement of $\cup_{k}A_{k}.$
The expectation operator $\mathbf{E}(\cdot|\mathcal{A}):L_{1}+L_{\infty}\to
L_{1}+L_{\infty}$ is defined by setting
$\mathbf{E}(x|\mathcal{A})=\sum_{k}\frac{1}{m(A_{k})}(\int_{A_{k}}x(s)ds)\chi_{A_{k}}.$
Note that we do not require $A_{\infty}$ to have finite measure.
Every expectation operator is a contraction both in $L_{1}$ and $L_{\infty}.$
Therefore,
$\mathbf{E}(x|\mathcal{A})\prec\prec x,\quad x\in L_{1}+L_{\infty}.$
It follows that $\mathbf{E}(\cdot|\mathcal{A})$ is also contraction in $E.$
It will be convenient to introduce the following notation. If $\mathcal{A}$ is
a discrete subset of the semi-axis (i.e. a subset without limit points inside
$(0,\infty)$), then the elements of $\mathcal{A}\cup\\{0\\}$ partition the
semi-axis. This partition consists of a (finite or infinite) sequence of sets
of finite measure. We identify this partition with the set $\mathcal{A}.$
Elements of $\mathcal{A}$ will be called nodes of the partition $\mathcal{A}.$
The corresponding averaging operator will be denoted by
$\mathbf{E}(\cdot|\mathcal{A}).$
Let $E$ be a symmetric Banach space either on the interval $(0,1)$ or on the
semi-axis. Define the sets
$\mathcal{D}_{E}={\rm Lin}(\\{x\in E:\ x=\mu(x)\\})=\\{\mu(a)-\mu(b),\ a,b\in
E\\},$ $Z_{E}={\rm Lin}(\\{x_{1}-x_{2}:\ 0\leq x_{1},x_{2}\in E,\
\mu(x_{1})=\mu(x_{2})\\}).$
Let $C$ be a Hardy operator defined by setting
$(Cx)(t)=\frac{1}{t}\int_{0}^{t}x(s)ds.$
The following theorem was proved in [13]. For convenience of the reader, we
give a new and simple proof in the appendix.
###### Theorem 8.
Let $E$ be a symmetric space on the semi-axis and let $x\in\mathcal{D}_{E}.$
We have $x\in Z_{E}$ if and only if $Cx\in E.$ A similar assertion is also
valid for the interval $(0,1)$ provided that $\int_{0}^{1}x(s)ds=0.$
The following uniform submajorization was introduced by Kalton and Sukochev in
[19].
Let $x,y\in L_{1}(0,1)$ (or $x,y\in(L_{1}+L_{\infty})(0,\infty)$). We say that
$y\lhd x$ if there exists $m\in\mathbb{N}$ such that
(6) $\int_{ma}^{b}\mu(s,y)ds\leq\int_{a}^{b}\mu(s,x)ds,\quad\forall ma\leq b.$
Let $x,y\in l_{\infty}.$ We say that $y\lhd x$ if there exists
$m\in\mathbb{N}$ such that
(7) $\sum_{k=ma+1}^{b}\mu(k,y)\leq\sum_{k=a+1}^{b}\mu(k,x)\quad\forall
ma+1\leq b.$
The following important theorem was proved in [19] (see Theorem 5.4 and
Theorem 6.3 there).
###### Theorem 9.
Let $x,y\in L_{1}(0,1)$ or $x,y\in(L_{1}+L_{\infty})(0,\infty)$ or $x,y\in
l_{\infty}$ be such that $y\lhd x.$ For every $\varepsilon>0,$ the function
$(1-\varepsilon)y$ belongs to a convex hull of the set $\\{z:\
\mu(z)\leq\mu(x)\\}.$
This theorem led to the following fundamental result (see [19]).
###### Theorem 10.
Let $E=E(0,1)$ (or $E=E(0,\infty)$ or $E=E(\mathbb{N})$) be a symmetric Banach
space either on the interval $(0,1)$ or on the semi-axis or on $\mathbb{N}.$
It follows that the corresponding set $E(\mathcal{M},\tau)$ is a symmetric
Banach space.
Also, the uniform submajorization permits us to prove the convexity of the
functional $\pi:\mathcal{E}\to\mathbb{R}$ defined in Section 1.
###### Lemma 11.
The functional $\pi:\mathcal{E}\to\mathbb{R}$ is convex on every symmetrically
normed operator ideal $\mathcal{E}.$
###### Proof.
Let $E$ be the corresponding symmetrically normed ideal of $l_{\infty}.$ For
every $A,B\in\mathcal{E},$ it follows from Proposition 8.6 of [19] that
$\mu(A+B)\lhd\mu(A)+\mu(B).$ Hence,
$\sigma_{m}\mu(A+B)\lhd\sigma_{m}(\mu(A)+\mu(B)).$ By Theorem 9, we have
$\|\sigma_{m}\mu(A+B)\|_{E}\leq\|\sigma_{m}(\mu(A)+\mu(B))\|_{E}\leq\|\sigma_{m}\mu(A)\|_{E}+\|\sigma_{m}\mu(B)\|_{E}.$
Note that $\|A^{\oplus m}\|_{\mathcal{E}}=\|\sigma_{m}\mu(A)\|_{E}.$ Dividing
by $m$ and letting $m\to\infty,$ we obtain
$\pi(A+B)\leq\pi(A)+\pi(B).$
∎
## 3\. Lifting of symmetric functionals
In this section, we explain a canonical bijection between symmetric
functionals and traces. In what follows, we require that a semifinite von
Neumann algebra $\mathcal{M}$ be either atomless or atomic with traces of all
atoms being $1.$
For an atomless von Neumann algebra $\mathcal{M},$ we have (see e.g. [12])
$\int_{0}^{t}\mu(s,A)ds=\sup\\{\tau(p|A|):\ p\in P(\mathcal{M}),\
\tau(p)=t\\},\quad A\in\mathcal{M}.$
For a atomic von Neumann algebra $\mathcal{M},$ we have (see e.g. [12])
$\sum_{k=1}^{m}\mu(k,A)=\sup\\{\tau(p|A|):\ p\in P(\mathcal{M}),\
\tau(p)=m\\},\quad A\in\mathcal{M}.$
In either case, this implies a remarkable inequality (see e.g. [12])
(8) $\mu(A+B)\prec\prec\mu(A)+\mu(B)\prec\prec 2\sigma_{1/2}\mu(A+B),\quad
0\leq A,B\in(L_{1}+L_{\infty})(\mathcal{M}).$
###### Lemma 12.
Let $E=E(0,1)$ (or $E=E(0,\infty)$ or $E=E(\mathbb{N})$) be a symmetric Banach
space either on the interval $(0,1)$ or on the semi-axis or on $\mathbb{N}.$
If $x,y\in E_{+}$ are such that $y\lhd x,$ then $\varphi(y)\leq\varphi(x)$ for
every positive symmetric functional $\varphi$ on $E.$
###### Proof.
Fix $\varepsilon>0.$ By Theorem 9, there exist $z_{k}\in E,$ $1\leq k\leq n,$
and positive numbers $\lambda_{k},$ $1\leq k\leq n,$ such that
$\mu(z_{k})\leq\mu(x)$ for every $1\leq k\leq n$ and
$(1-\varepsilon)y=\sum_{k=1}^{n}\lambda_{k}z_{k},\quad\sum_{k=1}^{n}\lambda_{k}=1.$
Since $\varphi$ is positive and symmetric, it follows that
$\varphi(z_{k})\leq\varphi(|z_{k}|)=\varphi(\mu(z_{k}))\leq\varphi(\mu(x))=\varphi(x).$
Therefore, $(1-\varepsilon)\varphi(y)\leq\varphi(x).$ Since $\varepsilon>0$ is
arbitrarily small, the assertion follows. ∎
The following assertion is essentially known. However, we provide the full
proof for readers convenience.
###### Lemma 13.
Let $\mathcal{M}$ be a semifinite atomless von Neumann algebra and let
$A,B\in(L_{1}+L_{\infty})(\mathcal{M},\tau)$ be positive operators.
$\int_{2a}^{b}\mu(s,A+B)ds\leq\int_{a}^{b}(\mu(s,A)+\mu(s,B))ds,\quad\forall
2a\leq b,$
$\int_{2a}^{b}(\mu(s,A)+\mu(s,B))ds\leq\int_{2a}^{2b}\mu(s,A+B)ds,\quad\forall
2a\leq b.$
Similar assertion is valid for atomic von Neumann algebra $\mathcal{M}.$
###### Proof.
Applying inequality (8) to the operators $A,B,$ we obtain that
$\int_{0}^{b}\mu(s,A+B)ds\leq\int_{0}^{b}(\mu(s,A)+\mu(s,B))ds$
and
$\int_{0}^{2a}\mu(s,A+B)ds\geq\int_{0}^{a}(\mu(s,A)+\mu(s,B))ds.$
Subtracting this inequalities, we obtain
$\int_{2a}^{b}\mu(s,A+B)ds\leq\int_{a}^{b}(\mu(s,A)+\mu(s,B))ds.$
Proof of the second inequality is identical. ∎
The following theorem answers Question 4 in the affirmative, as also does [20,
Theorem 5.2]. The proof below is very simple and based on a completely
different approach.
###### Theorem 14.
Let $E=E(0,1)$ (or $E=E(0,\infty)$ or $E=E(\mathbb{N})$) be a symmetric Banach
space either on the interval $(0,1)$ or on the semi-axis or on $\mathbb{N}$
and let $E(\mathcal{M},\tau)$ be the corresponding symmetric Banach operator
space.
1. (1)
If $\varphi$ is a positive symmetric functional on $E,$ then there exists a
positive symmetric functional $\mathcal{L}(\varphi)$ on $E(\mathcal{M},\tau)$
such that $\varphi(x)=\mathcal{L}(\varphi)(A)$ for all positive $x\in E$ and
$A\in E(\mathcal{M},\tau)$ such that $\mu(A)=\mu(x).$
2. (2)
If $\varphi$ is a positive symmetric functional on $E(\mathcal{M},\tau),$ then
there exists a positive symmetric functional $\mathcal{L}^{-1}(\varphi)$ on
$E$ such that $\varphi(A)=\mathcal{L}^{-1}(\varphi)(x)$ for all positive $x\in
E$ and $A\in E(\mathcal{M},\tau)$ such that $\mu(A)=\mu(x).$
###### Proof.
We will only prove (1). Proof of (2) is identical.
Let $A,B\in E_{+}(\mathcal{M},\tau).$ It follows from Lemma 13 that
$\mu(A+B)\lhd\mu(A)+\mu(B)\lhd 2\sigma_{1/2}\mu(A+B).$
It follows from Lemma 12 that
$\varphi(\mu(A+B))\leq\varphi(\mu(A)+\mu(B))\leq\varphi(2\sigma_{1/2}\mu(A+B))=\varphi(\mu(A+B)).$
It follows that $\mathcal{L}(\varphi)$ is additive on
$E_{+}(\mathcal{M},\tau).$ We than extend it to $E(\mathcal{M},\tau)$ by
linearity. ∎
Theorem 14 provides a very natural bijection between the set of all symmetric
functionals on $E$ and that on $E(\mathcal{M},\tau),$ observed first for the
case of fully symmetric functionals in [8]. Next corollary follows
immediately.
###### Corollary 15.
Let $E$ and $E(\mathcal{M},\tau)$ be as in Theorem 14. The functional
$\varphi$ is fully symmetric on $E$ if and only if $\mathcal{L}(\varphi)$ is a
fully symmetric functional on $E(\mathcal{M},\tau).$
We also need a lifting between sequence and function spaces. The following
space was introduced in [21].
Let $\mathcal{A}=\\{[n-1,n]\\}_{n\in\mathbb{N}}$ be a partition of the semi-
axis. Clearly, $\mathbf{E}(\cdot|\mathcal{A})$ maps $L_{1}+L_{\infty}$ into
the set of step functions which can be identified with sequences.
###### Proposition 16.
Let $E$ be a symmetric Banach sequence space and let $F$ be the linear space
of all such functions $x\in L_{\infty}$ for which
$\mathbf{E}(\mu(x)|\mathcal{A})\in E.$ The space $F$ equipped with the norm
$\|x\|_{F}=\|x\|_{\infty}+\|\mathbf{E}(\mu(x)|\mathcal{A})\|_{E}$
is a symmetric Banach function space.
The fact that the space $F$ is a Banach space is non-trivial. Proof of this
fact was missing in both [19] and [21]. We include it in the appendix.
Below, we assume that $E$ is embedded into $F.$
###### Theorem 17.
Let $E=E(\mathbb{N})$ be a symmetric Banach sequence space and let $F$ be the
corresponding function space.
1. (1)
If $\varphi$ is a positive symmetric functional on $E,$ then there exists a
positive symmetric functional $\mathcal{L}(\varphi)$ on $F$ such that
$\varphi(\mathbf{E}(\mu(x)|\mathcal{A}))=\mathcal{L}(\varphi)(x)$ for all
positive $x\in F.$
2. (2)
If $\varphi$ is a positive symmetric functional on $F,$ then its restriction
on $E$ is a positive symmetric functional. This restriction is an inverse
operation for the $\mathcal{L}$ in (1).
###### Proof.
Let us prove (1)
$\varphi(\sigma_{1/2}a)=1/2\varphi(a_{1},a_{3},\cdots)+1/2\varphi(a_{2},a_{4},\cdots)=$
$=1/2\varphi(a_{1},0,a_{2},0,\cdots)+1/2\varphi(0,a_{2},0,a_{4},\cdots)=1/2\varphi(a)$
for every $a\in E.$
Let $x,y\in F$ be positive. It follows from Lemma 50 that
$\mathbf{E}(\mu(x+y)|\mathcal{A})\lhd\mathbf{E}(\mu(x)+\mu(y)|\mathcal{A})\lhd
2\sigma_{1/2}\mathbf{E}(\mu(x+y)|\mathcal{A}).$
It follows from Lemma 12 that
$\varphi(\mathbf{E}(\mu(x+y)|\mathcal{A}))=\varphi(\mathbf{E}(\mu(x)+\mu(y)|\mathcal{A}))$
and (1) follows.
The first assertion of (2) is trivial. Clearly,
$\mu(x)-\mathbf{E}(\mu(x)|\mathcal{A})\in(L_{1}\cap L_{\infty})(0,\infty).$ If
$E\neq l_{1},$ then $\varphi(y)=0$ for every $y\in(L_{1}\cap
L_{\infty})(0,\infty)$ and every symmetric functional $\varphi$ on $F.$ If
$E=l_{1},$ then $F=(L_{1}\cap L_{\infty})(0,\infty)$ and the only symmetric
functional on both spaces is an integral. The second assertion of (2) follows.
∎
## 4\. Existence of symmetric functionals
In this section, we present results concerning existence of symmetric
functionals on symmetric function spaces. The main results of this section are
Theorem 23, Theorem 28 and Theorem 29.
We need the following variation of the Hahn-Banach theorem.
###### Lemma 18.
Let $E$ be a partially ordered linear space and let $p:E\to\mathbb{R}$ be
convex and monotone functional. For every $x_{0}\in E,$ there exists a
positive linear functional $\varphi:E\to\mathbb{R}$ such that $\varphi\leq p$
and $\varphi(x_{0})=p(x_{0}).$
###### Proof.
The existence of $\varphi$ follows from the Hahn-Banach theorem. We only have
to prove that $\varphi\geq 0.$ If $z\geq 0,$ then $\varphi(x_{0}-z)\leq
p(x_{0}-z).$ Therefore,
$\varphi(z)\geq\varphi(x_{0})-p(x_{0}-z)=p(x_{0})-p(x_{0}-z)\geq 0$
due to the fact that $z\geq 0$ and $p$ is monotone. ∎
Define operators
$M_{m}:(L_{1}+L_{\infty})(0,\infty)\to(L_{1}+L_{\infty})(0,\infty)$ (or,
$M_{m}:L_{1}(0,1)\to L_{1}(0,1)$) by setting
$(M_{m}x)(t)=\frac{1}{t\log(m)}\int_{t/m}^{t}x(s),\quad m\geq 2.$
###### Lemma 19.
If $0\leq x\in L_{1}+L_{\infty}$ (or, $0\leq x\in L_{1}(0,1)$), then
$\int_{a}^{b/m}x(s)ds\leq\int_{a}^{b}(M_{m}x)(s)ds\leq\int_{a/m}^{b}x(s)ds$
provided that $ma\leq b.$ In particular, $m^{-1}\sigma_{m}x\lhd M_{m}x\lhd x$
provided that $x=\mu(x).$
###### Proof.
Clearly,
$\int_{a}^{b}(M_{m}x)(s)ds=\frac{1}{\log(m)}\int_{a}^{b}\int_{t/m}^{t}x(s)ds\frac{dt}{t}=$
$=\frac{1}{\log(m)}\int_{a/m}^{b}\int_{\max\\{a,s\\}}^{\min\\{ms,b\\}}\frac{dt}{t}x(s)ds=\frac{1}{\log(m)}\int_{a/m}^{b}x(s)\log(\frac{\min\\{ms,b\\}}{\max\\{a,s\\}})ds.$
The integrand does not exceed $x(s)\log(m)$ and the second inequality follows
immediately. The integrand is positive and is equal to $x(s)\log(m)$ for
$s\in(a,b/m).$ The first inequality follows. ∎
###### Corollary 20.
If $E$ is a symmetric Banach function space either on the interval $(0,1)$ or
on the semi-axis, then $M_{m}:E\to E$ is a contraction for $m\in\mathbb{N}.$
###### Proof.
Let $x=\mu(x)\in E.$ It follows from Lemma 19 that $M_{m}x\lhd x.$ It follows
from theorem 9 that, for every $\varepsilon>0,$ the function
$(1-\varepsilon)M_{m}x$ belongs to a convex hull of the set $\\{z:\
\mu(z)\leq\mu(x)\\}.$ Therefore, $M_{m}x\in E$ and
$(1-\varepsilon)\|M_{m}x\|_{E}\leq\|x\|_{E}.$ Since $\varepsilon$ is
arbitrarily small, the assertion follows. ∎
###### Lemma 21.
Let $E$ be a symmetric Banach space either on the interval $(0,1)$ or on the
semi-axis. Let $p:\mathcal{D}_{E}\to\mathbb{R}$ be convex and monotone
functional. If $p=0$ on $Z_{E}\cap\mathcal{D}_{E},$ then $p$ extends to a
convex monotone functional $p:E\to\mathbb{R}$ by setting
$p(x)=p(\mu(x_{+})-\mu(x_{-})).$
Also, $p(x)=0$ for every $x\in Z_{E}.$
###### Proof.
If $x\in\mathcal{D}_{E},$ then $x-\mu(x_{+})+\mu(x_{-})\in
Z_{E}\cap\mathcal{D}_{E}.$ Therefore, $p(x-\mu(x_{+})+\mu(x_{-}))=0$ and, due
to the convexity of $p,$ $p(x)=p(\mu(x_{+})-\mu(x_{-})).$ This proves the
correctness of the definition.
For $x,y\in E,$ we have
$\mu((x+y)_{+})-\mu((x+y)_{-})-\mu(x_{+})+\mu(x_{-})-\mu(y_{+})+\mu(y_{-})\in
Z_{E}\cap\mathcal{D}_{E}.$
It follows that
$p(\mu((x+y)_{+})-\mu((x+y)_{-})-\mu(x_{+})+\mu(x_{-})-\mu(y_{+})+\mu(y_{-}))=0$
and
$p(x+y)=p(\mu((x+y)_{+})-\mu((x+y)_{-}))=$
$=p(\mu(x_{+})-\mu(x_{-})+\mu(y_{+})-\mu(y_{-}))\leq p(x)+p(y).$
Since $p$ is monotone on $\mathcal{D}_{E},$ then $p(y)\leq 0$ for every $0\geq
y\in\mathcal{D}_{E}.$ It follows that $p(y)=p(-\mu(y))\leq 0$ for $0\geq y\in
E.$ Therefore, $p(x+y)\leq p(x)+p(y)\leq p(x)$ for every $0\geq y\in E.$ ∎
###### Lemma 22.
Let $E$ be a symmetric Banach space either on the interval $(0,1)$ or on the
semi-axis. The functional
$p:x\to\limsup_{m\to\infty}\|(M_{m}x)_{+}\|_{E},\quad x\in\mathcal{D}_{E}$
satisfies the assumptions of Lemma 21. Also, for every $x\in\mathcal{D}_{E},$
we have $p(x)\leq\|x\|_{E}.$
###### Proof.
It follows from Corollary 20 that
$\|(M_{m}x)_{+}\|_{E}\leq\|M_{m}x\|_{E}\leq\|x\|_{E},\quad x\in E.$
It follows that
$p(x)=\limsup_{m\to\infty}\|(M_{m}x)_{+}\|_{E}\leq\|x\|_{E},\quad
x\in\mathcal{D}_{E}.$
Clearly, the mappings $x\to(M_{m}x)_{+}$ are convex and monotone. So are the
mappings $x\to\|(M_{m}x)_{+}\|_{E}.$ Therefore,
$p:\mathcal{D}_{E}\to\mathbb{R}$ is a convex and monotone functional.
If $x\in Z_{E}\cap\mathcal{D}_{E},$ then by Theorem 8 $|Cx|\in E.$ Therefore,
$(M_{m}x)(t)\leq\frac{1}{\log(m)}(|\frac{1}{t}\int_{0}^{t/m}x(s)ds|+|\frac{1}{t}\int_{0}^{t}x(s)ds|)\leq$
$\leq\frac{1}{\log(m)}(\frac{1}{m}\sigma_{m}|Cx|+|Cx|)(t).$
Since $\|\sigma_{m}\|_{E\to E}\leq m$ (see [22, Theorem II.4.5]), it follows
that
$\|(M_{m}x)_{+}\|_{E}\leq\frac{2}{\log(m)}\|Cx\|_{E}$
and $p(x)=0.$ ∎
###### Theorem 23.
Let $E=E(0,\infty)$ be a symmetric Banach space on the semi-axis. For a given
$0\leq x\in E,$ there exists a symmetric linear functional
$\varphi:E\to\mathbb{R}$ such that
$\varphi(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}.$
###### Proof.
Without loss of generality, $x=\mu(x).$ Let $p$ be the convex monotone
functional constructed in Lemma 22. It follows from Lemma 18 that there exist
a positive linear functional $\varphi$ on $E$ such that $\varphi\leq p$ and
$\varphi(x)=p(x).$ Since $p(z)=0$ for every $z\in Z_{E},$ it follows that
$\varphi(z)=0$ for every $z\in Z_{E}.$ Therefore, $\varphi$ is a symmetric
functional.
Since $\varphi(z)\leq p(z)\leq\|z\|_{E}$ for every $z=\mu(z)\in E,$ it follows
that $\|\varphi\|_{E^{*}}\leq 1.$ Therefore,
$\varphi(x)=\varphi(\frac{1}{m}\sigma_{m}x)\leq\frac{1}{m}\|\sigma_{m}x\|_{E}.$
Passing $m\to\infty,$ we obtain
$\varphi(x)\leq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}\mu(x)\|_{E}.$
On the other hand, It follows from Lemma 19 that $m^{-1}\sigma_{m}x\lhd
M_{m}x.$ Therefore,
$p(x)=\limsup_{m\to\infty}\|M_{m}x\|_{E}\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}\mu(x)\|_{E}.$
The assertion follows immediately. ∎
Consider the functional $\pi:E\to E$ (identical to the one defined in Section
1).
(9) $\pi(x)=\lim_{m\to\infty}\frac{1}{m}\|x^{\oplus m}\|_{E},\quad x\in E.$
Note that $\pi(-x)=\pi(x)$ for every $x\in E.$ If $p$ is a functionals defined
in Lemma 22, then $p(-x)=0$ for positive $x\in E.$ Therefore, $p\neq\pi.$
However, the assertion below follows from Theorem 23.
###### Lemma 24.
Let $E=E(0,\infty)$ be a symmetric Banach space on the semi-axis. Let $p$ and
$\pi$ be the convex functionals on $E$ defined in Lemma 22 and (9),
respectively. For every positive $x\in E,$ we have $p(x)=\pi(x).$
###### Proof.
For every $x\in E,$ consider the functional $\varphi$ constructed in Theorem
23. By construction, we have $\varphi(x)=p(x)=\pi(x).$ ∎
If $E\not\subset L_{1}(0,\infty),$ then the functional $\varphi$ constructed
in Theorem 23 is necessarily singular. The case $E\subset L_{1}$ requires more
detailed treatment.
###### Lemma 25.
Let $E$ be a symmetric (respectively, fully symmetric) Banach function space
either on the interval $(0,1)$ or on the semi-axis. Let
$\\{\varphi_{i}\\}_{i\in\mathbb{I}}\in E^{*}$ be a net and let $\varphi\in
E^{*}$ be such that $\varphi_{i}\to\varphi$ $*-$weakly.
1. (1)
If every $\varphi_{i}$ is symmetric, then $\varphi$ is symmetric.
2. (2)
If every $\varphi_{i}$ is fully symmetric, then $\varphi$ is fully symmetric.
###### Proof.
Let each $\varphi_{i}$ be symmetric. If $0\leq x_{1},x_{2}\in E$ are such that
$\mu(x_{1})=\mu(x_{2}),$ then
$\varphi(x_{1})=\lim_{i\in\mathbb{I}}\varphi_{i}(x_{1})=\lim_{i\in\mathbb{I}}\varphi_{i}(x_{2})=\varphi(x_{2}).$
Hence, $\varphi$ is symmetric.
Let each $\varphi_{i}$ be fully symmetric. Thus, $\varphi_{i}(x)\leq 0$ for
every $x\in\mathcal{D}_{E}$ such that $Cx\leq 0.$ Therefore,
$\varphi(x)=\lim_{i\in\mathbb{I}}\varphi_{i}(x)\leq 0$ for every
$x\in\mathcal{D}_{E}$ such that $Cx\leq 0.$
Let $x_{1},x_{2}\in E$ be positive elements such that $x_{1}\prec\prec x_{2}.$
Therefore, $z=\mu(x_{1})-\mu(x_{2})\in\mathcal{D}_{E}$ and $Cz\leq 0.$ It
follows from above that $\varphi(z)\leq 0.$ Hence, $\varphi$ is a fully
symmetric functional. ∎
###### Lemma 26.
Let $E$ be a symmetric (respectively, fully symmetric) Banach function space
either on the interval $(0,1)$ or on the semi-axis and let $\varphi$ be a
symmetric (respectively, fully symmetric) functional on $E.$ The formula
$\varphi_{sing}(x)=\lim_{n\to\infty}\varphi(\mu(x)\chi_{(0,1/n)}),\quad 0\leq
x\in E.$
defines a singular symmetric (respectivley, fully symmetric) linear functional
on $E.$
###### Proof.
If $x,y\in E$ are positive functions, then
$\mu(x+y)\chi_{(0,1/n)}\lhd(\mu(x)+\mu(y))\chi_{(0,1/n)}\lhd
2\sigma_{1/2}\mu(x+y)\chi_{(0,1/n)}.$
Taking the limit as $n\to\infty,$ we derive from Lemma 12 that
$\varphi_{sing}(\mu(x+y))=\varphi_{sing}(\mu(x)+\mu(y)).$
Since $\varphi$ is symmetric, it follows that
$\varphi_{sing}(x+y)=\varphi_{sing}(\mu(x+y))=\varphi_{sing}(\mu(x)+\mu(y))=\varphi_{sing}(x)+\varphi_{sing}(y).$
Hence, $\varphi_{sing}$ is an additive functional on $E_{+}.$ Therefore, it
extends to a linear functional on $E.$ Clearly, $\varphi_{sing}$ is symmetric.
Second assertion is trivial. ∎
In fact, the construction in Lemma 26 gives a singular part of the functional
$\varphi$ as defined by Yosida-Hewitt theorem.
###### Lemma 27.
Let $E=E(0,\infty)\subset L_{1}(0,\infty)$ be a symmetric Banach function
space on the semi-axis and let $\varphi$ be a symmetric functional on $E.$ If
$\varphi_{sing}$ is a functional constructed in Lemma 26, then
$\varphi-\varphi_{sing}$ is a normal functional (that is, an integral).
###### Proof.
It is clear that
$0\leq\varphi_{sing}(z)\leq\|z\|_{\infty}\lim_{n\to\infty}\varphi(\chi_{(0,1/n)})=0$
for every positive $z\in(L_{1}\cap L_{\infty})(0,\infty).$ It follows that
$\varphi_{sing}(\mu(x)\chi_{(1/n,\infty)})=0$ for every $x\in E.$ Therefore,
(10)
$(\varphi-\varphi_{sing})(x)=\lim_{n\to\infty}\varphi(\mu(x)\chi_{(1/n,\infty)})=\lim_{n\to\infty}(\varphi-\varphi_{sing})(\mu(x)\chi_{(1/n,\infty)}).$
On the other hand, for every positive $z\in(L_{1}\cap L_{\infty})(0,\infty)$
with $\|z\|_{\infty}=1,$ we have $z\prec\chi_{(0,\|z\|_{1})}.$ It is proved in
[30, Theorem 23] that $z$ belongs to the closure (in the topology of
$L_{1}\cap L_{\infty}$) of the set $\\{u\geq 0:\
\mu(u)=\chi_{(0,\|z\|_{1})}\\}.$ Thus,
$(\varphi-\varphi_{sing})(z)=(\varphi-\varphi_{sing})(\chi_{(0,\|z\|_{1})})=\|z\|_{1}(\varphi-\varphi_{sing})(\chi_{(0,1)}).$
By linearity,
(11)
$(\varphi-\varphi_{sing})(z)=(\varphi-\varphi_{sing})(\chi_{(0,1)})\cdot\int_{0}^{\infty}z(s)ds,\quad\forall
z\in(L_{1}\cap L_{\infty})(0,\infty).$
It follows that from (10) and (11) that
$(\varphi-\varphi_{sing})(x)=\lim_{n\to\infty}\int_{1/n}^{\infty}\mu(s,x)ds\cdot(\varphi-\varphi_{sing})(0,1)=\int_{0}^{1}x(s)ds\cdot(\varphi-\varphi_{sing})(\chi_{(0,1)})$
for every positive function $x\in E.$ The assertion follows immediately. ∎
###### Theorem 28.
Let $E\subset L_{1}(0,\infty)$ be a symmetric Banach space on the semi-axis.
For a given $0\leq x\in E,$ there exists a singular symmetric linear
functional $\varphi_{sing}$ such that
$\varphi_{sing}(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}.$
###### Proof.
Apply Theorem 23 to the function $\mu(x)\chi_{(0,1/n)}.$ It follows that there
exists a symmetric linear functional $\varphi_{n}$ such that
$\|\varphi_{n}\|_{E^{*}}\leq 1$ and
$\varphi_{n}(\mu(x)\chi_{(0,1/n)})=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x)\chi_{(0,1/n)})\|_{E}\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}.$
Since the unit ball in $E^{*}$ is $*-$weakly compact (Banach-Alaoglu theorem),
there exists a convergent subnet $\psi_{i}=\varphi_{F(i)},$ $i\in\mathbb{I},$
of the sequence $\varphi_{n},$ $n\in\mathbb{N}.$ Let $\psi_{i}\to\varphi.$ It
follows from Lemma 25 that $\varphi$ is a symmetric functional.
By the definition of a subnet (see [26, Section IV.2]), for every fixed
$n\in\mathbb{N},$ there exists $i_{n}\in\mathbb{I}$ such that $F(i)>n$ for
every $i>i_{n}.$ Thus, for every $i>i_{n},$ we have
$\psi_{i}(\mu(x)\chi_{(0,1/n)})\geq\varphi_{F(i)}(\mu(x)\chi_{(0,1/F(i))})\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}.$
The subnet $\psi_{i},$ $i_{n}<i\in\mathbb{I}$ converges to the same limit
$\varphi.$ Therefore,
$\varphi(\mu(x)\chi_{(0,1/n)})\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}.$
Now, taking the limit as $n\to\infty,$ we obtain the inequality
$\varphi_{sing}(x)\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E},$
where $\varphi_{sing}$ is a singular symmetric functional defined in Lemma 26.
The opposite inequality is trivial. ∎
###### Theorem 29.
Let $E$ be a symmetric Banach space on the interval $(0,1).$ For a given
$0\leq x\in E,$ there exists a singular symmetric linear functional
$\varphi_{sing}$ such that
$\varphi_{sing}(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}.$
###### Proof.
Let $F$ be a symmetric Banach space on the semi-axis with a norm given by the
formula
$\|x\|_{F}=\|\mu(x)\chi_{(0,1)}\|_{E}+\|x\|_{1},\quad\forall x\in F.$
Clearly, $F\subset L_{1}(0,\infty).$ Applying Theorem 28, we obtain a
symmetric singular functional $\varphi$ on $F$ such that
$\varphi(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{F}=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}.$
∎
## 5\. Existence of fully symmetric functionals
In this section, we present results concerning existence of fully symmetric
functionals on fully symmetric function spaces. The main results of this
section are Theorem 33, Theorem 35 and Theorem 36.
###### Lemma 30.
Let $E$ be a symmetric Banach function space either on the interval $(0,1)$ or
on the semi-axis. If $x,z\in\mathcal{D}_{E}$ are such that $Cx\leq Cz,$ then
$CM_{m}x\leq CM_{m}z.$
###### Proof.
Let $x=\mu(a)-\mu(b)$ and $z=\mu(c)-\mu(d)$ with $a,b,c,d\in E.$ It follows
from assumption $Cx\leq Cz$ that $C(\mu(a)+\mu(d))\leq C(\mu(b)+\mu(c))$ or,
equivalently, $\mu(a)+\mu(d)\prec\prec\mu(b)+\mu(c).$
Arguing as in Lemma 19, we have
$\int_{0}^{t}(M_{m}z)(s)ds=\int_{0}^{t}z(s)h(s,t)ds$
with
$h(s,t)=\left\\{\begin{aligned} 1,\quad 0\leq s\leq t/m\\\
\frac{\log(t/s)}{\log(m)},\quad t/m\leq s\leq t\end{aligned}\right.$
It is now clear that
$\int_{0}^{t}M_{m}(\mu(a)+\mu(d))(s)ds=\int_{0}^{t}(\mu(s,a)+\mu(s,d))h(s,t)ds,$
$\int_{0}^{t}M_{m}(\mu(b)+\mu(c))(s)ds=\int_{0}^{t}(\mu(s,b)+\mu(s,c))h(s,t)ds.$
Clearly, $h$ is positive and decreasing with respect to $s.$ It follows from
[22, Equality 2.36] that
$M_{m}(\mu(a)+\mu(d))\prec\prec M_{m}(\mu(b)+\mu(c))$
and the assertion follows. ∎
###### Lemma 31.
Let $E$ be a fully symmetric Banach function space either on the interval
$(0,1)$ or on the semi-axis and let $x=\mu(x)\in E.$ If $z\in\mathcal{D}_{E}$
is such that $Cx\leq Cz,$ then $p(x)\leq p(z).$
###### Proof.
Since $M_{m}x$ is decreasing, it follows from Lemma 30 that
$\int_{0}^{t}\mu(s,M_{m}x)ds=\int_{0}^{t}(M_{m}x)(s)ds\leq\int_{0}^{t}(M_{m}z)_{+}(s)ds\leq\int_{0}^{t}\mu(s,(M_{m}z)_{+})ds.$
Therefore, $(M_{m}x)_{+}=M_{m}x\prec\prec(M_{m}z)_{+}.$ The assertion follows
now from the definition of the functional $p.$ ∎
###### Lemma 32.
Let $E$ be a fully symmetric Banach function space either on the interval
$(0,1)$ or on the semi-axis. Let $p$ be the functional constructed in Lemma
22. The functional
$q(x)=\inf\\{p(z):\ z\in\mathcal{D}_{E},\ Cx\leq Cz\\},\quad
x\in\mathcal{D}_{E}$
satisfies the assumptions of Lemma 21.
###### Proof.
It is clear from the definition of $q$ that $q\leq p$ and that $q$ is a
positive functional.
We claim that $q$ is convex on $\mathcal{D}_{E}.$ Let
$x_{1},x_{2}\in\mathcal{D}_{E}.$ Fix $\varepsilon>0$ and select
$z_{1},z_{2}\in\mathcal{D}_{E}$ such that $Cx_{i}\leq Cz_{i}$ and
$p(z_{i})\leq q(x_{i})+\varepsilon$ for $i=1,2.$ Thus, $C(x_{1}+x_{2})\leq
C(z_{1}+z_{2})$ and
$q(x_{1}+x_{2})\leq p(z_{1}+z_{2})\leq p(z_{1})+p(z_{2})\leq
q(x_{1})+q(x_{2})+2\varepsilon.$
Since $\varepsilon$ is arbitrarily small, the claim follows.
We claim that $q$ is monotone on $\mathcal{D}_{E}.$ Let
$x_{1},x_{2}\in\mathcal{D}_{E}$ be such that $x_{1}\leq x_{2}.$ Fix
$\varepsilon>0$ and select $z\in\mathcal{D}_{E}$ such that $Cx_{2}\leq Cz$ and
$p(z)\leq q(x_{2})+\varepsilon.$ Thus, $Cx_{1}\leq Cx_{2}\leq Cz$ and
$q(x_{1})\leq p(z)\leq q(x_{2})+\varepsilon.$ Since $\varepsilon$ is
arbitrarily small, the claim follows.
For $x\in Z_{E}\cap\mathcal{D}_{E},$ we have $0\leq q(x)\leq p(x)=0$ and,
therefore, $q(x)=0.$ s ∎
The following theorem is the first main result of this section.
###### Theorem 33.
Let $E=E(0,\infty)$ be a fully symmetric Banach space on the semi-axis. For a
given $0\leq x\in E,$ there exists a fully symmetric linear functional
$\varphi:E\to\mathbb{R}$ such that
$\varphi(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}.$
###### Proof.
Without loss of generality, $x=\mu(x).$ Let $q$ be the convex monotone
functional constructed in Lemma 32. It follows from Lemma 18 that there exist
a positive linear functional $\varphi$ on $E$ such that $\varphi\leq q$ and
$\varphi(x)=q(x).$
It is clear that $\varphi\leq q\leq p.$ Since $p(z)=0$ for every $z\in Z_{E},$
it follows that $\varphi(z)=0$ for every $z\in Z_{E}.$ Therefore, $\varphi$ is
a symmetric functional. For every $z\in\mathcal{D}_{E}$ with $Cz\leq 0,$ we
have $\varphi(z)\leq q(z)\leq p(0)=0.$
Let $x_{1},x_{2}\in E$ be positive elements such that $x_{1}\prec\prec x_{2}.$
Therefore, $z=\mu(x_{1})-\mu(x_{2})\in\mathcal{D}_{E}$ and $Cz\leq 0.$ It
follows from above that $\varphi(z)\leq 0.$ Hence, $\varphi$ is a fully
symmetric functional.
Since $\varphi(z)\leq q(z)\leq p(z)\leq\|z\|_{E}$ for every $z=\mu(z)\in E,$
it follows that $\|\varphi\|_{E^{*}}\leq 1.$ Therefore,
$\varphi(x)=\varphi(\frac{1}{m}\sigma_{m}x)\leq\frac{1}{m}\|\sigma_{m}x\|_{E}.$
Passing $m\to\infty,$ we obtain
$\varphi(x)\leq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}\mu(x)\|_{E}.$
On the other hand, $q(x)=p(x)$ by Lemma 31. By Lemma 19, we have
$m^{-1}\sigma_{m}x\lhd M_{m}x.$ Therefore,
$\varphi(x)=q(x)=p(x)=\limsup_{m\to\infty}\|M_{m}x\|_{E}\geq\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}\mu(x)\|_{E}.$
The assertion follows immediately. ∎
If $\pi:E\to E$ is a convex functional defined in (9), then $\pi(-x)=\pi(x)$
for every $x\in E.$ If $q$ is a functional defined in Lemma 32, then $q(-x)=0$
for positive $x\in E.$ Therefore, $q\neq\pi.$ However, the assertion below
follows from Theorem 33.
###### Lemma 34.
Let $E=E(0,\infty)$ be a fully symmetric Banach space on the semi-axis. Let
$q$ and $\pi$ be the convex functionals on $E$ defined in Lemma 32 and (9),
respectively. For every positive $x\in E,$ we have $q(x)=\pi(x).$
###### Proof.
For every $x\in E,$ consider the functional $\varphi$ constructed in Theorem
33. By construction, we have $\varphi(x)=q(x)=p(x)=\pi(x).$ ∎
The proofs of the two following theorems are very similar to that of Theorem
28 (respectively, Theorem 29) and are, therefore, omitted. The only difference
is that the reference to Theorem 23 (respectively, Theorem 28) has to be
replaced with the reference to Theorem 33 (respectively, Theorem 35).
###### Theorem 35.
Let $E\subset L_{1}(0,\infty)$ be a fully symmetric Banach space on the semi-
axis. For a given $0\leq x\in E,$ there exists a singular fully symmetric
linear functional $\varphi_{sing}$ such that
$\varphi_{sing}(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}.$
###### Theorem 36.
Let $E\subset L_{1}(0,1)$ be a fully symmetric Banach space on the interval
$(0,1).$ For a given $0\leq x\in E,$ there exists a singular fully symmetric
linear functional $\varphi_{sing}$ such that
$\varphi_{sing}(x)=\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}.$
## 6\. The sets of symmetric and fully symmetric functionals are different
In this section, we demonstrate that the sets of symmetric and fully symmetric
functionals on a given fully symmetric space $E$ are distinct (provided that
one of these sets is non-empty). The main results are Theorem 47 and Theorem
48.
Let $x=\mu(x)\in(L_{1}+L_{\infty})(0,\infty)$ (or $x=\mu(x)\in L_{1}(0,1)$)
and let $X(t)=\int_{0}^{t}x(s)ds.$ For every $\theta>0,$ let $a_{n}(\theta)$
be such that
$X(a_{n}(\theta))=(3/2)^{n}\theta$
for every $n\in\mathbb{Z}$ such that $a_{n}(\theta)$ does exist. Given a
sequence
$\kappa=\\{\kappa_{n}\\}_{n\in\mathbb{Z}}\in(\mathbb{N}\cup\\{\infty\\})^{\mathbb{Z}},$
let
$\mathcal{B}_{\kappa,\theta}=\\{\kappa_{n}a_{3n}(\theta),\mbox{ where
}n\in\mathbb{Z}\mbox{ is such that
}\kappa_{n}^{2}a_{3n}(\theta)<a_{3n+1}(\theta)\\}.$
If $\kappa_{n}=m$ for all $n\in\mathbb{N},$ we write $\mathcal{B}_{m,\theta}$
instead of $\mathcal{B}_{\kappa,\theta}.$ Also, set
$\mathcal{A}_{m}=\\{ma_{n}(1):\quad m^{2}a_{n}(1)<a_{n+1}(1),\
n\in\mathbb{Z}\\}.$
###### Lemma 37.
If $x=\mu(x)\in L_{1}+L_{\infty}$ and if $\mathcal{C}_{i},$ $1\leq i\leq k,$
are discrete sets, then
$\mathbf{E}(x|\cup_{i=1}^{k}\mathcal{C}_{i})\prec\prec\sum_{i=1}^{k}\mathbf{E}(x|\mathcal{C}_{i}).$
###### Proof.
It is sufficient to verify
$\int_{0}^{t}\mathbf{E}(x|\cup_{i=1}^{k}\mathcal{C}_{i})(s)ds\leq\sum_{i=1}^{k}\int_{0}^{t}\mathbf{E}(x|\mathcal{C}_{i})(s)ds$
only at the nodes of $\mathbf{E}(x|\cup_{i=1}^{k}\mathcal{C}_{i}),$ that is at
the nodes of $\mathbf{E}(x|\mathcal{C}_{i})$ for every $i.$ However, if
$t\in\mathcal{C}_{i}$ for some $i,$ then
$\int_{0}^{t}\mathbf{E}(x|\cup_{i=1}^{k}\mathcal{C}_{i})(s)ds=X(t)=\int_{0}^{t}\mathbf{E}(x|\mathcal{C}_{i})(s)ds$
and we are done. ∎
We will need the following lemma.
###### Lemma 38.
If $x=\mu(x)\in L_{1}+L_{\infty}$ and if $\kappa\geq\kappa^{\prime}$ (that is
$\kappa_{n}\geq\kappa_{n}^{\prime}$ for every $n$), then
(12)
$\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})\prec\prec\frac{3}{2}\mathbf{E}(x|\mathcal{B}_{\kappa^{\prime},\theta}).$
###### Proof.
Let $n\in\mathbb{Z}$ be such that
$\kappa_{n}^{2}a_{3n}(\theta)<a_{3n+1}(\theta).$ It follows that
$\kappa_{n}^{\prime 2}a_{3n}(\theta)<a_{3n+1}(\theta).$ Therefore,
$\int_{0}^{\kappa_{n}a_{3n}(\theta)}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds\leq\int_{0}^{a_{3n+1}(\theta)}x(s)ds=3/2\int_{0}^{a_{3n}(\theta)}x(s)ds\leq$
$\leq
3/2\int_{0}^{\kappa_{n}^{\prime}a_{3n}(\theta)}x(s)ds=3/2\int_{0}^{\kappa_{n}^{\prime}a_{3n}(\theta)}\mathbf{E}(x|\mathcal{B}_{\kappa^{\prime},\theta})(s)ds.$
Hence, we have
(13) $\int_{0}^{t}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds\leq
3/2\int_{0}^{t}\mathbf{E}(x|\mathcal{B}_{\kappa^{\prime},\theta})(s)ds$
for every $t$ being a node of the partition $\mathcal{B}_{\kappa,\theta}.$
Thus, (13) holds for every $t>0.$ ∎
###### Remark 39.
The inequality (12) holds if $\kappa_{n}\geq\kappa_{n}^{\prime}$ only for such
$n\in\mathbb{Z}$ that satisfy the inequality
$\kappa_{n}^{2}a_{3n}(\theta)<a_{3n+1}(\theta).$
###### Lemma 40.
Let $E$ be a fully symmetric Banach function space either on the interval
$(0,1)$ or on the semi-axis. Let $x=\mu(x)\in E$ and $y=\mu(y)\in E$ be such
that $\varphi(y)\leq\varphi(x)$ for every positive symmetric functional
$\varphi\in E^{*}.$ There exists $0\leq u_{m}\in E$ such that $u_{m}\to 0$ in
$E$ and
$\int_{ma}^{b}y(s)ds\leq\int_{a}^{mb}(x+u_{m})(s)ds,\quad\forall ma\leq b.$
###### Proof.
Let $p$ be a convex positive functional considered in Lemma 22. By Lemma 18,
there exists a positive functional $\varphi\in E^{*}$ such that $\varphi\leq
p$ and $\varphi(y-x)=p(y-x).$ We have $p(z)=0$ for every $z\in Z_{E}$ and,
therefore, $\varphi(z)=0$ for every $z\in Z_{E}.$ Therefore, $\varphi$ is a
positive symmetric linear functional on $E.$
By the assumption, $\varphi(y-x)\leq 0$ and, therefore, $p(y-x)=0.$ Hence, by
the definition of $p,$ we have $u_{m}=(M_{m}(y-x))_{+}\to 0$ in $E.$ Clearly,
$M_{m}y\leq M_{m}x+u_{m}.$ It follows from Lemma 19 that
$\int_{ma}^{b}y(s)ds\leq\int_{ma}^{mb}(M_{m}y)(s)ds\leq\int_{ma}^{mb}(M_{m}x+u_{m})(s)ds\leq\int_{a}^{mb}(x+u_{m})(s)ds.$
∎
For each sequence $\kappa$ and $\lambda>0,$ we define the sequence
$\kappa^{\lambda}$ by setting
$\kappa^{\lambda}_{n}=\begin{cases}\kappa_{n},\qquad\kappa_{n}\geq\lambda\\\
\infty,\qquad\kappa_{n}<\lambda.\end{cases}$
###### Lemma 41.
If $m\in\mathbb{N},$ $x=\mu(x)\in L_{1}+L_{\infty}$ and $0\leq u\in
L_{1}+L_{\infty}$ are such that
$\int_{ma}^{b}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds\leq\int_{a}^{mb}(x+u)(s)ds,\quad\forall
ma\leq b\in\mathbb{R},$
then
(14)
$m^{-1}\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})\prec\prec
30\mu(u).$
###### Proof.
If $\kappa^{100m}_{n}=\infty$ for every $n\in\mathbb{Z},$ then
$\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})=0$ and the assertion is
trivial.
Let $n\in\mathbb{Z}$ be such that
$\kappa_{n}^{2}a_{3n}(\theta)<a_{3n+1}(\theta)$ and $\kappa_{n}\geq 100m.$ It
follows that
(15)
$\int_{0}^{m\kappa_{n}a_{3n}(\theta)}u(s)ds\geq\int_{a_{3n}(\theta)}^{m\kappa_{n}a_{3n}(\theta)}(x+u)(s)ds-\int_{a_{3n}(\theta)}^{m\kappa_{n}a_{3n}(\theta)}x(s)ds.$
By the assumption, we have
(16)
$\int_{a_{3n}(\theta)}^{m\kappa_{n}a_{3n}(\theta)}(x+u)(s)ds\geq\int_{ma_{3n}(\theta)}^{\kappa_{n}a_{3n}(\theta)}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds.$
Note that $m\kappa_{n}a_{3n}(\theta)<a_{3n+1}(\theta).$ It follows from (15)
and (16) that
(17)
$\int_{0}^{m\kappa_{n}a_{3n}(\theta)}u(s)ds\geq\int_{ma_{3n}(\theta)}^{\kappa_{n}a_{3n}(\theta)}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds-\int_{a_{3n}(\theta)}^{a_{3n+1}(\theta)}x(s)ds.$
Let $n^{\prime}$ be the maximal integer number such that $n^{\prime}<n$ and
$\kappa_{n^{\prime}}^{2}a_{3n^{\prime}}(\theta)<a_{3n^{\prime}+1}(\theta).$ It
is clear that
$\kappa_{n^{\prime}}^{2}a_{3n^{\prime}}(\theta)<a_{3n^{\prime}+1}(\theta)\leq
a_{3n-2}(\theta)<ma_{3n}(\theta)$
and
(18)
$\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})=\frac{X(\kappa_{n}a_{3n}(\theta))-X(\kappa_{n^{\prime}}a_{3n^{\prime}}(\theta))}{\kappa_{n}a_{3n}(\theta)-\kappa_{n^{\prime}}a_{3n^{\prime}}(\theta)}\geq\frac{X(a_{3n}(\theta))-X(a_{3n-2}(\theta))}{\kappa_{n}a_{3n}(\theta)}$
on the interval $(ma_{3n}(\theta),\kappa_{n}a_{3n}(\theta)).$
If $\kappa_{n^{\prime}}^{2}a_{3n^{\prime}}(\theta)\geq a_{3n+1}(\theta)$ for
every $n^{\prime}<n,$ then
(19)
$\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})=\frac{X(\kappa_{n}a_{3n}(\theta))}{\kappa_{n}a_{3n}(\theta)}\geq\frac{X(a_{3n}(\theta))}{\kappa_{n}a_{3n}(\theta)}$
on the interval $(ma_{3n}(\theta),\kappa_{n}a_{3n}(\theta)).$
It follows from (17) and (18) (or (19)) that
$\int_{0}^{m\kappa_{n}a_{3n}(\theta)}u(s)ds\geq\frac{\kappa_{n}-m}{\kappa_{n}}\cdot(1-\frac{4}{9})X(a_{3n}(\theta))-\frac{1}{2}X(a_{3n}(\theta)).$
Since $\kappa_{n}\geq 100m,$ it follows that
$\int_{0}^{m\kappa_{n}a_{3n}(\theta)}u(s)ds\geq((1-\frac{1}{100})(1-\frac{4}{9})-\frac{1}{2})X(a_{3n}(\theta))=\frac{1}{20}X(a_{3n}(\theta))=$
$=\frac{1}{30}X(a_{3n+1}(\theta))\geq\frac{1}{30}X(\kappa_{n}a_{3n}(\theta))=\frac{1}{30}\int_{0}^{\kappa_{n}a_{3n}(\theta)}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})(s)ds.$
It follows immediately that
(20) $\int_{0}^{t}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})(s)ds\leq
30\int_{0}^{mt}u(s)ds\leq 30\int_{0}^{mt}\mu(s,u)ds$
for every $t$ being a node of the partition
$\mathcal{B}_{\kappa^{100m},\theta}.$ Therefore,
$\int_{0}^{t}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})(s)ds\leq
30\int_{0}^{mt}\mu(s,u)ds,\quad t>0$
or, equivalently,
$\int_{0}^{t/m}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})(s)ds\leq
30\int_{0}^{t}\mu(s,u)ds,\quad t>0.$
The assertion follows immediately. ∎
###### Lemma 42.
Let $E$ be a fully symmetric Banach function space either on the interval
$(0,1)$ or on the semi-axis. If $x=\mu(x)\in E$ is such that
$\varphi(y)\leq\varphi(x)$ for every positive symmetric functional $\varphi$
on $E$ and every $0\leq y\prec\prec x,$ then
$\lambda^{-1}\sigma_{\lambda}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta})\to
0$ as $\lambda\to\infty.$
###### Proof.
Since $\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})\prec\prec x,$ it follows from
the assumption and Lemma 40 that there exists $0\leq u_{m}\to 0$ such that
$\int_{ma}^{b}\mathbf{E}(x|\mathcal{B}_{\kappa,\theta})(s)ds\leq\int_{a}^{mb}(x+u_{m})(s)ds,\quad\forall
ma\leq b\in\mathbb{R}.$
For every $\lambda\geq 100m,$ we have $\kappa^{100m}\leq\kappa^{\lambda}.$ It
follows from Lemma 41 that
$\frac{1}{\lambda}\sigma_{\lambda}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta})\prec\prec\frac{1}{m}\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta})\stackrel{{\scriptstyle
Lemma\ref{majorant
lemma}}}{{\prec\prec}}\frac{3}{2m}\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{100m},\theta})\stackrel{{\scriptstyle\eqref{kappa
estimate}}}{{\prec\prec}}45\mu(u_{m}).$
The assertion now follows immediately. ∎
###### Proposition 43.
Let $E$ be a fully symmetric Banach function space either on the interval
$(0,1)$ or on the semi-axis equipped with a Fatou norm. If $x=\mu(x)\in E$ is
such that $\varphi(y)\leq\varphi(x)$ for every positive symmetric functional
$\varphi$ on $E$ and every $0\leq y\prec\prec x,$ then
$m^{-1}\sigma_{m}\mathbf{E}(x|\mathcal{B}_{m,\theta})\to 0$ as $m\to\infty.$
###### Proof.
For every $m,r\in\mathbb{N},$ set
$\kappa^{m,r}_{n}=\begin{cases}m\qquad 0\leq|n|<r\\\ \infty\qquad
r\leq|n|\end{cases}$
and $\kappa^{m,r}=\\{\kappa^{m,r}_{n}\\}_{n\in\mathbb{Z}}.$ Clearly,
$\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r},\theta})\to\mathbf{E}(x|\mathcal{B}_{m,\theta})$
almost everywhere when $r\to\infty.$ It follows from the definition of Fatou
norm that
$\lim_{r\to\infty}\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r},\theta})\|_{E}=\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{m,\theta})\|_{E}.$
Select $r_{m}$ so large that
(21)
$\frac{1}{m}\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r_{m}},\theta})\|_{E}>\frac{1}{2m}\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{m,\theta})\|_{E}.$
Now define the sequence $\kappa=\\{\kappa_{n}\\}_{n\in\mathbb{Z}}$ by setting
$\kappa_{n}=\inf_{m\geq 1}\kappa^{m,r_{m}}_{n}=\inf_{r_{m}>|n|}m,\quad
n\in\mathbb{Z}.$
Clearly, $r_{\kappa_{n}}\geq|n|$ and, therefore, $\kappa_{n}\rightarrow\infty$
as $|n|\rightarrow\infty.$ In particular, the set $\\{n:\kappa_{n}<\lambda\\}$
is finite for every $\lambda\in\mathbb{N}.$ Set
$M(\lambda)=\max\\{\lambda,\max_{\kappa_{n}<\lambda}(\frac{a_{3n+1}(\theta)}{a_{3n}(\theta)})^{1/2}\\}.$
If $m>M(\lambda),$ then $m^{2}a_{3n}(\theta)\geq a_{3n+1}(\theta)$ whenever
$\kappa_{n}<\lambda.$ Thus, $\kappa_{n}\geq\lambda$ whenever
$m^{2}a_{3n}(\theta)<a_{3n+1}(\theta).$ Hence,
$\kappa_{n}^{\lambda}=\kappa_{n}$ whenever
$(\kappa_{n}^{m,r_{m}})^{2}a_{3n}(\theta)<a_{3n+1}(\theta).$ Therefore,
$\kappa_{n}^{\lambda}\leq\kappa_{n}^{m,r_{m}}$ for every $n\in\mathbb{Z}$ such
that $(\kappa_{n}^{m,r_{m}})^{2}a_{3n}(\theta)<a_{3n+1}(\theta).$ According to
Remark 39, it follows that
$\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r_{m}},\theta})\prec\prec\frac{3}{2}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta}).$
Since $m\geq\lambda,$ it follows that
(22)
$\frac{1}{m}\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r_{m}},\theta})\prec\prec\frac{3}{2\lambda}\sigma_{\lambda}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta}).$
By Lemma 42, for every $\varepsilon>0,$ there exists $\lambda$ such that
(23)
$\frac{1}{\lambda}\|\sigma_{\lambda}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta})\|_{E}<\frac{1}{3}\varepsilon.$
It follows that
$\frac{1}{m}\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{m,\theta})\|_{E}\stackrel{{\scriptstyle\eqref{qss17}}}{{\leq}}\frac{2}{m}\|\sigma_{m}\mathbf{E}(x|\mathcal{B}_{\kappa^{m,r_{m}},\theta})\|_{E}\stackrel{{\scriptstyle\eqref{qss18}}}{{\leq}}\frac{3}{\lambda}\|\sigma_{\lambda}\mathbf{E}(x|\mathcal{B}_{\kappa^{\lambda},\theta})\|_{E}\stackrel{{\scriptstyle\eqref{qss19}}}{{<}}\varepsilon$
for every $m>M(\lambda).$ Since $\varepsilon>0$ is arbitrarily small, the
assertion follows. ∎
###### Lemma 44.
Let $E$ be a fully symmetric Banach space either on the interval $(0,1)$ or on
the semi-axis equipped with a Fatou norm. If $x=\mu(x)\in E$ is such that
$\varphi(y)\leq\varphi(x)$ for every positive symmetric functional $\varphi$
on $E$ and every $0\leq y\prec\prec x,$ then
$m^{-1}\sigma_{m}\mathbf{E}(x|\mathcal{A}_{m})\to 0$ as $m\to\infty.$
###### Proof.
It is clear that $a_{k}(3/2)=a_{k+1}(1)$ and $a_{k}((3/2)^{2})=a_{k+2}(1)$ for
every $k\in\mathbb{N}.$ It follows that
$\mathcal{B}_{m,1}\cup\mathcal{B}_{m,3/2}\cup\mathcal{B}_{m,(3/2)^{2}}=\mathcal{A}_{m}.$
Therefore, by Lemma 37, we have
(24)
$\mathbf{E}(x|\mathcal{A}_{m})\prec\prec\mathbf{E}(x|\mathcal{B}_{m,1})+\mathbf{E}(x|\mathcal{B}_{m,3/2})+\mathbf{E}(x|\mathcal{B}_{m,(3/2)^{2}}).$
The assertion follows now from Proposition 43. ∎
###### Lemma 45.
Let $x=\mu(x)\in L_{1}+L_{\infty}(0,\infty)$ be a function on the semi-axis.
If $x\notin L_{1}(0,\infty),$ then, for every $t>0$ and every
$m\in\mathbb{N},$ we have
(25)
$X(t)\leq\frac{2}{3}X(m^{4}t)+\frac{3}{2}\int_{0}^{m^{4}t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds.$
###### Proof.
For a given $t>0,$ there exists $n\in\mathbb{Z}$ such that
$t\in[a_{n}(1),a_{n+1}(1)].$ If $a_{n+1}(1)>m^{2}a_{n}(1),$ then
$\int_{0}^{m^{4}t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds\geq\int_{0}^{ma_{n}(1)}\mathbf{E}(x|\mathcal{A}_{m})(s)ds=X(ma_{n}(1))\geq\frac{2}{3}X(t).$
If $a_{n+1}(1)\leq m^{2}a_{n}(1)$ and $a_{n+2}(1)>m^{2}a_{n+1}(1),$ then
$\int_{0}^{m^{4}t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds\geq\int_{0}^{ma_{n+1}(1)}\mathbf{E}(x|\mathcal{A}_{m})(s)ds=X(ma_{n+1}(1))\geq
X(t).$
If $a_{n+2}(1)\leq m^{2}a_{n+1}(1)$ and $a_{n+1}(1)\leq m^{2}a_{n}(1),$ then
$X(m^{4}t)\geq X(a_{n+2}(1))=\frac{3}{2}X(a_{n+1}(1))\geq\frac{3}{2}X(t)$
and the assertion follows. ∎
The situation in the case that $x\in L_{1}$ is slightly more complicated.
###### Lemma 46.
If $x=\mu(x)\in L_{1}(0,1)$ or $x\in L_{1}(0,\infty),$ then there exists
constant $C$ such that for every $t>0$
(26)
$X(t)\leq\frac{2}{3}X(m^{4}t)+\frac{3}{2}\int_{0}^{m^{4}t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds+C\int_{0}^{m^{4}t}\chi_{[0,1]}(s)ds.$
###### Proof.
Consider first the case of the semi-axis. Fix $n_{0}$ such that
$X(a_{n_{0}})\leq 4/9X(\infty).$ For a givne $t\in[a,a_{n_{0}}],$ there exists
$n\in\mathbb{Z}$ such that $n<n_{0}$ and $t\in[a_{n},a_{n+1}].$ Then, the
argument in Lemma 45 applies mutatis mutandi. For every $t\geq a_{n_{0}}$ we
have
$X(t)\leq\frac{X(\infty)}{\min\\{a_{n_{0}},1\\}}\min\\{m^{4}t,1\\}=\frac{X(\infty)}{\min\\{a_{n_{0}},1\\}}\int_{0}^{m^{4}t}\chi_{[0,1]}(s)ds.$
Setting $C=X(\infty)/\min\\{a_{n_{0}},1\\},$ we obtain the assertion.
The same argument applies in the case of the interval $(0,1)$ by replacing
$X(\infty)$ by $X(1).$ ∎
The following two theorems are crucial for the proof of the implication
$(3)\Leftrightarrow(4)$ in Theorem 5.
###### Theorem 47.
Let $E$ be a fully symmetric Banach space either on the interval $(0,1)$ or on
the semi-axis and let $x\in E.$ Suppose that the norm on $E$ is a Fatou norm.
If $\varphi(y)\leq\varphi(x)$ for every positive symmetric functional on $E$
and every $0\leq y\prec\prec x,$ then
(27) $\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\|_{E}=0$
provided that one of the following conditions is satisfied
1. (1)
$E=E(0,1)$ is a space on the interval $(0,1).$
2. (2)
$E=E(0,\infty)$ is a space on the semi-axis and $E(0,\infty)\not\subset
L_{1}(0,\infty).$
###### Proof.
Without loss of generality, $x=\mu(x).$ If $x\notin L_{1},$ then by Lemma 45,
$\int_{0}^{t/m^{4}}x(s)ds\leq\frac{2}{3}\int_{0}^{t}x(s)ds+\frac{3}{2}\int_{0}^{t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds,\quad\forall
t>0$
or, equivalently,
$\frac{1}{m^{4}}\sigma_{m^{4}}x\prec\prec\frac{2}{3}x+\frac{3}{2}\mathbf{E}(x|\mathcal{A}_{m}).$
Applying $m^{-1}\sigma_{m}$ to the both parts, we obtain
$\frac{1}{m^{5}}\sigma_{m^{5}}x\prec\prec\frac{2}{3}\frac{1}{m}\sigma_{m}x+\frac{3}{2}\frac{1}{m}\sigma_{m}\mathbf{E}(x|\mathcal{A}_{m}).$
Take norms and let $m\to\infty.$ It follows from Lemma 44 that
$\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}x\|_{E}\leq\frac{2}{3}\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}x\|_{E}.$
This proves (27).
If $x\in L_{1}$ and $C$ are as in Lemma 46, then it follows from Lemma 46 that
$\int_{0}^{t/m^{4}}x(s)ds\leq\frac{2}{3}\int_{0}^{t}x(s)ds+\frac{3}{2}\int_{0}^{t}\mathbf{E}(x|\mathcal{A}_{m})(s)ds+C\int_{0}^{t}\chi_{[0,1]}(s)ds,\quad\forall
t>0$
or, equivalently,
$\frac{1}{m^{4}}\sigma_{m^{4}}x\prec\prec\frac{2}{3}x+\frac{3}{2}\mathbf{E}(x|\mathcal{A}_{m})+C\chi_{(0,1)}.$
Applying $m^{-1}\sigma_{m}$ to the both parts, we obtain
$\frac{1}{m^{5}}\sigma_{m^{5}}x\prec\prec\frac{2}{3}\frac{1}{m}\sigma_{m}x+\frac{3}{2}\frac{1}{m}\sigma_{m}\mathbf{E}(x|\mathcal{A}_{m})+C\frac{1}{m}\sigma_{m}\chi_{(0,1)}.$
Take norms and let $m\to\infty.$ For every symmetric space $E$ on the interval
$(0,1)$ and for every symmetric space $E$ on the semi-axis such that
$E\not\subset L_{1}(0,\infty)$ we have
$m^{-1}\|\sigma_{m}\chi_{(0,1)}\|_{E}\to 0.$ It follows from Lemma 44 that
$\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}x\|_{E}\leq\frac{2}{3}\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}x\|_{E}$
and again (27) follows. ∎
###### Theorem 48.
Let $E=E(0,\infty)$ be a fully symmetric Banach space on the semi-axis
equipped with a Fatou norm such that $E(0,\infty)\subset L_{1}(0,\infty).$ If
$\varphi(y)\leq\varphi(x)$ for every positive symmetric functional on $E$ and
every $0\leq y\prec\prec x,$ then
(28) $\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}=0.$
###### Proof.
Fully symmetric Banach space $F$ on the interval $(0,1)$ consists of those
$z\in E$ supported on the interval $(0,1).$ Let $x_{1}=\mu(x)\chi_{(0,1)}\in
F.$ Suppose that
$\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x))\chi_{(0,1)}\|_{E}>0.$
It clearly follows that
$\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}(\mu(x_{1}))\|_{F}>0.$
By Theorem 47, there exists $0\leq y_{1}\prec\prec x_{1}$ and a positive
symmetric functional $\varphi\in F^{*}$ such that
$\varphi(y_{1})>\varphi(x_{1}).$ Let $\varphi_{sing}$ be a singular part of
the functional $\varphi$ constructed in Lemma 26. It follows from Lemma 26
that $\varphi_{sing}$ is symmetric. By Lemma 27, the difference
$\varphi-\varphi_{sing}$ is a symmetric normal functional on $F$ (that is, an
integral). Therefore, $\varphi_{sing}(y_{1})>\varphi_{sing}(x_{1}).$
Now we show that the functional $\varphi_{sing}$ can be extended from $F$ to
$E$ by setting
$\varphi_{sing}(z)=\lim_{n\to\infty}\varphi_{sing}(\mu(z)\chi_{(0,1/n)}),\quad
0\leq z\in E.$
Repeating the argument in Lemma 26, we prove that the extension above is
additive on $E_{+}.$ Thus, the functional $\varphi_{sing}\in E^{*}$ is
positive and symmetric. Since $y_{1}\prec\prec x$ and
$\varphi_{sing}(y_{1})>\varphi_{sing}(x_{1})=\varphi_{sing}(x),$ the assertion
follows. ∎
## 7\. Proof of Theorem 5
In this section, we prove an assertion more general then that of Theorem 5.
The assertion of Theorem 5 follows from that of Theorem 49 by setting
$\mathcal{M}=B(H).$
In what follows, the semifinite von Neumann algebra $\mathcal{M}$ is either
atomless or atomic so that the trace of every atom is $1.$
###### Theorem 49.
Let $E(\mathcal{M},\tau)$ be a symmetric operator space. Consider the
following conditions.
1. (1)
There exist nontrivial positive singular symmetric functionals on
$E(\mathcal{M},\tau).$
2. (2)
There exist nontrivial singular fully symmetric functionals on
$E(\mathcal{M},\tau).$
3. (3)
There exist positive symmetric symmetric functional on $E(\mathcal{M},\tau)$
which are not fully symmetric.
4. (4)
If $E(\mathcal{M},\tau)\not\subset L_{1}(\mathcal{M},\tau),$ then there exists
an operator $A\in E(\mathcal{M},\tau)$ such that
(29) $\lim_{m\to\infty}\frac{1}{m}\|\sigma_{m}\mu(A)\|_{E}>0.$
If $E(\mathcal{M},\tau)\subset L_{1}(\mathcal{M},\tau),$ then there exists an
operator $A\in E(\mathcal{M},\tau)$ such that
(30) $\lim_{m\to\infty}\frac{1}{m}\|(\sigma_{m}\mu(A))\chi_{(0,1)}\|_{E}>0.$
1. (i)
The conditions (1) and (4) are equivalent for every symmetric operator space
$E(\mathcal{M},\tau).$
2. (ii)
The conditions (1), (2) and (4) are equivalent for every fully symmetric
operator space $E(\mathcal{M},\tau).$
3. (iii)
The conditions (1)-(4) are equivalent for every fully symmetric operator space
$E(\mathcal{M},\tau)$ equipped with a Fatou norm.
###### Proof.
Implications $\eqref{second main condition}\Rightarrow\eqref{first main
condition}$ and $\eqref{third main condition}\Rightarrow\eqref{first main
condition}$ are trivial.
$\eqref{first main condition}\Rightarrow\eqref{fourth main condition}$ Let
$E(\mathcal{M},\tau)$ be a symmetric operator space with a singular symmetric
functional $\varphi.$ Let $A\in E(\mathcal{M},\tau)$ be an operator such that
$\varphi(A)\neq 0.$ Without loss of generality, $A\geq 0.$
If $E(\mathcal{M},\tau)\not\subset L_{1}(\mathcal{M},\tau),$ then
$|\varphi(A)|=\frac{1}{m}|\varphi(\underbrace{A\oplus\cdots\oplus
A}_{\mbox{$m$
times}})|\leq\|\varphi\|_{E^{*}(\mathcal{M},\tau)}\cdot\frac{1}{m}\|\sigma_{m}\mu(A)\|_{E}.$
Passing $m\to\infty,$ we obtain the required inequality (29).
Let now $E(\mathcal{M},\tau)\subset L_{1}(\mathcal{M},\tau).$ If $\mathcal{M}$
is atomic, then $E(\mathcal{M},\tau)=L_{1}(\mathcal{M},\tau)$ and the
assertion is trivial. Let $\mathcal{M}$ be atomless. Since $\varphi$ is a
singular functional and
$A-AE_{A}(\mu(\frac{1}{m},A),\infty)\in(L_{1}\cap
L_{\infty})(\mathcal{M},\tau),\quad\forall m\in\mathbb{N},$
we infer that
$|\varphi(A)|=|\varphi(AE_{A}(\mu(\frac{1}{m},A),\infty))|=$
$=\frac{1}{m}|\varphi(\underbrace{AE_{A}(\mu(\frac{1}{m},A),\infty)\oplus\cdots\oplus
AE_{A}(\mu(\frac{1}{m},A),\infty)}_{\mbox{$m$ times}})|\leq$
$\leq\|\varphi\|_{E^{*}(\mathcal{M},\tau)}\cdot\frac{1}{m}\|\sigma_{m}\mu(AE_{A}(\mu(\frac{1}{m},A),\infty))\|_{E}\leq\|\varphi\|_{E^{*}(\mathcal{M},\tau)}\cdot\frac{1}{m}\|(\sigma_{m}\mu(A))\chi_{(0,1)}\|_{E}.$
Passing $m\to\infty,$ we obtain the required inequality (30).
$\eqref{fourth main condition}\Rightarrow\eqref{first main condition}$
Firstly, we assume that the algebra $\mathcal{M}$ is finite. Without loss of
generality, $\tau(1)=1.$ Let $E(\mathcal{M},\tau)$ be a symmetric operator
space and let $E(0,1)$ be the corresponding symmetric function space. By the
assumption, there exists an element $x=\mu(A)\in E(0,1)$ such that
$m^{-1}\sigma_{m}x\not\to 0$ in $E(0,1).$ By Theorem 29, there exists a
positive singular symmetric functional $0\neq\varphi\in E(0,1)^{*}.$ Let
$\mathcal{L}(\varphi)$ be a functional on $E(\mathcal{M},\tau)$ defined in
Theorem 14. Clearly, $\mathcal{L}(\varphi)$ is a nontrivial positive symmetric
functional on $E(\mathcal{M},\tau).$
The case when $\mathcal{M}$ is an infinite atomless von Neumann algebra can be
treated in a similar manner. The only difference is that the reference to
Theorem 29 has to be replaced with the reference to either Theorem 28 or
Theorem 23.
Let $E(\mathcal{M},\tau)$ be a symmetric operator space on a atomic von
Neumann algebra $\mathcal{M}$ and let $E(\mathbb{N})$ be the corresponding
symmetric sequence space. It follows from the assumption that
$E(\mathcal{M},\tau)\neq L_{1}(\mathcal{M},\tau)$ or, equivalently,
$E(\mathbb{N})\neq l_{1}.$ By the assumption, there exists an element
$x=\mu(A)\in E$ such that $m^{-1}\sigma_{m}x\not\to 0$ in $E.$ Let
$F(0,\infty)$ be a symmetric function space constructed in Proposition 16.
Since $E(\mathbb{N})\neq l_{1},$ it follows that $F(0,\infty)\not\subset
L_{1}(0,\infty).$ Recall that the space $E(\mathbb{N})$ is naturally embedded
into the space $F(0,\infty)$ and that the norms $\|\cdot\|_{E}$ and
$\|\cdot\|_{F}$ are equivalent on $E(\mathbb{N}).$ We have $x\in F$ and
$m^{-1}\sigma_{m}x\not\to 0$ in $F(0,\infty).$ By Theorem 23, there exists a
positive symmetric functional $0\leq\varphi\in F(0,\infty)^{*}.$ The
restriction of the functional $\varphi$ to $E(\mathbb{N})$ is a nontrivial
positive symmetric functional on $E(\mathbb{N}).$ Let $\mathcal{L}(\varphi)$
be a functional on $E(\mathcal{M},\tau)$ defined in Theorem 14. Clearly,
$\mathcal{L}(\varphi)$ is a nontrivial positive symmetric functional on
$E(\mathcal{M},\tau).$
$\eqref{fourth main condition}\Rightarrow\eqref{second main condition}$ The
proof is very similar to that of the implication $\eqref{fourth main
condition}\Rightarrow\eqref{first main condition}$ and is, therefore, omitted.
The only difference is that references to Theorem 29, Theorem 28 or Theorem 23
have to be replaced with references to Theorem 36, Theorem 35 or Theorem 33,
respectively.
$\eqref{fourth main condition}\Rightarrow\eqref{third main condition}$
Firstly, we assume that the algebra $\mathcal{M}$ is finite. Without loss of
generality, $\tau(1)=1.$ Let $E(\mathcal{M},\tau)$ be a symmetric operator
space and let $E(0,1)$ be the corresponding symmetric function space. By the
assumption, there exists an element $x=\mu(A)\in E(0,1)$ such that
$m^{-1}\sigma_{m}x\not\to 0$ in $E(0,1).$ By Theorem 47, there exists a
positive symmetric but not fully symmetric functional $\varphi\in E(0,1)^{*}.$
Let $\mathcal{L}(\varphi)$ be a functional on $E(\mathcal{M},\tau)$ defined in
Theorem 14. Clearly, $\mathcal{L}(\varphi)$ is a symmetric but not fully
symmetric functional on $E(\mathcal{M},\tau).$
The case when $\mathcal{M}$ is an infinite atomless von Neumann algebra can be
treated in a similar manner. The only difference is that the reference to
Theorem 47 has to be replaced with the reference to either Theorem 47 or
Theorem 48.
Let $E(\mathcal{M},\tau)$ be a symmetric operator space on a atomic von
Neumann algebra $\mathcal{M}$ and let $E(\mathbb{N})$ be the corresponding
symmetric sequence space. It follows from the assumption that
$E(\mathcal{M},\tau)\neq L_{1}(\mathcal{M},\tau)$ or, equivalently,
$E(\mathbb{N})\neq l_{1}.$ By the assumption, there exists an element
$x=\mu(A)\in E$ such that $m^{-1}\sigma_{m}x\not\to 0$ in $E.$ Let
$F(0,\infty)$ be a symmetric function space constructed in Proposition 16.
Since $E(\mathbb{N})\neq l_{1},$ it follows that $F(0,\infty)\not\subset
L_{1}(0,\infty).$ Recall that the space $E(\mathbb{N})$ is naturally embedded
into the space $F(0,\infty)$ and that the norms $\|\cdot\|_{E}$ and
$\|\cdot\|_{F}$ are equivalent on $E(\mathbb{N}).$ We have $x\in F$ and
$m^{-1}\sigma_{m}x\not\to 0$ in $F(0,\infty).$ By Theorem 47, there exists a
positive symmetric functional $\varphi\in F(0,\infty)^{*}$ and a function
$0\leq y\prec\prec x$ such that $\varphi(y)>\varphi(x).$ Set
$z=\mathbf{E}(\mu(y)|\\{(n-1,n)\\}_{n\in\mathbb{N}}).$ Clearly, $z\in
E(\mathbb{N})$ and $\varphi(z)=\varphi(y)>\varphi(x).$ Hence, the restriction
of the functional $\varphi$ to $E(\mathbb{N})$ is a positive symmetric but not
fully symmetric functional on $E(\mathbb{N}).$ Let $\mathcal{L}(\varphi)$ be a
functional on $E(\mathcal{M},\tau)$ defined in Theorem 14. Clearly,
$\mathcal{L}(\varphi)$ is a positive symmetric but not fully symmetric
functional on $E(\mathcal{M},\tau).$ ∎
## 8\. Appendix
In this appendix, we set $\mathcal{A}=\\{(n-1,n)\\}_{n\in\mathbb{N}}.$
###### Lemma 50.
If $x,y\in(L_{1}+L_{\infty})(0,\infty)$ are positive functions, then
$\mathbf{E}(\mu(x+y)|\mathcal{A})\lhd\mathbf{E}(\mu(x)|\mathcal{A})+\mathbf{E}(\mu(y)|\mathcal{A})\lhd
2\sigma_{1/2}\mathbf{E}(\mu(x+y)|\mathcal{A}).$
###### Proof.
Recall that
$\mu(x+y)\prec\prec\mu(x)+\mu(y)\prec\prec 2\sigma_{1/2}\mu(x+y).$
It follows that
$\int_{0}^{b}\mu(s,x+y)ds\leq\int_{0}^{b}(\mu(s,x)+\mu(s,y))ds,$
$\int_{0}^{2a}\mu(s,x+y)ds\geq\int_{0}^{a}(\mu(s,x)+\mu(s,y))ds.$
Let now $a,b$ be positive integers. Subtracting the above inequalities, we
obtain
$\int_{2a}^{b}\mathbf{E}(\mu(x+y)|\mathcal{A})(s)ds=\int_{2a}^{b}\mu(s,x+y)ds\leq$
$\leq\int_{a}^{b}(\mu(s,x)+\mu(s,y))ds=\int_{a}^{b}\mathbf{E}(\mu(x)+\mu(y)|\mathcal{A})(s)ds.$
Similarly, we have
$\int_{2a}^{b}\mathbf{E}(\mu(x)+\mu(y)|\mathcal{A})(s)ds\leq\int_{2a}^{2b}\mathbf{E}(\mu(x+y)|\mathcal{A})(s)ds.$
∎
###### Corollary 51.
The quasi-norm in Construction 16 is a norm.
###### Proof.
It follows from Lemma 50 that
$\mathbf{E}(\mu(x+y)|\mathcal{A})\lhd\mathbf{E}(\mu(x)+\mu(y)|\mathcal{A})$
provided that $x,y$ are positive functions. By Theorem 9,
$\|\mathbf{E}(\mu(x+y)|\mathcal{A})\|_{E}\leq\|\mathbf{E}(\mu(x)|\mathcal{A})\|_{E}+\|\mathbf{E}(\mu(y)|\mathcal{A})\|_{E}.$
∎
###### Lemma 52.
Let $y=\mu(y)\in(L_{1}+L_{\infty})(0,\infty).$ It follows that
$\int_{2^{-k}\lambda
a}^{b}y(s)ds\leq\frac{\lambda}{\lambda-1}\int_{a}^{b}(\sigma_{2^{k}}y)(s)ds$
provided that $b\geq\lambda a.$
###### Proof.
Let $\alpha$ be the average value of $y$ on the interval $[2^{-k}\lambda
a,2^{-k}b].$ Clearly, $y\leq\alpha$ on the interval $[2^{-k}\lambda a,b]$ and
$y\geq\alpha$ on the interval $[2^{-k}a,2^{-k}b].$ Thus,
$\sigma_{2^{k}}y\geq\alpha$ on the interval $[a,b].$ Therefore,
$\int_{2^{-k}\lambda a}^{b}y(s)ds\leq(b-2^{-k}\lambda
a)\alpha\leq\frac{\lambda}{\lambda-1}(b-a)\alpha\leq\frac{\lambda}{\lambda-1}\int_{a}^{b}(\sigma_{2^{k}}y)(s)ds.$
∎
###### Theorem 53.
If $\\{x_{n}\\}_{n\in\mathbb{N}}$ be a Cauchy sequence in $F,$ then there
exists $x\in F$ such that $x_{n}\to x$ in $F.$
###### Proof.
For every $k>0,$ there exists $m_{k}$ such that $\|x_{m}-x_{m_{k}}\|_{F}\leq
4^{-k}$ for $m\geq m_{k}.$ Set $y_{k}=x_{m_{k+1}}-x_{m_{k}}.$ Clearly,
$\|y_{k}\|_{F}\leq 4^{-k}$ for every $k\in\mathbb{N}.$ In particular, the
series $\sum_{k=1}^{\infty}y_{k}$ converges in $L_{\infty}(0,\infty).$
Set $z_{n}=\sum_{k=n}^{\infty}\sigma_{2^{k}}\mu(y_{k}).$ We claim that
$z_{n}\in F$ and $z_{n}\to 0$ in $F.$ Indeed,
$\mu(y_{k})\leq\|y_{k}\|_{\infty}\chi_{(0,1)}+T\mathbf{E}(\mu(y_{k})|\mathcal{A}).$
Here, $T$ is a shift to the right. It follows that
$\mathbf{E}(\mu(z_{n})|\mathcal{A})\leq\sum_{k=n}^{\infty}\sigma_{2^{k}}(\|y_{k}\|_{\infty}\chi_{(0,1)}+T\mathbf{E}(\mu(y_{k})|\mathcal{A})).$
Therefore,
$\|z_{n}\|_{F}\leq\|z_{n}\|_{\infty}+\sum_{k=n}^{\infty}2^{k}\|\|y_{k}\|_{\infty}\chi_{(0,1)}+T\mathbf{E}(\mu(y_{k})|\mathcal{A})\|_{E}\leq$
$\leq\|z_{n}\|_{\infty}+\sum_{k=n}^{\infty}2^{k+1}\|y_{k}\|_{F}\leq\frac{1}{3}\cdot
4^{1-n}+2^{2-n}=o(1).$
It follows from Lemma 8.5 of [19] that
$\int_{\lambda
a}^{b}\mu(s,\sum_{k=n}^{\infty}y_{k})ds\leq\sum_{k=n}^{\infty}\int_{2^{-k}\lambda
a}^{b}\mu(s,y_{k})ds.$
It follows from Lemma 52 that
$\int_{2^{-k}\lambda
a}^{b}\mu(s,y_{k})ds\leq\frac{\lambda}{\lambda-1}\int_{a}^{b}(\sigma_{2^{k}}\mu(y_{k}))(s)ds.$
Therefore,
$\int_{\lambda
a}^{b}\mu(s,\sum_{k=n}^{\infty}y_{k})ds\leq\frac{\lambda}{\lambda-1}\int_{a}^{b}z_{n}(s)ds.$
Hence,
$\sum_{k=n}^{\infty}y_{k}\lhd\frac{\lambda}{\lambda-1}z_{n}.$
Since $\lambda>1$ is arbitrarily large, it follows from Theorem 9 that
$\|\sum_{k=n}^{\infty}y_{k}\|_{F}\leq\|z_{n}\|_{F}\to 0.$
Thus, the series $\sum_{k=1}^{\infty}y_{k}$ does converge in $F.$ The
assertion follows immediately.
∎
## 9\. Proof of Figiel-Kalton theorem
The proof of Theorem 8 follows from the combinations of Lemmas below.
###### Lemma 54.
Let $E$ be a symmetric Banach space either on the interval $(0,1)$ or on the
semi-axis. If $x\in Z_{E},$ then $C(\mu(x_{+})-\mu(x_{-}))\in E.$
###### Proof.
Let $x=\sum_{k=1}^{n}(x_{k}-y_{k})$ with $x_{k},y_{k}\in E_{+}$ and
$\mu(x_{k})=\mu(y_{k}),$ $1\leq k\leq n.$ Set
$z=x_{+}+\sum_{k=1}^{n}y_{k}=x_{-}+\sum_{k=1}^{n}x_{k}.$
It follows from the definition of $C$ and (8) that
$C\mu(z)\leq
C(x_{+})+\sum_{k=1}^{n}C\mu(y_{k})=C(\mu(x_{+})-\mu(x_{-}))+C\mu(x_{-})+\sum_{k=1}^{n}C\mu(x_{k}).$
Using the second inequality in (8), we obtain
$\int_{0}^{t}(\mu(s,x_{-})+\sum_{k=1}^{n}\mu(s,x_{k}))ds\leq\int_{0}^{(n+1)t}\mu(s,z)ds\leq\int_{0}^{t}\mu(s,z)ds+nt\mu(t,z).$
Therefore,
$C\mu(z)\leq C\mu(z)+C(\mu(x_{+})-\mu(x_{-}))+n\mu(z).$
It follows that $C(\mu(x_{-})-\mu(x_{+}))\leq n\mu(z).$ Similarly,
$C(\mu(x_{+})-\mu(x_{-}))\leq n\mu(z)$ and the assertion follows. ∎
###### Lemma 55.
Let $E$ be a symmetric Banach space either on the interval $(0,1)$ or on the
semi-axis. If $x\in\mathcal{D}_{E},$ then $C(\mu(x_{+})-\mu(x_{-}))\in Cx+E.$
###### Proof.
Since $x\in\mathcal{D}_{E},$ it follows that $x=\mu(a)-\mu(b)$ with $a,b\in
E.$ Set $u=\mu(a)-x_{+}\geq 0.$ Clearly, $\mu(a)=u+x_{+}$ and
$\mu(b)=u+x_{-}.$ It follows from the definition of $C$ and (8) that
$C\mu(a)\leq
C\mu(u)+C\mu(x_{+})=C(\mu(x_{+})-\mu(x_{-}))+C\mu(u)+C\mu(x_{-}).$
Using the second inequality in (8), we obtain
$C\mu(x_{-})+C\mu(u)\leq C\mu(b)+\mu(b).$
It follows that
$Cx\leq C(\mu(x_{+})-\mu(x_{-}))+\mu(b).$
Similarly,
$Cx\geq C(\mu(x_{+})-\mu(x_{-}))-\mu(a)$
and the assertion follows. ∎
###### Lemma 56.
Let $E=E(0,\infty)$ be a symmetric space on the semi-axis. If
$x\in\mathcal{D}_{E}$ is such that $Cx\in E,$ then $x\in Z_{E}.$
###### Proof.
Define a partition $\mathcal{A}=\\{(2^{n},2^{n+1})\\}_{n\in\mathbb{Z}}$ and
set $x_{1}=\mathbf{E}(x|\mathcal{A}).$ If $x=\mu(a)-\mu(b)$ with $a,b\in E,$
then $x_{1}=\mathbf{E}(\mu(a)|\mathcal{A})-\mathbf{E}(\mu(b)|\mathcal{A}).$
Clearly,
$\mathbf{E}(\mu(a)|\mathcal{A})\leq\sigma_{2}\mu(a)\in
E,\quad\mathbf{E}(\mu(b)|\mathcal{A})\leq\sigma_{2}\mu(b)\in E$
are decreasing functions. It follows that $x_{1}\in\mathcal{D}_{E}.$ It is
easy to see that
$|Cx_{1}-Cx|\leq 2\sigma_{2}(\mu(a)+\mu(b)).$
Therefore, $Cx_{1}\in E.$ Define a function $z\in E$ by setting
$z(t)=(Cx_{1})(2^{n+1}),\quad t\in(2^{n},2^{n+1}).$
Clearly, $x_{1}=2z-\sigma_{2}z\in Z_{E}.$
Consider the function $x-x_{1}$ on the interval $(2^{n},2^{n+1}).$ By Kwapien
theorem [23], there exist positive equimeasurable functions $y_{1n},y_{2n}$
supported on $(2^{n},2^{n+1})$ such that
$\mu(y_{1n})=\mu(y_{2n}),\quad\|y_{1n}\|_{\infty},\|y_{2n}\|_{\infty}\leq
6\|(x-x_{1})\chi_{(2^{n},2^{n+1})}\|_{\infty}.$
Set $y_{1}=\sum_{n\in\mathbb{N}}y_{1n}$ and
$y_{2n}=\sum_{n\in\mathbb{N}}y_{2n}.$ It follows that $y_{1},y_{2}\in E_{+}.$
Since $x-x_{1}=y_{1}-y_{2}$ and $\mu(y_{1})=\mu(y_{2}),$ it follows that
$x-x_{1}\in Z_{E}.$ The assertion follows immediately. ∎
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|
arxiv-papers
| 2011-08-12T10:14:23 |
2024-09-04T02:49:21.510335
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Sukochev, D. Zanin",
"submitter": "Dmitriy Zanin",
"url": "https://arxiv.org/abs/1108.2598"
}
|
1108.2603
|
arxiv-papers
| 2011-08-12T11:16:06 |
2024-09-04T02:49:21.518592
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongbi Li",
"submitter": "Yongbin Li",
"url": "https://arxiv.org/abs/1108.2603"
}
|
|
1108.2910
|
# A Study of Reactor Neutrino Monitoring
at Experimental Fast Reactor JOYO
H. Furutaa,1,∗, Y. Fukudab, T. Harac, T. Harunad,2, N. Ishiharae, M.
Ishitsukaa, C. Itof, M. Katsumatag, T. Kawasakig, T. Konnoa, M. Kuzea, J.
Maedaa,3, T. Matsubaraa, H.Miyatag, Y. Nagasakah, K. Nittaa,4, Y. Sakamotoi,
F. Suekanej, T. Sumiyoshid, H. Tabataj, M. Takamatsuf, N. Tamurag a Department
of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan b Department
of Physics, Miyagi University of Education, Sendai 980-0845, Japan c
Department of Physics, Kobe University, Kobe 657-8501, Japan d Department of
Physics, Tokyo Metropolitan University, Hachioji 192-0397, Japan e Institute
of Particle and Nuclear Studies, High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801, Japan fExperimental Fast Reactor Department, Oarai
Research and Development Center, Japan Atomic Energy Agency (JAEA), Oarai,
311-1393, Japan g Department of Physics, Niigata University, Niigata 950-2181,
Japan h Department of Computer Science, Hiroshima Institute of Technology,
Hiroshima 731-5193, Japan i Department of Information Science, Tohoku Gakuin
University, Sendai 981-3193, Japan j Department of Physics, Tohoku University,
Sendai 980-8578, Japan
###### Abstract
We carried out a study of neutrino detection at the experimental fast reactor
JOYO using a 0.76 tons gadolinium loaded liquid scintillator detector. The
detector was set up on the ground level at 24.3 m from the JOYO reactor core
of 140 MW thermal power. The measured neutrino event rate from reactor on-off
comparison was 1.11$\pm$1.24(stat.)$\pm$0.46(syst.) events/day. Although the
statistical significance of the measurement was not enough, the background in
such a compact detector at the ground level was studied in detail and MC
simulation was found to describe the data well. A study for improvement of the
detector for future such experiments is also shown.
###### keywords:
Reactor neutrino; Neutrino oscillation; Cosmic ray; Radioactivity; Low
background
††journal: Nuclear Instruments and Methods in Physics Research A
## 1 Introduction
11footnotetext: Corresponding Author. Tel.: +81 22 795 6727. Email address:
furuta@awa.tohoku.ac.jp (H. Furuta).11footnotetext: Present Address:
Department of Physics, Tohoku University, Sendai 980-8578,
Japan22footnotetext: Present Address: Canon Inc., Tokyo 146-8501,
Japan33footnotetext: Present Address: Department of Physics, Tokyo
Metropolitan University, Hachioji 192-0397, Japan44footnotetext: Present
Address: National Institute of Radiological Sciences, Chiba 263-8555, Japan
Reactor neutrinos have been playing an important role since its first
discovery in 1956 [1] for the progress of elementary particle physics and to
deepen our understanding of the nature. Now the reactor neutrino detection
techniques have become mature after a number of reactor neutrino experiments
so far performed [2][3]. Research and development of compact reactor neutrino
detector utilizing the up-to-date technologies have become active recently [4]
with an idea of using it as a monitor for Plutonium breeding in reactor cores
[3][5] and as a very near detector to calibrate reactor neutrino flux for long
baseline reactor neutrino oscillation experiments.
### 1.1 Reactor neutrinos
In operating reactors, 235U, 238U, 239Pu and 241Pu perform fission reaction
after absorbing a neutron. The fission products are generally neutron-rich
unstable nuclei and perform $\beta$-decays until they become stable nuclei.
One $\bar{\nu}_{e}$ (anti-electron neutrino) is produced in each
$\beta$-decay. The energy of the reactor neutrinos corresponds to
$\beta$-decay energy of a few MeV. Roughly $6~{}\bar{\nu}_{e}^{\prime}s$ are
produced in a fission reaction along with $\sim 200$ MeV of energy release,
resulting in $6\times 10^{20}~{}\bar{\nu}_{e}^{\prime}s$ production per second
in a 3 GWth power reactor.
### 1.2 Nondestructive Plutonium Measurement
Main components of reactor neutrinos come from 235U and 239Pu fissions, and
contributions of 238U and 241Pu are much smaller than those nuclei. Along with
the burn-up of the core, 235U is consumed and 239Pu is ’breeded’ from 238U
through neutron absorption and $\beta$-dacays. Because 239Pu can be used for
nuclear explosion, it is an important object of strict safeguard regulations.
Therefore, it is important to monitor reactor operation and track the
plutonium breeding. International Atomic Energy Agency (IAEA) watches reactors
in the world with surveillance cameras, reviewing operation record, etc.
Because it is impossible to hide the neutrinos, it could be a powerful tool to
monitor the reactor operation, in addition to the traditional monitoring
methods [6].
The reactor neutrino monitoring has a potential to non-destructively measure
the plutonium amount in the core.
Table 1 shows the energy releases and expected number of emitted
$\bar{\nu}_{e}$’s above 1.8 MeV per fission, and average ratio of fission in
the JOYO core for major isotopes in nuclear reactors. As shown in the Table 1,
235U produces significantly more neutrinos than 239Pu. Combining the neutrino
flux and thermal power generation, there is a possibility to measure Plutonium
amount in the core. This is simply depicted by the following equations
assuming the fuel is made up only from 235U and 239Pu.
$\displaystyle q_{235}F_{235}+q_{239}F_{239}=P_{th},$ (1)
$\displaystyle\nu_{235}F_{235}+\nu_{239}F_{239}=N_{\bar{\nu}_{e}}$ (2)
where, 235 and 239 represent 235U and 239Pu. $F_{x}$ is the fission rate of
the nucleus-$x$ in the core, $q_{x}$ is the energy release per fission.
$\nu_{x}$ is the expected number of emitted $\bar{\nu}_{e}$’s per fission,
$N_{\bar{\nu}_{e}}$ is the total emission rate of $\bar{\nu}_{e}$. A small
contribution from 238U and 241Pu is ignored to simplify the calculation.
The fission rate of 239Pu is calculated from those relations and the values of
the parameters, and the 239Pu amount in the core can be calculated from the
fission rate.
Isotope | $\nu$ ($>$1.8 MeV) | q (MeV) | Contribution |
---|---|---|---|---
| | | @JOYO(%) |
235U | 1.92$\pm$0.02 | 201.7$\pm$0.6 | 37.1 |
238U | 2.38$\pm$0.02 | 205.0$\pm$0.9 | 7.3 |
239Pu | 1.45$\pm$0.02 | 210.0$\pm$0.9 | 51.3 |
241Pu | 1.83$\pm$0.02 | 212.4$\pm$1.0 | 4.3 |
Table 1: Number of $\bar{\nu}_{e}$ per fission with the energy above 1.8MeV
[7] and energy release per fission for major isotopes in nuclear reactors [8].
### 1.3 Compact neutrino detectors
As R&D of compact neutrino detectors, an experimental program led by Lawrence
Livemore National Laboratory (LLNL) and Sandia National Laboratories (SNL)
measured neutrino energy spectrum at a short distance from a 235U-rich reactor
with a thermal power of 3.4 GWth, San Onofre Nuclear Generation Station
(SONGS), and indicated feasibility of the neutrino monitoring [9]. On the
other hand, further R&D studies of detector design and materials are still
necessary to realize a compact detector operation above ground for practical
use as a reactor monitor with the neutrino detection. Considering the neutrino
interaction cross-section on proton target (inverse $\beta$-decay,
$O(10^{-43})\,{\rm cm}^{2}$, see Section 3) and compact detector size, the
detector must be set at a short distance (less than a few tens of meters) from
the reactor core to accumulate enough statistics for monitoring. In addition,
feasibility of the measurement at ground level is required for the monitor
considering limited access to the reactor site, while the previous
measurements of neutrinos were operated at underground to reduce cosmic-ray
muon background. Therefore, the detector must be designed to be able to reduce
external backgrounds, e.g. cosmic-ray muons and fast neutrons.
We constructed a 0.76 tons gadolinium loaded liquid scintillator detector as a
prototype of KASKA detector [10] and we reused it to take part in such R&D
efforts [11]. The detector was set up at 24.3 m from Joyo experimental reactor
core whose thermal energy was 140 MW [12]. Unique points of this experiment
are, (1) the reactor power is much smaller compared with the ones so far used
to measure the neutrinos, (2) the detector is located above ground, (3) the
reactor was a fast reactor, so that the neutrinos came mainly from Plutonium.
The main goal of this experiment was to distinguish reactor-on and off by
neutrinos under this unfavorable conditions. One of the possible safeguard
applications is to monitor small reactors to prevent them to be hiddenly
operated to make plutonium. The points (1) and (2) of this experiment are
useful to study such a possibility. As for (3), neutrinos from 235U-rich light
water reactors have been measured [13][14], while observation of neutrinos
from 239Pu-rich fast reactor has not been reported yet and this experiment
could have been the first detection of the fast reactor neutrinos. If energy
spectrum of fast reactor neutrinos is measured in the future, $\nu_{235}$ and
$\nu_{239}$ can be determined separately by comparing the 239Pu-rich neutrinos
and 235U-rich neutrinos. This experiment is a good practice to perform an
experiment at a larger fast reactor in the future to measure the Plutonium-
rich neutrino spectrum.
## 2 Experimental fast reactor JOYO
The experimental fast reactor JOYO, whose thermal power is 140 MW, is located
in Japan Atomic Energy Agency (JAEA) Oarai Research and Development Center in
Ibaraki prefecture, Japan. The JOYO reactor is a sodium-cooled fast reactor
built as an experimental reactor to promote commercialization of fast breeder
reactor development [12]. The reactor fuel is plutonium-uranium mixed oxide
(MOX) which consists of enriched uranium dioxide UO(2) to 18 w% in 235U and
plutonium dioxide PuO(2). Fraction of fissile Pu content
($(^{239}Pu+^{241}Pu)/all$) is about 16 w% at the inner core and about 21 w%
at the outer core.
JOYO reactor operates for 60 days then stops for a few weeks in its
operational cycle. Therefore, we could collect data in both the reactor-on and
reactor-off conditions. The data taken under reactor-off condition were used
to measure the background. Thermal power of the rector was stable at 140 MW
during its operation. Figure 1 shows time variations of fission rates of main
isotopes (235U, 238U, 239Pu and 241Pu) in the fuel. Neutrino flux from the
reactor core was calculated from available measurements of $\beta$-decay
spectra with 2.5 % systematic uncertainty [7].
Figure 1: Fission rate of each fissile element as a function of time from the
4th to 6th operational cycles of experimental fast reactor JOYO. Four lines
correspond to 239Pu, 235U, 238U and 241Pu as indicated in the figure. A period
used for data analysis is also shown.
## 3 Neutrino detection principle
Reactor neutrinos are detected with a liquid scintillator formulated from
organic oils. Organic oils are abundant in free protons and the reactor
$\bar{\nu}_{e}$ react with the proton through inverse $\beta$-decay reaction.
$\bar{\nu}_{e}+p\rightarrow e^{+}+n$ (3)
Figure 2 shows the reactor neutrino flux at JOYO experimental site and the
cross-section of inverse $\beta$-decay reaction together with a shape of the
energy spectrum in the detector. Number of interactions in the detector is
determined as a multiplication of the flux, interaction cross-section and the
number of free protons in the detector.
Figure 2: Shape of neutrino energy spectrum (arbitrary unit) via inverse
$\beta$-decay reactions expected in the detector (solid line). Overlaid curves
show the reactor neutrino flux (dashed line) and cross-section of inverse
$\beta$-decay reaction (dotted line).
Energy threshold of the inverse $\beta$-decay interaction is 1.8 MeV. Cross-
section of inverse $\beta$-decay reaction is associated with the lifetime of
free neutrons and calculated precisely with 0.2 % accuracy [15].
The detector contains gadolinium-loaded liquid scintillator (Gd-LS), in which
neutrino signals are detected by using delayed coincidence technique. A
positron kinetic energy and $\gamma$’s from its annihilation are observed as
the prompt signal. Since the recoil energy of neutron is small, neutrino
energy can be measured from the energy of the prompt signal.
$E_{signal}=E_{\nu}-1.8\,{\rm MeV}+2m_{e}c^{2}$ (4)
Neutrons from inverse $\beta$-decay reactions are captured by gadolinium or
hydrogen in the Gd-LS mostly after thermalization, and $\gamma$-rays are
emitted. Those $\gamma$-rays are detected as delayed signal. In our detector,
we expect 76.9 % of neutron captures are on 155Gd or 157Gd, which have more
than $10^{5}$ times larger thermal neutron capture cross-section than hydrogen
[16]. $\gamma$-rays with total energy of approximately 8 MeV are emitted from
a neutron capture on Gd. The mean time difference ($\Delta t$) between the
prompt and delayed signals is estimated to be 46 $\mu$sec. The background
events are strongly suppressed by requiring coincidence of two signals.
## 4 The detector
### 4.1 Experimental setup
Figure 3 shows a schematic view of the detector. The detector was constructed
at Tohoku University as one of the R&D programs for the KASKA reactor neutrino
oscillation experiment [10]. The detector was moved to the Joyo reactor site
in September 2006 after the KASKA R&D studies, and was set up on the ground
floor of the reactor building near a delivery entrance, just at the west
outside of the reactor containment vessel. The distance to the Joyo core was
24.3 m. The location of the detector is shown in Figure 4. Because Joyo uses
sodium coolant, water was not allowed to be brought in the building and water
shield was not possible.
The data taking period was from January 2007 until December 2007.
Unfortunatelly the liquid scintillator deteriorated during the operation and
only net 38.9 days reactor-on data and 18.5 days of reactor-off data were used
for the analysis. The reason of the deterioration is not clear but we assume
the high temperature environment and N2 bubbling were possible reasons.
Details of detector design and components are shown in the following sections.
Figure 3: Schematic view of experimental setup of the detector. Figure 4:
Location of the detector at experimental fast reactor JOYO.
### 4.2 Main detector
The main detector consisted of 0.76 tons of Gd-LS filled in a transparent
acrylic spherical vessel with inner diameter of 1.2 m. The vessel was made of
UV transparent acrylic ACRYLITE(000) of MITSUBISHI RAYON Inc.. Two acrylic
hemispheres were made from 15 mm thick acrylic plates by vacuum forming. The
two hemispheres were put together sandwiching Viton O-ring at the equator to
form a sphere. There is a 30 cm diameter chimney at the top of the sphere. The
acrylic sphere was supported by an aluminum stand which stood in a oil pan.
The liquid scintillator was formulated by diluting the commercial Gd-loaded
liquid scintillator BC521 (Saint-Gobain) by Paraffine oil and Pseudocumene.
The compositions of the liquid scitillator were, 12.6 weight% (w%)
Pseudocumene (1,2,4-Trimethylbenzene: C9H12), 76.3 w% Paraol 850, 11.2 w%
BC521, and 1.52 g/liter of PPO (2,5-Diphenyloxazole: C15H11NO) as the fluor.
The Gd concentration was 0.05 w% (as contained in BC521). Paraol 850 is heavy
isoparaffin, one of Shell products. The scintillation light yield was measured
to be 56 % of Anthracene scintillator, which is equivalent to 9,400
photons/MeV. The Gd-LS was purged by N2 bubbling with a flowing rate of 100
cc/min during operation to reduce the oxygen quenching effect. Properties of
Gd-LS used in our detector are summarized in Table 2.
Parameter | Value
---|---
Density (20 ∘C) | 0.838 g/cm3
H/C ratio | 1.94
Number of Protons (H) | 6.22$\times$ 1028
Light yield | 9,400 photon/MeV
Gd concentration | 0.05 w%
Neutron capture time | 46.4 $\mu$sec
Table 2: Properties of gadolinium-loaded liquid scintillator used in our
detector
The scintillation lights from Gd-LS were measured by 16 Hamamatsu R5912 8-inch
photomultiplier tubes (PMTs) mounted on the surface of the acrylic vessel.
Each PMT was covered by a mu-metal skirt which was used for Kamiokande PMT
long time ago. Figure 5 shows a picture of the acrylic vessel with 16 PMTs on
the surface. The PMT was put in a acrylic housing cylinder and the space
between PMT surface and acrylic sphere were filled with RTV rubbers (Shin-Etsu
Silicones KE103, KE1052). The photo-cathode coverage was approximately 10 % .
Figure 5: Picture of sphere shape acrylic vessel equipped with 16 PMTs on the
surface. Gd-LS is not filled at the time of this picture.
### 4.3 Cosmic-ray veto counter and detector shielding
Since the detector was set up at ground level, cosmic-ray muon flux was large.
In order to reduce the cosmic-ray muon background, the main detector was
surrounded by a veto counter system. This system consisted of two layers of 1
cm thick plastic scintillator plates equipped with wavelength shifter and PMT
for the readout. Top of the detector and the north and south sides were fully
covered by scintillator layers, while only the upper half were covered for the
east and west sides. Veto efficiency of cosmic-ray muons by the counter system
was estimated to be 92 % from MC simulation including the acceptance. The
cosmic-ray veto signal rate was about 2 kHz. 5 cm thick lead blocks covered
the bottom area of the oil pan and 6 mm thick lead sheets backed by wooden
boards covered the side of the detector housing. 5 cm thick paraffin blocks
were arranged outside of the detector to suppress fast neutrons induced by
cosmic muons. Figure 6 shows a picture of the detector at the site. The size
of the detector was roughly 2.5 m$\times$2.5 m$\times$2 m(H).
Figure 6: A picture taken at the experimental site. The detector is in the
black cottage at the left hand side. The reactor containment vessel is behind
the concrete wall at the right hand side.
### 4.4 Data acquisition system
CAMAC standard electronics modules were used for data taking. Figure 7 shows
the schematic view of the data acquisition system. Signal from each PMT was
divided into four. The first one was fed into ADC, by which integrated charge
within 200 nsec time window was measured. The second signal was used to make a
common trigger for ADC and TDC, which was made from a discriminated analog sum
of all PMT signals. The threshold for the common trigger was set at 3.5 MeV.
If cosmic-ray veto counter had a signal within 100 $\mu$sec before the common
trigger was made, the trigger was canceled. The third signal provided a stop
signal to TDC, which measured the timing of the PMT hits. The last signal was
fed into another ADC for pulse shape discrimination (PSD) study aiming to
identify fast neutron background, although the PSD was not used for the study
described in this paper. In order to collect delayed coincidence signals from
neutrino interactions, lower trigger threshold at 2.5 MeV was applied to the
delayed signals for 100 $\mu$sec after a prompt trigger was created. Time
interval between the first and second triggers was measured by counting a 100
MHz clock signal by a CAMAC scaler and the data were saved along with the ADC
and TDC data for each trigger. If a trigger for delayed signal was not
generated within 100 $\mu$sec, the data acquisition system was back to the
normal mode with 3.5 MeV threshold. In addition to this delayed trigger, we
also took data with single trigger at 0.6 MeV threshold for the background
study. During the data taking at JOYO fast reactor site, the single trigger
rate was about 300 Hz. Mean dead time of the data taking inclusive of the
cosmic muon veto time was 38 %.
The readout and monitoring system in this experiment needed to be simplified
due to limited access to the experimental area. Therefore, we constructed our
DAQ software system in a CAMAC CC/NET [17] to read the data from ADC and TDC
modules and used a trigger system installaed into a NIM FPGA module.
As JOYO is a fast reactor and uses sodium as moderator, the experimental area
also needed to be kept off water and high humidity. In addition, the detector
using liquid scintillator, which generates organic gas, was placed in a large
box sealed with black vinyl sheets. In order to keep safety of the
experimental area during the operation of the experiment, we constructed a
monitoring system in a Linux computer and kept watching temperature, humidity
and density of oxygen and organic gases. The experiment and monitoring data
were automatically sent to a 220 km distance remote site, Tohoku Gakuin
University. We built a secure network on the internet by IPsec VPN
architecture over IPv4 protocol, which enabled an experiment shift person to
check the condition of DAQ and experiment area remotely [18].
Figure 7: The schematic view of the data acquisition system.
### 4.5 Monte Carlo simulation
Figure 8: Reconstructed energy spectrum from data taken with ${}^{60}Co$
$\gamma$-ray source at the detector center.
The collected data were compared with the Monte Carlo (MC) simulation based on
Geant4 (version 4.9.0.p1). Geant4 is a toolkit which provides a calculation of
particle tracking in materials [19]. For the hadronic interaction process,
QGSP_BIC_HP model [20] was employed in Geant4. It comprehends from low energy
region under 20 MeV such as behavior of thermal and fast neutron to high
energy region such as interactions between cosmic-ray muons and materials
around it. Trajectory of optical photons emitted in the Gd-LS was simulated
considering the optical process including attenuation and scattering.
Corrections for PMT responses and energy calibration were carried out by
putting a 60Co $\gamma$-ray source inside the detector. Energy was
reconstructed from the total observed charge by 16 PMTs in which correction to
the acceptance and attenuation length in the liquid scintillator were taken
into account. Figure 8 shows a reconstructed energy spectrum from the data
taken with a 60Co $\gamma$-ray source at the detector center. The 60Co source
mainly emits two gamma rays with 1.17 MeV and 1.33 MeV energies. A large peak
in Figure 8 is made from the gamma rays with 2.5 MeV total energy. The energy
resolution estimated from the peak at 2.5 MeV was 20 %/$\sqrt{E({\rm MeV})}$.
In addition, the measured data with 241Am-9Be ($\alpha$, n) neutron source at
the detector center were used to tune the quenching effects of protons
recoiled by neutrons parametrized by Birks’ constant $k_{B}$ [21] and evaluate
the neutrino MC simulation. The Birks’ constant of our Gd-LS was estimated to
be 0.07 mm/MeV from a comparison of the measured energy spectrum to the MC
simulation.
Not only the neutrino signal events, but also various background events were
generated by the MC simulation and compared with the observed data. Those
background events included cosmic-ray muons and the muon decay, fast neutrons
and environmental $\gamma$-rays from decay chains of 238U and 232Th series and
40K decays. In addition to the fast neutron and environmental $\gamma$-rays
generated inside the detector, those from outside of the detector were also
considered in the MC simulation.
## 5 Measurement of background spectrum
Figure 9: Comparison between reconstructed energy spectra of the prompt
trigger events for the reactor-on and off. Black and gray histograms show the
observed data for a day live-time under reactor-on and off conditions,
respectively. There is an excess around 8 MeV of the distributions attributed
to thermal neutron capture on Gd.
Figure 10: Reconstructed energy spectra of the prompt trigger events above
0.6MeV of the threshold level with different energy ranges. Points show the
observed data for 309 sec live-time taken in reactor-off condition. Overlaid
histograms show the expected neutrino signal and background energy spectrum
with the contributions from each background source.
Major background sources in this experiment were environmental $\gamma$-rays
and cosmic-ray muons. The environmental $\gamma$-rays are emitted by
radioactive isotopes contaminated in the detector and materials around the
detector. These $\gamma$-rays are produced through the decay chains of 238U
and 232Th series, and decay of 40K. The energy of $\gamma$-rays ranges up to
2.6 MeV. However, there were $\gamma$-ray contaminations above the
discriminator threshold level of 3.5 MeV due to the energy resolution tail.
The main source of the $\gamma$-rays was considered to be concrete walls
surrounding the detector. Cosmic-ray muons have wide energy range over GeV
scale. High energy muons generate fast neutrons by interactions in materials
composing the experimental site and fast neutrons turned out to be the
severest background for neutrino signals in this experiment. Most of the
background events produced by muons were excluded by the delayed coincidence
technique but there were still remaining backgrounds even after requiring it.
Those background events were further reduced by the data analysis as explained
in later sections.
Huge number of neutrons were produced in the reactor core, and a very small
fraction of them could reach the neutrino detector passing through materials
constructing the Joyo or crevices in the materials. Low energy neutrons are
detected as gamma-rays emitted via neutron capture on Gd. Figure 9 shows a
comparison of the reconstructed energy spectra for the reactor-on and off. An
excess in reactor-on was found around 8 MeV, which was considered to be made
by the low energy neutrons from the core. The excess rate between 4 MeV to 14
MeV was 6.1 Hz. Because energy distribution of the thermal neutron was
unknown, it was impossible to precisely calculate the detection efficiency.
Therefore we carried out rough estimation of the thermal neutrino flux.
Assuming the detection efficiency of 8 MeV gamma rays was 50% and considering
naively the total cross section for the neutrons to the detector was
equivalent to the surface area of the acrylic sphere, the neutron flux at the
detector site can be estimated to be approximately 10-4/cm2/sec. This neutron
flux is too small to be detected by usual neutron counters.
The background events which satisfy the delayed coincidence condition were
classified into two categories, namely accidental and correlated backgrounds.
The accidental background consists of two independent background events which
accidentally occur within the delayed coincidence time window. The main source
of such background events are environmental $\gamma$-rays followed by cosmic-
ray muons. The correlated backgrounds are caused by a continuous physics
process. Those processes include decays of cosmic-ray muons inside the
detector and fast neutrons induced by cosmic-ray muon followed by neutron
capture on Gd in the detector. In the former case, cosmic-ray muon causes
background to the prompt signal, and an electron from the muon decay (Michel
electron) is identified as the delayed signal. In the latter case, recoil
protons caused by a fast neutron are detected as a prompt signal, and the
$\gamma$-rays from neutron capture on Gd are identified as a delayed signal.
Some of radioactive isotopes produced by cosmic-ray muon interactions in the
detector cause coincidence signals in its decay chain and can be considered as
background source [22]. However, the production rate of such isotopes was
negligibly small compared to the other background sources in this experiment.
Figure 10 shows the reconstructed energy spectrum of the observed data with
0.6ṀeV of the threshold level together with the expected reactor neutrino
signal and background events from the MC simulation, in which delayed
coincidence cut condition was not required yet. Simulation of cosmic-ray muon
background was based on flux measured in [23], and correction factor 0.72 was
applied from a fit to the data for the high energy region between 20 MeV and
140 MeV. Fast neutron flux was obtained as 0.63 neutrons/cm2/sec above 10 MeV
(equivalent to visible electron energy of 3 MeV in the detector) from a fit to
the distribution of time interval ($\Delta t$) between the prompt and delayed
signals. As is shown in Figure 10, environmental $\gamma$-ray background is
dominant for the energy below 6 MeV before the delayed coincidence condition
is applied. We assumed typical concentration of radioactive isotopes in
concrete materials around the detector, 2.1 ppm of 238U, 5.1 ppm of 232Th and
1.4 ppm of 40K [24]. The observed energy spectrum was in reasonable agreement
with the expected background spectrum from the MC simulation after
normalization corrections were applied to the cosmic-ray muon and fast neutron
flux. As is shown in this figure, the background level is $10^{5}\sim 10^{7}$
times higher than the neutrino signals before the delayed coincidence. In
order to measure reactor neutrino signals significantly over the background,
reduction of the background events by the delayed coincidence technique and
further selections are necessary.
## 6 Neutrino event selection and results
Figure 11: Reconstructed energy spectra for the prompt (left) and delayed
(right) signal candidate events. Points show the observed data for 7.4 hours
live-time under reactor-off condition. Overlaid histograms show the expected
spectra of neutrino signals and background events from MC simulation.
We used the data taken with delayed coincidence trigger to search for neutrino
events. In order to reduce the backgrounds, following selection criteria were
applied to the data and the MC simulation.
* 1.
$4.5\leq E_{prompt}\leq 7\,{\rm MeV}$ and
$4.5\leq E_{delayed}\leq 11\,{\rm MeV}$,
where $E_{prompt}$ and $E_{delayed}$ are the reconstructed energy for the
prompt and delayed signals, respectively. Figure 11 shows the reconstructed
energy spectra for the prompt and delayed signals. The lower cut value at 4.5
MeV was set to reject environmental $\gamma$-rays, while the higher cut values
were set at 7 MeV and 11 MeV to select prompt signal shown in Figure 2 and
total 8 MeV $\gamma$-rays from neutron capture on Gd, respectively.
Especially, the delayed energy selection is very effective for rejection of
Michel electron events with 53MeV of end point, in contrast to the fast
neutron events induced by cosmic muons which have a similar distribution to
neutrino delayed signals.
* 1.
$2.5\leq\Delta t\leq 60\,\mu$sec
Figure 12 shows the time difference between the prompt and delayed signals
($\Delta t$) for the events after the energy cuts. Michel electron events have
distribution following 2.2 $\mu$s of muon life time, while the fast neutron
and neutrino events have corresponding distributions with the decay time of
46.4 $\mu$s, which is determined by Gd concentration in liquid scintillator.
Although the coincidence condition within 100 $\mu$s time window was required
in the data acquisition, further cut was applied in the analysis. The lower
limit was used to reject remaining Michel electron events after the energy
cuts, while the upper limit was set to collect enough neutrino events with
46.4 $\mu$s decay time.
In order to further reduce background events remained after the energy and
$\Delta t$ cuts were applied, charge balance ($CB$) is defined as follows:
$CB=\sqrt{\dfrac{16\left(\sum^{16}_{i=1}{\left(Q^{cor}_{i}\right)^{2}}\right)}{\left(\sum^{16}_{i=1}{Q^{cor}_{i}}\right)^{2}}-1},$
(5)
where $Q^{corr}_{i}$ is observed charge from $i$-th PMT after gain correction
was applied. This variable becomes large for the external background events,
such as environmental gamma rays and fast neutrons, with the vertex position
close to the surface of the detector while it is smaller for events occuring
near the center. So $CB$ cut corresponds to a kind of fiducial volume cut.
However when the vertex position is too close to the surface and between the
near PMT surfaces, the $CB$ becomes smaller and the vertex position mimics a
place around the center of the detector, because solid angles to the near PMT
surfaces from the vertex position are narrow and number of photoelectrons for
the near PMTs becomes less. Figure 13 shows the $CB$ distributions of reactor
neutrino and background events after energy and $\Delta t$ cuts were applied.
Distributions for the background events have a valley at $CB\sim 1.3$. To
maximize signal over noise ratio (S/N), cut conditions for $CB$ were defined
as:
* 1.
$0.8\leq CB_{prompt}\leq 1.4$ and $0.8\leq CB_{delayed}\leq 1.4$
Figure 12: $\Delta t$ distributions for events which satisfy the energy cut: $4.5\leq E_{prompt}\leq 7$ MeV and $4.5\leq E_{delayed}\leq 11$ MeV. Points show the observed data for 7.4 hours live-time taken in reactor-off condition. Overlaid histograms show the expectations of neutrino signal and background events from MC simulation. Selection Criterion | Event rate (/day) | S/N ratio
---|---|---
Trigger level | 162 | -
Energy cut | 8.89 | 1/1197
Coincidence cut | 7.01 | 1/1009
Charge balance cut | 0.988 | 1/128
Vertex $\phi$ cut | 0.494 | 1/34.6
Table 3: Effect of selection criteria on the reactor neutrino events in the
detector. Each cut condition is described in Section 6. Figure 13: Charge
balance ($CB$) distributions for the events which satisfy energy and $\Delta
t$ cuts. Left-hand and right-hand figures show the distributions for prompt
and delayed signals, respectively. Points show the observed data for 7.4 hours
live-time taken in reactor-off condition. Overlaid histograms show expected
distributions of neutrino signal and background events from MC simulation.
Even after the energy, $\Delta t$ and charge balance cuts were applied, the
remaining background events were still hundred times larger than the reactor
neutrino events. The dominant component of the remaining background events was
fast neutrons generated from muons in materials around the detector. Figure 14
shows the vertex $\phi$ distributions for the prompt and delayed signals after
energy, $\Delta$t and charge balance cuts were applied. $\phi$ is the
azimuthal angle in spherical polar coordinates as the $z$-axis vertical to the
ground. Then $\phi=0$ was defined as north of the detector. The vertex
position is reconstructed by a fit with expected charge of each PMT based on
the scintillation light yield and a solid angle to the PMT from the vertex
position. It is expected that the vertex positions of the neutrino
interactions distribute uniformly in the detector. On the other hand, the
vertex $\phi$ distributions of the data are not flat due to asymmetric
arrangement of building materials and paraffin shields. Therefore, we applied
the following cut based on the vertex $\phi$ position to maximize S/N ratio:
* 1.
$-100^{\circ}\leq\phi_{prompt}\leq 100^{\circ}$ and
$-100^{\circ}\leq\phi_{delayed}\leq 100^{\circ}$
The neutrino event selection cuts and expected event rates are summarized in
Table 3.
### 6.1 Result of the reactor neutrino event selection
The result of the neutrino event selection is summarized in Table 4. The
accidental background event rates were estimated from the measurements of
single background event rates and subtracted from the total event rates as
shown in Table 4. Then, the difference between the event rates for reactor-on
and reactor-off was calculated. Errors in Table 4 are only the statistical
ones. After reactor neutrino event selection and subtraction of accidental
background and reactor-off data, event rate for the neutrino candidate events
from 38.9 days of reactor-on data and 18.5 days of rector-off data was
obtained as 1.11$\pm$1.24(stat.)$\pm$0.46(syst.) events/day, while the
expected neutrino signal event rate from the MC simulation was 0.49
events/day. The systematic uncertainties were estimated considering the
uncertainties in energy resolution, energy scale and vertex reconstruction.
Figure 15 shows the prompt energy spectrum after all selections except for the
prompt energy cut. The measured excess rate was consistent with the expected
neutrino rate from the MC simulation, but also consistent with zero within the
systematic error. So the observation of neutrinos from experimental fast
reactor JOYO has not been statistically established in this measurement. A new
design of the detector is described in the following section, in which
sensitivity to the measurement of reactor neutrinos was estimated based on the
observed background event rates shown in this paper.
Figure 14: Reconstructed vertex $\phi$ distribution after the cuts of energy, $\Delta t$ and $CB$ were applied. Points shows the data for 18.5 days live-time taken in reactor-off condition. An overlaid histogram shows the expectation of neutrino signals. | Reactor-on | Reactor-off | $\Delta$(on $-$ off)
---|---|---|---
Live-time | 38.9 days | 18.5 days | -
Total | 19.0$\pm$0.7 | 17.1$\pm$1.0 | 1.93$\pm$1.19
Accidental | 2.34$\pm$0.24 | 1.52$\pm$0.29 | 0.82$\pm$0.38
Correlated | 16.7$\pm$0.7 | 15.6$\pm$1.0 | 1.11$\pm$1.24
Reactor $\nu$ MC | - | - | 0.494
Table 4: Observed event rates (events/day) and the statistical uncertainties
after the neutrino event selection criteria were applied. Accidental
background event rates were estimated by single background event rate.
Correlated event rates were obtained by subtracting the accidental BG event
rate from the total event rate. Figure 15: Reconstructed energy spectrum after
neutrino event selection criteria except for the prompt energy cut was
applied. Points show the data with the statistical errors in which energy
spectrum measured for reactor-off is subtracted from that taken for reactor-
on. Accidental background event rates were estimeted from the data and
subtracted. Boxes show the expected reactor neutrino energy spectrum from MC
simulation with the MC statistical errors.
## 7 New detector design for the next experiment
Problems found in the JOYO experiment were following.
* (1)
Statistics of the data was limited because the long term data taking was
impossible due to degradation of Gd-LS.
* (2)
Energy threshold level could not be sufficiently lowered due to large amount
of environmental $\gamma$-rays from outside of the detector. Therefore, the
neutrino detection efficiency was obliged to be low.
* (3)
Performance of event vertex reconstruction was not enough to distinguish
external background events entering the detector due to the same problem
caused by the detector structure as in the description for $CB$ in Section 6.
* (4)
Fast neutron background level was too high against the neutrino signals, for
which S/N ratio was 0.029 .
We designed a new detector for the next experiment solving these problems, and
estimated the sensitivity to reactor neutrino measurements based on the MC
simulation.
We are considering a liquid scintillator with high Pseudocumene concentration
above 99 w% as possible candidate to solve the Gd-LS degradation problem. The
high aromatic concentration is supposed to stabilize the Gd-LS. Experimental
studies of the long-term stability and characteristics of the proposed
Pseudocumene based Gd-LS are necessary. In addition, the detector design needs
to be improved to suppress fast neutron background. The new detector will
consist of two concentric sphere vessels. The inner vessel contains the Gd-LS
as target of reactor neutrinos. The outer vessel is filled with paraffin oil
with no scintillation light emission, and works as shield against fast
neutrons. In the MC simulation, we assumed the same target volume with Gd-LS,
surrounded by 20 cm layer of buffer oil. Environmental $\gamma$-rays are also
reduced by the outer layer, by which we estimate that the energy threshold can
be lowered to 3 MeV with the same trigger rate as the measurement at JOYO. The
scintillation lights are viewed by 24 10-inch PMTs isotropically arranged on
the surface of the outer vessel, providing 11 % photo-cathode coverage close
to JOYO detector. The PMT surfaces are away from the target vessel with the
interval buffer region by which performance of vertex position reconstruction
can be improved especially for those close to the surface of the target
vessel. The reconstructed vertex radii in the polar coordinate system are used
for rejection of the external background events.
According to the study using the MC simulation, we expect the S/N ratio is
improved from 0.029 to 0.093, with about 16 times larger neutrino selection
efficiency by the new detector design. If we put the same detector at MONJU
rector site [25], of which thermal power is approximately five times higher
than JOYO reactor, the S/N ratio is further improved to 0.48 with 41
events/day of neutrino observations. Significance of fast reactor neutrino
observation reaches 2 standard deviation after 12 days of reactor-on and off
live-times at JOYO reactor site or 1 day at MONJU reactor site.
## 8 Conclusions
We carried out an experimental study of fast reactor neutrino detection and
measured the background spectrum at fast reactor JOYO using a compact
detector. The observed reactor neutrino candidate signal was
1.11$\pm$1.24(stat.)$\pm$0.46(syst.) events/day after subtraction of
background events while the expected reactor neutrino event rate from the MC
simulation was 0.49 events/day. As a result, the first observation of fast
reactor neutrinos was not statistically established in this measurement. On
the other hand, various background sources at the ground level nearby the
reactor were studied in detail and those backgrounds were found to be
reproduced well by our MC simulation. These background studies will be useful
not only for the R&D of future reactor neutrino oscillation experiments but
also for the development of compact reactor neutrino detector as a remote
monitor for plutonium breeding.
A design concept of a new detector and its sensitivity to the observation of
fast reactor neutrinos were also described in this paper based on the
knowledge acquired by the measurement at JOYO. We expect the S/N ratio will be
improved from 0.029 to 0.093 by the new detector design, and it is further
improved to 0.48 if we put the detector at MONJU reactor site at the same
distance from the core. Expected reactor neutrino signal by the new detector
design is 8.0 events/day and 41 events/day at JOYO and MONJU reactor site,
respectively.
## 9 Acknowledgements
This work was supported by a grant-in-aid for scientific research (#16204015)
of Japan Ministry of Education, Culture, Sports, Science and Technology
(MEXT). This work was performed in cooperation with Japan Atomic Energy Agency
(JAEA). Especially, we would like to thank T. Aoyama and T. Kuroha for
supporting us in various ways. We thank K2K experimental group for providing
the muon veto counters for this measurement.
## References
* [1] F. Reines, C. L. Cowan, Jr., Nature 178, 446 (1956).
* [2] K. Anderson et al., arXiv:hep-ex/0402041.
* [3] A. Bernstein et al., arXiv:nucl-ex/0908.4338.
* [4] The 6th International Workshop on Applied Anti-neutrino Physics (AAP2010), ’http://www.awa.tohoku.ac.jp/AAP2010’ and links therein.
* [5] M. Cribier, arXiv:nucl-ex/0704.0891.
* [6] ’Focused Workshop on Antineutrino Detection for Safeguard Applications’ IAEA Headquarters, Vienna, Oct.28-30,2008.
* [7] 235U: K. Schreckenbach et al., Phys. Lett. B 160, 325 (1985), 239,241Pu : A. A. Hahn et al., Phys. Lett. B218, 365 (1989), 238U: P. Vogel et al., Phys. Rev. C 24, 1543 (1981).
* [8] M. F. James, J. Nucl. Energy 23, 517-536 (1969).
* [9] N. S. Bowden et al., Nucl. Instrum. Meth. A572 (2007) 985-998.
* [10] N. Aoki et al., arXiv:hep-ex/0607013.
* [11] H. Furuta, PhD thesis, Tokyo Institute of Technology (2009).
* [12] T. Aoyama et al., Nuclear Engineering and Design 237 (2007) 353-368.Oct. 28-29, 2008.
* [13] B. Achkar et al., Nucl. Phys. B 434, 503-532 (1995).
* [14] V. I. Kopeikin, L. A. Mikaelyan, and V. V. Sinev, Phys. At. Nucl. 60, 172 (1997).
* [15] P. Vogel and J. F. Beacom, Phys. Rev. D60, 053003 (1999).
* [16] M. Apollonio et al., Eur. Phys. J. C 27, 331 (2003).
* [17] Y. Yasu et al., Proceedings of 13th IEEE-NPSS Real Time Conference 2003 (2003).
* [18] Y. Sakamoto et al., Proceedings of 15th IEEE NPSS Real Time Conference 2007 (2007).
* [19] S. Agostinelli et al., Nucl. Instrum. Meth. A506 (2003) 250-303: J. Allison et al., IEEE Transactions on Nuclear Science 53 No. 1 (2006) 270-278.
* [20] M. G. Marino et al., Nucl. Instrum. Meth. A582 (2007) 611-620.
* [21] J. B. Birks, Theory and Practice of Scintillation Counting, Pergamon Press, 1964.
* [22] S. Abe et al., arXiv:hep-ex/0907.0066.
* [23] S. Haino et al., Phys. Lett. B594 (2004).
* [24] H. Furuta et al., Nucl. Instrum. Meth. A568 (2006) 710-715.
* [25] ’http://www.jaea.go.jp/04/monju/EnglishSite/index.html’ and links therein.
|
arxiv-papers
| 2011-08-14T23:10:57 |
2024-09-04T02:49:21.527675
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Furuta, Y. Fukuda, T. Hara, T. Haruna, N. Ishihara, M. Ishitsuka,\n C. Ito, M. Katsumata, T. Kawasaki, T. Konno, M. Kuze, J. Maeda, T. Matsubara,\n H. Miyata, Y. Nagasaka, K. Nitta, Y. Sakamoto, F. Suekane, T. Sumiyoshi, H.\n Tabata, M. Takamatsu, N. Tamura",
"submitter": "Hisataka Furuta",
"url": "https://arxiv.org/abs/1108.2910"
}
|
1108.3061
|
# Min-type Morse theory for configuration spaces of hard spheres
Yuliy Baryshnikov Departments of Mathematics and ECE, UIUC, Urbana, IL
ymb@uiuc.edu , Peter Bubenik Department of Mathematics, Cleveland State
University p.bubenik@csuohio.edu and Matthew Kahle School of Mathematics,
Institute for Advanced Study, Princeton NJ 08540 mkahle@math.ias.edu We
dedicate this paper to the memory of Boris Lubachevsky.
###### Abstract.
In this paper we study configuration spaces of hard spheres in a bounded
region. We develop a general Morse-theoretic framework and show that
mechanically balanced configurations play the role of critical points. As an
application, we find the precise threshold radius for a configuration space to
be homotopy equivalent to the configuration space of points.
The third author thanks IAS and NSA Grant # H98230-10-1-0227
## 1\. Introduction
Configuration spaces of $n$ points in ${\mathbb{R}}^{d}$ are well studied [4].
In this article we are interested in a natural generalization, configuration
spaces of non-overlapping balls in a bounded region in ${\mathbb{R}}^{d}$.
Besides their intrinsic mathematical interest, the study of these spaces is
motivated by physical considerations. For example, in statistical mechanics
“hard spheres” (or in two dimensions “hard disks”) are among the most well-
studied models of matter. Computer simulations suggest a solid-liquid phase
transition for hard spheres [17], but this is not well understood
mathematically.
A number of papers in statistical mechanics have explored the hypothesis that
underpinning phase transitions are changes in the topology of the underlying
configuration space or equipotential submanifolds [24, 16, 1, 9]. Franzosi,
Pettini, and Spinelli show that under fairly general conditions (smooth,
finite-range, confining potentials), the Helmholtz free energy cannot pass
through a phase transition unless there is a change in the topology of the
underlying configuration space [11, 10]. This theorem unfortunately does not
apply to configuration spaces of hard spheres, since the potential function is
not smooth — but the Morse-theoretic methods developed here may be a step in
the direction of extending it to include hard spheres.
Several other papers have investigated configuration spaces as models of
motion planning for robots [8, 13]. For example, Farber’s “topological
complexity” can be thought of as measuring the difficulty of designing an
algorithm for navigating the space. As Deeley recently pointed out when he
studied “thick particles” on metric graphs, the assumption that robots are
points is not physically realistic, and giving the points thickness wildly
complicates the topology of the underlying configuration space [6].
Let ${\mathcal{B}}$ be a bounded region in ${\mathbb{R}}^{d}$. Define
${{\mathtt{Conf}}}(n,r)$ to be the configuration space of $n$ non-overlapping
balls of radius $r$ in ${\mathcal{B}}$. We are especially interested here in
understanding when the topology changes if $n$ is fixed and $r$ is varying .
First we consider the extreme cases. For $r$ sufficiently small, one expects
that ${{\mathtt{Conf}}}(n,r)$ is homotopy equivalent to the configuration
space ${\mathtt{Conf}}(n)$ of $n$ distinct points in ${\mathcal{B}}$ — for a
survey of configuration spaces of points see Cohen [4]. On the other hand for
$r$ sufficiently large, ${{\mathtt{Conf}}}(n,r)$ is empty. Indeed finding the
smallest such $r$ is the sphere packing problem in a bounded region — see for
example Graham et al. [15, 2, 18, 19] and Melissen [22, 21].
In this note we develop a Morse-theoretic framework which provides a necessary
condition for the topology to change — mechanical balanced configurations play
the role of critical points (and submanifolds). As an illustration of the
method, we find the precise threshold radius below which
${\mathtt{Conf}}(n,r)$ is homotopy equivalent to ${\mathtt{Conf}}(n)$.
## 2\. Tautological Morse function
Fix $n$, and define ${\mathtt{Conf}}(n)$ to be the set of ordered $n$-tuples
of distinct points in a bounded domain ${\mathcal{B}}\subset{\mathbb{R}}^{d}$:
${\mathtt{Conf}}(n)=\\{{\vec{x}}=(x_{1},\ldots,x_{n})\mid
x_{i}\in{\mathcal{B}},x_{i}\neq x_{j}\text{ for }i\neq j\\}.$
As an open subset of ${\mathbb{R}}^{dn}$, ${\mathtt{Conf}}(n)$ has the
structure of a smooth manifold. Let
${\mathbf{\tau}}:{\mathtt{Conf}}(n)\to{\mathbb{R}}$ be defined by
(1) ${\mathbf{\tau}}({\vec{x}}):=\min\left(\frac{1}{2}\min_{i\neq
j}d(x_{i},x_{j}),\min_{i}\min_{p\in\partial{\mathcal{B}}}d(x_{i},p)\right),$
where $\partial{\mathcal{B}}$ denotes the boundary of ${\mathcal{B}}$. We call
$\tau$ the tautological function. Then by definition the configuration space
of $n$ balls of radius $r$ in ${\mathcal{B}}$ is given by
${{\mathtt{Conf}}}(n,r)={\mathbf{\tau}}^{-1}[r,\infty).$
This observation suggests using a “Morse”-type theory of ${\mathbf{\tau}}$ to
study the topology of ${{\mathtt{Conf}}}(n,r)$ and especially how the topology
changes as $r$ varies. One obvious trouble on that route is the fact that
${\mathbf{\tau}}$ typically is not smooth, so that we need a general framework
which allows us to work with non-smooth functions.
In the next section we will discuss the properties of min-type functions, that
is functions on a manifold $M$ given as the minimum of a parametric family of
real valued functions
${\mathbf{\tau}}(x):=\min_{p}f(p,x),x\in M,p\in P,$
where $P$ is a compact parameter space, and $f$ is continuously differentiable
in $x$ for every fixed $p\in P$. We note that the function given by (1) falls
within this category, if one considers as $P$ the disjoint union of the
discrete set corresponding to pairs $(i,j),1\leq i<j\leq n$ and of $n$ copies
of the boundary, formed by the pairs $(i,p),1\leq i\leq
n,p\in\partial{\mathcal{B}}$.
It should be remarked that the Morse-type theory of the min (or even min-max)
type functions has appeared in the literature (compare [20, 3, 12]), but in a
much more restrictive context (with essentially finite parameter space).
## 3\. Min-type Morse theory
Let us start with some notation and definitions.
For a manifold $M$ and function $f:M\to{\mathbb{R}}$, let $M^{c}$ denote the
superlevel set at $c$, i.e. $M^{c}=f^{-1}[c,\infty)$.
We say that a function $h:(s,t)\to{\mathbb{R}}$, is increasing with speed at
least $v>0$ if $h(t^{\prime})-h(s^{\prime})\geq v(t^{\prime}-s^{\prime})$ for
any $s^{\prime}<t^{\prime}$ in the interval $(s,t)$. We note that such a
function does not have to be even continuous.
We record for later use the following (immediate) result:
###### Lemma 3.1.
Let $f:M\times{\mathbb{R}}\to{\mathbb{R}}$ be a continuous proper function,
(strictly) increasing along each fiber $\\{x\\}\times{\mathbb{R}}$. Then the
fiber-wise inverse $\phi:(x,c)\mapsto\inf(t:f(x,t)=c)$ is continuous on
$M\times[c_{1},c_{2}]$ for any interval $[c_{1},c_{2}]$ which belongs to the
ranges of all the functions $f_{x}(\cdot):=f(x,\cdot)$.
###### Proof.
As $f$ is proper, then the $t$-projection of the preimage of a compact
interval $[c_{1},c_{2}]$ is compact as well. Further, if a sequence
$(x_{i},c_{i})$ converges to $(x_{*},c_{*})$, yet $t_{i}:=\phi(x_{i},c_{i})$
fails to converge to $t_{*}:=\phi(x_{*},c_{*})$, we can, using the boundedness
of the sequence $(t_{i})$, choose a subsequence such that along it $t_{j}\to
t\neq t_{*}$. By continuity, $\lim f(x_{j},t_{j})=f(x_{*},t)$ (as $x_{j}$
converge to $x_{*}$). The fact that $f(x_{j},t_{j})=c_{j}$ implies that
$f(x_{*},t)$ equals $c_{*}$, which by the continuity of $f$ also equals
$f(x_{*},t_{*})$. This contradicts the assumption that $f$ increases fiber-
wise. ∎
For a smooth vector field $V$ on $M$ we will denote the time $t$ shift along
the trajectories of $V$ as $S_{t}^{V}$. We will say that the function $f$
increases along the trajectories of $V$ with non-zero speed, if for some
common $v>0$, and for all $x\in M$, $h_{x}:t\mapsto f(S^{V}_{t}x)$ increases
with speed at least $v$.
###### Lemma 3.2.
Let $M$ be a smooth manifold and $f:M\to{\mathbb{R}}$ a continuous function,
such that $M^{a}$ is compact. Suppose that $M$ admits a smooth vector field
$V$ non-vanishing on $f^{-1}[a,b]$, and such that $f$ is increasing along the
trajectories of $V$ on the set $f^{-1}[a,b]$ with non-zero speed. Then $M^{b}$
is a deformation retract of $M^{a}$.
Lemma 3.2 can be seen as a generalization of Theorem 3.1 in [23] for non-
smooth functions $f$. We remark here that one can drop here the non-zero speed
condition, requiring only that $f$ is increasing along the trajectories of
$V$, but we do not need this strengthened form in this paper.
###### Proof.
For $x\in M^{a}$ set a partially defined function on $M^{a}\times{\mathbb{R}}$
by $g(x,t):=f(S_{t}^{V}x)$. The non-zero speed condition implies that $[a,b]$
is in the range of $g(x,\cdot)$ for any $x\in f^{-1}([a,b])$, and together
with the compactness of $M^{a}$ implies that $g$ is proper. Hence, by Lemma
3.1, $\tilde{\phi}(x,c)=\inf(t:f(S_{t}^{V}x)\geq c)$ is well defined and
continuous, as well as $\phi(x,c)=\max(\tilde{\phi}(x,c),0)$. As for any
$c\leq f(x)$, $\phi(x,c)=0$, $\phi$ vanishes on $M^{b}\times[a,b]$.
Now define the homotopy
$H:M^{a}\times[0,1]\to M^{a}$
as
$H:(x,\tau)\mapsto S^{V}_{\phi(x,(1-\tau)a+\tau b)}x.$
Continuity of $\phi$ implies continuity of $H$; the facts that
$H(x,0)=\mathit{id}_{M^{a}}$, $H(x,\tau)|_{M^{b}}=\mathit{id}_{M^{b}}$ for
$0\leq\tau\leq 1$ and $H(x,1)\in M^{b}$ are immediate. ∎
### 3.1. Regular values of min-type functions
Next we will use the fact that the tautological function ${\mathbf{\tau}}$ is
the minimum of a compact family of smooth functions. We want to establish
conditions when a value $c$ is topologically regular, that is for which there
exists some $\epsilon>0$ such that $M^{c+\epsilon}$ is a deformation retract
of $M^{c-\epsilon}$.
We give a general condition for topological regularity, as follows.
Let $P$ be a compact metric space, $M$ a compact smooth manifold with boundary
and
$f:P\times M\to{\mathbb{R}}$
a continuous function such that the $x$-derivative of $f$ (that is, the
gradient of $f_{p}$, where $f_{p}(x)=f(p,x)$) is continuous on $P\times M$. We
will be talking of $P$ as the parameter space.
We denote by ${\mathbf{\tau}}:=\min_{p\in P}f_{p}$ the min-function of the
family $f$. The set $N\subset P\times M$ defined by
$N:=\\{(p,x):f(p,x)={\mathbf{\tau}}(x)\\}$
is compact, and the slices
$N_{x}:=\\{p\in P:(p,x)\in N\\}$
are upper semi-continuous: for any $x\in M$ and any open neighborhood
$UN_{x}\supset N_{x}$ there exists an open neighborhood $Ux\ni x$ such that
for $x^{\prime}\in Ux$, $N_{x^{\prime}}\subset UN_{x}$.
Next we show that if one can perturb each $x$ to increase $\tau$ then we can
do so globally with a minimum speed.
###### Lemma 3.3.
Assume that for any $x\in M$, there exists a tangent vector $V_{x}\in T_{x}M$
such that $L_{V_{x}}f_{p}>0$ for all $p\in N_{x}$. Then
* •
for some positive $v$ there exists smooth vector field $V$ on $M$ such that
$L_{V}f_{p}\geq v>0$ in some open vicinity of $N$, and
* •
along the trajectories of $V$, the min-function ${\mathbf{\tau}}$ increases
with speed at least $v$.
###### Proof.
For any $x\in M$, we can extend the vector $V_{x}\in T_{x}M$ to a smooth
vector field on $M$ (which we still denote as $V_{x}$), such that
$L_{V_{x}}f>0$ in some open vicinity $UN_{x}\times Ux$ of
$N_{x}\times\\{x\\}$. By compactness, there exists a finite collection of
points $\\{x_{i}\\}$ in $M$ such that the open sets $U_{i}:=UN_{x_{i}}\times
Ux_{i}$ cover $N$ (and the open sets $Ux_{i}$ cover $M$), and $v>0$ such that
$L_{V_{i}}f_{p}\geq v$ on $U_{i}$ (here $V_{i}:=V_{x_{i}}$). Using a partition
of unity we arrive at the first conclusion. The second conclusion is
immediate. ∎
For $x\in M$ consider the intersection of the open half-spaces
$H_{x}(p):=\\{v\in T_{x}M:\langle df_{p}|_{x},v\rangle>0\\}$
over all $p\in N_{x}$. This is an open convex cone
$C^{o}_{x}:=\bigcap_{p\in N_{x}}H_{x}(p)$
in $T_{x}M$.
Upper semicontinuity of $N_{x}$ implies lower semicontinuity of $C^{o}_{x}$;
for any $x\in M$ and any open set $V\subset TM$ intersecting $C^{o}_{x}$,
there exists an open neighborhood $Ux\ni x$ such that for $x^{\prime}\in Ux$,
$C^{o}_{x^{\prime}}$ intersects $V$. In particular, if $C^{o}_{x}$ is non-
empty, it remains such in a vicinity of $x$.
Combining Lemmata 3.2 and 3.3 we obtain the following
###### Corollary 3.4.
If the cones $C^{o}_{x}$ are non-empty over the level set
${\mathbf{\tau}}^{-1}(c)$, then $c$ is topologically regular.
For general min-type functions this is essentially the best possible condition
for the regularity of the critical values. If the functions $f_{p}$ are quasi-
convex, i.e. have convex lower excursion sets $\\{f_{p}\leq c\\}$, then
Corollary 3.4 can be considerable strengthened. Thus, one can show (we will do
it in a follow-up paper) that a critical value is topologically regular, if
for all points at the level set, the intersection of the closed half-spaces is
a cone over a contractible base. This observation relies on stratified Morse
theory due to Goresky and MacPherson[14], but is in a nutshell close to the
elementary result used by Connelly in his work on the existence of continuous
“unlocking” deformations of hard ball configurations, see [5].
Corollary 3.4 implies that unless the level set of the tautological function
$\tau^{-1}(r)$ contains a point $x$ with $C^{o}_{x}=\varnothing$, the homotopy
type of ${{\mathtt{Conf}}}(n,r)$ is locally constant at $r$.
By Farkas’ lemma, the emptiness of the cone $C^{o}_{x}$ implies that there
exists a finite collection of points $p_{i}\in N_{x},i=1,\ldots,I\leq\dim
M+1$, and positive weights $w_{i}>0$ such that
(2) $\sum_{i}w_{i}{df_{p_{i}}|_{x}}=0.$
## 4\. Critical points and stress graphs
In our hard spheres setting, the vanishing of the convex combination (2) has a
clear geometric interpretation.
For ${\vec{x}}\in{\mathtt{Conf}}(n,r)$, define a stress graph of ${\vec{x}}$
to be a graph embedded in ${\mathbb{R}}^{d}$ whose vertices are the points
$x_{1},\ldots,x_{n}$ and boundary points $y\in{\partial}{\mathcal{B}}$ where
$d(x_{i},y)=r$ for some $i$. The edges are the pairs $\\{x_{i},x_{j}\\}$ where
$d(x_{i},x_{j})=2r$ and $\\{x_{i},y\\}$ where $d(x_{i},y)=r$. Each edge $k$ is
assigned a positive weight $w_{k}$. The points $x_{i}$ are referred to as
internal points and the points $y$ are referred to as boundary points. We
interpret this graph as a system of mechanical stresses, with (repulsive)
forces acting on the endpoints of a segment $k$ equal to $w_{k}$ times the
unit vector in the direction of $k$. Call the mechanical stresses acting on
boundary points boundary mechanical stresses. Call a connected component
trivial if it consists of a single point. Call ${\vec{x}}$ trivial if
$\Gamma({\vec{x}})$ has no edges.
A stress graph is said to be balanced if it satisfies the following condition.
* •
The mechanical stresses at each internal point sum to zero.111This result is
similar to the necessary conditions for “locking”, see e.g. [5].
* •
The boundary mechanical stresses on each connected component sum to zero.
Say that the configuration is ${\vec{x}}$ is balanced if it has a balanced
stress graph.
Call an internal point isolated if it is not in the boundary of any edges. For
each point $x_{i}$ call the intersection of the stress graph with the points
on the sphere $d(x_{i},x)=r$ kissing points of $x_{i}$. Call a kissing point
that is also a boundary point a boundary kissing point.
###### Lemma 4.1.
Assume that ${\vec{x}}\in{\mathtt{Conf}}(n,r)$ is balanced. Then
1. (1)
each non-isolated internal point is in the convex hull of its kissing points,
and
2. (2)
each non-trivial connected component is contained in the convex hull of its
boundary kissing points.
Now consider ${\vec{x}}\in{\mathtt{Conf}}(n)$ that is a critical point of
$\tau$ with critical value $r$. In (2), the parameters $p_{i}$ correspond
either to the pairs of touching hard spheres, $d(x_{a_{i}},x_{b_{i}})=2r$, or
to the hard sphere $x_{c_{i}}$ touching the boundary, $d(x_{c_{i}},y_{i})=r$,
at a point $y_{i}\in\partial{\mathcal{B}}$. Let $\Gamma({\vec{x}})$ be the
corresponding stress graph for ${\vec{x}}$ with weights given by the
coefficients $w_{i}$ in (2).
###### Theorem 4.2.
If ${\vec{x}}\in{\mathtt{Conf}}(n)$ is a critical point of $\tau$ with
critical value $r$, then ${\vec{x}}$ is balanced and nontrivial as a point in
${\mathtt{Conf}}(n,r)$.
###### Proof.
Consider ${\vec{x}}\in{\mathtt{Conf}}(n,r)$ and consider $\Gamma({\vec{x}})$.
From (2) it follows that the mechanical stresses at each internal point sum to
zero.
The sum of mechanical stresses in each connected component equals the sum of
mechanical stresses on internal points and the sum of external mechanical
stresses. From (2) and the first observation it follows that the boundary
mechanical stresses on each connected component sum to zero.
Finally, since the sum in (2) is nontrivial, ${\vec{x}}$ is nontrivial. ∎
## 5\. Hard spheres in a box
Consider now in more detail the case of hard spheres in a rectangular box with
sides $L:=L_{1}\leq\ldots\leq L_{d}$, given, for definitiveness sake, by
${\mathcal{B}}=\\{0\leq f_{m}\leq L_{m},m=1,\ldots,d\\}.$
(Here $\\{f_{m}\\}$ is the orthonormal coordinate system on
${\mathbb{R}}^{d}$.)
### 5.1. Initial interval
We show that Theorem 4.2 implies a lower bound on the length of the initial
interval of values of $r$, where the homotopy type of ${{\mathtt{Conf}}}(n,r)$
remains constant.
###### Theorem 5.1.
For the rectangular box ${\mathcal{B}}$, there are no critical values of
${\mathbf{\tau}}$ in $(0,L/2n)$, and therefore,
${{\mathtt{Conf}}}(n,r)\simeq{\mathtt{Conf}}(n)$
for $r<L/2n$.
###### Proof.
Assume that ${\vec{x}}\in{\mathtt{Conf}}(n)$ is a critical value for $\tau$
with critical value $r$. Then by Theorem 4.2, ${\vec{x}}$ is balanced and
nontrivial. A connected component of $\Gamma({\vec{x}})$ contains at most $n$
internal points and thus has diameter at most $2nr$. Since ${\vec{x}}$ is
nontrivial, it has at least one nontrivial connected component. It is
contained in the convex hull of its boundary points, so it contains at least
one boundary point. Since the boundary mechanical stresses of this connected
component sum to zero, it must contain a pair of boundary points from opposing
faces. Thus the diameter of this connected component is at least $L$.
Therefore $r\geq L/2n$. ∎
We remark that the balanced stress graph of minimal diameter is not
necessarily a segment, for non-rectangular boxes. For example, for the
“concave triangle”, it is a cone over three points, see Figure 1.
Figure 1. Minimal length stress graph
### 5.2. First perestroika
A natural question is now to ask, whether there is a topology change as $nr$
goes above the minimal length of the stress graph. We concentrate in the rest
of the note on the case of the rectangular box with the shortest side of
length $L$, and will investigate, whether
$i:{{\mathtt{Conf}}}(n,r^{\prime})\to{{\mathtt{Conf}}}(n,r),r^{\prime}=L/2n+\epsilon,r=L/2n-\epsilon$
is a homotopy equivalence, for small enough $\epsilon$.
We argue that it is not, by presenting explicit nontrivial $(dn-n-d)$-cycles
in $\ker(Hi)\subset H_{dn-n-d}({{\mathtt{Conf}}}(n,r^{\prime}),{\mathbb{Z}})$.
Indeed, let $0<\epsilon<L/2n(n-1)$ (so that $(n-1)$ disks of radius
$r^{\prime}$ would fit within the box when arranged in a vertical column, and
$n$ would not), and consider the set $S_{\epsilon}$ of $n$-point
configurations in ${\mathcal{B}}$ given by the conditions
* •
$x_{1}$ fixed is at distance $r^{\prime}$ from the center of the face
$\\{f_{1}=0\\}$,
* •
$|x_{i+1}-x_{i}|=2r^{\prime}$ for $i=1,\ldots n-1$, and
* •
$x_{n}$ is at distance $r^{\prime}$ from the face $\\{f_{1}=L_{1}\\}$.
In other words, we consider the configurations for which the $n$ disks touch
each other and the opposite horizontal faces, forming a chain, see Figure 2.
Figure 2. A configuration in $S_{\epsilon}$.
An immediate computation shows that $S_{\epsilon}$ is diffeomorphic to a $(nd-
n-d)$-dimensional sphere. Orient it in some way, obtaining a class $s\in
H_{nd-n-d}({{\mathtt{Conf}}}(n,r^{\prime}))$.
Next we show that $s$ is nontrivial by constructing a cohomology class with
which it has a nontrivial pairing.
Consider now the set $\Sigma$ of configurations in ${\mathcal{B}}^{n}$ given
by
* •
all points $x_{1},\ldots,x_{n}$ have the same coordinates
$f_{3},\ldots,f_{d}$;
* •
all points $x_{2},\ldots,x_{n}$ have the same coordinates
$f_{2},\ldots,f_{d}$;
* •
the $f_{1}$ coordinates of $x_{2},\ldots,x_{n}$ satisfy
$f_{1}(x_{1})\geq r;f_{1}(x_{i+1})-f_{1}(x_{i})\geq
2r,\mathrm{for\,}i=2,\ldots,n-1;f_{1}(x_{n})\leq L-r,$
and
* •
$f_{2}(x_{1})\leq f_{2}(x_{2})$.
In other words, the configuration consists of $n-1$ vertically aligned
nonoverlapping $r$-disks constrained to have the same $(f_{1},f_{2})$-plane
(with the same $f_{3},\ldots_{d}$ coordinates as the disk $x_{1}$), see Figure
3.
Figure 3. A configuration in $\Sigma$.
The conditions above are given by a finite collection of linear equalities and
inequalities, and therefore define a convex polyhedron of dimension $d+n$. The
boundary of this polyhedron is in
${\mathcal{B}}^{n}-{{\mathtt{Conf}}}(n,r^{\prime})$, whence, upon orientation
it defines a relative class $\sigma\in
H_{n+d}({\mathcal{B}}^{n},{\mathcal{B}}^{n}-{{\mathtt{Conf}}}(n,r^{\prime}))$.
We notice that the space ${\mathcal{B}}^{n}$ of $n$-tuples of points in
${\mathcal{B}}$ can be embedded into the $nd$-dimensional sphere $S^{nd}$
(consider a large ball containing ${\mathcal{B}}^{n}$ and contract its
boundary to a point). By excision and the long exact sequence for a pair,
$H_{n+d}({\mathcal{B}}^{n},{\mathcal{B}}^{n}-{{\mathtt{Conf}}}(n,r^{\prime}))\cong
H_{n+d}(S^{nd},S^{nd}-{{\mathtt{Conf}}}(n,r^{\prime}))\cong
H_{n+d-1}(S^{nd}-{\mathtt{Conf}}(n,r^{\prime}))$. By Alexander duality the
class $\sigma$ can be identified with a class (which we still denote by
$\sigma$) in $H^{nd-n-d}({{\mathtt{Conf}}}(n,r^{\prime}))$.
###### Lemma 5.2.
The pairing between the classes $s$ and $\sigma$ is non-trivial:
$s\cdot\sigma=\pm 1$.
###### Proof.
Indeed, the manifolds $S_{\epsilon}$ and $\Sigma$ intersect transversally at a
single point. ∎
As one can observe, there exists a retraction of $S_{\epsilon}$ to a point
staying within ${{\mathtt{Conf}}}(n,r)$, implying that the class $s$ is in the
kernel of $Hi$. Indeed, we first can reduce all the distances between by
shrinking the differences between the adjacent chain centers so that
$x_{i+1}-x_{i}\mapsto\frac{tr-(1-t)r^{\prime}}{r^{\prime}}(x_{i+1}-x_{i}),t=[0,1];i=1,\ldots,n-1$
and $x_{1}$ remains fixed (clearly, this homotopy keeps the configuration in
${{\mathtt{Conf}}}(n,r)$). Then one can pull all the vectors $(x_{i+1}-x_{i})$
so that they point vertically upwards222We think of $f_{1}$ as height. (as not
one was initially pointing downwards).
For each permutation $\pi$ of indices $1,\ldots,n$ one obtains different
classes $s_{\pi}$, and one can easily see that the pairing with the
corresponding $n!$ classes $\sigma_{\pi}$ is non-degenerate (because
corresponding $S_{\epsilon}$ and $\Sigma$’s are all geometrically distinct).
Hence, the rank of the kernel of $Hi$ is at least $n!$.
We notice that the sphericity of the set $S_{\epsilon}$ does not depend on the
fact that the stress graph is a chain. For the configuration on Figure 1, the
corresponding set is diffeomorphic to a sphere as well: it is just the
corollary of the first critical value coming from a topologically Morse
critical point, compare [20].
### 5.3. Betti numbers
We can also compute how the Betti numbers change across the first threshold.
Set $r_{*}=L/2n$, and note that as the tautological function is semi-algebraic
for semi-algebraic regions, its critical values are isolated. As the only
balanced stress graphs in the case of a rectangular domain are the chains
spanning the shortest dimension, for some small $\epsilon$ there are no other
critical values in $(r_{*}-\epsilon,r_{*}+\epsilon)$.
It is well known [4] that the configuration space ${\mathtt{Conf}}(n)$ of $n$
(labeled) points in ${\mathbb{R}}^{d}$ has Poincaré polynomial
$\displaystyle P(t)$ $\displaystyle:=\sum_{i\geq 0}\beta_{i}t^{i}$
$\displaystyle=\prod_{i=1}^{n-1}\left(1+it^{d-1}\right)$
$\displaystyle=1+\dots+(n-1)!H_{n-1}t^{(n-2)(d-1)}+(n-1)!t^{(n-1)(d-1)},$
where
$H_{n-1}=\sum_{i=1}^{n-1}1/i.$
This tells us the Betti numbers of ${\mathtt{Conf}}(n,r_{*}-\epsilon)$, since
we have already shown that ${\mathtt{Conf}}(n,r_{*}-\epsilon)$ is homotopy
equivalent to ${\mathtt{Conf}}(n)$. We wish to compute the Betti numbers of
${\mathtt{Conf}}(n,r_{*}+\epsilon)$.
Let $N=(n-1)(d-1)$. As we shrink the disks across the critical value
$r_{*}=L/2n$, to the configuration space we attach $k\,n!$ cells of dimension
$N$, where $k$ is the largest number such that $L_{k}=L$, whose boundaries are
representatives for the homology classes $s$ defined in Section 5.2. Each of
these cells either increments $\beta_{N}$ or decrements $\beta_{N-1}$.
The first observation is that
$\beta_{i}[{\mathtt{Conf}}(n,r_{*}+\epsilon)]=\beta_{i}[{\mathtt{Conf}}(n,r_{*}-\epsilon)]$
for $i\leq N-2$. As (the proof to appear in a follow-up paper) one can show
that $\beta_{i}({\mathtt{Conf}}(n,r))=0$ for $i\geq N$ and $r>r_{*}$, so in
particular
$\beta_{i}[{\mathtt{Conf}}(n,r_{*}+\epsilon)]=0$
for $i\geq N$. Thus $(n-1)!$ of the $N$-cells increase $\beta_{N}$. This
leaves only $\beta_{N-1}$ to compute.
Every $N$-cell that does not contribute to $\beta_{N}$ decreases
$\beta_{N-1}$. Since we know that $kn!$ cells are added, $(n-1)!$ of them
contributing to $\beta_{N}$, we have
$\beta_{N-1}[{\mathtt{Conf}}(n,r_{*}+\epsilon)]=\beta_{N-1}[{\mathtt{Conf}}(n,r_{*}-\epsilon)]+k\,n!-(n-1)!\;,$
and since
$\beta_{N-1}[{\mathtt{Conf}}(n,r_{*}-\epsilon)]=\left\\{\begin{array}[]{ll}(n-1)!H_{n-1}&:d=2\\\
0&:d\geq 3,\end{array}\right.$
we have
$\beta_{N-1}[{\mathtt{Conf}}(n,r_{*}+\epsilon)]=\left\\{\begin{array}[]{ll}(H_{n-1}+kn-1)(n-1)!&:d=2\\\
(kn-1)(n-1)!&:d\geq 3.\end{array}\right.$
## 6\. Concluding remarks
In a future article we will discuss non-degeneracy of critical points, which
is closely related to the question of making our necessary condition for a
change in the topology sufficient. We also discuss defining and computing the
index of critical points, and especially investigate more of the asymptotic
properties of ${\mathtt{Conf}}(n,r)$ as $n\to\infty$. In particular we obtain
bounds on the rate of growth of Betti numbers.
An important special case for which little seems known is: What is the
threshold radius $r=r(n)$ for connectivity of ${\mathtt{Conf}}(n,r)$? This is
an important question physically, since for example ergodicity of any Markov
process hinges on connectivity of the state space. Diaconis, Lebeau, and
Michel noted that $r\leq c/n$ is sufficient to guarantee connectivity of
${\mathtt{Conf}}(n,r)$ [7] and this is best possible for certain regions. It
would be interesting to know if connectivity of the configuration space ever
extends into the thermodynamic limit, i.e. are there any bounding regions so
that ${\mathtt{Conf}}(n,r)$ is connected for $r\leq Cn^{-1/d}$ and some
constant $C>0$?
## Acknowledgments
We thank AIM and for hosting the workshop on, “Topological complexity of
random sets” in August 2009, where we started discussing some of these
problems. Y.B. was supported in part by the ONR grant 00014-11-1-0178. M.K.
thanks IAS for hosting him this year and Robert MacPherson for several helpful
conversations.
## References
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* [6] Kenneth Deeley. Configuration spaces of thick particles on a metric graph. submitted, 2011.
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* [14] Mark Goresky and Robert MacPherson. Stratified Morse theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988.
* [15] R. L. Graham and B. D. Lubachevsky. Repeated patterns of dense packings of equal disks in a square. Electron. J. Combin., 3(1):Research Paper 16, approx. 17 pp. (electronic), 1996.
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* [19] Boris D. Lubachevsky, Ron L. Graham, and Frank H. Stillinger. Patterns and structures in disk packings. Period. Math. Hungar., 34(1-2):123–142, 1997. 3rd Geometry Festival: an International Conference on Packings, Coverings and Tilings (Budapest, 1996).
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* [21] Hans Melissen. Densest packings of eleven congruent circles in a circle. Geom. Dedicata, 50(1):15–25, 1994.
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|
arxiv-papers
| 2011-08-15T19:12:24 |
2024-09-04T02:49:21.535503
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuliy Baryshnikov, Peter Bubenik, Matthew Kahle",
"submitter": "Peter Bubenik",
"url": "https://arxiv.org/abs/1108.3061"
}
|
1108.3186
|
# Summed Parallel Infinite Impulse Response (SPIIR) Filters For Low-Latency
Gravitational Wave Detection
Shaun Hooper shaun.hooper@uwa.edu.au Australian International Gravitational
Research Centre, School of Physics, University of Western Australia, 35
Stirling Hwy, Crawley, WA 6009, Australia ICRAR-Fairway M468, School of
Physics, The University of Western Australia, Crawley, WA 6009, Australia
Shin Kee Chung Australian International Gravitational Research Centre, School
of Physics, University of Western Australia, 35 Stirling Hwy, Crawley, WA
6009, Australia ICRAR-Fairway M468, School of Physics, The University of
Western Australia, Crawley, WA 6009, Australia Jing Luan Theoretical
Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125,
USA David Blair Australian International Gravitational Research Centre,
School of Physics, University of Western Australia, 35 Stirling Hwy, Crawley,
WA 6009, Australia Yanbei Chen Theoretical Astrophysics 350-17, California
Institute of Technology, Pasadena, CA 91125, USA Linqing Wen
linqing.wen@uwa.edu.au Australian International Gravitational Research Centre,
School of Physics, University of Western Australia, 35 Stirling Hwy, Crawley,
WA 6009, Australia ICRAR-Fairway M468, School of Physics, The University of
Western Australia, Crawley, WA 6009, Australia
###### Abstract
With the upgrade of current gravitational wave detectors, the first detection
of gravitational wave signals is expected to occur in the next decade. Low-
latency gravitational wave triggers will be necessary to make fast follow-up
electromagnetic observations of events related to their source, e.g., prompt
optical emission associated with short gamma-ray bursts. In this paper we
present a new time-domain low-latency algorithm for identifying the presence
of gravitational waves produced by compact binary coalescence events in noisy
detector data. Our method calculates the signal to noise ratio from the
summation of a bank of parallel infinite impulse response (IIR) filters. We
show that our _summed parallel infinite impulse response_ (SPIIR) method can
retrieve the signal to noise ratio to greater than 99% of that produced from
the optimal matched filter. We emphasise the benefits of the SPIIR method for
advanced detectors, which will require larger template banks.
General Relativity, Gravitational Waves
###### pacs:
04.25.Nx, 04.30.Db, 04.80.Cc, 04.80.Nn, 95.55.Ym, 95.85.Sz
## I Introduction
The interferometric gravitational wave (GW) detectors LIGO LIG , and Virgo VIR
have reached a sensitivity at which the detection of GWs is possible. The LIGO
detectors are currently undergoing a major upgrade to Advanced LIGO, for which
the sensitivity will be improved ten fold relative to Initial LIGO Smith and
for the LIGO Scientific Collaboration (2009). Hence Advanced LIGO will be able
to detect GW (GW) sources within a volume of space one thousand times larger
than that of initial LIGO, out to $\sim$200-300$\,\mathrm{Mpc}$ Adv (2007).
The emission of GWs produced by compact binary coalescence (CBC) can be
modelled with a high degree accuracy Abbott et al. (2009). When two compact
bodies, such as neutron stars or black holes are in orbit, Einstein’s
equations predict the generation of GWs. As the bodies spiral towards each
other a GW is created that increases in frequency over time until the bodies
merge, following what is known as the inspiral waveform. Ground based
detectors have frequency passbands that allow them to be sensitive to the
final stages of such events up to a total system masses of several hundred
$M_{\odot}$.
Neutron star binary mergers are widely thought to be the progenitors of short
hard gamma-ray bursts (short GRBs) Fox et al. (2005); Nakar (2007). The delay
between the final GW emission and the onset of the GRB is estimated to be as
short as 0.1 seconds or as long as tens to hundreds of seconds van Putten
(2009); Zhang and Mészáros (2004). The electromagnetic emission of the GRB
event is not well understood. Related to the initial GRB there is thought to
be a prompt emission in the X-ray and optical wavelengths followed by a
delayed afterglow of cascading wavelengths. Prompt optical emission may occur
tens to hundreds of seconds after the initial burst. The low-latency detection
of the GW associated with a neutron star merger could lead to the localisation
of a GRB source event on the sky, enabling fast moving telescopes to observe
the prompt optical emission. Data collected from a multitude of sources — GWs,
gamma-rays, X-rays and optical counterparts of the GRB — will lead to maximum
insight into these highly energetic events.
The standard strategy for searching for the existence of inspiral waveforms in
the detector data is based on matched filtering Abbott et al. (2009) (and
references therein). This method, based on Wiener optimal filtering, is a
correlation of an expected inspiral waveform template and the detector data,
weighted by the inverse noise spectral density of the detector Wainstein and
Zubakov (1962). In order to save computational costs, this correlation is
performed in the frequency domain, via a Fourier transform of a finite segment
of detector data. In previous LIGO searches, the detector data is split up
into “science blocks”, which are further divided into “data segments” chosen
to be at least twice the length of the longest waveform in the template bank
Abbott et al. (2008). Each proceeding data segment is chosen to overlap the
previous one by 50%. Each segment therefore must be matched filtered in a time
that is half the length of the segment for a real-time analysis, that is, the
filter output rate is equal to the data input rate. In this case, the matched
filter process has a minimum latency (from signal arrival to signal detection)
that is proportional to the longest template (see Luan (2011) for more
details). Advanced LIGO will have an increased bandwidth over Initial LIGO,
with the lower bound dropping from 40$\,\mathrm{Hz}$ to 10$\,\mathrm{Hz}$ Adv
(2007). GW signals from CBC events spend much more time at these lower
frequencies. Hence waveforms used for matched filtering in Advanced LIGO will
be much longer (1000s of seconds). This in turn means the segment length will
be increased, further increasing the latency. The latency of this method to
produce GW triggers is longer than the time to onset of prompt optical
emission after coalescence (10s to 100s of seconds). After this amount of
time, the early electromagnetic counterpart of a GRB event will be
significantly faded, and may be missed by telescopes altogether.
A low-latency GW detection method is required to trigger follow-up
electromagnetic observations of the prompt optical emission. So far two
frequency domain methods have been developed to solve this issue. The VIRGO
group has produced a low-latency pipeline based on _Multi-Band Template
Analysis_ (MBTA) Buskulic et al. (2010), and LIGO is also working on a new
method, _Low-Latency On-line Inspiral Data_ analysis (LLOID) method. In MBTA
the matched filtering technique is split over two frequency bands, and the
output is coherently added, reducing latency. A latency of less than 3 minutes
until the availability of a trigger using this method has been achieved
Buskulic et al. (2010). Low-latency in the LLOID method is achieved by first
down-sampling the incoming data into multiple streams and then applying
frequency domain finite impulse response (FIR) filters gst . The computational
cost of this pipeline is reduced by decreasing the number of templates via
singular value decomposition Cannon et al. (2010).
We introduce a new method to detect CBC signals in the time domain using
infinite impulse response (IIR) filters. Approximating an inspiral waveform by
a summation of time shifted exponentially increasing sinusoids enables us to
construct a bank of parallel single-pole IIR filters. Each IIR filter acts as
a narrow bandpass filter. When each appropriately delayed IIR filter is added
the coherent output approximates the matched filter output of the exact
waveforms. We call this the _summed parallel infinite impulse response_
(SPIIR) method. Figure 1 visually demonstrates the idea of using a bank of IIR
filters as narrow bandpass filters.
Figure 1: A schematic overview of the SPIIR method. The input is split into
different channels, time delayed by an amount $d$, then passed through a
narrow bandpass IIR filters, each with a different central frequency $f$.
Finally the output of each individual IIR filter is summed, giving the output
of the SPIIR method.
For a full explanation of the mathematical principles, see Luan (2011). In
this follow up paper, we numerically address the issues essential to the
practical use of this method for the upcoming advanced detectors. We calculate
the filter coefficients and demonstrate via numerical simulations how well our
method approximates the optimal matched filter as a function the number of
filters per bank using a range of parameters. We also show that the detection
rate of the SPIIR method is very similar to that of the matched filter method.
It has been shown theoretically that in order to get the same latency as the
SPIIR method, the frequency domain matched filter method would require greater
computational resources Luan (2011).
The structure of this paper is as follows: In section II we will go through
the formal introduction of the inspiral waveform and matched filtering, and
how to get from the continuous frequency domain matched filter to the time
domain discrete matched filter. This will lead to a demonstration on how it is
possible to approximate an inspiral signal by a sum of exponentially
increasing sinusoids. The methodology is explained in Section III and will
cover how we set up our simulation to test the efficiency of the SPIIR method
as opposed to the frequency domain matched filter. Section IV will analyse the
results of the simulation and Section V will discuss the implications of these
results for advanced detectors.
## II Methodology
Gravitational wave interferometers output the strain induced by gravitational
waves incident on the detector, as well as inherent noise. In unitless strain,
the detector output will be,
$\displaystyle s(t)=\begin{cases}n(t)&\text{if signal is absent}\\\
n(t)+h(t)&\text{if signal is present}\end{cases}$ (1)
where $n(t)$ is the noise inherent in the detector. The sensitivity of the
instrument can be characterized by the (one-sided) strain power spectral
density $S_{n}(f)$,
$\displaystyle\left\langle\tilde{n}(f)\tilde{n}^{*}(f^{\prime})\right\rangle=\frac{1}{2}S_{n}(f)\delta(f-f^{\prime})$
(2)
where the tilde represents the forward Fourier transform,
$\displaystyle\tilde{q}(f)=\int_{-\infty}^{\infty}q(t)e^{-2\pi ift}dt.$ (3)
### II.1 The Inspiral Waveform
The gravitational-wave strain incident at the interferometer is given by
$\displaystyle
h(t)=F_{+}(\theta,\phi,\psi)h_{+}(t)+F_{\times}(\theta,\phi,\psi)h_{\times}(t)$
(4)
where the detector response functions $F_{+}$ and $F_{\times}$ are functions
of $(\theta,\phi)$ \- the standard spherical polar coordinates measured with
respect to the Earth’s fixed frame, and $\psi$ is the polarisation angle. The
detector response function can be found in Anderson et al. (2001). The $+$ and
$\times$ polarisations of the waveform are,
$\displaystyle h_{+}(t)$
$\displaystyle=\left(\frac{1+\cos^{2}\iota}{2}\right)A(t)\cos\phi(t)$ (5)
$\displaystyle h_{\times}(t)$
$\displaystyle=\left(\cos\iota\right)A(t)\sin\phi(t)$ (6)
For non-spinning binaries with a chirp mass
$\mathcal{M}=((m_{1}m_{2})^{3}/(m_{1}+m_{2}))^{1/5}$ in the range of
$1-3M_{\odot}$ — we will hereafter assume — the waveforms can be modelled to
very high accuracy using the Restricted post-Newtonian (PN) expansion Abbott
et al. (2004); Blanchet et al. (1995, 1996) in the LIGO band (assumed to be
10-1500 $\,\mathrm{Hz}$ for advanced LIGO). For restricted waveforms, only the
leading order of the amplitude $A(t)$ is taken,
$\displaystyle A(t)$
$\displaystyle=\frac{G\mathcal{M}}{Dc^{2}}\left(\frac{t_{c}-t}{5G\mathcal{M}/c^{3}}\right)^{-1/4}$
(7)
and the post-Newtonian phase $\phi(t)$ is given by
$\displaystyle\phi(t)$
$\displaystyle=\phi_{c}-2\left(\frac{t_{c}-t}{G\mathcal{M}/c^{3}}\right)^{5/8}+\mbox{higher
order terms}$ (8)
In addition to the source masses $m_{1},m_{2}$, there are several unknown
parameters; the time of coalescence $t_{c}$, the phase at coalescence
$\phi_{0}$, distance from observer to source $D$, the inclination angle of the
binary’s orbital plane relative the line of sight $\iota$, and the
polarisation angle $\psi$. However by using the linear combination
trigonometric identity, one can re-express the strain (4) by splitting the
scaling factor due to distance, sky location and orientation to the mass
dependant time evolution of the waveform Fairhurst and Brady (2008),
$\displaystyle h(t)$ $\displaystyle=\frac{1\,\mathrm{Mpc}}{D_{\rm
eff}}\left[h_{c}(t)\cos\phi_{0}+h_{s}(t)\sin\phi_{0}\right]$ (9)
where the scalar factor $D_{\rm eff}$ is,
$\displaystyle D_{\rm
eff}=\frac{D}{\sqrt{F_{+}^{2}\left(1+\cos^{2}\iota\right)^{2}/4+F_{\times}^{2}\left(\cos\iota\right)^{2}}}$
(10)
which gives $\phi_{0}$, an unknown phase as,
$\displaystyle\phi_{0}=\phi_{c}+\arctan\frac{F_{\times}\left(2\cos\iota\right)}{F_{+}\left(1+\cos^{2}\iota\right)}$
(11)
We now define the terms $h_{c}$ and $h_{s}$ as the waveform at $\phi_{0}=0$
and $\phi_{\pi/2}$, scaled at 1$\,\mathrm{Mpc}$ as the so called “cosine” and
“sine” phases Brady and Fairhurst (2008),
$\displaystyle h_{c}(t)$ $\displaystyle=A_{1\,\mathrm{Mpc}}(t)\cos\phi(t)$
(12) $\displaystyle h_{s}(t)$
$\displaystyle=A_{1\,\mathrm{Mpc}}(t)\sin\phi(t)$ (13)
### II.2 The Matched Filter
The matched filter $Q$ is a linear operator that maximises the ratio of
“signal” to “noise” present in the detector data $s$ Allen et al. (2005). It
is denoted by,
$\displaystyle z(t)$
$\displaystyle=2\int_{-\infty}^{\infty}\frac{\tilde{s}(f)\tilde{Q}^{*}(f)}{S_{n}(|f|)}e^{2\pi
ift}df=\left(s(t)\left|Q\right)\right.$ (14)
Where we have also defined the inner product $\left(a\left|b\right)\right.$.
The signal to noise ratio (SNR) is _generally_ defined as the ratio of
observed filter output to it’s expected root-mean square flucations or
standard deviation,
$\displaystyle\mathrm{SNR}$
$\displaystyle=\frac{z}{\sqrt{\left\langle(z-\left\langle
z\right\rangle)^{2}\right\rangle}}=\frac{z}{\sqrt{\left\langle
z\right\rangle^{2}}}=\frac{z}{\sqrt{\left(Q\left|Q\right)\right.}}$ (15)
Note that in the absence of a signal,
$\left\langle\mathrm{SNR}\right\rangle=0$ and
$\left\langle(\mathrm{SNR})^{2}\right\rangle=1$ independent of the
normalisation of the filter $Q$.
### II.3 Two-Phase Filter
A convenient way to search for the unknown phase constant $\phi_{0}$ is to
filter both phases $h_{c}$ and $h_{s}$ separately and then combined to form a
complex signal. The two-phase filter is defined as,
$\displaystyle z(t)$
$\displaystyle=\left(s(t)\left|h_{c}\right)\right.+i\left(s(t)\left|h_{s}\right)\right.$
(16a)
$\displaystyle\begin{split}&=2\int_{-\infty}^{\infty}\frac{\tilde{s}(f)\tilde{h}_{c}^{*}(f)}{S_{n}(f)}e^{2\pi
ift}df\\\
&\qquad\qquad+i2\int_{-\infty}^{\infty}\frac{\tilde{s}(f)\tilde{h}_{s}^{*}(f)}{S_{n}(|f|)}e^{2\pi
ift}df\end{split}$ (16b)
The advantage of using the phases $h_{c,s}$ is that in the stationary phase
approximation Droz et al. (1999), $h_{c}$ and $h_{s}$ are exactly orthogonal
($\left(h_{c}\left|h_{s}\right)\right.=\left(h_{s}\left|h_{s}\right)\right.$,
$\left(h_{c}\left|h_{s}\right)\right.=0$). It then follows,
$\tilde{h}_{c}(f)=i\tilde{h}_{s}(f)$ for $f>0$. Generally, this is applied to
(16) to give the two-phase matched filter as,
$\displaystyle z(t)$
$\displaystyle=4\int_{0}^{\infty}\frac{\tilde{s}(f)\tilde{h}_{c}^{*}(f)}{S_{n}(|f|)}e^{2\pi
ift}df$ (17)
However in this paper, we prefer to maintain the form of the two-phase filter
in (16). In convention with the field, the amplitude signal to noise ratio of
the (quadrature) matched filter is defined as the absolute value of the two-
phase filter, divided by a normalisation constant that is equal to standard
deviation of the real and imaginary parts of the two-phase filter,
$\displaystyle\rho(t)=\frac{|z(t)|}{\sigma}$ (18)
where $\sigma^{2}$ is,
$\displaystyle\sigma^{2}=2\int_{-\infty}^{\infty}\frac{\left|\tilde{h}_{c}(f)\right|^{2}}{S_{n}(f)}df=\left(h_{c}\left|h_{c}\right)\right.$
(19)
Note that in the in the absence of a signal (just noise), the SNR $\rho$ (18)
is Rayleigh distributed with mean $\sqrt{\pi/2}$ and variance $1$, which is
identical to the Chi-distribution with two degrees of freedom (one for each of
the phases). This of course implies that the SNR squared, $\rho^{2}$ is Chi-
square distributed with two degrees of freedom. Hence the probability of
finding an SNR value greater than $\rho_{*}$ is Brady and Fairhurst (2008),
$\displaystyle P(\rho^{2}>\rho_{*}^{2})=e^{-\rho_{*}^{2}/2}.$ (20)
### II.4 Digital Time Domain Filtering
The two-phase matched filter 16 is a cross correlation of phase $h_{c,s}(t)$
and the detector output $s(t)$, weighted by the inverse noise spectral density
$S_{n}(f)$. By defining the quantity $x$ as the _over_ -whitened strain data,
$\displaystyle x(t)=\int_{-\infty}^{\infty}\frac{\tilde{s}(f)}{S(f)}e^{2\pi
ift}df$ (21)
we can use the cross-correlation theorem to define the two-phase matched
filter in the time domain,
$\displaystyle z(t)$
$\displaystyle=2\int_{-\infty}^{t}x(t^{\prime})h_{c}(t^{\prime}-t)dt^{\prime}+i2\int_{-\infty}^{t}x(t^{\prime})h_{s}(t^{\prime}-t)dt^{\prime}$
(22)
$\displaystyle=2\int_{-\infty}^{t}x(t^{\prime})\hat{h}(t^{\prime}-t)dt^{\prime}$
(23)
where $\hat{h}=h_{c}(t)+ih_{s}(t)=A(t)e^{i\phi(t)}$.
The discrete form of the continuous time domain matched filter (23) is,
$z_{k}=2\sum_{j=-\infty}^{k}x_{j}\hat{h}_{j-k}\Delta t$ (24)
where $t=k\Delta t$. In practise, the inspiral waveform template $h_{i}$ is
bounded (because the detector is only sensitive over a bandwidth), and the
summation becomes finite, making this a _finite impulse response_ (FIR)
filter.
### II.5 Infinite Impulse Response Filter
Now let us introduce an alternative digital filter, the _infinite impulse
response_ (IIR) filter. The difference equation of a general IIR filter is,
$\displaystyle y_{k}=\sum_{n=1}^{N}a_{n}y_{k-n}+\sum_{m=0}^{M}b_{m}x_{k-m}$
(25)
where $y_{k}$ is the filter output at time step $k$, ($t=k\Delta t$), $x_{k}$
is the filter input, and $a$’s and $b$’s are complex coefficients.
Examples of IIR filters in common usage are Chebyshev, Butterworth and
elliptic filters. IIR filters use much less computational resources than an
equivalent FIR filter. This is because they have “memory” — the previous
outputs are fed back into the filter. However _digital_ IIR filter design is a
more complex process than FIR design. Obtaining the coefficients is usually
done by first constructing an equivalent analog filter and applying well-known
methods, such as the bi-linear transform or impulse invariance. Multiple IIR
filters used together have different forms, such as direct form I & II,
cascade (series) and parallel. In a series configuration, the overall transfer
function is the multiplication of each IIR filter transfer function. In a
parallel bank of IIR filters, where the output is summed together, the overall
transfer function is the summation of the different transfer functions.
First, let’s analyse the simplest single-pole IIR filter. The difference
equation of this filter is
$y_{k}=a_{1}y_{k-1}+b_{0}x_{k}.$ (26)
Figure 2: A signal processing schematic showing the flow of data through a
digital single-pole IIR filter. The input, $x_{k}$ is multiplied by a complex
constant $b_{0}$, then added to the previous output that has been multiplied
by another complex constant $a_{1}$, resulting in the current output $y_{k}$.
It should be noted that this filter, in principle, should be have been run
forever.
A solution to this first-order linear inhomogeneous difference equation is
$\displaystyle y_{k}=\sum_{j=-\infty}^{k}x_{j}b_{0}a_{1}^{k-j}.$ (27)
By defining the complex coefficient $a_{1}$ in the form,
$\displaystyle a_{1}=e^{-(\gamma+i\omega)\Delta t}$ (28)
and comparing (24) and (27), it is easy to see that the output of the simple
filter (26) is the cross-correlation of $x_{k}$ and complex sinusoid $u_{n}$
with frequency $\omega$ and a magnitude that increases with an exponential
factor $\gamma$ for $n<0$:
$\displaystyle u_{n}=b_{0}e^{(\gamma+i\omega)n\Delta t}\Theta(-n)$ (29)
where $\Theta(-n)$ is the Heaviside function.
### II.6 Approximation to an inspiral waveform
Since $\phi(t)$ is not linear in time, a complex sinusoid (29) cannot
approximate the $h_{c,s}$ phases of the inspiral waveform
$\hat{h}(t)=A(t)e^{i\phi(t)}$. However we can easily linearise the phases by a
first-order Taylor expansion about the time $t_{l}^{*}$:
$\displaystyle A(t)e^{i\phi(t)}\simeq
A(t_{l}^{*})e^{i\phi(t_{l}^{*})+i\dot{\phi}(t_{l}^{*})(t-t_{l}^{*})};$ (30)
since the amplitude $A(t)$ does not increase at the same rate as $\phi(t)$,
only a linear expansion of $\phi(t)$ is required. Multiplying by the window
function $e^{\gamma_{l}(t-t_{l})}\Theta(t_{l}-t)$ makes this approximation an
exponentially increasing constant frequency complex sinusoid with cutoff time
$t_{l}$:
$\displaystyle u_{l}(t)$
$\displaystyle=A(t_{l}^{*})e^{i(\phi(t_{l}^{*})+\dot{\phi}(t_{l}^{*})(t_{l}-t_{l}^{*}))}e^{(\gamma_{l}+i\dot{\phi}(t_{l}^{*})(t-t_{l})}\Theta(t_{l}-t).$
(31)
The expansion point $t_{l}^{*}$ is chosen to be near the cutoff time,
$t_{l}^{*}=t_{l}-\alpha T_{l}$, where $\alpha$ is a tunable parameter and the
interval $T_{l}$ is the duration in which the approximation is valid:
$\displaystyle|\frac{1}{2}\ddot{\phi}(t_{l})T_{l}^{2}|=\epsilon<1$ (32)
and $\epsilon$ is a tunable parameter chosen to be to small. Equation (31)
implies that the coefficient $b_{0}$ for the $l$th complex sinusoid is,
$\displaystyle b_{0,l}$
$\displaystyle=A(t_{l}^{*})e^{i(\phi(t_{l}^{*})+\dot{\phi}(t_{l}^{*})(t_{l}-t_{l}^{*}))}$
(33)
and the frequency $\omega_{l}=\dot{\phi}(t_{l}^{*})$.
In this paper, we chose the cutoff time $t_{l}$ of the first sinusoid to
correspond to the time at which the waveform has the highest frequency
detectable by the LIGO detector band. The next sinusoid is chosen by moving to
an earlier time, $t_{l+1}=t_{l}-T_{l}$. Since we want the $l$th sinusoid to be
mostly present on the interval $t_{l}-T_{l}<t<t_{l}$, we choose the damping
factor to be $\gamma_{l}=\beta/T_{l}$, where $\beta$ is a tunable parameter.
This procedure is repeated until the time $t_{l}$ corresponds to a time in the
waveform that has frequency below the LIGO detector band. Hence the number of
sinusoids is dependent on the value of $\epsilon$, the rate of frequency
change ($\ddot{\phi}(t)$), which is dependent on the masses of the system, and
the detector bandwidth. For more information on this procedure, see Luan
(2011).
We can now approximate the phases $\hat{h}(t)=A(t)e^{i\phi(t)}$ by an addition
of a series of damped sinusoids $u(t)$ with cutoff times $t_{l}$:
$\displaystyle A(t)e^{i\phi(t)}\simeq U(t)$ $\displaystyle=\sum_{l}u_{l}(t)$
$\displaystyle=\sum_{l}b_{0,l}e^{(\gamma_{l}+i\omega_{l})(t-t_{l})}\Theta(t_{l}-t).$
(34)
Figure 3 shows an illustration of how damped constant frequency sinusoids can
add to give an inspiral like waveform.
Figure 3: An illustrative diagram demonstrating the ability to linearly sum
exponentially increasing constant frequency sinusoids to approximate an
inspiral like waveform. The top three panels (a-c) show three example
sinusoids with different damping, frequency and cutoff time factors. Panel (d)
shows the linear addition of all the sinusoids (at different scales). Panel
(e) shows the exact inspiral-like waveform. Note that this figure is only for
illustrative purposes.
### II.7 Summed Parallel IIR filtering
Each complex sinusoid $u_{l}(t)$ in equation (II.6) can be searched for in the
data $x$ using the single pole IIR filter (26). Here the cutoff time is
incorporated by running each filter on a delay, $d_{l}=t_{l}/\Delta t$. The
output of the $l$th filter at time $k$ is
$\displaystyle y_{k,l}=a_{1,l}y_{k-1,l}+b_{0,l}x_{k-d_{l}}.$ (35)
The linear summation of the output of _all_ filters is the cross-correlation
of the data $x$ and the approximate waveform $U(t)$ in (II.6):
$\displaystyle z_{k}\simeq 2\Delta t\sum_{l}y_{k,l}.$ (36)
Here $z$ is equivalent to the value computed by the discrete time domain two
phase filter (24) when using a template $\hat{h}(t)=U(t)$. From equation (18),
it follows that the absolute value of the summation (36) divided by
$\sigma_{U}$ is the SNR, which we term the output of the _Summed Parallel
Infinite Impulse Response_ (SPIIR). The normalisation factor $\sigma_{U}$ is
defined as
$\displaystyle\sigma_{U}^{2}=4\int_{0}^{\infty}\frac{\left|\tilde{U}_{\Re}(f)\right|^{2}}{S_{n}(f)}df.$
(37)
Where $\tilde{U}_{\Re}(f)$ is the Fourier transform of the real part of
$U(t)$, (which approximates $h_{c}(t)$). The similarity of the SPIIR output
and the matched filter output will depend on how well $U(t)$ approximates the
given template.
## III Implementation for Performance Testing
### III.1 IIR bank construction
To confirm the ability of the SPIIR method to recover a good SNR, it is first
required to show that the approximate inspiral waveform (II.6) is a good
“match” to the theoretical inspiral waveform (9). We define the _overlap_
$\Delta$ as the inner product of the approximate waveform $U$ and the template
$h$:
$\displaystyle\Delta=\left(\frac{h}{\sqrt{\left(h\left|h\right)\right.}}\left|\frac{U}{\sqrt{\left(U\left|U\right)\right.}}\right)\right.=\frac{\left(h\left|U\right)\right.}{\sqrt{\left(h\left|h\right)\right.\left(U\left|U\right)\right.}}$
(38)
We initially approximate a canonical 2PN 1.4-1.4 $M_{\odot}$ inspiral waveform
band limited to 10-1500$\,\mathrm{Hz}$ using the value of the tunable
parameters $\epsilon$, $\alpha$ and $\beta$ to be consistent with the high
overlap results of Luan (2011). With some minor variation of their values, we
aim to recover the highest overlap possible. Once a good choice of $\alpha$
and $\beta$ is found for the 2PN 1.4-1.4 $M_{\odot}$ template, we use the same
values for other templates, but vary the value $\epsilon$ (and consequently
the number of IIR filters in each bank) to see the effect on overlap.
### III.2 Detector Data Simulation
To test the detection efficiency of the SPIIR method compared to the frequency
domain matched filter, we will filter two mock signals, one for which the
input data is just LIGO-like noise, and the other with the same noise plus an
inspiral waveform injection scaled to represent a source at a chosen effective
distance $D_{\rm eff}$.
For this test, we need to construct a finite segment of detector data to
filter. Because of the IIR filters should in principle should be run for an
infinite length of the input data, we need to run the IIR bank for a finite
“warm-up” period before the output is consistent with that of an IIR filter
that has been running for an infinite amount of time. In practise, we choose
to run each filter for 2 $e$-foldings of time before we accept the output as
being identical to one which has run for an infinite amount of time.
Additionally, since each IIR filter in the bank runs on a delay, the summed
output _of all the IIR filters_ will not be produced until after the longest
delay time ($d_{\rm max}$) has passed. The filter that has the longest delay
($d_{\rm max}$) is also the one that has the longest decay rate $\gamma_{\rm
max}$. In total, the input data must at least $d_{\rm max}+2\gamma_{\rm
max}^{-1}$ in length before any output is produced. Hence the length of the
input data is,
$N_{\rm input}=d_{\rm max}+2\gamma_{\rm max}^{-1}+N_{\rm analysis}$ (39)
where $N_{\rm analysis}$ is the length of analysis period, which we choose to
be 4 seconds. Hence the 4$\,\mathrm{s}$ SPIIR output will tell us whether
there is an injection that ended somewhere within those 4 seconds. At a sample
rate of 4096$\,\mathrm{Hz}$, the analysis period is $N_{\rm analysis}=16834$
data points long. In our simulation, we find $d_{\rm max}=4081683$ and
$2\gamma_{\rm max}^{-1}=149432$, resulting in $N_{\rm input}=4247499$.
#### III.2.1 Noise generation
The LIGO-like noise data is produced by creating a normally distributed white
noise time series of length $N_{\rm input}$, then colouring it by the
theoretical advanced LIGO noise spectrum $S_{n}(f)$ A. We then over-whiten
this time series using equation (21) to produce the waveform-free noise input
data $x$:
$\displaystyle x_{\rm noise}(t)=n^{\rm ow}(t).$ (40)
#### III.2.2 Waveform injection
We create our waveform injections by first producing an inspiral waveform
band-limited between 10 and 1500$\,\mathrm{Hz}$. The injection is padded with
zeros so that it has the length $N_{\rm input}$. The end of the waveform is
chosen so that it finishes somewhere after $d_{m}+2\gamma_{m}^{-1}$ data
points. The injection signal is then over whitened using equation (21). The
over-whitened injection can then be placed in the over-whitened noise signal,
$\displaystyle x_{\rm noise+injection}(t)=x_{\rm noise}(t)+h^{\rm ow}(t).$
(41)
#### III.2.3 Matched filter comparison
As a comparison, we will also perform a frequency domain correlation matched
filter. For this process, since the input data is already over-whitened, it
only needs to be cross-correlated with the waveform. Section II.2 outlines how
this is done. The cosine phase $h_{c}(t)$ gets pre-padded with enough zeros to
get to length $N_{\rm input}$. This ensures that $\tilde{h}_{c}(f)$ has the
same spectral resolution as $\tilde{s}(f)$. The matched filter (17) produces a
time series of $N_{\rm input}$ length. However the first $N_{\rm input}-N_{\rm
analysis}$ data points are erroneous wrap-around caused by the FFT. Only the
interval $[N_{\rm input}-N_{\rm analysis}+1,N_{\rm analysis}]$ is used to
determine if a waveform is present.
### III.3 Detection Efficiency
To test the detection efficiency of the SPIIR method compared to the
traditional matched filter method we will construct several receiver operating
characteristic (ROC) curves for 2PN 1.4-1.4 $M_{\odot}$ waveforms injected for
different effective distances $D_{\rm eff}$. To create each ROC curve, we
first find the false alarm rate. The false alarm rate is found by realising an
$N_{\rm input}$ length LIGO-like noise time series, filtering this input data,
and analysing the output of the 4$\,\mathrm{s}$ analysis period (the SNR). We
will count this realisation as a false positive if at any point within the 4
seconds the SNR goes over a given SNR threshold. Several thresholds will be
chosen, giving the false positive as a function of threshold. After $>10^{6}$
noise realisations, the false alarm rate is simply the ratio of total number
of false positives to number of noise realisations. Likewise, to see if the
IIR filter doesn’t miss too many true positives, we inject a 2PN 1.4-1.4
$M_{\odot}$ waveform using the prescribed method in III.2.2 for a given
$D_{\rm eff}$ into LIGO-like noise. After filtering, if at any point within
the analysis period the SNR is above a given threshold, this realisation is
counted as a true positive. Again, after $>10^{6}$ noise realisations, we
calculate the detection rate as a ratio of the total number of true positives
to number of realisations. The plot of false alarm rate versus detection rate
gives the ROC curve.
## IV Results
### IV.1 Inspiral Waveform Overlap
Starting with the canonical 1.4-1.4 $M_{\odot}$ second order post-Newtonian
binary waveform band limited to be between 10 and 1500 $\,\mathrm{Hz}$ we
found, using the parameters $\epsilon=0.04$, $\alpha=0.99$, $\beta=0.25$ in
the procedure outlined in Section II.6, that can recover an overlap of 99%
using 687 IIR filters.
We find that increasing the value of $\epsilon$ will in general increase the
overlap, as the frequency space is more finely sampled. However there seems to
be a limit, as the damping factor $\gamma$ causes the adjacent IIR filters to
run into each other.
With this choice of $\alpha$ and $\beta$ we are able to recover a high overlap
for different mass pairs as well. Figure 4 shows the overlap as a function of
number of IIR filters for six different mass pairs.
Figure 4: The overlap between the exact inspiral waveform and the approximate
inspiral waveform as a function of number of damped sinusoids. In general the
greater the number of sinusoids per waveform, the greater the overlap. However
the choice of $\gamma_{l}$s greatly affects the overlap.
### IV.2 Ability to Recover SNR
Figure 5 shows the SNR produced from both the matched filter technique and the
SPIIR method. The input time series is constructed following Section III.2.
The injection of a 2PN 1.4-1.4 $M_{\odot}$ waveform scaled for an effective
distance of 250$\,\mathrm{Mpc}$ is added to LIGO-like noise. The $x$-axis of
the plot is centred about end of the injection ($t=\tau_{c}$), which is
directly in the middle of the analysis period. Around this time, the SNR peaks
to 8.2, which is near the expected value of 7.9 for an injection at this
distance.
Figure 5: The SNR output of both the SPIIR method and a traditional matched
filter method. The plot is centred on $t-\tau_{c}$ where $\tau_{c}$ is the
time at which the injection ends. From the two curves, it is clear that the
SPIIR method can return a very similar SNR to that from the optimal filter.
This plot shows that the SPIIR method is capable of recovering a very similar
SNR to the matched filter at all times.
### IV.3 Detection Efficiency
We analysed over $10^{6}$ independent noise realisations, for which the
waveform had been injected at $D_{\rm eff}$ of 250, 300, 350, 400
$\,\mathrm{Mpc}$. We performed both IIR filtering and traditional matched
filtering. Figure 6 shows that the SPIIR method recovers most of the same
events as the traditional matched filter method. At false alarm rates of
greater than $10^{-5}$, the SPIIR method recovers greater than 99% of the
injections recovered by the matched filter when searching for injections at an
effective distance of 250$\,\mathrm{Mpc}$ (SNR$\sim$7.9). Even in the worst
case, at a false alarm rate of $10^{-6}$, the SPIIR method catches 4.5% of
injections scaled at an extreme 400$\,\mathrm{Mpc}$ (SNR$\sim$5), whereas the
matched filter catches 5% of injections at this scale.
Figure 6: The receiver operating characteristics (ROC) of both the IIR filter
method and the traditional matched filter method. The x-axis shows the false
alarm rate, and the y-axis the detection rate. A one-to one relationship,
which is the worst case scenario, is shown by the boundary of the shaded area.
We show four different ROC curves, where the each curve represents the
detection rate as a function of false alarm rate for waveform injected at
effective distances of 250, 300, 350 and 400 $\,\mathrm{Mpc}$ (SNR $\sim$ 8,
6.6, 5.7 and 5 respectively).
## V Summary and Discussion
The use of a bank of simple IIR filters for each template as opposed to the
matched filter method enables us get two extra processes for a minimal
additional cost. The first is that the individual IIR filter outputs can be
arranged into groups, such that their total summed output is roughly
independent and orthogonal to each other. This enables, with minimal extra
overhead, the calculation of a $\chi^{2}$ distributed statistic, giving a
secondary method of verification. We will demonstrate this in an upcoming
paper. The second natural advantage of using a parallel bank of single-pole
IIR filters is that they can easily be executed in parallel using multi-
threaded processors, such as graphics processing units (GPUs). Indeed, a side
study has shown that this is possible Liu (2011). This leads to the future
possibility that a single personal computer may be able to process the
detection of GWs.
A further way to reduce the computation of the IIR calculation is to split the
incoming data into differently down-sampled channels. The output of each IIR
filter in the bank is the correlation of a fixed frequency sinusoid and the
incoming data. For the sinusoids that have frequencies $<$124$\,\mathrm{Hz}$,
the incoming data need only be sampled at 256$\,\mathrm{Hz}$. The current
pipeline of LLOID uses a similar multi-channel down-sampling in their
detection pipeline. Their pipeline consists of the integration of the open-
source real-time multimedia handling software gstreamer and the LIGO Algorithm
Library (LAL) gst . This software library is an ideal platform to integrate
the SPIIR method. The total computation can also be further reduced by sharing
IIR filters (via interpolation) between different templates Luan (2011).
Although the design of the IIR filter so far only applies to chirping, post-
Newtonian approximation inspirals, we have performed preliminary tests using
more complicated combinations of single-pole IIR filters to replicate the
waveform of an inspiral with spin. If the amplitude/frequency beating of a
spinning inspiral waveform can be simulated by the linear addition of two
different non-spinning inspirals with different masses, then it can be
approximated by a linear addition of damped sinusoids. In this case, the SPIIR
method can produce the SNR for the beating waveform. There is also the
possibility of using higher order IIR filters, although designing the
coefficients can be very difficult.
## VI Conclusion
We have demonstrated that the through the use of a parallel bank of single
pole IIR filters, it is possible to approximate the SNR derived from the
matched filter with greater than 99% overlap. The main advantage of our SPIIR
method is that it operates completely in the time domain, and in principle it
has zero latency (not taking into account whitening or computational time).
The SPIIR method recovers most of the injections the optimal matched filter
recovers.
We foresee that the use of IIR filters for time domain filtering of Advanced
LIGO will be ideal, as the waveforms will be much longer. The frequency domain
matched filter will take more time to calculate GW triggers, essentially
ruling out the possibility of triggering the detection the prompt optical
emission related to neutron star mergers (GRBs). We have shown that the use of
a parallel bank of IIR filters requires less computational cost, with minimal
detection rate loss, and most importantly can be calculated in the time-domain
with near zero latency.
## VII Acknowledgements
We would like to thank Kipp Cannon, Drew Keppel and Chad Hanna for detailed
discussion on the design and implementation of low-latency detection
algorithms. This work was done in part during the LIGO Visiting Student
Researcher program, which was partially funded by the 2009 UWA Research
Collaboration Award. This research was supported by the Australian Research
Council. SH gratefully acknowledges the support of an Australian Postgraduate
Award.
## References
* (1) URL http://www.ligo.org.
* (2) URL http://www.virgo.infn.it/.
* Smith and for the LIGO Scientific Collaboration (2009) J. R. Smith and for the LIGO Scientific Collaboration, Classical and Quantum Gravity 26, 114013 (2009), eprint 0902.0381.
* Adv (2007) _Advanced LIGO Reference Design_ , LIGO Tech. Rep. M060056 (2007), URL http://www.ligo.caltech.edu/docs/M/M060056-08/M060056-08.pdf.
* Abbott et al. (2009) B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, et al. (LIGO Scientific Collaboration), Phys. Rev. D 79, 122001 (2009).
* Fox et al. (2005) D. B. Fox, D. A. Frail, P. A. Price, S. R. Kulkarni, E. Berger, T. Piran, A. M. Soderberg, S. B. Cenko, P. B. Cameron, A. Gal-Yam, et al., Nature (London) 437, 845 (2005), eprint arXiv:astro-ph/0510110.
* Nakar (2007) E. Nakar, Phys. Rep. 442, 166 (2007), eprint arXiv:astro-ph/0701748.
* van Putten (2009) M. H. P. M. van Putten, ArXiv e-prints (2009), eprint 0905.3367.
* Zhang and Mészáros (2004) B. Zhang and P. Mészáros, International Journal of Modern Physics A 19, 2385 (2004), eprint arXiv:astro-ph/0311321.
* Wainstein and Zubakov (1962) L. A. Wainstein and V. D. Zubakov, _Extraction of Signals from Noise_ (Prentice-Hall, 1962).
* Abbott et al. (2008) B. Abbott, R. Abbott, R. Adhikari, J. Agresti, P. Ajith, B. Allen, R. Amin, S. B. Anderson, W. G. Anderson, M. Arain, et al. (The LIGO Scientific Collaboration, http://www.ligo.org), Phys. Rev. D 77, 062002 (2008).
* Luan (2011) J. Luan, Phys. Rev. D (2011), (to be submitted).
* Buskulic et al. (2010) D. Buskulic, Virgo Collaboration, and LIGO Scientific Collaboration, Classical and Quantum Gravity 27, 194013 (2010).
* (14) URL https://www.lsc-group.phys.uwm.edu/daswg/projects/gstlal.html.
* Cannon et al. (2010) K. Cannon, A. Chapman, C. Hanna, D. Keppel, A. C. Searle, and A. J. Weinstein, Phys. Rev. D 82, 044025 (2010).
* Anderson et al. (2001) W. G. Anderson, P. R. Brady, J. D. Creighton, and É. É. Flanagan, Phys. Rev. D 63, 042003 (2001), eprint arXiv:gr-qc/0008066.
* Abbott et al. (2004) B. Abbott, R. Abbott, R. Adhikari, A. Ageev, B. Allen, R. Amin, S. B. Anderson, W. G. Anderson, M. Araya, H. Armandula, et al., Phys. Rev. D 69, 122001 (2004), eprint arXiv:gr-qc/0308069.
* Blanchet et al. (1995) L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett. 74, 3515 (1995), eprint gr-qc/9501027.
* Blanchet et al. (1996) L. Blanchet, T. Damour, and B. R. Iyer, Phys. Rev. D 54, 1860 (1996).
* Fairhurst and Brady (2008) S. Fairhurst and P. Brady, Class. Quant. Grav. 25, 105002 (2008), eprint 0707.2410.
* Brady and Fairhurst (2008) P. R. Brady and S. Fairhurst, Classical and Quantum Gravity 25, 105002 (2008), eprint 0707.2410.
* Allen et al. (2005) B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown, and J. D. E. Creighton (2005), eprint gr-qc/0509116.
* Droz et al. (1999) S. Droz, D. J. Knapp, E. Poisson, and B. J. Owen, Phys. Rev. D 59, 124016 (1999), eprint arXiv:gr-qc/9901076.
* Liu (2011) Y. Liu, J. Comput. Phys. (2011), (in preparation).
## Appendix A Noise Spectral Density
We use an algebraic expression for the noise spectral density of Advanced LIGO
detectors defined by,
$\displaystyle\begin{split}S_{h}(f)=S_{0}\left\\{\left(\frac{f}{f_{0}}\right)^{-4.14}-5\left(\frac{f_{0}}{f}\right)^{2}+\right.\\\
\left.111\left(\frac{1-\frac{f}{f_{0}}^{2}+0.5\frac{f}{f_{0}}^{4}}{1.+0.5\frac{f}{f_{0}}^{2}}\right)\right\\};\end{split}$
(42)
where, $f_{0}=215$Hz and $S_{0}=10^{49}$.
|
arxiv-papers
| 2011-08-16T09:19:50 |
2024-09-04T02:49:21.543095
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaun Hooper (UWA), Shin Kee Chung (UWA), Jing Luan (Caltech), David\n Blair (UWA), Yanbei Chen (Caltech), Linqing Wen (UWA)",
"submitter": "Shaun Hooper",
"url": "https://arxiv.org/abs/1108.3186"
}
|
1108.3206
|
# Modeling and frequency domain analysis of nonlinear compliant joints for a
passive dynamic swimmer
Juan Pablo Carbajal, Rafael Bayma, Marc Ziegler and Zi-Qiang Lang
(August 0 d , 2024)
###### Abstract
In this paper we present the study of the mathematical model of a real life
joint used in an underwater robotic fish. Fluid-structure interaction is
utterly simplified and the motion of the joint is approximated by Düffing’s
equation. We compare the quality of analytical harmonic solutions previously
reported, with the input-output relation obtained via truncated Volterra
series expansion. Comparisons show a trade-off between accuracy and
flexibility of the methods. The methods are discussed in detail in order to
facilitate reproduction of our results. The approach presented herein can be
used to verify results in nonlinear resonance applications and in the design
of bio-inspired compliant robots that exploit passive properties of their
dynamics. We focus on the potential use of this type of joint for energy
extraction from environmental sources, in this case a Kármán vortex street
shed by an obstacle in a flow. Open challenges and questions are mentioned
throughout the document.
## 1 Introduction
How much of the diverse behavior we observe in animals is a direct expression
of the dynamics of the individual’s body? In the last two decades, many
characteristics of animal locomotion on land were successfully linked to the
mechanical properties of the legs. The springy behavior observed in the
trajectories of the center of mass during running, walking and jumping can be
explained by the mechanical properties of the limbs and its tunning (Farley et
al., (1993); Farley, (1996); Ferris et al., (1998); Dickinson et al., (2000);
Kerdok et al., (2002); Moritz and Farley, (2003); Roberts and Azizi, (2011)).
The hypothesis that running, walking and jumping is tuned to the resonance of
the underlying mechanical system is strongly supported by experimental
evidence and by machines constructed based on this idea, the passive dynamic
walkers (Alexander, (1990); Ahlborn and Blake, (2002); McGeer, (1990);
Thompson and Raiber, (1989); Collins et al., (2005)). The consequence of such
setting is energy efficient performance and alleviation of the controller.
Due to similarities between running and swimming (Bejan and Marden, (2006);
Kokshenev, (2010)), it is not surprising (but not less exiting) that efficient
locomotion in fluids was reported to relay on the dynamics of the body of the
animal. Living trouts have been observed to exploit the energy in the flow
they inhabit to reduce their swimming efforts (Liao et al., (2003)). Later,
euthanized trouts performed passive self-propulsion when placed in the von
Kármán vortex street shed by an obstacle in a flow (Liao, (2007)). The
experimental results are supported by mathematical models, analytical and
numerical (Eldredge and Pisani, (2008); Kanso and Newton, (2009); Alben,
(2009)). These models do not fully agree with each other, however this is
expected due to the mathematical complexity of the interaction between
structures and fluids, of which the passive case is the worst scenario:
unprescribed motion of the interface boundary. This alone represents an open
challenge for the mathematical modeling community. Lest the challenge remains
unsolved for too long, robotic researchers design their machines with less
fluid-dynamic rigor, pursuing the first passive dynamic swimmer. The final
objective is to build a swimming machine that can perform at least as well as
fish, and at comparable power ratings (Harper et al., (1997); Lauder et al.,
(2007)).
For a robot to extract the energy in the surroundings, its mechanical
properties have to be tuned to the environmental energy storage. Stated this
way, the problem is one of energy harvesting, were nonlinear properties are
believed to be beneficial (Cottone et al., (2009)). Therefore, we need to pin
down the resonance characteristic of the actuators and joints to be used in
the robot that, as their biological counterparts, are generally nonlinear. As
if difficulty was lacking, the study of resonance of nonlinear systems has
suffer a very slow developmental process that started early in the 1960’s and
has yet not overcome its infancy. In a technical report from 1958 by
Brilliant, (1958) we read:
> Sometimes nonlinearity is avoided, not because it would have an undesired
> effect in practice, but simply because its effect cannot be computed.
Nowadays the situation is not completely different. However, we have more
powerful computers, new simulation methods and some novel uses of classical
tools promise to open the path ahead Peng et al., (2007); Vakakis et al.,
(2009).
In this paper we present the study of the joint of a robot fish from a
mathematical point of view. The amplitude of the oscillations of the joint in
response to periodic forcing is studied. In section 2 we briefly introduce the
real mechanical device and we move to its mathematical model in section 3. In
the same section the model for fluid-structure interaction is briefly
described. Approximated solutions methods are introduced in section 4. Results
are presented in section 5. Finally, we close the paper in section 6 with a
discussion on the implications of the results and the relevance of the
approach to the design of robots and the test of controllers based on
resonances.
## 2 A simple compliant joint
To extract energy from the environment, the robotic platform has to be
optimally driven by interaction forces. In this case, by the interaction
between the rigid body of the robot and the surrounding turbulent flow. The
feat can not be performed if the angle trajectories of the joints are fully
prescribed by the controller. Therefore, the joints of our robot are
compliant, i.e. the motion of the joint is not only defined by the controller,
but also by external actions. In Figure 1a we see the details of the joints of
the robot fish used in Ziegler et al., (2011). Each joint behaves as a
rotational spring. The restoring torque is generated when the relative angle
between the two connected bodies is not zero. The force producing the torque
is given by the extension of a linear spring fixed to the first body. The
spring is connected via an inelastic thread to an appendage of the second
body. When the deflection angle is zero, the extension of the spring is
minimum as well as the force it exerts. We call tension to this minimum force
value and it is referred with the letter $F$. Measurements of the torque for
$F=$0.73\text{\,}\mathrm{N}$$, are given in Figure 1b. The values of the
parameters used throughout this paper are given in Table 1. The two parameters
$r,d$ are distances that can be seen in the figures. The elastic constant of
the linear spring is $K$ and $I$ denotes the moment of inertia of the joint
around the axis of rotation. The linear specific damping coefficient of the
joint is denoted with $\zeta$. The parameters $\mathcal{Q}_{0}$ and $\Omega$
correspond to the amplitude and frequency of the external forcing,
respectively.
Name | Value
---|---
$r$ | $20.24\pm 0.02\text{\,}\mathrm{m}$ m
$d$ | $27.68\pm 0.02\text{\,}\mathrm{m}$ m
$K$ | $81\pm 1\text{\,}\mathrm{N}$/ m
$I$ | $(3.1\pm 0.1)\times 10^{-5}\text{\,}\mathrm{kg}$ m ^ 2
$\zeta\cdot I$ | $(2.2\pm 0.1)\times 10^{-4}\text{\,}\mathrm{N}$ m s
$\mathcal{Q}_{0}\cdot I$ | $1\times 10^{-4}\text{\,}\mathrm{N}$ m
$\nicefrac{{\Omega}}{{2\pi}}$ | $\left(0,3\right]\text{\,}\mathrm{Hz}$
Table 1: Value of the parameters used here and in the model studied in Ziegler
et al., (2011) . Figure 1: Description of the joints in WandaX. a) Details of
the joints used in Ziegler et al., (2011), note the axis of rotation and the
appendage. b) Measured torques applied to the joint under controlled
deflection. Replacing the parameters given in Table 1 into Eq. (1) the fit
provides $F=$0.73\pm 0.05\text{\,}\mathrm{N}$$.
## 3 Mathematical model
A geometrical representation of the joint is given in Figure 2. The parameters
$r$,$d$ and $K$ are fixed at construction time and always $d>r$ (Table 1). The
tension in the spring at its shortest length ($F$), is the controlled
parameter and a servomotor can change it dynamically.
Figure 2: Schematic of the rotational spring used to derive Eq. (1).
The torque applied to the bodies connected to the joint is thus,
$\begin{split}\tau=&\\\
&\frac{K\left(\sqrt{\epsilon^{2}+4S\sin^{2}\frac{\theta}{2}}-\epsilon\right)+F}{\sqrt{\epsilon^{2}+4S\sin^{2}\frac{\theta}{2}}}S\sin\theta.\end{split}$
(1)
Where we have defined the parameters $\epsilon=d-r$ and $S=rd$. The formula is
obtained by calculating the deformation of the spring as a function of the
deflection angle of the appendage ($\theta$). The deformation is inside
parenthesis in the numerator of equation (1) and when multiplied by the
stiffness $K$, it gives the force due to the deformation of the linear spring.
The outer factors come from the product of the force and the moment arm. Note
that the the torque $\tau$ is linear in the controlled input $F$.
For $\theta\ll 1$ the third order Taylor expansion gives
$\tau(\theta,F)=\kappa(F)\theta+\alpha(F)\theta^{3}+\mathcal{O}(\theta^{5}).$
(2)
Where
$\displaystyle\kappa(F)$ $\displaystyle=$ $\displaystyle\frac{SF}{\epsilon},$
(3) $\displaystyle\alpha(F)$ $\displaystyle=$
$\displaystyle\frac{SF}{\epsilon}\left[\frac{S}{2\epsilon^{2}}\left(\frac{\epsilon
K}{F}-1\right)-\frac{1}{6}\right].$ (4)
And the equation of motion of the deflection angle is
$\begin{split}\ddot{\theta}+\zeta\dot{\theta}+\frac{\tau(\theta,F)}{I}&\approx\\\
\ddot{\theta}+\zeta\dot{\theta}+k(F)\theta&+a(F)\theta^{3}\;=\;\Gamma.\end{split}$
(5)
Where $I$ denotes the moment of inertia around the axis of rotation. The
specific damping is given by $\zeta$, and we have defined
$k=\nicefrac{{\kappa}}{{I}}$, $a=\nicefrac{{\alpha}}{{I}}$. $\Gamma$ is the
specific net effect of all other external torques acting on the joint. The
approximating equation of motion is the well studied Düffing’s equation.
To quantify the error introduced by the approximation, we calculated the angle
at which the difference between the torque produced by equation (1) and
equation (2) is equal to a reference error given by $\Delta\tau=r\Delta F$,
where $\Delta F=$0.05\text{\,}\mathrm{N}$$ is a reasonable resolution for a
force sensor working in a $10\text{\,}\mathrm{N}$ range. These angles are
plotted in Figure 3 for different values of the tension.
Figure 3: Approximation error. Values of the deflection of the joint for which
the error reaches the reference value given in the text. The maximum
deflection has a peak close to $F_{3}=$(5.90\pm 0.01)\times
10^{-1}\text{\,}\mathrm{N}$$ meaning that terms of degree $>3$ almost cancel
each other.
### 3.1 Hardening, linear and softening spring
As can be seen from equation (2), $\alpha(F)$ modulates the intensity of the
cubic non-linear term. This term vanishes when
$F=F_{*}=\frac{\epsilon K}{\cfrac{\epsilon^{2}}{3S}+1}.$ (6)
rendering a linear spring for the angles where the third order approximation
is valid. For bigger values of $F$ the spring will be softening ($\alpha<0$)
and for smaller values it will be a hardening spring ($\alpha>0$) (Fig. 4).
However, the full expression of the torque (Eq. (1)) contains higher order
terms. Hence, the linear behavior will be even more evident if the higher
order terms cancel each other. In Figure 4 curves of torque versus angle for
several values of $F$ are shown. In particular we show the curve for which up
to seventh order nonlinearities give a minimum contribution (found via
optimization $F_{0}=0.9F_{*}$) together with the curve at $F=F_{*}$. These
illustrates the power of the actuation chose, since we can control the
dynamical properties of a virtual rotational spring at the joint (more details
in Ziegler et al., (2011)).
Figure 4: Plots of the torque function for values of tension
$F=[0.5,\,1,\,0.9,\,1.5]\cdot F_{*}$. Note the hardening behavior for small
tension and softening behavior for higher tension. Almost linear curves are
found for $F=F_{*}$ and $F=F_{0}$, as a reference the linear curve is shown.
### 3.2 Forcing model
As explained before, the external torques acting on the joint are due to
fluid-structure interaction. This kind of interaction still poses a great
challenge for the mathematical modeling community. In the search for
simplified models of the forces generated in the interaction between flexible
bodies and turbulent flows, we found the work of Kanso and Newton, (2009) and
Alben, (2009) instructive. Using equation (3.8) given by Alben, (2009), we can
calculate the pressure difference on the boundary of a slender body in a
vortex street. If the width of the body is bigger than the separation among
vortices and it is placed in the middle of the vortex street, the forces on
the body can be approximated by a sine function $f(t)=f_{0}\sin(\Omega
t+\phi)$, where $t$ is time and $\phi$ is an initial offset that sets the
balance between the sine and cosine behavior of the forcing. The frequency
$\Omega$ is proportional to the speed of the flow plus the vorticity of the
vortices (assumed to be equal for each vortex) scaled by a factor that depends
on the geometrical properties of the wake. The amplitude $f_{0}$ is
proportional to the density of the fluid and the square of the vorticity of
each vortex. With a few additional assumptions, the torque acting on the joint
can be made proportional to this force. Here we adopt this over-simplified
forcing model to avoid diverting the attention of the reader from the core
ideas of our work. In this manner we postpone a detailed study with a more
elaborated forcing model.
## 4 Solution methods
In this section we present two independent methods to estimate the amplitude
of oscillations of the joint under periodic forcing. The first method turns
out to be excellent for estimating the amplitude of the first harmonic. The
second method is based on a Volterra series expansion of the Duffing equation.
It allows to study the response of the joint to more general forcing
conditions, but the tension range for which it is valid is smaller.
### 4.1 Harmonic solutions of Düffing’s equation
Under periodic forcing $\Gamma=\mathcal{Q}_{0}\sin(\Omega t)$, equation (5)
has been extensively studied (see Holmes and Rand, (1976); Luo and Han, (1997)
and references therein). Following these analyses, we show here how to
maximize the amplitude of the periodic response of the joint by tunning the
tension parameter.
The key point of the analysis in Luo and Han, (1997), is that we search for
the amplitude of solutions of (5) that are periodic (this rules out sub-
harmonics, supra-harmonics and chaotic motion), $\theta(t)=A\sin(\Omega
t+\psi)$. Under this assumption it can be shown that the amplitude $A$ of such
solutions is given by the roots of the polynomial,
$\begin{split}A^{3}\frac{9a^{2}}{16}+A^{2}\frac{3\left(k-\Omega^{2}\right)a}{2}+&\\\
A\left[\left(\Omega^{2}+\zeta^{2}-2k\right)\Omega^{2}+k^{2}\right]&-\mathcal{Q}_{0}^{2}=0.\end{split}$
(7)
The roots of this polynomial can be obtained analytically using a computer
algebra system as Maxima, (2009) and they establish the relation between the
amplitude of the oscillations and the parameters of the equation,
$A\left(k,a,\Omega,\mathcal{Q}_{0}\right)$. In the case at hand, we have
$k(F)$ and $a(F)$, therefore $A\left(F,\Omega,\mathcal{Q}_{0}\right)$. For a
given value of the parameters, only one root corresponds to the observed
amplitude (there are unstable amplitudes). This implies, that is not enough to
look at the roots, but we must also check their stability. This adds some
complexity to the evaluation of the results that adds to the limitation of
pure harmonic inputs.
### 4.2 Volterra series expansion
When the amplitude of the forcing is sufficiently small, the behavior of
Duffing’s oscillator in the neighborhood of the origin can be described by
polynomial Volterra functionals(Theorem 3.1 of Rugh, (1981)). This means that
the angle of the oscillations can be described by an expansion of the form
$\theta(t)\approx\sum_{i=1}^{n}y_{i}(t),$ (8)
where $n$ is the order of the expansion and each term is the multi-dimensional
integral
$y_{i}(t)=\int_{-\infty}^{+\infty}h_{i}(t-\sigma_{1},\ldots,t-\sigma_{i})\\\
\Gamma(\sigma_{1})\ldots\Gamma(\sigma_{i})\mathrm{d}\sigma_{1}\ldots\mathrm{d}\sigma_{i}.$
(9)
where $\Gamma(t)$ is the forcing signal and we wish to determine the kernel
functions $\left\\{h_{i}\right\\}_{i=1}^{n}$. However, for the objective at
hand, it is more useful to calculate the Fourier transform $Y(\omega)$ of
these functionals. We have not found this process described in the literature,
hence we describe it succinctly. We used the inverse Fourier transform
definition on the input signal and substitute it into the functional
definition Eq. (9). By inverting the order of integration and splitting
variables, we make the kernel transform $H(\omega)$ appear. Then, we compare
this expression to the formal definition of the inverse Fourier transform of
$Y(\omega)$ and isolate the desired result. This process can be applied for
the first three orders, but it gets cumbersome for higher ones. Therefore,
before showing the results, we will briefly introduce some notation used here
to simplify the presentation.
An ordered set of arguments $(s_{1},s_{2},\ldots,s_{n})$ will be denoted
$s_{1:n}$. In general we have,
$(s_{k},s_{k+1},\ldots,s_{n})=s_{k:n}\quad k\leq n.$
For example, the fifth order kernel $H_{5}(s_{1},s_{2},s_{3},s_{4},s_{5})$
will be written
$\displaystyle H_{5}(s_{1:5})=$ $\displaystyle-3a\,H_{1}\left(\Sigma
s_{1:5}\right)H_{1}(s_{1})H_{1}(s_{2})H_{3}(s_{3:5}),$
where $\Sigma s_{1:5}=\sum_{i=1}^{5}s_{i}$. For the integration variable in
multiple integrals we will write
$\mathrm{d}s_{1:n}=\mathrm{d}s_{1}\mathrm{d}s_{2}\cdots\mathrm{d}s_{n}$. For
multiple summations over $n$ indexes we will write
$\sum_{k_{1:n}}=\sum_{k_{1}}\ldots\sum_{k_{n}}$. Having defined the notation,
we continue with our presentation.
For zero initial conditions, $(\theta,\dot{\theta})=0$, and working in the
frequency domain, each kernel can be obtained recursively based on lower order
kernels (details of the calculation are given in Peyton-Jones and Billings,
(1989)). The first order kernel derived from the model in Eq. (5), is
equivalent to the frequency response function (also known as transfer
function) of a linear second order system,
$H_{1}(s)=\frac{1}{s^{2}+\zeta s+k}.$ (10)
The following equations present the Volterra kernels up to seventh order using
recurrent relations,
$\displaystyle\begin{gathered}\begin{split}H_{3}(s_{1:3})=-a&H_{1}\left(\Sigma
s_{1:3}\right)\cdot\\\ &H_{1}(s_{1})H_{1}(s_{2})H_{1}(s_{3}),\end{split}\\\
\begin{split}H_{5}(s_{1:5})=-3a&H_{1}\left(\Sigma s_{1:5}\right)\cdot\\\
&H_{1}(s_{1})H_{1}(s_{2})H_{3}(s_{3:5}),\end{split}\\\
\begin{split}H_{7}(s_{1:7})=&-3aH_{1}\left(\Sigma s_{1:7}\right)\cdot\\\
&\Big{[}\;H_{1}(s_{1})H_{1}(s_{2})H_{5}(s_{3:7})+\\\
&H_{1}(s_{1})H_{3}(s_{2:4})H_{3}(s_{5:7})\;\Big{]}.\end{split}\end{gathered}$
(14)
Due to the symmetry of the eq. (5), all even order kernels are zero. Note that
these kernels are valid for any second order cubic oscillator. The recursive
formulas were obtained using the probing method of Peyton-Jones and Billings,
(1989), which proceeds as follows. We start from a polynomial nonlinear
ordinary differential equation (as Eq. (5)) and assume the response can be
represented by Volterra series, i.e. Eq. (8). We substitute this into the
system’s differential equation, and equate similar terms. This last step
produces equations relating different order kernels and the expressions are
always recursive (we obtain Eqs. (14) in our case). The factorial in the
recursive relations appear from the multinomial expansion of nonlinear
polynomial terms.
We proceed to calculate the Fourier transform of the output given by Eq. (9)
subject to harmonic forcing. We will use Eqs. (14) in order to obtain an
explicit relation between the parameters of our model
$k,a,\zeta,\mathcal{Q}_{0},\Omega,F$ and the amplitude of the first harmonic
of the output.
(a)
(b)
Figure 5: Amplitude of periodic response in the plane $(F,\Omega)$.
LABEL:sub@fig:JointORFLuo Amplitude of the oscillations according to Eq. (7).
The line of maxima is marked with dashes. The vertical dotted lines indicate
selected values of $\Omega$. LABEL:sub@fig:ORFError Relative differences
between the predicted amplitudes by the method described in section 4.1 and
the Volterra expansion described in section 4.2.
We recall that the Fourier transform of a general bandwith limited periodic
function is a scaled Dirac Comb (Shah function, Impulse train, Dirac train,
etc.) (see Schetzen, (1980)),
$\displaystyle\gamma\left(\omega\right)=\sum_{l=-L}^{+L}X_{i}\delta\left(\omega-\Omega
l\right)$ (15)
where $L$ is a positive integer, $X_{i}$ is the complex amplitude of the
$i$-th harmonic and $\delta(\cdot)$ is the impulse function (Dirac delta
function). The Fourier transform of each of the output terms in (8) is
$\begin{gathered}Y_{i}(\omega)=\left(2\pi\right)^{1-i}\int_{-\infty}^{\infty}H_{i}\left(\Delta\omega_{i}\right)\cdot\\\
\gamma(w_{1})\cdots\gamma(w_{i-1})\gamma\left(\omega-\Sigma
w_{1:i-1}\right)\mathrm{d}w_{1:i-1}=\\\
(2\pi)^{1-i}\sum_{l_{1:i}=-L}^{+L}B_{l_{1}:l_{i}}\delta\left(\omega-\Omega\Sigma
l_{1:i}\right),\end{gathered}$ (16)
where we have used Eq. (15) and
$\Delta\omega_{i}=(w_{1},\ldots,w_{i-1},\omega-\Sigma w_{1:i-1})$ and,
$\displaystyle B_{l_{1}:l_{i}}=H_{i}\left(\Omega l_{1:i}\right)X_{l_{1}}\cdots
X_{l_{i}}.$ (17)
The integral in (16), has an interesting geometric interpretation in terms of
convolutions in hyperplanes, we refer the interested reader to Lang and
Billings, (1996). Therein, the output frequency range of nonlinear systems
that are representable by Volterra series is analytically calculated.
Additionally, in Eq.(16) of that paper the frequency spectrum of the output
signal is represented as the superposition of contributions from the
nonlinearities. In the case studied herein, the input has only one frequency,
therefore $L=1$, $X_{0}=0$, $X_{\pm 1}=\pm j\mathcal{Q}_{0}\pi$ with $j$ the
imaginary unit. Replacing these values in all the equations and using the
relations in Eq. (14) we obtain the desired result,
$Y(\Omega)=\frac{-j\pi}{64}\Big{\\{}3a^{3}\mathcal{Q}_{0}^{7}\big{[}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaa}\\\
2H_{1}(3\Omega)H_{1}(-3\Omega)H_{1}^{3}(-\Omega)H_{1}^{5}(\Omega)\\\
+6H_{1}(3\Omega)H_{1}^{4}(-\Omega)H_{1}^{5}(\Omega)\phantom{aaaaaaa}\\\
+6H_{1}^{2}(3\Omega)H_{1}^{3}(-\Omega)H_{1}^{5}(\Omega)\phantom{aaaaaaa}\\\
+3H_{1}(-3\Omega)H_{1}^{4}(-\Omega)H_{1}^{5}(\Omega)\phantom{aaaaaa}\\\
-15H_{1}(3\Omega)H_{1}^{3}(-\Omega)H_{1}^{6}(\Omega)\phantom{aaaaaaa}\\\
+45H_{1}^{3}(-\Omega)H_{1}^{7}(\Omega)\phantom{aaaaaaaaaaaa}\\\
+45H_{1}^{4}(-\Omega)H_{1}^{6}(\Omega)+18H_{1}^{5}(-\Omega)H_{1}^{5}(\Omega)\;\big{]}\phantom{aaaaaaaaaaa}\\\
-12a^{2}\mathcal{Q}_{0}^{5}\big{[}H_{1}(3\Omega)H_{1}^{2}(-\Omega)H_{1}^{4}(\Omega)\phantom{aa}\\\
+6H_{1}^{2}(-\Omega)H_{1}^{5}(\Omega)\phantom{aaaaaa}\\\
+3H_{1}^{3}(-\Omega)H_{1}^{4}(\Omega)\;\big{]}\phantom{aaaaa}\\\
+48a\mathcal{Q}_{0}^{3}H_{1}(-\Omega)H_{1}^{3}(\Omega)-64\mathcal{Q}_{0}H_{1}(\Omega)\Big{\\}}.$
(18)
Where $H_{1}(x)$ is given by Eq. (10) with $s=jx$. The amplitude of the
oscillation is obtanied by taking the double of the modulus of the complex
number $Y(\Omega)/\pi$.
## 5 Results
Figure 5a shows the amplitude of periodic oscillations in the $(F,\Omega)$
plane, according to Eq. (7). The line of maxima is shown with dashes. This
line describes the value of the tension that produces maximum amplitude for a
given forcing frequency. In Fig. 5b we plot the logarithm of the relative
difference between the amplitudes given by (7) and (18). The two approximation
differ for regions of low frequency and low tension where the system is most
nonlinear (Luo and Han, (1997)). To compare with the observed amplitude
(obtained by simulating (5) without any approximation), we extracted the
curves of amplitude against tension for
$\nicefrac{{\Omega}}{{2\pi}}=$0.5,1.5\text{\,}\mathrm{Hz}$$ (vertical dotted
lines in Fig. 5). These curves are shown in Figure 6 together with the
amplitude calculated using Eq. (18) corresponding to the Volterra kernel
expansion. For the $1.5\text{\,}\mathrm{Hz}$ frequency, both models predict
the simulated amplitude accurately. For the $0.5\text{\,}\mathrm{Hz}$
frequency, the amplitude predicted by Eq. (7) drifts away from the simulated
value for lower tensions. The amplitude calculated using the Volterra
expansion diverges for low tensions at this frequency.
Figure 6: Amplitude of periodic response of the joint for forcing with
frequencies $\Omega=[$0.5,1.5\text{\,}\mathrm{]}$\mathrm{Hz}$. Amplitudes
according to Eq. (7) in solid line, amplitudes obtained from the Volterra
expansion Eq. (18) in dashes and amplitudes from simulations without
approximation in circles.
In the same figure we show the amplitudes measured in simulations using the
exact expression for the torque (1) (Octave (Eaton, (2002)) function ode45,
absolute and relative tolerances, $10^{-6}$). The amplitude of the
oscillations is obtained using the absolute value of the analytic signal of
the angle (Octave function hilbert). The agreement between the approximated
results and the simulated ones is noteworthy.
These results show that the approximation from Luo and Han, (1997) can be used
for a tension controller designed for a joint of this kind, to obtain periodic
responses with the same frequency as the forcing. The Volterra approximation
fails when the system becomes strongly nonlinear, however these expansion can
be used for more general responses or when the inputs have a more complicated
frequency spectrum (as in a realistic scenario).
## 6 Discussions and conclusion
Though the quote from Brilliant, (1958) remains valid, we used the knowledge
about Düffing’s equation to understand analytically a compliant joint. This
gives a corner stone to study more complicated setups (e.g. chain of joints,
as in the robotic fish) and define the robust engineering design of robots
with the desired resonance properties. Additionally, the results showed here
provide a reference solution to the problem of finding the right pretension
for a given external forcing, related to the idea of adaptive controllers. For
a forcing with a sufficiently slow varying frequency, the tension could be
adjusted to maintain the amount of energy transferred into the system to its
maximum possible value, i.e. keep the system close to the line of maxima in
Fig. 5a. With the results herein the adjustment could be done by direct
calculations. However, other methods like the frequency oscillators(AFO)
(Buchli et al., (2006)) were used to achieve a similar objective. Applications
of the latter has been reported in Buchli and Ijspeert, (2008), however, due
to lack of theoretical results on the resonance frequency of the nonlinear
platform studied therein, the theoretical solution to the problem was unknown
and a grounded evaluation of the performance was not possible. Approximations
as the one presented here could be used to verify the results generated by the
adaptive oscillators.
In our results we exclude any information about oscillations with frequencies
different from that of the forcing since that is the case where Eq. (7) is
valid. The higher order harmonics in the system’s output could be determined
and compared with the simulated results. Traditional harmonic balance method
can not be used to find these results. Additionally, a more complete panorama
of the frequency response of the system could be obtained with a nonlinear
analysis method as the NOFRF (proposed in Lang and Billings, (2005) and
showcased in Peng et al., (2007)), which is based on Volterra series
expansions as the one presented in this work. NOFRF give the response of the
system to a given set of inputs, therefore the natural ensemble of forcing
signals must be known to acquire useful information. This ensemble has to be
compiled from (or modeled based on) fluid-structure interactions data.
Comparing the present results with the information provided by NOFRF will be
the objective of our next work. Proved that such analysis is helpful for the
design of compliant robots, we will extend the analysis to biped and quadruped
robots. In that situation the forcing comes from the ground reaction forces,
that are generally not smooth (impacts) and contain a rich frequency spectrum.
We consider those scenarios as highly complex problems to be addressed with a
mature toolbox of methods.
Concluding, we have provided a method that allows the construction of a
controller that would maximize the harvesting of environmental energy, under
the circumstances defined herein. The method will be extended to cover more
realistic situations than the simplified periodic forcing, including forcing
with wider frequency spectrum. Additionally, the same methodology presented
here can be used to verify results obtained using heuristic methods as AFO or
neural networks. Our results can be easily validated and we invite our
colleagues to do it.
## Acknowledgements
The authors would like to thank Cecilia Tapia Siles for sharing insights and
information about passive swimmers.
#### Funding.
The research leading to these results has received funding from the European
Community’s Seventh Framework Programme FP7/2007-2013-Challenge 2 –Cognitive
Systems, Interaction, Robotics– under [grant agreement No 248311-AMARSi]. RB
is founded by the CAPES Foundation, Ministry of Education of Brazil. MZ
received funding for this work from the SNSF [project no. 122279] (From
locomotion to cognition).
#### Author contributions.
JPC, RB and ZQL worked on the mathematics of the models and in the
simulations. MZ designed and built the joint studied in this work. All authors
collaborated for the production of the manuscript.
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|
arxiv-papers
| 2011-08-16T11:26:29 |
2024-09-04T02:49:21.549463
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan Pablo Carbajal and Rafael Bayma and Marc Ziegler and Zi-Qiang\n Lang",
"submitter": "Juan Pablo Carbajal",
"url": "https://arxiv.org/abs/1108.3206"
}
|
1108.3233
|
arxiv-papers
| 2011-08-16T13:06:33 |
2024-09-04T02:49:21.555372
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Limiao Lin and Zhen Guo",
"submitter": "Shaowei Chen",
"url": "https://arxiv.org/abs/1108.3233"
}
|
|
1108.3267
|
ORLICZ SPACES ASSOCIATED WITH A SEMI-FINITE VON NEUMANN ALGEBRA
Sh. A. Ayupov${}^{1},$ V.I. Chilin${}^{2},$ R. Z. Abdullaev3
1 _Institute of Mathematics and Information Technologies, Uzbekistan Academy
of Science, Dormon yoli, 29, 100125, Tashkent. Uzbekistan_
and
_The Abdus Salam International Centre for Theoretical Physics. Triest. Italy._
Coresponding author. e-mail: _sh_ayupov@mail.ru_
2 _National University of Uzbekistan,_ e-mail: _chilin@usd.uz_
3 _Tashkent State Pedagogical University,_ e-mail: _arustambay@yandex.ru_
Introduction
Construction and investigation of various classes of symmetric spaces of
measurable operators affiliated with a von Neumann algebra $M$ is one of
important applications of the non commutative integration theory for a
faithful normal semi-finite trace on the von Neumann algebra $M.$ Examples of
such spaces are given, in particular by non commutative $L_{p}$-spaces
$L_{p}(m,\tau)$ [16] and by Orlicz spaces $L_{\Phi}(M,\tau)$ associated with
an $N$-function $\Phi$ [5],[6],[7]. All these spaces are realized as ideal
subspaces of the $*-$algebra $S(M)$ of measurable operators affiliated with
$M.$
Investigations based on the modular theory for von Neumann algebras enable to
consider non commutative versions of $L_{p}$-spaces associated with states and
weights (see e.g. the survey [14]).But in these cases in general
$L_{p}$-spaces can not be realizes as ideal subspaces of $S(M).$ This fact
explains in particular why in their attempt to introduce non commutative
Orlicz spaces for states in [1] as a subspaces of $S(M),$ the authors we
unable to prove the completeness of these spaces with respect to the Luxemburg
norm.
In the present paper we introduce a certain class of non commutative Orlicz
spaces, associated with arbitrary faithful normal locally-finite weights on a
semi-finite von Neumann algebra $M.$ We describe the dual spaces for such
Orlicz spaces and, in the case of regular weights , we show that they can be
realized as linear subspaces of the algebra of $LS(M)$ of locally measurable
operators affiliated with $M.$
For the terminology and notations from the von Neumann algebras theory we
refer to [10] and from theory of measurable and locally measurable operators
refer to [10],[15].
Preliminaries
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$ with
$\mathbf{1}$-the identity operator on $H,$ and let $P(M)=\\{p\in
M:p=p^{2}=p^{*}\\}$ be the lattice of all projection from $M$. Denote by
$S(M)$ (respectively by $LS(M)$) the ∗-algebra of all measurable
(respectively, locally-measurable) operators affiliated with $M$. It is well-
known that $S(M)$ is a ∗-subalgebra in $LS(M)$, and $M$ is a ∗-subalgebra of
$S(M)$ ([8], Ch.2).
If $x\in LS(M)$ and $x=u|x|$ is its polar decomposition, where
$|x|=(x^{*}x)^{1/2}$ and $u$ is a partial isometry, then we have that $u\in M$
and $|x|\in LS(M).$ It is also known that the spectral family of projections
$e_{\lambda}(x)$ for a self adjoint operator $x\in LS(M)$, always belongs to
$P(M).$
Given a subset $A\subset LS(M)$, put
$A_{h}=\\{x\in A:x=x^{*}\\},$ and
$A_{+}=\\{x\in A:(x\xi,\xi)\geq 0\\}$ for all $\xi\in D(x),$
where $D(x)$ is the domain of the operator $x\in LS(M),$ and $(\cdot,\cdot)$
is the inner product in the Hilbert space $H.$
Let $\tau$ be a faithful normal semi-finite trace on $M.$ For each real number
$p\geq 1$ consider the set
$L_{p}(M,\tau)=\left\\{x\in
S(M):\int\limits_{0}^{\infty}\lambda^{p}d\tau(e_{\lambda}(|x|))<\infty\right\\}.$
It known [16] that $L_{p}(M,\tau)$ is a linear subspace in $S(M)$ and the
function
$\|x\|_{p}=\left(\int\limits_{0}^{\infty}\lambda^{p}d(\tau(e_{\lambda}(|x|))\right)^{1/p}$
is a norm, which turns $L_{p}(M,\tau)$ into a Banach space.
A map $\varphi:M_{+}\rightarrow[0,\infty]$ is said to be _a weight_ if
$\varphi(x+y)=\varphi(x)+\varphi(y),\ \ \varphi(\lambda x)=\lambda\varphi(x),\
\ (x,y\in M_{+},\lambda\geq 0,\textrm{where, }0\cdot\infty=0).$
A weight $\varphi$ is said to be
— _normal_ , if $\varphi(x)=\sup\varphi(x_{i})(x_{i}\nearrow x;x_{i},x\in
M_{+});$
— _faithful_ , if $\varphi(x)=0,$ $x\in M_{+}$ implies that $x=0;$
— _semi-finite_ , if the linear span $m_{\varphi}$ of the cone
$m_{\varphi}^{+}=\\{x\in M_{+}:\varphi(x)<\infty\\}$ is dense in $M$ with
respect to the ultra-weak topology;
— _locally finite_ , if
$\forall x\in M_{+}\ \ (x\neq 0)\ \ \exists y\in M_{+}:y\leq
x,0<\varphi(y)<\infty;$
— _regular_ , if
$\forall\omega\in(M_{*})_{+}\,\,(\omega\neq 0)\
\exists\omega^{\prime}\in(M_{*})_{+}\,\,(\omega^{\prime}\neq
0):\omega^{\prime}\leq\omega,\omega^{\prime}\leq\varphi,$
where $(M_{*})_{+}$ is the set of all positive ultra-weakly continuous linear
functionals on $M.$
If the weight $\varphi$ is a trace, i.e. when
$\varphi(x^{*}x)=\varphi(xx^{*})$ for all $x\in M,$ the properties of semi-
finiteness and locally finiteness (and respectively of faithfulness and
regularity) of $\varphi$ coincide with each other [13].
For a faithful normal semi-finite weight $\varphi$ on $M$ there exists a
uniquely defined non singular self-adjoint positive operator $h$, affiliated
with $M$ such that $\varphi(\cdot)=\tau(h\cdot)$ , and which is called the
Radon-Nikodym derivative of the weight $\varphi$ with respect is the trace
$\tau$ [9].
Recall the following result
Theorem 1.[13] _Let $\tau$ be a faithful normal semi-finite trace on $M$ and
let $\varphi=\tau(h\cdot)$ be a faithful normal semi-finite weight on $M,$
where $h$ is the Radon-Nikodym derivative of $\varphi$ with respect to $\tau.$
Then_
$(i)$ _the weight $\varphi$ is locally finite if and only if the operator $h$
is locally measurable;_
$(ii)$ _the weight $\varphi$ is regular if and only if the operator $h^{-1}$
is locally measurable._
Now let $\varphi(\cdot)=\tau(h\cdot)$ be a faithful normal locally finite
weight on $M.$ For real numbers $p\geq 1$ and $\alpha\in[0,1]$ put
$m_{\alpha}^{1/p}=\\{x\in M:h^{\alpha/p}xh^{(1-\alpha)/p}\in
L_{p}(M,\tau)\\};$ $\|x\|_{p,\alpha}=\|h^{\alpha/p}xh^{(1-p)/p}\|_{p}.$
In [12] it has been proved that $m_{\alpha}^{1/p}$ is a linear subspace in
$M,$ and $\|\cdot\|_{p,\alpha}$ is a norm on $m_{\alpha}^{1/p}.$ The
completion of the normed space $(m_{\alpha}^{1/p},\|\cdot\|_{p,\alpha})$ is
denoted by $L_{p}(M,\varphi).$ In [12] it is proved that the Banach space
$(L_{p}(M,\varphi),\|\cdot\|_{p,\alpha})$ is isometrically isomorphic to the
space $(L_{p}(M,\tau),\|\cdot\|_{p})$ for all $\alpha\in[0,1].$
In order to define the Orlicz space associated with a weight, we need the
notion of $N$-function.
A continuous non-negative convex monotone increasing function $\Phi$ on the
set of real numbers $\mathbb{R}$ is called $N$-_function_ [4], if
$\Phi(t)=\int\limits_{0}^{|t|}p(s)ds,$
where $p(s)$ is a non-decreasing function, positive for $s>0$ and right
continuous for $s\geq 0$ , which satisfies the conditions
$p(0)=0,\ \ p(\infty)=\lim\limits_{s\rightarrow\infty}p(s)=\infty.$
For each $N$-function $\Phi(t)$ a complementary $N$-function $\Psi(t)$ is
defined as
$\Psi(t)=\int\limits_{0}^{|t|}q(s)ds,$
where $q(s)=\sup\\{t\geq 0:p(t)\leq s\\}.$ It is clear that the complementary
$N$-function for the $N$-function $\Psi(t)$ coincides with the initial
function $\Phi(t),$ and moreover the following Young inequality is valid
$ts\leq\Phi(t)+\Psi(s)\ \ \textrm{for all}\ \ t,s\geq 0.$
We say that an $N$-function $\Phi(t)$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, if given any real $k>0$ there exists a
positive number $r(k)$ such that $\Phi(kt)\leq r(k)\Phi(t)$ for all $t\geq 0.$
Examples of $N$-function which satisfy the $(\delta_{2},\Delta_{2})$-condition
are given by the function $\Phi(t)=\frac{1}{p}|t|^{p},$ $p>1.$
Let $\Phi(t)$ be an $N$-function and let $x\in LS_{h}(M),$
$x=\int\limits_{-\infty}^{\infty}\lambda de_{\lambda}(x).$ It is known ([8],
§2.3) that one can define a self-adjoint operator
$\Phi(x)=\int\limits_{-\infty}^{\infty}\Phi(\lambda)de_{\lambda}(x),$ and
moreover $\Phi(x)\in LS(M).$
Let us extend the faithful normal semi-finite trace $\tau$ from $M_{+}$ to
operators from $LS_{+}(M)$ as
$\tau(x)=\sup\limits_{t\geq 1}\tau\left(\int\limits_{0}^{t}\lambda
de_{\lambda}(x)\right)=\int\limits_{0}^{\infty}\lambda d\tau(e_{\lambda}(x)).$
It is known (e.g. [8], §4.1), that
$\tau(x)=\sup\\{\tau(y):y\in M_{+},y\leq x\\}$
for all $x\in LS_{+}(M).$
It is clear that $\tau(|x|)<\infty$ for $x\in LS(M)$ if and only if $x\in
L_{1}(M,\tau);$ in this case $\tau(\mathbf{1}-e_{\lambda}(|x|))<\infty$ for
all $\lambda>0.$ Further we shall need the following result.
Proposition 1. [3] _If $x,y\in LS_{+}(M),$ then_
$(i)$ $\tau(f(x))\leq\tau(f(y))$ for $x\leq y$ _for each continuous monotone
increasing function $f:[0,\infty)\rightarrow\mathbb{R}$ with $f(0)=0;$_
$(ii)$ $\tau(f(\lambda
x+(1-\lambda)y))\leq\lambda\tau(f(x))+(1-\lambda)\tau(f(y))$ _for all
$\lambda\in[0,1]$ and each convex monotone increasing function $f$ with
$f(0)=0.$_
Let $\Phi$ be an $N$-function. The set
$K_{\Phi}=\\{x\in S(M):\tau(\Phi(|x|))\leq 1\\}$
is an absolutely convex subset in $S(M)$ [5]. The linear subspace
$L_{\Phi}(M,\tau)=\bigcup\limits_{n=1}^{\infty}nK_{\Phi}$ equipped with the
norm
$\|x\|_{\Phi}=\inf\left\\{\lambda>0:\frac{x}{\lambda}\in K_{\Phi}\right\\},$
$None$
is a Banach space [5] which is called the Orlicz space associated with
$M,\tau$ and $\Phi$. If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then
$L_{\Phi}(M,\tau)=\\{x\in S(M):\tau(\Phi(|x|))<\infty\\},$
moreover the linear subspace
$m_{\Phi}^{\tau}=\\{x\in M:\tau(\Phi(|x|))<\infty\\}$
is dense in $\left(L_{\Phi}(M,\tau),\|\cdot\|_{\Phi}\right).$
Note that
$m_{\tau}=\\{x\in M:\tau(|x|)<\infty\\}\subset m_{\Phi}^{\tau}.$ $None$
Indeed, from the equalities
$\lim\limits_{t\downarrow 0}\frac{\Phi(t)}{t}=\lim\limits_{t\downarrow
0}p(t)=0$
it follows that $\Phi(t)\leq t$ for sufficiently small $t>0.$
Therefore for $x\in m_{\tau}$ there exists $t_{0}>0$ such that
$\tau(\Phi(|x|e_{t_{0}}(|x|)))=\int\limits_{0}^{t_{0}}\Phi(\lambda)d\tau(e_{\lambda}(|x|))\leq\int\limits_{0}^{t_{0}}\lambda
d\tau(e_{\lambda}(|x|))=\tau(|x|e_{t_{0}}(|x|))<\infty.$
Since $\tau(\mathbf{1}-e_{t_{0}}(|x|))<\infty,$ we have that
$\tau(\Phi(|x|(\mathbf{1}-e_{t_{0}})))\leq\Phi(\|x\|_{M})\tau(\mathbf{1}-e_{t_{0}}(|x|))<\infty,$
where $\|\cdot\|_{M}$ is the $C^{*}$-norm on $M.$ Therefore
$\tau(\Phi(|x|))<\infty,$ i.e. $x\in m_{\Phi}^{\tau}.$
Proposition 2. _If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then $m_{\tau}$ is dense in
_$L_{\Phi}(M,\tau).$
_Proof._ Since $m_{\tau}\subset m_{\Phi}^{\tau}$ (see (2)) and
$m_{\Phi}^{\tau}$ is dense in $L_{\Phi}(M,\tau),$ it sufficient to prove that
$m_{\tau}$ is dense in $m_{\Phi}^{\tau}.$ Moreover since each element of
$m_{\phi}^{\tau}$ is a finite linear combination of positive elements from
$m_{\Phi}^{\tau}$ it sufficient to show that every element from
$x\in\left(m_{\Phi}^{\tau}\right)_{+}$ belongs to the closure of $m_{\tau}$ in
$L_{\Phi}(M,\tau).$ First, let us show that
$x_{n}=x(\textbf{1}-e_{\frac{1}{n}})\in m_{\tau},$
where $e_{\lambda}=e_{\lambda}(x),$ $\lambda>0$ is the spectral family of $x.$
From
$\Phi\left(\frac{1}{n}\right)\tau\left(\textbf{1}-e_{\frac{1}{n}}\right)=\tau\left(\Phi\left(\frac{1}{n}\left(\textbf{1}-e_{\frac{1}{n}}\right)\right)\right)\leq\tau\left(\Phi\left(x\left(\textbf{1}-e_{\frac{1}{n}}\right)\right)\right)\leq\tau(\Phi(x))<\infty,$
it follows that $\tau\left(\textbf{1}-e_{\frac{1}{n}}\right)<\infty$ and the
inequality $0\leq
x\left(\textbf{1}-e_{\frac{1}{n}}\right)\leq\|x\|_{M}\left(\textbf{1}-e_{\frac{1}{n}}\right)$
implies that
$x_{n}=x\left(\textbf{1}-e_{\frac{1}{n}}\right)\in m_{\tau}.$
Since $0\leq xe_{\frac{1}{n}}\downarrow 0$ when $n\rightarrow\infty,$ it
follows that
$\tau\left(\Phi\left(\frac{1}{\varepsilon}xe_{\frac{1}{n}}\right)\right)\downarrow
0$ for any $\varepsilon>0.$ In particular, there exists $n(\varepsilon)$ such
that
$\tau\left(\Phi\left(\frac{1}{\varepsilon}xe_{\frac{1}{n}}\right)\right)<1$
for $n\geq n(\varepsilon),$ i.e.
$\left\|xe_{\frac{1}{n}}\right\|_{\Phi}<\varepsilon.$ This means that
$\|x-x_{n}\|_{\Phi}\rightarrow 0$, i.e. $m_{\tau}$ is dense $m_{\Phi}^{\tau}.$
The proof is complete. $\Box$
Let $\Psi$ be the complementary $N$-function for the $N$-function $\Phi$
satisfying the $(\delta_{2},\Delta_{2})$-condition. In this case given any
$y\in L_{\Psi}(M,\tau)$ the function $f_{y}(x)=\tau(xy),x\in
L_{\Phi}(M,\tau),$ defines the general form of continuous linear functionals
on $L_{\Phi}(M,\tau)$ [5], moreover
$\|f_{y}\|=\sup\\{|\tau(xy)|:x\in L_{\Phi}(M,\tau),\|x\|_{\Phi}\leq
1\\}=\|y\|_{\Psi}.$
Further we shall need also two inequalities from the following proposition.
Proposition 3. _Let $\tau$ be a faithful normal semi-finite trace on a von
Neumann algebra $M$. Then_
$(i)$([8], §3.4). _Given any $x,y\in LS(M)$ there exist two partial isometries
$u,v\in M$ such that_
$|x+y|\leq u^{*}|x|u+v^{*}|y|v.$
$(ii)$ [2]. _For every $N$-function $\Phi$, arbitrary operator $z\in M$ with
$\|z\|_{M}\leq 1$, and for each $x\in LS_{+}(M)$ we have the following
inequality_
$\tau(\Phi(z^{*}xz))\leq\tau(z^{*}\Phi(x)z).$
Orlicz spaces associated with a weight
In this section an approach is suggested for the construction of Orlicz spaces
associated with a faithful normal locally finite weight on a semi-finite von
Neumann algebra for an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition. For these spaces the dual spaces are
described. In the case of regular locally finite normal weights the
constructed Orlicz spaces are represented as spaces of locally measurable
operators.
Let $\tau$ be a faithful normal semi-finite trace on a von Neumann algebra
$M$. From now on $\varphi$ denotes a faithful normal locally finite weight on
$M$. Therefore the Radon-Nikodym derivative $h$ of the weight $\varphi$ with
respect to $\tau$ is a positive locally measurable non-singular operator.
Given an $N$-function $\Phi$ and a real number $\alpha\in[0,1]$ put
$U(x)=U_{\Phi,\alpha}^{\varphi,\tau}(x)=(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha},\
\ x\in LS(M).$
It is clear that $U(x)\in LS(M)$ and $\Phi(|U(x)|)\in LS(M).$
Consider the functional on $LS(M)$ defined by
$O_{\Phi,\alpha}^{\varphi,\tau}(x)=\tau(\Phi(|U(x)|)),$
and put
$m_{\Phi,\alpha}^{\varphi,\tau}=\left\\{x\in
M:O_{\Phi,\alpha}^{\varphi,\tau}(x)<\infty\right\\}.$
Consider on the set $m_{\Phi,\alpha}^{\varphi,\tau}$ the functional
$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda>0:O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq
1\right\\}.$
Theorem 2. _If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then the set
$m_{\Phi,\alpha}^{\varphi,\tau}$ is a linear subspace in $M$._
In order to prove this theorem we need the following inequality.
Lemma 1. _For the $N$-function $\Phi$ and real number $\lambda\in[0,1]$ the
following inequality is valid_
$\tau(\Phi(|U(\lambda x)|))\leq\lambda\tau(\Phi(|U(x)|))$ $None$
_for all_ $x\in m_{\Phi,\alpha}^{\Phi,\tau}.$
_Proof._ By the linearity of the map $U$ we have
$\tau(\Phi(|U(\lambda x)|))=\tau(\Phi(\lambda|U(x)|)).$
From the inequality $(ii)$ in Proposition 1 with $y=0$, we obtain
$\tau(\Phi(|U(\lambda x)|))\leq\lambda\tau(\Phi(|U(x)|)).$
The proof of lemma is complete. $\Box$
_Proof of the theorem 2._ The inequality (3) above implies that
$m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under the multiplication by complex
number $k$ with $|k|\leq 1.$ Let us show that for $x\in
m_{\Phi,\alpha}^{\varphi,\tau}$ and any complex number $k$ with $|k|>1$ we
have that $kx\in m_{\Phi,\alpha}^{\varphi,\tau},$ i.e.
$\tau(\Phi(|U(kx)|))<\infty.$
Since $\Phi$ satisfies the $(\delta_{2},\Delta_{2})$-condition, given any
positive number $|k|$ there exists a positive number $r(|k|)$ such that
$\Phi(|k|t)\leq r(|k|)\Phi(t)$ for all $t\geq 0$. Therefore
$(\delta_{2},\Delta_{2})$-condition implies that
$\tau(\Phi(|U(kx)|))=\tau(\Phi(|k||U(x)|))\leq$ $\leq
r(|k|)\tau(\Phi(|U(x)|))<\infty,$
i.e. the set $m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under multiplication
by any complex number.
Now let us prove that the sum of any two operators from
$m_{\Phi,\alpha}^{\varphi,\tau}$ also belongs to
$m_{\Phi,\alpha}^{\varphi,\tau}$. Let $x,y\in m_{\Phi,\alpha}^{\varphi,\tau},$
i.e. $\tau(\Phi(|U(x)|))<\infty$ and $\tau(\Phi(|U(y)|))<\infty.$ The
inequalities $(i)$ and $(ii)$ from proposition 3, the linearity of the
operator $U$, the convexity of $\Phi$, the tracial property of $\tau$ and the
fact that $m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under the multiplication
by complex numbers imply:
$\tau(\Phi(|U(x+y)|))=\tau(\Phi(|U(x)+U(y)|))\leq\tau(\Phi(u^{*}|U(x)|u+v^{*}|U(y)|v))\leq$
$\leq\tau\left(\frac{1}{2}(\Phi(2u^{*}|U(x)|u)+\Phi(2v^{*}|U(y)|v))\right)=$
$=\frac{1}{2}(\tau\left(\Phi(u^{*}2|U(x)|u))+\tau(\Phi(v^{*}2|U(y)|v)\right)\leq\frac{1}{2}\left(\tau(u^{*}\Phi(2|U(x)|)u)+\tau(v^{*}\Phi(2|U(y)|)v)\right)\leq$
$\leq\frac{1}{2}(\tau(\Phi(2|U(x)|))+\tau(\Phi(2|U(y)|)))=\frac{1}{2}(\tau(\Phi|U(2x)|)+\tau(\Phi|U(2y)|))<\infty,$
i.e. $x+y\in m_{\Phi,\alpha}^{\varphi,\tau}.$
The proof is complete. $\Box$
Theorem 3. _The set_
$K_{\Phi,\alpha}^{\varphi,\tau}=\\{x\in
M:O_{\Phi,\alpha}^{\varphi,\tau}(x)\leq 1\\}$
_is absolutely convex and absorbing in $m_{\Phi,\alpha}^{\varphi,\tau}.$_
_Proof._ Let us prove the convexity of $K_{\Phi,\alpha}^{\varphi,\tau}$. Let
$x,y\in K_{\Phi,\alpha}^{\varphi,\tau}$ and $\lambda\in[0,1].$ In view of
Proposition 3$(i)$ there exist partial isometries $u$ and $v$ in $M$ such that
$|\lambda U(x)+(1-\lambda)U(y)|\leq\lambda
u^{*}|U(x)|u+(1-\lambda)v^{*}|U(y)|v.$
From the inequalities of Proposition 1 and 3 and from the tracial property of
$\tau$ we obtain
$\tau(\Phi|\lambda
U(x)+(1-\lambda)U(y)|)\leq\lambda\tau(\Phi(u^{*}|U(x)|u))+(1-\lambda)\tau(\Phi(v^{*}|U(y)|v))\leq$
$\leq\lambda\tau(u^{*}\Phi(|U(x)|u))+(1-\lambda)\tau(v^{*}\Phi(|U(y)|)v)\leq\lambda\tau(\Phi(|U(x)|))+(1-\lambda)\tau(\Phi(|U(y)|)),$
i.e.
$O_{\Phi,\alpha}^{\varphi,\tau}(\lambda x+(1-\lambda)y)\leq\lambda
O_{\Phi,\alpha}^{\varphi,\tau}(x)+(1-\lambda)O_{\Phi,\alpha}^{\varphi,\tau}(y),$
which implies the convexity of $K_{\Phi,\alpha}^{\varphi,\tau}$.
The inequality (3) shows that the set $K_{\Phi,\alpha}^{\varphi,\tau}$ is
balanced, and hence is absolutely convex.
Finally let us move that $K_{\Phi,\alpha}^{\varphi,\tau}$ is absorbing in
$m_{\Phi,\alpha}^{\varphi,\tau}$.
If $x\in m_{\Phi,\alpha}^{\varphi,\tau},$ then there exists $t>1$ such that
$O_{\Phi,\alpha}^{\varphi,\tau}(x)<t.$ Let $\lambda\in\mathbb{C}$ and
$|\lambda|\geq t$ By Lemma 1 we have that
$O_{\Phi,\alpha}^{\varphi,\tau}(\frac{x}{\lambda})\leq\frac{1}{|\lambda|}O_{\Phi,\alpha}^{\varphi,\tau}(x)\leq\frac{1}{t}O_{\Phi,\alpha}^{\varphi,\tau}(x)<1,$
i.e. $\frac{x}{\lambda}\in K_{\Phi,\alpha}^{\varphi,\tau}(x).$ The proof is
complete. $\Box$
Corollary 1. _The Minkovsky functional of the set
$K_{\Phi,\alpha}^{\varphi,\tau}$ defined as_
$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda>0:\frac{x}{\lambda}\in
K_{\Phi,\alpha}^{\varphi,\tau}\right\\},$ $None$
_is a norm on the linear space $m_{\Phi,\alpha}^{\varphi,\tau}$._
_Proof._ It is sufficient to prove that $\|x\|_{\Phi,\alpha}^{\varphi,\tau}=0$
implies that $x=0$. Indeed, if $\|x\|_{\Phi,\alpha}^{\varphi,\tau}=0$ then
$O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq 1$ for all
$\lambda\in(0,1)$. By Lemma 1 we obtain that
$\frac{1}{\lambda}O_{\Phi,\alpha}^{\varphi,\tau}\left(x\right)\leq
O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq 1$ for all
$\lambda\in(0,1),$ i.e. $O_{\Phi,\alpha}^{\varphi,\tau}(x)=0.$ Faithfulness of
$\tau$ then implies that $\Phi^{-1}(h)^{\alpha}x\Phi^{-1}(h)^{1-\alpha}=0$.
Since $h\in LS_{+}(M)$ (see theorem 1$(i)$) and $h$ is a non singular
operator, we have that $\Phi^{-1}(h)^{\alpha},$ $\Phi^{-1}(h)^{1-\alpha}\in
LS_{+}(M)$ and $\Phi^{-1}(h)^{\alpha},$ $\Phi^{-1}(h)^{1-\alpha}$ are non
singular operators too. Let
$\Phi^{-1}(h)=\int\limits_{0}^{\infty}\lambda de_{\lambda},$
$x_{n}=\int\limits_{\frac{1}{n}}^{n}\left(\frac{1}{\lambda}\right)^{\alpha}\lambda
de_{\lambda},$
$y_{n}=\int\limits_{\frac{1}{n}}^{n}\left(\frac{1}{\lambda}\right)^{1-\alpha}\lambda
de_{\lambda}.$
Using
$x_{n}\Phi^{-1}(h)^{\alpha}=e_{n}-e_{\frac{1}{n}}=\Phi^{-1}(h)^{1-\alpha}y_{n}$
we see that
$\left(e_{n}-e_{\frac{1}{n}}\right)x\left(e_{n}-\frac{1}{n}\right)=0.$ Since
$\Phi^{-1}(h)$ is a non singular operator, it follows that
$\left(e_{n}-e_{\frac{1}{n}}\right)\uparrow\mathbf{1}$ for all $n=1,2,\ldots$
when $n\rightarrow\infty.$ Consequently, $x=0.$ The proof is complete. $\Box$
Denote by $L_{\Phi,\alpha}(M,\varphi,\tau)$ the Banach space obtained as the
completion of $m_{\Phi,\alpha}^{\varphi,\tau}$ in the norm
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and call this completion _the Orlicz
space_ constructed by the $N$-function $\Phi$ on the von Neumann algebra $M$
with respect to the faithful normal locally finite weight $\varphi$. It is
clear that if $\varphi$ is a trace or $M$ is a commutative von Neumann
algebra, then the norm $\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ do not depend on $\alpha\in[0,1].$
Note also that in the case where $\Phi(t)=\frac{1}{t}|t|^{p},\ p>1,$ the norm
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ do not depend on the choice of the faithful
normal semi-finite trace $\tau$ and of $\alpha\in[0,1]$ [12].
For general $N$-functions $\Phi$ this is not true even in the commutative
case.
Example. Take $M=l_{\infty},$ $f_{i}=\\{0,...,0,1,0...\\},$ where 1 is on the
$i$-th position, and put $\Phi(t)=|t|^{\beta}(\ln|t|+1),\ t\neq 0,\ \beta>1,\
\Phi(0)=0.$ In ([4], Ch. I, §4) it is proved that $\Phi$ is an $N$-function
satisfying the $(\delta_{2},\Delta_{2})$-condition. Consider the trace $\nu$
on $l_{\infty}$ defined as
$\nu(f_{i})=\frac{1}{i^{2}}((e^{i^{2}})^{2\beta}(2i^{2}+1))^{-1}.$ Put
$h=\\{e^{\beta i^{2}}+(i^{2}+1)\\}_{i=1}^{\infty},$
$f=\Phi^{-1}(h)=\\{e^{i^{2}}\\}_{i=1}^{\infty}.$
Now define the trace $\mu$ on $l_{\infty}$ as $\mu(\cdot)=\nu(h\cdot).$
Let us show that in this case the norms $\|\cdot\|_{\Phi,1}^{\mu,\nu}$ and
$\|\cdot\|_{\Phi,1}^{\mu,\mu}$ are not equivalent on the ideal $E$ of all
finite sequences from $l_{\infty}$ (it is clear that $E\subset
m_{\Phi,\alpha}^{\mu,\nu}$ and $E\subset m_{\Phi,\alpha}^{\mu,\mu}$). For this
it is sufficient to find a sequence $\\{x_{n}\\}$ of elements from
$(K_{\Phi,1}^{\mu,\nu})\cap E$ such that
$\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda K_{\Phi,1}^{\mu,\mu}$ for all
$\lambda>0.$ Let $x_{n}=\sum\limits_{i=2}^{n}e^{i^{2}}f_{i}.$ It is clear that
for commutative algebras one has
$O_{\Phi,1}^{\mu,\nu}(x)=\nu(\Phi(|\Phi^{-1}(h)x|))$
and
$O_{\Phi,1}^{\mu,\mu}(x)=\mu(\Phi(|x|)).$ $None$
Therefore
$O_{\Phi,1}^{\mu,\nu}(x_{n}f_{i})=\nu(\Phi(fx_{n}f_{i}))=\nu(\Phi((e^{2i^{2}}f_{i})^{2}))=(e^{i^{2}})^{2\beta}(2i^{2}+1)\nu(f_{i})=\frac{1}{i^{2}}.$
Hence
$O_{\Phi,1}^{\mu,\nu}(x_{n})=\sum\limits_{i=2}^{n}\frac{1}{i^{2}}<1,\textrm{i.e.}\,x_{n}\in
K_{\Phi,1}^{\mu,\nu},$ $None$
for all $n.$ Let us show that $\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda
K_{\Phi,1}^{\mu,\mu}$ for all positive real $\lambda.$ From (5) we have
$O_{\Phi,1}^{\mu,\mu}(x_{n}f_{i})=\mu(\Phi(x_{n}f_{i}))=\nu(h\Phi(x_{n})f_{i})=\nu(\Phi(\Phi^{-1}(h))\Phi(x_{n})f_{i})=\nu(\Phi(f)\Phi(x_{n})f_{i})=$
$=(e^{i^{2}})^{2\beta}(i^{4}+2i^{2}+1)\nu(f_{i})>(e^{i^{2}})^{2\beta}i(2i^{2}+1)\nu(f_{i})=\frac{1}{i}.$
Therefore $O_{\Phi,1}^{\mu,\mu}(x_{n})>\sum\limits_{i=2}^{n}\frac{1}{n},$ and
hence
$\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda K_{\Phi,1}^{\mu,\mu}$ $None$
for all positive $\lambda.$ From (6) and (7) it follows that the norms
$\|\cdot\|_{\Phi,1}^{\mu,\nu}$ and $\|\cdot\|_{\Phi,1}^{\mu,\mu}$ are not
equivalent on $E.$ In particular the identity mapping from $E$ into $E$ can
not be extended to an isomorphism between
$L_{\Phi,\alpha}(l_{\infty},\mu,\nu)$ and
$L_{\Phi,\alpha}(l_{\infty},\mu,\mu).$
At the same time by following theorem the Orlicz spaces
$L_{\Phi,\alpha}(l_{\infty},\mu,\nu)$ and
$L_{\Phi,\alpha}(l_{\infty},\mu,\mu)$ are isometrically isomorphic.
Theorem 4. _Let the $N$-function $\Phi$ satisfy the
$(\delta_{2},\Delta_{2})$-condition, $\alpha\in[0,1].$ Then the Banach space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is isometrically isomorphic to the Banach
space_ $L_{\Phi}(M,\tau)=L_{\Phi,1}(M,\tau,\tau)$.
_Proof._ For every $x\in m_{\Phi,\alpha}^{\varphi,\tau}$ we have
$U(x)=(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\in L_{\Phi}(M,\tau).$
Therefore from definitions (1) and (4) of the norms we obtain
$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\|(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\|_{\Phi}.$
This means that the map $U$ defined as
$m_{\Phi,\alpha}^{\varphi,\tau}\ni
x\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!U}\
(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\in L_{\Phi}(M,\tau)$ $None$
is a linear isometry. Let us show that the
$U(m_{\Phi,\alpha}^{\varphi,\tau})=(\Phi^{-1}(h))^{\alpha}m_{\Phi,\alpha}^{\varphi,\tau}(\Phi^{-1}(h))^{1-\alpha}$
is dense in $L_{\Phi}(M,\tau)$.
Let $h=\int\limits_{0}^{\infty}\lambda de_{\lambda}$ and
$q_{n}=\int\limits_{\frac{1}{n}}^{n}de_{\lambda}\,\,,(n=1,2,...)$. Consider
the set
$\mathcal{F}=\bigcup\limits_{m,n=1}^{\infty}q_{m}m_{\tau}q_{n}.$
Since $q_{n}\leq q_{n+1},$ it follows that $\mathcal{F}$ is a linear subspace
in $m_{\tau}$ and by (2) $\mathcal{F}\subset L_{\Phi}(M,\tau).$
First. Let us prove that $\mathcal{F}$ is dense in $L_{\Phi}(M,\tau)$. From
the $(\delta_{2},\Delta_{2})$-condition it follows that for $y\in
L_{\Psi}(M,\tau)$ (where $\Psi$ is the complementary $N$-function for $\Phi$)
the functional $f(x)=\tau(xy),$ $x\in L_{\Phi}(M,\tau),$ defines the general
form of continuous linear functional on $L_{\Phi}(M,\tau)$.
Let $y\in L_{\Psi}(M,\tau)$ and suppose that
$f(q_{m}xq_{n})=\tau((q_{m}xq_{n})y)=0$ for all $x\in m_{\tau}$ and
$m,n=1,2,....$ In order to prove that $\mathcal{F}$ is dense in
$L_{\Phi}(M,\tau)$ it is sufficient to show that $y=0.$
From the tracial property of $\tau$ we have that $\tau(xq_{n}yq_{m})=0$ for
all $x\in m_{\tau}.$ By proposition 2 $m_{\tau}$ is dense in
$L_{\Phi}(M,\tau)$ and hence $q_{n}yq_{m}=0$ for all $m,n=1,2,...$. Since
$q_{n}\nearrow\textbf{1}$ as $n\rightarrow\infty,$ this implies that $y=0$.
Therefore $\mathcal{F}$ is dense $L_{\Phi}(M,\tau)$.
Now let us show that $\mathcal{F}\subset U(m_{\Phi,\alpha}^{\varphi,\tau}).$
For this it is sufficient to prove that given any $x\in m_{\tau}$ and
$m,n=1,2,...,$ there exists $y\in m_{\Phi,\alpha}^{\varphi,\tau}$ such that
$q_{m}xq_{n}=U(y).$
Since the operators $(\Phi^{-1}(h))^{-\alpha}q_{m}$ and
$(\Phi^{-1}(h))^{\alpha-1}q_{n}$ belong to $M,$ the operator
$y=U^{-1}(q_{m}xq_{n})=(\Phi^{-1}(h))^{-\alpha}(q_{m}xq_{n})(\Phi^{-1}(h))^{\alpha-1}$
also belongs to $M.$ From (2) and from $\tau(|q_{m}xq_{n}|)<\infty$ we obtain
that $\tau(\Phi(|U(y)|))=\tau(\Phi(|q_{m}xq_{n}|))<\infty,$ i.e. $y\in
m_{\Phi,\alpha}^{\varphi,\tau}.$ This implies that $\mathcal{F}\subset
U(m_{\Phi,\alpha}^{\varphi,\tau}).$
Now since $m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(L_{\Phi,\alpha}(M,\varphi,\tau),\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau})$ and
$U\left(m_{\Phi,\alpha}^{\varphi,\tau}\right)$ is dense in
$(L_{\Phi}(M,\tau),\|\cdot\|_{\Phi})$ the isometry
$U:m_{\Phi,\alpha}^{\varphi,\tau}\rightarrow L_{\Phi}(M,\tau)$ defined in (8)
can be uniquely extended to an isometric isomorphism between
$L_{\Phi,\alpha}(M,\varphi,\tau)$ and $L_{\Phi}(M,\tau).$ The proof is
complete. $\Box$
Since every faithful normal semi-finite trace $\tau_{1}$ on $M$ is a locally
finite weight [13] the theorem 4 implies the following
Corollary 2._If $\tau_{1}$ and $\tau$ are faithful normal semi-finite traces
on a von Neumann algebra $M,$ $\Phi$ is an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition, then the Orlicz spaces
$L_{\Phi}(M,\tau_{1})$ and $L_{\Phi}(M,\tau)$ are isometrically isomorphic._
Theorem 4 and Corollary 2 together imply the following theorem
Theorem 5._Let $\tau_{1}$ and $\tau$ be faithful normal traces on a von
Neumann algebra $M,$ and let $\varphi_{1},\varphi_{2}$ be faithful normal
locally finite weights on $M.$ Suppose that $\Phi$ is an $N$-function
satisfying the $(\delta_{2},\Delta_{2})$-condition, $\alpha,\beta\in[0,1].$
Then the Orlicz spaces $L_{\Phi,\alpha}(M,\varphi_{1},\tau_{1})$ and
$L_{\Phi,\beta}(M,\varphi_{2},\tau_{2})$ are isometrically isomorphic._
Theorem 4 implies also the following
Corollary 3. _Let $\Phi$ be an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition and let $\Psi$ be the complementary
$N$-function for $\Phi$, and $\alpha,\beta\in[0,1].$ Then the dual space
$(L_{\Phi,\alpha}(M,\varphi,\tau))^{*}$ for the Orlicz space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is isometrically isomorphic to the space
$L_{\Psi}(M,\tau).$ If moreover $\Psi$ also satisfies the
$(\delta_{2},\Delta_{2})$-condition then
$(L_{\Phi,\alpha}(M,\varphi,\tau))^{*}$ is isometrically isomorphic to
$L_{\Psi,\beta}(M,\varphi,\tau)$ and the Banach space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is reflexive._
Now let us give a representation of the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ by locally measurable operators in the case
where $\varphi$ is a regular locally finite weight, and the $N$-function
$\Phi$ satisfies $(\delta_{2},\Delta_{2})$-condition.
Consider the following subset in the algebra $LS(M)$ of locally measurable
operators affiliated with the von Neumann algebra $M$:
$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)=\\{x\in
LS(M):O_{\Phi,\alpha}^{\varphi,\tau}(x)<\infty\\},$
and for each $x\in\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)$ put
$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda\geq
0:O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq
1\right\\}.$
It is clear that
$m_{\Phi,\alpha}^{\varphi,\tau}=M\bigcap\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau).$
Repeating the proof of the Theorems 2 and 3 and of Corollary 1 we obtain that
$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)$ is a linear subspace of $LS(M)$
and that $\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ is a norm on
$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau).$
Theorem 6._Let $\varphi$ be a regular locally finite normal weight on $M$ and
suppose that $\Phi$ is an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition and $\alpha\in[0,1].$ Then
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau})$ is a Banach space and
$m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$_
_Proof._ Let $h$ be the Radon-Nikodym derivative of the weight $\varphi$ with
respect to the trace $\tau$, and suppose that
$h=\int\limits_{0}^{\infty}\lambda de_{\lambda}$ is the spectral resolution of
the operator $h$. From Theorem 1 it follows that the operators $h$ and
$h^{-1}$ are locally measurable. Therefore the operators
$\Phi^{-1}(h)=\int\limits_{0}^{\infty}\Phi^{-1}(\lambda)de_{\lambda}$ and
$(\Phi^{-1}(h))^{-1}=\int\limits_{0}^{\infty}((\Phi^{-1}(\lambda)))^{-1}de_{\lambda}$
are also locally measurable. There exists a linear isometry $U$ from
$m_{\Phi,\alpha}^{\varphi,\tau}$ into $L_{\Phi}(M,\tau)$ (see (8)), in
particular $U$ is injective. Thus there exists the converse map $U^{-1}$ for
the map $U.$ Since $U(m_{\Phi,\alpha}^{\varphi,\tau})$ is dense in
$L_{\Phi}(M,\tau)$ (see the proof of theorem 4), the converse map $U^{-1}$ can
be extended to a linear isometry from $L_{\Phi}(M,\tau)$ to the closure of
$m_{\Phi,\alpha}^{\varphi,\tau}$ in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$
Similar to the proof of Theorem 4 we obtain that
$m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$ Therefore $U^{-1}$ can be extended
to a linear isometry from $(L_{\Phi}(M,\tau),$ $\|\cdot\|_{\Phi})$ onto
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}),$ i.e.
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}),$ is a Banach space. The proof is
complete. $\Box$
Theorem 6 implies that in the case where $h$ and $h^{-1}$ are locally
measurable operators and the $N$-function $\Phi$ satisfies
$(\delta_{2},\Delta_{2})$-condition, the Orlicz space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ can be described by locally measurable
operators in the following form
$L_{\Phi,\alpha}(M,\varphi,\tau)=\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)=(\Phi^{-1}(h))^{-\alpha}L_{\Phi}(M,\tau)(\Phi^{-1}(h))^{\alpha-1}\subset
LS(M).$
Acknowledgements
_The final version of this work was done within the framework of the
Associateship Scheme of the Abdus Salam International Centre for Theoretical
Physics (ICTP), Triest. Italy. The first author thanks ICTP for providing
financial support and all facilities during his stay in ICTP (July-August,
2011). This work is supported in part by the DFG AL 214 136-1 project
(Germany). The first and the third authors would like to thank the Institute
of Applied Mathematics of the Bonn University for hospitality (April-May,
2011)._
## References
* [1] M.H.A. Al-Rashed , B. Zegarlinski, _Noncommutative Orlicz spaces associated to a state,_ Studia Math., 180 (2007), 199-209.
* [2] L.G. Brawn , H. Kosaki ,_Jensen’s inequality in semi-finite von Newmann Algebras,_ J. Operator Theory.,23 (1990), P. 3-19.
* [3] T. Fack , H. Kosaki , _Generalized $s$-number of $\tau$-measurable operators,_ Pacif. J. Math., 123 (1986), 269-300.
* [4] M.A. Krasnosel’sky and Ya. B. Rutitskii, _Convex function and Orlicz Spaces,_ Noordhoff (1961). (Translated from Russian).
* [5] W. Kunze, _Noncommutative Orlicz Spaces and Generalized Arens Algebras,_ Math. Nachr.,147 (1990), 123-138.
* [6] M.A. Muratov, _Non commutative Orlicz Spaces,_ Doklady AN RUz. 6. (1978), p. 11-13.
* [7] M.A. Muratov, _Luxemburg norm in Orlicz Spaces,_ Doklady AN RUz. 1. (1979), p. 5-6.
* [8] M.A.Muratov, V.I.Chilin, _Algebras of measurable operators and locally measurable operators_. Kyev. Institute of Math. Ukrainian Academy of Sciences. 2007.
* [9] G. Pedersen, M. Takesaki, _The Radon-Nikodym theorem for von Newmann algebras,_ Acta math., 130:1-2 (1973), 53-87.
* [10] M. Takesaki, _Theory of operator algebras I,_ Springer-Verlag-New-York. 1979.
* [11] N.V. Trunov, On non commutative analogue of $L_{p}$-space - Izv. Vuzov. Math. 11, 1979, p. 69-77.
* [12] N.V. Trunov, The $L_{p}$-spaces associated with a weight on a semi-finite von Neumann algebra. Constructive theory of functions and functional analysis. Kazan, 3, 1981, p. 88-93.
* [13] N.V. Trunov, To the theory normal weights on von Neumann algebras. Izv. Vuzov. Math, 8, 1982, p. 61-70.
* [14] N.V. Trunov, A.N. Sherstnev, Introduction to the theory of non commutative integration. N. Soviet Math., 37. Translation from Itogi Nauki i Tekhniki. Sovr. Probl. Math. 27 (1985), 127-190.
* [15] F.J. Yeadon, _Convergence of measurable operators,_ Proc. Camb. Phil. Soc., 74 (1973), 257-268.
* [16] F.J. Yeadon, _Non-commutative $L^{p}$-spaces,_ Math. Proc. Cambridge Phil. Soc., 77:1 (1975), 91-102.
|
arxiv-papers
| 2011-08-16T15:24:44 |
2024-09-04T02:49:21.559557
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sh. A. Ayupov, V.I. Chilin, R. Z. Abdullaev",
"submitter": "Karimbergen Kudaybergenov",
"url": "https://arxiv.org/abs/1108.3267"
}
|
1108.3352
|
# Emission spectra of _p-Si_ and _p-Si:H_ models generated by _ab initio_
molecular dynamics methods
E. R. L. Loustauac, Ariel A. Valladaresb
a Centro de Investigación en Energía,
Universidad Nacional Autónoma de México (UNAM)
A. P.34, CP.62580 Temixco, Morelos, México.
b Instituto de Investigaciones en Materiales (UNAM)
A. P.72, CP.04510 D.F., México.
c Centro de Ciencias de la Complejidad (UNAM)
A. P.70472, CP.04510, D.F., México.
###### Abstract
We created 4 _p-Si_ models and 4 _p-Si:H_ models all with 50% porosity. The
models contain 32, 108, 256 and 500 silicon atoms with a pore parallel to one
of the simulational cell axes and a regular cross-section. We obtained the
densities of states of our models by means of ab initio computational methods.
We wrote a code to simulate the emission spectra of our structures considering
particular excitations an decay conditions. After comparing the simulated
spectra with the experimental results, we observe that the position of the
maximum of the emission spectra might be related with the size of the silicon
backbone for the _p-Si_ models as the quantum confinement models say and with
the hydrogen concentration for the _p-Si:H_ structures. We conclude that the
quantum confinement model can be used to explain the emission of the _p-Si_
structures but, in the case of the _p-Si:H_ models it is necessary to consider
others theories.
## 1 Introduction
Since the discovery of the _p-Si_ photoluminiscence (PL) at room temperature
realized by L.T. Canham in 1990 [1], a lot of studies have been carried out to
understand the origin of this property. The possibility that _p-Si_ could be
applied to photonic and optoelectronic devices, motivated the study of the
dependency of its PL to the temperature, chemical attack variables and
molecules on its surface. Despite the big amount of scientific reports about
the _p-Si_ PL, until now, we have not a model that reproduce the
characteristics of all experimental PL spectra reported in the bibliography.
During the research of the origins of the _p-Si_ PL, some models about the
nature of the PL were proposed; Cullis _et al._[2] say that all these models
could be reorganized in 6 mayor groups. We only review the fundamentals of the
confinement quantum model (QC), and of the hydrogen-groups model (HS) because
we will use them to interpret our results.
The QC model proposed by Canham in 1990 was the first model created with the
aim to explain the efficient PL of the _p-Si_ at room temperature. In this
model, because of the effects of the quantum confinement, the energy gap of
the silicon nanostructures increase, and their photoluminescence appear in the
visible range of the electromagnetic spectrum. In accord to the QC model, if
the size of the silicon nanostructures (where the recombination process take
place) decrease, then the energy of the photons created increase. The QC model
is the most successful theory about the nature of the _p-Si_
photoluminescence, because it explain various experimental observations, with
an exception: The PL spectra of oxidized _p-Si_ structures.
In the other hand, the HS model consider that the hydrogen atoms on the _p-Si_
surface (which satisfy the dangling bonds), could be involved in the origin of
PL of this material. The HS model was proposed after observing that if the
hydrogen atoms are removed from the _p-Si_ surface, by a thermic procedure,
then the intensity of the PL decrease dramatically (but still exist because of
the QC model argument). Since the _p-Si_ with a rusty surface present an
intense PL, we understand that, the porous silicon with an hydrogen passivated
surface (_p-Si:H_) is just one of the luminescent forms of the _p-Si_ (Cullis
_et al._).
Photoluminescence is a difficult phenomenon to simulate, since the dynamics of
the excitation and decay of the electrons require precise calculations of the
excited states and decaying mechanisms. Also, metastable states may take part
in the process with short, but relevant, mean free lives that require time-
dependent quantum mechanical calculations [3], [7]. Instead, we have decided
to initially investigate the photoemission spectrum as a crude approximation
to the photoluminescence phenomenon.
Bearing in mind the QC and HS models, we propose a simple photoemission model
in which the excited electrons of the conduction band, decay successively into
the unoccupied electronic states of the valence band (see figure 4). The
emission spectra that we obtain for the _p-Si_ and _p-Si:H_ structures are
comparable to the experimental spectra reported by Cullis _et al_.
We are aware that our simulated _p-Si_ structures do not have experimental
counterpart, because the number of silicon atoms involved in each simulation
is finite due to the computational cost of the ab initio calculations. Also,
in our simulations is not possible to reproduce all the chemical molecules
that generate after the chemical attack of the _p-Si_ surface, because
depending of the atmosphere where the attack was realized, molecules or atoms
besides hydrogen atoms, could satisfy the dangling bonds of the _p-Si_ surface
modifying the form of its PL spectrum.
However, until now the doubts about the origins of the _p-Si_ PL remains, so,
our work contributes to calculate the photoemission spectra of _p-Si_ and _p-
Si:H_ structures with ab initio molecular dynamics methods. In the following
section, we describe the construction method used to generate the _p-Si_ and
_p-Si:H_ structures, and the way we obtain their photoemission spectra.
## 2 Method
The _p-Si_ structures were created using the _Cerius2_ interface [8]. We
reproduced the silicon cell diamond-like which contains 8 silicon atoms 2
times along each of the coordinate axes to obtain a 64 silicon-atom supercell,
3 times to increase the number of atoms to 216, 4 times for a supercell with
512 atoms and finally, 5 times along each direction to obtain 1000 atoms in
the silicon supercell.
Once the supercells were created, we eliminated 50% of their silicon atoms
carving a central pore parallel to the z-axis and with a regular cross
section. After eliminating 50% of the silicon atoms contained in each
supercell, we obtained 4 _p-Si_ structures with 32, 108, 256 and 500 silicon
atoms. The pore of the _p-Si_ structure with 32 silicon atoms has an irregular
transversal section because the great number of atoms removed from the
supercell. In the other hand, the pore of the supercell with 500 silicon atoms
shows a regular cross section because even if we remove 50% of their atoms,
its silicon backbone is bigger than the one for the 32 silicon atoms model.
The 4 structures of _p-Si_ were subjected to a geometry optimization process
with the objective of finding their equilibrium configurations (see figures
1(a) to 1(d)). We perform the geometry optimization process with _FAST
STRUCTURE_ a code included in the _CERIUS2_ interface based in the Lin and
Harris approach [5], [4], and the VWN [6] functional of correlation energy.
In table 1 we indicate the computational parameters used for the geometry
optimizations of the 4 _p-Si_ structures. The integration grid refers to the
space were the atomic functions, atomic densities, Coulomb potential, exchange
and correlation potentials are represented. To calculate the hamiltonian
matrix elements it is necessary to put each atom in the center of the
integration grid. A coarse integration grid leads to a poor representation of
the atomic functions, whereas a fine grid is the best approximation but its
computational cost is elevated. The minimal bases indicate that we use the
valence atomic orbital to construct the wave function, and the minimal density
indicates that the electronic density should be obtained just using the
valence atomic orbital as well. Finally, the cut-off parameter corresponds to
the distance at which the wave function vanish.
At the end of the geometry optimizations of our _p-Si_ structures, the
interface brings forth their radial distribution functions (figures 2(a) to
2(d) and 3) and densities of states. We used part of the values of the
densities of states of the _p-Si_ models as the input information for our
emission code (EDOxEDO.f). We consider that the electronic states of the
conduction density are unoccupied and those of the valence density are
occupied. Following the decay order proposed by our code (figure 4), the
excited electrons of the conduction states decay into the valence density
states emitting a photon, whose energy is equal to the energetic difference
between the two electronic states involved. The energies of the photons are
computed as an histogram and then, we adjust a gaussian curve to the histogram
to compare it with the _p-Si_ PL experimental spectrum reported by Cullis _et
al._ (Cullis _et al._). To calculate the emission of all our structures, we
only consider the electronic states that belong to a 2 eV interval in the
valence and conduction densities of states. We decide to work out with a 2 eV
interval states because experimentally, the majority of the PL excitation
sources have energies of this order of magnitude.
The _p-Si:H_ models were constructed taking the _p-Si_ optimized structures as
the initial atomic arrangement. We saturated with hydrogen the _p-Si_
optimized structures. After the saturation with hydrogen, we applied a
geometry optimization to the _p-Si:H_ structures using the parameters of table
1. In figures 5(a) to 5(d) we present the 4 optimized structures of _p-Si:H_.
We obtain the valence and conduction states from the densities of states of
the _p-Si:H_ models in the same way than for the _p-Si_ structures. Again, to
obtain the emission spectra of the _p-Si:H_ models we just consider the
electronic states, in the valence and conduction densities, that are in an 2
eV interval. The order of decay of the excited electronic states into the
unoccupied valence states is shown in figure 4. The distribution of the photon
energies is obtained after a gaussian fit.
In the results section we exhibit the photoemission spectra of the _p-Si_ and
_p-Si:H_ models. We compare our photoemission spectra with the PL spectrum
reported by Cullis _et al._.
(a) pSi32
(b) pSi108
(c) pSi256
(d) pSi500
Figure 1: . _p-Si_ optimized models with 32 (a), 108 (b), 256 (c) y 500 (d) silicon atoms. All the models have 50 % porosity. Parameter | Option
---|---
Integration grid | Coarse
Bases | Minimal
Density | Minimal
Cut off radius | 5 Å
Table 1: . Computational parameters used for the geometry optimization process
implemented by the _FAST STRUCTURE_ code for the _p-Si_ and _p-Si:H_ models.
(a) FDRpSi32
(b) FDRpSi108
(c) FDRpSi256
(d) FDRpSi500
Figure 2: Radial distribution functions of the _p-Si_ optimized models with 32
(a), 108 (b), 256 (c) and 500 (d) silicon atoms. Figure 3: . Interatomic
distance variation of the _p-Si_ optimized models, as a function of the size
of their silicon backbones. Figure 4: . Order (considered in the EDOxEDO.f
code.), of the excited electrons decay from the conduction band to the valence
band.
(a) pSi32H42
(b) pSi108H68
(c) pSi256H153
(d) pSi500H196
Figure 5: . _p-Si:H_ optimized structures with 32 silicon atoms and 42 of
hydrogen (a), 108 silicon atoms and 68 of hydrogen (b), 256 silicon atoms and
153 of hydrogen (c) and 500 silicon atoms and 196 of hydrogen (d).
## 3 Results and discussion
As we can see from figures 2(a) to 2(d), the radial distribution functions of
the _p-Si_ structures become similar to the radial distribution function of a
crystalline silicon structure, until the number of silicon atoms reach 500
(figure 2(d)). The variation of the interatomic distance of the _p-Si_
structures as a function of their silicon backbones size is shown in figure 3.
The backbone of the _p-Si_ structure with 500 silicon atoms is thicker than
the one of the model with just 32 silicon atoms, therefore, the thinner
silicon backbone is really deformed after a geometry optimization process
because of the presence of the porous in the simulation cell, then the most
probable distance between silicon neighbors increase to 2.55 Å. In the other
hand, the backbone of the _p-Si_ structure with 500 silicon atoms, is thick
enough to remain as a crystalline silicon structure despite the central
porous, so then the distance between first neighbors is 2.35 Å, as for a
crystalline silicon structure.
Figures 6(a), 6(b), 6(c) and 6(d) show the gaussian curves fitted to the
histogram of each _p-Si_ model obtained by our EDOxEDO.f code. In figure 7(a)
we present the emission spectra (just the gaussian curves) of the 4 _p-Si_
optimized structures; for these 4 structures their photoemission energy range
from 1.2 eV to 2.8 eV, its mean maximum photoemission energy is 1.94 eV, and
as we can observe, the intensity of the emission increase if the size of the
backbone increase, but its maximum energy decrease in accord to the results of
Cullis _et al._.
In figure 7(b) we compare the _p-Si_ PL spectrum reported by Cullis _et al._
and the emission spectrum of our _p-Si_ model with 500 silicon atoms; from
this figure we can appreciate that our spectrum is larger than the Cullis _et
al._ but both have a similar form. Also, the maximum photoemission energy of
our spectrum is 0.34 eV to the blue of the electromagnetic spectrum
considering the va
In figure 8 we superpose the emission spectra of the 4 models of _p-Si:H_
calculated by the EDOxEDO.f code. We did not find experimental photoemission
spectra for the _p-Si:H_ to compare to our results.
(a) pSi32
(b) pSi108
(c) pSi256
(d) pSi500
Figure 6: Gaussian fitting of the _p-Si_ model histograms (a) with 32, (b)
108, (c) 256, and (d) 500 silicon atoms.
(a) Emission spectra of the _p-Si_ models
(b) Simulation and experiment
Figure 7: (a) Emission spectra of the 4 _p-Si_ structures calculated with the
EDOxEDO.f code. (b) _p-Si_ PL spectrum reported by Cullis _et al._ (black
curve) and the emission spectrum of our _p-Si_ model with 500 silicon atoms
(red curve). Figure 8: . Emission spectra of the four _p-Si:H_ models
calculated with de EDOxEDO.f code.
## 4 Conclusions
We observed that the form of the emission spectra, obtained after a gaussian
fitting of the histograms calculated by the EDOxEDO.f code, is similar to the
form of the Cullis _et al._ spectrum. The maximum width of our spectra is 1.6
eV, their minimal energy is 1.2 eV and the maximal one is 2.8 eV. The mean
emission energy calculated was 1.94 eV, this is, 0.34 eV toward the blue of
the electromagnetic spectrum, with respect the Cullis _et al._ maximum energy.
Is clear from the figure 7(a) that decreasing the number of silicon atoms in
the simulation cell, moves toward bigger energies the maximal energy of the
emission spectra. We think that this fact is due to the size of the quantum
confinement region in the silicon backbone; for example, in the 32 _p-Si_
structure, the silicon backbone is smaller than the one of the 500 silicon
atoms model and, in accord with the quantum confinement model, his emission
energy should be bigger than the one of the _p-Si_ 500 atoms model.
From figure 7(a) we conclude that decreasing the number of silicon atoms in
the _p-Si_ supercells results in a smaller emission intensity. We think that
because the _p-Si_ 500 silicon atoms model has a crystalline structure (2(d))
compared with the 32 atoms structure (2(a)), the radiative recombination of
the charges dominates in the crystalline structures of the _p-Si_ and is not
significant in the amorphous ones.
We observe from figure 7(b) that the emission spectra of our _p-Si_ 500
silicon atoms model is toward the right of the electromagnetic spectrum, with
respect to the PL maximum reported by Cullis _et al._. We consider that the
size of the supercell is the direct cause and probably, if we could be able to
increase the number of atoms, to thousand of they to construct the _p-Si_
models, theirs emission maximus will be near the experimental ones reported.
## 5 Acknowledgements
Emilye Rosas Landa Loustau acknowledges the financial support of CONACyT
during his PhD studies; she also wants to thank Dr. Lorenzo Pavesi, and Dr.
Juan Carlos Alonso Huitrón because of their fruitful discussions.
## References
* [1] L. T Canham, Appl. Phys. Rev.57 (1990) 1046-1048.
* [2] A. G. Cullis, L. T. Canham and P.D. J. Calcott, Appl. Phys. Rev.82 (1997) 909-965.
* [3] T. V. Torchynska, J. Palacios Gomez, G. P. Polupan, F. G. Becerril Espinoza, A. Garcia Borquez, N. E. Korsunskaya, L. Yu. Khomenkova, Appl. Surf. Sci. 167 (2000) 197-204.
* [4] J. Harris, Phys. Rev. B 31 (1985) 1770.
* [5] Zijing Lin and J. Harris, J. Phys: Condens. Matter 4 (1992) 1055.
* [6] S.H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58 (1980) 1200.
* [7] P. D. J. Calcott, Mater. Sci. Eng.,B 51 (1998) 132-140.
* [8] FastStructure_SimAnn User Guide, Release 4.0.0, Molecular Simulations, Inc., San Diego, September 1996.
|
arxiv-papers
| 2011-08-16T20:56:50 |
2024-09-04T02:49:21.565844
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. R. L. Loustau, Ariel A. Valladares",
"submitter": "Emilye Rosas Landa Loustau Dra.",
"url": "https://arxiv.org/abs/1108.3352"
}
|
1108.3368
|
# Generalizing the Converse to Pascal’s Theorem via Hyperplane Arrangements
and the Cayley-Bacharach Theorem
Will Traves Department of Mathematics, U.S. Naval Academy, traves@usna.edu
###### Abstract
Using a new point of view inspired by hyperplane arrangements, we generalize
the converse to Pascal’s Theorem, sometimes called the Braikenridge-Maclaurin
Theorem. In particular, we show that if $2k$ lines meet a given line, colored
green, in $k$ triple points and if we color the remaining lines so that each
triple point lies on a red and blue line then the points of intersection of
the red and blue lines lying off the green line lie on a unique curve of
degree $k-1$. We also use these ideas to extend a second generalization of the
Braikenridge-Maclaurin Theorem, due to Möbius. Finally we use Terracini’s
Lemma and secant varieties to show that this process constructs a dense set of
curves in the space of plane curves of degree $d$, for degrees $d\leq 5$. The
process cannot produce a dense set of curves in higher degrees. The exposition
is embellished with several exercises designed to amuse the reader.
Dedicated to H.S.M. Coxeter, who demonstrated a heavenly syzygy:
the sun and moon aligned with the Earth, through a pinhole.
(Toronto, May 10, 1994, 12:24:14)
## 1 Introduction
In Astronomy the word syzygy refers to three celestial bodies that lie on a
common line. More generally, it sometimes is used to describe interesting
geometric patterns. For example, in a triangle the three median lines that
join vertices to the midpoints of opposite sides meet in a common point, the
centroid, as illustrated in Figure 1. Choosing coordinates, this fact can be
viewed as saying that three objects lie on a line: there is a linear
dependence among the equations defining the three median lines. In Commutative
Algebra and Algebraic Geometry, a syzygy refers to any relation among the
generators of a module.
Figure 1: The three medians of a triangle intersect at the centroid.
Pappus’s Theorem, which dates from the fourth century A.D., describes another
syzygy. It is one of the inspirations of modern projective geometry.
###### Theorem 1 (Pappus).
If 3 points $A,B,C$ lie on one line, and three points $a,b,c$ lie on another,
then the lines $Aa$, $Bb$, $Cc$ meet the lines $aB$, $bC$, $cA$ in three new
points and these new points are collinear, as illustrated in the left diagram
of Figure 2.
Pappus’s Theorem appears in his text, Synagogue [14, 15], a collection of
classical Greek geometry with insightful commentary. David Hilbert observed
that Pappus’s Theorem is equivalent to the claim that the multiplication of
lengths is commutative (see e.g. Coxeter [5, p. 152]). Thomas Heath believed
that Pappus’s intention was to revive the geometry of the Hellenic period [12,
p. 355], but it wasn’t until 1639 that the sixteen year-old Blaise Pascal
generalized Pappus’s theorem [6, Section 3.8], replacing the two lines with a
more general conic section.
###### Theorem 2 (Pascal).
If 6 points $A,B,C,a,b,c$ lie on a conic section, then the lines $Aa$, $Bb$,
$Cc$ meet the lines $aB$, $bC$, $cA$ in three new points and these new points
are collinear, as illustrated in the right diagram of Figure 2.
Figure 2: Illustrations of Pappus’s Theorem (left) and Pascal’s Theorem
(right).
Pascal’s theorem is sometimes formulated as the Mystic Hexagon Theorem: if a
hexagon is inscribed in a conic then the 3 points lying on lines extending
from pairs of opposite edges of the hexagon are collinear, as in Figure 3.
Figure 3: The Mystic Hexagon Theorem.
It is not clear why the theorem deserves the adjective mystic. Perhaps it
refers to the case where a regular hexagon is inscribed in a circle. In that
case, the three pairs of opposite edges are parallel and the theorem then
predicts that the parallel lines should meet (at infinity), and all three
points of intersection should be collinear. Thus, a full understanding of
Pascal’s theorem requires knowledge of the projective plane, a geometric
object described in some detail in Section 2.
Pascal’s Theorem has an interesting converse, sometimes called the
Braikenridge–Maclaurin theorem after the two British mathematicians William
Braikenridge and Colin Maclaurin.
###### Theorem 3 (Braikenridge–Maclaurin).
If three lines meet three other lines in nine points and if three of these
points lie on a line then the remaining six points lie on a conic.
Braikenridge and Maclaurin seem to have arrived at the result independently,
though they knew each other and their correspondence includes a dispute over
priority.
In 1848 the astronomer and mathematician August Ferdinand Möbius generalized
the Braikenridge–Maclaurin Theorem. Suppose a polygon with $4n+2$ sides is
inscribed in a nondegenerate conic and we determine $2n+1$ points by extending
opposite edges until they meet. If $2n$ of these $2n+1$ points of intersection
lie on a line then the last point also lies on the line. Möbius’s had
developed a system of coordinates for projective figures, but surprisingly his
proof relies on solid geometry. In Section 3 we prove an extension of Möbius’s
result, using the properties of projective plane curves – in particular, the
Cayley-Bacharach Theorem.
The Cayley–Bacharach Theorem is a wonderful result in projective geometry. In
its most basic form (sometimes called the 8 implies 9 Theorem) it says that if
two cubic curves meet in 9 points then any cubic through 8 of the nine points
must also go through the ninth point. For the history and many equivalent
versions of the Cayley-Bacharach Theorem, see Eisenbud, Green and Harris’s
elegant paper [8]. A strong version of the Cayley–Bacharach Theorem, described
in Section 3 , can be used to establish another generalization of the
Braikenridge–Maclaurin Theorem. The following existence theorem is well-known
(see Kirwan’s book on complex algebraic curves [17, Theorem 3.14]) but we also
claim a uniqueness result. The statement of Theorem 4 is inspired by the study
of hyperplane arrangements – in this case, by collections of colored lines in
the plane.
###### Theorem 4.
Suppose that $2k$ lines in the projective plane meet another line in $k$
triple points. Color the lines so that the line containing all the triple
points is green and each of the $k$ collinear triple points has a red and a
blue line passing through it. Then there is a unique curve of degree $k-1$
passing through the points where the red lines meet the blue lines off the
green line.
When the red and blue lines have generic slopes, they meet in $k^{2}-k$ points
off the green line. Since $\binom{k+1}{2}-1=\frac{k^{2}+k-2}{2}$ points in
general position determine a unique curve of degree $k-1$ passing through the
points, it is quite remarkable that the curve passes through all $k^{2}-k$
points of intersection off the green line. The Braikenridge–Maclaurin Theorem
is just the instance $k=3$ of Theorem 4. The case where $k=4$ is illustrated
in Figure 4.
Figure 4: An illustration of Theorem 4 when $k=4$.
We use the Cayley-Bacharach Theorem to prove Theorem 4 in Section 3. In
Section 4 we consider the kinds of curves produced by the construction in
Theorem 4. For instance, we use the group law on an elliptic curve to give a
constructive argument that, in a way that will be made precise, almost every
degree-3 curve arises in this manner. More generally, almost every degree-4
and degree-5 curve arises in this manner. A simple dimension argument is given
to show that most curves of degree 6 or higher do not arise in this manner.
The proofs for degree 4 and 5 involve secant varieties – special geometric
objects that have been quite popular recently because of their applications to
algorithmic complexity, algebraic statistics, mathematical biology and quantum
computing (see, for example, Landsberg [18, 19] ).
The last section contains some suggestions for further reading. As well,
Sections 2 and 5 contain amusing exercises that expand on the topic of the
paper.
A paper generalizing a classical result in geometry cannot reference all the
relevant literature. One recent paper by Katz [16] is closely related to this
work. His Mystic 2$d$-Gram [16, Theorem 3.3] gives a nice generalization of
Pascal’s Theorem; see Exercise 16.6. He also raises an interesting
constructibility question: which curves can be described as the unique curve
passing through the $d^{2}-2d$ points of intersection of $d$ red lines and $d$
blue lines that lie off a conic through $2d$ intersection points?
Acknowledgements: The author is grateful for conversations with my colleagues
Mark Kidwell, Amy Ksir, Mark Meyerson, Thomas Paul, and Max Wakefield, and
with my friend Keith Pardue. My college algebra professor Tony Geramita’s work
and conduct has been an inspiration to me. I have much to thank him for, but
here I’ll just note that he pointed me in the direction of some key ideas,
including Terracini’s lemma. Many computations and insights were made possible
using the excellent software packages Macaulay2, GeoGebra, Sage and Maple.
## 2 Projective Geometry
The general statement of Pascal’s Theorem suggests that parallel lines should
meet in a point and that as we vary the pairs of parallel lines the collection
of such intersection points should lie on a line. This is manifestly false in
the usual Cartesian plane, but the plane can be augmented by adding points at
infinity, after which Pascal’s Theorem holds. The resulting projective plane
$\mathbb{P}^{2}$ is a fascinating object with many nice properties.
One powerful model of the projective plane identifies points in
$\mathbb{P}^{2}$ with lines through the origin in 3-dimensional space. To see
how this relates to our usual plane, consider the plane $z=1$ in 3-dimensional
space as a model for $\mathbb{R}^{2}$ and note that most lines through the
origin meet this plane. The line passing through $(x,y,1)$ is identified with
the point $(x,y)\in\mathbb{R}^{2}$. But what about the lines that don’t meet
this plane? These are parallel to $z=1$ and pass through $(0,0,0)$ so they are
lines in the $xy$-plane. Each of these lines can be viewed as a different
point at infinity since they’ve been attached to our copy of $\mathbb{R}^{2}$.
In 1827 Möbius developed a useful system of coordinates for points in
projective space [23], later extended by Grassmann. If we consider the
punctured 3-space $\mathbb{R}^{3}\setminus\\{(0,0,0)\\}$ and the equivalence
relation
$(x,y,z)\sim(\lambda x,\lambda y,\lambda z)\Leftrightarrow\lambda\neq 0,$
then each equivalence class corresponds to a line in $\mathbb{R}^{3}$ through
the origin. We denote the equivalence class of points on the line through
$(x,y,z)$ by $[x:y:z]$. This is a sensible notation since the ratios between
the coordinates determine the direction of the line. Returning to our earlier
model of $\mathbb{P}^{2}$, the points with $z\neq 0$ correspond to points in
our usual copy of $\mathbb{R}^{2}$, while the points with $z=0$ correspond to
points at infinity.
If points in $\mathbb{P}^{2}$ correspond to lines through the origin, then
what do lines in $\mathbb{P}^{2}$ look like? Considering a line in
$\mathbb{R}^{2}$ as sitting in the plane $z=1$ we see that the points making
up this line correspond to lines through the origin that, together, form a
plane. Any line in $\mathbb{R}^{2}$ can be described by an equation of the
form $ax+by+c=0$; the reader should check that this determines the plane
$ax+by+cz=0$. Thus, lines in $\mathbb{P}^{2}$ correspond to dimension-2
subspaces of $\mathbb{R}^{3}$. In particular, the line in $\mathbb{P}^{2}$
whose equation is $z=0$ is the line at infinity.
We can also add points at infinity to $\mathbb{R}^{n}$ to create
$n$-dimensional projective space $\mathbb{P}^{n}$. Again, points in
$\mathbb{P}^{n}$ can be identified with 1-dimensional subspaces of
$\mathbb{R}^{n+1}$ and each point is denoted using homogeneous coordinates
$[x_{0}:x_{1}:\ldots:x_{n}]$. Similarly, we can construct the complex
projective spaces $\mathbb{P}^{n}_{\mathbb{C}}$, the points of which can be
identified with 1-dimensional complex subspaces of $\mathbb{C}^{n+1}$.
###### Exercise 5.
If this is the first time you’ve met projective space, you might try these,
increasingly complicated, exercises.
1. 1.
Show that if $ax+by+cz=0$ and $dx+ey+fz=0$ are two lines in $\mathbb{P}^{2}$
then they meet in a point $P=[g:h:i]$ given by the cross product,
$\langle g,h,i\rangle=\langle a,b,c\rangle\times\langle d,e,f\rangle.$
2. 2.
Show that the line $ax+by+cz=0$ in $\mathbb{P}^{2}$ consists of all the points
of the form $[x:y:1]$ such that $ax+by+c=0$, together with a single point at
infinity (the point $[b:-a:0]$). We say that the line $ax+by+cz=0$ is the
projectivization of the line $ax+by+c=0$.
3. 3.
Show that the projectivizations of two parallel lines $ax+by+c=0$ and
$ax+by+d=0$ in $\mathbb{R}^{2}$ meet at a point at infinity.
4. 4.
The projectivization of the hyperbola $xy=1$ in $\mathbb{R}^{2}$ is the set of
points in $\mathbb{P}^{2}$ that satisfy $xy-z^{2}=0$. Show that whether a
point $[x:y:z]$ lies on the projectivization of the hyperbola or not is a
well-defined property (i.e. the answer doesn’t depend on which representative
of the equivalence class $[x:y:z]$ we use). Where does the projectivization
meet the line at infinity?
5. 5.
(a) By picking representatives of each equivalence class carefully, show that
$\mathbb{P}^{2}$ can be put into 1-1 correspondence with the points on a
sphere $S^{2}\subset\mathbb{R}^{3}$, as long as we identify antipodal points,
$(x,y,z)\sim(-x,-y,-z)$.
(b) Considering only the top half of the sphere, show that the points in
$P^{2}$ can be identified with points in a disk where antipodal points on the
boundary circle are identified.
(c) Considering a thin band about the equator of the sphere from part (a),
show that $P^{2}$ can be constructed by sewing a Möbius band onto the boundary
of a disk. How many times does the band twist around as we go along the
boundary of the disk?
(d) Blowing up is a common process in algebraic geometry. When we blow up a
surface at a point we replace the point with a projectivization of its tangent
space (that is, the space of lines through the base point in the tangent
space). Show that if we blow up a point on the sphere
$S^{2}\subset\mathbb{R}^{3}$ we get $\mathbb{P}^{2}$. Show that if we blow up
a second point we get a Klein bottle, the surface obtained by sewing two
Möbius bands together along their edges.
6. 6.
(a) Viewing $\mathbb{P}^{2}$ as the set of lines in $\mathbb{R}^{3}$ through
the origin, check that the map
$\iota:\mathbb{R}^{3}\setminus\\{(0,0,0)\\}\rightarrow\mathbb{R}^{4}$ given by
$\iota(x,y,z)=\left(\frac{x^{2}}{x^{2}+y^{2}+z^{2}},\frac{xy}{x^{2}+y^{2}+z^{2}},\frac{z^{2}}{x^{2}+y^{2}+z^{2}},\frac{(x+y)z}{x^{2}+y^{2}+z^{2}}\right)$
induces a well-defined embedding of $\mathbb{P}^{2}$ into $\mathbb{R}^{4}$.
(b) Samuelson [28] gives an elegant argument to show that $\mathbb{P}^{2}$
cannot be embedded in $\mathbb{R}^{3}$. Maehara [20], exploiting work of
Conway and Gordon [4] and Sachs [27], gives another simple argument for this
fact based on the observation that any embedding of the complete graph $K_{6}$
in $\mathbb{R}^{3}$ contains a pair of linked triangles. Complete the argument
by drawing $K_{6}$ on $\mathbb{P}^{2}$ in such a way that no triangles are
homotopically linked.
(c) Amiya Mukherjee [24] showed that the complex projective plane
$\mathbb{P}_{\mathbb{C}}^{2}$ can be smoothly embedded in $\mathbb{R}^{7}$.
Research Question: Can $\mathbb{P}_{\mathbb{C}}^{2}$ be holomorphicly embedded
in $\mathbb{C}^{3}$? Can $\mathbb{P}_{\mathbb{C}}^{2}$ be smoothly embedded in
$\mathbb{R}^{n}$ with $n<7$?
### 2.1 The Power of Projective Space
Projective space $\mathbb{P}^{2}$ enjoys many nice properties that Euclidean
space $\mathbb{R}^{2}$ lacks. Many theorems are much easier to state in
projective space than in Euclidean space. For instance, in Euclidean space any
two lines meet in either one point or in no points (in the case where the two
lines are parallel). By adding points at infinity to Euclidean space, we’ve
ensured that any two distinct lines meet in a point. This is just the first of
a whole sequence of results encapsulated in Bézout’s Theorem.
Each curve $C$ in the projective plane can be described as the zero set of a
homogeneous polynomial $F(x,y,z)$:
$C=\\{[x:y:z]\;:\;F(x,y,z)=0\\}.$
The polynomial needs to be homogeneous (all terms in the polynomial have the
same degree) in order for the curve to be well-defined (see Exercise 5.4). It
is traditional to call degree-d homogeneous polynomials degree-d forms. The
curve $C$ is said to be a degree-$d$ curve when $F(x,y,z)$ is a degree-$d$
polynomial. We say that $C$ is an irreducible curve when $F(x,y,z)$ is an
irreducible polynomial. When $F(x,y,z)$ factors then the set $C$ is actually
the union of several component curves each determined by the vanishing of one
of the irreducible factors of $F(x,y,z)$. If $F(x,y,z)=G(x,y,z)H(x,y,z)$ then
removing the component $G(x,y,z)=0$ leaves the residual curve $H(x,y,z)=0$.
###### Theorem 6 (Bézout’s Theorem).
If $C_{1}$ and $C_{2}$ are curves of degrees $d_{1}$ and $d_{2}$ in the
complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$ sharing no common
components then they meet in $d_{1}d_{2}$ points, counted appropriately.
Bézout’s Theorem requires that we work in complex projective space: in
$\mathbb{P}^{2}_{\mathbb{R}}$ two curves may not meet at all. For instance,
the line $y-2z=0$ misses the circle $x^{2}+y^{2}-z^{2}=0$ in
$\mathbb{P}^{2}_{\mathbb{R}}$; the points of intersection have complex
coordinates.
To say what it means to count appropriately requires a discussion of
intersection multiplicity. This can be defined in terms of the length of
certain modules [10], but an intuitive description will be sufficient for our
purposes. When two curves meet transversally at a point $P$ (there is no
containment relation between their tangent spaces) then $P$ counts as 1 point
in Bézout’s Theorem. If the curves are tangent at $P$ or if one curve has
several branches passing through $P$ then $P$ counts as multiple points. One
way to determine how much the point $P$ should count is to look at well-chosen
families of curves $C_{1}(t)$ and $C_{2}(t)$ so that $C_{1}(0)=C_{1}$ and
$C_{2}(0)=C_{2}$ and to count how many points in $C_{1}(t)\cap C_{2}(t)$
approach $P$ as $t$ goes to $0$. For instance, the line $y=0$ meets the
parabola $yz=x^{2}$ in one point $P=[0:0:1]$. Letting $C_{1}(t)$ be the family
of curves $y-t^{2}z=0$ and letting $C_{2}(t)$ be the family consisting only of
the parabola, we find that $C_{1}(t)\cap
C_{2}(t)=\\{[t:t^{2}:1],[-t:t^{2}:1]\\}$ and so two points converge to $P$ as
$t$ goes to 0. In this case, $P$ counts with multiplicity two. The reader
interesting in testing their understanding could check that the two concentric
circles $x^{2}+y^{2}-z^{2}=0$ and $x^{2}+y^{2}-4z^{2}$ meet in two points,
each of multiplicity two. More details can be found in Fulton’s lecture notes
[9, Chapter 1].
It is traditional to call this result Bézout’s Theorem because it appeared in
a widely-circulated and highly-praised book111Both the MathSciNet and
Zentralblatt reviews of the English translation [2] are entertaining and well-
worth reading. The assessment in the MathSciNet review is atypically colorful:
“This is not a book to be taken to the office, but to be left at home, and to
be read on weekends, as a romance”, while the review in Zentralblatt Math
calls it “an immortal evergreen of astonishing actual relevance”. . Indeed, in
his position as Examiner of the Guards of the Navy in France, Étienne Bézout
was responsible for creating new textbooks for teaching mathematics to the
students at the Naval Academy. However, Issac Newton proved the result over 80
years before Bézout’s book appeared! Kirwan [17] gives a nice proof of
Bézout’s Theorem.
Higher projective spaces arise naturally when considering moduli spaces of
curves in the projective plane. For instance, consider a degree-$2$ curve $C$
given by the formula
$a_{0}x^{2}+a_{1}xy+a_{2}xz+a_{3}y^{2}+a_{4}yz+a_{5}z^{2}=0.$ (1)
Multiplying the formula by a nonzero constant gives the same curve, so the
curve $C$ can be identified with the point
$[a_{0}:a_{1}:a_{2}:a_{3}:a_{4}:a_{5}]$ in $\mathbb{P}^{5}$. More generally,
letting $S=\mathbb{C}[x,y,z]=\oplus_{d\geq 0}S_{d}$ be the polynomial ring in
three variables, the degree-$d$ curves in $\mathbb{P}^{2}$ are identified with
points in the projective space $\mathbb{P}(S_{d})$, where we identify
polynomials if they are nonzero scalar multiples of one another. A basis of
the vector space $S_{d}$ is given by the $D=\binom{d+2}{2}$ monomials of
degree $d$ in three variables, so the degree-$d$ curves in $\mathbb{P}^{2}$
are identified with points in the projective space
$\mathbb{P}(S_{d})\cong\mathbb{P}^{D-1}$. Returning to the case of degree-2
curves in $\mathbb{P}^{2}$, if we require $C$ to pass through a given point,
then the coefficients $a_{0},\ldots,a_{5}$ of $C$ must satisfy the linear
equation produced by substituting the coordinates of the point into (1). Now
if we require $C$ to pass through $5$ points in $\mathbb{P}^{2}$ the
coefficients must satisfy a homogeneous system of 5 linear equations. If the
points are in general position (so that the resulting system has full rank),
then the system has a one-dimensional solution space and so there is just one
curve passing through all 5 points. In general, we expect a unique curve of
degree $d$ to pass through $D-1$ points in general position and we expect no
curves of degree $d$ to pass through $D$ points in general position.
## 3 A Generalization of the Braikenridge–Maclaurin Theorem
The following Theorem is a version of the Cayley-Bacharach Theorem that was
first proven by Michael Chasles. He used it to prove Pascal’s Mystic Hexagon
Theorem, Theorem 2. Because of its content, the theorem is often called the
$8\Rightarrow 9$ Theorem.
###### Theorem 7 ($8\Rightarrow 9$ Theorem).
Let $C_{1}$ and $C_{2}$ be two plane cubic curves meeting in $9$ distinct
points. Then any other cubic passing through any $8$ of the nine points must
pass through the ninth point too.
Inspired by Husemöller’s book on Elliptic Curves [13], Terry Tao recently gave
a simple proof of the $8\Rightarrow 9$ Theorem in his blog222See Tao’s July
15, 2011 post at terrytao.wordpress.com..
###### Proof.
(After Tao) The proof exploits the special position of the points in
$C_{1}\cap C_{2}=\\{P_{1},\ldots,P_{9}\\}$. Let $F_{1}=0$ and $F_{2}=0$ be the
homogeneous equations of the curves $C_{1}$ and $C_{2}$. We will show that if
$F_{3}$ is a cubic polynomial and $F_{3}(P_{1})=\cdots=F_{3}(P_{8})=0$ then
$F_{3}(P_{9})=0$. To do this it is enough to show that there are constants
$a_{1}$ and $a_{2}$ so that $F_{3}=a_{1}F_{1}+a_{2}F_{2}$ because then
$F_{3}(P_{9})=a_{1}F_{1}(P_{9})+a_{2}F_{2}(P_{9})=0$. Aiming for a
contradiction, suppose that $F_{1}$, $F_{2}$ and $F_{3}$ are linearly
independent elements of $S_{3}$.
To start, no four of the points $P_{1},\ldots,P_{8}$ can be collinear
otherwise $C_{1}$ intersects the line in $4>(3)(1)$ points so the line must be
a component of $C_{1}$ by Bézout’s Theorem. Similarly, the line must be a
component of $C_{2}$. But $C_{1}$ and $C_{2}$ only intersect in 9 points so
this cannot be the case.
Now we show that there is a unique conic through any 5 of the points
$P_{1},\ldots,P_{8}$. If two conics $Q_{1}$ and $Q_{2}$ were to pass through 5
of the points then by Bézout’s Theorem they must share a component. So either
$Q_{1}=Q_{2}$ or both $Q_{1}$ and $Q_{2}$ are reducible and share a common
line. Since 4 of the points cannot lie on a line, the common component must
pass through no more than three of the five points. The remaining two points
determine the residual line precisely so $Q_{1}=Q_{2}$.
Now we argue that in fact no three of the points $P_{1},\ldots,P_{8}$ can be
collinear. Aiming for a contradiction suppose that three of the points lie on
a line $L$ given by $H=0$ and the remaining 5 points lie on a conic $C$. Since
no 4 of the points $P_{1},\ldots,P_{8}$ lie on a line, we know that the 5
points on $C$ do not lie on $L$. Pick constants $b_{1}$, $b_{2}$ and $b_{3}$
so that the polynomial $F=b_{1}F_{1}+b_{2}F_{2}+b_{3}F_{3}$ vanishes on a
fourth point on $L$ and at another point $P\not\in L\cup C$. Since the cubic
$F=0$ meets the line $L$ in 4 points, $L$ must be a component of the curve
$F=0$. But the residual curve given by $F/H=0$ is a conic going through 5 of
the 8 points so it must be $C$ itself. So $F=0$ is the curve $L\cup C$. But
$F(P)=0$ by construction and $P$ does not lie on $L\cup C$, producing the
contradiction.
Now note that no conic can go through more than $6$ of the points
$P_{1},\ldots,P_{8}$. Bézout’s Theorem shows that the conic cannot go through
$7$ of the points, else $C_{1}$ and $C_{2}$ would share a common component. So
suppose that a conic $C$ given by $G=0$ goes through 6 of the points. Then
there is a line $L$ going through the remaining 2 points. The polynomial $G$
does not vanish at either of these 2 remaining points because no $7$ of the
points lie on a conic. Pick constants $b_{1}$, $b_{2}$ and $b_{3}$ so that the
cubic $F=b_{1}F_{1}+b_{2}F_{2}+b_{3}F_{3}$ vanishes on a seventh point on the
conic $C$ and another point $P\not\in L\cup C$. Now $F/G$ is a linear form
that vanishes on the two remaining points so $F/G=0$ must determine the line
$L$. It follows that $F=0$ is the curve $L\cup C$. Again, $F(P)=0$ by
construction and $P$ does not lie on $L\cup C$, producing the contradiction.
Now let $L$ be the line through $P_{1}$ and $P_{2}$ and let $C$ be the conic
through $P_{3},\ldots,P_{7}$. From what we’ve proven above, $P_{8}$ does not
lie on $L\cup C$. Pick constants $c_{1}$, $c_{2}$ and $c_{3}$ so that the
cubic $F=c_{1}F_{1}+c_{2}F_{2}+c_{3}F_{3}$ vanishes on two more points $P$ and
$P^{\prime}$, both on $L$ but neither on $C$. Since $F=0$ meets $L$ in four
points, $F=0$ contains $L$ as a component. Since $L$ cannot go through any of
$P_{3},\ldots,P_{7}$, the residual curve mut be $C$, so $F=0$ is the curve
$L\cup C$. But then $F(P_{8})=0$ by construction and $P_{8}$ does not lie on
$L\cup C$, producing the final contradiction. ∎
Note that we proved slightly more: any cubic curve passing through $8$ of the
nine points must be a linear combination of the two cubics $C_{1}$ and
$C_{2}$.
Pascal’s Mystic Hexagon Theorem, Theorem 2, follows from an easy application
of the $8\Rightarrow 9$ Theorem. Let $C_{1}$ be the cubic consisting of the 3
lines formed by extending an edge of the hexagon and its two adjacent
neighbors. Let $C_{2}$ be the cubic consisting of the 3 lines formed by
extending the remaining, opposite, edges. $C_{1}$ and $C_{2}$ meet in 6 points
on the conic $Q$ and in three points off the conic. Let $L$ be the line
through two of the three points of intersection not on $Q$. Then $Q\cup L$ is
a cubic curve through $8$ of the $9$ points of $C_{1}\cap C_{2}$. By the
$8\Rightarrow 9$ Theorem, $Q\cup L$ must contain the ninth point too. The
point cannot lie on $Q$ so it must lie on $L$. That is, the three points of
intersection not on $Q$ are collinear.
To prove the Braikenridge–Maclaurin Theorem, Theorem 3, using the Cayley-
Bacharach Theorem, just observe that each collection of three lines is a cubic
curve (it is determined by the vanishing of a degree-3 polynomial) and if
three of the points lie on a line $L$ and five of the remaining six points lie
on a conic $C$ then $L\cup C$ is a cubic curve passing through 8 of the nine
points and so it must pass through all nine points. However, the ninth point
cannot lie on the line $L$ if the original cubics meet only in points,
otherwise $L$ would meet each of the original cubics in more than three
points. So the ninth point must be on the conic $C$.
A more powerful version of the Cayley-Bacharach Theorem can be found in the
last exercise of the Eisenbrick333An affectionate name for David Eisenbud’s
excellent (and mammoth) tome on Commutative Algebra. [7, p. 554] (also see
Eisenbud, Green and Harris [8, Theorem CB5]). Before stating this result, we
introduce some notation. Requiring a degree-$d$ curve in $\mathbb{P}^{2}$ to
go through a point $p\in\mathbb{P}^{2}$ imposes a non-trivial linear condition
on the coefficients of the defining equation of the curve. If a set $\Gamma$
of $\gamma$ points imposes only $\lambda$ independent linear conditions on the
coefficients of a curve of degree $d$, then we say that $\Gamma$ fails to
impose $\gamma-\lambda$ independent linear conditions on forms of degree $d$.
For example, 9 collinear points fail to impose 5 independent linear conditions
on forms of degree 3 – any cubic that passes through 4 of the points must pass
through them all. More generally, any set of $k$ collinear points fails to
impose $k-(d+1)$ conditions on forms of degree $d\leq k-1$.
###### Theorem 8 (Cayley-Bacharach).
Suppose that two curves of degrees $d_{1}$ and $d_{2}$ meet in a finite
collection of points $\Gamma\subset\mathbb{P}^{2}$. Partition $\Gamma$ into
disjoint subsets: $\Gamma=\Gamma^{\prime}\cup\Gamma^{\prime\prime}$ and set
$s=d_{1}+d_{2}-3$. If $d\leq s$ is a non-negative integer then the dimension
of the space of forms of degree $d$ vanishing on $\Gamma^{\prime}$, modulo
those vanishing on $\Gamma$, is equal to the failure of
$\Gamma^{\prime\prime}$ to impose independent conditions on forms of degree
$s-d$.
We restate our generalization of the Braikenridge-Maclaurin Theorem and give a
proof using the Cayley-Bacharach Theorem. Kirwan [17, Theorem 3.14] gives a
simple proof for the existence part of the Theorem that is well-worth
examining.
###### Theorem 4.
Suppose that $2k$ lines in the projective plane meet another line in $k$
triple points. Color the lines so that the line containing all the triple
points is green and each of the $k$ collinear triple points has a red and a
blue line passing through it. Then there is a unique curve of degree $k-1$
passing through the points where the red lines meet the blue lines (off the
green line).
###### Proof.
The cases $k=1$ and $k=2$ are trivial so assume $k\geq 3$. Suppose that the
red lines are cut out by the forms $L_{1},\ldots,L_{k}$, the blue lines are
cut out by the forms $M_{1},\ldots,M_{k}$ and the green line is cut out by the
form $G$. Let $\Gamma$ be the points of intersection of the two degree $k$
forms $L_{1}\cdots L_{k}$ and $M_{1}\cdots M_{k}$. Note that there are no
degree-$(k-1)$ curves that pass through all the points of $\Gamma$: any such
curve meets the line $M_{i}=0$ in $k$ points so each $M_{i}$ divides the
equation of the curve, leading to a contradiction on the degree of the
defining equation. Now let $\Gamma^{\prime}$ be the points of $\Gamma$ that
lie off the green line and $\Gamma^{\prime\prime}$ the points of $\Gamma$ on
the green line. The $k$ collinear points in $\Gamma^{\prime\prime}$ impose
$k-1$ independent conditions on forms of degree $k-2$. So
$\Gamma^{\prime\prime}$ fails to impose $k-(k-1)=1$ condition on forms of
degree $k-2$. Because there are no curves of degree $k-1$ going through all of
$\Gamma$, the Cayley-Bacharach Theorem says that the dimension of the space of
forms of degree $k-1$ vanishing on $\Gamma^{\prime}$ is equal to the failure
of $\Gamma^{\prime\prime}$ to impose independent conditions on forms of degree
$k+k-3-(k-1)=k-2$, which is one. So up to scaling, there is a unique equation
of degree $k-1$ passing through the points of $\Gamma$ off the green line. ∎
In the generic case, just one red and one blue line pass through each point of
intersection, and the curve $S$ passes through all $k^{2}-k$ points of
intersection between the red lines and the blue lines that do not lie on the
green line. If we are not in the generic case the uniqueness claim needs
further interpretation. For those that know about intersection multiplicity,
the curve $S$ is the unique curve whose intersection multiplicity with the
union of the red lines at the points of intersection off the green line equals
the intersection multiplicity of the union of the blue lines with the union of
the red lines at those same points (and we can replace red with blue in this
statement).
Möbius [22] also generalized the Braikenridge-Maclaurin Theorem, but in a
different direction. Suppose a polygon with $4n+2$ sides is inscribed in a
irreducible conic and we determine $2n+1$ points by extending opposite edges
until they meet. If $2n$ of these $2n+1$ points of intersection lie on a line
then the last point also lies on the line. Using the Cayley-Bacharach Theorem
allows us to extend Möbius’s result, relaxing the constraint on the number of
sides of the polygon.
###### Theorem 9.
Suppose that a polygon with $2k$ sides is inscribed in an irreducible conic.
Working around the perimeter of the polygon, color the edges alternately red
and blue. Extending the edges to lines, consider the $k^{2}-2k$ points of
intersection of the red and blue lines, omiting the original $2k$ vertices of
the polygon. If $k-1$ of these points lie on a green line, then in fact
another of these points lies on the green line as well.
The theorem is illustrated when $k=4$ in Figure 5: both the green and purple
lines contain 3 of the 8 points off the conic, so they must each contain a
fourth such point too.
Figure 5: An illustration of Theorem 9 when $k=4$.
###### Proof of Theorem 9..
First compute the dimension of the degree-$(k-2)$ curves that go through all
the points off the conic. Since there are no degree-$(k-2)$ curves through all
the $k^{2}$ points of intersection of the extended edges, the Cayley-Bacharach
Theorem gives that this dimension equals the failure of $2k$ points that lie
on a conic to impose independent conditions on curves of degree
$k+k-3-(k-2)=k-1$. Because the conic is irreducible, these degree-$(k-1)$
curves must contain the conic as a component. So the failure is $2k$ minus the
difference between the dimension of the space of degree $k-1$ curves in
$\mathbb{P}^{2}$ and the dimension of the space of degree $k-3$ curves in
$\mathbb{P}^{2}$. The failure is thus
$2k-\left[\binom{k+1}{2}-\binom{k-1}{2}\right]=1.$
So, up to scaling, there is a unique curve of degree $k-2$ through all the
points off the conic. Since this curve meets the green line in at least $k-1$
points, Bézout’s Theorem shows that it must contain the line as a component.
Taking the union of the residual curve (of degree $k-3$) with the conic gives
a curve of degree $k-1$ through all the points on both the red and blue lines
that do not lie on the green line. Now the Cayley-Bacharach Theorem says that
the dimension of all degree-$(k-1)$ curves through all the points off the
green line equals the failure of the points on the green line to impose
independent conditions on forms of degree $k-2$. There are at least $k-1$
points on the green line, so the failure equals
$\text{(\\# points on the
line)}-\left[\binom{k}{2}-\binom{k-1}{2}\right]=\text{(\\# points on the
line)}-(k-1).$
But there is such a curve so this number must be at least one, in which case
the number of points on the green line must be at least $k$. Of course, the
number of points on the green line is bounded by the number of points on the
intersection of the green line with the $k$ red lines so there are precisely
$k$ intersection points on the green line. This shows that the last point must
also lie on the green line and establishes Möbius’s result. ∎
## 4 Constructible curves
Let’s take a constructive view of Theorem 4, our extension of the
Braikenridge-Maclaurin Theorem. Say that a curve $X$ of degree $d$ is
constructible if there exist $d+1$ red lines $\ell_{1},\ldots,\ell_{d+1}$ and
$d+1$ blue lines $L_{1},\ldots,L_{d+1}$ so that the $d+1$ points
$\\{\ell_{i}\cap L_{i}:1\leq i\leq d+1\\}$ are collinear and the other
$d(d+1)$ points $\\{\ell_{i}\cap L_{j}:i\neq j\\}$ lie on $X$.
We turn to the question of which curves are constructible. In particular, we
aim to show that for a certain range of degrees $d$ almost all curves of
degree $d$ are constructible and for degrees outside of this range, almost no
curves of degree $d$ are constructible. One way to make such statements
precise is to introduce the Zariski topology on projective space.
The Zariski topology is the coarsest topology that makes polynomial maps from
$\mathbb{P}^{m}$ to $\mathbb{P}^{n}$ continuous. More concretely, every
homogeneous polynomial $F$ in $n+1$ variables determines a closed set in
$\mathbb{P}^{n}$
$\mathbb{V}(F)=\\{P\in\mathbb{P}^{n}\;:\;F(P)=0\\},$
and every closed set is built up by taking finite unions and arbitrary
intersections of such sets. Closed sets in the Zariski topology are called
varieties. The nonempty open sets in this topology are dense: their complement
is contained in a set of the form $\mathbb{V}(F)$.
We’ll say that the construction is dense for degree-$d$ curves if there is a
nonempty Zariski-open set of degree-$d$ curves $U$ such that each $X\in U$ is
constructible.
###### Question 10.
For which degrees is the construction dense?
The construction is clearly dense for degree-1 curves (lines). Pascal’s
Theorem, Theorem 2, shows that the construction is dense for degree-2 curves.
We give a simple argument to show that the construction cannot be dense if
$d\geq 6$. Consider the number of parameters that can be used to define an
arrangement of $2d+3$ lines so that there are $d+1$ triple points on one of
the lines. Two parameters are needed to define the green line and then we need
$d+1$ parameters to determine the triple points and $2(d+1)$ parameters to
choose the slopes of pairs of lines through these points. So a
$\left(3d+5\right)$-dimensional space parameterizes the line arrangements. The
space of degree-$d$ curves is parameterized by a
$\left(\binom{d+2}{2}-1\right)$ projective space. Since
$\binom{d+2}{2}-1=\frac{d^{2}+3d}{2}>3d+5,$
when $d\geq 6$, it is impossible for the line arrangements to parameterize a
nonempty Zariski-open set of dimension $\binom{d+2}{2}-1$ when $d\geq 6$.
Using the group law on elliptic curves allows us to show that the construction
is dense for degree-3 curves.
###### Theorem 11.
The construction is dense for degree-3 curves.
###### Proof.
The set of smooth plane curves of degree 3 is a nonempty Zariski-open set in
the space $\mathbb{P}^{9}$ parameterizing all degree-3 curves [29, Theorem 2
in Section II.6.2]. Such curves are called elliptic curves and their points
form a group: three distinct points add to the identity element in the
elliptic curve group if and only if they are collinear444If two of the points
are the same then the line must also be tangent to $X$ at this point, while if
all three points are the same then the tangent line to $X$ at the point must
intersect $X$ with multiplicity 3.. Given an elliptic curve $X$ we pick 5
points, $p_{1}$, …, $p_{5}$ on the curve, no three of which are collinear. We
will construct the red, blue and green lines; the reader may wish to refer to
the schematic diagram in Figure 6 as the construction proceeds.
Figure 6: A Schematic Illustration of the Construction in the Proof of Theorem
11.
We draw a red line connecting points $p_{1}$ and $p_{2}$, meeting $X$ in the
third point $-(p_{1}+p_{2})$. A blue line joining $p_{1}$ and $p_{4}$ meets
$X$ at $-(p_{1}+p_{4})$ and a blue line joining $p_{2}$ and $p_{3}$ meets $X$
at $-(p_{2}+p_{3})$. A red line joining $p_{4}$ and $p_{5}$ meets $X$ at
$-(p_{4}+p_{5})$. A red line joins $-(p_{1}+p_{4})$ and $-(p_{2}+p_{3})$ and
meets $X$ in the point $p_{1}+p_{2}+p_{3}+p_{4}$. A blue line joins
$p_{1}+p_{2}+p_{3}+p_{4}$ to $p_{5}$, meeting $X$ in
$-(p_{1}+p_{2}+p_{3}+p_{4}+p_{5})$. A red line joins
$-(p_{1}+p_{2}+p_{3}+p_{4}+p_{5})$ to $p_{3}$, meeting $X$ in the point
$p_{1}+p_{2}+p_{4}+p_{5}$. A blue line through $-(p_{1}+p_{2})$ and
$-(p_{4}+p_{5})$ also hits $X$ at $p_{1}+p_{2}+p_{4}+p_{5}$. The four red
lines meet the four blue lines in 16 points, 12 of which lie on the elliptic
curve $X$. We will prove that the other 4 points, circled in the schematic
Figure 6 are collinear (lying on the green line) using the Cayley-Bacharach
Theorem. Indeed, let $\Gamma$ be the 16 points where the red lines meet the
blue lines and let $\Gamma^{\prime\prime}$ be the 12 points lying on the cubic
$X$. Let $\Gamma^{\prime}=\Gamma\setminus\Gamma^{\prime\prime}$ be the
residual set of the four circled points. Since there are no degree-1 curves
vanishing on the 16 points of $\Gamma$, the Cayley-Bacharach Theorem says that
the dimension of the space of degree-1 curves vanishing on all four points of
$\Gamma^{\prime}$ equals the failure of $\Gamma^{\prime\prime}$ to impose
independent conditions on curves of degree $4+4-3-1=4$. The failure equals
$12$ minus the codimension of the degree-4 forms vanishing on
$\Gamma^{\prime\prime}$ in the space of all degree-4 forms. This is equal to
three less than the dimension of the vector space of degree-4 forms vanishing
on $\Gamma^{\prime\prime}$:
$\displaystyle 12-\left[\binom{6}{2}-\text{dim degree-4 forms vanishing on
}\Gamma^{\prime\prime}\right]$ $\displaystyle=$ $\displaystyle\text{dim
degree-4 forms vanishing on }\Gamma^{\prime\prime}-3.$
Now any linear form times the equation of the cubic $X$ gives a degree-4 form
vanishing on $\Gamma^{\prime\prime}$ so the degree-4 forms vanishing on
$\Gamma^{\prime\prime}$ is a vector space of dimension at least 3. However,
the defining ideal of the four red lines also vanishes on
$\Gamma^{\prime\prime}$, so in fact the dimension is at least 4. It follows
that the failure is at least 1 so the four points in $\Gamma^{\prime}$ are
collinear. So the construction produces all elliptic curves and is dense in
degree 3. ∎
To establish that the construction is dense in degrees 4 and 5, we use
Terracini’s beautiful lemma about secant varieties [3, Lemma 3.1] (stated
below in a restricted form, though it holds for higher secant varieties too).
If $X$ is a subvariety of $\mathbb{P}^{n}$ and $p_{1}\neq p_{2}$ are two
points on $X$ then the line joining $p_{1}$ to $p_{2}$ is a secant line to
$X$. The secant line variety $\text{Sec}(X)$ is the Zariski-closure of the
variety of points $q\in\mathbb{P}^{n}$ that lie on a secant line to $X$.
###### Lemma 12 (Terracini’s Lemma).
Let $p$ be a generic point on $\text{Sec}(X)\subset\mathbb{P}^{n}$, lying on
the secant line joining the two points $p_{1}\neq p_{2}$ of $X$. Then
$T_{p}(\text{Sec}(X))$, the (projectivized) tangent space to $\text{Sec}(X)$
at $p$, is $\langle T_{p_{1}}(X),\;T_{p_{2}}(X)\rangle$, the projectivization
of the linear span of the two vector spaces $T_{p_{1}}(X)$ and $T_{p_{2}}(X)$.
In particular,
$\text{dim}\;\text{Sec}(X)=\text{dim}\;\langle
T_{p_{1}}(X),\;T_{p_{2}}(X)\rangle.$
We apply this in the setting where $X=\mathbb{X}_{1^{5}}$, the variety of
completely reducible forms of degree $5$ on $\mathbb{P}^{2}$. Letting
$S=\mathbb{C}[x,y,z]=\oplus_{d\geq 0}S_{d}$, $\mathbb{X}_{1^{5}}$ is a
subvariety of the parameter space $\mathbb{P}(S_{5})$ of all degree-$5$
curves:
$\mathbb{X}_{1^{5}}=\\{[F_{1}\cdots F_{5}]:\;\text{each}\;F_{i}\in S_{1}\\}.$
Fortunately, Carlini, Chiantini and Geramita recently described the tangent
space to $\mathbb{X}_{1^{d}}$.
###### Lemma 13 ([3, Proposition 3.2]).
The tangent space to a point $p=[F_{1}\cdots F_{d}]$ in
$\mathbb{X}_{1^{d}}\subset\mathbb{P}(S_{d})$ is the projectivization of the
degree-$d$ part of the ideal
$I_{p}=\langle G_{1},\ldots,G_{d}\rangle,$
where $G_{i}=(F_{1}F_{2}\ldots F_{d})/F_{i}$.
If the forms $F_{1},\ldots,F_{5}$ are distinct then $(I_{p})_{5}$ has
dimension $11$. To see this, note that
$\text{dim}\;(I_{p})_{5}=(\text{dim}S_{1})(\text{dim}(I_{p})_{4})-\text{dim}(\text{Syz}(G_{1},\ldots,G_{5})_{1})=3(5)-\text{dim}(\text{Syz}(G_{1},\ldots,G_{5})_{1})$,
where
$\text{Syz}(G_{1},\ldots,G_{5})_{1}=\\{(L_{1},\ldots,L_{5})\in(S_{1})^{5}:\;L_{1}G_{1}+\cdots+L_{5}G_{5}=0\\}$
is the degree-$1$ part of the syzygy module. However, if
$L_{1}G_{1}+\cdots+L_{5}G_{5}=0$ then
$L_{1}G_{1}=-(L_{2}G_{2}+\cdots+L_{5}G_{5})$. Since $F_{1}$ divides each of
the terms on the right-hand side of this equality, $F_{1}$ must divide
$L_{1}G_{1}$. But $F_{1}$ does not divide $G_{1}$ so $F_{1}|L_{1}$. Similarly,
$F_{i}|L_{i}$ for $i=1,\ldots,5$. It follows that the only degree-$1$ syzygies
are generated by the $4$ linearly independent syzygies $F_{1}e_{1}-F_{i}e_{i}$
($i=2,\ldots,5$). As a result, $T_{p}(X_{1^{5}})$ is the projectivization of
an $11$-dimensional vector space.
Terracini’s Lemma shows that if $p$ is a generic point on the line between
$p_{1}$ and $p_{2}$, $T_{p}(\text{Sec}\mathbb{X}_{1^{5}})$ is the
projectivization of the linear span of the vector spaces $(I_{p_{1}})_{5}$ and
$(I_{p_{2}})_{5}$. This span has dimension $2(11)-\text{dim}\;(I_{p_{1}}\cap
I_{p_{2}})_{5}$. If $I$ is an ideal in $S$, let $\mathbb{V}(I)$ denote the set
of points $P\in\mathbb{P}^{2}$ such that $F(P)=0$ for all $F\in I$. Now if
$p_{1}=[F_{11}\cdots F_{15}]$ and $p_{2}=[F_{21}\cdots F_{25}]$ and if all the
lines $\mathbb{V}(F_{ij})$ are distinct, then $\mathbb{V}(I_{p_{1}}\cap
I_{p_{2}})=\mathbb{V}(I_{p_{1}})\cup\mathbb{V}(I_{p_{2}})$ is a collection of
20 points (counted with multiplicities): 10 of the points are given by the
intersections of the $\binom{5}{2}$ pairs of lines $F_{1i}(x,y,z)=0$ and 10 of
the points are given by the intersections of the $\binom{5}{2}$ pairs of lines
$F_{2i}(x,y,z)=0$. A polynomial in $(I_{p_{1}}\cap I_{p_{2}})_{5}$ is a curve
that goes through these 20 points. If the 20 points were in general position,
we would expect only 1 curve to go through all 20 points and so
$\text{dim}\;(I_{p_{1}}\cap I_{p_{2}})_{5}=1$. However, the 20 points are in
special position – for example, many collections of four of the points are
collinear – so we cannot trust our intuition blindly. Note that an element $H$
of $(I_{p_{1}}\cap I_{p_{2}})_{5}$ corresponds to a solution to a system of
equations $A{\bf v}={\bf 0}$ where $A$ is a 20 $\times$ 21 matrix whose
columns correspond to the monomials of $S_{5}$ and whose rows correspond to
the 20 points in $\mathbb{V}(I_{p_{1}})\cup\mathbb{V}(I_{p_{2}})$. The 21
entries of a solution ${\bf v}$ are the coefficients of $H(x,y,z)$. The
entries of $A$ in the column corresponding to a given monomial are obtained by
plugging in the coordinates of a point into the monomial. The entries of the
point at the intersection of $F_{ij}(x,y,z)=a_{ij}x+b_{ij}y+c_{ij}z=0$ and
$F_{ik}(x,y,z)=a_{ik}x+b_{ik}y+c_{ik}z=0$ are given by the cross product
$\langle a_{ij},b_{ij},c_{ij}\rangle\times\langle
a_{ik},b_{ik},c_{ik}\rangle$. So the entries in the matrix $A$ are degree-$10$
polynomials in the coefficients of the $F_{ij}$. This matrix will have full
rank unless all the $20\times 20$ minors are zero. This means that the matrix
has full rank off the closed set where all the maximal minors vanish. So the
matrix has full rank (and dim $(I_{p_{1}}\cap I_{p_{2}})_{5}=1$) on an open
set. To show that this open set is dense we just need to show that it is
nonempty by exhibiting an example.
The hilbertFunction command in Macaulay2 [11] can be used to compute the
dimension of $(I_{p_{1}}\cap I_{p_{2}})_{5}$. Checking a randomly selected
example shows that $\text{dim}\;(I_{p_{1}}\cap I_{p_{2}})_{5}=1$ generically
and so for a generic point $p$ of $\text{Sec}(X_{1^{5}})$, we see that the
tangent space at $p$ is the projectivization of a $21$-dimensional space; that
is, $\text{Sec}(X_{1^{5}})\subset\mathbb{P}^{20}$ is a $20$-dimensional
projective variety and so $\text{Sec}(X_{1^{5}})=\mathbb{P}^{20}$. We’re now
ready to tackle the constructibility question for curves of degrees 4 and 5.
###### Theorem 14.
The construction is dense for curves of degree 4.
###### Proof.
We’ll show that there is a dense open subset of constructible irreducible
curves of degree 4. The irreducible curves of degree 4 are themselves dense
and open in the set of all curves of degree 4; see Shafarevich’s book [29,
Section 5.2] for details. First we note that the set of degree-$5$ forms
$Z\in\mathbb{P}(S_{5})$ such that there exist $p_{1}\in X_{1^{5}}$ and
$p_{2}\in X_{1^{5}}$ so that $Z$ is a linear combination of $p_{1}$ and
$p_{2}$ and the 10 lines in $\mathbb{V}(p_{1})\cup\mathbb{V}(p_{2})$ are not
distinct is contained in a 19-dimensional subvariety $W$ of
$\mathbb{P}(S_{5})$. The dimension count is easy: there are two parameters for
each of the 9 (possibly) distinct lines determining $p_{1}$ and $p_{2}$ and 1
parameter to reflect where $Z$ lies on the line joining $p_{1}$ and $p_{2}$.
Now fix a linear form $L$ and consider the map
$\phi_{L}:\mathbb{P}(S_{4})\rightarrow\mathbb{P}(S_{5})$ given by
multiplication by $L$. The inverse image $\phi_{L}^{-1}(W^{c})$ of the
complement of $W$ is open in $\mathbb{P}(S_{4})$. Given an irreducible
degree-4 form $F$ in this open set, there exist $p_{1}$ and $p_{2}$ in
$X_{1^{5}}$ so that $FL$ is a linear combination of $p_{1}$ and $p_{2}$ and
the 10 lines in $p_{1}$ and $p_{2}$ are distinct. Then $\mathbb{V}(FL)$
contains the 25 points of intersection between the lines in $p_{1}$ and the
lines in $p_{2}$. We claim that 5 of the 25 points in
$\mathbb{V}(p_{1})\cap\mathbb{V}(p_{2})$ lie on $\mathbb{V}(L)$ and the
remaining 20 points lie on $\mathbb{V}(F)$. If more than 5 points lie on
$\mathbb{V}(L)$ then Bézout’s Theorem shows that $L$ must divide $p_{1}$.
Similarly, $L$ must divide $p_{2}$. This is impossible because the 10 lines in
$\mathbb{V}(p_{1})$ and $\mathbb{V}(p_{2})$ are distinct. Similarly, if
$\mathbb{V}(F)$ goes through more than 20 points of
$\mathbb{V}(p_{1})\cap\mathbb{V}(p_{2})$ then $F$ and $p_{1}$ must have a
nontrivial common divisor. But $F$ is irreducible so this cannot occur. It
follows that $\mathbb{V}(F)$ is constructible (the red lines are the lines in
$\mathbb{V}(p_{1})$, the blue lines are the lines in $\mathbb{V}(p_{2})$ and
the green line is the line $\mathbb{V}(L)$).
It follows that an open set of irreducible degree-$4$ curves is constructible.
We give an example to show that this open set is nonempty. Take the green line
to be $y=0$, the red lines to be $x+2z=0$, $x+z=0$, $x=0$, $x-z=0$, and
$x-2z=0$, and the blue lines to be $x-y+2z=0$, $x-y+z=0$, $x-y=0$, $x-y-z=0$,
and $x-y-2z=0$. The red lines intersect the blue lines in 20 distinct points
off the green line and the polynomial
$5x^{4}-10x^{3}y+10x^{2}y^{2}-5xy^{3}+y^{4}-15x^{2}+15xy-5y^{2}+4$
vanishes on each of the 20 points. You can, for example, dehomogenize the
polynomial (set $z=1$) and use Maple’s evala(AFactor($\cdot$)) command to
check that the polynomial is irreducible. ∎
###### Theorem 15.
The construction is dense for curves of degree 5.
###### Proof.
First we note that $X_{1,V}$, the subvariety of $\mathbb{P}(S_{6})$ consisting
of degree 6 forms that factor into a linear form times an irreducible
degree-$5$ form, is in fact a subvariety of ${\text{S}ec}(X_{1^{6}})$. If $L$
is a linear form and $Q$ is an irreducible degree-$5$ form then
$Q\in\mathbb{P}(S_{5})={\text{S}ec}(X_{1^{5}})$ so there are completely
reducible forms $p_{1}$ and $p_{2}$ of degree $5$ so that $Q$ is a linear
combination of $p_{1}$ and $p_{2}$. It follows that the form $LQ\in X_{1,V}$
is a linear combination of $Lp_{1}$ and $Lp_{2}$ so
$X_{1,V}\subseteq{\text{S}ec}(X_{1^{6}})$. Moreover, $X_{1,V}$ is a closed set
in $\mathbb{P}(S_{6})$ since it is the image of the regular map
$\mathbb{P}(S_{1})\times V\rightarrow\mathbb{P}(S_{6})$, where the map is
given by multiplication and $V$ is projectivization of the irreducible forms
of degree 5. This shows that $X_{1,V}$ is a subvariety of
${\text{S}ec}(X_{1^{6}})$.
Now pick $Z\in V$ constructible so that $L$ is the defining equation of the
green line (for some set of distinct red and blue lines). For example, we can
take the green line to be $y=0$, the red lines to be $x+3z=0$, $x+2z=0$,
$x+z=0$, $x=0$, $x-z=0$ and $x-2z=0$, and the blue lines to be $x-y+3z=0$,
$x-y+2z=0$, $x-y+z=0$, $5x-y=0$, $5x-y-5z=0$ and $5x-y-10z=0$. The red lines
intersect the blue lines in 30 distinct points off the green line and the
irreducible polynomial
$\displaystyle 450x^{5}$
$\displaystyle-615x^{4}y+396x^{3}y^{2}-123x^{2}y^{3}+18xy^{4}-y^{5}+675x^{4}z-150x^{3}yz$
$\displaystyle-234x^{2}y^{2}z+93xy^{3}z-9y^{4}z-2400x^{3}z^{2}+2250x^{2}yz^{2}-504xy^{2}z^{2}$
$\displaystyle+29y^{3}z^{2}-2025x^{2}z^{3}-375xyz^{3}+141y^{2}z^{3}+2400xz^{4}-460yz^{4}+300z^{5}$
vanishes on each of the 30 points. Fixing $L$, consider the map
$\phi_{L}:V\rightarrow{\text{S}ec}(X_{1^{6}})$ given by sending an irreducible
degree-5 form $F$ to $FL\in X_{1,V}\subset{\text{S}ec}(X_{1^{6}})$. The set of
points in ${\text{S}ec}(X_{1^{6}})$ that lie on the line connecting completely
reducible forms $p_{1}$ and $p_{2}$ where the 12 lines forming
$\mathbb{V}(p_{1})\cup\mathbb{V}(p_{2})$ are not distinct is a closed set $W$
of dimension no larger than 23. We leave it to the reader to check that
${\text{S}ec}(X_{1^{6}})$ has dimension 25; the proof is similar to the
argument given above that ${\text{S}ec}(X_{1^{5}})$ has dimension 20\. It
follows that the inverse image $\phi_{L}^{-1}(W^{c})$ of the complement of $W$
is open in $V$. Since $\phi_{L}(Z)\not\in W$, the open set is nonempty. Now if
$F$ is a degree-5 irreducible form with $FL\not\in W$, there exist $p_{1}$ and
$p_{2}$ in $X_{1^{6}}$ so that $FL$ is a linear combination of $p_{1}$ and
$p_{2}$ and the 12 lines in $p_{1}$ and $p_{2}$ are distinct. Then
$\mathbb{V}(FL)$ contains the 36 points of intersection between the lines in
$p_{1}$ and the lines in $p_{2}$. Now, as in the proof of Theorem 14, Bézout’s
Theorem shows that 6 of the 36 points in
$\mathbb{V}(p_{1})\cap\mathbb{V}(p_{2})$ lie on $\mathbb{V}(L)$ and the
remaining 30 points lie on $\mathbb{V}(F)$. This allows us to use the lines in
$V(p_{1})$ as our red lines, the lines in $\mathbb{V}(p_{2})$ as the blue
lines and the line $\mathbb{V}(L)$ as our green line to construct the curve
$\mathbb{V}(F)$.
We’ve shown that a nonempty open subset of the irreducible degree-5 curves
consists of constructible curves. The result follows because the collection of
irreducible degree-5 curves form an open set in the parameter space
$\mathbb{P}(S_{5})$. ∎
We have not provided an example of a curve of degree less than 6 that is not
constructible. It may be that the set of constructible curves is Zariski-
closed. In this case, every curve of degree less than 6 would be constructible
because projective spaces are connected in the Zariski-topology: the only sets
in projective space that are both open and closed are the empty set and the
whole space.
## 5 Further Reading and Exercises
Pappus’s Theorem inspired a lot of amazing mathematics. The first chapter of a
fascinating new book by Richter-Gebert [26] describes the connections between
Pappus’s Theorem and many areas of mathematics, including cross-ratios and the
Grassmann-Plücker relations among determinants.
The history and implications of the Cayley-Bacharach Theorem is carefully
considered in Eisenbud, Green and Harris’s amazing survey paper [8]. They
connect the result to a host of interesting mathematics, including the
Riemann-Roch Theorem, residues and homological algebra. Their exposition
culminates in the assertion that the theorem is equivalent to the statement
that polynomial rings are Gorenstein.
My approach to the Braikenridge-Maclaurin Theorem was inspired by thinking
about hyperplane arrangements. A good introduction to these objects from an
algebraic and topological viewpoint is the book by Orlik and Terao [25]. For a
more combinatorial viewpoint, see Stanley’s lecture notes [30].
One way to view what we’ve done is to note that if $\Gamma$ is a complete
intersection – a codimension $d$ variety (or, more generally, scheme) defined
by the vanishing of $d$ polynomials – and $\Gamma$ is made up of two
subvarieties, then special properties of one subvariety are reflected in
special properties of the other subvariety. This point of view leads to the
beautiful subject of liaison theory. The last chapter of Eisenbud [7]
introduces this advanced topic in Commutative Algebra; more details can be
found in Migliore and Nagel’s notes [21].
###### Exercise 16.
The following exercises are roughly in order of increasing difficulty.
1. 1.
Pascal’s Theorem says that if a regular hexagon is inscribed in a circle then
the 3 pairs of opposite edges lie on lines that intersect in 3 collinear
points. Which line do the three points lie on? Is it surprising that it
doesn’t matter where in the plane the circle is centered?
2. 2.
When working with lines in $\mathbb{P}^{2}$ it is desirable to have a quick
way to compute their intersection points. Show that the lines
$a_{1}x+b_{1}y+c_{1}z=0$ and $a_{2}x+b_{2}y+c_{2}z=0$ meet in the point
$[a_{3}:b_{3}:c_{3}]$ where
$\langle a_{3},b_{3},c_{3}\rangle=\langle
a_{1},b_{1},c_{1}\rangle\times\langle a_{2},b_{2},c_{2}\rangle.$
Interpret the result in terms of the geometry of 3-dimensional space. Also
describe how to use this result to compute the intersection of two lines in
$\mathbb{R}^{2}$.
3. 3.
There is an interesting duality between points and lines in $\mathbb{P}^{2}$.
Fixing a nondegenerate inner product on 3-dimensional space, we define the
dual line $\check{P}$ to a point $P\in\mathbb{P}^{2}$ to be the
projectivization of the 2-dimensional subspace orthogonal to the 1-dimensional
subspace corresponding to $P$. Similarly, if $L$ is a line in
$\mathbb{P}^{2}$, it corresponds to a 2-dimensional subspace in 3-dimensional
space and we define the dual point $\check{L}$ to be the projectivization of
the 1-dimensional subspace orthogonal to this subspace.
(a) Show that a line $L$ in $\mathbb{P}^{2}$ goes through two points
$P_{1}\neq P_{2}$ if and only if the dual point $\check{L}$ lies on the
intersection of the two dual lines $\check{P_{1}}$ and $\check{P_{2}}$.
(b) Use part (a) and Exercise 16.2 to develop a cross product formula for the
line through 2 points in $\mathbb{P}^{2}$. Extend the formula to compute the
equation for a line through 2 points in $\mathbb{R}^{2}$.
(c) It turns out that the duals of all the tangent lines to an irreducible
conic $C$ form a collection of points lying on a dual irreducible conic
$\check{C}$, and vice-versa (see Bachelor, Ksir and Traves [1] for details).
Show that dualizing Pascal’s Theorem gives Briançon’s Theorem: If an
irreducible conic is inscribed in a hexagon, then the three lines joining
pairs of opposite vertices intersect at a single point555 Like Bézout, Charles
Julien Briançon (1783-1864) was a professor at a French military academy. The
French military of the $19^{\text{th}}$ century seems to have played an
interesting role in supporting the development and teaching of mathematics..
4. 4.
Provide a proof for one of the assertions in the paper: any set of $k$
collinear points fails to impose $k-(d+1)$ conditions on forms of degree
$d\leq k-1$.
5. 5.
Establish the following result due to Möbius [22] using the Cayley-Bacharach
Theorem. Consider two polygons $P_{1}$ and $P_{2}$, each with $m$ edges,
inscribed in a conic, and associate one edge from $P_{1}$ with one edge from
$P_{2}$. Working counterclockwise in each polygon, associate the other edges
of $P_{1}$ with the edges of $P_{2}$. Extending these edges to lines, Möbius
proved that if $m-1$ of the intersections of pairs of corresponding edges lie
on a line then the last pair of corresponding edges also meets in a point on
this line.
6. 6.
Establish the following result due to Katz [16, Theorem 3.3], his Mystic
$2d$-Gram Theorem. If $d$ red lines and $d$ blue lines intersect in $d^{2}$
points and if $2d$ of these points lie on an irreducible conic then there is a
unique curve of degree $k-2$ through the other $d^{2}-2d$ intersection points.
Katz’s interesting paper [16] contains several open problems.
7. 7.
Use the Cayley-Bacharach Theorem to show that if two degree-5 curves meet in
25 points, 10 of which lie on an irreducible degree-3 curve, then there is a
unique degree-4 curve through the other 15 points. Also convince yourself that
the hypotheses of this exercise can actually occur.
8. 8.
If a degree-8 curve meets a degree-9 curve in 72 points and if 17 of these
points lie on an irreducible degree-3 curve, then what is the dimension of the
family of degree-9 curves through the remaining 55 points? Convince yourself
that the hypotheses of this exercise can actually occur.
9. 9.
Use the $8\Rightarrow 9$ Theorem to show that the group law on an elliptic
curve is associative.
10. 10.
In general you might expect that if $X\subset\mathbb{P}^{n}$ then $\text{dim
}{\text{S}ec}(X)=\text{ min}(2\text{dim}(X),n)$. Varieties $X$ where this
inequality fails to hold are called defective. Check that
${\text{S}ec}(X_{1^{6}})$ is not defective: it has dimension 25.
## References
* [1] Andrew Bashelor, Amy Ksir, and Will Traves. Enumerative algebraic geometry of conics. Amer. Math. Monthly, 115(8):701–728, 2008.
* [2] Etienne Bézout. General theory of algebraic equations. Princeton University Press, Princeton, NJ, 2006. Translated from the 1779 French original by Eric Feron.
* [3] Enrico Carlini, Luca Chiantini, and Anthony V. Geramita. Complete intersections on general hypersurfaces. Michigan Math. J., 57:121–136, 2008. Special volume in honor of Melvin Hochster.
* [4] J.H. Conway and C. McA. Gordon. Knots and links in spatial graphs. J. Graph Theory, 7:445–453, 1983.
* [5] H. S. M. Coxeter. Projective geometry. Blaisdell Publishing Co. Ginn and Co. New York-London-Toronto, 1964\.
* [6] H. S. M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America, Washington, D.C., 1967. Volume 19 of the New Mathematical Library.
* [7] David Eisenbud. Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
* [8] David Eisenbud, Mark Green, and Joe Harris. Cayley-Bacharach theorems and conjectures. Bull. Amer. Math. Soc. (N.S.), 33(3):295–324, 1996.
* [9] William Fulton. Introduction to intersection theory in algebraic geometry, volume 54 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1984.
* [10] William Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, 1998.
* [11] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
* [12] Thomas Heath. A history of Greek mathematics. Vol. II. Dover Publications Inc., New York, 1981. Corrected reprint of the 1921 original.
* [13] Dale Husemöller. Elliptic curves, volume 111 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen.
* [14] Alexander Jones. Book 7 of the Collection, part 1. Springer-Verlag, Berlin, 1986. Introduction, text, translation.
* [15] Alexander Jones. Book 7 of the Collection, part 2. Springer-Verlag, Berlin, 1986. Commentary, index, figures.
* [16] Gabriel Katz. Curves in cages: an algebro-geometric zoo. Amer. Math. Monthly, 113(9):777–791, 2006.
* [17] Frances Kirwan. Complex algebraic curves, volume 23 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1992.
* [18] J. M. Landsberg. Geometry and the complexity of matrix multiplication. Bull. Amer. Math. Soc. (N.S.), 45(2):247–284, 2008.
* [19] J. M. Landsberg. $P$ versus $NP$ and geometry. J. Symbolic Comput., 45(12):1369–1377, 2010.
* [20] Hiroshi Maehara. Why is ${\bf P}^{2}$ not embeddable in ${\bf R}^{3}$? Amer. Math. Monthly, 100(9):862–864, 1993.
* [21] J. C. Migliore and U. Nagel. Liaison and related topics: notes from the Torino workshop-school; arXiv:math/0205161v1. Rend. Sem. Mat. Univ. Politec. Torino, 59(2):59–126 (2003), 2001\.
* [22] A. F. Möbius. Verallgemeinerung des pascalschen theorems, das in einen kegelschnitt beschriebene sechseck betreffend. J. Reine Angew. Math. (Crelle’s Journal), 36:216–220, 1848.
* [23] August Ferdinand Möbius. Der barycentrische Calcul. Georg Olms Verlag, Hildesheim, 1976. Ein neues Hülfsmittel zur analytischen Behandlung der Geometrie, Nachdruck der 1827 Ausgabe.
* [24] Amiya Mukherjee. Embedding complex projective spaces in Euclidean space. Bull. London Math. Soc., 13:323–324, 1981.
* [25] Peter Orlik and Hiroaki Terao. Arrangements of hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992.
* [26] Jürgen Richter-Gebert. Perspectives on projective geometry. A guided tour through real and complex geometry. Springer, Berlin, 2011.
* [27] Horst Sachs. On a spatial analogue of Kuratowski’s theorem on planar graphs—an open problem. In Graph theory (Łagów, 1981), volume 1018 of Lecture Notes in Math., pages 230–241. Springer, Berlin, 1983.
* [28] Hans Samuelson. Orientability of hypersurfaces in $\mathbb{R}^{n}$. Proc. Amer. Math. Soc., 22:301–302, 1969.
* [29] Igor R. Shafarevich. Basic algebraic geometry. 1. Springer-Verlag, Berlin, second edition, 1994. Translated from the 1988 Russian edition and with notes by Miles Reid.
* [30] Richard P. Stanley. An introduction to hyperplane arrangements. In Geometric combinatorics, volume 13 of IAS/Park City Math. Ser., pages 389–496. Amer. Math. Soc., Providence, RI, 2007.
|
arxiv-papers
| 2011-08-16T22:24:30 |
2024-09-04T02:49:21.571407
|
{
"license": "Public Domain",
"authors": "Will Traves",
"submitter": "William Traves",
"url": "https://arxiv.org/abs/1108.3368"
}
|
1108.3393
|
# Spin-$1/2$ $J_{1}-J_{2}$ Heisenberg antiferromagnet on a square lattice:
a plaquette renormalized tensor network study
Ji-Feng Yu Center for Quantum Science and Engineering, National Taiwan
University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan Ying-Jer Kao
Department of Physics, and Center for Advanced Study in Theoretical Science,
National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan
###### Abstract
We apply the plaquette renormalization scheme of tensor network states [Phys.
Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg
${J}_{1}$-${J}_{2}$ model on an $L\times L$ square lattice with $L$=8,16 and
32. By treating tensor elements as variational parameters, we obtain the
ground states for different $J_{2}/J_{1}$ values, and investigate staggered
magnetizations, nearest-neighbor spin-spin correlations and plaquette order
parameters. In addition to the well-known Néel order and collinear order at
low and high ${J}_{2}/{J}_{1}$, we observe a plaquette-like order at
${J}_{2}/J_{1}\approx 0.5$. A continuous transition between the Néel order and
the plaquette-like order near $J_{2}^{c_{1}}\approx 0.40J_{1}$ is observed.
The collinear order emerges at ${J}_{2}^{c_{2}}\approx 0.62J_{1}$ through a
first-order phase transition.
###### pacs:
75.10.Jm, 75.40.Mg, 03.67.-a
## I Introduction
The search for exotic states in quantum magnets has been the topic of
intensive research for the past decades. An extremely important question is
when the conventional Néel order is destroyed, what kind of states can emerge.
Frustrated antiferromagnetic spin systems, where the frustration from either
the lattice geometry, or the presence of competing interactions, are candidate
systems to study these states. It is proposed that when the Néel order is
destroyed by quantum fluctuations, only short-range correlations will survive,
and the system enters a quantum paramagnetic state which can be described as a
resonant valence bond (RVB) state.Anderson (1973) The RVB state can either be
a valence bond solid (VBS) phase, where some of the lattice symmetries are
broken,Read and Sachdev (1991) or a featureless spin liquid with strong short-
range correlations without any broken spin symmetry.Figueirido _et al._
(1990); Capriotti _et al._ (2001) One archetypical model to study the effect
of frustration from competing interactions is the antiferromagnetic (AF)
${J}_{1}$-${J}_{2}$ Heisenberg model on a square lattice.Chandra and Doucot
(1988); Dagotto and Moreo (1989); Figueirido _et al._ (1990); Schulz and
Ziman (1992); Zhitomirsky and Ueda (1996); Trumper _et al._ (1997); Bishop
_et al._ (1998); Siurakshina _et al._ (2001); Singh _et al._ (2003);
Mambrini _et al._ (2006); Richter and Schulenburg (2010); Reuther and Wölfle
(2010) The Hamiltonian is given by,
$H={J}_{1}\sum_{\langle
ij\rangle}{\mathbf{S}_{i}\cdot\mathbf{S}_{j}}+{J}_{2}\sum_{\langle\langle
ij\rangle\rangle}{\mathbf{S}_{i}\cdot\mathbf{S}_{j}},$ (1)
where ${J}_{1}>0$ and ${J}_{2}>0$ are the nearest-neighbor (NN) and next-
nearest-neighbor (NNN) couplings, and the sums $\langle ij\rangle$ and
$\langle\langle ij\rangle\rangle$ run over NN and NNN pairs, respectively.
Recent interests of this model have been revived by the discovery of Fe-based
superconducting materialsKamihara _et al._ (2008) where a weakened AF order
can be described by this model with $S>1/2$.Yildirim (2008); Si and Abrahams
(2008); Ma _et al._ (2008)
Properties of this model for $S=1/2$ in 2d have been studied extensively by a
variety of methods, such as spin wave theory,Chandra and Doucot (1988) exact
diagonalization(ED), Dagotto and Moreo (1989); Schulz and Ziman (1992);
Richter and Schulenburg (2010) series expansion,Gelfand _et al._ (1989);
Kotov _et al._ (1999); Singh _et al._ (1999); Sushkov _et al._ (2001);
Singh _et al._ (2003) large-$N$ expansion,Read and Sachdev (1991) functional
renormalization group,Reuther and Wölfle (2010) Green’s function
method,Siurakshina _et al._ (2001) projected entangled pair states,Murg _et
al._ (2009) etc. It is generally believed that in the region
${J}_{2}/{J}_{1}\lesssim 0.4$, the ground state (GS) of the model is the Néel
phase with magnetic long-range order (LRO). In the region
${J}_{2}/{J}_{1}\gtrsim 0.65$, spins in the GS are ordered at wave vector
$(\pi,0)$ or $(0,\pi)$, showing so-called collinear magnetic LRO. The GS in
intermediate region is proposed to be a quantum paramagnet without magnetic
LRO, but the properties of this phase are still under intensive debate. There
are several proposals for the GS, such as a columnar dimer state,Murg _et
al._ (2009); Kotov _et al._ (1999) a plaquette VBS order,Zhitomirsky and Ueda
(1996); Mambrini _et al._ (2006); Capriotti and Sorella (2000) or a spin-
liquid.Figueirido _et al._ (1990); Capriotti _et al._ (2001) In the mean
time, precise determination of the phase transition points is also not
conclusive. Earlier series expansion studiesKotov _et al._ (1999) estimate
the quantum paramagnetic region is between
$0.38\lesssim{J}_{2}/{J}_{1}\lesssim 0.62$. Recent ED studyRichter and
Schulenburg (2010) using results of up to $N=40$ to perform finite-size
extrapolation estimates the transition points at $J_{2}^{c_{1}}\simeq
0.35J_{1}$ and $J_{2}^{c_{2}}\simeq 0.66J_{1}$. Meanwhile, studies by
combination of random phase approximation and functional renormalization group
find this nonmagnetic phase begins near $J_{2}/J_{1}\approx 0.4\sim 0.45$ and
ends around $0.66\sim 0.68$.Reuther and Wölfle (2010)
Numerical studies of frustrated quantum spin systems present great challenges
in dimensions greater than one. The ED method is hampered by the limitation of
system size one can simulate. At present, the largest system size on the
square lattice that can be simulated is $N=40$.Läuchli _et al._ (2005);
Richter and Schulenburg (2010) Due to the minus sign problem,Loh _et al._
(1990) the powerful quantum Monte Carlo (QMC) method is not applicable to
highly frustrated systems. In 1d, the density matrix renormalization group
(DMRG)White (1992) algorithm, which generates matrix product states (MPS), can
reach very high accuracy even for frustrated spin systems; however, direct
extension of the algorithm to higher dimensions remains difficult. One
promising proposal is to generalize the MPS to higher dimensions, the tensor
network states (TNS),Verstraete and Cirac (2004); Nishino _et al._ (2000);
Vidal (2007) which can serve as potential candidates for studying these
systems. In the TNSs, the matrices are replaced by tensors of rank
corresponding to the coordination number of the lattice. On a 2d square
lattice, the tensor $T^{s}_{ijkl}(\sigma_{s})$ on site $s$ has four indices,
in addition to the physical index, which in the current case corresponds to
the $z$-component $\sigma_{s}$ of a spin.
Here, we should mention, according to the TNS representation, the rank of
tensors is chosen according to the coordination number instead of the
interaction pattern. In this way, the area law of entanglement entropy can be
satisfied well if bond dimension D is big enough, especially when $J_{2}$ not
very large.
Contracting over all bond indices gives the wave function coefficient for a
given spin state $\sigma_{1},\ldots,\sigma_{N}$.Gu _et al._ (2008); Jiang
_et al._ (2008); Wang _et al._ (2011a) In these tensor network based methods,
one of the major obstacles is the computational complexity involved in the
tensor contraction, then usually some type of approximation is required to
make the computation manageable. Several schemes have been proposed to
facilitate the contraction of the tensor networks.Gu _et al._ (2008); Jiang
_et al._ (2008); Wang _et al._ (2011a); Xie _et al._ (2009); Zhao _et al._
(2010) In particular, a contraction scheme based on the plaquette
renormalization with auxiliary tensors is proposed to retain the variational
nature of the method, and it is shown that for the transverse Ising model,
even with the smallest possible bond dimension ($D=2$), non-mean-field results
can be obtained.Wang _et al._ (2011a)
In this paper, we use the TNS with the plaquette renormalization scheme to
study the $J_{1}-J_{2}$ Heisenberg model on a square lattice. We find that
even with a small bond dimension $D=2$, it already provides a useful way to
study the nature of the transition and estimate the value of the transition
points. The rest of this paper is organized as follows. In the following
section, we review the plaquette renormalization scheme of TNS, and how to
apply the scheme to the current model. Main results will be presented in Sec.
III, as well as some discussions. Sec. IV will give a brief summary.
## II method
We investigate the ground state of frustrated Heisenberg ${J}_{1}$-${J}_{2}$
model on a square lattice, using the plaquette renormalized tensor networkWang
_et al._ (2011a). The trial wave function is written as
$|\Psi\rangle=\sum_{\\{\sigma\\}}{tTr(T_{1}^{\sigma_{1}}\otimes
T_{2}^{\sigma_{2}}\cdots)|\sigma_{1}\sigma_{2}\cdots\rangle},$ (2)
where $tTr$ indicates the tensor trace that all the tensor indices are summed
over. $T_{s}$ is rank-4 tensor on site $s$, with bond dimension $D$ for each
rank and $\sigma_{s}=\uparrow$ or $\downarrow$ is the physical spin state.
Figure 1: (a) Direct contraction of four connecting rank-4 tensors $T$ with
bond dimensions $D$ results in a new tensor $T^{\prime}$ with bond dimensions
$D^{2}$ ; (b) Plaquette renormalized tensor contraction via additional
auxiliary rank-3 tensors $A$ with bond dimensions $D$. The resulting tensor
$T^{\prime}$ has the same bond dimension $D$ as the original tensor $T$.
Explicit contraction of the tensor network is computationally intensive. To
keep the computational complexity from growing exponentially, auxiliary rank-3
tensors $A^{n}_{ijk}$ are added to each level of the contraction process (Fig.
1), each transforms and truncates a pair of indices. A sequence of plaquette
renormalizations, $n=1,2,\ldots$, is carried out and the bond dimension of
each rank is thus kept constant after every plaquette contraction.Wang _et
al._ (2011a) In order to compute physical expectation values based on a TNS,
one has to contract the tensors of a bra and ket state over their physical
(e.g., spin) indices in addition to the bond indices of the tensors. Normally,
one would first construct the double tensors by performing the sum over the
physical indices,
$\mathbb{T}^{s}_{abcd}=\sum_{\sigma_{s},\sigma^{\prime}_{s}=\uparrow,\downarrow}T^{s*}_{i_{2}j_{2}k_{2}l_{2}}(\sigma^{\prime}_{s})T^{s}_{i_{1}j_{1}k_{1}l_{1}}(\sigma_{s}),$
(3)
where the labels $a,b,c,d$ is a suitable combination of the indices of the bra
($T^{s*}$) and ket ($T^{s}$) tensors, i.e., $a=i_{1}+D(i_{2}-1)$, etc. In the
calculation of the matrix element $\langle\Psi|\hat{O}|\Psi\rangle$ of some
operator involving one or several sites, similar tensors are constructed for
the sites at which operators act weighted with a local expectation value
$\langle\sigma^{\prime}_{s}|\hat{O}_{s}|\sigma_{s}\rangle$. In addition, the
renormalization double tensors can be also formed
$\mathbb{A}^{n}_{abc}=A^{n*}_{i_{2}j_{2}k_{2}}A^{n}_{i_{1}j_{1}k_{1}}$ (4)
The bond dimension of each rank in the resulting double tensor becomes
$\mathbb{D}=D^{2}$. This renormalization scheme reduces the maximum
computational complexityWang _et al._ (2011a) to $\mathbb{D}^{8}=D^{16}$ for
a double tensor network.
The ground state wave function can be obtained by optimizing the elements of
tensors $\mathbb{T},\mathbb{A}$ for the ground state energy. Since the
plaquette renormalization is introduced at the wave function level, instead of
the constructed double tensor network, the method remains variational and the
final energy will give a upper bound for the true ground state energy. We
optimize the wave function using the derivative-free Brent’s method.Brent
(2002) Compared to previous methods involving singular value decomposition
(SVD),Jiang _et al._ (2008); Gu _et al._ (2008) the environment of a given
tensor is fully taken into account in the current scheme. However, the
introduction of the renormalization $A$ tensors at the wave function level
effectively reduces the maximum support of the entanglement entropy area law
in this tensor network. To reduce the number of free parameters, we impose
symmetries on the trial wave function. We use a single plaquette, i.e.
$2\times 2=4$ sites as a unit cell (Fig. 1), wherein tensors $T$ on each site
and auxilliary tensors $A_{0}$ are assumed to be different. This unit is
translated to generate a $4\times 4$ unit and another set of auxilliary
tensors $A_{1}$ are added. This procedure is repeated until the full lattice
is generated. Finally, the periodic boundary condition is applied.Wang _et
al._ (2011a)
Figure 2: (Color online) (a) The ground state energy per site as a function of
$J_{2}/J_{1}$. The curves for $L=8$ and 16 are shifted up by $0.05$ and $0.10$
for clarity; (b) The square of staggered magnetization as a function of
${J}_{2}/{J}_{1}$.
## III results and discussions
We obtain the ground state wave function by varying the elements in the
tensors $T$ and $A$ with $D=2$, which describes a slightly entangled state
beyond the product (mean-field) state ($D=1$). Figure 2(a) shows the ground
state energy with system sizes $L=8,$ 16, and 32. A clear cusp near
$J_{2}/J_{1}=0.62$ is observed, signaling a first-order phase transition. A
continuous change of the slope is found near ${J}_{2}/{J}_{1}=0.4$, probably
indicating a continuous phase transition there.
To study the details of the magnetic orders and the transition points, we
compute the magnetic structure factor, or the square of staggered
magnetization at wave vector $\mathbf{q}$, defined as
$M^{2}(\mathbf{q})=\frac{1}{N^{2}}\sum_{ij}e^{i\mathbf{q}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}\left\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right\rangle,$
(5)
where $\mathbf{r}_{i}=(x_{i},y_{i})$, and $\mathbf{q}=(\pi,\pi)$ for the Néel
order, and $(0,\pi)$ or $(\pi,0)$ for the collinear order. $M^{2}(\mathbf{q})$
tends to the square of the order parameter in the thermodynamic limit if there
is magnetic ordering at wave vector $\mathbf{q}$, and scales like $1/N$ in a
magnetically disordered phase.
Figure 3: (Color online) (a) Extrapolated order parameters $m_{0}$ and
$m_{1}$ as a function of $J_{2}/J_{1}$. (b) Finite-size scaling of
$M^{2}(\pi,\pi)$ and $M^{2}(\pi,0)$ at $J_{2}/J_{1}=0.5$, where both order
parameters $m_{0}$ and $m_{1}$ scale to zero in the thermodynamic limit.
Figure 2(b) shows the results of the square of staggered magnetizations
$M^{2}(\pi,\pi)$ and $M^{2}(\pi,0)$. From the small $J_{2}/J_{1}$ side, the
Néel order is smoothly suppressed as $J_{2}$ increases, until
$J_{2}/J_{1}\simeq 0.40$, where a discontinuous jump of the Néel order is
observed for $L=8$, and the jumps become less pronounced as the system size
increases. This strong size dependence of the jump is another example that in
a finite-size tensor network state with finite bond dimensions, there exists
two energy minima near the transition, rendering the transition first-order at
small $N$. For a putative continuous transition, these two minima move closer
to each other with increasing $N$ and the transition becomes continuous at
$N\rightarrow\infty$.Liu _et al._ (2010)
Figure 4: (Color online) The $z$(black) and $xy$(red) components of the
square of staggered magnetization and the sum of the two (green) as a function
of ${J}_{2}/{J}_{1}$. (Inset) Same quantities in the regime of
${J}_{2}/{J}_{1}=0.45\sim 0.65$.
From the large $J_{2}/J_{1}$ side, the collinear order also decreases
smoothly, until $J_{2}/J_{1}\simeq 0.6$ where a clear first-order transition
occurs. Unlike the previous case, the jumps in $M^{2}(\pi,0)$ remain robust
upon increasing $N$, strongly suggesting against a continuous transition here.
This transition to the collinear order is consistent with previous numerical
calculations.Gelfand _et al._ (1989); Kotov _et al._ (1999); Singh _et al._
(1999); Sushkov _et al._ (2001); Singh _et al._ (2003); Richter and
Schulenburg (2010)
Figure 5: (Color online) The NN spin-spin correlations
$\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle$ (black numbers near bond)
and the plaquette order parameter (red numbers in italic) for
${J}_{2}/{J}_{1}=$ (a) 0.10 and (b) 0.50, with system size $L=32$. We show
only one corner ($4\times 4$) of the entire lattice as the pattern is repeated
periodically.
We now use our data from different sizes to extract the order parameters in
the thermodynamic limit. This allows us to estimate the transition points
between the Néel/collinear state and the non-magnetic (disordered) phase. The
finite-size extrapolation rules for the two-dimensional antiferromagnetic
Heisenberg model are well-knownSandvik (1997); Hasenfratz and Niedermayer
(1993); Schulz _et al._ (1996). Following Refs. Schulz _et al._ , 1996;
Richter and Schulenburg, 2010, we define the Néel order parameter as
$m_{0}=2\lim_{N\rightarrow\infty}M(\pi,\pi)$. This normalization is chosen so
that $m_{0}=1$ in a perfect Néel state. The finite-size behavior of
$M^{2}(\pi,\pi)$ is given by,Schulz _et al._ (1996); Richter and Schulenburg
(2010)
$M^{2}(\pi,\pi)=\frac{m_{0}^{2}}{4}\left(1+\frac{0.62075\ c}{\rho
L}+\cdots\right)$ (6)
where $c$ is the spin-wave velocity and $\rho$ is the spin stiffness. The
order parameter for the collinear order is defined as
$m_{1}=\sqrt{8}\lim_{N\rightarrow\infty}M(\pi,0)$. The finite-size behavior of
$M(\pi,0)$ is given by,Schulz _et al._ (1996); Richter and Schulenburg (2010)
$M^{2}(\pi,0)=\frac{1}{8}m_{1}^{2}+\frac{\rm const.}{L}+\cdots$ (7)
The extra $1/2$ factor comes from the fact that the ground state has an extra
two-fold degeneracy $\mathbf{q}=(\pi,0),(0,\pi)$, and this symmetry is broken
in the thermodynamic limit. Figure 3(a) shows the extrapolated results for
$m_{0}$ and $m_{1}$ as a function of $J_{2}/J_{1}$. We find that the GS near
$J_{2}/J_{1}=0.5$ is magnetically disordered, i.e., both $m_{0}$ and $m_{1}$
vanish. Figure 3(b) shows the finite-size scaling of $M^{2}(\pi,\pi)$ and
$M^{2}(\pi,0)$ at $J_{2}/J_{1}=0.5$, which both shows a $1/N$ scaling with the
zero intercept as $N\rightarrow\infty$. The transition points are estimated to
be $J_{2}^{c_{1}}=0.40J_{1}$ and $J_{2}^{c_{2}}=0.62J_{1}$, consistent with
estimates from series expansion Gelfand _et al._ (1989); Kotov _et al._
(1999); Singh _et al._ (1999); Sushkov _et al._ (2001); Singh _et al._
(2003); Sirker _et al._ (2006) where $J_{2}^{c_{1}}\approx 0.38J_{1}$ and
$J_{2}^{c_{2}}\approx 0.62J_{1}$, and slightly different from ED results
$J_{2}^{c_{1}}\approx 0.35J_{1}$ and $J_{2}^{c_{2}}\approx 0.66J_{1}$. Richter
and Schulenburg (2010) Near $J_{2}^{c_{1}}$, we fit the Néel order parameter
$m_{0}$ to a power law $m_{0}\sim(J_{2}-J_{2}^{c_{1}})^{\beta}$, and an
asymptotic mean-field behavior consistent with $\beta=1/2$ is also
observed.Liu _et al._ (2010) For $J_{2}=0$, we obtain $m_{0}=0.592$ which is
slightly lower than the best estimate from the quantum Monte Carlo
($m_{0}=0.6140$).Sandvik and Evertz (2010) Although it is also possible to
extract $c$ and $\rho$ from our data based on Eq. (6), it is argued that
determination of these quantities by fitting the prefactors of the leading
finite-size corrections ($O(1/L)$) can not reach the same accuracy as the
magnetic order parameters.Richter and Schulenburg (2010)
Analogous to how mean-field theory produces symmetry-broken states, this
method can produce solutions which break spin-rotation symmetry on a finite
lattice.Sandvik and Evertz (2010); Liu _et al._ (2010) We examine the spin-
rotation symmetry of the ground state, with the focus in the nonmagnetic
phase. Figure 4 shows $z$ and $xy$ components of the square of staggered
magnetization at $q=(\pi,\pi)$ for $L=32$, defined as
$\displaystyle
M^{2}_{z}(\pi,\pi)=\frac{1}{N^{2}}\sum_{ij}e^{i\pi[(x_{i}-x_{j})+(y_{i}-y_{j})]}\left\langle
S_{i}^{z}S_{j}^{z}\right\rangle,$ $\displaystyle
M^{2}_{xy}(\pi,\pi)=\frac{1}{N^{2}}\sum_{ij}e^{i\pi[(x_{i}-x_{j})+(y_{i}-y_{j})]}\left\langle
S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}\right\rangle.$
For reference, the sum of the two is also included. In the Néel phase, the
spin-rotational symmetry is clearly broken.Sandvik and Evertz (2010)
Increasing $J_{2}$ through a phase transition to the strongly frustrated
regime (i.e., $0.45\lesssim{J}_{2}/{J}_{1}\lesssim 0.60$), the spin-rotation
symmetry is restored with $M_{z}^{2}=\frac{1}{2}M_{xy}^{2}=\frac{1}{3}M^{2}$,
as expected.
In order to clarify the possible new phase in the highly frustrated region
around $J_{2}/J_{1}=0.5$, we calculate the nearest-neighbor spin-spin
correlations for $L=32$. Figure 5 shows the results for $J_{2}/J_{1}=0.10$,
which is deep inside the Néel phase, and $J_{2}/J_{1}=0.50$, which is in the
magnetically disordered phase. The numbers in black near the bond are the NN
spin-spin correlation, and the thickness of the bond is proportional to its
magnitude. For $J_{2}/J_{1}=0.50$ [Fig. 5(b)], the NN spin-spin correlations
within a single plaquette are much stronger than those between plaquettes. On
the other hand, deep inside the Néel phase ${J}_{2}/{J}_{1}=0.10$ [Fig. 5(a)],
the NN spin-spin correlations shows a more uniform pattern, although weaker
correlations are present in some bonds between plaquettes. Overall, it is
clear that the correlations inside a $2\times 2$ plaquette become stronger
upon increasing $J_{2}/J_{1}$, which indicates a possible plaquette order in
the magnetically disordered phase.
We also investigate the plaquette order parameter, which distinguishes clearly
a Néel ordered phase from a plaquette order, defined asMurg _et al._ (2009)
$\displaystyle Q_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\textstyle{\frac{1}{2}}(P_{\alpha\beta\gamma\delta}+P^{-1}_{\alpha\beta\gamma\delta})=2\big{[}(\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\beta})(\mathbf{S}_{\gamma}\cdot\mathbf{S}_{\delta})$
(8) $\displaystyle+$
$\displaystyle(\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\delta})(\mathbf{S}_{\beta}\cdot\mathbf{S}_{\gamma})-(\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\gamma})(\mathbf{S}_{\beta}\cdot\mathbf{S}_{\delta})\big{]}$
$\displaystyle+$
$\displaystyle\textstyle{\frac{1}{2}}(\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\beta}+\mathbf{S}_{\gamma}\cdot\mathbf{S}_{\delta}+\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\delta}+\mathbf{S}_{\beta}\cdot\mathbf{S}_{\gamma})$
$\displaystyle+$
$\displaystyle\textstyle{\frac{1}{2}}(\mathbf{S}_{\alpha}\cdot\mathbf{S}_{\gamma}+\mathbf{S}_{\beta}\cdot\mathbf{S}_{\delta}+\textstyle{\frac{1}{4}}).$
The results of the plaquette order parameter are shown also in Figs. 5
(numbers in red italic) for $J_{2}/{J}_{1}=0.10$ and $0.50$. In the most
frustrated region, we observe signature of the plaquette order. For
$J_{2}/J_{1}=0.50$, the plaquette order parameter is much stronger within a
plaquette, consistent with observation from the spin-spin correlations. This
order parameter is small in Néel phase ($J_{2}/J_{1}=0.10$), although some
traces of the plaquette order is still present. This might be due to the
inherent structure of the renormalization scheme, which explicitly breaks the
translational invariance, or possibly the plaquette correlations already start
to build up in this regime. It remains to further explore whether this
plaquette order is favored due to our renormalization scheme. The plaquette
renormalization scheme reduces the amount of entanglement support between
plaquettes by a factor of $D$ compared with the exact contraction. This may
bias toward those correlations compatible with the plaquette structure.
## IV Conclusion
We use the plaquette renormalization scheme to study spin-1/2 frustrated
Heisenberg ${J}_{1}$-${J}_{2}$ model on a square lattice with different sizes
of $L=8,16$, and 32. Using the smallest possible bond dimension $D=2$ for the
underlying tensors, we are already able to obtain results beyond the mean-
field theory. Since our method is variational, and the calculations are done
on finite lattices, we are able to perform finite-size scaling to extrapolate
the order parameters in the thermodynamic limit. We observe signatures of a
continuous transition at $J_{2}^{c_{1}}\simeq 0.40J_{1}$, and a first-order
phase transition at $J_{2}^{c_{2}}\simeq 0.62J_{1}$, consistent with previous
numerical calculations.Richter and Schulenburg (2010); Kotov _et al._ (1999)
Our calculations on the NN spin-spin correlation and the plaquette order
parameter indicates a possible plaquette VBS order for
$J_{2}^{c_{1}}<J_{2}<J_{2}^{c_{2}}$. The effects of the plaquette
renormalization scheme and the bond dimension $D$ dependence of the physical
observables require further studies and will be presented in a future work.Yu
_et al._ (2011); Yu and Kao
## V acknowledgements
We thank A. Sandvik for useful conversation and collaboration on related work.
We are grateful to National Center for High-Performance Computing Computer and
Information Networking Center, NTU for the support of high-performance
computing facilities. This work was partly supported by the National Science
Council in Taiwan through Grants No. 100-2112-M-002 -013 -MY3,
100-2120-M-002-00 (Y.J.K.), and by NTU Grant numbers 99R0066-65 and 99R0066-68
(J.F.Y., Y.J.K.). Travel support from National Center for Theoretical Sciences
is also acknowledged.
_note added._ \- After submitting this manuscript, we recently learned of the
DMRG work by Jiang et al.Jiang _et al._ (2011) and the tensor product state
approach by Wang et al.Wang _et al._ (2011b) on the same model, which argue
that the ground state in the nonmagnetic regime near $J_{2}/J_{1}\sim 0.5$
could be a $Z_{2}$ spin liquid.
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|
arxiv-papers
| 2011-08-17T04:30:02 |
2024-09-04T02:49:21.579865
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ji-Feng Yu and Ying-Jer Kao",
"submitter": "Ji-Feng Yu",
"url": "https://arxiv.org/abs/1108.3393"
}
|
1108.3443
|
# Gravitational deflection of light in Rindler-type potential as a possible
resolution to the observations of Bullet Cluster 1E0657-558
Xin Li lixin@ihep.ac.cn Zhe Chang changz@ihep.ac.cn Institute of High Energy
Physics
Theoretical Physics Center for Science Facilities
Chinese Academy of Sciences, 100049 Beijing, China
###### Abstract
The surface density $\Sigma$-map and the convergence $\kappa$-map of Bullet
Cluster 1E0657-558 show that the center of baryonic matters separates from the
center of gravitational force, and the distribution of gravitational force do
not possess spherical symmetry. This hints that a modified gravity with
difference to Newtonian inverse-square law at large scale, and less symmetry
is worth investigating. In this paper, we study the dynamics in Randers-
Finsler spacetime. The Newtonian limit and gravitational deflection of light
in a Rindler-type potential is focused in particular. It is shown that the
convergence in Finsler spacetime could account for the observations of Bullet
Cluster.
###### pacs:
04.50.Kd,04.25.-g,95.35.+d
## I Introduction
In 1933, ZwickyZwicky analysed the velocity dispersion for the Coma cluster.
His analysis implied that the Coma cluster is full of invisible matters. There
are a great variety of observations which show that the rotational velocity
curves of all spiral galaxies tend to constant valuesTrimble . These include
the OortOort discrepancy in the disk of the Milky WayBahcall , the velocity
dispersions of dwarf Spheroidal galaxiesVogt , and the flat rotation curves of
spiral galaxiesRubin . These facts violate sharply the prediction of Newton’s
inverse square law of gravitation.
The most widely adopted way to resolve these difficulties is the dark matter
hypothesis. It is assumed that all visible stars are surrounded by massive
nonluminous matters. Though it explains the flat rotation curves of spiral
galaxies, the hypothesis has its own weakness. No theory predicts these
matters, and they behave in such ad hoc way. There are a lot of possible
candidates of dark matter (such as axion, neutrino et al), but none of them
satisfactory. Up to now, all of them either undetected or excluded by
observation.
Because of these troubles induced by dark matter, some models have been built
for alternative of the dark matter hypothesis. Their main ideas are to assume
that the Newtonian gravity or Newton’s dynamics is invalid in galactic scale.
The famous one is Milgrom’s Modified Newtonian Dynamics (MOND)Milgrom . MOND
assumes that the Newtonian dynamics does not hold in galactic scale. As a
phenomenological model, MOND explains well the flat rotation curves with a
simple formula and a new parameter. In particular, it deduces naturally a
well-known global scaling relation for spiral galaxies, the Tully-Fisher
relationTF . By introducing several scalar, vector and tensor fields,
BekensteinBekenstein rewrote the MOND into a covariant formulation. He showed
that the MOND satisfies all four classical tests on Einstein’s general
relativity in solar system.
However, MOND still face challenges. The strong and weak gravitational lensing
observations of Bullet Cluster 1E0657-558Bullet could not be explained well
by MOND and Bekenstein’s relativistic version of MONDAngus . Takahashi et al.
Takahashi have investigated weak gravitational lensing of three galaxy
clusters ((A1689, CL0024+1654, CL1358+6245) in terms of the MOND. They found
that MOND could not explain the data of three galaxy clusters unless a dark
matter halo is added. Problems still remain in fitting X-ray temperature of
galaxy clusters with MOND. In the framework of MOND, a plausible resolution of
these issues is the “marriage” of MOND with neutrino of mass $m_{\nu}\geq
2$eVAngus ; Takahashi . However, the WMAP7 data give an upper limit of 1.2eV
for the sum of neutrino mass Hannestad .
The main feature of the Bullet Cluster is that the magnitude of the
gravitational force approximately equals five times of what produced by
baryonic matters. Beside the main feature, there are two particular features
too. The surface density $\Sigma$-map reconstructed from X-ray imaging
observations gives the center of baryonic matters. The convergence
$\kappa$-map reconstructed from strong and weak gravitational lensing
observations gives the center of gravitational force. The first particular
feature is that the center of baryonic matters separates from the center of
gravitational force. The convergence $\kappa$-map manifests that the
distribution of gravitational force do not possess spherical symmetry. This is
the other particular feature. These two particular features are the reasons
why MOND could not explain well the observations of Bullet Cluster.
Due to the reasons given above, the observations of Bullet Cluster have been
regarded as a direct evidence of dark matter. The dominance of dark matter has
lasted almost six decades. However, up to now, no direct evidence claim that
dark matter is detected. It is very interesting to search an alternative to
dark matter. One famous and successful model is modified gravity (MOG)Moffat .
MOG assumes that some non geometrical fields couple to the gravitational field
of general relativity. MOG explains well the flat rotation curves of
galaxiesBrownstein . The MOG’s prediction for the $\kappa$-map results in two
baryonic components distributed across the Bullet Cluster 1E0657-558 with
averaged mass-fraction of 83% intracluster medium (ICM) gas and 17%
galaxiesBrownstein1 . One should notice that the locally measured value of
Newton’s constant $G$ varies spatially in MOG. In dark matter hypothesis, the
galaxy’s mass involve baryonic mass and the mass of dark matter. As for MOG,
the galaxy’s mass only involve baryonic mass, and the effective acceleration
depends on the running gravitational coupling $G(r)$.
In this paper, we try to introduce a possible alternative to dark matter,
which could account for the observations of Bullet Cluster. In 1912, A.
Einstein proposed his famous general relativity which gives the connection
between Riemann geometry and gravitation. In general relativity, the effects
of gravitation are ascribed to spacetime curvature instead of a force. By
mimicking Einstein, we may investigate the gravity in Finsler spacetimeBook by
Bao . Finsler geometry as a natural generalization of Riemann geometry could
provide new sight on modern physics.
The Finsler gravity modifies the Newtonian inverse-square law at large scale.
It reduces to the Newtonian inverse-square law at small distance. The center
of gravitational force may separate from the center of baryonic matters at
large scale. In general, Finsler spacetime admits less Killing vectors than
Riemann spacetime. In Finsler spacetime, the spherical symmetry may be broken
at large scale. Finsler spacetime possesses the two particular features of
Bullet Cluster. It could account for the observations of Bullet Cluster.
Recently, Grumiller Grumiller constructed an effective model for gravity of a
central object at large scales. In Grumiller’s model, to leading order in the
large radius expansion, all terms are expected from general relativity, except
for the Rindler one. The Rindler term leads to an anomalous acceleration Wald
, which could account for the Pioneer anomaly Anderson in solar system and
the rotational curve of galaxies. In this paper, we will show that the
Rindler-type potential could account for the observations of Bullet Cluster
within the framework of Finsler gravity.
This paper is organized as follows. In Sec.2, we present the vacuum field
equation in Finsler spacetime. In Sec.3, by making use of the post-Newtonian
approximation and the viewpoints of Zermelo navigation problem, we investigate
the dynamics in Randers-Finsler spacetime. The emphasis is focused on the
Newtonian limit and gravitational deflection of light in a Rindler-type
potential. The concluding remarks are given is Sec.4.
## II Vacuum field equation in Finsler spacetime
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given asBook by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(2)
The Finsler structure represents the length element of Finsler space.
The parallel transport has been studied in the framework of Cartan
connectionMatsumoto ; Antonelli ; Szabo . The notation of parallel transport
in Finsler manifold means that the length $F\left(\frac{dx}{d\tau}\right)$ is
constant. The geodesic equation for Finsler manifold is given asBook by Bao
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+G^{\mu}=0,$ (3)
where
$G^{\mu}=\frac{1}{2}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (4)
is called geodesic spray coefficient. Obviously, if $F$ is Riemannian metric,
then
$G^{\mu}=\tilde{\gamma}^{\mu}_{\nu\lambda}y^{\nu}y^{\lambda},$ (5)
where $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the Riemannian Christoffel
symbol. Since the geodesic equation (3) is directly derived from the integral
length of $\sigma$
$L=\int F\left(\frac{dx}{d\tau}\right)d\tau,$ (6)
the inner product
$\left(\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}=F\left(\frac{dx}{d\tau}\right)\right)$
of two parallel transported vectors is preserved.
In Finsler manifold, there exists a linear connection - the Chern
connectionChern . It is torsion freeness and almost metric-compatibility,
$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(7)
where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as
$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu},$
(8)
where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{\mu}\frac{\partial}{\partial y^{\nu}}$.
The gravity in Finsler spacetime has been investigated for a long timeTakano ;
Ikeda ; Tavakol1 ; Bogoslovsky1 . In this paper, we introduce vacuum field
equation from the extension of an analogy, which was discussed first by
PiraniPirani ; Rutz . In Newton’s theory of gravity, the equation of motion of
a test particle is given as
$\frac{d^{2}x^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial x^{i}},$ (9)
where $\phi=\phi(x)$ is the gravitational potential and $\eta^{ij}$ is
Euclidean metric. For an infinitesimal transformation $x^{i}\rightarrow
x^{i}+\epsilon\xi^{i}$($|\epsilon|\ll 1$), the equation (9) becomes, up to
first order in $\epsilon$,
$\frac{d^{2}x^{i}}{dt^{2}}+\epsilon\frac{d^{2}\xi^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial
x^{i}}-\epsilon\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial x^{j}\partial
x^{k}}.$ (10)
Combining the above equations(9) and (10), we obtain
$\frac{d^{2}\xi^{i}}{dt^{2}}=\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial
x^{j}\partial x^{k}}\equiv\xi^{k}H^{i}_{k}.$ (11)
In Newton’s theory of gravity, the vacuum field equation is given as
$H^{i}_{i}=\bigtriangledown^{2}\phi=0$. It means that the tensor $H^{i}_{k}$
is traceless in Newton’s vacuum.
In general relativity, the geodesic deviation gives similar equation
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}H^{\mu}_{\nu},$ (12)
where
$H^{\mu}_{\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}$ is Riemannian curvature tensor,
$D$ denotes the covariant derivative alone the curve $x^{\mu}(\tau)$. The
vacuum field equation in general relativity gives
$\tilde{R}_{\mu\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\lambda\nu}=0$Weinberg . It
implies that the tensor $H^{\mu}_{\nu}$ is also traceless,
$H=H^{\mu}_{\mu}=0$.
In Finsler spacetime, the geodesic deviation givesBook by Bao
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}H^{\mu}_{\nu},$ (13)
where
$H^{\mu}_{\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $R^{~{}\mu}_{\lambda~{}\nu\rho}$ is Finsler curvature tensor defined in
(8), $D$ denotes covariant derivative
$\frac{D\xi^{\mu}}{D\tau}=\frac{d\xi^{\mu}}{d\tau}+\xi^{\nu}\frac{dx^{\lambda}}{d\tau}\Gamma^{\mu}_{\nu\lambda}(x,\frac{dx}{d\tau})$.
Since the vacuum field equations of Newton’s gravity and general relativity
have similar form, we may assume that vacuum field equation in Finsler
spacetime holds similar requirement as the case of Netwon’s gravity and
general relativity. It implies that the tensor $H^{\mu}_{\nu}$ in Finsler
geodesic deviation equation should be traceless, $H=0$.
The notion of Ricci tensor in Finsler geometry was first introduced by Akbar-
ZadehAkbar
$Ric_{\mu\nu}=\frac{\partial^{2}\left(\frac{1}{2}F^{2}R\right)}{\partial
y^{\mu}\partial y^{\nu}},$ (14)
where $R=\frac{y^{\mu}}{F}R^{~{}\kappa}_{\mu~{}\kappa\nu}\frac{y^{\nu}}{F}$.
And the scalar curvature in Finsler geometry is given as
$S=g^{\mu\nu}Ric_{\mu\nu}$. Constructing a physical Finslerian theory of
gravity in arbitrary Finsler spacetime is a difficult task. However, it has
been pointed out that constructing a Finslerian theory of gravity in Finlser
spacetime of Berwald type is viable Tavakol . A Finsler spacetime is said to
be of Berwald type if the Chern connection (7) has no $y$ dependenceBook by
Bao . In light of the research of Tavakol et al. Tavakol , the gravitational
field equation in Berwald-Finsler space has been studied in Ref. Finsler DM ;
Lixin . In Berwald-Finsler space, the Ricci tensor reduces to
$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu}).$
(15)
It is manifestly symmetric and covariant. Apparently the Ricci tensor will
reduce to the Riemann-Ricci tensor if the metric tensor $g_{\mu\nu}$ does not
depend on $y$. We starts from the second Bianchi identities on Berwald-Finsler
spaceBook by Bao
$R^{~{}\alpha}_{\mu~{}\lambda\nu|\beta}+R^{~{}\alpha}_{\mu~{}\nu\beta|\lambda}+R^{~{}\alpha}_{\mu~{}\beta\lambda|\nu}=0,$
(16)
where the $|$ means the covariant derivative. The metric-compatibility
$g_{\mu\nu|\alpha}=0~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}g^{\mu\nu}_{~{}~{}|\alpha}=0,$
(17)
and contraction of (16) with $g^{\mu\beta}$ gives that
$R^{\mu\alpha}_{~{}~{}\lambda\nu|\mu}+R^{\mu\alpha}_{~{}\nu\mu|\lambda}+R^{\mu\alpha}_{~{}\mu\lambda|\nu}=0.$
(18)
Lowering the index $\alpha$ and contracting with $g^{\alpha\lambda}$, we
obtain
$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]_{|\mu}+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}_{|\mu}=0,$
(19)
where
$B_{\mu\nu\alpha\beta}=-A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F.$
(20)
Thus, the counterpart of the Einstein’s field equation on Berwald - Finsler
space takes the form
$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}.$ (21)
In Eq. (21), the term in “[ ]” is symmetrical tensor, and the term in “{}” is
asymmetrical tensor. By making use of the equation (21), the vacuum field
equation in Finsler spacetime of Berwald type implies
$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})=0.$
(22)
It means that the tensor
$H^{\mu}_{\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$
is traceless in Finsler spacetime of Berwald type. Therefore, the analogy from
the geodesic deviation equation is valid at least in Finsler spacetime of
Berwald type. For this reason, we may suppose that this analogy is valid in
general Finsler spacetime. For the rest of the paper, we adopt $H=0$ as a
requirement of vacuum field equation in Finsler spacetime.
## III The dynamics in Randers-Finsler spacetime
### III.1 The post-Newtonian approximation in Randers-Finsler spacetime
The Randers space is a special kind of Finsler geometry with Finsler structure
$F$ on the slit tangent bundle $TM\backslash 0$ of a manifold $M$,
$F(x,y)\equiv\alpha(x,y)+\beta(x,y),$ (23)
where
$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}},$ (24)
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu},$ (25)
and $\tilde{a}_{ij}$ is Riemannian metric.
The geodesic spray coefficient $G^{\mu}$ (4) in Randers-Finsler spacetime
readsBook by Bao
$\displaystyle
G^{\mu}=(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})y^{\nu}y^{\lambda}+(\tilde{a}^{\mu\nu}-l^{\mu}\tilde{b}^{\nu})(\tilde{b}_{\nu|\lambda}-\tilde{b}_{\lambda|\nu})\alpha\left(\frac{dx}{d\tau}\right)y^{\lambda},$
(26)
where $l^{\mu}\equiv y^{\mu}/F$, $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the
Christoffel symbol of Riemannian metric $\tilde{a}$ and
$\tilde{b}_{\nu|\lambda}$ denotes the covariant derivative with respect to the
Riemannian metric $\tilde{a}$
$\tilde{b}_{\nu|\lambda}=\frac{\partial\tilde{b}_{\nu}}{\partial
x^{\lambda}}-\tilde{\gamma}^{\mu}_{\nu\lambda}\tilde{b}_{\mu}.$ (27)
In the rest of the paper, we just consider the case that $\beta$ is a closed
1-form. Thus, the geodesic equation of such Randers spacetime is given as
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})y^{\nu}y^{\lambda}=0.$
(28)
In post-Newtonian approximation, to first order ($GM/r$), the non vanish
Christoffel symbols of Riemannian metric $\tilde{a}$ are
$\tilde{\gamma}^{i}_{00},~{}\tilde{\gamma}^{0}_{0i},~{}\tilde{\gamma}^{i}_{jk}$.
Here, $|GM/r|$ denotes the typical value of Newtonian potential. Deducing from
the definition of $l^{\mu}$, we find that $l^{0}$ is the same order with $1$
and $l^{i}$ is the same order with $GM/r$. Therefore, we obtain the
approximation formula of $G^{\mu}$
$\displaystyle G^{0}$ $\displaystyle=$ $\displaystyle
2\tilde{\gamma}^{0}_{0i}y^{0}y^{i}-l^{0}\tilde{b}_{i}(\tilde{\gamma}^{i}_{00}y^{0}y^{0}+\tilde{\gamma}^{i}_{jk}y^{j}y^{k}),$
(29) $\displaystyle G^{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}^{i}_{00}y^{0}y^{0}+\tilde{\gamma}^{i}_{jk}y^{j}y^{k}.$
(30)
In Finsler geometry, one can get the Chern connection from the geodesic spray
coefficient $G^{\mu}$Book by Bao
$N^{\mu}_{\nu}=\frac{1}{2}\frac{\partial G^{\mu}}{\partial
y^{\nu}},~{}~{}\Gamma^{\mu}_{\nu\lambda}=\frac{\partial
N^{\mu}_{\nu}}{\partial
y^{\lambda}}+\frac{1}{2}g^{\mu\rho}y_{\kappa}\frac{\partial^{2}N^{\kappa}_{\rho}}{\partial
y^{\nu}\partial y^{\lambda}}.$ (31)
By making use of the approximation formula of $G^{\mu}$ (29) and (30), we
obtain the approximation formula of the non vanish Chern connection
$\Gamma^{i}_{00}=\tilde{\gamma}^{i}_{00},~{}\Gamma^{0}_{0i}=\tilde{\gamma}^{0}_{0i},~{}\Gamma^{i}_{jk}=\tilde{\gamma}^{i}_{jk}.$
(32)
Therefore, to first order ($GM/r$), the Chern connection is none other than
the Christoffel symbols of Riemannian metric $\tilde{a}$. It implies that
curvature tensors approximately reduce to Riemannian, to first order ($GM/r$).
Following the deduction in Weinberg , we obtain the result
$\tilde{a}_{00}=-1+\delta a_{00},~{}~{}\tilde{a}_{ij}=\delta_{ij}(1+\delta
a_{00}),$ (33)
where $\delta a_{00}$ is a perturbation term with order of $GM/r$.
### III.2 The Zermelo navigation problem
Zermelo aimed to find minimum time trajectories in a Riemannian manifold
($M,h$), which under the influence of a wind represented by a vector field
$W$Zermelo . Shen Shen proved that the minimum time trajectories are exactly
the geodesic of a particular Finsler geometry-Randers spaceRanders , if the
wind is time independent. As a particular case of Finsler geometry, the
geometrical properties of Randers space have been studied Bao . Recently,
Randers space has drawn physicists’s attention. Gibbons et al. described
connection between the Randers spacetime and the Zermelo viewpoints by casting
the former in a Painlevé-Gullstrand formGibbons . It is interesting to study a
particle moving in Randers-Finsler spacetime.
In the following, we study the Zermelo navigation problem which is regarded as
an alternative of Randers-Finsler spacetime. For a given Riemannian metric $h$
and a wind $W$, one can set up a connection between Riemannnian data $(h,W)$
and Randers-Finsler structure $F$,
$\tilde{a}_{ij}=\frac{\lambda
h_{ij}+W_{i}W_{j}}{\lambda^{2}},~{}\tilde{b}_{i}=-\frac{W_{i}}{\lambda},~{}\lambda=1-h_{ij}W^{i}W^{j},$
(34)
where $W^{i}=h^{ij}W_{j}$ and $\tilde{a}^{ij}=\lambda(h^{ij}-W^{i}W^{j})$.
Schwarzschild metric as an exact solution of Einstein’s vacuum field equation
has been used to study four classical tests of general relativity. However,
the problems we mentioned in the beginning of the paper can not be solved in
the framework of Schwarzschild spacetime, if the gravitational source only
involves baryonic matters. Here, we investigate the equation of motion in
Finsler spacetime, which may be regarded as a candidate to solve the problems.
First, we start with the space part of Schwarzschild metric
$h_{ij}dx^{i}dx^{j}=\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2},$
(35)
and consider the influence of a radial wind $W(r)\equiv W_{r}dr$. Then, the
solution of Zermelo navigation problem gives the Randers metric, whose Finsler
structure is given as
$\displaystyle Fd\tau$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{-1}\left(\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2}\right)+\lambda^{-2}W_{r}^{2}dr^{2}}-\lambda^{-1}W_{r}dr$
(36) $\displaystyle=$
$\displaystyle\sqrt{\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}(r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2})}-\lambda^{-1}W_{r}dr,$
where we have used the formula of $\lambda$ (34) to get the second equation of
(36).
Next, we extend the structure (36) into relativistic form. By making use of
the approximate solution (33) of Finslerian vacuum field equation, we find
that the Randers-Finsler structure is of this form
$Fd\tau=\sqrt{-\lambda^{2}\left(1-\frac{2GM}{r}\right)dt^{2}+\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}(r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\varphi^{2})}-\lambda^{-1}W_{r}dr,$
(37)
while $|GM/r|\ll 1$.
The Schwarzschild metric is time independent and spatial isotropy, it means
that there are Killing vectors which correspond to the conserve quantities
$p_{t}=\left(1-\frac{2GM}{r}\right)\frac{dt}{d\tau}$ and
$p_{\varphi}=r^{2}\sin^{2}\theta\frac{d\phi}{d\tau}$. The Killing equations of
Randers space are given as Finsler PF
$\displaystyle K_{V}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(V_{\mu|\nu}+V_{\nu|\mu})y^{\mu}y^{\nu},$ (38)
$\displaystyle K_{V}(\beta)$ $\displaystyle=$
$\displaystyle\left(V^{\mu}\frac{\partial\tilde{b}_{\nu}}{\partial
x^{\mu}}+\tilde{b}_{\mu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\right)y^{\nu},$ (39)
where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. It is obvious that the vectors $V^{0}=C^{0}$ and
$V^{\varphi}=C^{\varphi}$ are solutions of (38) in Rander-Finsler spacetime
with structure (37), where $C^{0}$ and $C^{\varphi}$ are constant. Such
solutions $V^{0}=C^{0}$ and $V^{3}=C^{\varphi}$ are also satisfy the equation
(39). Thus, $V^{0}=C^{0}$ and $V^{3}=C^{\varphi}$ are Killing vectors in
Rander-Finsler spacetime with structure (37). This symmetry is the same with
Schwarzschild spacetime, it implies that conserve quantities exist in Rander-
Finsler spacetime with structure (37), like $p_{t}$ and $p_{\varphi}$ in
Schwarzschild spacetime.
### III.3 The equation of motion
One should notice that the Rander-Finsler structure (37) got in above
subsection depends only on $r,dx^{\mu}$. Therefore, $\tilde{b}_{\mu}dx^{\mu}$
is a closed 1-form, the geodesic equation in such spacetime (37) is of the
form
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+(\tilde{\gamma}^{\mu}_{\nu\lambda}+l^{\mu}\tilde{b}_{\nu|\lambda})\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0.$
(40)
It is convenient to denote $B(r)\equiv\lambda^{2}\left(1-\frac{2GM}{r}\right)$
and $A(r)\equiv\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}$. By making use
of the nonvanishing components of the connection
$\tilde{\gamma}^{\mu}_{\nu\lambda}$, we find from (40) that
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}t}{d\tau^{2}}+\frac{B^{\prime}}{B}\frac{dt}{d\tau}\frac{dr}{d\tau}+\frac{dt}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(41) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}r}{d\tau^{2}}+\frac{A^{\prime}}{2A}\left(\frac{dr}{d\tau}\right)^{2}+\frac{B^{\prime}}{2A}\left(\frac{dt}{d\tau}\right)^{2}-\frac{1}{2A}\left(\frac{d\theta}{d\tau}\right)^{2}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)-\frac{\sin^{2}\theta}{2A}\left(\frac{d\varphi}{d\tau}\right)^{2}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)+\frac{dr}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(42) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}\theta}{d\tau^{2}}+\frac{\lambda}{r^{2}}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)\frac{d\theta}{d\tau}\frac{dr}{d\tau}-\sin\theta\cos\theta\left(\frac{d\varphi}{d\tau}\right)^{2}+\frac{d\theta}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(43) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}\varphi}{d\tau^{2}}+\frac{\lambda}{r^{2}}\frac{d}{dr}\left(\frac{r^{2}}{\lambda}\right)\frac{d\varphi}{d\tau}\frac{dr}{d\tau}+2\cot\theta\frac{d\varphi}{d\tau}\frac{d\theta}{d\tau}+\frac{d\varphi}{d\tau}f\left(x,\frac{dx}{d\tau}\right),$
(44)
where
$f\left(x,\frac{dx}{d\tau}\right)\equiv\tilde{b}_{\nu|\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}/F$
and a prime denotes $d/dr$. Furthermore, we know that structure (37) only
depends on $r,dx^{\mu}$. Thus, the field is isotropic. It is convenient to
consider the orbit of particle confined to the equatorial plane
$\theta=\pi/2$. Then, the equation (43) is satisfied. Solving the equations
(41) and (44), we get
$\displaystyle\frac{d}{d\tau}\left(\ln\frac{dt}{d\tau}+\ln B\right)$
$\displaystyle=$ $\displaystyle\frac{d\ln J_{1}}{d\tau},$ (45)
$\displaystyle\frac{d}{d\tau}\left(\ln\frac{d\varphi}{d\tau}+\ln\frac{r^{2}}{\lambda}\right)$
$\displaystyle=$ $\displaystyle\frac{d\ln J_{1}}{d\tau},$ (46)
where we involved a new quantity $J_{1}$. It is defined as$\frac{d\ln
J_{1}}{d\tau}\equiv-f\left(x,\frac{dx}{d\tau}\right)$. Deducing from the
equations (45) and (46), we obtain two constants $(E,J)$ of motion. It
satisfies
$\displaystyle\frac{B}{J_{1}}\frac{dt}{d\tau}$ $\displaystyle=$ $\displaystyle
E,$ (47) $\displaystyle\frac{r^{2}}{\lambda J_{1}}\frac{d\varphi}{d\tau}$
$\displaystyle=$ $\displaystyle J.$ (48)
By multiplying the equation (42) with $2Adr/d\tau$ and making use of the
equations (47,48), we obtain
$B\left(\frac{dt}{d\tau}\right)^{2}-A\left(\frac{dr}{d\tau}\right)^{2}-\frac{r^{2}}{\lambda}\left(\frac{d\varphi}{d\tau}\right)^{2}=CJ_{1}^{2},$
(49)
where $C$ is a constant.
In the following, the value of $C$ and the formula of $J_{1}$ will be derived.
The derivative of the term $\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}$ gives
$\displaystyle\frac{d}{d\tau}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)$
$\displaystyle=$ $\displaystyle\frac{dx^{\nu}}{d\tau}\frac{\partial}{\partial
x^{\nu}}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)=\frac{dx^{\nu}}{d\tau}\left(\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)_{|\nu}$
(50) $\displaystyle=$
$\displaystyle\tilde{b}_{\alpha|\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}+\tilde{b}_{\mu}\left(\frac{d^{2}x^{\mu}}{d\tau^{2}}+\tilde{\gamma}^{\mu}_{\nu\lambda}\right)\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}$
$\displaystyle=$
$\displaystyle\left(1-\frac{\tilde{b}_{\mu}}{F}\frac{dx^{\mu}}{d\tau}\right)\tilde{b}_{\alpha|\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau},$
where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. Here, we have used the fact that the term
$\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}$ is a scaler in Riemannian spacetime
with metric $\tilde{a}_{\mu\nu}$, to get the second equation of (50). And we
have used the geodesic equation (40) to get the last equation of (50).
Noticing that $F$ is constant along the geodesic, we find from equation (50)
that
$\frac{d\ln\left(F-\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau}\right)}{d\tau}=-\tilde{b}_{\nu|\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\nu}}{d\tau}/F=-f\left(x,\frac{dx}{d\tau}\right).$
(51)
It implies that
$J_{1}=F-\tilde{b}_{\mu}\frac{dx^{\mu}}{d\tau},$ (52)
with normalization of $\tau$. Combining the equations (49) and (52), we find
$C=1$ for massive particles. The equations (49) and (52) can not determine the
value of $C$ for photons. While $\tilde{b}$ vanishes, the equation of motion
in Randers-Finsler spacetime must return to the one in Riemannian spacetime.
This physical requirement implies $C=0$ for photons.
At last, we list what we got. In Rander-Finsler spacetime with structure (37),
there are two constants of motion $E,J$, which again support our discussion
about Killing vectors in subsection B. There is equation of motion (49)
corresponding to the constancy of $F$. All equations (47,48,49) involve the
term $J_{1}$, which could be regarded as deviation from the equation of motion
in Schwarzschild spacetime. The constant $C$ in equation of motion equals 1 or
0, corresponds to massive particles and photons, respectively. Start with
these results, we could find the trajectory of particles moving in Randers-
Finsler spacetime.
### III.4 The Newtonian limit and gravitational deflection of light
In the above subsection, we derived the equation of motion for particles in
Randers-Finsler spacetime. In the following, we will study the orbit of
particles. Recently, Grumiller constructed an effective model for gravity of a
central object at large scalesGrumiller . It predicted a Rindler-type
acceleration Wald . We will show that if the Finslerian parameter $\lambda$ is
of the form
$\lambda=1+\frac{GM}{r_{s}^{2}}r,$ (53)
it will deduce a Rindler-type acceleration, where $r_{s}$ is a constant which
denotes the physical scale of gravitational system. First, we investigate the
Newtonian limit. By making use of the equations (47,48,49), we obtain the
relation between the radial distant $r$ and time $t$
$\frac{AE^{2}}{B^{2}}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}\lambda}{r^{2}}-\frac{E^{2}}{B}=-1~{}.$
(54)
In Newtonian limit, particles move slowly in a weak field. Thus, the
quantities
$\frac{J^{2}}{r^{2}},\left(\frac{dr}{dt}\right)^{2},E^{2}-1,\frac{GM}{r}$ all
are small. And to first order in these quantities, the equation (54) reduces
to
$\frac{1}{2}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}}{2r^{2}}-\left(1-\frac{r^{2}}{r_{s}^{2}}\right)\frac{GM}{r}=\varepsilon,$
(55)
where $\varepsilon\equiv\frac{1}{2}(E^{2}-1)$ is the energy in Newton’s
theory. The equation (55) implies the effective Newtonian potential is
modified
$\phi_{M}=-\left(1-\frac{r^{2}}{r_{s}^{2}}\right)\frac{GM}{r}~{}.$ (56)
Therefore, the effective acceleration is Rindler-type acceleration
$a_{M}=-\nabla\phi_{M}=-\left(\frac{GM}{r^{2}}+\frac{GM}{r_{s}^{2}}\right)~{}.$
(57)
Then, the velocity of galaxy rotational curve have the approximate relation
$v\approx\sqrt{\frac{GM}{r}+\frac{GM}{r_{s}^{2}}r}~{}.$ (58)
For a dwarf galaxies with mass scale $10^{8}$ solar masses, and
$r_{s}=1$kiloparsec (a typical length scale of dwarf galaxies), we find that
the term $\frac{GM}{r_{s}^{2}}$ in (58) equals $10^{-62}$ in natural units
($c=\hbar=G=1$). For a spiral galaxies with mass scale $10^{11}$ solar masses,
and $r_{s}=30$kiloparsec (a typical length scale of spiral galaxies), we find
that the term $\frac{GM}{r_{s}^{2}}$ in (58) also equals $10^{-62}$ in natural
units ($c=\hbar=G=1$). Therefore, for a reasonable parameter $r_{s}$, our
result is compatible with Grumiller’s results of galaxies rotational curve
Grumiller . It means that our result (58) could account for galaxies
rotational curve.
Next, we investigate the gravitational deflection of light. We limit us to the
case of weak gravitational field. It implies $\frac{GM}{r}\ll 1$. By making
use of the equations (47,48,49), and noticing that $C=0$ for photons, we
obtain the relation between the radial distant $r$ and angle $\varphi$
$\displaystyle\left(\frac{1}{r^{2}}\frac{dr}{d\varphi}\right)^{2}$
$\displaystyle=$
$\displaystyle\left(\frac{E}{J}\right)^{2}\frac{1}{AB\lambda^{2}}-\frac{1}{Ar^{2}\lambda}$
(59) $\displaystyle=$
$\displaystyle\left(\frac{E}{J\lambda}\right)^{2}-\frac{\lambda}{r^{2}}\left(1-\frac{2GM}{r}\right).$
In terms of variable $u=\frac{GM}{r}$, the equation (59) changes as
$\left(\frac{du}{d\varphi}\right)^{2}=\left(\frac{EGM}{J\lambda}\right)^{2}-\lambda
u^{2}(1-2u).$ (60)
At the closest approach to the gravitational source $M$,
$\varphi=\varphi_{m},~{}u=u_{m},~{}\lambda=\lambda_{m}$ and $du/d\varphi$
vanishes, thus
$\left(\frac{EGM}{J}\right)^{2}=\lambda_{m}^{3}u_{m}^{2}(1-2u_{m}).$ (61)
Substituting the equation (61) into (60), we obtain
$\frac{d\varphi}{du}=\frac{\lambda}{\sqrt{\lambda_{m}^{3}u_{m}^{2}(1-2u_{m})-\lambda^{3}u^{2}(1-2u)}}.$
(62)
Changing variable to $x=u/u_{m}$, to first order in $u_{m}$, we obtain from
(53,62) that
$\displaystyle\varphi_{m}-\varphi_{\infty}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}-2u_{m}(1-x^{3})+3u_{s}^{2}/u_{m}-2u_{s}^{2}/u_{m}x-xu_{s}^{2}/u_{m}}}$
(63) $\displaystyle=$
$\displaystyle\int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}}}\left(1+u_{m}\frac{1-x^{3}}{1-x^{2}}-\frac{3u_{s}^{2}/u_{m}-2u_{s}^{2}/u_{m}x-xu_{s}^{2}/u_{m}}{2(1-x^{2})}\right)$
$\displaystyle=$
$\displaystyle\frac{\pi}{2}+2u_{m}+\frac{u_{s}^{2}}{u_{m}}\left(\frac{3\sqrt{1-x^{2}}}{2(1+x)}-\log\left(\frac{1+\sqrt{1-x^{2}}}{x}\right)\right)\bigg{|}^{1}_{0}$
The term $\frac{\pi}{2}+2u_{m}$ is what we expect in general relativity, and
the third term in (63) tends to infinity at $x=0$. This infinity is due to
that the Rindler potential (56) is infinity at infinity distance $(u=0)$. It
is reasonable to consider that the Rindler-type potential vanishes outsider a
given cut off scale $r_{\rm out}$. Therefore, we deduce from (63) that the
deflection angle for the Rindler-type acceleration in Finsler spacetime is of
the form
$\alpha(r_{m})=\frac{GM}{r_{m}}\left(4+2\frac{r_{m}^{2}}{r_{s}^{2}}\left[\log\left(\frac{r_{\rm
out}+\sqrt{r_{\rm out}^{2}-r_{m}^{2}}}{r_{m}}\right)-\frac{3\sqrt{r_{\rm
out}^{2}-r_{m}^{2}}}{2(r_{\rm out}+r_{m})}\right]\right).$ (64)
While $r_{m}\ll r_{s}$, the deflection angle (64) reduces to the familiar one
in general relativity.
The strong and weak gravitational lensing survey of Bullet cluster
1E0657-558Bullet obtained a convergence $\kappa$-map. The convergence
$\kappa$ is defined asPeacock
$\kappa=\frac{D_{LS}}{2D_{S}}\nabla_{\theta}\alpha(\theta_{I})=\frac{D_{LS}D_{L}}{2D_{S}}\nabla_{\vec{\xi}}~{}\alpha(\vec{\xi}),$
(65)
where $\alpha(\vec{\xi})$ is deflection angle, $D_{LS}$ is the angular
distance from the lens plane to a source galaxy, $D_{L}$ is the angular
distance to the lens plane, $D_{S}$ is the angular distance to a source
galaxy, $\theta_{I}$ specify the observed position of the source galaxy,
$\vec{\xi}$ is the two dimensional vector in lens plane and
$\nabla_{\vec{\xi}}$ is two dimensional gradient operator. The surface density
$\Sigma(\vec{\xi})$ is derived as
$\Sigma(\vec{\xi})=\int\rho(\vec{\xi},z)dz,$ (66)
where $\rho(\vec{\xi},z)$ is the density of gravitational source $M$ and $z$
is the direction perpendicular to the lens plane. In weak field , the
deflection angle can be obtained as the superposition of the deflections
$\alpha(\vec{\xi})=4G\int\bar{\Sigma}(\vec{\xi}^{\prime})\frac{\vec{\xi}-\vec{\xi}^{\prime}}{|\vec{\xi}-\vec{\xi}^{\prime}|^{2}},$
(67)
where
$\bar{\Sigma}(\vec{\xi})=\Sigma(\vec{\xi})\left(1+\frac{|\vec{\xi}|^{2}}{2r_{s}^{2}}\left[\log\left(\frac{r_{\rm
out}+\sqrt{r_{\rm
out}^{2}-|\vec{\xi}|^{2}}}{|\vec{\xi}|}\right)-\frac{3\sqrt{r_{\rm
out}^{2}-|\vec{\xi}|^{2}}}{2(r_{\rm out}+|\vec{\xi}|)}\right]\right).$ (68)
Substituting (67) into (65), we obtain
$\kappa=\frac{4\pi
GD_{LS}D_{L}}{D_{S}}\bar{\Sigma}(\vec{\xi})\equiv\frac{\bar{\Sigma}(\vec{\xi})}{\Sigma_{c}}~{}.$
(69)
It is obvious from (68,69) that the convergence $\kappa$ does not reach its
maximum at the center of $\Sigma(\xi)$. It also implies that the position
where $\kappa$ reaches its maximum value is separated from the center of
$\Sigma(\xi)$. Thus, the convergence $\kappa$ deduced in Finsler gravity
satisfies the features of Bullet Cluster. The Rindler-type potential in
Finsler spacetime could account for the observations of Bullet Cluster.
## IV Conclusions
In this paper, we presented the vacuum field equation in Finsler spacetime. By
making use of the post-Newtonian approximation and the viewpoints of Zermelo
navigation problem, we investigated the dynamics in Randers-Finsler spacetime.
The Newtonian limit and gravitational deflection of light was obtained
explicitly. Within the framework of Finsler spacetime, the deflection angle
and the convergence $\kappa$ in Rindler-type potential were given.
The surface density $\Sigma$-map and the convergence $\kappa$-map of Bullet
Cluster 1E0657-558Bullet show that the center of baryonic matters separate
from the center of gravitational force, and the distribution of gravitational
force do not possess spherical symmetry. The formula (57) manifests that the
gravity in Finsler spacetime modified the Newtonian inverse-square law at
large scale. It is obvious from (68,69) that the convergence $\kappa$ does not
reach its maximum at the center of $\Sigma(\xi)$. It also implies that the
position where $\kappa$ reaches its maximum value is separated from the center
of $\Sigma(\xi)$. Thus, our model satisfies the first particular feature of
Bullet Cluster. The particular feature of spherical symmetry broken implies
that all modified gravity models with central potential need improvements for
accounting the observations of Bullet Cluster. However, for simplicity, the
central potential could regarded as the zero order term of the final modified
gravity model for Bullet Cluster. The spherical symmetry broken may be deduced
by the next leading order term of the final modified gravity model.
The convergence $\kappa$ in Rindler-type potential could account for
observations of Bullet Cluster. The numerical analysis is in progress. In
future work, we will consider the non-central potential in Finsler spacetime,
and investigate the effect of spherical symmetry broken.
###### Acknowledgements.
We would like to thank Prof. C. J. Zhu, M. H. Li and S. Wang for useful
discussions. The work was supported by the NSF of China under Grant No.
10875129 and 11075166.
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|
arxiv-papers
| 2011-08-17T10:48:33 |
2024-09-04T02:49:21.586000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1108.3443"
}
|
1108.3517
|
# Integer quantum Hall effect in a square lattice revisited
Santanu K. Maiti santanu@post.tau.ac.il School of Chemistry, Tel Aviv
University, Ramat-Aviv, Tel Aviv-69978, Israel Moumita Dey Theoretical
Condensed Matter Physics Division, Saha Institute of Nuclear Physics,
Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India S. N. Karmakar
Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
###### Abstract
We investigate the phenomenon of integer quantum Hall effect in a square
lattice, subjected to a perpendicular magnetic field, through Landauer-
Büttiker formalism within the tight-binding framework. The oscillating nature
of longitudinal resistance and near complete suppression of momentum
relaxation processes are examined by studying the flow of charge current using
Landauer-Keldysh prescription. Our analysis for the lattice model corroborates
the finding obtained in the continuum model and provides a simple physical
understanding.
###### pacs:
73.23.-b, 73.43.-f, 73.43.Cd
## I Introduction
The phenomenon of the integer quantum Hall effect (IQHE) is one of the most
significant discoveries in condensed matter physics klit . At much low
temperatures, typically below $1\,$K, this phenomenon occurs in two-
dimensional ($2$D) electron systems in presence of a strong perpendicular
magnetic field. It has been observed that the Hall resistance is quantized in
units of $h/2e^{2}$ with a great accuracy, specified in parts per million, and
due to this impressive accuracy it is utilized as a resistance standard hart .
Such a precise quantization of the Hall resistance comes from almost complete
suppression of momentum relaxation processes in the quantum Hall regime. In a
finite width conductor at high magnetic fields, the states carrying current in
one direction are spatially separated from those carrying currents in the
opposite direction, and as a result the overlaps between these two groups of
states get significantly reduced which results in a suppression of
backscattering. The near complete suppression of backscattering between the
forward and backward propagating states leads to a truly ballistic conductor
even in the presence of impurities. The quantized nature of Hall resistance
looks like the quantized resistance of conventional ballistic conductors, but
in these conductors precise quantization cannot be achieved since the
backscattering processes are not fully removed datta .
It has also been noticed that in a Hall bar the longitudinal resistance
oscillates as a function of the Fermi energy $E_{F}$. Naturally it may occur
to our mind that when the Fermi energy matches with anyone of the Landau
levels i.e., with a peak in the density of states (DOS) profile, the
resistance should have a minimum. But the real fact is completely opposite to
that. The resistance becomes minimum whenever the Fermi energy lies between
two Landau levels where the density of states also has a minimum, and, it
leads to an immense curiosity and gives rise to the question about the current
carrying states through the sample. It has been observed that the states those
are situated near the edges of the sample, the so-called edge states, play an
important role for carrying current when the longitudinal resistance has a
minimum halp ; macd . It is quite surprising that at the minima, the
longitudinal resistance is almost zero and the electrons are able to traverse
a long distance without dissipating their momentum. It emphasizes that
something special must be taking place which provide near complete suppression
of the momentum relaxation processes in this regime.
Several studies already exist in the literature which deal with the quantized
nature of the Hall resistance, oscillating behavior of the longitudinal
resistance, existence of the edge
Figure 1: (Color online). Four probe set-ups for determining (a) the Hall
resistance and (b) the longitudinal resistance.
states and almost complete suppression of the backscattering processes in a
Hall bar based on the continuum model lau ; thou1 ; thou2 ; pran ; avr ; klit1
. However, no rigorous effort has been made so far, to the best of our
knowledge, to unravel all these features within a discrete lattice model. In
the present work we consider a two-dimensional square lattice in presence of a
perpendicular magnetic field and establish them describing the system by the
tight-binding (TB) model. Using the Landauer-Büttiker formalism datta we find
the longitudinal and transverse conductivities. On the other hand, to
investigate the behavior of the current carrying states when Fermi energy lies
between two Landau levels or coincides with a Landau level we use Landauer-
Keldysh prescription niko1 ; niko2 . From this study we can clearly describe
the momentum relaxation processes. The physical picture about QHE that emerges
from our present study based on the discrete lattice model of the Hall bar is
exactly the same as obtained in the continuum model and provides much insight
to the problem under realistic condition. This approach could be much more
useful in further investigation of the unconventional quantum Hall effect in
graphene where the Hall conductivity is half-integer quantized novo ; kim ,
showing plateaus at $\sigma_{xy}=(4e^{2}/h)(n+1/2)$, $n=0$, $\pm 1$, $\pm 2$,
$\dots$, since this lattice model might be adapted with more convenience in
graphene and other topologically insulating materials hata ; long . It is our
first step towards this direction.
In what follows, we present the results. In Section $2$, we describe the model
and theoretical formulation. Section $3$ contains the numerical results and
related discussions, and we draw our conclusions in Section $4$.
## II The model and Theoretical formulation
The quantum Hall system is shown schematically in Fig. 1, where a square
lattice, placed in a magnetic field $\vec{B}$ perpendicular to its plane, is
attached to four finite width leads. The TB Hamiltonian for a square lattice
with $N$ atomic sites in both the $x$ and $y$ directions reads,
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{m,n}\epsilon_{m,n}c_{m,n}^{{\dagger}}c_{m,n}+t\sum_{m,n}\left(c_{m+1,n}^{{\dagger}}c_{m,n}\right.$
(1) $\displaystyle+$
$\displaystyle\left.c_{m,n+1}^{{\dagger}}c_{m,n}\,e^{i\theta_{m}}+\mbox{h.c.}\right).$
Here $\epsilon_{m,n}$ represents the site energy of an electron at the lattice
site ($m$,$n$), $m$ and $n$ being the $x$ and $y$ co-ordinates of the site,
respectively. $c_{m,n}^{\dagger}$ ($c_{m,n}$) corresponds to the creation
(annihilation) operator of an electron at the site ($m$,$n$) and $t$ is the
nearest-neighbor hopping integral. The phase factor $\theta_{m}=2\pi mBa^{2}$
($a$ being the lattice spacing) arises due to the magnetic field $B$ with the
choice of the vector potential $A$$\,=$($0,Bx,0$).
The side-attached leads are taken as semi-infinite square lattice ribbons,
which are also described by TB Hamiltonians similar to Eq. 1, where the
hopping terms do not invoke the phase factor $e^{i\theta_{m}}$ since
$\vec{B}=0$ in the leads. The lead Hamiltonians are parametrized by constant
on-site potential $\epsilon_{0}$ and nearest-neighbor hopping strength
$t_{0}$. The hopping integrals between the boundary sites of the leads and the
quantum Hall system are parametrized by $\tau$. The number of sites in the
transverse direction of the leads is denoted by $M_{y}$.
In order to find the Hall resistance using Landauer-Büttiker formalism, we
connect the leads to the sample as prescribed in Fig. 1(a) and restrict
ourselves within the coherent transport regime datta . The voltage leads,
namely, lead-2 and lead-3 are attached on the opposite sides of the sample. In
this formalism one can treat all the leads (i.e., current and voltage leads)
on equal footing and express the current in the lead-p as
$I_{p}=\sum_{q=1}^{4}G_{pq}\left[V_{p}-V_{q}\right]$, $G_{pq}$ being the
conductance coefficient related to transmission coefficient $T_{pq}$ between
the lead-p and lead-q by the expression $G_{pq}=(2e^{2}/h)T_{pq}$ (factor $2$
accounts for the spin of the electrons). Here $V_{p}$ is the applied bias
voltage in the lead-p. Since the currents in the various leads depend only on
voltage differences among them, we can set one of the voltages to zero without
loss of generality. Here we set $V_{4}=0$. It allows us to write the currents
in three other leads in the form of matrix equation, $V$=$\mathcal{G}$-1$I$,
where $\mathcal{G}$ is a $3\times 3$ matrix whose elements are expressed as
follows: $\mathcal{G}_{ii}=\sum_{k}^{k\neq i}G_{ik}$ and
$\mathcal{G}_{ij}=-G_{ij}$, where $k$ runs from $1$ to $4$. The above matrix
equation simplifies in the form:
$\left\\{\begin{array}[]{c}V_{1}\\\ V_{2}\\\ V_{3}\\\
\end{array}\right\\}=\left\\{\begin{array}[]{ccc}R_{11}&R_{12}&R_{13}\\\
R_{21}&R_{22}&R_{23}\\\ R_{31}&R_{32}&R_{33}\\\
\end{array}\right\\}\left\\{\begin{array}[]{c}I_{1}\\\ I_{2}\\\ I_{3}\\\
\end{array}\right\\}$ (2)
where, the matrix $R$ is the inverse of $\mathcal{G}$. In our four-terminal
set-up (Fig. 1(a)) the Hall resistance is determined from the relation,
$R_{H}=\left[\frac{V_{2}-V_{3}}{I_{1}}\right]_{I_{2}=I_{3}=0}=R_{21}-R_{31}.$
(3)
On the other hand, to determine the longitudinal resistance we connect the
leads to the conductor as shown in Fig. 1(b), attaching the voltage leads
lead-2 and lead-3 on the same side of the conductor. For this set-up also the
above expressions remain invariant, and the longitudinal resistance is given
by $R_{L}=R_{21}-R_{31}$. Obviously, for this set-up the numerical values of
$R_{21}$ and $R_{31}$ are different from those of the previous set-up (Fig.
1(a)).
With these $R_{L}$ and $R_{H}$ we can determine the longitudinal and Hall
conductivities where the conductivity tensor $\sigma_{\mu\nu}$ is the inverse
of the resistivity tensor $\rho_{\mu\nu}$. Mathematically it is expressed as
$\sigma_{\mu\nu}=\left(\rho_{\mu\nu}\right)^{-1}$, in which $\rho_{xx}$ and
$\rho_{xy}$ correspond to the longitudinal and Hall resistivities,
respectively. For a square lattice, $\rho_{xx}=\rho_{yy}=R_{L}$ and
$\rho_{xy}=-\rho_{yx}=R_{H}$. Therefore, we can write the final expressions of
the conductivities as hajdu ,
$\sigma_{xx}=\sigma_{yy}=\frac{R_{L}}{R_{L}^{2}+R_{H}^{2}};\,\sigma_{yx}=-\sigma_{xy}=\frac{R_{H}}{R_{L}^{2}+R_{H}^{2}}.$
(4)
Next, we find the bond charge current between the atomic sites $n$ and
$n^{\prime}$ by the Landauer-Keldysh method using the relation niko1 ; niko2 ,
$\langle\hat{J}_{nn^{\prime}}\rangle=\frac{et}{h}\int\limits_{E_{F}-eV_{0}/2}^{E_{F}+eV_{0}/2}\mbox{Tr}_{s}\left[G_{nn^{\prime}}^{<}(E)-G_{n^{\prime}n}^{<}(E)\right]dE.$
(5)
It actually describes the flow of charges which start at the site $n$ and end
up at the site $n^{\prime}$. In this expression $G^{<}$ is the conventional
lesser Green’s function niko1 and $V_{0}$ is the applied potential difference
between the biased leads. Here trace Trs is performed in the spin Hilbert
space, writing the Hamiltonian Eq. 1 including the spin of the electrons.
Throughout the numerical calculations we set $\epsilon_{0}=0$, lattice spacing
$a=1$ and fix the hopping integrals ($t$, $t_{0}$ and $\tau$) at $1$. All
energies are measured in units of $t$ and we choose $c=e=h=1$.
## III Numerical results and discussion
In the left panel of Fig. 2 we show the quantized nature of the Hall
conductance together with the density of states for an ordered ($W=0$, $W$
measures the disorder strength in the sample) square lattice. The sharp peaks
in the DOS are associated with the Landau bands those are produced in the
presence of magnetic field $B$. For a fixed sample size, the total number of
Landau bands and the number of states in each band strongly depend on the
strength of the applied magnetic field. Here we choose the uniform magnetic
field $B$ in the form $1/Q$, $Q$ being an
Figure 2: (Color online). Left panel: Hall conductance (red line) as a
function of Fermi energy and the electronic density of states (green line) for
a perfect ($W=0$) square lattice of size $20\times 20$ considering $B=0.05$.
Right panel: Longitudinal resistance as a function of Fermi energy for the
same parameter values.
integer, and it generates $Q$ number of Landau bands in the energy spectrum.
This choice of the magnetic field makes each Landau band populated by
$N^{2}/Q$ states for a square lattice of size $N\times N$. Thus, for the set
of parameter values chosen in Fig. 2, we will get twenty Landau bands at the
peaks of the DOS spectrum and each band accommodates twenty states. Since here
we present the results only for a particular energy range, few Landau bands
are visible. From the spectrum it is observed that the Hall conductance
increases in discrete steps whenever the Fermi energy crosses peaks in the DOS
profile. This is what we would expect from an ordinary ballistic waveguide
where two-terminal conductance changes in integer steps associated with the
integer number of sub-bands or transverse modes at the Fermi energy $E_{F}$
datta . On the other hand, plateaus appear in the Hall conductance when the
Fermi energy lies between two Landau bands. At these plateaus the Hall
conductance gets the value $(2e^{2}/h)M$, where $M$ corresponds to the total
number of Landau bands below the Fermi energy. This is exactly equal to the
number of edge states at the Fermi energy and these states play the equivalent
role similar to the transverse modes in a conventional ballistic waveguide.
Though the quantized Hall conductance resembles to the quantized conductance
in traditional ballistic waveguides, but the strange precise quantization can
never be achieved in ordinary ballistic conductors since the backscattering
processes are not fully suppressed. The almost complete elimination of the
backscattering processes can only be achieved in the quantum Hall regime which
provides surprisingly precise quantized Hall conductance. To reveal this fact,
we examine the flow of charge current in a perfect conductor when the Fermi
energy lies within a plateau region. The result is shown in the left panel of
Fig. 3, where $E_{F}$ is set at $-2.8$, and it predicts that the charge
current flows only through the edges of the conductor and no current is
available in the bulk. Very nicely we notice that the states carrying current
from the left side of the sample to the right one are spatially separated from
those carrying current in the reverse direction. As a result the overlap
between the forward and backward propagating states is almost reduced to
Figure 3: (Color online). Flow of charge current in a perfect ($W=0$) square
lattice of size $20\times 20$ considering $B=0.1$, where the left and right
panels correspond to $E_{F}=-2.8$ and $-1.46$, respectively. Left panel:
Charge current flows only through the edges and no current is obtained in the
bulk. Right panel: Charge current flows through the bulk.
zero leading to near complete suppression of the backscattering processes and
thereby the momentum relaxation process. In the continuum model these states
are referred as the positive and negative $k$-states and they are commonly
named as the edge states. At high magnetic field we get practically zero
overlap between these states. Due to this complete elimination of momentum
relaxation, the edge states carrying current towards the right side of the
sample are in equilibrium with the left lead, while the other states carrying
current in the opposite direction are in equilibrium with the right lead.
Therefore, the difference in voltage measured by two voltage probes placed
anywhere on the same side (the so-called longitudinal voltage) of the
conductor gets zero, while a non-zero value of the Hall voltage is obtained
when it is measured by two voltage probes placed anywhere on the opposite
sides of the conductor.
The almost entire suppression of the momentum relaxation process, when Fermi
energy lies in the plateau regions, gives rise to essentially zero
longitudinal resistance exactly what we have obtained numerically (see the
right panel of Fig. 2). The resistance of a conductor is governed by the rate
at which the electrons can loose their momentum. To relax the momentum an
electron has to be scattered from the left side to the right side of the
conductor through the allowed energy eigenstates in the bulk of the conductor.
This is practically zero (see the left panel of Fig. 3) as long as the Fermi
energy lies between two Landau levels. The non-zero value of the longitudinal
resistance is obtained exclusively when the backscattering process takes place
and it would be maximum when the Fermi energy synchronizes with a Landau band
(right panel of Fig. 2). To articulate the authenticity, in the right panel of
Fig. 3, we expose the nature of bond charge current in the conductor when the
Fermi energy is fixed at a Landau band ($E_{F}=-1.46$).
Figure 4: (Color online). Left panel: Hall conductance (red line) as a
function of Fermi energy and the electronic density of states (green line) for
a disordered ($W=0.5$) square lattice of size $20\times 20$ considering
$B=0.05$. Right panel: Longitudinal resistance as a function of Fermi energy
for the same parameter values.
It predicts a continuous flow of charge form one edge to the other, and
certainly, electrons can scatter from the left to the right side of the
conductor through its bulk. This scattering leads to the longitudinal
resistance. Thus, each time the Fermi energy crosses a Landau band, we get
maximum backscattering and hence a peak in the longitudinal resistance (see
the right panel of Fig. 2).
We now present the results for a disordered square lattice of size $20\times
20$ in Fig. 4. Disorder is introduced via a random distribution (width
$W=0.5$) of values of the on-site potentials (diagonal disorder), and results
averaged over hundred disorder configurations are presented. In the presence
of disorder sharp Landau levels are broadened (green line in the left panel of
Fig. 4), but it does not affect the quantized nature of the Hall conductance.
However, in contrast to the perfect conductor, the sharpness of the Hall
conductance gets reduced as observed in the left panel of Fig. 4 (red curve).
At the plateau regions the Hall conductance is again precisely quantized in
units of $2e^{2}/h$ and such conductance quantization, even in the presence of
impurities, comes due to the complete spatial separation of the forward and
backward propagating states which becomes apparent in Fig. 5 (left panel). On
the other hand, when the Fermi energy lies on a bulk Landau band, electrons
can easily scatter through the interior of the conductor, giving rise to non-
zero bond charge currents throughout the conductor (right panel of Fig. 5).
This scattering yields the longitudinal resistance (see the right panel of
Fig. 4).
It is important to note that with the increase of the disorder strength Landau
bands get broadened more by virtue of disorder. As a result of disorder
localized states are appeared in the tails of the broadened Landau bands and
they do not contribute to the transport. But, as we get the non-vanishing Hall
conductance, all the states within the Landau bands cannot be localized. At
the band centers extended states exist which carry the Hall current and the
amount of current which is lost due to the formation of localized states in
the tail of a Landau band is exactly compensated by the remaining extended
states prange ; ando . Since the localized states within the Landau band do
not contribute to the current the Hall plateaus become broadened which has
been clearly explained by Kramer et al. kramer . It
Figure 5: (Color online). Flow of charge current in a disordered ($W=1$)
square lattice of size $20\times 20$ considering $B=0.1$, where the left and
right panels correspond to $E_{F}=-2.8$ and $-1.46$, respectively. Left panel:
Charge current flows only through the edges and no current is obtained in the
bulk. Right panel: Charge current flows through the bulk.
is also to be noted that for large enough disorder strength and for weak
magnetic field, quantum Hall plateaus gradually vanish from the higher energy
side. We confirm it numerically. However, it was not the main motivation of
our present work, and the disappearance of the integer quantum Hall effect due
to disorder has already been reported in the literature sheng1 ; sheng2 .
Before we end this section, we would like to point out that the disappearance
of edge states and the appearance of finite resistance associated with the
backscattering processes, as a result of movement of the Fermi level $E_{F}$,
strongly depend on the system size, magnetic field and also on the impurity
strength. Here we describe very briefly about the nature of quantum phase
transition associated with the above mentioned factors, to make the present
communication a self contained study. The near zero value of the longitudinal
resistance is obtained only when the Fermi energy lies anywhere within the
plateau regions and in this situation the backscattering processes are almost
suppressed which result edge currents in the sample. While, for all other
choices of $E_{F}$, the finite value of longitudinal resistance is obtained
i.e., the backscattering processes are dominated which provide bulk current.
For a finite system size and fixed disorder strength, the appearance of
distinct Landau levels, and hence the quantized nature of Hall plateaus,
depends on the choice of the magnetic field. If the field is too weak, then
the Hall plateaus will no longer appear and the backscattering processes are
also strong enough to produce bulk current. On the other hand, if we fix the
system size and choose the magnetic field to a moderate one then the
appearance of the edge currents or bulk current upon the movement of the Fermi
level is significantly controlled by the impurity strength. For our system
size ($20\times 20$) we examine that when $W>2$, the Hall plateaus almost
disappear and we get bulk current for any choice of $E_{F}$ and this feature
would be visible for much less value of $W$ in the thermodynamic limit. Thus,
we can emphasize that the quantum phase transition in the phenomenon of
integer quantum Hall effect in a square lattice geometry strongly depends on
the system size, disorder strength and the applied magnetic field. For a
particular disorder strength and in presence of a moderate magnetic field, a
functional dependence of the Fermi energy $E_{F}$ with system size may be
obtained for the crossover between the edge currents and bulk currents. Since
at this stage it is really too complicated to compute it numerically we can
try to establish this functional form in our future work, but, we believe, the
main essence of all these features are quite well understood from our present
study.
## IV Conclusion
In conclusion, we have investigated the phenomenon of integer quantum Hall
effect in a discrete lattice model through Landauer-Büttiker formalism. The
almost complete suppression of momentum relaxation processes and the
oscillating nature of longitudinal resistance are clearly explored by
examining the flow of charge current using Landauer-Keldysh prescription. The
results presented in this communication are worked out for absolute zero
temperature. However, they should remain valid even in a certain range of
finite temperatures ($\sim 300$ K). This is because the broadening of the
energy levels of the conductor due to the conductor-lead coupling is, in
general, much larger than that of the thermal broadening datta . Our present
study about QHE based on the discrete lattice model provides much better
insight to the problem under realistic condition.
## References
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* (6) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
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* (10) J. E. Avron and R. Seiler, Phys. Rev. Lett. 54, 259 (1985).
* (11) K. von Klitzing, Rev. Mod. Phys. 58, 519 (1986).
* (12) B. K. Nikolíc, L. P. Zârbo, and S. Souma, Phys. Rev. B 73, 075303 (2006).
* (13) L. P. Zârbo and B. K. Nikolíc, Europhys. Lett. 80, 47001 (2007).
* (14) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature (London) 438, 197 (2005).
* (15) Y. Zhang, Y. -W. Tan, H. L. Stormer, and P. Kim, Nature (London) 438, 201 (2005).
* (16) H. Hatami, N. Abedpour, A. Qaiumzadeh, and R. Asgari, Phys. Rev. B 83, 125433 (2011).
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* (18) M. Janssen, O. Viehweger, U. Fastenrath, and J. Hajdu, Introduction to the theory of the integer quantum Hall effect, Ed. by J. Hajdu, VCH (Federal Republic of Germany) (1994).
* (19) R. E. Prange, Phys. Rev. B 23, 4802 (1981).
* (20) H. Aoki and T. Ando, Solid State Commun. 18, 1079 (1981).
* (21) B. Kramer, S. Kettemann, and T. Ohtsuki, Physica E 20, 172 (2003).
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|
arxiv-papers
| 2011-08-17T16:29:20 |
2024-09-04T02:49:21.592761
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti, Moumita Dey and S. N. Karmakar",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1108.3517"
}
|
1108.3560
|
# Resonant Behavior of an Augmented Railgun
Thomas B. Bahder William C. McCorkle Aviation and Missile Research,
Development, and Engineering Center,
US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A.
###### Abstract
We consider a lumped circuit model of an augmented electromagnetic railgun
that consists of a gun circuit and an augmentation circuit that is inductively
coupled to the gun circuit. The gun circuit is driven by a d.c. voltage
generator, and the augmentation circuit is driven by an a.c. voltage
generator. Using sample parameters, we numerically solve the three non-linear
dynamical equations that describe this system. We find that there is a
resonant behavior in the armature kinetic energy as a function of the
frequency of the voltage generator in the augmentation circuit. This resonant
behavior may be exploited to increase armature kinetic energy. Alternatively,
if the presence of the kinetic energy resonance is not taken into account,
parameters may be chosen that result in less than optimal kinetic energy and
efficiency.
## I Introduction
The goal of the design of an electromagnetic launch system is often to
maximize the kinetic energy of the armature (projectile) while keeping certain
design criteria fixed. For example, in a simple electromagnetic railgun (EMG),
we may want to maximize the armature kinetic energy while keeping the length
of the rails fixed. An example of such a system is the high-performance EMGs
planned by the navy for nuclear and conventional warships Walls et al. (1999);
Black (2006); McNab and Beach (2007), where the armature velocity must be
increased while keeping the length of the rails fixed. One approach to
increasing the armature velocity is to use some sort of augmentation to the
EMG circuit. Various types of augmentation circuits have been considered Kotas
et al. (1986), including hard magnet augmentation fields Harold et al. (1994)
and superconducting coils Homan and Scholz (1984); Homan et al. (1986).
In this paper, we consider a simple augmentation scheme consisting of a gun
circuit that is inductively coupled to an augmentation circuit. The gun
circuit, contains the rails connected to a d.c. voltage source, $V_{g}(t)$,
that powers the rails and armature. (We assume a d.c. voltage source for the
gun circuit because an a.c. voltage would have a lower average current and
hence lead to a lower armature velocity.) The augmentation circuit has its own
a.c. voltage generator, $V_{a}(t)$. Magnetic flux from the augmentation
circuit couples to the gun circuit. See Figure 1 for a schematic layout.
Figure 2 shows the equivalent lumped-circuit model that we are considering,
including the switches that control the currents. A simplified model of
augmentation has been previously considered, where the gun circuit was
augmented by a constant external magnetic field Harold et al. (1994). In our
work, we assume that a real augmentation circuit produces the magnetic field
that couples to the gun circuit, and hence, the gun circuit is interacting
with the augmentation circuit through mutual inductance, see Figure 2. This
coupling leads to a “back action” on the augmentation circuit by the gun
circuit, resulting in a non-constant $B$-field acting on the gun circuit.
Motion of the armature leads to variations of the self inductance and
resistance in the gun circuit, leading to a complex interaction between the
three degrees of freedom: the gun circuit, the augmentation circuit, and the
mechanical degree of freedom (the armature).
The resulting dynamical system is described by three non-linear differential
equations that are derived in Section II. We neglect the details of the
velocity skin effect (VSE) that is believed to be responsible for limiting the
performance of solid armatures, and is still the subject of research Young and
Hughes (1982); Drobyshevski et al. (1999); Stefani et al. (2005); Schneider et
al. (2007, 2009); Knoepfel (2000). However, the impact of the VSE is included
on the dynamical system through the use of position-dependent inductance and
resistance, $L_{g}(x)$ and $R_{g}(x)$, in the gun circuit. In this work, we
use a lumped circuit model of an augmented EMG. We find that there are
resonances in the magnitude of kinetic energy of the armature as a function of
the frequency of the driving voltage generator in the augmentation circuit.
These resonances depend on the switching time delay between augmentation and
gun circuits and other parameters.
## II Dynamical Equations
Consider an augmented railgun composed of an augmentation circuit with voltage
generator $V_{a}(t)$ and a gun circuit with voltage generator $V_{g}(t)$. We
assume that the circuits are inductively coupled, but have no electrical
connection, see Figure 1. The equivalent circuit for the augmented railgun is
shown in Figure 2. The motion of the solid armature leads to resistance of the
gun circuit, $R_{g}(x)$, that changes with armature position $x(t)$, and can
be written as
$R_{g}(x)=R_{g0}+R_{g}^{\prime}\,\,x(t)$ (1)
where $R_{g0}$ is the resistance of the gun circuit when $x=0$, and
$R_{g}^{\prime}$ is the gradient of resistance of the gun circuit at $x=0$.
Table 1: Electromagnetic gun and augmentation circuit parameters. Quantity | Symbol | Value
---|---|---
length of rails (gun length) | $\ell$ | 10.0 m
mass of armature | $m$ | 20 kg
coupling coefficient | $k$ | 0.80
self inductance of rails at $x=0$ | $L_{g0}$ | 6.0$\times$10-5 H
self inductance of augmentation circuit | $L_{a}$ | 6.0$\times$10-5 H
self inductance gradient of rails | $L_{g}^{\prime}$ | 0.60$\times$10-6 H/m
resistance of augmentation circuit | $R_{a}$ | 0.10 $\Omega$
resistance of gun circuit at $x=0$ | $R_{g0}$ | 0.10 $\Omega$
resistance gradient of gun circuit at $x=0$ | $R_{g}^{\prime}$ | 0.002 $\Omega/$m
voltage generator amplitude in gun circuit | $V_{g0}$ | 8.0$\times$105 Volt
voltage generator amplitude in augmentation circuit | $V_{a0}$ | 8.0$\times$105 Volt
open switch resistance in augmentation circuit | $r_{a0}$ | 30 $\Omega$
open switch resistance in gun circuit | $r_{g0}$ | 30 $\Omega$
Two dynamical equations for the augmented railgun are obtained by applying
Ohm’s law to the gun circuit and to the augmented circuitBahder and McCorkle
(2011):
$\displaystyle-V_{a}(t)+I_{a}R_{a}+I_{a}r_{a}(t)$ $\displaystyle=$
$\displaystyle-\frac{d}{dt}\phi_{a}$ (2) $\displaystyle-
V_{g}(t)+I_{g}R_{g}(x)+I_{g}r_{g}(t)$ $\displaystyle=$
$\displaystyle-\frac{d}{dt}\phi_{g}$ (3)
where
$\displaystyle S_{a}(t)$ $\displaystyle=$ $\displaystyle I_{a}(t)\,r_{a}(t)$
(4) $\displaystyle S_{g}(t)$ $\displaystyle=$ $\displaystyle
I_{g}(t)\,r_{g}(t)$ (5)
and $S_{a}(t)$ and $S_{g}(t)$ are the voltage drops across the time-dependent
resistances, $r_{a}(t)$ and $r_{g}(t)$, introduced into the augmentation and
gun circuits, respectively, by the switches $S_{a}$ and $S_{g}$, see Figure 2.
These switches allow introduction of an arbitrary time delay between the
current in the augmentation and gun circuits. We define the switching-on of
the currents by two time-dependent resistances
$\displaystyle r_{a}(t)=\begin{cases}r_{\text{a0}},&t<t_{\text{a0}}\\\
0,&t\geq t_{\text{a0}}\end{cases}$ (6) $\displaystyle
r_{g}(t)=\begin{cases}r_{\text{g0}},&t<t_{\text{g0}}\\\ 0,&t\geq
t_{\text{g0}}\end{cases}$ (7)
where $t_{a0}$ and $t_{g0}$ are the times at which the switches are closed,
and $r_{a0}$ and $r_{g0}$ are the switch resistances before the switches are
closed, in the augmented and gun circuits, respectively. The total flux in the
gun circuit, $\phi_{g}$, and the total flux in the augmentation circuit,
$\phi_{a}$, can be written as
$\displaystyle\phi_{g}$ $\displaystyle=$ $\displaystyle
L_{g}I_{g}+M_{\text{ga}}I_{a}$ (8) $\displaystyle\phi_{a}$ $\displaystyle=$
$\displaystyle L_{a}I_{a}+M_{\text{ag}}I_{g}$ (9)
where $I_{a}$ and $I_{g}$ are the currents in the augmentation circuit and gun
circuit, respectively, $L_{a}$ and $L_{g}$, are the self inductances of the
augmentation and gun circuits, respectively, and $M_{\text{ga}}$ and
$M_{\text{ag}}$ are the mutual inductances, which must be equal,
$M_{\text{ga}}=M_{\text{ag}}=M(x)$. As mentioned previously, the self
inductance of the gun circuit, $L_{g}(x)$, changes with armature position
$x(t)$. Also, the area enclosed by the gun circuit changes with armature
position, and therefore, the coupling between the augmented circuit and gun
circuit, represented by the mutual inductance, $M(x)$, changes with armature
position, $x(t)$. Furthermore, in order for the free energy of the system to
be positive, the self inductances and the mutual inductance must satisfy
Landau et al. (1984)
$M(x)=k\sqrt{L_{a}L_{g}(x)}$ (10)
for all values of $x$. Here, the coupling coefficient must satisfy $|k|<1$. We
can write the self inductance of the gun circuit as
$L_{g}(x)=L_{g0}+L_{g}^{\prime}\,\,x(t)$ (11)
where $L_{g0}$ is the inductance when $x=0$ and $L_{g}^{\prime}$ is the
inductance gradient of the gun circuit. Similarly, the mutual inductance
between augmented circuit and gun circuit can be written as
$M(x)=M_{0}+M^{\prime}\,\,x(t)$ (12)
where $M_{0}$ is the mutual inductance when $x=0$ and $M^{\prime}$ is the
mutual inductance gradient. For $x=0$ and $x=\ell$, where $\ell$ is the rail
length, Eq. (10) gives
$\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle
k\sqrt{L_{a}L_{\text{g0}}}$ (13) $\displaystyle M^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell}\left[k\sqrt{L_{a}\left(L_{\text{g0}}+L_{g}^{\prime}\,\ell\right)}-M_{0}\right]$
(14)
The coupling coefficient, $k$, can be positive or negative, and as mentioned
above, must satisfy $|k|<1$. The sign of $k$ determines the phase of the
inductive coupling between augmentation and gun circuits. Choosing the
coupling coefficient $k$, and the two self inductances, $L_{g0}$ and $L_{a}$,
Eq. (13) then determines the value of the mutual inductance, $M_{0}$. Then,
choosing a value for the rail length $\ell$, and the self inductance gradient,
$L_{g}^{\prime}$, Eq. (14) determines the mutual inductance gradient,
$M^{\prime}$. See Table 1 for parameter values used.
Two dynamical equations for the augmented railgun are obtained from Eq.
(2)-(3) and the third dynamical equation is obtained from the coupling of the
electrical and mechanical degrees of freedomMcCorkle and Bahder (2008).
Therefore, the three non-linear coupled dynamical equations for $I_{g}(t)$,
$I_{a}(t)$ and $x(t)$ are given by:
$\displaystyle-V_{a}(t)+I_{a}(t)(R_{a}+r_{a}(t))$ $\displaystyle=$
$\displaystyle-
L_{a}\frac{dI_{a}}{dt}-\left(M_{0}+M^{\prime}x(t)\right)\frac{dI_{g}}{dt}-M^{\prime}I_{g}(t)\frac{dx(t)}{dt}$
(15) $\displaystyle-
V_{g}(t)+I_{g}(t)\left(R_{\text{g0}}+R_{g}^{\prime}\,x(t)+r_{g}(t)\right)$
$\displaystyle=$
$\displaystyle-\left(L_{\text{g0}}+L_{g}^{\prime}\,x(t)\right)\frac{dI_{g}}{dt}-L_{g}^{\prime}\,\text{
}I_{g}(t)\frac{dx(t)}{dt}-\left(M_{0}+M^{\prime}x(t)\right)\frac{dI_{a}}{dt}-M^{\prime}\,I_{a}(t)\frac{dx(t)}{dt}\hskip
18.06749pt$ (16) $\displaystyle m\frac{d^{2}x(t)}{dt^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}L_{g}^{\prime}\,\,I_{g}{}^{2}(t)$ (17)
where $V_{g}(t)$ and $V_{a}(t)$ are the voltage generators that drive the gun
and augmentation circuits. From Eq. (17), we see that the EMG armature has a
positive acceleration independent of whether the gun voltage generator is a.c.
or d.c. because the armature acceleration is proportional to $I_{g}{}^{2}(t)$.
The armature velocity is essentially the integral of $I_{g}{}^{2}(t)$, and
therefore a higher final velocity will be achieved for d.c. current, and
associated d.c. gun voltage, $V_{g}(t)=V_{g0}$, where $V_{g0}$ is a constant.
Of course, the actual current will not be constant in the gun circuit because
of the coupling to the moving armature and to the augmentation circuit. We
want to search for solutions where the armature velocity is higher for an EMG
with the augmentation circuit than for an EMG without an augmentation circuit.
In order to increase the coupling between the augmentation circuit and the gun
circuit, we choose an a.c. voltage generator in the augmentation circuit:
$V_{a}(t)=-V_{a0}\cos(2\pi ft)$ (18)
where $V_{a0}$ is the amplitude and $f$ is the frequency of the augmentation
circuit voltage generator.
Figure 1: Schematic diagram of an inductively augmented EMG with its
surrounding augmentation circuit. Magnetic flux from the augmentation circuit
inductively links to the gun circuit. Figure 2: The equivalent circuit is
shown for the augmented railgun in Figure 1. Magnetic flux from the
augmentation circuit couples to the gun circuit through mutual inductance
$M(x)$. The self inductance of the gun circuit, $L_{g}(x)$, the mutual
inductance, $M(x)$, and the resistance, $R_{g}(x)$, are functions of the
armature position, $x(t)$. The voltage drops of the switches, $S_{a}(t)$ and
$S_{g}(t)$, are defined in Eq. (4) and (5).
As an example of the complicated coupling between augmentation circuit and gun
circuits, we will also obtain solutions for a d.c. voltage generator for the
augmentation circuit. As we will see, the gun circuit causes a back action on
the augmentation circuit, leading to a non-constant current in the
augmentation circuit.
We need to choose initial conditions at time $t=0$. We assume that there is no
initial current in the gun and augmentation circuits and that the initial
position and velocity of the armature are zero:
$\displaystyle I_{g}(0)$ $\displaystyle=$ $\displaystyle 0$ (19)
$\displaystyle I_{a}(0)$ $\displaystyle=$ $\displaystyle 0$ (20)
$\displaystyle x(0)$ $\displaystyle=$ $\displaystyle 0$ (21)
$\displaystyle\frac{dx(0)}{dt}$ $\displaystyle=$ $\displaystyle 0$ (22)
For the special case when $L_{g}^{\prime}=0$, $M^{\prime}=0$, and
$R_{g}^{\prime}=0$, the mechanical degree of freedom described by Eq. (17)
decouples from Eqs. (15)–(17). In this case, Eqs (15)–(16) describe a
transformer with primary and secondary circuits having voltage generators,
$V_{a}(t)$ and $V_{g}(t)$, respectively. The solution for the mechanical
degree of freedom is then $x(t)=0$ for all time $t$.
In what follows, we solve the dynamical Eqs. (15)–(17) numerically for the
case when the EMG and augmented circuits are coupled.
## III Numerical Solution
As described above, if we choose the parameters $k$, $L_{a}$, $L_{g0}$, then
$M_{0}$ is determined from Eq. (13). Next, if we choose $L_{g}^{\prime}$ and
$\ell$, then the inductance gradient, $M^{\prime}$, is determined by Eq. (14).
See Table 1 for values of parameters used in the calculations below. The sign
of the coupling coefficient, $k$, affects the interaction of the augmentation
and gun circuits in subtle ways.
In order to get a large increase in armature kinetic energy, the flux from the
augmentation circuit must induce a large rate of change of magnetic flux in
the gun circuit, leading to a large externally induced emf in the gun circuit.
Using values of parameters shown in Table 1, using a d.c. generator in the gun
circuit of magnitude $V_{g0}=800$ kV, and using an a.c. generator in the
augmentation circuit given by Eq. (18) with amplitude $V_{a0}=800$ kV, we
numerically integrate the dynamical Eqs. (15)–(17) to obtain the current in
the augmentation circuit, $I_{a}(t)$, the current in the gun circuit,
$I_{g}(t)$, and the armature position, $x(t)$, at a given frequency $f$ of
augmentation circuit voltage generator. At time $t_{f}$, the armature reaches
the end of the rails and attains its highest velocity, which provides the
boundary condition relating $t_{f}$ and the length of the rails, $\ell$:
$x(t_{f})=\ell$ (23)
## IV Kinetic Energy Resonance
When the armature reaches the end of the rails at $t=t_{f}$, energy is stored
in three places. Energy is stored in the magnetic field in the augmentation
circuit:
$E_{a}=\frac{1}{2}L_{a}(I_{a}(t_{f}))^{2}$ (24)
Energy is stored in the magnetic field in the gun circuit:
$E_{g}=\frac{1}{2}\left[L_{g0}+L_{g}^{\prime}\,x(t_{f})\right](I_{g}(t_{f}))^{2}$
(25)
and energy is stored in the armature kinetic energy,
$E_{k}=\frac{1}{2}\,m(\dot{x}(t_{f}))^{2}$ (26)
Energy is also stored in the mutual inductance between the augmentation
circuit and gun circuit:
$E_{m}=\left[M_{0}+M^{\prime}\text{
}x\left(t_{f}\right)\right]I_{a}\left(t_{f}\right)\text{
}I_{g}\left(t_{f}\right)$ (27)
The EMG shot is a transient phenomenon, not a steady state phenomenon.
Furthermore, the dynamical Eqs. (15)–(17) are non-linear, and hence do not
have a simple resonant condition. Never-the-less, we found that the armature
kinetic energy has a resonant behavior as a function of the frequency $f$ of
the driving voltage in the augmentation circuit, see Eq. (18). Figure 3 shows
a plot of the kinetic energy of the armature, $E_{k}$, as a function of the
frequency $f$ of the voltage generator of the augmentation circuit, see Eq.
(18). The integration of the dynamical Eqs. (15)–(17) is started at initial
time $t=0$. In Figure 3, the switches in the gun circuit and augmentation
circuit were closed at the same time: $t_{g0}=0$ and $t_{a0}=0$. For the
values of parameters in Table 1, the armature kinetic energy has a maximum of
201.6 kJ at frequency $f=204$ Hz, and a minimum of 107.2 kJ at frequency
$f=98$ Hz, which is an 88% variation in kinetic energy with driving frequency
$f$.
Typically, resonant phenomena occur in a steady state. The EMG shot is a
transient phenomena. However, Figure 3 shows that armature kinetic energy has
an intrinsic resonance as a function of the driving frequency of the
augmentation voltage generator. It is clear that a resonant condition exists
for the armature kinetic energy.
Figure 3: The armature kinetic energy is plotted as a function of the
frequency, $f$, of the driving voltage, $V_{a}(t)$, of the augmentation
circuit, see Eq. (18). The kinetic energy has strong oscillations indicating
that there is a resonant behavior. The gun circuit switch was closed at
$t_{g0}=0$ and the switch in the augmentation circuit was closed at $t_{a0}=0$
.
At time $t=t_{f}$, the energy stored in the inductance in the gun circuit,
$E_{g}$, and the energy stored in the augmentation circuit, $E_{a}$, are
plotted as a function of driving frequency, $f$, in Figure 4 and 5,
respectively. It is clear that when the armature kinetic energy is a minimum,
the energy stored in the gun circuit inductor is not a maximum. Instead there
is a complicated partition between energy stored in the gun circuit inductor,
in the augmentation circuit inductor, and in armature kinetic energy.
Figure 4: The energy stored in the gun circuit inductance, $E_{g}$, is plotted
as a function of the driving frequency, $f$, of the augmentation circuit
voltage, $V_{a}(t)$, see Eq. (18). The gun circuit switch was closed at
$t_{g0}=0$ and the switch in the augmentation circuit was closed at
$t_{a0}=0$. Figure 5: The energy stored in the augmentation circuit
inductance, $E_{a}$, is plotted as a function of the driving frequency, $f$,
of the augmentation circuit voltage, $V_{a}(t)$, see Eq. (18). The gun circuit
switch was closed at $t_{g0}=0$ and the switch in the augmentation circuit was
closed at $t_{a0}=0$.
In Figure 6, we plot the time-dependence of the dynamical variables at the
frequency $f=98$ Hz at which the kinetic energy has a minimum value
$E_{k}=107.24$ MJ. The time for the shot is $t=t_{f}=5.52$ ms. For this case,
the armature velocity is 3.27 km/s.
Figure 6: For the first kinetic energy minimum at frequency $f=98$ Hz in
Figure 3, the armature position, $x(t)$, the armature velocity,
$v(t)=\dot{x}(t)$, augmentation circuit current, $I_{a}(t)$, and gun circuit
current, $I_{g}(t)$, is plotted as a function of time. In the current plots,
red line is gun current $I_{g}(t)$ and blue line is augmentation circuit
current $I_{a}(t)$. These quantities correspond to Figure 3, where the gun
circuit switch is closed at $t_{g0}=0$ and the switch in the augmentation
circuit is closed at $t_{a0}=0$. For this case, $t_{f}=5.52\times 10^{-3}$ s.
Note that the plots are only valid for $0\leq t\leq t_{f}$.
For the kinetic energy maximum that occurs at $f=204$ Hz in Figure 3, the
time-dependence of the dynamical variables is shown in Figure 7.
Figures 6 and 7 show that there is a complicated interaction between the
currents in the augmentation and gun circuits.
Figure 7: For the kinetic energy maximum at frequency $f=204$ Hz in Figure 3,
the armature position, $x(t)$, the armature velocity, $v(t)=\dot{x}(t)$,
augmentation circuit current, $I_{a}(t)$, and gun circuit current, $I_{g}(t)$,
is plotted as a function of time. In the current plots, red line is gun
current $I_{g}(t)$ and blue line is augmentation circuit current $I_{a}(t)$.
These quantities correspond to Figure 3, where the gun circuit switch is
closed at $t_{g0}=0$ and the switch in the augmentation circuit is closed at
$t_{a0}=0$. For this case, $t_{f}=5.52\times 10^{-3}$ s. Note that the plots
are only valid for $0\leq t\leq t_{f}$.
## V Energy Conservation and Efficiency
The total energy that is input into the EMG system, $E^{\text{in}}$, during
the shot time $0\leq t\leq t_{f}$, is given by the sum of energy input into
the augmentation circuit and gun circuit,
$E^{\text{in}}=E_{a}^{\text{in}}+E_{g}^{\text{in}}$, where
$\displaystyle E_{a}^{\text{in}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{f}}I_{a}(t)V_{a}(t)\text{ }dt$ (28) $\displaystyle
E_{g}^{\text{in}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{f}}I_{g}(t)V_{g}(t)\text{ }dt$ (29)
During the shot time, energy is dissipated in the augmentation circuit
resistance
$Q_{a}=\int_{0}^{t_{f}}\left(I_{a}(t)\right){}^{2}R_{a}\text{ }dt$ (30)
and in the gun circuit resistance, which depends on armature position:
$Q_{g}=\int_{0}^{t_{f}}\left(I_{g}(t)\right){}^{2}\left(R_{\text{g0}}+R_{g}^{\prime}\text{
}x(t)\right)\text{ }dt$ (31)
Conservation of energy is expressed by
$E^{\text{in}}=E_{a}+E_{g}+E_{m}+E_{k}+Q_{a}+Q_{g}$ (32)
where the terms are defined in Eq. (24)–(31). We have verified that our
numerical solutions satisfy energy conservation to an accuracy
$\left[E^{\text{in}}-(E_{a}+E_{g}+E_{m}+E_{k}+Q_{a}+Q_{g})\right]/E^{\text{in}}\approx
2\times 10^{-4}$.
The efficiency, $\eta$, of the augmented EMG is given by the ratio of armature
kinetic energy to input energy:
$\eta=\frac{E_{k}}{E_{a}^{\text{in}}+E_{g}^{\text{in}}}$ (33)
When the armature kinetic energy is a minimum, at $f=98$ Hz, the EMG
efficiency is $\eta=0.00245$, see Figure 3. When the armature kinetic energy
is a maximum, at $f=204$ Hz, the efficiency $\eta=0.00453$. So the efficiency
at the kinetic energy maximum is 1.84 times the efficiency at the kinetic
energy minimum.
## VI Large Kinetic Energy Resonance
When designing an augmented EMG, care must be taken in the choice of
parameters. Certain parameter values lead to a strong kinetic energy
resonance, see Figure 8. For example, if the frequency of the augmentation
voltage was chosen to be $30.8$ Hz rather than d.c. , then we would obtain a
kinetic energy that is 5.7 times larger, see Figure 8. Alternatively, since we
do not know the precise values of the parameters in our experiments, we may
find that we are in a regime of strong kinetic energy resonance, and that the
armature kinetic energy is non-optimal. Also, in the regime of a strong
kinetic energy resonance, the efficiency of the EMG varies strongly with
frequency. For example, in Figure 8, we calculated the efficiency at $f=0$
(defined by Eq (33)) to be $\eta=1.3\times 10^{-4}$, while at $f=30.8$ Hz,
the efficiency is $\eta=9.0\times 10^{-4}$.
Figure 8: The armature kinetic energy is plotted as a function of the
frequency $f$ of the driving voltage, $V_{a}(t)$, of the augmentation circuit
(see Eq. (18)) for parameter values given by $L_{a}=6.0\times 10^{-3}$ H,
$L_{g0}=6.0\times 10^{-3}$ H, $L_{g}^{\prime}=0.5\times 10^{-6}$ H/m,
$R_{a}=0.1$ Ohm, $R_{g0}=0.1$ Ohm, $R_{g}^{\prime}=6.0\times 10^{-6}$ Ohm/m,
and the other parameters are taken as shown in Table 1. For this parameter
set, the kinetic energy has strong oscillations indicating that there is a
resonant behavior. The gun circuit and the augmentation circuit switches were
closed simultaneously at $t_{g0}=0$ and $t_{a0}=0$ .
## VII Time Delayed Switching
Changing the switch-on time of the gun circuit and the augmentation circuit,
by changing $t_{a0}$ and $t_{g0}$, causes small variations in the position of
the first minimum and maximum of armature kinetic energy. For example, when we
switch on the gun circuit at $t_{g0}=0$ and delay switching on the
augmentation circuit to $t_{a0}=3.0\times 10^{-3}$ s, the resulting armature
kinetic energy, $E_{k}$, is plotted in Figure 9. For this case, the armature
has kinetic energy maximum, $E_{k}=206.658$ MJ, which occurs at $f=0$ Hz,
i.e., which is a d.c. driving voltage in the augmented circuit. The first
armature kinetic energy minimum, $E_{k}=125.773$ MJ, occurs at $f=98$ Hz. The
next kinetic energy maximum, $E_{k}=200.18$ MJ, occurs at $f=202$ Hz.
Figure 9: The armature kinetic energy is plotted as a function of the
frequency $f$ of the driving voltage, $V_{a}(t)$, of the augmentation circuit,
see Eq. (18). The kinetic energy has strong oscillations indicating that there
is a resonant behavior. The gun circuit switch was closed at $t_{g0}=0$ and
closing the augmentation circuit was delayed by $t_{a0}=3.0\times 10^{-3}$ s.
Compare this figure with Figure 3.
At $f=0$, the d.c. driving voltage of the augmentation circuit in Figure 9,
the time-dependence of the dynamical variables is given in Figure 10. The
armature attains velocity is 4.47369 km/s, and the armature kinetic energy is
200.139 MJ.
Figure 10: For the kinetic energy maximum at d.c. frequency $f=0$ in Figure
9, the armature position, $x(t)$, the armature velocity, $v(t)=\dot{x}(t)$,
augmentation circuit current, $I_{a}(t)$, and gun circuit current, $I_{g}(t)$,
are plotted as a function of time. In the current plots, red line is gun
current $I_{g}(t)$ and blue line is augmentation circuit current $I_{a}(t)$.
These quantities correspond to Figure 9, where the gun circuit switch is
closed at $t_{g0}=0$ and the augmentation circuit is closed at
$t_{a0}=3.0\times 10^{-3}$ s. For this case, $t_{f}=5.479\times 10^{-3}$ s.
Note that the plots are only valid for $0\leq t\leq t_{f}$.
## VIII Improvement Due to Augmentation
When the coupling coefficient is set to zero, $k=0$, the augmentation circuit
is decoupled from the gun circuit. The augmentation circuit is then simply an
$L-R$ circuit driven by a voltage source. The gun circuit has no interaction
with the augmentation circuit. The results of integrating the dynamical Eqs.
(15)–(17) is shown in Figure 11. The augmentation circuit has the standard
current oscillations of an a.c. driven $L-R$ circuit. The gun circuit has a
current that initially increases and then decreases as energy is transferred
to the armature. For this case when the circuits are decoupled, $k=0$, the
armature reaches the end of the rails at time $t_{f}=5.54$ ms and has velocity
$v(t)=\dot{x}(t)=3.994$ km/s and kinetic energy, $E_{k}=159.556$ MJ. For the
augmented EMG, the kinetic energy of the armature for a 3 ms delay was
$E_{k}=206.658$ MJ, see Section VII. Without augmentation, the kinetic energy
of the armature for was $E_{k}=159.556$ MJ. Therefore, the improvement in
kinetic energy for these parameters is 29.5%.
The efficiency, defined by Eq. (33) is $\eta=0.00388679$. Note that the energy
of the augmentation circuit is in the denominator in Eq. (33), thereby making
the efficiency of the gun circuit seem smaller for the case when the circuits
are decoupled. If we define the efficiency for a decoupled gun to be
$\eta^{\prime}=\frac{E_{k}}{E_{g}^{\text{in}}}$ (34)
which only includes energies of the gun circuit, then
$\eta^{\prime}=0.005316$, which is larger than for the augmented gun case, see
the discussion following Eq. (33).
Figure 11: When the coupling constant $k=0$, the circuits are decoupled. For
the frequency $f=200$ Hz, the armature position, $x(t)$, the armature
velocity, $v(t)=\dot{x}(t)$, augmentation circuit current, $I_{a}(t)$, and gun
circuit current, $I_{g}(t)$, are plotted as a function of time. In the current
plots, red line is gun current $I_{g}(t)$ and blue line is augmentation
circuit current $I_{a}(t)$. These quantities correspond to Figure 9, where the
gun circuit switch is closed at $t_{g0}=0$ and the augmentation circuit is
closed at $t_{a0}=3.0\times 10^{-3}$ s. For this case, $t_{f}=5.479\times
10^{-3}$ s. Note that the plots are only valid for $0\leq t\leq t_{f}$.
## IX Summary
We have considered a lumped circuit model of an augmented electromagnetic gun
having a single augmentation circuit driven by an a.c. generator. The
augmentation circuit is inductively coupled to the gun circuit, which is
driven by a d.c. voltage generator. Using example numerical parameters, we
have solved the three non-linear dynamical equations for the augmentation
circuit current, the gun circuit current, and the armature position and
velocity as a function of time. We have found that the armature kinetic energy
has oscillations in magnitude as a function of the driving frequency of the
voltage generator in the augmentation circuit. These oscillations constitute a
resonance in armature kinetic energy, which may be exploited to increase
armature energies. For some values of parameters, the augmentation circuit
only provides a small increase in armature kinetic energy over an EMG with no
augmentation, see Section VIII, and therefore one may, or may not, want to use
such an augmentation circuit in an EMG design. However, for some values of the
parameters in the augmented EMG, we find that the resonance leads to an
armature kinetic energy that is 5.7 times larger at the peak than at the
minimum of the resonance curve. If augmentation is used, the presence of the
kinetic energy resonance should be taken into account, otherwise parameters
may be chosen that result in less than optimal EMG kinetic energy and
efficiency.
We have demonstrated that a kinetic energy resonance exists in an EMG with a
single augmentation circuit, however, we suspect that there will exist similar
resonances in kinetic energy for other augmentation schemes. The detailed
physics of such resonances should be carefully explored in order to optimize
the armature kinetic energy and system efficiency.
## References
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* Knoepfel (2000) H. E. Knoepfel, _Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use_ (Wiley, New York, 2000).
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* McCorkle and Bahder (2008) W. C. McCorkle and T. B. Bahder, 27th Army Science Conference, Nov.-Dec. 2010, Orlando, Florida, USA (2008), URL http://arxiv.org/abs/0810.2985.
|
arxiv-papers
| 2011-08-17T19:57:18 |
2024-09-04T02:49:21.598470
|
{
"license": "Public Domain",
"authors": "Thomas B. Bahder and William C. McCorkle",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/1108.3560"
}
|
1108.3590
|
Predictability of evolutionary trajectories in fitness landscapes
Alexander E. Lobkovsky1, Yuri I. Wolf1, Eugene V. Koonin1,∗
1 National Center for Biotechnology Information, National Library of Medicine,
National Institutes of Health, Bethesda, MD 20894
$\ast$ E-mail: koonin@ncbi.nlm.nih.gov
## Abstract
Experimental studies on enzyme evolution show that only a small fraction of
all possible mutation trajectories are accessible to evolution. However, these
experiments deal with individual enzymes and explore a tiny part of the
fitness landscape. We report an exhaustive analysis of fitness landscapes
constructed with an off-lattice model of protein folding where fitness is
equated with robustness to misfolding. This model mimics the essential
features of the interactions between amino acids, is consistent with the key
paradigms of protein folding and reproduces the universal distribution of
evolutionary rates among orthologous proteins. We introduce mean path
divergence as a quantitative measure of the degree to which the starting and
ending points determine the path of evolution in fitness landscapes. Global
measures of landscape roughness are good predictors of path divergence in all
studied landscapes: the mean path divergence is greater in smooth landscapes
than in rough ones. The model-derived and experimental landscapes are
significantly smoother than random landscapes and resemble additive landscapes
perturbed with moderate amounts of noise; thus, these landscapes are
substantially robust to mutation. The model landscapes show a deficit of
suboptimal peaks even compared with noisy additive landscapes with similar
overall roughness. We suggest that smoothness and the substantial deficit of
peaks in the fitness landscapes of protein evolution are fundamental
consequences of the physics of protein folding.
## Author Summary
Is evolution deterministic, hence predictable, or stochastic, that is
unpredictable? What would happen if one could “replay the tape of evolution:”
will the outcomes of evolution be completely different or evolution is so
constrained that history will be repeated? Arguably, these questions are among
the most intriguing and most difficult in evolutionary biology. In other
words, the predictability of evolution depends on the fraction of the
trajectories on fitness landscapes that are accessible for evolutionary
exploration. Because direct experimental investigation of fitness landscapes
is technically challenging, the available studies only explore a minuscule
portion of the landscape for individual enzymes. We therefore sought to
investigate the topography of fitness landscapes within the framework of a
previously developed model of protein folding and evolution where fitness is
equated with robustness to misfolding. We show that model-derived and
experimental landscapes are significantly smoother than random landscapes and
resemble moderately perturbed additive landscapes; thus, these landscapes are
substantially robust to mutation. The model landscapes show a deficit of
suboptimal peaks even compared with noisy additive landscapes with similar
overall roughness. Thus, the smoothness and substantial deficit of peaks in
fitness landscapes of protein evolution could be fundamental consequences of
the physics of protein folding.
## Introduction
One of the most intriguing questions in evolutionary biology is: to what
extent evolution is deterministic and to what extent it is stochastic and
hence unpredictable? In other words, what happens if “the tape of evolution is
replayed:” are we going to see completely different outcomes or the
constraints are so strong that history will be repeated [1, 2, 3, 4]? If
evolution is envisaged as movement of a population across a fitness landscape,
the question can be reworded more specifically: among the numerous
trajectories connecting any two points on the landscape, what fraction is
accessible to evolution? Until recently, these remained purely theoretical
questions as experimental study of fitness landscapes in the actual sequence
space was impractical, due both to the technical difficulty of producing and
assaying numerous expressed sequence variants and to the more fundamental
problem of defining an adequate quantitative measure of fitness. However,
recent experimental studies of fitness landscapes could potentially shed light
on the problem of evolutionary path predictability.
The most thoroughly characterized feature of empirical fitness landscapes is
the structure near a peak. In experiments that examine the peak structure, a
high fitness sequence is typically subjected to either random mutations or an
exhaustive set of mutations at a small number of important sites. The
resulting library of mutants is then assayed to measure a proxy of fitness [5,
6, 7, 8, 9]. Significant sign epistasis (a situation in which the fitness
effect of a particular mutation can be either positive or negative depending
on the genetic context) near the peak has been typically observed whereas
local deviations from the additive model have been found to be uncorrelated
with the genetic context and derived from a nearly normal distribution [10,
11, 12, 13]. Because these studies characterize only a small region of the
landscape, they cannot be used to address the question of path predictability.
Another broad class of experiments probes the evolutionary trajectories from
low to high fitness. Usually, in such experiments, a random peptide is
subjected to repeated rounds of random mutagenesis and purifying selection [8,
14, 15, 16, 17]. During this process fitness grows with each generation and
eventually stagnates at a suboptimal plateau. The characteristics of the
fitness growth as well as the dependence of the plateau height on the library
size can be used to classify landscapes [18]. A quantitative comparison to the
$NK$ model of random epistatic landscapes ($N$ is the number of sites in an
evolving sequence and $K$ is the number of sites that affect the fitness
contribution of a particular site through epistatic interactions) can even
yield quantitative estimates of $N$ and $K$ [19, 20]. The directed evolution
studies explore the evolutionarily accessible portion of the landscape and
could in principle be used to shed light on the question of path
predictability. However, the inaccessible regions of the landscape remain
unexplored and the volume of data at this point is insufficient to obtain
quantitative conclusions regarding path predictability.
A different type of landscapes has been explored in various microarray
experiments where protein-DNA(RNA) binding affinity serves as the proxy for
fitness [21, 22]. These experiments produce vast, densely sampled landscapes.
A comparison with a sophisticated Landscape State Machine model of a
correlated fitness landscapes yields estimates of the model parameters [23,
24]. The DNA binding landscapes, in principle, contain the information
required for the analysis of path statistics, and could be a valuable resource
for advancing the understanding of evolutionary path predictability.
Empirical studies that exhaustively sample a region of the fitness landscape
allow one to actually assess the accessibility of the entire set of
theoretically possible evolutionary trajectories in a particular (small) area
of the fitness landscape. For example, all mutational paths between two states
of an enzyme, e.g., the transition from an antibiotic-sensitive to an
antibiotic resistant form of $\beta$-lactamase [25, 26, 27] or the transition
between different specificities of sesquiterpene synthase [28] have been
explored. The results of these experiments, which out of necessity explore
only short mutational paths of $<10$ amino acid replacements, suggest that
there is a substantial deterministic component to protein evolution: only a
small fraction of the possible paths are accessible for evolution [25, 29, 30,
31].
Theoretical analyses of available empirical fitness data reveal a tight link
between genetic and molecular interactions which are responsible for the
landscape ruggedness and ubiquitous sign epistasis [13, 32]. The emerging
quantitative analysis of fitness landscapes can shed light on some of the most
fundamental aspects of evolution but the interpretation of the currently
available experimental results requires utmost caution as only a minuscule
part of the sequence space can be explored, and that only for a few more or
less arbitrarily selected experimental systems.
Here we focus on the question of the predictability of mutational paths which
is intimately tied to the ruggedness/smoothness of the fitness landscapes. The
study of random landscapes of low dimensionality revealed an intuitively
plausible negative correlation between the roughness of a landscape and the
availability of pathways of monotonic fitness [33]. In the same study,
Carneiro and Hartl showed that experimentally characterized landscapes are
significantly smoother than their permuted counterparts and exhibit greater
peak accessibility [33].
To gain insights into the structure of the fitness landscapes of protein
evolution and in particular the accessibility of mutational paths we used a
previously developed simple model of protein folding and evolution [34]. The
key assumption of this model, which is based on the concept of misfolding-
driven evolution of proteins [35, 36, 37], is that the fitness of model
proteins is determined solely by the number of misfolded copies that are
produced before the required abundance of the correctly folded protein is
reached. We have previously shown that this model accurately reproduces the
shape of the universal distribution of the evolutionary rates among
orthologous protein-coding genes along with the dependencies of the
evolutionary rate on protein abundance and effective population size [34].
These results appear to suggest that our folding model (described in detail
the Methods section) is sufficiently rich to reproduce some of the salient
aspects of evolution. The model is also simple enough to allow exhaustive
exploration of the fitness landscapes, which prompted us to directly address
the problem of evolutionary path predictability.
We build on the efforts of Carneiro and Hartl [33] who examined the statistics
of evolutionary trajectories. Although counting monotonic fitness paths
reveals important features of the landscapes, we argue that reliable
retrodiction of the evolutionary past is possible (i.e., evolution is quasi-
deterministic) only when the available monotonic paths are similar to each
other in a quantifiable way. We therefore propose a measure of path divergence
to quantify the difference between the available monotonic paths. Our aims are
to investigate the structure of the fitness landscapes of protein evolution
and to elucidate the connection between the roughness of landscapes and the
predictability of mutational trajectories. We analyze three classes of fitness
landscapes: landscapes in which fitness is derived from the folding robustness
of model polymers; additive random landscapes perturbed by noise; and
experimental landscapes derived from the combinatorial mutation analysis of
drug resistance and enzymatic activity. We show that all three classes of
landscapes are markedly smoother than their randomly permuted counterparts and
all exhibit a similar qualitative connection between roughness and path
predictability. However, at the same level of path predictability, the folding
landscapes have substantially fewer fitness peaks. Given that the statistical
properties of the model landscapes can be directly traced to the constraints
imposed by the energetics and kinetics of a folding heterpolymer, we
hypothesize that the relative smoothness and the suppression of suboptimal
peaks in fitness landscapes of protein evolution are fundamental consequences
of protein folding physics.
## Results
### Quantitative characterization of fitness landscapes
Carneiro and Hartl compared small random landscapes to several empirical
fitness landscapes using deviation from additivity as a measure of roughness
[33]. They found that empirical landscapes were significantly smoother than
their random counterparts and that the degree of smoothness was correlated
with the number of monotonic paths to the main summit. Deviation from
additivity of a landscape is computed by fitting an additive model in which
the fitness of each sequence is different from the peak fitness by the sum of
contributions of the substitutions that differentiate it from the peak
sequence. The negative fitness contributions of the substitutions to the peak
fitness are adjusted to minimize the sum $S$ of squares of the differences
between the actual fitnesses in the landscape and the fitnesses predicted by
the additive model. Deviation from additivity is defined as $\sqrt{S/L}$,
where $L$ is the number of points in the landscape.
Because roughness of a multidimensional landscape with variable degree
connectivity is not an intuitive concept, we introduce three additional
quantitative measures to probe alternative facets of the concept of roughness.
First, local roughness is the root mean squared difference between the fitness
of a point and its neighbors, averaged over the entire landscape. As defined,
local roughness conflates the measures of roughness and “steepness.” For
example, a globally smooth landscape, in which fitness depends only on the
distance from the peak, will have a non-zero local roughness. However, because
there is a large number of directions that change the distance from the peak
by one, the local roughness of a globally smooth landscape will be vanishingly
small. In addition, our landscapes tend to be globally flat–so that the
average decrease in fitness due to a single mutation step away from the main
peak is much smaller than the local fitness variability–everywhere except a
small region around the main peak (see Fig. 7). Therefore, the landscape-
average local roughness in our case is a true measure of the local fitness
variability.
Second, the fraction of peaks is the number of points with no fitter neighbors
divided by the total number of points in the landscape. A strictly additive
landscape has a single peak [30] whereas the peak fraction in landscapes
derived from the folding model as well as the corresponding randomized
landscapes depends on the method of landscape construction, alphabet size and
sequence length.
Third, the roughness of a landscape can be assessed by identifying its tree
component. The tree component is the set of all nodes with no more than one
neighbor of higher fitness. Thus, the tree component includes peaks and
plateaus. Monotonic fitness paths along the tree component form a single or
several disjoint tree structures without loops. In the limit of high selection
pressure, a mutational trajectory that finds itself on the tree component has
a single path to the nearest peak or plateau, i.e. evolution on the tree
component is completely deterministic. We use the mean distance to the tree
component, i.e. the distance to the tree component averaged over the
landscape, as a measure of roughness. In a fully additive landscape, only the
peak sequence and its immediate neighbors belong to the tree component and
therefore the mean distance to the tree component is a measure of the diameter
of an additive landscape (which, for example, could be defined as the maximum
pairwise distance between points on the landscape). Kauffman and Levin have
shown that in a large class of correlated random landscapes, the mean distance
to the tree component grows only logarithmically with the number of points in
the landscape [19].
We utilize two quantitative measures of the predictability of evolutionary
trajectories. First is fraction of monotonic paths to the main peak $F_{m}$
which is computed by counting the number $n_{i}$ of simple (without reverse
substitutions or multiple substitutions at the same site) monotonic paths to
the main peak from each point $i$ on the landscape, dividing it by the total
number of simple paths $h_{i}!$ (where $h_{i}$ is the Hamming distance from
point $i$ to the peak), and averaging over the landscape via
$F_{m}=\frac{1}{L}\sum_{i}\frac{n_{i}}{h_{i}!},$ (1)
where $L$ is the number of points in the landscape and the sum excludes the
main peak. The monotonic path fraction $F_{m}$ measures the scarcity of
accessible evolutionary paths when selection is strong. When the monotonic
path fraction is small, evolution is more constrained.
Second, the mean path divergence, is a fine-grained measure of evolutionary
(un)predictability. We first define the divergence $d(p_{1},p_{2})$ of a pair
of paths $p_{1}$ and $p_{2}$, as the average of the shortest Hamming distances
from each point on one path to the other path. Suppose that we have a way of
generating stochastic evolutionary paths. The outcome of a large number of
evolutionary dynamics simulations is a collection of paths with their
associated probabilities of occurrence. In general, the probability of
occurrence of an evolutionary path is proportional to the product of fixation
probabilities of its constituent mutation steps. Given a bundle of paths with
the same starting and ending points, we define its mean path divergence to be
$\bar{d}=\sum_{p_{1}\neq p_{2}}d(p_{1},p_{2})\,O(p_{1})\,O(p_{2}),$ (2)
where $O(p)$ is the probability of occurrence of path $p$ in the ensemble. In
other words, if two paths were drawn from the bundle at random with
probabilities proportional to $O(p)$, their expected divergence would be
$\bar{d}$. Alternatively, if we were to fix one path to be the most likely
path in the bundle and to select the second path at random with probability
proportional to $O(p)$, the divergence would be proportional to $\bar{d}$ as
well.
In an additive landscape, the mutational trajectory is maximally ambiguous. As
every substitution that brings the sequence closer to the peak increases
fitness, substitutions can occur in any order and all shortest mutational
trajectories to the peak–without reverse substitutions or multiple
substitutions at the same site–are monotonic in fitness. In the strong
selection limit of our model defined below, all monotonic trajectories have
roughly the same probability of occurrence, so the mutational path cannot be
predicted.
The mean path divergence is a better measure of the predictability of
evolutionary trajectories than the number or fraction of accessible paths.
Even when only a small fraction of paths are monotonic in fitness, these paths
could potentially be quite different, perhaps randomly scattered over the
landscape. In such a case, prediction of the evolutionary trajectory would be
inaccurate despite the scarcity of accessible paths which will be reflected in
a high value of path divergence.
Equation (2) introduces the mean path divergence of a bundle of paths with the
same starting and ending points. The landscape-wide mean path divergence is
measured by constructing representative path bundles with all possible [start,
peak] pairs including suboptimal peaks as trajectory termination points. Path
divergence is averaged over all bundles with the starting and ending points
separated by the same Hamming distance. To construct the path bundles, we
employed a low mutation rate model in which the attempted substitutions are
either eliminated or fixed in the population before the next mutation attempt
occurs.
We invoke the misfolding-cost hypothesis to assign a fitness to a sequence
that folds with probability $P$ to a particular structure. To produce an
abundance $A$ of correctly folded copies, an average of $A(1-P)/P$ of
misfolded copies are produced. The “fitness” of a sequence should be a
monotonically decreasing function of the cost incurred by the misfolded
proteins. Previously we showed that qualitative conclusions drawn from the
average population dynamics on the fitness landscape did not depend on the
precise functional relationship between the number of misfolded copies and
fitness [34]. We use simply the negative of the number of misfolded copies and
assign a fitness $w=-A/P$, to a sequence whose probability of folding to the
reference structure is $P$. Because the exact population dynamics model is not
important, we use diploid population dynamics in the low mutation rate limit.
Therefore, the probability of fixation of a mutant $j$ in the background of
$i$ is given by
$\pi(i\rightarrow
j)=\frac{1-e^{-2(w_{j}-w_{i})}}{1-e^{-4N_{e}(w_{j}-w_{i})}},$ (3)
where $N_{e}$ is the effective population size [38] which in all simulations
was fixed at $N_{e}=10,000$. The required abundance $A$ is a measure of the
strength of selection. In the limit of large $A$, the probability of fixation
of a beneficial mutation is unity whereas neutral and deleterious mutations
are never fixed. In this limit, all uphill steps in the fitness landscape are
equally likely and all monotonic uphill trajectories have equal evolutionary
significance.
In the analysis that follows, we study the association between landscape
roughness and path predictability for the folding landscapes and their
randomized (also referred to as permuted or scrambled) versions. In the
scrambled landscapes, the topology (i.e. connectivity) of the landscape is
preserved but the fitness values are randomly shuffled. We also compare the
roughness and path predictability characteristics of the model and the
experimental landscapes for $\beta$-lactamase [25] and sesquiterpene synthase
[28] to those for noisy additive landscapes with a continuously tunable amount
of roughness.
### Evolutionary path predictability in fitness landscapes
#### Deviation from additivity, local roughness, peak fraction, and monotonic
paths
We first establish that the folding and the experimental landscapes are
significantly different from their randomly permuted counterparts. The
deviation from additivity of the folding landscapes is typically several
standard deviations below the mean of their scrambled counterparts. Although
the additivity hypothesis accounts for less than 40% of the fitness
variability (computed by comparing the sum of the squares of the fitnesses in
the landscape to the sum of the squares of the residuals of the additive
fitness model fit) in all but one of the folding landscapes, the deviation
from additivity of the permuted landscapes is substantially greater (Fig. 1A).
The experimental landscapes follow the same pattern, in agreement with the
earlier findings of Carneiro and Hartl [33]. Furthermore, both in the folding
and in the experimental landscapes, the fraction of monotonic paths to the
main peak is several standard deviations greater than in the respective
scrambled landscapes (Fig. 1B). An even more striking disparity exists between
the fraction of peaks in the folding landscapes and their permuted versions:
the folding landscapes contain at least an order of magnitude fewer peaks than
their scrambled counterparts; the experimental landscapes resemble the folding
landscapes more closely than their own randomized versions (Fig. 1C).
To further characterize the deviation of the folding and experimental
landscapes from their permuted counterparts, each landscape metric was
measured and the mean and standard deviation were computed among 100 randomly
permuted landscapes. We then compute the Z-score (deviation from the mean
measured in the units of the standard deviation) of the original non-permuted
landscape compared to the ensemble of the permuted landscapes. This Z-score
shows how much more correlated the original landscape is, as measured by the
chosen characteristic, compared to its scrambled counterparts (Figure 2).
Notably, despite the considerable scatter of the Z-score values for the
folding landscapes, they all showed extremely large difference (mean Z-score
greater than 20 standard deviations) from the scrambled landscapes for all
measures, with the sole exception of the monotonic path fraction (Figure 2).
The two experimental landscapes also significantly differed from the scrambled
landscapes albeit less so than the folding landscapes, again with the
exception of the monotonic path fraction in which case the two classes of
landscapes had similar Z-scores (Figure 2).
Aside from the significant correlation (Pearson $\rho=-0.68$) between peak
fraction and mean distance to the tree component, there was little or no
correlation between the four measures of landscape roughness (Fig. 3).
Roughness of landscapes of high and variable dimensionality is impossible to
capture by a single quantity. Therefore, the different measures seam to reveal
distinct aspects of landscape architecture. The strong negative correlation
between the peak fraction and mean distance to the tree component is due to
the fact that each peak spawns a distinct subset of the tree component. The
higher the density of peaks on the landscape, the larger fraction of the
landscape that is covered by the tree component. Therefore the average
distance to the tree component declines with the increasing density of peaks.
#### Path divergence
Starting from a random non-peak sequence in the landscape, we introduced
random mutations and accepted or rejected them according to equation (3) until
the trajectory arrived at a fitness peak. This procedure was repeated a large
number of times, and path bundles were constructed for all pairs of starting
and ending sequences. Then the mean path divergence was computed for each path
bundle using equation (2) and averaged over all bundles for which starting and
ending points were separated by the same Hamming distance. When selection is
weak, all mutations which do not result in a sequence with zero folding
probability are accepted. Thus, evolution is a random walk on the landscape
and the statistical properties of evolutionary trajectories are fully
determined by the topology of the landscape (i.e. the connectivity of each
node). Conversely, in the strong selection limit, only mutations that increase
fitness are fixed. The mean path divergence varies smoothly between the two
limits (Fig. 4) and saturates at high selection pressure. In our analysis, we
focus on the strong selection limit plateau. In the weak selection limit, the
diversity of trajectories stems solely from the number of neighbors of each
point; by contrast, in the strong selection limit, the statistics of the
monotonic trajectories depend on the roughness of the landscape. Thus, the
weak selection limit probes only the topology of the landscape whereas the
strong selection limit also exposes its topography which appears to be
critical for assessing predictability of evolution.
#### Predictors and correlates of path divergence and monotonic path fraction
All four measures of landscape roughness can serve as predictors of path
divergence and monotonic path fraction to some degree (Fig. 5), in agreement
with the notion that each of these measures reflects salient properties of
fitness landscapes. The properties of the folding and empirical landscapes are
consistent with those of additive landscapes that were perturbed by a moderate
amount of noise (see Methods for details). A striking exception is the dearth
of peaks and monotonic paths in folding landscapes all other characteristics
being similar. Deviation from additivity and fraction of peaks are negatively
correlated with path divergence. This relationship captures the intuitive
notion that in rough landscapes there are fewer accessible evolutionary paths
than in smooth landscapes, and furthermore, in rough landscapes, even those
paths that are accessible show the tendency to aggregate within small areas on
the landscape. Indeed, in both the folding model-derived landscapes and the
experimental landscapes, the mean path divergence for all Hamming distances
between the starting and ending points was dramatically greater than in
scrambled landscapes (Fig. 6). Interpreting these findings in terms closer to
biology, the fitness landscapes derived from the model as well as experimental
landscapes show greater robustness to mutations than random landscapes: a
random mutation in a model-derived or experimental fitness landscape is more
likely than expected for random landscapes to have no deleterious effect,
leading to another monotonic path to the peak. Consequently, evolution on the
model-derived and experimental landscapes is less predictable (deterministic)
than it would be on uncorrelated random landscapes.
In contrast to deviation from additivity, the mean distance to the tree
component is positively correlated with path divergence. When the tree
component comprises a large fraction of the landscape, the mean distance to
the nearest tree branch is small. Consequently, the path divergence is reduced
as the paths that reach the tree component do not deviate from each other from
that point onward. By the same token, when the tree component is large, there
are fewer monotonic paths.
The origin of the positive correlation between the local roughness and path
divergence (Fig. 5) is less obvious. Paradoxically, greater noise results in
lower mean local roughness of noisy additive landscapes. The lowering of the
overall mean fitness with noise and, more importantly, the flattening of the
mean fitness dependence on the distance from the peak (Fig. 7) appear to
provide an explanation for this counter-intuitive result. Indeed we found that
in noisy additive landscapes there is a characteristic fitness value of
approximately 0.2 above which roughness increases with increasing noise and
below which roughness declines with increasing noise. Given that roughly 75%
of the points on the landscape have fitnesses below 0.2, the landscape-
averaged local roughness declines with increasing noise amplitude.
## Discussion
Here we examined the fraction of monotonic paths and introduced mean path
divergence as quantitative measures of the degree to which the starting and
ending points determine the path of evolution on fitness landscapes. The lower
the mean path divergence value, the more deterministic (and predictable)
evolution is. Global measures of landscape roughness correlate with path
divergence in the three analyzed classes of fitness landscapes: additive
landscapes perturbed by noise, landscapes derived from our protein folding
model and two small empirical landscapes. The folding landscapes are
substantially smoother than their permuted counterparts. As a result, although
in all analyzed landscapes only a small fraction of the theoretically possible
evolutionary trajectories is accessible, this fraction is much greater in the
folding and experimental landscapes than it is in randomized landscapes. In
addition, the mean path divergence in the randomized landscapes is
significantly smaller than in the original landscapes. Thus, the model and
empirical landscapes possess similar global architectures with many more
diverged monotonic paths to the high peaks than uncorrelated landscapes with
the same distribution of fitness values. Consequently, evolution in fitness
landscapes is substantially more robust to random mutations and less
deterministic (less predictable) than expected by chance. These findings are
compatible with the concept that might appear counter-intuitive but is
buttressed by results of population genetic modeling, namely, that robustness
of evolving biological systems promotes their evolvability [39, 40, 41].
Additionally, the folding landscapes exhibit a substantial deficit of peaks
compared to perturbed additive landscapes and experimental landscapes, a
property that translates into a substantially greater fraction of paths
leading to the main peak.
When it comes to the interpretation of the properties of fitness landscapes
described here, an inevitable and important question is whether the folding
model employed here is sufficiently complex and realistic to yield
biologically relevant information. In selecting the complexity of our folding
model, we attempted to construct the simplest model which exhibits 1) a rich
spectrum of low energy conformations across the sequence space, and 2) a non-
trivial distribution of substitutions effects on the low energy conformations.
An important choice is whether the location of monomers is confined to a
lattice or can be varied continuously. When the configuration space is
continuous, the distribution of energy barriers between energetically optimal
conformations can extend to zero. Therefore, the subtlety of distinctions
between conformations can lead to a richer structure of the fitness landscape.
We chose not increase the complexity of the model further and treated monomers
as point-like particles in a chain where the distance between nearest
neighbors is fixed but the angle between successive links in the chain in
unrestricted. Our level of abstraction is therefore somewhere between lattice
models and all-atom descriptions of proteins [42, 43, 44, 45, 46, 47, 48, 49,
50, 51].
Another important choice is the number of the model monomer types. Again, we
opted for an intermediate level of abstraction and chose four types of
monomers: hydrophobic, hydrophilic, and charged. This choice drastically
reduces the size of the sequence space while retaining some of the
substitution complexity whereby hydrophilic and charged monomers can be
swapped under some conditions without radically altering the native state. The
intermediate level of abstraction in our approach has its pros and cons.
Although the model reproduces key features of protein folding such as the
existence of the hydrophobic folding nucleus and two-stage folding kinetics
[52, 53], compact conformations certainly do not represent proteins. Rather,
we might think of our monomers as representing structurally grouped regions
several (perhaps up to a dozen) amino-acids in length. Compact conformations
in the model might therefore be analogous to tertiary structures of proteins.
Representing sequence space with only four monomer types and treating
mutations without reference to the underlying DNA or genetic code does not
accurately reflect the natural mutation process. However, our goal was to
isolate the features of fitness landscapes which could be traced directly to
the constraints imposed by the heteropolymer folding kinetics and energetics.
We therefore used a simple sequence space and a homogeneous mutation model to
avoid compounding the fitness landscape structure by the complexity derived
from the mutation process.
Most importantly, our folding model has been shown to reproduce the observed
universal distribution of the evolutionary rates of protein-coding genes as
well as the dependencies of the evolutionary rate on protein abundance and
effective population sizes [34]. Therefore, despite its simplicity, the
behavior of this model might reflect important aspects of protein evolution.
In particular, the conclusions drawn from the analysis of the model landscapes
exhaustively explored here could also apply to the fitness landscapes of
protein evolution. In the previous work, we concluded that the universal
distribution of evolutionary rates and other features of protein evolution
follow from the fundamental physics of protein folding [34]. The results
presented here suggest that the (relative) smoothness and a substantial
deficit of peaks in the fitness landscapes of protein evolution that lead to
mutational robustness and the ensuing evolvability could similarly follow from
the fact that proteins are heteropolymers that have to fold in three
dimensions to perform their functions.
The experimental landscapes considered here are decidedly incomplete. Due to
experimental limitations, only the analysis of binary substitutions at a
handful of sites is feasible at this time. The incompleteness of the empirical
landscapes analyzed in this work could be the cause of the observed lack of
peak suppression. This proposition will be put to test by the study of larger
parts of experimental landscapes that are becoming increasingly available.
## Materials and Methods
### Folding model
The goal of this study is to explore the relationship between roughness and
path divergence in realistic fitness landscapes. Our polymer folding model
provides a simple way of constructing such landscapes. The model has been
described in detail previously [34].
In brief, the model polymer is a flexible chain of monomers in which the the
nearest neighbors interact via a stiff harmonic spring potential with rest
length $a=1$. The angles between the successive links in the chain are
unrestricted. There are four types of monomers: hydrophobic H, hydrophilic P,
and charged + and –. Next nearest neighbors $i$ and $j$ in the chain and
beyond interact via a pairwise potential
$U_{ij}(r_{ij})=\frac{A_{ij}}{r_{ij}^{12}}-\frac{C_{ij}}{r_{ij}^{6}}+\frac{q_{i}q_{j}e^{-D\,r_{ij}}}{r_{ij}},$
(4)
where $r_{ij}$ is the distance between monomers $i$ and $j$, $q_{i}$ is the
monomer’s charge, $D$ is the Debye-Hückel screening length, and $A_{ij}$ and
$C_{ij}$ depend on the pair in question. The interaction parameters are chosen
to mimic the essential features of the amino-acid interactions. To emulate the
effects of solvent, we assign a stronger attraction to the HH pair than to the
PP, ++, and – – pairs. There is also a long range repulsion between H and P
and even stronger repulsion between H and the charged monomers. The values of
the parameters are $q_{\pm}=\pm 2$, Debye-Hückel screening length $D=3$. The
Lennard-Jones coefficients $A_{ij}$ and $C_{ij}$ are
$\displaystyle A_{HH}=4,\ A_{HP}=A_{H+}=2,\ A_{PP}=A_{P+}=A_{++}=1,$
$\displaystyle C_{HH}=8,\ C_{HP}=-1,\ C_{H+}=-3,\ C_{PP}=C_{P+}=C_{++}=2.$ (5)
Note that a $+$ can be substituted by a $-$ in the subscripts and the
coefficients are symmetric with respect to the interchange of the indices.
The energy of the chain is
$E=\sum_{|i-j|>1}U_{ij}+\frac{b\,T}{2}\sum_{i=1}^{N-1}(r_{i,i+1}-a)^{2},$ (6)
where the first term is the sum of the pairwise energies given by Eq. (4) over
non-nearest neighbor pairs, and the second term reflects the springs
connecting nearest neighbors. The spring constant is proportional to
temperature $T$. The parameters are fixed for all simulation runs at $b=300$,
and the quench temperature $T=1$. To mimic the observed tendency of the $N$
and $C$ termini to be in close proximity, we fixed the endpoint monomers of
the model sequences to be of $+$ and $-$ types.
Dynamics of folding are simulated via overdamped Brownian kinetics which are
appropriate when inertial and hydrodynamic effects are not important. Units
are chosen so that each component $\alpha$ of the $i$’th monomer’s coordinates
$x_{\alpha i}$ is updated according to
$x_{\alpha i}(t+\Delta t)=x_{\alpha i}(t)-\frac{\Delta t}{T}\,\frac{\partial
E}{\partial x_{\alpha i}}(t)+W_{\alpha i}(t),$ (7)
where $\Delta t$ is the time step and $W_{\alpha i}(t)$ is a random variable
with zero mean, variance $2\Delta t$, uncorrelated with $W$ for other times,
monomers and spatial directions.
### Native structure ensemble and correct folding probability
The “native structure” of a particular sequence is represented by an
equilibrium ensemble of conformations. The ensemble is constructed by
identifying the typical folded conformation and measuring the characteristic
RMSD $D$ due to thermal fluctuations in the folded state. Three thousand
quenches are then performed and the resulting folded conformations are
accumulated. The equilibrium ensemble that represents the native structure is
defined as the largest cluster of quenched conformations within RMSD distance
$D$ from each other. Thus, each conformation in the ensemble differs from any
other by an amount comparable to the differences introduced by thermal
fluctuations alone.
The concept of the native structure ensemble allows us to compute the
probability that a sequence folds to a particular structure in a natural,
physically plausible fashion. Given a native structure ensemble we assess its
conformation space density by computing the distance $d_{i}$ between each
member $i$ of the ensemble and its closest neighbor. Given the set
$\\{d_{i}\\}$ of these shortest distances we compute the median $Q$ and the
median absolute deviation (MAD) $V$. A new conformation is deemed to belong to
the ensemble if the shortest distance from this conformation to the members of
the ensemble is smaller than $R=Q+3V$.
Given a native structure ensemble of some sequence $s_{1}$ we compute the
probability $P$ that sequence $s_{2}$ (which could be $s_{1}$ itself) folds to
the this structure by accumulating $M=100$ equilibrated quenched conformations
of $s_{2}$ and using the above criterion to determine the fraction $P$ that
belong to the native structure ensemble of $s_{1}$. Because $M=100$ sample
conformations are computed, the smallest measurable $P$ is $1/M=0.01$. The
sample size used to measure $P,$ dictated by the computational demands of the
model, introduces a random component to the model fitness landscapes. As we
report below, model landscapes turn out to be substantially smoother than
random. Therefore the underlying global structure of the model landscapes
appears to survive the modest amount of randomness introduced by the
relatively small sample size used for measuring $P$.
### Search for compact robust folders
Robust folders (sequences with a high probability of correct folding) tend to
have large linear regions stretched by repulsive Coulomb interactions. Because
the linear regions have no contacts with other monomers, we focused our
attention on compact conformations with a high monomer contact density.
Substitutions in these higher complexity conformations were more likely to
exhibit non-trivial effects. To find compact robust folders in the vast
available sequence space of $23$-mers (the sequences are of length $N=25$ but
the endpoint monomer types are fixed) with $4$ monomer types, we implemented a
simulated annealing search which optimized the correct folding probability $P$
divided by the cube of the native conformation’s radius of gyration. The
search produced over 800 sequences with $P>0.5$ and at least two distinct
regions of the polymer in mutual contact.
### Assembly of the folding fitness landscapes
We examined each single substitution mutant of a robustly folding sequence and
computed the folding probability $P$ to the structure of the original
sequence. All mutants with $P>0$ were added to the landscape and if $P\geq
0.1$ their mutants were also examined. This process is repeated until all
mutants of the last sequence under consideration have $P<0.1$.
From our study of complete landscapes we estimate that on average for each
sequence with $P>0$ which is included into the landscape, roughly 6 others
with $P=0$ need to be examined. Since each quench and equilibration takes
about 2–4 seconds, landscape construction takes roughly 30 minutes to an hour
per included sequence. Thus landscapes larger than 10,000 sequences take
months to compile.
At the time of submission, 39 complete landscapes have been constructed, the
largest comprising 12969 sequences.
### Additive landscapes perturbed by noise
The organization of the folding fitness landscapes and experimental landscapes
were compared with perfectly additive landscapes perturbed by noise
constructed as follows. Each substitution to the peak fitness sequence was
assigned a negative fitness differential drawn at random from an exponential
distribution with parameter $\lambda=3$. The sum over the fitness
differentials of a particular set of substitution was modified by either
additive of multiplicative noise [54]. Additive noise is drawn from a Gaussian
distribution with zero mean and standard deviation $\nu$ which was varied
between $0.05$ and $0.5$. The multiplicative perturbation is achieved by
multiplying the fitness by a number drawn from a uniform distribution $[0,1)$
raised to a positive power $\mu$ varied between $0.1$ and $10.$ When $\mu$ is
small, multiplicative factors are close to unity and the perturbation is small
as well. If the perturbed fitness was positive, the mutant was included into
the landscape. The noise amplitude was varied to obtain a family of landscapes
of continuously varying roughness. Only the data for the additive landscapes
with multiplicative noise were included in this manuscript. Landscapes
perturbed by other types of noise exhibited essentially the same qualitative
behavior.
### Experimental landscapes
The studies on experimental fitness landscapes typically involve constructing
a library of all possible combinations of binary mutations at a small number
of sites. The first study included in the present analysis measured the
minimum inhibitory concentrations (MIC) of an antibiotic for a complete
spectrum of mutants with modified TEM $\beta$-lactamases; the transition from
the antibiotic-sensitive to the antibiotic-resistant form requires five
mutation, so the landscape encompassed 120 mutational trajectories between the
most distant points on the landscape (or 32 sequences) [25]. The logarithm of
MIC was used as the proxy for fitness. In the second study, catalytic activity
of 419 sesquiterpene synthase mutants that differed by at most 9 substitutions
was measured [28]. We used the catalytic specificity (propensity for producing
a particular reaction product rather than a broad spectrum of products) of the
mutant enzymes as the proxy for fitness. Before performing the analysis, the
fitnesses in the experimental landscapes are mapped onto the $[0.01,1)$
interval to enable meaningful quantitative comparisons of the roughness
measures.
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## Figure Legends
Figure 1: Roughness, monotonic paths and suboptimal peak suppression in
folding and experimental landscapes. (A) Deviation from additivity for the
folding landscapes (larger symbols), their scrambled versions (smaller
symbols) and the two experimental landscapes. Error bars show one standard
deviation within the ensemble of permuted landscapes. (B) Fraction of
monotonic paths to the main peak in folding, scrambled and experimental
landscapes. (C) The number of peaks is vastly greater in scrambled landscapes
than in folding or experimental landscapes (with the exception of the
sesquiterpene synthase landscape). Figure 2: The Z-scores of different
characteristics of the original folding and experimental landscapes measured
with respect to the ensembles of their randomly permuted counterparts. Figure
3: Correlations between different quantitative characteristics of the folding
landscapes. Each panel quotes the Spearman rank correlation coefficient
between the particular pair of characteristics. Figure 4: Mean path divergence
as a function of selection pressure for a folding landscape with 5936 nodes
and 65 peaks. Solid lines are labeled by the Hamming distance between the
pairs of starting and ending points of the trajectory bundles over which the
path divergence is averaged. Figure 5: The dependence of the path divergence
(top row) and the monotonic path fraction (bottom row) on the measures of
landscape roughness. The dots of different color correspond to noisy additive
landscapes with differing amounts of multiplicative noise: low (red), two
intermediate levels (green smaller than blue), and high (magenta). Yellow
circles represent the folding landscapes, the cyan squares–the
$\beta$-lactamase landscape, and the red triangles–the sesquiterpene synthase
landscape. Figure 6: Mean path divergence in folding and experimental
landscapes (larger symbols) landscapes, as well as their scrambled versions
(smaller symbols) as a function of Hamming distance from the main peak. Figure
7: Fitness averaged over all points at a particular distance $H$ from the peak
for folding landscapes, additive landscapes with the same three levels of
multiplicative noise used in Fig. 5 and the sesquiterpene synthase landscape.
|
arxiv-papers
| 2011-08-17T22:23:03 |
2024-09-04T02:49:21.606358
|
{
"license": "Public Domain",
"authors": "Alexander E. Lobkovsky, Yuri I. Wolf, Eugene V. Koonin",
"submitter": "Eugene Koonin",
"url": "https://arxiv.org/abs/1108.3590"
}
|
1108.3614
|
11institutetext: Research School of Computer Science, CECS, ANU
nmphuong@cecs.anu.edu.au
peter.sunehag,marcus.hutter@{anu.edu.au}
# Feature Reinforcement Learning in Practice
Phuong Nguyen Peter Sunehag Marcus Hutter
###### Abstract
Following a recent surge in using history-based methods for resolving
perceptual aliasing in reinforcement learning, we introduce an algorithm based
on the feature reinforcement learning framework called $\Phi$MDP [14]. To
create a practical algorithm we devise a stochastic search procedure for a
class of context trees based on parallel tempering and a specialized proposal
distribution. We provide the first empirical evaluation for $\Phi$MDP. Our
proposed algorithm achieves superior performance to the classical U-tree
algorithm [21] and the recent active-LZ algorithm [6], and is competitive with
MC-AIXI-CTW [28] that maintains a bayesian mixture over all context trees up
to a chosen depth. We are encouraged by our ability to compete with this
sophisticated method using an algorithm that simply picks one single model,
and uses Q-learning on the corresponding MDP. Our $\Phi$MDP algorithm is much
simpler, yet consumes less time and memory. These results show promise for our
future work on attacking more complex and larger problems.
## 1 Introduction
Reinforcement Learning (RL) [27] aims to learn how to succeed in a task
through trial and error. This active research area is well developed for
environments that are Markov Decision Processes (MDPs); however, real world
environments are often partially observable and non-Markovian. The recently
introduced Feature Markov Decision Process ($\Phi$MDP) framework [14] attempts
to reduce actual RL tasks to MDPs for the purpose of attacking the general RL
problem where the environment’s model as well as the set of states are
unknown. In [26], Sunehag and Hutter take a step further in the theoretical
investigation of Feature Reinforcement Learning by proving consistency
results. In this article, we develop an actual Feature Reinforcement Learning
algorithm and empirically analyze its performance in a number of environments.
One of the most useful classes of maps ($\Phi$s) that can be used to summarize
histories as states of an MDP, is the class of context trees. Our stochastic
search procedure, the principal component of our $\Phi$MDP algorithm
GS$\Phi$A, works on a subset of all context trees, called Markov trees. Markov
trees have previously been studied in [22] but under names like FSMX sources
or FSM closed tree sources. The stochastic search procedure employed for our
empirical investigation utilizes a parallel tempering methodology [7], [12]
together with a specialized proposal distribution. In the experimental
section, the performance of the $\Phi$MDP algorithm where stochastic search is
conducted over the space of context-tree maps is shown and compared with three
other related context tree-based methods.
Our $\Phi$MDP algorithm is briefly summarized as follows. First, perform a
certain number of random actions, then use this history to find a high-quality
map by minimizing a cost function that evaluates the quality of each map. The
quality here refers to the ability to predict rewards using the created
states. We perform a search procedure for uncovering high-quality maps
followed by executing $Q$-learning on the MDP whose states are induced by the
detected optimal map. The current history is then updated with the additional
experiences obtained from the interactions with the environment through
Q-Learning. After that, we may repeat the procedure but without the random
actions. The repetition refines the current “optimal” map, as longer histories
provide more useful information for map evaluation. The ultimate optimal
policy of the algorithm is retrieved from the action values Q on the resulting
MDP induced from the final optimal map.
Contributions. Our contributions are: extending the original $\Phi$MDP cost
function presented in [14] to allow for more discriminative learning and more
efficient minimization (through stochastic search) of the cost; identifying
the Markov action-observation context trees as an important class of feature
maps for $\Phi$MDP; proposing the GS$\Phi$A algorithm where several chosen
learning and search procedures are logically combined; providing the first
empirical analysis of the $\Phi$MDP model; and designing a specialized
proposal distribution for stochastic search over the space of Markov trees,
which is of critical importance for finding the best possible $\Phi$MDP agent.
Related Work. Our algorithm is a history-based method. This means that we are
utilizing memory that in principle can be long, but in most of this article
and in the related works is near term. Given a history $h_{t}$ of
observations, actions and rewards we define states $s_{t}=\Phi(h_{t})$ based
on some map $\Phi$. The main class of maps that we will consider are based on
context trees. The classical algorithm of this sort is U-tree [21], which uses
a local criterion based on a statistical test for splitting nodes in a context
tree; while $\Phi$MDP employs a global cost function. Because of this
advantage, $\Phi$MDP can potentially be used in conjunction with any
optimization methods to find the optimal model.
There has been a recent surge of interest in history based methods with the
introduction of the active-LZ algorithm [6], which generalizes the widely used
Lempel-Ziv compression scheme to the reinforcement learning setting and
assumes $n$-Markov models of environments; and MC-AIXI-CTW [28], which uses a
Bayesian mixture of context trees and incorporates both the Context Tree
Weighting algorithm [31] as well as UCT Monte Carlo planning [16]. These can
all be viewed as attempts at resolving perceptual aliasing problems with the
help of short-term memory. This has turned out to be a more tractable approach
than Baum-Welch methods for learning a Partially Observable Markov Decision
Process (POMDP) [4] or Predictive State Representations [24]. The history
based methods attempt to directly learn the environment states, thereby
avoiding the POMDP-learning problem [15], [20] which is extremely hard to
solve. Model minimization [8] is a line of works that also seek for a minimal
representation of the state space, but focus on solving Markovian problems
while $\Phi$MDP and other aforementioned history-based methods target non-
Markovian ones. It is also worthy to note that there are various other
attempts to find compact representations of MDP state spaces [18]; most of
which, unlike our approach, address the planning problem where the MDP model
is given
Paper Organization. The paper is organized as follows. Section 2 introduces
preliminaries on Reinforcement Learning, Markov Decision Processes, Stochastic
Search methods and Context Trees. These are the components from which the
$\Phi$MDP algorithm (GS$\Phi$A) is built. In Section 3 we put all of the
components into our $\Phi$MDP algorithm and also describe our specialized
search proposal distribution in detail. Section 4 presents experimental
results on four domains. Finally Section 5 summarizes the main results of this
paper, and briefly suggests possible research directions.
## 2 Preliminaries
### 2.1 Markov Decision Processes (MDP)
An environment is a process which at any discrete time $t$, given action
$a_{t}\in\mathcal{A}$ produces an observation $o_{t}\in\mathcal{O}$ and a
corresponding reward $r_{t}\in\mathbb{R}$. When the process is a Markov
Decision Process [27]; $o_{t}$ represents the environment state, and hence is
denoted by $s_{t}$ instead. Formally, a finite MDP is denoted by a quadruple
$\langle\mathcal{S},\mathcal{A},\mathcal{T},\mathcal{R}\rangle$ in which
$\mathcal{S}$ is a finite set of states; $\mathcal{A}$ is a finite set of
actions; $T=(T_{ss^{\prime}}^{a}:s,s^{\prime}\in\mathcal{S},\
a\in\mathcal{A})$ is a collection of transition probabilities of the next
state $s_{t+1}=s^{\prime}$ given the current state $s_{t}=s$ and action
$a_{t}=a$; and $R=(R_{ss^{\prime}}^{a}:s,s^{\prime}\in\mathcal{S},\
a\in\mathcal{A})$ is a reward function
$R_{ss^{\prime}}^{a}=\textbf{E}[r_{t+1}|s_{t}=s,a_{t}=a,s_{t+1}=s^{\prime}]$.
The return at time step $t$ is the total discounted reward
$R_{t}=r_{t+1}+\gamma{}r_{t+2}+\gamma^{2}r_{t+3}+\ldots$, where $\gamma$ is
the geometric discount factor ($0\leq\gamma<1$).
Similarly, the action value in state $s$ following policy $\pi$ is defined as
$Q^{\pi}(s,a)=\textbf{E}_{\pi}[R_{t}|s_{t}=s,a_{t}=a]=\textbf{E}_{\pi}[\sum^{\infty}_{k=0}\gamma^{k}{}r_{t+k+1}|s_{t}=s,a_{t}=a]$.
For a known MDP, a useful way to find an estimate of the optimal action values
$Q^{*}$ is to employ the Action-Value Iteration (AVI) algorithm, which is
based on the optimal action-value Bellman equation [27], and iterates the
update
$Q(s,a)\leftarrow\sum_{s^{\prime}}T_{ss^{\prime}}^{a}[R_{ss^{\prime}}^{a}+\gamma\max_{a^{\prime}}Q(s^{\prime},a^{\prime})].$
If the MDP model is unknown, an effective estimation technique is provided by
$Q$-learning, which incrementally updates estimates $Q_{t}$ through the
equation
$Q(s_{t},a_{t})\leftarrow Q(s_{t},a_{t})+\alpha_{t}(s_{t},a_{t})err_{t}$
where the feedback error
$err_{t}=r_{t+1}+\gamma\max_{a}Q(s_{t+1},a)-Q(s_{t},a_{t})$, and
$\alpha_{t}(s_{t},a_{t})$ is the learning rate at time $t$. Under the
assumption of sufficient visits of all state-action pairs, Q-Learning
converges if and only if some conditions of the learning rates are met [2],
[27]. In practice a small constant value of the learning rates
($\alpha(s_{t},a_{t})=\eta$) is, however, often adequate to get a good
estimate of $Q^{*}$. Q-Learning is off-policy; it directly approximates
$Q^{*}$ regardless of what actions are actually taken. This approach is
particularly beneficial when handling the exploration-exploitation tradeoff in
RL.
It is well known that learning by taking greedy actions retrieved from the
current estimate $\widehat{Q}$ of $Q^{*}$ to explore the state-action space
generally leads to suboptimal behavior. The simplest remedy for this
inefficiency is to employ the $\epsilon$-greedy scheme, where with probability
$\epsilon>0$ we take a random action, and with probability $1-\epsilon$ the
greedy action is selected. This method is simple, but has shown to fail to
properly resolve the exploration-exploitation tradeoff. A more systematic
strategy for exploring the unseen scenarios, instead of just taking random
actions, is to use optimistic initial values [27], [3]. To apply this idea to
$Q$-Learning, we simply initialize $Q(s,a)$ with large values. Suppose
$R_{\max}$ is the maximal reward, $Q$ initializations of at least
$\frac{R_{\max}}{1-\gamma}$ are optimistic as
$Q(s,a)\leq\frac{R_{\max}}{1-\gamma}$.
### 2.2 Feature Reinforcement Learning
Problem description. An RL agent aims to find the optimal policy $\pi$ for
taking action $a_{t}$ given the history of past observations, rewards and
actions $h_{t}=o_{1}r_{1}a_{1}\ldots{}o_{t-1}r_{t-1}a_{t-1}o_{t}r_{t}$ in
order to maximize the long-term reward signal. If the problem satisfies an
MDP; as can be seen above, efficient solutions are available. We aim to attack
the most challenging RL problem where the environment’s states and model are
both unknown. In [13], this problem is named the Universal Artificial
Intelligence (AI) problem since almost all AI problems can be reduced to it.
$\Phi$MDP framework. In [14], Hutter proposes a history-based method, a
general statistical and information theoretic framework called $\Phi$MDP. This
approach offers a critical preliminary reduction step to facilitate the
agent’s ultimate search for the optimal policy. The general $\Phi$MDP
framework endeavors to extract relevant features for reward prediction from
the past history $h_{t}$ by using a feature map $\Phi$:
$\mathcal{H}\rightarrow\mathcal{S}$, where $\mathcal{H}$ is the set of all
finite histories. More specifically, we want the states $s_{t}=\Phi(h_{t})$
and the resulting tuple $\langle{}\mathcal{S},\ \mathcal{A},\
\mathcal{R}\rangle$ to satisfy the Markov property of an MDP. As
aforementioned, one of the most useful classes of $\Phi$s is the class of
context trees, where each tree maps a history to a single state represented by
the tree itself. A more general class of $\Phi$ is Probabilistic-Deterministic
Finite Automata (PDFA) [29], which map histories to the MDP states where the
next state can be determined from the current state and the next observation.
The primary purpose of $\Phi$MDP is to find a map $\Phi$ so that rewards of
the MDP induced from the map can be predicted well. This enables us to use MDP
solvers, like AVI and Q-learning, on the induced MDP to find a good policy.
The reduction quality of each $\Phi$ is dictated by the capability of
predicting rewards of the resulting MDP induced from that $\Phi$. A suitable
cost function that measures the utility of $\Phi$s for this purpose is
essential, and the optimal $\Phi$ is the one that minimizes this cost
function.
Cost function. The cost used in this paper is an extended version of the
original cost introduced in [14]. We define a cost that measures the reward
predictability of each $\Phi$, or more specifically of the resulting MDP
induced from that $\Phi$. Based on this, our cost includes the description
length of rewards; however, rewards depend on states as well, so the
description length of states must be also added to the cost. In other words,
the cost comprises coding of the rewards and resulting states, and is defined
as follows:
$\textbf{Cost}_{\alpha}(\Phi|h_{n}):=\alpha\textbf{CL}(s_{1:n}|a_{1:n})+(1-\alpha)\textbf{CL}(r_{1:n}|s_{1:n},a_{1:n})$
where $s_{1:n}=s_{1},...,s_{n}$ and $a_{1:n}=a_{1},...,a_{n}$ and
$s_{t}=\Phi(h_{t})$ and $h_{t}=ora_{1:t-1}r_{t}$ and $0\leq\alpha\leq 1$. For
coding we use the two-part code [30], [10], hence the code length (CL) is
$\textbf{CL}(x)=\textbf{CL}(x|\theta)+\textbf{CL}(\theta)$ where $x$ denotes
the data sampled from the model specified by parameters $\theta$. We employ
the optimal codes [5] for describing data
$\textbf{CL}(x|\theta)=\log(1/Pr_{\theta}(x))$, while parameters are uniformly
encoded to precision $1/\sqrt{\ell(x)}$ where $\ell(x)$ is the sequence length
of $x$ [10]: $\textbf{CL}(\theta)=\frac{m-1}{2}\log{\ell(x)}$, here $m$ is the
number of parameters. The optimal $\Phi$ is found via the optimization problem
$\Phi^{optimal}=\operatornamewithlimits{argmin}_{\Phi}\textbf{Cost}_{\alpha}(\Phi|h_{n})$.
Denote $\textbf{n}_{\bullet}:=[n_{1}\ n_{2}\ldots\ n_{l}]$ ($l$ is determined
in specific context); $n_{+}:=\sum_{j}n_{j}$ ($n_{j}$s are components of
vector $\textbf{n}_{\bullet}$); $|\bullet|$ cardinality of a set;
$n_{ss^{\prime}}^{ar^{\prime}}:=|\\{t:(s_{t},a_{t},s_{t+1},r_{t+1})=(s,a,s^{\prime},r^{\prime}),\
1\leq{}t\leq{}n\\}|$; and
$\textbf{H}(\textbf{p})=-\sum_{i=1}^{l}p_{i}\log{}p_{i}$ Shannon entropy of a
random variable with distribution $\textbf{p}=[p_{1}\ p_{2}\text{\ldots}\
p_{l}]$ where $\sum_{i=1}^{l}{}p_{i}=1$. The state and reward cost functions
can, then, be analytically computed as follows:
$\displaystyle\textbf{CL}(s_{1:n}|a_{1:n})=\sum_{s,a}\textbf{CL}(\textbf{n}^{a+}_{s\bullet})=\displaystyle\sum_{s,a}n^{a+}_{s+}\textbf{H}\left(\frac{\textbf{n}^{a+}_{s\bullet}}{n^{a+}_{s+}}\right)+\frac{|\mathcal{S}|-1}{2}\log{n^{a+}_{s+}}$
$\displaystyle\textbf{CL}(r_{1:n}|s_{1:n},a_{1:n})=\sum_{s,a,s^{\prime}}\textbf{CL}(\textbf{n}^{a\bullet}_{ss^{\prime}})=\sum_{s,a,s^{\prime}}n^{a+}_{ss^{\prime}}\textbf{H}\left(\frac{\textbf{n}^{a\bullet}_{ss^{\prime}}}{n^{a+}_{ss^{\prime}}}\right)+\frac{|\mathcal{R}|-1}{2}\log{}n^{a+}_{ss^{\prime}}$
As we primarily want to find a $\Phi$ that has the best reward predictability,
the introduction of $\alpha$ is primarily to stress on reward coding, making
costs for high-quality $\Phi$s much lower with very small $\alpha$ values. In
other words, $\alpha$ amplifies the differences among high-quality $\Phi$s and
bad ones; and this accelerates our stochastic search process described below.
We furthermore replace $\textbf{CL}(x)$ with
$\textbf{CL}_{\beta}(x)=\textbf{CL}(x|\theta)+\beta\textbf{CL}(\theta)$ in
$\textbf{Cost}_{\alpha}$ to define $\textbf{Cost}_{\alpha,\beta}$ for the
purpose of being able to select the right model given limited data. The
motivation to introduce $\beta$ is the following. For stationary environments
the cost function is analytically of this form $C_{1}\times u(\alpha)\times
O(n)+C_{2}\times v(\alpha)\times t(\beta)\times O(\log(n))$ where
$C_{1},C_{2}$ are constants, and $u,v,t$ are linear functions. The optimal
$\Phi$ should be the one with the smallest value of $C_{1}\times u(\alpha)$,
however, the curse here is that in practice $C_{2}\times v(\alpha)$ is often
big, so in order to obtain the optimal $\Phi$ with limited data, a small value
of $\beta$ will help. We assert that with a very large number of samples $n$,
$\alpha$ and $\beta$ can be ignored in the above cost function (use
$\alpha=0.5,\ \beta=1$ as the cost in [14]). The choice of small $\alpha$ and
$\beta$ helps us more quickly to overcome the model penalty and find the
optimal map. This strategy is a quite common practice in statistics, and even
in the Minimum Description Length (MDL) community [10]. For instance, AIC [1]
uses a very small $\beta=2/\log{}n$.
The interested reader is referred to [14] for more detailed analytical
formulas, and [26] for further motivation and consistency proofs of the
$\Phi$MDP model.
### 2.3 Context Trees
The class of maps that we will base our algorithm on is a class of context
trees.
Observation Context Tree (OCT). OCT is a class of maps $\Phi$ used to extract
relevant information from histories that include only past observations, not
actions and rewards. The presentation of OCT is mainly to facilitate the
definitions of the below Action-Observation Context Tree.
Definition. Given an $|\mathcal{O}|$-ary alphabet
$\mathcal{O}=\\{o^{1},o^{2},\ldots,o^{|\mathcal{O}|}\\}$, an OCT constructed
from the alphabet $\mathcal{O}$ is defined as a $|\mathcal{O}|$-ary tree in
which edges coming from any internal node are labeled by letters in
$\mathcal{O}$ from left to right in the order given.
Given an OCT $\mathcal{T}$ constructed from the alphabet $\mathcal{O}$, the
state suffix set, or briefly state set
$\mathcal{S}=\\{s^{1},s^{2},\ldots,s^{m}\\}\subseteq\mathcal{O}^{*}$ induced
from $\mathcal{T}$ is defined as the set of all possible strings of edge
labels forming along a path from a leaf node to the root node of
$\mathcal{T}$. $\mathcal{T}$ is called a Markov tree if it has the so-called
Markov property for its associated state set, that is, for every
$s^{i}\in\mathcal{S}$ and $o^{k}\in\mathcal{O}$, $s^{i}o^{k}$ has a unique
suffix $s^{j}\in\mathcal{S}$. The state set of a Markov OCT is called Markov
state set. OCTs that do not have the Markov property are identified as non-
Markov OCTs. Non-Markov state sets are similarly defined.
Example. Figure 1(a)(A) and 1(a)(B) respectively represent two binary OCTs of
depths two and three; also Figures 1(b)(A) and 1(b)(B) illustrate two ternary
OCTs of depths two and three.
(a) Binary context trees
(b) Trinary context trees
Figure 1: Context Trees
As can be seen from Figure 1, trees 1(a)(A) and 1(b)(A) are Markov; on the
other hand, trees 1(a)(B) and 1(b)(B) are non-Markov. The state set of tree
1(a)(A) is $\mathcal{S}^{(a)(A)}=\\{00,01,01,11\\}$; and furthermore with any
further observation $o\in\mathcal{O}$ and $s\in\mathcal{S}^{(a)(A)}$, there
exists a unique $s^{\prime}\in\mathcal{S}$ which is a suffix of $so$. Hence,
tree 1(a)(A) is Markov. Table 1(a) represents the deterministic relation
between $s,\ o$ and $s^{\prime}$.
| | $s$ | 00 | 01 | 10 | 11 | 00 | 01 | 10 | 11 | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| | $o$ | 0 | 1 | | |
| | $s^{\prime}$ | 00 | 10 | 00 | 10 | 01 | 11 | 01 | 11 | | |
(a) Markov property of $\mathcal{S}^{(a)(A)}$
| $s$ | 0 | 001 | 101 | 11 | 0 | 001 | 101 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|---
| $o$ | 0 | 1 | |
| $s^{\prime}$ | 0 | 0 | 0 | 0 | 101 or 001 | 11 | 11 | 11 | |
(b) Non-markov property of $\mathcal{S}^{(a)(B)}$
Table 1: Markov and Non-Markov properties
However, there is no such relation in tree 1(a)(B), or state set
$\mathcal{S}^{(a)(B)}=\\{0,001,101,11\\}$; for $s=0$ and $o=1$, it is
ambiguous whether $s^{\prime}=$101 or 001. Table 1(b) clarifies the non-Markov
property of tree 1(a)(B).
Similar arguments can be applied for trees 1(b)(A) and 1(b)(B) to identify
their Markov property.
It is also worthy to illustrate how an OCT can be used as a map. We illustrate
the mapping using again the OCTs in Figure 1. Given two histories including
only past observations $h_{5}=11101$ and $h^{\prime}_{6}=211210$, then
$\Phi^{(a)(A)}(h_{5})=01,\Phi^{(a)(B)}(h_{5})=101,\Phi^{(b)(A)}(h^{\prime}_{6})=10,\
\text{and}\ \Phi^{(b)(B)}(h^{\prime}_{6})=210$.
Action-Observation Context Tree (AOCT). AOCTs are extended from the OCTs
presented above for the generic RL problem where relevant histories contain
both actions and observations.
Definition. Given two alphabets,
$\mathcal{O}=\\{o^{1},o^{2},\ldots,o^{|\mathcal{O}|}\\}$ named observation
set, and $\mathcal{A}=\\{a^{1},a^{2},\ldots,a^{|\mathcal{A}|}\\}$ named action
set, an AOCT constructed from the two alphabets is defined as a tree where any
internal node at even depths has branching factor $|\mathcal{O}|$, and edges
coming from such nodes are labeled by letters in $\mathcal{O}$ from left to
right in the order given; and similarly any internal node at odd depths has
branching factor $|\mathcal{A}|$, and edges coming from these nodes are
labeled by letters in $\mathcal{A}$ also from left to right in the specified
order.
The definitions of Markov and non-Markov AOCTs are similar to those of OCTs
except that a next observation is now replaced by the next action and
observation. Formally, suppose $\mathcal{T}$ is an AOCT constructed from the
above two alphabets; and
$\mathcal{S}=\\{s^{1},s^{2},\ldots,s^{m}\\}\subseteq(\mathcal{A}\times\mathcal{O})^{*}\cup\mathcal{A}\times(\mathcal{A}\times\mathcal{O})^{*}$
is the state suffix set of the tree, then $\mathcal{T}$ is defined as a Markov
AOCT if it has the Markov property, that is, for every $1\leq{}i\leq{}m$,
$1\leq j\leq|\mathcal{A}|$, and $1\leq k\leq|\mathcal{O}|$ there exist a
unique $1\leq l\leq m$ such that $s^{l}$ is a suffix of $s^{i}a^{j}o^{k}$.
AOCTs that do not have Markov property are categorized as non-Markov AOCTs.
The total number of AOCTs up to a certain depth $d$, $K(d)$, can be
recursively computed via the formula
$K(d+2)=\\{[K(d)]^{|\mathcal{A}|}+1\\}^{|\mathcal{O}|}+1$ where
$K(0)=1,K(1)=2$. As can be easily seen from the recursive formula, the total
number of AOCTs is doubly exponential in the tree depth.
An important point to note here is that in our four experiments presented in
Section 4, the $\Phi$ space is limited to Markov AOCTs, since as explained
above, the state suffix set induced from a non-Markov AOCT does not represent
an MDP state set; to put it more clearly, in non-Markov AOCTs, from the next
action and observation, we cannot derive the next state from the current one.
The Markov constraint on AOCTs significantly reduces the search space for our
stochastic search algorithm. In the U-tree algorithm [21], no distinction of
Marov and non-Markov trees is identified; the algorithm attempts to search for
the optimal tree over the whole space of AOCTs.
### 2.4 Stochastic search
While we have defined the cost criterion for evaluating maps, the problem of
finding the optimal map remains. When the $\Phi$ space is huge, e.g. context-
tree map space where the number of $\Phi$s grows doubly exponentially with the
tree depth, exhaustive search is unable to deal with domains where the optimal
$\Phi$ is non-trivial. Stochastic search is a powerful tool for solving
optimization problems where the landscape of the objective function is
complex, and it appears impossible to analytically or numerically find the
exact or even approximate global optimal solution. A typical stochastic search
algorithm starts with a predefined or arbitrary configuration (initial
argument of the objective function or state of a system), and from this
generates a sequence of configurations based on some predefined probabilistic
criterion; the configuration with the best objective value will be retained.
There are a wide range of stochastic search methods proposed in the literature
[23]; the most popular among these are simulated-annealing-type algorithms
[19], [25]. An essential element of a simulated-annealing (SA) algorithm is a
Markov Chain Monte Carlo (MCMC) sampling scheme where a proposed new
configuration $\tilde{y}$ is drawn from a proposal distribution
$q(\tilde{y}|y)$, and we then change from configuration $y$ to $\tilde{y}$
with probability
$\min\\{1,\frac{\pi_{T}(y)q(y|\widetilde{y})}{\pi_{T}(\widetilde{y})q(\widetilde{y}|y)}\\}$
where $\pi_{T}$ is a target distribution. In a simulated-annealing (SA)
algorithm where the traditional Metropolis-Hasting sampling scheme is
utilized, $\pi_{T}$ is proportional to $e^{-f(x)/T}$ if $f$ is an objective
function that we want to minimize, and $T$ is some positive constant
temperature. $\frac{q(y|\widetilde{y})}{q(\widetilde{y}|y)}$ is called the
correction factor; it is there to compensate for bias in $q$.
The traditional SA uses an MCMC scheme with some temperature-decreasing
strategy. Although shown to be able to find the global optimum asymptotically
[9], it generally works badly in practice as we do not know which temperature
cooling scheme is appropriate for the problem under consideration. Fortunately
in the $\Phi$MDP cost function we know typical cost differences between two
$\Phi$s ($C\beta\times\log(n)$), so the range of appropriate temperatures can
be significantly reduced. The search process may be improved if we run a
number of SA procedures with various different temperatures. Parallel
Tempering (PT) [7], [12], an interesting variant of the traditional SA,
significantly improves this stochastic search process by smartly offering a
swapping step, letting the search procedure use small temperatures for
exploitation and big ones for exploration.
Parallel tempering. PT performs stochastic search over the product space
$\mathcal{X}_{1}\times\ldots\ \
\times\mathcal{X}_{I}(\mathcal{X}_{i}=\mathcal{X}\ \forall 1\leq i\leq I)$,
where $\mathcal{X}$ is the objective function’s domain, and $I$ is the
parallel factor. Fixed temperatures $T_{i}$ ($i=1,\ldots\ ,I$, and
$1<T_{1}<T_{2}<\ldots<T_{I}$) are chosen for spaces $\mathcal{X}_{i}$
$(i=1,\ldots\ ,I)$. Temperatures $T_{i}$ ($i=1,\ldots,I$) are selected based
on the following formula
$(\frac{1}{T_{i}}-\frac{1}{T_{i+1}})|\Delta{}H|\approx-\log{}p_{a}$ where
$\Delta{}H$ is the “typical” difference between function values of two
successive configurations; and $p_{a}$ is the lower bound for the swapping
acceptance rate. The main steps of each PT loop are as follows:
* •
$(x^{(t)}_{1},\ldots\ ,x^{(t)}_{I})$ is the current sampling; draw $u\sim$
Uniform[0,1]
* •
If $u\leq\alpha_{0}$, update every $x^{(t)}_{i}$ to $x^{(t+1)}_{i}$ via some
Markov Chain Monte Carlo (MCMC) scheme like Metropolis-Hasting (Parallel step)
* •
If $u>\alpha_{0}$, randomly choose a neighbor pair, say $i$ and $i+1$, and
accept the swap of $x^{(t)}_{i}$ and $x^{(t)}_{i+1}$ with probability
$\min\\{1,\frac{\pi_{T_{i}}(x^{(t)}_{i+1})\pi_{T_{i+1}}(x^{(t)}_{i})}{\pi_{T_{i}}(x^{(t)}_{i})\pi_{T_{i+1}}(x^{(t)}_{i+1})}\\}$
(Swapping step).
The full details of PT are given in Algorithm 1.
Algorithm 1 Parallel Tempering (PT)
0: An objective function $h(x)$ to be minimized, or equivalently the target
distribution $\pi_{C}\ \alpha\ e^{-h(x)/C}$ for some positive constant $C$
0: Swap probability parameter $\alpha_{0}$
0: A proposal distribution $q(y|x)$
0: Temperatures $T_{1},T_{2},\ldots,T_{L}$, and number of iterations $N$
1: Initialize arbitrary configurations $x^{(1,1)},...,x^{(L,1)}($
{$x^{(k,i)}$: represents the $i^{th}$ value of $x$ for temperature $T_{k}$;})
2: $x_{opt}\leftarrow\operatornamewithlimits{argmin}_{x=x^{(\cdot,1)}}h(x)$
3: for $i=1$ to $N$ do
4: for $k=1$ to $L$ do
5: $\widetilde{y}\leftarrow x^{(k,i-1)}$
6: Sample $y$ from the proposal distribution $q(y|\widetilde{y})$
7:
$r\leftarrow\min\\{1,\frac{\pi_{T_{k}}(y)q(y|\widetilde{y})}{\pi_{T_{k}}(\widetilde{y})q(\widetilde{y}|y)}\\}$
(Metropolis Hastings)
8: Draw $u\sim$ Uniform[0,1] and update
9: if $u\leq r(\widetilde{y},y)$ then
10: $x^{(k,i)}\leftarrow y$
11: else
12: $x^{(k,i)}\leftarrow\widetilde{y}$
13: end if
14: if $h(x_{opt})>h(x^{(k,i)})$ then
15: $x_{opt}\leftarrow x^{(k,i)}$
16: end if
17: end for
18: Draw $u\sim$ Uniform[0,1]
19: if $u\geq\alpha_{0}$ then
20: Draw $a$ Uniform $\\{1,...,L-1\\}$ and let $b=a+1$
21:
$r\leftarrow\min\\{1,\frac{\pi_{T_{a}}(x^{(b,i)})\pi_{T_{b}}(x^{(a,i)})}{\pi_{T_{a}}(x^{(a,i)})\pi_{T_{b}}(x^{(b,i)})}\\}$
22: Draw $v\sim$ Uniform[0,1]
23: if $v\leq r$ then
24: Swap $x^{(a,i)}$ and $x^{(b,i)}$
25: end if
26: end if
27: end for
Return $x_{opt}$
If its swapping phase is excluded, PT is simply the combination of a fixed
number of Metropolis-Hastings procedures. The central point that makes PT
powerful is its swapping step where adjacent temperatures interchange their
sampling regions. This means that a good configuration can be allowed to use a
cooler temperature and exploit what it has found while a worse configuration
is given a higher temperature which results in more exploration.
## 3 The $\Phi$MDP Algorithm
We now describe how the generic $\Phi$MDP algorithm works. The general
algorithm is shown below (Algorithm 2). It first takes a number of random
actions ($5000$ in all our experiments). Then it defines the cost function
$Cost_{\alpha,\beta}$ based on this history. Stochastic search is then used to
find a map $\Phi$ with low cost. Based on the optimal $\Phi$ the history is
transformed into a sequence of states, actions and rewards. We use optimistic
frequency estimates from this history to estimate probability parameters for
state transitions and rewards. More precisely, we use
$\frac{R_{\max}+r_{1}+...+r_{m}}{m+1}$ instead of the average
$\frac{r_{1}+...+r_{m}}{m}$ to estimate expected reward, where
$r_{1},...,r_{m}$ are the rewards that have been observed for a certain state-
action pair, and $R_{\max}$ is the highest possible reward. The statistics are
used to estimate Q values using AVI. After this the agent starts to interact
with the environment again using $Q$-learning initialized with the values that
resulted from the performed AVI. The switch from AVI to Q-Learning is rather
obvious, as Q-Learning only needs one cheap update per time step, while AVI
requires updating the whole environment model and running a number of value
iterations. The first set of random actions might not be sufficient to
characterize what the best maps $\Phi$ look like, so it might be beneficial to
add the new history gathered by the Q-Learning interactions with the
environment to the old history, and then repeat the process but without the
initial sampling.
Algorithm 2 Generic Stochastic $\Phi$MDP Agent (GS$\Phi$A)
0: $Environment$; $initialSampleNumber$, $agentLearningLoops$,
$stochasticIterations$ and $additionalSampleNumber$
1: Generate a history $h^{initial}$ of length $initialSampleNumber$
2: $h\leftarrow h^{initial}$
3: repeat
4: Run the chosen stochastic search scheme for the history $h$ to find a
$\hat{\Phi}$ with low cost
5: Compute MDP statistics (optimistic frequency estimates $\hat{R}$ and
$\hat{T}$) induced from $\hat{\Phi}$
6: Apply AVI to find the optimal $Q^{*}$ values using the computed statistics
$\hat{R}$ and $\hat{T}$.
7: Interact with environment for $additionalSampleNumber$ iterations of
Q-Learning using $Q^{*}$ as initial values; the obtained additional history is
stored in $h^{additional}$
8: $h\leftarrow[h,h^{additional}]$
9: $agentLearningLoops\leftarrow agentLearningLoops-1$
10: until $agentLearningLoops=0$
11: Compute the optimal policy $\pi^{optimal}$ from the optimal $\Phi$ and $Q$
values
Return [$\Phi^{optimal}$, $\pi^{optimal}$]
In the first four experiments in Section 4, PT is employed to search over the
$\Phi$ space of Markov AOCTs.
### 3.1 Proposal Distribution for Stochastic Search over the Markov-AOCT
Space
The principal optional component of the above high-level algorithm, GS$\Phi$A,
is a stochastic search procedure of which some algorithms have been presented
in Section 2.4. In these algorithms, an essential technical detail is the
proposal distribution $q$. It is natural to generate the next tree (the next
proposal or configuration) from the current tree by splitting or merging
nodes. It is possible to express the exact form of our proposal distribution,
and based on this to explain how the next tree (next configuration) is
proposed from the current tree (current configuration). However, the
analytical form of the distribution is cumbersome to specify, so for better
exposition we opt to describe the exact behavior of the tree proposal
distribution instead.
The stochastic search procedure starts with a Markov AOCT where all of the
tree nodes are mergeable, and splittable. However, in the course of the
search, a tree node might become unmergeable, but not the other way round; and
a splittable node might turn to be unsplittable and vice versa. These specific
transfering scenarios are described as follows. A mergeable tree node of the
current tree becomes unmergeable if the current tree is proposed from the
previous tree by splitting that node, and the cost of the current tree is
smaller than that of the previous tree. A splittable leaf node of the current
tree becomes unsplittable if the state associated with that node is not
present in the current history; however, an unsplittable leaf node might
revert to splittable when the state associated with that node is present in
the future updated history. The constraint on merging is to keep good short-
term memory for predicting rewards, while the other on splitting is simply
following the Occam’s razor principle.
Merge and split permits. Given some current tree at a particular point in time
of the stochastic search process, when considering the generation of the next
tree proposal, most of the tree nodes, though labeled splittable and/or
mergeable, might have no split, or merge permit, or neither. A node has split
permit if it is a leaf node with splittable label. When a leaf node has been
split, we simply add all possible children for this node, and label the edges
according to the definition of AOCTs. As mentioned above, the newly added leaf
nodes might be labeled unmergeable if the cost of the new tree is smaller than
that of the old one; and these nodes might also be labeled unsplittable if the
states associated with the new leaf nodes are not present in the current
history. A node has merge permit if it is labeled mergeable, and all of its
children are leaf nodes. When a tree node is merged, all the edges and nodes
associated with its children are removed.
Markov-merge and Markov-split permits. Since our search space is the class of
Markov OACTs, whenever a split or merge occurs, extra adjustments might be
needed to make the new tree Markov. After a split, there might be nodes that
make the tree violate the Markov assumption, and therefore, need to be split.
After we split all of those we have to check again to see if any other nodes
now need to be split. This goes on until we have a Markov AOCT again. The same
applies to merging.
Figure 2: AOCT proposals
When a node is Markov-split, it and all of the leaf nodes that need to be
split (including recursive splits) as a consequence in order to make the tree
Markov, are split. A tree node is said to have Markov-split permit if it, and
all the other nodes that would be split in a Markov-split of the node, have
split permits. This notion is best illustrated with an example. First we
define Markov and Non-Markov states of an AOCT. A state of an AOCT is Markov
if given any next action-observation pair, the next state is determined;
otherwise it is labeled as non-Markov. Now in Figure 2(A), suppose the current
Markov AOCT is the tree without dashed edges. Then after splitting the leaf
node marked by * (the node associated with state 00101), the state 001 becomes
non-Markov so this associated node needs to be split. However, after splitting
this node (node associated with state 001), state 0 becomes non-Markov, hence
it needs splitting as well. In short, to split the node marked by *, the two
nodes associated with states 001 and 0 have to be split as well so as to
ensure the resulting tree is Markov after splitting. Similarly, a tree node
has Markov-merge permit if it, and all of the tree nodes that minimally and
recursively need to be merged after the original node is merged in order to
make the tree Markov, have merge permits. For example, in Figure 2(B), suppose
the current tree is the tree including both solid and dashed edges, then the
node marked by * has Markov-merge permit, if it itself, and the nodes
associated with paths $001$, $021$ and $00101$ that need to be merged, have
merge permits. When a node with Markov-merge permit is Markov-merged, it and
its Markov-merge-associated nodes are merged.
Our procedure to generate the next tree from the current tree (draw sample
from $q(y|\cdot)$) in the space of Markov AOCTs consists of the following main
steps:
* •
From the given tree, identify two sets: one is $N_{S}$ containing nodes with
Markov-split permits, and the other $N_{M}$ containing nodes with Markov-merge
permits.
* •
Suppose that either $N_{S}$ or $N_{M}$ is non-empty otherwise the algorithm
(GS$\Phi$A) must stop; then if either $N_{S}$ or $N_{M}$ is empty, select a
node uniformly at random from the other set; otherwise select $N_{S}$ or
$N_{M}$ randomly with probability $\frac{1}{2}$ each, and after that choose a
tree node randomly from the selected set.
* •
Markov-split the node if it belongs to $N_{S}$, otherwise Markov-merge it
Once we have drawn the new tree $\tilde{y}$, the Metropolis Hastings
correction factor can be straightforwardly calculated via the formula
$\frac{q(y|\widetilde{y})}{q(\widetilde{y}|y)}=\begin{cases}\frac{|\widetilde{N}_{M}|}{|N_{S}|}&\text{if}\
\widetilde{y}\ \text{is proposed from $y$ by Markov-splitting}\\\
\frac{|\widetilde{N}_{S}|}{|N_{M}|}&\text{if}\ \widetilde{y}\ \text{is
proposed from $y$ by Markov-merging}\end{cases}$
here $\widetilde{N}_{S}$ and $\widetilde{N}_{M}$ are respectively the set of
nodes with Markov-split permits, and the set of nodes with Markov-merge
permits of $\widetilde{y}$.
Sharing. If the stochastic search algorithm utilized is PT, we apply another
trick to effectively accelerate the search process. Whenever a node is labeled
unmergeable, that is, by splitting this node the cost function decreases, or
in other words a good additional relevant short-term memory for predicting
rewards is found, the states associated with the new nodes created by the
splitting are replicated in the trees with the other temperatures.
## 4 Experiments
### 4.1 Experimental Setup
Parameter | Component | Value
---|---|---
$\alpha$ | $Cost_{\alpha,\beta}$ | 0.1
$\beta$ | $Cost_{\alpha,\beta}$ | 0.1
$initialSampleNumber$ | GS$\Phi$A | 5000
$agentLearningLoops$ | GS$\Phi$A | 1
Iterations | PT | 100
$I$ | PT | 10
$T_{i},\ i\leq I$ | PT | $T_{i}=\beta\times i\times\log(n)$
$\alpha_{0}$ | PT | 0.7
$\gamma$ | AVI, Q-Learning | 0.999999
$\eta$ | Q-Learning | 0.01
Table 2: Parameter setting for the GS$\Phi$A algorithm
Below in this section we present our empirical studies of the $\Phi$MDP
algorithm GS$\Phi$A described in Section 3. For all of our experiments,
stochastic search (PT) is applied in the $\Phi$ space of Markov AOCTs.
For a variety of tested domains, our algorithm produces consistent results
using the same set of parameters. These parameters are shown in Table 2, and
are not fine tuned.
The results of $\Phi$MDP and the three competitors in the four above-listed
environments are shown in Figures 3, 4 7, 8 and LABEL:fig:relaymazeplot. In
each of the plots, various time points are chosen to assess and compare the
quality of the policies learned by the four approaches. In order to evaluate
how good a learned policy is, at each point, the learning process of each
agent, and the exploration of the three competitors are temporarily switched
off. The selected statistic to compare the quality of learning is the averaged
reward over 5000 actions using the current policy. For stability, the
statistic is averaged over 10 runs.
As shown in more detail below, $\Phi$MDP is superior to U-tree and active-LZ,
and is comparable to MC-AIXI-CTW in short-term memory domains. Overall
conclusions are clear, and we, therefore, omit error bars.
### 4.2 Environments and results
We describe each environment, the resulting performance, and the tree that was
found by $\Phi$MDP in the cheese maze domain.
$4\times 4$ Grid.
Figure 3: $4\times 4$ Grid
The domain is a 4$\times$4 grid world. At each time step, the agent can move
one cell left, right, up and down within the grid world. The observations are
uninformative. When the agent enters the bottom-right corner of the grid; it
gets a reward of 1, and is automatically and randomly sent back to one of the
remaining 15 cells. Entering any cell other than the bottom-right one gives
the agent a zero reward. To achieve the maximal total reward, the agent must
be able to remember a series of smart actions without any clue about its
relative position in the grid.
The context tree found contains 34 states. Some series of actions that take
the agent towards the bottom-right corner of the grid are present in the
context tree. As shown in the $4\times$4-grid plot in Figure 3, after 5000
experiences gathered from the random policy, $\Phi$MDP finds the optimal
policy, and so does MC-AIXI-CTW and U-Tree. Active-LZ, however, does not
converge to an optimal policy even after 50,000 learning cycles.
Tiger. The tiger domain is described as follows. There are two doors, left and
right; an amount of gold and a tiger are placed behind the two doors in a
random order. The person has three possible actions: listen to predict the
position of the tiger, open the right door, and open the left door. If the
person listens, he has to pay some money (reward of -1). The probability that
the agent hears correctly is 0.85. If the person opens either of the doors and
sees the gold, the obtained reward is 10; or otherwise he faces the tiger,
then the agent receives a reward of -100. After the door is opened, the
episode ends; and in the next episode the tiger sits randomly again behind
either the left or the right door.
Figure 4: Tiger
Our parallel tempering procedure found a context tree consisting of 39 states
including some important states where the history is such that the agent has
listened a few times before opening the door. It can be seen from the tiger
plot in Figure 4 that the optimal policy $\Phi$MDP found after 5,000 learning
experiences does yield positive reward on average, while from time point
10,000 on, it achieves as high rewards as MC-AIXI-CTW. U-Tree appears to learn
more slowly but eventually manages to get positive averaged rewards after
50,000 cycles like $\Phi$MDP and MC-AIXI-CTW. Active-LZ is performing far
worse. The optimal policy that $\Phi$MDP, MC-AIXI-CTW, and U-Tree ultimately
found is the following. First listen two times, if the listening outcomes are
consistent, open the predicted door with gold behind; otherwise take one more
listening action, and based on the majority to open the appropriate door.
Cheese Maze.
Figure 5: Cheese-maze domain
This domain, as shown in Figure 5, consists of a eleven-cell maze with a
cheese in it. The agent is a mouse that attempts to find the cheese. The
agent’s starting position for each episode is at one of the eleven cells
uniformly random. The actions available to the agent are: move one cell left
(0), right (1), up (2) and down (3). However, it should be noticed that if the
agent hits the wall, its relative position in the maze remains unchanged. At
each cell the agent can observe which directions among left, right, up and
down the cell is blocked by a wall. If wall-blocking statuses of each cell are
represented by 1 (blocked), and 0 (free) respectively; then an observation is
described by a four-digit binary number where the digits from left to right
are wall-blocking statuses of up, left, down and right directions. For
example, 0101 = 5, 0111 = 7, … as described in Figure 5. The agent gets a
reward of -1 when moving into a free cell without a cheese; hitting the wall
gives it a penalty of -10; and a reward of 10 is given to the agent when it
finds the cheese. As can be seen, some observations themselves alone are
insufficient for the mouse to locate itself unambiguously in the maze. Hence,
the mouse must learn to resolve these ambiguities of observations in the maze
to be able to find the optimal policy.
Figure 6: Cheese-maze tree
Our algorithm found a context tree consisting of 43 states that contains the
tree as shown in Figure 6. The tree splits from the root into the $6$ possible
observations. Then observations $5$ and $10$ are split into the four possible
actions; and some of these actions, the ones that come from a different
location and not a wall collision, are split further into the $6$ “possible”
observations before that. This resolves which $5$ or which $10$ we are at. The
states in this tree resolve the most important ambiguities of the raw
observations and an optimal policy can be found. The domain contains an
infinite amount of longer dependencies among which our found states pick up a
small subset. The cheese-maze plot in Figure 7 shows that after the initial
5000 experiences, $\Phi$MDP is marginally worse than MC-AIXI-CTW but is better
than U-Tree and Active-LZ. From time point 10,000, there is no difference
between $\Phi$MDP and MC-AIXI-CTW. U-Tree and Active-LZ remain inferior.
Figure 7: Cheese maze
Kuhn Poker.
Figure 8: Kuhn poker
In Kuhn poker [17] a deck of only three cards (Jack, Queen and King) is used.
The agent always plays second in any game (episode). After putting a chip each
into play, the players are dealt a card each. Then the first player says bet
or pass and the second player chooses bet or pass. If player one says pass and
player two says bet then player one must choose again between bet and pass.
Whenever a player says bet they must put in another chip. If one player bets
and the other pass the better gets all the chips in play. Otherwise the player
with the highest card gets the chips. Player one plays according to a fixed
but stochastic Nash optimal strategy [11]. $\Phi$MDP finds $89$ states. It can
be observed from the Kunh-poker plot in Figure 8 that $\Phi$MDP is comparable
to MC-AIXI-CTW and much better than U-Tree and Active-LZ, who loose money.
## 5 Conclusions
Based on the Feature Reinforcement Learning framework [14] we defined actual
practical reinforcement learning agents that perform very well empirically. We
evaluated a reasonably simple instantiation of our algorithm that first takes
$5000$ random actions followed by finding a map through a search procedure and
then it performs Q-learning on the MDP defined by the map’s state set.
We performed an evaluation on four test domains used to evaluate MC-AIXI-CTW
in [28]. Those domains are all suitably attacked with context tree methods. We
defined a $\Phi$MDP agent for a class of maps based on context trees, and
compared it to three other context tree-based methods. Key to the success of
our $\Phi$MDP agent was the development of a suitable stochastic search method
for the class of Markov AOCTs. We combined parallel tempering with a
specialized proposal distribution that results in an effective stochastic
search procedure. The $\Phi$MDP agent outperforms both the classical U-tree
algorithm [21] and the recent Active-LZ algorithm [6], and is competitive with
the newest state of the art method MC-AIXI-CTW [28]. The main reason that
$\Phi$MDP outperforms U-tree is that $\Phi$MDP uses a global criterion
(enabling the use of powerful global optimizers) whereas U-tree uses a local
split-merge criterion. $\Phi$MDP also performs significantly better than
Active-LZ. Active-LZ learns slowly as it overestimates the environment model
(assuming $n$-Markov or complete context-tree environment models); and this
leads to unreliable value-function estimates.
Below are some detailed advantages of $\Phi$MDP over MC-AIXI-CTW:
* •
$\Phi$MDP is more efficient than MC-AIXI-CTW in both computation and memory
usage. $\Phi$MDP only needs an initial number of samples and then it finds the
optimal map and uses AVI to find MDP parameters. After this it only needs a
Q-learning update for each iteration. On the other hand, MC-AIXI-CTW requires
model updating, planning and value-reverting at every single cycle which
together are orders of magnitude more expensive than Q-learning. In the
experiments $\Phi$MDP finished in minutes while MC-AIXI-CTW needed hours.
Another disadvantage of MC-AIXI-CTW is that it is a memory-hungry algorithm.
$\Phi$MDP learns the best tree representation using stochastic search, which
expands a tree towards relevant histories. MC-AIXI-CTW learns the mixture of
trees where the number of tree nodes grows (and thereby the memory usage)
linearly with time.
* •
$\Phi$MDP learns a single state representation and can use many classical RL
algorithms, e.g. Q-Learning, for MDP learning and planning.
* •
Another key benefit is that $\Phi$MDP represents a more discriminative
approach than MC-AIXI-CTW since it aims primarily for the ability to predict
future rewards and not to fully model the observation sequence. If the
observation sequence is very complex, this becomes essential.
On the other hand, to be fair it should be noted that compared to $\Phi$MDP,
MC-AIXI-CTW is more principled. The results presented in this paper are
encouraging since they show that we can achieve comparable results to the more
sophisticated MC-AIXI-CTW algorithm on problems where only short-term memory
is needed. We plan to utilize the aforementioned advantages of the $\Phi$MDP
framework, like flexibility in environment modeling and computational
efficiency, to attack more complex and larger problems.
## Acknowledgement
This work was supported by ARC grant DP0988049 and by NICTA. We also thank
Joel Veness and Daniel Visentin for their assistance with the experimental
comparison.
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|
arxiv-papers
| 2011-08-18T03:50:35 |
2024-09-04T02:49:21.614699
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Phuong Nguyen, Peter Sunehag, and Marcus Hutter",
"submitter": "Phuong Nguyen",
"url": "https://arxiv.org/abs/1108.3614"
}
|
1108.3679
|
Fundamental Solution of Laplace’s Equation in Hyperspherical Geometry
Fundamental Solution of Laplace’s Equation
in Hyperspherical Geometry
Howard S. COHL †‡
H.S. Cohl
† Applied and Computational Mathematics Division, Information Technology
Laboratory,
† National Institute of Standards and Technology, Gaithersburg, Maryland, USA
howard.cohl@nist.gov
‡ Department of Mathematics, University of Auckland, 38 Princes Str.,
Auckland, New Zealand http://hcohl.sdf.org
Received August 18, 2011, in final form November 22, 2011; Published online
November 29, 2011
Due to the isotropy of $d$-dimensional hyperspherical space, one expects there
to exist a spherically symmetric fundamental solution for its corresponding
Laplace–Beltrami operator. The $R$-radius hypersphere ${\mathbf{S}}_{R}^{d}$
with $R>0$, represents a Riemannian manifold with positive-constant sectional
curvature. We obtain a spherically symmetric fundamental solution of Laplace’s
equation on this manifold in terms of its geodesic radius. We give several
matching expressions for this fundamental solution including a definite
integral over reciprocal powers of the trigonometric sine, finite summation
expressions over trigonometric functions, Gauss hypergeometric functions, and
in terms of the associated Legendre function of the second kind on the cut
(Ferrers function of the second kind) with degree and order given by $d/2-1$
and $1-d/2$ respectively, with real argument between plus and minus one.
hyperspherical geometry; fundamental solution; Laplace’s equation; separation
of variables; Ferrers functions
35A08; 35J05; 32Q10; 31C12; 33C05
## 1 Introduction
We compute closed-form expressions of a spherically symmetric Green’s function
(fundamental solution of Laplace’s equation) for a $d$-dimensional Riemannian
manifold of positive-constant sectional curvature, namely the $R$-radius
hypersphere with $R>0$. This problem is intimately related to the solution of
the Poisson equation on this manifold and the study of spherical harmonics
which play an important role in exploring collective motion of many-particle
systems in quantum mechanics, particularly nuclei, atoms and molecules. In
these systems, the hyperradius is constructed from appropriately mass-weighted
quadratic forms from the Cartesian coordinates of the particles. One then
seeks either to identify discrete forms of motion which occur primarily in the
hyperradial coordinate, or alternatively to construct complete basis sets on
the hypersphere. This representation was introduced in quantum mechanics by
Zernike & Brinkman [38], and later invoked to greater effect in nuclear and
atomic physics, respectively, by Delves [6] and Smith [32]. The relevance of
this representation to two-electron excited states of the helium atom was
noted by Cooper, Fano & Prats [5]; Fock [11] had previously shown that the
hyperspherical representation was particularly efficient in representing the
helium wavefunction in the vicinity of small hyperradii. There has been a rich
literature of applications ever since. Examples include Zhukov [39] (nuclear
structure), Fano [10] and Lin [25] (atomic structure), and Pack & Parker [29]
(molecular collisions). A recent monograph by Berakdar [3] discusses
hyperspherical harmonic methods in the general context of highly-excited
electronic systems. Useful background material relevant for the mathematical
aspects of this paper can be found in [23, 34, 36]. Some historical references
on this topic include [18, 24, 30, 31, 37].
This paper is organized as follows. In Section 2 we describe hyperspherical
geometry and its corresponding metric, global geodesic distance function,
Laplacian and hyperspherical coordinate systems which parametrize points on
this manifold. In Section 3 for hyperspherical geometry, we show how to
compute ‘radial’ harmonics in a geodesic polar coordinate system and derive
several alternative expressions for a ‘radial’ fundamental solution of the
Laplace’s equation on the $R$-radius hypersphere.
Throughout this paper we rely on the following definitions. For
$a_{1},a_{2},\ldots\in{\mathbf{C}}$, if $i,j\in{\mathbf{Z}}$ and $j<i$ then
$\sum\limits_{n=i}^{j}a_{n}=0$ and $\prod\limits_{n=i}^{j}a_{n}=1$. The set of
natural numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and the set
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$
## 2 Hyperspherical geometry
The Euclidean inner product for ${\mathbf{R}}^{d+1}$ is given by $({\bf
x},{\mathbf{y}})=x_{0}y_{0}+x_{1}y_{1}+\cdots+x_{d}y_{d}$. The variety $({\bf
x},{\bf x})=x_{0}^{2}+x_{1}^{2}+\cdots+x_{d}^{2}=R^{2}$, for ${\bf
x}\in{\mathbf{R}}^{d+1}$ and $R>0$, defines the $R$-radius hypersphere
${\mathbf{S}}_{R}^{d}$. We denote the unit radius hypersphere by
${\mathbf{S}}^{d}:={\mathbf{S}}_{1}^{d}$. Hyperspherical space in
$d$-dimensions, denoted by ${\mathbf{S}}_{R}^{d}$, is a maximally symmetric,
simply connected, $d$-dimensional Riemannian manifold with positive-constant
sectional curvature (given by $1/R^{2}$, see for instance [23, p. 148]),
whereas Euclidean space ${\mathbf{R}}^{d}$ equipped with the Pythagorean norm,
is a Riemannian manifold with zero sectional curvature.
Points on the $d$-dimensional hypersphere ${\mathbf{S}}_{R}^{d}$ can be
parametrized using subgroup-type coordinate systems, i.e., those which
correspond to a maximal subgroup chain $O(d)\supset\cdots$ (see for instance
[19, 21]). The isometry group of the space ${\mathbf{S}}_{R}^{d}$ is the
orthogonal group $O(d)$. Hyperspherical space ${\mathbf{S}}_{R}^{d}$, can be
identified with the quotient space $O(d)/O(d-1)$. The isometry group $O(d)$
acts transitively on ${\mathbf{S}}_{R}^{d}$. There exist separable coordinate
systems on the hypersphere, analogous to parabolic coordinates in Euclidean
space, which can not be constructed using maximal subgroup chains.
Polyspherical coordinates, are coordinates which correspond to the maximal
subgroup chain given by $O(d)\supset\cdots$. What we will refer to as standard
hyperspherical coordinates, correspond to the subgroup chain given by
$O(d)\supset O(d-1)\supset\cdots\supset O(2).$ (For a thorough discussion of
polyspherical coordinates see Section IX.5 in [36].) Polyspherical coordinates
on ${\mathbf{S}}^{d}_{R}$ all share the property that they are described by
$(d+1)$-variables: $R\in[0,\infty)$ plus $d$-angles each being given by the
values $[0,2\pi)$, $[0,\pi]$, $[-\pi/2,\pi/2]$ or $[0,\pi/2]$ (see [19, 20]).
In our context, a useful subset of polyspherical coordinate are geodesic polar
coordinates $(\theta,{\mathbf{\widehat{x}}})$ (see for instance [28]). These
coordinates, which parametrize points on ${\mathbf{S}}_{R}^{d}$, have origin
at $O=(R,0,\ldots,0)\in{\mathbf{R}}^{d+1}$ and are given by a ‘radial’
parameter $\theta\in[0,\pi]$ which parametrizes points along a geodesic curve
emanating from $O$ in a direction
${\mathbf{\widehat{x}}}\in{\mathbf{S}}^{d-1}$. Geodesic polar coordinate
systems partition ${\mathbf{S}}_{R}^{d}$ into a family of $(d-1)$-dimensional
hyperspheres, each with a ‘radius’ $\theta:=\theta_{d}\in(0,\pi),$ on which
all possible hyperspherical coordinate systems for ${\mathbf{S}}^{d-1}$ may be
used (see for instance [36]). One then must also consider the limiting case
for $\theta=0,\pi$ to fill out all of ${\mathbf{S}}_{R}^{d}$. Standard
hyperspherical coordinates (see [22, 26]) are an example of geodesic polar
coordinates, and are given by
$\displaystyle\begin{array}[]{@{}l}x_{0}=R\cos\theta,\\\
x_{1}=R\sin\theta\cos\theta_{d-1},\\\
x_{2}=R\sin\theta\sin\theta_{d-1}\cos\theta_{d-2},\\\
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\\
x_{d-2}=R\sin\theta\sin\theta_{d-1}\cdots\cos\theta_{2},\\\
x_{d-1}=R\sin\theta\sin\theta_{d-1}\cdots\sin\theta_{2}\cos\phi,\\\
x_{d}=R\sin\theta\sin\theta_{d-1}\cdots\sin\theta_{2}\sin\phi,\end{array}$ (8)
$\theta_{i}\in[0,\pi]$ for $i\in\\{2,\ldots,d\\}$, $\theta:=\theta_{d}$, and
$\phi\in[0,2\pi)$.
In order to study a fundamental solution of Laplace’s equation on the
hypersphere, we need to describe how one computes the geodesic distance in
this space. Geodesic distances on ${\mathbf{S}}_{R}^{d}$ are simply given by
arc lengths, angles between two arbitrary vectors, from the origin in the
ambient Euclidean space (see for instance [23, p. 82]). Any parametrization of
the hypersphere ${\mathbf{S}}_{R}^{d}$, must have $({\bf x},{\bf
x})=x_{0}^{2}+\cdots+x_{d}^{2}=R^{2}$, with $R>0$. The distance between two
points ${\bf x},{{\bf x}^{\prime}}\in{\mathbf{S}}_{R}^{d}$ on the hypersphere
is given by
$d({\bf x},{{\bf x}^{\prime}})=R\gamma=R\cos^{-1}\left(\frac{({\bf x},{{\bf
x}^{\prime}})}{({\bf x},{\bf x})({{\bf x}^{\prime}},{{\bf
x}^{\prime}})}\right)=R\cos^{-1}\left(\frac{1}{R^{2}}({\bf x},{{\bf
x}^{\prime}})\right).$ (9)
This is evident from the fact that the geodesics on ${\mathbf{S}}_{R}^{d}$ are
great circles, i.e., intersections of ${\mathbf{S}}_{R}^{d}$ with planes
through the origin of the ambient Euclidean space, with constant speed
parametrizations.
In any geodesic polar coordinate system, the geodesic distance between two
points on the submanifold is given by
$d({\bf x},{{\bf x}^{\prime}})=R\cos^{-1}\left(\frac{1}{R^{2}}({\bf x},{{\bf
x}^{\prime}})\right)=R\cos^{-1}\bigl{(}\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos\gamma\bigr{)},$
(10)
where $\gamma$ is the unique separation angle given in each polyspherical
coordinate system used to parametrize points on ${\mathbf{S}}^{d-1}$. For
instance, the separation angle $\gamma$ in standard hyperspherical coordinates
is given through
$\cos\gamma=\cos(\phi-\phi^{\prime})\prod_{i=1}^{d-2}\sin\theta_{i}{\sin\theta_{i}}^{\prime}+\sum_{i=1}^{d-2}\cos\theta_{i}{\cos\theta_{i}}^{\prime}\prod_{j=1}^{i-1}\sin\theta_{j}{\sin\theta_{j}}^{\prime}.$
(11)
Corresponding separation angle formulae for any hyperspherical coordinate
system used to parametrize points on ${\mathbf{S}}^{d-1}$ can be computed
using (9) and the associated formulae for the appropriate inner-products.
One can also compute the Riemannian (volume) measure $d{\rm vol}_{g}$ (see for
instance Section 3.4 in [17]), invariant under the isometry group $SO(d)$, of
the Riemannian manifold ${\mathbf{S}}_{R}^{d}$. For instance, in standard
hyperspherical coordinates (8) on ${\mathbf{S}}_{R}^{d}$ the volume measure is
given by
$\displaystyle d{\rm
vol}_{g}=R^{d}\sin^{d-1}\theta\,d\theta\,d\omega:=R^{d}\sin^{d-1}\theta\,d\theta\,\sin^{d-2}\theta_{d-1}\cdots\sin\theta_{2}\,d\theta_{1}\cdots
d\theta_{d-1}.$ (12)
The distance $r\in[0,\infty)$ along a geodesic, measured from the origin, is
given by $r=\theta R$. To show that the above volume measure (12) reduces to
the Euclidean volume measure at small distances (see for instance [22]), we
examine the limit of zero curvature. In order to do this, we take the limit
$\theta\to 0^{+}$ and $R\to\infty$ of the volume measure (12) which produces
$d{\rm vol}_{g}\sim R^{d-1}\sin^{d-1}\left(\frac{r}{R}\right)drd\omega\sim
r^{d-1}dr\,d\omega,$
which is the Euclidean measure in ${\mathbf{R}}^{d}$, expressed in standard
Euclidean hyperspherical coordinates. This measure is invariant under the
Euclidean motion group $E(d)$.
It will be useful below to express the Dirac delta function on
${\mathbf{S}}_{R}^{d}$. The Dirac delta function on the Riemannian manifold
${\mathbf{S}}_{R}^{d}$ with metric $g$ is defined for an open set
$U\subset{\mathbf{S}}_{R}^{d}$ with ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{S}}_{R}^{d}$ such that
$\displaystyle\int_{U}\delta_{g}({\bf x},{{\bf x}^{\prime}})d{\rm
vol}_{g}=\begin{cases}1&\mathrm{if}\ {{\bf x}^{\prime}}\in U,\\\
0&\mathrm{if}\ {{\bf x}^{\prime}}\notin U.\end{cases}$ (13)
For instance, using (12) and (13), in standard hyperspherical coordinates on
${\mathbf{S}}_{R}^{d}$ (8), we see that the Dirac delta function is given by
$\delta_{g}({\bf x},{{\bf
x}^{\prime}})=\frac{\delta(\theta-\theta^{\prime})}{R^{d}\sin^{d-1}\theta^{\prime}}\frac{\delta(\theta_{1}-\theta_{1}^{\prime})\cdots\delta(\theta_{d-1}-\theta_{d-1}^{\prime})}{\sin\theta_{2}^{\prime}\cdots\sin^{d-2}\theta_{d-1}^{\prime}}.$
### 2.1 Laplace’s equation on the hypersphere
Parametrizations of a submanifold embedded in Euclidean space can be given in
terms of coordinate systems whose coordinates are curvilinear. These are
coordinates based on some transformation that converts the standard Cartesian
coordinates in the ambient space to a coordinate system with the same number
of coordinates as the dimension of the submanifold in which the coordinate
lines are curved.
The Laplace–Beltrami operator (Laplacian) in curvilinear coordinates
${\mathbf{\xi}}=(\xi^{1},\ldots,\xi^{d})$ on a Riemannian manifold is given by
$\displaystyle\Delta=\sum_{i,j=1}^{d}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial\xi^{i}}\biggl{(}\sqrt{|g|}g^{ij}\frac{\partial}{\partial\xi^{j}}\biggr{)},$
(14)
where $|g|=|\det(g_{ij})|,$ the metric is given by
$\displaystyle ds^{2}=\sum_{i,j=1}^{d}g_{ij}d\xi^{i}d\xi^{j},$ (15)
and
$\sum_{i=1}^{d}g_{ki}g^{ij}=\delta_{k}^{j},$
where $\delta_{i}^{j}\in\\{0,1\\}$ is the Kronecker delta
$\displaystyle\delta_{i}^{j}:=\begin{cases}1&\mathrm{if}\ i=j,\\\
0&\mathrm{if}\ i\neq j,\end{cases}$ (16)
for $i,j\in{\mathbf{Z}}$. The relationship between the metric tensor
$G_{ij}=\mathrm{diag}(1,\ldots,1)$ in the ambient space and $g_{ij}$ of (14)
and (15) is given by
$g_{ij}({\mathbf{\xi}})=\sum_{k,l=0}^{d}G_{kl}\frac{\partial
x^{k}}{\partial\xi^{i}}\frac{\partial x^{l}}{\partial\xi^{j}}.$
The Riemannian metric in a geodesic polar coordinate system on the submanifold
${\mathbf{S}}_{R}^{d}$ is given by
$\displaystyle ds^{2}=R^{2}\big{(}d\theta^{2}+\sin^{2}\theta\
d\gamma^{2}\big{)},$ (17)
where an appropriate expression for $\gamma$ in a curvilinear coordinate
system is given. If one combines (8), (11), (14) and (17), then in a geodesic
polar coordinate system, Laplace’s equation on ${\mathbf{S}}_{R}^{d}$ is given
by
$\displaystyle\Delta
f=\frac{1}{R^{2}}\left[\frac{\partial^{2}f}{\partial\theta^{2}}+(d-1)\cot\theta\frac{\partial
f}{\partial\theta}+\frac{1}{\sin^{2}\theta}\Delta_{{\mathbf{S}}^{d-1}}f\right]=0,$
(18)
where $\Delta_{{\mathbf{S}}^{d-1}}$ is the corresponding Laplace–Beltrami
operator on ${\mathbf{S}}^{d-1}$.
## 3 A Green’s function on the hypersphere
### 3.1 Harmonics in geodesic polar coordinates
The harmonics in a geodesic polar coordinate system are given in terms of a
‘radial’ solution (‘radial’ harmonics) multiplied by the angular solution
(angular harmonics).
Using polyspherical coordinates on ${\mathbf{S}}^{d-1},$ one can compute the
normalized hyperspherical harmonics in this space by solving the Laplace
equation using separation of variables. This results in a general procedure
which, for instance, is given explicitly in [19, 20]. These angular harmonics
are given as general expressions involving trigonometric functions, Gegenbauer
polynomials and Jacobi polynomials. The angular harmonics are eigenfunctions
of the Laplace–Beltrami operator on ${\mathbf{S}}^{d-1}$ which satisfy the
following eigenvalue problem (see for instance (12.4) and Corollary 2 to
Theorem 10.5 in [33])
$\displaystyle\Delta_{{\mathbf{S}}^{d-1}}Y_{l}^{K}({\mathbf{\widehat{x}}})=-l(l+d-2)Y_{l}^{K}({\mathbf{\widehat{x}}}),$
(19)
where ${\mathbf{\widehat{x}}}\in{\mathbf{S}}^{d-1}$,
$Y_{l}^{K}({\mathbf{\widehat{x}}})$ are normalized angular hyperspherical
harmonics, $l\in{\mathbf{N}}_{0}$ is the angular momentum quantum number, and
$K$ stands for the set of $(d-2)$-quantum numbers identifying degenerate
harmonics for each $l$ and $d$. The degeneracy
$(2l+d-2)\frac{(d-3+l)!}{l!(d-2)!}$
(see (9.2.11) in [36]), tells you how many linearly independent solutions
exist for a particular $l$ value and dimension $d$. The angular hyperspherical
harmonics are normalized such that
$\int_{{\mathbf{S}}^{d-1}}Y_{l}^{K}({\mathbf{\widehat{x}}})\overline{Y_{l^{\prime}}^{K^{\prime}}({\mathbf{\widehat{x}}})}d\omega=\delta_{l}^{l^{\prime}}\delta_{K}^{K^{\prime}},$
where $d\omega$ is the Riemannian (volume) measure on ${\mathbf{S}}^{d-1}$,
which is invariant under the isometry group $SO(d)$ (cf. (12)), and for
$x+iy=z\in{\mathbf{C}}$, $\overline{z}=x-iy$, represents complex conjugation.
The angular solutions (hyperspherical harmonics) are well-known (see Chapter
IX in [36] and Chapter 11 [9]). The generalized Kronecker delta symbol
$\delta_{K}^{K^{\prime}}$ (cf. (16)) is defined such that it equals 1 if all
of the $(d-2)$-quantum numbers identifying degenerate harmonics for each $l$
and $d$ coincide, and equals zero otherwise.
We now focus on ‘radial’ solutions of Laplace’s equation on
${\mathbf{S}}_{R}^{d}$, which satisfy the following ordinary differential
equation (cf. (18) and (19))
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+(d-1)\cot\theta\frac{du}{d\theta}-\frac{l(l+d-2)}{\sin^{2}\theta}u=0.$
(20)
Four solutions of this ordinary differential equation
$u_{1\pm}^{d,l},u_{2\pm}^{d,l}:(-1,1)\to{\mathbf{C}}$ are given by
${\displaystyle
u_{1\pm}^{d,l}(\cos\theta):=\frac{1}{(\sin\theta)^{d/2-1}}{\mathrm{P}}_{d/2-1}^{\pm(d/2-1+l)}(\cos\theta)},$
and
$\displaystyle
u_{2\pm}^{d,l}(\cos\theta):=\frac{1}{(\sin\theta)^{d/2-1}}{\mathrm{Q}}_{d/2-1}^{\pm(d/2-1+l)}(\cos\theta),$
(21)
where
${\mathrm{P}}_{\nu}^{\mu},{\mathrm{Q}}_{\nu}^{\mu}:(-1,1)\to{\mathbf{C}}$ are
Ferrers functions of the first and second kind (associated Legendre functions
of the first and second kind on the cut). The Ferrers functions of the first
and second kind (see Chapter 14 in [27]) can be defined respectively in terms
of a sum over two Gauss hypergeometric functions, for all
$\nu,\mu\in{\mathbf{C}}$ such that $\nu+\mu\not\in-{\mathbf{N}}$,
$\displaystyle{\mathrm{P}}_{\nu}^{\mu}(x):=\frac{2^{\mu+1}}{\sqrt{\pi}}\sin\left[\frac{\pi}{2}(\nu+\mu)\right]\frac{\Gamma\left(\frac{\nu+\mu+2}{2}\right)}{\Gamma\left(\frac{\nu-\mu+1}{2}\right)}x(1-x^{2})^{-\mu/2}{}_{2}F_{1}\left(\frac{1-\nu-\mu}{2},\frac{\nu-\mu+2}{2};\frac{3}{2};x^{2}\right)$
$\displaystyle\phantom{{\mathrm{P}}_{\nu}^{\mu}(x):=}{}+\frac{2^{\mu}}{\sqrt{\pi}}\cos\left[\frac{\pi}{2}(\nu+\mu)\right]\frac{\Gamma\left(\frac{\nu+\mu+1}{2}\right)}{\Gamma\left(\frac{\nu-\mu+2}{2}\right)}(1-x^{2})^{-\mu/2}{}_{2}F_{1}\left(\frac{-\nu-\mu}{2},\frac{\nu-\mu+1}{2};\frac{1}{2};x^{2}\right)$
(cf. (14.3.11) in [27]), and
$\displaystyle{\mathrm{Q}}_{\nu}^{\mu}(x):=\sqrt{\pi}2^{\mu}\cos\left[\frac{\pi}{2}(\nu+\mu)\right]\frac{\Gamma\\!\left(\frac{\nu+\mu+2}{2}\right)}{\Gamma\\!\left(\frac{\nu-\mu+1}{2}\right)}x(1-x^{2})^{-\mu/2}{}_{2}F_{1}\left(\frac{1-\nu-\mu}{2},\frac{\nu-\mu+2}{2};\frac{3}{2};x^{2}\right)$
$\displaystyle{}-\sqrt{\pi}2^{\mu-1}\sin\left[\frac{\pi}{2}(\nu+\mu)\right]\frac{\Gamma\left(\frac{\nu+\mu+1}{2}\right)}{\Gamma\left(\frac{\nu-\mu+2}{2}\right)}(1-x^{2})^{-\mu/2}{}_{2}F_{1}\left(\frac{-\nu-\mu}{2},\frac{\nu-\mu+1}{2};\frac{1}{2};x^{2}\right)$
(22)
(cf. (14.3.12) in [27]). The Gauss hypergeometric function
${}_{2}F_{1}:{\mathbf{C}}\times{\mathbf{C}}\times({\mathbf{C}}\setminus-{\mathbf{N}}_{0})\times\\{z\in{\mathbf{C}}:|z|<1\\}\to{\mathbf{C}}$,
can be defined in terms of the infinite series
${}_{2}F_{1}(a,b;c;z):=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}z^{n}$
(see (15.2.1) in [27]), and elsewhere in $z$ by analytic continuation. On the
unit circle $|z|=1$, the Gauss hypergeometric series converges absolutely if
$\mbox{Re}\,(c-a-b)\in(0,\infty),$ converges conditionally if $z\neq 1$ and
$\mbox{Re}\,(c-a-b)\in(-1,0],$ and diverges if
$\mbox{Re}\,(c-a-b)\in(-\infty,-1]$. For $z\in{\mathbf{C}}$ and
$n\in{\mathbf{N}}_{0}$, the Pochhammer symbol $(z)_{n}$ (also referred to as
the rising factorial) is defined as (cf. (5.2.4) in [27])
$(z)_{n}:=\prod_{i=1}^{n}(z+i-1).$
The Pochhammer symbol (rising factorial) is expressible in terms of gamma
functions as (5.2.5) in [27]
$(z)_{n}=\frac{\Gamma(z+n)}{\Gamma(z)},$
for all $z\in{\mathbf{C}}\setminus-{\mathbf{N}}_{0}$. The gamma function
$\Gamma:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$ (see Chapter 5
in [27]) is an important combinatoric function and is ubiquitous in special
function theory. It is naturally defined over the right-half complex plane
through Euler’s integral (see (5.2.1) in [27])
$\Gamma(z):=\int_{0}^{\infty}t^{z-1}e^{-t}dt,$
$\mbox{Re}\,z>0$. The Euler reflection formula allows one to obtain values of
the gamma function in the left-half complex plane (cf. (5.5.3) in [27]),
namely
$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z},$
$0<\mbox{Re}\,z<1,$ for $\mbox{Re}\,z=0$, $z\neq 0$, and then for $z$ shifted
by integers using the following recurrence relation (see (5.5.1) in [27])
$\Gamma(z+1)=z\Gamma(z).$
An important formula which the gamma function satisfies is the duplication
formula (i.e., (5.5.5) in [27])
$\displaystyle\Gamma(2z)=\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma\left(z+\frac{1}{2}\right),$
(23)
provided $2z\not\in-{\mathbf{N}}_{0}$.
Due to the fact that the space ${\mathbf{S}}_{R}^{d}$ is homogeneous with
respect to its isometry group, the orthogonal group $O(d)$, and therefore an
isotropic manifold, we expect that there exist a fundamental solution on this
space with spherically symmetric dependence. We specifically expect these
solutions to be given in terms of associated Legendre functions of the second
kind on the cut with argument given by $\cos\theta$. This associated Legendre
function naturally fits our requirements because it is singular at $\theta=0$,
whereas the associated Legendre functions of the first kind, with the same
argument, is regular at $\theta=0$. We require there to exist a singularity at
the origin of a fundamental solution of Laplace’s equation on
${\mathbf{S}}^{d}_{R}$, since it is a manifold and must behave locally like a
Euclidean fundamental solution of Laplace’s equation which also has a
singularity at the origin.
### 3.2 Fundamental solution of the Laplace’s equation on the hypersphere
In computing a fundamental solution of the Laplacian on
${\mathbf{S}}_{R}^{d}$, we know that
$\displaystyle-\Delta{\mathcal{S}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=\delta_{g}({\bf x},{{\bf x}^{\prime}}),$ (24)
where $g$ is the Riemannian metric on ${\mathbf{S}}_{R}^{d}$ (e.g., (17)) and
$\delta_{g}$ is the Dirac delta function on the manifold
${\mathbf{S}}_{R}^{d}$ (e.g., (13)). In general since we can add any harmonic
function to a fundamental solution of Laplace’s equation and still have a
fundamental solution, we will use this freedom to make our fundamental
solution as simple as possible. It is reasonable to expect that there exists a
particular spherically symmetric fundamental solution ${\mathcal{S}}_{R}^{d}$
on the hypersphere with pure ‘radial’, $\theta:=d({\bf x},{{\bf x}^{\prime}})$
(e.g., (10)), and constant angular dependence due to the influence of the
point-like nature of the Dirac delta function in (24). For a spherically
symmetric solution to the Laplace equation, the corresponding
$\Delta_{{\mathbf{S}}^{d-1}}$ term in (18) vanishes since only the $l=0$ term
survives in (20). In other words we expect there to exist a fundamental
solution of Laplace’s equation on ${\mathbf{S}}_{R}^{d}$ such that
${\mathcal{S}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})=f(\theta)$ (cf. (10)),
where $R$ is a parameter of this fundamental solution.
We have proven that on the $R$-radius hypersphere ${\mathbf{S}}_{R}^{d}$, a
Green’s function for the Laplace operator (fundamental solution of Laplace’s
equation) can be given as follows.
###### Theorem 3.1.
Let $d\in\\{2,3,\ldots\\}.$ Define ${\mathcal{I}}_{d}:(0,\pi)\to{\mathbf{R}}$
as
${\mathcal{I}}_{d}(\theta):=\int_{\theta}^{\pi/2}\frac{dx}{\sin^{d-1}x},$
${{\bf x}},{{{\bf x}^{\prime}}}\in{\mathbf{S}}_{R}^{d}$, and
${\mathcal{S}}_{R}^{d}:({\mathbf{S}}_{R}^{d}\times{\mathbf{S}}_{R}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{S}}_{R}^{d}\\}\to{\mathbf{R}}$ defined such
that
${\mathcal{S}}_{R}^{d}({\bf x},{\bf
x}^{\prime}):={\displaystyle\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}R^{d-2}}{\mathcal{I}}_{d}(\theta)},$
where $\theta:=\cos^{-1}\left([{\widehat{\bf x}},{{\widehat{\bf
x}^{\prime}}}]\right)$ is the geodesic distance between ${\widehat{\bf x}}$
and ${{\widehat{\bf x}^{\prime}}}$ on the unit radius hypersphere
${\mathbf{S}}^{d}$, with ${\widehat{\bf x}}={\bf x}/R$, ${{\widehat{\bf
x}^{\prime}}}={{\bf x}^{\prime}}/R$, then ${\mathcal{S}}_{R}^{d}$ is a
fundamental solution for $-\Delta$ where $\Delta$ is the Laplace–Beltrami
operator on ${\mathbf{S}}_{R}^{d}$. Moreover,
$\displaystyle{\mathcal{I}}_{d}(\theta)=\begin{cases}\displaystyle\frac{(d-3)!!}{(d-2)!!}\biggl{[}\log\cot\frac{\theta}{2}+\cos\theta\sum_{k=1}^{d/2-1}\frac{(2k-2)!!}{(2k-1)!!}\frac{1}{\sin^{2k}\theta}\biggr{]}&\mathrm{if}\
d\ \mathrm{even},\vspace{2mm}\\\
\left\\{\begin{array}[]{l}\displaystyle\left(\frac{d-3}{2}\right)!\sum_{k=1}^{(d-1)/2}\frac{\cot^{2k-1}\theta}{(2k-1)(k-1)!((d-2k-1)/2)!},\vspace{2mm}\\\
\mathrm{or}\\\
\displaystyle\frac{(d-3)!!}{(d-2)!!}\cos\theta\sum_{k=1}^{(d-1)/2}\frac{(2k-3)!!}{(2k-2)!!}\frac{1}{\sin^{2k-1}\theta},\end{array}\right\\}&\mathrm{if}\
d\ \mathrm{odd},\end{cases}\vspace{2mm}$
$\displaystyle\phantom{{\mathcal{I}}_{d}(\theta)}{}=\begin{cases}\displaystyle\cos\theta\
{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right),\vspace{2mm}\\\
\displaystyle\frac{\cos\theta}{\sin^{d-2}\theta}\
{}_{2}F_{1}\left(1,\frac{3-d}{2};\frac{3}{2};\cos^{2}\theta\right),\vspace{2mm}\\\
\displaystyle\frac{(d-2)!}{\displaystyle\Gamma\left(d/2\right)2^{d/2-1}}\frac{1}{(\sin\theta)^{d/2-1}}{\mathrm{Q}}_{d/2-1}^{1-d/2}(\cos\theta).\end{cases}$
In the rest of this section, we develop the material in order to prove this
theorem.
Since a spherically symmetric choice for a fundamental solution satisfies
Laplace’s equation everywhere except at the origin, we may first set
$g=f^{\prime}$ in (18) and solve the first-order equation
$g^{\prime}+(d-1)\cos\theta\;g=0,$
which is integrable and clearly has the general solution
$\displaystyle g(\theta)=\frac{df}{d\theta}=c_{0}(\sin\theta)^{1-d},$ (25)
where $c_{0}\in{\mathbf{R}}$ is a constant. Now we integrate (25) to obtain a
fundamental solution for the Laplacian on ${\mathbf{S}}_{R}^{d}$
$\displaystyle{\mathcal{S}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=c_{0}{\mathcal{I}}_{d}(\theta)+c_{1},$ (26)
where ${\mathcal{I}}_{d}:(0,\pi)\to{\mathbf{R}}$ is defined as
$\displaystyle{\mathcal{I}}_{d}(\theta):=\int_{\theta}^{\pi/2}\frac{dx}{\sin^{d-1}x},$
(27)
and $c_{0},c_{1}\in{\mathbf{R}}$ are constants which depend on $d$ and $R$.
Notice that we can add any harmonic function to (26) and still have a
fundamental solution of the Laplacian since a fundamental solution of the
Laplacian must satisfy
$\int_{{\mathbf{S}}_{R}^{d}}(-\Delta\varphi)({{\bf
x}^{\prime}}){\mathcal{S}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})\,d{\rm
vol}_{g}^{\prime}=\varphi({\bf x}),$
for all $\varphi\in{\mathcal{S}}({\mathbf{S}}_{R}^{d}),$ where ${\mathcal{S}}$
is the space of test functions, and $d{\rm vol}_{g}^{\prime}$ is the
Riemannian (volume) measure on ${\mathbf{S}}_{R}^{d}$ in the primed
coordinates. Notice that our fundamental solution of Laplace’s equation on the
hypersphere ((26), (27)) has the property that it tends towards $+\infty$ as
$\theta\to 0^{+}$ and tends towards $-\infty$ as $\theta\to\pi^{-}$. Therefore
our fundamental solution attains all real values. As an aside, by the
definition therein (see [15, 16]), ${\mathbf{S}}_{R}^{d}$ is a parabolic
manifold. Since the hypersphere ${\mathbf{S}}_{R}^{d}$ is bi-hemispheric, we
expect that a fundamental solution of Laplace’s equation on the hypersphere
should vanish at $\theta=\pi/2$. It is therefore convenient to set $c_{1}=0$
leaving us with
$\displaystyle{\mathcal{S}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=c_{0}{\mathcal{I}}_{d}(\theta).$ (28)
In Euclidean space ${\mathbf{R}}^{d}$, a Green’s function for Laplace’s
equation (fundamental solution for the Laplacian) is well-known and is given
by the following expression (see [12, p. 94], [13, p. 17], [4, p. 211], [7, p.
6]). Let $d\in{\mathbf{N}}$. Define
$\displaystyle{\mathcal{G}}^{d}({\bf x},{\bf
x}^{\prime})=\begin{cases}\displaystyle\frac{\Gamma(d/2)}{2\pi^{d/2}(d-2)}\|{\bf
x}-{\bf x}^{\prime}\|^{2-d}&\mathrm{if}\ d=1\mathrm{\ or\ }d\geq
3,\vspace{1mm}\\\ \displaystyle\frac{1}{2\pi}\log\|{\bf x}-{\bf
x}^{\prime}\|^{-1}&\mathrm{if}\ d=2,\end{cases}$ (29)
then ${\mathcal{G}}^{d}$ is a fundamental solution for $-\Delta$ in Euclidean
space ${\mathbf{R}}^{d}$, where $\Delta$ is the Laplace operator in
${\mathbf{R}}^{d}$. Note that most authors only present the above theorem for
the case $d\geq 2$ but it is easily-verified to also be valid for the case
$d=1$ as well.
The hypersphere ${\mathbf{S}}_{R}^{d}$, being a manifold, must behave locally
like Euclidean space ${\mathbf{R}}^{d}$. Therefore for small $\theta$ we have
$e^{\theta}\simeq 1+\theta$ and $e^{-\theta}\simeq 1-\theta$ and in that
limiting regime
${\mathcal{I}}_{d}(\theta)\approx\int_{\theta}^{1}\frac{dx}{x^{d-1}}\simeq\begin{cases}-\log\theta&\mathrm{if}\
d=2,\vspace{1mm}\\\ {\displaystyle\frac{1}{\theta^{d-2}}}&\mathrm{if}\ d\geq
3,\end{cases}$
which has exactly the same singularity as a Euclidean fundamental solution.
Therefore the proportionality constant $c_{0}$ is obtained by matching locally
to a Euclidean fundamental solution
$\displaystyle{\mathcal{S}}_{R}^{d}=c_{0}{\mathcal{I}}_{d}\simeq{\mathcal{G}}^{d},$
(30)
in a small neighborhood of the singularity at ${\bf x}={{\bf x}^{\prime}},$ as
the curvature vanishes, i.e., $R\to\infty$.
We have shown how to compute a fundamental solution of the Laplace–Beltrami
operator on the hypersphere in terms of an improper integral (27). We would
now like to express this integral in terms of well-known special functions. A
fundamental solution ${\mathcal{I}}_{d}$ can be computed using elementary
methods through its definition (27). In $d=2$ we have
${\mathcal{I}}_{2}(\theta)=\int_{\theta}^{\pi/2}\frac{dx}{\sin
x}=\frac{1}{2}\log\frac{\cos\theta+1}{\cos\theta-1}=\log\cot\frac{\theta}{2},$
and in $d=3$ we have
${\mathcal{I}}_{3}(\theta)=\int_{\theta}^{\pi/2}\frac{dx}{\sin^{2}x}=\cot\theta.$
In $d\in\\{4,5,6,7\\}$ we have
$\displaystyle{\mathcal{I}}_{4}(\theta)=\frac{1}{2}\log\cot\frac{\theta}{2}+\frac{\cos\theta}{2\sin^{2}\theta},$
$\displaystyle{\mathcal{I}}_{5}(\theta)=\cot\theta+\frac{1}{3}\cot^{3}\theta,$
$\displaystyle{\mathcal{I}}_{6}(\theta)=\frac{3}{8}\log\cot\frac{\theta}{2}+\frac{3\cos\theta}{8\sin^{2}\theta}+\frac{\cos\theta}{4\sin^{2}\theta},\qquad\mathrm{and}$
$\displaystyle{\mathcal{I}}_{7}(\theta)=\cot\theta+\frac{2}{3}\cot^{3}\theta+\frac{1}{5}\cot^{5}\theta.$
Now we prove several equivalent finite summation expressions for
${\mathcal{I}}_{d}(\theta)$. We wish to compute the antiderivative
$\mathfrak{I}_{m}:(0,\pi)\to{\mathbf{R}}$, which is defined as
$\mathfrak{I}_{m}(x):=\int\frac{dx}{\sin^{m}x},$
where $m\in{\mathbf{N}}$. This antiderivative satisfies the following
recurrence relation
$\displaystyle\mathfrak{I}_{m}(x)=-\frac{\cos
x}{(m-1)\sin^{m-1}x}+\frac{(m-2)}{(m-1)}\mathfrak{I}_{m-2}(x),$ (31)
which follows from the identity
$\frac{1}{\sin^{m}x}=\frac{1}{\sin^{m-2}x}+\frac{\cos x}{\sin^{m}x}\cos x,$
and integration by parts. The antiderivative $\mathfrak{I}_{m}(x)$ naturally
breaks into two separate classes, namely
$\displaystyle\int\frac{dx}{\sin^{2n+1}x}=-\frac{(2n-1)!!}{(2n)!!}\left[\log\cot\frac{x}{2}+\cos
x\sum_{k=1}^{n}\frac{(2k-2)!!}{(2k-1)!!}\frac{1}{\sin^{2k}x}\right]+C,$ (32)
and
$\displaystyle\int\frac{dx}{\sin^{2n}x}=\begin{cases}\displaystyle-\frac{(2n-2)!!}{(2n-1)!!}\cos
x\sum_{k=1}^{n}\frac{(2k-3)!!}{(2k-2)!!}\frac{1}{\sin^{2k-1}x}+C,\qquad\mbox{or}\vspace{1mm}\\\
\displaystyle-(n-1)!\sum_{k=1}^{n}\frac{\cot^{2k-1}x}{(2k-1)(k-1)!(n-k)!}+C,\end{cases}$
(33)
where $C$ is a constant. The double factorial
$(\cdot)!!:\\{-1,0,1,\ldots\\}\to{\mathbf{N}}$ is defined by
$n!!:=\begin{cases}\displaystyle n\cdot(n-2)\cdots 2&\mathrm{if}\ n\
\mathrm{even}\geq 2,\\\ \displaystyle n\cdot(n-2)\cdots 1&\mathrm{if}\ n\
\mathrm{odd}\geq 1,\\\ \displaystyle 1&\mathrm{if}\
n\in\\{-1,0\\}.\end{cases}$
Note that $(2n)!!=2^{n}n!$ for $n\in{\mathbf{N}}_{0}$. The finite summation
formulae for $\mathfrak{I}_{m}(x)$ all follow trivially by induction using
(31) and the binomial expansion (cf. (1.2.2) in [27])
$(1+\cos^{2}x)^{n}=n!\sum_{k=0}^{n}\frac{\cot^{2k}x}{k!(n-k)!}.$
The formulae (32) and (33) are essentially equivalent to (2.515.1–2) in [14],
except (2.515.2) is in error with the factor $28^{k}$ being replaced with
$2^{k}$. This is also verified in the original citing reference [35]. By
applying the limits of integration from the definition of
${\mathcal{I}}_{d}(\theta)$ in (27) to (32) and (33) we obtain the following
finite summation expression
$\displaystyle{\mathcal{I}}_{d}(\theta)=\begin{cases}\displaystyle\frac{(d-3)!!}{(d-2)!!}\left[\log\cot\frac{\theta}{2}+\cos\theta\sum_{k=1}^{d/2-1}\frac{(2k-2)!!}{(2k-1)!!}\frac{1}{\sin^{2k}\theta}\right]&\mathrm{if}\
d\ \mathrm{even},\vspace{1mm}\\\
\left\\{\begin{array}[]{l}\displaystyle\left(\frac{d-3}{2}\right)!\sum_{k=1}^{(d-1)/2}\frac{\cot^{2k-1}\theta}{(2k-1)(k-1)!((d-2k-1)/2)!},\vspace{2mm}\\\
\mathrm{or}\\\
\displaystyle\frac{(d-3)!!}{(d-2)!!}\cos\theta\sum_{k=1}^{(d-1)/2}\frac{(2k-3)!!}{(2k-2)!!}\frac{1}{\sin^{2k-1}\theta},\end{array}\right\\}&\mathrm{if}\
d\ \mathrm{odd}.\end{cases}$ (34)
Moreover, the antiderivative (indefinite integral) can be given in terms of
the Gauss hypergeometric function as
$\displaystyle\int\frac{d\theta}{\sin^{d-1}\theta}=-\cos\theta\;{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right)+C,$
(35)
where $C\in{\mathbf{R}}$. This is verified as follows. By using
$\frac{d}{dz}\;{}_{2}F_{1}(a,b;c;z)=\frac{ab}{c}\;{}_{2}F_{1}(a+1,b+1;c+1;z)$
(see (15.5.1) in [27]), and the chain rule, we can show that
$\displaystyle-\frac{d}{d\theta}\cos\theta\;{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right)$
$\displaystyle\qquad=\sin\theta\left[{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right)+\frac{d}{3}\cos^{2}\theta\;{}_{2}F_{1}\left(\frac{3}{2},\frac{d+2}{2};\frac{5}{2};\cos^{2}\theta\right)\right].$
The second hypergeometric function can be simplified using Gauss’ relations
for contiguous hypergeometric functions, namely
$z\;{}_{2}F_{1}(a+1,b+1;c+1;z)=\frac{c}{a-b}\bigl{[}{}_{2}F_{1}(a,b+1;c;z)-{}_{2}F_{1}(a+1,b;c;z)\bigr{]}$
(see [8, p. 58]), and
${}_{2}F_{1}(a,b+1;c;z)=\frac{b-a}{b}{}_{2}F_{1}(a,b;c;z)+\frac{a}{b}\;{}_{2}F_{1}(a+1,b;c;z)$
(see (15.5.12) in [27]). By applying these formulae, the term with the
hypergeometric function cancels leaving only a term which is proportional to a
binomial through
${}_{2}F_{1}(a,b;b;z)=(1-z)^{-a}$
(see (15.4.6) in [27]), which reduces to $1/\sin^{d-1}\theta$. By applying the
limits of integration from the definition of ${\mathcal{I}}_{d}(\theta)$ in
(27) to (35) we obtain the following Gauss hypergeometric representation
$\displaystyle{\mathcal{I}}_{d}(\theta)=\cos\theta\;{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right).$
(36)
Using (36), we can write another expression for ${\mathcal{I}}_{d}(\theta)$.
Applying Eulers’s transformation
${}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}\;{}_{2}F_{1}\left(c-a,c-b;c;z\right)$
(see (2.2.7) in [2]), to (36) produces
${\mathcal{I}}_{d}(\theta)=\frac{\cos\theta}{\sin^{d-2}\theta}\;{}_{2}F_{1}\left(1,\frac{3-d}{2};\frac{3}{2};\cos^{2}\theta\right).$
Our derivation for a fundamental solution of Laplace’s equation on the
$R$-radius hypersphere in terms of Ferrers function of the second kind
(associated Legendre function of the second kind on the cut) is as follows. If
we let $\nu+\mu=0$ in the definition of Ferrers function of the second kind
${\mathrm{Q}}_{\nu}^{\mu}:(-1,1)\to{\mathbf{C}}$ (22), we derive
${\mathrm{Q}}_{\nu}^{-\nu}(x)=\frac{\sqrt{\pi}}{2^{\nu}}\frac{x(1-x^{2})^{\nu/2}}{\Gamma\left(\nu+\frac{1}{2}\right)}\;{}_{2}F_{1}\left(\frac{1}{2},\nu+1;\frac{3}{2};x^{2}\right),$
for all $\nu\in{\mathbf{C}}$. If we let $\nu=d/2-1$ and substitute
$x=\cos\theta$, then we have
$\displaystyle{\mathrm{Q}}_{d/2-1}^{1-d/2}(\cos\theta)=\frac{\sqrt{\pi}}{2^{d/2-1}}\frac{\cos\theta\sin^{d/2-1}\theta}{\Gamma\left(\frac{d-1}{2}\right)}\;{}_{2}F_{1}\left(\frac{1}{2},\frac{d}{2};\frac{3}{2};\cos^{2}\theta\right).$
(37)
Using the duplication formula for gamma functions (23), then through (37) we
have
$\displaystyle{\mathcal{I}}_{d}(\theta)=\frac{(d-2)!}{\Gamma(d/2)2^{d/2-1}}\frac{1}{\sin^{d/2-1}\theta}{\mathrm{Q}}_{d/2-1}^{1-d/2}(\cos\theta).$
(38)
We have therefore verified that the harmonics computed in Section 3.1, namely
$u_{2+}^{d,0}$ (21), give an alternate form for a fundamental solution of the
Laplacian on the hypersphere.
Note that as a result of our proof, we see that the relevant associated
Legendre functions of the second kind on the cut for $d\in\\{2,3,4,5,6,7\\}$
are (cf. (34) and (38))
$\displaystyle{\mathrm{Q}}_{0}(\cos\theta)=\log\cot\frac{\theta}{2},$
$\displaystyle\frac{1}{(\sin\theta)^{1/2}}{\mathrm{Q}}_{1/2}^{-1/2}(\cos\theta)=\sqrt{\frac{\pi}{2}}\cot\theta,$
$\displaystyle\frac{1}{\sin\theta}{\mathrm{Q}}_{1}^{-1}(\cos\theta)=\frac{1}{2}\log\cot\frac{\theta}{2}+\frac{\cos\theta}{2\sin^{2}\theta},$
$\displaystyle\frac{1}{(\sin\theta)^{3/2}}{\mathrm{Q}}_{3/2}^{-3/2}(\cos\theta)=\frac{1}{2}\sqrt{\frac{\pi}{2}}\left(\cot\theta+\frac{1}{3}\cot^{3}\theta\right),$
$\displaystyle\frac{1}{(\sin\theta)^{2}}{\mathrm{Q}}_{2}^{-2}(\cos\theta)=\frac{1}{8}\log\cot\frac{\theta}{2}+\frac{\cos\theta}{8\sin^{2}\theta}+\frac{\cos\theta}{12\sin^{4}\theta},\qquad\mbox{and}$
$\displaystyle\frac{1}{(\sin\theta)^{5/2}}{\mathrm{Q}}_{5/2}^{-5/2}(\cos\theta)=\frac{1}{8}\sqrt{\frac{\pi}{2}}\left(\cot\theta+\frac{2}{3}\cot^{3}\theta+\frac{1}{5}\cot^{5}\theta\right).$
The constant $c_{0}$ in a fundamental solution for the Laplace operator on the
hypersphere ${\mathbf{S}}_{R}^{d}$ (28) is computed by locally matching up,
through (30), to the singularity of a fundamental solution for the Laplace
operator in Euclidean space (29). The coefficient $c_{0}$ depends on $d$ and
$R$. For $d\geq 3$ we take the asymptotic expansion for
$c_{0}{\mathcal{I}}_{d}(\theta)$ as $\theta\to 0^{+}$, and match this to a
fundamental solution for Euclidean space (29). This yields
$\displaystyle\displaystyle c_{0}=\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}}.$
(39)
For $d=2$ we take the asymptotic expansion for
$c_{0}{\mathcal{I}}_{2}(\theta)=-c_{0}\log\tan\frac{\theta}{2}\simeq
c_{0}\log\|{\bf x}-{{\bf x}^{\prime}}\|^{-1},$
as $\theta\to 0^{+}$, and match this to $\displaystyle{\mathcal{G}}^{2}({\bf
x},{{\bf x}^{\prime}})=(2\pi)^{-1}\log\|{\bf x}-{{\bf x}^{\prime}}\|^{-1},$
therefore $\displaystyle c_{0}=(2\pi)^{-1}$. This exactly matches (39) for
$d=2$. The $R$ dependence of $c_{0}$ originates from (27), where $x$ and
$\theta$ represents geodesic distances (cf. (10)). The distance
$r\in[0,\infty)$ along a geodesic, as measured from the origin of
${\mathbf{S}}_{R}^{d}$, is given by $r=\theta R$. To show that a fundamental
solution (28) reduces to the Euclidean fundamental solution at small distances
(see for instance [22]), we examine the limit of zero curvature. In order to
do this, we take the limit $\theta\to 0^{+}$ and $R\to\infty$ of (27) with the
substitution $x=r/R$ which produces a factor of $R^{d-2}$. So a fundamental
solution of Laplace’s equation on the Riemannian manifold
${\mathbf{S}}_{R}^{d}$ is given by
${\mathcal{S}}_{R}^{d}({\bf x},{\bf
x}^{\prime}):={\displaystyle\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}R^{d-2}}{\mathcal{I}}_{d}\left(\theta\right)}.$
The proof of Theorem 3.1 is complete.
Apart from the well-known historical results in two and three dimensions, the
closed form expressions for a fundamental solution of Laplace’s equation on
the $R$-radius hypersphere given by Theorem 3.1 in Section 3.2 appear to be
new. Furthermore, the Ferrers function (associated Legendre) representations
in Section 3.1 for the radial harmonics on the $R$-radius hypersphere do not
appear to be have previously appeared in the literature.
### Acknowledgements
Much thanks to Ernie Kalnins, Willard Miller Jr., George Pogosyan, and Charles
Clark, and for valuable discussions. I would like to express my gratitude to
the anonymous referees and an editor at SIGMA whose helpful comments improved
this paper. This work was conducted while H.S. Cohl was a National Research
Council Research Postdoctoral Associate in the Information Technology
Laboratory at the National Institute of Standards and Technology,
Gaithersburg, Maryland, USA.
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|
arxiv-papers
| 2011-08-18T08:24:00 |
2024-09-04T02:49:21.622939
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Howard S. Cohl",
"submitter": "Howard S. Cohl",
"url": "https://arxiv.org/abs/1108.3679"
}
|
1108.3685
|
# Dickson algebras are atomic at $p$
Nondas E. Kechagias Department of Mathematics, University of Ioannina,
Greece, 45110 nkechag@uoi.gr http://www.math.uoi.gr/~nondas_k
###### Abstract.
The notion of atomicity defined by Cohen, Moore and Neisendorfer is studied
for the Dickson algebras. Not any ring of invariants respects this property.
It depends on the property of the Dickson algebra that given any monomial $d$
there exists a sequence of Steenrod operations $P\left(\Gamma,d\right)$ such
that $P\left(\Gamma,d\right)d$ becomes a $p$-th power of the top Dickson
algebra generator.
###### Key words and phrases:
Dickson algebra, Steenrod algebra, atomic objects
###### 2000 Mathematics Subject Classification:
Primary 13A50; Secondary 55P10
This paper is in final form and no version of it will be submitted for
publication elsewhere.
## 1\. Statement of results
The term atomic was introduced by Cohen, Moore and Neisendorfer in ([1]) to
answer the question of whether a given space admits a nontrivial product
decomposition up to homotopy. We consider the analogue question for the
Dickson algebra. Let $V^{n}$ denote an $n$-dimensional vector space over
$\mathbb{Z}/p\mathbb{Z}$, then
$H^{\ast}\left(BV^{n};\mathbb{Z}/p\mathbb{Z}\right)\cong
E(x_{1},\cdots,x_{n})\otimes P[y_{1},\cdots,y_{n}].$
$P[y_{1},\cdots,y_{n}]^{GL_{n}}=\mathbb{Z}/p\mathbb{Z}\left[d_{n,1},\cdots,d_{n,n-1},d_{n,n}\right]$
denotes the classical Dickson algebra which is a graded polynomial algebra and
$D_{n}:=\left(E(x_{1},\cdots,x_{n})\otimes
P[y_{1},\cdots,y_{n}]\right)^{GL_{n}}$
the extended Dickson algebra studied by Mui ([11]).
In a series of papers, Campbell, Cohen, Peterson and Selick studied self maps
on certain important spaces in topology. In order to prove homotopy
equivalence at $p$, they considered the corresponding $mod-p$ homology
homomorphisms. So they had to use the Dyer-Lashof algebra and (or) quotients
of it as the main ingredient. It is well known that the Dyer-Lashof algebra is
closely related with Dickson algebras. Hence, they studied and used properties
of the Dickson algebras, especially papers [2] and [3]. The advantage is that
the Steenrod algebra action on Dickson algebras is better understood than the
Nishida relations on the Dyer-Lashof algebra.
Motivated by topological questions regarding the cohomology of an infinite
loop space and strongly influenced by the work of Campbell, Cohen, Peterson
and Selick in [2] and [3] we study the problem under which conditions is a
degree preserving $\mathcal{A}$-endomorphism of $D_{n}$ an isomorphism. Here
$\mathcal{A}$ stands for the Steenrod algebra.
Theorem. 27 The extended Dickson algebra $D_{n}$ is atomic as a Steenrod
algebra module.
The proof depends on a remarkable property that $D_{n}$ satisfies with respect
to its Steenrod algebra action.
Theorem. 11 Let $d^{K}=\prod\limits_{1}^{n}d_{n,i}^{k\left(i\right)}$. There
exists a sequence of Steenrod operations $P\left(\Gamma,K\right)$ such that
$P\left(\Gamma,K\right)d^{K}=ud_{n,n}^{p^{m}}\text{.}$
for some natural number $m$ and unit $u$.
In particular,
Corollary. 29 $\overline{D_{n}}$ is not directly decomposable as an
$A$-module.
Here $\overline{D_{n}}$ denotes the augmentation ideal of $D_{n}$ and
corollary implies that the only direct summands are $0$ and
$\overline{D_{n}}$.
Finally we apply Theorem 27 to the study of self maps between $Q_{0}S^{0}$.
Theorem. 30 Let $f:Q_{0}S^{0}\rightarrow Q_{0}S^{0}$ be an $H$-map which
induces an isomorphism on $H_{2p-3}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$. Let
$p>2$ and
$f_{\ast}(Q^{\left(p,1\right)}[1])=uQ^{\left(p,1\right)}[1]+others$
for some $u\in(\mathbb{Z}/\mathbb{Z}p)^{\ast}$. Then $f_{\ast}$ is an
isomorphism.
Last Theorem has been proved by Campbell, Cohen, Peterson and Selick in [3]
for $p=2$.
We recently informed that Pengelley and Williams have studied similar
properties of the Dickson algebra in [13].
## 2\. Classical Dickson algebras are atomic at $p$
The term atomic was defined by Cohen, Moore and Neisendorfer. Let us recall
from [2] that a CW complex $X$ is atomic at $p$, if given any map
$f:X\rightarrow X$ such that $f$ induces an isomorphism on
$H_{r}(X,\mathbb{Z}/p\mathbb{Z})$, then $f_{(p)}$ is a homotopy equivalence.
Here $r$ is the lowest degree such that
$\overline{H}_{r}(X,\mathbb{Z}/p\mathbb{Z})\neq 0$.
In this section we study the action of the Steenrod algebra on monomials of
the Dickson algebra. We prove that its augmentation ideal is directly
indecomposable. First, we recall definitions and basic properties for the
benefit of the reader. The main result of this section is Theorem 11.
The following theorem is well known:
###### Theorem 1.
[5] The classical Dickson algebra is
$P[y_{1},\cdots,y_{n}]^{GL_{n}}=\mathbb{Z}/p\mathbb{Z}\left[d_{n,1},\cdots,d_{n,n-1},d_{n,n}\right]\text{.}$
It is a polynomial algebra and
$\left|d_{n,i}\right|=2\left(p^{n}-p^{n-i}\right)$
($\left|d_{n,i}\right|=2^{n}-2^{n-i}$ for $p=2$).
As $n$ varies, we get Dickson algebras of various length. In the last section,
Dickson algebras of different lengths will be considered in connection with
the $mod-p$ cohomology of the base point of
$QS^{0}=\lim\Omega^{n}\Sigma^{n}S^{0}$.
We shall recall some well known results concerning the action of the Steenrod
algebra on Dickson algebra generators.
###### Proposition 2.
[7] (Th. 30, p. 169)
$P^{p^{k}}(d_{n,i}^{p^{j}})=\left\\{\begin{array}[]{lll}d_{n,i+1}^{p^{j}}\text{,}&\text{if
}k=n+j-i-1\text{ and }i<n&\text{(1)}\\\
-d_{n,i}^{p^{j}}d_{n,1}^{p^{j}}\text{,}&\text{if }k=j+n-1&\text{(2)}\\\
0\text{, }&\text{otherwise}&\end{array}\right.\text{.}$
###### Definition 3.
Let $c$ and $j$ be natural numbers such that $j\leq c+1$. Let $P(c,j)$ stand
for the Steenrod iterated operation
$P\left(c,j\right)=P^{p^{c-j+1}}...P^{p^{c}-j+j}\text{.}$
###### Lemma 4.
Let $i=1,...,n$ and $k\left(i\right)$ a natural number. Let
$c=k\left(i\right)+n-i-1$ for $i<n$ and $c=k\left(n\right)+n-1$. Then
$P\left(c,j\right)(d_{n,i}^{p^{k\left(i\right)}})=\left\\{\begin{array}[]{ll}ud_{n,n}^{p^{k\left(i\right)}}\text{,}&\text{if
}0<n-i=j\text{ and }i<n\\\ ud_{n,n}^{2p^{k\left(i\right)}}\text{,}&\text{if
}i=j=n\\\ d_{n,i+j}^{p^{k\left(i\right)}}\text{, }&i+j<n\\\
0,&\text{otherwise}\end{array}\right.\text{.}$
Here $u$ is a unit.
###### Proof.
This is a repeated application of last proposition. If
$k\left(i\right)+n-1=k\left(n\right)+n-1$, then
$k\left(i\right)=k\left(n\right)$ and
$k\left(i\right)+n-i-1<k\left(n\right)+n-1$ for $1\leq i<n$. Hence case (2) of
last proposition applies only for $i=n$.
###### Lemma 5.
Let $i=1,...,n$ and $k\left(i\right)$ natural numbers such that
$c=k\left(i\right)+n-i-1$ and $c=k\left(n\right)+n-1$. Let
$m\left(i\right)=a_{i}p^{k\left(i\right)}$ with $0\leq a_{i}\leq p-1$ and
$j=\max\left\\{n,n-i\ |\ a_{n},a_{i}\neq 0\right\\}$. Let
$d=\prod\limits_{1}^{n}d_{n,i}^{m\left(i\right)}$. Then
$P\left(c,j\right)(d)=\left\\{\begin{array}[]{ll}ud_{n,n}^{p^{k\left(n\right)}}d\text{,}&\text{if
}j=n\\\
ud_{n,n}^{p^{k\left(n-j\right)}}dd_{n,n-j}^{-p^{k\left(n-j\right)}}\text{,}&\text{if
}j<n\end{array}\right.\text{.}$
Here $u$ is a unit. Moreover, $P\left(c,t\right)(d)=0$, if $j<t$.
###### Proof.
The proof depends on the Cartan formula and last lemma. Two cases shall be
considered: $j=n$ and $j<n$.
First case: $j=n$.
$P^{p^{c}}d=u_{n}d_{n,1}^{p^{k\left(n\right)}}d+\sum\limits_{1}^{n-1}u_{i}d_{n,i+1}^{p^{k\left(i\right)}}dd_{n,i}^{-p^{k\left(i\right)}}\text{.}$
Here $u_{i}=0$, if $a_{i}=0$. Since
$k\left(n\right)<k\left(1\right)<...<k\left(n-1\right)$, we have
$P^{p^{c-1}}\left(d_{n,1}^{p^{k\left(n\right)}}d\right)=u_{n}^{\prime}d_{n,2}^{p^{k\left(n\right)}}d\text{.}$
And
$P\left(c-1,n-1\right)\left(d_{n,1}^{p^{k\left(n\right)}}d\right)=u^{\prime}d_{n,n}^{p^{k\left(n\right)}}d\text{.}$
For the rest of the terms we have
$P^{p^{c-1}}\left(d_{n,i+1}^{p^{k\left(i\right)}}dd_{n,i}^{-p^{k\left(i\right)}}\right)=u_{i}^{\prime}d_{n,i+2}^{p^{k\left(i\right)}}dd_{n,i}^{-p^{k\left(i\right)}}\text{.}$
Since $n>i$, last lemma implies
$P\left(c-1,n-1\right)\left(d_{n,i+1}^{p^{k\left(i\right)}}dd_{n,i}^{-p^{k\left(i\right)}}\right)=0\text{.}$
The proof of the case $j<n$ follows a similar pattern.
For the last statement of the lemma, proposition 2 is applied: For $j<t\leq n$
and $c-j=k\left(n-j\right)-1<k\left(n-j\right)+j-1=k\left(i\right)+n-i-1$, we
have
$P^{p^{c-j}}\left(d_{n,n}^{p^{k\left(n-j\right)}}dd_{n,n-j}^{-p^{k\left(n-j\right)}}\right)=0\text{.}$
We are ready to proceed to the main Theorem of this section which is the
building block to construct an algorithm turning a monomial $d$ to
$d_{n,n}^{p^{l}}$. Let us firstly demonstrate our method.
###### Example 6.
Let $p=2\ $and $n=3$. Instead of $d_{3,1}$, $d_{3,2}$ and $d_{3,3}$ we write
$d_{1}$, $d_{2}$ and $d_{3}$ respectively. Let
$K=(k_{1}=2^{2}+2^{3},k_{2}=2^{3}+2^{4},k_{3}=2^{1}+2^{2})$ and
$d^{K}=d_{1}^{2^{2}+2^{3}}d_{2}^{2^{3}+2^{4}}d_{3}^{2+2^{2}}\text{.}$
Let $J$ be the sequence consisting of the first terms of those of $K$.
$J=(2^{3},2^{2},2^{1})\text{.}$
For each exponent $m_{i}$ in $J$ consider the minimum of $m_{n}+n-1$ and
$m_{i}+n-i-1$ for $1\leq i\leq n-1$:
$\min\left(J\right)=\left\\{3+0,2+1,1+2\right\\}=3\text{.}$
Let $i_{\left(J\right)}$ be the maximum of the following set
$\left\\{n-i\text{, }n|m_{i}+n-i-1=\min\left(J\right)\text{ and/or
}m_{n}+n-1=\min\left(J\right)\right\\}\text{.}$
Which is $3$ in this case.
We apply $i_{\left(J\right)}=3$ squaring operations, namely:
$Sq^{2^{\min\left(J\right)}}\text{, }Sq^{2^{\min\left(J\right)-1}}\text{, and
}Sq^{2^{\min\left(J\right)}-2}\text{.}$
$Sq^{2^{\min\left(J\right)}}d^{K}=d_{3}^{2+2^{2}}d_{1}^{2}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}+d_{3}^{2+2^{2}}d_{3}^{2^{3}}d_{2}^{2^{4}}d_{1}^{2^{2}+2^{3}}+d_{3}^{2+2^{2}}d_{2}^{2^{2}+2^{3}+2^{4}}d_{1}^{2^{3}}\text{.}$
$\displaystyle
Sq^{2^{\min\left(J\right)-1}}\left[d_{3}^{2+2^{2}}d_{2}^{2^{3}+2^{4}}d_{1}^{2+2^{2}+2^{3}}+d_{3}^{2+2^{2}+2^{3}}d_{2}^{2^{4}}d_{1}^{2^{2}+2^{3}}+d_{3}^{2+2^{2}}d_{2}^{2^{2}+2^{3}+2^{4}}d_{1}^{2^{3}}\right]$
$\displaystyle=d_{3}^{2+2^{2}}d_{2}^{2}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}+d_{3}^{2+2^{2}+2^{2}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{3}}$
$Sq^{2^{\min\left(J\right)-2}}\left[d_{3}^{2+2^{2}}d_{2}^{2+2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}+d_{3}^{2+2^{2}+2^{2}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{3}}\right]=d_{3}^{2^{3}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}\text{.}$
Finally,
$Sq^{2^{3-2}}Sq^{2^{3-1}}Sq^{2^{3}}d^{K}=d_{3}^{2^{3}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}\text{.}$
Let $K=(2^{3}+2^{4},2^{2}+2^{3},2^{3})$. Then $K=(2^{3},2^{2},2^{3})$ ,
$\min\left(J\right)=3$, and $i_{\left(J\right)}=2$.
$Sq^{2^{2}}Sq^{2^{3}}d^{K}=d_{3}^{2^{2}+2^{3}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{3}}\text{.}$
Applying the procedure described above five more times we get:
Let $Sq(\Gamma,K)$ be the following operation
$Sq(2^{6},3)Sq(2^{4},1)Sq(2^{4},2)Sq(2^{4},3)Sq(2^{3},1)Sq(2^{3},2)Sq(2^{3},3)\text{,}$
then
$Sq(\Gamma,K)d_{3}^{2+2^{2}}d_{2}^{2^{3}+2^{4}}d_{1}^{2^{2}+2^{3}}=d_{3}^{2^{6}}\text{.}$
###### Definition 7.
Let $K=(k\left(1\right),...,k\left(n\right))$ and
$d^{K}=\prod\limits_{1}^{n}d_{n,i}^{k\left(i\right)}$. For each
$k\left(i\right)$, let $a_{i}p^{m\left(k\left(i\right)\right)}$ be its lowest
non-zero term in its $p$-adic expansion. Here $a_{i}=0$, if
$k\left(i\right)=0$. Let
$J=(a_{1}p^{m\left(k\left(1\right)\right)},...,a_{n}p^{m\left(k\left(n\right)\right)})\text{,}$
and $\min\left(J\right):$
$\min\\{m\left(k\left(n\right)\right)+n-1,m\left(k\left(i\right)\right)+n-i-1\;|1\leq
i<n\text{ and \ }a_{n},a_{i}\neq 0\\}\text{.}$
Let $i_{\left(J\right)}$ stand for the maximum of the $n-i$’s and/or $n$ such
that $m\left(k\left(i\right)\right)+n-i-1=\min\left(J\right)$ and/or
$m\left(k\left(n\right)\right)+n-1=\min\left(J\right)$.
###### Theorem 8.
Let $d^{K}$, $\min\left(J\right)$ and $i_{\left(J\right)}$ as in the
definition above, then
$P\left(\min\left(J\right),i_{\left(J\right)}\right)(d^{K})=\left\\{\begin{array}[]{ll}ud_{n,n}^{p^{m\left(k\left(n\right)\right)}}d^{K}\text{,}&\text{if
}i_{\left(J\right)}=n\\\
ud_{n,n}^{p^{m\left(k\left(n-i_{\left(J\right)}\right)\right)}}d^{K}d_{n,n-i_{\left(J\right)}}^{-p^{m\left(k\left(n-i_{\left(J\right)}\right)\right)}}\text{,}&\text{if
}i_{\left(J\right)}<n\end{array}\right.\text{.}$
Here $u$ is a unit.
###### Proof.
Let $b_{n-i_{\left(J\right)}}=a_{n-i_{\left(J\right)}}$, and $b_{t}=0$,
otherwise. Let $B=\left(b_{1},...,b_{n}\right)$. The second statement of lemma
5 implies that
$P\left(\min\left(J\right),i_{\left(J\right)}\right)(d^{K})=\left(P\left(\min\left(J\right),i_{\left(J\right)}\right)(d^{B})\right)d^{K-B}\text{.}$
The first claim of lemma 5 provides the claim.
###### Remark 9.
Let
$d^{K^{\prime}}=P\left(\min\left(J\right),i_{\left(J\right)}\right)(d^{K})$,
where $K$, $\min\left(J\right)$, and $i_{\left(J\right)}$ as in the Theorem
above. Let
$K^{\prime}=(k^{\prime}\left(1\right),...,k^{\prime}\left(n\right))$, then
$k\left(n\right)<k^{\prime}\left(n\right)$ and $k\left(i\right)\geq
k^{\prime}\left(i\right)$ for $i<n$.
###### Corollary 10.
Let $d^{K}=\prod\limits_{1}^{n}d_{n,i}^{k\left(i\right)}$ such that
$\sum\limits_{1}^{n-1}k\left(i\right)>0$. Then there exists a sequence
$P\left(K\right)$ of Steenrod operations such that
$P\left(K\right)d^{K}=d^{L}$
Here $d^{L}=\prod\limits_{1}^{n}d_{n,i}^{l\left(i\right)}$ satisfies
$k\left(n\right)<l\left(n\right)$, $k\left(i\right)>l\left(i\right)$ for some
$i<n$ and $k\left(t\right)=l\left(t\right)$ for $t\neq i,n$.
###### Proof.
The hypothesis $\sum\limits_{1}^{n-1}k\left(i\right)>0$ implies that case (2)
of Theorem above will be applied at some stage of the procedure. So, for some
$i$, the corresponding exponent will be smaller in the new monomial.
###### Theorem 11.
Let $d^{K}=\prod\limits_{1}^{n}d_{n,i}^{k\left(i\right)}$. There exists a
sequence of Steenrod operations $P\left(\Gamma,K\right)$ such that
$P\left(\Gamma,K\right)d^{K}=ud_{n,n}^{p^{m}}\text{.}$
for some natural number $m$ and unit $u$.
###### Proof.
Last corollary is applied repeatedly, so the sequence of the exponents of the
resulting monomial will be $\left(0,...,0,l\left(n\right)\right)$. If
$l\left(n\right)$ is not a $p$-th power, applying case (1) of last Theorem
repeatedly the exponent of $d_{n,n}$ will become a $p$-th power.
Before proceeding to the proof that the classical Dickson algebra is atomic,
let us consider an example which illuminates the key ingredient for the proof.
###### Example 12.
Let $p=2\ $and $n=3$. Let
$d^{K}=d_{1}^{2^{2}+2^{3}}d_{2}^{2^{3}+2^{4}}d_{3}^{2+2^{2}}$ and
$d^{K^{\prime}}=d_{1}^{2^{2}+2^{4}}d_{3}^{2+2^{2}+2^{4}}$. Then
$|d^{K}|=|d^{K^{\prime}}|$. As in the last example there exist sequences of
Steenrod operations $Sq\left(\Gamma,K\right)$ and
$Sq\left(\Gamma,K^{\prime}\right)$ such that
$Sq(\Gamma,K)d^{K}=d_{3}^{2^{6}}=Sq\left(\Gamma^{\prime},K^{\prime}\right)d^{K^{\prime}}.$
We recall that $Sq\left(\Gamma,K\right)=$
$Sq(2^{6},3)Sq(2^{4},1)Sq(2^{4},2)Sq(2^{4},3)Sq(2^{3},1)Sq(2^{3},2)Sq(2^{3},3)$
and
$Sq\left(\Gamma^{\prime},K^{\prime}\right)=Sq(2^{6},3)Sq(2^{5},2)Sq(2^{4},3)Sq(2^{3},2)Sq(2^{3},3).$
But
$Sq(\Gamma,K)d^{K^{\prime}}=0.$
###### Definition 13.
A graded module $\mathcal{M}$ is called atomic, if given any degree preserving
module map $f:\mathcal{M}\rightarrow\mathcal{M}$ which is an isomorphism on
the lowest positive degree, then $f$ is an isomorphism.
It turns out that the classical Dickson algebra is atomic as a Steenrod
algebra module.
###### Corollary 14.
Let $f:D_{n}\rightarrow D_{n}$ be an $\mathcal{A}$-linear map of degree $0$
such that $f(d_{n,1})\neq 0$. Then $f$ is an isomorphism.
###### Proof.
By hypothesis and proposition 2, $f(d_{n,i})=\lambda d_{n,i}$ for $i=1,...,n$
after applying a suitable Steenrod operation.
Let $d^{K}$ and $f(d^{K})=0$, then according to last Theorem there exists a
sequence of Steenrod operations such that
$P\left(\Gamma,K\right)d^{K}=ud_{n,n}^{p^{m}}$. Thus
$P\left(\Gamma,K\right)f\left(d^{K}\right)=uf\left(d_{n,n}^{p^{m}}\right)$ and
$0=ud_{n,n}^{p^{m}}$.
Let homogeneous monomials $d^{K(t)}$ for $1\leq t\leq l$ and $f(\sum
a_{t}d^{K\left(t\right)})=0$. Let
$P\left(\Gamma\left(t\right),K\left(t\right)\right)$ be the appropriate
corresponding sequences of Steenrod operations as in last Theorem:
$P\left(\Gamma\left(t\right),K\left(t\right)\right)=\prod\limits_{1}^{m_{t}}P\left(c_{s}\left(t\right),k_{s}\left(t\right)\right).$
Without lost of generality, we suppose that at least one of the
$P\left(c_{s}\left(1\right),k_{s}\left(1\right)\right)\text{'s}$
is different. Otherwise, the common part is applied on the $d^{K}$’s and we
consider the new terms. Let
$P\left(\Gamma\left(1\right),K\left(1\right)\right)=\prod\limits_{1}^{m_{t}}P\left(c_{s}\left(1\right),k_{s}\left(1\right)\right)$
satisfy the properties $c_{1}\left(1\right)=\min\left\\{c_{1}\left(t\right)\
|\ 1\leq t\leq l\right\\}$ and if there are more than one, then
$k_{1}\left(1\right)$ is the biggest among the equal ones.
If $c_{1}\left(1\right)<c_{1}\left(t\right)$, then
$P\left(\Gamma\left(1\right),K\left(1\right)\right)f(\sum
a_{t}d^{K\left(t\right)})=a_{1}P\left(\Gamma\left(1\right),K\left(1\right)\right)d^{K\left(1\right)}$
and the first part of the proof provides a contradiction.
If $c_{1}\left(1\right)=c_{1}\left(2\right)$, then
$k_{1}\left(1\right)>k_{1}\left(2\right)$ according to our assumption. Thus
(please consider last example)
$P\left(c_{1}\left(1\right),k_{1}\left(1\right)\right)d^{K\left(2\right)}=0\neq
P\left(c_{1}\left(1\right),k_{1}\left(1\right)\right)d^{K\left(1\right)}.$
This is because
$P\left(c_{1}\left(1\right),k_{1}\left(1\right)\right)=P\left(c_{1}\left(1\right)-k_{1}\left(2\right)+1,k_{1}\left(1\right)-k_{1}\left(2\right)\right)P\left(c_{1}\left(2\right),k_{1}\left(2\right)\right)$
and $c_{1}\left(1\right)-k_{1}\left(2\right)+1<c_{2}\left(2\right)$. By lemma
5:
$P\left(c_{1}\left(1\right)-k_{1}\left(2\right)+1,k_{1}\left(1\right)-k_{1}\left(2\right)\right)P\left(c_{1}\left(2\right),k_{1}\left(2\right)\right)d^{K\left(2\right)}=0.$
And the first part of the proof provides a contradiction.
Next we show that the ring of upper triangular invariants does not have this
property.
###### Example 15.
Let $p=2$ and $H_{2}=P[y_{1},y_{2}]^{U_{2}}$ the ring of upper triangular
invariants. Here $U_{2}=\left\\{\left(\begin{array}[]{cc}1&a\\\
0&1\end{array}\right)\ |\ a=\mathbb{Z}/2\mathbb{Z}\right\\}$. It is known that
it is a polynomial algebra on $h_{1}=y_{1}$ and $h_{2}=y_{2}^{2}+y_{2}y_{1}$
([11]). Let $f:H_{2}\rightarrow H_{2}$ be an $\mathcal{A}$-linear map such
that $f(h_{1})=h_{1}$. Since $Sq^{1}h_{1}=h_{1}^{2}\neq h_{2}$,
$f\left(h_{2}\right)$ can be defined independently of $h_{1}$:
$f\left(h_{2}\right)=ah_{2}+bh_{1}^{2}$ with $a,b\in\mathbb{Z}/2\mathbb{Z}$.
Even if $f\left(h_{2}\right)=h_{1}^{2}$, $f$ is not an isomorphism:
$f\left(d_{2,1}\right)=f(h_{2}+h_{1}^{2})=0=f(d_{2,0})=f(Sq^{1}d_{2,1})$.
## 3\. Dickson algebras are atomic
In this section the previous results are extended to the extended Dickson
algebra.
Mui gave an invariant theoretic description of the cohomology algebra of the
symmetric group and calculated rings of invariants involving the exterior
algebra $E(x_{1},\cdots,x_{n})$ as well in [11]. We recall that $|x_{i}|=1$
and $\beta x_{i}=y_{i}$.
###### Theorem 16.
[11]The extended Dickson algebra
$D_{n}:=(E(x_{1},\cdots,x_{n})\otimes P[y_{1},\cdots,y_{n}])^{GL_{n}}$
is a tensor product of the polynomial algebra $P[y_{1},\cdots,y_{n}]^{GL_{n}}$
and the $\mathbb{Z}$$/p$$\mathbb{Z}$-module spanned by the set of elements
consisting of the following monomials:
$M_{n;s_{1},...,s_{m}}L_{n}^{p-2};\ 1\leq m\leq n,\ \text{and }0\leq
s_{1}<\dots<s_{m}\leq n-1.$
Its algebra structure is determined by the following relations:
a) $(M_{n;s_{1},...,s_{m}}L_{n}^{p-2})^{2}=0$ for$\ 1\leq m\leq n,\ $and
$0\leq s_{1}<\dots<s_{m}\leq n-1$.
b)
$M_{n;s_{1},...,s_{m}}L_{n}^{(p-2)}d_{n,1}^{m-1}=(-1)^{m(m-1)/2}M_{n;s_{1}}L_{n}^{p-2}\dots
M_{n;s_{m}}L_{n}^{p-2}$.
Here $1\leq m\leq n$, and $0\leq s_{1}<\dots<s_{m}\leq n-1$.
The elements $M_{n;s_{1},...,s_{m}}$ above have been defined by Mui in [11] as
follows:
$M_{n;s_{1},...,s_{m}}=\frac{1}{m!}\left|\begin{array}[]{ccccc}x_{1}&&\cdots&&x_{1}\\\
\vdots&&&&\vdots\\\ x_{1}&&\cdots&&x_{n}\\\ y_{1}&&\cdots&&y_{n}\\\
\vdots&&&&\vdots\\\
y_{1}^{p^{n-1}}&&\cdots&&y_{n}^{p^{n-1}}\end{array}\right|\text{,}L_{n}=\left|\begin{array}[]{ccccc}y_{1}&&\cdots&&x_{n}\\\
y_{1}^{p}&&\cdots&&y_{n}^{p}\\\ \vdots&&&&\vdots\\\
y_{1}^{p^{n-1}}&&\cdots&&y_{n}^{p^{n-1}}\end{array}\right|$
Here there are $m$ rows of $x_{i}$’s and the $s_{i}$-th’s powers are omitted,
where $0\leq s_{1}<\dots<s_{m}\leq n-1$ in the first determinant.
The degrees of elements above are
$|M_{n;s_{1},...,s_{m}}(L_{n})^{p-2}|=m+2((p^{n}-1)-(p^{s_{1}}+\dots+p^{s_{m}}))$.
Next, some important subalgebras of $D_{n}$ are defined.
###### Definition 17.
Let $SD_{n}$ be the subalgebra of $D_{n}$ generated by:
$d_{n,s+1}\text{, }M_{n;s}(L_{n})^{p-2}\text{ and
}M_{n;s_{1},s_{2}}(L_{n})^{p-2}\text{. }$
Here $0\leq s\leq n-1$. $0\leq s_{1}<s_{2}\leq n-1$.
$D_{n}$ and $SD_{n}$ are $\mathcal{A}$-algebras. It is known that $SD_{n}$ is
related with the $\hom$-dual of the length $n$ coalgebra $R[n]$ of the Dyer-
Lashof algebra $R$ ([8]).
###### Definition 18.
Let $I_{n}$ stand for the ideal of $SD_{n}$ generated by
$\\{d_{n,n}\text{, }M_{n;i}(L_{n})^{p-2}\text{ and }M_{n;0,i}(L_{n})^{p-2}\ |\
0\leq i\leq n-1\\}.$
The ideal $I_{n}$ is related with the $\hom$-dual of the length $n$ module of
indecomposable elements of $H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$
([2]).
First, we recall the next proposition concerning the action of the Steenrod
algebra generators on exterior generators on the extended Dickson algebra. For
the benefit of the reader we also recall that $P^{p^{k}}L_{n}^{p-2}=0$ for
$0\leq k\leq n-2$.
###### Proposition 19.
[8]
$\beta M_{n;0}L_{n}^{n-2}=d_{n,n};$ $\beta
M_{n;0,s}L_{n}^{n-2}=-M_{n;s}L_{n}^{n-2}\text{, for }n-1\geq s>0;$
$P^{p^{s-1}}M_{n;s}L_{n}^{n-2}=M_{n,s-1}L_{n}^{n-2}\text{, for }n-1\geq s>0.$
###### Remark 20.
Propositions 2 and the last one imply that $I_{n}$ is closed under the
Steenrod algebra action.
In the next lemmata we explain step by step how a monomial is transformed in
to a power of $d_{n,n}$. We start with an application of the proposition
above.
###### Lemma 21.
$\beta
P\left(i-1,i\right)M_{n;i,s_{1},...,s_{l}}L_{n}^{n-2}=-M_{n;s_{1},...,s_{l}}L_{n}^{n-2}\text{.}$
Here $0\leq i<s_{1}<...<s_{l}\leq n-1$.
$\beta
P\left(t-1,t\right)M_{n;i}L_{n}^{n-2}=\left\\{\begin{array}[]{cc}d_{n,n}&\text{for
}t=i\\\ 0&\text{for }t\neq i\end{array}\right.\text{.}$
$\beta\underbrace{P^{p^{0}}\beta}...\underbrace{P^{p^{l-2}}...P^{p^{0}}\beta}M_{n;0,1,...,l-1}L_{n}^{p-2}=ud_{n,n}\text{.}$
###### Definition 22.
Let $M=M_{n;s_{1},...,s_{l}}L_{n}^{p-2}$ and $\left(t_{1},...,t_{n-l}\right)$
be the support of $\left(s_{1},...,s_{l}\right)$ in
$\left\\{0,1,...,n-1\right\\}$. Let $P\left(B\left(M\right)\right):=$
$\beta\underbrace{P^{p^{0}}\beta}...\underbrace{P^{p^{l-2}}...P^{p^{0}}\beta}\underbrace{P^{p^{l-1}}...P^{p^{t_{1}}}}...\underbrace{P^{p^{l+k-2}}...P^{p^{t_{k}}}}...\underbrace{P^{p^{n-2}}...P^{p^{t_{n-l}}}}\text{.}$
###### Lemma 23.
Let $M=M_{n;s_{1},...,s_{l}}L_{n}^{p-2}$, then
$P\left(B\left(M\right)\right)M=ud_{n,n}\text{.}$
###### Proof.
We recall that $M_{n;s_{1},...,s_{l}}$ is a sum of monomials of the form
$x_{1}...x_{l}y_{l+1}^{p^{t_{1}}}...y_{n}^{p^{t_{n-l}}}$. Let $s_{l}=n-1$.
Then
$P^{p^{n-2}}...P^{p^{t_{n-l}}}y_{l+1}^{p^{t_{1}}}...y_{n}^{p^{t_{n-l}}}=y_{l+1}^{p^{t_{1}}}...y_{n-1}^{p^{t_{n-l-1}}}y_{n}^{p-2}.$
Hence
$P^{p^{n-2}}...P^{p^{t_{n-l}}}M=M_{n;s_{1},...,s_{l-1},t_{n-l}}L_{n}^{p-2}$.
Let $s_{l}<n-1$ and $k$ maximal such that $t_{k}<t_{k+1}-1$ and $k<n-l$. In
this case
$P^{p^{t_{k+1}-2}}...P^{p^{t_{k}}}y_{l+1}^{p^{t_{1}}}...y_{n}^{p^{t_{n-l}}}=y_{l+1}^{p^{t_{1}}}...y_{n}^{p^{t_{n-l}}}y_{k}^{p^{t_{k+1}-1}}y_{k}^{-p^{t_{k}}}.$
Hence
$P^{p^{t_{k+1}-2}}...P^{p^{t_{k}}}M=M_{n;s_{1},...,t_{k},\widehat{t_{k+1}-1},...,s_{l}}L_{n}^{p-2}$.
Here $\widehat{t_{k+1}-1}$ means that the index $t_{k+1}-1$ is missing. Now
the claim follows.
The main point of this section is to prove that, if $M\neq M^{\prime}$, then
$P\left(B\left(M\right)\right)M^{\prime}=0$. The next example demonstrates the
idea.
###### Example 24.
1) $P^{p^{7}}P^{p^{6}}M_{10;4,7,8}L_{10}^{p-2}=M_{10;4,6,7}L_{10}^{p-2}$.
$P^{p^{7}}P^{p^{6}}M_{10;3,5,7}L_{10}^{p-2}=P^{p^{7}}M_{10;3,5,6}L_{10}^{p-2}=0$.
2) $P^{p^{8}}M_{10;3,5,9}L_{10}^{p-2}=M_{10;3,5,8}L_{10}^{p-2}$.
$P^{p^{8}}M_{10;3,5,7}L_{10}^{p-2}=0$.
###### Proposition 25.
Each element $M=M_{n;s_{1},...,s_{l}}L_{n}^{p-2}$ uniquely determines
$P\left(B\left(M\right)\right)$ such that
$P\left(B\left(M\right)\right)M=ud_{n,n}$ and
$P\left(B\left(M\right)\right)M^{\prime}=0$ for
$M^{\prime}=M_{n;s_{1}^{\prime},...,s_{l^{\prime}}^{\prime}}L_{n}^{p-2}\neq
M$.
###### Proof.
Let $\left(t_{1},...,t_{n-l}\right)$ and
$\left(t_{1}^{\prime},...,t_{n-l}^{\prime}\right)$ be the corresponding
supports. We recall that $M_{n;s_{1},...,s_{l}}$ is a sum of monomials of the
form $x_{1}...x_{l}y_{l+1}^{p^{t_{1}}}...y_{n}^{p^{t_{n-l}}}$.
Let $s_{l}=n-1>s_{l}^{\prime}$. Then
$P^{p^{n-2}}...P^{p^{t_{n-l}}}y_{l+1}^{p^{t_{1}^{\prime}}}...y_{n}^{p^{t_{n-l}^{\prime}}}$
is either zero (if $t_{n-l}\neq t_{q}^{\prime}$) or contains two identical
powers $p^{m-1}$ and in either case the determinant
$P^{p^{n-2}}...P^{p^{t_{n-l}}}M^{\prime}$ is zero.
Let $s_{l}^{\prime}<s_{l}<n-1$ and $k$ maximal such that $t_{k}<t_{k+1}-1$ and
$k<n-l$.
If $t_{k}\notin\left(t_{1}^{\prime},...,t_{n-l}^{\prime}\right)$, then
$P^{p^{t_{k}}}M^{\prime}=0$ (please recall the proof of the last lemma). Let
$t_{k}\in\left(t_{1}^{\prime},...,t_{n-l}^{\prime}\right)$ and
$t_{k}^{\prime}=t_{k}$. If $t_{k+1}^{\prime}=t_{k}^{\prime}+1$, then
$P^{p^{t_{k}}}y_{l+1}^{p^{t_{1}^{\prime}}}...y_{n}^{p^{t_{n-l}^{\prime}}}$
contains two identical powers $p^{t_{k}^{\prime}+1}$ and the determinant
$P^{p^{t_{k}}}M^{\prime}$ is zero.
If $t_{k+1}^{\prime}>t_{k}^{\prime}+1$, then
$t_{k+1}-t_{k}>t_{k+1}^{\prime}-t_{k}^{\prime}$. Again for the same reason
$P^{p^{t_{k+1}-2}}...P^{p^{t_{k}}}M^{\prime}=0$.
Let $s_{l}<s_{l}^{\prime}$, then
$P\left(B\left(M\right)\right)=(\beta
P^{p^{0}}\beta...P^{p^{l-2}}...P^{p^{0}}\beta)P^{p^{i_{q}}}...P^{p^{i_{1}}}$
and
$P\left(B\left(M^{\prime}\right)\right)=(\beta
P^{p^{0}}\beta...P^{p^{l-2}}...P^{p^{0}}\beta)P^{p^{i_{q^{\prime}}^{\prime}}}...P^{p^{i_{1}^{\prime}}}$
such that $i_{1}<i_{1}^{\prime}$ (please see lemma 23). Now
$P^{p^{i_{q}}}...P^{p^{i_{1}}}M=M_{n;0,1,...,l-1}L_{n}^{p-2}=P^{p^{i_{q^{\prime}}^{\prime}}}...P^{p^{i_{1}^{\prime}}}M^{\prime}$
If $P^{p^{i_{q}}}...P^{p^{i_{1}}}M^{\prime}\neq 0$, then
$P^{p^{i_{q}}}...P^{p^{i_{1}}}M^{\prime}\neq M_{n;0,1,...,l-1}L_{n}^{p-2}$,
because $i_{1}<i_{1}^{\prime}$. Thus
$P\left(B\left(M\right)\right)M^{\prime}=0$.
If $s_{l}=s_{l}^{\prime}$, then by applying a suitable sequence of Steenrod
operations the case is reduced to one of the previous ones.
###### Corollary 26.
Let $M=M_{n;s_{1},...,s_{l}}L_{n}^{p-2}d^{K}$ be a monomial in $D_{n}$, then
$P\left(B\left(M\right)\right)M=ud_{n,n}d^{K}$. Let
$M^{\prime}=M_{n;s_{1}^{\prime},...,s_{l^{\prime}}^{\prime}}L_{n}^{p-2}d^{K^{\prime}}$
such that $s_{t}\neq s_{t}^{\prime}$ for some $t$, then
$P\left(B\left(M\right)\right)M^{\prime}=0$.
###### Proof.
For the first statement, last lemma implies that
$P\left(B\left(M\right)\right)M=ud_{n,n}d^{K}+\left(others\right).$
Last proposition implies that $\left(others\right)=0$. The second statement is
an application of last proposition.
Now we are ready to proceed to our main results of this section.
###### Theorem 27.
The extended Dickson algebra $D_{n}$ is atomic.
###### Proof.
Let $g:D_{n}\rightarrow D_{n}$ be an $\mathcal{A}$-linear map such that
$g(M)\neq 0$. We will prove that $g$ is an isomorphism. Here
$M=\prod\limits_{1}^{n}x_{i}L_{n}^{p-2}$. Applying corollaries 26 and 14, the
claim is obtained.
###### Proposition 28.
a) $SD_{n}$ is atomic as a Steenrod algebra module.
b) $I_{n}$ is atomic as a Steenrod algebra module.
###### Proof.
a) Let $f:SD_{n}\rightarrow SD_{n}$ satisfy
$f(M_{n;n-2,n-1}L_{n}^{p-2})=uM_{n;n-2,n-1}L_{n}^{p-2}\text{.}$
Applying corollaries 26 and 14, the claim is obtained.
b) Let us recall that $I_{n}$ is the ideal generated by
$\\{d_{n,n},M_{n;s_{1}}L_{n}^{p-2},M_{n;0,s_{1}^{\prime}}L_{n}^{p-2}\\}.$
Here $0\leq s_{1}<n$ and $0<s_{1}^{\prime}\leq n-1$. Let $f$ satisfy
$f(M_{n;0,n-1}L_{n}^{p-2})=uM_{n;0,n-1}L_{n}^{p-2}.$
The proof follows the same pattern as above.
Proposition b) above is a reformulation of Theorem 4.1 in [2].
We close this section by observing a property of the Steenrod algebra. We
recall that an $\mathcal{A}$-module is indecomposable, if it is not a non-
trivial direct sum.
Let $\overline{D_{n}}$ denote the augmentation ideal of $D_{n}$.
###### Corollary 29.
$\overline{D_{n}}$ is not directly decomposable as an $\mathcal{A}$-module.
###### Proof.
Assume $\overline{D_{n}}=\bigoplus\limits_{i\in I}(D_{n})_{i}$ such that
$(D_{n})_{i}\neq 0$. If $d\left(i\right)$ and $d\left(j\right)$ are
homogeneous polynomials in $(D_{n})_{i}$ and $(D_{n})_{j}$ respectively, then
there exist $P^{\Gamma}$ and $P^{\Gamma^{\prime}}$ such that
$a_{i}P^{\Gamma}d\left(i\right)=d_{n,n}^{p^{l}}=b_{j}P^{\Gamma}d\left(j\right)$.
## 4\. $Q_{0}S^{0}$ is $H$-atomic at $p$
We close this work by applying the main result in the $mod-p$ homology of
$QS^{0}$. $H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$ is described in terms
of Dyer-Lashof operations. For their properties please see May [4].
Iterates of the Dyer-Lashof operations are of the form
$Q^{\left(I,\varepsilon\right)}=\beta^{\epsilon_{1}}Q^{i_{1}}\dots\beta^{\epsilon_{k}}Q^{i_{k}}$
where $(I,\varepsilon)=((i_{k},\dots,i_{1}),(\epsilon_{k},...,\epsilon_{1}))$
with $\epsilon_{j}=0\ $or$\ 1$ and $i_{j}$ a non-negative integer for $j=1,\
\dots\ ,k$. If $p=2$, $\epsilon_{j}=0\ $for all $j$. The degree is defined by
$|\left(I,\varepsilon\right)|:=|Q^{\left(I,\varepsilon\right)}|=2(p-1)\left(\sum\limits_{t=1}^{k}i_{t}\right)-\left(\sum\limits_{t=1}^{k}e_{t}\right)$
[$|Q^{I,\varepsilon}|=\left(\sum\limits_{t=1}^{k}i_{t}\right)$, for $p=2$].
Let $l(I,\varepsilon)=k$ denote the length of $\left(I,\varepsilon\right)$ or
$Q^{I,\varepsilon}$ and let the excess of $(I,\varepsilon)$ or
$Q^{\left(I,\varepsilon\right)}$, denoted by
$exc(Q^{\left(I,\varepsilon\right)})=i_{k}-\epsilon_{k}-|Q^{(I\left(k-1\right),\varepsilon\left(k-1\right))}|$,
where
$(I\left(t\right),\varepsilon\left(t\right))=((i_{t},\dots,i_{1}),(\epsilon_{t},\dots,\epsilon_{1}))$.
$exc(Q^{I,\varepsilon})=i_{k}-\epsilon_{k}-2(p-1)\sum\limits_{1}^{k-1}i_{t}\text{,
[}exc(Q^{I})=i_{k}-\sum\limits_{1}^{k-1}i_{t}\text{].}$
The excess is defined $\infty$, if $I=\emptyset$ and we omit the sequence
$(\epsilon_{1},...,\epsilon_{k})$, if all $\epsilon_{i}=0$. We refer to
elements $Q^{I}$ as having non-negative excess, if
$exc(Q^{(I\left(t\right),\varepsilon\left(t\right))})$ is non-negative for all
$t$.
There are relations among the iterated operations called Adem relations, so an
operation can be reduced to a sum of admissible operations after applying Adem
relations. A sequence $(I,\varepsilon)$ is called admissible, if
$pi_{j}-\epsilon_{j}\geq i_{j-1}$ ($2i_{j}\geq i_{j-1}$) for $2\leq j\leq k$.
The Kronecker pairing and the left $\mathcal{A}$-module on
$H^{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$ induces a right
$\mathcal{A}_{\ast}$-module structure on
$H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$. We follow Cohen and May in
writing the Steenrod operations on the left.
The Dyer-Lashof algebra can be decomposed as opposite Steenrod coalgebras with
respect to length
$R=\bigoplus\limits_{k\geq 0}R[k]\text{.}$
Let $R^{+}$ be the positive degree elements of $R$ and $R_{0}$ be the ideal
generated by positive degree elements of excess zero
$R_{0}=<Q^{\left(I,\varepsilon\right)}\ |\ exc(I,\varepsilon)=0>\text{.}$
Let $(I,\varepsilon)$ be an admissible sequence such that
$|Q^{\left(I,\varepsilon\right)}|>0$, then $Q^{\left(I,\varepsilon\right)}[1]$
corresponds to
$[1]_{\left(I,\varepsilon\right)}:=Q^{\left(I,\varepsilon\right)}[1]\ast<Q^{\left(I,\varepsilon\right)}[1]>^{-1}\in
H_{\ast}(Q_{0}S^{0})\text{.}$
Here $<Q^{\left(I,\varepsilon\right)}[1]>^{-1}=[-p^{l(I)}]$.
According to Madsen and May, $H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$ is
the free commutative graded algebra generated by
$Q^{(I,\varepsilon)}[1]\ast[-p^{l(I)}]$. Here $(I,\varepsilon)$ are admissible
sequences of positive excess and $\ast$ denotes Pontryagin multiplication.
There exists an $\mathcal{A}_{\ast}$-module isomorphism between the generators
of $H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$ and the quotient $R/Q_{0}R$
where $Q_{0}R=\\{Q^{(I,\varepsilon)}|exc(I,\varepsilon)=0\\}$. It is known
that $R[k]^{\ast}\cong SD_{k}$ as Steenrod algebras ([12], [8]) and
$(R/Q_{0}R)[k]^{\ast}\cong I_{k}$ as Steenrod modules ([2]).
Next Theorem has been given in [3] as Theorem 2.3 for $p=2$ by a similar
method.
###### Theorem 30.
Let $f:Q_{0}S^{0}\rightarrow Q_{0}S^{0}$ be an $H$-map which induces an
isomorphism on $H_{2p-3}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)$. Let $p>2$ and
$f_{\ast}(Q^{\left(p,1\right)}[1])=uQ^{\left(p,1\right)}[1]+others$
for some $u\in(\mathbb{Z}/\mathbb{Z}p)^{\ast}$. Then $f_{\ast}$ is an
isomorphism.
###### Proof.
The case $p>2$ shall be considered. We shall prove that $f_{\ast}$ is an
isomorphism on $Q\left(H_{\ast}(Q_{0}S^{0};\mathbb{Z}/\mathbb{Z}p)\right)$,
the module of indecomposable elements. There is an $\mathcal{A}_{\ast}$ module
isomorphism between the previous module and $R/Q_{0}R$. The last isomorphism
provides an $\mathcal{A}$-isomorphism between $\left(R/Q_{0}R\right)^{\ast}$
and $I=\oplus I_{k}$. Let $\overline{f}_{k}$ be the induced map of $f^{\ast}$
in $I_{k}$. It suffices to prove that $\overline{f}_{k}$ is an isomorphism for
each $k$ and this is true as long as $\overline{f}_{k}(d)=ud$ for
$d=M_{k;0,s}L_{k}^{p-2}$ or $d_{k,k}$ and $0<s\leq k-1$ according to
proposition 28. Here $u$ is a unit.
Given $f_{\ast}\left(\beta Q^{1}[1]\right)=u\beta Q^{1}[1]$ we have
$\beta f_{\ast}\left(Q^{1}[1]\right)=f_{\ast}\left(\beta
Q^{1}[1]\right)=u\beta Q^{1}[1]\text{.}$
Thus $f_{\ast}\left(Q^{1}[1]\right)=uQ^{1}[1]$, for degree reasons. Moreover,
$f_{\ast}\left(Q^{1}[1]\right)^{p^{m}}=u\left(Q^{1}[1]\right)^{p^{m}}\approx
uQ^{\left(p^{m-1}-p^{m-2},...,p-1,1\right)}[1].$
Dually ([9]), $\overline{f}_{1}\left(d_{1,1}\right)=ud_{1,1}$. Given
$f_{\ast}\left(Q^{(p,1)}[1]\right)=u^{\prime}Q^{(p,1)}[1]+others$, we have
$\overline{f}_{2}\left(d_{2,2}\right)=u^{\prime}d_{2,2}+others$. Induction on
the length $k$ is applied. Suppose that
$f_{\ast}\left(Q^{\left(p^{k-1},...,p,1\right)}[1]\right)=uQ^{\left(p^{k-1},...,p,1\right)}[1]+others.$
Now,
$f_{\ast}\left(Q^{\left(p^{k-1},...,p,1\right)}[1]\right)^{p}=u\left(Q^{\left(p^{k-1},...,p,1\right)}[1]\right)^{p}+others$.
And
$f_{\ast}\left(Q^{\left(p^{k}-1,p^{k-1}...,p,1\right)}[1]\right)=uQ^{\left(p^{k}-1,p^{k-1}...,p,1\right)}[1]+others.$
Since $k>2$,
$P_{\ast}^{1}\left(Q^{\left(p^{k}-1,p^{k-1}...,p,1\right)}[1]\right)=Q^{\left(p^{k},p^{k-1}...,p,1\right)}[1]$
uniquely by Nishida relations. Thus
$f_{\ast}\left(Q^{\left(p^{k},p^{k-1}...,p,1\right)}[1]\right)=uQ^{\left(p^{k},p^{k-1}...,p,1\right)}[1]+others.$
Dually ([9]),
$\overline{f}_{k+1}\left(d_{k+1,k+1}\right)=u^{\prime}d_{k+1,k+1}+others$.
Proposition 28 implies that $\overline{f}_{k+1}$ is an isomorphism for all
$k$. Now the Theorem follows.
### 4.1. Acknowledgement
We express our profound thanks to H.-W. Henn, P. May and L. Schwartz.
## References
* [1] F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Exponents in homotopy theory. Algebraic topology and algebraic $K$K-theory (Princeton, N.J., 1983), 3–34, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987.
* [2] H. E. A. Campbell, F. P. Peterson and P. S. Selick, Self-maps of loop spaces I, Trans. A. M. S. 293 no1 (1986),1-39.
* [3] H. E. A. Campbell, F. R. Cohen, F. P. Peterson and P. S. Selick, Self-maps of loop spaces II, Trans. A. M. S. 293 no1 (1986), 41-51.
* [4] F. Cohen, T. Lada , and J. P. May P, The homology of iterated loop spaces. Lecture notes in Mathematics, 533, (1975).
* [5] L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. A. M. S. 12 (1911), 75-98.
* [6] N. H. V. Hung and F. P. Peterson, $\mathcal{A}$-generators for the Dickson algebra, Trans. A. M. S. 347 no1 (1995), 4687-4728.
* [7] Nondas E. Kechagias, The Steenrod algebra action on generators of rings of invariants of subgroups of $GL_{n}(\mathbb{Z}/p\mathbb{Z})$. Proc. Amer. Math. Soc. 118 (1993), no. 3, 943–952.
* [8] Nondas E. Kechagias, Extended Dyer-Lashof algebras and modular coinvariants. Manuscripta Math. 84 (1994), no. 3-4, 261–290.
* [9] Nondas E. Kechagias, Adem relations in the Dyer-Lashof algebra and modular invariants. Algebr. Geom. Topol. 4 (2004), 219–241.
* [10] I. Madsen, On the action of the Dyer-Lashof algbera in $H_{\ast}(G)$, Pacific J. Math. 60, 1975, 235-275.
* [11] Huyhn Mùi, Modular invariant theory and the cohomology algebras of the symmetric groups, J. Fac. Sci. Univ. Tokyo, IA (1975), 319-369.
* [12] Huynh Mùi, Homology operations derived from modular coinvariants. Algebraic topology, Göttingen 1984, 85–115, Lecture Notes in Math., 1172.
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|
arxiv-papers
| 2011-08-18T08:42:45 |
2024-09-04T02:49:21.629084
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nondas E. Kechagias",
"submitter": "Nondas Kechagias E",
"url": "https://arxiv.org/abs/1108.3685"
}
|
1108.3810
|
H. Atik and E. Ulualan
Induced Quadratic Modules
We give the constructions of pullback (or co-induced) and induced quadratic modules.
A. M. S. Classification: 18D35, 18G30, 18G50, 18G55.
Keywords: Induced Crossed modules, Quadratic Modules
§ INTRODUCTION
Algebraic models for homotopy connected 3-types can be thought as extended version of crossed modules which models for 2-types introduced by Whitehead in [19]. Some of these are 2-crossed modules [13], braided regular crossed modules [4], crossed squares [16] and quadratic modules [3]. For the categorical relations among these structures see [2].
Some universal constructions for crossed modules, for example, the notions of pullback and induced crossed modules have been worked in [5, 6, 7]. Furthermore, for Lie algebra cases of these constructions see [12], and for commutative algebras see [17]. By extending these constructions for two dimensional case of crossed modules, Arslan, Arvasi and Onarli in [1], have defined the notions of pullback and induced 2-crossed module. Brown and Sivera in [9] gave a construction of the induced crossed square. This gives another view of a presentation of the induced crossed square in [8] and which is applied to free crossed squares in [14] for homotopy type calculations. For another applications of higher homotopy van Kampen theorem see also [15]. In this work, by using a similar way given in these cited works, we have constructed the pullback and induced quadratic modules. More precisely, if $\sigma: B\rightarrow C_0$ is a monomorphism of groups, then there is a `pullback' or restriction functor $\sigma^*: \mathbf{Quad}/C_0 \rightarrow \mathbf{Quad}/B$, where $\mathbf{Quad}/C_0$ is the subcategory of the category of quadratic modules $\mathbf{Quad}$ made up by the quadratic $C_0$-modules. We have also constructed a functor, and the image by this functor of a quadratic $Q$-module is called the induced quadratic module.
Acknowledgements. The authors wishes to thank Professor Z. Arvasi for his helpful comments.
§ QUADRATIC MODULES
Quadratic modules of groups were initially defined by Baues in [3] as models for connected 3-types.
In this section we will give a construction of a pullback quadratic module. Firstly, we recall some basic definitions from [3].
Recall that a pre-crossed module is a group homomorphism
$\partial :M\rightarrow Q$ together with an action of $Q$ on $M$,
written $m^q $ for $q\in Q$ and $m\in M$, satisfying the
condition $\partial (m^q)=q^{-1}\partial (m)q$ for all $m\in M$
and $q\in Q$.
A nil(2)-module (cf. [3]) is a pre-crossed module $\partial
:M\rightarrow Q$ with an additional “nilpotency”condition. This condition is
$P_{3}(\partial )=1$, where $P_{3}(\partial )$ is the subgroup of
$M$ generated by Peiffer commutator $\left\langle
x_{1},x_{2},x_{3}\right\rangle $ of length $3$. The Peiffer commutator in a pre-crossed module $\partial :M\rightarrow Q$ is defined by
\begin{equation*}
\left\langle x,y\right\rangle =x^{-1}y^{-1}x(y)^{\partial_1 x}
\end{equation*}
for $x,y\in M$. For a pre-crossed module $\partial:M\rightarrow Q$, if $\langle M, M \rangle=1$, then it is called a crossed module. That is, a nil(1)-module is a crossed module.
A morphism between two nil(2)-modules $\partial:M\rightarrow Q$ and $\partial':M'\rightarrow Q'$ is a pair $(g,f)$ of homomorphisms of groups $g:M\rightarrow M'$ and $f:Q\rightarrow Q'$ such that $f\partial = \partial' g$ and the actions preserved, i.e. $g(m^q)=g(m)^{f(q)}$ for any $m\in M, q\in Q$. We shall denote the category of nil(2)-modules by $\mathbf{Nil}(2)$.
Now we can give the following definition from [3].
A quadratic module $(\omega ,\partial_2 ,\partial_1 )$ is a diagram
\begin{equation*}
\xymatrix{ & C\otimes C\ar[dl]_{\omega} \ar[d]^{w} \\ C_2
\ar[r]_{\partial_2} & C_1 \ar[r]_{\partial_1} & C_0 }
\end{equation*}
of homomorphisms between groups such that the following axioms are satisfied.
$\mathbf{QM1)}$ The homomorphism $\partial_1 :C_1\rightarrow C_0$ is a
nil(2)-module with Peiffer commutator map $w$ defined above. The
map $C_1\twoheadrightarrow C=(C_1^{cr})^{ab}$ is given by $x\mapsto \{x\}%
, $ where $\{x\}\in C$ denotes the class represented by $x\in C_1$ and $%
C=(C_1^{cr})^{ab}$ is the abelianization of the associated crossed module $%
C_1^{cr}\rightarrow C_0$.
$\mathbf{QM2)}$ The boundary
homomorphisms $\partial_2 $ and $\partial_1 $ satisfy $\partial_1 \partial_2 =1$
and the quadratic map $\omega $ is a lift of the Peiffer commutator
map $w$, that is $\partial_2 \omega =w$.
$\mathbf{QM3)}$ $C_2$ is a
$C_0$-group and all homomorphisms of the diagram are equivariant with
respect to the action of $C_0$. Moreover, the action of $C_0$ on $C_2$
satisfies the formula ($a\in C_2,x\in C_1$)
\begin{equation*}
a^{\partial_1 x}=\omega (\left( \{x\}\otimes \{\partial_2
a\}\right) \left( \{\partial_2 a\}\otimes \{x\}\right) )a.
\end{equation*}
$\mathbf{QM4)}$ Commutators in $C_2$ satisfy the formula ($a,b\in C_2$)
\begin{equation*}
\omega (\{\partial_2 a\}\otimes \{\partial_2 b\})=[b,a].
\end{equation*}
A morphism $\varphi :(\omega ,\partial_2 ,\partial_1 )\rightarrow (\omega ^{\prime
},\partial_2 ^{\prime },\partial_1 ^{\prime })$ between quadratic modules is given
by a commutative diagram, $\varphi =(f_2,f_1,f_0)$
\begin{equation*}
\xymatrix{ C\otimes C\ar[d]_{\varphi_{\ast}\otimes\varphi_{\ast}}
\ar[r]^-{\omega} & C_2 \ar[d]_{f_2} \ar[r]^{\partial_2} & C_1 \ar[d]_{f_1}
\ar[r]^{\partial_1} & C_0 \ar[d]_{f_0} \\ C^{\prime }\otimes C^{\prime }
\ar[r]_-{\omega^{\prime}} & C_2^{\prime} \ar[r]_{\partial_2^{\prime}} & C_1^{\prime}
\ar[r]_{\partial_1^{\prime}} & C_0^{\prime} }
\end{equation*}
where $(f_1,f_0)$ is a morphism between nil(2)-modules which
induces $\varphi _{\ast }:C\rightarrow C^{\prime }$ and where $f_2$
is an $f_0$-equivariant homomorphism. We shall denote the category of quadratic modules by $\mathbf{Quad}$.
A simplicial group $\mathbf{G}$ consists of a family of groups
${G_{n}}$ together with face and degeneracy maps
$d_{i}^{n}:G_{n}\rightarrow G_{n-1}$, $0\leq i\leq n$ $(n\neq 0)$
and $s_{i}^{n}:G_{n}\rightarrow G_{n+1}$, $0\leq i\leq n$
satisfying the usual simplicial identities. In [2], the first and third authors have defined a functor from the category of simplicial groups with Moore complex of length 2 to that of quadratic modules. Therefore we can say that the Moore complex of a 2-truncated simplicial group gives a quadratic module.
§.§ Examples of Quadratic Modules
Porter in [18] has given the relations between 2-crossed complexes with the trivial Peiffer lifting map and crossed complexes. Here we give the similar relations about quadratic modules. The construction of quadratic modules from simplicial groups given in [2] gives a generic family of examples.
Any nil(2)-module $\partial:M\rightarrow N$ yields a quadratic module $\overline{\partial}:(1,w,\partial)$ given by
\begin{equation*}
\xymatrix{ & C\otimes C\ar[dl]_{1} \ar[d]^{w} \\ L
\ar[r]_{w} & M \ar[r]_{\partial} & N }
\end{equation*}
where $L=C\otimes C$. This quadratic module is called the quadratic module associated to the nil(2)-module $\partial$. This is of course, functorial and $\mathbf{Nil}(2)$ can be considered to be a full subcategory of $\mathbf{Quad}$. It is a reflective subcategory since there is a reflection functor given by Baues in [3].
A nil(2)-complex of groups is a positive chain complex of groups
\begin{equation*}
\xymatrix{\mathbf{C}:&\cdots\ar[r]&C_2 \ar[r]^{\partial_2}&C_1
\ar[r]^{\partial_1}&C_0}
\end{equation*}
in which
$(i)$ $\partial_1: C_1 \rightarrow C_0$ is a nil(2)-module
$(ii)$ For $n\geqslant 2$, $C_n$ is Abelian and for $n\geqslant 1$, $C_n$ is
a $C_0$-group and $\partial_1 (C_1)$ acts trivially on $C_n$ for $n \geqslant 2$,
$(iii)$ for $n\geqslant 1$ $\partial_{n} \partial_{n+1} =1$.
Any nil(2)-complex of length 2, that is one of form
\xymatrix{\cdots 1\ar[r]&1\ar[r]&C_2 \ar[r]^{\partial_2}&C_1 \ar[r]^{\partial_1}&C_0}
in which $\partial_1:C_1\rightarrow C_0$ is a nil(2)-module, gives us a quadratic complex
\begin{equation*}
\xymatrix{& & C\otimes C\ar[dl]_{\omega} \ar[d]^{w} \\
\cdots 1\ar[r]& L
\ar[r]_{\partial_2} & M \ar[r]_{\partial_1} & N }
\end{equation*}
on taking $L=C_2$, $M=C_1$, $C=(C_1^{cr})^{ab}$, $N=C_0$ with $\omega(\{x\}\otimes \{y\})=1$ for all $x,y\in M$.
This is of course functorial and we can say that there is a functor from the category of nil(2)-complexes of length 2 to that of full subcategory of quadratic complexes of length 1 in which the quadratic modules with trivial quadratic map.
Exploration of trivial quadratic map
Suppose we have a quadratic module
\begin{equation*}
\xymatrix{ & C\otimes C\ar[dl]_{\omega} \ar[d]^{w} \\ L
\ar[r]_{\partial_2} & M \ar[r]_{\partial_1} & N }
\end{equation*}
with the extra condition that $\omega(\{x\}\otimes \{y\})=1$ for all $x,y\in M$. The obvious thing to do is to see what each of the defining properties of a quadratic module give in this case.
$(i)$ $\omega $ is a lifting of the Peiffer commutator, so if $\omega(\{x\}\otimes \{y\})=1$, the Peiffer identity holds for $\partial_1$, i.e., that is a crossed module. Indeed, from axiom $\mathbf{QM2)}$ we have
1=x^{-1}y^{-1}xy^{\partial_1 x}=w(\{x\}\otimes \{y\})=\partial_2\omega(\{x\}\otimes \{y\})
for all $x,y\in M$.
$(ii)$ From axiom $\mathbf{QM4)}$, if $l_0, l_1 \in L$, then
$$1=[l_0, l_1]=\omega(\{\partial_2 l_0\}\otimes \{\partial_2 l_1\}),$$
so $L$ is Abelian and
$(iii)$ as $\omega$ is trivial, from $\mathbf{QM3}$, we have $l^{\partial_1(x)}=l$, so $\partial_1(M)$ has trivial action on $L$.
This is functorial and we can say that there is a functor from the full subcategory of quadratic complexes of length 1 given by those quadratic complexes with trivial quadratic map to the category of crossed complexes of length 2. For further work about crossed and quadratic complexes see [3] and [7].
§ PULLBACK QUADRATIC MODULE
In this section we give a construction of a pullback quadratic module. Firstly, we should give the construction of a pullback nil(2)-module.
§.§ Pullback (Co-induced) Nil(2)-Module
Suppose that
$\partial :M\rightarrow Q$ is a nil(2)-module and $\sigma :P\rightarrow Q$
is a homomorphism of groups. We give a construction of a pullback nil(2)-module
hence we define a functor which changes the base of $\partial $ from $Q$
to $P$. That is, we shall define a functor
$$\lambda :\mathbf{Nil}(2)/Q\longrightarrow \mathbf{Nil}(2)/P$$
where $\mathbf{Nil}(2)/Q$ is the subcategory of $\mathbf{Nil}(2)$ whose objects are nil(2)-modules with the common codomain $Q$.
Consider the following diagram
\begin{equation*}
\xymatrix{&M\ar[d]^{\partial}\\ P\ar[r]_{\sigma}&Q.}
\end{equation*}
Take $\sigma ^{\ast }(M)=\{(p,m):\partial (m)=\sigma (p)\}$ as the fiber product
of $\partial $ and $\sigma $. Thus we have the following pullback diagram
\begin{equation*}
\xymatrix{\sigma^*(M)\ar[d]_{\beta_{1}}\ar[r]^-{\sigma_{1}}&M\ar[d]^\partial%
\\P\ar[r]_\sigma&Q}\tag{1}
\end{equation*}
where $\sigma _{1}:\sigma ^{\ast }(M)\rightarrow P$ is given by $\sigma
_1(p,m)=m$ and $\beta _{1}:\sigma ^{\ast }(M)\rightarrow P$ is
given by $\beta _{1}(p,m)=p$ for all $(p,m)\in \sigma ^{\ast }(M)$. The action of $p^{\prime }\in P$ on $(p,m)\in
\sigma^*(M)$ can be given by
\begin{equation*}
(p,m)^{p^{\prime }}=(p'^{-1}pp',m^{\sigma(p')}).
\end{equation*}
This action obviously is a group action of $P$ on $\sigma^*(M)$ and according to this
action, $\beta_1$ becomes a nil(2)-module. Indeed, $\beta_1$ is a
pre-crossed module since,
\begin{align*}
\beta_1((p,m)^{p^{\prime }})&=\beta_1(p'^{-1}pp',m^{\sigma(p')}) \\
&=p'^{-1}pp^{\prime }\\
&=p'^{-1}\beta_{1}(p,m)p^{\prime },
\end{align*}
for all $(p,m)\in \sigma^*(M)$ and for $(p_1,m_1),(p_2,m_2),(p_3,m_3)\in \sigma^*(M)$, we have
\begin{align*}
\langle\langle(p_1,m_1),(p_2,m_2)\rangle,(p_3,m_3)\rangle
m_1)},(p_3,m_3)\rangle \\
^{-1})(p_1,m_1)(p_2,m_2)^{p_1},(p_3,m_3)\rangle \\
=&\langle(1,{m_1}^{-1}{m_2}^{-1}{m_1}{m_2}^{\sigma(p_1)}),(p_3,m_3)\rangle \\
\\
^{-1}) \\
&\quad(1,{m_1}^{-1} {m_2}^{-1}{m_1}{m_2}^{\partial(m_1)})(p_3,m_3)^{%
\beta_{1}(1,{m_1}^{-1}{m_2}^{-1}{m_1}{m_2}^{\partial(m_1)})} \\
\\
&\quad \quad(1,{m_1}^{-1}{m_2}^{-1}{m_1}{m_2}^{\partial(m_1)})(p_3,m_3) \\
=&(1,{\langle m_1,m_2 \rangle }^{-1})({p_3}^{-1},{m_3}^{-1})(1,{\langle
m_1,m_2 \rangle })(p_3,m_3) \\
=&(1,{\langle m_1,m_2 \rangle }^{-1}{m_3}^{-1}\langle m_1,m_2 \rangle m_3) \\
=&(1,{\langle m_1,m_2 \rangle }^{-1}{m_3}^{-1}\langle m_1,m_2 \rangle {m_3}%
^{\partial_{1}(\langle m_1,m_2 \rangle)})\\
=&(1,\langle \langle m_1,m_2 \rangle ,m_3 \rangle).
\end{align*}
Since $\partial : M \rightarrow Q$ is a nil(2)-module, we have $\langle
\langle m_1,m_2 \rangle ,m_3 \rangle=1$ and then we have
$$\langle(p_1,m_1),(p_2,m_2)\rangle,(p_3,m_3)\rangle=(1,1)\in \sigma^*(M).$$
Similarly, it can be shown that $\langle(p_1,m_1),\langle(p_2,m_2),(p_3,m_3)\rangle\rangle=(1,1).$
Thus $\beta_1 : \sigma^*(M)\rightarrow P$ is a nil(2)-module. In diagram (1), the pair of homomorphisms ($\sigma_1,\sigma$) is a nil(2)-module morphism. This diagram is commutative since
\partial\sigma_1(p,m)=\partial(m)=\sigma(p)=\sigma\beta_1(p,m)
for $p\in P$ and $m\in M$. We have
\begin{align*}
\sigma_1((p,m)^{p^{\prime }}) =& \sigma_1((p')^{-1}pp',m^{\sigma(p')}) \\
=&m^{\sigma(p')} \\
=&\sigma_1 (p,m)^{\sigma (p')}
\end{align*}
for all $(p,m)\in \sigma^*(M)$ and $p\in P$.
Thus we have a nil(2)-module with the base $P$. Obviously this is functorial and we can define a functor by
$$\lambda (\partial:M\rightarrow Q)=(\beta_1 : \sigma^* (M)\rightarrow P)$$
which changes the base of the nil(2)-module $\partial$ from $Q$ to $P$ and where $\beta_1$ is the pullback nil(2)-module of $\partial$ by the homomorphism $\sigma$.
§.§ Construction of a Pullback Quadratic Module
\begin{equation*}
\xymatrix{&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\ C_2
\ar[r]_{\partial_2}&C_1 \ar[r]_{\partial_1} &C_0}
\end{equation*}
be a quadratic module of groups and $\sigma : B \rightarrow C_0$ a
homomorphism of groups. We try to construct a pullback quadratic
module by the homomorphism $\sigma : B \rightarrow C_0$. Given any homomorphisms of groups
\begin{equation*}
\xymatrix{{C_2} \ar[r]^{\partial_2}&{C_1}\ar[r]^
{\partial_1}& {C_0}}
\end{equation*}
and $\sigma: B \rightarrow C_0$, for the pullback $\langle B_1 , \beta_1 , \sigma_1
\rangle$ of $\partial_1$ by $\sigma$ and the pullback $\langle B_2 ,
\beta_2 , \sigma_2 \rangle$ of $\partial_2$ by $\sigma_1$. Then $\langle B_2 , \beta_1\beta_2 , \sigma_2 \rangle$ is a pullback of $\partial_1\partial_2$ by $\sigma$. Now, consider the diagram
\begin{equation*}
\xymatrix {&C_2 \ar[d]^{\partial_1\partial_2}\\ {B/{\ker\sigma}}
\ar[r]_-{\sigma^*} & C_0}
\end{equation*}
where $\sigma^* : {B/{\ker\sigma}} \rightarrow C_0$ given by $\sigma^*(b\ker\sigma) = \sigma(b)$ for $b\in B$ and $\partial_1\partial_2=1$. Thus the following diagram
\begin{equation*}
\xymatrix{B_{21} \ar[d]_{\beta}\ar[r]^p &C_2 \ar[d]^{\partial_1\partial_2}\\
\end{equation*}
is a pullback diagram where
\begin{align*}
B_{21}=& \{(b\ker\sigma, c_2): \sigma (b)= \partial_1\partial_2(c_2) = 1 \}
\\
=&\{(b\ker\sigma, c_2): b\in \ker\sigma\} \\
=&\{(\ker\sigma, c_2): c_2 \in C_2\}
\end{align*}
and where $\beta$ and $p$ are given by $(\ker\sigma,c_2)\mapsto \ker\sigma$ and $(\ker\sigma,
c_2) \mapsto c_2$ for $(\ker\sigma, c_2)\in B_{21}$ respectively.
Now consider the following diagram
\begin{equation*}
\xymatrix {&C_1 \ar[d]^{\partial_1}\\ {B/{\ker\sigma}} \ar[r]_-{\sigma^*} &
\end{equation*}
in which $\partial_1$ is a nil(2)-module. The pullback nil(2)-module by the homomorphism $\sigma^*:B/\ker\sigma\rightarrow C_0$ can be given by a diagram
\begin{equation*}
\xymatrix{B_1 \ar[d]_{\beta_1}\ar[r]^{\overline\sigma_1} &C_1
\ar[d]^{\partial_1}\\ B/{\ker\sigma}\ar[r]_-{\sigma^*}&C_0} \tag{1} \\
\end{equation*}
where $B_1= \{(b\ker\sigma, c_1): \sigma^* (b\ker\sigma)= \sigma(b)=
\partial_1(c_1)\}$ is the fiber product of $\partial_1$ and $\sigma^*$. $\beta_1$ , $\overline{\sigma_1}$
are given by $\beta_1: B_1 \rightarrow B/\ker\sigma$, $\beta_1(b\ker\sigma,c_1)=b\ker\sigma$ and
$\overline{\sigma_1}: B_1\rightarrow C_1$, $\overline{\sigma_1}(b\ker\sigma,c_1)=c_1$ for all $b\ker\sigma \in B/\ker\sigma$ and $c_1\in C_1$. Then diagram (1) becomes a pullback diagram. Furthermore, from the following diagram
\begin{equation*}
\xymatrix {&C_2 \ar[d]^{\partial_2}\\ B_1 \ar[r]_-{\overline\sigma_1} & C_1}
\end{equation*}
and since the pullback of a pullback is again a pullback, we can define a pullback of $\overline{\sigma_1}$ and $\partial_2$ as given in the following diagram
\begin{equation*}
\xymatrix{B_2 \ar[d]_{\beta_2}\ar[r]^{\sigma_2}&C_2 \ar[d]^{\partial_2}\\
\end{equation*}
in which
\begin{align*}
B_2=& \{((b\ker\sigma,c_1), c_2): \sigma^* (b\ker\sigma)=\sigma(b)=
\partial_1(c_1), \overline{\sigma_1}(b\ker\sigma,c_1)=c_1=\partial_2(c_2) \}
\\
=&\{((b\ker\sigma,c_1), c_2):
\sigma(b)=\partial_1(c_1)=\partial_1(\partial_2(c_2)) = 1\} \\
=&\{((b\ker\sigma,c_1), c_2): b\in \ker\sigma\} \\
=&\{(\ker\sigma,\partial_2(c_2), c_2): c_2 \in C_2\}
\end{align*}
and $\beta_2$ is given by $\beta_2(\ker\sigma,\partial_2(c_2),
c_2)=(\ker\sigma,\partial_2(c_2))$ and $\sigma_2$ is given by $\sigma_2(\ker\sigma,\partial_2(c_2), c_2)=c_2$ for all $(\ker\sigma,\partial_2(c_2))\in B_2$. Since for all $(\ker\sigma,\partial_2(c_2),
c_2)\in B_2$
\begin{align*}
\partial_2\sigma_2(\ker\sigma,\partial_2(c_2), c_2)=&\partial_2(c_2) =c_1 =\overline{\sigma_1}(\ker\sigma, c_1) =\overline{\sigma_1}\beta_2(\ker\sigma,\partial_2(c_2), c_2),
\end{align*}
the diagram is commutative, and is also a pullback diagram. We can define an isomorphism
$\Phi:B_2 \rightarrow B_{21}$ by $\Phi(\ker\sigma,\partial_2(c_2),
c_2)=(\ker\sigma,c_2)$ for $(\ker\sigma,\partial_2(c_2),
c_2)\in B_2$.
By using this isomorphism, we have the following diagram
\begin{equation*}
\xymatrix{B_2\ar@/^1.1pc/[drr]^{\sigma_2}\ar[dr]^\Phi\ar@/_1.1pc/[ddr]^{%
\beta_2}&&\\
&B_1\ar[d]^{\beta_1}\ar[r]^{\sigma_1} &C_1\ar[d]^{\partial_1}\\
&B/{\ker\sigma}\ar[r]_-{\sigma^*}&C_0} \\
\end{equation*}
where $\overline{\partial_2}: B_{21}\rightarrow B_1$ is given by
c_2)=(\ker\sigma,\partial_2(c_2))$ and $\beta_1(b\ker\sigma,c_1)=b\ker\sigma$, for $(\ker\sigma, c_2)\in B_{21}$ and for $(b\ker\sigma,c_1)\in B_1$.
Then we have $\beta_1\overline{\partial_2}(\ker\sigma,c_2)=\beta_1(\ker%
\sigma,\partial_2(c_2))=\ker\sigma \in {B/\ker\sigma}$, hence,
\begin{equation*}
\xymatrix{B_{21} \ar[r]^{\overline{\partial_2}}&B_1\ar[r]^-{\beta_1}&B/\ker\sigma}
\end{equation*}
becomes a complex of groups.
But, we want to construct the pullback quadratic module by the homomorphism $\sigma: B\rightarrow C_0$ instead
of $\sigma^*:B/\ker\sigma \rightarrow C_0$. To construct it by $\sigma$, we must have ${B/\ker\sigma} \cong B$. This is possible only if $\ker\sigma =\{1\}$ which means $%
\sigma$ is a monomorphism. That is, the homomorphism $\sigma:B\rightarrow C_0$ must be a monomorphism.
We construct the pullback quadratic module by taking $\sigma$ as a monomorphism.
Then we have $\sigma^*=\sigma$ and the following isomorphisms:
B/\ker\sigma \cong B
\begin{align*}
B_1=& \{(b\ker\sigma,c_1): \sigma^* (b\ker\sigma)=\sigma(b)=
\partial_1(c_1)\} \\
\cong &\{(b,c_1): \sigma(b)=\partial_1(c_1)\} \quad (\because \ker\sigma=\{1\})\\
\end{align*}
and where $\sigma^*(C_1)$ is the usual fiber product. Thus we have the following commutative diagram
\begin{equation*}
\xymatrix{B_1 \ar[d]_{\beta_1}\ar[r]^{\cong}&{\sigma^*(C_1)}
\ar[d]^{\overline{\beta_1}}\ar [r]^-{\mu_1}&C_1 \ar[d]^{\partial_1}\\
B/\ker\sigma \ar[r]_-{\cong}&B\ar[r]_\sigma &C_0}
\end{equation*}
in which the right square is a pullback square of $\partial_1$ by the monomorphism $\sigma$ and where the maps $\overline{\beta_1}$ and $\mu_1$ are given by
$\overline{\beta_1}(b,c_1)= b$ and $ \mu_1(b,c_1)=c_1$ for all $(b,c_1)\in \sigma^*(C_1)$ and then we have
Thus we have that $\overline{\beta_1}:\sigma^*(C_1)\rightarrow B$ is a
pullback nil(2)-module by the homomorphism $\sigma$ as constructed in section <ref>.
Furthermore, since $\ker\sigma =\{1\}$, we have an isomorphism
$$B_{21}=\{(\ker\sigma,c_2):c_2\in C_2\}= \{(1,c_2):c_2\in C_2\} \cong \{1\}\times
C_2\cong C_2$$
and we have the following diagram
\begin{equation*}
\xymatrix{B_{21} \ar[d]_{\overline{\partial_2}}\ar@{.>}[dr] \ar[r]^{\cong}&{C_2}
\ar[d]^{\overline \beta_2}\ar@{=} [r]^{id}&C_2 \ar[d]^{\partial_2}\\ B_1
\ar[r]_-{\cong}&{\sigma^*(C_1)}\ar[r]_{\mu_1} &C_1}
\end{equation*}
where $\overline{\beta_2}:C_2\rightarrow \sigma^*(C_1)$ is given by $\overline{\beta_2}(c_2)=(1,\partial_2(c_2))$ for $c_2 \in C_2$. Thus the right square is a pullback square of $\partial_2$ by the homomorphism $\mu_1$. Consequently, we have the following commutative diagram
\begin{equation*}
\xymatrix{C_2 \ar[d]_{\overline{\beta_2}}\ar@{=}[r]^{id}&C_2
\ar[d]^{\partial_2}\\
&C_1\ar[d]^{\partial_1}\\ B\ar[r]_\sigma &C_0,}
\end{equation*}
\begin{equation*}
\xymatrix{C_2\ar[r]^-{\overline\beta_2}&{\sigma^*(C_1)}\ar[r]^-{\overline%
\beta_1}&B}
\end{equation*}
is a complex of groups, since for all $c_2\in C_2$
\begin{equation*}
\overline{\beta_1}\overline{\beta_2}(c_2)=\overline{\beta_1}%
\end{equation*}
Now, we must define the quadratic map. Let $C^{^{\prime }}=((\sigma^*(C_1))^{cr})^{ab}$. The quadratic map
$\omega^{\prime }:C^{\prime }\otimes C^{\prime }\longrightarrow C_2$
can be given by
\begin{equation*}
\omega^{\prime }(\{(b_1,c_1)\}\otimes \{({b'_1},c'_1)\})=\omega (\{c_1\}\otimes \{{c'_1}\})
\end{equation*}
for all $\{(b_1,c_1)\}\otimes \{(b'_1,c'_1)\}\in C' \otimes C'$ and $\{c_1\}\otimes \{c'_1\}\in C\otimes C$ where $C=({C_1}^{cr})^{ab}$ and where $\omega$ is the quadratic map of the first quadratic module.
Thus we have
The diagram
\begin{equation*}
\xymatrix{&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\ C_2
\ar[r]_-{\overline\beta_2}&\sigma*(C_1)\ar[r]_-{\overline\beta_1} &B}
\end{equation*}
is a quadratic $B$-module.
QM1: In section <ref> we have showed that $\overline\beta_1 :
\sigma^*(C_1)\rightarrow B$ is a nil(2)-module and $\overline{\beta_1} \overline{\beta_2}=1$.
QM2: For all $(b_1,c_1)$,$(b'_1,c'_1)\in \sigma^*(C_1)$, we
\begin{align*}
\overline\beta_2\omega^{\prime }(\{(b_1,c_1)\}\otimes\{({b'_1},{c'_1})\})
=&\overline\beta_2(\omega\{c_1\}\otimes\{c'_1\}) \\
=& (1,\partial_2\omega(\{c_1\}\otimes\{c'_1\})) \\
=& (1,w(\{c_1\}\otimes\{c'_1\}))
\intertext{and since}
\langle(b_1,c_1),(b_2,c_2)
\rangle=&(b_1,c_1)^{-1}(b_2,c_2)^{-1}(b_1,c_1)(b_2,c_2)^{\overline%
\beta_1(b_1,c_1)} \\
=&({b_1}^{-1},{c_1}^{-1})({b_2}^{-1},{c_2}^{-1})(b_1,c_1)(b_2,c_2)^{b_1} \\
=& ({b_1}^{-1}{b_2}^{-1}b_1{b_1}^{-1}b_2{b_1},{c_1}^{-1}{c_2}^{-1}{c_1}{%
c_2}^{\sigma(b_1)}) \\
=& (1,{c_1}^{-1}{c_2}^{-1}{c_1}{c_2}^{\partial_1(c_1)})\qquad (\because
\sigma(b_1)=\partial_1(c_1)) \\
=& (1,w({c_1}\otimes{c_1^{\prime }})),
\end{align*}
we have
\begin{align*}
\overline\beta_2\omega^{\prime }(\{(b_1,c_1)\}\otimes\{({b'_1}^,{c'_1})\})=&(1,w(\{c_1\}\otimes\{c'_1\})) \\
=&w^{\prime }(\{(b_1,c_1)\}\otimes\{({b'_1}^,{c'_1}^)\}).
\end{align*}
The verification of the other axioms of quadratic module is routine, so, we leave it to the reader.
The constructed quadratic module
\begin{equation*}
\xymatrix{&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\ C_2
\ar[r]_-{\overline\beta_2}&\sigma^*(C_1) \ar[r]_-{\overline\beta_1} &B}
\end{equation*}
is a pullback quadratic module of
\begin{equation*}
\xymatrix{&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\ C_2
\ar[r]_{\partial_2}&C_1 \ar[r]_{\partial_1} &C_0}
\end{equation*}
by the monomorphism $\sigma:B\rightarrow C_0$.
Firstly, we will show that in the following diagram
\begin{equation*}
\xymatrix{&C_2
\ar[d]_{\overline\beta_2} \ar@{=}[r]^{id} &C_2\ar[d]^{\partial_2}\\
C'\otimes C' \ar@/^1.2pc/[ur]_{\omega'}\ar[r]_{w'}&\sigma^*(c_1)\ar[r]_{\mu_1}\ar[d]_{\overline\beta_1}&C_1
\ar[d]^{\partial_1}& C\otimes
C\ar[l]^w \ar@/_1.2pc/[ul]^\omega \\ &B\ar[r]_\sigma&C_1&}
\end{equation*}
$(id,\mu_1,\sigma)$ is a quadratic module morphism. In the above
construction, we have showed that this diagram is commutative. Now, we show that
$(id,\mu_1,\sigma)$ preserves the actions of $B$. The actions of $B$ on $\sigma^*(C_1)$ and $C_2$ are given by
$(c_2)^b=(c_2)^{\sigma (b)}$ and $(b_1,c_1)^b=(bb_1b^{-1},c_1^{\sigma (b)})$ for $c_2\in C_2$ and $(b_1,c_1)\in \sigma^*(C_1)$. We have $ id ({c_2}^b)={c_2}^b={c_2}^{\sigma(b)},$
\begin{align*}
\mu_1((b_1,c_1)^b)= &\mu_1(bb_1b^{-1},{c_1}^{\sigma(b)}) \\
=&{c_1}^{\sigma(b)} \\
=& \mu_1(b_1,c_1)^{\sigma(b)}
\intertext{and }
\omega(\{\mu_1(b_1,c_1)\}\otimes\{\mu_1(b_1^{\prime },c_1^{\prime })\})
=&\omega(\{c_1\}\otimes\{c_1^{\prime }\}) \\
=&\omega'(\{(b_1,c_1)\}\otimes\{(b_1^{\prime },c_1^{\prime })\}) \\
=&id\omega^{\prime }(\{(b_1,c_1)\}\otimes\{(b_1^{\prime },c_1^{\prime })\})
\end{align*}
for all $c_2\in C_2, c_1\in C_1$ and $b,b_1, b'_1\in B$. Thus $(id,\mu_1,\sigma)$ is a quadratic module morphism. Now, we check the universal property.
Start with the quadratic module
\begin{equation*}
\xymatrix{&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\
C_2\ar[r]_{\partial_2}&C_1 \ar[r]_{\partial_1} &C_0}
\end{equation*}
and the homomorphism $\sigma:B\rightarrow C_0$, and the pullbacks $\langle\sigma^*(C_1),\overline\beta_1,\mu_1\rangle$ and $\langle C_2,\overline\beta_2,id\rangle$
constructed above and any quadratic module
\xymatrix{&C''\otimes C'' \ar[dl] \ar[d] \\
E\ar[r]_{\delta}&\sigma^*(C_1) \ar[r]_{\overline{\beta_1}} &B.}
Let $f:E\rightarrow B$ and $g:E\rightarrow C_2$ be two morphisms as given in the following diagram
\begin{equation*}
\xymatrix{E\ar[ddr]^{\delta}\ar@/_1.2pc/[dddr]_f \ar@/^1.2pc/[drr]^g&& \\ &C_2
\ar[d]^{\overline\beta_2}\ar@{=}[r]&C_2 \ar[d]^{\partial_2}\\
\\ &B\ar[r]_\sigma&C_0}
\end{equation*}
where $\sigma f= \partial_1 \partial_2 g =1$. But then $\partial_2 g$ and the
universal property of the pullback nil(2)-module $\langle\sigma^*(C_1),\overline\beta_1,\mu_1\rangle$
gives a unique morphism $h=\delta:E\rightarrow \sigma^*(C_1)$ with $%
f=\overline\beta_1 h$ and $\partial_2 g=\mu_1 h.$ There are two commutative diagrams
\begin{tabular}{cc}
\xymatrix{E\ar@/_1.2pc/[ddr]_f \ar@/^1.2pc/[drr]^g&& \\ &C_2
\ar[d]^{\overline\beta_1\overline\beta_2}\ar@{=}[r]^{id}&C_2
\ar[d]^{\partial_1\partial_2}\\ &B\ar[r]_\sigma&C_0}
\xymatrix{E\ar@/_1.2pc/[ddr]_f
\ar@{.>}[dr]^h\ar@/^1.2pc/[drr]^{\partial_2g}&& \\
\\ &B\ar[r]_\sigma&C_0}
\end{tabular}
Using the isomorphism $B_{21}\cong C_2$ given above and the universal property for the pullback of a pullback
$\langle C_2,id,\overline\beta_2\rangle$, for $g:E\rightarrow C_2$ and $h:E\rightarrow \sigma^*(C_1)$ with $\partial_2 g=\mu_1 h$ gives a map $\varepsilon:E\rightarrow C_2$ with $h=\overline\beta_2
\varepsilon$ and $g=\varepsilon$. Thus, there are two commutative
diagrams, where the second one is obtained by gluing two commutative
diagrams together along $\mu_1:\sigma^*(C_1)\rightarrow C_1$.
\begin{tabular}{cc}
\xymatrix{E\ar@/_1.2pc/[ddr]_h\ar@{.>}[dr]^{\varepsilon=g}%
\ar@/^1.2pc/[drr]^g&& \\ &C_2 \ar[d]^{\overline\beta_2}\ar@{=}[r]&C_2
\ar[d]^{\partial_2}\\ &\sigma^*(C_1)\ar[r]_{\mu_1}&C_1}
\xymatrix{E\ar@/_1.2pc/[dddr]_f
\ar@/^1.2pc/[drr]^g\ar@{.>}[dr]^{\varepsilon=g}\ar[ddr]^h&& \\ &C_2
\ar[d]^{\overline\beta_2}\ar@{=}[r]&C_2 \ar[d]^{\partial_2}\\
\\ &B\ar[r]_\sigma&C_0.}
\end{tabular}
In particular $f=\overline\beta_1\overline\beta_2\varepsilon$ and $g=\varepsilon$,
and the existence part of the proof has been accomplished.
Suppose now that $\eta:E\rightarrow C_2$ is another map with $%
f=\overline\beta_1\overline\beta_2\eta $. Then both $%
\overline\beta_2\varepsilon,\overline\beta_2\eta:E\rightarrow
\sigma^*(C_1).$ Furthermore, the commutativity of the last diagram gives
that $f=\overline\beta_1(\overline\beta_2\varepsilon)=\overline\beta_1(%
\overline\beta_2\eta)$ and $\partial_2g=\partial_2\varepsilon=\mu_1(%
\overline\beta_2\varepsilon)=\mu_1(\overline\beta_2\eta).$ By the
uniqueness property for the pullback $\sigma^*(C_1),$ we have
$\overline{\beta_2} \varepsilon=\overline\beta_2\eta$. We thus have two commutative diagrams:
\begin{tabular}{cc}
\xymatrix{E\ar@/_1.2pc/[ddr]_f\ar@<1ex>[dr]^{\varepsilon}\ar@<0ex>[dr]_{%
\eta}\ar@/^1.2pc/[drr]^g&& \\ &C_2 \ar[d]^{\overline\beta_2}\ar@{=}[r]&C_2
\ar[d]^{\partial_2}\\ &\sigma^*(C_1)\ar[r]_-{\mu_1}&C_1}
\xymatrix{E\ar@/_1.2pc/[ddr]_{h}
\ar@<1ex>[dr]^{\overline\beta_2\varepsilon}\ar@<0ex>[dr]_{\overline\beta_2%
\eta}\ar@/^1.2pc/[drr]^{\partial_2g}&& \\
\\ &B\ar[r]_\sigma &C_0.}
\end{tabular}
Finally, the uniqueness for the pullback $\langle
C_2,\overline\beta_2,id\rangle$ constructed by using the isomorphism $B_{21}\cong C_2$ in the second diagram yields that $%
\eta=\varepsilon$.
This construction can be expressed functorially
\sigma^*:\mathbf{Quad}/C_0\longrightarrow \mathbf{Quad}/B
which is a pullback functor. This functor has a left adjoint
\sigma_*:\mathbf{Quad}/B\longrightarrow \mathbf{Quad}/C_0
which gives an induced quadratic module as follows.
§ INDUCED QUADRATIC MODULE
In this section we give a construction of an induced quadratic module. We start by giving the construction of an induced nil(2)-module.
§.§ Induced Nil(2)-Module
Let $\mu : M \rightarrow P$ be a nil(2)-module and $f: P
\rightarrow Q$ be a homomorphism of groups. Let $f^*(M) = F(M\times Q)$ be a free
group generated by the set $M\times Q$. Let $S$ be a subgroup of $f^*(M)$ generated by the following relations:
($m,m'\in M$, $q\in Q$)
* $(m,q)(m^{\prime },q)(mm',q)^{-1}\in S$
* $(m^p,q){(m,f(p)q)}^{-1}\in S$
Now, consider the following diagram
\begin{equation*}
\xymatrix{M \ar[d]_{\mu}\ar[r]^-{\theta}& {f^*(M)}/S\ar[d]^{\overline\mu}\\ P\ar[r]_f &Q}
\end{equation*}
in which $\overline{\mu} : f^*(M)/S \rightarrow Q$ is given by
\overline{\mu}((m,q)S)= q^{-1}f\mu (m)q
and $\theta : M\rightarrow f^*(M)/S$ is given by
\theta (m)= (m,1)S
for $m\in M $ and $q\in Q.$ This diagram is commutative, since
\begin{equation*}
\overline{\mu}\theta (m)= \overline{\mu}((m,1)S)=f \overline{\mu}(m).
\end{equation*}
for all $m\in M$.
The action of $Q$ on $f^*(M)/S$ can be given by
\begin{equation*}
((m,q)S)^{q^{^{\prime }}}=(m,qq^{^{\prime }})S
\end{equation*}
for $m\in M$ and $q, q'\in Q$. By using this action, we have the following result.
The homomorphism $\overline{\mu} : f^*(M)/S \rightarrow Q$ given by $\overline{\mu}((m,q)S)=
q^{-1}f\mu (m)q$, as defined above, is an induced nil(2)-module by the homomorphism of groups $f:P\rightarrow Q$ of the nil(2)-module $\mu:M\rightarrow P$.
\begin{align*}
\overline{\mu}{(((m,q)S)}^{q^{^{\prime }}})=&\overline{\mu}((m,qq^{^{\prime
}})S) \\
=&{(qq^{^{\prime }})}^{-1}f\mu(m)qq^{^{\prime }} \\
=&(q^{^{\prime }})^{-1}(q^{-1}f\mu(m)q)q^{^{\prime }} \\
=&(q^{^{\prime }})^{-1} \overline{\mu}((m,q)S)q^{^{\prime }},
\end{align*}
for all $m\in M$ and $q,q'\in Q$, $\overline{\mu}$ is a pre-crossed module.
Further, for all $(m,q)S,(m',q)S,(m'',q)S \in f^*(M)/S,$
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
\langle\langle(m,q)S,(m',q)S\rangle,(m'',q)S\rangle&=& \langle(m,q)S(m',q)S(m,q)S^{-1}((m',q)S^{-1})^{\overline\mu(m,q)S},(m'',q)S\rangle\\
&=& \langle(m,q)S(m',q)S(m^{-1},q)S((m'^{-1},q)S)^{q^{-1}f\mu(m)q},(m'',q)S\rangle \\
&=& \langle(mm'm^{-1},q)S((m'^{-1},qq^{-1}f\mu(m)q)S,(m'',q)S\rangle \\
&=& \langle(mm'm^{-1},q)S((m'^{-1})^{\mu(m)},q)S,(m'',q)S\rangle\\
&=& \langle(mm'm^{-1}(m'^{-1})^{\mu(m)},q)S,(m'',q)S\rangle \\
&=& \langle(\langle m,m'\rangle ,q)S,(m'',q)S\rangle \\
&=& (\langle m,m'\rangle ,q)S(m'',q)S(\langle m,m'\rangle ,q)^{-1}S((m'',q)S^{-1})^{\overline\mu(\langle m,m'\rangle ,q)S}\\
&=& (\langle m,m'\rangle ,q)S(m'',q)S(\langle m,m'\rangle ,q)^{-1}S((m'',q)S^{-1})^{q'{-1}f\mu(\langle m,m'\rangle)q}\\
&=& (\langle m,m'\rangle ,q)S(m'',q)S(\langle m,m'\rangle^{-1} ,q)S((m''^{-1},q)S)\\
&=& (\langle m,m'\rangle m''\langle m,m'\rangle^{-1}(m''^{-1},q)S) \\
&=& (\langle m,m'\rangle m''\langle m,m'\rangle^{-1}(m''^{-1})^{\mu(\langle m,m'\rangle)},q)S \\
&=& (\langle\langle m,m'\rangle m''\langle,q)S\\
&=& (1,q)S \cong S
\end{eqnarray*}
Similarly, it can be shown that
$$\langle(m,q)S,\langle (m',q)S,(m'',q)S\rangle \rangle =S.$$
Thus we have that $\overline{\mu}$ is a nil(2)-module. Now, we will show that $(\theta,f)$ is a nil(2)-module morphism.
We have
\begin{align*}
\theta (m^p)=&(m^p,1)S \\
=&(m,f(p)1)S \\
=&((m,1)S)^{f(p)} \\
=&\theta (m)^{f(p)}
\end{align*}
and $\overline{\mu}\theta (m)=\overline{\mu}((m,1)S)=f\mu(m)$ for all $m\in M$ and $p\in P$.
Now we check the universal property. Consider the following diagram
\begin{equation*}
\xymatrix{&&N\ar@/^1.1pc/[ddl]^\upsilon\\
Q)/S\ar@{.>}[ur]_{h'}\ar[d]^{\overline\mu}&\\ P\ar[r]_f&Q&}
\end{equation*}
in which $N\rightarrow Q$ is any nil(2) $Q$-module and
\begin{equation*}
(h,f) : (M\rightarrow P) \rightarrow (N \rightarrow Q)
\end{equation*}
is any nil(2)-module morphism. The homomorphism $h^{\prime }:f^*(M)/S \rightarrow N$ given by $(m,q)S\mapsto h(m)^q$ for $(m,q)S\in f^*(M)/S $ is the necessary unique morphism extending the commutativity of the diagram. Indeed, we have on generators
\begin{equation*}
h^{\prime }\theta(m)= h'((m,1)S)=h(m)
\end{equation*}
\begin{equation*}
\upsilon h'((m,q)S)=\upsilon(h(m)^q)=q^{-1}\upsilon (h(m))q=q^{-1}f\mu (m)q=\overline {\mu}%
\end{equation*}
Thus we have $\upsilon h^{\prime }=\overline {\mu}$. Therefore, $\overline{\mu}$ is an induced nil(2)-module by the homomorphism of groups $f:P\rightarrow Q$.
§.§ Construction of an Induced Quadratic Module
For any quadratic module
\begin{equation*}
\xymatrix{&&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\
\left(\sigma=(\omega,w,\partial_2)\right): &L\ar[r]_{\partial_2}&M\ar[r]_{\partial_1} &P}
\end{equation*}
and a morphism $\phi :P\rightarrow Q,$ the induced quadratic module can be
given by
(i) a quadratic module
\begin{equation*}
\xymatrix{&&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\
\phi_{\ast}(L)\ar[r]_{\overline{\partial_2}}&\phi_{\ast}(M) \ar[r]_-{\overline{\partial_1}}&Q}
\end{equation*}
(ii) given a quadratic module morphism
\begin{equation*}
(f_{1},f_{2},\phi ):(\sigma:(\omega ,w,\partial _{2}))\longrightarrow
\end{equation*}
then there is a unique quadratic module morphism
$$(f_{1\ast },f_{2\ast },id_{Q}):\phi_{*}(\omega,w,\partial_2)\longrightarrow (\sigma'':(\omega'',w'',\partial''_{2}))$$
such that commutes the following diagram
\begin{equation*}
\xymatrix{(\sigma:(\omega,w, \partial_2))\ar[d]_{(f_1,f_2,\phi)}\ar[drr]\\
(\sigma'':(\omega'',w'', \partial''_2))&&\phi_{*}(\omega,w,\partial_2).%
\ar@{.>}[ll]^-{(f_{1*},f_{2*},id_Q)}}
\end{equation*}
For a homomorphism of groups $\phi :P\rightarrow Q$ and a quadratic module
\xymatrix{&&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\
\sigma:& L\ar[r]_{\partial_2}&M\ar[r]_{\partial_1} &P,}
let $F(L\times Q)$ be a free group generated by elements of $L\times Q$. Let $S'$ be a normal subgroup of $F(L\times Q)$ generated by following relations:
* $(l,q)^{(m,q)}(l^m,q)^{-1}\in S'$
* $(l,q)(l',q)(ll',q)^{-1}\in S'$
* $(l^{p},)(l,\phi (p)q)^{-1}\in S'$
* $(l,q)^{q'}(l,qq')^{-1}\in S'$
for all $l_1 ,l_2 \in L$, $q_1 , q_2 \in Q$ and $p\in P$.
Recall from section <ref> that if $\partial _{1}:M\rightarrow P$ is a nil(2)-module and $\phi :P\rightarrow Q$ is a homomorphism of groups, we constructed the induced nil(2)-module $\phi _{\ast
}(M)\rightarrow Q$ where $\phi _{\ast }(M)=F(M\times Q)/S$ and $S$ is the
normal subgroup of the free group $F(M\times Q)$ generated by elements of
the forms given in section <ref>.
We define $\phi _{\ast }(L)=F(L\times Q)/S^{\prime }$ and $\phi _{\ast
}(M)=F(M\times Q)/S$. From section <ref>, $\overline{\partial }_{1}:{F(M\times Q)}/S\rightarrow Q$ can
be given by on generators $\overline{\partial }_{1}((m,q)S)=q^{-1}(\phi
\partial _{1}(m))q$ for all $m\in M$ and $q\in Q$. The map $\overline{\partial }_{2}:\phi _{\ast }(L)\rightarrow {F(M\times Q)}%
/S$ can be given by on generators $\overline{\partial }_{2}((l,q)S^{\prime
})=(\partial _{2}l,q)S$ for $l\in L$ and $q\in Q.$ Since for all $l,l' \in L$ and $q\in Q$ and $\partial _{2}(S^{\prime })=S$, we have
\begin{align*}
\overline{\partial }_{2}((l,q)S^{\prime }(l',q)S^{\prime })=&
\overline{\partial }_{2}((ll',q)S^{\prime }) \\
=& ((\partial _{2}l)(\partial _{2}l'),q)S \\
=& (\partial _{2}l,q)S(\partial _{2}l',q)S
\end{align*}
so this is a well-defined group homomorphism.
Let $C^{\prime }=((\phi _{\ast }(M))^{cr})^{ab}.$ The quadratic map $\omega ^{\prime }:C^{\prime }\otimes C^{\prime }\rightarrow \phi _{\ast }(L)$ can be given by on generators
\begin{equation*}
\omega ^{\prime }(\{(m,q)S\}\otimes \{(m',q)S\})=(\omega
(\{m\}\otimes \{m'\}),q)S^{\prime }
\end{equation*}
where $(m,q)S, (m',q)S \in {F(M\times Q)/S},\{(m,q)S\},\{(m',q)S\}\in C^{\prime
},\{m\},\{m'\}\in C$. Then we have
The diagram
\begin{equation*}
\xymatrix{&&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\
\phi_{*}(\omega,w,\partial_2):&\phi_{\ast}(L)\ar[r]_{\overline\partial_2}&\phi_{\ast}(M)
\ar[r]_-{\overline\partial_1}&Q}
\end{equation*}
is a quadratic $Q$-module.
QM1: In theorem <ref>, we have proven that
$\overline\partial_1:\phi_*(M)\rightarrow Q$ is a nil(2)-module and we have
\partial_1(\partial_2 l)q=q^{-1}\phi(1)q=1,$ for all $(l,q)S'\in \phi_*(L)$.
For all $\{(m,q)S\}\otimes \{(m',q)S\}\in C^{\prime }\otimes
C^{\prime },$ we have
\begin{align*}
\overline\partial_2\omega^{\prime }(\{(m,q)S\}\otimes
\{(m',q)S\})=&\overline\partial_2((\omega\{m\}\otimes\{m'\},q)S^{%
\prime }) \\
=&(\partial_2\omega(\{m\}\otimes\{m'\}),q)S^{\prime } \\
=&(w(\{m\}\otimes\{m'\}),q)S \\
\end{align*}
On the other hand,
\begin{align*}
\{(m',q)S\})=& \langle (m,q)S,
(m',q)S\rangle \\
=&(m,q)S(m',q)S((m,q)S)^{-1}(((m',q)S)^{-1})^{\overline\partial_1((m,q)S)} \\
=& (mm'm^{-1},q)(m'^{-1},qq^{-1}\phi\partial_1(m)q)S\\
=& (mm'm^{-1},q)((m'^{-1})^{\partial_1(m)},q)S\\
=& (mm'm^{-1}(m'^{-1})^{\partial_1(m)},q)S
\end{align*}
then we have
\begin{align*}
\overline\partial_2\omega^{\prime }(\{(m,q)S\}\otimes \{(m',q)S\})=&({%
m}^{-1},{q}^{-1}) ({m'}^{-1},{q}^{-1}) (m,q)(m',q{q}^{-1}\phi\partial_1 (m)q)S \\
=&w^{\prime }(\{(m,q)S\}\otimes \{(m',q)S\}).
\end{align*}
QM3: For all $(\{\overline\partial_2(l,q)S^{\prime }\}\otimes
\{(m,q)S\} \in C'\otimes C'$.
\begin{multline*}
\omega^{\prime }(\{\overline\partial_2(l,q)S^{\prime }\}\otimes \{(m,q)S\}
\cdot\{(m,q)S\}\otimes\{\overline\partial_2(l,q)S^{\prime }\}) \\
\begin{aligned} &=\omega'(\{(\partial_2l,q)S\}\otimes
\{(m,q)S\}\cdot\{(m,q)S\}\otimes\{(\partial_2l,q)S\})\\
\end{aligned}
\end{multline*}
QM4: For all $(\{\overline\partial_2((l,q)S^{\prime
})\}\otimes\{\overline\partial_2((l',q)S^{\prime })\}) \in C' \otimes C'$.
\begin{align*}
\omega^{\prime }(\{\overline\partial_2((l,q)S^{\prime
})\}\otimes\{\overline\partial_2((l',q)S^{\prime })\})=&\omega^{\prime
}(\{(\partial_2l,q)S\}\otimes\{(\partial_2l',q)S\}) \\
=&(\omega\{\partial_2l\}\otimes\{\partial_2l'\},q)S' \\
}.(l',q)S^{\prime }.({l}^{-1},{q}^{-1})S'({l'}^{-1},{q}^{-1})S'\\
},(l',q)S^{\prime }].
\end{align*}
Now, we check the universal property for the constructed quadratic module $\phi_*(\sigma)=\phi_*(\omega, w, \partial_2)$. In the construction of the quadratic module $\phi_*(\sigma)$, we have the following diagram
\begin{equation*}
\xymatrix{C\otimes C \ar[dr]_w \ar[r]^\omega&L
\ar[d]_{\partial_2} \ar[r]^{\sigma_2} &\phi_{*}(L)\ar[d]^{\overline\partial_2}& C'\otimes
C'\ar[l]_{\omega'}\ar[dl]^{w'}\\ &
\ar[d]^{\overline\partial_1}&\\&P\ar[r]_\phi &Q&}\tag{1}
\end{equation*}
in which $\sigma_1$ and $\sigma_2$ are given by $\sigma_1(m)=(m,1)S$ and $\sigma_2(l)=(l,1)S'$ for all $l\in L$ and $m\in M$.
Now, we shall show that in this diagram $(\phi,\sigma_1,\sigma_2)$ is a quadratic module morphism.
We first show that the diagram (1) is commutative. For all $m\in M,$ we have
and, for all $l\in L$
\begin{align*}
\overline\partial_2\sigma_2(l)=&\overline\partial_2((l,1)S')\\
\end{align*}
Furthermore, for all $p\in P$ and $l\in L$, we have
\begin{align*}
\sigma_2(l^p)=&((l^p,1)S')\\
=&(l,\phi(p))S'\quad (\because \text{generators of } S')\\
=&{((l,1)S')}^{\phi(p)}\quad (\because \text{generators of } S')\\
\end{align*}
and for $m\in M$ we have
\begin{align*}
\sigma_1(m^p)=&((m^p,1)S)\\
=&(m,\phi(p))S \quad (\because \text{generators of } S)\\
=&{((m,1)S)}^{\phi(p)}\quad (\because \text{generators of } S)\\
\end{align*}
Thus $\sigma_1$ and $\sigma_2$ preserve the actions of $P$ and diagram $(1)$ is a commutative diagram.
Now we show that $\sigma_1,\sigma_2$ and $\phi$ commute with the quadratic maps $\omega$ and $\omega'$.
For $m,m'\in M,$ $\sigma_1(m)=(m,1)S\in \phi_*(M),$ $\sigma_1(m')=(m',1)S\in \phi_*(M)$ and $\{\sigma_1(m)\}\otimes \{\sigma_1(m')\}\in C'\otimes C',$ we have
\begin{align*}
\omega'(\{\sigma_1(m)\}\otimes \{\sigma_1(m')\})=&\omega'(\{(m,1)S\}\otimes \{(m',1)S\})\\
=&(\omega \{m\}\otimes \{m'\},1)S'\\
=&\sigma_2(\omega\{m\}\otimes \{m'\}).
\end{align*}
Consequently as shown in the following diagram
\begin{equation*}
\xymatrix{C\otimes C \ar[d]_{\Phi^{*}} \ar[r]^\omega&L\ar[r]^{\partial_2}\ar[d]^{\sigma_2}&M\ar[d]_{\sigma_1} \ar[r]^{\partial_1} &P\ar[d]^\phi\\
C'\otimes C'\ar[r]_{\omega'}&\phi_{*}(L)\ar[r]_{\overline\partial_2}& \phi_{*}(M)\ar[r]_{\overline \partial_1} &Q }
\end{equation*}
$(\phi,\sigma_1,\sigma_2)$ becomes a quadratic module morphism.
Now, suppose that
\begin{equation*}
\xymatrix{&&C''\otimes C'' \ar[dl]_{\omega''} \ar[d]^{w''} &&\\
(\sigma'':(\omega'', w'', \partial''_2)):&B_2\ar[r]_{{\partial''_2}}&B_1
\ar[r]_{{\partial''_1}}&Q}
\end{equation*}
is any quadratic $Q$-module and
$$(\phi,f_1,f_2):(\sigma:(\omega, w, \partial_2))\longrightarrow (\sigma'':(\omega'', w'',\partial''_2))$$
is any quadratic module morphism. Since the constructions of $\phi_{*}(L)$ and $\phi_{*}(M)$, there is a unique morphism $(id_Q,f_{1*},f_{2*})$ as showed in the following diagram
\begin{equation*}
\xymatrix{C'\otimes C' \ar[d]\ar[r]^{\omega'}&\phi_{*}(L)\ar[d]^{f_{2*}}\ar[r]^{\overline\partial_2}&\phi_{*}(M)\ar[d]^{f_{1*}}\ar[r]^{\overline\partial_1} &Q\ar[d]^{id}\\
C''\otimes C''\ar[r]_{\omega''}&B_2\ar[r]_{{\partial_2}'}&B_1
\ar[r]_{{\partial_1}'}&Q}
\end{equation*}
given by
$$f_{2*}:\phi_*(L)\rightarrow B_2;\quad f_{2*}((l,q)S')=f_2(l)^q$$
$$f_{1*}:\phi_*(M)\rightarrow B_1;\quad f_{1*}((m,q)S')=f_2(m)^q$$
for $l\in L$, $m\in M$ and $q, q'\in Q$. For example, we have
\begin{align*}
\end{align*}
\begin{align*}
\end{align*}
for $l\in L$, $m\in M$ and $q, q'\in Q$.
Consequently, the constructed quadratic module
\begin{equation*}
\xymatrix{&&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\
\phi_{*}(\omega,w,\partial_2):&\phi_{\ast}(L)\ar[r]_{\overline\partial_2}&\phi_{\ast}(M)
\ar[r]_-{\overline\partial_1}&Q}
\end{equation*}
is an induced quadratic module by the homomorphism $\phi:P\rightarrow Q$.
\begin{equation*}
\xymatrix{&&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\
&L\ar[r]_{\partial_2}&M\ar[r]_{\partial_1} &P}
\end{equation*}
be a quadratic module, $\phi:P\rightarrow Q$ be an epimorphism with $\ker\phi=K$ then
$$\phi_{*}(L)\cong L/[K,L] \quad\text{and}\quad \phi_{*}(M)\cong M/[K,M]$$
where $[K,L]$ is the subgroup of $L$ generated by $\{l^{-1}l^k: k\in K, l\in L\}$ and $[K,M]$ is the subgroup of $M$ generated by $\{m^{-1}m^k: k\in K, m\in M\}.$
As $\phi:P\rightarrow Q$ is an epimorphism, $Q\cong P/K.$ Since $Q$ acts on $L/[K,L]$ and $M/[K,M],$ $K$ acts trivially on $L/[K,L]$ and $M/[K,M],$ $Q\cong P/K$ acts on $L/[K,L]$ by $(l[K,L])^q=(l[K,L])^{pK}=l^p[K,L]$ and $M/[K,M]$ by $(m[K,M])^q=(m[K,M])^{pK}=m^p[K,M]$ respectively.
\begin{equation*}
\xymatrix{&&C'\otimes C' \ar[dl]_{\omega'} \ar[d]^{w'} \\
&L/[K,L]\ar[r]_{{\partial_2}_*}&M/[K,M]\ar[r]_-{{\partial_1}_*} &Q}
\end{equation*}
is a quadratic module where${{\partial_2}_*}=\partial_2l[K,M]$,${{\partial_1}_*}=\partial_1mK,$
the action of $M/[K,M]$ on $L/[K,L]$ by $(l[K,L])^{m[K,M]}=l^m[K,L].$
As ${{\partial_1}_*}{{\partial_2}_*}(l[K,L])={{\partial_1}_*}(\partial_2l[K,M])=\partial_1\partial_2lK=K\cong 1_Q,$
$$\xymatrix{L/[K,L]\ar[r]_{{\partial_2}_*}&M/[K,M]\ar[r]_-{{\partial_1}_*} &Q}$$
is a complex of groups.
The quadratic map $\omega':C'\otimes C'\rightarrow L/[K,L]$ is given by
$$\omega'(\{m[K,M]\}\otimes \{m'[K,M]\})=\omega(\{m\}\otimes\{m'\})[K,L]$$
We know that $\partial_1:M\rightarrow Q$ is a nil($2$)-module
for $m[K,M],m'[K,M],m''[K,M] \in M/[K,M]$
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
\langle m[K,M],\langle m'[K,M],m''[K,M]\rangle\rangle &=&\langle m[K,M], m'm''m'^{-1}[K,M](m''[K,M]^{-1})^{{\partial_1}_*(m'[K,M])}\rangle \\
&=& \langle m[K,M], m'm''m'^{-1}[K,M](m''[K,M]^{-1})^{{\partial_1}m'K}\rangle\\
&=& \langle m[K,M], m'm''m'^{-1}[K,M](m''^{-1})^{{\partial_1}m'}[K,M]\rangle\\
&=& \langle m[K,M], \langle m',m''\rangle[K,M]\rangle \\
&=& m[K,M]\langle m',m''\rangle[K,M]m^{-1}[K,M](\langle m',m''\rangle^{-1}[K,M])^{{\partial_1}_*(m[K,M])} \\
&=& m[K,M]\langle m',m''\rangle[K,M]m^{-1}[K,M](\langle m',m''\rangle^{-1}[K,M])^{{\partial_1}mK}\\
&=& m[K,M]\langle m',m''\rangle[K,M]m^{-1}[K,M](\langle m',m''\rangle^{-1})^{{\partial_1}m}[K,M]\\
&=& \langle m,\langle m',m''\rangle\rangle[K,M]\\
&\cong & [K,M]
\end{eqnarray*}
So, ${{\partial_1}_*}:M/[K,M]\rightarrow Q$ is a nil($2$)-module.
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
{\partial_2}_*\omega'(\{m[K,M]\}\otimes \{m'[K,M]\})&=&{\partial_2}_*(\omega(\{m\}\otimes\{m'\})[K,L]) \\
&=& {\partial_2}\omega(\{m\}\otimes\{m'\})[K,L]\\
&=& w(\{m\}\otimes\{m'\})[K,M]\\
&=& w'(\{m[K,M]\}\otimes\{m'[K,M]\})
\end{eqnarray*}
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
\omega'(\{{\partial_2}_*(l[K,L])\}\otimes \{{\partial_2}_*(l'[K,L])\})&=&\omega'(\{{\partial_2}l[K,M]\}\otimes \{{\partial_2}l'[K,M]\}) \\
&=& \omega(\{{\partial_2}l\}\otimes \{{\partial_2}l'\})[K,L] \\
&=& [l,l'][K,L] \\
&=& [l[K,L],l'[K,L]]
\end{eqnarray*}
$\omega'(\{{\partial_2}_*(l[K,L])\}\otimes \{m[K,M])\} \{m[K,M])\}\otimes\{{\partial_2}_*(l[K,L])\})$
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
&=& \omega'(\{\partial_2l[K,M]\}\otimes \{m[K,M])\} \{m[K,M])\}\otimes\{{\partial_2}l[K,M]\}) \\
&=& \omega(\{\partial_2l\}\otimes \{m\})[K,M]\omega (\{m)\}\otimes\{{\partial_2}l\})[K,M]\\
&=& \omega(\{\partial_2l\}\otimes \{m\}\{m)\}\otimes\{{\partial_2}l\})[K,M] \\
&=& l^{\partial_1m}[K,M]\\
&=& (l[K,L])^{{\partial_1}_*(m[K,M])}
\end{eqnarray*}
Additionally, universal property can be shown as in proposition $3.3$
If $\phi:P\rightarrow Q$ is an injection and
\begin{equation*}
\xymatrix{&&C\otimes C \ar[dl]_{\omega} \ar[d]^w \\
&L\ar[r]_{\partial_2}&M\ar[r]_{\partial_1} &P}
\end{equation*}
is a quadratic module, let $T$ be the right transversal of $\phi(P)$ in $Q$ and let $B$ be the free product of groups $L_T(t\in T)$ each isomorphic with $L$ by an isomorphism $l\rightarrow l_t(l\in L)$ and $C$ be the free product of groups $M_T(t\in T)$ each isomorphic with $M$ by an isomorphism $m\rightarrow m_t(m\in M).$ Let $q\in Q$ acts on $B$ by the rule $(l_t)^q=(l^p)u$ and similarly $q\in Q$ acts on $C$ by the rule $(m_t)^q=(m^p)u,$ where $p\in P, u\in T$ and $qt=\phi(p)u.$
$\gamma:B\rightarrow C$ and $\delta:C\rightarrow Q$
$l_t\mapsto \partial_2(l)_t$ $m_t\mapsto t^{-1}\phi\partial_1mt$
and the action of $C$ on $B$ by $(l_t)^{m_t}=(l^m)_t.$ Then
$\phi_*(L)=B $ and $\phi_*(L)=C$
and the quadratic map $C\otimes C\rightarrow L$ is given by
$\omega(\{m_t\}\otimes \{m'_t\})=(\omega(\{m\}\otimes \{m'\})_t$
Since any $\phi:P\rightarrow Q$ is the composite of a surjection and an injection, an alternative description of the induced quadratic module can be obtained by using the construction methods of previous two propositions.
Now consider an arbitrary push-out square
&&&\qquad\quad (1)\\
\{L_2,M_2,P_2,\omega,\partial_2,\partial_1\}\ar[rr]&&\{L,M,P,\omega,\partial_2,\partial_1\}}$$
of quadratic modules. In order to describe $\{L,M,P,\omega,\partial_2,\partial_1\}$, we first note that $P$ is the push-out of the group morphisms $P_1\leftarrow P_0\rightarrow P_2.$ This is because the functor
$$\{L,M,P,\omega,\partial_2,\partial_1\}\rightarrow \{L/\omega(M/w(\{\langle x,y\rangle \}\otimes\{z\}),P,\partial_1\}$$
from quadratic module to nil($2$)-module has a right adjoint
$(N,P,\partial)\longrightarrow (1,N,P,1,1,\partial)$
and the forgetful functor $(M/w(\{\langle x,y\rangle \}\otimes\{z\}),P,\partial_1)\rightarrow P$ from nil($2$)-module to group has a right adjoint $P\rightarrow (P,P,id). $ The morphisms $\phi_i:P_i\rightarrow P (i=0,1,2)$ in $(1)$ can be used to form induced quadratic $Q$-modules $B_i=(\phi_i)_*L_i$ and $C_i=(\phi_i)_*M_i.$ Clearly $\{L,M,P,\omega,\partial_2,\partial_1\}$ is the push-out in $\mathbf{Quad/P}$ of the resulting $P$-morphisms
$$(B_1\rightarrow C_1\rightarrow P)\longleftarrow(B_0\rightarrow C_0\rightarrow P)\longrightarrow(B_2\rightarrow C_2\rightarrow P)$$
can be described as follows.
\begin{equation*}
\xymatrix{&&D_i\otimes D_i \ar[dl]\ar[d] \\
&B_i\ar[r]&C_i\ar[r] &P}
\end{equation*}
be a quadratic $P$-module for $i=0,1,2$ where $D_i=({C_i}^{cr})^{ab}$ and let
\begin{equation*}
\xymatrix{&&C\otimes C \ar[dl] \ar[d] \\
&L\ar[r]&M\ar[r] &P}
\end{equation*}
be the push-out in $\mathbf{Quad/P}$ of $P$-morphisms
$$\xymatrix{(B_1\rightarrow C_1\rightarrow P)&&(B_0\rightarrow C_0\rightarrow P)\ar[ll]_{(\alpha_1,\beta_1,id)}\ar[rr]^{(\alpha_2,\beta_2,id)}&&(B_2\rightarrow C_2\rightarrow P)}$$
Let $(B\rightarrow M)$ be push-out of $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$ in the category of $\mathbf{Nil(2)}$, equipped with the induced morphism $\xymatrix{B\ar[r]^\mu&C\ar[r]^\nu&P}$, the quadratic map $\omega:({C}^{cr})^{ab}\otimes ({C}^{cr})^{ab}\rightarrow B$ and the induced action of $P$ on $B$ and $C$. Then $L=B/S$, where $S$ is the normal subgroup of $B$ generated by the elements of the form
$$\omega(\{\mu b\}\otimes\{\mu b'\})[b,b']^{-1}$$
$$\omega(\{\mu b\}\otimes\{c\}\{c\}\otimes\{\mu b'\})(b^{-1})^{\nu(c)}b$$
and $M=C/R$ where $R$ is the normal subgroup of $C$ generated by the elements of the form
$$\mu \omega(\{c\}\otimes\{c'\})c^{\nu(c)}cc'^{-1}c^{-1}$$
for $b,b'\in B, c.c'\in C$ and $p\in P$.
In the case when $\{L_2,M_2,P_2,\omega,\partial_2,\partial_1\}$ is the trivial quadratic module $\{1,1,1,id,id,id\}$ the push-out quadratic module $\{L,M,P,\omega,\partial_2,\partial_1\}$ in $(1)$ is the cokernel of the morphism
Cokernels can be described as follows
$Q/\overline P$ is the push-out of the group morphisms $1\leftarrow P\rightarrow Q$. Let $\{A_*,G_* ,Q/\overline P,\omega,\partial_2,\partial_1\}$ be the induced from $\{A,G,P,\omega,\partial_2,\partial_1\}$ by $P\rightarrow Q/\overline P$. If $\{1,1,Q/\overline P,id,id,\partial_1\}$ and
$$\{B/[\overline P,B],H/[\overline P,H],Q/\overline P,\omega ,\partial_2,\partial_1\}$$
are induced from $\{1,1,1,id,id,id\}$ and$\{B,H,Q,\omega,\partial_2,\partial_1\}$ by $1\rightarrow Q/\overline P$ and the epimorphism $Q\rightarrow Q/\overline P$ then the cokernel of a morphism
is $\{coker(\beta_*,\lambda_*),Q/\overline P,\omega,\partial_2,\partial_1\}$ where $(\beta_*,\lambda_*)$ is a morphism of
$$(A_*,G_*)\longrightarrow (B/[\overline P,B],H/[\overline P,H]).$$
[1] U. E. Arslan, Z. Arvasi and G. Onarli, Induced two-crossed modules, arXiv:1107.4291v1 [math.AT] 21 Jul 2011.
[2] Z. Arvasi̇ and E. Ulualan, On algebraic models for homotopy 3-types, Journal of Homotopy and Related Structures Vol.1, No 1, pp.1-27, (2006).
[3] H.J. Baues, Combinatorial homotopy and 4-dimensional complexes, Walter de Gruyter, 15, 380 pages, (1991).
[4] R. Brown and N.D. Gilbert,
Algebraic models of 3-types and automorphism structures for
crossed modules. Proc. London Math. Soc., (3) 59,
51-73, (1989).
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[6] R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc., (3) 36 (2) (1978)193-212.
[7] R. Brown, P. J. Higgins and R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical higher homotopy groupoids,
http://www.bangor.ac.uk/ mas010/pdffiles/rbrsbookb-e231109.pdf.
[8] R. Brown and J.-L. Loday, Homotopical excision, and Hurewicz theorems for
n-cubes of spaces, Proc. London Math. Soc. (3) 54 (1) (1987).
[9] R. Brown and R. Sivera, Algebraic colimit calculations in homotopy theory using fibred and cofibred categories, Theory and Applications of Categories, 22 (2009) 222-251.
[10] R. Brown and C. D. Wensley, Computation and homotopical applications
of induced crossed modules, Journal of Symbolic Computation 35, 2003,
[11] R. Brown and C. D. Wensley, On finite induced crossed modules, and
the homotopy 2-type of mapping cones, Theory and Applications of Categories, (3) 1 (1995), 54-71.
[12] J.M. Casas and M. Ladra, Colimits in the crossed modules category in Lie algebras, Georgian Mathematical Journal, V7 N3, 461-474, 2000.
[13] D. Conduché, Modules croisés généralisés de longueur 2, Journal of Pure and Applied Algebra, 34, pp 155-178, (1984).
[14] G.J. Ellis, Crossed squares and
combinatorial homotopy , Math.Z. , 214, 93-110,
[15] G.J. Ellis and R. Mikhailov, A colimit classifying spaces,
arXiv:0804.3581v1 [math. GR] 22 Apr 2008.
[16] J.L. Loday, Spaces with finitely many
non-trivial homotopy groups , J. Pure and Applied
Algebra, 24, 179-202, (1982).
[17] T. Porter, Some categorical results in
the theory of crossed modules in commutative algebras,
Journal of Algebra , 109, pp 415-429, (1987).
[18] T. Porter, The crossed menagerie: an introduction to crossed gadgetry and
cohomology in algebra and topology,
[19] J.H.C. Whitehead, Combinatorial homotopy
II, Bull. Amer. Math. Soc., 55, pp 453-496,
\begin{array}{lll}
\text{H.Atik} & \text{E. Ulualan}\\
\text{Melik\c{s}ah University}, & \text{Dumlup\i nar University} \\
\text{Science and Art Faculty} & \text{Science and Art Faculty}\\
\text{Mathematics Department} & \text{Mathematics Department}\\
\text{38280, Kayseri, TURKEY} & \text{K\"{u}tahya, TURKEY}\\
\text{e-Mail: hatik@meliksah.edu.tr} & \text{e-Mail: eulualan@dumlupinar.edu.tr} \\
\end{array}%
|
arxiv-papers
| 2011-08-18T18:51:36 |
2024-09-04T02:49:21.636860
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Atik and E. Ulualan",
"submitter": "Erdal Ulualan",
"url": "https://arxiv.org/abs/1108.3810"
}
|
1108.3915
|
# City on the Sky: Flexible, Secure Data Sharing on the Cloud
Dinh Tien Tuan Anh, Wang Wenqiang, Anwitaman Datta
{ttadinh,wqwang,anwitaman}@ntu.edu.sg
###### Abstract
Sharing data from various sources and of diverse kinds, and fusing them
together for sophisticated analytics and mash-up applications are emerging
trends, and are prerequisites for grand visions such as that of cyber-physical
systems enabled smart cities. Cloud infrastructure can enable such data
sharing both because it can scale easily to an arbitrary volume of data and
computation needs on demand, as well as because of natural collocation of
diverse such data sets within the infrastructure. However, in order to
convince data owners that their data are well protected while being shared
among cloud users, the cloud platform needs to provide flexible mechanisms for
the users to express the constraints (access rules) subject to which the data
should be shared, and likewise, enforce them effectively. We study a
comprehensive set of practical scenarios where data sharing needs to be
enforced by methods such as aggregation, windowed frame, value constrains,
etc., and observe that existing basic access control mechanisms do not provide
adequate flexibility to enable effective data sharing in a secure and
controlled manner. In this paper, we thus propose a framework for cloud that
extends popular XACML model significantly by integrating flexible access
control decisions and data access in a seamless fashion. We have prototyped
the framework and deployed it on commercial cloud environment for experimental
runs to test the efficacy of our approach and evaluate the performance of the
implemented prototype.
##### Keywords
cloud computing, access control, flexible sharing, fine-grained policies,
XACML
## I Introduction
The emergence of cloud computing in recent years is rapidly changing the way
businesses and government agencies, as well as individuals, are storing and
managing their data as well as workflows. Instead of developing and
maintaining individual data management infrastructures and data sharing
mechanisms, data owners now leverage on the cloud services to make their data
available to users. The fact that data from multiple sources now reside in one
logical place, i.e., the cloud, makes it much easier than ever before to
develop large scale applications that require data and knowledge from multiple
domains and sources. These applications could include environmental study,
city infrastructure planning, disaster monitoring, and many more. In an era
when the cloud infrastructure was non-existent, to develop such applications,
the developer would have to first talk to individual data owners to
specifically provide the data to them, which is likely to involve tedious
administration procedures such as signing documents regarding the privileges
and responsibilities of each parties, apart from the cumbersome process of
actually shipping the data. Then the developer would have to develop software
that work with the individual data exchange interfaces/protocols provided by
different owners to collect and reformat the data before they could be fed
into the applications for analysis or real-time monitoring tasks.
On the multitenant cloud, such data from diverse sources are naturally
collocated, making it much easier and much more efficient for the application
developers to obtain what they need for their work. More specifically, the
storage and data exchange can be handled efficiently by the cloud providers.
This means data owners need not worry about how to share, but what and who to
share. Putting one’s proprietary data online on the cloud raises concerns
regarding data security, privacy and ownership. Even if the cloud service
provider is trusted, and legally obliged (through service level agreements and
law enforcement) to prevent illegal access of data and information leakage,
there needs to be meaningful, comprehensive and flexible ways for the data
owners to express their sharing preferences, in a manner which can readily be
interpreted and enforced by the cloud service provider. This paper discusses
how this can be achieved. One can further argue how this can be realized if
the cloud service provider is not even trusted, but that is an issue outside
the scope of this work, and is part of our future work.
The objective of this work is to propose and showcase a framework for sharing
data on the cloud. The framework, called eXACML, facilitates sharing in an
_easy-to-use_ , _secure_ , _flexible_ and _scalable_ manner. For security, we
make use of/extend XACML [21] — the XML-based and popular framework for access
control. XACML has become a standard for specifying and enforcing access
control policies. It evaluates requests for resources against a set of
policies and returns _permit_ or _deny_ decision, which does not involve
accessing any data. In eXACML, we extend XACML to support more fine-grained
policies as well as to handle data processing. We demonstrate eXACML’s
flexibility by using it in different access control scenarios with different
levels of granularity. For usability, eXACML provides an intuitive, easy-to-
use interface for data owners and data users to specify and enforce security
policies and to access shared data. Finally, we carry out experiments to
evaluate the framework performance in a cloud-like environment, the results of
which suggests that eXACML is scalable. We motivate our work with scenarios
from ongoing works on better city planning, specifically related to weather
and traffic information, and the evaluations are also based on datasets, part
of which are real, while the rest is synthetic.
In summary, the main contributions of this work are as follows:
1. 1.
We demonstrate the needs for secure and flexible data sharing with practical
examples involving city planning and management based on data from weather and
traffic monitoring stations. We discuss scenarios in which access control with
different levels of granularity of data access are needed.
2. 2.
We extend the XACML framework to support fine-grained policies. In particular,
fine-grain access control policies (which require data filtering) are
expressed within _obligations_ that are passed from the Policy Decision Point
(PDP) to the Policy Enforcement Point (PEP), which connects to the database
and processes the data queries embedded in the obligations. We refer to this
implementation as
XACML*. We discuss why this approach could perform better than the traditional
approach based on views.
3. 3.
We implement a prototype of the framework
(eXACML), providing additionally, an easy-to-use user interface. The prototype
allows data owners to easily add and modify their policies. Data users can
query meta data and details of access policies at remote servers. They can
also specify aggregated data from multiple sources in single requests.
Responses to data requests contain information of matching policies, enabling
flexible conflict resolutions.
4. 4.
We evaluate the performance of our prototype in cloud-like settings. Our
experiments illustrate that the framework incurs low overhead. We attribute
this scalability to the framework’s ability to cache responses and perform
aggregation of responses from multiple sources prior to returning them to the
data users.
The rest of this paper is organized as follows: Section 2 describes practical
scenarios that motivates our framework. Section 3 details our extensions to
XACML, followed by the logical design of our framework in Section 4. The
prototype and its evaluation are presented in Section 5. We discuss related
and future works in Section 6 and Section 7 and conclude in Section 8.
Before proceeding further, we’ll like to make a final note on the scope of the
current work and implementation. Broadly speaking, there are two kinds of data
- data already stored in the system (which we refer to as archived/archival
data), and data stream, where live data is flowing into the system. Likewise,
the queries could be ‘on demand’, typically on the stored data, or continuous
queries, to be evaluated on the incoming data streams. The current
implementation deals with on demand queries on stored data. This is summarized
in Table I.
Query/Database | Archival (relational) databases | Stream databases
---|---|---
On demand query | current implementation | n/a
Continuous query | n/a | Future work
TABLE I: Scope of eXACML, regarding database and query type
## II Motivating Example
As increasing portion of the world population is rapidly moving to the cities,
while the resources at our disposal are shrinking at an alarming rate,
numerous research and industrial initiatives (e.g., IBM’s smart cities
initiative
111http://www.ibm.com/smarterplanet/us/en/smarter_cities/overview/index.html)
are focusing in realizing what are being termed as ‘smart(er) cities’ in order
to manage resources efficiently at the city scale. Enabling such a move
towards smarter cities are cyber-physical systems aggregating data and
actuating the necessary resource management actions at the edge, while the
necessary data storage and analytics is carried out on cloud based back-end.
In this section, we use some scenarios of road congestion analysis to showcase
the need among data owners for flexible data sharing.
### II-A Settings.
Noticing that one of the major expressways in the city suffers serious
congestion during every monsoon season, Singapore’s Land Transport Authority
(LTA) has, after preliminary studies, hypothesized that such congestion is
mainly caused by three factors, (1) large number of vehicles on the road, (2)
slow speed of vehicles, (3) bad weather.
To validate such preliminary conclusions and build a traffic condition model
during the monsoon season, researchers need more data. Fortunately, many
organizations have been collecting related data: LTA itself has a number of
sensors deployed along the road side to record traffic volume, i.e., the
number of vehicles passing by at unit time; furthermore, another independent
entity, a large local taxi company, collects the speed and location data from
their taxis’ GPS devices. At almost any time, there are a number of such taxis
running over the whole stretch of the express way. Likewise, the national
environmental agency (NEA) has several weather stations deployed close to the
congested areas, that record weather parameters such as temperature, humidity,
rain rate, etc.
If all these different data owners use a shared cloud infrastructure222Note
that we are unaware of the current practice of the individual organizations
mentioned above, and what follows is a hypothetical scenario. to store and
process the above mentioned data-sets for their individual needs, then when
complex analytics involving multiple such datasets become necessary, the data
is readily available on the infrastructure thanks to such collocation on the
multi-tenant cloud.
Suppose the data are stored in relational tables as shown in Table II for
traffic volume information, Table III for cab’s location and speed information
and Table IV for weather information.
SamplingTime | TrafficVolume
---|---
2011-06-06 10:00:00 | 60
2011-06-06 10:05:00 | 67
2011-06-06 10:10:00 | 50
… | …
TABLE II: Table TrafficInfo: Traffic volume data from road side sensors SamplingTime | Speed (km/hr) | latitude | longitude
---|---|---|---
2011-06-06 10:00:00 | 100 | x1 | y1
2011-06-06 10:05:00 | 80 | x2 | y2
2011-06-06 10:10:00 | 40 | x3 | y3
… | … | |
TABLE III: Table VehicleInfo: Vehicle speed and location data from GPS devices SamplingTime | Temperature(C) | Humidity (%) | RainRate (mm/hr) | …
---|---|---|---|---
2011-06-06 10:00:00 | 27.2 | 70 | 0.0 | …
2011-06-06 10:01:00 | 27.5 | 70 | 0.0 | …
2011-06-06 10:02:00 | 27.5 | 73 | 0.0 | …
2011-06-06 10:03:00 | 27.4 | 72 | 0.0 | …
2011-06-06 10:04:00 | 27.3 | 75 | 0.0 | …
2011-06-06 10:05:00 | 27.3 | 76 | 0.0 | …
2011-06-06 10:06:00 | 27.0 | 77 | 0.1 | …
2011-06-06 10:07:00 | 27.1 | 80 | 5.0 | …
2011-06-06 10:08:00 | 26.8 | 81 | 14.0 | …
2011-06-06 10:09:00 | 26.6 | 82 | 20.0 | …
2011-06-06 10:10:00 | 26.5 | 85 | 34.4 | …
… | … | … | … | …
TABLE IV: Table WeatherInfo: Weather data from weather stations
### II-B Example 1
Suppose that NEA decides to share (possibly for a price) only the rain rate
data with LTA researchers, since other weather parameters such as temperature
and humidity are not expected to affect traffic condition as much as rainfall
does in the context of Singapore, and hence LTA does not want pay for the
temperature or humidity information. Furthermore, even if the original
collected data available with NEA is for one minute interval, it may want to
expose only the data corresponding to five minute averages to LTA. It may also
expose the more detailed data to its own employees or to other customers.
The first constraint corresponds to the projection operation in the relational
database model and a sample SQL query will be something like ” _select
RainRate from WeatherInfo_ ”. The second constraint can be considered as a
sliding window query over a data stream, i.e., the time series rain rate data.
Standard SQL does not support these kind of queries well, hence additional
operations need to be implemented on top of the RDBMS query engine. To specify
a sliding window query on a time series data sequence in our scenario, five
parameters are needed, namely, the _starting time_ , _ending time_ , _window
size_ , _window advance step_ and _aggregation function_. The _starting time_
and _ending time_ are the general temporal constraints that specify the
segment of the data stream to be returned. The _window size_ and _window
advance step_ decide the length of the query window and how fast the window is
moving along the data stream. The _aggregation function_ includes numerical
functions such as _average()_ , _max()_ , _min()_ , _count()_ , etc., which
are applied to the data records to summarize the portion of the data stream
within the window.
### II-C Example 2
Consider that the taxi company agrees to help the researchers by providing
their taxis’ location and speed data, but the company only wants to share such
information for taxis within some specific regions in the vicinity of the
congested areas being studied, instead of exposing the information about its
whole fleet, which it deems important business secret not to be exposed to
third parties. To enforce such a constraint, a selection operator is applied
to the longitude and latitude columns to filter out those records that are not
supposed to be shared with the researchers. For the sake of simplicity, assume
that this range is specified by a rectangle with the geographical coordinate
of the upper left vertex as (a1,b1) and of the lower right vertex as (a2,b2),
we can have the corresponding SQL query: _select SamplingTime, Speed from
VehicleInfo v where v.longitude $=>$ a1 and v.longitude $<=$ a2 and v.latitude
$>=$ b2 and v.latitude $<=$ b1_.
To enable the above access contraints in XACML, we make use of the
_obligation_ element in _policy_ element to specify the constraints. Fig. 4
and Fig. 5 present two examples of XACML obligations that embed these
constraints. In Figure 4, line 2 indicates that the permission to perform the
sliding window query if the decision returned from PDP is ‘permit’. Line 3
indicates that the aggregation function to be used in the sliding window query
is average calculation. Lines 5 to 8 specify that _starting time_ is zero
o’clock of June 6th, 2011, _ending time_ is zero o’clock of June 7th, 2011,
_window size_ is 5 minutes and _window advance step_ is also of 5 minutes.
Line 9 indicates that the sliding window is applied on SamplingTime column as
well, besides on the actual rain rate data column, which is not shown here
within the obligation part. Line 3 in Figure 5 shows the selection predicate
to be included in the SQL query to be evaluated on the data table, which only
allows vehicle information to be returned if the vehicle’s location is within
a given boundary.
### II-D Fine-grained Policies
The examples above demonstrate real needs for an access control model that
supports fine-grained policies involving fine-grained data processing. At a
high level, the models need to be able to express and enforce the following
types of policies:
1. 1.
Aggregated data: Only results of aggregation functions over raw data such as
_average_ ,_sum_ , _min_ , _max_ are shared.
2. 2.
Trigger-based: a row of data is accessible only if the value of a column
satisfies a certain predicate: exceeds a specific threshold, or is contained
within a range. As an example, a taxi company is granted access to temperature
reading only if the temperature is over $30^{o}$C.
3. 3.
Sliding window: a sliding window is specified by its starting time, ending
time, window size and advance step. Only aggregated data (average, for
instance) over the windows are accessible.
4. 4.
Approximation: only data whose values approximate those given in the requests
are accessible. For example, a request includes a value $X$, and the policies
is specified such that a row of data is returned only if the column $c$’s
value $V$ satisfies $|V-X|<\epsilon$ for some distance function.
We next explore how such fine-grained policies can be flexibly supported.
## III Flexible Sharing Through Fine-Grained Policies
Existing frameworks, such as XACML, do not natively support different levels
of granularity to support fine-grained access control. Nevertheless, XACML has
emerged in recent years as a mature and widely used model for expressing and
enforcing access control policies. Therefore, we extend XACML in order to
support fine-grained policies, including those described in Section 2.
For the rest of this paper, we assume relational databases (SQL types) are
used for managing data in the back-end. Without loss of generality, but for
the purpose of simplicity of exposition, we consider that each database
consists of a single table indexed by time values. When requesting for data,
the user provides his credentials (for example, name and role) and specifies
the location of data. The response contains either a _deny_ decision (i.e. no
access to the data), or permit decision together with the returned data as
specified in the policies.
### III-A XACML
XACML is an OASIS framework for specifying and enforcing access control [21].
It is XML based and the latest version is $3.0$. XACML allows administrators
to control their resources by writing policy files, which are then loaded into
a Policy Decision Point (PDP) module. An user wishing to access a specific
resource sends request to a Policy Enforcement Point (PEP) where the decision
is made by consulting the PDP. XACML specifies standards for writing policies,
requests and interpreting the response.
1. 1.
_Subjects_ , _Resources_ and _Actions_. A _subject_ in XACML has a set of
credentials such as its name, role, etc. The subject wishes to perform certain
_actions_ (read, write, for example) on a set of system _resources_.
2. 2.
_Requests_. Request for accessing system resources are written in XML. The
subject credentials, system resources and actions are specified in one or more
_Attribute_ elements included in the Subject, Resource and Action elements
respectively. Fig. 1 shows an example of an XACML request from a subject with
role _admin_ to perform _read_ action the _temperature_ column from
_weather_data_ database.
Ψ<Subject>
Ψ <Attribute AttributeId = ‘‘exacml:subject:role-id’’
DataType={http://wwww.w3.org/2001/XMLSchema#string}>
Ψ <AttributeValue>admin</AttributeValue>
Ψ </Attribute>
Ψ</Subject>
Ψ
Ψ<Resource>
Ψ <Attribute AttributeId = ‘‘exacml:rdmb-database-id’’
DataType={http://www.w3.org/2001/XMLSchema#string}>
<AttributeValue>weather_data</AttributeValue>
</Attribute>
Ψ <Attribute AttributeId = ‘‘exacml:rdmb-column-id’’
DataType={http://www.w3.org/2001/XMLSchema#string}>
<AttributeValue>temperature</AttributeValue>
</Attribute>Ψ
Ψ</Resource>
Ψ<Action>
Ψ <Attribute AttributeId = ‘‘exacml:action-id’’
DataType={http://www.w3.org/2001/XMLSchema#string}>
<AttributeValue>read</AttributeValue>
</Attribute>
Ψ</Action>Ψ
Ψ
Figure 1: Example of a well-formed XACML request, in which the user with the
role _admin_ requests _read_ access to the column _temperature_ of the
database _weather_data_
3. 3.
_Policies_. A policy contains a _Target_ , a set of _Rules_ each of which has
at most one _Condition_ , and a set of _Obligations_. Multiple policies can be
grouped into a _policy set_ , which has its own Target element. The policy is
indexed by its Target element, which consists of a number of conditions needed
to be satisfied by the request before the rest of the policy can be evaluated.
Conditions are essentially boolean expressions over the values included in the
request. The policy returns access control decision which is either _Permit_ ,
_Deny_ , _Not Applicable_ or _Intermediate_. The last two are used when there
is no applicable policy or an error occurred during evaluation. Fig. 2
illustrates an example of an XACML policy that grants access to subjects with
_government_ role to the _samplingtime_ and _temperature_ columns of
_weather_data_.
<Target>
<Subjects>
<Subject>
<SubjectMatch MatchId="urn:oasis:names:tc:xacml:1.0:function:string-equal">
<AttributeValue DataType="http://www.w3.org/2001/XMLSchema#string">
government
</AttributeValue>
<SubjectAttributeDesignator AttributeId="exacml:subject:role-id"
DataType="http://www.w3.org/2001/XMLSchema#string"/>
</SubjectMatch>
</Subject>
</Subjects>
<Resources>
<Resource>
<ResourceMatch MatchId="urn:oasis:names:tc:xacml:1.0:function:string-equal">
<AttributeValue DataType="http://www.w3.org/2001/XMLSchema#string">
weather_data
</AttributeValue>
<ResourceAttributeDesignator AttributeId="exacml:rdbms-database-id"
DataType="http://www.w3.org/2001/XMLSchema#string"/>
</SubjectMatch>
</Resource>
<Resources>
<Actions>
<AnyAction/>
</Actions>
</Target>
<Rule RuleId="example" Effect="Permit">
<Target> <Subjects> <AnySubject/> </Subjects>
<Resources> <AnyResource/> </Resources>
<Actions> <AnyAction/> </Actions>
</Target> ΨΨ
<Condition FunctionId="urn:oasis:names:tc:xacml:1.0:function:string-subset">
<ResourceAttributeDesignator AttributeId="exacml:rdbms-column-id"
DataType="http://www.w3.org/2001/XMLSchema#string"/>
<Apply FunctionId="urn:oasis:names:tc:xacml:1.0:function:string-bag">
<AttributeValue DataType="http://www.w3.org/2001/XMLSchema#string">
samplingtime
</AttributeValue>
<AttributeValue DataType="http://www.w3.org/2001/XMLSchema#string">
temperature
</AttributeValue>
</Apply>
</Condition>ΨΨΨΨ
</Rule>
Figure 2: Example of a well-formed XACML policy which grant access to column
_samplingtime_ or _temperature_ of the database _weather_data_ to any subject
with role _goverment_
When more than one rules are applicable to a particular request, they are
evaluated according to _rule combination algorithm_ specified in the policy.
Similarly, multiple applicable policies in a policy set are evaluated
according to a specified _policy combination algorithm_. Examples of combining
algorithms (for both policies and rules) are _Permit-overrides_ where a permit
policy or rule is evaluated, and _First-applicable_ where the first applicable
policy is evaluated.
4. 4.
_Policy Enforcement Point (PEP)_. User requests first go through the PEP,
which translates them into canonical forms before passing to the PDP.
Additionally, PEP also interprets responses and obligations returned from the
PDP. In summary, PEP deals with application logics and acts as the access
control enforcement mechanism. Our framework extends PEP to provide support
for more fine-grained policies.
5. 5.
_Policy Decision Point (PDP)_. Data owners’ policies are _loaded_ into the
PDP, which evaluates requests received from the PEP against the active
policies. Its main task is to efficiently find applicable policies for a given
request and to quickly evaluate their rules and conditions to determine the
access control decision. It sends back to PEP a well-formed response
containing a decision and a set of obligations.
### III-B View-Based vs Obligation-Based
The traditional access control model in relational databases is based on
_view_ [24]. Basically, a view is the result of a SQL query on existing
tables, to which read/write access are specified. The database management
systems maintain the views and enforce access control rules on them.
A simple approach based on view to support fine-grained policies with XACML
can be realized as follows. First, views are created with no access control
restriction, and assigned with unique resource IDs. This can handle all types
of policies discussed earlier. PEP maintains a mapping between the IDs and
actual views. Next, the IDs are used to specify the resources in XACML
policies, as well as to construct data requests. Once PDP returns a permit
decision, PEP retrieves and returns the corresponding views.
However, there are a number of weaknesses with this approach:
* •
Views need to be created prior to policies or requests. They must also be
removed explicitly by the data owner.
* •
Views are static and may be very large in number (potentially infinite number
of views for trigger-based and sliding window policies). Maintaining these
views are inefficient at best and impossible at worst.
* •
An user requesting for data must also maintain a mapping of all the view IDs
they wish to access. Not only is such a requirement undesirable for data
users, but also it is expensive to implement.
Figure 3: Extensions to XACML that support more flexible access control
policies.
Fig. 3 illustrates the obligation-based approach (extensions to XACML is
highlighted in bold). The basic idea is to embed queries for creating views
into obligations. The PEP, upon receipt of the obligations, executes the
embedded queries on the database and returns the results in a well-formed
response. Unlike the view-based approach, the size of data (views) maintained
by PEP is bounded. Furthermore, popular queries can be cached by the database
management system or the PEP. In the experiment section, we demonstrate the
benefit of caching in improving request time.
### III-C Implementations
#### III-C1 Obligations.
Description | ObligationId
---|---
Column aggregation | exacml:obligation:column-aggregation
Simple selection | exacml:obligation:simple-selection
Sliding window | exacml:obligation:column-sliding-window
Approximation | exacml:obligation:column-approximation
TABLE V: Obligation types
Using obligation-based approach, policy writers utilize different types of
obligations to specify different database queries. Our current implementation
supports four types of obligations (Table V):
1. 1.
_Column aggregation_ : consists of a string attribute with ID
exacml:obligation:aggregation-id. The string represents an aggregation
function, such as average (Fig. 4, line 2-3), min, max, count or sum.
2. 2.
_Simple selection_ : consists of a string attribute with ID
exacml:obligation:selection-id. The string is a boolean expression that will
be used as the WHERE clause when constructing the database query. An example
of this obligation is shown in Fig. 5, in which the policy restricts access to
data to within a certain geographical region.
3. 3.
_Sliding window_ : we assume that the column from which the sliding windows
are based is of type DateTime (although sliding windows could be constructed
from any other sortable types). The obligation consists of a number of
attributes:
* •
Sliding window column: string attribute with ID
exacml:obligation:sliding-window-column-id specifies the column of type
DateTime from which sliding windows are constructed.
* •
Start and End: time attributes with IDs
exacml:obligation:sliding-window-start-id and
exacml:obligation:sliding-window-end-id respectively.
* •
Window size: integer attribute with ID
exacml:obligation:sliding-window-size-id specifies the window size (in hours).
* •
Advance step: integer attribute with ID
exacml:obligation:sliding-window-step-id specifies how the sliding window
advances, i.e. the number hours between starting time of two consecutive
windows.
Fig. 4 (line 4-10) shows an example of a sliding window based on
_SamplingTime_ column. The window’s size is 5 hours, starting from 2011-06-06
00:00:00, advancing in 5-hour steps until 2011-06-07 00:00:00.
4. 4.
_Approximation_ : this obligation specifies the acceptable distance between
the column values with respect to the values included in the request.
Attributes containing column IDs are specified in both the requests and the
policies. Specifically:
* •
In the request: string attribute with ID
exacml:data-value-id is of the form
<columnId>:<value> which represent the value of the specified column.
* •
In the policy: string attribute with ID
exacml:obligation:approximation-param-id contains the column IDs. Columns
specified in the requests must be a subset of what is specified in the
policies. Also required is a double attribute with ID
exacml:obligation:approximation-value-id which represents the distance between
the vector of column values in the database and that included in the request.
<Obligations>
<Obligation ObligationId="exacml:obligation:column-aggregation" FulfillOn = "Permit">
<AttributeAssignment AttributeId="exacml:obligation:aggregation-id"
DataType = "http://www.w3.org/2001/XMLSchema#string">
avg
</AttributeAssignment>
</Obligation>
<Obligation ObligationId="exacml:obligation:column-sliding-window" FulfillOn = "Permit">
<AttributeAssignment AttributeId="exacml:obligation:sliding-window-start-id"
DataType = "http://www.w3.org/2001/XMLSchema#time">
2011-06-06 00:00:00
</AttributeAssignment>
<AttributeAssignment AttributeId="exacml:obligation:sliding-window-end-id"
DataType = "http://www.w3.org/2001/XMLSchema#time">
2011-06-07 00:00:00
</AttributeAssignment>
<AttributeAssignment AttributeId="exacml:obligation:sliding-window-size-id"
DataType = "http://www.w3.org/2001/XMLSchema#integer">
5
</AttributeAssignment>
<AttributeAssignment AttributeId="exacml:obligation:sliding-window-step-id"
DataType = "http://www.w3.org/2001/XMLSchema#integer">
5
</AttributeAssignment>
<AttributeAssignment AttributeId="exacml:obligation:sliding-window-column-id"
DataType = "http://www.w3.org/2001/XMLSchema#string">
samplingtime
</AttributeAssignment>
</Obligation>
</Obligations>
Figure 4: Obligation portion of the XACML policy for Example II-B
<Obligations>
<Obligation ObligationId="exacml:obligation:simple-selection" FulfillOn = "Permit">
<AttributeAssignment AttributeId="exacml:obligation:selection-id"
DataType = "http://www.w3.org/2001/XMLSchema#string">
longitude >= a1 and longitude <= a2 and latitude >= b2 and latitude <= b1
</AttributeAssignment>
</Obligation>
</Obligations>
Figure 5: Obligation portion of the XACML policy for Example II-C
#### III-C2 Handling obligations.
PEP extracts attributes embedded in the obligations and constructs
corresponding queries to be executed on the database. It is not uncommon for a
policy to have more than one types of obligations, which allows for more
expressive, fine-grained conditions for accessing data. Essentially, PEP
creates queries of the following form:
select f(column_1), f(column_2),..,f(column_n) from Table_name where
Where_Condition (1)
where column_i $(1\leq i\leq n)$ and Table_name are extracted from the
Resources element of the request. When no obligation is returned, f and
Where_Condition are set to empty strings. In this case, the query becomes:
select column_1, column_2,..,column_n
from Table_name
PEP obtains f from the string attribute in the column aggregation obligation.
When a simple selection obligation is returned, Where_condition is taken
directly from its string attribute. For approximation obligations, the PEP
first retrieves a vector of values from the request, namely
$(x_{1},x_{2},..,x_{k})$ from columns
$c_{1},c_{2},..,c_{k}$. It then obtains the distance value $\delta$ in the
obligation, and sets Where_condition as:
sqrt$((c_{1}-x_{1}).(c_{1}-x_{1})+..+(c_{k}-x_{k}).(c_{k}-x_{k}))<\delta$
Handling sliding-window obligations are more complex. First, the tuple
$(start,end,window\\_size,advancing\\_step)$ are extracted from the
obligation. The total number of windows are:
$nW=\lfloor\frac{end-start-window\\_size+1}{advancing\\_step}\rfloor+1$
For every window, PEP creates a different query. More specifically, let $c$ be
the column (of type DateTime) from which the sliding windows are constructed,
a query $i$ $(0\leq i<nW)$ is of the form:
select f(column_1), f(column_2),..,f(column_n) from Table_name where
Where_Condition AND c $\geq$ start+step*i AND c < start+step*i+size
where Where_Condition are constructed from simple selection and approximation
obligations.
## IV The Logical Framework
Figure 6: eXACML framework. XACML* denotes the extended XACML described in
Section 3.
This section presents our design of the framework that enables secure, easy-
to-use, flexible and scalable data sharing. The security comes from the use of
XACML for specifying and enforcing access control. The flexibility property is
the result of our enhancement to XACML which supports a wider range of access
control policies. Usability and scalability are achieved through a simple
client interface and the use of a proxy server, whose details are described
below.
### IV-A Entities
Fig. 6 illustrates the main entities and how they interact in our framework.
_Clients_ consist of data owners who wish to share and enforce access control
on their datasets, and of data users who are interested in accessing the data.
A data owner can have more than one datasets and a data user can request
access to multiple datasets. _Databases_ are database servers which manage
clients’ datasets. Access to the database is controlled by at least one
instance of XACML* (discussed below). These servers are likely to be remote
and maintained by a third party (cloud) provider.
Our framework — eXACML — is positioned in between clients and databases (Fig.
6). Its roles are to mediate their interactions and to safeguard the
databases. Essentially, eXACML is made up of a client interface, a proxy
server, cloud servers and XACML* instances.
* •
Clients interact with the databases through a local _client interface_ that
parses inputs into request messages and forwards them to the proxy server. It
waits and interprets response messages before returning them back to the
clients. This interface abstracts out the complexity of exchanging well-formed
messages with the proxy server. It allows clients to share and query data in
an intuitive manner.
* •
A cloud server (or _server_), usually located in the same machine as the
databases, accepts and processes client requests. We will refer to this
component as _server_. It manages and responses to meta queries concerning
XACML* instances. For data requests, it forwards them to the appropriate
XACML* instances and sends the results to the proxy in well-formed messages.
* •
XACML* is an implementation of the extended XACML model described in Section 3
(Fig. 3). It processes data requests (received from the cloud server) by first
asking PDP for the access decision. If permitted, it executes the obligations,
which involves querying the database. The result is forwarded back to the
cloud server.
* •
Communications between clients and servers go through a proxy server (or
_proxy_). It processes requests from clients before forwarding them to the
servers, and combines the results into client response messages. As an
example, suppose a request from a data user requires accessing data from
multiple datasets, the proxy first creates multiple requests and sends to the
corresponding servers. It waits for all the responses from servers, then
combines the results into a single response message for the data user.
The benefit of having the proxy server is two-fold:
1. 1.
_Improved performance_ : Combining data before returning to the users reduces
communication costs. Caching at the proxy can also improve response time and
reduce both computation and communication costs for the database servers. We
demonstrate this effect in the evaluation section.
2. 2.
_Additional level of abstraction_ : The proxy server acts like a DNS service
mapping datasets into to global, easy-to-remember names, achieving network
data independence, which makes it easier for clients to manage and query data.
### IV-B Trust and Data Model
We assume cloud severs and the proxy server are _honest_. This means that they
are trusted to run the correct, latest eXACML framework. They are also trusted
not to violate data privacy. More specifically, the proxy is trusted not to
tamper with the data received from database servers, and not to violate data
privacy. The only _adversaries_ are rouge clients who can collude in attempt
to gain unauthorized access to the datasets belonging to honest data owners.
We remark that these assumptions (particularly, that of trusted service
providers) are reasonable since cloud service providers are striving to gain
reputation to run their business, and furthermore have legal obligations based
on Service Level Agreements [23].
We assume that datasets are managed by relational database systems. For
simplicity, each data owner has at most one dataset. This assumption can be
relaxed by _virtualizing_ the data owner, so that it has multiple identities,
each of which possesses a different dataset.
### IV-C Cloud Model
We now discuss different ways to connect the database, XACML* and cloud server
components. As seen in Fig. 6, the number of servers, the number of databases
and XACML* instances do not have to match. In particular, multiple databases
may share the same XACML* instance, while a cloud server may handle multiple
XACML* instances.
A server represents a logical, addressable machine to which the proxy
connects. One server can handle requests for multiple datasets, but we assume
each server is connected to one dataset. This assumption is reasonable since
each data owner has at most one dataset, and it is likely that data owners use
independent virtual machines.
Next, we consider the question of how XACML* instances are shared among
databases. At one extreme, a single XACML* instance is sufficient to deal with
all access requests. In this case, the servers connect to the the same XACML*
instance, and policies are added to the same PDP. The PEP has access to
multiple databases at different machines. However, this approach introduces a
single point of failure, and data owners may prefer to have their access
control systems separated from each other. Moreover, extra layers of
authorization is required to prevent rouge clients from uploading policies
associated with datasets of honest data owners. At the other extreme, the
server maintain one XACML* instance per dataset. Since data requests can be
processed in parallel, this approach could lead to significant improvement in
performance. However, a potential drawback is the overhead in maintaining a
large number of XACML* instances, especially if many are idle.
When multiple datasets share the same physical machine (but are in separate
virtual machines), it makes more sense for them to share one XACML* instance.
This approach benefits from the parallelism in processing requests, while
having reduced overhead in maintenance. However, sharing an XACML* instance
experience the same problem with single point of failure and extra layer of
authorization as with a single XACML* instance.
Considering the above trade-offs, in this paper, we finally adopted the
simple, no-sharing approach, i.e. one server connects to one XACML* that
safeguards one database (illustrated in Fig. 7). This model does not require
another layer of authorization and therefore is easy to implement.
Figure 7: Interaction model of the cloud server, XACML* and database
### IV-D One or Multiple Proxies?
Having multiple proxies addresses the trust problem associated with a single
proxy. It could also improve client throughputs, since requests can be
processed in parallel. However, joining data — one of the proxy’s main
features — across multiple proxies is more complex. Since proxies also
maintain data caches, a mechanism for cache coherence among distributed
servers is also required. Therefore, trade-offs between efficiency and
maintenance overhead must be carefully considered. Our current framework
employs only one proxy. We defer the protocols with multiple proxies for
future work.
### IV-E Initialization
In the beginning, a data owner creates a database for its datasets and
initializes an XACML* instance at a remote data server. The XACML* instance
starts with an _initial policy_ specifying who can add and remove data and
policies. This process is done by invoking
{success,fail}
<- initDatabase(host, port, dataID,
databaseType, credentials)
where host, port are the address of the server, dataID is the unique
identifier of the dataset, databaseType is name of the database management
system (MySQL, for example), and credentials consists of the data owner’s
name, role and other authentication information for accessing the server. The
client interface wraps these parameters into a message forwarded to the proxy,
then sends it to the specified server. After authenticating the data owner,
the server creates the database, starts an XACML* instance and connects its
PEP to the database. Finally, the server uploads a _root policy_ to the newly
created XACML* instance. The root policy specifies that only users with
credentials can add new data, upload new and remove existing policies. This
policy prevents other clients from adding their own policies to this XACML*
instance.
If successful, the proxy creates a new mapping from dataID to the dataset, as
explained next.
### IV-F Data and Policy Management.
Once a database is initialized successfully, it can be identified uniquely by
its dataID. The proxy maintains a mapping dataID_to_desc, which is a list of:
dataID:<host, port, database name>
All client requests contain dataIDs. The proxy resolves locations of the
dataset using its mapping, before forming new requests and forwarding them to
the appropriate database servers.
##### Adding and removing data.
To add or remove new data from a dataset, the data owner invokes
{succses, fail}
<- addData(data file, dataID, credentials)
{success, fail}
<- removeData(remove query, dataID,
credentials)
where data file contains data to be added to dataID using the given
credentials. remove query is the query to remove records from the database.
The client interface sends a request to the proxy, which in turn constructs
and forwards a well-formed XACML request together with the file hash or query
hash to the server. The server keeps the hash as the _pending add_ or _pending
removal_ token. Only if the access control decision is ‘permit’ does the
client interface sends data file or remove query to the server, which verifies
that the content hash matches with the _pending add_ or _pending remove_
before performing the query. In this protocol, the hash value is used to
prevent other data owners from adding rouge data or remove unauthorized data.
##### Loading and removing policy.
Every loaded policy is identified uniquely by its ID of the form
dataID:policyID where policyID is the integer index of the policy. The XACML*
instance maintains an index counter which advances whenever a new policy is
added.
To add or remove a policy, a data owner invokes
{policyID, fail}
<- loadPolicy(policy file, dataID,
credentials)
{success, fail}
<- removePolicy(policyID, dataID,
credentials)
where policy file contains the XACML file to be uploaded to dataID using the
given credentials. The policy to be removed is identified by the tuple
(dataID, policyID). The client interface forwards a request to the proxy,
which creates a well-formed XACML request (for loading or removing policy)
using dataID and the credential. Once arrived at the server, the request is
evaluated by the appropriate XACML* instance. Only if the decision is permit
is the policy file added or the policy dataID:policyID is removed from the
corresponding PDP. In case of policy addition, the new policy ID — the current
index counter’s value — is forwarded back to the data owner. We assume that
policy is small, thus there is no need for the 2-step protocols as in adding
and removing data.
##### Querying policy.
Both data owner and the server keep track of the policy IDs associated with
the dataset. One can query about the loaded policies for a dataset, using
{{(policyID, description)}, fail}
<- queryPolicy(dataID, credentials}
which returns a set of tuples (policyID, description) where description is the
Description element of the corresponding policy.
### IV-G Data Request.
A data user issues a request for data through the client interface. The
request may involve accessing multiple datasets. The data user knows dataIDs,
but may not know of the detailed structure of the datasets.
#### IV-G1 Querying meta data.
A data user can issue a query for the dataset’s meta data prior to requesting
the raw data. Typical meta data includes table names and schemas. Data owners
can restrict access to such information through a set of policies. To query
meta data, the data user invokes:
{{tableID}, fail}
<- queryTables(dataID, credentials)
{(columnID, type)}, fail}
<- queryColumns(dataID, tableID,
credentials)
The proxy translates the client request into a well-formed, standard XACML
request in which the _Action_ attribute is set to show_table or show_column
respectively. If the PDP returns a permit decision, the PEP retrieves and
returns the database’s metadata accordingly. The result for queryTables (if
permitted) is a set of tableIDs, which can later be used in requesting raw
data. The result for queryDataScheme is a set of tuples (columnID, type)
representing the column name and type.
#### IV-G2 Querying data.
Clients can request data by invoking:
{{data record}, {matching policies}, fail}
<- queryData(requested resources,
joining condition)
where
requested resources
= {<credentials, dataId, {columns},
{actions}, {constraints}>}
represents the resources requested from different datasets. joining condition
specifies how the results from those datasets are joined. These results are
returned separately if joining condition is null. constraints contains
conditions that are applied to the returned data. For example,
$column_{i}>\theta$ where $column_{i}\in\\{\mbox{{columns}\\}}$ indicates that
the request is only for data whose $column_{i}$ values are greater than
$\theta$. The protocol proceeds as follows:
1. 1.
For every requested resource, the proxy creates a well-formed XACML request
using dataId, columns as _Resources_ and actions as _Actions_ attributes. The
request is then forwarded to the server specified by dataId.
2. 2.
The XACML* instance returns access control decision, the accompanied data (if
decision permitted), and IDs of the matching policies.
3. 3.
The proxy, on receipt of non-empty data, applies conditions specified in
contraints. Depending on the value of joining column, it performs data joining
(discussed next) before sending the final response to the client.
### IV-H Data Joining.
The joining condition parameter used in queryData specifies how the results
are joined before returning to the client. In particular:
$\mbox{{joining condition}}\in\\{\mbox{{null}},\\{c_{1},c_{2},..,c_{k}\\}\\}$
where $k$ is the number of requested resources and $c_{i}$ $(1\leq i\leq k)$
are the joining columns of the returned data. When joining column = null, the
proxy forwards what it receives from the server directly back to the client.
Otherwise, it waits until getting data from all requested servers, then
constructs a client response by joining the results using normal database join
operations.
### IV-I Conflict Resolution.
It is possible for clients to receive empty data for their requests,
especially when the requests involve more than one datasets. This arises
because different policies associated with different datasets are enforced. We
refer to this as _policy conflict_ , which happens in one of the two cases:
1. 1.
There is at least one policy that denies the client’s access.
2. 2.
All policies permit access, but the joined data still results in an empty set.
For example, one policy allows access to data where $column_{i}>\theta$
whereas another policy allows access to data where $column_{i}\leq\theta$.
Another example is when two policies specify different sliding windows, as a
consequence the joining columns do not have values in common.
We provide a simple mechanism for dealing with policy conflict. Responses from
queryData includes IDs of the matching policies. When conflict occurs, the
client is aware of the cause and is able to contact the dataset owner to
resolve the conflict. We assume that such resolution is done out-of-band and
is not within the scope of the framework.
### IV-J Caching.
The proxy maintains a cache of data received from the servers. Since
operations in the cloud server are slow, especially when involving database
access, caching can improve the response time. It is also reasonable to expect
a cache-friendly request pattern from clients, as popular data are frequently
requested.
We consider a simple design, in which data cache is the map <request>:<data>
where request is the XACML request with the corresponding data.
* •
_Cache replacement_ : when full, an old entry is evicted in a random fashion.
* •
_Cache coherence_ : stale entries can lead to security violation. For
instance, a new policy update denies a client access to a dataset, but the
cache contains data of previous access which will be served by the proxy at
the client’s next request. We address this problem by simply purging entire
cache every time a policy is loaded or removed.
## V Prototype and Evaluation
### V-A Prototype
We have implemented a prototype of eXACML, which consists of over $3400$ lines
of Java code. Database accesses are provided by JDBC API, while communications
between clients, proxy and servers are done through Socket interface. For
XACML*, we extended Sun’s XACML implementation [28] — an open source, Java
project that supports XACML 2.0 standard. We instrumented its PEP module to
handle more obligations (Section 3). The prototype supports all the features
discussed in the previous section: a client is able to load, remove, query
data and policies.
(a) Data view
(b) Query form
Figure 8: User interface for querying data
(a) Policy view
(b) Policy upload
Figure 9: User interface for managing access control policies.
Our prototype provides an easy-to-use graphical interface for querying and
managing data. A query form (Fig. 8b) takes in user credentials and requests.
A response from the server includes the data server information, matching
policies and the data (if applicable), which are displayed in the data view
window (Fig. 8a). Policies are updated and queried using similar GUI, as shown
in Fig. 9.
### V-B Evaluation
We evaluated our prototype’s performance, and its ability to support dynamic,
fine-grained access control policies. The system performance is measured by
the time taken to fulfill user requests. We compare our prototype’s
performance against that of a system that executes the requests directly, i.e.
without the access control layer. We refer to the later as _direct-query
system_.
#### V-B1 Methodologies.
##### Setup.
We emulate a cloud-like environment running our prototype, as shown in Fig. 6.
More specifically, we make use of four machines, two running servers, on
running the proxy and the other represents a client. The machines belong to
the PDCC cluster333http://pdcc.ntu.edu.sg/content/128-cores-linux-cluster-
pdccsce, each has one Xeon processor 3.0Ghz, running OCS5.1 (2.6.18-53El5smp)
operating system with 4GB of RAM. The machines are connected via InfiniBand
20Gbps.
The servers maintain two databases: a weather database and a traffic database.
The former contains four tables with real data taken from four different
weather stations collected in a 5-day duration and with one-minute sampling
interval. We synthesize the traffic database with two tables containing
records of traffic volume and vehicle speed that match with the weather
datasets.
##### Workloads.
We generate synthetic workloads that include large numbers of policies and
requests. Since our prototype is compared against a direct-query system, the
workloads also contain a large number of direct database queries, each
corresponds to a request in our prototype. A _direct query_ is forwarded to
the server, which executes and returns the same data as when executing the
corresponding request in our system. The parameters used in generating
workloads are shown in Table. VI. The workloads and source code for generating
them can be found at http://sands.sce.ntu.edu.sg/trac/exacml/
First, we use $nDirectQueries$ and $directQueryDist$ to create a set $DQuery$
of direct queries of five different types: selection, approximation,
aggregation, sliding window and data joining. The first three types are
ordinary database SELECT query, which is forwarded by the server directly to
the database engine. Sliding window queries are first converted into multiple
SELECT queries, one for every window, which are then sent to the database
engine. Data joining queries contain two sub-queries (of the other four types)
chosen at random and for different data servers. Each data server processes
and returns the result independently. Next, $nPolicies$ unique XACML policies
are generated, each with different exacml.subject:role-id. Every policy
corresponds to a direct query whose type is either selection, approximation,
aggregation or sliding window. Therefore, the set of policy obligations and
$DQuery$ represent the same set of SELECT queries to be executed by the
database engines.
Variable | Value | Description
---|---|---
$nDirectQueries$ | 1000 | number of direct queries
$directQueryDist$ | 248:248:248:156:100 | distribution of direct queries (selection:approximation:aggregation:sliding window:joining request)
$nPolicies$ | 900 | number of unique policies
$nRequests$ | 1500 | number of matching requests
$\alpha$ | 0.223 | skew parameter for Zipf distribution
$maxRank$ | 300 | maximum rank of unique requests from which Zipf distribution is generated
TABLE VI: Summary of parameters used in setting up experiments
Next, we generate a set of requests. For every policy, we construct one
matching and one non-matching request. The matching request contains
credentials, resources and actions as specified in the policy. For the non-
matching request, we use a different
exacml:rdbms-database-id from the weather and traffic database names. For each
data joining direct query, we create corresponding (matching and non-matching)
requests made up of two sub-requests. Each sub-requests from the matching
request corresponds to a sub-query in the data joining direct query. In
summary, a matching request executed in our prototype returns the same data as
the corresponding query evaluated in the direct-query system.
Finally, we create a workload of $nRequests$ requests following Zipf
distribution with skew parameter $\alpha$. This workload models a realistic
use of the prototype, in which a small number of popular data are requested
frequently. Such request pattern is found in many other systems, such as P2P
file-sharing and web caching [3, 16]. We select $maxRank$ unique queries from
$DQueries$ at random, then assign them with random ranks. A sequence of
queries is generated from the selected set with Zipf distribution, using
$\alpha=0.223$ (as in [16]). For every direct query, this workload also
contains the corresponding policy, matching and non-matching request.
#### V-B2 Metrics.
In the following experiments, we investigate our prototype’s effectiveness in
granting data access to authorized requests and denying unauthorized ones. We
also measure its performance in terms of the time taken to fulfill authorized
data requests. This is compared against the direct-query system, i.e. one
without eXACML. We also provide quantitative analysis of the proxy, especially
its caching and data joining features.
#### V-B3 Experiments and Results.
We first load $nPolicies$ unique policies onto the data server. The measured
time is reasonably small, with mean of $0.034s$ and standard deviation of
$0.016$ per loading operation.
We then run two sets of experiments:
1. 1.
The workload consisting of $nDirectQueries$ unique queries and the
corresponding unique requests. We enable the data joining option at the proxy
in the first run, and disable it in the second. To disable cache, we simply
change the proxy configuration file. To run without the joining option, we re-
generate the workload without data joining queries and requests. We measure
the time taken to fulfill direct queries and data requests.
2. 2.
The workload contains $nRequests$ queries and the corresponding requests,
which follow the Zipf distribution.
Figure 10: Overall performance, with vs without exacmlXACML. Caching and
joining options are on Figure 11: Overall performance with exacmlXACML when
the joining and caching options are disabled
In both experiments, non-matching requests are denied access. Fig. 10 and Fig.
11 compare the performance of our prototype against direct-query system, using
measurements of matching requests. In both figures, there is a number of
requests taking over $5s$ to finish. They are sliding window requests, which
translates into a large number of SELECT queries to be executed by the
database engines. That the server needs to wait and aggregate the results into
a single client message, and that JDBC implementation incurs non-significant
overhead for executing a SELECT query both contribute to the noticeable delay.
Fig. 10 illustrates eXACML’s overhead when both caching and data joining
options at the proxy are enabled. For unique queries and requests, there is no
overhead from the $99^{th}$ percentile. $80\%$ of the requests incurs less
than $10\%$ overhead. The largest overhead is less than $0.4s$ and is observed
from between $87\%$ to $90\%$ percentile. An interesting pattern in which
eXACML outperforms the direct-query system can be seen at lower percentiles.
Besides network and computational variations, this can be attributed to the
data joining feature at the proxy (discussed later). For requests and queries
following Zipf distribution, eXACML performs better most of the time (up until
the $89^{th}$ percentile). This is thanks to the caching mechanism at the
proxy, whose benefit will be analyzed in more detail later.
Fig. 11 shows how the overhead changes when the proxy performs neither caching
nor data joining. The overhead is more discernible: for unique requests, the
overhead starts from $20^{th}$ percentile, as compared to $45^{th}$ percentile
in Fig. 10. Similarly, for queries following Zipf distribution, the overhead
is seen from $10^{th}$ percentile, as compared to $89^{th}$ percentile in Fig.
10. This implies that caching and data joining at the proxy are most effective
when the query distribution is heavy-tailed.
Figure 12: Benefit of caching on performance. Queries follow Zipf distribution
We proceed to analyze benefits of caching at the proxy. Request times for
Zipf-distribution queries with and without cache are extracted from the
experiments and plotted in Fig. 12. We show the results with and without data
joining queries. In both cases, caching results in better performance. By
itself, i.e. without the joining data feature, caching leads to $50\%$
improvement for more than $80\%$ of the requests. For the workload including
data joining queries, a similar pattern can be seen, although the improvement
is not as noticeable.
Figure 13: Benefit of proxy performing data joins. All queries require data
joining
Finally, we analyze the benefit of the data joining feature at the proxy. We
run the same experiments as before, but with workloads consisting of only data
joining queries and requests. The results shown in Fig. 13 are for both unique
and Zipf-distribution requests. It can be seen that eXACML outperforms the
direct-query system up until $65^{th}$ percentile for unique queries and
$70^{th}$ percentile for Zipf-distribution queries. This is because for most
requests, eXACML helps reducing the data size substantially (by joining the
results from two servers) before transferring it back to the client. In
contrast, without eXACML, the client has to wait for all data to come back
individually before performing joining by itself. Notice that some requests in
eXACML still experience longer delay (after $70^{th}$ percentile), because
extra communication between client and proxy (as opposed to the direct
communication between client and server) and computation overhead at the proxy
are not fully discounted.
## VI Related work
There exists cloud-based systems that enable data sharing from multiple
sources. SenseWeb [26], SensorBase [10] are examples of cloud services that
let users upload and share their sensor data. They support coarse-grained
access control model in which an user either makes its dataset public, shares
it with a list of collaborators or keeps it private. Similarly, Google’s
Fusion Table [14] allows user to upload generic data and to perform simple
analysis such as data visualization on the cloud. Recently, companies such as
Okta [22] have started implementing cloud-brokerage models that provide
centralized service for management of enterprises’ resources, including access
control. However, these access control model is also coarse-grained, which
means it cannot deal with the access scenarios we consider in this paper. In
addition, data owners in these systems upload their datasets onto a
centralized cloud, whereas our work does not make such assumption (we consider
multi-cloud environment in which different data owner uses its own cloud
provider).
There are also numerous works focusing on access control and data privacy on
the cloud. Airavat [27], for example, assumes the cloud is trusted in
enforcing access control. It uses a simple mandatory access control system
available in SELinux [1], and provides a trusted environment for executing
MapReduce [11] jobs while guaranteeing differential privacy [12]. Our work
makes the same assumption about clouds’ trustworthiness, but aims at improving
the access control aspect of the system, which is complementary to Airavat.
Other works [29, 15, 23] assume the cloud is untrusted and employ
cryptographic approach for access control. In [29], data is encrypted with
attribute-based encryption [13, 7] by a proxy using a proxy re-encryption
technique. Embedded in the ciphertext are conditions that must be met when
decrypting. Plustus and CloudProof [15, 23] use broadcast encryption [19] to
protect the data, while key management [15] is done using key rolling and lazy
revocation techniques. These cryptographic approaches provide strong
guarantees for data security, but they cannot express fine-grained access
control policies as described in our work. Thus the focus in these works is
also complimentary to ours. In addition, key management and revocation
protocols are complex and incur much overhead in such an untrusted
environment.
Multiple policies matching in XACML is usually resolved by the top-level
policy combining algorithms. XACML supports only a limited number of combining
algorithms. Ninghui et al. [20] and Rao et al. [25] propose a formal language
for expressing more fine-grained policy composition. The language can deal
with evaluation errors and combining of obligations. Mazolleni et al. [17]
propose a method for combining policies based on their similarity and users’
preferences.
Time-series data — similar to those considered in our paper — could arrive at
the system in continuous streams, for which relational databases such as MySQL
and Postgresql are not ideal. Aurora [2] is a popular data stream management
system that addresses limitations of relational databases when it comes to
stream data. Carminati et al. [9, 8] are among the first to propose a model
and implementation of access control for data streams based on Aurora. The
model supports four access scenarios: column-based, value-based, general
window and sliding window. Our framework supports all of these scenarios for
on-demand queries over archival databases. The extension to eXACML that deals
with continuous queries over stream databases is left for future work.
## VII Future work
We have implemented a simple prototype and carried out preliminary evaluation
of our framework. The next step would be to improve the prototype and perform
more comprehensive evaluations. More specifically, the cloud-like environment
set up in the experiment contains only two data servers. In addition, only one
dataset comes from real monitoring stations, and the workloads are synthetic.
Therefore, we plan to acquire more realistic datasets and workloads, and to
evaluate the prototype with larger numbers of data servers. We also plan to
export our prototype into real cloud environments such as Amazon EC2 and
Microsoft’s Azure [4, 18], and benchmark it with real data mining applications
accessing real datasets.
We assumed that each dataset is guarded by an independent XACML* instance. We
have acknowledged the trade-offs in having multiple datasets sharing one
XACML* instance, especially when datasets reside in the same physical machine.
Another trade-off is the number of proxy servers. It would be interesting to
investigate these trade-offs further by extending the framework with XACML*
sharing and distributed proxies.
As shown in Table I, eXACML only deals with archival databases and queries.
The immediate extension will be to support stream databases and continuous
queries. Relational databases are not the best tool for handling stream data,
for which other models have been proposed [2]. We will examine the design and
compare performance of the extended eXACML to that of the existing works on
access control for stream data [9, 8].
Regarding data sharing, access control only addresses the problem of
authorization. We have so far made an assumption that authentication is
implicit, that is, clients are given static credentials and the servers always
accept the given credentials. We plan to incorporate an authentication model
into our framework. It is an interesting challenge in decentralized settings,
of which our multi-cloud scenario is an example, since authentication may
depend not only on static credentials but also on previous interactions
between parties and the states of the entire system. Authentication is also an
important when the cloud provider has to log and notify data owners of access
to their data (for billing purposes, for example). We plan to use other access
control languages such as DynPal [6] or SecPal [5], because they are more
suitable for handling dynamic authentication than XACML.
Finally, we have always assumed the cloud is trusted in enforcing access
control policies and not to violate user’s data security and privacy. However,
users with sensitive data or data that have been expensive to collect will
demand highest level of security. As a consequence, they cannot assume the
cloud is trusted in handling their data. Existing works have taken the
cryptographic approach that encrypt data and attempts to outsource the key
management to the cloud. Nevertheless, the range of access control policies
supported by the existing systems has been limited. For future work, we aim to
find practical cryptographic protocols that can handle more fine-grained
access control scenarios. Since eXACML contains two components belonging to
third parties: the proxy server and the cloud servers, we will investigate
relaxing the trust assumption for these components one by one.
## VIII Conclusion
In this paper, we have proposed a framework (eXACML) that allows users to
share their data on the cloud in a secure, flexible, easy-to-use and scalable
manner. We considered a trusted cloud environment, in which data are
maintained in relational databases. The cloud environment makes it easy for
data owners to share and benefit from mining the aggregated data. The main
challenge is how to let users control access to their data in most flexible
ways. We achieved security and flexibility by extending the XACML framework,
allowing users to specify fine-grained access control policies. Our framework
contains a proxy server residing in between clients and the cloud servers. It
processes requests from the clients, joins and caches responses from the
servers before sending back to the client. We have implemented a prototype and
carried out preliminary experiments to evaluate its performance. The results
suggested that the framework is scalable, as the overhead incurred is small,
thanks to the caching and data joining features at the proxy. In addition, the
prototype provides a graphical user interface that lets users share and manage
their data in an easy-to-use manner.
We believe that in order to take full advantage of cloud computing, having a
framework such as ours is very important. Our paper has taken the first steps
towards realizing a practical and usable sharing-friendly cloud environment.
We have also identified many avenues for future work, such as improving
scalability with more proxies, adding support for stream data and other policy
languages, and relaxing assumptions on the trustworthiness of the cloud.
##### Acknoledgments.
This work has been supported by A*Star TSRP grant number 1021580038 for
‘pCloud: Privacy in data value chains using peer-to-peer primitives’ project.
The authors will like to thank Dr. Lim Hock Beng for providing access to the
weather data sets used in some of the experiments.
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2024-09-04T02:49:21.646260
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dinh Tien Tuan Anh and Wang Wenqiang and Anwitaman Datta",
"submitter": "Tien Tuan Anh Dinh",
"url": "https://arxiv.org/abs/1108.3915"
}
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1108.3987
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# The Identification of the X-ray Counterpart to PSR J2021+4026
Martin C. Weisskopf11affiliation: NASA Marshall Space Flight Center, Space
Science Office, VP62, Huntsville, AL 35812 USA. , Roger W.
Romani22affiliation: Department of Physics, Stanford University, Stanford, CA
94305 , Massimiliano Razzano33affiliation: Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, I-56127 Pisa, Italy 44affiliation: Santa Cruz
Institute for Particle Physics, Department of Physics, University of
California at Santa Cruz, Santa Cruz, CA 95064, USA 55affiliation:
Dipartimento di Fisica, Università di Pisa, 56127 Pisa, Italy , Andrea
Belfiore44affiliation: Santa Cruz Institute for Particle Physics, Department
of Physics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA
66affiliation: INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, I-20133
Milano, Italy 77affiliation: Universitá di Pavia, Dipartimento di Fisica
Teorica e Nucleare (DFNT), I-27100 Pavia, Italy , Pablo Saz
Parkinson44affiliation: Santa Cruz Institute for Particle Physics, Department
of Physics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA
, Paul S. Ray88affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA , Matthew Kerr99affiliation: Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall,
Stanford, CA 94305, USA , Alice Harding1010affiliation: Astrophysics Science
Division, NASA Goddard Space Flight Center, Code 663, Greenbelt, MD 20771 USA.
, Douglas A. Swartz1111affiliation: Universities Space Research Association,
NASA Marshall Space Flight Center, Space Science Office, VP62, Huntsville, AL
35812 USA. , Alberto Carramiñana1212affiliation: Instituto Nacional de
Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Tonantzintla, Puebla
72840, México. , Marcus Ziegler44affiliation: Santa Cruz Institute for
Particle Physics, Department of Physics, University of California at Santa
Cruz, Santa Cruz, CA 95064, USA , Werner Becker1313affiliation: Max-Planck-
Institut für extraterrestrische Physik, 85741 Garching bei München, Germany. ,
Andrea De Luca66affiliation: INAF-Istituto di Astrofisica Spaziale e Fisica
Cosmica, I-20133 Milano, Italy 1414affiliation: IUSS-Istituto Universitario di
Studi Superiori, Viale Lungo Ticino Sforza 56, 27100 Pavia, Italy
1515affiliation: INFN-Istituto Nazionale di Fisica Nucleare, sezione di Pavia,
via A. Bassi 6, 27100 Pavia, Italy , Michael Dormody44affiliation: Santa Cruz
Institute for Particle Physics, Department of Physics, University of
California at Santa Cruz, Santa Cruz, CA 95064, USA , David J.
Thompson1616affiliation: Astroparticle Physics Laboratory, NASA Goddard Space
Flight Center, Code 661, Greenbelt, MD 20771 USA. , Gottfried
Kanbach1313affiliation: Max-Planck-Institut für extraterrestrische Physik,
85741 Garching bei München, Germany. , Ronald F. Elsner11affiliation: NASA
Marshall Space Flight Center, Space Science Office, VP62, Huntsville, AL 35812
USA. , Stephen L. O’Dell11affiliation: NASA Marshall Space Flight Center,
Space Science Office, VP62, Huntsville, AL 35812 USA. , Allyn F.
Tennant11affiliation: NASA Marshall Space Flight Center, Space Science Office,
VP62, Huntsville, AL 35812 USA.
###### Abstract
We report the probable identification of the X-ray counterpart to the
$\gamma$-ray pulsar PSR J2021+4026 using imaging with the Chandra X-ray
Observatory ACIS and timing analysis with the Fermi satellite. Given the
statistical and systematic errors, the positions determined by both satellites
are coincident. The X-ray source position is R.A. 20h21m30s.733, Decl.
$+$40°26′46.04″ (J2000) with an estimated uncertainty of 1.″3 combined
statistical and systematic error. Moreover, both the X-ray to $\gamma$-ray and
the X-ray to optical flux ratios are sensible assuming a neutron star origin
for the X-ray flux. The X-ray source has no cataloged infrared-to-visible
counterpart and, through new observations, we set upper limits to its optical
emission of $i^{\prime}>23.0$ mag and $r^{\prime}>25.2$ mag. The source
exhibits an X-ray spectrum with most likely both a powerlaw and a thermal
component. We also report on the X-ray and visible light properties of the 43
other sources detected in our Chandra observation.
pulsars: individual (PSR J2021+4026) — supernova remnants: individual (SNR
78.2+2.1 = $\gamma$-Cygni SNR) — X rays: general
## 1 Introduction
Pulsars are often depicted as relatively simple objects: a highly magnetized,
fast rotating neutron star (NS) emitting radiation along its poles. Most
emission models start from this basic picture, in principle common to all
isolated pulsars. The rotation of the magnetic dipole induces an electric
field near the polar caps in vacuum, which is then short circuited by an e±
pair plasma (Ruderman & Sutherland 1975), or in space-charge limited flow,
which is screened at a pair formation front (Arons & Scharlemann 1979). Models
seek then suitable regions of particle acceleration. Although the radio
emission is thought to originate in the polar regions of the magnetosphere
(e.g Michel 1991 and references therein), the high-energy emission is now
thought to originate from the outer magnetosphere, with the original polar cap
concept (Daugherty & Harding 1996) having been superseded by outer-gap (Cheng,
Ho & Ruderman 1986; Romani 1996) and slot-gap (Arons 1983, Muslimov & Harding
2004) models. The picture that has emerged is that radiation is emitted in a
continuous, very broad, spectral range, with curvature radiation producing
most of the GeV emission (Romani 1996) and synchrotron and/or Compton
scattering by cascade products producing the near-infrared to soft
$\gamma$-ray emission (Takata & Chang 2007, Harding et al. 2008).
The high energy emission properties of pulsars were revealed in the 1990s when
use of the EGRET experiment on the Compton Gamma-Ray Observatory (e.g.
Thompson, 2001 and references therein) led to the multiwavelength spectral
characterization of half a dozen $\gamma$-ray pulsars, including the discovery
of the radio-quiet pulsar Geminga (Halpern & Holt 1992), the second brightest
steady GeV source in the $\gamma$-ray sky. Amongst the EGRET legacy was a
sample of 170 unidentified sources, 74 of which are at $|b|<10^{\circ}$
(Hartmann et al. 1999). It has now been found that $\approx 43$ EGRET
unidentified sources have counterparts in the Fermi Large Area Telescope (LAT)
first year catalog (Abdo et al. 2010b; 1FGL) and a large fraction of those
lying on the Galactic plane turned out to be pulsars (Abdo et al. 2008, 2009a,
2009b, 2010a, 2010b), a result anticipated by several authors (Yadigaroglu &
Romani 1995, Cheng & Zhang 1998, Harding & Muslimov 2005).
In fact, it is very plausible that many more radio-loud or radio-quiet pulsars
are hidden in the unidentified Galactic LAT sources, although estimates are
highly dependent on details of the different emission mechanisms. At this
point it is still unclear what makes some $\gamma$-ray pulsars radio-loud and
some radio-quiet. Understanding the distinct properties of the individual
sources will surely lead to a better understanding of models for the emission
mechanisms, for example, the connection between the radio and the near-
infrared to the $\gamma$-ray component.
The source 2CG 078+2 was one of about twenty $\gamma$-ray sources detected by
the COS-B satellite 30 years ago (Swanenburg et al. 1981). The source is in
the line of sight towards the supernova remnant G78.2+2.1. The remnant
comprises a 1-degree-diameter, circular, radio shell with two bright and broad
opposing arcs on its rim (Higgs, Landecker & Roger 1977, Wendker, Higgs &
Landecker 1991). The remnant has a kinetic distance of about 1.5 kpc
(Landecker, Roger, and Higgs 1980; Green 2009) and is estimated to have an age
of 5400 yr (Sturner & Dermer 1995). 2CG 078+2 is often cited as $\gamma$-Cygni
due to its proximity to the second magnitude foreground star (mV = 2.2,
spectral type F8Iab) that lies on the eastern edge of the remnant although
there is no other physical association between the supernova remnant (SNR) and
the star. A small HII region, located close to the star, forms the
$\gamma$-Cygni nebula.
The $\gamma$-ray source, next cataloged as 3EG J2020+4017 (Hartmann et al.
1999), was suspected to be either extended emission from the SNR or a pulsar
formed in the supernova event (Sturner & Dermer 1996). Soon after its June
2008 launch, the Fermi $\gamma$-ray Space Telescope highlighted the discovery
of a radio-quiet pulsar in the CTA-1 SNR (Abdo et al. 2008) as a first light
result. This was followed by the discovery of twenty-three other $\gamma$-ray
pulsars using blind search techniques (Abdo et al. 2009b, Saz Parkinson et al.
2010), among them PSR J2021+4026 lying within the EGRET error circle of 3EG
J2020+4017. PSR J2021+4026 is a 265-ms pulsar with a spin-down age of 77 kyr
and a total spin-down luminosity of 1.1$\times 10^{35}$ erg/s. Interestingly,
most of the pulsars discovered by the LAT with blind searches have not been
seen at other wavelengths. In fact, only three of the 26 discovered to date
have been detected in radio (Camilo et al. 2009; Abdo et al. 2010c)
Even before the Fermi discovery of pulsations, and because the EGRET
$\gamma$-ray properties of the source matched those of known $\gamma$-ray-
pulsars (hard, steady and in the Galactic plane), the search for the radio and
X-ray counterpart of this source began (Brazier et al. 1996; Becker et al.
2004, Weisskopf et al. 2006). No radio pulsar is associated with PSR
J2021+4026 with searches having now been conducted down to $100\mu$Jy at 1665
MHz (Trepl et al. 2010), 40 $\mu$Jy at 820 MHz (Becker et al. 2004), and to
$11\mu$Jy at 2 GHz (Ray et al. 2011). There is an extended radio source (GB6
J2021+4026) in the vicinity which appears to be positioned more or less
symmetrically (see Figure 8 of Trepl et al. 2010) about our best position for
the X-ray counterpart but there is no evidence that it is associated with the
$\gamma$-ray pulsar.
We have previously (Becker et al. 2004, Weisskopf et al. 2006) performed
Chandra observations (ObsIDs 3856 & 5533) aimed at different portions of the
$\gamma$-Cygni field and these pointings were based on the best known
$\gamma$-ray positions available at the time. The latter Chandra observation,
ObsID 5533, overlapped with about 3/4 of the current Fermi-LAT 99%-confidence
positional error circle and detected several potential X-ray counterparts,
including the source designated as ”S21” as reported in Weisskopf et al.
(2006). Subsequently, Trepl et al. (2010) reanalyzed the available Chandra
data and also used XMM-Newton data to search for a counterpart. They identify
source 2XMM J202131+402645, at virtually the same location as the Chandra
source “S21”, as the likely counterpart because the X-ray source position fell
within the $\sim$4′-radius Fermi-LAT 95%-confidence positional error circle at
the time (0FGL, Abdo et al. 2009b). We note that the most recent Fermi-LAT
error circle radius (2FGL, Abdo et al. 2011) is 0.6′ (§3) and no longer
includes “S21”. We re-evaluate the situation using both the greatly improved
LAT source localization and the position measured from timing of PSR
J2021+4026.
The current work describes a new observation that completes the Chandra
coverage of the field. As a result of these observations, we conclude that the
source originally labelled “S21” by Weisskopf et al. (2006), 2XMM
J202131+402645 by Trepl et al. (2010), and source #20 in the new observation,
remains the most probable X-ray counterpart to PSR J2021+4026. Moreover, we
also show that the X-ray source is dominated by thermal, not powerlaw,
emission. In this regard, it is interesting to compare this source with
Geminga, one of the best studied radio-quiet $\gamma$-ray pulsars, which is
older (340 ky) and less luminous ($\sim 3\times 10^{34}$ erg/s) than PSR
J2021+4026 but has a similar period and period derivative.
§2.1 describes the analysis of the X-ray image; §2.2 describes the analysis of
the X-ray spectrum, especially of the X-ray source #20; §2.3 compares the
X-ray spectral properties of source #20 to Geminga and CTA 1; §3 describes new
Fermi-LAT pulsar timing of PSR J2021+4026; §4 describes our search for an
optical counterpart; and §5 discusses properties of the other Chandra sources
in the field. We provide summary conclusions in §6.
## 2 Chandra X-Ray Observations and Data Analysis
We obtained a 56-ks Chandra observation (ObsID 11235, 2010 August 27) using
the Advanced CCD Imaging Spectrometer (ACIS). Here we report the data taken
with the back-illuminated CCD ACIS S3 in faint, timed-exposure mode with
3.141-s frame time. Background levels were nominal throughout the observation.
Standard Chandra X-ray Center (CXC) processing provided accurate aspect
determination.
Starting with level-1 event lists, we reprocessed the data using the CIAO v4.2
tool acis_process_events to remove pixel randomization in order to improve the
on-axis point spread function (PSF), thus enhancing the source-detection
efficiency and positional accuracy. In searching for sources, we utilized
events in pulse-invariant channels 35-550 corresponding to 0.5 to 8.0 keV.
To verify the Chandra position accuracy, we compared the pointing parameters
(RA, Dec, roll) given by the Chandra data-processing to that using the 19
2MASS sources (Section 5.3.2) with high probability X-ray counterparts. For
these calculations we used $0.075\arcsec$ per axis for the 2MASS position
uncertainty111http://www.ipac.caltech.edu/2mass/releases/allsky/doc/explsup.html.
Assuming no change to the Chandra parameters yielded an acceptable positional
fit for these 19 sources with $\chi^{2}$ of 24 for 38 degrees of freedom.
Allowing the three pointing parameters to vary marginally improved $\chi^{2}$
to 21 for 35 degrees of freedom and would imply the following corrections:
$(0.14\pm 0.10)\arcsec$ (RA), $(-0.04\pm 0.08)\arcsec$ (Dec), and $(43\pm
78)\arcsec$ (roll). However, these corrections, even allowing for errors,
being negligible we did not correct the Chandra positions.
### 2.1 X-Ray Image Analysis
We searched for point-like X-ray sources employing techniques described in
Tennant (2006), using a circular-Gaussian approximation to the PSF and setting
the signal-to-noise (S/N) threshold for detection to 2.4. The resulting
background-subtracted point-source detection limit is about 7 counts, with
fewer than 1 accidental detection expected over the field. Based upon tests on
Chandra deep fields, this approach finds all X-ray sources in those fields
down to 10 counts, which we thus regard as the completeness limit. Figure 1
shows the ACIS-S3 image with a small circle at the position of each Chandra-
detected source.
Table 1 tabulates the X-ray properties of the 44 Chandra-detected sources,
with the source number in column 1. Columns 2–5 give, respectively, right
ascension RA, declination Dec, extraction radius $\theta_{\rm ext}$, and
approximate number of X-ray counts $C_{x}$ detected from the source. Column 6
lists the single-axis RMS error $\sigma_{x}=[(\sigma_{\rm
PSF}^{2}/C_{x})+\sigma^{2}_{\rm sys}]^{1/2}$ in the X-ray-source position,
where $\sigma_{\rm PSF}$ is the dispersion of the circular Gaussian that
approximately matches the PSF at the source location and $\sigma_{\rm sys}$ is
a systematic error. Uncertainties in the plate scale222See
http://cxc.harvard.edu/mta/ASPECT/aca_plate_scale/ imply $\sigma_{\rm
sys}\approx 0\farcs 13$: To be conservative, we set $\sigma_{\rm sys}=0\farcs
2$ (per axis). Column 7 gives the radial uncertainty
$\epsilon_{99}=3.03\>\sigma_{x}$ in the X-ray position — i.e.,
$\chi^{2}_{2}=9.21=3.03^{2}$ corresponds to 99% confidence on 2 degrees of
freedom — for inclusion of the true source position. Columns 8 & 9 are color
ratios defined and discussed in § 5.1.
In view of the spin-down age and energetics of PSR J2021+4026, the possibility
exists that a pulsar wind nebula (PWN) may also be present in the X-ray image
of the field. We searched for moderately extended sources in the field and
identified three features of interest. Two are located near the edge of the S3
CCD which increases the likelihood of their being false positives. The third
feature lies $\sim$6″ west of source #20. Simultaneously fitting the
combination of this feature and source #20 to circular Gaussians (plus a
constant background) results in a Gaussian width of 2.2″ and a total of 22
X-ray events within this extended feature.
We also compared the spatial distribution of events in and around source #20
to a model of the Chandra/ACIS PSF valid for its location relative to the
aimpoint and its characteristic energy using the PSF library psflib v4.1
(CALDB v4.2). Source #20 is consistent with being pointlike.
### 2.2 X-Ray Spectral Analysis
We used the XSPEC (v.12.5) spectral-fitting package (Arnaud, 1996) to perform
spectral fits to the 44 X-ray point sources. We treat source #20 separately in
§2.2.1. Data were binned to obtain at least 10 counts per spectral bin before
background subtraction. The background was determined after masking off a
circular region around each of the 44 detected X-ray sources corresponding to
a circle of radius 25 times the uncertainty listed in column 6 of Table 1 and
then averaging over the remaining pixels. Individual response matrix and
ancillary response files appropriate to each source position were created for
this analysis using the mkacisrmf and mkarf tools available in CIAO version
4.2.
#### 2.2.1 The X-Ray Spectrum of Source #20
A sufficient number of counts were detected from source #20 to perform more
extensive spectral analysis. Figure 2 shows the X-ray spectrum of source #20
with background subtracted. Data were again binned to obtain at least 10
counts per bin before background subtraction. The background is less than 3%
of the signal plus background from the region that includes source #20.
We begin our spectral analysis by first considering single-component spectral
models (Table 2) with a multiplicative absorption component: an absorbed power
law (powerlaw in XSPEC); an absorbed black body (bbodyrad); and three
different absorbed neutron star atmosphere models,333A NS radius of 12.996 km
and mass of 1.358 M⊙ is assumed throughout for purposes of computing effects
of gravitational redshift in the neutron star atmosphere models namely, nsa
(Pavlov et al. 1995), nsmax-1260 and nsmax-130190 (Ho, Potekhin & Chabrier
2008).
For computing absorption, we utilized abundances (XSPEC’s wilm) from Wilms,
Allen, & McCray (2000) with cross-sections (vern) from Verner et al.(1993) and
allowed for interstellar extinction by grains using the model (tbabs) of
Wilms, Allen, & McCray (2000).
All of these models provide statistically adequate fits to the data in the
0.5-8.0 keV range (Table 2). (We note that so did a fit to an absorbed mekal
model.) From the Hi in the Galaxy (Dickey & Lockman, 1990), one infers a
column density $N_{\rm H}\approx 1.4\times 10^{22}$ cm-2 through the Galaxy in
the direction of source #20, implying that values below this should be
expected if there is no circumstellar absorption, and this is indeed the case
(Table 2). Next we posit that the very steep power law index of almost 5 is
not physical, but indicative of a soft, thermal component. The bbodyrad
model’s normalization, $(R^{2}_{\rm km}/D^{2}_{10{\rm kpc}})=0.80$, where
$R_{\rm km}$ is the radius of the emitting area in km and $D_{10{\rm kpc}}$ is
the distance to the source in units of 10 kpc, corresponds to an emitting area
of $R_{\rm km}=0.13$, assuming the association of the pulsar with G78.2+2.1 at
a nominal distance of $D_{10{\rm kpc}}=0.15$. This is much smaller than a
typical NS radius, but not very different than the standard polar cap radius,
$\approx R_{NS}sin\theta\approx R_{NS}^{3/2}(2\pi/cP)^{1/2}$ which for
$P=0.265$ s and $R_{NS}=13$ km is 0.42 km.
Conversely, if we assume the emission comes from a NS with a characteristic
radius of 13 km, then the bbodyrad norm implies a source distance of $\approx
145$ kpc, well outside the Galaxy. This distance estimate drops for the
different NS-atmosphere models, e.g. down to 13 kpc (Table 2), still somewhat
distant to remain within the Galaxy. In addition, the temperature estimates of
the (cooling) NS, range from $\log(T_{\infty})\geq 6.5$ to $6.1$, with the
precise value depending on the model (Table 2). These estimates are consistent
with those expected for a pulsar of an estimated age somewhere between 5400
and 77000 years, depending of course on the equation of state, composition of
the heat-blanketing envelope, and the degree of superfluidity in the star’s
core. (See Yakovlev et al. 2010 and references therein for recent details on
the subject of cooling NSs.)
We then ask whether or not combining a power law with the other models is
indicated by the data. Table 3 tabulates the change in $\chi^{2}$, the
f-statistic, and the probability that combining models has significantly
improved the quality of the fit. The table also tabulates he derived spectral
parameters. In all cases combining various thermal models with a power law
does improve the quality of the fit with a confidence better than 2-sigma, but
not 3-sigma. Moreover, these two-component models allow a wide latitude for
the uncertainties of the best-fit parameters and hence these parameters are
not as constrained as one might wish. This follows from the fact that all of
the single models (Table 2) themselves provide statistically adequate fits.
Thus, the 3-sigma contours for any two-component model allows for one or the
other of the component models to have a zero norm – e.g. a power law component
is not completely required by the data at this level of significance. Keeping
this proviso in mind, we continue to examine the two-component models.
Certainly the physical interpretation of the data is perhaps more sensible
when both components are introduced, especially in light of the strong
$\gamma$-ray emission which cannot arise from any single-component “thermal”
model spectrum. In addition, both the inferred NS temperatures and distances
implied by the two-component models (Table 3), with the possible exception of
the bbodyrad+powerlaw model, are consistent with what one might expect for a
young cooling neutron star within the galaxy.
#### 2.2.2 Discussion of the Spectrum of Source #20
The considerations of the single-component models in the previous section lead
one to conclude that we are perhaps seeing a hot spot of size comparable to
the polar cap rather than thermal emission from the entire surface of the
neutron star and with a temperature higher than one expects from cooling of a
neutron star at an age $>5000$ years. It is thus possible that the bulk of the
emission comes from heating of the polar cap by backflowing accelerated
particles. The expected luminosity and temperature from heating by positrons
produced by curvature radiation of primaries in a space-charge limited flow
model is $L_{+}\sim 10^{31}\,\rm erg\,s^{-1}$ and $\log(T_{+})\simeq 6.3$
(Harding & Muslimov 2001). This luminosity is deposited over an area roughly
that of the polar cap and radiated on a timescale less then the heat diffusion
timescale across field lines to other areas of the NS. The temperature $T_{+}$
is close to that determined from the bbodyrad and nsa model fits allowing for
the gravitational redshift. It is therefore quite possible that the X-ray
emission from PSR J2021+4026 is dominated by polar cap heating.
On the other hand, many well-studied neutron stars that exhibit both X-ray and
$\gamma$-ray emission have composite X-ray spectra, showing non-thermal power
law magnetospheric emission and/or hot polar cap emission in addition to the
lower-temperature full surface thermal emission (e.g. Geminga, PSR B0665+14,
PSR B1055-52, see De Luca et al., 2005). While the present data quality do not
demand such two-component models, the inferred distances from such two-
component fits, e.g., $\sim 6$ kpc for the NS atmosphere models (Table 3)
become reasonable for plausible stellar radii assuming full surface emission.
This distance is larger than the kinematic distance to the SNR 78.2$+$2.1 of
1.5 kpc but is comparable to the distance to the Cygnus arm at Galactic
longitude $\sim$78°. Thus, it is possible that a two-component model, with
full-surface cooling and magnetospheric power law emission present at lower
levels, might be needed to accurately describe the emission physics. In
practice, decomposing such complex X-ray spectra requires good statistics and
phase-resolved X-ray spectroscopy. For a target this faint, extremely long
observations or next-generation X-ray satellites are clearly required.
### 2.3 Comparison of the Spectrum of #20 to Geminga and CTA 1
There are similarities and differences between the X-ray spectrum of PSR
J2021+4026 and two of the other radio-quiet $\gamma$-ray pulsars with detected
X-ray emission, Geminga and CTA 1. First PSR J2021+4026, like Geminga (Jackson
and Halpern, 2005) and CTA 1 (Caraveo et al. 2010), may also be characterized
by two spectral components, a thermal component and power law. In Geminga,
however, the power law component begins to dominate above about 0.5 keV and
$\log(T_{\infty})$ is about 5.7. For PSR J2021+4026 the power law dominates
above 2.5 keV and $\log(T_{\infty})$ is higher, as one would expect as PSR
J2021+4026, based on its spin-down age, is younger.
The spectral indices for the power law components for source #20 and Geminga
are not dissimilar, but one needs to recognize the large uncertainty in the
measurements reported here. Another, possibly important, spectral difference
between the two X-ray spectra is that, in the case of Geminga, the blackbody
component gives an emission radius that is plausible for a NS radius. This is
not so for PSR J2021+4026. In this case, the neutron star atmosphere models
seem to yield more physically reasonable parameters than the bbodyrad+powerlaw
model. If we assume that the younger and hotter star still has an atmosphere
while the older Geminga does not, then these results are sensible. Finally,
there is a weak extended emission feature near source #20 that may be
indicative of a PWN. If so, then it extends no more than 0.04 to 0.17 pc from
source #20 (assuming a distance of 1.5 to 6.0 kpc, respectively) and
contributes $\sim$7% of the X-ray counts detected from source #20 and its
surroundings. Emission associated with the Geminga PWN has a similar extent
and contributes 10% of the non-thermal X-ray flux of the pulsar, but only
about 1% of the total flux (Pavlov et al. 2010).
For CTA 1, the measured temperature, powerlaw index and emission radius are
$\log(T_{\infty})=6.08$, $\Gamma=1.3$ and $r=0.64$ km for a powerlaw+bbodyrad
model, and $\log(T_{\infty})=5.78$, $\Gamma=1.25$ and $r=4.92$ km for
powerlaw+nsa, with slightly lower $\chi^{2}$ for the former (Caraveo et al.
2010). In both cases, the emitting radius is significantly smaller than a
standard NS radius. Thus, similar to PSR J2021+4026, CTA-1 shows a possible
heated polar cap component, with a temperature very close to the model
prediction of $\log(T_{\infty})\simeq 6.2$ (Harding & Muslimov 2001), however,
the emitting radius in this case is a factor of 2.5 larger than the polar cap
radius. CTA 1 does not show evidence for a cool component, with the upper
limits making it unusually cool for its age.
## 3 Fermi-LAT Localization and Timing Analysis
The Fermi-LAT normally localizes $\gamma$-ray sources using the incident
$\gamma$-ray photon directions. The LAT has a point spread function that is
strongly energy dependent, with a resolution of about 0.8∘ at 1 GeV. For a
bright source, however, localization with arcminute accuracy is possible. The
source in the second LAT catalog (Abdo et al. 2011) that corresponds to PSR
J2021+4026 is 2FGL J2021.5+4026. The catalog position for this source is R.A.
20h21m34s.1, Decl. $+$40°26′28″ (J2000), with a 95% confidence radius of
0.60′.
For pulsars, one can use timing techniques better to localize the source,
independent of the photon direction measurements, as described in Ray et al.
(2011). The position determination from timing of this pulsar is hampered by
the large contribution from rotational instabilities common in young pulsars
and manifest as “timing noise”. Therefore, we have taken two different
approaches to try to confirm the association between PSR J2021+4026 and source
#20. For this analysis, we used $\gamma$ rays detected by the LAT from 2008
Aug 4 to 2011 Mar 12, selecting only those within $0.8^{\rm o}$ from the
previous best position (Ray et al. 2011), and with energies greater than 400
MeV. These “cuts” were chosen to maximize the significance of the pulsation.
We chose only photons belonging to the most restrictive “diffuse” class
according to the “Pass 6” instrument response functions (see Atwood et al.
2009), which have the lowest background contamination. Furthermore, we
selected only photons with a zenith angle of $<105^{\rm o}$, to reduce
contamination due to secondary-atmospheric $\gamma$ rays.
As a first test, we evaluate the significance of the $\gamma$-ray pulsations
by assuming the pulsar is at each of the candidate X-ray source locations seen
in our Chandra observation (Table 1). For each candidate location, we
transform arrival times to the barycenter using the X-ray position and then
use the prepfold routine from the PRESTO pulsar package (Ransom, Eikenberry &
Middleditch 2002) to find the frequency ($f$), and its first and second
derivatives, $\dot{f}$ and $\ddot{f}$, which maximize the statistical
significance of the pulsation. Figure 3 shows the results from this
exploratory search, where it is clear that source #20 gives the maximum
significance for pulsation using this algorithm. This indicates that of the
possible X-ray sources in the field, source #20 is the most likely X-ray
counterpart.
Next, we use pulsar timing to fit for the position of the pulsar, as described
in Ray et al. (2011). We measured 55 times of arrival (TOAs) based on 22-day
integrations spanning the data set described above. The typical uncertainty on
each TOA measurement was 4.7 ms. Using Tempo2 (Hobbs et al. 2006), we fit the
TOAs to a timing model including $f$, $\dot{f}$, and $\ddot{f}$. With only
these terms in the model, we observe very large residuals and the $\chi^{2}$
of the fit is very poor. This poor model fit means that the statistical errors
in the fitted right ascension and declination reported by Tempo2 are
unreliable. To get an estimate of the systematic error in the position fit, we
use the following procedure. We added 5 so-called “WAVE” terms to the timing
model to account for the timing noise using harmonically-related sinusoids
(Hobbs et al. 2004). We then perform a fine scan over a positional grid around
the location of source #20. Holding the position fixed at each grid point, we
fit for the spin and WAVE parameters. The grid position with the lowest
resulting $\chi^{2}$ for the fit is R.A. 20h21m29s.683, Decl. $+$40°26′54.61″
(J2000). This new timing position is $10\arcsec$ away from the one reported in
Ray et al. (2011) which was based on 14 fewer months of data. Based on the
$\chi^{2}$ map over the grid, we estimate the 95% confidence region of the new
timing position to be an ellipse of dimensions 26″$\times$10″, as shown in
Figure 4. The separation between the position of source #20 and the refined
timing analysis position obtained here is $14.7\arcsec$ and source #20 lies
outside the 95% confidence region.
A precise evaluation of systematic timing errors is complicated by the
(erratic) timing behavior of the pulsar itself and the relatively short data
span with respect to the 1-year modulation that is introduced by an incorrect
position. That is, the position can be perturbed by any component of the
timing noise that appears to be a 1-year sinusoid. The magnitude of this
effect is difficult to estimate because we have just one realization of the
stochastic timing noise process in our data. To see the potential
contribution, we have plotted an estimate of the timing noise contributions in
Figure 5. Here, we assume that the parameters $f$ and $\dot{f}$ are dominated
by the secular spindown of the pulsar, while any higher order frequency
derivatives and all of the WAVE parameters are dominated by timing noise.
Clearly, this is a red-noise stochastic process. Note that a position error of
10$\arcsec$ will introduce a sinusoidal term with a 1-year period and an
amplitude of 24 ms. To get another estimate of the systematic error, we fit a
timing model including the pulsar position to three overlapping segments of
data, the first half of our data span, the middle half, and the last half.
These three fitted positions are separated by 7–9$\arcsec$, giving another
estimate of the systematic error resulting from timing noise.
Based on these considerations, we adopt $10\arcsec$ as an estimate of the
systematic error in the timing position. Combining, in quadrature, this
systematic error estimate with the error estimated in Ray et al. (2011)
results in the smaller, 10.3″ radius, circular 95% confidence region depicted
in Figure 4. Source #20 lies within this region.
In summary, when using pulsar timing to derive a position at the few arcsec
level it is important to allow for low frequency (year timescale) timing
noise. We have done this two ways: First, by adding WAVE terms to the solution
and allowing these terms to vary when deriving an error in our grid search.
Second, by time slicing the data and looking at how the derived position
changes. As seen in Figure 4, each of these methods gave similar sized error
regions that overlap.
## 4 Search for an Optical Counterpart to the X-Ray Source
As reported in Weisskopf et al (2006) and repeated in §5.3, there are no
cataloged optical counterparts for source #20. This is not surprising as, with
the exception of the $m\approx 16$ Crab, most optically detected pulsars have
magnitudes $\geq 25$. In addition the field of PSR J2021+4026 is crowded and
optical observations are further hampered by the presence of the 8th magnitude
star BD+39 4152 (=V405 Cyg) one arcmin away to the east.
We present here observations of the field taken on October 31, 2008 with the
OPTIC orthogonal frame-transfer camera on the Kit Peak National Observatory,
3.6m, Wisconsin, Yale, Indiana, & NOAO (hence WIYN) telescope. The OPTIC
camera, with a 10′ field and plate scale 0.141′′/pixel, allows improved image
quality through “Orthogonal Transfer” (OT) rapid electronic guiding following
motions of a reference star (Tonry et al. 2004). We used BD+39 4152 itself as
the guide star and were able to correct at 50 Hz, collecting $3\times 180$ s
dithered exposures in $r^{\prime}$ and $i^{\prime}$. These frames were subject
to standard calibrations, except for the flat field frames which were
assembled by applying image shifts matching those of the OT guiding during the
individual science exposures. The resulting image stacks have final PSF widths
of 0.87′′ ($r^{\prime}$) and 0.62′′ ($i^{\prime}$) near the guide star; the
PSF width increases by $\sim 30$% toward the edge of the frame. We estimate
that the frame is aligned to the Chandra coordinates with
$<0.2^{\prime\prime}$ precision.
Figure 6 shows a portion of the OPTIC frames in $r^{\prime}$ and $i^{\prime}$
centered near the position of source #20. Note the secondary reflections of
BD+39 4152 and the strong scattering background, especially in $i^{\prime}$.
Magnitudes were corrected to the SDSS photometry scale using observations of
the calibration star Ru 149F (Smith et al. 2002). We measured the fluxes of
the faintest detectable stellar sources in the vicinity of our target position
and used these to estimate upper limits ($\sim 95$% confidence) on the
undetected optical flux for source #20 of $i^{\prime}>23.0$ mag (the
sensitivity is severely limited by scattered flux) and $r^{\prime}>25.2$ mag.
Some of the diffuse emission toward the right (west) of the $r^{\prime}$ image
is part of larger scale filamentary structure visible over several arcminutes.
This is likely H$\alpha$/[NII] associated with the $\gamma$ Cygni SNR itself.
We note that this remnant has been poorly studied in the optical. Mavromatakis
(2003) described extensive diffuse line emission over a $\sim 1^{\circ}$
region, but found little filamentary emission and was not able to detect the
very faint filaments seen in our data. No corresponding X-ray structure is
seen, supporting the claim in Mavromatakis (2003) that the $\gamma$ Cygni
remnant is dominated by low velocity shocks. Deep narrow band imaging to trace
this structure could be useful in testing the connection, if any, between PSR
J2021+4026 and the $\gamma$ Cygni SNR.
## 5 The Other 43 X-Ray Sources in the Field
### 5.1 Spectral Analysis
There is a drop in the Chandra energy response above the mirror coating’s
iridium-M edges ($\approx\\!2$ keV), thus any source with a substantial
fraction of its detected photons above 2 keV is indicative of a very hard
spectrum. Figure 7 shows the X-ray color-color diagram for the 21 sources that
have more than 15 source counts. The diagram comprises 3 bands: S (soft)
covering 0.5 to 1.0 keV; M (medium) covering 1.0 to 2.0 and H (hard) 2 to 8
keV with T (total) simply the sum of S, M and H. The color ratios that
comprise Figure 7 are given in columns 8 & 9 of Table 1. The X coordinate in
Figure 7 (H-S)/T measures how hard the spectrum is and the y coordinate (M/T)
measures how centrally peaked the spectrum is. Positive source counts requires
data points to be inside the triangle, but background subtraction causes a few
to appear slightly outside. Sources with (H-S)/T greater than 0.5 are
spectrally very hard and are likely background AGNs shining through the
galactic plane. Only one of this group of sources, #42, has a cataloged
optical counterpart (Table 5). Sources with negative values of (H-S)/T likely
have thermal spectra and are plausibly lightly absorbed foreground stars. Note
that source #20 has the highest fraction of counts in the 1-2 keV band. Two
X-ray sources, other than source #20, have sufficient counts to warrant a
spectral analysis: #3, the brightest in the field; and #32.
#### 5.1.1 The X-Ray Spectrum of Source #3
We first fit the source #3 data using absorbed powerlaw, bbodyrad, and mekal
models. None of these models provided acceptable fits to the data, $\chi^{2}$
being 48.3, 57.2, 70.8, respectively, for 32 degrees of freedom. The only two-
component+ model that provided an acceptable fit was a two-temperature mekal
model with $\chi^{2}$ of 23.4 for 30 degrees of freedom, further indicating
that this is a foreground star. The results of our spectral analysis,
following the procedures discussed in §2.2.1, are in Table 4.
#### 5.1.2 The X-Ray Spectrum of Source #32
The data for source #32 are well-fit by an absorbed powerlaw model ($\chi^{2}$
of 15.7 for 19 degrees of freedom), but, as with source #20, the powerlaw
index is very steep being 3.4. In this case, however, neither a single-
temperature mekal model nor a bbodyrad model provide as compelling fits
($\chi^{2}$ of 46.9 & 23.9 respectively), although the latter is statistically
acceptable. There is simply too much uncertainty to firmly classify this
source on the basis of the X-ray spectroscopy alone.
### 5.2 Temporal Variability
The general paucity of counts also precludes a sensitive time-variability
analysis for almost all these sources. Nonetheless, one of the three X-ray-
brightest sources, #32, shows evidence for a significant temporal variation,
suggestive of stellar coronal emission. The existence of both a likely 2MASS
and USNO candidate counterpart (Table 5) reinforces this interpretation in
which case simple spectral models might not be expected to fit the data, as we
have seen (§5.1.2)
### 5.3 Candidate Catalog Optical and Near-Infrared Counterparts
We searched for candidate optical counterparts to the detected X-ray sources.
We used HEASARC’s BROWSE444See http://heasarc.gsfc.nasa.gov/db-
perl/W3Browse/w3browse.pl. feature to search for cataloged objects within the
99%-confidence radius ($\epsilon_{99}$) of X-ray source positions in Table 1.
Table 5 tabulates results of a cross correlation of the X-ray positions of the
Chandra-detected sources (column 1) with optical sources (columns 2–7) in the
USNO-B1.0 catalog (Monet et al. 2003) and with near-infrared sources (columns
8–12) in the 2MASS catalog (Skrutskie et al. 2006).
#### 5.3.1 USNO-B1.0
For fourteen (14) X-ray sources, we found a USNO-B1 (optical) source within
the 99%-confidence radius $\epsilon_{99}$ of the Chandra position (Table 1).
Table 5 columns 2–4 list, respectively, the USNO-B1 right ascension RA,
declination Dec, and RMS positional error $\sigma_{o}$ in the form
($\sigma_{o}({\rm RA})$, $\sigma_{o}({\rm Dec})$). Column 5 gives the angular
separation $\delta_{ox}$ between optical and X-ray positions; column 6, the
I-band magnitude. Column 7 estimates the probability $p_{o}(\delta_{ox},{\rm
I})$ for a chance coincidence within the observed separation of an object as
bright or brighter than the I magnitude of the optical candidate. We
determined this probability from the I-magnitude distribution of the 893 USNO
sources within $6\arcmin$ (slightly larger than the 8$\times$8′ Chandra field
of view) of the X-ray pointing direction. We designate a potential optical
counterpart to an X-ray source as a ‘strong candidate’ only if the sample
impurity—i.e., probability of chance coincidence—$p_{o}(\delta_{ox},{\rm
I})<1\%$. All the candidate USNO-B1 sources satisfy this criterion.
#### 5.3.2 2MASS
For nineteen (19) X-ray sources, we found a 2MASS (near-infrared) source
within the 99%-confidence radius $\epsilon_{99}$ of the Chandra position
(Table 1). Table 5 columns 8 and 9 list, respectively, the 2MASS right
ascension RA and declination Dec, each with an RMS positional error
$\sigma_{i}\approx 0\farcs 080$. Column 10 gives the angular separation
$\delta_{ix}$ between near-infrared and X-ray positions; column 11, the Ks-
band magnitude. Column 12 estimates the probability $p_{i}(\delta_{ix},{\rm
K}_{s})$ for a chance coincidence within the observed separation of an object
as bright or brighter than the Ks magnitude of the infrared candidate. We
determined this probability from the $K_{s}$-magnitude distribution of the
1188 2MASS sources within $6\arcmin$ of the X-ray pointing direction. We
designate a potential near-infrared counterpart to an X-ray source as a
‘strong candidate’ only if the sample impurity—i.e., probability of chance
coincidence—$p_{i}(\delta_{ix},{\rm K}_{s})<1\%$. Eighteen (18) sources
satisfy this criterion, the exception being source #44. Note that the 2MASS
set of 18 strong candidate counterparts includes 13 of the 14 USNO-B1 set of
strong candidates (§5.3.1).
Table 6 tabulates the 2MASS near-infrared photometry (columns 2–7) of the 18
strong-candidate optical (visible–near-infrared) counterparts to Chandra-
detected X-ray sources. Examination of the near-infrared color–color diagram
for all 2MASS sources within 6′ of the pointing direction indicates those that
are strong-candidate counterparts to X-ray sources are distributed as the
field sources—i.e., as reddened main-sequence stars. Although most Galactic-
plane 2MASS objects are normal stars, the objects identified with the X-ray
sources need not be normal stars: For example, the X-ray emission may
originate in an accreting compact companion. Figure 8 shows the X-ray hardness
ratio of the 9 brighter X-ray sources versus the infrared color J-H of the
corresponding strong 2MASS counterparts. The x-ray-soft and bluer infrared
sources in the lower left corner of the figure are likely foreground stars.
The x-ray-hard and reddened sources in the upper right hand corner may be
X-ray binaries and/or AGN.
#### 5.3.3 WIYN observations
The images we describe in Section 4 allow us to measure or limit the optical
magnitudes of several other X-rays sources. A few sources were not covered in
all sub-frames of the image stack and so were measured from individual
exposures. Three additional optical counterparts to the X-ray sources were
detected in the low exposure guide sector allowing improved frame
registration. Table 7 gives the detected magnitudes and upper limits.
## 6 Further Discussion and Summary
Using the Chandra X-ray Observatory, we continued our search (Becker et al.
2004, Weisskopf et al. 2006) for possible X-ray counterparts to the intriguing
$\gamma$-ray source now known as PSR J2021+4026. We found 44 X-ray sources in
a field centered on the PSR J2021+4026 position, located along the line-of-
sight toward the $\gamma$-Cyg SNR. Only one of these sources, #20, can
reasonably and with high confidence be taken as the X-ray counterpart to the
$\gamma$-ray source.
There are a number of reasons supporting this conclusion. First and foremost,
our X-ray source #20, is only 14.7″ distant from the best-fitting $\gamma$-ray
timing position. In addition, this separation is within the combined
statistical and systematic errors on that position. There are also no other
X-ray sources detected within 66″ of the $\gamma$-ray source position to a
Chandra source-detection limit of $\sim$10-15 erg cm-2 s-1 in the 0.5$-$8.0
keV bandpass making it highly unlikely that any of the other X-ray sources in
the field are candidate counterparts. Furthermore, the spectrum of source #20
has a shape consistent with soft ($\log(T_{\infty})\sim 6.0$ to $6.5$) thermal
emission as expected from a young neutron star though perhaps somewhat higher
than expected from the spin-down age of 77000 yr estimated for PSR J2021+4026.
There is also a hint of extended diffuse X-ray emission in the vicinity of
source #20 that may be an associated PWN.
With source #20 as the counterpart, we infer $F_{\gamma}/F_{\rm X}\sim
1.1\times 10^{4}$, not atypical of young isolated neutron stars (e.g., Becker
2009). If source #20 is not the X-ray counterpart, the flux ratio is at least
30$\times$ larger, which would be substantially larger than the observed ratio
for other $\gamma$-ray pulsars.
A similar argument using the optical data also supports source #20 as the
counterpart: our r${}^{\prime}>25.2$ limit implies a lower limit of $F_{\rm
X}/F_{\rm V}\approx 250$, with some uncertainty due to extinction. This is
already larger than the maximum value for X-ray binaries ($\sim$15) or BL Lacs
($\sim$100) and is approaching typical values for isolated neutron stars
($\sim 10^{3-4}$) (e.g. Schwope et al. 1999). Thus, based on the X-ray/optical
evidence alone, source #20 is likely an isolated neutron star and is the
likely counterpart for PSR J2021+4026.
Finally, the X-ray spectrum has a shape consistent with the soft thermal
emission expected from a young neutron star. This emission likely represents a
fraction of the stellar surface heated by back-flowing particles generated by
magnetospheric activity (eg. Harding & Muslimov 2001). At present the fitted
parameters suggest that this thermal component implies a relatively large
distance, $\approx 6$ kpc, incompatible with an association with SNR G78+2.1.
However, the fits also indicate a complex spectrum with at least two
components; much higher S/N data will be needed to extract strong spectral
constraints. Of course, a heated polar cap suggests that sensitive
observations should also be able to detect X-ray pulsations at the 265 ms spin
period, the definitive test of the counterpart’s association.
The work of MCW, DAS, RFE, SLO, and AFT is supported by the Chandra Program.
The Chandra data was obtained in response to proposal number 11500575 by the
Chandra X-ray Observatory Center, which is operated by the Smithsonian
Astrophysical Observatory for and on behalf of the National Aeronautics Space
Administration under contract NAS8-03060. Support for this work was also
provided to PSP in response to this proposal through Chandra Award Number
GO0-11086A issued by the Chandra X-ray Observatory Center. The work of RWR was
supported in part by NASA grant NNX08AW30G. The Fermi LAT Collaboration
acknowledges generous ongoing support from a number of agencies and institutes
that have supported both the development and the operation of the LAT as well
as scientific data analysis. These include the National Aeronautics and Space
Administration and the Department of Energy in the United States, the
Commissariat à l’Energie Atomique and the Centre National de la Recherche
Scientifique / Institut National de Physique Nucléaire et de Physique des
Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale
di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports,
Science and Technology (MEXT), High Energy Accelerator Research Organization
(KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A.
Wallenberg Foundation, the Swedish Research Council and the Swedish National
Space Board in Sweden. Additional support for science analysis during the
operations phase is gratefully acknowledged from the Istituto Nazionale di
Astrofisica in Italy and the Centre National d’Études Spatiales in France. Our
analyses utilized software tools provided by the Chandra X-ray Center (CXC) in
the application package CIAO and from the High-Energy Astrophysics Science
Archive Research Center (HEASARC, operated by the NASA Goddard Space Flight
Center, Greenbelt MD, and by the Smithsonian Astrophysical Observatory,
Cambridge MA). We also thank Mike Wolff for a careful critique of the
manuscript.
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Table 1: Chandra X-ray sources on ACIS-S3 in the Fermi-LAT PSR J021+4026 field. Source | RA(J2000) | Dec(J2000) | $\theta_{\rm ext}$ | $C_{x}$ | $\sigma_{x}$ | $\epsilon_{99}$ | $(H-S)/T$ | $M/T$
---|---|---|---|---|---|---|---|---
| ${}^{\rm h}\>\ {}^{\rm m}\ \ {}^{\rm s}\>$ | $\ \arcdeg\>\ \ \arcmin\>\ \ \arcsec$ | $\arcsec$ | | $\arcsec$ | $\arcsec$ | |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
1 | 20 21 9.043 | 40 27 3.28 | 1.8 | 17 | 0.40 | 0.81 | 0.14 $\pm$ 0.20 | 0.66 $\pm$ 0.13
2 | 20 21 10.472 | 40 28 44.47 | 3.3 | 12 | 0.64 | 1.28 | |
3 | 20 21 11.429 | 40 28 4.87 | 1.9 | 497 | 0.31 | 0.61 | -0.41 $\pm$ 0.03 | 0.47 $\pm$ 0.02
4 | 20 21 12.731 | 40 28 31.91 | 1.4 | 14 | 0.38 | 0.76 | 0.10 $\pm$ 0.25 | 0.41 $\pm$ 0.10
5 | 20 21 13.315 | 40 26 3.13 | 0.9 | 10 | 0.35 | 0.70 | |
6 | 20 21 13.576 | 40 25 55.80 | 1.4 | 25 | 0.34 | 0.69 | 0.18 $\pm$ 0.13 | 0.67 $\pm$ 0.09
7 | 20 21 13.665 | 40 28 59.60 | 1.7 | 9 | 0.46 | 0.91 | |
8 | 20 21 14.337 | 40 25 20.37 | 0.9 | 7 | 0.36 | 0.72 | |
9 | 20 21 16.969 | 40 25 17.18 | 1.4 | 18 | 0.36 | 0.72 | -0.37 $\pm$ 0.23 | 0.13 $\pm$ 0.06
10 | 20 21 19.524 | 40 25 32.86 | 1.4 | 12 | 0.39 | 0.78 | |
11 | 20 21 20.694 | 40 24 1.86 | 1.3 | 13 | 0.37 | 0.74 | |
12 | 20 21 21.132 | 40 27 46.35 | 0.8 | 11 | 0.34 | 0.68 | |
13 | 20 21 22.317 | 40 28 50.75 | 3.7 | 12 | 0.71 | 1.43 | |
14 | 20 21 25.164 | 40 28 13.34 | 1.3 | 33 | 0.33 | 0.67 | 0.90 $\pm$ 0.08 | 0.04 $\pm$ 0.04
15 | 20 21 25.750 | 40 27 44.10 | 1.1 | 7 | 0.39 | 0.78 | |
16 | 20 21 26.087 | 40 23 5.22 | 1.0 | 7 | 0.38 | 0.76 | |
17 | 20 21 28.658 | 40 24 15.88 | 1.0 | 8 | 0.37 | 0.74 | |
18 | 20 21 29.773 | 40 24 55.09 | 0.8 | 11 | 0.33 | 0.67 | |
19 | 20 21 30.342 | 40 29 48.31 | 3.2 | 52 | 0.40 | 0.81 | 0.78 $\pm$ 0.10 | 0.18 $\pm$ 0.07
20 | 20 21 30.733 | 40 26 46.04 | 1.3 | 281 | 0.31 | 0.61 | 0.02 $\pm$ 0.03 | 0.75 $\pm$ 0.02
21 | 20 21 30.801 | 40 25 16.38 | 1.3 | 20 | 0.35 | 0.70 | 0.9 $\pm$ 0.13 | 0.09 $\pm$ 0.08
22 | 20 21 31.385 | 40 22 56.47 | 1.4 | 26 | 0.34 | 0.69 | 0.14 $\pm$ 0.12 | 0.71 $\pm$ 0.09
23 | 20 21 31.889 | 40 24 25.51 | 1.4 | 11 | 0.39 | 0.79 | |
24 | 20 21 32.659 | 40 28 21.38 | 2.2 | 11 | 0.50 | 1.01 | |
25 | 20 21 32.905 | 40 24 20.88 | 0.8 | 14 | 0.33 | 0.66 | |
26 | 20 21 33.031 | 40 23 0.62 | 2.3 | 11 | 0.52 | 1.05 | |
27 | 20 21 33.650 | 40 29 8.97 | 2.8 | 48 | 0.39 | 0.78 | 0.80 $\pm$ 0.11 | 0.25 $\pm$ 0.09
28 | 20 21 34.097 | 40 25 26.50 | 1.3 | 12 | 0.38 | 0.76 | |
29 | 20 21 34.559 | 40 23 19.16 | 2.0 | 33 | 0.37 | 0.73 | 0.86 $\pm$ 0.10 | 0.09 $\pm$ 0.07
30 | 20 21 35.268 | 40 28 35.85 | 2.0 | 16 | 0.43 | 0.87 | |
31 | 20 21 35.485 | 40 28 13.57 | 1.9 | 6 | 0.55 | 1.10 | |
32 | 20 21 37.579 | 40 29 58.09 | 3.6 | 248 | 0.33 | 0.67 | -0.21 $\pm$ 0.05 | 0.55 $\pm$ 0.03
33 | 20 21 38.401 | 40 29 35.49 | 2.8 | 53 | 0.38 | 0.76 | 0.74 $\pm$ 0.11 | 0.22 $\pm$ 0.08
34 | 20 21 38.431 | 40 24 42.77 | 1.7 | 94 | 0.32 | 0.64 | -0.45 $\pm$ 0.07 | 0.43 $\pm$ 0.05
35 | 20 21 38.579 | 40 24 14.88 | 1.3 | 7 | 0.43 | 0.86 | |
36 | 20 21 39.214 | 40 27 9.90 | 1.0 | 9 | 0.36 | 0.73 | |
37 | 20 21 40.083 | 40 24 9.43 | 2.5 | 11 | 0.55 | 1.11 | |
38 | 20 21 43.107 | 40 23 53.76 | 2.6 | 48 | 0.38 | 0.76 | -0.08 $\pm$ 0.10 | 0.72 $\pm$ 0.07
39 | 20 21 44.548 | 40 29 34.34 | 3.3 | 19 | 0.55 | 1.10 | |
40 | 20 21 47.294 | 40 24 54.84 | 3.2 | 28 | 0.47 | 0.95 | 0.99 $\pm$ 0.16 | 0.01 $\pm$ 0.12
41 | 20 21 47.584 | 40 26 57.50 | 1.9 | 18 | 0.41 | 0.82 | -0.53 $\pm$ 0.44 | 0.20 $\pm$ 0.13
42 | 20 21 52.529 | 40 25 7.72 | 4.2 | 66 | 0.44 | 0.87 | 0.79 $\pm$ 0.09 | 0.16 $\pm$ 0.07
43 | 20 21 57.170 | 40 26 24.36 | 7.8 | 29 | 0.93 | 1.87 | 0.94 $\pm$ 0.23 | 0.15 $\pm$ 0.18
44 | 20 21 57.751 | 40 26 46.78 | 5.8 | 105 | 0.46 | 0.92 | -0.42 $\pm$ 0.12 | 0.31 $\pm$ 0.04
Table 2: Fits to single models with absorption.
parameter | powerlaw | bbodyrad | nsaa | nsmax-1260b | nsmax-130190b
---|---|---|---|---|---
$\rm n_{H}$ | 1.58 | 0.79 | 1.03 | 1.08 | 1.08
$\sigma_{nH}$ c | (-0.20,+0.22) | (-0.15,+0.18) | (-0.19,+0.21) | (-0.19,+0.21) | (-0.18,+0.21)
pl-index | 4.86 | | | |
$\sigma_{pl_{i}ndex}$ c | (-0.48,+0.55) | | | |
$\log(T_{\infty}$ | | 6.52d | 6.16 | 6.12 | 6.10
$\sigma_{\log(T_{\infty})}$ c | | (-0.05,+0.04) | (-0.08.+0.07) | (-0.08,+0.08) | (-0.08,+0.08)
norm | 1.06 $\times 10^{-4}$ | 0.80 | 1.8$\times 10^{-9}$ | 0.56 | 0.83
$\sigma_{norm}$ c | (-0.32,+0.51)$\times 10^{-4}$ | (-0.37,+0.83) | (-1.2,+4.5)$\times 10^{-9}$ | (-0.38.+1.44) | (-0.19,+0.21)
D(kpc)e | | 174. | 23 | 16 | 13
$\sigma_{D}$ c | | (-52,+63) | (-4,+17) | (-8,+12) | (-1.4,+1.8)
Rem(km)f | | 0.1 | 0.9 | 1.2 | 1.5
$\chi^{2}$ | 23.3 | 30.6 | 29.8 | 28.5 | 27.7
degrees of freedom | 22 | 22 | 22 | 22 | 22
Flux $\times 10^{14}$ (${\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$) | 43 | 4.9 | 8.0 | 8.9 | 2.0
Note. —
a The mass, radius, and magnetic field were fixed at $1.358M_{\odot}$,
12.996km, and $1.0\times 10^{13}$ gauss respectively.
b The mass and radius of the neutron star were chosen as for the nsa model so
that the input, the gravitational redshift 1+z, was fixed at 1.15.
c Uncertainties based on considering only 1 interesting parameter, i.e. the
bounds indicated by the minimum $\chi^{2}$$+1$.
d Assumes M/R=1.358/12.996 $M_{\odot}$/km.
e Derived from the different normalizations assuming a NS mass and radius
$1.358M_{\odot}$, 12.996km.
f Rough estimate of the size of the emitting region by scaling the distance to
1.5 kpc.
Table 3: Fits to dual models with absorption.
parameter | bbodyrad | nsaa | nsmax-1260b | nsmax-130190b
---|---|---|---|---
$\chi^{2}$ | 16.38 | 16.24 | 16.36 | 16.38
degrees of freedom | 20 | 20 | 20 | 20
f statistic | 4.23 | 8.34 | 7.03 | 6.93
probability | 0.029 | 0.0023 | 0.0049 | 0.0052
$\rm n_{H}$ | 0.76 | 0.97 | 0.98 | 0.98
$\sigma_{nH}$ e | (-0.16,+0.19) | (-0.18,+0.22) | (-0.19,+0.23) | (-0.18,+0.23)
$\log(T_{\infty}$ | 6.41 | 6.01 | 5.98 | 6.00
$\sigma_{\log(T_{\infty}}$ e | (-0.08,+0.07) | (-0.12.+0.11) | (-0.13,+0.11) | (-0.14,+0.10)
Dd(kpc) | 91. | 6.8 | 6.0 | 5.7
$\sigma_{D}$ e | (-15,+39) | (-4.7,+9.5) | (-4.3,+9.4) | (-4.4,+6.2)
Rem(km)g | 0.2 | 2.8 | 3.3 | 3.4
$\Gamma$ | 1.2 | 1.10 | 0.73 | 0.73
$\sigma_{\Gamma}$ e | (-1.5,+1.2) | (-1.6,+1.3) | (-2.10,+1.44) | (-2.2,+1,5)
PL norm | 1.5 | 1.26 | 0.69 | 0.68
$\sigma_{norm}\times 10^{6}$ e | (-1.5,+6.0) | (-1.13,+5.88) | (-0.57,+1.05) | (-0.56,+7.02)
Fluxf $\times 10^{14}$ (${\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$) | 8.4 | 15.8 | 16.3 | 16.8
Note. —
a The mass, radius, and magnetic field were fixed at $1.358\rm M_{\odot}$,
12.996km, and $1.0\times 10^{13}$ gauss respectively.
b The mass and radius of the neutron star were chosen as for the nsa model so
that the gravitational redshift 1+z was fixed at 1.15.
d Derived from the different normalizations assuming, where necessary, a NS
mass and radius of $1.358\rm M_{\odot}$, 12.996km respectively.
e Uncertainties based on considering only 1 interesting parameter, i.e. the
bounds indicated by the value of the parameter in question at the minimum
$\chi^{2}$$+1$.
f Unabsorbed flux.
g Rough estimate of the size of the emitting region by scaling the distance to
1.5 kpc.
Table 4: Fit to a two-temperature mekal model for source #3.
$\rm n_{H}$ | $kT_{1}$ | $kT_{2}$ | $norm_{1}$ | $norm_{2}$
---|---|---|---|---
($10^{22}cm^{-3}$) | (keV) | (keV) | $\times 10^{4}$ | $\times 10^{5}$
$0.44$ | $0.20$ | $1.03$ | $1.37$ | $2.36$
$(-0.13,+0.11)$ | $(-0.02,+0.04)$ | $(\pm 0.07)$ | $(-0.9.+1.9)$ | $(-0.32,+0.25)$
Note. —
Uncertainties based on considering only 1 interesting parameter, i.e. the
bounds indicated by the value of the parameter in question at the minimum
$\chi^{2}$$+1$.
Table 5: Candidate cataloged counterparts to X-ray sources in the PSR
J2021+4026 field.
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
---|---|---|---|---|---|---|---|---|---|---|---
| USNO (optical) candidate counterpart | 2MASS (infrared) candidate counterpart
X-ray Source | RA(J2000) | Dec(J2000) | $\sigma_{o}^{a}$ | $\delta_{ox}$ | I | $p_{o}(\delta_{ox},{\rm I})$ | RA(J2000)b | Dec(J2000)b | $\delta_{ix}$ | Ks | $p_{i}(\delta_{ix},{\rm K}_{s})$
| ${}^{\rm h}\>\ {}^{\rm m}\ \ {}^{\rm s}\>$ | $\ {}^{\circ}\>\ \ \arcmin\>\ \ \arcsec$ | $\arcsec$ | $\arcsec$ | mag | % | ${}^{\rm h}\>\ {}^{\rm m}\>\ \ {}^{\rm s}\>$ | $\ \arcdeg\>\ \ \arcmin\ \ \arcsec$ | $\arcsec$ | mag | %
1 | 20 21 09.036 | 40 27 03.51 | (0.04,0.07) | 0.24 | 18.58 | 0.36 | 20 21 09.035 | 40 27 03.30 | 0.12 | 13.625 | 0.18
3 | 20 21 11.433 | 40 28 05.01 | (0.04,0.10) | 0.18 | 13.15 | 0.07 | 20 21 11.424 | 40 28 04.71 | 0.18 | 11.155 | 0.01
5 | | | | | | | 20 21 13.324 | 40 26 02.91 | 0.24 | 14.501 | 0.27
6 | | | | | | | 20 21 13.596 | 40 25 55.80 | 0.24 | 14.175 | 0.21
8 | 20 21 14.343 | 40 25 20.63 | (0.04,0.03) | 0.24 | 14.91 | 0.12 | 20 21 14.348 | 40 25 20.37 | 0.12 | 12.064 | 0.04
9 | 20 21 06.967 | 40 25 17.44 | (0.04,0.04) | 0.24 | 13.71 | 0.10 | 20 21 16.967 | 40 25 17.17 | 0.06 | 12.559 | 0.06
10 | 20 21 19.557 | 40 25 33.02 | (0.04,0.03) | 0.42 | 18.19 | 0.30 | 20 21 19.549 | 40 25 32.59 | 0.36 | 13.869 | 0.20
11 | 20 21 20.702 | 40 24 02.14 | (0.04,0.07) | 0.30 | 14.58 | 0.12 | 20 21 20.700 | 40 24 01.97 | 0.12 | 12.249 | 0.05
15 | 20 21 25.749 | 40 27 44.39 | (0.04,0.08) | 0.30 | 17.21 | 0.22 | 20 21 25.767 | 40 27 44.06 | 0.18 | 13.722 | 0.18
17 | 20 21 28.657 | 40 24 15.97 | (0.04,0.02) | 0.12 | 19.07 | 0.35 | 20 21 28.679 | 40 24 15.50 | 0.42 | 15.083 | 0.44
18 | | | | | | | 20 21 29.779 | 40 24 55.53 | 0.42 | 15.002 | 0.35
22 | 20 21 31.425 | 40 22 56.71 | (0.04,0.16) | 0.54 | 17.47 | 0.18 | 20 21 31.422 | 40 22 56.41 | 0.42 | 13.188 | 0.09
32 | 20 21 37.566 | 40 29 58.32 | (0.04,0.07) | 0.30 | 14.54 | 0.09 | 20 21 37.575 | 40 29 57.90 | 0.18 | 13.302 | 0.09
34 | 20 21 38.410 | 40 24 42.97 | (0.04,0.08) | 0.30 | 13.11 | 0.08 | 20 21 38.427 | 40 24 42.63 | 0.12 | 11.476 | 0.02
37 | | | | | | | 20 21 40.056 | 40 24 09.63 | 0.36 | 14.465 | 0.65
38 | 20 21 43.117 | 40 23 54.22 | (0.04,0.08) | 0.48 | 13.68 | 0.11 | 20 21 43.116 | 40 23 54.05 | 0.30 | 11.967 | 0.04
39 | 20 21 44.515 | 40 29 34.03 | (0.04,0.03) | 0.48 | 18.00 | 0.55 | 20 21 44.538 | 40 29 33.44 | 0.90 | 14.003 | 0.45
41 | 20 21 47.628 | 40 26 57.94 | (0.04,0.32) | 0.66 | 17.33 | 0.24 | | | | |
42 | | | | | | | 20 21 52.555 | 40 25 08.16 | 0.54 | 13.863 | 0.25
43 | | | | | | | 20 21 57.240 | 40 26 23.10 | 1.50 | 14.723 | 2.28
Note. — a USNO RMS positional uncertainty in each axis (RA, Dec)
b 2MASS RMS positional uncertainty $\sigma_{i}=0\farcs 08$ per axis
Table 6: 2MASS infrared photometry of strong-candidate counterparts to X-ray sources. (1) | (2) | (3) | (4)
---|---|---|---
X-ray Source | J | J$-$H | H$-$Ks
1 | $15.583\pm 0.043$ | $1.436\pm 0.068$ | $0.522\pm 0.073$
3 | $11.927\pm 0.018$ | $0.675\pm 0.028$ | $0.097\pm 0.027$
5 | $16.031\pm 0.068$ | $1.285\pm 0.107$ | $0.245\pm 0.133$
6 | $15.772\pm 0.056$ | $1.195\pm 0.087$ | $0.402\pm 0.099$
8 | $13.113\pm 0.021$ | $0.698\pm 0.030$ | $0.351\pm 0.030$
9 | $13.045\pm 0.021$ | $0.387\pm 0.030$ | $0.099\pm 0.032$
10 | $15.278\pm 0.040$ | $1.177\pm 0.059$ | $0.232\pm 0.073$
11 | $13.154\pm 0.022$ | $0.713\pm 0.033$ | $0.192\pm 0.034$
15 | $15.133\pm 0.038$ | $0.991\pm 0.064$ | $0.420\pm 0.074$
17 | $16.021\pm 0.089$ | $0.945\pm 0.121$ | $-0.007\pm 0.190$
18 | $16.046\pm 0.136$ | $0.442\pm 0.164$ | $0.602\pm 0.180$
22 | $14.761\pm 0.027$ | $1.238\pm 0.041$ | $0.335\pm 0.047$
32 | $13.887\pm 0.021$ | $0.534\pm 0.031$ | $0.051\pm 0.043$
34 | $12.045\pm 0.021$ | $0.517\pm 0.030$ | $0.052\pm 0.028$
37 | $15.822\pm 0.064$ | $1.119\pm 0.093$ | $0.238\pm 0.124$
38 | $12.582\pm 0.021$ | $0.480\pm 0.030$ | $0.135\pm 0.028$
39 | $15.477\pm 0.108$ | $0.974\pm 0.129$ | $0.500\pm 0.123$
42 | $15.995\pm 0.062$ | $1.307\pm 0.097$ | $0.825\pm 0.101$
43 | $16.249\pm 0.075$ | $1.280\pm 0.128$ | $0.246\pm 0.160$
Table 7: WIYN Optical Magnitudes
Source | $i^{\prime}$ | $r^{\prime}$ | Source | $i^{\prime}$ | $r^{\prime}$ | Source | $i^{\prime}$ | $r^{\prime}$ | Source | $i^{\prime}$ | $r^{\prime}$
---|---|---|---|---|---|---|---|---|---|---|---
1 | 18.89 | 21.04 | 2 | $\sim$24.4 | $>^{a}$ | 3 | 15.52 | 15.77 | 4 | 17.96 | 19.95
5 | 19.30 | 21.36 | 6 | 19.02 | 19.02 | 7 | $>$ | $>$ | 8 | 15.63 | 17.07
9 | 15.15 | 15.50 | 10 | 18.39 | 20.39 | 11 | 15.63 | 15.75 | 12 | $>$ | $>$
13 | 22.40 | $>$ | 14 | $>$ | $>$ | 15 | 17.94 | 19.82 | 16 | $>$ | $>$
17 | 19.37 | 21.83 | 18 | 18.13 | 20.15 | 19 | $\cdots$ | $\cdots$ | 21 | $>$ | $>$
22 | 17.82 | 19.94 | 23 | 20.92 | 23.29 | 24 | $\sim 24.4$ | $\sim 25.5$ | 25 | $>$ | $>$
26 | $>$ | $>$ | 27 | 22.45 | $\cdots$ | 28 | $\cdots$ | $\cdots$ | 29 | $>$ | $>$
aafootnotetext: Magnitude limits: $>$ = $>24.2$ ($i^{\prime}$), = $>25.3$
($r^{\prime}$)
Figure 1: The field showing the most recent Chandra observation, ObsID 11235.
For this figure a nearest neighbor smooth has been applied. The rotated square
shows the extent of the ACIS S3 chip, i.e., the region searched. The color bar
on the right shows the number of (smoothed) counts detected in a pixel during
the observation. The numbers on the left and bottom show the ACIS coordinates
in pixels. Sources are numbered in order of increasing right ascension. See
Table 1 for source X-ray properties.
Figure 2: The spectrum of source #20 with background subtracted and fit to a
nsa+powerlaw model. The dotted line in the upper panel is the nsa component,
the dashed line is the powerlaw component and the solid line is the two
components combined. The lower panel shows the contributions to $\chi^{2}$.
Figure 3: Significance of pulsation detection, as measured by the $\chi^{2}$
test, versus source number. The data were binned into 20 pulse phase bins and
fit to a constant flux model (no pulsations). The worse the fit (higher
$\chi^{2}$), the more likely the X-ray source is the $\gamma$-ray pulsar
counterpart.
Figure 4: Same as Fig. 1 but now just showing the region around source #20.
The large circle denotes the most recent LAT imaging position 95% confidence
error circle (2FGL, Abdo et al. 2011). The small ellipse is the 95% confidence
timing ellipse from the current work using WAVE terms to estimate the impact
of timing noise. The small circle is the 95% confidence region obtained by
combining the timing solution determined by Ray et al. (2011) added in
quadrature with our new (10″) estimate of the systematic error. See text for
further details.
Figure 5: Illustration of the magnitude of the timing noise observed in PSR
J2021+4026. The dotted (blue) curve shows the timing residuals that can be
attributed solely to timing noise. This is the sum of two components: The thin
upper (red) curve is the contribution from the polynomial terms $\ddot{f}$ and
$\dddot{f}$, neither of which can be attributed to the secular spin down of
the pulsar. The lower (green) curve is the combined sinusoidal WAVE components
(see text). Note that the pulsar period is 0.265 seconds, so timing noise
causes several complete phase wraps over this interval.
Figure 6: Portions of the WIYN 3.6m/OPTIC $r^{\prime}$ (left) and
$i^{\prime}$ (right) images of the field of PSR J2021+4026. Reflected light
and scattering from the 8th magnitude star BD+39 4152, used here as a guide
star and located to the left of each image, dominates the fields. The position
of Chandra source #20 is marked with a circle of radius 2 arcsec. There is no
detection visible in either band with upper limits as given in the text.
Figure 7: The X-ray color-color diagram for the 21 X-ray sources that have
more than 15 source counts. The Chandra band is divided into 3 bands with S
(soft) covering 0.5 to 1.0 keV, M (medium) covering 1.0 to 2.0 and H (hard) 2
to 8 keV. The Total, T, is the sum of S, M and H. The solid curves within the
triangle represent powerlaw spectra with photon indices of 1, 2, 3 and 4. The
dotted curves correspond to lines of constant absorbing columns of (bottom to
top) $1\times 10^{20}$, $1\times 10^{21}$, $2\times 10^{21}$,$5\times
10^{21}$, and $1\times 10^{22}$, respectively. The triangle encloses the
physically-meaningful range of colors. See text for the discussion
Figure 8: X-Ray hardness ratio (H-S)/T defined in the caption to figure 7
versus the near infrared color J-H for X-ray sources with 15 or more counts
and likely 2MASS candidate counterparts.
|
arxiv-papers
| 2011-08-19T15:48:59 |
2024-09-04T02:49:21.655791
|
{
"license": "Public Domain",
"authors": "Martin C. Weisskopf, Roger W. Romani, Massimiliano Razzano, Andrea\n Belfiore, Pablo Saz Parkinson, Paul S. Ray, Matthew Kerr, Alice Harding,\n Douglas A. Swartz, Alberto Carraminana, Marcus Ziegler, Werner Becker, Andrea\n De Luca, Michael Dormody, David J. Thompson, Gottfried Kanbach, Ronald F.\n Elsner, Stephen L. O'Dell, Allyn F. Tennant",
"submitter": "Martin C. Weisskopf",
"url": "https://arxiv.org/abs/1108.3987"
}
|
1108.4173
|
# Vector dark energy and high-$z$ massive clusters
Edoardo Carlesi,1 Alexander Knebe,1 Gustavo Yepes,1 Stefan Gottlöber,3 Jose
Beltrán Jiménez,4,5 Antonio L. Maroto,2
1Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049,
Cantoblanco, Madrid, Spain
2Departamento de Física Teórica, Universidad Complutense de Madrid, 28040,
Madrid, Spain
3Leibniz Institut für Astrophysik, An der Sternwarte 16, 14482, Potsdam,
Germany
4Institute de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai E. Ansermet, 1211 Genève, Switzerland
5Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
E-mail: edoardo.carlesi@uam.es
(Accepted XXXX . Received XXXX; in original form XXXX)
###### Abstract
The detection of extremely massive clusters at $z>1$ such as SPT-CL
J0546-5345, SPT-CL J2106-5844, and XMMU J2235.3-2557 has been considered by
some authors as a challenge to the standard $\Lambda$CDM cosmology. In fact,
assuming Gaussian initial conditions, the theoretical expectation of detecting
such objects is as low as $\leq 1\%$. In this Letter we discuss the
probability of the existence of such objects in the light of the Vector Dark
Energy (VDE) paradigm, showing by means of a series of $N$-body simulations
that chances of detection are substantially enhanced in this non-standard
framework.
###### keywords:
methods:$N$-body simulations – galaxies: haloes – cosmology: theory – dark
matter
††pagerange: Vector dark energy and high-$z$ massive
clusters–References††pubyear: 2011
## 1 Introduction
Present day cosmology is still failing to explain satisfactorily the nature of
dark energy, which is supposed to dominate the energetic content of the
universe today and to be responsible for the current accelerated expansion. In
the standard $\Lambda$CDM model, this cosmic acceleration is generated by the
presence of a cosmological constant. However, the required value for that
constant turns out to be tiny when compared to the natural scale of gravity,
namely the Planck scale. Thus, the gravitational interaction would hence be
described by two dimensional constants differing by many orders of magnitude,
and this poses a problem of naturalness. This is the so-called “cosmological
constant problem” and it motivated to consider alternative explanations for
the current acceleration of the universe by either modifying the gravitational
interaction at large distances or introducing a new dynamical field.
Indeed, one of the main challenges of observational cosmology is exactly to
devise new tests which could help discriminating between the constant or
dynamic nature of dark energy. In this regard, several authors have recently
pointed out that the observation of extremely massive clusters at high
redshift, such as SPT-CL J2106-5844 (Foley et al. (2011), $z\simeq 1.18$,
$M_{200}=(1.27\pm 0.21)\times 10^{15}M_{\odot}$), SPT-CL J0546-5346 (Brodwin
et al. (2010), $z\simeq 1.07$, $M_{200}=(7.95\pm 0.92)\times
10^{14}M_{\odot}$), and XMMU J2235.3-2557 (Jee et al. (2009), $z\simeq 1.4$,
$M_{200}=(7.3\pm 1.3)\times 10^{14}M_{\odot}$) may represent a major
shortcoming of the $\Lambda$CDM paradigm, where the presence of such objects
should be in principle strongly disfavoured (see, for example, Baldi &
Pettorino, 2011; Mortonson et al., 2011).
While, on the one hand, this tension could be solved keeping the standard
scenario and relaxing the assumption of Gaussianity in the initial conditions
(as proposed in Hoyle et al. (2011) and Enqvist et al. (2011)), it could be as
well possible to use this observations as a constraint for different
cosmological models. In this work we look at the VDE model, where the role of
the dark energy is played by a cosmic vector field (Beltrán Jiménez & Maroto,
2008). By means of a series of $N$-body simulations, we study the large scale
clustering properties of this cosmology, computing the cumulative halo mass
functions at different redshifts and comparing them to the predictions of the
standard model. In this way, we are able to show that the VDE cosmology does
indeed predict a higher abundance of massive haloes at all redshifts, thus
enhancing the probability of observing such objects with respect to
$\Lambda$CDM.
## 2 Vector Dark Energy
The action of the vector dark energy model (see Beltrán Jiménez & Maroto
(2008)) can be written as:
$\displaystyle S=\int d^{4}x\sqrt{-g}\left[-\frac{R}{16\pi
G}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right.$
$\displaystyle\left.-\frac{1}{2}\left(\nabla_{\mu}A^{\mu}\right)^{2}+R_{\mu\nu}A^{\mu}A^{\nu}\right].$
(1)
where $R_{\mu\nu}$ is the Ricci tensor, $R=g^{\mu\nu}R_{\mu\nu}$ the scalar
curvature and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. This
action can be interpreted as the Maxwell term for a vector field supplemented
with a gauge-fixing term and an effective mass provided by the Ricci tensor.
It is interesting to note that the vector sector has no free parameters nor
potential terms, being $G$ the only dimensional constant of the theory.
For a homogeneous and isotropic universe described by the flat Friedmann-
Lemaître-Robertson-Walker metric:
$ds^{2}=dt^{2}-a(t)^{2}d\vec{x}^{2}$ (2)
we have $A_{\mu}=(A_{0}(t),0,0,0)$ so that the corresponding equations read:
$\ddot{A}_{0}+3H\dot{A}_{0}-3\left[2H^{2}+\dot{H}\right]A_{0}=0$ (3)
$H^{2}=\frac{8\pi G}{3}\left[\rho_{R}+\rho_{M}+\rho_{A}\right]$ (4)
with $H=\dot{a}/a$ the Hubble parameter and
$\rho_{A}=\frac{3}{2}H^{2}A_{0}^{2}+3HA_{0}\dot{A}_{0}-\frac{1}{2}\dot{A}_{0}^{2}$
(5)
the energy density associated to the vector field, while $\rho_{M}$ and
$\rho_{R}$ are the matter and radiation densities, respectively. During the
radiation and matter eras in which the dark energy contribution was
negligible, we can solve Eq. (3) with $H=p/t$, where $p=1/2$ for radiation and
$p=2/3$ for matter eras respectively, that is equivalent to assume that
$a\propto t^{p}$. In that case, the general solution is:
$A_{0}(t)=A_{0}^{+}t^{\alpha_{+}}+A_{0}^{-}t^{\alpha_{-}},$ (6)
with $A_{0}^{\pm}$ constants of integration and $\alpha_{\pm}=-(1\pm 1)/4$ in
the radiation era, and $\alpha_{\pm}=(-3\pm\sqrt{33})/6$ in the matter era.
After dark energy starts dominating, the equation of state abruptly falls
towards $w_{DE}\rightarrow-\infty$ as the Universe approaches $t_{end}$, and
the equation of state can cross the so-called phantom divide line (Nesseris &
Perivolaropoulos (2007)), so that we can have $w_{DE}(z=0)<-1$.
Using the growing mode solution in Eq. (6) we can obtain the evolution for the
energy density as:
$\rho_{A}=\rho_{A0}(1+z)^{\kappa},$ (7)
with $\kappa=4$ in the radiation era and $\kappa=(9-\sqrt{33})/2\simeq-1.63$
in the matter era. Thus, the energy density of the vector field scales like
radiation at early times so that the ratio $\rho_{A}/\rho_{R}$ is constant
during such a period. Moreover, the value of the vector field $A_{0}$ during
that era is also constant hence making the cosmological evolution insensitive
to the time at which we impose the initial conditions (as long as they are set
well inside the radiation dominated epoch). Also, the required constant values
of such quantities in order to fit observations are $\rho_{A}/\rho_{R}|_{\rm
early}\simeq 10^{-6}$ and $A_{0}^{\rm early}\simeq 10^{-4}M_{p}$ which can
arise naturally during the early universe, for instance, as quantum
fluctuations. Furthermore, they do not need the introduction of any unnatural
scale, thus, alleviating the naturalness or coincidence problem. On the other
hand, when the Universe enters the era of matter domination, $\rho_{A}$ starts
growing relative to $\rho_{M}$ eventually overcoming it at some point so that
the dark energy vector field becomes the dominant component.
Figure 1: Equation of state of the vector dark energy model for the best fit
to SNIa data, shown in the range of our simulations.
Once the present value of the Hubble parameter $H_{0}$ and the constant
$A_{0}^{early}$ during radiation (which fixes the total amount of matter
$\Omega_{M}$) are specified, the model is completely determined. In other
words, this model contains the same number of parameters as $\Lambda$CDM, i.e.
the minimum number of parameters of any cosmological model with dark energy.
Notice however, that in the VDE model the present value of the equation of
state parameter $w_{0}=-3.53$ is radically different from that of a
cosmological constant (cf. Fig. 1, where the redshift evolution of $\omega(z)$
is shown our the range of our simulations). Despite this fact, VDE is able to
simultaneously fit supernovae and CMB data with comparable goodness to
$\Lambda$CDM (Beltrán Jiménez & Maroto, 2008; Beltrán Jiménez et al., 2009).
## 3 The Data
### 3.1 Simulations
We wanted to estimate the probability of finding massive clusters at $z>1$ in
the VDE scenario compared to the $\Lambda$CDM one by means of CDM only
$N$-body simulations. For this purpose, we chose to use a suitably modified
version of the publicly available GADGET-2 tree-PM code (Springel, 2005),
which had to take into account the different expansion history that
characterizes the two cosmologies. In Table 1 we show the cosmological
parameters used in the different simulations. For the VDE model, we have used
the value of $\Omega_{M}$ provided by the best fit to SNIa data; the remaining
cosmological parameters have been obtained by a fit to the WMAP7 CMB data of
the model. $w_{0}$ denotes the present value of the equation of state
parameter of dark energy. For $\Lambda$CDM we used the Multidark Simulation
(Prada et al., 2011) cosmological parameters with a WMAP7 $\sigma_{8}$
normalization (Larson et al., 2011). In addition, we also simulated a
$\Lambda$CDM-vde model, which implements the VDE values for $\Omega_{M}$ and
$\sigma_{8}$ in an otherwise standard $\Lambda$CDM picture. Although this
model is certainly ruled out by current cosmological constraints, it provides
nonetheless an interesting case study that allows us to disentangle the
effects of these two cosmological parameters on structure formation in the VDE
model.
We chose to run a total of eight $512^{3}$ particles simulations summarized in
Table 2 and explained below:
* •
a VDE (and a $\Lambda$CDM started with the same seed for the phases of the
initial conditions) simulation in a 500 $h^{-1}\,{\rm Mpc}$ box,
* •
a second VDE (and again corresponding $\Lambda$CDM) simulation in a 1
$h^{-1}$Gpc box,
* •
two more VDE simulations with a different random seed, one in a 500
$h^{-1}\,{\rm Mpc}$ and one in a 1 $h^{-1}$Gpc box, as a check for the
influence of cosmic variance,
* •
two $\Lambda$CDM-vde simulations in a 500 $h^{-1}\,{\rm Mpc}$ and a 1000
$h^{-1}\,{\rm Mpc}$ box.
The full set of simulations will be presented and analyzed in more detail in
an upcoming companion paper; in this Letter, instead, we chose to focus on
some of them only in order to gather information on large scale structures and
massive cluster at high redshift, respectively, in the two cosmologies. To
this extent, the use of the same initial seed for generating the initial
conditions in the coupled $\Lambda$CDM vs. VDE simulations allows us to
directly compare the structures identified by the halo finder.
Table 1: Cosmological parameters for $\Lambda$CDM $\Lambda$CDM-vde and VDE. Model | $\Omega_{m}$ | $\Omega_{de}$ | $w_{0}$ | $\sigma_{8}$ | h
---|---|---|---|---|---
$\Lambda$CDM | 0.27 | 0.73 | -1 | 0.8 | 0.7
$\Lambda$CDM-vde | 0.388 | 0.612 | -1 | 0.83 | 0.7
VDE | 0.388 | 0.612 | -3.53 | 0.83 | 0.62
Table 2: $N$-body settings used for the GADGET-2 simulations, the two 500$h^{-1}\,{\rm Mpc}$ and the two 1$h^{-1}$Gpc have the same initial random seed and starting redshift $z_{\rm start}=60$ in order to allow for a direct comparison of the halo properties. The number of particles in each was fixed at $512^{3}$.The box size $B$ is given in $h^{-1}\,{\rm Mpc}$ and the particle mass in $h^{-1}{\rm{M_{\odot}}}$. Simulation | $B$ | $m_{p}$
---|---|---
$2\times$VDE-0.5 | 500 | $1.00\times 10^{11}$
$2\times$VDE-1 | 1000 | $8.02\times 10^{11}$
$\Lambda$CDM-0.5 | 500 | $6.95\times 10^{10}$
$\Lambda$CDM-1 | 1000 | $5.55\times 10^{11}$
$\Lambda$CDM-0.5vde | 500 | $1.00\times 10^{11}$
$\Lambda$CDM-1vde | 1000 | $8.02\times 10^{11}$
As a final remark, we underline here that the choice of the boxes was made in
order to allow the study of clustering on larger scales, without particular
emphasis on low mass objects, e.g. dark matter haloes with
$M<10^{14}$$h^{-1}{\rm{M_{\odot}}}$. This means that even though our halo
finder has been able to identify objects down to $\sim
10^{12}{{h^{-1}{\rm{M_{\odot}}}}}$ in the 500$h^{-1}\,{\rm Mpc}$ box and $\sim
10^{13}$ $h^{-1}{\rm{M_{\odot}}}$ in the 1$h^{-1}$Gpc one (which corresponds
to a lower limit of 20 particles, see below), we are not comparing the mass
spectrum at this far end. Therefore, since we are only interested in studying
the behaviour of the mass function of these models at the very high mass end,
in the following sections we will mostly refer to the $\Lambda$CDM-1,
$\Lambda$CDM-vde and VDE-1 simulations, where we have a larger statistics for
the supercluster scales.
### 3.2 Halo Finding
In order to identify halos in our simulation we have run the MPI+OpenMP hybrid
halo finder AHF described in detail in Knollmann & Knebe (2009). AHF is an
improvement of the MHF halo finder (Gill et al., 2004), which locates local
overdensities in an adaptively smoothed density field as prospective halo
centres. The local potential minima are computed for each of these density
peaks and the gravitationally bound particles are determined. Only peaks with
at least 20 bound particles are considered as haloes and retained for further
analysis, even though here we focus on the most massive objects only.
The mass of each halo is then computed via the equation
$M(r)=\Delta\rho_{c}4\pi r^{3}/3$ where we applied $\Delta=200$ as the
overdensity threshold. Using this relation, particular care has to be taken
when considering the definition of the critical density $\rho_{c}=3H^{2}/8\pi
G$ because it involves the Hubble parameter, that differs substantially at all
redshifts in the two models. This means that, identifying the halo masses, we
have to take into account the fact that the value of $\rho_{c}$ changes from
$\Lambda$CDM to VDE. This has been incorporated into and taken care of in the
latest version of AHF where $H_{VDE}(z)$ is being read in from a precomputed
table.
We would like to mention that we checked that the objects obtained by this
(virial) definition are in fact in equilibrium. To this extent we studied the
ratio between two times kinetic over potential energy $2T/|U|$ confirming that
at each redshift under investigation here this relation is equally well
fulfilled for the $\Lambda$CDM and – more importantly – the VDE simulations
(not presented here though). We therefore conclude that our adopted method to
define halo mass in the VDE model leads to unbiased results and yields objects
in equilibrium – as is the case for the $\Lambda$CDM haloes.
## 4 The Results
Figure 2: Mass functions (and their ratios) as computed for the VDE-1,
$\Lambda$CDM-1 and $\Lambda$CDM-1vde simulations at $z=1.4,1.2,1.1,$ and $0$.
These redshifts have been chosen in order to overlap with the aforementioned
observed massive clusters.
### 4.1 Mass Function
With the halo catalogues at our disposal, we computed the cumulative mass
functions $n(>M)$ at various redshifts. We show in Fig. 2 the results for the
1$h^{-1}$Gpc simulations at redshifts $z=1.4$, $z=1.2$, $z=1.1$ and $z=0$.
This plot is accompanied by Table 3 where we list the masses of the most
massive haloes found in each model and the redshifts under consideration.
We notice that the mass function for objecst with
$M>10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$ is several times larger in VDE than in
$\Lambda$CDM at all redshifts, i.e. the number of high mass haloes in this
non-standard cosmological model is significantly increased. In particular, at
this mass scale the VDE mass function is about three times larger at the
relevant redshifts $z=1.4,1.2,$ and $1.1$ – and even larger at today’s time.
In order to verify that this feature of the VDE model is not a simple
reflection of cosmic variance (which should affect in particular the high mass
end, where the statistics is small) we compared the results presented in Fig.
2 to the mass functions of the set of two additional simulations started from
a different random seed for the initial conditions confirming aforementioned
results.
An interesting remark we would like to add here, is that the physical mass
(obtained dividing by the corresponding $h$ values the values quoted in
$h^{-1}{\rm{M_{\odot}}}$units) of the largest haloes in the VDE-1 simulation
at $z=1.4$, $z=1.2$ and $z=1.1$ are perfectly compatible with the ones of the
above clusters referred to in the Introduction, whereas the corresponding
$\Lambda$CDM candidates are outside the $2\sigma$ compatibility level. And
again, similar massive clusters have also been found in the duplicate VDE-1
simulation with a different initial seed.
Table 3: The most massive halo found in the three 1$h^{-1}$Gpc simulations (in units of $10^{14}$$h^{-1}{\rm{M_{\odot}}}$) as a function of redshift. z | $\Lambda$CDM-1 | VDE-1 | $\Lambda$CDM-1vde
---|---|---|---
1.4 | 4.16 | 5.63 | 6.47
1.2 | 5.13 | 6.51 | 8.16
1.1 | 6.01 | 7.63 | 10.2
0 | 18.1 | 31.6 | 35.1
Comparing the $\Lambda$CDM-1vde to the VDE-1 simulation at different
redshifts, we note that while the two mass functions are almost
indistinguishable for $M<10^{14}\times{{h^{-1}{\rm{M_{\odot}}}}}$, on the
higher mass end the former even outnumbers the latter by approximately $\sim
3$. In the hierarchical picture of structure formation, we can attribute this
relative difference to dynamical effects caused by the different expansion
histories (based upon different $H(z)$) at later times $z\approx 1$, when the
most massive structures actually start to form. In general, however, the
$\Lambda$CDM-vde analysis shows that the enhancement seen in the VDE mass
function with respect to $\Lambda$CDM is clearly driven by the higher values
of $\Omega_{M}$ and $\sigma_{8}$, a result expected since the abundance of
clusters sensitively depends on the product of these two parameters (cf.
Huterer & White, 2002). On the one hand, this complicates the issue of model
selection, since (although disfavoured by the WMAP7 data) we could invoke a
(slightly) larger $\Omega_{M}$ or a higher $\sigma_{8}$ normalization at $z=0$
for $\Lambda$CDM trying to alleviate the current tension with the high-$z$
massive clusters observations. On the other hand, the distinct expansion
history that characterizes and differentiates between the two $\Lambda$CDM and
VDE models would still leave a clear imprint on structure formation, which
could be detected by, for instance, measuring $\sigma_{8}$’s dependence on
redshift. Such a test would indeed provide invaluable information for the
study of $\Lambda$CDM and for any cosmological model beyond it such as VDE.
### 4.2 Probability
Figure 3: Numerical cumulative number densities of objects with $M>5\times
10^{14}$$h^{-1}{\rm{M_{\odot}}}$ for VDE, $\Lambda$CDM and $\Lambda$CDM-vde.
In order to provide a more quantitative estimate of the the relative
probability of observationally detecting such massive clusters at the
indicated redshifts we used $n(>M,z)$ – the expected cumulative number density
of objects above a threshold mass $M$ as a function of redshift as given by
our simulations – and integrated it over the comoving volume $V_{c}$ of the
survey
$N(>M)=\int_{\Delta z,\Omega_{\rm survey}}n(>M,z)dV_{c}(z)$ (8)
where $\Delta z$ and $\Omega_{\rm survey}$ are the redshift interval and the
fraction of the sky covered by the survey to which we want to compare our
theoretical expectations.
While $n(>M,z)$ can be readily calculated in $\Lambda$CDM cosmologies (e.g.
Press & Schechter, 1974; Sheth & Tormen, 1999; Jenkins et al., 2001; Tinker et
al., 2008), in VDE we have to devise a strategy to compute it based upon our
numerical results only. We chose to adjust the formula of Sheth & Tormen
(1999) as follows:
* •
we calculated the cumulative number densities in the desired redshift
intervals $\Delta z$ based upon our simulation data,
* •
we adjusted the parameters of the Sheth-Tormen mass function fitting the
numerical cumulative number densities derived from the VDE-1 and VDE-0.5
simulations,
* •
we used these best fit estimates to analytically compute $n(>M,z)$ now having
access to masses even outside our numerically limited range to be used with
Eq. (8).
The results of the numerical integration over the comoving volumes (obtained
using the limits quoted in the observational papers by Jee et al. (2009),
Brodwin et al. (2010) and Foley et al. (2011)) are listed in Table 4 for the
VDE, $\Lambda$CDM-vde and $\Lambda$CDM model. We can clearly see that the
chances are substantially larger in the VDE model to find such massive objects
than in $\Lambda$CDM; the numbers for VDE are in fact comparable to our
fiducial $\Lambda$CDM-vde model confirming their relation to the enhanced
values of $\Omega_{M}$ and $\sigma_{8}$. However, note that while the VDE
model is compliant with both SN-Ia and CMB data the $\Lambda$CDM-vde is
obviously ruled out observationally. We complement these results with Fig. 3
where we plot the abundance evolution of clusters with mass $M>5\times
10^{14}$$h^{-1}{\rm{M_{\odot}}}$ computed utilizing above described procedure
again. This plot confirms our previous analysis of the mass functions and
shows that the expectation of massive objects is amplified in VDE by a factor
$\sim 3$ to $\sim 10$ over the considered redshift range, a factor which is
even higher for the discretionary $\Lambda$CDM-vde. We would like to remark
here that while our $\Lambda$CDM estimate for XMMU J2235.3-2557 is in
agreement with the result quoted by Jee et al. (2009) (obtained using the same
approach as here), the calculation done for SPT-CL J2106-5844 leads to an
estimate substantially smaller than the one quoted by Foley et al. (2011),
calculated using a Monte Carlo technique. However, this does not affect our
conclusions, which are based on the comparison of results obtained in a
consistent manner for the two models.
Table 4: Expected number of objects $N(>M)$ in excess of mass $M$ and inside a certain (comoving) volume in the $\Lambda$CDM and VDE for different mass thresholds and survey volumes. Solid angles $\Omega$ are measured in deg2 and masses are measured in $10^{14}$$h^{-1}{\rm{M_{\odot}}}$. $M$ | $\Delta z$ | $\Omega_{\rm survey}$ | $N_{\Lambda\rm CDM}$ | $N_{\rm VDE}$ | $N_{\Lambda\rm-VDE}$
---|---|---|---|---|---
$>10$ | $>1$ | 2500 | 0.007 | 0.02 | 0.04
$>7$ | $>1$ | 2500 | 0.03 | 0.31 | 0.56
$>5$ | $1.38-2.2$ | 11 | 0.005 | 0.06 | 0.07
## 5 Conclusions
The observation of massive clusters at $z>1$ provides an additional, useful
test for the $\Lambda$CDM and other cosmological models beyond the standard
paradigm. In this Letter we have shown that the Vector Dark Energy (VDE)
scenario (Beltrán Jiménez & Maroto, 2008) might account for such observations
better than the $\Lambda$CDM concordance model, since the relative abundance
of extremely massive clusters with $M>5\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$
is at all redshifts higher in this non-standard cosmology: the expected number
of massive clusters is enhanced in VDE by at least a factor of $\sim 3$ to
find an object such as SPT- CL J2106-5844 at redshift $z\approx 1.2$ (Foley et
al., 2011) and a factor of $\sim 10$ for the other two observed clusters SPT-
CL J0546-5346 (Brodwin et al., 2010) and XMMU J2235.3-2557 (Jee et al., 2009).
Of course, these results might as well simply point in the direction of
modifying the standard paradigm, for example including non-Gaussianities in
the initial conditions or either using a higher $\sigma_{8}$ or $\Omega_{M}$
value for the $\Lambda$CDM as the comparison to the $\Lambda$CDM-vde model
seems to suggest.
Nonetheless, this first results on the large scale clustering in the case of
VDE cosmology point in the right direction, significantly enhancing the
probability of producing extremely massive clusters at high redshift as recent
observations seem to require. For a more elaborate discussion and comparison
of the VDE to the $\Lambda$CDM model (not solely focusing on massive clusters)
we though refer the reader to the companion paper (Carlesi et al., in
preparation).
## Acknowledgements
EC is supported by the MareNostrum project funded by the Spanish Ministerio de
Ciencia e Innovacion (MICINN) under grant no. AYA2009-13875-C03-02 and
MultiDark Consolider project under grant CSD2009-00064. AK acknowledges
support by the MICINN’s Ramon y Cajal programme as well as the grants AYA
2009-13875-C03-02, AYA2009-12792-C03-03, CSD2009-00064, and CAM
S2009/ESP-1496. GY acknowledges support from MICINN’s grants
AYA2009-13875-C03-02 and FPA2009-08958. JBJ is supported by the Ministerio de
Educación under the postdoctoral contract EX2009-0305 and also wishes to
acknowledge support from the Norwegian Research Council under the YGGDRASIL
programme 2009-2010 and the NILS mobility project grant UCM-EEA-ABEL-03-2010.
We also acknowledge support from MICINN (Spain) project numbers FIS
2008-01323, FPA 2008-00592 and CAM/UCM 910309.
## References
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* Brodwin et al. (2010) Brodwin M., et al., 2010, ApJ, 721, 90
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|
arxiv-papers
| 2011-08-21T09:55:35 |
2024-09-04T02:49:21.669566
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edoardo Carlesi, Alexander Knebe, Gustavo Yepes, Stefan Gottloeber,\n Jose Beltran Jimenez, Antonio L. Maroto",
"submitter": "Edoardo Carlesi Mr",
"url": "https://arxiv.org/abs/1108.4173"
}
|
1108.4214
|
# LAMMPS Framework for Dynamic Bonding and an Application Modeling DNA
Carsten Svaneborg science@zqex.dk Center for Fundamental Living Technology,
Department of Physics and Chemistry, University of Southern Denmark, Campusvej
55, DK-5320 Odense, Denmark
###### Abstract
We have extended the Large-scale Atomic/Molecular Massively Parallel Simulator
(LAMMPS) to support directional bonds and dynamic bonding. The framework
supports stochastic formation of new bonds, breakage of existing bonds, and
conversion between bond types. Bond formation can be controlled to limit the
maximal functionality of a bead with respect to various bond types.
Concomitant with the bond dynamics, angular and dihedral interactions are
dynamically introduced between newly connected triplets and quartets of beads,
where the interaction type is determined from the local pattern of bead and
bond types. When breaking bonds, all angular and dihedral interactions
involving broken bonds are removed. The framework allows chemical reactions to
be modeled, and use it to simulate a simplistic, coarse-grained DNA model. The
resulting DNA dynamics illustrate the power of the present framework.
###### keywords:
Dynamic directional bonds, coarse-grain DNA models, chemical reactions,
molecular and dissipative particle dynamics
††journal: Computer Physics Communications
### 1 Introduction
When performing molecular dynamics simulations, we distinguish between bonded
and non-bonded interactions.[1, 2] Effectively, this means that the
interactions have been coarse-grained on the energy scale of the simulation.
Certain degrees of freedom are frozen, and we describe them as being permanent
bonded. Other degrees of freedom remain dynamic, and we describe them with
relatively weak non-bonded interactions. However, this situation is less clear
when simulating systems undergoing chemical reactions where bonds are created
or broken. Another example is DNA molecules where hybridization bonds are
broken at high temperatures and reformed when cooling the system. For such
systems, it can be computationally more efficient to model these degrees of
freedom as being dynamically bonded.
The problem of bond dynamics is closely related to the question of how to
represent chemical reactions in a molecular dynamics simulation. Reactive
force fields such as ReaxFF and empirical valence bond (EVB) can be used to
model chemical reactions.[3] Bond order potentials are interesting since they
allow three body interactions in the neighborhood of a bond to modify the
strength of the bond.[4] When coarse-graining systems capable of chemical
reactions, it is important to note that the reaction radius and probability
also has to be appropriately coarse-grained.[5] When the bonds become dynamic,
this also induces a dynamic for the angular and dihedral interactions. When
breaking a bond, all angular and dihedral interactions involving that bond
become invalid, and should be removed. Similarly, when creating a bond, we
have to identify which angular and dihedral interactions to create in the bond
neighborhood. This ensures that after melting and renaturing of a system, it
is again governed by the same set of interactions and return to the same
equilibrium structure.
DNA molecules are comprised of the four bases adenine (A), cytosine (C),
guanine (G), and thymine (T). The bases are attached to a 2-deoxyribose sugar
ring. For naturally occurring DNA, sugar rings are linked to each other
through phosphodiester bonds, that connect the 3’ to 5’ carbons in consecutive
sugar rings. This builds a molecular directionality into the back bone of a
DNA strand, which will have a 3’ and a 5’ end. The strand is also
characterized by a specific sequence of bases. Together the phosphate
backbone, the sugar and the base is denoted a nucleotide, which is the repeat
unit of a single DNA strand. A-T and C-G are Watson-Crick pairs and can form
hydrogen bonds with each other. The energetically favorable stacking
interactions allow two complementary single strands to form 3’-5’/ 5’-3’ anti-
parallel aligned double strands. Double stranded DNA can be melted and
renatured by repeated cycling the temperature around the melting point or by
varying solvent conditions.
DNA is a very complex molecule and numerous models exists to describe behavior
from atomistic properties to mesoscopic mechanical properties. The molecular
structural details of short DNA oligomers can be studied with atomistic
molecular dynamics simulations such as Amber[6, 7] and Charmm[8, 9]. However,
when we want to understand the large scale properties of DNA molecules or
materials in which DNA molecules are a component, coarse-grained DNA models
are essential. Coarse-graining is the statistical mechanical process by which
uninteresting microscopic details are systematically removed, leaving a
coarse-grained, effective model that is described by an effective free energy
functional.[10, 11, 12, 13] A major advantage of coarse-grain models are that
we can use them to simulate the interesting large scale dynamics of a system
directly without wasting time on uninteresting details. This allows larger
systems to be studied for longer times which paves the way for studying e.g.
the properties materials rather than single molecules.
A number of coarse-grain DNA molecular dynamics models exists. In the “three
site per nucleotide” model of de Pablo and co-workers, a single nucleotide is
represented by a phosphate backbone site, a sugar group site, and a base site,
respectively[14, 15, 16, 17]. The model uses an implicit representation of
counter ions at the level of Debye-Hückel theory, but has recently been
generalized to explicit counter ions.[18] A version of this model has also
been generalized to include non-Watson-Crick base pairing such as Hoogsteen
pairing.[19] There is also a number of “two site per nucleotide” models where
one site represents the back bone and the sugar ring. The other site
represents the base.[20, 21, 22, 23, 24] One challenge to “one site per
nucleotide” models are to represent the DNA double helix. Savelyev and
Papoian[25, 26] does this by special “fan” shaped pair-interactions between a
bead and a large number of beads on the opposite strand. This model does not
allow for DNA melting. Trovato and Tozzini[27] produce a helical structure
using angular and dihedral interactions along the double strand. In the case
where the large scale DNA mechanical properties are of interest, it can be
advantageous to coarse-grain a whole base-pair to a single rigid ellipsoidal
or plate-shaped object and regard DNA as a latter-like chain of such
objects.[28, 29] Here the coarse-graining has eliminated the melting and
renaturation dynamics all together. Other types of coarse-grain DNA models are
applied to study behavior of DNA functionalized nano-particles. The DNA
molecules can e.g. be modeled as rigid rods with a single sticky site on one
end and tethered to the surface of the nano-structure by the other end[30], as
semi-flexible polymers with attractive sites on the monomers[31], or the whole
DNA molecule can be modeled as a single sticky site that can be hybridized to
complementary free sticky sites.[32] Here the coarse-graining has completely
eliminated the chemical structure, while the melting, renaturing, and sequence
specificity has been retained in the dynamics.
The two most prevalent statistical mechanical models of RNA and DNA melting
are the Poland-Scheraga[33, 34] (PS) and the Dauxois-Peyrard-Bishop[35] (DPB)
models. The Poland-Scheraga model describes DNA as a 1D lattice model where a
base-pair can either be hybridized or open. The free energy expression for the
PS model contains empirical stacking free energies each stack of hybridized
base-pairs as well as contributions from the strand configuration entropy due
to internal bubbles, frayed ends and empirical initiation terms. The DPB model
also describes DNA as a 1D lattice model, but where each base-pair is
characterized by a continuous base-pair distance. Contrary to the PS model,
the DPB model has a Hamiltonian where the base-base potential is described by
an anharmonic potential representing hydrogen bonding, and deviations between
nearest neighbor base-pair extensions are penalized by a harmonic term. A
generalization of the PS model exists, where the strand conformations are
represented explicitly as lattice polymers. This provides a conceptual
simplification since the conformational entropy of bubbles and frayed ends
emerges naturally from the polymer model. This real space lattice PS model has
been studied using exact enumeration techniques[36], a version of the model
has also been applied to study RNA folding using Monte Carlo simulations.[37]
The dynamic bonding framework allows us to study classes of DNA models where
hybridization bonds, angular bonds, and dihedral bonds are created and broken
dynamically. These dynamic bonding DNA models are intermediates between the
real space lattice PS models, the coarse-grained molecular dynamics models,
and the sticky DNA models described above. In the PS model, base pairs can
either be hybridized or open and are characterized by a corresponding free
energy. In a dynamic bonding model, base pairs will be either hybridized or
open and a free energy will also characterize this transition. In the coarse-
grained molecular dynamics models and the DPB model, base pairs are
represented by a continuous non-bonded pair-potential. In the dynamic bonding
DNA models, base pairs are characterized by a continuous bond potential. The
dynamic bond DNA models can also be regarded as being off-lattice
generalizations of the real space lattice PS model, where a single strand is
described as a semi-flexible bead-spring polymer where complementary monomers
will form hybridization bonds when they are close. The dynamic bonded DNA
models are “one site per nucleotide” models, but we can also lump sequence of
nucleotides into a single coarse-grained bead. In this case, we can as a first
approximation assume that only beads representing complementary sequences can
hybridize, and that the breaking of a hybridization bond corresponds to the
creation of a DNA bubble. This would be a “many nucleotides per site” dynamic
bonding DNA model more akin to the sticky site DNA models used to study DNA
functionalized nano-particles.
The dynamic bonded DNA models ensure anti-parallel strand alignment in the
double strand state, through the interplay between the dihedral interactions
and the directional bonds. Such degrees of freedom are absent from both the PS
and DPB 1D lattice models. The coarse-grained models use angular and dihedral
interactions to ensure a structure resembling the real chemical structure of
DNA molecules. In dynamic bonded DNA models, the angular and dihedral
interactions are dynamically introduced when hybridization bonds are formed to
promote a zipper-like closing dynamic. Similarly angular and dihedral
interactions are dynamically removed as hybridization bonds are broken to
promote zipper-like opening dynamic. Hence in dynamic bond DNA model, we
utilize the interplay between dynamic bonded, angular, and dihedral
interactions to model cooperative effects in the DNA bubble and zippering
dynamics, rather than to model chemical structure.
The simplicity and success of the PS model in predicting sequence specific DNA
melting temperatures suggests that the essential physics of DNA hybridization,
melting and renaturing can, in fact, be accurately captured in a model without
chemical details, and where the key property is the dynamics of hybridization.
This is our motivation for developing the dynamic bonding framework. We will
use it to develop and apply models to study the properties of hybrid materials
containing both DNA molecules and soft-condensed matter.
We have implemented directional bonds and dynamic bonding in the Large-scale
Atomic/Molecular Massively Parallel Simulator[38] (LAMMPS). LAMMPS is a
versatile, parallel, highly optimized, open source code for performing
Molecular Dynamics (MD) and Dissipative Particle Dynamics (DPD) simulations of
coarse-grained models. Due to the modular design, LAMMPS is easy to extend
with new interactions and functionality. The dynamic bonding implementation is
also modular and easy to extend with new functionality. Our extension is by no
means limited to modeling DNA, but could equally well be used for simulations
of chemical reactions such as living polymerization, cross-linking of stiff
polymers, coarse-grained dynamics of worm-like micelles and active driven
materials. A snapshot of the LAMMPS code with the directional bonds and
dynamic bonding implementation can be obtained from the CPC Program Library.
Included with the code is also the documentation necessary for porting the
directional and dynamic bonding framework to future LAMMPS versions.
Sect. 2 is a summary of the implementation of directional bonds and the
dynamic bonding framework. We present a simplified DNA model based on the
dynamic bonding framework in sect. 3, which is provides the examples of DNA
dynamics shown in sect. 4. We conclude with our conclusions in sect. 5, and
present the details of the directional bonds and dynamic bonding
implementation in an appendix.
### 2 Implementation
Double stranded DNA only exists in a state where the two strands are aligned
anti-parallel 3’-5’/5’-3’. In order distinguish between parallel and anti-
parallel strand alignment, we regard the 3’-5’ back bone structure as a
property of the back bone bonds, which become directional. This is necessary
since the chemical structure of the nucleotides have been coarse-grained to a
single structureless site. The directional bonds will also play a crucial role
when introducing angular and dihedral interactions in a double stranded DNA
molecule, since this affects the stability, zippering dynamics, and mechanical
properties.
To implement directional bonds in LAMMPS, we make use of the fact that Newtons
3rd law is optional when calculating bond forces. When Newtons 3rd law is
enabled, each bond force is only calculated once, but subsequently has to be
communicated to the bond partner. When it is disabled, LAMMPS calculates the
bond force twice, once for each of the two bond partners. In this case, each
of the two bond partners store information about the bond type and the
identity of the other bond partner. We can denote this situation by
$A\begin{subarray}{c}t\\\ \rightarrow\end{subarray}B$ and
$A\begin{subarray}{c}t\\\ \leftarrow\end{subarray}B$, which shows that the $A$
bead stores $t$ as the type of the bond to $B$, and the $B$ bead stores $t$ as
the type of the bond to $A$. With a few modifications, LAMMPS will load and
store different bond types in the two bond partners. Hence, we can have
$A\begin{subarray}{c}t\\\ \rightarrow\end{subarray}B$ and
$A\begin{subarray}{c}s\\\ \leftarrow\end{subarray}B$, where the bond type $s$
from $B$ to $A$ and the bond type $t$ from $A$ to $B$ differs. When the two
bond types refer to the same bond potential, Newtons 3rd law still applies,
and the dynamics is unaffected. However, we can interpret the pattern of bond
types as the directionality of the strand. Note that if we instead use
different bond potentials in the two directions or only a “half” bond, the
result would be a net force along the bond, which can be used to model driven
active matter. We shall not pursue this situation further in the present
paper.
Define a dynamic bond fix:
fix fixid beadgroup bonddynamics _everystep_ [paircheck13] [paircheck14] ¡list
of rules¿
Each dynamic bonding rule is one of:
createbond _bondtype_ _beadtype1_ _beadtype2_ maxdistance probability
createdirbond _bondtype1_ _bondtype2_ _beadtype1_ _beadtype2_ maxdistance
probability
breakbond bondtype mindistance probability
convertbond bondtype1 bondtype 2 probability
killbond _bondtype_ mindistance
createangle _angletype_ _beadtype1_ _beadtype2_ _beadtype3_ _bondtype1_
_bondtype2_
createdihedral _dihedraltype_ _beadtype1_ _beadtype2_ _beadtype3_ _beadtype4_
_bondtype1_ _bondtype2_ _bondtype3_
maxbond _bondtype_ _beadtype_ _maxnumber_
Figure 1: LAMMPS syntax for the dynamic bonding fix, and the types of rules
currently implemented.
The dynamic bonding framework allows a number of rules to be specified, that
completely define the bond dynamics. These rules are applied to a specified
group of reactive beads with a specified frequency. The application of the
rules is conditional on the types of beads, types bonds, distance between
beads and length of bonds involved. In particular, we have implemented rules
for stochastic creation of symmetric and directional bonds within a certain
reaction distance, stochastic removal of symmetric bonds larger than a
breaking distance, removal of all symmetric bonds exceeding a certain length,
and stochastic conversion of a symmetric bond from one type to another.
Furthermore, all bond creation rules ensure that a bead can never have more
than a specified number of bonds of a given type. The implementation is
structured such that it is easy to implement new types of rules.
Besides the bond dynamics, the consistency of the angular and dihedral
interactions should be ensured at all times. After bonds have been broken, all
invalid potential angular and dihedral interactions involving broken bonds
should also be removed. After bonds have been formed, all triplets or quartets
of beads that could be connected by at least one new bond are checked to see
if they require the creation of an angular or dihedral interaction. We discard
cyclic triplets and quartets where the same bead appears more than once.
An angular creation rule specifies which angular interaction can be introduced
between a triplet of connected beads $A$, $B$, and $C$. Since the triplets are
not ordered, the rule should match either $ABC$ or $CBA$. To test if the $ABC$
bead order matches, we first compare the types of the $ABC$ beads with the
bead types the rule specifies. We then compare the two bond types $t$ and $s$
with the bond types the rule specifies, where the bond types are defined
directionally as $A\begin{subarray}{c}t\\\ \leftarrow\end{subarray}B$ and
$B\begin{subarray}{c}s\\\ \rightarrow\end{subarray}C$. If the $ABC$ bead order
did not match, it is repeated $CBA$ bead order, where the bonds types are
defined directionally as $C\begin{subarray}{c}t\\\ \leftarrow\end{subarray}B$
and $B\begin{subarray}{c}s\\\ \rightarrow\end{subarray}A$. If a rule match,
then the specified angular interaction is introduced between the three beads.
A creation rule for a dihedral interaction specifies four bead types and three
bond types. Again we test both $ABCD$ and $DCBA$ ordered bead quartets. First
the bead types of the quartet are compared to the bead types specified by the
rule, subsequently the bond types are compared, the bond types are defined
directionally as $A\begin{subarray}{c}r\\\ \leftarrow\end{subarray}B$,
$B\begin{subarray}{c}s\\\ \leftarrow\end{subarray}C$,
$B\begin{subarray}{c}s^{\prime}\\\ \rightarrow\end{subarray}C$, and
$C\begin{subarray}{c}t\\\ \rightarrow\end{subarray}D$. The bond types match if
$r$, $s$ or $s^{\prime}$, and $t$ matches the three bond types specified by
the rule. If the $ABCD$ bead order did not match, it is repeated $DCBA$ order.
If a dihedral rule match a quartet of beads, the specified dihedral
interaction is introduced between the four beads. These rules allows us to
selectively and dynamically introduce angular and dihedral interactions taking
both bead types and directional bond types into account. Note that the same
directionality applies to matching the bead type and bond type patterns.
To have an efficient parallel implementation, we implement the bond creation
and breaking by an pair matching algorithm inspired from the bond/break and
bond/create fixes already implemented in LAMMPS. In the dynamic bonding fix,
preferred bond creation/breakage partners are identified in each simulation
domain. This information is communicated between and aggregated across
neighbor simulation domains. Afterwards, the bonds selected for breakage are
removed. The local neighborhood of all reactive beads are checked for angular
and dihedral interactions, that should be removed because they cross broken
bonds. Then bonds are created between partners selected for bonding. Again, we
check the local neighborhood of all reactive beads to introduce angular and
dihedral interactions. After this final step, we broadcast bond statistics to
all simulation domains. Note that due to the pair matching algorithm, each
bead can maximally have one bond created and broken at each call to the
dynamic bonding fix. All rules are applied to a bead pair (in the specified
order) when identifying if they are eligible for matching. If multiple rules
apply to the same bead pair, the last matching rule will be always be chosen.
Hence, if this last rule has a very low reaction probability, it will
completely shadow more probable rules specified earlier. These shadowing
issues does not apply to the DNA model below, and will not play a role at low
concentrations of reacting beads. The details of the implementation and
shadowing issues are discussed in Appendix Appendix.
The LAMMPS syntax of the dynamic bond fix is shown in fig. 1. The first line
defines the name of the particular instance of the fix, the group of reactive
beads (_beadgroup_), and how often the bond dynamics fix is applied
(_everystep_). By default creation rules only apply to potential bonding bead
pairs, that are further than 4 bonds apart or not bonded. The optional
_Paircheck13_ and _Paircheck14_ switches includes 1-3 and 1-4 chemically
distant beads in the search of potential bonding partners. The line is
followed by a number of dynamic bonding rules. _Createbond_ rules specify
pairs of bead types, that can be bonded, if they are within a certain maximum
reaction distance from each other. If a bead has more than one potential bond
partners, the closest partner is chosen, and a bond with the specified type is
then created with the given probability. _Createdirbond_ rules does the same
as _createbond_ , but creates a directional bond with the two specified bond
types between the two bead types. _Breakbond_ rules identifies bonded bead
pairs with bonds longer than the specified minimum distance and breaks the
bond with the specified probability. If a bead has more than one potential
bond break partner, then the most distant partner is chosen. Since only a
single bond can be removed per bead per call to the dynamic bonding fix, a
breakbond rule with unit probability does not ensure that all bonds longer
than the minimum distance are broken. Hence, we have also implemented
_killbond_ rules. These rules operate directly on the bond structures, and are
not limited by the pair matching algorithm. _Convertbond_ rules stochastically
convert symmetric bonds of one type into another type. This is implemented as
nominating the bond pair for removal of the old bond, followed by creation of
the new bond. The dynamic bonding framework ensures that angular and dihedral
interactions across the bond are also converted accordingly. _Createangle_ and
_Createdihedral_ rules defines which angular and dihedral interaction types
should be created between triplets and quartets of beads with the specified
types of bead, and types of bonds between the beads as discussed above.
_Createangle_ and _createdihedral_ rules do not specify a probability, since
they are created as required by the local neighborhood around new bonds. Note
that angular and dihedral interactions are only introduced as a consequence
bond creation events, they are not introduced between already bonded beads
even though the bead types and bond types match the rule. When checking
potential beads for bond creation, all _Maxbond_ rules are checked to discard
beads that already have the maximal number of the specified bond types.
### 3 DNA model
We have chosen the present DNA model has been chosen because it produces a
simple ladder like equilibrium structure, which allows us to illustrate the
power of the dynamic bonding framework, and to visualize all the interactions
that are dynamically introduced and removed. Real DNA molecules performs a
whole twist every 10.45 base pairs, and to model twist we need a somewhat more
complex force field, but exactly the same dynamic bonding rules. Because we
are interested in studying DNA programmed self-assembly, we choose to use
Dissipative Particle Dynamics (DPD)[39, 40]. DPD is given by a force field
comprising a conservative soft pair-force $F^{C}$, a dissipative friction
force $F^{D}$, and a stochastic driving force $F^{R}$ given by
${\bf F}_{ij}=\left(F^{C}+F^{R}+F^{D}\right)\frac{{\bf
r}_{ij}}{r}\quad\mbox{for}\quad r=|{\bf r}_{ij}|<r_{c}$
where the forces contributions are given by
$F^{C}=aw(r)\quad\quad F^{D}=-\frac{\gamma w^{2}(r)}{r}\left({\bf
r}_{ij}\cdot{\bf v}_{ij}\right)\quad\quad F^{R}=\frac{\sigma
w(r)\xi}{\sqrt{\Delta t}}.$
Here ${\bf r}_{ij}={\bf r}_{i}-{\bf r}_{j}$ and ${\bf v}_{ij}={\bf v}_{i}-{\bf
v}_{j}$ denotes the separation and relative velocity between two interacting
beads $i$ and $j$, respectively. $\xi$ denotes a Gaussian random number with
zero mean and unit variance, and the thermostat coupling strength is
$\sigma=\sqrt{2k_{B}T\gamma}$. The weighting function is
$w(r)=1-\frac{r}{r_{c}}$. We integrate the DPD dynamics with a Velocity Verlet
algorithm with a time step $\Delta t=0.01\tau$. The unit of energy is
$\epsilon=k_{B}T$, where we chose to set Boltzmann’s constant to unity, such
that temperature is measured in energy units. We use $T=1\epsilon$ in all of
the simulations except the DNA bubble simulation where $T=5\epsilon$. The unit
of length $\sigma$ is defined by the pair force cut-off $r_{c}=1\sigma$. The
mass is $m=1$ for all beads, this allows us to define the unit of time as
$\tau=\sigma\sqrt{m/\epsilon}$. The DPD pair-force parameter is
$a=25\epsilon\sigma^{-1}$ between all species of beads. The viscosity is
$\eta=100\epsilon\tau\sigma^{-2}$. Non-bonded pair interactions are switched
off between beads in molecules that are less than 3 bonds apart. The DNA
molecule is simulated in an explicit solvent at a density $\rho=3\sigma^{-3}$.
We represent a nucleotide by a single DPD bead, and let the four ATCG
nucleotides correspond to bead types 1-4, respectively. They are colored red,
green, blue, and magenta, respectively, in figures below. Red and green beads
(A-T) are complementary as are blue and magenta (C-G) beads. A single strand
of DNA is represented as a string of beads joined by permanent directional
back bone bonds. The two 3’ to 5’ and 5’ to 3’ backbone bond potentials (bond
type 2 and 3, respectively, colored green and blue in the bond visualizations)
are given by the same potential
$U_{backbone}(r)=\frac{U_{min}}{(r_{l}-r_{0})^{2}}\left((r-r_{l})^{2}-(r_{0}-r_{l})^{-2}\right),$
with $U_{min}=10.0\epsilon$, $r_{l}=0.3\sigma$, and $r_{0}=0.6\sigma$. The
hybridization bond potential (bond type 1, colored red in the bond
visualizations) is given by
$U_{hyb}(r)=\begin{cases}\frac{U_{min}}{(r_{h}-r_{0})^{2}}\left((r-r_{0})^{2}-(r_{h}-r_{0})^{-2}\right)&\quad\mbox{for}\quad
r<r_{c}\\\ 0&\quad\mbox{for}\quad r\geq r_{c}\end{cases},$
with $U_{min}=1.0\epsilon$, $r_{h}=0.6\sigma$, and $r_{c}=1.0\sigma$.
Besides the DNA interactions, we need to define the bonding dynamics of the
DNA beads. The corresponding dynamic bonding fix command is shown in fig 2.
Hybridization bonds are created with probability one when two complementary
beads are within a distance of $r_{h}$. Bead type $2$ and $3$ are able to form
a 5’ 3’ backbone bond when they are within a distance of $r_{l}=0.3\sigma$
from each other. The probability of creation of a back bone bond is $0.1$.
This is a simplification for the oligomer-template simulation below. Only
hybridization bonds can be broken, and they are removed if they are longer
than $r_{c}=1\sigma$. To control hybridization, we only allow all bead types
($*$) to have maximally one hybridization bond (type 1), one 3’ end (type 2)
and one 5’ end (type 2) of a back bone bond. In the model all nucleotides has
the same interactions, hence use $*$ for all the bead types rule
specifications.
1: fix dnadyn dna bonddynamics 1 paircheck14
2: createbond 1 1 2 0.6 1.0
3: createbond 1 3 4 0.6 1.0
4: createdirbond 2 3 2 3 0.3 0.1
5: killbond 1 1.0
6: maxbond 1 * 1
7: maxbond 1 * 2
8: maxbond 1 * 3
9: createangle 1 * * * 2 3
10: createangle 2 * * * 1 2,3
11: createdihedral 1 * * * * 1 2,3 1
12: createdihedral 2 * * * * 2 1 3
13: createdihedral 3 * * * * 2 1 2
14: createdihedral 3 * * * * 3 1 3
Figure 2: LAMMPS dynamic bonding fix for producing the DNA dynamics shown in
figs. 3-7. Bond types are shown with plain digits (hybridization: red 1, back
bone 3’ bonds: green 2, and back bone 5’ bonds: blue 3). Bead types are shown
with bold digits represent nucleotides (A:red 1, T:green 2, C:blue 3, G:
magenta 4). Angular and dihedral bond types are shown italic digits
corresponding to the interaction type numbers. The bead and interaction type
colors correspond to those used in the visualizations. * is the wild card and
is used to match any bead or bond type.
The model has two angular interactions, which are described by the potential
$U(\theta)=K(\theta-\theta_{0})^{2}$, where $K$ defines the angular spring
constant and $\theta_{0}$ the equilibrium angle. The first angle interaction
(type 1) promotes a straight angle between back bone bonds. This interaction
is shown as red angles in the angle visualizations, and it has parameters
$K=20\epsilon$ and $\theta_{0}=180\text{\textordmasculine}$. Type 1 angles are
dynamically introduced for bonding patterns $A\begin{subarray}{c}3^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}5^{\prime}\\\
\rightarrow\end{subarray}C$ and $A\begin{subarray}{c}5^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}3^{\prime}\\\
\rightarrow\end{subarray}C$ (i.e. for model bonds types $2$ $3$, since $CBA$
order matches $3$ $2$). The second angle interaction (type 2) promotes a right
angle between back bone and hybridization bonds. This interaction is shown as
green angles in the angle visualizations, and it has $K=1\epsilon$ and
$\theta_{0}=90\text{\textordmasculine}$. Type 2 angles are dynamically
introduced for bonding patterns $A\begin{subarray}{c}H\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}3^{\prime}/5^{\prime}\\\
\rightarrow\end{subarray}C$ and $A\begin{subarray}{c}3^{\prime}/5^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}H\\\
\rightarrow\end{subarray}C$ (i.e. model bond types $1$ and $2,3$, since $CBA$
order matches the reverse pattern).
The DNA model has three dihedral interactions, which are described by the
potential $U(\phi)=K(1+d\cos(\phi))$. We use dihedral spring constant
$K=1.0\epsilon$, and $d=+1$ ($-1$) for promoting trans (cis) conformations.
The first dihedral interaction (type 1, shown red in dihedral visualizations)
promotes a cis conformation when a back bone bond connects two hybridized
nucleotide pairs. This corresponds to the bonding patterns
$A\begin{subarray}{c}H\\\ \leftarrow\end{subarray}B$,
$B\begin{subarray}{c}3^{\prime}\\\ \leftarrow\end{subarray}C$,
$B\begin{subarray}{c}5^{\prime}\\\ \rightarrow\end{subarray}C$,
$C\begin{subarray}{c}H\\\ \rightarrow\end{subarray}D$ and
$A\begin{subarray}{c}H\\\ \leftarrow\end{subarray}B$,
$B\begin{subarray}{c}5^{\prime}\\\ \leftarrow\end{subarray}C$,
$B\begin{subarray}{c}3^{\prime}\\\ \rightarrow\end{subarray}C$,
$C\begin{subarray}{c}H\\\ \rightarrow\end{subarray}D$, where $H$ denotes a
hybridization bond (i.e. model bond numbers $1$ $2,3$ $1$). The second
dihedral interaction (type 2, shown green in the dihedral visualizations)
promotes a cis conformation of the two beads that are connected by back bone
bonds to a hybridized bead pair and is located on the same side of the bead
pair. The bonding pattern is $A\begin{subarray}{c}3^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}H\\\
\leftarrow\end{subarray}C$, $B\begin{subarray}{c}H\\\
\rightarrow\end{subarray}C$, $C\begin{subarray}{c}5^{\prime}\\\
\rightarrow\end{subarray}D$ (i.e. model bond numbers $2$ $1$ $3$). The third
interaction (type 3, shown blue in the dihedral visualizations) promotes a
trans conformation of the two bead that are connected by back bone bonds to a
hybridized bead pair but are localized on opposite sides of the bead pair. The
bonding patterns are $A\begin{subarray}{c}3^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}H\\\
\leftarrow\end{subarray}C$, $B\begin{subarray}{c}H\\\
\rightarrow\end{subarray}C$, $C\begin{subarray}{c}3^{\prime}\\\
\rightarrow\end{subarray}D$ and $A\begin{subarray}{c}5^{\prime}\\\
\leftarrow\end{subarray}B$, $B\begin{subarray}{c}H\\\
\leftarrow\end{subarray}C$, $B\begin{subarray}{c}H\\\
\rightarrow\end{subarray}C$, $C\begin{subarray}{c}5^{\prime}\\\
\rightarrow\end{subarray}D$ (i.e. model bond numbers $2$ $1$ $2$ and $3$ $1$
$3$). Note that without the directional bond, we would be unable to
distinguish between these two last types of dihedrals. The examples belows are
included as test cases with the dynamic bonding code submitted to the CPC
Program Library, and require less than a CPU hour of computational effort.
### 4 Example DNA dynamic
Figure 3: Oligomer - DNA template hybridization (rows 1-4) showing the
dynamics of bond, angular, and dihedral interactions (columns a-c) for times
$t=0$, $0.01\tau$, $0.04\tau$, and $0.23\tau$ into the simulation. Bead and
interaction colors match those in fig. 2. Note that back bone bond
directionality is only shown in the first row for simplicity.
To illustrate the dynamic bonding framework with the DNA model, we simulate a
$5^{\prime}-ATCGATCG-3^{\prime}$ template in the presence of two
$3^{\prime}-TAGC-5^{\prime}$ oligomers. The first oligomer is already
hybridized with the template, while the second is placed in the vicinity of
the template. Fig. 3 shows snapshots along the trajectory where the remaining
oligomer hybridizes with the template. The top left visualization shows the
initial designed configuration. The blue-green pattern of the hybridized
oligomer backbone shows it has 3’ 5’ direction, while the green-blue pattern
of the template backbone shows the 5’ 3’ direction. The top center
visualization shows the angular interactions of the initial configuration. The
back bone stiffness is controlled by the red angular interactions between back
bone bond pairs, which promote a straight back bone configuration. The green
angular interactions promote hybridization bonds that are perpendicular to the
strand axis. The top right visualization shows the dihedral interactions of
the initial configuration. The hybridized template shows red and green
dihedral interactions which are promote cis arrangement of stacked bead pairs,
while the blue dihedral interaction promotes trans arrangement. Together they
stabilize the ladder-like structure of the double strand. Without the bond
directionality, we would have no way to distinguish between green and blue
dihedral interactions, and hence control over the stiffness of the double
strand relative to that of the single strands.
As we let the simulation run (left column top to bottom) initially two
hybridization bonds are introduced between the two beads at right most end of
the template. Later a third and a fourth hybridization bond are also
introduced. Along with the hybridization bonding dynamics, angular and
dihedral interactions (center and right columns) are also created. The angular
interactions cause the free oligomer to align with the template, while the
dihedral interactions creates a torque that ensures that the alignment is
anti-parallel.
Figure 4: Back bone ligation reaction by addition of directional back bone
bond (rows 1-3) showing dynamic of bond, angular and dihedral interactions
(columns a-c) for the simulation in fig. 3 continued to times $11.50\tau$,
$12.46\tau$, and $12.48\tau$, respectively.
Fig. 4 shows how the nick in the DNA molecule is closed by forming a back bone
bond. The interactions between the two oligomers and the template ensures that
they are both aligned anti-parallel to the template backbone axis. The single
red dihedral interaction across the nick promotes a cis configuration, and
twists the two oligomers towards the same side of the template. Finally the
missing back bone bond is created following the 3’-5’ directionality of the
strand, along with all the angular and dihedral interactions to produce a
double stranded configuration. Together figs. 3 and 4 simulates a chemical
reaction where a DNA template and two complementary oligomers first hybridize
due to their complementary sequences, and then ligate to produce the
complementary template sequence.
Figure 5: DNA unzipping by a weak vertical force $f=28\epsilon\sigma^{-1}$
applied to the left most bead pair (rows 1-4) for bond, angular and dihedral
interactions (columns a-c). The rows corresponds to times $1.72\tau$,
$1.84\tau$, $3.03\tau$, $3.22\tau$, respectively, starting from a straight
double strand conformation at $t=0\tau$.
To melt the double strand, we can e.g. apply an external force to tear the two
strands apart[41] or increase the temperature to let thermal fluctuations do
the work. Fig. 5 shows the result of applying an external opposing force to
left most nucleotide pair. Progressively the left most hybridization bond
snaps. Along with the breakage of hybridization bonds, we also see the gradual
removal of green angular interactions and all the dihedral interactions. The
external force is opposed by a single left most hybridization bond along with
the angular and dihedral interactions across the gab. During the unzipping
process, often the hybridization bonds are transiently reformed just after
breakage if thermal fluctuations pull them within the hybridization reaction
distance.
Figure 6: Time series of DNA unzipping by a strong horizontal force
$f=100\epsilon\sigma^{-1}$ applied to the left and right most beads of the two
strands (a-h). The snapshots corresponds to times $0.21\tau$, $0.30\tau$,
$0.59\tau$ (top row), $0.81\tau$, $0.89\tau$, $0.92\tau$ (middle row), and
$0.95\tau$, $1.20\tau$, $1.37\tau$ (bottom row) starting from a straight
double stranded conformation at $t=0\tau$.
In fig. 6 we perform another pulling experiment, where a much stronger
horizontal force is applied to the left most bottom strand and right most top
strand beads of the double strand. Initially the whole molecule is sheared, as
all the green angular interactions cooperate in opposing the deformation.
Gradually bonds snaps from either end towards the center. Interestingly, since
the two molecules have a 4-nucleotide long repeating sequence, when the
hybridization bonds are broken, they very rapidly reform with the
complementary beads one repeat sequence further down the molecule. The shear
process repeats for the second hybridization sequence until it too is broken,
and two single strands are formed.
Figure 7: Time series showing bubble opening and closing dynamics for DNA at
an elevated temperature $T=5\epsilon$ (a-h). The snapshots are from times
$t=55.40\tau$, $55.44\tau$, $55.48\tau$ (top row), $55.55\tau$, $55.85\tau$,
$55.89\tau$ (middle row), and $55.96\tau$, $56.05\tau$, $56.09\tau$ (bottom
row) starting from a straight double stranded conformation at $t=0\tau$.
DNA can be molten by raising the temperature. The melting temperature depends
on the sequence, the length of the strands as well as the strand
concentration.[33, 42] Prior to melting, bubbles of open nucleotide sequences
appear since they contribute configuration entropy and hence lower the free
energy similar to vacancies in crystals. At increased temperatures, the number
and the size of these bubbles grow and cause the two strands to melt.[43, 44,
45, 46] In fig. 7 we show a time series of a bubble, that is created by
breaking a single hybridization bond, the bubble grows until it breaks the
last hybridization bond. However, the two frayed strands form a hybridization
bond at the end, and progressively the bubble closes again. Simulating the
chain for sufficiently long time at an elevated temperature will cause the
double stands to melt with a transition very much like the one shown in fig.
7.
### 5 Conclusions
We have implemented a versatile framework for studying the effects of dynamic
bonding of ordinary and directional bonds in coarse-grained models within the
context of the Large-scale Atomic/Molecular Massively Parallel Simulator
(LAMMPS)[38]. The dynamic bonding framework ensures that angular and dihedral
interactions are kept consistent during bond breakage and creation. The code
has been parallelized and optimized to the case where the bond formation or
breakage probability for each bead is relatively low. Since the dynamic
bonding code is very modular it will be easy to extend with other types of
bonding rules. The dynamic bonding framework was written with the aim of
developing a new type of coarse-grained models of DNA dynamics. We have
illustrated a dynamic bonding DNA model using DNA hybridization and ligation,
as well as two geometries of force induced unzipping and bubble dynamics.
Clearly the present DNA model is very simple, nonetheless it qualitatively
captures some of the fundamental phenomena of DNA molecules. The dynamic
bonding framework will allow us to build DNA models, that we expect will
provide quantitative predictions as good as the Poland-Scheraga model[33, 34],
while we can use these DNA models as components in Molecular Dynamics and
Dissipative Particle Dynamics simulations of hybrid materials containing both
soft-condensed matter and DNA molecules.
### 6 Acknowledgements
C.S. gratefully acknowledges discussions with H. Fellermann, R. Everaers,
P.-A. Monnard, M. Hanczyc, and S. Rasmussen.
## Appendix
### Implementation details
When Newtons 3rd law is not applied to bonded interactions, LAMMPS has an bond
interaction table for each bead listing the other beads it is bonded to and
the type of the bond. Similar angular and dihedral interaction tables exists
for each bead. LAMMPS also has a neighbor structure where bonded neighbors,
next nearest neighbors, and third nearest neighbors are stored. This
information is derived from the bonding structure, and used to enable or
disable non-bonded interactions between beads connected by up to three bonds.
Initially when LAMMPS reads the control file to set up a simulation, the
dynamic bonding fix is called to parse the entire set of rules such as those
in fig. 2. The rules and their parameters are sanity checked and stored
internally in the fix. When the simulation is initialized, the dynamic bonding
framework starts by having each simulation domain count how many bonds of each
type each reactive bead has.
Then at a specified frequency the code does:
1. 1.
Communication. Forward communication of ghost particle positions to
neighboring nodes and the table of bond counts. This is required for testing
distances and for applying maximum rules.
2. 2.
Creation nomination. Each reactive bead can nominate a single preferred
bonding partner. The search for partners is performed over all beads in the
reactive group and each creation rule is tested in succession. The test of
rules is done in the order they are specified, and if more than one rule match
the same bead pair, the last matching rule will apply. The search is over all
non-bonded beads and optionally over beads 2 or 3 bonds away from the current
bead. For each bead pair and creation rule, their types are tested and if they
within the maximum reaction distance. Beads that already have the maximal
number of bonds of the type, that would be produced by the current rule are
discarded. Of all the potential bonding partners, the closest partner in the
same simulation domain (if any) is nominated for bonding.
3. 3.
Bond breakage nomination. Each reactive bead can nominate a single preferred
partner to break an existing a bond to. The search for partners is performed
over all beads in the reactive group and each bond break rule is tested in
succession. The test of rules is done in the order they are specified, and if
more than one rule match the same bead pair, the last matching rule applies.
For each bead pair and bond breakage rule, it is tested if the bond between
them has the specified type, and if they are further apart than the minimum
bond breakage distance. Of all the potential bond breakage partners, the
partner most distant in the same simulation domain (if any) is nominated for
bond breakage. Bond conversion is internally represented as a bond pair that
nominates each other for a bond breakage and creation of the new bond. Hence
bond conversion over rules both bond breakage and creation in case they occur
simultaneously.
4. 4.
Communication. The nominated partners are distributed to and aggregated across
neighboring simulation domains and the closest partner is chosen for creation
and the most distant partner is chosen for bond breakage. Information about
which rule lead the nomination of each partner are also distributed along with
a random number for stochastic bond breakage and a random number for
stochastic bond creation.
5. 5.
Bond breakage. If any killbond rules are defined, all beads check, if they are
part of a bond longer than the cut-off distance, and if that is the case then
the bond is marked for removal. If two bonds nominate each other as bond
breakage partners, then bond breakage is attempted. Each bead contributes a
uniform random number for bond breakage, these are averaged and compared to
the specified bond breakage probability. In case the random number is smaller
than the probability, the bond is marked for removal. This ensures that beads
on different simulation domains makes the same random choice. When bonds are
marked for removal the bond type in the corresponding entry in the bond
interaction tables is set to -1. If a maximum rule applies to that particular
bond and bead type, the table of bead functionalities is also updated. The
outdated neighbor structure is retained.
6. 6.
Removing angular and dihedral interactions. To ensure parallelism, each
reactive bead is alone responsible for all its angular and dihedral
interactions. If a bond has been broken in its local neighborhood, the bead
has to remove any angular and dihedral interactions involving that bond. This
is done by generating all non-cyclic paths of length three and four either
starting at or crossing the present bead using the outdated neighbor structure
(which still contains the broken bonds). The beads checks each path for bond
breakage events (using the bond interaction tables, which shows if a bond has
been marked for breakage). If a path involves a broken bond, then the bead
removes the corresponding entry in its angular and dihedral interaction
tables, if they exist.
7. 7.
The LAMMPS neighbor structure is updated, and the broken bond entries are
removed from the bond interaction tables. If no bonds are to be created, we
can jump directly to 10.
8. 8.
Bond creation. If two bonds nominate each other as bond creation partners,
then an attempt is made at creating the bond. Each bead contributes a uniform
random number for bond creation, these are averaged and compared to the
specified bond creation probability. Again this ensures the same random choice
for beads residing in different simulation domains. The new bond is added to
the bond interaction table for the bead. The neighbor structure is also
updated. If a maximum rule applies to the bond and bead type, the table of
bead functionalities is also updated.
9. 9.
Creating angular and dihedral interactions. Again each reactive bead is
responsible for determining if a bond was created in their local neighborhood.
This is done the same way as angular and dihedral interactions are removed.
Since the neighbor structure now contains the new bonds, we can generate non-
cyclic paths of length three and four starting at or crossing the present bead
using the updated neighbor structure (which now contains the new bonds). Each
path is checked for bond creation events using the bond interaction tables. If
the bead determines that it is part of a new triplet or quartet of beads, then
it compares the bead types and directional bond types with all the angular and
dihedral creation rules. If a match is found, then the bead adds the
corresponding interaction to its interaction table.
10. 10.
Statistics. Distribution of statistics of the total number of bonds, angles,
dihedrals introduced and removed in the current time step.
Since bond creation requires a distance check, the LAMMPS pair communication
distance should be at least the longest reaction distance, otherwise bonds
will only be created between bead pairs within the communication distance from
each other. Since the implementation also depends on all beads knowing about
all their bonded, angular, and dihedral interactions, it will not work without
Newtons 3rd law being disabled for bonded interactions. This is also required
for the implementation of directional bonds. The dynamic bonding framework
transparently handles symmetric bonds, hence they are just special cases of
directional bonds.
The dynamic bonding code is optimized to the situation, where the density of
reacting beads is so low that at most one bond breakage and bond creation
event is likely to occur per bead per time step. For instance, the match
making algorithm does not attempt to make matches between rejected partners,
that could still be eligible for bond breakage or bond creation rules. Nor
does the match making algorithm attempt to pick the most likely of multiple
possible reaction path ways. For instance, if multiple bond creation rules
applies to a single bead, then only the last nominated bond creation partner
is stored. Hence a creation rule with a low reaction probability can overwrite
the bonding partner nominated by a prior creation rule with much higher
reaction probability. In this case, the high probability reaction will never
happen. Similar issues apply when multiple bond break rules involve the same
bead. Since the bond conversion rules are implemented as bond deletion
followed by bond creation, these can interfere with both bond creation and
bond breakage rules. killbond rules are completely safe, since they are not
implemented using the match making algorithm. For the DNA model, none of these
caveats apply.
PROGRAM SUMMARY
Manuscript Title: LAMMPS Framework for Dynamic Bonding and an Application
Modeling DNA.
Authors: Carsten Svaneborg
Program Title: LAMMPS Framework for Directional Dynamic Bonding
Journal Reference:
Catalogue identifier:
Licensing provisions: GPL
Programming language: C++
Computer: Single and multiple core servers
Operating system: Linux/Unix/Windows
RAM: 1Gb
Number of processors used: Single or Parallel
Keywords: Dynamic bonding, directional bonds, molecular dynamics.
Classification: 16.11 Polymers, 16.13 Condensed-phase Simulations
Nature of problem: Simulating coarse-grain models capable of chemistry e.g.
DNA hybridization dynamics.
Solution method: Extending LAMMPS to handle dynamic bonding and directional
bonds.
Unusual features:Allows bonds to created and broken while angular and dihedral
interactions are kept consistent.
Running time:hours to days
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|
arxiv-papers
| 2011-08-21T21:42:35 |
2024-09-04T02:49:21.675579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carsten Svaneborg",
"submitter": "Carsten Svaneborg",
"url": "https://arxiv.org/abs/1108.4214"
}
|
1108.4329
|
# The behavior of noise-resilient Boolean networks with diverse topologies
Tiago P. Peixoto Institut für Theoretische Physik, Universität Bremen, Otto-
Hahn-Allee 1, D-28359 Bremen, Germany tiago@itp.uni-bremen.de
###### Abstract
The dynamics of noise-resilient Boolean networks with majority functions and
diverse topologies is investigated. A wide class of possible topological
configurations is parametrized as a stochastic blockmodel. For this class of
networks, the dynamics always undergoes a phase transition from a non-ergodic
regime, where a memory of its past states is preserved, to an ergodic regime,
where no such memory exists and every microstate is equally probable. Both the
average error on the network, as well as the critical value of noise where the
transition occurs are investigated analytically, and compared to numerical
simulations. The results for “partially dense” networks, comprised of
relatively few, but dynamically important nodes, which have a number of inputs
which greatly exceeds the average for the entire network, give very general
upper bounds on the maximum resilience against noise attainable on globally
sparse systems.
###### pacs:
05.40.-a, 05.40.Ca, 05.70.Fh, 02.50.Cw, 02.30.Sa, 87.16.Yc, 87.18.Cf,
89.75.-k, 89.75.Hc
## 1 Introduction
An essential feature of many self-organized and artificial systems of several
interacting elements is the ability of to function in a predictable fashion
even in the presence of stochastic fluctuations, which are inherent to the
system itself. Good examples are biochemical signaling networks and gene
regulation in organisms [1], as well as artificial digital circuits, and
communication networks [2]. In such systems, it is often the case that the
source of the fluctuations cannot be entirely removed, and the system must be
able to deal with them, by incorporating appropriate error-correction
measures. These may include specific dynamical properties [3, 4], choice of
functional elements and structural properties [5, 6], which one way or another
result in enough information redundancy, which can be used to counteract the
deviating effects of noise. In this work, the focus is turned on optimal
bounds which can be attained by a wide class of such systems, when many
parameters can be freely varied. More precisely, we consider a paradigmatic
system of dynamically interacting Boolean elements, regulated by Boolean
functions, where noise is introduced by the probability that at any time, any
input of a given function can be “flipped” to its opposite value, before the
output of the function is computed. The networks considered are regulated by
optimal majority functions, and can possess arbitrary topological structures.
The choice of majority functions corresponds to the limiting case where the
trade-off between robustness against noise and fitness for a given task is at
a maximum for every function in the network.
We obtain – both analytically and numerically – relevant properties of the
system, such as the average probability of error as a function of noise, and
critical value of noise, for which reliability is no longer possible. At this
noise threshold, the system undergoes a dynamic phase transition from a non-
ergodic regime, where a memory of its past states is preserved, to an ergodic
regime, where no such memory exists and every microstate is equally probable.
We identify the most relevant topological properties which can confer more
robustness to the system, namely the existence of a more densely connected
subset of the network, which is responsible for the dynamics of a significant
portion of the system. The properties of such optimal topologies serve as
general optimal bounds on the maximum resilience against noise which is
attainable by this class of system.
The behavior of similar systems under noise has been studied previously by a
number of authors. The dynamics of random Boolean networks (RBNs) with noise
(random functions and topology, not necessarily aiming at robustness [7]) was
studied in [8, 9, 10, 11, 12, 13, 14, 15]. The early works presented in [8, 9,
10] considered only small networks with $N\leq 20$ nodes, and focused on the
average crossing time between trajectories in state space which started from
different initial states. It was found that the trajectories must cross over
“barriers,” which correspond to the attractor basin boundaries. However, the
probability of crossing is always non-vanishing in such small systems. It was
further shown in [15] that the dynamics of RBNs is always ergodic for any
positive value of noise, and thus cannot preserve any memory of its past
states. However, this is not true for random networks composed of threshold or
majority functions, as shown in [11, 16]. These networks undergo the
aforementioned phase transition between ergodicity and non-ergodicity at a
critical value of noise. The same type of transition has also been observed
for Boolean systems composed of majority functions, but having acyclic and
stratified topology (i.e. Boolean _formulas_) [17, 18]. It was also shown in
[5] that this transition has a general character, since any Boolean network
can be made robust by introducing an appropriate restoration mechanism with
majority functions.
Boolean networks with majority functions share some similarities with the so-
called majority voter model [19, 20], which is usually defined on undirected
regular lattices. This system also undergoes a phase-transition based on
noise, which belongs to the universality class of the Ising model [21].
The issue of reliable computation under noise has also been tackled by the
mathematical community, starting with von Neumann [22], who was the first to
notice an important difference between reliable computation of noisy Boolean
circuits and the more general scenario of reliable communication considered by
Shannon [2], namely that it is not possible to guarantee an arbitrarily small
error rate, if the a given circuit has a fixed number of inputs per function.
He also pointed out that reliable computation is not at all possible for
Boolean functions with three inputs after a given noise threshold. His results
were later improved by Evans and Pippinger [23], who proved a similar bound
for Boolean formulas with two inputs per node, and finally Evans and Schulman
[24] who proved the bound for Boolean formulas with any odd number of inputs
per node. Recently, an extension to these bounds which are also valid for
functions with even number of inputs was derived in [25].
This paper is divided as follows. In section 2 we describe the model and in
section 3 we analyse the phase transition based on noise for several different
topological models: In 3.1 we consider random networks with a single-valued
in-degree distribution, and in 3.2 we extend the model to arbitrary in-degree
distributions. In 3.3 we consider a more general stochastic blockmodel, which
represents a much larger class of possible topological structures. We finalize
in section 4 with some concluding remarks.
## 2 The model
A Boolean Network (BN) [26, 27] is a directed graph of $N$ nodes representing
Boolean variables $\mathbf{\sigma}\in\\{1,0\\}^{N}$, which are subject to a
deterministic update rule,
$\sigma_{i}(t+1)=f_{i}\left(\bm{\sigma}(t)\right)$ (1)
where $f_{i}$ is the update function assigned to node $i$, which depends
exclusively on the states of its inputs. It is also considered that all nodes
are updated in parallel.
Here, noise is included in the model by introducing a probability $P$ that at
each time-step a given input has its value “flipped”: $\sigma_{j}\to
1-\sigma_{j}$, before the output is computed [15]. This probability is
independent for all inputs in the network, and many values may be flipped
simultaneously. The functions on all nodes are taken to be the majority
function, defined as
$f_{i}(\\{\sigma_{j}\\})=\begin{cases}1\text{ if
}\sum_{j}\sigma_{j}>k_{i}/2,\\\ 0\text{ otherwise, }\end{cases}$ (2)
where $k_{i}$ is the number of inputs of node $i$. The definition above will
lead to a bias if $k_{i}$ is an even number, since if the sum happens to be
exactly $k_{i}/2$ the output will be $0$, arbitrarily. Alternative definitions
could be used, which would remove the bias [28]. Instead, for the sake of
simplicity, in this work all values of $k_{i}$ considered will be odd, making
this bias a non-issue.
Starting from a given initial configuration, the dynamics of the system
evolves and eventually reaches a dynamically stable regime, where (for
sufficiently large systems) the average value $b_{t}$ of $1$’s no longer
changes, except for stochastic fluctuations [11]. In the absence of noise
($P=0$) there are only two possible attractors (if the network is sufficiently
random and not disjoint), where all nodes have the same value, which can be
either $0$ or $1$. We will consider these homogeneous attractors as being the
“correct” dynamics, and denote the deviations from them as “errors”. More
specifically, without loss of generality, we will name the value of $1$ as an
“error”, and the value of $b_{t}$ as the average error on the system.
We note that the above model has an optimal character regarding robustness
against noise, for the following two reasons: 1. It is known that the majority
function as defined in Eq. 2 is optimal in the case of fully redundant inputs
(i.e. in the absence of noise, they all have simultaneously the same value),
which have an uniform and independent probability of being “flipped” by noise.
In this situation, the output of the majority function will be “correct” with
greater probability than any other function with the same number of inputs
[22, 24]. 2. The existence of only two possible attractors with uniform values
can be interpreted as an extremal trade-off between dynamical function and
increased resilience against noise: A network with more elaborate dynamics in
the absence of noise, composed of many attractors with smaller basis of
attraction, would be invariably more difficult to stabilize if noise is
present, since it would become harder to distinguish between dynamical states.
## 3 Dynamical phase transition based on noise
As previously defined, the average “error” on the network is characterized by
the average value of $1$’s in the network at a given time, $b_{t}$. In this
section we will obtain the value of $b_{t}$ for networks with different
topological characteristics. We will focus first on uniform random networks
with all functions having the same in-degree, and networks with arbitrary in-
degree distributions. We then move to an arbitrary blockmodel, which can
incorporate more general topological features.
### 3.1 Single-valued in-degree distribution
In this session, we compute the value of $b_{t}$ for networks composed of
nodes with the same number $k$ of inputs per node, which are randomly chosen
between all possible nodes. This type of system has been studied before by
Huepe et al [11] and is essentially equivalent to the same problem posed for
Boolean _formulas_ by Evans et. al [24], since the presence of short loops can
be neglected for large networks. For the sake of clarity, we shortly reproduce
the analysis developed in [24], and we extend it by calculating the critical
exponent of the transition. We then proceed to generalize the approach to more
general topologies in the subsequent sections.
In order to obtain an equation for the time evolution of $b_{t}$ we employ the
usual annealed approximation [29], which assumes that at each time step the
inputs of every function are randomly re-sampled, such that any quenched
topological correlations are ignored, and all inputs will have the same
probability $b_{t}$ of being equal to $1$. If the inputs of a majority
function have a value of $1$ with probability $b$ (independently for each
input), the output will also be $1$ with a probability given by
$m_{k}(b)=\sum_{i={\lceil k/2\rceil}}^{k}{k\choose i}b^{i}(1-b)^{k-i}.$ (3)
The time evolution of $b_{t}$ can then be written as
$b_{t+1}=m_{k}\left((1-2P)b_{t}+P\right),$ (4)
where $P$ is the noise probability, as described previously. The right-hand
side of Eq. 4 is symmetric in respect to values of $b_{t}$ around $1/2$ (as
can be seen in Fig. 1), such that the dynamics for values of
$b^{\prime}_{t}>1/2$, can be obtained from $b^{\prime}_{t}=1-b_{t}$, with
$b_{t}<1/2$. Thus, without loss of generality, we will only consider the case
$b_{t}\leq 1/2$ throughout the paper.
Given any initial starting value $b_{0}\leq 1/2$, the dynamics will always
lead to a fixed point $b^{*}\leq 1/2$, which is a solution of Eq. 4, with
$b_{t+1}=b_{t}\equiv b^{*}$. This is in general a solution of a polynomial of
order $k$, for which there are no general closed-form expression. However,
since the right-hand side of Eq. 4 is a monotonically increasing function on
$b_{t}$, we can conclude there can be at most two possible fixed points:
$b^{*}=1/2$ (ergodic regime) or $b^{*}<1/2$ (non-ergodic regime). Furthermore,
considering the right-hand side of Eq. 4 is a convex function (for $b_{t}\leq
1/2$, as is always assumed), if the fixed point $b^{*}=1/2$ becomes stable,
i.e. $\frac{db_{t+1}}{db_{t}}|_{b^{*}=1/2}\leq 1$, the other fixed point
$b^{*}<1/2$ must cease to exist, since in this case $b_{t+1}>b_{t}$ for any
$b_{t}<1/2$. Thus, the value of $P$ for which $b^{*}=1/2$ becomes a stable
fixed point marks the transition from non-ergodicity to ergodicity. In order
to obtain this value, we need to compute the the derivative of the right-hand
side of Eq. 4 in respect to $b_{t}$. Using the derivative of Eq. 3 (see [24]
for a detailed derivation of this expression),
$m^{\prime}_{k}(b)\equiv\frac{dm_{k}(b)}{db}=\frac{k}{2^{k-1}}{k-1\choose{\lfloor
k/2\rfloor}}[1-(1-2b)^{2}]^{{\lfloor k/2\rfloor}}$ (5)
we have that $(1-2P^{*})m^{\prime}_{k}(1/2)=1$, where $P^{*}$ is the critical
value of noise. Thus, a full expression for $P^{*}$ is given by
$P^{*}=\frac{1}{2}-\frac{2^{k-2}}{k{k-1\choose{\lfloor k/2\rfloor}}}.$ (6)
Taking the limit $k\gg 1$, one obtains
$P^{*}\approx\frac{1}{2}-\frac{1}{2}\sqrt{\frac{\pi}{2k}}$ using the Stirling
approximation. Eq. 6 is the main result of [24]. We note however that a
slightly less explicit but more general expression was derived previously in
[11], for the case where the majority function accepts inputs with different
weights.
For a given value of $k$, the value of $b^{*}$ increases continuously with $P$
until it reaches $1/2$ for $P\geq P^{*}$ (see Fig. 1), characterizing a
second-order phase transition. One can go further and obtain the critical
exponent of the transition by expanding Eq. 3 near $b=1/2$,
$m_{k}(b)=\frac{1}{2}-\frac{1}{2}m^{\prime}_{k}(1/2)(1-2b)+\frac{1}{6}{\lfloor
k/2\rfloor}m^{\prime}_{k}(1/2)(1-2b)^{3}+O\left((1-2b)^{5}\right)$ (7)
and using it in 4, and solving for $b^{*}=b_{t+1}=b_{t}$, which leads to
$b^{*}\approx\frac{1}{2}-\left[\frac{3}{2}\frac{m^{\prime}_{k}(1/2)^{3}}{{\lfloor
k/2\rfloor}}\widetilde{P}\right]^{1/2}$ (8)
where $\widetilde{P}=P^{*}-P$. From this expression it can be seen that the
critical exponent is $1/2$, corresponding to the mean-field universality
class.
Figure 1: The dynamic map of Eq. 4 for different values of $P$ (left), and the
value of the stable fixed-point $b^{*}\leq 1/2$, as a function of $P$ (right).
The values of $b^{*}$ and $P^{*}$ can be understood as general bounds on the
minimum error level and maximum tolerable noise, respectively, which must hold
for random networks composed of functions with the same number of inputs.
These are rather stringent conditions, and it is possible to imagine
interesting situations where they are not fulfilled. Therefore, for more
general bounds, one needs to relax these restrictions. We proceed in this
direction in the following section, where we consider the case of arbitrary
in-degree distributions, but otherwise random connections among the nodes.
### 3.2 Arbitrary in-degree distributions
We turn now to uncorrelated random networks with an arbitrary distribution of
inputs per node (in-degree), $p_{k}$. Here it is assumed that the inputs of
each function are randomly chosen among all possibilities, and that the in-
degree distribution $p_{k}$ provides a complete description of the network
ensemble. This configuration was also considered in [16], for a more general
case where the inputs can have arbitrary weights. We analyse here the special
case with no weights in more detail, and obtain more explicit results.
The annealed approximation can be used in the same manner as in the previous
section: One considers simply that at each time step the inputs of each
function are randomly chosen 111Note that this input “rewiring” has no effect
on the in-degree distribution.. The time evolution of $b_{t}$ now becomes,
$b_{t+1}=\sum_{k}p_{k}m_{k}\left((1-2P)b_{t}+P\right).$ (9)
Like for Eq. 4, there are only two fixed points $b^{*}\leq 1/2$, and the
transition can be obtained by analysing the stability of the fixed point
$b^{*}=1/2$. In an entirely analogous fashion to Eq. 6, using the derivative
of the right-hand side of Eq. 9 one obtains the following expression for the
critical value of noise,
$P^{*}=\frac{1}{2}-\left[\sum_{k}p_{k}\frac{k{k-1\choose{\lfloor
k/2\rfloor}}}{2^{k-2}}\right]^{-1}.$ (10)
Considering the limit where all $k\gg 1$, one has
$P^{*}\approx\frac{1}{2}-\left[\sqrt{\frac{8}{\pi}}\sum_{k}p_{k}\sqrt{k}\right]^{-1}$.
Note that the above expression only holds if $p_{k}=0$ for every $k$ which is
even, as is assumed throughout the paper. The critical exponent can also be
calculated in an analogous fashion, and is always $1/2$, unless $p_{k}$ has
diverging moments. In this case the critical exponents will depend on the
details of the distribution (see [16] for a more thorough analysis).
With this result in mind, one can ask the following question: What is the best
in-degree distribution, for a given average in-degree ${\left<k\right>}$, such
that either $b_{t}$ is minimized or $P^{*}$ is maximized? As it will now be
shown, in either case the best distribution is the single-valued distribution,
already considered in the previous section. For simplicity, let us consider
the case where ${\left<k\right>}$ is discrete and odd. We begin with the
analysis of $b_{t}$. We can observe that for $b\leq 1/2$, $m_{k}(b)$ is a
convex function on $k$ (see Fig 2),
$m_{k}(b)\leq\frac{m_{k-2}(b)+m_{k+2}(b)}{2},$ (11)
and thus by Jensen’s inequality we have that
$m_{{\left<k\right>}}(b)\leq{\left<m_{k}(b)\right>}$. Since the equality only
holds only for the single-valued distribution
$p_{k}=\delta_{k,{\left<k\right>}}$ (assuming $b\notin\\{0,1/2\\}$), the
right-hand side of Eq. 9 will always be larger for any other distribution
$p_{k}$. The same argument can be made for the value of $P^{*}$: Since we have
that $(1-2P^{*})^{-1}=\sum_{k}p_{k}m^{\prime}_{k}(1/2)$, and
$m^{\prime}_{k}(1/2)$ is a concave function on $k$,
$\displaystyle\frac{m^{\prime}_{k-2}(1/2)+m^{\prime}_{k+2}(1/2)}{2}$
$\displaystyle=\frac{m^{\prime}_{k}(1/2)}{2}\left[\frac{k-1}{k}+\frac{k+2}{k+1}\right]$
(12) $\displaystyle=m^{\prime}_{k}(1/2)\left[1-\frac{1}{2k(k+1)}\right]$ (13)
$\displaystyle<m^{\prime}_{k}(1/2)$ (14)
we have that
$m^{\prime}_{{\left<k\right>}}(1/2)\geq{\left<m^{\prime}_{k}(1/2)\right>}$.
Again, the equality only holds only for $p_{k}=\delta_{k,{\left<k\right>}}$,
which is therefore the optimal scenario.222Of course, this argument does not
hold if ${\left<k\right>}$ is not discrete and odd, since in this case the
distribution cannot be single-valued. But the above argument should make it
sufficiently clear that in this case the optimal distribution should also be
very narrow, and similar to the single-valued distribution.
Figure 2: Convexity of $m_{k}(b)$, as stated in Eq. 11.
One special case which merits attention is the scale-free in-degree
distribution
$p_{k}\propto k^{-\gamma},$ (15)
which occurs often in many systems, including, as some suggest, gene
regulation [30]. It is often postulated that networks with such a degree
distribution are associated with different types of robustness, due to their
lower percolation threshold [31] which can be interpreted as a resilience to
node removal “attacks”. However, in the case of robustness against noise Eq.
15 by itself does not confer any advantage. For instance, from Eq. 10, using
Stirling’s approximation one sees that the expression within brackets will
diverge only if $\gamma\leq 3/2$, leading to $P^{*}=1/2$. This means that for
$3/2<\gamma\leq 2$, we have that the average in-degree diverges
(${\left<k\right>}\to\infty$) but the critical value of noise is still below
$1/2$. This is considerably worse, for instance, than a fully random network
with in-degree distribution given by a slightly modified Poisson, which is
defined only over odd values of $k$,
$p_{k}=\frac{1}{\sinh\lambda}\frac{\lambda^{k}}{k!},$ (16)
with ${\left<k\right>}=\lambda/\tanh\lambda$. For this distribution, we have
that $P^{*}\to 1/2$ for ${\left<k\right>}\to\infty$, as one would expect also
for the single-valued distribution. A comparison between these two
distributions is shown in Fig. 3.
Figure 3: Critical value of noise $P^{*}$ as a function of $\gamma$ for the
scale-free in-degree distribution given by Eq. 15 and for the Poisson
distribution given by Eq. 16, where $\lambda$ is chosen such that the average
in-degree is the same for both distributions.
The above analysis shows that the single-valued in-degree distribution
$p_{k}=\delta_{k,{\left<k\right>}}$ is the best one can hope for with a given
average in-degree ${\left<k\right>}$, as long as the inputs of each function
are randomly chosen. However, this is a restriction which does not need be
fulfilled in general. In order to obtain more general bounds, one needs to
depart from this restriction, and consider more heterogeneous possibilities,
which is the topic of the next section.
### 3.3 Arbitrary topology: Stochastic blockmodels
We now consider a much more general class of networks known as stochastic
blockmodels [32, 33, 34], where it is assumed that every node in the network
can belong one of $n$ distinct classes or “blocks”. Every node belonging to
the same block has on average the same characteristics, such that we need only
to describe the degrees of freedom associated with the individual blocks. In
particular we use the degree-corrected variant [35] of the traditional
stochastic blockmodel, which incorporates degree variability inside the same
block. Here, we define $w_{i}$ to be the fraction of the nodes in the network
which belong to block $i$, and $p^{i}_{k}$ is the in-degree distribution of
block $i$. The matrix $w_{j\to i}$ describes the fraction of the inputs of
block $i$ which belong to block $j$. We have therefore that $\sum_{i}w_{i}=1$,
$\sum_{j}w_{j\to i}=1$ and $\sum_{i,k}kw_{i}p^{i}_{k}={\left<k\right>}$. Since
the out-degrees are not explicitly required to describe the dynamics, they
will be assumed to be randomly distributed, subject only to the restrictions
imposed by $w_{i}$ and $w_{j\to i}$.
In the limit where the number of vertices $Nw_{i}$ belonging to each blocks
$i$ is arbitrary large, we can use a modified version of the annealed
approximation to describe the dynamics: Instead of randomly re-assigning
inputs for each function, we choose randomly only amongst those which do not
invalidate the desired block structure. In other words, we impose that after
each random input rewiring, the inter-block connections probabilities are
always given by $w_{j\to i}$. In this way, we maintain the dynamic
correlations associated with the block structure, and remove those arising
from quenched topological correlations present in a single realization of the
blockmodel ensemble. Due to the self-averaging properties of this ensemble,
for sufficiently large networks the annealed approximation is expected to be
exact, in the same way it is for random networks without block structures.
With this ansatz, we can write the average value of $b_{i}$ for each block
over time as
$b_{i}(t+1)=\sum_{k}p_{k}^{i}m_{k}\left((1-2P)\sum_{j}w_{j\to
i}b_{j}(t)+P\right),$ (17)
which is a system of $n$ coupled maps. It is easy to see that $b^{*}_{i}=1/2$
is a fixed point of Eq. 17. In order to perform the stability analysis we have
to consider the Jacobian matrix of the right-hand side of Eq. 17,
$J_{ij}=\frac{\partial b_{i}(t+1)}{\partial b_{j}(t)}=(1-2P)w_{j\to
i}\sum_{k}p^{i}_{k}m^{\prime}_{k}\left((1-2P)\sum_{j}w_{j\to
i}b_{j}(t)\right).$ (18)
At the fixed-point $b_{i}(t)=1/2$ we can write the Jacobian as
$\bm{J^{*}}=(1-2P)\bm{M},$ (19)
where matrix $\bm{M}$ is given by
$M_{ij}=w_{j\to i}\sum_{k}p^{i}_{k}m^{\prime}_{k}(1/2).$ (20)
The largest eigenvalues of $\bm{J^{*}}$ and $\bm{M}$, $\lambda$ and $\xi$
respectively, are related to each other simply by $\lambda=(1-2P)\xi$. Since
the fixed-point in question will cease to be stable for $\lambda=1$, we have
that the critical value of noise is given by
$P^{*}=\frac{1}{2}-\frac{1}{2\xi}.$ (21)
Thus, for $P>P^{*}$ the fixed point $b_{i}(t)=1/2$ becomes a stable fixed-
point, and this marks the transition from non-ergodicity to ergodicity, as in
the previous cases.
We note that the sizes of the blocks $w_{i}$ play no role in Eq. 21, and only
the correlation probabilities $w_{i\to j}$ and the in-degree distributions
$p^{i}_{k}$ define the value of $P^{*}$. For this reason, the average error
$b^{*}=\sum_{i}w_{i}b_{i}$ on the network may not be always a suitable order
parameter to identify the aforementioned phase transition, since the blocks
which are responsible for the value of $P^{*}$ may be arbitrarily small in
comparison to the rest of the network. However, these are obviously corner
cases, since the most interesting situations are those where all blocks are
relevant to the dynamics (or a given block could be otherwise ignored).
Given any desired many-block structure, one could find the largest eigenvalue
$\xi$ of the matrix $\bm{M}$ and then determine the critical value of noise
with Eq. 21. In the following, we will focus on the simplest nontrivial block
structure which is composed only of two blocks. Such 2-block systems are fully
accessible analytically, and are sufficient to obtain more general upper and
lower bounds on the values of $P^{*}$ and $b^{*}$, respectively.
### 3.4 2-block structures
Here we consider networks composed of two blocks, where the block with the
largest average in-degree will be labeled “core”. The size and average in-
degree of the core block are $w_{c}$ and $k_{c}$ respectively, and for the
non-core block $w_{r}=1-w_{c}$ and
$k_{r}=({\left<k\right>}-w_{c}k_{c})/(1-w_{c})$. For simplicity, we will
consider that the in-degree distribution of each block is the single-valued
distribution $p^{i}_{k}=\delta_{k,k_{i}}$, where $k_{i}$ is the average in-
degree of the block.
The matrix $w_{j\to i}$ has the general form
$\bm{w_{\to}}=\left(\begin{array}[]{cc}w_{c\to c}&w_{c\to r}\\\ w_{r\to
c}&w_{r\to r}\end{array}\right)=\left(\begin{array}[]{cc}m_{c}&m_{r}\\\
1-m_{c}&1-m_{r}\end{array}\right),$ (22)
with only two free variables $m_{c}$ and $m_{r}$, denoting the fraction of
inputs which belong to the core block, for both blocks. Instead of considering
all possible values of $m_{c}$ and $m_{r}$, we consider the following
parametrization
$\displaystyle m_{c}$ $\displaystyle=\begin{cases}4a(1-a)w_{c}&\text{if }a\leq
1/2\\\ m_{r}&\text{if }a>1/2\end{cases}$ (23) $\displaystyle m_{r}$
$\displaystyle=1-4a(1-a)(1-w_{c}),$
where the single parameter $a\in[0,1]$ allows for the topology to be
continuously varied between three distinct topological configurations (see
Fig. 4): For $a=0$ we have a “restoration” topology, where the network is
bipartite, and all inputs from the non-core block belong to the core block and
vice-versa; for $a=1/2$ the inputs are randomly selected; and for $a=1$ we
have a “segregated core” structure, where all the inputs of both blocks belong
exclusively to the core block.
$a=0$
Restoration
$a=1/2$
Random
$a=1$
Segregated core
Figure 4: Three distinct 2-block structures possible with the parametrization
given by Eq. 23, for different values of the parameter $a$.
For this system we can write the matrix $\bm{M}$ from Eq. 20 as
$\bm{M}=\left(\begin{array}[]{cc}w_{c\to c}m^{\prime}_{k_{c}}(1/2)&w_{r\to
c}m^{\prime}_{k_{c}}(1/2)\\\ w_{c\to r}m^{\prime}_{k_{r}}(1/2)&w_{r\to
r}m^{\prime}_{k_{r}}(1/2)\end{array}\right),$ (24)
from which we can extract the largest eigenvalue $\xi$,
$\xi=\frac{1}{2}(w_{c\to c}m^{\prime}_{k_{c}}(1/2)+w_{r\to
r}m^{\prime}_{k_{r}}(1/2))\quad+\\\ \frac{1}{2}\sqrt{4w_{r\to c}w_{c\to
r}m^{\prime}_{k_{c}}(1/2)m^{\prime}_{k_{r}}(1/2)+\left(w_{c\to
c}m^{\prime}_{k_{c}}(1/2)-w_{r\to r}m^{\prime}_{k_{r}}(1/2)\right)^{2}}.$ (25)
From $\xi$, the critical value of noise can be obtained by Eq. 21.
The general behaviour of the asymptotic average error
$b^{*}\equiv\lim_{t\to\infty}{\left<b_{i}(t)\right>}$, computed from Eq. 17 as
a function of $a$ is shown in Fig. 5 for ${\left<k\right>}=5$ and $k_{r}=3$,
and several values of $k_{c}$ (and $w_{c}$ chosen accordingly). In the same
figure are shown results from numerical simulations of quenched networks with
$N=10^{5}$ nodes, evolved according to Eq. 1, showing perfect agreement. On
the right of Fig. 5 are shown the values of $b^{*}$ according to the reduced
noise $P-P^{*}$, with $P^{*}$ computed according to Eqs. 25 and 21. The
calculated values of $P^{*}$ for several values of $k_{c}$ are plotted on the
right of Fig. 6. The nature of the phase transition is systematically the
same, as can be seen in the right of Fig. 6, where the slope of the curves
correspond to mean-field critical exponent $1/2$.
Figure 5: Average error $b^{*}$ as a function of $a$ for different values of
noise $P$ (left) and as a function of the reduced noise $P-P^{*}$, with
$P^{*}$ computed according to Eqs. 25 and 21, for several values of $a$
(right). All curves are for ${\left<k\right>}=5$, $k_{r}=3$ and $k_{c}=19$.
The symbols are results of numerical simulations of quenched networks with
$N=10^{5}$ nodes, and the solid lines are numerical solutions of Eq. 17.
Figure 6: Critical value of noise $P^{*}$ as a function of $a$, for several
values of $k_{c}$, with $k_{r}=3$ and ${\left<k\right>}=5$ (left), and value
of $1-2b^{*}$ as a function of $P-P^{*}$ close to the critical point, for
different values of ${\left<k\right>}$, $a$, $k_{r}$ and $k_{c}$ (right). The
dashed line corresponds to a function proportional to $(P-P^{*})^{1/2}$.
It is interesting to compare the performance of the restoration ($a=0$) and
segregated core ($a=1$) topologies. Both outperform the random topology
($a=1/2$), but the segregated core is always the best possible, having both
the lowest values of $b^{*}$ and largest values of $P^{*}$. This is not
surprising, since the segregated core is nothing more than an isolated
network, which is more densely connected than the whole network, to which the
remaining nodes are enslaved. On the other hand it is rather interesting how
the restoration topology ($a=1$) is only marginally worse than the segregated
core, since in this situation every node is dynamically relevant. We note that
the relative advantage of the partially random topologies ($0<a<1$) may depend
on the actual value of noise. This can be seen in Fig. 5 (right), where the
curves for $b^{*}$ with different values of $a$ cross each other when
$P-P^{*}$ is varied (the same is also observed when the curves are plotted
against $P$). The reason for this is that the relative advantage of the
segregated core topology in respect to restoration may manifest itself only as
the value of noise approaches the critical point. For lower values of noise it
is possible, for instance, for a full restoration topology with $a=0$ to
outperform a partial segregated core structure with $a=0.9$, since it will
perform comparably to a full segregation, $a=1$ (see Fig. 5, left). However,
as noise is increased the relative advantage of the segregated topology makes
up for this difference. In the general case, therefore, the optimal topology
will depend on the value of noise.
Either with the restoration and segregated core topologies, the values of
$b^{*}$ and $P^{*}$ become increasingly better for larger values of $k_{c}$,
as can be seen in Figs. 6 and 7. One can therefore postulate that an optimum
bound can be achieved for $k_{c}\to\infty$. Let us consider the situation
where $w_{c}\propto 1/k_{c}$, such that
$\lim_{k_{c}\to\infty}{\left<k\right>}=k_{r}$. For both $a=0$ and $a=1$ the
value of $b^{*}$ approaches asymptotically $m_{{\left<k\right>}}(P)$, for
$k_{c}\to\infty$, as can be seen in Fig. 7. This means that the average error
of the core nodes will eventually vanish, and the remaining nodes will
encounter the optimal scenario where the inputs are affected by the noise $P$
alone, and the error does not accumulate over time. It is therefore safe to
conclude that
$b_{\text{min}}=m_{{\left<k\right>}}(P)$ (26)
is a general lower bound on the average error on a network with average in-
degree ${\left<k\right>}$ and an arbitrary topology, which is asymptotically
achieved for both the restoration and segregation topologies, for
$k_{c}\to\infty$.
Figure 7: Values of $b^{*}$ as a function of $P$ for 2-block structures with
$a=1$ (left) and $a=0$ (right), with $w_{c}=1/(100\times k_{c})$, $k_{r}=5$
and several values of $k_{c}$. The dashed curves are given by Eq. 26 with
${\left<k\right>}=k_{r}$.
## 4 Conclusion
We have investigated the behaviour of optimal Boolean networks with majority
functions and different topologies in the presence of stochastic fluctuations.
The dynamics of these networks undergo a phase transition from ergodicity to
non-ergodicity. The non-ergodic regime can be can be interpreted as robustness
against noise, since there is a permanent global memory of the initial
condition. The ergodic phase, on the other hand, represents a situation where
the effect of noise has destroyed any possible long-term dynamical
organization of the system. We obtained, both analytically and numerically,
the average error and the critical value of noise for networks composed of
arbitrary in-degree distributions and for a more general stochastic
blockmodel, which can accommodate a wide variety of network structures. We
showed that both the average error level as well as the critical value of
noise are improved both for the segregated core and restoration topologies,
where the dynamics is dominated by a smaller subset of nodes, which have an
above-average in-degree. In the limit where the average in-degree of these
“core” nodes diverges, the network achieves an optimum bound, which
corresponds to the maximum resilience attainable.
In a separate work [6], we show that segregated core structures emerge
naturally out of an evolutionary process which favors robustness against
noise.
As was discussed, the networks considered are made from optimal elements,
which in isolation have the best possible behaviour. Because of this, the
results obtained have a general character, and show the best scenario which
can in general be achieved, under the constraints considered. However, it is
important to point out that there are different types of stochastic
fluctuations which can be considered in Boolean systems. Other than the type
of noise considered in this work, it is possible for instance to incorporate
fluctuations in the _update schedule_ of the nodes [36]. It has been shown in
[37], for random networks, that even if the update schedule is completely
random, ergodicity is preserved, and the dynamics eventually leads to distinct
attractors. Furthermore, it was shown in [3] that it is possible to obtain
absolute resilience against noise in the update sequence, where the
trajectories are always the same, independent of the update schedule used. In
[4] this type of resilience has been coupled with single-flip perturbations,
which correspond to very small values of the noise parameter $P$ considered in
this work, and it was shown that arbitrary mutual resilience is also possible.
The broader question of how a single system can be simultaneously robust
against many different types of perturbations, and which features become more
important in this case, still needs to be systematically tackled.
## References
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|
arxiv-papers
| 2011-08-22T14:35:37 |
2024-09-04T02:49:21.684192
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tiago P. Peixoto",
"submitter": "Tiago Peixoto",
"url": "https://arxiv.org/abs/1108.4329"
}
|
1108.4335
|
,
# The Quantification of Quantum Nonlocality by Characteristic Function
Wei Wen ( ΰ) Corresponding author:chuxiangzi@semi.ac.cn Shu-Shen Li ( ) State
Key Laboratory for Superlattices and Microstructures, Institute of
Semiconductors, Chinese Academy of Sciences, China
###### Abstract
Quantum nonlocal correlation (QNC) is thought to be more general than quantum
entanglement correlation, but the strength of it has not been well defined. We
propose a way to measure the strength of QNC basing on the characteristic
function. The characteristic function of QNC in a composite system is defined
as a response function under the local quantum measurement. It is explored
that once a characteristic function is given, the state of a composite system,
with just a local trace-preserving quantum operation uncertainty, will be
determined. We show that the strength of QNC basing on the characteristic
function is a half-positive-definite function and does not change under any LU
operation. For a two-partite pure state, the strength of QNC is equivalent to
the quantum entanglement. Generally, we give a new definition for quantum
entanglement using the strength function. Furthermore, we also give a
separability-criterion for $2\times m$-dimensional mixed real matrix. This
letter proposes an alternate way for QNC further research.
###### keywords:
Quantum Nonlocality, Characteristic Function, Strength of QNC, Quantum
Entanglement, Schrödinger Steering
###### PACS:
03.67.Mn; 03.65.Ud
One of the most subtle phenomena in quantum theory is quantum nonlocal
correlation (QNC). Although a large amount of research on QNC has been done,
it has mainly arisen from the view that nonlocality cannot be described by any
local hidden variable theory (LHV). Based on this, some new concepts have been
proposed, such as Bell nonlocality[1], quantum entanglement [2, 3, 4],
Schrödinger’s steerability [5, 6, 7, 8] and so on, which are all defined by
different forms of the local joint quantum measurement (LJQM) probability
$P(a,b|A,B;W)$ [7, 8]. However, the question about what is the QNC is still
far from being solved. Till now, although much attention is paid on the
information perspective, the physical aspect of the QNC is also worth
studying.
Recently, some other researchers have paid attention to other unorthodox
methods and put forward that nonlocality can be more general [9, 10, 11, 12,
13]. For example, Bandyopadhyay present that the nonlocality can be redefined
by local indistinguishability of a set of orthogonal quantum states, and show
that more nonlocality may be with less purity [12]. Luo and Fu point out that
the measurement can induce the nonlocality nonlocality [13]. These works all
try to find a new way to study quantum nonlocality and inspirit the motivation
to reinspect the physical action played in the QNC.
In this Letter, we will further study the quantum nonlocal correlation. We
regard the QNC as one sort of elements in the state of a composite system, of
which the other element is the set of subsystems. We think the QNC in every
special composite system has its own character to be distinguished from
others. We express the relation between QNC and the state of a composite
system with the mathematical language as follows,
$\mathrm{\mathbb{B}}:\rho_{ABC\cdots}\rightarrow\\{\\{\rho_{A},\rho_{B},\rho_{C},\cdots\\},\\{\mathrm{QNC}s\\}\\}.$
(1)
The function $\mathbb{B}$ is a bijection and every composite state is mapped
into the set of subsystems and the QNC between them. Our task is to find a
mathematical expression to describe the QNC
$\mathrm{\mathbb{B}}^{\prime}:\rho_{ABC\cdots}\rightarrow\\{\\{\rho_{A},\rho_{B},\rho_{C},\cdots\\},\mbox{\boldmath$\mathrm{F}$}_{QNC}\\}.$
(2)
We call the function $\mathrm{F}$ that maps the abstract physical quantity
$\mathrm{QNC}$ to a mathematica quantity as a characteristic function of QNC.
This is because, just like a special fingerprint corresponds a special man, a
special function $\mathrm{F}$ corresponds a special composite state once the
its sub-states are fixed on. Therefore, we can use it to analysis QNC just
like using density matrix to analysis the state of system.
Additionally,theoretically speaking, we can redefine the concepts Bell
nonlocality, quantum entanglement, Schrödinger’s steerability and so on basing
on the characteristic function. In fact, in the following text, we find a new
formulation of quantum entanglement.
Before expatiating on the characteristic function, we will introduce some
intuitive-right but unobvious conclusions first. An unlimited quantum
measurement is defined as a physical process in which the projection operator
can be arbitrarily chosen and the number of copies of the unknown state
measured is sufficient. Considering a general situation, an arbitrary
projection operator in a finite $n$ Hilbert space is expressed as
$\hat{M}(\Theta,\Psi)=|\psi(\Theta,\Psi)\rangle\langle\psi(\Theta,\Psi)|$,
where $\Theta$ and $\Phi$ are the sets of variables $\\{\psi_{k}\\}$ and
$\\{\theta_{k}\\}$. $|\psi(\Theta,\Phi)\rangle$ is a pure state in
$n$-dimensional space,
$|\psi(\Theta,\Psi)\rangle=\sum_{k=1}^{n}a_{k}|k\rangle$, where
$a_{k}=\prod_{l=1}^{k-1}\sin(\theta_{l})\cos(\theta_{k})e^{\mathrm{i}\phi_{l}}$
when $k<n$ and $a_{n}=\prod_{l=1}^{n-1}\sin(\theta_{l})$. Under the unlimited
quantum measurement, any unknown state $\rho$ can be distinguished out and
this is shown in the following lemma.
###### lemma 1
If $\rho_{1}$, $\rho_{2}$ are density matrixes and $\hat{M}(\Theta,\Phi)$ is
the projection operator in a finite $n$-dimensional Hilbert space, we say
$\rho_{1}=\rho_{2}$ when
$\mathrm{Tr}(\hat{M}(\Theta,\Phi)\rho_{1})=\mathrm{Tr}(\hat{M}(\Theta,\Phi)\rho_{2})$
for $\forall\theta_{i}\in(0,\pi)$ and $\forall\phi_{i}\in(0,2\pi)$.
###### Proof 1
It is known any $n\times n$ density matrix $\rho_{x}$ can be decomposed in an
orthonormal basis ${\Gamma_{n}^{(k)}}$ of traceless generators of group
$SU(n)$ [14, 15]
$\rho_{x}=\frac{1}{n}(1+\sum_{k=1}^{n^{2}-1}r_{x}^{(k)}\Gamma_{n}^{(k)}),$ (3)
where $r_{x}^{(k)}=\mathrm{Tr}(\rho_{x}\Gamma_{n}^{(k)})$ and generators
$\Gamma_{n}^{(k)}$ have the property of
$\mathrm{Tr}(\Gamma_{n}^{(i)}\Gamma_{n}^{(j)})=n\delta_{ij}$. The $n^{2}-1$
real parameters $r_{x}^{(k)}$ will uniquely determine a density matrix
$\rho_{x}$. Therefore, we can use vector
$\mbox{\boldmath$r$}_{x}=(r_{x}^{(1)},r_{x}^{(2)},\ldots,r_{x}^{(n^{2}-1)})$
to represent the density matrix. The length of vector
$|\mbox{\boldmath$r$}|=\sqrt{n-1}$.
According to Eq. (3), when $n=2$, $\rho_{1}$ and $\rho_{2}$ can be replaced
with vectors $\mbox{\boldmath$r$}_{1}$ and $\mbox{\boldmath$r$}_{2}$ in the
Bloch-Sphere and $\hat{M}$ with $\mbox{\boldmath$r$}_{M}(\theta,\phi)$ on the
surface of the Bloch-Sphere. Consequently,
$\mathrm{Tr}(\hat{M}\rho_{1})=\mathrm{Tr}(\hat{M}\rho_{2})$ means that
$\mbox{\boldmath$r$}_{M}(\theta,\phi)\cdot(\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2})=0$.
The last equation holds iff
$\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}=0$ , namely
$\rho_{1}=\rho_{2}$.
For $n>2$, we can always reduce it to $n(n-1)/2$ $2$-dimensional subsystems
and proof any corresponding subsystems equal. To detail this process, we
define an operation $V(i,j)_{n,m}=\delta_{n,m}(\delta_{n,i}+\delta_{n,j})$
first. It is known that
$R_{s}(i,j)=V(i,j)RV(i,j)^{{\dagger}}$ (4)
satisfies $R_{i,j}=(R_{s}(i,j))_{i,j}$, where $R$ is an $n\times n$ matrix.
Moreover, $\hat{M}_{s}(i,j)=V(i,j)\hat{M}V(i,j)^{{\dagger}}$ is an equivalent
$2$-dimensional projection operator and can be realized by setting some
variables $\phi_{k}$ and $\theta_{k}$ to zero.
$\mathrm{Tr}(\hat{M}_{s}(i,j)\rho_{1})=\mathrm{Tr}(\hat{M}_{s}(i,j)\rho_{2})$
for every $i$ and $j$, namely
$V(i,j)\rho_{1}V(i,j)^{{\dagger}}=V(i,j)\rho_{2}V(i,j)^{{\dagger}}$. Therefore
$\rho_{1}=\rho_{2}$.
This theorem is very important in this letter to get the characteristic
function. It is shows us that any states, mixed or pure, can be distinguished
by unlimited quantum measurements. It is anti-intuition because the general
opinion is that $n^{2}-1$ parameters are needed to fix on an arbitrary
$n\times n$ density matrix, but we show here that quantum measurements with
$2n-1$ parameters are just enough. Following this, we will give a corollary
about the local quantum measurement. It will be shown that unlimit local
quantum measurement can also explore the whole information of the composite
system.
###### Corollary 1
If $\rho_{AB}$, $\rho_{AB}^{\prime}$ are density matrixes in $n_{A}\otimes
n_{B}$-dimensional Hilbert space and $\hat{M}_{A}(\Theta,\Phi)$ is the
projection operator of system $A$, we say $\rho_{AB}=\rho_{AB}^{\prime}$ when
$\mathrm{Tr}_{A}(\hat{M}_{A}(\Theta,\Phi)\rho_{1})=\mathrm{Tr}_{A}(\hat{M}(\Theta,\Phi)\rho_{2})$
for $\forall\theta_{i}\in(0,\pi)$ and $\forall\phi_{i}\in(0,2\pi)$.
###### Proof 2
For a composite system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, the
orthonormal basis
$\\{\Gamma_{AB}^{k}\\}=\\{\Gamma_{A}^{i}\otimes\Gamma_{B}^{j}\\}$; $n_{B}\cdot
i+j=1,2,\ldots,n_{A}n_{B}$. For convenience, $\Gamma^{(0)}=1$ here. Therefore,
it is shown that
$\displaystyle\hat{M}_{A}\otimes I_{B}=\frac{I_{A}\otimes
I_{B}}{n_{A}}+\sum_{i=1}^{n_{A}}r_{M}^{(i)}\Gamma_{A}^{(i)}\otimes I_{B};$
$\displaystyle\rho_{AB}=\frac{I_{A}\otimes
I_{B}}{n_{A}n_{B}}+\sum_{j+k=1}^{n_{A},n_{B}}r_{AB}^{(j,k)}\Gamma_{A}^{(j)}\otimes\Gamma_{B}^{(k)}.$
(5)
It is noted that only these terms $\Gamma_{A}^{(0)}\Gamma_{B}^{(i)}$ remain
when a partial trace is done under the system $A$, and then we take the form
$\displaystyle\mathrm{Tr}_{A}(\hat{M}_{A}\rho_{AB})-\mathrm{Tr}_{A}(\hat{M}_{A}\rho_{AB}^{\prime})$
$\displaystyle=\sum_{k}\mbox{\boldmath$r$}_{M}^{\prime}\cdot\Delta\mbox{\boldmath$r$}_{AB}^{(k)}\Gamma_{B}^{(k)},$
(6)
where, $\mbox{\boldmath$r$}_{M}^{\prime}=(1,\mbox{\boldmath$r$}_{M})$.
Therefore, it is gotten that
$\mbox{\boldmath$r$}_{M}^{\prime}\cdot\Delta\mbox{\boldmath$r$}_{AB}^{(k)}=0$
if
$\mathrm{Tr}_{A}(\hat{M}_{A}\rho_{AB})-\mathrm{Tr}_{A}(\hat{M}_{A}\rho_{AB}^{\prime})=0$.
According to the Lemma 1, $\Delta r_{AB}^{(jk)}=0$, namely
$\rho_{AB}=\rho_{AB}^{\prime}$.
Let us return to the study of QNC. A local quantum measurement is under a
subsystem $\rho_{A}$ of a composite system $\rho_{AB}$ spanning in the
$n_{A}\times n_{B}$ Hilbert space. After a local quantum measurement,
subsystem $\rho_{A}$ will collapse into $|\psi(\Theta,\Phi)\rangle_{A}$, and
$\rho_{B}$ will correspondingly change into
$\rho_{B}^{M}=\frac{\mathrm{Tr}_{A}(\hat{M}_{A}\rho_{AB})}{\mathrm{Tr}(\hat{M}_{A}\rho_{AB})}.$
(7)
$\rho_{B}^{M}$ is a functional of $\hat{M_{A}}$. Considering the projection
operator has a minimal variety
$\hat{M}_{A}\rightarrow\hat{M}+\delta\hat{M}_{A}$ (which just like the
stimulation input), the sub-state $\rho_{B}^{M}$ will correspondingly change
into $\rho_{B}^{M}\rightarrow\rho_{B}^{M}+\delta\rho_{B}^{M}$ (which is the
response output). Therefore, we define the following equation about the ratio
of change $\delta\rho_{B}^{M}/\delta\hat{M}$
$\mbox{\boldmath$\mathrm{F}$}(\Theta,\Psi;\rho_{AB})_{A\rightarrow
B}=\sum_{i=1}^{n-1}\frac{\mathrm{Tr}(|\delta^{\theta_{i}}\rho_{B}^{M}|)}{\mathrm{Tr}(|\delta^{\theta_{i}}\hat{M}_{A}|)}\mbox{\boldmath$e$}_{\theta_{i}}+\frac{\mathrm{Tr}(|\delta^{\phi_{i}}\rho_{B}^{M}|)}{\mathrm{Tr}(|\delta^{\phi_{i}}\hat{M}_{A}|)}\mbox{\boldmath$e$}_{\phi_{i}}.$
(8)
According to Eq. (3), we get that
$\mathrm{Tr}(|\rho_{1}-\rho_{2}|)=|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|$,
therefore the equation above can be rewritten as
$\mbox{\boldmath$\mathrm{F}$}(\Theta,\Psi;\rho_{AB})_{A\rightarrow
B}=\sum_{i=1}^{n-1}\frac{|\delta^{\theta_{i}}\mbox{\boldmath$r$}_{B}^{M}|}{|\delta^{\theta_{i}}\mbox{\boldmath$r$}_{M}|}\mbox{\boldmath$e$}_{\theta_{i}}+\sum_{i=1}^{n-1}\frac{|\delta^{\phi_{i}}\mbox{\boldmath$r$}_{B}^{M}|}{|\delta^{\phi_{i}}\mbox{\boldmath$r$}_{M}|}\mbox{\boldmath$e$}_{\theta_{i}},$
(9)
where $\delta^{x}f=(\partial f/\partial x)\delta x$.
$\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$ is a vector in
$2n-1$-dimensional space. It forms a surface in this spaces when $\Phi$, and
$\Theta$ each change from 0 to $2\phi$. We call
$\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$ the characteristic function of
QNC in $\rho_{AB}$ because it is defined totally by the character of QNC.
Every special QNC corresponds to a unique characteristic function. This
character of $\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$ can be clearly
seen in the following theorem.
###### Theorem 1
Any two density matrixes, $\rho_{AB}^{(i)}$ and $\rho_{AB}^{(j)}$ with the
same characteristic function $\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$
can be transformed into each other by a local unitary transformation under
system $B$, namely, $\rho_{AB}^{(i)}=(I_{A}\otimes
U_{B}^{(ij)})\rho_{AB}^{(j)}(I_{A}\otimes U_{B}^{(ij)})^{{\dagger}}$.
The proof of the corollary is not complication when one notes that
$\mathrm{Tr}(|(\delta\rho_{B}^{M})^{(i)}|)=\mathrm{Tr}(|(\delta\rho_{B}^{M})^{(j)}|)$,
which is equivalent to
$(\rho_{B}^{M})^{(i)}=U_{B}^{(ij)}(\rho_{B}^{M})^{(j)}(U_{B}^{(ij)})^{{\dagger}}$,
if $\rho_{AB}^{(i)}$ and $\rho_{AB}^{(j)}$ have a same characteristic
function. Basing on the conclusion of corollary 1, we get the theorem above.
The definition of $\mbox{\boldmath$\mathrm{F}$}_{B\rightarrow A}$ is analogous
with the $\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$ and will not be
repeated again. Both $\mbox{\boldmath$\mathrm{F}$}_{B\rightarrow A}$ and
$\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow B}$ can act as the characteristic
function in a composite state. According to this theorem, we can also conclude
that
$|\mbox{\boldmath$\mathrm{F}$}(\Theta,\Psi;\rho_{AB}^{\prime})_{A\rightarrow
B}|=|\mbox{\boldmath$\mathrm{F}$}(\Theta^{\prime},\Psi^{\prime};\rho_{AB})_{A\rightarrow
B}|$ if $\rho_{AB}^{\prime}=U_{A}\otimes
U_{B}\rho_{AB}U_{A}^{{\dagger}}\otimes U_{B}^{{\dagger}}$, where
$U_{A}\hat{M}(\Theta,\Psi)U_{A}^{{\dagger}}=\hat{M}(\Theta^{\prime},\Psi^{\prime})$.
This is because
$\rho_{B}^{M^{\prime}}=\mathrm{Tr}_{A}(M_{A}^{\prime}\rho_{AB})$, where
$M_{A}^{\prime}=U_{A}^{{\dagger}}M_{A}U_{A}$. Moreover, $\mathrm{Tr}(|\delta
M_{A}^{\prime}|)=\mathrm{Tr}(|\delta M_{A}|)$ because $\delta
M_{A}^{\prime}=U(\delta Q-\delta S)U^{{\dagger}}$ ( $\delta Q$ and $\delta S$
are infinitesimal positive operators with orthogonal support). Hence
$\mathrm{Tr}(|\delta M_{A}^{\prime}|)=2\mathrm{Tr}(\delta
Q)=\mathrm{Tr}(|\delta M_{A}|)$.
Let us show examples, for a pure qubit system,
$|\psi\rangle_{AB}=\cos\alpha|0,0\rangle+\sin\alpha\exp(\mathrm{i}\gamma)|1,1\rangle$,
the characteristic function can be expressed as
$\mbox{\boldmath$\mathrm{F}$}(\theta,\phi;|\psi\rangle_{AB})_{A\rightarrow
B}=\frac{2|\sin
2\alpha|(\mbox{\boldmath$e$}_{\theta}+\mbox{\boldmath$e$}_{\phi})}{2+\cos
2(\theta-\alpha)+\cos 2(\theta+\alpha)}.$ (10)
To show that local transformation cannot change the shape of
$|\mbox{\boldmath$\mathrm{F}$}|$, we let
$|\psi\rangle_{AB}^{\prime}=U_{A}|\psi\rangle_{AB}$, where
$U_{A}|0\rangle=\sqrt{3}/2|0\rangle+1/2|1\rangle$, and
$U_{A}|1\rangle=-\sqrt{3}/2|1\rangle+1/2|0\rangle$. We show the pictures of
the characteristic function of the state $|\psi\rangle_{AB}$ and
$|\psi\rangle_{AB}^{\prime}$ in the Bloch sphere. As can be seen in Fig. 1,
these characteristic functions can be transformed into each other through a
rotation.
Figure 1: The absolute value of the characteristic function in Bloch Sphere.
The picture $A$ shows the shape of
$|\mbox{\boldmath$\mathrm{F}$}(\theta,\phi;|\psi\rangle_{AB})_{A\rightarrow
B}|$ and the picture $B$ explores the shape of
$|\mbox{\boldmath$\mathrm{F}$}(\theta,\phi;|\psi\rangle_{AB}^{\prime})_{A\rightarrow
B}|$. We choose $\alpha=\pi/3$ here.
According to this character, we definite a physical quantity that is
independent with the form of unlimited quantum measurement as:
$G(\rho_{AB})_{A\rightarrow
B}=\frac{\sqrt{n_{A}}}{\Omega}\int_{\Omega}|\mbox{\boldmath$\mathrm{F}$}_{A\rightarrow
B}|\mathrm{Tr}(\rho_{A}\hat{M}_{A})\mathrm{d}_{\mathbb{R}^{2n-2}}\Omega,$ (11)
where $\mathrm{d}_{\mathbb{R}^{2n-2}}\Omega$ is
$\displaystyle\mathrm{d}_{\mathbb{R}^{2n-2}}\Omega=\prod_{l=1}^{n-1}\sin(\theta_{l})^{n-l-1}\mathrm{d}\theta_{l}\mathrm{d}\phi_{l}.$
(12)
For a two-particle pure state, it can be proven that
$G(\rho_{AB})_{A\rightarrow B}=G(\rho_{AB})_{B\rightarrow A}$, but for an
arbitrary state, these two terms are not necessarily equal each other. Hence
we define the strength of QNC as the average value of these two terms:
$G(\rho_{AB})=\frac{1}{2}(G(\rho_{AB})_{A\rightarrow
B}+G(\rho_{AB})_{B\rightarrow A}).$ (13)
###### Theorem 2
These density matrixes which can be transformed each other by local operation
have the same nonlocal strength. Namely, $G(\rho_{AB})=G(U_{A}\otimes
U_{B}\rho_{AB}U_{A}^{{\dagger}}\otimes U_{B}^{{\dagger}})$.
Theorem 2 is obvious because according to the analysis above
$|\mbox{\boldmath$\mathrm{F}$}(\Theta,\Psi)^{\prime}|=|\mbox{\boldmath$\mathrm{F}$}(\Theta^{\prime},\Psi^{\prime})|$
and the integrating range of Eq. (11) is $SU(2n-2)$ by symmetry.
In terms of the definition of Eq. (13), some separable states, such as
$\rho_{AB}=1/2(|00\rangle\langle 00|+|11\rangle\langle 11|)$ have a nonlocal
correlation although without entanglement (In fact, $G(\rho_{AB})=1/2$ here).
It can be also seen that $G(\rho_{AB})$ is not monotonic under LOCC, but it is
monotonic under local trace-preserving quantum operation.
###### lemma 2
Suppose $\varepsilon_{p}$ is a partial local trace-preserving quantum
operation and let $\rho$ be a density operator. Then
$G(\rho_{AB})\geq G(\varepsilon_{P}^{x}(\rho_{AB})).$ (14)
$\varepsilon_{p}^{A}(\rho_{AB})=\sum_{i}\lambda_{i}U_{A}\otimes
I_{B}\rho_{AB}U_{A}^{{\dagger}}\otimes I_{B}$ and
$\varepsilon_{p}^{B}(\rho_{AB})=\sum_{i}\lambda_{i}I_{A}\otimes
U_{B}\rho_{AB}I_{A}\otimes U_{B}^{{\dagger}}$.
To prove this lemma, we should use the previous conclusion $D(\rho,\sigma)\geq
D(\varepsilon(\rho),\varepsilon(\sigma))$, where $D(\rho,\sigma)$ is the trace
distance[16]. Basing on this lemma, we will get a more important conclusion as
follows.
###### Corollary 2
Suppose $\rho_{AB}=\sum_{i}{\gamma_{i}}\rho_{AB}^{(i)}$ is a pure state
decomposition of $\rho_{AB}$ and
$\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}=\sum_{i}{\gamma_{i}}\rho_{A}^{(i)}\otimes\rho_{B}^{(i)}$,
where $\rho_{A}^{(i)}=\mathrm{Tr}_{B}(\rho_{AB}^{(i)})$ and
$\rho_{B}^{(i)}=\mathrm{Tr}_{A}(\rho_{AB}^{(i)})$. Then,
$G(\rho_{AB})\geq G(\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}).$ (15)
The mark $\mbox{\boldmath$\gamma$}\otimes$ here is just an illustration of a
direct product decomposition of $\rho_{AB}$ and we call
$\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}$ the productization of
$\rho_{AB}$ for concision. It is not difficult to understand this corollary
because the $\rho_{AB}$ contains more information than
$\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}$. We can obtains
$\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}$ from $\rho_{AB}$ but not the
reverse. Then, we define a half-positive-definite quantity
$E(\rho_{AB})=C\inf\\{\mbox{\boldmath$\gamma$}:G(\rho_{AB})-G(\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes})\\},$
(16)
where $\mbox{\boldmath$\gamma$}=\\{\gamma_{i},\rho_{AB}^{(i)}\\}$ is a symbol
of the set of productization. We can determine if $\rho_{AB}$ is separable,
$E(\rho_{AB})=0$ and else $E(\rho_{AB})>0$.
Let us look at a special example. The nonlocal correlation strength of a pure
qubit state
$|\psi\rangle_{AB}=\cos\alpha|0,0\rangle+\sin\alpha\exp(\mathrm{i}\gamma)|1,1\rangle$
is $G(|\psi\rangle)=|\sin 2\alpha|$. Hence, the maximum of strength of QNC
appears where the maximum of entanglement appears. This is not surprising
because for an arbitrary two-partite pure state, the form of the
productization is determined, equaling to zero. Therefore
$E(|\psi_{AB}\rangle)=G(|\psi_{AB}\rangle)$, namely the strength of QNC can be
used as the measurement form of quantum entanglement in two-partite pure
state.
For a totally mixed state
$\rho_{AB}=1/2(|\psi_{+}\rangle\langle\psi_{+}|+|\psi_{-}\rangle\langle\psi_{-}|)$,
where $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$ are the Bell states, We get
$\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}=1/2(|00\rangle\langle
00|+|11\rangle\langle 11|).$ (17)
Therefore, $G(\rho_{AB})=\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}$and the
entanglement $E(\rho_{AB})=0$.
We should note that Eq. (16) is usually hard to calculated because we still
have not an efficient way to find out the supremum of
$G(\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes})$ for a general state
$\rho_{AB}$. However, it does not mean this definition is useless.
Historically, the entanglement of formation had been also hard to calculated
initially until the concurrence was proposed. Eq. (16) supports an alternate
way to research the quantum entanglement and its values needed further
studied. In fact, it is different from previous theories, it is the
formulation of the function integral and clearly shows the relationship
between QNC and quantum entanglement.
The reason why we take the Eq. (2) is that we are inspired by the words of
Schrödinger. In terms of Schrödinger’s words, in a composite correlated state,
a subsystem will be _steered_ or _piloted_ into one or the other type of state
if a local quantum measurement is done on the other subsystem [17]. We think
this “steering” can be seen as the corresponding change of one sub-state when
other one be locally measured. It is namely the trace of
$\mbox{\boldmath$r$}_{B}^{M}$. In fact, the trace of
$\mbox{\boldmath$r$}_{B}^{M}$ will form a surface in a
$(n_{B}^{2}-1)$-dimensional space when the projection operator
$\hat{M}(\Theta,\Phi)$ ranges though the parameter-space
$\Theta|=\\{[0,\pi]\\}$, $\Phi=\\{0,2\pi\\}$. Studying the surface of
$\mbox{\boldmath$r$}_{B}^{M}$ can result in some new conclusions. For example,
consider a $n_{A}\otimes 2$ composite separate state
$\rho_{AB}=\sum_{i}^{m}a_{i}\rho_{A}^{(i)}\otimes\rho_{B}^{(i)}$ and with
$\langle\psi_{A}^{(i)}|\psi_{A}^{(j)}\rangle=\delta_{ij}$.
$\mbox{\boldmath$r$}_{B}^{M}$ will form a $m$-polyhedron in the Bloch sphere.
This is very interesting, because for an inseparable state,
$\mbox{\boldmath$r$}_{B}^{M}$ is usually smooth. In fact, the converse result
is also correct under this situation. Namely, if $\mbox{\boldmath$r$}_{B}^{M}$
forms a $m$-polyhedron, the $\rho_{AB}$ must be expressed by the formula above
(To get this conclusion we should use corollary 1). Additionally, a more
general conclusion is shown as follows.
Figure 2: The surfaces of $\mbox{\boldmath$r$}_{B}^{M}$. We show the
$\mbox{\boldmath$r$}_{B}^{M}$ of
$\rho_{AB}=\sum_{i}^{m}a_{i}\rho_{A}^{(i)}\otimes\rho_{B}^{(i)}$ forms the
$m$-polyhedron in the Bloch sphere. In this figure,
$|\psi_{A}^{(i)}\rangle=|i\rangle_{A}$ and
$|\psi_{B}^{(i)}\rangle=\cos(i\pi/m)|0\rangle_{B}+\sin(i\pi/m)|1\rangle_{B}$.
$m$ are chosen as $4$,$8$,$12$ and $20$ in the picture $A$, $B$, $C$ and $D$
respectively.
###### Theorem 3
The sufficient and necessary condition for the $2\otimes n_{B}$ composite
state $\rho_{AB}$ to be decomposed into
$\rho_{AB}=\sum_{i}a_{i}\rho_{A}^{i}\otimes\rho_{B}^{i}$, where $\rho_{A}^{i}$
is real density matrix, is that the main normal line of
$\mathrm{Tr}(\rho_{A}\hat{M})\mbox{\boldmath$r$}_{B}^{M}$ is constant in
$n_{B}$-dimensional space.
###### Proof 3
Let
$\mbox{\boldmath$r$}_{b}=\mathrm{Tr}(\rho_{A}\hat{M})\mbox{\boldmath$r$}_{B}^{M}=\sum_{k}a_{k}\lambda_{k}\mbox{\boldmath$r$}_{B}^{(k)}$,
where
$\lambda_{k}=\mathrm{Tr}(|\psi_{A}^{k}\rangle\langle\psi_{A}^{k}|\hat{M})$. If
$\rho_{AB}=\sum_{i}a_{i}\rho_{A}^{i}\otimes\rho_{B}^{i}$, $\lambda_{k}$ can be
expressed as
$\lambda_{k}=\cos(\theta-\alpha_{A}^{k})^{2}+\sin 2\theta\sin
2\alpha_{A}^{k}\sin(\varphi/2-\varphi_{A}^{k}/2)^{2}.$ (18)
Moreover, $\rho_{A}$ is real, $\varphi_{A}^{k}=0$, hence
$\parallel\mbox{\boldmath$r$}_{b}^{\theta}\times\mbox{\boldmath$r$}_{b}^{\psi}\parallel=\parallel\sum_{i,j}\sin
2(\alpha_{i}-\alpha_{j})\mbox{\boldmath$r$}_{B}^{(i)}\times\mbox{\boldmath$r$}_{B}^{(j)}\parallel,$
(19)
where $\parallel\mbox{\boldmath$r$}\parallel$ means the normalization of $r$.
$\mbox{\boldmath$r$}_{B}^{(k)}$ is independent of $\theta$ and $\phi$,
therefore
$\parallel\mbox{\boldmath$r$}_{b}^{\theta}\times\mbox{\boldmath$r$}_{b}^{\psi}\parallel=cons.$
Namely, the direction of the main normal line does not change with the
variables $\phi$ and $\theta$. Conversely, if the main normal line
$\mbox{\boldmath$r$}_{\mathbf{n}}$ is constant, it can be rewritten as
$\mbox{\boldmath$r$}_{\mathbf{n}}=\parallel\sum_{i,j}\sin
2(\alpha_{i}-\alpha_{j})\mbox{\boldmath$r$}_{B}^{(i)}\times\mbox{\boldmath$r$}_{B}^{(j)}\parallel$
when $\alpha_{i}$ and $\alpha_{j}$ are appropriately chosen. Therefore,
$\mbox{\boldmath$r$}_{b}$ can be determined. consequently, according to
corollary 1, $\rho_{AB}$ is a separable state.
## 1 Discussion and conclusion
In this above work, we calculate the characteristic function and the strength
of QNC. We regard the characteristic function as a corresponding function and
through the trace distance, differential theory is brought into QNC research.
It supports an alternate way to analysis and categorize the QNC. In fact, as a
special sort of QNC, quantum entanglement is defined and shown in Eq. (16).
This definition is shown a more definite and clear relationship between the
quantum entanglement and QNC. We also show a way to distinguish whether a real
mixed composite density matrix is separable or not.
However, it should be reminded that the monotonic character of our new
definition of quantum entanglement under LOCC has not rigorously proven yet
although we have done some computation to show that it is correct. The
productization density matrix $\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes}$ is
brought into this Letter. It seems that the supremum of it only possesses the
“local” correlation of $\rho_{AB}$. It is can be seen that the definition
$E_{s}(\rho_{AB})=\mathrm{inf}\\{\mbox{\boldmath$\gamma$}:S(\rho_{AB}^{\mbox{\boldmath$\gamma$}\otimes})-S(\rho_{AB})\\}$,
is also a possible entanglement measurement definition, where $S(\rho_{AB})$
is the Shannon entropy of $\rho_{AB}$ here. This work was supported by the
National Basic Research Program of China (973 Program) grant No.
G2009CB929300.
## References
* [1] J.S. Bell, Physics (Long Island City, N.Y.), 1, 195 (1964);
* [2] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, Phys. Rev. A , 54, 3824 (1996);
* [3] J. Walgate, A.J. Short, L. Hardy and V. Vedral, Phys. Rev. Lett , 85, 4972 (2000);
* [4] R.F. Werner, Phys. Rev. A , 40, 4277 (1989);
* [5] R.W. Spekkens, Phys. Rev. A , 75, 032110 (2007);
* [6] K.A. Kirkpatrick, Found. Phys. Lett , 19, 95 (2006);
* [7] H.M. Wiseman, S.J. Jones, A.C. Doherty Phys. Rev. Lett. , 98, 140402 (2007);
* [8] S.P. Walborn, A. Sallers, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, Phys. Rev. Lett. 106, 130402 (2011);
* [9] Horodecki, M, SenDe, A and Horodecki, K, Phys. Rev. Lett. , 90, 047902 (2003);
* [10] Oxley, MP, Cosgriff, EC and Allen, LJ, Phys. Rev. Lett. , 94, 203906 (2005);
* [11] Popescu, S, Short, AJ and Winter, A, Phys. Rev. Lett., 99, 180502 (2007);
* [12] Somshubhro Bandyopadhyay, Phys. Rev. Lett. , 106, 210402 (2011)(and references therein);
* [13] Shunlong Luo, and Shuangshuang Fu, Phys. Rev. Lett, 106, 120401 (2011);
* [14] D.F.V. James, P.G. Kwiat, W.J. Munro and A.G, Phys. Rev. A, 64, 052312 (2001);
* [15] M.G.A. Paris and J. Řeháček, in _Quantum State Estimation_ (Lect. Notes Phys. ), 649 (Springer, Berlin Heidelberg, 2004);
* [16] Michael A. Nielsen and Isaac L. Chuang, in _Quantum Computation and Quantum Information_ (Cambridge University Press, University of Cambridge, Cambridge, England, 2000);
* [17] E. Schrödinger, Proc. Cambridge Philos. Soc., 31, 553 (1935); 32, 446 (1936).
|
arxiv-papers
| 2011-08-22T14:56:50 |
2024-09-04T02:49:21.690239
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Wen, Shu-Shen Li",
"submitter": "Wei Wen",
"url": "https://arxiv.org/abs/1108.4335"
}
|
1108.4341
|
# Emergence of robustness against noise: A structural phase transition in
evolved models of gene regulatory networks
Tiago P. Peixoto tiago@itp.uni-bremen.de Institut für Theoretische Physik,
Universität Bremen, Otto-Hahn-Allee 1, D-28359 Bremen, Germany
###### Abstract
We investigate the evolution of Boolean networks subject to a selective
pressure which favors robustness against noise, as a model of evolved genetic
regulatory systems. By mapping the evolutionary process into a statistical
ensemble and minimizing its associated free energy, we find the structural
properties which emerge as the selective pressure is increased and identify a
phase transition from a random topology to a “segregated core” structure,
where a smaller and more densely connected subset of the nodes is responsible
for most of the regulation in the network. This segregated structure is very
similar qualitatively to what is found in gene regulatory networks, where only
a much smaller subset of genes — those responsible for transcription factors —
is responsible for global regulation. We obtain the full phase diagram of the
evolutionary process as a function of selective pressure and the average
number of inputs per node. We compare the theoretical predictions with Monte
Carlo simulations of evolved networks and with empirical data for
Saccharomyces cerevisiae and Escherichia coli.
###### pacs:
87.18.-h 05.40.Ca 05.70.Fh 02.50.Cw
## I Introduction
Many large-scale dynamical systems are composed of elementary units which are
_noisy_ , i.e., can behave non-deterministically, but nevertheless must behave
globally with some degree of predictability. A paradigmatic example is gene
regulation in the cell, which is a system of many interacting agents — genes,
mRNA, and proteins — which are subject to stochastic fluctuations. What makes
gene regulation particularly interesting is that it is assumed to be under
evolutionary pressure to preserve its dynamic memory against stochastic
fluctuations Kitano (2004); Raser and O’Shea (2005); Maheshri and O’Shea
(2007). Many important cellular processes require such reliability, such as
circadian oscillations Raser and O’Shea (2005). Furthermore, in multicellular
organisms, errors in signal transduction can potentially lead to catastrophic
consequences, such as embryo defects or cancer Kitano (2004); Gutkind (2000).
Since the source of noise cannot be fully removed Lestas _et al._ (2010), a
gene regulation system must adopt characteristics which compensate for the
unavoidably noisy nature of its elements. Since they are a product of natural
selection, these characteristics must emerge from random mutations and
subsequent selection based on fitness. A central question concerns the general
large-scale features which are likely to emerge in this scenario that result
in reliable function under noise. In this work, we study the emergence of
robustness against noise in networks of Boolean elements which are subject to
selective pressure, functioning as a model for evolved gene regulatory
systems. We show that the system undergoes a structural phase transition at a
critical value of selective pressure, from a totally random topology to a
“segregated-core” structure, where a smaller and more densely connected subset
of the network is responsible for the regulation of most nodes in the network.
This characteristic is present to a significant degree in gene regulatory
systems of organisms such as yeast and _Escherichia coli_ , in which all the
regulation is done by a much smaller (and denser) subset of the network,
comprised of transcription factor genes.
Boolean networks (BNs) have been used extensively to model gene regulation
Kauffman (1969a, b); Bornholdt (2005); Drossel (2008). The Boolean value on a
given node represents the level of concentration of proteins encoded by a
gene, which in the simplest approximation can be either “on” or “off.” The
regulation of genes by other genes is represented by Boolean functions
associated with each node, which depend on the state of other nodes called the
_inputs_ of the function. The dynamics on these networks serve as a model for
the mutual regulation of genes which control the metabolism of cells in an
organism. Gene regulation is composed of specific steps involving the
production of proteins and other metabolites, which need to be carried out in
specific sequences and under certain conditions. During each of these steps
the dynamics is subject to stochastic fluctuations Raser and O’Shea (2005);
Maheshri and O’Shea (2007); Raj and van Oudenaarden (2008), since the number
of proteins involved can be very low Raser and O’Shea (2005); Eldar and
Elowitz (2010), and the whole process lacks an inherent synchronization
mechanism. In order for the regulation process to work reliably, the network
must possess some degree of robustness against these perturbations Kollmann
_et al._ (2005). Indeed, the investigation of real regulatory networks
modelled as BNs, such as the one responsible for the yeast cell cycle Li _et
al._ (2004), revealed a remarkable degree of robustness, where most
trajectories in state space lead to the same attractor, regardless of the
initial conditions. Similar results were also obtained for the segment
polarity regulatory network in _Drosophila melanogaster_ Albert and Othmer
(2003); Chaves _et al._ (2005), which showed that wild-type attractors are
significantly robust to different initial conditions and perturbations, and
seem to depend only on general topological characteristics of the network,
instead of specific functional details. However, the general features which
make BNs robust against different types of perturbations are still being
identified.
Perhaps the simplest form of perturbation one can consider is a “flip” of a
single node in the network, and the propagation of flips which result from it.
This corresponds to the situation where the stochastic noise is very weak, and
can be modeled as a single flip event. After the perturbation, the system has
an arbitrary amount of time to recover (if it recovers), and different
perturbations do not build up. Many authors have considered the robustness
against perturbations of this type, including Kauffmann Kauffman (1969a, b)
who was the first to propose random Boolean networks (RBNs) — networks with
fully random topology and functions — as a model of gene regulation. According
to this type of perturbation, the dynamics of RBNs Drossel (2008) can belong
to one of two phases, depending on the number of inputs per node $K$: A frozen
phase ($K<2$) where the perturbation propagates sub-linearly in time and
eventually dies out; and a “chaotic” phase ($K>2$) where the perturbation
grows exponentially and eventually reaches the entire system. A critical line
exists at $K=2$, where the perturbation grows algebraically, and features from
both phases are simultaneously observed.
Although RBNs in the frozen phase and on the critical line show features which
can be interpreted as robustness in some sense, they fall short of being
convincing models for gene regulation. Actual gene regulation networks are not
random and show a high degree of topological Maslov and Sneppen (2005) and
functional Harris _et al._ (2002) organization which are not present in
simple RBNs. Conceivably, these features arise out of stringent requirements
to perform specific tasks and of types of robustness which are more demanding
than the containment of single flip perturbations. As an attempt at producing
more complete models, many authors investigated the evolution of BN systems,
where the fitness criterion is some form of robustness against perturbation
which is not inherent to RBNs. The majority of authors assumed single flips as
the only type of noise, but considered different types of responses as fitness
criteria Bornholdt and Sneppen (1998, 2000); Stern (1999); Bassler _et al._
(2004); Aldana _et al._ (2007); Szejka and Drossel (2007); Mihaljev and
Drossel (2009); Pomerance _et al._ (2009), most of which are related to the
capacity of the network to display the same dynamical pattern after a single-
flip. In particular, in Szejka and Drossel (2007) it was found that if the
fitness criterion is the ability to return to the same attractor after the
perturbation, the evolved networks always achieve maximum fitness. Furthermore
these networks with maximum fitness span a huge portion of configuration
space, and show a high degree of variability. This means not only that this
type of robustness can evolve, but also that it is not a very demanding task
for the evolutionary process.
In this work, we consider the arguably more realistic situation where the
perturbations are caused by transcriptional noise, which can be arbitrarily
frequent 111Another realistic source of noise are perturbations in the update
sequence of nodes, since gene regulation lacks a global synchronizing clock
Klemm and Bornholdt (2005). However, it can be shown that absolute robustness
against this type of noise can be achieved in an independent manner, and with
a very small effect to the global topological characteristics of the system
Peixoto and Drossel (2009b).. In this scenario, the effects of noise can
overlap and build up in time. The appropriate fitness criterion remains
whether or not the network is capable of performing some predefined dynamical
pattern, but this is a task which becomes much more complicated. In fact, it
can be shown that for networks which are sparse, i.e., the average number of
inputs per node is some finite number, perfect robustness can never be
achieved, and some amount of deviations, or “errors,” in the dynamics are
always going to exist Peixoto (2010, 2012). Instead, one measures robustness
not only by the amount of existing errors, but also by the ability of the
system to not be overtaken by them and consequently lose all memory of its
dynamical past — i.e., to become ergodic. This type of robustness is much
stronger than, and not necessarily related to the ability of the system to
contain single-flip perturbations. This was shown in Peixoto and Drossel
(2009a) for RBNs subject to transcriptional noise, for which neither phase
(chaotic or frozen) is robust, and both display ergodic behavior, for any non-
zero value of noise.
Furthermore, unlike Bornholdt and Sneppen (1998, 2000); Stern (1999); Bassler
_et al._ (2004); Szejka and Drossel (2007); Mihaljev and Drossel (2009), in
this work we also consider the _cost_ which is associated with different
levels of robustness. It is generally the case that improved robustness can be
obtained by introducing redundancy or some other mechanism that counteracts
the effect of noise, which increases the overhead in the system. This added
overhead can impact negatively on the fitness of the organism, which needs to
spend more energy or more time to perform the same task. Therefore the trade-
off between overhead and robustness is also driven by the evolutionary
process. In this work, this overhead is controlled by fixing the average in-
degree during the evolutionary process, which becomes an external parameter.
By selecting the appropriate value, one automatically determines a selective
pressure that yields the corresponding trade-off.
Our main result is that under transcriptional noise, the selective pressure
can have a very noticeable effect on large-scale properties of the system: If
it is large enough, it triggers a structural phase transition, where networks
change from a random topology to a segregated-core structure, with most nodes
being regulated by a smaller and denser subset of the network. This observed
segregated-core topology is strikingly similar (even if qualitatively so) to
what is observed in most real gene regulation networks; namely, genes are
separated into two classes: target genes, and those which regulate
transcription factors. Only transcription factor genes are responsible for
regulation of other genes, and they are usually orders of magnitude smaller in
number than target genes Nimwegen (2006).
This work is divided as follows. We begin in Sec. II by presenting the model,
and in Sec. III we define the evolutionary process and map it into an
equivalent Gibbs ensemble. We then parametrize the topological characteristics
of the system as a stochastic blockmodel in Sec. IV, and obtain an expression
for its entropy. In Sec. V we describe the technique used to minimize the free
energy. We follow in Sec. VI with the characterization of the existing phase
transition and obtain the phase diagram. We perform comparisons with Monte
Carlo simulations in Sec. VII and with the gene regulatory networks of yeast
and _E. coli_ in Sec. VIII. Finally, we conclude with a discussion.
## II The model
A Boolean network Kauffman (1969a); Drossel (2008) is a directed graph of $N$
nodes representing Boolean variables $\mathbf{\sigma}\in\\{1,0\\}^{N}$, which
are subject to a deterministic update rule,
$\sigma_{i}(t+1)=f_{i}\left(\bm{\sigma}(t)\right)$ (1)
where $f_{i}$ is the update function assigned to node $i$, which depends
exclusively on the states of its inputs. At a given time step all nodes are
updated in parallel.
We include noise in the model by introducing the probability $P$ that at each
time step a given input has its value “flipped”: $\sigma_{j}\to 1-\sigma_{j}$,
before the output is computed Peixoto and Drossel (2009a). This probability is
independent for all inputs in the network, and many values may be flipped
simultaneously. The functions on all nodes are taken to be the majority
function, defined as
$f_{i}(\\{\sigma_{j}\\})=\begin{cases}1\text{ if
}\sum_{j}\sigma_{j}>k_{i}/2,\\\ 0\text{ otherwise, }\end{cases}$ (2)
where $k_{i}$ is the number of inputs of node $i$. It is assumed throughout
the paper that the values of $k_{i}$ are always odd 222 The definition above
will lead to a bias if $k_{i}$ is an even number, since if the sum happens to
be exactly $k_{i}/2$ the output will be $0$, arbitrarily. Alternative
definitions could be used, which would remove the bias Szejka _et al._
(2008). However, for the analysis presented here, this is not an issue since
$k_{i}$ is always odd.. This is so chosen because odd-valued majority
functions are optimal, since no other function performs better against noise
Evans and Schulman (2003). By using Eq. 2, we essentially remove the choice of
functions from the evolutionary process and concentrate solely on topological
aspects.
Starting from an initial configuration, the dynamics of the system evolves and
eventually reaches a dynamically stable regime, where the average fraction
$b_{t}$ of nodes with value $1$ no longer changes, except for stochastic
fluctuations which vanish for a large system size Huepe and Aldana-González
(2002); Peixoto (2012). In the absence of noise ($P=0$) there are only two
possible attractors (if the network is sufficiently well connected) where all
nodes have the same value, which can be either $0$ or $1$. We will consider
these homogeneous attractors as representing the “correct” dynamics, and
denote the deviations from them as “errors.” More specifically, and without
loss of generality, we will name the value of $1$ as an “error,” and the value
of $b_{t}$ as the average error on the system.
The steady-state fraction of errors $b^{*}\equiv\lim_{t\to\infty}b_{t}$ (for
$b_{0}=0$) will increase with $P$. For any network with a finite average in-
degree there will be a critical value of noise $P^{*}$ for which the dynamics
undergoes a phase transition, and the value of $b^{*}$ reaches $1/2$, and
remains at this value for $P>P^{*}$ Peixoto (2010). The value $b^{*}=1/2$ is
special, since it means that the dynamics lost the memory of its initial
state, since any other initial value of $b_{0}$ (including $b_{0}>1/2$) would
lead eventually to this same value of $b^{*}$. Therefore, the value of $P^{*}$
marks the transition from a nonergodic to an ergodic dynamics. Robustness
against noise is synonymous with nonergodicity, since only in this regime are
dynamical correlations not destroyed over time.
BNs with majority functions serve as a paradigmatic model for networks robust
against noise, since they are composed of optimal elements, and they show a
minimal dynamical behavior in the absence of noise, namely two homogeneous
attractors with $\\{\sigma_{i}\\}=0$ or $1$. If robustness cannot be attained
for such a system, it is much less likely to be possible for a different
system with a another choice of Boolean functions, or displaying a more
elaborate dynamical pattern Peixoto (2012).
In this work we will consider the value of the steady-state average error
$b^{*}$ as the main fitness criterion governing the survival probability of an
organism, since it directly measures the deviation from the situation without
noise. Although the phenotype itself, i.e. the dynamics without noise, does
not change during the evolutionary process considered here, its _stability_ ,
as measured by $b^{*}$, does. This translates into an actual fitness
criterion, since it is not enough for phenotypes to exist; they must also be
stable against perturbations. If they are not, they are not viable in
practice, and thus should not be observed.
## III Evolutionary dynamics
We suppose that a given BN represents the genotype of a full organism, which
can self-replicate and belongs to a population that is subject to an
evolutionary pressure. The number of individuals in the population is assumed
to be sufficiently large and constant. Individuals replicate a given number of
times with a constant rate. Parents die the moment they replicate. The
offspring are always initially identical to their parents, but are
individually subject to point mutations represented by the matrix $\mu_{ij}$,
which defines the probability of mutating from genotype (i.e. network) $i$ to
$j$. The offspring survive with probability $a_{i}$, given by the Boltzmann
selection criterion
$a_{i}\propto e^{\beta f_{i}},$ (3)
where $f_{i}$ is the fitness of genotype $i$. The parameter $\beta$ controls
the _selective pressure_ : For large values of $\beta$ only the very best
networks survive, whereas for smaller values most networks do. As mentioned
previously, the fitness of a network will be given by the fraction of ones
(“errors”) after a sufficiently long time,
$b^{*}_{(i)}\equiv\lim_{t\to\infty}b_{t}^{(i)}$, for $b_{0}^{(i)}=0$, as
$f_{i}=-Nb^{*}_{(i)}.$ (4)
Thus, the largest fitness a network can have is $f_{i}=0$, which should be
possible only if there is no noise ($P=0$).
We suppose that the global offspring mortality rate is such that the size of
the population always remains constant. If we consider that the dynamics
occurs in discrete time steps, we can write the probability $\pi_{i}(t)$ of
finding an individual in the population with genotype $i$ at time $t$ as a
Markov chain,
$\pi_{i}(t)=a_{i}\sum_{j}\pi_{j}(t-1)\mu_{ji}.$ (5)
The mutation probabilities $\mu_{ji}$ have a decisive effect on what
topologies emerge. Mutations in actual biological systems may result in
topological bias, such as gene duplications, which are not reversible and
result in networks with broad degree distributions Ispolatov _et al._ (2005);
Enemark and Sneppen (2007). However, the central aim here it to obtain the
most likely topology that arises due to the selective pressure _alone_. For
this reason we are more interested in mutations which will lead to all
possible networks with equal probability in the absence of selective pressure
(i.e. ergodicity). A simple and conventional choice is reversible mutations,
$\mu_{ij}=\mu_{ji}$, for which the steady state
$\pi_{i}\equiv\lim_{t\to\infty}\pi_{i}(t)$ obeys the detailed balance
condition: $\pi_{i}\mu_{ij}a_{j}=\pi_{j}\mu_{ji}a_{i}$. This is a sufficient
condition for the desired ergodicity property, but it is not strictly
necessary, since other types of mutations may also fulfill it. However, from
this condition we easily obtain that the steady-state probability of finding
an individual with genotype $i$ is given by its survival probability,
$\pi_{i}=a_{i}=e^{\beta f_{i}}/\mathcal{Z},$ (6)
where $\mathcal{Z}=\sum_{i}e^{\beta f_{i}}=\sum_{i}e^{-\beta Nb^{*}_{(i)}}$.
This corresponds exactly to a Gibbs ensemble of all possible genotypes, with a
partition function given by $\mathcal{Z}$, where $Nb^{*}_{(i)}$ plays the role
of the “microstate energy” and $\beta$ is the “inverse temperature” (these are
of course only mathematical analogies, since these quantities do not actually
represent a physical energy and temperature, respectively). The average
intensive “energy” in the ensemble is thus
$b^{*}=\sum_{i}b^{*}_{(i)}e^{-\beta Nb^{*}_{(i)}}/\mathcal{Z},$ (7)
and the canonical entropy is
$\mathcal{S}=-\sum_{i}\pi_{i}\ln\pi_{i}=\ln\mathcal{Z}+\beta Nb^{*}.$ (8)
The objective is to obtain not only $b^{*}$ for a given $\beta$, but also the
network topologies which characterize the ensemble. Instead of considering all
microstates individually (i.e., all possible networks) and computing Eqs. 7
and 8 directly, we may parametrize the whole ensemble via some macroscopic
variables $\\{x_{j}\\}$ which sufficiently describe its topological
properties. These variables must be chosen so that it is possible to write
both $b^{*}(\\{x_{j}\\})$ and $\mathcal{S}(\\{x_{j}\\})$ as functions of these
variables alone. The entropy can, for instance, be obtained via the
microcanonical formulation
$\mathcal{S}(\\{x_{j}\\})=\ln\Omega(\\{x_{j}\\}),$ (9)
where $\Omega(\\{x_{j}\\})$ is the number of different networks given a
macroscopic parametrization $\\{x_{j}\\}$. The values of $\\{x_{j}\\}$ which
correspond to thermodynamic equilibrium [i.e., the steady state of Eq. 5] can
be obtained by minimizing the “free energy”,
$\mathcal{F}(\\{x_{j}\\})=Nb^{*}(\\{x_{j}\\})-\mathcal{S}(\\{x_{j}\\})/\beta,$
(10)
with respect to $\\{x_{j}\\}$. This stems from the principles of maximum
entropy and minimum energy, for closed systems with fixed energy and entropy,
respectively, which need to hold in thermodynamic equilibrium Callen (1985).
It should again be emphasized that the theory so far is only a mathematical
tool, which, although exact, says nothing about the actual physical
thermodynamical properties of the evolved systems, i.e., they have no relation
to an actual measurable energy or temperature. Instead, the minimization of
Eq. 10 is entirely analogous to obtaining the steady state of Eq. 5 by any
other means. However, this approach, together with an appropriate topological
parametrization, allows us to obtain the outcome of the evolutionary process
on the population, without having to actually implement any dynamics. As will
be described in detail in the next section, we will parametrize the ensemble
as a general _stochastic blockmodel_ , which allows for a wide range of
topological configurations, while at the same time allowing for a tractable
computation of $b^{*}$ and $S$, which then can be used to minimize
$\mathcal{F}$.
It should also be mentioned at this point that we are interested in the
properties of typical networks in the ensemble when the selective pressure
$\beta$ is varied, under the restriction that the average number of inputs per
node (the average in-degree) ${\left<k\right>}$ is always the same. As
mentioned in the introduction, this restriction originates from the assumption
that a larger number of inputs increases the putative cost for the organism of
realizing a regulatory mechanism which depends on more elements. Thus, the
value of ${\left<k\right>}$ should on its own impact the fitness of the
organism, and should also be subject to natural selection. For simplicity, we
do not describe the fitness landscape which depends on ${\left<k\right>}$ and
its evolution, in order to emphasize the effects of robustness against noise
alone. Instead, we consider ${\left<k\right>}$ as an external parameter, which
essentially means that the fitness pressure on ${\left<k\right>}$ supersedes
that of the other parameters, such that it cannot change during evolution. In
this way, we are implicitly considering the cost associated with the
robustness achieved by increasing ${\left<k\right>}$: the smaller is the value
of ${\left<k\right>}$ chosen, the larger is the implied fitness penalty of
having more connections.
## IV Stochastic blockmodel
Simultaneous consideration of all possible networks with a given
${\left<k\right>}$ is a tremendous task, due to the gigantic number of diverse
configurations which are possible. For arbitrary networks the computation of
$b^{*}$ according to Eq. 1 may be very cumbersome, since it may depend on many
degrees of freedom. Therefore, we narrow down the allowed subset of possible
network topologies to those which can be accommodated in a _stochastic
blockmodel_ Holland _et al._ (1983); Faust and Wasserman (1992); Karrer and
Newman (2011) 333Blockmodels are essentially equivalent to the hidden-variable
model Boguñá and Pastor-Satorras (2003), when the hidden variables are
discrete, and their multiplicity is smaller than the number of nodes.. As will
become clear in the following, we do so without sacrificing the generality of
the approach, since we can progressively add to this model as many degrees of
freedom as we desire, and in this way obtain arbitrarily elaborate structures
in a controlled fashion.
Figure 1: (Color online) Example of a network corresponding to a blockmodel
with five blocks. The vertices of each block are labeled with the same color.
A stochastic blockmodel assumes that the nodes in the network can be
partitioned into discrete _blocks_ , such that every node belonging to the
same block has (on average) the same characteristics. Hence, for very large
systems, we need only to describe the degrees of freedom associated with the
individual blocks (see Fig. 1). By considering a system composed of many
blocks, we can describe a wide array of possible topological configurations.
More precisely, a (degree-corrected Karrer and Newman (2011)) stochastic
blockmodel is a system of $n$ blocks, where $w_{i}$ is the fraction of nodes
in the network which belong to block $i$ (we have therefore that
$\sum_{i}w_{i}=1$), and $p^{i}_{k}$ is the in-degree distribution of block
$i$. Thus, the average in-degree is given by
${\left<k\right>}=\sum_{k,i}kp^{i}_{k}w_{i}$. The matrix $w_{j\to i}$
describes the fraction of the inputs of block $i$ which belong to block $j$
(we have therefore that $\sum_{j}w_{j\to i}=1$). Since the out-degrees are not
explicitly required to describe the dynamics (see Eq. 11 below), they will be
assumed to be randomly distributed, subject only to the restrictions imposed
by $w_{i}$ and $w_{j\to i}$.
We note that, although we have diminished the class of networks which will be
accessible by the evolutionary algorithm, we still allow a very large array of
possible configurations, which can in principle incorporate arbitrary in-
degree distributions, degree correlations Newman (2003a), assortative or
disassortative mixing Newman (2003b), and community structure Girvan and
Newman (2002), to name only a few properties. As will become clear in the
following section, this blockmodel is sufficient to characterize the most
important topological property that is relevant for robustness against noise,
which is the formation of densely connected central subgraphs.
### IV.1 The value of $b^{*}$ for a blockmodel
Supposing that the number of vertices $Nw_{i}$ belonging to each block $i$ is
arbitrarily large, we can compute the value of $b^{*}$ using an heterogeneous
version of the annealed approximation Derrida and Pomeau (1986), by supposing
that at each time step the inputs of each function are randomly chosen, such
that the specified block structure given by $w_{i\to j}$ is always preserved
Peixoto (2012). If the number of vertices in each block is large enough, we
can expect this approximation to become an exact description for quenched
networks as well. We can then write the average value of $b_{i}$ for each
block over time as
$b_{i}(t+1)=\sum_{k}p_{k}^{i}m_{k}\left((1-2P)\sum_{j}w_{j\to
i}b_{j}(t)+P\right),$ (11)
which is a system of $n$ coupled maps, where $m_{k}(b)$ is the probability
that the output of a majority function will be $1$, if the inputs are $1$ with
probability $b$, and is given by
$m_{k}(b)=\sum_{i={\lceil k/2\rceil}}^{k}{k\choose i}b^{i}(1-b)^{k-i}.$ (12)
A fixed point of Eq. 11 represents the solution of a polynomial system of
arbitrary order, and therefore cannot be written in closed form. However, it
can be obtained numerically by starting the system at $b_{i}=0$, and iterating
Eq. 11 until a fixed-point $\\{b^{*}_{i}\\}$ is reached. The value of $b^{*}$
can then be obtained as $b^{*}=\sum_{i}w_{i}b^{*}_{i}$.
### IV.2 Blockmodel entropy
We obtain the entropy of the stochastic blockmodel ensemble Bianconi (2009);
Peixoto (2011) by enumerating all possible networks which are compatible with
a given choice of $w_{i}$, $p_{k}^{i}$ and $w_{i\to j}$. To make the counting
simpler, we ignore the difficulty of forbidding parallel edges, and consider
the ensemble of _configurations_ , since the occurrence of parallel edges
should vanish for large network sizes (see Peixoto (2011) for more details).
Later we compare the results obtained with Monte-Carlo simulations with
parallel edges forbidden, and we find very good agreement.
We begin by enumerating all possible in-degree sequences of each block which
correspond to the prescribed in-degree distributions,
$\Omega_{d}=\prod_{i}\frac{(Nw_{i})!}{\prod_{k}{(Nw_{i}p_{k}^{i})!}}.$ (13)
For a given block $i$ with a fixed in-degree sequence, we can count the number
of different input choices as
$\Omega^{i}_{e}=\frac{E_{i}!}{\prod_{j}E_{j\to i}!}\prod_{j}(Nw_{j})^{E_{j\to
i}},$ (14)
where $E_{i}=Nw_{i}k_{i}$ is the total number of inputs belonging to block $i$
and $E_{j\to i}=w_{j\to i}E_{i}$ is the total number of inputs from block $i$
which belong to block $j$. Since the set of inputs of each function is
unordered, we still need to divide the whole number of input combinations by
$\prod_{k}(k!)^{N_{k}}$, where $N_{k}=N\sum_{i,k}w_{i}p_{k}^{i}$ is the total
number of vertices with in-degree $k$. Putting it all together we have
$\Omega=\Omega_{d}\frac{\prod_{i}\Omega^{i}_{e}}{\prod_{k}(k!)^{N_{k}}}.$ (15)
Taking the logarithm of this expression, and the limit $N\gg 1$, and using
Stirling’s approximation, we obtain the full entropy (up to a trivial constant
term, which is not relevant to the minimization of the free energy),
$\mathcal{S}/N={\left<k\right>}\ln N+\sum_{i}w_{i}\mathcal{S}_{i}^{k}\\\
-\sum_{i}w_{i}k_{i}\sum_{j}w_{j\to i}\ln\left(\frac{w_{j\to
i}}{w_{j}}\right),$ (16)
where $\mathcal{S}_{i}^{k}$ is an entropy term associated with the degree
distribution of block $i$, and is given by
$\mathcal{S}_{i}^{k}=-\sum_{k}p_{k}^{i}(\ln p_{k}^{i}+\ln k!).$ (17)
### IV.3 Choice of single-block in-degree distribution
We want to constrain the number of degrees of freedom in the model, such that
only the average in-degree $k_{i}$ of each block is specified, not the entire
distribution. In this way, graphs with many different global in-degree
distributions are still possible by composing different blocks with different
$k_{i}$’s, but we have a finite number of degrees of freedom per block. In
order to obtain the in-degree distribution of the individual blocks, we
maximize the entropy $\mathcal{S}$, with the restriction that the average in-
degrees are fixed. For that, we construct the Lagrangian,
$\Lambda=\mathcal{S}-\sum_{i}\lambda_{i}^{\prime}\left(\sum_{k}kp_{k}^{i}-k_{i}\right)-\sum_{i}\mu_{i}\left(\sum_{k}p^{i}_{k}-1\right),$
(18)
where $\\{\lambda_{i}\\}$ and $\\{\mu_{i}\\}$ are Lagrange multipliers which
keep the averages and the normalizations constant. We note that the sum over
$k$ is made only over _odd_ values of $k$, due to the imposed restrictions on
the majority function. Obtaining the critical point
$(\\{\frac{\partial\Lambda}{\partial
p^{i}_{k}}\\},\\{\frac{\partial\Lambda}{\partial\lambda^{\prime}_{i}}\\},\\{\frac{\partial\Lambda}{\partial\mu_{i}}\\})=0$,
and solving for $\\{p^{i}_{k}\\}$ one obtains,
$p_{k}^{i}=\frac{1}{\sinh\lambda_{i}}\frac{\lambda_{i}^{k}}{k!},$ (19)
where $k_{i}=\lambda_{i}/\tanh\lambda_{i}$. Equation 19 is a Poisson
distribution, which is defined and normalized only for odd values of $k$.
This choice of $p^{i}_{k}$ is not necessarily the optimal one. In fact, it is
possible to show that single-valued distributions with zero variance tend to
provide the best error resilience Peixoto (2012). Nevertheless, the
improvement over a Poisson distribution is _very_ small, and the definition of
Eq. 19 allows for the average $k_{i}$ to be continuously varied, which is very
practical for the optimization of the free energy.
### IV.4 Block splitting, decrease of entropy and the necessary number of
blocks
For the blockmodel defined in this section, there are $2n+n^{2}$ variables
which define the topology, where $n$ is the number of blocks. In order for
arbitrary topologies to be faithfully represented by the model, one would need
to make $n\to\infty$, which would render this approach impractical. However,
we will show that for the purpose at hand, only a minimal number of _two_
blocks is sufficient to fully characterize the evolutionary process, without
relying on any approximations. This is due the following two facts: 1. Any
possible value of $b^{*}$ can be obtained with only two blocks; 2. Any other
topology with the same $b^{*}$ will invariably have a lower entropy, and thus
a larger free energy. Thus the minimum of the free energy will always lie on a
two-block structure.
The first fact can be shown by construction: Consider a system of two blocks,
where one of them (the “core”) is smaller and much denser, and the remaining
block has an average in-degree close to the global average. The inputs of the
core block belong mostly to the core itself, as well as the inputs of the
remaining block. By changing the density of the core block, as well as the
degree of input segregation, it can be shown Peixoto (2012) that any possible
value of $b^{*}$ can be achieved 444This is not the only two-block structure
which can generate arbitrary values of $b^{*}$. In Peixoto (2012) it is shown
how a bipartite “restoration” structure also achieves this, albeit less
efficiently..
The second fact can be shown by considering a system of many blocks, and
selecting any two blocks, $l$ and $m$. If all other blocks are kept intact, it
can be shown that the entropy will always be larger if these two blocks are
merged into an effective single block. This can be shown by partially
maximizing the entropy $S$ via the Lagrangian,
$\Lambda=\mathcal{S}-\mu\left(\sum_{i}k_{i}w_{i}-{\left<k\right>}\right)-\sum_{i}\gamma_{i}\left(\sum_{j}w_{j\to
i}-1\right),$ (20)
where $\mu$ and $\\{\gamma_{i}\\}$ are Lagrange multipliers which keep the
average in-degree and the normalization of $w_{j\to i}$ fixed, respectively.
Obtaining the critical point $(\frac{\partial\Lambda}{\partial w_{l\backslash
m}},\frac{\partial\Lambda}{\partial k_{l\backslash
m}},\\{\frac{\partial\Lambda}{\partial w_{{l\backslash m}\to
j}}\\},\\{\frac{\partial\Lambda}{\partial w_{j\to{l\backslash m}}}\\})=0$, and
solving for $w_{l\backslash m},k_{l\backslash m},\\{w_{j\to l\backslash
m}\\},\\{w_{l\backslash m\to j}\\}$ one obtains,
$\displaystyle k_{l}=k_{m}$ (21) $\displaystyle\frac{w_{l\to
j}}{w_{l}}=\frac{w_{m\to j}}{w_{m}}$ (22) $\displaystyle w_{j\to l}=w_{j\to
m}.$ (23)
This corresponds to the situation where the nodes from blocks $l$ and $m$ are
topologically indistinguishable, i.e. the outgoing and incoming edges are
randomly distributed among the nodes of both blocks. Since any arbitrary many-
block structure can be converted into a single-block by successive block
merges, it follows directly that any arbitrary many-block structure can be
constructed by starting from a single block, and successively splitting
blocks. Thus, since the merging of blocks always increases entropy, and the
splitting decreases it, the entropy of the final structure must be smaller
than that of both the initial single block and the succeeding two-block
network.
In order for a many-block structure to have a lower free energy than the two-
block structure with the same value of $b^{*}$, it needs to have a larger
entropy. But according to the above argument, networks with a larger number of
blocks tend to have _smaller_ entropy. Networks with larger entropy and number
of blocks would have to be more randomized than the two-block structure, which
would invariably result in a larger value of $b^{*}$. We can therefore
conclude that the global minimum of the free energy always occurs for a two-
block structure, and thus we need to deal with only eight variables 555We have
empirically verified this by minimizing the free energy with up to 10 blocks,
and the solutions were always _identical_ to that of the two-block case
presented in the following section. The general character of the two-block
topology was also verified by Monte Carlo simulations with up to $20$ blocks,
as discussed later in the text (see also Fig. LABEL:fig:mc-multiblock)..
## V Minimization of the free energy
Although we have an analytical expression for the entropy $\mathcal{S}$, the
value of $b^{*}$ cannot be obtained in closed form, and thus the same is true
for $\mathcal{F}$. Therefore we must resort to a numerical computation of
$b^{*}$, via the iteration of Eq. 11, and minimize $\mathcal{F}$ with a
gradient descent algorithm, using finite differences. Many of these methods
work only for unconstrained optimization problems, and we need to impose
several constraints: The average in-degree must be fixed, and the $w_{i}$ and
$w_{j\to i}$ distributions must be normalized. However we can make the problem
unconstrained by using the following transformations,
$\displaystyle
w_{i}=\frac{e^{\widetilde{w}_{i}}}{\sum_{j}e^{\widetilde{w}_{j}}}$ (24)
$\displaystyle w_{j\to i}=\frac{e^{\widetilde{w}_{j\to
i}}}{\sum_{l}e^{\widetilde{w}_{l\to i}}},$ (25)
where $\widetilde{w}_{i}$ and $\widetilde{w}_{j\to i}$ are unconstrained real
variables. Likewise we can transform $\lambda_{i}$ as
$\displaystyle\widetilde{k}_{i}=\frac{e^{\widetilde{\lambda}_{i}}}{\tanh
e^{\widetilde{\lambda}_{i}}}$ (26) $\displaystyle
k_{i}=c\widetilde{k}_{i}+\gamma$ (27)
$\displaystyle\lambda_{i}=k_{i}\tanh\lambda_{i},$ (28)
where $\widetilde{\lambda}_{i}$ is also an unconstrained real variable. To
obtain $\lambda_{i}$ from Eq. 28 it is simply iterated until it converges to
the correct value, within some desired precision. The values of $c$ and
$\gamma$ are chosen to force $k_{i}\geq 1$ and the average in-degree to a
prescribed value ${\left<k\right>}$,
$\displaystyle c$
$\displaystyle=\frac{{\left<k\right>}}{\sum_{i}\widetilde{k}_{i}w_{i}},$
$\displaystyle\gamma$ $\displaystyle=0,$ $\displaystyle\text{ if
}\widetilde{k}_{m}\geq 1$ (29) $\displaystyle c$
$\displaystyle=\frac{{\left<k\right>}-1}{\sum_{i}\widetilde{k}_{i}w_{i}-\widetilde{k}_{m}},$
$\displaystyle\gamma$ $\displaystyle=1-c\widetilde{k}_{m},$ otherwise,
where $\widetilde{k}_{m}=\min(\\{\widetilde{k}_{i}\\})$. Thus we have obtained
an unconstrained minimization problem of function $\mathcal{F}$ with respect
to the variables $\\{\widetilde{w}_{i}\\}$, $\\{\widetilde{w}_{j\to i}\\}$,
$\\{\widetilde{\lambda}_{i}\\}$.
In order to find the minimum of Eq. 10, we employed the L-BFGS quasi-Newton
algorithm Byrd _et al._ (1995), with the gradient computed by finite
differences.
## VI Structural phase transition
The minimization of the free energy leads to one of two possible structures
(see Fig. 2): 1. For low values of $\beta$ the topology is a fully random
graph; 2. For larger values of $\beta$ there is the emergence of a _segregated
core_ structure, where one of the blocks has a larger in-degree density and is
more responsible for the regulation of the whole network.
Random topology
$\xrightarrow{\text{increasing }\beta}$ Segregated core
Figure 2: (Color online) Structural phase transition observed when varying the
selective pressure $\beta$, as described in the text.
In order to precisely characterize the phase transition, we define the
following order parameter,
$\phi=\frac{b^{*}-b_{r}}{b_{\text{min}}-b_{r}},$ (30)
where $b_{r}$ is the value of $b^{*}$ for a fully random network, and
$b_{\text{min}}$ is the smallest possible value of $b^{*}$ for a given
${\left<k\right>}$ Peixoto (2012), given by
$b_{\text{min}}=\sum_{k}p_{k}m_{k}(P),$ (31)
where $p_{k}$ is given by Eq. 19 with $k_{i}={\left<k\right>}$. We have
therefore that $\phi\in[0,1]$ and if $\phi=0$ the network ensemble must be
fully random, and if $\phi=1$ it has the largest possible value of fitness.
As can be seen in Fig. LABEL:fig:diag-op, there is a second-order phase
transition at a critical value $\beta^{*}$, which depends on the noise level
$P$. The dependence of $\beta^{*}$ on $P$ divides the $\beta\times P$ phase
diagram into an upper and lower branch, as can be seen in Fig. LABEL:fig:diag-
phi. The branches are divided at a value of $P=P^{*}_{r}$, which is the
critical value of noise of a fully random network (see Peixoto (2012) for an
exact calculation of $P^{*}_{r}$). At this value of noise, a random network
undergoes a dynamic phase transition, where the steady state error fraction
reaches the maximum level $b^{*}=1/2$, and the dynamics become ergodic, as was
described previously. For $P<P^{*}_{r}$, random networks are intrinsically
robust, since $b^{*}<1/2$, and the critical value $\beta^{*}$ becomes larger
with smaller $P$. In other words, the smaller is the value of noise $P$, the
better is the behavior of fully random networks, such that the entropic cost
of providing further improvement by creating a segregated core becomes larger,
which therefore occurs only at larger values of selective pressure. The
situation changes for $P>P^{*}_{r}$. Since random networks are no longer
resilient, and have collapsed onto $b^{*}=1/2$ (see Fig. LABEL:fig:diag-pc), a
segregated core provides a significant improvement, for a relatively low
entropic cost. This cost increases with $P$, since the core needs to be either
denser, smaller or more isolated to provide the same benefit under larger
noise. Thus the value of $\beta^{*}$ also increases with $P$. Interestingly,
in the vicinity of $P=P^{*}_{r}$, the value of $\beta^{*}$ tends to zero. For
this value of noise, the dynamics of the fully random topology lies exactly at
the critical point where $b^{*}=1/2$, and even the smallest (structural)
perturbation can move the fixed point appreciably. Since such small structural
perturbations have negligible entropic cost, the value of $\beta^{*}$ vanishes
to zero. Thus, networks with ${\left<k\right>}$ such that $P^{*}_{r}=P$ are
the most easily evolvable.
The value of $b^{*}$ itself can be seen in Fig. LABEL:fig:diag-pc. The upper
branch $P>P^{*}_{r}$ corresponds to transitions from $b^{*}=1/2$ (ergodic
dynamics) to $b^{*}<1/2$ (nonergodic dynamics), whereas the lower branch
$P<P^{*}_{r}$ shows $b^{*}<1/2$ for both phases.
The topological properties of each phase can be seen in detail in Fig.
LABEL:fig:diag-top, where are shown the average in-degrees $\\{k_{i}\\}$,
block sizes $\\{w_{i}\\}$ and the total fraction of inputs which lead to the
segregated core, $E_{c}=\sum_{j}w_{c\to j}w_{j}k_{j}/{\left<k\right>}$, where
$c$ is the core block. The core block emerges at $\beta=\beta^{*}$, with an
infinitesimal size, but with a value of $k_{i}$ which is usually significantly
above average. For $P>P^{*}_{r}$ the segregated core usually has a
significantly larger average in-degree than for $P<P^{*}_{r}$. The dominance
and segregation of the core block increases continuously with $\beta$,
reaching values close to $E_{c}=1$, for larger values of $\beta$, especially
for values of $P>P^{*}_{r}$.
Each network on the evolved ensemble has a critical value of noise $P^{*}$
(different from the value of $P$ for which it was evolved), for which its
dynamics undergoes the aforementioned ergodicity transition and which
represents the maximum tolerable noise (see Peixoto (2012) for an exact
calculation of $P^{*}$ for arbitrary blockmodels). Interestingly, the
evolution of $b^{*}$ does not automatically result in larger values of
$P^{*}$, as is shown in Fig. LABEL:fig:diag-pc: Some ensembles evolved under
larger selective pressure possess a lower value of $P^{*}$ than others evolved
under lower selective pressure (for the same value of $P$ under evolution).
This means the evolution is reasonably specialized for the level of noise it
is under, and the behavior of the networks under larger values of noise for
which they were evolved is not automatically better than that of other
networks with smaller fitness. However, despite these deviations, the general
tendency is that, for larger values of $\beta$, $b^{*}$ and $P^{*}$ are
decreased and increased, respectively.
## VII Monte-Carlo simulations
We have also performed Monte Carlo simulations to observe the phase-transition
obtained in the previous section. We employed the Metropolis-Hastings
Metropolis _et al._ (1953); Hastings (1970) algorithm, starting from a random
network with $N$ vertices, with a given average in-degree ${\left<k\right>}$
and a partition into $n$ blocks, represented by assigning block labels to each
vertex (which is initially randomly chosen). At each iteration, one of the
following moves is attempted with equal probability:
1. 1.
_Block label move_ : A random vertex $v$ is chosen, and its block label is
randomly chosen among all $n$ possible values.
2. 2.
_Input move_ : A vertex $v$ is chosen with probability $p\propto k(k-1)$,
where $k$ is the in-degree of $v$. Another vertex $u$ is randomly chosen with
uniform probability. Two random inputs from $v$ are deleted and moved to $u$.
3. 3.
_Source move_ : A random vertex $v$ is chosen. A random input from $v$ is
deleted and replaced by a randomly chosen one.
A move is rejected if it generates parallel edges or self-loops. The
difference $\Delta b^{*}$ of the value of $b^{*}$ after and before the move is
computed. The move is then accepted with a probability $p_{a}$ given by
$p_{a}=\begin{cases}1&\text{ if }\Delta b^{*}\leq 0,\\\ e^{-\beta N\Delta
b^{*}}&\text{ otherwise.}\end{cases}$ (32)
The probability $p\propto k(k-1)$ in move (2) is chosen to correspond to _two_
independent single-edge moves affecting the same vertices $v$ and $u$, where
in each move a random edge is chosen, and its target is moved to a randomly
chosen vertex. This guarantees that there is no topological bias, and that the
in-degrees are always odd.
The value of $b^{*}$ is computed by obtaining the values of $\\{w_{i}\\}$,
$\\{w_{j\to i}\\}$ and $\\{k_{i}\\}$, and iterating Eq. 11. This is much
faster than actually measuring the error level on the network, and produces
deterministic values 666One could argue that this may overlook the buildup of
correlations in the network, since it assumes that the blocks are homogeneous
and have an random in-degree distribution as given by Eq. 19. However, we are
_only_ interested in networks which have this property, which, as discussed in
the text, correspond to a partial maximization of entropy when the remaining
constraints are in place, so this should not be an issue. To be sure, we have
compared the value of $b^{*}$ computed this way with the actual empirical
value and found a very good agreement (see Fig. LABEL:fig:mc-trans)..
Since we have employed the block label move (1), which tends to partition the
network evenly into $n$ blocks of equal sizes, we have included an entropic
cost associated with the size of a block, which did not exist in the original
blockmodel above. In the original model, the partitions themselves are not
relevant, and only the resulting graph topology contributes to the entropy.
However, move (1) makes the algorithm very efficient and easy to implement,
and it should not fundamentally change the results. But in order to compare
with the theory, we need to include the following correction in the number of
possible networks:
$\Omega^{\prime}=\Omega\times\frac{N!}{\prod_{i}(Nw_{i})!}$ (33)
which leads to the slightly modified entropy,
$\mathcal{S}^{\prime}/N=\mathcal{S}/N-\sum_{i}w_{i}\ln w_{i}.$ (34)
In Fig. LABEL:fig:mc-trans we can see the same phase transition observed
previously, which matches very well the theoretical predictions. In Fig.
LABEL:fig:mc-top the topology can be assessed more closely, and the emergence
of the segregated core is clear. Due to the partition entropy introduced in
Eq. 34, the core does not vanish at the transition; it merges continuously
with the other block instead. However, the critical value $\beta^{*}$ is
identical with the non-modified model.
The inclusion of the partition entropy also introduces the fact that different
solutions are obtained for a different number of blocks, since this has a
direct effect on the preferred sizes of the blocks (see Fig. LABEL:fig:mc-
multiblock, left). However, this _does not_ change the fact that for any
number of blocks the preferred topology will always be an effective two-block
structure. This follows from the argumentation presented previously based on
the reduction of entropy resulting from block splits, and can be observed in
simulations with many blocks, as shown in Fig. LABEL:fig:mc-top, which shows a
comparison between the topologies obtained with two and three blocks, as well
as the outcome of a typical simulation with $20$ blocks, which shows the
eventual collapse of into an effective two-block structure.
## VIII Gene regulatory networks
Here we make a comparison with some features observed in actual gene
regulatory networks. We consider the networks for _Saccharomyces cerevisiae_
(yeast) and _Escherichia coli_ , extracted from the YEASTRACT Abdulrehman _et
al._ (2010) and RegulonDB Gama-Castro _et al._ (2010) databases,
respectively. We are interested in extracting the “functional core” of the
network, i.e. those nodes which are solely responsible for global regulation,
like those belonging to the _segregated core_ which emerges in the phase
transition observed in the evolutionary process above. We will characterize
the core nodes in two ways: 1. Nodes which have an out-degree larger than
zero; 2. Nodes which belong to a _strongly connected component_ (SCC) of the
graph (i.e. the maximal set of nodes which can directly or indirectly regulate
each other). The first criterion is a necessary condition, since if the out-
degree is zero, then a node is not a regulator. The second criterion is
stronger, since even if a node is a regulator, it can have its dynamics
completely enslaved by other nodes. The nodes in the SCC are exactly those
which are not necessarily enslaved, and can mutually regulate each other.
Without a least one SCC in the network, an autonomous behavior with dynamical
attractors other than simple fixed-points is not possible.
The yeast network is composed of $N=6402$ nodes, with an average in-degree of
${\left<k\right>}\approx 7.51$. The _E. coli_ network is smaller and sparser,
with $N=1658$ and ${\left<k\right>}\approx 2.32$. In both networks the number
of transcription factor (TF) genes is much smaller than the total number:
$N_{\text{TF}}=182$ in yeast, and $N_{\text{TF}}=154$ in _E. coli_. These are
core genes according to the first criterion, since they have an out-degree
larger than zero, as can be seen in Fig. 10.
Figure 10: (Color online) Gene regulatory networks for _Saccharomyces
cerevisiae_ (left) and _Escherichia coli_ (right), extracted from the
YEASTRACT Abdulrehman _et al._ (2010) and RegulonDB Gama-Castro _et al._
(2010) databases, respectively. The nodes in purple (towards the middle) are
transcription factor genes, and are the only ones with out-degree larger than
zero.
In yeast, the average in-degree of the core nodes is higher than average,
${\left<k\right>}_{c}\approx 10.03$, as observed in the segregated core phase
of the evolved networks obtained. For the SCC, the number of nodes decreases
slightly to $N_{\text{SCC}}=146$, and the average in-degree changes negligibly
${\left<k\right>}_{CC}\approx 10.48$ (if one counts only edges between
vertices of the SCC, this value is virtually identical,
${\left<k\right>}_{CC}\approx 10.42$). This is similar to what was found
previously in Maslov and Sneppen (2005) for the yeast network (using an older,
and less complete dataset with only $837$ genes). They have also found that
the TF genes have different connection patterns, and those with the largest
out-degree tend to regulate genes with lower than average in-degree. However
they did not find that the TF genes form a _denser_ subgraph, with an larger
than average in-degree, which is most likely due to the incompleteness of the
dataset used. Very similar numbers to those presented here were obtained more
recently in Balaji _et al._ (2006a), using a more complete dataset (which is
not identical to the one used in this work).
For _E. coli_ the situation changes somewhat: The average in-degree of the
transcription factor nodes is ${\left<k\right>}_{c}\approx 1.97$, which is in
fact _lower_ than the global average. However, if one extracts the largest
SCC, the number of nodes drops dramatically to $N_{\text{SCC}}=8$. These nodes
are responsible for the regulation of $411$ genes. A majority of $1093$ genes
are instead enslaved to the dynamics of SCC with only two mutually regulating
nodes. Although the largest SCC _does_ have an average in-degree
${\left<k\right>}_{CC}=6$, the core topology seems significantly more sparse
than for yeast and the evolved networks, at least with the data currently
available Cosentino Lagomarsino _et al._ (2007). Arguably, such a sparse
regulating core is suspect from the point of view of data set completeness,
since it would mean that the range of dynamical behavior for the regulatory
network is very restricted. As previously mentioned, and older and less
complete data set for yeast also did not reveal a denser regulating core
Maslov and Sneppen (2005). Nevertheless, one should also consider that such
real networks are simultaneously under different, possibly competing selective
pressures which also influence the resulting topology, robustness against
noise being only one of them. These other factors, which are neglected in the
model, could be one reason for such a discrepancy. We emphasize, however, that
although apparently it is not denser, a regulating core certainly exists in
the measured _E. coli_ network, which is at least in partial qualitative
agreement with what is observed in the model.
There are other factors that may contribute to this observed segregation which
are not in principle related to noise resilience. For instance, non-regulating
genes exist mostly to transcribe proteins which have some specific metabolic
or structural function in the cell, and it may be difficult for these proteins
to have a dual role as transcription factors, and therefore become specialized
(although non-specialized proteins are not impossible, since a protein can in
principle bind both to DNA and to other proteins). Nevertheless, there are
good reasons to consider robustness to noise as a very plausible driving force
toward this type of topology. This is corroborated, for instance, by evidence
that core TF genes tend to be less noisy Fraser _et al._ (2004); Jothi _et
al._ (2009), and that the vast majority of TF genes in yeast are not vital for
the survival of the cell if repressed in isolation Balaji _et al._ (2006b).
This is fully compatible with the idea of a highly redundant functional core,
which provides robustness for the rest of the network.
Another feature which is commonly investigated in empirical networks is the
in- and out-degree distributions. The in-degree distribution is often narrow,
while the out-degree distribution is broader, and as some suggest Maslov and
Sneppen (2005), compatible with a power law. The model considered in this work
is parametrized as a stochastic blockmodel, where each block has in- and out-
degrees that are Poisson distributed. When the segregated core emerges, the
system is composed of only two blocks, thus both the in- and out-degree
distributions are bimodal. The in-degree distribution is indeed narrower,
since the difference between the average in-degree of the two blocks is not
very large for most networks obtained. The out-degree distribution is also
much broader, since the average out-degree of the non-core block tends to
zero, while for the core block it tends very rapidly to infinity, when the
selective pressure is increased. However, the homogeneous and seemingly scale-
free properties of the empirical distributions are not reproduced by the
model. This implies that these features are not a direct result of evolved
robustness against noise, and may, for instance, be due simply to mutational
bias caused by gene duplication, which is known to qualitatively reproduce
these types of in- and out-degree distributions Ispolatov _et al._ (2005);
Enemark and Sneppen (2007).
## IX Conclusion
We have investigated the effect of selective pressure favoring robustness
against noise on the structural evolution of Boolean networks with optimal
majority functions, functioning as a conceptual model for gene regulation. We
have mapped the evolutionary process onto a Gibbs ensemble, and obtained its
outcome by minimizing the associated free energy. We showed that the
structural properties of the system undergo a phase transition at a critical
value of selective pressure, from a random topology to a segregated-core
structure, where a smaller fraction of the nodes form an isolated core, which
is denser than the rest of the network and is responsible for most of the
regulation. Since the core is denser, its nodes can profit from more
regulatory redundancy, which greatly diminishes the effect of noise. This
robustness is propagated to the rest of the network, which relies on the core
for most of the regulation. The segregated core becomes denser, smaller, and
more isolated as the selective pressure increases. We have compared the
theoretical predictions with Monte-Carlo simulations of actual networks, and
found perfect agreement.
We have also shown that this segregated-core structure is present in the gene
regulatory network of yeast and _E. coli_. Both networks are composed of a
much smaller fraction of transcription factor genes which are responsible for
all regulation. In yeast, the existing core structure is very similar
qualitatively to the outcome of the evolutionary process considered, with
transcription factor genes forming a denser subgraph, with an average in-
degree above the average for the whole network. In _E. coli_ the isolated
transcription factor core is composed of few very small regulating cores
(strongly connected components), the largest of which has only eight nodes. We
conjecture that such a sparse regulating core is possibly due to data set
incompleteness, since it would severely restrict the range of possible
dynamical behavior for the network. A less complete data set for yeast also
did not show a denser regulating core structure Maslov and Sneppen (2005),
although it is clearly seen with more up-to-date datasets including more genes
and interactions Balaji _et al._ (2006a). However, one should not rule out
other selection criteria which are not incorporated in the model.
It should also be noted that regulating cores of transcription factors are a
common feature of other organisms, such as _Mycobacterium tuberculosis_
Balazsi _et al._ (2008). Additionally, a similar (but not identical) “bow-
tie” structure was also observed in the mammalian signal transduction network
Ma’ayan _et al._ (2005); Ma’ayan (2009), where most pathways are funneled
through a central core.
It is possible to formulate other reasons for the existence of such a core
structure, such as a forced specialization of genes into either transcription
factors or target genes. Furthermore one should mention that the effects of
noise are not always detrimental, and can in some circumstances even be
beneficial Blake _et al._ (2006); Eldar and Elowitz (2010). Nevertheless
there is compelling evidence that the core genes provide a degree of
robustness to the cell. Not only are the best-connected TF nodes less noisy
Fraser _et al._ (2004); Jothi _et al._ (2009), they are usually found — if
removed individually — not to be vital for cell survival Balaji _et al._
(2006b). This corroborates the idea that one of the major functions of the
regulating core is to provide robustness via redundancy.
Furthermore, aside from the direct applicability to gene regulation, we have
identified a fundamental mechanism of robustness against noise, which emerges
naturally when networks are randomly selected for that purpose 777Other
evolutionary, non-equilibrium pathways are also possible, see e.g. Perotti
_et al._ (2009).. Although most interesting systems require more than just
robustness for their functioning, it is reasonable to conclude that the
emergence of regulating cores is to be expected when there is enough selective
pressure favoring noise resilience.
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|
arxiv-papers
| 2011-08-22T15:11:50 |
2024-09-04T02:49:21.696874
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tiago P. Peixoto",
"submitter": "Tiago Peixoto",
"url": "https://arxiv.org/abs/1108.4341"
}
|
1108.4432
|
# Exploiting the Passive Dynamics of a Compliant Leg to Develop Gait
Transitions
Harold Roberto Martinez Salazar http://ailab.ifi.uzh.ch/martinez/
martinez@ifi.uzh.ch Juan Pablo Carbajal http://ailab.ifi.uzh.ch/carbajal/
carbajal@ifi.uzh.ch, both authors can be contacted regarding the content of
the paper Artificial Intelligence Laboratory, Department of Informatics,
University of Zurich
Andreasstrasse 15 8050 Zurich Switzerland
###### Abstract
Abstract
In the area of bipedal locomotion, the spring loaded inverted pendulum (SLIP)
model has been proposed as a unified framework to explain the dynamics of a
wide variety of gaits. In this paper, we present a novel analysis of the
mathematical model and its dynamical properties. We use the perspective of
hybrid dynamical systems to study the dynamics and define concepts such as
partial stability and viability. With this approach, on the one hand, we
identified stable and unstable regions of locomotion. On the other hand, we
found ways to exploit the unstable regions of locomotion to induce gait
transitions at a constant energy regime. Additionally, we show that simple
non-constant angle of attack control policies can render the system almost
always stable.
###### pacs:
Valid PACS appear here
††preprint: APS/123-QED
## I Introduction
One of the most accepted mathematical models for bipedal running is the spring
loaded inverted pendulum (SLIP, for an extensive review seeHolmesSIAM.06 ). In
a similar fashion, the rigid inverted pendulum has been extensively used to
model bipedal walkingMochon1980 . In 2006, Geyer et al.Geyer2006 propose the
SLIP model as a unifying framework to describe walking as well as running. The
unified perspective proves useful for accurately explaining data from human
locomotionGeyer2006 . Additionally, it allows describing both gaits (walking
and running) in terms of dynamical entities observed in a discrete map,
obtained by intersecting the trajectories of the system with a predefined
section of lower dimension. Geyer associates these entities with limit cycles
of the hybrid dynamical systemGuckenheimer95 ; Cortes2008 and named their
attracting behavior as self-stabilization. Though the nature of the observed
dynamical properties is not yet clarified, those results emphasize that
bipedal locomotion may be dictated solely by the mechanics of the system. As a
consequence, the control necessary for locomotion is thus reduced to the swing
phase of the leg, showed in Fig. 1 between points A and B. The most popular
control policy is to produce touchdowns at constant angle of attack $\alpha$
($\operatorname{CAAP}\left(\alpha\right)$), i.e. the angle spanned by the
landing leg and the horizontal.
In the last decade, many energy-efficient bipedal walking machines have been
developed. Through careful design, they exploit the passive dynamics of their
own body to move forward, requiring little control or noneMcGeer1990 ;
Collins2001 ; MartijnWisseandJanvanFrankenhuyzen2006 ; Collins2005 ; Geng2006
. However, the construction of bipedal machines capable of exploiting passive
dynamics in different gaits remains an unsolved engineering challenge. In this
context, Geyer et al.Geyer2006 report that, in the SLIP model, it is not
possible to have multiple gaits at the same energy. The results are based on
simulations that do not cover all possible initial conditions of the system.
In addition, Rummel et al.Rummel2010 prove that walking and running is
possible at the same energy level. They use a new map that allows comparing
different gaits with ease. The map is defined at the vertical plane crossing
the landing point of the foot (Fig. 1). In this way, they find the self-stable
regions, but their intersection is empty. To concretize these ideas, let us
describe this region for the running map $\operatorname{\mathcal{R}}$.
$E_{\infty}^{R}=\\{x|\;x\in\mathcal{S}\wedge\left(\exists\alpha|\;x=\operatorname{\mathcal{R}}_{\alpha}\left(x\right)\right)\\},$
(1)
where the subscript in $\operatorname{\mathcal{R}}_{\alpha}$ denotes running
using $\operatorname{CAAP}\left(\alpha\right)$ and $\mathcal{S}$ denotes the
section where the map is defined. Therefore, if for different gaits these
stable regions do not intersect, e.g. $E_{\infty}^{R}\cap
E_{\infty}^{W}=\emptyset$, we conclude that a transition between the two gaits
cannot occur if the system is to remain in these regions. In other words,
$\begin{split}x\in E_{\infty}^{R}\>\wedge\>&y\in
E_{\infty}^{W}\quad\Rightarrow\\\
&\operatorname{\mathcal{R}}_{\alpha}\left(y\right)\notin
E_{\infty}^{R}\>\wedge\>\operatorname{\mathcal{W}}_{\beta}\left(x\right)\notin
E_{\infty}^{W}\quad\forall\>\alpha,\,\beta.\end{split}$ (2)
Figure 1: (Color online) Illustration of the evolution of the SLIP model for
running and walking. The mass is represented with a filled circle. The color
of the fill indicates touchdown event (black), takeoff event (white), and the
crossing of the section (pink (grey)). The landing leg is pictured with a
thick solid line, and the leg at takeoff is represented with a blurred line.
Due to the passive properties of these models, control is necessary only
during the swing of the leg, i.e. during free fall while running and from
point A to B while walking.
In this study, we will show how transitions between gaits are found at points
outside these stable regions. The transitions require the selection of the
angle of attack; therefore CAAP’s are not suitable for this task. We will also
show evidence indicating that it is possible to find an angle of attack
$\theta$ that maps a point into a stable region, e.g. $x\notin
E_{\infty}^{R}\>\wedge\>\left(\exists\>\theta,y\;|\>y\neq x,\;y\in
E_{\infty}^{R},\;y=\operatorname{\mathcal{R}}_{\theta}(x)\right)$.
Additionally, we introduce the concepts of partial stability and viability
that will be useful in the construction of the transitions presented herein.
This paper is organized as follows. In section II, we describe the models used
for our simulations, their representation in state variables and the
definition of the discrete map. Next, in section III, we introduce the new
concepts, and we show the regions where the transitions between gaits exist.
Later, in section IV, we discuss about the requirements of a controller for
the system and the implications for robot design and bipedal locomotion. We
conclude the paper in section V with our conclusion.
## II Methods
As explained previously, we use the SLIP model to study bipedal gaits. We
adopt the framework in JuergenRummel2009 , which is described in the language
of hybrid dynamical systems. Therefore, we reintroduce some notation and
definitions.
To represent the different phases of a gait, the model is segmented into three
sub-models. We will call these sub-models chartsGuckenheimer95 or phases see
Fig.1. Each chart represents the motion of a point mass under the influence
of: only gravity (ff-chart or flight phase), gravity and a linear spring
(s-chart or single stance phase), gravity and two linear springs (d-chart or
double stance phase). The point mass represents the body of the agent and the
massless linear springs model the forces from the legs (Fig.1). A trajectory
switches from one chart to another when some real valued functions evaluated
on it cross zero (event functionsGuckenheimer95 ; Piiroinen08 ). We define a
running gait as a trajectory that switches from the s-chart to the ff-chart
and back to the s-chart. A walking gait is defined as a trajectory that
switches from the s-chart to the d-chart and back again to the s-chart.
Switches from the ff-chart to d-chart or vice versa are not included in this
study.
### II.1 Equations of motion in each chart
The motion in all the charts is governed by a system of ordinary differential
equations:
$\dot{\vec{X}}=\vec{F_{i}}\left(\vec{X}\right),$ (3)
where $\vec{X}$ is the vector of state variables and $\vec{F_{i}}$ is a force
function characteristic of each chart. Since all forces are conservative, the
energy of the system is constant. For the ff-chart the state is described by
the Cartesian coordinates of the position of the point mass and its velocity
$\vec{X}_{ff}=\left(x,y,v_{x},v_{y}\right)^{T}$,
$\dot{\vec{X}}_{ff}=\begin{pmatrix}v_{x}\\\ v_{y}\\\ 0\\\ -g\end{pmatrix},$
(4)
where $g$ is the acceleration due to gravity.
The state in the s-chart is represented in polar coordinates
$\vec{X_{s}}=\left(r,\theta,\dot{r},\dot{\theta}\right)^{T}$, where $r$ is the
length of the spring and $\theta$ is the angle spanned by the leg and the
horizontal, growing in clockwise direction. Thus, the equations of motion are:
$\dot{\vec{X}}_{s}=\begin{pmatrix}\dot{r}\\\ \dot{\theta}\\\
\frac{k}{m}\left(r_{0}-r\right)+r\dot{\theta}^{2}-g\sin\theta\\\
-\frac{1}{r}\left(2\dot{r}\dot{\theta}+g\cos\theta\right)\end{pmatrix}.$ (5)
It is important to note that $\theta(t_{TD})=\alpha$, i.e. the angular state
at the time of touchdown is equal to the angle of attack. The parameter
$r_{0}$ defines the natural length of the spring.
In the d-chart the state is also represented in polar coordinates
$\vec{X_{d}}=\left(r,\theta,\dot{r},\dot{\theta}\right)^{T}$, with the origin
of coordinates in the new touchdown point. The motion is described by:
$\displaystyle\dot{\vec{X}}_{d}=\begin{pmatrix}\dot{r}\\\ \dot{\theta}\\\
\begin{split}\frac{k}{m}\left[(r_{0}-r)+\left(1-\frac{r_{0}}{r_{{\tiny\mars}}}\right)(x_{{\tiny\mars}}\cos\theta-r)\right]\\\
+r\dot{\theta}^{2}-g\sin\theta\end{split}\\\
\begin{split}-\frac{1}{r}\left[\frac{k}{m}\left(1-\frac{r_{0}}{r_{{\tiny\mars}}}\right)x_{{\tiny\mars}}\sin\theta+2\dot{r}\dot{\theta}+g\cos\theta\right]\end{split}\end{pmatrix}$
(6) $\displaystyle
r_{{\tiny\mars}}=\sqrt{r^{2}+x_{{\tiny\mars}}^{2}-2rx_{{\tiny\mars}}\cos\theta},$
(7)
where $x_{{\tiny\mars}}$ is the horizontal distance between the two contact
points and $r_{{\tiny\mars}}$ is the length of the back leg.
### II.2 Event functions
Event functions are functions on the phase space of the system. An event
occurs when the trajectory of the system intersects a level curve of the event
function. At the time of the event, the current state of the system is mapped
to the state of another chart. Some event functions are parameterized with the
angle of attack and the natural length of the springs.
Switches from the ff-chart to the s-chart are defined by:
$\mathcal{F}_{ff\rightarrow
s}\left(\vec{X}_{ff},\alpha,r_{0}\right):\begin{cases}y-r_{0}\cos\alpha=0\\\
v_{y}<0\end{cases},$ (8)
which means that the mass is falling and the leg can be placed at its natural
length with angle of attack $\alpha$. Therefore, the motion is now defined in
the s-chart. The switch in the other directions is simply:
$\mathcal{F}_{s\rightarrow ff}\left(\vec{X}_{s},r_{0}\right):r-r_{0}=0.$ (9)
These are the only two event functions involved in the running gait. The map
from one chart to the other is defined by:
$\quad x=-r\cos\theta\qquad y=r\sin\theta.$ (10)
It is important to have in mind that the origin of the s-chart is always at
the touchdown point.
For the walking gait, we have to consider switches between single and double
stance phases. From the s-chart to the d-chart, we have:
$\mathcal{F}_{s\rightarrow
d}\left(\vec{X}_{s},\alpha,r_{0}\right):\begin{cases}r\sin\theta-
r_{0}\cos\alpha=0\\\ \theta>\frac{\pi}{2}\end{cases},$ (11)
which is similar to (8) with the additional condition that the mass is tilted
forward. Additionally, if we consider the sign of the radial speed, we
differentiate between walking gait $\operatorname{\mathcal{W}}$ with
$\dot{r}<0$ and Grounded Running gait $\operatorname{\mathcal{GR}}$, with
$\dot{r}>0$.
The switch from the double stance phase to the single stance phase is defined
by:
$\mathcal{F}_{d\rightarrow
s}\left(\vec{X}_{d},r_{0}\right):r_{{\tiny\mars}}-r_{0}=0,$ (12)
with $r_{{\tiny\mars}}$ as defined in (7). The map from the d-chart to the
s-chart is the identity. In the other direction we have:
$\displaystyle r_{d}=r_{0}\quad\theta_{d}=\alpha,$ (13) $\displaystyle
x_{{\tiny\mars}}=r_{0}\cos\alpha-r_{s}\cos\theta_{s},$ (14)
where the subscripts indicate the corresponding chart.
If the system falls to the ground ($y\leq 0$), attempts a forbidden transition
(e.g. d-chart to ff-chart), or renders $v_{x}<0$ (motion to the
left,“backwards”), we consider that the system fails.
### II.3 Simulation of the dynamics
The state of the model is observed when the trajectory of the system
intersects the section defined by $\mathcal{S}:\theta=\nicefrac{{\pi}}{{2}}$.
In this way, the map
$\operatorname{\mathcal{R}}_{\alpha}:\mathcal{S}\rightarrow\mathcal{S}$
transforms points through the evolution of the system from the s-chart to the
ff-chart and back again to the s-chart using an angle of attack $\alpha$.
Similarly, the map
$\operatorname{\mathcal{W}}_{\alpha}:\mathcal{S}\rightarrow\mathcal{S}$
transforms points through the evolution of the system from the s-chart to the
d-chart and back again to the s-chart using an angle of attack $\alpha$.
All initial conditions are given in the $\mathcal{S}$ section and in the
s-chart, i.e. only one leg touching the ground and oriented vertically.
Moreover, all the initial conditions are given at the same total energy. The
results are visualized using the values of the length of the spring $r$ and
the radial component of the velocity which, in $\mathcal{S}$, equals the
vertical speed $\dot{r}=v_{y}$ ($v_{x}$ is obtained from these values and the
equation of constant energy). It is important to note that all possible values
of $r$, $v_{y}$ and $v_{x}$, for a given value of the total energy $E$, lay on
an ellipsoid. Besides, there is a transformation that maps the ellipsoid to a
sphere. This can be shown as follows: the total energy in the section is,
$E=\frac{1}{2}k\left(r_{0}-r\right)^{2}+\frac{1}{2}m\left(v_{x}^{2}+v_{y}^{2}\right)+mgr$
(15)
Defining the parameters
$\displaystyle L$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{k}\left[E-mg\left(r_{0}-\frac{mg}{2k}\right)\right]},$
(16) $\displaystyle\omega$ $\displaystyle=$ $\displaystyle\sqrt{\frac{k}{m}},$
(17)
the new variables
$\displaystyle\hat{v}_{x}$ $\displaystyle=$
$\displaystyle\frac{v_{x}}{\omega},$ (18) $\displaystyle\hat{v}_{y}$
$\displaystyle=$ $\displaystyle\frac{v_{y}}{\omega},$ (19)
$\displaystyle\hat{r}$ $\displaystyle=$ $\displaystyle
r-\left(r_{0}-\frac{mg}{k}\right),$ (20)
transform equation (15) into,
$L^{2}=\hat{v}_{x}^{2}+\hat{v}_{y}^{2}+\hat{r}^{2}$ (21)
which defines a sphere. Therefore, all initial conditions of $\hat{r}$ and
$\hat{v}_{y}$ with constant energy, are defined inside a circle. A Delaunay
triangular mesh was created in the circle with $65896$ initial conditions as
vertices ($131245$ triangles). Each vertex was transformed using
$\operatorname{\mathcal{R}}_{\alpha}$, $\operatorname{\mathcal{GR}}_{\alpha}$
and $\operatorname{\mathcal{W}}_{\alpha}$ with $400$ values of
$\alpha\in[55\mathrm{\SIUnitSymbolDegree},90\mathrm{\SIUnitSymbolDegree}]$. To
compute the evolution of an arbitrary initial condition, we used bilinear
interpolation in the triangles of the mesh.
The model implementation and data analysis were carried out in MATLAB(2009,
The MathWorks), GNU Octaveoctave2002 and Matplotlibmatplotlib . Simulations
were run for constant energy, using the step variable integrator ode45
(relative tolerance: $1\times 10^{-6}$ and absolute tolerance: $1\times
10^{-8}$). Table 1 shows the values of the parameters used.
Table 1: Values used for the simulations presented in this paper. Description | Name | Value
---|---|---
Mass | $m$ | $80\text{\,}\mathrm{kg}$
Elastic constant of linear springs | $k$ | $15\text{\,}\mathrm{k}$ N m
Rest length of linear springs | $r_{0}$ | $1\text{\,}\mathrm{m}$
Total energy | $E$ | $820\text{\,}\mathrm{J}$
Acceleration due to gravity | $g$ | $9.81\text{\,}\mathrm{m}$/^ 2 s
Angle of Attack | $\alpha$ | from $55\text{\,}\mathrm{\SIUnitSymbolDegree}$to $90\text{\,}\mathrm{\SIUnitSymbolDegree}$
## III Results
In this section, we present the results of the analysis on the data collected
from the models as described in section II.3. Aiming to define a controller,
we introduce some important properties of the dynamics of each gait, namely
finite stability for a given CAAP and viability.
### III.1 Finite stability and Viability
Finite stability describes the set of initial conditions where the system can
do a maximum amount of steps (sequential applications of the map) before
failing, using CAAP. For example, we can define for
$\operatorname{\mathcal{W}}$
$E_{n}^{W}=\\{x|\;x\in\mathcal{S}\wedge\left(\exists\alpha|\;y=\operatorname{\mathcal{W}}_{\alpha}^{n}\left(x\right),\;n\geq
1,\;y\in\mathcal{S}\right)\\}.$ (22)
That is, at a given state $x=(r,v_{y})$ in $\mathcal{S}$ there is a
$\operatorname{CAAP}\left(\alpha\right)$ such that the system can do at most
$n$ steps before failing. The region $E_{0}^{W}$ are all the points in the
section where applying $\operatorname{\mathcal{W}}$ produces a failure. The
existence of $E_{n}^{W}$ implies that a controller of the system may not need
to take a decision at each step. In addition, the controller may exploit this
alleviation by planning future angles of attack. Viability describes how easy
is to choose the future angle of attack. The level of ease is measured in
terms of the size of the interval of angles that can be chosen to avoid a
failure of the system. For the running gait this region is defined as:
$\begin{split}V^{R}\left(\Delta\alpha\right)=&\\{x|\;x\in\mathcal{S}\wedge\\\
&\left(\exists\alpha\in
I_{\alpha},\;\|I_{\alpha}\|\geq\Delta\alpha\;|\;y=\operatorname{\mathcal{R}}_{\alpha}\left(x\right),\;y\in\mathcal{S}\right)\\},\end{split}$
(23)
where $I_{\alpha}$ denotes a real interval and $\|\cdot\|$ measures its
length. In a real system, it is required that a viable angle of attack exists
for a definite interval, since real sensors and actuators have a finite
resolution and are affected by noise.
Figure 2: (Color online) Finite stability regions. The figures show initial
conditions for $\operatorname{\mathcal{R}}$, $\operatorname{\mathcal{GR}}$ and
$\operatorname{\mathcal{W}}$ that can do multiple steps under CAAP before
failing. A region in white corresponds to $E_{0}^{i}$ for gait $i$.
Fig. 2 shows the finite stability regions for each gait. The stable region of
$\operatorname{\mathcal{R}}$, as reported in JuergenRummel2009 ($v_{y}=0$) is
not visible. Although $E_{\infty}^{R}$ may have some area of attraction, due
to the resolution we used for the angles of attack (described in section II.3)
we do not see it in our results. Based on results not presented here, we
estimate that the resolution in the angle of attack to detect such basin for
the current energy is $\sim 10^{-4}$. In despite of the low resolution in the
angles, the system can perform an average of $10$ steps in
$\operatorname{\mathcal{R}}$, and at least $25$ steps (maximum calculated) in
$\operatorname{\mathcal{GR}}$ and $\operatorname{\mathcal{W}}$. This means
that running is more difficult at this energy level than the other two gaits.
Particularly for $\operatorname{\mathcal{GR}}$ and
$\operatorname{\mathcal{W}}$, we see that there is a plateau with the maximum
number of steps. This is the evidence of the self-stable regions of these
gaits, and the plateau is related to the basing of attraction of that region.
Fig. 3 shows the $V^{i}(\Delta\alpha)$ regions for each gait $i$. Comparing
with Fig. 2, we see that in general long partial stability implies wider
options for the angle of attack. Particularly, transitions are found near
these regions of high viability and long partial stability, as will be
described in the next section.
Figure 3: (Color online) Viability regions for each gait. The figures show
the range of angles of attack that can be selected in each initial condition
that allows the system give at least one more step. Colors indicate the size
of the window, spanning from $0\text{\,}\mathrm{\SIUnitSymbolDegree}$to
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$.
Fig. 4 shows one of the strongest results presented here. For each gait $i$,
there is at least one angle of attack that maps the current state of the
system into $E_{\infty}^{i}$, and this angle exists for an extense region of
$\mathcal{S}$. This implies that if we consider control policies with variable
angle of attack, almost any point in the section can be rendered stable. For
this region the optimal control policy requires two angles: the first one maps
the point to $E_{\infty}^{i}$; the second angle, keeps the system in this
region.
Figure 4: (Color online) Points that can be mapped to stable regions in one
step. The figures show the initial conditions that can be mapped to a small
neighborhood of the stable region $E_{\infty}^{i}$, $|v_{y}|<1\times 10^{-3}$
($v_{y}=0$, dashed horizontal lines). Color indicates the angle chosen.
Regions $V^{i}(2\mathrm{\SIUnitSymbolDegree})$ are marked with solid lines.
### III.2 Transition regions
As it was shown in the previous section, the only way of producing transitions
between gaits is to put the system in a region with finite stability (due to
the empty intersection of the $E_{\infty}^{i}$ regions reported in
JuergenRummel2009 , see Fig 4). In Fig. 5 we show transitions starting at
$E_{n}^{i}$ and arriving at $V^{j}\left(2\mathrm{\SIUnitSymbolDegree}\right)$
for $i\neq j$ and $(i\rightarrow j)=\\{(R\rightarrow GR),(GR\rightarrow
W),(W\rightarrow GR),(W\rightarrow R)\\}$. We show the transitions that will
be used in the next example, however transitions between two any gaits are
possible. It shall be noticed that wherever two regions of different gaits
intersect, the transition is trivial.
Figure 5: (Color online) Transitions regions landing in $\Delta\alpha\geq
2\mathrm{\SIUnitSymbolDegree}$. All the initial conditions that have a future
inside the region with $\Delta\alpha\geq 2\mathrm{\SIUnitSymbolDegree}$ of the
objective gait are plotted with black dots. The same region of the starting
gait is given as a reference and appears shaded. Colors in the objective
region indicate the angle of attack used to perform the transition. Wherever
two regions of different gaits intersect, the transition is automatically
given.
Finally, Fig. 6 and Fig. 7 show one example of three transitions for a given
initial condition. The trajectory has a total of $26$ steps and the angle
sequence is
$\begin{split}\alpha=\left(81.886^{5},88.500,62.400,72.350,71.100^{3},71.000,\right.\\\
\left.74.400,72.130,74.000^{4},78.000^{2},76.500,69.000,81.728^{4}\right)&\end{split}$
(24)
where the exponent indicates how many times the angle was used. The path of
the center of mass in the Cartesian plane is also shown in the figures.
Figure 6: (Color online) Transition sequence. The plot shows a trajectory
with three transitions. The Regions
$V^{i}\left(2\mathrm{\SIUnitSymbolDegree}\right)$ are shown shaded with self-
stable regions in dotted line. The arrows indicate the order of the sequence
and the step number is given. The angle of attack sequence is given in (24).
Figure 7: Transition time series. The figure shows the motion of the point
mass222An animation of these transitions can be seen in
http://www.ifi.uzh.ch/arvo/ailab/people/hamarti/GaitT.avi in the plane is
shown together with the crossing of the section (filled circles 6). Transition
points are indicated with a vertical line.
All together we have shown that the SLIP model can be easily controlled to
present transitions between gaits. To find transitions we must search for an
intersection between the future of the starting region and the desired
objective region. Depending how these regions are defined, it may be the case
that multiple steps are required to achieve a successful transition.
## IV Discussion
There are two important aspects regarding the viability regions. First, it is
important to notice that $V^{i}(\Delta\alpha)$ enclose the $E_{\infty}^{i}$
region, and the points that can be mapped to stable regions in one step (Fig.
4) . Second, as it can be seen in Fig. 3, the bigger the range of the angle of
attack is, the smaller the viability region is. We can take advantage of these
properties to stabilize the system more easily. The selection of an
appropriate $\Delta\alpha$ e.g. $2\mathrm{\SIUnitSymbolDegree}$ defines a set
of $V^{i}(\Delta\alpha)$ inside the section $\mathcal{S}$, where the
controller has at least a range of $2\mathrm{\SIUnitSymbolDegree}$ to select
an appropriate angle of attack. Moreover, the agent can select conservative
angles, step by step, to bring itself to the $E_{\infty}^{i}$ region (Fig. 5).
Despite the relief to the controller induced by the viability region, the
selection of the $\Delta\alpha$ can generate regions that do not intersect;
e.g. in Fig. 4 we can see that $V^{i}(2\mathrm{\SIUnitSymbolDegree})$ does not
intersect any other region, which makes the gait transition more difficult to
carry out. In order to cope with this situation, we look at the future of all
the initial conditions in $E_{n}^{i}$. As it is presented in Fig. 5, we found
that there are some initial conditions, that under a set of angles of attack,
are mapped from $E_{n}^{i}$ to $E_{n}^{j}$ (e.g. $E_{n}^{R}$ to $E_{n}^{GR}$).
What is also important is that the region where we can find these initial
conditions are inside the viability region (Fig. 5).
In these terms, the controller has two purposes. First, based on the state on
the $\mathcal{S}$ section, it has to select the gait, and the angle of attack
to keep the agent stable. Thus, the controller needs to have the knowledge of
all the $V^{R}(\Delta\alpha)$, and the desired $\Delta\alpha$ to identify
which gait has to be selected; the angle of attack can be selected based on
the gait model. Second, the controller has to be able to produce gait
transition when it is needed. Hence, the transition regions should be known by
the controller and with a model of the gait, the angle of attack required can
be selected. We expect that this approach can be used to handle uneven
terrain, given that these irregularities can be modeled (under certain
restrictions) as a change in energy.
All these results are conditioned to the selection of the $\mathcal{S}$
section. This means that we are analyzing the system in only one point in the
whole trajectory. From what we see in these results, in some regions the
trajectories are very close. It would not be a surprise that these
trajectories of $\operatorname{\mathcal{R}}$, $\operatorname{\mathcal{W}}$,
and $\operatorname{\mathcal{GR}}$ cross each other in another point along
their continuous evolution, but given that we are looking just at the
$\mathcal{S}$ section, this cannot be anticipated. Nevertheless, the selection
of this section establishes the angle of attack as a natural control action to
stabilize the system and to generate the transitions.
## V Conclusion
In the present study we have taken advantage of the perspective of hybrid
dynamical systems to represent locomotion as a process generated by several
charts. Although, this view makes evident a bigger set of connections among
the charts, in this paper we take into account a small subset (s-chart to ff-
chart, and s-chart to d-chart) which allow us to discover new alternatives to
perform gait transitions. The development of the maps
$\operatorname{\mathcal{W}}_{\alpha}^{1}$,
$\operatorname{\mathcal{GR}}_{\alpha}^{1}$,
$\operatorname{\mathcal{R}}_{\alpha}^{1}$ is fundamental to identify important
regions in the $\mathcal{S}$ section that bring the system to stable
locomotion and to a gait transition. The present results bring new ideas about
plausible mechanisms that biped creatures could use to carry out gait
transitions and stable locomotion. These mechanisms exploit the passive
dynamics of the system, which reduces the amount of energy needed to control
the system. These features are also present in biped machines with compliant
legs, and as suggested in this paper, these mechanisms can be exploited to
develop stable gaits and gait transitions.
## Acknowledgments
Funding for this work has been supplied by SNSF project no. 122279 (From
locomotion to cognition), and by the European project no. ICT-2007.2.2
(ECCEROBOT). Additionally, the research leading to these results has received
funding from the European Community’s Seventh Framework Programme
FP7/2007-2013-Challenge 2-Cognitive Systems, Interaction, Robotics- under
grant agreement No 248311-AMARSi.
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|
arxiv-papers
| 2011-08-22T20:26:42 |
2024-09-04T02:49:21.705455
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Harold Roberto Martinez Salazar and Juan Pablo Carbajal",
"submitter": "Juan Pablo Carbajal",
"url": "https://arxiv.org/abs/1108.4432"
}
|
1108.4440
|
# Promoting scientific thinking with robots
Juan Pablo Carbajal,Dorit Assaf, Emanuel Benker
###### Abstract
This article describes an exemplary robot exercise which was conducted in a
class for mechatronics students. The goal of this exercise was to engage
students in scientific thinking and reasoning, activities which do not always
play an important role in their curriculum. The robotic platform presented
here is simple in its construction and is customizable to the needs of the
teacher. Therefore, it can be used for exercises in many different fields of
science, not necessarily related to robotics. Here we present a situation
where the robot is used like an alien creature from which we want to
understand its behavior, resembling an ethological research activity. This
robot exercise is suited for a wide range of courses, from general
introduction to science, to hardware oriented lectures.
## 1 The Braitenberg vehicle exercise
A simple self-made robotic platform built by the authors was used for the
activity. The robot had two wheels, each one actuated by a DC motor. Two light
sensors [1] were attached to the robot. The robot was controlled by a simple
on-board program that defined a relation between inputs coming from the
sensors and output signals sent to each motor.
We provided the robot with the behavior of Valentino Braitenberg’s vehicle
number 3 [2]. The light sensors of the robot commanded the rotational speed of
the two motors. The connection was inhibitory, meaning that when the sensor
measured light, the speed of the motor connected to it was reduced
proportionally to the sensor’s output. This sensor-motor configuration
generates a light following behavior (Figure 1). More details about the robot,
the control program and how to reproduce this exercise are explained in later
sections.
Next we describe how we used the robot to engage students in scientific
thinking. This exercise was part of a class on modeling mechatronics systems
that took place at the Baden-Wuerttemberg Cooperative State University
Loerrach, Germany. The students were mainly 3rd year bachelor students. The
objective of the activity was to let students find out the sensor-motor
relationship by means of hands-on experimentation and free exploration. The
students had to create a hypothesis about the controller implemented in the
robot and later verify it through experiments.
Figure 1: Braitenberg vehicle 3, the light lover. Each sensor reduces the
speed of the motor on its side proportionally to the measured light intensity.
The figure shows the qualitative behavior of the robot: it moves towards the
light and tends to stop close to it.
### Introducing the robot.
The activity started with the presentation of the robot and a demonstration of
its behavior when a light was placed in front of it. The robot moved by
default in a straight line, and when it passed close to the light it turned
towards it. The robot was even able to track the light (this depends on the
sensor gain and motor speed, therefore it requires calibration prior to
demonstration). This light loving behavior, though simple, always captivates
the audience as well as the teachers.
### The assignment.
After several playful tests with the light, the students were asked to give
explanations, in the simplest possible way, about the controller implemented
in the robot such that it shows this behavior. Additionally, they were asked
to propose an experiment that tests their explanation. In other words, they
were asked to develop a model of the internal works of the robot and to
produce a hypothesis verifiable through experimentation. The robot allowed us
to create a complete and interesting research situation. At this point, to
avoid diverging explanations, we suggested to the students to focus on the
role of the sensors.
### Hands on.
The production of models and tests was done in small groups (3-4 people) and
we let the students form the groups by themselves. During this phase, we
visited each group and discussed their ideas to assure the experiments will
help deciding whether a given model should be discarded or not. It is
important to remark that we did not correct the models, since any model is
just an approximation. Thus, we just suggested changes in the model to
simplify the verification process. After several minutes of group discussion,
the groups presented their models, the experiment to be conducted on the robot
and what they expected to observe. Since the number of available robots was
enough, the students were able to perform their experiments. Otherwise the
teacher could select a few experiments and try them in the robot.
### The closure.
The conclusion of the activity is left to the criteria of the teacher. In our
case, due to the lack of time, we explained the controller and introduced
Braitenberg’s ideas. In other circumstances, we would have requested the
students to produce a short report of the experience and postpone the
explanation to the next class.
## 2 Robot hardware
The custom robotic platform is shown in Figure 2. Next, we describe the
hardware that is needed to reproduce the robot exercise just described.
As mentioned above, the robot has two motors that can rotate individually at
different speeds. Light sensors are placed at the right and left front side of
the robot. These sensors can detect a light source within a range of about 10
cm and were previously calibrated by the students by measuring the output
voltage as a function of the distance to a light source. The robot control
program was implemented such that each light sensor is connected (via the
controller) to one motor and influences its speed directly. Whenever a light
sensor measures light the speed of the motor is reduced proportionally to the
sensor’s measurement. The less light a sensor detects, the faster the motor
rotates and vice versa.
A commercial Arduino control board (http://www.arduino.cc) was used to control
the robot. Figure 2 shows the components of the robot. Six rechargeable
batteries are used for power supply. An USB communication unit is used for
programming and monitoring the control board. Two light sensors provide
sensory input to the control board which controls the two motors and wheels
through the motor driver component. Since the robot was designed to be used in
different experiments [5], it can actually be equipped with many more sensors
and therefore the controller board is more powerful than what would be
required for the exercise presented here.
Figure 2: The robot and its components. Six rechargeable batteries are used
for power supply. An USB communication unit is used for programming and
monitoring the control board. Two light sensors provide sensory input to the
control board which controls the two DC motors and wheels through the motor
driver component.
Nowadays materials to build these robots are abundant. For example, ready-to-
use chassis can be acquired from online retailers such as Maker SHED
(http://www.makershed.com) or Dwengo (http://www.dwengo.com). Tutorials on how
to build robots are easily accessible as in Make magazine
(http://makezine.com) or any of the many blogs on robotics. The approximate
material cost for the robot presented here is EUR 140. Information about how
to rebuild the robot and the required software libraries is available on Dorit
Assaf’s website (http://www.embed-it.ch).
## 3 Robot software
The Arduino project provides open source programming libraries and software
development kits. Alternatively, the MATLAB language offers the
ArduinoIO111MATLAB is a widespread scientific computing language, almost a
standard in the scientific research community nowadays,
http://www.mathworks.com., an easy to use programming interface. Below we show
a snippet of the C code used for a controller that produces Braitenberg’s
vehicle 3 behavior. Lest the unexperienced user find the source code daunting,
the Arduino project offers very easy tutorials to get started.
The digital output that controlled the wheels had an 8-bit resolution (it can
produce 256 different values), therefore the speed of the motor is given by a
number between 0 and 255, being 127 the middle value or half-speed. The
preamble of the code includes our custom libraries needed and initializes
sensors and motors. Next a function to set up the robot is defined, it
initializes the default robot speed (127 = half-speed) and forward direction.
After this function is executed, the continuous loop() routine starts. There,
the sensor values of light sensor 1 and light sensor 2 are read and saved in
the variables sensorValue1 and sensorValue2. The sensor values range from 0
(dark) to 1023 (bright). The map() function, as its name indicates, maps the
first two arguments (the sensor range [0,1023]), to the range [255,0]. This
value will replace the default speed of the robot via the setSpeed() function,
therefore, bright light will slow down the robot.
// Include libraries with functions// for the specific sensors and
motors#include <LightSensor.h>#include <DCMotor.h>// Define sensors and
motors// Two sensors connected to pins 1 and 2.LightSensor lightSensor(1,
2);// Connect pins to motor driver componentDCMotor motor1(12, 8, 10);DCMotor
motor2(18, 19, 11);void setup(){// This function is loaded// at startup and
after each reset // Set default speed of the motors motor1.setSpeed(127);
motor2.setSpeed(127); // Set default direction of rotation
motor1.setDirection(FORWARD); motor2.setDirection(FORWARD);}void loop(){//
This function runs while the robot is alive // Read sensor values int
sensorValue1 = lightSensor.readSensorValue1(); int sensorValue2 =
lightSensor.readSensorValue2(); // Convert sensor values to motor speed int
newSpeed1 = map(sensorValue1, 0, 1023, 255, 0); int newSpeed2 =
map(sensorValue2, 0, 1023, 255, 0); // Apply new speed values to motors
motor1.setSpeed(newSpeed1); motor2.setSpeed(newSpeed2);}
Litle more is needed to get the robot running. The source code is available on
the website http://www.embed-it.ch together with some programming guidelines.
## 4 Discussion and conclusion
During the class we observed that students were fully engaged and were having
fun. Based on their feedback, we attribute this to the presence of the robot,
a non-standard tool for teaching.
The students produced creative models (with a tendency to complicated
schemes), hypotheses, and interesting experiments. No group actually found the
correct solution (Braitenberg’s vehicle 3) or anything equivalent. However,
one group proposed a feedback controller that, despite its complexity, seemed
aligned with Braitenberg’s ideas. Nevertheless, our goal was to challenge the
students and allow them to build their hypothesis based on hands-on evidence,
therefore this goal was met. Using robots as a learning tool allows to prepare
fully customizable class activities with different levels of difficulty and
which can emulate real research situations.
The validation process, given that expectations were clearly stated, resulted
to be fairly simple: either the model predicted the behavior or not. Several
students showed determination to find a working model and automatically
reworked theirs without being told to do so. We were surprised to note that
the cycle: build a model, test it, rework the model; emerged naturally after
the little push given when we described the activity to the students.
From the teacher’s perspective this activity requires some extra work
especially in the preparation phase. However, the effort was worth it and we
encourage other teachers to try. A caveat of this kind of exercise is the
difficulty to define criteria to grade a student’s performance, due to the
unstructured nature of the activity and the variety of possible solutions.
This could be avoided by complementing the activity with a written report or a
presentation. In the case where parallel activities are also performed such as
calibration of sensors or construction of a speedometer222Students could build
wheel speed sensors (speedometers) and verify the discussion in [4], grading
could be simplified.
A more physics based experience could be to set up a robotic car crash and
engage students in a forensic physics experience, where they could determine
initial speeds and directions or maneuvers made by the artificial car drivers
[3].
We invite other teachers to try similar activities. We offer our support for
the programming and assembly of the robot and invite the reader send us
feedback.
## References
* [1] Mickey Kutzner, Richard Wright and Emily Kutzner, _An inexpensive LED light sensor_ , Phys. Teach. 48, 341 (May 2010)
* [2] Valentino Braitenberg, _Vehicles, experiments in synthetic psychology_ ,(MIT Press, Cambridge Massachusetts, 1986), p. 10.
* [3] Arthur C. Damask, _Forensic physics of vehicle accidents_ , Physics Today 3, 40 (March 1987).
* [4] Clifton Murray, _Wheel Diameter and Speedometer Reading_ , Phys. Teach. 48, 416 (September 2010).
* [5] Assaf, D. and Pfeifer, R. (2011). _Robotics as Part of an Informatics Degree Program for Teachers_ In Proceedings of Society for Information Technology & Teacher Education International Conference 2011 (pp. 3128-3133). Chesapeake, VA: AACE.
|
arxiv-papers
| 2011-08-22T21:03:32 |
2024-09-04T02:49:21.711371
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan Pablo Carbajal, Dorit Assaf, Emanuel Benker",
"submitter": "Juan Pablo Carbajal",
"url": "https://arxiv.org/abs/1108.4440"
}
|
1108.4448
|
††thanks: Artificial Intelligence Laboratory.
University of Zürich.
e-mail: {carbajal,naveenoid}@ifi.uzh.ch
# Magneto-mechanical actuation model for fin-based locomotion
Juan Pablo Carbajal and Naveen Kuppuswamy
(August 27, 2024)
###### Abstract
In this paper, we report the results from the analysis of a numerical model
used for the design of a magnetic linear actuator with applications to fin-
based locomotion. Most of the current robotic fish generate bending motion
using rotary motors which implies at least one mechanical conversion of the
motion. We seek a solution that directly bends the fin and, at the same time,
is able to exploit the magneto-mechanical properties of the fin material. This
strong fin-actuator coupling blends the actuator and the body of the robot,
allowing cross optimization of the system’s elements.
We study a simplified model of an elastic element, a spring-mass system
representing a flexible fin, subjected to nonlinear forcing, emulating
magnetic interaction. The dynamics of the system is studied under unforced and
periodic forcing conditions. The analysis is focused on the limit cycles
present in the system, which allows the periodic bending of the fin and the
generation of thrust. The frequency, maximum amplitude and center of the
periodic orbits (offset of the bending) depend directly on the stiffness of
the fin and the intensity of the forcing; we use this dependency to sketch a
simple parameter controller. Although the model is strongly simplified, it
provides means to estimate first values of the parameters for this kind of
actuator and it is useful to evaluate the feasibility of minimal actuation
control of such systems.
## 1 Introduction
In the last two decades underwater fin-based propulsion has been a topic of
intense research. Theoretical models and simulations based on a multiplicity
of numerical methods have been developed and reported [6, 15, 8, 12, 11, 18,
9, 3] as well as experiments performed on artificial platforms and with
animals [1, 13, 10, 7]. Theoretical studies such as[6, 15] focus on models of
the viscoelastic body, while[8, 12, 11] treat the body-fluid coupling and the
emergence of locomotion. In [18, 9, 2] the main interest is to understand how
passive thrust is generated in vortex wakes, a question that remains open.
Despite the considerable work done, experimental results are not fully
understood and several numerical simulations are yet to be validated.
Influenced by the activity on the field, we address a parallel problem related
to robotics, _actuation_.
Fins as a tool for locomotion offer several appealing properties with respect
to propellers. From an environmental point of view fins reduce sound pollution
characteristic of propellers[16]. Additionally, a flexible body offers the
possibility of extracting energy from the environment, as shown in the
technological study presented in [4]. This idea is related to the fact that
trouts exploit vortex wakes (shed by obstacles in the flow) to reduce the cost
of swimming[5] (see [14] for a review). These aspects are of primary relevance
in situations where low environmental impact and mimicry are important, as in
pipes maintenance routine, or for underwater life observation (scientific
naiveness may make us forget about military applications, we provide this
short caution.). Moreover, in environments where moving parts may be clogged
up due to fouling, rotatory propellers may be unfeasible for locomotion.
In the field of bio-inspired robotics, actuation for swimming robots using
fin-like propulsion is usually implemented by the use of rotatory electric
motors to control the fin swing angle. However, The presence of motors hinders
miniaturization and integration of actuators into the robot structure, a
requirement for flexible machines with deformable body. At the same time this
imposes a mechanical conversion of rotations into oscillatory linear motion,
the complexity of which becomes an issue at smaller size. Moreover, minimizing
or eliminating the number of moving parts required for actuation simplifies
maintenance to a large extent. To tackle the mentioned difficulties we are
directing our research towards new ways of actuation. Herein we report a
simple mathematical model and the numerical analysis of a possible
alternative.
We start by considering a robot composed of a hull and a fin attached to it.
The fin is modeled as an elastic beam (see [15] or [6] for more detailed
models), which we want to set into oscillatory motion. In the setup shown in
Fig. 1, we choose to support the beam at two points. The first support is at
one edge of the beam and stands for the hull of the robot. The second support
is placed at some intermediate point in the beam. The section of the beam
beyond this second support is meant to generate the thrust by interacting with
the surrounding fluid. The actuation is done in the section of the beam
between the two supports by means of a combination of permanent magnets (one
of them attached to the beam) and solenoids. In the configuration chosen, the
permanent magnets serve to increase the compliance of the system and to reduce
the force that needs to be actively applied by the solenoid. The distance
between the supports defines the rigidity of the actuated section and could be
tuned for optimal energy transfer. Similar working principles are described in
patents of electric razors, and of active dampers of oscillations for digital
cameras lenses (in these contexts the actuator is often called motor or
electromagnetic spring). Similarly, the control of the resonant modes of a
structure is a commonplace problem in structural dynamics[19]. It is
noteworthy that all these techniques exploit (to be more efficient) or require
(to be implemented) knowledge of the resonant modes of the system under study.
Figure 1: Schematic of the system described by equations (1). A beam is used
to model the fin and it is simply supported at two points. The actuation is
done by means of a combination of permanent magnets and solenoids. The
distance between the supports defines the rigidity of the actuated section.
## 2 Dynamic Model
The displacement of the magnet in the fin (we will refer to this point as the
fin magnet), can be modeled by a spring-mass system under the effect of an
external force field. Considering only one dimensional motion, the system is
written as,
$\begin{split}\dot{x}&=v\\\
\dot{v}&=\frac{F_{2}(x)+F_{1}(x)+F_{s1}(x)+F_{s2}(x)}{m}-\frac{\mathcal{K}}{m}x-\frac{\Gamma}{m}v,\end{split}$
(1)
where $x$ is the displacement of the fin magnet, $m$ is an effective moving
mass, $\mathcal{K}$ represents an effective elastic constant of the fin setup
and $\Gamma$ is used to include dissipation. The $F_{i}$ and $F_{si}$ terms
are forces acting on the fin due to the external magnets and the solenoids,
respectively.
Magnetic forces can be highly complex; to keep our model as simple as possible
we approximate each magnet as a point magnetic dipole, which is a good
approximation when the distances are significantly bigger than the size of the
magnet in the direction of the magnetization[20]. In this situation the force
can be expressed as follows,
$F_{i}(x)=-\frac{\mathcal{C}_{i}}{\left(x-x_{i}\right)^{\alpha}}\operatorname{sign}(x-x_{i}),$
(2)
where $\mathcal{C}_{i}$ is a constant that depends on the magnetic moments of
the magnets and their geometry, positive values represent attractive forces
and negative values repulsive forces. In the case of the solenoids, this
constant depend also on the current, i.e. $\mathcal{C}_{si}(I)=C_{si}I(t)$
(the index $s$ refers to solenoid). The position of the external magnet (or
solenoid) measured from the rest position of fin magnet is $x_{i}$. Henceforth
we define $k=\nicefrac{{\mathcal{K}}}{{m}}$,
$\gamma=\nicefrac{{\Gamma}}{{m}}$, $c_{i}=\nicefrac{{\mathcal{C}_{i}}}{{m}}$
and $c_{si}=\nicefrac{{C_{si}}}{{m}}$. Additionally, we assume that the
deflected fin does not reach the external magnets, in mathematical terms this
is expressed as $x\in(x_{1},x_{2})$.
Next we study the dynamics of the system without actuation, $I(t)\equiv 0$.
The expression for the fixed points $x^{*}$ is obtained by equating system (1)
to zero. The second equation yields
$c_{2}\left(x^{*}-x_{1}\right)^{\alpha}-c_{1}\left(x^{*}-x_{2}\right)^{\alpha}-kx^{*}\left[\left(x^{*}-x_{1}\right)\left(x^{*}-x_{2}\right)\right]^{\alpha}=0,$
(3)
where the assumption $x\in(x_{1},x_{2})$ was used to determine the signs.
### Linear Stability Analysis.
To classify the fixed points, we calculate the trace and determinant of the
$2\times 2$ Jacobian matrix $J$ of (1). These are given by
$\displaystyle\operatorname{Tr}(J)$ $\displaystyle=$ $\displaystyle-\gamma$
(4) $\displaystyle\operatorname{Det}(J)$ $\displaystyle=$ $\displaystyle
k+\alpha\left[\frac{c_{2}}{\left(x-x_{2}\right)^{\alpha+1}}-\frac{c_{1}}{\left(x-x_{1}\right)^{\alpha+1}}\right].$
(5)
In general the fixed points of the system will be saddle-nodes, centers or
spirals, depending on the value of the parameters $\gamma$, $k$, $c_{i}$ and
$x_{i}$. However, the position of the fixed point (i.e. the solutions of (3))
are independent of $\gamma$.
To proceed with the analysis we introduce further assumptions. The exponent
$\alpha$ depends on the arrangement of magnets[20]. Here we will consider
identical cylindrical magnets placed symmetrically with respect to the rest
position of the fin magnet and with dipoles parallel to it (attracting); hence
$\alpha=4$, $c_{1}=c_{2}=c>0$ and $x_{2}=-x_{1}=x_{0}>0$. By neglecting
dissipation, i.e. $\gamma=0$, we set the trace of the Jacobian to zero.
Consequently, the fixed points are either saddle-nodes or centers, depending
on the sign of (5). Using these assumptions to simplify the equality (3) we
obtain,
$x^{*}\left[8cx_{0}\left(x^{*2}+x^{2}_{0}\right)-k\left(x^{*2}-x^{2}_{0}\right)^{4}\right]=0,$
(6)
rendering evident that $x^{*}=0$ is one of the fixed points, in consequence of
the symmetry of the problem. The determinant (5) at this point is,
$\operatorname{Det}(J)|_{x^{*}=0}=k-8\frac{c}{x_{0}^{5}},$ (7)
which is positive for $\nicefrac{{c}}{{k}}<\nicefrac{{x_{0}^{5}}}{{8}}$, and
the origin is a center. Although any real system will not show centers without
actuation (due to dissipation) their position will match the pole of the
spirals observed.
It can be shown that the nonzero solutions of (6) are saddle-nodes.
Displacements beyond the saddle-nodes will bring the fin magnet into a region
where the attraction is stronger than the elastic restitution, causing the fin
to stick to the closest magnet. The saddles establish a natural limit for the
maximal amplitude of the orbits of the system. To illustrate these ideas, we
show in Fig. 2 three plots of the polynomial defined by (6) for different
values of the ratio $\nicefrac{{c}}{{k}}$, together with phase portraits of
the system. The figure depicts the trade-off between the rigidity of the fin
and the interaction of the fin magnet and the permanent magnets. Keeping the
$x_{0}$ fixed, the stronger the magnets (or the more compliant the fin), the
smaller the region where the system can present stable orbits. At the critical
ratio $\nicefrac{{c}}{{k}}=\nicefrac{{x_{0}^{5}}}{{8}}$, the saddle-nodes
collide at the origin and the center is transformed into a saddle-node.
Figure 2: Plot of the polynomial defined in (6) for different values of
$\nicefrac{{c}}{{k}}$. The star symbols mark the position of the fixed points.
The phase portraits to the right show that the saddle-nodes define a limit for
the amplitude of the orbits. The figure illustrates the trade-off between the
rigidity of the fin and the intensity of the magnetic interaction.
## 3 Actuation and Control
As mentioned before, dissipation will reduce the amplitude of the
oscillations. Therefore, to keep the system close to the desired trajectory we
need to pump energy into it. To do this, we have placed a solenoid surrounding
both magnets such that they can increase or decrease the interaction with the
fin magnet. Both solenoids are constructed similarly but arranged anti-
parallel to each other ($c_{s2}=-c_{s1}=c_{s}>0$). We place them as close as
possible to the fin magnet, for example near the saddle-nodes of the system.
In order to determine the parameters of the solenoid needed to drive and
control the system, we use a simple PID controller which can regulate the
applied force. To this end, we rearrange the terms of (1) and write them as,
$\begin{split}&\dot{v}=\frac{c}{\left(x-x_{0}\right)^{4}}-\frac{c}{\left(x+x_{0}\right)^{4}}+F_{c}(t)-kx-\gamma
v\\\ &F_{c}(t)=k_{p}e(t)+k_{d}\dot{e}(t)+k_{i}\int_{0}^{t}e(s)ds\\\
&e(t)=x_{d}(t)-x(t).\end{split}$ (8)
Where $x_{d}(t)$ is the desired displacement. The effort required to drive the
system into steady oscillations, depends on the appropriate choice of $k_{p}$,
$k_{d}$, $k_{i}$.
## 4 Parameter values
In the following sections we define the values of the parameters used for the
numerical simulation of the platform. We give a brief description on the
assumptions and criteria used to select them. In Table 1 we summarize the
information.
### Magnets.
The constant for the force (2) in the case of cylindrical magnets magnetized
along their length $\ell$ is,
$\mathcal{C}=\frac{3\mu_{0}}{2\pi}\left(\frac{B_{r}}{\mu_{0}}\right)^{2}\left(\pi
R^{2}\ell\right)^{2},$ (9)
where $\mu_{0}$ is the permeability of vacuum, $R$ is the radius of the magnet
and $B_{r}$ is the remanence of the magnet (a value available from
manufacturers). The factors in parenthesis represent the magnetization and the
volume of the magnet, they are squared because we are assuming both
interacting magnets are equal. Using values from commercially available
Neodymium magnets we have calculated $\mathcal{C}=2.460\times 10^{-10}N\cdot
m^{4}$.
### Elastic constant.
As can be seen in Fig. 1 we model the fin using a beam simply supported in two
points. The supports are separated by a distance $L$. The fin magnet is placed
at the point $y$ and the elastic stiffness there can be written as,
$\mathcal{K}=EJ\frac{3L}{y^{2}(L-y)^{2}},$ (10)
where $E$ is the Young’s modulus of the material and $J$ is the second moment
of area of the beam. As discussed before, the behavior of the system depends
on the relation between the elastic constant of the fin and the strength of
the magnets. The current setup allows tuning the elasticity of the fin by
setting different materials and profiles of the fin, or by moving the fin
magnet, or by changing the distance between the supports. If needed, the setup
could be transformed into a cantilever by removing the second support. To
provide good ranges of elasticity we use Low-density polyethylene plastic with
$E\approx 0.2\times 10^{9}Pa$ and $L=30mm$. The fin has width and thickness of
$10mm\times 0.5mm$, respectively. The fin magnet is placed in the middle of
the two supports. This values yield $\mathcal{K}=37.03N\cdot m^{-1}$.
### Damping and mass.
When a body moves in a liquid, it transfers kinetic energy to the surrounding
fluid reducing the acceleration it presents corresponds to the one observed on
a body with higher mass. This phenomena, known as added mass, can be estimated
using models as the one presented in[21]. However, we postpone a detailed
description for future work and simply consider a total mass 300 times bigger
than the mass of the fin and the magnet together, $m=84.3g$. Estimation of the
damping $\Gamma$ without an experimental setup is not straightforward,
therefore we use damping ratios in the range $Q\in[0,0.5]$. Where $Q=0$ means
no damping and a value $Q=1$ corresponds to critical damping.
The table below summarizes the value of the parameters. Though the values are
reasonable, we do not expect them to correspond to any real device and
corrected ones will come from a future validation process.
Parameter | Value | Units
---|---|---
$c$ ($\nicefrac{{\mathcal{C}}}{{m}}$) | $2.919\times 10^{-9}$ | $N\cdot m^{4}\cdot kg^{-1}$
$k$ ($\nicefrac{{\mathcal{K}}}{{m}}$) | $439.3$ | $N\cdot m^{-1}\cdot kg^{-1}$
$d$ ($\nicefrac{{\gamma}}{{m}}$) | $[0,20.96]$ | $N\cdot s\cdot m^{-1}\cdot kg^{-1}$
$x_{0}$ | $1\times 10^{-2}$ | $m$
Table 1: The table summarizes the values of the parameters used for the
numerical results reported in the text. Values for $d$ correspond to the range
of damping ratio $Q\in[0,0.5]$.
## 5 Results
### Phase space and time series.
Numerical results for the undamped system are presented in Fig. 3. We take
three initial conditions on the region of the phase space to study. All the
initial conditions start with zero velocity, i.e. they lay on the horizontal
axis. It is important to note that the time series of the fin displacement
clearly show different frequencies. This is due to the attraction of the
magnets, the higher the initial displacement the lower the frequency of the
orbit. These results are shown in detail in Fig. 4. For each initial condition
we plot the power spectrum of the signal and it is visible how the main
component decreases at higher amplitudes.
Figure 3: Trajectories in phase space and time series for the undamped system
starting from three different initial condition. The frequency of the signal
decreases with the amplitude due to the interaction of the magnets. Figure 4:
Variation of the natural frequency with the amplitude of the oscillations. The
power spectrum of the orbits is plotted, the behavior of the main component is
shown in detail in the inset.
The offset of the oscillations corresponds to the position of the center. By
breaking the symmetry of the system, either by setting $c_{1}\neq c_{2}$ or by
feeding constant current to the solenoids, we can move the center off the
origin. This could be required for turning maneuvers or useful for initiating
oscillations. In Fig. 5 we show how the center and the saddles move for
different values of $c_{1}$ and $c_{2}$ (or increasing current).
Figure 5: Control of the position of the center and saddles. The center moves
symmetrically around the origin for differences $\Delta c$ between the
magnetic constants. The maximum amplitude is also compromised, because the
saddle on the side of the stronger magnet come closer to the center.
### Actuated system.
The controller is provided with reference signal of the form $A\cos(wt+\phi)$.
The amplitude $A$ is taken inside the bounds defined by the saddle-nodes.
Given a value for the amplitude, we use the curve shown in the inset of Fig. 4
as a frequency lookup table. The initial phase $\phi$ is calculated from the
initial conditions. We took the same initial conditions as in the previous
section, namely $x(0)={A,0.5\cdot A,1.25\cdot A}$. However simple, the control
technique shows a remarkable performance as can be appreciated in Fig. 6. In
the figure the reference trajectory is the dashed line, for the case without
damping, the actual trajectories of the system overlap with. On the left
panels the time series of the displacement are shown and compared with the
reference amplitude.
Figure 6: Trajectory in phase space of the actuated system for three different
initial conditions close to the desired amplitude. On Top the results without
damping and below them, with damping ratio of $Q=0.5$, which gives an
unfeasible solenoid. In the second case the reference trajectory in phase
space is shown in dashed line. On the left we show the time series of the
displacement of the fin.
### Solenoid.
If we take the maximum of the output from the controller, $F_{c}(t)$, and
limit the maximum current fed to the solenoids to $I\leq 20mA$, we can find a
suitable expression for calculating the ideal number of turns $N$ of a coil.
The net force generated on the fin magnet is given by the sum of forces due to
the 2 coils, the wiring of which ensures that the force generated by each on
the fin magnet is additive. This force can be expressed as,
$\begin{split}F_{m}=\frac{3}{2}\left(B_{r}R^{2}\ell\right)\left(NIA\right)\operatorname{G}(x_{m})\\\
\operatorname{G}(x_{m})=\frac{\left(x_{m}+x_{0}\right)^{4}+\left(x_{m}-x_{0}\right)^{4}}{\left(x_{m}^{2}-x_{0}^{2}\right)^{4}},\end{split}$
(11)
where $B_{r}$,$R$ and $\ell$ are as in 9, and $x_{m}$ is the displacement
corresponding to the maximum force $F_{m}$. Solving for $N$ and replacing with
the corresponding values we get $N\in[600,1000]$, for $Q\in[0,0.2]$. Higher
values of damping impose too many turns on the solenoid. Although the model is
not yet validated this result is encouraging.
## 6 Discussion and conclusion
In this short report we have presented the first step towards the design and
construction of a novel actuator for small swimming machines. Though we have
used rough models, the results show that we are pointing in the right
direction. The design shown here is not necessarily the best in reducing the
actuation needed. For example, one could think of using the instability of the
center to initiate motion, by forcing the system through its bifurcation. This
could be achieved by on-line modification of the distance between the supports
or by bringing the permanent magnets closer. Additionally, placing a permanent
magnet perpendicular to the plane at the origin could be used to further
reduce the frequency of the orbits or to control the offset in a more
sensitive way than the one shown here.
Our model includes dissipation proportional to the velocity and therefore the
role of dissipation is marginal. More detailed models of the fluid dynamics
and the bending of the fin will surely bring dissipation into a more primary
role in the behavior of the system. In addition, thrust, heat dissipation and
energy consumption could be estimated in such multi-physics models.
We have shown how a simple PID controller could perform reasonably when
information about the phase portrait of the system, like the dependence of
frequency with amplitude, is included. The use of adaptable frequency
oscillators[17] or standard model-based controllers (like feedback
linearization), could improve performance and reduce these requirements.
Additionally, a system that is too flexible does not possess orbits without a
controller. Such a controller would requires large amount of actuation, since
it is _forcing_ the system to behave unnaturally. Therefore existence of
orbits can be exploited to reduce energy consumption. This stresses the fact
that passive dynamics are a key to improve the way we control and design our
robots. Controlling the force between solenoids and moving magnets, brings
several challenges on the design of the electrical circuits due to the changes
in impedance. Another interesting aspect of the problem that will be addressed
in further studies. We understand that results obtained solely from
simulations are as ’words without actions’, however the use of simple models
can help us evaluate the feasibility of certain designs. In our particular
case, a device with low friction could be driven with a tuned PID, a frequency
lookup table and a solenoid with $\leavevmode\nobreak\ 800$ turns, consuming
about $20mA$.
## Acknowledgments
We want to thank our lab coworkers Dr. Max Lungarella, Dr. Hugo G. Marques,
Tao Li, Cristiano Alessandro and Dr. Lijin Aryananda for their comments and
constructive criticism. We thank Dr. Rolf Pfeifer for his continuous support
to our research. Also for the continuation of the A.I. Lab., the friendly
human environment which makes our work possible. Funding for this work has
been supplied by SNSF project no. 122279 (From locomotion to cognition) and by
the European project no. FP7-231608 (OCTOPUS).
### Author Contributions
Both authors contributed equally to the work presented in this paper.
## References
* [1] B. Ahlborn, S. Chapman, R. Stafford, and R. Harper. Experimental simulation of the thrust phases of fast-start swimming of fish. J. Exp. Biol., 200(17):2301–2312, 1997.
* [2] Silas Alben. On the swimming of a flexible body in a vortex street. J. Fluid Mech., 635(-1):27–45, 2009.
* [3] Silas Alben. Passive and active bodies in vortex-street wakes. J. Fluid Mech., 642(-1):95–125, 2010.
* [4] J. J. Allen and A. J. Smits. Energy harvesting eel. J. Fluids Struc., 15(3-4):629 – 640, 2001.
* [5] D. N. Beal, F. S. Hover, M. S. Triantafyllou, J. C. Liao, and G. V. Lauder. Passive propulsion in vortex wakes. J. Fluid Mech., 549(-1):385, 2006.
* [6] J.-Y. Cheng, T. J. Pedley, and J. D. Altringham. A continuous dynamic beam model for swimming fish. Philos. Trans.: Bio. Sci., 353(1371):981–997, 1998.
* [7] X. Deng and S. Avadhanula. Biomimetic micro underwater vehicle with oscillating fin propulsion: System design and force measurement. In Proc. 2005 IEEE Int. Conf. on Rob. Autom., pages 3312–3317, 2005\.
* [8] Jeff D. Eldredge. A reconciliation of viscous and inviscid approaches to computing locomotion of deforming bodies. Exp. Mech., 2009.
* [9] Jeff D. Eldredge and David Pisani. Passive locomotion of a simple articulated fish-like system in the wake of an obstacle. J. Fluid Mech., 607, 2008.
* [10] Brenden P. Epps, Pablo Valdivia y Alvarado, Kamal Youcef-Toumi, and Alexandra H. Techet. Swimming performance of a biomimetic compliant fish-like robot. Exp. Fluids, 47(6):927, 2009.
* [11] Eva Kanso. Swimming in an inviscid fluid. Theor. Comp. Fluid Dyn., 2009.
* [12] Eva Kanso and Paul K. Newton. Passive locomotion via normal-mode coupling in a submerged spring–mass system. J. Fluid Mech., 641:205, 2009.
* [13] George V. Lauder, Erik J. Anderson, James Tangorra, and Peter G. A. Madden. Fish biorobotics: kinematics and hydrodynamics of self-propulsion. J. Exp. Biol., 210(Pt 16):2767–80, 2007.
* [14] James C. Liao. A review of fish swimming mechanics and behaviour in altered flows. Phil. Trans. R. Soc. B, 362(1487):1973, 2007.
* [15] T McMillen and P Holmes. An elastic rod model for anguilliform swimming. J Math Biol, 53(5):843–86, 2006.
* [16] W.J. Richardson, C.R. Greene, C.I. Malme, and D.H. Thomson. Marine Mammals and Noise. Academic Press, London and San Diego, 1995.
* [17] L. Righetti, J. Buchli, and A.J. Ijspeert. Dynamic hebbian learning in adaptive frequency oscillators. Physica D, 216(2):269–281, 2006.
* [18] Ratnesh K. Shukla and Jeff D. Eldredge. An inviscid model for vortex shedding from a deforming body. Theor. Comp. Fluid Dyn., 21(5):343, 2007.
* [19] A.F. Vakakis, O.V. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen, and Y.S. Lee. Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, volume 156 of Solid Mech. Appl. Springer Netherlands, 2009.
* [20] D. Vokoun, M. Beleggia, and P. Sittner L. Heller. Magnetostatic interactions and forces between cylindrical permanent magnets. J. Magn. Mag. Mat., 321(22):3758–3763, 2009.
* [21] Y. Yadykin, V. Tenetov, and D. Levin. The added mass of a flexible plate oscillating in a fluid. J. Fluids Struc., 17(1):115 – 123, 2003.
|
arxiv-papers
| 2011-08-22T21:44:30 |
2024-09-04T02:49:21.719574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan Pablo Carbajal and Naveen Kuppuswamy",
"submitter": "Juan Pablo Carbajal",
"url": "https://arxiv.org/abs/1108.4448"
}
|
1108.4560
|
# Partially Screened Gap - general approach and observational consequences
, George I. Melikidze and Janusz Gil
J.Kepler Institute of Astronomy, University of Zielona Góra, Zielona Góra,
Poland
E-mail E.Kharadze Georgian National Astrophysical Observatory, Georgia
###### Abstract:
Observations of the thermal X-ray emission from radio pulsars implicate that
the size of hot spots is much smaller then the size of the polar cap that
follows from the purely dipolar geometry of pulsar magnetic field. Most
plausible explanation of this phenomena is an assumption that the magnetic
field at the stellar surface differs essentially from the purely dipolar
field. We can determine magnetic field at the surface by the conservation of
the magnetic flux through the area bounded by open magnetic field lines. Then
the value of the surface magnetic field can be estimated as of the order of
$10^{14}$ G. On the other hand observations show that the temperature of the
hot spot is about a few million Kelvins. Based on these observations the
Partially Screened Gap (PSG) model was proposed which assumes that the
temperature of the actual polar cap (hot spot) equals to the so called
critical temperature.
We discuss correlation between the temperature and corresponding area of the
thermal X-ray emission for a number of pulsars. The results of our analysis
show that the PSG model is suitable to explain both cases: when the hot spot
is smaller and larger then conventional polar cap. We argue that in the second
case structure and curvature of field lines allow pair creation in the closed
field lines region thus the secondary particles can heat the stellar surface
outside the actual polar cap.
We have found that the Curvature Radiation (CR) plays dominant role in
avalanche pair production in the PSG. We studied dependence of the PSG
parameters on the pulsar period, the magnetic field strength and the curvature
of field lines.
## 1 Introduction
The Standard model of radio pulsars assumes that there exists the Inner
Acceleration Region (IAR) above the polar cap where the electric field has a
component along the opened magnetic field lines. In this region particles
(electrons and positrons) are accelerated in both directions: outward and
toward the stellar surface. Consequently, outflowing particles are responsible
for generation of the magnetospheric emission (radio and high-frequency) while
the backflowing particles heat the surface and provide required energy for the
thermal emission. In such scenario analysis of X-ray radiation is an excellent
method to get insight into the most intriguing region of the neutron star.
### 1.1 Observations
About 30 years ago the first X-ray telescope called `Einstein` was put into
space thus opening the possibility for direct investigation of thermal
emission from isolated neutron stars. A significant contribution to this study
was provided by `ROSAT` in 1990’s. Currently operating observatories such as
`Chandra` and `XMM-Newton` have greatly increased the quality and availability
of observations of thermal radiation from neutron star surfaces.
The X-ray radiation from an isolated neutron star in general can consist of
two distinguishable components: the nonthermal emission and the thermal
emission. The nonthermal component is usually described by a power-law
spectral model and attributed to radiation produced in the pulsar
magnetosphere while the thermal emission can originate either from the entire
surface of cooling neutron star or the small hot spots around the magnetic
poles on stellar surface (polar caps and adjacent areas).
Thermal X-ray emission seems to be a quite common feature of radio pulsars.
The black body fit allows us to obtain directly the temperature ($T_{s}$) of
the hot spot. Using the distance ($D$) to the pulsar and the luminosity of
thermal emission ($L_{\rm bol}$) we can estimate the area ($A_{\rm BB}$) of
the hot spot. In the most cases $A_{\rm BB}$ differs from the conventional
polar cap area $A_{\rm pc}\approx 6.2\times 10^{4}P^{-1}$ m2, where $P$ is
pulsar period. We use parameter $b=A_{\rm pc}/A_{\rm BB}$ to describe the
difference between $A_{\rm pc}$ and $A_{\rm BB}$.
### 1.2 The case with $b<1$
In the most cases observed hot spot area ($A_{\rm BB}$) is larger then the
conventional polar cap area. We can distinguish two types of pulsars in this
group, with $b\ll 1$ and $b\lesssim 1$.
The first type is associated with observations of thermal emission from the
entire stellar surface and can be used to test cooling models. Although we
have to remember that for young pulsars ($\tau=1$ kyr) the nonthermal
component dominates, making it impossible to measure accurately the thermal
flux. As a pulsar becomes older, its nonthermal luminosity decreases faster
then the thermal luminosity up to the end of the neutrino-cooling era
($\tau\sim 1$ Myr). Thus, the thermal radiation from the entire stellar
surface can dominate at soft X-ray energies for middle-age pulsars ($\tau\sim
100$ kyr) and some younger pulsars ($\tau\sim 10$ kyr) [2].
Some observations show that the hot spot area is larger then the conventional
polar cap area but still significantly less then the area of the star. In this
case the radiation comes from the surface adjacent to polar cap. Therefore, it
can not be explained by cooling of the surface and some addition heating
mechanism is needed. The model of such heating is based on the assumption that
the pulsar magnetic field near the stellar surface differs significantly from
the pure dipole one. The calculations show that it is natural to obtain such
geometry of magnetic field lines that allows the pair creation in the closed
field lines region. The pairs move along closed magnetic field lines and heat
the surface beyond the polar cap on the opposite side of the star (Fig. 1). In
such scenario heating energy is generated in IAR (outward particles) hence the
luminosity should be same order as the nonthermal one but it is hard to
predict which of them would prevail in X-ray flux of an old neutron star.
However, it cannot be ruled out that the thermal one may be dominant as
suggested by some observations. In most cases area of a such heated surface
may be larger (but not necessarily) then the conventional polar cap area. That
makes the estimation of parameters of black-body radiation even harder. More
detailed description of this phenomenon can be found in [3].
Figure 1: Cartoon of the magnetic field lines in the polar cap region. Red
lines are open field lines and green dashed lines correspond to the dipole
field. The blue arrows show direction of the curvature photons emission.
### 1.3 The case with $b>1$
In some cases observed hot spot area ($A_{\rm BB}$) is less then the
conventional polar cap area ($b>1$). The model mentioned in 1.2 can be used in
order to explain this phenomenon, because the size of modified polar cap is
much smaller then size of conventional polar cap (see Fig. 1). The surface
magnetic filed can be estimated by the magnetic flux conservation law as
$b=A_{\rm pc}/A_{\rm BB}$ = $B_{s}/B_{d}$. Where $B_{d}=2.02\times
10^{12}(P\dot{P}_{-15})^{0.5}$, $P$ is the pulsar period in seconds and
$\dot{P}_{-15}=\dot{P}/10^{-15}$ is the period derivative.
Medin & Lai [4] calculated the condition for the formation of a vacuum gap
above neutron star surface. In Fig. 2 we present positions of pulsars with
derived surface temperature ($T_{s}$) and hot spot area ($A_{\rm BB}$) on the
$B_{s}-T_{s}$ diagram where $B_{s}$ is estimated as $B_{s}=bB_{d}$. Red line
represents dependence of $T_{\rm crit}$ on $B_{s}$. We can see that in most
cases the pulsars positions follow with $B_{s}-T_{\rm crit}$ theoretical
curve. Few cases which do not coincide with the theoretical curve can be
explained by heating the surface outside the polar cap (see 1.2).
Figure 2: Diagram of the surface temperature ($T_{6}=T_{s}/10^{6}$) vs. the
surface magnetic field ($B_{14}=B_{s}/10^{14}$). The red line is the critical
temperature ($T_{\rm crit}$) evaluated from [4] and dashed lines corresponds
to uncertainties in calculations. Left panel includes all pulsars with $b>1$
while right panel is zoom of the upper right part of the graph. Error bars
corresponds to $1\sigma$.
According to our model the actual surface temperature equals to the critical
value ($T_{s}\sim T_{\rm crit}$) which leads to the formation of Partially
Screened Gap (PSG) above the polar caps of neutron star [5]. Hot spot
parameters derived from X-ray observations of isolated neutron stars are
presented in Table 1.
## 2 The Model
The PSG model assumes existence of heavy (Fe56) ions with density near but
still below corotational charge density ($\rho_{\rm GJ}$), thus the actual
charge density causes partial screening of the potential drop just above the
polar cap. The degree of shielding can be described by parameter
$\eta=1-\rho_{i}/\rho_{\rm GJ}$ where $\rho_{i}$ is the charge density of
heavy ions in the gap. The thermal ejection of ions from surface causes
partial screening of the acceleration potential drop
$\Delta V=\eta\Delta V_{max}=2\pi\eta B_{s}h^{2}/cP,$ (1)
where $\Delta V_{max}$ is the potential drop in vacuum gap, $h$ is the gap
height, $B_{s}=bB_{d}$ surface magnetic field (applicable only if $b>1$).
Using calculations of Medin & Lai [4] we can express the dependence of the
critical temperature on pulsar parameters as
$T_{\rm crit}=1.6\times
10^{4}\left(\left(\left(P\dot{P}_{-15}\right)^{0.5}b\right)^{1.1}+17.7\right)$
(2)
or $T_{\rm crit}=1.1\times 10^{6}\left(B_{14}^{1.1}+0.3\right)$, where
$B_{14}=B_{s}/10^{14}$.
The actual potential drop $\Delta V$ should be thermostatically regulated and
there should be established a quasi-equilibrium state, in which heating due to
electron/positron bombardment is balanced by cooling due to thermal radiation
(see [5] for more details). The necessary condition for formation of this
quasi-equilibrium state is
$\sigma T_{s}^{4}=\eta e\Delta Vcn_{\rm GJ},$ (3)
where $\sigma$ is the Stefan-Boltzmann constant, $e$ \- the electron charge,
$n_{\rm GJ}=\rho_{\rm GJ}/e=1.4\times 10^{11}b\dot{P}_{-15}^{0.5}P^{-0.5}$ is
the corotational number density. Using equations (2) and (3) we can express
the acceleration potential drop as
$\Delta V=7.3\times
10^{5}\eta^{-1}B_{14}^{-1}\left(B_{14}^{1.1}+0.3\right)^{4}P,$ (4)
and finally using equations (1) and (4) we can estimate the gap height in PSG
model as
$h\eta=6B_{14}^{-1}\left(B_{14}^{1.1}+0.3\right)^{2}P$ (5)
As we see both $\Delta V$ and $h$ depend on shielding factor $\eta$ which
actually depends on the details of the avalanche pair production in the gap.
First we need to determine which process, Curvature Radiation (CR) or Inverse
Compton Scattering (ICS), is responsible for the pair production. We will need
following parameters: $l_{\rm acc}$ \- the distance which a particle should
pass to gain Lorentz factor equal to $\gamma_{\rm acc}$, $l_{e}$ \- the mean
length an electron (or positron) travels before a gamma-photon is emitted, and
$l_{\rm ph}$ \- the mean free path of gamma-photon before being absorbed by
the magnetic field.
### 2.1 Acceleration path
In the frame of PSG model, using formalism described in [6], we estimate the
component of electric field along the magnetic field line in the gap as
$E\approx\eta\frac{4\pi B_{s}}{cP}(h-z)$ (6)
which vanishes at the top $z=h$. The Lorentz factor of particles after
distance $l_{\rm acc}$ can be calculated as follows
$\gamma_{\rm
acc}=\frac{e}{m_{e}c^{2}}\int_{z_{1}}^{z_{2}}Edz\approx\eta\frac{4\pi
B_{s}e}{m_{e}c^{3}P}(z_{2}-z_{1})(h-\frac{z_{1}+z_{2}}{2})$ (7)
where $m_{e}$ is mass of an electron and $z_{2}-z_{1}=l_{\rm acc}$. Then we
can approximate $z_{1}+z_{2}\approx 2z$ and assume $z\approx h/2$, thus
$l_{\rm acc}=\gamma_{\rm acc}\frac{m_{e}c^{3}P}{2\pi h\eta B_{s}e}.$ (8)
### 2.2 Electron or positron mean free paths
The mean free path of electron or positron ($l_{e}$) can be defined as the
mean length that a particle passes until a gamma-photon is emitted. In the
case of CR electron mean free path can be estimated as a distance that
particle with Lorentz factor $\gamma$ travels during the time which is
necessary to emit curvature photon (see [7])
$l_{e,cr}\sim
c\left(\frac{P_{cr}}{E_{\gamma,cr}}\right)^{-1}=\frac{9}{4}\frac{\hbar\Re
c}{\gamma e^{2}},$ (9)
where $P_{cr}=2\gamma^{4}e^{2}c/3\Re^{2}$ is the power of the CR,
$E_{\gamma,cr}=3\hbar\gamma^{3}c/2\Re$ is the characteristic photon energy,
$\Re$ \- the curvature radius of magnetic field lines.
For the ICS process, calculation of the electron mean free path $l_{e,ics}$,
is not as simple as that of the CR process. Although we can define $l_{e,ics}$
in a same way as we defined $l_{e,cr}$ but it is difficult to estimate a
characteristic frequency of emitted photons. We have to take into account
photons of various frequencies with various incident angles. An estimation of
the mean free path of electron (or positron) to produce a photon is [8]
$l_{e,ics}\sim\left[\int_{\mu_{0}}^{\mu_{1}}\int_{0}^{\infty}{\sigma^{\prime}(\epsilon,\mu_{i})(1-\beta\mu_{i})n_{\rm
ph}(\epsilon)d\epsilon d\mu_{i}}\right]^{-1}$ (10)
where $\epsilon$ is the incident photon energy in units of $m_{e}c^{2}$,
$\mu_{i}=\cos{\psi_{i}}$ is the cosine of the photon incident angle,
$\beta=v/c$ is the velocity in terms of speed of light,
$n_{\rm
ph}=\frac{4\pi}{\lambda_{c}^{3}}\frac{\epsilon^{2}}{\exp{(\epsilon/\theta)}-1}d\epsilon$
(11)
represents the photon number density distribution of a semi-isotropic
blackbody radiation, $\theta=kT_{s}/m_{e}c^{2}$, $k$ is the Boltzmann
constant, and $\lambda_{c}=h/m_{e}c=2.424\times 10^{-10}$ cm is the electron
Compton wavelength. Here $\sigma^{\prime}$ is the cross section of scattering
in the particle rest frame. In the Thomson regime cross section of scattering
in the particle rest frame can be written as [9]
$\sigma^{\prime}=\frac{\sigma_{T}}{2}\left[\frac{u^{2}}{(u+1)^{2}}+\frac{u^{2}}{(u-1)^{2}+a^{2}}\right],$
(12)
where $\sigma_{T}$ is the Thomson cross section,
$u=\epsilon^{\prime}/\epsilon_{B}$, $a=\frac{2}{3}\alpha_{f}\epsilon_{B}$,
$\epsilon^{\prime}=\gamma\epsilon(1-\beta\mu_{i})$ is the incident photon
energy in the particle rest frame in units of $m_{e}c^{2}$,
$\epsilon_{B}=B_{s}/B_{q}$ is the electron cyclotron resonance energy in units
of $m_{e}c^{2}$, $B_{q}=m_{e}^{2}c^{3}/e\hbar=4.413\times 10^{13}$ G, and
$\alpha_{f}=e^{2}/\hbar c$ is the fine-structure constant.
Equation (12) however does not include the quantum relativistic effects of a
strong magnetic field ($B_{s}>0.1B_{q}$), therefore, it can be used only at
altitudes where magnetic field is relatively weaker. For strong magnetic
fields we can use an approximation proposed in [10] and then an approximate
averaged (polarization-independent) cross section can be written as follows
$\sigma^{\prime}=\frac{3\sigma_{T}}{8}\frac{\epsilon\epsilon^{\prime
2}(1+\mu_{i}^{2})}{2\epsilon-\epsilon^{\prime}}\left[\frac{1}{(\epsilon-\beta_{q})^{2}}+\frac{1}{(\epsilon+\beta_{q})^{2}}\right],$
(13)
where $\beta_{q}=B_{s}/B_{q}$. Equation (13) represents the exact cross
section when the particle after scattering falls on the zero Landau state (see
[10] for more accurate results).
We should expect two modes of ICS process, resonant and non-resonant. The
resonant ICS takes place if the photon frequency in particle rest frame equals
to the cyclotron electron frequency. Non-resonant ICS includes all scattering
processes of photons with frequencies around the maximum of the thermal
spectrum. At resonance frequency equation (13) has a singularity, so it could
be used only for non-resonant case. For resonant case, one should use approach
proposed in [11], thus the cross section in the particle rest frame is
$\sigma_{res}^{\prime}\simeq
2\pi^{2}\frac{e^{2}\hbar}{m_{e}c}\delta(\epsilon^{\prime}-\epsilon_{b})$ (14)
Equation (14) can be also used for strong magnetic fields ($\beta_{q}>0.1$),
since the resonant condition $\epsilon^{\prime}=\epsilon_{b}$ holds regardless
of field strength. Although one have to remember that this equation represents
the upper limit for cases with very high magnetic fields (see [10] for more
details). The particle mean free path above a polar cap for the resonant ICS
process is
$l_{e,rics}\sim\left[\int_{\mu_{0}}^{\mu_{1}}\int_{0}^{\infty}{\sigma_{res}^{\prime}(1-\beta\mu_{i})n_{\rm
ph}(\epsilon)d\epsilon d\mu_{i}}\right]^{-1}=\left[\frac{2\pi
e^{2}\hbar}{m_{e}c\gamma}\int_{\mu_{0}}^{\mu_{1}}n_{\rm
ph}\left(\frac{\epsilon_{b}}{1-\beta\mu_{i}}\right)d\mu_{i}\right]^{-1}$ (15)
For the altitudes of the same order as the polar cap size we can use
$\mu_{0}=1$, $\mu_{1}=0$ as incident angle limits for outflowing particles,
and $\mu_{0}=0$, $\mu_{1}=-1$ as incident angle limits for backflowing
particles.
### 2.3 Photon mean free path
A photon with energy $E_{\gamma}>2m_{e}c^{2}$ and propagating with nonzero
angle $\psi$ with respect to an external magnetic field can be absorbed by the
field and as a result electron-positron pair is created. The optical depth of
a such process can be defined as [11]
$\tau=s_{\rm ph}R_{\|,\perp},$ (16)
where $s_{\rm ph}$ is a distance traveled by an photon,
$R_{\|,\perp}=R^{\prime}_{\|,\perp}\sin{\psi}$ is the attenuation coefficient
for the $\|$ or $\perp$ polarized photons, and $R^{\prime}$ is the attenuation
coefficient in the ”perpendicular” frame (the frame where the photon
propagates perpendicular to the local magnetic field).
The total attenuation coefficient for pair production is given by
$R^{\prime}=\sum_{jk}R^{\prime}_{j,k}$, where $R^{\prime}_{j,k}$ is the
attenuation coefficient for process (channel) in which the photon produces an
electron in Landau level $j$ and positron in Landau level $k$, and the sum is
taken over all possible states for the electron-positron pair. Since pair
production is symmetric with respect to the electron and positron,
$R^{\prime}_{kj}=R^{\prime}_{kj}$ for simplicity we will use $R^{\prime}_{jk}$
to represent the combined probability of creating the pair in either $(jk)$ or
$(kj)$ state. For a given channel $(jk)$, the threshold condition for pair
production is:
$E_{\gamma}^{\prime}>E_{j}^{\prime}+E_{k}^{\prime},$ (17)
where $E_{\gamma}^{\prime}=E_{\gamma}\sin{\psi}$ is the photon energy in the
perpendicular frame and $E_{n}^{\prime}=m_{e}c^{2}\sqrt{1+2\beta_{q}n}$ is the
minimum energy of an electron/positron in Landau Level $n$. In dimensionless
form this condition can be written as
$x=\frac{E_{\gamma}^{\prime}}{2mc^{2}}=\frac{E_{\gamma}}{2m_{e}c^{2}}\sin{\psi}>\frac{1}{2}\left[\sqrt{1+2\beta_{q}j}+\sqrt{1+2\beta_{q}k}\right]$
(18)
The first nonzero attenuation coefficients for both polarizations are given in
[12]:
$R^{\prime}_{\|,00}=\frac{1}{2a_{0}}\frac{\beta_{q}}{x^{2}\sqrt{x^{2}-1}}e^{-2x^{2}/\beta_{q}},\hskip
28.45274ptx>x_{1}=1;$ (19)
$R^{\prime}_{\perp,01}=2\times\frac{1}{2a_{0}}\frac{\beta_{q}}{2x^{2}}\frac{2x^{2}-\beta_{q}}{\sqrt{x^{2}-1-\beta_{q}+\frac{\beta_{q}^{2}}{4x^{2}}}}e^{-2x^{2}/\beta_{q}},\hskip
28.45274ptx>x_{2}=\left(1+\sqrt{1+2\beta_{q}}\right)/2;$ (20)
where $a_{0}$ is Bohr radius (let us note that $R^{\prime}_{\perp,00}=0$ and
higher orders of attenuation coefficients should be used if
$x>x_{3}=\left(1+\sqrt{1+4\beta_{q}}\right)/2$).
The optical depth is defined as:
$\tau=\int_{0}^{s_{\rm ph}}R(s)ds=\int_{0}^{s_{\rm
ph}}R^{\prime}(s)\sin{\psi}ds$ (21)
We can assume $\psi\ll 1$, because all high energy photons ($x>1$) will
produce pairs much earlier then $\psi$ reaches value near unity. In this limit
$\sin{\psi}\simeq s_{\rm ph}/\Re$ so relation between $x$ and $s_{\rm ph}$ can
be expressed by
$x\simeq\frac{s_{\rm ph}}{\Re}\frac{E_{\gamma}}{2m_{e}c^{2}}.$ (22)
The equation (21) can be rewritten as
$\tau=\tau_{1}+\tau_{\|,2}+\tau_{\perp,2}+...$
$\tau_{1}=\int_{s_{1}}^{s_{2}}R_{\|,00}ds,\hskip
14.22636pt\tau_{\|,2}=\int_{s_{2}}^{s_{3}}\left(R_{\|,00}+R_{\|,01}\right)ds,\hskip
14.22636pt\tau_{\perp,2}=\int_{s_{2}}^{s_{3}}R_{\perp,01}ds$ (23)
where $s_{1}$ and $s_{2}$ are distances which the photon should pass in order
to have energy $x_{1}$ and $x_{2}$ respectively (in perpendicular frame of
reference). Let us note that $s_{1}$, $s_{2}$, and $s_{3}$ are of the same
order and if $s<s_{1}$ attenuation coefficient is zero. As shown in [11] for
strong magnetic fields ($\beta_{q}\gtrsim 0.1$) $\tau_{1}$, $\tau_{\|,2}$, and
$\tau_{\perp,2}$ are much larger then one. Therefore, the pair production
process happens according two scenarios. If $\beta_{q}\gtrsim 0.1$ photons
produce pairs almost immediately upon reaching the first threshold, so the
created pairs will be in low Landau levels ($n\lesssim 2$). If
$\beta_{q}\lesssim 0.1$ photons will travel longer distances to be absorbed
and created pairs will be in higher Landau levels.
Thus, for strong magnetic fields ($\beta_{q}\gtrsim 0.1$) electron mean free
path can be calculated as
$l_{\rm ph}\sim s_{1}=\Re\frac{2m_{e}c^{2}}{E_{\gamma}},$ (24)
and for relatively weak magnetic fields ($\beta_{q}\lesssim 0.1$) we can use
asymptotic approximation derived by Erber [13]
$l_{\rm ph}=\frac{4.4}{(e^{2}/\hbar
c)}\frac{\hbar}{m_{e}c}\frac{1}{\beta_{q}\sin{\psi}}\exp{\left(\frac{4}{3x\beta_{q}}\right)}$
(25)
### 2.4 Partially Screened Gap height
As we mentioned above, PSG can exist if equation (3) is satisfied. On the
other hand for the heating of the stellar surface the high enough flux of
back-streaming particles is required. Thus, we need to estimate shielding
factor $\eta$ and gap height $h$ which are the main parameters of PSG. Let us
note that $\eta$ and $h$ are connected with each other by equation (5).
Generally gap height can by defined as $h\approx l_{e}+l_{ph}$ with the
necessary condition that $l_{e}>l_{acc}$. Latter corresponds to demand that
the particle should gain the energy that is required for photon emission by
either ICS or CR processes. The Fig. 3 shows dependence of free paths on the
particles Lorentz factor $\gamma$ for some particular pulsar parameters
(dependence on pulsar parameters will be discussed in section 3). Let us note
that this free paths do not depend on the gap height $h$ (see equations (9),
(15), (24)). As it follows from equation (8) $l_{acc}$ also does not depend on
a gap height because $h$ and $\eta$ are connected by equation (5).
Results presented in the Fig. 3 do not allow us to define the gap height
unambiguously but we can already find that CR is the dominant pair creation
process for PSG scenario. Although $l_{e,ics}<l_{e,cr}$ for particles with
$\gamma\sim 10^{3}-10^{4}$ the gamma-photon production by ICS process is not
effective because $l_{\rm acc}<<l_{e,ics}$, i.e. the particles will be
accelerated to higher energies ($\gamma\sim 10^{5}-10^{6}$) before they would
upscatter X-ray photons emitted from the hot polar cap.
Figure 3: Diagram of the electron/positron mean free path for both CR and ICS
processes. The left panel (a) corresponds to calculations for $B_{14}=1.5$,
$T_{6}=1.9$, $\Re_{6}=0.1$, $P=1$ while the right panel (b) corresponds to
calculations for $B_{14}=3.5$, $T_{6}=4.4$, $\Re_{6}=1$, $P=1$. Acceleration
path was calculated for the same gap parameters ($h=20$ m, $\eta=0.03$) in
both panels (see discussion in section 2.4). Although $l_{e,ics}<l_{e,cr}$ for
particles with $\gamma\sim 10^{3}-10^{4}$ the $\gamma$-photon production by
ICS process is not effective because for these particle $l_{\rm
acc}<<l_{e,ics}$. That means that they will be accelerated to higher energies
($\gamma\sim 10^{5}-10^{6}$) before they would upscatter X-ray photons emitted
from the stellar surface.
As soon as we determine the dominant process, i.e. CR, which is responsible
for gamma-photons emission we can estimate the gap height. Curvature emission
by a particle is effective for Lorentz factors $\gamma\sim 10^{5}-10^{6}$
(when $l_{e,cr}\leq l_{\rm acc}$). The reaction force although is not high
enough to stop the acceleration by electric field. Equilibrium between
acceleration and deceleration (by reaction force) would be established when
the CR power would equal to ”electric power” ($P_{cr}=P_{\rm acc}$, where
$P_{\rm acc}=4\pi\eta vB_{s}(h-z)/cP$ is work done by electric field in unit
of time). Using the characteristic Lorentz factors of radiating particles we
can find characteristic frequencies of CR photons and thus, we can check
whether the necessary condition ($l_{e}+l_{\rm ph}\leq h$) for cascade
formation is satisfied.
The Fig. 4 shows dependence of the CR photon mean free path ($l_{ph}$), the
particle mean free path ($l_{e}$), and the acceleration path ($l_{acc}$) on
the energy expressed in units of $m_{e}c^{2}$. Top axis shows the Lorentz
factors of particles which emit the curvature photons with energy shown on
bottom axis. While the energy of particle falls in blue region in the Fig. 4
there will be no CR because the particle will be accelerated to higher
energies before it travels distance enough to emit one curvature photon
($l_{e}>l_{\rm acc}$). For the energies in green region CR is most efficient.
The particle in the PSG never can reach energies falling in red region because
reaction force caused by CR process is bigger then acceleration force. Thus,
the characteristic Lorentz factor $\gamma_{c}$ of particles (and
characteristic photon energy $E_{c}$) corresponds to value defined by the
border between green and red region. Therefore, we can define gap height for
the characteristic energies $\gamma_{c}$ and $E_{c}$ as
$l_{e}(\gamma=\gamma_{c})+l_{ph}(E_{\gamma}=E_{c})<h$ (26)
Then we have to calculate $l_{e}$, $l_{ph}$ for different values of $h$ and
check which value satisfies condition (26). The Fig. 4 shows two cases: panel
(a) that corresponds to $h=10$ m and panel (b) which corresponds to $h=40$ m.
From panel (a) we can see that condition (26) is not satisfied while for gap
height $h=40$ m cascade process will form. This approach allows us to
calculate minimum height at which gap sparking breakdown is possible and thus,
we can estimate unambiguously the gap height in the frame of the PSG model.
Figure 4: Diagram of the photon mean free path, the particle mean free path,
and the acceleration path vs. the photon energy and the Lorentz factor of
accelerated particles (top axis) for specific pulsar ($B_{14}=3.5$,
$T_{6}=4.39$, $\Re_{6}=0.3$, $P=1$). See description of blue, green and red
regions in the text above equation (26). The black horizontal line corresponds
to the gap height. Calculations for panel (a) were done using $h=10$ m,
$\eta=0.03$ while for panel (b) using $h=40$ m, $\eta=0.01$.
## 3 Observational consequences
In order to compare the model with observations we need to define main
parameters of PSG model for a given pulsar with known period and period
derivative.
### 3.1 PSG model parameters
We can distinguish two types of PSG parameters: observed and derived. As we
mention above in some cases when X-ray observations are available we can
directly estimate surface magnetic filed $B_{s}$. On one hand $B_{s}$ can be
calculated using the size of the hot spot $A_{\rm BB}$, and on the other hand
we can find $B_{s}$ using estimation of the critical temperature and
assumption that $T_{s}=T_{\rm crit}$. One of the most important requirements
of PSG model is that these two estimations should coincide with each others.
As it is clear from Fig. 2 in most cases when the hot spot parameters are
available this requirement is fulfilled. Thus, we can assume the
characteristic values of $B_{s}$ vary in range of $(1-4)\times 10^{14}$ which
corresponds to critical/surface temperature in range of $(1.1-4.5)\times
10^{6}$ (see Table 1). Using these values we can estimate derived parameters
of PSG such us the gap height $h$, the shielding factor $\eta$ and the Lorentz
factor of primary particles $\gamma_{c}$. Let us note that these parameters
depend also on the curvature radius of magnetic field lines $\Re$. The
curvature can not be neither observed or derived but modeling of surface
magnetic field (see Fig. 1) indicates that the curvature radius varies in the
range of $(0.1-1)\times 10^{6}$ cm. Below we will discuss influence of pulsar
parameters such as the magnetic field, the curvature of field lines and the
period on derived PSG parameters.
### 3.2 Influence of the magnetic field
The conditions in PSG are mainly defined by the surface magnetic field. In
Fig. 5 panel (a) we present dependence of the gap height on the surface
magnetic field calculated according to approach described in section 2.4. It
is clear that the gap height decreases as the surface magnetic field
increases. The Fig. 5 panel (b) shows dependence of the shielding factor on
the surface magnetic field calculated using equation (5). We can see that for
stronger magnetic fields $\eta$ increases which means that the density of
heavy ions above the polar cap decreases. Let us note that the surface
temperature ($T_{s}$) stays very near to the critical temperature ($T_{\rm
crit}$) which is shown on the top axis of the diagrams. In Fig. 5 panel (c) we
present dependence of Lorenz factors of the particles accelerated in PSG. The
Green line ($\gamma_{0}$) presents the values corresponding to the boundary
between blue and green region in the Fig. 4 while the red line corresponds to
the boundary between the green and the red region which is the characteristic
value ($\gamma_{c}$) for primary particles. We see that $\gamma_{c}$ is very
slightly affected by the magnetic field strength.
One can expect that for higher temperatures (which correspond to stronger
magnetic fields) the gap breakdown could be dominated by ICS process. However
this is not the case because the particle mean free path is much higher then
the acceleration path ($l_{e,ics}>>l_{\rm acc}$) even for strong magnetic
fields.
Figure 5: Dependence of the gap height (panel a), the shielding factor (panel
b), and the particles Lorentz factor (panel c) on the surface magnetic field.
Solid lines correspond to calculations for $\Re_{6}=0.1$ while dashed lines
correspond to calculations for $\Re_{6}=1$. For both cases the pulsar period
$P=1$ was used. The green line presents the values corresponding to the
boundary between blue and green region in the Fig. 4 while the red line
corresponds to the boundary between green and red region. Corresponding
critical temperature is shown on top axis of diagrams.
### 3.3 Influence of the field lines curvature radius
The curvature of magnetic field lines significantly affects the PSG parameters
since the CR process is responsible for PSG breakdown. As it can be expected
the gap height decreases for smaller curvature radius of magnetic field lines
(Fig. 6 panel a) since CR is more efficient for smaller values of the
curvature radius. Consequently the shielding factor decreases with increasing
curvature radius of magnetic field lines (Fig. 6 panel b).
The Lorentz factor of primary particles increases for bigger radius of
curvature (Fig. 6 panel c), since both conditions $l_{e}\sim l_{\rm acc}$ and
$P_{cr}\sim P_{\rm acc}$ are satisfied for higher Lorentz factors.
Figure 6: Dependence of the gap height (panel a), the shielding factor (panel
b), and the particles Lorentz factor (panel c) on the curvature radius of
magnetic field lines. Solid lines correspond to calculations for $B_{14}=3.5$
while dashed lines correspond to calculations for $B_{14}=1.5$. For both cases
the pulsar period $P=1$ was used.
### 3.4 Influence of the pulsar period
As we can see from figure 7 panel (a) and panel (c) neither the gap height nor
the Lorentz factor of primary particles depend on pulsar period. This is one
of the most important result of our calculations. The shielding factor is just
proportional to period (Fig. 7 panel b) as is expected from equation (5).
Figure 7: Dependence of the gap height (panel a), the shielding factor (panel
b), and the particles Lorentz factor (panel c) on the pulsar period. Solid
lines correspond to calculations for $B_{14}=3.5$ while dashed lines
correspond to calculations for $B_{14}=1.5$. For both cases the radius of
curvature $\Re_{6}=0.1$ was used.
## 4 Conclusions
X-ray observations of pulsars show that temperature of the hot spot is about
few million kelvins while its area is much smaller then the conventional polar
cap surface which is calculated assuming purely dipolar geometry of pulsar
magnetic field. Such observations can be easily explained in the frame of PSG
model which assumes that the geometry of magnetic field near the stellar
surface differs significantly from dipolar one, actually the field is much
stronger and curved. At the same time the surface temperature is about the
critical temperature which depends only on the magnetic field strength.
In this paper we calculated dependence of the gap parameters such as the gap
height, the shielding factor, and characteristic energies of primary particles
on the surface magnetic field, the curvature radius of field lines, and the
pulsar period in the frame of PSG model. The surface magnetic field can be
calculated using the X-ray observations (if such data are available), however
the curvature radius can be only estimated by simulations of different
geometry of magnetic field lines. Let us note that the gap height is the most
important parameter for all models of the Inner Acceleration Region in
pulsars. We can define the gap height as a sum of the particle mean free path
and the photon mean free path, $h\approx l_{e}+l_{ph}$. In order to estimate
the PSG height we discuss two processes responsible for cascade pair
production: the Inverse Compton Scattering and the Curvature Radiation. We
found that CR is the dominant process for the sparking breakdown of the PSG.
Although ICS is more efficient for particles with $\gamma\sim 10^{3}-10^{4}$,
the particles are accelerated to higher energies $\gamma\sim 10^{5}-10^{6}$
before they upscatter X-ray photons emitted from the polar cap. As soon as
particles Lorentz factor $\gamma>10^{5}$ CR is more efficient then ICS. In
section 3 we show the dependence of PSG parameters ($h$, $\eta$, $\gamma_{c}$)
on the pulsar parameters ($B_{s}$, $\Re$, $P$). Since CR is the dominant
process for gamma-photon emission the PSG parameters strongly depend on
curvature radius of magnetic field lines. Pulsars with smaller curvature
radius should have the lower gap height and also the higher shielding factor,
consequently the density of heavy ions should be lower. In pulsars with
stronger surface magnetic field gap heights should be smaller and also
shielding factor should be smaller. Our calculations show that on one hand $h$
and $\gamma_{c}$ do not depend on pulsar period, but on the other hand $\eta$
increases along with increase pulsar period. The evaluated PSG parameters
should play decisive role in pulsar emission models.
We found that PSG model is suitable to explain both cases: when the hot spot
is smaller then the conventional polar cap, and vice versa when the hot spot
is larger then conventional polar cap. In the latter case the surface is
heated by particles created in the closed magnetic filed lines region by
photons emitted in the open magnetic filed lines region. Let us note that in
the purely dipolar magnetic field geometry photons emitted tangent to magnetic
field lines always stay in the open field lines region.
###### Acknowledgments.
This paper was partially supported by research Grants N N 203 2738 33 and N N
203 3919 34 of the Polish Ministry of Science and Education. GM was partially
supported by the GNSF grant ST08/4-442.
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name | $P$ | $D$ | $T_{s}$ | $R_{\rm BB}$ | $L_{\rm bol}$ | $\chi$ | $b$ | $B_{s}$ | $T_{\rm crit}$ | age | ref.
---|---|---|---|---|---|---|---|---|---|---|---
| $($ s $)$ | $($ kpc $)$ | $($ $10^{6}$ K $)$ | | $($ erg$/$s $)$ | | | $($ $10^{14}$ G $)$ | $($ $10^{6}$ K $)$ | |
J0108-1431 | $0.808$ | $0.18$ | $3.2^{+0.4}_{-0.3}$ | $6^{+5}_{-4}$ m | $7\times 10^{27}$ | $1\times 10^{-3}$ | $768$ | $3.9$ | $5.2$ | $166$ Myr | [14]
B1929+10 | $0.227$ | $0.36$ | $4.5^{+0.3}_{-0.5}$ | $27^{+5}_{-4}$ m | $5\times 10^{29}$ | $1\times 10^{-4}$ | $129$ | $1.3$ | $1.8$ | $3$ Myr | [15]
J0633+1746B | $0.237$ | $0.16$ | $2.3^{+0.1}_{-0.1}$ | $36^{+9}_{-9}$ m | $8\times 10^{28}$ | $2\times 10^{-6}$ | $68$ | $2.2$ | $3.0$ | $342$ kyr | [17]
| | | $0.5$ | $10$ km | | | | | | |
B0943+10 | $1.098$ | $0.63$ | $3.1^{+1.1}_{-1.1}$ | $18^{+40}_{-15}$ m | $5\times 10^{28}$ | $5\times 10^{-4}$ | $62$ | $2.5$ | $3.3$ | $5$ Myr | [16]
B0950+08 | $0.253$ | $0.26$ | $2.3^{+0.3}_{-0.3}$ | $37^{+25}_{-25}$ m | $7\times 10^{28}$ | $1\times 10^{-4}$ | $59$ | $0.3$ | $0.6$ | $18$ Myr | [20]
B1133+16 | $1.188$ | $0.36$ | $3.2^{+0.5}_{-0.4}$ | $18^{+14}_{-12}$ m | $7\times 10^{28}$ | $8\times 10^{-4}$ | $52$ | $2.2$ | $3.0$ | $5$ Myr | [16]
B0834+06 | $1.274$ | $0.64$ | $2.0^{+0.8}_{-0.6}$ | $28^{+56}_{-15}$ m | $2\times 10^{28}$ | $2\times 10^{-4}$ | $21$ | $1.2$ | $1.7$ | $3$ Myr | [18]
B0628-28 | $1.244$ | $1.45$ | $3.3^{+1.3}_{-0.6}$ | $59^{+65}_{-46}$ m | $7\times 10^{29}$ | $5\times 10^{-3}$ | $5$ | $0.3$ | $0.6$ | $3$ Myr | [19]
J2043+2740 | $0.096$ | $1.80$ | $1.5^{+0.4}_{-0.7}$ | $467^{+200}_{-200}$ m | $2\times 10^{30}$ | $3\times 10^{-5}$ | $1$ | – | – | $1$ Myr | [21]
| | | $0.6$ | $10$ km | | | | | | |
B1055-52 | $0.197$ | $0.75$ | $1.8^{+0.1}_{-0.1}$ | $460^{+60}_{-60}$ m | $4\times 10^{30}$ | $1\times 10^{-4}$ | $0.5$ | – | – | $535$ kyr | [22]
| | | $0.8$ | $12$ km | | | | | | |
J0538+2817 | $0.143$ | $1.20$ | $2.8^{+0.1}_{-0.1}$ | $666^{+38}_{-38}$ m | $5\times 10^{31}$ | $9\times 10^{-4}$ | $0.3$ | – | – | $30$ kyr | [25]
J1809-1917 | $0.083$ | $3.50$ | $2.0^{+0.4}_{-0.4}$ | $951^{+920}_{-693}$ m | $2\times 10^{31}$ | $1\times 10^{-5}$ | $0.3$ | – | – | $51$ kyr | [23]
J0821-4300 | $0.113$ | $2.20$ | $6.3^{+0.2}_{-0.2}$ | $1.2^{+0.1}_{-0.1}$ km | $4\times 10^{33}$ | $1\times 10^{-1}$ | $0.1$ | – | – | $2$ Myr | [24]
| | | $3.2$ | $6.0$ km | | | | | | |
B0833-45 | $0.089$ | $0.30$ | $1.9^{+0.1}_{-0.1}$ | $2.0^{+0.2}_{-0.2}$ km | $3\times 10^{31}$ | $5\times 10^{-6}$ | $0.06$ | – | – | $11$ kyr | [2]
| | | $0.9$ | $14$ km | | | | | | |
J1357-6429 | $0.166$ | $2.50$ | $2.2^{+0.3}_{-0.3}$ | $1.9^{+0.4}_{-0.4}$ km | $2\times 10^{32}$ | $5\times 10^{-5}$ | $0.03$ | – | – | $7$ kyr | [26]
B1916+14 | $1.181$ | $2.10$ | $1.5^{+0.1}_{-0.1}$ | $800^{+100}_{-100}$ m | $6\times 10^{30}$ | $1\times 10^{-3}$ | $0.03$ | – | – | $88$ kyr | [27]
B1706-44 | $0.102$ | $2.50$ | $2.2^{+0.2}_{-0.2}$ | $2.8^{+0.7}_{-0.7}$ km | $3\times 10^{32}$ | $9\times 10^{-5}$ | $0.03$ | – | – | $18$ kyr | [28]
B0656+14 | $0.385$ | $0.29$ | $1.2^{+0.03}_{-0.03}$ | $1.8^{+0.2}_{-0.2}$ km | $1\times 10^{31}$ | $4\times 10^{-4}$ | $0.02$ | – | – | $111$ kyr | [22]
| | | $0.7$ | $21$ km | | | | | | |
B2334+61 | $0.495$ | $3.10$ | $1.7^{+0.5}_{-0.8}$ | $1.7^{+0.6}_{-0.4}$ km | $1\times 10^{31}$ | $2\times 10^{-4}$ | $0.02$ | – | – | $41$ kyr | [29]
J1119-6127 | $0.408$ | $8.40$ | $3.1^{+0.4}_{-0.3}$ | $2.6^{+1.4}_{-0.2}$ km | $1\times 10^{33}$ | $5\times 10^{-4}$ | $0.008$ | – | – | $2$ kyr | [30]
Table 1: Observed and derived parameters of isolated neutron stars with
available X-ray observations. $P$ \- the pulsar period, $D$ \- the distance
used for calculation of $L_{\rm bol}$, $T_{s}$ \- the spot temperature,
$R_{\rm BB}$ \- the radius of the spot obtained from the black-body fit,
$L_{\rm bol}$ \- the bolometric luminosity of the hot or warm spot ($L_{\rm
bol}=\pi R_{\rm BB}^{2}\sigma T_{s}^{4}$), $\chi=L_{bol}/L_{sd}$ \- the
efficiency of X-ray emission where $L_{sd}$ is the spin down luminosity,
$b=A_{\rm pc}/A_{\rm BB}=B_{s}/B_{d}$, $B_{s}$ \- the surface magnetic field
strength, $T_{\rm crit}$ \- the critical temperature. Pulsars are sorted by
decreasing parameter $b$.
|
arxiv-papers
| 2011-08-23T11:55:19 |
2024-09-04T02:49:21.726320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrzej Szary (1), George I. Melikidze (1 and 2) and Janusz Gil (1)\n ((1) J.Kepler Institute of Astronomy, University of Zielona G\\'ora, Poland\n (2) E.Kharadze Georgian National Astrophysical Observatory, Georgia)",
"submitter": "Andrzej Szary M.Sc.",
"url": "https://arxiv.org/abs/1108.4560"
}
|
1108.4586
|
# Universality of striped morphologies
E. Edlund erik.edlund@chalmers.se M. Nilsson Jacobi mjacobi@chalmers.se
Complex Systems Group, Department of Energy and Environment, Chalmers
University of Technology, SE-41296 Göteborg, Sweden
###### Abstract
We present a method for predicting the low-temperature behavior of spherical
and Ising spin models with isotropic potentials. For the spherical model the
characteristic length scales of the ground states are exactly determined but
the morphology is shown to be degenerate with checkerboard patterns, stripes
and more complex morphologies having identical energy. For the Ising models we
show that the discretization breaks the degeneracy causing striped
morphologies to be energetically favored and therefore they arise universally
as ground states to potentials whose Hankel transforms have nontrivial minima.
The study of pattern formation in simple systems has received much attention
the last 20 years. Not only are the pictures visually arresting, producing
reviews with considerable artistic qualities Seul and Andelman (1995); Bowman
and Newell (1998), but spatially inhomogeneous phases present difficulties for
standard theoretical methods, calling for new principles to describe the
physics of systems exhibiting them Emery _et al._ (1999). Whatever weight can
be assigned to the former as an explanation for the interest, the latter is
justification enough, especially as the experimental evidence of stripes,
spots and checkerboards in practically important materials abound. Examples
include lipid monolayers Keller and McConnell (1999), adsorbates on metals
Kern _et al._ (1991), and various magnetic fluids Rosensweig _et al._
(1983); Seul and Wolfe (1992). Striped phases are also hypothesized to play a
role in the high-temperature superconductivity of transition metal oxides
Tranquada _et al._ (1995); Orenstein and Millis (2000).
Many experimental systems displaying heterogeneous patterns involve a
competition between long- and short-range interactions De’Bell _et al._
(2000) and most theoretical work concentrate on specific examples of such
interactions Garel and Doniach (1982); Löw _et al._ (1994); Grousson _et
al._ (2000); Stoycheva and Singer (2000); Giuliani _et al._ (2006), e.g. spin
models with Hamiltonians on the form
$H=K\sum_{j}s_{j}^{2}-L\sum_{\langle
i,j\rangle}s_{i}s_{j}+\frac{Q}{2}\sum_{i\neq
j}\frac{s_{i}s_{j}}{r_{ij}^{\alpha}},$ (1)
where the spins typically represent some coarse-grained feature of the system
of interest, for example local charge density in a Mott insulator Emery _et
al._ (1999) or phases in a Langmuir film Weis and McConnell (1984). However,
the qualitative success of such models may have little to do with the
underlying physics as noted by Zaanen in the context of Mott insulators Zaanen
(1998). Indeed, the same general behavior can be observed in models with for
example only short-ranged, purely repulsive forces Malescio and Pellicane
(2003). An explanation for the universality of striped morphologies must
therefore be independent of specific details of the involved forces. The aim
of this Letter is to present such a general treatment. As expected our method
shows that stripes appear naturally for large classes of models, but the added
generality also leads to new tools allowing us to design potentials with
desired properties.
Here we study a generic Hamiltonian with isotropic pairwise interactions
$H=\sum_{ij}^{N}V_{ij}s_{i}s_{j}$ (2)
where $V_{ij}=V(\left|i-j\right|)$ is a matrix representation of the potential
that only depends on the distance between spins $i$ and $j$, here denoted
$\left|i-j\right|$, with (1) as a special case. Depending on considerations
regarding experimental fit or theoretical ease, one may take the spins in (2)
to assume continuous values with the restriction $\sum_{i}s_{i}^{2}=N$,
corresponding to a spherical model Baxter (1982), or take values from some
finite set, where $s\in\\{\pm 1\\}$ and $s\in\\{0,\pm 1\\}$ are the most
common choices, equivalent to different Ising models Löw _et al._ (1994);
Giuliani _et al._ (2006).
Consider now the spherical model. Due to the quadratic form of the Hamiltonian
(2) and the constraint, ${\bf s}^{T}V{\bf s}$ and ${\bf s}^{T}{\bf s}=N$ in
matrix notation, the ground state is given by an eigenvector of the
interaction matrix $V$ corresponding to the lowest (energy) eigenvalue Lay
(2003). The central observation for our analysis is the existence of a common
basis of eigenvectors for all radial potentials, namely the Fourier basis.
Figure 1: Reflection symmetry of triangles on a lattice causes the
interaction matrices from any two radial potentials to commute.
To prove this we start by recalling the fact that if two matrices commute,
then it is possible to find a set of eigenvectors that simultaneously
diagonalize them Horn and Johnson (1990). Consider the commutator for two
interaction matrices $V$ and $W$:
$\sum_{k^{\prime}}V_{ik^{\prime}}W_{k^{\prime}j}-\sum_{k}W_{ik}V_{kj}.$ (3)
Each term, $V_{ik^{\prime}}W_{k^{\prime}j}$, in the first sum can be
represented by a triangle, A in Fig. 1. Assuming that the lattice is periodic
or infinite, there will for each such triangle exist a unique triangle B,
constructed as a reflection of A in the line equidistant from point $i$ and
$j$ (dashed in the figure), corresponding to the term $W_{ik}V_{kj}$ in the
second sum. From the reflection symmetry and the pure radial dependence of the
potential it follows that $V_{ik^{\prime}}W_{k^{\prime}j}-W_{ik}V_{kj}=0$,
which proves that $V$ and $W$ commute.
It now suffices to find a set of eigenvectors for a particular potential.
Perhaps the simplest choice is a nearest neighbor interaction, $V_{ij}=1$ if
$\left|i-j\right|=1$ and $V_{ij}=0$ if $\left|i-j\right|>1$. If we in addition
make an appropriate choice of self-interaction, which only shifts the
eigenvalues and do not affect the eigenvectors, $V_{ii}=-2d$ where $d$ is the
lattice dimension, $V_{ij}$ becomes a discrete finite difference Laplacian on
the lattice. It is well known that both the discrete and continuous Laplacian
have harmonic eigenfunctions, e.g.
$f_{\vec{k}}(\vec{x})=C_{\vec{k}}\prod_{i}^{d}\cos{(2\pi
k_{i}x_{i}/L+\phi_{i})}$ which is an orthogonal eigenbasis in $d$ dimensions
when $\vec{k}$ goes over all distances on the reciprocal lattice, $L$ is the
linear size of the lattice, $\phi_{i}=\pm\pi/4$ and $C_{\vec{k}}$ is an
appropriate normalization constant. We have thus shown that all interaction
matrices have a Fourier eigenbasis. An alternative, more direct but for our
purposes less illustrative, argument for the common Fourier basis is to note
that the structure of $V$ implies that it is a so called circulant matrix
Davis (1994), for which the result is known in the signal processing
literature. That the Fourier base effectively diagonalize the Hamiltonian in
the spherical model with translationally invariant interactions has also been
pointed out by Nussinov Nussinov .
This result has two important consequences. First, it helps us to understand
why systems with different interactions are expected to have similar ground
states. Second, knowledge of the universal eigenbasis allows us to compute the
energy spectrum for any particular system using a linear transform of the
potential. From (2) it follows that the energy per spin for a harmonic
eigenfunction with wave vector $\vec{k}$ are given by
$E(\vec{k})=\frac{1}{N}\sum_{\vec{x},\vec{y}}V(\left|\vec{x}-\vec{y}\right|)f_{\vec{k}}(\vec{x})f_{\vec{k}}(\vec{y})$.
Using various trigonometric identities and the radial structure of $V$ this
expression can be reduced to a radial Fourier transform
$E(\vec{k})=\sum_{\vec{r}}V(\left|\vec{r}\right|)\prod_{i=1}^{d}\cos{(2\pi
k_{i}r_{i}/L)}$ (4)
where the sum goes over all distances $\vec{r}$ on the lattice. Note that the
energy of a configuration can be computed through a fast Fourier transform
over the lattice.
Figure 2: Examples of eigenmodes of the two dimensional spherical model. a-c,
Any phase shift of a ground state is a new ground state, so checkerboards,
stripes and everything between can be produced by the same model. Linear
combinations of $f_{(2,2)}$ with different phase shifts are shown, all having
the same energy. d, Exchanging the elements of $\vec{k}$ gives a new ground
state and linear combinations of them give rise to complex morphologies. Shown
is $\frac{1}{2}f_{(3,4)}+\frac{1}{2}f_{(4,3)}$.
The ground state of the spherical model is the eigenvector $f_{\vec{k}}$
corresponding to the minimum of $E(\vec{k})$. The simplest ground state
patterns in two dimensions are checkerboards and stripes with the
corresponding wavelength, exemplified in Fig. 2a and c. Further, the subspace
of the eigenbasis corresponding to the minimum can contain two kinds of
degeneracies. First, any change of the phases $\phi_{i}$ leaves the energy
invariant. In two dimensions this means that anything between checkerboards
and stripes can be produced, as illustrated in Fig. 2a-c. Second, the energy
is similarly unaffected by arbitrary permutations of the elements of
$\vec{k}$, reflecting that the energy only depends on the magnitude of the
wave vector (seen most clearly in the continuous limit (5)). Linear
combinations of vectors with different permutations give rise to complex
morphologies, exemplified in Fig. 2d.
Figure 3: Predicting ground states in different two dimensional Ising models.
(Top) Potentials (inset) and their energy spectra in $|\vec{k}|$-space from
the transform (4). a, a purely repulsive potential Malescio and Pellicane
(2003), b, two competing interactions Löw _et al._ (1994) (see equation (1)),
and c, an RKKY-like interaction Fischer and Hertz (1993). For small $k$ (long
wavelengths) the spectra only depend on the magnitude of $\vec{k}$, but for
large $k$ (short wavelengths) lattice effects breaks the independence on the
direction of $\vec{k}$ and $E(|\vec{k}|)$ becomes multivalued. To illustrate
this, we link series with constant $k_{y}$ with lines and in a examples of
such states are shown. (Below) Local minima as arrived by through Monte Carlo
annealing as well as ground states of Ising spin-1/2 models with corresponding
potentials.
The eigenmode analysis is exact for the spherical model but also has
implications for the discrete Ising models. It is not directly applicable as
in general an eigenvector of the interaction matrix cannot be constructed in
the restricted discrete space of the Ising spins. However, continuous
eigenvectors are often used to approximate solutions to discrete optimization
problems, for example graph coloring Aspvall and Gilbert (1984) and
partitioning networks into modules with minimal intra-connectivity Fiedler
(1973); Newman (2006). Here we use the same strategy to predict ground states
for Ising spin-$1/2$ models with corresponding potentials by mapping the spins
in the spherical model to $-1$ or $+1$ depending on their sign:
$\hat{f}_{\vec{k}}(\vec{x})=\mbox{sign}(f_{\vec{k}}(\vec{x}))$. 111This
discretization of the Fourier modes to approximate the ground states of Ising
models was also discussed in Nussinov where it is argued that striped ground
states should be energetically favored if the minimum in the energy spectrum
is sharp. The argument we present does not have this assumption which is
important as many commonly used potentials have broad energy minima and
striped ground states, see e.g. Fig. 3 a and b. The discretization breaks the
energy degeneracy and stripes become energetically favorable compared to
checkerboards and more complex patterns. To see why we note that in each group
of degenerate eigenmodes, with wavelength $|\vec{k}|$, there exist linear
combinations that produce stripes, for example $\cos(\vec{k}\cdot\vec{x})$.
The error introduced by the discretization,
$\|\hat{f}_{\vec{k}}(\vec{x})-f_{\vec{k}}(\vec{x})\|_{2}$ with the standard
$L^{2}$ norm, always increases the energy in the discrete configurations when
compared to the continuous ground state. Due to the $\pm$-symmetry of the
harmonic functions, the difference between $\hat{f}_{\vec{k}}(\vec{x})$ and
$f_{\vec{k}}(\vec{x})$ (with appropriate scaling) is largest in regions where
the continuous function is close to $0$, i.e. at the interface between $+$ and
$-$ regions. From this argument it follows that the errortends to be smallest
for the striped eigenmode since the interface is minimized (assuming that the
width of the stripes is large compared to the lattice spacing).
There are two exceptions when the ground state does not have stripes. Energy
spectra with minimum at the boundary produce ground states that are either a
uniform ferromagnet (the zero frequency mode) or a checkerboard pattern (the
highest frequency mode allowed on the lattice) associated with an anti-
ferromagnet. These two cases can be viewed as degenerate cases of stripes with
infinite respective infinitesimal width.
In Fig. 3 some examples of Ising spin-$1/2$ models with different potentials
are shown together with their energy spectra in $|\vec{k}|$-space, examples of
local minima 222Local minima of the spin systems are found by a standard
Metropolis algorithm on lattices with periodic boundaries. The results shown
in Fig. 3 where generated by simulating 50$\times$50-lattices (in a and b) or
a 150$\times$150-lattice (in c) with the potentials shown at $T=0.1$ for
$2.5\times 10^{5}$ trial flips and then with $T=0$ until convergence. In Fig.
4 a 300$\times$200-lattice with $6\times 10^{5}$ trial flips at $T=0.2$ was
used. and their ground states. First is a short-ranged, purely repulsive
potential related to the model studied in Malescio and Pellicane (2003).
Second is a nearest neighbor ferromagnet with long-range repulsive Coulomb
interaction on the form (1) from Löw _et al._ (1994). Last is an attenuated
Bessel function, $J_{0}(r)/(r+1)$, chosen for its similarity to the RKKY
interaction in spin glasses Fischer and Hertz (1993). We see that, while the
potentials are qualitatively very different, the ground states are defined
only by the minima in the energy spectrum, i.e. by a single length scale.
Through rescaling, the potentials can be adjusted to have identical ground
states. This illustrates how little observing striped behavior tells us about
the interactions in a system. The local minima do however show a qualitative
difference between the potentials in a and b and the RKKY-like potential in c,
probably related to the difference in localization in energy space.
Equation (4) also has implications for molecular self-assembly. The Fourier
basis in the transformation is orthogonal and can be inverted to find the
potential corresponding to a given energy spectrum. This allows us to design,
from an observed striped state, families of potentials that generate similar
patterns at low-temperature by identifying the dominant wavelength and invert
an energy spectrum with a minimum at this wavelength. A demonstration of the
procedure is shown in Fig. 4: the Fourier power spectrum of a pixelised image
of a metastable state in an experimental system Rosensweig _et al._ (1983)
was calculated; an energy spectrum was constructed with gaussian minimum at
the same wavelength as the experimental system; and finally the corresponding
potential was found using the inverse transform of (4). The constructed system
has striped metastable states similar to those found in the experimental
system. We conclude that it is relatively easy to construct families of
potentials with desired metastable striped morphologies.
Figure 4: Designing interactions to imitate observed striped patterns. a,
Stripes in a ferrofluid confined between two glass plates in a magnetic field
[from Rosensweig _et al._ (1983); Seul and Andelman (1995), reprinted with
permission from AAAS and Elsevier] b, the (negative) radial power spectrum
(blue) of previous picture together with a gaussian (red), (inset) a pair
potential $V(r)$ constructed as the (inverse) transform (4) of said gaussian
and c, metastable state of an Ising spin-$1/2$ model with potential $V(r)$.
Note that the chosen energy spectrum is not unique. Many potentials having a
spectrum minimized at the same wavelength will show similar low-temperature
behavior.
In the continuous limit the transformation (4) becomes a Hankel transform, in
two dimensions defined as
$E(\vec{k})=2\pi\int_{0}^{\infty}rdrV(r)J_{0}(2\pi|\vec{k}|r)$ (5)
where $J_{0}$ is a Bessel function of the first kind. For the general
expression in higher dimensions, see Folland (1992). Note that in the
continuum limit the energy only depends on the magnitude of the wavevector
since the effects of the principal lattice directions disappear. As noted in
Fig. 3, this independence holds true on the lattice as well for small
wavevectors and sufficiently long-range interactions. Equation (5) allows us
to use the analytical properties of the Hankel transform to qualitatively
understand for example why the Bessel function of Fig. 3c has such a sharp
spectrum: the Hankel transform of a Bessel function is a Dirac delta function.
In summary we have shown that the energy spectrum of spherical spin systems
with isotropic interactions can be derived directly from the Fourier transform
of the potential. Due to a degeneracy in the energy eigenstates the spherical
model has ground states with various patterns such as stripes, checkerboards,
and more complicated morphologies. In discrete spin models the degeneracy is
broken leading to striped ground states being energetically favored. We
suggest that this can offer a generic explanation to why striped patterns are
so frequently observed in various experimental and natural systems.
###### Acknowledgements.
The authors would like to thank Olle Häggström for pointing out how purely
repulsive potentials can give rise to striped ground states.
## References
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* Note (1) This discretization of the Fourier modes to approximate the ground states of Ising models was also discussed in Nussinov where it is argued that striped ground states should be energetically favored if the minimum in the energy spectrum is sharp. The argument we present does not have this assumption which is important as many commonly used potentials have broad energy minima and striped ground states, see e.g. Fig. 3 a and b.
* Note (2) Local minima of the spin systems are found by a standard Metropolis algorithm on lattices with periodic boundaries. The results shown in Fig. 3 where generated by simulating 50$\times$50-lattices (in a and b) or a 150$\times$150-lattice (in c) with the potentials shown at $T=0.1$ for $2.5\times 10^{5}$ trial flips and then with $T=0$ until convergence. In Fig. 4 a 300$\times$200-lattice with $6\times 10^{5}$ trial flips at $T=0.2$ was used.
* Folland (1992) G. B. Folland, “Fourier Analysis and Its Applications,” (Brooks/Cole Pub Co, Pacific Grove, 1992) Chap. 7.5.
|
arxiv-papers
| 2011-08-23T13:15:52 |
2024-09-04T02:49:21.732641
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Erik Edlund, Martin Nilsson Jacobi",
"submitter": "Erik Edlund",
"url": "https://arxiv.org/abs/1108.4586"
}
|
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